
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 an ellipse $E$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{1}{2}$, the minimum distance from a point $P$ on the ellipse to the right focus is $1$.

(1) Find the equation of the ellipse $E$;

(2) A line passing through the point $(0,2)$ with a slope angle of $60^{\circ}$ intersects the 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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