
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
Let \( p, \alpha, m \in \mathbf{Z}_{+} \), and let \( p^{\alpha} \| m \) denote that \( p^{\alpha} \mid m \), but \( p^{\alpha+1} \nmid m \). For any positive integer \( n \), find the maximum value of the positive integer \( \alpha \) satisfying \( 2^{\alpha} \| \left( 3^{n} + 1 \right) \).

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