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You don't navigate back to your own computer. Once your EXA is in the correct host, figure out a way to communicate the correct numbers back to another EXA in your own host that's holding the correct file. Communicate over the M registry.
you just have to figure out what info each of them carries or communicates in which direction via the M register
when you solved it, i have a good solution for it so you can compare, just comment if you want it
Create a "sentinel" exa in front of the exit, and COPY 0 M in local mode with it. Now, if you want to find if you're in front of the exit, just use TEST MRD in local mode : if it's true, you're in the exit room.
It's kinda elegant (in my opinion), but overkill for this puzzle.
You could also count the number of time you travelled before finding the tape
all you really have to do is have the homehost guy copy the target location into m 2 times.
then the other guy goes to the first area, checks if host is M, if it is it reads from m again to free the home guy. if not go to next area and read from m again to compare to host, if its not here either, go to last area.
M is your friend.