Stealth
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Another alternative to saving Kent is to take the stealth approach. When the elevator opens, immediately crouch using the R3 button and take it slow. You can remain hidden and kill Sinjin’s men to the left and right of the platform if you’re careful enough. With the raiders taken care of, take aim for Avery, as if you kill Sinjin first, she will kill Kent immediately.
Once she is taken care of, pop Sinjin with a few powerful shots to the head. Remember, don’t use explosives as they’re not your best friend in situations that require a careful hand in Fallout 4. After all, Kent isn’t made of titanium.