
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 $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, and $\{b_n\}$ is a geometric sequence, where $a_1=2$, $b_1=1$, $a_2=b_2$, $2a_4=b_3$, and there exist constants $\alpha$ and $\beta$ such that $a_n=\log_{\alpha}b_n+\beta$ holds for every positive integer $n$, then $\alpha^{\beta}=$ ?

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