Solving the problem with the authorization request on the site?

Question

PHP

Hello. I work with the site https://freedom-stat.com/ ...
Below there is a registration, a field for entering mail with a button "try for free", after clicking there should be an account login, which I understand ..
The bottom line is that first, when loading the page, the browser receives several cookies, one of the key ones is laravel_session. Why key? Because through the browser inspector (f12) I messed around with cookies, deleted different ones in order to find out without which one I would not log into my account .. and this is the most important one. It is stored in the browser after the initial download, and when the registration button is clicked, it is sent to the server in the headers.
And now, the problem is that the server then returns json - {"message": ""}, but in the browser, upon successful registration, something else is returned.
I made a test script first: https://pastebin.com/DGiii28c.
I took the headers from the fidler, but the fact is that the laravel_session cookies change dynamically with each page load, and logically, the last one should be substituted in the registration request ..
Who will have time to deal with the site - help, please, tell me how the script is correct compose. I would be very grateful, and Santa Claus will come to you for the New Year :)