Olympics Emojis & Symbols

⠀⠀⠀⢀⡤⠖⣛⣉⣉⣉⣛⠲⢤⣀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢚⣋⣉⣉⣛⠒⠦⣄⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣋⣉⣉⣙⡓⠦⣄
⠀⢀⡞⣩⠴⠋⠉⠀⠀⠀⠉⠙⠲⣌⠳⡄⠀⠀⢀⡴⢋⡴⠚⠉⠀⠀⠀⠈⠙⠲⣌⠳⣄⠀⠀⠀⡴⢋⡴⠚⠉⠁⠀⠀⠈⠉⠳⢬⡙⢆
⢠⠏⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⡄⢀⡞⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡘⣆⠀⡼⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⢧
⡟⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢻⣼⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⣼⠃⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘
⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⢸⢩⠭⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⢸⣯⡭⢥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⡘⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣹⣳⠃⡟⠙⣒⡶⢬⡙⢦⡀⠀⠀⠀⠀⣠⠞⣩⢧⠏⡼⠘⣓⣲⢬⣙⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⡟⣸
⠀⢳⡘⢦⡀⠀⠀⠀⠀⢠⠞⣰⢋⡴⢃⡞⠁⠀⠹⣍⠳⣙⢦⣙⣄⠀⢀⡞⣡⢞⡱⢋⡼⠁⠀⠙⣌⠳⣌⠳⣘⣦⠀⠀⠀⠀⠀⣠⠞⣰⠃
⠀⠀⠙⠦⣍⡓⠦⢤⣄⡏⣸⣱⣋⠴⠋⠀⠀⠀⠀⠈⠳⣌⡓⠿⠭⡄⡾⢰⢳⣋⡴⠋⠀⠀⠀⠀⠈⠳⢬⡙⠺⠭⠥⠤⠤⠖⣋⡥⠞⠁
⠀⠀⠀⠀⠈⠉⠓⠚⢻⡇⡇⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣒⣃⡇⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣒⣒⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡏⣇⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡼⠁⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠦⣀⣀⠀⠀⢀⣀⡤⠞⣡⠞⠀⠀⠀⠘⢦⡙⠦⣄⣀⠀⠀⠀⣀⡠⠖⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣭⣉⣉⣩⡥⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⠓⠦⣭⣉⣉⣩⡥⠴⠚⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠿⡏⠟⠰⣶⣿⣾⣿⣷⣿⣄⣤⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⠀⠀⠀⣠⣤⣿
⣿⣷⠄⠴⢸⣿⡿⢟⣿⣿⡟⠱⢦⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠈⠁⠀⠀⢠⣿⣿⣿
⣿⠉⠀⢠⣾⣿⠀⠀⠙⠿⣶⠀⣻⣿⣷⢀⣤⠖⠀⠀⠀⣀⣤⣴⡆⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠒⣿⣿⠙⠁
⠉⣐⣖⣄⠛⠛⠀⢀⡴⣂⡀⠙⠛⣿⣿⣾⢥⣶⣷⣿⣾⣯⣻⣾⣧⣸⢀⣇⣱⣀⡇⢸⢸⣧⣟⢸⠀⣶⣶⠀⠁⠁⠀⠀⠀⢿⡀⠀⡀⣴⠾⠇⠀⣼⠟
⣿⣿⣿⢿⣿⣿⣿⣿⣷⣿⣿⢶⣒⠓⢿⣿⣧⡙⠻⣿⣿⣿⢭⣄⣿⣷⣿⣿⣿⣷⣷⣿⣾⡅⣽⢲⣾⣿⣯⣶⣂⠰⠦⣶⣶⢚⡳⢬⣁⣨⣱⡀⣠⢤⣤⡴
⣿⣿⣿⣿⡟⠿⡿⠋⠑⠛⠋⠹⠟⢂⡎⠻⠋⡏⠲⢌⣙⡃⣾⠇⠿⠉⠈⠉⠉⠉⠙⠋⣷⣾⠋⢉⣘⠛⢿⣾⣿⣷⠐⡾⢿⡿⠿⠎⠳⠀⠙⠷⠿⣿⣭⣄⣚
⣨⡿⣿⣿⣧⣼⡇⣶⡻⠀⠆⠀⠀⠘⣧⣰⡦⡔⠁⠀⠉⠳⣾⠁⠀⢰⡽⣦⣀⣀⣀⢘⣿⣇⣀⢠⣿⣦⣿⣿⢿⡿⠃⠀⢀⠀⠶⠂⠀⠱⢤⣶⠀⠹⣯⢹⠿
⠽⣿⣤⣿⣿⠋⠀⠈⠃⢿⠀⠀⠀⠀⢘⣈⣷⣾⠃⠀⠀⠀⠙⢿⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣧⠹⠶⣤⠾⠐⠂⠀⠀⠀⣾⣿⣧⡖⠫⡄
⠀⣾⠁⢀⣿⡄⣶⣴⣿⣾⣇⠀⠀⠀⠈⠀⠡⠀⠀⠀⠀⠀⠀⠀⠉⣻⡿⢿⣿⣿⣿⣿⣿⠋⠀⣤⣅⣈⣿⣿⣿⠆⠉⠀⠀⠡⢄⢀⣰⣾⢿⣿⡟⣿⣆⢸⣦
⠀⣿⣆⣈⢙⣿⣿⣿⣿⣿⡇⣀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣵⡦⠬⠟⠙⠻⣿⣷⣼⢟⣿⠟⠛⠟⠁⠀⠀⠀⠀⠀⠠⣾⣿⢾⣷⡘⢿⣿⡟⢻⣷
⠙⢥⣽⣶⣶⣿⡶⣤⡿⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠋⠀⠀⠀⠀⠀⠈⠉⠉⣸⣿⣤⠤⠄⠤⠄⠀⣄⣠⣄⣴⠻⣿⣾⣯⣷⠌⠛⡷⠀⠉
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⠀⠀⠉⢛⣿⣿⣶⣹⠟⠀⠀⠀⠀⠀⠀⠀⣄⠀⣙⣲⠈⠉⢡⡁⠀⠀⢡⣩⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣌⣿⣷⣖⠺⣿⣿⣿⣏⠔⠏⠀⠀⢀⣿⣶⣶⡆⠰⣿⡀⠿⠉⠁⠀⢸⣿⡆⠀⢸⣿⡇
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⣿⣿⠿⠏⢹⣻⠟⠈⠉⠛⠛⠒⠀⠀⢘⣿⣷⣿⣿⣿⣄⠈⣀⣤⣶⣀⡀⢸⣿⣷⠀⠘⢿⡷
⠀⠀⠀⡀⢀⣠⣤⣤⡤⢴⣶⠾⠿⠿⠿⠹⠿⠿⠿⣿⣿⣿⣿⣭⣤⣤⣄⠛⠀⠀⠀⠀⠀⠀⠄⣴⣾⣿⣿⣏⣁⣠⣠⡈⣿⣿⡿⠛⢿⣿⠚⣿⣿⣆⠀⠈⣇
⢸⡏⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣼⣿⡿⠛⠉⠁⠀⠀⠈
⠘⠀⠀⠐⠻⢶⢶⡶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⡏⠉⠉⠀⠀⠀⠀⠂
⣾⣦⣴⣮⣷⣴⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣤⣤⣤⣶⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠐⠈⢂⡀⠀⣴⠀⠀⠀⠀⠨⠅⠀⢰⡄⡀⠀⠀⠀⠀⠀⣄⣹⠆⠀⠀⠀⠀⠀⢀⣤⣷⡶⠶⠶⠀⠀⠀⠀⠀⠀⣠⣧⡄⠀⠀⠀⠀⠀⠀⠀⣄⡆⣿⡀
⠀⠀⠀⠈⠀⢣⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⠀⠀⠀⠀⣠⠞⠻⠇⣀⠀⠀⠀⠀⠘⠦⠘⠫⣶⡄⠀⠀⠀⠀⠀⠀⠙⠚⠷⠄⠀⠀⠀⠀⢀⣴⡿⣠⡛⢹
⠀⠀⡀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⠀⠀⠀⠀⠁⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⠀⠀⠀⠀⠀⠀⠀⠸⢧⡀⠀⠀⠀⠘⠀⢢⣻⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠐⢣⣤⣤⡄⠀⠀⠀⠀⠀⠴⢴⢿⡷⠂⠀⠀⠀⠀⢀⣨⠿⣏⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡃⠀⠀⠀⠀⠲⣤⣦⢄⡀⠀⠀⠀⠀⠀⣠⣠⣞⣦
⠀⠀⠀⢈⡷⣫⠾⣇⡀⠀⠀⠀⠠⠞⠙⠄⠀⠀⠀⠀⠀⠀⠀⠰⠚⠳⣄⣀⠀⠀⠀⠀⢀⣤⣴⣾⠫⡟⠀⠀⠀⠀⠀⡇⠰⡋⢳⣀⠀⠀⠀⠙⠶⠋⢉⣵⡇
⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⣊⡠⠤⠒⠉⠁⠀⠀⠀⠠⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢦


⠀⠀⠙⢲⡖⢲⡄⠀⠀⠀⠀⠀⠀⣠⡴⣶⣀⠀⠀⠀⠀⠀⠠⠤⢤⣼⠦⠀⠀⠀⠀⠀⠠⣤⣔⣦⣦⣀⠀⠀⠀⠀⠀⣀⠐⢀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠋⡠⠊
⠀⠀⠀⠈⠙⠳⡇⠀⠀⠀⠀⠠⢮⠉⠉⠁⡩⣄⠀⠀⠀⠀⠀⠀⢀⠛⡀⠀⠀⠀⠀⠀⢀⣀⢨⣄⣀⠉⠀⠀⠀⠠⠞⠁⡦⡆⠱⠄⠀⠀⠀⠉⠉⠙⠛⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠇⠈⠂⠀⠀⠀⢀⠔⠁⠀⢹⡆⠀⠀⠀⠀⠈⠛⠓⠋⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣀⣤⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡴⠛⠉⣸⡇⠀⠀⠀⠀⠀⣄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠰⠤⣦⣀⠀⠀⠀⠀⠀⢀⣀⣴⣋⢀⡴⠂⠀⠀⠀⠀⠰⣦⣼⣤⠀⠀⠀⠀⠀⠀⠀⠀⢦⡼
⠀⠸⣧⣀⣴⠛⠽⣾⡇⠀⠀⢨⠿⠻⠏⢩⠿⠻⠆⠀⠀⠀⠐⠒⠉⣩⠶⣄⠀⠀⠀⠈⠁⠩⠵⠿⢖⡤⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⠀⠀⠠⠼⢛⡄
⠀⠀⠈⣛⣧⡴⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠀⢀⣤⠞⡷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠳⣄
⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡭⡿⠟⠿⠿⠿⠿⠟⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣿⣿⡟⢀⣿⣧⠀⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⣠⣾⣿⣿⠿⠛⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠟⠁⠀⠀⠀⠀⠠⠶⠾⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠁⠀⠀⠀⠰⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠺⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⣿⠃⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⣿⣿⣶⣆⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢣⣭⣽⣭⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣿⣿⡮⣻⣿⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢻⡇⢿⣿⡿⠟⠉⢭⡍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡾⢶⡁⠀⣿⡇⢰⣮⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣼⣿⣐⢿⠟⢀⣿⣇⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠀⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⠀⠀⠸⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣟⣥
⠀⠀⠀⠀⠀⢀⡴⣫⣾⠟⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠞⠀⠀⠈⠻⢳⣿⡿⠃⠀⠹⡆⣄
⠀⠀⠀⠀⠀⣾⣱⡟⠁⣰⣿⢃⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢹⠶⠞⠀⠀⠀⢀⣈⣋⡀⠀⠀⠙⠶⠟⢠
⠀⠀⠀⠀⠀⡇⣿⠀⣼⠟⣡⣾⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠾⠖⠁⠀⠀⢠⣼⣿⣿⣧⡄⠀⠀⠳⠶⠟
⠀⠀⠀⠀⠀⢷⢻⠸⢡⣿⡿⠋⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡴⠏⠀⠀⠀⠈⢻⣿⣿⠛⠁⠀⠀⠘⠦⡵
⠀⠀⠀⠀⠀⠈⠂⠀⢿⠏⣠⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⠆⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠘⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣁⣰⠂⠀⠀⠀⠀⣿⡇⠀⠀⠀⡀⢧⣀⣁
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢉⣣⣸⡇⠠⣄⠀⢻⡇⢀⡄⠰⣇⣼⣉⠁
⠧⠽⠀⣠⠤⠤⡀⠀⡠⠤⢄⠀⢀⠤⠤⡀⠰⠭⠝⠀⠀⠀⠀⠀⠀⠀⠉⠉⢙⣲⡿⢤⣸⡧⠾⣶⣊⠉⠉⠁
⠀⠀⠀⡇⠀⢀⣼⢾⣄⠀⣠⡷⣧⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠙⠋⠳
⠀⠀⠀⠙⠒⣟⠁⠈⢻⠾⡋⠀⠈⡳⠒⠁
⠀⠀⠀⠀⠀⠈⣓⡒⢉⣀⠙⢒⣊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡆
⠀⠀⠀⠀⠀⢸⠀⢱⢸⠀⡇⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢟⣷
⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣿⡿⠁
⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣁⣀
⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⢿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⢰⡄⣗⢰⠐⣆⡆⢲⠘⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢧⠀⡇⢸⠀⡄⣤⢠⠠⡆⠇⢸⣿⣿
⠀⠀⠀⠀⠀⢸⠀⢸⢸⠀⡇⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⠈⠁⠉⠈⠀⠁⠈⠉⠈⠁⠀⠈⣿⣿
⠀⢀⠤⡀⠀⠸⣀⡸⢸⣀⡇⣇⣀⠇⠀⠠⢄⠀⠀⠀⠀⠀⠀⡧⠌⡠⢄⢠⠀⡦⢠⠠⡄⡄⢠⢸⡇⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠜⣄⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠄⣀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠆⠤⠊⠀⠘⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⣸⡟⠻⠻⠿⠿⠿⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⠀⠀⠒⠒⠒⠀⡠⢊⡁
⠀⠀⠀⠀⠀⢰⡿⠁⢀⠀⠀⠠⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡞⠁⠒⠀⠂⠠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠏⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⠿⠿⢿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⡁⠀⠄⠀⠀⠀⠀⠁⣀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠀⠀⢄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠈⠀⠀⠠⠈⠛⠻⡀
⠀⠀⠀⠀⠀⠀⠉⠲⢄⠐⠀⣀⣀⣈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠠⠴⢾⠃⠀⠀⠀⠀⠀⠀⠑⢦⡀⢀⠀⠀⠄⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣏⠀⡽⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⣦⠀⣀⣈⡧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣷⠄⢀⡀⠀⢀⣡⠴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣴⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠉⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡗⠒⢯⠀⣴⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣉⣉⣉⠉⢉⠠⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⡀⠀⠀⠀⠀⣿⠃⣤⠎⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⡀⠀⠀⠱⡀⠀⠀⠀⠀⠀⣀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠂⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠙⢒⡖⠒⠊⠳⠖⢲⡛⠁⠀⠀⠀⢸⠍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⢀⡞⠁⠀⠛⠂⠛⠃⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠤⣀⣀⡼⠀⠀⣀⠠⠇
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠘⠉⡏⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡴⠁⠀⠀⠀⠀⠹⡄
⠀⠀⠀⡤⠤⠵⠶⠴⠦⠤⠤⠤⠤⠤⣤⣤⣬⣥⣤⣤⡽⠀⠀⠀⠀⠀⠀⣠⡼⡷⠖⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣠⣃⣀⣀⣀⠀⠀⠀⠀⣛⣀⣀⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⠶⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠶⠛⠓⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣳⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠛⠉⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠴⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠙⠻⣆⠀⠀⢀⣰⠟⠁⠀⠀⠈⠙⢿⣄⠀⠀⠀⣼⠋⠀⠀⠀⠈⠙⢿⣄
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⣴⢾⡿⠛⣿⣶⣦⠀⠀⠀⠀⢀⡴⢾⡟⠛⣿⣶⣆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢠⡞⢁⡾⠃⠀⠘⣦⡙⢷⡀⠀⣰⡞⢡⡾⠃⠀⠙⣆⡙⣷⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣿⠞⠋⠀⠀⠀⠀⠈⠻⢾⣷⣤⣿⠚⠋⠀⠀⠀⠀⠘⠻⢾⣧⣤⡶⠚⠋
⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠙⢻⡷⣷⡀⠀⠀⠀⣠⡶⣿⡟⠙⢻⡷⣦⡀⠀⠀⠀⢠⡶⣿⡏⠙⠛⢾⣦
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠻⣎⣻⡄⠀⣰⣏⡴⠏⠀⠀⠈⢳⣌⣷⡄⠀⣴⣯⡼⠋⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣁⡀⠀⠀⠀⠀⠀⣈⣽⣿⣿⣿⣡⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⣸⠏⣹⡇⠀⢹⡍⠻⣆⠀⠀⢀⡺⠋⣼⠇⠀⢻⡝⢻⣄⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠹⣶⣄⣀⣀⣼⣥⠾⠋⠀⠀⠀⠻⣦⣽⣇⣀⣿⣥⠾⠋⠀⠀⠀⠻⣦⣽⣆⣀⣀⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠿⣿⣤⣄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢠⡼⣣⡿⠃⠀⠘⣯⡻⣦⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⠴⣿⠛⠉⠀⠀⠀⠀⠈⠻⢻⡷⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⢀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠒⠒⠚⠋⠁


