
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
For a geometric sequence $\{a_{n}\}$, the sum of the first $n$ terms is $S_{n}$. The correct conclusions are______$(fill in the numbers)$.<br/>① If $a_{1}=1$ and the common ratio is $\frac{1}{2}$, then $S_{n} \lt 2$;<br/>② The sequence $\{a{}_{n}^{2}\}$ must be a geometric sequence;<br/>③ The sequence $\{a_{n+1}+a_{n}\}$ must be a geometric sequence;<br/>④ For any positive integer $n$, $a{}_{n}^{2}+a{}_{n+2}^{2}\geqslant 2a{}_{n+1}^{2}$;<br/>⑤ The sequences $S_{n}$, $S_{2n}-S_{n}$, $S_{3n}-S_{2n}$, $\ldots$ must be a geometric sequence.

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