Fix rounds management bugs and invitation flow

- Fix rounds list showing 0 projects by adding _count to program.list query
- Fix round reordering by using correct cache invalidation params
- Fix finalizeResults to auto-advance passed projects to next round
- Fix member list not updating after add/remove by invalidating user.list
- Fix invitation link error page by correcting path from /auth-error to /error
- Add /apply, /verify, /error to public paths in auth config

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

src/app/(admin)/admin/rounds/page.tsx

@@ -161,17 +161,19 @@ function RoundRow({
 
   const reorder = trpc.round.reorder.useMutation({
     onSuccess: () => {
-      utils.program.list.invalidate()
+      utils.program.list.invalidate({ includeRounds: true })
     },
   })
 
   const moveUp = () => {
+    if (index <= 0) return
     const ids = [...allRoundIds]
     ;[ids[index - 1], ids[index]] = [ids[index], ids[index - 1]]
     reorder.mutate({ programId, roundIds: ids })
   }
 
   const moveDown = () => {
+    if (index >= totalRounds - 1) return
     const ids = [...allRoundIds]
     ;[ids[index], ids[index + 1]] = [ids[index + 1], ids[index]]
     reorder.mutate({ programId, roundIds: ids })
@@ -179,14 +181,14 @@ function RoundRow({
 
   const updateStatus = trpc.round.updateStatus.useMutation({
     onSuccess: () => {
-      utils.program.list.invalidate()
+      utils.program.list.invalidate({ includeRounds: true })
     },
   })
 
   const deleteRound = trpc.round.delete.useMutation({
     onSuccess: () => {
       toast.success('Round deleted successfully')
-      utils.program.list.invalidate()
+      utils.program.list.invalidate({ includeRounds: true })
     },
     onError: (error) => {
       toast.error(error.message || 'Failed to delete round')