
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
Regarding the function $f(x)=x\ln |x|$, there are five propositions:  
$(1) f(x)$ is a monotonically increasing function on the interval $(-\infty, -\frac{1}{e})$;  
$(2) f(x)$ has only a minimum point, no maximum point;  
$(3) f(x) > 0$ has a solution set of $(-1,0)\cup(0,1)$;  
$(4)$ The equation of the tangent line to the function $f(x)$ at $x=1$ is $x-y+1=0$;  
$(5)$ The function $g(x)=f(x)-m$ has at most $3$ zeros.  
Among these, the true propositions are __________ (Please fill in the numbers of the true propositions on the line).

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