⣿⣿⠻⠿⢻⠿⡿⢿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⡀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⡎⠀⣀⡴
⠀⠀⢀⡤⠶⠦⡀⠀⢉⣹⠶⠀⢱⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⣿⣷⣦⣄⣀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⡓⠂⣠⠞⢹⣷⣦⣤⠴⠂
⠀⠀⠸⠀⠀⠀⢀⡴⠋⠁⠀⠀⠨⠗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣭⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡇⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠀⢸⣶⠶⠦⣤⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣠⣤⣶⣿⣟⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣼⡇⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢿⣄⡀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠸⣄⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣶⣤⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠲⣀⣴⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢻⣿⣿⡏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠏⠁⢀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠿⠷⢶⣤⡀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⢻⣿⡇⠀⠀⠘⣆⠀⠀⠀⠀⣠⠴⠒⠒⠒⠚⠉⢠⡞⢋⡽⠒⠒⠂
⠀⠀⠀⠀⠀⠤⢴⡿⠟⢿⣿⠟⠒⠒⠒⠀⠽⠦⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⠀⢰⠏⠀⢀⠀⠀⠀⠀⠘⢠⡏⠀⠀⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠐⡌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢸⣿⡇⠀⠀⣠⡿⠀⠀⠀⢸⡄⠀⠈⠀⠀⠀⠀⠀⠸⣇⠀⠈⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠐⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⢀⣿⣿⢀⣠⣴⡿⠁⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠙⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⢿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠐⢒⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣾⠟⠁⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠣⣀⣠⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⡧
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⣼⣿⣿⠀⢀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠆⢹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣸⣿⣿⣿⠞⠛⠁
⠀⠀⠀⠀⠀⠞⠁⢿⣿⣤⣶⣶⣤⣤⣬⣿⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⡇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣀⠴⢾⣿⣿⡿⢿⣿⣿⣷⠶⣀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠈⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠐⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣤⣶⣦⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⣿⣿⣿⡿⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣴⣶⣶⣄⠈⢿⣷⣤⣹⣿⣟⣥⣴⣿⠇⠀⢀⣀⣀⡀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠸⣿⣿⣿⡿⠀⠀⠉⣿⣿⡿⣿⣿⠋⠁⠀⠀⣾⣿⣿⣿⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣬⣿⣯⣤⡀⠀⠀⣿⣿⣶⣿⣿⠀⠀⠀⠀⠙⣿⣿⠟⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⣿⣿⣿⡄⠀⣿⣿⣿⣿⣿⠀⠀⢀⣼⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⠀⣿⣿⡟⣿⣿⠀⠀⣾⣿⣿⣏⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⣿⡏⠋⠀⣿⣿⡇⣿⣿⠀⠸⠿⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⣠⣿⣿⣧⣿⣿⣤⣄⠀⢸⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⣿⣿⣿⡆⣿⣿⣿⣿⠀⢸⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣧⣿⣿⣧⣤⣿⣿⣿⣷⣿⣿⣿⣿⣀⣸⣿⣟⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢞⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣠⣴⣾⡿⠿⠿⠿⠿⢿⣿⣶⣄⠀⠀⠀⠀⠀⣠⣴⣿⡿⠿⠿⠿⠿⢿⣿⣦⣄⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠿⠿⠿⣿⣷⣦⣀
⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧
⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣺⣿⣧⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣼⣿⢿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⣣⣿⡟⠀⢲⣾⣝⠻⣿⣷⡀⠀⠀⢠⣾⣿⠟⣱⣿⡟⠀⢲⣾⣝⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠙⢿⣷⣦⣀⠀⠀⢀⣿⡿⣣⣾⣿⠟⠀⠀⠀⠹⣿⣷⣬⣻⠿⠄⢠⣿⡿⣥⣾⣿⠏⠀⠀⠀⠻⣿⣷⣌⣻⠷⠄⠀⠀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣼⣿⡷⠟⠋⠁⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⡿⣾⣿⠳⠟⠋⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣶⣶⣶⣶⣶⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣶⣶⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

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




⣿⣟⢻⡟⠿⠿⠿⠟⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⠠⠤⠄⢀⠀⠈⠐⣄⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡄⠊⠀⢀⡀⠤⠤⠄⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠘⠁⠀⠀⠀⡠⠘⢂⣉⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣈⡁⠚⠡⡀⠀⠀⠀⠈⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⠦⣀⠘⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⢀⠀⡇⠀⠀⠀⠀⠈⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠀⠀⠤⠤⠤⠀⠀⠀⠐⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢄⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠔⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠐⠄⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠄⢊⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠌⡐⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠢⢐⣨⣴⣶⡏⠀⠀⠀⠀⠀⠁⠒⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⡎⠀⠀⠀⠀⠀⠀⠂⠉⣦⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⡀⠀⠀⣀⣠⠞⠈⡿⠿⠟⠛⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠏⣠⠏⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⣷⣶⣶⣶⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⣀⣴⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠊⠀⠀⣈⠀⠀⣿⣿⣿⣷⣄⡠⢀⣠⡤⠾⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠂⠶⠍⡀⢠⠀⠄⠆⠐⠀⡄⣀⣿⣿⣿⣿⡿⠁⠈⡄⠀⠀⠈⠻⣿⣿⠿⠛⠉⠀⠙⠶⣄⣠⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠂⠈⠉⠀⠀⠄⢈⠁⠁⠀⠀⠀⠀⠀⠀⠈⠐⢈⡡⢞⠟⠁⠀⠀⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠀⠐⠒⠀⠀⠲⠖⠚⣋⠥⠖⠁⠀⠀⠂⠀⠉⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⡠⠀⠀⢰⣤⣴⣶⣾⣷⡀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢡⡀⠀⠀⠀⠀⠀⣀⠀⢠⠥⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡀⢀⠸⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠒⠀⠉⠐⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠙⠀⠀⠀⠀⠀⠈⠉⠉⠉



⣶⣶⣶⣶⣶⡿⣾⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⣶⣿⢷⡶⠾⠿⠶⢶⣾⡿⠶⠶⠶⢾

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠
⠀⠀⠲⢶⣿⣄⠀⠀⠀⠀⠈⢓⣶⣖⠀⡀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠤⣾⡧⠤⠀⠀⠀⣿⡶⠀⠀⠀⠀⠠⠌⠓⢶⣿⠀⠀⠀⣾⡧
⠀⠀⠀⣾⣯⠀⠉⠀⠀⠀⢀⣼⣿⠉⠉⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢿⣷⡄⠀⠀⠀⠀⠀⣠⣾⡿⠀⠀⠠⣿⡁
⠀⢀⡴⠃⠈⢳⡀⠀⠀⣠⠋⠀⠈⠳⣄⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⢰⠇⠘⠇⠀⠀⠀⢰⠋⢿⡀⠀⠀⠀⠀⢻⠏⠁⠀⠀⣀⡿⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠈⠀⠈⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⢈⢀
⠀⠤⠤⢤⣾⣇⡀⠀⠀⢠⣿⠖⠒⠂⢀⡴⣿⡶⠒⠲⠀⠄⠴⣾⡶⠊⠁⠀⣼⣾⡉⠀⠀⠀⠲⣶⣿⡄⠀⠀⣾⣿⡋⠁⠐⠳⣾⡦⡤
⠀⠀⠀⣼⣏⠀⠉⠀⠀⣹⣿⣄⠀⠀⠈⠀⣹⡟⠀⠀⠀⠀⠐⣿⡄⠀⠀⠀⣹⣿⡇⠀⠀⠀⢸⣿⠋⠉⠀⠀⢰⣿⠁⠀⠀⠘⣿⡇
⠀⠀⡠⠞⠘⡇⠀⠀⢀⡏⠀⠸⡂⠀⠀⠀⠙⡦⠀⠀⠀⠀⢐⠃⡇⠀⠀⢀⡏⠈⠻⡄⠀⠀⡟⡿⠀⠀⠀⠀⣸⢸⠀⠀⠀⠀⠏⡇
⠀⠀⠁⠀⠀⠁⠀⠀⠈⠀⢄⠀⠁⠀⠀⠀⠀⠁⢀⣀⣀⣀⠈⠁⠁⣀⠀⠈⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠈⠓⢲⣾⡿⡆⠀⢰⡿⠻⣫⠋⠀⢀⣼⡃⠀⢢⣠⡶⡶⣶⠆⠀⠀⠀⡄⣀⠤⠀⠀⣐⠄⠀⠀⠐⢤⣤⣤
⠀⠀⠹⣤⡏⣀⠀⣀⣠⠀⠀⠀⢸⠃⠀⠁⠀⠀⠀⠀⠋⠀⠐⣿⣻⡃⠀⠘⠟⠉⠻⠫⠀⠀⠀⢀⡿⠋⠀⠀⠀⢻⣧⡀⢸⠛⣿⠉⠈
⠀⠀⠀⠙⢿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⠻⡇⠀⠀⠀⠀⠀⡟⡇⠐⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⣤⡶⠀⠀⠀⠀⣠⣆⠀⠼⠀⠙⠢⠀⢠⡄⠀⠀⢻
⠀⠀⢀⣶⡟⢀⠀⠀⠀⢸⣧⣄⠀⠀⠀⠀⡠⢾⣿⠢⠀⠀⠀⠈⢼⣿⣭⠂⠀⠀⢠⣿⠿⢦⡄⠀⠀⠀⢰⣿⠃⠀⠀⠈⠀⠐⢺⣿⠶
⠠⢄⣸⡟⠛⠳⣄⡄⠀⢊⣿⠀⠀⠀⠀⢀⣑⣋⢿⡄⠀⠀⠀⠶⠒⠼⢧⠄⠀⠐⠛⠁⠀⠀⠀⠀⠀⠉⠛⢫⠭⠖⠞⠀⠤⢴⡿⠍⠄
⠀⠀⠘⠛⠳⠆⠀⠀⢀⡟⠟⠉⠁⠀⠀⢀⠀⠀⠠⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠁
⠀⢀⡀⠀⠀⠀⠀⠀⠛⠁⠀⠈⠀⠀⠀⠈⠃⠒⠒⠉⠀⠀⠢⠧⠖⠚⠉⠀⠀⠐⠒⠒⠀⠉⠁⠀⠀⠠⠄⠺⠒⠂⠀⠀⠐⠀⠀⠀⠂
⠀⣧⡄⠀⠀⠀⡜⣠⡀⠀⢀⡄⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⢰⣿⠳⡀⠀⠀⠘⣿⡖⠂⠀⢹⣷⠃⠀⠀⡼⣿⡿⠀⠀⣰⣿⡀⠀⠀⢿⣿⠓⠄⠀⠀⠀⢀⣀⣀⠀⢻⠿⣶⡄⠀⠀⠀⠀⣠⣄⡀
⢸⡏⠷⠈⠁⣀⠼⣿⠇⠀⠀⢸⡿⢷⠀⠀⠀⣿⣿⠀⠈⣰⢿⡆⠀⠀⣾⢿⡈⠀⢀⢀⣾⣟⡿⠉⠀⠈⠀⢾⣧⡀⠀⠀⠀⣯⣿⣿⡀
⠈⡇⠀⠀⠀⠈⠀⠙⠋⠁⠀⠛⠃⢸⡀⠀⠀⢹⠁⠀⠀⣻⠈⠀⠀⠀⠈⠂⠙⣄⠉⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠃⠒⠀⠀⠀⠋⠉
⠀⠈⣀⣴⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠀⠀⠀⠀⡄⠀⠀⠀⣤⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠎⣽⣿⠂⠀⠀⡴⣿⠾⠀⠀⠀⢴⣿⡆⠀⠀⣾⣦⠀⠀⡔⣿⣷⠀⠀⠀⣿⠄⠀⠀⣾⡿⡆⠀⢞⣿⠦⠀⠀⢶⣾⡇⠀⠡⣿⡶⠄
⠀⢠⣿⠃⠀⠀⠀⣹⠟⢶⡄⠀⠀⢸⢿⡆⠀⠀⡿⣧⠀⠀⢸⡿⠀⠀⠀⢰⡿⣄⠀⠀⣿⡟⠁⠀⠀⣿⠁⠀⠀⢸⣿⡁⠀⡴⠾⣷⠃
⠀⠰⠁⠀⠠⠒⠉⠁⠀⠀⠦⠀⠀⠉⠘⡃⠀⠀⢱⠹⠀⠀⠸⠀⠀⠀⣠⠻⠞⠁⠀⠀⠉⡇⠀⠀⠰⠹⡀⠰⠚⠉⠀⡃⠀⡵⠀⠈⠓

