NGINX reverse_proxy with subpath and regex

I have a site running on localhost:8001 and I would like it to be accessible on different paths. I need to extract part of the path using regex.

This does work:

server {
    location /user/amy/ {
        proxy_pass http://localhost:8001/;
    }
}

(but it breaks if I remove the slash after 8001).

However, the username can be dynamic, for example:

mydomain.com/user/amy/ --> localhost:8081/
mydomain.com/user/bob/foo --> localhost:8081/foo

In both cases I want to get the username out in order to do a subrequest. Here is what I have tried:

server {
    location ~ ^/user/([a-z]+)/(.*)$ {
        set $user $1; # do stuff with this later
        proxy_pass http://localhost:8001/$2;
    }
}

It doesn't work: I get a too many redirects error. I've tried many things and nothing seems to work. I want it to work exactly like the prefix example I gave first, except that I extract the username for further processing. Any help would be massively appreciated.

I suggest you try Nginx Proxy Manager.