
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
Assume that \( x_{1}, \ldots, x_{n} \) and \( y_{1}, \ldots, y_{n} \) are monotonically increasing sequences of non-negative numbers, each with \( n \) terms summing to 1.

a) What is the maximum value of \( \min _{1 \leq i \leq n}\left|x_{i} - y_{i}\right| \)?

b) What is the maximum value of \( \sum_{i=1}^{n}\left|x_{i} - y_{i}\right| \)?

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