Crustacean Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡈⠉⠛⠻⢿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣼⡿⠛⠉⠉⠉⠃⠀⠀⠀⠘⢳⡄⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⣤⣤⣤⣄⡄⠀⢦⣤⣤⣶⡿⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡺⠁⣴⣁⠻⠛⠋⠉⠀⠀⠀⠈⠉⠙⠒⠀⢬⣿⣦⣆⣵⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠳⣿⡟⠁⠀⠀⠀⠀⠀⡄⢀⠀⡀⠤⣀⡀⠀⠙⠻⣿⡏⠀⣃⢀
⠀⠀⠀⠀⠀⠀⡄⢠⠀⣴⠉⠀⠀⠀⠀⠄⠱⠹⣿⠁⠈⠀⢿⠇⠐⠂⡄⠀⠀⠀⠀⣿⡟⠀⢰
⠀⠀⠀⠸⠀⠀⢱⣿⣿⡟⠀⢐⠱⠈⠀⠀⠰⠦⠀⠀⠀⠀⠸⠟⠃⠀⠀⠑⡞⣯⣧⡿⠃⢀⡽
⠀⠀⠀⠀⠢⣀⡀⠉⠛⠸⣷⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡸⠋⠀⣠⠞⠀⣠
⠀⠀⠀⠰⣄⠉⢷⣶⣤⣀⣀⣀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠⣤⡴⠟⠁⣠⠜⠁
⠀⠀⠀⠀⠈⠲⠦⣄⣈⣉⣩⣉⢷⣷⣶⣶⣾⣿⡦⠀⠀⠀⠰⠟⢻⠿⣿⣷⣯⠤⠶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠄⣀⣉⡀⣉⣭⣍⣼⣿⣯⡉⠀⠀⠀⠀⠀⠀⡵⠶⠶⠀⠀⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⢠⢄⣦⡼⢥⣀⣀⠀⠀⠀⠺⠳⠶⠦⣆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠚⠁⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⣿⡿⠻⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠒⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⣶⠶⢶⣉⣩⣽⣷⠶⠟⠛⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠛⠁⡠⣽⣋⣩⡦⠴⢲⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢠⣴⣾⠁⠙⣟⡀⠀⢀⡀⢐⢻⣙⣡⠔⠊⠁⠀⣠⡤⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⣽⡂⠀⠀⠙⣿⣟⣉⣄⣷⠞⠋⠀⠀⠀⣠⣾⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠈⠹⣝⡦⣶⡚⠋⠉⠉⠉⡀⠀⠀⠀⢠⣾⣿⢿⣵⡿⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣰⣖⣁⣀⡀⠙⣷⣀⠐⣋⠴⠂⣠⡖⢯⣬⣿⣾⣿⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡾⠛⠛⡉⠩⠍⠛⠋⠛⠛⠋⢉⡟⢿⡍⠀⣾⠟⣠⡞⠀⠀⠀⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠻⣿⠁⠀⠀⠀⠀⠀⠀⢀⡠⠌⢰⣶⣴⠟⠛⢳⣴⠉⣿⣿⡈⠀⠀⢰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡽⠟⠀⠀⠉⠳⢤⠀⠀⠀⠀⠢⠤⣤⣀⣈⣱⣿⣦⡄⣾⠈⢧⡀⠘⣿⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣤⢚⣧⠖⠊⠛⠛⠈⠳⣦⠀⠀⠀⠈⣩⣷⣿⠿⢋⣠⠁⣶⣀⣙⣦⠜⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣿⣻⢿⣿⠛⢿⣿⣿⠑⠠⠀⠀⠀⠀⠀⣈⢷⡜⠧⣾⠿⢻⣆⡐⠿⣤⣿⣾⣿⡷⡶⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠁⠻⣿⡄⢿⣿⣅⠀⡀⣀⣤⠎⢀⡼⢲⡄⠰⢶⣶⠌⣙⣿⣿⣷⣬⣉⣉⡀⠄⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⣿⣼⣿⠄⠀⠙⢿⡆⠀⠩⣿⢸⣿⡿⢷⠛⠛⠛⠰⢿⣿⣯⣧⣶⠙⢿⣿⣿⣦⣌⠛⠿⣮⠛⠛⠦⠽⡛⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡜⣿⠧⠙⣇⣠⠀⠛⠋⢀⣀⡟⣿⣯⣷⣼⣿⣿⣦⣴⡄⢿⣿⡟⠿⠳⣆⡉⠉⣉⡙⠳⣦⡀⢧⡀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣯⠹⣿⡇⡈⠻⡀⢈⢿⠰⢿⣦⡨⠶⣾⣿⡿⡿⠉⠁⠁⠛⢾⣃⡜⠙⠛⠷⢦⣄⡻⢶⡙⠙⠳⢮⣥⣌⠻⡄⠀⠀⠀⠀⠀⠀⠱
⠀⠀⢀⣠⡶⢒⣶⣾⣿⡗⠙⠳⣥⢠⣵⣘⢾⠿⠆⠋⠁⠀⠀⠛⠻⠧⠾⣄⣤⣀⠀⣿⡆⠀⠀⠀⠀⠀⠈⠓⠿⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠘⡌⠓⣹⣯⠉⠉⠛⣆⢀⠚⣷⠽⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⢰⣏⣀⠷⠏⣀⡴⢣⡇⢾⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠄
⠀⠀⠈⢿⣿⣦⣤⣴⡴⡿⠃⡘⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠈⠙⠛⠿⢿⣄⣀⣰⡟⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠱⡄⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⢻⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⡇⠀⡜
⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⣷⠈⢯⠘⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠹⣄⠙⢶⣴⣿⣷⣄⡀⠀⠲⣄⠤⢤⡀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠙⢳⣄⠑⢬⡙⠻⢭⣕⢲⡈⠑⠶⣦⣤⣀⠀⠀⠈⠁⠒⠖⠠⠤⠤⠀⠀⠀⣼⠀⠠⣤⢤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣬⣽⣒⡛⠳⠦⣭⣒⢦⣰⠟⣻⣅⠒⠂⠀⡀⠀⠀⠀⠀⠀⢀⠡⣦⡀⠀⠀⢁⡃⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢛⣛⡛⠿⠷⣿⣷⣿⣞⢁⣖⠀⠙⠿⣷⣷⡾⢿⣿⣶⣿⣿⣿⣿⠘⣿⣶⣠⣄⡴⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢩⣭⣿⣿⣷⣾⣯⣿⣿⣇⠜⠢⣀⠈⠁⠛⠚⠛⢛⣿⣿⣿⠯⢸⣿⣿⣿⣿⣶⣦⣄⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡠⠖⢫⣄⣦⣬⣭⣭⢭⣦⢤⣬⣽⣶⣷⣶⣶⣴⡦⠶⣾⣿⢟⣀⣠⡾⠿⡋⢸⡏⠀⠉⢈⣥⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⡾⡡⣪⠞⢋⣵⡶⠟⣛⣯⣽⣿⡿⠿⠛⠟⣛⣫⣭⠿⠟⠁⣩⠟⣫⣤⡴⠟⠶⣏⣀⣠⣎⠱⠉⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠋⣥⣴⠟⠉⢀⡜⣿⠉⣴⣦⡶⣛⣿⠟⣻⡿⠊⠀⢀⡜⠁⡠⠞⠁⣀⣦⠴⢿⣿⣅⣿⠂⢀⡴⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⠕⣺⠅⠀⠀⢀⠌⠀⣿⠀⢇⢄⠟⣉⡴⠋⠁⠀⠀⢀⠟⢀⡞⢀⡴⠋⠀⣠⠖⠛⠛⢛⢿⣶⡎⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣨⠃⠘⠁⠀⠀⠀⠋⠀⠀⢧⠀⣼⡋⠀⠀⡇⠀⠀⠀⠀⠋⠀⠈⠀⠊⠀⠠⠊⣠⠴⠒⠚⢛⢷⡂⠁⢠⣤⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⠁⠀⠀⠀⣿⡰⠟⠳⢦⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠴⠛⠉⠛⠛⣇⣤⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠘⣿⠀⠀⠀⢹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠇⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣑⠀⠀⠘⢦⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣆⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣻⡿⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣵⣾⠟⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⣿⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡨⣀⣔⣶⢶⣿⡍⠿⠦⠀⢀⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠱⢟⡽⢈⡹⡿⢟⣱⠶⠓⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⢀⠀⡀⠀⠋⠀⠀⢸⡏⢰⣋⣴⢾⠏⡀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠐⠁⠀⠀⠠⠀⠒⠇⠈⠀⠠⠀⠁⠀⢐⠁⠀⠊⣳⡼⠟⠉⠈⠰⡈⠁⠠⡀⠘⣷
⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⣀⡤⠞⣁⣤⡀⠀⠀⠀⠀⠄⠀⠈⠀⢽⡇
⠀⠀⡠⠂⡀⠀⠊⠀⢃⠀⢀⣀⠀⠀⠑⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢊⠡⠔⣈⡅⢠⡉⠠⠀⠀⠀⠈⢄⠀⠀⢸⡧⠄
⠀⠸⣠⠈⠀⠀⠀⠀⢸⣦⣀⠀⠁⠸⣄⢟⠀⠀⠀⠀⠀⠀⣀⣤⢶⢿⣿⠷⠒⠋⠉⠀⠀⠉⠲⢬⡆⠀⠀⠈⠄⠀⢠⡿⠀⠠⡀
⠀⠀⡌⠑⢄⡒⠛⠛⠛⢻⣿⣷⣶⣾⡿⠛⠳⣶⣶⢶⣖⣿⣽⣿⣯⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⢆⠀⠀⠈⡀⢸⡇⠀⠀⠀⢀
⠀⣰⠀⠀⠀⠙⣶⣴⣾⣾⡿⡮⠥⣬⣿⢶⡦⠈⠻⣿⣿⣿⠿⡻⢿⣿⣿⣷⣖⠦⣄⠀⠀⠀⠀⣸⠀⠀⠳⡄⠀⠁⣼⠃
⡇⡇⠀⢀⡽⠋⣩⡿⣿⡟⣒⡒⠿⣧⣹⣦⣽⢶⡄⠈⠻⣝⠳⣌⡓⠬⣙⢋⣻⢧⣼⠆⠀⠀⠀⠃⠀⠀⠀⠹⠄⢰⡟⠀⠀⠀⠀⠀⠀⠁
⢠⠁⠐⠢⠀⢾⡾⣿⣿⣦⡘⠛⠓⢦⣙⢍⠿⠜⠇⠀⠀⠈⠳⡌⠻⣆⠈⡆⢻⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢀⡞⠠⢸⣿⣿⣿⡟⠛⠓⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⣷⠘⡆⠀⠀⢿⣸⣇⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⢀⠀⠀⢻⠹⣽⠌⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣷⠀⡀⢸⣿⢹⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠋⢀⠀⠘⡤⢎⢶⡀⣌⠛⠶⢤⣤⣤⣀⡀⢀⣤⣀⣀⠀⡟⣿⠀⡤⠎⠈⡎⡆⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠑⠲⠤⣁⣁⠑⣏⣛⣃⣾⠿⢅⣁⣙⣻⣻⣿⠟⠀⡇⢻⠀⠇⠂⣀⣠⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣾⣿⢟⡛⠟⠻⠶⠶⠷⣿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠰⠁⠂

⠀⠀⠀⠀⠀⣮⠽⣆⠀⢀⣴⣟⡋⠉⠉⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠻⣄⣹⡄⠿⣧⣈⣰⡆⢀⠀⠠⣭⡴⡄⣠⢀⣀⡤⣠⣤⣄⣾⡦⠀⢀⣀⠤⣄
⠀⠀⢠⡾⠿⠶⣬⣁⠀⠀⠌⠉⡉⠉⠁⠀⠀⠀⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢀⠁⠀⠊⣭⡀⢀⡀
⠀⠀⠀⠙⠳⠖⠛⠉⠈⠀⠐⠀⠙⠒⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⣠⠞⠇⠈⡟⡩⢑⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠁⠀⠀⠀⠀⢀⣄⣈⣠⠄⠀⠀⠀⠀⠀⠈⡰⠇⢀⣀⡞⠁⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠒⢀⡹⠷⠂⠀⠀⠀⠀⡴⠁⡠⠛⠋⠄⠊
⠀⠀⠀⣠⠤⠔⠒⠀⠀⠀⠀⠀⠀⠀⠠⠛⠛⠉⠀⠛⢉⠀⠈⠀⠀⠀⠐⢞⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⢀⢀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢊⡄⠁⡈⢲⡀
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠛⠒⠒⠗⠀⠀⠠⠔⠋⠁⠵⠀⠁⠀⠧⠃⠐
⠀⠀⡄⠐⡇⠀⠀⣆⠀⠀⠘⢀⡀⠀⠀⠀⠻⠶⠖⠒⠁⢀⡶⠲⢄⠀⠀⠐⢼⡗⠒⡂⡴⡋⠀⠀⠠⠦⢀
⠀⠀⣀⠔⠀⡄⠐⡆⠮⠀⢘⡂⣈⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⠀⡸⠀⠀⠀⠈⠀⠀⠛⠁⠀⠂⡡⠀⠀⢰⠇
⠀⠀⠈⢤⣸⠦⢀⠀⠀⢰⡉⢾⡃⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠆⠶⠘⠁
⠀⠀⠀⠀⢺⠃⢸⠐⠀⠈⡧⠘⠣⠀⠀⠐⠉⠈⠛⠛⣳⠀⢀⣤⡶⣿⣿⣿⠖⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠘⢧⡤⢴⡟⠀⠀⠀⠀⠐⠖⠶⠛⠋⡁⠀⢿⣟⡿⠿⠛⠁⠀⠀⠀⠐⣻⣦⠴⠒⠛⣲⣦⠈
⠀⠀⠀⠀⠀⠀⡀⠀⡥⠀⡇⠀⠀⠀⠀⠀⢀⠠⠒⣹⠇⣠⣄⠀⣠⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣀⠨⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⢹⠀⠀⠀⠀⢀⠥⡶⠂⠊⠀⠋⠈⢣⠘⣦⢀⠑⢦⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠠⡧⠼⠂⣀⠆⠀⠀⠀⠀⠀⠀⠀⠸⣀⣸⠄⠈⠙⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠈⠒⠂⠐⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⣀⣀⣠⣴⣶⠿⠿⠛⢛⡂⠀⠀⢠⡀⣠⣤⣶⣶⣶⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠤⣄⡀
⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣞⢧⡿⣫⡤⣶⣿⣾⢗⣦⣄⣰⣮⣬⣛⡻⣿⣿⣿⣷⣦⣄⡀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⢀⣠⣽⣿⣆⠀⠀⢠⠴⣿⣿⣿⣿⢫⣽⣿⣿⣿⣍⡙⣿⣷⣼⣿⣿⣿⣿⡟⠉⢩⣿⣿⡿⢫⣿⢿⣿⣿⣿⣿⣶⡄⣐⢦⣿⣀
⠀⠀⠀⠀⢻⣯⢠⠻⣷⡄⣰⣿⣶⣻⣽⡟⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣟⣴⣴⣿⣿⣿⣴⣿⣿⣧⡹⣿⢻⣿⣵⣷⡀⣞⠋⣸⣿⠄
⠀⠀⠀⠀⠈⢿⣧⠳⡙⢻⣿⣿⣿⣿⣟⣿⣿⢟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⣿⣿⣧⠿⢯⣿⣿⣿⣷⠿⢁⣿⠟
⠀⠀⠀⠀⠀⢦⣿⣷⣥⣼⢿⣿⣿⣻⡷⢾⣿⣿⣆⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣻⣿⣿⣿⢎⣴⡝⣿⣿⣯⣴⡾⡷
⠀⠀⠀⠀⠀⠀⠙⢳⣿⣿⣿⣿⣿⣿⣧⣿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣼⠷⢿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣙⣛⣻⣷⢇⠒⢭⣝⠿⢿⣿⣿⣞⢿⡿⣿⣿⣿⢿⡿⢣⣿⣿⣿⣿⠿⢟⣽⢻⣷⣾⣿⣿⣿⣿⣿⣾⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣿⣿⣿⡿⠟⠩⣿⣷⣝⣷⠌⠛⣿⣿⣦⣴⣮⣿⣧⣶⣴⣿⣿⣿⡿⣣⢶⣴⣿⣷⠉⠻⠿⠿⠿⠿⠿⠛⠻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣼⣿⣿⣿⠟⠙⠛⠛⠉⠀⠀⠀⠙⠻⠟⠿⠻⣿⡝⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢋⣴⣿⣮⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠈⠛
⠀⠀⠀⢸⠿⠃⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⣠⠤⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⢼⣿⣿⣭⣍⣝⠀⠀⠀⠐⠲⢶⣶⣾⡿⣳⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠛⢷⣿⣿⣿⣿⣿⣷⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⡿⠋⠀⣀⡀⠀⠀⣀⠀⠀⠀⢀⡄⠀⠀⠙⠿⣿⣿⢩⣷⣿⣦⡀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⣾⣿⣿⢷⢛⣋⡉⠁⢀⣠⣴⣼⢻⣿⣿⣟⣿⣿⣿⣿⣶⣦⡀⠀⢀⣀⣬⣄⣿⣿⣿⠇⠀⠀⣠⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡈⠿⣿⣿⣿⣿⣿⣴⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣾⣿⣿⣿⣿⣯⡙⠁⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣛⠿⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⢿⠿⢿⣿⣤⣾⡿⠋⠁
⠀⠀⢠⡴⠻⠿⠿⠛⠛⠻⠿⢿⣷⣶⣮⡟⣿⣷⣷⢼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣛⣛⣭⣶⣶⣿⡿⠿⠛⠿⠿⠿⠿⢧⣄
⠀⠀⠛⠀⠀⠀⠀⠀⣀⣀⢀⣤⣤⣄⣤⣌⣛⣛⠿⠶⠿⣿⣿⣻⠿⢿⣏⢻⣿⣿⢿⣿⣿⣟⣽⡿⢋⣵⣿⣿⣿⣿⣿⣧⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⣴⠿⠿⠛⠋⠛⠋⠉⠈⢙⣛⣛⠛⣿⡿⠟⠸⠏⣛⢷⣿⣿⣿⡿⣿⣷⠿⣛⣷⣟⠿⣿⡟⢻⣿⠛⠋⠉⠉⠉⠙⠛⢿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣤⣶⣦⠺⣿⠟⠿⢿⣿⡧⠿⠆⠛⠃⠹⠿⠟⠿⠿⠇⠚⠙⠿⢿⣿⡿⠿⠙⣻⣴⣿⣿⣿⢦⡄⠀⠀⠉⠉⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣞⠿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢟⣫⣿⣿⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠉⠛⠻⣿⣙⣤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣀⣠⠶⠛⠉⠈⠑⢀⣒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⣟⣾⣷⡀⠁
⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⣠⠶⠂⡹⢧⣤⠀⠀⠀⠂⣸⠿⠿⣿⣿⣌⡂⡠⣴⣤⣤⣤⣀⠉⠙⢿⣿⣿⢻⠷⢄⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣿⡟⢁⣼⡇⠀⠀⢀⡸⢿⡆⠀⣾⠀⠿⣦⣾⣟⣘⡘⡇⣇⠛⠉⠉⠉⠛⠿⣶⠤⣙⣷⣿⣷⡈⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⣸⣿⢹⣿⣿⣇⣀⡴⡋⢡⣤⣿⣶⣘⡇⠐⣻⣿⣿⣧⠀⢀⣿⡄⠀⠀⠀⠰⠄⠁⣰⠏⠛⢿⣿⣷⡀⠀⠀⠀⣼⡍
⠀⠀⠀⠀⠀⠉⣿⣈⣿⣿⣿⢁⠈⣿⣿⣿⣟⡄⠑⢤⣌⠻⣿⣿⣿⡷⢝⠃⠀⣲⣶⣀⠀⠀⢧⣻⣧⣠⠆⠙⣿⡇⠀⠀⣴⠛⡇
⠀⣀⠀⠀⠀⠀⠹⣯⡘⣿⣁⣨⣿⣿⣿⣿⣇⣬⣿⡯⠉⣀⣿⠿⣿⣷⡀⢀⣴⣿⣿⣿⡧⢠⡀⠈⠺⡛⢦⠖⠈⢡⠄⣼⠇⠐⠁
⠀⢟⣤⣄⣀⠀⠀⠙⢷⣤⣀⣽⣿⣿⣿⣿⣿⢿⡇⢠⡾⠏⠀⠀⣼⣿⡿⠿⣿⣿⣿⡿⢦⡄⠀⠀⠀⠉⠀⠂⠀⠉⣴⠏⡼⠁
⠀⠈⢻⣿⣿⣿⣦⣀⠀⢩⠈⢹⣿⣿⣿⣿⣧⡄⠁⡈⠁⠀⠃⢸⣟⢹⠅⣠⣿⣇⣿⡆⠀⠓⠀⠀⠀⠀⠄⣀⡀⢻⡍⠈⣀⡠⠤⠤⡔⠂⠐⠐⠐
⠀⠀⠀⠙⢿⣛⣿⣌⡉⠛⣿⣶⣿⣻⣿⣿⠯⣗⡤⠀⠀⠀⠀⣴⠛⠁⠀⠙⡆⠿⣿⣄⡀⠂⠀⠀⠀⠤⠂⣙⣶⣤⠶⠟⢃⠤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣧⣽⣿⣿⣦⣄⡀⡵⠇⠄⠀⠀⠀⢨⡇⢀⢸⡇⠀⠘⠡⠃⠀⠀⢀⣤⣖⠫⠽⠹⠒⠂⠀⠁
⠀⠀⠀⠀⠀⢀⣠⢶⠞⢛⣿⠂⣉⠉⠙⠛⢿⣿⣦⢀⠀⠀⠀⠦⠜⠓⠈⠈⠀⠀⠀⠀⢀⣤⠾⠛⢉⣉⣸⣏⣗⣐⠠⡀
⠀⠀⠀⢀⣴⣋⣤⠾⠛⠟⠀⠛⡻⣿⣿⣷⣾⣿⡿⡷⣶⢾⣯⣤⣤⡬⣤⡭⣤⣦⡶⣖⡛⠷⠶⠾⢿⣥⣀⡀⠈⠙⠷⣄⡳⢦
⠀⠀⠀⣿⠋⠀⠀⠀⠀⢀⣠⣦⡅⠀⠀⠀⣀⠽⠗⠛⠚⠋⠛⠗⠛⠓⠛⠓⠛⠻⠿⠻⢷⣄⡀⠶⢶⣾⣿⡃⣀⠀⠀⠈⠉⠚⠷⡄
⠀⠀⠀⠛⠀⠀⠀⠠⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢳⣬⢆⠀⠀⠀⠀⠑⠯⡄
⠀⠀⠀⠀⠀⣠⣶⡟⢁⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⡄⠃
⠀⠀⠀⠀⣴⠟⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢊⣡⠞⠁⠀⠀⠀⠀⠀⠀⢨⣱⣍⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⣠⠴⢛⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⡍⠳⡄⠉⠒⢄⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣈⠇⠀⠀⠹⡄⠀⠀⠀⠘⡿⡄
⠀⠀⠀⠀⠀⣾⡄⠀⠀⣿⢠⣠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⣤⣤⢽⡄⠀⠀⠀⡇⢹
⠀⠀⠀⠀⢸⠹⡇⠀⠀⡇⠀⠈⢢⡀⠀⠀⠀⣤⡄⢀⡀⢀⡀⢤⣀⣀⣀⣰⡆⠀⠀⣠⠊⠀⠀⡰⠀⠀⠀⠀⣇⠘
⠀⠀⠀⠀⢸⠀⡇⠀⠀⢻⣆⠀⠀⢳⣀⠀⠔⠛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠙⠓⠦⣼⣧⡀⠀⡴⢃⡠⠴⠒⠋⣙⠶
⠀⠀⠀⠀⢸⡄⣿⣀⣀⠈⢿⢷⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⣏⡠⠤⠚⠋⠁
⠀⠀⠀⠀⠈⢧⣿⡈⠉⠙⠛⣷⠋⠀⠀⢠⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⣏⣀
⠀⠀⠀⠀⠀⠀⣈⣉⣹⣶⣶⢿⣐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣴⣿⣉⣉⣙⣓⣲⣤
⠀⠀⠀⠀⡐⢫⠥⠤⠤⢤⣤⣤⣼⣷⣮⣑⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠒⠋⠛⠶⣄⠈⠹⣮⢳
⠀⠀⠀⡜⡠⠋⠀⢀⡠⠔⢚⣉⣀⣠⠾⠛⢿⣶⣿⣶⣶⣖⣒⣒⣢⡭⠴⠖⠻⠿⣿⣤⠌⠛⠷⣖⠲⡞⠳⡀⠘⣧⣧
⠀⠀⣸⠜⠁⠀⢀⡞⠀⠉⢩⠽⠋⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⡰⢊⡷⢿⡀⣇⠀⠈⢻⡆
⠀⠀⠋⠀⠀⠀⣼⢀⠎⠀⢸⡄⢖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡞⠁⠈⣧⣿
⠀⠀⠀⠀⠀⠀⣧⠃⠀⠀⠀⠹⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠋⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠓⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⢀⣀⣤⠖⠂⠀⠒⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⡀⠸⠻⡍⠀⠀⠤⣶⠉⠀⠀⠀⠀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠐⠞⠛⣶⢶⡀⠀⠈⢳⣄⠈⠻⣆⠀⠀⣩⠀⠀⢰⠏⠉⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⢀⠉⠀⠀⠀⠙⢟⣦⡀⣀⣿⣦⣠⣿⣆⣴⣯⣠⣴⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣈⣄
⠀⠀⠀⠄⠀⠀⠀⠾⢦⡀⣀⠿⣯⣾⡿⣿⣿⣿⣿⠟⢟⣛⣛⡻⠿⣿⢿⣿⢶⣶⣶⣤⣤⣴⣤⣤⣤⣀⣤⣿⣹⢿⡟⠉⢷
⠀⠀⠀⠀⠉⠒⠀⠀⠈⣟⢩⠛⠛⠂⠀⠀⠣⣿⣿⠀⠀⠁⠀⠟⠲⠀⣿⠀⠈⣿⠀⢿⡇⠉⠿⡏⠹⣿⠈⣿⡅⠩⡗⠒⠺⠄
⠀⠀⠀⠀⣀⠀⠀⠀⢀⣯⡘⣤⣄⠀⠀⠀⡔⣿⣿⠀⠀⡂⠀⣦⠶⠀⣿⠀⢠⣿⠀⣼⡇⣠⣼⡇⣠⣿⢀⣿⡃⣐⡧⠤⢴⠂
⠀⠀⠀⠊⠀⠀⠀⢴⠞⠁⠉⣶⣟⠿⣶⣿⣿⣿⣿⣦⣾⣭⣭⣵⣶⣿⣾⣿⣿⠿⢷⠛⠛⠿⠟⠙⠛⠉⠛⣿⣻⣾⣧⣀⡴
⠀⠀⠀⠀⠀⠀⢀⠘⢷⠀⠀⠀⣠⣾⠟⠉⠉⣿⠟⠙⣿⠟⠻⣟⠙⠻⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠈⠃⠤⣦⣤⠾⠿⠃⠀⢀⡼⠋⢀⣴⠏⠀⠀⣸⠀⠀⠸⣄⣀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠁⢰⣴⡃⠀⠀⠒⠿⣀⠀⠀⠈⠀⡉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠈⠉⠳⠦⠀⢀⠤⠁⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠄⠀⣤⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣦⣠⣠⠞⠀⠰⣀⣀⣤⠴⠲⠎⠀⠉⠁⠀⢸⡀⢀⠀⢸⡀⢀⡋⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢁⣤⡾⣿⢿⣿⡿⣤⡞⠋⠁⠀⠀⠀⡀⠀⠀⠀⡀⢄⢳⣿⣦⡘⣿⡌⣿⡞⡏⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⢠⣤⣿⣿⣿⠇⢰⣹⡞⡁⢁⢀⡀⠘⠀⠁⠸⣦⣿⣷⣿⣿⣷⣽⣿⠶⠿⠷⢿⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣀⡾⢿⣿⣿⠟⠁⠀⢀⠋⡔⢰⠾⠧⠠⣀⠀⠀⢸⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣷⣿⡟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣤⡾⢋⣴⣿⡟⠁⠀⠀⢀⢴⣶⣧⠬⣼⣾⣿⣼⣧⣠⣿⣿⡿⠿⣋⠱⣆⠀⣾⠉⢿⢿⡿⠃
⠀⠀⠀⠀⠀⠀⢰⣿⣽⠟⣡⣾⡿⠋⠀⠤⠤⠤⢴⣾⣿⡿⠿⠿⠿⠿⣽⠷⣾⠉⠻⣇⠀⢸⣧⢸⠀⠙⠃⠐⠋
⠀⠀⠀⠀⠀⠀⣾⡿⢏⣴⡿⠋⠀⠀⠀⠀⠀⢀⠍⠁⠀⠀⠀⠀⠀⠀⣴⣶⣿⣇⢀⡟⠀⣸⠇⣾
⠀⠀⠀⠀⠀⠀⢻⡥⠟⠋⠀⠀⠀⢀⣀⡀⠀⠈⠀⣀⢀⡄⢠⡄⣤⣴⣾⣿⣿⡟⢈⡇⢠⡍⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠛⢛⢉⠛⢻⣿⣿⠏⠀⠀⠀⠀⠘⢿⡟⣿⡿⠃⡞⠁⡸⠁⠈
⠀⠀⠀⠀⠀⢀⣴⣬⣵⡶⢶⣾⣿⣿⣿⣿⡿⠛⠋⣀⣄⣤⣤⣒⠋⢸⠿⠋⡱⠟⠀⠀⠁
⠀⠀⢠⡆⢀⠚⣩⠠⠄⠐⠊⠛⠒⠛⠉⠉⠀⣠⣾⣿⣿⣿⢟⠓⢋⡼⠀⠈
⠀⠀⠀⣿⣦⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⢙⡤⠔⠋
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠝⠑⠋⠉⠁
⠀⠀⠀⠀⠀⠈⠚⠹⠟⠿⠙⠻⠛⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠄⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⡀⠀⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠭⢳⡆⢤⢀⡀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠁⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠟⠙⠀⠉⣦⠄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠊⠀⢀⠊⢁⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠂⢀⠀⢠⡀⠔⠒⠀⠀⠀⠀⠀⠋⠄⠀⢐⠀⠠⠂⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠐⠂⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀⡀⠤⠒⠀⡉⠀⠀⠀⠀⠈⠸⠃⠀⠀⠀⠀⢀⡀⠀⠂⠐⡔⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠄⢀⠀⠁⠡⠊⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡌⠆⠂⠤⢀⡀⠁⢱
