
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 $\{a\_n\}$ where all terms are positive, let $S\_n$ represent the sum of the first $n$ terms of the sequence. For any $n \in \mathbb{N}^*$, it is known that $2S\_n = 2a\_n^2 + a\_n - 1$.

(I) Find the general formula for the terms of the sequence $\{a\_n\}$;
(II) Let $b\_n = 2^n \cdot a\_n$. Find the sum $T\_n$ of the first $n$ terms of the sequence $\{b\_n\}$.

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