# Similar Curves

Algebra Level 5

Let two curves $$f(x)$$ and $$g(x)$$ be similar in the range $$[a,b]$$ if for all $$x$$ in the range $$[a,b]$$ we have $$|f(x)-g(x)|\le 1$$.

Let $$f(x)$$ be the half-hyperbola $$y=\sqrt{x^2+1}$$ and let $$g(x)$$ be a parabola with equation $$y=mx^2+n$$.

If $$f(x)$$ and $$g(x)$$ are similar in the range $$[p,q]$$ where $$p\le q$$, then the largest possible value of $$q-p$$ can be expressed as $a\sqrt{b+c\sqrt{d}}$ for positive integers $$a,b,c,d$$ with $$d$$ square-free. Find $$a+b+c+d$$.

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