How to sync Many To Many relation in pagekit?

Question

I'm working with PageKit CMS. I have 2 tables with Many To Many relation (item and type).

Item model:

class Item implements \JsonSerializable
{
    ...
    /**
     * @ManyToMany(targetEntity="Type", tableThrough="@prefix_item_type", keyThroughFrom="item_id", keyThroughTo="type_id")
     */
    public $types;
    ...
}

Type model:

class Type implements \JsonSerializable
{
    ...

    /**
     * @ManyToMany(targetEntity="Item", tableThrough="@prefix_item_type", keyThroughFrom="type_id", keyThroughTo="item_id")
     */
    public $items;
    ...
}

In backend interface on item edit page I created multi select with all types. When I send item save request, I get type ids.

My save item method have a look:

public function saveAction($data, $id = 0, $selected_types = [])
{
    /*
     * $selected_types = array(2) {
     * [0]=>int(1)
     * [1]=>int(2)
     * }
     */
    if (!$id || !$item = Item::query()->related(['types'])) {
        if ($id) {
            App::abort(404, __('Item not found'));
        }
        $item = Item::create();
    }

    if (!$data['slug'] = App::filter($data['slug'] ?: $data['title'], 'slugify')) {
        App::abort(400, __('Invalid alias'));
    }

    if(!App::user()->hasAccess('ext_name: manage all items')) {
        $data['user_id'] = App::user()->id;
    }

    if(!App::user()->hasAccess('ext_name: manage all items') && !App::user()->hasAccess('ext_name: manage own items') && $item->user_id !== App::user()->id) {
        App::abort(403, __('Access denied'));
    }

    $item->save($data);

    /*
     * Here I need to sync $item->types with $selected_types ids
     */

    return [
        'message' => 'success',
        'entity' => $item,
    ];
}

How can I sync this relation if I have current item id and new type ids?