
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 the sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $a_1=2$ and $a_{n+1}=2S_n+2$.
1. Find the general term formula for the sequence $\{a_n\}$.
2. If each term of the sequence $\{b_n\}$ is positive, and $b_n$ is the geometric mean of $\frac{n}{a_n}$ and $\frac{n}{a_{n+2}}$, find the sum of the first $n$ terms of sequence $\{b_n\}$, denoted as $T_n$.

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