
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
There are positive integers  $x$  and  $y$  that satisfy the system of equations 
\begin{align*}
\log_{10} x + 2 \log_{10} (\gcd(x,y)) &= 60  \log_{10} y + 2 \log_{10} (\text{lcm}(x,y)) &= 570.
\end{align*}
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  $3m+2n$ .

Present the answer in LaTex format: \boxed{Your answer}
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