my car ran yesterday, won't start today
repair shop says they think the timing belt is broken, and car needs to be towed to hyundai dealer. If it is, did I wreck the engine trying to start it, with a broken timing belt?
A timing belt replacement is a DIY job for someone with basic skills or higher. Since your car is a 2004 and likely no longer under warranty, why couldn't the people in the repair shop handle it? Just thinking out loud...
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