Here is the IRS's modus operendi. Let's take as an example someone who made $100,000 in salary working in Israel and paid $40,000 in Israeli taxes with 6 kids. The taxes that they paid in Israel more then cover what their US tax liability and since the child tax credit is refundable they would get between $5000 - $6000 as a refund. What the IRS does is they believe that you made $100,000 but throw everything else away. (Very similar to teh idea of פלגינן דיבורא in halacha). They deny that the foreign taxes were paid and the fact that you have 6 children. From their perspective it is all a fiction to cheat the US government. Therefore, from the IRS's perspective you made $100,000, paid no taxes, and in fact got $6000 in illegitimate child tax credits. Oh, and by the way you have 30 days to answer them otherwise they start taking your assets.
Proving that I made what I said I made and paid taxes in Israel like I said, is not difficult, I have my pay stubs and year end (tofes 106) statement. However, how do you prove that your children are your children and that they live with you? What kind of proof can you bring? The accountant suggested going to our family doctor and getting a letter from him attesting that these are our children. We did that and it turns out we were not the first to ask him for it.
The accountant needs to get my Israeli pay stubs etc. translated and then he will respond to the IRS. Hopefully it will go well.
One lesson that I learned, never throw out things like pay stubs, etc. You never know when you may need them.
After submitting all my supporting documentation (pay stubs, children's passports, letters from schools, doctors, etc.) the IRS has withdrawn their claim and sent me an official letter stating that I don't owe anything.