Humiliated Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢰⢄⣴⣿⣇⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠀⢠⣮⣿⣾⣿⣿⠋⠹⡄⠀⠸⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠠⠄⠀⡼⣽⠆⡄⠀⠀⠀⠀⢸⣛⠙⣿⣖⣙⣤⣽⡁⢀⡄⡄⢰⠖⡉⠀⠛⢉⡅
⠀⠀⠀⠀⠁⠀⠀⠀⢾⣷⠶⠀⠐⠀⠀⠁⠀⢸⣿⣾⣿⣿⣿⣿⠋⣹⣿⡄⣛⢾⣿⣟⣷⣤⠉
⠀⠐⠂⠀⠀⠀⠀⠀⠜⠉⠀⠀⠐⠀⠀⠁⠀⢸⠿⢿⠋⠻⣿⣷⣾⣿⣿⣷⣣⣿⣿⣿⡁⢈⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⡀⢸⡄⣼⣿⣟⣛⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠘⡽⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠿⡟⠇⠀⠀⠀⠀⠀⢠⣤⣄
⠆⡶⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠠⠀⠀⢠⠀⠀⠀⣸⣷⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣜⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀
⣀⢂⣄⡐⠂⢀⠂⢺⠠⠘⠀⡆⠀⢐⡇⢨⢸⡆⣸⡿⣇⣿⣿⣥⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣦⣀⢀⣙⠿⣿⣷⣴⠂
⠋⢸⣟⣧⠀⠈⡀⠈⠀⢠⡿⣷⠠⠋⠧⡼⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠛⢁⣤⣾⣿⣶⠈⠙⢿⣿⣿⡧⠾⠿⠿⠷
⠀⢛⢿⣁⠀⠀⠈⠀⠀⠀⠙⠛⠄⠀⠀⠀⠀⣿⣟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⡀⠀⢹⡍⣢⣤⠀⢲⢶
⠀⠀⢀⣀⡠⣴⡶⠒⣆⡤⡤⠀⠀⢄⣀⠄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠦⠄⢹⣿⣿⣿⣿⣾⡶⢲⡶⣦⣄⣻⣿⣀
⠀⠀⠀⠈⣡⣤⣥⣤⣀⣀⣁⣀⡒⠚⠉⢡⣿⣿⣿⡿⠙⣿⣿⡿⢻⣿⣿⠛⣿⣿⠃⠀⢀⡄⠀⠘⢿⡿⠿⣿⣿⣿⡛⢿⣿⡾⠉⢻⣿
⠤⣀⣲⣄⣿⣷⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⣿⠇⣰⣿⡿⠁⢆⣿⣿⢴⣿⡟⣀⠐⠁⠟⣠⣤⣼⠀⠂⠠⠌⣿⣿⠍⢠⡬⢶⣶⠟
⠰⠾⠟⠟⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣐⣿⠷⣿⡏⠀⠉⠀⢿⡿⢘⣿⣧⣤⣼⣾⣿⡟⠃⢸⠾⡅⢈⡙⠋⢡⣶⣆⡈⡁⢿⣶
⣴⣶⣶⣶⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣾⣷⣴⣿⣦⣤⣿⣿⣽⣿⣿⡏⣀⣀⣤⣾⣿⠄⣩⣵⡿⠋⢹⣿⣿⣶⣾
⣿⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣿⣿⣶⣶⣶⣿⣿⣷⣾⣾⣿⣿⣾⣷
⣿⣿⣦⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⠀⠱⠃⢸⠀⠀⠀⠀⠀⠀⠀⠲⣄⡴
⠀⠀⠀⠀⠀⠀⠀⣀⢸⣯⣭⣍⣉⣉⣩⣿⠀⠀⠠⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠴⠋⠳
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠠⡶⢸⡿⠤⠬⣿⠀⠀⠀⢀⣀⣀⠀⠀⠀⣤⡀⠀⢠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣷⣶⣿⣿⣿⣿⣿⠀⠀⣴⣿⣿⣿⣿⡄⠀⠤⠭⠀⢸⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⢿⣿⣿⣿⣿⡇⠀⠖⠋⠀⠸⠟⠛⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣿⣟⠻⠟⣻⣿⣇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇⣦⡀
⠀⠀⠀⠀⠀⢸⣿⡟⣡⣌⢻⣿⡟⠁⠀⣠⣿⣿⠟⣵⣿⣿⣿⣿⣿⣧⢻⣷⣦⣄
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠇⠀⠀⣿⣿⠇⠀⢹⣿⣿⣿⣿⣿⣿⠈⠙⠻⠿⣿⡖⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠘⣛⣛⣛⣭⣭⣭⡄⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⣿⣿⣿⡟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢻⣿⣿⡇

⠀⠀⠀⠀⠀⠰⠿⢆⣴⠶⠂⠲⣄
⠀⠀⠀⡨⠑⠂⠀⠀⠉⠉⠀⢀⠹⣧
⠀⠀⡘⣣⡆⠀⠀⠀⠀⠀⠀⠈⣶⠾⠄⢀⡀
⠀⠀⣿⡟⡄⠀⠀⠀⠀⠀⠀⠀⢻⣠⣶⡏⢉⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⢻⣷⢴⢀⠀⠀⢠⠀⠀⠀⢲⣿⣿⣧⣠⠉⢉⠽⠛⠿⠿⠿⠏⠉⠉⠉⢛⠓⠶⢀⣀
⠀⠀⢸⣿⠟⢸⡇⠀⢸⢠⠀⠀⢸⣦⣿⠛⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠐⠞⠁⢱⡀⠆⠚⠇⠂
⠀⠀⢸⣿⡇⢸⠃⠀⠸⢸⠀⠀⠈⣿⣿⠋⠻⣆⢠⣱⠆⠀⠀⠙⣦⠀⠀⠀⠉⠻⠷⣶⡆⣀⠑⠊⢷⣄⡀
⠀⠀⢸⣿⡇⢸⠁⠀⠀⡿⠀⠀⠀⢿⡟⠀⠀⠛⠒⠛⠀⠀⠀⠀⠀⠑⠂⠥⣄⣀⣀⣀⢈⣿⠟⠲⢄⠈⡙⢦⣀
⠀⠀⢸⣿⡇⢸⠠⠀⠀⠈⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⢦⠤⣄⣄⣀⣀
⠀⠀⢸⣿⠇⠈⢡⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⣠⡶⠾⠛⠓⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢤⡒⠝⠛⣳⣶⣤⣀
⠀⠀⢸⣷⣷⠐⠎⠀⠀⠁⠀⠀⠀⠸⣿⣿⣧⠄⠀⠀⠀⢰⠏⣴⣿⡿⠷⡄⣿⢃⣶⢒⣆⣰⣒⣶⣂⣀⡤⠬⠝⠑⣲⠟⣁⣴⣶⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⡏⢰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⣿⠀⣿⣿⡖⣒⣴⣯⣱⠿⠛⠛⠋⠁⠀⠀⠀⠀⢀⣴⣿⣵⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⠂
⠀⠀⢸⣿⡇⣿⡇⠀⠀⠀⠀⢀⠀⠀⢹⣿⣿⣿⣧⡀⠀⠘⢧⣘⠛⠿⠟⢋⣼⣷⣾⣿⣿⣿⣷⣶⣤⣀⠔⣛⣽⣿⠟⠁⠈⠉⠉⠛⠻⣷⣽⣿⠏
⠀⠀⢸⣿⡷⠏⡇⠀⠀⠀⠀⠸⠀⠀⡄⢻⣿⣿⣿⣷⠀⠀⠀⠙⠻⠟⢿⣿⣿⣿⣟⣍⢻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁
⠀⠀⢸⣷⣛⠀⠁⠀⠀⠀⠀⠀⠀⠀⣏⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣯⡟⡽⠁⣸⠇⠻⣿⣿⣿⣿⣿⣷⣿⠀⠀⠀⠀⠀⡀⢀⠀⣾⡿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⠀⠀⡇⠀⣽⡀⠘⣿⣿⣿⡇⠆⠀⠀⠀⢀⣿⣿⡀⣷⣿⡏⠁⠀⠹⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠹⣬⡜
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠘⠀⠀⡇⠀⠘⡇⠀⣿⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⡯⢿⠟⠀⠀⠀⠀⠻⣿⠟⠉⠀⠀⠀⠀⠀⢀⡤⢳⡼⠁
⠀⠀⢈⣿⣿⠀⡄⠂⠀⠀⠀⠄⠀⠀⠀⠇⢰⣿⣿⣿⢇⠀⠀⠀⠀⢸⣿⣿⡷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠚⡋⠵⡿⠁
⠀⠀⠀⢿⣯⠀⡃⢠⠀⠀⠀⠀⢀⢰⡞⢠⣿⣿⣿⠋⠞⠀⠀⠀⢠⡿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣶⡏⠀⠌
⠀⠀⠀⠀⠉⠛⠷⠇⠠⠢⠿⠋⠉⠁⣠⣿⣿⣿⠋⠀⢀⣀⣤⡶⠋⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢘⡇⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡉⢩⣿⣿⣿⠿⠟⠉⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⢀⣈⣉⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣍⡙⠁⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣴⣿⣶⣷⢶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⠫⢿⣷⣦⣄⡀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣿⢿⡛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣯⢟⠊⠙⠻⣿⣿⣷⡄⠀⢻⣷⣀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣍⠀⠀⠠⠀⠙⠻⢿⣦⡼⠟⠛⠿⣶⣤⣄⣀⣠⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠿⠾⣟⣿⠿⣉⠟⠻⣿⣦⣄⡀⠀⠈⣾⡟⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢸⣿⡟⢾⣿⣿⣿⣿⣿⣦⣅⣦⠀⢀⠋⠿⠳⢶⣏⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣜⠢⠉⣯⣿⣿⣿⢿⣿⣄⡀⠀⠁⢂⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣤⣁⠺⠿⣧⣈⣿⣿⠮⢀⠀⠈⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢰⣿⣿⠛⠛⠛⠾⢽⣿⣳⣶⠤⠄⣀⣠⡾⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⡄⠘⠟⡷⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡆⠀⠀⣿⠀⠰⠏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡇⠠⠤⠤⠤⠤⠤⠤⠤⠈⠃⠀⠀⢠⠀⠐⠂⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⢰⠖⢲⡆⠠⠤⠤⠤⢀⡁⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠸⠶⠾⠃⠐⠒⠒⠒⢈⡇⠀⢠⡞⠉⠁⠈⠙⠳⡄⠀⠘⠳⠦⠀⡇⠀⠉⠉⢻⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⡟⠀⠀⠀⠀⠀⠀⢹⠀⠛⠛⣃⢀⡇⠀⠀⠀⢸⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢷⠀⠀⠀⠀⠀⠀⣸⠀⠘⠋⠁⠠⣇⣤⠴⠖⠛⠉⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠈⠳⣄⣀⣀⣀⣴⣃⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠉⠉⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢧⣤⠞⠁⠀⠻⣤⡀⠀⢀⡘⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠞⠙⠶⠀⠀⣾⠋⠁⣠⠏⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⡿⠀⠀⣿⠀⠀⢰⡏⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡶⢤⣀⡀⠀⠈⢹⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠈⣇⠀⣀⣀⣀⣠⣤⡤⠼⡇⠀⠈⠙⠓⠦⣿⣀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⣿⠉⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠶⠶⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣰⠇⠀⠀⢸⠀⠀⠀⠀⢠⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⡄⠀⠀⠀⢸⡄⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠉⠁⠀⠀⠀⠈⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠈⠛⠿⠿⠟⠛⠛⠛⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠻⠿⠛⠛⠛⠻⢿⣿⣿⠿⠿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣄⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⢫⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣁⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⣀⣀⣦⣤⣿⣦⡉⢻⣿⣿⣿⡿⠿⠙⠉⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣤⣾⣿⣿⣶⣾⣠⡀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠋⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⣀⠄⣄⣴⠏⠀⠀⢀⠀⠀⣠⢶⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣨⣴⡿⠛⢁⣀⣲⠭⠴⠖⠋⠀⠸⣡⡖⡄⠰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⡄⠀⡴⠋⠀⠉⠿⠛⠋⠁⠀⠈⠓⠲⠤⠀⠀⠙⢣⠃⡤⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣣⣤⣇⠑⠶⠶⠶⠖⠁⠀⠀⠀⠑⠲⠶⠶⠖⠃⡿⣸⣻⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣇⣸⠀⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠈⢧⣮⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠿⠛⢁⠼⠛⢋⣡⣴⣤⣴⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⣉⣛⠛⠿⢿⣿⡒⠻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠉⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢄⠀⠠⠴⠚⠩⠭⠙⠲⣄⡀⠀⣠⠞⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠓⠆⠈⠙⠀⠀⠙⣿⣿⣿⣿
⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠛⠩⠅⠀⣧⠉⢗⠦⣀⠀⠀⠀⠀⢀⡠⠔⠋⡇⠀⢀⠇⠠⠙⠻⣿⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⣿⡋⠁⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢸⡄⠘⣆⠀⠉⠒⠖⠚⠉⠀⠀⡼⠀⠀⡜⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿
⣿⣷⣀⣴⣶⣶⣤⣴⣾⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢣⡀⠘⢦⡀⠀⠀⠀⠀⣠⠞⠀⢀⠜⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⣀⡀⢀⣠⣀⠀⠀⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠉⠒⠒⠒⠋⠁⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⢃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠘⡄⠀⠀⣀⠄⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢸⠁⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡷⠀⠈⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⠐⠀⢸⡌⠐⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠌⣠⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣯⣟⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⢿⣽⣻⣟⣷⣿⣿⡿
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⢹⣽⣧⣿⣳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⡆⠀⠀⠀⢰⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠘⣿⣷⣯⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢰⡇⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠃⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢼⡇⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢇⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⡀⣀⡀⠀⠀⠀⣀⢀⣀⢀⣀⡀⢀⣀
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡀⠀⢀⣀⣠⣤⣤⣴⣾⣁⣀⣀
⠀⠀⠀⠀⠰⣶⣄⡀⠀⠀⠀⠀⠀⠘⠀⠀⣠⢊⣴⣾⢟⣏⣉⠙⠋⠻⠿⠿⠿⣿⣿⣖⠤⣀
⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄⠀⠀⠀⠀⣀⣀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣗⣢⣤⡸⠉⠛⣛⣿⣮⣕⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠖⠛⢻⡿⠋⣩⣥⣴⣶⣾⣶⣿⣿⣿⣿⣶⣶⡤⠄⠹⢿⣿⣷⡝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠛⠛⢉⣉⣛⣀⣈⡛⠙⡿⣿⣿⣿⣿⣿⣿⣶⣤⣐⠬⣿⣿⣷⣷⡀
⠀⠀⠀⣤⣤⣤⣶⡶⠶⠖⠀⢠⣾⠀⠴⠚⠛⠙⠛⠉⠉⠋⠙⣷⠷⢾⠀⠙⣿⠻⢿⣿⣿⣦⡔⢬⣿⣿⣿⣿⡄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡠⢣⡏⠠⠴⠶⠒⠛⠛⠉⠟⢛⣿⣗⠒⠋⢰⣶⠟⠀⢐⣿⣿⣿⣟⣀⣬⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠸⠀⣀⣠⣴⠶⢿⣿⡿⣿⣿⣏⡙⠷⠀⠀⠀⠀⢰⣿⣿⣏⠈⠙⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣿⠋⠉⠀⣀⠚⠋⢳⡈⠻⠟⠳⡶⠄⠀⠀⠠⣿⣟⠛⠀⠠⣤⣹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣰⣿⣷⠙⠶⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠧⠠⣠⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡞⠁⢀⢀⣶⣿⣿⣓⣒⡀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⣠⣾⠀⣴⣿⣷⡾⠿⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠙⢆⡈⢻⣿⣿⣯⣤⣤⣭⣑⣄⠀⡿⠀⠀⠀⢀⣤⡾⠟⠓⠀⠙⠛⠉⣾⣿⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣻⠷⠦⠤⠤⠄⠈⠿⠁⣡⠀⠀⢠⠞⠉⠀⠀⠀⣀⣤⢾⣟⠿⠟⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⡀⠀⠀⠀⠐⠒⠛⠁⠀⠀⠁⠀⠀⠀⣀⣴⡟⠻⠦⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣤⣀⣀⣤⠀⡀⣀⣀⣀⣀⣤⣤⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠈⠁⠀⠀⣀⡤⠶⠒⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣋⠉⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣀⡴⠖⠋⠁⠀⢀⣀⣀⣀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠉⠉⣀⡤⢹⡆⠠⣤⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣤⠞⠉⠀⠀⠰⠖⠛⠉⠉⠁⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠳⡌⠁⢀⣸⠀⠠⣤⣄⣀⠓⠀⠀⠀⢀⡈⢁⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠋⠁⠀⠀⠀⠀⠉⠓⢦⡀⠀⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣤⣄⠀⠀⠀⠀⠀⠙⢦⣴⠋⠀⣠⣤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠈⢳⡀⣀⣀⣤⣄⣌⣁⠀⡾⠁⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢸⠇⠉⠀⠀⠀⠀⠉⠘⣇⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣰⠟⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠙⣦⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⡼⠋⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡴⠋⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠙⠳⣍⠛⠶⠦⣤⣤⠤⠶⠚⣩⠾⠋⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣶⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴
⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡸⠛⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⢀⡄⢠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢁⣠⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⢓⠺⠟⠂⢀⣠⣶⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⠹⣃⣌⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⢠⠼⣵⡄⣿⣿⣿⣶⣽⣿⣿⣿⣿⡿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣧⣾⡿⣶⣤⣦⣴⣶⡤⣌⣿⣇⣟⣖⣸⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣽⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⡏⢿⣿⣿⡇⢿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢾⣿⣿⡇⢸⠇⣤⣿⣿⠟⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣾⣿⣿⣿⣿⠀⠁⠀⠀⠈⢩⣿⣇⣴⣼⣟⣿⡏⠀⢹⣿⣿⣿⣿⣿⣯⣿⣿⡿⢿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠴⠠⢼⣿⣿⣿⣿⣽⢿⡷⣿⣿⣿⣿⣿⣿⣿⣟⠉⠀⠀⡀⠁⠀⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⠈⣿⠁⣿⢿⣿⠃⢴⡖⠲⣤⣤⣬⣿⢿⣿⢻⣿⠃⢄⠙⠻⣿⣿⣿⣿⣿⣿⢀⠀⠀⣿⣿⠀⢀⣠⣘⢃⣼⣿⣿⣿⣿⣿
⡏⠻⠃⠀⠀⠀⠘⡘⣻⡄⢠⣤⣴⣾⣻⣿⣿⣷⣿⣾⠃⠀⣼⣧⣦⣄⣙⢿⣷⣻⣿⡇⠀⠀⣾⡿⣰⣨⣯⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠲⠀⠀⠋⠉⢻⡇⢸⣿⣿⣿⣿⢹⣿⣿⣿⣿⡀⠀⣹⣿⣿⣿⣿⣷⣯⣿⣿⡷⣀⣀⠿⣷⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠄⠀⠈⢿⠃⠐⠟⢿⣿⠇⠞⣿⣿⣿⣿⠘⣄⢹⣿⣿⣿⣿⠟⣟⢻⣿⣶⣿⠏⠀⣼⣿⣿⣿⣿⣿⠋⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣤⣶⣄⠈⠁⠘⢸⣿⣾⣿⣿⡅⣸⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣾⣿⡟⠁⠀⠀⢸⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠠⢤⣾⣾⣿⣿⣧⣴⣆⣤⣴⣿⣿⣿⣍⣽⡟⠛⠛⢛⣛⣿⣿⣧⣼⣿⡶⣶⣿⣿⣿⣿⣇⣤⣤⣤⣴⣾⣿⣤⣾
⠀⠀⡔⠀⠐⠀⣠⣾⣯⣿⠛⢡⠈⡙⣿⣷⣷⣾⣿⣭⣼⣿⣿⡙⣿⣿⣿⣭⣽⣿⣿⣿⣿⣻⡿⠟⠻⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⣗⠛⢻⠶⣿⣿⣿⣧⡀⢰⣾⣿⣾⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣟⡛⠖⠒⣤⣤⡤⠝⣭⣿⢻⡟⣛⣿
⠀⠀⢿⣿⣶⣴⣾⣿⣿⢙⣟⣾⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡤⠀⠀⠀⠉⣿⣿⣿⣿⣾⣷⣟⣛⠯⣤⣿⡭⠝⠉
⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠙⣿⣿⣿⣿⣿⡛⠋⠉⠁⠀⠀⠐⣂⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣯⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠉⠉⠁⠈⠀⠀⠀⠀⠀⢀⣠⡽⢧⣾⠃⣽⣇⣼⢸⣿⡿⢁⢻⢻⣿⣿⣿⡇
⠀⣿⣿⠯⠍⠻⠿⢿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠶⠶⠖⠀⠄⠉⠡⠞⠛⠻⡾⢿⣿⣧⣿⣿⢠⣿⠰⠘⣿⣻⣿⣿
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠤⢤⣤⣿⣤⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡹⠛⠻⣯⣼⣿⣠⣼⡏⣌⣿⢹
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠠⠬⣀⡀⠂⠀⠀⠈⢉⣁⣤⣤⠴⠶⠋⠁⠀⠀⠀⠀⣤⠿⢻⣿⣵⡿⠁⢨
⠀⠀⠀⠀⠠⠤⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠚⠋⠀⠀⠀⠀⠀⣿⠏⠉⠀⡐⠊⠠⠒⠀⠀⠤⡄⠀⠸⠇⠀⠜
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠈⠉⠁⠀⠀⠀⠘⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣼⣷⡤⠤⠀⠀⠀⠀⠀⣠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⣤⠙⡟⠁⠀⠀⠀⠈⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⢨⠃⠀⠈⠁⣇⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠷⠟⠀⠀⢣⡼⠟⠀⠀⠀⣠⣿⠋⠂⠁⠉⠃⠰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡦⠰⠤⣤⣤⣤⣤⣤⣠⣀⣀⡀⠀⠀⣴⡟⠓⠒⠒⠒⠶⠒⠺⢿
⠀⠀⠀⠀⠀⠀⠀⠂⢠⣼⠏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁⣠⣾⣿⣇⢒⠂⠒⠙⠂⢐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⣶⣿⣷⣿⣿⣿⣿⠋⠀⣴⣥⣿⣿⣿⣯⣽⣿⡉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠂⠒⠶⠖⠴⠀⠀⠚⠻⠿⠿⡿⠿⠃⢠⣞⣾⣿⠿⠏⠈⠿⡀⠙⢿⠄⠰⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠖⠒⠂⠠⢠⠤⠀⠀⠐⠀⠒⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⡙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢸⡇⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠸⠇⠀⠀⠈⢧⡀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠆⣿⠀⠈⠉⠙⠛⠶⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣄⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢸⡇⠀⠀⠀⢀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠈⣷⠀⢠⣟⣀⣀⣀⣀⣀⣘⣁⣀⣀⣀⣀⣈⣷⠀⠀⢸⡇⠀⣠⠞⠉⠉⠙⢦⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⣠⠏⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡥⢦⣜⢷⣤⣠⡴⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣴⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣽⠀⣀⡴⠶⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⠀⠀⣿⠞⠉⢀⣠⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢰⡏⠀⣼⠂⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠸⡇⠀⠀⣀⡴⠛⠁⠀⠀⠀⣿⠀⣀⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡾⠀⢠⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠻⠶⠞⠉⠀⠀⠀⠀⠀⠀⣿⠀⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠸⣇⣀⣼⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠁⣿⠀⠀⣴⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡾⢧⡀⠀⠹⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣸⠃⠘⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡟⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⡏⠀⠀⣿⠀⠀⢹⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣇⣀⣸⠇⢿⣀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⢀⡾⠁⠀⠀⢹⣄⢀⣸⠃⠀⠀⠀⠀⣿⠀⢀⣀⣤⡤⠶⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣆⣠⣤⠶⠞⠛⠛⠉⠉⠉⠛⠛⠲⢦⣄⡀
⠀⠀⠀⠀⢀⣴⡟⠋⠁⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⣤⡈⠙⢦⡀
⠀⠀⠀⠀⢩⣿⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⣀⠀⢀⠀⠈⢿⡄⠀⠹⣆
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⢀⠎⣡⢾⠁⠀⢨⢷⡄⢄⠀⠀⠀⠹⣆
⠀⠀⠀⢀⣿⣿⡀⠀⠀⡔⠀⠀⢜⣴⣿⠀⡇⠀⡌⢰⣿⣼⡄⠀⠀⠀⢻⡀
⠀⠀⠀⢸⣿⡄⠁⠀⣸⡇⠀⣠⡞⠿⠋⢰⠇⠀⢇⠈⢿⠿⣧⡀⠀⠀⠸⣧⡴⢦
⠀⠀⢀⣤⢿⡇⠀⢠⣻⠀⠀⡀⢧⠀⢀⡼⠀⠰⡾⡄⠀⢀⠏⠁⠀⡄⠀⡟⣴⢺⠇
⠀⠀⢾⠁⢄⠙⠆⠈⠋⢀⠀⠹⢦⣉⠋⠁⠀⠀⢻⡜⠛⣫⠞⠀⠀⣧⠀⣧⠻⡼
⠀⠀⠸⡾⠌⣧⠀⠀⠀⣼⠀⠘⠒⠀⠀⠘⡉⠀⠀⠙⢦⡐⠋⠀⠘⣿⠀⢹⣄⠇⠀⠀⠀⠚⣉⣡⣤⣄⡀
⠀⠀⠀⠹⣄⠻⠇⠀⢠⢻⠀⠀⠀⣰⠆⠘⠛⠛⠶⢤⣀⠙⠆⠀⠀⠉⠀⢸⡏⠀⠀⠀⠀⢈⣥⠴⠶⠦⣄
⠀⠀⠀⠀⠈⠳⣤⠆⠈⣽⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠛⢋⠀⠀⠀⠀⢸⡇⠀⠀⣠⠞⠉⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢸⡆⠈⠁⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣾⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣏⣠⡤⠴⠶⠶⠦⠤⣤⣀⠀⣿⠀⢀⣼⠏⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣗⠘⢷⡄⠀⠰⡍⠁⠀⠀⠘⠛⠛⠀⠀⠉⠓⠼⣠⣾⠏⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⣼⡇⠀⠈⠻⢦⣄⡐⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠻⠁⠀⠰⡟⢢⠈⢉⡹⠿⣶⣦⣤⣤⣴⡾⠟⠀⠀⠀⠀⠀⠀⠈⣳⢶⡀⠀⠀⣀⡴⠟⢁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠞⠋⣉⣠⣴⣴⣀⣠⣝⢍⡷⣤⠀⠀⠀⢰⡞⠳⣴⡷⢻⠏⠛⠋⢁⣠⡴⠋
⠀⠀⠀⠀⣠⠴⠶⢾⡇⠀⠀⠀⢹⠴⡿⢺⠷⣌⢩⡞⠀⣞⣧⡀⢰⣏⠳⡶⠎⠳⣾⡀⠀⣈⣩⡥
⠀⠀⠀⢸⡷⠚⠛⠋⠀⠀⠀⠀⠀⣨⠇⣸⠀⠈⠀⠀⠀⡉⠋⣻⣼⡏⠹⠃⠀⠀⠙⣿⠀⠀⠁
⠀⠀⠀⠸⠁⠀⠀⢀⣠⠶⠶⠖⣿⠁⢠⣿⣇⣀⠀⠀⠘⠳⡾⠁⠠⢷⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠰⢦⣤⠶⠛⠀⠀⠀⠀⣹⡶⣿⣿⣿⠭⠭⣭⣽⣿⡇⢸⠃⠀⠛⣦⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠒⠶⠶⣿⣄⣈⠛⠁⠀⠀⠀⠀⣀⣹⣼⣧⡤⠴⠿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣌⠉⠃⠀⠀⠀⠀⠀⠉⠁⢠⡆⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠙⣷⠚⠛⠛⠛⠛⠛⠓⢶⡿⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⣤⣤⣿⠙⣿⡟⠛⠛⢿⡷⠈⣇⣀⣀⣀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠾⠤⠽⣯⠀⣿⠁⠀⠀⢸⡇⠀⣹⠋⣯⡉⠁

⠀⠀⠀⠀⠀⠀⠈⠻⢤⣭⣤⠴⠲⠶⠟⠀⠈⠀⠀⠀⠈⠁⢸⣌⣉⠉⠙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠦⣤⣬⣽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠻⠿⣾⣶⣾⣿⣿⣿⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⡶⢟⣉⠉⠉⢻⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡫⠿⢞⣓⣾⣿⣿⡿⣿⣿⣿⣿⡗⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣖⣚⡙⠩⢍⣛⠻⠙⠊⠓⠁⠁⠑⣄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⢩⣤⣄⡙⠂⠉⠁⢶⣦⣄⣀⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⡟⠀⠠⠿⠿⠬⠟⠀⠀⠀⠀⠙⠟⠋⡛⢣⡀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠲⢶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢰⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢾⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠤⢺⡟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⣠⡶⠂⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⠒⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣋⣡⣤⠤⠤⠴⠒⠒⠒⠦⠄
⣀⣤⣴⣶⣾⣿⣿⣿⣗⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢟⡻⠟⠉
⠋⠉⠉⠈⠀⠀⠀⠉⠉⠛⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣡⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠛⠂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠁⠈⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠁⠀⠀⠀⠀⠨⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡀⠀⢠⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⣰⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠔⠛⠳⣶⣿⣿⣷⠀⠀⠀⠀⠀⣤⢄⡐⢶⣤⣾⣿⣿⣇⣼⡇⠀⠀⠀⠀⣠⢺⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢻⡟⠻⣿⡿⢿⡿⢿⡿⠛⠛⠛⣻⣿⣿⠆⠻⢿⣿⣦⡀⠀⡜⢱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⢸⣿⢆⠀⣀⣾⠀⠀⢣⣀⣴⡾⠛⠻⣿⣇⠀⠀⠉⣿⣿⡈⠁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⣠⣿⠃⠀⣹⠟⠉⣳⠶⠉⠨⣿⡁⠀⡀⠹⣿⣷⣤⣴⣿⣿⠉⠙⢻⣦⡀⠀⠀⠀⠀⠀⠀⡠⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠋⠀⢠⠇⠀⣼⡷⠀⠉⠘⣿⡅⠀⠘⠀⠈⠛⣯⣍⣽⢧⠀⣀⣼⡏⢳⡀⠀⠀⠐⢤⣜⣣⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢯⡿⣤⣄⡆⠀⢸⣧⣿⠀⠀⠀⢸⣿⣤⠁⠀⢠⠞⠋⢻⣻⣧⣴⣿⠟⠁⠈⠀⠀⠀⠀⣐⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⢛⠋⠀⠘⠛⣷⠀⠈⡏⣹⡷⠚⠷⣟⠟⢻⡀⡰⠂⠀⠀⠸⣿⣿⠋⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠻⣧
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⠀⠀⡠⠸⣿⡟⠿⢻⣿⣆⠀⠀⢰⠦⣾⣷⡖⠀⠀⠀⠀⣼⠿⣧⣀⣀⣀⣠⣤⠟⣿⣿⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⢠⣿⡟⠻⣆⠀⠀⠀⢀⡄⠰⣿⣀⠀⢉⣿⣯⡤⠀⣠⡟⣠⠟⠱⢀⣤⡤⢾⡟⠀⠄⠉⠋⠩⡝⣣⣾⣿⣿⣿⠁⠀⣾⣿
⠀⠀⠀⠀⢀⣿⠏⠀⠀⠈⢻⣦⣴⡟⠀⠀⠹⡧⣼⣿⡟⢿⣿⡿⠋⠁⠘⣦⠶⠫⠁⠀⢸⡇⠘⠀⣴⣷⣷⣾⠟⠉⣰⣿⣿⠄⢀⣿⣿⠂
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣿⡏⠀⠀⠟⣧⡀⢀⣼⠇⠀⠀⠀⠀⢸⡇⠀⠘⢋⣿⠋⠑⠖⠚⣹⡿⠙⠀⣸⣿⡟
⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣾⣿⣿⡇⠀⠀⠀⠀⣿⠀⡀⠀⠀⢸⣿⢌⡁⢈⡤⠀⣀⡴⠻⣧⠀⣼⠋⠀⠀⠀⣠⡾⠋⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⢸⣷⣤⠶⠛⠉⠉⢻⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡇⠀⠀⡞⠻⣧⡿⠀⠀⢿⣌⠁⠀⠀⢰⡿⠋⠀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⢻⠀⠀⠀⠀⢻⣅⠀⠀⣧⡴⠪⡟⠲⡄⠈⣿⠀⠀⢠⡟⠀⣠⡾⠛⠉⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⣰⣿⣿⡄⠀⠀⠀⠀⢀⣿⠿⢿⣆⠀⠀⢸⠀⠀⠀⠀⠀⠙⠛⣬⣿⣆⣀⣾⡀⢹⡄⠘⣆⠀⢸⠇⣼⣿⣶⣶⣶⣦⣤⣞⣛⣻⣿⣽⣿
⠀⠀⣼⣿⡿⠋⢻⡆⠀⢀⣤⠞⠁⠀⠀⠹⣧⡀⠀⡆⠀⠀⠀⠀⢀⣴⠟⠁⣿⠛⠉⢉⣿⡇⠀⠙⡄⠈⠀⣿⣿⡿⣶⣤⣾⡉⠙⠛⠛⠛⠛⠉
⢀⡼⢿⣿⠁⠀⡘⢧⣶⣿⠃⠀⠀⠀⠀⠀⠈⢿⣄⣿⠀⠀⠀⠀⣼⠛⠳⣄⡿⠀⠀⢸⣿⡇⠀⠀⠘⣆⣀⠘⣿⣀⡼⠋⠙⢯⣷⣄⠀⠀⠀⢀
⣼⠁⠀⡇⠀⠀⢀⡿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢧⠀⠀⢰⣿⠀⠀⠀⠉⠳⣤⣿⣿⣿⡀⠀⠀⢿⣿⢀⣿⣯⣴⣶⣶⣦⠉⠛⠿⠿⠿⠋
⡇⠀⠀⠃⠀⠀⢿⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⠀⠈⣿⡀⠀⠀⠀⢀⣿⠟⢙⢻⣷⣤⣄⣤⡿⠿⠿⡿⣟⠀⠈⣿
⠀⠀⠀⢸⠀⠀⠸⡇⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⠀⢿⡇⠀⠀⢀⣾⠏⠀⠀⠑⠿⠷⢿⣧⡄⠀⠀⢈⣿⠀⠀⠏
⠀⠀⠀⠀⠁⠀⠀⣽⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⣇⢀⣠⣿⡿⠀⠀⠀⠀⠀⣠⣾⣿⣁⠀⣀⣼⡇
⠀⠀⠀⠀⠘⣤⡾⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢻⡇⠀⠀⠀⠀⡼⠏⢀⣼⢏⣸⣿⠋
⠀⢀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡇⠻⣆⣀⣀⣤⣼⠷⠞⠋⠑⣦⡿⠁
⠺⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⣶⣶⠾⠛⠛⠛⠋⠉⠀⣷⠄⡀⢹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠿⠋⠑⠀⠀⠀⠀⠀⠀⠀⢠⠤⠐⠋⠀⠙⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠾⠿⡋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡤⣔⡉⠉⢉⣒⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡜⡠⠔⠛⠈⣙⠶⠄⠐⠚⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣏⣀⠤⠒⠉⢀⣠⢒⠘⣿⡶⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠤⠤⣼⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⠤⠐⢲⡉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⠠⠐⠈⢈⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡊⣤⠀⠀⢀⣼⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠁⠀⠀⠀⠀⠀⠀⣠⠤⣰⠂⢉⣉⣉⠁⢻⡃⣁⠊⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠆⠈⠁⡐⠋⠉⠩⠳⡀⡉⠈⠀⡧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢇⠙⣀⠷⣇⣇⡄⣄⣄⡷⢇⡜⢁⠋⠢⠈⢒⠢⣄⣀
⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠱⡉⠢⢠⡜⡷⡟⣤⢠⠋⡳⠁⠀⢰⠀⠸⠀⠘⠀⠑⠆⠤⡀
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠑⠲⢌⡖⢅⠕⣯⠴⠊⠀⠀⢀⠆⢠⠃⠀⠸⠀⠀⠀⠀⠀⠑⢄
⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢀⠎⢠⠃⠀⠀⠆⠀⠀⠀⡇⠀⠀⠀⠡⡀
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡐⠁⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⢡
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠊⠀⠠⠃⢠⠀⠀⢀⠃⠰⠀⠀⠀⠀⠀⠀⡆
⢃⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⢀⠔⠁⠀⡰⠁⢀⠃⠀⠀⡌⠀⠇⠀⠀⢰⠀⠀⠀⢰
⠀⠢⡀⠀⠀⠀⡀⠴⠚⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⡠⠒⠁⠀⢀⠊⠀⠠⠃⠀⠀⡐⠀⡌⠀⠀⠀⠨
⣤⣤⣬⣥⣬⣥⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣿⣶⣶⣶⣶⣷⣶⣶⣷⣶⣶⣾⣶⣶⣷⣶⣶⣶⣷⣶⣶⣶⣶
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⡿⠟⠋⠉⠉⠉⠉⠙⠻⣆⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠋⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣠⣀⢰⠀⢀⡀⠀⠀⠀⢸
⠀⡇⠀⣴⣿⠿⠋⠉⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟⢉⣤⣶⣾⣿⣷⣿⣾⣿⡅⡀⠀⠀⢸
⠀⡇⢰⠟⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⠋⢰⡋⠙⣻⡿⡇⠈⣻⡿⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠰⣤⣾⣭⣧⣤⣤⣍⣀⣠⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠁⠤⠤⠖⠀⢀⡿⠉⠀⠀⠀⢸
