
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 the line segment $CD=2 \sqrt {3}$, and the midpoint of CD is O. The moving point A satisfies $AC+AD=2a$ ($a$ is a positive constant).
(1) Find the equation of the curve on which point A lies;
(2) If there exists a point A such that $AC \perpendicular AD$, try to find the range of values for $a$;
(3) If $a=2$, and the moving point B satisfies $BC+BD=4$, and $AO \perpendicular OB$, try to find the maximum and minimum values of the area of $\triangle AOB$.

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