
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 $xOy$, it is known that a circle $C$ with a center in the second quadrant and a radius of $2\sqrt{2}$ is tangent to the line $y=x$ at the origin $O$.  
$(1)$ Find the equation of circle $C$;  
$(2)$ Investigate whether there exists a point $Q$ on circle $C$, other than the origin, such that the distance from $Q$ to the fixed point $F(4,0)$ is equal to the length of the line segment $OF$. If it exists, find the coordinates of point $Q$; if not, explain the reason.

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