
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 a moving circle C passes through point A(-2, 0) and is internally tangent to the circle M: $(x-2)^2+y^2=64$  
(1) Find the equation of the trajectory of the center of circle C;  
(2) Suppose the line $l: y=kx+m$ (where $k, m \in \mathbb{Z}$) intersects the trajectory obtained in (1) at two distinct points B and D, and intersects the hyperbola $\frac{x^2}{4} - \frac{y^2}{12} = 1$ at two distinct points E and F. Is there any line $l$ such that $\overrightarrow{DF} + \overrightarrow{BE} = \overrightarrow{0}$? If yes, how many such lines are there? If no, please explain why.

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