
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 the function $f\left(x\right)=\ln x-k\sin x-1$, $x\in \left(0,\pi \right]$, the following four statements are provided:<br/>① For any real number $k$, $f\left(x\right)$ must have a critical point;<br/>② When $k\geqslant 0$, $f\left(x\right)$ must have a zero point;<br/>③ When $k≥\frac{2}{π}$, $f\left(x\right)$ must have two critical points in the interval $\left(0,\pi \right)$;<br/>④ There exist infinitely many real numbers $k$ such that $f\left(x\right)$ has a maximum value.<br/>The correct statement numbers are ____.

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