
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 a Cartesian coordinate system, a circle with center at the origin O and radius r is tangent to the line $\sqrt{3}x - y + 4 = 0$.
(1) Find the equation of circle O.
(2) The circle O intersects the x-axis at points A and B (where point B is on the positive half of the x-axis). A moving point P satisfies $|PA| + |PB| = 4r$. Find the equation of the trajectory of point P.
(3) There is a line l passing through point B, l is parallel to the line $\sqrt{3}x - y + 4 = 0$ and intersects the trajectory of the moving point P at points C and D. Find the area of triangle OCD.

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