Playwright TS Captcha Fix -- 3

I am stuck on a single Playwright + TypeScript test that fails at the captcha step. The blocker is a Math Captcha (custom type) and I already have the vendor’s API documentation for solving it, but my current script is not wiring the request-response cycle correctly.

Here’s what I need:
• Patch or rewrite the section of the test that calls the captcha-solving API, parses the response, and submits the answer before the page times out.
• Keep the rest of the flow unchanged; the test should run headless and pass with `npx playwright test`.
• Add brief inline comments so I understand what was changed.

Acceptance: running the updated test on my machine with `npm test` completes end-to-end without manual input and the captcha is solved every time.

You are free to introduce lightweight helpers or Playwright test fixtures as long as they stay in TypeScript and require no extra paid services beyond the existing API.