fix(teacher): group day slots by real lesson start time

slotIndex is a per-department index of non-empty intervals, not a time.
Merging all departments of a period by slotIndex and deriving the slot
time from the first inserted lesson produced shifted times (e.g. Meltsov
10:00 shown at 11:45) depending on sync order. Group by lesson start
minute instead so each department's real times are kept.

Add debug logging for teacher schedule sync/build and debounce teacher
search in picker.

app/src/main/java/agzam/vyatsu/ui/picker/PickerViewModel.kt

@@ -18,6 +18,7 @@ import kotlinx.coroutines.Dispatchers
 import kotlinx.coroutines.flow.MutableStateFlow
 import kotlinx.coroutines.flow.StateFlow
 import kotlinx.coroutines.flow.asStateFlow
+import kotlinx.coroutines.flow.debounce
 import kotlinx.coroutines.launch
 import kotlinx.coroutines.withContext
 
@@ -160,16 +161,18 @@ class PickerViewModel(
 
         teachersJob?.cancel()
         teachersJob = viewModelScope.launch {
-            teacherScheduleRepository.observeTeachers(period).collect { teachers ->
-                val query = _uiState.value.teacherQuery
-                val isLoading = _uiState.value.teachersLoading
-                _uiState.value = _uiState.value.copy(
-                    teachers = teachers,
-                    filteredTeachers = filterTeachers(teachers, query),
-                    teachersLoading = isLoading && teachers.isEmpty(),
-                    error = null
-                )
-            }
+            teacherScheduleRepository.observeTeachers(period)
+                .debounce(300)
+                .collect { teachers ->
+                    val query = _uiState.value.teacherQuery
+                    val isLoading = _uiState.value.teachersLoading
+                    _uiState.value = _uiState.value.copy(
+                        teachers = teachers,
+                        filteredTeachers = filterTeachers(teachers, query),
+                        teachersLoading = isLoading && teachers.isEmpty(),
+                        error = null
+                    )
+                }
         }
     }