
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 that the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = -a_n - \left(\frac{1}{2}\right)^{n-1} + 2$, where $n$ is a positive integer.

(I) Find $a_1, a_2, a_3, a_4$, and conjecture the general formula for the $n$th term of the sequence $\{a_n\}$ (proof is not required).

(II) Compare the size of $S_n$ and $2 - \frac{1}{n-1}$ for $n \geq 4$, and provide a proof.

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