
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 at the origin, one of its foci coincides with the focus of the parabola $y^{2}=4x$, and the point $(1, \frac {3}{2})$ lies on the ellipse $C$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let the left and right foci of the ellipse be $F_{1}$ and $F_{2}$, respectively. A line $l$ passing through $F_{2}$ intersects the ellipse $C$ at points $A$ and $B$. If the area of $\triangle AF_{1}B$ is $\frac {6 \sqrt {3}}{5}$, find the equation of the circle with $F_{1}$ as its center and tangent to the line $l$.

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