
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 an ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left(a > b > 0\right)$ with eccentricity $e = \dfrac{1}{2}$, and the distance from the right focus to the right vertex is $1$.

$(1)$ Find the equation of ellipse $C$;

$(2)$ Points $A$ and $B$ are the left and right vertices of ellipse $C$, respectively, and point $P$ is a point on the ellipse different from $A$ and $B$. Let the slopes of lines $PA$ and $PB$ be $K_{PA}$ and $K_{PB}$, respectively. Find the value of $K_{PA} \cdot K_{PB}$.

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