GeeehaaaaaaAAAaaa!
My first about a few minutes long look told me: this is impossible. :)))
A bit later I got an idea, but it was not yet exactly right, so I got stuck. After that I had an ''eureka'' moment when I became pretty sure of what needs to be done, but was still not able to make it happen. ***************************************************************************************************************************************************************************** My first touch then finally happened (at 728 and it took 45 minutes). I'll try to optimize, but reading Zach's comment I already have fears. :/
Magnificent structure!
EDIT1 (some minutes later): My second touch was 600+, my third is 588. But now not many ideas on how to cut, left.
Incredible design! Really enjoyed the process of reaching the exit here. Optimizing is really tricky as well. Lots of easy to miss 2-cuts even when you’ve mostly found the right route