Bamboo Emojis & Symbols

⠀⠀⠀⣠⣤⣶⠾⢛⡄⠀⣠⠖⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢠⣤⣀
⠀⠀⠛⢉⡉⠀⠀⣾⠗⠚⠛⠒⢀⠀⢀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣰⣿⡖⢿⣿⣾⣧
⠀⢀⣴⣿⣷⣧⢸⡏⠀⢸⠀⠀⠘⣧⠀⠻⠄⠀⠀⠠⠄⡀⠀⠀⠀⠀⠐⠃⣰⣿⣭⣿⣆⢻⣿⣿⡀
⢀⣾⠟⡿⢻⣿⠈⠁⠀⣾⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠉⠛⠦⣀⠀⠀⠸⣯⣀⣸⣯⢿⣼⣿⣿⡇
⠞⠁⠜⠀⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣦⣀⣼⣿⡍⣿⡎⢹⡇⢸⣇
⠀⠀⠀⠀⢿⣿⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠉⠛⠿⢮⣿⣿⣧⠀⣿⠈⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣔⣤⡴⣾⢿⣿⠀⣿⣶⣿
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠋⠁⡼⠃⠘⣿⠀⠉⣿⣿⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⡄⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣿⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣸⣿⣿⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡴⢙⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣼⠇⢸⣿⣿⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠀⠀⠀⢹⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠘⡇⢸⣿⡿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⣧⠈⣿⡇⣿⠀⡼⠋⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⡟⢀⣿⣷⢟⣴⢿⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠆⠀⣟⠸⣿⣾⠿⣧⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣿⠀⣿⣿⡆⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢹⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢸⠃⠸⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⣿⠀⠀⢸⡀⠀⣿⡇⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⢀⡔⢸⡇⠀⢿⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠈⡇⠀⢸⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⡠⣿⠀⢸⣿⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⡏⢠⡾⠁⢸⡀⢸⣿⡎⣿⢻⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⡿⠁⠀⠘⡇⢸⣿⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⢸⡏⠀⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⣷⠈⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠸⠆

⠀⠀⠹⣄⠀⠀⠉⠳⠶⠌⢷⠂⠀⠀⠠⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠰⣿⠂⢀⠀⠀⢀⡰⠟⠀⠀⢀⠀⠃⠀⠀⠀⠀⠸⠠⠀⠀⠀⠀⠀⠈⠍⠉
⡀⠀⡶⣽⣅⠈⠉⠀⠀⠀⠀⠀⠀⡸⢦⠶⠀⠀⠀⠀⠀⣠⠔
⠙⢿⡯⠉⠃⠀⠀⠀⠀⠀⠀⣀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⡠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⣠⡗⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢿⡟⠀⢀⣆⣄⠀⠀⠀⠀⠀⡛⠃⠀⠀⠀⠀⢀⣸⣿⡿⠂⢀⡤⠄⢠⣾⡓⠋⠀⢀⣀
⠀⠀⠀⠀⠈⣷⠀⠨⡧⢘⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠭⠀⠐⡀⠐⢿⣿⡶⠕⢾⡿⢦⠈⢭⡅
⠀⠀⠀⢀⡀⢸⡄⢰⣷⠋⠑⠀⠀⠀⣧⣠⠃⠀⠀⣠⣤⣤⠊⢀⣤⠔⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣤⢀⡀
⠀⠀⡀⠀⡙⢿⣧⣾⣷⠎⠀⠀⠠⣄⣘⡵⠂⠣⠤⣏⠻⣯⣴⠛⢉⠐⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠙
⠀⠀⠸⠆⢱⣺⣿⠛⣿⠖⠁⠀⠀⠘⣧⠆⠀⠀⠀⠈⠑⠈⠧⠀⠈⠳⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠘⣫⣭⣿⣿⡰⣿⡤⠀⠀⠐⠃⠉
⠀⠀⠀⠘⠁⣿⠻⣿⠛⡀⠀⡀
⠀⠀⠀⠀⢰⡿⠀⣿⣄⠜⠹⡇⠀⠀⣠⡀
⠀⠀⠀⠀⣸⡇⠘⣿⡏⠀⠀⡇⠀⠀⡿⠁
⠀⠀⠀⠀⣿⡇⠀⣿⣿⡀⢰⣷⠀⢰⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⣺⣇⣴⣄⡀⣀⠆⡀⠂⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢴⣺⣸⣟⣿⣿⡿⢿⡧⠠⣗⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⢴⣷⣀⢲⣟⠷⢰⣤⣤⠯⠸⠛⠁⠈⠀⠀⠀⠀⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⢯⣷⣶⣿⣯⣾⣿⠥⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⡀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡙⠿⢛⣿⣿⣿⡗⣾⠖⠀⣀⣀⣢⣠⣗⣶⡆⠽⢿⠿⠿⠻⠙⢿⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⢿⣿⡬⡉⠜⢃⢚⣿⠎⠙⠉⠙⠋⠙⠛⠋⠐⠉⠃⠃⠀⠉⢷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣽⣿⣾⣧⠤⡔⠛⠁⢠⢂⡀⡀⠀⠀⠀⠀⠠⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⣽⠟⠋⠝⠛⠁⠀⠀⠀⣨⣾⣷⣷⣧⣤⣷⣼⣷⣤
⠀⠀⠀⠀⠀⠀⢈⣛⣍⢟⣷⣶⣴⡄⠀⠀⠀⠀⢰⠬⢏⣝⠻⡛⠿⣿⡿⣽⡛⠛⠛⢀⣀⠀⠄
⠀⠀⠀⠀⠰⣾⣿⣿⣻⣧⣿⣿⣿⣿⣾⣷⣿⣿⣿⡃⠀⠀⠀⠀⠀⠈⠁⠈⠷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠛⡛⠿⢿⣽⡏⢙⡚⠻⣯⣿⣿⣿⢿⣤⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠁⠈⠕⢫⠀⠁⠀⠉⢦⣿⣼⣿⣿⣿⣿⣿⣿⣧⣰⣀⣨⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⡟⣿⣿⣿⣉⡉⣡⣁⣨⣉⣉⡙⣉⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣷⡏⣾⣿⠓⠦⣶⣴⠿⣿⣟⣷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡇⣿⠓⣿⡗⠈⠀⠏⠘⠂⠉⠧⢻⠛⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠝⠟⠉⠁⠿⠛⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡋⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠟⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⢀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣰⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣼⣿⡇⠀⢀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⣸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⣇⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠧⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠂⠀⠀⠀⠀⣹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⠀⠀⠀⠀⢀⣐⣶⣤⣤⡟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⠀⠐⠒⠚⠛⠉⠉⢩⠏⠹⣦⡀
⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⠻⣆
⠀⠉⠒⣄⠀⠷⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠈⠳⣀⠆⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣆⣠⡂⠀⢠⡟
⠀⠀⠀⣀⣤⡶⠟⠙⠿⣦⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡟⢻⣿⡰⡎
⠀⠀⠉⠁⠉⠀⠀⠇⠀⠈⣿⣿⣿⠀⣸⣿⣿⢿⣿⣿⡇⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⣿⣿⣿⠀⣿⣿⣿⠘⣿⣿⡇⢠⣾⢁
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⣿⣿⣿⣴⣿⣿⡟⢀⣿⣿⣇⡿⠃
⠀⠀⠀⠀⠐⠒⣲⡶⠞⢾⣿⣿⣧⣼⣿⣿⡇⢸⣿⣿⡿⠃
⠀⠀⠀⠠⠶⠛⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠃⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣀⣰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠉⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢀⠁⠈⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠈⠀⠀⣿⣿⡇⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢰⠀⠀⠀⠀⢸⠆⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠷⣤⣤⣿⡄⠀⠀⠀⢻⠀⠆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠈⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⠀⠀⣠⣶⣿⣀⣀⣠⣽⡆⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⢙⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢘⣿⣿⣿⠉⠉⠁⢸⣿⡿⠿⢷⡄⠀⠀⠀⢸⡗⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⣿⠏⠀⣿⠀⠘⠀⢸⡷⣄⠀⠀⠀⠀⠀⠀⣿⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⢸⡄⠋⠀⣀⣿⠰⠀⠀⢸⣧⠾⢷⡄⠀⠀⠀⢠⡏⠀⢀⡏⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⣴⣿⣭⣿⠀⠀⠀⢸⣧⠀⠀⢻⣆⠀⠀⢸⡇⠀⢸⡇⠀⣀⣤⣾⣫⣥⣤⠶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣇⠀⠀⣠⣿⠀⠀⠀⢸⣟⣧⠀⠀⢹⣧⡀⣾⠀⠀⢸⣷⣿⡿⠶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⣠⣴⠖⠀⠀⠀⠀⠸⣿⠿⠟⠹⡟⠀⣰⣫⣿⠀⠀⠀⢸⡿⣿⡀⠀⠀⢻⣿⣟⠀⣠⣾⡉⠉⠛⣛⣷⠶⠶⢽⣷⡀
⠀⠀⠀⠀⠘⠛⠛⠛⠷⣭⣳⡄⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠿⠋⣿⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠙⣿⣹⢻⣟⣿⣴⠿⣿⡶⠶⢾⣿⡛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢿⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⠀⠀⢀⣰⣿⣿⣿⣏⠁⠀⠀⠀⠙⠳⢦⣌⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⣼⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣸⡄⠀⠀⣿⠀⠀⠀⢿⡇⠀⢀⣤⡾⠟⣿⠁⠀⣾⠉⠛⢦⣄⣀⣠⡤⠶⠾⠿⢿⣿
⠀⢀⣤⣶⣿⣿⣿⣾⣟⠛⠛⠋⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⣿⡇⠀⠀⣿⠀⠀⠀⢸⣷⣾⠟⠉⠀⠀⡿⠀⠀⡿⠀⠀⠀⡿⣿⠻⣿⣿⡛⠛⠉⠁
⠀⢛⣭⠾⣻⣿⣿⡟⠙⠛⠛⠶⣶⣤⣽⣦⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⣿⠛⠛⠛⢻⡿⠁⠀⠀⠀⢸⡇⠤⢸⡇⠀⠀⢸⡇⢹⡄⠻⣎⠙⢦⡀
⣰⣿⡷⠛⢡⣏⣼⣃⣤⣤⣶⠛⠉⠁⠀⠈⠙⠳⣄⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠰⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⢸⠇⠀⠀⠀⣷⢸⡇⠀⠘⠷⣌⠻⣄
⠉⠁⠀⠀⣿⡿⢛⣿⡏⢹⣿⡆⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⢸⠇⠀⠀⣿⠀⠀⣿⠰⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣾⠀⠀⠀⠀⠸⣾⡇⠀⠀⠀⠙⠳⣿
⠀⠀⠀⣰⣟⡴⢻⡿⡇⢸⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢸⡄⠀⣀⣸⣦⠀⣿⡆⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠛⠋⠀⣸⣷⡇⠈⢷⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡛⠛⠋⢹⡇⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⣰⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⣠⣾⡿⣿⠿⠶⢶⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣧⠀⣿⠀⠀⠰⢸⣿⣯⣅⠀⣿⠀⠀⢸⡷⠛⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣗⠁⠀⠀⣿⠀⣿⠀⠀⠀⢸⡟⠷⢬⣿⣿⠀⠀⣼⣡⡴⢞⣛⣩⣭⣽⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿⣀⠀⠀⢸⣷⣄⠀⣸⡇⠀⠀⣿⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢹⡀⣟⠉⠀⠀⢸⣛⡿⠟⢹⡇⠀⠀⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠃⠀⢸⣧⣿⠀⠀⠀⢸⠏⠀⠀⣸⠃⠀⠸⣿⣦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⢀⣀⣸⡀⠀⠀⣿⢀⠀⢀⡏⠈⠛⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⢸⣿⡿⠿⠟⠛⣻⠇⠀⠀⣿⠈⠀⢸⡇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣤⣤⣤⣿⡇⠀⠚⠂⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢠⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢀⣿⣂⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠃⠀⠀⠀⣿⠀⠀⢀⡿⠛⠛⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢰⣿⢄⡀⠀⠀⣿⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⠘⠂⠀⠀⣿⠀⠀⠸⠇⠀⠈⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣬⠉⠉⡉⠁⠀⡇⠀⡏⠀⠀⢸⡏⣡⡀
⠀⠀⠀⢀⣀⠄⠀⠠⠅⠛⣿⡆⣴⣏⡀⠀⠃⠀⡇⠀⠀⢸⡇⢃⡈⠳⢤⡀
⠀⠀⠖⠛⠉⠀⠀⠀⠀⢀⣿⠆⣈⠉⠹⠷⢲⡶⣧⠀⠀⢸⡇⠀⠀⡄⠏⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢿⣿⣴⡦⠰⡟⢀⣼⠀⡷⣾⣧⢰⠞⠁⡀⠀⠚⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡿⠋⡟⠳⢀⢢⣠⢸⣧⠐⠛⠉⠉⠼⠠⢨⢄⠄⠈⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡏⡀⣷⠠⠷⡀⠘⠳⢟⠃⠻⠠⣤⣄⡷⣴⣧⡾⠃⠀⡆
⠀⠀⠀⠀⠀⢠⡴⢾⠃⢀⣿⣀⣠⡇⠀⠀⡾⠀⢠⡀⠈⠻⡇⠘⠟⠁⠀⢀⠃
⠀⠀⠀⠰⢻⠀⡗⠈⠀⠈⣷⠅⠀⠀⠀⠀⠁⠀⢆⠘⣄⠘⡇⠀⠀⠀⠀⢈⢀⣇⡇
⠀⠀⠀⡄⢀⠀⡇⠀⠋⠢⣿⠦⣤⣤⡀⠀⡀⠀⠌⣣⢼⣤⣇⠀⢢⡀⠀⠉⢹⣿⠁⠒
⠀⠀⡜⠀⠘⠀⢉⠀⠀⠀⠺⡀⠈⢿⣤⢉⡟⢻⠏⠽⢶⠻⡇⠀⠀⠳⠀⢣⡜⠍⢰⡀⢀⡤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠃⠌⠀⠘⣦⣴⢃⠌⠀⢹⡇⠀⠀⠀⠠⣌⠻⠾⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⡀⠆⠀⡇⢹⠮⡀⠀⠀⡟⠳⠊⠈⠷⣌⠑⠂⠐⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣏⣀⡗⠃⠀⡇⢸⡆⠀⠀⢀⣇⣸⠖⠀⠀⣀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠹⠘⣇⠀⠀⡝⢨⡇⠀⠀⠀⡇⢈⣀⢴⡉⠛⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⣨⠉⠀⠀⡇⠀⡇⠀⠀⢰⡇⡏⢠⠀⢳⡈⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⡇⡇⠙⣆⠀⠀⡇⠀⣧⡆⠀⣩⠀⠁⠀⠀⠀⠣⠀⠙⠇
⠀⠀⠀⠈⠀⢀⠀⠈⠉⢳⡇⠀⠀⠀⠀⠀⣇⣴⣯⠄⠊⡡⠄⣴⣤⡤
⠀⠀⠀⠀⣊⠤⠂⠀⡠⠘⡇⠀⠆⠀⠀⠀⣯⡜⠃⢿⠋⠐⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣧⣠⣄⡄⠀⠀⣿⣯⡀⠀⠙⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠁⡁⠀⠀⣿⢀⡇⠀⠀⠀⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡀⠀⠀⠀⠀⣿⢨⡇⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣀⣀⠀⠀⠹⡆⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⠉⠉⠀⠀⢸⣧⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⡇⠀⠀⢀⣇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⠆⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡀⠀⣼⡄⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⢸⡇⠀⣿⡇⡇⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⠀⠀⣼⣡⠄⠀⠀⠀⣠⡇⠀⡰⡿⣏⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠏⣤⣼⠀⠀⣿⠋⢁⣦⣄⠀⠈⢀⡞⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⡏⣀⣈⠉⠁⠀⠀⠏⢠⡮⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠐⡟⠘⠟⠁⠀⢀⡟⠀⠀⢲⢲⣤⣌⡻⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⢨⡅⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢅⣸⡇⢸⡇⡀⡴⠁⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣻⢻⡇⣾⣇⡳⣤⠀⡟⣀⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⢿⠘⠉⠋⣸⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢦⢨⡇⣿⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣼⡇⣿⠐⠶⢹⡇⠀⢠⠺⣉⣰⠠⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡇⣿⠀⠰⣿⢰⠿⠢⠀⠒⠋⣀⡠⣮⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠄⢰⠃⠈⠀⢀⠔⠀⠀⢻⠛⠜⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⡿⠀⠀⠐⠁⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⢰⡇⠀⠊⠀⡸⠻⠻⡄⢠⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠿

⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢰⣿⠋⠿⠉⢹⣿⠃
⠀⣸⣿⣠⣾⣰⣷⡆⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠇
⠀⣽⣿⣿⣿⣿⢻⡄⢸⠀⢸⠇⠀⠰⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀
⠘⡿⢹⠉⠹⠛⠀⠃⠈⡆⣸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠇
⠀⡇⠈⠀⠀⠀⠀⠀⠀⣧⣿⠌⠀⣀⡠⠀⢠⡏
⠀⡇⠀⠀⠀⠀⠀⠀⡄⢹⣿⡖⠉⠉⠀⠀⣾⡅⠀⠀⠀⠀⠁
⠀⡇⠀⠀⠀⠀⠀⠀⢿⠀⠙⡌⠳⣄⠀⣸⣿⠅⠀⠀⠀⡀
⠀⡇⡄⠀⠀⠀⡆⠀⠸⣶⠀⢻⠀⠈⢳⣿⣿⠀⢀⡀⢰⠃
⠀⣇⡇⠀⠀⠀⡇⠀⠀⠿⠀⠈⠄⠀⣾⣿⣿⣦⣿⣧⡿⠀⠀⠀⠀⠀⡼
⠀⡏⡇⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⠇⢠⠎
⣰⣇⣷⣆⣶⣿⡇⠀⢠⠀⠀⠀⠀⣿⡟⠉⠛⠻⠃⠿⠀⠀⠀⠀⠀⡟⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁
⢹⣿⣿⣿⣿⣿⠱⢀⡇⠀⠀⠀⢠⣿⠁⠀⠀⠀⠠⡇⠀⠀⠀⠀⣰⠀⠈⠀⠀⠀⢀⡀
⢸⣿⢿⣿⡿⣿⢀⣄⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠜⠀⢀⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠒
⢸⣏⠈⠋⠀⡇⢸⣿⠀⢀⠀⠀⣾⠀⠀⠀⠀⢀⠀⠀⣠⡶⢋⠁⠀⠰⠋⠀⠀⠀⢠⣄⠀⠀⣀⡄⢐⣥⣀⡈
⢸⡏⠀⠀⠀⡅⠀⣿⡞⠁⢠⢠⡟⠀⠀⠀⠀⠋⢀⡼⠋⡴⠀⡀⢀⡆⠀⠀⠀⠀⠀⠉⢂⠀⠀⠀⠀⠁⠀⠉⠙⠒⠤
⠈⡇⠀⠀⢰⡇⡀⠹⠀⢀⠋⣼⠁⠀⠀⠀⠀⠀⠊⣡⠊⢀⠌⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠑
⢸⡇⠀⠀⠀⡇⠁⠄⢠⡎⢰⡇⠔⠀⡘⢀⡆⠀⡰⠁⣠⠎⠀⡜⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠄
⢸⡇⠀⠀⠀⣇⠸⣧⠟⢠⣿⣷⣹⣿⣿⣿⠃⡔⢁⠀⠁⠀⡼⠁⠀⠀⠀⠀⠱⠀⠀⠀⡴⠃
⢸⡇⠀⠀⠀⣹⠀⢹⠀⣼⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⢀⡜⣡⠀⢀⠀⠀⠀⠀⢀⡴⠚
⢸⣤⢠⢹⡀⣿⡇⠈⠀⣿⠿⠃⠀⣿⣿⡏⠀⠀⠀⠀⠊⢰⠇⢀⠋⠀⠀⣠⠴⠋
⢾⣿⣿⣶⣿⣯⠀⢸⣴⡏⠀⠀⠈⢯⣿⠁⠀⢀⠆⠀⠀⣾⣠⠉⠀⠠⠺⡟⢿⣿⠦⠤⣤⡐⣦⣤⣤⣤⣀⡀
⢰⣿⡟⠿⣿⣿⠃⠸⡏⠁⠀⠀⠀⢸⠇⠀⠀⡀⠀⠀⠀⣹⡅⠂⠁⠀⠀⠹⡄⠀⠑⠠
⠀⡏⠁⠀⠀⠀⡀⢠⠇⠀⠀⠀⠀⣼⠀⠀⡔⠀⢀⠠⠂⠀⠀⣠⠖⠀⠀⠀⠙⢦
⠀⢃⠀⠀⠀⠀⢃⠘⠀⠀⠀⠀⠀⡏⢀⡞⢀⠶⢋⡀⠐⠘⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀
⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼⢔⡡⠴⠋⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⣴⡿⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⣿⡆⢫⠀⠀⠀⢸⠀⠀⢺⣼⡇⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿⡇⠸⡖⣿⣷⠀⠑⠀⠀⠸⡄⠀⣬⢿⡃⠀⢀⣀⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢾⣇⠀⠀⢹⣿⠀⠀⠃⠰⣶⡇⢀⣿⣎⠃⠀⠀⡬⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⡇⠀⢤⣤⣿⡆⠀⠀⠀⢸⡇⠈⣿⣿⠁⠀⠰⣥⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣈⠀⢸⠁⣿⡇⠀⠀⠀⠸⡇⢠⣋⣅⠀⠀⣄⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠈⣿⠀⠈⠀⠹⡇⠀⠀⠀⠀⠇⢼⣿⡉⠀⢰⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⢸⠀⠀⢻⠀⠐⠶⢶⣿⠀⠀⠂⢲⣷⢸⣿⡇⠀⠘⢻⡟⠀⠀⠀⠠⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠰⠀⢦⣀⣀⣠⡀⠀⠃⢘⣿⠀⠀⠀⠀⣿⢸⣿⡇⢠⣇⣸⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⣟⠃⠀⡄⣈⣻⠀⠀⠀⠀⣿⢸⣿⠅⣸⢻⡿⠀⠁⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⠇⠉⡄⡋⣿⣿⢇⠀⠀⠀⠻⣿⣿⡀⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠀⠝⢱⡇⢸⡏⢺⡖⠀⠰⣾⣿⣿⡿⡇⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠠⣾⣧⣤⠁⢋⣴⠀⠀⣿⣷⣿⠎⣿⣿⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⣀⠔⢨⣿⡿⠀⣀⠞⢀⠀⠸⣿⡏⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⣾⣆⠀⡀⠀⠙⢻⣿⣷⣶⣿⣶⡞⠀⠀⢹⠓⠜⣫⣿⣿⣿⣿⡷⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⣧⣠⠟⠋⣼⣿⣿⣿⣿⡇⠀⠀⣿⠆⣨⣿⣿⣿⣟⣉⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠐⡾⢿⣿⠋⣿⡟⠠⠠⢿⢿⣀⣛⣿⣿⣿⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠋⣰⣿⣿⣄⣲⣏⡁⠈⣡⣄⠈⢻⡿⠋⠁⣼⣿⡿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣠⡀⠓⢤⣶⡿⠋⠉⣿⣿⣿⣼⣿⣿⣿⣿⣷⣾⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣥⣢⣀⡀⠛⣁⣐⣶⣿⣿⡟⠋⠋⠉⠈⠈⠁⠉⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠉⠉⠙⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣄⣀⡀⠀⠠⠤⣴⣶⣾⣿⠿⢿⣷⣶⣶⣶⣤⡀
⠀⠀⡀⠀⠀⠉⣛⣿⣶⣦⣼⡀⠀⣦⢰⣿⡇⢻⣿⢿⣟⠻⢧⡀
⠀⠀⠐⠀⠘⠋⠉⠉⣩⣴⣿⣷⣦⡇⢸⡏⡁⠀⠹⠄⠉⠣
⠀⠀⠀⠀⠀⠀⠀⡴⢟⣿⣿⠿⣿⡇⠘⠉⠙⠀⣀⣠⣤⣤⡤⠤
⠀⠀⠀⢀⠀⡀⠈⠀⣼⣿⠏⠀⢿⡇⠀⢀⣰⣿⣿⡯⠭⠄
⠀⠀⠀⠸⠀⠱⠀⣸⣿⡧⣀⠀⣾⠀⣴⣿⣿⣏⠻⣷⣄
⠀⠀⠀⠀⠡⠈⣴⠟⠀⠀⠀⠀⢻⣼⣿⣿⣿⣿⠀⠀⢉⣣⣤⣤⣤⠄
⠀⠀⠀⠀⠀⢱⠛⠀⠸⠀⠀⠀⣸⣿⠏⢹⠙⣿⡇⢀⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠋⣤⡄⣀⠀⠀⠀⡿⠁⠀⠀⠀⢸⡇⢼⡟⣿⣿⣧⠙⠳⡀
⠀⠀⠀⠀⠀⢀⣾⢳⣿⢿⣦⡰⠁⠀⠀⠀⠀⠈⣷⢸⣷⠘⣏⢻⣧
⠀⠀⠀⠀⠀⣿⡟⢸⣿⡘⣿⣿⡀⠀⠀⠀⠀⢠⡿⠀⡟⠀⠹⡄⠙⡄
⠀⠀⠀⠀⣸⡿⠁⢸⣿⣇⠘⣿⣧⠀⠀⠀⣄⣸⡇⠀⠀⠀⠀⠁
⠀⠀⠀⢀⡿⠁⠀⣜⣿⡇⠀⠈⠻⡄⠀⠀⡏⠀⠆
⠀⠀⠀⡼⠁⠀⠀⡏⢻⡇⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠃⠀⠀⠀⠁⢸⢿⢀⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠅⠶⠗
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⡗⠤⢄⠀⣇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠀⠐⡄⠀⡇⠀⠀⠀⠀⠀⣴⣟⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠔⠚⠂⠀⡇⠀⡆⠀⠀⣸⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡆⠉⠉⠹⠀⣸⣿⡟⢿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⠀⠀⠃⡆⠀⠀⢠⣿⣿⣿⡄⠙⢧
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢀⡁⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢘⣇⢠⣀⣸⡃⠀⣼⢹⣿⡟⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠋⢩⢐⠀⠀⡇⠀⣿⠀⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⠀⢰⠢⣿⠀⠃⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⠀⡇⠀⢸⠀⠰⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⡆⢠⠀⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⢀⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣆⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣤⣄⡀⠀⠉⠉⠉⠉⠙⠛⠛⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣌⡻⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⣴⠿⠛⠉⠀⢠⣾⡿⢻⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣿⣄⡀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⢸⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣧⠐⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⢸⣿⡇⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⣠⣶⣿⣿⣿⣿⠃⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⢸⣿⠃⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⣡⣿⣿⣧⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠸⣿⠀⠀⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠋⢠⡾⢻⣿⣟⠻⣷⣄⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⣆⠀⠀⣸⡏⢸⠀⠀⠙⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠟⠀⢸⡀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢸⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢸⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈⣿⣾⣇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⣦⣿⣟⡀⠀⠰⠶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠟⠛⠛⣹⣿⣿⠀⠀⢿⣿⣿⣏⣼⡿⠟⠛⢿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠈⠉⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠉⠀⠀⠀⣠⣾⣿⠿⣿⣧⢰⣿⣿⣿⡏⠁⠀⠀⣰⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣰⣿⡿⠋⠰⣿⠙⣿⣿⣿⣿⠁⠀⢀⣼⠿⣿⣿⠹⣿⡏⢿⣿⣧⡀⠀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠁⠀⠀⣿⠀⠘⢿⣿⣿⠀⣠⡾⠃⢸⣿⡇⠀⣿⡇⠀⠹⣿⣿⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣶⡏⠀⠀⢸⣿⠁⠀⢹⣿⢀⣴⠟⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢸⣿⣿⢿⣿⣦⡀⢸⣿⠀⠀⢸⣿⠟⠁⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⠙⢿⣿⣾⡏⠀⣠⡾⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⠀⠀⠙⣿⣇⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⡇⠀⠀⢠⣿⡇⠀⠀⠀⠀⠶⠾⠿⠿⠷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣿⣿⡇⠀⠀⢸⣿⣧⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠁⠀⣠⣿⣿⡏⣸⡟⢻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⣰⡟⢻⣿⣿⠏⠀⠸⣿⡌⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⠏⠀⣿⣿⡏⠀⠀⠀⢻⣇⠀⠀⠉⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⠆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣿⣿⣿⡆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣿⣿⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠘⠿⠿⠿⠇⠀⠐⠿⠿

⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⣿⣀⣱⣾⣠⣤⣼⡧
⢀⣿⠀⠀⢸⡇⢀⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣽⠉⢹⠏⣀⠀⣘⡁
⢸⣿⣦⠀⣼⡤⠸⡄⢀⣿⠁⠀⢸⡇⠀⠀⠀⢰⣯⡇⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⣿⢀⠂⠀⠈⠑⢿⠇
⢘⣿⠃⠀⢿⣇⣀⣻⣾⣃⣀⣀⣼⣧⠀⠀⠀⠈⣿⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⢿⠁
⠨⣿⠀⠀⢨⡟⠛⢹⡟⠉⠉⠉⣿⡗⠀⠀⠀⠀⣿⡀⠀⠀⣿⡖⠐⢄⢀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼
⠀⣿⡀⠀⢸⡄⢀⠎⠈⠙⢦⡀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⠇⠀⠨⡻⣧⡆⠸⣿⠀⠀⠀⠀⠀⣿⡄
⠀⣿⣿⠀⢸⡇⠊⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠠⣿⠳⠀⠀⢸⡄⠒⠀⠈⠻⡀⠀⣿⠀⠀⠀⠀⠀⣾⡗
⠀⣿⡏⠀⣿⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠁⠀⠀⣼⡄⠀⠀⠀⠀⠸⣄⠏⠀⠀⠀⠀⠀⣿⡇
⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣿⠀⠀⠀⣹⣇⠀⠀⠀⠀⠀⢸⣅⡀⠀⠀⠀⠀⣿⡇
⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⣿⣃⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⣿⡥⠀⣼⡄⡰⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⣿⣼⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⡿
⠀⣿⡇⠀⣿⣿⠅⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡏⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣁⡀
⠀⣿⡇⠀⣾⡯⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡅⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⣿⠃
⠀⣿⡇⢀⣿⠃⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⡗⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⣯⣁⠀⠀⠀⠀⣿
⠀⣿⣧⠎⢹⠀⠀⠀⠀⠀⠀⠀⢸⣭⡀⠀⠀⠀⣿⣿⡀⢠⣿⡖⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢻⡀
⠀⣿⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡏⠀⣀⣀⣠⣿⣹⣧⣞⣿⠤⠤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇
⠀⡿⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⣯⠛⢻⠛⢿⠷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⣁
⠀⠂⠀⠀⢸⣶⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⠿⢋⠉⢇⢸⡆⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢸⡏
⠠⣦⡤⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠐⢱⣾⡏⠀⠈⠾⠟⠀⠀⠀⠀⠀⢀⣏⡀⠀⠀⣠⠀⢸⠇
⠀⣿⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠀⠀⣿⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠈⣿⠸⡄⢰⡏⠀⢸

⠀⠀⢠⣸⣼⣿⢸⣿⠀⠀⠀⢸⣿
⠀⠀⢤⣿⣿⣿⢾⣿⠀⠀⠀⢸⡿
⠀⠀⠀⠘⣿⡟⢿⣿⡷⠀⠀⣿⡇
⠀⠀⠀⠀⣿⣧⣼⣿⣧⣤⠖⣿⡏
⠀⠀⠀⠈⣿⡟⢻⣿⠷⣦⣀⣿⡇
⠀⠀⠀⠀⣿⡇⢸⣿⡀⠀⢨⣿⡇
⠀⠀⠀⠀⣿⡇⢸⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⣿⡇⢸⣿⡇⠀⢸⣿⠀⠀⠀⡴
⠀⠀⠀⠀⣿⡇⠸⣿⡇⠀⣼⣿⣦⣠⣼⡡
⠀⠀⠀⠀⣿⡇⢠⣿⡆⠀⣿⣿⠀⣿⠿⠖
⠀⠀⠀⠀⢿⣇⠰⣿⡇⠀⣿⣟⡜⠁
⠀⠀⠀⢠⣾⣿⡀⣿⡇⢀⣿⡟
⠀⣀⡀⢸⣿⣿⠃⣿⡇⢸⣿⡇
⠀⠈⠛⢿⣿⣿⡶⣿⣧⢸⣿⠁
⠠⠖⠛⠙⣿⣿⠉⣿⣿⣾⣿⠀⠀⠀⡀
⠀⠀⠀⠀⢻⣿⠀⢸⣿⣿⡏⠀⠀⣸⠀⠀⢀⡀
⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⡷⢶⡄⣿⣠⡶⠋
⠀⠀⠀⠀⢸⣿⠀⢸⣿⣿⡛⠛⢿⣿⣯⣄⣀
⠀⠀⠀⠀⢸⣿⡄⢸⣿⣿⣇⠔⠻⣦⠉⠉⠉⠃
⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⠁⠀⠀⠹⣇
⠀⠀⠀⠀⠈⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⣿⣇⣸⣿⣿
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠐⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠲⠾⢿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠴⠖⠚⣷⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄⠀⣆⠀⣿⢋⠕⠛⠓⢯⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡷⠼⠀⠀⠀⢈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢰⡇⠀⠀⢀⡠⢎⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣾⡴⣒⣫⣥⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣩⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠉⠉⢳⠀⠀⠀⣶⣀⠀⠀⠀⢀⣆⣸⣁⣉⣽⠿⠛⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠉⠁⠈⠣⢤⣄⣙⣒⣒⣪⣭⣭⠷⠛⠈⠁⠈⠉⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢶⣖⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⢖⣲⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠒⠒



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣄⢠⣤⣤⣤⣤⣄⠀⠀⠘⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠠⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠞⣿⡎⣿⡆⣿⣿⣿⣿⣿⠀⠀⠀⢀⠎⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⠋⠈⠉⠉⠉⠉⠁⠀⠀⢀⠂⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⢠⣶⣶⣴⣶⣶⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠁⠀⠀⢀⣠⡤⣤⣀⠀⠀⠈⠀⠀⠀⠐⠀⠃⠈⠀⠀⠀⣴⡿⠃
⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⣾⡿⠓⠻⠿⣧⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡉⡉⢙⠛⡛⢛⠛⠛⢛⠛⠛⠛⠉⠉⠀⠘⠃⠠⠀⠀⠀⠉
⠀⠀⠠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠳⡼⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣾⣿⠇
⠀⠀⠀⠀⠀⢐⣢⣀⠀⠀⠠⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡱⢋⠠⠀⡆⠠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋
⠀⠀⠀⡀⠀⡨⠻⢏⣿⢶⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢠⣀⡄⡤⣄⢠⣶⠄⣴⠆⠀⣴⡀⠀⠀⢀⠄⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⣿⣿⣻⣻⣿⣿⡆⢸⠛⠆⠧⠞⠸⠛⠂⠻⠶⠘⠛⠃⠀⣀⡀⠁⠀⠀⠀⢲
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢰⣿⣷⣠⣤⣀⣤
⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⡀⢠⠤⣄⠺⣏⡀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⣿⣿⣟⡉⣿⡿
⠀⠀⠀⢰⡆⠐⣶⠅⠀⠀⠀⠀⠀⠀⠀⠻⣏⢙⡶⠚⠘⠛⠁⠀⠀⠈⠀⠀⠂⢠⡅⠀⠀⠋⠀⠀⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢰⠏⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠘⢷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠸⠇⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠶⠶⠶⠶⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⡶⠛⠉⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠉⠙⠳⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣴⣶⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣻⣭⣭⠹⣦⠀⠙
⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠈⣿⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⣿⠿⠿⠛⠛⠛⠋⠉⠁⣸⠏⠁⠀⣸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⣤⠶⣿⢦⣄⠀⠀⠀⠀⠀⢠⣿⣠⡴⠟⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⣆⠄⠀⠀⠀⠀⠀⣿⠀⠋⠀⣿⠀⠀⠀⠀⣠⣿⣩⣥⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣙⣧⡀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⢀⣴⣟⡉⠁⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉⠉⠙⠷⢦⣤⣤⣬⣥⣭⣭⣤⡤⠶⠋⠉⠉⠙⠛⣦⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⢤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⡟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⣠⣄⠀⠀⡾⠁⠀⠀⠀⡇⢀⣤⡄⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡇⢸⡇⢿⠀⢠⡇⠀⠀⠀⢿⢈⡇⠰⡷⠄⠀⠀⠀⡇⠸⣷⣃⡸⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠶⠛⠀⠻⠞⠁⠸⡗⠋⠀⠀⠀⠀⠈⠋⠀⠀⣷⠀⠀⠀⠀⠃⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣥⣤⣓⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢫⠳⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢚⣣⣤⡅
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⣛⣉⡓⠂⠀⠒⠉⢁⣠⡇⢧⣀⠉⠘⠂⠀⠒⣉⣙⡉⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢉⣩⡇⠊⠀⠙⢋⡏⣜⣷⣟⡜⣏⠋⠁⠀⠀⣮⣉⡁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠋⢀⢘⡈⡁⠀⣠⢘⣴⢷⣼⣼⢷⡜⢠⡀⠀⡉⣘⢀⠈⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠀⣹⣟⡿⡀⡸⣟⣿⡁⠀⠈⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠬⢿⡉⢐⠈⣹⠯⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⠐⠐⠙⠌⠜⠙⠐⠀⢠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢸⡇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠸⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⢀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠑⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⣀⠀⠐⠈⠉⠁⠀⠒⠒⠒⠛⠓⠲⠶⠀⠉⣉⣀⠴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠀⠀⠐⠐⠒⠒⠛⠋⠉⠉⠉⠁⠀⠀⣸⡥⠔⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠄⢠⡏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠉⠀⠀⢀⡾⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⡖⠁⠀⠈⠀⢹⡆⠀⠀⠀⢀⡾⠁⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠳⣄⠀⠀⡠⠞⠁⠀⠀⢀⣾⠡⠤⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣄⣀⣀⣀⣀⣀⡤⠘⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠰⣶⡄⠀⠂⠐⠂⠁⠤⠀⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈⣿⡀⢹⡿⠀⠺⠷⠷⠏⠀⢰⡋⠀⡼⠫
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⢈⡁⠘⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⣀⠂⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠃
⠀⠀⠀⠀⠐⢐⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠈⠀⠀⡂⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠖⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠂⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⠂⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡘⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⢀⡔⠙⣦⠀⠀⠀⣀⡀⠀⠀⢰⡶⠿⣦⠈⠁⠀⠀⠰⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢡⠶⡁⠀⡀⠈⠀⠁⢀⠀⠘⠋⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠋⠀⠀⠀⠀⠀⠉⠑⠈⠀⠈⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣿⣿⣿⣿⠿⠛⠛⠛⠿⢿⣍⡻⢿⣿⣿⠿⣷⣦⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⠈⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠌⠻⣿⣷⠌⠿⣿⣦⣀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠈⠻⣿⣷⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠹⡝⣿⡄
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡀⠀⠀⢹⣜⣿⡄
⠀⠀⠀⣹⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⢿⣻⣿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠸⣏⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢂⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⣠⡀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢠⣿⣿⣿⠇
⠀⠀⠀⠈⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⣿⣿⡿
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆⢠⣧⣎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣤⠈⠙⠣⠀⠀⠀⠐⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢰⣬⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⢀⣀⣠⣤⡤⢰⣆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠿⣶⣤⣤⣤⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣽⣽⣯⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣿⣿⣽⣿⣯⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⢠⡿⠁⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⠀⠘⢷⣄⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠈⢻⡆⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠟⠁⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⣠⣶⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⠀⣸⣷⣤⡶⠶⠶⣶⣄⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⡶⠶⠿⣶⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠤⣤⣤⣤⡄⠀⠀⠀⢻⣇⠀⠀⠈⠉⠙⠛⠛⠛⠛⠻⠿⠿⢿⡿⠿⠛⠛⠛⠛⠋⠉⠉⠀⠀⢃⣿⠃⠀⠀⢀⣾⠇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⢄⣼⡏⣀⣤⡶⠟⣁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠸⡗⠈⣿⡇⠀⠀⠀⠀⠀⠀⣴⡿⠟⠛⣉⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠄⠀⠀⠀⠙⣷⣎⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣼⣿⡷⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣦⡀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣿⡇⠀⠀⢀⣠⣾⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠛⠉⠉⠉⠻⢷⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡶⠟⠋⠉⠉⠙⠛⠷⣦⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠿⠿⠿⠷⠶⠶⠶⠾⠿⠿⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⡟⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡶⠀⠀⠀⠀⢸⡇⠀⢀⣀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡄⠀⣴⠀⣾⠛⠛⢻⡇⠀⠀⠀⠀⣼⠛⢷⡀⠀⣿⠀⠀⠀⠀⠀⢸⡇⢀⡟⣹⠆⢀⡾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢀⣀⠈⣧⢀⣿⡀⢻⡄⢀⣼⠃⠀⠀⠀⠀⢸⣆⣼⠃⠘⣿⠛⠀⠀⠀⠀⢸⡇⠈⢿⣯⣤⠌⠷⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡶⠶⠛⠁⠀⠻⠟⠉⠁⠘⣿⠛⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠶⠶⠾⠇⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠡⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⢀⡽⠀⠀⠰⠶⠀⠀⠀⢰⣿⠍⠂⠀⢹⣇⡀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠤⠀⠉⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⣆⠀⠀⠀⠀⣤⡉⠃⠀⠀⠀⠀⠐⠠⠀⠉⠀⠁⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠁⠀⠀⠳⠬⣀⣼⠃⠀⠉⠁⠀⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣶⢚⣛⣿⣭⡄⠀⠀⠀⠀⠀⣴⣶⣖⣦⠀⠀⠀⠀⠀⢀⠀⢀⣀⣴⣾⣷⣾⣾⣶⡶⡖⠲⣤⣀⡀
⠀⠀⠀⠀⡀⣀⠀⢱⣿⣷⡄⠀⠀⠀⠚⣛⣛⠉⠁⠘⣛⣛⣃⡀⠀⠀⣿⣻⣛⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⣰⣿⣿⡄⣿⡏⠉⠁⠀⣀⣀⣔⣟⣉⢦⣤⠁⠈⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠊⠉⠉⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣿⠀⢈⣷⠈⠁⠫⠄⠀⠉⠉⠉⠉⠉⠘⣿⣦⡀⠀⠀⠀⠀⠀⠰⠶⠶⠆⠀⣤⣀⣄⣤⣄⣀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠃
⠀⠀⠀⠀⠀⡄⠁⢀⢁⠀⠀⠀⠀⠀⢀⣤⣴⣶⣼⣩⣧⣤⡀⠀⠴⠶⠶⠄⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠏⠀⠰⢶⡶⠂⠀⠀⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⠀⠂⠀⠀⠀⠀⠀⠿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⡀⠀⢰⠒⠛⠣⠄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠄⠀⠀⢀⡄⠀⠦⡤⠐⠀⠀⠀⡀⠀⠀⠐⠀⡀⠀⠂⠁⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢀⡰⠁⠀⠀⠀⠀⠀⠘⠀⢹⠈⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠈⠃⠀⠀⠀⢀⢠⠎⠀⠀⠁⠀⠀⠀⢠⡄⠀⠀⠘⠂⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢁⣀⠛⠂⢀⡀⠄⠀⠀⠀⠀⠀⢀⡴⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣀⣠⡤⠶⠛⠛⠃⠈⠐⢠⣀⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠉⠁⠀⠀⠀⠀⠀⢰⣄⡔
⠀⠀⠀⠀⠀⠀⠀⣠⠆⠘⠋⠉⠀⠀⠀⠀⠛⠁⠀⠀⠀⠉⠁⠀⠀⠀⢲⠀⠀⠀⠀⠘⠀⠀⠀⢰⣿⠷⠀⠀⠀⠀⣁⠄⠀⢠⣼⡉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠛⠀⢠⣤⠶⠿⠀⠀⠀⠀⠆⠀⠀⠄⠀⠀⠀⠇⢤⡄⠐⠗⠀⠀⢠⠈⠉⠀⠉⠀⠀⠈⠛⠀⠀⠈⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠀⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⣼⣦⣤⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⡀⣿⢏⢌⣽⣷⣦⠐⠁⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⡀⠰⠿⠃⠀⠀⠀⠀⢀⡀⠀⠀⠐⠃⢀⣤⣀
⠀⠀⠀⠘⣻⠿⣶⣾⡟⠡⢈⣷⡀⢀⡀⠀⠀⠀⠾⠂⠀⠀⢀⣀⠀⠀⢹⡆⠀⢰⠆⠀⠀⠆⠀⠀⠀⠀⠀⠀⢱⣿⡇⢀⣠⡀⠰⣿⠿⣿⡇
⠀⠀⠀⠀⢻⡀⠴⢙⣿⣴⣿⣿⣗⣻⣧⡄⠀⠀⠀⣀⣠⣴⣿⡿⠛⢦⡀⡇⠀⠈⠀⠀⠀⠀⠠⠦⠤⣀⡀⠀⢾⠿⠇⠀⠡⣌⠚⢯⠀⣹⠇
⠀⠀⠀⠀⠈⢣⡀⠘⠸⣿⠛⠹⣿⢫⣿⣿⣿⣿⣿⠿⠻⠿⠛⠁⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡛⣷⡦⢿⣆⣸⡇⣖⡶
⠀⠀⠀⠀⠀⠀⠑⣄⠀⠸⣧⣶⡟⣣⠈⠉⠉⠉⣀⣀⡀⠐⠁⠀⡴⡟⢴⠊⠀⠈⢴⡀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⣁⠟⠋⠀⠀⠀⠀⠉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠈⢻⠞⠉⠙⢶⡆⠁⠀⠉⠀⠀⣀⢰⠁⢏⠈⢦⣀⠀⠖⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢣⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⢋⠀⠀⠈⢃⠓⠷⠆⠀⠀⡀⠉⠛⠀⠈⠁⢀⡉⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠀⠠⠒⠲⠄⠀⠀⠀⠁⠀⠀⠐⠀⢘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⢀⡄⠀⠀⢀⠉⠀⠖⠀⠀⠀⠘⠁⠀⠀⠀⢰⠃⠀⢡⠀⠀⠀⠀⠐⠂⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣠⠄⠀⠛⠁⢀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⠀⠀⠘⠀⠈⠀⡸⠄⠀⠀⠀⢀⠀⡄⠀⠀⠀⢀⣆⢀⣨⣷⣄⡀⠀⠀⠀⠁⠀⠀⠀⢀⠗⢀⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣄⡀⠀⠀⠠⠋⠀⠒⠢⠀⠁⣀⡄⠑⠀⠀⠀⡾⢿⠿⠟⠁⢠⣿⣶⠤⡀⠀⠀⠀⢈⣠⣶⡿⢛⣝⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣤⡁⠀⠀⠸⠟⠀⠀⠀⠈⠠⠀⠀⠘⠀⠀⠀⣠⣴⣿⠟⠁⠀⢀⣿⡿⠖⠋⠉⠉⠈⡥⠍⣠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢼⡛⢳⣦⠉⠛⠋⠁⠤⠀⠀⠀⢈⣃⣀⣀⠀⠀⢹⣦⣶⣶⣿⡿⠟⠁⠀⢀⣴⠿⠋⠀⠀⢰⡦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣛⠓⠾⠁⠀⢀⣤⣴⠴⠶⠒⠉⢉⠉⠹⠟⠀⠀⢀⠏⠛⣍⣁⣤⣤⣶⣾⠟⠁⠀⠰⠟⣆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣡⡀⣩⠋⠀⠁⠀⠈⠉⠁⠀⡤⠰⠆⠀⠀⠀⠀⢠⠴⠉⠀⠀⠀⠉⢋⡩⢛⣁⣀⣤⡀⠀⢀⠈⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠂⠀⠀⠀⠈⠀⠛⠀⠀⠀⢜⠡⠀⠀⠂⠀⠀⠄⠈⢀⠄⠈⠋⠉⠉⠁⠀⠸⠀⡇⢀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠈⠁⢀⣀⡀⢠⡔⢂⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠐⠂⡿⠠⣤⠆⢀⣿⡿⠿⠿⠿⠿⠷⠶⠶⠖⠒⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠦⠀⠀⢀⠐⢠⠄⠂⠀⠉⢀⠀⠀⠀⠀⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣀⣠⣴⣦⣤⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣇⣶⣿⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡟⠉⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⢻⣿⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⣉⡁⠀⠰⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠈⠹⢳⢦⣀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠀⣱⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⡄⠁⢄⠀⠀⠀⠈⠲⣀⢬⣷⡴⠃⠀⠀⠀⠀⢺⢹⢸⠀⠀⠀⠚⣲⢾⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠐⠛⠢⠀⡀⠤⠉⠒⠦⠀⠀⠀⠀⠀⠀⠸⠼⠞⠀⠀⠀⠀⠈⠫⢤⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⡀⠀⠐⠀⡀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠎⢀⠀⠠⠀⠰⢠⡆⠀⠜⠀⠁⠇⠉⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⡀⢀⠀⢀⣔⣁⠠⠠⠌⡘⢰⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠐⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠀⢀⠀⠡⠄⠂⠀⠀⠀⢶⣦⡶⠀⡄⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⠛⠃⠀⠈⠐⠄⠀⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⣀
⠀⠀⠀⠀⠰⠀⠒⠢⢄⡀⣿⡟⠛⠇⠠⠀⡀⠀⠀⠀⡴⠃⠀⠳⡄⢀⠂⠀⣀⠤⠒⠓⠀⠃⠀⣀⠤⠐⠒⠶⠀⠀⠀⠀⠀⠀⠀⠳⡻⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠈⢢⡴⠋⠀⠀⠀⠀⠈⠳⣀⠞⠀⠀⠀⠀⠀⢸⠊⠁⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢤⣶⡄
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡎⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⡱⣄⡀⠀⠀⠐⢄⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⠞⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⢀⡠⠖⠉⠀⠀⠉⣓⠦⠤⣄⣑⣠⣀⠀⠘⢄⠀⠀⠀⠀⠀⠀⣀⠔⢁⣀⡤⠞⢁⣀⠤⠴⢋⠁⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⠋⠁
⠲⢎⣁⡀⠀⢀⠤⠂⠁⠀⠀⠀⣠⠈⠉⠉⠉⠙⡛⠛⠒⠒⠀⠚⠻⢟⡉⠉⠉⠉⠉⠠⠀⠀⠀⠀⠀⠢⣀⠂⣀⡈⣓⡤
⠀⠀⠀⢁⠜⠃⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠠⠀⠈⣄⡀⠀⠀⠀⠈⠣⣌⡀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠉⠐⠒⠒⠒⠀⢉⠏⠀⠀⠀⠀⣀⠴⠳⣀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣀⠀⠀⠀⠀⠘⡀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠒⠒⠒⠒⠉⠁⠀⠀⠀⠁⠢⡄⢀⠔⠊⠁⠀⠀⠀⠉⠓⠒⠖⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣮⣤⣤
⣿⣿⣏⣉⣛⣉⣙⣙⣏⣋⣽⣻⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠱⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠟⢠⠂
⠀⢠⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣇⠀⡀⠹⢦⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠈⠁⠉⠀⠀⠀⣦⣤⡄⡀
⠀⣏⠀⢱⣶⠀⣧⠀⢸⠁⣄⠀⠀⠀⠲⢿⣷⣶⣾⣿⣿⣛⣛⣛⣻⣛⣿⣿⣷⣶⡾⠟⣃⠸
⠀⠹⣦⡄⢹⣶⡻⠀⡖⣠⡘⢆⠀⢴⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣻⣿⠁⣆⣲⣦⣄⢲⡀
⣀⣀⣈⠳⣄⣹⡇⢰⠃⣿⡇⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢸⠋⠁⠀⡼⣸⠇
⠘⡍⢶⣄⣩⣭⣽⣿⣦⣿⣧⠟⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠇⣎⣠⠴⢊⣼⠟
⠀⠈⠢⣀⠜⠋⠹⠛⣿⠿⣁⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⣸⢋⣤⠾⠛⠁
⠀⠀⠀⠈⠐⠒⠒⠁⠀⠀⠙⠷⠆⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⠏⡰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢾⣒⠁⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣫⣞⠓⢤⣤⣐⡢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠀⠉⠙⠋⣆⠰⢍⡛⠷⠶⠶⠶⠶⠾⡿⢿⣿⠿⠋⠻⢂⠽⠂⢉⣆⡕⠀⠀⢀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⡴⣞⡋⠠⡆⠀⢀⠀⢀⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣻⣲⣿⣻⣿⣻⣻⣿⣷⣿⣿⣿⠿⣿⠛⠋⠉⠀⠀⠉⠙⠣⡀⠸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣏⠙⠶⣤⣤⠤⠴⠒⠚⣷⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣿⡀⠀⠀⠀⢀⣽⣬⠿⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠮⣿⣷⣠⠤⠚⠋⠁
⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⠖⣰⣖⡲⢒⢖⣰⣀⣶⣤⣔⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡽⠚⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⣿⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣼⢀⣤⣶⢶⣦⣄⠀⠀⢀⣤⣶⣶⣶⣿⢻⣦⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣼⡇⠚⢁⣀⣀⣈⡹⠃⠀⣾⣋⣀⣀⣘⠛⠁⢹⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⠀⠐⠋⠿⠟⠛⠛⠀⠀⠸⠟⠛⠻⠟⠙⠤⠈⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡭⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⡦⣀⠀⠀⠀⠀⠀⣼⡋
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠀⠀⠀⣠⣶⣆⣴⢿⣿⣿⣦⣿⣿⣦⠀⠀⠀⠛⣾⠃
⠀⠀⠀⢀⡀⣠⣾⠇⢾⡏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⣿⣿⡟⠳⠶⠿⠶⠟⣿⣿⣿⡄⠀⠀⢀⠁
⠀⠀⢠⣾⢃⣼⣿⣴⣿⡇⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣴⣶⣶⣶⣤⣿⣿⣿⣧
⠀⠀⣸⣿⡞⠻⣿⡏⣹⣧⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠀⠈
⠀⠀⠙⢿⠇⠠⠿⠀⠻⣿⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣐⣿⡀⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠿⠁⠀⠀⠀⠚⠿⠿⠷⠶⠛⣿⡿⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡏
⠀⠐⣿⣷⣶⣤⠀⢀⣠⣴⣾⣿⣿⡿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣁⣀⣀⡀⠀⢠⡇⣴⢠⡄⣸⢀⠆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⣿⣿⣿⣿⣿⣶⣿⡇⡇⣾⣧⡇⣸⡄

