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⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡀
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⣰⡶⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⢶⣆
⠀⠀⠀⢻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡟
⠀⠀⠀⠀⢀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀
⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⢠⡶⢦⡄⠀⣤⢶⡄⠀⣴⠶⠄⠠⢤⡤⠄⠀⠀⣤⠶⣄⠀⣤⠤⠤⢠⡤⠤⠄⣤⠀⣠⡶⡤⢀⡤⠤⠄⠀⠀⠀⠈⣿⢹⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⣇⣸⡇⢸⡇⠀⣿⠀⢷⣄⠀⠀⢸⡇⠀⠀⢠⡏⠀⢻⡀⣿⣀⣀⢸⣇⣀⡀⣿⠀⣿⠀⠀⢸⣧⣀⡀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡏⠉⠀⢸⡇⠀⣿⠀⠀⠙⣷⠀⢸⡇⠀⠀⠸⡇⠀⣼⠁⣿⠉⠁⢸⡏⠉⠀⣿⠀⣿⠀⠀⢸⡏⠉⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠸⠃⠀⠀⠈⠻⠶⠋⠐⠶⠶⠋⠀⠸⠇⠀⠀⠀⠻⠶⠛⠀⠻⠀⠀⠘⠇⠀⠀⠻⠀⠙⠷⠶⠘⠿⠶⠆⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⢰⡶⠶⠶⠶⠶⠶⢶⡶⠶⠶⠶⠶⠶⢶⡆⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠂⠀⣿⠀⢰⡆⠀⠀⢰⣆⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢶⡀⠀⠀⢰⡆⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠈⣿⠀⠀⠀⣿⡀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⣇⠀⠀⠘⣧⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠀⢹⡆⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⣿⠀⠀⠀⢿⡄⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⠀⠘⣧⠀⠀⠈⣿⠀⢸⡇⠀⢸⡇⠀⢠⣤⡤⠀⢸⡇⠀⢤⣤⡄⠀⢸⡇⠀⢸⡇⠀⠀⢹⡇⠀⠀⠸⡇⠀⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⣠⣼⣧⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣼⣧⣄
⠀⠀⠀⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠙⢶⣄⡀⢀⣤⠾⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡴⠟⠉⠛⠋⠉⠳⣤⡀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⢿⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣼⡷⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⡤⠠⡄⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡷⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⣶⠀⢰⡶⠶⠶⠶⠶⠶⠶⢾
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⢠⡤⠤⠤⠤⠤⣤⡄
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⠸⣧⠤⠤⠤⢤⣾⠇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⡇⠀⠀⠀⠀⣿⠠⠤⠤⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⣇⣀⣀⣀⣀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⣿⠀⠈⠉⠉⠉⠉⠉⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⢀⣿⣒⣒⣒⣺⣇
⠀⡀⠀⣀⣀⡀⣸⣟⡛⡛⠛⡛⢛⣛⣛⣛⣛⣛⣛⣛⠛⠛⠛⣛⣀⢛⢂⠛⡛⣻⡏⣉⣉⣉⣉⣹⡇⣀
⠀⠉⠀⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⣰⠿⣧⡀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡟⠁⢀⡼⠋⠀⠈⢷⣄⠀⠹⣿⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠛⠳⣦⡀
⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⣠⡟⠁⠀⠀⠀⠀⠹⣦⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠙⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡟⠁⢀⡾⠃⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠈⢻⣄⠀⠹⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠛⠃
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⠁⠀⢸⡟⠳⣤⡀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣿⠀⠀⠙⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⠀⠀⠀⢸⡇⠀⠈⠛⣦⣄⠀⢀⣴⡞⠋⠀⠀⣿⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣠⡾⠋⠙⠻⠛⠉⠻⣦⡀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣧⡞⢉⡀⣀⣀⣀⣀⣀⡀⣈⣛⣦⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣹⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⢻⠀⠀⣧
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⣶⡶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢸⡇⠀⠀⠉⠻⢦⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠛
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣀⣀⣀⣈⡀
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⡇⠀⠀⢠⡶⠛⠉⠉⠉⠉⠛⢶⣄
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢰⡏⠀⠀⠀⠀⠀⠀⣼⠃⠹⣦⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣴⠾⠷⠶⠶⠾⢷⡄⠀⠀⠀⠀⣿⠀⠀⢸⡇⠰⣾⡷⠶⠶⠶⠶⠶⠶⠶⠶⢶⣷⠶
⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⢀⣤⣤⣤⣤⣤⣤⣀⠀⣿
⠀⠀⢸⡇⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣠⡿⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⠀⢸⣇⣀⣀⣀⣀⣀⣿⠀⣿
⠀⠀⢸⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠁⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⣿
⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣇⣀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢾⣀

⠀⠀⠀⠀⠀⠀⠀⠀⢐⡋⠀⢸⠈⠳⠀⡆⠘⠋⢀⠺⠀⢰⣋⡶⢰⠀⠀⣏⠁⠀⠰⣎⠀⠈⡛⠀⣊⣑⠄⠞⠈⢢⢰⣠⠂⢸⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣅⣀⣸⣀⡔⢁⡆⢀⡀⣜⠃⣳⣘⣅⡷⣸⣀⣀⣏⣀⡀⢀⣈⡗⠀⣃⢀⣋⢙⡄⣇⣀⡜⢸⡉⢆⣸⣅⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣶⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⣾⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣧
⠀⠀⠈⢉⣽⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣯⡉⠁
⠀⠀⠀⢹⢹⠀⠀⠀⡖⢲⡄⡴⢲⡀⡖⠂⠐⣶⠂⠀⣰⠲⡄⣴⠒⢠⡖⠂⣦⢠⠖⠂⡶⠒⠀⠀⠀⡏⡟
⠀⠀⠀⣼⢸⠀⠀⠀⡗⠚⠁⡇⢸⡇⠙⢦⠀⣿⠀⠀⣿⠀⡇⣿⠒⢸⡗⠂⣿⢸⠀⠀⡿⠒⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠀⠀⠃⠀⠀⠓⠚⠀⠒⠚⠀⠛⠀⠀⠙⠚⠃⠛⠀⠘⠃⠀⠛⠘⠓⠂⠓⠒⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢀⣤⣤⣤⣤⣤⣤⣤⣄⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⣦⠀⠀⣆⠀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⢰⡄⠀⢰⡀⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⢹⡀⠀⢹⡀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⠈⣇⠀⠘⡇⠀⣿⠀⡇⣿
⠀⠀⠀⣼⢸⠀⣿⠀⠘⡇⠀⠘⡇⢸⡇⠀⡇⠀⣤⡄⢸⡇⢠⣤⠀⢸⠀⢸⡇⠀⢻⠀⠀⢻⠀⣿⠀⡇⣏
⠀⠀⠀⣿⢸⠀⠻⠶⠶⠶⠶⠶⠶⠾⠇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠸⠷⠶⠶⠶⠶⠶⠶⠿⠀⡇⣯
⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⣤⣿⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣼⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣿⣤
⠀⠀⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠒⠒⠒⠚⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⢠⠈⢶⠒⠒
⠀⠀⠀⡟⢍⠉⠉⠉⠉⢉⠝⡇⠀⠀⠀⠀⠀⢸⠣⡀⠀⠀⠀⠀⡠⢻⠀⠀⠀⠀⠀⠀⡇⠋⠙⠘⠉⠋⡏⡙⡃⠀⠀⠀⠀⡾⠍⣀⠀⠀⢀⠦⣀⡈⣇⠁
⠀⠀⠀⡇⠀⡱⢄⣀⠴⡁⠀⡇⠀⠀⠀⠀⠀⢸⠀⢈⠖⠈⠐⢎⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠒⠃⡇⠀⠀⠀⠀⠘⡄⠀⡝⠘⠉⣀⡠⠔⠛⠐
⠀⠀⠀⣧⣊⣀⣀⣀⣀⣈⣢⡇⠀⠀⠀⠀⠀⢸⣔⣁⣀⣀⣀⣀⣑⣼⠀⠀⠀⠀⠀⠀⣥⣀⣄⣄⣠⣠⣀⣄⡇⠀⠀⠀⠀⠀⠸⣴⣤⣖⣋⣉⣀⣀⣀⣀


⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢘⡆⠟⠒⠓⠚⠇⣞⠀⠀⠀⠀⡴⠋⣀⠤⢄⡈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠲⠖⠒⠲⣶⠶⡄⠀⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⢨⠆⠄⠀⠀⠀⡅⢮⠀⠀⠀⢸⠀⡌⠀⠀⠀⠘⡀⢸⠆⠬⢉⣈⣀⡩⠀⠀⠀⣇⣀⣸⣃⣆⣀⣈⣣⡇⠀⠀⠀⠀⢰⠋⠉⢉⠉⠉⠉⡹⢛⡆⢸
⠀⠀⠀⢨⡃⠀⠀⠀⠀⠄⣫⠀⠀⠀⠸⡀⠡⡀⠀⠀⡰⠀⣸⣓⠒⠒⠀⠀⠒⠀⠀⠀⡇⠈⠚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠉⠈⡇⢸
⠀⠀⠀⢰⣃⣷⣒⣖⣢⣃⣳⠀⠀⠀⠀⠙⢦⣀⣉⣁⣠⠾⠁⠤⠄⣈⣐⣂⡉⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⡇⠀⠀⠀⠀⠸⣀⣀⣈⣀⣀⣀⣀⣀⡗⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢈⠍⢹
⠀⠀⠀⡞⠥⢴⡷⠶⠶⠠⠝⡆⠀⠀⠀⠀⠀⠀⠊⢸⠘⠀⠚⢹⠈⠂⠀⠀⠀⠀⠀⠀⡏⣀⡀⢀⣀⠀⢀⡈⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠁⠀⡸
⠀⠀⠀⡇⠀⠘⠂⠤⠖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⡏⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⠋⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀
⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⡇⣀⣀⣀⡀⠈⠀⠀⡇⠀⠀⠀⠀⠀⡰⠋⢉⠉⠉⠉⠉⠉⡉⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⡇⣅⣀⣀⡸⠀⠀⠀⡇⠀⠀⠀⠀⣰⢡⠁⢠⠀⢀⡄⠂⢉⠙⡆⠘⡇⠀⠀⠀⡞⠉⠹⠉⣉⣉⣉⡙⡆⠀⠀⠀⠀⢀⣤⠾⣭⣽⠉⡏⠉⠉⠉⢢
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⢣⣀⠤⠖⠉⠀⡇⠀⠀⠀⡇⠀⢀⠀⠀⠀⠈⠃⡇⠀⠀⠀⠀⢸⠉⣢⣀⣰⡄⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⡀⡀⣀⡀⢀⡇⠀⠀⠸⣄⡴⣿⢦⡤⠤⠤⠤⡤⣤⣮⢷⣄⡽⠀⠀⠀⠧⠤⠤⣤⢤⡤⠤⠤⠇⠀⠀⠀⠀⠘⠾⠤⠤⠴⣤⠤⡤⠤⠤⠼
⠀⠀⠀⠉⠈⠁⡿⠨⡇⠉⠁⠁⠀⠀⠀⠀⠙⠓⠋⠀⠀⠀⠀⠀⠈⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠇⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡴⠃⣰⠏⢳⡀⠻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡼⠁⠀⠀⠹⣆⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠏⠀⠀⠀⠀⠀⠈⢧⡈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⣀⣀⣀⣀⣀⣀⣀⣀⡞⢁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⣆⣀⣀⣀⣀⣀⣀⣀⣀⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠉⠉⠉⠉⠉⠉⠻⣤⠞⠁⣾⡻⢿⣿⣿⣿⣿⡿⣛⡆⠙⢦⣼⠏⠉⠉⠉⠉⠉⠉⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡖⠒⠒⠒⠒⠒⡆⠀⠀⣿⣿⣦⠙⠿⠟⠡⣾⣿⡇⠀⠀⢰⠖⠒⠒⠒⠒⠲⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⣿⢋⣴⣿⣷⣿⣷⣌⡻⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡟⠛⠛⠛⠛⠛⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⠛⠛⠛⠛⠛⠛⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠞⠋⢹⠋⠛⠲⣤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⣰⠏⠀⠀⠀⠘⠀⠀⠀⠈⢳⡄⠀⢸⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⡏⠀⠀⠀⢀⡞⢧⡀⠀⠀⠀⣷⠀⢸⠀⢠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢰⠟⠒⠚⢳⡆⠀⠀⣿⠀⢸⠀⣿⠛⠛⠛⠛⠛⠛⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣇⣀⣀⣀⣀⣀⡇⠀⡇⠀⠀⠘⠓⠒⠒⠚⠃⠀⠀⣿⠀⢸⠀⣿⢰⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠁⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠛⠒⠒⠂⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣤⣤⣤⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠋





