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 $$$x^{\beta}+\frac{1}{x^{\beta}}~\mbox{mod}~m$$$ for given positive integers $$$\beta$$$ and $$$m$$$.