If the state canvassing board, the county canvassing board, secretary of state, county clerk or any member of a precinct board [election board] fails or refuses to do or perform any of the acts required of them pertaining to recounts or rechecks, the applicant for recount or recheck may apply to any district court, the court of appeals or the supreme court of New Mexico for writ of mandamus to compel the performance of the required act and such court shall entertain such application.