⠀⡇⠀⠀⠘⢾⣶⣿⠟⠋⡟⠛⢻⣿⡟⠳⡆⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠂⠀⣤⠄⠲⠖⠲⠖⠀⢀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠢⠿⣿⠀⠀⠓⢄⣉⠥⠁⡀⠙⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠐⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠈⠉⢷⠦⠶⡤⠄⠂⠀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠑⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡇⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣶⠶⣦⣄⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢶⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣶⣥⣀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⡀⢻⣿⣿⣿⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⢟⡿⠉⢿⡀⢹⡇⠈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡿⢋⡞⠀⠀⢸⣧⠀⠇⠸⣿⣿⣿⣿⣿⣿⣿⣶⡆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣹⡏⢰⡞⠀⣰⡇⠈⠿⠀⠂⢀⣿⡿⣿⡏⢾⣿⣦⣦⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡬⠙⢻⣿⣿⣿⣯⣿⣿⣿⣾⣿⣿⠃⢸⠀⢰⣿⠁⠀⠀⠀⢀⠞⣿⣿⣟⡹⣿⡄⢀⡙⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠳⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⠿⠀⠈⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⠟⠉⠛⠟⠙⠆
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢉⢐⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢰⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠓⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣦⡊⠀⠀⠀⢹⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣼⣿⣿⣿⡧⠀⠀⠀⠀⢻⣿⣿⣷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⠙⠛⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢀⠘⢿⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣹⣿⣿⣿⠃⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣓⣦⣼⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠛⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠙⡦⠄
⡞⠁⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠤⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠁⠀⠀⠀⠀⢀⠊⡠⠖⠚⠉⠉⣉⠂⠀⡥⢤⡤⠤⠤⡄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡭
⡂⣷⠤⠀⠀⣤⠘⡥⠀⠂⣥⠖⠁⡠⢾⣤⠂⢂⠄⢀⠈⣄⣶⢰⡟⢸⠁⠀⠄⠀⠀⠀⣠⢠⡀⠀⠂⠀⠀⠂⡈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰
⠛⠀⠀⠀⢀⣿⢢⠑⢀⡴⠋⣠⠊⠠⣄⠈⠦⢓⠀⣀⣿⠸⣿⢸⡇⣼⢾⣅⣀⠠⠊⠀⠀⠀⠑⠀⠀⢈⡉⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⣼⠁⣼⣷⠎⠠⡾⢏⠂⠀⡍⠀⣞⠶⠎⠀⠈⢀⠛⠘⢫⣹⣿⣲⢗⠾⠿⣿⡦⠀⠘⠀⣇⠈⠀⠀⠀⡘⢠⠀⠀⠀⠂⠘⠙⠎⠑⣦
⢀⣦⡀⢰⣿⢸⣿⡇⢠⠁⢱⡦⢉⠂⡍⠉⠁⡀⠀⠀⠀⡀⢥⠰⠙⠓⠘⠗⢩⣿⠿⠿⡇⢤⡠⡄⢸⣕⡋⢤⣡⠄⠈⡌⣀⠀⠀⠲⢀⠀⢸⣟⣷
⡉⠀⠀⢸⣿⡜⣿⠀⡆⠀⣄⣹⡾⠍⠀⠀⢀⣴⠝⠈⡐⢏⡌⠄⠸⠇⣴⢳⣤⠯⡄⣄⠚⡆⠳⣀⠋⡳⣴⠆⠑⠓⠀⠀⡭⠐⠂⠀⡈⠀⢀⠹⣾⡇
⠧⠄⠀⠈⢿⣷⡘⡆⠘⢄⠘⠁⠀⠀⢀⣴⢩⣱⣦⠀⠀⢄⠁⢄⠀⠀⡙⢮⣏⢑⠻⡃⢚⢡⡆⠻⠆⢹⣝⣿⣷⠟⡀⠠⠀⠀⠀⠀⠀⣠⣾⣷⣿⢻⡀
⡀⠀⠀⠀⠀⠙⢿⣬⣦⣀⠉⠁⢀⠀⢮⣶⣿⣿⣿⣦⠐⡈⢲⣦⡑⡀⡘⠉⠻⡬⠷⣀⣿⠇⣿⣿⣣⢘⡮⢈⡟⢎⢀⠀⠀⠀⢀⠀⡐⢿⣏⠂⣽⣿⠇
⠷⠀⠀⠀⠀⠀⣸⣿⡏⣭⠀⠀⡪⠶⢀⡉⠁⠋⠟⠟⢡⢨⠮⣧⣷⠋⠀⣾⠛⡆⣿⢮⣵⠀⢿⣌⣉⠀⢐⠟⢘⢄⡼⢀⣾⣿⡏⠀⠌⣠⣿⣤⣿⠃⣆
⠀⠀⠀⠀⠀⣰⣿⠛⠋⢓⡆⠀⠀⡷⢆⠉⢉⠛⡔⠒⠴⠟⠋⢠⠃⠀⣾⣫⣶⡧⣝⢶⣎⣄⢽⣟⣿⠀⢘⠓⣰⠃⠀⠘⠋⠉⠀⣡⡆⠈⡷⣸⡏⠀⠿
⠀⠀⠀⠀⢰⣿⣿⣆⠘⠿⢷⠀⠀⣏⣀⡈⠙⡳⠼⢷⣶⣯⠀⠎⠀⠀⣿⣽⣞⣎⠁⣐⢹⡄⣍⠁⢿⡆⢩⡜⢪⡀⢘⠆⠀⠀⣰⠿⠇⣌⢳⣟⠁
⠀⠀⣠⠀⠀⣿⣿⣿⡆⠀⢸⣇⢠⣬⣩⣵⣒⡀⠙⠛⠍⣿⡿⠀⠀⠘⢤⣿⣿⡇⢣⡉⠸⠇⢹⡗⢿⡷⠈⠖⠉⣠⣇⠴⢄⣴⣯⠴⢦⠤⠘⠈⠁
⣁⣴⠏⠀⠀⠹⢿⣿⣿⡄⠈⠋⣿⣇⣱⣥⡛⣥⠀⡗⣀⣩⣦⣼⣷⡦⢌⡻⣿⣵⢾⠉⠛⠶⢤⣤⣤⣤⡴⢶⠛⠁⠉⠳⣤⣈⣁⣀⣤⡔⠁
⣿⣃⣠⡾⢁⣼⠃⠈⣻⣷⢀⠀⠉⠁⠀⠀⣿⠦⣍⠛⠻⠯⠀⠚⠋⢰⣟⣛⡻⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠒⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀
⣎⡿⣿⢛⣾⣃⣃⢠⡟⣽⣧⣁⠩⠔⢚⣭⣶⡿⠛⢿⣶⣤⣴⡶⠗⠁⠀⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⣿⣿⣿⣿⠃⢉⢱⡿⠳⠿⣿⣷⣾⣿⠟⠋⣁⠠⠐⠁
⣿⣿⣿⣷⣵⢾⠬⣧⣀⣀⣹⣿⡭⠴⠚⢋⡡⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣞
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣀⣠⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣽⣿⣯⠛⣻⣿⣷

⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠐⣶⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠉⠙⠻⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣸⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠇⡠⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣷⣶⣶⣦⡀⠀⠀⢹⣿⣿⣿⣟⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠁⠀⠻⠿⠛⠛⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣼⣿⣷⡶⢶⣆⢀⣀⠀⠀⠀⡼⢿⣽⡿⠡⠀⣄⣀⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣻⣽⣿⣻⣿⣿⣿⣿⣿⣿⣿⡻⠶⡟⠃⠀⠀⠀⡸⣷⠟⠻⡷⠀⠈⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣟⡋⠁⠊⢡⣰⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⡆⣠⣿⡷⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣙⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⠛⠿⠛⠻⠟⢉⣀⠀⣤⡾⠛⢉⡽⠟⠛⠉⠙⠛⠿⢿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠠⠶⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣶⣴⣦⣾⠯⣭⣵⣿⣫⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣭⣿⣿⣿⣿⣿⣿
⢠⣼⣿⣿⣿⣿⣿⣧⡄⣀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿
⣾⣿⣿⣿⣿⣿⣿⡟⠓⠢⠀⠩⠥⠀⠀⠄⠠⠀⠀⠀⢀⡀⣀⢀⣀⣀⣀⡀⢀⡀⠀⠀⠀⠀⡀⠀⡾⠛⠁⠙⣉⣨⣷⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣀⡀⢦⣀⠤⠀⠀⠀⠀⠀⠙⠂⢺⠃⠀⠈⠁⠀⢀⠀⠠⠀⣀⢴⠀⠒⠁⡄⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⣀⣀⡀⣈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣀⠋⢿⠉⣿⠶⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⢿⢿⣿⣯⡹⡟⠞⠊⠉⠙⠉⠟⠍⠉⠀⠀⠀⠀⠀⣿⠀⠛⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣭⣿⡿⠟⢛⣉⣧⣿⣿⣿⡟⢉⠀⣀⠀⠀⠙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⡟⣰⠏⡴⠁⡔⢀⣴⠿⠃⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⡀⠀⠀⠀⣾⡀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣇⣾⣡⣾⣡⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠙⠗⢤⢠⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠸⣾⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢉⣄⣤⣤⡀⢄⣸⣿⣦⣄⣰⣿⣿⣤⣀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡁⠈⠁⠀⠀⢉⣹⣽⣁⣝⢻⣩⡅⣸⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠙⠛⣛⣛⣻⣛⣛⣛⣿⣿⣿⣿⣿⣟⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⠉⠉⠀⠀⣈⣉⣻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠋⠀⣿⣿
⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⣿⣧⡼⠛⠀⠀⣰⣿⣿
⣿⣿⣿⣿⣿⣺⢿⣿⣿⣿⣿⣼⡁⠘⣍⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣫⣿⣿⣿⡇⣽⣿⣷⡆⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠆⢿⣿⣿⣿⣿⠛⠀⠙⡇⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣛⣻⢛⣿⣿
⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣭⣥⣠⡟⠀⠀⠵⠾⠿⠿
⣿⣿⣍⣭⣩⣍⡉⠉⠿⠿⠿⠟⠳⢤⠿⡄⠀⠀⠀⠉⠈⠉⢭⣍⣉⣉⣉⠉⠉⠀⠀⠈⠉⠛⣿⣿⡿⠿⢿⡏⠙⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⢀⣴⣾⣿⢭⣓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣭⣽⣿⣿⣷⣶⣶⣿⣯⣭⣍⣙⠺⣷⣝⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⠛⠛⠉⠉⠀⢠⣾⠋⠀⠀⠀⠙⢿⡄⠉⠉⠛⢷⣮⡻⢟⡺⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⠿⢸⡇⠀⠀⠀⠀⠙⢿⣆⠈⢮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠈⢻⣆⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣭⣛⠛⠛⠋⠀⢀⣦⡀⠀⠀⠀⠀⠈⣿⡘⡏⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠂⢀⡞⠀⢿⠀⠀⠀⠀⠀⢻⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠿⠀⠀⠀⠀⢀⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠛⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠶⣶⣶⣶⡶⠶⠶⠾⢿⣿⠛⠛⠉⠉⠁⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⡤⠶⠛⣋⡉⠉⠛⠳⣦⡀⠀⠀⠀⠸⣷⣶⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠠⠞⠉⠙⢦⠀⠀⢸⣷⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢷⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢹⡄⠈⠉⣽⢻⠙⠛⢛⡿⡟⠛⠛⢻⡟⠉⠉⠉⠉⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠈⢷⣀⡼⠉⣆⣧⣠⣼⡴⢳⡀⢠⡞⠁⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠈⠛⠁⠘⢧⡀⣠⠴⠃⠈⢷⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⢇⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⢆⡽⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⡀⠀⠀⠀⠀⠀⢀⣏⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡟⠛⠛⡛⠛⠛⢻⣿⠷⢦⡤⠾⠿⠾⠾⠿⠿⠿⢿⣿⠿⣿⡿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⣠⣤⠶⠒⠚⠛⠒⠒⠒⠶⢤⣄⡀⠀⠀⠀⠸⣧⣀⣰⣿⡿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⡇⠀⢴⣷⣶
⠀⠀⠀⠀⠘⠿⢧⣀⣀⠀⠀⠀⢀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠈⣉⣉⣉⣀⣀⣸⣿⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣬⣿⡀⠈⠛⠛⠛⠛⠁
⠀⠀⣀⣤⠤⠤⠤⠤⠄⣉⡩⠉⠁⠈⠉⠁⠐⠒⣲⡿⠂⣴⡋⠉⠁⠀⠉⠉⣩⡙⢶⡀⠀⢠⡞⠉⠉⠀⠀⢀⣀⡈⢳⡄
⣴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁⠀⠀⠈⠻⠶⠶⠶⠞⠛⠙⠷⠾⠃⠀⠈⠛⠶⠶⠷⠛⠋⠉⠷⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣛⣭⣭⣭⣛⡛⠿⠿⠿⠿⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣼⣿⣿⠿⢿⣿⣿⡿⠿⣿⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠲⢄⣿⣿⡸⠆⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣆⢿⣶⣭⣭⣭⣭⣶⡿⣸⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣮⡻⢿⣿⣿⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢹⣿⡿⠿⣛⣂⢿⣿⣿⣿⣿⣿⣿⡿⣑⣛⠿⢿⣿⡏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢡⣾⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣷⡎⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⢿⣿⣿⣿⣿⡜⣿⣿⣿⣿⢣⣿⣿⣿⣿⡿⢱⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⡿⠿⠿⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⡀⠨⠭⠀⠀⢀⡀⠀⠀⠀⠀⠈⠉⡉⡉⠉⠁⠩⠅⣀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡏⠀⢨⠀⠔⠈⠀⡀⠚⠛⠁⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠢⠀⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠊⠀⣠⠨⠐⠀⢁⣶⣤⠀⠀⣤⣶⡈⠀⠀⠀⣄⠀⠑⠂⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠴⠦⡍⠽⠀⠀⠯⢩⠴⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⢾⣷⡄⢠⣿⡷⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢁⠝⠀⠀⠫⡈⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠠⣀⠀⠀⣀⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠄



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠒⠒⠒⠒⠲⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡠⢦⣠⠤⡀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠤⠤⣀⣠⢷⡀⠹⡄⠱⡄⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠃⠀⠀⠀⣹⠀⠱⡀⠱⡀⢹⢡⢾⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢻⡎⠀⠀⠀⢸⠈⢣⠀⢣⠀⢇⠈⣧⠘⣤⠗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢺⡇⠀⠀⠀⣈⣆⠀⢧⠘⠂⠀⠀⠛⠀⡿⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡇⠀⠐⠢⢄⣸⠀⠘⠀⠀⠀⠀⠀⢠⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣵⢢⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡀⠉⠂⠀⠀⡇⠀⠀⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠑⣄⠀⠀⠀⠱⣄⠀⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡏⠱⣄⠈⠩⠭⠵⡌⠁⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⢧⠀⠀⠀⠀⠀⠈⢯⠉⠒⠲⠤⢄⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠘⡄⢀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢣⠀⠣⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⠁⠈⠐⢤⠀⠂⠐⠀⠠⢀⠀⠀⠀⠑⢄⠀⠊⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡁⠠⣘⢁⠀⠀⠱⢀⠀⠀⠀⠑⠀⠀⠀⠀⢱⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠒⠷⠃⠈⠉⠀⠆⠐⠛⠦⠡⠀⠀⢣⠀⠀⠀⢀⡞⠀⠀⠀⠁⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡰⠁⠠⡀⢀⡰⠃⠀⠀⠀⡀⢀⡀⠀⢀⣠⠴⠯⠔⠒⢤⠀⢑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠛⠭⣉⠉⠀⠈⠂⠀⠒⣤⠀⠀⠀⠁⠀⠀⠈⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢇⠀⠀⣼⣀⢀⠄⠀⠂⡉⠜⠀⠀⠀⠃⠈⡠⠔⠉⠀⣀⣀
⠀⠀⠀⠀⠀⣠⠤⣀⠀⠸⡀⠈⢇⠀⠀⢀⡈⠧⡀⠈⠂⠀⠂⠀⠀⠈⢠⠀⢀⡴⢏⠁⠀⠀⠀⡜⠀⢠⠁⠀⣠⠤⡀
⠀⠀⠀⠀⠀⢣⡀⠈⠑⢤⣣⠀⠸⡄⠀⡤⠢⣔⢻⣦⡀⠠⠈⠃⠀⠋⣡⣔⣅⠒⡨⠷⢄⠀⣰⠀⠀⣎⡴⠊⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠉⠀⠀⢘⡾⠀⢠⠏⠉⠁⢈⠖⠦⠴⠖⠉⠁⢳⠈⠉⢳⡀⠈⢶⡁⠀⠀⠁⠀⡠⠞⠁
⠀⠀⠀⠀⢸⡉⠉⠙⠋⠀⠀⠀⠀⠁⠀⠀⣞⡠⠴⠒⢻⠀⠀⠀⠀⠁⠀⠈⣷⠦⣄⣧⠀⠀⠁⠀⠀⠀⠈⠓⠋⠉⢉⠆
⠀⠀⠀⠀⠀⠉⠑⠒⠒⢦⡀⠀⢀⡤⠴⠊⠁⠀⠀⠀⢳⢄⠀⠀⢀⡀⠀⣠⡇⠑⢄⠈⠓⠤⢄⠀⠀⢀⠴⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⡏⠀⠑⠴⠁⠑⠊⠀⡇⠀⠀⠑⢄⠀⠈⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢲⢲⠂⠀⠀⡇⠀⢂⠀⠀⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠁⠀⠀⣾⢸⠀⠀⠀⢷⣠⠤⠷⢄⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠀⠀⠀⠀⠀⢻⢸⠂⠂⠀⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠋⠘⠒⠒⠲⠶⠶⠶⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠂⠉⠉⠉⠉⠉⠉⠐⠒⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠎⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠉⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⡘⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⣰⣶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⣇
⡇⠀⠀⠀⠀⡰⠃⠀⢀⠊⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠱⠈⠙⠢⣀⠀⠀⠀⠀⠉
⡇⠀⠀⠔⠊⠀⠀⠀⠈⠄⠈⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠈⠀⡰⠀⠀⠀⠈⠑⢢⠀⠀⢳⠄⠄⠤⠤⠤
⢓⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠁⠀⡿⠀⠀⠀⠈
⠘⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠐⠀⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⡇
⠀⢇⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡼
⠀⠘⡄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⠃
⠀⠀⠱⡄⠀⠀⠀⠙⢄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⡰⠃⠀⠀⠀⢠⠃
