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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣥⣶⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆⣀⡤⠤⠴⣶⠐⠻⠛⠿⣅⣶⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣴⣤⠖⠻⣿⣦⠀⠰⣶⠦⠀⠀⠄⠠⠀⠉⠓⢄⡀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⣛⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠲⡈⠛⠉⠁
⠀⠀⠀⠀⠀⡀⠀⠰⠀⢰⣊⣄⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⣼⣄⠀⠃
⠀⠀⠀⠀⠀⢿⣷⡀⢰⠛⠛⠉⠁⠀⢀⡄⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⡆
⠀⠀⠀⠀⠀⠈⠙⠀⠇⣀⠈⠁⠀⠀⢸⡇⢰⠄⢸⣴⢼⢠⠢⡠⣤⢤⢄⣤⠄⠀⠀⠈⠁⡶⢸⣶⠶⠄
⠀⠀⠀⠀⠀⢰⡄⢸⠟⠋⠁⠀⠀⠀⠀⢧⣸⣀⠎⠙⠋⠈⠛⠙⠉⠈⠈⠙⠀⠀⠀⠀⠀⠀⠸⡿⠂
⠀⠀⠀⠀⠀⢸⣿⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⣇⡤⠀⣰⣤⡤⣄⠀⠀⠀⠀⠀⣀⣤⡧⣄
⠀⠀⠀⠀⣀⣈⠻⠿⡄⠐⣀⠀⠀⠀⠀⢀⣸⣀⡇⢀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠚⠛⠋⡇⠈
⠀⠀⠀⠀⠈⠻⣿⣶⣧⠀⡁⠀⠀⠀⠀⠈⢹⠉⡇⣍⡷⢸⢹⠉⡇⣗⣡⠆⠀⠀⠀⣰⣶⠀⡘
⠀⠀⠀⠀⠀⠀⠬⠛⠿⢿⡇⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢠⡄⠀⠘⠣⡀⠁⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⢀⣼⣿⣀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⡷⠀⠀⠀⠀⠀⣀⠀⠀⡀⢀⡀⢴⠀⠀⢀⣴⡁⣿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠑⢲⣶⣤⡻⠀⠻⣷⣄⠁⢀⣙⠷⠖⢪⣿⡿⠃⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠁⠉⠉⠉⠉⠉⣨⣄⠀⠐⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡿⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⠙⣿⡿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢰⣿⠿⠿⣿⣿⢿⣿⣿⢿⡏⢹⡟⣉⢻⣿⠟⣋⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣯⢸⣷⡌⣿⢸⣿⡇⢸⣿⣿⠇⢡⣾⡏⢘⣛⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣌⣛⣁⣿⣄⠛⡁⢿⣿⣿⣀⣄⢛⣃⠸⠿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢻⣿⠟⣿⠛⣿⠟⡛⠿⣿⢿⣿⣿⠿⢛⠻⣿⠟⢛⣛⠻⣿⣿⡿⣿⢿⣿⣿⣿⠿⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠘⡟⠀⠏⣸⠁⣘⣉⣤⣿⢸⣿⠁⣾⣿⣷⡟⢠⣿⣿⣧⢸⡇⢠⠀⣶⢘⣿⠃⠾⠿⢸⣿⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣼⣧⣴⣿⣦⣭⣭⣼⣇⣘⣱⡄⠻⠿⢛⣷⣈⣛⣛⣵⣾⣇⣿⣸⣿⠸⢛⡀⢿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⣀⣤⡆⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢹⣿⠁⣸⡟⠀⣠⡶⠲⡄⣠⠄⢠⣴⠀⠀⢀⣠⣤⣴⡄⠀⠀⠀⠀⠒⠤⡤⠤⠒
⠀⠀⢸⣿⡴⣿⡇⠀⢿⣇⣠⠇⢿⠀⣼⣿⡀⠀⠀⣻⠏⠛⢁⣾⣛⣷⡀⠀⠀⠂
⠀⠀⢀⣀⣰⡟⠀⠀⠀⠈⠁⠀⠘⠉⠈⠉⠀⠀⠠⠿⠂⠀⠈⠿⠯⠥
⠀⠰⣷⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⣴⣤⠀⠊⠉⠉⠑⢮⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡴⠞⠁⠉⠁⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣶⣄⠀⠀⠀⠀⠀⡠⠂⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢌⠿⡿⣸⢄⠀⠀⠊⣀⠔⠁⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⢀⠋⠉⠉⢀⠠⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠀⠐⡁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠚⠁⢢⡘⡔⠊⠉⢢
⠀⠀⠀⠀⠀⠀⣠⡴⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣴⢸⡇⠂⠀⠀⢱⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀⠀⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⡿⠛⣿⣿⣿⣿⣷⣾⣿⠿⣿⣿⣿⠛⢿⣿⣦⣴⣿⡿⢿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠸⣿⡏⠉⠘⣿⠁⢸⠟⣥⠴⣴⣈⠻⣿⠀⣿⠀⠈⣆⡏⠉⠉⠻⣿⡇⠀⠈⠉⢻⣿⡆⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣛⢿⣿⠿⣿⣿⣿⡄⠀⠁⠀⡏⣸⠺⢋⢽⠌⡄⠻⠀⣿⠀⠀⣿⣥⠶⠦⡀⠈⡿⠀⠠⣦⣼⣿⣷⣿⠿⣿⢟⠻⣿⡆
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡐⠗⣷⡧⢧⣿⣿⣿⣿⡄⠀⢸⢀⡋⠇⠓⠂⢈⠷⢀⠀⢹⠀⠀⣿⣿⠀⠀⠇⠀⡀⠀⠀⠀⣿⣿⣿⣿⡶⣾⡆⢇⣿⣿⣄
⠀⠀⠀⠀⠀⢰⣿⡯⣶⠶⠾⣷⣽⣷⣾⣷⣿⣿⡇⠀⢸⡆⣤⠘⠁⠀⣹⠃⣾⠀⠘⠀⠀⣿⡯⠀⠀⠀⢰⣿⠀⠀⣿⣿⣿⣶⣯⣾⣷⡿⠶⢶⢽⣿⡆
⠀⠀⠀⠀⠀⠘⢿⣷⡏⡾⣽⣷⣾⣷⣿⣷⣿⣿⡇⠀⠘⣿⣄⠓⠖⠒⢉⣼⣿⣆⠀⠀⣰⣿⡇⠀⢰⠀⠸⣋⠀⠀⠀⣿⣿⣿⣿⠟⠷⢾⡵⢸⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡏⠉⢹⡟⢻⠇⠀⢠⣧⣤⣤⣿⠋⠉⣶⣿⡿⠋⠛⣿⣿⣿⡿⠿⠿⢶⣿⣦⣄⣿⣿⣿⣿⡿⠿⠿⢿⠀⠀⣸⣷⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⣸⠀⠀⣿⡿⠁⠀⠀⢈⣿⢋⡴⢛⠳⣆⢻⡟⠛⠻⠉⢹⣇⠀⠀⣀⣸⠇⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⣸⠃⠀⠛⣿⠀⠀⣿⡇⠀⠀⣶⣾⡇⢀⡞⣩⣱⢸⡆⡇⠀⠀⠀⠈⡿⠀⠀⠛⢿⡆⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⣿⡆⠀⣶⣿⠀⢀⣿⡇⠀⠀⠻⠻⡇⢸⡇⢏⣉⡼⠀⠁⠀⠀⠀⠀⢻⡀⠀⣤⣼⣧⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢀⣇⣀⣠⣟⣁⠀⢈⡿⠀⠀⠁⢰⡄⠀⠀⢀⣶⡈⢿⣤⡭⢃⣼⠀⢀⣆⣰⡀⢸⡇⠀⠉⢻⣧⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣾⣿⠿⠿⠿⠿⠿⣿⣷⣤⣤⣤⣼⣿⣦⣤⣾⡿⣿⣶⣤⣶⣿⣧⣤⣼⣿⠿⣿⣿⣤⣤⣤⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠛⠋⠉⠁⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣀⠀⠀⢠⠤⡀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢸⡇⠀⢹⡇⠀⣰⠆⢸⢀⣿⠃⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣼⢀⠀⠘⣿⠰⣿⠀⡸⠸⣧⠎⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠠⢸⡇⠀⠙⢚⣁⠀⠀⠀⢿⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣼⠃⣠⠤⠜⠪⡧⣤⠀⡤⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢤⠀⠀⢠⠎⠀⠀⢰⡟⢰⣇⢴⡆⢀⣿⡇⣸⡧⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⠀⠀⠘⣇⠀⢣⠀⣞⠀⠀⢠⡟⠀⠀⠀⢸⣧⠎⠘⠂⠈⠛⠒⠃⠀⠀⢀⣀⣠⡤⠤⠶⠶⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢸⡆⠀⠀⠈⠳⣌⡇⠙⠦⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠐⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⣴⡄⠈⠹⠓⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢸⣷⠀⢰⢹⣇⠀⠀⡄⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠰⡒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⡞⢸⣿⠀⠀⠃⠀⢠⡶⢢⠀⣾⠃⢀⣴⠦⠀⢠⡌⢻⠀⡶⠀⢴⠆⠰⠶⢀⠶⠶⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠸⣿⠀⠀⢸⢠⡟⠀⠁⣸⡏⠀⣿⠃⢀⣴⣿⠁⡜⢸⣿⡞⣿⡼⣹⡏⣿⠇⠚⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣴⡇⠀⢿⡆⠀⡼⠸⣇⡠⠊⣿⠀⡸⢿⠴⠋⠸⣧⡴⠃⠸⠿⠘⠿⠁⣿⠁⠻⢧⣤⠴⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠻⠶⠃⠀⠀⠀⠀⢿⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⡼⣫⡶⣦⡄⣴⡆⢠⣶⠀⢫⣤⡴⣦⣠⡴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢰⣿⠀⣿⡇⣿⡇⢸⣿⠀⢰⣿⠀⠰⣿⠷⠞⠃
⠀⠀⠀⠀⣀⣀⡀⣀⣀⡐⠛⣓⡀⠙⠚⢋⣠⡝⠛⠉⠋⠀⠘⠛⠀⠀⠙⠓⠋⠁
⠀⠀⠀⠀⢹⣿⠀⢸⣿⡅⢀⡟⣡⡴⣦⡄⣿⡇⣠⡤⢦⡄⣠⡴⢦⡄⣤⣄⣤⣦⣤⣴⡄⢀⡤⢶⣤
⠀⠀⠀⠀⠀⣿⣧⠏⢿⣧⡞⢸⣿⠴⠟⢹⣿⢰⣿⠁⠈⠡⣿⠁⣸⡗⣾⡇⢸⣿⠀⣿⠇⣿⡧⠾⠋
⠀⠀⠀⠀⠀⠙⠋⠀⠘⠛⠀⠈⠛⠒⠋⠘⠛⠀⠙⠓⠚⠀⠙⠒⠋⠀⠛⠁⠘⠋⠀⠛⠃⠙⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠤⠤⠤⠴⠚⠃⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠬⡠⠄⠒⠀⠀⠀⠠⠒⠐⠒⠀⠄⠀⠀⠈⠲⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠄⠐⡁⡰⠁⠀⣀⠀⣄⢀⡀⢀⡀⣀⠀⢄⣄⣀⢀⣀⠐⡄⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢿⣀⣿⣿⢁⢾⡇⣿⡄⢸⡟⠈⣿⠊⠀⡠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢄⢃⠡⡀⠐⠄⠀⢙⡟⢈⡉⡈⢉⠙⣃⢀⡋⠀⠙⠂⢠⠃⠐⢡⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢄⡊⠀⠀⠉⡢⠄⡀⠈⠁⢸⠀⡟⣸⠺⡏⠀⡇⠀⠀⠔⠁⠀⠀⠀⢁⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠑⠀⠀⢠⡸⠀⣇⡿⢠⣸⠀⡇⢀⠀⠔⠄⠂⠤⠾⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⣛⡐⣒⢒⢒⠒⠒⢊⡗⣒⠛⣀⣑⠒⢒⡖⠒⣲⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡧⣿⣿⢿⣼⡇⠀⣿⡐⣟⢸⣿⡾⠀⢾⣡⢸⡗⠀⣿⡉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⡼⠀⠀⠈⠀⠀⢸⡏⠀⠀⠀⠀⣸⠇⢀⣧⠼⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠐⠒⣶⠖⠒⣴⡒⠀⢠⣶⡄⠀⠀⠿⠀⣸⠷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⣾⠤⣿⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⢿⡁⠀⢻⣇⠀⡟⠀⢹⣇⣠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⢠⠖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀⢰⠋⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⣀⡀⢸⢇⣤⣄⠀⢠⡾⠋⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⢦⣷⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡄⠀⢹
⠀⠀⠀⠀⠀⢠⠤⠤⠤⠾⠦⠴⠦⠶⠷⠽⠭⠯⠿⠽⠿⠷⠿⠧⠽⠿⠿⠷⠶⠾⠤⠤⢤
⠀⠀⠀⠀⠀⢸⠀⢀⡆⠀⡞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⢰⠀⢰⢃⡄⠀⡀⡠⠀⠰⢧⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⡆⠀⣸⠘⢠⠸⢠⡃⠀⢠⡃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠈⠀⠇⠈⢀⡤⢤⠁⠀⠠⡁⠈⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⡇⠈⠀⠀⠀⢀⠆⠀⡇⠀⠁⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡞⠀⠘⡇⠀⠀⡜⢅⡎⣰⠋⣰⠁⢠⡇⡷⠆⡴⠍⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡡⠂⠀⠙⠆⠀⠃⠈⠂⠑⠀⠑⠈⠘⠀⠘⠂⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠉⢹⠛⠛⠋⠛⣩⢟⣿⠛⠙⣿⠛⠛⠛⠛⢻⡟⡹⠋⠉⣉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣠⡾⢣⣾⡇⠀⠀⢿⡀⠀⠀⣰⣿⢧⠇⠀⡜⠁⡴
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠁⠀⣾⢸⡆⠀⠀⠀⠙⠒⠻⡟⠋⡸⠀⠀⢀⢀⠃⠀⡸⡠⢪⠀⡄⣰
⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠈⡦⠀⠔⠁⠁⠫⠘⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠈⠓⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⠀⣀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⣀⣴⣶⠂⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠘⢻⣿⡟⠀⢸⣿⠃⠀⢀⣤⡤⢤⡀⠀⣀⠀⢀⣀⣄⠀⠀⠀⠀⢀⡀⣀⣤⡀⠀⠀⠀⠀⠈⠢⢤⣀⣀⣀⣀⡤⠔⠁
⠀⠀⠀⣾⣿⠃⣠⣿⣿⠀⢠⣾⡏⠀⢈⡇⢸⣿⠀⢠⣿⡇⠀⠀⠀⠚⣿⡟⢿⡿⠃⢀⣴⣴⣄⠀⠀⠀⠀⡉⠁
⠀⠀⠀⠻⣿⡷⢟⣿⡏⠀⠈⠻⢧⣤⠾⠁⠸⣿⢠⡼⣿⣦⠀⠀⠀⠀⣿⡇⠀⠀⢠⣿⣿⣨⣿⡇⠀⠀⠀⡇
⠀⠀⠀⢀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠤⠿⠇⠀⠀⠈⢿⣿⣤⣤⠄⠀⠀⠀⡇
⠀⠀⣰⣫⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠙⠛⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⣾⣿⣶⡀⠈⠀⠀⠀⠀⠈⠳⣖⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣹⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢀⣤⠾⠛⠛⠋⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⢸⣿⣷⣌⢂⠀⠀⠀⠀⠀⠀⡰⠊⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠹⡌⢿⣿⣿⠀⣇⡀⠀⠀⠠⠊⠀⣀⠌⠐⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈⠢⠬⠥⠞⠀⢹⠦⠶⠦⠔⠋⠁⢀⣀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⡠⠒⠉⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢠⠼⠃⠢⡄⠹⢀⠴⠊⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣦⠁⡀⠀⠀⠘⢆⡋⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠉⠀⣀⠠⣤⣤⣤⠄⠀⠀⠀⠀⠀⢀⣠⣴⣾⣉⣿⡀⠥⠀⠀⠀⢸⣇⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣠⣾⣿⡆⢰⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁⣿⡿⠀⢀⣱⣦⣤⣦⠔⢫⣾⠾⢷⣤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⣿⣷⡿⠁⠀⠀⣰⣿⠿⣦⡀⠀⠀⣿⡇⠀⣾⣿⠃⠾⠃⠠⣿⡟⠀⢀⡟⠠⠶⣿⡷⢺⣿⠒⣿⡇⢀⣴⡿⣶⣄⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢿⣿⠁⠀⠀⢰⣿⣷⠶⠿⠃⠀⠀⣿⡇⠀⢿⣿⣀⣀⡀⠀⠙⠳⠶⠛⠀⠀⠀⣿⡇⢨⣿⢀⣿⡇⣾⣿⠷⠾⠛⠀⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠶⠾⠂⠀⠴⠿⠛⠂⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠇⠚⠛⠀⠛⠁⠘⠿⠷⠶⠖⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⡄⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠰⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣤⣐⣠⣭⣤⣤⡶⠒⠒⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢠⠃⠀⠀⡜⢁⠇⠀⣠⣄⠀⠤⠤⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡎⢀⠀⣠⡄⣤⠴⡄⢀⢆⠃⠀⣠⠀⢀⡎⠀⠀⣸⠶⠋⠘⣹⠋⣠⡀⣠⣤⢤⣤⢤⣄⡠⠤⣀⠀⣀⡠⠔⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠺⠕⠛⠊⠉⠉⠀⠀⠀⠀⢸⠀⡰⡟⢺⠟⡠⠐⠊⡝⠳⠞⠖⠃⠐⠉⠉⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣺⡿⠿⠷⠛⠿⠿⠿⣫⣾⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣵⢯⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢘⣿⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣳⠞⢄⠀⠀⠀⠁⠒⠀⠒⠁⠀⠀⢸⣯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⠀⠀⠁⠠⠄⣀⠀⠀⠀⠀⢀⣠⣾⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢇⠀⠀⡀⠀⠀⠀⠈⡏⠉⢹⡷⢀⡿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣵⣎⠀⠀⠑⠤⡀⠀⠀⡇⠀⣼⠃⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠕⠁⠀⠀⠀⠀⠈⠂⢤⡵⠊⠀⢰⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢦⡠⠖⠋⠀⠀⠀⠀⠀⠈⠑⢤⡜⠉⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠋
⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⢀⠔⡡⠀⠤⠀⢀⠄⠀⠀⠀⢀⡄⠄⠀⢀⠤⠀⠀⠀⠀⢀⠆⠀⠀⠀⠠⠔⡂⠀⡄⠀⠀⠄⠀⠀⢀⡀⠀⢀⣀⡄⠀⠤
⠀⠀⠀⠀⠀⡠⠁⠘⠀⠀⢀⠇⠈⠀⠀⡄⠀⡾⠢⠃⠀⠞⠂⠀⠀⠀⠀⠸⢀⠀⠠⠀⠛⠉⠀⢰⢀⡀⠀⡀⠀⠀⠃⢀⠄⠘⠉⠀⠀⠆
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠂⠁⠀⠈⠀⠈⠀⠀⠃⠀⠂⠈⠁⠀⠀⠀⠀⠀⠘⠉⠈⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⣴⣶⠆⣴⠟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠃
⠀⠀⠀⠀⠀⢀⡴⣿⡿⠁⠀⠲⡔⠚⢛⣋⠀⢰⡏⣠⠖⣒⠉⠉⢛⣒⠒⠒⣒⠒⡚⠛⠛⠛⠉⢁⡀
⠀⠀⠀⠀⢠⣿⢰⣿⠁⢠⣶⠀⢡⣰⡏⡹⢠⣿⠘⢂⣾⠋⠃⣠⡾⣉⡧⢾⣯⢺⣿⡔⣿⠇⣼⡟⡹
⠀⠀⠀⠀⢿⣿⢸⡇⠀⢸⡏⠀⡎⠘⠧⠔⠁⠻⠤⠊⠳⠤⠔⠹⠤⠜⠁⠼⠃⠸⠟⠀⠻⠤⠜⠯⠤⠆
⠀⠀⠀⠀⠀⠀⠘⣇⣠⠃⠳⠜⠀⣀⣠⣤⣴⣶⠶⠶⠞⠛⠛⠓⠒⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣠⡴⠶⠛⠛⠻⠿⣷⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⢠⡿⠋⠀⣀⣠⣤⣀⠀⠀⠉⠙⠻⢿⣿⣶⣤⣀⡀⠀⠀⠀⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⣿⡇⣰⣿⠟⠁⠀⢻⡆⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⢤⡤⠤⠴⠞⠋⠀⠀⠀⠹⣷⣿⡟⠀⠀⣀⡾⠁⠀⠀⠀⠀⣴⠖⠲⣦⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⣀
⠀⠀⠀⣼⣿⠏⠀⢀⣶⡆⠀⢻⡄⠀⣠⣶⡶⢶⣦⢠⣿⣿⠷⠒⠛⠉⣀⣤⣤⣤⡀⢸⣯⣤⠄⢹⣀⢤⣤⠀⡼⠁⢸⣿⢀⡴⠋⣽⣿⢀⣴⣶⠟⣶⡄
⠀⠀⢰⣿⡟⠀⠀⢸⣿⡇⠀⠀⣷⢰⣿⣟⣀⡼⠋⢸⣿⡟⠀⠀⠀⣴⣿⡟⠁⠘⠃⣾⣿⠗⠶⣿⠁⣾⣿⡼⠁⢀⣿⣿⠟⠀⣼⣿⠏⣼⣿⣇⡤⠟⢁⡀
⠀⠀⣿⣿⠃⠀⠀⣼⣿⡇⠀⠀⣿⠘⢿⣧⠀⠀⢀⣽⣿⡇⠀⠀⢀⣿⣿⡇⠀⢀⡼⣿⣿⣀⡼⠃⠰⣿⣿⠃⠀⢸⣿⡏⠀⣼⣿⠏⠀⣻⣿⡇⠀⠀⣸⠇
⠀⠀⠹⣿⡀⢀⡜⠁⠻⠿⠤⠾⠃⠀⠀⠉⠉⠋⠉⠸⣿⣇⠀⢀⡾⠁⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⣿⡟⠀⢠⡿⠉⠙⠒⠋⠁
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣶⡄⠀⠈⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⢿⣿⡇⣀⡾⠁⠀⠀⢀⣠⣤⣤⣤⣀
⠀⢀⣤⣶⣾⣿⣷⣶⣄⠀⠀⢰⣿⣿⡿⠁⣿⣿⡏⢀⣤⣤⣤⣤⡀⠀⠀⢀⣴⡾⣿⣿⣶⡀⠀⢠⣾⠛⣿⣿⣧⠀⠀⠀⠈⡛⠛⠋⠀⠀⢠⣶⣿⡿⠛⠙⢻⣿⣷
⣰⣿⣿⡟⠁⠈⣿⣿⣿⡄⠀⣿⣿⣿⣃⣾⡿⢏⣴⣿⣿⠏⠀⠙⣿⣆⣰⡿⠃⠀⣿⣿⣿⡇⢠⣿⠃⠀⣿⣿⣿⠀⣠⣾⣿⣿⣿⣆⠀⣰⣿⣿⡿⠁⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⣿⣿⣿⡇⢸⣿⣿⣿⠟⠋⠀⣾⣿⣿⣯⣀⣀⣠⣽⣿⠋⠀⠀⢀⣿⣿⣿⢀⣿⠏⠀⢰⣿⣿⣿⣰⣿⠋⢸⣿⣿⣿⢀⣿⣿⣿⠇⠀⠀⠀⢠⣿⣿
⠘⠿⣿⣧⠀⠀⣿⣿⣿⡧⢾⣿⣿⣿⠀⠀⢸⣿⣿⣿⠉⠉⠉⠉⢹⣿⠀⠀⠀⣼⣿⣿⡿⣸⡟⠀⠀⢸⣿⣿⣿⡿⠁⠀⣼⣿⣿⡏⢸⣿⣿⣿⠀⠀⠀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⢸⣿⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⡟⠀⠀⠀⣿⣿⣿⣧⣿⠁⠀⠀⢸⣿⣿⣿⠁⠀⢀⣿⣿⣿⠃⢸⣿⣿⣿⣤⠴⠞⠋⣡⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠈⣿⣿⣿⡀⠀⢸⣿⣿⣧⠀⠀⢀⣼⡿⠃⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⣾⣿⣿⡇⠀⠀⢸⣿⣿⡿⠀⠸⣿⣿⣿⡀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⢻⣿⣿⣇⠀⠀⠻⣿⣿⣦⣶⣿⠟⠁⠀⠀⠀⠀⢿⣿⣿⣟⠀⠀⠀⠀⠿⣿⡿⠃⠀⠀⣿⣿⣿⡇⠀⠀⠙⢿⣿⣷⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⢀⣀⣤⣭⣭⣉⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣀⣀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠉⠙⢻⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠛⠉⠙⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⢿⣿⣿⡆⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋⠀⠈⠻⣿⣿⣦⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⢾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣷⣶⣤⣤⣤⣤⣤⣶⣶⡿⠿⠛⠋⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⡀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷
⠀⠀⠀⠄⣀⣼⣿⣷⠞⠛⠲⢿⣷⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡞⠉⠋⠀⠀⢀⡴⠦⡶⠆
⠀⡀⣰⠋⠀⣠⡄⠐⢲⣷⣦⠀⠀⢳⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢸⣟⢰⠇⠈⠓⡀
⢀⣷⠃⠀⣄⣿⣝⣷⣶⡁⣿⣽⠀⠀⢻⣀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢀⠋⢹⣿⣤⣛⣾⡃
⢸⡇⠀⠀⠹⣿⣛⠏⠈⠀⣄⠃⡰⡶⠋⠉⢱⠇⠀⠀⠀⠀⠀⠀⢠⠙⢦⣀⠀⠀⠀⣀⣈⣿⡷⣽⠛⡇
⠀⢷⠆⠙⣦⠈⠙⠛⠄⠠⡤⣶⠏⢸⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⣦⠀⠀⠀⠈⠿⠆
⠀⠈⠀⠛⠁⣀⣀⠐⢄⣨⠇⠉⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⣿⡀⠀⠀⠀⠀⢀⠀⠀⣀⣴⡄
⠀⠀⠀⠠⡈⠛⠠⡄⠉⠀⢠⠀⠀⠈⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⣿⠉⠀⠀⢀⠀⣴⡃⠀⢿⡽⡀⠹⠂
⠀⠀⠀⠀⣹⣦⡴⣿⣄⠀⢁⠀⠀⢠⣾⡽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⠛⠁⠀⠴⣦⠛⢧⡀⠀⠀⠀⠂
⠀⠀⠀⠈⠉⠁⠐⠋⡉⠁⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢋⣠⠀⠀⠀⠂⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣦⣠⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣿⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢤⡶⠦⣤⣄⣛⣁⣩⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣴⣧⣙⣿⠀⠀⠀⠰⠋⠀⠀⡀
⠀⠀⠀⠀⣹⡾⡇⠀⠋⢹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠏⣼⣿⣿⣿⡇⠀⠀⠀⠿⠶⣢⣿⣄⠀⡀
⠀⠀⠀⠀⢨⠟⡠⣷⢖⣼⡷⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⣿⣿⡝⠋⠹⣆⢀⣤⣤⡼⠋⠀⠀⠉⠁
⠀⠀⠀⣰⠃⣾⢃⠻⣿⣿⡟⢷⣤⠀⠀⢀⢀⢄⠀⠀⠀⠀⠀⠀⠈⢫⠟⠓⠢⠀⠘⠛⠻⠟
⡀⢀⠠⠋⢝⡻⠈⢇⡈⢻⣷⣤⣿⠀⠀⠀⢿⡙⡇⠀⠀⣠⣤⠀⡟⣿⣠⡀⠀⠀⠀⠘⠁⠄
⠈⠉⠀⠀⠈⠀⢀⡺⠃⢀⢻⣿⣌⢧⠀⠀⢀⢻⡇⠀⠀⢛⠖⣠⡇⢿⡟⠛⣀
⠀⠀⠀⠀⠀⠀⣮⠀⢠⣶⣿⣿⣿⢀⠀⢀⡼⢫⠃⠀⠀⠘⣄⣇⠁⣼⣡⢸⠏⠀⠀⠀⠰⣶⡀
⠀⠀⠀⠀⠀⠀⠻⣷⠷⢿⣿⣿⣿⢆⣀⣤⡿⠋⠀⠀⠀⠀⠈⢺⡤⢿⠗⢉⣦⣤⣄⣠⣶⠟⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣬⣤⣬⣥⣶⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⡶⠶⠿⠿⠿⠛⠛⠿⠿⠷⠶⢶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠿⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⢷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⣀⠀⢠⣴⣖⠀⠀⠀⢀⣴⡟⠉⠉⠲⠀⣠⣴⠖⠰⣄⡀⠀⢀⠀⠀⠀⠉⠙⠻⠷⣶⣤⣀
⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⢀⡀⢴⣶⡄⠀⠀⠁⠀⣿⣿⠀⠀⠀⢾⣿⡁⠀⠀⠀⢸⣿⡏⠀⠀⣿⣷⠀⣼⣿⡇⠀⠀⣀⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⣠⡴⠞⠋⠁⠀⠀⢀⣠⣄⠀⠘⡄⠘⣿⣷⠔⠚⠃⠀⢹⣿⡄⠀⢀⠘⠿⣧⡀⢀⡀⠘⢿⣇⠀⢰⣿⠟⢰⠉⣿⣇⡠⣺⣿⠟⠀⣤⡀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠙⣷⡄⠀⢀⣠⣦⡈⠛⣿⣷⣄⡇⠀⠸⣿⣧⣀⠠⠆⠘⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠁⠠⠎⠀⡿⠋⣰⣿⡏⠀⣰⣿⡟⠁⠠⡀⠀⢠⣾⠋
