
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
There are  $2022$  equally spaced points on a circular track  $\gamma$  of circumference  $2022$ . The points are labeled  $A_1, A_2, \ldots, A_{2022}$  in some order, each label used once. Initially, Bunbun the Bunny begins at  $A_1$ . She hops along  $\gamma$  from  $A_1$  to  $A_2$ , then from  $A_2$  to  $A_3$ , until she reaches  $A_{2022}$ , after which she hops back to  $A_1$ . When hopping from  $P$  to  $Q$ , she always hops along the shorter of the two arcs  $\widehat{PQ}$  of  $\gamma$ ; if  $\overline{PQ}$  is a diameter of  $\gamma$ , she moves along either semicircle.

Determine the maximal possible sum of the lengths of the  $2022$  arcs which Bunbun traveled, over all possible labellings of the  $2022$  points.

*Kevin Cong*

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