var bigInt = (function (undefined) {
    "use strict";

    var BASE = 1e7,
        LOG_BASE = 7,
        MAX_INT = 9007199254740992,
        MAX_INT_ARR = smallToArray(MAX_INT),
        LOG_MAX_INT = Math.log(MAX_INT);

    function Integer(v, radix) {
        if (typeof v === "undefined") return Integer[0];
        if (typeof radix !== "undefined") return +radix === 10 ? parseValue(v) : parseBase(v, radix);
        return parseValue(v);
    }

    function BigInteger(value, sign) {
        this.value = value;
        this.sign = sign;
        this.isSmall = false;
    }
    BigInteger.prototype = Object.create(Integer.prototype);

    function SmallInteger(value) {
        this.value = value;
        this.sign = value < 0;
        this.isSmall = true;
    }
    SmallInteger.prototype = Object.create(Integer.prototype);

    function isPrecise(n) {
        return -MAX_INT < n && n < MAX_INT;
    }

    function smallToArray(n) { // For performance reasons doesn't reference BASE, need to change this function if BASE changes
        if (n < 1e7)
            return [n];
        if (n < 1e14)
            return [n % 1e7, Math.floor(n / 1e7)];
        return [n % 1e7, Math.floor(n / 1e7) % 1e7, Math.floor(n / 1e14)];
    }

    function arrayToSmall(arr) { // If BASE changes this function may need to change
        trim(arr);
        var length = arr.length;
        if (length < 4 && compareAbs(arr, MAX_INT_ARR) < 0) {
            switch (length) {
                case 0: return 0;
                case 1: return arr[0];
                case 2: return arr[0] + arr[1] * BASE;
                default: return arr[0] + (arr[1] + arr[2] * BASE) * BASE;
            }
        }
        return arr;
    }

    function trim(v) {
        var i = v.length;
        while (v[--i] === 0);
        v.length = i + 1;
    }

    function createArray(length) { // function shamelessly stolen from Yaffle's library https://github.com/Yaffle/BigInteger
        var x = new Array(length);
        var i = -1;
        while (++i < length) {
            x[i] = 0;
        }
        return x;
    }

    function truncate(n) {
        if (n > 0) return Math.floor(n);
        return Math.ceil(n);
    }

    function add(a, b) { // assumes a and b are arrays with a.length >= b.length
        var l_a = a.length,
            l_b = b.length,
            r = new Array(l_a),
            carry = 0,
            base = BASE,
            sum, i;
        for (i = 0; i < l_b; i++) {
            sum = a[i] + b[i] + carry;
            carry = sum >= base ? 1 : 0;
            r[i] = sum - carry * base;
        }
        while (i < l_a) {
            sum = a[i] + carry;
            carry = sum === base ? 1 : 0;
            r[i++] = sum - carry * base;
        }
        if (carry > 0) r.push(carry);
        return r;
    }

    function addAny(a, b) {
        if (a.length >= b.length) return add(a, b);
        return add(b, a);
    }

    function addSmall(a, carry) { // assumes a is array, carry is number with 0 <= carry < MAX_INT
        var l = a.length,
            r = new Array(l),
            base = BASE,
            sum, i;
        for (i = 0; i < l; i++) {
            sum = a[i] - base + carry;
            carry = Math.floor(sum / base);
            r[i] = sum - carry * base;
            carry += 1;
        }
        while (carry > 0) {
            r[i++] = carry % base;
            carry = Math.floor(carry / base);
        }
        return r;
    }

    BigInteger.prototype.add = function (v) {
        var n = parseValue(v);
        if (this.sign !== n.sign) {
            return this.subtract(n.negate());
        }
        var a = this.value, b = n.value;
        if (n.isSmall) {
            return new BigInteger(addSmall(a, Math.abs(b)), this.sign);
        }
        return new BigInteger(addAny(a, b), this.sign);
    };
    BigInteger.prototype.plus = BigInteger.prototype.add;

    SmallInteger.prototype.add = function (v) {
        var n = parseValue(v);
        var a = this.value;
        if (a < 0 !== n.sign) {
            return this.subtract(n.negate());
        }
        var b = n.value;
        if (n.isSmall) {
            if (isPrecise(a + b)) return new SmallInteger(a + b);
            b = smallToArray(Math.abs(b));
        }
        return new BigInteger(addSmall(b, Math.abs(a)), a < 0);
    };
    SmallInteger.prototype.plus = SmallInteger.prototype.add;

    function subtract(a, b) { // assumes a and b are arrays with a >= b
        var a_l = a.length,
            b_l = b.length,
            r = new Array(a_l),
            borrow = 0,
            base = BASE,
            i, difference;
        for (i = 0; i < b_l; i++) {
            difference = a[i] - borrow - b[i];
            if (difference < 0) {
                difference += base;
                borrow = 1;
            } else borrow = 0;
            r[i] = difference;
        }
        for (i = b_l; i < a_l; i++) {
            difference = a[i] - borrow;
            if (difference < 0) difference += base;
            else {
                r[i++] = difference;
                break;
            }
            r[i] = difference;
        }
        for (; i < a_l; i++) {
            r[i] = a[i];
        }
        trim(r);
        return r;
    }

