Complex Reverse tax de calculation

I've got a formula which was provided by a tax brazilian federal agency to calculate the taxes.

Note: the tax agency did not want to provide the decalculation excel formula so it's up to me to figure it out.

The formula works great to calculate the value after tax from the value without tax.

However I need to decalculate the tax to make sure we are aligned.

There are 3 taxes and they are linked to each other in such a way that despite my best efforts I have failed to come up with something that correctly decalculates it.

IPI is calculated on the value without tax, ICMS and PIS/Cofins are each calculated on the value of the item including IPI.

I have attached the Excel file and I hope someone here can think of something that would work!