
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 sum of the first $n$ terms of a sequence $\{a_{n}\}$ is $S_{n}$, and $\{a_{n}\}$ is not a constant sequence, which of the following statements is correct? <br/>① If the sequence $\{a_{n}\}$ is an arithmetic sequence, then $\left\{{{2^{{a_n}}}}\right\}$ is a geometric sequence; <br/>② If the sequence $\{a_{n}\}$ is an arithmetic sequence and $S_{n} \gt 0$ always holds, then $\{a_{n}\}$ is a strictly increasing sequence; <br/>③ If the sequence $\{a_{n}\}$ is a geometric sequence, then $\{\log _{2}a_{n}\}$ is an arithmetic sequence; <br/>④ If the sequence $\{a_{n}\}$ is an arithmetic sequence, $a_{1} \gt 0$, and $S_{6}=S_{11}$, then the maximum value of $S_{n}$ is achieved when $n$ is $8$ or $9$.

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