
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
It is known that the difference between the largest and smallest of the numbers \( x_{1}, x_{2}, x_{3}, \ldots, x_{9}, x_{10} \) is 1. What is the a) largest; b) smallest possible difference between the largest and smallest of the 10 numbers \( x_{1}, \frac{1}{2}(x_{1} + x_{2}), \frac{1}{3}(x_{1} + x_{2} + x_{3}), \ldots, \frac{1}{10}(x_{1} + x_{2} + \ldots + x_{10}) \)?

c) What will be the answer if there are \( n \) numbers instead of 10?

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