Integrate OpenAI Responses API for image-based solving with gpt-5-nano

[brawliscool]

Motivation

• Replace the previous hard-coded demo solver with a live model call so users get actual AI-generated step-by-step solutions from uploaded images.
• Allow users to supply their own OpenAI API key at runtime rather than baking credentials into source.
• Ensure the solver returns structured data (problem + steps) so the UI can render reliable results.

Description

• Added OPENAI_API_KEY_STORAGE_KEY and OPENAI_MODEL constants and a getOpenAiApiKey helper that prompts once and caches the key in localStorage.
• Implemented generateSolutionWithOpenAI(imageDataUrl, apiKey) which calls https://api.openai.com/v1/responses using fetch, sends the image as input_image, and parses data.output_text as JSON containing problem and steps.
• Converted the solve flow to an async handler for the solve button and replaced the hard-coded demo solution with a call to generateSolutionWithOpenAI using the configured model gpt-5-nano-2025-08-07.
• Added failure handling that restores a spent credit, clears the loading state, shows an error toast, and leaves the UI ready for a retry.

Testing

• Ran node --check script.js to validate there are no syntax errors, and it completed successfully.

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[chatgpt-codex-connector]

Re-prompt for API key after auth failures

getOpenAiApiKey always returns the cached key once it exists, so if a user initially enters a mistyped/revoked key, every future solve attempt keeps reusing that invalid credential and fails (typically 401) without ever prompting for a replacement. Since this flow has no in-app reset for OPENAI_API_KEY_STORAGE_KEY, users can be stuck in a permanent failure loop until they manually clear local storage.

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