
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
There are the following propositions:
$(1)$ If the function $f(x)$ is both an odd and an even function, then the range of $f(x)$ is $\{0\}$;
$(2)$ If the function $f(x)$ is an even function, then $f(|x|) = f(x)$;
$(3)$ If the function $f(x)$ is not monotonic within its domain, then $f(x)$ does not have an inverse function;
$(4)$ If the function $f(x)$ has an inverse function $f^{-1}(x)$, and $f^{-1}(x)$ is not identical to $f(x)$, then the intersection points of the graphs of $f(x)$ and $f^{-1}(x)$ must lie on the line $y = x$;
List the numbers of all true propositions.

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