Understanding action of $(\mathbb Z_p/(p^n))^ \times$ in Lubin Tate formal group laws context on $\mathbb{Q}_p[\zeta_{p^r}]$

Question

I want to understand that the isomorphism $$(\mathbb Z_p/(p^n))^\times\ \stackrel{\sim}{\longrightarrow}\ \text{Gal}(\mathbb Q_p[\zeta_{p^r}]/\mathbb Q_p),$$ is the standard one.

So how does element $a\in(\mathbb Z_p/(p^n))^ \times $ act on $x \in \mathbb Q_p[\zeta_{p^r}]$? By Lubin-Tate formal group laws description it acts as $[a]_f(x) $, so for example on $\zeta_{p^r}$ as $[a]_f(\zeta_{p^r}) $. We come back to this but first we figure how $a$ acts via standard isomorphism.

I suppose it does as $a\cdot\zeta_{p^r}=\zeta_{p^r}^{a} $ (note that $(a,n)=1$).

So back to $[a]_f $ it is $[a]_f=aT+(\text{deg} \geq 2) $ and $[a]_f \circ f=f \circ [a]_f $. We had started with $f(T)=(1+T)^p-1$.

I do not see how these two actions will be same?