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That all elements in a symmetric group with a specified arbitrary generating set can be reached in a polynomial amount of steps is a known open problem, and has been investigated for some time now. This long-standing conjecture has been proven for most choices of generating sets. Heuristically one expects this to be true for the reasons mentioned by Colin Reid above. That is, if certain conjectures are true, then every Cayley graph has a Hamiltonian cycle and so is expected to have exponentially many such cycles. So one expects to be able to reach any element of the group pretty fast.

For the symmetric group the problem of determining tight bounds on the diameter of its Cayley graph has been studied (and partially resolved) by L. Babai and coauthors. HereIn the paper "On the diameter of cayley graphs of the symmetric group" it is proved that one can reach any elements using words of length at most $e^{\sqrt{n\log n}(1+o(1))}$ for any generating set, while the optimal bound should be $O(n^{c})$. Here and here more partial results are proved, showing evidence that polynomial bounds are in fact the true asymptotic. I will remark again that one expects this behavior for all nonabelian finite simple groups too.

These papers are a good survey of what is currently known about this problem, I don't know if restricting your target to specific conjugacy classes of elements (such as n-cycles) makes the problem easier so I will think about it a bit more.
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That all elements in a symmetric group with a specified arbitrary generating set can be reached in a polynomial amount of steps is a known open problem, and has been investigated for some time now. This long-standing conjecture has been proven for most choices of generating sets. Heuristically one expects this to be true for the reasons mentioned by Colin Reid above. That is, if certain conjectures are true, then every Cayley graph has a Hamiltonian cycle and so is expected to have exponentially many such cycles. So one expects to be able to reach any element of the group pretty fast.

For the symmetric group the problem of determining tight bounds on the diameter of its Cayley graph has been studied (and partially resolved) by L. Babai and coauthors. Here it is proved that one can reach any elements using words of length at most $e^{\sqrt{n\log n}(1+o(1))}$ for any generating set. , while the optimal bound should be $O(n^{c})$. Here and here more partial results are proved, showing evidence that polynomial bounds are in fact the true asymptotic. I will remark again that one expects this behavior for all nonabelian finite simple groups too.

These papers are a good survey of what is currently known about this problem, I don't know if restricting your target to specific conjugacy classes of elements (such as n-cycles) makes the problem easier so I will think about it a bit more.
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This is a known open problem, and has been investigated for some time now. Heuristically one expects this to be true for the reasons mentioned by Colin Reid above. That is if certain conjectures are true, then every Cayley graph has a Hamiltonian cycle and so is expected to have exponentially many such cycles. So one expects to be able to reach any element of the group pretty fast.

For the symmetric group the problem of determining tight bounds on the diameter of its Cayley graph has been studied (and partially resolved) by L. Babai and coauthors. Here it is proved that one can reach any elements using words of length at most $e^{\sqrt{n\log n}(1+o(1))}$ for any generating set. Here and here more partial results are proved, showing evidence that polynomial bounds are in fact the true asymptotic. I will remark again that one expects this behavior for all nonabelian finite simple groups too.