
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 numbers, let $S\_n$ denote the sum of the first $n$ terms. The sequence satisfies the equation $2 \sqrt {S_{n}}=a_{n}+1$ for all positive integers $n$.

1. Find the general formula for the $n$-th term of the sequence $\{a\_n\}$.
2. If $b\_n=(a\_n+1) \cdot 2^{a\_n}$, 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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