AT_abc398_g [ABC398G] Not Only Tree Game

You are given a simple undirected graph with $ N $ vertices and $ M $ edges, with vertices labeled $ 1 $ to $ N $ and edges labeled $ 1 $ to $ M $ . The $ i $ -th edge connects vertices $ U_i $ and $ V_i $ . Initially, $ G $ does not contain an odd cycle. Takahashi and Aoki will play a game using this graph $ G $ . With Aoki going first, they take turns performing the following operation: - Choose a pair of integers $ (i,j) $ with $ 1 \leq i < j \leq N $ that satisfies both of the following conditions, then add an edge connecting vertices $ i $ and $ j $ to $ G $ . - $ G $ does not already have an edge connecting vertices $ i $ and $ j $ . - Adding an edge connecting vertices $ i $ and $ j $ does not create an odd cycle. A player who cannot perform this operation loses, and the other player wins. Determine who wins when both players play optimally. What is an odd cycle?A sequence of vertices $ (v_0,v_1,\ldots,v_k) $ of $ G $ is called an odd cycle if and only if all of the following conditions are satisfied: - $ k $ is odd. - $ v_0=v_k $ . - For every $ 1\leq i \leq k $ , there is an edge connecting $ v_{i-1} $ and $ v_{i} $ .

The input is given from Standard Input in the following format: > $ N $ $ M $ $ U_1 $ $ V_1 $ $ U_2 $ $ V_2 $ $ \vdots $ $ U_M $ $ V_M $

If Aoki (the first player) wins, print `Aoki`; otherwise, if Takahashi (the second player) wins, print `Takahashi`.

### Sample Explanation 1 If Aoki (the first player) adds the edge $ (1,4) $ , Takahashi (the second player) cannot move. Thus, Aoki wins. ### Sample Explanation 2 No matter how Aoki plays, Takahashi wins. ### Constraints - $ 1 \leq N \leq 2\times 10^5 $ - $ 0 \leq M \leq 2\times 10^5 $ - $ 1 \leq U_i < V_i \leq N $ - The given graph does not contain an odd cycle. - The given graph does not contain multi-edges. - All input values are integers.