AIME 2025b Problem 12

Let $A_1A_2\dots A_{11}$ be a non-convex $11$-gon such that • The area of $A_iA_1A_{i+1}$ is $1$ for each $2 \le i \le 10$, • $\cos(\angle A_iA_1A_{i+1})=\frac{12}{13}$ for each $2 \le i \le 10$, • The perimeter of $A_1A_2\dots A_{11}$ is $20$. If $A_1A_2+A_1A_{11}$ can be expressed as $\frac{m\sqrt{n}-p}{q}$ for positive integers $m,n,p,q$ with $n$ squarefree and $\gcd(m,p,q)=1$, find $m+n+p+q$.