Permutation Groups · Hard problem

In $S_5$, let $\sigma=(1\,2)(3\,4)$ and $\rho=(1\,3\,5)$. Suppose $\sigma x\rho=\upsilon$ where $\upsilon$ sends $1\to2,\,2\to5,\,3\to4,\,4\to1,\,5\to3$ (and $xy$ means "apply $y$, then $x$"). Solve for the permutation $x=\sigma^{-1}\upsilon\rho^{-1}$ and state where $x$ sends $1$.