2023 Cayley Problem 20

A robotic grasshopper jumps 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After every fourth jump, the grasshopper restarts the sequence of jumps: 1 cm to the east, then 2 cm to the north, then 3 cm to the west, then 4 cm to the south. After a total of n jumps, the position of the grasshopper is 162 cm to the west and 158 cm to the south of its original position. The sum of the squares of the digits of n is

\textbf{(A)}\ 22\quad \textbf{(B)}\ 29\quad \textbf{(C)}\ 17\quad \textbf{(D)}\ 14\quad \textbf{(E)}\ 13

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.