from typing import Union


def short_bt_sig_uuid_to_long(short: Union[str, int]) -> str:
    """
    Returns the long-format UUID of a Bluetooth SIG-specified short UUID
    :param short: Short-form BT SIG-specified UUID (as a hex string or a raw integer)
    :return: The fully-qualified long-form BT SIG UUID
    """
    hex_form = short
    if type(short) == str:
        pass
    elif type(short) == int:
        hex_form = hex(short)[2:]
    else:
        raise TypeError

    return f'0000{hex_form}-0000-1000-8000-00805f9b34fb'