A problem by Breno Linhares

Level pending

Evaluate: \[\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{...+\frac{1}{2005}}}}}+\frac{1}{1+\frac{1}{1+\frac{1}{3+\frac{1}{4+\frac{1}{...+\frac{1}{2005}}}}}}\]

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