2017 Fermat Problem 25

Let r = \sqrt{\frac{\sqrt{53}}{2} + \frac{3}{2}}. There is a unique triple of positive integers (a, b, c) such that r^{100} = 2r^{98} + 14r^{96} + 11r^{94} - r^{50} + ar^{46} + br^{44} + cr^{40} What is the value of a^2 + b^2 + c^2?

\textbf{(A)}\ 11\,421\quad \textbf{(B)}\ 20\,229\quad \textbf{(C)}\ 16\,291\quad \textbf{(D)}\ 15\,339\quad \textbf{(E)}\ 17\,115

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Fermat contests here.