
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
The function $$f(x)=\lg \frac {x^{2}+1}{|x|}$$ (where $x\neq 0$, $x\in\mathbb{R}$) has the following propositions:  
(1) The graph of $y=f(x)$ is symmetric about the y-axis.  
(2) When $x>0$, $f(x)$ is an increasing function, and when $x<0$, $f(x)$ is a decreasing function.  
(3) The minimum value of $f(x)$ is $\lg 2$.  
(4) $f(x)$ has neither a maximum value nor a minimum value.  
Among these, the correct propositions are \_\_\_\_\_\_.

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