
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 $A$ and $B$ are any two points on the graph of the function $f(x)=\frac{1}{2}+\log_{2}\frac{x}{1-x}$, and $M$ is the midpoint of $AB$ with coordinates $(\frac{1}{2},m)$.

1. Find the value of $m$.
2. If $S_{n}=f(\frac{1}{n})+f(\frac{2}{n})+...+f(\frac{n-1}{n}), n\in\mathbb{N}^{*}$ and $n\geqslant 2$, find $S_{n}$.
3. Given that $a_{n}=\begin{cases} \frac{1}{2},n=1 \\\\ S_{n},n\geqslant 2\end{cases}$ where $n\in\mathbb{N}^{*}$, $T_{n}$ is the sum of the first $n$ terms of the sequence ${a_{n}}$, and $T_{n} > λ(S_{n+1}+1)$ holds for all $n\in\mathbb{N}^{*}$, find the range of $λ$.

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