Hardwood Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⠏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠐⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠐⢠⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⡠⠀⠠⢠⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠐⠁⠀⠀⠀⠀⠔⠁⠀⠀⠀⠐⠁⠀⢠⠏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢠⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡡⠀⢀⠎⠀⠀⠀⠀⢠⠂⠠⠀⠀⢀⠊⠀⠄⢠⡟⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⡠⠃⠀⠀⠀⠀⢀⠇⠀⠁⠀⢠⠃⠀⠀⢠⡟⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠜⠀⡠⢀⠀⠀⢀⡞⠀⠀⠀⢠⠃⠀⠂⢠⡟⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢠⠊⣠⠎⠀⠔⠀⠀⠀⢀⠞⠀⠎⠀⣰⠃⠀⠄⢀⡞⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡴⢁⡴⠁⢠⠎⡀⠀⠀⢠⠏⠀⠈⠀⣰⠃⠀⠈⢀⡞⢠⠏
⠀⠀⠀⠀⠀⠀⢀⡴⢃⠀⠈⣠⠋⠀⡐⢁⠜⠀⠀⢠⠃⢀⠀⠀⢠⠃⠀⠀⢀⡾⢡⠏
⠀⠀⠀⠀⠀⣠⠎⡠⠉⢀⠜⠁⢀⠎⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⠇⠀⠀⢀⡾⢡⡏
⠀⠀⠀⢀⡼⢁⠈⢀⠔⢡⠀⣠⠋⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⡎⠀⠀⢀⡾⢠⡟
⠀⠀⣰⠏⠀⢀⡴⠃⢀⠆⠰⠁⠀⠀⠀⠀⢀⡞⠀⠀⠄⠀⣸⠁⠀⢀⡾⢁⡞
⢀⡞⠁⠀⢠⠏⠀⠀⡞⠀⠀⠀⡠⠁⠀⣰⠋⠀⠀⡆⠀⣰⠃⠀⢀⡾⢁⡞
⣿⠶⠶⠲⡿⠶⠶⠾⠷⠶⠶⠾⠷⠶⢾⠗⠶⠶⢾⠶⠶⡿⠶⠶⣾⢅⡾⠁
⣿⠀⠀⠀⣆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠁⠀⠀⣿⡾⠁
⣷⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣶⣶⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣽⣿⣭⣿⣿⣿

⠀⠀⠀⠰⠶⠶⠖⠂⠀⠀⠀⠀⠠⠦⠤⠀⠤⠀⠀⠀⠤⠤⠤⠤⠀⠀⠤⠀⠀⠠⠴⠀⠀⠀⠀⡀
⠀⠀⠀⠣⣭⣉⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠄⢤⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⠤⠄
⠀⠀⠀⠀⠫⡶⠤⠭⠉⠉⠁⠀⠀⠀⠀⠀⠀⣒⠒⠂⠀⠤⠀⠈⠉⠼⠒⠒⢓⠒⠒⣘⣃⡦⡀
⠀⠀⠀⠀⠼⠿⠿⢗⣒⣶⣒⣒⣶⠤⢤⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀
⠀⠀⠀⠀⠈⠉⢑⡖⠒⠒⠄⢤⣠⠤⠄⠒⢒⡊⠂⠀⢀⠡⠐⠈⠉⠁⠈⠀⠀⠀⠀⠈⠉⠩⠉⠉⠁
⠀⠀⠀⢒⣀⣈⠉⢉⣈⣉⡁⠀⠀⠐⠂⠈⠀⠀⡀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⣠⣴⣄
⠀⠀⠀⠈⠋⠀⠠⠶⠶⠖⠲⠤⠘⠀⠀⠤⠶⠶⠆⠀⠀⠀⠀⠀⠐⠀⠰⠶⠶⠤⠠⠄⠀⠤⠴⠶⠂
⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⣤⢤⠤⠤⠤⢤⠤⢤⣤⠤⠀⠠⢤⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⡤⠤⠄⠂⣀⢀⡤⠤⠤⠄⠀⠈⠁⣭⣀⡀⠀⢑⡀⠀⠀⠀⠀⠀⣀⡀⣀⠠⢼
⠀⠀⠀⠀⠀⠈⡄⠒⠋⠁⡠⠖⢒⣛⣁⡐⠀⠄⢂⠄⠈⠀⠀⠨⡙⠒⠤⣒⣒⠂⠀⣠⠖⡈
⠀⠀⠀⠀⠀⠀⠡⠀⠑⠀⠀⠙⠾⠾⠷⠜⠀⠉⠠⠂⡀⠠⠂⠇⢸⠀⠀⠀⠈⠉⠠⠤⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠸⠶⠢⠦⠤⠉⡉⠒⠀⠀⠈⣉⣀⣀⣶⣒⡛⠂⠁⠠⡤⣖⢚⣁⣀⣀⣀⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠒⠚⠛⠓⠒⠒⠒⠐⠒⠒⠒⠒⠚⠛⠛⠛⠚⠃⠂⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠉⠍⠉⠙⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢨⠋⠒⠒⢲⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠌⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡜⠀⠀⠀⠀⠇⠀⢀⡯⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠶⠶⣖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⢍⠀⠀⠀⠀⠀⠀⡸⠁⢀⠆⠀⢸⠀⠀⣸⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⣦⡀
⠀⠀⠀⢀⡴⠋⢀⣀⠠⠄⣈⠙⠶⡖⠒⠒⠒⠒⠂⠀⠀⠀⠀⠠⠄⠙⠒⠒⠂⠀⠀⠡⠒⠉⠀⢠⠃⠀⠀⡿⣀⡀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠤⠤⠤⠽⣆
⠀⠀⣰⠋⢀⠔⠁⢀⣀⡀⠀⠑⢄⠙⣆⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⢧
⠀⣰⠁⠠⠃⠔⠉⠀⡀⠈⠑⢄⠈⡇⠸⡆⠀⠀⠀⠀⠀⠐⠂⠙⠋⠉⠉⠉⠉⠥⠤⠤⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠬⠥⠥⠒⠛⠛⠉⠉⠉⠀⠀⢸
⢰⠃⠀⠀⡌⠠⠊⢀⠀⠀⠀⠈⠀⠰⠀⢷⠀⠐⠂⠀⠀⠒⠒⣤⠤⠖⠒⠋⢉⣉⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⡾⠀⠀⢰⠀⢀⠎⡔⠂⢤⢣⠱⠀⠀⠀⢸⡉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠒⡉⣁⠤⠤⠬⠭⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⡇⠀⠄⠸⠀⠈⠀⠑⠒⢃⠌⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣂⣉⣁⡀⠠⠤⠀⠁⢁⣉⡠⠤⠀⠀⠀⠀⠀⠀⠒⠤⣀⣀
⡇⠀⢀⠀⠀⠳⣄⡈⠉⡡⠆⠀⠂⢠⠀⡾⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⢐⡒⠒⠂⠤⠄⠒⠀⣀⣀⠉⢉⣉⣀⠀⠄⠀⠀⠀⠀⠂⢀⣈⣉⣉⣉⠤⠀⠀⠀⢠
⢻⡀⠘⡀⠐⢄⡀⠈⠁⠀⡀⠀⠀⡌⢠⠇⠒⠒⠒⠀⠀⠀⠀⢀⣀⣀⣀⣈⣉⣀⡒⠒⠊⠉⠉⠉⠉⠀⠀⠀⠀⠒⠒⠀⠤⠤⠄⠒⠚⠉⠁⠀⠒⠒⠉⣩⠏
⠈⢣⡀⠳⡀⠀⠈⠉⠈⠁⠀⣀⠔⢀⡞⠐⠀⠀⠀⠀⠀⠀⠀⠀⢐⡲⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠤⠤⠤⠶⠖⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠙⢤⡈⠓⠂⠀⠐⠂⠉⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠤⠤⠤⠤⠭⠭⠭⠉⠉⠉⠉⠉⣉⣛⣋⣉⣀⣀⣀⣀⣀⣀⣨⣭⣭⠥⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠚⠋⠁⢻
⠀⠀⠀⠀⠀⠀⣠⡴⠾⢛⣉⠉⠉⠙⠛⢆⠐⠀⠀⠘⣇⣀⡀
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠉⠉⠀⢈⡆⠀⠀⠀⠹⣍⣉⡙⠛⠛⠛⠓⠶⠶⠤⣤⣤⣀⣀⣀
⠀⠀⠀⢀⣿⠁⠉⠁⠐⠀⠀⠀⠀⢒⠀⡾⠀⠀⢀⠀⠀⠀⠖⠦⡉⠉⠐⠀⠀⠀⠀⠤⠤⠄⠈⢉⡉⠉⠙⠛⠛⠻⠛⣛⠒⠲⣶⠶⠶⣦⣤⡀
⠀⠀⠀⠘⣿⠀⠀⣀⠀⠀⠤⠤⠀⣉⣂⣠⣄⠀⣈⡀⠀⢀⣀⣀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⡀⠄⢺⠃⠙⢦
⠀⠀⠀⠀⢻⣅⡀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠈⠉⠉⠙⠛⠓⠠⠠⠤⠀⣈⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡰⢋⠔⠒⠉⠀⠀⠘⣧
⠀⠀⠀⠀⠘⣷⣮⣭⣭⣁⣐⡒⠐⠂⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⢈⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡟⢐⣣⠋⠐⠁⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠙⣧⡉⠁⠒⠀⠈⠁⢐⡒⠒⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠦⠤⠤⠄⢀⠤⠤⠤⣼⠁⡜⡇⢀⣃⢸⢀⠎⡄⠀⣼
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⢦⣤⣀⡀⠈⠁⠀⠂⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣷⢞⡇⢸⠸⠀⡇⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠲⠶⠤⣤⣤⣄⣉⡁⠒⠂⠀⠄⠀⠀⠀⠀⠠⠌⠁⠁⢀⡒⠒⠒⠶⣷⠸⠀⠃⡜⠷⢸⠀⡇⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠄⢻⠀⡀⡀⠱⣠⠎⢀⣷⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠦⣤⣀⡀⠀⠀⠀⠀⠀⢺⡆⠡⠘⠤⠠⢔⡡⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⣤⣄⣹⣦⣤⣀⣂⣥⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⢀⣤⣤⣄⡀⣀⡤⠀⢀⣠⠤⠀⠀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡀⠀⢠⣤⣀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢤⣤⡄⠀⠠⠄
⠀⢸⣿⣿⣿⡯⠥⠀⠀⠁⣀⣐⣈⣁⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠤⣄⣀⣀⣉⡉⠉⠙⠲⢦⡀⠈⠙⠳⢶⢶⠾⠿⢃⣀⠤
⠀⢸⣿⣿⣿⣿⣿⣯⣭⡭⠴⠶⠶⠒⢚⣛⣻⣟⣛⣛⠛⠀⠒⠂⢐⢒⣈⣁⠀⠀⠐⠓⠒⠚⠩⢿⣗⣦⣄⣈⡉⠉⢉⣛⠻⠛⠉⠉
⠀⠰⣿⣿⣿⣿⣷⣿⣿⣠⡀⠒⠿⣿⣿⣿⣿⣿⣿⣿⠿⠗⠂⠀⠐⠂⠀⠒⠒⠒⠒⠒⠸⢯⣽⣖⠢⢌⠑⠀⠩⠍⣹⠛
⠀⢠⣿⣿⣿⣿⣯⣭⣤⣤⣨⣭⣼⣶⣿⣿⣷⣿⣿⣿⣿⣯⣤⣤⣬⣤⣤⣤⣥⣤⣥⣷⣶⣦⣬⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⠃
⠀⠀⠀⠀⠀⠀⣐⣒⣂⠐⠒⠒⠒⠂⠀⠀⠀⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⠤⠀⠀⢀⣬⣭⡅
⠀⠀⠀⠀⠤⠤⠤⠿⠤⠤⠤⣤⣄⣀⣀⡀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠒⠂⠀⢾⣿⡿⠆
⠀⠀⠀⠤⠭⢿⣿⣛⡓⠒⠲⠦⠤⣄⠈⠉⠑⠲⢤⡤⠤⡆⠀⠀⢠⠖⠛⠱⣦⠀⠀⢤⡀⠙⠲⢶⣤⣄⣀⠀⠀⠀⠙⠓⠒⠒⠾⢿⣿⣿⡆
⠀⠀⣿⣶⣶⡄⠀⠀⠈⠉⠁⠀⠢⡀⠙⠲⠶⠦⠤⠶⡿⢱⢦⠀⢿⣀⣙⠴⠂⢀⠀⢲⡌⢳⣶⣤⠈⠙⢿⣿⣀⣴⣿⣶⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠿⠿⠟⠟⠑⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⣀⠀⠲⡈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⡿⠁⠀⢀⣼⠟⠿⠲⣾⣿⠿⢿⣿⡿⢿⠿⠃
⠀⠀⠐⣒⣛⡛⣿⡟⣋⣉⢉⡀⠀⢀⣀⣀⣀⣀⣀⣈⣁⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢁⠀⠀⠒⠉⠀⠀⠀⢀⣠⣤⣠⣤⣶⣶⣿⣿⠂
⠀⠀⣩⡭⠈⠁⠠⠤⠤⠤⠀⠠⠤⢤⣤⣤⣤⣤⣤⣄⣈⣉⣉⣉⣁⣀⣠⡤⠤⠤⠀⠀⢀⣀⡀⠠⠤⠔⢒⣂⣉⣉⣭⣭⣽⣿⣿⣿⣿⡿⠆
⠀⢠⣶⣦⣤⣤⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣤⣤⣤⣤⣤⣴⣶⣶⣤⠀⢰⣶⣤⣤⣶⣦⣶⣴⣶⣶⣶⣶⣶⣾⣿⣿⣭⡄
⠀⠐⠃⠸⠿⠓⠿⠿⠿⠛⠛⠻⠟⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣋⣉⠁
⠀⠀⠀⣠⣤⣄⣀⢀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣻⣟
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣁⣛⣛⣛⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠉⠉⠁




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡸⠆⠀⣀⣠⣤⣤⣄⣀⡀⠀⢲⡀⠠⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⢀⣸⣿⡇
⠀⠀⠠⠴⠒⠛⠉⠁⠀⠀⠀⠙⣏⠀⠞⠉⠰⠑⠂⠁⠀⠀⠀⠀⠀⠁⠀⠁⡀⠃⢀⠴⠟⠋⠁⣀⡤⠾⠛⢋⣁⡀
⠀⠀⠀⣒⡀⢉⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠊⠁⠀⠀⠀⠁⠤⠒⣚⣻⣭⠭⠄
⠀⠀⠉⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣷⡀
⠀⠀⡶⢾⡷⢾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⡇
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠁⠈⠉⠈⠀⠀⠀⠀⠀⠈⠉⢉⡀
⠀⠀⣶⣤⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣂⣀⡀⠀⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⠶⠶⠄
⠀⠀⣿⠅⢣⡀⠀⡍⠀⠀⣦⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠈⢁⣀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠀⢀
⠀⠀⣿⣢⠭⠅⠐⠁⠀⠚⠉⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠀⠀⠺⠏⠤⠄⡀⠀⠐⠏⣉⣉⣉⣿⠆⠀⠁
⠀⠀⣿⣿⣯⣤⣄⣂⡀⢀⠀⠀⠬⠩⠥⠬⠭⠉⠀⠀⠈⠉⠉⠉⠉⠉⠂⠀⠒⠢⠄⣈⠁⠀⠤⠌⠭⠭⠄⠒⠁
⠀⠀⣿⠿⣿⣷⣎⡛⠀⠀⠀⠄⠠⠿⡿⠴⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠃
⠀⠀⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣿⣿⣷⣿⣮⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣤⣶⣶⣦⣴⣤⣶⣦⣦⡄
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠡⠄
⠀⠀⠀⠀⠀⣀⣲⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⠆⠀⠀⣰⣿⡆
⠀⠀⠀⢀⣀⣒⡛⠠⠤⠉⠉⠉⠁⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠠⠤⠤⣼⣿⡄
⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠂⠀⠈⠓⠂⠀⠀⠀⠲⠶⠶⠶⠿⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⠀⠀⡇⠀⢘⣏⣻⢸⣿⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢽⡗⣾
⠀⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⣶⠀⠀⠀⡇⠀⠀⠟⢿⣽⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠀⠀⠀⡇⠀⢰⡀⢘⡿⡏⠈⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⣼⠀⠠⠰⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣿⡇⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⡟⠀⣿⠀⣾⡔⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⠋⢻⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⠀⠀⠀⢘⡇⠀⠃⠀⣿⢸⣻⡇⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢡⣾⣮⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⠃⠀⠀⠀⡇⠀⡆⡇⢿⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠟⣵⣮⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡘⡇⠀⣇⣷⢸⣼⠄⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⣿⡗⣿
⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣼⠆⠀⠀⠛⡇⠀⣿⡇⢸⠻⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢰⣞⣿⣩⣿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠻⠇⠀⠀⡀⡟⠀⢸⠁⠸⡳⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⢿⣣⠁⣏
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡅⡇⠀⢸⠀⠀⣧⢀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⢻⢠⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⣷⠀⠀⠀⣴⠘⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠼⠀⢸⣿
⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠳⣜⠆⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠁⢣⡸⣿
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠟⣰⣷⡈⡆⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡴⣄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⢋⣤⡑⠅⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⠙⠸⡷⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠒⠋⠀⠈⠃⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠀⠀⣌
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⣗⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⢠⢴⣇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣽⠀⢸⡇⠀⠴⣿⢿⡆⡆
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⢀⢸⡗⠀⠰⢣⣼⡇⣗
⠀⠀⠸⠀⠀⠀⠀⣿⠀⠀⢠⠀⠀⠀⠀⣿⠀⠀⠄⢠⢰⣤⣶⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠟⢸⢸⡇⠀⠀⢟⡛⡁⢸
⠀⢀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠰⢸⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠜⢧⡀⠈
⠀⠀⠀⣌⠀⠀⢠⣿⠀⠀⣤⣷⠀⠀⠀⣿⠸⠀⣠⣤⢾⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡰⡏
⠀⠀⠀⣉⠀⠀⢸⣿⠀⢸⣟⣥⡀⠀⠀⡿⠀⠘⠈⠁⠆⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⠘⠒
⣀⣐⣀⣿⣀⣀⣘⣟⣀⣙⣫⣝⣇⣀⣘⣿⣀⣐⣀⣐⣘⣂⣿⣿⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣘⣇⣀⣀⣫⣆⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡀⢀⢀⠀⣀⣀⣀⡀⣀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀⢀⡀⢀⡀⣀⡀⢀⣀⡀⠀⠀⣀⣀⠀⠀⡀⢀⣀⢀⣀⢀⡀⠀⡀
⠀⡇⠀⠀⠀⠀⠀⢸⣷⠈⠦⣅⣐⡦⢨⡝⠓⠒⠘⠟⣾⣶⡋⢘⠗⢭⠚⠀⠃⠸⠃⠈⠙⠭⡛⠆⠓⠒⠁⠈⠻⢻⡅⠀⠘⠆⠛⠰⠀⠀⠇
⠀⡇⠀⠀⠀⠀⢸⡈⡯⠰⠁⠀⠂⣧⣼⠑⠄⠂⠀⡾⢧⠀⡿⠊⠣⠀⢀⠀⢀⢀⢰⡀⠀⡇⢸⠒⢴⢦⡀⠀⠀⢸⢰⠄⠀⠈⣇⠀⢀⡇⡇
⠀⡇⠀⠀⠀⠀⢸⠁⣇⠠⠟⡸⡇⢸⡇⠀⠀⠃⠀⠃⢨⠆⠖⡂⠀⠀⢸⡀⢀⠀⢸⠉⡔⠇⢘⠀⢀⢹⠆⠀⠀⠃⣤⠄⠀⠀⢻⠼⠀⠁⡆
⠀⡇⠀⠀⠀⠀⠸⠈⡧⠂⠠⠄⠧⠾⢄⡀⠀⠀⠰⡆⠁⠐⠶⣏⠈⠡⢤⡀⠀⠀⢸⠰⠀⣦⠙⣦⣀⠀⠀⠀⠀⠀⣧⠐⠀⠀⡜⢀⠀⡄⠆
⠀⡇⠀⠀⠀⡀⠀⢈⣧⣯⣭⢤⣅⠉⠏⢡⣀⠀⠘⡗⠨⣬⠏⡗⠀⠒⢘⡁⠀⠀⢸⠀⠀⡏⢰⠀⠃⠠⠀⠀⢸⡄⣧⡈⠀⠀⡇⠐⣦⠣⡆
⠀⡇⠀⠀⠀⠀⠀⠀⠁⠄⠀⢸⣿⠆⡀⠥⠘⠁⠀⡇⢠⠏⡆⡀⠀⡄⠸⠃⠀⠀⠘⢠⠀⡇⠀⠹⠂⢰⠀⠀⠸⠀⣟⠀⠠⠀⡇⠀⠉⢷⡇
⠀⡇⠀⠀⠀⡀⠀⢰⠀⡆⢠⠘⡟⠀⠀⠀⢀⠀⢰⣇⡾⠀⡇⢇⠀⡇⠀⡆⠀⡀⠰⢸⠀⠇⠀⣀⠈⢸⠀⠀⢀⠀⢸⠁⠀⠀⡀⠀⠀⠙⡇
⠀⡇⠀⠀⠀⠀⠀⢸⡀⡇⢸⢠⡇⠀⠀⠀⠘⠀⢸⡏⢀⠀⠆⢸⠀⡇⠀⠁⠂⡀⢠⠀⠀⠀⠀⡋⠀⠹⠀⠀⠘⠁⢸⠀⠲⠀⡇⠀⠀⠀⡇
⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⢸⡇⠸⠀⠀⠂⠀⠃⠇⠀⠀⠀⢸⠀⠀⠀⠀⡅⠀⠀⠀⠀⢀⠀⢀⡆⢰⠀⡇⠀⠀⠀⡇
⠀⡇⠀⡇⠀⠀⠀⢀⡆⠀⡀⢸⡀⠂⠀⠀⠆⠀⠀⡇⠈⡘⠀⢸⠀⠐⠀⠀⠀⠀⢀⡄⢀⠀⠀⠇⠀⠀⠀⠀⠸⠀⠈⡇⠈⠀⡃⠀⠀⠀⡇
⠀⡇⠀⡇⠀⠀⠀⠈⠁⠀⠁⠘⠑⠀⠠⠠⠀⠀⠀⣇⠀⠠⠄⠠⢄⡀⠀⠀⠀⠀⠀⢁⠈⠀⠀⠄⠸⠀⠀⠀⠠⠀⠀⣿⡀⠀⡇⠈⠀⢃⡅
⠀⡇⠀⡇⠀⠀⢐⠐⠀⠀⠀⠀⠀⠤⠀⠀⠄⠀⠈⠑⠓⠀⠀⠈⣸⠇⠄⠘⠀⠀⠀⢸⠃⠂⢰⢠⡆⠀⡀⠀⠀⠀⠀⢸⡇⢀⢠⠀⠀⢸⡇
⠀⡇⠀⡇⠀⠀⢀⠀⠀⠀⠀⡀⢀⡀⠒⠀⠘⠐⠂⠀⠀⠀⠀⠀⢸⡇⠃⠀⠈⠀⠀⢸⡷⠀⠀⣾⠁⡆⠁⠀⠀⠀⠀⡀⡆⠀⢸⡀⠀⠈⡇
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡇⠀⠀⠀⠀⡄⢘⢡⠠⡄⡟⠀⢰⠀⠀⠀⢀⠀⠀⠁⠀⢸⠇⠀⠀⡁
⠀⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⣀⠒⢉⠀⠀⢀⡀⢀⠀⠧⠀⠀⣀⠀⡀⠀⠘⠀⠀⠀⠈⠀⠀⠰⠃⠘⠃⠀⠁⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⠐⠋⡀⠕⠀⠀⠐⠀⠀⡀⠀⠂⠠⠤⠀⠀⡔⢁⠌⢀⠂⠀⠀⠀⠀⠀⠀⠆⠃⠇⡿⠀⠰⠈⡇
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠠⠌⢐⡀⢌⠁⠀⠄⡀⠘⠊⢡⣀⠂⠀⠂⠀⠀⡆⠄⢁⠀⠀⡤⢁⠀⠄⠀⡇
⠀⡇⠀⡆⠀⠀⠆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠀⠀⠀⠀⠀⠉⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠶⠔⢠⠂⠠⠀⠰⡇
⠀⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠁⠀⠘⠁⡠⢀⠀⠀⠂⢰⡇
⠀⡇⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠒⠀⠄⠁⠀⡊⠀⠈⠀⠀⠀⠀⡅
⠀⡇⠀⠀⠀⠀⠀⠀⠈⣀⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠀⠂⠐⠂⠤⠠⠀⠠⠤⠀⠉⠀⠒⠒⠀⠀⢀⠀⠈⣉⠀⠒⠒⡇
⠀⡇⠀⠀⠀⠀⢀⠰⠀⢨⠐⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠂⠀⠂⠀⠒⠂⠐⠀⠀⠀⠠⠄⠀⠀⠀⠈⠉⠈⠇
⠀⡇⠀⠀⠀⠀⠸⡆⠄⠂⠖⡥⠄⠃⢀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠐⡃
⠀⡇⠀⠀⠀⠀⠀⣠⠐⠐⠘⣐⡰⢀⡞⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃
⠀⡇⠀⠀⠀⠀⠀⠈⠆⠀⠁⢉⢡⡀⠙⠘⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⠀⢺⠀⠐⡀⢆⠳⠒⢼⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠃⠀⠀⠁⠀⠀⢁⠠⠁⠘⡔⢀⡁⠀⡀⠀⠆⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⡇⠀⠠⠀⠀⠀⠀⠀⢁⢂⢀⡀⢡⡼⢁⣶⠒⢲⣆⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠂⠐⠂⠐⠒⡇
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⢀⠀⠱⡀⠀⣩⣜⠡⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⡇⠀⠀⠀⠀⠀⢡⠐⠀⠈⠄⢪⢓⠀⠱⡞⢀⠈⠉⡁⠈⠈⠊⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⡇⠀⠀⠀⠀⠈⡀⠂⠠⠀⠈⠀⢠⠠⠄⠐⢠⠢⢠⣷⣧⢎⡔⡀⡀⡄⠈⠀⠄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⣧⣰⠀⠀⣀⢀⢰⣈⡀⡄⣀⢈⡄⢁⢈⢢⡄⢣⣌⡟⢳⣿⡟⣇⣁⣤⣁⣤⡌⣠⣅⣠⡂⣀⣀⣀⣀⠀⢀⡀⠀⣀⣀⣀⣀⣀⣀⣀⡀⡆

⠀⠀⢀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠁
⣉⣀⣤⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠁⠀⠀⠀⠄⠀⠀⠀⢀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀
⠀⣴⣄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠴⠺⣟⣉⠁⠀⠁⠦⠀⠀⠀⢀⠀⠈⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⣽⠀⠀⠄⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀⠀⠤⠄⠀⠀⠀⠀⢀⠀⣖⣤⣾⠇⠀⠀⠀⠀⢀⠘⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⣉⠐⠒⠆⠄⣴⣇⠰⠶⣷⡾⠟⠋⠀⠀⠀⠀⣤⠞⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠺⠦⡤⢦⣦⣄⣀⡤⠄⢠⡄⠐⠀⠶⡖⠂⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠰⢤⣀⡀⠀⢀⡉⣀⠉⠙⠓⠊⠂⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠶⠟⠁⠉⠉⠁⠉⠉⠻⣿⢶⣾⣿⠞⠛⠼⠍⠠⢀⡠⣀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠀⠀⠀⠀⠀⠁⠀⠠⠀⠤⢤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠤⣤⣀⡠⠄⡀⢄⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⢛⢛⣀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠤⣭⣉⡓⣦⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠿⠉⠙⠉⠙⡟⠅
⣴⣶⣤⣴⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣷⣷⣶⣶⣶⣶⣦⣤⣤⣤⣤⣴⣦⣤⣶⣶⣶⣶⣦⣴⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⣰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠈⠃⠀⠀⠀⠀⡀⠋⠀⡆⠀⣰⠇⠀⠂⠀⠀⢀⠆⠀⠀⢰⠃
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠈⠀⠀⠀⢐⠁⠀⢠⠀⠀⠀⠀⣴⠀⠀⠐⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⡌⠀⢀⡎⠀⣠⠀⢠⠃⡤⢀⢆⠀⠀⠀⠀⢸⠀⠀⠑
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⠁⠀⣸⠀⠘⠐⣴⢡⣞⡄⢀⠀⠀⢸
⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠈⠀⣿⢃⡀⣦⣦⢡⣿⡏⠀⣡⠀⣰⣇
⠁⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠇⠰⠃⣾⣴⡟⡟⠈⣵⣽⣿⣾⣿⣷⡶⢡⣶⣿⠇
⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⠀⠀⠀⠼⠙⠁⠀⠖⡸⠟⡟⣴⣧⣶⣿⣿⣿⣿⣿⣿⢣⣿⣿⡇
⠀⠀⠀⠈⡀⠀⠀⠀⡀⠀⠈⠁⡀⠀⠀⠀⠀⢀⠀⠀⣴⠎⠔⠁⠀⣴⣿⣿⡿⡿⢙⢿⣿⠟⠁⣾⢏⡝⠁⠄⠁
⠀⠀⠀⠀⢱⡀⠀⠀⠀⢳⣤⣤⡀⠀⢠⠅⠀⠀⢰⠠⡇⡀⠂⠏⢀⣾⣹⣏⢁⢀⣿⡿⠞⠄⢠⣧⢺⠀⠚⠐
⠀⠀⠀⠀⠀⠃⠀⠂⠀⢸⡿⡏⡄⠀⣰⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⣏⡇⢩⣷⣿⣿⣷⣿⣦⠞⢳⠎⠀⣠⠀⣀
⠀⠀⠀⠀⠈⢆⠀⠀⢀⠀⢫⢣⡀⠐⠙⡅⠀⠀⠇⠀⡂⠀⢠⠀⣿⢻⡧⣼⣿⣿⣿⣿⣿⡏⢰⠃⠀⠊⢸⣷⠟
⠀⠀⢀⠀⠀⠀⠳⣄⠀⠀⢄⣿⣿⠠⡀⢻⠀⠀⢁⠀⣿⣶⣼⡇⣿⣅⣿⣽⣿⣿⣿⣿⣿⠀⡏⢯⠀⠁⠈⠂
⠀⠀⢠⡀⠀⠀⠀⠈⠳⣤⠜⢿⣷⡟⢹⣿⢆⠀⢸⣠⣹⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠈⠀⠘⠀⡄⠀⠀⣀
⠈⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⡁⠠⠉⠈⠀⢈⣿⢛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡄⠀⠀⠀⠀⢾⠐⠀⠘⡣
⣴⡶⢶⣶⣶⣶⣶⣶⣶⣴⣶⣶⣷⣾⣾⣶⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣦⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⢀⡴⠚⠉⠛⠙⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠳⣤⡟⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⢶⣦⡀⠀⠀⠀⠀⠉⠳⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⠀⠀⠀⠀⠀⠀⠈⠓⢬⡳⣄
⠀⠀⠀⠀⢀⡤⠶⠖⠒⠻⢦⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣷⣄
⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣀⣤⣤⡦⢽⣿⡷⣄
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⣿⣦⡀⠀⢠⣾⠟⠉⠀⠀⠈⢳⣿⡈⠑⢦⣀
⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠙⣿⡆⣿⡇⠀⠀⠀⠀⠀⣸⣿⣧⢴⣶⠾⢷⣦⣄
⠀⣠⠤⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡠⡿⡇⠀⠀⠀⠀⠀⠀⢸⣿⢻⣇⠀⠀⠀⠀⣰⣿⣿⠟⠁⠀⠀⠀⠘⡿⡆
⣼⠁⠀⠀⠘⢧⡀⢠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣳⡀⠀⠀⠀⠀⢀⣼⣏⣛⣿⣷⣶⣶⠿⢻⣷⠃⠀⠀⠀⠀⠀⠀⣧⡇
⣟⠀⠀⠀⠀⠀⠹⣼⣄⠀⢹⠀⠀⠀⠀⣀⣤⣶⣖⣲⣿⢿⣶⣶⣶⣾⣿⠿⠛⠛⠛⢿⣮⠳⣄⠺⣿⡆⠀⠀⠀⠀⠀⢰⣿⠁
⠻⡄⠀⠀⠀⠀⠀⠈⠻⣟⠋⠀⠀⢠⣾⠟⠁⠀⠀⠀⠈⠙⣽⡆⢀⣿⠃⠀⠀⠀⠀⢨⣿⠀⠀⠙⢿⣽⣦⣀⣀⣀⣴⣿⣩⣷⣤⡀
⠀⠙⢆⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣌⣿⣇⠀⠀⠀⢀⣼⣿⣖⣺⠿⠿⢿⣛⣟⠛⣻⡿⠟⠉⠉⠙⢻⣦
⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡽⣄⠀⠀⠀⠀⠀⢠⢿⣏⣬⡽⠿⣷⣶⣶⢿⣿⠋⠁⠀⠀⠀⠀⠈⣞⣷⣿⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣶⣤⣴⣶⡾⣵⣿⠟⠋⠉⠉⠒⢽⢶⣿⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣄⠀⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⣠⣶⠿⠯⠯⣿⣿⣧⡰⡇⡇⠀⠀⠀⠀⠀⠀⡇⣿⣜⣦⣀⠀⠀⠀⣀⣴⣿⠃⠉⠛⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⢰⣿⠁⠀⠀⠀⠀⢹⣿⡻⣷⣳⡄⠀⠀⠀⠀⣠⣿⠏⠈⠙⠻⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⣯⠀⠀⠀⠀⠀⢸⣿⠁⠈⠛⠾⣷⣖⣾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣤⣤⣤⣴⡿⠋

