
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 functions $f(x)=\frac{e^x-1}{e^x+1}$, $g(x)=f(x-1)+1$, and the sequence $a_n=g(\frac{1}{n})+g(\frac{2}{n})+g(\frac{3}{n})+\ldots+g(\frac{2n-1}{n})$ where $n \in \mathbb{N^+}$, let $S_n$ denote the sum of the first $n$ terms of the sequence ${a_n}$.

1. Find the general term formula for the 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 $T_n$ denotes the sum of the first $n$ terms of the sequence ${c_n}$, find the maximum positive integer value of $k$ such that the inequality $T_n>\frac{k}{57}$ holds for all $n \in \mathbb{N^+}$.

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