Adding Cookie in asmx not working

Question

I have a asmx webservice which has a webmethod that I'm calling it via jquery ajax and it returns a string, I want to read and edit cookies from this method, I know that I can read the cookie like this:

HttpContext.Current.Request.Cookie["cookie_name"];

but when I'm trying to change the value of this cookie, this doesn't work:

HttpContext.Current.Response.Cookie["cookie_name"].Value = "some_value";

so let's say this is my function:

[ScriptMethod(ResponseFormat = ResponseFormat.Json)]
[WebMethod(EnableSession = true)]
public string myMethod()
{
    //...some actions 
    if(HttpContext.Current.Response.Cookie["cookie_name"] != null)
    {
        HttpContext.Current.Response.Cookie["cookie_name"].Value = "some_value";
    }
    return "";
}

and here is my ajax call

    $.ajax({
        type: "POST",
        url: "/route-to/myMethod",
        data: JSON.stringify({ }),
        contentType: "application/json",
        charset: "utf-8",
        dataType: "json",
        async: true,
        cache: false,
        success: function (msg) {

        }
    });

What's exactly is wrong in this code?