
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
A sequence $\{a\_n\}$ with positive terms has a sum of the first $n$ terms denoted as $S\_n$. Given that $S\_1=2$, $a\_7=20$, and $2(a+b)S\_n=(a\_n+a)(a\_n+b)$ for all positive integers $n$, where $b > \frac{3}{2} > a$.

1. Find the values of $a$ and $b$.
2. If $b\_n = \frac{a\_n+1}{3 \cdot 2^{n}}$, let the sum of the first $n$ terms of the sequence $\{b\_n\}$ be denoted as $T\_n$. Find $T\_n$.

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