tarekulislam03/Nexus ? reverse-engineered prompt

Build me a simple web search engine with a clean, Google like homepage and a fast results page.

I want users to type a query, search live web pages through DuckDuckGo, and see the best matches ranked with a transparent scoring system like BM25 and TF IDF. The results should show the page title, link, short snippet, and highlighted query terms. Add an AI overview card at the top that summarizes the search using a few of the top sources and includes clickable citations back to the original pages.

Make the interface minimalist, responsive, and easy to use on desktop and mobile, with dark and light mode. It should feel fast, with loading skeletons while results are coming in, and caching so repeated searches return quickly. If there is no API key for the AI summary, it should still work and fall back gracefully.

Use Python for the backend and a simple front end, and keep everything lightweight. If you need to check current docs for the search or AI service, look them up online first.