Sickness Emojis & Symbols

⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣷⣶⣟⡛⣿⣿⣿⣇⠀⢰⣤⣴⡶⢠⣤⣤⡄
⠀⠀⠀⠀⣿⣿⣿⡿⠿⣿⣿⡿⠀⠘⠉⣿⠁⣿⣚⣿⠃⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣿⣿⣿⣤⡖⣟⣟⠋⠀⠀⠀⠏⠘⠱⠿⠏⠀⡿⢛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣷⣦⣦⣵⣦⡀
⠀⠀⠀⠀⣿⣿⣿⣿⡉⢸⣿⡁⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⠉⢹⠟⣿⣾⣿⣟⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣧⣸⣿⣗⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣱⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣼⣿⣷⡇⣿⢢⣿⣿⣿⣿⡗
⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸⠇⠀⠀⠀⠀⢺⡗⢰⢿⡅⣋⠀⠸⣿⡄⣀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡆⠀⠀⠀⠈⢹⣿⣿⣿⣗⠈⢻⣿⣿⣿⣿⣿⣷⠀⠀⣿⣍⠀⠀⠀⠀⠀⠈⠙⠿⠟⢀⡀⠙⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⠀⠈⢁⡀⠀⠀⢰⠛⠻⠿⠿⠛⠀⢠⡏⠈⣿⣿⣿⣿⡆⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⡘⠛⠂⣀⠀⡀⠀⠀⢰⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣀⣴⣿⣿⣷⡄⢸⣼⣿⣦⣶⣾⣶⣬⣅⡴⢻⣿⣿⣿⣿⠟⠛⠛⠂⠀⠀⠀⠀⠀⠀⠁⠀⣇⣄⠚⠁⠻⣿⡆⠀⣼⣧
⠀⠀⠀⠐⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣯⠀⣴⣁⡽⢛⣩⣠⡠⠆⠀⢢⠀⠀⠀⢰⡆⠀⠐⠉⢉⣭⣤⣶⣿⡿⠇⢰⣿⣷
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣇⣤⡄⠀⠀⠉⠙⠛⠿⣯⡇⣿⣙⣯⠘⢃⣴⣶⣤⡐⠒⢦⠀⡁⣀⢀⠀⠈⢥⠀⢻⣿⠁⢠⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⠃⠒⣽⣼⣿⣿⣿⡿⠀⠀⣀⣀⣀⣶⣿⠛⠉⠉⠈⡃⣿⣿⣿⡟⢀⡀⠀⠀⣧⠙⣿⣦⣾⡏⠀⣾⠃⢀⣾⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠋⠉⢰⣵⣿⣻⣿⣿⣿⣇⢠⣿⣿⣿⣿⣧⣤⣷⣶⣾⡿⢿⠍⠛⠋⠉⠛⠀⣀⣤⣿⣼⣿⣿⣿⠃⣸⣃⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠿⠿⠿⢿⣿⣟⠀⣦⡀⣼⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠢⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⢧⣿⠃⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣲⣶⣶⣶⣶⠆⠀⠀⠏⠐⣿⣿⣿⣿⣿⡿⣿⣿⣿⡄⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠈⠿⣿⣿⣿⣿⣾⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠋⢉⣩⣥⠀⠀⠀⠀⣶⣤⣽⣿⢿⣿⣿⡇⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⣴⣿⡿⢿⠿⢿⣿⣿⣿⣿⣿
⠀⣤⣿⣿⡇⠀⠀⠀⠀⠹⣿⣧⠁⢾⣿⣿⡇⣿⣿⣿⣿⣿⣄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠟⠟⠿⠿⢫⣿⡟⠰⢞⣿⣿⣿⣿⡿
⢸⣿⢿⣿⣷⣶⣶⣴⣶⣶⣿⣿⣦⣾⡿⠛⢁⣿⣇⣼⣿⣿⠅⣿⠀⢻⣯⢹⣿⣿⡿⠿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣧⣤⠶⠶⠶⣿⡟⢳⣾⣿⣿⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠉⠉⠀⠘⠙⠛⠛⠁⠀⠀⠁⠉⠛⠛⠓⢾⣿⣿⣿⣧⣾⣷⣄⠙⣿⣿⣿⣿⣿⣿⣿⠷⠆⠐⣉⡤⠞⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠿⠿⠿⠷⠼⠟⠉⠁⠀⠀⠀⠀⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣶⣶⣖⣒⣻⣭⣝⣒⡲⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢼⠉⠙⠛⠓⠒⠒⠒⠒⠒⣯⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡜⠀⡠⠞⠀⠀⠀⠳⠤⠀⢣⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣿⠁⢀⡤⠤⡄⠀⢠⠤⢤⡀⠈⣿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠸⣀⣂⡽⠀⢯⣐⣀⠇⠀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⡇⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⢸⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢳⣀⡼⠁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⡜⠀⢳⡀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⣀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣧⠀⠈⠺⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⣀⣦⣀⣀⠁⡀⠀⢿⣿⣰⣇⣶⣿⣿⣫⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠄⠠⠢⣀⠀⠀⠘⠉⠰⣧⡦⡄⢀⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡄⣦⡏⠒
⠀⠀⠀⠀⠀⠀⠀⠐⠥⣦⠀⢀⠀⠀⠀⠱⣀⡂⡄⢾⠻⡟⣬⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⡉⡐⠀⠀⠆⡹⣤⠉⣪⣼⣶⣺⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⣚⣟⢁⣠⡃⠀⠀⠠⢿⣍⡗⣶⣿⡿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠹⣮⣀⣀⣀⣮⢻⣛⣿⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⠹⠝⡁⠹⣿⣸⣿⣿⣿⡿⣿⣷⣿⣛⢽⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣍⣭⡀
⠀⠀⠸⠍⣿⣷⣤⠀⠀⠀⠀⢀⣀⠠⣀⠈⠉⠀⢀⡿⣠⣽⠿⣻⣿⢿⣿⣶⣚⣻⢺⣿⣿⣿⣿⣿⣿⣿⣿⡏⠘⠁
⠀⢀⣀⠀⢠⣾⣻⠀⣠⣤⣠⡾⣿⣻⣿⡑⣶⢒⣿⣷⠏⠙⠁⠈⠈⡏⠻⡿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⡦
⠀⣶⣾⣿⣾⣽⣿⣾⡿⣿⡿⣧⣽⠿⢻⣷⣡⣿⣽⠟⢠⠀⠈⠁⠤⠴⠊⢄⣸⣋⣽⣿⣿⣿⣏⣽⣿⣿⣿⠶⠃
⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡧⣂⢸⠸⣷⣿⣿⡦⠀⠻⠶⣤⡄⠀⠈⠈⠻⠟⠡⠴⠏⣰⠻⢿⠏
⢹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⢷⡈⠀⠈⢻⠿⡦⢁⣠⠀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣷⣦⣽⣿⡾⣿⠎⠀⠀⣰⡞⠀⠈⡧⣼⣳⡜⣷⠇
⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣏⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⡟⠁⠉⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡄⠀⠀⠀⡀⣐⠣⠔⠻⠟⠀⢠⣤
⠀⠀⠉⠿⣻⣿⣿⣿⣿⣍⠉⠀⣠⣴⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣬⡇⠀⠰⣼⡿⢉⢧⣠⣤⠀⠀⢘⣣⡤
⠀⠀⠀⠀⠉⣿⣿⣿⡿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣟⣿⣿⣿⣟⡿⠀⠀⠰⠏⠑⠈⣾⣟⣥⢠⠀⢟
⠀⠀⠀⠀⠀⠁⠻⡿⠃⠀⠀⠀⢻⣿⣧⣽⣿⣿⣟⠳⠀⠀⠙⣿⠿⠟⣉⠀⠀⢠⣶⣀⣤⡼⣯⣤⣿⣧⠀⠛⠡⠂⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⡿⣯⣿⣿⣿⡘⠝⠶⣤⢀⡘⢯⣃⣭⣥⡷⢯⣽⣿⣿⣶⣿⣾⣞⠀⠐⠳⠐⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣸⣠⡝⠂⠘⣿⢱⣎⡦⣬⡆⠹⣧⣄⠠⠉⢿⡚⣃⣀⠛⢷⡌⠻⣟⣌⠁⣾⡵⠀⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡿⠲⢈⣆⣄⣿⣿⣷⡋⣁⣶⣧⠿⣷⡤⢑⣦⡉⢳⣶⠾⡿⢤⣤⣾⠿⠿⠧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡇⠒⠺⣿⠛⠩⣿⣾⣑⣿⢁⣿⣟⣽⣿⣽⡳⣷⣆⡤⣠⣴⡎⠉⠻⠇⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⠤⠀⠻⠳⢦⣅⣾⣿⡏⠽⣌⡙⣿⠧⠜⢻⡀⠚⠅⠉⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠆⠀⠹⢧⠸⣷⢴⠛⠓⠀⠘⠻⢤⡀⠀⠁

⠀⠀⠀⠀⠀⢀⠀⣴⢦⣰⢷⣦⣟⣆⣀⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢯⣧
⠀⠀⠀⢠⠴⢿⡟⣿⣹⣟⣶⣯⠿⢿⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⣁⠀⣈
⠀⠀⣆⣡⡁⠹⢿⠟⠿⠋⢛⣦⢑⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⣨⡞⠋
⠀⠠⠽⠏⠖⠻⠦⠀⠻⠦⡴⠦⠈⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⢰⠀⠶⠓⠂⠀⢠⡶⣿⣿⣆⠀⠀⠀⠀⡂⢀
⠀⣀⡐⢲⣶⠒⣲⣶⣶⠶⡿⣷⣶⣖⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣰⡃⠢⡀⠀⡚⠈⠀⠀⠀⠄⠀⠀⣿⢻⡟⣿⣿⡄⠀⠀⠀⢻⣄⣀
⠀⠈⠛⡛⢛⡰⢮⣝⣀⠀⣇⣊⣈⣉⠀⠈⠈⠀⠀⠀⠀⠀⠀⣨⡽⣯⡟⠸⡄⣈⡀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠸⡟⣠⡀⠀⠀⠀⢻⣷⠄
⠠⣿⣿⣿⣿⠁⠸⣿⡙⠦⡘⠀⢭⣿⠀⠛⠃⠀⠀⠀⠀⠀⢐⠋⠉⡙⢿⠀⠷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣿⠋⢡⡜⠀⢀⡀⢀⣀⣀
⠀⢸⣿⣿⡿⠀⢰⣿⣷⡀⠈⠂⣄⣋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⣶⠖⢽⣶⣶⠾⢿⣆⠀⠀⠀⠀⠀⠀⢹⢣⡼⢿⠿⠟⠁⣬⣿⣟
⢸⣿⣿⣿⣯⣀⢸⣿⣿⣧⣾⣿⣾⠛⠛⢻⡀⠀⠀⠀⠀⡀⠀⠀⠀⢻⣦⣤⠀⠉⠈⢿⣦⣾⡟⠀⠀⠀⠀⠀⠀⠙⣹⡧⠁⢀⡀⣠⣬⣿⣟
⢨⣿⢻⣾⢿⣿⣿⣿⡿⠟⢋⠿⠻⡷⣦⡈⣧⠀⠀⠀⢠⠎⠀⠀⢀⠀⠈⣿⣷⣀⣶⣿⣯⡍⣿⠀⠀⠀⠀⠀⠀⠀⣿⣡⡀⠈⠉⠉⠉⠉⠁
⢸⣿⣿⣷⣿⡷⠛⡏⣤⣠⣿⣉⠀⣾⠻⣇⣿⣶⣴⣶⣧⢃⣤⣬⣿⡄⠀⠉⠉⠉⠛⠛⠛⠛⠿⢦⡀⠀⠀⠀⠀⠀⢿⡩⠽⠆
⠙⣿⣿⣯⣿⣃⡾⢹⣿⣿⡿⣿⣳⡿⢰⣿⡏⠉⠙⢯⠕⠉⠀⠈⠛⡇⠀⠐⢰⡖⠒⠻⠟⠛⢻⡏⠀⠀⠀⠀⠀⠀⠸⠙⣆
⠀⣿⣽⣽⣿⡿⢉⠈⠀⠀⠀⢈⢹⣷⣾⣿⠀⠀⢀⣤⣿⡀⠀⠀⠀⣷⠀⠈⠈⣿⡀⠀⠀⠀⣼⣧⡀⠀⢀⣀⣀⠀⣀⡆⠘⣷
⠀⠛⢿⡟⣿⡃⣴⣄⠀⠀⠀⠀⣿⣿⣿⠗⠀⢠⣿⣿⣿⣷⣦⣄⠀⠘⣇⣀⣰⣿⣷⣟⣿⣶⣿⡌⠷⠀⡀⠈⠉⠙⠛⠓⠄⠈⠣⡀
⠀⠀⢛⣾⣯⡇⣿⣿⣿⣄⣴⢷⣿⣿⡇⠀⠀⣾⣿⣿⠯⠍⠉⠙⡄⠀⠙⠃⠈⠃⠀⠙⠁⠈⠹⣧⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠷⠒
⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠟⣾⣿⠟⠀⠀⢀⣿⣿⣿⣶⠂⠀⠀⠈⣶⣦⣲⡶⠂⣤⣄⣀⣀⢰⡿
⠠⣾⡷⣶⣿⡿⠛⠛⠋⣰⣶⡿⠇⢀⣤⣾⣿⣿⣿⣿⣿⠀⠀⠀⢸⡿⠿⠲⣤⣤⠈⢉⡉⠙⠛⠛⠓
⠀⠈⠙⠛⠉⠑⠓⣿⣾⠿⠛⢳⠘⠟⠿⠟⠻⣟⣿⣿⣿⡀⢀⠖⠋⠀⠀⣠⡼⠁⠉⠀⠸⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠸⣧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣧⡀⣀⣦⣶⣿⣡⡴⠾⠓⠒⠛⠒⠛⠚⠶⠆⠆

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣄⡀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡀
⠀⠀⠀⠀⠀⣠⣤⣰⡯⣿⣿⣙⣿⠷⣿⣶⡶⠒⠛⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⠛⠏
⠀⠀⠀⢈⣫⢙⣿⣼⣿⣿⣵⣿⠷⢟⠛⢿⠛⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠰⠄⣠⡦
⠀⠀⢷⣴⢇⢐⡚⠟⠈⣛⠀⢩⠿⠢⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠥⠀⠀⠀⢰⣾⡅
⠀⠐⠶⠋⠸⠉⠟⠂⠈⠛⠓⣟⡗⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠆⠛⠘⠚⠁⠃⠀⢠⡿⣿⣿⣿⣄⠀⠀⠀⠀⡅
⠀⢠⣍⣉⣿⡟⠉⣟⣿⡿⠿⡟⣿⣿⣯⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⡤⣇⡙⣄⠈⠉⡉⠈⠁⠂⠀⠀⠀⠀⢸⣟⣿⢻⣿⣿⠂⠀⠀⠀⠹⣆⣠⡄
⠀⠈⠛⣛⠛⣓⠲⢶⣗⣀⠈⣇⣉⣀⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣯⣿⠁⢧⢀⣭⠬⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢻⣋⡄⡀⢀⠀⠀⠹⣷⡇
⠠⣿⣽⣿⣿⡏⠀⢿⣏⠻⢄⡙⠀⣍⣿⡇⠙⠻⠀⠀⠀⠀⠀⠀⢰⠏⠉⡉⢿⡅⠹⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢘⡟⡉⢥⠝⠀⠀⡀⠀⢀⢀
⠀⠨⣿⣿⣿⠇⠀⣷⣿⣇⠀⠙⠢⣀⠚⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡤⢶⡶⠬⣷⣤⣴⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⣿⣡⣿⢶⢿⠟⠉⣀⣽⡿
⠸⣿⣿⣿⣯⡀⠀⣻⣿⣿⣦⣼⣷⣥⠿⠿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣧⣨⡂⠐⠃⠙⣿⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠗⣿⡏⠃⠀⠀⣀⣈⣻⣿
⠘⣿⣿⣏⣿⢿⣷⣿⣿⣿⡿⠟⣿⢿⣦⣤⡀⢿⡀⠀⠀⠀⢈⡗⠀⠀⠀⠀⠙⢻⣷⡄⢠⣤⣿⣏⡛⣷⠀⠀⠀⠀⠀⠀⢀⢸⣿⣀⠀⠘⠓⠛⠟⠛⠛
⢸⣻⣶⣛⣐⣟⣻⣿⠛⡁⢀⣾⠖⠀⣩⣾⣿⢸⣗⣀⣤⣤⠎⠄⢀⣐⣶⡆⠀⠹⠿⠿⠿⠿⠿⢿⣷⣿⣀⠀⠀⠀⠀⠀⠀⠸⣿⢳⣲⡄
⠼⣿⣿⡿⢿⡿⢁⡴⣯⣿⣿⣿⣷⣄⣺⠇⣽⣿⠟⠻⢿⠿⣋⠞⠛⠛⢿⣧⠀⠀⠀⣀⣤⣤⣤⣤⣤⣬⡭⠆⠀⠀⠀⠀⠀⠀⠻⢯⠁⠈
⠀⣿⢿⡿⣿⣷⡾⡡⠛⠛⠙⠋⠻⢾⣿⢰⣿⡏⡀⠀⠀⠙⡀⠀⠀⠀⠀⣯⠀⠠⠄⣿⡔⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠈⢷⣄
⠀⣿⣾⣿⣿⣿⡁⢂⠀⠀⠀⠀⠠⣼⣿⣹⣿⠋⠀⢀⣶⣿⣿⣄⠀⠀⠀⢹⣄⠀⠀⢸⣿⣆⣠⣄⠀⣿⠿⡆⠀⠀⠶⠶⣦⣴⣾⠃⠀⢳⡄
⠀⠀⣩⢭⣾⣶⢰⣿⣦⣀⠀⠀⡀⣿⣿⣿⠏⠀⠀⣿⣿⣿⣿⣿⣷⣦⠀⠀⣿⣤⠤⣿⠟⠻⣴⡿⢿⣿⡧⠙⠂⠶⢤⣤⣤⣀⣈⣉⣁⣠⡏⠂
⠀⠠⣨⣿⣿⣾⣈⣿⣿⣿⣤⣿⢻⣿⣿⠗⠀⠀⢸⣿⣿⡟⠓⠂⠀⠈⢦⡀⠀⠉⠀⠈⠀⠀⠈⠀⠀⠘⣷⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠉
⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⠟⠀⠀⣀⣸⣿⣿⣿⣾⠁⠀⠀⢀⣷⣷⡮⣽⡛⠰⣤⣤⣤⣄⣰⣿
⠠⣾⣶⣷⣿⣿⠛⠛⠛⢉⣶⣶⡟⠇⠀⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⠿⠗⢦⣤⢄⣀⡀⡈⢉⠙⠛⠛
⠀⠈⠝⠛⠋⠙⠚⢼⣷⡾⠿⠟⢳⡌⠟⠿⠿⠛⣿⣻⣿⣿⣿⠀⢀⠖⠋⠀⠀⢀⣤⡍⠈⠁⠀⢱⠈⣦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⡀⠠⣦⣴⣾⣏⣡⣴⠶⠶⠒⠓⠒⠛⠾⠦⡦⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣤
⠀⠀⠀⣄⣠⣿⣧⣴⡄⠀⠀⠀⠀⠀⣤⡾⢛⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⢠⡴⣻⠟⠁⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⡿
⠀⠀⢰⣿⠿⠿⠟⢿⠇⠀⠀⠀⠀⠀⠙⠛⣧⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡇
⠀⠀⢸⡇⠘⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠃⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠃
⠀⠀⠈⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐
⠀⠀⠀⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⣰⣟⣸⣿⣇⣻⣆⠀⠀⠀⠀⠀⠀⠀⣾⠿⣷
⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⢠⣾⣯⣤⣽⣶⣆
⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⡇⣿⠂
⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋

