
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 $(xOy)$, there is an ellipse $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$. The distance from the upper vertex of the ellipse to the focus is $2$, and the eccentricity is $\frac{\sqrt{3}}{2}$.

(1) Find the values of $a$ and $b$.
(2) Let $P$ be a moving point on the major axis of the ellipse $(C)$, and draw a line $(l)$ with a slope of $k$ through point $P$, intersecting the ellipse $(C)$ at points $A$ and $B$.
   (i) If $k=1$, find the maximum value of the area of $\triangle OAB$.
   (ii) If $PA^{2}+PB^{2}$ is independent of the position of point $P$, find the value of $k$.

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