
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 \((b_1, b_2, b_3, b_4, b_5, b_6, b_7)\) of \((1, 2, 3, 4, 5, 6, 7)\) such that 
\[
\frac{b_1 + 1}{2} \cdot \frac{b_2 + 2}{2} \cdot \frac{b_3 + 3}{2} \cdot \frac{b_4 + 4}{2} \cdot \frac{b_5 + 5}{2} \cdot \frac{b_6 + 6}{2} \cdot \frac{b_7 + 7}{2} \leq 7!.
\]

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