AI algorithm for "RaceTrack" game

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does anyone know (or can suggest) a good algorithm for an AI for the RaceTrack pencil-paper game?

since you have 9 possible choices in each step and you need to look at least 6-10 steps ahead to decide on a good strategy, bruteforce is getting very expensive even if you can rule out some choices because of intersection with the boundary.

Currently I'm trying to assign each choice some quality value in order to decide which choices to rule out - but I don't know good rules yet on how to assign such a quality value.

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