
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 $\{n\}$ with the first term $a_1$ and a common difference $d_0$. Also, $a_1$, $a_2$, and $4$ form a geometric sequence. Let the sequence $\{b_n\} = \frac{1}{a_n^2 - 1}$. Find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $S_n$.

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