= Solution
Regard the given <short exact sequence> as a degreewise short exact sequence of <chain complexes> concentrated in degree $n$. The inverse functor in the <Dold–Kan correspondence> is exact, so it produces a degreewise short exact sequence of <simplicial abelian groups>
$$
0\longrightarrow K(A_1,n)\longrightarrow K(A_2,n)\longrightarrow K(A_3,n)\longrightarrow0.
$$
The last map is degreewise surjective and hence a Kan fibration. Its strict fiber is $K(A_1,n)$, and a strict fiber of a fibration computes the homotopy fiber. This proves the asserted <homotopy fiber sequence> of pointed <Kan complexes>.
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