
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 $C$ with its center of symmetry at the origin $O$, its foci on the $x$-axis, and eccentricity $\frac{1}{2}$, it is known that the point $(1, \frac{3}{2})$ lies on this ellipse.  
(Ⅰ) Find the equation of ellipse $C$.  
(Ⅱ) A line $l$ passing through the left focus $F_1$ of the ellipse $C$ 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 with center at the origin $O$ and tangent to the line $l$.

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