Prepare sc!, sv!, pp!, and dg! with fast I/O for the supplied reader and writer.
prepare_io!(buffered; reader, writer) reads incrementally for large numeric inputs.
Every read must satisfy FastInput’s token requirements.
vertices; tree, root, paths; |u| add, |u| remove, |i| answer, or edges; ....
paths yields (u, v). Callbacks receive vertex or original edge IDs and input query IDs.
Start with empty state; answer preserves it. The final state is unspecified.
Path elements are unordered; answers are visited in Mo order.