Consider the following three lines in the Cartesian plane:

\[
\begin{cases}
\ell_1: & 2x - y = 7\\
\ell_2: & 5x + y = 42\\
\ell_3: & x + y = 14
\end{cases}
\]

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 \( f_1(f_2(f_3(X)))= Y \). Let \( t \) be the length of segment \( XY \); what is the sum of all possible values of \( t^2 \)?

Let's think step by step and output the final answer within \boxed{}.
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