
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
A line passing through the focus F of the parabola $y^2 = 2px$ ($p > 0$) intersects the parabola at points M and N. The product of the y-coordinates of M and N is -4.

1. Find the equation of the parabola.
2. Find the value of $\overrightarrow{OM} \cdot \overrightarrow{ON}$ (where O is the origin).
3. Given point A(1, 2), determine if there exist two points B and C on the parabola such that AB is perpendicular to BC. If they exist, find the range of possible y-coordinates for point C. If not, explain why.

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