
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 2023 cups numbered from 1 through 2023. Red, green, and blue balls are placed in the cups according to the following rules:
- If cups \( m \) and \( n \) both contain a red ball, then \( m - n \) is a multiple of 2.
- If cups \( m \) and \( n \) both contain a green ball, then \( m - n \) is a multiple of 3.
- If cups \( m \) and \( n \) both contain a blue ball, then \( m - n \) is a multiple of 5.

What is the smallest possible number of empty cups?

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