
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
At the beginning of the school year, $40\%$ of all students in Ms. Smith's class answered "Yes" to the question "Do you love math?", $30\%$ answered "No", and $30\%$ answered "Maybe." At the end of the school year, $60\%$ answered "Yes", $20\%$ answered "No", and $20\%$ answered "Maybe." Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$?

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