feat: export LLM synthesis context to cache for debugging

Save full LLM context to _cache/llm_context_TIMESTAMP.txt:
- System prompt
- User prompt with evidence
- Raw evidence items
- LLM's final answer

Helps debug why LLM fails despite having correct evidence.

Co-Authored-By: Claude <noreply@anthropic.com>

src/agent/llm_client.py

@@ -1107,6 +1107,44 @@ Examples of bad answers (too verbose):
 
 Extract the factoid answer from the evidence above. Return only the factoid, nothing else."""
 
+    # ============================================================================
+    # SAVE LLM CONTEXT TO CACHE - For debugging and comparison
+    # ============================================================================
+    from pathlib import Path
+    import datetime
+
+    cache_dir = Path("_cache")
+    cache_dir.mkdir(exist_ok=True)
+
+    timestamp = datetime.datetime.now().strftime("%Y%m%d_%H%M%S")
+    context_file = cache_dir / f"llm_context_{timestamp}.txt"
+
+    with open(context_file, "w", encoding="utf-8") as f:
+        f.write("=" * 80 + "\n")
+        f.write("LLM SYNTHESIS CONTEXT\n")
+        f.write("=" * 80 + "\n")
+        f.write(f"Timestamp: {datetime.datetime.now().isoformat()}\n")
+        f.write(f"Question: {question}\n")
+        f.write(f"Evidence items: {len(evidence)}\n")
+        f.write("\n" + "=" * 80 + "\n")
+        f.write("SYSTEM PROMPT:\n")
+        f.write("=" * 80 + "\n")
+        f.write(system_prompt)
+        f.write("\n" + "=" * 80 + "\n")
+        f.write("USER PROMPT:\n")
+        f.write("=" * 80 + "\n")
+        f.write(user_prompt)
+        f.write("\n" + "=" * 80 + "\n")
+        f.write("EVIDENCE ITEMS:\n")
+        f.write("=" * 80 + "\n")
+        for i, ev in enumerate(evidence):
+            f.write(f"\n--- Evidence {i+1}/{len(evidence)} ---\n")
+            f.write(ev)
+        f.write("\n" + "=" * 80 + "\n")
+
+    logger.info(f"[synthesize_answer_hf] LLM context saved to: {context_file}")
+    # ============================================================================
+
     logger.info(f"[synthesize_answer_hf] Calling HuggingFace for answer synthesis")
 
     messages = [
@@ -1137,6 +1175,15 @@ Extract the factoid answer from the evidence above. Return only the factoid, not
     answer = response.choices[0].message.content.strip()
     logger.info(f"[synthesize_answer_hf] Generated answer: {answer}")
 
+    # Append answer to context file
+    with open(context_file, "a", encoding="utf-8") as f:
+        f.write("\n" + "=" * 80 + "\n")
+        f.write("LLM ANSWER:\n")
+        f.write("=" * 80 + "\n")
+        f.write(answer)
+        f.write("\n" + "=" * 80 + "\n")
+    logger.info(f"[synthesize_answer_hf] Answer appended to context file")
+
     return answer