
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
There is a regular  $17$ -gon  $\mathcal{P}$  and its circumcircle  $\mathcal{Y}$  on the plane. 
The vertices of  $\mathcal{P}$  are coloured in such a way that  $A,B \in \mathcal{P}$  are of different colour, if the shorter arc connecting  $A$  and  $B$  on  $\mathcal{Y}$  has  $2^k+1$  vertices, for some  $k \in \mathbb{N},$  including  $A$  and  $B.$  
What is the least number of colours which suffices?

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