CF1698C.3SUM Closure

You are given an array $a$ of length $n$ . The array is called 3SUM-closed if for all distinct indices $i$ , $j$ , $k$ , the sum $a_i + a_j + a_k$ is an element of the array. More formally, $a$ is 3SUM-closed if for all integers $1 \leq i < j < k \leq n$ , there exists some integer $1 \leq l \leq n$ such that $a_i + a_j + a_k = a_l$ .

Determine if $a$ is 3SUM-closed.

The first line contains an integer $t$ ( $1 \leq t \leq 1000$ ) — the number of test cases.

The first line of each test case contains an integer $n$ ( $3 \leq n \leq 2 \cdot 10^5$ ) — the length of the array.

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ( $-10^9 \leq a_i \leq 10^9$ ) — the elements of the array.

It is guaranteed that the sum of $n$ across all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output "YES" (without quotes) if $a$ is 3SUM-closed and "NO" (without quotes) otherwise.

You can output "YES" and "NO" in any case (for example, strings "yEs", "yes" and "Yes" will be recognized as a positive response).

In the first test case, there is only one triple where $i=1$ , $j=2$ , $k=3$ . In this case, $a_1 + a_2 + a_3 = 0$ , which is an element of the array ( $a_2 = 0$ ), so the array is 3SUM-closed.

In the second test case, $a_1 + a_4 + a_5 = -1$ , which is not an element of the array. Therefore, the array is not 3SUM-closed.

In the third test case, $a_i + a_j + a_k = 0$ for all distinct $i$ , $j$ , $k$ , and $0$ is an element of the array, so the array is 3SUM-closed.