Circumcision Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠑⠂⠉⠑⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠞⠛⠋⠐⠒⠒⠂⠐⠀⠙⠚⠛⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⣟⣛⣛⣟⣻⣛⣻⣹⣯⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠆⠴⠤⠤⠠⠄⠤⠤⠦⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⢰⣶⣆⠀⠐⢶⡀⠀⢀⠀⠀⠀⠦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣴⡆⠰⡂⠀⠰⠦⣶
⠀⠀⢸⣿⣯⡀⠜⢧⡑⠀⠂⠠⠀⠀⠀⠠⠋⠔⢀⠀⠀⠀⠀⠐⠀⠀⢖⠀⠀⠂⠀⠀⠀⡌⠋⠁⠚⠛⠀⣠⣦⣴⣿⡇
⠀⠀⢸⣿⣿⣷⡄⢠⣇⠀⢨⢣⠁⠈⡀⡀⢀⡀⠘⠒⠢⠀⠁⠀⠄⢋⢀⠀⣀⡤⠁⢙⢧⡲⠀⠔⠦⠀⣤⠾⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣇⠈⠀⠘⢈⡀⢠⡌⠀⡡⠁⣐⠀⠐⠀⠈⠀⢆⢉⡊⢄⢀⣥⠀⣉⠰⠽⠆⠀⠀⣠⣷⣼⣿⣯⢻⡇
⠀⠀⢸⣿⡿⣿⣿⣿⣷⣦⣇⣄⡀⠀⣙⠈⠈⠈⠋⠉⠁⢐⠀⠀⠀⠉⠁⠘⠀⠀⠆⠀⢀⣀⣠⣤⣼⣿⣿⡏⠀⡇⢸⡇
⠀⠀⢸⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣴⣶⣤⣶⣶⣤⣤⣤⣤⣀⡠⢄⣐⣶⣴⣴⣯⣿⣿⣿⣿⣿⠃⣸⠁⢀⡃
⠀⠀⢸⣿⡏⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⡇
⠀⠀⢸⣿⡇⣟⢈⣻⣿⢻⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣷⣿⣿⣿⣿⠛⠟⢻⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣫⣿⠋⢨⡆
⠀⠀⢸⣟⠁⠀⢸⣿⡿⠇⠸⣉⣿⡟⠁⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣽⣾⣿⠀⠸⡇
⠀⠀⠘⡧⠀⢯⣿⣿⠙⡃⠠⣤⠿⠅⣤⣤⢽⣿⣿⣿⣿⣿⢋⣉⣿⣿⣿⣿⣿⡟⢣⣼⡄⢀⠤⠙⣿⡟⠁⠀⠹⡧⠀⠃
⠀⠀⠀⠀⠀⢺⡏⢷⣿⣿⠈⣀⣀⢰⡃⠠⣾⣿⣿⣿⣿⣿⡿⠀⠈⣿⣿⣿⣿⣷⣿⣿⣷⢄⠄⢁⠙⡃⢠⣷⣶⡟⠀⡆
⠀⠀⠀⠀⡂⣰⣿⣏⣿⡃⠀⢸⣿⣿⣿⣿⣿⣿⣶⣿⡟⠁⢻⡄⢾⣿⠴⢿⣿⣿⣿⡿⠁⠀⠀⠀⢐⣥⣾⣀⣩⣿⣷⡇
⠀⠀⢀⡐⡇⢸⣿⣿⣿⠂⠀⠈⢿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠁⠀⠀⢱⠀⢸⣿⣿⣿⠁⡰⠓⢀⠄⠀⠍⠀⣈⣿⣿⣿⡇
⠀⠀⠸⠇⣷⣎⠙⢿⣿⣶⣶⠄⠐⠛⠻⢿⣿⣟⣿⣿⠷⠂⠀⢘⣦⣄⣀⡾⣿⣿⠃⠀⢀⣶⡄⢸⠃⠀⣴⣿⣿⣿⣸⡇
⠀⠀⠘⢿⣿⣿⣷⣾⣿⡅⠙⣿⣶⣶⣤⣤⣤⡍⢀⣴⡦⠀⣬⠙⠋⠁⣨⣿⣿⣿⠀⣶⡿⣏⡧⣧⠀⢰⣿⣿⣿⢿⢻⡇
⠀⠀⠘⣤⣠⣿⣿⣟⠻⢿⡒⡿⣼⠿⠻⢏⣉⡒⠚⠉⡏⣤⣰⣶⣾⣿⣿⣿⡿⢿⢸⡟⢰⡟⣵⠇⠀⣸⣿⣿⡏⡎⣼⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣼⣿⣿⣧⣿⣀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢉⣉⣤⣤⣴⣶⣿⣿⡧⣸⣧⠋⣠⣾⡿⢛⡟⣰⣷⣿⡇
⠀⠀⢀⣻⣿⣿⡏⠁⣸⣿⣾⣿⡇⢸⠀⣴⣿⡆⠀⠀⠀⠀⠀⣸⣤⣿⣿⣿⣿⣿⣿⣧⣿⡇⣼⣿⢳⣷⣬⣾⣿⣿⣿⡇
⠀⠀⠈⢙⣿⣿⣿⣿⣿⠁⣹⣿⡇⢸⢰⣿⣿⡇⠀⠀⠀⡀⠀⣻⣧⣿⣿⣿⣿⣿⣿⣿⡏⠁⠋⡻⢸⠋⠻⣿⣿⢿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⡇⣿⣿⡇⢸⢸⣿⣿⡇⠀⠀⢠⡅⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣶⠿⣿⡆⠈⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣇⣻⣈⣿⣿⣗⡂⠀⢼⠀⣸⣿⡇⢼⣿⣿⣿⣿⣿⣿⣿⣟⣿⠀⣿⣧⠈⠂⠹⣿⣿⡇
⠀⠀⢨⣿⣿⣿⣿⣿⣿⡗⢿⣿⣿⣿⢹⣿⣿⣿⠶⢴⣿⣦⣼⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⡟⠀⢩⣿⣃⠰⣏⠘⣿⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠟⠙⣥⣾⣿⣿⡟⠒⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⣿⣇⢦⣬⣀⠹⡇
⠀⠀⢸⣿⣿⣿⣸⣿⣿⣾⣯⣷⣿⣿⣿⣿⡿⠛⠋⠈⠉⠛⠋⠉⠉⢉⣉⣡⣼⣿⣿⣿⣿⡏⣏⡆⠰⣿⡏⢡⢻⣿⣿⡇
⠀⠀⢸⣿⣷⣶⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠲⠶⠶⠶⠶⠿⠛⠛⠛⠋⣉⣀⣀⣈⣑⣛⣷⣄⣻⣿⠀⣦⣻⣿⡇
⠀⠀⠈⠛⠛⠛⠛⠎⠹⠏⠻⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢿⢿⣿⣷⣿⣿⣿⣿⡇⢸⣧⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⠋⠀⠀⠒⠛⠛⠛⠛⠓⠒⠒⠘⠛⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠀⠀⢀⣤⠞⢋⣠⡶⠛⣁⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠷⡼⠋⢻⣋⡵⠞⢋⣤⠶⢋⣡⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣤⣄⡀⠀⠀⠀⠀⠀⢰⡞⠛⠃⠀⡆⢠⠞⠉⠉⢠⡶⢋⣡⠶⠋⠁
⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠈⠻⣄⠰⡄⠑⢻⣄⣤⡴⣋⣯⣭⣄⡀
⠀⠀⠀⠀⠀⢸⡆⠀⢠⡶⠳⢦⣄⠀⠈⠻⣆⠀⠀⠀⠈⢷⡀⢀⣀⣭⠶⠛⠉⠀⢀⣈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⢻⣄⠈⢻⣄⠀⠉⠳⡦⠀⠸⡆⠀⣠⡴⠶⠛⠋⠉⣠⠞⠓⠶⠶⠋⠉⠛⠓⣀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⡙⠛⠶⠚⠁⠀⢀⡿⠛⠉⠀⣠⡴⠶⠞⣃⣠⣤⠶⡶⠚⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⠤⣴⠋⣤⣄⠀⢰⣧⡴⡞⠋⠉⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⣈⣤⠞⠛⡓⠤⣼⠀⡇⢀⡏⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣀⠀⢠⡟⠁⠈⠳⣤⡉⠛⢾⣦⡇⣸⢁⡟⠒⠶⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡶⢋⡼⠀⣾⠀⠀⠀⠀⠀⠙⠲⢸⣿⠇⠃⣸⡃⠚⠛⠳⠶⠟⣹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠘⠛⠋⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⡿⠙⠛⠶⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣄⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠖⠀⠀⠀⠦⠖⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⣲⣦⡀⣠⣤⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢉⣠⣤⣄⣀⠈⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢸⣋⣫⣿⢿⣞⠙⡆⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠁⠀⠀⠈⠋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢈⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢁⡠⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣀⠀⠈⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⢎⡼⢡⠞⢠⠀⡆⢸⠀⡆⢱⡄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠸⠀⠃⠀⠙⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠒⠀⠀⠀⢨⣙⠋⣻⡏⠀⠀⠀⠀⠰⠶⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠟⣶⠄⠀⠀⠀⠀⠀⢀⡇⠉⠀⣿⣟⠀⠀⠀⠀⠀⠈⠉⣀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣭⡿⢆⣸⣿⢤⡄⠀⠀⠀⠀⢼⠁⢸⡆⣹⡟⠀⠀⠀⠀⠀⠀⣿⣏⣀⣴⠶⠛⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢴⢿⠿⣿⡟⣫⡤⠟⣻⣟⡃⠈⠇⠀⠀⠀⠀⣀⣤⣎⣁⣿⣿⡀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⣤⡶⠚⠉⠉⠠⣿⡦⢄⡀
⠀⠀⠀⠀⠀⠀⢡⣤⠘⣧⣌⣹⠋⠀⠘⢻⣧⠀⣶⠄⠀⣀⣀⣀⣀⡀⠀⠀⢹⣯⡀⢀⣀⢠⡤⠰⢶⡶⠉⠉⠃⣸⢛⣄⠀⠀⣠⣿⠷⢀⡟⠢⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠉⡀⣬⣄⠀⠀⣼⣿⠀⠉⠀⠘⠋⠉⠙⢻⠃⠀⠀⠀⢹⣷⡄⠋⠀⠀⠀⣿⡇⠀⠀⠀⠉⠀⠁⠀⠠⣿⡿⠀⣎⡑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠥⠀⠀⡟⠁⠀⠀⠀⢀⣷⠤⣤⣿⡇⠀⠀⠀⠀⢿⣿⣶⣿⠃⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡗⠀⠀⠀⠀⠀⠀⢐⡟⠉⠀⠀⠀⠀⢠⡄⢸⣿⣿⣷⣴⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⡇⠀⣀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢸⠄⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠿⣼⣿⣿⣀⡀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣴⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡿⠿⠀⠀⠀⢠⠀⠰⡏⣀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠉⠉⠉⠛⠛⠛⢻⡟⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠿⣿⠁⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠁⢀⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣏⢀⠘⠛⠻⠀⠀⠀⠤⠀⠀⣠⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡄⠀⠈⡝⠃⠀⠄⢾⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣀⠐⢢⣄⠀⠀⠀⢂⣽⣿
⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢸⡿⠀⠀⠀⠔⠀⠀⠀⠀⣸⡀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠘⠿⣇⠀⠀⠸⣿⡿⠗⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⡌⠀⠀⠀⠀⡀⢠⠀⠀⠀⢻⡇⣀⠀⠉⠠⠀⠀⠀⠀⡀⠀⠋⠁⢶⡾⣿⡇⠀⠀⠄⠈⠀⠀⠀⠈⠁⠸⠏⠈⢻⣦⡀
⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⢠⣄⠀⠀⠰⡟⠑⠀⠀⢀⣼⡏⠁⠀⠀⠀⠘⢇⡀⠀⠀⠀⠀⠀⡀⠀⣿⣿⡿⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⣤⣾⣿⣧⠀⠀⠀⠀⠀⠘⢷⣄⢠⠄⠀⠈⠁⠀⢸⠟⠁⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠈⠻⣿⣿⣿⡇⠀⠀⠀⠀⡀⠰⣯⠀⠀⠀⠀⣦⣤⡞⠀⠀⢀⠶⠺⢿⡀⠀⠀⠀⠀⠀⠀⡀⠛⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠹⠏⠹⢏⣀⠀⠀⠀⠀⢞⣷⣄⠀⠀⠀⠘⠙⠁⠀⣠⠊⠀⢠⢼⣧⠀⠀⠀⠀⠀⣸⠃⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠳⡄⠀⠀⠀⠀⠀⠀⠀⠦⢰⠄⠀⢘⠁⠀⠀⠀⠰⣿⡿⢿⠆⠀⠀⠀⠀⠀⣠⡏⠀⢀⣀⣾⣿⣧⡀⠐⠀⠀⠀⠐⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣤⠂⠀⠀⠀⠀⠀⢠⠀⠚⠂⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠈⢻⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠰⣿⠖⢠⡆⠀⠀⠀⠀⠀⠈⣤⣄⣤⠀⡤⠀⠀⠀⣤⣤⣶⠀⠂⠀⠀⣤⣤⣣⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⠀⢀⠀⢀⠑⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠇⢀⠸⡇⠀⠀⠀⣄⠀⠀⡿⢡⠄⣄⠀⠀⠀⢠⣿⣿⠿⢀⠀⠀⠀⢹⣿⡇⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⡇⠀⡁⠈⠀⢀⣿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡇⠀⠀⢴⡧⢠⣄⠀⢸⡆⠛⠀⠀⠀⠘⢀⣀⠀⠀⠶⠀⠰⠟⠛⠋⠁⠘⠁⠙⢻⣿⡿⢻⣿⣿⣷⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠛⠀⡘⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿⡇⢸⣿⣿⣿⠀⠀⠀⠀⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⠀⠸⣿⣿⣿⡆⠀⠀⠀⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠁⠸⠀⠀⢹⣿⣿⣇⢀⠈⢠⡟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣆⠀⠘⢙⣿⣿⣿⠀⠀⢠⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠀⡀⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠛⠀⠀⢸⣿⣿⣿⠀⠀⡈⢻⣿⡇
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠁⢀⡀⢲⣄⠁⣰⣮⣓⠴⠒⠦⠀⠀⠀⠀⠀⢰⠶⢶⣤⣶⣿⣿⡷⣿⡴⠆⣾⣿⡿⠟⠀⠀⠙⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠀⠀⠀⡖⠀⠎⠀⢺⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⣿⣠⣿⠻⡄⠘⠃⠘⢻⣾⠇⢀⣵⣤⣄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⢠⠚⠀⠀⠀⠀⠾⣿⠛⠏⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⠀⠀⢀⠄⠀⠼⣋⡀⢸⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠿⠀⠸⠋⠀⠀⠀⡄⢸⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⡀⠉⠰⠏⣴⡍⠁⣄⡈⠛⠋⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⢀⣸⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⡰⢀⡈⠂⠀⠘⠛⠛⠛⣹⣿⡇
⠀⠀⠀⠀⠀⠐⠉⠀⠂⠀⢠⡞⠀⠐⠂⠀⠠⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⣤⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡖⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⠟⠁⢀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠺⠛⠇
⠀⠀⠀⠀⢀⣀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠁⠀⠈⠁⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠘⠳⠂⠐⠞⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠛⠙⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣄⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡠⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠺⠁⠀⠘⡆⠀⠀⠀⠀⠀⠄⠄⠀⠀⣦
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠃⠀⠀⠀⠸⡄⠀⠀⠀⠀⠑⠈⢠⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠓⡄⠀⠐⠀⠀⠀⠀⠀⣃⠀⠀⡒⠀⢀⡄⢠⣼⣿⣃
⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠙⣿⣜⣦⠐⠀⠀⠀⠀⠄⠀⢢⣍⠓⢿⠎⠂⣄⠀⣤⠸⡦⣤⣤⠀⠀⠁⠉⠻⢿⣿⣷⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠐⠋⠁⡴⠻⠛⢁⣤⣴⡀⡀⠄⣤⣿⢛⠿⠿⠆⠀⠿⠀⡋⢀⢁⣷⢦⡀⠀⠀⡄⠀⠘⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠢⡀⠀⢀⣼⣷⣶⣾⠏⠘⣷⣉⠋⠀⠀⠳⠀⣰⠞⠉⠀⠐⣾⣷⣿⠀⠁⠶⠁⠀⠘⡅⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢈⣀⡸⣏⣴⣬⣿⣶⣰⣻⣿⣼⠶⠧⡈⢾⠁⠀⠀⠀⠀⠘⣿⣧⠀⠀⡂⠀⢵⡌⢺⠀⣀⣠⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠐⠊⠟⠁⠀⠀⠀⠉⠙⠲⣤⠈⢻⡀⠀⢀⠼⠂⠀⠀⡀⠒⢀⠙⠋⠀⠀⠁⣠⡾⢹⡿⠚⠁⠀⢸⡎⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣼⡇⠀⠀⠀⠀⠀⠀⠉⠛⠛⢰⣾⡟⠛⠉⠉⠀⠈⢸⠀⠀⠀⠈⡇⢻⣇
⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⣿⡇⠀⠒⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡯
⠀⠀⠠⠀⠁⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⣜⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⣇⡀
⠀⠀⠈⠀⢀⣀⣤⣤⠖⠉⠀⠀⢀⠀⠀⠈⠀⠙⣿⡿⣷⡀⠀⠀⣀⡠⠤⠖⠉⣠⣾⣿⡿⠁⠀⠀⠀⣀⡀⠀⠀⠤⢮⣸⡿⢿⣿⣿⣯
⠀⠀⠀⠀⠥⠞⠛⠛⠂⠀⠀⢀⡞⠀⠀⠀⠀⢒⣮⣽⡿⠟⠂⠊⠩⠓⠒⠂⠈⠁⠀⠀⠀⠀⠀⠒⡨⠋⠀⠀⠀⠀⣶⢾⣷⠟⠈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠪⠂⠐⠀⠀⠒⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠠⠄⠐⠒⠈⠀⠀⠴⠛⠉

⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⢠⡤⠤⡤⠤⣤⠤⢤⠤⢤⠤⢤⠤⠤⡤⠤⡤⠤⣤⠤⢤⠤⢤⡤⢤
⠀⠀⠀⠀⠱⡄⠀⣖⣻⣚⣻⢓⣻⣓⣻⢛⣿⣛⣖⣛⣖⣛⡖⣛⣗⣛⣗⣻⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡸⠀⡏⠙⠋⠉⠛⠉⠛⠉⠙⠋⠉⠛⠉⠛⠉⠙⠋⠙⠋⠉⠛⠉⠛⠉⠙⠋⠙⠋⠉⠛⠉⠛⠉⠙⠋⠙⠋⠉⠛⠉⡇
⠀⠀⠀⢀⡞⠁⣀⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢰⣆⣰⠀⣰⣆⢀⡶⣦⢰⢶⡄⣆⡴⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠈⠉⠉⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠹⠀⠟⠻⠀⠏⠁⠸⠉⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢧⣀⣂⣀⣒⣀⣰⣀⣐⣂⣀⣂⣀⣒⣀⣰⣀⣐⣂⣀⣆⣀⣐⣀⣐⣀⣀⣂⣀⣆⣀⣐⣀⣐⣂⣀⣂⣀⣂⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠾⠤⠮⠴⠯⠴⠯⠴⠧⠴⠥⠾⠥⠾⠤⠾⢤⠮⠴⠯⠴⠧⠴⠧⠴⠥⠾⠥⠾⠤⠾⠤⠮⠴⠯⠴⠯⠧⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⣀⡀⢀⠀⣀⡀⠀⣀⡀⢀⠀⣀⠀⡀⠀⡀⠀⣀⡀⢀⠀⣀⡀⣀⠀⣀⣀⠀⣀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡇⠀⠀⠀⢸⣉⡁⢸⠀⡿⣟⢸⣏⡁⢸⡀⣿⢸⡿⡼⣿⢸⣍⡁⢸⠈⣻⡄⣿⢸⡉⣹⠆⡿⢦⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⠀⠀⡀⠉⡁⠈⢀⠁⢈⡀⠉⡁⠀⡉⠁⢈⠀⢀⡈⠀⡉⠁⣈⠀⢉⠀⠉⡀⠉⡉⠀⣁⠀⢁⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⢒⡶⢲⠖⢲⠖⣲⠖⡲⠖⡶⠒⡶⢒⡶⢒⠶⢲⠖⢲⠖⣲⠖⡲⠖⡶⠒⡶⢒⡶⢒⠶⢲⠖⢲⠖⣲⢾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠭⠉⠩⠍⠩⠍⠉⠭⠉⠭⠉⠩⠍⠩⠍⠉⠭⠉⠭⠉⠩⠍⠩⠍⠉⠭⠉⠭⠉⠩⠍⠩⠭⠉⠭⠉⠩⠍⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠶⣦⢠⡶⠀⠀⢶⡶⢠⣄⣰⡄⣶⠆⠀⢰⡄⢠⡴⢦⡀⣴⣦⠀⡶⢦⡀⠀⣴⠶⡄⢀⣦⠀⣆⣰⠀⢸⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠤⠞⠸⠏⠀⠀⠸⠇⠸⠏⠹⠃⠿⠅⠀⠸⠧⠘⠦⠼⠃⠟⠻⠐⠧⠾⠃⠀⠿⠴⠃⠞⠛⠇⠸⠇⠀⢸⠉⢀⡼⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⣀⣠⣄⣠⣄⣀⣤⣀⣤⣀⣠⣄⣠⣄⣀⣤⣀⣤⣀⣠⣄⣠⣄⣀⣤⣀⣤⣀⣠⣄⣠⣄⣀⣤⣀⣠⣄⣸⠀⡎
