
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 $\left\{a_{n}\right\}$ satisfying $a_{1}=2$, for any $n \in \mathbf{N}$, $a_{n}>0$, and $(n+1) a_{n}^{2}+a_{n} a_{n-1}-n a_{n-1}^{2}=0$. Also given the sequence $\left\{b_{n}\right\}$ satisfying $b_{n}=2^{n-1}+1$.

1. Find the general term $a_{n}$ of the sequence $\left\{a_{n}\right\}$ and its sum of the first $n$ terms $S_{n}$.
2. Find the sum of the first $n$ terms $T_{n}$ of the sequence $\left\{b_{n}\right\}$.
3. Conjecture the relationship between $S_{n}$ and $T_{n}$ and provide a justification.

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