
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
A set of balls contains  $ n$  balls which are labeled with numbers  $ 1,2,3,\ldots,n.$  We are given  $ k > 1$  such sets. We want to colour the balls with two colours, black and white in such a way, that

(a) the balls labeled with the same number are of the same colour,

(b) any subset of  $ k\plus{}1$  balls with (not necessarily different) labels  $ a_{1},a_{2},\ldots,a_{k\plus{}1}$  satisfying the condition  $ a_{1}\plus{}a_{2}\plus{}\ldots\plus{}a_{k}\equal{} a_{k\plus{}1}$ , contains at least one ball of each colour.

Find, depending on  $ k$  the greatest possible number  $ n$  which admits such a colouring.

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