use std::cmp::*; // https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8 macro_rules! input { ($($r:tt)*) => { let stdin = std::io::stdin(); let mut bytes = std::io::Read::bytes(std::io::BufReader::new(stdin.lock())); let mut next = move || -> String{ bytes .by_ref() .map(|r|r.unwrap() as char) .skip_while(|c|c.is_whitespace()) .take_while(|c|!c.is_whitespace()) .collect() }; input_inner!{next, $($r)*} }; } macro_rules! input_inner { ($next:expr) => {}; ($next:expr, ) => {}; ($next:expr, $var:ident : $t:tt $($r:tt)*) => { let $var = read_value!($next, $t); input_inner!{$next $($r)*} }; } macro_rules! read_value { ($next:expr, [ $t:tt ; $len:expr ]) => { (0..$len).map(|_| read_value!($next, $t)).collect::>() }; ($next:expr, $t:ty) => { $next().parse::<$t>().expect("Parse error") }; } /// Verified by https://atcoder.jp/contests/abc198/submissions/21774342 mod mod_int { use std::ops::*; pub trait Mod: Copy { fn m() -> i64; } #[derive(Copy, Clone, Hash, PartialEq, Eq, PartialOrd, Ord)] pub struct ModInt { pub x: i64, phantom: ::std::marker::PhantomData } impl ModInt { // x >= 0 pub fn new(x: i64) -> Self { ModInt::new_internal(x % M::m()) } fn new_internal(x: i64) -> Self { ModInt { x: x, phantom: ::std::marker::PhantomData } } pub fn pow(self, mut e: i64) -> Self { debug_assert!(e >= 0); let mut sum = ModInt::new_internal(1); let mut cur = self; while e > 0 { if e % 2 != 0 { sum *= cur; } cur *= cur; e /= 2; } sum } #[allow(dead_code)] pub fn inv(self) -> Self { self.pow(M::m() - 2) } } impl Default for ModInt { fn default() -> Self { Self::new_internal(0) } } impl>> Add for ModInt { type Output = Self; fn add(self, other: T) -> Self { let other = other.into(); let mut sum = self.x + other.x; if sum >= M::m() { sum -= M::m(); } ModInt::new_internal(sum) } } impl>> Sub for ModInt { type Output = Self; fn sub(self, other: T) -> Self { let other = other.into(); let mut sum = self.x - other.x; if sum < 0 { sum += M::m(); } ModInt::new_internal(sum) } } impl>> Mul for ModInt { type Output = Self; fn mul(self, other: T) -> Self { ModInt::new(self.x * other.into().x % M::m()) } } impl>> AddAssign for ModInt { fn add_assign(&mut self, other: T) { *self = *self + other; } } impl>> SubAssign for ModInt { fn sub_assign(&mut self, other: T) { *self = *self - other; } } impl>> MulAssign for ModInt { fn mul_assign(&mut self, other: T) { *self = *self * other; } } impl Neg for ModInt { type Output = Self; fn neg(self) -> Self { ModInt::new(0) - self } } impl ::std::fmt::Display for ModInt { fn fmt(&self, f: &mut ::std::fmt::Formatter) -> ::std::fmt::Result { self.x.fmt(f) } } impl From for ModInt { fn from(x: i64) -> Self { Self::new(x) } } } // mod mod_int macro_rules! define_mod { ($struct_name: ident, $modulo: expr) => { #[derive(Copy, Clone, PartialEq, Eq, PartialOrd, Ord, Hash)] struct $struct_name {} impl mod_int::Mod for $struct_name { fn m() -> i64 { $modulo } } } } const MOD: i64 = 1_000_000_007; define_mod!(P, MOD); type MInt = mod_int::ModInt

