
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 terms {a_n} with the sum of the first n terms denoted by S_n, such that $\sqrt{S_n}$ is the arithmetic mean between 1 and $a_n$.  
(i) Find the general term formula for the sequence {a_n};  
(ii) Find the sum of the first n terms, T_n, for the series { $\frac{2}{a_{n}a_{n+1}}$ }.

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