
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 parabola \( y = x^2 \) and the ellipse \( (x-3)^2 + 4y^2 = 1 \), where the focus of the parabola is \( F \), and \( A \) is a point on the ellipse. The line \( FA \) intersects the parabola at points \( B \) and \( C \). Tangents to the parabola at points \( B \) and \( C \) are constructed. Find the locus of the intersection point of these tangents.

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