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

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

$(2)$ Points $A$ and $B$ are the left and right vertices of the ellipse $C$, and $P$ 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 ${k}_{PA} \cdot {k}_{PB}$.

$(3)$ Based on this, what general conclusion can you draw? (No proof required)

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