
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 is divided into  $13$  segments, numbered consecutively from  $1$  to  $13$ . Five fleas called  $A,B,C,D$  and  $E$  are sitting in the segments  $1,2,3,4$  and  $5$ . A flea is allowed to jump to an empty segment five positions away in either direction around the circle. Only one flea jumps at the same time, and two fleas cannot be in the same segment. After some jumps, the fleas are back in the segments  $1,2,3,4,5$ , but possibly in some other order than they started. Which orders are possible ?

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