Consider the following three lines in the Cartesian plane:
\begin{cases}\ell_{1}: & 2 x-y=7 \\ \ell_{2}: & 5 x+y=42 \\ \ell_{3}: & x+y=14\end{cases}
and let f_{i}(P) correspond to the reflection of the point P across \ell_{i}. Suppose X and Y are points on the x and y axes, respectively, such that f_{1}\left(f_{2}\left(f_{3}(X)\right)\right)=Y. Let t be the length of segment X Y; what is the sum of all possible values of t^{2}?