P15921 [TOPC 2023] Exponentiation

Exponentiation is a mathematical operation that involves raising a base number to a certain exponent to obtain a result. In the expression $a^n$, where $a$ is the base and $n$ is the exponent, it means multiplying $a$ by itself $n$ times. The result of this operation is called the *exponentiation* of $a$ to the $n$-th power. For examples, $2^3 = 2 \times 2 \times 2 = 8$ and $5^2 = 5 \times 5 = 25$. In these examples, 2 is the base, 3 is the exponent in the first case, and 5 is the base, and 2 is the exponent in the second case. Exponentiation is a fundamental operation in mathematics and is commonly used in various contexts, such as solving equations, and cryptography. In many cryptographic algorithms, particularly those based on number theory like RSA (Rivest-Shamir-Adleman) and Diffie-Hellman, modular exponentiation is a fundamental operation. Modular exponentiation involves raising a base to an exponent modulo a modulus. This operation is computationally intensive but relatively easy to perform, even for very large numbers. Let $x + \frac{1}{x} = \alpha$ where $\alpha$ is a positive integer. Please write a program to compute $x^\beta + \frac{1}{x^\beta} \mod m$ for given positive integers $\beta$ and $m$.

The input has only one line, and it contains three space-separated positive integers $\alpha$, $\beta$ and $m$.

Output $x^\beta + \frac{1}{x^\beta} \mod m$. If there are multiple solutions, you may output any of them in the range from $0$ to $m - 1$.

### Note $x$ can be a complex number. For example, $x$ is either $\frac{1+\sqrt{3}i}{2}$ or $\frac{1-\sqrt{3}i}{2}$ if $\alpha = 1$. However, $x^\beta + \frac{1}{x^\beta}$ is always an integer in this problem. $\alpha$, $\beta$ and $m$ are positive integers less than $2^{64}$. You may assume $x^\beta + \frac{1}{x^\beta}$ is an integer.