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

两个版本的唯一区别在于，本版本的约束条件更高。

初始时，数组 $a$ 仅包含数字 $1$。你可以执行若干次操作来改变该数组。每次操作中，你可以选择 $a$ 的某个子序列$^{\dagger}$，并将该子序列所有元素之和作为一个新元素添加到 $a$ 中。

现给定一个最终数组 $c$。请判断：是否存在一系列（可能为零次）操作，使得从初始数组 $a$ 出发能够得到 $c$。

$^{\dagger}$ 序列 $b$ 是序列 $a$ 的一个子序列，当且仅当 $b$ 可通过从 $a$ 中删除若干（可能为零个，但不能全部）元素而得到。换言之，选取 $k$ 个（$1 \leq k \leq |a|$）互不相同的下标 $i_1, i_2, \dots, i_k$，并在 $a$ 中任意位置插入一个值为 $a_{i_1} + a_{i_2} + \dots + a_{i_k}$ 的新元素。

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 $2 \cdot 10^5$.

输入的第一行包含一个整数 $t$（$1 \leq t \leq 1000$），表示测试用例的数量。随后是各测试用例的描述。

每个测试用例的第一行包含一个整数 $n$（$1 \leq n \leq 2 \cdot 10^5$），表示最终数组 $c$ 应包含的元素个数。

每个测试用例的第二行包含 $n$ 个以空格分隔的整数 $c_i$（$1 \leq c_i \leq 2 \cdot 10^5$），即应由初始数组 $a$ 得到的最终数组 $c$ 的各个元素。

保证所有测试用例的 $n$ 值之和不超过 $2 \cdot 10^5$。

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).

对于每个测试用例，如果存在这样的一系列操作，则输出 "YES"（不带引号）；否则输出 "NO"（不带引号）。

您可以以任意大小写形式输出答案（例如，字符串 "yEs"、"yes"、"Yes" 和 "YES" 均被视为肯定回答）。

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).

对于第一个测试用例，初始数组 $a$ 已经等于 $[1]$，因此答案为 “YES”。

对于第二个测试用例，执行任意次数的操作都会使 $a$ 变为长度至少为 2 的数组，且该数组不可能仅包含元素 $2$，因此无法得到数组 $[2]$，答案为 “NO”。

对于第三个测试用例，我们可以按以下顺序执行操作，以获得最终给定的数组 $c$：

• 初始时，$a = [1]$。
• 选择子序列 $[1]$，并在数组中插入 $1$，$a$ 变为 $[1, 1]$。
• 选择子序列 $[1, 1]$，并在数组中间插入 $1+1=2$，$a$ 变为 $[1, 2, 1]$。
• 选择子序列 $[1, 2]$，并在数组第一个 $1$ 之后插入 $1+2=3$，$a$ 变为 $[1, 3, 2, 1]$。
• 选择子序列 $[1, 3, 1]$，并在数组开头插入 $1+3+1=5$，$a$ 变为 $[5, 1, 3, 2, 1]$（即我们需要得到的数组）。