
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 $f(x)=\log_a(1-x)+\log_a(x+2)$, where $a > 1$, let the domain of $f(x)$ be $D$.

(1) Determine the domain $D$ of the function $f(x)$.

(2) If the maximum value of the function $f(x)$ is $2$, find the value of $a$.

(3) If the inequality $x^2 - mx + m + 1 > 0$ holds for any real number $x$ within the domain $D$, determine the range of the real number $m$.

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