Split and combine pdf using API

You do not "raise a case" by using these forums. You talk to a community of Adobe customers who might share their experience and expertise. Or nothing. There is no case to follow up, and no requirement to answer. As suggested, if you want to do this with an Adobe product, you need to pay for Acrobat, not use the free Acrobat Reader; be sure you know the difference.  (You can raise a case with Adobe, but they cannot help programmers write code).

When the reply says "a script can do that" the meaning is you (as a script programmer) can write a script to do this. The place to start is the Acrobat SDK, with thousands of pages of detailed information for script writers and programmers.  https://opensource.adobe.com/dc-acrobat-sdk-docs/acrobatsdk/index.html

You can't find it. You need to write it, or have someone write it for you.

If the number of pages in the file will always be divisible by 3 then you can use this script to do it:

for (var p=0; p<this.numPages; p+=3) {
	this.extractPages(p, p+2, this.path.replace(this.documentFileName, "F"+((p/3)+1)+".pdf"));
}

You can run it from the JS Console or via a Custom Command, for example.