
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
If the sequence $\{a_{n}\}$ satisfies: there exists a positive integer $T$, for any positive integer $n$, $a_{n+T}=a_{n}$ holds, then the sequence $\{a_{n}\}$ is called a periodic sequence with period $T$. Given that the sequence $\{a_{n}\}$ satisfies $a_{1}=m\left(m \gt 0\right)$ and $a_{n+1}=\left\{\begin{array}{l}{{a}_{n}-1,{a}_{n}＞1}\\{\frac{1}{{a}_{n}},0＜{a}_{n}≤1}\end{array}\right.$, the following conclusions can be drawn:<br/>① If $a_{3}=4$, then $m$ can take $3$ different values;<br/>② If $m=\sqrt{2}$, then the sequence $\{a_{n}\}$ is a periodic sequence with period $3$;<br/>③ For any $T\in N^{*}$ and $T\geqslant 2$, there exists $m \gt 1$ such that $\{a_{n}\}$ is a periodic sequence with period $T$;<br/>④ There exists $m\in Q$ and $m\geqslant 2$ such that the sequence $\{a_{n}\}$ is a periodic sequence.<br/>Among these conclusions, the correct ones are ____.

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