A8399 | Edges in MST

You are given a connected weighted undirected graph without any loops and multiple edges.

Let us remind you that a graph's spanning tree is defined as an acyclic connected subgraph of the given graph that includes all of the graph's vertexes. The weight of a tree is defined as the sum of weights of the edges that the given tree contains. The minimum spanning tree (MST) of a graph is defined as the graph's spanning tree having the minimum possible weight. For any connected graph obviously exists the minimum spanning tree, but in the general case, a graph's minimum spanning tree is not unique.

Your task is to determine the following for each edge of the given graph: whether it is either included in any MST, or included at least in one MST, or not included in any MST.

The first line contains two integers $n$ and $m$ ( $2<=n<=10^{5}$ , ) — the number of the graph's vertexes and edges, correspondingly. Then follow $m$ lines, each of them contains three integers — the description of the graph's edges as " $a_{i}$ $b_{i}$ $w_{i}$ " ( $1<=a_{i},b_{i}<=n,1<=w_{i}<=10^{6},a_{i}≠b_{i}$ ), where $a_{i}$ and $b_{i}$ are the numbers of vertexes connected by the $i$ -th edge, $w_{i}$ is the edge's weight. It is guaranteed that the graph is connected and doesn't contain loops or multiple edges.

Print $m$ lines — the answers for all edges. If the $i$ -th edge is included in any MST, print "any"; if the $i$ -th edge is included at least in one MST, print "at least one"; if the $i$ -th edge isn't included in any MST, print "none". Print the answers for the edges in the order in which the edges are specified in the input.