
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 three points \( A\left(0, \frac{4}{3}\right) \), \( B(-1,0) \), and \( C(1,0) \) in the Cartesian coordinate system \( xOy \), the distance from point \( P \) to the line \(BC\) is the geometric mean of its distances to lines \( AB \) and \( AC \).

(1) Find the equation of the locus of point \(P\).

(2) If the line \(l\) passes through the incenter (denoted as \( D \)) of triangle \( ABC \) and intersects the locus of \( P \) at exactly three points, determine the possible range for the slope \( k \) of the line \( l \).

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