
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 $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, 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 $\{b_n\}$ be $T_n$, 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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