Is the second act performed by a Muslim as a precaution after doubting the validity of the first act, and then becoming certain of the invalidity of the first, sufficient? Or does this fall under the issue of "whoever performed ablution and then doubted ritual impurity, so he performed ablution as a precaution, and then became certain of his impurity"? And can the hadith "you will have two rewards" be used to infer the sufficiency of the first repetition?

If a Muslim doubts the validity of his purification (wudu) or the correctness of his prayer, and he performs ablution or repeats the prayer out of precaution, and then it becomes clear to him that the first one was invalid, his second ablution or prayer is sufficient. This is indicated by the principle that precaution fundamentally entails performing an action as an obligation. If the intention of precaution were not sufficient, then the very concept of seeking precaution would not have been legislated in the first place.

As for the hadith (definition) of Abu Sa'id (may Allah be pleased with him) concerning the prayer of one who performed tayammum (definition) and then found water, it is not evidence for our issue. Rather, it is evidence that the first prayer is sufficient, that repeating it is not recommended, and that whoever repeats it, acting as a mujtahid, will receive a double reward.