
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:
\[
\begin{align*}
\log_{10} x + 3 \log_{10} (\text{gcd}(x,y)) &= 90\\
\log_{10} y + 3 \log_{10} (\text{lcm}(x,y)) &= 630.
\end{align*}
\]
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 $2m+3n$.

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