Generalized cohomology theory with coefficients and long exact sequence

Question

Let $E$ be a spectrum, $A$ be an abelian group, and $M$ be the Moore-spectrum corresponding to $A$, i.e. $M_n = M(A, n)$, where $M(A, n)$ is the Moore space of $A$ in degree $n$. Then we get a new spectrum $E_A := M \wedge E$, and a corresponding cohomology theory $E(-; A)$, which is the cohomology theory $E$ with coefficients in $A$. I want to prove the following: Suppose $q \in \mathbb{N}$ with $q > 2$, and $X$ is a topological space. Then there is a long exact exact sequence \begin{align*} ... \to E^n(X) \stackrel{\cdot q}{\to} E^n(X) \to E^n(X; \mathbb{F}_q) \to E^{n+1}(X) \to ... \end{align*} Does anybody know a source for this statement or know how to prove it?