
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 \(a+b+c=2012\), and \(a!b!c!=m\cdot 10^n\), where \(m\) and \(n\) are integers and \(m\) is not divisible by \(10\). What is the smallest possible value of \(n\)?
A) 495
B) 498
C) 501
D) 504
E) 507

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