
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 $f(x) = \lg(1-x) + \lg(p+x)$, where $p > -1$  
(1) Find the domain of $f(x)$.  
(2) If $p=1$, when $x \in (-a, a]$ where $a \in (0, 1)$, and $a$ is a constant, does the function $f(x)$ have a minimum value? If it exists, find the minimum value of $f(x)$; if not, please explain why.

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