
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
Let  $C$  be a circle with radius  $R$  and centre  $O$ , and  $S$  a fixed point in the interior of  $C$ .  Let  $AA'$  and  $BB'$  be perpendicular chords through  $S$ . Consider the rectangles  $SAMB$ ,  $SBN'A'$ ,  $SA'M'B'$ , and  $SB'NA$ .  Find the set of all points  $M$ ,  $N'$ ,  $M'$ , and  $N$  when  $A$  moves around the whole circle.

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