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For $500-700 I would definitely be looking at used bikes. I don't think a new $500-700 mountain bike is going to be a significant improvement, it will still be a hardtail with cross country geometry and an underwhelming fork. The only potential improvement will be disc brakes, and entry level discs can often be extremely underwhelming. The used market should have some older bikes that aren't "cool" or the hot new thing anymore that still ride great.

This is a complex question. A mountain bikes first purpose is to get you excited about riding so that you ride more. Are you going to stay excited about a $700 used bike for very long? probably not, but if you absolutely hate riding this bike than by all means do it. But understand you'll probably want to start saving for your next bike immediately. The sweet spot in this market seems to be between $2500 and $5000 before tax. This will get you into a bike that won't get in your way any time soon... so long as you...

The used market is pretty flooded right now as is the clearance new bike market. Might be able to sweet talk yourself into a decent bike from someone who has something sitting around especially if you stretch your budget by a couple hundred.