bookshelf

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 $cnt_i(0 \le cnt_i \le N)$ 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 $stable_i = f[cnt_i]$.

4. Define the beauty value of i-th layers $beauty_i = 2^{stable_i} - 1$.

5. Define the whole beauty value of ZYG $score = gcd(beauty_1, beauty_2, ..., beauty_k)$(Note: $gcd(0, x) = x$).

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

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 $n, k(0 < n, k \le 10^6)$.

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

Formally, it is guaranteed that under given constraints the probability is always a rational number $\frac{p}{q}$ (p and q are integer and coprime, q is positive), such that q is not divisible by $10^9 + 7$. Output such integer a between 0 and $10^9+6$ that $p-aq$ is divisible by $10^9+7$.

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2018 Multi-University Training Contest 6