⠀⢀⣀⣀⣀⡀⠀⠀⣠⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡆⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠿⠿⢿⡇⠀⣠⣮⣿⠟⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠤⠄⠀⠤
⠀⠀⠀⢸⠁⢠⣽⠟⠁⠀⠀⠀⠀⠀⠀⢉⣶⣶⣆⠀⢀⢺⣿⠟⠀⠀⠀⢀⠄⣀⠠⠄⠘⠛⡀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢰⣷⣷⢸⣶
⠀⠀⠀⢸⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡗⢸⠟⠁⠀⠀⠀⠀⢠⡀⠶⠀⣴⠢⠶⠄⣁⠀⠀⠀⠀⢀⡀⠀⠀⢐⣧⣕⢸⣨
⠀⠀⠀⠘⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠃⠃⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠖⠊⠁⠀⠀⠀⠠⠟⠁⠀⠀⠘⠿⠟⠸⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⢀⣀⢀⣀⣠⡤⡀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣼⣧⣶⠆⠀⠀⠀⠀⠀⠀⠀⡞⠛⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣿⡆
⠏⠉⠉⠉⣀⠀⠃⢀⠉⠛⠒⠲⠀⠀⠀⠀⣰⠋⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣻⣿⣟⣀⣠⣀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⣴⣶
⢸⠈⠁⠀⠀⠀⢑⢈⡀⠐⠀⠌⠀⠀⠀⠀⣏⠀⣀⡀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡫⣽⢿⣿⡇⠀⠀⠀⠀⠀⠈⠛⢿⣿⠛⠁⠈⠉
⠈⠢⡐⠶⠞⡰⠣⡈⢉⣠⠎⠀⠀⠀⠀⠀⠙⢻⣿⣿⡛⠋⠀⠀⠀⠀⠀⠀⢸⣿⣷⣝⣻⣾⣿⡇⠀⠀⠀⠀⠀⠀⣠⠏⠙⣆
⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢰⣷⠄⠀⣀⠀⠀⠀⠀⣀⠀⢀⣀⣘⣿⣀⡀⣀⠀⠀⠀⠀⠀⢀⣙⣲⠤⣞⣋⡀
⠀⠀⡇⠀⠀⣽⣿⡀⠀⢘⠀⠀⠀⠀⠀⠉⠛⠿⣷⣇⢸⡟⠁⠀⠀⠀⠀⠈⠙⠶⡄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣶⠀⠉⠁
⠀⠀⡇⣤⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢾⣽⡶⠶⠶
⠀⠀⣃⣸⣿⣿⣿⣿⣟⣐⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧
⢰⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⡿⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⣿⣿⠇


⠀⢀⡠⢀⣀⠀⠀⣀⣀⢠⣀⠀⠀⠀⠀⠀⢀⡴⠒⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠆⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿
⠀⣿⠀⠀⠀⠙⡏⢸⡇⠀⣿⡇⠀⠀⠀⠀⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢠⠲⡤⢾⣂⡛⢠⠋⠀⠀⠀⠀⠀⠀⠘⡿⣿⡿⠂
⠀⣿⠀⠀⠀⠀⡇⠈⠁⠀⣿⡇⠀⠀⠀⠀⠀⢣⢀⣀⡎⠀⠀⠀⠀⠀⠀⢺⠉⣠⣦⡈⣹⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⣿⣤⣤⣀⣀⣇⣀⣤⣤⣿⡇⠀⠀⠀⠀⠀⠈⣾⣿⠁⠀⠀⠀⠀⠀⠀⠺⢤⠈⡋⢠⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡄
⠀⠉⠉⠉⠉⠙⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠤⢢⣙⡁⠀⣀⠀⠀⠀⠀⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⣀⡀⠀⠹⠿⣷⡀⠠⣌⠶⣠⡀⠐⠁⠘⣧⢴⣩⣄⢀⢀⠵⣃⢨⣄⠐
⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠈⠛⠿⠦⢦⡴⢼⣷⣤⣀⣼⣿⣧⠀⣩⣉⡿⢶⡅⠉⢂⣰⣊⡽⠶⢷⠀⠀⠍⢦⡥⠠⢀⡤⣀⡒
⠀⠀⠀⠀⠠⣴⣶⣶⣮⢶⡔⢦⡐⠒⣶⢲⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣶⣤⣿⣿⣿⣿⣿⣿⣮⣳⣤⠀⢀⣀⡐⠈⠀⡈⠲⢀⣀⣀
⠀⠀⠀⠀⠀⠉⠉⠀⣀⣄⠙⢿⣷⣿⣿⣿⣿⠛⣿⣿⣿⣿⣯⣁⢘⡟⠉⢙⣟⣛⣿⣿⣿⢦⡟⠛⠛⠛⠻⢿⣥⣽⣿⣫⣿⣷⣿⣟⣿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠵⣶⣦⣽⣿⣿⣿⣟⣽⠛⠿⣿⣿⣿⣿⡋⣿⣭⣿⢮⣙⣿⣼⡿⠿⠿⠗⠶⣼⣷⣶⣤⣧⣼⣿⣿⣟⣙⢃⡀
⠀⠠⠀⣴⣶⣷⣶⣦⣦⣾⣿⣛⣿⡯⣉⢿⣿⣷⣼⣷⠓⣻⣿⣁⢠⣿⠛⠿⣿⣿⣟⣷⣛⣀⣤⣦⣶⣿⣿⣿⣿⣿⡿⣿⡟⠛⣛⠛⠛⢀
⠀⠸⠀⠀⠊⠋⠋⠉⠹⣿⣿⡟⢻⣿⣿⣿⣏⣁⣽⣿⣿⣿⣿⣏⣻⣯⣦⣷⣾⣿⣿⣟⢻⣿⡛⠋⣩⣹⣿⣿⣿⣶⡿⠿⣿⣿⡟⠛⠁⠎⠰
⠀⠀⠀⠀⠂⣶⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢉⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣋⣙⣶⣿⣿⣿⣶⣿⣿⣇⣄⣀⣀⡀
⠀⠀⣀⣠⣄⣰⣿⡿⢻⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⢻⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠿⣾⣿⣿⣿⣿⠿⣿⣿⣋⣙⣻⢿⡿⠷⢤⣀⣀
⠀⠀⣐⣾⣿⡟⣀⣶⡿⠛⠋⣡⠿⠿⢿⡿⣿⠉⠙⠀⢿⠛⢿⣿⢻⡟⣿⡿⢻⣿⣿⣿⡯⣶⢿⣿⣧⣁⢀⣜⢳⣿⢿⣿⡿⢿⣷⡀⠀⠉⠟⠻⠢
⠀⠀⠛⠛⠈⠁⠺⠻⠛⠀⠀⠎⠈⠀⠈⡇⣼⠃⠀⢄⣿⠿⢣⠀⣽⠃⣽⠁⢸⡟⢹⣿⡦⠀⠠⣿⢹⠛⠀⠐⠀⠻⣿⣿⡄⠂⠻⣽⣧⣄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⣀⠀⠀⣧⠐⠀⠀⠀⣾⠀⠀⢀⡿⢀⣿⠀⢸⠇⢸⠏⠃⠀⠀⢸⡿⠿⠟⢻⣄⣄⠀⠛⢿⠆⠀⠈⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠉⠵⠂⣿⠀⠀⠀⠀⢸⠀⠀⣸⠃⠏⣿⠃⢸⠀⣿⣴⠀⠀⠀⢸⠁⠀⠀⠈⠑⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠤⠖⠀⢸⠀⠀⡿⠀⢀⡇⣰⣾⠀⡏⠈⠀⢰⠆⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⢸⡇⢸⡇⠀⢸⡇⠉⣿⢰⡇⠀⠀⠈⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⢸⡇⠀⠊⠁⠈⡇⣾⢀⣤⣼⡇⠀⡇⢸⡇⠀⢲⠄⢸⢹⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣷⡧⠠⠁⢸⠇⢀⡇⢸⠀⠁⠘⠀⡎⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣻⠀⠀⠀⠀⣿⠁⠀⠀⢸⠀⢨⡇⣸⠀⠀⠀⢸⠃⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢳⣸⠂⠀⠳⣞⣿⠀⠀⠀⣿⣤⢈⡁⡏⠀⠀⠀⡿⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿⠀⠀⠰⣧⡟⢸⢷⡇⠀⠀⢰⡇⠈⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠀⡺⢹⡆⠀⠀⣿⠀⡾⠃⡇⠀⠀⣼⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢰⡇⠘⡇⠀⠀⡏⠀⡇⢸⠃⠀⢀⡟⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣿⠀⣼⠃⠀⡇⠀⢸⡇⢸⡇⢸⠀⠀⢸⠇⠀⠀⢾⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣽⡀⡿⠀⠘⡟⠀⢸⡇⢸⠇⣾⠆⠀⡾⠀⠀⠀⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡇⠀⠀⣷⠀⣸⠃⢺⢀⡟⠀⢰⡇⠀⠀⠀⣼⠀⠀⠀⠀⡀⠢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢄⣤⢀⡞⣿⠃⠀⠀⣿⠀⣿⠀⣿⢸⡿⢦⣿⣧⠀⠀⠀⣿⠀⠀⠀⡀⠰⠀⠦⠂
⢀⠴⠶⠖⠀⠀⠀⠀⠀⠁⠛⠓⠀⠀⠊⠀⠛⢁⣿⠰⠀⠀⢻⣸⣿⢠⣿⣿⣷⣀⣿⠀⠀⠀⠀⣿⣀⣠⡤⠶⣿⣶⢦⡁⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠁⠔⠈⠁⠀⠔⠳⠀⠀⠀⠘⢻⢿⡟⠣⢸⣿⡀⠀⠀⣸⣿⡟⢠⡿⣿⡋⢸⡟⣳⣿⢷⠄⣿⡏⠀⣱⣂⣛⣁⠀⠀⠀⠸⠋⢁⣠⠠⢤⠄
⠀⠀⠀⡴⠚⠗⠀⢀⢄⣀⣤⡄⠀⢀⡈⢋⡀⣿⣿⡧⠂⠀⣹⣿⡗⢺⡧⣿⢃⣾⣅⣃⠀⠊⠀⣿⡇⠉⠎⢹⡝⡁⠻⠄⠀⠠⠤⢤⠁⠀⠁
⠀⠀⠀⠀⠀⠉⠱⠾⢿⠉⠋⠁⠀⠈⣼⠛⣿⡟⠻⣷⣦⠀⢿⣿⡁⢘⣱⣿⢙⣿⢿⡿⣶⣄⠙⣿⣇⠾⠶⣾⡟⢿⣿⣮⠀⠀⣀⡈⠀⢀⡤
⠀⠀⠀⠀⢀⡀⠀⠀⢸⠛⠀⠓⠂⠀⢈⢦⣿⡃⢀⣟⢿⡇⠸⣿⡇⣿⣿⡟⣸⣿⣦⠈⣉⣿⣢⣹⡏⢠⣦⣽⣇⠠⡾⠃⢘⠋⢉⠙⣀⡌
⠀⠀⠀⣀⣤⡇⠸⢦⣰⣿⠮⣷⣾⣷⣇⣼⣿⡇⠀⣿⣿⣧⡞⣛⢷⣿⣿⣿⣿⣿⣾⡯⠋⣸⢌⣹⣿⣯⣶⣿⣿⡞⢱⢧⡀⠀⢁⣾⣹⠈⠃⡀
⠀⠰⣤⣱⣦⣗⠰⡆⢉⣿⣾⡫⣿⣿⣿⣿⣿⢷⣨⡿⣿⣿⢿⣿⡇⢈⣱⣿⣽⣿⣿⣽⣖⡁⣸⣿⣿⠿⠿⠿⣿⡷⣼⢿⣹⣦⠷⣈⠂⡤⠀⠂
⠀⣠⡦⢿⡦⠙⠓⠀⠈⠛⠙⠋⠋⠉⢫⠛⠛⠙⠃⠘⠛⠛⠛⢿⣜⡿⠿⠿⣿⣿⠿⡿⠿⣿⣿⢿⣿⣷⣴⣿⣿⣿⣿⡟⣯⣉⠈⠁⣈⠗⣁⣼⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠑⠀⠀⠘⠃⠚⠘⠉⠛⠋⠁⠰⠾⠦⠤⠄⠀⠘⠏⠙⠟⠻⠻⠟⠍⠛⢛⠛⠿⠉⠷⠓⠖⠦⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⡷⠀⠀⠀⠈⠺⡿⢿⣿⠿⣿⠀⠠⠠⢀⠀⠀⠀⠀⠀⢀⠠⢠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣻⣿⡄
⠀⠀⣊⠰⠆⠲⠂⣟⢂⠀⠀⠀⠀⣿⣿⣹⣿⡷⠀⠦⠘⠁⠀⠀⠀⠙⠀⣀⣤⢤⠉⣿⡟⡀⠀⠀⠀⢀⣴⣾⣷⠜⠛⠘
⠀⡆⣍⠁⠀⠀⠀⠿⢀⠀⠀⠀⠀⠘⠷⠶⠿⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃⠟⠀⠀⠀⢀⠴⠛⠛⠉
⠀⠀⢉⠲⢀⠠⠂⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁
⠀⠠⣶⣖⣲⣶⠀⡁⠀⠀⠀⠀⣀⣤⣵⣾⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⡯⣝⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢹⣾⠀⠀⠀⠀⠀⠀⡀⢹⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⠠⢀
⠀⠀⠸⠤⠩⠄⠀⡀⠀⠀⠀⠀⣿⠸⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⡏⠸⢄⡀⠀⡀⠀⠀⠀⠫⡒⢟⠛⠛⠀⡀⠜
⠀⢀⠐⠀⠀⠂⡀⠀⠀⠀⠀⠀⠙⠓⠒⠒⠒⠒⠛⠛⠀⠀⠀⠀⠀⢸⣾⠃⠀⠀⠙⢧⣼⣆⠀⠀⠀⠈⠐⠎⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⠀⠀⢠⢴⠀⢐⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣠⣀⡴⠐⠒⠂
⠀⢀⣶⠘⡂⢃⢈⡀⡀⠀⠀⠀⠰⠈⠀⠈⠒⡇⠀⠀⠀⠀⠀⠀⢀⡤⢢⠎⣦⢳⡐⢤⡀⠀⠀⠀⠀⢠⠏⡏⠉⢷⠃⠐
⠀⢰⣾⣾⣷⣿⣾⣷⡇⠀⠀⠀⠠⠀⠀⠶⣶⣶⣶⣶⠆⠀⠀⠀⢛⡤⠛⣤⠛⣴⠛⣴⠋⠀⠀⠀⠀⣾⡀⡇⠀⢸⡆
⠀⢸⣿⣏⣽⣟⣿⣿⡇⠀⠀⠀⠀⢁⠀⢠⠈⢹⡉⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⠉⠀⢹⡇⠀⠀⠀⠀⠘⠓⣿⣖⠛⠁
⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣰⣾⣿⣷⣤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠗⠉⣿⣿⣿⣿⣿⡇⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠹⣿⡿⠛⠉⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠛⠛⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣠⣥⣤⠄⠀⢀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠞⣡⢛⣛⣽⣿⣯⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢱⠋⣰⣧⣾⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣾⣿⣿⣿⡄⠈⠻⠿⠿⣿⡻⡿⢿⣿⠿⢿⣿⣿⣿⣿⣿⠟⠉⠹⡄
⠀⠈⠙⠲⢚⠷⠛⣲⠾⣧⣳⣤⣿⣿⣿⣿⣿⣿⣟⣛⣷⠒⠈⠗⠀⠈⠻⢯⣳⣯⣝⣿⣿⣿⡿⠟⠁⠀⠀⠀⢧
⠀⠀⣤⣄⣙⣿⡗⣛⡾⢳⣿⢻⣿⣿⣿⣿⣿⣻⡿⣯⠽⠿⠟⠁⠀⠹⠃⠀⠙⠪⠿⠉⠉⣀⣤⣤⣄⣀⡀⠀⠘
⠀⠀⠀⡶⢴⡅⣘⣽⣖⢿⡇⢸⠮⢋⣿⣿⣿⣿⣿⣷⣾⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⡀⡴⠁⣏⢼⡞⠈⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠘⠂⠀⠘⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣿⡿⠿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠓⠒⠀⠤⣶⡶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⡀⠀⠀⢀⣠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠛⠾⠿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⡀⠀⠻⣿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢞⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣋⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣵⣯⣫⣿⠀⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣟⠁⢿⣿⣏⣾⣿⣟⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣛⠽⡵⢺⡄⠘⣿⣿⣿⣿⡏⠄⢀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠈⠵⣚⣵⠗⢶⡀⠘⢿⣿⣿⣿⢫⢴⡓⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⢻⡀⠀⠀⠘⡥⠖⣿⡳⢦⡀⠈⢻⣿⣯⡞⣻⣬⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⠀⠀⠈⣧⠀⠀⠀⠐⣋⡼⢟⡰⢦⡄⣸⣿⣾⣻⣿⠿⠛⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠻⢿⡿⠀⠀⠀⠘⣧⠀⠀⠀⠡⠞⣩⠟⢰⣭⡿⠛⠛⠉⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⣸⣇⠀⣀⡀⠀⠘⣧⠀⠀⠀⠉⡴⢚⡵⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇⠀⠀⠀⠀⠛⠛⠻⣿⡇⠀⠀⠈⢧⠀⠠⠀⠘⢩⢔⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢹⡀⠀⠀⠀⠀⠀⢀⣿⡇⢀⡀⠀⠈⢷⡀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡷⠀⠀⢳⡄⠀⠀⠀⠀⠘⠿⠛⢿⣿⠀⠀⠀⠻⣄⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠃⢠⡀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⠀⠀⠙⣦⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠞⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠂⢀⠡⡀⠀⠘⢦⡀⠀⠀⠀⠀⠈⠉⠀⣿⣀⡀⢀⣞⢀⣀⣀⣀⣠⡤⢶⡞⠋⢉⡏⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠀⡊⠀⡱⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢿⠿⢿⡾⠉⣿⠋⠁⣽⠇⠀⠸⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠀⠀⠀⡄⠈⣆⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡼⠁⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠁⠀⠀⠑⣄⠀⠀⠈⠓⢦⡀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣠⣤⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠰⠀⢤⠀⠀⠀⠈⠑⣄⠀⠀⠀⠉⢳⡾⢻⠟⣶⠞⡟⢲⠞⠻⠛⠞⠉⠉⠙⠁⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠘⡀⠀⢀⠀⠢⠀⢻⠢⣄⣠⢮⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠶⣺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣤⡀⠢⣀⠑⣦⣇⡼⠃⠀⠀⠉⠉⠉⠋⠙⠉⠉⠉⠉⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣡⠟⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⠒⠛⢻⠛⠛⣿⠛⠛⠛⠒⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣄⣀⣀⣠⣤⡤⠤⠤⠤⠤⠴⠦⠤⠤⠤⢤⣤⣄⣀⡀⣙⣿⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠳⣶⠒⠒⠛⠋⠉⠁⣿⠶⢦⣄⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⡧⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣯⣄⠀⠀⠀⠀⠀⢰⣇⣀⡤⠔⢁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⣴⡉⣿⠀⠀⠀⢠⠿⠤⠶⠖⠚⠉
⠀⠀⠀⠀⠀⢀⣀⣠⣤⡼⠿⣦⡀⠀⠀⠀⠀⠀⠀⣿⢾⡓⠀⢸⡄⢀⡴⠓⠶⠤⢤⣀⡀
⠀⠀⠀⠀⣾⠋⠉⠀⠀⠀⠀⢨⣿⣶⣤⣀⠀⠀⠀⠻⠾⠶⢚⣫⣴⣿⡄⠀⠀⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣶⣶⣶⣿⣿⠿⠟⠛⠁⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠓⠶⠶⠶⠶⠶⠒⠚⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣴⠉⠹⠟⠀⣀⢀⡠⡄⢤⠒⠆⢲⣄⣶⢡⠂⢢⠰⣄⢀⢀⡈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢴⢿⠒⠀⠃⢹⠈⠐⡇⣇⢸⣷⠆⠼⢹⣽⠘⠦⠜⠠⡟⣮⠀⢱⠞⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⡇⠞⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠘⠄⠛⠄
⠀⠀⠀⠀⢰⣶⡖⠀⡤⠉⠀⠀⠀⠀⡴⠛⠛⠳⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶
⠀⠀⠀⠀⢸⣿⡇⠐⠀⠀⠀⠀⠀⠀⠙⠂⠀⠐⠒⠟⠈⢉⣉⡭⢤⠤⠤⠒⠒⠞⠓⠒⡿⣄
⠀⠀⠀⠀⢸⣿⣇⠀⣷⠂⠉⠐⠒⠂⠉⠉⠉⠁⠀⠀⡈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⢶⡀
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢰⢧⡉⠐⠒⠤⣀
⠀⠀⠀⠀⠘⡄⠙⢄⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⢧⠀⠀⠑⣄⠀⠀⠀⠀⠈⠒⢄⡀
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠈⠢⡀⠀⠀⠀⡴⣏⠀⠈⣑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠈⠢⡀⢰⠁⠈⠉⠈⠀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢀⢴⡈⠲⣧⡄⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⡏⠀⠉⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⡤⣀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠈⢻⢷⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⠷⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⡀⠓⢿⣦⡀⢱⡒⢤⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⢯⣧⠀⠙⣯⣷⠄⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢈⠻⠆⠠⠃

