In isosceles $\vartriangle ABC, AB = AC, \angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$, respectively, so that $AE = 10, AF = 15$. The area of $\vartriangle AEF$ is $60$, and the area of quadrilateral $BEFC$ is $102$. Find $BC$.

Let's think step by step and output the final answer within \boxed{}.
user