
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 a sequence of positive numbers $\{a\_n\}$ with the sum of the first $n$ terms denoted as $S\_n$, and it is known that $a\_n^2 + a\_n - 2S\_n = 0$. Let $c\_n = a\_n \cdot b\_n$.

(1) Find the general formula for the sequence $\{a\_n\}$.
(2) If $b\_1 = 1$, $2b\_n - b\_{n-1} = 0$ for $n \geq 2$, 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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