Update homepage routes to include language and mode

apps/homepage/src/components/auth/AuthShell.tsx

@@ -1,7 +1,8 @@
 'use client'
 
-import { localizedPath, type Locale } from '@/lib/localization/config'
+import { localizedModePath, modeFromPathname, type Locale } from '@/lib/localization/config'
 import Image from 'next/image'
+import { usePathname } from 'next/navigation'
 import type { ReactNode } from 'react'
 import styles from './AuthShell.module.css'
 
@@ -13,6 +14,7 @@ interface AuthShellProps {
 }
 
 export function AuthShell({ locale, title, subtitle, children }: AuthShellProps) {
+  const mode = modeFromPathname(usePathname())
   return (
     <main
       id="main-content"
@@ -21,7 +23,7 @@ export function AuthShell({ locale, title, subtitle, children }: AuthShellProps)
     >
       <div className={styles.container}>
         <div className={styles.header}>
-          <a href={localizedPath('home', locale)}>
+          <a href={localizedModePath('home', locale, mode)}>
             <Image
               src="/rentaldrivego.png"
               alt="RentalDriveGo"