Update web search prompt guidance in GPT-5.2 prompting guide

[dmitry-openai]

Summary

• Replace the optional detailed web search guidance in the GPT-5.2 prompting guide with the new comprehensive research prompt.

Motivation

• Aligns the example web search prompt with the latest guidance for research persona and behavior.

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For new content

• I have added a new entry in registry.yaml (and, optionally, in authors.yaml) so that my content renders on the cookbook website.
• I have conducted a self-review of my content based on the contribution guidelines:
  • Relevance: This content is related to building with OpenAI technologies and is useful to others.
  • Uniqueness: I have searched for related examples in the OpenAI Cookbook, and verified that my content offers new insights or unique information compared to existing documentation.
  • Spelling and Grammar: I have checked for spelling or grammatical mistakes.
  • Clarity: I have done a final read-through and verified that my submission is well-organized and easy to understand.
  • Correctness: The information I include is correct and all of my code executes successfully.
  • Completeness: I have explained everything fully, including all necessary references and citations.

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Codex Task

[msingh-openai]

approved.