⠀⠀⠀⠀⣺⣿⡦⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠺⣿⡿⣻⡇⠀⠀⠀⠀⠀⠀⡰⠋⢠⣶⡄⠉⢦⠀⠀⠀⠀⠀⠀⢀⣬⣿⠋⠀⠀⣿⡇⠀⠀⠀⣠⣶⣞⣿⡿⢶⣄⡀
⠀⠀⠀⠀⣿⣿⣯⠁⠀⠀⠀⠀⠀⠀⡇⠸⢿⣿⡿⠇⢸⠀⠀⠀⠀⠀⠰⣿⣟⠛⠓⠶⢓⠻⠃⠀⠀⠀⠛⠿⣿⣿⣿⡿⠿⠃
⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠛⢃⡠⠋⠀⠀⠀⠀⠶⠾⠿⠿⠇⠀⠩⠿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠻⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠄
⠀⠀⠀⠀⢾⣷⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠢⡿⠂⢲⣄⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣷⣿⠛⠋⠀⠀⠀⠀⠀⡴⠻⣠⡷⢄⠀⠀⠀⠀⠀⠀⣶⢠⣄⣤⣄⣀⣀
⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢶⡋⣣⡾⠓⣄⠀⠀⠀⠀⢛⣾⣸⣿⣿⣿⣿⣿⣶
⠀⠀⠐⢥⡏⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣨⠾⠋⠀⠀⠀⠀⠈⠏⠉⠉⠉⠉⠉⠉⠏
⠀⠀⠀⠈⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠿⠿⠾⠿⠶⠶⠶⠶⠶⠶⠿⠵⠿⠽⠿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢄⣿⣾⣿⣶⣦⣶⣴⣿⣷⣮⡄⢀⠀⢠⡀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣰⣿⣿⣿⣯⣿⣿⣿⡟⠁⠀⠀⠀⠈⠛⢴⡦⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠉⣿⣿⣿⣿⠟⢉⣅⣀⣀⡀⠀⠤⠀⣁⣀⠀⠀⢀⢠⡶⠧⠀⠀⠀⢰⣿⣯⠐⠋⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠉⠉⠉⠹⠖⠂⠀⠀⠀⠈⠉⠙⠻⠿⠿⠷⠓⠋⠀⢀⣰⡆⠀⠀⣮⡿⠁⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⡿⠗⠋⠉⠀⠀⠀⠀⠀⢁⣴⣿⠛
⠀⠀⠀⠐⠂⠀⠀⠀⢀⡞⣿⣿⢁⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣿⣿⣿⣿⡟⣁⣴⣄⡀⠀⠀⢠⣀⣠⣤⣼⣿⠁
⠀⠀⠀⠀⠀⠈⠀⠀⣺⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣚⠓⠐⠚⠺⠟⠛⠛⠛⠛⠛⠛⠓⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠃⠠
⠀⠀⠀⠈⠑⠉⠂⠉⠉⠈⠀⠀⠀⠈⠙⠛⠋⠉⠙⠓⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠋⠉⠉
⠀⠈⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠲⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠒⠒⠷⣿⡖⠐⠢⠀⠀⠀⢀⠀⢀⣀⣀⠀⠀⠀⠀⡠⠒⢒⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠙⠲⣶⣶⣦⣀⢈⡿⠶⠴⠁⠀⠀⢻⣦⣤⣤⣄⣍⣑⠖⠀⣠⣤⡿⠋⠁⣠⡑⣀⢀⡀
⠀⠀⠀⠀⠀⡠⠊⠉⠉⠛⠲⢶⣿⣀⠀⠈⢻⣿⢿⠟⠁⠀⠀⠀⠀⠀⠘⠿⢿⠿⠟⠻⡇⠀⣸⡿⠋⢀⣼⣿⡟⠋⢉⣁⣌⠃
⠀⠀⠀⢀⠊⢀⣼⣿⣶⣤⣤⣾⣿⣿⣷⣶⠾⢱⢺⠀⠀⠀⠀⠀⠀⠀⣴⣤⣾⣦⣼⠘⠀⣠⣿⠇⠀⣸⣿⠏⠀⣰⣿⣿⣿⡀⠆
⠀⠀⠀⢘⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⡧⣼⡦⠤⣄⠀⠀⠀⠀⡖⠒⠀⠀⢺⠘⠶⠋⢰⠀⣰⣿⣿⣤⣾⣿⣿⣿⣟⣳⠈
⠀⠀⠀⢸⠈⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⠄⣰⠊⠀⠀⣠⠞⠀⠀⠀⠀⡆⠀⠁⠀⠙⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢡
⠀⠀⠀⢘⠈⠋⠹⢛⣿⣿⣿⣿⣿⢿⣿⣶⣯⣤⣴⡿⠁⠀⠀⠀⠀⠀⢷⣶⣶⣶⣾⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⡆
⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⡿⣳⣿⣿⣿⣿⣿⢻⠃⠀⠀⠀⠀⠀⠀⢾⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢱
⠀⠀⠀⠠⠀⢿⣿⣿⣿⡟⣿⠁⣿⣿⣿⣿⡟⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠛⠛⠛⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠸⡆
⠀⠄⠀⠘⣣⣉⣿⣿⣿⣇⢻⡄⠻⡟⠋⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡘⡻⣿⣿⣿⣿⣿⣿⣯⣤⡈⡇
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⢾⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⢿⡿⠃⠀⠀⠀⠀⣾⣿⡆⠀⠀⠰⣞⣿⣻⡇⠀⠀⠀⠀⣸⣿⣿⣂⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣰⣶⣦
⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⣸⣿⣿⣿⡀⠀⠀⠈⣿⣿⣿⡆⠀⠀⢨⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿⣿⡄⠀⠀⠀⢀⣴⣦⠿⣿⡟⠀⠀⠀⠘⠛⠋⣶⣦
⠀⠀⢸⢻⣿⣿⡏⣷⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢹⣿⣿⡏⠀⠀⠈⢻⣿⣿⡏⠁⠀⠀⢠⣿⣿⡇⠀⢠⡄⠀⠀⠀⠀⠀⢀⣿⣿⡇⢀⣤⣤
⠀⠀⠈⢸⣿⣿⡇⠁⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⡿⢿⡇⠀⠀⠀⠈⠻⢿⣿⠆⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣃⣸⣿⣿
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⠀⠀⣼⡟⠀⠀⣻⣆⠀⠀⠀⠈⣿⠟⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠋⠙⠁⠀⠀⠀⡈⠛⠘⠃⠀⠀⠀⠀⠛⠙⠃⠀⠀⠀⠀⠈⠋⠙⠁⠀⠀⠀⠙⠁⠈⠋⠀⠀⠀⠀⠀⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣤⣄
⠀⠀⠀⠀⠀⢰⣶⣦⠀⠁⠀⠤⠀⠀⠀⠸⣯⣿⠀⠀⠀⠀⢸⣿⣿⠈⠀⠀⠀⠀⣶⣷⡆⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⡆
⠀⠀⠀⢠⣾⣿⣷⣯⣤⠀⠀⠀⠀⢀⣴⣿⣿⣶⠀⠀⠀⣤⣶⣿⣶⡄⠀⠀⠀⣠⣿⣿⣷⠀⠀⢰⡄⠘⣿⣿⣃⡀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠈⣿⣿⣿⠻⠇⠀⠀⠀⠴⠟⢹⣿⣿⣿⡄⠀⠀⠀⣿⣿⣿⣿⠀⢀⡼⠋⢿⣿⣿⠀⠀⠸⡇⣾⣿⣿⡿⠙⣛⣻⣃⣐⣶⣖⣀⣀⣀
⠀⠀⠀⠰⣿⣿⣯⡀⠀⠀⠀⠀⠀⢠⣿⢿⣿⠈⠓⠀⠀⠀⣾⣿⣿⠟⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⣿⣿⣿⣿⣧⠀⠙⣟⠛⠛⠛⠛⠛⣻⠋
⠀⠀⠀⢀⣿⡇⢹⡿⠀⠀⠀⠀⢀⣿⠏⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢠⡟⠛⣿⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠐⠿⠋⠀⠻⠇⠀⠀⠀⠀⠘⠋⠀⠈⠿⠀⠀⠀⠀⠀⠻⠟⠿⠀⠀⠀⠀⠀⠻⠟⠿⠀⠀⠸⠋⠘⠿⠻⠟⠀⠀⠿⠀⠀⠀⠀⠀⠿⠀⠀⣾⣿⣷
⠀⠀⠀⢀⣤⣄⠀⢤⠀⠀⠀⠀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢋⣤⣄
⠀⠀⠀⠹⢿⣟⠀⠀⠀⠀⠀⠸⣟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠏⠀⠀⠀⠀⠀⠀⠀⣶⣷⡆⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⢶⣤⣿⢿⣿⣧
⠀⠀⠈⣿⣿⣧⢰⡄⠀⠀⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⢠⣤⣝⣛⣁⣀⣀⣼⣿⣷⡄⠀⠀⠀⣠⣾⣿⣶⢤⣤⣴⡾⠛⠁⠈⢻⣿⡇
⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⠈⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢀⣿⣿⡅⠉⠉⠛⠋⣷⣹⣿⣿⡄⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠈⣿⡟⣷⡄⠀⠀⠀⣿⡿⣿⡇⠀⠀⣀⣤⣤⣶⡀⠉⣿⣿⣿⠁⠀⠀⠀⢈⣿⣿⢿⣿⠀⢘⣙⠛⠛⠛⡛⠃⠀⢀⣸⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡏
⠀⠀⠀⣼⡟⠁⣿⡇⠀⠀⠀⣿⡇⣿⡇⠀⠀⢹⣿⢿⡿⠃⠀⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⢸⣿⣛⣻⣿⣏⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠋⠀⠀⢛⡀⠀⠀⠀⠛⠁⠙⠃⠀⠀⠀⠀⠀⠁⠀⣀⣛⣉⣋⣀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠈⠛⠉⠛⠙⠋⠀⠀⠈⠉⠉⠉⠀⠀⢠⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⢲⡆
⠀⠀⠀⢀⣤⣄⠉⢩⣶⣶⣄⠀⠀⠀⠀⠀⢠⣶⣦⠀⢠⣤⡀⠀⣴⣶⡄⢀⣤⣄⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢹⢀⡀
⠀⠀⠀⣸⣿⣿⡀⢈⣿⣿⣿⣧⠀⠀⠀⠀⣈⣿⣏⡀⠻⣿⠏⢀⣻⣿⣁⠘⢿⡟⢀⣘⣿⣏⡀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣿⣿⣿⣿⡾⣿⣿⣿⣿⣧⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣄⢀⣸⣿⣿⣡⠀⠀⠀⠀⠀⠐⢸
⠀⠀⢸⣿⣿⣿⣿⠐⠿⣿⠿⠟⠋⠀⠀⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠇⠀⠀⣷⣼⣿⣿⣿⣿⡄⠀⠀⠀⣀⣀⣸⠀⠀⠀⢰⣶⡄⡇
⠀⠀⠈⢹⣿⣿⠉⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡾⣿⣿⡿⢷⣿⣿⡿⢿⣿⣿⠿⣾⣿⣿⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠇⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⣿⣿⡷⢸⣿⣿⡇⢸⡇⣿⠀⣿⣿⣿⠀⠀⠀⠀⡿⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⢄⡀⠀⠀⠀⢠⡷⣄
⠀⠀⠀⠈⠁⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠈⠀⠁⠀⠉⠈⠁⠀⠁⠉⠀⠈⠀⠉⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠋⠀⠀⠀⠈⠛⠁

⠀⠀⠀⢀⣠⣈⣶⣦⣠⣀⠀⠀⠀⠀⠀⢠⣶⢆⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠸⣯⣤⣿⣯⣬⡿⠇⠀⠀⠀⠀⠿⣧⣤⣽⣧⣤⠀⠀⠀⠀⠀⠀⠀⣘⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄
⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⢿⣿⡿⣇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⢸⣿⣇⣻⠀⠀⠀⠀⠀⠼⢫⣿⣿⣿⣿⠹⠇
⠀⠀⠀⠀⠀⢸⣏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⡇⠀⠀⠀⠀⠀⠀⠁⢸⠀⣿⠈⠀⠀⠀⠀⠀⠀⠈⣿⡇⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣿⡟
⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁

⠀⠀⠀⠀⠀⢀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤
⠀⠀⠀⠀⠀⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣦⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠃⠀⠀⠀⠀⠀⢀⠀⣼⠛⣿⣶⡀
⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⠿⢿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⢸⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿


⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⢠⣶⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⢠⣿⡁
⠀⠀⠀⠀⣿⣯⣿⡄⣷⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣷⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣀⡀
⠀⠀⠀⠀⠀⢸⣿⠁⣹⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⢿⣇⠀⠀⠀⠀⠀⠀⣹⡿⢉⣿⠇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⡄
⠀⠀⠀⠀⠀⠘⠃⠀⠛⠂⠀⠀⠀⠀⠀⠀⠚⠛⠛⠈⠋⠀⠀⠀⠀⠀⠀⠛⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⠛⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡙⠀⠖⠂⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⠊⡁⢉⡇⠀⠀⠀⠀⠀⠀⠀⡞⠁⠉⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠂⢻⢌⡟⣄⠉⠁⠴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡻⠄⠴⢴⡇⠀⠀⠀⠀⠀⠀⠨⡗⠂⠐⣿⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠾⢥⡤⠲⠞⣸⠋⠀⢸⠀⣠⡶⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢙⡇⢣⡤⠚⠃⠀⠀⠀⡀⠀⠀⠀⠙⢀⡎⢘⢁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣧⢾⠆⢳⣴⣻⣟⢋⣴⠞⠁⠈⡆⠀⠀⠀⠀⢀⡞⠙⣅⣤⣾⠑⢄⠀⠀⠘⢏⢺⠀⡴⢋⣹⣦⣾⠛⢿
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⡞⠑⢠⣯⠀⢸⠟⠁⠀⣠⠈⣷⠀⠀⠀⠀⢸⠀⢰⣯⣛⠅⣧⢸⠀⠀⠀⠀⡟⠷⢁⣜⣿⡾⣿⣧⠀⡇
⠀⠀⠀⠀⣤⠒⢆⢛⡿⢤⡀⢰⠃⠀⣾⠸⡇⠚⠃⣠⡾⠁⠀⣿⠀⠀⠀⠀⠸⠀⢸⡏⣄⠄⣿⠀⡆⠀⠀⠀⠙⠖⢁⠏⢠⠰⡽⢯⡀⠇
⠀⠀⠀⠀⡇⠙⠲⣭⠕⢺⠀⠘⠒⠚⣏⠀⠻⠶⠞⠋⠀⠀⠀⣻⠀⠀⠀⠀⠀⡃⠰⣇⣈⢸⣿⠀⡇⠀⠀⠀⠀⠀⢸⣀⣁⣑⣚⢘⡄
⠀⠀⠀⠀⡇⠀⠀⣿⠀⢸⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠣⠀⠱⠼⠶⠿⠰⠇⠀⠀⠀⠀⠀⠀⡇⠈⡅⠉⢹⡁
⠀⠀⠀⠀⠉⠓⠶⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡵⠻⠟⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠂⣱⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠠⡷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠤⠤⣄⠀⠀⡌⡿⡉⢖⢡⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠒⠒⢲⣶⢾⡂⢸⣄⣹⠎⢳⡀
⠀⠀⠀⠀⠀⡔⢻⢸⠆⠠⠤⣼⢀⠜⠂⣿⢛⢻⣧⠡⣌⣱⣦⣀⣄⡀⠀⠀⠀⠀⣾⣍⣥⠶⢶⣟⠉⢻⡇⢸⣇⡴⠦⠴⢧⠀⡀
⠀⠀⠀⠀⠀⡇⢸⠈⠑⠒⠒⢲⣎⡠⢊⣟⣰⡢⠝⠂⣇⠉⢹⣈⣀⠉⠀⠀⠀⠈⢓⣾⡇⠀⢀⣽⡶⠼⣅⣸⠻⠆⢸⣶⣮⣏⠟
⠀⠀⠀⠀⠀⠿⠟⠀⠀⡐⠂⠃⠲⠤⡀⠀⠉⠉⠉⠛⠓⠒⠾⠤⠤⡄⠀⠀⠀⠀⠓⢢⡿⡺⣿⠿⢿⣦⣌⡉⠛⠲⢤⣝⣲⠈⠳⡄
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠉⠀⠉⠀⢀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠼⠻⣽⡛⠓⢦⣄⡈⠛⠶⠄⠀⠀⠉⠁⠀⢹⡀
⠀⠀⠀⠀⠒⠲⢭⣭⣽⣒⣒⠦⠤⢤⣀⣸⠀⠈⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢡⡶⠿⣷⣦⣄⡙⠓⠂⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠠⠻⠌⣆⣿⣿⡩⠛⠛⠓⡶⠾⡆⠀⠀⠀⢸⣧⠠⠤⠄⠀⠀⠀⠀⠀⠀⠙⣶⠈⠳⣤⡀⠉⠛
⠀⠀⠀⠀⠸⣵⣞⣍⣿⡿⢙⣿⡿⠝⠊⠀⡏⠉⡐⠖⠺⣯⢉⣉⡒⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⣴⠉⠙⠒⠛⠲⠶⠶⠶⠶⠚⠶⢤⡀
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⣿⣯⡤⣀⣀⣿⣠⡾⢿⠀⠀⠀⠀⠀⠀⠰⠴⠆⠀⢷⣀⠀⣠⣆⠀⠀⣤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠓⠒⠃⠀⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⢟⣡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⢟⣼⣷⣮⣭⣭⣍⡩⣵⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⡍⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⢩⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠿⠿⢛⠿⣿⣿⣿⣿⠿⡛⠿⠿⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣴⣿⣿⢟⡛⢡⡌⠿⣿⣿⣿⣦⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣸⡇⣿⢱⡗⢼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⣿⢃⣿⢸⡏⣾⢃⡆⣽⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠻⣸⣧⣿⣱⡟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣿⣿⣿⣿⣿⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡄⣿⣿⣿⣿⣿⣿⣿⢣⡿⢟⣴⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⢹⣿⣿⣿⣿⣿⡟⣨⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠁⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⠿⣿⢿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠛⠳⣶⣄
⠀⠀⠀⠀⢠⣾⣿⠉⢲⣶⣶⣾⡝⣿⣿⣷⡄⠀⠀⠀⠀⢠⣾⢫⣿⢟⣼⣿⣿⣷⡝⣿⣷⡀⠀⠀⠀⠀⢠⣾⠻⠿⢋⠊⡀⠧⢤⣻⣿⣷⡀
⠀⠀⠀⢀⣿⣿⣿⣔⣚⣛⣛⣛⣵⣿⣿⣿⠇⠀⠀⠀⢀⣿⣿⣿⣻⣶⠉⣽⣭⣿⣾⣿⣿⠇⠀⠀⠀⢀⣿⣿⢋⡀⣢⢚⣵⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠉⢰⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠉⢰⠀⠀⠀⠀⢸⣿⣷⣍⣴⣶⣿⡿⠛⠛⠉⠉⢸
⠀⠀⠀⠀⣿⣿⣿⠋⠉⠛⠃⠀⠀⢤⠄⠀⢇⠀⠀⠀⠀⣿⣿⣿⠋⠉⠛⠃⠀⠀⢤⠄⠀⢇⠀⠀⠀⠀⣿⣿⣿⠋⠉⠛⠃⠀⠀⠤⠄⠀⣆
⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠘⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠈⢻⠛⠁⠀⠀⡀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠈⢻⠛⠁⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠈⢻⠛⠁⠀⠀⠀⠀⠀⢈⠏
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠉⠉⠉⠉
⠀⠀⠀⢀⣤⣶⣿⣤⣤⣤⣼⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣤⣤⣤⣼⣷⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣷⣤⣤⣤⣼⣷⣤⡀
⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠛⠿⠿⣿⣿⣿⣿⣿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⣿⣿⣿⡿⠿⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠈⡧⠀⠀⠀⠀⠀⣀⣴⠾⠛⣉⣥⣤⣴⠶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠀⠀⢀⣠⠾⢋⣥⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠿⠛⠛⠛⠛⠛⢷⡶⠶⠛⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡈⠳⠶⠶⠶⠶⠖⢚⣡⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠉⠛⠳⠶⠶⠶⠾⠛⠋⠉⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠋
⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢶⡆
⠀⠀⠸⣧⠒⠂⠒⠒⠐⠒⠐⠒⠂⠒⠒⠐⠒⠀⠒⠂⠒⠂⠐⠒⠂⠲⠖⠐⠶⠂⠶⠰⠆⣼⠃
⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣶⠶⠶⠶⠶⠶⠿⠿⠟⠿⠛⠛⠛⠛⠛⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

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



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⢀⣀⣀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠻⣦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠾⢡⢀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡾⠋⠠⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢉⠙⣦⠀⠀⠄⠀⢀⣾⠋⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠛⠀⢸⡇⠀⠀⢠⡟⠁⠀⠀⠀⠐⢯⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡟⠀⢠⡟⠀⠀⠀⠊⣀⡴⠞⠉⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⠤⠢⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠁⠀⠊⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⣿⠀⡀⣼⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣀⠽⣇⠸⢿⠀⢿⡇⠞⠒⠒⠒⠢⠤⠤⣤⣒⠤⠊⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⡤⠖⠚⠉⢁⠀⠀⠘⢦⠻⢇⢻⡔⢀⡀⠒⠀⠠⠄⠀⠀⠁⣠⠞⠻⢷⡦⡀
⠀⠀⠀⠀⠀⠀⢀⠊⠞⠁⢀⠄⠀⠨⠀⠀⠀⠀⢂⡙⢦⣿⣗⠀⡀⠀⠈⠂⠀⠀⢀⡼⠋⣀⠠⠀⠉⢪⢢
⠀⠀⠀⠀⠀⠀⠈⢆⠢⢀⠀⠀⠈⠀⠠⠀⠀⠐⠀⠁⠄⠛⠿⠷⠀⠒⠀⠠⢁⠐⠸⠃⠀⠀⠀⠀⠂⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠈⣟⠮⣵⣀⠀⢀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠂⡀⠈⣀⡯⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠈⠉⠛⠒⠾⠥⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠤⠗⠒⠊⠉⠀⢰⠇
⠀⠀⠀⠀⠀⠀⣠⠶⠖⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⠿⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠱⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢡⠞⠁⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⣘⣿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⡾⠃⣀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢍⣉⠛⠓⠶⠤⠤⠤⣤⣤⣤⣤⣤⠤⠴⠶⠶⠟⣛⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠒⠒⠉⠁

