
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

How many solutions in natural numbers \(x, y\) does the system of equations have?

$$
\left\{\begin{array}{l}
\text{GCD}(x, y) = 20! \\
\text{LCM}(x, y) = 30!
\end{array}\right.
$$

(where \(n! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot n\))

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