Problem involving drawing a diagram with belts and elevators and specific directions on a given floor/belt

Today, while reading A Practical Guide to Problem Solving in Mathematics by Carol Meyer and Tom Sallee, I came across a problem I was unsure of how to solve except by brute forcing possibilities and eliminating them if they did not fit the given criteria. I was wondering if there was possibly a diagram or something else that could explicitly show/help show the route one needed to take. It seems to be a much more efficient strategy, but I was not able to draw a diagram that was of very much use. The problem in whole is listed below:

"Most Hrunkla lived in giant, $12$ story apartment houses, and their homes were large square rooms bounded on four sides by corridors. Each room had a single door which opened halfway along the corridor. On even numbered floors: the doors opened onto the east corridors; on odd numbered floors, the doors opened onto the north corridor. At each intersection of corridors, there was something like an elevator which could be ridden up or down. Half of the corridors had moving belts on the floor, and no self respecting Hrunkla would walk if they could ride the belts. The belts were so arranged that those on floors $1$, $5$, and $9$ ran to the east; those on floors $2$, $6$, and $10$ ran to the south; those on floors $3$,$7$,and $11$ ran to the west; and those on the floors $4$,$8$, and $12$ ran to the north. Describe how a Hrunkla who lived on floor $10$ could use these moving belts and elevators to visit a friend who lived in the room directly below his."

After quite a while, I found a solution that works:

a. Ride belt to SE corner.

b. Ride to eighth floor (or fourth or twelfth).

c. Ride North to NE corner.

d. Ride to seventh floor (or third or eleventh).

e. Ride W to NW corner.

f. Ride to ninth floor

g. Ride E to friend's floor.

I assume there are many different routes possible, but I was not sure if I found the shortest route possible (even though the question does not ask for that). I am mainly looking to see if there is an "easy" way to solve the problem via a diagram or something of the sorts, so when I come across future problems such as this, I have a better plan of attacking it.

Any help would be greatly appreciated.

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