Problem:
Score : 600 points

Problem Statement

You are given N integers. The i-th integer is a_i.
Find the number, modulo 998244353, of ways to paint each of the integers red, green or blue so that the following condition is satisfied:

Let R, G and B be the sums of the integers painted red, green and blue, respectively. There exists a triangle with positive area whose sides have lengths R, G and B.

Constraints

3 \leq N \leq 300

1 \leq a_i \leq 300(1\leq i\leq N)

All values in input are integers.

Input

Input is given from Standard Input in the following format:

N
a_1
:
a_N

Output

Print the number, modulo 998244353, of ways to paint each of the integers red, green or blue so that the condition is satisfied.

Sample Input 1

4
1
1
1
2

Sample Output 1

18

We can only paint the integers so that the lengths of the sides of the triangle will be 1, 2 and 2, and there are 18 such ways.

Sample Input 2

6
1
3
2
3
5
2

Sample Output 2

150

Sample Input 3

20
3
1
4
1
5
9
2
6
5
3
5
8
9
7
9
3
2
3
8
4

Sample Output 3

563038556

Solution in C++14 (GCC 5.4.1) (judge verdict: Time Limit Exceeded)

#include <bits/stdc++.h>

using namespace std;

#define int long long

int mod = 998244353;
int a[305];
map < vector < int >, int > one[305];
vector < int > c;
vector < int > c1;

main(){
    ios_base::sync_with_stdio(0);
    cin.tie(0);
    cout.tie(0);
    int n;
    cin >> n;
    for (int i = 1; i <= n; ++i) cin >> a[i];
    one[1][{0, 0, a[1]}] = 1;
    for (int i = 2; i <= n; ++i){
        for (auto key : one[i - 1]){
            c = key.first;
            for (int j = 0; j < 3; ++j){
                c1 = c;
                c1[j] += a[i];
                sort(c1.begin(), c1.end());
                one[i][c1] += key.second;
                if (one[i][c1] >= mod) one[i][c1] -= mod;
            }
        }
    }
    int ans = 0;
    for (auto key : one[n]){
        if (key.first[0] * 2 + key.first[1] > key.first[0] + key.first[2] && key.first[0] != 0) ans += key.second;
        if (ans >= mod) ans -= mod;
    }
    cout << ans * 3 % mod << '\n';
    return 0;
}