How to generate this map in sage

edit

Okay this is my code so far.

F.<x,y,z> = FreeGroup()
A = F.algebra(QQ)
Y = A(y)
Yinv = A(y^-1)
Z = A(z)
Z2 = A(Z)
X = A(x)
X2 = A(X)
X2inv = A(X^-1)
Xinv = A(x^-1)
Y2 = A(Y)

def functionMap(X,Y,Z,Xinv):
    X = A(X2 * Y2 * X2inv)
    Y = A(X2 * Z2 * X2inv)
    Z = A((Z2 * Y2 + 1)*X2inv)
    print(X)
    print(Y)
    print(Z)
    X2 = A(X)
    X2inv = A(X2^-1)
    Y2 = A(Y)
    Z2 = A(Z)


for i in range(6): 
    functionMap(X,Y,Z,Xinv)

As you can see, I'm trying to repeatedly take the map x,y,z to xyx^-1, xzx^-1, (zy+1)x^-1 repeatedly. I'm getting a lot of errors though. Does anyone know how to fix my code? I think It might have something to do with the multiple variables calling the symbol 'x' but idk how else to not cause issues when I'm doing the three different parts of my map (like if I change X then the middle part has the wrong iteration of x)