2024 Mock AMC 12 - Problem 7

Let A(0,0) and B(10,0) be two adjacent vertices of a regular dodecagon (12-sided polygon). There exists a third vertex C \neq A that is adjacent to B. The coordinates of C can be either \left(a + b\sqrt{c}, d\right) or \left(a + b\sqrt{c}, e\right), where a, b, c, d, e are integers and c is not divisible by the square of any prime. Find a + b + c + d + e.

\textbf{(A) }12\qquad\textbf{(B) }28\qquad\textbf{(C) }23\qquad\textbf{(D) }18\qquad\textbf{(E) }13