
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 \( a_{1} = 2 \),
\[ a_{n+1} = \frac{2^{n+1} a_{n}}{\left(n+\frac{1}{2}\right) a_{n} + 2^{n}} \text{ for } n \in \mathbf{Z}_{+}, \]

(1) Find the general term formula of the sequence \(\{a_{n}\}\).

(2) Given that \( b_{n} = \frac{n^{3} + 2n^{2} + 2n + 2}{n(n+1)(n^{2} + 1)a} \), find the sum of the first \( n \) terms of the sequence \(\{b_{n}\}\).

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