AMO-Bench 2025 Problem 8

Given that $a$ is a positive integer and $b, c$ are positive real numbers, find the maximum value of $$\frac{16(\cos a)^5 - \cos(5a)}{2\cos a + \cos(3a)} \cdot \frac{a^2 b + b^2 c}{a^3 + b^3 + c^3}.$$