⠀⠀⡇⠀⠀⠀⠀⢠⡇⠆⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸⢸⡇⡆⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣏⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡟⠀⠀⠀⠰⢀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠁⠀⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠘⠀⠀⠀⠀⣿⠀⠰⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡷⠀⠀⠀⠀⠀⡇⠀⠀⠀⡀⢸⡇⠀⠀⠀⠀⢸⡇⠐⠀⠠⢢⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣧⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠘⠀⢠⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡯⠠⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠇⠀⠎⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣧⠐⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⡆⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⡄⠀⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⡇⠀⡆⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠠⠀⣿⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠁⠀⠇⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠂⠀⡇⠀⠀⠀⠀⠀⡇⢠⠀⠀⠤⢸⡇⢠⠀⠄⠀⢸⡇⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣇⠀⠀⠀⠀⠀⡇⠘⡀⠀⠀⢸⡇⠘⠀⠀⢰⢸⡇⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠀⣿⠀⡎⠀⢸⠀⣿⠀⡆⠀⠀⠃⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⠀⠇⠀⢸⠀⣿⠀⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠆⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⢰⡇⠀⠀⠀⠀⠀⡇⢀⠃⠀⡆⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⣼⡇⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⢸⡇
⠀⠋⡇⠀⠀⠀⠀⠀⡇⠸⠀⠐⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢸⡇
⠀⠀⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡇⢐⠀⠐⠀⢸⡇⠀⠀⠀⠀⠀⡇⢰⢀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠀⢀⠀⠀⠀⢸⡇
⢀⡀⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣇⠀⠀⠀⢰⠀⣿⠀⡄⠀⠀⠀⢸⡇⠃
⢸⠃⡇⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⢸⡇⢠⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡟⠀⡀⠀⠈⠀⣿⠀⠱⠀⠀⠀⢸⡇
⢸⡇⡇⠀⠀⠀⠀⠀⡇⠄⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⣿⠀⠀⠀⠀⠀⣷⠀⠇⠀⠀⠀⣿⠀⠀⠀⢀⠀⢸⡇
⠨⡁⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⢸⡇
⠀⡟⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⣿⠀⠀⠈⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⢸⡇
⠀⡇⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣯⠀⠀⠀⠐⡆⣿⠀⠀⠀⠀⠠⢸⡇
⠀⠃⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠰⡆⠀⠈⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠔⠒⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣀⠀⠤⠔⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⠁⠀⣀⠠⠄⠂⠀⠉
⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⠀⠀⡀⠔⠂⠉
⢀⡀⠤⠀⠒⠀⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠒⠈⠁⠀⠀⣀⠤⠒⠁
⠀⠀⠀⣀⠀⠀⠀⠤⠄⠐⠒⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⠊⠁
⠀⢀⡀⠤⠠⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠚⠋

⠀⢸⡇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⢆⠀⠀⠀⡄⠀⠀⡜⠀⠀⠀⢻⣷⡁⣦⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸
⠀⠸⠁⠀⠀⢸⠀⠃⠀⠀⠀⠀⠐⠁⠈⣆⠀⢄⠀⠀⠈⢧⡢⠀⠀⠇⠀⠀⣸⢰⠀⠀⠸⣿⡧⢸⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⡀
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢀⣤⡘⣆⠈⠀⠐⡀⠈⠳⣔⣔⢸⠀⠀⢹⠀⠁⠀⠀⣿⠀⢸⡄⠰⠛⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠁⠸⠀⠰⠀⠀⠀⠀⠺⠈⣧⣹⡆⠀⢲⣾⢿⣦⠘⣯⡿⠀⢠⢸⡆⢀⠀⠀⠘⢧⣸⣷⡀⢢⡀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⠁
⠀⠀⠈⠀⠀⠀⡀⢀⠀⠀⡀⠀⡀⢰⠿⣯⢸⣷⣠⠀⠙⠎⢻⡀⠈⠓⠂⠀⡌⢷⡄⠂⠀⠀⠈⡇⠻⣇⢈⠇⠀⣼⣿⠀⠀⠸⠀⠀⠀⢸⡇⠀⡞
⠀⠀⠃⠀⡄⢠⠃⠆⠀⢰⢰⡟⢠⢁⢿⣿⡇⢻⡽⣧⠀⠀⠐⡟⣧⠀⠀⠀⢻⣎⢧⡀⢀⢀⠀⡇⠀⢻⢸⡇⢰⢿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢰⠇⡎⠰⠀⠀⡜⣾⠃⠀⢇⢲⣿⣇⠈⡏⢘⣷⠀⣄⠘⣮⣣⠀⠀⠀⢻⣦⣳⣤⣀⡀⠀⠀⠘⣯⠁⣟⡾⠻⠀⡄⢠⠀⠀⠀⢸⡇
⠀⠀⠀⣸⠰⠀⠂⠀⡕⡇⣿⠀⠘⠈⣹⣿⣿⡄⢿⡌⠙⢀⣿⠀⢸⣿⠇⠀⠀⢄⢻⡿⠹⣿⣷⠠⠀⣄⠹⢰⣿⡇⠀⠀⠁⢸⠀⠀⠀⢸⠀⢸⢠⡀
⠀⠀⠀⣿⡆⠀⠆⢀⢷⢻⣸⡄⠐⡈⣿⣿⡟⡇⢸⢃⠆⣼⣿⠀⣸⢹⢃⠄⠀⠀⢢⠣⠀⢹⣿⡆⠄⠈⣇⠸⣿⣷⠀⠀⠀⠘⠀⠄⠀⣸⠀⡇⢸
⠀⠀⠀⣿⡇⡀⢸⠈⠜⣎⢧⡿⡃⢣⣻⢻⠟⠦⣸⡾⠀⣿⡿⠀⠏⠀⡈⠀⢠⣦⡀⢳⡄⠘⣿⣷⢸⠀⢸⠀⢿⣿⡄⢄⠀⠀⡇⠀⠀⡏⢠⣇⢘⠃⠀⠀⢀
⠀⠀⠀⡇⣇⠹⣄⠀⠀⠈⠳⠁⠘⠀⢻⣦⠇⢩⢻⡇⣸⠏⠀⠎⠀⡀⠀⠀⡄⣿⣷⠈⣷⠀⢻⣿⣾⡇⠀⡄⠘⣇⣇⠈⠃⢠⣿⡀⠀⡇⢸⢹⠸⠀⠀⠀⢘
⠀⢨⠀⠂⢹⠀⠈⠓⠀⠀⠀⠑⡀⢀⠀⢬⠀⢫⡟⠀⣿⠀⠈⠀⠀⠀⠀⠈⠀⣽⣿⠀⢸⡆⢸⣿⣿⡀⠀⢧⡀⢻⣾⡆⠀⠀⢿⣧⠰⠇⠸⢸⣰⠀⠀⠀⢸
⣀⠀⢃⠀⣼⠀⣄⠀⠀⠀⠀⠸⠀⣆⢂⠀⡰⢻⡏⢰⠃⠀⢀⠀⠂⠀⠰⠀⠀⢸⣿⠀⠀⣷⡄⣿⡟⡇⠀⠀⠱⣄⢻⣿⠰⡄⠀⠻⡄⡄⠀⢸⣿⠀⠀⠀⣸
⢙⠀⢠⠀⠟⢠⡙⣆⠀⢰⠀⠀⠰⡘⣆⠳⠆⡟⠀⠀⠀⢴⡟⢰⡠⠀⡇⠀⠀⢸⣿⠀⠀⢹⡇⢻⡇⠰⠈⡄⠀⠈⢆⠹⡄⠙⢄⠈⡐⠀⠀⣾⠈⠀⠀⠀⡇
⠀⠀⠸⠀⠘⢸⢻⠸⡆⠈⠁⠀⠀⠀⠘⢆⢸⣅⠀⣷⠀⠘⣷⣸⡇⠀⠁⠀⠀⠀⡿⠀⠀⢸⡇⠈⣇⠀⠀⠀⢄⠀⠘⡄⠈⠄⠈⠆⢡⠀⠀⣿⡀⠀⠀⢸⠃
⠀⠀⠀⡄⢣⢸⡆⠀⢻⡔⡄⣌⣆⠀⠀⠀⠙⠟⣆⢿⠀⢃⢻⢻⡇⠀⠀⠀⠀⠀⣇⠀⠀⢸⡇⠀⠘⠀⠀⣴⠈⢄⠀⠃⠀⢸⠀⠀⠀⡆⠀⢿⢧⠀⠀⠰
⠀⠀⠀⢳⠀⠘⣷⠀⠸⣷⢷⡌⠘⠄⠀⢀⠀⠐⠀⠸⠀⢸⣾⡏⡇⠀⠈⠀⠠⡁⣿⠀⠀⠸⣧⠀⠀⠀⠀⢹⡇⠈⣆⠀⠀⠀⡄⠀⠀⣇⠀⢸⣾⠀⠀⠂
⠀⡇⠀⠸⡆⡀⢻⡆⠀⠹⣧⢻⡀⠀⠀⠠⠀⠀⡇⠀⠀⠨⣇⢻⡁⠀⠀⠁⠀⣿⣿⠀⠀⡇⣿⠀⠀⠀⠀⢼⣷⠀⢹⡀⠀⠀⠀⠀⠀⢿⠀⢸⣿⠀⠀⢠
⠀⢃⠀⠀⡇⢹⠘⣿⡄⠀⡹⡇⡇⠀⠀⠀⡀⠀⢡⠀⢀⡀⠉⠃⠁⠀⠀⠘⡀⢸⣿⡆⠀⢧⣿⡄⠀⠀⠀⢸⣿⡆⠸⡇⠀⠀⢰⠀⠀⢸⡆⠈⣿⣇⠀⠀⠄
⠀⢸⠀⠀⣷⠘⡆⠘⣷⠀⡇⠇⠰⠀⠀⠀⠀⠀⢸⠀⠘⢷⡀⠀⡀⠀⠀⠀⢃⠀⢻⡇⢀⢾⣿⠇⠀⠀⠀⢸⣿⣷⠀⢷⠀⠀⠘⠀⠀⡄⣧⠀⠸⡿⡄⠀⠘
⡇⠀⠀⠀⠸⡄⠹⡄⠈⠀⠐⢦⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⢷⡀⠘⡆⠀⠀⠀⡆⠈⠁⣸⢸⣿⠀⠀⠀⠀⢸⣿⣿⠀⠸⠀⠀⠀⠀⠀⢣⣿⠀⠀⢳⣃
⣇⠀⠀⠀⠀⢱⠀⢹⡄⠀⠀⠐⣆⠠⠀⠀⠀⠀⠀⡇⠀⠀⠘⢷⠀⢸⠄⢀⠀⠀⢠⣴⡇⣾⡿⠀⠀⠂⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠂⣿⠀⠀⢸⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠲⢾⣿⣦⣀⢈⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣌⠙⠿⣿⣷⡤⠝⠛⠉⠁⠀⠠⠀⠁⠊⠒⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀⠲⢿⣷⣦⣈⠛⠿⠛⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢉⡙⢿⣿⠶⠄⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⣀⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⡜⠅⣤⣤⡄⢀⣶⣶⠂⣰⣶⡖⠀⣶⣶⡖⠀⣰⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡌⠡⠽⠿⠏⠠⠾⠿⠁⠴⠿⠏⠀⠼⠿⠏⠀⠼⠿⠏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⣀⣀⣀⡀⣤


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠂


⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠒⠒⠋⠉⠁⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠜
⠀⠀⢠⡄⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠏⠐⠊
⠀⠀⢸⠈⠁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠛⠁⠀⠀⠀⠀⣀⠴
⣀⣤⣼⣦⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⠴⠊⠉⠈⠀⠀⠀⠀⠀⣠⠔⠊⠁
⡇⠀⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⢧⠀⠀⠀⠘⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠂⠀⠘⠉⠀⠀⡀⠀⡀⠀⢀⡤⠖⠋⠁
⢸⡀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉
⠀⠓⢄⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉
⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⠀⠐⣂⣠⠶⠊⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠈⠙⠷⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⢀⠀⠀⣀⡤⠖⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣀⠠⣶⡦⠒⠋⠋⠉⠉⠉⠉⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠴⠒⠋⠁⠀⠀⠀⠀⠀⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡠⠄⡲⣒⣀⠀⠀⢀⣀⣀⣀⠀⢹⣿⡆
⠀⠀⠀⠀⢠⣿⣿⣠⡄⠀⠤⠀⡠⠶⠟⠃⠨⠭⣑⣷⣄⣬⣊⣠⣶⣭⣬⣉⠒⡦⡤⠤⠤⠤⠼⣿⣷
⠀⠀⠀⠀⣾⣿⠃⠀⢀⣌⣥⣤⣶⣶⣾⣿⣿⣿⣿⡿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣦⣤⣤⡤⢿⣿⡆
⠀⠀⠀⢰⣍⣿⣿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠙⠛⠶⠛⠛⢋⣿⠉⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠓⠒⠉⢷
⠀⠀⠀⠀⠀⣠⡴⠾⠛⣉⠉⠉⠛⠳⡐⠀⠀⠸⣆⣀⣀
⠀⠀⠀⠀⣼⢋⡐⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠬⠍⢉⣉⠉⠛⠓⠲⠶⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⡄⠘⠁⠀⠠⠀⠀⠈⠐⠢⠀⠈⠀⠀⠀⠀⠒⠂⠀⠀⠄⠈⠉⠉⠉⠉⠩⠍⣙⡿⠛⠛⣶⢦⡀
⠀⠀⠀⢸⡇⠀⠤⠀⠀⠐⠒⠒⠀⠔⠲⠤⠴⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡠⠀⣀⠏⠀⠙⣆
⠀⠀⠀⠀⣿⣒⠤⠤⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠉⠀⠀⠒⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⡜⡴⢁⠄⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠘⣿⠛⠻⠝⢒⠂⠀⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⣀⣀⡀⠀⠀⠀⠀⢀⡏⢠⡞⠀⠄⠰⢀⠆⠀⢸⠇
⠀⠀⠀⠀⠀⠈⠓⠶⠦⣤⣄⡀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⢸⠃⡛⢠⠸⠃⠂⠂⢸⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠦⠤⣤⣀⣈⣉⠀⠒⠠⠤⠤⠀⠀⠠⠤⠄⠀⠀⠐⠒⠢⠤⢼⡀⠇⠌⡸⡧⢸⢸⢈⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⢴⣤⣀⣀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠸⡇⠀⠀⠡⡠⠇⣸⡖⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⠦⢤⣀⣀⠀⠀⠀⠐⣧⠐⠌⠠⠤⡨⠜⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠦⠬⣷⣦⣬⣥⡴⠞⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠴⠖⠒⠛⠛⠛⠋⠙⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠒⠚⠋⠉⠉⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠒⠛⠛⠉⠉⢁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠒⠢⠌⠳⣟⢦⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⡠⠀⡰⠇⠉⠉⠲⡄⢨⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠰⠁⡎⡠⠀⠀⠈⢳⡘⠀⡈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣀⣀⣤⠴⠿⠶⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠀⡇⠀⢉⠇⠀⣃⠀⡇⣿⡇⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠒⠛⠉⠁⠀⠀⢀⡠⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⡄⠀⠳⡀⠀⠀⢀⡜⡸⢡⣿⠇⠀⠀⣀⣀⠀⠀⢀⣀⣠⡤⠶⠚⠋⢉⠥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠙⣌⠲⠮⠭⠴⠋⠴⣡⣿⡟⢀⣀⣬⡥⠶⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸
⠀⠀⠀⢀⣠⣴⣖⣋⣉⣻⣷⣈⠓⠂⡄⠀⣀⣴⣿⣿⣾⣯⣭⣤⣠⣤⡤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⣼
⠀⢀⣴⣻⢿⣟⣛⣛⣛⠻⢿⣿⣿⠾⠿⠿⢿⣿⡽⢒⣋⣉⣉⣹⠿⣗⢦⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⢸
⢀⣾⢏⢕⠽⠗⠒⠂⠀⠙⠆⠙⣿⣦⠖⣰⡿⢻⠞⠁⠀⠒⠦⣍⠀⠈⢳⣳⣤⠄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⣼⡇⠀⠋⠐⠉⠙⢆⠀⣀⣀⠀⠘⣿⣦⣿⠁⠁⡀⡴⠊⠉⠑⡎⠆⢀⠀⣯⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⣀⣠⡴⠟⠁
⣿⡇⠀⢠⡀⠒⠀⠈⢈⠇⠀⡄⠀⣿⣿⣿⠀⠀⡇⠇⠚⠱⢀⡟⠀⢸⡆⣿⢸⠀⠀⠀⠀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠈⠁⣀⣠⡤⠶⠚⠛⠉
