
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 a right angle, a quarter-circle arc is drawn with the vertex $O$ as the center. At any point $P$ on the arc, a tangent is constructed. The tangent intersects the legs of the angle at points $Q_{1}$ and $Q_{2}$. Consider the centers $C_{1}$ and $C_{2}$ of the incircles of triangles $O P Q_{1}$ and $O P Q_{2}$, respectively. What path do $C_{1}$ and $C_{2}$ trace as $P$ traverses along the quarter-circle arc?

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