In a case where the Department is subrogated to the rights of the recipient, the Department has a claim to the proceeds of the recovery from the persons liable, whether the proceeds of the recovery are by way of judgment, settlement, or otherwise and shall, to the extent permitted by federal law, be satisfied in full. The claim is in the amount of all Program benefits paid minus any deductions required by law. The claim is effective when: Filed with a court of competent jurisdiction in the State; and Notice of filing of the claim is served by the Department upon the third party, personally or by registered, certified, or insured mail, return receipt requested.