⠀⠀⠀⠹⡄⠀⠀⠀⠈⠳⣄⠀⠡⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠄⠌⠀⡠⠞⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠙⡉⠐⠒⣲⠀⢠⡖⠒⠂⢉⠉⠀⢀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⣹⠻⠯⡍⠉⠀⢀⠤⠚⠀⠈⠗⢤⡀⠈⠉⡩⠭⠛⣍⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⡘⢢⡓⠂⠴⠃⣄⠀⢊⠃⠀⣆⠙⠤⠐⢳⡐⢣⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠑⠒⠊⠀⠇⠀⠀⢸⠀⠑⠒⠈⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠒⠒⠒⠚⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⢀⡞⠀⠀⠀⠀⠀⠀⢣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢤⠤⠤⠤⠤⠤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣴⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⢠⠋⠛⠁⠘⠉⠀⢙⠦⠦⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⣠⠒⠚⠓⠒⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠘⡄⠀⢀⢘⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠧⠘⠁⠉⠀⡠⢃⠎⢨⠓⣦⡀⠀⠀⠀⣀⠖⢶⠒⡤⡸⢤⠊⠑⡆⢠⠏
⠀⠀⠀⠀⠀⡠⠘⡤⢤⠊⠑⠊⡠⠊⣠⠃⠀⠀⠀⠀⠈⣆⠸⡀⠃⠘⣄⠤⣠⠃⢿
⠀⠀⠀⠀⠀⡇⡐⠱⠇⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⢘⡆⠁⠀⠀⠘⣴⢃⠇⡯⣄
⠀⢀⡴⠚⠛⢣⡰⢂⠆⠀⠀⠀⡜⠁⠀⠀⡄⠀⢀⡆⠀⠄⢱⠀⠀⠀⠀⢩⠈⡏⠀⠈⠳⡄
⠀⡞⠀⠀⠀⢤⡗⡏⠀⠀⠀⣠⠃⠈⠀⠀⠈⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⣴⢴⡗⠂⠀⠀⣷
⠀⡇⠀⠀⠀⠞⠀⡇⠀⡰⠊⡹⠀⣀⠀⠀⠀⠉⠀⠁⠀⠀⢹⠦⡀⠀⢀⠇⠀⠈⠂⠀⢀⡇
⠀⠘⣄⠀⠀⠀⠀⢱⡞⠀⢠⠷⠤⠴⠤⣀⣀⠀⠀⣀⡠⠔⠋⢣⡈⢢⠎⠀⠀⠀⠀⠀⣾
⠀⠀⠹⡀⠀⠀⠀⠀⢳⣤⡏⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠔⠛⠧⣄⣀⠀⠀⠀⢀⠏
⠀⠀⠀⠈⠓⠒⠒⠈⠉⠀⣱⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠊⠈⡆⢳⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⢤⣀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⣾⡠⠤⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠐⠠⣀⣀⣀⡅⠀⠀⠀⠀⠀⠀⠀⠜⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⡤⠖⠋⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣀⣠⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠉⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡼⠉⠢⢄⣀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣠⠜⠀⠀⠀⠀⠀⠩⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢲⠋⠀⠰⡀⠀⠊⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡲⡞⡶⣶⢢⢤⠀⠀⠀⠀⠀⢏⠁⠀⢧⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠹⠌⠃⠁⠀⠀⠈⢸⠀⠀⠀⠀⠀⠀⠢⡀⠈⢁⡀⠀⠀⡸⠱
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡒⢄⠴⠀⢆⣀⣀⣠⢤⣄⣀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⣀⠜⠀⠀⠱⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠦⠳⠈⠁⣇⠀⠀⠀⢙⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⡠⠁⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⢔⡛⢓⣀⢀⣎⢹⣷⡀⢀⢎⣜⣰⠷⢀⣀⣀⠀⠀⠀⠀⠀⡠⠔⠂⠈⠑⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢹⡀⢪⣷⢡⡾⠛⠁⠀⢀⣬⡡⣌⠑⠤⡀⠎⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣒⠷⡀⠉⠖⠰⠋⠀⣠⣾⣏⡿⠬⠍⠠⢭⣶⣖⣼⣯⣁⠒⢌⠙⠛⠬⠒⠂⠀⠀⢀⡀⠀⠀⠀⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠢⣅⡢⣿⡐⣇⡆⢰⡏⠟⣿⣅⠀⠀⠀⠀⠈⢻⣈⠉⠁⠀⠀⠱⢄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣀⠀⣿⠀⢠⡑⢌⠃⣼⡸⢀⣿⣟⣆⣚⠃⡾⠀⠀⣿⠄⠀⠀⠀⣠⠊⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡹⢑⢼⣇⣿⣇⣀⣧⢸⣧⢊⢻⣧⣿⣷⡿⠁⠀⠀⠉⠀⠉⠉⠉⡇⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡶⠷⣿⣿⡏⠀⠉⣆⡏⣈⣼⣍⣸⠯⢟⡆⠀⢻⠦⡀⠀⠀⠀⢡⠀⠀⠀⠑⣄⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡑⠒⠉⠩⠭⠭⠭⠭⠭⠉⠐⠒⠒⠂⠉⠀⡁⠀⠀⠑⠸⠀⠀⠀⠈⡄⠀⠀⠀⠈⢆⠀⠁⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠤⢤⣀⠀⠀⠀⠀⢀⣤⠴⠖⠂⢰⣇⡄⠀⣀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⢸⠏⡅⠀⡿⡀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠙⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣹⠀⠀⠉⠉⠀⠀⢸⠀⠀⠀⠉⢉⡁⠀⢸⡰⠀⠀⢥⠃⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡟⠇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠻⠖⠂⠀⠈⠛⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠰⠶⠶⠶⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠔⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠻⢿⠛⡟⠻⡿⢛⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠛⠋⠙⠙⠛⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠴⠞⠐⠨⣝⠙⠲⠤⣄⣠⠤⡀⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠃⠀⠀⠀⠀⠀⢣⡐⠂⠉⠁⠢⠘⢦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣰⢛⠀⠀⠀⠀⠐⠂⢸⡂⠒⠀⠀⠀⡇⠈⢳⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡽⠁⠈⠳⣄⡀⠀⣀⠴⠏⠷⣤⣀⡤⠞⠁⠀⠀⠹⣝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠉⢩⠃⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⢀⣀⣰⣋⣳⡀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣰⣋⣳⣀⣀
⣸⠥⠄⡍⠀⠀⠙⡆⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡞⠁⠀⠈⡅⠤⢽
⢳⠖⠒⠊⠁⠁⠀⣿⡤⠚⠙⠚⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠋⠙⠢⣼⡇⠀⠁⠉⠲⠰⢶
⠘⠒⠒⠲⠤⣤⣾⠟⠀⠀⢀⠀⠀⠏⠓⢦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢤⠖⠋⡇⠀⢀⠀⠀⠘⢾⣦⡤⠴⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠑⠦⠤⠖⢿⠀⠀⠀⠀⠘⠒⠤⢉⣉⠝⠯⣙⠉⢉⣩⠟⢏⣉⠡⠔⠚⠀⠀⠀⠀⢸⡓⠤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⠀⠤⠤⠀⠐⠈⠻⠋⠑⠀⠀⠤⠄⢀⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠤⣄⣀⡈⠀⠐⠂⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠒⠀⠈⣀⣀⡤⠄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡉⠉⠐⠒⠒⠒⠒⠂⠘⠐⠒⠒⠒⠒⠒⠈⠉⡁⠀⠄⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠉⠀⠀⠀⠀⠐⠂⠒⠀⠀⠀⠀⠈⠁⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠓⠲⠦⠤⣤⣤⣄⣤⣤⡤⠤⠶⠒⠋⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠉⠉⠉⢹⡋⠀⠀⠀⠈⣻⠉⠉⠉⠙⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠉⠉⠉⢹⡂⠀⠀⠀⠀⣺⠉⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣴⣾⣶⣿⣾⠀⠀⠀⠀⠀⢸⣿⣷⣷⣶⣤⣤⣤⣤⣵
⣶⣶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠉⠉⠉⠑⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣀⣀⣀⠐⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⠲⢅⠀⠙⠢⡀⠀⠀⢀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡳⣄⠈⠑⢦⡀⠈⠲⣄⢁⡤⢤⠀⠀⠻⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⢸⡄⠈⠓⢄⡀⠙⠂⠀⠈⢿⠀⡏⠀⢠⡶⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡟⠦⣄⠀⠉⠀⠈⠀⠀⠀⠀⡇⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡼⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⢹⣦⡈⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠀⠀⠀⢠⠇⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡇⠙⠻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠡⣤⣤⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡄⠈⢳⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠳⣄⠀⠀⠀⠀⠀⠀⢿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⢀⡠⠔⢒⣉⣀⢀⣀⣀⢀⣀⠤⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄⠀⠀⠀⠀⢘⠷⡄⠀⠈⠉⠛⠛⡿⣿⣿⣿⣶⡄
⠀⠀⠉⢢⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⠤⠚⠁⠀⢹⠀⠀⠀⡰⠋⠀⠈⢻⣿⣿⡇
⠀⠀⢀⡀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡯⠞⠊⠁⠀⠀⠀⠀⠈⡇⢀⠎⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠙⣄⢣⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠸⡎⠀⠀⠀⢀⣀⡀⠀⠘⠇
⠀⠀⠀⠀⠈⢆⡇⠀⠀⢀⠴⠒⠉⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⣧⣴⣶⠿⠛⠓⠒
⠀⠀⠀⠀⠀⠘⡇⡠⠞⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠑⢆⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠞⠛⠛⠛⠛⠛⠶⠶⣤⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣼⠋⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠋⣉⣽⠟
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⢰⡏⠀⠀⡇⠀⢸⠀⠀⠀⢀⠀⠀⢰⡀⠀⠀⢳⡀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⣿⠀⠀⣰⡇⠀⢸⡆⠀⠀⢸⠀⠀⠀⢧⠀⠀⠈⢷⡀⠀⠀⠀⢳⡀⠀⠀⠀⢀⣧⣀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⣿⠀⢰⠁⣧⠀⠈⡇⠀⠀⠸⡇⠀⠀⠘⣆⠀⠀⠀⢳⡀⠀⠀⠀⢷⡀⠀⣴⣭⣤⡜
⠀⢾⣉⣙⣿⠀⡏⠀⠹⣄⠀⣷⠀⠀⠀⢻⡀⠀⠀⠹⡄⠀⠀⠈⣷⡀⠀⠀⢸⡷⣄⠀⠀⣿⢡
⠀⡀⠀⠀⣿⣄⡃⠀⠀⢈⣉⠁⢧⡀⠀⠀⡷⣄⠀⠀⢸⠦⣄⣠⠇⠙⠦⣤⡼⠃⠈⠓⠶⠋⢸⠀⠀⠀⠀⠀⢀
⢯⡤⠟⠀⢺⣄⡁⢀⡞⢩⠭⣍⠲⣝⠲⠖⠃⠈⠙⠚⠋⠀⠀⢀⡞⢉⣽⣟⠢⣄⠀⡀⠀⢀⡏⢀⡴⠴⢦⡀⣼
⠀⠀⠀⠀⠈⣿⠀⠀⠑⡞⠉⢉⠖⠋⠀⠀⢀⡤⠖⠢⢤⡀⠀⠘⠲⣏⠉⠙⡶⠞⠀⠀⠀⠈⠛⠋⡖⢀⡼⢁⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣗⠀⠈⣆⠀⠀⢠⠏⢀⣤⠤⣀⣱⡀⠀⠀⣸⠀⠀⢣⡀⠀⠀⢠⡤⣀⡴⠛⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠹⣆⠀⠀⡏⠀⠠⡇⠀⠀⣾⠀⠙⢲⠦⠀⠈⢷⠀⠀⣸⠂⠀⣞⣡⢤⡀⢀⣧⠼⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡟⠀⠀⣹⠀⠀⠻⠶⠴⠯⣤⠶⠦⡼⠀⠀⢹⠀⢠⠏⠀⠀⡷⠋⠀⠀⡀⠀⠐⣆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠙⠳⢦⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡴⠛⢦⣤⣨⣱⣦⣤⣴⠷⣤⣤⠟⠷⠞
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡼⢷⣤⣦⣩⡙⣛⡷⠲⣖⠒⠂⢀⡤⠞⢙⡟⠁⠀⠀⢰⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠛⢦⣀⠈⣹⠶⣏⣀⡴⠛⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⢀⡀⠀⠈⠙⠁⠀⠈⠉⠁⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣧⠀⣠⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡤⠾⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⢹⡟⠛⠛⠛⠛⠛⠻⣏⠉⠁⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠸⣇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣧⣾⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣳⣤⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⠀⠰⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣛⣛⣛⣛⣛⣛⣛⣿⠀⠀⠀⢠⠀⠐⠂⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡏⣩⡍⢹⣟⣛⣛⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⣇⣉⣁⣼⣯⣭⣭⣭⣿⠀⠀⠀⣠⣶⣾⣷⣦⣄⠀⠀⠘⠳⠦⠀⢸⣿⣷⣶⡄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆⠀⠛⢛⣓⠀⢸⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⣿⣿⣿⣿⣿⣿⠇⠀⠘⠋⠁⠀⠸⠛⠋⠉⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡙⠛⣡⣿⣿⣧⣀⠀⠀⠀⢠⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣡⣶⣙⣿⣿⡏⠁⠀⠀⣴⣿⣿⡿⠋⣾⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⡟⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠛⠿⣿⣿⣿⡆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠸⣿⡿⠿⠿⠿⠛⢛⣃⠀⠀⠀⠀⠈⠙⠐⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢟⣹⣿⣿⣿⢷⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠉⠉⠉⠛⢷⣮⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢠⣆⣠⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣿⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢻⣇⠀⠀⠀⢠⡟⣿⠃⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣿⣷⣄⡀⠈⠁⠀⣀⣴⡿⣿⣿⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣴⡿⠋⢻⡆⢿⣿⣿⠿⠛⠛⠉⣼⡿⢸⣿⠀⠀⠀⣋⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⢸⣇⠀⠉⠙⣧⡀⠀⣰⡿⠁⢸⣿⠀⡴⠟⣁⡀⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠻⣶⡿⠁⠀⢸⣿⠀⢈⣛⣉⠁⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣀⣀⣤⣤⣤⣼⣿⠀⠈⠉⠉⢡⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⣾⣿⠟⠛⠋⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠸⣿⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣿⠇⣶⣶⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢈⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⢁⠀⣄⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠐⠤⠀⠉⠻⢕⣀⣀⠀⠀⠂⢠⠆
⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣴⠞⣡⠆⠀⣁⠠⢖⠀⠠⠄⡛⢿⣷⣇⠀⠦⠀⠀⢀⠞⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣧⠋⢤⢖⡠⢄⠦⠠⣤⡶⠦⡉⠚⢻⣬⣳⣧⠔⠒⠌⢀⠀⠀⣠⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⢀⡾⢃⣠⠖⠀⠉⠛⠂⠁⠀⠀⠈⠀⡩⢴⣿⣮⣿⣦⣐⣀⡘⡀⠀⢀⡀⢀⡠⠰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⢸⢋⠀⠁⠀⢈⣤⣶⣶⣶⣃⠀⠀⡘⡴⠃⠙⣿⣷⣝⣿⠖⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠈⢦⠀⠀⢀⢴⡿⣊⣤⣡⣾⣿⡟⢹⠉⠉⠉⠛⠀⠴⠁⢄⠀⡞⣿⣜⣧⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⣸⡟⢛⣋⣈⢸⠟⡉⠀⠀⡀⠀⠀⢀⠂⢰⠆⠀⣠⢹⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⠈⠁⠀⣠⣿⠛⠟⠛⠙⠉⠀⠃⠀⠀⠀⠀⠠⠁⠀⠋⠰⣰⢧⣿⣿⣳⣤⡄⡀⠀⠀⠀⠀⡊⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣡⣿⢿⢳⠀⠀⠀⠉⠀⠠⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠠⠀⠀⠀⠁⠀⠢⢀⠀⠀⣤⣷⡆⠀⠀⠀⠀⠀⣠⢾⠟⠀⣈⡸⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠖⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡿⣦⣧⠀⠀⠀⠀⢾⡟⢠⡇⣿⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡀⠠⠄⠀⢀⠀⠀⠀⠙⢿⡟⣧⠀⠀⠀⠘⠆⣾⡏⢸⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⣎⠆⠀⠀⠄⢲⣽⡿⠘⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⣠⣜⠞⠀⠀⠀⠈⢿⠇⠀⠀⠁⢀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠘⢿⠝⠻⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠙⢄⡃⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠚⠶⢄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠠⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠙⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣉⠀⢈⣑⠂⢻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⢰⡁⠀⠀⠀⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠤⠆⠀⢀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⣠⠔⠋⠀⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⣿⣿⣿⢿⠀⠀⠀⣟⢿⣿⣿⠃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠖⠚⢉⡏⡎⠀⠀⠀⢈⠈⢫⠙⠒⠲⢄⠀⠘⠄
⠀⠀⠀⠀⠀⠀⢀⣞⢿⣲⡶⣄⣀⣀⣀⠀⠀⠀⠀⠀⢸⠀⡀⡇⢰⠁⠀⠀⠀⠸⠀⠛⡄⠀⠀⠘⠀⠀⠀⣀⠔⣲⡷⡶⡿⡶
