At any time after commencement of the action a defending party, as a third-party plaintiff, may cause a summons and complaint to be served upon a person not a party to the action who is or may be liable to the third-party plaintiff for all or part of the plaintiff’s claim against him. The third-party plaintiff need not obtain leave to make the service if the third-party complaint is filed not later than ten (10) days after service of the third party plaintiff’s original answer. Otherwise the third-party must obtain leave on motion upon notice to all parties to the action. The person served with the summons and third-party complaint, hereinafter called the third-party defendant, shall make his or her defenses to the third-party plaintiff’s claim as provided in Rule 12 and counterclaims against the third-party plaintiff and cross-claims against other third-party defendants as provided in Rule 13. The third-party defendant may assert against the plaintiff any defenses which the third-party plaintiff has to the plaintiff’s claim. The third-party defendant may also assert any claim against the plaintiff arising out of the transaction or occurrence that is the subject matter of the plaintiff’s claim against the third-party plaintiff. The plaintiff may assert any claim against the third-party defendant arising out of the transaction or occurrence that is the subject matter of the plaintiff’s claim against the third-party plaintiff, and the third-party defendant thereupon shall assert any defenses as provided in Rule 12 and his or her counterclaim and cross-claims as provided in Rule 13. Any party may move to strike the third-party claim, or for its severance or separate trial. A third-party defendant may proceed under this rule against any person not a party to the action who is or may be liable to the third-party defendant for all or part of the claim made in the action against the third-party defendant.