
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
Given are  $n$  different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points  $A,B$ .We consider  $k$  distinct lines passing through  $A$  and  $m$  distinct lines passing through  $B$ .There is no line passing through both  $A$  and  $B$  and all the lines passing through  $k$  intersect with all the lines passing through  $B$ .The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.

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