#line 1 "main.cpp" #include #line 2 "/home/user/Library/utils/macros.hpp" #define REP(i, n) for (int i = 0; (i) < (int)(n); ++ (i)) #define REP3(i, m, n) for (int i = (m); (i) < (int)(n); ++ (i)) #define REP_R(i, n) for (int i = (int)(n) - 1; (i) >= 0; -- (i)) #define REP3R(i, m, n) for (int i = (int)(n) - 1; (i) >= (int)(m); -- (i)) #define ALL(x) std::begin(x), std::end(x) #line 4 "/home/user/Library/modulus/modpow.hpp" inline int32_t modpow(uint_fast64_t x, uint64_t k, int32_t MOD) { assert (/* 0 <= x and */ x < (uint_fast64_t)MOD); uint_fast64_t y = 1; for (; k; k >>= 1) { if (k & 1) (y *= x) %= MOD; (x *= x) %= MOD; } assert (/* 0 <= y and */ y < (uint_fast64_t)MOD); return y; } #line 5 "/home/user/Library/modulus/modinv.hpp" inline int32_t modinv_nocheck(int32_t value, int32_t MOD) { assert (0 <= value and value < MOD); if (value == 0) return -1; int64_t a = value, b = MOD; int64_t x = 0, y = 1; for (int64_t u = 1, v = 0; a; ) { int64_t q = b / a; x -= q * u; std::swap(x, u); y -= q * v; std::swap(y, v); b -= q * a; std::swap(b, a); } if (not (value * x + MOD * y == b and b == 1)) return -1; if (x < 0) x += MOD; assert (0 <= x and x < MOD); return x; } inline int32_t modinv(int32_t x, int32_t MOD) { int32_t y = modinv_nocheck(x, MOD); assert (y != -1); return y; } #line 6 "/home/user/Library/modulus/mint.hpp" /** * @brief quotient ring / 剰余環 $\mathbb{Z}/n\mathbb{Z}$ */ template struct mint { int32_t value; mint() : value() {} mint(int64_t value_) : value(value_ < 0 ? value_ % MOD + MOD : value_ >= MOD ? value_ % MOD : value_) {} mint(int32_t value_, std::nullptr_t) : value(value_) {} explicit operator bool() const { return value; } inline mint operator + (mint other) const { return mint(*this) += other; } inline mint operator - (mint other) const { return mint(*this) -= other; } inline mint operator * (mint other) const { return mint(*this) *= other; } inline mint & operator += (mint other) { this->value += other.value; if (this->value >= MOD) this->value -= MOD; return *this; } inline mint & operator -= (mint other) { this->value -= other.value; if (this->value < 0) this->value += MOD; return *this; } inline mint & operator *= (mint other) { this->value = (uint_fast64_t)this->value * other.value % MOD; return *this; } inline mint operator - () const { return mint(this->value ? MOD - this->value : 0, nullptr); } inline bool operator == (mint other) const { return value == other.value; } inline bool operator != (mint other) const { return value != other.value; } inline mint pow(uint64_t k) const { return mint(modpow(value, k, MOD), nullptr); } inline mint inv() const { return mint(modinv(value, MOD), nullptr); } inline mint operator / (mint other) const { return *this * other.inv(); } inline mint & operator /= (mint other) { return *this *= other.inv(); } }; template mint operator + (int64_t value, mint n) { return mint(value) + n; } template mint operator - (int64_t value, mint n) { return mint(value) - n; } template mint operator * (int64_t value, mint n) { return mint(value) * n; } template mint operator / (int64_t value, mint n) { return mint(value) / n; } template std::istream & operator >> (std::istream & in, mint & n) { int64_t value; in >> value; n = value; return in; } template std::ostream & operator << (std::ostream & out, mint n) { return out << n.value; } #line 5 "/home/user/Library/number/karatsuba.hpp" /** * @brief Karatsuba method ($O(n^{\log_2 3})$) */ template std::vector karatsuba_convolution(const std::vector & x, const std::vector & y) { int n = x.size(); int m = y.size(); if ((int64_t)n * m <= 100) { std::vector z(n + m - 1); REP (i, n) REP (j, m) { z[i + j] += x[i] * y[j]; } return z; } int half = (std::max(n, m) + 1) / 2; std::vector x0(x.begin(), x.begin() + std::min(n, half)); std::vector y0(y.begin(), y.begin() + std::min(m, half)); std::vector z0 = karatsuba_convolution(x0, y0); std::vector x1(x.begin() + std::min(n, half), x.end()); std::vector y1(y.begin() + std::min(m, half), y.end()); std::vector z2 = karatsuba_convolution(x1, y1); assert (x1.size() <= x0.size()); std::vector dx = x0; REP (i, x1.size()) dx[i] -= x1[i]; assert (y1.size() <= y0.size()); std::vector dy = y0; REP (i, y1.size()) dy[i] -= y1[i]; std::vector dz = karatsuba_convolution(dx, dy); std::vector z(n + m - 1); REP (i, z0.size()) { z[i] += z0[i]; if (half + i < (int)z.size()) z[half + i] += z0[i]; } REP (i, dz.size()) { if (half + i < (int)z.size()) z[half + i] -= dz[i]; } REP (i, z2.size()) { z[half + i] += z2[i]; if (2 * half + i < (int)z.size()) z[2 * half + i] += z2[i]; } return z; } #line 5 "main.cpp" using namespace std; template std::vector karatsuba_special_convolution(const std::vector & x) { int n = x.size(); if ((int64_t)n * n <= 100) { std::vector z(2 * n - 1); REP (i, n) REP3 (j, i + 1, n) { z[i + j] += x[i] * x[j]; } return z; } int half = (n + 1) / 2; std::vector x0(x.begin(), x.begin() + std::min(n, half)); std::vector x1(x.begin() + std::min(n, half), x.end()); std::vector z0 = karatsuba_special_convolution(x0); std::vector z1 = karatsuba_convolution(x0, x1); std::vector z2 = karatsuba_special_convolution(x1); std::vector z(2 * n - 1); REP (i, z0.size()) { z[i] += z0[i]; if (half + i < (int)z.size()) z[half + i] += z0[i]; } REP (i, z1.size()) { if (half + i < (int)z.size()) z[half + i] += z1[i]; } REP (i, z2.size()) { if (2 * half + i < (int)z.size()) z[2 * half + i] += z2[i]; } return z; } constexpr int64_t MOD = 1000000007; // the commutative ring (F, *, +) for the finite field (F, +, *) struct cring_t { mint value; cring_t() : value(1) {} }; cring_t & operator += (cring_t & a, cring_t b) { a.value *= b.value; return a; } cring_t & operator -= (cring_t & a, cring_t b) { a.value /= b.value; return a; } cring_t & operator *= (cring_t & a, cring_t b) { a.value += b.value; return a; } cring_t operator + (cring_t a, cring_t b) { return a += b; } cring_t operator - (cring_t a, cring_t b) { return a -= b; } cring_t operator * (cring_t a, cring_t b) { return a *= b; } mint solve(int n, const vector &a) { mint x = 1; { vector b(n); REP (i, n) { b[i].value = a[i]; } auto c = karatsuba_special_convolution(b); for (auto c_i : c) { x *= c_i.value; } } mint y = 1; { int64_t sum_a_j = 0; REP_R (i, n) { y *= mint(a[i]).pow(sum_a_j); sum_a_j += a[i]; } } mint z = 0; { double log_z = INFINITY; int64_t a_j = a[n - 1]; REP_R (i, n - 1) { double log_a_i_a_j = log(a[i] + a_j) + a_j * log(a[i]); if (log_a_i_a_j < log_z) { log_z = log_a_i_a_j; z = mint(a[i] + a_j) * mint(a[i]).pow(a_j); } a_j = min(a[i], a_j); } } return x * y / z; } int main() { int n; cin >> n; vector a(n); REP (i, n) { cin >> a[i]; } auto ans = solve(n, a); cout << ans << endl; return 0; }