    function subtractAny(a, b, sign) {
        var value;
        if (compareAbs(a, b) >= 0) {
            value = subtract(a,b);
        } else {
            value = subtract(b, a);
            sign = !sign;
        }
        value = arrayToSmall(value);
        if (typeof value === "number") {
            if (sign) value = -value;
            return new SmallInteger(value);
        }
        return new BigInteger(value, sign);
    }

    function subtractSmall(a, b, sign) { // assumes a is array, b is number with 0 <= b < MAX_INT
        var l = a.length,
            r = new Array(l),
            carry = -b,
            base = BASE,
            i, difference;
        for (i = 0; i < l; i++) {
            difference = a[i] + carry;
            carry = Math.floor(difference / base);
            difference %= base;
            r[i] = difference < 0 ? difference + base : difference;
        }
        r = arrayToSmall(r);
        if (typeof r === "number") {
            if (sign) r = -r;
            return new SmallInteger(r);
        } return new BigInteger(r, sign);
    }

    BigInteger.prototype.subtract = function (v) {
        var n = parseValue(v);
        if (this.sign !== n.sign) {
            return this.add(n.negate());
        }
        var a = this.value, b = n.value;
        if (n.isSmall)
            return subtractSmall(a, Math.abs(b), this.sign);
        return subtractAny(a, b, this.sign);
    };
    BigInteger.prototype.minus = BigInteger.prototype.subtract;

    SmallInteger.prototype.subtract = function (v) {
        var n = parseValue(v);
        var a = this.value;
        if (a < 0 !== n.sign) {
            return this.add(n.negate());
        }
        var b = n.value;
        if (n.isSmall) {
            return new SmallInteger(a - b);
        }
        return subtractSmall(b, Math.abs(a), a >= 0);
    };
    SmallInteger.prototype.minus = SmallInteger.prototype.subtract;

    BigInteger.prototype.negate = function () {
        return new BigInteger(this.value, !this.sign);
    };
    SmallInteger.prototype.negate = function () {
        var sign = this.sign;
        var small = new SmallInteger(-this.value);
        small.sign = !sign;
        return small;
    };

    BigInteger.prototype.abs = function () {
        return new BigInteger(this.value, false);
    };
    SmallInteger.prototype.abs = function () {
        return new SmallInteger(Math.abs(this.value));
    };

    function multiplyLong(a, b) {
        var a_l = a.length,
            b_l = b.length,
            l = a_l + b_l,
            r = createArray(l),
            base = BASE,
            product, carry, i, a_i, b_j;
        for (i = 0; i < a_l; ++i) {
            a_i = a[i];
            for (var j = 0; j < b_l; ++j) {
                b_j = b[j];
                product = a_i * b_j + r[i + j];
                carry = Math.floor(product / base);
                r[i + j] = product - carry * base;
                r[i + j + 1] += carry;
            }
        }
        trim(r);
        return r;
    }

    function multiplySmall(a, b) { // assumes a is array, b is number with |b| < BASE
        var l = a.length,
            r = new Array(l),
            base = BASE,
            carry = 0,
            product, i;
        for (i = 0; i < l; i++) {
            product = a[i] * b + carry;
            carry = Math.floor(product / base);
            r[i] = product - carry * base;
        }
        while (carry > 0) {
            r[i++] = carry % base;
            carry = Math.floor(carry / base);
        }
        return r;
    }

    function shiftLeft(x, n) {
        var r = [];
        while (n-- > 0) r.push(0);
        return r.concat(x);
    }

    function multiplyKaratsuba(x, y) {
        var n = Math.max(x.length, y.length);

        if (n <= 30) return multiplyLong(x, y);
        n = Math.ceil(n / 2);

        var b = x.slice(n),
            a = x.slice(0, n),
            d = y.slice(n),
            c = y.slice(0, n);

        var ac = multiplyKaratsuba(a, c),
            bd = multiplyKaratsuba(b, d),
            abcd = multiplyKaratsuba(addAny(a, b), addAny(c, d));

        var product = addAny(addAny(ac, shiftLeft(subtract(subtract(abcd, ac), bd), n)), shiftLeft(bd, 2 * n));
        trim(product);
        return product;
    }

    // The following function is derived from a surface fit of a graph plotting the performance difference
    // between long multiplication and karatsuba multiplication versus the lengths of the two arrays.
    function useKaratsuba(l1, l2) {
        return -0.012 * l1 - 0.012 * l2 + 0.000015 * l1 * l2 > 0;
    }

    BigInteger.prototype.multiply = function (v) {
        var n = parseValue(v),
            a = this.value, b = n.value,
            sign = this.sign !== n.sign,
            abs;
        if (n.isSmall) {
            if (b === 0) return Integer[0];
            if (b === 1) return this;
            if (b === -1) return this.negate();
            abs = Math.abs(b);
            if (abs < BASE) {
                return new BigInteger(multiplySmall(a, abs), sign);
            }
            b = smallToArray(abs);
        }
        if (useKaratsuba(a.length, b.length)) // Karatsuba is only faster for certain array sizes
            return new BigInteger(multiplyKaratsuba(a, b), sign);
        return new BigInteger(multiplyLong(a, b), sign);
    };

