feat(mobile): collapsed pill composer and mobile input redesign

Mobile composer redesign: when the keyboard is closed the input collapses
into a narrow pill (sessions, attach, placeholder, mic) with a round
new-session button that fades away on the draft screen. Model and agent
selectors move into a row above the textarea; the draft project/branch
pickers and the attachment menu become searchable bottom sheets reusing
MobileOverlayPanel; a drag handle (also available while dictating) swipes
the composer into and out of a fullscreen mode.

Keyboard-lifecycle hardening: composer controls (agent cycle, dictation
and its overlay controls) no longer steal focus and dismiss the keyboard;
overlays reopen the keyboard on close via a debounced restore chain that
survives menu-to-picker handoffs and skips the native file picker; open
overlays and dictation keep the composer expanded. Dictation starts
directly from the pill and its overlay content fades in after the shape
settles. The keyboard slide compensates the pill-to-full height change in
one motion, and the mobile highlight mirror is disabled so the caret
always matches the text layout.

packages/ui/src/lib/i18n/messages/ko.ts

@@ -1920,6 +1920,9 @@ export const dict: Record<I18nKey, string> = {
   'chat.chatInput.previewContextRemove': '미리보기 컨텍스트 제거',
   'chat.chatInput.projectRoot': '프로젝트 루트',
   'chat.chatInput.branch': '브랜치',
+  'chat.chatInput.draftPicker.projectTitle': '프로젝트',
+  'chat.chatInput.draftPicker.searchProjects': '프로젝트 검색...',
+  'chat.chatInput.draftPicker.searchBranches': '브랜치 검색...',
   'chat.chatInput.worktrees': '워크트리',
   'chat.chatInput.worktreeNew': '+ 새로 만들기',
   'chat.chatInput.drop.insertMention': '여기에 놓아 멘션으로 추가',