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Essentially the temperature does not change because the energy is used to actually change the phases and not to heat the substance. So for example if ice is going to liquid the energy is used to melt the ice not warm the mixture so the temp remains constant.
The heat is simply being use to get the substance through its phase change. After the phase change is completed then any addition of heat will result in higher temperature, until of course another phase change is begun and completed.
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