
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 in the geometric sequence $\{a_n\}$, $a_1=1$, and $a_1$, $a_2$, $a_3-1$ form an arithmetic sequence.

(I) Find the general formula for the sequence $\{a_n\}$;

(II) If the sequence $\{b_n\}$ satisfies $b_n=2n-1+a_n$ ($n\in \mathbb{N}^*$), and the sum of the first $n$ terms of the sequence $\{b_n\}$ is $S_n$, compare the size of $S_n$ and $n^2+2^n$.

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