ERJ-175 First Officer reported receiving a low altitude alert from ATC when the aircraft descended below the Final Approach Fix (FAF) as a result of the First Officer having set an incorrect FAF altitude.

Narrative

Abeam the airport level at 8000 ft.; the Captain prompted me to start slowing; I activated the approach speeds in preparation for configuration and descent. I added a last second briefing item about configurations and we got a descent clearance in the middle of that briefing. I misunderstood the descent clearance as 4000 ft.; instead of 5000; I consequently set 4000 in the altitude selector. We got a vector during the descent as expected; we got ICE Speeds as we broke out of the overcast; about 5000 ft. and the Captain was entering those and cleaning up the FMS for the approach. I was heads-up and saw the city lights so I had no cause for concern. At 4500 ft. ATC gave us an ALT Alert; low altitude; climb 5000. I used the TCS (Touch Control Steering) button to override the automation and asked the captain to set 5000 ft. and we climbed back up. There was no GPWS alert and the ground was in visual reference. The rest of the approach proceeded without incident. As the aircraft was slowing I remembered the that I had not briefed when I would call for 'gear down/flap 3; and when I would be fully configured for landing; due to the non-standard RA readings to [Runway] XXR - where the TDZE is elevated from the terrain west of the threshold. I asked the Captain to verify my math and confirm if by 4300 ft. (1500AFE) would be gear-down/f3 altitude; which is notably before to FAF ZZZZZ; and landing config just after ZZZZZ. A similar briefing as I would give in ZZZ1; but I had forgotten to do this in the original briefing. As the Captain was verifying this for me we got a descent clearance; 'descend 5000'; that I heard at the time as 'descend 4000'. We began descending and I was expecting a vector any moment. We were leaving 8000 ft. and we were NW of ZZZ; and the vector would take us away from terrain; so in the task saturation of entering ice speeds at the same time as getting a vector; I failed to recognize that the altitude I had set was 4000 and the final approach altitude was actually 4100. That would usually make me question what I thought I heard. We broke out of the clouds it was VMC and I had even less concern because our vector was taking us inside of ZZZZZ1. I am pretty sure that when I was asking the Captain earlier to double check the configuration altitudes; we both got 4000 stuck in our heads; because that was about where the configs would be taking place. Seeing the city and having the tasks completed helped me relax; but I still did not catch that we had set the selector 100 ft. lower than the final segment altitude; which would usually be a red-flag that I had misunderstood something and need double check. The Tower alerted us that we were low and we promptly corrected as the Captain set 5000 ft. and I TCS'd to initiate the climb. Adding somewhat complex briefings during a final segment of STAR to approach caused misunderstanding of the clearance and at the same time degraded my situational awareness of clues that would alert to a misunderstanding. As so often does; it went from a gradual deceleration for configuration; to everything happening at once; distracting us with ice speeds; vectors and FMS management; especially the PM (Pilot Monitoring) can become severely task saturated in an instant. Even though I was intending to cover the late addition to the briefing while things were still relatively quiet; I distracted the captain and myself by doing so at that time. If I recognize that I have omitted something from the briefing; I should transfer controls; and abbreviate the points I need to address. At the [time] I brought it up; I was trying to abbreviate it and I thought we had sufficient time; which quickly changed. Verifying altitudes that are assigned with charted altitudes as a cross-reference needs to become a habit of mine; so that I don't ever set an altitude 100 ft. lower than FAF altitude again.