CF1637C.Andrew and Stones

Andrew has $n$ piles with stones. The $i$-th pile contains $a_i$ stones. He wants to make his table clean so he decided to put every stone either to the $1$-st or the $n$-th pile.

Andrew can perform the following operation any number of times: choose $3$ indices $1 \le i \lt j \lt k \le n$, such that the $j$-th pile contains at least $2$ stones, then he takes $2$ stones from the pile $j$ and puts one stone into pile $i$ and one stone into pile $k$.

Tell Andrew what is the minimum number of operations needed to move all the stones to piles $1$ and $n$, or determine if it's impossible.

安德鲁有 $n$ 堆石头。第 $i$ 堆包含 $a_i$ 颗石头。他希望将桌面清理干净，因此决定将所有石头全部移至第 $1$ 堆或第 $n$ 堆。

安德鲁可以执行以下操作任意多次：选择满足 $1 \le i \lt j \lt k \le n$ 的三个下标，且要求第 $j$ 堆中至少有 $2$ 颗石头；然后他从第 $j$ 堆取走 $2$ 颗石头，并分别向第 $i$ 堆和第 $k$ 堆各放入 $1$ 颗石头。

请告诉安德鲁：将所有石头全部移至第 $1$ 堆和第 $n$ 堆所需的最少操作次数是多少？若不可能完成，请予以判定。

The input contains several test cases. The first line contains one integer $t$ ($1 \leq t \leq 10\,000$) — the number of test cases.

The first line for each test case contains one integer $n$ ($3 \leq n \leq 10^5$) — the length of the array.

The second line contains a sequence of integers $a_1, a_2, \ldots, a_n$ ($1 \leq a_i \leq 10^9$) — the array elements.

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

输入包含多个测试用例。第一行包含一个整数 $t$（$1 \leq t \leq 10\,000$），表示测试用例的数量。

每个测试用例的第一行包含一个整数 $n$（$3 \leq n \leq 10^5$），表示数组的长度。

第二行包含 $n$ 个整数 $a_1, a_2, \ldots, a_n$（$1 \leq a_i \leq 10^9$），表示数组的元素。

保证所有测试用例的 $n$ 值之和不超过 $10^5$。

For each test case print the minimum number of operations needed to move stones to piles $1$ and $n$, or print $-1$ if it's impossible.

对于每个测试用例，输出将石子移动到第 $1$ 堆和第 $n$ 堆所需的最少操作次数；如果无法实现，则输出 $-1$。

In the first test case, it is optimal to do the following:

1. Select $(i, j, k) = (1, 2, 5)$. The array becomes equal to $[2, 0, 2, 3, 7]$.
2. Select $(i, j, k) = (1, 3, 4)$. The array becomes equal to $[3, 0, 0, 4, 7]$.
3. Twice select $(i, j, k) = (1, 4, 5)$. The array becomes equal to $[5, 0, 0, 0, 9]$. This array satisfy the statement, because every stone is moved to piles $1$ and $5$.

There are $4$ operations in total.

In the second test case, it's impossible to put all stones into piles with numbers $1$ and $3$:

1. At the beginning there's only one possible operation with $(i, j, k) = (1, 2, 3)$. The array becomes equal to $[2, 1, 2]$.
2. Now there is no possible operation and the array doesn't satisfy the statement, so the answer is $-1$.

In the third test case, it's optimal to do the following:

1. Select $(i, j, k) = (1, 2, 3)$. The array becomes equal to $[2, 0, 2]$. This array satisfies the statement, because every stone is moved to piles $1$ and $3$.

The is $1$ operation in total.

In the fourth test case, it's impossible to do any operation, and the array doesn't satisfy the statement, so the answer is $-1$.

在第一个测试用例中，最优操作如下：

1. 选择 $(i, j, k) = (1, 2, 5)$。数组变为 $[2, 0, 2, 3, 7]$。
2. 选择 $(i, j, k) = (1, 3, 4)$。数组变为 $[3, 0, 0, 4, 7]$。
3. 两次选择 $(i, j, k) = (1, 4, 5)$。数组变为 $[5, 0, 0, 0, 9]$。该数组满足题意，因为所有石子均被移至第 $1$ 堆和第 $5$ 堆。

总共进行了 $4$ 次操作。

在第二个测试用例中，无法将所有石子移入编号为 $1$ 和 $3$ 的堆中：

1. 初始时唯一可行的操作是 $(i, j, k) = (1, 2, 3)$。数组变为 $[2, 1, 2]$。
2. 此时已无可行操作，且数组不满足题意，因此答案为 $-1$。

在第三个测试用例中，最优操作如下：

1. 选择 $(i, j, k) = (1, 2, 3)$。数组变为 $[2, 0, 2]$。该数组满足题意，因为所有石子均被移至第 $1$ 堆和第 $3$ 堆。

总共进行了 $1$ 次操作。

在第四个测试用例中，无法执行任何操作，且数组不满足题意，因此答案为 $-1$。