## DELTACAT - Delta catheti (hard)

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(3, 4, 5) is a famous Pythagorean triple, it gives a quick answer to the question:
For a given integer d, is there a Pythagorean triple (a, b, c) such that b - a = d?

A solution is (3d, 4d, 5d), and in fact one can easily prove that the set of solutions is infinite, and that there is an obvious total order on those solutions.
Given n, you'll have to find the nth term of the sequence of solutions.

Geometrically, it is the study of right triangles for which the difference of the catheti are equal to d.

### Input

The first line of input contains an integer T, the number of test cases.

Each of the next T lines contains three integers n, d, m.

### Output

For each test case, compute the nth term amongst the solutions (a, b, c) for the problem :
a2 + b2 = c2 with b - a = d and 0 < a < b < c .

### Example

```Input:
3
1 1 235813
3 21 1000
9 119 11
```
```Output:
3 4 5
63 84 105
5 3 1
```

### Explanations

For the first case, the first solution is (3, 4, 5), as 4 - 3 = 1.

For the second case, the firsts solutions are:
(15, 36, 39), (24, 45, 51), (63, 84, 105), (144, 165, 219), (195, 216, 291), (420, 441, 609), ...
The third one is (63, 84, 105).

For the third case, the first solutions are:
(24, 143, 145), (49, 168, 175), (57, 176, 185), (85, 204, 221), (136, 255, 289), (180, 299, 349), (196, 315, 371), (261, 380, 461), (357, 476, 595), (481, 600, 769), (616, 735, 959), ...
The 9th solution is (357, 476, 595), reduced modulo 11, we get (5, 3, 1).

### Constraints

```0 < T < 10^5
0 < n < 10^7
0 < d < 10^8
1 < m < 10^9
```

n, d, m : Uniform randomly chosen in their range.

Constraints allow some interpreted languages to get AC, but it could be hard.
For your information, I have two distinct solutions.
My first python3 code get AC under 17s. My second python3 code get AC under 8s.
(Timing updated, 2017-02-11 ; after compiler changes)
With a compiled language, it should be at least 20× faster. Have fun ;-)

 Added by: Francky Date: 2013-04-19 Time limit: 30s Source limit: 50000B Memory limit: 1536MB Cluster: Cube (Intel G860) Languages: All except: ASM64 Resource: Own problem