feat: 강아지 등록 중성화 필수 선택 + AI 추천 중성화 조건 옵션

- 등록 폼: 중성화 여부(완료/안 함) 필수 토글, 미선택 시 등록 불가
- AI 추천: 중성화 찾기 옵션(상관없음/완료/안 함) → 조건 강아지 중에서만 추천

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

src/pages/ProfilePage.vue

@@ -102,6 +102,22 @@
             emit-value
             map-options
           />
+          <div>
+            <div class="text-caption text-grey-7 q-mb-xs">중성화 여부 <span class="text-negative">*</span></div>
+            <q-btn-toggle
+              v-model="form.neutered"
+              spread
+              unelevated
+              no-caps
+              toggle-color="primary"
+              color="grey-3"
+              text-color="grey-8"
+              :options="[
+                { label: '중성화 완료', value: true },
+                { label: '안 함', value: false },
+              ]"
+            />
+          </div>
         </q-card-section>
         <q-card-actions align="right">
           <q-btn flat label="취소" v-close-popup />
@@ -110,7 +126,7 @@
             unelevated
             label="등록"
             :loading="creating"
-            :disable="!form.name"
+            :disable="!form.name || form.neutered === null"
             @click="onCreate"
           />
         </q-card-actions>
@@ -141,10 +157,16 @@ const sizeOptions = [
   { label: '중형견', value: 'MEDIUM' },
   { label: '대형견', value: 'LARGE' },
 ]
-const form = reactive<{ name: string; breed: string; size: string | null }>({
+const form = reactive<{
+  name: string
+  breed: string
+  size: string | null
+  neutered: boolean | null
+}>({
   name: '',
   breed: '',
   size: null,
+  neutered: null,
 })
 
 onMounted(async () => {
@@ -208,13 +230,20 @@ function openCreate() {
   form.name = ''
   form.breed = ''
   form.size = null
+  form.neutered = null
   createOpen.value = true
 }
 
 async function onCreate() {
+  if (form.neutered === null) return
   creating.value = true
   try {
-    await dogs.createDog({ name: form.name, breed: form.breed || null, size: form.size })
+    await dogs.createDog({
+      name: form.name,
+      breed: form.breed || null,
+      size: form.size,
+      neutered: form.neutered,
+    })
     createOpen.value = false
     $q.notify({ color: 'primary', message: `${form.name} 등록 완료!`, icon: 'pets', position: 'top' })
   } catch (e) {