
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
Given that the sequence \( \{a_n\} \) is an arithmetic sequence with a nonzero common difference, \( \{b_n\} \) is a geometric sequence, where \( a_1 = 3 \), \( b_1 = 1 \), \( a_2 = b_2 \), \( 3a_5 = b_3 \), and there exist constants \( \alpha \) and \( \beta \) such that for every positive integer \( n \), \( a_n = \log_\alpha b_n + \beta \). Find \( \alpha + \beta \).

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