Question: How to evaluate a parametric function?

I want to evaluate a parametric function to find intervals in which a condition is satisfied. For example, in the following instance, I know that 0<=c<=1 and 0<r<1. How can find intervals for r such that the following expression is less than 1?

-r+c+1 + sqrt(  r^2 - 2*r*c -r +2*c  ) <1,


0<=c<=1 and 0<r<1  and 

r in (0, min(1,2c) )

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