A15342 | Robot in a Hallway

There is a grid, consisting of $2$ rows and $m$ columns. The rows are numbered from $1$ to $2$ from top to bottom. The columns are numbered from $1$ to $m$ from left to right.

The robot starts in a cell $(1, 1)$ . In one second, it can perform either of two actions:

- move into a cell adjacent by a side: up, right, down or left;
- remain in the same cell.

The robot is not allowed to move outside the grid.

Initially, all cells, except for the cell $(1, 1)$ , are locked. Each cell $(i, j)$ contains a value $a_{i,j}$ — the moment that this cell gets unlocked. The robot can only move into a cell $(i, j)$ if at least $a_{i,j}$ seconds have passed before the move.

The robot should visit all cells without entering any cell twice or more (cell $(1, 1)$ is considered entered at the start). It can finish in any cell.

What is the fastest the robot can achieve that?

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 $m$ ( $2 \le m \le 2 \cdot 10^5$ ) — the number of columns of the grid.

The $i$ -th of the next $2$ lines contains $m$ integers $a_{i,1}, a_{i,2}, \dots, a_{i,m}$ ( $0 \le a_{i,j} \le 10^9$ ) — the moment of time each cell gets unlocked. $a_{1,1} = 0$ . If $a_{i,j} = 0$ , then cell $(i, j)$ is unlocked from the start.

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

For each testcase, print a single integer — the minimum amount of seconds that the robot can take to visit all cells without entering any cell twice or more.