
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  $r_1$ ,  $r_2$ ,  $\ldots$ ,  $r_m$  be a given set of  $m$  positive rational numbers such that  $\sum_{k=1}^m r_k = 1$ .  Define the function  $f$  by  $f(n)= n-\sum_{k=1}^m \: [r_k n]$  for each positive integer  $n$ .  Determine the minimum and maximum values of  $f(n)$ .  Here  ${\ [ x ]}$  denotes the greatest integer less than or equal to  $x$ .

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