Build an HTML Formatter tool using HTML, CSS, and JavaScript. The tool should allow users to paste raw or unformatted HTML code into a text area. Upon clicking a "Format HTML" button, the code should be beautified with proper indentation and structure. Use a client-side library like js-beautify for formatting. Include a “Copy to Clipboard” button and an optional “Clear” button. Make sure the interface is responsive and minimal, suitable for both desktop and mobile users.