⠀⠀⢤⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢀⡀⢀⣤⡀⢀⡀⠀⣠⣄
⠀⠀⠈⠓⢶⣿⣦⣀⣠⠄⠶⣤⣀⣻⣗⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢻⣷⢤⣤⡀⠀⠀⠀⠀⢨⣿⣿⣭⡾⠋⠉⠙⢷⣭⣿⣿⣏
⠀⠀⠀⠀⣸⣿⣇⡀⠀⠀⠀⠀⠀⠉⣿⣯⠉⠙⠀⠀⠀⠀⠀⠰⣿⡉⠙⠛⠋⢹⣿⠆⠀⠀⠀⠉⠉⠙⡿⠀⠀⠀⠀⠀⢻⡏⠉⠉
⠀⠀⠀⢸⡏⠀⠘⣷⡀⠀⠀⠀⠀⣿⠉⠙⠷⣤⡀⠀⠀⠀⠀⣼⠏⣿⠀⠀⢠⡟⢿⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣈⠛⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣦⠀⠀⢲
⠀⢸⡗⠂⢲⣦⣿⣷⡆⠒⠺⡇⠀⠀⠀⠀⠀⣠⣶⣞⠛⠀⠀⠀⠈⠓⠦⣽⣏⣠⡴⠃⠀⠤⣤⣤⣼⡛⠀⣠⡀⠀⢘⠋⠺⣷⣼
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⡼⣿⣥⡌⣿⣂⡀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⣾⣿⢛⣒⣋⡀⠀⠈⠛⠋⣿⣷
⠀⠀⠀⠀⠀⣸⠏⣧⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⠾⢿⠋⠇⠙⡆⠀⢀⡾⠋⣷⠀⠀⠀⠀⠀⢠⡟⠻⣆⠀⠀⠀⠀⠀⠀⢀⣼⠛⣷
⠀⠀⠀⠀⠀⠻⠀⠿⠀⠀⠀⠀⠀⠘⠻⠿⠋⠈⠀⠈⠳⠤⠞⠁⠀⠀⠿⠀⠈⠳⠀⠀⠀⠰⠏⠀⠀⠟⠀⠀⠀⠀⠀⠰⠟⠁⠀⠟

⠀⠀⠀⠀⢀⣄⠀⢂⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠀⣤⡄
⠀⠀⠠⢶⣾⣿⣤⣼⠀⠀⠀⣤⣿⣷⣶⡶⠀⠀⠀⢀⣼⣯⣤⡤⠖⠒⠀⠀⢀⣀⡀⣶⡄⠀⠀⠀⠀⢿⠇⢀⠀⠀⢠⡴⣶⡷⠖⠋
⠀⠀⠀⠀⣿⣿⠀⠈⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⡟⣿⣿⠀⠀⠀⠀⠀⠸⢿⣼⡿⣌⠀⠀⠀⠀⣾⡟⠦⡜⠃⠀⠀⠀⢻⡷⣦
⠀⠀⠀⢠⡟⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠁⣿⣿⠀⠀⠀⠀⠀⠀⢸⡿⣦⠈⠁⠀⠀⠈⣿⣶⣄⠀⠀⠀⠀⢀⡾⠃⠟
⠀⠀⠀⠸⠇⠹⠆⠀⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠀⠀⠿⠸⠇⠀⠀⠀⠀⠴⠟⠀⠹⠆⠀⠀⠀⠰⠟⠀⠹⠆⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠠⠁⢴⡆⠀⠈⠀⠀⠀⠀⠰⣶⣠⡤⠶⣢⡾⠃⠀⠀⠀⠀⢀⣺⣟⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠠⣶
⠀⠀⠘⠲⣿⠚⢁⡞⠀⠀⠀⠀⠀⣼⠿⣿⣾⣋⠀⠀⠀⠀⠘⠋⠉⣿⣿⠉⠉⠃⠀⠀⣴⠞⠛⣿⠁⠀⠀⠀⠀⠀⠀⠓⢾⡟⠋⣰⠇
⠀⠀⠀⢀⡼⠛⠋⠀⠀⠀⠀⠀⠀⣟⠀⠀⠉⢉⡛⠛⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠁⢀⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠁
⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠙⣿⠛⠛⠛⠛⠛⠻⡟⠃⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣤⣤⣬⣭⣿⣤⣤⣤⣄⠀⠀⠀⢰⡏

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

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⣠⣾⠟⠛⠋⠙⠛⠻⣷⣄⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠙⠛⠿⣶⣄⠀⠀⠀⠀⢀⣴⡿⠛⠛⠉⠛⠛⢿⣦⡀
⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⣀⣚⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⣾⡇⠙⣻⡻⢿⣦⡀⠀⠀⠀⠀⣠⣾⠟⣿⠏⠙⣿⡻⢷⣦⡀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠈⢿⣦⡀⠀⠀⠀⠀⣰⡿⠁⣴⡿⠁⠀⠘⣿⣄⠙⣿⡆⠀⠀⣼⡟⢁⣼⡟⠀⠀⠹⣷⣄⠹⣷⡄⠀⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠙⢿⣦⣤⣄⣠⣿⢳⡿⠋⠀⠀⠀⠀⠈⠻⢷⣮⣿⣄⢸⣿⣴⠿⠋⠀⠀⠀⠀⠈⠻⣷⣽⣧⣀⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡍⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣤⣤⣀⣤⣴⣾⠟⠁⠀⠀⠀⠀⠙⢿⣶⣤⣄⣀⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⣿⣿⣿⣿⣷⠙⣿⠧⠖⠛⠁⠀⠀⠀⠀⠈⠀⢨⠁⠂⠀⡴⠀⠀⠀⠰⠋⠀⠀⠀⠸⣧⠀⢠⠼⠉⠙⠃⠀⠸⠁⠀⠀⠀⣁
⣟⡉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠤⠻
⠋⠁⠁⠀⢀⣤⣀⢠⣤⣀⣀⡀⠀⠀⠀⢀⠀⠨⠔⠀⠀⠀⣟⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠤⢄⣸⣀
⢣⣈⡀⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠃
⠸⢿⡏⠀⢹⣿⣿⣿⣟⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣾⡟⠁
⠠⣶⠃⠀⠈⣯⣿⣿⣿⡿⡟⠀⠠⠎⠀⠀⠀⠀⠀⠀⠰⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠰⠃⠀⠀⠀⢸⣽⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣶⣶⣦⣄⠀⣼⣿⣿⣿⡆⠈⠀⠀⠀⣀⣀⠀⣾
⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⢻⣿⢸⣿
⢀⣤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣛⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣸⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣠⣼⣿⣿
⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⣄⢀⣤⣾⣿⣿⣯⡞
⠀⢸⣶⣗⠄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣾⣿⣿⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠋⠀⠙
⠀⣈⡁⣉⠂⠀⠀⠀⠀⠀⠀⠛⠻⠛⠛⠻⣿⣿⡆⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⠋
⠀⡈⠻⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣁⣀⣀⣀⣀⣀⣀⣔⣤⣤⣤⣤⣤⣤⡄
⣀⠀⠀⠷⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⣵⣶⠀⠀⢠⣿⣿⣿⣷⣤⣶⣶⣦⣤⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⣠⣮⣿⣻⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣯⣿⢿⠿⣟⣁⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣤⣀⣤⣤⣤⣀⣀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠋⣛⢠⢸⣿⣿⣿⣿⠉⢻⡏⠙⣿⣿⠟⠙⠻⣿⣿⡟⠉⠉⠻⣿⡏⠉⠙⣿⣟⠉⣙⣿⣿
⣿⡉⠀⢸⣿⣿⣿⣿⡇⠈⡁⠀⣿⣿⠀⠀⠀⣿⣿⡀⠀⠀⠀⣿⣇⢠⣾⣿⣿⠆⢻⣿⣿⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿
⣿⣂⠀⠘⣿⣿⣿⣿⣇⣸⣿⣀⣿⣿⣦⣤⣴⣿⣿⣷⣤⣤⣾⣿⣿⣸⣿⣯⣤⣤⣼⣿⡿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣌⣍⠉⢉⠉⡉⢉⣩⣭⢉⡍⢩⡍⠉⡉⣭⣍⣉⣋⠉⠉⠉⡉⠉⠉⠀⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠟⠋⢲⠂⠰⠆⢰⣼⣿⢿⡼⣿⡌⠀⠴⠶⠻⠟⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠤⣶⡶⠶⠶⠒⣚⣻⣿⣯⡨⠙⣀⣀⢀⡀⠠⠀⣀⣾⣷⣠⣦⣀⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠈⣿⣷⠓⢂⣫⢶⣛⣽⣿⣿⡿⣧⣤⣤⣀⢘⣻⣿⣿⣾⣷⣶⣿⣿⣿⣿⣿⣇⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢲⡄⢹⣿⠟⣯⡿⠿⠟⢿⡏⣿⢿⢿⠿⠯⢿⠿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢻⣆⣀⡀⠀⣃⣼⣿⣀⣀⣤⣄⣤⣤⣤⣠⣦⣤⡄⢠⣀⠀⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣬⡿⠋⢻⣾⣿⣿⣿⣅⣿⡇⣿⣿⢸⣿⡇⠸⣿⡇⢸⣿⣰⣿⣿⡿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣟⡛⠛⠋⠛⠒⣢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢻⠰⢆⣨⣿⣿⣿⣟⠋⢻⣧⣿⡿⢸⣿⡁⠀⣿⠆⣾⣿⠻⣿⣿⣷⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠁⠉⢹⣿⡿⡷
⠀⠀⠀⠀⠀⠀⠠⢼⡌⠈⣻⣿⣍⢿⠃⠡⣼⣿⣿⣿⣼⣿⣥⣾⣿⣿⣿⣿⡀⠘⠦⢉⣟⣤⣱⣽⣿⣿⣿⣿⢻⣿⢿⣷⣦⡀⢸⣿⣷⠆⣀
⠀⠀⠀⠀⠀⠀⠀⣿⡒⢸⡟⢁⣿⣯⣵⣬⣯⣽⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⡀⠀⠀⠘⠋⠙⣿⡜⠙⣂⣈⠛⠝⠓⡸⠿⠋⠝⠋⠋⠉⠛⠈
⠀⠀⠀⠀⠀⠀⠈⣬⡁⠺⠟⡊⠉⢻⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣠⠲⠀⣬⠚⢰⠉⣿⢿⡏⢁⠀⢀⠘⣿⣦⣄⠙⢲⠂⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡤⣝⢻⠫⣀⣹⣿⣿⡟⡸⡽⢟⣻⠿⣿⣿⣿⣿⣿⣿⠋⡅⢱⡦⠿⠿⠀⣿⡿⣿⡸⠄⠚⠆⣸⣿⠿⣗
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡧⡜⠁⠉⡿⣿⠾⠷⠾⠶⢤⢠⣾⣿⣿⣿⣿⣧⣤⢀⠀⠀⢓⡲⢾⠛⠛⠛⠟⢿⡟⠛⠉⠉⣑⣬⢿⠤
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠂⢤⣀⣽⣿⣇⣂⣠⣄⣻⣤⣿⣿⣿⣿⣿⣿⠿⠀⠄⠀⣠⢠⠀⣀⠀⣀⣴⣧⠄⣀⣤⣶⡏⠻⣿⢶⣶⡶⣦
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠻⠛⡀⠿⡈⠛⠛⠉⠻⢏⣉⠉⠉⠙⠀⣼⢋⠉⠀⠙⠓⠲⢤⢄⡅⠈⠀⢿⣍⣉⢿⡣⣠⣿⣿⣿⣥⣤⣄⠠⣈⣿⠹⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⡄⣲⡀⢳⣄⡀⣠⣤⣤⣀⣶⣿⣷⣆⠈⣧⣤⣤⣌⣤⣤⣀⡈⣷⢢⣄⣤⡟⣿⣿⣿⣷⡈⢿⣮⣿⣿⣿⣿⣿⡟⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠿⢹⣿⣿⣿⣟⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣼⣿⣿⣿⣿⣎⣿⣿⣿⣿⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢄⠀⠀⠀⠈⣉⢉⣉⠋⠅⠈⠁⠁⠐⠃⠚⠤⢹⡿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠛⢛⣿⠛⠿⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠻⠷⣧⣿⣿⣾⣿⣷⣄⢀⣤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠛⠛⠛⠛⠛⠛⠒⠐⠐⠒⠐⠀⠀⠀⠐⠒⠒⠂⠀⠂⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⢻⡈⠃⢙⣿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⣠⣶⠟⢻⣠⠖⡀⠈⠙⢳⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠅⠀⣾⡿⠁⢤⣽⣿⠂⠈⠀⠤⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡯⢀⠀⢸⡟⠛⠒⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⣾⣿⣦⣹⣿⠀⠠⠤⠀⠀⠀⠙⢲⣿⣿⡆⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣴⣧⣍⣀⠀⠀⣀⣡⣤⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣿⣿⠙⠿⠏⢛⡿⡟⠙⣿⡋⠻⠏⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣻⣿⣧⣤⢴⡍⢰⡗⠀⠈⠁⢠⣴⣿⣿⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⣿⣿⣿⡃⢁⡿⠶⠊⠁⠀⣺⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡇⢻⣿⢿⡇⢨⣛⡽⠇⠒⠀⠉⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣍⢰⣶⣿⣿⡇⢸⣿⡿⡗⠀⠁⠀⠀⠀⠐⠀⣠⣿⠃⠀⠸⠀⢀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡀⠠⡌⠻⣿⣿⣿⡇⣸⣿⣿⣦⣤⣁⣀⡀⣠⣤⣴⣿⣿⡆⠀⠀⠀⣸⠀⠖⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡉⠛⠲⣄⡙⢿⡛⢾⣿⣿⣿⣿⡽⣿⡏⢀⣿⣟⣿⣿⣿⡇⠀⢀⣴⡃⠀⣠⣧⠇⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠶⠤⣠⡀⠀⢠⡈⠻⣦⣙⠲⡿⣿⢿⡛⢷⣹⣿⣿⣟⣼⠿⢛⣿⣿⠦⠟⡉⣠⡴⠋⠁⠀⣴⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀⠀⠛⢿⣦⣄⠻⣦⡘⢿⣟⠲⣌⠑⠙⠦⣭⣋⣛⠛⠷⠟⣋⣩⣤⡶⠟⠚⡁⠀⠀⠀⡸⠋⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⢀⡙⢧⣹⣟⢦⡙⠛⢦⡙⠶⣤⣈⡙⠉⠭⢭⣉⣋⣉⣁⣠⣤⣎⣀⣀⡴⠋⠀⣠⠔⡻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣘⠿⣾⣷⡙⢦⡳⢦⡀⠙⠦⣄⠈⠙⠻⢶⣄⠀⠀⠉⠉⠉⣉⣁⣀⣤⠤⢖⣺⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢽⣷⣦⣍⡛⠶⣬⡢⠹⢶⣤⣌⣙⠶⢤⣤⣬⣤⡤⠶⠿⠛⣉⣉⣡⣤⣶⣯⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠻⣿⣿⣾⣿⣮⣢⢽⣻⣿⣿⣿⣿⣾⣿⣿⡶⢟⣿⣿⣿⡛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠈⠀⠉⠉⠉⠉⠀⠀⠀⠈⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⠶⠶⠶⣶⣶⣶⡶⠶⠶⠶⠶⣶

⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡆⣸⠃⠀⠀⠀⠀⡤⢤⣠⠘⠀⠀⠀⠀⠀⢠⠖⠀⠂
⠀⠀⠤⡄⠹⠋⠁⠀⠀⠀⠀⠀⢠⠲⣌⡗⠢⠬⠀⠀⠀⠀⢠⡇⢚⠈⠃⠀⠀⠀⠀⠀⠀⣴⡄⠘⠶
⠀⠀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠠⠏⠠⠇⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠔⠀⠀⠀⠀⠼⠙⠁

⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢀⡀
⠀⠀⠘⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠠⣆⡤⠖⠀⠀⠀⠀⠀⡘⢃⡀⠀⠀⠀⠀⠀⠀⢠⣁⢞⡴⠋
⠀⠀⠠⠚⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⡉⢥⠀⠀⠀⠀⠀⠀⠀⠉⠒⠓⠒⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠓⠀⠀⠀⠙⠂⠀⠀⠀⠀⠰⠚⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⢀⢀⣴⠀⠀⠀⠀⠀⠀⢤⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⢢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠄
⠀⠀⠀⠀⠈⠏⡾⠀⠀⠀⠀⠀⠀⠈⣙⠲⢤⡀⠀⠀⠀⠀⢬⡳⣄⣙⠆⠶⠀⠀⠀⠀⠀⢶⠳⠿⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡞⡩⠃⠀⠀⠀⠀⠀⠀⣠⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢉⡴⠃
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣘⣀⡀⠀⠀⠀⠀⠀⠀⠀⢨⡨⡄⠀⠀⠀⠀⠀⠀⢆⠀⣀⠀⠀⠀⠀⠀⠀⠐⠚⠳⡴⠂
⠀⠀⠒⡦⠈⠉⡀⢀⠀⠀⠀⠀⠀⣴⡦⠙⠲⠄⠀⠀⠀⠀⠀⠈⠙⠻⠦⠄⠀⠀⠀⠀⠀⡖⣦
⠀⠀⠚⠁⠀⠀⡽⢪⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠸⠃⠹


⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢡⠖⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠀⠚⣙⠐⣂⣀
⠀⠀⠀⠀⠰⢾⡶⠀⠀⠀⠀⠀⣠⠞⠲⡈⠉⠉⠀⠀⠀⠘⡽⣯⠀⠙⡽⣯⠁⠀⠀⠀⢠⢇⠽⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠉

⠀⠀⠀⠀⣔⠢⣄⠀⠀⠀⠀⠀⠀⠤⣤⣴⣆⠀⠀⠀⠀⠀⠀⠰⡤⠖⠳⠀⠀⠀⠀⠀⠀⢀⢐⢆⡀
⠀⠀⠀⠀⠮⡟⠊⠀⠀⠀⠀⠀⠀⣠⣶⠄⠬⠃⠀⠀⠀⠀⠀⢀⣈⠓⠂⠀⠀⠀⠀⠀⠠⠬⡯⠠⣧⢤⡀
⠀⠀⢠⠟⢠⠏⠀⠀⠀⠀⠀⠀⢠⠞⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢧⡀⠀⠀⠀⠀⠰⠃⠀⡟⠐⣆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠆⠀⠄




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠠⣀⠐⢄⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⠈⣷⣤⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡄⠉⠀⠈⠢⠀⠀⠀⡠⠊⠉⠹⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣈⠀⠀⠀⠀⣰⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢸⣿⣷⠄⠄⠒⠻⣿⣿⡿⠣⢄⠀⣠⡄⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣄⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠻⢏⠀⠀⠀⠀⢨⠁⠀⠀⠀⢸⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣠⣶⣶⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠠⣤⣤⣶⡀⠀⠀⢀⠌⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⡀
⣄⠀⠀⠘⣿⠃⠀⢀⣾⠀⠀⠀⣠⣴⣾⠿⠟⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣶⣤⡀
⢹⡷⠀⠀⢹⡇⠀⠀⢹⣇⣴⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⡀
⠸⣧⠀⠀⢸⣷⠀⣠⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⣿⡄⠀⠈⣿⣾⡿⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡷⡼⠈⢦⢳⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⢻⣇⠀⣠⣿⣟⣀⣴⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣣⣤⣿⣤⣘⡓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠘⢿⣿⣿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⠏⣤⡀⣀⡘⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⢠⣿⠏⠙⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣂⣀⣤⡍⢁⢼⣸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⣾⡿⢀⡴⠟⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣽⣿⣯⢽⣋⣒⣤⣼⣿⣿⣱⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⢠⣿⡇⠯⢠⠴⠊⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⢋⡜⢡⠞⣽⣿⡿⣿⣇⢻⣿⣿⣿⣿⡳⣄⠀⠀⠀⠀⠀⠀⢹
⠀⢸⣿⠀⠘⠄⠒⣉⣼⠃⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⢻⣼⡇⣿⢸⠟⣸⢇⣯⡟⣼⠛⠀⠀⠉⠹⢎⢧⣀⠀⠀⠀⠀⢸
⠀⢸⣿⠀⠀⢡⣈⣡⣾⢧⡀⠀⠀⠀⠀⠀⣰⠃⢠⡀⠀⠀⠙⠹⣧⣼⡄⣿⢸⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠳⣄⠀⠀⢸
⠀⢸⣿⡄⠀⠀⠈⢻⣿⡆⠙⣦⠀⠀⣠⡶⠋⠀⢈⣷⡀⠀⠀⠀⠈⠟⠿⡷⠉⠀⠀⠀⠀⣴⢄⣿⣆⡀⠀⠀⢿⣮⢦⡀⢸
⠀⠈⣿⡇⠀⠀⠀⢸⣿⣿⡇⠈⠳⡾⠋⠀⢀⡇⣾⣿⡷⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⣠⣿⣫⣿⣷⣭⣿⡤⠀⠀⠈⣷⣷⣼
⠀⠀⢻⣿⡀⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⣀⡾⢃⣟⣼⡏⢿⣿⡶⠞⠋⢹⠀⠈⢓⠒⠲⢿⡟⣻⠏⠈⠛⠻⠶⣤⠄⠀⠙⠟
⠀⠀⠈⢿⣧⠀⠀⠀⢻⣿⢿⣟⠀⠰⢚⣩⡴⠞⠋⠀⠹⡏⠸⡇⠀⣀⣬⠓⠀⠀⡗⠠⣾⢰⠏⠀⠀⠀⠀⣰⠋⠀⢠⣶⡼
⠀⠀⠀⠈⢿⣧⠀⠀⢸⣿⡀⠛⠷⠒⠉⠁⠀⠀⠀⠀⣾⡿⣄⡇⠀⠀⡸⣆⠀⠀⠁⣰⢇⡆⠀⠀⠀⠀⣰⠇⣠⣶⣿⠏
⠀⠀⠀⠀⠈⢻⣷⡄⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡇⠈⠙⠻⢶⣧⣤⣀⣀⣰⣿⣼⡇⠀⠀⠀⠐⠓⣤⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣦⣿⣷⠀⠀⠀⠀⠀⢀⣠⣿⠐⣿⣿⣄⢲⣝⢶⣬⣙⣳⣌⡉⣀⣸⣿⡄⠀⠀⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄⠀⢀⣤⠖⠉⠸⣿⣦⡘⢿⡌⢷⣝⢯⣙⠿⣿⣿⣿⣿⣿⣿⣿⣖⣰⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣤⣶⣄⣻⣿⣿⣦⣻⣦⡙⣷⣿⣷⣶⣭⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠈⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣐⡤⠶⣔⣀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡴⠊⠃⠓⠚⠃⠉⠉⡷⢦⡀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡷⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠱⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡆⡃⠀⠀⠀⠀⠀⢸⠄⢊⡤⢞⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡛⢷⡤⡤⠤⢤⠔⣿⠋⢁⡠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⢹⡶⡧⢤⣴⢲⡗⣿⣥⠖⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣦⠷⣶⡚⠋⠀⣿⠁⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠇⠀⠈⠙⠦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢦⡴⠶⠶⣤⣄⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣀⣀⣀⠈⠀⠀⠀⠀⠀⠘⠾⠳⢶⣒⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣭⣭⡭⣛⡶⠂⡀⠀⠀⠀⡆⠀⡐⠁⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠊⠁⠀⠐⢄⠀⡠⠁⠐⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣶⣶⣂⠂⠀⡀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⡠⠋⠀⡐⠁⠀⠀⠀⠀⠀⣀⣀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠉⠻⢿⣦⡈⠢⠀⢀⠔⠁⡐⠀⠰⡐⠂⠐⠈⠀⡠⠚⢤⣴⠖⠶⠿⠛⠛⠉⠉⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⢿⣦⣰⠁⢀⠄⠀⠀⠠⠈⠀⠀⡠⠊⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠙⣿⠖⠁⠀⠀⠈⠛⢷⠄⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣕⡲⠲⠟⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠐⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠸⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠸⠀⠀⠀⠀⠀⠈⠦⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠔⢂⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠄⠀⠀⠀⠀⠀⢰⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠛⣦⠀⠀⠀⠀⢸⣁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡼⠋⠀⠀⠀⠀⠀⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢪⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡇⢀⠞⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡄
⠀⠀⠀⠀⠀⢰⣶⣿⣶⣷⣅⡀⣀⠀⣐⡒⠶⠶⠶⠶⠤⠤⠀⠀⠀⠀⠤⠶⡄⠹⡽⡄
⠀⠀⠀⠀⠀⣼⢣⡌⢻⠿⡟⠀⠀⠀⠀⠉⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆⠀⢻⢟⡡
⠀⠀⠀⠀⢀⣧⣾⢧⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶⠶⠆⣦⣸⡷⡇
⠀⠀⠀⡐⡻⣻⠃⠀⢻⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣆⣀⡀⠀⣤⣤⣄⣠⢣⠇⠹⣭⣭⣽⡄
⠀⠀⡼⣫⡵⠃⠀⠀⡀⠹⣄⠈⠳⣄⡀⠀⠀⠀⠠⣾⢻⣟⠛⠿⢿⣤⠄⢐⣴⠃⡞⠀⢀⠀⠀⢻⡇
⠀⡸⠉⠁⠀⠀⠀⣰⠇⠐⣎⢦⣄⠈⠻⣦⣄⠘⢿⣿⣿⢛⣿⠀⡾⠁⣠⠞⠁⡼⠁⢀⣉⡁⠀⠈⠇
⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⣷⡄⠀⠈⡙⠻⢿⣿⣿⣿⣿⣷⣾⠋⠀⣼⠃⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠷⠀⠙⢷⣤⣴⣷⠀⠈⠿⢿⣿⣿⣿⣇⣀⣼⠇⠀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠀⠀⣤⣨⡏⠉⣇⠘⣿⠏⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣬⣵⣿⢠⣷⣿⠀⣿⠀⠀⠀⢢⣐⣒⣒⣒⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣿⣿⣿⣾⣿⣿⠀⡿⠀⠀⣤⣈⣿⣿⢟⣛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⠟⠛⠛⠛⠿⣿⣇
⠀⠀⠀⠀⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⡟⠁⠀⠀⠀⣠⣤⣌⣻⣶
⠀⠀⠀⠀⣠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠉⠛⣿⠛⢛⣣⣿⠀⠀⢸⡿⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿
⠀⢀⣠⣾⣿⣿⣿⠀⢀⠀⠘⡆⠀⠀⣴⠟⣁⣴⣾⣿⣿⠿⠛⣛⣿⡄⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣦⣼⡟⠀⠀⠠⠶⠶⠶⠖⠛⢁⣠⣴⣶⣿⣿⣿⣿⣦⣼⣿⣆⠀⠀⠀⠀⠀⣰⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣯⣍⠛⠷⣤⣤⣴⣶⣶⣶⣿⠿⠛⠉⢁⣤⣴⣶⣿⠟⠋⠁⠀⠀⠀⠀⠀⢾⣿⣿⣿⡿
⢸⣿⣿⣿⣿⣿⣿⣿⣏⣁⠀⠀⠉⠻⣿⣿⠟⠁⠀⠀⠐⠻⢿⣿⠋⠀⠀⠀⠀⠀⣠⣶⣾⣿⣦⣿⣿⣿⣷
⣸⣿⣿⣭⠍⠉⠙⠋⠉⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠛⠛⠿⣿⣿⣿⣿
⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⣰⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣧⠀⠀⢿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣭⣭⣭⣭⣥⣤⣬⣿⣿⣿⣿⣿⣿
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠰⢻⠀⠀⠄⡄⠀⠀⠀⢴⠢⠀⠀⣌⠠⢠⡀⡇⠀⢀
⠀⠀⠀⢀⡏⠰⠓⢤⠛⠧⣌⠀⠐⢠⢀⣆⠀⡅⠁⡬⠁⢡⠠⠌⡠⣄⠆⣶⠀⡴⡀⠀⡀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠑⡆⠀⣈⠀⢰⠧⠂⠘⠤⠤⡕⠀⠸⡐⠂⠄⢀⢡⡍⠧⢋⢘⠐⡤⠠⠀⡴⡄⣒⢃⠤⡁⠄⢁⢠⡤⠶
⠀⠀⠀⠀⠀⠃⣿⠉⠃⠐⠀⣾⠀⠀⠄⢣⠀⡀⣉⡍⢁⡉⠁⢠⡌⠀⠨⠟⡇⡀⠀⢰⠙⠛⠅⠨⠭⣤⣼⢾⠃
⠀⠀⠀⠀⠨⠧⣿⣄⣸⠀⠀⣿⣼⣴⣃⠉⠀⠁⠀⠀⠀⣉⠀⠀⣉⡆⠉⠓⣶⣆⠂⢀⣠⣤⡀⠀⠄⠙⠀⠘
⠀⠀⠀⢠⣄⣀⣬⠩⣍⠁⠍⠁⣮⠋⢨⠀⣬⠉⠉⠁⣦⡿⠿⠀⡧⠀⡦⠠⠽⠛⠃⠀⠥⡌⠃
⠀⠀⠀⠀⢻⣿⠋⠀⠀⠀⠉⠃⠓⠈⠀⠀⠃⠀⠀⠀⠁⢈⢁⠀⠀⠐⢧⠀⠠⠧⢦⠐⠢⡦⢀
⠀⠀⠀⠀⠘⠻⠀⠀⠀⠀⠀⣀⠀⢠⣾⣞⣶⠀⠀⠀⠀⠀⢘⠳⣀⠐⠀⠈⠀⠀⠀⠨⠛⣛⠲⣦⠜⢂
⠀⠀⠀⠀⠸⠸⡀⠀⠀⠀⠀⠀⠀⢈⠟⠛⠉⠉⠈⠁⠀⠈⠀⠜⠋⢠⠖⠀⠀⠀⠄⠀⠀⠀⠈⠻⢦⠘⠀⠐⡀
⠀⠀⠀⠤⠄⣀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠒