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

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

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢃⠀⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢰⠁⠈⠀⠀⡎⠈⢳⠀⢳⠀⡌⠀⢸⠀⠀⡏⠳⡄⢀⣀⡀⠀⢸⠀⢸⡠⡇
⠀⠀⠀⠀⠀⠸⢤⠤⠂⠀⠳⠤⠞⠀⠈⠟⠀⠀⠸⠀⠀⠧⠚⠁⠀⠀⠀⠀⠸⠀⠀⠠⠃
⠀⠀⠐
⣤⣄⣀⣤⣤⣤⣄⣀⣤⣤⣄⣤⣤⣄⣀⣤⣄⣤⣤⣤⣤⣀⣀⣤⣀⣠⣤⣀⣤⣤⣄⣠⣤⣤⣄⣀⣤⣄⣠
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡀
⠀⠀⠀⢀⡏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⡆
⠀⠀⠀⠈⢧⣄⣠⠆⠀⢸⠇⠀⠀⠀⠀⠀⠀⢿⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⣳⣄⣠⣤⣄⣀⣀⣀⡀⣠⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣧⣤⣶⡾⠿⠟⠛⠛⠛⠒⠛⠛⢻⡋⠉⠙⠛⠛⠛⡿⠿⢷⣶⣦⣤⣀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⢦
⠀⠀⠀⠀⣠⢴⣫⠝⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠙⠛⢿⣶⣄⣠⠶⠋⢁⣀⣤⣶⡿
⠀⢀⡴⡿⠟⠉⠀⠀⠈⠣⣀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣾⣤⣤⣀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⢨⠟⠁⢠⣾⠿⠛⠉⠁
⢠⣯⠎⠀⠀⠀⠀⢀⣠⠴⠟⠻⠯⠉⠀⣤⡴⠀⢈⡁⠀⠀⣀⠀⠀⠀⠈⠙⣿⠒⠦⣤⣀⠀⣰⠋⠀⣰⣿⣇
⣾⣿⠀⠀⢀⡤⠞⠉⠀⠀⠐⠲⠆⣠⠤⢤⡄⠀⠫⠭⠇⠘⠾⠃⡴⡆⠀⠐⠯⠗⠀⢠⡼⠛⠁⠀⣰⣻⣿⣿
⣿⡹⡄⢠⠞⢡⢺⡆⠀⠀⠀⠀⢀⢬⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠖⣻⢆⠀⠸⡆⠀⢀⣴⣿⢫⢏⣿
⢻⡻⣿⣯⣀⠈⢉⢤⠸⣹⡆⠀⠈⢦⡗⠀⠀⠀⣴⣶⠆⠀⣴⣻⠇⠀⠈⠉⢁⣀⠀⢀⣽⣶⠟⢛⡷⣫⠟⡿
⠈⣧⣄⠙⠻⢿⣶⣯⣤⣀⣀⣀⡀⠀⠐⠏⡷⠀⠀⠀⠀⠀⠀⠀⠠⣒⣶⡆⠓⠚⠀⣈⣑⣿⣾⠿⠋⠁⣸⠇
⠀⠸⣞⡳⣦⣄⡀⠉⠙⠛⠛⠻⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣒⣛⣺⣿⡭⠽⠟⠛⠉⠁⢀⣤⢾⡏
⠀⠀⠹⣤⠇⣠⣿⢷⠲⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠚⣽⣩⡟
⠀⠀⠀⠹⣗⠿⣯⡾⢡⡞⡭⢹⡆⣩⢿⡻⣟⠛⢒⡶⢶⡖⠒⢲⡶⢶⠒⢛⡿⠿⡍⢡⣿⡏⣷⣹⠞
⠀⠀⠀⠀⠘⠧⣄⠉⠙⠓⠿⠯⣤⣇⣻⣥⠟⢰⡏⣞⣃⡿⠀⡯⣞⣸⠇⠸⡼⣷⣃⣸⢦⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠈⢷⠦⣄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠚⠛⠛⠛⠋⠉⠉⣉⡥⠖⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠉⠛⠓⠶⠦⠤⣤⣤⣀⣀⣀⣀⣤⣤⡤⠴⠖⠚⠋⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠚⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠺⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠆
⠀⠀⠀⠀⠀⠀⠰⣦⡀⠈⢷⣄⢀⣀⣠⣿⣄⡀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢰⡁⣎⡖⣦⢲⢸⣱⠀⢰⠐⣶
⠀⠀⠀⠀⠋⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠙⠛⠲⣦
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠉⣿⠿⠿⠿⠟⠛⠿⣧
⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠈⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣀⣀⣀⣀⣤⣤⡤⠶⠖⠛⢻⣄⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣼⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⠁⢠⣤⡤⠶⠆⠀⠀⠰⠶⢤⣤⡄⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣠⠀⠀⣀⣀⣠⠄⠀⠀⢠⣄⣀⣀⠀⢠⡍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣸⡀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⢸⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠀⠠⢦⠶⠀⠀⠀⠀⢀⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠹⣦⡀⠀⢀⣀⣀⣀⡀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣧⠀⢸⡟⢦⣈⠉⠙⠻⣿⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣸⠃⠀⠙⠳⠶⠶⠛⠙⣯⡉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠉⠳⣆⣀
⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠛⠉⠉⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⠶⠋⠉⠙⠛⠓⢶⣄
⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡴⢊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⠂⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣀⡁⠔⠈⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣿⡿⡆⠀⡿⡇⢸⠶⢶⣶⣶⣤⡤⠃⢀⡔⠉
⠀⠀⠀⠀⢀⣤⣶⣿⠿⠛⠋⢉⡉⠀⠀⠤⠀⠀⠀⠀⡇⡐⡲⢃⠆⠀⠀⠠⢠⠜⠁⡰⠛⠻⢷⣶⣄
⠀⠀⠀⢰⣿⡟⠉⠠⠐⠈⠀⠀⠐⠀⣠⣥⡀⢋⠀⠞⠨⠾⢁⠁⠒⠚⠀⠀⠁⠀⠌⠀⠀⠠⠀⣘⣿⡇
⠀⠀⠀⠘⢿⣿⣦⣦⣀⣘⣛⡁⠈⠤⠄⠐⢀⠲⠖⠈⠠⠤⠀⠀⠛⠡⠀⢶⣂⣀⣠⣥⣴⣾⠿⠟⡟
⠀⠀⠀⠀⠈⢦⠉⠙⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⡜
⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⣠⠖⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⢄⡀
⠀⠀⠀⢠⡞⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠱⡄
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁
⣿⡿⢻⠿⠻⠿⡿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣤⠔⠀⠀⠈⠀⠀⠀⠀⠉⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⠀⠀⠀⠁⠀⠀⠄⠠⢠⣄⠀⡆⢺⠁⢰⠀⢿⢄⡇⢿⠒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⢸⠀⢀⠰⡄⡄⢰⠸⢇⠹⡄⠀⠀⠀⠀⠉⠐⠨⠄⢸⠀⠘⠈⠃⠘⠉⠁⠀⠀⠀⠐⡶⠤⣀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡞⢇⡸⡏⠘⡎⠀⠀⠀⠀⢘⠦⣄⠀⠀⠀⠈⠀⠒⠤⠄⡀⠀⢠⡀⠱⡄⠹⡆⠈⠁⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⠀⢢⠀⠀⡀⠁⠀⠀⠀⢠⠿⠤⠬⢃⡴⡆⠀⠀⡀⠀⠀⠀⠉⠀⠛⠄⡘⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠠⠀⠁⢀⠄⠀⠑⣄⠢⢄⡀⠀⠀⠀⠀⠀⠴⠋⠓⡇⠀⡰⢷⠀⣰⠀⢤⡀⠀⠀⠀⠈⠑⠂⠤⢀⠀⠈⠀⢀⠀⡄⠀⠳⡀
⠀⠀⠀⠀⠀⢇⠀⠀⣴⣋⣤⡤⠄⠈⠢⡀⠈⠈⠑⠂⠤⢀⡀⠀⠀⠋⠰⠃⠸⣤⠃⢠⠏⠹⡄⢐⡶⢤⡄⠀⠀⠀⠈⠉⠒⠬⠿⡄⠒⠀⠙⡄
⠀⢀⠉⢀⡀⠀⠣⠀⠀⠀⠀⢠⡔⠃⣄⠙⢄⠀⠀⠀⠀⠀⢈⠁⣒⠤⢄⡀⠀⠀⠲⠾⣤⡤⢃⡞⠲⠀⠀⢀⢶⡀⠀⣀⠀⠀⠀⠀⠀⠐⠢⠄⣀
⠀⠠⠀⣠⠈⠂⠶⡡⠀⠀⢀⠀⠈⠉⠙⠇⣀⠀⠀⠀⢠⡀⠀⢱⡈⠦⠀⠀⠁⠒⠠⢄⡀⠀⠈⠙⠓⠂⠀⠎⠀⠛⢉⡇⠀⡜⠀⣀⣤⣀⠀⠀⠀⠉⠐
⠀⡇⠀⠁⠀⠰⠔⠀⠀⡔⠁⠀⢀⡠⢤⡀⠀⠁⠊⠤⣀⠙⠦⠤⠃⠀⠀⠐⠀⠀⠀⠠⠀⠁⠒⠠⢄⡀⠀⠀⠀⠀⠚⠀⣸⠁⣎⠀⠀⠁⠀⠀⠀⠀⢀
⠀⠁⠐⠂⠀⠌⢢⣄⡀⠉⠓⠀⢇⣀⢀⣟⡠⠤⠤⠀⠀⠉⡒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠈⠉⢒⠤⢄⡀⠀⠀⠀⠈⠓⠶⠖⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠤⡀⠈⡉⠊⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⢰⡖⢢⢰⡀⢹⡀⣳⣦⠄⠀⠀⠀⡇⠀⢈⣍⠒⠤⣄⡀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠔⠒⠀⠀⠁⠀⠀⢀⠀⢀⠀⢰⣄⠐⢦⣸⠀⢳⠀⠿⠓⠂⠓⠊⠀⠈⠀⢀⠀⠀⡠⠂⠀⠊⠀⠉⡏⠀⠈⡉⠒⠠⢀⡰⠁
⡀⠄⠐⠂⠀⠁⠀⠀⠀⠀⣀⣀⠀⡠⠐⢢⡘⡷⢼⡄⠟⠚⠓⠀⠛⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠂⠤⣀⡈⠑⠂⠤⣀⠀⠀⠀⡰⠁
⢀⠀⠀⠀⡴⠆⢀⡎⠙⣆⢸⣼⣁⠻⣤⠜⠁⠃⠀⠀⠀⠀⠀⡀⠀⢄⠀⠈⠙⠋⠀⣇⠤⡎⣠⠄⢀⠀⠀⠈⠑⠢⢄⠀⠈⠑⠒⠁
⠀⠀⠀⢸⠁⢀⡘⢧⠤⠃⠈⠀⠈⠀⠀⢀⠀⠀⢀⠴⠈⠀⠁⠀⠀⠀⠈⠑⠢⠄⡀⠀⠀⠛⠁⡰⢹⡆⡠
⠀⣇⠀⠀⠉⠉⠀⢀⡀⠀⡄⠀⠀⠀⣆⢈⢆⠀⠁⠀⢢⢃⠀⡆⠀⢀⠔⠒⠀⡔⠛⢷⠒⠤⣀⠀⠀⠛⠁⠀⠠⡀⣀⡀⠀⠚⡇⡎⢽
⠀⠣⡀⠀⠀⠠⡈⠀⠹⡀⠘⣄⡀⠄⠈⠁⠀⠡⡀⠀⢛⡡⣄⠃⠀⢮⣀⡠⠀⠧⠤⠊⠀⠹⡔⠉⠐⠤⠀⠘⠒⠁⠀⠀⠀⠀⠃⠀⠜⠀⢨
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠓⠀⠀⠀⠠⠐⠈⠀⠀⠘⣄⠈⢣⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠸
⠀⠀⠀⠀⠀⠀⠈⢇⡀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠋⠩⠉⠉⠙⠻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⣤⡀⢸⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣙⠛⠶⠶⠶⠶⠶⠛⣩⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⡇⢸⠀⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣸⡇⢸⣿⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⡾⠉⣠⣼⣿⣧⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⠀⣼⣿⣿⣿⣿⣿⣇⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢿⠀⢻⣿⣿⣿⣿⣿⡏⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢷⣀⠙⠛⠿⠛⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠉⠓⠲⠶⠖⠛⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣿⣻⣽⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣯⣭⣭⣭⣭⣿⣯⣭⣯⣽⣍⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⠃
⠀⠀⠀⠀⠐⢶⡐⣿⠈⡇⢰⡖⠀⣰⣦⠀⠀⠀⠀⣤⠀⠸⣦⢀⢰⡆⠀⠀⠀⠀⡄⢀⡀⠐⢢⡀⡇⣿⡀⠀⣠
⠀⠀⠀⢠⣄⢈⣷⠞⠛⣛⠻⢦⡞⠁⣀⠀⠀⢠⣄⣉⣷⣿⣿⣿⣿⣶⣃⡤⡄⠀⠀⢀⠹⣦⡀⣷⣿⣿⣇⡼⠀⣰⠛
⠀⠀⣀⠀⠈⣿⢁⡄⠀⡋⠀⣀⢻⡚⠉⣀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⡤⠀⠀⣄⡈⢦⣹⣷⣿⣿⣿⣿⣵⠾⠋
⠀⠀⠛⣉⣉⣿⡐⢀⡀⠋⣀⠉⣼⡋⠉⠋⠀⠛⢚⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠀⠀⢉⣉⣽⣿⣿⣿⣿⣿⣿⣿⡀⠰⠆
⠀⠀⠀⠙⠁⣨⠷⣬⣄⣀⣭⠾⣏⠙⠛⠀⠀⠀⢉⣹⣿⣿⣿⣿⣿⣿⡿⠧⠀⠀⣰⠟⢋⡽⡿⣿⣿⣿⢿⡝⡝⢷⡀
⠀⠀⠀⠀⠙⠃⢴⠇⢸⠃⢸⡆⠈⠻⠂⠀⠀⠀⠉⠡⠞⣹⡏⢻⠛⠱⡆⠀⠀⠀⠛⠀⢸⣧⣿⠟⡟⣿⡀⠑⠬⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠓⠘⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢐⢀⣀⢀⠀⡀⣀⣀⠀⠀⣀⡀⢀⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡟⢸⢸⡇⢸⣀⡇⣓⣦⠀⢘⣳⡅⣿⣚⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠁⠈⠁⠀⠀⠉⠀⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣠⡀
⠀⠀⠀⠀⠸⢦⣤⣽⢶⣦⣏⣤⠀⠀⠀⠀⠀⠀⠀⢲⣀⣼⠧⣤⣟⠀⣰⠆⠀⠀⠀⠀⠀⠘⢷⣴⠾⠷⠶⢯⣦⠀⣀
⠀⠀⠀⠛⢲⢞⡙⡶⠾⡁⣌⢳⣞⠉⠀⠀⠀⣠⣹⠞⠙⠘⢁⡘⢈⠻⣅⣤⠀⠀⠀⠀⢙⣳⡟⡀⢀⠈⣀⠐⠛⣏⡉
⠀⠀⢠⣬⣿⢬⡀⠲⠿⢡⢡⡍⣿⢥⡄⠀⠶⠤⡏⢥⡜⠓⣀⠀⣁⠈⢹⣄⣄⠀⠀⠠⣬⣿⠀⡁⠤⠀⡍⠀⠃⣸⢷⡆
⠀⠀⠀⠰⠿⣆⡣⢈⣁⠨⢌⣼⣗⠀⠀⠀⠀⠒⢻⡂⡴⢢⢀⡀⢤⢒⡾⡄⠁⠀⠀⠀⠰⠿⣧⡀⠶⢀⠰⠆⣰⣿⡂
⠀⠀⠀⠠⠾⢩⠿⣶⢿⣶⢿⣋⠉⠁⠀⠀⠀⠰⠞⠑⢦⣎⣀⣸⡴⠚⢦⡀⠀⠀⠀⠀⠐⠾⢫⢿⣷⣶⣶⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠉⠘⠃⠈⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣿⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠈⠋⠀⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡆⠀⠀⠒⠀⠈⠉⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠂⠤⠐⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠠⠒⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠚⢀⠀⠁⠀⠈⡃⠒⠀⢠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢀⠓⠄⠀⠀⠀⠀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠛⢒⣄⠀⠀⠀⠀⠀⠐⠒⠢⠤⠔⠀⠀⠑⠰⠄⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢨⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡌⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⡟⠀⠀⠀⠀⠀⠀⢀⢀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠰⠁⠈⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂
⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠔⠀⠀⠀⠀⠀⢉⡀⠂⠠⠤⠤⠤⠀⠲⠂⠉⠀⠀⠀⠒⠒⠲⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠁⠀⠠⠄⠤⠄⠀⠉⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠠⠤⠤⠀⠀⠀⠉⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠁⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣟⣛⣻⣛⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣽⣯⣭⣿⣿⣽⣿⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢰⣿⣿⣷⠆⠀⠀⢠⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡟⠁⣴⣶⡦⢸⣿⡀⠸⣿⠷⠀⣽⣿⠛⡁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡆⢠⣴⡶⢿⣷⠀⠀⢿⠄⢈⣫⣄⡀⠟⠀⣘⡿⠃⢘⣿⠷⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣦⡄⠙⣷⣀⠁⢶⣿⠇⠀⠈⢹⡟⠁⠀⠈⣻⡟⠂⠘⠋⠀⢠⣾⡟⣉⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⣾⣷⡀⠙⠃⠰⠿⣿⡀⠀⠹⠇⠀⠀⠈⠁⠀⠀⠀⠛⠁⠀⠀⣻⣿⠆⢩⣴⠿⡷⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣤⣅⠙⢛⣤⡀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢠⣦⡀⠀⢶⣿⢟⠻⠟
⠀⠀⠀⠀⠀⠀⠘⠛⣭⡙⠛⠁⠸⠟⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠃⠀⢀⣤⡿⣷⢀⣴⣄
⠀⠀⠀⠀⢀⣴⣄⠘⠿⠻⠷⣴⡆⠀⠀⢰⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⠀⠀⣤⣼⣷⠀⠁⢴⣾⡿⣿⣿
⠀⠀⠀⠀⢸⣿⡿⠿⡿⠆⠘⠛⠛⠗⠀⠈⢿⡏⢀⢸⣷⣦⡄⠀⠀⠀⠀⣠⣶⣿⢀⠈⣿⠏⠀⠀⠉⠀⠉⠀⢀⣀⣴⣆⠈⠉
⠀⠀⠀⠀⠀⢩⣦⣀⣀⠀⣤⡀⡀⠀⠀⠀⠀⠙⠈⠞⠛⠋⠀⠀⠀⠀⠀⠈⠛⠛⠎⠘⠁⠀⠀⠀⢀⣀⣠⣦⠈⠉⢙⣋⣀⣴⣦
⠀⠀⠀⠀⢀⡘⠟⠉⠉⠘⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠈⠉⠙⠟⢀⡀⠘⡿⠟⣿⣿
⠀⠀⠀⠀⣾⣿⣶⣾⣷⠀⢠⣀⣀⡀⠀⠀⠀⣠⣾⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣄⠀⠀⠀⠀⠀⢀⡀⠈⠛⣿⡟⠀⠈⠁
⠀⠀⠀⠀⠘⠿⢃⣤⣀⣄⠹⡿⠉⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠘⠛⣿⡇⠀⠀⣾⣦⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠈⠿⠋⠉⠀⠀⣀⢀⣀⠀⠸⣿⣿⣿⣿⣿⡏⠻⠸⠋⠋⣿⣿⣿⣿⣿⣿⡿⢀⠀⠀⠀⠈⣤⣄⣀⡈⠉⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣾⣷⣶⣿⠗⠀⢿⣿⠉⠀⢀⡙⠻⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⠙⢻⣿⠇⠀⠀⠹⠿⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠙⢿⠇⣶⣶⠟⠃⠀⠀⣴⣿⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⢷⣀⡀⠀⣡⡀⠀⢾⣷⣄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⣀⣾⡷⠀⢈⡉⠁⠀⣀⣿⠀⠀⠀⢰⡗⠀⠀⢈⣿⣤⠀⠺⠟⠁⠀⠈⣿⡿⠀⢹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣯⠠⣶⣿⠃⢀⣀⠙⠛⢃⠀⠐⠿⠿⠂⠀⠈⠛⠉⠰⣷⡀⠀⣿⣦⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⣀⣾⡿⠀⣠⣿⠁⠀⣤⣄⣸⣇⠀⢰⣶⡘⠿⠛⠀⢨⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠙⠛⠁⢀⣿⣟⠛⠛⠁⣨⣿⣧⡄⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠿⢿⡿⠇⠀⠀⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⢀⣠⣴⣦⡀⢀⣴⣶⣦⡀⢠⣶⣶⡄⠀⣠⣾⣿⣿⣦⣰⣾⣶⣤⠀⢀⣴⣾⣷⣦⠀⢰⣶⣄⠀⣠⣄⣤⣦⡀⣴⣿⣿⠧⣾⣶⣰⣦⠀⢰⣿⣿⣿
⣾⣿⠙⠛⢁⣿⡏⢀⣿⡇⣽⣷⣿⠃⢠⣿⣿⣻⣿⣿⣿⣿⡇⣿⣇⣿⣿⣟⣿⣿⡆⢸⣿⣿⣾⣿⠟⣿⣿⠇⣿⣿⠁⠀⣿⣿⣿⣿⠀⠘⢿⣿⣟
⢻⣿⣿⡿⠘⣿⣷⣿⡿⠁⣿⡟⢿⣿⣶⢿⣿⣿⡿⠋⢻⣿⡇⠿⠿⠻⣿⣿⡿⠿⣿⠆⢿⣿⠟⠁⠀⣿⣿⠃⣿⣿⠀⠀⠻⠿⠿⣿⢶⣿⣿⣿⣿
⣤⣬⣥⣤⣤⣤⣭⣥⣤⣤⣬⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣬⣥⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣭
⣿⣿⣉⣭⣋⣬⣉⣽⣩⣿⣹⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠒⠀⠀⠔⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠰⠁⠀⠀⠀⠀⠀⠀⠈⠰⠒⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠤⠀⠀⠒⠀⠀⠐⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢠⠆⠀⠀⠀⣠⣾⠇⢰⣴⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡤⡀⠀⠸⠀⠀⠀⠚⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡌⠆⢀⠁⠀⠀⠀⢀⣠⠀⢰⠴
⠀⠀⠀⣀⡤⠤⠤⠤⢀⡀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠝⠋⠀⠀⠢⡀⠀⠀⠀⠀⠀⡀⠤⠐⣠⡙⠆
⡞⠙⠛⠁⠀⠀⠀⠀⠀⠈⢩⠒⠂⠀⠀⠀⠀⠀⠀⠸⣇⡴⢷⡴⠃⠀⢠⢐⣠⠥⠔⠂⠈⠭⠐⠁⠀⠀⠈
⠘⠦⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢼⣶⠾⠏⠃⠁⠈⠁
⠺⡄⠀⠀⠀⠀⠀⠀⢐⣺⣿⡇⠀⢀⣀⣀⠀⠀⠀⠋⠁⠰⠆⢠⠊
⠼⠁⠀⠀⠀⠀⠀⠀⠶⠟⠏⠀⠀⠀⠀⠈⠑⢦⡀⠀⢀⣀⡤⠾⡀
⢠⠀⠀⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢆⢀⡤⠼⠖⠒⠶⠤⠤⣀⡀
⢸⠆⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢢⡀⠀⠀⠀⠀⠳⣴⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣤⣀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⠉⠉⠒⠒⡴⠚⠂⠀⢀⡀⠀⠀⠀⠈⠙⠓⠶⢤⣄⣀
⠳⠀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⢤⣄⣀
⠠⡄⠙⠀⠀⠀⠀⠀⠀⠴⢺⣇⠀⠀⢀⡼⠃⠈⠓⠢⠤⢤⣤⡤⠶⠛⠉⠒⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠤⣄⡀
⠀⢳⠈⠆⠀⠀⠀⠀⢀⣴⣿⣿⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦
⠀⠀⠒⠒⠒⠂⠈⠉⠀⠀⡀⠝⠋⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠶⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣶⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣀⣀⣀⣐⣒⣒⣒⣶⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣒⣚⣉⣁⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠉⠉⠓⢦
⠀⠀⣀⠴⠖⠒⠂⠄⡀⡖⠛⠉⠑⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠒⠭⣄⡀⠀⠀⠀⠀⠈⠃⠀⠀⢠⠤⡴⠖⠒⠖⠛⠛⠛⠛⠓⠒⠒⠒⠤⠤⣠⡔⠉⠀⠀⠀⠀⢀⣀⠤⠒⠋⠉⠁
⠀⠀⠘⠑⠂⠠⠀⠀⢀⣀⣀⡀⠀⣾⡶⠔⠂⠈⠉⠉⠛⠛⢲⣤⣤⣄⣀⣀⡀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠈⣧⠤⠄⠀⠀⠂⠈⠀⠀⠀⡉⠉⠀⠉⠙⠻⠶⢦⣌⠑⣄
⠀⠰⢤⣤⣤⣶⣤⣤⠴⠿⠛⠒⠛⠛⠛⠃⠁⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡆
⠀⠀⢠⠸⣿⠯⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠣⠤⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠸⡦⡙⠷⢤⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢄⢀⠀⠀⠀⠉⠑⢒⡠⠔⣫⢾⠃
⠀⠀⠀⢧⠙⠢⢄⣈⠙⠛⠲⠶⠤⠤⢤⡀⣀⣀⣈⣀⣀⣀⣀⣀⣤⠤⠴⠖⠒⣋⣡⠴⠚⠁⡼
⠀⠀⠀⠘⣆⡀⠀⠀⠉⠁⠐⠂⠀⠠⠄⠤⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⠒⠋⠉⠉⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⠴⠶⠚⠻⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡾⠃⢀⣤⣤⠶⠆⠀⠀⠰⠶⢤⣤⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢡⡀⠀⣀⣀⣠⡄⠀⠀⠠⣄⣀⣀⠀⠀⣌⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣸⡇⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠁⠀⢰⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⢦⡴⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⢧⡀⠀⠀⣀⣀⣀⡀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣷⠀⢸⡟⢦⣄⠉⠉⠻⣿⣷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⢸⡇⠀⠉⠳⠦⠴⠞⠙⢷⡉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠈⠛⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠛⠋⠉⠁⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠶⠟⠉⠙⠛⠛⠶⣤⡀
⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠟

