  Here we propose a different perspective of the deep factorisation in
Kyprianou (2015) based on determining potentials. Indeed, we factorise the
inverse of the MAP-exponent associated to a stable process via the Lamperti-Kiu
transform. Here our factorisation is completely independent from the derivation
in Kyprianou (2015) , moreover there is no clear way to invert the factors in
Kyprianou (2015) to derive our results. Our method gives direct access to the
potential densities of the ascending and descending ladder MAP of the
Lamperti-stable MAP in closed form.
  In the spirit of the interplay between the classical Wiener-Hopf
factorisation and fluctuation theory of the underlying Levy process, our
analysis will produce a collection of of new results for stable processes. We
give an identity for the point of closest reach to the origin for a stable
process with index $\alpha\in (0,1)$ as well as and identity for the point of
furthest reach before absorption at the origin for a stable process with index
$\alpha\in (1,2)$. Moreover, we show how the deep factorisation allows us to
compute explicitly the stationary distribution of stable processes
multiplicatively reflected in such a way that it remains in the strip [-1,1].
