
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  $k>1$  be a positive integer. A set  $S{}$  is called *good* if there exists a colouring of the positive integers with  $k{}$  colours, such that no element from   $S{}$  can be written as the sum of two distinct positive integers having the same colour. Find the greatest positive integer  $t{}$  (in terms of  $k{}$ ) for which the set \[S=\{a+1,a+2,\ldots,a+t\}\]is good, for any positive integer  $a{}$ .

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