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠠⠄⢠⡀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⡀⠄⠀⠄⠈⢡⠁⠐⠠⠂⠀⠄⠁⠈⠃⠀⠞⠁⠈⠀⠀⠐⠶⠾⠄⢄⢐⡌⠂⠀⠀⠀⢀⡠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡈⠀⠀⠀⠀⠀⠀⠈⠗⢤⡰⠀⠉⠁⠉⠉⠐⠒⠤⡉⠉⠉⠀⠀⢂⡽⣀⠀⠄⠒⠐⢀⠂⠠⢀⠀⠄⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⢨⠁⠠⠀⠀⠀⠆⠀⢤⠀⡈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⡀⠀⠀⠀⢀⡀⢀⠀⠥⡀⢈⠲⡈⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢢⠠⠁⠈⠂⠀⠁⠐⠀⠀⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠲⡐⠈⠉⠽⣁⠄⠀⠱⣙⠊⢸⠀⢠⠋⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠭⠔⠚⠛⢦⡀⠀⠀⠂⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠸⡂⠱⡄⡀⠀⠸⡄⠀⠀⠘⡑⠘⠛⠃⠘⠄⠠⠂⠁⡀
⠀⠀⠀⠃⠐⠀⠤⣴⠞⠀⠀⠀⠀⢈⠇⠛⡤⠰⠉⠠⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⢱⡀⢱⢀⡐⡁⢂⡄⠀⠀⠘⠉⠙⠦⣀⠃⠀⡀⢀⠑
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠄⠐⢀⡀⠀⠀⣟⠀⠀⠀⠀⠀⠀⢀⡀⠄⠊⠁⠀⠀⡀⠀⠀⠀⠀⠱⡜⠁⠉⠒⠀⢈⡀⠀⡀⠘⢢⠄⡀⠌⠅⠁⠠⢰
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠁⢀⠀⠀⠀⠀⠀⡧⠀⠠⠰⠐⠂⠉⠀⠀⠀⠀⠰⢀⢃⠀⠀⠀⠀⢀⠤⠃⢀⠈⠀⠘⠂⠗⠀⠀⠈⠤⠀⠹⠘⠂⣀⡀
⠀⠀⠀⡠⠊⠐⠀⡈⢳⡤⠤⠤⡤⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⠀⡴⠀⢡⡀⡠⠊⠁⠀⢆⢈⠂⢀⠀⠀⠰⣊⠍⠢⠀⠀⢀⠩⠖⠁⠠
⠀⠀⢰⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠱⣄⠀⠀⠀⠀⠀⠀⠀⠘⠓⠢⡀⠀⠀⢠⢠⠞⢠⡘⢦⠀⣨⠀⠀⠀⢀⡍⠐⠀⢀⡀⢀⠀⠀⠀⠠⠄⠀⠁
⠀⠀⠀⢙⠤⣀⢀⠀⠀⠀⠈⠐⠄⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⡀⣱⠋⠰⡦⠌⠠⡼⠁⠀⠀⠠⡄⡠⠀⠀⠀⠁⢤⡀⠒⠒⢆⠀⠈⠀⠂
⠀⠀⠐⠀⢀⠀⠍⢃⠀⢀⠀⠁⠀⠀⠐⠦⡐⠀⠈⠀⠀⠀⠀⣀⠺⣉⠀⠰⢠⢃⡈⠻⡟⠀⠀⠀⠀⠁⠀⢠⣠⠀⠀⠀⠀⠀⠀⠀⣠⠄⠋⠀⠠⠁⠁
⠀⠀⠀⢃⠈⠀⠨⠄⠑⠂⢜⠠⠄⠀⠀⠀⠀⠒⢦⡄⠀⠈⠛⠤⠠⠄⠀⠀⢀⣤⡉⠒⠀⠀⠀⠀⠤⠀⡐⠉⠀⠀⠀⠀⢀⡠⢔⢉⡁⢈⣈⠰⡛⢦⠂
⠀⠀⠐⠌⠀⠀⢀⢤⡠⠀⠈⠀⡀⠤⠳⠀⢀⠀⠊⠹⠆⠙⠒⠲⡆⢀⠔⢠⡶⣌⠛⠠⢁⠀⠀⠂⠀⠀⠉⠰⠄⠀⠀⠈⠀⠰⠀⢀⠈⢈⠴⠆⠃⠀⠂
⠀⢠⠀⠀⠂⠤⠄⠀⠀⠀⠄⠁⠀⢀⠀⠉⠑⠀⠀⠀⠙⠀⠀⣁⠔⠁⢴⡦⡙⠂⡀⢣⠄⠠⠀⠁⡠⠉⡢⠄⠀⠀⠀⠀⠀⣀⡠⢀⡤⡉⠒⠀⠀⠀⠄
⠀⠀⣀⣀⠀⠀⡀⠀⠀⠀⠷⢄⠂⠉⠄⣀⠀⣠⡄⢼⡀⠉⠀⠀⠈⢻⡀⠉⡠⠞⢩⠀⣄⠀⠀⡴⠁⠐⡠⠂⠁⠀⠀⠀⠀⢁⡴⠋⠀⢙⠦⠚⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⣀⢈⠷⢀⠥⠜⠈⠀⡅⣌⢳⡀⠀⠀⠀⠀⡠⠌⢄⢀⠁⠂⠀⡙⠀⠡⠀⠉⠐⠀⠢⠀⠀⣀⠞⠁⠀⡠⠔⡁⠀⠀⢀⡇
⠀⠀⠀⠐⠃⠃⠛⠂⢀⠐⡈⡇⢀⠌⢈⠴⢚⠉⠅⠨⠬⠀⠉⣈⣫⠋⠀⠐⠄⠀⠈⡂⠘⠡⠆⠀⠀⡀⡀⠀⠀⠠⠊⠁⢀⠴⠊⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠐⠃⡍⠀⠀⠈⡃⡇⣋⠀⠃⠀⠀⠀⠃⠐⠀⠤⠤⡴⠁⠀⢀⠚⠁⠃⠀⠄⡈⠃⠹⢦⠎⢤⠈
⠀⠀⠀⢀⡀⠛⠢⢈⠃⠀⠃⠁⠈⠑⠢⡄⢠⣀⠐⠁⢀⣀⡈⠙⠢⢄⣤⡀⠒⠂⢀⠼⠈⢈⡐⠋⠀⠀⡘⠡⡀⠈⠀⠀⠀⠀⠀⣀⠘⠁⠀⠰⠀⠀⡄
⠀⠀⠀⠈⠀⢄⠀⠀⠉⠒⠀⠃⠀⠀⠠⠀⠐⣀⠀⠀⠀⠠⠀⠙⠲⡈⠀⠈⠀⠐⠁⠈⠀⢨⠁⠀⠀⢀⣤⡄⠀⣀⠀⠀⡀⠀⠖⠁⠁⠆⠠⠄⢀⠀⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄⠁⠌⢀⡠⠆⠀⠀⠈⠈⠣⠀⠡⡀⠀⠀⠁⠀⠀⠀⡀⠀⣀⠀⠀⠠⣄⡠⠈⠀⣁⠡⠟⠂⡁⠀⠀⠀⠐⣄⡸⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠘⣄⠂⠀⣀⣀⠁⠀⠀⠂⠀⠀⢀⡆⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠕⢀⠋⠁⠀⠀⠀⢀⡀⣀⠀⠘⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠀⡁⠙⠒⠄⡁⠘⠓⠜⢀⠲⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⣼⠋⠀⢂⡀⠤⠐⠎⠁⣽⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠄⠀⠈⠁⠄⢂⠐⠀⠀⢀⢄⠣⡀⠀⠀⠀⠀⠀⡀⠀⢀⠐⠃⠠⡀⠀⠀⠀⡀⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠁⡀⢈⢱⠐⠀⠁⠀⠀⠀⠈⠀⠀⠐⠈⠀⠄⠀⠈⠀⠁⠂⠁⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠒⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⢀⡀⠤⠀⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠠⢈⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⢀⠠⠖⢋⣁⣀⣤⣤⠶⠖⠀⠀⠀⠀⠐⠂⠠⠴⣶⣦⣤⣤⣤⣤⣀⡀⠈⠑⠢⢈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢄⣪⣵⣶⣿⣿⡿⠿⠉⢁⣠⠴⠶⠖⠒⠀⠰⠦⢤⡀⠀⠝⢬⡿⠿⣿⡿⠿⠿⠿⣶⣦⣅⣠⡙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣰⣿⣿⣿⣿⡿⠻⠉⠀⢐⡊⠉⠍⠈⠈⠭⠁⢒⣀⠤⠀⢀⠀⠀⠀⠃⠂⠁⠶⠒⠒⠒⠲⢿⣿⣿⣿⣆⠙⣆
⠀⠀⠀⠀⠀⢀⣀⣼⠁⣼⣿⠿⠏⠁⠫⠥⠈⠉⠁⠤⠤⠤⠄⠀⠀⠠⠤⠀⣀⠀⠘⡏⠉⠉⠉⠐⠢⡀⣠⣤⠀⠀⠀⢄⠙⢬⡛⣛⣷⠘⣆
⠀⠀⣠⠖⠋⠉⠁⡿⠀⣿⠃⠀⠀⠀⠀⠀⠀⣤⡄⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⢨⡋⠀⠀⠀⠀⢀⣠⣴⡈⠁⢀⠚⠂⠀⠀⠉⠉⣩⣿⡆⣿⡀
⠀⢸⠃⠀⠀⠀⣴⢿⠀⢷⣀⣀⣀⣠⣤⠀⠀⠉⠁⣠⣤⡍⠀⣤⠄⠀⠀⠀⠠⠄⠀⠀⠧⠴⠚⠁⡀⠊⠙⡀⠻⠿⠃⠀⠀⠀⠀⣾⣷⡇⢸⢧
⠀⠈⢧⠀⠀⣼⠋⠈⣧⡘⣿⣯⣿⣯⣍⣀⠀⠀⠘⠛⠡⣬⣤⠰⣿⠏⣠⠀⠀⠰⠛⠀⢀⡀⠀⠀⢱⠀⠠⠴⠀⢀⣄⡠⠔⢣⣾⢿⡿⢠⣾⡀⠱⣄
⠀⠀⠀⢳⡼⠃⠀⠀⠈⢷⣜⠻⣿⣿⣿⣿⣷⣤⠀⠀⠀⠙⡋⠀⠀⢾⣿⣿⣿⡷⠀⠐⠿⠓⠀⠀⠈⠀⠀⠀⠀⠛⣉⣠⣶⣛⡷⠋⣠⡿⢿⠍⣤⠘⣆
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⠷⣌⠙⠿⣮⣍⣙⠉⠀⢈⣿⣿⣿⣦⠀⠙⠿⠿⠇⠀⠀⠀⠀⠐⠒⠐⠀⠀⣀⣤⣾⣿⡿⠟⢉⣤⠞⠁⠀⠀⣸⠈⠳⢿⣀
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡈⠙⠛⠲⠿⢭⣽⣿⣿⣴⣦⣤⣀⣤⣤⣶⣾⣿⣷⣶⣾⣿⠿⠶⠚⠉⣁⣤⠾⠋⠁⠀⠀⢀⡴⠋⠀⠀⠀⠉⢷⣄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⢤⣄⣀⣀⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣠⣤⡶⠞⠛⠉⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⣻⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠓⠒⠒⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜⣹⠃
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠶⠖⠒⠶⠶⠶⠖⠒⢛⣡⠾⠋
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣤⠶⠶⠻⣿⣿⣿⠋
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣦⠀⠀⠀⢀⣀⡤⠤⠾⠷⠤⢤⣀⡀⠀⠀⠀⣴⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢦⣠⠞⢉⣁⠀⠀⣰⠲⣆⠀⠈⡙⠳⣄⣠⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢿⣈⡇⠀⠙⠒⠋⠀⣏⣉⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⢤⡀⠀⠀⠀⢰⡏⢳⠄⠀⠈⠉⠀⡴⠲⡼⡆
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡟⠀⠷⠴⠃⢰⡋⣳⠀⠉⠉⠀⠀⣞⢙⡆⠙⠚⠁⢻⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢸⣉⡷⠶⠶⡇⠀⣀⡀⠀⠈⠉⠁⢰⡏⣹⠄⢀⣈⠉⢀⡶⢶⡀⢸⠷⠶⢾⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣇⠸⣅⡽⠀⡞⢻⡆⠀⢉⣉⡀⢯⣨⠇⠈⠓⠚⠁⣸⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣀⣍⠉⠀⠀⢻⣀⡟⠀⠀⢰⡋⣳⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢧⣼⠇⢀⡤⢤⠀⠀⣠⢤⡈⠙⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠏⠙⢦⣄⡀⠘⠧⠞⠁⠀⠳⢴⣣⡴⠋⠳⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠟⠀⠀⠀⠈⠉⠛⠒⢶⡶⠒⠛⠉⠁⠀⠀⠀⠻⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠼⠃

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

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





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

⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⡠⣄⠀⢀⣸⢾⣁⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣀⣀⠽⢄⣀⠀⢠⣄
⠀⠀⠀⠓⣫⠚⠁⠀⠀⠀⠉⠒⡗⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠊⢵⠆⢺⠃⣴⠕⣏⠁
⠀⢀⡀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⡀⠀⠀⣀⠀⢀⣀⠽⠧⣀⡀⠀⣀⠀⠀⢀⣀⢠⠓⠷⢰⡦⣛⠀⣴⠜⠛⣆⢀⡀
⠀⠣⠞⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠇⠀⠈⣻⠛⠁⡿⠄⠀⠀⣩⡻⡏⠀⠀⠈⠋⠙⡴⠆⢴⠄⣻⡰⠶⣘⢃⡏⠙⠁
⠀⠀⠀⠈⡦⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢰⠃⢴⡦⠄⠀⠀⠈⠋⠀⢹⡀⠀⠀⠀⠀⢱⣔⠷⢠⣌⢡⣌⣋⡞
⠀⠀⠀⢏⡟⠉⠒⠤⣠⠔⠚⠉⣏⠕⠀⠀⠛⢿⣀⡔⠈⠀⠸⡱⠀⠠⣀⢠⠟⠛⠀⠀⠘⠏⠀⠙⠒⣶⠒⠋⠁⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠺⠭⠆⠀⠀⠀⠀⠀⠀⠀⠈⢟⠉⠀⠰⢀⡄⠈⠙⢛⡟⠀⠀⠀⠀⠀⠀⢀⣀⠀⠛⠀⣖⣲
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡖⠀⠀⠀⣀⠀⠀⠀⠠⡾⠓⠤⣸⣿⣀⡠⠔⠻⡷⠀⠀⠀⠀⠀⠀⠣⣾⠤⢴⡶⡋⠣⢄⠀⢠⢦⡀
⠀⠀⠀⠀⠈⠑⣄⠤⠴⣳⠦⢄⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⠋⠀⠙⡖⢇⠀⠀⠉⡝⠚⠁
⠀⠀⣠⣀⢀⡜⡁⠿⢂⣟⠑⠷⢉⣆⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢸⡩⢶⠁⠀⠀⠀⠀⠀⠀⢀⠀⢸⡀⢀
⠀⠀⠀⠀⢹⠘⠃⠾⢎⣍⠐⠷⣙⡉⡅⠀⠀⠀⠀⠀⣀⠀⢀⣀⡤⠇⠧⣄⣀⠀⢀⡀⠀⠀⡇⠀⣀⣆⡄⠀⠀⠲⣏⣯⠽⣍⠇
⠀⠀⣀⣠⢼⡛⠓⡿⠀⣥⠟⢃⣝⢱⠥⣀⡀⠀⠀⠈⢫⠓⠉⠀⠀⠀⠀⠀⠈⠑⢫⠋⢠⠤⡯⢄⡳⠼⠀⠀⠀⠀⡴⠃
⠀⠀⠈⠀⠀⠑⣜⡃⢴⠆⢴⣎⡵⠃⠀⠈⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠙⠁⠀⠉⣳⡴⠒⠊⠉⣟⡲
⠀⠀⠀⠀⠠⡜⠁⠈⠉⡟⠉⠁⠘⡦⠀⠀⠀⢠⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠀⠀⠀⠳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⡴⢤⠀⠀⠀⠀⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢮⡹⣀⣠⣤⠟⢯⡀⠀⢀⣄⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⡀⠰⣆⣐⣂⣰⠆⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠺⡥⡇⠀⠈⠓⢪⠴⠃⠐⠯⠚⠢⢄⣀⠀⣀⣀⠤⠚⠪⠗⠀⠠⣌⡽⢪⣍⢠⣢⢉⣕⢮⣁⣄
⠀⠀⠀⠀⢰⣛⣒⠋⠀⠀⠀⠁⠈⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢧⠇⠀⠀⠀⠀⠀⠰⢤⢎⢤⢈⣁⢴⡆⢈⣋⣴⣷⡤⡆
⠀⠀⠀⠀⠀⠁⠈⡇⠀⠀⡄⣀⠀⠀⢪⡗⣏⣳⣊⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⢀⣉⢘⡋⢴⡆⣈⡚⣤⡄⡧⠶
⠀⠀⠀⠀⠀⢀⡤⢕⣀⠫⣹⠅⠀⠀⠀⢩⠏⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⡌⢋⣙⠃⠰⡶⠙⢱⡎⣰⠧⡆
⠀⠀⠀⠀⠀⠈⠳⠃⠈⠑⢢⣠⠤⠖⠒⡜⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠋⣪⣛⠰⠷⠀⢾⣦⡔⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠽⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⠏⢹⡋⠹⠄⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠐⡫⠀⡀⠀⠄⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠀⠒⣄⠀⠈⢂⠀⠀⠀⠊⠈⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⢰⠁⠨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡠⠀⠀⠀⠀⠀⠁⠀⠀⢀⠄⢀⣀⡿⠋⠀⠀⢀⠂⠀⠀⠀⠀⠀⡀⠂⠁⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡄⠀⢀⠄⠁⢠⠜⠁⠀⠀⠠⠀⠁⠀⠀⠀⠀⠠⠀⠀⠈⢀⠔⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠁⠀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠸⣬⠀⢨⣴⠁⡰⠁⠀⠀⠀⠀⠀⢀⠀⠀⡠⠈⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣠⠥⠔⢺⡀⢛⡆⢘⡟⠘⠉⠉⠉⠉⠉⠓⠒⢶⠼⠊⢀⣴⡉
⠀⠀⠀⠀⠀⠀⠠⣂⠥⠒⠋⠁⡀⠀⠠⠀⠻⣜⠿⣞⣷⠋⣠⠈⠉⡁⠒⠒⠀⠀⢀⡔⠉⠑⠫⣒⢄
⠀⠀⠀⠀⠀⢊⡞⠁⢀⠄⠂⠁⠂⠀⠀⠀⠘⣈⠻⣾⣿⡄⠠⠀⠀⠀⢈⠉⡄⠀⠔⠁⠍⠀⢄⠈⠣⠑⡀
⠀⠀⠀⠀⠈⢆⠢⣀⠁⠀⠀⠀⠠⠄⠀⡀⢂⠈⠀⠘⠿⠿⠦⠐⠂⠠⠌⡤⠀⠶⠀⠀⠀⠄⠀⠆⡠⡰⠃
⠀⠀⠀⠀⠀⠘⣷⡮⣕⡠⠀⠀⡀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠠⠀⠈⠂⠀⠈⣁⠭⣾⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣶⣯⣝⣳⠶⢬⢥⣄⣦⣀⣀⣀⣀⣀⣀⣀⣀⣐⣂⣤⡤⢤⠤⣶⣚⣯⡵⢾⠏
⠀⠀⠀⠀⠀⠀⠠⠘⢿⠆⣉⠛⠛⠛⢶⠶⠾⢾⣥⣭⣬⣽⣼⣵⣾⣷⡾⠶⠿⠻⡟⡛⠉⣇⣸⠟⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⠈⠻⣭⡀⠥⢀⡀⡃⠀⠀⠀⢀⡇⠠⣢⣳⠅⠶⠦⣘⠶⠸⣧⣂⣀⡴⠃⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠁⠢⣄⡀⠀⠈⠻⣟⠁⠙⣦⣼⣀⡠⠞⠉⠈⢹⣏⠀⠀⠀⢺⢱⣦⣈⡽⠋⠀⠀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠂⠈⠷⢾⣱⠠⣂⣐⣈⠈⢀⣌⣿⣀⠁⣀⣘⣮⡾⠉⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⣀⡀⠈⠉⠉⠉⠉⠉⠉⠉⠀⢉⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠒⠚⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⣶⣶⣤⣄⠀⠀⠀⢹⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣿⣾⣿⣿⣿⣿⣿⣆⠀⠀⢳⡀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⡇⠀⠀⠀⠀⢠⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⣵⣭⣛⡇⣩⣭⣭⣭⡹⡇⠀⡞⠁⠀⠀⠀⠀⢸⡇⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡅⠘⠇⠀⠤⠤⠤⠤⣼⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⢀⣸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣟⣭⣝⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢺⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢼⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠀⠀⠀⢸⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⠀⠀⢀⣴⣿⠇⣟⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⢸⣿⡇⣿⣿⡧⣻⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⠘⢿⣷⣬⣭⣵⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠛⠛⠛⠛⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⡀⢀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠸⠇⠹⠇⠿⠸⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⢠⣴⣶⣮⣿⣦⠀⢠⣶⣶⣶⣴⡄⠀⠘⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠉⠉⠉⠉⠋⠀⠈⠛⠛⠛⠛⠁⠀⠀⠸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠃⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠻⣗⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀
⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠘⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄

⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠏⠋⠀⠀⠀⠀⠀⠒⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠙⢿⣿⣿⣿⣿⣿
⣿⣿⠟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠻⣿⣿⣿⣿⣿⣿⣿⡇⢹⡏⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⣿⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢱⡈⠙⠿⠿⠿⠿⠛⣁⠮⠥⡯⢴⢶⣦⣀⠀⠀⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠋⠒⢘⣯⠟⡀⠒⣶⣶⢾⡶⠾⢋⠝⠭⠫⡉⠉⠙⠺⢿⣿⣿⣿⣿
⣧⢀⠠⠀⠀⠀⠀⠀⠈⢀⡀⡀⠀⠀⠀⠀⣸⢻⠀⠐⠋⣹⣿⣥⠱⡔⡍⠘⢂⢠⣼⠦⡄⠀⠀⠙⣿⣿⣿
⣿⡄⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⠃⠛⠂⠀⠀⢸⣟⠻⠒⢜⡥⣶⡬⡔⣾⠄⠚⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣌⠆⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠒⠒⠚⠶⣿⢿⣷⠀⢈⣧⣭⣷⣃⡂⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠠⣿⡍⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣿⣿⣿⣿⣿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢄⠐⠶⢬⣅⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⡇⣤⡄⠀⠀⠀⠀⠀⢛⣿⣿⣶⣤⣀⠀⠤⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⣭⠢⡀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⠀⠀⠀⠀⠘⠟⠛⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡍⣢⡀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⡰⣖⠄⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿

⠀⠀⠀⠀⣴⣶⣶⣶⣦⠀⣶⣶⣶⣦⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠉⣿⣿⣿⣿⣷⡙⣿⣟⠁⠀⣾⣿⣿⠋⢻⣿⣿⡆⠀⠀⠀⠰⠎⠲⠂⠰⡇⠰⠀⠶⠀⠇⠰⠇⠸⠂⠀⠀⠸⠀⠷⠘⠇⠸⠆
⠀⠀⠀⠀⠀⣿⣿⠙⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⡀⣼⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⠘⣿⣿⣿⡧⠀⠀⠈⠻⢿⣿⣿⠿⠋⠀⠀⠀⠈⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒
⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⢀⢀⣀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠿⣿⣿⣿⠿⣿⣿⣿⡆⠀⠀⠸⣿⣿⣿⡿⠇⠀⣠⣾⣿⣿⢿⣿⣿⡇⠀⠰⢿⣿⣿⣿⠿⢿⣿⣿⣿⠇⠀⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣶⣾⣿⣿⠃⠀⠀⠀⢸⣿⣿⡇⠀⠠⣿⣿⣿⢡⣤⣙⣿⣡⡀⠀⠀⣿⣿⣿⣶⣾⣿⣿⡇⠀⠀⠻⠿⠇⣿⣿⣿⠸⠿⠇
⠀⠀⠀⠀⣠⣿⣿⣿⣄⢹⣿⣿⣧⣄⠀⢠⣼⣿⣿⣧⡀⠀⠻⣿⣿⣬⣿⣿⣿⡏⠁⠀⣤⣿⣿⣿⣤⣼⣿⣿⣧⣄⠀⠀⢀⣤⣿⣿⣿⣤
⠀⠀⠀⠀⠛⠛⠛⠛⠛⠈⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠁⠀⠀⠉⠛⠛⠛⠉⠛⠃⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠈⠛⠛⠛⠛⠛
⠀⠀⠀⠀⣠⣤⣤⣤⢀⣤⣤⣄⠀⠀⠀⢀⣤⡄⠀⢤⣤⣤⣤⡄⠀⠀⢠⣤⣤⣤⡄⢀⣤⣤⡀⢤⣤⣤⣤⠄⣤⣤⣤⡄⢠⣤⣤⠄⣤⣤⣤⣄
⠀⠀⠀⠀⠈⣼⣿⡀⠘⢿⣼⠟⠀⠀⠀⣼⣿⣿⡄⢰⣿⢿⣿⠀⠀⠀⢠⣿⣿⣿⡆⠸⣧⡿⠇⢰⡿⢿⡿⠀⣸⣿⣿⡇⢸⣿⣿⠆⠁⣿⣧⠁

