This problem is a harder version of FIBPWSUM.

The Fibonacci numbers is defined by

$$ f_0=0, f_1=1, $$

and

$$ f_n = f_{n-1}+f_{n-2} $$

for $n > 1$. 

Given three integers $N$, $C$ and $K$, compute the summation

$$ \sum_{n=0}^N f_{Cn}^K. $$

Since the answer can be huge, output it modulo $10^9+7$.

Input

The first line contains an integer $T$, denoting the number of test cases. Each test case contains three space separated integers in the order: $N$, $C$ and $K$.

Constraints

• $1 \leq T \leq 100$
• $1 \leq N, C \leq 10^{18}$
• $1 \leq K \leq 10^5$

Output

For each test case, output a single line in the format "Case X: Y" without the quotes. Here, X is the case number and Y is the desired answer denoting the sum of the series.

Example

Input:
5
10 1 1
5 2 2
3 3 4
1000000007 7 9
996969696969696 9 6

Output:
Case 1: 143
Case 2: 3540
Case 3: 1340448
Case 4: 880410497
Case 5: 689328397

Credits

• sgtlaugh - FIBPWSUM
• ZOJ 3774. Power of Fibonacci

Information

There are two test files. The first file is randomly generated while the second file is not.

@Speed Adicts: My solution runs in 1.94s. (approx less than 1s per file)