
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
Let $(a_1,a_2,a_3,\ldots,a_{12})$ be a permutation of $(1,2,3,\ldots,12)$ for which
$a_1>a_2>a_3>a_4 \mathrm{\ and \ } a_4<a_5<a_6<\ldots<a_{12}.$
An example of such a permutation is $(5,4,3,1,2,6,7,8,9,10,11,12).$ Find the number of such permutations.

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