
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 sequence $\{a_n\}$, we have $a_1 = 1$ and $a_{n+1} = \left(1+ \frac {1}{n}\right)a_n+ \frac {n+1}{2^{n}}$.  
(1) Find the sum of the first $n$ terms of the sequence $\{a_n\}$, denoted as $S_n$.  
(2) In the sequence $\{a_n\}$, does there exist three consecutive terms that form an arithmetic sequence? If such terms exist, write down all the terms that satisfy the condition; if not, explain why.

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