feat(memory): async LLM context summary injection on trim

- Unified flush + context injection into a single async LLM call
  (flush_from_messages accepts context_summary_callback)
- Fixed response parsing bug: handle generator returns and Claude-format
  dicts from bot.call_with_tools, which previously caused all LLM
  summaries to silently fail (falling back to rule-based extraction)
- Removed standalone context summary prompts and methods; reuse the
  existing [DAILY]/[MEMORY] summarization pipeline
- Updated docs (zh/en/ja) to reflect the new injection behavior

docs/en/memory/context.mdx

@@ -39,14 +39,15 @@ When conversation turns exceed `agent_max_context_turns`:
 
 - The **oldest half** of complete turns is trimmed (preserving tool call chain integrity)
 - Trimmed messages are summarized by LLM and **written to the daily memory file**
-- Remaining turns stay intact
+- Once the LLM summary is ready, it is also **injected into the first user message** of the retained context, helping the model maintain conversational continuity
+- Summary injection runs asynchronously in the background and takes effect from the next turn onward
 
 ### 3. Token Budget Trimming
 
 After turn trimming, if tokens still exceed the budget:
 
 - **Fewer than 5 turns**: All turns undergo **text compression** — each turn keeps only the first user text and last Agent reply, removing intermediate tool call chains
-- **5 or more turns**: The **first half** of turns is trimmed again, with discarded content also written to memory
+- **5 or more turns**: The **first half** of turns is trimmed again, with discarded content written to memory and a context summary injected
 
 ### 4. Overflow Emergency Handling