
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 \( n \) be a positive integer, \( [x] \) be the greatest integer less than or equal to the real number \( x \), and \( \{x\}=x-[x] \).
(1) Find all positive integers \( n \) satisfying \( \sum_{k=1}^{2013}\left[\frac{k n}{2013}\right]=2013+n \);
(2) Find all positive integers \( n \) that make \( \sum_{k=1}^{2013}\left\{\frac{k n}{2013}\right\} \) attain its maximum value, and determine this maximum value.

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