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

⠀⠀⠀⣼⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⡄
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⢀⣀⣾⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣆⣀
⣾⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠙⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛
⠀⠀⣶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⡆
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⠛⢷⡄⢠⡾⠻⣦⠀⣾⠛⠓⠀⠛⣿⠓⠀⠀⢠⡾⠻⣦⠀⣾⠛⠛⠀⣿⠛⠓⢰⡆⢀⡾⠛⠂⢰⡟⠛⠂⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⣤⣼⠇⢸⡇⠀⣿⠀⠻⣦⡀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⣿⣤⣤⠀⣿⣤⡄⢸⡇⢸⡇⠀⠀⢸⣧⣤⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⠀⠈⣻⡆⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠟⠀⠀⠀⠈⠻⠾⠃⠀⠳⠾⠛⠀⠀⠛⠀⠀⠀⠈⠻⠾⠋⠀⠻⠀⠀⠀⠛⠀⠀⠘⠃⠀⠛⠿⠃⠘⠷⠶⠆⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡏
⠀⠀⣿⢸⡇⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣦⠀⠀⣶⠶⠶⠶⠶⠶⠶⣶⡶⠶⠶⠶⠶⠶⢶⡆⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡂⠀⣶⠀⠀⠀⢰⡆⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠰⣦⠀⠀⠀⣶⡄⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡇⠀⢹⡇⠀⠀⠘⣿⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⡀⠀⠀⢸⣇⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡇⠀⠘⣿⠀⠀⠀⢻⡆⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⡃⠀⠀⢿⡆⠀⠀⠸⣷⠀⢸⣿⠀⠀⣿⠀⠀⣀⣀⡀⠀⣿⡇⠀⣀⣀⡀⠀⢸⡇⠀⠀⣿⠀⠀⠈⣿⠀⠀⠀⢹⡇⠀⢸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⣿⣁⣀⣀⣈⣁⣀⣀⣀⣉⣀⣸⣿⠀⠀⣿⠀⠀⠉⠉⠁⠀⢻⠃⠀⠉⠉⠁⠀⢸⡇⠀⠀⣿⠀⠀⠀⠉⣀⣀⣀⣈⣁⣀⣸⡇⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠁⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⣶⠾⠿⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠶⠶⠶⠶⠿⠷⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠾⠿⢶
⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠉⠉⣉⣉⢉⣉⡉⣉⡉⣉⣉⠉⢉⣉⠉⣉⡉⣉⡉⡉⢉⣉⢉⣉⠙⠉⢹⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⣿⠞⢹⣀⡇⠙⣦⠀⡇⠀⢾⣈⡇⡷⠂⡷⠂⡇⢯⡀⣽⡂⠀⠀⢸⣿⣿⣷⣄
⠀⠀⠀⠀⠻⠿⠿⠿⡿⠧⠤⠤⠤⠤⠤⠭⠤⠬⠥⠤⠤⠤⠤⠭⠤⠤⠤⠤⠤⠤⠤⠭⠬⠥⠤⠤⠼⢿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢲⠀⠀⣶⠒⠒⢲⠀⠀⡖⠒⠒⣶⠀⠀⡖⠒⠒⡆⠀⢰⡖⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⣿⠀⠀⢸⠀⠀⡇⠀⠀⣿⠀⠀⡇⠀⠀⡇⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣸⠀⠀⣿⣀⠀⢸⠀⠀⡇⠀⠀⣿⠀⠀⣇⡀⠀⡇⠀⢸⡆⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⢹⠉⠉⠙⡏⠉⠉⣷⢸⣿⣿⣿⣿⡇⣿⠉⠉⢹⡏⠉⠉⠏⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢀⡇⠀⠀⣿⢸⣿⣿⣿⣿⡇⡟⠀⠀⢸⠁⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢠⡇⠀⠀⣿⢸⣿⣿⣿⣿⡇⡇⠀⠀⢸⡆⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⡇⠀⠀⣏⢸⣿⣿⣿⣿⡇⡇⠀⠀⢸⡇⠀⠀⡇⠀⠀⣸
⠀⠀⠀⠀⢀⡀⠀⢀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣙⣉⣁⣸⣿⣿⣿⣿⣇⣉⣉⣉⣉⣉⣙⣉⣉⣉⣉⣉⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣶⠿⠶⣦⡀⠀⣠⡶⠿⢶⣄⠀⢀⣴⠾⠷⣦⠀⠾⢿⣿⠿⠆⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣀⣀⣼⠇⢀⣿⠀⠀⠈⣿⠀⠸⢿⣤⣀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⠛⠉⠁⠀⠀⣿⡀⠀⢀⣿⠀⢀⡀⠈⠙⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⢠⣤⣶⣾⠿⣿⣿⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠙⠿⠶⠿⠋⠀⠘⠻⠶⠾⠋⠀⠀⠸⠟⠀⠀⠀⠀⠀⣿⡿⠿⣷⣶⣤⡄
⠀⠀⢸⣿⠁⠀⠀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠈⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⣿⡇
⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣶⣶⣶⣶⣶⣶⠀⠀⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⠀⠀⣶⣶⣶⣶⣶⣶⣦⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⠀⣸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠻⠿⠿⢿⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠻⠿⠿⣿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣤⣤⣤⣼⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣧⣤⣤⣤⣿⣿⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠛⠛⠛⠛⠛⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⡀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣷
⠀⠀⠅⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠋⠉⠉⣻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⡶⢿⡉⠻⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⣀⡀⠀⠐⢳⢄⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⣁⡴⠂⠀⠀⡜⡉⠉⠢⣄
⢀⠴⡂⢶⠚⡼⠴⠶⣤⣶⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣿⡚⠃⠴⠃⠀⠀⠀⠸⣆
⠈⠀⠉⠻⡍⠁⢠⡌⢍⡍⠁⠙⢅⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠁⠋⢻⠏⠀⢠⡄⠀⠀⠀⠀⠘⢧⡀
⢸⠀⠀⠀⢹⠤⠤⠤⠤⠤⠤⠦⢾⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⢸⠠⠀⣠⠁⢠⠂⢰⣶⡦⠄⠀⣹⣆
⢸⠀⠀⠀⠸⠀⢰⠶⠒⠶⡶⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣠⡘⢀⡀⠉⠀⠘⠣⠈⢽⡷⠨⢷⢾⣿
⢸⠀⠀⠀⢸⠀⢙⣈⣩⣀⣐⠁⢨⠀⠀⠀⠀⠀⠀⠀⢠⣀⣸⡏⢳⡉⠀⡎⠑⠤⢴⣄⠬⣁⠀⢤⣾⣏⡈⢤⣀
⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠔⠊⠉⢙⡿⢌⣈⠁⣽⡠⠀⣠⢁⡀⢠⣽⠟⠁⠀⠁⠀⣩⡃
⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⡤⠸⢄⣠⠤⠴⡈⠀⠱⣤⣭⣽⢇⣸⢉⠁⢉⠁⠠⢄⠀⣷⠞⠁⠀⠀⣠⡴⠞⠁⠈⢂
⠈⠉⢳⡶⣾⣷⣶⣿⠃⠨⣷⣀⠂⠀⠉⢒⣦⢱⣠⡾⠟⠉⠐⠀⣾⠿⣶⣤⡀⠀⢀⣴⠋⠉⠛⠛⠛⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠉⠲⠖⠋⠀⠀⠀⠀⠀⢠⠀⢠⡿⠀⠀⠻⣿⣿⠁⠻⣷⣄⣀⠀⢂⠀⠀⠀⠀⠀⢀
⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣼⠃⠀⠀⠀⡏⠿⣦⠀⠹⣌⡛⠀⠀⠢⡀⠀⢀⡴⠋
⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠃⠀⠀⠀⢀⡦⠀⠙⠿⢿⣿⣿⣶⣆⣠⠴⠒⠉
⠀⠀⢠⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠀⠀⠀⣤⡞⠀⢣⠀⢠⠀⢹⣿⡇⠉
⠀⠀⢸⠀⢀⠀⠸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡂⠀⢀⡜⣿⠀⠀⠀⡄⠈⡇⠀⠛⣷

⠀⠀⠀⢀⡞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣇
⠀⠀⢀⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹⣀
⠀⠀⢿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⠇
⠀⠀⠀⣶⣿⠀⠀⠀⣀⠀⠀⣀⠀⢀⡀⢀⣀⠀⠀⢀⡀⠀⣀⡀⢀⣀⠀⠀⢀⠀⢀⡀⠀⠀⢸⣷⡆
⠀⠀⠀⣿⣿⠀⠀⢸⣉⡇⣾⠙⡇⢯⡁⠈⣿⠁⠀⡏⢹⡀⣏⡁⣿⣉⢸⢰⡏⠁⣯⣉⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⠀⠀⢸⠁⠀⢻⣠⠇⣀⣹⠀⣿⠀⠀⣇⣸⠁⡇⠀⣿⠀⢸⠸⣇⡀⣯⣀⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⢰⡟⢛⠛⠛⣛⠛⠛⡇⠀⡟⠛⠛⠛⢻⠛⠛⠛⠛⡇⠀⡟⢛⡛⠛⢛⠛⠛⣷⢸⣿⡇
⠀⠀⠀⣿⣿⢸⡇⠘⡇⠀⢸⡄⠀⡇⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⡇⠀⣇⠀⠸⡇⠀⣿⢸⣿⡇
⠀⠀⠀⣿⣿⢸⡇⠀⢿⠀⠀⣇⠀⡇⠀⡇⠀⣀⡀⢸⠀⣀⡀⠀⡇⠀⡇⠀⢹⡀⠀⢷⠀⣿⢸⣿⡇
⠀⠀⠀⣿⣿⠸⣧⣤⣬⣤⣤⣭⣤⡇⠀⡇⠀⠉⠁⢸⠀⠉⠁⠀⡇⠀⢧⣤⣬⣥⣤⣬⣤⡿⢸⣿⡇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣠⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣧⣀⣀⣀⣼⣤⣀⣀⣠⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣧⡀
⠀⠀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠻⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣰⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣷⣦⡀⢀⣴⣾⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣆
⠀⠀⠀⠀⣿⡿⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠻⢿⡿⠟⠉⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿⠈⠉⠉⠉⠉⠉⠉⢿⣿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣀⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⡀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⢠⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⡄
⠀⠀⢸⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇⠀⣿⣿⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⠀⣿⣿⠀⠀⣠⣾⡿⠿⠿⢿⣷⣄⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢰⣿⡏⢠⣤⣤⣄⢹⣿⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇⠘⠛⠛⠋⢸⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⣶⣶⣶⡄⠀⣿⣿⠀⢠⣶⣶⣶⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠉⠙⠛⠃⠀⣿⣿⠀⠘⠛⠋⠉⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠸⣿⣿⣷⣶⣶⣶⣾⣿⠇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣀⣿⣿⡀⣀⣀⣀⠀⣿⣿⢀⣀⢀⡀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⣿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢀⣤⣤⣿⣿⣤⣼⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣤⣿⣿⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣤⣿⣿⣤⣤⡀
