
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
In the Cartesian coordinate system $xOy$, a moving point $M$ has a sum of distances to $F_1(-\sqrt{3}, 0)$ and $F_2(\sqrt{3}, 0)$ equal to 4.  
(1) Find the equation of the trajectory $C$ of the moving point $M$;  
(2) Suppose a line $l$ passing through point $P(3, 0)$ intersects the trajectory $C$ at points $A$ and $B$. Is there a fixed point $Q$ such that $\overrightarrow{QA} \cdot \overrightarrow{QB}$ is a constant? If it exists, find the coordinates of point $Q$ and the constant value; if not, explain why.

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