
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 an arithmetic sequence $\{a_n\}$ where $S_n$ represents the sum of the first $n$ terms, and given that $a_2 = 2$, $S_5 = 15$, consider another sequence $\{b_n\}$ defined by $b_1 = \frac{1}{2}$ and $b_{n+1} = \frac{n+1}{n}b_{n}$ for $n\in\mathbb{N}^{*}$. Let $T_n$ denote the sum of the first $n$ terms of $\{b_n\}$.
(1) Find the general term $a_n$ and the sum of the first $n$ terms $S_n$ of the sequence $\{a_n\}$.
(2) Find the general term $b_n$ and the sum of the first $n$ terms $T_n$ of the sequence $\{b_n\}$. Determine the extremum of $T_n$ and find the corresponding index $n$.

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