feat(frontend): add scroll to bottom button

frontend/src/routes/[mnemonic]/+page.svelte

@@ -11,6 +11,27 @@
 	const usePolling = getContext<boolean>('convex-use-polling') ?? false;
 	let mnemonic = $derived(page.params.mnemonic);
 
+	let lastMessageElement: HTMLDivElement | null = $state(null);
+	let showScrollButton = $state(false);
+
+	$effect(() => {
+		if (!lastMessageElement) return;
+
+		const observer = new IntersectionObserver(
+			([entry]) => {
+				showScrollButton = !entry.isIntersecting;
+			},
+			{ threshold: 0, rootMargin: '0px 0px -90% 0px' }
+		);
+
+		observer.observe(lastMessageElement);
+		return () => observer.disconnect();
+	});
+
+	function scrollToLastMessage() {
+		lastMessageElement?.scrollIntoView({ behavior: 'smooth', block: 'start' });
+	}
+
 	const chatDataWs = usePolling
 		? null
 		: useQuery(api.chats.getWithUser, () => (mnemonic ? { mnemonic } : 'skip'));
@@ -118,12 +139,22 @@
 		<div class="py-4 text-center text-xs text-neutral-500">Not found</div>
 	{:else}
 		<div class="space-y-1">
-			{#each messages as message (message._id)}
+			{#each messages as message, i (message._id)}
+				{#if i === messages.length - 1}
+					<div bind:this={lastMessageElement}>
 						<ChatMessage
 							role={message.role}
 							content={message.content}
 							isStreaming={message.isStreaming}
 						/>
+					</div>
+				{:else}
+					<ChatMessage
+						role={message.role}
+						content={message.content}
+						isStreaming={message.isStreaming}
+					/>
+				{/if}
 			{/each}
 		</div>
 
@@ -145,4 +176,13 @@
 			</div>
 		</div>
 	{/if}
+
+	{#if showScrollButton}
+		<button
+			onclick={scrollToLastMessage}
+			class="fixed bottom-12 right-3 z-50 flex h-8 w-8 items-center justify-center rounded-full bg-blue-600 text-white shadow-lg animate-pulse"
+		>
+			↓
+		</button>
+	{/if}
 </div>