
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 \(\{a_n\}\), where \(a_n = a \cdot 2^n + b \cdot n - 80\) and \(a, b \in \mathbf{N}_{+}\), it is known that the sum of the first \(n\) terms, \(S_n\), attains its minimum value if and only if \(n = 6\), and \(a_{36}\) is divisible by 7. Find the value of \(\left|a_1\right|+\left|a_2\right|+\cdots+\left|a_{12}\right|\).

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