For hard problems you’ll have to use the GPT 5.5 pro model (available via api if you don’t want to spend $100 on the monthly subscription)
I have that but don’t see any ‘pro’ option.
GPT 5.5 Pro is only in chat/API, not Codex.
From https://openai.com/index/introducing-gpt-5-5/
In Codex, GPT‑5.5 is available for Plus, Pro, Business, Enterprise, Edu, and Go plans with a 400K context window.
It's just the $20 a month sub (for chat), or else use the API.