def add_time(start: str, duration: str, day: str=""):
    days: list[str] = ["monday", "tuesday", "wednesday", "thursday", "friday", "saturday", "sunday"]
    day = day.casefold()
    time, am_pm = start.split(" ")
    time_hour, time_minute = [int(part) for part in time.split(":")]
    add_hours, add_minutes =  [int(part) for part in duration.split(":")]
    new_time_minute = (time_minute + add_minutes)%60
    new_time_hour = (time_hour + add_hours + int((time_minute + add_minutes)/60))%12
    if new_time_hour == 0:
        new_time_hour = 12
    am_pm_revolutions = int((time_hour + add_hours + int((time_minute + add_minutes)/60))/12)
    days_past = int(am_pm_revolutions/2)
    new_am_pm = am_pm
    if (am_pm_revolutions%2):
        days_past += 1 if am_pm == "PM" else 0
        new_am_pm = "AM" if am_pm == "PM" else "PM"

    if days_past == 1:
        days_str = " (next day)"
    elif days_past > 1:
        days_str = f" ({days_past} days later)"
    else:
        days_str = ''

    if day:
        new_day = f", {days[int((days.index(day)+days_past)%len(days))].capitalize()}"
    else:
        new_day = ""
    new_time = f'{new_time_hour}:{str(new_time_minute).zfill(2)} {new_am_pm}{new_day}{days_str}'

    return new_time