
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 $f(x) = \frac{e^x - 1}{e^x + 1}$, let $g(x) = f(x - 1) + 1$. Define the sequence $\{a_n\}$ such that $a_n = g\left(\frac{1}{n}\right) + g\left(\frac{2}{n}\right) + g\left(\frac{3}{n}\right) + \dots + g\left(\frac{2n - 1}{n}\right)$, where $n$ is a positive integer. The sum of the first $n$ terms of sequence $\{a_n\}$ is denoted by $S_n$.  
(1) Find a general formula for the terms of sequence $\{a_n\}$.  
(2) If the sequence $\{b_n\}$ is an arithmetic sequence, and $b_n = \frac{2S_n - n}{n + c}$, find the non-zero constant $c$.  
(3) Let $c_n = \frac{1}{a_n a_{n+1}}$. If the sum of the first $n$ terms of sequence $\{c_n\}$ is denoted by $T_n$, find the largest positive integer $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all positive integers $n$.

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