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G - Edge Elimination

Score : $600$ points

Problem Statement

Solve the following problem for $T$ test cases.

We have a perfect $K$-ary tree of depth $D$ (with $1+K+K^2+\dots+K^D$ vertices).
Your objective is to cut some of the edges to obtain a connected component with exactly $X$ vertices.
At least how many edges must be cut to achieve the objective?

Constraints

• All values in the input are integers.
• $1 \le T \le 100$
• $1 \le D$
• $2 \le K$
• $\displaystyle 1 \le X \le \sum_{i=0}^{D} K^i \le 10^{18}$

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Input

The input is given from Standard Input in the following format:

$T$

$case_1$

$\vdots$

$case_T$

Here, $case_i$ denotes the $i$-th test case.
Each test case is given in the following format:

``` $D$ $K$ $X$ ```

Output

Print $T$ lines.
The $i$-th line should contain the answer to the $i$-th test case as an integer.

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11
2 2 1
2 2 2
2 2 3
2 2 4
2 2 5
2 2 6
2 2 7
1 999999999999999999 1
1 999999999999999999 2
1 999999999999999999 999999999999999999
1 999999999999999999 1000000000000000000

1
2
1
1
2
1
0
1
999999999999999998
1
0