
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 $f(x)$ is a linear function, $f[f(1)] = -1$, and the graph of $f(x)$ is symmetric about the line $x - y = 0$ is $C$. If the point $\left(n, \frac{a_{n+1}}{a_{n}}\right)$ $(n \in \mathbf{N}^{\cdot})$ lies on the curve $C$, and given $a_{1} = 1$, $\frac{a_{n+1}}{a_{n}} - \frac{a_{n}}{a_{n-1}} = 1 (n \geqslant 2)$:
1. Find the analytic expression of $f(x)$ and the equation of the curve $C$.
2. Let $S_{n} = \frac{a_{1}}{3!} + \frac{a_{2}}{4!} + \frac{a_{3}}{5!} + \cdots + \frac{a_{n}}{(n+2)!}$. For all $n \in \mathbf{N}^{\cdot}$, if $S_{n} > m$ holds, find the maximum value of the natural number $m$.

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