Solve the following math problem and provide a step-by-step ...

Prompt

Solve the following math problem and provide a step-by-step explanation. Find the number of pairs $(x; y)$ such that $x, y \in \mathbb{Z}$ and: $$\begin{cases} \sin\left(\frac{\pi y}{2}\right) + \cos\left(\frac{\pi x}{2}\right) = 0, \\ \sin\left(\frac{\pi x}{12}\right) = \cos\left(\frac{\pi x - 2\pi y}{12}\right), \\ |x + 52| + |x - 2y| \le 52. \end{cases}$$