= Solution
One form of the <Hensel lemma> is the following. Let $R$ be a complete <discrete valuation ring> with maximal ideal $\mathfrak m$, and let $f\in R[X]$. If $a_0\in R$ satisfies
$$
f(a_0)\equiv0\pmod{\mathfrak m},
\qquad f'(a_0)\not\equiv0\pmod{\mathfrak m},
$$
then there is a unique $a\in R$ with $f(a)=0$ and $a\equiv a_0\pmod{\mathfrak m}$.
Define $a_{n+1}=a_n-f(a_n)/f'(a_n)$. Since $f'(a_n)$ remains a unit, Taylor expansion gives
$$
f(a_{n+1})\equiv0\pmod{f(a_n)^2},
$$
so the valuations of the errors at least double. The corrections tend to zero, making $(a_n)$ a <Cauchy sequence>; completeness gives a limit $a$, and continuity gives $f(a)=0$. If $a,b$ are two such roots, then
$$
0=f(a)-f(b)=(a-b)(f'(a_0)+u)
$$
with $u\in\mathfrak m$, so the second factor is a unit and $a=b$.
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