
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
In general, the locus of a moving point $F$ in a plane, where the ratio of the distances from two fixed points $P$ and $Q$ is a constant $\lambda (\lambda\ \ \gt 0$ and $\lambda \neq 1)$, is a circle, which is famously known in the history of mathematics as the "Apollonian circle." Based on the above fact, complete the following questions:  
$(1)$ Given points $A_{1}(1,0)$ and $A_{2}(-2,0)$, if $\frac{|MA_{1}|}{|MA_{2}|}=\frac{\sqrt{2}}{2}$, find the equation of the locus of the moving point $M$;  
$(2)$ Given that point $N$ moves on the circle $(x-3)^{2}+y^{2}=4$, and point $A_{3}(-1,0)$, investigate: Is there a fixed point $A_{4}$ such that $\frac{|NA_{3}|}{|NA_{4}|}=2$? If it exists, find the coordinates of the fixed point $A_{4}$; if it does not exist, please explain the reason.

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