
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
14 students attend the IMO training camp. Every student has at least  $k$  favourite numbers. The organisers want to give each student a shirt with one of the student's favourite numbers on the back. Determine the least  $k$ , such that this is always possible if: $a)$  The students can be arranged in a circle such that every two students sitting next to one another have different numbers. $b)$   $7$  of the students are boys, the rest are girls, and there isn't a boy and a girl with the same number.

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