
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
\[\left(\frac{a_1 + 1}{2}\right)^3 \cdot \left(\frac{a_2 + 2}{2}\right)^3 \cdot \left(\frac{a_3 + 3}{2}\right)^3 \cdot \left(\frac{a_4 + 4}{2}\right)^3 \cdot \left(\frac{a_5 + 5}{2}\right)^3 \cdot \left(\frac{a_6 + 6}{2}\right)^3 > 6!. \]

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