
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 $C: x^{2}+y^{2}-8y=0$, a moving line $l$ passing through point $P$ intersects circle $C$ at points $A$ and $B$, with the midpoint of segment $AB$ being $M$, and $O$ being the origin.
$(1)$ Find the equation of the trajectory of $M$;
$(2)$ When $|OP|=|OM|$, find the equation of $l$ and the area of $\triangle POM$.

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