
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 the sequence $\{a\_n\}$ that satisfies $a_{n+1} - a_n = 4n + 1$ ($n \in \mathbb{N}^*$), and $a_1 = 1$,
(I) Find the general formula for the sequence $\{a\_n\}$;
(II) If $b\_n = (-1)^n \frac{4n^2(n+1)}{a_n a_{n+1}}$ ($n \in \mathbb{N}^*$), find the sum of the first $n$ terms, $S_n$, of the sequence $\{b\_n\}$.

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