
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 given positive integer. A set  $S$  of positive integers is called *good* if we can colour the set of positive integers in  $k$  colours such that each integer of  $S$  cannot be represented as sum of two positive integers of the same colour. Find the greatest  $t$  such that the set  $S=\{a+1,a+2,\ldots ,a+t\}$  is *good* for all positive integers  $a$ .

*A. Ivanov, E. Kolev*

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