If the Mayor refuses or fails to sign any ordinance, resolution, contract, warrant, demand, or other document or instrument requiring his or her signature for five days consecutively, then a majority of the members of the Council may, at any regular or special meeting, authorize the Vice Mayor or, in his or her absence, an Acting Mayor to sign the ordinance, resolution, contract, warrant, demand, or other document or instrument which, when so signed, shall have the same force and effect as if signed by the Mayor.