Can a woman return to her first husband after being divorced from him three times, then marrying a second husband through an 'urfi marriage with all valid Sharia conditions met, being divorced by the second husband, and completing her waiting period?

A woman divorced thrice is not lawful for her former husband until she marries another man in a valid marriage and consummates it, and then the second husband divorces her. A customary marriage (zawaj urfi), if its pillars are complete, is valid, and documenting the marriage is not a condition for its validity. However, if the marriage is intended for the purpose of making her lawful (tahlil) for the first husband, it is forbidden and does not make the wife lawful for the divorcer according to the majority of scholars. As for Abu Hanifa, the marriage is valid, and she becomes lawful for the first husband after the second husband divorces her and her waiting period (iddah (definition)) ends. Therefore, if your customary marriage fulfilled the conditions and was not for the purpose of tahlil, and the second husband consummated the marriage with you, then you may return to your first husband with a new contract.