⢿⣇⠈⢀⠁⠀⢀⡠⠊⠀⡸⠀⣰⢿⠟⣿⡀⢫⡁⠀⢀⣀⣉⠄⠀⣸⠁⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡤⠴⠚⠋⠉
⠈⢿⣦⠊⠀⠈⠉⢀⣠⠞⢅⣰⣣⡏⢀⣽⣷⡀⣙⠲⠤⠄⠀⠀⠀⡠⢰⢧⡏⠀⠀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠻⢷⣤⣀⣉⣉⣤⣴⣺⣿⡿⠟⠉⠀⠹⣿⣮⣓⠂⠀⠀⠀⣈⣽⣿⠿⣖⣠⡤⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠒⠲⠶⠶⡤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠤⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠘⣧⡞⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢹⣿⠋⣇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢿⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⢸⡈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠈⡇⢿
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⢳⢸⡄
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⠀⠘⡇⣇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡏⠀⣻⠀⠀⢹⣹⡄
⠀⠀⠀⠀⢠⣟⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣾⣿⡄⠀⠀⢷⢷
⠀⠀⠀⢠⣤⣼⣏⣉⣉⣛⣓⣒⣶⣦⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠈⣧⣀⣀⣘⣟⣧
⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠿⠿⠭⢭⣟⣛⢛⣿⡆⢻⠀⠀⢿⠀⠀⠉⠉⠁
⠀⠀⠀⠀⣹⠿⣿⡿⣿⢹⣹⡏⠉⢿⠛⠒⠲⠶⠤⢤⣤⣀⣀⣸⣿⣿⠇⣇⢸⡄⠀⢸⡄
⠀⠀⠀⠀⡟⠀⣼⢀⡇⠈⣧⣷⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢻⠈⣇⠀⠀⣧
⠀⠀⠀⢸⡇⠀⡏⢸⠃⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢨⡏⣿⠀⠸⡆⢻⡀⠀⢹⡀
⠀⠀⠀⠈⠉⠛⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⠇⡟⠀⠀⠳⠼⣧⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⢼⣤⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢠⡟⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣿⣠⠴⠖⠋⠉⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⢉⣁⣤⣭⣉⣉⣛⠿⣿⡛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠞⠀⠀⠀⠸⣇⣠⠀⠀⠒⠉⠁⠀⠈⠳⣄
⠀⠀⠀⠀⢀⣴⢯⡀⣠⠞⠋⠉⠀⠀⠀⠉⠉⠻⣦⠙⢷⡔⠒⠒⠒⠂⠀⠠⠤⠤⠤⠀⠀⠒⠒⠒⠂⠀⠈⠛⣷⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⢀⡞⠁⢀⡽⠋⠀⠀⠀⠀⣀⣀⡀⠀⠀⠘⣷⡀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠄⢿⡀
⠀⠀⠀⢸⠇⢠⡏⠀⠀⠀⠀⠀⡘⠀⠀⠘⡆⠀⠀⠘⣷⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠐⠒⠒⠒⠛⠉⠉⠉⠉⠉⠁⠀⠀⢀⣠⡞⠁
⠀⠀⠀⢸⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⡇⠘⣷⠒⠛⠋⠉⠉⠉⢉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢠⣴⠶⠒⠒⠊⠉⠉⠉⠉⠉⠉⠛⢶⡀
⠀⠀⢀⡼⠃⡟⠀⠀⠀⠀⠀⣀⠤⠦⣄⠀⠸⡆⠀⠀⠀⣿⠀⠹⣟⠉⠉⠉⠉⣉⣁⣤⡀⠀⠀⠀⠀⠀⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⣾⠁⢸⠇⠀⠀⠀⢀⠞⠁⠀⠀⠘⡄⠀⣧⠀⠀⠀⢹⡄⠀⣿⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡀⣸⡆⠀⠀⢀⠏⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡏⢹⡇⠀⠀⢸⠀⠀⡄⠀⠀⣰⠃⠀⡟⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠖⠛⣿
⠀⠀⢸⡇⠘⡇⠀⠀⢸⡀⠀⠧⠤⠖⠃⠀⢀⠇⠀⠀⠀⢸⡇⣸⠇⠠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠒⠒⠚⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⡿
⠀⠀⠀⣧⠠⣷⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣾⠁⣿⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠖⠂⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠤⢴⡇
⠀⠀⠀⢹⡆⠸⣇⠀⠀⢱⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢰⡏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⣉⣉⣁⣀⣤⣤⡤⠴⠶⠒⣾
⠀⠀⠀⠈⣷⠠⢻⡄⠀⠀⠳⠤⠒⠋⠁⠀⠀⠀⠀⣠⠏⢤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⠉⠉⠉⠉⢉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢹⡄⠀⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣤⠞⠓⠒⠒⠒⠒⠒⠒⠛⠛⠉⠉⣉⣉⣡⣤⠤⠔⠒⠊⠉⠉⠉⠉⢉⣀⣠⡤⠶⠖⠚⠛⠛⠁
⠀⠀⠀⠀⠈⢷⣀⠀⡏⠛⠶⢤⣤⣄⣠⣤⠶⣋⡴⠿⠥⠤⠤⠀⠀⠐⠒⠖⠒⠛⠛⠛⠉⠉⢉⣀⣀⣤⣤⡤⠶⠶⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣴⣤⣤⣤⣶⣚⣉⣀⣀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠶⠖⠚⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⣫⣿⠿⣻⣿⡿⢿⣿⣿⣿⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠟⣋⣭⠶⣛⡭⢖⣫⡭⠖⠠⣚⡵⢚⣵⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢛⣩⡵⠾⣛⣭⠶⣛⣵⣾⣿⣡⡤⠶⢛⣭⣾⣿⡿⠛⠁⠠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⢛⡩⠔⢚⣩⡵⢖⣋⡥⠖⣛⢉⣈⣩⡭⠶⢒⣻⣽⡿⢿⠵⠂⣀⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⢿⠋⠁⠠⢚⣡⡴⢞⣫⡴⢞⡫⢕⣚⣍⣙⠋⢉⣀⣘⣿⣿⡟⠉⢀⣠⣴⡿⢟⣫⣵⣿
⠀⠀⠀⣀⣤⣶⣿⠟⢋⣡⠞⠁⢀⣴⠾⣋⣵⠞⠫⢕⣪⡵⢚⣭⣬⡭⠔⣀⣼⣿⠟⣛⣭⡴⠾⣛⣭⠶⢞⣿⣿⠟⠁
⣤⣶⣿⣿⣯⣿⣋⣴⡋⢀⣠⢞⣭⠶⡛⠉⣠⣴⣾⠟⣫⠾⣋⣥⣶⣶⣿⣿⠿⠛⠛⣋⡥⠖⠛⣩⣴⣾⣿⠟⠁
⣿⣿⠽⢟⣛⣛⣿⣿⣿⡿⠿⠿⢿⣿⣶⣿⣿⣿⣥⣾⣵⣏⣩⣾⠿⠛⠉⠀⢀⣤⡾⠋⢀⡴⢋⣵⡿⠋
⣿⣟⢉⣀⣤⣶⠶⠶⠶⣾⣭⣤⣌⣉⣉⣉⣉⣙⡛⠻⠯⣻⣿⣁⣤⣤⣤⣴⠿⠋⠀⠰⢋⣴⡿⠋
⣿⡿⢿⣿⠁⠠⣶⣾⡷⢖⡺⠭⠙⠛⠛⠋⣙⠛⠻⢷⣦⣸⣿⠟⠛⠉⠁⢀⡠⠔⣠⣾⡿⠋
⣿⣷⢤⣉⠳⢤⣈⠻⣷⡙⢏⠩⣤⠤⠤⢤⣌⠉⠀⢀⠀⢹⣿⡤⠴⠒⢉⣡⣴⣿⠟⠉
⣿⣿⠶⣭⣛⡢⠍⣓⠦⣝⡳⠥⢄⡀⠉⠉⠵⠋⠴⣋⡴⢻⣿⣠⡴⠞⣫⣿⠟⠁
⢿⣿⣿⣶⣿⣿⣿⣮⣽⣶⣍⣙⣓⠶⠶⠶⠶⠶⢛⣁⡤⢼⣿⢁⣴⣾⠟⠁
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⢿⣿⣷⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⣛⣛⣛⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠤⠤⠤⠤⠶⠶⠋⠀⠀⠀⠉⠁⠠⣧⣤⡤⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠉⠉⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣴⣁⡀
⠀⠀⠀⠀⢀⣤⣶⣿⣧⣀⣀⣠⣾⣿⡶⠶⠶⠶⠒⠒⠒⠒⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢰⡿⠉⠀⠙⣿⡟⢛⣿⠟⠙⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⠁⠀⠀⠀⢸⣿⣾⠃⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⢿⢧⠀⠀⠀⣸⡟⣿⣆⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠋
⠀⠀⠀⠀⠻⣷⣤⣴⡟⠁⠈⢿⣦⣤⣾⠏⣀⣀⣀⣀⣠⣤⡤⠤⠤⠶⠖⠒⠒⠚⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⣿⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⢳
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣠⣟⣒⣒⣒⣒⣒⣒⣚⣓⣒⣖⣒⣒⣒⣒⣒⣒⣒⣲⣜⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⣄
⠀⠀⠀⠀⢰⣿⠭⠵⠖⠲⣌⡷⣄
⠀⠀⠀⠀⢸⢫⠄⠀⠚⠛⢻⣇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣿⠀⠞⠂⣉⣉⠙⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋⠉⢁⣀⣀⣀⣠⡴⠊
⠀⠀⠀⠀⡇⠀⠀⠐⠻⣭⡼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⡇⢰⡂⠀⠀⣿⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣀
⠀⠀⠀⠀⢷⡈⠙⢛⣯⡵⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠰⣫⠿⠭⠋⠁
⠀⠀⠀⠀⠈⣇⢀⠀⠈⠘⣆⠻⡄⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠙⠛⠾⣷⡶⣄⠀⢤⡀
⠀⠀⠀⠀⠀⣿⠀⢷⡀⠀⠈⠃⠹⣦⡀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢷⡀⠹⣦⡀
⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠈⠻⣦⡿⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡈⠻⣦⣀
⠀⠀⠀⠀⠀⣿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠈⠻⣶⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠳⣄⠉⠛⠛⠒⠒⠒
⠀⠀⠀⠀⠀⢻⡰⡀⠘⣽⢦⠀⠀⠀⠀⠀⠀⢹⡍⢷⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⡇⣸⠀⠛
⠀⠀⠀⠀⠀⠘⣧⠫⠀⣿⠙⠿⠟⠂⠀⠀⠀⢨⣷⢾⠠⢤⣤⣤⣤⠞⠋⣠⣴⠄⠀⣧⡏
⠀⠀⠀⠀⠀⠀⠸⣆⠀⠹⣆⡀⠀⠀⠀⠠⠴⠚⠸⡇⠀⠀⠀⠀⣀⢄⡰⡝⠁⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠨⡉⠛⠛⠒⠒⠀⠀⠘⢁⡼⠀⠀⠀⠀⠑⠒⠺⠀⢀⣼⠋⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠘⢦⣀⣀⣀⠀⠔⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⡴⠟⠁⠀⠈⠙⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣉⢁⣠⡴⠶⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠒⠂⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠋⡀⠀⠐⠖⠊⠀⠀⠀⠀⠀⠀⠀⠤⢖⠆⠐⠠⡀
⠀⠀⠀⠀⠀⠐⠁⠀⡀⠐⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡀⠀⠁
⠀⠀⠀⢀⡀⠧⢉⡌⠣⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠤⠒⠉⠉⣈⡤⠀⠀⡄
⠀⠀⠂⠚⠑⠋⠀⠲⠂⠰⠀⠀⠀⠀⢀⣠⡄⠀⠀⢀⡀⠀⠀⠀⠀⣀⠤⡄⠀⠉⠁⢀⣬⢗⡋⠁⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⢚⢴⠂⠊⠁⠀⠀⠀⠠⠛⠙⠀⢀⡠⣔⠮⠓⠊⠁⠀⠀⠠⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣦⣤⠀⣠⣀⠉⠀⠀⠴⠓⠀⣠⠔⢊⣥⡖⠚⠉⠈⡅⠀⠁⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠨⣬⠽⠀⣌⠉⡧⠀⠀⠀⢀⣨⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⣴⣋⡩⢶⡄⢴⡃⠉⡛⠓⣀⣤⣾⠿⠋⠀⠤⢔⡊
⠀⠀⠀⠀⢠⠃⣈⡀⠀⣈⠷⢤⣤⣿⣶⡽⢀⢚⡥⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠈⠀⣡⣠⠀⣄⠀⣿⡼⠿⠅⢼⠌⠉⠃⢘⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠄⣙⣖⢊⣠⠼⠇⠰⡁⣘⡴⠃⣠⢬⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠹⣽⡐⣫⠷⠊⠈⠀⠀⠀⠀⠀⠀⠀⢘⡡⠔⠊⠁⠀⠀⣀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣔⣀⡂⢠⠴⠚⠉⠀⣀⠤⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⢀⡴⢋⣱⡸⣍⠀⠉⠀⣀⠤⣶⠿⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⣩⠌⢠⠞⣸⣦⣤⣤⠚⢧⡄⠂
⠀⠀⠀⠀⠉⣧⣀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠨⣗⠰⡟⣗⣙⠋⠟⠁⠀⠀⢻
⠀⠀⠀⠂⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠈⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠒⣤⡤⠔⠊⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⣀⠤⠒⠋⠁⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⡥⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⡯⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡯⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡮⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣾⣿⠋
⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⡪⠓⣵⡟⠁
⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡪⠗⠉⣠⢪⠎
⠈⢪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡪⠕⠉⠀⢀⢔⡵⠁
⠀⠀⠙⢮⡢⡀⠀⠀⠀⢀⣤⡪⠑⠉⠀⠀⠀⡰⡣⠊
⠀⠀⠀⣔⡉⠪⣓⣄⡪⠑⠉⠀⠀⠀⠀⢠⢎⠜⠁
