
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 $f(x) = 1 + \ln \left( \frac{x}{2-x} \right)$ where $0 < x < 2$.

(1) Does there exist a point $M(a, b)$ such that any point $P$ on the graph of $y = f(x)$ has its symmetric point $Q$ also on the graph of $y = f(x)$? If it exists, find the coordinates of point $M$; if not, explain why.

(2) Define $S_{n} = \sum_{i=1}^{2n-1}f\left( \frac{i}{n} \right) = f\left( \frac{1}{n} \right) + f\left( \frac{2}{n} \right) + \cdots + f\left( \frac{2n-1}{n} \right)$, where $n \in \mathbb{N}^{*}$, find $S_{2013}$.

(3) Under the condition of (2), let $S_{n} + 1 = 2a_{n}$, if the inequality $2^{a_{n}} \cdot (a_{n})^{m} > 1$ holds for all $n \in \mathbb{N}^{*}$ and $n \geq 2$, find the range of the real number $m$.

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