
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 $xOy$, it is known that the ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passes through the point $P(1, \dfrac{3}{2})$, and its eccentricity is $\dfrac{1}{2}$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let the line $l$ intersect the ellipse $C$ at points $A$ and $B$.  
$(i)$ If the line $l$ passes through the right focus of the ellipse $C$, denote the product of the slopes of the lines containing the sides of $\triangle ABP$ as $t$, find the maximum value of $t$;  
$(ii)$ If the slope of the line $l$ is $\dfrac{\sqrt{3}}{2}$, investigate whether $OA^{2} + OB^{2}$ is a constant value. If it is, find this constant value; if not, please explain why.

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