
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  $n$  be positive integer and fix  $2n$  distinct points on a circle. Determine the number of ways to connect the points with  $n$  arrows (oriented line segments) such that all of the following conditions hold:  	
- each of the  $2n$  points is a startpoint or endpoint 	of an arrow;
- no two arrows intersect; and
- there are no two arrows  $\overrightarrow{AB}$  and 	 $\overrightarrow{CD}$  such that  $A$ ,  $B$ ,  $C$  and  $D$  	appear in clockwise order around the circle 	(not necessarily consecutively).


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