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠈⠉⠉⠉⠉⠉⣯⢽⣯⢼⣭⠼⣭⠼⣭⠼⣥⡽⢭⣿⢥⣯⢥⣯⠥⣯⠽⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠓⠒⠓⠒⠓⠒⠚⠒⠚⠒⠒⠓⠒⠓⠒⠓⠒⠚⠒⠚⠒⠚⠃


⠀⠁

⠀⠈⠙⠛⠿⠭⠭⠭⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡲⠀⣦
⠀⠀⠀⠀⠀⠀⠀⡀⡆⢀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢤⠀⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠃
⠀⠀⠀⠀⠀⠀⠀⡗⠀⠸⣧⠈⣧⠱⠀⠀⠀⠀⣦⠀⠸⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⢤⣤⣄⢠⣄⡀
⠀⠀⡀⣠⣀⣶⣾⠀⣺⠔⢻⡗⠿⠐⢖⡿⠀⠸⣿⡀⠃⠄⠹⢧⡤⠀⠀⠀⠀⠀⠀⢀
⠀⢘⣿⣿⣿⣿⣿⣾⣿⣀⠙⠓⠂⠀⠛⢻⢖⡖⣰⠀⡀⣀⢐⠆⣲⢀⡆⢰⠄⡇⢰⠌⣠⣁⣄⣀⠄⡀
⠀⠐⣿⣿⣿⣿⡿⡿⠿⠿⠿⠶⠶⠤⠄⠀⠀⠀⠀⠈⠀⠁⠈⠀⠉⠈⢁⠈⠀⡁⠈⠐⠁⠟⡜⠬⣲⡀⠐⠠⠄
⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡄⢠⠀⠀⠀⡞⠀⠀⢦⡀⠀⠀⠀⠐⢧⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣽⠀⣄⡆⠀⡄⠀⢀⠚⠀⠂⠰⠀⠀⠀⢸⠇⠀⠉⡈⢁⠀⠀⢤⢬⡜⡃⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠾⡱⣿⡇⣼⣇⢀⣿⡂⢘⣷⠀⢀⣮⡀⠀⢓⠀⠀⠘⡀⠁⡅⢨⢸⡇⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢏⠀⢻⢃⠐⡟⠃⠨⡿⠀⠈⢿⠁⠐⢼⠃⠀⠄⠇⠷⡇⠈⠄⠄⠁
⠀⠀⣿⣿⣾⣶⣶⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠈⠈⠁⠁⠀⠀⠑⠒⠀⠘⠁⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠠⠤⠤⠾⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡄⡀
⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠄⠀⠀⠀⠀⣰⡾⡿⠿⠟⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⡀⠀⠀⠀⡰⠊⠋⠀⠀⠀⣸⣿⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡃⡇
⠀⢀⠁⠀⢀⠞⠁⠀⠠⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⢫⣽⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⡇⡇
⢠⣾⠀⢠⠏⠀⠀⠀⣴⣤⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⡇
⢸⡟⠀⡏⠀⠀⠀⢀⣿⣗⣴⠶⠎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡇
⢸⡇⢸⠁⠀⠀⠀⣼⢩⣽⣿⣆⣠⡶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡇
⢸⣿⡟⠀⠀⠀⣀⣿⢿⢻⣿⣿⣭⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇
⢸⣿⡇⠀⠀⠀⢿⣼⣿⣮⣿⡯⢱⣟⣻⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⢟⣿⡆⡇
⠘⣿⠷⠀⠀⠀⠻⣿⢿⣭⢿⣿⣿⣯⣽⣿⢿⣿⣿⣽⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢙⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠁⡇
⠀⠟⠀⠀⠀⠠⡄⢙⣾⣻⠹⢻⣟⠛⢻⣿⡿⢾⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡶⠸⢿⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠄⣸⠀⠙⠂⡈⠙⠓⢼⣿⣷⣿⣿⣿⣿⣿⣿⣧⣻⣿⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠀⢸⡇⠀⡇
⢸⡿⢻⠛⠛⠈⡷⢸⠀⣸⠊⠁⠀⠐⣋⣭⣿⣭⣽⣿⣏⡹⣿⣿⣿⣿⡿⠻⡿⢿⣿⣿⣿⣿⣿⣿⣿⠀⠢⠀⢠⣿⠀⡇
⢸⣟⠘⠀⠀⣀⡀⠀⠨⠋⢠⠀⣤⣿⣿⣿⣿⣿⣿⣿⣦⡳⣜⣿⣿⣿⣿⠀⡗⢸⡟⣿⠿⣿⣿⣿⣿⠀⡇⠀⢸⣿⠀⡇
⢸⣿⠀⠀⡆⠀⠘⠺⠀⣼⠇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⣿⠀⣷⢸⣿⡇⣾⣿⣿⣿⣿⠀⡅⠀⢸⣿⠀⡇
⢸⣿⠀⠀⡇⠀⠀⠀⠀⡄⢀⣿⣿⣿⡿⢿⣿⣿⣿⣿⡟⢿⡇⣿⠞⣿⣿⣷⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⡀⠀⢸⣿⠀⡇
⢸⠿⠀⠀⠄⠐⣿⠷⠀⠒⠀⣿⠿⣿⣧⣬⢿⣿⣿⣟⡻⢿⣿⣿⣧⣿⣹⣭⣿⣆⠌⠻⣿⣿⣾⣿⣿⢀⠀⠀⢸⣴⠀⡇
⠘⣦⠀⠀⠀⠀⣿⡀⠀⡁⢨⣿⣷⠻⣿⣷⡤⢿⣿⢻⣽⣿⣿⣿⣷⠜⣿⣿⣿⣿⠼⣅⢻⣿⣿⣿⠿⢬⣁⡄⢸⣯⠀⡇
⢸⣻⠀⠀⠃⢸⣟⣦⠀⢇⣰⡿⣾⣿⠉⣻⡅⢸⣿⣾⣋⠉⢻⡷⣿⠄⣿⣿⣿⣿⣷⣿⣿⣟⣡⠀⣶⠊⠉⢿⣿⣉⠀⡇
⢸⢉⠀⣴⡿⣿⡟⢧⠞⢿⣿⡀⢸⣿⣷⣿⡾⠛⣿⢿⣿⣷⣼⣧⣵⡀⠛⣿⣿⣿⣿⣿⣿⣿⣧⡘⣿⣷⢀⠸⢿⣿⠀⡇
⢸⡏⢸⣿⠀⢸⣇⡈⠰⣏⢿⣴⣿⣻⣾⣿⠀⠀⡛⣸⣧⣾⠎⢿⣿⣏⢸⣿⣿⢿⣿⣿⣿⣿⣿⣇⣽⣽⣿⣴⠀⠘⠀⡇
⢈⡄⣼⠏⠀⢸⠁⣼⣇⠈⢫⠉⢻⣿⣧⣿⡀⢨⣿⣾⣿⣿⣆⣼⣿⢿⣇⣙⣷⠄⢻⣿⣿⣿⣿⣿⡛⢻⣏⣈⠖⠀⠀⡇
⢨⣇⣯⠨⣦⣿⠐⣿⣿⣷⣾⣦⡘⣿⣿⡿⣧⣴⣯⣸⣿⠁⣾⢁⣛⡀⠀⢻⡷⢠⣾⣿⣿⣿⣿⣿⣿⣦⡙⡛⠀⠀⠀⡇
⢸⣿⣿⡀⠸⣟⢠⣬⣿⣉⣿⡷⠻⣿⣿⣿⣿⡿⢻⣯⣼⣶⣾⣿⣿⣿⠎⠻⢷⣾⣣⣿⣿⣿⣿⣿⡿⣿⣆⠇⠀⠠⠀⡇
⢸⣿⣿⣠⡔⠯⣘⠏⠈⢻⠻⣧⢸⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⡿⠻⠿⡀⠀⡆⠙⣿⣹⣿⣿⣿⣟⣻⣿⡿⠷⡦⢀⠀⡇
⢸⣿⣿⣿⣷⣶⣼⠀⠀⠈⣼⣿⣿⣿⡧⣿⣿⣿⣿⣿⠇⢾⠟⢅⡀⢰⣿⣄⡀⠀⣿⡷⣄⠉⣿⣿⣿⠩⡇⠘⠷⠊⠀⡇
⢨⠅⣿⣿⣿⣿⡅⠀⠀⡆⢿⢹⣿⣿⣿⢿⣿⢻⣿⣿⢠⣂⣠⣿⣗⢮⡽⣇⠈⠛⣬⢳⠟⣰⣾⣿⣿⠀⡇⣲⠀⠀⠀⡇
⠀⠀⢸⣿⣿⣻⣟⠀⢀⣌⣿⣿⡘⣿⡟⣯⣿⣨⣿⣿⣤⣭⣥⣿⣿⠟⠀⠻⠀⡖⢛⣿⢿⣿⣿⣿⡇⡐⡇⠁⠀⠀⠀⡇
⠀⠀⠈⣿⣿⣿⣟⠋⣹⡟⠿⢿⣷⣒⣛⣛⣿⣷⣽⣿⣿⢹⣿⣿⣿⠟⠁⠀⠀⠈⢧⣿⣼⣇⣩⣿⢧⠚⢣⠀⠀⢰⠀⡇
⠀⠀⠀⢸⣿⢻⡿⠀⠻⠃⠀⠀⠈⠋⠙⠛⢿⣿⣿⣿⣿⣿⣏⣿⣞⢧⠀⠀⠀⠀⠀⢻⣿⣿⡷⣿⢸⠀⢸⠇⢀⠘⠀⡇
⠀⠀⠀⢸⣿⢃⣿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢻⡿⣹⣿⣿⠈⢓⢺⣦⣀⠀⢀⠸⡿⣿⣿⡏⣼⡇⠸⡆⠀⢸⠀⡇
⠀⠀⠀⠀⠸⠘⠟⠡⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣇⣿⡟⣻⣿⣿⣿⣽⡛⠐⠀⣴⣾⣿⣿⣁⣧⣿⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠠⠶⠶⠄⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠁⠟⠻⠯⣽⣿⡇⢉⡀⠛⡛⢉⠛⣿⣿⣿⣇⠉⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠣⠁⠘⠻⠀⠁⠈⠁⠘⢿⣟⣿⣄⣿⣷⠀⡇
⠀⠀⠀⠈⠉⠁⠀⠉⠉⠀⠉⠀⠉⠀⠀⠀⠀⠐⠒⠒⠒⠒⠐⠒⠒⠒⠐⠀⠀⠀⠄⠠⠤⠐⠆⠖⠛⠟⠟⢿⠷⠻⢀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠲⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣯⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⢸⠈⢹⠉⠉⣻⠛⠛⠛⡟⠛⠛⢿⠛⠛⢻⠛⠛⢻⠛⠛⢹⡏⠉⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢾⣦⣼⣄⣀⣽⣀⣀⣠⣇⣀⣀⣼⣀⣀⣸⣀⣀⣸⣄⣀⣼⣄⣤⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠤⢤⣉⣉⣉⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣁⡠⠿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⠷⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠘⠛⠛⣶⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣷⠀⠀⠀⠀⣴⠟⠛⠃⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠀⣼⠋⠉⠉⠉⡏⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⠟⠀⠀⢿⣤⣤⣤⣤⣿⣤⣤⣤⠾⠁
⠀⠀⠀⠀⢀⣤⠶⠶⢶⣤⣤⣭⣍⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⡴⠶⢶⣤⡀
⠀⠀⠀⠀⣟⠀⠀⠀⡼⠀⠀⠀⠈⠙⣦⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⠁⠈⢹⣧⠀⢀⡼⠛⢶⣄
⠀⠀⠀⠀⠙⢦⣄⣀⣳⡀⠀⠀⢀⣰⠟⢀⣀⠀⠀⣿⠁⠀⠀⠙⣳⣦⣀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠙⢦⣠⡾⠋⠳⣮⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⢛⣻⣷⠞⠋⠉⢻⡄⠹⣦⡀⢠⠞⠁⠀⠙⢷⣄⠀⠀⠀⢿⡀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠻⢦⣄⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠈⢧⠀⠀⣠⠟⠀⠈⠻⣧⡀⠀⠀⠀⠀⢹⡇⠀⠀⠈⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣀⣼⠷⠛⠁⠀⠀⠀⠀⠈⠻⢦⣀⣀⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⣶⣶⣶⡶⠶⣶⣶⣶⣶⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡿⠿⢰⣇⠸⠿⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣇⣛⢻⡿⢛⣃⣿⣿⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⣿⣿⣘⣃⣸⣿⣿⣿⡇⠀⠀⣿⡴⠖⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣡⣴⣾⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣧⡴⠶⠶⠶⠶⣿⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠹⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠙⠳⠶⠶⠦⠴⢿⣿⣿⣿⡿⠟



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠀⠈⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⣀⡤⠖⢒⣶⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠘⠁⢀⠤⠚⣛⡛⢟⣟⣻⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⡄⠀⠀⠀⠁⡰⠟⢛⣿⣮⣭⣧⣤⣤⣝⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢰⢱⣤⣆⡴⢊⠔⢺⣿⣏⡙⣿⣿⣿⣿⣿⣿⢦⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠛⢿⡴⠁⠀⠀⢻⠀⠀⢹⣿⣿⣿⣿⣿⡇⡟⢶
⠀⠀⠀⠀⢀⣠⡤⠶⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠺⠤⠈⠷⡦⠾⠆⠀⣠⣿⣿⣿⣿⣟⡏⢱⣾
⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣤⣤⠄⠀⠄⠀⠘⠻⣿⣿⣿⣿⡟⠀⣾⡟
⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⣉⢿⣿⣀⠀⢀⠀⠀⡀⠀⠈⠙⣿⣿⣷⡦⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠆⠀⠀⠀⠀⠀⠀⠻⢆⣿⡿⠋⠉⠁⠉⠁⢀⠀⠀⠘⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢈⠳⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠈⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈
⠈⠳⣦⣌⡉⠙⠛⠛⠛⠛⠛⢉⡉⠤⠀⡾⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⡀⠀⠀⠀⠐⡆⠀⠀⠀⢹⡤⡤⠐⠁⠀⠀⠀⣆⡀⠀⠀⠀⢸⠤
⠀⠀⠈⠙⠒⠷⠠⠀⠠⠈⠄⠄⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⢀⣧⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣦⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠐⢶⠶⠆⠀⠠⠀⠀⠀⠀⠙⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠞⠻⢦⣤⣀⣉⣉⠉⠉⠉⠉⠉⠉⠉⣉⣉⣁⣬⠟⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠶⣶⣤⣬⣭⣭⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣶⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⢸⡇⢸⡏⢹⡏⢹⡏⢹⡏⢹⡏⠉⡏⠉⡇⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣄⣸⡇⢸⡇⢸⡇⢸⡇⢸⡇⢸⡇⠀⡇⢀⣇⣠⣧⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠙⠛⠛⠻⠿⠷⠾⠷⠾⠷⠾⠿⠿⠛⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⡾⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⣦⣤⣤⣀⣀⣀⣀⣤⣤⣤⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣤⣤⣤⣀⣀⣤⣤⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣉⣉⣉⣛⣛⡋⠉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡇⠀⠀⠀⠀⠀⠀⢀⣴⠶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠿⢿⣶⣶⣤⣤⣤⣤⣤⣴⣶⣶⠿⠟⠛⢻⡇⠀⠀⠀⠀⢀⣴⣟⡱⠋⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠛⠻⡿⠷⠶⣦⡀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢰⡿⠃⠈⠹⣦⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⣠⡇⠉⠛⢸⡇⠀⠀⢻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡇⠀⠸⣧⣀⣠⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⣿⣷⣾⠟⠁⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢟⣛⡛⣟⠛⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠉⣀⣿⣀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡏⢹⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⡄⢠⡤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣤⣤⡀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⢠⣤⣤⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇
⣰⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣤⠤⠤⡄⠀⠀⠀⣤⠤⠤⣄⠀⠀⠀⢠⠤⠤⣤⠀⠀⡇
⡇⠀⠀⠀⣾⠛⠛⣷⠀⠀⠀⢸⠀⠀⣿⣀⣀⡇⠀⠀⠀⣿⣀⣀⡿⠀⠀⠀⢸⣀⣀⣿⠀⠀⡇⠀⠀⠀⣾⠛⠛⡇
⡇⠀⠀⠀⠻⠤⠤⠟⠀⠀⠀⢸⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡇⠀⠀⠀⠿⠤⠤⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⠒⠒⡆⠀⠀⠀⣴⠒⠒⣦⠀⠀⠀⢰⠒⠒⣶⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⣤⣤⡇⠀⠀⠀⢿⣤⣤⡟⠀⠀⠀⢸⣤⣤⡿⠀⠀⡇
⡇⠀⠀⠀⣴⠶⠶⣦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣴⠶⠶⡆
⡇⠀⠀⠀⢿⣀⣀⡿⠀⢀⡴⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⢦⡀⠀⢿⣀⣀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠴⣯⡤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡽⠦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣤⣤⡄
⡇⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣤⣤⣤⣤⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⡇
⡇⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠋⠙⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⠿⡿⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠈⢉⣉⣭⡛⠃⠀⠀⠛⣫⣍⣉⠉⠀⢸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠻⣆⠀⠈⠸⠿⠙⢂⣀⣀⠛⠽⠿⠈⠀⢀⡞⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⡏⡷⠶⠶⠚⠋⠁⠀⠈⠉⠛⠲⠶⠶⡏⣏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣷⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⠀⠀⠀⠀⠀⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠒⠤⠔⠚⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⣶⠞⠉⠩⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠻⡝⠒⠲⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⠀⠀⠀⠀⣇⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢹⡀⠀⠀⠀⠀⠈⠉⠓⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⢶⣄⠈⢆⣀⠀⠀⠀⠀⠀⢀⣠⠎⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡄⠘⣇⠉⠉⠉⠉⠉⢁⡟⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠘⡇⠀⠘⡄⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢿⠀⠀⠸⡄⠀⠀⡼⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⡄⠀⠀⠹⡄⡼⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⡇⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⡇⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⠞⠓⠒⠚⠁⠀⠀⠀⠀⠀⣠⡤⠛⢵⡄⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣄⠀⡠⡃⠀⠀⠀⠀⠀⠀⠀⡇
⠉⠓⠲⠤⣤⣀⡀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣀⣠⡤⠴⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣄⣀⣄⣀⣀⣀⣄⣀⣀⣀⣠⣀⣀⣄⣀⣀⣄⣀⣀⣀⣀⣀⣠⣄⣀⣄⣤⣀⣠⣠⣄⣠⣤⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⣀⠀⠀⢀⣀⠀⠀⢀⠀⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡇⠀⠀⠀⠘⠋⠃⠀⠘⠤⠇⠀⠀⠽⠂⠀⠸⠈⠀⠀⠛⠀⠀⠸⠀⠀⠘⠻⠀⠀⠻⠄⠀⠀⢰⡇
