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>

src/ChatAgent.Client/Pages/Chat.razor

@@ -75,8 +75,21 @@
     </div>
 
     @* Input area: pinned at the bottom of the chat container.
-       Disabled attribute prevents interaction while the assistant is streaming. *@
+       Disabled attribute prevents interaction while the assistant is streaming.
+       The "New Chat" button clears the conversation to start fresh. *@
     <div class="input-area">
+        @if (_messages.Count > 0)
+        {
+            <div class="input-actions">
+                <MudButton Variant="Variant.Text"
+                           Size="Size.Small"
+                           StartIcon="@Icons.Material.Filled.Add"
+                           OnClick="NewChat"
+                           Disabled="_isStreaming">
+                    New Chat
+                </MudButton>
+            </div>
+        }
         <MudTextField @bind-Value="_userInput"
                       Placeholder="@(_isStreaming ? "Waiting for response..." : "Type a message...")"
                       Variant="Variant.Outlined"
@@ -106,6 +119,15 @@
     // Used to disable input and show the thinking indicator.
     private bool _isStreaming = false;
 
+    /// <summary>
+    /// Clears all messages to start a new conversation.
+    /// </summary>
+    private void NewChat()
+    {
+        _messages.Clear();
+        _userInput = string.Empty;
+    }
+
     /// <summary>
     /// Handles the Enter key press to submit the message.
     /// </summary>
@@ -154,14 +176,16 @@
 
         try
         {
-            // Build the request with the current user message.
-            // Future phases will include full conversation history for multi-turn.
+            // Build the request with the full conversation history for multi-turn context.
+            // We include all messages except the last one (the empty assistant placeholder
+            // that is waiting to be filled by the stream). This gives the AI the full
+            // conversation so it can reference prior exchanges.
             var request = new ChatRequest
             {
-                Messages = new List<ChatMessage>
-                {
-                    new ChatMessage { Role = "user", Content = userText }
-                }
+                Messages = _messages
+                    .Where(m => !string.IsNullOrEmpty(m.Content))
+                    .Select(m => new ChatMessage { Role = m.Role, Content = m.Content, Timestamp = m.Timestamp })
+                    .ToList()
             };
 
             // Stream tokens from the API. IAsyncEnumerable yields each text delta

src/ChatAgent.Client/Pages/Chat.razor.css

@@ -77,3 +77,10 @@
     border-top: 1px solid var(--mud-palette-lines-default);
     background-color: var(--mud-palette-background);
 }
+
+/* Action row above the text input (New Chat button) */
+.input-actions {
+    display: flex;
+    justify-content: flex-end;
+    margin-bottom: 0.25rem;
+}