⠀⢀⣠⣬⢿⣆⠈⠙⠻⢿⣷⣼⠉⠻⣧⣤⡄⠉⠁⣀⣠⣤⡴⢶⣾⣿⠉⠉⠀⠀⠀⠀⠉⠉⠛⠓⠶⢦⣤⣄⣀⠀⠀⠐⠻⠏⠀⣼⣿⠟⠤⣀⠀⠈⣰⡿⣥⣄⡀
⣴⣟⠉⠀⠚⠽⣧⡀⠀⠀⠉⠻⠂⠀⣸⣿⡷⠾⠛⠉⠁⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣀⠈⠛⠏⡀⠀⠈⢀⣼⣟⡓⠢⠉⣻
⠉⠙⠳⣦⡀⠀⠙⢷⡀⠀⣀⣤⣶⢿⣿⣿⢷⣄⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⢟⡿⣶⣤⣈⠛⢀⣾⡛⠂⢈⣴⠞⠋
⠀⠀⢀⣾⠋⠀⠀⠀⠻⣾⣯⣉⣈⣠⣿⣏⣀⣙⣦⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⣀⠀⠀⠀⠀⠀⣴⣋⣀⣁⣈⣀⣁⣉⣽⣷⠟⠁⠀⠀⠙⣷⡀
⠀⠀⢸⡇⣀⣤⣤⣤⣤⣤⣤⣉⣉⣹⣿⣋⣉⣽⠿⠃⠀⠀⣾⣿⠁⠀⠀⠀⠀⣤⣦⣾⣿⣿⣿⣿⣿⠄⠀⠀⠘⠿⣯⣉⣉⣉⣉⣉⣉⣤⣤⣤⣤⣤⣤⣀⢸⡇
⠀⠀⠸⠟⠉⠉⠀⠀⠀⠀⠀⠀⠈⣿⡿⠉⠁⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠻⠇
⠀⠀⠀⠀⣤⢠⠆⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⢯⡿⠀⠀⠀⠀⠀⠀⣠⣿⣇⣠⣤⣤⣴⣶⡶⣿⣿⠛⠛⠀⢀⣠⣄⠀⠀⠀⣤⠀⠀⣠⣤⡀⠀⠀⣠⣤⣤⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠠⠄⣧⣾⠗⠀⠀⠸⠿⢿⣿⠟⠛⠋⠉⠁⠀⠀⣿⡇⠀⠀⣰⣿⠃⢹⣇⠀⢸⣿⠀⣼⡿⣿⡇⢀⣾⡿⢻⣿⠀⠀⢀⣾⡿⣿⡆⠀⠀⠘⠁
⠀⠀⠀⡤⣄⡏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⣿⠀⢸⣿⠀⢸⣿⣼⣿⠁⣿⣧⣿⡟⠀⢸⣿⠀⢀⣿⡟⢠⣿⠃⠀⢀⡀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠙⠺⣧⠈⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠰⣿⡿⢦⣼⣿⠀⣾⣿⣿⡏⠀⣿⣿⡟⠀⠀⣿⣿⠀⢸⣿⣤⡿⠁⠀⢀⡾⠀⢠⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠦⠸⣾⡟⠁⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⢀⣿⡏⠀⣿⣿⣿⠁⠀⢿⡿⠁⠀⠀⣿⡿⠀⢸⣿⡄⠀⠀⢀⣾⠁⠀⢠⡈⠁
⠀⠀⠀⠀⢠⡤⣿⠠⠄⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢿⣷⣼⣿⠃⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠻⣿⣦⣶⡿⠁⣄⠂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠙⠛⢧⣰⣿⠆⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠙⠛⠁⠀⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠉⠁⠀⠀⠸⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠆⠻⣿⢠⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠛⣷⡀⠀⠀⣤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣗⣿⣦⣸⣿⠋⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⢠⡴⠛⠉⠉⠉⠙⠛⠶⣤⣀⠀⠀⢸⡇⠀⢀⣿⠃⠀⢸⡇⠀⢸⡇⣴⡗⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠿⣄⠰⠀⣠⡄⠀⠀⢻⣿⣄⠀⢠⡟⠀⢰⣾⠛⠻⣆⠀⠀⠈⠉⠛⠶⠾⣷⡶⠛⠁⠀⠀⠀⣷⣴⠟⠀⠛⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⣷⣦⣽⣿⠀⠀⠈⠻⠟⠀⠀⢷⡀⠈⠛⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣤⣴⠾⠛⠛⠶⠶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠒⠦⠀⠀⠀⠀⠀⠈⠻⢦⣤⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⡇⠀⢀⣴⣶⣆⠀⠀⢠⣄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡞⠛⢷⡀⠀⠉⠋⣠⠟⠀⣰⣿⡿⢃⡟⠀⠀⠈⠉⠀⠀⠀⠀⣾⣧⡈⣷
⠀⠀⠀⠀⠀⢠⡞⠛⣿⣷⡄⠀⠀⠘⠷⣤⣼⣧⣤⡴⠞⠁⠀⣼⣿⣿⢡⡞⠀⠀⠐⠀⣠⠖⠒⠲⣄⠈⠉⢀⡟⠀⠀⣀⣤⣄⠀⣀⣤⣶⣄⠀⠠
⠀⠀⠀⡀⠀⣿⣷⡀⣿⣿⣷⠀⢠⣤⡀⠀⢈⡇⣠⣶⡿⢳⣤⣿⣿⣧⢏⣴⣾⡿⢷⡄⣿⣠⣦⡄⠸⣆⣠⣾⣄⣠⠞⣻⣿⣿⠞⠉⣼⣿⡿⠀⣀⣀
⠀⠀⠀⠀⠀⠙⠛⠀⢿⣿⣿⡀⠈⣿⣿⠀⢸⡇⣿⣿⠗⠋⣸⣿⣿⠁⣾⣿⡟⠀⠿⠇⣼⣿⣿⠓⢚⡏⠁⣿⣿⠋⢠⣿⣿⠋⢀⣼⣿⡿⢣⣾⣿⣋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿⣿⡀⣾⠀⠻⣿⣆⡤⣿⣿⡏⠀⢿⣿⣧⣀⣠⠞⣿⣿⡏⠀⣸⠃⢰⣿⡏⠀⠘⠿⠇⢀⣾⣿⡿⠁⢾⣿⠃⢀⣄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣾⠻⣿⣿⠃⠀⠀⠈⠁⠀⢹⣿⣇⠀⢀⣙⠛⠋⡁⠀⠹⠿⡦⠾⢃⣀⣀⡙⠃⠀⠀⠀⠀⣸⣿⣿⣥⣀⠘⠿⠤⠾⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠋⠁⠀⠈⠁⠀⢀⣀⠀⠈⠀⠈⢿⣿⠀⠿⠋⡇⠀⠁⠀⠀⠠⠀⣴⠋⣁⣬⣿⠀⠀⠀⣀⡴⠟⢿⣇⣠⠝⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠈⠛⠀⠀⠀⠀⠈⠙⠷⠶⠚⠁⠀⠰⠶⠀⠀⠀⣧⠀⠻⠟⢋⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠛⠛⠛⠉⠁⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠀⠉⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠉⠂⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠰⡀⠀⠀⠀⠘⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠙⢦⠀⠀⠙⣤⠀⠀⢀⡤⠒⠉⡛⢦⠀⠀⠤⢄⡀⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⣤⣀⠀⠀⠸⣄⠀⠀⠈⣇⠀⠀⠀⠙⢲⡾⠤⠐⠀⠀⠀⢸⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠒⠒⠒⠲⣤⡀
⠀⠀⠐⠁⠀⣿⠀⠀⠀⠈⠉⠒⠀⢻⠀⠀⢆⠀⡾⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠀⠂⠈⠙⠢⢄⣀⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣿⠀⠀⢠⡄⠀⠀⠀⡼⢀⣠⡀⢸⡇⠂⠀⠀⣀⡀⠀⡴⠚⠳⡄⠀⠀⠀⢤⡄⠀⠈⠁⠀⠀⢀⣀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⢠⢃⡏⠀⠅⢸⠁⠀⢠⡎⠀⠁⠀⡷⣯⠤⣿⢲⡆⠀⠀⣇⡠⣦⠀⢠⡞⠉⠉⡇⠀⢠⠇
⠀⠀⠀⠀⠀⢿⠀⠀⠜⢷⣄⠴⠃⠘⣯⣀⠀⠸⣧⣀⡸⣧⣀⣀⠼⣧⡀⣠⠇⢸⡇⠀⢠⡟⠀⡿⠀⢸⣷⠀⠈⠀⢠⠎
⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢀⠤⠒⠲⡅⠀⠀⠀⠁⠀⠈⠉⠀⠀⠈⠉⠁⠀⠈⠁⠀⠈⠁⠀⠳⠄⠀⠙⠳⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢀⡤⢤⡄⢀⠖⢹⡄⠀⠀⣧⠴⣦⠀⠀⣼⡁⠀⠨⠃⠀⠀⠄
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣿⠀⠀⠲⣄⠘⣶⠧⢿⡅⠀⢸⠇⠀⠀⡏⠀⢻⡀⠀⢿⣍⠉⠀⠀⠀⠘
⠀⠀⠀⠀⢆⡀⠀⠀⠀⠀⠹⣦⠀⠀⢹⡀⢿⣄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⣉⣁⣀⠙⠓⠖⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠃⠀⢸⡇⠀⠀⠀⡰⠊⠛⠲⣄⠀⡠⠂⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠹⣄⠀⢀⠹⡎⠀⠀⠀⢀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠈⠉⠀⢠⡇⠀⠀⠀⢧⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠊⠳⠄⠀⠀⠀⠀⠲⠉⠊⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢀⠙⠀⡫⠀⡀⠀⡀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣉⡡⠀⠈⠒⠀⠁⠐⠠⠂⠀⠀⢀⠈⠈⠈⠊⠁⠠⡉⣩⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠒⠒⠚⠈⠉⣀⡀⠀⣀⣀⠀⠈⠉⠀⠇⠒⠐⠁
⠀⠀⠀⠀⠀⢠⠀⠠⠄⠄⠄⠆⠠⠠⠀⢤⠀⠀⠰⠀⠀⠀⠀⠀⢠⠂⢀⠀⠀⠀⠀⠠⠒⠈⠈⠁⠐⠄⠄
⠀⠀⠀⠀⠀⠏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠌⠐⠀⠀⠒⠀⠀⠈⠁⠀⠀⠈⠈⡁⡀⠀⢀⢈⠈⠁⠀⠈⠈
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢧⡁⢀⠽⠠⠀⠀⢔⠒⠒⠒⡊⢠⢧⢰⢿⡈⡒⠒⠒⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠄⠀⠤⠤⠀⠈⠶⡬⡉⡣⢲⡤⢏⡞⠀⢇⠳⢼⡲⡜⡁⡤⠎⠀⠀⠠⠄⠀⠤
⠀⠀⠀⠄⠠⠃⠂⠀⠀⢄⡀⢀⠠⠀⠀⠀⠣⠐⣀⠀⠀⠑⠒⠈⣡⢔⣀⡀⢀⢠⢭⠴⠀⠀⠀⠀⠰⠮⡭⢀⡀⡀⣐⢠⡈⠐⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢍⠉⠁⠁
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠠⣄⣀⣠⠄⣴⠀⠀⢀⣀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣼⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢉⢼⠩⡧⡘⡡⠀⢀⣸⡈⢦⠀⠀⢸⣻⡇⠀⠀⢸⢟⡇⠀⠀⡰⠃⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢸⠀⡇⡛⠁⠀⢸⡀⠳⡀⠱⡀⠀⣿⣟⣠⣆⣻⣿⠄⠀⠜⠀⡜⢁⡏
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⡵⠀⠀⠀⢣⠙⢆⠙⠢⡀⢠⡙⠙⠃⠘⠛⢙⡠⢈⡄⠊⡠⠊⡼
⠀⠀⠀⠀⠠⣶⣀⠈⢂⠀⠀⠌⠁⣠⡶⠄⠀⠀⠀⠀⠀⠀⢀⠸⢲⠥⢴⡛⠀⠄⣗⠦⠤⣽⣏⠛⡈⠳⡄⢠⡖⢁⠗⣩⣮⠤⠤⢺⠁
⠀⠀⠀⠰⠲⣌⢉⢷⡶⠂⠲⣶⡾⠉⣰⠾⠀⠀⠀⠀⠀⠀⣤⡴⣹⠇⠸⢧⠣⠶⠈⢟⠒⣚⠣⣄⠘⡀⡽⢻⡀⡪⢀⠬⣓⠒⣺⠁
⠀⠀⢀⣀⡀⡘⠻⡖⠁⠀⠀⠘⠾⠋⢁⢀⣠⠀⠀⠀⠀⠠⡤⠖⢹⠂⠀⣧⠙⠲⠂⠀⠉⠣⢖⣁⡍⢲⡁⢈⡗⢸⣈⣒⠄⠋
⠀⠀⢸⠀⠈⠙⠒⡗⠀⠀⠀⠀⢓⠚⠋⠀⠈⡀⠀⠀⠤⠀⣀⡠⠋⠀⠀⠈⠓⣠⠤⣹⡄⠀⠀⠀⠈⠉⡀⠀⠈⠉
⠀⠀⠳⢆⢤⢇⡐⡏⠀⠀⠀⠀⢟⣂⡤⠤⠰⠃⠀⠀⠙⠉⠈⠀⠀⠀⣠⠞⢉⠍⡓⢀⢦⠀⠶⡆⢒⡩⡉⠳⣄
⠀⠀⠠⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⢸⡰⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡜⠁⠐⢅⠒⠤⢏⢨⠀⡑⠁⠥⠒⡈⠃⠈⡑⡄
⠀⠀⠈⢱⡊⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⢱⠈⠀⠀⠀⠀⠀⠀⠀⠰⡠⢃⠸⡼⠐⠢⣀⠍⠀⠈⣇⡔⠂⢣⠜⢘⢄⡠
⠀⠀⠀⢻⡟⢋⠀⠀⠀⠀⠀⠀⠀⢈⠛⣿⡃⠀⠀⠀⣀⠤⠂⣡⠔⡀⠀⡀⠀⡌⢠⠀⠀⠀⠀⠀⢇⢰⡀⢀⠀⢀⠲⣄⠐⠤⣀⡀
⠀⠀⠀⠈⠓⠬⢧⠀⠀⠀⠀⠀⠀⡾⠴⠋⠁⠀⠺⠩⢤⡶⢛⣡⠞⣡⢾⣁⠜⠑⠋⠀⠀⠀⠀⠀⠈⠊⠑⣄⡷⢄⠳⣬⣙⠶⡤⠌⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⣀⡀⠀⠀⣼⠃⠀⣀⡀⣴⢋⣹⠇⢰⡟⢀⣀⠀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣴⡿⢠⡦⠤⣼⠿⠃⣠⣴⡇⢀⡾⠉⣉⣿⢯⣿⠤⣾⣵⢿⣏⡾⣿⣁⣼⠿⠋⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡾⠩⣧⠟⠀⠀⠙⠒⠋⠁⣿⣰⠿⠷⠛⠁⢿⠞⠁⠰⠿⠁⠻⠋⠀⠈⠉⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⢠⣶⡄⠀⣶⡄⠀⣠⣿⣷⣤⡀⠀⣤⡀⠀⠀⣠⡆⠀⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⡇⢸⣿⠁⠀⢹⣿⠀⣿⣿⣦⣾⣿⡇⠀⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⣿⡇⠘⣿⣦⣤⣾⠟⠀⣿⡇⠙⠋⣿⡇⠀⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠉⠉⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠉⠉⠉⠁


⣿⣿⢛⠻⣛⡛⢿⢟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⡴⠁
⠀⠀⠀⣠⠞⠋⠉⠉⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢠⠞⠁
⠀⠀⠀⠈⠁⠀⡠⠀⠀⠀⠀⢳⠀⠀⠀⣀⠀⢠⡟⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⢸⡇⢠⡞⡽⠃⣾⠕⠉⠀⢀⣴⠖⠢⠀⠀⠀⠀⢀⡖⢀⡴⣲⢀⡤⠞⣩⠟⣠⠤⡀
⠀⠀⠀⢠⠏⢀⡴⢳⠀⠀⠀⢸⢇⡿⠋⠀⢴⡏⠀⠀⣰⠟⠁⠀⢀⡴⠒⢰⡄⣾⡴⠋⣰⡿⠋⢠⡾⢣⠞⣁⠔⠁
⠀⠀⢀⡟⣰⠟⠀⠈⠣⣤⡴⠏⠘⣇⠀⠀⢸⡇⠀⡴⣿⠀⢀⣠⣾⠁⣀⡾⢹⡟⠀⠰⠏⠀⣰⡟⠀⣿⠉
⠀⠀⠈⠛⠁⠀⠀⠀⠀⢀⡤⣀⠀⠈⠁⠀⠸⣇⠞⠁⠈⠁⠉⠀⠈⠛⠉⠀⠘⠀⠀⠀⠀⢠⣿⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣼⡴⠋⠀⠀⣀⠀⠀⢀⣴⠟⣦⠀⢰⡄⢀⡴⢺⡷⣠⠞⢉⣼⠏⢀⣴⢶⡆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⡴⢋⡼⠃⣰⠏⠙⣴⠧⢶⡿⣠⠎⢠⣿⠟⠁⢀⣾⠋⢠⣿⡵⠞
⠀⠀⠀⠀⠀⠀⢠⡇⣠⠋⢀⡾⠁⡰⣿⢀⡴⠃⠀⣼⡟⠁⢀⣿⠋⠀⢀⣿⠃⠀⣿⢁⣠⠔
⠀⠀⠀⠀⠀⠀⠀⣷⠃⢀⡾⢁⡜⠀⠙⠉⠀⠀⠀⠋⠀⠀⠼⠃⠀⠀⣼⠇⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣧⠞⠀⠀⠀⠀⠀⠀⠀⢀⡤⣦⠀⠀⠀⠀⠻⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⡤⠋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠉⠉⠉⠑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢀⣤⡤⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⠃⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠰⣿⣿⠃⠀⣾⡏⠀⠀⣠⣠⣀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠢⢄⣀⠀⠀⢀⣀⠤⠂
