
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
Given a convex  $2010$  polygonal whose any 3 diagonals have no intersection points except vertices. Consider  *closed broken lines* which have  $2010$  diagonals (not including edges) and they pass through each vertex exactly one time. Find the possible maximum value of the number of self-crossing. Note that we call *closed broken lines* such that broken line  $P_1P_2\cdots P_nP_{n+1}$  has the property  $P_1=P_{n+1}.$ 

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