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 PyPy3 (2.4.0) (judge verdict: Time Limit Exceeded)

N = int(input())
A = [int(input()) for i in range(N)]
from functools import lru_cache
# @lru_cache(maxsize=None)
# def solve(i, s):
#     if i==-1:
#         if s==0:
#             return 1
#         else:
#             return 0
#     if s<0:
#         return 0
#     return solve(i-1, s) + 2 * solve(i-1, s-A[i])

mod = 998244353
@lru_cache(maxsize=None)
def solve(i, s, m):
    if i==-1:
        if s==0 and m==0:
            return 1
        else:
            return 0
    if s<0 or m<0:
        return 0
    return (solve(i-1, s, m) + solve(i-1, s-A[i], m) + solve(i-1, s, m-A[i])) % mod

ans = 0
su = sum(A)
for sm in range(su//2+1, su*2//3+1):
    l = su - sm
    for s in range(1, sm//2+1):
        m = sm - s
        if not (s <= m <= l):
            continue
        if s==m==l:
            mul = 1
        elif s==m or m==l:
            mul = 3
        else:
            mul = 6
        ans += solve(N-1, s, m) * mul
        ans %= mod
print(ans)