
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 $f(x)=\lg \frac {x^{2}+1}{|x|} (x\neq 0)$, the following statements are given:  
$(1)$ Its graph is symmetric about the $y$-axis;  
$(2)$ When $x > 0$, $f(x)$ is an increasing function; when $x < 0$, $f(x)$ is a decreasing function;  
$(3)$ $f(x)$ is an increasing function on the intervals $(-1,0)$ and $(2,+\infty)$;  
$(4)$ The minimum value of $f(x)$ is $\lg 2$;  
$(5)$ $f(x)$ has neither a maximum nor a minimum value.  
The correct statement(s) is/are \_\_\_\_\_\_.

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