
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
Given a positive integer \( n \) and real numbers \( x_{1}, x_{2}, \cdots, x_{n} \) such that \( \sum_{k=1}^{n} x_{k} \) is an integer, define 

\[ d_{k}=\min _{m \in \mathbf{Z}}\left|x_{k}-m\right| \text{ for } 1 \leq k \leq n. \]

Find the maximum value of \( \sum_{k=1}^{n} d_{k} \).

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