
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
What is the largest integer \( n \) such that the quantity
\[
\frac{50!}{(5!)^{n}}
\]
is an integer?
Note: Here \( k! = 1 \times 2 \times 3 \times \cdots \times k \) is the product of all integers from 1 to \( k \). For example, \( 4! = 1 \times 2 \times 3 \times 4 = 24 \).

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