⠀⢀⡼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⢇⣻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⡀⠀⠀⠀⠰⡷⠀⠀⣀⣀⡀⣴⡆⠀⠐⣿⡟⠳⠤
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢴⣶⣦⠾⠇⠀⠰⢻⣿⠛⠁⠀⠀⢾⣿⠓⠒⠚⠁⣠⡿⣇⡀⠀⠦⠽⢟⣛⡳⠆
⠀⠀⢸⣿⡇⠀⠀⢠⣾⣿⡟⠉⠀⠀⣠⡾⢿⡌⠉⠁⠀⢀⣼⠟⣶⠀⠀⢀⣼⠛⣦⠀⣤⣤⡿⣷⠈⠁⠀⠀⠀⠀⡇
⠀⠀⠘⠛⠃⠀⠲⠚⠓⠋⠛⠀⠐⠛⠁⠀⠸⠃⠀⠀⠰⠛⠁⠀⠟⠀⠐⠛⠁⠈⠃⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠐⠗
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⢀⣺⢗⣀⣿⠃⣀⡟⢃⣸⠟⣀⣸⠛⣀⡻⢇⣀⡟⢃⣸⡟⠀⠀⠀⠀⢀⣀⣀⠠⣶⠀⠀⠀⢾⠖⢾⣿⡋⡉⠛⠇
⠀⠀⠀⢿⡟⠛⢿⡿⠼⣿⡟⢻⣿⠋⢿⣿⠛⢿⡟⠻⣿⡛⠾⣿⠧⠀⠀⠀⠀⠀⢾⣿⡟⠀⠀⠀⠀⠀⠀⡏⠛⠀⡇
⠀⠀⢀⣸⠿⣦⡼⢿⡄⣼⣻⣦⡿⣦⢠⣟⣷⣸⢿⡄⣿⣻⣦⠿⣧⠀⠀⠀⢤⣤⡼⢻⡆⠀⠀⠀⣀⣀⣀⣇⣀⡀⡇
⠀⠀⠉⠉⠀⠛⠀⠘⠓⠋⠉⠉⠀⠙⠛⠉⠉⠁⠀⠛⠋⠉⠀⠀⠛⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠻⠛⠟⠛⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⢀⡾⣇⡤⠖⠂⠀⠀⢠⣦⣤⣠⣦⠀⠀⠀⠀⣄⠀⠿⣦⠤⠤⠀⠀⠘⢛⣤⡤⠆
⠀⠀⠘⠛⢿⣧⠀⠀⠀⠀⠀⢠⣾⡟⡇⠀⠀⠀⠀⠈⠙⠛⣿⣆⠀⠀⠀⠀⢹⣧⣀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢶⢰⣶⠛⠁
⠀⠀⠀⢀⣼⠋⠻⣦⠀⠀⠀⣈⣿⣆⠃⠀⠀⠀⠀⠀⠀⢠⡿⠙⠳⣆⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠔⠉⠀⠀⠈⢻⣧
⠀⠀⠀⠛⠁⠀⠀⠈⠁⠀⠈⠉⠘⠃⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠉⠀⠀⠀⠀⠛⠀⠀⢀⠄⠀⠀⠀⠀⢰⡶⣾⠇
⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢶⢀⣤⠀⡶⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠈⠃⣿
⠀⢀⡄⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣺⡇⠈⢫⣿⡞⠁⣠⡄⢰⠇⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢯⣟⡁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢸⣿⡇⢠⣾⣷⠋⠀⠀⠀⠂⠀⠀⠀⣷⣶⣾⣿⣿⡇
⠀⠀⢸⣿⣿⠀⠀⠀⣠⣤⣟⡁⠀⠀⠀⠀⠀⠀⣿⣿⠉⢸⣿⡇⠿⣿⣿⠀⠀⠀⠰⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀
⠀⠀⢸⣿⡇⠀⠀⠸⠿⠶⣬⣥⡀⠀⠀⠀⠀⠀⣿⣿⢠⣼⣿⣧⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣿⣿⣿⣷
⠀⢀⣘⡛⢃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣿⣿⣆⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⡇⠛⠋⠉⠁
⠀⠈⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣾⠀⡄⠀⠀⠀⠀⠀⢠⡆⣿⣆⡄⠀⠀⠀⠀⣬⡓⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⢀⢰⣄
⠀⠀⠀⢰⣿⣿⣿⣆⠀⠀⠀⢰⣼⣿⢿⣿⣿⠀⠀⠀⠀⣎⣿⣰⠄⠀⠀⢠⣿⡿⣿⡆⠀⠀⠀⣴⣿⡿⢻⣧⡄⠀⠀⢀⣸⣿⢿⣠
⠀⠀⠀⠀⠉⡋⠊⢻⠆⠀⠀⠈⠿⠉⠘⠫⠟⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠈⢋⣀⣙⠃⠀⠀⠀⠋⠀⠀⠛⠟⠀⠀⠀⠘⠟⠀⠘⠟
⠀⠀⠀⠀⠀⣻⣦⠈⠀⠀⠀⠀⠐⣷⣶⡿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠈⣿⣿⠃⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⣿⠇⠀⠀⠀⠀⠀⢹⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠙⣿⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⢶⣷⠖⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣰⡆⢰⡆⠀⠀⠀⠀⠀⠀⡙⢦⡀⠀⠀⠀⠀⣠⣸⣧⠀⠀⠀⠀⠀⢃⣿⠀⠀⠀⠀⢸⣷⡀⢰⣄⠀⠀⠀⠀⠀⣌⣿⣆⡀
⠀⠀⠀⠈⠻⣎⢷⢰⡄⠀⠀⠀⣼⣷⣿⡧⠀⠀⠀⢰⣿⡟⣿⠆⠀⠀⠀⠰⣫⡵⠂⠀⠀⠀⢮⣿⣿⣿⣿⣷⡀⠀⠀⠀⡿⠟⣿⣧
⠀⠀⠀⠀⢀⣨⠙⠼⠃⠀⠀⠀⠿⢉⡉⠀⠀⠀⠀⠈⣀⡘⠋⠀⠀⠀⠀⢀⣥⠀⠀⠀⠀⠀⠈⠙⠻⠀⠀⠹⠃⠀⠀⠀⢰⣤⡈⠏
⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠀⠀⠀⠀⠀⡀⠀⡤⢀⠄
⠀⠀⠑⡄⠀⠠⠢⡀⠀⠀⠡⡀⢈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⢄⠔⠀⠀⢀⠎⠴⠆⡠⠃
⠀⠀⠀⠀⡄⠀⠀⡘⣄⠀⠀⠑⡀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠠⠀⢀⠊⠀⠀⡰⠁⠔⢀⠌
⠀⠀⠀⠀⠈⢄⠠⠀⠉⢄⠀⠀⠐⠄⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠄⡠⠁⠀⢀⠜⠀⠊⡰⠁
⠀⠀⠀⠀⠀⠀⢂⠀⢀⠈⢆⠀⠀⠈⢆⠂⡌⢄⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠌⡒⠁⠀⡠⠂⠄⢃⠔
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡈⢃⠀⠀⠀⠡⠀⢈⢆⠀⠀⠀⠀⢠⠊⠀⢀⢠⠊⠀⠀⠔⠁⠂⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠡⡄⠀⠀⠐⠄⠀⢡⠀⠀⡐⠁⠀⠀⡰⠁⠀⢀⠈⠀⢃⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠠⠂⠘⢆⠀⠀⠈⠂⠠⢱⠊⠀⠐⣀⠜⠀⠀⡐⠁⠀⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠠⠀⠡⠣⡀⠀⢀⡵⠁⠀⠀⡠⠋⠀⢀⠌⠀⠂⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠱⣬⠎⠀⠀⢀⠜⠁⠀⡠⠁⠀⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠀⡴⠃⡀⠀⡡⠂⠀⢀⠌⠀⠊⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠶⠴⠴⠤⡼⡥⠤⠤⠮⠴⠤⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠲⠺⣥⣤⣤⠽⠶⠤⠤⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣀⠖⠶⠭⠩⠽⠶⢦⢤⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⡝⡧⡶⠀⠀⠀⠀⠀⠀⣀⢈⣽⣗⡽⢶⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠻⢳⣼⠟⣧⠷⠀⢀⠤⠤⠤⣄⢈⡖⣻⣿⣜⡦⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡝⣈⢡⣝⢧⡝⠁⠀⠈⢀⡀⠀⠀⣧⢀⡩⠙⠶⣛⣷⡿⡜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠠⣴⠗⣻⠀⠀⡄⠀⠀⡅⠀⠀⢹⠌⠁⡅⠀⢩⣽⣧⠿⡌⣧⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⡣⣼⣯⠾⠀⠀⠀⠀⠀⠁⠀⢠⣿⠀⠈⠐⠀⢠⠶⣾⢼⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡆⢡⢼⡦⣷⡀⠠⠤⠀⠀⢀⢔⠞⣝⠀⠀⡔⢈⣟⣿⡷⣿⠧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡰⡌⣶⣻⣏⡦⠀⠀⠀⠀⢿⠖⠖⠟⠀⠈⣠⡛⣼⣟⣾⠏⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠢⡡⢿⣳⣬⣍⡄⠦⠁⠌⠈⠠⡄⢳⣬⠷⢿⡽⣻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⠥⣛⢉⡨⣽⡿⠷⠲⢾⡞⣿⣯⡿⠛⣣⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠷⠶⢥⣤⢤⣬⣤⢴⠶⠟⠋⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡠⠀⢀⠀⡀⡀⢀⠀⡀⢀⢀⢄⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠉⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⢸⡌⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡈⠁⠀⠀⢰⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠈⡝⠀⠀⢀⠝⠣⢀⣀⣀⠸⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢠⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⢸⡀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠰⠲⠀⠄⡌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠢⡀⠀⡀⠄⠂⠁⠀⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢀⠔⠙⠉⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⡐⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠢⠤⣔⠁⠀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⢀⠇⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠂⠐⠊⠁
⠀⠀⠀⠀⠠⠊⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⡀⠀⠀⠬⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠡⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠀⠄⢀⣀⣀⠀⠠⠀⠇⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢆⠀⠀⠀⠙⠿⠷⣶⣦⣤⡀⠀⠉⠂⢄⡀
⠀⠀⣰⡷⠶⢠⣀⡀⠤⠄⠒⢀⠂⢣⠀⠀⠀⠈⠀⠒⠒⠒⡶⠤⡈⠛⢿⣦⡀⠀⠀⠰
⠀⠀⣸⡠⢾⠁⠀⠀⠀⠀⠀⢸⠀⠈⢆⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⢆⠀⠙⣿⣄⠀⢀⡄
⠀⣴⢳⢃⣾⠟⠉⠉⠁⠒⠒⠛⠂⠠⠬⠷⠤⠠⠐⠉⠀⠀⠀⠀⠀⠈⠢⠄⠜⠛⠉⠁⢡⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠂⠀⠒⢠⣄
⣸⠃⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠠⣀⣲⡆⠀⠀⠀⠀⠀⢠⡾⠋⠙⢆⠀⠀⣠⠛⠣
⠛⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡈⠁⠀⠸⡀⠀⠀⠀⠀⣾⡁⠀⠀⣼⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⡇⠀⠀⠀⠀⢿⡟⠉⠉⠙⢿⠛⠙⠢⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢰⣶⣶⣤⣤⣬⣷⣦⣀⣠⣏⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⡙⠻⠤⣴⣚⣩⣤⣤⠟⠙⠛⠛⠃⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣞⣉⣁⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣴⠖⣶⣦⡀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠐⠒⠤⡀
⠀⠀⠀⠀⢠⣿⡟⠤⠝⣿⣷⠀⠀⠀⠀⠀⢸⣇⡀⠀⠒⡗⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠄⠀⠀⡀⣿⡄
⠀⠀⠀⠀⠈⠿⢷⠒⠦⠎⠏⠀⠀⠀⠀⠀⠈⠛⢿⠙⠲⠃⠠⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⡘⣿⣿⣿⡇
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈⡗⠒⣶⢰⡀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⣼⣿⡿⠁
⠀⠀⠀⠙⠒⠤⢤⣀⣀⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠃⠄⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡤⣘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⢭⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠜⢓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⡆⠀⢸⣿⣖⣶⣤⠀⠀⠀⠀⠀⠀⠀⢪⠇
⠀⠀⠀⡀⢰⡷⣽⢻⡟⣿⢛⢿⣀⡜⠀⠀⠀⠘⣿⠉⠉⠁⠀⠈⠉⠉⢹⡿⠁⠀⠀⠀⠀⠀⠆⠹
⠀⠀⠀⠈⠙⠿⠿⠾⠷⠿⠾⠿⠉⠁⠀⠀⠀⠀⠀⠢⠄⣀⡀⢀⣄⣠⠞⠁⠀⠀⠀⠀⠀⢲⢠⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣯⣝⢿⣿⠿⣿⣟⣛⢾⢖⣛⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠄⢀⡀⣀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⢢⣶⣈⣧⢹⡄⠀⠀⠄⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⠦⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⣷⠀⠀⠀⡅⢸⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣥⢿⡀⢩⣜⠔⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢿⡿⠛⣉⣌⠀⢀⡀⡇⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣴⣿⣿⣶⠾⠛⢹⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠸⣿⣿⢍⣢⠠⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠷⠦⡀⡆⢸⣿⣿⣿⣇⣬⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣸⣹⣿⣿⡏⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣿⢼⣿⣿⣿⣿⣮⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠐⠀⠂⠀⠀⠀⠀⠀⠀⠄⠀⠤⠀⠀⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠖⠂⠀⠤⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠉⠉⠉⠙⠛⢶⣄⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠉⠉⠙⠻⢶⣄⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⣟⠛⠛⡷⢶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⠶⡟⠛⢛⡷⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⣸⠏⠀⠀⢿⡄⠈⠻⣆⠀⠀⠀⢀⡾⠋⠀⣼⠇⠀⠈⢿⡀⠈⢻⣆⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⣾⠁⣠⡾⠋⠀⠀⠀⠈⠻⣦⣀⢹⣆⠀⢀⣾⢁⣠⡾⠋⠀⠀⠀⠈⠻⣦⡀⢻⡆⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⠶⢾⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣿⠶⢾⡇⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠶⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⣤⣤⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣤⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⠐⠈⢰⠳⣄
⠀⠀⠀⠀⣶⠖⣶⠀⣠⣟⢴⠁⠢⠄⠀⢀⢿⡀
⠀⠀⣀⡀⠘⢈⡏⢰⡏⢑⠾⢀⠐⠀⠀⡂⠸⣷
⠀⢼⡁⢻⠀⡇⡇⢸⡔⡨⠄⢀⠀⠀⢀⡂⣸⡏
⠀⠀⠹⣄⢧⣥⣷⠘⣇⢋⠀⠀⠀⠀⠀⠀⡽⠁
⠀⠀⠀⠘⣌⢻⣼⠀⢻⡼⡚⠦⠀⡠⢀⣼⠃
⠀⠀⠀⠀⠈⣆⢿⡆⠀⢛⢦⣬⣰⣐⡿⠁
⠀⠀⠀⠀⠀⡟⢶⠘⣆⠀⠃⠀⡔⠁
⠀⠀⠀⠀⠀⡇⠘⣆⠈⢳⡄⢰
⠀⠀⠀⠀⠀⠃⠀⠸⡄⠀⠙⢾⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⠤⡤⠤⣀
⠀⠀⠀⠀⠀⡇⠀⠀⠻⡀⠀⠀⠣⡀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠁⠀⠨⡆⣷⡀⠀⠀⠱⡀⠀⠀⡰⠁⠀⠀⡀⠂⠀⠀⠀⢀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣷⡀⠀⠀⠱⣄⡠⢅⠀⠐⠈⠀⠀⠀⢠⠊⠁⠀⢀⠴⡚⠉⠉⠀⠈⢇⠢⢄
⠀⠀⠀⠀⠀⢠⠀⠀⢹⡿⠛⢷⡄⠀⠀⠘⠅⠀⠱⡄⠀⠀⠀⠀⠀⠀⢄⡰⠁⠀⣴⣦⣤⠀⠀⠀⠁⢰⣧
⠀⠀⠀⠀⠀⢸⠀⢠⠜⠃⣦⡀⠹⢆⠀⠀⠈⢆⠀⠹⡀⠀⠠⠀⠀⠀⢠⣄⡀⠘⢿⣿⡿⠀⠀⠀⠀⠸⢿
⠀⠀⡔⠊⠃⢹⡀⠽⠄⠀⠀⠙⠢⠈⢂⠀⠀⠈⢢⡰⠓⡆⠉⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠂
⠀⢸⠀⠀⠀⠈⣧⣀⠀⠈⠑⠂⠀⠀⠘⢧⡀⠀⠀⠡⣄⡙⠄⣀⣀⠀⠘⠣⢄⣀⣤⣆⠀⠀⠀⡄
⠀⠀⠢⠀⠀⣀⣇⠹⡀⢀⡀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠘⡜⡄⠀⡘⠒⠈⢡⡀⢻⣿⣿⡗⠀⠀⠀⠀⣶⡖
⠀⠀⠀⠀⠀⠀⠈⠒⠚⠋⠀⠀⠀⠀⠀⠀⢀⡚⠢⢴⠴⠓⠃⠀⠀⠀⠀⠀⠉⠢⢈⠉⢢⡀⠀⠀⠄⠋

