
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 Cartesian coordinate system \( xOy \), given three points \( A\left(0, \frac{4}{3}\right) \), \( B(-1,0) \), \( C(1,0) \). The distance from point \( P \) to line \( BC \) is the geometric mean of the distances from that point to lines \( AB \) and \( AC \).
(1) Find the equation of the locus of point \( P \);
(2) If line \( l \) passes through the incenter (denoted as \( D \)) of triangle \( ABC \) and intersects the locus of point \( P \) at exactly 3 points, find the range of the slope \( k \) of line \( l \).

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