⠀⠀⠀⠀⡤⠀⠐⠂⠀⡘⠀⠀⠆⠁⠠⠀⢀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠰⠀⠀⠀⡀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢠⠒⠀⠀⠀⠀⠅⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠠⣷⡀⠀⠀⠀⢀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡠⠎⠀⠀⠀⢀⠨⣤⣀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⣠⡀⠀⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠚⢛⠀⠀⠀⠀⠀⠐⣢⡌⢉⡁⠘
⠀⠀⠀⠀⠀⠀⠘⠄⠄⢀⡄⠤⠀⢀⠠⠀⠸⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⢐⠐⣾⠀⠀⠀⠀⢀⠞⠁⢍
⠀⠀⠀⠀⢀⠀⢃⠀⠌⠠⠀⠀⠀⢢⠀⢠⠇⠀⡼⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠈⠀⢈⣽⠀⠀⠀⢠⡞⠀⢠⡀⠐⠒⠐⠒⠁
⠀⠀⠀⠀⠈⢆⠀⠑⠂⠼⠀⠀⠀⠀⢋⡁⠀⠒⠁⠀⡠⠀⠀⠀⠀⠁⠀⠀⡐⠀⠀⢨⣧⠀⡀⣀⠆⠀⠐⠀⠀⠑⠆
⠀⠀⠀⠀⢀⡀⠁⠒⠀⠚⡄⠀⠀⠀⣁⠂⠐⠐⠂⠀⠀⠀⠀⠀⡀⠀⡴⠸⡁⠀⠀⠈⠘⠤⠒⠋⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠈⠀⠈⠉⡠⠋⡄⠀⠘⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠳⠀⠀⠃⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡎⡁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠠⠀⢀⡠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠂

⠀⢸⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣴⡒⠖⢶⡤⣄⠀⣀⣀
⠀⢀⠀⠀⢂⠀⠀⠀⠑⣠⣤⠤⠤⠶⠻⣿⡟⠉⠁⠒⠉⠂⠈⠉⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⠹⣦⡀
⠀⠈⢄⠀⠀⠑⢤⣀⠀⠈⠁⠴⢤⣀⣀⣀⡀⡀⠀⠠⠴⠒⠒⠀⠂⠄⢀⠀⠀⠀⡘⠀⠀⠀⠀⠈⣷⢤⡀
⠀⠀⠀⠑⠤⣀⠀⠀⠉⠙⠒⢒⣚⣻⠭⠭⠛⠛⠓⣢⠶⡦⣴⣁⡒⠤⡀⠀⠀⢰⡇⠀⠀⠠⡾⢀⡇⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⡇⠏⠘⠛⠛⠙⠃⠐⡾⣇⡀⠀⠈⣠⠞⠀⠀⢨⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢦⡣⡣⡀⠀⠀⠀⠀⣼⣾⣿⣿⣿⠉⠀⠀⠀⠀⢈⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠛⠎⠢⠀⠀⠀⠀⠀⣙⢿⣿⡷⠤⠄⢒⡂⠁⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠾⣿⣧⣀⠀⠀⠀⢀⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢏⠀⠉⠐⠂⠤⠟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⡠⢊⠔⠀⠱⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⢭⣭⣓⣦⣄⡐⠉⠀⠁⠀⠀⠀⠢⢶⣦⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣄⣤⣭⣿⣯⡀⠀⠀⢀⣀⣤⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠘⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢀⣀⣠⡤⢤⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠲⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣾⠒⣿⠆
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠁⠀⠀⠀⠀⠀⠀⡍⢨⡝⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠿⠀⠀⢸⣦⡀⡀⠀⠀⠳⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣹⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢸⡱⠊⠀⠀⠘⠠⡻⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⡄⢠⠀⠀⢀⠀⡄⠀⠰⡋⠘⢻⡆⠀⣸
⠀⠀⢤⡔⠚⠁⠀⠀⠙⠶⣎⡤⠀⠀⠀⠀⠀⠀⢠⠉⠃⢀⡰⠼⢠⣞⠉⡃⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇⢸⠀⠀⠈⠠⡈⢠⣾⡇⠀⢸⡍⢋⣁⠔
⠀⠐⠊⢁⡀⠀⠠⠀⠀⢀⠈⠦⠀⠀⠀⠀⠀⠀⠈⢆⠲⡈⡄⠀⣀⠟⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢼⠀⠀⠐⠤⢼⢀⡼⣷⣀⣾⣄⢰⠅
⠀⠀⠀⠈⠉⠅⠈⠁⠀⠄⠈⠀⠀⠀⣴⠀⣦⠀⠀⠀⠀⠈⠂⠀⠋⠐⠁⠀⠀⠀⠀⠀⠀⠀⢠⣬⡏⠀⠁⣶⣄⠀⠀⠈⠻⠔⠋⠛⠻⠈⠆
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⣀⠟⣀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠰⠃⠀⠀⠃⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⠀⠀⣰⠀⠀⠀⢀⣀⡷⣤⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠊⠀⢀⠁⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡍⠀⠀⠀⠀⢹⠰⡆⢠⣄⣤⣬⣄⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⣤⠄⡠⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠳⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⣠⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢸⡀⢸⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⠀⢸⠀⣸⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠳⠜⠀⠈⠃⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠉⢂⠀⠀⠋⠀⢘⠴⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⢀⡀⠀⠊⠀⠀⠀⠂⠄⠀⣀⠀⠀⢠⠎⠀⠀⠈⠃⠀⣀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠘⠐⠏⠀⠀⠀⠀⠀⠀⢀⣀⣔⣻⡄⠈⣜⠀⠀⠀⠀⠀⠀⠈⠗⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠠⠀⠀⢐⠔⠹⠃⠿⠘⠃⠀⠀⠀⠀⠀⠈⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠊⠁⠀⡀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠙⠂⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠃⠀⠀⠠⠤⡞⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠂⠀⠀⠀⠀⢊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠈⣁⠀⠀⠀⠀⠀⠀⠀⢈⡁⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠊⠉⡄⠀⠀⠀⠀⠀⢀⡌⠉⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠠⠁⠀⠀⠀⠀⠁⠨⠂⠀⠠⠀⠠⠆⠈⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠲⠀⠸⠤⠄⠤⠄⠀⣄⠀⠰⠣⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⢄⡀⠀⠀⠀⠀⢀⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠓⠀⠤⣤⡤⠄⢊⠀⠀⠀⠀⠀⠛⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠠⠠⠄⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠉⠀⠄⠀⠀⠀⠤⠽⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠂⠐⠳⠄⠺⠤⠤⠤⠄

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⡿⠿⠓⠀⠀⠢⠀⠔⠁⠐⠒⢘⡛⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⢀⣴⣿⣯⣼⣷⣦⠴⠒⠒⠂⠀⠉⠛⠓⠺⢿⣷⣤⣙⣿⢷⡄⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⣏⢻⣿⢿⠿⠋⢁⣠⣴⣶⣦⡤⣶⣾⣦⣤⣄⡈⢙⣻⡧⣾⣿⠀⠀⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠙⢝⣦⡀⠀⠹⣋⣿⣿⣶⡾⢻⣿⢈⣽⣿⣇⠈⣻⣿⠛⢿⣿⣿⣟⣿⠟⠁⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢬⣦⣀⡀⠙⠿⣿⡁⢸⣿⣿⣿⠃⣿⣿⣿⡁⣠⣽⣿⣿⠟⠁⣠⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠄⣠⡬⣮⣙⠶⣄⡈⢿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣷⠶⣶⣶⣤⣤⣀
⠀⠀⣠⣴⣾⣿⡿⠗⠈⠁⢙⣻⢽⠳⣦⡙⠻⢿⣿⣿⡿⢻⠿⣿⣿⣿⣿⣿⣟⣻⣿⣾⣿⣿⣥⣤⡀⠈⠙⠛⠛⠛⠷⣄
⠀⣼⠛⠁⠀⠀⠀⣀⣤⣺⡿⠛⠾⣛⠓⣹⠿⣿⣿⡷⠿⠾⠾⠿⠿⠿⠛⠻⠿⠿⣿⢿⡟⠛⠉⢿⣾⣶⣶⣄⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⢠⣾⠿⠛⠉⢀⣐⣻⠟⠚⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠈⠙⣿⣷⣄⠈⠉⠛⢿⡆
⠀⠀⠀⠀⠀⣾⠃⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣧⡀⠀⠈⠷
⠀⠀⠀⠀⠘⠇⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⡦⠶⠂⠀⠀⠀⠀⠲⢶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠋⠀⠀⢀⣀⣀⡄⡀⠀⠀⠈⠳⣤⣤⡇⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠸⣌⠳⡄⠀⠀⠀⠀⡄⣿⡤⠶⠂⠀⠚⠛⠥⠉⠝⠒⡶⠶⣤⣸⣇⡀⠀⠀⠀⢀⠴⣃⠞
⠀⠀⢄⠀⠀⠈⠳⡛⠦⣄⠀⠀⠙⡏⠀⠐⠈⠓⡠⠘⠂⠀⠀⠼⠿⠤⠿⢿⠟⠀⠀⣠⠴⢚⠟⠁⠀⠀⣠⠄
⠀⠀⠀⠑⠦⣤⣀⠈⠉⠛⠻⠽⢿⢇⣀⠀⠤⠀⡀⢦⣴⡀⠀⠀⢐⢤⢄⣤⡿⠿⠜⠛⠉⠁⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠓⠮⠴⢶⣤⣤⣶⣄⡀⠒⠠⢈⡓⢀⠠⠌⣥⣴⣾⣯⣶⡶⠶⠽⠛⠛⠉
⠀⠀⠀⠀⠀⠀⢀⣠⣴⡦⠶⠶⠶⣶⣯⣭⣿⣿⣍⣙⡃⣈⣛⣽⣿⣯⣽⣿⣷⠦⠗⠲⠶⣦⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠉⠁⢀⣠⣶⠟⠓⠛⠻⠿⠟⠙⠛⠛⠛⠛⠛⠻⠿⠿⠓⠛⠳⣶⣄⠀⠀⠉⠙⠦⡄
⠀⠀⠀⠀⠁⠀⠀⠀⢀⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⡀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣶⢷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠻⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡴⢫⡴⠯⡼⠃⡠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⡄⢿⠹⣎⢳
⠀⠀⠀⠀⠀⠀⣼⡷⠟⠁⣟⣠⠎⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠩⡈⡇⠹⣅⣇
⠀⠀⠀⠀⠀⣼⣟⠋⠡⡐⢯⣤⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⠼⣃⠐⣸⣿
⠀⠀⠀⠀⢠⣿⣏⡁⠂⠟⠠⣶⣿⡄⠀⠀⠈⠄⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⢿⣿⡗⠺⠅⠀⢴⣿
⠀⠀⠀⠀⠸⣻⡀⠕⠀⣤⣼⣿⣟⠁⠀⠀⠀⠈⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠻⣿⣷⡄⠀⠈⢻⣿
⠀⠀⠀⠀⠀⢿⣿⠾⠷⡿⠛⠉⠀⠀⠀⠀⠀⠀⢰⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⢷⣼⡿⠃
⠀⠀⠀⠀⠀⠈⢻⡧⠘⠻⣦⡄⠀⠀⠀⢀⠀⠀⠘⡆⣰⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⢼⡟
⠀⠀⠀⠀⠀⠀⠘⢧⡀⣿⣿⣿⣶⣠⡶⠛⢷⡄⢀⣿⣿⡀⢠⡿⠛⢶⣄⣾⣿⣿⣿⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⡾⣹⠿⣷⣤⡈⣿⡟⢻⠛⢛⡿⢁⣤⣾⢿⣏⢦⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠻⠿⣿⣿⣿⠆⠞⠀⠠⣽⣿⣿⠿⠛⢿⡅
⠀⠀⠀⢀⣴⡶⠤⠾⠿⣯⣥⣾⡦⠀⠀⠈⢻⣯⣞⡄⠀⣲⣿⡟⠁⠀⠀⣼⣷⣭⡿⠿⠧⠤⣶⣦⡀
⠀⠀⣴⡿⢋⣀⢠⣤⣦⡀⢬⣏⡙⠛⠷⣦⣼⠟⠻⢷⡟⠿⢿⣇⣴⠾⠛⢋⣽⡤⢠⣦⣤⡄⣈⡙⢷⣄
⠀⠈⠋⣀⣿⠟⠉⠁⠉⠛⢿⡷⢦⣵⣦⣹⢇⣾⡀⠈⠀⠀⢠⢹⣁⣴⣮⡶⢿⡟⠛⠉⠉⠉⠻⣿⣢⡙⠁
⠀⠰⠻⠟⠁⠀⠀⢀⣀⣤⣼⣧⣤⣄⣉⣻⠸⠛⢡⢸⠀⠀⠰⢹⣛⣉⣠⣤⣼⣦⣤⣀⡀⠀⠀⠈⠻⠝⠆
⠀⠀⠀⠀⠀⢀⣾⠿⠋⠉⣽⠉⠉⠉⠉⢻⠀⠀⠀⠘⠀⠀⠀⢸⡏⠉⠉⠉⢹⡏⠉⠛⠿⣷⡀
⠀⠀⠀⠀⣰⠿⠋⠀⢀⣤⣾⠿⠞⠛⠋⠻⣦⣶⣥⣤⣤⣤⣄⣼⠛⠛⠛⠷⢿⣷⣄⠀⠀⠙⠿⡄
⠀⠀⠀⠀⠁⠀⠀⢀⣾⠞⣿⠀⠀⠀⠀⠀⣿⣿⣟⡉⠉⢹⣿⡯⠀⠀⠀⠀⠀⡟⠿⣷⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣿⠀⠀⠀⠀⠀⣹⣿⡿⠛⠛⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡇⠀⠀⠀⠀⠀⢹⣿⡿⠛⠳⣾⣿⠃⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⡿⠛⠲⣿⣿⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⣿⡿⠷⠶⣿⡟⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⣿⡷⠖⠶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⣠⣿⣧⣄⣠⣿⣧⡀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡠⢪⡿⠛⣿⡄⢈⣿⠛⢿⡓⣄⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢀⣁⣼⣁⣀⣽⣷⣼⣧⣀⣸⣧⢈⠀⠀⠀⠀⠀⠈⢆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣹⣟⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠿⡋⠀⠀⠀⠘⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⡠⠚⠁⠀⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠙⠂⡈⠠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⠔⠁⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠈⠀⢠⡄⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⡀⢀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠘⢷⣤⠀⠀⠀⣸⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⢀⠀⠈⣿⠀⠀⢀⠠⠄⠂⠐⠈⠉⠁⠀⠀⠀⠈⠙⠘⠀⠂⠀⠀⠀⠙⠖⠉⠙⣿⠇⡄⢀⣿⢰⡀
⠀⠀⠀⠀⠀⠀⡼⠀⣈⠀⠈⡟⠛⣯⣠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠰⡉⠈⣻⡟⠀⠉⣤⠇⢨⡇
⠀⠀⠀⠀⢸⠀⠘⣆⠈⠀⠀⠳⡄⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠃⠀⢠⡌⠠⢺
⠀⠀⠀⠀⢀⠀⠀⠘⢷⠂⠓⠦⣙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡄⢀⣴⠟⠠⢀⡼
⠀⠀⠀⠀⠈⡉⠀⡀⠈⠓⠶⢦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢌⠿⠛⠁⠀⠃⡩
⠀⠀⠀⠀⠀⠐⢄⣐⡄⡀⠀⠀⠈⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⣙⠀⣒⣄⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠃⢄⣈⣓⣦⠐⢀⣱⣦⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣽⡶⠊⠉⠉⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣉⠁⣀⣭⣿⠾⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⣐⡿⢏⠉⠙⠒⠶⠶⠬⢦⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⡴⠗⠋⠁⠄⣀⣠⣼⠟⠶⣦⣀⣶⣤⣤⣴⣶⡾⠿⠛⠶⣦⣉⢒⡄⠀⠀⠀⠀⠈⠛⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠋⠀⠀⠀⠀⢹⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠚⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡴⠆⡀⣾⠀⢀⣤⠀⠠⣤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣹⣾⠋⢀⠴⢡⡟⠀⠘⡀⠳⣄⠀⠙⢤⡙⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⣴⠟⠁⢀⠂⣰⡿⠁⠀⠀⠡⡄⠘⣧⡀⠠⠷⣄⠹⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⢀⣾⡏⠀⡴⠁⣰⡿⠁⠀⠀⠀⠀⠸⠀⠸⣿⣄⠀⠸⣦⠙⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣾⣿⠃⠀⠀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠠⡀⠸⣿⣧⡈⠻⡄⠹⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢰⡏⠑⢶⣀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⢿⣿⣷⡄⠳⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⢳⡶⠎⠉⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠿⠉⣲⠀⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⠂⠀⢀⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠘⠃⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠘⠁⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠄⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠂⢠⣤⣴⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⣀⡴⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡿⠾⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⢀⣤⣸⡿⠀⣠⡄⠀⠀⣄⠀⠀⠀⠀⠀⣰⡄⠈⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⣿⣇⣤⠞⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⣧⠀⠀⠈⠻⣦⡦⠀⠀⢿⡇⠀⣻⡇⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⢀⣿⡯⠃⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠳⣦⡀⠀⢸⣇⣤⣿⡗⠀⢺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣤⡀⠀⢀⡾⠿⣧⣤⠀⠀⠀⣼⣿⣿⣧⡿⠉⠇⢿⣿⣿⣷⡀⠀⣶⣿⣛⠂⠸⣿⣿⡿⠁⠀⢸⣿⡇
⠀⠀⢀⡾⠀⢀⣄⠀⠀⠉⠁⠉⢀⣴⣿⣯⣴⠀⣼⣿⣿⡟⠀⢰⡆⠀⣿⣿⣿⣿⣿⣿⡟⣭⣿⣿⣿⣿⠏⠀⠀⠀⢸⣿⢃
⠀⠀⢸⡇⠀⠘⠉⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣾⠻⣿⣿⠁⠀⠉⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣭⣿⠏⠀⠀⠀⣠⠜⣏⠘⣷
⠀⠀⢰⡇⠀⠀⠐⠁⠀⠀⠙⢷⣶⣾⣿⣿⣿⣏⣶⡿⠧⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣯⡿⠃⠀⠀⠀⡰⢋⣾⠇⠀⠸
⠀⠀⢨⡇⠀⠀⠀⢠⡔⢰⣄⠀⠹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠖⠀⠀⠀⣀⠉⣳⠛⠁⠀⠀⢸⠁
⠀⠀⠀⠻⣄⠀⠀⠈⠀⠘⣿⡄⠀⠈⠻⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⠀⠀⠀⣰⣧⠾⠉⠀⠀⠀⠀⢸⣀
⠀⡼⠀⠀⠈⢢⠀⠀⠀⠀⢌⢻⣆⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⣧⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⢨⡏⠀⢀
⠀⢿⠀⠀⠰⣦⡿⢦⡀⠀⠀⠀⠙⢧⡄⠀⣽⡿⠿⣿⣾⡀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡀⣠⣾⠟⠁⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠈
⠀⠛⡀⠀⠀⠈⠂⢤⠉⢠⣀⠀⠀⠈⢻⣶⣿⠁⠀⢹⣶⡿⣦⣿⣿⣿⡿⣿⠋⣼⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢰⠎⠁
⠀⠀⠳⡀⠀⠀⠀⠀⠀⠐⠟⠳⢦⣀⠈⠻⣿⡇⠀⠀⠙⣇⠀⠉⠉⠀⠀⠏⠸⣿⣿⣿⣿⠏⠁⠀⠀⣀⣴⢞⠀⠀⠀⠀⠀⠀⢀⣰⠆
⠀⠀⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣙⢿⣿⣄⠀⠀⠀⠛⠀⠀⠀⠀⢰⠀⠘⣿⣿⣿⣿⣀⣤⣤⡾⡟⠛⠁⠀⠀⠀⠀⠀⣠⣾⠋
⠀⠄⠀⠰⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠏⠉⠈⠀⠀⠀⢠⣄⠀⣀⣤⡿⠁⣶⡀
⡜⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠉⠀⠀⢀⡀⢂⣠⣬⡛⠛⠉⠀⠀⠀⢹⡇
⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣀⣹⣦⣸⣿⣿⣄⣀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣶⣶⠖⢉⣷⠔⠛⠁⠀⠀⠀⠀⢀⣤⡤⠊
⠀⠀⠹⢾⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢿⣿⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠋⠁⠀⣀⠀⡄⠀⠀⠀⠴⣿⠟⠁
⠀⠀⠀⠀⠙⠴⠦⠀⢤⣤⣤⣠⣤⣤⣀⣤⣤⣴⣤⡄⣡⡀⠉⠉⠉⠉⠉⣿⣿⣿⣿⣶⣶⣴⣿⣿⣿⣤⣷⡿⠞⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠉⢁⠀⠀⠀⠀⢸⣿⣿⣿⡏⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣠⣤⡴⣾⡆⠀⠀⢀⣤⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⡯⠤⠬⠤⠶⠶⠶⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣤⡶⣶⡄⠀⠀⠀⣨⣿⣯⣽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠷⠬⠀⠀⠴⠦⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣶⣾⢶⡄⠀⠀⢀⣼⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣩⡾⠀⠀⠀⢈⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠿⠁⠀⠀⠀⣨⣿⣿⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣧⡤⠀⠀⠀⠀⢉⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚⢁⣤⣿⡞⣳⡄⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡟⣿⣿⡿⣿⡆⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⣽⣿⣿⣦⣦⣤⣤⣼⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠿⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣶⣾⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⣿⣿⣿⣿⠿⠾⠿⡯⣿⣿⣿⠙⠋⣿⠿⠇⠀⠀⣀⢤⣤⣤⣴⣤⡶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⠙⠉⠀⡀⠀⠀⠰⣿⣷⣿⡏⠀⠀⠈⠙⡭⠆⠀⢸⠛⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣾⣿⡉⠛⠁⠀⠀⠀⢀⠂⢀⣠⣼⢇⣴⢀⡰⠛⠉⠁⢰⡄⠀⠀⠀⠀⠀⠚⠓⠿⣂⡄⠈
⠀⠀⠀⠀⠀⠀⠀⣤⢾⠟⠒⠛⠛⠓⢪⣿⣿⣿⡿⣡⣿⢷⡶⠶⣶⡶⠥⠴⢞⠛⠛⠿⠉⠁⣠⠤⠤⠞⠛⢓⣒⣒⢲⣄
