atode, mazu

hello,
1.when i was reading i encountered a statement saying in below sentence the ato de cannot be used,
"okita ato de ha wo migakimasu" , i want to know if ato de cannot be used then what shall be used.

2. pls tell me the difference between ato de and ato.(with the help of sentence)
3. pls tell me the difference between mazu and sakki(with the help of sentence)

4. sousu wo tsukete, tabemasu
gohan o tabete, kaisha e ikimasu.
in the above first sentences te + verb pattern indicates that WITH sauce i eat food but in the second sentence the sense of WITH is not clearly understandable.

pls clarify the above doubts asap.

1. I don't understand why "ato de" cannot be used there. Was it mentioned in a textbook?

2. "ato de" is "later," as in
- Ato de kuwashiku hanashimashou. (Let's talk in details later.)
"Ato" by itself is "after (something)."
- Asa-gohan no ato, kaisha e ikimasu. (After breakfast, i go to my company.)
- Asa-gohan wo tabeta ato, kaisha e ikimasu. (After eating breakfast, I go to my company.)

3. "Mazu" is "first of all," when you talk about doing things one after another.
(For example, when you are a cooking teacher and explain how to make a cake)
- Mazu, komukigo wo hakarimasu. (First, you measure/weigh the flour.)
Sorekara, mizu wo 200ml iremasu. (Then, you add 200ml water.)

"Sakki" is "earlier," "a while ago."
- Tom wa doko ni imasuka? (Where is Tom?)
- Sakki honya de mimashita. (I saw him at the bookstore a while ago.)

So "mazu" and "sakki" are quite different - do you mean "saki"?

4.
- Sousu wo tsukete, tabemasu.
Means "I PUT sauce (on the food), and eat."
"... wo tsukeru/tsukemasu" is a verb that means "to put something," or "to add something."

This "te form" of the verb talks about things that you do one after another. It does NOT have the meaning of "with."

So the sentence:
- Gohan wo tabete, kaisha e ikimasu
simply means "I eat, then go to my company."

I can't explain why, but we just don't say, "okita ato de ha wo migakimasu" when we want to say, "After I wake up, I brush my teeth." If I want to say that brushing teeth is something I do every morning right after I wake up, I would say, "okitaRA ha wo migakimasu."

Even if we wanted to say, "(I don't brush my teeth at night, but) I brush my teeth after I wake up," or "(I won't brush my teeth yet, but I'm going to take a nap now and) I'll brush my teeth after I wake up," we would say something like, "neta ato de ha wo migakimasu," or "okite kar ha wo migakimasu."

"Okita ato de ha wo migakimasu" sounds like, "I wake up, and then I would brush my teeth," or "Once I wake up, I would brush my teeth," as if though the "waking up" happens and then something else starts happening, as opposed to waking up and almost automatically doing something, if you know what I mean.

"okita ato de" may seem natural at first glance, but when you imagine yourself explaining your morning routine, you realize you just wouldn't say it. Those who can read Japanese can Google "起きたあとで" or "目覚めたあとで" to get a bigger picture.