
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 that the distance from point $P$ to the line $y=-3$ is $2$ units more than the distance from point $P$ to point $A(0,1)$.
$(1)$ Find the equation of the locus of point $P$;
$(2)$ The moving line $l$ passing through point $Q(0,2)$ intersects the locus of point $P$ at points $M$ and $N$. Is there a fixed point $R$ such that $\angle MRQ=\angle NRQ$? If yes, find the coordinates of point $R$; if not, explain the reason.

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