Two Versus Three

y+12+y4+18+y16+132+=1+x3+19+x27+181+x243+ \begin{array} { l l l l l } & \color{#D61F06}{y} & +\frac { 1 }{ 2 } & +\frac { \color{#D61F06}{y} }{ 4 } & +\frac { 1 }{ 8 } & +\frac { \color{#D61F06}{y} }{ 16 } & +\frac { 1 }{ 32 }& + \dots\\ = & 1 & +\frac { \color{#3D99F6} {x} }{ 3 } & +\frac { 1 }{ 9 } & +\frac { \color{#3D99F6} {x} }{ 27 } & +\frac { 1 }{ 81 } & +\frac { \color{#3D99F6} {x} }{ 243 } & +\dots \end{array}

Find the smallest positive integer x\color{#3D99F6} {x}, such that there exists an integer y\color{#D61F06}{y} which satisfies the above equation.

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