feat: add mobile application structure with emotion picker and API integration

- Added favicon and icon assets for the mobile app.
- Implemented EmotionPicker component for selecting emotions with child options.
- Created hooks for fetching emotions, feeds, and rapport data.
- Established API service for handling requests and authentication.
- Integrated Zustand for state management of user authentication.
- Defined color palette for the application.
- Set up TypeScript types for various entities including User, Emotion, and Feed.
- Configured Babel and TypeScript for the mobile project.
- Added package.json with necessary dependencies for Expo and React Native.

mobile/app/_layout.tsx

@@ -0,0 +1,56 @@
+import { useEffect } from 'react';
+import { Stack, useRouter, useSegments } from 'expo-router';
+import { StatusBar } from 'expo-status-bar';
+import { QueryClient, QueryClientProvider } from '@tanstack/react-query';
+import { useAuthStore } from '@/lib/auth-store';
+import { View, ActivityIndicator } from 'react-native';
+
+const queryClient = new QueryClient();
+
+function AuthGuard({ children }: { children: React.ReactNode }) {
+  const { isAuthenticated, isLoading, loadToken } = useAuthStore();
+  const segments = useSegments();
+  const router = useRouter();
+
+  useEffect(() => {
+    loadToken();
+    // eslint-disable-next-line react-hooks/exhaustive-deps
+  }, []);
+
+  useEffect(() => {
+    if (isLoading) return;
+
+    const inAuthGroup = segments[0] === 'auth';
+
+    if (!isAuthenticated && !inAuthGroup) {
+      router.replace('/auth/login');
+    } else if (isAuthenticated && inAuthGroup) {
+      router.replace('/(tabs)');
+    }
+    // eslint-disable-next-line react-hooks/exhaustive-deps
+  }, [isAuthenticated, isLoading, segments]);
+
+  if (isLoading) {
+    return (
+      <View style={{ flex: 1, justifyContent: 'center', alignItems: 'center', backgroundColor: '#fff' }}>
+        <ActivityIndicator size="large" color="#fce117" />
+      </View>
+    );
+  }
+
+  return <>{children}</>;
+}
+
+export default function RootLayout() {
+  return (
+    <QueryClientProvider client={queryClient}>
+      <AuthGuard>
+        <Stack screenOptions={{ headerShown: false }}>
+          <Stack.Screen name="auth" />
+          <Stack.Screen name="(tabs)" />
+        </Stack>
+      </AuthGuard>
+      <StatusBar style="auto" />
+    </QueryClientProvider>
+  );
+}