Enqueueing JQuery or not for this script

Hi:
I have a theme I built a few years ago running. At the time my programmer advised we use the google libraries for jquery. There’s a custom script I’m using to open/close the mobile menu.

/*registering google jquery and mobile menu script*/
/*--------------------------*/
function my_init() {
       if (!is_admin()) {
               wp_deregister_script('jquery');
               wp_register_script('jquery', ("https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"), false, '1.8.3', false);
               wp_enqueue_script('jquery');
               // load custom scripts
		 wp_enqueue_script('mobilemenu', get_bloginfo('template_url').'/js/mobilemenu.js', false, '', true);
       }
}
add_action('init', 'my_init');

I have to change my form plugin and one I was trying out gives me a message that my version of jquery is outdated.

So: can you tell me what version I should call these days?

OR, can I rely on the WordPress’ jquery? In which case, how would I change the above code?

Thanks!!

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