SILK-ROAD 2022 Problem 2

Distinct positive integers $A$ and $B$ are given $.$ Prove that there exist infinitely many positive integers that can be represented both as $x_{1}^2+Ay_{1}^2$ for some positive coprime integers $x_{1}$ and $y_{1},$ and as $x_{2}^2+By_{2}^2$ for some positive coprime integers $x_{2}$ and $y_{2}.$ (Golovanov A.S.)