⠀⠀⠀⠀⠀⣴⢽⣧⡜⠀⠀⠀⠀⠀⣾⣿⣿⣟⣡⢛⣫⣤⣭⣥⣼⣥⣤⠴⠿⠒⠒⠺⠯⠛⠉⠀⠀⠈⠉⠉⠉⠀⢸⣿⡄
⠀⠀⠀⠀⡾⠁⢸⣿⠀⠀⠀⠀⠀⢀⡟⢉⢘⣿⣿⡟⠛⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣿⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠃⠸⣶⠁⠈⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣸⣷⡄⢻⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡿⡇⠙⠢⢉⣢⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢷⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⢧

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⡀⢠⠀⠀⢀⠃⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠁⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢀⡞⠀⠬⠮⣤⠨⠅⠀⢀⠉⡜⠈⡠⡪⠃
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠈
⠀⣤⠤⡤⠷⣦⡀⡄⡠⠀⠀⠀⢀⠀⠀⠀⢀
⠸⣿⣷⣷⣀⡔⠙⢿⣡⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠛⠙⠋⠉⠳⣤⡼⠿⣿⣶⠀⠠⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⣾⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢷⣤⡿⢿⡞⠀⠀⠀⠀⠈⠙⢦⣤⠲⠒⠠⣼⡏⠃⢔⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣧⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠲⠛⠳⣍⠣⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠻⣷⣀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠉⠉⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⠀⠀⠈⢻⣾⣤⣄⠀⠀⠀⣧⠀⠀⠀⠐⠂⠀⠀⠀⠀⣿⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠀⠀⣠⠋⠻⢾⡷⣶⣿⣦⣀⡀⣴⡀⠀⡄⠀⠀⢻⣆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣤⠀⠀⠘⠠⣌⢻⠿⠿⠧⠿⣇⡁⠀⠀⠀⢿⠀⢴⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠸⣆⠠⠀⠡⡈⠲⡀⠀⠀⠀⠀⠹⡄⠀⠀⠸⡀⠈⠇⠀⣯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⢰⡶⠞⠉⠒⠀⠀⠀⠐⣄⠀⠀⠙⡄⠐⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣼⣇⠀⠀⠀⠀⢒⠀⠀⠈⠇⠀⠀⠈⢦⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⡄⠀⠀⠀⠀⢣⠀⠀⠀⠙⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⢰⡀⠀⠀⠀⠀⠆
⡀⢰⢲⣠⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠱⡄⠠⣄⡠⠃
⠓⠉⠄⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠩⡀
⠀⢸⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡍⣠⣀⡀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⢀⣠⡤⠭⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣱⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣜⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠉⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠠⠀⡀⠖⠱⠄
⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣠⣄⠀⠠⠀⠀⠀⠐⠛⠃⠀⠄⡀⠀⠀⠀⠀⠁⠀⣤⣤⠄
⠀⠀⠀⢀⠁⠷⠀⣀⡀⠐⠀⠀⠀⡠⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡔⠀⠀⠀⡠⠊
⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠂⠀⣀⠤⠒⢢
⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠉⠠⢤⡞⠁
⠀⠀⠀⠀⢠⠄⠠⠤⠤⠤⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠗⠒⠒⢓⠳
⠀⠀⠀⠀⠀⠙⠶⠖⢂⣀⣀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢿⡐⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠊⠥⢤⣴⠤⢺⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠔⠛⠁⠀⠈⠐⠂⠀⠠⠄⠀⠐⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠻⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⠦⠶⣴⢀⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡄⢘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⣄⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣆⣿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⡸⠳⠛⢯⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡏⠀⠈⣿⡆⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠙⠿⠒⢀⡀⢀⠀⡀⢀⠠⠠⢤⢀⣀⣀⣀⠀⠀⠀⠏⠀⠀⢀⣿⡇⠀⠀⢀⣡⡇
⠀⠀⠀⠀⠀⢷⠃⠀⠀⣷⠀⠀⢀⡼⠄⢃⣈⠀⠀⠤⠤⠤⠶⠀⠴⠶⣿⣧⣭⣤⡳⠶⣷⣶⣤⣾⡿⠁⠀⠀⠈⢹
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⣤⣴⠛⠀⣰⠎⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠸⣿⡋⠻⣿⣷⣍⢻⣿⠟⠁⠀⠀⠀⢠⡞⠀⠀⠀⠀⡀
⠀⠸⡆⠀⠀⠈⣦⠶⠀⠀⠈⠙⠂⣴⠃⠀⠀⠀⠀⣿⠀⠀⠀⡆⠀⠀⢀⣿⡃⠀⠈⣿⣿⠛⡇⠀⠀⢀⣼⣷⡟⠀⠀⠀⢀⡾
⠀⠀⢳⡴⢄⠀⠈⢦⠈⢠⡤⠀⢠⡿⢆⣀⢀⡀⣀⣼⡷⣄⠀⡇⢀⣀⣼⣷⡁⠀⢀⣿⣿⡆⢿⣤⡎⣠⣿⠋⠀⠀⢀⣜⡽⠁
⠀⠀⠀⠑⢆⠐⠄⠀⠉⠃⠇⠀⠸⠁⠀⠈⠉⠀⡼⠉⠀⠀⠀⠇⠉⠉⠙⢿⡛⠛⠛⣿⣿⢇⣼⠿⠛⠋⠁⠀⢀⠔⣼⡏⠁
⠀⠀⠀⠀⠀⠳⢤⣹⡇⠐⢾⣐⡆⡆⠀⠀⠀⠀⠳⣄⠀⠀⠀⡆⠀⠀⠀⡸⠁⠀⢀⣿⣿⣾⣿⡄⣤⣴⠒⠒⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠂⡙⣄⠀⠀⠀⠀⠀⠛⠒⡄⡅⠀⠀⢾⣅⡀⢀⣾⣿⣿⠿⠿⠛⠋⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠊⠀⠉⣻⣮⡛⢶⣖⠀⠀⠀⢈⡅⠁⠀⠀⠀⠈⣹⣿⣿⣿⣧⠤⠤⣀
⠀⠀⠀⠀⠀⠀⣠⣅⡘⠉⠉⠉⠉⣀⡀⡬⣦⡙⢦⣄⣀⣺⡄⠆⠀⠀⣀⣤⣿⣿⡉⠛⠛⠻⠶⠾⣿⠶⡄
⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⢰⣁⠀⠀⠐⠋⠙⠿⠶⢒⣉⣁⣀⣰⣿⠿⠿⠟⠻⠿⣍⣐⣄⠀⠀⠘⣷⠛
⠀⠀⠀⠀⠀⠰⣞⡀⠀⠀⠀⣈⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠇⠀⠀⠘⣧⡇
⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠟⠀⠀⣠⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠀⠀⠀⣿⣿⣿⣿⡿⢶⡄⢠⠾⣛⣿⣿⣿⣿⠂⠀⢀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⠀⠀⣿⣿⡇⠈⠙⠓⠁⠀⠉⠉⠁⢸⣿⣿⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⢿⣿⣷⣄⢸⣿⣶⡀⠀⠀⣤⡠⡀⣀⣴⣾⣿⣡⣾⣿⣿⣿⡿⠇
⠀⠀⠀⠀⢠⣼⣷⣦⣼⣿⡌⢻⣿⣿⣷⡙⢿⣿⣾⣿⣿⣿⣾⣿⡿⢟⣵⣿⣿⡿⠁⣿⣷⣤⣾⣧⣄
⠀⠀⠀⠀⠘⣿⣿⡿⣿⣿⣷⣀⠙⠿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣦⠸⣿⡿⠏⣀⣤⣿⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠘⢻⣶⡌⢻⣿⣿⣷⡄⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢀⣶⣿⣿⡿⢋⣶⡟⠋
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣦⣿⣿⡿⡿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⡿⣿⣿⣥⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠙⢿⣿⣟⡻⣿⣿⣿⣿⣯⣭⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣩⣽⣿⣿⣿⣿⠟⣛⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣾⣿⣯⣍⠹⠛⠃⠈⠙⠛⠿⠿⠿⠟⠛⠋⠈⠛⠟⣉⣭⡿⣫⣾⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣮⡉⠛⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠛⢋⣭⣾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⡶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣶⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠁⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠶⠀⠀⢨⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢡⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠁⠀⡀⠀⠀⠀⠈⠉⠀⠀⠈⠑⠀⠀⠀⠀⠀⠙⠒⠀⢻⠂⠀⠸⢠⡀
⠀⠀⠀⠀⠀⡒⠠⠀⠈⡍⠹⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⣹⡏⠀⠠⠖⢠⠇
⠀⠀⠀⠘⠀⠘⣆⡠⢀⠑⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⢀⡔⠀⢀
⠀⠀⠀⠐⠂⠀⠈⠢⢤⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠋⠠⢀⠉
⠀⠀⠀⠀⠀⣀⠀⢀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⣂⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠈⠀⠤⠵⠆⠀⠴⠶⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣵⣖⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠠⠔⠚⠋⢱⣄⣀⠀⠀⠀⠀⣀⣠⣤⢾⡑⠀⠀⠉⠉⠛⠓⠦⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠰⠊⠀⠀⠀⡶⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠙⠓⠞⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢦⡀⠀⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⠀⠀⢰⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⢦⣷⠀⠀⠀⠀⠀⣰⣷⣬⡇⠀⠀⠀⠀⢀⣼⣶⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣸⣿⠀⠀⠀⠀⢸⣿⢀⣼⡏⠀⠀⠀⠀⢾⣏⣀⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣫⠿⣇⠀⠀⠀⠀⢀⣏⡹⠿⡄⠀⠀⠀⠀⢸⣛⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡠⠖⡏⠀⠀⠀⠀⢸⣁⡤⢖⡇⠀⠀⠀⠀⢸⣁⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠖⢋⣿⠀⠀⠀⠀⢸⡧⠒⢉⡇⠀⠀⠀⠀⢸⠵⠚⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢚⣩⣧⠀⠀⠀⠀⢸⡶⢚⣩⡇⠀⠀⠀⠀⢸⠔⣋⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢉⡤⡟⠀⠀⠀⠀⢸⠋⣉⡤⡇⠀⠀⠀⠀⢸⢋⣡⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡥⠒⣧⠀⠀⠀⠀⢸⣏⠥⠚⡇⠀⠀⠀⠀⢸⣩⠴⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠴⢚⡏⠀⠀⠀⠀⢸⡥⠴⢚⡇⠀⠀⠀⠀⢸⡤⠖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠷⠾⠥⠿⠤⠖⠒⠒⠚⠓⠚⠛⠓⠒⠒⠒⠒⠺⠶⠯⠽⢄⣀⣀
⠀⠀⠀⠀⠀⣀⡤⠴⠒⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠢⢤⣀
⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⢀⡤⠒⠢⣄⠀⠀⠀⣀⠀⠀⡞⠉⢧⠀⠀⢰⠋⠀⠉⢇⠀⡞⠁⠹⡄⠀⢀⡞⠉⠉⢳⡀⠀⣠⢤⡀⠀⢀⠀⠀⣷
⠀⠀⠀⠘⡇⠀⠀⡾⠀⠀⠀⢸⠀⠠⡏⠉⡇⠀⡇⠀⢸⠀⠀⡅⠀⠀⠀⣀⠀⢻⠀⠀⡇⠀⢸⠀⢀⠀⠀⡇⠀⡇⢰⡇⠀⢸⢳⣤⠏
⠀⠀⠀⠀⠳⣀⣠⠃⠂⠀⠀⢸⠀⠀⡇⠀⠣⠤⠃⠀⡞⠀⠀⢹⠀⠀⠀⡇⠀⠸⣤⠀⡇⠀⠸⡀⠀⠀⢸⠁⠀⡇⠀⠓⠲⠋⡼
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀⠀⣹⠠⠀⠀⠀⠘⡇⠀⠀⠘⡆⠀⠰⡇⠀⠀⡇⢰⠇⠀⠀⢧⢀⠀⠘⢤⡤⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⡇⠀⠈⠀⠀⠀⠙⡚⠁⠀⠀⠀⠀⠸⡅⠀⠀⢀⡇⠀⠒⠙⠒⠚⠁⢸⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠈⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠤⠴⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠂⢸
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⢀⣀⡤⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢤⣀⡀
⢀⡠⠖⠋⠉⠀⢀⣠⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⣄⡀⠀⠈⠙⠲⢤
⡎⠀⠀⠀⠀⢼⠉⠀⠘⣆⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠉⣷
⠙⠦⣀⠀⠀⠈⠓⠦⢤⣀⣉⠓⠒⠲⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠴⠒⠒⣋⣁⡤⠴⠚⠁⠀⠀⣀⡴
⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡈⠉⠉⠙⠒⠒⠒⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠒⠒⠋⠉⠉⢁⣀⣀⡤⠤⠖⠚⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣙⣓⣒⣒⣶⣲⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣖⣶⣒⣒⣒⣋⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⢾⠇⢀⡄⠀⢠⡀⠀⢀⡄⠀⢠⠀⢸⡷⣄⠀⠀⠀⠀⢀⣴⡆⠀⡀⠀⠠⡄⠀⠀⡤⠀⠀⡀⠀⣶⣄
⠀⠀⢰⠃⢸⢀⡼⣿⠀⠈⡇⠀⣼⠀⠀⣿⢧⢀⡇⠘⡆⠀⠀⠀⣾⣿⡀⢠⣿⠀⠀⢻⠀⢰⠇⠀⢸⣧⡀⢸⣿⡆
⠀⠀⢸⠀⠈⠋⢠⠇⠀⣠⣿⣀⡿⣄⠀⠹⡄⠛⠁⢀⡇⠀⠀⠀⣿⣿⣷⣿⠏⠀⢀⣼⡀⣸⣄⠀⠈⢿⣷⣿⣿⡇
⠀⠀⠈⠳⢤⡀⠻⣄⠀⡷⠈⠉⠀⢯⠀⣠⠟⢀⡤⠞⠁⠀⠀⠀⠙⠿⣿⣿⣄⠀⢨⣿⣿⣿⣿⠀⢀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠙⠲⢬⣙⠁⢹⢠⡏⠈⣋⡥⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣷⢻⢳⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠈⣇⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣼⡜⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠤⠤⠴⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣛⣛⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣓⣒⣚⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣭⣭⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠬⣭⠭⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣛⣛⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⠆⠀⠀⠀⠸⠁⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡼⠛⠰⣊⠉⢹⣄⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⢠⡇⡀⣿⣦⣤⣼⠿⠽⠤⠤⠴⠦⢤⣀⡀⠈⣆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢘⠉⠀⣿⣄⠀⣾⠀⠀⢀⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠐⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠆⠀⣷⡀⠘⣯⣤⣿⡖⠟⠤⠷⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣲⡤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡔⠈⠳⣤⣈⣿⡯⢉⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠉⠉⠄⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⣤⡀⢌⠙⣿⣇⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠺⣇⣠⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠨⡙⠶⣥⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢾⣻⣿⣧⣤⣴⣾⡿⠋⠀⠀⣀⡿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠀⠈⠳⢿⣶⣤⣽⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢐⣿⣟⣛⠛⠉⠁⠀⢮⠴⠛⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠀⠀⠀⠀⠀⠀⡈⠙⠛⣿⡽⠀⠰⠀⠀⠰⠂⡁⣷⣾⣿⡟⠙⠛⠶⠶⠶⠟⠛⠉⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠜⣶⠲⠶⠚⠛⠳⣅⠊⠀⣀⣢⣾⣿⣿⡟⠉⠙⠳⠶⠦⠤⠴⠴⠷⣦⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⡏⠀⠉⠙⠳⠶⢤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⠄⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠰⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡰⠟⠃⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠒⠒⠒⠐⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠉⠉⠉⠉⠉⠙⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢹⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⢀⡜⠀⠀⠀⠆⠀⠴⠀⠄⠀⠀⠇⠀⠀⠰⣄⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠉⢹⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠂⠀⢀⠢⢀⠀⠀⡘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⣀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⢺⢁⠘⠓⠰⢰⣿⢷⡄⠀⠀⠀⠀⣾⢿⡇⠀⠈⠇⠌⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠗⠀⠀⣾⠿⡆⠀⠀⠀⠐⣰⢿⣷⠀⠀⡞⠀⠁
⠀⠀⠀⠀⠀⠘⠦⣀⢲⡀⠈⠉⠉⡀⠀⠀⠀⠀⠈⠉⠁⢀⡰⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠁⠘⣄⠀⠙⠋⠁⠀⠀⠀⠀⠈⠙⠁⠀⡠⠂⢀⠎
⠀⠀⠀⠀⠀⣀⣀⣀⡤⠚⠓⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠋⠠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠀⣠⠄⠓⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠂⠠⡀⠘⡄⡀
⠀⠀⠀⠀⠀⡇⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠂⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⡇
⠀⠀⠀⠀⠀⠇⡜⠀⠀⠊⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠠⠴⡥⢲⠁⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘⠂⡽⠦
⢀⣀⡀⠀⢀⡀⡀⠀⠄⢀⡐⠀⠐⠀⠑⡀⠀⢀⡀⠠⠬⠁⠀⠀⠀⣇⠀⠀⢀⣀⣀⡀⠀⠀⣇⡀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣀⣀⠀⣸⠀⠀⠀⠀⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠀⡀⠆⠀⠀⢀⡀⡀⢠⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡅⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⢀⣀⣀⣰⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠼⢀⠠⠂⠀⠀⠀⢜⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠐
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠄⠄⠒⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠂⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒
⠀⠀⠀⠀⣄⡀⢁⣀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠸⠀⢀⠔⠀⠘⡀⠀⢀⢆⠘⠉⠉⠉⠉⠉⠉⠁⠒⡀⡀⡀⠂
⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⡁⠈⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠀⠈⠁⠾⣜⠂⠀⠀⠀⠀⠀⠀⠀⠐⠒⡃⠇⠉⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠈⠉⠁⠀⠠⠂⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠀⠀⠐⠒⡒⠐⠒⠒⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⡆⠀⠀⠂⠐⠂⠀⠅⠀⠂⠠⣀⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠂⠀⠄⢰⠀⠄⠀⠀⠀⠰⠂⢀⡆⢀⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠄⠄⠀⠌⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠋⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡄⠤⠤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡎⠀⠀⢰⠃
⠀⠀⠀⠀⠲⠬⠥⡤⣬⣭⣬⣭⡵⡇⠀⠀⠀⠀⠀⢀⠠⢼⠤⠀⢰⡧⠤⢤⡧⣀⠀⠀⠀⠀⠀⠀⠀⢀⣁
⠀⠀⣠⡒⠓⠶⠶⠤⠶⠶⠶⠾⠟⢓⣠⠀⠀⠀⠀⠀⠒⠶⠶⠦⠤⠤⠴⠶⠶⠖⠇
⠀⠀⢨⠉⠙⠛⠛⠳⠚⠻⠛⠛⠙⠋⢽⠀⠀⠀⠀⡏⠁⠚⢀⠆⠀⠀⠜⠀⠒⠀⡆
⠀⠀⠈⡄⠈⠀⠀⠀⠐⠚⠀⠃⠃⠀⣼⠀⠀⠀⠀⢷⠠⠤⠄⡀⢀⡠⢂⠤⠤⢤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠙⠛⠒⠒⠶⠖⠶⠖⠓⠒⠋⠉⠀⠀⠀⠀⠀⠉⠛⠓⠒⠒⠒⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉

⠀⠀⠀⠀⠀⢶⣈⣹⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⢸⠀⠀⠀⡆⠀⡄
⠀⠀⠀⠐⣛⣉⡥⢴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠔⠺⠖⠺⠛⠈⠿⠉⠏⠉⠏⠉⠟⠐⣢