; // FFT (in-place, verified as NTT only) // R: Ring + Copy // Verified by: https://judge.yosupo.jp/submission/53831 // Adopts the technique used in https://judge.yosupo.jp/submission/3153. mod fft { use std::ops::*; // n should be a power of 2. zeta is a primitive n-th root of unity. // one is unity // Note that the result is bit-reversed. pub fn fft(f: &mut [R], zeta: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let mut m = n; let mut base = zeta; unsafe { while m > 2 { m >>= 1; let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m); *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = w * (u - d); w = w * base; } r += 2 * m; } base = base * base; } if m > 1 { // m = 1 let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } } } } pub fn inv_fft(f: &mut [R], zeta_inv: R, one: R) where R: Copy + Add + Sub + Mul { let n = f.len(); assert!(n.is_power_of_two()); let zeta = zeta_inv; // inverse FFT let mut zetapow = Vec::with_capacity(20); { let mut m = 1; let mut cur = zeta; while m < n { zetapow.push(cur); cur = cur * cur; m *= 2; } } let mut m = 1; unsafe { if m < n { zetapow.pop(); let mut r = 0; while r < n { let &u = f.get_unchecked(r); let d = *f.get_unchecked(r + 1); *f.get_unchecked_mut(r) = u + d; *f.get_unchecked_mut(r + 1) = u - d; r += 2; } m = 2; } while m < n { let base = zetapow.pop().unwrap(); let mut r = 0; while r < n { let mut w = one; for s in r..r + m { let &u = f.get_unchecked(s); let d = *f.get_unchecked(s + m) * w; *f.get_unchecked_mut(s) = u + d; *f.get_unchecked_mut(s + m) = u - d; w = w * base; } r += 2 * m; } m *= 2; } } } } /// Depends on ModInt.rs /// Finds x modulo M1*M2 s.t. x = a (mod M1), x = b (mod M2). /// Verified by https://yukicoder.me/submissions/303386. fn garner2(a: mod_int::ModInt, b: mod_int::ModInt) -> i64 { let factor2 = mod_int::ModInt::new(M1::m()).inv(); let factor1 = mod_int::ModInt::new(M2::m()).inv(); ((b * factor2).x * M1::m() + (a * factor1).x * M2::m()) % (M1::m() * M2::m()) } mod arbmod { use crate::mod_int::{self, ModInt}; use crate::fft; const MOD1: i64 = 1012924417; const MOD2: i64 = 1224736769; const G1: i64 = 5; const G2: i64 = 3; define_mod!(P1, MOD1); define_mod!(P2, MOD2); // f *= g, g is destroyed fn convolution_friendly(a: &[i64], b: &[i64], gen: i64) -> Vec { let d = a.len(); let mut f = vec![ModInt::

::new(0); d]; let mut g = vec![ModInt::

::new(0); d]; for i in 0..d { f[i] = a[i].into(); g[i] = b[i].into(); } let zeta = ModInt::new(gen).pow((P::m() - 1) / d as i64); fft::fft(&mut f, zeta, ModInt::new(1)); fft::fft(&mut g, zeta, ModInt::new(1)); for i in 0..d { f[i] *= g[i]; } fft::inv_fft(&mut f, zeta.inv(), ModInt::new(1)); let inv = ModInt::new(d as i64).inv(); let mut ans = vec![0; d]; for i in 0..d { ans[i] = (f[i] * inv).x; } ans } // Precondition: 0 <= a[i], b[i] < mo pub fn arbmod_convolution(a: &[i64], b: &[i64], ret: &mut [i64]) { let d = a.len(); assert!(d.is_power_of_two()); assert_eq!(d, b.len()); let x = convolution_friendly::(&a, &b, G1); let y = convolution_friendly::(&a, &b, G2); for i in 0..d { ret[i] = super::garner2(ModInt::::new(x[i]), ModInt::::new(y[i])); } } } // https://yukicoder.me/problems/no/856 (4.5) //A_i^{A_j} の積は累積和で簡単。(A_i + A_j)A_i^{A_j} の最小値については、i について調べるべき j が 1 通りに定まるので、対数などを用いて N 通り調べれば良い。A_i + A_j の積については、A_i が小さいことから、A_i を添字とした畳み込みで A_i + A_j の頻度がわかる。 // Tags: arbitrary-modulo-convolution fn main() { input! { n: usize, a: [usize; n], } const W: usize = 1 << 18; let mut ret = vec![0; W]; let mut f = vec![0; W]; for &a in &a { f[a] += 1; } arbmod::arbmod_convolution(&f, &f, &mut ret); for &a in &a { ret[2 * a] -= 1; } for i in 0..W { ret[i] /= 2; } let mut prod = MInt::new(1); let mut acc = vec![0; n + 1]; for i in (0..n).rev() { acc[i] = acc[i + 1] + a[i] as i64; } for i in 0..n { prod *= MInt::new(a[i] as i64).pow(acc[i + 1]); } for i in 0..W { if ret[i] > 0 { prod *= MInt::new(i as i64).pow(ret[i]); } } let mut mi = (1.0 / 0.0, MInt::new(0)); let mut ami = a[n - 1]; for i in (0..n - 1).rev() { let x = a[i] as f64; let y = ami as f64; let val = y * x.ln() + (x + y).ln(); let key = (val, MInt::new(a[i] as i64).pow(ami as i64) * (a[i] + ami) as i64); if mi > key { mi = key; } ami = min(ami, a[i]); } println!("{}", prod * mi.1.inv()); }