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

(1) When the chord $AB$ is the shortest, find the equation of line $l$ and the length of the chord $AB$;
(2) Find the equation that represents the trajectory of $M$.

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