    BigInteger.prototype.times = BigInteger.prototype.multiply;

    function multiplySmallAndArray(a, b, sign) { // a >= 0
        if (a < BASE) {
            return new BigInteger(multiplySmall(b, a), sign);
        }
        return new BigInteger(multiplyLong(b, smallToArray(a)), sign);
    }
    SmallInteger.prototype._multiplyBySmall = function (a) {
            if (isPrecise(a.value * this.value)) {
                return new SmallInteger(a.value * this.value);
            }
            return multiplySmallAndArray(Math.abs(a.value), smallToArray(Math.abs(this.value)), this.sign !== a.sign);
    };
    BigInteger.prototype._multiplyBySmall = function (a) {
            if (a.value === 0) return Integer[0];
            if (a.value === 1) return this;
            if (a.value === -1) return this.negate();
            return multiplySmallAndArray(Math.abs(a.value), this.value, this.sign !== a.sign);
    };
    SmallInteger.prototype.multiply = function (v) {
        return parseValue(v)._multiplyBySmall(this);
    };
    SmallInteger.prototype.times = SmallInteger.prototype.multiply;

    function square(a) {
        var l = a.length,
            r = createArray(l + l),
            base = BASE,
            product, carry, i, a_i, a_j;
        for (i = 0; i < l; i++) {
            a_i = a[i];
            for (var j = 0; j < l; j++) {
                a_j = a[j];
                product = a_i * a_j + r[i + j];
                carry = Math.floor(product / base);
                r[i + j] = product - carry * base;
                r[i + j + 1] += carry;
            }
        }
        trim(r);
        return r;
    }

    BigInteger.prototype.square = function () {
        return new BigInteger(square(this.value), false);
    };

    SmallInteger.prototype.square = function () {
        var value = this.value * this.value;
        if (isPrecise(value)) return new SmallInteger(value);
        return new BigInteger(square(smallToArray(Math.abs(this.value))), false);
    };

    function divMod1(a, b) { // Left over from previous version. Performs faster than divMod2 on smaller input sizes.
        var a_l = a.length,
            b_l = b.length,
            base = BASE,
            result = createArray(b.length),
            divisorMostSignificantDigit = b[b_l - 1],
            // normalization
            lambda = Math.ceil(base / (2 * divisorMostSignificantDigit)),
            remainder = multiplySmall(a, lambda),
            divisor = multiplySmall(b, lambda),
            quotientDigit, shift, carry, borrow, i, l, q;
        if (remainder.length <= a_l) remainder.push(0);
        divisor.push(0);
        divisorMostSignificantDigit = divisor[b_l - 1];
        for (shift = a_l - b_l; shift >= 0; shift--) {
            quotientDigit = base - 1;
            if (remainder[shift + b_l] !== divisorMostSignificantDigit) {
              quotientDigit = Math.floor((remainder[shift + b_l] * base + remainder[shift + b_l - 1]) / divisorMostSignificantDigit);
            }
            // quotientDigit <= base - 1
            carry = 0;
            borrow = 0;
            l = divisor.length;
            for (i = 0; i < l; i++) {
                carry += quotientDigit * divisor[i];
                q = Math.floor(carry / base);
                borrow += remainder[shift + i] - (carry - q * base);
                carry = q;
                if (borrow < 0) {
                    remainder[shift + i] = borrow + base;
                    borrow = -1;
                } else {
                    remainder[shift + i] = borrow;
                    borrow = 0;
                }
            }
            while (borrow !== 0) {
                quotientDigit -= 1;
                carry = 0;
                for (i = 0; i < l; i++) {
                    carry += remainder[shift + i] - base + divisor[i];
                    if (carry < 0) {
                        remainder[shift + i] = carry + base;
                        carry = 0;
                    } else {
                        remainder[shift + i] = carry;
                        carry = 1;
                    }
                }
                borrow += carry;
            }
            result[shift] = quotientDigit;
        }
        // denormalization
        remainder = divModSmall(remainder, lambda)[0];
        return [arrayToSmall(result), arrayToSmall(remainder)];
    }

    function divMod2(a, b) { // Implementation idea shamelessly stolen from Silent Matt's library http://silentmatt.com/biginteger/
        // Performs faster than divMod1 on larger input sizes.
        var a_l = a.length,
            b_l = b.length,
            result = [],
            part = [],
            base = BASE,
            guess, xlen, highx, highy, check;
        while (a_l) {
            part.unshift(a[--a_l]);
            trim(part);
            if (compareAbs(part, b) < 0) {
                result.push(0);
                continue;
            }
            xlen = part.length;
            highx = part[xlen - 1] * base + part[xlen - 2];
            highy = b[b_l - 1] * base + b[b_l - 2];
            if (xlen > b_l) {
                highx = (highx + 1) * base;
            }
            guess = Math.ceil(highx / highy);
            do {
                check = multiplySmall(b, guess);
                if (compareAbs(check, part) <= 0) break;
                guess--;
            } while (guess);
            result.push(guess);
            part = subtract(part, check);
        }
        result.reverse();
        return [arrayToSmall(result), arrayToSmall(part)];
    }

    function divModSmall(value, lambda) {
        var length = value.length,
            quotient = createArray(length),
            base = BASE,
            i, q, remainder, divisor;
        remainder = 0;
        for (i = length - 1; i >= 0; --i) {
            divisor = remainder * base + value[i];
            q = truncate(divisor / lambda);
            remainder = divisor - q * lambda;
            quotient[i] = q | 0;
        }
        return [quotient, remainder | 0];
    }

