Let $\Gamma$ be a $r$-transitive group of permutations of a set $E$ with $n$ elements. Let $\sigma\in\Gamma$ be a permutation of $E$ such that $\sigma\ne id_E$ such that $$|\{x\in E\ |\ \sigma(x)=x\}|=n-s.$$ If $s>r$, show that there exists a permutation $\tau\in\Gamma$ such that $\sigma^{-1}\tau\sigma\tau^{-1}\ne id_E$ and $$|\{x\in E\ |\ \tau(x)=x\}|\ge n-2(s-r+1).$$
This exercise comes with the following hint: "use the decomposition of $\sigma$ into its component cycles".
Let $\langle\sigma\rangle$ be the subgroup generated by $\sigma$ and lets $\mathcal{S}$ be the set of $\langle\sigma\rangle$-orbits not consisting of a single element. For $Y\in \mathcal{S}$, let $\zeta_Y(x)=\sigma(x)$ if $x\in Y$ and $\zeta_Y(x)=x$ if $x\not\in Y$. The family of cycles $(\zeta_Y)_{Y\in\mathcal{S}}$ is the unique family such that $$\sigma=\prod_{Y\in\mathcal{S}}\zeta_Y$$ and $\text{supp}(Y)\cap\text{supp}(Y')=\emptyset$ for all $Y\ne Y'\in\mathcal{S}$. I am not sure how to use this cycle decomposition to construct $\tau$.
Edit: some deductions
We can easily deduce that $|\{x\in E\ |\ \sigma(x)\ne x\}|=s$ and that $$\{x\in E\ |\ \sigma(x)\ne x\}= \bigcup_{Y\in\mathcal{S}} Y.$$ Since $s>r$ there exists a $k\in\mathbb{N}$ such that $k\ne 0$ and $s=r+k$. At this point I most likely have to utilize the assumption of $r$-transitivity.