
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 the center of a circle C, which is in the second quadrant and has 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) Determine whether there exists a point Q on circle C, different from the origin O, such that the distance from Q to the fixed point F(4, 0) is equal to the length of segment OF. If such a point exists, find the coordinates of Q; if not, explain why.

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