
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
Let $M$ be a moving point on the unit circle $x^{2}+y^{2}=1$, and let $N$ be a vertex forming an equilateral triangle with the fixed point $A(3,0)$ and $M$. The orientation is such that $M \rightarrow N \rightarrow A \rightarrow M$ is in a counterclockwise direction. As the point $M$ moves, determine the equation for the locus of point $N$.

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