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How did simplify_full() manipulate this expression?

This is sort of a follow up to a previous question of mine:

I wanted to prove a certain combinatorial equality. Now, to prove this identity in Sage, I can define the two function $\text{LHS}(n,k)$ and $\text{RHS}(n,k)$ representing the left- and righthand side and make sure that $\text{LHS}(n,k)-\text{RHS}(n,k)=0$ with the following code (actually the code provides a proof of a slightly more general identity, which holds for other values of $n, k$ as well.):

var('n k')

def LHS(n,k):
    return n/2*(

def RHS(n,k):
    return (n/k)*(2**(n/2-k))*binomial(n/2-2,k-2)*binomial(k-2,k/2-1)


This outputs


which is what I want but I would like to prove this identity with pen-and-paper and this seems to be a bit tricky. I wonder if there is a way to see how exactly simplify_full() were able simplify the expressions?