
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 circle $M$: $(x+1)^{2}+y^{2}= \dfrac {1}{4}$, circle $N$: $(x-1)^{2}+y^{2}= \dfrac {49}{4}$, and a moving circle $D$ that is externally tangent to circle $M$ and internally tangent to circle $N$. The trajectory of the center of circle $D$ is curve $E$.  
$(1)$ Find the equation of curve $E$;  
$(2)$ If the right focus of the hyperbola $C$ is the right vertex of curve $E$, and the line $y= \sqrt {3}x$ is one of the asymptotes of $C$.  
  $(i)$ Find the equation of the hyperbola $C$;  
  $(ii)$ A line $l$ passing through point $P(0,4)$ intersects the hyperbola $C$ at points $A$ and $B$, and intersects the $x$-axis at point $Q$ (point $Q$ does not coincide with the vertex of $C$). When $\overrightarrow{PQ}=\lambda_{1} \overrightarrow{QA}=\lambda_{2} \overrightarrow{QB}$, and $\lambda_{1}+\lambda_{2}=- \dfrac {8}{3}$, find the coordinates of point $Q$.

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