⠀⠀⠀⠑⠪⢕⡪⢍⡀⠀⠀⠀⠀⠀⠔⡵⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠬⢁⡒⠤⡤⡪⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣽⣿⣿⣿⣦⡀⢀⣾⣿⣿⣿⣿⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣟⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⢧⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⣟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣝⣿⣿⣿⣿⣻⣿⣯⣵⣶⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣽⣿⣿⣷⣝⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣾⢿⣿⣿⣿⣿⣿⢻⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡷⠶⢟⣼⣿⣿⣭⣭⣭⡾⢿⣾⣻⣶⣿⣽⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⡇⣿⣿⣿⣯⣿⢧⣿⣿⣻⣿⣿⣿⡿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⣻⣵⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣿⣻⢿⣾⣽⣿⣿⢋⣾⣿⡿⣿⣿⣿⣿⣴⢿⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣎⣿⣿⣿⣶⣯⣿⡷⣾⢯⣿⣿⣿⡎⣿⣇⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣿⣷⢿⣧⣟⣻⣿⡿⣯⣾⣷⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣞⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣽⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣯⣽⣟⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⢰⢖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⢲⡆
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢍⠻⣸⡇
⠀⠀⠀⠀⠀⠀⠸⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠏⠉⠉⠹⠷⠦⡀⠪⡑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠠⠀⠤⠌⠒⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⠄⠈⠄⠀⠀⠀⠀⠀⠀⠀⡾⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠌⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠈⡀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⠉⠈⠉⠟⠀⠀⠀⠀⠈⠀⡀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠉⠀⠀⠀⠀⠀⠀⣠⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢀⣤⣤⠀⡆⠀⢀⠔⠀⠀⠀⠀⠔⣠⣤⣄⢠⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⡑⠛⠀⢀⣤⣾⣦⣠⣶⣦⡀⠀⠙⢂⣿⡆⠄⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠙⠻⠶⢿⠟⠋⠉⢈⡁⠉⠙⠻⡿⢶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠂⠄⠜⠣⠀⠄⠊⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣈⡄⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠘⠛⠛⠛⠢⠀⠀⠀⠀⠀⠀⠀⡀⠘⠛⠛⠛⡄⠀⠀⠀⡀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣀⣀⣀⣀⣀⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⠛⠛⠛⠛⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠸⠉⠉⢹⠀⠯⠉⠉⠽⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠠⡀⠀⡈⠀⠰⠀⠀⡄⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣷⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣠⣄⣠⣄⣀⣀⣥⣀⣇⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠸⠟⠋⠁⠀⠀⠀⠈⠙⠻⠷⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠟⠛⠋⠁

⣿⠉⠉⠉⠉⠩⠩⠉⣉⣉⣉⣋⡉⠉⠛⠛⠉⠙⠉⠉⠉⠉⠋⠩⢝⠫⠭⠍⠉⠉⠭⠭⠩⢭⣭⠭⠽⠉⠛⠻⢿⣿⠿⠟⠉⠍⠽
⣿⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠤⠤⠀⠀⠀⣀⠠⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠐⠀⠒⠂⠐⠀⠀⠀⢀
⣇⠀⠀⠰⠦⠤⠬⠁⠀⠀⠀⠀⢠⠀⠀⠒⠒⠂⠀⠭⣭⣭⣀⠀⠀⠀⠀⠐⠶⠶⣬⡄⠀⣀⡀⠀⠤⠀⠠⣄⣖⣀⣀⡀⠀⠈⣉
⣿⠃⠂⠀⠀⠀⠠⠠⠶⠖⠒⠂⢀⠀⠀⠀⠀⠀⠍⠉⠉⠉⠉⠉⠉⠉⠉⠀⠂⠀⠀⠀⠀⠒⠒⠠⠤⠤⠼⢤⢶⣶⡞⠓⠛⠰⠶
⣿⠉⠉⠬⠭⠭⠥⠾⠟⠛⠋⠉⠙⠭⠙⠣⠠⠭⠫⠭⠭⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠛⠛⠯⠉⠸⠥⠂⠀⠠⠾
⣷⠀⠀⠀⠀⠀⠀⠀⣉⠉⠈⠈⣉⣭⢱⣆⡀⡀⣀⠀⠀⣠⠀⠀⣀⣀⣤⣷⣆⣀⣀⣠⡀⢂⣀⡀⢀⣀⡀
⣿⠑⠚⠊⠋⠛⣿⠻⠛⠛⠙⠻⠍⠿⠾⠉⠭⠭⠉⠉⠉⠻⠟⠻⠉⠉⠉⠉⠉⠉⠋⠈⠛⠛⠉⠉⢙⣉⣁⣀⡀⠀⠈⣩⠉⠍⠉
⣇⣀⡀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠠⠤⠠⠀⠀⠀⠄⠈⠉⠀⢀⡀⠀⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣬
⣧⠤⠈⠁⠠⠤⠤⠭⠥⠤⠴⠶⠀⠠⢀⠀⠠⠽⠶⠒⠂⠀⣀⡀⠀⣠⣬⡽⠷⠒⠈⠛⠒⠀⠠⡤⠀⠀⠀⠀⠈⠭⠴⠖⠂⠀⢀
⣷⡖⠂⠉⠙⣻⣷⣶⣔⡷⠶⠄⠐⠛⠒⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠁⠉⠉⠉⠩⠀⠐⠀⠀⠀⡀⠀⠒⠲⠾⠄⠤⠀⠀⠀⠉⠉
⣿⠖⠂⠀⠒⠲⠒⠀⠸⠁⠀⠂⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠽⠛⠚⠒⠒⠚⠛⢙⠋⠉⠙⠻⠿⠿⠌⠽⠐⠒⠚⠛
⣏⡀⠀⢀⠀⠀⠀⣀⣈⣀⣀⣀⣀⣁⡐⢀⣤⣀⣀⣰⣾⣶⣀⣀⣀⣀⣆⡀⠀⣀⣠⢀⣴⡆⣬⣉⣁⢀⡀⣀⡀⠀⣀⣀⡀⠀⢀
⡏⠉⠀⠈⠉⠉⠁⠉⠉⠉⠁⣉⣉⡛⠐⠟⠁⠋⠉⠀⠈⠹⠉⠉⠉⠙⠟⣿⠏⠉⠉⠛⠁⡌⠉⠉⠉⠉⠁⠁⠉⠀⠀⠀⠁
⣿⣤⡤⠤⠄⣰⡲⢴⣦⣦⣄⣀⣨⣁⣤⡤⠤⠤⣤⣤⣂⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⣠⡖⠀⢴⠶⠔⠀⠐⢴
⣿⣀⠂⠀⠀⠐⠒⠒⠶⠶⠄⠄⠀⠀⠀⠀⠀⠀⣒⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⠀⠀⠀⠀⠦⠤⠀⠒⠒⢾⠝⠿⢿⣧⣤⣤⠤⠶
⡏⠀⠀⠰⠖⠒⢒⣀⠀⠀⠀⠀⠘⠀⠀⠤⠤⠄⠀⣖⣛⡛⠛⠀⠀⠀⠀⠠⠤⠶⣞⠂⠀⠁⠁⠀⠒⠒⠒⢒⡊⠁⠀⠀⠀⢀⣈
⣿⠟⠛⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠀⠐⠀⢀⠀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢀⡀⠀⠀⠉
⣷⣠⣰⣄⣀⣐⣀⣀⣀⣀⣈⣄⣀⣄⣴⣤⣀⣡⣁⣀⣀⣀⣤⣀⣺⣐⣒⣂⣀⣀⣒⣒⣐⡒⣗⣐⣶⣄⣤⣴⣶⣶⣶⣆⡀⠆⢰
⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠄⠰⠈⠉⠉⠉⠹⠿⠛⠁⠀⠀⠀⠃⠀⠀⠀⢀⡀⠙⢁⣉⣁⡄⠀⠀⡀⠀⠀⣀
⣿⣶⠂⠀⠠⢒⡆⣀⣲⣤⣤⠀⠄⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⠶⠶⠠⠒⠒⠲⠶⠖⡆⠠⠤⠶⠶⠶⠲⠗⠒⠒⠒⠖
⡿⠶⠆⣤⣤⣼⠿⠷⠓⠖⠒⠒⠒⠤⠤⠀⠀⠀⠀⡠⠀⣄⣀⣀⣠⣀⣀⣀⢀⣒⠂⠀⠀⠀⠀⠉⠁⠠⠴⠤⠄⠉⠀⠀⠠⠤⣤
⣟⣁⣀⠀⠈⠈⠍⢑⡒⠒⠀⠒⠀⠈⠈⠀⠘⡛⠶⠶⠄⠀⠀⠀⠀⠋⣛⣛⣒⠂⠠⠤⠤⠀⠀⠁⠀⠀⠀⠀⣀⡐⠒⠲⠆
⣏⠁⠀⠀⠀⡀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠉⠀⡀⠀⠒⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛
⣿⣄⣠⣠⣄⣴⣶⣶⣆⣠⣀⣶⣒⣿⣒⣂⣒⣒⣀⣀⣀⣒⣲⣕⣀⣀⣀⣀⣀⣀⣄⣀⣠⣤⣀⣈⣙⣉⣈⣀⣀⣐⣂⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠒⢉⣉⣭⣍⣉⣑⠒⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⢖⠉⠉⠀⠴⠄⠛⢻⡍⠑⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⡇⠸⣇⣙⣓⣒⣂⣀⣀⠤⢀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣶⣤⠭⣭⣭⣭⣭⡴⠲⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⠛⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣿⡇⣰⣉⣼⡏⣿⠀⢰⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡴⢶⣿⣿⡿⠛⠲⣆⠘⣇⠀⠀⠀⠀⠀⠀⠀⠸⢀⠂⠀⣿⡿⢣⢹⡿⢸⣿⠀⢸⣇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠰⣿⣶⣆⣹⣿⡶⣁⣴⠏⠀⣿⠀⠀⠀⠀⠀⠀⠀⣦⢸⠀⠀⣿⠀⢸⡆⣿⢸⡏⠀⣼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢙⣿⠿⡏⠉⢹⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⢀⣿⠀⡌⠀⢹⢸⡇⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣿⠀⢧⡀⢸⣿⡀⠀⡏⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⢸⡏⠀⠀⠀⢸⢸⡇⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠏⡿⠀⠈⢷⣼⣿⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡟⢸⠀⠈⠁⠀⠀⠀⣸⢸⡇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠛⢸⡿⠋⢳⡀⢿⣿⠀⣾⠀⠀⠀⠀⢀⣀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⡇⢸⠇⢀⡆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢸⡇⡀⠸⡇⣸⡟⠀⠟⠀⠀⠀⢠⡿⣿⣧⣸⠅⢸⠀⠀⠀⢰⠀⢸⡇⣼⠀⢸⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⡟⢱⡇⠀⢰⣿⡇⢰⠃⠀⠀⠀⠀⣿⡳⠈⣿⠄⢸⠀⠀⠀⣸⠀⠀⠀⣿⠀⡜⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠸⠃⠀⢸⣿⡇⢸⠀⠀⠀⠀⠀⠘⣷⣶⣹⠂⢸⠀⠀⠀⣿⠀⠀⠀⣿⢀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⡇⠀⠀⢺⠀⡄⢿⡇⡘⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⡆⠀⠀⠀⡿⠀⣇⠀⠇⣼⠀⡿⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣯⣌⣷⡆⠀⠀⢸⡇⢸⡇⠀⡇⢸⠀⡇⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⡇⠀⠀⢠⡇⢠⡟⠀⠀⡟⢠⡇⠀⠀⠀⠀⣰⣯⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⠛⢉⠿⣥⠀⠀⢸⡇⢸⠇⡀⡇⢸⡀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡇⠀⠀⢸⡇⢸⢁⡄⢰⡇⢸⡇⠀⠀⠀⢠⣟⡼⠟⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⡎⠻⣿⣆⠀⢸⡇⠸⠀⡇⣁⢸⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢸⠁⠘⠀⠇⠘⠃⢸⠀⠀⠀⠀⡼⠛⣴⠆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠶⠿⣿⢆⢸⡇⠀⠀⡇⣿⠘⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢠⢸⣶⡇⠀⢠⠀⠀⣾⠀⠀⠀⢠⠁⡸⡞⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⢻⡜⣾⡇⠀⠀⡇⣿⠀⠇⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸⠈⣿⡇⠀⠘⠀⠀⢸⡀⠀⠀⠎⣠⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⣮⠁⠀⠀⠃⠻⠞⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡆⠙⢀⢸⡇⠀⠀⢸⡇⢠⠖⢠⠋⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠸⠇⢠⡀⢙⠀⠀⣰⠁⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠘⠊⡧⠀⢸⡇⠘⠷⠋⣰⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⢸⣄⡴⠃⢀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢿⠀⢠⠀⠀⢰⣧⠀⠀⠜⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠇⠀⠉⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠸⠀⠘⠀⠀⣿⣿⡀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⠀⢂⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⡇⠀⠀⠀⠇⠀⡼⢡⡀⡇⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠋⡇⢸⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⣸⠃⢸⡇⢃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣴⣧⠘⠀⠛⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡇⠀⠀⠀⠀⡿⠀⢸⠇⢸⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠹⣿⠀⡆⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠀⠀⠀⠀⢸⠇⠀⠸⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣻⡆⢱⠀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⠀⠀⠇⠀⠀⡇⠀⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⡆⠈⣆⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⡄⠀⣾⠀⣴⡄⡇⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⡀⣿⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⡇⢠⡇⢸⠸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢦⠀⠈⡆⣿⠀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠘⡇⢸⠁⠈⡀⠀⠀⢀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢰⠀⠘⡆⠀⡇⠁⣸⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠸⠀⢃⣧⠀⠀⣼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡉⠀⠀⡇⠀⡇⢸⠇⡈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⢸⣿⠀⠀⢹⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠃⠀⠀⡇⠀⠃⠟⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⢸⣯⠀⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣦⠀⡇⠀⠸⡄⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠸⡄⠀⠘⣷⠀⢁⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