AMO-Bench 2025 Problem 14

Let $\{a_n\}$ be a sequence of positive integers, and let $[x]$ denote the greatest integer less than or equal to $x$. Consider the system of equations in $x,y$: $$\big([y] \cdot x^2 + 2n + 2\big)\left(\sum_{k=0}^{n-1} \frac{1}{(k+n+1)(2n-k)}\right) = \sum_{k=1}^{n} \frac{1}{k(2k-1)},$$ $$4\big([x]y^2\big)^5 + 3n(4-n)\big([x]y^2\big)^3 - n^2(6+n)\big([x]y^2\big)^2 + 2n^2(4-3n)\big([x]y^2\big) = 8n^3.$$

Suppose there are $a_n$ distinct real solutions $(x,y)$ to this system. Determine whether there exist three positive prime numbers $p,k,q$ with $p > k > q$ such that $$a_p < a_k < a_q.$$ If no such primes exist, explain why; if they do exist, find the minimum value of $p+k+q$.