
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 a point \( A(1,1) \) on the parabola \( y=x^2 \), the tangent to the parabola at this point intersects the \( x \)-axis at point \( D \) and the \( y \)-axis at point \( B \). Point \( C \) lies on the parabola, and point \( E \) lies on segment \( AC \) such that \( \frac{AE}{EC}=\lambda_1 \). Point \( F \) lies on segment \( BC \) such that \( \frac{BF}{FC}=\lambda_2 \), with \( \lambda_1+\lambda_2=1 \). Line segment \( CD \) intersects \( EF \) at point \( P \). As point \( C \) moves along the parabola, find the equation of the locus of point \( P \).


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