How can a wife be taken back after two valid divorces, and is the presence of witnesses required? What is the ruling on the remaining number of divorces if she is taken back after her waiting period has ended with a new contract and dower?

If a man divorces his wife with the first or second pronouncement, he has the right to take her back during her iddah (waiting period) without her consent, or her guardian's consent, or a new mahr (dowry), or a new contract. The raj’ah (revocation of divorce) can be achieved verbally, such as by saying "I take you back," or by action, such as engaging in intercourse with the intention of taking her back. It is recommended to have witnesses for the raj’ah, but it is not obligatory. If her iddah expires, she becomes irrevocably separated from him with a minor separation (baynunah sughra), and she does not become lawful for him again except through a new contract, mahr, guardian, witnesses, and her consent. The minor separation does not annul the previous divorce; rather, any subsequent divorce pronouncements are added to it. If the pronouncements reach three divorces, she becomes irrevocably separated with a major separation (baynunah kubra), and she does not become lawful for him thereafter until she marries another husband. After the raj’ah, the man only possesses one pronouncement of divorce, and thereafter, the wife becomes irrevocably separated with a major separation.