AMO-Bench 2025 Problem 47

Assume that graph $G$ has $50$ vertices, and for any two non-adjacent vertices, they have exactly two common neighbors. Vertices $v_1$ and $v_2$ are adjacent and have no common neighbors, the degree of $v_1$ is $d(v_1)=11$. The parameter $M$ is defined as: if such a graph $G$ exists, then $M=max(d(v_2)-d(v_1))$; otherwise, $M=0$. Let $a$ be the smallest positive real root of the equation $x^{6}-4x^{5}+7x^{4}-25x^{3}+38x^{2}-6x+34=M(2x^3-4x^2+3x-19-M)$, find the remainder of the closest integer to $a^{25}$ divided by $1000$.