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/specs/chat-ui/spec.md

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+## MODIFIED Requirements
+
+### Requirement: Streaming AI response
+
+The assistant SHALL reply with a real AI response streamed from the backend API, using the full conversation history as context. Tokens appear incrementally as they arrive.
+
+#### Scenario: Bot replies with streamed AI response
+
+- **WHEN** the user sends any message
+- **THEN** the assistant message appears and grows token by token as the stream delivers text
+
+#### Scenario: Full history sent with each request
+
+- **WHEN** the user sends a message after prior exchanges
+- **THEN** all previous user and assistant messages are included in the API request so the AI has conversational context
+
+## ADDED Requirements
+
+### Requirement: New chat button
+
+The chat page SHALL provide a button to clear the current conversation and start a new one.
+
+#### Scenario: User starts a new chat
+
+- **WHEN** the user clicks the "New Chat" button
+- **THEN** all messages are cleared and the empty state is shown
+
+#### Scenario: New chat button disabled during streaming
+
+- **WHEN** the assistant is currently streaming a response
+- **THEN** the "New Chat" button is disabled