    function divModAny(self, v) {
        var value, n = parseValue(v);
        var a = self.value, b = n.value;
        var quotient;
        if (b === 0) throw new Error("Cannot divide by zero");
        if (self.isSmall) {
            if (n.isSmall) {
                return [new SmallInteger(truncate(a / b)), new SmallInteger(a % b)];
            }
            return [Integer[0], self];
        }
        if (n.isSmall) {
            if (b === 1) return [self, Integer[0]];
            if (b == -1) return [self.negate(), Integer[0]];
            var abs = Math.abs(b);
            if (abs < BASE) {
                value = divModSmall(a, abs);
                quotient = arrayToSmall(value[0]);
                var remainder = value[1];
                if (self.sign) remainder = -remainder;
                if (typeof quotient === "number") {
                    if (self.sign !== n.sign) quotient = -quotient;
                    return [new SmallInteger(quotient), new SmallInteger(remainder)];
                }
                return [new BigInteger(quotient, self.sign !== n.sign), new SmallInteger(remainder)];
            }
            b = smallToArray(abs);
        }
        var comparison = compareAbs(a, b);
        if (comparison === -1) return [Integer[0], self];
        if (comparison === 0) return [Integer[self.sign === n.sign ? 1 : -1], Integer[0]];

        // divMod1 is faster on smaller input sizes
        if (a.length + b.length <= 200)
            value = divMod1(a, b);
        else value = divMod2(a, b);

        quotient = value[0];
        var qSign = self.sign !== n.sign,
            mod = value[1],
            mSign = self.sign;
        if (typeof quotient === "number") {
            if (qSign) quotient = -quotient;
            quotient = new SmallInteger(quotient);
        } else quotient = new BigInteger(quotient, qSign);
        if (typeof mod === "number") {
            if (mSign) mod = -mod;
            mod = new SmallInteger(mod);
        } else mod = new BigInteger(mod, mSign);
        return [quotient, mod];
    }

    BigInteger.prototype.divmod = function (v) {
        var result = divModAny(this, v);
        return {
            quotient: result[0],
            remainder: result[1]
        };
    };
    SmallInteger.prototype.divmod = BigInteger.prototype.divmod;

    BigInteger.prototype.divide = function (v) {
        return divModAny(this, v)[0];
    };
    SmallInteger.prototype.over = SmallInteger.prototype.divide = BigInteger.prototype.over = BigInteger.prototype.divide;

    BigInteger.prototype.mod = function (v) {
        return divModAny(this, v)[1];
    };
    SmallInteger.prototype.remainder = SmallInteger.prototype.mod = BigInteger.prototype.remainder = BigInteger.prototype.mod;

    BigInteger.prototype.pow = function (v) {
        var n = parseValue(v),
            a = this.value,
            b = n.value,
            value, x, y;
        if (b === 0) return Integer[1];
        if (a === 0) return Integer[0];
        if (a === 1) return Integer[1];
        if (a === -1) return n.isEven() ? Integer[1] : Integer[-1];
        if (n.sign) {
            return Integer[0];
        }
        if (!n.isSmall) throw new Error("The exponent " + n.toString() + " is too large.");
        if (this.isSmall) {
            if (isPrecise(value = Math.pow(a, b)))
                return new SmallInteger(truncate(value));
        }
        x = this;
        y = Integer[1];
        while (true) {
            if (b & 1 === 1) {
                y = y.times(x);
                --b;
            }
            if (b === 0) break;
            b /= 2;
            x = x.square();
        }
        return y;
    };
    SmallInteger.prototype.pow = BigInteger.prototype.pow;

    BigInteger.prototype.modPow = function (exp, mod) {
        exp = parseValue(exp);
        mod = parseValue(mod);
        if (mod.isZero()) throw new Error("Cannot take modPow with modulus 0");
        var r = Integer[1],
            base = this.mod(mod);
        while (exp.isPositive()) {
            if (base.isZero()) return Integer[0];
            if (exp.isOdd()) r = r.multiply(base).mod(mod);
            exp = exp.divide(2);
            base = base.square().mod(mod);
        }
        return r;
    };
    SmallInteger.prototype.modPow = BigInteger.prototype.modPow;

    function compareAbs(a, b) {
        if (a.length !== b.length) {
            return a.length > b.length ? 1 : -1;
        }
        for (var i = a.length - 1; i >= 0; i--) {
            if (a[i] !== b[i]) return a[i] > b[i] ? 1 : -1;
        }
        return 0;
    }

    BigInteger.prototype.compareAbs = function (v) {
        var n = parseValue(v),
            a = this.value,
            b = n.value;
        if (n.isSmall) return 1;
        return compareAbs(a, b);
    };
    SmallInteger.prototype.compareAbs = function (v) {
        var n = parseValue(v),
            a = Math.abs(this.value),
            b = n.value;
        if (n.isSmall) {
            b = Math.abs(b);
            return a === b ? 0 : a > b ? 1 : -1;
        }
        return -1;
    };

