
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 \((b_1, b_2, b_3, b_4)\) of \((1,2,3,4)\) that satisfy
\[
\left(\frac{b_1^2 + 1}{2}\right) \cdot \left(\frac{b_2^2 + 2}{2}\right) \cdot \left(\frac{b_3^2 + 3}{2}\right) \cdot \left(\frac{b_4^2 + 4}{2}\right) \geq 4!.
\]

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