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/server/routers/program.ts

@@ -21,7 +21,12 @@ export const programRouter = router({
             select: { rounds: true },
           },
           rounds: input?.includeRounds ? {
-            orderBy: { createdAt: 'asc' },
+            orderBy: { sortOrder: 'asc' },
+            include: {
+              _count: {
+                select: { roundProjects: true, assignments: true },
+              },
+            },
           } : false,
         },
       })