    BigInteger.prototype.compare = function (v) {
        // See discussion about comparison with Infinity:
        // https://github.com/peterolson/BigInteger.js/issues/61
        if (v === Infinity) {
            return -1;
        }
        if (v === -Infinity) {
            return 1;
        }

        var n = parseValue(v),
            a = this.value,
            b = n.value;
        if (this.sign !== n.sign) {
            return n.sign ? 1 : -1;
        }
        if (n.isSmall) {
            return this.sign ? -1 : 1;
        }
        return compareAbs(a, b) * (this.sign ? -1 : 1);
    };
    BigInteger.prototype.compareTo = BigInteger.prototype.compare;

    SmallInteger.prototype.compare = function (v) {
        if (v === Infinity) {
            return -1;
        }
        if (v === -Infinity) {
            return 1;
        }

        var n = parseValue(v),
            a = this.value,
            b = n.value;
        if (n.isSmall) {
            return a == b ? 0 : a > b ? 1 : -1;
        }
        if (a < 0 !== n.sign) {
            return a < 0 ? -1 : 1;
        }
        return a < 0 ? 1 : -1;
    };
    SmallInteger.prototype.compareTo = SmallInteger.prototype.compare;

    BigInteger.prototype.equals = function (v) {
        return this.compare(v) === 0;
    };
    SmallInteger.prototype.eq = SmallInteger.prototype.equals = BigInteger.prototype.eq = BigInteger.prototype.equals;

    BigInteger.prototype.notEquals = function (v) {
        return this.compare(v) !== 0;
    };
    SmallInteger.prototype.neq = SmallInteger.prototype.notEquals = BigInteger.prototype.neq = BigInteger.prototype.notEquals;

    BigInteger.prototype.greater = function (v) {
        return this.compare(v) > 0;
    };
    SmallInteger.prototype.gt = SmallInteger.prototype.greater = BigInteger.prototype.gt = BigInteger.prototype.greater;

    BigInteger.prototype.lesser = function (v) {
        return this.compare(v) < 0;
    };
    SmallInteger.prototype.lt = SmallInteger.prototype.lesser = BigInteger.prototype.lt = BigInteger.prototype.lesser;

    BigInteger.prototype.greaterOrEquals = function (v) {
        return this.compare(v) >= 0;
    };
    SmallInteger.prototype.geq = SmallInteger.prototype.greaterOrEquals = BigInteger.prototype.geq = BigInteger.prototype.greaterOrEquals;

    BigInteger.prototype.lesserOrEquals = function (v) {
        return this.compare(v) <= 0;
    };
    SmallInteger.prototype.leq = SmallInteger.prototype.lesserOrEquals = BigInteger.prototype.leq = BigInteger.prototype.lesserOrEquals;

    BigInteger.prototype.isEven = function () {
        return (this.value[0] & 1) === 0;
    };
    SmallInteger.prototype.isEven = function () {
        return (this.value & 1) === 0;
    };

    BigInteger.prototype.isOdd = function () {
        return (this.value[0] & 1) === 1;
    };
    SmallInteger.prototype.isOdd = function () {
        return (this.value & 1) === 1;
    };

    BigInteger.prototype.isPositive = function () {
        return !this.sign;
    };
    SmallInteger.prototype.isPositive = function () {
        return this.value > 0;
    };

    BigInteger.prototype.isNegative = function () {
        return this.sign;
    };
    SmallInteger.prototype.isNegative = function () {
        return this.value < 0;
    };

    BigInteger.prototype.isUnit = function () {
        return false;
    };
    SmallInteger.prototype.isUnit = function () {
        return Math.abs(this.value) === 1;
    };

    BigInteger.prototype.isZero = function () {
        return false;
    };
    SmallInteger.prototype.isZero = function () {
        return this.value === 0;
    };
    BigInteger.prototype.isDivisibleBy = function (v) {
        var n = parseValue(v);
        var value = n.value;
        if (value === 0) return false;
        if (value === 1) return true;
        if (value === 2) return this.isEven();
        return this.mod(n).equals(Integer[0]);
    };
    SmallInteger.prototype.isDivisibleBy = BigInteger.prototype.isDivisibleBy;

    function isBasicPrime(v) {
        var n = v.abs();
        if (n.isUnit()) return false;
        if (n.equals(2) || n.equals(3) || n.equals(5)) return true;
        if (n.isEven() || n.isDivisibleBy(3) || n.isDivisibleBy(5)) return false;
        if (n.lesser(25)) return true;
        // we don't know if it's prime: let the other functions figure it out
    }

    BigInteger.prototype.isPrime = function () {
        var isPrime = isBasicPrime(this);
        if (isPrime !== undefined) return isPrime;
        var n = this.abs(),
            nPrev = n.prev();
        var a = [2, 3, 5, 7, 11, 13, 17, 19],
            b = nPrev,
            d, t, i, x;
        while (b.isEven()) b = b.divide(2);
        for (i = 0; i < a.length; i++) {
            x = bigInt(a[i]).modPow(b, n);
            if (x.equals(Integer[1]) || x.equals(nPrev)) continue;
            for (t = true, d = b; t && d.lesser(nPrev) ; d = d.multiply(2)) {
                x = x.square().mod(n);
                if (x.equals(nPrev)) t = false;
            }
            if (t) return false;
        }
        return true;
    };
    SmallInteger.prototype.isPrime = BigInteger.prototype.isPrime;