⠀⠀⠀⣿⡿⢀⣼⣿⠀⢠⣾⡏⠀⢹⠆⣿⠁⠀⣿⡇⠀⠀⠀⢾⣿⠾⣿⠇⢀⣤⣤⡀⠀⠀⠀⢸⠉⠁
⠀⠀⠀⠿⠿⠞⣿⡟⠀⠈⠻⢧⡤⠏⠀⢿⣀⡼⣿⡷⠀⠀⠀⢸⣿⠀⠀⠀⣿⣯⣼⡿⠆⠀⠀⢸
⠀⠀⠀⣠⣄⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠼⠟⠀⠀⠀⠻⠷⠦⠶
⠀⠀⣮⣼⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⢀⣀⣀⠀⠠⠒⠚⠓⠲⢤⡃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⢹⣤⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⣠⡿⢿⡿⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⣾⣦⠢⡀⠀⠀⠀⠀⠀⡠⠚⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠱⡘⣿⣿⡇⣣⡀⠀⠀⠠⠊⠀⡠⠂⠚⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠑⠢⠭⠵⠃⠘⡦⠴⠦⠔⠉⠀⣀⣸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⠔⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡄⠀⠐⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠠⠊⠉⠑⢦⠠⡔⠉⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡦⢼⡇⢆⠀⠀⠈⢢⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠐⢻⣿⠀⣤⣶⠈⣻⡟⠁⠀⠀⠀⠀⠀⠐⠛⣿⡗⠛⠣⣈⣀⣀⣀⢟⣦⣤⣀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⣿⣧⡿⠀⠀⢀⣤⣤⣄⠀⠀⠀⣿⠁⣰⣿⡟⣹⠏⢡⣾⡏⠈⢹⠁⣠⣴⣦⢤⣦⢤⣶⠀⢀⣤⣤⡀⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⣿⣿⠁⠀⠀⣾⣿⣤⡿⠇⠀⢸⣿⠀⢹⣿⠃⠀⡀⠈⠻⢧⡴⠋⠀⠀⣿⡇⢸⣿⢸⣿⢠⣿⣧⣼⠿⠀⣀⣥
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠀⠀⠀⠹⠿⠶⠶⠂⠀⠼⠿⠒⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠘⠛⠈⠛⠀⠻⠷⠶⠶⠀⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠈⠀⡀⠀⠀⠀⣀⡤⠄⢀⣰⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠉⠀⠀⢪⠤⢴⠍⠀⢀⣼⡁⠀⠀⣼⠃⢐⣳⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢤⠀⢠⠂⠂⠀⡋⠀⣈⢻⡿⠖⠛⠛⢳⣦⣾⣿⡇⢳⠀⣠⠆⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠘⢀⣧⢀⡄⠀⠀⠀⠉⠀⠇⠀⠃⠶⣬⣿⢟⣿⡟⠀⠲⠁⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣯⠈⠄⠀⣿⡆⠎⠀⣰⠀⣾⣦⣾⣿⣿⡿⢆⠀⠀⠀⡠⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣥⢼⡄⠈⣻⠄⠀⠀⠀⠀⣿⣿⣿⢾⣿⢨⡿⣢⠄⠘⠑⣠⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠘⠳⠤⠇⠀⠈⢱⠀⡔⠀⠑⠁⣸⣿⣿⣾⡟⠀⠀⠀⢇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢹⠀⠎⠀⠀⠀⡄⠸⣧⠀⠘⢋⠿⣼⣿⠇⠀⠂⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⠀⠀⠀⣄⣧⠀⠉⢄⣤⣯⣤⠸⠃⠀⠐⢡⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡆⠄⢄⠆⠀⠀⠘⠙⢧⠀⠈⠠⣸⡿⠁⣀⠀⡤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢅⠀⠰⠦⠄⢴⣤⣢⣄⣠⢿⠅⣀⠄⠈⠄⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠎⢀⠀⠀⠀⡩⠋⣉⣸⠅⠈⡀⢄⡾⠃⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⣠⡈⣆⠀⠀⡤⠾⠋⠀⠀⢤⡤⠎⡀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂⡀⠈⠈⠉⠁⠀⣀⣀⡴⠟⠋⣠⡔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⢑⢖⠒⣛⢚⣿⠏⠃⣠⡴⣠⠍⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠚⣇⠀⠄⢈⠤⠟⠉⠁⠀⠀⠀⠀⠁⠔⠐⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠴⠊⠁⠀⠀⠀⠀⢀⠴⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠈⢇⣁⣶⠒⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢑⠁⠀⢄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠣⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⣼⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⢀⣿⡇⠀⠃⠃⠃⠀⠀⠀⢀⣀⡿⣹⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣧⠀⠀⠀⢸⣿⠇⣠⡀⣰⣦⢰⣰⣧⣾⣿⣟⠁⢠⡿
⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⢸⣇⠀⣿⠀⣠⡀⢸⡿⢰⡏⢱⡿⡿⢻⡏⠻⠈⠋⠻⠶⠟
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⢀⣄⠻⣆⣿⣰⡿⠃⢸⡇⣼⣧⠾⠿⠁⠈⠀⠀⡆⢀⡀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⢻⡄⠈⣿⠙⣇⣠⠏⠛⠁⠀⠀⠀⠀⢀⣤⢠⣷⡿⠋⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⡇⠈⢿⣀⡇⠀⠈⠁⢀⣤⡀⠀⡴⢶⠀⣿⠀⣼⡟⢷⣄⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣸⠃⠀⠘⣿⡇⠀⢰⠀⡞⣼⠃⢸⣃⢿⣴⠻⠞⠸⠇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⣿⣼⢠⡿⠟⣾⠉⠀⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⢁⠀⢹⡆⠀⢸⡇⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡏⠀⣸⡇⠀⣸⠇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠁⠀⣿⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⠏




⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣛⣿⣿⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣀⣠⣤⠤⠤⢄⣶⠆⠀⢠⡿⠑⠀⢀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⡖⢀⠀⠀⠀⢀⡀⠀⡀⠀⠀⣀⣤⡀
⠀⠀⠀⠛⠁⠀⠀⠀⣾⠃⢠⡶⠀⠙⢠⣾⠋⠁⢰⡿⠀⢀⣴⠋⠇⣀⣾⠂⠔⠀⣾⠇⣼⠟⣸⡟⢠⣾⠏⠔⢁⣠⠴⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢿⡇⠀⠀⠈⠛⠀⠀⢿⡃⠀⠘⠇⠀⠀⠻⡇⠀⠀⠘⠋⠈⠋⠀⠟⠀⠙⠇⠄⠒⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢙⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂
⠀⠀⠀⠀⣠⣽⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⡯⡀
⠀⠀⠀⠀⠈⡻⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠵
⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠈⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠈⢩⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣠⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣸⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣸⣿⣁⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠋⠟⠋⠉⠻⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⡿⠟⠉⠻⠏⠳⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣼⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠙⣿⣿⣿⣿⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⣿⣿⣿⣿⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠿⣿⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠠⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢿⣿⠀⢸⡇⠀⣠⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠋⠹⡆⠀⠀⢉⣰⠟⠀⣼⣿⡟⣰⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢹
⠀⠀⠀⠀⠀⣴⡋⢹⣿⣷⡀⠀⠀⠙⠳⠦⣿⠴⠖⢋⡁⠀⣼⣿⡟⣰⠋⣀⣀⠀⢀⡶⠋⠉⠳⡄⠀⢀⡾⠀⢀⣠⣴⣦⢀⣤⢶⣶⣦
⠀⠀⠀⠀⠀⣿⣷⠈⣿⣿⡇⠀⣶⣦⡀⠀⣿⢠⣾⣟⣹⢶⣿⣿⡟⢡⣾⣿⠋⣷⡜⣷⣾⣷⢀⣻⡴⢻⣦⣴⠏⣹⣿⣿⠋⢀⣾⣿⡏⣠⣴⣤⡀
⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣷⠀⢸⣿⡇⢠⡇⢿⣿⡏⠁⣸⣿⡿⠀⣿⣿⡇⠘⠛⣡⣿⣿⡏⠉⣿⠀⣼⣿⠃⢠⣿⡿⠁⢠⣿⣿⡟⣸⣿⡿⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⣸⣿⣧⣾⠁⠈⠿⠷⠞⢻⣿⡇⠀⠻⣿⣷⡶⠞⠙⣿⣿⠀⣰⠇⠀⢿⣿⠀⠀⠙⠃⢠⣿⣿⠟⠀⢿⣿⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠋⠻⠿⠃⠀⠀⠀⠀⠀⠸⣿⣇⠀⣤⣦⡀⠀⠀⠀⠉⠛⠛⣡⠴⠶⢦⠀⠀⠀⠀⠀⣸⣿⡿⠲⢦⠀⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⢉⣰⠇⠀⠀⠀⠀⠀⢸⠃⢰⣾⣿⠃⢀⣠⠴⠛⠁⠙⠓⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣭⡤⠖⠋⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣿⣵⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣶⣿⣿⣷⣿⣿⣷⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣤⡶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢿⣿⠀⢸⡇⠀⣠⣶⣿⡆⠀⠀⣴⡄⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠉⠹⡆⠀⠀⢉⣠⠟⠀⣼⣿⡟⣰⠇⠀⠀⠈⠀⠀⠀⠀⢸⣿⡆⢹
⠀⠀⠀⠀⠀⣴⡋⢹⣿⣷⡀⠀⠀⠙⠳⠦⣿⠴⠖⢋⡁⠀⣼⣿⡟⣰⠋⣀⣀⠁⢀⡶⠋⠉⠳⡄⠀⢀⡾⠀⢈⣠⣴⣦⢀⣤⢶⣶⣦⠀⠐
⠀⠀⠀⠄⠀⣿⣷⠈⣿⣿⡇⠀⣶⣦⡀⠀⣿⢠⣾⣟⣹⢷⣿⣿⡟⢡⣾⣿⠋⣷⡜⣷⣾⣷⢀⣻⡴⢻⣦⣴⠏⣹⣿⣿⠋⢀⣾⣿⡏⣠⣴⣦⡀
⠀⠀⠀⠀⠀⠈⠁⠀⣿⣿⣷⠀⢸⣿⡇⢠⡇⢿⣿⡏⠁⣸⣿⡿⠀⣿⣿⡇⠘⠛⣡⣿⣿⡏⠉⣿⠀⣼⣿⠃⢠⣿⡿⠁⢠⣿⣿⡟⣸⣿⡿⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⣸⣿⣧⣾⠁⠈⠻⠷⠞⢻⣿⡇⠀⠻⣿⣷⡶⠞⠹⣿⣿⠀⣰⠇⠀⢿⣿⠀⠀⠙⠃⢠⣿⣿⠟⠀⢿⣿⠀⣴⡆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠋⠻⠿⠃⠀⠀⠀⠀⠄⠸⣿⣇⠀⣤⣦⡀⠀⠀⠀⠉⠛⠛⣡⠴⠶⢦⠀⠀⠀⠁⠀⣸⣿⡿⠒⢦⠀⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣶⠄⠀⠀⠀⠹⣿⣄⢉⣰⠇⠀⢀⡀⠀⠁⢸⠇⢰⣾⣿⠃⢀⣠⠴⠛⠁⠙⠓⠒⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠉⠀⠀⠀⠈⠁⠀⠀⠘⢧⣤⣭⡤⠖⠋⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣿⣽⣽⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣷⣶⣷⣿⣿⣷⣿⣿⣷⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠎⢁⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠉⠁⠀⠈⠙⢦
⠀⠀⠀⠀⠀⢠⠏⡠⠀⠀⠀⠑⠤⣤⣤⡤⠖⠋⠁⠀⢀⡔⠉⡆⢀⠔⠄⠀⠘
⠀⠀⠀⠀⠐⡾⠉⠀⠀⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠐⢠⠋
⠀⠀⠀⠀⢸⡇⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⡀⢠⠏⠀⠀⠀⠀⡠⠒⠂⠄
⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⠀⣼⢁⠔⠀⡞⠀⠀⠀⣠⢤⠱⢄⡀⣸⠤⢰⡆⠀⡶⠀⢠⡀⢀⡤⠦⠄
⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⢠⠃⣴⠋⡘⢰⡧⠂⠀⢸⠇⠀⠀⣾⠁⠈⢰⠂⢀⡎⠀⣾⠀⢰⡧⢊⡼⣱⠏
