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I think I figured out Quirrell's plan, or at least what the big challenge for Harry is going to be.

Quirrell spoke of Voldemort learning Slytherins secrets from the basilisk, tempting Harry. Oh but the snake is dead, too bad all those secrets are lost.

Now Harry figures out that his mysterious dark side is Voldemort, has all those secrets, and apparently can be 'reformed'.

So he can have all those secrets (and at some point he is likely to be made to need them) but to get them he has to strengthen the horcrux-voldemort talk to it and listen to it.