feat: add multi-turn conversation support within a session

Send full conversation history with each API request so the AI maintains
context across exchanges. Add "New Chat" button to clear the conversation
and start fresh. No persistent storage — session resets on page reload.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

openspec/changes/multi-turn-conversations/design.md

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+## Context
+
+Chat.razor currently constructs a `ChatRequest` with only the latest user message (line 161-164). The backend and Responses API already support multi-message input — no server changes needed. This is purely a client-side change.
+
+## Goals / Non-Goals
+
+**Goals:**
+- Send full conversation history with each request for multi-turn context
+- Add a "New Chat" button to reset the session
+
+**Non-Goals:**
+- Persistent conversation storage (explicitly deferred)
+- Multiple conversation tabs/sidebar
+- Token limit management or history truncation (small conversations for now)
+
+## Decisions
+
+### Decision 1: Send full _messages list minus the empty placeholder
+
+When building the `ChatRequest`, include all messages from `_messages` except the last one (which is the empty assistant placeholder waiting for streaming). This gives the AI the full conversation context.
+
+### Decision 2: New Chat button in the AppBar
+
+Place a "New Chat" icon button in the `MudAppBar` (in MainLayout.razor or Chat.razor). Clicking it clears `_messages` and resets to the empty state. Disabled during streaming.
+
+Alternative considered: putting it in the input area. Rejected — the AppBar is more natural and matches ChatGPT/Gemini placement.
+
+## Risks / Trade-offs
+
+- [No token limit management] → For long conversations, the full history could exceed the model's context window. Acceptable for now; truncation can be added later.