
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 $f(x)=\log_{2}\frac{\sqrt{2}x}{a-x}$, it passes through the fixed point $A(\frac{1}{2},\frac{1}{2})$. The line intersects the graph of $f(x)$ at two points $B$ and $C$. Also, $\overrightarrow{AB}+\overrightarrow{AC}=\overrightarrow{0}$.
(1) Find the value of $a$;
(2) If $S_{n}=f(\frac{1}{n})+f(\frac{2}{n})+\ldots+f(\frac{n-1}{n})$, $n\in\mathbb{N}^{*}$, and $n\geqslant 2$, find $S_{n}$;
(3) The sequence $\{a_{n}\}$ satisfies $a_{1}=\frac{2}{3}$ and $\frac{1}{a_{n}}=(S_{n}+1)(S_{n+1}+1)$, where $n\in\mathbb{N}^{*}$. Let $T_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$. If $T_{n}<\lambda(S_{n+1}+1)$ holds for all $n\in\mathbb{N}^{*}$, find the range of $\lambda$.

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