⠀⠀⠀⢸⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠛⢿⣦⣄⠀⠀⠀⢀⣴⣾⠟⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⣻⣷⣦⣾⣿⡋⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⣿⡇⢀⣴⣾⠟⠉⠙⠉⠛⢿⣶⣄⠀⣿⡇⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣿⡏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣇⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣁⣸⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢸⣿⠿⠿⢿⠿⠿⠿⡿⠿⠿⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣷⣶⣴⣶⠀⢸⣿⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢸⣿⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⢸⣿⠀⣿⡏⠉⠉⠉⠉⠉⠉⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⠉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠁⠀⠀⠀⠀⢸⣿⠀⠛⠃⠀⠀⠀⠀⠀⠀⠛⠁⢸⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣶⣶⣶⣶⣶⣶⣶⡆⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⢸⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣷⣶⣶⣶⣶⣿⡇⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠆⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⡆⠀⢀⣀⣀⣸⣿⣀⣿⣇⣀⣀⣀⣀⣀⢀⣿⣇⣸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣥⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣿⡇⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⢰⣶⣶⣾⣿⠋⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀
⣠⣿⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣿
⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⣿⠀⠀⠀⣴⠶⣤⠀⣤⢦⡄⢀⡴⠦⠀⠴⣶⠦⠀⠀⢠⡴⣤⡀⢠⡶⠦⠀⣶⠶⠄⣠⡄⣠⡴⣤⠀⣴⠶⠦⠀⠀⠀⣿
⣿⣿⠀⠀⠀⣿⠴⠟⢸⡇⢸⡇⠈⠻⢶⡄⠀⣿⠀⠀⠀⢸⡇⢸⡇⢸⡷⠂⠀⡿⠖⠀⣻⠄⣿⠀⣈⠀⣿⠶⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠛⠀⠀⠀⠛⠚⠃⠘⠓⠛⠁⠀⠛⠀⠀⠀⠈⠛⠛⠁⠘⠃⠀⠀⠋⠀⠀⠙⠁⠘⠓⠛⠀⠛⠛⠓⠀⠀⠀⣿
⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⢶⡶⠀⠠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⣿⣿⠀⢸⣿⡿⠿⠿⠿⠿⢿⣿⡿⠿⠿⠿⠿⠿⣿⣷⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⠀⣿
⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿
⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄⣴⣾⣿⣆⠀⠀⢸⣿⡇⠀⣿
⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⣠⣤⡀⣼⣿⠋⠉⢻⣿⡟⠉⢹⣿⣆⠀⢸⣿⡇⠀⣿
⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠙⠿⣿⣿⣿⣦⣤⣾⣿⣷⣶⣿⠿⠃⠀⢸⣿⡇⠀⣿
⣿⣿⠀⢸⣿⡇⠀⢀⣀⠀⠀⣿⡇⠀⢀⣀⠀⠀⣿⣿⠀⢸⣿⡇⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⢸⣿⡇⠀⣿
⣿⣿⠀⢸⣿⡇⠀⣿⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⣿⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇⠀⣿
⣿⣿⠀⢸⣿⡇⠀⠙⠋⠀⠀⣿⡇⠀⠈⠋⠀⠀⣿⣿⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⣿
⣿⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⣿⣿⠉⢹⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀
⠀⠀⢸⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿
⠀⠀⠀⢀⣬⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣄
⠀⠀⠀⢸⡇⡇⠀⠀⠀⣶⠲⡆⢰⠖⣦⢠⡖⠒⠐⢶⠒⠀⠀⡶⢲⡄⣴⠒⠂⣶⠒⠀⡆⢰⠖⠂⣴⠒⠂⠀⠀⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⠀⠀⣿⠶⠃⣾⠀⢸⠀⠛⣦⠀⢸⠀⠀⢸⡇⢸⡇⣿⠶⠂⣿⠶⠀⡇⢸⠀⠀⣿⠶⠀⠀⠀⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠿⠀⠀⠸⠦⠟⠠⠤⠼⠀⠸⠀⠀⠀⠷⠼⠃⠹⠀⠀⠿⠀⠀⠇⠸⠦⠄⠿⠤⠄⠀⠀⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⣿⠛⣛⠛⠛⢛⠛⠛⢻⡇⠀⣿⠛⠛⠛⠛⢻⠛⠛⠛⠛⢻⡇⠀⣾⠛⢛⠛⠛⢛⡛⠛⠛⡇⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⣿⠀⢻⠀⠀⢸⡆⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡆⠀⠈⣧⠀⠀⡇⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⣿⠀⠸⡇⠀⠀⣷⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⣧⠀⠀⢻⡀⠀⡇⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⣿⠀⠀⠿⠀⠀⠹⠄⢸⡇⠀⣿⠀⠴⠶⠀⢸⠀⠰⠶⠄⢸⡇⠀⣿⠀⠀⠹⠄⠀⠘⠇⠀⡇⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⠻⠶⠶⠶⠶⠒⠶⠶⠾⠃⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠻⠶⠶⠶⠖⠲⠶⠶⠶⠃⠀⡇⣿
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⢀⣸⣇⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣸⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣇⣿⣀
⠀⠀⢸⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣽



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢹⣏⣛⣛⣙⣋⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣟⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣟⣿⢻⣿⣿⣿⡏⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢀⠠⡉⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢿⠟⠛⡄⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁⠈⠀⠈⠙⣿⡏⠀⠄⠀⠀⠉⢿
⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⠀⠀⣀⣀⡠⠀⠀⣿⠉⠉⡄⠀⠰⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠠⠇⠀⠸⠄
⠀⠀⠀⠀⢀⠉⠉⠀⠀⠉⠉⡀⢐⠀⠀⢠⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠿⢦⣴⡇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠃⠀⠠⠀⠸
⠀⠀⠀⡠⠚⣁⣀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄
⠀⠀⡀⠀⣈⠀⠈⠊⡵⠒⠉⠉⠢⡀⠀⠀⠀⠀⠀⡠⠀⠀⠄⠤⠤⡀⠀⠀⠀⠀⠠⠤⠆⢄⠀⠀⠀⠀⠀⠀⢀⠀⠆
⠀⠀⢉⡁⠀⠀⢀⡚⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠱⠀⠀⠀⠀⠀⠀⢀⢠⢀⠀⠀⠀⠀⢸⢀⣰⠒⠀⠀⠀⢸⣀⣠⠀⠀⠀⠀⢠⣄
⠀⠀⠈⠍⠛⠢⠀⡇⢀⡀⡀⣀⠀⡇⠀⡀⢀⣠⠀⠤⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠈⡀⠉⠀⠀⠃⠀⠈⠀⠈
⠀⠀⠀⠀⠈⠀⢰⠇⠈⠉⠉⣀⠀⡇⠠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣇⡄⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⡄
⠀⠀⠀⠄⠀⠀⠘⡂⠀⠀⣀⡠⠀⡛⠀⠀⠐⡁⠀⠀⠸⠀⠂⠂⠄⠂⠀⠀⢌⠀⠀⢀⠆⠀⠈
⠀⠀⠀⠀⠀⠀⢰⠘⠉⠁⠀⠀⠀⡇⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⢀⠴⢂⣈⣐⠲⡆⠀⠀⠀⠀⢀⡀⠐⠀⠄⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⢸⡠⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢃⠆⠀⠀⠀⠙⠘⡆⠀⠀⠀⠀⠙⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠦⠠⠀⠀⠒⠆⠀⠀⠀⠢⡀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⣧⠠⡄⠀⠀⠀⠀⠀⡄⢠⠃⠀⠀⠀⠀⠈⠀⠀⠀⠘⠦⣀⢀⠀⠔⠓⠀⠀⠀⠀⠋⠉⠉⠉
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠁⠀⠀⠀⠀⠀⣀⣀⢀⡀⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡏⠀⡇⠀⠀⠀⡔⠂⠀⠀⠴⠪⠒⠁⠀⠠⠀⠁
⠀⠀⠀⡇⢸⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⡇⠀⡇⠀⠀⠀⡇⠀⠀⣀⡀⠀⠛⣧⡄
⠀⠀⠀⠧⠼⠀⠀⠀⠈⠀⠀⢹⠀⠀⠀⠳⠤⠇⠀⠀⠀⡧⠔⠋⠁⠀⠘⣶⣤⠀⠠
⠀⠀⠀⠀⠀⠾⠐⠀⠀⠀⠀⠺⠀⠂⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠐⠀⠀⠀⠐⠂⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⡆
⠀⠀⢘⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⠟⠛⠛⢿⣿⣿⣇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣴⠞⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⡍⠛⠖⠀⠁⣸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣤⡤⠀⠀⠁⠛⠲⢤⣠⣄⡀
⠀⠀⠐⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣴⠋⠀⠇⠀⡌⠈⠀⠈⠲⣄
⢀⠔⢖⠴⠷⠛⠼⠶⠶⠶⠴⣖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣿⣻⣗⢹⠀⢴⠁⢀⠀⠀⠀⠼⣄
⢸⠀⠈⠛⢏⠉⠀⢈⠈⢉⢉⠉⠉⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠁⠙⢻⣿⠊⠀⢠⡠⠀⠀⠀⠀⠀⠹⣦⡀
⢸⠀⠀⠀⠈⣧⣀⣈⣁⣀⣉⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠀⠁⠀⠀⠋⢀⢀⠀⣤⣤⣀⠂⠀⢹⣄
⢸⠀⠀⠀⠀⠁⠀⢀⣀⣀⣀⣀⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣧⢀⠸⠏⠁⠠⠏⠀⣘⡎⠀⠟⣿⣂⠤⣂⣸⣿⡆
⢸⠀⠀⠀⠀⠠⠀⣠⡏⠥⠬⠙⠃⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡼⣆⢤⡖⠀⢀⢘⠀⠀⠁⠈⡿⠃⠈⠹⢝⣿⠇⢄
⢸⠀⠀⠀⠀⢰⠀⠠⠥⠤⠦⠴⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣧⠉⡆⣀⠸⡆⠈⠉⠙⢷⠖⠲⢤⣀⣻⡿⠿⢴⠐⠢⣤⡀
⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⣠⡔⠊⠀⢀⣉⡿⠆⣥⡀⢀⣧⠰⠀⢠⠤⣀⠀⣤⣾⠟⠀⠀⠀⠀⢀⣼⣅
⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡠⠼⢅⣀⡠⠠⢴⡉⠀⠙⢆⣨⣭⣿⢇⢠⠏⡉⠈⠉⠁⠀⠄⠀⢸⡧⠟⠁⠀⠀⢀⣤⠶⠏⠁⠈⢣
⠀⠓⢲⣶⣶⣾⣦⣶⣶⡖⠒⢲⣍⠀⠐⠠⠌⠒⠢⣄⢣⢀⣤⣾⠟⠉⠐⠀⣸⣿⣦⣄⣈⠀⠂⠈⣠⡿⠉⠛⠷⠶⠾⠋⠁⠀⠀⠀⠀⠀⣧
⠀⠀⠈⠉⠀⡇⠉⠙⣿⡇⠀⠈⠉⠉⠒⢤⣀⡴⠚⠉⠈⠙⠋⠀⠀⠐⠀⠀⣿⠃⠈⠹⣿⣶⣴⠊⠻⣷⣄⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⡇⠀⢰⠈⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢸⡿⠀⠀⠀⢈⠿⣿⡄⠀⠙⡟⠷⣶⡀⠈⢆⠀⠀⠀⠀⢀⡠⠋
⠀⠀⢸⡀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡿⠁⠀⠀⠀⠈⡂⠛⢿⣄⣀⣹⣶⣄⣀⠀⠈⠡⠀⢀⡤⠎
⠀⠀⠐⠂⠀⠇⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠁⠀⠀⠀⠀⣰⠳⡀⠀⢛⠛⠿⣿⣿⡿⣏⣠⠖⠋⠁
⠀⠀⠀⡄⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣾⠏⠀⢱⠀⠈⡀⠀⢿⣿⡇
⠀⠀⠀⡇⠀⡀⠀⠀⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠆⠀⠀⣰⢻⡇⠀⠀⠀⡇⠀⢹⠀⠀⠟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠙⠲⢄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⢀⠀⠀⠀⠀⠀⠙⠲⣄⡴⠋⢿⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⡠⠂⠁⠀⠀⠀⠀⠀⢂⡴⠋⠀⠀⠀⠙⢦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡸⠂
⠀⠀⠀⠀⣠⠊⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠞⠣⡀
⠀⠀⠀⢸⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡴⠋⠀⠀⣸⡆
⠀⠀⠀⡏⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣠⠋⠀⣀⠴⠊⠁⣇
⠀⠀⠀⡇⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠞⣡⠔⠋⠁⠀⠀⠀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠐⠀⡀⠉⠳⢤⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣴⠗⠋⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣡⠖⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠲⢄⡀⠀⢀⡠⢤⣀⠀⣀⡤⠚⠁⠀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠽⠛⠉⡀⠄⡈⠙⠣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠁⠀⠐⠀⠀⠀⠀⠐⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡀⠀⠀⠀⠀⡏
⠀⠀⠀⢸⡄⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⡇
⠀⠀⠀⠈⠷⣮⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣽⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠐⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⣰⠖⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠛⠒⠛⠛⠛⠛⠛⢱⡆
⠀⠀⠛⠀⠂⠒⠀⠂⢐⡀⠘⠀⡂⠀⢂⠀⠂⠛⠀⠃⠐⠀⢻⠁⠀⠐⠂⠃⠐⠃⠀⢂⡀⠃⠀⡀⢀⢀⠈⠀⠐⠂⠐⠃⠚⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡉⠀⠘⠀⠈⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡀⢀⠀⠈⠀⠓⠀⠃⡀⠈⣁⢒⠀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣁⣑⣀⣘⣀⣀⣀⣀⣀⣀⣀⠀⢸⠀⠀⠀⠀⢐⣁⣉⣘⣂⣊⣙⣋⣂⣅⣀⣋⣊⣃⣙⣋⣁⠀⠀⡇
⠀⠀⠀⠀⣏⣡⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⠀⠀⠀⠀⢸⣿⣵⣧⡆⡨⡠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇
⠀⠀⡇⠀⣿⢾⠓⠥⠄⠆⠠⠠⠴⠐⠠⠀⢰⠦⠴⠆⠾⠀⢸⠀⠀⠀⠀⢸⡿⡟⠧⠿⠖⢲⠶⠰⡶⠶⡖⠀⡖⠲⡶⠶⠀⠀⡇
⠀⠀⡇⠀⡻⢾⠖⠲⠒⠖⠖⠒⠶⠒⢺⠒⢶⠒⢲⠒⠶⠀⢸⠀⠀⠀⠀⢸⠶⠬⠦⠶⠤⢤⠤⠤⠦⠤⡥⠤⡤⠤⠤⠶⠀⠀⡇
⠀⠀⠁⠀⡇⠺⣿⣾⣷⡆⠁⠀⡽⠀⢸⠀⢸⠀⢸⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⢼⠀⢵⠀⢸⠀⠀⡃⠨⡇⠀⡇⠀⡇⢰⠀⠀⡇
⠀⠀⠀⠀⣅⣸⣯⣩⣼⡷⣷⠀⣼⣿⢽⢢⢸⠀⢸⠀⢠⠀⢸⠀⠀⠀⠀⢸⠀⠡⠀⠨⢰⢸⠞⢒⡁⠀⡇⠀⡇⠀⡇⠘⠀⠀⡇
⠀⠀⠀⢸⣿⣿⣿⣥⣶⠶⡇⠀⢹⡟⢻⠈⢸⠀⠘⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⢘⠀⠀⠘⢿⠋⡸⠀⠀⡇⠀⡇⠀⡇⢠⣀⠀⡇
⠀⠀⠀⠀⣀⣽⣟⣟⣁⠀⠇⢞⡋⠰⣹⢠⢰⡮⢠⠀⢸⠀⢸⠀⠀⠀⠀⢸⠀⠀⢀⡰⠒⢹⢄⠒⡅⠀⡇⠀⣧⡀⢹⣧⣌⣀⡇
⠀⠀⡤⠈⠁⠀⠀⠀⠁⠱⣕⠀⣓⠀⢹⠠⢸⡗⢱⠀⠸⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⢸⠈⢐⠆⠀⠃⠀⡇⠙⣆⢩⠈⠛⠀⡀
⠀⢠⠗⠀⠀⠀⠀⠀⢘⠀⠀⠸⡿⣾⡼⢤⡬⠰⠎⠀⣨⠀⢸⠀⠀⠀⠀⢸⠀⠰⠀⢘⠀⢸⡗⠀⠄⠀⡃⠀⡇⠀⠜⠻⠒⣶⣾⠃⠀⠐
⠀⢸⠰⡀⠀⠀⠀⢘⣿⡗⣮⡀⢹⠖⠛⣉⡈⡟⢛⡛⠻⠀⢸⠀⠀⠀⠀⢸⠛⠛⠚⡛⠛⠺⡗⠚⠓⣒⣓⡲⠟⠀⠀⠀⠘⣿⠂⡌
⠀⠀⢻⠛⠖⠂⠋⣠⣾⡇⠀⢉⣥⣶⣴⡴⠶⠿⠓⠀⠈⠀⢸⠀⠀⣀⠀⠘⠊⠁⠀⠁⠈⠉⠙⠛⠐⠬⢈⡀⣠⡞⠛⠀⠀⢿⡱⣧
⠀⠄⣼⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠦⠷⠧⠴⣧⠞⠀⠈⠣⠥⠥⠭⠍⠁⠬⠭⠉⠌⠉⠩⠽⠿⣿⠃⡀⠀⠀⢹⠿⡯
⠀⢰⡇⠀⠀⠀⠀⠀⢿⡧⣤⠤⣬⣤⡤⡤⡤⠤⠤⠤⠤⠤⠏⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠾⠼⢦⡵⠶⠾⣶⣷⠂
⠀⢸⠁⠀⠀⠀⢰⠀⢸⠆⠀⢠⢟⣿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢸⣻⣧
⠀⡘⢰⠀⠀⡇⠀⠀⢸⡆⠀⡀⣾⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠅⣿⡇
⠀⠇⠈⠀⠀⡇⠀⠀⢸⠃⢰⣧⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠀⢹⣧
⠀⡴⠀⠀⠅⠇⠀⠀⣰⣀⣭⣿⣿⢟⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃⠘⠸⡆
⠀⡇⠀⢠⡀⠀⠀⢰⣿⣿⢃⣿⢯⣿⣣⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣄⣈⠀⠁
⠀⡷⠶⢰⣤⣴⣶⣿⣿⣿⢸⡷⢱⢻⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⢿⣿⣿⣿⣿⠟⠃⡄
⠠⡇⠀⠀⣻⠘⣿⠹⡟⣿⠃⢿⣷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠃⣽⣿⣿⠶⡋
⠰⠇⠀⠀⠅⢀⢿⡞⣦⣿⠉⢻⠋⣽⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡿⣿⣏⢠⠖⠁
⠀⠀⢤⣀⣀⣀⣞⣣⣿⣾⣿⣿⠿⠛⠛⣛⣛⣛⡛⠛⠒⣛⣿⡟⢛⣛⣛⠛⣛⣓⣛⣛⣛⡛⠛⠓⠛⠛⠛⠛⠛⢛⢽⣿⣿⣿⢿⣽⡿⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣶⣶⡶⠾⠿⠟⠛⢻⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠾⠿⢶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⣀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣾⠿⠷⠶⣶⠀⣿
⠀⠀⢴⣶⠿⠟⠛⢻⣿⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣫⣽⣿⣛⡋⠀⣿⣄⣿
⠀⠀⢸⣷⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⢾⣿⣭⡥⢀⣿⢹⣿
⠀⠀⠈⣿⡀⠀⠈⢹⣿⠀⠀⠀⣉⣉⣉⣉⣉⣉⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠿⠿⠿⠟⢸⣿
⠀⠀⠀⢿⡇⠀⠀⢹⣿⠀⠀⠀⣄⣀⣀⣈⣉⣉⡀⠀⠀⢸⡇⠀⠀⠀⠚⠒⠒⠲⠀⠠⠤⠤⠤⠤⠤⠤⢤⣀⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠈⣿⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⢸⠇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠛⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢹⣇⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣤⣀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠘⣿⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢻⡇⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⢀⡀⢀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠘⣿⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢀⣀⣠⣤⣤⣶⣶⠾⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠛⠛⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡒⠒⢲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⢾⡿⠀⣄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣷⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣛⣀⣀⡤⠛⠁⠠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⣠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣶⣶⣒⣒⣒⣶⣶⣶⣖⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠂⠤⣝⣦⣄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⣭⣭⠭⣭⣭⣤⠄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢬⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠘⠂⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⠞⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠁⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⢀⣄⠀⠂⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣀⣴⡆⠀⠸⢤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧
⠀⠀⠀⠀⠀⣾⠙⠆⠀⠄⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠘⠙⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⣹⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⢀⡀⣠⠾⠛⠁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣧⢠⣀⣀⢀⣀⣀⣀⢀⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠌⠙⠛⠘⠛⠛⠋⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣴⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⣾⣿⣿⣿⣷⣶⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⡿⢻⡿⠟⠿⢿⣿⣿⣿⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡟⢠⣌⠁⣤⠀⠀⠹⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⡇⠈⢍⣉⣿⠆⢠⣀⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣿⣿⢹⠀⠀⠂⠡⠀⠀⣿⣍⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⡏⠀⠀⢀⣀⣐⣠⠟⠀⠃⠀⣄⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠆⠉⢉⡳⢷⠀⠉⠙⠛⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣦⣖⣾⠇⣿⣿⣿⣷⣆⣉⣸⠟⣨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣯⣻⣭⢿⣿⣋⢐⠻⣿⣿⣿⣿⣿⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡟⣿⠻⣷⢉⣹⣟⣿⢿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢞⡻⢿⣷⢿⡄⣸⠟⡿⣟⣿⣿⣿⣷⢿⣮⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠌⠙⠁⠹⡴⡿⠛⠿⢿⡿⠻⠿⠿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠒⠒⠒⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠑⠂⠈⠠⡆⠀⠀⠀⠀⠀⠀⢀⡠⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⡀⢳⡀⣀⡤⠤⠇⠀⠀⠀⠀⡠⠖⠉⠀⠀⠸⡄⠀⠀⡰⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠐⠚⣀⡬⠇⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢀⡠⠔⠊⠁⠀⢰⡀⠐⠄⢀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠊⠁⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⠈⠓⠉⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠙⢆⠀⠀⠀⠀⠀⠀⣀⡤⢴⡄
⠀⠀⠀⠀⠀⠀⣠⠔⢛⢡⡄⡜⠳⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠘⡀⠀⠀⣀⠴⠚⠁⠀⡼⣇⣀⠤⠤⠤⢤
⠀⠀⠀⠀⠀⡴⠁⠀⠘⠒⠒⠃⠀⢹⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠀⠀⠀⢸⠀⡇⠀⠀⢳⠀⠤⢤⠜⠁⢻⡀⠀⠀⠀⡞
⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡗⢤⢾⠀⡞⢇⠀⡄⡇⠀⠀⠀⢸⡀⢱⠀⠀⠀⢣⠀⣠⠴⢲⠉⠀⠀⠀⢠⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⡇⠀⠈⣄⠗⠚⡄⠇⠧⠖⠛⠀⢸⠄⢸⠀⠀⠀⠀⠉⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢸⠀⠉⠉⠀⠒⠒⠐⢢⠀⡟⠀⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⠰⠖⠁⠀⠀⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠘⣯⠉⠉⠁⣀⠒⠀⠌⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⢰⠊⠉⠁⠀⣠⠎⣷
