
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  $a_1, a_2, ... , a_7, b_1, b_2, ... , b_7$  satisfy  $2 \leq a_i \leq 166$  and  $a_i^{b_i} \cong a_{i+1}^2$  (mod 167) for each  $1 \leq i \leq 7$  (where  $a_8=a_1$ ). Compute the minimum possible value of  $b_1b_2 ... b_7(b_1 + b_2 + ...+ b_7)$ .

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