The department is authorized to cancel any lease issued as provided in this section for nonpayment of rentals or royalties or nonperformance by the lessee of any provision or requirement of the lease. However, before any such cancellation is made, the department shall mail to the lessee by registered mail, addressed to the post office address of such lessee shown by the records of the department, a notice of intention to cancel such lease specifying the default for which the lease is subject to cancellation. If lessee shall, within thirty days after the mailing of said notice to the lessee, commence and thereafter diligently and in good faith prosecute the remedying of the default specified in such notice, then no cancellation of the lease shall be entered by the department. Otherwise, the cancellation shall be made and all rights of the lessee under the lease shall automatically terminate, except that lessee shall retain the right to continue its possession and operation of any well or wells in regard to which lessee is not in default. Further, failure to pay rental and royalty required under leases within the time prescribed therein shall automatically and without notice work a forfeiture of such leases and of all rights thereunder. Upon the expiration, forfeiture, or surrender of any lease, no new lease covering the lands or any of them embraced by such expired, forfeited, or surrendered lease, shall be issued for a period of ten days following the date of such expiration, forfeiture, or surrender. If more than one application for a lease covering such lands or any of them shall be made during such ten-day period the department shall issue a lease to such lands or any of them to the person offering the greatest cash bonus for such lease at a public auction to be held at the time and place and in the manner as the department shall adopt by rule.