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Professor Zhang has $n$ kinds of characters, and the quantity of the $i$-th character is $a\_i$. Professor Zhang wants to use all the characters to build several palindromic strings. He also wants to maximize the length of the shortest palindromic string.

For example, let there be $4$ kinds of characters denoted as 'a', 'b', 'c', 'd', and let their quantities be $\\{2, 3, 2, 2\\}$, respectively. Professor Zhang can build ("acdbbbdca"), or ("abbba" and "cddc"), or ("aca", "bbb" and "dcd"), or ("acdbdca and "bb"). The first is the optimal solution where the length of the shortest palindromic string is $9$.

Note that a string is called palindromic if it can be read the same way in either direction.

There are multiple test cases. The first line of input contains an integer $T$ indicating the number of test cases. For each test case:

The first line contains an integer $n$ ($1 \le n \le 10^5$): the number of kinds of characters. The second line contains $n$ integers $a\_1, a\_2, \ldots, a\_n$ ($0 \le a\_i \le 10^4$).

There are at most $110$ test cases, and the total size of the input is at most $6$ mebibytes.

For each test case, output an integer denoting the answer.