
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 two circles M: $(x+1)^2 + y^2 = \frac{1}{4}$ and N: $(x-1)^2 + y^2 = \frac{49}{4}$, a moving circle D is externally tangent to circle M and internally tangent to circle N. The locus of the center of circle D is curve E.  
(1) Find the equation of curve E;  
(2) If the right focus of hyperbola C is the right vertex of curve E and the line $y = \sqrt{3}x$ is an asymptote of C,  
(i) find the equation of C;  
(ii) A line l passes through point P(0, 4) and intersects the hyperbola C at points A and B, and intersects the x-axis at point Q (Q point does not coincide with the vertex of C). When $\vec{PQ} = \lambda_1 \vec{QA} = \lambda_2 \vec{QB}$, and $\lambda_1 + \lambda_2 = -\frac{8}{3}$, find the coordinates of point Q.

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