    BigInteger.prototype.isProbablePrime = function (iterations) {
        var isPrime = isBasicPrime(this);
        if (isPrime !== undefined) return isPrime;
        var n = this.abs();
        var t = iterations === undefined ? 5 : iterations;
        // use the Fermat primality test
        for (var i = 0; i < t; i++) {
            var a = bigInt.randBetween(2, n.minus(2));
            if (!a.modPow(n.prev(), n).isUnit()) return false; // definitely composite
        }
        return true; // large chance of being prime
    };
    SmallInteger.prototype.isProbablePrime = BigInteger.prototype.isProbablePrime;

    BigInteger.prototype.modInv = function (n) {
        var t = bigInt.zero, newT = bigInt.one, r = parseValue(n), newR = this.abs(), q, lastT, lastR;
        while (!newR.equals(bigInt.zero)) {
            q = r.divide(newR);
            lastT = t;
            lastR = r;
            t = newT;
            r = newR;
            newT = lastT.subtract(q.multiply(newT));
            newR = lastR.subtract(q.multiply(newR));
        }
        if (!r.equals(1)) throw new Error(this.toString() + " and " + n.toString() + " are not co-prime");
        if (t.compare(0) === -1) {
            t = t.add(n);
        }
        if (this.isNegative()) {
            return t.negate();
        }
        return t;
    };

    SmallInteger.prototype.modInv = BigInteger.prototype.modInv;

    BigInteger.prototype.next = function () {
        var value = this.value;
        if (this.sign) {
            return subtractSmall(value, 1, this.sign);
        }
        return new BigInteger(addSmall(value, 1), this.sign);
    };
    SmallInteger.prototype.next = function () {
        var value = this.value;
        if (value + 1 < MAX_INT) return new SmallInteger(value + 1);
        return new BigInteger(MAX_INT_ARR, false);
    };

    BigInteger.prototype.prev = function () {
        var value = this.value;
        if (this.sign) {
            return new BigInteger(addSmall(value, 1), true);
        }
        return subtractSmall(value, 1, this.sign);
    };
    SmallInteger.prototype.prev = function () {
        var value = this.value;
        if (value - 1 > -MAX_INT) return new SmallInteger(value - 1);
        return new BigInteger(MAX_INT_ARR, true);
    };

    var powersOfTwo = [1];
    while (2 * powersOfTwo[powersOfTwo.length - 1] <= BASE) powersOfTwo.push(2 * powersOfTwo[powersOfTwo.length - 1]);
    var powers2Length = powersOfTwo.length, highestPower2 = powersOfTwo[powers2Length - 1];

    function shift_isSmall(n) {
        return ((typeof n === "number" || typeof n === "string") && +Math.abs(n) <= BASE) ||
            (n instanceof BigInteger && n.value.length <= 1);
    }

    BigInteger.prototype.shiftLeft = function (n) {
        if (!shift_isSmall(n)) {
            throw new Error(String(n) + " is too large for shifting.");
        }
        n = +n;
        if (n < 0) return this.shiftRight(-n);
        var result = this;
        while (n >= powers2Length) {
            result = result.multiply(highestPower2);
            n -= powers2Length - 1;
        }
        return result.multiply(powersOfTwo[n]);
    };
    SmallInteger.prototype.shiftLeft = BigInteger.prototype.shiftLeft;

    BigInteger.prototype.shiftRight = function (n) {
        var remQuo;
        if (!shift_isSmall(n)) {
            throw new Error(String(n) + " is too large for shifting.");
        }
        n = +n;
        if (n < 0) return this.shiftLeft(-n);
        var result = this;
        while (n >= powers2Length) {
            if (result.isZero()) return result;
            remQuo = divModAny(result, highestPower2);
            result = remQuo[1].isNegative() ? remQuo[0].prev() : remQuo[0];
            n -= powers2Length - 1;
        }
        remQuo = divModAny(result, powersOfTwo[n]);
        return remQuo[1].isNegative() ? remQuo[0].prev() : remQuo[0];
    };
    SmallInteger.prototype.shiftRight = BigInteger.prototype.shiftRight;

    function bitwise(x, y, fn) {
        y = parseValue(y);
        var xSign = x.isNegative(), ySign = y.isNegative();
        var xRem = xSign ? x.not() : x,
            yRem = ySign ? y.not() : y;
        var xDigit = 0, yDigit = 0;
        var xDivMod = null, yDivMod = null;
        var result = [];
        while (!xRem.isZero() || !yRem.isZero()) {
            xDivMod = divModAny(xRem, highestPower2);
            xDigit = xDivMod[1].toJSNumber();
            if (xSign) {
                xDigit = highestPower2 - 1 - xDigit; // two's complement for negative numbers
            }

            yDivMod = divModAny(yRem, highestPower2);
            yDigit = yDivMod[1].toJSNumber();
            if (ySign) {
                yDigit = highestPower2 - 1 - yDigit; // two's complement for negative numbers
            }

            xRem = xDivMod[0];
            yRem = yDivMod[0];
            result.push(fn(xDigit, yDigit));
        }
        var sum = fn(xSign ? 1 : 0, ySign ? 1 : 0) !== 0 ? bigInt(-1) : bigInt(0);
        for (var i = result.length - 1; i >= 0; i -= 1) {
            sum = sum.multiply(highestPower2).add(bigInt(result[i]));
        }
        return sum;
    }

