Make Triangle
Description
You are given $n$ positive integers $x_1, x_2, \ldots, x_n$ and three positive integers $n_a, n_b, n_c$ satisfying $n_a+n_b+n_c = n$.
You want to split the $n$ positive integers into three groups, so that:
- The first group contains $n_a$ numbers, the second group contains $n_b$ numbers, the third group contains $n_c$ numbers.
- Let $s_a$ be the sum of the numbers in the first group, $s_b$ be the sum in the second group, and $s_c$ be the sum in the third group. Then $s_a, s_b, s_c$ are the sides of a triangle with positive area.
Determine if this is possible. If this is possible, find one way to do so.
Each test contains multiple test cases. The first line contains an integer $t$ ($1\le t\le 100\,000$) — the number of test cases. The descriptions of the $t$ test cases follow.
The first line of each test case contains the integers $n, n_a, n_b, n_c$ ($3 \leq n \leq 200\,000, 1\leq n_a,n_b,n_c \leq n-2, n_a+n_b+n_c = n$) — the number of integers to split into three groups, and the desired sizes of the three groups.
The second line of each test case contains $n$ integers $x_1, x_2, \ldots, x_n$ ($1 \leq x_i \leq 10^{9}$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $200\,000$.
For each test case, print $\texttt{YES}$ if it is possible to split the numbers into three groups satisfying all the conditions. Otherwise, print $\texttt{NO}$.
If such a split exists, then describe the three groups as follows.
On the next line, print $n_a$ integers $a_1, a_2, \ldots, a_{n_a}$ — the numbers in the first group.
On the next line, print $n_b$ integers $b_1, b_2, \ldots, b_{n_b}$ — the numbers in the second group.
On the next line, print $n_c$ integers $c_1, c_2, \ldots, c_{n_c}$ — the numbers in the third group.
These $n_a+n_b+n_c=n$ integers should be a permutation of $x_1, x_2, \ldots, x_n$, and they should satisfy the conditions from the statement.
If there are multiple solutions, print any of them.
Input
Each test contains multiple test cases. The first line contains an integer $t$ ($1\le t\le 100\,000$) — the number of test cases. The descriptions of the $t$ test cases follow.
The first line of each test case contains the integers $n, n_a, n_b, n_c$ ($3 \leq n \leq 200\,000, 1\leq n_a,n_b,n_c \leq n-2, n_a+n_b+n_c = n$) — the number of integers to split into three groups, and the desired sizes of the three groups.
The second line of each test case contains $n$ integers $x_1, x_2, \ldots, x_n$ ($1 \leq x_i \leq 10^{9}$).
It is guaranteed that the sum of $n$ over all test cases does not exceed $200\,000$.
Output
For each test case, print $\texttt{YES}$ if it is possible to split the numbers into three groups satisfying all the conditions. Otherwise, print $\texttt{NO}$.
If such a split exists, then describe the three groups as follows.
On the next line, print $n_a$ integers $a_1, a_2, \ldots, a_{n_a}$ — the numbers in the first group.
On the next line, print $n_b$ integers $b_1, b_2, \ldots, b_{n_b}$ — the numbers in the second group.
On the next line, print $n_c$ integers $c_1, c_2, \ldots, c_{n_c}$ — the numbers in the third group.
These $n_a+n_b+n_c=n$ integers should be a permutation of $x_1, x_2, \ldots, x_n$, and they should satisfy the conditions from the statement.
If there are multiple solutions, print any of them.
4
6 2 2 2
1 1 1 1 1 1
5 3 1 1
1 1 1 1 1
6 2 2 2
1 1 1 1 1 3
8 1 2 5
16 1 1 1 1 1 1 12
YES
1 1
1 1
1 1
NO
NO
YES
16
12 1
1 1 1 1 1
Note
In the first test case, we can put two $1$s into each group: the sum in each group would be $2$, and there exists a triangle with positive area and sides $2$, $2$, $2$.
In the second and third test cases, it can be shown that there is no such way to split numbers into groups.
In the fourth test case, we can put number $16$ into the first group, with sum $16$, numbers $12$ and $1$ into the second group, with sum $13$, and the remaining five $1$s into the third group, with sum $5$, as there exists a triangle with positive area and sides $16, 13, 5$.