
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 table near the sea, there are  $N$  glass boxes where  $N<2021$ , each containing exactly  $2021$  balls. Sowdha and Rafi play a game by taking turns on the boxes where Sowdha takes the first turn. In each turn, a player selects a non-empty box and throws out some of the balls from it into the sea. If a player wants, he can throw out all of the balls in the selected box. The player who throws out the last ball wins. Let  $S$  be the sum of all values of  $N$  for which Sowdha has a winning strategy and let  $R$  be the sum of all values of  $N$  for which Rafi has a winning strategy. What is the value of  $\frac{R-S}{10}$ ?

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