
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
In the plane rectangular coordinate system \( xOy \), given three points \( A\left(0, \frac{4}{3}\right) \), \( B(-1, 0) \), and \( C(1, 0) \), 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) Suppose that the line \( l \) passes through the incenter \( D \) of \( \triangle ABC \), and it intersects the locus of point \( P \) at exactly 3 points. Find the range of the slope \( k \) of the line \( l \).

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