⠀⣀⡀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠠⢄⠀⡀⢲⣾⣻⣍⣯⣿⣦⣾⣤⣌⡓⣒⣠⣤⡤⢤⣀⠀⠀⠀⠀⠀⠀⣰⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣺⡿⠻⠩⠿⣯⡿⠛⠀⠀⠀⠠⠤⢀⣤⣦⣲⠀⠀⠉⠙⠻⢯⠿⠷⠶⠿⢿⣶⣤⡀⡞⡵⠚⠉⠉⢝⠺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣧⣶⠐⠗⠃⠀⠀⠀⠀⣤⣄⢸⣎⡙⣣⠜⣻⠠⣴⠒⡦⣀⠀⠀⠀⠀⠀⢠⡄⢙⣿⡿⣇⢿⡀⠀⠀⠀⠀⢹⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡯⡋⡷⢻⣿⣦⠀⠀⠐⣠⣦⡽⣿⣷⡿⢿⣧⠲⣌⠶⢏⠁⠀⠀⠀⠀⠀⢀⣴⢎⡴⢣⣿⡷⡁⢸⣷⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡻⡃⠑⡘⢛⣩⢈⠀⠀⠀⠸⠾⠃⠛⣀⠨⣅⡐⠛⠭⠈⠁⠀⠀⠀⠀⠀⠀⠙⠉⢀⡠⡾⣔⣻⠿⠶⠀⠀⢀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢶⣥⣬⣉⣊⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠆⠀⠀⠀⠀⡠⢤⣲⣖⣠⣥⠶⠿⠋⠀⠖⡦⢧⠀⠧⠄⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠟⠃⢀⠀⠈⠉⠙⠛⠛⠓⠿⠿⠛⠛⠖⠲⠟⠷⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠐⡦⠔⠀⣁⡤⠐⢫⡀
⠀⠀⠀⠀⣠⣮⠀⠀⢀⠰⡇⣀⣂⠀⠄⠀⠀⠀⠄⡀⠀⢠⣄⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣄⣦⡀⠀⠀⠀⠀⠠⢋⠴⢂⣉⣉⠀⠀⠀⢻⣚⣀
⠀⠀⠀⢠⣿⡇⠀⠀⠋⡐⢛⠉⠼⣣⠀⠀⠀⠀⣼⣇⡀⢈⣁⡜⣠⠤⣶⠚⣯⢣⢄⠀⡀⠀⠛⢾⠏⡀⠀⠀⠀⠀⢀⠊⠈⣋⢬⠏⠉⠀⠀⢀⠉⢹⣷
⠀⠀⠀⢾⡏⡀⠀⠀⠒⣤⣿⡛⣾⣀⢀⠁⠀⠘⠛⠻⠛⢀⣍⣤⢧⣉⡰⢇⡿⡞⠂⡞⣉⡶⠗⠛⠛⠀⠀⠀⠀⢾⣋⠈⠁⠉⠅⠀⠀⣂⢢⢾⣴⢎⡿
⠀⠀⠀⠘⢿⣿⠿⠦⠀⠈⠓⠉⠀⠁⢴⠆⠑⡀⠀⠀⠒⠁⠠⠀⠸⣿⠊⠁⠀⠉⣡⠟⠛⠇⠀⠀⠀⠀⠀⡀⠐⠯⣭⠔⠀⠀⠀⠀⠀⣤⣴⣿⠔⠋
⠀⠀⠀⠀⠈⠙⢶⣢⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⢠⢀⡤⠀⠀⣀⣈⣤⠼⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠾⣦⢶⣤⣀⣀⡀⠀⠀⠀⢫⡑⠂⠤⢤⣤⣠⣤⣤⣔⣶⣴⡬⠴⠖⢋⣝⣀⣤⣬⣭⠬⠖⢛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⡀⠉⠉⠓⠒⠒⠛⠒⠒⠶⠦⠴⠶⠒⠒⠒⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⢶⣶⣤⣦⣤⣤⣤⣴⣰⣶⠤⠤⠆⣲⠖⠲⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⡠⠂⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡙⣆⡾⠁⡀⢸⢃⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣾⠀⠘⢯⡁⠀⣄⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⢀⡏⢹⠳⢌⠋⠢⠈⠀⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⣀⠳⠐⢱⡇⠸⡦⠀⠀⠀⡁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⢢⡀⠑⢤⣇⠄⠀⠈⠉⠀⢻⡇⠀⡀⠀⢠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢉⠃⠸⡆⠠⠀⣀⡈⢢⠰⢸⡇⣾⠟⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠄⢶⢹⡀⠀⠀⠉⡥⠀⢸⡁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢸⠆⠀⢣⠀⠠⣞⠁⠀⢸⠁⠀⢠⠔⠁⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠔⠋⢤⠤⠀⠈⡇⠨⣅⠀⡀⢸⠀⠀⢛⡒⡉⠀⠈⠀⠈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢹⠀⢸⡀⢷⢸⠇⢄⡈⢻⠀⠀⠠⠀⠐⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⣇⠸⣿⠾⢼⡟⠒⣸⡏⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⢼⡤⠼⠀⠈⠀⣸⣿⠀⣾⡆⠀⠀⢰⣿⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣑⠆⠀⠀⣇⠀⠛⠀⢀⣿⣿⡀⠈⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⢹⠀⠀⠀⠘⣿⣿⣧⠀⢠⡶⠀⢰⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⡄⠀⣿⣿⣿⣶⣄⡀⢤⢀⣤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⣶⣿⣿⣿⣿⣿⡟⢲⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣝⣿⣿⣿⣿⠀⡈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠛⠋⠙⠿⠿⠇⠀⠀⠻⠿⠟⠁


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

⠀⠀⠀⠀⠀⠂⠀⠂⠀⢄⠰⡀⢁⠄⢄⢅⢍⠀⠃⡄⢐⡣⢀⠄⡰⡦⠤⢴⢂⣄⡀⢠⠆⠀⠀⡀⠀⠀⡤⣤
⠀⠀⣀⠀⢀⠀⠈⠀⠌⠐⠀⢧⡈⠄⡀⠚⡾⢀⠬⠔⠒⠒⠒⢊⠇⠈⠒⢲⠀⠀⠉⠓⢍⠂⠀⠀⠀⡰⣀
⢁⠈⠀⢀⡄⢡⡀⠀⠐⠂⠁⢢⠉⠢⠔⠠⢰⣡⠒⡆⠀⢤⠔⣫⠔⠒⠲⣼⣄⡤⠒⠲⣄⠇⠀⠐⡆
⠄⠀⠀⠀⠙⠉⠉⠂⠀⠰⡠⠤⠥⢘⠩⡉⢹⣧⡶⢥⡀⢸⢰⠁⢀⡤⠤⠬⣟⡗⠀⠀⣸⠇⠃⠀⠀⠀⠀⢀
⢀⠀⠐⡠⠒⣢⠶⢦⠰⣄⡱⢄⣀⠀⠑⣜⡌⠉⠀⢀⣹⢁⢸⣴⡡⠚⣉⣉⡉⠛⠛⠋⡤⢁⠁⠀⠀⠰
⠀⠀⡞⠀⢸⠁⠀⠀⣷⠀⠉⠉⠓⢝⢦⠸⣵⡰⢊⡭⢖⡏⢇⢏⢷⡫⠤⣤⠈⡇⠀⢰⡱⠋⢱⠀⠀⠀⢀⢨⠔
⠀⠸⠀⠀⠀⣳⣾⣲⡟⢀⠖⠒⠦⡀⢣⣷⠙⡱⢁⠔⠉⡏⡸⡘⢸⣀⣴⣣⠜⠁⡠⣫⠃⠀⡘⣈⠀⠁
⠀⡌⢸⠉⢩⠟⠁⡇⡇⢸⠀⡟⠲⣽⣼⠏⢀⣇⢺⣀⣞⡴⣳⣣⠤⠄⠻⢶⣶⣭⠖⣁⠤⣊⠜⢸⠄⠀⠉⠀⠄
⠰⢗⣁⡀⠸⡀⠀⠳⡘⢄⣳⡵⠤⠼⠀⡆⡇⢜⡿⠛⣛⣙⠛⢶⡩⣉⡦⠤⢄⡺⢟⡊⠉⠰⣶⠾⣖⠪⡀⠀⢀
⠈⢶⢱⠈⠳⡙⠢⠤⠬⠿⣓⡪⠭⢕⣲⢽⣬⠋⡴⢋⣥⣌⡙⢦⠙⣿⡖⠓⠢⡈⢦⢱⡀⠀⡹⠣⠤⠇⠘⡐⠈
⠂⢱⢸⠀⠀⠋⠉⢉⡦⢬⢳⡲⡗⠒⢬⣳⣯⢸⡁⢿⠀⠀⣷⢈⡇⣻⢯⡓⠤⢼⠮⢧⡑⠺⣁⣀⣠⠀⠀⡇⡐
⠀⠈⡌⡇⢰⠒⠢⣎⠀⠈⢇⠳⡉⠲⠤⠼⣿⣄⠳⣌⡙⢋⣡⠞⣠⡛⣗⠫⢕⣒⡪⢭⣶⡒⠒⠢⣌⢦⡀⢇⠠
⠠⠀⠀⡈⠢⠽⡶⠿⠇⣀⡨⣵⡮⠑⠒⠺⣉⣊⠷⣤⣍⣭⣤⣾⡕⢸⠸⠀⡖⠒⢞⢯⠑⡌⢢⠀⠈⡆⠈⢉⢵
⠀⡄⣁⠁⠀⠁⡅⡔⡩⠚⢉⡥⣚⠿⠷⣦⠐⠒⢩⢯⠞⡽⠉⡇⢹⠁⣰⡟⣟⠦⣼⠀⡇⢸⢸⢀⣴⣃⣀⡇⡘
⠈⠀⠈⠀⠀⠀⡉⡜⠀⢠⣫⠊⢀⠔⢫⠟⠉⡇⡌⡎⣸⣀⠔⢁⢎⣄⣿⢧⠈⠲⠤⠴⠁⣼⠏⡿⢯⠀⠀⠀⡇
⠀⠀⢰⠈⠀⠈⢃⢇⡠⢎⠇⠀⢸⡀⠛⠒⣪⢷⣱⢱⣸⠵⣚⡡⠎⢟⡄⠣⣕⢤⣀⡀⠀⢿⠀⠀⢀⡇⠀⡴⠁
⠀⠈⠇⡀⢀⠀⠈⡗⢀⠚⣠⣤⣤⣈⣉⣉⡤⢊⠟⡇⢁⣏⠁⠀⢀⠘⡜⢄⠀⠉⠑⢮⠙⠀⠳⢶⣫⢤⠪⢐⡐
⠀⠀⢜⠁⠤⡒⠈⠀⣰⡏⠀⠀⢬⣽⡒⠒⠒⠁⢀⠇⡇⠈⢓⠞⢻⣇⣈⣒⡍⢒⠒⠊⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠁⠁⠘⠀⠀⠀⠀⢁⢰⠙⠦⠤⠚⠙⡟⠦⠤⠔⣫⠔⠓⠀⠸⠤⢋⠇⢌⢀⠀⠠⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠁⠀⣀⠄⠋⠆⡑⢤⣀⠀⠀⠧⠤⡀⢰⡡⢤⠤⠤⠖⣒⡥⠂⠀⢂⠋⠀⠀⠡⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠀⠐⠁⡐⢉⠀⠄⠉⠉⠁⠒⠺⠎⠌⢠⠔⠄⠀⠂⠤⠄⠄⡀⢂⠒⠀⠁⠄⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⡀⠄⠀⠶⠂⠖⠒⠒⠒⣶⣶⣦⣤⣄⣀⡜⠲⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡤⠀⠀⠀⡀⠠⣍⣀⣀⣤⠈⠉⠉⠂⠛⢛⡛⢿⣷⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠰⣇⠀⠀⠀⠀⣡⠀⢩⡀⠈⠹⠷⣤⣈⣁⣼⡿⢃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⢤⣒⡐⠀⠀⠀⠀⣠⣶⡀⢼⠭⢽⢟⠉⢰⣏⣹⡷⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠴⠖⠀⠉⠙⠀⠋⡿⣿⠃⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⠶⠒⠋
⠀⠀⠀⠀⠀⢀⡔⠟⠛⠉⢛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣶⣤⣀
⠀⠀⠀⠀⠐⠈⠀⠀⠀⣰⠻⡿⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⣻⣿⠵⣺⡀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢒⣛⣷⠋⡷⢲⣦⣧⣀⡀⠀⡀⠀⢀⢃⣠⣶⣟⢓⣏⣹⣿⣧⠆⠀⠀⣘⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢋⠥⢠⠈⢉⡛⠛⠛⠿⠛⠿⠛⠻⣿⣯⣯⣯⠋⠁⠀⢀⣠⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠛⠁⠀⣀⣀⠀⠀⠀⣀⠀⠈⢀⣤⠄⢰⠿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠉⠉⠁⠀⠓⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠠⠀⠀⠂⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠖⠀⡠⠲⠶⣀⠈⠁⡀⠘⠀⠀⠐⠀⠀⠡⠀⠯⠀⢠⡠⠀⢀⡀⠙⠀⡟⠲⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⣴⣦⣀⠎⠺⣏⠁⠄⠀⠀⠀⡀⡀⢈⡽⠁⣾⣿⣿⣷⡄⡅⠀⠀⠀⠀⣴⠀⠀⠀⠀⡴⠀⢀
⠀⠀⠀⠀⠀⠀⡐⠠⠄⠀⣡⡌⠨⡀⢀⣍⠀⠄⠈⢉⡦⠀⢠⣤⠀⣀⣤⠦⠀⠄⠀⢿⣿⣗⠁⠀⠀⠉⠀⠀⢀⠀⠀⠀⡅⠀⠐
⠀⠀⠀⠀⠀⠐⠋⠀⠀⠐⠉⠀⠑⠂⣀⡉⠀⡙⠃⢀⠀⣤⠐⠄⠀⠃⠀⠀⠰⢿⣷⣶⣿⠟⡀⡀⠀⠀⣀⣰⣿⣦⣀⠂⠀⠀⠉⣄
⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡌⠀⠐⠋⡂⠄⡊⠀⠈⠐⠛⠉⣀⢠⡘⢀⠠⡈⠙⢹⣿⣿⣷⢾⡠⠀⣀
⠀⠀⠀⠀⠀⠈⡡⠒⢄⠠⠚⢢⡠⠄⠰⡌⢃⡀⠔⢶⡆⠀⠀⠆⡉⠁⡤⢑⡦⡀⠀⠀⢀⠇⠐⠀⠘⠛⢷⢴⣼⣿⣿⣿⣾⡄⣠
⠀⠀⠀⠀⠀⠀⠤⣠⡖⠒⡤⡔⠒⣤⡦⠬⣉⡉⣆⠀⡀⠰⠤⣋⠁⠀⣄⢀⡀⢀⣀⠦⠐⢀⠐⠦⠀⠀⠀⣠⡍⠙⠛⠛⠙⣢⠐
⠀⠀⠀⠀⠀⡘⠁⡅⠀⠀⣄⢠⢀⡀⠐⢀⠃⣀⡄⢀⢤⠂⠒⠉⠉⢰⠠⠄⠀⠀⠀⠀⠀⢞⢸⣸⠀⠀⠀⠉⠀⠀⠀⠀⠐⠊⠘⠚⠄
⠀⠀⠀⠀⠀⠈⡀⣷⠀⠀⠱⣶⢹⠃⢳⡏⡀⢑⡼⠃⠺⠑⠄⢀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠾⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡅⢹⣠⠀⣂⣿⣲⠀⡁⠈⠀⣿⡗⠈⠀⠐⠤⢌⠁⡀⢘⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⢐⠘⢿⢻⢟⡟⣷⢒⢵⣠⠄⡙⢷⣤⣄⡀⠘⠉⠈⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡖⠂⠀⠀⠀⡘⠉⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡌⣾⣿⣯⡟⠓⠄⠐⠋⠆⢈⠨⠑⡄⠀⡀⢁⡰⠀⠀⠀⠶⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠴⣀⠌⢽⠰⡖
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠁⠈⠉⠉⠀⠓⠀⠀⢤⣄⣈⣷⡀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠸⠀⠀⢠⣠⠄⠀⠀⠂⠀⠀⠀⠀⡼⣆⣻⣎⠌⢧⠁
⠀⠀⠀⠀⠀⠀⠀⠤⡁⢫⠵⢦⡄⠐⢣⡄⠀⢀⡄⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠠⠁⠀⠀⣐⠄⠀⠒⠉⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⢉⣋⣈⠃⢈⠣⠤⠤⠤⠤⢤⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣀⠀⠀⠀⢉⢷⢖⣸⠃⢈⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡿⠿⢍⠙⣿⠿⣿⣯⣟⠻⢿⠿⠂⣠⡄⠀⠀⠀⠀⠀⠉⠁⠨⠀⠀⠀⠀⠠⣴⢺⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢳⢧⣌⣉⠈⠈⠀⠀⠀⠀⠉⠳⡄⠔⢨⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡿⠻⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠉⠚⠳⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠉⠉⠙⠛⠋⠉⠛⠟⠛⠿⣕⣊⠅⠀⠈
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠛⡁
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⣸⣿⢷⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠻⡿⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡇⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠃⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠀⣼⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣰⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⢧⡣⠀⠹⣿⣿⣿⣿⡝⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠛⠛⢯⣑⠀⠀⠈⠓⠢⠤⣤⣀⣀⣀⣀⣤⠤⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣀⣰⣶⣶⣶⣶⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⢿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠤⠄⠀⠀⠀⠀⠀⠀⣬⣷⣶⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡴⠺⠿⣄⡀⠐⠒⢦⣤⣴⣇⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⡀
⠀⠀⠀⢀⣠⡐⣀⢀⣉⣀⠀⠀⠀⠀⠀⠈⣶⣿⣿⣯⣽⡿⣻⣿⣶⣆⣉⠉⠍⠀⠁⠀⠒⠤⣀⡠⣤⡩⠝⠋
⠀⠀⠀⠋⠀⢀⡴⠛⠉⠉⠉⣿⠿⠛⠉⠉⠉⠉⠛⢻⡽⠸⣿⠛⠟⠁⠀⠈⠀⠀⠀⠤⣀⠀⠀⠉⠄
⠀⠀⠀⠀⡴⠃⠀⠀⢀⣤⠖⠁⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠈⢿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⢦⡀
⠀⠀⢀⡾⣆⣴⣴⣾⠟⠁⠀⠀⠀⣀⣤⣶⠟⠹⢿⠀⠀⠀⠀⢸⡟⠛⠛⠻⣶⣤⣀⠀⠀⠈⠀⠚⠀⠉⠁⠛⠛
⠀⢀⣾⣿⣿⣿⣿⣏⣠⣤⡶⣶⣾⢻⠏⠀⠀⠀⢸⡆⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣟⣳⣦⣀⣀⠀⠀⠠⡀
⠀⢸⣿⣿⣿⣽⣿⢿⣿⣿⣧⣮⣝⠋⠀⠀⠀⢀⣾⣧⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⢘⣟⣻⠿⣿⣿⣶⣶⣤⢄
⢀⣿⣿⣿⣿⢃⣿⣿⣿⣁⠻⣷⣜⠀⡀⠀⢠⡾⣿⣿⣆⠀⣼⣿⡼⣿⣷⣄⣄⠀⠈⣿⣷⣳⠀⠙⢋⣌⠻⣧⡀
⢸⣿⣿⣿⣏⢿⡿⣿⣿⣾⠴⣛⣷⣠⣰⠞⢻⠆⠸⣶⣿⣽⡿⣿⡩⢻⣿⣿⣻⣡⠔⢿⡟⠋⢰⡂⢸⡿⢲⣹⡇
⣾⣿⣻⣿⣿⢹⣷⣿⣿⡦⠐⣯⡧⠛⠁⠀⢧⠀⠀⡉⠀⠩⠁⠉⠀⠀⢿⠉⣹⣿⣿⠾⣧⣤⠸⠃⢀⡤⡙⠃⠻
⢸⣿⣿⣿⣿⣾⡟⣿⡧⢡⣦⡟⠁⠘⢁⠀⠀⠀⠀⣅⠀⠀⠀⠜⠀⠄⢸⡁⢰⣿⣾⠀⠈⣿⠀⡦⠈⣷⠇⣀⣿
⢸⣿⣷⣿⣿⢸⣿⣿⣿⣼⡷⠁⣀⡄⠉⠀⣠⢠⠀⠀⠀⡀⡄⠀⠠⡆⠀⠅⠚⠊⠁⢰⡶⡄⠘⠿⠀⠀⢸⣾⣹
⢰⣿⣿⣻⠿⢃⣽⣿⣷⣿⣯⡀⢻⡇⢀⣀⠻⠏⢀⢄⠈⠿⠃⣴⢆⠀⢠⡤⣴⠷⠀⠸⣿⠀⣀⡀⣼⣡⠀⣀⣷
⠀⣿⣿⣿⣤⣺⢿⣿⣿⡿⣷⠇⡘⣁⠸⠂⣀⠀⢸⡶⠀⣀⡄⢿⡏⠀⢸⡶⢿⡟⠀⢠⡀⠀⣷⠇⢙⡃⢸⣯⣿
⠀⠹⣿⣿⣿⣟⠻⣿⣧⡿⣙⡁⢸⣽⠆⢼⣱⠀⣠⠀⠈⣧⠇⠈⢠⣞⠀⣿⡀⠀⠀⣧⡕⢀⠁⠀⣿⣽⢘⣿⣿
⠀⠸⣿⣿⣿⣿⣟⣿⣿⢧⣹⡖⠈⠏⠀⠘⠋⠐⣿⠀⣤⠁⢀⣀⠘⠏⠀⡿⢡⣶⡆⠋⠀⣿⡱⢀⣉⠁⣸⣿⠃
⠀⠀⠈⣿⣿⣿⣿⣿⣟⣯⣿⣿⢠⡟⠂⢰⡱⠄⠁⢸⡤⠀⢸⣼⠀⠀⣾⡍⠘⣉⣤⡀⣠⡘⠁⣿⡽⢠⣾⠇
⠀⠀⠀⢸⣿⣿⣾⣾⡟⢻⣿⣦⣼⠿⠀⠘⠃⣤⡄⠈⢠⣄⠈⠁⣾⠆⠉⣀⠀⣿⠞⢁⡿⠃⣠⣜⠋⣿⡏
⠀⠀⠀⠈⠻⣿⣿⣿⣧⣾⣿⣷⣷⢰⣟⠆⠀⢳⡇⠀⠻⣾⢀⣀⠹⠃⢸⣽⠆⢸⢶⠀⢀⣀⣿⣽⣾⠋
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢹⣿⣼⠋⢠⣟⠦⠀⡤⡄⠀⢸⣯⡇⠀⠀⣽⡄⠸⠟⢻⣻⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣷⣻⣷⢆⠿⡇⣠⣬⢛⡵⣴⢻⣷⡀⣾⣷⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣎⣴⡽⣷⣾⡽⣷⣿⡿⣿⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡞⢿⣿⣿⣻⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⢿⡀⠀⢀⡴
⠀⢲⣄⠀⠀⠀⠘⣧⢠⡿⠁
⠀⣤⣽⣆⡀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⣀⡹⠿⠶⠘⣭⡄⠀⠀⡀⠀⢀⠄⠀⠀⠀⠘⠁⠀⠀⠀⢀⡄⢀⣤⣶⠟⠁
⠀⠀⠈⠉⠀⠐⠚⠋⢹⣇⠀⣸⢃⡾⠋⠀⠀⠐⢦⡀⠀⢀⠆⣠⢏⣴⡿⠛⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣌⣿⠀⠁⠶⠿⠿⠛⠂⠀⠀⠁⠠⢋⣼⠏⠀⠀⠘⢷⣄⠀⠀⠀⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣗⠂⣰⠇⠀⢢⡀⠀⠀⠀⢀⣾⠃⠈⠙⠓⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡠⠏⠀⠀⠈⢻⡄⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⢸⡉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢰⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢠⡟⠀⢸⡀
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢽⡆⢀⡿⠀⠀⠘⡇
⠀⠀⢰⡿⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⣼⠁⠀⡆⠀⣠⣴⠆
⣦⣤⣘⠀⢠⡆⠈⢳⣄⠀⠀⢠⣿⣿⡇⠀⡄⣷⠚⠛⠁
⠈⠉⠙⢓⢺⢇⡆⠀⠙⣷⣄⣼⡿⣿⡇⠀⢹⠘⢂⡄
⠀⠀⠹⣮⡁⡾⢡⡆⠀⢣⡙⢿⣵⣿⣿⢸⠈⣧⡞
⠀⠀⠀⠈⠻⢅⣿⠁⢻⣦⡵⣻⣿⢸⣿⠘⡇⠋
⠘⣆⠀⡆⠀⣼⠃⠀⠀⡟⢷⣿⡇⢸⣿⠀⢳⠀⠠⠀⡄

⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣤⡶⠃⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡄
⠀⠀⠀⡖⠒⠚⠛⠛⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢹⠃⠀⠀⠀⠀⠀⠈⣿⡏⠀⢀⡀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣯⣿⣽⡁
⠀⠀⣴⡇⠀⢿⣿⣿⠇⢀⣷⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠻⢷⣤⣤⠀⠀⠀⢠⣶⢦⢸⡛⡟⣿⢻⣃⣴⣄
⠀⠀⡇⠈⠓⢤⣉⣡⠔⠋⠠⡅⠀⠀⠀⠀⠀⣶⡆⢸⣿⣿⢠⣶⠄⠀⠀⠀⠀⠀⠀⣴⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢧⢸⣿⠋⢻⣿⠁⡆
⠀⠀⣇⣀⣀⣀⣈⣀⣀⣀⣠⠆⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠾⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠛⠛⠛⠛⠳⠛⠛⠟⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠉⠉⠉⠉⠉⠈⠀⠁
⠀⠀⠀⠀⢀⣠⡤⣤⣀
⠀⠀⠀⠀⣾⣿⣷⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠋⢩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠀⡆
⠀⠀⠀⢰⡏⣿⠟⢿⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣠⠌⢥⣄⠀⠀⠀⠀⠀⢀⣤⣶⣶⡤⢀⣤⣶⡶⣤⣄
⠀⠀⠀⠀⠀⣵⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⢟⢧⠀⠀⠀⢠⣿⡟⠁⠀⢰⣿⢟⠁⠀⠈⣟
⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⠀⢀⠇⡼⠀⠀⠀⢸⣿⠀⠀⠀⢿⣿⢸⠀⠀⢀⠏
⠀⠀⠀⠀⠀⢸⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⣔⣫⠜⠁⠀⠀⠀⠀⠻⢦⣤⣴⣈⠻⣦⣤⣔⣫⠜
⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡴⠒⠒⠲⢤⠴⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⢰⡶⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣵⣷⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⡞⢠⣿⣿⣿⠀⣰⣶⣶⡄⢸⡆⠀⠀⠀⠀⠀⢰⣧⠀⣴⣧⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⠀⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠇
⠀⠳⡈⢻⣿⡿⠃⣿⣿⣿⡇⡸⠃⠀⠀⠀⠀⠀⢸⡿⣧⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡈⣀⣇⣰⣲⣆⣦⣀⠀⠀⠀⣶⠟⢆⠀⠀⠈⠁⠀⢀⡶⣦
⠀⠀⠙⢿⣿⣧⠀⢀⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⡹⣿⣿⣿⠛⠁⠀⠀⠀⠈⢸⡀⠀⠀⠀⠀⠀⠀⣸⠘
⠀⠀⠀⠀⠙⠻⣧⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠉⠉⠁⠀⠀⠀⠀⠰⣶⣾⣷⣶⣶⣶⣶⣶⣶⣿⣶
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢛⣃
⠀⠀⠀⠀⠀⢀⡤⠶⠶⠦⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡿⡿⡇
⠀⡀⣷⡦⡰⠃⠀⠀⠀⠀⠀⠱⡄⣾⢠⡄⠀⠀⠀⠀⡼⣭⢷⢠⣍⣡⠀⠀⠀⠀⠀⠀⢠⣤⣤⣭⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⣐⣀⣐⣀⣐
⢹⡇⢸⡁⡇⠀⠀⠺⣾⠂⠀⠀⡇⢻⢸⡇⠀⠀⠀⠀⡃⠀⢀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢏⣠⠞⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛
⢸⡇⢸⡇⠱⡀⠀⠀⠀⠀⠀⡰⠃⢸⠀⡆⠀⠀⠀⠀⣿⣿⣿⠀⢙⠋⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⢰⣶⣾⣶⣶⣶⣶⣷⡆
⠘⠃⠸⠃⠀⠈⠓⠦⠤⠖⠊⠁⠀⠸⠐⠃⠀⠀⠀⠀⠦⠶⠴⠀⠸⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠩⠉⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣠⣸⠒⠒⠒⠒⠒⠚⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡅⠀⠀⠀⠀⠀⠠⠲⣶⠶⠶⣿⢿⠀⠀⠀⠀⡀⢸⠼⣿⠶⠶⣶⠖⠄⠀⠀⠀⠀⠀⢈⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡇⠀⠀⠀⠀⠀⠀⠘⣭⠃⢚⣍⢻⠀⣇⣾⣦⡇⢸⢸⣩⡓⠘⣭⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠈⠸⣿⠠⠾⣿⢿⡖⣻⣿⣿⡛⣾⠸⣿⠷⠀⣿⠇⠁⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣟⡃⠀⠀⠀⠀⠀⠨⠈⣭⠃⢚⣭⢻⣳⡏⣭⣍⣿⣿⢸⣭⡓⠘⣭⠁⠁⠀⠀⠀⠀⠀⢚⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣿⣾⣶⣷⣿⣷⣶⣶⣶⣿⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣾⣷⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣿⣿⣷⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⢹⣿⣿⡿⠛⠟⠉
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣾⣷⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⡔⠀⠈⠀⠀⠠⠀⢀⡀⠀⠂⣀⣼⡇⠀⡰⠒⢬⠛⠻⣶⣦⣙⠒⠆⠀⠘⢧⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢻⣷⡶⡽⠋⠉⣠⠄⠀⠀⠀⠀⣀⠰⣸⠿⣿⣿⣿⣿⣧⣶⣾⡟⠃⠀⠿⣿⣿⠁⠀⠀⡀⠄⠑⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⢫⣧⣸⣷⠾⣿⣦⡤⠀⠀⠀⠠⠘⠀⣸⣶⣿⠟⣿⣿⣿⣷⣥⡀⣀⠠⣄⠘⣿⣿⠀⣀⠾⡆⡤⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠉⠀⠴⣤⢬⣹⣿⣟⡄⠉⠵⠚⢁⡰⠦⣶⣰⣿⣿⣏⠏⢀⣿⣿⡿⢁⡠⠀⠤⠀⠀⠉⠈⠻⣷⣡⠔⠀⠁⢠⡀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠴⣒⢤⡈⢿⣿⣿⣯⠖⠐⠖⣀⣷⢦⣿⣿⣿⣷⠈⣰⣿⢟⡁⣡⣼⣷⣶⢶⣷⣮⣤⠀⠦⣿⣿⣟⡄⠀⠸⠁⠺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡙⡿⢿⣾⡏⠀⠀⠀⠆⢱⣶⣿⣿⣿⣏⣰⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣯⡁⠀⢀⣾⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠔⣦⣷⣼⣿⣿⠁⠀⠀⠃⢠⣿⣿⣿⣿⣋⣴⣿⣭⣿⣟⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠉⠙⣻⣿⣭⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⣤⣷⣿⣿⣿⣿⣿⠂⠀⢠⣠⣽⣿⣿⣩⣿⣿⣿⣟⠿⣹⣿⡿⡿⢿⣿⣿⣿⣿⣿⢿⣹⡖⣶⣿⣿⣿⣧⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣼⡿⢿⣟⡄⢀⠀⣼⣿⣻⠟⢩⣿⣿⣽⣿⣷⣿⣷⣴⣾⣿⣿⣿⣿⣿⠿⢳⣿⣿⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣾⣣⣤⣿⢆⠽⣿⣿⣏⡇⢲⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣻⣯⣽⣶⣾⣛⠉⢹⢿⠉⢺⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⢟⣿⣿⣟⣥⣴⣿⣻⠉⢀⠉⣿⣿⣿⣿⣿⡍⠑⠲⣶⠦⠉⠉⠛⠛⠛⠛⠃⠀⠀⢼⠖⢻⣷⡏
⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⣿⣿⣿⣿⡏⠈⣿⠗⣀⡤⠀⣿⣿⣿⣿⣿⣏⠳⣤⠀⠐⠈⠀⠈⠀⠒⠀⠀⠀⡇⢾⣤⣾⢿⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⣿⣿⣿⣷⠀⣰⠟⠉⠀⣄⣿⣿⣿⣿⣿⡿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⢑⣾⣿⡗⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠹⢿⡿⣉⢛⡡⠆⡀⣵⣿⣿⣿⣿⣿⣿⡆⠀⠡⠀⠀⠀⠀⠀⠐⠀⠠⠟⣠⣿⡟⣼⡔⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡰⢾⡿⠁⠀⠀⢻⣆⠤⢻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠂⠀⠀⣐⣂⣾⣥⣯⠟⠈⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⢱⢾⣃⡀⠀⣀⠉⠘⠦⣽⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠡⣀⣠⣶⣟⣿⣿⢟⡁⢎⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⣀⣮⡴⢀⣽⠄⠀⣦⠾⣿⣿⣿⣿⢿⠀⠀⠀⠀⠄⢶⠳⠎⠛⣿⡿⢯⠟⠀⠀⠠⠀⠀⢡
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠀⠉⠃⠀⠶⠀⠀⠀⠈⢿⣾⣿⣿⣿⣾⢷⣄⢤⠀⡐⠂⠀⣳⡈⠟⠃⠀⠀⠐⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠠⠀⠀⠠⠐⢔⢀⣴⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⣄⣿⣿⣇⠀⠀⡿⠂⠀⠂⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣁⡈⠀⠀⠀⠀⠐⠃⢀⡒⠊⡀⠤⠨⠝⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⠄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠐⠒⠋⢁⠀⣈⠏⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠇⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⣤⠀⠖⠈⠀⠀⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⢿⣿⣿⣿⡀⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠄⠀⡀⠀⠁⠀⠄⠀⠀⠀⢀⠠⣄⣶⣿⣿⣿⣿⡏⣿⣿⣧⠋⠱⢠⠈⢻⣿⣿⡀⠄⠀⠀⠀⢀⣶⠟⠃⠀⠀⠃
⠀⠀⠀⠠⢱⣦⣥⠋⠀⠀⠀⠀⢀⣠⣡⣾⡿⠛⢟⣹⣿⣿⣾⣿⡿⠟⣛⠠⠞⠀⠈⠉⣿⣿⣁⡄⠀⢀⣤⡾⡹⠂⠀⡀
⠀⠀⠀⢹⣿⣿⣿⣮⣿⣶⣴⣬⣽⣭⣼⣥⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠂⠤⠀⠀⢠⣾⡿⠿⠄⣰⡟⢋⠀⠀⠀⡀
⠀⠀⠀⣻⠟⠙⠻⠿⠿⢿⣿⣿⣿⣿⡿⠟⣻⠁⢘⣿⣬⣿⣿⠃⠀⠈⠁⠀⠀⠀⠀⠀⣼⣿⢶⡾⡿⠁⠀⠀⠀⡐⠁
⠀⠀⠠⢿⣿⣿⣿⣷⣄⢠⣤⡀⢿⠟⣠⡾⣡⣾⣿⣿⣿⣿⣿⠋⠀⠂⠄⠀⠀⠠⠀⣠⡿⢧⣭⣿⠀⠀⠂⠀⠖
⠀⠀⠀⠈⠉⢡⣿⣿⣿⣷⣬⣤⣾⣷⣶⣾⣿⢿⣿⠿⠟⠻⡁⠀⠀⠀⠀⠀⠀⠀⢤⣿⣀⡀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠉⠈⠈⢉⣽⣿⣿⣿⣿⣿⡿⠿⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣼⠇⠈⠐⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡿⣿⣿⣿⠇⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣬⡁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⠀⠀⠀⢄⠀⠈⠑⠠⠀⠤⡀⢰⣮⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⢹⡏⠋⠙⠀⠀⠀⠀⠈⠀⠄⢠⡀⠀⠀⠀⠛⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡍⢳⢨⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢹⠀⡇⢸⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⡞⢠⠇⡜⡿⠁⠀⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣏⠝⣡⣿⣟⣷⣏⡼⢱⣷⠞⣛⣫⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⣿⣡⢞⣿⣿⣿⣿⣿⣶⣻⢷⠋⡏⢹⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠈⣿⣇⠞⣿⣿⣿⣿⣿⢿⡏⣸⢠⡇⢸⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⠘⠃⠀⠘⣿⣤⢿⡟⣿⢿⢻⣼⢡⣧⣾⣿⠿⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⡀⢀⣤⠀⠘⣿⡜⣿⣷⣾⣾⡿⠛⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⡸⣿⣧⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⢁⡤⠖⠒⠒⠒⢤⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢅⣯⢧⢹⣿⣆⠀⠺⠀⠀⠀⠻⣿⣿⠟⠁⠶⣀⣤⣜⠁⠀⠈⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⡘⣆⢻⣿⣆⠀⠰⠆⠀⠀⢸⡏⠀⣀⠀⠃⠠⠈⢧⠘⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡌⡷⠛⠋⠻⣿⣧⡀⠀⠰⢠⡟⢷⡀⠀⠀⠶⠀⢀⣿⢠⢀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡏⢀⣤⡞⣉⣻⣿⣿⣆⣠⠟⣧⢸⠙⢳⠶⢶⠖⣻⠃⢀⡞⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣸⣠⠞⣡⡴⣯⣟⣿⠏⠀⠈⠛⠶⠾⠦⠞⠋⠑⢢⠞⣇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣞⢆⣇⣇⡼⠃⠰⡶⠀⠠⡶⠀⠀⠋⣠⠖⢧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣏⣀⣀⣀⣀⣠⣤⠤⢶⠞⠉⡇⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠦⣴⣃⣠⡇⢀⣇⣀⣾⣤⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣞⣀⣀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⠛⠫⠭⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡞⡀⠀⢀⣤⣞⣋⣭⣾⣯⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⡌⠀⠀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⣷⢸⡿⠁⠀⢈⣿⠿⠿⠥⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣛⣛⡃⠀⡇⠀⡸⠁⢻⣟⢦⣄⠁⠀⠀⠀⠀⠀⠹⣿⠇⣡⡾⠋⠁⠀⠀⠀⣀⣤⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢰⠇⠐⠁⠀⠀⠻⣆⠙⢦⡀⠀⠀⠀⠀⣠⣵⠞⠁⠀⠀⠀⢀⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⢸⢠⠃⠀⠀⠀⠀⠹⡀⠀⠑⠀⠀⣠⣾⠟⠁⢀⡀⠄⠠⣶⣿⣥⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣼⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠿⠒⠊⠉⠀⠀⠀⢸⡿⣿⣿⣦⡉⠉⠛⠛⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣿⡝⢷⡄⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⠇⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⣰⠁⠀⠀⢸⠃⠀⠘⢿⡀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢰⡄⠀⣰⠇⠀⢀⠄⠘⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠈⢿⢰⠏⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⣼⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⢸⣿⣿⡇⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⢀⠆⠀⡠⠊⢁⣿⢻
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢿⠀⠀⢸⣿⣿⣟⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⣜⡴⠋⣠⡏⣸⠏⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡘⡇⠀⠸⣿⣿⣿⣾⡿⠁⠀⠀⠀⠀⠀⠀⡼⠋⠀⡴⠋⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⠀⠀⣿⣿⣯⣽⠇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢻⣿⣿⡏⠀⠀⠀⠀⢠⣾⣋⠤⠄⠒⠒⠒⠒⠒⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣇⠀⠀⠀⣴⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣌⠉⢲⣤⢤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⣿⣛⣿⡇⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠹⡄⢸⡎⢷⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⣼⡟⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠁⠿⠃⠸⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣆⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣻⠿⠋⠉⠉⣉⣉⠉⠙⠻⠿⣿⣿⣷⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⠟⠉⠈⠀⠀⠀⢀⣀⣤⣼⣿⡆⠀⠀⠈⠙⢿⣿⣷⣵⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣿⠃⠀⠀⠀⢀⡀⠀⠈⠉⠉⠉⠙⣷⣄⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⢠⠏⠀⣠⡶⠶⠶⠶⢿⣿⣿⣷⡄⠀⠀⠀⢿⣷⣿⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⣿⢀⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠸⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢹⡿⠱⠶⢶⡄⠀⢀⣤⣤⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⠘⣇⠐⠢⠤⠁⠀⣿⣥⣬⣿⡟⠀⠀⠀⠀⠀⣿⢻⡿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠀⠀⢠⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡄⠀⠀⠀⣾⣧⠀⠀⠀⠼⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄⠀⠈⠠⢻⠀⠀⠀⠴⠿⣿⣟⠛⠃⠀⠀⢀⣴⢟⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠇⠀⢀⣠⠞⠀⠀⠀⠀⠀⣿⣿⣤⡄⠀⠴⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠥⠒⠚⠉⠀⠀⠀⠀⠀⠀⣼⠟⣹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠁



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢸⣿⡀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣵⠀⠀⠀⠀⢸⣿⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣀⣤⣶⣶⣦⣤⣀⠀⠀⠀⡀⣠⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢠⠀⠀⠐⠀⢀⣀⣿⡇⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣷⣄⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠀⠀⠈⠀⠈⢰⣦⣾⣿⣯⡄⠀⠀⠀⠙⠛⠛⢻⣿⣿⠛⣿⣿⣿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⡄⠀⠀⠀⠀⠀⠀⠸⣿⢁⣀⣿⣿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡿⠷⣬⣼⣿⣿⣿⠧⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⠂⠐⠲⠆⠀⢠⣶⠀⣿⡟⡇⠀⢸⣿⣿⣿⡿⣿⠀⠀⠀⠀⢀⣿⣾⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠁⢹⡇⠀⠀⠀⠀⠀⣸⡿⠀⣿⡇⠷⠀⢈⣿⣿⣿⣥⠤⠀⢠⡦⠤⡾⣿⣿⣿⣿⣟⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⣉⣀⣀⢛⣀⠛⢛⠋⡃⠀⠀⠺⡷⠾⣿⣿⡛⠘⣿⡏⠀⣽⠓⠐⣶⣤⡀⠀⠀⠀⠤⠤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⣄⠀⠀⠀⠀⣤⣴⣤⣿⣯⢉⣿⣿⠹⣏⣿⣿⣯⡇⠀⠀⠶⢰⣶⣿⣿⠁⢀⣿⣧⡤⠌⠀⢠⡟⠓⠾⠟⣫⣀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢛⣣⣔⢶⣞⣿⣿⣿⣿⠇⢸⣿⣧⠐⢸⣡⣿⣿⠄⠀⠀⠐⠘⠛⠿⠿⠿⠿⠿⠿⣄⠂⠀⢸⡇⠂⠀⠌⠙⢡⣶⠦⣀⠀⠎
⠀⠀⣀⡀⠀⠀⠂⡗⠀⣨⣾⠟⠛⣿⢿⡿⣿⣿⡿⠿⢿⠀⠿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⣀⢰⣤⠌⡓⢯⣿⡄
⠀⠈⠉⠉⣁⣤⠔⢋⡉⣍⢻⠁⣤⢩⣈⣧⠉⠁⢁⣠⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⢀⠀⠀⠠⠤⠀⠀⢀⡀⠈⠚⢇⣘⡇⣦⡿
⢤⠀⠒⠀⢙⡷⢢⣯⣵⣶⣾⡿⠛⡃⢀⣠⣴⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠄⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⠈⠋⠀⠀⠀⠉⠳⢿⡿
⣤⠀⠀⢠⣶⣶⣿⠿⠟⠋⢱⣶⠷⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠒⠀⢨⠈⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠁
⠀⠀⠀⡄⢻⡉⠀⣴⠿⠟⠋⠡⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⢰⣶⣦⣴⣷⣴⣄⣠⣶⣃⣠⠒⠾⢒⣤⠀⠢⡽⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢀⣠⡤⠈⢋⡁⠔⢲⠀⡖⠀⢙⠀⢀⠀⠀⠀⣠⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⢸⠀⠀⣤⡀⠙⠛⠁⠀⠁⠀⠀⠀⠈⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢾⡆⢸⡤⠘⠹⡟⠈⠃⡀⠀⠀⡀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠻⡆⠀⠀⡷⠀⠀⣇⣀⣸⢻⣶⣿⣿⣿⣿⣿⣿⣿⡿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢰⡀⠠⠆⠀⠀⡇⡟⣴⢿⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣉⢀⣾⡿⠿⢿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⠀⠀⠀⠀⡇⣿⣯⢸⠀⠀⠀⣷⢻⣿⣿⣿⠏⠉⠻⠿⣿⣥⠃⠀⠀⠀⠀⠀⠀⠈⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠀⢀⡀⠀⡇⣿⣿⣼⠀⠀⠀⣿⢸⣿⣿⣿⢦⡈⠀⠀⢡⣼⣤⣄⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠨⠁⢠⡟⠈⠛⣿⠀⠀⠀⣿⣾⣿⣿⣿⣧⣸⣋⣰⠿⣿⣿⡏⠀⠀⠀⠀⢸⣿⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠚⠂⠰⡷⣄⠀⠋⠀⠀⠀⣿⣽⣿⣿⣷⣿⣿⣿⣧⣼⣿⣤⣍⣀⡀⢀⣀⣼⡏⠙⠿⣿⣿⣽⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣿⢾⣿⣿⣿⣾⣿⡟⠯⣿⣿⣿⣿⠋⠉⠉⠀⠸⠃⠀⠰⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⡄⠀⣿⡟⠳⣿⣿⣧⣿⣿⣷⣶⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⣿⣧⡞⢻⣿⣿⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣿⡟
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠆⠀⠀⣿⡏⠀⠀⢿⣿⣿⣷⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣹⣿⠿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⢴⣿⢳⣄⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠤⣆⠀⠀⠀⠀⠈⢉⠀⠙⢦⡀⠀⠀⢹⡛⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣽⣿⣿⣿⣿⣿⣝⣻
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢷⡀⠈⠻⣄⠀⠀⣷⠈⢻⣯⡷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⡇⠐⣆⠀⠀⠀⠀⠀⠙⣦⠀⠙⢷⡀⠸⡇⢸⣯⠁⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠆⠀⠀⠈⠱⠶⠟⣦⠀⠀⠀⠀⠀⠈⢧⡀⠈⠻⣦⣹⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠇⠀⠀⠀⠀⠀⠀⠻⣤⠀⠘⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⢿⡿⠿⡿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⣾⠀⠀⠀⠿⠟⠛⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢠⣇⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠁⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⣼⠏⠀⠸⣿⣿⣿⣿⣿⣦⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⢿⣿⣿⠆⠀⠀⠀⠀⢹⡟⢉⣿⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⢼⣿⣿⣿⠀⠀⠀⢰⢲⣥⣿⣿⠇⠸⠇
⠀⠀⠀⠀⠀⠀⢀⠀⢲⠀⠈⠙⠀⢰⡇⣿⣟⠀⢸⣿⣿⠛⠀⣀⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠃⠈⠀⠀⠀⠀⠚⠁⣉⣀⣀⡸⢟⠏⠁⠀⡿⠾⣿⠟⢿⡗⠐⡖⢠⣄⡀⠀⠀⠤⢤⣤
⠀⠀⠀⠀⠀⠀⢀⠄⢠⡀⢀⣠⣶⣶⡿⢸⣿⢛⢞⣿⡟⠀⠘⣱⣾⣿⣁⣼⡷⠂⠀⢸⠏⠛⢵⠄⡀⠀⢀
⠀⠀⡀⠀⠀⠂⣠⡶⢶⣿⡿⣾⣿⠽⠇⠾⣿⣶⠾⠛⠁⠀⠀⠀⠀⠈⠉⠁⠀⠁⠀⠘⠠⢄⢀⢈⡣⢳⣄
⠀⢈⠉⣅⣊⠥⣈⣧⣦⠄⠒⢉⣠⡴⠖⠋⠁⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⢠⡄⠉⠪⢇⣿⡇
⡍⠀⢀⣯⣼⡿⠟⣋⣥⡴⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢙⠛⠿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠙⠁
⠀⠀⠌⠯⣁⠖⠛⠁⠀⠀⠀⠀⠀⠀⣀⡰⣶⣴⣤⣤⣤⣤⣀⣀⣀⣀⣈⣛⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠲⣨⡖⠘⡟⠀⡄⠀⣀⢀⣤⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢀⠨⡇⠀⠀⣄⣷⠼⠹⣿⣿⣿⣿⣿⠛⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢨⡆⠀⠀⠀⣿⡿⠀⢀⣟⣿⣿⡟⠙⠿⣿⡍⠉⠁⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⠀⡴⢿⣷⠀⠀⡏⣿⣾⣷⢠⠄⣴⣷⠦⠀⠀⠀⣤⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠒⠀⠧⡀⠋⠀⠀⣿⣿⣿⣿⣿⣿⣽⣯⣁⣀⣀⣀⡟⠻⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢢⠀⠀⣿⣿⣿⣿⣿⣛⣛⣿⡿⠉⠀⠀⠃⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢘⠉⠀⣿⡴⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⡀⠀⠀⠙⠀⠀⣿⠀⠈⠻⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⣟
⠀⠀⠀⠀⢠⡀⠀⠀⠘⠃⠑⣄⠀⠈⡿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠿
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢦⠈⢦⡀⠱⡈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⢠⡄⠀⠀⠀⠳⡀⠹⣄⢳⣿⠁⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢤⡟⠀⠀⠀⠀⠹⣆⠈⢻⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣴⣦⣤⣷⣦⣤⣤⣤⣤⣴⣿⣷⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣍⣭⣯⣬⣉⣽⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢦⡙⠀⠀⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣻⣴⣆⣸⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⢀⡨⠛⠊⠀⢺⠅⠀⠠⣤⠀⠐⠂⠈⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡄⠀⠀⠀⠠⡧⢂⠀⠀⠀⡸⠶⡀⠀⠐⠮⠁⠂⠀⢄⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠄⠚⡄⠀⠀⣉⡀⠀⠉⢦⠋⢠⣈⣁⡀⠀⡠⢄⠠⢀⡴⠥⠹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢀⣷⡀⠚⠛⠉⢁⣠⠾⣆⡨⠉⠙⠑⢴⠀⣴⠀⢸⠠⡀⠀⠀⡠⠄⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⣄⠀⠀⠀⠀⢠⡌⠥⣹⠧⣴⠀⠀⠀⠤⢰⣃⣸⡷⠀⠀⠀⡔⠸⠃⢠⠎⠙⢡⠆⢈⡀⠁⠂⢰⡄⡁
⠀⠀⠀⠀⠀⠀⢀⡀⣠⢞⠟⣉⢙⠆⠀⢀⣶⡌⠁⠀⠙⡈⠗⣀⡄⠀⠴⠾⠁⠘⣆⠀⠀⢀⡇⠀⡃⠰⠀⠀⢀⠀⠀⠻⢮⢤⣊⠜
⠀⠀⠀⠀⠀⠀⢇⢀⡞⡏⡔⡽⠉⢢⡀⠁⠈⠓⡄⠀⠀⢼⣷⠴⠛⠚⠛⠉⠋⠉⠉⠉⠙⢛⠿⡦⣟⡓⣀⣆⠞⢳⣤⡂⠄⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢚⢱⠀⢭⠠⠀⣼⢇⠀⠀⣀⠃⢺⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣄⣈⡀⢀⡀⠒⠊⢹⢿⣉⣶⠄⠉⠀⠀⠘⢯⣔⣠⣤⡀⢔⠢⢄
⠀⠀⠀⠀⠀⠀⠀⢘⡤⠀⢉⠖⠈⢱⡠⢀⠴⠜⢪⣹⡇⠀⠠⠂⠀⠀⠀⠀⠀⠀⢀⣠⠤⠍⣈⠐⣢⠈⠀⢹⡅⠀⢠⠀⠀⠀⢀⠜⢣⠎⠁⠘⣅⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠈⠀⠀⣨⣦⠅⠁⠀⠘⡏⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠉⠁⠀⠀⠀⣷⡀⢆⠀⠀⠀⢷⡀⠘⣦⡁⠀⢈⡒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠁⠀⠀⠀⢨⡅⠀⠀⠀⠀⠄⢄⣀⣀⠀⠀⢠⠀⣀⡀⠀⠀⠀⠀⠀⢨⣿⠶⠛⢆⠀⢀⠜⠉⠑⠅⡼⠛⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⠀⣲⠰⠷⠖⢀⠅⢿⠃⠈⢳⠒⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⡇⠐⠀⠀⠉⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠠⠀⡽⠶⠒⠺⠠⠀⢿⡧⠀⠀⣄⡀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣿⢃⠉⢻⡁⡐⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⠁⠀⢀⡀⠀⠀⠀⠈⣧⡀⠀⠀⠉⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢺⠟⠚⠳⢇⣠⠁⠀⠀⡰⠋⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡋⠀⣠⠇⠲⡙⢃⡀⠀⠀⠀⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣈⠁⠀⠀⠉⠐⠀⠀⣅⠁⢈⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡉⠃⠀⠑⡀⢸⠎⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡷⢄⡡⡠⢀⠀⠀⣠⣶⡂⠈⠓⢼⡤⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡊⠁⠉⠃⠀⡀⠐⠐⠁⠜⠳⢭⣿⠶⠶⢦⣄⣰⣦⣭⣤⣟⢻⣇⠀⠈⢦⠊⠀⠀⠀⢀⠄⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠁⠀⢠⠏⠰⣦⡀⠀⠘⠀⠀⠀⡀⠀⠀⣀⡀⢠⠄⠀⠐⡋⡄⢁⠁⠀⠀⠡⠀⣻⠟⡉⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠠⠜⠀⠀⠙⠋⢀⢀⠀⠀⠀⠀⠀⠀⡥⠘⠀⡄⠀⠀⠰⠓⠈⡙⠄⣴⣆⠐⠩⡈⠀⢀⠜⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡁⢲⢤⠡⠲⣤⠄⢁⣲⢌⡀⠀⢶⠊⢦⠀⠛⠂⠉⠁⠀⠀⠀⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡸⠃⠀⠀⠀⠀⡸⠁⠀⠐⠆⠀⠀⠘⠠⠠⠐⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠈



