
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 point $P(\sqrt{3}, \frac{1}{2})$ is on the ellipse $E$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, $F$ is the right focus, and $PF$ is perpendicular to the $x$-axis. $A$, $B$, $C$, $D$ are four points on the ellipse, and $AC$, $BD$ intersect at the origin $O$.

$(1)$ Find the equation of the ellipse $E$;  
$(2)$ Let $A(x_{1},y_{1})$, $B(x_{2},y_{2})$, satisfying $\overrightarrow{OA} \cdot \overrightarrow{OB} = 5y_{1}y_{2}$. Determine whether the value of $k_{AB} + k_{BC}$ is constant. If so, find this constant value and the maximum area of quadrilateral $ABCD$; otherwise, explain why.

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