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠒⠻⠧⠴⠇⡄⠐⠈⠀⠁⠀⠀⠀⠀⢀⡠⣄⠀⠀⠀⢀⠀⠝⠒⠛⠃⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠊⠁⡀⠸⡄⠀⠀⠸⠧⠔⠒⠒⠒⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⢉⠁⠀⠀⢀⠔⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠙⠢⠊⠁⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠛⠓⠒⠒⠲⠦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠁⠀⡀⠠⠖⢀⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣲⣒⣀⠀⠀⠀⠀⠁⠓⡠⠆⠀⢈⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⣀⡶⠤⠤⠄⠐⠓⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⡛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⢸⡏⠓⡞⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠳⠮⣭⣝⣛⣳⠶⣦⣤⣬⣇⣀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⣤⣄⣀⡀⠉⠉⠙⠛⠶⠦⣭⣍⣛⡛⠻⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣉⣿⠛⣷⣶⠦⣤⣼⣛⡃⠉⠉⠛⠓⠶⢦⣭⣉⣛⠲⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠉⠉⠁⣠⠿⠽⠶⠏⢸⡏⢹⡟⠻⣶⣦⣤⣄⣀⣈⠉⠙⢿⠉⣿⣤⣤⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⢸⡇⠸⣇⣰⣿⣿⣀⡼⢻⣿⡛⠻⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠤⠤⣤⡤⠶⠶⠚⠋⠁⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⢸⣇⣴⡟⠡⣿⠈⠉⢀⡾⠻⠿⠟⠁⠈⠉⠛⠓⠶⠶⠶⠶⠶⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣋⣥⣤⣀⡀⠀⠀⠀⠀⢀⣼⡏⢹⡇⢸⡿⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠀⣿⠻⠥⣄⣉⠉⠛⢳⣶⣾⣏⣽⡇⢸⡇⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣄⣀⣬⠿⠻⠿⣤⣤⣈⡉⠙⢻⡷⢿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠙⣷⣍⣩⡿⢳⣶⣤⣀⣀⣉⣽⣿⠿⢦⣤⣈⡉⠛⠻⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠉⠉⠀⣸⣿⣦⣬⡿⠻⣷⣬⣄⣀⠀⠉⠙⠛⠳⢶⣤⣌⣉⢛⣿⣦⠀⠀⠀⠀⡀
⠀⠀⠀⠘⠛⠛⠒⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣿⠀⢨⡏⢻⣷⣶⢤⣄⣀⡀⠈⠉⣿⢁⣿⠦⠤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⣸⡿⠷⠶⠿⣏⣿⠛⣶⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⢶⡏⢸⣏⠀⠀⢀⡿⠛⠛⠁⠀⠉⠛⠲⠶⠶⠶⠶⠶⠶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣿⠁⢸⡇⢸⡟⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⣀⣤⠴⠞⠛⠉⠀⠀⣿⠀⢸⡇⣸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⢼⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠤⠶⠖⠛⠛⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⡃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣸⣇⣼⠃
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣖⢶⣶⣶⣶⣶⣶⡶⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡄⠀⠀⣠⣶⣶⣶⣄⠀⠀⢀⠤⠞⢸⣿⣷⣭⠻⣿⠟⣭⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠺⠿⠾⠿⠷⠿⠗⢀⠎⠀⠀⢸⡿⣫⣴⣿⣶⣿⣶⣝⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⣀⣀⣠⡔⠋⠀⠀⠀⢀⡙⠛⠛⠛⠛⠛⠛⠙⣫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢲⣿⣇⡈⠉⠀⣠⣿⡖⠲⢄⡀⢸⡇⠀⠀⡤⠤⠴⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠱⣼⡿⠿⣏⠉⠉⠻⣿⠇⠀⠀⠙⢾⣇⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠘⢦⠀⠈⢦⣀⡞⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠳⡀⠀⠹⡄⠈⠉⠉⠁⠀⠀⣇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⣆⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠚⠓⠒⠚⠓⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠯⢽⠻⢭⢟⠒⢤⣀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡊⠀⠀⠀⠉⣩⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣤⣤⠀⢰⢡⠀⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⢉⡠⠔⣊⣥⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⡴⣵⠛⢺⢱⠉⠓⢆
⠀⠀⠀⠀⢀⡠⠔⠚⣉⡭⠟⢻⠁⠀⠹⣇⡀⠈⢙⢖⡉⠑⢢⠤⢄⡀⠀⠀⡰⠃⠀⡄⠀⠙⠁⣀⣯⣇⣀⡀⠸
⣀⣤⣖⣋⣁⠤⢒⣩⠴⠀⠀⢸⠀⠀⣀⠭⠛⠷⣮⣅⣨⣧⠎⠀⠀⠉⠑⠶⠁⠀⠀⡇⠀⠀⢰⡟⠀⠀⠀⡏
⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣼⠖⠋⠠⠒⠁⠀⡇⠉⠉⠈⠑⠢⣄⡀⠀⠀⠀⢀⡴⡇⠀⡠⠿⠧⢄⡤⠴⠇
⡇⠀⠀⢸⡇⠀⣀⡤⠖⡋⠁⢸⠀⠀⠀⢀⡠⠔⠃⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠁⠀⢳⠀⢧⡀⢀⣀⣇⠀⢀⣀
⡇⠀⠀⢸⠖⠋⠁⠒⠉⠀⠀⢸⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠘⠛⡯⠤⠬⠏⠉⠈
⡇⠀⠀⢸⡇⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠀⣳
⡇⠀⠀⢸⣃⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⠉⠙⢦⡀⠀⠀⣠
⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⡆⠀⠉⠉⠉⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣥⣤⣠⣸⣿⣆⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣙⣛⣛⠛⡛⠛⠻⣿⡆
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣛⣛⠛⢿⣿⠀⣿⠇
⠀⠀⢠⣾⡿⠟⠛⠋⠉⣿⡇⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣥⣼⣿⣉⣉⠀⢸⢿⢸⣿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡤⣾⣿⣭⣭⠀⢸⡿⢸⣿
⠀⠀⠀⢻⣧⠀⠀⠘⠉⣿⡇⠀⠀⠀⠛⠛⠛⠛⠛⠓⠒⠒⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣼⢿⣽⣿⣿⠏⢸⣿
⠀⠀⠀⠸⣿⡀⠀⠀⠄⣿⡇⠀⠀⠀⠚⠛⠛⠒⠒⠒⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⣠⠤⣤⣤⣀⡀⠀⣀⣀⣀⣀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠉⣿⡇⠀⠀⠀⠛⠛⠒⠲⠶⠖⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠙⠛⠃⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢹⣧⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠶⠦⠤⠤⠤⠤⠤⢤⣤⣤⣄⣀⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠦⠴⠆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣤⢤⣄⠀⣀⣀⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠸⣿⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⠀⠉⠈⠀⠘⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠿⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣀⣀⣤⣤⣴⣶⣶⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠛⠛⠋⠉⠉


⣿⣿⣿⢿⣿⠿⢿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣼⣾⣶⣶⣷⣾⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⠂⠓⠙⠓⠀⠛⠛⠛⠋⠛⠘⠃⠋⠙⠋⠛⠈⠀⠉⠘⠛⠛⠋⠙⠂⠋⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⢀⣤⣤⣤⡤
⠀⠀⠀⠀⢀⣤⠀⠀⠀⢠⠀⡤⡤⣤⢤⡤⠠⡤⢠⣄⡄⢠⣤⠀⠀⠀⠀⠀⠀⣿⣛⣶⣖⣓⣒⣞⣿⡿⠛⠁⠉⠈⠁⠀⠀⢠⣦⠀⡀⠀⠒⠒⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⣬⡁⢭⠌⣭⢤⣤⣤⡌⣭⡍⠈⠀⠀⠀⠀⠀⠀⣿⠿⢛⣛⣛⠛⢫⠍⠘⠂⠀⡀⢀⡀⠀⠀⠘⠿⢁⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⢃⣀⣀⠙⠀⢈⣉⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⠀⠀⠀⢰⡗⢹⣿⠀⠀⢸⣿⡿⠿⠿⡿⠿⠿⢿⣿⣷⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣶⣶⣦⣶⣾⣻⣿⣿⣷⣦⢸⣯⣭⣽⣭⣭⣭⣽⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣙⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⠀⠸⠻⠛⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠓⠀⡙⠻⠿⠿⠿⠛⠋⠛⠛⠛⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠛⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣧⡀⣴⣦⠀⣒⡒⠒⠂⠀⠀⠀⠀⢀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢼⣿⠓⣿⡿⠀⠄⠖⠔⠆⠀⠀⠀⠀⠨⠭⠤⠉⠩⠥⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠘⠉⠙⠛⠛⠛⠀⠀⣠⡄
⠀⠀⠀⠀⣴⣶⣶⣿⣿⣿⣶⣶⣤⣠⣄⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⡖⢀⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⡀⣀⣀⡀⠀⣀⣠⣤⣤⣄⣀
⣶⣿⣿⣶⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣾⣷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⣀⣤⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⢀⠑⣦
⠀⠀⠀⠀⠀⠀⢸⠁⠀⢳⡀⠀⠀⠀⠀⣂⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⠀⠋⠁⡛⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⡧⠀⣀⡴⠋⠁⠀⠀⠀⢠⠐⠀⠀⠀⠀⢀⣼⡆⠀⠀⠀⠀⠈⠁⡆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣠⣧⠀⠀⠀⣿⠀⣁⠔⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠘⠈⠁⠀⠀⠀⠐⠨⠷⢄⠀⣠⣤⣠⢤⡞
⠀⠀⠀⡴⠊⠁⠀⠙⠲⣦⡿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠡⡀⡀
⠀⠀⠀⠀⢀⣠⡶⠴⠛⠁⠀⠀⠀⠀⠀⠀⠠⠖⠂⠀⠀⠀⠀⠀⠠⠀⠀⡀⠙⠀⠀⢐⠀⠀⠀⠀⠀⠀⠃⠇⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣔⢀⠂⢠⣾⠞⠀⠀⠀⠀⠀⠀⠉⠀⡀⣠⠏
⠀⠀⠀⠀⠀⡀⠐⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⠷⣄⠀⠈⠋
⠀⠀⠀⣰⠀⠀⠀⠰⣄⢠⠂
⠀⠀⠀⢀⠙⠓⠒⠂⠐⠛⡋⠀⠀⠀⠀⠀⠲⡀⠀⢀⣀⠀⠀⠄⠀⠀⡀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⣀⡀⠀⠄⠀⠀⠀⣤⠀⠀⠀⠀⢉⠀⠀⠘⠟⠁⠀⢸⠀⠀⢁⠀⢀⠄⠀⠀⢀⡄
⠀⠀⠀⠁⠀⣀⠖⠋⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠓⠁⠤⠠⠄⠈⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⢢⣀⡤⠖⠉⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠠⠤⠀⠀⣀⠱⡄
⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠇⠀⣀⠑⡄⠠⠠⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠠⡎⠀⠀⠀⠀⢠⠀⠀⠚⢀⡀⢀⠜⠁⠘⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠄⠊⢁⡆⠀⠈⠡⠉⠀⠐⠦⢤⣖⠁
⠀⠀⢀⡀⠤⠒⠉⠀⠀⠀⠠⠂⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡉⢶
⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠐⢄⠀⠠⠀⠁⠀⡇⠀⠀⠀⠀⣀⣀
⠙⠆⠀⠀⠀⠀⠀⠀⢀⠛⠀⠀⠀⠀⢨⠄⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢀⡴⠊⠸⠈⠑⢄
