
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  $n,k$  be arbitrary positive integers. We fill the entries of an  $n\times k$  array with integers such that all the  $n$  rows contain the integers  $1,2,\dots,k$  in some order. Add up the numbers in all  $k$  columns – let  $S$  be the largest of these sums. What is the minimal value of  $S$ ?

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