
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
Fix integers  $n\ge k\ge 2$ . We call a collection of integral valued coins  $n-diverse$  if no value occurs in it more than  $n$  times. Given such a collection, a number  $S$  is  $n-reachable$  if that collection contains  $n$  coins whose sum of values equals  $S$ . Find the least positive integer  $D$  such that for any  $n$ -diverse collection of  $D$  coins there are at least  $k$  numbers that are  $n$ -reachable.

*Proposed by Alexandar Ivanov, Bulgaria.*

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