
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 the first n terms being $S_n$, $a_1 = 3$, and $a_{n+1} = 2a_n + 2^{n+1} - 1$ for $n \in \mathbb{N}^*$.
(1) Find $a_2$ and $a_3$.
(2) Find the real number $\lambda$ such that $\left\{\frac {a_n+\lambda}{2^n}\right\}$ forms an arithmetic sequence, and from this, deduce $a_n$ and $S_n$.
(3) Determine all values of $n$ such that $\frac {S_n}{a_n} \in \mathbb{N}^*$, and provide the reasoning.

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