⠀⠀⠀⠀⠀⠀⠉⠚⠳⣗⡛⠑⠂⠀⠀⠑⠄⠀⠀⠀⢸⠀⠇⡇⡞⠙⠒⠒⢻⠃⢀⠜⢀⠄⠀⠀⡆⢠⠊⠠⠼⢳⢃⡵⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⠈⠉⠙⡄⠘⡄⠀⠀⢸⠀⠀⣷⡇⠀⠀⢀⡏⣠⠊⠀⢸⠀⠀⠀⣇⠃⠀⡖⠒⠛⠋⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢈⡁⠒⠂⠀⠀⠀⠀⢣⠀⠰⡀⠀⢸⠘⠀⢿⠁⠀⠀⢸⡷⠁⠀⠀⢸⠀⠀⠀⣿⠀⢰⠀⠀⠀⠠⠊⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠱⣀⡘⠀⡆⢸⡆⠀⠀⢸⠃⠀⠀⠀⠘⡀⠀⠀⠇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡅⡇⠀⠇⠈⡇⠀⠀⢻⠀⠀⠀⠀⠀⠇⢀⡾⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡇⠃⠀⢸⠀⡇⠀⠀⢸⡄⠀⠀⠀⠀⢸⠈⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠵⢵⠤⢻⠀⡗⣦⢷⡸⡇⠀⠀⠀⠀⠘⡅⣄⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⡏⠈⢧⡁⡇⠀⠀⠀⠀⣀⡹⡑⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠜⢸⣓⡖⢶⡖⣻⠀⠀⠲⣉⣁⠤⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⣴⣿⣿⣿⣿⣿⣿⣧⣀⣀⣠⣤⣴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⠿⠿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⠁⠀⠀⠀⠀⠀⠉⠙⠻⢿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⢰⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠋⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⡶⠾⠿⠿⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣀⣀⣠⣤⣤⣶⡿⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⡿⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠹⣿⣆
⠀⠀⠀⢀⣴⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠙⣿⣆
⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⣿⡆
⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⢤⣄⠀⠀⣿⡇
⠀⠀⢻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣿⡇
⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢻⣇
⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿
⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⡿⠿⠿⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢲⣄⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠹⣷⣄⠀⣰⣿⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠘⣿⣿⡏
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⣿⣿⡇
⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣦⣤⣄⣀⡀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⡶⠶⠀⠀⢀⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⣟⣀⣠⣴⣶⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠈⠉⠉⠉⠉⠀⠒⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⠠⠄⠐⠀⠐⠀⠒⠀⠀⠠⢄⡀⡠⠀⠒⠒⠀⠤⣀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⣀⣀⠀⠤⠤⠤⠀⢀⣀⡈⢆⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣒⣛⣉⡉⠀⠒⠒⠒⠀⠤⣀⡀⠀⠉⠐⠒⠤⢄⡀⠀⠀⠇⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⢄⡀⠀⠈⠑⠢⢄⡀⠀⠀⠈⠐⠤⣘⡀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢺⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣀⠀⠈⠑⠦⡀⠀⠀⠀⠑⠢⣄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠸⠸⡎⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⠈⠑⠢⣀⠀⠀⠀⠉⠢⢀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡄⠀⠙⣆⠀⠀⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣑⠢⠄⡀⠀⠀⠑⠦⡈⠆
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⠁⠀⠀⠈⠳⣄⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠱⢦⣤⣤⢘⡘⡄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⡰⠉⠛⢦⣄⡀⠀⠀⠀⠈⠐⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠛⠉⡙⠘⡄
⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⠜⠀⠀⠀⠀⠉⠊⠕⡒⠠⢄⣀⠀⠀⠀⠈⠁⠒⠀⠀⠠⠤⠤⠤⢠⣤⡤⠤⠐⠒⣛⣱⡚⠤⠚⢁⠇⠘⢄
⠀⠀⠀⡠⠊⠀⠀⠀⣀⠔⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠬⢉⣀⠒⠒⠒⠒⠐⠒⠒⠂⢈⣁⡠⠤⠴⠖⢉⣉⡀⠠⠤⠐⠃⢤⠀⠀⠉⠀⠈⠁
⢀⡴⠊⠀⠀⣀⠄⠊⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠒⠀⠤⠤⠀⠀⣠⠤⡄⠒⠁
⠈⠉⠐⢶⡭⢤⠤⠔⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠸⣄⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⡸⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢆⠉⠐⠒⠈⠁
⠀⠀⠰⠁⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢂
⠀⠀⢡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜
⠀⠀⠀⠑⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠉⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠊
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠠⠤⠀⣀⣁⣒⣀⣀⣒⣉⠀⠠⠤⠐⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡊⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⡶⠶⡏⢸
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⣀⠤⠤⠤⠧⠤⠦⠤⠧⠼⠤⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢄⡀⠉⠉
⢈⠉⠆⠀⠀⡴⠂⠉⠉⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠒⠢⡄⠀⠀⢀⠔
⠀⠈⠘⠀⢠⡀⠀⠀⠠⠤⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢀⡀⢐⠀⠀⠦⠔
⡠⠀⢠⠄⡰⢆⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⣼⠀⠀⣀⣀
⠑⠊⠁⢀⠇⠀⠈⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠁⠒⠀⠁⠈⡆⠀⠑⠤
⠀⢠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠉⠑⣦⣀⠀⠀⠀⠀⠀⠀⠈⠙⢆
⠀⠈⠦⣠⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣠⡠⠋
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠋⠀⠙⠋⠁⠉⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢸⠿⠿⠛⠛⠛⠛⠛⠒⠛⠂⠉⠉⠉⠉⠛⠛⠛⠻⠿⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋
⠀⠀⠀⠀⠀⢐⠑⠃⠀⠳⢝⣲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡫⠞⠁⠵⠤⡀
⠀⠀⠀⠀⠰⡋⠀⠊⠀⠀⠀⠀⠀⡈⠉⠓⠲⠦⠤⣤⣤⣤⣤⠤⠴⠖⢚⡉⠀⠀⠀⠀⠀⠂⠐⢧
⠀⠀⠀⠀⠀⠹⣅⡠⠆⠒⠒⠊⢩⡃⣠⠤⢄⠀⠀⠀⠀⠀⠀⡠⠤⢄⢸⠉⠉⠒⠒⠤⢄⠀⡲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠛⠉⠁⠀⢀⡧⠤⠤⠬⠥⠤⡇⠀⢀⠁⠙⢲⣄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠉⠀⠀⢈⡆⠀⠄⠁⠀⠀⡇⠀⠀⠈⠁⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡎⣩⣀⣀⡀⢀⣠⡛⡄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⡠⠃
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢠⣟⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⢠⡞⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡿⠀⠀⢰⣿⣿⣿⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⣿⣿⣿⣿⠁⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢠⣞⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀⣤⣤⣤⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠶⣤⣴⠖⠀⣷⠀⠀⠳⠶⠶⠃⠀⢹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠚⠉⠉⠉⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠉⣉⡇
⠀⠀⠀⠀⡠⠒⠉⠀⠉⠉⠑⢄⠀⠎⠀⠀⠀⠀⢀⠀⠀⠀⠀⡖⢒⡉⢉⠉⠑⠚⢆⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠸⢥⢄⣀⣀⣀⣀⢀⣸⠦⠀⠀⠀⠀⠀⠫⠴⠀⠀⠰⣳⠓⠑⠎⡷⢄⠀⢰⠁⠲⠆⠀⠀⢃⡼⠇
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣧⠟⠑⢶⡍⠁⡏⠉⣉⣲⠄⠀⠀⢹⠈⠐⠀⠁⣄⠳⡺⠀⣃⣀⣀⣀⠂
⠀⠀⠀⠀⠀⢑⡤⠤⣤⠮⢼⡀⠀⢸⠃⠀⢱⣸⢑⢵⠀⠀⠀⠈⠦⣀⡒⠀⢠⠊⡡⢊⣴⡄⣤⡀⠉⡷⡄
⠀⠀⠀⢠⠾⠭⠭⠭⣽⠷⢤⡇⠀⠸⡀⠀⢳⠉⢆⣸⠀⠀⠀⠀⡴⠻⠥⠒⠉⠓⡇⡎⠄⠛⢠⠙⣄⢀⢸
⠀⠀⠀⣼⡄⠀⢀⠔⠁⠀⠀⢹⠀⠀⣷⣤⣼⡠⠼⠚⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡙⡸⢸
⠀⠀⢸⢸⠈⠉⠁⠀⠀⡄⣠⡞⢀⠜⠉⠉⠚⠳⡄⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⡄⠀⡏⡳⢶⣀⣴⡟⠧⠎
⠀⠀⢣⢸⠀⠀⠀⠀⠀⡜⢡⢇⡘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠉⡦⣼⠃⡠⠀⠀⢹
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠧⢤⡞⠇⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⣸⡗⠤⠤⢀⡼⠻⣄⠀⠹⠀⠀⣠⢾
⠀⠀⠀⡾⢭⣹⠑⣒⣒⣒⢺⡵⡤⢤⠤⣀⣸⠀⢸⠀⠀⠀⠀⣟⡇⢠⣀⢰⡓⢲⠎⡗⠢⠴⠊⠀⢸
⠀⠀⠀⡇⢸⣿⠀⡇⠀⠈⡟⡜⠀⡜⠀⢸⡞⡄⢸⢹⠀⠀⠀⠀⢹⠀⡍⠀⠓⠊⣸⡤⢤⡞⣏⠉⠉⣳
⠀⠀⠀⢹⠉⠀⢣⠈⠉⢹⠁⠁⠀⠁⠀⠀⡇⡧⢚⣬⠂⠀⠀⠀⠈⣦⠃⠀⠀⢠⢻⠁⠀⢇⣈⣉⣉⣹
⠀⠀⠀⠸⣀⣀⣸⣀⣀⣘⣦⣤⣀⣀⣀⣀⣈⡱⠚⠁⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⢸⣀⣀⣀⣀⣀⣀⣈⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⣤⣤⣴⣶⣶⣦⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣆⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠁⢀⡀⠄⠀⠈⠉⠙⠛⠛⠛⠿⠹⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⠐⠁⢀⠄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃⠀⠐⠁⠀⠄⠀⠀⣠⠎⠀⣀⣀⣀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⣿⡿⠁⠀⠀⢀⠄⠀⠀⠔⠈⡹⠿⠿⠛⠋⠉⠁⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡜⠀⠀⠀⢀⠂⠀⡀⠁⢀⠔⠁⠀⡀⠀⠀⠀⠸⣿⡛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⠀⠀⠀⠀⠀⠀⠘⠀⢀⠊⠀⠀⠀⠈⠉⠀⠀⠀⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠠⠀⠀⠀⠀⠀⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⣠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⢀⡴⣇⠀⠀⠀⠀⠀⠀⣀⠜⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠉⠁⠁⠀⠀⠈⠁⠀⠀⢸⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⢰⠣⢀⣀⢀⡀⠀⠀⠀⢀⣀⡠⠴⠊⠁⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠈⠈⡏⠀⠀⠐⠉⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠲⠤⢤⣀⠃⠐⢀⠃⠈⣀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⢶⠋⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⢸⡇⢀⡇⠀⠀⠀⠀⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠁⠁⢼⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡞⠈⠀⠀⠀⠀⢠⣿⠙⢿⣿⣦⣄⠀⢸⠀⡈⠀⣠⣾⠀⠀⠀⠀⠀⢀⣠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠻⢿⣿⣿⣶⣶⡿⠟⢿⣷⣦⣤⣶⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⢠⣿⡏⠀⠀⣶⠚⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠾⠿⠧⣤⣀⣸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢳⣠⡞⠁⠀⣀⣀⣀⠀⠀⣹⠟⣧⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠀⢀⣼⣿⣀⣀⡀⠀⣶⢞⣧⡀⠀⠀⠀⡟⠀⠀⢿⣆⢠⡾⠟⣛⣿⠷⠾⠟⢲⣿⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢶⣶⣶⡾⠋⠉⠀⠀⠀⠉⠓⢿⣯⡅⠹⣦⠀⢰⡇⠀⠀⠈⣿⣺⡁⠘⣇⠀⠀⠀⠀⢸⣿⣇⣀⣸⡏⠒⠂
⠀⠀⠰⡦⠞⠆⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⣠⣤⣦⣤⡀⠀⠀⢻⡟⠄⠈⠳⣼⠁⠀⠀⠀⠘⣿⠳⣤⣿⡾⠟⠀⠀⣼⠛⣏⠁⠀⢻⣤⣤⣀⣀
⠀⠀⠰⠧⢄⡖⠀⠀⠀⠀⠀⠀⣿⣿⣟⣼⠟⠁⠀⠉⢻⡆⠀⠀⣿⣰⡄⠀⠉⠀⠀⣿⡀⠀⠸⣇⠀⠀⠀⢀⣠⡾⢫⡴⣿⣶⡶⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣯⣿⠀⠀⠀⠀⠀⡿⠀⠀⣸⣿⣿⣦⣀⢀⣸⡏⣷⣀⣀⣽⣿⡿⠿⠟⠋⢰⡟
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣽⢷⣤⣀⣠⠾⠁⠀⢀⠟⠻⠿⠝⢿⣷⡿⠇⠘⠟⠛⠋⢥⡐⢦⣀⣀⣈⢷⣄⠀⠀⠀⠀⣼⠛⢻⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡤⠊⢻⣿⣟⣃⡀⠈⠀⠀⠀⣠⣿⣼⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣟⠻⣯⡉⠉⠉⠓⠀⠀⠀⠀⠳⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠈⠛⣿⣷⣶⣶⡶⠟⢛⢿⣲⣀⠀⠀⠀⠀⢀⣀⣀⡄⢀⡀⢈⠀⠈⢅⡈⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠟⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠓⠙⢿⡆⠀⢀⡼⠛⠙⠛⣚⡙⠙⠻⡆⠁⠈⠛⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠘⡇⢀⡞⠁⢠⡞⠋⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠋⠈⢲⠄⠀⠀⠀⠀⠀⣿⡞⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⢤⣰⠃⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣷⣶⣤⣚⣓⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠆⣷⣾⣭⣟⣛⣛⣛⡋⡌⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣿⣿⣛⣿⣿⣿⣟⣻⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣒⣿⣿⣿⣿⣿⣕⣪⠇⠀⠀⢠⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⣒⠶⢖⣠⡿⠋⠀⠀⣴⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣈⣉⣛⣋⣉⡀⢀⣀⣾⣯⣟⡿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣮⡻⢿⣿⣿⢟⣱⣿⣿⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣁⣤⣶⣶⣤⣬⣐⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣴⣾⠿⠟⠛⡟⣙⣿⠛⣿⡿⠿⢿⣿⠀⠈⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣯⣤⣀⠀⠀⠡⡀⠿⠛⠻⠷⡀⢀⠀⠀⠀⠀⠀⣙⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠁⠈⣿⣶⣦⣬⣀⣀⣀⣀⣈⣀⣤⣤⣶⢶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠛⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⡇⠛⠀⣿⣿⣿⣿⢉⡏⢿⠁⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡿⠋⢻⠀⠁⢸⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣿⠀⠀⢸⠀⠀⢸⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢿⠀⠀⠈⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢸⠀⠀⠾⠇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡠⣦⣤⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡃⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⣿⠻⣷⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⣸⠞⢼⣿⣮⡓⢀⠀⠀⢰⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⡞⣿⠻⡏⠙⢦⣉⣲⣿⣍⢻⣿⣿⣗⣹⣿⣿⣿⣿⣿⡟⠉⠙⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣇⠀⣀⢻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡄⡀
⠀⠤⡤⢤⣄⠀⡀⠀⠀⠈⠁⣘⣽⣿⣟⣯⣿⣿⣿⣿⡿⢿⣷⣿⣯⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠈⠃⠤⣶⡿⣿⣿⣿⣿⣧⢽⣿⣿⣯⣿⣿⣿⣟⣿⣿⡀
⠀⠀⠀⠈⢳⣠⢠⣤⡈⠀⣤⡀⠀⢀⡐⢀⣈⡽⠛⠛⣿⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠈⠔⠁⠀⢊⠽⠿⢦⣴⣏⣙⣿⣧⡇⠴⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠈⠀⠀⠀⡠⠉⠀⠠⡀⢼⡷⠿⣩⣑⣠⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⣠⠤⠇⢠⡄⠀⠟⠋⠠⣠⣿⣿⣧⣿⣟⣿⣿⣿⣿⣿⣿⣿⣧⣿⣯⣿⠆
⠀⠀⠀⠀⠀⠘⠁⢸⣴⡥⠊⢁⣠⣤⣄⣤⣤⡵⣿⣿⣿⣿⢻⣿⡿⣿⡿⣿⣿⣿⣿⡏
