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D. Edges in MST

time limit per test

2 secondsmemory limit per test

256 megabytesinput

standard inputoutput

standard outputYou 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.

Input

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.

Output

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.

Examples

Input

4 5

1 2 101

1 3 100

2 3 2

2 4 2

3 4 1

Output

none

any

at least one

at least one

any

Input

3 3

1 2 1

2 3 1

1 3 2

Output

any

any

none

Input

3 3

1 2 1

2 3 1

1 3 1

Output

at least one

at least one

at least one

Note

In the second sample the MST is unique for the given graph: it contains two first edges.

In the third sample any two edges form the MST for the given graph. That means that each edge is included at least in one MST.

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