
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 circle $C: x^{2}+\left(y-2\right)^{2}=1$ and a fixed point $A(3,0)$, where $P$ is a point outside the circle $C$, the line $PQ$ is tangent to the circle $C$ at point $Q$, and $PQ=\sqrt{2}PA$.  
$(1)$ Find the equation of the locus of point $P$.  
$(2)$ A line $l$ passing through point $A$ intersects the locus of point $P$ at points $M$ and $N$, where $MN=8$. Find the equation of the line $l$.

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