
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 $C: x^{2}+y^{2}=4$, point $M(1,1)$, and a certain point $N$ in the plane (different from point $M$), for any moving point $A$ on circle $C$, the ratio $\frac{|AN|}{|AM|}$ is a constant value. The coordinates of point $N$ are ______.

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