
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
In the arithmetic sequence $\{a_n\}$, $a_2=4$, and the sum of its first $n$ terms $S_n$ satisfies $S_n=n^2+\lambda n (\lambda \in R)$.  
(I) Find the value of the real number $\lambda$, and derive the general formula for the sequence $\{a_n\}$.  
(II) If the sequence $\left\{ \frac{1}{S_n}+b_n \right\}$ is a geometric sequence with the first term $\lambda$ and common ratio $2\lambda$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

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