
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
Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:
<span style="color:#0000FF">Alice:</span> Are you going to cover your keys?
<span style="color:#FF0000">Bob:</span> I would like to, but there are only  $7$  colours and I have  $8$  keys.
<span style="color:#0000FF">Alice:</span> Yes, but you could always distinguish a key by noticing that the red key next to the green key was different from the red key next to the blue key.
<span style="color:#FF0000">Bob:</span> You must be careful what you mean by "*next to*" or "*three keys over from*" since you can turn the key ring over and the keys are arranged in a circle.
<span style="color:#0000FF">Alice:</span> Even so, you don't need  $8$  colours.**Problem:** What is the smallest number of colours needed to distinguish  $n$  keys if all the keys are to be covered.

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