⠀⠀⠀⠈⠁⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡆⣙⠒⠒⠒⠒⠒⠒⡒⠒⢒⣒⡋⡉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠃⠠⠀⠀⠀⠀⠀⠀⢁⠌⠁⠖⠁⡐⠃⠈⠀⠛⠀⠉⢙⣐⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⠉⠉⠒⠚⠛⠒⠚⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢨⠀⠀⠀⠖⠄⢀⠒⡒
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠶⠦⠶⠶⢾⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⡀
⠀⠀⠀⣴⡪⡩⣍⣉⡆⠀⠀⠀⠀⣼⠉⠈⠒⠒⠶⠶⠶⠘⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠠⠟⡉⠟⠫⠍⠟⢓⡀
⠀⠀⠀⡇⠐⠊⠁⠂⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⢘⠛⠛⠛⠛⠛⠛⣛⣛⡀
⠀⠀⠀⡗⢒⠶⠒⠚⡇⠀⠀⠀⠀⣿⣿⠠⠤⠴⠂⠖⠀⣦⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢘⣉⣉⣉⣙⣛⣛⣋⡉
⠀⠀⠀⢠⣀⣀⣠⣤⠄⠀⠀⠀⠀⢸⣌⡊⠀⠀⠀⠀⠀⠠⣼⠀⠀⠀⠐⠂⠀⠖⡆⠀⠀⠀⢘⠿⣷⣭⣮⣾⣿⣿⡷
⠀⠀⠀⠨⠌⣩⡥⠠⠀⠀⠀⠀⠀⠸⢦⣄⣉⣙⣓⣒⣒⣂⡏⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠈⢿⣿⣾⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠀⠠⡧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⡇⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡇⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⠀⡆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠳⢴⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣴⡴⠃⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⡟
⠀⠀⠀⠠⠤⠤⠶⠖⠒⠛⠛⠛⠙⠛⠛⠛⢿⡶⣶⣶⣛⣠⣤⢄⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⢤⣄⣘⣳⣴⡶⣾⠛⠛⠛⠛⠛⠛⠛⠓⠒⠲⠦⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠈⢻⡄⠀⠈⠉⠙⠛⣦⢀⢰⡞⠛⠉⠉⠀⠀⣼⠋⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢷⠀⠀⠀⠀⡐⠻⠋⠻⠗⡀⠀⠀⠀⢰⠇⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⢀⣸⡇⠀⠀⣰⠿⠂⠀⠀⠺⠷⡀⠀⠀⣾⣀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠉⠳⡄⢰⠃⠀⠀⠀⠀⠀⠀⢣⠀⡴⠋⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠈⡟⠀⠀⣰⡏⢀⠀⠀⠀⠀⠀⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣿⠀⠈⢷⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣰⠏⠀⢸⡆⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⢠⣿⠀⠀⠈⢳⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠋⠀⠀⢸⣇⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢸⡇⠀⠀⢀⡟⠹⣦⡀⠀⠀⠀⠀⠀⠀⢠⣾⠙⣇⠀⠀⢀⣿⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠘⣿⣆⠀⣸⠃⠀⠙⠿⣶⣤⣤⣤⣴⡾⠟⠁⠀⢹⡀⢀⣾⡟⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢻⣧⡄⠀⠀⠸⣟⢆⡿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⣇⢞⡾⠁⠀⠀⣴⣿⠃⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠙⢿⣦⣄⠀⠘⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠀⢀⣤⣾⡟⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠈⠙⢮⡳⢦⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣠⠶⣫⠞⠉⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠙⠳⣌⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⡴⠟⠁⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⢄⡀⠈⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⣀⢴⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣤⣀⡀⠀⠀⠀⠀⠈⠙⠺⢽⡒⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢗⣺⠽⠚⠉⠀⠀⠀⠀⠀⢀⣠⣤⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣤⣤⣤⣤⣉⡘⣷⣀⣤⠴⠶⠶⠶⠶⢦⣄⣠⡟⣉⣡⣤⣤⣤⣤⣤⣴⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠒⠒⠚⣋⣠⣤⣤⠤⠤⣤⣤⣈⡛⠒⠒⠒⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⢀⣀⣀⣀⣀⣀⣀⠈⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣉⣉⣉⣉⣉⣉⣉⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠋⠉⠉⠉⠉⠉⠉⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠤⠤⠤⠤⠤⠤⠤⢴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⡀⠀⠀⠀⠀⢀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠛⠉⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠙⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⡼⠀⠀⣦⠀⠀⠀⢠⡆⠀⠸⣄⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣾⡇⠀⠀⢹⡄⠀⠀⣼⠁⠀⠀⣿⣦⣤⡶⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣓⣚⣋⣛⣶⣾⣋⣛⣒⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⣾⠆⢀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⢿⠀⢸⡿⠁⢠⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠹⣧⡀⠘⡆⢸⡇⢠⡟⠁⣠⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠈⠳⡄⠡⠸⡇⡼⢀⣼⠟⠋⠉
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠈⠑⠂⠄⠙⢆⠀⠆⠠⢋⡠⠶⠿⠿⠂
⠀⠀⠀⠀⠀⠻⠿⣷⣦⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⡀⠒⠛⠛⠿⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣤⠴⠚⠁⠀⠀⢀⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠺⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⡀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⢀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⣠⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀
⢠⣿⣇⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠻⠿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢂⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢀⣘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠈⠳⣴⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠐⣾⣿⣿⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⣿⣿⠏⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡈⡑⠤⡀⠂⠀⠐⠓⢄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠾⠟⠋⠀⣼⠠⠤⠐⠒⠋⠉⠉⠉⠛⠛⠓⠒⠶⠤⣌⡓⠁⢠⠊⠀⠀⠀⠢
⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⡈⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⡗⠀⠀⠀⠀⢣
⠀⠎⠀⠀⠈⠒⠤⣄⡀⠀⠀⡀⠷⣤⣠⠟⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠊⠐⠢⡈
⠀⠀⠀⠁⠐⠢⢄⡀⠉⢢⠔⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣤⠼⠁⠀⠑⡄⢣
⠀⠀⡀⠀⠀⠀⠀⠉⠲⠎⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⠋⠙⠒⢤⡤⡀⠀⢡⢸
⠀⠀⠁⠀⢀⡠⠤⠤⢸⣀⣤⣀⠀⢸⠀⠀⠀⠀⢀⣴⣾⡿⠋⢉⠙⣿⣿⣿⡇⢰⣢⠀⣽⡿⣋⣀⠀⠀⢀⠜⠀⡇⠀⠀⡜
⠀⠀⠀⠀⠈⠀⠀⡠⠴⢄⠀⠈⠙⠉⠙⡦⠀⢠⣿⣿⣿⣷⣄⣻⣃⣾⣿⣿⣿⡶⢶⣻⠭⠚⠁⠀⠉⠙⠁⠀⢠⠃⠀⠰⠃
⠀⠈⠀⠀⠐⡀⢈⠀⠀⠀⠑⠲⠶⠶⣚⡥⠞⠛⠫⠭⠭⠉⠩⠭⠭⠭⠽⠒⣒⣈⡠⠤⠤⠤⣤⣀⡠⡞⢀⡴⠁
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⢄⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠱⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠉⠪⢦⡄⠀⠀⠀⠀⠀⢀⣤⣜⣤⣠⠤⣀⣀⠠⣲⣦⡴⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠙⠛⠛⠻⠿⠿⠿⠙⠋⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⡀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠉⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⡄⢀⠀⠁⠀⠰⠁⢀⠀⢠⣶⡢⢤⡀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣭⣿⣿⣿⣷⣿⡷⠀⠀⠄⢸⣿⡿⢿⣯⡭⢽⣧⡀⠀⠀⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⢀⣽⡏⠀⠀⠈⠻⣿⣯⡅⠁⠇⢸⣿⡷⠂⠀⠀⠀⢹⣧⠀⣠⢣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⡄⣸⢿⠁⠀⠀⠀⢠⣽⣯⢿⢟⠃⠜⢻⣤⠀⠀⠀⠀⠀⢿⣶⢛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠰⣿⡿⣆⠀⠀⠀⢤⣿⣻⣿⣷⢠⣢⣴⣿⣷⣇⠚⠀⠀⣠⡾⣿⠎⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠀⡀⠀⠀⠠⠓⣻⣿⡀⠀⠘⢟⣹⣿⠗⠋⢝⣿⣿⣿⡇⠀⢀⣄⣀⠛⢸⣆⡀⣀⣴⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠁⠀⠈⠻⢷⣄⣀⡊⢘⠻⠐⠃⢻⣿⣿⣿⣧⣠⣟⠝⠁⠀⠀⢈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⠀⠙⢿⢉⣶⢀⣡⣤⣿⣿⣿⣿⣿⠓⠁⠀⢀⡔⠃⠘⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠈⠑⢤⣀⡤⢪⡊⢄⢉⡭⣽⣽⣿⣿⣿⣀⣤⠔⠉⠀⠀⠀⠣⠀⡀⠀⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠑⠠⠒⠀⠀⠀⠀⠀⠀⠐⡀⠶⠀⣀⠀⠀⠙⣿⣿⠹⢾⣾⣿⣿⣿⣿⣿⣿⡟⢁⣀⣠⣤⡤⠶⢸⠉⠁⠀⠀⠀⠉⠙⠂⢤⣀
⠀⠀⠀⠀⠀⠘⠁⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣐⠸⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠶⠆⠶⠤⠴⠦⠴⠶⠶⣿⣿⣭⠽⠾⠆⠾⢿⣿⣿⣿⣷⣠⣤⣀⣀⣀⣀⣀⢡⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⣻⠵⠶⠶⡿⣾⣿⣿⣿⣏⢀⠀⠀⠀⠈⠉⠉⠁⢾⠉⠉⠁⠸⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣀⢀⣠⣤⠤⠶⠾⠛⡿⠚⠚⠛⢛⡛⣿⣿⣿⡟⠛⠳⠶⢦⣤⣀⣀⡀⢸⠆⠀⠀⠀⠀⠙⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠛⠋⠉⠀⠀⠀⠀⠀⢻⡒⣂⢀⣁⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠙⠂⢸⡇⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠉⢋⣟⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⢴⣿⡿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠉⠳⡄⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣀⣀⣄⢸⣧⣰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠙⢦⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠭⠽⣿⣿⣿⣶⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⢾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣀⣠⣤⢰⡿⢿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢾⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⠙⠷⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⡿⠿⠿⣿⣿⡏⠉⠛⠻⣄⣀⣤⠒⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡆⣴⣾⡿⠇⠀⠀⠙⠛⠉⠁⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒

⠀⠀⠀⠀⡠⠐⠂⠉⠁⠒⠠⢄⡀
⠀⠀⠠⠛⠋⠀⠀⡤⠄⡖⢒⠒⠇
⠀⡔⠟⠀⠛⠀⠈⡠⠊⠀⢸
⠸⡆⠘⠀⠀⠀⡟⠀⠀⡠⠁⢠⠔⡒⠢⡀⡠⠔⠒⣲⣄
⠀⠑⠢⢤⣀⣴⣁⡀⠰⠀⠀⢠⠔⢁⡄⣈⡥⢞⡉⠈⢦
⠀⠗⡆⢀⠀⠀⠈⠇⠀⠀⠀⠘⢄⠈⠁⣰⡄⠈⠉⢀⣼⣤⠒⣴⠤⡀
⠀⠈⢢⡌⠁⠀⠀⢠⠠⠤⢀⠀⢀⢟⣜⠿⣿⣦⠰⣿⡀⢀⠀⠀⢐⠾⢢⠀⠀⠀⢀⠀⠂⡆
⠀⠀⠀⠻⢠⡀⠀⠀⠀⢤⣀⣩⡴⠿⣿⣿⣿⣿⠿⢩⣿⣦⡀⠀⠘⠃⠈⢣⡠⠐⠁⢀⡴⠀⣀⠀⡀
⠀⠀⠀⠀⠑⠥⠰⢄⠀⠀⠈⠙⢷⣄⠈⠛⠻⠿⠿⠛⠩⠐⠳⠀⠀⠀⠀⠈⠀⢀⠔⠡⠰⡀⢋⡠⢃
⠀⠀⠀⠀⢐⠂⠀⠀⣉⣒⣀⣐⣈⣙⣅⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠰⣀⡴⠗⢁⣠⠐⣣⠸⣥
⠀⠀⠀⠀⠈⡀⠙⠢⣤⡤⣤⠀⡀⣀⡉⡙⠢⣀⠀⠀⠀⣤⡞⠁⠀⠀⠀⡉⡴⠖⢛⣉⠤⠜⢇⠻⢄⣆
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠀⠀⠀⠀⠀⠈⠙⢋⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠚⠉⠁⠀⠀⠀⠈⡦⣿⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠠⡰⠆⠀⣤⡄⠀⠀⠀⢀⡎⣠⡄⠐⠶⠀⢀⣤⠠⠃⠀⠀⠀⠀⠀⠀⢀⡞⣜⠝⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⠶⠄⠈⠁⠾⠋⠁⠒⠂⠈⠉⠑⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠤⠚⠓⡀⠒⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆⠀⠀⣀⠀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠾⠆⠀⢀⠟⡆⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢠⠎⠀⠱⠀⠀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⣃⠄⠀⡋⠀⢠⠀⡇⠀⡴⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠳⢄⡣⠜⡀⢘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⠀⠇⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠈⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⡉⠉⠉⠙⠛⠉⠉⠉⠉⠉⠉⠉⠙⠃⡅⠀⠀⡇⠀⠀⠀⠀⠤⠤⠀⠠⠀⠒⠒⠒⢂⣤⡔⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠡⣌⠉⢛⠀⠀⢠⠤⣀⣄⡀⡀⠀⣀⡀⠀⠀⠀⠲⠂⡀⠱⠁⠀⠀⠀⠀⠀⡀⠀⣠⡀⠦⠤⣇⣀⠄⠐⠁⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢧⢣⠀⠀⠉⠁⠒⠛⠓⠈⠁⠎⠤⠥⠤⠥⠤⠤⡤⠤⠤⠀⠠⠀⡀⠒⠒⢚⣂⠀⠀⠈⠉⠁⠀⠓⠀⠈⠀⠀⠀⠘⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣠⣤⣠⡤⣄⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠁⠈⠁⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣀⠀⢤⣼⢶⣄⣠⢏⢧⡨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠳⠙⠡⠋⠐⠩⣻⡷⠄⠀⢰⠹⡇⠠⣤⡾⣫⢛⡽⣲⠖⡲⢋⠔⣡⢜⣿⡂⠀⣰⢟⢟⡽⡫⠣⠢⠪⠃⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠈⡗⢈⣳⣒⠤⣰⣆⣀⡈⠛⣦⠠⠆⠈⠛⠹⠋⠈⠀⠉⠀⠀⠈⠀⠀⠀⠀⠉⠉⠘⡙⠁⢁⣀⡀⢠⡄⡀⠤⠰⠆⢺
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⠎⠱⠦⠴⡤⢬⣸⣗⣈⣲⠠⠿⠔⣂⡐⠤⣶⠴⠂⣲⠠⡿⠤⠒⣂⡒⠚⠓⢊⣡⣤⣍⣉⣉⡴⢲⡳⠖⢻
⠀⠀⠀⠀⠀⠀⠀⣠⡗⣦⢆⠄⡀⢀⠀⠉⠀⠈⠁⠽⠉⠙⠛⠋⢽⠉⠛⠋⠛⠉⣤⠈⠓⠚⠋⣭⠉⠛⠛⠉⠀⠛⠀⢀⢀⢀⡼⣥⣤⢨⣀
⠀⠀⠀⠀⠀⠀⡼⠀⠛⢶⣥⣊⣔⡡⡢⢚⠔⡠⣠⢆⡤⡠⢀⠴⡤⢂⠄⡀⣀⢀⡀⡠⢠⢀⣄⣀⡰⣢⢂⡖⣴⡿⠿⠛⠛⠛⠚⠘⠉⠀⢻⠳⡀
⠀⠀⠀⠀⠀⢰⢥⡶⢤⣤⠀⡀⢀⠉⠙⠓⠾⠾⠵⠯⠞⢛⣷⣛⡒⠓⠮⠶⠵⣫⡾⠾⠓⠋⠉⠉⠉⠉⠉⠙⠋⠀⠀⢀⠀⠀⠀⣴⣲⣄⠀⠀⡛
⠀⠀⠀⠀⠀⠀⠟⢧⡓⣷⡺⡫⣻⡀⠐⣧⣦⠀⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣀⠀⠀⠀⢀⣤⣶⡲⣤⣴⢿⣧⣀⡠⠛⠛⠛⠗⡞⠁
⠀⠀⠀⠀⠀⣶⠠⠤⠀⠀⠈⠈⠪⠿⠴⠾⣻⣤⣿⣟⣽⠉⠀⣾⡻⣧⣤⣶⣛⣿⣦⣶⣣⣮⣧⡀⣤⣼⠟⠓⠙⠘⠋⠁⠀⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⣿⡀⢠⠄⣀⣀⠀⣀⠀⣀⡀⠀⠀⠀⠈⠛⠳⠜⠛⠛⠛⠉⠈⢈⠁⠉⠀⠁⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀⢀⣲⡇
⠀⠀⠀⠀⠀⣿⣌⣙⣃⣬⣩⣝⠋⠋⣉⣉⠣⣐⣶⢀⣤⣀⣰⣄⢠⠄⠠⣀⣤⣀⣀⣄⡀⢠⡄⠀⠀⢠⡦⣀⠤⠤⠾⠷⠊⣈⣻⣉⣀⣤⡥⡇
⠀⠀⠀⠀⠀⣠⠈⠉⠉⠁⠳⠈⠙⠋⢩⡍⠱⠿⠿⠦⡶⠦⢤⣭⡤⠖⣓⠫⣭⣁⣠⠤⢬⣟⣋⣡⠬⠽⣯⣅⣤⠶⣶⠲⠛⠡⠮⠉⠉⠀⠁⡖
⠀⠀⠀⠐⣒⣻⡷⢹⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠸⠂⠀⠀⠀⠛⠀⠀⠀⠀⠀⣉⢀⠀⠄⠄⠂⠀⠀⣠⡴⠄⠀⣀
⠀⠀⣉⠤⠔⢂⡱⠦⢄⡉⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢄⡠⡠⢂⠄⡰⣠⢢⢂⠜⠜⠁⠀⠁⠀⠀⢀⣠⠄⢒⠽⠋⢀⡤⠊⠁⣀
⢀⡠⠄⠒⢉⡠⠔⠒⢉⡡⠝⠓⠒⣠⠤⠄⢀⣀⠀⠀⠁⠐⠁⠈⠊⠐⠋⠚⠱⠋⠊⠐⠁⠊⠈⠀⠀⢀⣀⡀⠤⣤⠖⠂⡩⠛⣁⠔⠁⡠⠖⠁⣀⠔⠋⣠⠖
⢠⠔⠂⠉⣀⡤⠖⠈⣁⡤⠒⠋⠁⣀⡤⠖⠉⠉⢀⣉⠍⠉⠀⡂⠴⣒⠴⠖⢒⡴⠖⠀⣉⠝⠋⢉⡴⠋⢁⠴⠋⢀⠔⢋⡤⠊⣀⠔⠋⢀⡴⠊⢁⡴⢊⡠⠖
⠀⠖⠊⢉⣀⠤⠖⠋⠁⢀⡤⠖⠋⠀⣀⡤⠖⠈⠁⣀⡤⢖⣩⠔⠊⢀⡤⠖⠁⣠⠔⠋⢀⣠⠞⢁⡠⠞⠁⡠⠞⣁⠔⠁⣠⠞⠁⡠⠚⠁⡠⠒⠁⠀⠋⠀⠈
⠀⠀⠊⠉⢀⣀⠤⠒⠊⢁⣀⠤⠐⠉⠁⢀⡠⠔⢋⣡⠖⠉⢀⡠⠖⠁⢀⡤⠊⠁⣠⠔⠉⢀⡴⠋⢀⠔⠊⡠⠞⢁⠴⠊⢠⠔⠉⠀⠀⠈
⠀⠀⠀⠀⠉⠀⠀⠤⠒⠉⠀⢀⡠⠔⠊⠁⡠⠔⠋⠀⣠⠔⠉⢀⡠⠒⠁⢀⡴⠋⠀⣠⠞⠁⢠⠖⠁⠀⠊⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢆⠀⠀⠀⢀⠀⠀⢣⠆⠀⠀⢠⠀⠀⠀⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠞⠀⠀⠠⣋⡆⠀⡆⡆⠀⢠⣀⠀⠀⠀⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡆⠀⠀⢠⢉⠀⠀⡇⢡⠀⢠⢃⠀⠀⢰⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠤⠄⢺⢸⠒⠒⣇⡜⠒⢺⠸⠒⠒⢺⠀⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠉⠣⠇⠀⠀⢸⢸⠀⠀⠀⠀⠀⢸⢰⠀⠀⠀⠒⠁⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠢⢄⣀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣀⠤⠔⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡆⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⢀⡀⠀⠀⣀⣀⠞⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣧⠏⠀⠀⠑⠒⠒⠒⠉⠙⢆⡴⠁⠀⠱⠀⡁⠈⠢⠴⠁⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠒⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣐⠄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠤⠀⠀⠀⠀⠀⠄⢈⠀⠀⣀
⠀⠀⠀⠀⠀⠐⠊⢠⡔⠉⠀⠀⠀⠀⠀⠀⠈⢻⣽⣯⡉⠀⠎⠀⢀⠀⠀⠀⠀⠀⠶⠤⣀⡋⢀⡄⠀⠀⠀⠀⠀⠀⠤⣄⠒⠀⣱⠂
⠀⠀⠀⢈⠥⠒⠒⠦⠿⣶⠶⠀⣀⣀⡠⢔⠴⠯⡀⠑⠂⠴⠤⠔⠁⠀⠀⠀⠀⠀⠀⡠⠤⠨⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣦⡀
⠀⠀⡴⠉⠁⢠⠈⡀⠀⠇⠀⠁⡉⠉⠁⠨⢠⠂⠈⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠔⠁⠑⠒⠀⣄⠀⠀⠀⠀⠀⠀⢀⠰⠦⠈⠛⢯⠂
⠀⠀⠂⠀⠀⠁⠀⡏⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡀⠀⠀⠀⠀⠀⠁⡀⠐⠀⣤⡤⢈⡁⠀⠀⠓⠀⠐⠋⠁⣀⠀⠈⡻⠉
⠀⠀⠀⠠⠁⠀⠀⢣⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠸⠋⠓⠒⠀⢐⡒⠴⠛⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⢔⠉⡐⠀⢈⡦⠒⡂⠀⠀⠀⢰⢆⡴⠂⢶⡍⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠀⠘⠀⠈⠀⠠⠉⠠⠤⠐⠁⠀⠑⠂⠾⠤⢾⢈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢘⠄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠁
⠀⠀⠀⡀⠀⠀⣀⢠⠀⠱⣰⡦⠂⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠐⠠⠤⣀⣀
⠀⠀⠀⠄⠀⠈⢤⡈⢢⣐⠋⠀⠀⠀⠀⠀⠘⣆⠀⠀⠋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠚⠉
⠀⠈⠀⠀⠀⠀⡀⢁⣄⣻⠀⠀⠀⠀⠀⠀⠀⡷⠐⠐⠃⡇⡀⠀⠀⠀⠀⠀⠠⠜⢋⡁⠒⣳⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠳⠢⡀
⠀⠀⠀⠀⡀⠊⠀⠀⠈⠙⠄⠀⠀⠀⠀⢀⠼⠋⠁⡘⠈⠃⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠈⠀⠀⠀⠡⣤⠶⠤⠶⠶⣚⡂⠀⡄⡁⠙
⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠠⠯⠔⠂⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⡀⠁⠀⣀⣼⡐
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠦⠁⠀⣀⢎⣠⠟⠁⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⣤⠴⡶⠛⠛⢻⡟⠉⢹⣿⠟⠟⠛⠚⠉⠉⣽⠋⠉⠁⡀⠀⠀⢠⡙⠛⠙⠒⢂⠠⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⢀⣴⡟⠋⠀⣠⠃⠀⠠⠺⣧⣠⣾⡷⠀⠐⠀⠀⠀⠀⠈⠑⠐⠋⢀⣀⣠⡾⣁⠢⠍⠻⢦⡤⠀⠐⠒⠒⠒⠒⠂⠠⠄⣀⠀⠀⠘
⠀⢀⡎⠙⢧⣔⠛⠙⡀⠀⠀⠀⠉⢸⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⣼⡯⠽⡠⣀⣀⣄⣤⡀⠀⠀⠀⡀⣀⡤⢀⣶⣶⣦⠵⠄
⠀⢸⡟⢾⣏⡈⢑⡤⠼⠒⠂⢠⡀⠀⠷⣦⣥⣀⣀⣀⣄⣀⣠⣶⡺⢧⣾⣿⠴⡦⢒⡭⠀⠀⣀⣀⣴⡷⠀⠀⠉⠳⠾⠳⣉⣉⣀⠽⢔⠟
⠀⠸⣧⣤⢀⣼⠯⣌⣀⠸⣆⠘⠁⠀⠀⠙⡄⠙⢍⠹⢧⡅⠙⠲⠾⢼⣆⣙⣛⠉⠉⣙⣿⣋⣁⣽⡿⣧⣄⣀⣀⣀⣀⣀⣀⣀⣤⠐⠋
⠀⠀⠹⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠠⣵⡆⢛⡒⠀⢠⣤⣄⡀⠁⠁⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⡄
⠀⠀⠀⠙⢶⡆⠈⠉⠉⠐⢲⠄⢰⠄⠀⠀⠀⠀⡛⡄⠀⠚⢿⡀⠈⠱⣤⡀⠀⠨⠖⠄⡀
