Lumen redirect from project root to public folder

Question

I want to create an API service powered by Lumen 5.4 which running in a shared hosting environment.

here's my folder structure

/public_html
    |- api
        |- public
            |- index.php
            |- .htaccess

the api folder is the root of Lumen project.

and now I need to access http://example.com/api/public to access my Lumen project. What I want is directly access to http://example.com/api

So I try to put a .htaccess in api folder:

<IfModule mod_rewrite.c>
    RewriteEngine On
    RewriteRule ^(.*)$ public/$1
</IfModule>

What I trying to do is redirect all requests from api folder to public folder. At the end, requests to public folder will all redirect index.php

The request was redirected to the public folder and Lumen can already process it. But I get NotFoundHttpException in RoutesRequests.php line 594 error.

I refer to many references and all of them having the same RewriteRule as I using. But how come it doesn't work at all on my shared hosting?

Any help would be appreciated.