fix signin signout and apply the new style

apps/dashboard/src/lib/dashboardPaths.ts

@@ -0,0 +1,19 @@
+const DASHBOARD_BASE_PATH = '/dashboard'
+
+export function toDashboardAppPath(path?: string | null): string {
+  const value = (path ?? '').trim()
+  if (!value || value === '/') return '/'
+
+  let normalized = value.startsWith('/') ? value : `/${value}`
+
+  while (normalized === DASHBOARD_BASE_PATH || normalized.startsWith(`${DASHBOARD_BASE_PATH}/`)) {
+    normalized = normalized.slice(DASHBOARD_BASE_PATH.length) || '/'
+  }
+
+  return normalized
+}
+
+export function toPublicDashboardPath(path?: string | null): string {
+  const appPath = toDashboardAppPath(path)
+  return appPath === '/' ? DASHBOARD_BASE_PATH : `${DASHBOARD_BASE_PATH}${appPath}`
+}