I first marked the dead ends, filling them with yellow.
Then the ways that only lead back to the same engine (orange).
That made it easier to mark the final useful ways, grey means alternative ways or shortcuts.
So, from the upper left to number 1 via the blue, grey and green
then back on the green and grey using the last part of the blue to number 2.
From 2 to 3 its the red one.
The ways between 3, 4 and 5 were the last I found ... unmarked. so it goes from 3 to 4 and on to 5.
And on with two blue marked ways to number 6 and then out on the lower right.
Thanks for a challenging maze.