
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 the hyperbola $C$ passes through the point $(2,3)$, and its asymptote equations are $y=\pm \sqrt {3}x$. The ellipse $C_{1}$ shares the same foci with the hyperbola $C$, and the length of the minor axis of the ellipse $C_{1}$ is equal to the length of the real axis of the hyperbola $C$.
$(1)$ Find the equations of the hyperbola $C$ and the ellipse $C_{1}$;
$(2)$ A line $l$ passing through the left focus $F$ of the ellipse $C_{1}$ intersects the ellipse at points $A$ and $B$. Is there a fixed point $D$ such that $\angle ADF=\angle BDF$ regardless of how $AB$ moves? If it exists, find the coordinates of point $D$; if not, explain why.

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