⠀⠀⠀⠀⠘⡇⠀⠀⣿⠀⠀⡰⠃⠠⣟⣀⡤⣾⠀⠀⠀⣾⠀⠀⠐⢧⣀⠄⠙⠶⠋⠀⠐⠏⣀⣀⣀⣾⠁⢿⣀⣀⠄
⠀⠀⠀⠀⠀⠙⠦⠞⠈⠓⠊⠀⠀⠀⠉⠉⠀⡏⠀⠀⠀⢻⣠⠆⠀⠀⠀⠀⠀⣀⡤⠖⠚⠉⠉⠀⢸⡏⡁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢤⠀⢧⡴⠂⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⢀⡆⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠴⠃⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠐⠂⠀⡾⠂⠉⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⣀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣸⠁⢜⡉⢑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢄⢀⡄⠀⡉⠛⠒⢀⠆⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⡸⢸⡃⠀⡇⡠⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠁⠈⠁⠀⠉⠀⢀⡥⡄⠀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢠⡞⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠰⠀⠀⠀⣠⠖⠢⠀⡴⠖⣶⠋⢀⡤⢤⡾⠀⠀⣰⠂⢠⠇⠀⢨⠆⢰⠏⢁⢽⡄
⠀⠀⠀⠀⠀⠀⢸⠀⢰⡀⢀⠂⠀⣏⠀⠀⢼⣁⠀⡟⠀⣾⢁⢼⠇⠀⠠⣇⡠⠺⠖⠀⡞⢀⠘⠶⠋⢸⠇
⠀⠀⠀⠀⠀⠀⠘⠒⠈⠓⠋⠀⠀⠈⠓⠁⠀⠀⠀⠳⠄⠈⠁⠈⠓⠂⠀⠀⠀⠀⠀⠀⠓⠃⡤⢦⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠤⣀
⠀⠀⠀⠀⢠⡄⠀⠀⠸⣏⢇⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣆
⠀⠀⠀⢠⣿⠁⣾⢳⠀⠹⣾⡀⢀⣴⠖⠲⡄⣿⡇⣠⠞⠀⢀⣤⣄⠀⠀⠀⣀⠀⢠⣿⢀⠔⣷⢀⠔⣻⡇⢀⣴⠚⢲⠀⠀⣸
⠀⠀⠀⢸⡇⠀⣿⣸⠀⠀⢸⠋⣿⡿⠲⠒⢡⣿⡋⠀⠀⢰⣿⠁⠀⢀⣾⠺⣭⡷⢻⣧⠏⣸⡿⠃⣰⡟⢀⣾⡧⠔⠋⠀⢀⡏
⠀⠀⠀⢸⡇⠀⠸⣿⠀⠀⡸⠀⠸⣿⣄⣠⠎⢻⣇⠀⢀⣿⣧⠀⣠⠛⢿⣀⣠⠃⢸⡟⢀⣿⠁⢠⡟⢠⠎⢻⣆⡀⢀⣠⠎
⠀⠀⠀⠀⠻⢤⠴⠋⠓⠚⠁⠀⠀⠀⠉⠁⠀⠀⠙⠓⠋⠀⠛⠛⠁⠀⠀⠉⠀⠀⠈⠃⠘⠃⠀⠘⠓⠃⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢼⡀⠀⠀⠀⠀⣼⣿⡄⣰⣿⡆⠀⢸⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣸⡇⠈⠰⢦⣸⣦⡴⠆⠁⠀⣿⠻⣷⣿⠹⣿⠀⢸⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠋⢻⡇⢀⢢⠴⣻⡗⠢⡄⡀⢸⡟⠀⠻⠇⠀⢿⡇⢸⣟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠉⠀⠁⢾⠃⠀⠀⠉⣾⡇⠀⠀⠀⠀⢸⣿⢸⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣹⠀⢸⡄⢀⡅⠂⣰⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢸⡇⢺⣀⡆⠻⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠇⢀⢩⣠⠀⠀⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⡔⠁⢀⡟⢰⣣⡅⢨⡿⢰⡇⠅
⠀⠀⠀⠀⠀⠀⠀⡠⢢⡀⠀⢳⡀⢂⢸⠀⢀⡼⠁⠀⠐⠧⠋⠓⠈⠓⠂⠀⣀⣠⠤⠤⠖⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⢰⠀⠘⡇⠀⢀⡙⠲⣌⣁⡉⠀⠀⠀⠀⣀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⡉
⠀⠀⠀⠀⠀⠀⠘⡀⠀⡇⠀⢼⡇⠀⠂⠀⠉⠉⠉⢩⡍⠁⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⢠⠀⣷⠀⢘⠀⡰⠂⢀⡾⢀⡴⠦⢀⡄⠓⣰⢂⣶⢀⣶⣠⠖⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠀⣿⠀⢸⢰⠇⢀⣼⠃⣼⣇⡴⣿⢁⠇⣿⠆⣿⢱⡇⣿⡈⢀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⢧⠜⠈⠉⠁⢿⡰⠉⠉⠀⠙⠋⠀⠈⠀⠁⠸⡇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒

⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⡀
⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⢠⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⣿⠌⡌⣷
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⢀⡀⠀⠀⠀⢠⣿⣿⢣⡼⣿⢈⣧⠘⣷
⠀⠀⢠⣄⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠂⠀⠀⢠⢳⣜⣧⠀⠀⣠⣿⣿⠟⣸⡇⣿⢸⣷⢲⠈⠃
⠀⠀⠉⡀⠀⠀⠀⠀⠦⡐⠐⠂⡀⠀⠀⠰⢊⣰⢃⠀⠀⠀⠘⣷⠻⠎⣡⣴⣿⡿⡃⣴⡘⢁⠏⢸⣿⢸⡆⢰⡄
⠀⢀⠁⠁⠀⠀⠈⠘⠀⢣⡠⠀⡞⠢⠄⣀⡀⠀⠀⠀⠁⠀⢿⣿⣿⡷⠿⢟⢻⣾⣷⡙⢡⣿⢠⢲⣏⠸⡇⣧⠁
⠀⠟⢐⠀⠀⠀⠈⠘⠀⠀⠀⠔⠁⡜⠀⠄⠈⠎⠒⠀⠀⠀⠀⢫⣶⣾⡏⣿⠿⡷⠉⣴⣼⣿⢸⢻⣿⢦⣌⢹⡀
⠀⡆⢸⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠖⢤⠴⠣⠀⠀⠀⠀⠀⠘⣣⣙⢃⡌⣴⢰⣿⣻⠆⡟⣾⣻⣿⢸⣿⠘⡇
⠐⠁⣤⠀⢨⠀⢠⠀⢡⡰⡀⡄⠠⠀⢧⠈⠂⠣⠀⠀⠀⠀⠀⠀⠫⠏⣿⢣⣿⢸⡇⡿⡐⢰⣿⢹⣿⢸⣿⠀⣧
⠀⢠⠹⠀⠀⠀⢨⠀⠀⠀⠛⠱⡧⠁⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠸⢹⠸⢡⡄⣿⠈⣿⣼⡏⣿⣿⢀⢹
⠀⡣⠈⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣻⠇⣿⢰⡏⣿⠇⣿⡋⣾⡌
⠀⠀⢠⠃⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⠘⢱⡟⠀⢿⠇⢫⡄
⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡄⡾⠁⠆⠘⠀⠈⡃
⠀⠀⠂⠀⡜⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡇⠀⡾⠀⠀⡼⡀⡁
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢠⣡⠀⡇⢽⡇
⠀⠀⠀⠈⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢏⣾⡏⣴⡆⠻
⠀⠀⠀⡀⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢃⣼⡟⣰⡟⠀⠰
⠀⠀⠘⠀⠀⠀⡀⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⣾⠟⢠⠟⣴⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⢀⠠⣾⣿⠘⣿⡇
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣽⣿⠸⣿⣿⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⣿⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠀⠘⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣤⡄
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠓⠲⣿⡿⢽⡿⠤⠤⣤⣤⣴⣿⣧⢤⣾⠿⢷⣤⠴⠺⣿⠓⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⠀⠀⠈⠻⣼⠁⠀⠀⠘⢿⣿⣿⡇⠀⣿⠄⠀⣸⠇⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠻⠇⠀⡿⠔⠛⠁⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣠⠞⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠑⡄
⠀⠀⠀⠀⢠⡟⢰⣷⠈⠻⣇⢠⡶⠒⣄⣿⡠⠞⢀⣤⠄⠀⠀⣀⠀⣾⢡⢶⣆⠔⣺⠂⡵⠒⡆⠀⡇
⠀⠀⠀⠀⢸⡇⠸⣿⠀⠀⡇⢻⡇⠈⡸⣿⠀⠀⣾⠃⢀⣼⡏⠓⠏⣿⠃⣾⠃⣼⢃⣼⡗⠊⠀⣠⠃
⠀⠀⠀⠀⠈⢧⣀⠝⠦⠔⠀⠈⠛⠛⠁⠙⠦⠔⠹⠧⠞⠈⠓⠊⠀⠛⠸⠇⠀⠧⠊⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠀⢰⡏⠀⠀⣀⡀⠀⣴⠆⢠⣤⠀⣠⡄⠀⣠⠤⡀⠀⣴⡶⣄
⠀⠀⠀⠀⠀⢀⡼⢷⠀⠈⢹⡟⠛⢻⠃⣴⠎⠧⣱⠤⣿⢰⢁⣿⠎⣼⠃⣼⣃⡠⠇⠀⣿⠀⠘⢲⡆
⠀⠀⠀⠀⠀⠻⠷⢬⡇⠀⢸⡇⠀⢿⠀⣿⡀⢀⠎⠀⣿⠃⣸⡏⢰⡇⢠⠻⣇⣀⡄⠀⣿⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠸⠀⠈⠉⠁⠀⠀⠉⠀⠙⠀⠘⠓⠁⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⠴⢶⠄⣼⠇⣾⠔⢚⡉⠉⠉⠛⠲⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⡏⠀⣠⡤⠀⣿⠘⠁⣼⡏⠉⢠⣾⠙⡂⢠⡬⠙⠻⢶⣦⣤⣀⣴⡄
⠀⠀⠀⠀⠀⢀⣤⣄⠀⢻⣧⡀⠹⡄⣿⡇⠀⣿⡄⠀⣿⣇⠀⣾⡇⢠⢁⣿⠗⣹⡇⢴⡎⠉⠛⡋⠁
⠀⠀⠀⠀⠀⢸⡀⠹⣷⡀⢻⣷⣄⢱⠻⣷⣠⠛⠧⠔⠈⢙⡋⠈⠓⠃⢾⠏⣰⡟⢡⡿⢡⣶⣟⣉⠇
⠀⠀⠀⠀⠀⠈⠛⠀⠘⣿⣌⡏⠙⠋⠀⠀⢀⣀⡤⠖⠊⠉⠈⠙⠒⠦⢤⣀⡈⠀⠿⠵⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢡⣄⠀⢠⣾⠛⢛⡶⣒⠒⣀⡀⠻⠟⢡⣄⢪⣿⠿⡄⠀⠈⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⣿⢠⣿⠀⢸⡆⣿⣇⠀⠀⣾⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⢸⣿⢸⣿⠀⢸⡇⣿⢻⣆⢸⢸⣿⢸⣿⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠸⣿⠀⢸⠇⢸⠀⢻⠇⢸⣿⢸⣿⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣭⣤⣬⣭⣤⣬⣭⣥⣤⣭⣤⣤⣤⣬⣭⣭⣭⣭⣍

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶
⠀⢰⣿⡷⠀⠀⢰⣄⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏
⢀⣿⣿⠁⠀⢠⣿⡿⠀⠀⣼⡟⢀⣠⣄⡀⠀⣰⣿⠇⢀⣤⡤⣄⠀⠀⡀⠤⣄⠀⢠⣄⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⢠⣿⠏
⣼⣿⠇⠀⣠⣿⣿⠁⠀⣰⡟⣰⣿⠉⢠⡿⣰⣿⠏⣴⣿⠋⢀⣼⣠⣾⠋⠀⣽⣧⣿⡟⣴⣿⡿⣣⣶⣿⡇⣰⣿⠋⢀⣿⠀⢼⡏
⣿⡿⠀⡴⢻⣿⠃⠀⣰⠏⠀⣿⡧⠴⢛⣵⣿⠏⠀⣿⡏⠀⢈⡴⣿⠻⣄⣰⡿⢾⣿⡞⢸⣿⠞⣽⣿⠋⢀⣿⡟⠤⢞⠁⢀⣤⡁
⠙⠓⠊⠀⠈⠛⠶⠚⠁⠀⠀⠙⠓⠒⢩⣿⡟⠀⠀⡟⠓⠒⠉⠀⠙⠒⠚⠁⠀⠈⠛⠀⠘⠃⢸⡿⠃⣠⠋⠈⠓⠐⠋⠀⠈⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣧⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣧⣜⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⢙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣻⣷⣦⣽⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⣀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠓⠶⣿⣿⣟⣿⣿⣿⣿⣿⣦⣤⡀⠀⢀⠠⢀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣷⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣦⢸⣷⣤⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⠻⢿⠿⣿⣿⣿⣿⣿⣿⠿⠷⠿⠛⠀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠟⢻⣿⣿⣷⣿⣦⣀⠀⠀⣸⣿⣿⡿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⢿⣿⣿⡤⠀⣼⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣨⠏⠀⣿⣿⣅⢀⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣰⣿⣿⣿⣿⣇⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣷⡟⠙⠁⠀⢹⣿⣿⣿⣿⣿⠀⠀⠹⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡀⠀⠀⠘⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠴⠶⠷⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠍⠀⠒⠀⢉⣙⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⢀⡀⠀⠀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣨⣷⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⢀⣀⣀⣀⣴⣤⠤⣾⣿⣿⣿⣛⠉⣴⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣫⣴⣞⡉⠉⠀⠈⣉⣛⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⡛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⣿⣾⠃⠙⠻⢿⣿⣆⠀⠈⠿⠿⠟⠀⠀⠀⠿⠀⠚⠀⠀⠀⠉⠓⣤⡀⠀⢀⣴⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠿⠛⠁⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⢶⡤⠘⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣠⠏⠐⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠃⠀⠀⠘⣦⠄⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠸⠃⠀⣴⣷⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⣸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⡼⠿⠟⠛⠉⠀⠀⠀⠀⣤⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠁⣼⠁⠀⠐⠷⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⢻⢀⣀⣸⠀⣀⡀⠀⡀⢀⡀⡀⠀⣀⠀⠀⠀⠀⠀⠟⠁⣠⡄⣧⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⡤⠀⠀⠀⠀⠀⠀⣿⠀⢸⡝⠀⡾⢿⣞⣿⢰⣇⣡⣎⣹⡏⣹⣹⣾⣝⡁⠀⠀⠀⠀⠀⠐⠛⠁⣿⣿⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢠⡟⠋⠁⠀⠠⠀⠀⠀⠀⠀⠸⣆⣸⣅⣴⠃⠈⠉⠀⠁⠉⠁⠉⠁⠁⠉⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣈⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⢠⡗⢀⣀⠀⣠⣄⢀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢰⡄⠀⠀⠈⠓⠙⠋⠀⠙⠛⠛⠛⠙⠛⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⡇⠻⢧
