AMO-Bench 2025 Problem 30

Let $p, q$ be positive integers satisfying the system of congruences: $$\begin{cases} p \equiv 3 \pmod{911}, \\ q \equiv 2 \pmod{997}, \\ p \equiv 738 \pmod{q}; \end{cases}$$ Let $\{ a_n \}, \{ b_n \}, \{ c_n \}$ be periodic sequences of rational numbers satisfying the equations: $$\left[ 4b_n + 1 - 3\sin(\pi a_n) \right]^2 + \left[ 4c_n - 1 + 3\sin(\pi a_n) \right]^2 = 0 (\forall n \in \mathbb{N})$$ and $$\left| \sum_{i=1}^{p} b_i^3 \right|+\left| \sum_{i=1}^{q} c_i^3 \right|+\left| p - 3 \sum_{i=1}^{p} b_i \right|+\left| q - 3 \sum_{i=1}^{q} c_i^2 \right| = 0.$$ Find the minimum value of $p + q + 2a_p^2 + b_q^2 + c_{p+q}^2 + (a_p + a_{q-1})^2 + (b_p + c_q)^2$.