
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 all terms are positive numbers, and $a_{n+1}^{2}-a_{n+1}=a_{n}$ for $n=1,2,3,\ldots$, the following four conclusions are provided:<br/>① For any $n\geqslant 2$, $a_{n} \gt 1$; <br/>② The sequence $\{a_{n}\}$ cannot be a constant sequence; <br/>③ If $0 \lt a_{1} \lt 2$, then the sequence $\{a_{n}\}$ is an increasing sequence; <br/>④ If $a_{1} \gt 2$, then for $n\geqslant 2$, $2 \lt a_{n} \lt a_{1}$. <br/>The correct conclusions are numbered as ______.

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