⠀⠿⠤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⢀⣀⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⣿⡟⠛⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠠⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠠⠀⠀⠀⠸⠀⡄⠤⠤⠤⠄⠤⠤⠤⠤⢤⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣆⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠳⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢠⠞⣧⠀⠀⢀⣀⡤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⠀⠀⣠⠋⠀⠘⠚⠉⠉⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠙⠲⢤⠞⠉⠉⠙⠣⡀⠀⠉⠳⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⡈⣧⠀⠀⠀⠀⡰⣋⣞⠉⣱⣄⡀⢸⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡼⠁
⠙⣄⠀⠈⠉⠉⠉⠀⠈⠉⠉⠀⠀⢠⡾⣽⠣⣼⠃⢌⣇⡜⡸⠀⢀⡤⠲⡈⠳⣄⠀⠀⠀⠀⠘⠧⠤⠤⢤⣤⣀⣀⣀
⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠒⠲⢧⡦⢴⠛⣬⣶⠁⡰⠛⢢⠰⢇⣠⠟⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠈⢾⣠⣿⠏⠻⣅⣀⠼⣀⡠⠃⢱⠛⡆⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⣨⠆⠀⠀⢠⡇⠀⠀⠀⠀⠠⠁⠀⣼⠟⠁⠀⠀⠘⠷⣤⢼⠒⠒⡇⢀⠇⠀⠀⠀⠀⠀⠐⠛⠲⠤⣄⣀
⠀⢀⡠⠖⠋⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠳⠿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡨⠽⠗
⠶⣉⡀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉
⠀⠀⠉⠙⠒⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡛⢏⣙⡦⢵⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠉⢫⠈⣇⣸⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡏⣧⣞⡁⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⣀⣈⣶⡄
⠀⠀⠀⣴⣁⣤⠤⠤⠶⠒⣶⠂⠀⠀⠀⡠⢿⡀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠉⠙⠲⢤⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣠⠔⠋⠀⠀⠙⠓⠒⠚⠋⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈⠑⠢⢌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡠⠞⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡽⠃⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⢀⡴⠋⠸⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⠀⠀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⢿⣿⡷⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⣹⣧⠀⠀⠀⠸⣿⡟
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠻⠿⣀⠀⠀⠀⠿⠿⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⢀⣴⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣴⣶⣗⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣀⣀⣶⣴⠚⢿⠿⠿⠟⠻⠁⠛⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣶⡦⢀⣠⣾⣿⣿⣿⣿⣿⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠛⠃⠙⠻⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠈⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡉⠀⠀⠁⢠⣤⡄⠀⠀⠀⠀⡤⣤⡄⠀⠀⠀⢛⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠻⠿⠃⠀⠀⠀⠀⠻⠿⠃⠐⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⣀⠀⠀⠁⠀⣠⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠳⣶⣶⡾⠋⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⠂⠠⢀⡀⠀⡀⠠⠔⢻⠆
⠀⠈⠁⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠂⠸⡅⠀⠀⣤
⣆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⠀⠀⢂⣀⠙⠁⠀⠀⠀⠈⠀⢀
⠷⠀⠆⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⢉⠐⠂⠰⠖⠉⠆⠀⠃
⠘⢧⠒⠀⢊⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠀⠀⠉⠲⡀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠁
⠀⠀⠀⠀⠀⠀⠻⢤⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⠰⠍⠀⠀⠀⠀⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠔⠠⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡻⠗⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢀⠂⡀⣀⠀⡀⣀⡀⣀⡤⠖⠉⢡⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠋⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣀⠀⠄⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣤⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⠀⠐⠄⢀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡀⠠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠺⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠀⠀⠀⢈⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠻⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣄⡤⠀⢠⠖⡄⠀⠠⣆⠀⠀⡶⠄⠀⢠⠀⠀⡶⠀⠀⣤⡀⠀⢠⠀⠀⠀⢸⡇
⠀⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡧⠤⠤⠤⠥⠤⠤⠤⠭⢤⠤⠬⠭⠤⠤⠥⠤⠤⠤⠤⠤⠤⡤⠤⠤⠥⠤⠬⠥⠤⠤⢼⠃
⠀⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠇⠐⠀⠐⠀⠀⠂⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⠿⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠂⠀⠂⠀⠐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠐⠀⠐⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇⢰⠒⠒⠒⠒⠒⠒⠒⠒⠂⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣀⠀⠀⢀⣀⠀⠀⢀⣃⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠋⠛⠸⠀⠀⢸⡏⠉⣻⣿⠋⠻⣿⡿⠿⠛⢻⣿⠉⢿⠹⡏⢻⠨⠃⠀⠈⡏⠉⣹⣿⠋⠻⣿⡿⠿⠛⢻⡟⠉⠏
⠀⠈⠀⠇⠀⠈⡇⢠⡇⠙⠀⠀⣼⠁⠀⠀⠸⣟⠐⠀⢸⠁⠀⠀⠃⠀⠀⡇⢠⡇⠙⠀⠀⣾⠁⠀⠀⢸⣇⠸
⠀⠐⠀⠀⠀⢠⠇⠸⠇⠀⠀⢰⡇⠀⠀⠀⠰⡏⠀⠀⠈⠀⠀⠀⠀⠃⢀⡇⠸⠇⠀⠀⢰⡇⠀⠀⠀⢸⠋
⠀⠀⠃⠀⠀⠿⠀⠀⠀⢀⠀⠸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠓⠀⠀⠺⠀⠀⠀⢀⠀⠸⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠉⠉⠙⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠉⠻⣿⣦⣄⣠⣤⣤⣴⣶⣶⣶⣾⡿⠿⠿⠿⠟⠛⠻⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠙⠻⣿⣯⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣽⣿⣷⣄⡀⠀⠀⠀⠙⠻⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣟⡛⠛⠋⠉⠉⠉⠙⠻⣿⣶⣤⣀⠀⠀⠀⠉⠛⠿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣻⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⠈⠙⠻⣿⣷⣤⣀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⣀⠀⠀⠀⠉⠙⠛⠿⢿⣿⣿⣷⣶⣶⣶⣶⣿⣿⠁
⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣄⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢉⣿⠃
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⡾⠋
⠀⠀⠀⠀⣼⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠉⠉⠉⠉⠁
⠀⠀⠀⠀⣿⣿⡿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣦⣄⣀⣠⣾⠟⣿⣿
⠀⠀⠀⠀⠘⣿⣧⡈⠙⠻⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣿⣿⣿⠟⠋⢠⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠈⠉⠙⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⢀⠄
⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣇⣀⣠⣤⣬⣄⣤⡤
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠙⠁⠀⠀⠀⠀⠀⢲⣶⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣀⡀
⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡏⠁
⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⡟⠸⠿⠟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⠟⢻⡿⢿⣿⣿⣟⡿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⣿⢸⣿⡇⠀⢸⣿⣿⣿⠏⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⠀⠀⢸⠁⠿⠀⠀⢹⣧
⠀⠀⠀⠀⠀⢻⣾⣿⠇⠀⢸⣿⣿⡟⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠘⠁⠀⢸⡆⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢸⣿⣿⡇⠀⠀⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⢸⡇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⣿⢿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠾⠀⠀⠀⠀⠈⠁⠀⠀⠀⠻⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠷⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠄⠀⠤⢠⣀⠀⠀⠈⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠢⣄⡀⡀⡀⢠⣷⣧⣠⣿⣇⠂⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⢈⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣋⡀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠄⠀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢞⠂⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠏⠁⠄
⠀⠀⠈⠉⠉⠉⢩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠠⠶⠿⠛⠛⠛⠛⠛⠋⠉⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⠠⢹⠏⡟⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠟⠀⠀⠸⠐⠀⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⣀⡉⢛⣤⡽⠭⣍⠛⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣳⠘⢋⢶⣶⣶⡆⠀⠹⢯⠁⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠙⠛⣻⠀⠘⠘⠛⠋⠀⠀⠀⢣⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢠⣉⣒⣀⣤⣴⡆⠀⠀⣰⣇⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡙⢦⣤⠶⠋⠀⣠⢾⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⢴⡞⠁⢸⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⣀⣤⠤⠔⠆⠳⠶⠍⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠄⠐⣶⣾⠇⠀⢈⣷⠦⣄⡀
⠀⠀⠀⠀⠀⠘⠯⣁⡀⠊⠀⠀⠀⣀⣀⡽⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢁⠄⣠⠞⣿⠏⠀⣠⡟⢹⡄⠀⠉⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠸⡋⠀⡠⠋⠢⣀⠀⠀⠀⠀⠀⣠⠊⠀⣠⢊⢴⠏⣸⣿⣤⣾⠟⠀⡿⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠀⠈⠑⢄⡀⡠⠞⠁⠀⡔⠁⢀⣿⠀⣿⣿⣿⠏⣤⠾⣟⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠙⠆⠀⠀⠀⡆⠀⢸⡇⢰⣿⣿⠏⡼⠡⠀⢸⡦⠀⠳⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⢀⡠⠖⢃⢰⣟⡇⢸⣿⠏⣸⠇⠀⠀⣸⠁⠀⣰⠻⡄⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤⠒⠉⠀⠀⠘⡛⠿⠃⢸⡟⠠⡿⡄⢀⣰⠏⠀⡴⠃⠀⠘⣄⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢸⡁⠀⠈⠁⠙⠋⢀⡜⠁⠀⠀⠀⡨⠒⠛⠁⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢸⢀⠀⠀⠀⠀⢠⠎⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠉⠀⠀⠀⡰⡇⢀⡴⠛⡁⠀⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠦⠁⠀⠀⠀⠀⡘⠉⠣⣀⣑⠶⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢃⢰⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡼⢇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠈⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣦⣄⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⢷⡶⢶⣶⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠛⠛⠻⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣄⡀
⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⡇⠀⠀⠀⠀⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶
⠀⠀⠀⠀⠛⠛⠛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⢛⡃⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠠⠤⠧⠼⠀⠀⠤⠤⠤⠄⠀⠠⠼⠧⠄⠀⠠⠤⠤⠤⠀⠀⠴⠤⠮⠄⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠰⠲⠒⠲⠀⠀⠖⠾⠖⠂⠀⠰⠖⠒⠆⠀⠐⠶⠒⠒⠀⠀⠖⠲⠷⠂⠀⠐⠲⠒⠆
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣄⣠⣠⣠⣤⣠⣠
⠀⠀⠀⠀⠀⠠⠤⠤⣬⠥⢤⠄⠀⠀⢰⡆⠁⠀⠁⠈⠀⠀⠀⢉⡉⠀⣶⠀⠀⠀⠤⢤⢧⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠜⠓⠐⠂⠀⠀⠐⠀⠀⠀⠐⠀⠛⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠇⠀⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣾⡆⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⢸⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣀⣀⣸⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡏⠉⠉⠉⢹⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣧⣤⣤⣤⣼⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠋⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡾⠥⠤⠤⣀⡀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠏⠀⠀⠀⠀⠀⠉⠓⣄⠀⢀⡼⠁
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡞
⢀⡴⢏⠉⠋⠉⠙⠓⠦⣄⠀⣷⣀⣀⣀⣀⣀⣤⡤⠤⠤⠶⠒⠚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⣾⡇⠀⠑⢦⡀⠢⡀⠀⠈⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⣯⠱⡀⠀⠀⠙⢦⡈⠢⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠹⡄⠱⣄⠀⠄⣀⠙⠢⡘⢦⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠰⠐⠀⢀⡾⠁
⠀⠙⣆⠈⠣⣄⠀⠀⠀⠈⠢⡙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠏
⠀⠀⠈⠛⢦⣀⠉⠒⠦⠤⠤⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠶⠴⠶⠚⠉⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠛⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢀⣠⠴⠚⠋⠉⠁⢀⡠⠞⠋⣻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⢀⣴⠋⠁⠀⠈⠁⡠⠖⠉⣠⠔⠋⠁⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠈⠛⠲⠶⠶⠒⠋⠁⠀⠀⠀⠀⣼⠃⠀⠀⣠⠔⠁⢀⡴⠊⠡⠀⠀⢀⢼
