§ 671. General duties of coroner, or coroner and coroner's physician, or medical examiner; additional duties may be directed. 1. The coroner, or if he is not a physician duly licensed to practice medicine in this state, the coroner and a coroner's physician, together, or in counties in which the office of coroner has been abolished, the medical examiner,

(a) shall make inquiry into unnatural deaths within his county as prescribed by law;

(b) shall make inquiry into all deaths whether natural or unnatural in his or her county occurring to an incarcerated individual of a correctional facility as defined by subdivision three of section forty of the correction law, whether or not the death occurred inside such facility. 2. The coroner shall perform such additional and related duties as may be prescribed by law and directed by the board of supervisors. 3. In those counties in which the office of coroner has been abolished, the medical examiner shall perform such additional and related duties involving exercise of professional skills and training as may be prescribed by the board of supervisors and the district attorney.