
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 function $f(x)$, if there exists an $x\_0 \in \mathbb{Z}$ that satisfies $|f(x\_0)| \leq \frac{1}{4}$, then $x\_0$ is considered a "near-zero point" of the function $f(x)$. For the function $f(x) = ax^2 + bx + c (a > 0)$ with four distinct "near-zero points", determine the maximum value of $a$.

A) $2$
B) $1$
C) $\frac{1}{2}$
D) $\frac{1}{4}$

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