
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
On a plane, there are grasshopper Kolya and his 2020 fellow grasshoppers. Kolya plans to jump over each of the other grasshoppers (in any order) such that the starting and ending points of each jump are symmetric with respect to the grasshopper being jumped over. Let's call a point a finish point if Kolya can reach it after the 2020th jump. What is the largest number $N$ for which there exists an initial arrangement of grasshoppers where there are exactly $N$ different possible finish points?

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