Create a menu that replicates my custom taxonomy hierarchy in wordpress

I have a custom post type (“products”) with an associated custom taxonomy (“product_type”). So when I create a product, I choose in which category it goes.

The taxonomy is hierarchical and it goes like this:

Category 1
    Sub-Category 1.1
    Sub-Category 1.2
    Sub-Category 1.3
Category 2
Category 3
    Sub-Category 3.1
    Sub-Category 3.2
etc, etc

What I want (and don’t know how to do it) is to create a menu that dynamically replicate my taxonomies, with the products associated to them.

The rendered html should be something like this:

<ul>
    <!--first level-->
    <li>
        <a href="#">Category 1</a>
        <ul>
            <!--second level-->
            <li>
                <a href="#">Sub-category 1.1</a>
                <!--third level-->
                <ul>
                    <li><a href="#">product</a></li>
                    <li><a href="#">product</a></li>
                </ul>
            </li>

            <!--second level-->
            <li>
                <a href="#">Sub-category 1.2</a>
                <!--third level-->
                <ul>
                    <li><a href="#">Product</a></li>
                    <li><a href="#">Product</a></li>
                </ul>
            </li>
        </ul>
    </li>

    <!--first level-->
    <li class="first-level">
        <a href="#">Category 2</a>
        <ul>
            <!--second level-->
            <li><a href="#">Product</a></li>

            <!--second level-->
            <li><a href="#">Product</a></li>

            <!--second level-->
            <li><a href="#">Product</a></li>
        </ul>
    </li>
</ul>

I know html, but I’m not too savvy in php.

Can anybody point me in the right direction to accomplish that?

Thanks!