⠀⠀⠀⠀⣠⡞⠻⢭⣀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣴⠋⠀⣀⠔⠁⠀⢀⣀⠤⠞⢁⡾
⠀⠀⢀⡾⠉⠙⠲⢤⣈⠉⠒⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡉⠉⠉⠋⠉⣉⣉⣀⡤⠶⠋
⠀⠀⢸⣧⣀⠀⠀⠀⠈⠙⠲⢤⣀⠉⠒⠤⣄⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⢿⡈⠓⠦⢄⡀⠀⠀⠀⠈⠙⠲⢤⣀⣉⣳⠖⢻⡆
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠉⠑⠒⠒⠒⠒⠚⠋⠉⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣶⣶⣶⣶⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣿⣇⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠛⣿⡟⠛⠃⠀⢸
⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠉⠡⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀
⠀⠢⠠⡤⠄⠠⠠⠠⠤⠄⠀⠠⠤⠄⠠⠀⠀⠠⠠⠀⠀⠀⠤⠠⠀⠠⠤⠄⠀⠀⠀⠠⠤⠤⠠⠀⠄⠤⡄⠠⠌
⠀⠀⠀⡆⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⠲⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠎⠀⠀⠀⠈⠇⠀⠀⠎⠀⠀⠀⠈⠇⠀⠀⠎⠀⠀⠀⠈⠇⠀⠀⠏⠀⠀⠀⠈⠇⠀⠀⡇
⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠸⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⣸⢀⢀⠀⢸⡄⠀⠀⣸⠀⠀⢀⢸⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⡇⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⢀⡔⠒⢲⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢘⣠⡮⠤⠤⠤⢥⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠞⠀⠀⠀⠘⠆⠀⠀⠞⠀⠀⠀⠘⠆⠀⠀⠞⠀⠀⣀⡾⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⣷⣶⣶⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠋⠛⠒⠐⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⣸⣀⣀⣀⣸⡀⠀⠀⠀⢸⠀⠀⠀⢰⣄⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠑⠀⠂⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⣿⢈⣽⡀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⢸
⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠍⠁⠀⠀⠁⣿⠿⠋⢡⡾⢋⣴⣦⣌⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⣴⣤⡙⢷⡄⢸⡀
⡖⠒⠒⠓⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠚⢇⠸⣿⣿⡿⢸⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⡇⢿⣿⣿⠇⠘⠓⠒⠚⠒
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠬⠥⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠬⠅⠂
⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠂⡆⢰⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣀⣁⣃⣘⣘⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⣀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⢸⠉⢉⣉⣉⣉⣈⡀⠀⠀⠀⡴⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⢨⡇⠀⢀⠎⠈⠢⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠘⣀⠤⡄⠒⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠤⠤⠤⠤⠤⠤⡇⢠⠊⠀⠀⢱⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠣⢀⣀⣀⣀⠠⠔⠊





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠻⠿⢿⡿⠋⠁⠈⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⠀⠔⣛⢻⣶⡄⢰⣿⣛⣳⠀⢸⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠉⠙⠉⠹⠃⠈⠉⠉⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠂⠰⠋⡈⠙⢊⠁⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠤⠤⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⡀⠀⠀⠀⣠⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣧⡽⠿⠶⠒⠂⠁⡀⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠻⢀⡿⠈⢇⠀⠀⢠⣿⣿⣷⡄⠀⠀⠀⢸⠀⠀⠀⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠃⠸⡇⠀⠈⢆⡠⠛⣿⣿⡏⠻⣀⠆⢀⡿⠤⠼⠀⠀⠀⠀⠈⠐⠒⠒⢄
⠀⠀⠀⠀⢀⠐⠉⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠈⢣⡀⣿⣿⣷⠀⡜⢀⣼⢥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⡞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠹⣿⣿⣿⡞⢁⡟⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢘⣇⡠⡀⠀⠹⣿⡟⠀⠸⡇⠀⠀⠀⠘⠇⠀⢠⡇
⠀⠀⠀⣼⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⢴⠿⠋⠀⠘⠄⠠⡏⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢦⣤⣤⠆⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣲
⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣸
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⡀⠀⣠⡦⣼⢧⡇
⠀⠀⠀⢢⠹⡆⠀⠀⠀⠀⠀⠈⣵⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⡶⡣⠊⡰⡀⠀⠀⠀⢀
⠀⠀⠀⠈⣇⠘⠀⠀⠀⠀⠀⠀⢹⠛⠒⠦⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⢀⠈⡰⠁⠀⠀⠀⠈
⠀⠀⠀⠀⢸⡄⠀⠀⠀⣀⣀⣀⣸⣀⠀⠀⠀⠀⢸⠋⠁⠀⠀⠀⠁⠀⠒⠒⠶⠏⠀⠀⠈⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣈⣭⣿⡶⢶⣤⠄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢐⣈⣢⣄⣀⣔⣁⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡁⠀⠀⠀⠀⡁⢀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀⠀⡀⠀⠀⢀⠀⡀⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⡄⢀⡀⡀⠀⣀⣀⠀⢀⢀⡀⢠⣨
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢂⠀⡀⠀⡀⢀⠀⢀⠀⡀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⡀⠅⠀⠀⡀⢸⢈⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠠⠀⠀⠀⡄⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢹⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡧⠯⠠⠤⠤⠤⠠⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠣⠀⠘⠓⠠
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠰⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡂⠀⠀⠠
⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠈⡄⠀⠐⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⡌
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣬⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠈⠻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣤⡄⠀⠀⠀⣤⣄⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⣠⣦⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠙⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣤⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⢀⣴⡆⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠻⢷⣤⣀⣀⠀⠀⢀⣀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⠀⠀⠀⠈⠙⠛⢻⣶⠊⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡼⠛⠁⢸⣿⣧⣤⣤⣀⠀⠀⠀⠀⠀⣸⡿⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⣿⣿⡟⠀⠀⠀⢸⣿⠋⠈⠉⠉⠀⠠⣶⣶⣾⣿⠄⠈⠻⣿⣿⡷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⣴⡆⠀⠀⠀⣿⣿⡇⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣿⣿⡇⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠈⢃⣄⠀⠀⣿⣿⡇⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⣿⣿⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠀⠀⠀⠀⠀⠸⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣿⣿⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⣿⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣠⣴⣿⣿⣷⣦⡀⠀⠹⠃
⠀⠀⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⢿⣿⠙⠻⣿⣦⡀
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠀⠹⣿⣿⣄⣀⡀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⣾⣿⣦⣤⣾⣿⡇⢸
⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⣼⣟⠛⠛⠉⢱⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠉⠛⠛⠉⠙⠻⣿⣿⣿
⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⣄⠈⠻⠿⠀⠘⠃⣨⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠘⣿⣀⣴⡄⠀⠀⠀⠻⣿⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠇⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠀⠀⠀⠀⠿⣏⣀⣴⣦⣄⣼⣿⡇⠀⠀⠀⠀⠀⣰⣿⡿⠇⠀⠀⣾⡀⠀⠀⠀⠀⢠⣿⣿
⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⣿⡟⠈⢻⡇⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⢸⣿⡇⠀⠀⠀⠀⢿⣿⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢩⣿
⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣶⡇⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣿⡇⠀⠀⠀⠀⠀⢿⣿⣦⣀⡀⣿⣿⡄⠀⠀⢸⣆⣠⣿⣿
⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠀⠀⣴⣾⣿⡿⠛⠃
⣿⡃⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇⠀⠀⠙⣿⡿
⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇
⠿⠁⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⠀⠀⢸⠃
⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠿⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸
⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡄
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠙⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⢤⣾⣿⠀⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣆⣀⣴⣶⡎⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⣴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣶⣾⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢠⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠶⠂⠀⠒⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠛⠓⠀⠀⠀⣀⣤⣤⣤⣤⣄⣀⣀⣀⡀⠀⠸⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⡸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⣸⣿⠀⠈⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢠⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣸⡆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣄⠀⠀⠀⠀⢻⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠈⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⢻⣄⠀⠀⠀⢸⣷⣤⡀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠙⢷⣄⠀⠸⠟⠉⠀⣶⣶⡄⠀⠀⢙⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠂⣀⣀⠀⠀⠈⢻⣷⣶⡾⠟⠁⢀⣴⣶⣶⡾⠟⠋⠘⠿⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣴⠛⠛⢀⣤⣤⣴⣿⠟⠁⣤⡄⠈⠉⠉⠉⠁⠀⣤⣤⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠿⠟⠛⠛⠀⠀⠀⠈⣱⡄⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠻⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣇⣀⣠⣀⣀⣀⣀⣀⣀⣠⣴⣤⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠠⡠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡛⠒⠂⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣶⡐⠠⢀
⠀⠀⠀⠀⢀⡞⠁⠰⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣶⣶⣦⣤⣀⣠⣄⣽⣿⣶⡄⠀⠀⠀⠀⣠⠀⠀⠀⠀⣴⣴⣿⠟⠀⣸⡇
⠀⠀⠀⠀⢸⣇⠀⠈⣿⣿⣿⣿⠏⠛⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⣿⣿⣿⡿⠃⠀⠀⠀⣴⡇⢠⣤⣴⠾⠟⠋⠁⢀⣠⣿⠇
⠀⠀⠀⠀⠀⢿⣧⣀⠘⢿⣿⣿⣧⠀⠀⢣⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣦⣠⣀⣠⣤⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠘⢿⣿⣶⣼⣿⣿⣿⣧⣄⣿⠀⠀⠀⢠⣾⣷⣾⣷⣶⣦⣦⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⣶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣀⣀⣂⣵⠞⠛⠉⠁⠀⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣴⣿⣿⡛⢉⣀⣀⣀⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⣀⠈⠙⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡠⠀⢀⣴⣶⣟⣿⣶⣄⡀⠀⢆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢠⣴⠿⠇⠀⠀⠀⠠⡀
⠀⠀⠀⠀⡼⠀⠀⠠⠁⠉⠁⠉⠙⠿⡅⠀⢸⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠁⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣸⣿⠀⠀⠀⠀⡔⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢸⣇⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠰⣧⡀⠀⠓⢦⣀⣀⣀⣀⠀⠀⠀⢨⡇⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⣀⣠⣤⣾⡿⠋⠀⠀⠀⠀⠀⠹⣿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣠⣾⠃⠀⠀⠀⠀⠻⢿⣿⣦⣤⣄⣤⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⢠⣿⠋⠀⠀⡏⢻⠀⠀⠙⣿⡄⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠾
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⢰⡞⢚⠃⠘⡓⣲⠀⢹⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣿⡀⠉⠉⡇⢸⠉⠉⢀⣿⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢿⣦⣀⣉⣉⣀⣴⡿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⢸⡿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣾⡇
⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⢸⣿⣀⣀⣸⣇⣠⣤⣤⣄⣼⣇⣀⣀⣿⣄⣤⣤⣤⣄⣄⣼⣧⣠⣤⣄⣄⣼⣧⣤⣤⣤⣤⣤⣤⣿⣄⣤⣸⣧⣠⣤⣤⣄⣼⣧⣠⣄⣿⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣟⣻⡿⣿⠿⠿⠿⣿⡿⢿⣟⣻⢿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠶⢮⣭⣭⣭⣭⣭⣭⣭⣭⡵⠶⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡶⠛⠛⢶⡞⠉⠀⠈⠉⢉⣭⣭⣿⣿⡛⠉⠉⠉⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣼⣷⡀⠀⠀⠹⢦⣤⣤⣤⣴⣿⣿⣿⣿⣇⣀⣀⣤⣿⡿⣿⠟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠻⢶⣄⣠⣶⣽⣶⡉⠁⢻⣿⣿⣿⡟⠉⠉⠀⣨⣽⣿⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢰⣀⠀⠀⠀⠀⣩⣛⣿⣿⣇⣠⣴⠿⠿⢿⣻⣿⣶⣤⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣙⠻⠶⣦⣤⣿⣿⣿⣿⡟⠉⠀⠀⠀⢻⣿⣿⣿⠇⠀⠉⣉⣽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣌⣉⡙⠛⠋⣄⣀⣠⣤⠶⠟⢛⣉⣀⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⠛⠛⠛⠛⠁⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⢠⡄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⡏⠙⠶⠶⠆⠀⠈⠉⠙⢷⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠶⠀⠀⠠⢦⠀⠀⠀⠀⠷⣀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠁⣤⡀⠀⠀⣀⡀⠀⠀⠀⠀⠙⠀⠀⣀⠋
⠀⠀⠀⠀⠀⠀⠀⠐⣯⠀⠈⢻⠁⠀⠀⠛⠁⠀⠀⠀⠀⢀⣤⣼⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⣽⠖⠘⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⣤⣤⡤⣤⣤⣤⡤⠀⠛⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⡄⠀⠀⠀⠀⢀⣾⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢶⣿⡏⢻⡷⠴⠶⠀⠉⠙⣿⠀⠀⠙⣷⢴⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⢠⣿⠇⢸⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⣿⡟⠉⠲⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⢸⣿⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⣿⡇⠀⠀⢈⡿⢦⡀
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⡇⠀⣿⡇⠀⢠⡞⠁⠀⠻⡆
⠀⠀⠀⠀⠀⡄⠻⠇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠇⠀⠘⠃⠀⠀⠀⠸⠆
⠀⠀⠀⠀⡀⠋⠀⠀⢀⣴⣿⠿⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⠀⢰⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⠇⠀⠀⠀⢸⣿⣵⣶⡦⣼⣿⣇⠀⠀⠀⠀⣴⠀⠀⠀⡟⠀⣸⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⡿⠀⠀⡇⠀⣿⡏⠀⢸⡁⢸⡟⣿⡆⠀⠀⠀⣿⠀⢀⣾⠿⠿⢿⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠃⠀⠀⡇⢸⣿⠁⠀⢸⡇⢸⠀⢻⣇⠀⠀⠀⣿⠀⣾⠄⠁⠀⠀⢼⡇⠀⢰⡆⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣷⠀⠀⠰⣇⢸⡿⠀⠀⠘⠁⢸⠃⠘⣿⡀⠀⠀⣿⠀⣸⡶⠟⠈⢻⡆⠁⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⣶
⠀⣠⠀⠀⠀⢠⡟⢻⣇⠀⠀⠀⠀⣸⠀⠀⣿⡇⠀⠀⣿⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⠀⠙
⠀⡿⠀⠀⠀⠀⡇⠈⠻⣶⣤⡀⢰⣿⣶⣾⠏⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣷
⣄⠀⠀⠀⠀⠀⡇⠀⠀⠈⠛⠁⣼⡟⠉⠁⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠙
⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆
⠟⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇
⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣤⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⠛
⠀⠀⠀⠀⢷⡀⠀⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢠⡜
⠀⠀⠀⠀⠀⠳⣴⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡀⠀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠘⢃⡈⠓⣦⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣴⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠓⠀⠼⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠛⠛⠛⠛⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠑⠲⠒⠒⠶⠒⢿⠀⠀⠀⠀⣨⠿⠖⠆⣀⣀⣀⡀⢴⡞⠁
⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⡴⠂⠁⠀⠀⠀⠀⠀⠀⣠⡖⠛⠋
⠀⠀⠀⠀⠉⠦⠤⠀⠀⢤⡤⠞⠋⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⣤⠞⠁
⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠤⠤⠤⠤⢤⢤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⣀⣸⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠉⠉⠉⠉⠈⠹⢿⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠤⠤⠤⠤⠶⠾⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣴⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠛⡌⠐⢲⡶⠲⠂⠒⠒⢒⣾⢻⠀⠀⠀⠀⠀⠀⠀⣧⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⡔⠺⣾⡟⢀⣦⣀⡸⠟⡀⣸⠀⠀⠀⠀⠀⠀⠀⣥⣿⡆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⢈⣿⡋⢁⣼⡁⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠓⠲⣶⠶⣾⠟⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠃⠀⠀⢤⣧⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢙⡆⠀⠀⣞⠉⣷⠀⠀⣚⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠈⠋⠁⣀⡀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⢸⡇⣼⣿⣿⣿⣿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⢀⠀⡤⠠⡄⠀⠀⣠⠒⢠⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠉⠉⠉⠉⠉⠉⢸⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠼⠃⠀⠀⠢⠴⠃⠀⠀⠙⠶⠞⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣋⣸⡇⠀⠀⢨⣀⣵⠀⠀⠺⣤⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠐⠀⠀⢠⣤⣤⣤⣤⣤⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰⡄⠀⠀⣔⠀⣆⠀⠂⢠⡁⣰⠄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢹⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠙⠛⠃⠀⠀⠈⠛⠋⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣈⣀⣀⣼⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁

⣷⠛⠛⣶⢶⡶⣶⠶⠶⣾⠶⢶⠶⠶⣾⠷⠶⢿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⢶⡶⡶⣾⢿⣿⣿⠾⠿⠷⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⠈⢧⣀⣴⣊⡀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣠⠏⠉⠯⣉⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡀⢤⡤⠀⠹⢄⡉⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠙⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠶⠌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠓⣖⠁⠀⢀⣶⢏⠷⡄
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠠⡰⠖⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣿⠔⠦⢤⣶⡿⠛⠁⡰⠉⠐⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠐⢇⠀⠀⠘⡆⠀⠀⠀⠀⠀⣠⢶⠋⡇⠀⠸⡇⢀⣾⠛⠒⠒⠉⡠⠀⢀⠤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢀⡀⠀⡾⣀⣀⣠⠶⠤⠚⠹⡼⢸⠇⠀⠀⣿⣼⠃⠀⠀⠀⡔⠠⣞⣡⡴⡶⡶⣿⡆
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢹⠁⠈⠉⠀⠘⢉⣁⣀⣀⣀⣀⣙⡏⠀⠀⢰⢿⡿⠂⠀⠀⡐⡴⢰⠋⢀⠀⠀⢀⢀⡿
⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⠛⠲⢦⣀⣀⣤⠴⠶⠬⣭⣉⢻⡇⢀⣰⢏⡾⠃⠀⠀⠌⡜⢠⡏⠔⢁⡠⠔⠉⣼⡇
⠀⠀⢠⡏⠀⠀⠀⢷⠀⠀⢸⣦⣤⢾⡍⣉⡉⠑⣞⠛⠉⠉⢻⠛⠿⠟⡻⠁⠀⠀⢀⠞⡴⠉⠁⠉⠀⠀⢀⣼⢻⡇
⠀⠀⣼⠀⠀⠀⠀⢸⠀⠀⢸⡏⠀⠈⠻⣶⣄⡀⠈⠒⠲⢤⣿⣀⣀⣰⠁⣀⣀⣰⠧⣼⣅⡀⠀⠀⢀⡴⣿⡧⣾⣀⡀
⠀⢠⠇⠀⠀⠀⠀⢸⡄⠀⠈⡇⠀⠀⠀⠀⠀⠙⠦⠤⢼⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠀⠰⢋⣾⠃⣠⠇⠈⠉⠛⣦
⠀⠸⡧⠄⠀⠭⣤⠰⢇⠐⠂⠙⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⠤⠤⢤⣤⣀⣀⠀⠀⠀⠀⠀⠉⣠⡾⠁⡼⠁⠀⠀⠀⢰⠇
⠀⢸⠃⠀⢒⠒⠛⣏⠈⠳⣄⠀⠈⢦⡀⠀⠀⣀⣀⡤⠾⣁⠀⢀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⡛⠋⢠⣦⣧⣄⣀⠀⢀⡞
⠀⢸⠈⠉⠉⠀⣀⠈⣆⢀⣨⠷⢦⣀⠙⠶⢋⡉⠁⠀⠠⢌⣑⡪⣭⠽⢒⣒⣒⠲⠄⠐⠊⠁⠀⣠⠎⠈⡇⠀⣹⠛⠛⠁
⠀⠀⠀⠀⠀⠉⣉⣀⣉⣐⡁⠀⠀⠈⠑⠲⠤⣼⡷⢤⠀⠀⠀⠀⣀⣀⡀⠀⠠⠤⠤⠄⢀⣀⡘⠁⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢦⣤⣄⣀⣀⢀⠀⠀⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠈⠀⠀⢀⡷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡶⠚⠁⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡀⠀⢀⣾⡁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠾⠉⠙⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣤⣤⣤⣄⡀
⠀⠉⠉⠛⢿⣿⠟⢻⡿⢿⡿⢿⣿⣿⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠻
⠀⠀⠀⠀⢸⡇⠀⠈⠁⢸⡇⠀⢹⠁⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⡀
⠀⠀⠀⠀⠺⠇⠀⠀⠀⠘⠃⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⡀
⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉⠉
⢀⣿⣿⡇⠀⣠⣿⣤⣶⣶⣶⣧⣦⣤⣤⣭⣤⣤⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⢈⡉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠈⣿⣿⣿⠀⠉⠙⠛⠟⠛⠛⠛⠛⠿⡟⠛⠻⠋⠉⡉⠉⠁⠐⠂⢀⣤⣾⣷⣿⣷⠀⢰⣿⣷⣇⠀⠀⠀⠲⠂⣀⣤
⠀⣽⣿⣿⠀⠀⠀⠂⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⣻⠿⣋⡅⠀⠈⠻⠿⠋⠀⠰⠂⠀⠐⡿⠿⠁
⠀⣽⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⠁⣠⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣋⣄
⠀⢻⣿⣿⠘⠀⠀⣠⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠛⠉⢀⣀⣤⣿⣿⣶⣦⡄⠀⠀⠸⠿⠁⠈⠛
⠀⢿⣿⣿⡆⠀⠐⣎⣹⣿⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠈
⠀⡴⠌⣯⡇⠀⠘⠻⡋⢿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠈⠻⣿⡿⠿⠛⠉⠀⠰⡦
⠀⢰⡆⣿⠇⠀⠀⠀⠀⠈⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠂⠈⠀⢠⠀⣀⣀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⣠
⠀⠘⢣⡏⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⢠⡀⢠⠜⣿⣿⠃⠀⠀⠀⢁⠀⠈⠋⠀⢠⣌⠀⠉
⠀⠀⠼⠃⠀⠀⠀⣀⣠⣶⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠐⠃⠀⠄⠐⠀⢀⡀
⠀⠀⠐⠀⠀⠀⠀⢻⣿⡿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⠀⠠⠖⠮⠀⠀⠀⠀⠀⠀⠀⠁⠈⠋
⠀⢀⡀⠀⠀⠀⠀⠀⠉⢵⣮⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣟⡢⠀⡀⠀⠀⠀⠀⠈⢩⣿⠋⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⠀⠀⠀⣀⣤⡀⢰⡶⠀⠀⠀⠘⠛⠈⠙⠛⠒⠌⣢⡤⢤⣀⠀⠈⠉⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⣤⠀⠀⣾⣿⣷
⠀⠀⠀⠀⠀⢉⣉⡁⡀⠀⠀⠀⣤⣤⠀⠀⠀⠂⠣⠀⠉⠛⠷⠿⠷⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠟⠋⠁
⠀⠀⠀⠀⠀⠘⠿⠇⠁⠤⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⣩⣍⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣠⣤⣬⣥⣤⣄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠉⠉⣉⣉⠀⠀⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡘⠃⡨⢁⠘⢃⡈⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠋⠉⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠄⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢡⡧⠀⠀⣬⠄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠐⠲⠔⠂⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⠀⠀⠀⠈⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣪⠕⢯⠑⢂⡔⠒⣹⠲⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⣤⣤⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡿⢧⡸⣤⠘⡏⢻⡟⢛⠇⢠⡼⠨⡏⠒⠦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡦⢄⡡⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⠀⠀⡇⠳⡄⢹⣼⣿⡜⢀⡜⢀⣴⣣⡄⠀⠸⡄⠀⠀⠀⠀⠀⠀⠘⣿⢿⣿⣀⠀⠀⣨⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠘⠀⣰⠁⠀⠘⢦⣻⣿⡡⠋⠀⠈⠻⠿⠃⠀⠀⡇⠀⠀⠀⠀⠀⣴⣿⣹⣿⣿⣿⣷⣾⡿⠿⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠭⠭⠤⠭⠤⠤⠤⠿⠿⠥⠤⠤⠤⠤⠤⠤⠤⠤⠿⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠋⠉⠉⠉⠛⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣖⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠦⠀⠀⢃⡝⠀⣸⡀⠀⠀⡄⠄⠀⠀⠀⠀⠸⣷⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢶⠚⠒⠢⢰⡆⠠⢳⡇⠀⠘⣶⡀⠰⠤⢄⡀⠀⠀⠉⠙⠛⠓⠲⢦⡀
⠀⠀⠀⠀⠀⠀⡌⠠⠐⣐⢟⣁⣐⣒⢪⣕⣢⣚⣑⠒⡒⠋⠳⠄⠀⢄⠙⠄⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⡇⠅⡐⠐⢉⡔⡠⠞⢙⠯⠋⡉⢠⠠⢄⣀⣉⠉⠓⣮⡃⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⡀⠄⠄⡄⠑⠨⠠⠒⠁⠤⠂⠁⠁⠔⠀⠐⠉⡑⢤⢹⡇⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⡟⢉⣉⠙⠛⠋⠊⠈⠛⠓⠒⠒⠒⠂⢼⢪⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠠⠄⠄⠄⠀⠀⠀⠀⢈⡍⠄⠀⢸⢼⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠀⠂⠀⠄⠈⡇⢠⢠⢠⠤⠤⠀⠀⠀⠀⠀⢠⠠⠀⢸⢾⠀⠀⠀⠀⠀⠘⢷⣄⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⣅⡁⢀⣤⡇⠀⠀⠀⠀⠀⠀⠠⠦⠈⠤⡄⠭⢤⢼⡀⡇⠂⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠆⠁⢁⡇⠘⠀⠘⠘⠚⠀⠀⠀⡒⠒⠒⠒⠂⢸⠞⡇
⠀⠀⠀⠀⠀⠀⡄⡆⡂⡾⡇⢀⡀⡀⣀⢀⠀⠀⠀⠉⠀⠙⠙⠀⢸⣠⡇⡄
⠀⠀⠀⠀⠀⠀⠆⠁⡇⠫⠣⠌⡁⡁⣁⣈⣀⣀⣀⣀⣀⣒⣀⣀⣸⣃⡇⡇
⠀⠀⠀⠀⠀⠀⠆⢃⢇⠘⠒⠢⣄⠠⠉⠀⠀⠂⠀⠉⠀⡤⠉⢁⠀⣸⡇⡇
⠀⠀⠀⠀⠀⠀⠰⣌⠛⠷⠯⠷⠥⠠⠤⠤⠤⠤⠤⠤⣬⣐⣠⠽⠦⢃⡔
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠢⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢰⡇⠀⠀⠀⠀⠉⢳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⡐⡇⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠻⣤⡀⠈⠳⣶⣤⣈⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡥⠀⠀⢀
⠀⠀⠀⠈⠙⠷⣾⣿⣿⣿⣷⣦⣉⣻⣟⠂⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣼⣷⣔⡁⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣟⣿⠡⣓⣤⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢜⠁⠀⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⣀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠉⠁⢀⣤⠏⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢷⠀⢀⠀⠀⢀⠀⣀⣴⣿
⣿⣶⠦⠄⠀⠂⠀⠈⠀⠀⣀⡠⠴⠚⠋⠁⠀⠀⠀⠀⢠⣄⣀⣀⣀⣤⣆⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣻⣿⣿⡿⠃
