2020 Cayley Problem 24

A cube with vertices FGHJKLMN has edge length 200. Point P is on HG, as shown. The shortest distance from G to a point inside \triangle PFM is 100. Which of the following is closest to the length of HP?

\textbf{(A)}\ 53\quad \textbf{(B)}\ 55\quad \textbf{(C)}\ 57\quad \textbf{(D)}\ 59\quad \textbf{(E)}\ 61

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 Cayley contests here.