
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 a sequence $\{a_n\}$ where $a_n = q^n$ ($q>0$, $n \in \mathbb{N}^*$), identify which of the following statements are correct:
1. $\{a_{2n}\}$ is a geometric sequence
2. $\left\{\frac{1}{a_n}\right\}$ is a geometric sequence
3. $\log a_n$ is an arithmetic sequence
4. $\{\log a_n^2\}$ is an arithmetic sequence.

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