⠙⠻⠶⣤⣤⠤⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠂⠄
⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠒⠐⠚⡆⠀⠀⠀⠀⠀⠀⢠⣾⣿⠞⠓⠂⠐
⠀⠀⠀⠉⢁⢤⠁⠀⠀⠀⠀⠀⠀⠘⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣟⡽⠁
⠀⠀⣾⡤⠴⠐⢦⠀⠀⠀⠀⠀⠀⠀⢻⡧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⡀⢸⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⣿⢳⡴⠀⠚⠇⠰⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠸⡇⢹⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⢹⣿⢀⣄⣄⣰⡶⠀⠀⠀⠀⠀⠀⢹⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⢸⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠻⣏⡭⠵⢿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⣭⣿⡇⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⡠⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠶⠦⠴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡶⠞⠛⠛⠛⠛⠛⠛⠷⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣶⣄⡀⠀⠀⠀⠀⢀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣯⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⡀
⠀⠀⠀⠀⣶⢦⣤⣨⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠘⣦⡈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⣠⡶⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡟⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠻⣇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⡿⠀⠀⠘⠋⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠛⠉⠉⠙⠃⠀⠀⠀⠀⣿⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⣿⠀⡷⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣠⣦⡄⠀⠀⠀⠀⠀⣿⠀⣸⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⣸⡇⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠸⣧⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠛⠋⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⠶⠾⣋⣼⣿⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠹⣿⣭⣭⣭⣭⣤⣶⣾⣋⣸⡇⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠉⠷⣤⣤⡟⠉⣁⣠⡿⠋⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣄⠀⠀⠀⠀⠀⢠⡔⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⠀⠀⠀⠀⠀⢹⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣭⣽⣟⠀⠀⠀⠀⠀⢈⣿⣭⣭⠛⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⣶⣿⡿⠋⢼⣿⡉⠈⠻⣦⡀⠀⠀⣠⡾⠋⠈⣹⣿⠀⠈⠻⣿⣶⣤⣠⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣤⠞⠛⠛⣿⠋⠉⠉⠉⠁⢀⣼⣿⠁⠀⠘⣿⢷⡀⠀⠈⠻⣤⡼⠋⠀⠀⣰⣿⡟⠀⠀⢠⡟⢿⣯⡈⠉⠉⠉⢻⠟⠛⠛⠳⣤⡀
⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⠀⢿⡘⢷⡀⣠⡾⠛⠳⣦⡀⣼⠏⣸⠇⠀⠀⣾⠇⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⡇⢸⣧⠀⠀⢸⡇⢨⡿⠋⠀⠀⠀⠈⠻⣏⢰⡏⠀⠀⢰⡏⠀⣸⣿⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⠇⠈⢿⡀⠀⠀⢻⡏⠻⣤⣀⣀⣀⣠⡾⠋⣿⠁⠀⠀⣸⠇⠀⣿⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⣆⠀⢸⡇⠀⠀⠘⣧⠀⣼⠋⠉⠙⢻⡄⢸⠟⠀⠀⢰⡿⠀⢰⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡿⠉⢿⣦⣼⣿⠷⡶⣦⣿⣴⡿⠀⠀⠀⠸⣧⣿⠀⢶⣶⠾⠃⠀⠈⣿⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⣿⣶⣿⣛⡿⠷⠶⠶⠛⠛⠛⢿⣛⢯⣿⣿⠁⠀⠀⠀⠀⣿⡇⠀⠀⣿⠇⠀⠀⠀⠙⢿⣷⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇
⠀⢀⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣏⣿⡆⠀⠀⠀⢠⡿⠀⢠⡾⠃⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡀⠀⠀⣸⠇⣠⡾⠁⢀⣤⡶⠶⠶⣤⣾⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⠀⢰⡏⣠⡟⠀⣰⡟⠁⠀⠀⠀⠀⠙⣧⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⣾⣿⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⠃⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⠳⣦⣄⣀⣠⣤⠾⠃⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⠛⠛⣷⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸⢻⣇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠁⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣤⣴⠟⠉⣳⡄⠀⠀⠀⠀⠀⠸⠷⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢻⡄
⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠉⠛⠛⠛⠲⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇
⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡆
⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡇
⣿⣀⣀⣀⣀⣀⣀⣸⡟⠀⣿⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣸⡇⠀⢸⣇⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠛⠋⠙⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣄⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⡰⠃⠂⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⢀⡜⠑
⡴⠃⠀⠀⡠⠂⠀⠁⠂⢄⠀⠀⠀⠀⠀⢸⠀⢀⠾⡷⠖⠋⠀⠀⢀⠞⠁⠀⢀⠔⢂⠉⠀⠢⡀⠀⠀⠀⠀⠀⠈⠀⢀⢿⡷⠞⠁
⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠡⡀⠀⠀⠀⠀⢔⢦⣀⠀⠀⡀⠠⠐⠁⠀⠀⠀⠈⠀⢠⡖⠀⠀⠈⢀⠀⠀⠀⠀⠘⣔⢄⣀⠀⢀⡀⠠⠊
⠀⠀⠀⠐⡀⠀⢳⡀⠀⣄⠠⠱⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢧⡀⠀⡄⠀⢃⠀⠀⠀⠀⠀⠉⠀⠈⠁
⠀⠀⠀⠀⠐⣄⠀⠻⠿⠿⡀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠈⠻⠿⢿⠀⡀⠄⠀⠀⠀⠀⠀⣀⣀⣀⡀
⢄⠀⠀⠀⠀⠈⠓⢤⣄⣙⣧⡳⠃⠀⢀⠔⠈⠉⠁⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠈⠳⢤⣄⣛⣞⡴⠀⠀⡠⠒⠁⠉⠀⠀⠀⠀⠀⠁
⠀⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡀⣠⠌⠁⠀⠂⠒⠠⣀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠔⣤⠄⠁⠐⠐⠂
⡙⢦⠀⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⢈⡓⣄⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⣘⣦⡇⢀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠈⠒⢤⣀⣀⣀⣀⣽⣮⡆⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠙⠲⢄⣀⣀⣀
⠛⠋⢀⣄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠁⣡⢢⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠈⠉⠛
⢴⣲⡵⠋⠀⠀⢀⠄⠒⠀⠉⠙⠳⢦⣄⠀⠀⠑⠠⢀⡀⠀⠠⠀⠠⢴⣢⠶⠋⠀⠀⡀⠄⠂⠉⠉⠙⠷⣦⡀⠀⠀⠐⠠⣀⡀⠀⠄
⠉⠀⠀⠀⢀⠄⠁⠀⠀⠀⠰⣶⠈⠈⢿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠠⠊⠀⠀⠀⠀⢰⣦⡁⠨⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡰⠫⠀⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⡼⡡⠀⠀⡏
⠀⠀⠀⠈⠁⢀⣶⣄⡤⠖⠁⠀⢀⡴⠁⠀⠀⡠⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⡆⠀⢠⣦⣤⡤⠊⠀⠀⢀⠞⠀⠀⢀⡠⠀⠀⢄
⠀⠀⠀⢠⢀⠑⠙⠉⠀⠀⢀⡠⠊⠀⠀⠀⠎⠀⢀⡀⠀⠈⠢⠀⠀⠀⠀⠀⠃⣀⠋⠉⠁⠀⠀⢀⠔⠋⠀⠀⠠⠃⠀⣠⡀⠀⠀⢄
⠀⠀⠀⠀⠑⠫⠤⠄⠒⠀⠁⠀⠀⠀⠀⠀⡁⠀⠹⡀⠀⠀⠀⠱⠀⠀⠀⠀⠈⠒⠩⠤⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠳⣶⣼⡆⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠈⢳⣦⣶⡄⢀
⠀⠀⠀⠀⠤⠒⠒⠀⠀⠒⠀⠠⡀⠀⠀⠀⠀⠈⠢⣀⢀⠉⣞⡰⡀⠀⠀⢀⠠⠤⠖⠂⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠑⢦⣀⢈⢻⣆⢌
⠀⢀⠉⠀⡀⠀⠄⠤⢠⠠⢀⠀⠈⠐⣄⠀⠀⠀⠀⠈⠙⠓⠓⠊⠁⠀⡰⠁⠀⠀⠀⠄⠤⡀⠄⠀⠀⠈⠢⡀⠀⠀⠀⠀⠉⠛⠚⠒⠋
⠀⡆⠀⠀⠰⡏⠉⠀⠀⠀⠀⠈⠦⢄⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢼⡋⠁⠀⠀⠀⠀⠘⠦⣀⠈⢦
⠀⠡⠀⠀⠀⠙⠢⣀⠀⠀⠀⠈⠑⢎⢣⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠲⣌⢆⠀⣆
⠀⠀⠡⡀⠀⠀⠀⠈⠙⠚⠶⠶⣾⡿⠛⢀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠈⠙⠳⠶⠶⣿⠟⠓⢀⡸
⣤⡀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⢀⡀⢰⡫⠋⠀⠀⠀⡀⠤⠠⠤⣤⣄⡀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣀⡠⢐⡷⠃⠀⠀⢀⡠⠤⠀⠤
⡀⢛⡇⠀⠀⠀⠈⠑⠒⠤⠥⠭⠿⠚⠉⠀⠀⢀⠔⠁⠀⠀⢀⣠⡀⠈⣻⡄⠀⠀⠀⠉⠐⠲⠤⠤⠥⠟⠚⠉⠀⠀⠀⠐⠁⠀⠀⢀⣤
⣃⣈⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣿⣓⣀⣨⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣁⣀⣀⣀⣀⣀⣟
⣿⣿⡋⡛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠉⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣀⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣶⣶⣶⣶⣶⣶⣿⡇⠀⣿⠀⢸⠁⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣉⣹⣋⣉⣿⣿⡟⠛⣿⠛⢻⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣉⣹⣏⣉⣿⣿⡷⠶⣿⠶⢾⠶⠶⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣉⣹⣉⣉⣿⣿⣇⣀⣿⣀⣸⣀⣀⣿⣿⡯⠤⢽⠥⢼⡧⠤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣉⣹⡏⣉⣿⡟⠛⠛⣛⠛⣛⠛⠛⠛⣿⡗⠒⣺⠒⢺⡗⠒⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⠀⠉⠀⠀⠀⣿⣯⣭⣽⣍⣽⣯⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣤⡄⣤⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠛⠛⠛⠛⠛⠛⠁⠀⠈⠉⠁⠉⠁⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢠⣶⣷⣶⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠸⣿⣿⣿⡟⠀⠀⠀⠀⡀⣰⣀⡀⠀⠀⠀⠸⣿⣿⣿⡇⠀⢦⣴⣿⣿⣿⣿⣆⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠈⠉⠁⠀⠀⠀⢀⣾⣿⣿⣿⣿⡗⠀⠀⠀⠈⠉⠁⠀⠀⠾⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢀⣠⣦⣄⠀⠀⠀⠸⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣴⣦⡀⢀⠿⢿⣿⢿⣿⠋⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠉⠉⠁
⠀⠀⢼⣿⣿⡄⠀⠀⠀⢻⠿⣿⡿⠿⠏⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠋⠀⠱⠀⠀⠀⠀⠀⠀⠙⢿⡿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⡀
⠀⠀⠘⠻⠛⠁⠀⠀⠀⠀⠀⠸⠅⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⢀⣴⣤⣤⡀
⠀⠀⠀⠀⢀⣴⣶⣶⣤⠀⠀⠀⠀⣾⣿⣿⡄⠀⠻⠿⡿⠿⠁⠀⠀⠀⠀⣼⣿⣿⣿⣶⡄⠀⠀⠸⣿⡿⠃⠀⠀⢀⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠂⠀⠀⠀⠙⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠻⣿⣿⣿⡿⠃⠁
⠀⠀⠀⠀⠠⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠙⢻⣿⣿⣿⣿⠟⠀⠀⠀⠀⣴⣶⣦⡀⠀⠰⠋⠈⠁⠙
⠀⠀⣠⣤⣄⠀⠀⠀⠀⠈⠀⠀⠀⠀⣴⣀⣰⠀⠀⠀⠀⠿⣿⠇⠀⠀⠀⠀⠈⠘⠏⠀⠀⠀⠀⠀⠀⢿⣿⡿⠇⠀⠀⠀⠀⠀⣠⣤⣤⡀
⠀⠀⣿⣿⣿⠆⠀⠀⠂⠀⠀⠀⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡷⠆⠀⠀⠀⢰⣿⣿⣿⡷⠀⠀⠀⢀⣠⣤⣦⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠉
⠀⠀⠀⢠⣶⣶⣶⣄⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⣼⣿⣿⣿⣿⣧⢄⠀⠀⢀⣶⣿⣿⣷⣦
⠀⠀⠀⢸⣿⣿⣿⣿⠀⢀⠈⠁⠘⣿⠿⡿⠉⠛⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠼⠛⠛⠛⠃⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⠀⠀⢹⠉⠁⠀⠀⠀⠀⠘⠻⠿⣿⠟⠻⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣷⠀⠀⠀⠀⠀⠻⣿⡿⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠐⠲⠤⠤⠤⠤⢤⠤⠤⠴⣶⢶⢾⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡽⠦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣾⣟⣲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠁⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠛⣛⣿⢿⣿⠑⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣙⡎⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠿⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠹⠀⡆⡆⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣱⢳⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡆⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠦⠴⠶⠞⠳⠻⠛⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡖⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠤⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡇⢰⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡼⠇⠸⣤⣄⣄⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣤⣤⣴⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣮⠉⢧⠀⠀⡎⠘⢿⣿⠉⠉⢏⠉⠉⠉⠉⢩⠉⣴⣦⡶⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢀⣡⡀⣷⠀⣸⣿⡖⠀⢈⢄⠀⠀⠀⣸⢴⡿⠈⢅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣿⡀⠉⠻⣿⡻⣿⡇⣠⢞⣺⣧⣤⡰⠣⠋⠁⡐⠕⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄⠀⠈⠻⣾⣿⡿⠋⠁⠀⣨⡧⠁⠀⡰⠋⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣟⣷⠤⣠⣿⣿⣀⡠⣤⣾⢟⣷⣀⣠⣦⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⠶⠾⠿⠛⠛⠛⠉


⡿⠿⠿⠿⡿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣛⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢀⣠⡏⢹⣄⡀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠼⠯⣿⠸⠦⡄⢠⠴⠇⣿⠽⠧⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠛⠛⣿⣀⣀⣛⣛⣀⣀⣿⠛⠛⠛⡟
