
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 $C$: $x^{2}+y^{2}-8y=0$, let the chord $AB$ intersect with circle $C$ at points $A$ and $B$. The midpoint of $AB$ is $M$, and $O$ is the origin of the coordinate system.
(1) Find the trajectory equation of $M$;
(2) When $|OP|=|OM|$, find the equation of line $l$ and the area of $\triangle POM$;
(3) Under the condition of (2), find the equation of the smallest circle that passes through the intersection points of circle $C$: $x^{2}+y^{2}-8y=0$ and line $l$.

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