
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  $m$  circles intersect in points  $A$  and  $B$ . We write numbers using the following algorithm: we write  $1$  in points  $A$  and  $B$ , in every midpoint of the open arc  $AB$  we write  $2$ , then between every two numbers written in the midpoint we write their sum and so on repeating  $n$  times. Let  $r(n,m)$ 
be the number of appearances of the number  $n$  writing all of them on our  $m$  circles. 

a) Determine  $r(n,m)$ ;

b) For  $n=2006$ , find the smallest  $m$  for which  $r(n,m)$  is a perfect square.

Example for half arc:     $1-1$ ;
 $1-2-1$ ;
 $1-3-2-3-1$ ;
 $1-4-3-5-2-5-3-4-1$ ;
 $1-5-4-7-3-8-5-7-2-7-5-8-3-7-4-5-1$ ...

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