⡀⠀⠀⠀⠀⡀⠀⢀⣀⡀⡀⠀⢀⣀⣄⣸⣿⣷⣶⡔⠂⠠⡴⠂⣶⣤⣶⠜⠋⠛⣿⣌⣉⣓⢒⣶⣤⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⣤⢋⣾⣿⣿⣻⡀⠀⠀⠀⠀⠀⢀⣙⣿⠋⠀⢀⣐⡛⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣿⠧⣿⣿⣿⠐⠲⣶⡖⣶⣬⠁⠀⣸⣶⣿⣿⣿⣿⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⣀⣀⣠⣤⡤⠶⠚⠋⢹⢠⣿⣿⣀⣿⡏⣿⣿⣷⣏⠁⠀⠁⠀⢀⠉⠉⠻⠿⠿⠋⠀⡐⠁⠀⠀⠄⡀⠀⠀⠀⠀⠈
⣿⣿⣿⠿⠛⠙⠀⠀⢰⣾⣿⢰⣿⣿⣿⣷⣿⣿⡿⠿⠛⠛⢛⣀⣉⣉⣀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⡷⣼⣿⣿⣿⣿⣷⣿⡷⠶⠾⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣧⣿⡋⣿⣿⣿⠻⣷⣿⡇⣴⡞⠻⠿⠿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠆⠀⢸⣿⣿⣇⢻⣿⣿⣶⢻⣿⡅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠹⢿⣦⡲⣾⣷⣖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣄⣿⣿⣿⣿⠟⠟⠀⠀⢼⡇⠀⠀⠘⢷⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣟⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⡟⠛⠛⠁⠀⠀⠀⠀⠈⠙⠀⡀⠀⢸⠏⠀⣩⡉⠐⠦⣄⠀⢠⣄⠀⠀⠀⢻⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⡀⠀⠀⢀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢠⣄⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠃⠀⠀⠀⠀⠉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠿⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⠆⢤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⣤⠜⠛⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⠚⠉⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⡘⠻⠛⠛⠛⠛⠉⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠜⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⡀⠰⡄⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⣠⠀⠈⠀⠀⠀⠂⠀⠈⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠠⡦⠀⠋⢆
⠀⣤⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣰⡇⠀⠀⠈⠅⢀⡄⠀⠒
⠰⠯⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣈⣥⠀⠋⠹⡄⠀⣀⠀⠸⠓
⢠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠤⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⡦⠗⠀⠀⠀⠀⠀⠈⠁
⢸⡆⠸⠆⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡝⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤
⠘⠁⠀⡄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⡇⠀⠀⠀⠀⡄⠀⠀⠀⡀⢀⣠⣄
⠀⠀⠀⡇⠘⢦⡀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠁⠀⠀⢠⣴⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠉⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠄⠀⠀⠀⠀⡸⠏⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣴⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠠⠂⠂⠀⠀⢀⡀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠰⣶⡿⠋⠁⠀⢀⠈⠀⠀⠀⠀⠀⠀⣴⢿⣿⠿⠋⠁
⠀⠀⠰⠶⠶⣶⣦⣷⣶⠆⠒⠶⠶⣂⡨⠿⠿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⢘⣆⣸⣿⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⡀⠉⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⡏⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⠟⠙⣿⡖⣀⣾⣿⠋
⠀⢸⣷⡴⢤⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠻⠟⣿⣿⡿⠉⠀⠀⠀⠀⢠⠆
⠀⠘⣿⡗⣫⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠈⠁⢺⣿⣿⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠈⣿⠷⣺⠀⢌⡦⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠄⠄⠀⢠⠀⠀⣄⢹⠀⠁⠞⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⣿⣟⣣⣾⡼⢇⠀⠀⣹⡏⠀⠂⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⠇⢄⠀⠰⣿⣿⠄⠀⠀⠀⠀⠀⠀⢡
⠀⣰⣿⡋⠻⣍⡃⠈⠀⠈⢉⡄⠀⢤⣴⢴⣂⠀⠀⠀⠀⠘⢅⠀⠀⠀⠀⢹⣧⡀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣜⣿⣇⣡⢋⣵⣧⡤⢨⠇⠀⣀⣨⣤⣯⣵⣿⣮⣤⣴⣿⣦⠀⠀⣀⣶⣿⣿⡷⣂⣘⣻⠀⠀⠀⠀⠀⠀⢀⡞
⡴⠉⠉⠛⢛⠓⠛⠛⠓⠚⠓⠈⠴⠿⠿⠿⠿⠟⠋⠛⠛⠙⠳⠶⠾⠛⠟⠛⠛⠛⠛⠛⠛⠂⠂⠰⠀⡀⢠⡉⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⣴⣾⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠇⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠉⠹⣏⣻⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣡⡤⠄⠀⣀⣤⠖⠋⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣄⣀⣉⣉⣉⣤⣴⡶⠟⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡿⠉⢀⣀⣀⠀⠀⢀⡿⠿⠄⠀⠀⠀⠀⠸⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢿⣿⡄⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⢦⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⠻⣄⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠀⠈⠙⠂⠠⢀⣠⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠉⠁⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡇⣀⣉⠓⠢⠭⣹⡗⠀⢀⣯⣤⡶⠖⠂⠆⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣧⠀⠉⠙⣻⡶⢀⣷⣿⢿⣿⣿⣯⣄⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠈⢉⡴⣿⣿⠟⠘⣿⠿⣿⠶⠤⠄⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠁⠴⠿⠿⠀⢀⡸⠀⠈⠂⠀⣠⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠟⠻⠿⠿⠿⠿⠛⠛⠿⠿⠿⠿⠶⠶⠶⠶⠊⠹⡿⠿⢿⣿⠏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⠛⠉⠘⢽⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡏⠙⠉⠀⡀⠀⣿⣿⣿⣥⣄⣀⣤⡈⣿⣷⠀⣰⣾⣿⡟⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠉⠳⣤⣭⣕⠈⢻⣿⣿⡙⡷⠨⠉⢠⣿⠁⢰⣿⣿⣿⣷⡾⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⠰⠶⠀⢿⣿⣄⠸⣿⣿⣿⣿⡿⣤⣾⣿⠀⠸⣿⣿⡻⠷⠶⠶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⡀⢀⣾⣾⡿⠀⢹⣿⣿⣿⣷⣾⡿⠁⠀⠀⠙⣿⣷⣼⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣜⣿⢋⣿⣧⣤⣾⣿⣿⣿⡿⠋⠈⣶⣆⡆⠸⣿⣿⣿⣿⣯⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⡵⢿⠟⢻⣿⣿⠿⠋⠻⣤⠀⢠⣿⡟⢀⣴⠟⠛⠋⠁⢿⠏⡿⣴⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⠃⠀⠨⣿⣽⡎⠿⢧⡀⠀⣀⠠⣾⣿⡶⠋⠀⣀⡴⡛⢰⣿⣼⢷⡟⢷⣯
⠀⠀⠀⠀⠀⠀⣼⠋⣽⣿⢿⢽⣿⣿⣄⠀⣼⡟⢿⠀⠀⠑⢌⠹⣋⣿⡿⢋⢔⣴⣿⡿⡇⢇⠀⠙⣵⢊⢀⣴⣟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠈⢹⣿⣿⣿⣿⣿⣧⡄⠠⠂⠀⠙⣿⣿⢟⣡⡶⢻⠟⢸⠀⣧⢸⡀⠀⠞⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦⣄⣾⣿⡿⢿⣿⣿⣿⣿⣷⣶⣤⡿⣋⠕⢉⡵⢡⡟⢀⣾⠈⣦⡟⠣⢰⠞⢛⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠟⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⢟⣵⣶⡏⣠⣠⢁⣿⣿⡀⢸⡇⠀⠀⣾⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣷⡿⠀⣸⣿⣿⣿⣿⣿⣽⣾⣿⣿⣭⡾⣫⠏⣼⣼⣿⢀⢸⣷⠀⠈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢸⣧⣤⠏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢡⣾⣿⡿⣿⣼⣼⣿⣆⡄⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣵⣿⣿⣿⡿⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠉⠙⠻⠛⠉⠉⠻⠿⠿⠟⠻⠏⠉



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

⠐⢆⣘⣟⠀⠀⠀⣀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠰⣦⡤⢖⣤⠶⢦⡲⢤⣴⡷
⠀⠀⢸⣿⡓⠚⠁⠈⠙⠒⢿⣿⠒⠦⠀⠀⠀⠀⢠⡿⠳⠿⠽⢿⣦⠀⠀⠀⠺⠛⢿⡏⠀⠀⠀⠹⣿⠟⠗
⠀⢰⡏⠉⢻⡄⠀⠀⠀⢰⡞⠛⢦⣄⡀⠀⠀⠀⣸⠻⡆⠀⠀⡾⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠁⠀⠁⠀⠀⠁⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⣄⠀⢀⢰⣶⢀⠀⢀⡄⠀⠀⠀⠀⠀⡰⣶⡄⠀⠀⢤⣀⢠⣦⠀⢀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢼⡷⣤⣸
⠉⠀⠘⢻⣿⠛⠀⠈⠁⠀⠀⠀⢠⣿⡛⢮⡤⠀⠀⠀⠈⢻⣿⠚⠉⠀⠈⠙⣻⣿⢦⡼⠃⠀⠰⠤⢿⣾
⠀⠀⠀⣼⢻⡀⠀⠀⠀⠀⣶⣿⣧⣿⣿⡾⠋⠓⡄⠀⢠⠟⣿⠀⠀⠀⠀⢀⡿⢷⡀⠀⠀⠀⠀⠀⢸⡿⣦
⠀⠀⠀⠿⠸⠇⠀⠀⠀⠀⠻⠿⠟⠉⠀⠙⠤⠔⠁⠀⠸⠆⠘⠷⠀⠀⠀⠞⠁⠸⠇⠀⠀⠀⠀⠰⠋⠀⠟

