
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
In the sequence \(\{a_n\}\), \(a_n = 2^n a + b n - 80\) \((a, b \in \mathbf{Z}_{+})\), it is known that the minimum value of the sum of the first \(n\) terms, \(S_n\), is achieved if and only if \(n = 6\) and \(7 \mid a_{36}\). Find the value of \(\sum_{i=1}^{12}\left|a_i\right|\).

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