
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 arithmetic sequence $\{a_n\}$, $S_3 = 12$, and $a_5 = 2a_2 - 1$.
(I) Find the general term formula for $a_n$;
(II) Calculate the sum of the first $n$ (where $n \geq 2)$ terms, $S_n$, for the sequence $\left\{ \frac{a_{n+2}-a_n}{a_n a_{n+2}} \right\}$.

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