
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 $\{a\_n\}$ whose first $n$ terms' sum is $S\_n=a\_n+\frac{1}{2}n^2+\frac{3}{2}n-2 (n\in\mathbb{N}^*)$.

1. Find the general formula for the sequence $\{a\_n\}$.
2. If $b\_n=\begin{cases} \frac{1}{(a\_n-1)(a\_n+1)}, &n \text{ is odd}\\ 4(\frac{1}{2})^{a\_n}, &n \text{ is even} \end{cases}$ and the first $n$ terms' sum of the sequence $\{b\_n\}$ is $T\_n$, find $T\_{2n}$.

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