
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 that point $P$ is any point on circle $G$: ${{(x+\\sqrt{6})}^{2}}+{{y}^{2}}=32$, and the perpendicular bisector of the line segment connecting point $P$ and point $A(\\sqrt{6},0)$ intersects with $GP$ at point $Q$.

(I) Find the equation of the trajectory $C$ of point $Q$;

(II) A line $l$ passing through the coordinate origin $O$ intersects trajectory $C$ at points $E$ and $F$, with line $EF$ not coinciding with the coordinate axes. $M$ is a point on trajectory $C$. If the area of $\\Delta EFM$ is $4$, determine whether the product of the slopes of lines $EF$ and $OM$ is a constant value. If it is, find this value; otherwise, explain the reason.

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