Determine the maximum value of c satisfying complex geometric and algebraic conditions involving line positioning, point relationships, and circle properties | Step-by-Step Solution

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I can see you're working with a fascinating multi-constraint optimization problem that combines geometry, algebra, and analysis! This involves understanding when two points lie on the same side of a line, which is a fundamental concept in coordinate geometry. Here's what I'd like you to think about first: if you have a line with equation ax + by + c = 0, what happens when you substitute the coordinates of two different points into this equation, and how do the signs of those results tell you about the relative positions of the points? Also, when you have multiple constraints like arithmetic progressions and circle properties, how might you use substitution to express everything in terms of just one variable? I'd encourage you to start by recalling the geometric condition for points being on the same side versus opposite sides of a line - this sign relationship is really the heart of the problem. Once you establish that key insight, you can systematically work through the other constraints (the arithmetic progression and circle conditions) to build a system that you can optimize. You've got all the tools you need from coordinate geometry and algebra to tackle this step by step!