
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 $C: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ passes through the point $P\left( 1, \frac{3}{2} \right)$, and its eccentricity is $\frac{1}{2}$.

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

$(2)$ If a line $l$ with a slope of $\frac{\sqrt{3}}{2}$ intersects the ellipse $C$ at points $A$ and $B$, investigate whether $OA^{2}+OB^{2}$ is a constant value. If it is a constant, find this constant; if not, please explain why.

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