
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
Let positive integers \(a\) and \(b\) satisfy the system of equations:
\begin{align*}
\log_{10} a + 3 \log_{10} (\text{gcd}(a,b)) &= 90, \\
\log_{10} b + 3 \log_{10} (\text{lcm}(a,b)) &= 870.
\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 \(4p + 3q\).

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