
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 $\{a\_n\}$ with the sum of the first $n$ terms denoted as $S\_n$ and a common ratio $q > 0$. It is known that $S\_2 = 2a\_2 - 2$ and $S\_3 = a\_4 - 2$.

1. Find the general formula for the $n$th term of the sequence.
2. Let $b\_n = \begin{cases} \frac{\log\_2 a\_n}{n^2(n+2)}, & \text{if $n$ is odd} \\ \frac{n}{a\_n}, & \text{if $n$ is even} \end{cases}$ and let $T\_n$ be the sum of the first $n$ terms of the sequence $\{b\_n\}$. Find $T\_{2n}$.

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