$x, y, z,$ and $w$ are distinct real numbers and $t$ is a positive real number such that $x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{w}=w+\frac{1}{x}=t$, find $t$.
As $y=t-\frac{1}{z}$, by substituting in $t-\frac{1}{z}$ for $y$, $x+\frac{1}{t-\frac{1}{z}}=t$. Thus, $x+\frac{1}{t-\frac{1}{t-\frac{1}{t-\frac{1}{x}}}}=t$, which simplifies to $\frac{t^3x-t^3-xt-3t+1}{t^2x-t-x}=t$. (I'm not sure if my expansion is wrong.) However, I cannot solve this equation. Can anyone solve it?