
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]


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Let {a<sub>n</sub>} be a sequence with all positive terms, and let S<sub>n</sub> be the sum of the first n terms of {a<sub>n</sub>}, such that for any n∈N<sup>*</sup>, it holds that $2\sqrt{S_{n}} = a_{n} + 1$.

(1) Find the general formula for the sequence {a<sub>n</sub>}.  
(2) Let $b_{n} = (-1)^{n-1}a_{n}$, find the sum of the first n terms of the sequence {b<sub>n</sub>}, denoted as T<sub>n</sub>.  
(3) Let $c_{n} = \frac{1}{\sqrt{a_{n}S_{2n+1}} + \sqrt{a_{n+1}S_{2n-1}}}$, find the minimum value of $\sum\limits_{i=1}^{n}[(\sqrt{2n+1}+1)c_{i}]$.

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