
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 $C$: $\dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)$ with eccentricity $\dfrac {1}{2}$, and a line passing through the right focus and perpendicular to the $x$-axis intersects the ellipse with a chord length of $3$.

$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Points $A$ and $B$ are the left and right vertices of the ellipse $C$, respectively, 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 $k_{PA} \cdot k_{PB}$.

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