
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 a regular 103-sided polygon. 79 vertices are colored red and the remaining vertices are colored blue. Let  $A$  be the number of pairs of adjacent red vertices and  $B$  be the number of pairs of adjacent blue vertices.
a) Find all possible values of pair  $(A,B).$ b) Determine the number of pairwise non-similar colorings of the polygon satisfying  $B=14.$  2 colorings are called similar if they can be obtained from each other by rotating the circumcircle of the polygon.

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