
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 $x$, $y$, and $z$ are positive integers such that $x+y+z=2026$, and $x!y!z!=p \cdot 10^q$, where $p$ and $q$ are integers and $p$ is not divisible by $10$. Find the smallest possible value of $q$.

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