
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 the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n=\frac{n(n+1)}{2}$.
(I) Find the general term formula of the sequence $\{a\_n\}$;
(II) Let $T\_n$ be the sum of the first $n$ terms of the sequence $\{b\_n\}$, where $b\_n=\frac{a_{n+1}}{2S\_n \cdot S_{n+1}}$, find $T\_n$;
(III) If there exists $n \in \mathbb{N}^*$, such that $T\_n - \lambda a\_n \geqslant 3 \lambda$ holds, find the range of values for the real number $\lambda$.

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