
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 a sequence of positive terms $\{a_{n}\}$, where the sum of the first $n$ terms $S_{n}$ satisfies $2S_{n}=a_{n}^{2}+a_{n}-2$.  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If $b_{n}=\frac{{2}^{n}(n-1)}{n{a}_{n}}$ for $n\in \mathbb{N}^{*}$, find the sum of the first $n$ terms $T_{n}$ of the sequence $\{b_{n}\}$;  
$(3)$ Determine whether there exists a real number $\lambda$ such that $T_{n}+2 > \lambda \cdot S_{n}$ holds for all $n\in \mathbb{N}^{+}$. If it exists, find the range of values for the real number $\lambda$; if it does not exist, provide a reason.

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