
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
Let A$(x_1, y_1)$ and B$(x_2, y_2)$ be any two points on the graph of the function $f(x) = \frac{1}{2} + \log_{2}\frac{x}{1-x}$. It is given that $\overrightarrow{OM} = \frac{1}{2}(\overrightarrow{OA} + \overrightarrow{OB})$, and the x-coordinate of point M is $\frac{1}{2}$.

(1) Find the y-coordinate of point M;

(2) If $S_n = f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + \ldots + f\left(\frac{n-1}{n}\right)$, where $n \in \mathbb{N}^*$ and $n \geq 2$,

   ① Find $S_n$;

   ② Given $a_n = \begin{cases} \frac{2}{3}, & n=1 \\ \frac{1}{(S_n+1)(S_{n+1}+1)}, & n \geq 2 \end{cases}$, where $n \in \mathbb{N}^*$, and $T_n$ is the sum of the first n terms of the sequence $\{a_n\}$. If $T_n \leq \lambda(S_{n+1}+1)$ holds for all $n \in \mathbb{N}^*$, find the minimum positive integer value of $\lambda$.

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