Random generation with cycle type restrictions

We study random generation in the symmetric group when cycle type restrictions are imposed. Given $\pi, \pi' \in S_n$, we prove that $\pi$ and a conjugate of $\pi'$ likely to generate at least $A_n$ provided only have not too many fixed points $2$-cycles. As an application, investigate following question: For which positive integers $m$ should expect two elements order $A_n$? Among other things, give answer for any having divisor $d$ range $3 \leq d o(n^{1/2})$.