⠀⠈⢦⠀⠀⡘⠀⠀⠁⠀⣀⠤⠒⠋⠓⠒⠒⠤⠤⠄⣀⣀⣰⠇⢀⡴⠋⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠱⣀⣁⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠒⠂⠀⠉⣩⣅⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠀⠀⠀⠤⠀⠈⠀⠀⣠⡞⣁⣀⣀⣀⣀⣀⠀⠀⣀⠉⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⢄⠀⠀⠀⠀⠀⠀⣠⠆⠉⠀⠀⠀⠀⢀⠔⢁⠔⠉⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⠊⠑⢄⣀⡀⣠⠎⠁⠀⠀⠀⠀⠀⢠⠃⠠⠁⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣤⣤⣄⣐⣂⣉⣉⡀⠀⠀⠀⠁⠀⠊⠀⠸⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣉⣉⣀⣈⣁⣀⣀⣈⣉⣉⣛⣛⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢲⠞⡄⠀⠒⡐⠏⡵⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡀⠈⠚⠀⡀⠇⠀⠠⣀⠀⡄⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠒⠀⠙⠧⠝⢼⡧⠤⠗⠤⠤⠀⡃⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠠⠤⠤⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⡁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⣤⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣤⡤⠶⠒⠛⠋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡆⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠘⢧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡀⣸
⣴⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣥⣤⣤⣤⣦⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣁⣠⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⣴⣾⡟⢿⢿⣽⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡜⠃⠀⠀⠀⠀⠀⠛⠙⠉⠋⠀⠀⠀⢲⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣀⡀⠀⠀⠀⠨⢧⠤⠒⠊⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⢿⡺⠃⠀⠀⠀⠀⠀⢄⡀⠒⠀⠀⠀⠘⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣸⠆⠀⠠⠜⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠉⠛⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⡇⠄⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⣀⡀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣆⡠⠰⣶⡆⠀⠈⢧⣀⠀⠉⠀⠐⠂⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠿⣽⠹⣷⣀⣠⣤⣄⠀⣤⣤⣄⣀⣤⣿⣿⣅⡤⣾⣒⡀⠀⠘⡆⠀⠀⠀⠂⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣶⣿⣿⣿⡿⡟⡟⢹⣿⣿⣿⣿⡿⠿⣧⠁⠈⠑⢀⠀⠀⠹⠂⠠⠄⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⢨⣯⠈⠷⢲⡤⢘⣻⣿⡋⢽⠻⢿⠉⠉⠀⠀⠂⠈⣀⣤⡖⠀⠀⠀⠀⠀⠀⣸⠛⠋⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢺⣿⠿⠛⢙⠛⠳⠀⠚⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠛⠿⠶⠦⠤⠀⠀⠃⠀⠀⠀⢀⣴⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠄⠀⠐⠒⠲⠶⠀⠀⠀⠂⠤⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠛⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉⠁⠢⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠤⠦⢤⣤⣤⣤⣤⡄⠈⠙⠳⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠶⣦⠄⠀⠀⠀⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠆⢀⣀⡀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠓⠀⠋⠓⠀⠈⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠠⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⠀⢀⣀⣀⣀⣀⠀⢀⣀⡀⢀⣀⠀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣄⠀⠀⠀⠙⠈⠙⠯⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⢻⣿⠟⠁⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠐⠀⠂⠐⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠃⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⠀⢡⠀⠉⠙⠉⠭⣀⡚⠛⣀⠀⠀⠒⠒⠃⠀⠠⠀⠀⠀⠈⠉⠁⠐⠒⠀⠀⢀⣠⢄⢄
⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⠒⠀⠀⠉⠀⠀⠀⠀⡐⠀⠈⠒⠤⠄⠀⣀⣀⣈⣁⣛⠲⠄⣀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡏⠉⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈⠀⠘⠒⠒⠀⠀⠴⠀⠤⠤⠀⡀⠓⠒⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠄⠤⠤⠤⠼⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠠⠀⠀⠀⠀⡠⢲⡯⡉⢀⠁
⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡞⡉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣌⠀⠀⠀⠈⠱⠤⠀⠂⠀⠀⠀⠉⠀⠀⢠⣤⣶⠪⢶⠈⣆⠴⠋⠀⠸⢤⡀⠘
⠀⠀⠀⠀⠀⠀⡈⠁⢈⠀⣇⠈⠀⠒⠋⠍⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣂⣰⣦⣝⡄⢹⠀⠀⠀⡮⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠇⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠫⠶⠃⠀⢸⠀⠀⢀⡏⠉⢀⠆
⠀⠀⠀⠀⠀⡀⠘⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⠄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣀⣠⣄⡤⠤⠰⠖⠒⠈⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠂⠀⣠⡐⣀⡀⠤⠄⢀⣤⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠎⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢙⣒⠀⠩⠍⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⡀⠀⠂⠈⠨⠥⠀⠂⠀⠀⠈⠁⠀⠀⠀⠀⢀⠀⠀⠤⠀⠐⠐⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠒⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡾⢿⡿⣟⣛⣫⣭⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣵⣿⣿⣿⣿⣿⡿⠋⠙⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣼⣿⣿⣿⣘⠛⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣽⣿⣿⡟⠉⠉⢿⣿⣿⠁⠀⣸⣿⡿⠟⠛⠛⢿⣿⡟⠉⢻⣿⠿⠿⣿⣿⠋⠀⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠈⣿⡏⠀⢠⣿⠏⠀⢀⣤⡀⠀⢻⡇⠀⠘⡟⠀⠀⢹⡟⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠇⠀⢸⡀⠀⢻⠇⠀⢸⡏⠀⠀⠈⠉⠀⢀⣼⣷⠀⠀⠁⠀⠀⠸⠇⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⣿⡇⠀⠘⠀⠀⣿⡇⠀⢠⣶⣶⣾⣿⣿⣿⠀⠀⠀⢠⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢰⣿⣷⠀⠀⠀⢠⣿⣇⠀⠈⠛⠛⠛⠙⢿⣿⣇⠀⣠⣿⣧⠀⢀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⡅⠀⣸⣿⣿⣧⣀⣀⣾⣿⣿⣷⣤⣄⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠴⢂⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠛⠋⠻⣿⠋⠉⠉⠁⠀⠀⢈⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣿⣿⠀⠀⣀⠀⠈⣿⠋⠀⠀⠀⠈⢿⡏⠀⢠⣤⣾⣿⣶⣶⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠋⠀⢀⡇⠀⢰⣿⡆⠀⢸⣧⠀⠈⠛⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⢀⣤⣴⣿⡇⠀⠸⠟⠀⢀⣾⣿⣿⡆⠀⠈⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⡇⠀⢸⣿⣿⣿⣿⣄⣀⣀⣠⣾⣿⠁⠀⠀⣀⣴⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⠷⣶⣿⣿⡏⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠺⢭⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠉⠙⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠦⣄⡀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠓⢦⣍⡓⠲⢤⣄⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠳⢭⣙⠒⣺⠋
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠛⠀⠀⠨⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⣄⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠽⠒⠻⣝⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹
⠀⠀⠀⠀⠀⢀⡴⣻⠓⢤⣀⡀⠙⠒⠆⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠘⡎⡆
⠀⠀⠀⠀⢠⣾⣾⠇⣿⠀⠈⢯⠟⠛⢿⡟⡿⢦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⣷⣷
⠀⠀⠀⠀⡞⣿⣟⡀⢻⡄⠀⠀⢀⡠⢊⠞⠁⠀⠉⠛⠓⠶⠶⢦⡶⠖⠱⣀⡀⠄⠐⢊⠝⠀⢸⣿⠀⠀⠀⠀⠀⣿⡟
⠀⠀⢀⣰⠛⠉⠉⠉⠉⢷⣤⢖⣫⢶⡏⢶⠂⠀⠤⠤⢀⣀⡀⠈⠓⠒⠒⠂⠤⠤⠤⠊⠀⠀⢨⠟⠶⠾⠿⣭⣭⣿⣧⠤⢤⣤⡤⠤⢤⠴⠶⠶⠶⠶⠶⠶⠶
⠀⢀⣾⡇⠀⠀⠀⢀⢄⡾⠿⣯⣤⣊⣀⡈⠣⠴⠒⠈⢁⠼⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣼⡇⠀⠘⣿⡇⠀⢸
⠀⢸⠉⢷⣄⡄⣀⣨⠏⠀⢰⡇⠉⣖⠒⠤⠥⢍⣀⣒⢊⣜⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣄⠀⠀⠀⣠⣾⠟⠀⠀⠀⣿⡇⠀⣸
⠀⠘⣆⠘⠿⠟⠉⠀⠀⠀⡾⠀⢠⠏⠀⠉⠀⠀⠐⠂⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣤⣶⠿⠛⠁⠀⠀⠀⠀⣿⡇⠀⡿
⠀⠀⠈⠓⠦⢤⣤⠤⠶⠾⢥⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⡇
⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣇⣄⣧⣶⠏
⠀⠀⠀⣠⣞⣛⣙⣳⡆⢀⣾⣛⣋⣛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⡟⠋⠛⢱⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⢄⣀⣀⣀⣀⣤⠴⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠁⠀⠀⠀⠀⠀⣀⣤⡶⠶⣄⣩⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣼⣯⣧⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢀⣀⣤⠴⡖⣿⣻⡭⠷⣚⣫⡭⠖⠛⠛⠛⠛⠓⢒⣲⡶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⢥⣤⣴⣶⣿⠯⠿⢺⣏⣿⣭⣔⣚⣉⣉⣀⣀⣠⣤⠤⢶⠻⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡯⠤⠶⣶⢻⣿⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⣨⡇⠈⠉⠉⠙⠓⠒⠶⠦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡼⡟⠺⣝⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠚⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⠛⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠐⣷⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠁⠹⢄⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⠆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⣰⡛⠋⠉⡇⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣠⣾⣻⢿⡿⢿⣭⣾⠉⠉⡻⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⢴⣶⢶
