
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 the angle $xOy$, two points $A$ and $B$ on the $Ox$ arm, and a point $C$ on the $Oy$ arm, draw two lines $CM$ and $CN$ from point $C$, which are antiparallel with respect to the given angle and intersect the $Ox$ arm at points $M$ and $N$. Form the equation of the circumcircle of the triangle $MNC$. Determine $M$ and $N$ such that $AM=-BN$.

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