
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
The equations for positive integers $x$ and $y$ are given by:
\[
\log_2 x + 3 \log_2 (\text{gcd}(x, y)) = 30
\]
\[
\log_2 y + 3 \log_2 (\text{lcm}(x, y)) = 270
\]
Let $p$ be the total number of (not necessarily distinct) prime factors in the prime factorization of $x$, and let $q$ be the total number of (not necessarily distinct) prime factors in the prime factorization of $y$. Find $4p + 3q$.

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