
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
A circle $O$ with radius $R$ and an internal point $A$ with $O A = a$ is drawn on a piece of paper. The paper is folded such that a point $A'$ on the circumference of the circle coincides with $A$. Each fold leaves a straight crease. When $A'$ takes all possible positions on the circumference of the circle, find the set of points lying on all the fold lines.

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