Add IME hotkey command with menu-key migration and conflict detection

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_01QKxGW1raAWaqeU8WdHsMsp

bin/omarchy-cn-doctor

@@ -11,7 +11,7 @@ module="all"
 fix="false"
 for arg in "$@"; do
   case "$arg" in
-  all | network | mirror | dev-mirror) module="$arg" ;;
+  all | network | mirror | dev-mirror | ime) module="$arg" ;;
   --fix) fix="true" ;;
   *)
     echo "Unknown argument: $arg" >&2
@@ -73,14 +73,20 @@ check_dev_mirror() {
   omarchy-cn-dev-mirror-doctor || failures=$((failures + 1))
 }
 
+check_ime() {
+  omarchy-cn-ime-status || failures=$((failures + 1))
+}
+
 case "$module" in
 network) check_network ;;
 mirror) check_mirror ;;
 dev-mirror) check_dev_mirror ;;
+ime) check_ime ;;
 all)
   check_network
   check_mirror
   check_dev_mirror
+  check_ime
   ;;
 esac