3SUM Closure
Description
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).
Input
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 $2 \cdot 10^5$.
Output
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).
Samples
4
3
-1 0 1
5
1 -2 -2 1 -3
6
0 0 0 0 0 0
4
-1 2 -3 4
YES
NO
YES
NO
Note
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.