
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 left and right foci $F_{1}(-c,0)$ and $F_{2}(c,0)$, respectively. The line $y=\frac{{2\sqrt{3}}}{3}x$ intersects $C$ at points $A$ and $B$, where $|AB|=2\sqrt{7}$, and the area of quadrilateral $AF_{1}BF_{2}$ is $4\sqrt{3}$. <br/>$(1)$ Find $c$; <br/>$(2)$ Find the equation of $C$.

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