
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
Let E be an ellipse given by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) with an eccentricity of $\frac{1}{2}$. The minimum distance from a point P on ellipse E to its right focus is 1.  
(1) Determine the equation of ellipse E.  
(2) A line passing through point (0, 2) at a $60^\circ$ angle intersects ellipse E at points A and B. Find the area of $\triangle AOB$.

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