AJAX to Sugar REST blocked by CORS

Question

I tried sending an AJAX request with jQuery to a SugarCRM REST endpoint, but it was blocked by Cross Origin Resource Sharing and returned HTTP 404 along with an error.

This is the AJAX request:

$.ajax({
        type: 'POST',
        url: 'https://example.sugarinstance.com/rest/v10/Cases/web/submit',
        data: jsonData,
        dataType: 'json',
        contentType: 'application/json',
        success: function(response) {
          console.log('SUCCESS')
          console.log(response)
        }
      })

And this is the error returned:

XMLHttpRequest cannot load https://example.sugarinstance.com/rest/v10/Cases/web/submit. Response to preflight request doesn't pass access control check: No 'Access-Control-Allow-Origin' header is present on the requested resource. Origin 'null' is therefore not allowed access. The response had HTTP status code 404.

This is the endpoint I created just to test without authentication since it was not working at all:

<?php

class Web2CaseApi extends SugarApi
{
    public function registerApiRest()
    {
        return array(
            'Web2CaseEndpoint' => array(
                'reqType' => 'POST',
                'noLoginRequired' => true,
                'path' => array('Cases', 'web', 'submit'),
                'method' => 'newCase',
                'shortHelp' => 'Receives JSON data for a new Case record',
            ),
        );
    }
    public function newCase($api, $args)
    {
        return $args;
    }
}

This endpoint now appears in the API help, so it definitely exists.

How am I supposed to use AJAX to send data to SugarCRM if it is blocked?
Should the server be set to allow CORS by default?

I also tried these options in the jQuery AJAX request to no avail:

crossDomain: true,
headers: {
  'Access-Control-Allow-Origin': '*'
},

Artis Plocins · Accepted Answer

I found out that CORS can be used by keeping the request to a bare minimum and adding a couple headers in the API endpoint file.
The request should have no headers, just data, url and method.
These two lines should be at the top of the custom API endpoint file (I put them above the class definition):

header('Access-Control-Allow-Origin: *');
header('Access-Control-Allow-Methods: POST');

Note that the origin URL should be specified as narrowly as possible when making the endpoint public so that random people on the internet cannot take the code and keep using it on their website. Unless that's exactly what you want, of course.