
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 $C_1$: $(x+2)^2+y^2= \frac{81}{16}$ and $C_2$: $(x-2)^2+y^2= \frac{1}{16}$, a moving circle $Q$ is tangent to both $C_1$ and $C_2$.  
(1) Find the equation of the trajectory of the center of circle $Q$.  
(2) Does there exist a fixed point $M$ on the negative x-axis such that $\angle QC_2M = 2\angle QMC_2$? If it exists, find the coordinates of $M$; if not, explain why.

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