
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 $a\_1=1$, $a\_n>0$, and the sum of the first $n$ terms is $S\_n$. If $$a_{n}= \sqrt {S_{n}}+ \sqrt {S_{n-1}}$$ (for $n\in\mathbb{N}^*$ and $n\geq2$), find:

(I) The general term formula for the sequence $\{a\_n\}$;
(II) Denote $c\_n=a\_n\cdot 2^{a\_n}$, find the sum of the first $n$ terms, $T\_n$, for the sequence $\{c\_n\}$.

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