⠀⣠⠞⣽⡄⠈⡇⠀⢸⡜⠒⢠⣧⣴⣿⣷⡦⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟⠀⠀⠀⠀⠀⣀⣧⠤⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣽⣾
⡀⡈⢢⣽⣧⣀⣧⣀⣼⣀⣉⣿⡟⠁⢀⠀⠉⡆⢹⠈⠉⠉⠙⡛⠻⣛⠛⠛⠛⢻⣿⡛⢹⣿⡀⠀⠀⠀⢀⣰⠏⣴⠶⠚⠁⢠⣤⠤⠤⠠⠄⠐⡒⠀⠀⠀⣿
⡇⠁⢹⠒⠒⠲⠿⠶⠬⠭⣽⠋⠀⢰⠏⠀⠀⡌⣸⠀⠀⠀⠀⠘⡄⠈⠑⢦⡔⠁⢸⣎⣻⢸⣿⠓⠦⢶⢯⠹⣄⣷⠀⠀⠀⠀⠚⢲⠩⠭⠭⠬⠥⠀⠀⠀⢹
⠁⠇⢸⡀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡟⠀⠀⡴⣡⠃⠀⠀⠀⠀⠀⠈⠦⠒⡄⠈⠳⢄⣿⣧⠟⢼⡀⠀⠀⢋⣣⡼⣧⣀⣠⣤⡤⠤⠤⠴⠶⠶⠶⠶⠶⠖⠒⠻
⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⣴⣪⠞⢡⣿⠷⡲⠦⣄⣀⠀⠀⠀⠉⠙⠓⠒⠻⡏⠀⠈⡏⠓⠒⠛⠋⠀⠰⣂⣒⠀⠐⠒⠒⠒⠂
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠸⣇⠀⠀⡸⢯⣀⣠⣞⣈⡙⠁⠀⢀⣼⣇⠀⠀⠀⠀⠀⠀⠀⡼⠂⠀⣿
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠹⣄⣔⠔⣱⠏⠩⣭⣤⡟⠁⢀⣾⣯⠏⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⣸⠟⠾⠣⣄⡐⠶⣶⠟⢣⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⡀⠀⣼⠋⠉⠙⠛⢡⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣧⣀⠀⠀⢀⣾⣿⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⢖⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡄⠀⠀⠀⠙⢯⣗⠺⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣼⠙⢦⣀⠀⠀⠀⣈⡽⠃⠀⣤⡀⠀⠀⣀⣠⡴⠞⠋⠁⠀⠀⢀⣠⢾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⠉⠛⠉⠁⠀⢀⡼⠃⢹⣦⡀⠙⠦⣄⠀⠀⠀⢀⡴⠛⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⠛⠲⠶⠶⠞⠋⠀⠀⠠⡀⠙⢦⣀⠈⠙⠒⠶⠋⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⠍⠻⢦⣄⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⠒⠒⠉⢉⠙⣀⣀⣀⣀⣀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⢠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⠂⠀⠀⠐⠉⡀⠀⠀⠀⠠⡴⢀⠀⠀⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠓⢀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠉⠒⡀⠀⠂⠀⠀⠀⢀⣀⡤⠃
⠀⢀⣠⠴⠒⠒⢈⡉⠭⠀⠀⠈⠋⠉⢻⠀⠁⢀⠀⠀⠠⠆⠀⠀⠐⠁⠀⠀⠀⠀⠁⢤⠒⠂⠉
⡰⠋⠀⠀⠀⠔⠁⠀⢀⡀⠀⡀⠀⠀⠰⣄⠂⠀⠀⠀⠀⠀⠀⠀⢄⠂⠒⠐⡀⠀⠀⠀⠑⣤
⡇⠀⠀⠀⠀⡀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠋⠈⠁⠀⠁⠉⠢⢄⠀⠀⠀⠈⠀⠀⡠⠋
⡇⠀⠀⠀⠀⠇⢀⢀⣀⠀⠠⢄⡀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⣠⠞⠀⠀⢀⣀⣤⡀
⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠐⠢⣤⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠄⠒⠈⠁⠀⠀⠱⡀
⡇⠀⠀⠀⠀⢠⠀⣤⠤⠀⠐⠂⠈⠁⠁⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⢰⡊⣁⠀⠀⠀⠀⠀⣌⠀⠀⠡
⡇⠀⠀⠀⠀⢸⠀⢀⡀⡀⢀⠤⡀⡀⠀⣴⡉⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠂⠀⠀⠀⠀⠉⡀
⡇⠀⠀⠀⠀⢸⠀⠁⠀⡸⠃⠀⠈⡵⠂⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣀⡤⠔⠊⠉
⢃⠀⠀⠀⠀⢸⠀⢀⡃⠀⠑⠀⠊⠈⠈⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁
⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠉⠁
⠀⠀⠀⠀⠙⠢⡤⠤⠤⡖⠂⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣤⣄⡇⠀⢻⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢫⠙⠙⠊⠀⣸

⠀⠀⠀⠀⠀⠀⢠⠶⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣛⡆⠀⠀⠀⠀⠀⠀⢸⣉⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠑⠫⡲⡈⢣
⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⠜⠂⠤⠤⣤⡼⠈⢺⡄⠀⠀⠀⠀⠀⢾⣙⠆
⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⢫⠈⠉⠀⢣⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⢤⣤⡀⠀⠀⠀⠀⠈⠢⣄⡠⠔⠋⣁⡼⡄
⠀⠀⠀⠀⠀⠀⠀⣤⣖⣋⣀⣀⣀⣀⡠⠔⣒⣍⣆⣇⣀⣀⡀⠀⠀⢸⣳⢒⣒⣋⡏⠀⠸⡄
⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡭⢰⠻⠯⢤⢻⣭⡂⠀⠈⢲⢤⡇⡏⠉⠀⠀⡀⠀⠀⢷
⠀⠸⠼⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢙⡦⠡⠒⠤⠮⡜⡇⡇⠀⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠸⠀⣀⠤⠊⡇⠀⠀⠀⠀⢹⡇⡇⠀⠀⠀⠸⡆⠀⠘⡇⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣧⠈⣉⣠⣆⠃⡇⠀⠀⠀⠀⢸⠃⡇⠀⠀⠀⠀⡇⠀⠀⢹⠀⠈⠚⠁
⠀⠀⠀⢀⣀⠀⠸⡇⠀⠀⠀⠀⠀⡿⠀⠋⠉⠁⠀⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢱⣒⣒⡞
⠀⠀⠀⠘⠚⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡤⣷⣀⠀⠀⢀⣇⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⢀⡠⠇⠀⠀⠀⠀⢸⠃⠀⠈⡟⡏⠉⠉⣷⡿⠏⠀⠀⠀⠀⢠⠤
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣀⡤⠚⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⠃⣇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠙
⣞⣲⠀⠀⠀⠀⠀⠉⠉⠉⠉⡏⠉⠉⢹⠀⠀⠀⢰⣒⡆⠀⠀⠀⠘⡦⠤⢽⠀⢸⠤⠤⢧
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⢧⠀⢸⠀⢸⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢸⡇⠀⣇⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠖⠚⠁⢠⣃⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⢒⡄⠀⠀⠀⠀⠀⠀⠀⠠⡖⡆
⠀⠀⠀⠹⠷⢟⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢸⠀⠀⡄⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠀⠀⠀⠀⠃⠁⠀⠀⠂
⠀⠀⠀⠀⠀⣼⠓⠆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡀⠀⢤⣄⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠆
⠀⠀⠀⣾⡟⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡄⡰⠋⠈⠻⢿⣅⠀⠀⠀⠀⠀⠀⠠⣾⡎⠉
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡧⠄⠀⣀⢀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠐


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⣼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧
⠀⠀⠈⢉⣽⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣯⡉⠁
⠀⠀⠀⣿⢹⠀⠀⠀⡶⢲⡄⡴⢶⡀⡶⠂⠐⣶⠂⠀⣰⠲⡄⣴⠒⢰⡖⠂⣦⢠⠶⠂⡶⠒⠀⠀⠀⡏⣿
⠀⠀⠀⣿⢸⠀⠀⠀⡗⠚⠁⡇⢸⡇⠙⢦⠀⣿⠀⠀⣿⠀⡇⣿⠒⢸⡗⠂⣿⢸⠀⠀⡟⠒⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠀⠀⠃⠀⠀⠓⠞⠀⠲⠚⠀⠛⠀⠀⠙⠞⠃⠛⠀⠘⠃⠀⠛⠘⠳⠂⠳⠖⠀⠀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⣦⠀⠀⣆⠀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⢰⡄⠀⢰⡀⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⢹⡀⠀⢻⠀⢸⡇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⢸⡇⠈⣇⠀⠘⡇⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⣿⠀⠘⡇⠀⠘⡇⢸⡇⠀⡇⠀⣤⡄⢸⡇⢠⣤⠀⢸⠀⢸⡇⠀⢻⠀⠀⢻⠀⣿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠿⠶⠶⠶⠶⠶⠶⠾⠇⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠸⠷⠶⠶⠶⠶⠶⠶⠿⠀⡇⣿
⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿
⠀⠀⣠⣿⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣤⣤⣤⣼⣧⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣿⣄
⠀⠀⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⣠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⢻⡅⠃⠻⠆⠀⠀⠀⠀⢀⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢠⣟⡍⢉⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⡄⠁⠐⠆⠀⢀⡀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⣀⣀⠀⠀⢈⠓⣄⠀⢹⣿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠊⠀⠶⣦⣀⠀⢸⡧⠃⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠾⢴⣒⣠⣄⣾⠿⠧⠴⢖⢁⣀⣤⣾⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⠀⠀⠐⢤⣤⡴⠛⠁⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣨⡇⢸⣷⠀⣀⢀⡀⠘⢥⡼⣿⣿⣿⣿⣿⠉⢡⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠀⢐⠀⠀⢀⡀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠉⢹⠃⠘⡟⠒⢛⣖⠁⠀⠀⠈⠘⡞⢙⣿⣷⣶⣿⣷⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠐⡀⢈⠀⠀⠈⡧⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠂⢸⣃⢰⡂⣀⡈⣋⠀⠀⠀⢀⣰⣉⠘⢿⣿⣿⡿⠇⠰⡄⠀⠀⠀⠀⠄⠘⠋⠀⠀⠀⠱⠆⠀⠀⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣺⣽⣦⡆⠀⠀⠀⠀⠀⠀⠀⠁⡏⡳⢞⣿⣿⣷⣄⢠⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢘⣮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡏⠠⠀⠀⠀⠀⠀⠀⠀⠁⠛⠁⣼⡟⣿⣏⣿⣾⣿⡀⠀⠀⠀⠒⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠵⢋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣧⣿⣿⡀⠀⠀⠀⠸⠂⢴⡀⢀⠀⠀⢸⠃⠁
⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⢂⣘⡅⢾⠛⠐⣨⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠻⡿⠿⠿⢛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⢇⣀⠁⠐⠎⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⣿⣿⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⠴⠒⠒⠛⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⡀
⣠⠤⠴⠋⠒⠒⠂⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠓⢲⠄⠀⠀⠀⠀⠀⠀⠀⢀⠎⠐⠀⠀⠁⠢⢄⡀
⠙⡄⠠⣠⠀⢈⡆⠈⣘⠀⠀⣀⡀⠠⠄⠀⠠⠌⠀⢬⠚⡔⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⠀⢘⡆⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠁⡀⠀⡀⠄⠐⠀⡝
