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I have been proposed in class to obtain the inverse of a given element in $\mathbb{Z}_2$ field with the Almost Inverse Algorithm (AIA). I do not understand very well how to obtain it since the result that the algorithm returns does not know what has to be applied to have the inverse form.

Let $ \mathbb{F} = \mathbb{Z}_{2}[X]/f(x) $ where $ f (x) = x ^ 4 + x ^ 3 + 1 $ calculate the inverse of $ x ^ 3 + x $
Almost Inverse Algorithm (AIA)

I would like to know if you could help me, showing me the iterations of the algorithm.

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