Please implement a web application in PHP that meets the following requirements:

1. The application must be a single, self-contained .php file.
2. When first accessed, it should display an HTML form with two input fields ("Username" and "Password") and a login button. [0.5p]
3. When the login button is clicked, a POST request should be submitted to the application, containing the entered username and password. [0.5p]
4. Before the form is submitted, client-side JavaScript must validate that both input fields are filled. If either is empty, the form should not be submitted, and a JavaScript alert() message should be shown to the user. [1.5p]
5. If the submitted username is "user" and the password is "pass", the application should treat the session as authenticated and display a "Hello user!" message along with a logout button. [1.5p]
   (Tip: use $_POST, session_start().)
6. The session cookie must use the __Host- prefix. [1p]
   (Tip: use session_name() and session_set_cookie_params().)
7. If the username or password is incorrect, the application should display "Login incorrect!" along with the initial login form. [1p]
8. Clicking the logout button should send a GET request with the query parameter action=logout. The application must then delete the session cookie (using the Set-Cookie header) and redirect the user (using the Location header) back to the login form. [1.5p]
   (Tip: use isset(), $_GET, setcookie() and header().)
9. The application must only handle GET and POST methods. Any other HTTP method must result in an error response with status code 405 and a message in the response body indicating that only GET and POST are supported. [0.5p]
   (Tip: use $_SERVER["REQUEST_METHOD"] and http_response_code().)

Please submit the link to your deployed PHP website and the source code of your .php file below.