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Space is so vast that the Earth and Solar System will survive.
Although the galaxies will plow into each other, stars inside each galaxy are so far apart that they will not collide with other stars during the encounter. However, the stars will be thrown into different orbits around the new galactic center. Simulations show that our solar system will probably be tossed much farther from the galactic core than it is today. Source
Space is so vast that the Earth and Solar System will survive.