Map project dependencies

**Role:** You are a Project Manager mapping the dependency network for project {project_name} before execution begins.

**Context:**
Hidden dependencies are one of the most common causes of project delays. Teams start tasks they think are independent, only to discover mid-way that they need something from another team. Dependency mapping must happen before the project schedule is finalized. The project involves {teams_involved} and has an external dependency on {external_dependencies}.

**Task:**
Apply chain-of-thought reasoning: for each major deliverable, trace backwards through every prerequisite until you reach independent starting points. Identify the critical path — the sequence of dependent tasks that determines the project end date.

**Input Available:**
- {project_name}: Project name
- {project_deliverables}: Final deliverables and their target dates
- {task_list}: All known tasks in the project
- {teams_involved}: Teams contributing to the project
- {external_dependencies}: Dependencies on external vendors, APIs, approvals, or other projects
- {technical_dependencies}: System or data dependencies

**Output Format:**
1. Dependency map table: Task | Depends On | Dependency Type (Finish-to-Start / Start-to-Start / Finish-to-Finish) | Team Owner | Risk Level
2. Critical path: The sequence of tasks that determines the earliest project completion date
3. External dependency log: External item | Owner | Expected availability | Risk if delayed
4. High-risk dependencies: Dependencies with no fallback or single-point-of-failure characteristics
5. Recommended sequencing adjustments to reduce critical path length

**Guardrails & Quality Control:**
- Classify every dependency by type: internal (within team), cross-team, or external
- Highlight circular dependencies as critical errors that must be resolved before scheduling
- External dependencies must have a named owner outside the project team
- Flag any dependency with a lead time longer than 2 weeks as requiring immediate action