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We investigate a coupled system of the Ricci flow on a closed manifold $M$ with the harmonic map flow of a map $\phi$ from $M$ to some closed target manifold $N$, $$\frac{\partial }{\partial t}g=-2\mathrm{Rc}+2\alpha \nabla \phi \otimes \nabla \phi ,\frac{\partial }{\partial t}\phi ={\tau }_g\phi ,$$ where $\alpha$ is a (possibly time-dependent) positive coupling constant. Surprisingly, the coupled system may be less singular than the Ricci flow or the harmonic map flow alone. In particular, we can always rule out energy concentration of $\phi$ a-priori by choosing $\alpha$ large enough. Moreover, it suffices to bound the curvature of ...