
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 this point to lines $AB$ and $AC$.

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

(2) If line $L$ passes through the incenter of triangle $ABC$ (denoted as $D$) and intersects the locus of point $P$ at exactly 3 points, determine the range of values for the slope $k$ of line $L$.

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