
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 $f(x) = \lg|x-1|$, which of the following statements are correct?  
(1) The domain and range of function $f(x)$ are both $\mathbb{R}$;  
(2) The function $f(x)$ is monotonically decreasing on $(-\infty, 1)$ and monotonically increasing on $(1, +\infty)$;  
(3) The graph of function $f(x)$ is symmetric about the y-axis;  
(4) The function $f(x+1)$ is an even function;  
(5) If $f(a) > 0$, then $a < 0$ or $a > 2$.

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