Problem:
Score : 500 points

Problem Statement

For two sequences S and T of length N consisting of 0 and 1, let us define f(S, T) as follows:

Consider repeating the following operation on S so that S will be equal to T. f(S, T) is the minimum possible total cost of those operations.

Change S_i (from 0 to 1 or vice versa). The cost of this operation is D \times C_i, where D is the number of integers j such that S_j \neq T_j (1 \leq j \leq N) just before this change.

There are 2^N \times (2^N - 1) pairs (S, T) of different sequences of length N consisting of 0 and 1. Compute the sum of f(S, T) over all of those pairs, modulo (10^9+7).

Constraints

1 \leq N \leq 2 \times 10^5

1 \leq C_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
C_1 C_2 \cdots C_N

Output

Print the sum of f(S, T), modulo (10^9+7).

Sample Input 1

1
1000000000

Sample Output 1

999999993

There are two pairs (S, T) of different sequences of length 2 consisting of 0 and 1, as follows:

S = (0), T = (1): by changing S_1 to 1, we can have S = T at the cost of 1000000000, so f(S, T) = 1000000000.

S = (1), T = (0): by changing S_1 to 0, we can have S = T at the cost of 1000000000, so f(S, T) = 1000000000.

The sum of these is 2000000000, and we should print it modulo (10^9+7), that is, 999999993.

Sample Input 2

2
5 8

Sample Output 2

124

There are 12 pairs (S, T) of different sequences of length 3 consisting of 0 and 1, which include:

S = (0, 1), T = (1, 0)

In this case, if we first change S_1 to 1 then change S_2 to 0, the total cost is 5 \times 2 + 8 = 18. We cannot have S = T at a smaller cost, so f(S, T) = 18.

Sample Input 3

5
52 67 72 25 79

Sample Output 3

269312

Solution in Rust (1.42.0) (judge verdict: Accepted)

#[allow(unused_imports)]
use itertools::Itertools;
#[allow(unused_imports)]
use num::*;
use proconio::input;
#[allow(unused_imports)]
use proconio::marker::*;
#[allow(unused_imports)]
use std::collections::*;

#[derive(Copy, Clone, Debug, Eq, PartialEq)]
pub struct ModInt(u32);

impl ModInt {
    pub const MOD: u32 = 1_000_000_007;

    pub fn inv(self) -> Self {
        if self.0 == 0 {
            panic!();
        }
        self.pow(Self::MOD - 2)
    }

    pub fn one() -> Self {
        Self(1)
    }

    pub fn pow(self, e: u32) -> Self {
        if e == 0 {
            return Self::new(1);
        }
        let mut res = self.pow(e >> 1);
        res *= res;
        if e & 1 == 1 {
            res *= self;
        }
        res
    }

    pub fn zero() -> Self {
        Self(0)
    }

    fn new(n: i64) -> Self {
        let mut n = n % (Self::MOD as i64);
        if n.is_negative() {
            n += Self::MOD as i64;
        }
        Self(n as u32)
    }
}

impl From<i32> for ModInt {
    fn from(n: i32) -> Self {
        ModInt::from(n as i64)
    }
}

impl From<i64> for ModInt {
    fn from(n: i64) -> Self {
        Self::new(n)
    }
}

impl From<isize> for ModInt {
    fn from(n: isize) -> Self {
        ModInt::from(n as i64)
    }
}

impl From<u32> for ModInt {
    fn from(n: u32) -> Self {
        ModInt::from(n as u64)
    }
}

impl From<u64> for ModInt {
    fn from(n: u64) -> Self {
        Self::new(n as i64)
    }
}

impl From<usize> for ModInt {
    fn from(n: usize) -> Self {
        ModInt::from(n as u64)
    }
}

impl Into<i32> for ModInt {
    fn into(self) -> i32 {
        self.0 as i32
    }
}

impl Into<i64> for ModInt {
    fn into(self) -> i64 {
        self.0 as i64
    }
}

impl Into<isize> for ModInt {
    fn into(self) -> isize {
        self.0 as isize
    }
}

impl Into<u32> for ModInt {
    fn into(self) -> u32 {
        self.0
    }
}

impl Into<u64> for ModInt {
    fn into(self) -> u64 {
        self.0 as u64
    }
}

impl Into<usize> for ModInt {
    fn into(self) -> usize {
        self.0 as usize
    }
}

impl std::fmt::Display for ModInt {
    fn fmt(&self, f: &mut std::fmt::Formatter<'_>) -> std::fmt::Result {
        write!(f, "{}", self.0)
    }
}

impl std::ops::Add for ModInt {
    type Output = Self;

    fn add(self, rhs: Self) -> Self {
        Self::new((self.0 + rhs.0) as i64)
    }
}

impl std::ops::AddAssign for ModInt {
    fn add_assign(&mut self, rhs: Self) {
        *self = *self + rhs;
    }
}

impl std::ops::Div for ModInt {
    type Output = Self;

    fn div(self, rhs: Self) -> Self {
        self * rhs.inv()
    }
}

impl std::ops::DivAssign for ModInt {
    fn div_assign(&mut self, rhs: Self) {
        *self = *self / rhs;
    }
}

impl std::ops::Mul for ModInt {
    type Output = Self;

    fn mul(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) * (rhs.0 as i64))
    }
}

impl std::ops::MulAssign for ModInt {
    fn mul_assign(&mut self, rhs: Self) {
        *self = *self * rhs;
    }
}

impl std::ops::Sub for ModInt {
    type Output = Self;

    fn sub(self, rhs: Self) -> Self {
        Self::new((self.0 as i64) - (rhs.0 as i64))
    }
}

impl std::ops::SubAssign for ModInt {
    fn sub_assign(&mut self, rhs: Self) {
        *self = *self - rhs;
    }
}

fn solve() {
    input! {
        n: usize,
        mut cv: [usize; n]
    };

    cv.sort();
    let all_perm_count = ModInt::from(2).pow(n as u32);
    let mut res = ModInt::zero();
    for i in 0..n {
        let c = ModInt::from(cv[i]);
        let len = n - i;
        let len_perm_count = ModInt::from(2).pow((len - 1) as u32);
        let change_count =
            len_perm_count + (len_perm_count / ModInt::from(2)) * ModInt::from(len - 1);
        let sum = c * ModInt::from(2).pow(i as u32) * change_count * all_perm_count;
        res += sum;
    }
    println!("{}", res);
}

fn main() {
    std::thread::Builder::new()
        .name("big stack size".into())
        .stack_size(256 * 1024 * 1024)
        .spawn(|| {
            solve();
        })
        .unwrap()
        .join()
        .unwrap();
}