
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 the first $n$ terms $S_n$ satisfies $S_n = n^2 + \lambda n$ ($\lambda \in \mathbb{R}$).  
   (I) Find the value of the real number $\lambda$, and derive the general formula for the sequence $\{a_n\}$.  
   (II) If the sequence is a geometric sequence with the first term $\lambda$ and common ratio $2\lambda$, find the sum of the first $n$ terms $T_n$ of the sequence $\{b_n\}$.

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