⡇⠀⢸⠁⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⢳⡀⠀⣿⠀⢸⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠘⡆⠃⠀⠘⡀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠘⣇⠀⠁⠀⣾⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⢀⠀⠀⡇⢳⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠻⡀⠀⠀⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⡇⣿⠃⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠃⠀⠀⠀⡇⠀⠀⠁⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠆⠛⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣤⣤⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢠⠖⠂⠀⠀⠀⠐⠲⠶⠶⠤⠤⠴⠶⠶⠖⠒⠂⣀⡴⠒⠀⠐⠀⠀⠀⠐⠀⠂⠀⠐⠲⠦⣀⠀⠂⠀⠀⠀⠀⠂⠐⠲⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠤⠤⠤⠤⠄⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠛⠲⢄⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⢔⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠞⠛⠛⠛⠛⠻⠛⠟⠟⠛⠷⡢⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡿⡄⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠘⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣿⡿⡆⠀⠀⠿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠔⠁⠀⠀⠳⠤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢀⢾
⠀⠀⠀⠀⠀⠀⠀⡇⡞⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⠎⢸
⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡌⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠁⢠⢻
⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⡟⢸
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⠀⠀⠀⠀⢸⠒⠻⡄⠀⠀⢠⡶⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣹⣦⠀⠀⠀⠸⣤⠴⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠗⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢷⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⠉⠙⠻⢍⣉⠉⠉⠉⣉⡩⠟⠉⠉⢉⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣀⠤⠒⠛⠉⠉⠉⠉⠙⠓⠢⣄⡉⠂⢄⡀⠀⠀⠀⠀⠈⠑⢤⣀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣠⠎⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠤⠤⠤⠤⣐⣒⣒⣒⣒⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉⢹⠋⠉⠉⠉⠉⠉⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠀⠸⠦⠤⠤⠤⠤⠄⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡰⢡⣢⠜⣓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⢀⠜⢔⠏⠀⠯⠀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⠔⡠⡊⡀⠡⠂⠈⡘⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠐⠁⠠⢈⢌⣴⠇⢀⣨⣾⡽⠗⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⡠⠂⢀⠔⠕⠟⢁⣄⣴⣹⢟⠔⢌⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⡠⢔⠀⠊⠔⠁⠞⠄⠐⣽⡻⢚⠕⡡⣪⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠕⡡⠊⠡⡀⢠⠀⢀⠔⣡⣺⠞⢕⠕⢉⣞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡋⡡⠊⢀⠀⣠⠔⠒⠈⣠⢞⡷⠉⠈⢊⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢀⠈⢀⡴⣠⠞⡡⠒⢁⡼⡽⢊⡉⢡⡼⣯⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⠔⠁⣠⢿⠎⠁⢀⡀⣠⣮⢋⠀⠀⣠⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣂⠉⣠⠞⡑⠁⠀⣄⣡⢷⠟⡑⠁⢀⡜⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠳⢭⣛⠷⢴⣴⣁⣰⢞⡱⡫⢪⣄⡴⣻⠞
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⠟⠟⠾⡾⠷⡪⢈⣴⣊⢾⡵⠁
⠀⠀⠀⠀⠀⠀⢠⣎⣀⠀⠌⡠⠄⡌⡼⢠⠟⣡⡿⠋
⠀⠀⠀⠀⠀⠀⠘⠛⠻⠹⠘⠃⠏⠞⡞⢲⣾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⣀⣀⣤⣤⣤⣄⣀⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣯⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⡿⠿⠛⠋⠉⣉⣀⣭⠍⠉⡙⠛⠛⠿⢿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠋⢁⣀⣤⣶⣾⣟⣫⣥⣤⣴⣶⣷⣶⣶⣶⣤⣬⣉⠻⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⢁⣠⣶⣿⠿⣟⣻⣭⣿⣿⣿⣭⣤⣼⣧⠀⠠⢤⣀⡀⠈⠁⢀⡿⠁⢠⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠒⣫⣽⣿⣿⣿⣿⠿⢛⣋⣉⣉⣠⣤⣤⣯⣀⣀⠀⠈⠙⠻⣶⡾⢁⣴⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣬⣷⣤⠶⠛⣩⡽⢟⣫⣵⣶⠿⠟⠛⣉⣉⣭⣥⣽⣯⣭⣉⣛⠳⠆⠀⣼⣥⠟⢁⣾⠟⠁⠹⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠉⠙⢿⣿⣄⠀⠁⠒⣩⠟⠉⠀⠀⠒⠛⢋⣉⣛⢻⡿⠿⢿⣍⠉⠙⢃⣼⣿⡁⠀⠘⠁⠀⡀⢀⡘⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⡰⠀⠀⠈⠻⢦⡀⠘⠁⣠⠞⢁⣴⠾⠛⠋⠉⠉⢹⠛⠓⠦⠈⠉⢀⣾⠟⠉⠳⠰⣄⠀⡄⠸⡄⣷⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⢰⡇⠀⢀⠀⣰⠆⠉⠀⠈⣠⣾⠟⢁⡄⢀⡴⠚⠛⠉⠇⠀⡀⠀⠠⠋⠀⠀⠀⢠⡀⠘⣧⢸⡄⢿⣸⣇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⣼⢠⡆⣾⠀⣿⠀⠀⠀⣰⣿⠃⣰⠏⢀⡾⠀⠠⠶⢦⣄⠀⠘⢦⠀⠀⢀⣦⣤⡴⢷⡛⣿⣯⣿⣼⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⣿⣼⡇⣿⠀⣿⠀⠀⡰⢻⡇⣰⡯⠄⣾⠁⠀⢀⡀⢀⣿⣷⠀⢸⡇⠘⢏⠉⣷⠀⠈⣧⢸⣿⣿⣿⣿⣿⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢠⣿⣿⣿⣿⠀⢹⣧⢀⣥⣿⡏⢻⠀⡀⣿⠀⢘⣿⣿⡿⠟⠋⣠⡿⢠⡇⣼⠀⣹⢀⡇⢻⡈⣿⣿⣿⣿⣿⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣇⠸⡟⣿⣿⣿⣶⣾⣿⡋⣄⠸⣧⠀⠀⢧⠙⢧⣾⡁⠀⠀⣠⡾⢋⣴⣟⣼⠏⢀⡿⢸⡇⢸⡇⣿⣿⣿⢿⣿⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⡀⣧⢹⣿⣿⣯⠀⠻⣧⠘⣦⠘⢷⣄⡘⣷⡟⠉⠛⠛⠛⠹⢞⣿⣿⡿⠋⣠⡞⢡⡟⠠⣾⢧⣿⣼⣿⣼⣷⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⢹⡜⣿⣿⡟⣇⠀⠘⢷⡈⠳⠤⠉⣿⣿⣀⡐⠲⢾⠶⠚⠛⣋⣥⠴⠟⢁⡴⠋⣠⡾⠃⡼⠃⣿⢻⡟⠉⣹⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡌⢷⣿⡿⣷⡘⣦⠀⢀⠙⠀⢀⣼⡟⠀⢉⣙⠛⣿⠛⠛⠋⠉⠀⣀⠀⢉⣠⣾⠟⠁⠀⠁⠸⢡⡿⠁⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡌⢿⣇⠙⣷⡈⠣⠈⠳⣦⣾⠟⠀⢀⣈⠉⠙⣿⠛⠛⠋⣁⣠⣼⣷⠿⠋⠁⢀⡼⠁⠀⣠⠟⠁⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣆⠻⣆⠈⠛⢦⣠⣾⣿⣥⣀⠀⠀⠉⠛⠛⢿⠛⠛⠋⠉⣁⡀⠹⣦⣤⡾⠋⠀⣠⠞⠁⢠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣈⠓⣀⣾⣿⣅⠈⠙⠛⠷⠦⣼⣤⣀⣸⣇⠀⠙⠉⢁⠀⠀⠙⣿⣦⠀⠘⠁⢀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡏⠀⠉⠻⠦⠄⠀⠲⣾⠤⣤⣤⣿⡄⠀⠀⠘⣧⡀⠀⠈⢿⣷⣤⣾⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⢸⣿⣿⣷⣦⣀⡀⠀⠀⣰⡇⠀⠀⠀⣿⠴⠶⠶⠚⢻⢿⣤⣤⣶⣿⡍⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⢿⣿⣿⣶⣿⣧⣤⣄⣴⣏⣤⣴⣶⣶⣾⡇⠙⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⡿⠿⣿⣿⡿⠿⠿⠛⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣤⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡔⢣⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠛⢀⠀⠁⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⢤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠁⠈
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠀⠄⠀⠒⣴⣾⣆⠀⢰⡶⠠⠀⠠
⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠀⠀⢀⠀⡔⢌⠉⠀⣤⠑⠌⢡⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣶⣦⣀⡀⠘⠉⠀⠐⠀⠒⠀⠑⠋⡄⠀⠉⠀⡀⢀⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣸⡋⠉⡈⠛⠁⠂⠀⠄⠚⢀⠀⠀⡐⠁⠀⠠⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠄⢤⠓⣄⠒⠀⠠⣧⠟⠠⡇⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠠⣤⠀⠛⠃⠀⠀⠀⠃⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠁⠈⠜
⠀⠀⠀⠠⡔⠀⢀⠀⣀⠀⠄⠠⠀⠰⠛⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠠⠀⠁⠀⠀⠀⠀⡗⠀⠠⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠐⠐⠂⠀⠀⠀⢠⡃⠠⠀⡡⠂⠀⠀⠀⠀⠀⠀⢀⠄⡀⠠⢀⣀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠁⠀⢀⠀⠀⠀⠀⣸⢏⣡⣄⠤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠐⠀⠀⠀⢠⣬⠙⡆⠛⠹⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠀⠀⠀⠀⠉⠀⠀⠁⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣾⣿⣿⣿⣧⣤⣤⣤⣤⣤⣴⣾⣿⣷⣤⣄
⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠉⠉⠙⠻⠿⠿⠿⠿⠿⠛⠋⠙⢿⣿⣿⣷⣄
⢀⣶⣶⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡀
⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⢿⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠘⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⣀⠀⢰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⢿⣷⡄⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⡇
⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⣿⡇
⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⣿⡇
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣧
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⣿⣿
⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⡆
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⣿⡇
⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣇
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⢸⣿⣿
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿
⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⡆
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⡇
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⡇
⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⡇
⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢸⣿⡇
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇
⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣷
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⡀
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⣿⣿
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠺⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⡆⠀⠀⣿⣇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢹⣿⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡆⠀⣿⣿
⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡆
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣧⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⣷
