
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 that the center of symmetry for ellipse C is the origin O, its foci are on the x-axis, and its eccentricity is $\frac{1}{2}$. Additionally, the point $(1, \frac{3}{2})$ lies on the ellipse.

(I) Determine the equation of ellipse C.

(II) A line l passes through the left focus $F_1$ of ellipse C and intersects the ellipse at points A and B. If the area of $\triangle AOB$ is $\frac{6\sqrt{2}}{7}$, find the equation of the circle centered at the origin O that is tangent to line l.

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