⠀⠘⠶⣤⠀⠀⠀⠐⠒⠒⠒⠚⠓⠒⠀⠀⠀⠀⠒⠒⠀⡟⠁⠀⠀⣀⡠⠤⠀⠦⡀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢸⠀⠀⠂⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢁⢀⡧⠄⠂⠉⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠉⠒⠤⣀⠀⡴⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⡟⠉⠉⠉⠙⠛⡍⠁⠀⢺⠡⠄⢀⠀⠀⠀⠀⠠⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠶⢦⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠐⠨⠥⠬⠭⠍⠙⠂⠂⠀⠈⢇⠀⠀⠀⠁⠀⠂⠀⠀⠀⢀⠼⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠌⠀⢱⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣀⣀⢀⠀⣀⣀⠀⠀⠀⠈⣦⠀⠀⠀⣀⠠⠐⠊⠉⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢳
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘⠁⠤⠀⠀⠀⠠⠄⡄⠀⠀⠀⡏⠙⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠒⠀⠤⠀⠀⡀⠀⠀⡠⠓⠂⠤⠄⡀
⠀⠀⠀⠀⢸⠀⢰⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⡠⠁⠈⠉⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡔⠁⠀⠀⣀⡀⠀⠒⠉⠁
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⠘⠒⠒⣒⡒⢢⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⡀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠗⠒⠀⠁⠉⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠄⠀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠰⠀⠤⠒⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠠⠤⠤⢤⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⡆⠀⡎⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡯
⠀⠀⠀⠀⠀⡴⠆⠠⠦⠄⠴⠤⠤⠤⠤⠤⠤⠶⠤⠐⠣⠷⡄
⠀⠀⠀⠀⠸⡇⢀⠀⣀⠀⢀⠀⠀⡀⠀⣀⠀⢀⡀⠀⡀⢀⢸
⠀⠀⠀⠀⠀⡇⠈⠒⠉⠂⠁⠁⠈⠙⠊⠀⠁⠁⠁⠁⠈⠁⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⢀⠔⠂⠄⠀⠤⠂⠢⠂⠐⠔⠈⠒⠀⠐⠀⢸⡇
⠀⠀⠀⠀⢀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅
⣀⣀⣀⣀⣀⣳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣋⠉⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⣀⣀⡀⠀⠈⣉⣉⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡡⠊⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠝⢁⠔⠋⠁⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⡜⠁⠀⠀⠸⠀⠀⠀⠀⠀⢀⡀⠀⠀⡰⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣇⣀⣀⡠⠤⠔⠒⠢⠴⠶⠶⠤⠤⠤⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠉⠙⠛⠛⠒⠒⠒⠒⠒⠒⠃⠀⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠠⠒⠒⠒⠒⠒⠒⠂⠀⠀⠐⠒⠒⠒⢆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⢥⡬⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⣀⣠⠀⠀⣀⠀⡠⠄⡀⣀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠘⠃⠃⠰⢀⠆⠁⢸⠁⡆⡇⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠶⢤⣼⢠⠘⡼⠸⢄⠇⠣⠄⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠉⠀⠈⠙⠃⠓⠓⠒⠚⠉⠒⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠦⠤⠤⠤⢬⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠖⣶⠶⠶⠶⠤⠤⠤⣤⣄⣀⣀⣀⣀⡀⠀⢀⠀⡤⠶⠒⠛⠉⠉⠀⢸⣤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠈⠿⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣘⣀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢫⡏⠀⠀⠄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢸⡄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠈⣧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡿⠆⢤⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⡠⠦⢤⣶⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⢠⡠⠈⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⠀⠀⠀⢀⠀⠀⠀⢄⣀⢀⣀⣀⣀⠠⠤⠔⠉⠀⡌⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢁⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡾⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢣⠀⠐⠆⠀⠀⠀⠀⠀⠀⡤⠄⠀⢀⠀⡜⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠛⠛⠚⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⢸⠀⣤⠔⠊⠁⠀⠂⢠⡁⠀⠀⠀⠀⠢⠀⣀⠙⠻⠿⠿⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠰⠘⡄⠀⠀⠀⠀⠀⠒⠒⠠⢄⡀⠆⠀⠃⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⣠⠈⠓⢒⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠁⠚⢙⡉⠉⠉⠉⢩⠎⠀⠀⠁⠀⠀⠀⠀⡤⠀⢀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⣠⡂⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠢⢄⡀⠀⠀⠂⠀⠈⠙⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠃⣧⣠⠞⠁⠀⢸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢢⠀⢨⠀⠀⠀⠀⠈⠻⣖⣀⣐⣂⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⢩⠈⡍⠉⠒⠒⠒⠒⢢⡤⠤⢤⠀⠀⠀⣸⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡏⠉⠉⠉⠹⡄⢹⠉⠉⠉⠉⠉⠉⠀⡔⠁⠀⠀⣴⠃⠀⢸⠀⠀⠀⠀⠀⠻⢮⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⡆⣎⣩⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⡜⠁⢣⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⡂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢉⡀⠀⠀⠀⠀⢠⠃⠀⣠⠞⠀⠀⠘⠄⢈⡆⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⢀⣜⣿⠒⠖⠶⢦⡤⢍⣛⠉⢡⣀⣀⣠⠞⠣⠄⡜⠁⠀⠀⠀⠂⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢀⠤⣚⣵⠾⠓⠊⠑⢦⡀⠀⠐⣶⡑⠒⠣⠀⠀⠀⠀⢠⡀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡵⠊⠁⠀⠀⠀⠀⠀⠀⢙⣦⡄⠇⠙⢦⣤⣓⣠⠀⠀⠈⠁⠀⠓⠒⠒⠂⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⡵⠏⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⠀⠉⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣇⣀⣠⠤⠤⠶⠒⠒⠒⠉⠉⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠋
⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠢⢄
⠀⠀⠀⠈⠑⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂⠒⠒⠒⠒⠒⠂⠠⠤⠤⠼⢤⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠄⠐⢲⠚⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣶⣦⣤⣤⣄⣀⣠⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⢦⣄⣀⣀⣀⣀⣤⣴⣶⠟⣫⣿⣿⠋⢀⣴⣿⠇⠀⣴⣿⠋⢩⣿⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⣉⣴⣿⣿⠟⠁⣠⣾⣿⡿⠀⣸⣿⡟⢠⣿⣿⢸⣿⡟⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⡿⠿⠟⠋⢁⣠⣾⣿⣿⡿⠁⣰⣿⣿⢃⣿⣿⡟⠸⣿⣷⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣶⣾⣿⣿⠿⠛⢁⣠⣾⣿⣿⠃⣼⣿⣿⠃⠀⣿⡏⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠾⠿⠿⠟⢋⣠⣾⡿⠟⠁⠀⠀⠟⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣰⣤⢴⡶⠶⢶⡾⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⡏⠀⢳⡀⢻⡀⠀⢳⡈⢯⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⢻⠀⠈⣧⣠⢿⡛⠺⣷⡈⢧⡀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⢧⠀⢸⣧⠀⠻⡄⣹⢳⡀⠳⣄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠄⠀⠀⠀⠈⠳⣼⡈⢧⡀⢹⣧⠀⠻⣄⠘⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⡀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠚⠻⣇⠈⠳⣄⠙⢦⣀⡿⠦
⠀⠀⠀⠀⠀⣠⠖⠛⠋⠉⣉⣉⣉⣉⣉⣉⣙⡆⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠘⢦⡀⠈⢷⡈⣿⣇
⠀⠀⢀⡴⠞⠛⠋⢉⣉⣉⣉⣉⣁⣉⣉⣉⣉⣉⡳⣆⠀⠀⠀⠀⠀⠘⣷⠖⠃⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠙⢦⣄⣹⢯⡘⠷
⠀⢰⡿⡖⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⢉⡏⢻⡆⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣠⡿⠻
⠀⡿⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠒⠒⠚⠛⠛⠛⢉⣿⣄⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣠⣤⣶⡛⠉⠓⠒
⢸⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⡟⠓⠶⠶⠶⠶⠶⠚⠋⣽⡏⣿⠁⠀⠈⣏⠙⢦⣀
⠈⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣠⣤⡷⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠋⠀⠀⠀⢻⡄⠀⠙⢦⣀⣀⡀
⠀⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠶⠖⣾⢻⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢷⡀⠀⢸⡟⠻⠿⠃
⢸⡷⠞⠒⠒⠒⣲⠖⢶⣖⣒⣶⡶⢶⠶⠶⠶⠿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢷⡀
⢸⡇⠀⠀⠀⠀⠛⠶⠶⠽⠿⠯⠤⠾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣀⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣦
⢸⡇⠀⠀⠀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⢻⡏⠉⣷⠚⣷⣄⠀⠀⠀⠀⠀⠀⢈⣦⣻⣧⡤⢦⣄
⢸⡇⠀⠀⣴⠋⠁⠀⠈⠀⠉⠉⠉⠉⠉⢻⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡄⠀⠈⢷⣉⢨⡟⠳⡶⠶⡞⠛⢿⡉⠹⣦⠙⠛⠹⣦
⢸⠇⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠺⡇⠀⠀⠀⠙⢿⣄⣰⠇⠀⣿⠀⢸⡇⢀⣹⢦⣠⠖⢻
⣸⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠤⠶⣿⠀⠀⠀⠀⠀⠈⠛⠳⠶⠷⠶⠞⠛⠋⠁⠀⠙⠦⡼
⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣠⡤⢾⡆
⣿⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣷⠀⠀⡀
⣿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠚⠉⠘⣷⢋⣟
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⠴⠾⣿⣭⡭⠇
⣿⠀⠀⡿⠶⠶⠶⠶⠶⠶⠶⠦⠤⠶⠦⠿⣦⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠁⠀⡞⣻
⠛⠒⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣵⣤⠴⠟⠋