⠀⠀⠀⠐⠀⠀⠴⡾⠋⠤⠌⠠⣶⣿⣭⡷⣬⢻⣿⠿⣿⣿⣿⣿⣲⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠣⡤⠀⠀⢶⣶⡾⡉⠐⠀⣌⣈⣰⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣈⣉⣙⣩⡷⠶⢭⡋⢰⠐⣻⣿⡿⠻⠿⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠳⡄⠀⣠⡿⠚⠛⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣴⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠸⠿⠿⠿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣮⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠉⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢇⠀⠀⠀⠈⠛⠠⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⠂⠀⠀⠀⠀⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⠤⠄⠤⢀⣔⠈⠀⢃⠀⢀⣾⠀⠈⠃⠀⠀⠀⠀⠈⢹⢻⠉⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠁⠀⠊⠕⢀⠀⠀⠀⠀⠀⠀⠀⠸⠻⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠈⠁⢀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠄⠂⠁⠂⠀⠀⠀⠀⠀⠠⠂⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⢀⠡⠀⠆⠃⠑⡀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢸
⠀⠀⠀⠈⠂⠈⡂⡀⠀⠀⣀⢠⠆⠀⢀⢀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠁⠀⢦⣁⡝⠡⠟⠁⠀⢀⡀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⡑⢔⠿⠉⣇⠀⠐⠒⠉⠀⠀⢀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡏⠀⠀⠀⠀⢀⡟⠁⠐⢀⡏⣀⠀⠀⠠⢤⡤⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢱⠖⠠⠀⠐⠁⠨⢦⡦⠖⠓⠀⠉⠀⠒⠆⠄⠀⠀⠐⢊⠀⠀⡠⠂⢀⡀⠀⠠⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⠁
⠀⢣⠀⠀⠀⠀⢀⣮⠥⣄⣁⠀⡀⢀⠠⠄⡀⠀⠀⡠⠊⠐⠨⠤⡌⠠⠈⡄⠃⠀⠌⠁⠄⠀⠄⠀⠀⠀⠀⠀⢸
⠀⠈⢣⠀⠀⠀⠎⣾⠄⠀⠈⠁⠠⠄⠂⠀⢘⢀⠉⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠠⣀⠄⣁⠈⣢⠀⠀⠀⠀⠰⣀⠗⣈⡇⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⢰⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⣼⠀⠘⠄⠀⠀⡀⠄⠒⠚⠉⠉⠉⠁⠀⠈⠉⠉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠡⢤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠲⠶⠶⠶⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⡇⠀⠀⢀⡴⠚⠛⠓⢦⣄⢀⣠⡤⠖⠚⠛⠛⠛⠶⣄⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣠⠴⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠓⠀⣴⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣰⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠿⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⢀⡴⠖⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠙⠳⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⢀⣴⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇⠀⠀⣠⠶⠒⢶⠖⢶⠖⠶⢶⠖⢚⣛⣫⣭⣤⣤⣤⣤⣤⣤⠴⠶⠛⠁⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣦⣀⠻⢤⣤⣼⣤⣼⠧⠶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠴⠖⢀⣠⣴⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣋⣉⣉⣉⣉⣩⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠁⠀⠑⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠒⠒⠂⢀⡖⠊⠁⠀⠀⠈⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠀⠁⠀⠀⠀⠀⠀⠀⣸⠠⠄⠐⠒⠒⠢⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀⠷⠀⠀⠉⠉⠉⠉⠉⠉⠓⠢⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠸⡄⠀⠀⢀⣀⠀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⢈⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡆
⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⠤⢤⣤⠃⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠓⠦⢄⣀⡀⣀⢀⣀⣀⣀⠠⠔⢣⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠠⡀⠃⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢣
⠀⠀⠀⡀⠑⠲⠦⠤⠤⠤⠤⠤⠄⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠀⠈⣸⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠈⠒⠤⢄⣀⣀⠀⠠⠖⠃⠀⠀⠐⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠀⡠⠔⠋



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠉⠻⣿⣷⣄⡀⢀⣀⣠⣤⢿⣿⣿⡦⣤⣄⣀⡀⢀⣠⣾⣿⠟⠋⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢠⣀⣈⣙⠻⢿⣿⣇⠀⠀⢸⣿⣿⠁⠀⠀⣼⣿⡿⠟⣋⣁⣀⡤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⣈⣉⡉⠀⠀⠈⠿⠀⠀⠀⣿⡟⠀⠀⠀⠟⠁⠀⠀⢉⣉⣁⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠛⠛⠉⠀⠀⠀⣀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⠀⠀⠀⠙⠛⠛⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⣀⡀⠀⠀⣀⣤⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣷⣤⣀⠀⠀⢀⣀⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⠛⠛⠃⠀⠉⢹⣃⣀⣵⡄⠀⠀⠀⠀⠀⠀⢰⣯⣀⣹⡍⠁⠀⠘⠛⠛⠛⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣠⠇⠀⢀⣀⣀⡀⠀⠹⣶⣆⡼⠁⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⠿⣿⣿⣷⡶⠶⠖⠀⠀⠀⠀⠉⣡⠀⠀⠀⠀⠀⠀⠀⠀⣌⠉⠀⠀⠀⠐⠲⠶⢶⣿⣿⡿⠿⠟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⣀⡤⠖⠒⠛⠒⠦⢤⡀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣀⡤⠖⠒⠛⠒⠲⢤⣀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡄⢠⠞⠁⠀⠀⠀⠀⠀⠀⡀⠉⠻⣶⣤⣄⣀⣀⣀⣀⣤⣴⣶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⣤⣤⣾⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠛⠀⢠⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠂⠀⠉⠛⢻⡟⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠀⢳⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⢠⠏⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⢦⡀⠀⠀⠀⠈⠓⢤⡤⣤⡤⠤⠞⠁⠈⠳⠤⢤⣤⢤⡴⠚⠁⠀⠀⠀⢀⡴⣋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠦⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡧⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣀⣀⣀⣀⡤⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⣄⠈⠉⠉⠀⣠⣴⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠛⠁⠀⠀⠀⠀⠉⠛⠿⣿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣁⣀⣀⣀⡀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢚⣉⠤⡤⠤⣄⣈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣷⠀⠀⠀⠈⡦⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⡞⠀⣎⣀⡜⠁⠀⠀⠈⢣⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠒⠒⠒⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢆⠈⣿⠉⡉⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠦⠆⠀⢣⠀⠀⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠈⠀⠀⠤⠶⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠐⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠐⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣴⠲⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⢰⠲⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠸⣄⡀
⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠢⠄⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⣛⡛⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣛⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⣤⣉⣵⣯⣿⣿⣧⠌⢳⡀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⢤⡘⢩⡷⠠⣤⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠃⣻⡾⣿⢿⣿⡟⠺⠿⠆⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢺⠀⢸⠀⢰⠋⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⠸⣹⠞⠛⠛⠑⠀⠀⡀⡀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⣀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠁⠀⠀⠀⠈⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⢤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠑⠏⣰⣿⣾⣿⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣼⡷⠾⠟⠻⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣀⡀⠀⠤⠤⠜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠉⠤⠀⠀⠀⠁⠀⠱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠘⢷⡤⠀⠀⡠⢾⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠇⠈⠂⠀⠀⠀⠀⠈⡆⢈⣿⣦⠀⢀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠐⠛⠻⢿⣿⣴⠾⠛
⠀⠀⠀⠀⠀⠳⠀⠀⠠⡄⣧⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡠⠂⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⢠⡏⠀⣀
⠀⠀⠀⠀⠰⠆⠀⠀⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣠⠟⠀⢠⣿⣷
⠀⠀⠀⠀⠀⠁⠀⣐⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡀⠀⢸⣿⣿
⠀⠀⠀⠀⠗⠀⠠⢽⣿⣯⢹⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⡾⠁⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⡆⢳⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠘⠿⣄⣀⡀⠀⠀⡀⠀⠀⠀⡀⠰⠀⠀⠄
⠀⢀⣈⣥⡆⠀⢀⠄⠀⠀⣿⡎⢇⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠨⠄⠀⠀⡦⢒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠆⠈⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⡸⠆
⠀⠀⠀⠀⣺⠂⣴⠀⠆⡄⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡇
⠀⠀⠀⠀⠀⠀⠁⠈⠈⣅⠀⠀⠀⠀⠀⠀⣄⠉⠀⠀⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠑⠰⣂⠤⢶⣼⣿⣆⣀⣤⣾⢿⠟⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠉⠉⠹⢻⠿⠿⠛⠁⣠⠴⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣂⠘⠥⠮⠤⣤⡤⠒⠛⢉⡠⠃⡦
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠸⣇⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⢸⣏⠀⠀⠀⢰⡏

⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⢿⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣿⣿⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⡿⠟⠋⠀⠀⠀⠀⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⡀⠀⠀⠀⠀⢀⣬⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠌⠁⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⠄⠀⠀⠠⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⢰⡆⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠋⣻⣿⣶⣶⣾⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠛⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣀⠀⠀⣤⣤⣤⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡅⠀⠀⣴⠟⠋⠀⠀⠀⡀⡉⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠟⡩⠀⠀⠀⠀⠐⠐⠀⠀⠄⠀⠰⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡽⣄⠢⠀⠀⠀⠈⠋⠀⡀⠀⢈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣮⣻⡿⣇⠀⠀⠀⠀⠃⠀⠀⠂⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣀⠠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⢸⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⢸⣿⣳⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⢿⣿⣷⣶⣶⣶⣾⣿⣿⡗⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣝⣿⣟⣽⣿⣾⣽⣛⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠃⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠞⠛⠛⠛⠛⠳⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠁⣸⠖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢸⠀⡟⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢸⢸⠀⡇⠀⢀⡟⢳⠶⠶⠶⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣿⠘⠀⠀⠇⠀⢸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡟⠀⠀⠀⠀⠀⠀⠁⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠒⠲⢄⠀⠀⠀⠀⠀⣰⢿⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠛⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣸⠃⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⣤⠞⠋⢀⡼⠃⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣀⣤⣾⣧⣤⣤⡶⢞⣋⣁⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠇⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⠀⠀⣀⣤⡴⠶⠞⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠳⣦⣤⣤⣤⣤⠶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⢰⣿⣿⡿⠿⢿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢸⣿⣿⡿⠻⠻⠟⠋⠀⠀⠉⠉⠉⢹⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠈⢿⣿⣇⣀⣠⣄⣀⣀⣀⣀⣀⣸⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣆⠀⢠⣦⣤⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠶⠶⠶⠶⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠉⠉⠓⣆
⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠠⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠉⠲⠤⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⡈⠀⣠⠊⢁⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢑⠶⠤⣄⣀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⡇⢠⠇⠀⢸⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠉⢙⠓⠦⢤⡖⣿⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣇⡼⠀⠀⠸⡆⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠠⠁⠀⠀⣀⠴⠚⠉⠑⠢⣄⠀⠸⣝⠒⠺⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⢳⡀⠀⣀⠀⠀⠀⠑⣄⠀⡸⠓
⠀⠀⠀⠀⠇⠀⠔⠋⠁⠀⠀⠀⠀⠀⠈⢣⠞⠈⠳⣄⠀⠀⠀⠀⠀⠉⣽⡃⠀⠀⠀⠀⠀⢹⡀⠀⠑⠦⠏⠣⣀⠀⠀⠀⠉⠀⣠
⠀⠀⠀⠘⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠺⣿⠀⠀⢀⣬⣳⠄⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠈⠁⠐⠂⠂⠈
⠀⠀⠀⡇⠀⠆⠈⠀⠙⠓⠦⠤⣤⣤⠤⢿⠀⠀⠈⠛⡟⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡄⠄⠊⠀⠀⠀⠀⠀⠷⠐⠒⠒⠒⢞⠉
⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⡜
