A13585. Segment Intersections

You are given two lists of segments $[al_1, ar_1], [al_2, ar_2], \dots, [al_n, ar_n]$ and $[bl_1, br_1], [bl_2, br_2], \dots, [bl_n, br_n]$ .

Initially, all segments $[al_i, ar_i]$ are equal to $[l_1, r_1]$ and all segments $[bl_i, br_i]$ are equal to $[l_2, r_2]$ .

In one step, you can choose one segment (either from the first or from the second list) and extend it by $1$ . In other words, suppose you've chosen segment $[x, y]$ then you can transform it either into $[x - 1, y]$ or into $[x, y + 1]$ .

Let's define a total intersection $I$ as the sum of lengths of intersections of the corresponding pairs of segments, i.e. $\sum\limits_{i=1}^{n}{\text{intersection_length}([al_i, ar_i], [bl_i, br_i])}$ . Empty intersection has length $0$ and length of a segment $[x, y]$ is equal to $y - x$ .

What is the minimum number of steps you need to make $I$ greater or equal to $k$ ?

The first line contains the single integer $t$ ( $1 \le t \le 1000$ ) — the number of test cases.

The first line of each test case contains two integers $n$ and $k$ ( $1 \le n \le 2 \cdot 10^5$ ; $1 \le k \le 10^9$ ) — the length of lists and the minimum required total intersection.

The second line of each test case contains two integers $l_1$ and $r_1$ ( $1 \le l_1 \le r_1 \le 10^9$ ) — the segment all $[al_i, ar_i]$ are equal to initially.

The third line of each test case contains two integers $l_2$ and $r_2$ ( $1 \le l_2 \le r_2 \le 10^9$ ) — the segment all $[bl_i, br_i]$ are equal to initially.

It's guaranteed that the sum of $n$ doesn't exceed $2 \cdot 10^5$ .

Print $t$ integers — one per test case. For each test case, print the minimum number of step you need to make $I$ greater or equal to $k$ .

In the first test case, we can achieve total intersection $5$ , for example, using next strategy:

- make $[al_1, ar_1]$ from $[1, 2]$ to $[1, 4]$ in $2$ steps;
- make $[al_2, ar_2]$ from $[1, 2]$ to $[1, 3]$ in $1$ step;
- make $[bl_1, br_1]$ from $[3, 4]$ to $[1, 4]$ in $2$ steps;
- make $[bl_2, br_2]$ from $[3, 4]$ to $[1, 4]$ in $2$ steps.

In result, $I = \text{intersection_length}([al_1, ar_1], [bl_1, br_1]) + \text{intersection_length}([al_2, ar_2], [bl_2, br_2]) + \\ + \text{intersection_length}([al_3, ar_3], [bl_3, br_3]) = 3 + 2 + 0 = 5$ In the second test case, we can make $[al_1, ar_1] = [0, 1000000000]$ in $1000000000$ steps and $[bl_1, br_1] = [0, 1000000000]$ in $1000000000$ steps.

In the third test case, the total intersection $I$ is already equal to $10 > 3$ , so we don't need to do any steps.