
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
Let \( x_1, x_2, \ldots, x_m \) be positive rational numbers with sum 1. What is the maximum and minimum value of \( n - \lfloor n x_1 \rfloor - \lfloor n x_2 \rfloor - \ldots - \lfloor n x_m \rfloor \) for positive integers \( n \)?

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