
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
The center of ellipse $C$ is at the origin, and its right focus is $F(\sqrt{3}, 0)$. The distance from point $F$ to one endpoint of the minor axis is equal to the focal distance.  
(Ⅰ) Find the equation of ellipse $C$;  
(Ⅱ) Suppose the intersection points of ellipse $C$ and the curve $|y| = kx$ ($k > 0$) are $A$ and $B$. Find the maximum value of the area of $\triangle OAB$.

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