## JDCC '15 Contest 2 P5 - Randomize

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Points: 15
Time limit: 5.0s
Memory limit: 256M

Author:
Problem types

For her new ICS assignment, Caroline needs to design a program that uses random numbers. However, she discovers that Ms. Dyke has forbidden using any built-in functions! Now, she needs to create a random number generator to make her assignment work. After checking online, she finds that random numbers can be generated using the following function:

$f(0) = SEED$

$F(N) = (A \times F(N-1) + B) \bmod P$

Where $$SEED$$ is some initial value between $$0$$ and $$P-1$$ inclusive. After some tinkering she finds that for most values of $$A$$, $$B$$, and $$P$$, the generated numbers quickly fall into a repeating cycle. She’d like to figure out which values of $$A$$, $$B$$, and $$P$$ produce the best results and has enlisted your help to find the average length of a cycle for one set of values.

Note: The cycle length for some value of $$SEED$$ is defined as smallest value $$N$$ for which $$F(N)$$ produces a number already in the sequence. For example, if $$SEED = 1, F(1) = 2, F(2) = 3,$$ and $$F(3) = 3$$, then the cycle length is $$3$$, as $$3$$ was already in the sequence. The average length of a cycle is defined as the average of the cycle lengths for every possible value of $$SEED$$.

#### Input Specification

The first line of the input provides the number of test cases, $$T\ (1 \le T \le 100)$$. $$T$$ test cases follow. Each test case contains 3 integers, $$A, B$$, and $$P\ (1 \le A, B, P \le 10^6)$$.

For the first $$20\%$$ of cases, $$A, B, P \le 10^3$$.

#### Output Specification

For each test case, your program should output one real number, rounded to 6 decimal places, the average length of a cycle.

#### Sample Input

2
3 2 5
4 5 3

#### Sample Output

3.400000
3.000000

#### Explanation for Sample

In the second test case, if you start with a $$SEED$$ of $$0$$, then

• $$F(1) = 4(0) + 5 \bmod 3 = 2$$
• $$F(2) = 4(2) + 5 \bmod 3 = 1$$
• $$F(3) = 4(1) + 5 \bmod 3 = 0$$

Since $$0$$ is already in the sequence, the cycle length is $$3$$. Starting with $$1$$ or $$2$$ will also result in a cycle length of $$3$$, so the average cycle length is $$3$$.