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For graph <inline-formula> <tex-math notation="LaTeX">$G=(V,E)$ </tex-math></inline-formula>, the open neighborhood of a vertex <inline-formula> <tex-math notation="LaTeX">$v$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">$N(v)=\{u\in V|uv\in E\}$ </tex-math></inline-formula>. A function <inline-formula> <tex-math notation="LaTeX">$f: V(G)\rightarrow \{0, 1, 2\}$ </tex-math></inline-formula> is called an Italian dominating function of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> if <inline-formula> <tex-math notation="LaTeX">$\sum _{u\in N(v)}f(u)\geq 2$ </tex-math></inline-formula>, for each vertex <inline-formula> <tex-math notation="LaTeX">$f(v)=0$ </tex-math></inline-formula>. The weight of <inline-formula> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> is <inline-formula> <tex-math notation="LaTeX">$w(f)=\sum _{v\in V(G)}f(v)$ </tex-math></inline-formula>. The minimum weight of an Italian dominating function of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula> is called the Italian domination number of <inline-formula> <tex-math notation="LaTeX">$G$ </tex-math></inline-formula>, denoted by <inline-formula> <tex-math notation="LaTeX">$\gamma _{I}(G)$ </tex-math></inline-formula>. In this paper, we present a bagging approach and a partitioning approach to investigate the Italian domination number of Cartesian product of circles and paths, <inline-formula> <tex-math notation="LaTeX">$C_{n}\Box \,P_{m}$ </tex-math></inline-formula>. We determine the exact values of the Italian domination numbers of <inline-formula> <tex-math notation="LaTeX">$C_{n}\Box \,P_{3}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$C_{3}\Box \,P_{m}$ </tex-math></inline-formula>. We also present some bounds on the Italian domination number of <inline-formula> <tex-math notation="LaTeX">$C_{n}\Box \,P_{m}$ </tex-math></inline-formula> for <inline-formula> <tex-math notation="LaTeX">$n, m\geq 4$ </tex-math></inline-formula>.