Please reason step by step, and put your final answer within \boxed{}.
In a plane, \( C \) is a circle, line \( L \) is a tangent to the circle, and \( M \) is a point on \( L \). Find the set of all points \( P \) with the following property: There exist two points \( Q \) and \( R \) on line \( L \) such that \( M \) is the midpoint of segment \( Q R \), and \( C \) is the incircle of triangle \( P Q R \).
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