A16025 | Fill the Matrix

There is a square matrix, consisting of $n$ rows and $n$ columns of cells, both numbered from $1$ to $n$ . The cells are colored white or black. Cells from $1$ to $a_i$ are black, and cells from $a_i+1$ to $n$ are white, in the $i$ -th column.

You want to place $m$ integers in the matrix, from $1$ to $m$ . There are two rules:

- each cell should contain at most one integer;
- black cells should not contain integers.

The beauty of the matrix is the number of such $j$ that $j+1$ is written in the same row, in the next column as $j$ (in the neighbouring cell to the right).

What's the maximum possible beauty of the matrix?

The first line contains a single integer $t$ ( $1 \le t \le 10^4$ ) — the number of testcases.

The first line of each testcase contains a single integer $n$ ( $1 \le n \le 2 \cdot 10^5$ ) — the size of the matrix.

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ( $0 \le a_i \le n$ ) — the number of black cells in each column.

The third line contains a single integer $m$ ( $0 \le m \le \sum \limits_{i=1}^n n - a_i$ ) — the number of integers you have to write in the matrix. Note that this number might not fit into a 32-bit integer data type.

The sum of $n$ over all testcases doesn't exceed $2 \cdot 10^5$ .

For each testcase, print a single integer — the maximum beauty of the matrix after you write all $m$ integers in it. Note that there are no more integers than the white cells, so the answer always exists.