
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Consider the following three lines in the Cartesian plane:   $$ \begin{cases}
\ell_1: & 2x - y = 7
\ell_2: & 5x + y = 42
\ell_3: & x + y = 14
\end{cases} $$  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  $f_1(f_2(f_3(X)))= Y$ . Let  $t$  be the length of segment  $XY$ ; what is the sum of all possible values of  $t^2$ ?

Present the answer in LaTex format: \boxed{Your answer}
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