
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
Six thousand students took an exam, and each received a score that is an integer between 0 and 8 (inclusive). Vincent decides to replace all scores of 1, 2, and 3 with 0, and all scores of 5, 6, or 7 with 8. The other scores remain unchanged. After this change, the average score increased by $\frac{1}{10}$.

Demonstrate that there exist two integers $a$ and $b$ such that the difference between the number of students who received score $a$ and the number of students who received score $b$ before Vincent's modifications is at least 100.

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