
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
The curve \( y = 2x^3 - 8x \) (\( G_3 \)) and the curve \( y = mx^2 - 8x \) (\( G_2 \)) have one more common point, \( A \), besides the origin. Determine the equation of the line \( e \) defined by the points \( O \) and \( A \).

The curve \( G_3 \) has two additional zeros besides the origin. Let the perpendicular foot from these points onto the line \( e \) be \( M_1 \) and \( M_2 \). What is the locus of points \( M_1 \) and \( M_2 \) as \( m \) takes all real values?

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