Nested Relations

Algebra Level 4

x=1y+1y+1y+1y+... \huge x=\frac{1}{y+\frac{1}{y+\frac{1}{y+\frac{1}{y+...}}}}

y=x+x+x+x+x+... \large y=\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+...}}}}}

Given that xx and yy are numbers that satisfy the relations given above, determine the exact value of

((1y)2(2014)+x2014)((1x)2014+(y)2(2014)) \left(\left(\frac{1}{y}\right)^{2\left(2014\right)}+x^{2014}\right)\left(\left(\frac{1}{x}\right)^{2014}+\left(y\right)^{2\left(2014\right)}\right)

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