
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
Suppose $a$, $b$, and $c$ are positive integers with $a+b+c=3000$, and $a!b!c! = m \cdot 10^n$, where $m$ and $n$ are integers and $m$ is not divisible by $10$. Find the smallest possible value of $n$.

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