
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, a grasshopper named Kolya and his 2020 friends are situated. Kolya is about to make a jump over each of his friends (in any order) such that the starting and ending points of each jump are symmetric relative to the grasshopper he is jumping over.

We call a point a finish point if Kolya can reach it after the 2020th jump. What is the largest number $\$ N \$$ for which there is an initial arrangement of grasshoppers such that there are exactly $\$ \mathrm{~N} \$$ different possible finish points?

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