
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 where $a+b+c=2003$, $a = 2 * b$, and $a!b!c! = m \cdot 10^n$ for integers $m$ and $n$, where $m$ is not divisible by $10$. What is the smallest possible value of $n$?
A) 398  
B) 400  
C) 402  
D) 404  
E) 406

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