Gcd and Lcm

You are given an integer $N$. You are asked to find total number of integer pairs $(A,B)$ such that - $ 1 \leq A,B \leq N$ - $A^2+B^2+gcd^2(A,B)+lcm^2(A,B)=N$. Note that $gcd^2(A, B)$ and $lcm^2(A, B)$ denote the square of [gcd](https://en.wikipedia.org/wiki/Greatest_common_divisor) and the square of [lcm](https://en.wikipedia.org/wiki/Least_common_multiple) of numbers $A$ and $B$ respectively.