Let G = S_n (the permutation group on n elements). Let A be a subset of G such that A generates G.

Is there an n-cycle g in G that can be expressed as

g = a_1 a_2 ... a_k

where a_i in A union A^{-1} and k is at most c_1 n^{c_2}, where c_1 and c_2 are constants?

What about 2-cycles, or elements of any other particular form?