CF1807G2.Subsequence Addition (Hard Version)

The only difference between the two versions is that in this version, the constraints are higher.

Initially, array $a$ contains just the number $1$ . You can perform several operations in order to change the array. In an operation, you can select some subsequence $^{\dagger}$ of $a$ and add into $a$ an element equal to the sum of all elements of the subsequence.

You are given a final array $c$ . Check if $c$ can be obtained from the initial array $a$ by performing some number (possibly 0) of operations on the initial array.

$^{\dagger}$ A sequence $b$ is a subsequence of a sequence $a$ if $b$ can be obtained from $a$ by the deletion of several (possibly zero, but not all) elements. In other words, select $k$ ( $1 \leq k \leq |a|$ ) distinct indices $i_1, i_2, \dots, i_k$ and insert anywhere into $a$ a new element with the value equal to $a_{i_1} + a_{i_2} + \dots + a_{i_k}$ .

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

The first line of each test case contains a single integer $n$ ( $1 \leq n \leq 2 \cdot 10^5$ ) — the number of elements the final array $c$ should have.

The second line of each test case contains $n$ space-separated integers $c_i$ ( $1 \leq c_i \leq 2 \cdot 10^5$ ) — the elements of the final array $c$ that should be obtained from the initial array $a$ .

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

For each test case, output "YES" (without quotes) if such a sequence of operations exists, and "NO" (without quotes) otherwise.

You can output the answer in any case (for example, the strings "yEs", "yes", "Yes" and "YES" will be recognized as a positive answer).

For the first test case, the initial array $a$ is already equal to $[1]$ , so the answer is "YES".

For the second test case, performing any amount of operations will change $a$ to an array of size at least two which doesn't only have the element $2$ , thus obtaining the array $[2]$ is impossible and the answer is "NO".

For the third test case, we can perform the following operations in order to obtain the final given array $c$ :

• Initially, $a = [1]$ .
• By choosing the subsequence $[1]$ , and inserting $1$ in the array, $a$ changes to $[1, 1]$ .
• By choosing the subsequence $[1, 1]$ , and inserting $1+1=2$ in the middle of the array, $a$ changes to $[1, 2, 1]$ .
• By choosing the subsequence $[1, 2]$ , and inserting $1+2=3$ after the first $1$ of the array, $a$ changes to $[1, 3, 2, 1]$ .
• By choosing the subsequence $[1, 3, 1]$ and inserting $1+3+1=5$ at the beginning of the array, $a$ changes to $[5, 1, 3, 2, 1]$ (which is the array we needed to obtain).