
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 that the graph of the function $y=f(x)$ is symmetric to the graph of the function $y=a^{x}-1(a > 1)$ about the line $y=x$.
(1) Find the analytic expression of $f(x)$;
(2) If the range of $f(x)$ in the interval $[m,n] (m > -1)$ is $[\log_{a} \frac{p}{m},\log_{a} \frac{p}{n}]$, find the range of the real number $p$;
(3) Let the function $g(x)=\log_{a}(x^{2}-3x+3)$, $F(x)=a^{f(x)-g(x)}$, where $a > 1$. If $w\geqslant F(x)$ holds for all $x\in(-1,+\infty)$, find the range of the real number $w$.

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