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 Ruby (2.3.3) (judge verdict: Runtime Error)

N = gets.chomp.to_i

a = []
sum = 0
x = 0
while x < N
  a[x] = gets.chomp.to_i
  sum += a[x]
  x += 1
end

ng = (sum+1)/2
max_a = a.max

arr = Array.new(N+1){ Array.new(max_a*N+1,0) }

arr[0][0] = 1
t=1
while t <= N
  r = (ng-max_a*(N-t) > 0 ? ng-max_a*(N-t) : 0)
  while r <= (ng+max_a < max_a ? ng+max_a : max_a*t)
    arr[t][r] = r >= a[t-1] ? (arr[t-1][r-a[t-1]] + 2*arr[t-1][r]) % 998244353 : arr[t][r] = (2*arr[t-1][r]) % 998244353
    r += 1
  end
  aa = ng+max_a
  bb = ng+max_a*2
  while r <= bb && r > aa
    arr[t][r] = (3*arr[t-1][r]) % 998244353
    r += 1
  end
  t += 1
end

arr2 = Array.new(N+1){ Array.new(max_a*N+1,0) }
arr2[0][0] = 1
t=1
while t <= N
  r = (ng-max_a*(N-t+1) > 0 ? ng-max_a*(N-t+1) : 0)
  while r <= ng
    arr2[t][r] = r >= a[t-1] ? (arr2[t-1][r-a[t-1]] + arr2[t-1][r]) % 998244353 : arr2[t][r] = (arr2[t-1][r]) % 998244353
    r += 1
  end
  t += 1
end

c1 = 0

x = ng
while x <= max_a*N
  c1 += arr[N][x]
  x += 1
end
c1 = c1 % 998244353 

c2 = sum % 2 == 0 ? arr2[N][ng] : 0

all = 1

x = 1
while x <= N
  all *= 3
  all = all % 998244353
  x += 1 
end

answer = (all - c1*3 + c2*3 + 998244353) % 998244353 

puts answer