S46 TRACON Controller reported miscommunication with another ATC facility resulted in them accepting an aircraft flying below their MVA and flight towards terrain.

Narrative

Working all the airspace at a scope; mid shift operations. Very light traffic; not complex. NUW initiated a H/O (Hand Off) on Aircraft X at 4000' in a 4400' MVA. I evaluated the situation and was immediately aware of two things: NUW's MVAs are different from S46 (so Aircraft X was obviously above their MVA at the time) and we have an LOA (Letter of Agreement) with Aircraft X that allows their a/c (Aircraft) to operate at 'ZK routes and altitudes' along predefined routes and often below the applicable MVAs. In light of this knowledge; I initiated a landline coordination with NUW to confirm that Aircraft X was assigned ZK routes/altitudes. The NUW controller answered that Aircraft X was assigned ZK routes/altitudes; so I accepted the H/O on Aircraft X. Aircraft X checked in on frequency and reported level at 4000'; so I asked the a/c if they were assigned ZK altitudes. Aircraft X indicated that they were not assigned Aircraft X altitudes and only assigned 4000' but that they could fly whatever was needed. Realizing that the information I had received from the NUW controller was inaccurate I immediately transitioned to recovery and asked Aircraft X if they were established on ZK-547. Aircraft X responded that they were established on ZK-547 so I instructed the a/c to maintain ZK altitudes to an appropriate point on their flight plan and then assigned an altitude after that point which would comply with my MVAs afterwards. After establishing this recovery; I initiated another landline coordination with NUW to determine what exactly was assigned to Aircraft X. It was at this point that the NUW controller indicated they had NOT assigned ZK altitudes to Aircraft X but had thought that 4000' was the ZK altitude. I informed the NUW controller that I would need to file a report for an MVA violation because I had accepted the H/O below my MVA due to the incorrect information they had previously provided. The NUW controller did inform me that Aircraft X was in a 3700' MVA for NUW; which at least confirmed that Aircraft X was not in any imminent danger due to terrain. Aircraft X subsequently continued to their destination without further incident and landed safely. As the CIC (Controller In Charge) ; I filled out an MOR pertaining to the incident and notified my Manager via a phone call; because I was unsure if filing the MOR alone was sufficient. The OM indicated as long as the incident was not a major incident then the MOR would suffice. I did not believe the incident qualified as severe; but I gave the OM a brief summary to be certain and they concurred that it was not severe.Recommendation: While I should be able to operate based on information received from controllers at an adjacent facility; I could have waited for Aircraft X to exit the 4400' MVA before accepting the H/O from NUW. Waiting to accept the H/O may have required coordination to allow NUW to enter my airspace prior to the completion of the H/O; but it would have averted the incident entirely. I will endeavor to be more vigilant in this regard in the future.