    BigInteger.prototype.not = function () {
        return this.negate().prev();
    };
    SmallInteger.prototype.not = BigInteger.prototype.not;

    BigInteger.prototype.and = function (n) {
        return bitwise(this, n, function (a, b) { return a & b; });
    };
    SmallInteger.prototype.and = BigInteger.prototype.and;

    BigInteger.prototype.or = function (n) {
        return bitwise(this, n, function (a, b) { return a | b; });
    };
    SmallInteger.prototype.or = BigInteger.prototype.or;

    BigInteger.prototype.xor = function (n) {
        return bitwise(this, n, function (a, b) { return a ^ b; });
    };
    SmallInteger.prototype.xor = BigInteger.prototype.xor;

    var LOBMASK_I = 1 << 30, LOBMASK_BI = (BASE & -BASE) * (BASE & -BASE) | LOBMASK_I;
    function roughLOB(n) { // get lowestOneBit (rough)
        // SmallInteger: return Min(lowestOneBit(n), 1 << 30)
        // BigInteger: return Min(lowestOneBit(n), 1 << 14) [BASE=1e7]
        var v = n.value, x = typeof v === "number" ? v | LOBMASK_I : v[0] + v[1] * BASE | LOBMASK_BI;
        return x & -x;
    }

    function max(a, b) {
        a = parseValue(a);
        b = parseValue(b);
        return a.greater(b) ? a : b;
    }
    function min(a, b) {
        a = parseValue(a);
        b = parseValue(b);
        return a.lesser(b) ? a : b;
    }
    function gcd(a, b) {
        a = parseValue(a).abs();
        b = parseValue(b).abs();
        if (a.equals(b)) return a;
        if (a.isZero()) return b;
        if (b.isZero()) return a;
        var c = Integer[1], d, t;
        while (a.isEven() && b.isEven()) {
            d = Math.min(roughLOB(a), roughLOB(b));
            a = a.divide(d);
            b = b.divide(d);
            c = c.multiply(d);
        }
        while (a.isEven()) {
            a = a.divide(roughLOB(a));
        }
        do {
            while (b.isEven()) {
                b = b.divide(roughLOB(b));
            }
            if (a.greater(b)) {
                t = b; b = a; a = t;
            }
            b = b.subtract(a);
        } while (!b.isZero());
        return c.isUnit() ? a : a.multiply(c);
    }
    function lcm(a, b) {
        a = parseValue(a).abs();
        b = parseValue(b).abs();
        return a.divide(gcd(a, b)).multiply(b);
    }
    function randBetween(a, b) {
        a = parseValue(a);
        b = parseValue(b);
        var low = min(a, b), high = max(a, b);
        var range = high.subtract(low).add(1);
        if (range.isSmall) return low.add(Math.floor(Math.random() * range));
        var length = range.value.length - 1;
        var result = [], restricted = true;
        for (var i = length; i >= 0; i--) {
            var top = restricted ? range.value[i] : BASE;
            var digit = truncate(Math.random() * top);
            result.unshift(digit);
            if (digit < top) restricted = false;
        }
        result = arrayToSmall(result);
        return low.add(typeof result === "number" ? new SmallInteger(result) : new BigInteger(result, false));
    }
    var parseBase = function (text, base) {
        var length = text.length;
		var i;
		var absBase = Math.abs(base);
		for(var i = 0; i < length; i++) {
			var c = text[i].toLowerCase();
			if(c === "-") continue;
			if(/[a-z0-9]/.test(c)) {
			    if(/[0-9]/.test(c) && +c >= absBase) {
					if(c === "1" && absBase === 1) continue;
                    throw new Error(c + " is not a valid digit in base " + base + ".");
				} else if(c.charCodeAt(0) - 87 >= absBase) {
					throw new Error(c + " is not a valid digit in base " + base + ".");
				}
			}
		}
        if (2 <= base && base <= 36) {
            if (length <= LOG_MAX_INT / Math.log(base)) {
				var result = parseInt(text, base);
				if(isNaN(result)) {
					throw new Error(c + " is not a valid digit in base " + base + ".");
				}
                return new SmallInteger(parseInt(text, base));
            }
        }
        base = parseValue(base);
        var digits = [];
        var isNegative = text[0] === "-";
        for (i = isNegative ? 1 : 0; i < text.length; i++) {
            var c = text[i].toLowerCase(),
                charCode = c.charCodeAt(0);
            if (48 <= charCode && charCode <= 57) digits.push(parseValue(c));
            else if (97 <= charCode && charCode <= 122) digits.push(parseValue(c.charCodeAt(0) - 87));
            else if (c === "<") {
                var start = i;
                do { i++; } while (text[i] !== ">");
                digits.push(parseValue(text.slice(start + 1, i)));
            }
            else throw new Error(c + " is not a valid character");
        }
        return parseBaseFromArray(digits, base, isNegative);
    };

    function parseBaseFromArray(digits, base, isNegative) {
        var val = Integer[0], pow = Integer[1], i;
        for (i = digits.length - 1; i >= 0; i--) {
            val = val.add(digits[i].times(pow));
            pow = pow.times(base);
        }
        return isNegative ? val.negate() : val;
    }

