
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 line $y=x$ is tangent to circle C at the origin O, and circle C is in the second quadrant with the same area as circle $C'$: $x^2+y^2-2x-2y-6=0$.  
(I) Find the standard equation of circle C;  
(II) Investigate 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 it exists, find the coordinates of point Q; if not, explain why.

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