Virtual contest is a way to take part in past contest, as close as possible to participation on time. It is supported only ICPC mode for virtual contests.
If you've seen these problems, a virtual contest is not for you - solve these problems in the archive.
If you just want to solve some problem from a contest, a virtual contest is not for you - solve this problem in the archive.
Never use someone else's code, read the tutorials or communicate with other person during a virtual contest.

No tag edit access

The problem statement has recently been changed. View the changes.

×
G. El Toll Caves

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputThe prehistoric caves of El Toll are located in Moià (Barcelona). You have heard that there is a treasure hidden in one of *n* possible spots in the caves. You assume that each of the spots has probability 1 / *n* to contain a treasure.

You cannot get into the caves yourself, so you have constructed a robot that can search the caves for treasure. Each day you can instruct the robot to visit exactly *k* distinct spots in the caves. If none of these spots contain treasure, then the robot will obviously return with empty hands. However, the caves are dark, and the robot may miss the treasure even when visiting the right spot. Formally, if one of the visited spots does contain a treasure, the robot will obtain it with probability 1 / 2, otherwise it will return empty. Each time the robot searches the spot with the treasure, his success probability is independent of all previous tries (that is, the probability to miss the treasure after searching the right spot *x* times is 1 / 2^{x}).

What is the expected number of days it will take to obtain the treasure if you choose optimal scheduling for the robot? Output the answer as a rational number modulo 10^{9} + 7. Formally, let the answer be an irreducible fraction *P* / *Q*, then you have to output . It is guaranteed that *Q* is not divisible by 10^{9} + 7.

Input

The first line contains the number of test cases *T* (1 ≤ *T* ≤ 1000).

Each of the next *T* lines contains two integers *n* and *k* (1 ≤ *k* ≤ *n* ≤ 5·10^{8}).

Output

For each test case output the answer in a separate line.

Example

Input

3

1 1

2 1

3 2

Output

2

500000007

777777786

Note

In the first case the robot will repeatedly search in the only spot. The expected number of days in this case is 2. Note that in spite of the fact that we know the treasure spot from the start, the robot still has to search there until he succesfully recovers the treasure.

In the second case the answer can be shown to be equal to 7 / 2 if we search the two spots alternatively. In the third case the answer is 25 / 9.

Codeforces (c) Copyright 2010-2021 Mike Mirzayanov

The only programming contests Web 2.0 platform

Server time: Jan/16/2021 17:16:15 (f2).

Desktop version, switch to mobile version.

Supported by

User lists

Name |
---|