docs: add guide for managing user context with Store

[jjhpka]

Add comprehensive documentation for managing user preferences using LangGraph Store and static runtime context pattern.

• Demonstrate Store + Static Context integration
• Cover loading, updating, and dynamic field patterns

Overview

This PR adds a new guide (manage-user-context.mdx) that demonstrates how to manage user-specific
settings (language preferences, timezone, communication style) using LangGraph Store combined with static
runtime context.

The guide fills a documentation gap by showing developers how to:

• Persist user preferences across conversations using Store
• Load preferences as immutable static context during agent runs
• Access user settings in middleware for dynamic prompts
• Support extensible field patterns with the `extra' dictionary

Type of change

Type: New documentation page

Related issues/PRs

• GitHub issue:

• Feature PR:

• Linear issue:

• Slack thread:

Checklist

• I have read the contributing guidelines
• I have tested my changes locally using docs dev
• All code examples have been tested and work correctly
• I have used root relative paths for internal links
• I have updated navigation in src/docs.json if needed
• I have gotten approval from the relevant reviewers
• (Internal team members only / optional) I have created a preview deployment using the Create Preview Branch workflow

Additional notes