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We prove that for $k\\geq 3$, there exists $\\varepsilon_{k}>0$ such that if a graph $G$ satisfies $\\sum_{0\\leq j\\leq k-1}c_{2j+1}(G-X)\\leq \\varepsilon_{k}|X|$ for all $X\\subseteq V(G)$, then $G$ has a $\\{P_{2},P_{2k+1}\\}$-factor, where $c_{i}(G-X)$ is the number of components $C$ of $G-X$ with $|V(C)|=i$. 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