
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
A grasshopper named Kolya and 2020 of his friends are on a plane. Kolya is about to make jumps over each of the other grasshoppers (in any order) such that the starting and ending points of each jump are symmetrical 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 maximum possible number \( N \) of different finish points for which there exists an initial arrangement of the grasshoppers?

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