
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 the circle $C: x^{2}+y^{2}-6y+8=0$, and $O$ is the origin.  
$(1)$ Find the equation of the line $l$ that passes through point $O$ and is tangent to circle $C$.  
$(2)$ If $P$ is a moving point on circle $C$, and $M$ is the midpoint of $OP$, find the equation of the trajectory of point $M$.

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