
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_1, a_2,\ldots  ,a_8$  be  $8$  distinct points on the circumference of a circle such that no three chords, each joining a pair of the points, are concurrent. Every  $4$  of the  $8$  points form a quadrilateral which is called a *quad*. If two chords, each joining a pair of the  $8$  points, intersect, the point of intersection is called a *bullet*. Suppose some of the bullets are coloured red. For each pair  $(i j)$ , with  $ 1 \le i < j \le 8$ , let  $r(i,j)$  be the number of quads, each containing  $ a_i, a_j$  as vertices, whose diagonals intersect at a red bullet. Determine the smallest positive integer  $n$  such that it is possible to colour  $n$  of the bullets red so that  $r(i,j)$  is a constant for all pairs  $(i,j)$ .

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