
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 $f(x)=\frac{{\sqrt{{x^2}-{x^4}}}}{{|{x-1}|-1}}$, the following four conclusions are provided:<br/>$(1)$ The domain of $f(x)$ is $\left[-1,0\right)\cup \left(0,1\right]$<br/>$(2)$ The range of $f(x)$ is $\left(-1,1\right)$<br/>$(3)$ $f(x)$ is an increasing function within its domain<br/>$(4)$ The graph of $f(x)$ is symmetric about the origin<br/>The correct conclusions are those numbered ______.

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