⠀⠀⠀⠀⠀⠙⠶⣤⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⣟⠲⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡿⠒⠒⠲⠶⠶⠒⠒⠚⠋⠉⢁⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣁⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⢶⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠦⠤⣤⠀⣤⣤⣤⣤⠤⠤⠴⠒⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⢄⣀⠀⢠⠀⠀⠀⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠤⠽⠿⠭⠝⠛⠓⠦⠭⢉⣛⣿⣃⡠⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠋⠉⠉⠉⠛⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢩⢣⠏⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠤⡄⠀⠀⠀⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⡤⠎
⠀⠀⠀⢀⢀⠀⣀⡤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⠲⡄⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⡤⡀

⠀⠀⠀⢏⠉⠛⠉⠀⠀⠀⠷⡄⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠛⠀⢸⠀⠀⠀⠀⠀⠀⢀⡞⠉⠁⠈⠙⠋⣁⠇
⠀⠀⠀⠈⠙⠒⠦⢤⠂⠀⠀⢳⠀⠀⠀⣠⠖⠛⠙⠁⠀⠀⠀⠀⠘⠛⠓⢄⠀⠀⢠⠏⠁⠀⢠⡤⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢣⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠎⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣠⠴⠋⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⡏⠓⠦⣄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⡀⠀⠀⡀⠀⢄⠀⠈⠙⠂⠈⢦
⠀⠀⠀⠀⠀⠈⢰⡄⠀⠀⠀⠢⡀⠘⠆⠀⠑⠀⠀⠳⡀⣄⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢳⠆⣈⢮⡁⠐⣄⡠⣲⣤⠿⠋⠀⠀⠀⡈⢆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠈⢇⠃⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠈⢆⠑⠤⣀
⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡄⠑⡄
⠀⠀⠀⠀⠀⠐⠀⠀⠃⢀⣀⣀⠀⠀⠀⠀⢵⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡘⡄⠘⣩⣭⣉⠐⠄⠀⠠⠒⠻⠿⠉⠓⠲⠤⡀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⢀⡼⠀⠀⠀⡏⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⠀⠀⠀⣠⠊⠀⠀⠀⣴⠁
⠀⠀⠀⠀⢸⡁⠀⠀⢀⠀⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⠀⢰⡃⠀⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⢸⠓⠀⠀⠈⠌⣄⠀⠀⠀⢀⣀⣠⣤⣄⠀⠀⠀⠀⢸⡷⠀⢀⡟⠀⠠⠂⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⡘⢆⠀⠀⢻⣯⢿⣿⠟⠀⠀⠀⢀⣾⡇⠀⠀⠀⡄⠀⠀⠀⢀⡔⠀⠀⠀⠀⠠⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠈⢲⣆⠀⠉⠈⠢⡀⠀⠈⠉⠁⠀⠀⢀⣴⣿⣿⣷⠀⠀⠀⠀⠠⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿
⠀⠀⠀⠀⢀⠀⠁⠊⡻⠀⠀⠀⠀⡽⠐⠤⣤⣤⣤⣶⣿⡿⠿⠟⠋⡄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠀⠀⢸⠄⢀⠀⠀⠀⠚⡠⠞⠁⡀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠈⢧⡈⠁⣀⠀⠀⠀⠀⠈⣠⢀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠸⣧⡀
⠀⠀⠀⠀⠀⠀⠉⠀⣼⡀⠀⠂⣀⡼⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢪⡽⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⢀⣤⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⢠⣿⣻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⣀
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⣠⣾⢿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⣄⠀⢲⢀⣠⡶⢶⣿⠣⣽⣆⣮⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⢰⠿⣄⡰⢣⢞⣽⣿⣿⣴⣿⣿⣡⡴⣣⠶⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢔⠫⣹⣮⡿⣷⣿⣿⣿⣤⣷⣿⣿⣿⣿⢟⣱⣦⡴⢤⣴⢟⣾⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣨⣿⣿⣻⣽⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡭⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢨⡙⢮⣻⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢱⣼⡼⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣯⣶⡟⢁⡔⣠⠀⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠋⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣭⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠼⠶⣾⣿⣿⣿⣿⡿⠻⢡⠛⠙⠟⢿⣿⣿⣿⣽⠿⠟⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⡴⠛⢻⣿⡽⣯⠁⠀⠀⠜⡀⠈⠈⣿⡟⣿⢿⣻⣦⡀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢒⡀⢹⣟⡇⠉⠀⠀⠀⢠⡀⠀⠀⠘⠃⡛⠒⠈⠙⠓⠵⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠭⠄⠱⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡌⢀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢤⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠂⠄⠀⡴⠂⠉⠀⠀⢣⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⢁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡆⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠊⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠋⠁⠀⢠⡉⠉⠃⠀⠀⠀⠀⠀⠉⠙⢻⠐⠟⠛⢶
⠀⠀⠀⠀⠁⡀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠒
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠄⠠⠎⠘⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠑⠂
⠀⢠⠀⣀⠀⡀⢀⠀⢀⡀⢀⠀⠀⣄⠀⠀⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⡀⡤
⠀⠀⠀⠉⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠘⠁⠂⠀⠁⠉⠁⠂⠀⠀⠀⠉⠁⠁⠈⠁⠂⠃⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠄⠠⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠓⠄⠠⠴⠚⠉⠁⠀⠀⠈⠉⠑⠀⠈⠉⠄⠀⠀⠀⠀⠀⠴⠛⠁⠀⠈⠋⠙⠆
⠀⠀⠀⢶⡦⠀⠀⠰⡶⠀⠀⠀⠀⠀⠒⠒⠠⠀⠐⠚⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠘⠃⠄⠀⠀⠀⢠⠠⠄⢀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠉⢀⣀⣀⠈⡓⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⡏⠀⠀⠈⠁
⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠁
⠀⠀⠀⠄⠀⢀⡀⠀⢀⠀⠀⠀⠀⠦⠀⢠⢠⢀⣠⠄⠀⠀⠀⢀⠀⡀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠂⠀⣀⠂⠀⣀⡀⠤
⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠈⠈⠈⠈⠉⠀⠉⠁⠀⠈⠉⠀⠀⠀⠉⠉⠈⠀⠁⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⠿⠛⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⢀⡾⠛⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⣸⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠓⠶⣼⢷⡀⠀⠀⠀⠀⠀⢠⠷⠤⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢰⠇⠀⠙⠦⣄⡀⠀⣰⠋⠀⠀⢨⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡟⠀⠀⠀⠀⠀⠉⣹⠃⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⡾⠀⠀⠀⣀⣤⠤⣶⠃⠀⠀⠀⣼⡽⠟⠉⠛⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⣼⠃⢀⡴⡟⠁⠀⣰⠃⠀⠀⠀⣰⠏⠀⠀⢀⡴⠞⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣰⣣⠶⠋⢸⡇⠀⣰⠇⠀⠀⠀⢠⠏⠀⢀⡴⠋⠀⠀⠀⠀⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⠟⠁⠀⠀⢸⡇⢠⡏⠀⠀⠀⢀⡟⢀⡴⠋⠀⠀⠀⠀⠀⣴⠃⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠠⠟⠀⠀⠀⠀⣸⣇⡞⠀⠀⠀⢀⣾⡴⠋⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣿⠃⠀⠀⠀⡼⠉⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⣿⠀⠀⠀⠼⠁⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠶⠋⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠰⠖⠐⠇⢸⠀⠐⠀⠀⣠⠀⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣇⠘⡄⣄⣧⣰⣇⣿⣀⣆⣄⣠⣠⡞⢁⡶⠃⢠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡀⢌⢲⣄⠢⣽⣶⣬⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣶⣁⣠⠄
⠀⠀⠀⠀⠀⠀⠈⠢⡑⢤⣀⠬⢓⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣔⠦
⠀⠀⠀⠀⠀⠀⣀⠀⠈⡲⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠛⠿⣿⣿⠛⢿⣷⣶⣂⡠
⠀⠀⠀⠀⢤⣀⣈⣙⣲⣶⣾⣿⣿⣿⠟⠁⠀⠀⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⣸⣿⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠉⠉⢍⠹⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⡀⠈⣿⣿⣿⣿⣿⣿⣿⠉⠊⠀⢀⣿⠇⠀⠀⠀⢠⣿⣿⣿⢷⡶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠂⣹⣷⣌⠻⣿⡄⠀⠀⠀⠀⠀⠻⣿⣾⣿⣿⣿⠿⠿⢻⠋⣧⣰⣴⣿⠏⠀⠀⠀⣠⣾⣿⣟⠑⠂⠐⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠟⢳⣦⣉⠳⢤⣀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣾⣄⣾⣿⡿⠟⢁⣀⢤⣖⣯⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠚⢉⡷⠶⣬⣙⠓⠲⠤⠤⢄⡉⠉⠙⠛⠫⠍⢁⣒⣯⠵⡞⠛⠍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⠚⠉⡹⠛⢻⠗⢶⠒⠖⢲⡶⠒⢿⠋⠻⠱⠙⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⡟⠾⠳⠶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⡠⡠⢀⣀⣄⣄⢣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢎⡟⠲⠀⠀⠀⠠⠠⡄⢠⣇⡀⠀⠀⠀⠀⠀⠠⣄⡀⠠⣼⠔⠚⠀⢈⠀⠁⢁⠠⣸⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠁⠄⠀⠀⠈⠀⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⢾⡥⣠⣘⠀⠀⠃⠊⠀⠀⡘⠒⢹⠗⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠒⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡀⠀⠀⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣒⡤⡄⢠⠀⣤⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⣀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡁⠀⠈⠈⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠑⠔⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠃⠀⠀⠀⠀⠢⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⣚⠀⠀⠀⠀⢠⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠁⠀⠀⠀⠀⠀⠱⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠙⠛⠘⠃⠈⠉⠧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠀⠀⠀⠀⠀⢀⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠈⠉⠀⠉⠑⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠀⠀⠂⠐⢂⡿⢠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠀⠠⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠉⠉⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠢⠀⠀⡤⢀⣀⡀⠠⠐⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠢⢤⠤⢤⠆⢰⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⡙⠉⠒⡖⠀⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡎⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡇⠀⠀⢃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣠⠾⠰⡇⠀⢀⡆⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢎⡠⣼⠇⠀⡰⠛⢐⡷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠘⠻⠅⠓⠒⠊⠀⠐⠾⠦⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠒⠶⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡟⠈⠀⢽⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣾⣿⠉⠉⠈⣹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡝⣿⣿⠐⠚⣳⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⡟⠀⣨⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⢸⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠄⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡀⣸⣿⡇⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡯⠁⠀⢀⡠⠐⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣼⣋⣉⣿⣿⢱⣶⣿⣟⣿⠿⠟⠿⢿⠛⢿⣿⣼⣃⠉⠒⠉⠀⠀⢀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣤⣻⣿⣿⡋⠀⡀⢀⠀⠀⠄⠜⠋⠁⠀⠀⣀⣤⡾⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⠓⠴⠐⠴⠆⠲⣚⠉⠀⠪⠥⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠛⣈⣉⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠭⠇⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠗⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠆⠀⠾⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠲⢾⠂⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠻⣿⣿⣿⣯⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⢀⣿⣿⣿⣿⣿⠿⠛⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣫⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿
⣿⣿⠀⠀⠀⠀⠀⠳⠢⠄⠤⣀⠠⠤⠤⠤⠴⠾⠿⠿⠟⠁⠀⠛⠋⠙⠃⠖⠋⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠠⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠿⠛⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠸⠀⠀⠀⠈⠻⣿⣿⡿⠃⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠲⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⠻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⢀⠉⠀⠀⠀⠀⠀⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢀⢁⡀⠈⠉⠀⠈⠉⠢⡀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠢⠄⠠⠅⠀⠐⠒⠲⡆⠀⠘⡦⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢃⠐⠂⠐⣃⠀⠀⡀⠀⢸⠀⠀⠱⠊⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⡥⠆⠀⠴⠒⠆⠀⠀⠁⠀⠀⠀⠀⣦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢀⠌⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠣⡀⠀⡀⠀⠠⠂⠐⠢⠖⠈⠀⠀⣰⠀⠀⢀⠀⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠈⠢⣅⣁⢂⠤⠄⠀⠀⠤⠒⠁⠅⢠⣀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠈⠉⠢⠄⣀⠤⠒⠒⢄⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⢀⠠⠤⠤⡀⡀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⡀⠀⠀⠀⢀⡇⢰⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠉⠉⠁⠈⠀⠈⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢤⣤⣤⣤⠤⠚⠦⠤⠤⠔⡩⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡄⠀⠀⠀⡄⠀⠀⠀⢂⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡀⠀⢀⡇⠀⠀⠀⠘⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡨⡉⠉⠗⢒⡖⠒⡰⠇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠈⠁⠐⠐⠚⡆⠈⠘⠑⠉⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⢀⠶⢄⡀⠀⠀⣀⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠋⠁⠀⠀⠈⠉⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠟⠛⠛⠛⠛⠉⠙⠛⠛⠛⠛⠻⠶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⡶⠶⠶⠶⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⣴⣦⣴⣶⣶⣶⣶⣄⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣿⡇⠀⢀⣤⡄⠈⣿⣶⣶⡶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣶⣿⣇⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⣿⡇⠀⠈⠛⠁⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠁⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣤⣴⣶⡶⠶⣶⣤⣿⠃⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠁⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠉⠀⠀⠀⠈⠁⠉⠙⣷⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣸⣯⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⣶⣶⣤⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠻⠷⠶⠶⣦⣤⣴⣿⠉⠉⠉⠙⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣤⣤⣤⣤⣶⠿⠃⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⡟⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁


⣿⣿⡿⠻⠟⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠷⠛⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣟⠛⢶⡆⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣿⣿⣿⣿⠀⢠⠇⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢠⡶⠛⣹⣿⣿⣿⣿⡟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⠋⠀⠁⠀⠀⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠘⢧⡀⠹⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⡶⠾⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠋⠁⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⢠⡿⠁⠀⠈⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠈⠛⠷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠀⣀⣤⣤⣤⣤⣤⣄⣀⣸⡇
⢶⡞⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠉⢻⡄⠀⠀⠀⠀⠉⠛⠻⠶⠶⠶⠶⠶⠟⠋⠁⠀⢠⡟⠉⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⢳
⠈⠛⢶⣤⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣤⡄⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢠⣤⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣤⡶⠛
⠀⠀⠀⠀⠉⠉⠙⠛⠛⣟⠋⠉⠉⠉⣿⠷⠶⠾⠾⠿⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠿⠶⠷⠶⠿⣿⠉⠉⠉⠙⣻⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢠⡟⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢻⡀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣼⣧⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣤⣤⣤⣤⣤⣤⣼⣧⣤⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⣠⡄⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⡛⠶⣶⠀⢻⡟⠀⣶⠶⢛⣁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠖⠂⠀⡈⠀⠘⠃⠀⢁⠀⠐⠲⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠶⠆⠐⣟⢣⠀⠀⠀⠀⡜⣳⠂⠰⠶⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣤⠄⠀⠘⠓⠀⠐⠂⠀⡛⠂⠀⠠⣤⣤⡶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⠔⠈⠀⠁⠂⢤⣀⣀⣀⣀⡤⠐⠈⠀⠀⠢⡀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠢⡀⠀⠀⢀⡀⣀⠜⠢⣀⢀⡀⠀⠀⠀⠔⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⢄⣀⠀⠀⠀⠀⠀⠀⣀⡠⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠈⢀⣑⡀⠀⠀⢀⣊⡀⢁⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢁⠀⢸⣿⣿⡷⢾⣿⣿⡇⠀⡈⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣤⡌⠀⠘⠛⠋⠀⠀⠙⠛⠃⠀⢡⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙



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

⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⡀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣉⣿⣤⠶⠛⠋⠉⠉⠉⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢠⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⠶⠾⠿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠸⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠿⠟⠛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠃⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠃⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⠶⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⠇⠀⠀⢻⡄⠀⠀⠀⠀⠉⢻⡿⠿⠛⠛⠿⠿⠟⠛⠛⠋⠉⠁
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣀⣠⡾⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠒⣤⠤⠴⠂⠉⠉⠉⠉⠓⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠤⡴⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠎⠀⠀⠀⠀⠀⠉⠉⠓⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⣤⡀⠀⠹⣿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠠⠤⠤⠂⠀⠀⠀⣀⠀⠤⠄⠐⠂⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠛⢦⡀⠀⠀⠀⢀⠄⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⡞⠉⠙⠓⠶⠤⠤⠤⠤⠶⠚⠉⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⢻⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⢧⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠙⢦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡸⠃⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠓⠒⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⢀⠠⡒⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠢⠤⣀⡀⠀⠀⠀⠀⢸⠀⢀⡴⠋⡐⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⣸⡔⣋⣀⠰⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠂⠀⠀⠀⠀⠀⢠⠁⠀⠈⠉⠲⣄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠈⢣⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠳⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡀⠀⡠⠂⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⢱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱
⠀⠀⠀⣄⢰⡆⡖⠀⡀⠀⢸⠀⠀⠀⢤⣀⠘⢰⡆⢲⣄⠀⣠⠆⠀⠠⣄⠄⠘⣁⠀⡀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠸⢀⡀⠀⠀⢃⡀⢇⠀⠐⠂⠀⠠⠔⢂⣀
⠀⠀⠀⠈⢛⠡⠤⠤⢀⣀⡄⠀⠀⠉⠂⠈⠳⠌⠉⠀⠀⠉⠉⠁⠉⠀⢀⠈⠁⠀⠈⠁⠀⠈⠀⠀⠀⠘⠒⠀⠀⠘⠉⠀⠀⠀⠈⠐⠈⢦⣐⠒⠒⠲⢤⡁
⢦⡀⢠⠀⠳⢤⣀⣀⡀⠤⢃⢤⠄⠀⠀⢤⡀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠈⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠁⠄⠠⣔
⠘⠷⠎⠳⠓⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⣆⣰⠛⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠈⢿⡀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⠋⠉⠛⠲⢤⣀⣼⠋⠙⢦⡀⠀⠀⢷⠀⠀⠈⣧⠀⣠⣤⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⢹⡄⠀⠈⢷⡀⠀⠈⣧⠀⠀⠸⡏⠁⢀⣬⣧⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣸⠃⠀⠀⠀⠀⠀⠀⣀⣤⣷⠀⠀⠈⢧⠀⠀⠘⣇⠀⠀⢻⢠⡏⠀⣿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠲⣷⢠⡇⠀⠀⠀⠀⠀⠀⢠⡏⠀⠘⣧⠀⠀⠘⣇⠀⠀⢹⡄⠀⠸⣿⡇⠀⢹⡀⢸⡧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠘⣆⠀⠀⢹⡄⠀⠈⣧⠀⠀⣿⡇⠀⠸⣇⡟⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠸⣆⠀⠈⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⡷⠓⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠈⣷⠀⠀⠀⠀⠀⠉⠉⠛⠓⢿⡀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢹⠀⠀⠀⠀⠶⠖⣄⣀⠀⠈⣧⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢸⡇⣀⠀⠀⠀⠀⠀⠉⠉⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣌⡇⠹⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠷⣄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⡆⠀⠀⠈⠃⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢻⡄⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠟⣦⠀⠀⠀⠐⠶⠶⠶⠿⠶⣄⠈⠳⠤⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣾⠀⠈⠳⣄⠀⠀⠀⠶⠤⠴⠶⠞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡙⠳⠦⣤⣀⣀
⠀⠀⢀⣀⡤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⠶⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠈⠉⠙⠛⠲⢦⣄⡀
⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⣿⣿⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠠⣤⣀⡛⣶⣶⣦⣄⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣆⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣴⣿⣿⣿⣭⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣻⣛⣙⣛⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⣠⡾⠿⠛⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠿⢷⣄
⠀⠀⠠⠚⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣿⣿⣿⣷⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣶⣴⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⡫⢉⠿⣿⣿⡿⢳⠙⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⡟⣿⣿⣿⢿⠉⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠉⢻⣮⣣⠀⠣⣍⣉⡥⠟⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠳⣌⣉⣩⠜⣀⣼⣷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠀⣶⣦⡤⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⣶⡆⠚⠟⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⢠⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠸⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠶⢤⣄⣀⣠⠤⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⠒⣁⡀⢒⠒⠒⠲⠦⣆⢍⣍⡃⣵⠀⣠⠂⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣴⣶⣿⠟⣻⣉⣉⣁⣄⡤⣀⠡⡙⢻⣮⣞⡵⡣⢀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣽⣿⣿⣿⣿⣿⣝⢮⣦⣻⣿⡾⣿⢶⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢣⢫⠙⠙⠙⠟⠛⠿⠻⢃⣼⢃⢀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠰⢾⣯⠰⡈⢦⡻⣤⠤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠽⠂⣭⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⣠⡾⣿⠟⡿⢿⡄⠹⣿⣿⠙⣿⠆⠀⡤⠖⠛⠛⠻⠿⣶⡤⡜⠀⣰⣠⣴⣶⡏⢷
⠀⠀⠀⠀⢰⣿⣾⣿⣤⣿⡀⢻⡄⠹⣿⠀⢸⡀⠀⠀⢀⣀⠀⠀⠰⣆⡉⠀⠀⡞⠁⠀⣠⠇⠸⡆
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣧⠀⢳⡀⠹⡇⠈⡇⠀⠀⠀⠙⠛⠛⠛⠉⠁⠀⠀⡏⠳⠶⢾⠖⠀⢧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢧⠀⢷⠀⠱⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠸⡆⠀⠸⡄
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣷⠸⣆⠈⢣⠀⢱⡀⠘⢦⡀⠀⠀⠀⠀⠀⠰⠖⢀⣴⠇⠀⢀⡇⠀⠀⣧
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣆⠘⡆⠈⠃⠀⠀⠀⠀⠻⡄⢐⡀⠀⢀⣀⡀⣀⡀⠀⠀⣸⠇⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⢾⡄⠇⠀⠀⠀⠀⠀⢻⠚⠓⠿⣿⣿⢿⣿⠟⠀⣰⠏⣤⠒⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣇⠈⢻⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠉⠙⠉⣞⣴⡋⠀⠸⠆⠂⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣦⣰⠁⢠⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⡾⠋⠳⣴⣦⠀⠀⠀⢀⣤⣬⣷⣄
⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡂⠏⣇⠀⠀⠀⠀⣿⣷⣶⣤⣤⣤⠾⠁⠀⠀⠈⠙⢳⡀⣰⣧⡽⠏⣀⣬⣷⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠀⣼⠛⠋⠀⢠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠹⣿⢃⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠀⠀⠀⠀⠀⢰⠇⠀⠀⢀⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⡀⠀⠀⢀⡟⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠃⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠒⠒⠒⠲⠤⣄⡀⣴⣢⠼⠻⣋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⡀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⢀⣾⣧⣰⣦⠀⠀⠀⣾⠋⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡜⠛⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣽⣼⣿⣌⢹⣿⣀⠀⣸⡟⠖⠉⠥⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢸⣿⣦⣶⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣌⠻⣿⣥⣼⢁⡬⢂⣠⣤⣤⠀⠀⠀⢾⣿⣿⣿⣿⡆⠀⣿⣿⡿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣽⣎⠱⡜⣿⣿⣾⣣⣿⣿⡾⣿⣷⡀⠀⠸⢯⡶⠛⠋⠑⠓⠲⣶⣦⣤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⣿⣾⣿⣿⠿⣞⡵⣶⣻⣿⣿⣄⣖⠐⢿⡅⠀⠀⠀⠀⠹⡟⢸⠀⠀⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⢿⣻⣵⠛⠋⠉⠙⣿⣿⣿⣿⣿⣷⣾⣿⠶⣄⠀⠀⠀⠃⢰⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⢴⣿⣿⣿⠂⠀⠀⢠⣿⣿⣿⣿⣿⣟⣿⡏⠀⠈⠀⠀⠀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢿⣿⣿⠳⠀⠀⣸⠀⣻⠟⠙⢉⣉⣀⣀⡙⢂⠀⠀⠀⡗⠂⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠙⣟⠛⣄⠆⢸⣾⠇⣴⣾⣿⡟⣿⣿⣗⣦⡄⢰⢠⠃⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⡿⢿⣿⣧⡹⣦⣿⠟⠛⢛⣁⣒⣺⡿⠃⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠩⣉⣛⣿⣿⣷⡘⢿⠷⠞⠋⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠒⠒⠒⣒⣒⣶⣿⣿⣿⡄⠃⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⣉⣩⡭⠥⠶⣲⣿⣿⣧⣤⣤⣴⣾⣿⡾⣟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⢀⣀⠴⠖⠋⣩⣶⢟⡿⢋⡿⢹⠛⡇⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⠀⠉⠀⢀⣤⡾⠋⣠⠞⢀⠟⠀⠸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠘⠁⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⢦⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠳⠄⠒⠀⠙⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠻⠿⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠿⣿⣿⠿⠋⠀⠀⠀⠀⣀⣀⣠⣤⠴⠂⠀⠀⠀⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⡆⣶⣶⡌⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣛⠆⠀⠀⠀⠀⠀⣴⠾⣿⣿⣿⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢧⣍⡽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠶⠿⠿⠿⢿⣿⣿⣿⣇⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠤⠤⠶⠶⠚⠋⠁