⠀⠀⠀⠀⠀⠙⠦⢄⣀⣠⠾⠄⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠃⠀⠀⠈⠿⡶⡄⠈⠐⡟⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⣰⠀⠀⠀⠀⠀⠘⣾⠆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡀⠀⢠⠀⠀⡐⠀⢀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⡴⠊⢿⣤⢸⡆⢀⡇⣤⡿⠑⢦⡙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⢀⡞⠀⢀⡼⣿⢸⡇⢸⠀⣿⢦⡀⠀⢳⡀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡼⠀⢀⡾⢠⡇⢸⠀⢸⠀⢹⡄⣷⡀⠀⢧⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢠⡇⢠⠏⠀⡞⠀⡏⠀⢸⠀⠀⢳⡀⠹⡄⢸⡄⠀⠀⠁⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢴⢁⠏⠀⢸⠁⢰⠃⠀⠘⡄⠀⠈⣇⠀⠹⡈⡆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢠⠀⠀⠸⣏⣀⣤⠇⠀⢸⠀⠀⠀⡇⠀⠀⠘⣦⣤⡹⠇⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⣧⠀⠀⠀⢀⣽⠃⠀⠀⡏⠀⠀⠀⣽⠀⠀⠀⠘⣯⡀⠀⠀⠀⣼⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢿⣷⣦⣾⢟⡜⠀⠀⠀⣧⠀⠀⠀⢸⠀⠀⠀⠀⢣⡻⣷⣴⣾⣿⡽
⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠾⠓⢏⠀⠀⠀⠀⢸⣆⢀⡀⣸⠀⠀⠀⠀⠀⡹⠚⠷⠒⣻⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⣽⠇⠀⠀⠀⠀⠀⠀⠀⢿⣦⠀⢀⡜⠀⠀⠀⠀⠐⣿⡾⢷⣿⠂⠀⠀⠀⠀⢣⡀⠀⣴⡟⠀⠀⠀⠀⠀⠀⠀⡸⣏
⠀⠀⠀⠀⠀⠙⣾⣄⠀⠀⠀⠀⠀⠀⠘⣿⣶⣗⣧⠀⠀⠀⠠⣿⣿⠁⠈⣿⣿⠀⠀⠀⠀⣼⣺⣶⣿⠃⠀⠀⠀⠀⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠠⣿⣿⠋⠘⣇⠀⠀⢠⠿⠋⠀⠀⠙⠿⡄⠀⠀⣸⠀⠙⢻⡿⠀⠀⠀⠀⠀⢀⣴⡷⠃
⢻⣆⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠻⣿⣦⡀⠙⢦⠀⡏⠀⠀⠀⠀⠀⠀⢹⠀⡴⠁⢀⣴⣿⠟⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⣰
⠀⠻⣷⣤⡀⠀⠀⠀⠀⠈⢿⣦⣀⠀⠀⠀⠈⠻⣝⣦⣀⣿⣇⠀⠀⠀⠀⠀⠀⣸⣿⣁⣼⣫⠟⠁⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⢀⣤⣾⠟
⠀⠀⠈⠙⠿⣶⣄⡀⠀⠀⠀⠙⠣⠿⣷⣶⣤⣀⡈⠙⣟⡾⢏⠷⣄⣀⣀⣠⠞⡽⢷⣻⠋⣁⣀⣤⣴⣾⠯⠟⠋⠀⠀⠀⢀⣠⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡤⣤⣄⣀⣀⣀⠉⠉⠛⠻⣽⡇⠀⠁⠈⠛⠛⠁⠈⠀⢹⣭⠟⠛⠉⠉⣁⣀⣀⣠⣤⢤⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠿⠿⢷⣖⣺⠃⠀⠀⠀⢀⠀⠀⠀⠀⠸⣗⣲⡾⠿⠿⠿⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡁
⠀⠀⠀⠀⢀⣀⣠⣴⣤⣴⣴⣶⣶⣶⣭⣭⡭⠭⠽⠿⢿⣧⣄⠀⠀⠀⠀⠀⠀⣠⢾⡿⠿⠯⠭⢭⣭⣭⣶⣿⣶⣶⣦⣤⣦⣄⣀⡀
⠀⣠⣴⠾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣷⡙⢷⣶⣤⣤⣶⡿⢋⣾⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠷⣦⣄
⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠁⠀⡘⢻⡶⠦⠤⠤⢴⢶⣟⢇⠀⠈⠙⠻⢿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠁⠘⠃⠀⠀⠀⠀⢈⣙⡼⠀⠀⠀⠀⠀⠈⠛⠻⢧⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⣄⡀⠉⠙⢛⣻⣯⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠙⠛⠦⠤⠤⠴⣾⡻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⠶⠀⠐⠂⠀⠲⣃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢦⡈⠙⠋⢉⣵⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⠀⠀⠉⠉⠉⣍⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠈⠉⠁⠈⠉⠁⠉⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⠀⠀⠀⠀⡌⠀⠘⡇⢸⠀⠀⢢⠀⠀⠀⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡘⠀⠀⠀⢹⡏⠀⠀⠀⢣⠀⢀⡴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣲⣤⣤⣞⣳⣤⣤⣖⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⠞⢙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠳⣄
⠀⠀⠀⠀⢀⡞⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢷
⠀⠀⠀⠀⡼⠀⠀⠀⢸⠀⠀⠀⢀⣤⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⣤⡀⠀⠀⠀⡇⠀⠀⠈⣧
⠀⠀⠀⢀⡇⠀⠀⠀⠈⢧⣠⠴⠋⢸⠀⠀⢠⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⡇⠙⢦⣀⡼⠁⠀⠀⠀⢸⡀
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠙⢦⡀⡸⠀⠀⠀⢀⠔⢉⣉⡑⢄⡠⠤⠤⠤⠤⢤⠖⣉⣉⠓⣄⠀⠀⠀⠀⢇⢀⡴⠋⠁⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⠰⣧⣼⣿⢸⠀⠀⠀⠀⠀⢎⢸⣅⣽⡷⢸⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣿⡦⠴⠖⠊⠢⢍⣋⠥⣫⠄⠀⠀⠀⠀⣜⡦⢍⣋⠵⠋⠙⠲⠦⢶⣿⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⡀⣹⠦⣄⣀⣀⣀⢴⣽⡗⢄⡰⠚⠙⠒⠤⠤⢅⣀⡀⠀⠀⢀⣀⠤⠄⠔⠊⠉⠲⡀⠤⢻⣹⡦⣀⣀⣀⣠⠴⣏⣀
⠀⢀⡔⠊⠉⡀⠷⢦⠤⠤⠽⠶⢎⣀⣵⠢⡀⠀⢠⠶⡂⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⣒⢦⡀⠀⢀⠔⣯⣀⡱⠶⠯⠤⠤⡴⠾⢀⠉⠑⢦⡀
⠀⡟⢽⠒⠹⡯⢽⠥⣾⡋⠉⠉⣉⣉⣺⣤⡇⠀⠀⠣⡈⠙⢲⣲⡶⠦⠴⢶⣶⡖⠋⢀⠆⠀⠀⢸⣤⣗⣉⣉⠉⠉⢹⣷⠬⡯⢽⠏⠒⡯⢻
⠀⡇⢸⠀⡤⠛⢁⡠⠛⠉⠉⠉⠁⠀⠀⠀⠙⢦⣀⠀⠙⠦⣏⡤⠬⢭⡭⠴⠦⣼⡴⠋⠀⣀⡴⠋⠀⠀⠀⠈⠉⠉⠉⠛⢄⡈⠛⢤⠀⡇⢸
⠀⢻⣸⣾⣳⡴⢻⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠦⣌⡙⠓⠒⠒⠒⢛⣩⠶⠓⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡟⢦⣞⣷⣇⡟
⠀⠀⠙⢱⡀⡅⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢨⢀⡎⠋
⠀⠀⠀⠈⣧⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣼⠁
⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠬⣧⠀⡏⡹⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⢤⣸⣸⡟⡡⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡉⣩⡿⣟⠒⠒⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠖⢆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠙⠒⠃⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡠⢄⠀⠀⢧⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⠦⠜⠀⢀⣀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢿⣠⠗⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡆⢈⡧⠀⠀⠀⠀⠀⡏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⠉⠉⠀⡰⠒⣦⠀⠀⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠙⠒⠃⠀⢠⠔⢦⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⡞⢑⡄⠀⠀⣀⠀⠀⠀⠘⠶⠊⠀⢨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠙⠋⠀⠀⣎⣁⡗⠀⠀⢀⡀⠀⠀⠹⠼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⡁⣨⠆⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢧⣠⠗⠀⢀⡤⢄⡀⠀⠀⠀⠉⠁⠀⢀⡴⠲⡄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠸⣄⡴⠀⠀⡠⠠⡄⠀⠀⠈⠓⠚⠁⠀⠘⣇
⠀⠀⠀⠀⠀⠀⢠⠇⣰⠋⣢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠴⠁⠀⢀⠤⠢⠀⠀⢠⠖⠻⣆
⠀⠀⠀⠀⠀⠀⡾⣦⣉⠚⠁⠀⠀⢰⠊⣲⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠚⠀⠀⢈⣳⡶⠛⡇
⠀⠀⠀⠀⠀⠀⢷⣄⡉⠛⠶⢤⣀⣈⠙⠁⠀⠀⢼⣉⡷⠀⠀⢠⣖⣦⣤⠴⠞⠋⣁⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣉⠉⠛⠒⠒⠚⠛⠓⠓⠛⠉⠉⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠒⠚⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠍⣉⠛⢷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠿⢿⣷⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢻⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡿⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠉⠈⠛⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⣿⡿⠛⠁⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠈⢿⣷⣦⣀⣤⣾⣿⣿⣿⣿⣿⣦⣄⡀⠀⣆⣠⠽⠋⠁⠀⣀⡼⠛⠁⠀⠀⠀⢀
⠀⠀⠈⢲⠄⣀⠀⠀⠀⠈⠁⠛⠒⠶⢦⣤⣼⣿⣫⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⡴⠖⠚⠛⠈⠉⠀⠀⠀⣀⡀⡴⠏
⠀⠀⠀⠀⠀⠉⠛⠁⠐⠤⡀⣠⠤⠤⠴⣶⡟⠉⢩⣽⣿⡳⣽⣿⣽⣿⣿⣿⣿⡿⠿⠶⠦⠤⠤⠀⢀⡤⠖⠛⠛⠉
⠀⠀⠀⠀⢀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠶⠞⠛⠢⣮⣺⣤⣿⣿⣮⣿⣿⠿⠛⠛⠓⠲⠤⢄⠀⠀⠀⣀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⠁⠛⠁⠀⠉⠉⠀⠀⢰⣄⣬⡷⠞⠚⠛⠛⠛⠛⠛⠛⠛⠻⢷⣾⣍⡄⣤⡄⠈⠙⠉⠙⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰⣦⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡄⠀⠀⣏⢸⠀⠀⠀⣼⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⠁⠀⠀⣹⣯⠀⠀⠀⠻⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡇⠀⠀⣧⢾⠀⠀⠀⣾⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⡇⠀⠀⣧⢾⠀⠀⠀⢿⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠴⢾⢹⡗⠒⠒⢿⡼⠓⠒⠒⢿⢹⡗⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⢤⣄⣀⣈⣉⣡⣈⣀⣀⣈⣀⣀⣀⣈⣉⣡⣤⡭⠴⠞⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡆⠀⢀⡭⢯⡉⠉⠉⣭⡍⣩⣍⠉⠀⢠⢤⡄⢸⡿⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣧⢀⡞⠀⠄⢧⣀⡼⠣⠙⡁⢹⡀⠀⡟⠐⠙⠚⠁⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣈⣉⣀⣁⣐⣀⣉⣀⣐⣀⣀⣂⣛⣚⣁⣈⣀⣂⣐⣀⣟⣀⡀
⠀⠀⠀⠀⠀⢀⣞⠉⠉⠉⣩⡭⢭⣉⠉⠉⠉⠉⠉⣩⡭⢭⣉⠉⠉⠉⠉⣩⠽⠿⢭⡉⠉⢳
⠀⠀⠀⠀⠀⢸⠛⠲⠶⠛⡁⠐⠀⠈⠳⠤⡤⠴⠚⠁⡒⠀⠉⠓⠶⠶⠛⠁⠀⡁⠀⠉⡙⢹⡇
⠀⠀⠀⠀⠀⢸⣄⠀⢀⣠⣥⣄⡉⢀⣂⣠⣈⡀⠠⠀⠁⢀⣨⣤⣤⣀⡀⠁⠀⣀⡤⠖⠒⢾⡇
⠀⠀⠀⠀⠀⢸⣏⣉⣉⣤⢤⣌⣉⣉⣡⠤⣤⣉⠛⠒⣛⣉⡤⠴⠦⢬⣉⣉⣉⣩⡴⠖⠚⢺⠇
⠀⠀⠀⠀⠀⠀⣧⠉⢉⢀⡀⣤⢉⡉⢀⢠⡀⢩⠉⡉⠉⠁⡀⢰⠀⡀⣈⠉⣉⠀⣀⠀⡄⢸
⠀⠀⠀⠀⠠⠤⠿⠶⠾⠾⠷⠿⠴⠧⢼⣼⣧⣼⣤⣷⠼⠦⠷⠾⠶⠷⠿⠶⠿⠶⠿⠴⠷⠟⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⢈⠉⠙⣳⣶⡤⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠰⢿⣶⠶⠂⠀⠀⠀⠈⠓⠦⣄⡉⢣⡀⡤⠋⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠑⢐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠁⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠰⠀⠀⢠⡦⠊⠀⠀⠀⠀⠀⢀⡀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢰⠀⠶⠏⠀⠀⠀⠀⠀⠂⠀⢠⠄⠃⠘⠈⢳⣬⡆⠀⠀⠀⠀⠀⠀⡄⠈⠉⠁⠻⣧
⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⢠⣶⡆⠁⠀⠀⠀⠈⠀⠀⠀⠀⠐⣿⡧⠠⣄⡀⠁⢠⢶⠆⠀⠀⠀⣠⡇⢀
⠀⠀⠀⠀⢻⡄⠈⢆⡀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠄⠈⡷⢀⣼⣿⠀⠀⣠⡾⠏⠀⡞
⠀⠀⠀⠀⠀⠡⣄⠀⠈⠀⠐⠯⡙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠛⠉⠉⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠢⠤⠤⣤⣀⡈⠩⢷⣄⣀⡀⣦⣄⣀⣦⣀⣤⣾⡿⠏⢀⣀⣐⡦⠀⠀⠋
⠀⠀⣠⣤⣤⣄⠀⣐⡀⠠⢧⠤⠤⠤⠄⢾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣧⣤⠤⠄⢠⡬⠁⢐⠂⠀⢠⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠰⡠⠴⡂⢀⣀⠀⠗⠁⠀⠀⠈⠁⠀⠲⠀⣀⣀⢁⠶⠴⠆⠀⠲⠄
⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠼⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠦⠀⠉⠀⠀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣤⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠉⠀⠀⣀⣤⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⢀⣤⠞⠟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⣿⣿⣠⣴⣶⣿⠟⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣀⣴⣿⣿⣿⣟⡟⢁⡠⠖⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠛⣻⣿⣿⣿⣿⣿⣿⠏⠉⠁⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⡔⠀⠀⣼⠞⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣀⠤⠖⠂⠉⢀⣤⣴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢦⡀⠀⠀⠀⣾⡇⠀⡏⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⠔⠊⠁⠀⠀⣠⠞⢉⣴⠏⠀⠀⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣿⡀⠀⣳⡖⣟⡿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢠⠞⠁⠀⠀⠀⠀⣼⠋⣰⣿⠏⠀⠀⢠⣾⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠘⣧⠀⢹⠀⠈⢿⣆⠀⠀⠀⠀⠀⢀⣤⣤⣴⣮⡄⠔⠁⠀⠀⠀⠀⠀⣸⡇⣠⣿⡿⠀⠀⣰⣿⠃⠀⡇
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⢤⣄⡀⠀⠈⢷⡀⠘⣦⡀⠣⡀⠈⢿⠀⣀⣤⠴⠞⢻⣿⡿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣻⣭⣽⡛⠃⣠⣾⡿⠃⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠛⢦⡈⠻⣆⠙⣶⡾⠛⠉⠀⠀⠀⠈⠁⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠿⠋⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠉⠳⣄⠙⠦⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠉⠉⠉⠉⠙⠳⢤⡈⢱⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⣀⡀⠀⠀⣈⣿⢿⡟⠀⠀⠀⠀⢀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠳⠶⢶⣮⢻⣿⡿⠛⠉⠉⢩⢻⣤⣤⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⠛⢿⣦⣤⣀⣠⣤⣶⣿⣿⣿⡿⠓⠄⡀⠀⠈⢓⡟⠁⠀⡠⠖⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣧⣄⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⠟⠲⣄⡀⠀⠉⠙⠛⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠙⢷⣦⣀⣴⣿⣿⣻⣿⣿⠟⣅⠙⢦⡈⠙⠶⢤⣀⣀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠈⠛⢶⣄⠀⠀⣻⣿⣿⣿⡿⠁⠋⠀⢦⡈⠳⣄⠙⢦⣀⠀⠈⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠷⣤⡀⠀⠙⣿⣾⣿⣿⠉⠉⠀⠀⠀⠀⠀⠹⡆⠈⢳⡀⠈⠙⠶⢦⣀
⠀⠀⠀⣔⡒⠂⠀⢀⠀⣀⢿⣄⠀⠈⢻⣶⣾⣿⡏⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠻⣄⠀⠀⠀⠙⠳⣄
⠀⠀⠈⢿⣿⣿⠷⠚⣿⡷⣄⠈⢳⣦⣾⣿⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢹⡄⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⣯⣤⣴⠾⠛⣻⣮⡳⣾⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠿⣁⣠⣾⣿⠟⠛⡗⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠘⠻⢟⣫⡇⣰⢻⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠋⣾⣷⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣤⣄⠀⠀⠘⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⣿⡟⠛⠛⠀⢀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⢨⣿⠃⠈⠓⠀⠀⠀⠉⠁⢸⡿⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡵⠾⢿⣿⣷⡀⠀⠀⠀⢸⣿⡄⠀⠀⣠⣄⠀⠀⢀⣾⡇⠀⠀⠀⢀⡾⣫⡼⠿⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⣿⣿⣅⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠻⣷⣄⠜⣹⣿⣧⣤⣿⠞⠁⠀⠀⢠⣞⣴⠏⠀⠀⠀⣨⣿⠿⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣵⡾⣯⡙⢷⣿⣷⣄⠀⠀⠹⣿⣿⣄⠀⠀⠀⢸⣿⢩⣿⣿⣿⣿⡅⠀⠀⠀⢠⣿⣿⠋⠀⠀⢠⣾⣿⡿⢊⣽⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⡿⠟⢉⣾⣿⣿⣦⣍⠻⣿⣦⡀⠀⠘⣿⣿⡦⠔⢊⣿⣿⣶⣿⣿⣿⡟⣼⣷⡶⣴⣿⣿⠃⠀⢀⣴⣿⠟⣡⣴⣯⣿⣷⡉⠻⢿⡀
⠀⠀⠀⠀⠀⢀⡾⠀⠀⣾⣿⡟⠙⠿⣿⣷⣌⠻⣿⣆⠀⢘⡿⢡⣾⣿⣿⣻⣿⣿⣿⣿⣻⣿⣾⣿⣿⡿⡇⠀⣰⡿⠟⣡⣾⣿⠞⠋⢻⣿⣷⠀⠀⢳⡀
⠀⠀⠀⠀⠀⣾⠇⠀⢰⣿⡿⠀⠀⠀⠀⠙⠿⣿⣦⡙⢷⣾⣶⠋⣼⢿⣟⣶⣥⣿⣿⣿⡽⣿⣿⡿⢹⣿⣷⡾⢋⣴⣿⠿⠋⠁⠀⠀⠀⢿⣿⡆⠀⠸⣷
⠀⠀⠀⠀⠀⡿⠀⠀⠸⣿⠁⠀⠀⣰⣶⣶⣤⣄⡙⠻⢿⣿⣿⣾⣿⡿⣷⣝⣿⣟⣷⢟⣟⣿⣾⡞⣯⣿⣾⡿⠟⢉⣠⣤⣶⣶⣆⠀⠀⠈⣿⠇⠀⠀⢿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣿⠇⠙⠻⣿⣿⣷⣼⣿⣿⠟⠛⠓⣿⣷⡾⢮⣯⡙⣿⣧⡞⢻⣼⣬⣟⣧⢞⣡⣶⠟⠁⠸⢿⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠁⠀⠀⢸⣇⠀⠀⠀⣿⠀⠀⠀⠀⠉⠛⢻⣿⣷⣄⣟⣰⡝⢿⣘⣨⣧⡖⢾⣦⣾⢿⣿⣸⠿⠟⠛⠉⠀⠀⠀⠀⣛⠀⠀⠀⣸⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠛⠿⣽⣮⣽⢏⢶⣭⣻⣿⣆⡀⡸⢿⣻⣿⠶⠛⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠚⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁







⣷⢶⣶⣾⣶⡿⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣶⡾⠷⠶⠶⠶⢶⣿⡿⠿⢿⠿⠷⠾⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠈⢫⠻⠢⡉⠢⢌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠈⠃⠀⢃⠱⠐⡄⠀⠡⡀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣃⠀⠉⠐⠃⠀⠈⡀⣧⠘⡀⠀⠱⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡀⠀⠈⠐⠒⡆⠀⠀⠃⠸⠀⠃⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠀⠀⠉⠒⠂⠠⡇⠀⠀⡄⠀⠀⠸⠀⠀⠰⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣀⠀⠀⠁⠒⠠⠄⢀⡇⠀⢠⠀⡀⠀⠀⠀⠀⠀⠣⡈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠈⠐⠂⠤⠀⠀⢀⠃⠀⡌⢠⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠈⢉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠉⠐⠂⠤⢀⣀⢀⣀⢠⠀⠀⠁⡈⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠀⠉⠉⠁⠐⠢⠄⣀⠀⠀⠀⣀⢠⠀⠘⠀⡅⠀⠈⢆⡑⠠⠄⣀⠀⠀⠀⠰⣁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⠉⠁⠀⠒⠠⠄⣀⠀⠀⠈⠉⠁⠀⢠⠀⠀⠑⢌⡑⠢⡀⠈⠁⠒⠠⡑⢄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡔⠋⠈⠉⠁⠐⠂⠠⢄⡀⠀⠉⠁⠐⠒⠒⠒⢸⠀⠀⠀⠀⠈⠐⡈⢆⠀⠀⢀⠇⢨
⠀⠀⠀⠀⠀⣠⣊⠤⠀⠠⠤⠄⣀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⡠⠃⡸⠀⡠⠊⡠⠃
⠀⠀⠀⢀⠔⢁⡀⠀⠄⠀⢀⡀⠀⠈⠁⠒⠠⠄⣀⡀⠀⢀⡀⠤⠼⠀⠀⠀⠰⡊⡠⠚⠀⠀⠈⠉
⠀⠀⠀⢏⠈⠉⠁⠐⠂⠠⠄⣀⠈⠑⠂⠤⢀⡀⠀⠀⠀⢀⡀⠄⠒
⠀⠀⠀⠀⠈⠒⠤⡀⠀⠀⠀⠀⠈⠁⠒⠠⢀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢀⠀⠀⠀⠀⠀⠀⠑⠂⠤⡀⠒⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠄⣀⡀⠀⠀⠈⠑⠧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠒⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⡄⠀⠀⠀⠀⠀⠀⠀⣠⠟⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠁⠀⢹⠀⠀⠀⠀⠀⠀⡞⠁⠀⢸⡇⠀⠀⠀⠀⠀⢀⡾⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⡞⠀⠀⠀⠀⠀⠀⠹⣄⣀⡼⠃⠀⠀⠀⠀⠀⠈⣧⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡭⣭⣀⠀⠀⠀⠀⠀⢠⣶⣛⣩⣤⡀⠀⠀⠀⠀⠀⢠⡤⠷⢋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣍⡁⠛⢻⠀⠀⠀⠀⠀⠠⣧⣉⠉⢸⡇⠀⠀⠀⠀⠀⢸⡟⠛⠋⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠙⠛⢻⠀⠀⠀⠀⠀⠀⡇⠀⠙⢻⡇⠀⠀⠀⠀⠀⢸⡋⠛⠶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸⠀⠀⠀⠀⠀⢠⣇⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠶⣤⣼⠁⠀⠀⠀⠀⠀⡏⠑⠦⣼⡇⠀⠀⠀⠀⠀⢸⠗⣤⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠀⢸⡅⠀⠀⠀⠀⢠⣇⠀⠀⠐⡇⠀⠀⠀⠀⠀⢸⠁⠀⠉⢹