⠀⠀⠀⠀⠀⠀⣤⣀⣈⠛⠟⢷⠀⠉⣠⡀⠀⠀⠀⠀⠀⠀⣀⣹⣀⣸⡇⢀⣀⠀⢀⣀⣀⣀⡀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢰⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣼⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉⢹⡇⡍⠙⣧⢸⡇⣿⠀⡗⢼⠤⠃⣠⡄⠀⠀⠀⠀⢠⣴⣶⠀⠀⡞⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣤⣾⠀⣶⣄⠀⠀⠀⠀⠀⠙⠀⠘⠃⠙⠛⠁⠘⠃⠛⠀⠃⠘⠓⠛⠋⠀⠀⠀⠀⠀⠸⠟⠃⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠈⠉⠛⢻⣿⠃⠻⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠈⠛⢆⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠃⠀⢠⣿⣿⣇⢸⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠈⠳⣄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⣠⠞⢹⣿⣿⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠙⠿⣿⠀⠀⠀⠀⢀⣄⠀⢸⣤⡀⠀⢰⡄⠀⢲⣤⡙⠇⠀⢀⣤⣶⣿⡧⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠷⠈⠛⣿⣿⣶⣄⡻⠀⠈⢿⣿⣷⡄⠀⠀⣀⣙⡿⠷⠚⢡⣾⣿⡿⠃⠀⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⣏⠉⠙⢓⠞⠛⠛⠛⠛⠉⠋⡀⠀⠀⢠⡿⠟⠋⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠄⠀⠀⠀⢠⡤⠀⠀⣾⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁

⠀⢀⡀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢣⣤⣄⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣼⣿⣷⣴⣤⣶⣖⡦⠤⠤⡤⡤⢤⡤⣤⢤⣤⡴⣼⣷⣷⣽⣦⣤⣭⣤⣤⣤⣠⣀⣀⣀⣀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷
⠀⠀⠀⠀⠠⡤⠀⠀⣿⣗⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠐⠂⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠐⢺⣿⠀⠀⠀⠄⢤⡀
⠀⠀⠀⢠⡃⠁⠁⡴⠈⢱⠆⢀⠶⢠⠖⠒⢲⠰⠂⠀⠀⣠⢖⠢⠀⣠⠤⢄⢀⡤⢄⠈⣠⢿⣢⠼⠥⢬⠋⡴⡆⠀⠘⢸⡇
⠀⠀⠀⠰⠁⠀⠀⢰⢀⡇⠀⡸⠀⢸⠨⣀⣀⡆⠀⠀⢠⡃⡎⠃⢰⠁⠀⡆⢰⠃⠀⣴⠁⠀⢸⢀⠀⠈⠀⡇⠀⠀⢈⢻⡇
⠀⠀⠀⢀⠀⠀⠀⠸⠈⠑⡇⠁⠀⢸⠐⣀⡁⠇⢀⣀⣸⡆⣀⡤⢸⠀⡀⡆⢸⠄⣷⠀⣴⠀⢼⠀⠛⠛⠀⢃⠂⠀⠴⢾⡇
⠀⠀⠀⠈⠏⡄⠀⠀⠁⠁⠘⠊⠀⠘⠂⠀⠀⠐⠂⠀⠀⠓⠨⠐⠀⠣⠭⠴⠘⠧⠁⠤⠸⠠⠸⠌⠒⢲⠐⠯⢁⣀⠀⢾⡇
⠀⠀⠀⠸⢸⠁⢀⣤⣤⣀⠀⢰⢀⣀⠀⠀⠀⠀⠈⠁⠀⣊⡁⢀⢆⠀⠀⠰⠒⠊⡁⠀⡀⠒⠀⠀⠀⠐⠒⠂⣹⣙⠉⣼⠇
⠀⠀⠀⢠⡘⠀⢇⡉⢑⡄⢙⣯⢟⡟⠀⠠⠀⣶⡶⠄⢂⡒⢧⠾⣀⣼⣰⠂⢤⣴⡦⣘⢿⠴⡂⠠⣄⠀⡈⣑⢆⠀⠐⣼⡇
⠀⠀⠀⠈⡅⠀⠀⠉⠙⠿⠟⣠⣷⣶⣅⣀⠀⢹⣼⣤⣔⣛⣱⣤⣞⢲⣶⠷⣾⠝⠀⠈⡰⢺⡷⣠⣞⡿⣧⠤⠞⠀⢥⣼⠇
⠀⠀⠀⠀⠁⠀⠂⢽⣴⠖⠰⠊⠙⠊⡀⡤⢉⡉⠻⣗⢮⡛⢿⣼⡿⢓⣥⣾⣮⠀⣀⠀⠈⠘⠉⢁⣁⣸⡤⢶⣖⠂⡄⣿⡆
⠀⠀⠀⠀⠀⠣⣤⣀⡀⠀⠀⠀⠀⠉⠉⠒⠚⠁⠋⠀⠉⢿⣶⣾⣿⢩⠀⠀⠀⠀⠭⠔⠠⠀⠒⠂⠴⠶⢤⠮⠴⠶⣥⣽⡧
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠦⠴⠶⠶⠲⠶⠶⠶⠶⠶⠶⠦⣶⡒⡔⠶⠦⠾⠧⠟⠛⠋⠛⠛⠛⠛⠛⠓⠛⠚⠛⠓⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠤
⠀⠈⠙⡦⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠀⠈⠀⢀⠰⠂
⠀⠀⠀⠈⠀⢠⠉⠓⠶⠠⢄⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠐⠪⠀⠢⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠠⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⣠⡧⣿⡄⠀⠀⠀⡘⠰⠁⠀⠀⠀⠀⢠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠑⠄⢹⡷⠁⢁⠘⢥⣶⠀⠀⠃⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠈⢷⡁⠀⠀⢰⡝⠒⠀⡀⠀⢀⠐⠀⠀⠂⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡠⠀⠀⠠⠀⢈⡖⠀⠀⡏⠀⡈⠁⠠⢀⢳⡨⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⢁⠘⠀⡌⠙⠒⠀⠀⠀⢰⡀⠀⠀⠑⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣿⡌⢸⠀⡇⠰⠀⠀⠀⠀⠀⠓⣼⠀⢰⠒⠲⠅⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠛⢹⠀⠀⣇⠀⠀⠀⠀⣀⣀⠀⠈⣕⢺⡧⠀⠀⠚⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡦⠴⠖⠀⠉⠁⠀⠀⡀⠤⡀⢄⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⡧⠀⠰⠀⠀⠀⠀⠀⠀⠀⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⠚⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣧⠀⠀⠐⢄⠀⠀⠀⠀⠀⠃⠀⠐⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⠳⡄⠀⠀⣼⣀⡤⠀⢷⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠤⠂⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⡀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠤⡀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠴⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣧⣠⣶⡂⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⣀⠞⡡⠂⠪⡉⠉⠉⠉⠲⠔⢒⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣑⣾⠃⠐⢧⣘⣁⠇⢀⡔⢉⡙⡆⠊⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⣀⠀⠀⢘⡶⠦⠈⠳⠬⠔⠁⠀⢸⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⢸⠉⠉⠉⠉⢳⣀⠀⠀⢀⣤⠤⢀⣈⣀⡀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⣀⣀⣀⣤⠊⠉⢀⣀⣈⣧⣄⣀⣀⣀⣈⣉⣋⣙⣮⣑⡾⠁⠀⣨⣦⣤⡄⣀⠀⠈⢧⡀
⠀⠀⠀⡇⠀⣀⠟⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠠⣇⠀⠀⠀⢿⠀⠀⠀⠳⡄
⠀⠀⠀⡇⠀⠓⢾⣤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢾⣠⠗⠋⠀⠀⠀⢸⠀⠀⠀⠀⢳
⠀⠀⠀⡇⠀⡀⠀⠀⠀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⣀⡀⠀⢸⠀⠀⠀⠀⠘⡇
⠀⠀⠀⡇⠈⡇⢰⣇⢠⠆⡷⠒⠀⢸⠀⠀⢰⠋⠁⠠⣎⡳⢰⠛⠞⣇⢸⠅⠀⠀⢸⠀⠀⠀⠀⠀⣧
⠀⠀⠀⡇⠀⠙⠏⠘⠚⠀⠙⠒⠀⠘⠉⠀⠈⠓⠈⠀⠈⠀⠈⠀⠀⠀⠈⠉⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⣀⠀⢸⠀⠀⠀⠀⠀⡟
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠈⠙⠺⠒⠒⠀⠀⣰⠃
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣤⣤⠤⠴⠚⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⢀⢐⠄⢀⡀⢀⡏⡌⠀⠀⠀⠈⠣⠤⠤⠤⠔⠒⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠄⠐⠊⢁⡞⠀⢀⠀⢆⡸⢠⡏⡜⣼⡜⠀⡔⠲⠄⠐⣏⣧⢠⢆⢺⡢⢺⢠⣏⠇
⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⣼⠁⡠⢿⠀⡸⠓⢾⠋⢀⣿⣀⡸⣇⢀⡠⣞⣀⠇⠘⠃⠘⠁⠧⠸⣅⡀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠙⠊⠀⠈⠉⠀⠀⠬⠭⠥⠤⠥⠤⠬⠥⠤⠬⠥⠄⠒
⠀⠀⠀⠘


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠶⠛⢳⡟⠛⠶⠀⠀⠀⠀⠲⠛⢻⣾⠛⠶⠀⠀⠀⠀⠲⠛⢻⡞⠛⠶⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⢸⡏⣭⠭⠭⠭⠭⠭⢭⡭⠭⠉⡵⠃⢄⠛⣽⡏⠛⡤⠚⢦⡴⠓⢤⠛⣹⣯⠛⣤⠐⢦⡴⠓⢤⠛⣽⣿⠛⣤⠐⢾⠉⠭⠭⢭⠭⠭⠭⠭⠭⣭⢹
⢸⡇⣷⠀⠀⢀⡀⠠⠊⠐⡀⠀⣄⠀⡿⠀⢼⡧⠀⢟⠀⣠⣄⠀⡻⠀⢸⣧⠀⢿⠀⣠⣄⠀⡻⠀⢼⡧⠀⢿⠀⣠⢀⡀⠰⠁⠠⠀⠀⠀⠀⣹⢸
⢸⡇⡗⡀⠤⠄⡿⠒⣤⠠⢷⣗⠈⠉⣈⠛⣻⡗⠛⣁⠉⠁⠈⠉⣀⡛⢻⣿⠛⣁⠉⠁⠈⠉⣀⠛⣻⡟⠛⣁⡉⠥⢬⠗⠢⣤⠴⣾⡃⠀⠀⣿⢸
⢸⡇⣷⠀⠄⡡⠲⣾⣿⡦⣎⠀⠈⡲⠈⠛⠋⠙⠛⠁⠀⠀⠀⠀⠈⠛⠛⠙⠛⠉⠀⠀⠀⠀⠈⠛⠋⠙⠛⠉⠑⢈⡰⢢⣿⣷⠤⡆⠀⠁⡢⢽⢸
⢸⡇⣯⠐⠛⠗⢠⡟⠛⡇⠼⣷⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠃⣴⠙⢹⡤⢼⣶⠉⠀⣿⢸
⠘⠃⣿⠀⠈⠀⠠⢾⡿⡈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⠇⡘⠂⡀⠀⠀⣿⠘
⠀⠀⣧⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣧⡄⢤⢤⢠⢤⣠⢠⠤⣠⢤⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠈⠈⡿⠈⠉⠈⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢲⡀⢀⣀⠀⢠⡆⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⢸⡇⡼⣿⢀⡞⣀⡀⣸⠇⢀⣀⡀⢀⡀⠀⢀⣀⣀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡾⢱⣧⠞⣼⠵⠃⡿⢠⡟⠈⢡⡏⢸⡇⣼⠃⣿⠁⡟⢰⡯⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠋⠀⠈⠓⠈⠙⠀⠙⠂⠀⠑⠊⠁⠋⠀⠃⠀⠓⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⣿
⢠⡄⣏⠀⢀⠀⠐⢾⣷⢁⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡆⢡⠄⠁⠀⠀⣿⢠
⢸⡇⣟⠠⣤⡦⠸⣦⣠⡇⢲⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⡄⠻⣠⣸⠃⢺⠿⣀⠀⢿⢸
⢸⡇⡿⠀⠂⡁⠴⢿⣿⠗⢏⠀⢀⠕⣀⣤⣄⣠⣤⡀⠀⠀⠀⠀⢀⣤⣤⣠⣤⣀⠀⠀⠀⠀⢀⣤⣄⣠⣤⣀⡠⢊⡱⠜⢿⡿⠒⠇⠀⣀⠕⢽⢸
⢸⡇⣧⠁⠒⠂⣷⠤⠛⠐⡾⡭⢀⣀⢉⣤⣼⡧⣤⡉⣀⡀⢀⣀⠉⣤⢼⣿⣤⡉⣀⡀⢀⣀⢉⣤⣼⣷⣤⡉⣁⡒⢚⡦⠔⠋⠲⢿⡅⠀⠀⣿⢸
⢸⡇⣿⠀⠀⠀⠁⠐⢄⠠⠁⠀⠋⠀⣷⠀⢺⡗⠀⣮⠀⠙⠋⠀⣵⠀⢸⡟⠀⣾⠀⠙⠋⠀⣵⠀⢺⡗⠀⣾⠀⠙⠈⠁⠰⣀⠐⠀⠀⠀⠀⢿⢸
⢸⣇⣛⣒⣒⣒⣒⣒⣚⣓⣒⣀⡷⠄⠛⣤⣿⣏⣤⠓⢤⠞⠳⡤⠚⣤⣸⣿⣤⠑⢠⠾⠳⡤⠋⣤⣹⣿⣤⠙⢠⢞⣀⣒⣒⣚⣒⣒⣒⣒⣒⣛⣸
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠶⣤⡾⣧⣤⠦⠀⠀⠀⠀⠴⣤⣼⢿⣤⠶⠀⠀⠀⠀⠶⣤⣼⢧⣤⠶⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⢨⠉⣙⠉⠉⢭⠉⠉⠉⠉⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠇⠈⢷⣀⠉⡿⢁⢀⠄
⠈⠀⠀⢼⣀⠛⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣿⠀⢐⠾⠀⣀
⠀⠈⢦⣄⠉⠶⠄⢢⣹⡉⣦⡔⠁⣼⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣿⢁⣉⠂⡽⢃
⠀⠶⠀⠙⢷⡄⠘⣪⠀⠇⠨⠀⠀⣿⠀⠀⠆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡤⠀⠉⠀⢀⠈⠸⠅⣢
⠀⠰⢆⠀⠀⠹⡆⠁⠈⠀⠄⠀⠀⠈⠃⠠⢻⢓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⡘⣭⠢⡷⡸⠃⡠⠀⠀⠄⠀⠀⠋⢀
