When a member of the governing board of a school district which is being reorganized and which will cease to exist takes office as a member of the initial or interim governing board of a newly formed school district, he or she shall cease to be a member of the governing board of the district being reorganized unless he or she elects to remain a member of that board. If the member does not elect to remain on the board of the district being reorganized, the county board of education shall then appoint another person who is eligible to serve on the governing board of the district being reorganized to the vacant position for the duration of the existence of the district being reorganized, but in no case for longer than 12 months.