
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 points $A(0,1)$, $B(0,-2)$, and a moving point $P$ such that $|PB|=2|PA|$. Let the locus of point $P$ be curve $C$.  
$(1)$ Find the equation of curve $C$.  
$(2)$ If there exists a point $M$ on the line $mx+y-3m-1=0$ such that $|MB|\geqslant 2|MA|$, find the minimum value of the real number $m$.

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