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())
MOD = 998244353
A = []
S = [0]
for i in range(N):
    A.append(int(input()))
    S.append(S[-1] + A[-1])
S = S[::-1]
A = sorted(A, reverse=True)
dp = [[] for i in range(N + 1)]
dp[0].append((0, 0, 0))
d = {}
d[(0, 0, 0)] = 1
for i in range(1, N + 1):
    k = A[i - 1]
    tmp = d.copy()
    d = {}
    for a, b, c in dp[i - 1]:
        for j in range(3):
            s = [a, b, c]
            s[j] += k
            s = tuple(sorted(s, reverse=True))
            if s[0] > s[1] + s[2] + S[i - 1]:
                continue

            if s in dp[i]:
                d[s] += tmp[(a, b, c)]
                d[s] %= MOD
            else:
                dp[i].append(s)
                d[s] = tmp[(a, b, c)]

num = 0
for a, b, c in dp[N]:
    if a < b + c:
        num += d[(a, b, c)]
        num %= MOD

print(num)