There are positive integers x and y that satisfy the system of equations

\begin{aligned}
& \log _{10} x+2 \log _{10}(\operatorname{gcd}(x, y))=60 \\
& \log _{10} y+2 \log _{10}(\operatorname{lcm}(x, y))=570
\end{aligned}

Let m be the number of (not necessarily distinct) prime factors in the prime factorization of x, and let n be the number of (not necessarily distinct) prime factors in the prime factorization of y. Find 3 m+2 n.