
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 $\{a_n\}$, the sum of its first $n$ terms $S_n$ satisfies: $S_n = t(S_n - a_n + 1)$ ($t > 0$), and $4a_3$ is the arithmetic mean of $a_1$ and $2a_2$.  
(I) Find the value of $t$ and the general formula for the sequence $\{a_n\}$.  
(II) Let $b_n = \frac{2n+1}{a_n}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, $T_n$.

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