CF1823E.Removing Graph

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 $k$ ($l \le k \le r$; $l \lt r$) 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.

爱丽丝和鲍勃正在一张图上进行一场游戏。他们拥有一张无向图，该图不含自环和重边。图中所有顶点的度数均为 $2$。该图可能由若干个连通分量组成。注意：若这样的图有 $n$ 个顶点，则它恰好有 $n$ 条边。

爱丽丝和鲍勃轮流进行操作，爱丽丝先行。在每一轮中，当前玩家可以选择 $k$（其中 $l \le k \le r$；且 $l < r$）个顶点，这些顶点需构成一个连通子图，并将这些顶点及其所有关联边从图中删除。

无法进行操作的玩家判负。

例如，假设他们在如下图所示的图上进行游戏，且给定 $l = 2$、$r = 3$：

爱丽丝在第一步中可选择的合法顶点集为以下之一：

• ${1, 2}$
• ${1, 3}$
• ${2, 3}$
• ${4, 5}$
• ${4, 6}$
• ${5, 6}$
• ${1, 2, 3}$
• ${4, 5, 6}$

假设爱丽丝选择了子图 ${4, 6}$。

则鲍勃在第一步中可选择的合法顶点集为以下之一：

• ${1, 2}$
• ${1, 3}$
• ${2, 3}$
• ${1, 2, 3}$

假设鲍勃选择了子图 ${1, 2, 3}$。

此时爱丽丝无法再进行任何操作，因此她输掉游戏。

现给定一张大小为 $n$ 的图，以及整数 $l$ 和 $r$。若爱丽丝与鲍勃均采取最优策略，谁将获胜？

The first line contains three integers $n$, $l$ and $r$ ($3 \le n \le 2 \cdot 10^5$; $1 \le l \lt r \le n$) — 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 $2$.

第一行包含三个整数 $n$、$l$ 和 $r$（$3 \le n \le 2 \cdot 10^5$；$1 \le l \lt r \le n$）—— 分别表示图中顶点的数量，以及 Alice 或 Bob 每次操作可选择的顶点数的约束范围。

接下来 $n$ 行描述图中的边：每行一条边。第 $i$ 行包含两个整数 $u_i$ 和 $v_i$（$1 \le u_i, v_i \le n$；$u_i \neq v_i$）—— 表示第 $i$ 条边。

保证所给图中每个顶点的度数均为 $2$。

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

如果 Alice 获胜（不区分大小写），则输出 Alice；否则输出 Bob。

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 $r = 2$.

第一个测试用例展示了图例中的相同输入。

第二个测试用例展示了图例中的相同图，但其中 $l = 1$ 且 $r = 2$。