
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n = n^2 + 2n$,

(1) Find the general formula for the sequence $\{a_n\}$;  
(2) Let $b_n = \frac{1}{S_n}$, and the sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$, find $T_n$;  
(3) If the sequence $\{c_n\}$ satisfies the condition: $c_{n+1} = a_{c_n} + 2^n$, and $c_1 = 3$, is there a real number $\lambda$ such that the sequence $\left\{ \frac{c_n + \lambda}{2^n} \right\}$ forms an arithmetic sequence?

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