Lifting a congruence $f(X) \equiv gh (\textrm{mod} \ p)$ to a higher power moduli I have the polynomial ring $R = \mathbb{Z}[X]/(X^N - 1)$ with $N = 11$ and $h \in R$ with
\begin{equation*}
h = 7 -11X - 7X^2 -12X^3 + 8X^4 - 11X^5 + -8X^6 + 11X^7 - 4X^8 + 2X^9 + 3X^{10}
\end{equation*}
Now I want to find the polynomial $u$ such that
\begin{equation*}
u\cdot h + v\cdot (X^N - 1) \equiv X - 1 \ (\textrm{mod} \ 32)
\end{equation*}
where we assume $\gcd(h, X^N - 1) = X - 1$ reducing coefficients $\textrm{mod} \  32$. If we also let $*$ denote the multiplication in $R$, I can easily find $u$ such that $u*h \equiv X - 1 \ (\textrm{mod} \ 2)$ using the extended Euclidean algorithm, and it's $u = 1+X+X^{4}+X^{5}+X^{9}$, but I'm having issues using this to construct the solution such that $u' * h \equiv X - 1 \ (\textrm{mod} \ 32)$. Can someone outline an algorithm or link me some reading material on how to do this?
One way to raise solutions to higher powers of the modulus is Hensel's Lemma.

First, you need to express your element as a root of a function: $g(u)= hu - (X-1)$
Find the solution modulo 2, which you already did.
Use the Lemma to raise this solution to higher powers of 2. Maybe it is useful to do this in several steps. Find the solutions mod 4,8,16 and then 32.