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⢻⠀⢀⣀⣉⠉⠉⣉⣉⠉⠉⣉⣀⡀⠀⡟⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⢹⠀⢸⠀⢸⡁⣿⠀⢸⡇⢸⡇⠀⣿⢈⡇⠀⡇⠀⡏⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠧⠼⠀⢸⠀⢈⣉⣉⠀⠀⣉⣉⠀⠀⣉⣉⡁⠀⡇⠀⠧⠼⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡶⢶⠀⢸⠀⢸⠉⣿⠀⢸⡏⢹⡇⠀⣿⠉⡇⠀⡇⠀⡶⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⠀⢸⠀⠈⠉⠉⠀⠈⠉⠉⠁⠀⠉⠉⠁⠀⡇⠀⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣉⣉⠀⢸⠀⠸⡿⠿⠿⠿⢯⡽⠿⠿⠿⢽⠇⠀⡇⠀⣉⣉⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⢹⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⡇⠀⡏⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠷⠾⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⡇⠀⠷⠾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠾⠤⠤⠧⠤⠤⠤⠼⠧⠤⠤⠤⠼⠤⠤⠷⠤⠤⠤⠤⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣏⡌⠀⠀⣠⠂⠀⠀⠀⠉⢛⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢿⠇⠈⠚⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠋⠁⢿⡇⠀⠙⢦⡀⠀⢤⡀⠀⠀⣰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠙⠓⠶⣬⣿⡀⠀⠀⠙⢦⡈⠙⠶⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠲⢄⠀⠀⠀⠀⢠⡇⠘⣷⣀⣠⡤⠔⠓⣾⢻⣷⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠈⢳⡀⠀⠀⠈⠳⣄⣹⡏⠈⢦⠀⢸⠃⢸⡿⠉⠛⣧⠓⢄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⠙⢿⡀⠈⢧⢸⡇⣿⣧⠀⠀⠈⢧⠀⣇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⣟⣦⡈⢿⣧⡟⢸⠀⠀⠀⠈⢇⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠠⡏⢌⢿⡝⣎⣿⡇⠘⣧⣀⡀⠀⠘⣾
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠙⠒⠋⠀⠈⢿⠁⠀⠀⠉⠁⠘⠓⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠘⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣧⢯⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠲⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⢧⣀⠀⠀⢀⣠⠖⠋⠀⠀⠀⣠⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣼⡀⢀⡠⠤⠶⠶⠖⣛⠲⢦⣨⣻⠟⣋⣠⣄⢀⠶⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡿⠁⠀⠀⠀⠰⠶⣦⣬⠿⠋⢰⡛⡇⢧⠈⡿⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠛⠳⢦⣤⣀⠀⠀⠀⢀⡞⢰⣇⣀⡤⠖⠋⣩⠿⠋⠁⣠⠴⢮⣁⣨⠤⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣄⣸⠉⢻⠃⠀⢠⡴⣋⣁⡤⠖⢻⠯⣦⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡿⢤⢤⣀⣀⣀⣀⣀⣀⣀⣠⡽⡇
⠀⠀⠀⠀⠀⠀⣀⠴⠚⠙⢧⡀⠻⣤⣀⣀⠀⠀⠀⠁⢈⣀⣀⡴⠃
⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠙⣆⡧⢀⣈⡉⠉⠉⠉⠉⠁⠀⣶⣛⣉⢉⣙⣦
⠀⠀⠀⢠⣶⠋⠙⢫⡳⡀⠀⠀⠘⡇⠀⠀⢩⢭⠉⢹⠀⠀⢸⡷⠤⠤⠤⠤⢾⣀
⠀⠀⠀⠇⡇⠀⣠⠤⣷⢹⠀⠀⠀⣷⠀⠀⠘⡫⠆⢸⠀⠀⢸⠀⠀⠀⠀⠀⠈⡇⠉⠓⠢⢄⡀
⠀⠀⠀⡇⢧⡠⠗⣶⣹⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⣾⠑⠒⠲⠶⠒⠊⡇⠀⠀⢀⠔⠉⡆
⠀⠀⠀⠘⢾⣤⣞⣡⡧⠏⠀⠀⠀⢸⠉⠒⠦⠤⠴⠚⠀⠀⢻⠀⠀⠀⠀⠀⠀⡇⡌⡀⡜⢰⠁⢩⣦
⠀⠀⠀⠀⠀⠈⢉⣈⣱⡦⠤⠤⠴⠚⠓⠤⣄⣀⣀⣀⡤⠔⢻⡄⣀⣀⢀⣀⣠⠏⠙⠓⠷⠞⠛⠛⠉⠑⠢⢄⡀
⠀⠀⡠⠤⢤⠴⢅⣀⡤⠟⠀⠀⠀⠀⠀⡴⠶⠶⣆⠀⠀⠀⠀⠉⠛⠛⠚⠛⠁⠀⢠⠤⠖⠢⠤⢤⡀⠀⠀⠀⠀⠁
⠀⠀⠑⠲⠦⠤⠤⠃⠀⠀⠀⡆⢉⠉⣹⡯⠭⢭⡿⠶⡉⣹⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠄⣤⡤⠤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠭⠭⠟⠁⠀⣎⠀⣀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣬⣭⣬⣭⣯⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠠⠄⠀⠀⢀⣀⡤⠀⠘⠉⠉⠉⠀⠂⠀⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⢣⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⠤⣤⠤⣤⣠⣀⠀⠀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣀⢀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣺⣿⡧⠤⢤⣧⢀⠻⠈⠳⣄⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠍⠐⠒⠂⠀⠀⠀⠀⢀⣀⣄⢀⠃⢸⡟⠁⠀⠀⠀⡇⠀⠀⠀⠸⣿⡆⠀⠸⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠉⠀⠀⠀⣀⠀⠀⠀⢀⠀⠹⣿⡿⢿⣶⣾⣶⠂⠀⠀⠀⠄⠀⠀⣠⣿⡏⠀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠁⠀⠀⣈⣄⣨⣺⣵⠀⠈⢋⣼⡿⡖⣿⣤⣶⣶⠷⢿⣿⣿⣿⣿⣿⢏⡽⢁
⠀⠀⢠⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣟⠙⠻⡿⣇⢁⣴⣶⣋⣀⣿⣿⡿⠉⠀⠀⠈⣍⣹⣿⣏⠘
⠀⠀⠚⢐⠂⠀⠀⠀⠀⡀⠀⠀⠀⣀⡀⢆⢰⣶⢶⢾⣿⡶⢤⡄⡗⢰⠟⠨⣿⠿⣿⣿⣿⣟⢁⣤⣶⣦⣤⣶⣄⣿⣿⣧⣶⣶⡦
⠀⠒⠐⠋⠁⠈⠉⠉⠀⢤⠄⠉⠀⠀⠁⠀⠀⠀⠈⠙⠛⠀⠘⠳⢿⢮⡄⢠⣧⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣯⢿⣿⣿⡟⠛⠋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠦⠄⢀⠀⠀⠀⠀⠀⠀⢛⣥⣠⠞⠙⢙⠉⠛⢷⣦⣿⣿⣿⣿⡏⣿⣤⣿⣿⣷⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠃⠀⠈⠉⠐⠀⢢⣶⣶⣻⣾⡏⠀⠠⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⠶⢶
⠀⠀⢠⠀⣀⣀⣀⠀⠀⠀⠀⣀⠀⠠⠀⡄⠀⠀⠀⠀⠀⠀⠁⠀⠉⢉⣱⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠤⠄⠄⠠⠀⠀⠤⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠈⠁⠛⠉⠛⠙⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⠿⣿
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠀⠙⠛⠙⣿⠿⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠤⠄⠤⠤⠠⣧⡀⠘⠿⠿⠿⣿⣿⣧⣤⣦⣠⠀⠀⡀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⢀⣛⠀⠒⢻⣿⣿⣧⣶⣿⣿⠤⠀⡄⠲⠐⠻⢿
⠀⠀⠀⠀⠀⢰⣶⣾⡿⢿⠿⠟⠋⠉⠌⠉⠁⠀⠀⣀⡀⣾⣧⠀⠀⠀⣀⣻⣾⣿⣿⣿⠂⢸⣿⣿⣧⣤⣄⠀⣠⣶⣆
⢠⠀⠀⠐⠉⠉⠁⠀⠐⠀⠈⣿⡆⠀⠀⢀⠀⠀⢠⣿⠿⠳⠛⠲⢤⣅⡀⠀⢠⣿⣿⡟⠐⣿⣿⢻⣿⣿⣿⣾⠿⢿⠿⠇⠀⠀⢠
⠀⠀⠀⠀⢀⠀⢀⣀⡀⠠⣴⣾⣧⣶⡤⠂⠁⠤⣤⠠⣶⣶⣴⣶⣄⡀⠀⣐⢾⣿⣿⡃⡀⢸⣋⣨⣽⡛⢿⠟⢀⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣀⠀⠀⠀⠀⠘⢿⢿⣿⡗⢰⣧⠽⢛⠁⣨⡿⠋⠉⠳⣎⠈⠛⣯⣵⣿⡀⢿⡇⠂⠀⠋⠉
⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢲⠄⠈⠀⠀⠠⣄⡀⢀⣀⡼⠟⠉⠀⠀⠉⠐⢿⠇⠀⠀⠀⣿⠀⣽⣿⡏⢿⡇⠠⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠞⣻⡏⡥⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠀⣿⣿⠁⣼⣅⢰⣆⠀⠘⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⢈⡉⠉⠉⠛⠉⢹⣇⠀⠀⠀⠀⣿⣀⣾⣏⡀⢸⡏⠉⠉⠀⠀⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣃⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣒⣁⣀⣀⣀⣼⣷⣬⣉⣁⣠⣘⣻⣿⣛⣿⣇⣘⣁⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣸⣿⣀⣀⣸⣇⣀⣀⣿⣇⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠾⠷⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣴⠟⠋⠉⠉⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣀⣀⣿⠀⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠉⠉⠉⠀⠀⠉⠉⠉⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠿⠶⠶⣶⠀⠀⣶⠶⠶⠿⠀⠀⠀⠀⠀⣿⣷⠀⠀⢀⣀⣀⣤⣤⣴⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⠛⠛⠋⠉⠉⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⣿⡀⢶⡄⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣧⠈⠻⣦⣤⣤⡀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠁⠀⠀⠙⢷⣤⣀⣀⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⡀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢧⢰⢻⡇⣾⣩⠆⣶⠙⡇⢰⢠⡎⠁⢐⣩⡆⢹⠋⢷⢠⠎⢱⡄⡖⠙⡆⢾⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠘⠃⠙⠒⠂⠙⠒⠃⠘⠀⠓⠒⠙⠒⠃⠘⠂⠙⠈⠓⠚⠁⠃⠀⠃⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢒⣛⣉⣉⣙⠛⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⣀⠴⠊⡀⠀⠀⠀⢳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⡀⢀⣠⡤⠶⠒⠋⠉⠀⠀⠀⠘⣆⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢔⣡⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢁⠖⡠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢠⢃⡼⢱⡇⠀⣀⣠⣀⡀⠀⠀⠀⠀⠀⣠⣤⠄⠀⣿⡇⠇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠇⡼⠁⢸⠇⠈⠘⠿⠎⠁⠀⡀⠀⠀⠀⠙⠛⠀⠀⡇⡇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⠃⡆⠘⡆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠸⠀⡇⠀⢣⠀⠀⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⣸⢰⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⣆⠀⠀⠀⠀⡀⠀⡀⠀⢄⡀⠀⠀⡏⣼⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢃⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠄⢠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠈⢆⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⢠⠇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈⢦⠸⡇⠈⠓⠦⠤⠤⠤⠀⠀⢠⡿⢠⣎⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠷⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢻⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡎⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡄⠀⡏⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⣆⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠸⡇⠀⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠤⠤⠴⢾⠁⠀⠀⢀⠓⠈⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⢀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⢄⠁⠀⠀⠀⠈⡆⠀⠀⠀⠈⠃⢀
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠤⠐⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠈⢧
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡼⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⣇
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠑⢄⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⠃⠀⠀⢀⡴⠊⢄⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠙⣦⠀⠀⠐⠤⡀⠀⠀⠈⡆⠀⠀⡎⠀⢀⠴⠋⠀⠀⠀⠣⡀⠀⡏⠀⠀⠀⡇⠀⠀⠀⢸⡆
⢀⡇⠀⠀⠀⢰⡄⠀⠀⠀⠈⢧⡀⢀⡿⠀⠀⠀⠀⠈⠀⠀⡀⠘⡄⢰⢀⠔⠁⠀⠀⠀⠀⠀⠀⠁⣾⡁⠀⠀⠀⡇⣀⠀⠀⠈⣧
⢸⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢰⡌⠀⠀⠀⠀⠀⠀⠀⠀⡀⡬⠁⠈⠀⠠⠆⣷⠃⠀⠀⠀⠻
⡎⠀⠀⠀⠘⡌⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⠏⠄
⣧⣤⣤⣤⣤⣷⣾⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣶⣦⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣧⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣶⣶⣾⣿⣿⣾⣿⣷⣾⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⡟⠛⡏⡒⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⡁⠀⣁⣹⣾⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠓⢒⡒⠒⢒⠂⠂⡀⢒⡒⠳⢄⣳⡤⢏⣿⣟⠒⣐⠒⢒⡒⢒⡂⠒⢻⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠛⠀⠘⠃⠘⠃⠀⠂⠐⢛⡋⠂⠘⠂⠛⠀⠘⠃⠘⠂⠀⢸⣀⣘
⠀⠀⠀⠀⠀⠀⠛⢩⠍⠈⡭⠉⢩⡄⠩⠅⢩⠉⢩⠉⠩⠉⣭⠉⡭⠉⢩⡄⢩⡭⠉⢹⠉⢹
⠀⠀⠀⠀⠀⠀⣄⡈⡉⠀⠉⠀⠀⠁⠠⠀⠠⠠⠀⠀⠈⠀⠉⠀⠀⠀⠈⠀⠀⠠⠤⣼⣶⣾
⠀⠀⠀⠀⠀⠀⠂⠀⠯⠀⠺⠄⠨⠅⠐⠇⠸⠅⠰⠄⠨⠀⠿⠀⠯⠀⠸⠂⠐⠇⠀⣠⣀⣸
⠀⠀⠀⠀⠀⠀⠙⠉⣥⠉⠡⡌⢠⠍⢠⡌⢩⡍⢩⡉⣩⠈⡌⠉⣬⠉⢩⡍⢩⡍⠁⢯⠉⢹
⠀⠀⠀⠀⠀⠀⢀⣀⠉⢁⠈⠁⠈⠁⠈⠁⠀⠁⠈⠁⠭⠤⠭⠤⠤⠤⠤⠤⠤⠤⠤⣿⣶⣽
⠀⠀⠀⠀⠀⠀⠈⠀⠔⡂⢐⡂⢒⡂⠀⠰⣄⣐⢂⣀⣄⣀⡆⠀⣒⠀⣶⡂⢰⣒⠀⢹⠀⣸
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠼⡇⠂⠐⠀⠀⠀⢡⠀⠠⠄⠈⠁⠄⠁⠀⣼⢤⢼
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠁⠉⠉⠉⠉⠉⠉⠉⠙⠐⠒⠐⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠠⣄⣀⡀⠀⢀⠀⢀⣴⢶⡚⠳⠀⢀⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠾⢶⣦⣽⠀⠈⠉⠙⠁⠂⠉⠁⢱⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢢⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠱⠀⠀⣸⢳⡖⠀⠀⠈⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⢀⠃⠀⠀⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡤⢺⠁⠉⠙⠛⠁⠀⠀⢀⡎⠀⢠⠴⠃⠀⢉⠓⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢠⠀⠛⠛⠟⠀⠀⠠⣀⠄⠀⠀⣸⠃⠀⠀⠀⠰⠋⠘
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠠⢟⡧⣀⣀⣀⠰⡄⠀⠀⠀⠀⠐⣾⠶⠴⠛⠈⣳⢄⡀⡄⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⡿⠇⠀⠀⢱⣌⣹⠋⠈⠙⠄⢰⣶⣄⣀⣅⠀⢀⣠⣴⢧⠀⠀⠈⠁
⠀⠀⠀⠠⢄⠀⠀⠀⠀⢀⠀⠀⠐⠤⡄⢶⠾⠛⠛⠀⠀⠀⠈⣟⠉⠉⢽⣛⣛⣯⡅⠀⢣⠁⣠
⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢻⣋⠙⠻⠿⢿⡄⠀⠈⠒⠁⠀⠀⡀
⠀⠀⠀⠤⠴⠂⠀⠀⠀⡠⠐⢢⣄⠀⣀⠀⠀⠀⠹⣀⠀⢠⠾⣿⢽⣯⡷⠄⠀⠀⠉⡄⠀⢠⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠹⠀⠈⠀⠒⠒⢂⠀⠉⡁⠈⠀⠀⠀⡳⠳⠾⠛⠋⠀⠉⠀⠈⠀⠀⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠀⡠⡗⠀⠀⢁⡀⠀⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⡔⠊⠀⠀⡀
⠀⠀⠀⠠⠂⠀⠀⠀⠹⠿⠄⠀⠀⠀⠀⠀⠀⠜⠀⠀⡁⠁⠀⠀⠀⣠⡴⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠁⠀⠠⡀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠋⠔⠉⠀⢠⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⡷⠾⠛⢹⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢇⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⠀⣴⡞⠛⠛⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⡈⠻⠶⠶⠾⠛⠉⠉⢉⠉⠉⠛⠛⠉⠉⢹⡇⠀⠀⠀⠀⢀⣠⣴⠶⢛⣩⡉⢁⣬⠟⠀⠉⠛⠳⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠸⠿⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠉⠀⠻⢧⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡶⠿⣦⠀⢸⡏⠁⠀⠀⠀⢀⣀⣤⣤⣤⡤⠤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠭⠙⢻
⠀⠀⠀⠀⠀⠘⠷⠾⢋⣠⣼⣷⣤⣤⡀⠸⣏⡇⠀⠈⠉⠉⠉⠉⠐⠒⠒⠢⠤⠭⠭⠭⣭⣉⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠶⠛⠋
⠀⠀⠀⠀⣀⣠⣤⣤⣿⠁⠀⠈⠉⠉⠀⠀⠀⠓⡖⠒⠶⠤⠤⠤⠤⠤⢤⣀⣀⣀⣀⡀⠀⠀⢀⠷⡀⠀⠀⠀⠀⠀⠀⠺⢯⣤⡀
⠀⠀⢠⡾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣬⣭⡉⠉⠉⠉⠓⠒⢦⢎⣹⠀⠣⠭⠭⠝⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⠉⣩⣩⡿⢵⡀⠀⠈⠓⠡⡄⠀⡎⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⠞⠃
⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⢀⣴⡞⠉⡽⠉⠁⠙⠉⠉⢯⠑⡄⡇⠀⠀⠀⠀⢳⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⢸⣌⣆⡀⣣⡀⣀⣰⣀⣀⣸⢤⡃⠸⠀⠀⠀⠀⠀⡆⣧⠀⠀⣼⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⣹⡆⠀⠀⠀⡇⠀⢸⠁⡝⠉⠁⢹⠀⠀⠹⡄⡇⠀⢆⠀⠀⠀⠀⠹⣘⡒⠒⣗⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⢠⡶⠛⠋⠀⠀⠀⠀⢣⣀⣸⣄⣸⣄⠠⣼⠤⠔⠺⡟⢣⠀⢸⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣿⣀
⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⢫⣀⢸⠁⠸⡇⠀⠈⡇⠀⣀⣧⡼⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠙⢷⡄
⠀⠀⠀⠀⠀⢹⡦⠀⠀⠀⠀⠀⢸⢹⡉⠉⡏⠉⠙⡏⠉⣧⠞⠀⠘⢤⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣼⡟
