
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  $ABCDEF$  regular hexagon of side length  $1$  and  $O$  is its center. In addition to the sides of the hexagon, line segments from  $O$  to the every vertex are drawn, making as total of  $12$  unit segments. Find the number paths of length  $2003$  along these segments that star at  $O$  and terminate at  $O$ .

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