The hydrogen fuel in the core eventually starts to run out, and an insufficient radiation pressure causes the star to collapse again. This heats the matter in the star again, and eventually a shell of matter around the core becomes hot enough for hydrogen nuclei to fuse into helium nuclei.
This exerts an outwards radiation pressure to hold up the outer layers, but the core continues to collapse. As the core collapses, the core and the shell heat further, and so the rate of fusion in the shell increases. This increases the outwards radiation pressure, pushing the outer layers of the star, and increases the power output of the star, making the star appear brighter. As the outer layers are pushed outwards they cool, and so the star turns redder in colour.
A red giant forms, in the shell hydrogen burning phase.
Eventually the core becomes hot enough for helium nuclei to fuse into carbon and oxygen nuclei. This exerts an even higher radiation pressure, heating the shell more and increasing the fusion rate. The radiation pressure increases again, the outer layers of the star are pushed out further. The star becomes larger, brighter and redder again. The star remains as a red giant in the core helium burning phase.
Once all the helium fuel runs out in the core the star collapses again. The shell heats and helium starts to fuse in the shell in the shell helium burning phase. During this phase a shell above the helium shell becomes hot enough for hydrogen to fuse into helium, increasing the radiation pressure even further.
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