Conjugacy and the Class Equation · Hard problem

In $S_7$, let $\tau = (1\,2\,3\,4)(5\,6)$ (so $7$ is fixed) and $\rho = (3\,5\,6\,7)(1\,2)$ (so $4$ is fixed) — both cycle type $(4,2,1)$. Matching corresponding cycles AND the fixed point in the order written gives $\sigma$ with $\sigma\tau\sigma^{-1}=\rho$. What is $\sigma(1)+\sigma(7)$?