
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
Find the number of permutations $(a_1, a_2, a_3, a_4, a_5, a_6)$ of $(1,2,3,4,5,6)$ such that 
\[\frac{a_1^2 + 1^2}{2} \cdot \frac{a_2^2 + 2^2}{2} \cdot \frac{a_3^2 + 3^2}{2} \cdot \frac{a_4^2 + 4^2}{2} \cdot \frac{a_5^2 + 5^2}{2} \cdot \frac{a_6^2 + 6^2}{2} > 5!.\]

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