Removing Graph
Description
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:
- $\{1, 2\}$
- $\{1, 3\}$
- $\{2, 3\}$
- $\{4, 5\}$
- $\{4, 6\}$
- $\{5, 6\}$
- $\{1, 2, 3\}$
- $\{4, 5, 6\}$
Then a valid vertex set for Bob to choose at the first move is one of the following:
- $\{1, 2\}$
- $\{1, 3\}$
- $\{2, 3\}$
- $\{1, 2, 3\}$
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.
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 $2$.
Print Alice (case-insensitive) if Alice wins, or Bob otherwise.
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.
6 2 3
1 2
2 3
3 1
4 5
5 6
6 4
6 1 2
1 2
2 3
3 1
4 5
5 6
6 4
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
Bob
Bob
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 $r = 2$.