⣿⣿⡿⠿⣿⡿⢿⢿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣄⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⢈⠉⠵⢀⣇⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢶⣠⠤⡯⢀⠀⠀⣼⡀⣰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡉⢀⡇⠀⢋⢠⣿⢻⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⣸⡇⣤⡇⢨⡏⠋⣿⢸⢻⠿⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠛⠛⣷⣧⢸⡇⠀⣿⢸⣿⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢀⠀⣀⠙⢿⣾⡇⠀⣿⣼⡿⠘⠏⠁
⠀⠀⠀⠀⠀⠸⠀⡇⣢⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠘⠀⠀⢹⣇⠀⣿⣻⡇⢸
⠀⠀⠀⠀⢠⠀⡇⢱⣿⠀⡇⢠⠷⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⣷⣾⣮⣸⠀⡿⣿⡇⡿⠾⠄
⠀⠀⠀⠀⠚⠆⢹⢸⠏⠀⡇⠰⠀⠀⠀⠀⠀⠀⣼⣿⡟⠉⠀⣀⠀⠉⠻⣿⣿⠀⣧⣿⢱⢻⠂
⠀⠀⠀⠀⠀⠀⠘⣿⣷⣤⡇⡘⠀⠀⠀⠀⠀⠐⣿⣿⣆⡀⠆⠉⠣⣴⣾⣿⣿⣶⣿⣿⣼⣾
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣠⠀⠀⠀⠀⠀⣈⡻⣿⣿⠀⠀⠀⠐⠒⣶⣙⣛⣿⣛⣻⡛⡀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀⣠⣿⣿⣿⣷⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠻⠿⠛⠻⠿⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣤⣴⣥⢀⠀⠀⠀⡘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣧⣼⡆⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⡇⣻⣿⠈⠀⠐⢀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢐⡃⢹⠁⣴⣿⣿⣦⠀⣼⣿⣿⣷⡀⠀⠀⠀⠀⢀⡇⣸⢿⣀⠀⠀⠀⢰
⠀⢀⡇⣼⡌⢿⣿⣿⠟⠀⢿⣿⣿⣿⣷⢄⠀⠀⠀⣼⡇⢸⠸⣿⣦⣀⠀⣘⣀
⠀⠈⢸⣿⡿⠀⠀⠀⠀⠀⠀⡼⠋⠀⠉⠈⣧⣀⣼⣿⠁⢹⢀⣏⣿⣿⠳⣯⢇⡀
⠀⣰⡆⣹⡀⠀⠀⠀⠀⠀⠀⢿⣀⣀⠀⣠⣿⣿⣿⡏⠀⣿⢀⠿⡿⠿⠂⠴⠿⠛⠓⠦⣀
⠀⣻⠗⣿⡇⠀⠀⠀⠀⠀⢸⠠⢽⣯⣿⣿⣿⣿⡏⠀⠀⢸⠰⣷⡇⠀⣴⡄⠀⠀⠀⠀⠈⠑⢄
⠀⣧⣄⢨⠀⠀⠀⠀⠀⠀⣸⣶⣿⣿⣿⣿⣿⢋⡀⣸⣥⣿⣿⣿⣇⠀⠉⠀⢘⠃⠀⠀⠀⠀⠀⢳⣤⡦
⠀⣿⣿⢻⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠛⡉⢠⣴⡿⠛⠁⣼⠀⡏⡿⢿⣶⡀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠲⢦⣠
⠀⣟⠁⣾⠀⠀⣰⣿⣿⣷⠟⠛⣋⢠⠠⣴⣿⣯⣶⣤⠀⠟⠀⡇⣇⠀⠙⣿⣦⠀⠈⢿⣿⣽⣿⣿⣿⣿⣅⣀⣹
⠀⣿⣧⡶⠀⣰⣿⣿⡟⠡⡞⣂⢗⣤⣿⣿⣿⣿⣿⣿⠇⠀⠀⣷⡏⠀⠀⠈⢟⣧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢐⡿⢃⡇⢠⣿⣿⣿⢖⠶⠘⢶⣾⣿⣿⢤⣝⣿⡏⠁⠀⢰⠀⣿⢣⠀⠀⠀⠈⣿⣆⢠⣿⢿⣿⣿⣿⣿⣿⣿⣿
⢰⡇⣸⣧⢸⣿⣿⣿⠰⠹⠔⡾⣿⣿⠇⠤⠀⠉⣷⠀⠀⢸⠀⣿⢨⢀⣴⣶⣦⣹⣯⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿
⠘⢷⣿⣿⣾⣿⣿⣿⣷⣿⣶⣶⣿⣿⣶⣶⣶⣾⣿⣦⣤⣾⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡕⡏⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⢁⠜⢸⡡⠼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⣚⠙⢧⠈⢣⣀⣿⣠⠒⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠎⢉⣸⢁⢍⡒⢼⣱⢄⣸⠧⢄⡙⢮⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡎⡈⢣⡁⣀⡆⡜⡴⢌⡒⢌⣙⡟⠒⢤⠱⡀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⠀⠱⡀⡤⠣⡸⠰⡰⢠⠢⢌⠒⣼⠉⢳⡨⠖⠋⣁⠽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠒⡟⢈⠟⠙⣄⠤⠑⣧⢁⢃⠖⢤⣱⠗⠈⣁⠠⡴⢋⣉⡉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡜⠉⢣⠀⢁⠊⠀⢠⢚⢦⡰⠈⢫⡎⡐⢄⣽⠖⠛⠲⢼⣧⣈⣤⣇⠀⢧
⠀⠀⠀⠀⠀⠀⡜⢸⢰⠈⢳⣄⣀⣧⡾⠵⡵⠵⣄⠆⠙⢄⡸⠁⢀⠤⡀⠀⡱⢁⡒⠀⣧⣼⡀
⠀⠀⠀⠀⠀⢀⡧⠋⣩⠭⠽⣧⠀⣮⠙⣄⠣⣠⠁⠳⣌⣰⡏⡍⢳⣈⣁⣴⡧⠤⠛⢉⣑⡴⡇
⠀⠀⠀⠀⠀⢸⡇⡜⢡⠒⢲⢈⣿⣧⠿⡺⡲⣿⡛⢏⣹⠧⠬⠖⠓⠌⣉⡢⢤⠗⡚⢁⠮⣙⡇
⠀⠀⠀⠀⠀⠘⣇⢱⡱⣀⡙⢉⠜⠈⢾⠋⠀⡧⣉⣾⣡⢥⡴⠒⠛⠛⠚⠴⣇⡌⢠⡎⠢⣨⠁
⠀⠀⠀⠀⠀⠀⠘⢦⡑⠤⡉⠀⠠⠀⠒⠑⢄⣠⢏⡱⢖⡩⠔⡊⡉⣉⡑⢦⠈⠳⡧⡈⠢⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣄⣒⣒⣢⣜⠝⡫⠖⡡⢒⡥⢎⡉⣱⢘⠄⡇⠀⣧⠈⡲⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⢎⠔⢉⠐⠣⢍⡒⠤⠄⠊⡰⠁⣰⢣⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠙⠾⣳⣄⡀⠈⠉⠉⠁⢀⣴⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠰⠞⠁⠀⠀⢀⣴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⠟⠓⢶⣄⠀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠟⠃⠀⠀⠀⠿⣧⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡿⠋⠀⣠⠄⣀⠖⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠟⠁⢀⣴⣾⣿⠟⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⡿⠋⠀⢀⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⣠⣾⣿⠟⠁⠀⠀⠐⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠋⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡅⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢰⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡖⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠉⠁⠀⠀⠀⠀⠙⠂⠀⠸⠃⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠠⠺⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠙⢶⢀⡀⠀⠀⠸⠋⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣷⣶⡀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⣲⠐⣀⢠⡤⠀⠀⣸⠁⡀⠀⠼⠀⠿⣆⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⠛⠀⠠⠀⠀⠀⢀⡈⣀⣤⣾⣿⣆⣿⢸⣿⡀⠀⣿⡄⠀⠀⠀⠀⠀⡉⠃⠀⠀⢠⣾⣿⣿⡇
⠀⢀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠘⠙⢻⣏⠀⠀⠇⠀⠐⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡹⠇⢀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠄⡀⣠⣀⠀⠀⠀⠈⣿⠀⠀⠀⢀⠀⠀⠀⢁⣠⣿⣿⣿⣿⣿⣿⠿⢿⡇⠀⢸⣷⠀⠀⠀⠁⠀⠀⡄⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠐⠱⠳⠁⠀⠀⠀⠘⠀⠀⠘⠁⢀⡀⠀⠸⣿⣿⣿⣿⣷⣧⣤⣤⡬⠃⠀⠀⢀⡄⠀⠀⠀⠀⣠⠇⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⡀⠀⠐⠺⡶⢄⠙⠿⣿⣿⣿⣿⣿⡿⠄⠁⠀⠀⠈⠁⠀⢈⠙⠲⣿⡀⠀⠀⠀⠘⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡀⠁⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠆⠀⠠⡈⠀⢰⣦⢴⠧⠀⡄⢸⡿⡁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⡿⢿⣿⣦⠀⢤⡄⠈⠁⢀⡄⢰⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠓⠀⠂⢁⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⡾⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣬⣿⣿⣿⣟⠻⣆⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠹⣿⣿⣧⣼⣷⡝⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠿⠇⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⣶⣶⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢿⣶⡀⢰⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⣿⣿⡇⢀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣴⣿⣿⣿⣿⣿⣄⠘⣿⡇⢸⣿⣿⣿⠿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣿⣿⣿⣿⠆⢻⢀⣼⠿⠿⠁⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⠛⠟⠙⠃⢀⣶⣶⣷⣆⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠛⠋⠀⠀⠀⢀⣈⣄⠛⣴⣦⠀⣾⣿⡟
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠇⣠⣖⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⢹⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢿⡀⢿⣿⡄⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣸⣿⣿⡿⠁
⠀⠀⠀⢀⣤⣾⣿⣿⣿⠿⢟⣤⣧⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣸⣿⠟⠁
⠀⠀⢰⣿⣿⣿⣿⠛⠋⠀⣿⣿⣿⣧⡀⢻⣧⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡟⠁
⠀⠀⠀⢻⣿⣿⣿⠀⢀⣀⠙⣿⣿⣿⣿⠈⠙⣃⣠⠀⠀⠀⠀⠘⠃⠀⢀⣠⣤⣶⣿⣿⣿⠷⠂⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠈⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡟⠁⢰⣿⣿⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⢰⣦⠀⠀⠈
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⢏⣿⣿⣷⠀⠹⠿⠿⠿⠀⠀⠀⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡄⢺⣿⣿⡟⢠⡆⠀⠀⠺⣶⡀⠀⣛⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣥⣾⣧⠀⠀⠀⠈⠛⣰⣧⡄⡀⠀⠘⠻⣿⣿⣿⣿⣿⡿⠗⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠧⠀⠀⠀⠀⠘⠟⠙⠛⠷⡶⢾⣿⣿⡏⠙⢁⡀⠀⠈⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣷⣾⠷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢛⣋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⠀⠀⠀⠀⠀⠀⠈⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠁⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠛⠁






⣤⣤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠀⠄⢸⣟⣙⣻⢙⣋⣛⣉⣛⣏⠙⣛⣉⣋⣹⠛⡉⢉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣉⣹⣟⣉⣿⣿⣏⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠄⢀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡹⡆⠀⠀⠀⠀⢠⡀⠀⣠⠄⠶⣇⠤⡨⠂⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⢀⠴⠚⠋⢙⠀⠀⠀⠈⠀⠀⢗⣆⠀⣨⠅⣀⠄⠀⠀⠀⠀⠀⢠⠏⠸⠄⠈⢊⠀⣄⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⢀⠠⠴⠚⠋⠀⠀⠀⠀⠢⠀⠈⠛⠫⠀⢂⠈⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢠⣎⣠⠄⢋⣘⠁⣄
⠀⠀⠀⠀⠀⠀⣀⣠⠄⠴⠋⠀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⣀⠀⠀⡤⢯⡇⠈⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠰⠛⣉⠔⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠦⠰⠚⠁⠸⣏⠲⠀⠀⠀⢘⡇
⠀⠀⠀⠀⣰⣎⣁⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⣒⠂⠀⠀⠈⠀⠀⠀⠐⠒⠂⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡔⢢⣤⠛
⠀⠀⠀⠀⠙⢿⡧⠊⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⠭⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠳⣄⡀⢀⠠⢄⡤⠤⣄⣠⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⣀⣴⡯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⣀⣀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠘⡆⠑⡶⠉⠑⠂⠛⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠍⠚⡅⢰⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠙⠷⠚
⠀⠀⠀⢠⣍⣤⠞⡙⠠⡤⡌⠄⠀⠄⠀⠀⢱⠘⡧⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⢣⠏⡠⠊⠙⡆⢳⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠚⢣⡼⠲⠔⠼⠀⠀⠀⠀⠀⢃⢧⠈⠑⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⡀⠤⠤⠒⠊⠉⢀⣮⠏⠉⠁⠀⠜⢁⠌⠀⠀⠀⣠⠘⢃⠀⠙
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣟⡠⠐⣀⠄⠂⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⣡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⣠⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣠⣞⠋⠉⠉⠉⠉⢉⣁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣟⣿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢹
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠀⠀⣀⣈⣉⣢⣶⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣫⡄⢸
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠛⠋⢉⡴⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⢐⡇⢸⡆
⠀⠀⠀⠀⢀⡃⣀⣀⣀⡰⠂⠀⢠⠎⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⢻⣟⠁⡈⠀⠈⡇⢸
⠀⠀⠀⠀⠈⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⣷⠿⣡⣶⡽⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⢠⠈⠁⠱⣈⠳⡴⢁⡎
⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡹⢉⣺⣷⣤⣄⠀⠀⠀⠀⠴⣒⠉⠡⣀⣆⠀⣬⣀⠀⣲⠿⠉⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢸⠅⠒⠀⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠛⣵⠿⣿⡍⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣉⣉⣀⣉⣉⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣠⠀⠀⢠⡦⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠋⠙⠂⠀⢠⠀⠀⠀⢸⠁⠀⠀⠀⢸⠀⠀⠐⠀⠀⣀⠈⠁⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⡀⢀⡨⠊⠀⠈⠻⠁⠐⢄⠀⠈⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠀⠀⠀⠀⠀⣸⢹⠃⠀⢈⠕⠋⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠆⠀⠈⠑⠦⠘⠁⣾⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⢿⣟⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠛⢃⣀⢀⡄⠘⠃⠀⠀⠀⠀⠀⢸⡇⠀⡠⠊⠐⠚⠉
⠀⠀⠀⠀⠀⠖⠚⠛⠉⠪⠌⣋⣭⡉⠀⠀⠀⠀⣘⡁⠜⠁⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠴⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⢺⡿⠒⠀⡀⠀⣀⡠⠤⠒⠂⢀⣀⠐⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢸⡇⠠⠊⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠁⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⢀⠄⢀⡀⠤⠤⠤⠄⠀⠀⠀⠀⠀⣬⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠤⢄⠐⠄
⠀⠀⠀⠀⢸⣤⣶⠒⡻⠿⢛⡇⠀⠀⠀⠀⠻⡿⠡⠄⠀⠀⡌⢿⡇⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⣿⣿⡧⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠍⠉⠩⠐⢷⣶⡇
⠀⠀⠀⣸⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⣴⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠘⠻⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠢⠤⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢹⣿⣿⣿⣿⣿⡇⢿⣿⡄⠀⠀⠀⠈
⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⢸⣿⣿⣿⣿⣿⣿⠘⠿⠗⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠉⢉⣉⡉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿
⠀⠀⠀⠀⠀⠐⠛⠛⠋⠛⠛⠓⠀⠀⠀⠀⠀⠚⠛⠛⠃⠛⠛⠓⠂⠀⠀⠀⠀⠀⠚⠛⠛⠘⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠀⠀⢀⠀⠠⠀⢀⠀⠀⠀⢀⠄⠐⠠⠀⠤⢂⠀⠠⠄⡀⠂⠀⠁⢀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠠⠀⣄⠀⠀⠀⠀⠀⣠⠤⡐⠃⡀⢀⠀⢠⣠⢃⣴⣶⣦⡌⢂⡀⠀⠈⡀⠄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠊⢀⠏⣱⠓⠀⣠⡄⢧⣄⣥⠀⢳⣤⣤⡾⠻⠸⣿⣏⠛⠁⣸⣿⣷⣶⠟⣩⣤⣤⣤⣄⠠⠰⣄⠂⡑
⠀⠀⠀⢱⠄⢠⠖⢂⡞⣡⣤⣤⣄⢀⡁⠂⠉⠻⢷⡊⠡⡙⠻⢷⣾⠃⠄⡻⠿⠘⢧⣹⣿⡿⠋⢄⠀⠊⡀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠀⡱⢭⣿⣯⡿⠇⠀⢠⡀⠂⠀⠳⠄⢠⠐⠀⠛⢀⠐⠄⣠⢀⡜⠈⠋⠠⠀⡄⠜⢠⠀⠈⠍⢂⠁⢀⠰⡁
⠀⠀⠢⡀⠀⠀⠘⡁⠀⠈⢻⡏⣴⠛⠲⡎⠇⠈⠀⢀⣧⠅⠀⠀⠂⠀⢠⠄⠈⠎⠀⠀⠈⠆⠀⠠⠀⠁⡈⢁⠄⠃⠀⠢⠑⠈
⠀⠀⠀⠀⣠⣴⣶⡆⠀⠀⠀⠠⣙⣀⠠⠐⠀⠀⢀⡌⠩⡔⠀⣀⠀⠁⡄⠀⠃⠀⠈⠀⠀⣼⠦⠠⠄⢀⡠⢺
⠀⠀⠊⠠⠛⣿⣿⡇⠀⢀⣠⠴⠚⠁⠀⠀⢀⢠⣚⠛⣂⠘⠒⠂⢴⣾⣧⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠠
⠀⠠⠀⠀⠸⣿⡿⠗⠊⢉⢁⠀⣀⠤⠖⢿⡿⢐⠀⠀⠀⣿⠿⠴⢿⠛⠛⠓⠚⠦⣀⠀⠀⠀⠀⣀⢐⠈⠀⠄⢀⢀⡄⠊⠔⠁
⠀⠀⠁⠀⠰⠏⠀⣴⢛⠻⢄⠀⠛⠛⠒⢃⢰⣿⠷⠶⠿⠣⠿⠏⢈⣴⡿⢟⠁⠐⢦⠌⣈⡉⠁⠉⢁⡤⠊⢉⣉⣀⣬⣦⠀⠐
⠀⠀⠔⠁⢘⠀⠀⠰⣶⡌⢂⢹⣶⣾⠟⢭⠀⠀⠀⠠⠰⠀⠀⢰⠐⠐⠒⠸⢳⣤⣀⠁⠈⡿⠌⠀⠁⠀⠀⠀⠌⠻⣿⣿⣷⠄
⠀⠀⠀⢰⡏⠠⣀⠐⠀⠀⠀⠜⢛⠿⢇⠀⠀⠀⠀⠐⠀⠀⠀⠌⠀⠀⢄⡄⠉⢉⡽⠻⠇⠘⠚⢧⠄⡈⠀⠐⣶⡀⠐
⠀⠁⠑⣬⠻⠄⠈⠑⠲⡾⠀⠐⠁⠀⠀⠑⠀⠀⣀⣀⣀⠄⠊⠀⠀⠀⠀⠑⢂⠉⠁⡀⠀⠀⠀⠣⠄⢈⣤⣁⡘⣿⣷⡀⠁
⠀⠀⠀⡤⡀⠀⢱⣤⡜⠁⠀⣄⠀⠀⠀⠀⠀⠀⢸⣟⣿⠘⠀⠀⠀⠀⢤⠚⢤⣀⣤⣈⣠⣀⣀⣴⣶⠞⠛⣫⣿⣿⣿⣏⢈⠂⡆
⠀⡄⢄⠳⢥⣴⡿⠟⠁⠀⠀⠈⡦⣄⣀⠀⢀⣠⠊⠀⠉⠓⢌⡂⠄⢀⠀⢀⠈⠟⠁⠉⠀⠈⠋⠁⢡⡴⠊⠀⠈⠛⣟⠋⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠈⠈⠇⠘⢏⠂⠁⢄⠈⢻⡦⢀⡿⢮⠈⠁⡀⠀⠀⠀⠀⠐⠋⠀⠈⠁⠠⠀⠀⣆⢠
⠀⠀⠀⡂⠀⠈⠀⠈⢛⠀⠀⢀⠀⠀⢠⢠⠀⡀⢸⠀⡀⡀⢂⠀⠰⡏⠁⠄⢳⡈⢀⡀⠀⠀⠤⠈⠢⡀⢀⠀⠀⡁⠀⠘
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠂⠀⠀⢚⣃⡌⠴⠁⢀⠀⠘⠀⢁⠐⢤⡀⠊⡄⠙⠄⠠⣠⣴⠟⠀⠀⡇⠠⣀⢤⠯⢢⡌⢀⠄
⠀⠀⠁⠠⡀⠨⠀⠀⠀⠀⢲⣶⣶⠀⠱⢠⢹⣦⡀⠄⠀⠀⠁⠀⠀⠀⡠⢄⠀⢄⠀⠂⠉⠁⠀⠀⠄⠁⡀⠀⠠⣀⠌⠀⢀⠄
⠀⠀⠀⠀⠉⠂⠀⢠⠀⢀⠀⠉⠉⠀⠀⠀⣾⣿⠿⢷⣶⣶⡶⠦⡂⠣⠈⠐⠠⢘⡀⠀⢀⢀⠀⠀⢀⠁⣄⡑⢀⣿⠄
⠀⠀⠀⡀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠈⠀⢀⠈⠉⠈⠐⠤⠙⢆⡀⠈⠀⠄⡀⠀⠢⢈⡐⠒⠀⠀⣠⣼⣦⠄⠀⣂⠄⢀
⠀⠀⠀⠀⠀⠀⠄⠂⠄⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠈⠁⢐⠉⠀⠀⠈⢂⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠁⠐⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣫⢟⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣱⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣿⡟⣰⠏⣼⣣⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢫⡟⣸⡟⣰⣿⣿⣿⣿⣿⣭⡶⠲⠶⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡞⣴⡿⣡⠿⣫⣷⠟⠛⠛⠋⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢻⡶⠖⣫⣾⠟⡴⠃⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣩⢿⣿⢋⠞⢀⣾⣿⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⢧⠏⢀⣾⡟⣿⠀⠀⡴⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⡏⠀⣼⡿⠀⠸⡆⢈
⠀⠀⠀⠀⠀⠀⠀⠇⢻⣏⠁⢀⣿⠁⠀⠀⠁⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠸⡜⣿⡄⠘⡏⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣷⡀⠙⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⠻⠿⠿⠛⠁⠀⠈⠳⢄⡀⢀⠀⠀⠀⠀⠀⡀⠔⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⢤⣀⡀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠀⠀⣀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣬⣑⡒⠶⠦⠤⠤⠤⠴⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡅⡜⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡶⢸⡋⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠯⢱⡾⠖⠦⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠤⣓⡒⢽⣀⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⡔⢋⣉⢙⡏⣁⠹⡀⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⠻⣎⠑⣚⣲⣶⡷⠚⠉⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠡⡈⢷⡩⢹⠏⠀⢀⣠⠴⡾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠀⠣⡘⢄⠹⣿⠀⢠⡿⡁⠂⠁⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠐⡈⢂⠑⡈⢆⢹⠀⢸⡓⠌⡶⢋⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠲⢬⣄⢂⠘⡌⢿⡆⠸⣧⠤⣇⠣⠥⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠁⠂⠬⢉⣛⣻⡾⣽⠖⠛⠛⠛⠛⠷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣈⠁⢒⡶⠫⡉⣀⣣⠊⣱⠮⠭⣖⠒⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣠⡿⢀⣰⠟⠉⣩⡇⢀⣀⣀⡀⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣸⠏⢠⡿⠋⢹⡋⢈⠉⢳⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠺⠧⣴⣿⣿⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠀⠀⠉⠉⠉⠉⠀⠀⠐⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⡨⠐⠂⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⡘⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠠⢐⠡⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡨⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠢⡀
⠐⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠐⠁
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣐⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡟⢻⣿⡿⠋⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⠁⠈⠉⠱⢶⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇⠀⠀⠀⣿⠀⣠⣄⠀⠈⠉⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⣚⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⡇⠀⠀⠀⣿⠀⣿⣿⠀⢰⣦⠀⠛⢻⣿⣿⡇⠀⠀⠀⣿⣿⣿⣀⣿⣄⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣷⣿⣿⣶⣿⣿⣶⣶⣾⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⠀⡰⢂⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣒⣢⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆
⠀⠀⠀⠀⠛⠋⣻⠉⣩⠥⡀⠀⠀⠀⠀⠀⠀⠘⣶⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠙⢳⡞⠋⢀⡷⠀⠀⠀⢀⣀⣋⠛⢿⣿⣿⡿⠟⠀⠀⠀⠀⠸⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⢦⣤⡾⠃⠙⠛⠁⠀⠀⠀⠉⣿⣿⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⣀⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢤⢤⠤⢤⠤⠀⠀⠀⠀⠀⠀⠀⣠⣼⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⢰⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⣾⣾⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⡶⢦⡞⠀⢳⡤⢶⠆
⠀⠀⠀⢀⣈⣿⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⠘⢻⣿⡿⠇⠀⠀⠀⠀⠀⠰⣿⣵⣿⣿⣬⣷⠂⠀⠀⢠⡤⠧⠀⠃⠀⠘⠀⠸⢤⡄
⠀⠀⠀⠙⠛⠻⠿⠿⠻⠿⠛⠀⠀⠀⠀⠀⠀⠈⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠛⠛⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⢷⣦⣶⣺⣿⣿⣧⣤⠀⠀⠀⠀⢠⣤⣶⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣽⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃
⠀⠀⠀⠚⠳⣿⣿⣿⣿⣿⡯⠀⠀⠀⢤⣼⣿⣿⣿⣿⣧⡤⠶⠀⠀⠀⠀⠦⠝⠾⢸⢠⣂⠀⠀⠀⢴⠁⠰⠀⠀⠓⠀⠀⠆⠘⡆
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠰⣶⣶⣾⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⢀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⢻⣿⣿⢸⣷⣶⣶⠂⠀⠀⠀⠀⠈⠉⠀⢩⡁⠀⠀⠀⠀⠀⣺⣿⣿⡟⠘⣿⣿⣿
⠀⠀⠀⠀⣀⣀⣐⣁⣀⣀⠀⠀⠀⠀⠀⠸⠟⠋⣾⣿⣿⣿⠀⠀⠀⠀⢠⡶⢦⣄⡴⠃⠀⠀⠀⠀⠀⠙⠟⠋⠀⠀⠈⠛⠟⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣧⢻⢯⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠘⡅⡰⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣷⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠒⠒⠒⠶⠶⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠛⣛⣉⣉⣭⣭⣭⣭⣭⣉⣉⣙⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣼⣿⣤⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠘⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⢹⡷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣠⡾⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⢁⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺
⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐


