
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  $A_1A_2A_3A_4A_5A_6A_7A_8$   be convex 8-gon (no three diagonals concruent).
The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of  $A_1A_2A_3A_4A_5A_6A_7A_8$  and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest  $n$  satisfying:
We can color n "button" such that for all  $i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k)$  are the same where  $s(i,k)$  denote the number of the  "sub quadrilaterals" has  $A_i,A_k$  be the vertices and the intersection of two its diagonals is "button".

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