
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 $a$ and $b$ that satisfy the system of equations\begin{align*} 
\log_{10} a + 2 \log_{10} (\text{gcd}(a,b)) &= 12\\ 
\log_{10} b + 2 \log_{10} (\text{lcm}(a,b)) &= 42. 
\end{align*}
Let $p$ be the number of (not necessarily distinct) prime factors in the prime factorization of $a$, and let $q$ be the number of (not necessarily distinct) prime factors in the prime factorization of $b$. Find $3p+2q$.

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