    function stringify(digit) {
        var v = digit.value;
        if (typeof v === "number") v = [v];
        if (v.length === 1 && v[0] <= 35) {
            return "0123456789abcdefghijklmnopqrstuvwxyz".charAt(v[0]);
        }
        return "<" + v + ">";
    }
    function toBase(n, base) {
        base = bigInt(base);
        if (base.isZero()) {
            if (n.isZero()) return "0";
            throw new Error("Cannot convert nonzero numbers to base 0.");
        }
        if (base.equals(-1)) {
            if (n.isZero()) return "0";
            if (n.isNegative()) return new Array(1 - n).join("10");
            return "1" + new Array(+n).join("01");
        }
        var minusSign = "";
        if (n.isNegative() && base.isPositive()) {
            minusSign = "-";
            n = n.abs();
        }
        if (base.equals(1)) {
            if (n.isZero()) return "0";
            return minusSign + new Array(+n + 1).join(1);
        }
        var out = [];
        var left = n, divmod;
        while (left.isNegative() || left.compareAbs(base) >= 0) {
            divmod = left.divmod(base);
            left = divmod.quotient;
            var digit = divmod.remainder;
            if (digit.isNegative()) {
                digit = base.minus(digit).abs();
                left = left.next();
            }
            out.push(stringify(digit));
        }
        out.push(stringify(left));
        return minusSign + out.reverse().join("");
    }

    BigInteger.prototype.toString = function (radix) {
        if (radix === undefined) radix = 10;
        if (radix !== 10) return toBase(this, radix);
        var v = this.value, l = v.length, str = String(v[--l]), zeros = "0000000", digit;
        while (--l >= 0) {
            digit = String(v[l]);
            str += zeros.slice(digit.length) + digit;
        }
        var sign = this.sign ? "-" : "";
        return sign + str;
    };

    SmallInteger.prototype.toString = function (radix) {
        if (radix === undefined) radix = 10;
        if (radix != 10) return toBase(this, radix);
        return String(this.value);
    };
    BigInteger.prototype.toJSON = SmallInteger.prototype.toJSON = function() { return this.toString(); }

    BigInteger.prototype.valueOf = function () {
        return +this.toString();
    };
    BigInteger.prototype.toJSNumber = BigInteger.prototype.valueOf;

    SmallInteger.prototype.valueOf = function () {
        return this.value;
    };
    SmallInteger.prototype.toJSNumber = SmallInteger.prototype.valueOf;

    function parseStringValue(v) {
            if (isPrecise(+v)) {
                var x = +v;
                if (x === truncate(x))
                    return new SmallInteger(x);
                throw "Invalid integer: " + v;
            }
            var sign = v[0] === "-";
            if (sign) v = v.slice(1);
            var split = v.split(/e/i);
            if (split.length > 2) throw new Error("Invalid integer: " + split.join("e"));
            if (split.length === 2) {
                var exp = split[1];
                if (exp[0] === "+") exp = exp.slice(1);
                exp = +exp;
                if (exp !== truncate(exp) || !isPrecise(exp)) throw new Error("Invalid integer: " + exp + " is not a valid exponent.");
                var text = split[0];
                var decimalPlace = text.indexOf(".");
                if (decimalPlace >= 0) {
                    exp -= text.length - decimalPlace - 1;
                    text = text.slice(0, decimalPlace) + text.slice(decimalPlace + 1);
                }
                if (exp < 0) throw new Error("Cannot include negative exponent part for integers");
                text += (new Array(exp + 1)).join("0");
                v = text;
            }
            var isValid = /^([0-9][0-9]*)$/.test(v);
            if (!isValid) throw new Error("Invalid integer: " + v);
            var r = [], max = v.length, l = LOG_BASE, min = max - l;
            while (max > 0) {
                r.push(+v.slice(min, max));
                min -= l;
                if (min < 0) min = 0;
                max -= l;
            }
            trim(r);
            return new BigInteger(r, sign);
    }

    function parseNumberValue(v) {
        if (isPrecise(v)) {
            if (v !== truncate(v)) throw new Error(v + " is not an integer.");
            return new SmallInteger(v);
        }
        return parseStringValue(v.toString());
    }

    function parseValue(v) {
        if (typeof v === "number") {
            return parseNumberValue(v);
        }
        if (typeof v === "string") {
            return parseStringValue(v);
        }
        return v;
    }
    // Pre-define numbers in range [-999,999]
    for (var i = 0; i < 1000; i++) {
        Integer[i] = new SmallInteger(i);
        if (i > 0) Integer[-i] = new SmallInteger(-i);
    }
    // Backwards compatibility
    Integer.one = Integer[1];
    Integer.zero = Integer[0];
    Integer.minusOne = Integer[-1];
    Integer.max = max;
    Integer.min = min;
    Integer.gcd = gcd;
    Integer.lcm = lcm;
    Integer.isInstance = function (x) { return x instanceof BigInteger || x instanceof SmallInteger; };
    Integer.randBetween = randBetween;

    Integer.fromArray = function (digits, base, isNegative) {
        return parseBaseFromArray(digits.map(parseValue), parseValue(base || 10), isNegative);
    };

    return Integer;
})();

// Node.js check
if (typeof module !== "undefined" && module.hasOwnProperty("exports")) {
    module.exports = bigInt;
}

//amd check
if ( typeof define === "function" && define.amd ) {
  define( "big-integer", [], function() {
    return bigInt;
  });
}