⠀⠀⠀⠀⠰⣮⣶⡄
⠀⠀⠀⠀⠘⣻⣿⣷
⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠐⡻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠠⡽⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢦⠀⠸⠀⠀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢣⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⠈⠠⠘⠛⠃⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⣀⢀⡈⢿⣧⢠⢧⠀⠀⠀⢰⣷⠀⠀⠐⢿⣿⣦⡀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠙⠧⠙⠸⣿⣧⠸⣧⠀⡀⢿⡿⠀⠀⠀⠀⠙⠛⢡⡾⠀⣴⣿⣿⣿⣿⣦⡀⠀⢠⡀⣀⠀⣷⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠈⣆⠳⣦⣄⠰⣶⣶⣃⡠⣶⣿⣧⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠑⢮⣱⢷⠈⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡤⠹⡄⠈⠻⣿⣿⣟⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠙⠿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣷⣄⡀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⣮⡓⢤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡛⣻⣦⣿⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠀⡗⠀⠀⠈⢿⣿⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠛⠢⢿⣿⣷⠻⠿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⠈⠀⢤⢰⡿⠻⢇⡀⠀⢹⣿⣿⡀⢀⣾⣿⣿⣿⣿⣿⣿⣿⢣⣶⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷⣶⣿⣿⣿⡄⢀⣿⣿⡷⢶⣿⣿⣿⣿⣿⣿⣿⠻⣿⡵⠾⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠂⠘⡐⢻⣿⢿⣿⣷⣿⡅⢠⣽⣿⣿⣿⣿⣿⣿⣿⣿⣫⠏⣻⣿⣌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⣴⠁⠶⣶⢷⣌⡏⢿⣿⣶⡏⢻⣻⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠳⠁⠀⣿⠘⢷⣧⠀⢿⣿⣦⡀⢸⣛⣻⣽⣿⣿⣿⣿⣿⢳⣍⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠉⠂⠀⢿⣷⠉⠋⠀⠉⠉⠻⣿⠻⢿⣿⣼⣿⣦⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢛⣻⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⣠⣴⣶⣶⣶⣶⣿⣿⣿⠛⠉⠁⠉⠀⠀⠈⠹⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⡴⢋⣵⣿⣿⣿⢏⣽⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⠳⣄⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣴⡏⠀⣼⣿⣿⣿⡟⣸⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠒⠲⢿⣿⣿⣿⣿⡇⢻⣷⡄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣾⣿⡇⠀⣼⣿⣿⣿⡇⣿⣿⣿⣫⣭⣭⣄⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡗⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢿⣿⣿⡄⢹⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠃⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⠻⣿⣿⣆⣿⣿⣿⡇⣿⣿⢿⣿⣿⣽⡿⠃⠀⠀⠀⠘⢿⣿⣿⣻⠛⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⡇⣿⣿⣷⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⢧⣿⣿⣿⣿⣿⡇⠀⠀⢀⣤⣤⣤⣄⠀⠸⠃⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⣿⣾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠙⢿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⣀⣀⣀⣤⣾⠷⠤⠤⠚⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠏⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢸⣿⣿⡟⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠈⣿⣿⣿⣌⠻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡙⠻⢿⣦⣌⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣉⣙⠂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣛⣛⡈⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣭⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣒⣿⣿⣾⡇⠀⠀⠀⣀⣦⣀⣀⣀⣀⣀⠀⠀⠀⢸⢷⣶⣆⠻⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡄⠀⢠⣿⣿⢟⣛⠿⣛⡻⣿⣿⡆⠀⠸⣿⣿⡟⣫⣾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡍⠉⠀⠀⢸⣿⣿⣜⢿⣿⡿⢃⣿⣿⡇⠀⠸⠿⠿⢣⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣻⣃⣤⡀⠀⠈⠻⠿⠿⠷⠾⠾⣿⡿⠟⠁⠀⠀⠀⣾⡟⠿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡟⣫⣭⡻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣾⣶⡭⠵⣾⣮⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣷⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢏⣾⣿⢎⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡸⣿⣿⣧⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢿⣿⢫⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⢻⣿⣿⣞⣃⣀⡀⠀⣀⣀⡀⣀⡀⠀⠀⠀⣰⣿⡷⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⢸⣿⣿⣿⣿⣿⠟⣑⣛⣛⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠒⠒⠒⠀⠀⠀⠀⠈⠉⠁⠉⠁⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⢄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠤⠀⠁⠀⠸⣀⠆⣃⣀⠀⠀⢀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡐⠃⠄⠀⠈⠈⡆⠠⡞⠀⠀⠑⠀⠆⠁⠁⠀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⢄⠀⠈⠀⠀⠁⠠⠃⠈⠀⠀⠁⠀⠁⡀⢠⡌⠆⡠⡠⠀⠀⠉⢅⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠙⠀⠀⠐⠀⠀⠀⠀⠄⠀⣀⠔⠀⠤⡈⠁⠈⠎⡠⠀⡂⠀⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⣁⣀⠑⡀⢰⠁⠀⠀⠀⠀⠀⠀⡰⣁⢀⠆⠀⢰⡠⢊⠞⠁⠀⠀⠁⠀⡀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠰⠀⠐⠁⠀⡇⠁⠸⠀⠀⠘⠠⠁⠈⠂⠀⠀⠀⠀⣀⢉⠊⠀⠀⠀⠀⠀⠀⣀⠊⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠑⠀⠀⠚⠁⠀⠂⡀⠀⡈⠀⠢⢄⣀⣀⣀⣀⠈⣀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢱⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⡡⠀⠐⠄⠲⡁⠘⠀⠦⠴⠖⢒⣶⣾⡧⣠⣤⣭⣄⢀⣤⣴⡄⠀⣠⣶⣶⡈⢆⠀⠁⢀⠆⡼⠔⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠁⡁⠀⠀⠁⠀⢀⡀⢀⣴⣿⣿⠟⢰⣿⣿⣏⠏⢸⣿⣿⡇⣼⠃⣿⣿⠇⠀⠉⠨⠅⠀⠀⡐⠠
⠀⠀⠀⠀⠀⠀⢀⠐⢞⠀⡀⢀⠀⠀⠀⢠⡡⢃⣴⣿⣿⠟⠁⠀⢿⣿⣿⠁⠀⢸⣿⣿⣿⠃⠀⣿⣿⠀⠠⣀⠀⡀⠈⣾⠖
⠀⠀⠀⠀⠀⠀⠀⡄⡉⠘⠀⠁⠪⠈⠀⠡⢠⣾⣿⣿⣋⣠⡤⠄⠘⢿⣿⣧⡤⠘⠿⠿⠇⠀⢸⣿⣿⠀⡌⢀⠈⠇⠈⢁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠘⡐⠄⠄⠁⠘⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣿⡟⠀⠈⢉⡜⠉⠩⠒⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⠆⠈⠀⣀⡀⠔⠀⡐⢁⠐⠉⠁⠂⠀⠀⢀⠔⠊⠁⢀⡀⠉⢢⠸⠿⠁⠐⠀⠸⢄⠈⠠⠤⠹⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠤⠂⠀⠸⠄⠈⢸⠀⠃⢀⠀⠀⠀⠀⡠⠒⠁⠀⢀⠐⢅⣀⡠⠊⠀⢐⠀⠂⠀⠲⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠐⡄⠠⠀⠀⠈⠊⠂⠤⠐⠀⠉⠀⠀⠀⠔⢰⠀⢀⠤⢄⢀⠀⠦⠆⠄⠁⠀⠠⢤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠄⢰⠀⠀⠀⠔⣀⡀⠀⠀⠂⠀⠐⠀⠐⠌⠜⠀⠁⠠⢀⡀⠀⢀⡈⠁⠁⠠⡑⠄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣤⠌⠀⠀⡈⡀⠅⢈⠆⠀⠀⠀⢀⠰⠀⠀⠠⠬⠒⠒⠉⢀⠤⡄⢻⢺⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡐⠊⠠⠭⡠⣄⠀⠱⠁⠀⠌⢀⣠⠴⠒⠦⠀⠘⢄⣀⠼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⢇⠢⠑⡒⠂⠉⠁⠐⠆⠠⡆⡛⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠥⠈⠊⠥⠴⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠈⠉⠉⠙⠛⢿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⢀⣀⣤⣤⣄⣀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⣀⠀⠀⠀⢀⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣾⣿⣿⣷⡀⣠⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠏⢨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡇⣾⡇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣦⡉⠻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠆⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣧⣘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣌⣉⠙⠛⠃⠀⠀⢀⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣧⣤⣴⣶⣦⡄⠀⠀⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢙⠟⠻⠿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⡙⠛⠿⢿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⡍⣇⣠⠶⢟⣿⣿⠿⠟⠃⠀⠀⠀⣸⣿⣷⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⠀⠀⠀⠐⠘⠌⡀⠁⠊⠛⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣧⡀⠀⠀⠘⠛⠁⡀⣼⠟⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⠄⠀⠘⠛⠛⠁⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠁⣰⠀⠀⠀⠀⣀⡄⢲⣾⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣴⣿⣀⣤⣶⣿⣿⠀⢸⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠀⢸⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠇⠴⠾⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡠⠲⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢣⠀⠈⢂⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠑⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠸⡆⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠼⢷⡀⠀⢹⡉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⢙⣦⡀⠹⣤⠤⠀⠒⠒⠀⠒⠒⠒⠂⢀⠀⠀⠀⠀⠀⠐⠂⠒⠒⠒⠠⠤⣀⣠⣾⣠⠴⠖⠒⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠳⠶⢯⣥⣄⣹⣆⣈⣳⣀⣀⣀⣀⣀⣀⣀⣴⣋⣉⣓⣦⣶⣶⣦⡤⠤⠤⠶⠖⠛⠉⠁⣿⣁⡤⠴⠶⣄⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⢠⡏⠁⠀⠀⠀⡿⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣠⠞⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡿⠀⣴⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠇⣼⠁⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⡿⠛⢁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗⣚⣉⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠚⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣀⣀⠀⠉⠉⠉⠉⠙⠛⠲⠦⣄
⠀⠀⠀⠀⢀⣤⠶⠛⠉⠁⠀⢀⣀⣀⣤⢽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⠉⠉⠉⠉⠒⠒⠠⠀⠀⠈⣷
⠀⠀⠀⣴⠟⠁⢀⡠⠔⠒⠉⠉⠀⠀⠀⠀⠙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠲⠶⠶⠶⠦⢾⠶⠶⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⢛⣽⠋
⠀⠀⠀⠹⣗⠲⠤⣤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⡀⠀⠀⣀⣀⣀⣠⣤⣤⠴⠶⠖⠚⠛⠋⠉⠉⣀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣿⠉⠉⠙⠛⢯⡉⠁⠀⢀⣀⣀⣠⣤⠤⠶⠖⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠲⠶⠶⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣏⠀⠀⠀⠀⠀⠙⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣋⣾⣿⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢿⣿⣿⣶⣿⣿⣌⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⠉⣾⡟⠁⢸⣿⣿⣿⡆⠙⣿⣧⡙⢿⡄⠀⠀⠀⡀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⣼⠋⠀⠀⢹⣿⣟⠹⣿⡄⠈⠻⢷⡀⠹⡄⠀⣸⠇⠀⠀⠀⠀⣸⢇⣠⣤⡶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⢹⡇⠀⠘⢧⠀⠀⠀⠻⡄⠀⢰⡿⠀⠀⠀⠠⣄⠉⢿⣥⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⢦⣄⠀⠀⠀⠀⠀⠿⣡⣶⣤⣾⣴⣿⣷⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠤⣤⣤⣬⡷⠀⠀⠀⢰⣿⠋⣿⣿⣿⣏⢻⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⠀⠀⠀⠀⣸⣇⠀⠀⠀⣀⠠⢚⣉⣀⡊⠀⠀⢹⠛⠇⠹⡟⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣻⣿⢿⣿⢿⣿⣿⡿⢶⣤⣀⣿⣿⠀⠰⠚⠹⣟⡛⠛⠛⠛⠛⠛⠻⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣰⡿⠃⢈⣿⠀⢈⣩⣿⡷⢾⣿⣿⣿⡁⣀⡀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢈⣿⠒⠋⠉⠀⣠⡾⠋⣻⣿⣿⡍⠻⣷⡄⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢸⡏⠀⠀⢀⠞⠉⠀⠀⣿⣇⠸⣷⡀⠈⠻⢦⡀
⠀⠀⠒⠲⠶⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣦⣙⣧⣄⣀⠀⠙⠄
⠀⠀⠴⠶⠟⠂⡀⣄⠻⣦⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠄⣠⣔⠶⣶⣌⠁⢸⣾⡏⠈⣿⠻⣯⠙⠳⣤⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⣾⠃⢻⣧⠀⣙⢷⣴⢿⣷⡦⢀⡤⠶⣿⡿⢻⣿⢘⣿⣿⣷⡌⠙⢷⣎⣸⡇⠀⣟⣠⣼⣿⣾⣶⢋⣴⣶⣾⣿⣿⣟⣱⣿⣟⠻⢿⣶⣄⡀
⠀⠀⠀⠀⣸⠏⠀⠀⠹⠆⠀⢀⡴⠚⠉⣴⢿⣟⢿⣯⣠⣿⡿⣿⣿⣿⣿⡛⣆⠀⠈⢹⡇⣠⡿⠛⢩⣿⡟⠁⣸⡟⣿⣷⡿⠛⠛⣿⣯⣿⣷⡀⠀⠉⠛⠦
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠏⠈⢿⡄⠈⣿⣿⠀⣿⠙⣧⠙⢧⠈⠲⠄⣸⣏⣿⠀⢠⡿⠋⠀⠀⣿⡇⢸⠿⣷⣄⠀⢿⠀⠉⠙⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⣿⡟⠀⡟⠀⠉⠂⠈⠀⠀⠈⣿⣿⣇⢠⣿⣤⣄⡀⠀⢻⠁⣾⠀⠀⠙⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢀⣴⡾⠿⠿⠷⠶⠶⢀⣿⣿⡟⣆⠹⣿⣝⠿⣷⡜⣶⣿⣷⢾⡻⢿⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⡟⢸⣟⠻⢷⣦⣠⣤⣤⣾⣿⣿⣧⣿⣦⣬⣻⣷⣌⣽⣿⣿⣿⣠⣿⡾⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⠀⠀⠀⢰⠏⢀⡿⠁⠀⢻⡟⠛⠉⠁⠀⣸⣿⣿⣿⣿⣏⣿⠛⠿⠿⣿⣧⣾⠋⣽⣿⠙⢿⣆⠀⠉⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠤⠒⠋⢩⣶⠟⢫⣿⣿⠳⣧⡀⢸⠃⠀⠀⠀⠃⠀⠀⠀⢰⣿⠋⣿⣿⠙⢿⣿⡄⠀⠀⠀⢼⣿⠀⠈⠿⠀⠀⢿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⣿⠃⢿⡆⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢷⣄⠉⠛⠦⠀⠀⢸⣿⠀⠀⢀⣀⣀⣈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⣼⠃⠀⠈⢿⡀⢰⣿⢓⡄⠀⠀⠀⠀⠀⢰⣿⠛⣿⣟⣿⠀⠉⠓⠄⠀⠀⠀⣼⠃⣠⣾⡛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣶⣦⠻⣿⡿⢶⣤⣀⠀⠸⠃⢰⡿⣯⣀⣤⣤⣦⣤⡀⠀⣠⣿⡇⢿⡟⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣰⡿⠋⣿⠏⣿⠀⠘⠻⣦⣈⠉⠛⠂⠀⡿⢰⡯⠈⢀⣴⠿⠛⣡⣶⡿⣿⠃⠈⠳
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀⠈⡿⢿⣄⠰⡟⠀⣿⠀⠀⠀⠀⠉⠓⠄⠀⢰⡇⢸⡇⠐⠋⠀⣀⣴⣿⣿⣧⣸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⡽⠿⢋⣠⣤⡽⢾⢁⣰⡏⠀⠀⠀⠀⠀⡀⠀⠀⣾⠀⢸⣿⠀⢚⣾⣿⣿⣿⠿⢿⡎⠙⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⣰⣿⣿⡿⣷⣄⣀⣼⠻⣶⣆⢠⡴⠟⣿⣿⣾⣯⣴⣟⣿⠟⣿⢻⣿⣿⡿⠀⢈⡁⡀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢻⣧⡼⢿⡏⣿⠀⠈⠛⢷⣄⡸⢋⡿⣿⣿⢿⣾⡿⣿⣅⢺⣟⣠⣾⣟⡛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢿⠀⠀⠹⣿⡆⠀⢀⡀⠸⣿⣆⣀⣿⠿⣿⣿⣾⣿⣿⣿⠿⢿⣧⣉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⠀⣿⡧⠐⠉⠀⠀⣹⣟⣻⠁⣼⠟⠁⣿⣿⡿⣧⡀⠀⠉⠙⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠶⠛⣻⠿⢁⣿⡇⠀⠀⠀⠀⠈⣿⠃⣰⠋⠀⠀⣿⣿⡇⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣼⣿⣇⠀⠀⠀⠀⣼⡿⠀⠁⣠⡾⢳⣿⢛⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣿⣀⣴⣶⡶⣿⢧⣤⠞⠉⠀⣼⠟⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢱⡟⢸⡟⠈⠻⣆⠀⢠⡟⠀⠘⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⠁⢸⠁⠀⠀⠹⡆⣾⠁⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢸⡆⠀⠀⠀⣸⡇⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠸⡷⠀⠀⠀⣿⠇⠀⠀⠀⠀⠻⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢹⠉⡋⢽⠋⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⡿⣿⠿⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠿⠿⠿⠿⠛⠋⠀⢀⣀⣤⣤⣤⣤⣄⣀⡀⠸⣿⣄⣻⣇⣏⣄⣿⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣾⢿⢿⡿⢿⡿⣿⡿⣿⠿⢦⡈⠉⠛⠛⠛⢿⣿⣿⠟
⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣦⣤⣀⠀⠀⠀⠹⣿⣨⣸⣁⣸⣇⣻⣇⣻⣀⣸⡇⠀⠀⠀⠀⠀⠋
⠀⠀⠀⣴⠟⢻⡟⣿⢻⡿⣿⠟⠛⢳⡀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⢿⣃⣹⣤⣼⣘⣇⣁⣿⣸⣿⠃⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⢼⣘⣁⣡⣜⣃⣁⣁⣫⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠗⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡄⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠟⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⣤⠀⠀⠀⢰⣿⠟⠛⠛⢿⣷⠀⠀⣠⣾⣿⠿⠟⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⣿⣿⡿⠀⠀⣿⡿⠁⠀⠀⠒
⠀⠀⣠⣶⣶⣿⣷⣤⣄⠀⠀⠀⠀⠀⢻⡱⣿⡆⣠⢃⣤⣶⣿⣷⡈⠁⣠⠶⣾⡆
⢀⣸⣿⣿⣿⡿⠛⠙⣿⣾⣿⣿⣿⣦⠀⢹⠖⢰⠁⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠃⠀⠀⠀⠀⠀⣀⣴⣶⣤⣀⡀
⢺⣿⣿⣿⠿⠰⠄⠰⣿⡟⠁⠀⠀⠈⢿⣿⣦⣥⡿⣻⣿⡿⠿⣿⡇⠈⠉⠉⠀⠀⠀⠀⠀⢀⣾⠿⠿⡿⠿⣿⣿
⢿⣿⣿⣿⣷⡀⠰⣢⣿⡗⠃⠀⠰⠀⢸⠇⠈⠁⠀⠛⣴⡦⢴⡶⡃⠀⠀⠀⠀⠀⢠⠀⢠⣼⡏⡀⠀⢀⠀⠼⣿
⠀⢿⣿⣿⡿⠀⣶⠿⠿⠿⣶⣿⣤⣤⠄⠀⠀⠀⠀⣄⡄⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡁⠀⠀⠀⠀⠋
⠀⠈⢹⣧⣄⣀⡀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⢀⡀⠀⣠⣼⣿⣿⣿⣿⠆
⠀⢠⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠙⠟⠛⠁⠀⣿⣿⣿⣿⣿⠟⠉⠁
