
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 that \( AB \) is a chord of length \( m \) of the parabola \( y = ax^2 \) with \( a > 0 \):

(I) When \( m \geq \frac{1}{a} \) (length of the latus rectum), the minimum distance from the midpoint \( M \) of \( AB \) to the \( x \)-axis is \( \frac{2ma - 1}{4a} \).

(II) When \( m < \frac{1}{a} \) (length of the latus rectum), the minimum distance from the midpoint \( M \) of \( AB \) to the \( x \)-axis is \( \frac{am^2}{4} \).

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