Using the new results on coherent elastic neutrino-nucleus scattering data in cesium-iodide provided by the COHERENT experiment, we determine a new measurement of the average neutron rms radius of $^{133}text{Cs}$ and $^{127}text{I}$. In combination with the atomic parity violation (APV) experimental result, we derive the most precise measurement of the neutron rms radii of $^{133}text{Cs}$ and $^{127}text{I}$, disentangling for the first time the contributions of the two nuclei. By exploiting these measurements we determine the corresponding neutron skin values for $^{133}text{Cs}$ and $^{127}text{I}$. These results suggest a preference for models which predict large neutron skin values, as corroborated by the only other electroweak measurements of the neutron skin of $^{208}text{Pb}$ performed by PREX experiments. Moreover, for the first time, we obtain a data-driven APV+COHERENT measurement of the low-energy weak mixing angle with a percent uncertainty, independent of the value of the average neutron rms radius of $^{133}text{Cs}$ and $^{127}text{I}$, that is allowed to vary freely in the fit. The value of the low-energy weak mixing angle that we found is slightly larger than the standard model prediction.
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