E. Removing Graph
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

Alice and Bob are playing a game on a graph. They have an undirected graph without self-loops and multiple edges. All vertices of the graph have degree equal to $$$2$$$. The graph may consist of several components. Note that if such graph has $$$n$$$ vertices, it will have exactly $$$n$$$ edges.

Alice and Bob take turn. Alice goes first. In each turn, the player can choose $$$k$$$ ($$$l \le k \le r$$$; $$$l < r$$$) vertices that form a connected subgraph and erase these vertices from the graph, including all incident edges.

The player who can't make a step loses.

For example, suppose they are playing on the given graph with given $$$l = 2$$$ and $$$r = 3$$$:

A valid vertex set for Alice to choose at the first move is one of the following:

  • $$$\{1, 2\}$$$
  • $$$\{1, 3\}$$$
  • $$$\{2, 3\}$$$
  • $$$\{4, 5\}$$$
  • $$$\{4, 6\}$$$
  • $$$\{5, 6\}$$$
  • $$$\{1, 2, 3\}$$$
  • $$$\{4, 5, 6\}$$$
Suppose, Alice chooses subgraph $$$\{4, 6\}$$$.

Then a valid vertex set for Bob to choose at the first move is one of the following:

  • $$$\{1, 2\}$$$
  • $$$\{1, 3\}$$$
  • $$$\{2, 3\}$$$
  • $$$\{1, 2, 3\}$$$
Suppose, Bob chooses subgraph $$$\{1, 2, 3\}$$$.

Alice can't make a move, so she loses.

You are given a graph of size $$$n$$$ and integers $$$l$$$ and $$$r$$$. Who will win if both Alice and Bob play optimally.

Input

The first line contains three integers $$$n$$$, $$$l$$$ and $$$r$$$ ($$$3 \le n \le 2 \cdot 10^5$$$; $$$1 \le l < r \le n$$$) — the number of vertices in the graph, and the constraints on the number of vertices Alice or Bob can choose in one move.

Next $$$n$$$ lines contains edges of the graph: one edge per line. The $$$i$$$-th line contains two integers $$$u_i$$$ and $$$v_i$$$ ($$$1 \le u_i, v_i \le n$$$; $$$u_i \neq v_i$$$) — description of the $$$i$$$-th edge.

It's guaranteed that the degree of each vertex of the given graph is equal to $$$2$$$.

Output

Print Alice (case-insensitive) if Alice wins, or Bob otherwise.

Examples
Input
6 2 3
1 2
2 3
3 1
4 5
5 6
6 4
Output
Bob
Input
6 1 2
1 2
2 3
3 1
4 5
5 6
6 4
Output
Bob
Input
12 1 3
1 2
2 3
3 1
4 5
5 6
6 7
7 4
8 9
9 10
10 11
11 12
12 8
Output
Alice
Note

In the first test the same input as in legend is shown.

In the second test the same graph as in legend is shown, but with $$$l = 1$$$ and $$$r = 2$$$.