If $$$1-\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{5+\frac{1}{6+\frac{1}{7}}}}}}=\frac{1}{x_1+\frac{1}{x_2+\frac{1}{x_3+\frac{1}{x_4+\frac{1}{x_5+\frac{1}{x_6+\frac{1}{x_7}}}}}}}$$$ where $$x_{1},x_{2},....,x_{7}$$ are positive integers, find $$x_{1}+x_{2}+x_{3}+x_{4}+x_{5}+x_{6}+x_{7}$$.

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