The canyons are where boss battles are settled.
There are two parts to a canyon duel, first the boss will be in front. Your job is to stay as close to him as possible without driving off the edge and collect as many points as you can. The closer you are, the more points. After that, you are in front and the boss will be following you. Do the same and take victory. You can win their car if you get a pink slip from the card selection at the end.
You might also notice a similarity to Initial D touge battle rules. In fact the only difference is that occasionally Initial D would allow a shotgun start sprint race to the finish while most of the mountain pass battles are either 4 car or chase follow races.
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