
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 the center of a circle lies on the line $y=4x$ and is tangent to the line $l$: $x+y-2=0$ at point $P(1,1)$.
(I) Find the equation of the circle;
(II) Line $kx-y+3=0$ intersects the circle at points $A$ and $B$. If point $M$ is on the circle and has a position vector $ \overrightarrow{OM}= \overrightarrow{OA}+ \overrightarrow{OB}$ (where $O$ is the origin), find the real number $k$.

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