⠀⠀⠀⠀⠀⡴⠋⠙⠲⣄⠀⢀⡤⠤⠤⠾⣿⡲⢤⣼⠧⠒⠒⠒⠦⣤⡏⣱⠖⠚⠛⠒⠲⣄⠀⣠⠼⠷⢤⣄⣼⣄⡤⠴⠒⠦⣤⡀⣠⠤⠤⢤⡀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⢹⡿⠁⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠉⣿⠏⠀⠀⠀⠀⠈⡿⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⢸⡏⠙⠲⠶⠖⠲⢤⣄⡀⢀⣀⣤⠶⣄⡀⠀⠀⠀⢀⣠⠴⣦⡀⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⢀⠾⣤⡀⠀⢀⡼⠉⠳⠶⠶⠖⢿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠑⠋⠉⠀⠀⠈⠑⠲⠶⠶⠞⠉⠀⠉⠓⠲⠚⠉⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢀⣀⣀⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⡸⠋⠉⠁⠙⠻⣇⣰⠟⠉⠀⠀⠀⠉⠙⣆⣀⡴⠞⠛⠛⠉⠙⠛⢦⡀⢀⡴⠛⠋⠉⠈⠉⠛⢦⣀⡴⠖⠛⠛⠓⠶⣄⣀⡴⠚⠛⠛⠛⠳⢦⣸⡶⠞⠛⠲
⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠟
⠙⢦⣄⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣠
⠀⠀⡏⠉⠉⠐⠋⠉⠓⠶⢤⣤⣤⣤⡤⠖⠚⠢⣤⣀⣀⣀⣀⣀⣤⠶⠚⢶⣀⠀⠀⠀⠀⢀⣠⠾⢧⣀⡀⠀⢀⣠⡞⠛⢦⣤⣀⣠⡴⠃⠈⠙⠒⠒⠋⢹
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠑⠓⠒⠒⠉⠁⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⡇
⠀⠀⡇
⢀⣠⢧⡤⣄⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀
⣞⠁⠀⠀⠈⠙⢦⣠⠞⠋⠉⠉⠉⠙⠓⣦⡴⠚⠉⠉⠁⠀⠈⠉⠳⣄⣠⠞⠉⠀⠀⠀⠀⠉⠳⣤⡜⠋⠉⠉⠙⠳⣄⢠⡞⠋⠉⠉⠉⠙⢳⣤⠞⠉⠉⠉
⣏⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠉
⠈⠳⠶⣤⣄⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡴⢶⣄⠀⢀⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠲⠶⠶⠶⠖⠉⠉⠛⠶⠦⣤⣤⣤⡴⠶⠋⠉⠻⣦⡀⠀⠀⢀⣤⠴⠋⠳⣤⣄⣀⣤⡴⠋⠉⠳⠶⠦⠶⠞⠁⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠋⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠈⢻⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠃⠀⢀⣠⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣀⠀⠀⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣯⠉⠁⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣦⣤⣄⡀⠀⠀⠀⠀⠉⢩⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠘⣿⣿⣿⣟⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⢘⣿⣿⣿⡟⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠹⢿⣿⣧⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢠⣿⣿⠿⠃⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁⠀⠀⠀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⢿⣿⣷⣶⣶⣶⢞⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⢶⣶⣶⣶⣿⡿⠟⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣤⣄⣀⣉⣉⣭⣶⡿⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠻⣷⣮⣭⣉⣁⣀⣠⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⠿⠿⣫⣴⣶⣶⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣬⡻⠿⢿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢛⣿⡿⠋⠁⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠈⠻⣿⣟⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠈⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠟



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢋⣿⠇⣿⢿⡙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣿⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡏⠀⣿⡇⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣹⣿⠃⠀⣿⣷⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⣿⡿⠀⠀⢸⣿⡀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣼⣿⡇⠀⠀⢸⣿⡇⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢰⣿⣿⠁⠀⠀⢸⣿⣿⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣿⣿⡟⠀⠀⠀⢸⣿⣿⡀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⣼⣿⣿⠇⠀⠀⠀⢸⣿⣿⡇⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢰⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢀⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⡀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⣾⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⠤⠀⣀⣀⣀⣀⣀⣸⣿⠿⠿⠿⠓⠂⠉


⣀⣀⣀⠀⡀⣀⣀⡀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⢀⣿⠹⣇⠀⠀⢠⡟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠘⣧⣤⡟⠀⠀⢸⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠁⠀⠀⠀⣬⣿⣤⠀⠀⣀⣹⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣻⣿⠀⠀⢠⣿⡴⣿⠀⠀⣿⣩⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠟⣡⣿⠀⠀⢸⣏⡴⣿⠀⠀⣿⡡⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢠⣿⠀⠀⢸⡏⡰⣿⠀⠀⣿⢁⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⢉⣿⠀⠀⣸⣟⣀⣿⣀⣀⣿⣁⣸⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡶⣦⣤⣤⣤⣤⢦⣤⣀⣀⡀⠀⠂⠀⠀⣀⣀⣨⣤⡶⢦⣤⣤⠶⠶⠾⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠒⠒⠉⠳⠤⠤⠔⠋⠻⣍⣛⣛⡿⠛⠻⢍⣉⣀⡤⢊⡁⠑⠢⠔⠊⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⣤⣤⣬⣤⣤⣌⣀⣬⣥⣤⣤⣄⣐⣀⣀⣒⣀⣀⣀⣀⣀⣤⣥⣤⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣿⣤⣧⣀⣍⡀⣹⠉⢹⠁⢹⠀⢹⠉⢹⠉⢹⠉⢹⠉⢹⠉⡏⢹⢀⣇⣰⣼⡷⠾⠛⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⢠⡄⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠳⠿⠶⠾⠶⠾⠶⠾⠶⠾⠶⠿⠾⠛⠛⣛⠉⠉⠀⠀⠀⠀⠛⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣠⣤⠀⢴⠄⣀⡀⠁⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⢀⢀⡀⠉⠠⠀⢀⣿⠛⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠻⣤⣤⡴⠟⠛⠛⢷⣤⣀⣀⣠⣴⠿⢷⣴⣦⡈⠈⠋⠀⣀⣴⠟⠛⠛⠶⠶⠶⠟⠁⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠄⠀⠀⠀⠀⠀⠃⠀⠀⠉⠉⠁⠀⣀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠠⠄⠀⠀⠀⠀⠀⠈⣁⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⣤⣤⣄⣀⣀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⢀⣀⣀⣠⣤⣤⢶⣶⠟⠛⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣷⣀⣠⡿⢿⡉⠛⠛⣿⣿⠲⠶⠶⣶⡶⠶⢶⣶⡶⠞⠛⢻⣿⠛⠉⣹⠿⣧⣀⣸⣟⣷⣾⠿⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⠛⠛⠳⠾⠿⢶⣾⣯⣽⣦⣤⣾⣋⣷⣤⣾⣍⣿⣦⣴⣿⣽⠷⠾⠿⠟⠛⠛⠉⣉⣠⣤⣶⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣽⢿⡟⠶⣶⣶⣤⣤⣤⣤⣤⣈⣉⣉⣉⣉⣉⣉⣀⣀⣠⣤⣤⣤⣤⣴⣶⠶⢻⡟⢿⣁⣿⠙⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⡈⣷⡾⠃⢸⣇⣴⠏⠹⣇⢠⡿⠙⣯⢉⣽⠋⢻⡍⣼⠏⠹⣇⣼⠃⠘⣧⣾⠁⠈⣿⣇⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⢦⣤⣿⣃⣀⡀⣿⡟⠀⠀⢻⣾⠃⠀⠈⣿⡏⠀⢀⣻⣿⣀⣤⣼⣿⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⣷⣦⣄
⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡇⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⡈⢻⣿⣿⡀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⣸⣿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢪⠍⠈⠍⠢⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢷⡀⢻⣿⣷⡀⠀⠀⠐⢸⣿⣿⡀⠀⠀⠀⣠⣿⣿⠏⣰⡇
⠀⠀⠀⠃⠠⡐⠰⡀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡰⠔⡀⠀⠀⠀⠀⠀⠀⢷⡈⠻⣆⠹⢿⣿⣷⣦⣴⣿⣿⣿⣿⣤⣶⣿⣿⠿⢁⣼⠏⣠⠇
⠀⠀⠂⠀⡄⠂⠀⡀⠠⠆⠠⠀⠀⠀⠀⠀⠀⠠⠐⠦⠀⡀⠀⠂⡀⠀⠆⠀⠀⠰⡌⠻⣦⡉⠻⣦⣍⠛⢻⣿⣿⣿⣿⣿⣿⠛⢋⣥⡶⠋⣡⡾⠋⣰
⠀⠀⠀⠀⠀⠄⠠⣐⠓⠠⢌⠀⠀⠀⠀⠀⠀⢨⠠⠐⢓⣠⠀⠄⠀⠀⠀⠀⠀⠀⠙⢶⣌⡙⠳⣤⣍⣛⢺⣿⣿⣿⣿⣿⣿⣞⣋⣥⡴⠟⣉⣴⠾⠁
⠀⢤⠀⠀⠁⠀⠀⣜⠉⠒⠒⠀⠀⠀⠀⠀⠀⠐⠒⠊⢙⡄⠀⠀⠁⠀⢀⠀⠀⡀⠀⠀⠉⠛⠳⠶⣬⣭⣹⣿⣿⣿⣿⣿⣿⣯⣭⣴⠶⠟⠋⠁⠀⠀⡀
⠀⠀⠀⠀⣐⠂⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠌⠀⠀⠠⠀⠀⢒⡀⠀⠀⠀⠀⠙⠳⠶⢦⣤⣤⣤⣤⡴⠾⣿⣿⣿⣿⣿⡿⠶⢤⣤⣤⣤⣤⠴⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠒⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣹⣿⣿⣿⣿⣴⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⢿⣿⣿⢿⣿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⡿⠋⠁⠀⢀⣤⠀⠀⠀⠀⠀⠀⠐⠛⠛⠻⠿⣿⣿⣿⣷⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣧⣿⣿⠁⣀⣠⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⣶⣄⣀⠀⠀⠉⣿⣯⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡿⢿⢟⣃
⠀⠀⠀⠀⠀⣀⣤⠶⠒⠀⢈⣭⣿⣛⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠲⢦⣄⡀
⠀⠀⠀⠀⠸⣯⡁⠀⠀⠀⣼⣿⣿⣿⡟⣻⣉⣀⠀⠀⠀⠀⠀⠀⣠⣸⣶⣶⣶⣶⣶⣶⣶⣾⣥⣀⠀⠀⠀⠀⠀⣠⣴⣷⡼⣿⣿⣿⠏⠀⠀⠀⠀⣨⡿
⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠹⣿⣿⣿⣻⣿⣿⣿⣿⣶⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣰⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠈⠙⠋⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢀⣠⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣶⣦⣤⣄⣉⣛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣀⣤⣤⣴⣶⠟⠉⠁
⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⢀⣀⣀⣈⣉⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠛⠛⠋⣉⣀⣀⣀⡀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⣠⠟⠁⠉⠙⠻⠶⠖⠛⠛⠉⠉⠛⠛⠻⠿⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣴⡶⠿⠟⠛⠛⠉⠙⠛⠿⠦⠶⠟⠛⠉⠈⢳⣄
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠿⠷⠶⣶⣶⣦⣤⣼⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣅⣤⣴⣶⡶⠶⠾⠷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⠋⠀⠀⠀⠀⠀⣠⣤⣴⣦⣶⣶⢻⣾⣝⢻⣿⣿⣿⣿⡿⣭⣾⢿⣶⣦⣼⣦⣤⣄⠀⠀⠀⠀⠀⠙⠷⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠋⠛⠉⠀⠈⠀⠈⠉⠉⠉⠈⠁⠀⠁⠀⠙⠋⠛⠛⢛⣿⣿⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄
⠀⠀⠀⠀⠀⠀⢺⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢴⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⡿⣷⣄⡀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣧⣴⣷⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⢿⣿⣧⣍⠛⢷⣆⠀⠰⠟⣉⣠⣴⣿⣿⡿⢿⣿⣿⣿⣟⠀⠀⠀⠀⢠⣾⣦⣼⣦⡀
⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣷⡀⠀⠀⠀⠐⣿⣿⣿⣿⡇⠀⠉⠙⠻⠷⠆⠉⠀⠀⠈⠉⠉⠉⠁⠀⠀⣿⣿⣿⣿⡏⠀⠀⢀⣶⣿⣿⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠻⣿⣿⣿⣿⣷⣄⠀⢠⣽⣿⣾⣿⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⣷⣾⣻⡛⠁⣠⣾⣿⣿⣿⣿⢟⣿⣿⣿⡅
⠀⠀⢀⣀⠀⠀⠀⠉⣿⣿⣧⠹⣿⣿⣿⣿⣿⣧⡀⠙⣿⣿⣿⣿⣤⡀⠀⣀⡤⣾⣷⣤⡀⠀⣠⣶⣼⣿⣿⣿⠿⢋⣾⣿⣿⣿⣿⣿⠏⣼⣿⣿⠉⠀⠀⠀⣠
⢀⣰⣾⣿⣿⣶⣦⣼⣿⣿⣿⠀⠙⣿⣿⣿⣿⣿⣿⣆⠈⠿⣿⣇⣿⣇⣠⣿⣶⣿⣿⣴⣿⣶⣿⣿⢿⣿⠟⠉⣸⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⣣⣴⣶⣿⣿⣧⣖
⠘⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⢀⣿⣿⣿⣿⣿⣿⠟⠁⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⢻⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣦⣀⠀⠸⠿⢿⣿⣿⡿⣃⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡼⢿⣿⣿⡿⠿⠅⠀⣀⣴⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡟
⠀⠀⢹⣿⡿⣡⣄⠙⠿⣿⣿⣿⣿⣿⣧⣤⠀⠀⣴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⠋⣠⣌⢿⣿⡏
⠀⠀⠀⠉⢹⣿⣿⣆⠀⠉⣿⣿⣿⣿⣿⣿⣧⢀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠀⣾⣿⣿⣿⣿⣿⣿⠁⠀⣼⣿⣿⡏⠉
⠀⣸⣷⣶⣶⣿⣿⣿⣄⣀⠸⢿⣯⣿⡿⣿⣻⣾⣧⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⣾⣿⣿⣿⢿⣿⣝⡿⠇⣀⣠⣽⣿⣿⣶⣶⣾⣇
⣷⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⠾⣛⡛⠱⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠏⢛⡻⠟⣾⣯⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠙⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢋⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⡛⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿
⠀⠀⠛⢿⣿⣯⣶⣤⡈⢻⣿⡿⣿⣿⣿⣿⣿⡿⣺⣧⣈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣁⣼⣇⢿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣤⣶⣹⣿⡿⠋
⠀⠀⠀⠈⠛⠹⣿⣯⣿⣦⠙⣿⣿⣦⣛⠉⠉⣿⠿⠛⠋⠀⠈⠙⠛⠿⠿⢿⡿⠛⠟⢿⡿⠿⠿⠛⠉⠁⠀⠙⠛⠿⣷⠉⠉⣫⣴⣿⣿⠋⣴⣿⣽⣿⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣌⠙⢿⣿⣷⣦⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣴⣾⣿⡿⠋⣠⣾⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⡀⠈⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠁⢀⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣄⠀⣀⣀⣀⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⣀⣀⡀⠀⣠⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⠻⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⢿⣻⣿⢻⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡎⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⡀⠈⡆⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠄⠼⠤⠚⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡙⠢⠼⠤⠼⡄
⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⡜⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠀⠘⡄⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⡏⡇⠀⠀⠀⠀⢰⠁⠀⢀⠞⠀⠀⠀⠀⢶⠆⠀⣀⣀⣀⡀⠀⢶⠆⠀⠀⠀⠘⢆⠀⠀⢱⠀⠀⠀⠀⠀⣏⡇
⠀⠀⠀⠀⢻⡉⢢⡀⠀⠀⣏⢉⡶⠯⣄⠀⢀⡠⠔⡊⠉⠉⠀⠀⠀⠈⠉⠉⠒⠤⣀⠀⢀⡬⠷⣎⢉⡇⠀⠀⣠⠊⡹⠃
⠀⠀⠀⠀⠀⠙⢄⡱⢄⡀⠈⠫⣀⡁⢊⠗⠩⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢏⠀⣁⡨⠋⠀⣀⠴⣁⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠤⣹⠒⠒⠢⠽⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠅⠒⠒⢺⡡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣉⣓⣒⣒⣛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣓⣒⣒⣋⣉⣉⣀⡀
⠀⠀⠀⠀⠀⢀⠴⢺⡥⠼⠦⠤⠤⠤⣤⡤⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⣤⡤⠤⠤⠤⠾⠤⣽⠢⢄
⠀⠀⠀⠀⠀⣷⠞⠁⠀⣀⠴⣋⣉⣉⣁⣠⠤⢜⣲⣄⣀⠀⠀⠀⠀⠀⢄⣂⣴⣛⡤⢤⣀⣉⣉⣉⡳⢄⡀⠀⠙⢶⡇
⠀⠀⠀⠀⠀⢿⠀⠀⢰⢧⠞⠁⠀⠀⡶⢿⡭⠥⠬⠷⠚⠋⠉⠉⠉⠉⠛⠲⠯⠤⠬⣭⢷⡄⠀⠀⠑⢦⢷⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠘⠇⠀⣼⢸⠀⠀⠀⠀⢱⠔⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠓⢴⠁⠀⠀⠀⢸⢸⡄⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢭⣢⣄⠀⠀⠀⠑⠮⠤⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠬⠖⠁⠀⠀⢀⣤⣪⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⡀⠀⠐⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⣀⠀⠀⣀⠤⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡘⠀⢀⠞⠳
⠀⠀⠀⠘⠀⠙⢦⡀⡇⠀⠀⠱⡀⠀⠀⠀⠀⡰⠉⢀⣴⣦⣄⠑⡎⢀⣴⣶⣤⠀⠈⢆⠀⠀⠀⠀⢠⠎⠀⠀⠀⣩⠔⠁⠀⠀⡇
⠀⠀⠀⢡⠀⠀⠀⠙⠃⠀⠀⠀⠘⡀⠀⠀⠀⡁⠀⣿⣿⠙⣿⡆⡇⢸⣿⡋⣻⡇⠀⢸⠀⠀⠀⠀⡈⠀⠀⠀⠀⠁⠀⠀⠀⡜
⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠱⣀⠘⢿⣿⠟⣡⠗⢌⡻⣿⠿⢁⡴⠃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠙⠢⠄⣀⡀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠁⣒⡲⠈⠧⠤⠤⠏⠉⠏⠠⠄⣀⠀⠀⠀⢀⠎⠀⢀⠤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠙⠢⣄⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠑⢤⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⢎⠀⠀⠀⢀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⣈⠑⠢⠤⣀⠇⠀⠀⠀⠀⠀⣀⣠⡤⠤⢴⠒⡎⢋⣹⠼⡟⠀⠀⠀⠀⠀⠈⠦⠔⠊⣡⠤⡄⠈⠢⡀
⠀⠀⠀⠀⠀⡴⠁⢀⠜⠀⠉⣒⣦⣄⠀⠀⠀⠀⠀⠀⠙⢗⠷⠺⠔⡖⢻⠙⣠⠞⠀⠀⠀⠀⠀⠀⢠⡴⠖⢧⠀⠀⠈⢆⠀⠱⡀
⠀⠀⠀⠀⡜⠀⢠⠎⠀⢀⠔⠁⡤⢉⣃⠀⠀⠀⠀⠀⠀⠀⠙⠲⠧⠵⠜⠊⠁⠀⠀⠀⠀⠀⠀⣀⠤⠐⠲⡀⠱⡀⠀⠈⢆⣀⠅
⠀⠀⠀⠀⠈⠒⠁⠀⢀⠎⠀⡜⠀⠀⡰⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠱⠀⠀⠀⢱⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠜⠀⠀⢠⠁⢠⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠀⠀⠀⢡⠀⢃⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⢼⣏⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠃⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡖⠀⠀⠀⠈⠹⠒⠋⠐⠶⠀⠐⢶⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⣤⠀⠀⣴⠯⠀⠀⠀⢀⣀⢀⣵⢶⣄⢀⣀⠀⠀⠀⠨⢗⡄⠀⢠⠄⠀⠀⠀⠀⠀⢸⣟⣿⡿⢻⣿⡆
⠀⠀⠀⠀⠀⡄⢀⣤⠈⠀⠀⠀⠀⠈⡭⠀⠁⠀⠁⠨⡍⠀⡀⠀⠀⠈⢠⣄⠀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢩⠛⠀⢱⡀⠀⢠⠁⠈⠫⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣿⣀⠀⠀⠨⡏⠇⠀⠀⠀⡡⠤⡁⠀⠀⠀⠏⡏⠀⠀⢀⡰⣇⡄⠀⠀⠀⠀⠀⢸⣿⡇⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠠⣠⡦⠀⠀⠀⡰⢂⣄⠀⠀⠅⠀⠁⠀⢀⡄⠲⣀⠀⠀⠠⢦⡠⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠂⠂⠂⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠸⠿⠟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢾⡷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⢸⡇⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⢰⡇⢸⡇⢸⡆⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣿⣀⠀⠀⠀⠀⠸⡇⢸⡇⢸⠇⠀⠀⠀⠀⣀⣿⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠙⠓⢶⣄⡀⠀⠀⢸⡇⠀⠀⢀⣤⡶⠚⠉⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠳⢦⣼⣧⡴⠞⠋⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⡾⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠋⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣧⠀⠀⢀⣠⡴⠞⠛⠉⢉⣉⣉⠛⠳⢦⣄⡀⠀⠀⣼⢷⣄
