Chef gives you a sequence A of length N.
Let X denote the MEX of the sequence A. Chef is interested in the count of positive values k, such that, if every element A_{i} of A is replaced by $\texttt{max}(A_{i}-k,0)$, the MEX of the sequence still remains X.

Find the count of such values. If there are infinite such values, print -1 instead.

As a friendly reminder, the MEX of a sequence is the smallest non-negative integer that does not belong to the sequence. For instance:

The MEX of [2,2,1] is 0 because 0 does not belong to the sequence.
The MEX of [3,1,0,1] is 2 because 0 and 1 belong to the sequence, but 2 does not.

------ Input Format ------

• The first line of the input contains a single integer T denoting the number of test cases. The description of T test cases follows.
• The first line of each test case contains a single integer N - the length of the sequence A.
• The second line of each test case contains $N space-separated integers A_{1}, A_{2}, \ldots, A_{N}$.

------ Output Format ------

For each test case print a single integer — the count of positive values k satisfying the given condition. If there are infinite such values, print -1 instead.

------ Constraints ------

$1 ≤ T ≤ 10^{4}$
$1 ≤ N ≤ 10^{5}$
$0 ≤ A_{i} ≤ 10^{9} $
- Sum of $N$ over all test cases does not exceed $2 \cdot 10^{5}$.

----- Sample Input 1 ------
3
4
1 2 3 5
1
0
7
5 10 2 4 0 6 1
----- Sample Output 1 ------
0
-1
1
----- explanation 1 ------
- Test case $1$: There doesn't exist any value of $k$ satisfying the given condition.
- Test case $2$: There are infinite values of $k$ satisfying the given condition.
- Test case $3$: There is only $1$ value of $k$ satisfying the given condition: $k=4$. The $MEX$ of the given sequence is $3$. When every element $A_{i}$ of the given sequence is replaced by $max(A_{i}-4,0)$, the sequence becomes $[1, 6, 0, 0, 0, 2, 0]$, which also has $MEX=3$.

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