Add DeepSeek integration for solving images and store API key

[brawliscool]

Motivation

• Replace the local mock solution flow with an actual model-based solver to return real step-by-step math solutions from uploaded images.
• Persist a user-provided DeepSeek API key so solves can be authenticated without exposing keys in code.

Description

• Added constants for DeepSeek configuration including DEEPSEEK_API_KEY_STORAGE_KEY, DEEPSEEK_MODEL, and DEEPSEEK_MAX_OUTPUT_TOKENS and updated billing/plan logic usage.
• Implemented getDeepSeekApiKey() to prompt for and persist the API key in localStorage under DEEPSEEK_API_KEY_STORAGE_KEY and validate presence before solving.
• Implemented parseSolutionPayload() to normalize and parse JSON responses from the model and validate the expected { problem, steps } shape.
• Implemented generateSolutionWithDeepSeek() which posts to https://api.deepseek.com/chat/completions with the configured model and returns a parsed solution.
• Replaced the previous mock setTimeout solver with an async call to generateSolutionWithDeepSeek() on solve, added error handling to restore credits on failure, and updated UI state and toasts accordingly.

Testing

• No automated tests were added or executed as part of this change.

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Codex Task

[chatgpt-codex-connector]

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[chatgpt-codex-connector]

Prevent stale solve responses from replacing newer results

The async solve callback unconditionally applies setSolutionResult(solution) after awaiting the API call, but it never verifies that this is still the most recent solve request. If the user triggers another solve while an earlier request is still in flight, an older response can arrive later and overwrite the newer result, so the UI can display the wrong solution for the current image.

Useful? React with 👍 / 👎.

[chatgpt-codex-connector]

Avoid embedding full image data URLs in chat prompt

This sends the entire base64 Data URL as plain prompt text, which is extremely large for typical uploaded photos and can exceed model context/request limits or cause frequent request failures and excessive token usage. The solve flow now depends on this path for every request, so reliability drops sharply unless the image is sent via a proper image input mechanism (or aggressively reduced first).

Useful? React with 👍 / 👎.