
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
Compute
$$
\sum_{\substack{a+b+c=12 \\ a \geq 6, b, c \geq 0}} \frac{a!}{b!c!(a-b-c)!},
$$
where the sum runs over all triples of nonnegative integers \((a, b, c)\) such that \(a+b+c=12\) and \(a \geq 6\).

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