⠀⢸⠛⠲⢦⣤⣀⠀⠋⠀⠘⣧⣴⠟⠁⠀⠀⠀⠀⠈⠉⠉⠙⠳⢦⣈⠻⣦⣼⠃⠀⠙⠀⣀⣤⡴⠖⠛⡇
⠀⣾⠀⠀⠀⠀⠉⠛⠳⠶⣤⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡈⢻⣤⠶⠞⠛⠉⠀⠀⠀⠀⣷
⢸⡇⠀⢀⣄⣀⣀⣀⡀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢻⡄⢀⣀⣀⣀⣠⡀⠀⢸
⣾⠁⢀⡿⠁⠉⠉⢉⣙⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣋⡉⠉⠉⠈⢿⡀⠈
⠀⠀⣼⠃⠀⣴⢶⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻⡶⣦⠀⠘⣧
⠀⠰⠏⠀⣸⠇⠀⠉⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⡟⠉⠀⠸⣇⠀⠹⠄
⠀⠀⠀⢠⡏⠀⠀⠀⠘⣷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣾⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠻⣦⡙⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠈⠛⢦⣌⡙⠁⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⡀⠀⠀⠀⣼⠛⠛⠓⠲⠖⠚⠛⠛⣧⠀⠀⠀⢀⣠⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡾⠃⢠⡗⠀⠀⠀⠀⢲⡄⠘⢷⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⡾⠁⠀⠀⠀⠀⠈⢷⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠶⠛⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣤⣀⣈⣉⣉⣥⣤⡴⠶⠛⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⡀
⠀⠈⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⡰⠊⠀⠀⠀⠈⢋⣉⠆⠠⠆⣠⡀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⣠⡀⠦⠀⢎⢉⠋⠀⠀⠀⠈⠲⡄
⠀⡌⠀⣀⡤⠖⡞⠀⠈⠀⠀⠀⠀⠌⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠐⠘⡖⠦⣄⡀⠈⡄
⠀⠛⠉⠀⢀⡌⢀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠌⠁⠀⠁⠄⠰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠈⣄⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠅⠂⠠⠤⠤⠤⠄⠐⠊⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠚⠢⣄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠱⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠐⣇⢀⡎⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣄⣠⣸⣶⠞⡵
⠀⠀⠀⠀⢠⠃⠀⠈⠙⠓⣅⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠈⠀⠫⠇
⠀⠀⠀⢀⡇⠀⠀⠀⠀⣀⡌⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢈⣰
⠀⠀⠄⢸⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣺⣻⠿⠓⠐⠻⠚⠻⢟⣻⡀⠀⠀⠀⠀⠀⠹⠇⠀⢄⣘⣿
⠀⢠⠀⢿⣶⠔⢀⡛⠟⠁⠀⠀⠀⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⢠⡇⠀⠐⠀⣬⡇⠀⠀⢰
⠀⢸⠆⢸⡇⠀⠀⠀⡴⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢡⠀⠸⡅⠀⠀⠀⢰⠀⠀⠀⠈
⠀⠘⡄⢻⡅⠀⠀⠉⡇⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡄⣷⠐⠀⢤⡞⠀⠀⠰⡞
⡇⠀⠘⣼⢿⣄⡀⢀⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣯⣴⣟⠀⠀⠀⣰⠃
⢧⡀⠀⠘⢜⢿⣿⣿⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣭⠎⠀⢤⡴⠃
⠈⢇⠀⠀⠈⠢⡙⣿⡿⠿⣧⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⡁⠈⣠⠞
⠀⠈⢳⡴⠤⠀⠉⠻⢶⣶⣾⣷⢴⣶⡆⣄⡀⠀⣢⠀⣴⡤⢦⡀⠀⠀⡀⢀⣠⡂⣼⠖⢲⣮⠴⠋
⠀⠀⠀⠙⠒⠒⠒⠒⠛⠓⠶⠾⣿⣿⣿⣧⣦⡟⠁⠀⢻⠇⠀⠑⣰⣾⣿⢻⣷⠞⠛⠛⠋⣡⠀⠀⠀⠀⣠⣤⠎
⠀⠀⠀⠀⠀⠀⠀⣠⣧⣬⣄⣀⣀⣬⣿⣿⣿⡄⣀⣁⣈⣁⠀⣠⣾⡿⣳⣿⡛⠉⠉⠉⣛⡍⠛⠒⠛⠋⠁
⠀⠀⠀⠀⣰⣦⡴⠋⠀⣠⣄⠈⠉⣉⣙⣻⣿⣻⢿⣿⣿⡿⢿⣿⣻⣽⣿⠛⠋⠉⣙⣿⠛⠷⡀
⠀⠀⠀⢐⣨⡟⠀⢀⡞⢹⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠛⠉⠉⠉⠻⣆⠀⠈⣶⠆
⠀⠀⠐⣿⡋⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢶⠀⠹⣄⡀
⠀⠀⠀⠈⠳⡀⠀⠈⣇⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠙⠀⠀⠘⠿⠶⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠄⠇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣌⢰⣤⡞⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢛⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠘⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠘⠲⡔⠚⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠁⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⡀⡐⡀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⡑⠀⠀⠉⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠄⢠⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠤⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠥⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠐⠀⠈⠀⠀⠀⠀⣀⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⢤⣄⣀⣀⣄⣥⣤⠴⠖⠛⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⢀⡼⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢣⡀⠀⠀⠈⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⡞⡄⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠻⡄⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁⠀⠀⠀⢟⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠄⠻⣆⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⣾⠎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⢆⢸⡀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⣸⡏⢠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢻⡇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠙⣤⡬⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠤⣤⠚⠋⠀⠀⡠⠚⡀⠀⢠⠞⡧⣄⠈⢀⣰⡼⢧⠀⠀⠀⢠⣄⠀⠀⠉⢦⣠⣄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⠀⠘⢦⠀⠀⣠⢶⠞⠁⡴⠁⠀⣴⣿⠁⢸⣷⡆⠀⠳⡀⠀⠈⠻⣷⢶⣀⠴⠁⠀⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠉⠐⣧⡌⢀⡜⠀⠀⡔⠉⠉⠀⠀⠉⠘⡄⠀⠱⡄⠀⢀⡜⠋⠁⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣄⡀⣠⡾⠷⣮⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⢱⠀⠀⢹⡖⠻⣦⣤⣠⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⢿⣉⣠⠞⠀⠙⢶⣮⠀⠀⠀⠀⠀⠀⠀⠘⣷⠂⠀⠘⣄⢀⣸⠟⢩⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⢻⡿⢦⣄⣠⣼⠿⠀⠀⠀⠀⠀⠀⠀⠐⢿⣤⠀⢤⠿⣯⠟⠰⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣴⡎⠁⠀⠁⠀⡼⠀⠂⣤⠀⠀⠀⢠⡤⠃⢸⣤⡀⠀⠀⠙⣤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠀⡀⠀⠈⡙⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣋⡀⠀⣀⠴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡬⠅⠀⠁⠒⠚⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⢉⠁⣋⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢸⠃⣀⠀⠀⠀⠰⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠈⠀⠔⠉⣀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡀⠇⠀⢁⠒⠄⠀⠀⠶⢿⣶⠶⠦⠤⠶⠶⣿⣾⠿⠶⠀⠀⠀⠀⠉⠀⡏⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢰⡔⠀⠀⠘⠃⠀⠀⠀⠀⠀⣯⣧⣤⡈⢀⣴⣿⣯⠀⠀⠀⠀⠀⠀⠃⠀⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠁⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⢠⠄⠘⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⢀⣿⣦⣤⠤⢀⠴⢾⣟⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠸⣇⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⠂⠀⠀⠒⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⣀⡀⢀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠒⠀⠀⠐⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠂⠁⠀⠀⡾⣬⣿⣤⠀⢠⣴⣿⣷⡆⠈⠀⠀⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠊⠀⠈⣧⠈⠉⠀⢀⠏⠀⠀⢢⡀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⢧⡀⠀⠀⢸⣄⠀⢀⣼⠀⠀⢀⣠⡿⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠛⠛⠉⠛⠛⠋⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣛⣛⣛⣛⣛⣉⣛⣿⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣉⣩⣍⣭⣭⣿⣯⣉⣉⣍⣻⣯⣉⣛⣛⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⢸⢠⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡎⠀⠀⢸⢸⢸⠀⠀⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠓⠤⣀⢸⢀⡠⠔⠊⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⠀⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⢸⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⣀⠤⠖⢚⣒⡢⢄⡀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠒⠦⢤⣸⠞⠁⠀⠀⠀⠀⠉⠳⣝⢾⣠⠤⠖⠊⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣠⠤⠤⢤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢧⠤⠤⢴⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠇⡰⢾⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⠶⡀⢧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠃⠀⢻⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠃⠀⢳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠈⠳⣝⠢⠀⠀⠀⠀⣀⡴⠋⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⢠⠟⡓⠒⠒⡙⢧⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣸⠃⠀⠀⢻⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⠀⠀⠀⢀⣀⣀⡤⢆⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠊⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣾⣾⣿⣿⣭⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠐⠃⠑⣼⣿⣿⠏⠉⠉⠉⠓⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⢉⡿⠷⢽⣷⡄
⠀⠀⠀⠀⠀⠀⠠⣧⣶⣮⣵⣿⣿⠿⠦⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡐⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣉⠁⠀⠀⠀⣤⣄⡀⠀⠀⠀⣀⣀⣠⡀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠲⡋⠀⣿⣯⣿⠀⠀⠀⢠⣿⡿⠃⣒⣤⣼⣿⣿⣿⣷⣤⣀⠀⠀⢠⢬⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣴⣿⣿⣶⣾⣔⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣞⢀⣿⣿⡷
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣈⣙⠻⣼⣻⣿⣿⣾⣏⠀⠀⠌⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⢀⣴⡿⠿⣿⣿⣿⣿⣾⣿⣿⣷⣯⣽⣟⠷⣶⣿⠿⣿⣿⣿⣿⣿⣿⣋⣭⣶⣄
⠀⠀⠀⢀⣿⡿⣡⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠈⣿⣄⣾⡟⠁⡞⢡⣿⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠻⢿⣿⣿⣿⡆⠉⢻⢿⣿⠀⢹⣧
⠀⠀⠀⠀⣿⣧⢹⣦⠘⠇⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠎⣸⣿⠃⠀⡿
⠀⠀⠀⠀⠘⣧⠀⢻⣧⡀⠈⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⣠⣿⠏⠀⢸⠇
⠀⠀⠀⠀⠀⠈⠃⠀⠉⠻⣤⣙⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⣴⣿⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠁⠀⣼⡿⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣟⠋⣽⡾⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⢦⣹⣰⣏⣤⣊⠥⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⣉⣿⡋⣹⠿⣷⣶⣤⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⠿⣛⣭⢿⠟⣶⣯⡀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠞⠉⢀⠏⢠⡏⣧⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠁⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⠀⣀⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢯⠙⢁⡝⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⠉⠁⠀⣠⢤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⡴⢲⡄⠀⠀⠀⠹⢄⡉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣄⡤⠊⠀⠀⢀⣄⠀⠀⠙⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⠙⢃⡼⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡥⢤⡤⣄⠀⠀⠉⠋⠀⠀⣠⠄⣤⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠙⢦⡀⡠⠋⠀⠀⠀⠀⠀⠀⠈⢢⡤⠊⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠉⠀⠀⢀⡤⢤⡤⡄⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⢣⣀⡰⠃⠀⠀⢀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢤⡀⠀⢏⠙⠋⡷⠀⠀⠉⠀⠀⠀⠀⢏⠐⠋⡽⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣥⠞⠁⠀⠈⠓⠋⠀⠀⠀⠀⢀⣀⠀⠀⠈⠳⠞⠁⢰⡂⠻⡄
⠀⠀⠀⠀⠀⠀⠀⣼⢧⡈⠙⠲⢤⣄⣀⡀⠀⠀⣎⠙⠋⡸⠁⠀⠀⠀⠀⣀⣠⠽⠞⢳⡀
⠀⠀⠀⠀⠀⠀⣸⣷⣄⡉⠓⠦⣤⣀⡈⠉⠙⠛⠛⠓⠚⠒⠒⠒⠛⠋⠉⣁⣠⠤⠖⢋⣷
⠀⠀⠀⠀⠀⠀⢿⣄⠈⠙⠲⠤⣄⣀⠉⠉⠙⠓⠒⠒⠒⠒⠒⠒⠛⣋⣉⣡⡤⠶⠚⢋⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⠀⠉⠉⠛⠓⠒⠒⠒⠒⠒⠛⠛⠉⠉⠉⢀⣠⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠤⠤⠤⠤⠴⠶⠖⠒⠚⠉⠁

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠟⠉⣠⠀⠀⠀⠀⠒⠿⠿⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⣴⡾⠋⠀⠀⠀⠀⠻⣶⣤⣬⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣅⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢾⡁⠀⢼⣿⣿⣻⣉⡀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠀⠀⣠⣽⣽⣿⡿⠀⠀⢉⡗
⠀⠀⠀⠀⠀⠀⠀⠙⣄⡈⠻⠻⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⠋⠀⣀⡴⠋
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠓⠶⠦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⠶⠶⠛⠉
⠀⠀⠀⠀⡴⠉⠙⠲⠖⠛⠉⢛⣓⣶⡬⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢥⣶⣚⡛⠉⠛⠢⠖⠛⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⣹⣧⣿⡽⣿⢿⣿⣿⡿⣿⢯⣿⣽⣋⠉⠉⠉⠳⣀
⠀⠀⠀⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠻⣯⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣽⠃⠀⠀⠀⠈⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢿⣿⣦⣄⠀⠀⠀⠀⠀⢀⣠⣴⣿⡟⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣶⣦⣀⠀⠀⢀⣠⣴⣶⣾⣶⣤
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣄⠀⠀⠀⣿⣿⡟⠻⣯⣙⠧⠀⠫⠥⠾⠟⢻⣿⣿⠀⠀⣠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⡀⠀⢿⣿⣃⠀⠀⠈⠀⠀⠀⠀⠀⢀⢻⣿⡟⢀⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⡀⠀⠚⣿⣟⢿⣿⣿⣆⠘⣿⣿⣆⡀⠀⣰⣆⠀⢀⣤⣾⣿⠷⣴⣿⣿⡿⣻⣿⠃⠀⢀
⠀⠀⠀⢠⣶⣿⣷⣶⣿⣿⠈⢻⣿⣿⣷⣬⢻⣿⣧⣾⣾⣷⣷⣿⣿⠟⣥⣾⣿⣿⡟⠁⣿⣿⣶⣾⣿⣶⡄
⠀⠀⠀⠀⢿⣿⡿⢿⣿⣿⣦⣄⠙⠿⣿⣿⣴⣾⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⠿⠋⣠⣴⣿⣿⡿⢿⣿⡿
⠀⠀⠀⠀⠀⠛⣿⣦⠙⢿⣿⣿⣷⡀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⢠⣾⣿⣿⡿⠋⣴⣿⠛
⠀⠀⠀⢠⣿⣿⣿⣿⣦⣼⣿⣿⣿⡿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⢿⣿⣿⣿⣧⣴⣿⣿⣿⣿⡄
⠀⠀⠀⠘⢿⣿⣿⠻⢿⣿⣿⣿⣿⣯⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⣹⣿⣿⣿⣿⡿⠟⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⠻⣿⣦⡻⣿⣽⠛⠻⠟⠒⠙⠛⠿⢿⠿⠿⡿⠿⠛⠋⠒⠻⠟⠛⣯⣿⢟⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⡛⠻⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⢛⣵⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣤⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⢚⣳⠶⠂⠀⠀⠀⠀⠐⠶⣟⡓⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⢀⡴⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⢦⡀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠁⠀⠀⢀⡴⠋⢀⡞⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⡀⠙⢦⡀⠀⠀⠈⠳⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢻⡁⠀⠀⠀⢀⡞⠀⣠⠟⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠻⣄⠀⢳⡀⠀⠀⠀⢀⠟⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠁⠀⠀⠘⠷⠞⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠳⠾⠃⠀⠀⠈⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⢀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢯⡖⠒⢦⡀⠀⠀⠀⠀⢀⡴⠒⢲⡽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⡀
⠀⠀⠀⠀⠀⠀⡿⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢸⠀⢀⣼⠃⠀⠀⠀⠀⠘⣧⡀⠀⡇⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢻
⠀⠀⠀⠀⠀⠀⡇⠀⠈⠒⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢸⡆⣸⠃⠀⠀⠀⠀⠀⠀⠘⣇⢰⡇⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠐⠁⠀⢸
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣾⠉⠁⠀⠀⠀⠀⣀⣀⣷⡼⠤⠶⠶⠶⠶⠶⠶⠦⠯⣾⣀⣀⠀⠀⠀⠀⠈⠉⣷⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⠀⠀⡽⣄⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣀⠀⣠⠏⠀⠀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠈⢻⠏⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠹⡟⠁⠀⢀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠓⠒⠦⣤⣀
⠀⠀⠀⠀⣠⠞⢉⡴⠷⠶⢶⣶⣶⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣶⣶⡶⠶⠾⢦⡉⠳⣄
⠀⠀⢀⡞⢁⡴⠋⣀⡴⠚⠉⠁⣀⣠⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⠈⠉⠓⢦⣀⠙⢦⡈⢳⡀