⠀⢸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢰⡂⠀⠀⠀⣶⠀⠀⣼⣹⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⡇⠰⣤⣼⣿⡀⠱⠀⠀⠀⢻⠀⠀⡽⣿⡇⠀⠀⠀⠐⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⣬⣧⠀⡇⢸⣿⡄⠀⢰⣀⣀⣸⠀⠀⣿⡜⡇⠀⢀⠓⢃⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⣿⠀⠀⠀⢻⣇⠀⠀⠀⠈⣿⠀⢰⣿⣿⠄⠀⢀⡘⢘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠻⠀⠸⡞⢿⣿⠀⠀⠀⠀⢸⡄⠀⣿⠟⠀⠀⢀⢷⣝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣴⠀⠀⡇⢸⣿⠀⠀⠀⠀⢸⡇⢾⣶⡷⠀⠀⡸⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠⠀⠀⣿⠀⠀⡀⠀⢻⡀⠀⡀⠀⢈⡃⢻⣿⡆⠀⢰⣅⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⡾⠀⠀⠹⠀⠀⡟⠛⣿⡇⠀⠀⠉⢻⡇⢸⣿⡇⠀⣘⢸⣿⠀⠀⠀⠀⠗⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠟⠀⠷⣀⣀⣤⡄⠀⡇⠀⣿⡇⠀⠀⠀⠘⡇⢸⣿⡇⢠⣏⣸⠏⢠⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⢱⠀⣅⣀⣝⣇⠀⠀⠀⠀⣿⢻⣿⠇⢸⠉⣿⠁⠈⠀⠀⠘⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢰⠇⠈⡄⠈⢻⣿⡿⡀⠀⠀⠀⢻⣼⣿⡆⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⠀⡸⠃⣶⠈⣿⠓⢵⡒⠐⢲⣾⣿⣿⡷⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⠈⠠⣴⣿⣀⡉⠙⢉⡇⠀⠈⣿⣿⣿⠅⣿⣾⣇⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⢠⡀⡠⠜⣿⣿⠇⠈⢩⠦⠀⠀⢹⣿⡟⠶⣽⣿⣿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣴⡀⠀⠀⠈⠳⢾⣿⣿⣤⣼⣧⣦⠟⠀⠀⢿⢧⣰⠿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠿⣿⣿⣦⣄⠀⣼⠾⢋⣿⣿⣿⣿⣿⡀⠀⢀⣾⡀⢑⣾⣿⣿⣿⠿⠛⠱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡟⠙⠡⣤⣾⣿⣿⢹⣿⡏⢃⣀⣼⣿⠑⢾⣿⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡤⠈⣩⣿⡇⢸⣷⠥⠉⠨⠉⠙⠷⣶⣿⡿⠟⢹⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⣗⠀⢀⣼⣿⠿⣿⣷⣾⡶⢤⣰⣾⣧⣤⣘⣿⣤⣤⣼⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⣦⠴⠀⠈⠑⢿⡿⠁⣀⣸⣿⣿⣿⣿⠿⡿⡿⠿⠿⠿⢿⣟⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣾⣵⣦⣤⣈⣠⣬⣾⣿⣿⠿⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠀⠀⠒⠀⠀⠐⠒⠀⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠁⠄⠀⠀⠊⠀⢀⢀⣵⣾⣀⡀⢁⠀⠐⠂⠀⣉⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢁⠀⠀⠁⢂⠀⠄⠀⠚⠀⠀⠀⠀⠄⢣⡀⠀⠉⠉⠐⠒⠂⢠⣀⡀⠐⠀⠐⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⠁⡀⠀⢀⠠⠂⠉⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄⠈⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⢀⡤⠛⠁⠀⠀⠀⠀⡀⠀⠀⣠⠔⠃⠀⠀⠀⠀⠀⠀⠁⢦⠀⠐⠲⠀⠀⠀⠀⠀⠈⠒⢄⣠⡐⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠰⡿⢯⠠⠤⠤⠤⠤⠤⢤⣀⠀⣜⡁⠤⠄⠐⠒⠒⠒⠂⠠⠠⠌⣣⣈⠴⠒⠒⠓⠒⠒⠦⠤⠄⢹⢧⡀⡀⡈⢦
⠀⠀⠀⠀⠀⠀⠀⡜⠑⠀⠀⠞⠀⢸⠀⠀⠀⢀⠀⠀⢀⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⡀⠀⠀⠀⠀⢀⠀⡎⠀⠑⡄⠀⠀⠳⡀
⠀⠀⠀⠀⠀⢀⠊⣠⠀⡸⠃⠀⠀⡇⠀⠀⠀⠋⡀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠈⠀⠀⣇⠀⠀⠈⢆⠈⠀⠱⡄
⠀⠀⠀⠀⢀⠊⠀⠀⡖⠀⠀⠀⢠⠃⠀⠀⡠⠈⠀⠀⠀⠀⠀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠈⠑⠠⢄⠀⠀⠀⠀⠀⠑⢄⠀⠀⢹⡀⠀⠀⠀⠃⠀⠰⠐⡀
⠀⠀⠀⠀⡤⠀⠁⣠⠁⠀⠀⠀⠘⡆⠀⠔⠁⠐⠀⠀⠀⠴⠊⠀⡠⠔⠂⠉⠉⠉⠉⠉⠀⠒⠄⡀⠀⠁⠢⡀⠀⠀⠀⠈⠢⢀⠏⡀⢠⠄⠀⠀⢂⠀⠀⢡
⠀⠀⠀⠰⠀⠄⠀⠁⠀⠀⠀⠁⠤⠡⡌⠀⠀⠀⠀⢀⠆⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠘⢆⠀⠀⠀⠀⠹⠒⢄⠀⠀⠀⠀⠀⠀⠈⠀⡆
⠀⠀⠀⡆⠀⠀⠙⠀⠀⠀⠀⣀⡤⢻⠀⠀⠀⠀⢀⠃⠀⡰⠁⠀⠀⠀⢀⠤⠒⠀⠀⠐⠠⢀⠀⠀⠀⠀⠂⠀⠈⠆⠀⠀⠀⠀⢁⠀⠐⢄⠀⠀⠀⠑⠀⠀⢀
⠀⠀⢀⠀⠈⠠⡌⠀⢀⡤⠚⠁⠀⡌⠀⠀⠀⠀⡄⠀⢠⠃⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢀⠱⡀⠀⠀⠘⡀⠀⢰⠀⠀⠀⠀⠘⠀⠀⠀⠱⡄⠀⢡⠀⠀⠸
⠀⠀⢰⠀⠐⣠⡧⠚⠉⠀⠀⡄⠀⡇⠀⠀⠀⢀⠁⠀⠘⠀⠀⠀⢐⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⡀⠈⠓⢸⣷
⠀⠀⠀⠈⠂⠙⡇⠑⢆⠀⠀⠀⠀⡇⠀⠀⠀⢈⠀⠀⢸⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠁⠈⠁⢀⡴⠋⢰⡍⠀⠈
⠀⠀⠀⠃⠠⠄⣡⠀⠈⠣⡀⠀⠀⢨⠀⠀⠀⠀⠆⠀⠈⢦⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠰⠁⠀⡘⠀⠀⠠⠀⠸⠀⠀⡴⠋⠀⠀⠸⠂⠃⢰
⠀⠀⠀⠈⡀⠀⠘⠀⠀⠀⠘⠦⣀⡈⡄⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠂⠉⠀⠀⠀⠀⠐⠁⠀⡔⠁⠀⠀⠀⢠⠧⠒⠊⠁⠀⠀⠀⡁⠀⠄⡅
⠀⠀⠀⠀⢛⠀⠀⣦⠀⠀⠀⡀⠠⠈⢹⡄⠀⠄⠀⠀⠃⡀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⣠⠊⠀⠀⠀⠀⢠⠂⢳⠈⠠⠀⠀⠀⡈⠃⡀⠰
⠀⠀⠀⠀⠀⠆⠀⠀⡄⠀⠀⠀⠀⢀⠎⠈⢆⠀⠀⠀⠀⠈⠐⢄⡀⠀⠉⠀⠒⠂⠐⠒⠀⠉⠀⠀⣀⠎⠀⠀⠀⠀⠀⡠⠃⠀⠐⡇⠠⠀⠀⡰⠁⠀⣠⠃
⠀⠀⠀⠀⠀⠘⡄⠀⠘⠄⠀⠀⠀⢸⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠁⠀⠀⠰⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠘⠄⠀⠈⠢⠀⠀⢸⠀⠀⠠⠧⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠔⠊⠀⠒⠀⠀⠀⣸⠀⠀⠀⠁⡀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠱⡄⠘⢀⣀⣀⣀⣀⣀⣀⣀⣁⡤⢄⡀⡁⠀⠀⠀⠀⠀⣀⢀⣀⠠⡴⠪⢤⣀⣀⣀⣀⣀⡀⠀⠀⢀⠈⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠺⡷⠄⠀⠀⠀⠀⠉⠉⠈⠀⠀⠸⣄⠀⠀⢀⡄⠁⠀⠀⠀⠀⡔⠁⠠⠀⠈⠈⠁⠀⠀⠉⣡⠾⠛⠃⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢁⠀⠐⠀⠀⢄⠀⠀⠀⠀⠀⠃⠀⠘⣆⠀⠀⠀⠂⠀⢀⡤⠊⠀⠀⠀⠐⠀⠀⠀⢠⠔⠊⢀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣂⠀⠀⡀⠐⠲⢤⣀⠀⠀⠀⠀⠳⢆⠀⢀⠔⠁⠀⠀⠀⠀⢀⢀⠄⠒⠁⠠⠀⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠠⠀⠈⠉⠙⠐⠖⠢⠤⢧⣤⡄⠀⠀⠲⠊⠃⡈⠀⠀⣐⠠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠁⠀⠀⠂⠀⠀⠈⠁⠀⠀⠀⠂⠠⠀⠐⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⢉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠄⠠⠄⣀⠀⢄⠈⠀⠀⢀⣀⣤⣀⣀⡈⠉⠉⠀⣀⣀⠀⠀⠀⠀⠄⢀⢤⣀⠀⠡⠀⠈⠀⠁⢀⠀⠀⠀⠈⢇⠀⢀
⠀⠀⢠⠛⠐⠀⡄⠀⢰⠋⠀⡛⠻⠿⠹⠂⠀⢠⠉⠉⠉⡆⠀⠀⠀⡎⠉⠉⡆⠀⠀⠀⢀⣀⢈⣁⡀⠀⠠⡆⠀⠀⢡
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠠⠄⠀⠈⠀⠀⠈⠀⠤⠀⠃⠀⠀⠀⠁⠠⠀⠁⠀⠀⠀⠸⠀⠄⠀⣃⣀⣀⣃⠀⠄
⠀⠀⠀⢠⣤⣤⠀⠀⢠⢀⠸⠀⠈⡆⠀⡠⠀⠀⢀⡀⣀
⠀⢰⠉⠀⠀⠀⠀⠠⡄⣯⡀⠀⠀⠀⣉⠀⠐⠀⠀⠀⠀⠀⠴⠄⣄⠀⠀⠀⠀⡼⠛⢢⠀⠃⠲⠋⣶⢶⣂⡄
⠀⠀⢂⠀⠀⠀⠠⠤⠤⠤⡇⠀⠀⠰⡄⠀⠀⡀⠀⠀⠀⡌⠀⠀⠈⡄⠀⠀⢰⠃⠀⠘⡆⠀⠀⠐⠀⠀⠀⠃
⠀⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈⠁⠐⠀⠁⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠘⢄⠀⣠⠁⠀⠀⠀⠠⠀⠠
⠀⠀⣀⣀⡀⣀⡀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠥⠀⠇⢀⠀⢀⠐⠃⠀⠘⠂
⠙⠋⠉⣋⠁⣀⡔⠀⠀⠀⠀⠫⠀⠀⢀⠀⠀⠀⠙⠉⢳⠀⠀⠀⠀⠟⠤⠀⣡⠀⠀⡀⠀⠠⠉⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠁⠠⠀⠁⠀⠀⠠⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠁⠀⠊⠀⡀⠀⠀⢢⠁⠀⠀⠃⢀⡀⠰
⠀⠀⠀⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠌⠁⠄
⢤⣄⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⡤⠀⠀⠈⠀⠀⠀⠀⠄⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⡀⡀⠀⢀⠀⢀
⠉⠐⡇⠀⠀⢐⡈⠃⠀⡄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⣶⠀⣩⡁⠀⠀⠀⣾⡁⢙⣷⠀⣀⣀⢐⠄⠀⠛⡄⠁⠀⠿
⠂⠀⠁⠀⠀⠈⡇⠠⠀⡐⠀⠀⠀⠠⡀⠀⡰⠀⠀⠀⠸⠃⠁⢈⠇⠀⠀⠀⡟⠓⠒⠙⠀⠀⠀⠘⠀⢠⠀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢸⣤⣼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠸⠂⠀⠀⠀⣟⠈⠁⢈⠀⠀⠀⠀⣀⠀⣀
⡀⠈⠀⠀⠀⠂⡀⠀⠀⠀⡠⣄⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠆⠖⢄⠀⠀⠀⠀⡠⠀⣀⠀⠉⠀⢠⠖⢒⠈
⠁⠀⠀⡀⠀⠀⢠⠀⠀⢶⡁⠀⠀⠉⡀⠀⠀⠀⡄⠀⠀⡰⠀⠀⠀⠸⠀⠀⠸⠈⠇⠀⢠⣦⠀⣫⡀⠀⠸⣧⠁⠈⠉
⠀⠀⠀⠘⡀⠀⣸⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠁⡘⢀⠀⠀⠀⠀⠀⠇⠃⠆⠀⠀⠀⡘⠉⠀⢨⠁⠀⠀⠈⠣⠀⠐
⠀⠀⠀⠂⠀⠈⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠌⠀⢀⠀⢀⠀⠀⣀⠀⠀⣀⠀⠐⠆⠀⠻⠂⠀⠀⠐⠂
⠀⢀⠑⠄⣐⣤⡈⠀⠈⠀⡄⠀⠀⢀⠀⠀⠶⣄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠈⠈⠀⠠⡆
⠀⠀⢀⠒⠀⠀⠸⡄⣀⠠⠆⠀⠀⠠⡀⠀⠀⡹⠀⠀⠀⠈⠀⡀⠀⡃⠀⠀⠊⠀⠀⠀⠏⠀⠀⠀⡆⠀⠀⡶⠁⠀⠰
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠃⠀⠘⠀⠀⠀⠀⢀⠀⢐⠀⠄⠀⠀⢢⠐⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⣿⣯⢳⡌⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣠⣾⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣼⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⣿⣿⣿⣿⡟⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⣿⣷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⣿⣿⣿⡿⠋⠀⠀⠀⠘⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⣆⠀⠀⣼⣿⣿⠋⣠⡶⠀⢀⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡤⠀⡠⠀⠀⠈⣿⣿⣆⣀⣿⣿⣷⢾⣯⢄⣴⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⠘⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣷⣀⣾⣿⠟⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠻⣷⣄⠀⠀⠀⢿⡄⠘⢦⡀⠀⠦⣠⡞⠛⠁⢀⣽⢃⣠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣄⠘⣿⣤⡈⢿⣤⣶⣿⣿⣿⣶⣿⣷⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣛⣿⣾⣿⣷⣾⣿⣿⣿⣿⣶⣿⣿⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠻⣿⢿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠽⢿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⡀⢀⣠⠿⠾⣿⣿⣶⣄⡙⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⠉⠀⠀⠸⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠈⢠⠀⠀⠀⣠⣄⣀⠻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣿⡤⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣼⣿⣷⣦⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠙⠛⠿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⣿⠀⠀⠀⠀⠀⠀⣿⡏⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡿⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢻⠇⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢁⣄⡀⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⠉⠻⢢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢓⠌⠙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣶⡾⠟⠛⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠯⠿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⣾⣍⣀⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⣉⡿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠙⠓⠾⢭⣭⣛⣓⣚⡶⠶⠶⠶⠤⠤⢤⣴⣶⣷⣾⡧⠿⠟⠛⣩⡧⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣧⣤⣤⣄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣰⠃⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠛⢋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠙⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢧⡈⠑⠲⠤⣤⣀⣀⣀⣀⣀⣤⠴⠖⢋⡡⠂⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣤⣤⣀⣉⠉⠒⠒⠒⠒⠶⠶⠶⠶⠒⠒⠋⢉⣀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠯⠭⣽⣿⣿⣶⣖⣚⣳⣾⡿⠿⠿⠛⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠟⡿⠿⡿⠿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠤⠄⢀⣀⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⣤⡀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣸⠀⠀⣾⣿⡿⢷⡞⠻⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣎⠪⣿⣡⣾⣡⡄⡀⠂⠀⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣦⣙⡷⣿⣿⣿⣃⣠⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡜⠻⣿⣿⢷⡿⠳⡖
⠀⠀⠀⠀⠀⠀⠀⣠⠺⠿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡝⢿⣧⡘⢧⡇⣆⣀⣀
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠇⠀⠹⣿⡟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣇⣜⣀⢿⣿⣧⡉⠻⣿⣫⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣦⣬⣥⣀⣀⣚⣉⣙⣻⣿⡿⠟⠩⠒⠀⠉⠿⠂⠁⠀⠝⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⣀⣛⣿⠝⠋⠁⠀⢀⣠⣠⣄⣀⣁⢠⣀⡆⢈⣦⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⡰⢛⡉⠉⠉⠉⢹⣾⣿⣿⣿⣿⣮⣧⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⠤⠞⠓⠏⠷⠞⠀⠀⠀⠛⠛⢿⣿⣿⣿⡿⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠤⡬⠆⠔⡠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠘⠁⠀⡄⡓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣈⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣅⣜⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡈⠡⠉⡀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠄⠨⠀⡀⠂⣄⠂⠰⡀⡄⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡔⠒⠈⡣⠘
⠀⠀⠀⠀⠀⠀⢀⣔⡎⠀⠃⠰⠀⠤⡀⠀⠀⠀⠑⠍⠀⠀⠐⢂⠀⡀⠀⠘⠢⠠⠀⠆⠔⠑⡠⠁
⠀⠀⠀⠀⠀⠀⠘⡉⠀⠰⠄⠀⠀⡈⠧⠈⠀⡂⠁⠀⡄⠛⠈⢑⠻⠂⠀⠋⠀⠈⠀⠉⣰⠦⠃
⠀⠀⠀⠀⠀⠺⠒⡀⠀⠂⠀⠊⢀⠁⠀⠁⠀⠀⢀⠀⢠⡀⠐⠀⠀⠂⢘⠀⠀⠀⠀⠐⠈⡌
⠀⠀⠀⠀⠀⢀⡶⠢⠘⡅⠁⠀⣈⠒⢀⠠⠒⢈⡡⠐⡀⠀⠀⠀⠄⠘⣈⠁⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠐⣰⢘⠂⢲⠐⠂⠌⠱⣠⡀⡿⠆⢐⢢⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠄⠁⠀⡄⠀⣀⠈⡀⠿⢀⠁⣐⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢢⢰⢈⣩⠀⠠⠁⠄⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠑⠀⠱⢑⡄⠀⠄⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠎⢩⢖⡄⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠅⠡⡄⠕⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠀⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⢠⣶
⠀⣿⣿⠀⠀⣤⣄⡀
⠀⣿⣿⣿⣤⣿⣿⣧⣀⡀
⠀⣿⣿⣟⣿⣿⣿⣿⣿⣷
⠀⣿⣿⡇⢹⣿⣿⣿⣿⣿⡆
⠀⣿⣿⠇⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⣿⣿⠀⣿⡟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠛⣿⡟⠀⢆⡦⡄
⠀⣿⣿⠀⢸⡗⢻⣿⣿⣿⠁⠀⠀⠀⠀⠀⣸⣦⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⣴⣤⠀⣷⠁⠰⣾⡗
⠀⣿⣿⠀⠀⠇⠘⢿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⢻⢿⣴⣿⡆⠀⣿⣇
⠀⣿⣿⠀⠀⠀⠀⠈⠀⠷⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠈⠸⠿⣯⢶⣿⣿⢷
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢿⡿⡿⠉⢸⡷⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡥⣧⡠⣿⣸⣿⢹⡀
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⣿⣷⢶⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⡟⣽⣷⢿⣿⣿⣿⣿⡿⠄
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠸⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢹⣅⠇⢸⠃⣿⣿⣿⢻⣟⠂
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡟⠈⠀⠈⠁⡶⢸⢿⡜⣿⣿⢻⣗⡀
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠃⠀⠀⠀⠀⠀⠁⠘⠞⣿⡧⣿⣿⣿⣧
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠾⢹⣿⢻⣾⡿⣯⡇
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢸⣷⣿⢿⠁
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⢀⡄⢀⠞⢟⢩⣿⣺⢶
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠴⢻⣾⡟⣿⡀
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣸⡇⣿⣇
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⣂⣀⣤⣿⣴⣶⠶⠂⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⣷⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣾⣿⣿⡿⠿⠛⠻⠙⠂⠀⣀⣤⣴⣶⣶⣿⣿⣿⡿⠿⠟⠛⠋⠁
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠁⠀⠁⠀⠀⠠⡴⣶⣿⡿⠿⠛⠛⠉⠉
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠚⠋⠉⠁
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠒⠚⠉⠉
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠖⠒⠋⠉⠉
⠀⣿⣿⠀⠀⠀⠀⣀⣀⣤⡤⠴⠒⠚⠋⠉⠁
⠀⠻⠿⠶⠒⠚⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⡁⠄
⠀⠀⠀⢈⡃⠀⠀⣐⣦⡶⢎⡿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠐
⠀⠀⠠⡌⠉⠁⣠⠉⠉⠁⡈⠀⠁⠀⠀⠀⠀⢨⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠄⢻⡤⠀⠀⡂⠀⠀⡄⣤⢀⢴⡾⡇⠀⠀⠀⢠⠄⠀⠀⠀⠦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠯⠻⠈⠘⠁⠀⠀⠀⠀⠈⠑⠋⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⡀⢰⠤⢀⠀⠀⠈⡁⢠⠀⠀⠀⠠⣰⠀⠳⠃⢀⠴⠆⠀⠀⠄⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠆⠰⠀⣀⠈⠀⠀⠀⢀⠀⠠⡀⠠⠀⢈⣡⠄⡁⠀⣀⡀⠴⠃⠀⠀⡼⠧⢄⢠⢴⠖⢲⡄⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠠⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⢒⠐⠀⠀⠀⠀⡀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠐⠄⠀⠀⣀⣤⣄⡀⠀⠀⠊⠩⠅⠈
⠀⠀⠀⠈⠀⠀⠐⠒⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠀⠄⠀⠁⢀⣀⣠⣄⡀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠒⠲⢶⡴⠛⢋⣠⠄⠀⠀⠚⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠯⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