
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
Positive integers \(a_{1}, a_{2}, \ldots, a_{7}, b_{1}, b_{2}, \ldots, b_{7}\) satisfy \(2 \leq a_{i} \leq 166\) and \(a_{i}^{b_{i}} \equiv a_{i+1}^{2} \pmod{167}\) for each \(1 \leq i \leq 7\) (where \(a_{8}=a_{1}\)). Compute the minimum possible value of \(b_{1} b_{2} \cdots b_{7} \left(b_{1}+b_{2}+\cdots+b_{7}\right)\).

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