
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 point $P(2, 2)$ and circle $C: x^2 + y^2 - 8x = 0$, any moving line $l$ passing through point $P$ intersects with circle $C$ at points $A$ and $B$. The midpoint of segment $AB$ is $M$, and $O$ is the origin of the coordinates.

(I) Find the trajectory equation of the moving point $M$;  
(II) Determine the equation of line $l$ when $|OP| = |OM|$.

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