⠀⠀⣿⣹⠋⢀⡾⢋⡾⠓⠛⣩⡽⠋⠉⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠉⠙⢯⣍⠛⠒⢷⡙⢷⡀⠙⣏⣻
⠀⠀⢿⠏⢠⠟⢀⡞⠁⢠⠞⣁⣠⢴⡾⠋⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡋⠙⢷⡦⣄⣈⠳⡄⠈⢳⡄⢻⡄⠹⡿
⠀⠀⠀⠀⡾⣴⠏⠀⣴⠋⢠⡏⢠⡞⣥⠖⠋⠈⠳⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⠞⠁⠙⠲⣬⠳⡄⢹⡄⠙⣦⠀⠹⣦⢷
⠀⠀⠀⠀⢷⣿⠀⡼⣃⡴⠋⢠⡏⢀⡟⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠒⠒⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⢻⡀⢹⡄⠙⢦⡘⢧⠀⣿⡾
⠀⠀⠀⠀⠈⠃⠀⣧⣸⠀⠀⡾⢤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡤⢷⠀⠀⣇⣼⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠻⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠟⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠀⠀⣀⣀⠀⠀⠀⠀⠂⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠁⠒⠤⠤⣬⣲⣤⣤⣤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣦⣤⣤⣤⣌⣑⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢻⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣌⣭⣭⣿⣿⣿⣿⣿⡿⣛⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡛⢿⣿⣾⣿⣽⣧⣿⣽⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣤⣤⣍⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⠿⢛⡿⢟⡽⣹⠏⢠⠏⠈⠋⠙⠃⠀⠈⢳⡄⢿⡌⠉⠈⠉⠉⠁⠉⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠿⠿⠿⠿⠿⠟⠋⠉⠀⣤⠶⠋⠀⢞⣵⠃⢠⠟⣀⣘⡃⠀⠙⡆⠀⠀⠙⠘⣃⠀⠀⠀⠀⠀⠀⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⣷⠀⠀⠀⣰⢟⡛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣧⢹⡆⠀⢀⣿⣾⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⢃⣿⡇⠀⢸⣿⣌⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⢀⣤⣤⣤⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⢸⣿⢟⡵⣡⣾⣿⣿⣷⣄⠿⠿⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢰⣿⢸⣿⣿⣿⣿⠘⠁⣫⣾⣿⣿⣿⣿⣿⣿⡆⣶⣶⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠸⣿⠘⣿⣿⣿⣿⡀⢴⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢹⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⡶⢉⡉⢛⡻⢿⣿⣿⣿⡇⣶⡄⠀⠙
⠀⠀⠀⠀⠀⠀⣠⣄⢿⣿⣿⣧⡹⣿⣿⢸⣿⢸⣿⣿⣿⡿⢫⣴⡿⣱⣿⣿⣦⡝⢿⡟⠐⢿⣷
⠀⠀⠀⠀⢠⣾⣿⣿⣎⢿⣿⣿⣿⣎⠻⠷⠶⢿⣿⡿⢟⣴⣿⡿⢱⣿⣿⣿⣿⣿⣦⠀⠀⠈⡿
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣾⣿⣿⢣⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠻⣿⣿⣿⣇⠀⠀⠀⣼⡿⠟⠁⣾⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢄⠀⠀⠀⠀⠀⠀⣠⠎⠱⡄⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠈⡆⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡋⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠜⠁⠀⠀⠀⠀⢀⣑⣒⡊⠀⠀⠀⠀⠀⠀⠀⠱⠄⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠯⠥⢶⡄⠀⠀⠀⠀⣿⡉⠒⠚⡇⠀⠀⠀⠀⠀⢸⠳⠮⠽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠒⠢⢼⡇⠀⠀⠀⠀⣿⠈⠉⠓⡇⠀⠀⠀⠀⠀⢸⠙⠒⠠⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⣿⣀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⠤⢤⡇⠀⠀⠀⠀⣿⠈⠙⠒⡇⠀⠀⠀⠀⠀⢸⠉⠲⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠧⠤⠔⠒⠓⠒⠒⠒⠒⠋⠀⠀⠀⠉⠈⠀⠐⠒⠒⠛⠒⠒⠂⠾⠤⠤⣀⣀
⠀⠀⢀⣠⠤⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠤⣄
⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳
⢰⠁
⡇⠀⠀⠀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠑⢦⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠦⡀
⢧⣀⢀⡠⠎⠈⢧⡀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠑⠤⣀⣀⢀⣀⣀⡠⠔⠋⠀⠀⠀⠀⠁⠀⠈⠒⠤⠤⠤⠔⠉⠀⠀⠈⠓⠤⠤
⠀⠀⢹⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⣀⠤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠶⠶⠦⠤⢄⡀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠲⠤⣀⠀⠀⠀⠀⢸
⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠙⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠈⠙⠒⠒⠚⢹
⠀⠀⢸⣀⡠⠤⠖⠒⠒⠒⢄⡀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠉⠉⠀⠀⠈⠑⠦⢄⣀⣀⠤⠤⠴⠒⠒⠒⠒⠒⠢⢄⣀⣠⠤⢚
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⢀⡸⠆⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡔⠒⠚⠢
⠀⠸⡀⠀⣰⠋⠁⠉⠦⠒⠒⠒⡤⠖⠒⢤⠤⠒⠢⡤⠔⠒⢤⡠⠔⠒⣤⠴⠒⢤⠤⠤⢤⡠⠔⠢⢤⠒⠒⠒⣎⠁⠀⠈⣆⣀⠴
⠀⠀⠈⠉⠉⠢⠤⠔⠳⣄⣀⡠⢇⣀⢀⡸⣄⠀⣀⣧⣀⢀⣠⢆⡀⢀⣧⣀⠀⣧⡀⠀⢀⣇⠀⢀⡸⢇⣀⣠⠞⠒⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠃⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⣀⠐⠀⠀⠀⠀⠀⡀⢴⣑⢃⢠⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠒⣾⣦⣦⣄⣀⠀⣼⣿⣿⠟⠁⣭⣐⠂⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠆
⠀⠀⠀⠀⠀⠀⠀⣶⣜⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠛⠇⠅⠀⠉⠻⠆⠻⠋⠀⠀⡉⠠⢈⠿⠂⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣴⡇⠄
⠀⠀⠀⠀⠀⠀⠐⠿⣦⡀⠀⠨⠀⢄⠀⠀⠀⠀⠀⠀⠀⠴⠒⠒⠀⠀⠀⢰⣶⡆⠀⠰⠉⢥⢸⠆⠀⠀⠀⠀⠀⠊⠄⠀⢀⠀⢻⡛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣸⣟⠠⡁⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠃⣀⣄⣴⠇⢼⠛⢳⡭⣿⣦⡴⠍⠀⠀⠀⠀⠔⠀⠀⠀⠐⠀⠀⣿⠙⠏⣠⠀⠄⢤⡀
⠀⠀⠀⠀⢠⣀⡀⠘⣿⣻⣆⠈⠀⠄⠀⠀⢰⠄⠀⠀⢀⣀⡀⣹⡏⠁⠀⠈⠀⠈⠁⠂⡻⠀⠀⠀⠀⣀⠀⠀⢀⠀⠋⠀⠀⢰⡿⠟⢀⠀⢀⠀⣨⡇
⠀⠀⠀⠀⣜⠛⠀⠌⠘⢿⣿⡀⠀⠀⠐⢠⣀⣤⣶⢂⠀⠒⠀⠈⠁⠀⠀⠀⠀⢈⡀⣾⠇⠉⢓⠤⡈⠻⣶⡖⠁⠀⠀⠤⠀⠉⢀⠀⠀⠐⣶⢺⡿⠁
⠀⠀⠀⠀⢛⠲⣄⡂⠀⠀⠀⠁⠠⡀⠀⠀⠛⠛⡇⡈⢀⠀⠀⠁⠠⠀⠀⢚⣀⣰⡶⠃⠀⠁⠀⢀⡙⡆⢹⠁⠀⠄⠀⢀⠁⣀⡌⠃⠀⡸⣶⡀
⠀⠀⠀⠀⠈⠁⢫⣿⣶⣤⣁⠀⠀⠈⠐⠀⠀⠠⠜⠇⠈⠁⠠⠄⠀⠂⠀⠛⠛⢻⠃⠒⠀⠀⠆⠀⣀⣿⢰⣴⣇⠀⢠⠀⣾⣿⠃⠀⣴⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠫⣟⣦⡹⣶⣄⠀⣀⣟⣶⣧⠆⠈⠀⠀⣀⠀⠀⠠⠘⡀⢛⡧⠀⢀⠀⠐⠀⠘⣏⠼⠳⠟⢁⣠⡾⠏⠉⢀⣼⣶⣃⡀
⠀⠀⠀⠀⠀⣤⡐⠂⠐⠘⠻⠿⠌⠻⣦⡘⠻⢋⠙⢶⣔⠶⣄⠀⢀⡀⠀⢋⡀⡉⠁⢀⡀⠀⡀⣈⣱⣅⡃⠰⣆⢉⠍⠐⠂⠀⠈⠉⠂⠀⠀⣿⠦
⠀⠀⠀⠀⠐⢧⣤⣒⣀⠁⠀⠂⠀⠉⠀⣀⠀⠻⣶⣾⣿⣶⣷⣦⣄⢀⣠⣬⣠⣥⣄⣀⣀⣠⣥⠝⡷⣿⣿⣶⠁⠀⠂⠀⠆⡀⠨⠀⠤⠖⠐⡻⠁
⠀⠀⠀⠀⠀⠈⠙⣿⡌⣹⠇⢅⠐⠲⣶⣦⠄⢴⣿⠟⠉⠁⠉⠙⠉⠛⠛⠟⠟⠟⠙⠟⠛⠏⡈⠈⠀⠈⠟⠛⠀⢶⣿⡿⠏⠀⡠⠼⢷⣀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⠲⣄⠸⠁⠬⡁⠀⠀⠈⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⢈⠀⠄⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢄⡘⠁⠒⠠⡀⢀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⣀⠀⡀⠠⠢⠉⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣀⣹⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡗⢂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢏⠁⠀⠀⠀⠀⠻⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢠⠞⠁⠀⠀⠁⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠘⠳⠠⠘⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⢠⡜⠁⠀⠀⠀⣠⠎⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠑⠀⠈⣄⠀⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠀⠀⣏⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠀⠀⠀⢀⢘⡄⠀⠈⠉⢳
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⡁⠂⠲⣴⠏⠀⠀⠀⠀⠀⣀⠀⠀⠶⠲⠆⠀⠀⠑⠐⠶⡤⠤⠀⡀⠀⠀⠈⠻⢧⣀⠤⠒⣾⣷⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢣⣠⣤⣿⠀⠀⣀⠴⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣄⢀⣀⣼⡇⠸⠤⠜⡛⠘⡇
⠀⠀⠀⠀⠀⢀⠀⢞⠁⠰⡄⠀⠀⡆⠀⠳⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠠⣼⡿⠀⠀⠀⣾⠃⢠⡇
⠀⠀⠀⠀⠀⡞⠀⠈⢧⡀⠀⠀⠀⠱⣄⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠃⠐⠀⣲⠆⠔⠉⡁
⠀⠀⠀⠀⠀⡄⠀⠀⠈⢿⠎⠈⠓⢄⣘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠠⡤⢀⣠⡾⠋⠈⠀⣼
⠀⠀⠀⠀⠀⠙⠁⠠⠀⠀⠑⠶⣤⣄⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿⠾⠋⢀⠠⠃⢉⠂
⠀⠀⠀⠀⠀⠀⢢⠀⠀⢄⠀⠀⠀⠉⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠀⠤⠄⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢗⠈⠐⠦⣄⡀⠀⠀⠲⣀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⣒⣋⡰⠶⠶⠶⠞⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠶⠾⠿⠤⠴⠶⠟⠷⠿⣅⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢾⣿⣏⣀⡀⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⢀⣠⠴⠶⠟⠛⠳⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠟⠑⢄⡀⠀⠉⠉⠛⠛⠛⠲⣤⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣁⠶⠋⠉⠀⢠⠀⣤⣤⡶⠟⠛⠶⢦⣤⣶⠶⠶⠶⠶⠿⠿⠛⠉⠻⢿⣄⣈⣱⡀⠀⠀⠀⠀⠀⠀⠙⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡹⠁⠀⠀⠀⠀⠀⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠋⣱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣠⣴⣶⣶⢤⠀⠀⠐⢶⣶⣶⣶⣤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡤⣶⡞⣿⣿⣿⣿⣿⣿⡟⠋⠉⠛⠷⣷⡀⠀⠀⠀⣸⡿⣛⣿⣿⠯⢤⣤⣄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢻⡇⠀⣷⡼⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠀⠀⡸⣷⣤⣀⣠⣾⣴⠛⠉⠉⠉⠛⠛⠩⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢲⠀⢿⡀⠀⣿⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⠃
⠀⠀⠀⠀⠀⡀⢃⢸⡆⢾⡅⣸⣿⣼⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⡯⢻⣾⡦⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠘⡁⣿⣶⡿⢻⠏⠻⠏⠈⠿⠹⠟⣿⡀⠹⣿⣉⢿⣟⢿⣟⣿⣯⣬⣝⣛⣿⣛⡋⠡⡯⠓⠽⢶⢾⣿⣴⣤⣤⣀⠢
⠀⠀⢀⣈⣿⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⠓⠳⠯⣿⣷⠾⠽⠯⣽⣾⣿⣟⡻⡶⠦⠷⣦⡈⣱⡏⠉⠉⠙⠻⣾⣟⠲⠾⠤⣤⣤⡀
⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠈⠣⣈⡙⢷⡄⠀⠈⠳⠿⠿⣿⣶⣿⣶⣾⣽⣿⣶⣦⣤⣀⡀
⠀⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠁⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠙⢷⡈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠺⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠈⣷⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡠⠀⠀⡀⠀⢀⠀⠀⢀⢀⠀⢀⣶⠆⠀⠀⠀⠀⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣆⠀⠰⢡⢣⢂⠎⠀⡘⢸⡄⠀⢸⢸⠀⣼⠏⢀⠀⡄⡄⠀⣏⡏⡀⠀⣀
⠀⠀⠀⠀⠀⠠⡀⢀⡀⠀⠀⠀⠀⠘⡜⠀⠀⠀⠏⠎⠀⢀⠃⢻⠀⠀⠘⠀⠰⠏⣰⠃⠀⣧⡇⠸⡻⡰⢁⢞⡁
⠀⠀⠀⠀⠀⠀⣈⠢⡙⢦⠀⠀⢀⠀⠘⠀⣿⡀⠀⠀⠀⢸⠀⠋⠀⠀⢀⣴⠀⠀⠁⠀⠀⠃⠀⢀⡁⠰⡣⠋⠀⠀⠀⢀⡤⣚⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢌⠳⠅⢀⣿⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣮⡟⠀⠀⠀⡄⠀⠀⠀⣞⡇⠈⠁⠀⠀⢀⠐⠉⢈⣁⡠⠤⠄
⠀⠀⠀⣀⣐⠢⠤⣀⡀⠙⠂⢘⣼⠀⠀⠀⡞⣷⠀⠀⠠⢳⡀⠀⠀⢰⢻⡆⠀⠀⢰⢹⡀⠀⢠⣾⡆⠀⠀⠀⢰⣻⠀⠈⠐⠒⠒⠘⠃
⠀⠀⠀⠀⠈⠉⠓⠒⠪⠄⠀⣴⣷⡆⠀⠀⡇⣿⠀⠀⠀⢻⡇⠀⠀⢸⢀⡇⠀⠀⠈⣾⡀⠀⢸⠀⡇⠀⠀⢠⣾⡇⠀⠐⠒⠲⠶⠭⠭⢥⣄⡀
⠀⠀⠀⠀⣒⡒⠫⠭⠙⡁⠀⢻⠀⡇⠀⠀⡇⣿⠀⠀⠀⡓⣿⠀⠀⢸⢸⡇⠀⠀⢸⠋⣿⠀⢸⠀⡇⠀⠀⡸⢰⡇⠀⠲⠭⣒⡢⠄
⠀⠀⠀⢀⣀⠤⠖⠊⠉⠀⠀⢸⠀⣷⠀⠀⡇⢿⣀⣠⠬⠄⣿⠤⠤⢾⢘⡧⠤⠤⢼⡆⣿⣀⣸⠄⡇⠀⠀⡇⣸⠀⠀⠀⠀⠀⠉⠓⠲⠤
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡼⡀⢿⠉⠹⣳⣞⡦⠀⠀⡇⣿⠀⠠⡞⣼⢷⡄⠀⠘⡇⣿⠀⡸⠀⣏⡒⢲⠁⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⣇⣸⣄⠀⠈⠉⠀⠀⠀⡇⢸⠀⠀⠉⠉⠉⠀⠀⠀⡇⢹⠀⠑⠿⠷⠃⣸⢰⡇⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠘⠞⠗⠋⠀⠀⠀⠀⠀⣖⠣⢾⣦⠀⠀⠀⠀⠀⠀⢮⣳⣞⡷⠀⠀⠀⠸⢜⣺⠽⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣃⡤⠞⡗⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣠⡀⠀⣀⣤⣀⡀⠀⠀⢠⠖⢦⣀⡟⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠑⠛⠉⠉⠑⢄⣀⡤⠞⠋⠉⠑⢄⣀⣠⡾⠁⠉⠁⠉⠛⠉⠀⠀⠉⠳⢤⡼⠀⠀⠀⠀⣻⣢
⠀⠀⠀⠀⠀⠀⠀⠘⣟⠣⣀⠀⠀⠀⠀⢀⣀⢶⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⢦⣠⠀⠀⢀⣀⣤⣾⢿⠁⡠⠤⠤⢤⣀⣀
⠀⠀⠀⣠⠴⠒⠋⠉⣸⠉⠻⢝⣓⡒⢖⣼⠂⠈⣽⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠮⡁⠀⢯⡽⠿⠻⠟⠉⠁⢸⡧⡄⠀⠀⠀⠀⠈⠙⢦
⠀⠀⣞⣁⣀⠀⠀⠀⢝⣄⡀⠀⠀⠈⠉⠉⠚⠑⠻⠦⠯⣽⣒⣒⣒⠶⡶⣖⢒⡒⣏⡭⢽⠴⠲⠉⠉⠁⠀⠀⠀⠀⢀⡼⡣⠇⠀⠀⠀⠀⣀⣀⡼
⡠⠊⠉⠉⠉⠀⠀⠀⠀⠙⠼⣶⡤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠙⠛⠋⠉⠛⠁⠀⠀⠀⠀⠀⢀⣀⠤⣶⣮⠽⠋⠁⠀⠀⠀⠀⠈⠉⢙⡇
⠙⠦⠤⠤⡤⠶⠃⠀⠀⠀⠀⠈⠉⠓⠽⠿⣗⢦⣤⣀⣤⠤⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⣤⣴⣒⣺⣭⠽⠛⠛⠁⠀⠀⠀⠀⠀⢠⣤⣤⡤⠶⠋
⠀⠀⠀⠀⠳⣄⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⠙⠓⠚⠛⠓⠋⠙⠛⠋⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠙⢯⣁⣀⣀⠀⣀⣠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢽⣷⠤⠶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠳⠦⣤⣤⣤⣀⣤⣤⡾⠗⠀⠀⠀⠀⣠⣾⠃⠀⠀⢀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢄⡴⣋⠳⢖⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⢱⢟⢻⣻⣭⠿⣿⣔⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠋⢹⣼⣯⣙⣿⣞⣧⡜⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢬⠟⣡⣶⢟⣽⣿⠽⣬⠙⣆⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡼⠏⣏⠸⡯⢹⡟⠛⣱⡯⠭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣏⣉⣹⣻⠶⠖⠋⢹⣟⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠈⢿⣧⡾⣳⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠋⠁⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠲⠤⣄⣀⣀⡠⠾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠓⠦⢄⣀⣀⣀⣀⠤⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⡀⠀⢀⣀⡀⠀⠀⣠⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡴⠃⠀⢫⣉⣹⠄⠀⠹⠦⠼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⣄⣉⣉⠉⠉⠉⠉⠉⠉⠉⠀⣀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⠦⢤⣀⣀⡉⠉⠉⠉⠉⠉⠉⣉⣉⣀⣀⡤⠴⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢳⠀⠀⣀⢠⡀⠈⠉⠉⠉⠉⠉⠉⠀⠀⣀⠀⠀⠀⡖⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠊⠀⠀⠧⣤⠟⠀⠀⢰⡎⠛⡦⠀⠀⠻⣀⣨⠂⠀⠙⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢤⣀⣈⠉⠑⠒⠒⠒⠒⠠⠤⠤⠤⠤⠒⠒⠛⠉⠉⢀⣀⠤⠒⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠈⠉⠓⠒⠶⠶⠤⠤⠤⠶⠶⠶⠶⠖⠒⠚⠋⠁⠀⠀⠀⠈⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣶⠈⠉⠙⠛⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠚⠛⠉⢡⣄⠀⢐⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣴⣶⠀⠀⣰⠆⠀⢀⡤⠀⠀⢀⣶⣆⣠⣄⠀⠀⢠⣶⡄⠴⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠉⠷⠶⠋⠙⠶⠾⠟⠁⠉⠁⠈⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡏⢹⢀⡴⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⢾⡇⢸⣻⠇⠀⣇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⢤⣀⣸⣵⠃⢸⡇⣜⣻⢀⡾⠁⣿⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠚⣉⣿⣍⣙⣶⣄⠀⠀⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⣾⠁⣠⠿⡛⠲⣾⡙⣆
⠀⠀⠀⠀⠀⢴⣿⣒⣋⣉⣁⡀⠀⠉⢧⠀⠳⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⡴⠁⢰⠛⢦⣬⠿⣾
⠀⠀⣀⣶⠋⠉⢉⣹⣤⣄⣈⣭⡽⠶⣌⠳⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣷⠖⠓⠲⠶⣽⡄⠈
⢠⢞⡥⠞⠋⠋⠉⠀⠀⠀⠀⠀⠙⢦⡀⠙⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡇⠀⠀⠀⣰⠃⠙⣶
⠿⠋⠀⠀⠀⠀⣀⣠⢤⡴⠶⠶⠶⠦⠬⠷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠑⣾⠷⠤⠤⡾⠁⠀⠀⣸
⠀⠀⠀⠀⣰⠊⢁⣠⠞⠓⢒⣒⣶⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢐⣽⣿⣿⣄⠤⣈⡿⠃⠀⠀⣰⠛⠒⠒⠚⡏
⠀⢀⡴⠚⣉⡿⠋⠀⣰⣞⣉⣁⣀⡤⢴⡃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢞⣽⣿⣷⣼⡻⠿⠟⢨⡧⠋⠀⠀⠀⣠⡇⠀⠀⠀⢰⠃
⠀⣏⣤⠞⠋⠀⠀⡜⠁⢸⠁⠀⠀⠀⡴⠙⢾⣁⣰⠃⢠⠄⢀⡄⠀⡄⢸⡙⠿⠛⢈⡗⢢⣶⡋⠀⣀⠴⢚⣉⡞⠙⠢⣤⣴⠏
⠀⡇⢸⠀⠀⠀⢠⣧⢴⡇⠀⠀⠀⢰⠇⠀⠀⠉⣿⠚⠿⠦⠾⠧⠴⠧⠬⠿⠶⠒⢻⣿⠉⠙⣣⣾⠵⠚⣉⡿⠁⠀⢀⠞
⠀⢧⢸⠀⠀⠀⠸⡀⢰⡇⠀⠀⠀⠘⣦⠤⠖⠚⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⣀⡴⠏⢀⣠⠴⠋
⠀⠈⠻⡄⠀⠀⠀⠱⡞⢧⡀⠀⠀⠀⢷⠀⠀⠀⢀⡿⠒⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣙⣦⣀⠀⠈⠳⠦⡖⠋⠀⠀⠀⠀⠀⣤⣭⣭⣭⣽⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠘⠢⢤⣀⣀⣀⣀⣀⣀⡤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡉⡁⠀⠐⢒⡦
⠀⠀⠀⠀⠀⠀⢀⠤⢶⠐⢆⠀⠀⠀⠀⠀⠀⣀⡀⢀⢊⢭⢭⣉⠖⠲⣤⡀⠀⠀⡘⠀⠈⠋⠙⠒⠼⢆⡀⠈⠲⢄
⠀⠀⠀⠀⡠⠊⠁⡰⠛⡔⠈⢃⠀⠀⠀⠀⡜⢓⡾⡣⣀⠉⠉⠉⠲⢶⠃⠈⡆⢀⠁⠀⠃⠀⠀⠀⣀⠀⠈⠐⢄⠀⠐⠂⡀
⠀⠀⢀⠈⠓⡦⠘⠀⠀⠈⣳⠈⣇⠀⠀⢀⠧⠔⡄⢎⠉⢀⠀⠀⡔⡏⠲⢴⠀⠘⠀⠀⠂⠀⢀⠊⠠⡏⠐⢄⡀⠙⠢⡀⠈⣴
⠀⢠⠂⢠⠊⢀⠤⢄⠀⠀⠸⡃⠈⢆⠀⠈⠢⣀⢣⢈⡿⠏⠸⢼⣰⢃⢰⠃⠀⠁⠀⡘⠀⡰⠁⡠⠊⠙⠶⣄⠈⣆⠀⠈⠐⢇⢣
⠀⠆⠠⠎⣠⠚⣄⠀⠁⢠⠀⠑⡆⠈⡂⠀⠀⡸⠙⠚⣡⠀⠀⠘⠙⠳⢄⢀⠤⣀⡐⢀⠜⠀⡰⠁⠀⠀⠀⠀⢋⠁⠰⡀⠀⠀⠱
⣌⡀⠊⢰⠁⠰⠁⠘⢌⡂⠑⢄⠘⢞⢉⠒⠉⠀⠀⠀⠆⠀⠀⠄⠀⠀⠀⠈⠲⡎⢀⠊⢀⡔⠀⠀⠀⠀⠀⠀⠀⠡⣠⠘⢤
⢈⡌⢡⢥⡀⠇⣠⠠⣀⠑⣅⠀⡡⣀⠃⠀⠀⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄⠀⢸⡙⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠈⠂
⠘⡇⡄⠀⡍⠰⡟⢦⡀⠑⢌⠓⢟⣸⠁⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⢍⠀⢀⣀⠀⠀⠠⠄⠤⠐⡄⠀⠀⠘⢦⣴
⠀⠠⡃⢰⠀⢸⠀⣰⠙⠲⣄⠑⢄⣐⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠃⡏⠀⠀⢀⣀⣀⣀⡠⠊⢠⠀⠀⠀⠀⢸
⠀⠀⠇⢨⠀⢸⠳⡆⠀⠀⠈⠑⠦⠷⣀⠔⠠⢄⠀⠠⠀⠀⠀⠠⠀⠀⡠⠂⠀⠈⠉⠁⠉⠁⠈⠀⠀⡜⠀⠀⡂⠀⠀⠀⣰⠃
⠀⠀⠀⠈⠀⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠑⠒⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⢦⡌
⠀⠀⢠⠔⠀⠸⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⡠⠊
⠀⠀⠀⠺⡀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⣀⠀⠈
⠀⠀⠀⠀⠈⠀⠱⢖⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⠓⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