Simple 4-cycle permutation 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.

I'm strictly shooting from the hip (i.e. this is just instinct), but it might help if you consider the following:
(1) The 4th powers of 4-cycles are unity. I.e. $(1234)^4 = e$ where $e$ is the identity.
(2) This means that inverses exist. I.e. $(1234)^3 (1234)^1 = e$.
(3) And this gives a suggestion for a way of walking around the 4-cycles.  
If you need another hint, come back and ask again tomorrow?
