
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 positive integers  $k,n$ . A candy vending machine has many different colours of candy, where there are  $2n$  candies of each colour. A couple of kids each buys from the vending machine  $2$  candies of different colours. Given that for any  $k+1$  kids there are two kids who have at least one colour of candy in common, find the maximum number of kids.

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