Subsequence Addition (Hard Version)
Contest: Codeforces Round 859 (Div. 4)
Difficulty rating: 1100
Tags: bitmasks, dp, greedy, implementation, sortings

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 $$$a_{i_1} + a_{i_2} + \dots + a_{i_k}$$$.

Input format:
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 $$$2 \cdot 10^5$$$.

Output format:
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).

Constraints:
Time limit: 2.0 s
Memory limit: 256.0 MB

Note:
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).

Example input:
6
1
1
1
2
5
5 1 3 2 1
5
7 1 5 2 1
3
1 1 1
5
1 1 4 2 1

Example output:
YES
NO
YES
NO
YES
YES

Editorial:
Let's prove that for an array $$$a$$$ that was created by using a number of operations, with a sum of elements $$$s$$$ we can add into $$$a$$$ any number $$$x$$$ ($$$1 \leq x \leq s$$$).
Suppose that it is true that in the array $$$a$$$ with some length $$$l$$$ we introduce a number $$$x$$$ ($$$1 \leq x \leq sum_a$$$). Then after introducing we can create using the initial elements of the array any number $$$b$$$ ($$$1 \leq b \leq sum_a$$$) and using the element $$$x$$$ and some subset of the initial elements we can create any number $$$b$$$ ($$$x \leq b \leq sum_a+x$$$), and because $$$x \leq sum_a$$$ we proved that for the new array of length $$$l+1$$$ we can still create any number between $$$1$$$ and $$$sum_a+x$$$.
Since it is true for the initial array, we can use induction and this fact to prove it is true for all arrays. So we just need to verify if our array satisfies this condition. We should sort the array and check for each $$$i$$$ ($$$2 \leq i \leq n$$$) if $$$c_i \leq \sum_{j = 1}^{i-1}c_j$$$.