⠀⠀⠀⠀⢿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿⠀⠸⣿⣿⣿⣿⢿⣿⣿⡇⠀⣿⣿⣿⣿⠇⠀⠺⣿⣿⣿⣿⣌⢿⣿⣿⠇⢀⣴⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣀⠀⠀⢸⣿⣿⡄⠀⠀⠈⢿⣿⣷⣼⣿⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⢿⣿⣿⣤⣿⡇⠀⣾⣿⣿⣇⣤⣽⣿⣤⡀
⠀⠀⠀⠀⣤⣿⣿⣷⣤⣿⣿⠀⢠⣼⣿⣿⣦⠀⠀⠀⠈⣿⣿⣿⠃⠀⠀⠀⣴⣿⣿⣷⡄⠀⢤⣾⣿⣤⠻⣿⣿⣿⡇⠀⠸⣿⣿⣧⣽⣿⣿⡏⠁
⠀⠀⠀⠀⠙⠋⠙⠛⠛⠙⠋⠀⠘⠛⠋⠛⠛⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠛⠛⠋⠛⠃⠀⠘⠛⠛⠛⠀⠙⠛⠛⠁⠀⠀⠈⠉⠙⠋⠉⠙
⠄⠀⠀⠠⣶⣤⣶⣤⣤⣦⣤⣤⣤⣤⣤⣤⣦⣤⣶⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣶⣶⣶⣶⣶⣤⣴⣶⣦⣶⡄

⠀⠀⠀⠀⢿⠇⢾⡷⠰⡿⠂⠰⣿⠒⡿⠓⢾⠆⠶⠿⠸⡷⠶⣶⠤⣷⠰⡷⠶⠿⠆⠰⠷⠶⠿⠃⠀⢿⡧⢾⠶⠾⠦⢿⠗⠺⠷⢾⠿⠀⠺⠷⠄
⠀⠀⠀⠠⣤⣤⣤⠀⣤⡄⠀⠀⣤⣄⠀⣠⣤⣤⢠⣤⣤⡤⠀⠀⢤⣤⣤⣤⣄⢀⣤⣤⡀⣤⣤⣤⡀⢠⣤⣤⡄⢠⣤⣤⡄⢤⣤⣤⡄⢠⣤⣤
⠄⠀⠀⢀⣿⣿⣿⠄⣿⣧⡶⢸⣿⣿⡆⠻⣷⠇⢸⣿⣿⡧⠀⠀⠀⢻⡿⡿⠃⠸⢷⡿⠇⣼⣿⢿⡄⢸⣿⣿⣆⢨⣿⣿⠇⣸⣿⣿⣧⣸⣿⡿⠄
⠀⠀⠀⠀⣤⠀⣤⣠⡀⠀⣤⣤⡄⣤⣤⡄⣤⣤⡄⠀⣠⣤⢠⡄⢠⣤⡄⣤⠀⣤⠀⠀⢠⣤⣤⡄⠀⣤⠀⢠⣤⡄⠀⠠⣤⣤⢠⣤⣄⠀⣤⡀
⠀⠀⠀⠀⠛⠐⠛⠛⠀⠀⠘⠟⠀⠛⠛⠇⠺⠛⠀⠀⠛⠗⠘⠗⠈⠛⠀⠛⠐⠛⠒⠀⠀⠛⠛⠁⠚⠛⠓⠚⠛⠗⠀⠐⠛⠛⠘⠛⠛⠒⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⠀⠀⠠⠁⠈⠩⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣠⡦
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣱⠏⠄⠈⣷⡄
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡝⢁⡄⠀⠀⣿⣿⡀
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹⣿⣟⠁⠬⢱⡆⢀⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢣⠄⠀⡾⠀⠸⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠀⠄⠉⠀⣤⣾⣿⣿⡇
⠄⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠘⠃⠀⠈⠡⢾⡟⢹⣿⣿⡇
⠂⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠾⠏⠀⠙⢶⣦⣄⡀⠀⠀⠀⠀⠀⣤⡀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠸⣿⠰⠇⣼⢓⣶⢈⣿⢸⣿⣿⡇
⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⠁⡄⠀⣴⠀⠀⡄⠙⠻⣿⣦⠕⣛⢷⠘⣿⣿⣿⣿⠀⠠⠀⠀⠀⠀⣠⣴⡄⣸⣿⣿⣿⠀⠖⢻⣟⣉⣤⣹⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠃⠀⠘⠀⠈⠃⠀⠀⠈⢻⣿⣧⠘⣿⣿⣿⣿⣿⡈⠓⠈⣄⠐⠀⠻⠏⢉⢹⣿⡿⠛⡈⠃⢨⣿⡛⠛⢿⣿⣿⣿⡇
⠀⠀⠈⣿⣿⣻⣟⣿⣿⣿⣿⡄⠄⠀⢀⠀⠀⠆⠀⣾⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠀⢈⢀⡀⠀⠐⠉⠀⡿⣺⣀⣠⣶⠛⠻⠿⣶⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠈⣀⠀⠻⢀⣈⡁⢿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠉⢁⣶⣤⣄⣰⣿⣿⣿⣿⣿⣶⡶⠶⡻⣿⣿⣿⣿⡇
⠀⠀⠈⣿⣿⣿⣿⢿⣿⣿⣿⠀⡘⢋⣀⣋⣿⠃⢸⣿⡇⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣾⣧⣉⣛⡃
⠀⠀⢠⣿⣿⢻⣴⣧⣹⣿⣿⣾⣿⡁⢈⠿⢿⣿⡋⢿⣿⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿
⠀⠀⠘⡿⢿⣺⣿⣿⣿⣿⣿⣭⣿⣿⣤⣤⣤⣿⣷⣄⣿⣿⣴⣿⣿⣿⣿⠟⢿⣿⢙⣉⣹⠛⣿⣻⠿⣾⣿⡯⢿⣿⣿⡿⣿⣿⠻⣿⣿⣿⢿⡏
⠀⠀⢠⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣯⣽⣤⣿⣿⣿⣷⣝⣿⠾⠿⠿⠛⠺⣤⣾⡄
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⠉⢛⠉⠉⠉⠉⠉⣙⣋⡉⠉⠉⢉⣩⣝⠉⠙⣋⣉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡖⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠴⠶⠶⠶⠦⠤⣤⣤⣤⣖⣶⠟⠀⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⣿⣭⣴⣶⡾⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠀⢀⣴⣿⣿⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢾⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠶⠆⠀⢠⣤⠀⠀⠚⠰⣠⠏⠈⠙⢷⣄⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⠋⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠘⣇⡆
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⡙⠃⠀⠀⢀⠀⠀⠾⠃⠀⡀⠀⠗⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠰⡍⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣀⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠿⣮⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣺⣿⣿⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣶⣝⡢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⣪⣵⣿⣿⣿⣿⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣯⣥⣐⣒⡀⠠⠄⠀⢀⣀⠀⠠⠤⠤⠤⣖⣚⣋⣭⣶⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠿⠇⠁⠉⠙⠻⡿⣿⣿⡿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⣇⠁⠀⠀⠀⠀⠀⠁⠐⠁⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣿⣻⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠿⠭⣯⣽⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣴⣾⣇⢁⣴⣄⣿⣿⣿⣽⣶⣶⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⡄⠀⠀⠀⠀⠀⠀⠳⡆⣠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣀⠰⣏⠀⠀⠀⣀⣀⣤⣼⠃⢿⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢳⠈⠳⠚⠋⣡⠤⠶⢤⠀⠈⠙⠓⢦⣀⠀⠀⢸⡍⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠸⣅⣀⣤⡞⠁⠀⠀⠀⠀⠈⠳⠞⢋⡴⢦⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⢹⠀⠳⡄⠀⠀⠀⠀⠀⠀⠐⣏
⠀⠀⠀⠀⠀⠀⢰⡏⠙⢧⣤⡴⠏⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠸⣇⣴⠖⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢠⣤⣈⣹⠞⠙⡆⠛⠶⠴⠏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⣀⡾⢀⣤⠆⠀⠀⠀⠀⠀⠈⣇⣀⡴⢳⠞⠛⢳⣄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠄⠀⠀⣏⠉⠙⢿⡁⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠏⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠳⠞⣡⠴⢦⡀⠈⠓⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⣘⡇⠀⠀⠉⠳⢤⣀⡀⠀⠀⠀⠀⣀⣠⡤⢦⡈⢷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⣷⢠⡟⠋⠉⠉⠀⠀⠀⣹⠆⠉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠸⢦⠀⠀⠀⠀⠀⠀⠙⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠞

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⢀⣀⢀⡀⣤⣄⣤⡀⠀⣀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢴⠼⡦⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⣼⣿⣸⣇⠼⠗⢻⣴⣆⣿⡁⠈⣷⡀⣿⡿⣤⠉⠩⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⢿⣿⡇⣆⠀⣺⠛⠙⣿⡷⢀⣿⠡⠏⣼⠿⢧⢼⣿⡃⠀⠐⠀⠀⠀⠈⠀⠀⠀⢀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⣄⣀⣬⣤⡀⠀⣙⠋⣘⢳⡏⠀⢱⣤⡀⠀⢀⢉⣀⡀⠀⠏⠀⢨⢃⣼⠗⠒⢢⣴⠆⠀⣶⠀⢘⣯⠈⢉⠉⠛
⠀⠀⠀⠀⠀⢀⡛⠓⠐⠛⠓⢸⣯⢹⡇⣻⡆⢷⡎⣿⡏⢸⣇⢸⣿⡟⠻⡇⣀⣸⠘⢷⠀⣠⣿⠀⠀⠜⠿⢧⠘⣿⡃⠁⠀⠠
⠀⠀⠀⠀⠀⠘⠋⠀⢀⣠⣴⣾⣿⣾⣇⣽⠁⠆⣧⣏⡇⠀⠙⣶⠛⡧⠀⠉⠁⠸⠄⠀⠠⠉⠛⢆⠀⠐⡟⡇⢸⢹⠐⡀
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣻⢿⡛⢿⠿⡷⠘⠁⠀⠈⠁⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠈⡆⠉⠁⠙⠿⠶⠄⠁⠐⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣾⣧⣶⣶⣶⣦⣤⣴⣆⣀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⢰⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠴⢡⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢫⢃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⡵⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡤⠤⠀⠂⠈⠛⠛⠉⠉⠉⠉⠉⠹⠃⡜⠓⠛⠛⠈⠉⠉⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⢠⣞⠑⠈⠀⠀⠀⠀⠀⣀⡀⠤⠄⠀⠀⠀⠠⠁⡸⠠⠄⠤⠤⠤⠄⠀⠀⠀⠀⠀⢸⣡
⠀⠀⠀⠀⠀⠻⣻⣀⢀⡠⠒⠂⠀⠤⠀⠠⠤⠀⠀⠒⠂⠈⠈⢁⣀⠀⠒⠂⠀⠐⠒⠂⠙⠶⠒⠉⡇
⠀⠀⠀⠀⠀⠀⢩⠉⠛⠋⠀⠀⠀⠤⠤⠠⠄⠀⠀⠀⠀⠀⠠⠌⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠋⠉⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⡀⣀⠀⠀⣀⡀⠀⠀⣀⣀⡀⢠⣶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠛⠓⠒⠒⠒⠒⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣇⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⠷⠶⠀⠀⠀⠀⠙⠛⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠛⠻⠟⠋⡡⠤⠀⠉⠀⣴⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢄⠀⠈⣧
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠁⠀⠠⠀⣼⣿⣶⠀⠀⠀⢀⠀⠀⠀⠀⡂⠀⠸⡆
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣄⠀⡠⠊⠀⠀⠀⠐⠀⠀⠀⢷
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣇⠀⠀⠀⣀⣀⡀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣷⣄⠈⠉⠉⠁⣀⣴⣿⡿⢿⣿⣿⠓⣇⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠀⠀⠀⠉⠋⢉⡀⠄⠁⠀⠀⠹⣆⢀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠆⠀⠀⠀⠀⠁⠉⡽⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⢉⢡⠐⠀⠀⠀⠀⡄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⢷⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡁⠈⠙⠛⠁⠀⠀⠀⠁⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⢠⠤⠤⢠⡀⢀⡄⣤⠤⠄⣠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠂⢸⠓⠒⠈⢧⡼⠀⣿⠒⠂⣾⠲⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠈⠁⠀⠉⠉⠁⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⠐⡦⢀⣤⡤⠐⠀⠀⣰⣆⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣡⣄⣀⠀⡇
⠀⠀⠀⣾⣋⡍⣱⣬⠋⣿⡇⣨⣿⣡⠟⠀⠀⠀⠀⢠⣿⡋⠀⢠⠴⣄⠀⣿⡿⢿⣿⠟⣿⣿⣿⣿⡇⣦⣤
⠀⠀⣸⣿⡿⢷⣿⣧⢰⣿⣿⣯⡴⢿⣿⣿⣿⠻⠿⣿⣿⣿⣷⡾⢷⣿⣿⣿⡏⠸⠛⢻⣶⣿⣿⣿⡟⠉⠏
⠀⠀⣾⠏⠔⣌⢹⣇⣸⣿⢧⠳⣠⣶⣾⠻⠿⠀⠀⣿⣿⣿⢿⣿⡑⠿⡿⣫⡄⠀⣠⣇⢋⣾⣿⣿⣟⣿⣃
⠀⠀⠹⢺⣶⣿⢸⣿⣿⣿⣌⣃⣻⣿⣧⠀⠀⠀⠀⣿⣿⣿⡄⢿⣏⡹⢿⣿⢰⡆⢩⣿⣿⣿⣿⣿⡿⣿⠑
⠀⠀⢴⢾⣿⣿⡸⣿⣿⣿⣷⣼⣿⢻⣿⡆⠀⠀⢀⣿⣿⣿⣇⠀⠀⠀⣿⣿⢿⣗⣠⣤⣿⡿⣿⣿⣧⢸⡆
⠀⠀⢸⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⡆⢻⣷⡀⠒⠾⣿⣿⣿⠋⠀⠀⠀⣿⠇⣾⣿⣯⣿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠸⣿⣏⡿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⡇⣀⠀⢸⣿⣿⠀⠀⠀⠀⣿⢰⣿⣿⣧⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣋⣭⣾⢿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⡿⠚⣿⣿⢀⣀⣠⣤⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⢸⣿⣿⢿⣿⣯⡿⠟⠋⠙⠛⠛⠉⠉⠁⠀⠀⠈⠉⠿⠟⠋⠹⠿⠾⠟⠛⠛⠛⠉⢙⣿⣏⣽⠿⢻⣯
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠤⠤⠤⠤⠤⠷⠶⠶⠦⠴⠶⠶⠶⠶⠒⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠰⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢳⠀⠀⠀⠀⠀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⣇⣿⠀⠀⠀⠀⠀⡷⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡀⠀⠀⠀⢻⠘⡄⠀⠀⠀⣰⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡙⣝⢦⣠⠶⣄⠀⠴⠤⠄⠘⠁⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠓⠌⠈⠀⠉⢢⠸⡀⠀⠀⠀⠀⠀⠀⠘⠲⢄⠀⠀⠀⠀⣠⢏⣨⠷
⠀⠀⠀⠀⠀⠀⠓⠒⢮⡒⠦⠤⠄⠐⠁⠀⠀⠘⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢉⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠤⢤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡭⠦⠴⠃⠤⢤⣠⠤⠖⣶
⠀⠀⠀⠀⠀⠀⠐⠊⠉⠒⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡤⠖⠚⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣠⢤⡀
⠀⠀⠀⠀⡠⠤⣄⡤⠔⠂⠀⠀⠀⠀⠀⠀⠀⢧⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠚⠳⠶⠃
⠀⠀⠀⠀⠙⠤⠖⠚⠛⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡙⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣀⠴⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠉⠲⣍⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠋⢁⠞⢅⣠⠀⠀⠈⠙⠒⠂⠐⣄⠀⡆⠈⠁⠀⣶⣄⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠸⡅⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡆⠰⣪⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⢽⢾⡀
⠀⠀⠀⠀⠀⣤⠀⡄⢠⡄⢀⠠⡄⢠⡄⢤⠀⢠⡄⠀⢴⠾⠃⣴⠾⣿⣦⠀⢤⠀⢠⡄⠤⢀⡄⢤
⠀⠀⠀⠀⠀⢹⡀⠀⢸⡇⠘⢷⡄⢸⡇⢰⠀⣊⣿⡚⢿⡇⠀⣿⠚⢻⡿⠐⣾⣀⣨⢻⡄⢸⡇⣶
⠀⠀⠀⠀⠀⠈⠇⠀⠸⠇⠰⠤⠃⠘⡧⠚⢶⣍⡿⠆⠼⠴⠀⠿⣀⣾⣷⡷⠿⢇⣽⠞⣿⠈⠣⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠐⣦⠤⠀⠉⣲⣄⠀⠀⢰⣧⡍⣁⡤⠔⠊⢠⠞⢃⡄
⠀⠀⠠⣴⡦⣄⠀⠀⠀⣿⣿⠀⠀⣤⠶⣦⡞⠓⣶⣿⣼⣄⠀⠀⣟⣹⡅⠀⢀⣶⠶⢻⠾⢷⡄
⠀⠀⠀⣿⡇⢸⣧⠀⠀⣿⡇⠀⠘⣷⣀⡘⣯⣤⣼⣯⣿⠈⠀⠀⢹⣿⣷⠀⠘⢷⣴⣤⣠⣈⠁⠀⢶⡖⢦
⠀⠀⠀⣿⡇⢸⣿⠀⠀⣿⣏⠀⠀⠙⠿⣿⣇⣠⣸⡟⡿⠇⠀⠀⣾⣿⣿⡆⢴⡆⢀⠞⠉⣹⣷⠀⢸⡇⣨
⠀⠀⠀⣿⡇⢸⡿⠀⠀⣿⣧⠀⠀⣆⠀⠘⣿⢀⣸⣟⣠⣦⠶⢱⣿⣿⣿⣷⡎⠛⠳⢶⠞⢛⠡⠀⢸⡇⢠
⠀⠀⠐⠛⠓⠛⠁⠀⢀⣿⣿⠀⠀⠛⠒⠚⠁⠈⢠⡏⠁⡠⠔⣾⡟⢹⣿⣿⣷⡀⠀⠈⢠⠇⠠⠀⠛⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠞⠀⠘⠛⠧⠅⠻⢿⣿⣿⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠈⣿⡟⢁⣤⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠒⠐⢤⣀⠀⡀⣨⠄⠀⢸⣿⡇⣾⣿⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠿⣿⣄⠀⠀⡟⠀⠴⠁⠀⠀⢺⣿⣿⠟⢿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣇⣀⠀⠉⠀⠰⠿⢶⠆⡀⠐⣾⣟⠉⠁⡠⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠀⠀⠀⣀⣴⣠⣤⣟⠻⢿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⡴⠛⠋⠻⣿⣿⣿⣗⣹⡛⢀⡄⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣿⣿⡷⠾⠶⠆⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⣷⡀⠀⠀⠈⠛⠋⡙⠓⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡖⠉⠀⢀⣀⠀⢠⣶⠃⠀⣀⣴⡞⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⣺⣿⣿⣿⡃⠀⠀⠻⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠈⠛⠷⠶⡄⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠐⠃⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠐⣦
⠂⠀⠀⠀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⣾⣅⠀⠀⠀⠀⠀⠀⣼⣁⠤⠿⡿⣆⢠⡀
⠉⠆⠐⠒⠀⠈⠫⠴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣋⡄⠀⠀⠀⠀⠀⢸⡍⢻⣄⢸⣿⣾⣷⣿⣷
⢧⠀⠀⠀⠀⠀⢼⣆⣍⠁⢀⣼⡿⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠁⠀⠀⠀⠀⠀⣻⣿⣄⢹⣿⣿⣿⣿⣿⣿
⠈⣆⠐⠀⣾⠀⣬⣿⢾⠀⣨⡙⠁⣠⣿⡟⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡸⣿⣄⠘⢧⢸⣿⢺⡄⣿⠁⣠⣈⠋⠀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⣴⣛⣳⡄⠀⠀⠀⠀⠀⠀⠀⠈⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠃⠰⠘⢿⣦⠈⢻⣿⠈⠇⡇⣰⣿⠏⢀⣤⡝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣏⣱⣯⡿⠀⠁⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠤⠦⠖⠈⠻⣷⡄⠹⣇⠀⣴⣿⠃⢠⣿⠏⠀⠀⠑⣾⣶⣤⣀⠀⠀⠀⠀⣀⣀⡀⣀⣿⠁⠉⠉⠀⣤⣀⣰⣶⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠻⠋⠀⠀⠀⠈⠙⠦⢘⣾⣿⠃⣰⣿⠋⠀⠀⠀⠀⠈⠙⠛⢿⣷⣿⣶⣶⣿⣿⣿⣿⣟⠀⠀⠀⢀⡁⢹⣿⣿⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⣼⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⠿⠿⣿⣿⣿⡇⠀⠀⠀⠋⠁⠀⣿⣿⣿⣧⣀⠻⣿⢻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⢿⣦⡀⠀⠀⢀⣀⡀⠀⠀⠼⣿⣿⣷⣶⣿⣿⣿⠇⡰⠂⠀⢀⡶⣦⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⡈⠳⢄⢰⣿⣿⣷⠀⠀⠀⢹⡿⣿⣿⣿⣿⡟⢰⡇⠀⠀⠤⠒⣨⡥⣈⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡝⢿⣷⣌⣿⣿⣿⣿⣿⣶⣄⣿⣾⣿⣿⣿⣿⣧⣿⣇⠀⠀⠑⢾⣧⣴⡏⠀⠀⠉⠋⠙⣿⣿⣿⣏⣯⣿⣿⣿
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⣿⣿⣧⠘⢿⡀⠀⠀⠀⠙⠛⠁⣀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿
⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⢿⣞⣿⣿⣿⣿⣧⡈⢷⠀⠀⠀⢀⣤⣾⣷⣶⣄⣤⠀⠀⢀⣹⣿⡿⣿⣿⣿
⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣘⣿⡇⣿⡘⣿⣿⣿⣿⣿⣿⡎⣿⣿⣿⣿⣿⣧⣿⣆⣇⠁⠀⣾⣿⣿⣿⢿⣿⣿⣷⣿⣿⣿⣿⣷⣿⣿⣿
⠈⠛⠁⠀⠀⠀⠀⣀⡀⠀⠀⡛⠘⢿⣿⣿⣶⣿⣧⠘⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣸⣿⣿⣿⣿⣿⣿⣿⡋⠘⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣦⣴⣿⣿⣿⣿⢩⣙⣟⣠⣾⣿⡁⣼⣿⠀⣿⣟⣿⣿⡿⣿⣿⣿⣧⣞⣿⣿⣿⣿⣿⣿⢿⣿⣇⠀⠈⢿⣿⣿⣿⣿⣿
⣩⣴⣿⣿⣿⣿⠿⢻⣿⡿⣿⣿⡟⣿⢻⡟⢛⣿⣿⡟⣿⣿⣁⢿⣿⣤⢻⣿⣿⡇⠀⢸⡏⠉⠉⠉⢿⣿⢿⣿⣿⡟⠸⣿⣿⡄⠀⠈⣿⣿⣿⣿⣿
⣿⠿⣟⣣⠤⠒⠉⠙⢿⣧⠘⠋⢽⣝⠒⣮⣉⡽⢻⣿⣿⣿⣏⣼⣿⡻⢸⣿⣿⣷⠀⢸⣿⣾⣾⣷⣾⣿⡜⣿⣿⡇⢇⢻⣿⣇⠀⢀⣾⣿⣿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠟⠀⠀⠀⠀⠐⠛⣷⡾⠂⠀⠀⠊⢿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡄⠀⠠⣴⠀⠀⢀⡀⡄⠉⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢋⢁⡠⣤⠁⢀⡀⠀⢛⡓⠀⠀⢻⣾⣧⠀⠀⠚⠟⠀⠀⠀⠀⠀⣆⢠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣐⣦⣬⣿⣾⡿⡔⣳⣶⡀⢹⣿⠀⠨⠷⠁⠀⣈⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠯
⠀⠀⠀⠀⠀⡀⠀⠈⢙⣶⣿⣿⣿⣿⣿⣿⣾⣼⡉⢻⣿⣿⢿⡇⣰⠠⡄⠚⠃⠀⠀⠀⠀⠐⢲⣄⣀
⠀⠀⠀⠀⠀⠉⠛⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣯⢕⡞⠓⠀⣤⡤⠀⢸⣦⠂⠉⠉⠛⠋⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⠀⠉⠁⠀⠀⠀⠀⠀⣄⢠⠀⠀⠀⠀⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠠⠿⠯⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠴⠞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠰⡞⠉⠀⢙⡿⢿⣿⣿⣿⣿⡿⠿⢿⡁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢸⡇⠀⠀⣷⠀⠈⠗

