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: Wrong Answer)

N = gets.to_i
a = []
N.times do
  a << gets.to_i
end

MOD = 998244353
total = 0
a.each do |x|
  total += x
end
HALF = (total + 1) / 2

cnt_all = 1
N.times do
  cnt_all *= 3
  cnt_all %= MOD
end

# p cnt_all

# AのコストでDP
cost = { 0 => 1 }
N.times do |i|
  next_cost = Hash.new(0)
  cost.each do |k, v|
    # Aで塗った時
    k1 = k + a[i]
    next_cost[k1] = (next_cost[k1] + v) % MOD

    # B, C で塗った場合
    next_cost[k] = (next_cost[k] + 2 * v) % MOD
  end
  cost = next_cost
end
cnt_a = 0
# p cost
cost.each do |k,v|
  cnt_a += v if k >= HALF
  cnt_a -= MOD if cnt_a > MOD
end
# p cost.values.inject(0, &:+)

# C=0でDP
cnt_ab = 0
if total % 2 == 0
  sum = 0
  cost = { 0 => 1 }
  N.times do |i|
    sum += a[i]
    next_cost = Hash.new(0)
    cost.each do |k, v|
      # Aで塗る場合、
      k1 = k + a[i]
      next_cost[k1] = (next_cost[k1] + v) % MOD

      # Bで塗る場合
      k2 = sum - k
      next_cost[k2] = (next_cost[k2] + v) % MOD

      # Cを塗る場合、カウントしない
    end
    cost = next_cost
  end

  cost.each do |k,v|
    cnt_ab += v if k == HALF
    cnt_ab -= MOD if k > MOD
  end
end

# p [total, cnt_all, cnt_a, cnt_ab]

ans = cnt_all
3.times do
  ans -= cnt_a
  ans += MOD if ans <= 0
end
# p [ans, cnt_ab]
3.times do
  ans += cnt_ab
  ans -= MOD if ans > MOD
end
puts ans