⠀⠀⢰⡦⠀⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣦⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣻⣃⣸
⠀⠙⣿⣟⢛⠇⠀⠀⢿⣿⠛⠋⠀⠀⢀⣶⣿⠚⠋⠁⠀⢀⣴⣲⣼⠛⠀⠀⠀⣨⣏⣠⡆⠀⠚⢹⣿⣉
⠀⠀⡿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⢸⣿⡀⠀⠀⠀⠀⢹⣿⡉⠒⠀⠀⠸⣿⣄⠉⠀⠀⠀⢀⡿⣽
⠀⠰⠇⢿⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⢠⠞⠁⠿⠀⠀⠀⢰⠏⠙⠧⠀⠀⠀⠋

⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠠⣅⣽⡣⢄⡀⠀⠀⠀⠘⣷⣾⣉⡿⠋⠀⠀⢠⡤⢤⣿⡤⠤⡄⠀⠀⣀⣤⣤⠞⠁⠀⠀⠀⠠⣄⣽⡣⢀
⠀⠀⣻⡦⠞⠀⠀⠀⠀⠸⡏⠙⠛⠶⠶⠆⠀⠀⠀⢸⣿⡇⠀⠀⠀⠘⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⣻⡦⠞
⠀⣰⠃⠀⠀⠀⠀⠀⠻⡿⠛⠛⠛⠛⢿⠟⠀⠀⠀⢸⣿⡇⠀⠀⢀⣀⣀⣛⣿⣀⣀⣀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠉⠉⠉⠩⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢀⡀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⡆⢡⡘⠁⠀⣤⣿⣤⣀⠀⠉⢀⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣦⡀⢠⣼⠃⠀⢸⠟⢡⣤⣭⠉⣻⡍⣉⣻⣿⡷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣾⣥⣦⣤⣾⣴⣯⣿⣿⣄⠈⢣⡀⠉⠙⠛⣒⣛⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣿⣿⠿⠿⠟⠋⠉⠀⠀⠀⠀⠈⠹⣿⢶⣤⣼⣿⠿⠋⠩⣭⡽⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⣉⡛⠳⠒⠀⠐⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡽⠀⠀⢀⠂⠀⢀⣀⣀⣀⣀⡀⠀⠻⣷⣀⣴⣦⡉⢤⣉⠋⠁⠞⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣧⣴⣤⣤⣶⣿⣿⣿⣿⣿⣿⣷⠶⠦⢭⣿⣿⣿⣿⣶⣤⣁⣴⠾⠛⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠶⢿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢹⡀⠙⣿⣿⣿⣿⣿⣷⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢠⣿⣿⠁⠀⠹⣿⣿⣿⣿⣿⣄⠀⠀⠠⣾⣍⣹⣯⡿⠟⣰⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢣⣤⣤⣤⣸⠦⡀⠉⠙⢿⣷⠀⢀⠈⠀⣾⣿⣀⣠⣾⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠦⢿⡟⠉⠋⠀⠀⠀⠀⠸⣇⠲⣈⠓⢦⣨⣿⢿⣿⣿⣿⣟⣳⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⣷⣾⠿⢷⣶⣷⣄⣀⣀⡉⢉⣷⡀⠀⠀⢉⣿⠀⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢿⣿⠗⠂⠈⠉⠛⢻⣿⠛⠻⡉⠀⣉⢻⣵⣿⡏⠀⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⡰⠁⠀⠀⢀⠙⠆⠀⠉⣄⠠⣤⣰⡉⣿⣿⣿⡇⠀⠁⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠛⠃⠈⠃⡀⡀⢱⡀⠀⠀⢸⡧⣸⣟⣰⣿⣿⣿⠀⠀⡘⠀⢻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠂⠀⢀⣀⣿⢿⣠⠀⢀⣴⢿⣿⣿⣿⣿⣿⣿⠇⠀⢰⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⣀⢰⡏⢁⢄⣼⣫⣤⣼⢃⣾⣝⣿⣿⣿⣿⠏⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣵⣮⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣽⣿⣿⠿⠿⠛⠻⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠋⠁



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

⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⢀⠔⠹⣄⠤⠔⠒⠚⠓⣄⠀⠀⠀⢀⣴⡇⢟⣫⣭⣵⣶⣶⡄⠀⠀⠀⣠⡎⠁⠀⠀⣀⣀⣉⢢⡀⠀⠀⡀⢾⣿⣿⡿⢟⣛⡻⢦⡀
⠀⠀⣏⠀⠀⠱⣄⠀⣀⣀⣀⠘⡆⠀⠀⢾⣿⣿⣜⢿⣿⠿⠿⠿⣿⡄⠀⢰⠃⠘⠒⠒⠋⠀⠀⠈⢣⢱⠀⢰⣿⣶⣭⣵⣾⣿⣿⣿⣎⢷
⠀⠀⡏⢦⡀⠀⢨⠏⠁⢸⡏⠉⡇⠀⠀⣮⡻⣿⣿⡗⣴⣿⣿⢸⣷⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸
⠀⠀⠱⡄⠉⢲⠞⠀⠀⣸⢀⡼⠁⠀⠀⠹⣿⣮⡟⣱⣿⣿⡏⣾⡟⠀⠀⠀⢣⡖⠉⠉⠒⠤⡤⠴⣣⠎⠀⠈⠟⣭⣭⣭⣝⡻⠿⢟⣵⠃
⠀⠀⠀⠈⠒⠥⣄⣀⠴⠗⠋⠀⠀⠀⠀⠀⠈⠋⠾⢿⠿⠟⠘⠉⠀⠀⠀⠀⠀⠙⠢⠤⣀⡠⠤⠚⠁⠀⠀⠀⠀⠙⠻⠿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀
⠀⠀⢀⠴⠿⠁⠀⠀⡹⠿⡷⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣦⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠂⠀⠀⠀⠀⣠⣾⣿⣷⣦⣉⠻⠄
⠀⠀⠀⠀⠀⣼⣾⣿⠀⠀⠀⡀⠀⠀⠀⣾⣿⣿⠁⠉⠈⣿⣿⣿⣿⡀⠀⠀⠀⠄⠀⠀⢀⣴⠗⠀⠙⠇⠀⠀⠀⣼⣿⣿⡟⠃⣸⣿⣷
⠀⠀⣷⡄⠈⠛⢿⠟⠁⠀⣴⣿⡇⠀⠀⠈⣻⣷⣤⣀⣤⣾⣿⠋⠀⠀⠀⠀⡀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⢀⣾⣿⣿⠏
⠀⠀⠹⠇⠀⠀⣀⠀⠀⠀⠟⠟⠁⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣦⠀⠀⠀⠀⠛⢦⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣈⠻⣷⣤⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠾⢿⣿⠯⠀⠈⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠹⠛⠁⠀⠀⠀⠀⠀⠀⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡉⠛⠋

⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⢀⡄⠂⠀⠐⠂⢄⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣆⠀⠀⠀⡔⢹⠃⠀⠀⣠⠖⠛⠑⠄⠀⢀⣼⠁⣿⣿⡿⢋⣥⣶⣶⡄
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⣿⣿⣿⣿⠀⠀⢰⢠⠏⠀⠀⣼⠁⠀⠀⠀⢸⠀⢸⢏⣼⣿⣿⢃⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣻⣿⣿⣿⣿⣿⣿⠀⠀⠘⡏⠀⠀⠀⢿⠀⠀⠀⠀⡘⠀⠠⣾⣿⣿⣿⠸⣿⣿⣿⣿⡏
⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠘⢄⡀⠀⠈⠧⣄⣀⠔⠀⠀⠀⠙⢿⣿⣿⣧⡙⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡈⡁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⣬⡉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠇
⠀⠺⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢘⠤⠔⠒⠠⢄⡀⠀⠀⠀⣸⣿⢇⣠⣤⣤⣤⣀
⠀⠀⠀⠹⡗⢄⠀⠀⠀⡀⠄⠠⡀⠀⠈⢱⣄⠀⠀⠀⢀⣠⣤⣶⡄⠀⠀⠀⠐⠀⣰⠁⠀⠂⣀⠀⠀⠑⡄⠀⢸⣿⢣⣿⣯⣽⢿⣿⣿⣧
⠀⠀⠀⠀⠐⡀⠁⠂⠁⢠⣆⠀⠉⢀⢠⠀⢻⣿⣶⣾⣿⡏⡶⠿⢿⠿⢶⠀⠐⠀⠇⠀⠘⠀⠁⠀⠀⠀⡇⠀⠸⣿⣿⣿⣶⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠄⢀⣀⣈⡀⠤⠄⠂⠂⠀⠀⠻⠿⠿⣿⣷⡭⠟⠛⠛⠋⠀⠀⠣⣸⣄⠀⠀⠀⠀⠀⡰⠃⠀⠀⠻⠜⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠒⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⡿⢿⢿⠿⠿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡄⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠞⠿⠁⠈⣇⠘⢻⣧
⠀⠀⠀⠀⠀⣀⣄⣸⣣⣿⠟⠶⡄⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣸⡿⠆⠈⢃
⠀⠀⠀⠀⠀⠙⠛⠉⠉⠻⠀⠀⣿⠀⣰⠀⣀⣀⠠⠀⠐⠶⣄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣠⣷⠠⢈⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣦⣁⡜⠙⢀⣸⣏⠻⣿⣿⢿⣿⠟⣶⠀⠀⢀⠄⠀⠀⠀⢐⣿⡿⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢋⣽⠃⠀⠀⣨⣾⣦⡈⠟⠳⠄⠛⡟⠑⠢⣄⣐⣦⣤⣆⡁⠀⣠⣴⠖⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠁⣠⠶⠈⠋⠉⠛⠉⠀⠠⣄⢀⣀⠀⣒⣺⣛⣿⣿⣿⣑⠈⣯⣴⠁⠀⠀⠀⣱⣄⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠙⠻⡿⢿⣿⣿⣿⣿⣿⠧⢿⠇⠀⠀⢀⣄⣻⠏⠉⠛⢿⣿⣧
⠀⠀⠀⠀⠰⠳⡀⠀⢀⣾⣴⣶⣠⣼⣄⣀⣠⣶⠀⠀⠀⠀⢒⣿⡄⠀⠁⠀⠀⠙⠿⠻⠻⠑⠤⠀⠀⣴⣿⣿⠏⠀⠀⠀⢠⣿⣿
⠀⣤⣶⡦⠴⠶⢷⡒⠋⠉⠉⠉⠉⠁⠉⠉⠉⠋⠉⠃⠠⡄⠀⢿⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠉⠻⣷⠀⠀⠀⠀⠈⠟⠃
⠀⠙⢧⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠠⡄⠡⣿⣷⡀⠀⠀⠀⠐⠚⠛⠋⠀⠴⠟⠃
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠒⠂⠀⠀⠀⠀⠴⠠⠤⠄⠂⠀⠀⠀⠙⠀⠀⠀⠰⣆
⡄⠲⠇⠺⠛⠁⡈⠀⡤⣀⣶⡆⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⣲⣚⣲⣖⣨⣁⣄⣠⣈⣉⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢰⠏⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⢸⣆⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣤⣾⣿⣿⠏⠀⠀⠀⠀⠙⠳⢶⠞⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠈⣿⠋⠉⠉⠉⠙⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠁
⠀⠀⠀⣠⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⡀
⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡇
⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣷⡄⠘⢿⣿⠟⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠁⠀⠀⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠃⠀⠀⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠃⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠰⠃