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢸⣿⠀⠈⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⣿⡀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⡆
⠀⠀⠀⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⣷
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⢻⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⢀⣀⣠⣴⣶⣄⣀⣀⣀⣀⣀⣀⣤⣤⣶⣦⣤⣤⣴⣶⣶⣶⣶⣦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣦⣤⣀⣀⣀⠀⣼⣿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣼⠥⠀⠀⠀⠈⠑⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣿⠾⠃⠀⠀⠀⠀⠀⢀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣀⣴⣶⣶⡶⠶⢶⢶⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⣼⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠺⠛⠛⠛⠻⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠇⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣠⣴⠶⠿⠛⠷⠶⣦⣀⡏⠈⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⢰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠞⠛⢧⣄⣀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢀⡴⠞⠛⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠺⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣙⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⡠⢊⠔⡠⢀⣴⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠄⠔⡀⠀⠀⠀⠀⠀⠀⠈⠀⠙⡸⠋⠀⠀⠀⢀⠜⠄⢁⠀⣴⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢠⡤⠊⠀⠂⠉⠀⠀⠀⡀⠀⠀⣠⡐⠋⠀⠀⠀⠀⠀⡀⠀⠔⣡⣾⠟⢏⠀⡇⡆
⠀⠀⠀⡄⣀⡀⠀⠘⠛⠛⠿⠗⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⢰⠜⠁⠀⠀⣀⣀⣀⠀⠀⢀⣾⠋⢿⠀⣮⣀⡴⠃
⠀⠀⠀⡭⣽⣆⣿⣿⡇⣾⣶⡆⡀⠁⠀⠀⢠⠞⠛⠺⣤⢪⡾⠃⠀⠀⠀⠀⠙⠋⠛⠛⠻⠿⠁⠀⢸⣶⣿⢾⠿⢾⡿⠇
⠀⠀⠀⠛⠋⠃⠙⠛⠋⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣰⠟⠀⢀⣀⣀⡤⢐⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠬⠥⠐⣶⠒⠂
⠀⠀⠀⠀⢠⣴⡿⠀⠀⢾⣥⡷⠿⠛⠁⢀⣤⣏⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣥⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⣿⢰
⠀⠀⠀⢠⢴⡟⠁⠀⢀⡸⡟⠀⠀⣀⣴⣷⣟⡿⣃⠀⠀⠀⠀⠀⡆⠀⠀⢸⠩⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⡄⢨⣿⣩
⠀⠀⠀⢀⣿⠁⠀⢠⣸⢓⣦⣸⣶⣿⢾⡽⠃⣠⠏⠀⠀⠐⢄⠀⠃⠀⠀⠀⣗⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠅⢠⡿⢮⡄
⠀⠀⠀⣼⡏⠀⠀⠈⠇⠈⠁⠹⣭⣋⠑⠞⠋⠀⠀⠀⠀⠀⠈⢲⡀⠀⠀⣀⡖⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⣸⡞⠁
⠀⠀⠘⠥⢷⣄⠀⠀⠀⠀⠀⠀⣿⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⣹⣿⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠖⠀⠀⠀⠀⠀⠈⠀⠀⠈⢹⠓⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⠀⢚⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣇⠀⠀⠉⠀⠀⠀⠘⠀⠀⠃⢿⡀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠂⣯⣿
⠀⠀⠰⠀⠲⡆⢀⡀⠀⠀⠈⣿⡄⠀⠀⠀⡟⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣥⣨⠀⠀⠀⠈⣿⡇⠀⠀⠀⡇⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠒⢒⠒⠒⠒⠆⠠⢄⣀
⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⢽⣷⠀⠀⠀⡟⠀⠀⠉⠀⠀⠀⠀⠀⣠⠞⠉⡤⠶⠶⣾⠛⢛⣃⣞⡣⠴⠶⠀⠀⠑⢢⡀
⠀⠀⠈⠀⢠⣿⡗⠀⠀⠀⠀⢸⣿⠀⠀⢀⣥⠀⠀⠆⠀⠀⠀⠀⢰⣾⣄⡀⠤⢄⠴⠒⠊⡽⠋⢏⣉⣉⣩⡚⠂⠀⢀⡇
⠀⠀⠃⠀⠈⠋⠃⠀⠀⢰⠖⠋⠘⠀⠀⢸⣤⣄⣀⡀⠀⠀⠀⠀⠘⣿⣾⣿⣶⣦⣤⣀⣈⣀⣀⣉⣉⣁⢤⡤⣶⣾⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠋⠁⠀⠀⠀⠀⠀⠈⠙⠻⢿⡏⣷⣿⣿⣾⣶⣿⠯⡟⣾⡿⡭⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠋⠉⠉⠉⠙⠓⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣹⣀⣀⣀⣀⣀⣀⣀⣀⠀⠼⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⣀
⡖⠒⠒⠒⠊⠉⠉⠉⠉⠉⠉⠉⠀⠀⢠⢤⠀⠀⠀⢠⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠻⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀
⢸⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⢧⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⣶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⢸⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⢸⣄⣀⣀⣀⣀⣀⣤⣄⣤⣄⣄⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⣀⡇
⣶⣶⠶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⡉⠉⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠘⠉⠉⢉⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⠀⠀⣜⢀⡀
⠀⠀⠀⠀⠀⣀⣠⣤⣤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⡒⠉⠉⠉⠉⠉⡟⠀⡀⠀⡌⠀⢠⠯⠍⠉⠉⠉⠉⠉⠉⠛⠓⠦⣀
⠀⠀⠀⡠⠊⢁⡀⠄⢈⠑⢦⡒⠒⠒⠒⠂⠀⠀⠀⠠⠈⠒⠀⠀⠀⠈⠄⠊⠀⠠⠃⠀⡼⡄⠀⠀⠀⠀⠀⠀⠉⠀⠠⠤⠌⢧⡀
⠀⢀⠎⢀⠎⡁⠠⠄⣀⠑⢄⠹⡄⠀⠀⠀⠀⠠⠄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⢀⣐⣐⣂⡀⠀⢹
⢀⠏⠀⠀⠈⡠⠄⠀⠀⠣⠀⠆⢻⠀⢀⠀⠀⢀⡁⠀⠀⢀⣀⠤⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠁
⡼⠀⠀⡀⠀⡄⡂⢀⢀⢀⠀⠀⢸⡖⠂⠀⠀⠀⠐⠋⠉⠁⠤⠒⢒⣒⠒⠠⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⡇⠀⠀⠀⠀⠀⠢⠜⡸⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠉⠉
⡇⠀⡀⠀⠑⢄⡀⣈⠔⠀⠂⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠒⠂⠤⠐⢀⣀⣀⣉⣉⠁⠀⠄⠀⠀⠀⠀⢀⣈⣉⡉⠥⠀⠀⣀
⢱⡀⢡⠀⠢⠄⡀⣀⠀⠀⢀⠀⡞⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠤⠭⠉⠉⠉⠁⠀⠀⢀⡀⠀⠀⠀⠒⠒⠒⠂⠉⠀⠀⠀⠉⢡⠟
⠀⠳⡄⠣⢄⡀⠀⣀⠠⠐⢁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠤⠄⠀⠀⠤⠤⠤⣤⠤⠄⠀⠀⠉⠉⠉⠭⠤⠤⠀⣀⡴⠋
⠀⠀⠈⠓⠤⣄⣀⣀⣠⣴⣋⣀⣤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠿⠿⠿⠿⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⢘⣤⣀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠤⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠐⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⡥⠤⠤⠀⢤⣤⣭⣽⡯⠤⠠⠤⠤⠤⠤⣴⣶⠦⠤⠤⠀⠠⠤⣤⣤⠤⠤⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠤⠤⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⡀⠀⠇
⠀⠀⠀⠀⠀⡱⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢁⠀⠀⣀⣀⣀⣀⠀⠀⠐⠒⠂⠀⣀⣀⡉⠉⠀⣀⣀⣀⣉⣉⣁⣀⣀⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⢈⠀⠀⠐⠚⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠴⠦⢤⡍⠉⠉⠉⠉⠀⠀⢀⣠⡞
⠀⠀⠀⠀⠀⣅⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠲⠒⠀⠀⠀⠀⠀⠠⠄⠀⠘⠙⠛⠀⠐⠀⠠⠤⠤⠤⠤⠤⠔

⣴⣶⣤⣴⢤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣦⣴⣤⣤⣦⣤⣤⣤⣤⣤⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣤⡶⢿⣭⣿⣽⡿⢶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⡓⠀⢼⣿⠿⠿⠿⠟⠛⢶⣿⣻⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⢀⠇⠃⠀⠀⠛⠀⠀⠀⣀⣀⠀⠻⣯⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢛⢿⡄⠚⠙⢶⠤⠐⡿⠟⢿⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠍⠀⠀⠀⠀⠀⢐⣀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣘⣃⠀⠀⠀⠀⣻⣦⣿⠁
⠀⠀⠀⠀⡄⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⣀⣤⠞⣿⢇⡽⣿⣦
⠀⠀⠀⡈⠀⠀⠀⢠⡇⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⡀⠀⠙⣾⠿⠋⠀⠀⠸⣼⣛⣯⡟
⠀⠀⠀⡧⠀⠀⠐⠀⠉⠉⠉⠉⠉⠉⠉⠛⠻⣝⣟⠁⠈⢹⣷⣾⣿⣄⠀⠀⠀⠀⠙⣿⢯
⠀⠀⠀⣧⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢹⡆⠀⠈⠻⠿⠉⠛⢦⠀⠀⢀⡴⠃⣰⠷⣄
⠀⠀⠀⠀⠉⠙⠲⠤⣄⣀⣀⠀⠀⠀⢠⣴⣾⡟⡼⠁⠀⠀⠠⣤⠴⢻⡞⣷⣞⣉⡀⢰⡇⠀⠈⠙⠲⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢻⣷⣶⣿⣿⣦⡆⣸⠇⠀⠀⠀⠀⠘⢻⡿⠁⠀⠳⣸⠁⡠⠂⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠻⠛⠿⢾⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠠⣿⡀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠇⠀⠘⠟⠒⠒⣤⠶⣦⣶⣶⣿⣷⢔⠤⣄⣄⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⡄⠀⢚⠃⠀⠀⠀⠀⠀⠐⠟⣿⢛⣻⣿⣿⣯⣗⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠙⠄⢻⡅⠀⠀⠀⠀⠉⠉⣼⢣⣏⣭⣵⣾⣯⣟⠭⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠠⡄⠸⡘⡇⠀⠀⠀⠀⠀⢸⣧⠜⠋⠁⠀⠀⠀⠙⢷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡈⠣⠐⢧⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⢂⣀⡨⠤⠖⠋⠁⠀⠀⠀⠀⢀⣠⡴⠖⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠁⠀⠀⠀⣀⠀⠤⠤⠖⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⣠⠆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂

⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠋⠉⠉⠉⠙⡀⠀⢠⡀⠀⣄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⣠⣤⣤⡀⠀⠀⣀⣠⡴⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠙⠞⠙⠷⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠙⠃
⠊⠀⠀⠀⠈⠻⠿⣿⡿⣿⠿⠛⠻⣿⣿⢿⣿⡒⢾⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠃⠀⠘⣷⣯⠀⠀⠀⠸⣿⣟⡿⢿⣮⡿⠿⣿⣿⣿⣦⣤⣀⠴⠒⠒⠒⠒⢚⢹⡧⠤⠤⠤⣒⠉⠂
⠀⠠⢤⣴⣆⣠⢴⡔⠺⣿⠣⠂⠀⢰⣿⣿⣾⣿⣰⣿⣿⡻⣿⣿⣿⢿⣿⣧⣴⣿⣿⣿⣿⣶⣦⣤⡠⠾⣿⣀⣀⠀⠀⠀⣀⡤⡀
⠀⠀⠀⠀⠀⠀⠈⡍⠚⠻⠦⠀⠀⠘⢿⣿⢿⣿⣿⣿⣿⠇⠉⡏⠉⠁⠺⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠓⠯⢤⣬⠠⠛⠛⠛⠛⠒⠒⠾⠙⡆
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠁⠐⠀⡀⣿⡿⢿⣟⠋⣉⠀⠀⣷⠀⠀⢂⠜⣷⣦⡈⠻⣿⣿⣿⣿⡟⠀⠀⠀⠀⢠⣄⠤⣴⠒⠒⢒⣖⢸⣇
⠀⣀⣀⡀⠁⠀⡀⠃⠀⠀⢀⠀⠀⠸⠀⠉⠁⠈⡏⠀⡹⣄⠀⢸⣄⠀⠀⢸⣿⣿⣧⢀⠿⠋⠙⠛⠒⠶⠒⠚⠉⠉⠀⠘⠚⠋⠀⠸⠋⠸⠟
⠸⣿⡆⠈⠒⠚⠓⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠙⠓⠷⠿⡆⠀⢸⡿⠟⠋
⠀⠉⠓⠒⠚⠛⠛⠛⠛⠓⠒⠒⠛⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⣀⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠈⠙⣓⠖⠀⠁⠀⠀⡑⣃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠀⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣿⠿⢿⡿⠿⠿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⣠⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦⣄⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⣿⡟⠛⣿⡆⠀⠀⠀⣴⡿⠟⠛⠛⠻⠇⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⡿⠻⣷⡀⠀⠀⠀⣿⣇⢀⣼⡟⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣻⣷⠀⠀⠀⣿⡟⠛⠻⣷⡄⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛⢻⣧⠀⠀⣿⡇⠀⣀⣿⡇⠀⠘⣿⣦⡀⠀⠀⣀⡄⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠰⠿⠃⠀⠀⠀⠀⠈⠻⠇⠀⠿⠿⠿⠛⠋⠀⠀⠀⠈⠙⠻⠿⠿⠟⠃⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣸⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠙⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡯⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣧⠶⠋⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣛⣉⣙⣙⣙⣉⣛⣁⣀⣀⣀⣀⣀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠴⢦⡄⡀⠀⠤⠀⠀⠀⠐⠒⢲⡄⠂⠀⠲⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⣠⠖⡀⠘⠰⠒⠂⡀⠀⢙⡯⠁⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⣿⠿⠿⠖⢀⠀⠀⠀⠀⠠⠖⠒⠉⠈⠘⡋⠀⢠⣄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠚⣛⣒⣒⣲⣾⣳⡄
