
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
Considering the sequence ${a_n}$ that satisfies $a_1 = 3$, and $a_{n+1} = a_n^2 - 2na_n + 2$ where $n \in \mathbb{N}^*$.  
(1) Find the values of $a_2$, $a_3$, and $a_4$, and conjecture a general formula for the sequence ${a_n}$ (no proof required);  
(2) Let $S_n$ be the sum of the first $n$ terms of the sequence ${a_n}$. Determine the smallest positive integer $n$ such that $2^n > S_n$, and provide a proof.

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