D. Segment Intersections
time limit per test
2 seconds
memory limit per test
256 megabytes
standard input
standard output

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 $$$k$$$.

3 5
1 2
3 4
2 1000000000
1 1
999999999 999999999
10 3
5 10
7 8

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.