F. Closest Pair
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

There are $$$n$$$ weighted points on the $$$OX$$$-axis. The coordinate and the weight of the $$$i$$$-th point is $$$x_i$$$ and $$$w_i$$$, respectively. All points have distinct coordinates and positive weights. Also, $$$x_i < x_{i + 1}$$$ holds for any $$$1 \leq i < n$$$.

The weighted distance between $$$i$$$-th point and $$$j$$$-th point is defined as $$$|x_i - x_j| \cdot (w_i + w_j)$$$, where $$$|val|$$$ denotes the absolute value of $$$val$$$.

You should answer $$$q$$$ queries, where the $$$i$$$-th query asks the following: Find the minimum weighted distance among all pairs of distinct points among the points in subarray $$$[l_i,r_i]$$$.

Input

The first line contains 2 integers $$$n$$$ and $$$q$$$ $$$(2 \leq n \leq 3 \cdot 10^5; 1 \leq q \leq 3 \cdot 10^5)$$$ — the number of points and the number of queries.

Then, $$$n$$$ lines follows, the $$$i$$$-th of them contains two integers $$$x_i$$$ and $$$w_i$$$ $$$(-10^9 \leq x_i \leq 10^9; 1 \leq w_i \leq 10^9)$$$ — the coordinate and the weight of the $$$i$$$-th point.

It is guaranteed that the points are given in the increasing order of $$$x$$$.

Then, $$$q$$$ lines follows, the $$$i$$$-th of them contains two integers $$$l_i$$$ and $$$r_i$$$ $$$(1 \leq l_i < r_i \leq n)$$$ — the given subarray of the $$$i$$$-th query.

Output

For each query output one integer, the minimum weighted distance among all pair of distinct points in the given subarray.

Example
Input
5 5
-2 2
0 10
1 1
9 2
12 7
1 3
2 3
1 5
3 5
2 4
Output
9
11
9
24
11
Note

For the first query, the minimum weighted distance is between points $$$1$$$ and $$$3$$$, which is equal to $$$|x_1 - x_3| \cdot (w_1 + w_3) = |-2 - 1| \cdot (2 + 1) = 9$$$.

For the second query, the minimum weighted distance is between points $$$2$$$ and $$$3$$$, which is equal to $$$|x_2 - x_3| \cdot (w_2 + w_3) = |0 - 1| \cdot (10 + 1) = 11$$$.

For the fourth query, the minimum weighted distance is between points $$$3$$$ and $$$4$$$, which is equal to $$$|x_3 - x_4| \cdot (w_3 + w_4) = |1 - 9| \cdot (1 + 2) = 24$$$.