  In this paper, it is explained that a topological invariant for 3-manifold
$M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopy
theoretic approach for Chern--Simons perturbation theory to a local system on
$M$ of rational functions associated to the free abelian covering of $M$. Our
invariant takes values in Garoufalidis--Rozansky's space of Jacobi diagrams
whose edges are colored by rational functions. It is expected that our
invariant gives a lot of nontrivial finite type invariants of 3-manifolds.