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

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⢀⡤⠶⠛⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠏⢀⣀⣀⣀⡼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⠘⠋⠩⠔⣚⣿⣿⣿⣿⣿⠟⠋⢻⣿⣿⣿⣿⣿⣧⠀⠐⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠡⢖⣻⠿⠛⠉⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⠀⠀⣀⡤⠴⠚⠉⠀⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠏⠀⠀⣰⠁⠀⠀⠴⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻⠤⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢨⣿⡆⠀⠀⢠⠤⠀⠀⡆⠀⠉⠁⠀⡀⡠⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣾⣿⣧⠀⢀⢀⠀⠀⠈⠃⠀⠀⠀⠈⠐⠁⣹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠘⢿⣿⣿⡗⠁⠂⠀⠀⢔⣉⠭⠵⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠉⠻⣿⢿⣿⢷⡀⠀⠀⢸⠛⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⢶⠲⠔⠃⠀⠀⠀⠙⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⣠⠞⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠙⢶⣶⣶⡶⠖⠋⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠰⡔⣀⡀⠄⠠⠞⠙⠀⣼⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡝⣿⣦⠀⠘⢞⣷⡇⣸⠁⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠄⠀⠒⢒⣦⢠⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⠀⠸⣿⣷⠁⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠓⠀⢀⠀⠎⠀⠀⣸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⠀⢀⣀⡀⠀⠋⠐⠷⣄⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣆⡀⠙⣿⢰⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣄⠀⢠⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣼⠡⠸⠏⠀⠀⠀⠀⠀⡀⠸⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠙⢦⡘⣾⡄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢦⢄⡆⠈⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢠⠉⠀⢠⡄⠀⠰⠂⠀⠘⠗⠀⣏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣿⡇⠀⢳⣹⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠐⢼⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⡏⢰⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢿⣷⠀⠘⣿⣷⡇
⠀⠀⠀⠀⠀⠉⠂⠀⡸⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⡊⠈⣡⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢻⡟⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠠⢨⡿⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⠁⡇⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠟⠡⣾⠧⠝⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⠀⠀⢰⣠⠎
⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠰⠦⠦⡄⠀⢨⣿⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠘⣇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠈⠣⠀⡈⠀⠘⣿⣿⣿⣇⠀⠀⠀⠀⢰⡿⠖⢀⡀⠀⠀⠀⠀⠉⠀⠈⠈⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⣿⣰⡏⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⣠⡇⠀⠁⠀⠀⠉⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⣿⡟⢀⠞⠁
⠀⠀⠀⠀⠠⠀⡀⠀⠁⣠⡄⠀⠸⣿⣿⡇⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⢠⢱⡄⠀⠌⠣⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⢰⣿⡴⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢻⡆⠀⢻⣿⣿⡄⠀⠀⢰⡿⠀⠀⠀⠀⠀⢧⣸⡿⠝⢄⠈⠆⠀⠀⡆⠁⠀⠀⠀⠀⠀⢀⣾⣿⡿⠀⢀⣾⢯⠆⢀⣠⡤⠶
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠃⠀⣽⠀⠀⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⣀⠀⠸⡆⠠⣧⡴⠃⠀⠀⠀⢀⣾⣿⡿⠁⠀⡼⢱⣯⡶⠛⠁
⠀⠀⠀⠀⠤⠀⠂⠄⠠⡀⠠⠀⠰⠄⠈⢿⣿⣷⡄⠀⡄⠀⠀⠀⠀⠀⢸⢇⢠⠏⠇⡤⡤⢶⣾⢻⡇⠀⠀⠀⣠⣿⣿⡿⠁⢀⠞⣴⠟⢁⣤⠤⠤⠄
⠀⠀⠀⠀⠈⠂⠀⠈⠈⠀⠀⠐⠈⠈⣦⣌⠻⣿⣿⣦⡁⠀⠀⠀⠀⢀⡿⠇⠀⠀⠀⠀⠃⠘⡟⠘⠁⠀⢀⣴⣿⣿⠟⠐⣠⣿⣮⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣧⣹⣿⣿⣿⣦⣄⡀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠃⢠⣾⠟⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⠌⠻⢿⣿⣿⣷⣶⣦⣤⣤⣀⣠⣤⣤⣴⣶⣿⣿⣿⠟⠉⣠⡾⠋⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⡙⠛⠿⠿⢿⣿⣿⡿⠿⠿⠟⠛⠉⠋⠁⠄⠊⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡶⠄⡀⢂⡀⠠⣀⣀⣠⡤⠒⠉⠀⠀⠀⠀⢚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢝⣦⣽⠀⠀⠀⣿⣿⡯⠀⠀⠀⡊⠠⠤⠎⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣄⠀⠀⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣴⡀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠠⠾⠃⠀⠀⠢⡀⠈⠉⠋⠛⠛⣆⣤⡀
⠀⠀⠀⠀⠀⠈⠋⠐⠻⠿⣦⣄⠈⠉⠀⠀⠀⠀⠜⠀⠀⢀⣤⣴⣦⣼⣶⣄⣀⠀⢸⣿⣿⣿⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣦⣄⡀⠀⠀⢔⠒⠬⣄⡴⢿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⡉⠆⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡾⠁⣤⣓⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⡀⣝⢿⣿⣿⠎⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢠⣄⣀⣠⡤⠖
⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢻⡗⠹⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣠⡵⠿⠋⠁
⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣼⠢⢡⡀⠀⠀⠀⢀⣼⣷⣷⡀⣐⣾⠷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠄⠲⠛⣫⣷⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⠀⠸⣿⣿⣿⡿⠿⢛⣽⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢆⢀⠀⠀⣠⡴⠛⢋⣠⣾⣿⣿⣇⣀⣣⡠⠭⢹⣩⣩⣷⡶⠲⠶⣶⣟⡶⠛⠋⠑⠁⠀⠀⠀⠀⠀⠀⠀⠰⠞⠉⠉⠉
⠿⠿⠛⠛⠛⠛⠛⢓⡛⢛⡷⠟⠛⠿⡛⢏⡂⠖⣋⡿⣻⣿⣿⡢⠀⠍⢏⣸⣿⡿⣿⡎⠀⡤⠀⡾⣽⣷⡤⢆⣤⣤⡤⣄⠤⣀⢄⣀⡠⡀⠀⡀⠀⠠⣤⠀⣠
⠀⠀⠈⠀⠀⠤⠶⣿⣿⡷⠶⠀⠀⠀⡋⠛⢟⣷⣍⢠⣿⣿⣿⡵⠁⠔⠤⣤⣻⣾⣯⣶⡟⢠⣮⠘⠛⠟⠑⠈⠙⠋⠁⠉⠉⠉⣀⡈⠅⠛⠚⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⣿⠿⠹⠇⠈⠑⠀⠀⠀⠈⠁⠑⠛⠻⣳⣿⣟⠿⣿⡀⠀⠀⠈⠛⠟⣋⢗⣱⣿⠉⠠⠀⠀⡀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⢾
⣶⣾⣂⠐⠟⠁⠀⠀⠀⠀⠀⠐⡀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣗⠞⣷⣤⠀⠀⠀⠀⣡⣾⣿⣯⠶⠀⡀⠀⠀⠈⠀⠃⢄⠈⡉⡰⠆⠀⡴⠤⡆⢠⡨⢽⢦⣀⣈
⡁⠈⣾⡟⠀⠀⠀⠀⠀⣠⣄⣠⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⠕⠉⣿⡟⠑⠦⡤⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈⠒⠀⠰⠿⣽
⣿⣾⣿⣷⡄⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡩⠋⢹⡿⢇⡟⠛⣷⣦⠀⠀⢘⠷⠀⠀⠀⠈⠈⠙⠻⠶⠖⣤⢀⠀⡀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⡜⠁⠀⣘⣵⡿⠃⠀⠸⡟⠀⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠣⠀⠀⡃⢰⠀⣆⢠⡄⢀⡤⢤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣶⣶⣶⣶⣦⣤⣤⣄⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠉⠉⣿⣿⣦⢆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀⠀⠲⣿⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣿⣭⣭⣽⣿⣷⣾⣷⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣽⣿⣿⣿⠟⠻⣿⣿⣿⣿⡟⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⣿⣿⡿⣿⠀⠀⢸⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⣑⠠⡼⠏⢻⣿⣿⣵⢧⠀⢀⡞⢹⣿⠇⣾⣏⣿⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣤⣄⡽⠶⠋⠉⠀⠀⠀⠀⠈⠀⠀⣀⠋⠀⡜⠀⢿⢏⡼⢻⣿⠇
⠀⠀⠀⠀⢀⣴⣭⣿⠶⠶⠶⠾⠿⠿⠶⣶⣶⣶⣶⣤⣤⣠⠞⠁⢀⡠⠞⠁⣈⣭
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣶⣒⣚⡭⠄⢀⣴⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣯⣯⣭⣶⣾⠿⠿⠷⠦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⠶⠶⠶⠒⣻⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⠿⠛⠀⠀⣠⢔⣽⣿⡟⠉⠉⠉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠃⠀⠙⠻⢿⣿⣿⣿⣷⡐⠄⠃⠀⠀⢰⡞⣳⣿⡿⠋⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣄⠀⠀⠀⠘⡟⣿⣿⣿⠀⠀⠀⠀⣾⣸⣿⠟⠀⠀⠀⢀⣀⣀⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣍⠛⠿⠂⠀⠀⠀⠀⠀⠻⣟⣧⠀⠲⣶⡟⣿⣏⣠⣤⣶⣾⣿⣿⣿⣿⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠿⠷⣦⣴⣦⡀⠀⠀⠀⠀⠉⠈⠓⠞⠋⣰⣯⣿⣿⣿⣿⣿⡿⠿⣿⡁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣈⠙⢃⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢹⣟⣿⣿⣿⣏⣵⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠾⢿⣿⡿⠆⡀⠀⠀⠀⠀⠀⠛⠛⠻⠿⠟⠙⠋

⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠢⣀⣀⣀
⠀⠀⢀⠔⠉⠉⠺⠀⠀⠀⠈⠁⠀⠀⠱
⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄
⠀⢀⠜⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡀
⢰⠷⠀⣠⣾⠿⠀⠀⠀⠀⠀⠀⠠⠾⠃⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠸⡀⢾⣿⡿⡄⣤⣀⣀⣠⠞⡗⠤⣀⢀⡀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣬⣅⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠑⠤⠭⠥⢔⠣⣘⠈⢇⣼⣠⣶⣿⣎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⣽⣿⠆⠀⠀⠀⠀⠀⣬⣧⡀
⠀⠀⠀⠀⠀⠀⠑⢌⠀⣈⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠋⠀⠀⠀⠀⠀⢰⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠈⡆⢸⠻⠼⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠰⢿⣇⢸⡇⣸⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢐⣷⣤⣽⣿⣿⣧⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⡀⠻⣿⣿⣿⣿⡿⠃⣀⠀⠀⢡⣤⡞⢻⡟⢷⣤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣰⣿⣿⣦⣌⣉⣉⣉⣤⣾⣿⣿⣦⠀⠙⠁⠾⠷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠀⣿⣿⣿⠻⣿⣿⣿⣿⣿⡋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣿⣿⣿⠀⣿⡿⠿⣿⣿⣿⣿⣿⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⣿⣿⣿⣶⣿⣿⢠⣄⡉⠛⠿⠿⠟⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠿⠿⠿⠿⠿⢸⣿⣿⣷⣦⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣄⠰⣶⠆⣠⡀⠀⠀⠀⣶⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⢀⠛⠻⣦⣿⣤⡿⠛⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⣿⡁
⠀⠀⠀⠀⠀⠀⢸⠿⠟⣿⣿⣿⡿⠻⡯⠀⠀⣼⣿⣿⣿⠿⣿⣿⣿⡆⠀⠐⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢶⣾⠋⣿⠉⣿⡶⠀⠀⢀⣿⣿⣿⣿⠀⢿⣿⣿⣷⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠃⠈⠀⠀⠀⢸⣿⣿⣿⡇⠀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠘⠷⣦⡀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⣿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢰⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⠀⠉⠀⠂⣿⠃⠒⠛⠀⠀⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢛⣀⠀⢀⠒⢂⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⢀⠐⠀⠄⠠⠦⢤⡄⠀⠀⠐⠀⠀⠀⠀⠀⢀⡦⠀⠴⠦⠀⠠⠟⠀⠢⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠆⠁⠀⠀⠀⠁⠈⢁⡀⠀⠐⠛⣁⡀⠀⠀⠈⠙⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⠀⠉⠁⠀⠀⠀⠀⠀⠠⠸⠆⠀⠨⠀⠀⠊⠀⠀⠀⠀⡀⠐⠊⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠒⠒⠂⠀⠤⠤⠤⠤⠄⠒⠒⠒⠈⠀⠈⠁⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠶⠖⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

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

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


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

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

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

⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⢀⣀⣀⣀⣀⢀⣀⣀⡀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀
⠀⢸⣿⣿⣿⠟⢻⣿⣿⣿⣿⡧⣼⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡿⠻⠋⠀⢀⡤⠄
⠀⠸⣿⣿⣥⣤⣬⣩⣿⡿⠿⢡⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⠟⠀⣀⣀⡔⠛⠛⠃
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⢿⣿⠟⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⠉
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⣉⣤⡶⠛⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠉⠉⠁⣄⣀⣤⣤⠄⠀⠀⠀⠀⠀⠀⢢⠄
⠀⢀⡈⢹⣿⣿⣿⣿⣿⠟⢛⣼⣿⠋⠁⠀⣀⠀⠀⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⢲⠛⠛⠛⠋⠁⠀⢀⣀⡀⠀⢀⣼⣶⣷⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣮⣀⣀⠀⠀⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠉⢛⣿⣿⣿⡟⠛⠁
⠀⢸⣾⣿⣿⣿⣿⣿⣿⡶⣶⠛⠛⢻⡆⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢈⠉⠉⢩⠉⡍⠀⠀⠬⠷⠛⠛⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⠓⢺⠆⢀⣼⡇⣄⠀⣠⠄⠀⠀⠀⠂⠀⠀⠀⡄⣤⣤⣄⠀⠀⠀⢀⣆⣶⠀⠘⠀⡢⠈⠁⢦⠀⠄⡀⣤⠄⠀⠂
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣗⠛⣄⡈⢹⡇⠫⣤⡽⣯⣠⣼⣀⣄⠀⠀⡀⢻⣿⣿⣟⠀⠠⡄⠀⠹⠿⢿⠀⠀⠀⢀⣤⣿⣶⠀⢸⣏⠀⠒⠳⠂
⠀⢸⣿⣿⣿⣿⣿⡿⠉⠋⠀⠙⠳⢶⣷⣶⣿⡏⢻⣿⣿⣿⢿⠟⣿⣷⣿⣿⣏⢸⣷⣄⠰⣆⢀⣀⡀⢀⢀⠄⣉⣤⣿⣿⣖⠉⠀⣼⢶⡜⠁
⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⣠⣇⣿⣿⣧⣸⣘⣿⣿⣾⣦⣿⣿⣿⣿⢟⣿⡿⡆⢑⠀⠀⣉⡄⠈⠁⠀⣸⣿⠟⠛⣿⣶⣦⣏⢘⣩⣠
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⢿⣴⣤⣾⣇⣼⣿⣧⣴⠀⠹⣻⡧⠀⢠⣿⣿⣻⣿⣿⣿⣿⠆
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⢿⣿⣿⣿⣿⣬⣽⣿⣿⣿⣿⡟⣿⣷⣿⣿⣿⡿⢿⣿⣿⣧⣤⠘⣿⣿⣦⠙⣿⣿⣷⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡏⠈⣿⠛⠿⣿⣿⣿⣿⣿⣿⠿⠔⣿⠿⢿⣿⣿⣇⣻⣿⣿⣿⣿⡷⠯⠋⡉⠉⠙⠙⠛⠏⠋⠙⠃
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣉⣁⣼⠁⣠⣾⠂⠀⠀⠀⠀⠀⣉⣀⠀⣰⡦⠴⠀⠈⠉⠉⠉⠉⠙⠛⠛⠓⠀⠀⠀⠀⣀⣀⣀⣁⣶⣶⣤
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠛⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⠟
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⢷⠞⠚⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠇⠀⠂⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠⣦⠀⢠⣆⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣘⠋⠀⣘⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⣀⣀⠀⢀⡀⠀⢀⡀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠁⠀⠈⠉⠁⠉⠉⠀⠉⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠒⠀⠠⠀⠤⠤⠤⠤⠶⣅⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠢⣀⠀⠀⠤⠀⠀⢰⠊⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣊⢢⠠⡈⠠⠈⢂⣀⡠⠜⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢀⠅⢠⠈⠋⠐⠠⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⣨⠉⠉⠀⠀⠈⠉⢳⢠⠄⢤⡀⠈⢦⡀⠀⠀⠀⠀⢀⠗⠊
⠀⡀⢀⡀⠀⣦⠀⠂⣠⣚⡥⠤⠐⠒⢻⣉⢳⡀⠀⠀⠀⠀⠀⢸⡀⣀⠗⠢⢤⣙⣆⠀⡀⠀⣸⠶⠒⡆⣀
⠀⡯⢩⠁⣄⣉⣁⡈⡄⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠸⠑⠉⠀⠀⠀⠀⠀⠉⠃⠀⠀⢈⡀⣀⠃
⠀⣏⣹⠀⣟⣁⣀⠊⠇⠀⠀⠀⠀⠀⢀⠸⣀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠠⡄⠀⡀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠉⠒⠒⢀⣠⣤⠴⠖⠀⡉⢁⠀⠀⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣦⣬⣥⣥⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣴⠀⣴⠀⠀⡀⢠⠀⣀⣀⣤⢀⣀⠀⣀⠀⣤⢀⣀⣀⣀⢀⣀⠀⡀⡄⠀⢀⣀⢀⠀⡄⣀⡀⢀⡀⢀⠀⠀⢀⠀⠀⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⡀⠀⠈⡀⡀⢁⣀⡀⡀⠀⡤⠀⣀⣀⠁⢀⡈⠈⣉⠈⠁⠈⠉⡀⠀⠁⡀⢉⠉⢈⣁⠁⠉⡁⢈⡉⠉⠈⠁⠈⠉⠀⠈⢁⠉⠉⠉⠉⠉⢀
⠀⠀⠀⠁⢀⡀⡉⠁⠈⠁⠀⠀⢀⣠⡀⠁⠈⢠⣄⠉⠀⢈⠀⢀⡁⠀⢀⠀⠈⣁⠀⠉⢈⣁⡈⠉⣉⠈⡁⠈⠈⣉⡉⠈⠛⠉⠉⡀⠁⠀⠉⠉⠉
⠀⠀⠀⡉⠉⠁⣉⠀⠉⠈⠈⠁⣈⢉⠉⡉⠋⠈⣉⠉⠁⠘⠁⣉⠉⠃⠉⢀⡈⠃⠉⠉⠉⠛⠙⠁⠉⠀⠉⠃⠀⠋⠈⠉⠁⠉⠁⠉⠉⠈⠁⠉⠁
⠀⠀⠀⡁⠹⢆⠉⠂⡉⠀⠉⠁⠉⠘⠀⠋⠈⠁⠃⠈⠘⠁⠘⠀⠘⠃⠐⠂⠓⠀⠛⠃⠐⠃⠓⠚⠃⠛⠈⠙⠛⠁⠛⠀⠛⠀⠛⠓⠘⠃⠛⠛⠛
⠀⠀⠀⣉⣀⣬⣉⣩⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠈⠉⠁⠀⠀⢁⠆⠀⠀⡀⠈⠥⠀⠌⣉⠀⠈⢻⣿⡯⠀⠁⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⠀⢀⠀⠀⠀⠀⠀⡀⢼⠳⢦⡄⠲⠆⠀⠾⣀⠀⠀⠤⠀⠐⠀⠛⢠⡏⠑⢀⣿⣿⠀⠀⠀⠘⠋⠃⠁⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⢮⣹⠶⣀⠐⡞⠂⠾⠖⣧⠀⠉⠁⣀⠴⠒⠢⢀⣼⠁⠀⣰⣿⡙⠛⠀⡀⠀⠀⠀⠀⠀⠘⣣⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⡇⢠⠞⠁⠀⠀⢠⠀⡜⠉⠀⢉⡿⠃⠠⠀⠀⠀⠣⡀⠀⢻⣤⠀⡄⡉⡸⠅⣰⣟⡉⠛⢆⠀⠀⠳⠐⠒⠀⡀⠀⢠⠐⠶⣄⠀⠀⠀⠀⢀
⠀⠀⠀⣷⣄⠀⣠⡄⠀⢠⢸⡆⠀⠀⢺⡕⠀⢸⡆⠀⠀⠀⠸⢦⢤⣏⣤⠀⠸⣇⣴⠿⠋⠓⠀⣀⠃⠀⠀⠘⠀⠀⠣⠀⢈⠂⠀⢨⠆⠀⠀⠀⢸
⠀⠀⠀⣿⢻⣿⡟⠀⠠⠚⢯⣉⣱⡖⠋⠀⣀⡈⢿⡀⠀⠀⠀⠀⠀⢙⡯⠁⠀⠻⠇⣠⣤⡠⢄⠈⠳⣄⡐⠛⣄⠀⢠⡀⠀⠁⠃⠀⣿⠀⠀⠀⢸
⠀⠀⠀⣿⡄⣿⠀⠀⠰⣀⡾⠛⠴⠁⣠⣿⡿⢞⡇⠹⣦⠄⠀⡞⢠⠈⠅⣀⠀⣴⠂⠛⠁⡉⡸⠇⣸⠃⠀⠀⢠⠀⠰⠀⠀⠀⡄⠀⢻⠀⠀⠀⢸
⠀⠀⠀⣿⣷⣿⡀⠀⠰⡏⣰⠏⠀⠘⣴⣥⠤⣻⠛⢄⣰⣐⣐⠂⠀⢀⢸⡟⣟⣳⠜⠛⠤⠢⠬⢨⣴⣆⠀⠀⠀⢀⠄⠀⠀⠀⣠⠼⠋⠀⠀⠀⢸
⠀⠀⠀⢉⠉⢿⡇⠀⢀⣿⠹⣆⠀⠀⠻⣿⣿⣿⠱⣎⡉⠉⠀⣀⣴⠀⠈⡃⠹⠀⣀⠀⢀⠀⡄⠀⣌⢻⢯⣅⣷⠀⢛⡀⠀⠀⠻⣇⡘⡄⠀⠀⢸
⠀⠀⠀⣿⡀⢿⡗⢠⠀⢻⣧⠹⡦⢤⠀⣭⣉⣿⢶⡆⣉⣁⣍⣽⣿⠀⠀⣇⡆⣼⡏⠀⠀⡤⠳⡄⠉⠿⣺⣿⡟⠀⣼⣡⠀⠀⢀⣿⠇⡆⠀⠀⢸
⠀⠀⠀⣿⠏⢨⡙⢧⠀⢸⡄⠀⣀⠀⣦⣿⢉⣿⡾⠁⢸⣿⣿⣿⠃⠀⢀⡯⠛⢻⣇⠀⡈⠻⢤⣶⡄⢀⣯⣌⣠⣴⣿⡤⣄⡀⢸⣿⣷⡤⣀⠀⢸
⠀⠀⠀⣿⠀⣤⢃⡀⢀⠀⢶⣶⠄⠀⠈⣹⣯⠹⣷⡀⠀⣿⣿⣡⠀⠂⣸⡇⡻⠀⣠⣤⣀⣤⣾⣿⣷⣽⣿⣿⣿⡟⠳⠟⠦⠀⠛⣥⠀⠃⠈⠀⢸
⠀⠀⠀⣿⡆⢿⠀⠁⢦⡄⠈⠁⢈⣣⢶⣿⣿⣀⠀⢅⠀⠛⢾⡋⠀⣰⣟⠁⠃⠰⠟⠛⢲⣾⣿⣿⡛⣿⠅⠉⠈⠉⠀⠀⠁⠀⠂⠀⠀⠀⠀⡆⢸
⠀⠀⠀⣿⠳⢄⠀⠸⣃⠴⠉⢉⠉⠁⢈⡁⢀⢹⣇⠘⣷⡉⠈⠁⠉⠉⠉⠷⠐⠀⠀⣀⠈⠁⠿⣿⣿⡹⢆⠀⠀⡀⠀⠀⠀⠀⠰⠁⠀⠳⠀⠀⣾
⠀⠀⠀⣷⣀⠈⠋⠀⣿⣶⣿⣏⠀⠀⠘⠑⡾⣦⢹⡖⣏⠙⠁⡶⠒⢲⣶⠖⡶⠉⠥⠐⠂⢙⠀⢹⣿⣇⠀⠀⠀⠉⠀⠀⠀⠀⡏⠀⢃⡀⠀⠠⢿
⠀⠀⠀⡏⠙⣷⣦⣀⠙⠛⣷⡤⣀⠄⠀⠀⢐⣸⡇⣐⣾⠀⠀⠘⢣⠘⢳⡆⠅⠀⠀⡼⣆⠀⡷⠄⠻⣯⠀⠀⠀⠀⡀⠂⠠⣶⡀⢠⡜⠉⠛⠠⣾⡆
⠀⠀⠀⣇⣠⡭⠉⠉⠛⠄⢻⣇⠂⠀⣀⣤⣮⣥⡾⢻⣿⠀⠀⠀⠂⣶⣭⣥⠐⠀⠀⢸⣿⡈⠇⢈⡷⣿⢂⡀⠶⣤⠀⢀⡀⠈⢁⡄⠉⢢⣤⣀⣾⡇
⠀⠀⠀⣿⠹⣆⠀⡀⢷⢤⣾⠿⡴⠾⠿⢛⣿⣿⣇⣸⡇⡆⠀⠀⢠⢹⡇⢼⠀⠀⠀⠘⠋⢳⣶⣾⠟⣾⣿⠀⡀⣰⣆⣼⠁⢀⡖⠀⠠⠀⠀⢻⣿⠇
⠀⠀⠀⡷⠂⠈⢀⠙⠛⢧⡇⠀⠀⠄⠀⠈⣹⣿⢿⡿⡇⠁⠀⠀⠀⡀⢀⣼⠀⠀⠀⠀⠠⣸⣟⣿⠘⣿⣍⣼⣷⣤⣀⠈⢀⣌⢁⡀⣤⠀⠀⠸⢻⠃
⠀⠀⠀⣿⠘⠛⢡⡆⣧⠐⣻⠗⠂⠒⣾⠁⠸⡟⠂⠀⣄⣲⣶⠂⠴⠉⢠⡀⡃⠀⠀⠀⠴⣟⠈⣟⢰⣏⢿⡟⣙⠁⠉⢻⣾⣷⠘⣇⠀⠀⠀⠈⢸
⠀⠀⠀⣼⡀⠀⠘⢿⡟⢿⢸⡄⠀⠀⣟⠀⠀⡗⠙⠛⠛⡛⢡⡝⠂⠈⠃⠈⠀⠀⣀⣠⡾⢧⣰⣻⠐⣿⡎⠁⠿⠀⠀⠈⢣⡝⣿⣿⢀⠀⠀⠷⢸⡄
⠀⠀⠀⣿⣋⣉⡁⢘⢻⠷⣿⣴⠆⢺⣭⣄⣒⠁⠀⠘⠘⠻⠐⠫⡥⠤⣤⣤⡤⢾⡟⢁⡀⠠⠈⠧⡾⣿⠀⠐⠀⠀⠀⠀⢺⡇⢛⣿⣇⠱⣀⡒⣿
⠀⠀⠀⣿⣦⣤⣄⣀⣀⣐⠻⡏⠈⢻⠀⠀⠫⣄⠀⢀⠀⢠⡤⢄⡀⠀⠠⠀⠀⣸⣅⠘⡛⠷⣇⡶⣧⣌⠀⠀⠈⠀⠀⠀⢈⠁⠈⢷⣽⢎⣹⠿⢿
⠀⠀⠀⠃⢌⡉⠊⠛⠿⠛⠛⡷⠶⢿⣷⣉⣦⠈⠛⠿⠔⢠⠀⠀⡄⠀⠀⠉⠰⠬⣟⠛⠳⠀⣠⡅⠘⢯⠀⠀⢠⠀⠀⠀⠀⣄⠀⠘⢿⡀⠁⠀⢼
⠀⠀⠀⡇⠤⠄⠀⢀⡀⣀⣸⠗⠀⠤⢌⣙⢿⣟⡵⠖⠛⠋⠀⠀⠈⠻⢀⣄⡤⢄⡀⡙⠃⠸⠃⣀⣴⣿⠐⡀⠸⡆⠀⠀⠀⣤⠾⠄⠘⣧⠀⠀⣸⠅
⠀⠀⠀⣃⠀⣶⠀⣾⠀⣨⡘⠀⠀⠀⠀⠈⠙⢾⠟⣿⣷⡶⣶⣶⣶⣤⣤⣬⣤⣤⣍⣥⣴⣶⠋⠁⢈⠺⢃⣴⡀⠈⡄⠀⠀⠀⠀⠀⠀⢉⠐⠒⣹
⠀⠀⠀⢻⠀⠛⢷⣿⣶⠟⠠⠀⠀⠀⢀⠠⠤⣀⠀⠀⠈⠉⠉⠓⠒⠟⠛⠛⠋⠐⠋⠉⠉⠌⠁⠀⠀⠀⠈⠿⢻⢤⡀⠀⡀⠀⠁⠀⠀⣠⠛⠛⢹
⠀⠀⠀⢸⠀⢠⠀⢿⡇⡰⠀⠀⠀⢠⠀⠆⠈⠈⢆⠀⢀⡀⢤⣤⣄⡀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠓⠾⢿⣦⣤⠊⢀⣀⣤⣾
⠀⠀⠀⢻⠀⠀⠀⠐⠈⠀⠀⠀⠀⠈⢾⠇⣀⡀⢈⠙⠋⠀⠀⠉⠀⠀⠀⠒⠒⠚⢶⣦⡨⣀⡀⠀⠤⢄⣀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠉⠉⠁⠀⢸
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠟⠃⠀⠀⣀⡀⠀⠉⠶⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⡀⠑⠳⠖⠶⠒⠂⠠⣤⡰⢿⣤⡤⠴⠊⠉⠀⠑⠒⠢⢄⠑⠀⠂⢰⠆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣷⣶⣶⡦⠤⠤⠤⠤⠽⠤⠤⠤⠤⠤⠴⠶⠦⠤⡶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠼

⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⢢⠀⠀⠀⠀⠀⠀⢀⡤⠤⣀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⣧⠀⠀⠀⠀⠀⢸⠀⢀⣀⡼⠃⠀⠀⠀⠀⠀⢸⣀⡀⢘⡆
⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⡏⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠘⣦⣀⣀⣀⠀⠀⠀⢀⡤⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⣤⣶⠶⠿⠟⠛⠛⠛⠛⣿⠛⠛⠛⠛⢿⠿⣶⣦⣤⣀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡄
⠀⠀⠀⠀⠀⣠⣴⡿⠟⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⢾⠀⠀⠀⠉⠙⠻⣶⣤⠴⠋⣁⣤⣶⡿⠃
⠀⠀⠀⣠⣾⠛⠁⠀⠀⠈⢳⣄⡤⠤⠴⠶⠒⢶⠞⠓⠒⠲⠶⠤⢼⣄⣀⡀⠀⠀⠀⡼⠁⣠⡾⠛⠉
⠀⠀⢰⣿⠁⠀⠀⡠⠖⠋⠉⣤⡍⢀⣈⡛⠁⢴⢢⡀⣶⡆⢀⡀⠀⣽⡎⠉⢓⡦⠞⠁⣰⣿⣿
⠀⠀⢸⣿⡀⣠⠊⠴⡄⠀⠀⠀⢀⣙⠐⠋⠀⠀⠈⠀⠀⠀⠛⠁⡠⢶⡄⠀⢯⠀⠀⣰⡿⣻⣽
⠀⠀⠸⡟⢿⣧⣀⠉⢥⠪⡷⠀⠈⢬⠇⡀⠀⣴⡶⠀⢠⣾⠆⠀⠉⢉⣀⠀⣨⢷⡟⣻⣾⠟⣹
⠀⠀⠀⢷⣤⡉⠛⠿⣷⣦⣤⣤⣄⣀⣘⣝⣀⣀⠀⠀⠀⠀⠐⣽⣶⣙⣋⣤⣬⣽⠿⠛⠁⣰⠇
⠀⠀⠀⠈⢿⢻⠓⣦⣄⣀⡈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⢀⣠⣴⢯⡏
⠀⠀⠀⠀⠈⢷⢼⣿⡹⢠⢟⡻⡗⢲⣶⣶⠤⠤⣤⣤⠤⢤⣤⡤⠤⣴⣶⡒⣻⣟⣦⢯⠟
⠀⠀⠀⠀⠀⠈⠳⣌⠙⠚⠾⠾⣅⣗⣯⡼⢀⡏⣗⣸⠂⡮⣖⡽⠀⣷⣦⣃⣿⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡦⣤⣀⡀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⣉⡥⠔⢊⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡈⠉⠛⠓⠒⠒⠶⠶⠶⠖⠒⠒⠋⠉⠁⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠶⠒⠋⠉

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


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠶⠁⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⡐⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠔⠀⠀⠐⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠑⢆⠀⡆⠀⠠⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⠄⠀⠈⡄⢃⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣀⣀⣧⣨⡄⣤⡄⢀⣄⡀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⢔⠒⠺⢋⠉⢃⠈⠃⡈⠀⠐⠰⢸⡏⠀⠻⠀⡀⠑⠀⠀⠓⠚⠻⣍⡙⢟⠭⣖⣲⡦⢤⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⢀⣤⣖⠒⠉⠅⠀⠀⠉⠀⠀⠀⠠⠀⢀⠀⠠⢀⡥⢀⠀⡞⡀⠄⡀⠀⡀⠡⠈⠒⢄⡠⢄⢀⠉⠓⢦⣀⣁⣿⢿⠧⣝⡷⣦
⠀⠀⠀⣴⠏⠁⠒⠂⠀⠀⠐⢤⡠⢠⡂⠄⠈⠒⠀⠛⠒⠓⠀⢀⡝⠉⠉⠀⢀⠀⠈⢩⡉⠉⠀⠁⠐⡛⠚⠗⣹⣾⣟⣳⣄⠰⢬⣃⡾
⠀⠀⠀⠙⣟⣢⣤⣤⣽⠔⠫⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠰⠂⠀⠀⠀⠆⠀⡀⠀⠐⠀⠀⢀⣴⡾⠛⠄⠀⣈⣻⡷⣾⠋
⠀⠀⠀⠀⠈⢿⣛⣛⠛⠛⠶⠶⣤⣤⣤⣀⣀⢀⣚⠀⡀⠐⠀⠂⠀⠀⣀⣈⣁⣀⣀⣀⣀⣀⢠⡤⠾⠿⠿⠶⠚⠛⠋⠉⠁⡶⡏
⠀⠀⠀⠀⠀⢸⠃⠊⠁⠒⠒⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣇
⠀⠀⠀⠀⠀⢸⠁⣰⢶⠶⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡤⢢⠀⠀⠘⢢⠔⢡⣏
⠀⠀⠀⠀⠀⢸⢶⠈⡷⠊⢀⡀⠐⢄⣠⠜⠀⠀⠀⠰⡐⠖⡐⠀⠀⠀⣄⠶⣰⠀⠀⠀⠦⣉⠜⠀⣀⡉⠒⠁⠀⠞⣱⡄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣏⠀⠀⠲⠯⢌⠆⠀⠀⠀⢀⢞⡱⡄⠈⠋⠀⢠⠖⢄⠈⠛⠁⢀⢔⡒⡄⠀⠀⠠⡱⠖⠂⠀⠀⠉⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠘⢦⣠⣤⣤⡄⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠑⠊⠈⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⣭⣀⣒⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠭⣭⣛⣓⣒⡦⠤⠤⠤⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⠶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⠒⠒⠒⠒⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⣀⡀⠀⠀⠘⢧⣼⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢹⡋⠀⠀⠀⠀⣀
⠀⠀⠀⠸⣥⡧⣤⠔⠚⠃⠙⠓⠦⣤⢼⣬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⡴⢚⡉⢩⣭⠉⣓⢦⡼⠛⠁
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⡀⠙⠋⣤⣍⠘⠛⢠⣽⡄
⠀⢠⢤⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⡤⡄⠀⠀⣠⣄⢀⣠⢤⠚⠛⠲⠤⣄⢀⣤⡄⠀⠀⢠⢤⣀⡎⠙⠋⠸⠿⣨⣅⠘⠿⢈⡉⢹⣀⣤⡄
⠀⠙⠞⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠲⠃⠀⠀⢈⡿⠉⠀⠻⠳⠀⠀⢀⣤⡝⢎⠁⠀⠀⠀⠉⠀⢳⠪⠇⢾⠆⢉⣥⠺⢇⣙⠋⡼⠀⠉⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡜⠀⣴⡦⠤⠀⠀⠀⠉⠟⠀⠈⡇⠀⠀⠀⠀⠀⠈⢧⡰⠷⢀⣬⡉⣠⡜⢛⡼⠁
⠀⠀⠀⢰⡒⡕⠒⠦⣀⡀⢀⣠⠴⠒⢾⢒⡆⠀⠀⢶⢾⡇⢀⡤⠙⠂⠀⡖⣦⠀⢀⠀⠀⢸⣶⡆⠀⠀⠀⣶⠏⠉⠒⠤⢭⡤⠽⠛⠉⠹⣶
⠀⠀⠀⠀⠙⠁⠀⠀⢀⡼⢯⡀⠀⠀⠈⠋⠀⠀⠀⠀⠀⢿⠿⠒⠀⠀⡀⠉⠁⠤⠤⣿⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡦⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡀⠀⢀⣧⡜⠀⠀⠀⣠⣞⠀⠀⠀⠀⠀⠀⠀⢠⡔⢲⠀⠀⠀⢈⡧⡞
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡇⠀⠀⠀⢠⣄⠀⠀⠀⠀⠙⠋⠉⠓⠪⢿⣥⠤⠖⠋⠉⠻⠃⠀⠀⠀⠀⠀⠀⠈⠙⣯⠓⢚⠽⢻⡄⠙⠲⣄⣀⣟⢳⡀
⠀⠀⠀⠀⠀⠀⠘⣦⡤⠖⢚⣓⠒⠦⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡼⠁⠀⠈⢻⠛⡄⠀⠀⠀⠡⡏⠉
⠀⠀⢀⣄⠀⠀⡔⠁⠸⠿⢈⡛⠁⢯⠎⣳⡄⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠰⣇⣷⣚⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣷
⠀⠀⠈⠈⠉⣻⠸⠷⢠⣢⠘⡊⠠⣖⡜⠛⢻⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⣏⣀⠀⠀⠀⠀⠀⠈⠀⢸⡀⠀⠀⢠⢀⡀⠀⠐⠢⡟⢹⣏⣶⠋⡇
⠀⠀⠀⠀⠀⢧⣴⡦⣠⣅⠘⣟⢰⣦⠸⠿⢸⠀⠀⠀⠀⠀⠀⠰⣟⢦⡴⠚⠉⠉⠀⠈⠉⠙⠳⣤⠜⡳⠀⢀⣈⡇⠀⢾⡛⣎⠀⠀⠀⠀⠙⡹⠃⠀⠉⠁
⠀⠀⠘⠖⠊⠙⢆⢠⣬⠁⣘⠿⢀⡐⠿⢢⠟⠙⠲⠞⠀⠀⠀⠀⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢧⣹⠏⠙⠦⣍⡁⢀⣀⣀⣤⣸⣅⡀
⠀⠀⠀⠀⠀⠀⠈⡳⢭⣀⣻⢃⣸⣿⢶⡋⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣠⠷⡏⠉⠀⠀⠿⠤⠃
⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⠉⡏⠀⠀⠀⢹⠆⠀⠀⠀⠀⣶⣒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⡆⠀⠀⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⣠⠤⣄⠀⠀⠀⠀⠈⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡊⣷⠀⢀⣀⣨⢲⡋⠀⠀⠀⢀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣤⠀⢲⠆⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡏⠉⢩⠛⣳⠀⠉⠓⢤⡴⣋⣱⠀⢠⡜⡤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⣘⢤⠀⠀⣀⠐⢫⡤⠞⠒⣛⡓⠚⠦⣬⠓⢀
⠀⠀⠀⠀⠀⢀⣤⡀⢠⠏⠀⠀⠈⢻⠱⠄⠀⠀⠀⢸⡁⠀⠀⠀⠉⠀⠀⠉⠓⠲⡄⡴⠒⠊⠁⠀⠈⠉⠀⡀⠙⡶⠁⠰⡽⠀⣙⠋⠰⣷⠎⣳⡞⠃
⠀⠀⠀⠀⠀⠸⣤⠽⡅⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⣇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠐⠋⣾⢱⣳⢀⡤⡄⠳⠃⢠⢤⡸⠽⢻⠞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡴⣠⠀⠀⠘⠲⣋⣽⡬⢏⣩⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⣇⣠⣄⢀⣉⠀⢯⠆⡬⡍⢴⣲⢸⠷⠾
⠀⠀⠀⠀⠀⠀⠀⣠⡼⣃⡀⢯⣙⡇⠀⠀⠀⠀⢉⠞⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡶⢻⡙⠋⣙⠚⠀⢶⡦⠙⢡⢦⠁⡼⢦⡆
⠀⠀⠀⠀⠀⠀⠈⠧⡼⠁⠙⠲⢄⠀⣀⣠⠤⠤⡸⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠛⢌⡻⠃⣔⣦⠀⣖⡆⣩⠞⢧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠹⡄⠀⠀⠀⠣⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠉⡶⠖⣤⠴⢾⡋⢹⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢠⡀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⡀⠀⠀⢹⠂⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠨⠃⠀⠀⡘⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠊⠐⡑⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠐⠠⠀⠀⠀⠀⢠⠊⠈⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠤⡄⠈⡄⠀⡐⡑⠀⠀⡠⢈⠊⡁⡈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⢀⠄⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠻⢰⠀⡆⢐⠄⠀⢀⠊⠀⠘⣸⢧⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡐⠁⡠⠂⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠒⠒⢺⠀⠟⡟⠀⠠⠋⠉⠉⣉⡽⠋⠀⠀⠘⠲⠲⠤⠤⢄⣀⠀⠀⠀⡀⠊⠀⠐⠒⠈⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⢠⠁⠀⢠⠀⠉⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢉⠳⠂⠀⠀⠀⢀⠐⠊⠉
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠄⠾⡀⡆⣇⠐⠂⢲⠃⠀⠚⠉⠀⢀⡁⠀⠒⠀⠠⢀⡀⠐⠁⠀⠀⠀⠠⢐⠵⡁
⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠀⠀⡠⢤⡀⠀⢷⡁⠸⣆⠀⣟⠐⠁⠈⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⢀⢀⡊⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠐⡁⠀⠀⠈⣁⠀⠀⠀⠀⠀⠡⠤⡙⠀⠙⠀⠋⣀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠓⢀⠠⡲⠃⠁⠀⠑⣄⣠⢿
⠀⠀⠀⠀⠀⠀⢯⠛⠶⣧⣀⠀⠀⠁⣀⠀⠀⠀⠁⠈⠁⠤⠐⠖⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⣀⣤⠶⠛⠁⡇
⠀⠀⠀⠀⠀⠀⠘⡆⢀⡴⠟⠛⡿⠶⣦⣤⣤⣄⣀⣈⣂⣀⠀⠀⠀⠐⠘⠂⠀⠀⠀⠀⠁⢀⣓⣀⣀⣀⣤⣤⠴⠶⡛⢟⣉⣀⠀⠘⡺
⠀⠀⠀⠀⠀⠀⠀⠘⣾⠂⠄⠀⡔⢠⢀⡄⠁⠈⡍⢩⠏⡭⢿⠿⡛⠛⡻⡛⣟⠛⠛⣛⠛⠛⠻⡝⢉⡩⠋⡄⣀⠴⣠⠊⠈⡉⢳⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠛⠚⡃⠀⣀⠀⢧⢠⠖⠇⣺⠈⣤⠄⢰⣠⡆⠘⡀⣑⣤⢀⡡⣄⣈⢧⣄⣐⡃⣴⣡⡔⠹⣔⣀⣨⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡡⠋⢀⡈⠙⡿⢯⡂⡬⠘⢶⡭⢐⣉⣆⡑⣾⠅⣆⣰⡾⢀⢀⠈⣺⠀⢃⣰⡟⠋⠉⠑⠬⡅⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠁⠸⢰⠘⣿⠗⡊⠱⣴⡋⣐⠀⠘⡞⢻⡿⡑⢄⡀⣀⣰⠧⢻⣾⠿⡄⡠⠂⠑⠀⣹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢲⡂⢏⢉⣩⣾⣆⡚⣴⣿⣇⡡⣄⡴⠁⢾⣿⠈⣨⠁⢘⠀⠁⢐⣏⠼⠒⡻⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⢨⠻⠺⢷⣌⣵⣠⡎⢫⢁⣾⣏⡰⠧⣺⣞⡴⠟⠛⡒⠈⢹⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢎⡘⠢⠸⠙⡋⢻⣷⠙⣿⡾⡟⢫⣀⣴⡿⠙⢲⢄⢀⡦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢦⣦⣭⣾⣿⠉⢹⡇⢡⣾⠟⢛⣓⣤⡽⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣉⣙⠛⠛⠛⠛⠛⠛⠛⣉⠉⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⡿⣿⢟⠿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣛⣛⣿⣛⣛⣟⣛⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⡀⠀⠀⠀⠀⠀⣰⣿⠟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣰⣿⠃⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣿⡇⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⣿
⣤⣄⠀⠀⠠⣤⡄⠀⠀⢠⣤⡀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣦⣤⣤⣾⡿
⢘⣿⡆⠀⠀⢹⣿⠀⠀⠈⣿⣷⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠉⠉⠁
⣾⡟⠁⠀⢠⣿⡏⠀⠀⢰⣿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣀⣀⡀
⢻⣿⡄⠀⠀⢿⣷⠀⠀⠘⣿⣆⠀⣿⡇⠀⠀⠀⠀⠀⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠷⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡄
⣼⣿⠃⠀⢀⣾⡿⠀⠀⢠⣿⠏⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣽⣿⡿⠗⠀⠀⠀⠐⠿⣿⣿⣭⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠉⠁⠀⠀⠀⠉⠀⠀⠀⠈⠉⠀⠀⣿⡇⠀⠀⠀⠀⠐⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠿⢿⣷⣶⣶⣶⣶⣾⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣯⣍⠛⠻⣇
⠀⠀⠀⠀⠀⠀⠀⣶⣦⠀⣀⡤⠟⠀⠀⠚⠛⠛⠛⣻⣿⣿⣿⣿⣿⡀⣀⣀⠀⠀⠀⣀⣤⠀⣰⣿⣿⣿⣿⣟⣛⠛⠛⠛⠁⠀⠀⠀⠀⠀⣴⣶
⠀⠀⠰⠷⠀⢀⣀⠛⠋⠀⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠉⠉⣀⠀⠠⡶
⠀⠀⠀⠀⠀⠜⠋⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⠛⠛⢩⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣏⠉⠛⢿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠙⠄
⠀⠀⠐⠿⠂⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠉⠁⠰⠶⠒⠋⠁⠈⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⠁⠈⠉⠀⠉⠛⠒⠶⠂⠀⠉⠻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠾⡇
⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢋⣼⠟⠁⠆⠀⢹⣿⡗⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⡀⢰⣿⡏⠀⠐⠈⠛⠯⠓⠀⠀⠀⠂⠀⠀⢠
⢀⣿⠇⠀⠀⠀⠀⢸⣦⠀⠋⢁⡀⢤⣤⣶⠟⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣿⣿⣿⣶⣿⡷⣶⣤⡤⣠⠀⠀⢀⣶⣯⠀⠀⠀⢸⣷
⠈⡿⠀⠀⠀⠀⠀⢸⣿⣷⣄⣾⣟⣸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣇⣿⣇⣤⣿⣿⡏⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠁⠈⠙⠛⠻⣿⣿⣿⣿⣿⠏⠛⠛⠛⠉⢉⣀⣢⡀⠀⠀⣠⡂⣀⠈⠙⠋⠉⠻⣿⢿⣿⣿⡿⠟⠛⠉⠀⠀⠈⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠿⠀⠸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠲⠶⣖⢿⣿⣿⣿⣿⣿⣤⣶⡾⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⠀⠛⢛⣛⡫⣽⣿⣯⡿⣥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠿⠿⣡⣾⣏⣩⠿⠞⠦⠍⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣷⠄⡈⠀⠀⡢⠠⣤⣤⣀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡠⠟⣿⡿⣷⣾⣶⠶⠂⠀⠉⠉⠉⠓⠶⠄⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠆⢈⡙⠺⢿⣷⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢓⡘⣿⣿⣶⣿⢻⣯⣿⣽⣷⡄⠀⠀⠀⢀⡐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠘⠀⡏⣼⣿⣿⢠⣿⡽⣿⢟⣿⣀⠀⢲⣜⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⣶⣧⣽⢿⣿⣶⣿⣿⣾⣿⣿⣾⣷⣿⠿⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⢿⣿⣿⣿⡇⠈⠉⠉⠀⢠⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⣾⣿⣿⡿⣫⣾⣿⣿⣿⡇⠀⠀⠀⢀⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣁⣿⣿⣧⣿⣿⡇⠀⠀⠀⢸⠀⠘⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⠀⠀⡸⢀⣠⣄⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⢳⣿⣿⡟⠁⢋⣠⣿⣿⡇⠀⠀⢙⠊⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠈⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⣿⠟⣻⣿⣽⣿⢿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣩⠖⣾⣿⠉⠿⢀⣾⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⣁⣠⣴⣿⣿⣹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣶⣾⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⣿⠏⣿⣇⣿⣿⣯⣿⣿⣟⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢘⣿⣞⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⡻⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣄⣍⣛⣽⣿⣿⣿⣦⣽⣿⣧⣿⣿⣿⣿⡿⠟⠀⠉⠀⠀⠀⡀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠘⢻⡏⣿⡿⡿⢹⣿⡿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⡌⠀⠀⠀⡀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣍⠊⡼⢳⣿⡇⣷⣾⣿⣧⢽⣿⣿⣿⣿⠀⠀⠀⠀⡼⣿⡃⣶⣶⠙⢿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣭⣻⣥⣾⣋⣿⣷⣿⣿⣟⣏⡻⣿⣿⡟⠀⠀⠀⠀⢑⠈⢃⣝⣛⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⠛⠟⠟⠛⠫⠭⣭⠽⠿⠏⠉⠏⠛⠻⠿⠿⠿⠉⠛⠛⠶⠾⠿⠟⠻⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⢀⣀⣤⣤⣼⣧⣤⣤⣀⡀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣴⠾⠛⠉⠉⠀⣀⣤⣄⠈⠉⠛⠷⣦⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣴⠶⣦⡀⠀⣿⣁⣽⠇⢀⣴⢶⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡂⠀⠀⠀⠀⢀⣾⠏⠀⠀⠈⠿⣶⠿⠁⠀⣈⣉⠁⠀⠸⣧⣤⡿⠀⠀⠹⣧⡀⠀⠀⠀⠀⢐⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠻⠶⣦⣤⣾⠃⣤⡶⣦⠀⠀⠀⠀⠀⢾⡏⣹⡇⠀⠀⠀⠀⠀⢰⡟⢻⣿⣷⣤⣴⠶⠟⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢿⣤⡿⠃⣴⡟⢻⡆⠈⠛⠋⠁⠀⢰⡟⠛⣷⠘⠻⠞⠃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠘⠻⠟⠁⣰⡞⢷⡄⠀⠈⠻⠾⠋⣠⣤⣄⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣰⡟⢻⡆⢀⣤⣤⣄⠀⠙⠷⠾⠃⣰⡟⢻⣆⠀⢿⣅⣽⠇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣸⣧⠘⠷⠾⠃⠸⣧⣠⡿⠀⠀⣴⠶⣦⡘⠷⠾⠋⣀⣀⡉⠁⠀⣼⣧⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡶⠟⠛⠉⠉⠹⣦⠀⠀⢀⣤⣬⡉⠀⠀⠀⢿⣤⡿⠃⠀⠀⢸⣏⣹⡷⠀⣼⠟⠈⠉⠛⠻⢶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣧⡀⢺⣧⣼⡗⠀⣠⣤⣤⡀⠀⢀⣤⣤⡀⠙⠋⢁⣼⠏⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣌⠁⠀⠀⢿⣤⣼⠇⠀⢺⣧⣼⡟⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠙⠻⠶⣦⣤⣬⣀⣀⣠⣤⣼⠿⠟⠋⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡿⠁⠀⠀⠀⠀⠀⠀⠉⢹⡏⠉⠀⠀⠀⠀⠀⠀⠈⢿⣄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉


⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⡀
⣿⣏⢭⣟⡛⣛⣛⠛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠝⠐⢠⡇⢀⣆⠀⢸⡇⠉⠀⠀⣿⠁⣽⠀⠀⠛⠁⠀⢀⣀⠀⡀⠶⢦⠀⡿⡏⢼⠆⠘⣛⠀⠀⠀⣤⠀⣀⡀
⡀⡿⠀⢾⡇⢸⣯⠀⢨⡁⠀⡀⠁⠭⠀⠧⠀⣶⣆⠀⠀⠘⢟⣀⠀⡤⠤⠰⠏⠀⠘⠀⢸⣿⠀⣿⠁⢉⠀⠿⠁
⢃⡁⠐⠲⠁⣸⢹⡆⠈⠁⣀⠀⠀⠀⣿⠀⠀⠉⠁⠀⣿⠀⠈⠁⠈⡗⠀⠀⠀⠰⠄⠀⠀⢹⠀⢀⠀⢰
⠂⠀⠀⠀⢱⡿⠰⠇⠖⠀⠿⣰⣦⠀⠛⠸⠇⠀⠾⠗⠛⠀⠀⣠⡀⠀⠀⠀⠀⠀⠂⠀⠀⣾⠀⠈⢠⡾⡇
⠘⠃⢀⠀⠀⠀⠄⢼⣿⠀⡐⠿⠿⠀⡤⠀⣶⠐⠓⠀⣸⠀⠀⠈⠓⠄⠀⠀⠀⢠⣤⠀⠀⠖⠀⠿⠀⠀⠰⣿
⠀⠀⠂⠀⠀⠀⠀⠸⠻⢰⣧⠀⠀⠀⡷⠀⣿⠀⠲⠂⠋⠀⣼⡄⢸⣿⠀⠐⣵⣸⢹⠠⠀⠰⠶⠀⠀⣭⠀⠉⣰
⣰⠀⣿⠀⠄⠻⠀⠰⠐⠈⠉⠀⠀⠀⠁⠀⠀⡄⣶⠀⡄⡇⢹⠇⠀⡿⠀⣸⠈⠀⢠⣠⡄⠑⣶⣶⣀⡉⠀⠰⠟
⡉⠀⣿⢈⡀⠀⢰⣿⠆⠀⠀⠀⠀⢷⠀⠀⠀⠀⢀⠀⠀⣤⢀⡀⠀⠂⠀⠈⠰⡄⢺⡇⠁⠐⡿⣿⠀⠀⠀⣯⡄
⠀⠀⠅⠸⠇⠀⠸⠏⡄⠐⠛⢤⡤⠀⣀⡄⢰⣿⡏⣤⠀⡟⠃⠀⡀⠀⣸⡀⡐⡗⠘⠀⢠⣤⠡⠸⡀⠀⠀⣿⠉
⠀⢀⢀⡈⣤⠀⣤⠀⠀⠀⠀⢸⡇⡎⠹⠳⠠⠃⠀⣿⢀⣤⡆⠀⣿⠀⢹⡇⠉⠛⠐⠀⢸⡟⠂⠀⡇⠀⠀⠁⡀
⠀⣸⠀⠀⣿⠀⠏⠂⣀⠀⣀⣀⠀⡀⠈⠈⣴⠀⠰⠋⠀⢈⢀⠀⡒⠀⠸⠇⠀⠀⢀⠀⠈⣀⣀⢸⡇⠀⠈⠘⠛

⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⢠⠖⣆⣀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡠⠄⡛⠀⠀⠀⡖
⠀⠀⠈⢹⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠫⢀⠇⠰⠁⡈⠁
⠀⡖⠦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠤⣔⠛⠓⠀⠀⣴⠂⠀⠀⣀⣴⣋⡃⠺⢨⡇⣸⢍⡅⠀⣠⠶
⠀⠓⠋⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠙⠃⠀⠰⡇⠈⠀⠀⠀⠈⠈⢟⢂⠗⠀⡇⠓⣴⠁
⠀⠀⢠⠼⡠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠘⠺⠁⣠⠄⠀⠀⠀⢀⡴⠄⠀⠀⢸⠻⢄⣻⣀⠫⠀⠰⠤⡄
⠀⠀⠈⠓⠃⠀⢈⠂⠄⠀⠀⠀⠀⠀⠈⠉⣧⠾⠀⠀⡈⠀⠀⠾⠛⠂⠀⠀⠀⠀⠉⠀⠀⠀⢳⢄⠤
⠀⠀⠀⠀⠀⠀⠀⠉⢰⠀⠀⠀⠀⠀⠀⠀⠨⣦⣀⠀⢳⡇⠀⠀⢴⢤⠀⠀⠀⠀⠀⣎⡹⣀⣀⣤
⠀⠀⠀⠀⠈⢦⣀⠤⠼⠀⠀⢠⠏⠀⠀⠀⠈⠁⠀⠉⠹⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⢋⣈
⠀⠀⣀⣀⣠⠊⠶⢀⡇⠰⡆⣁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⣋⣓⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢨⡟⠃⠲⢈⡇⢰⡆⡃⠀⠉⠁⠀⠀⠀⣀⡀⢀⣀⠠⠇⠄⠀⠀⠀⠀⠉⠈⡇⠀⠀⢆⡀⠀⠘⢊⣸
⠀⠀⡠⠴⣟⢀⡮⠀⣗⠛⣠⠃⠠⢄⣀⠀⠀⠀⢳⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢫⢄⡣⠤
⠀⠀⠀⠀⠈⣺⣂⠺⠀⠾⠈⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢈⡇⠄
⠀⠀⠀⠀⠰⠁⠀⠀⡏⠀⠀⠈⠦⠀⠀⠀⣞⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠤⡉⠀⠀⡆⠰⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠙⣦⠑⢲⠞⠅⠀⠀⠀⠀⠀⠀⡹⢂⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢢⡨⠗⢘⠂⠼⢀⡆
⠀⠀⠀⠀⡤⢄⡰⠁⠀⠀⠆⠂⠀⠀⠀⠀⠀⠀⠑⠁⠉⠒⠲⡄⠀⠀⠀⠀⠀⢀⣈⡵⠁⠮⠀⡳⠀⠆⢀⡀⠚⢀
⠀⠀⠀⠀⠳⠖⡆⠀⠀⠀⠀⠀⠀⡰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣸⠙⠅⢶⠀⡥⠀⣆⢘⠂⠈⢉
⠀⠀⠀⠀⠀⣀⣹⠀⢄⠘⠂⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡾⠄⣆⠀⡥⠸⢅⡙⠀⢀⣉
⠀⠀⠀⠀⠀⠣⡰⠋⠪⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠳⣜⠅⢰⠈⢠⠊⠁⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢹⠒⢶⠐⣇⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⡞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⢀⡾⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⢠⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢾⡉⠻⣦⡀⠀⠀⠀⠀⠀⢀⣠⣼⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⡀⠀⠀⠀⠀⠀⢀⣴⠟⢉⡷
⠀⠀⠀⠀⠀⠙⢦⡈⠻⣦⡀⢀⣴⣾⣿⣿⣿⣀⣈⣉⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⡉⠉⣉⣁⣀⣿⣿⣿⣷⣦⡀⢀⣴⠟⣁⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⠟⢹⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠙⢿⣿⣿⣿⡏⠻⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠛⢁⡄⢸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡇⢠⡈⠛⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣴⣿⡇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢸⣿⣦⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⡄
⢀⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⠉⢻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟⠉⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤
⢿⣅⣤⣠⣤⣄⣤⣄⠙⣿⠇⣰⡄⢻⣿⣿⣿⣿⣿⣿⣿⠋⢸⣿⣿⣿⣿⡟⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣆⠸⣿⠋⣠⣤⣤⣤⣤⣤⣨
⠀⠀⠀⠀⠀⠀⣿⣿⣧⠈⢀⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⣼⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⡿⠟⠁⢀⣤⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⣤⣀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⢿⠛⠉⠉⠀⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⠂⠀⠀⢨⡿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠀⣺⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⢸⡧⠀⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⣼⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⡇⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⠗⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣄⣀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣤⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠶⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠶⠾⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠰⠖⠆⠰⠀⠠⠤⢀⣀⠀⠀⣠⡄⢤⠄⢤⠀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣋⣀⣤⣤⣤⣤⣧⣤⣤⣠⣀⣀⣀⠀⠉⠑⠓⠿⡞⢿⡛⣷⠺⣶⠷⠶⡄
⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⡦⣤⣀⠀⣝⣯⣙⣏⣙⣟⢻⡛⣄
⠠⡤⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣸⣿⣿⣟⣿⣿⣿⢏⠿⠂⠙⣷⣿⣿⣿⣭⣬⣽⣏⣹⣉⣆
⢂⠣⠀⠀⠀⠈⠙⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣴⣶⣮⣶⣴⢟⠟⠋⢻⣿⣴⣤⠼⣦⣽⣌⣦
⠈⢆⠠⠉⠌⠁⠀⠈⠿⣆⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⣯⡷⠣⢊⣤⣾⣿⣿⣿⣷⣾⣦⠬⣤⣵⡀
⠀⠈⢢⠐⠀⠆⠀⠤⠠⣼⣷⣤⡀⠀⠀⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⢁⣴⣷⠿⠂⠀⡋⠀⠀⠻⣿⣿⡿⣿⠿⣦⠤⢦⡼⣄
⠀⠀⠀⠈⠉⠓⠻⣿⡛⠛⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣯⣄⣀⣀⣀⣹⣆⠀⣚⣛⣭⡼⠿⡦⠼⢦⠤⢦⡬⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠳⡤⠀⠀⠀⠀⠀⠈⢉⣠⣶⣶⠾⠿⠛⠛⠛⠉⠭⢚⡩⠔⠋⠀⠀⢡⠿⣖⣻⣗⢻⣖⠚⢶⠖⢶⡖⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⣾⣻⣿⠿⠿⠉⠀⠀⠀⠠⠤⠤⠐⠒⠈⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇
⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⡤⠤⠤⠤⠤⠤⠤⠤⡤⢤⢤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣿⣶⣷⣶⣿⣿
⢰⣶⣿⣭⣷⣶⣾⣷⣿⣿⡷⢿⣟⣑⠦⡤⠙⣗⠀⠸⢷⣤⡈⠳⢠⣈⠿⠁⢄⣿⣅⣾⣿⣯⣿⡿⣿⣿⣿⣿⣿⣿⣿
⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣄⣀⠙⠳⢤⣀⣻⣷⡴⠮⢀⣆⣷⣿⣿⣿⣛⣿⣿⡿⢿⣿⣟⣿⣿⣿⣿⣿
⢸⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢉⡻⢿⣦⣀⣴⣿⣿⣿⣿⣻⣿⣿⣇⣰⣿⠿⢛⣻⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣝⣛⠿⢿⣦⣍⣯⢿⣿⣿⣿⣮⣿⣿⣿⣯⣽⣿⣿⣿⣿⡏⣩⣿⣷⣶⣾⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣷⣾⣽⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠋⠛⠙⠉⠉⠉⠉⢋⠿⠉⢙⣛⡛⣙⡛⡛⣋⣉⣛⣿⣏⡙⠛⢋⣿⡯⠵⣴⣿⣿⣿⣿⣿⣿⣿⣟⣿⠿⠉⠉⠛⠻
⣰⣶⣾⣿⣿⣿⣿⣿⣿⡿⢟⠉⠽⡯⠶⠲⣿⣿⠋⠉⠉⡉⠌⠉⠙⡿⣶⣷⣶⣄⣀⡉⠙⠛⠛⠻⠿⣿⣿⣷⣶⣶⣶
⣿⣿⣿⣿⢟⠫⠙⣨⡿⣺⣩⣖⡳⠾⢾⣿⣿⡟⢮⣵⠾⠁⠀⡀⠀⠘⠉⣈⠿⣿⣿⣿⣷⣓⣾⠷⣶⣶⣭⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣶⠖⠢⣀⣀⣶⠲⠼⢷⡾⠥⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⣤⡾⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⣒⡄⠀⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⢀⡴⢩⠏⠀⠀⢰⢀⡼⢇⡄⠀⠀⠀⠀⠀⠈⢾⠁⠘⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠼⢀⡴⠴⠻⠊⠀⡟⢡⡷⣠⡄⠀⠀⠀⢾⠀⠀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⢸⢳⡀⠀⢀⣤⡖⠀⠀⢀⠀⠈⠑⢤⡄⠀⡾⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣟⣲⣾⣾⡇⠀⠀⠀⣔⡀⠀⠺⠣⠄⠀⠀⠘⠦⠧⡄⠀⠀⠉⠳⡆⠀⠀⢀⣤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠷⣄⣈⠉⠙⡿⢷⡶⠶⠾⣥⠀⠀⠀⠘⠷⠀⣠⠀⣰⠃⢀⡴⠋⢱⠇⠀⠀⣶⠘⠛⠛⢹⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢉⡟⣋⠉⠛⠶⣄⠀⠉⠙⠒⠷⡀⠀⠀⠀⠀⠓⠀⠀⢀⣠⠞⠛⠋⠁⡴⠋⠀⣰⠋⠀⠀⠀⢻⠀⠀⠀⢸⠀⠀⡟⠛⢻
⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠘⢧⣏⣧⣄⡶⠼⣶⣶⣒⠦⣤⠞⠲⢤⣤⣀⣤⠴⢾⡋⠀⠀⠀⠀⡼⠁⢀⠞⠁⠀⠀⠀⠀⢸⡀⠀⠀⡟⠀⣈⡗⠒⢾
⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠙⢦⣄⠀⠀⠠⢯⡀⠀⠀⠉⠓⠦⣤⣘⡡⠖⠉⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⢰⠃
⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠴⣿⠋⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠒⢦⡀⠸⣄⠀⠀⣀
⠞⠁⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⣿⡴⢀⡟⠒⠒⠒⠲⠦⣄⠀⠀⠀⢀⡘⢟⣦⣖⠽⠃⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋⠀⠀⠀⠀⣠⠀⠘⣦⠀⠀⠾⠱⠆⠹⠷
⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⣇
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢤⣤⡀⠀⣀⠠⣠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠉⠀⠀⢀⣀⠉⠁⢸⠿⠆⣸⡛⠓⢶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⠾⣟⠀⠀⠀⠙⢋⣤⡰⠃⢀⣠⠿⢁⠀⠀⠈⣹⡯⢷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠋⠀⣷⡀⠀⢀⣀⣸⣇⠀⠐⠋⠁⠀⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⣟⣀⡀⢈⣿⡿⣳⡄⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠠⡐⠗⣴⣿⣿⠿⢿⣯⠀⠀⠁⠈⠙⢃⡴⠁⢀⡿⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠈⠻⢿⡶⣷⣤⡄⠘⢀⣀⣼⣇⡉⠁⣀⣏⣀⠀⡀⣠⡶⠊⠀⣠⣿⣀⣰⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠛⠃⠀⠀⠙⢶⣟⠉⠀⠀⠀⢁⡀⢲⣾⣥⣤⡾⢋⣻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣤⣈⣛⣷⣟⣿⣭⣉⠉⠉⠉⠻⠿⠟⢶⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⠻⣿⣿⣶⠾⠟⢋⣭⣷⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣶⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢁⣾⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠖⠒⠒⢀⣀⣐⡒⠒⠒⠦⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⣴⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣩⣥⣶⣶⣞⡃⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⢻⣿⣓⡶⢤⣀⠀⠀⠀⠀⠀⣠⡶⣺⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⣵⣾⣿⣿⡿⠿⠟⠓⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⡷⣮⣝⢦⣀⣠⣬⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣫⣾⣿⡿⠛⠉⣁⡀⠀⣠⡄⠀⠔⠒⠓⠶⢴⢠⣶⡆⠐⠂⠒⠒⠂⠀⠀⠈⠙⠻⣿⣷⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣸⢹⣿⠟⠁⠀⠀⠘⠛⢡⡖⠁⣠⣄⠀⢀⣠⣴⣶⣶⠤⡄⠊⠀⠀⠠⣒⠍⠉⡐⢂⣬⠾⠛⡉⣼
⠀⠀⠀⠀⠀⠀⠀⢿⣖⠍⠀⠀⣀⠁⡀⠀⠘⠇⠘⢩⡤⠑⢦⣽⡟⠙⣾⣄⣉⣱⡆⠀⠈⠿⣦⡄⣉⠿⠖⠀⢘⡵⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠑⢤⠀⠈⠀⠈⠂⠀⢤⡤⠟⠁⠤⢄⠉⠀⠀⣀⣝⡄⠘⢛⣀⡀⠰⠁⠀⠀⢀⣠⣴⡿⢱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⣀⣠⣴⣶⣿⡿⣟⢂⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠶⠶⠶⠖⠒⠚⠿⣿⣿⣿⣿⣿⣿⣿⡋⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣿⣿⣽⣿⣿⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠟⣁⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⣛⣿⠟⠉⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⢰⠠⢄⣀⣀⡀⠀⠀⠀⠀⣈⣛⣉⣭⣿⡟⣩⠀⢀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⡋⠉⠉⠉⠉⠙⠻⣿⣿⠿⣟⣧⣾⣿⣿⣿⡿⠂⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⣶⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⣾⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⢔⣩⣾⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣵⣾⣿⣿⡿⣟⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠶⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣷⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠭⠭⠽⠯⠿⠿⠿⠿⠛⠛⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠄