
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 an arithmetic sequence $\left\{a_n\right\}$ satisfying $a_4 - a_2 = 2$, and $\{a_1, a_3, a_7\}$ forms a geometric sequence.

$(I)$ Find the general formula for $\left\{a_n\right\}$.

$(II)$ Let $b_n = \frac{1}{a_n^2 - 1}$, calculate the sum of the first $n$ terms of the sequence $\left\{b_n\right\}$.

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