
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
How many positive integers $n$ are there such that $n$ is a multiple of 8, and the least common multiple of $8!$ and $n$ equals $8$ times the greatest common divisor of $12!$ and $n?$
A) 128
B) 192
C) 256
D) 336
E) 400

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