I call a 4-cycle permutation simple if I can write it as $(i,i+1,i+2,i+3)$ so $(2,3,4,5)$ is a simple 4-cycle but $(1,3,4,5)$ is not. I want to write $(1,2,3,5)$ as a product of simple 4-cycles. So this is what I did: $ (1,2,3,5)=(1,2)(1,3)(1,5) $ but $\begin{align} (1,3)&=(2,3)(1,2)(2,3)\\ (1,5)&=(4,5)(3,4)(2,3)(1,2)(2,3)(3,4)(4,5) \end{align}$ So now $(1,2,3,5)=(1,2)(2,3)(1,2)(2,3)(4,5)(3,4)(2,3)(1,2)(2,3)(3,4)(4,5)$ Can you please give me a hint on how I can express $(1,2)(2,3)(1,2)(2,3)(4,5)(3,4)(2,3)(1,2)(2,3)(3,4)(4,5)$ as a product of simple 4-cycles.
Note: We do permutation multiplication from left to right.