
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 sequence $\left\{a_{n}\right\}$ with its sum of the first $n$ terms $S_{n}$ satisfying $2 S_{n}-n a_{n}=n$ for $n \in \mathbf{N}^{*}$, and $a_{2}=3$:
1. Find the general term formula for the sequence $\left\{a_{n}\right\}$.
2. Let $b_{n}=\frac{1}{a_{n} \sqrt{a_{n+1}}+a_{n+1} \sqrt{a_{n}}}$ and $T_{n}$ be the sum of the first $n$ terms of the sequence $\left\{b_{n}\right\}$. Determine the smallest positive integer $n$ such that $T_{n}>\frac{9}{20}$.

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