
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 a positive sequence $\{a\_n\}$ is $n$, and $2 \sqrt {S\_n}=a\_n+1$.
(1) Find the general term formula of the sequence $\{a\_n\}$.
(2) If the sequence $b\_n= \frac {a\_n+3}{2}$, and $T\_n$ is the sum of the first $n$ terms of the sequence $\{\frac {1}{b\_n b\_{n+1}}\}$, find $T\_n$.
(3) If $T\_n \leqslant \lambda b\_{n+1}$ holds for all $n \in \mathbb{N}^*$, find the minimum value of the real number $\lambda$.

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