Mayer–Vietoris sequence from Brown's conditions

Exercise 3.4.7 (Mayer–Vietoris sequence). Let \(F\colon \Ho (\An _{*,\geq 1})\catop \to \Ab \) satisfy the wedge axiom and the Mayer–Vietoris property, and consider a pushout square of connected pointed animae

Show that the cofiber of \((i,j)\colon A\vee B\to X\) is isomorphic to \(\Sigma C\), and use the coPuppe sequence to construct a long exact Mayer–Vietoris sequence \[ \cdots \to F(\Sigma X)\to F(\Sigma A)\times F(\Sigma B)\to F(\Sigma C)\to F(X)\to F(A)\times F(B)\to F(C). \] Determine the maps and signs explicitly.