
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 circle $O$ with the origin as its center and passing through point $(\frac{1}{2}, \frac{\sqrt{3}}{2})$, $M$ and $N$ are two points on the plane symmetric about the origin. The coordinates of $N$ are known to be $(0, -\frac{\sqrt{3}}{3})$. A line is drawn through $N$ intersecting the circle at points $A$ and $B$.

(1) Find the equation of circle $O$;

(2) Determine the range of possible values for the area of triangle $ABM$.

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