bookshelf

Problem Description

Patrick Star bought a bookshelf, he named it ZYG !!

Patrick Star has N book .

The ZYG has K layers (count from 1 to K) and there is no limit on the capacity of each layer !

Now Patrick want to put all N books on ZYG :

1. Assume that the i-th layer has cnti(0≤cnti≤N) books finally.

2. Assume that f[i] is the i-th fibonacci number (f[0]=0,f[1]=1,f[2]=1,f[i]=f[i−2]+f[i−1]).

3. Define the stable value of i-th layers stablei=f[cnti].

4. Define the beauty value of i-th layers beautyi=2stablei−1.

5. Define the whole beauty value of ZYG score=gcd(beauty1,beauty2,…,beautyk)(Note: gcd(0,x)=x).

Patrick Star wants to know the expected value of score if Patrick choose a distribute method randomly !

Input

The first line contain a integer T (no morn than 10), the following is T test case, for each test case :

Each line contains contains three integer n,k(0<n,k≤106).$(0<n,k\le {10}^6).$

Output

For each test case, output the answer as a value of a rational number modulo 109+7.${10}^9+7.$.

Formally, it is guaranteed that under given constraints the probability is always a rational number pq (p and q are integer and coprime, q is positive), such that q is not divisible by 109+7. Output such integer a between 0 and

Sample Input

1 6 8

Sample Output

797202805

題意：把N本書放到K層的書架上，每一層的美麗值為bi=2fib[cnt]−1$b_i=2^{fib[cnt]}-1$，其中cnt是這一層書的數量，fib[x]為斐波那契數列，整個書架的美麗值為gcd(b1,b2,...,bk)$gcd(b_1,b_2,...,b_k)$，問整個書架的美麗值的期望

題解：考試的時候因為本人實在是太菜了。對這個題目完全沒有一點思路，考完試以後聽了dls直播算是對這個題目有了一點思路程式碼很短，但是從學習各種姿勢到寫完用了一天，感覺一下子掌握了好多神奇的姿勢，開心。

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