
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
The unit-length rods $OA$ and $AB$ are joined at a pivot at $A$. Another rod, $BC$, of the same length as $AB$, is attached at $B$ such that point $C$ lies in the plane defined by $O$, $A$, and $B$, and the angle $\angle ABC = 90^\circ$. Keeping point $O$ at the origin, we move the structure in the coordinate plane such that point $B$ moves along the line passing through $O$ with the direction vector $(1, 1 - \sqrt{2})$. What path does point $C$ trace while $A$ moves around a unit circle centered at the origin?

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