## Instruction 
Inspired by the provided math content, develop a math problem suitable for the AMC 12 competition, targeting high-school students with capabilities in advanced areas like algebra, geometry, trigonometry, counting, probability, and number theory. The problem should challenge students to utilize sophisticated problem-solving skills and mathematical reasoning. Focus on concepts from the given content, but adapt them into an original problem. Avoid delving into higher-level college mathematics topics beyond the AMC 12 scope.

## Math Content
<EXTRACT>

## Guidelines
[Problem]: Formulate a compelling and complex math problem for AMC 12 participants inspired by the provided math content. The problem should encourage students to employ advanced logical reasoning and problem-solving strategies related to the given concepts. Craft it to be solvable in the AMC 12 format and difficulty level.

[Solution]: Present a comprehensive, step-by-step solution that both solves the problem and educates the student, around 250-350 words long. Clearly articulate the reasoning and methods used at each step, providing insight into the problem-solving process. Use language that challenges yet instructs high school students looking to improve their skills. Take care to format any equations properly using LaTeX or appropriate notation.

## New Problem