Is it possible to place the search page component inside the Bitrix catalog component so that the latter works on standard Ajax?

Question

1C-Bitrix

It's not that it was some kind of problem, it's just that there is an interest in what's going on. It is clear that the problem can be solved if you rewrite AJAX yourself, however, I was just interested in the question of whether the standard one can be made to work.
The situation is the following. There is a Bitrix catalog component, it contains a call to the search page component (if it is important, the infoblocks of the catalog and the one that is searched in another component are different and the types are also different).
Ultimately, this search page is displayed for example at the URL /uslugi/?SECTION_ID=49&ELEMENT_ID=2447 to the entire search page is displayed in the second tab
Now, if I include AJAX_MODE => "Y" in the settings of the AJAX search page component, it does not work. Moreover, if the same component is moved to a separate page, for example, test3.php, then AJAX with the same settings is applied to both the search form and pagination.
In addition, if the component has a directory after all this, replace the component call with echo file_get_contents(" http://domain_name/test3.php "); then opening the second tab at the same address (/uslugi/?SECTION_ID=49&ELEMENT_ID=2447) the search page works correctly from the tab on the directory, connecting via AJAX to the test3.php file
Maybe someone knows how to force the search page component with the minimum number of movements all the same to work out on standard AJAX from inside the catalog component?
In case of thoughts - why is it needed at all (well, it’s actually needed for convenience, there are a lot of elements related to a section or directory element - a really convenient solution to let you search for them in a separate tab).

And I’ll ask a separate sub-question right there , is there such a GET parameter as ? PAGEN_N=X , where X is the number of the paginated page that you want to display, and N is some serial number, for example 1,2,3,4... So, the question is: how is this N formed at all? Why do I have on the test3.php page on which the only called component that is on the page - the search page, has the PAGEN_4=X parameter, why 4? After all, I repeat, only 1 component is called from this page...