⠀⠀⢀⣠⡶⠟⠁⠰⠀⠀⠀⠀⠀⠙⠓⠒⠓⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠿⠿⠿⠿⠟⠋
⢀⣴⠟⠁⣀⣤⡴⠶⠶⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣴⠏
⣼⡇⢀⣾⠋⠀⠀⣠⡴⠞⠁⠀⠀⠀⢠⣄⠀⠀⠀⠀⣠⡴⠟⠋⠉⢩⣿⠀⠀⠀⢀⣤⡾⠋
⠙⠷⠟⠃⠀⠀⠀⣿⡀⠀⠀⠀⠘⠀⢈⣁⣀⣤⡴⠟⠉⠀⠀⠀⠀⠘⠿⠷⠶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⣀⣤⣤⡀⠈⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣁⣽⠇
⠀⠀⠀⣿⣰⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⣶⣶⡶⠿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⠚⠛⠛⠛⠛⠛⠛⠒⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠲⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⢿
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠫⣉⠀⠸⣄⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⠟
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡄⢸⠙⠛⠒⠒⠶⠦⠴⠤⠶⠶⠖⠚⠛⡃⣠⡤⠤⠤⢤⣄⡀
⠀⠀⠀⠀⣜⣀⣤⣀⡀⠀⠀⠀⠀⠹⣼⡇⠂⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⢰⣿⠧⠤⠤⠤⠤⠴⢿
⠀⠀⡴⡿⠋⠉⠙⢦⡹⣄⠀⠀⠀⠀⢹⡇⠀⠀⠀⡤⠤⡄⠀⣿⠀⠀⠀⢸⡿⠶⠤⠤⠤⠴⠶⣏⣀
⠀⡼⢰⡇⠀⠀⢀⣈⣧⠘⡆⠀⠀⠀⠀⡇⠀⠀⠀⠷⢮⡤⠀⣿⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⡏⠉⠓⠦⢤⣀
⠀⣇⢸⡇⠀⢾⣁⣈⢿⡄⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠉⠀⢸⠀⠀⠀⣿⠤⣀⣀⠀⠀⢀⣀⡤⡇⠀⠀⠀⠀⠈⣹⠦⡄
⠀⢹⡀⢧⠔⠋⢩⠿⣼⢡⡏⠀⠀⠀⠀⣧⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠉⠉⠉⠉⠁⠀⡧⠀⡀⠀⠀⡞⢁⡀⣿⡀
⠀⠀⠳⣜⡦⢴⣋⣴⣣⠞⠀⠀⠀⠀⠀⢿⠈⠙⠓⠶⠶⠶⠖⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠁⡄⣸⠀⡏⠀⢀⣽⡆
⠀⠀⠀⠈⠉⠉⠉⠉⠳⢤⣀⣀⣀⣀⡤⠾⠦⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⣿⡀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠲⢧⣿⣤⠽⠳⠾⠋⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠤⠒⠻⠉⢹⡄⠉⠉⠀⠀⠀⠀⠀⣉⣉⣉⡛⠛⠋⠉⠀⠀⠈⠳⣭⣓⣒⣒⣒⡪⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄
⡞⠋⠉⡛⠛⠓⢦⡴⠒⠋⠀⠀⠀⢀⣀⣀⡀⡾⣒⣂⣉⣹⡆⠀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⢹⠉⠓⢦⡀
⠉⠒⠲⠧⠤⠤⠼⠁⠀⠀⠀⢠⡎⠁⡸⠉⢉⡷⠦⠭⠭⣾⠖⠫⡁⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣼⡤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣙⣛⣛⡩⠇⠀⠀⡼⠁⠀⠀⣰⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣿⡟⢻⣿⣀⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣿⡟⠛⠁⠈⠛⢻⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣼⣇⠀⠀⠀⠀⠀⠻⠿⣿⡆⢰⣿⠿⠟⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⣿⠛⣿⡆⢸⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣿⡇⠀⣿⠛⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠿⠿⠿⠃⢸⡇⠀⠀⣿⣁⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣿⡇⠀⣿⡇⠀⠿⠿⠿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⠛⣿⡆⢸⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⣿⡇⠀⣿⠛⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠿⠿⠿⠃⢸⡇⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠿⠿⠿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠛⢻⣿⠛⠛⢻⣿⠛⠛⢻⡿⠛⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⣿⠛⣿⡆⢸⡇⠀⠀⣿⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡇⠀⣿⡇⠀⣿⡇⠀⣿⠛⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⠿⠿⠿⠃⢸⡇⠀⠀⣿⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡇⠀⣿⡇⠀⣿⡇⠀⠿⠿⠿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⠀⢸⡇⠀⠀⣿⠀⢸⣿⠀⠐⢿⣿⠃⠀⢸⡇⠀⣿⡇⠀⣿⡇⠀⢀⣀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⣿⠛⣿⡇⢸⡇⠀⠀⣿⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡇⠀⣿⡇⠀⣿⡇⠀⣿⠋⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⠀⣿⡇⢸⡇⠀⠀⣿⠀⢸⣿⠀⠀⢸⣿⠀⠀⢸⡇⠀⣿⡇⠀⣿⡇⠀⣿⠀⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠾⠿⠿⠿⠷⠾⠿⠶⠾⠿⠷⠾⠿⠶⠶⠿⠿⠶⠷⠿⠿⠶⠿⠿⠶⠿⠷⠾⠿⠿⠿⠷⠾⠟






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⢤⢤⣴⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⢠⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣼⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠀⠁⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣿⣿⠿⠿⠿⣼⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡷⢶⣦⢉⡉⡋⠉⠁⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⡧⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠉⠓⢥⡀⠄⣪⣽⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠶⠴⡶⢶⠒⣻⠂⠀⠀⠀⢀⣹⡥⡀⠀⠀⠹⡿⠿⠿⠿⢿⣶⣶⡶⢶⣤⣤⣇⡀⢸⠉⠙⠓⠒⢲⠀⠀⢱⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣀⠸⠀⠿⡂⠀⢠⣾⡿⠛⠿⣖⢄⠀⣠⣷⣄⠀⠀⠀⠀⠈⠉⠲⡄⠈⠙⣿⣿⡄⠀⠀⠀⢸⠀⠀⢸⢸⣿⣿⣇
⠀⠀⠀⠀⢀⣤⠟⢳⣯⡿⠯⢿⣄⣶⣛⣦⣼⣿⣦⣤⣤⣾⣿⣟⢋⣡⣾⡁⠀⢀⣹⣿⣿⣷⣷⣦⣶⣿⣿⠁⠀⠀⠀⢸⠀⠀⢸⣼⣿⣿⡟
⠀⠀⠀⠀⡿⢥⣀⣀⠈⠉⠉⠉⢡⣿⣤⣤⣠⡤⣼⣟⣭⢫⡉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⣽⢫⣿⡏⢸⠀⠀⠀⠀⢸⠀⠀⢸⣿⣿⣿⣷
⠀⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⢹⣿⣿⡆⡇⠀⢸⡟⣸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣿⡇⠸⠀⡀⠀⠀⢸⠀⠀⢸⡟⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡏⠓⠒⠒⠒⠒⠚⣿⢛⣿⠻⠛⣿⡇⣿⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⣿⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠀⣠⣾⣛⠋⣘⣃⢨⡳⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠂⠉⠉⠉⠀⠒⣦⡀⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣞⣿⣿⣿⣾⣿⣿⣗⠙⡘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠛⠛⠒⠚⠋⢹⡇⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠳⠤⠤⢤⣤⠤⠴⠚⡇⠀⠀⠀⠀⡇⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⠀⢉⠉⠉⠉⢉⣉⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⠛⠒⠤⠶⠶⠶⠖⠙⢙⣆⡠⠤⢤⣧⣿⣿⣿
⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡇⠀⠘⢒⡶⠻⣟⠛⠀⠀⣿⣝⡢⢤⣄⣀⣀⣀⣀⣀⡤⣿⣝⣲⠲⠤⠤⠤⠤⠒⣎⣭⡏⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⢤⣄⣀⡀
⠀⠀⠀⠀⣧⠀⠓⠤⠤⠤⢤⡇⠀⢰⡋⠀⠀⠈⣱⠆⠀⣿⣿⡿⠋⠓⠓⠒⠒⠒⠒⣋⣿⠀⠉⠉⠛⠒⠛⠋⠉⠁⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠽
⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⢀⡇⠀⢀⠙⢦⣤⠞⢁⠀⠀⢿⣿⡇⠒⠢⢤⠖⠂⠛⢿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⡇⠀⠈⠙⠋⠛⠋⠋⠀⠀⣸⠿⣇⣀⡀⢸⣶⣶⣲⠷⡳⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣦⣤⣀⣀⠴⠊⠁⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣷⣶⣦⣤⣤⣴⣶⡆⢹⣿⠀⠀⠈⢉⡹⣻⡉⢈⠇⢱⣿⣄⣀⠀⠀⠀⠀⠀⠀⣀⡤⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⡤⠞⠉
⠀⠀⠀⢠⠶⠶⠖⠒⠒⠒⠒⠛⠉⠉⠉⠙⣿⣟⣛⣡⠴⠃⣴⣆⣢⠿⠿⣿⣏⣁⣠⣿⣽⠀⠉⠙⠛⠒⠒⠛⠋⠁⠀⣿⠉⠙⠻⠿⢿⣿⣿⣇⡤⠞⠉
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠄⠀⡄⠀⡈⠉⠛⠂⠀⠼⠈⠉⠉⠀⠘⠧⣄⣀⣀⠀⠀⢀⣀⣠⠴⠃⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⠺⢿⡶⠦⠛⠲⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠸⡧⠤⠤⠶⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⢻⠀⠈⠳⠖⠚⠉⠁
⠀⠀⠀⠀⣧⡤⠤⠤⠤⠴⠶⠶⠒⠒⠒⠒⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠻⠃⠂⠀⠰⠀⠀⠀⠀⢲⣶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⣀⡀⠀⠈⣣
⠀⠀⠀⠀⢀⣤⣼⣿⢇⠀⠀⠀⡀⢀⠀⢁⣴⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣽⠀⠀⠀⠤⠤⠒⣃⣿⡆
⠀⠀⣠⣾⡟⠹⢿⣿⣇⠄⢀⡫⠹⢣⣴⣾⠇⢄⠀⢀⢀⠌⢆⠀⠀⠠⠐⠀⠀⠀⠀⠈⢀⣤⠿⠋⢄⠀⠀⠀⠀⠀⡈⠉⠁
⢰⣶⣽⣿⣄⠀⠀⠉⠛⠲⠤⠁⢀⡀⣿⡿⠀⠘⢋⠖⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢺⠆⢀⣠⡔⢏⠁⠇
⠀⠻⣿⣿⣿⣷⣦⠀⠀⠀⠀⠃⠐⠒⠛⢄⡄⠠⠄⠂⠕⠀⠀⠀⠀⠉⠂⠀⠤⢀⡀⡀⠀⠀⠀⠀⢸⡇⠉⠸⠆
⠀⠸⢶⣽⣿⣿⣿⣿⣿⣶⣤⣤⣤⣤⡀⠀⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⠠⠋⢀⠎⠀⠈⠉⢉⠖⠤⢾⡇⢰⡟⠂⠀⠂
⠀⠀⠈⡽⢿⣿⣿⣻⠋⠥⠀⠀⠱⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢸⣧⠸⡿
⠀⠀⠀⡠⢃⣽⣿⣿⣿⣿⣿⡇⢫⠳⠀⠀⠀⠀⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢟⠀⠃⡔
⠀⠀⠈⠰⠈⠋⠘⠹⣿⣿⣿⡆⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⢀⠰⣧⡄⢂⣀⣀⠂⣤⡷⢘⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢋⡀⠈⢀⡈⢀⠌⠠⠐⢄⠉⡀⠌⠀⡌⢛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⣰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠇⠀⠀⠀⠀⠀⠘⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⣾⠏⢘⠀⠀⠀⠀⠀⠀⠀⢐⠋⣿⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣁⣤⣶⣿⣷⣼⣆⠀⠀⠀⠀⠀⢀⡾⠀⣿⠀⠀⠙⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠⣿⠋⠀⠀⠈⢻⣿⣦⠀⠀⠀⣠⣾⠇⠀⣿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⣿⠀⠀⡀⠀⠀⣿⣿⣿⣶⣾⣿⡟⠀⢰⡏⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠂⠀⣴⣿⢹⣿⣿⣿⡿⠁⠀⣾⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⠁⠀⠀⠠⡏⠉⠀⢻⣿⡿⠁⡔⠌⢉⠀⠀⠀⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰⡇⢂⠀⠀⠐⠀⠀⠀⠻⡁⠀⢄⠄⢀⠀⠀⠀⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠀⠈⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠠⠃⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠣⠀⠀⠀⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢛⣉⡛⠶⣄
⠀⠀⠀⠀⠀⠀⢠⡤⣤⣤⣤⣤⡤⣼⢁⣀⣼⢸⣇⣀⠹⣦⣤⣤⣤⣤⣤⡤⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣿⠘⠒⢲⢰⡖⠚⢡⣇⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠉⠉⠳⣄⣘⠚⣃⡤⠏⠉⠉⠉⠉⠉⠉⢩⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣀⣀⡉⠉⠁⣀⣀⣀⣀⣀⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢨⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⢘⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠉⠉⠉⠙⣷⠀⠀⣿⠉⠉⠉⠉⢹⡆⠀⢸⡇⠀⣿⠉⠉⠉⡿⠀⢾⠉⠉⠉⣿⠀⢸⠉⠉⠉⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠿⠤⠤⠤⠟⠀⠹⠤⠤⠤⠿⠀⠸⠤⠤⠤⠾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠷⠶⠶⠶⠶⠟⠀⠀⠻⠶⠶⠶⠶⠾⠃⠀⠨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠉⠉⠉⣗⠀⣼⠉⠉⠉⣿⠀⢸⠉⠉⠉⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡶⠶⠶⠶⠶⣦⠀⠀⣴⠶⠶⠶⠶⢶⡄⠀⢸⡇⠀⣿⣤⣤⣤⡟⠀⢺⣤⣤⣤⣿⠀⢸⣤⣤⣤⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣠⣤⣄⢸⡇⠀⢸⣇⣀⣀⣀⣀⡿⠀⠀⣿⣀⣀⣀⣀⣸⠇⠀⢸⡇⠀⣾⠛⠛⠛⡧⠀⢼⠛⠛⠛⣷⠀⢸⠛⠛⠛⢻⠀⢸⡇⣀⣀
⠀⠀⣠⠞⠁⠀⠈⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⣀⣀⣀⡿⠀⢽⣀⣀⣀⣿⠀⢸⣀⣀⣀⣸⠀⣼⠟⠁⠈⠛⠳⡆
⣤⠞⠁⠀⠀⠀⠀⠀⠈⢹⡆⠀⠀⠀⠀⡟⠛⠛⠛⠛⢻⡆⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉⠁⠀⠀⠉⠉⢉⡥⠞⠁⠀⠀⠀⠀⠀⠛
⠻⣆⠀⢀⠀⣠⡀⣀⣀⡿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡇⠀⣶⠒⠒⠒⣦⠀⣴⠶⠶⠶⢶⠀⢰⠖⠒⠚⢧⡀⠀⠀⠀⡀⠀⠀⢐
⠀⠈⠉⠉⣿⢹⠉⢹⡏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠨⡇⠀⣿⠀⠀⠀⣿⠀⣿⠀⠀⠀⢸⠀⢸⠀⠀⠀⣸⠛⢺⣿⡟⣿⠛⠉⠋
⠀⠀⠀⠀⡿⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢨⡇⠀⠉⠉⠉⠉⠁⠀⣿⠀⠀⠀⢸⠀⠈⠉⠉⠉⠉⠀⢸⣿⡇⢿
⠀⢠⡶⠶⡷⠾⠷⡾⠷⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⢾⠷⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⢿⢶⠶⡶⠾⠶⠶⠶⠶⠶⠶⠶⠾⢿⣶⢾⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣒⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠿⠴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠂⣠⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠼⠃⣠⠀⠀⣠⣄⡀⠀⠀⠀⣄⠀⠊⠃⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣁⠄⠈⠉⠐⠆⢢
⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⢿⣀⡉⠁⠈⣀⡜⠀⠀⠻⠷⠟⠊⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠞⠻⠻⠛⡠⢪⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠬⠍⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠤⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠦⠴⠿⠓⠁⢈⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⣀⠀⠤⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠊⢠⠐⠒⠒⣒⣒⠶⣦⠐⣄⠀⠀⠀⠀⠀⠐⠤⠤⠴⠶⠶⠊⠁⠀⠀⠀⠀⠀⠀⠒⠲⠢⠀⠀⠸⡧⣀⠁⠐⠒⠛⠛⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣷⡀⠀⠤⠐⠂⠀⠂⠓⠀⡰⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢃⠀⠀⠑⠪⢉⣐⣒⠒⠂⠈⠡⠚⠁
⠀⠀⠀⠀⠀⠀⠘⠿⣷⠶⠦⣤⣲⡶⢞⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⡯⠂⠀⡀⠀⠊⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠀⠀⢰⠁⠀⠔⠁⠀⠀⡱⣆⠀⡀⠀⠀⢀⡄⠈⠁⠂⠈⠀⠀⠚⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈⠉⠘⢁⡴⡷⠀⠀⢸⢀⡀⠀⠔⠚⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⡀⣀⠀⠀⢀⠀⠀⠙⠢⢬⣭⣿⣶⠟⠛⠃⠀⠀⠈⠑⠒⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢤⠀⠐⠁⠠⣌⠩⠯⡂⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⡀⢠⡶⠚⠛⠻⠀⠘⢉
⠀⠀⠀⠀⠀⠀⢻⠂⠀⠁⠈⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠁⠈⠁⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢒⠒⠒⢀⣀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠗⡦⠄⢤⡀⠀⠀⠀⠀⠀⠀⠓⢵⣶⣤⡴⠖⠢⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠚⠗⠂⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣦⣄⣠⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