Determine all triples $(a,b,c)$ of positive integers for which $ab-c,bc-a$, and $ca-b$ are powers of $2$. Explanation:$A$ power of $2$ is an integer of the form $2^n$, where $n$ denotes some nonnegative integer. 
How many triples $(a, b, c)$ satisfy the conditions?

Let's think step by step and output the final answer within \boxed{}.
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