⠀⠂⠶⢠⠶⡆⠀⠀⢀⢀⠀⡀⠃⠰⢯⢃⠯⠀⡁⣀⠓⢰⠀⠷⡀⠀⠀⠀⠀⠋⠀⠄⠠⢇⢏⠀⠙⣦⣀⠀⠈⠀⠀⢀⠀⢸⡿
⠈⠓⢦⣄⣀⠀⡆⠀⠀⠀⠈⠠⠟⠁⠲⡍⠀⠠⠋⠁⠀⣿⠀⠀⠁⠀⠀⣄⠈⢳⡄⠀⢀⠀⢹⡀⢠⠇⠀⣠⣤⠀⠡⣤⡆⠀⢀
⠑⢂⡄⢈⠙⠛⠛⢄⠀⢀⡀⠓⠲⢆⠀⢡⠀⠀⢀⠄⠐⠙⡷⠀⠁⠀⠈⠻⣶⣈⡧⢀⡈⠑⣠⠷⢂⡤⠾⠛⠀⠃⢐⠈⢿⠀⠈
⠀⠀⠀⢹⠐⠀⢋⢊⡤⠶⢛⡢⠤⠤⢴⠋⢺⣖⣡⠀⠀⠻⣧⠀⠀⡄⠀⢠⡿⠟⠃⠉⠉⠉⣩⠤⣾⠓⠦⣤⡴⠆⠶⠾⡲⢀
⠀⠀⠀⢸⠀⠀⠞⠃⣤⣶⠇⢀⣠⡄⠊⠗⢛⠉⠇⠀⠀⢀⠘⠂⢀⡀⠀⠊⣀⠀⠀⢈⣋⡀⠀⢰⠀⢡⣄⠀⢡⣤⠀⡈⣤⡉⠃
⠀⠀⠀⢸⠀⠀⠀⠀⠄⠔⠋⠉⠁⠀⠸⠀⠘⢧⡀⠐⠀⠀⠀⠀⢄⠀⠀⢀⡀⣀⣀⣈⡁⠁⠀⠞⠂⠀⣙⠣⣌⢢⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠚⠾⠂⠀⠐⠀⢠⣤⡇⠀⠀⠈⠀⠀⠀⠀⠄⠰⠀⠉⠉⠙⠙⠻⠶⣄⠈⠃⠘⠋⠉⠉⠓⠇⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⠀⠀⠠⠤⠴⠛⠛⠉⠒⢀⡀⢠⠂⠀⠁⠀⠈⡄⠀⡀⢀⡀⠀⠀⠀⠀⠈⠀⠈⠂⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢀⡀⠀⠤⠀⠴⡃⠋⠦⡀⠀⡀⢤⠉⠏⠈⢢⠘⡇⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⣟⠀⠈⠀⠘⢧⣸⠀⠘⠒⡠⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢰⡆⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⣽⠏⣰⡗⠀⣼⢃⡄⢀⣾⠁⣠⡀⢀⣄⡀⢀⠀⣄⠀⣀⠀⣠⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠋⠀⢰⡏⢰⣿⢁⡼⢱⣟⠼⣾⠃⣾⠋⢳⡟⢠⢧⡿⢡⡿⢰⡏⣼⢏⠟⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄⠸⠷⠋⠀⠛⠀⠘⠛⠘⠇⠀⠛⠐⠋⠘⠁⠘⠀⠛⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⢀⣀⡀⢀⠀⠀⡀⠀⠀⡀⣀⡀⠀⣀⡀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⡄⢀⠀⢀⡀⢀⢀⡀⠀⡀⣠⡄⡀⢀⠀⣠⡀⠀⠀⢠⠀⠀⠀⣀⢀⠀⣠⣀⠀⠀⠀⠀⠀⡇
⠀⠂⠀⢸⠀⠀⠀⠀⠀⠄⠠⠤⠀⠄⠀⠤⠄⠀⠤⠤⠀⠠⠤⠀⠤⠄⠤⠀⠀⠠⠤⠤⠀⠤⠄⠤⠀⠤⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠒⠀⠀⠐⠀⠒⠀⠐⠂⠐⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠄⡇
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡿⠋⠉⠉⣹⡿⠓⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢹⡇⣴⡟⣳⢀⣿⠁⢀⣤⢦⡄⣠⣤⡀⣠⣄⣀⣄⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⡶⢿⡄⢰⡇⠈⣿⠛⣿⣟⣡⢾⣿⣀⣿⡃⠘⣡⡷⢤⡿⣿⡟⣹⡿⢫⡿⢠⣾⣋⡗
⠀⠀⠀⠀⠀⠻⠇⢸⣇⣿⣷⣴⠏⣀⣬⣭⠁⠈⠛⠋⠙⠿⠛⠻⠧⠞⠁⠻⠀⡿⠁⣿⣡⢿⣇⡴
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣃⣨⣭⣴⡿⠤⠞⠁⣴⠛⢲⣴⣿⡶⣿⡶⣶⡆⣰⡾⢲⠀⣉⣵⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠀⣿⣵⢾⡧⢰⣟⢋⡿⣸⡏⣰⡟⢰⡟⣸⣿⠛⣩⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⣿⡇⣾⣇⡼⠛⠋⠁⠈⠁⠉⠁⠻⠟⠉⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⣠⠞⡁
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⠃⠀⠀⢰⡏⠀⠀⠰⠁⠀⠀⠀⠀⠀⣰⢃⠄
⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠀⠀⢠⡟⠀⠀⢠⡿⠀⠀⢀⠇⠀⡠⠂⡀⠀⣰⠗⠀⠀⣠⠄⠠⠀⢀⡴⠖⠉⡆⠀⠀⣠⡆⠀⡆⠀⠀⡦⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢹⠃⠀⠀⡘⠀⣼⠁⠚⠁⢠⡏⠀⠀⣰⠏⠀⠀⠀⣾⠀⠀⢠⠃⠀⢰⡟⠁⢰⠇⠀⡼⠃⠀⣴⠃⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢟⠀⢀⠔⠁⠀⠻⠀⠀⠂⠸⣀⠀⠀⠘⠄⠀⠐⠀⠹⠄⠰⠂⠀⠀⠞⠀⠠⠏⠀⠘⠇⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⡀⣠⣤⣬⣥⣥⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⣠⠀⠀⠀⢀⡄⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⢀⡇⠀⠀⣼⠀⠀⠀⢰⠏⠀⠀⠀⠙⢷⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⠀⡇⠀⢻⠁⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⢠⠞⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⡇⠀⢸⣷⡶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⡰⠋⠀⠀⠀⠀⣿⠀⠀⠀⠘⢷⣄⣀⣀⣴⠟⠀⠀⠀⠀⠀⠻⣦⣀⣀⣠⡼⠃⠀⠹⣆⣀⣀⣾⡇⠀⢸⠉⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣰⡟⣠⠎⢀⣀⠀⠀⠀⠀⠉⠀⢀⣀⡤⠶⣬⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⡉⠉⠁⠀⠀⣀⡀⠈⠉⠁⠀⠁⠀⢈⡀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⡿⢚⡵⠚⠁⣼⠇⠀⠀⠀⢠⡾⢉⡏⠀⠀⢸⡇⠀⠀⠀⠀⢠⣾⢏⡴⠋⠀⣿⠀⢀⠔⠉⢹⡟⠀⠀⠀⠀⣠⡾⠋⠀⣶
⠀⠀⠀⠀⢀⡿⡠⠋⠀⠀⣴⠏⠀⠀⠀⢀⣿⠁⢸⠀⠀⠀⣸⠇⠀⠀⠀⡰⣻⣿⠋⠀⠀⣸⡏⠔⠁⠀⣠⡿⠁⠀⠀⠀⣴⡟⠀⢀⡼⠃⠀⠀⢀
⠀⠀⠀⠀⣾⠏⠀⠀⠀⣼⠏⠀⠀⠀⡠⢻⡇⠀⢸⠀⠀⢀⡟⠀⠀⢀⠞⣰⡿⠃⠀⠀⢰⡿⠃⠀⠀⣰⡿⠀⠀⠀⢀⢾⣿⠁⠚⠉⠀⠀⠀⡰⠃
⠀⠀⠀⣸⠏⠀⠀⠀⢸⡟⠀⠀⣠⠞⠁⠸⣇⠀⠀⠣⣠⣞⣀⡠⠔⠁⢠⡿⠁⠀⠀⢀⣿⠁⠀⠀⠀⣿⠀⠀⢀⠔⠁⠸⣧⠀⠀⠀⢀⡤⠊
⠀⠀⠐⠛⠀⠀⣀⣀⠘⠓⠒⠊⠀⠀⠀⠀⠙⠳⠖⠚⠉⠉⠁⠀⠀⠀⠛⠁⠀⠀⠀⠚⠃⠀⠀⠀⠀⠙⠒⠈⢀⣀⠀⠀⠉⠓⠒⠚⠉
⠀⠀⠀⠀⣤⣤⣽⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣶⣤⣤⡤
⠀⠀⠀⠀⠀⠠⠾⠿⣿⣿⣿⣿⣿⠟⠛⠛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠛⣻⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡛⠰⣄⠀⢀⣤⠀⠀⢠⡄⠀⣠⡾⠅⠰⣬⠉⠙⣿⠁⠲⢦
⠀⠀⠀⠀⢰⡖⢒⣊⣉⠠⠾⠿⣿⣿⣏⠀⠪⣅⠀⢤⣟⠒⠉⢛⡠⠴⠒⠂⡭⠍⠉⡁
⠀⠀⠀⠀⠀⡹⠛⢿⡿⠀⢸⠀⡀⢨⢩⠷⠒⡔⠒⢶⣾⡒⡿⢣⠀⣦⡴⡇⡧⠀⠀⡏⣭⣙⣓⠒⣶⠄
⠀⠀⠀⠀⠸⠷⠶⠾⠥⢶⡈⠟⠳⠏⢸⠖⠀⡇⠀⢸⠀⠀⡇⢸⠀⡇⠀⣇⣳⣶⣤⡏⣿⡿⠋⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠉⢰⡒⠶⢼⣧⣄⣓⣂⣸⣴⣦⡭⠭⠒⠊⣿⣿⣿⣿⠿⢁⠁⠐⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⣢⠬⠷⠄⣀⣐⣶⣄⣤⣽⡦⠄⠂⠫⠄⠀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠚⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠋⠉⠉⠁⠉⠉⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⣷⠀⠀⠀⠀⠀⢸⠀⣿⠀⣿⢸⡏⠉⢸⡇⠀⢰⡏⢻⡆⣾⠋⢻⡀⣿⡀⢠⣿⢸⡟⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣷⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⡟⡇⡇⢸⣧⡄⢸⡇⠀⢸⠀⠈⠁⣿⠀⢸⡇⣿⡇⢸⣿⢸⣧⡤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡶⣦⠀⠀⣰⠏⢀⡾⠁⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⠃⣿⡇⢸⡇⠀⢸⡇⠀⢸⡀⢰⡆⣿⠀⢸⠇⣿⢻⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠙⢦⡈⠳⡾⠃⣰⠏⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠛⠀⠘⠀⠘⠓⠒⠘⠓⠒⠈⠳⠛⠁⠙⠶⠛⠀⠛⠘⠃⠛⠘⠓⠒
⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⠻⣆⢀⡼⠃⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⡁⠀⠀⠉⠀⢀⡀⠁⢈⡁⠀⣀⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠐⢀⠀⡀⠀⠒⣂⠐⠂⣀⣐⣀⠂⠐⠐⠀⠂⡀⠐⠂⢀⠀⢀⠒⠂⢀⠀⢂⡐⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠂⠀⠀⠀⠐⢢⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⣄⠀⠀⠰⡀⠀⠀⡀⠃⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠹⡄⠀⠈⠶⣀⠀⣠⠖⠉⠉⠛⠄⠀⠀⠠⢄⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠠⠶⣄⠀⠀⠑⢄⣀⣀⢹⠀⠀⠀⠈⣹⠓⠂⠀⠀⠀⠘⣄⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠉⠉⠲⢦⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠑⢤⡇⠀⠀⠀⠀⠀⠀⢈⣉⠉⠉⠀⠀⠈⠑⠲⠤⠤⠤⠀⠀⠀⠀⠹
⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⡸⣠⠒⢆⢸⠁⠀⢀⡴⠲⠄⢸⣁⠈⢳⣀⠀⠀⢹⡆⠀⠀⠀⢀⡤⠶⣄
⠀⠀⠀⣏⠀⠀⣿⡀⠀⡰⠃⣿⠀⠀⢸⡀⠀⢸⡀⠀⠀⣾⠙⠊⡹⢸⡇⠀⢸⡧⢉⡇⠀⣿⠀⠀⠎⠀⡸
⠀⠀⠀⠹⡄⠀⠈⠛⠋⢀⣀⣙⡓⠀⠈⠛⠒⠉⠓⠂⠀⠙⠷⠖⠁⠸⠇⠀⠘⠃⠸⣇⠀⠹⢧⣄⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣀⣀⡀⢀⠔⢦⠀⠀⣿⣀⣤⠀⠀⣰⠋⠀⢹⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣄⠐⣧⣎⢷⠁⠀⣿⠀⠀⢿⠁⢸⡇⠀⢿⡓⠂⠁⠀⠀⠌
⠀⠀⢀⠀⠀⠀⠀⠈⣷⡀⠀⢸⡆⢿⣄⡼⠀⠀⠁⠀⠀⠀⠀⠀⠙⣂⡈⠙⠢⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠈⠛⠀⠈⣇⠀⠀⠀⡐⠚⠒⢦⡀⢀⠔⠈⠀⠀⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠳⣀⢀⡀⢿⠁⠀⠀⢀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠸⡀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣀⠔⠁⠀⠀⠀⠀⠈⠉
⣿⡷⠻⠗⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠂⠀⠀⢀⠐⠁⠀⠀⠘⠀⠁⠽⠀⠀⠀⠀⡋⠀⠀⠀⠀⠀⠐⠁⠀⡀⠀⠠⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠸⡀⠀⠀⠄⠀⠀⠧⢼⠁⠀⠀⠀⠀⠀⠀⠀⢸⡥⢤⡴⠀⠀⣤⠃⠀⠈⡀⠀⠀⠀⠹⠀⠀⠀⠀⡆⠀⠀⠣⠥⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠃⠀⢀⠀⠀⠀⠀⢈⢠⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠀⣖⣸⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⡇⠀⢠⠀⠀⣔⡒⠂⡀⣒⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠐⢀⡄⠀⠐⠃⠀⡀⠀⠀⠈⠀⠀⢀⠀⠀⠀⡁⠀⠀⠰⠁⠀⢸⠀⠀⠈⠀⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⡀⢤⡼⡁⠠⡠⣠⠀⠀⠀⠠⠀⣠⡇⠠⣀⠀⢐⣼⠈⠤⡀⣀⡴⢃⠀⠢⣀⢀⡄⠚⠈⡀⣈⡃⠀⢈⡏⢠⣤⣽⣭⣤⣀⣴⣿⡿⣬⣣⡔
⠀⠀⠀⠀⠀⠀⠃⠂⠂⠀⠐⠒⠒⠁⠀⠂⠂⠀⠂⠘⠃⠀⠉⠐⠋⠀⠁⠒⠒⠉⠀⠀⠙⠒⠒⠋⠁⠈⠉⠉⠁⠉⠙⠃⠛⠛⠛⠛⠛⠻⠻⠛⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⣴⡞⠋⣷⡀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⣀⡤⡀
⠀⠀⠀⠀⠀⣼⠏⠀⢰⣿⠃⠀⠀⠀⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⢀⣾⠋⢀⠇
⠀⠀⠀⠀⠀⠛⠀⢀⣿⡟⠀⠀⠀⣼⣿⠁⠀⠀⢸⠁⠀⠀⠀⢀⣿⠏⢠⠎
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⣰⣿⠇⠀⠀⢠⢃⣤⡤⡄⠀⣼⣿⠔⠁⢀⣠⡤⡄⢀⣠⠦⡄⠀⣠⡄⢀⣀⡀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⠔⢹⣿⠀⠀⡴⠃⣾⡟⠒⣁⠔⣿⡇⠀⣠⣿⡏⠀⢁⣿⠏⢒⡟⣺⣿⠟⢹⣿⠟⢹⣿⠃⣾⠧⠜⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁⠀⠀⠙⠒⠋⠀⠀⠈⠑⠉⠁⠀⠈⠓⠊⠀⠙⠓⠊⠁⠙⠒⠊⠀⠙⠋⠀⠙⠋⠀⢻⡇⠀⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠷⣶⣚⠋⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡆⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⣆⢀
⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⢰⣦⣺⣞⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣟⣿⣢⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⡁⠈⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⣻⣿⣷⣄⡀⠀⣀⣴⣿⣿⣻⣿⣿⣿⣿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⣿⣿⣿⠋⠀⠈⠊⠀⠈⢻⣿⣿⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠟⣿⡿⣿⣦⡀⠀⠀⠀⣠⣾⡿⣿⡿⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣦⣠⡾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢄⣤⠆⠰⡶⠈⣿⡾⡎⢡⡧⡿⡆⠶⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢖⣤⠻⠘⠃⠻⢦⣷⡋⠸⣿⣃⢸⡄⣼⣡⣿⣦⡟⣧⢖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⡴⡆⣌⠩⣷⠘⠋⠾⢧⣤⠨⠝⣂⣈⣽⣤⣙⡃⢘⣿⣿⢆⡴⢟⡷⡐⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⣥⣱⣱⠹⠽⠼⠴⠶⠴⣬⡙⠏⠀⠸⣿⣿⢿⣿⣿⡿⣈⡻⢟⢰⡿⠟⡀⣿⣾⢁⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡾⣦⡘⢵⢯⣿⣱⠘⠚⡚⠚⠺⠗⢼⣿⣷⡿⣻⣿⢷⢹⣿⣿⣟⠘⢫⢾⣿⢿⣯⣿⠠⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠛⢓⣛⠀⠶⣌⣻⠀⡎⣿⢻⡬⠅⢸⣿⣿⣧⡻⣾⣾⣾⣿⣿⡟⠀⣿⣯⣟⡽⠛⠉⢠⢬⠘⠂
⠀⠀⠀⠀⠀⠀⢿⣧⣬⡖⢩⡿⠀⠀⣙⣁⣈⣴⡿⠞⢠⣿⠻⠿⠿⣿⣿⣿⣿⣿⡍⣙⢁⣞⠾⣚⠋⡀⠄⠓⣟⢉⡁⣱⡄
⠀⠀⠀⠀⠀⡘⣡⡄⣼⢱⡟⠀⣾⣤⠈⡏⢉⣠⣤⡉⠈⣿⠘⢂⣤⣻⠟⣫⣽⡻⢤⡄⠊⠀⡜⣽⡃⣠⢶⠶⣄⠈⢿⡿⠗
⠀⠀⠀⠀⠀⣀⣿⡇⢻⣿⣇⡼⣿⡿⠀⡟⢈⣯⡴⢇⣷⡇⢀⣇⠋⠛⣸⣿⠋⡇⣿⣧⢾⣾⢱⣿⠁⣇⣏⡠⠏⣤⡈⢠⠶⡀
⠀⠀⠀⠀⠀⠻⡌⠓⣠⣭⣍⡀⠙⠷⠴⢃⠈⠳⠤⠊⣘⠷⠊⠙⠂⢊⡀⠛⠚⡄⠘⠛⠈⣃⠸⡧⠞⣩⡳⠤⠶⢋⡀⡎⢸⡇
⠀⠀⠀⠀⢀⠛⡴⢋⣴⣿⣿⡿⠀⣓⠺⠅⠠⠾⣿⢲⡈⠽⢅⣾⢹⡆⠟⠼⢐⠐⢫⣽⡆⣨⢭⣥⡐⢙⡟⢻⢷⠌⡁⢳⠋
⠀⠀⠀⠀⠈⢸⡇⢸⣿⡙⠋⠥⣸⣹⣧⢾⣿⡆⣿⣼⢃⣀⠞⣿⡾⢠⢲⢄⣭⡰⣿⣸⠱⣇⣾⣩⠟⣸⢣⣏⣞⡈⢏⡇⠶
⠀⠀⠀⠀⠀⠈⠻⢾⣿⣿⣷⣾⣅⠙⣯⠀⠻⠷⠘⠿⠉⠟⠀⢿⣇⡞⠛⠋⢘⣁⢻⣯⡴⠙⠻⠶⠾⡁⠸⣿⠿⢠⡿⣁⠂
⠀⠀⠀⠀⠀⠀⠸⢣⡶⣰⠃⠈⢿⡧⢠⣖⣶⣦⡀⣻⣴⣶⣄⢀⡶⣀⣾⢰⠈⣉⢡⣄⣄⣦⣴⣀⣴⣞⠰⢎⣉⣁⣄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⣄⣠⠿⣡⢼⣿⣶⣽⣿⣯⡍⣿⠏⣠⣞⡫⠵⠋⣠⠿⠿⣿⠏⠻⠙⠂⣍⠀⠺⠿⠥⠳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡦⣠⣤⡶⣖⣻⡇⠉⣋⣹⢾⣅⠘⢩⡍⢠⣤⠈⣴⣶⡆⠠⠰⠆⡴⣷⢿⢶⣤⣛⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣼⣿⣿⣴⣤⣟⣿⣾⣿⠀⣂⣴⣿⣏⣾⣵⣟⣴⣾⡦⠀⣧⢻⣼⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⢿⣭⡾⢋⣿⣉⣁⠎⢷⢬⣭⢡⣌⣛⡱⢿⣍⡀⠲⣌⠛⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠙⠿⣟⡇⢸⡿⣿⡿⢹⣽⡇⠙⠻⠗⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣰⠲⡄⠀⠀⠀⠀⠀⡼⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⢸⠃⣠⠏⠀⠀⠀⡴⠛⠲⡄⠀⠀⣠⣤⡼⢉⡇
⠀⠀⠀⠀⠀⠀⡇⠀⢠⡄⠀⣇⢸⠃⢠⢶⡆⠀⣿⡴⠃⢠⠖⠆⠀⣧⠀⠀⣇⣠⣾⠇⠸⠁⣸⠁⢠⠶⣆
⠀⠀⠀⠀⠀⠀⣇⠀⣼⠁⠀⢸⡿⠤⢾⠋⢀⣴⡏⠀⢀⣿⣀⣠⠞⡏⠉⣽⠋⠀⠾⠀⠀⢠⡇⠀⣿⢞⣡⠆
⠀⠀⠀⠀⠀⠀⠈⠛⢹⠀⢠⠏⠀⠀⠈⠛⠉⢸⡇⢀⡞⠁⠉⠀⠀⠳⠞⠁⠀⠀⠀⠀⠀⣾⠀⣰⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠀⠀⠀⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠞⠁




⣶⣶⣶⣶⠶⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⡶⣶⣶⠶⢶⣶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣠⣦⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣰⢿⡿⠀⠀⡴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⡟⢸⠇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⣸⠃⡟⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢀⡿⣸⠁⠀⠀⠀⢠⡏⠀⠀⢀⣤⠄⠀⠀⢸⠃⠀⠀⠀⠀⣠⣶⠀⠀⠀⢀⣴⣆⠀⠀⠀⣴⣿⢀⣾⣿⠀⠀⢀⣴⣾⡗
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⡞⠀⠀⡰⠟⠟⠀⠀⢀⡿⠀⠀⠀⠀⡜⠛⠃⠀⠀⣠⠛⢿⣿⡄⠀⢸⣿⣿⣾⢹⣿⠀⢠⠞⠉⡿⠁
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⡘⠁⠀⡼⢠⠎⠀⠀⡀⣸⠇⠀⠀⠀⣼⠁⠀⠀⠀⢰⡇⠀⠘⣿⠇⠀⣿⢻⣿⡏⢸⡇⢠⡇⢀⡔⠁
⠀⠀⠀⠀⣿⣧⠀⠀⠀⡴⣿⣧⡀⠀⢀⠜⠀⠀⣸⣷⡀⠀⢀⡜⠀⣿⠀⠀⡀⢰⣿⡀⠀⣴⠀⣾⣇⠀⢀⡟⠀⢀⣿⢸⣿⠁⣾⡇⣿⣧⣈⣀⣀⣀
⠀⠀⠀⠀⢿⣿⣷⣶⠟⠀⢻⣿⣿⡾⠋⠀⠀⠀⢻⣿⣿⡿⠋⠀⢸⣿⣷⡞⠁⠸⣿⣿⡿⠃⠀⢿⣿⣿⠏⠀⠀⢸⠇⢸⠏⠀⠟⠀⢿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠛⠁⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣰⣾⣿⣿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⠿⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣶⣶⡶⠶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠁⠀⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠀⣀⠤⢀⣬⣕⣦⠀⠀⡜⠀⢠⣄⣀⠄⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠠⠈⠤⣰⠏⠀⠳⡝⢳⣄⡁⠀⡞⠘⡇⠊⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢰⠁⠐⠀⡌⠀⠀⠀⠹⡄⠹⡇⢸⠁⠀⢱⠀⠐⢁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡆⠀⠀⣜⠀⠀⠶⠂⠀⠙⣄⢻⠇⠀⢴⠞⡆⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠄⠈⠢⣀⠀⠀⠀⠀⢈⡿⡄⠀⠀⠀⢳⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡀⠀⠀⠀⠈⠉⠁⠒⠚⠉⡇⠈⠲⢄⡠⠞⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠲⠤⣄⡀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢄⣀⣀⣀⡀⢀⣀⠉⠓⢄⠀⠀⠀⠈⠓⠦⠤⠤⢤⡀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠊⠀⠀⠀⠸⠠⠒⠒⠒⠒⠒⠒⠊⠁⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⢠⠇⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠣⣄⡀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⡤⠶⡎⠁⠀⠀⠀⠀⠀⠀⠉⠙⡏⠉⢷⢀⣠⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⣆⠀⠀⠀⡀⠀⠰⡤⠤⣷⠀⢸⠉⠈⡇⠾⢥
⠀⠀⠀⠀⠀⢀⡠⢶⠒⠒⠈⢸⠀⠀⠀⠀⠀⠈⢷⠉⠙⣷⠄⢞⡇⠀⢸⠀⠘⣇⡠⠷⠒⠁
⠀⠀⠀⠀⢰⣋⠀⢸⠦⡤⣄⣸⠀⠀⠀⠀⠀⠀⠀⠳⡴⠋⠀⠀⠈⠉⡽⠀⠀⡇
⠀⠀⠀⠀⠀⠛⠚⠁⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢠⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠀⠀⠀⠘⢤⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠤⢤⣀⣀⣀⣀⠤⠖⠋⠀⠀⠀⠀⠀⠈⡅⢨⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡄⢀⠀⠀⡰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠀⠈⠒⠁⠀⠂⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⢤⣴⣶⡚⠭⠡⠃⠒⠂⠄⡀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢒⣚⣭⠴⢒⣋⠉⣁⠽⣬⣼⣅⣒⠀⠀⠀⡁⠀⢀⣾⡿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠤⣶⣲⡿⠿⡿⠿⢟⣛⡿⢷⣶⣚⣩⠶⢋⡵⢊⡕⣠⢟⣦⣁⢛⢸⠁⠠⣾⣿⣿⣾⣿⣿⠷⣒⡦
⠀⢀⡠⢔⣒⡩⠽⢚⡊⠥⠐⣀⡡⠴⢛⣫⡽⠖⣫⠔⢋⡽⡋⠿⣐⣯⣞⡵⣋⣴⡟⣿⠸⡀⠀⢿⣥⡟⣯⣡⣤⡾⠋
⢓⡞⡚⡩⠠⠖⣊⣉⡤⠔⣛⠉⠀⢔⠋⢁⢐⡋⠀⠖⠫⡠⢊⠔⢉⡤⠙⣟⣥⡞⢁⠘⢻⡟⢃⣛⣿⣯⣿⣿⠃⠀⢄
⠀⠉⠀⠋⠉⠈⠀⠈⠈⠉⠀⠈⠁⠀⠀⠈⠉⠀⠉⠁⠀⠁⠈⠈⠃⠀⢀⠀⠒⠚⣏⣼⣼⢴⣿⣯⣼⡥⣸⠃⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢁⡊⠄⡴⣱⠞⣵⣿⠟⣹⠿⢿⡓⠁⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠜⢡⠞⡵⢅⡜⠁⣈⡝⢇⣰⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠁⠀⠓⠒⡶⣖⡥⠄⠀⠉⠉⣻⣿⡲⡤⣦⣼⡵⠵⠞⢛⣣⣼⣯⠞⠓⡿⠀⠐⣀⡠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠤⠔⢊⠁⠀⠁⠀⠀⠀⠀⢀⢙⡿⣃⠂⢁⠨⠁⠀⢁⣾⣏⡉⠁⠪⣤⣝⡞⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠀⠀⠀⠀⢀⣴⣾⠟⠡⠐⠁⠰⢡⢠⢤⣄⠌⢡⣬⠛⠛⢛⡁⣤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⢞⠩⢊⠞⢁⣴⣥⠤⠚⣠⠪⡠⡠⢫⠔⠀⣀⢠⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣤⠅⠪⠳⠊⠅⠈⠳⠄⠀⠉⠓⠛⠋⠁⠀⢞⡵⠎⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠁⠀⠠⠀⠀⠀⠔⠉⠼⠡⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⡀⢈⡥⠶⠊⡁⠄⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣭⠫⠝⠻⢀⡤⠊⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡁⠀⠀⣀⡔⠈⠒⠤⢀⣀⡠⠄⠒⠀⠚⠋⠠⡤⠤⡀⠏⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠈⠑⠢⠀⠀⠈⢉⠥⠀⣄⣀⢀⡀⠁⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠤⢤⣀⣈⠀⣂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠒⠉⢱⠀⡌⠉⠉⠁⢀⣠⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡰⣐⡊⠅⠈⠑⠇⠀⢀⣀⠠⠤⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠉⠀⢒⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢠⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢠⡿⣰⣃⣀⣀⣀⣠⣤⣤⣴⠆
⠀⠀⠀⠀⠀⠀⣠⣴⡾⢿⣷⠀⠀⣠⡀⢸⣿⠀⠀⠀⣾⡿⠛⠉⡉⣩⣭⠉⣵⢠⡄⣠⡄⣴⣶⠀⢸⡆
⠀⠀⠀⠀⣠⡾⠋⠁⠀⢀⡿⢀⣴⣿⠁⣾⣏⣴⢶⣶⡏⢀⡾⢻⢳⡯⣼⢾⡿⣿⣿⡟⣼⡏⠁⢀⣾⠇
⠀⠀⠀⣼⡟⠁⠀⠀⠀⣼⣷⢟⣿⠃⢰⡿⣿⠗⢚⣾⡇⣼⣇⣠⠾⠴⠋⠻⠃⠿⠸⠟⠘⠷⠴⠞⠋
⠀⠀⢸⣿⠀⠀⠀⢀⣾⡿⠃⢸⡏⢀⣿⠃⠻⠖⠋⠘⣛⣀⣉⣁⣤⣤⣤⣤⣤⣤⣴⡶⠞⠁
⠀⠀⠘⠿⠟⠁⠀⣾⡟⠁⠀⢻⣥⠞⢁⣤⣶⡾⠿⠛⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢠⠎⠉⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠈⢳⠏⢀⠀⠀⠀⣠⡷⠖⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢹⣰⣿⣷⣦⣿⣾⣿⣿⣦⣰⣯⣴⣶⣄⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⡉⠉⣿⠻⠿⠿⢿⣿⢿⣿⣟⣿⣟⠿⣿⣿⣿⣀⠀⣸
⠀⠀⠀⠀⠀⠀⡏⠉⠉⡉⠉⢉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⡇⠀⢸⠃⢀⡇⠀⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⣸⢡⠖⠀⡄⢰⠀⠀⠙⠆⣠⠒
⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⡼⢌⠰⠼⡀⣏⠀⠀⢼⠰⡃
⠀⠀⠀⠀⠀⠀⡇⠀⡀⢀⠃⠀⢀⡴⠶⠆⢀⠀⢤⠀⠀⠀⠀⡴⠂
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠈⠀⠀⠀⠀⡜⠀⢸⠀⠀⡆⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⡸⠀⠀⢿⠀⠀⢀⣞⠅⡎⢠⠎⢡⠏⠀⣼⢸⢵⠀⡔⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢰⡇⡀⠁⠘⢦⠄⠸⠤⠘⠄⠻⠄⠘⠔⠂⠇⠈⠨⠘⠧
⠀⠀⠀⠀⠀⠀⠓⠒⢲⠶⠶⠶⠶⠷⡶⣶⡶⠶⠶⡶⠶⠶⠶⠶⢶⡶⢶⡶⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣠⠞⣱⡿⠀⠀⠸⡇⠀⠀⠀⢀⣾⡇⡞⠁⠀⠶⠂⣠⠈⠳
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠞⠁⢰⢿⡇⠀⠀⠀⠹⢦⣀⣴⣾⡿⢃⡇⠀⠘⠀⢸⠃⠀⢸⢁⡤⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠛⠀⣷⠀⠀⠀⠀⠀⠀⠈⠃⠀⡼⠀⠀⠀⢆⠀⠀⢀⠎⠎⣼⢀⠏⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠘⠁⠒⠁⠀⠉⠈⠀⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠈⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⣀⣀⣠⡤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