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.