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BIG TIP: If you can't immediately surpass the other chocobos with a long dash, your chocobo isn't fast enough to beat Teioh. You always need to get ahead of everybody at the start of the race 'cause that is when they move the slowest . . . or the stupidest. Something like that.
By the way, breeding can help you get a better chocobo. Sure gold is fast, but every new offspring gets a boost in max speed and stamina from their parents. So mate your gold with any other chocobo and you may get a faster one. The key is to ask Chole right after the new chocobo is born and she'll tell you if it is faster than the others.