⣿⣿⣷⣾⣷⣿⣾⠶⠖⠛⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣄
⣤⣥⣟⡻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢸⡿⠟⠋⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡟⠿⠄
⠛⠀⢱⡦⠁⢹⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣡⡄⢀⡀⠀⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⡀
⠀⠀⠈⡇⢀⡸⠀⠀⣶⣶⣦⡄⠒⠄⣀⣀⣀⡀⠀⠈⠁⠀⠀⠠⠐⠛⢻⣸⠁⢀⠀⠀⠀⠀⠀⠰⣄⢀⣀⣀⠀⠀⠀⠻⠃⡀⢰⣾⣧
⠀⠀⠀⠀⢹⣿⣞⣀⣹⣿⡯⠁⢀⣀⣠⣤⡭⣿⠂⠀⢀⣠⠀⠀⠀⠀⠀⠁⠀⠘⠁⠰⣿⣦⣾⣿⣿⠿⡿⠮⠁⠀⢼⣿⣿⣿⣿⣿⣿
⢀⣾⣀⣿⢈⣿⣽⣦⡘⠹⣿⠛⢛⠿⢿⣁⣴⣃⣤⡔⠓⠇⠀⢀⡠⠄⠀⠀⠀⠈⠀⠶⢿⠿⠟⠛⠃⠀⠀⠀⠀⠀⢰⣾⣿⡿⠛⠋⠙
⣟⣿⣿⣿⣿⣿⣿⠾⡏⣺⡗⠾⠋⠒⠿⠝⣿⠀⡛⠓⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⣿⣿⡿⣿⣿⣿⣿⢿⣾⣿⠖⠀⠀⠀⠀⠀⠉⠀⠄⢰⢯⢶⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣴⠄⡠⠂⠀⠀⠀⠀⢀⠄⢸⣿⡇⠀⠀⠐⢶
⣿⣿⣿⣿⣿⣿⡿⠂⣺⢳⣎⠀⠐⢆⡀⠀⠀⠀⠀⣿⡖⠆⠳⠶⢾⣷⣦⡀⠀⢠⣿⡟⣿⣯⠀⣸⣤⣄⡀⠀⠀⠀⠸⣿⡁⠀⡄⣶⣄
⣿⣿⣟⣻⣿⣹⣿⡷⢤⣼⢛⣦⡄⠈⠹⢆⣠⣴⣾⣿⣧⣀⡴⠀⠀⠹⣿⡟⠢⢸⣿⣿⣿⡿⠠⣿⣿⢿⣏⣶⣫⡷⡦⠀⠀⠀⠁⣿⡇
⢿⡹⣿⣿⣿⣿⡿⢇⠀⢿⠈⣿⣇⡔⣊⣿⣿⣿⣷⣬⣿⡿⣭⣤⣶⡶⢹⣿⣤⠞⣿⣿⣿⣇⠀⢹⠛⠒⣾⡉⠛⠃⠀⠀⠀⠀⠀⣿⡇
⣿⣧⣹⣿⣿⣿⣿⣿⣷⣿⣶⣿⣷⣾⣿⣿⢿⣿⣿⣮⣿⣥⣽⣿⣿⢷⡶⠍⢋⣴⣿⣿⣿⣿⡶⣏⡀⢸⣿⣿⠆⠀⠀⠀⠀⠂⠀⠉⣡
⠛⣛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣯⢧⣹⡝⣯⣀⣤⣿⣿⣿⣿⣿⡟⠚⡿⠛⠀⣿⣿⣟⠋⠐⠦⣀⠰⢺⣟⣿
⣿⣿⣿⣿⣿⣿⣿⢿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢙⠚⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣾⠿⠛⣷⣦⠀⠀⢤⣴⣆⡄⠉⠈
⣿⣿⣝⣿⣿⣿⣾⣿⣿⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣼⣿⣶⣀⣀⣑⡂⠀⠀⠉⣩⣤⣴⣶
⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣿⣿⣿⣿
⣿⣿⣥⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡄⠀⣀⣀⡀
⠀⣠⣶⣶⣶⣾⠀⠘⣿⣿⡄⠀⡀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⣀⣀⣠⣀⣄⡈⢸⣿⣿⠀⠸⣿⣿⡿
⢸⣿⣿⠋⠉⠹⡆⠀⢸⣿⣷⣿⣿⣷⡀⠀⢸⣿⣿⣻⣿⣧⠀⢹⣿⣿⣿⣿⣿⣸⣿⡏⠀⢶⣦⣴⠖⠀⢀⣀⣀⡀
⢼⣿⣿⠀⠀⠀⠀⠀⠘⣿⣿⠈⢿⣿⡇⠀⣠⣶⡿⢿⣿⣿⠀⢀⣿⡏⠈⠋⠁⣿⣿⠃⠀⢸⣿⡏⢀⣴⣿⡿⠻⣿⣧
⠸⣿⣿⣇⠀⠀⢠⣤⠀⣿⣿⠀⢸⣿⣇⢸⣿⣿⣤⣾⣿⣧⣄⣸⣿⣧⠀⠀⢰⣿⣿⠀⠀⣾⣿⠁⣼⣿⡿⣷⣶⣿⣿
⠀⠹⣿⣿⣷⣤⣴⣿⢠⣿⠿⠧⠿⠛⠛⠃⠉⠛⠛⠁⠉⠉⠁⠉⠉⠉⠁⠐⠛⠛⠿⠆⠼⣿⣿⡄⢿⣿⣇⣲⣿⣯⠁
⠀⠀⠈⠛⠛⠛⠛⠁⠈⠀⠀⠠⣆⡴⡄⢠⣷⣸⢷⣾⣛⡆⠘⡏⣯⣶⣾⡅⢠⢤⣠⠀⠀⠀⠈⠙⠈⠻⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⠀⠀⠀⣀⠈⠈⠉⢈⣉⠉⠀⢀⡁⠁⠉⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⣼⠋⣽⠇⣿⠀⢸⣧⣾⠋⠻⣆⣸⡏⢽⡇⣼⡟⠙⣷⣼⡇⠈⣿⢿⡜⢻⡟⠛⣿⠛⠛
⠀⠀⠀⠀⠀⢿⣆⣠⡖⣿⠛⢻⡟⣿⣀⣰⡟⢻⣇⣠⡶⢻⣄⣀⡿⣿⣇⣸⣿⠾⣷⢸⣇⠀⣿⣛⣃
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠈⠁⠈⠉⠉⠀⠀⠉⠉⠀⠀⠉⠉⠀⠈⠉⠉⠉⠀⢉⡈⠉⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠋⣼⢿⡛⢰⡟⢻⡆⠛⣿⠛⣴⠟⠻⣦⢸⡟⢛⣷⢻⡇⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢠⣿⢼⣧⢻⣆⣠⡦⠀⣿⠀⣿⣄⣰⡿⢸⡟⣿⡋⣤⢻⡟

⠀⠀⠀⠀⠀⣸⣿⣿⡿⠋⠉⣿⠫⡛⣿⣇⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣀⣀⠀⠀⠀⠸⣇⢸⢤⣶⡟⣻⡟⠋⠀⠘⣿⣿⡧
⠀⠀⠀⠀⢠⣿⣿⡟⡷⠖⣶⠓⡦⠼⢺⣿⠀⠀⢰⡏⠉⢻⡄⣷⠀⡾⠉⢩⡇⠀⠀⠀⢿⡀⢦⡿⠋⠳⣿⠃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣾⣿⣿⠁⡇⣠⠟⠦⣑⡰⢸⡿⠀⠀⠘⢧⡤⠟⠀⡇⡆⠹⠶⠞⠁⠀⠀⠀⠀⠛⠄⣬⡉⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⣿⣿⡇⠀⠈⠛⠠⠀⠈⢁⡾⠃⠀⠀⠀⢠⣶⠒⠚⡇⠻⠂⣀⠖⠀⠀⠀⣀⣀⡀⠀⠈⠙⠳⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⢀⡴⠚⠉⠀⣀⣤⣤⣀⣀⡈⠙⠻⣇⣠⠞⠁⣀⣤⡞⠻⠛⠉⢹⣿⣾⣄⠀⢹⡄⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⡆⠀⠀⢠⡟⠀⣰⣷⣺⣿⣿⡅⡠⠍⠻⣷⣄⠘⠃⢠⡾⢋⠁⠀⠀⠤⣴⠃⣿⣾⣿⣷⡀⣷⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⢹⣿⡇⠀⠀⣼⠀⣸⣿⡛⢟⣽⡇⠹⣆⠥⢀⠈⢻⣇⢴⡟⠀⠀⠀⠄⠀⣴⠃⠀⣿⡷⣯⣻⡇⠸⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠈⣿⣿⠀⠀⣿⠀⣿⡏⣰⠿⣣⣧⠀⠹⡎⠒⠀⡀⢿⣾⠃⠁⢀⣀⢡⡞⠁⠀⢰⣿⡿⣯⢹⡇⢐⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣇⠀⢹⡆⠸⣷⣩⣖⡶⣿⠀⠀⠹⣆⣀⠀⢼⣿⠒⢉⣩⣰⠿⠀⠀⠀⣼⣯⡻⢮⢿⡇⣼⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⢳⠀⢻⣎⡿⣷⠋⠀⠀⠀⠈⡙⠓⠾⠿⠞⠋⣉⣤⠆⠀⠀⠀⠻⣝⣏⢢⣼⢡⠇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠤⠷⠄⠙⣿⠃⢀⣀⣀⠈⠻⡙⠛⠛⢲⡶⠊⢁⡏⣠⠶⠶⢶⣄⠈⠻⣿⠇⣾⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠰⢖⠒⠒⠛⠉⠉⡿⡢⠀⣴⠋⣉⣙⠳⡄⠙⣦⢰⠋⢹⡆⡾⢱⢃⡤⢤⠀⠹⡄⠀⡼⡶⠒⠒⠒⣂⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⠀⢰⡇⢷⠀⣇⠀⠧⢸⠀⠹⡄⠘⢷⡑⠚⣽⠃⢸⡈⢁⡸⠀⠀⣇⠀⡇⢧⠀⢤⡚⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⣀⣈⣧⠼⠆⠘⠳⠒⠋⢶⡆⢻⠀⡈⢷⣰⡏⢀⠀⠙⠋⢁⠀⣰⠏⢀⡓⠷⠀⠠⠭⠷⢤
⠀⠀⠀⠀⠀⠀⠼⠕⠚⠋⠉⢠⠤⠲⡔⢏⡆⡴⠒⡲⠀⢸⣌⠉⠀⠿⠀⠙⠀⠛⠀⠙⣰⡏⢠⠊⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⢎⡜⢀⡼⢱⡢⢸⠦⣍⣉⡐⠒⠂⠤⠤⠤⣤⠏⢠⠃⡰⠄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡌⢡⠞⠀⣤⡄⢸⣇⣒⡒⠂⠠⢄⣒⣒⣿⠃⣠⣁⣰⢱⣜⣄⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⣨⡁⢸⢓⣀⣁⣀⣐⣒⣶⠿⠁⠀⠉⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⢃⡮⣵⣂⠄⣉⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣸⡏⠢⠭⣳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⢠⣿⢭⠛⢿⠏⣀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢸⣿⣿⣻⡏⠘⣛⠤⣍⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠸⣿⣯⣿⠀⣞⣀⠀⢸⠄⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡁⢻⣿⣿⡀⠘⠋⢀⡟⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣠⡍⢻⢷⣦⠶⢋⣄⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣈⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣈⣀⡀⢀
⠀⠀⠀⠀⠀⢀⠔⣡⣾⣿⣿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⣿⣿⣦⡑⠄
⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⡡⠀⠀⠀⠄⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠄⠀⠀⠠⡙⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠸⠿⠋⠁⠊⠀⠀⠀⠀⠀⢄⢀⡀⠢⢀⠀⠈⠂⠀⠀⠀⠀⠈⠀⢀⠠⠂⢀⢀⠀⠀⠀⠀⠀⠈⠀⠙⠻⠿
⠀⠀⠀⠀⠀⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢠⠀⠄⠀⠀⠄⢠⠀⠀⠠⠤⡄⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⢀⣀⠠⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠜⠋⠉⠑⢄⠀⠀⠀⢀⠜⠛⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣶⣶⣶⡎⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⣀⠠⣿⣿⣿⣿⣧⡀⠀⠀⢀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣼⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠙⡟⠉⠀⠈⠢⣾⡿⠀⠀⡀⠮⠒⠑⠀⠀⠂⠁⠁⠀⢠⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⣄⠀⠀⢀⣐⠀⠀⠀⢀⡀⠋⢄⠢⠁⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⣿⣿⣿⣦⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠾⠟⠻⠛⠙⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⣴⡶⣿⣿⠉⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⣤⣤⣤⣶⣿⠙⠿⠃⠉⠉⠀⠀⠄⠁⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠝⠙⠛⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡌⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣦⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⢀⡄⢤⠀⠀⠀⠀⠀⠀⣤⠰⠦⣶⠆⡀⠀⠀⠀⠀⣠⣒⣶⡖⢄⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⣠⡆⣄⣶⣸⣘⣇⡆⣤⢀⠀⠀⠀⢀⡚⣡⣙⠲⡄⠀⠀⠀⢘⣽⡃⠂⣠⣀⡓⠀⠀⠀⠀⢖⢏⠀⠈⣼⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠭⠃
⠀⠀⠀⠀⠻⠿⠭⠿⠭⠽⠯⠭⠿⠃⠀⠀⠀⢰⡤⠑⠏⢽⡇⠀⠀⠀⠈⢯⡉⣌⢁⡿⠃⠀⠀⠀⠀⠈⢺⠌⣴⠏⠀⠀⠀⠀⠀⣿⣿⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠈⠹⠏⠀⠀⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⣹⡁⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣤⠑⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠲⣀⠀⠤⠿⠶⠀⠒⠛⠓⠒⠰⠺⠷⠄⠀⣠⠓⠀⠀⠀⠀⢀⠼⡟⠁⡀⠀⢰⣿⣿⣿⣿⣿⡆⠉⠀⠀⠀⠀⢹⣿⣿⣿⡆⣤⠀⡄
⠀⠀⠀⠀⠀⣧⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣦⣤⣄⠀⠀⠀⠁⠀⠀⠀⠀⣴⣤⠈⠙⣿⣿⡿⠃⢰⡷⠂⣀⡀⠉⠛⠛⠈⠉⠋⣁⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⢙⢿⣿⠽⠿⠿⣿⣻⣿⣿⣿⠏⢀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢀⣀⠈⢿⡿⠀⠀⠀⠀⠰⢾⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⣿⣾⣶⢾⣶⣶⡿⣶⣿⣿⣿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⣾⣿⣿⣧⣤⡄⠀⠀⠀⠙⠆⣸⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣸⣾⣝⣣⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠶⡤⠈⠟⠃⠈⠗
⠀⠀⠀⠀⠀⠀⢀⣿⡿⢸⣿⠉⠉⠉⠹⣿⡜⣿⣧
⠀⠀⠀⠀⠀⠀⠘⠿⠃⠈⠉⠀⠀⠀⠀⠉⠁⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⣷⡶⠶⢂⡀
⠀⠀⠀⢀⡀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠆⢠⣌⡀⠀⠀⠤⢤
⠀⠀⠀⠸⣤⣤⣤⣤⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢿⣿⣿⣿⠁⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠂⠀⠀⠀⠀⠁⠀⠀⠈⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠦⠶⠀⠆
⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠠⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠸⠇⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠀⠁⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠈⠢⠄⠀⣴⣷⠀⠀⠀⠀⠙⢠⣴⣼⡃⠂⠀⠀⠀⠀⠀⡆⢠⠋⠰⠀⠀⠀⠀⠈⠀⢠⢶⢆⡄⠀⠀⠀⠰⢅⣟⠀⠀⢀
⠀⠀⠀⠀⠀⠰⠛⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⠂⠀⠀⠘⠀⠃⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃⠀⠀⠀⠀⠀⠈⠀⢡⠄
⠀⠀⠀⠀⠀⠀⠀⣢⡂⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⡤⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣥⡇⠀⠀⠀⠀⠄⢤⠚⠋
⠀⠀⠀⠀⠀⠀⣐⣿⠆⠀⠀⠀⠀⠐⡆⢸⡇⠀⠀⠀⠀⠀⠀⠃⢨⡇⠀⠀⠀⠀⠀⠀⠀⢤⣿⡇⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⣿⠇⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠠⢤⠀⠀⠀⣀⠀⠘⡀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⣤⡀⠀⠀⢸⡀⠀⠘⡿⠀⠀⠀⠀⢡⠀⠀⢠⡀
⠀⠀⠀⣀⠀⠻⡀⠀⠈⠁⠀⠀⠛⠶⠘⠛⣆⠀⠀⠀⠀⠀⠀⠹⢛⠉⠁⠀⠀⠀⠈⠛⠓⢐⣻⣇⠀⠀⠀⠈⠛⡉⠒
⠀⠀⠉⠚⡦⠴⡀⠖⠀⠀⠀⠀⠀⢸⡇⠸⠟⠇⠀⠀⠀⠀⠠⠾⠟⠁⠀⠀⠀⠀⠀⠀⢨⠘⣿⠘⢣⡄⠀⠀⠀⣉⢹⣎
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢸⠇⠀⢂⠄⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠠⠀⠀⠀⠀⠀⠀⣇⡿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⢄⢸⠄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢨⢤⣷⠀⠀⠀⠀⠀⠀⠀⢷⠃⠀⠀⠀⠀⠀⠀⠀⠈⠷⠄⠀⠀⠀⠀⠀⠀⠀⠈⠼⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠘⠁⠘
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠂