
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-geometric sequence $\{a\_n\}$ where all terms are positive, and $a\_1 + 2a\_2 = 5$, $4a\_3^2 = a\_2a\_6$.
1. Find the general term formula for the sequence $\{a\_n\}$.
2. If sequence $\{b\_n\}$ satisfies $b\_1 = 2$ and $b\_{n+1} = b\_n + a\_n$, find the general term formula for sequence $\{b\_n\}$.
3. Let $c\_n = \frac{a\_n}{b\_nb\_{n+1}}$, find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $T\_n$.

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