
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  $P_1$  be a regular  $n$ -gon, where  $n\in\mathbb{N}$ . We construct  $P_2$  as the regular  $n$ -gon whose vertices are the midpoints of the edges of  $P_1$ . Continuing analogously, we obtain regular  $n$ -gons  $P_3,P_4,\ldots ,P_m$ . For  $m\ge n^2-n+1$ , find the maximum number  $k$  such that for any colouring of vertices of  $P_1,\ldots ,P_m$  in  $k$  colours there exists an isosceles trapezium  $ABCD$  whose vertices  $A,B,C,D$  have the same colour.

*Radu Ignat*

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