
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
There is a circle \( \odot O \) with radius \( R \) and a point \( A \) within the circle such that \( OA = a \). When folding the paper such that a point \( A' \) on the circumference coincides with point \( A \), each fold leaves a crease line. Find the set of points on all the crease lines when \( A' \) takes all possible positions on the circumference.
(2003 National High School Mathematics Competition)

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