Multi-Level Permalinks on Custom Post Type

Hi-
Let’s say I have a custom post type called portfolio, and set it up with the following code.

function create_portfolio_post_type() {
	$portfolio_labels = array(
		'name' => 'Projects',
		'singular_name' => 'Project',
		'add_new' => 'Add New',
		'all_items' => 'All Projects',
		'add_new_item' => 'Add New Project',
		'edit_item' => 'Edit Project',
		'new_item' => 'New Project',
		'view_item' => 'View project',
		'search_items' => 'Search projects',
		'not_found' => 'No projects found',
		'not_found_in_trash' => 'No projects found in Trash',
		'parent_item_colon' => 'Parent Project',
		'menu_name' => 'Portfolio'
	);

	$portfolio_definition = array(
		'labels' => $portfolio_labels,
		'description' => 'Projects',
		'public' => true,
		'exclude_from_search' => false,
		'show_ui' => true,
		'show_in_nav_menus' => true,
		'show_in_menu' => true,
		'show_in_admin_bar' => true,
		'menu_position' => 5,
		'capability_type' => 'post',
		'hierarchical' => true,
		'supports' => array('title', 'editor', 'author', 'thumbnail', 'excerpt', 'trackbacks', 'custom-fields', 'revisions', 'page-attributes', 'post-formats'),
		'taxonomies' => 'project_type',
		'has_archive' => true,
		'rewrite' => array('slug', 'portfolio'),
		'can_export' => true
	);
	register_post_type ('portfolio', $portfolio_definition);
}
add_action ('init', 'create_portfolio_post_type');

Now let’s say I added a taxonomy to my post type with this code:

function create_portfolio_taxonomy() {
	$project_type_labels = array(
		'name' => 'Project Types',
		'singular_name' => 'Project Type',
		'search_items' => 'Search Project Types',
		'popular_items' => 'Popular Project Types',
		'all_items' => 'All Project Types',
		'parent_items' => 'Parent Project Type',
		'parent_item_colon' => 'Parent Project Type',
		'edit_item' => 'Edit Project Type',
		'update_item' => 'Update Project Type',
		'add_new_item' => 'Add New Project Type',
		'new_item_name' => 'New Project Type Name',
		'menu_name' => 'Project Types'
	);

	$project_type_definition = array(
		'labels' => $project_type_labels,
		'public' => true,
		'show_in_nav_menus' => true,
		'hierarchical' => true,
		'rewrite' => array('slug' => 'projects'),
		'capabilities' => array('manage_terms', 'edit_terms', 'delete_terms', 'assign_terms')
	);
	register_taxonomy ('project_type', 'portfolio', $project_type_definition );
}

What I want to do is set up my permalink structure so that if someone wants to get to a project, thy can type in:

https://www.example.com/post_type(portfolio)/taxonomy(projects)/(taxonomy_item)animations/video1(post_name)/

and get to the right post.

However, when I use

'rewrite' => array('slug' => 'portfolio/projects')

in my taxonomy register function, the site returns a 404. What can I do to fix this?

Also, yes, I did flush the rewrites.