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠐⠛⠉⠉⠀⠀⠐⢡⣶⣶⠆⠀⠠⣐⣀⢤⡿⠂⠀⠀⠀⠀⠀⠀⠁⢉⡉⣩⣭⣭⡭⡩⠭
⠀⠀⠀⠀⠘⠛⠋⠉⠀⣠⣤⣤⣤⣶⣶⣦⡄⠀⠀⠀⠀⠀⠙⠃⠀⠘⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⢈⡛⠂⠀⠀⠈⠛⠛⠻⠿⠿⠿⠧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀⠀⠀⠠⠴⠶⣦⣤⣦⣤⣔⣤⣄⡀
⠀⠀⠀⠀⠘⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠉⣻⠏⣛⣉⡀⠀⠀⠀⠀⠈⠉⠁⠛⠛⠛⠋⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⠤⠀⠀⠀⠀⠀⠀⠀⣦⠀⢤⡆⠀⠀⠀⢚⣩⣥⣄⣀⣀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠐⠀⠀⠂⠀⠐⠒⠦⠠⠜⠉⠯⠀⢤⠤⠀⠈⠛⠀⠀⠒⠲⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁


⠀⠀⠀⠀⣰⣾⣿⣶⣶⣶⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⣐⣒⣒⠒⡒⣒⣒⡒⣂⣤⣠⣤⣶⣶⠶⠄⢀⣄⠀⣠⣴⣶⣶⣶⣶⡆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣶⠂⠀⠐⠒⠛⠛⠛⠁⠈⠉⠉⠉⠉⠩⠍⠩⠥⠤⠬⠭⠿⠍⠉⠛⠋⠁⠈⠉⡀⠈⠉⠿⠿⠿⠋⠁
⠀⠀⠀⠀⣘⣛⣙⣛⣛⣃⣘⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡤⣤⣦⣴⣶⠿⠦⡆⠀⠤⠤⠀⡄
⠀⠀⠀⠀⣹⣯⣭⣭⣉⢉⣉⠉⠉⠉⠉⠉⠉⠉⠉⢁⠈⢉⡉⢉⣉⣉⣩⣥⣤⣶⣴⣷⣶⣾⣿⣿⢿⣿⡿⠿⠿⢿⣿⣷⣶
⠀⠀⠀⠀⠿⣿⣿⡿⣿⣿⣿⣷⣧⣴⡦⠄⠘⠙⠋⠉⠉⠉⠉⠈⠉⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⠀⣠⣿⣿⡿⠦
⠀⠀⠀⠀⣀⡀⠀⡀⣀⣀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣄⣀⣠⡀⠤⠄⠀⠻⠿⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⣀⣀⠀⠀⠈⠉⠁⢤⣤
⠀⠀⠀⠀⣸⣿⣿⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣶⣶⣷⣶⣷⣶⣿⣿⣷⣶⣶⣶⣶⣦⣴⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⡿⠷⠾⠶⠶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⠟⡿⠛⠛⠿⠿⠿⠟⠻⠿⠛⢻⡏⠭⠉⠉⢉⣭⣏


⠀⠀⠀⠀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠠⠤⠀⢀⣀⣀⣀⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⣀⣀⣀⣠⣀⣀⣤⡀⢀⣀⣀⡀
⠀⠀⠀⠀⣻⣿⣿⣿⣿⣯⣭⡭⠽⠷⠶⢖⣒⣛⣫⣭⠭⠷⣖⣒⣢⡴⠦⡝⣓⡀⠡⢤⡰⣦⣯⠤⢙⣿⣿⣿⣿⣿⣶⣿⣿⣭⣭⣭
⠀⠀⠀⠀⠛⣉⣩⣭⣽⣿⣶⡶⠶⠛⣛⣛⣩⣥⣴⡶⠟⣛⣭⣝⣻⣵⣾⣿⡝⠒⣍⠢⢼⣿⣶⣾⣿⣿⣷⣮⡙⢿⡛⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠿⠛⠋⠉⠁⠀⣠⣴⣶⠿⠟⠛⢉⡥⠶⠞⣫⠽⣺⡿⢫⠟⡕⣩⠖⠉⠀⠀⢼⡟⢻⣝⣿⠙⣿⣿⣿⣦⣍⠙⠛⠛⠲⠶
⠀⠀⠀⠀⣠⣀⠀⠀⠸⣿⣿⣿⣀⠀⠀⠰⣿⡀⠀⢸⣯⢀⣿⣆⣾⢘⡆⣇⢀⡠⡤⢤⣼⣿⣿⡿⢡⣾⡿⠛⣉⣉⣉⣥⣤⣤⣤⣤
⠀⠀⠀⠀⠿⣿⣿⣤⣄⡙⠻⠿⣿⣿⣶⣀⠻⣿⣦⡘⢿⣾⣿⣿⣹⣿⣯⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⢿⣷⡾⣿⣿⡿⠟⠛⠛⠛⠛
⠀⠀⠀⠀⣿⣷⣾⣭⣙⣻⠿⣦⣬⣉⡛⠿⠿⣶⣭⣛⠲⢮⣙⡛⠿⣿⣿⡿⢿⣵⠏⠀⠙⠛⠛⠣⠄⠲⣛⣠⡿⠟⢀⣠⠴⠶⠶⠗
⠀⢀⠀⠀⣿⣿⣿⣿⢿⣽⣛⣒⠶⠯⢭⣟⣀⣲⣮⣭⣙⣓⡾⢷⣦⣴⣒⣤⠀⠀⡈⠁⠀⠀⠀⠄⠀⠀⠩⠥⠶⠚⣋⣡⣤⣴⠀⠂
⠀⠈⠀⠀⢿⣿⣿⣷⣶⣯⣭⡭⠽⠙⢛⣶⠶⠿⠿⣿⣿⡭⠿⠖⢲⣭⠽⠟⠛⡛⠶⠾⣤⠴⠠⠄⠨⠥⠟⠃⠴⠮⠭⠥⠾⠣⢶⡥
⠀⠈
⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠭⠍⢙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⢀⣀⠀⠤⠤⠤⠄⠀⢀⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⡷⠀⠀⠀⠀⠀⣀⠤⠒⢈⠩⠄⠐⠂⠀⠀⠀⠀⠀⠉⠹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠝⠀⠀⠀⠀⠐⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠴⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠌⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢀⡀⣴⠀⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠺⠘⠻⠓⠏⠹⠟⠇⢀⡇⠃⠇⠺⠺⠞⠟⠟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠁⠁⠉⣉⣉⣁⣀⣀⣈⣉⠉⣉⠉⠭⠭⠭⠭⠉⠉⠉⠉⠉⠉⠀⠐⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣛⣉⣁⣀⡀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠓⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⠉⠉⠉⠉⠁⠀⠀⣀⣀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣀⣀⣠⣤⠤⣒⣚⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⣤⣶⠶⠂⠀⠘⠛⠓⠒⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠤⠤⠄⣀⣉⡉⠁⠀⠀⣀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠉⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠉⣁⣀⣀⣀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠛⠋⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠉⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢠⡟⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣿⣠⠴⠖⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⢉⣁⣤⣭⣍⣉⣛⠷⣾⡛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠞⠀⠀⠀⠸⣇⣠⠀⠀⠒⠉⠁⠀⠈⠳⣄
⠀⠀⠀⠀⢀⣴⢋⡀⣠⠾⠋⠉⠀⠀⠀⠉⠉⠻⣦⠙⢷⡔⠒⠒⠒⠂⠀⠠⠤⠤⠄⠀⠀⠒⠒⠒⠂⠀⠈⠛⣲⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⢀⡞⠁⢀⡽⠋⠀⠀⠀⠀⣀⣀⡀⠀⠀⠘⣷⡀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠤⠄⢿⡀
⠀⠀⠀⢸⠇⢠⡏⠀⠀⠀⠀⠀⡜⠀⠀⠘⡆⠀⠀⠘⣷⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠒⠒⠒⠛⠉⠉⠉⠉⠉⠀⠀⠀⢀⣠⡞⠁
⠀⠀⠀⢸⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢸⡇⠘⣷⠒⠋⠋⠉⠉⠉⣉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢠⡴⠶⠒⠒⠊⠉⠉⠉⠉⠉⠉⠛⢶⡀
⠀⠀⢀⡼⠃⡟⠀⠀⠀⠀⠀⣀⠤⠦⣄⠀⠸⡆⠀⠀⠀⣿⠀⠹⣟⠉⠉⠉⠉⣉⣁⣤⡀⠀⠀⠀⠀⠀⠒⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⣾⠁⢸⠇⠀⠀⠀⢀⠞⠁⠀⠀⠘⡄⠀⣧⠀⠀⠀⢹⡄⠀⣿⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡀⣸⡄⠀⠀⢀⠏⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⡇⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡏⢹⡇⠀⠀⢸⠀⠀⡄⠀⠀⢠⠃⠀⡟⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⠖⠛⣿
⠀⠀⢸⡇⠘⡇⠀⠀⢸⡀⠀⠧⠤⠖⠃⠀⢀⠇⠀⠀⠀⢸⡇⣸⠇⠀⠄⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠒⠒⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⡿
⠀⠀⠀⣧⠠⣷⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣼⠁⣿⢤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠴⠖⠂⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⢤⠤⢴⡇
⠀⠀⠀⢹⡆⠸⣇⠀⠀⢱⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⢰⠏⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⣉⣉⣀⣀⣤⣤⠤⠴⠶⠒⣾
⠀⠀⠀⠈⣷⠠⢻⡄⠀⠀⠳⠤⠒⠋⠁⠀⠀⠀⠀⣠⠏⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⠉⠉⠉⠉⢉⣉⣉⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⢹⡄⠀⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣤⠞⠓⠒⠒⠒⠒⠒⠒⠛⠛⠉⠉⣉⣉⣡⣤⠤⠄⠒⠈⠉⠉⠉⠉⢉⣀⣠⡤⠶⠖⠚⠛⠛⠁
⠀⠀⠀⠀⠈⢷⣀⠀⡏⠛⠶⢤⣤⣄⣠⣤⠶⣋⡴⠿⠥⠤⠤⠤⠀⠀⠒⠒⠒⠛⠛⠋⠉⠉⢉⣀⣀⣤⣤⡤⠶⠶⠖⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣴⣤⣤⣤⣶⣚⣉⣀⣀⣀⣀⣀⣤⣤⣤⡤⠴⠶⠶⠖⠚⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⠖⠛⣇
⠀⠀⠀⠀⢀⡴⠶⠛⠉⠉⠙⠛⡄⠀⠀⠸⣄⣀⣀⣀⣀
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠑⡀⠭⠉⠉⠉⠙⠓⠒⠲⠦⠤⠤⣤⣄⣀⣀⣀⣀⡀⠀⠀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠐⢐⠁⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣠⠟⠉⢘⡟⢦⡀
⠀⠀⠀⢻⣀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠚⠓⠒⠩⠄⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢃⢂⠅⠐⠁⠀⠹⡀
⠀⠀⠀⠘⣷⣶⡤⠭⠍⡁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣤⠃⡈⠐⢀⠀⠀⡇
⠀⠀⠀⠀⠘⠧⢤⣄⣀⠀⠀⠈⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠂⠐⡏⢰⢹⢰⠄⠀⡆⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⣤⣀⣀⣀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⡇⠌⠘⠀⡄⠀⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠴⣤⣀⣀⠀⠀⠀⠀⠀⠉⠒⠄⣧⠀⠀⠠⠠⠃⣴⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠦⠤⣀⣀⠀⠀⠸⣄⠂⠀⠀⠴⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠻⠷⠶⠶⠞⠋





⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⢀⣠⣿⠀⢹⣇⣀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢺⣏⠉⠛⠋⠉⠀⠀⠀⠈⠉⠛⠋⢉⣿⠂⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠛⠛⠶⠶⠶⣤⣤⡟⠀⠀⢀⣿⠃⠀⠀⣠⡴⠶⢶⣤⡀⠀⠀⢻⡇⠀⠀⠀⢿⣤⣤⠶⠶⠶⠛⠛⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⣠⣤⠾⠇⠀⠀⣾⠋⠀⠀⠀⠈⢻⡄⠀⠀⠿⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⢿⣤⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⣿⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⢷⣄⣀⣠⣴⠟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣿⠧⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠾⣧⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠳⢶⣤⠀⢠⡴⠾⠛⠷⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠉⠻⣦⠀⠀⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢳⣄⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠛⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣶⠏⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⣿⣤⡀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⢸⣷⣄⡀⠀⠀⠀⠀⣀⣤⣿⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠀⢿⣤⣤⣤⠿⠈⠙⠳⣦⣤⠶⠛⠉⠸⣧⣤⣤⣼⠇⠈⠛⠶⣤⣴⠞⠋⠁⠿⣤⣤⣤⠿⠀⠙⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⢀
⣿⡟⡋⣻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣤⣤⡄⢠⢤⣤⣴⣿⣿⣶⣿⣶⠀⣾⣿⣾⣿⣿⣿⠀⣿⣿⡆
⠀⢀⣀⣀⣤⠀⠀⢠⣤⣤⣤⣶⣶⠀⡆⠀⠀⣿⣿⣿⢺⣿⣿⣿⡿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⡇
⠀⢸⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⠀⡇⠀⢀⣿⣿⣿⢸⣿⣿⣿⡇⣿⣿⡇⢸⢻⡏⢻⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣧⡄
⠀⢸⣿⣿⣿⠀⠀⢸⢿⣿⣿⣿⣿⢸⡇⠀⢸⣿⣿⣿⢸⣿⠙⣿⡇⣿⣿⡇⢸⢸⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣯⠉⢹⣿⠀⣿⣿⣿⡇
⠀⢸⣿⣿⣿⠀⡇⢀⣸⣿⣿⣿⣿⢸⡇⢰⣿⣿⣿⣿⢸⣿⢰⣿⣿⣿⣿⣷⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⠀⢸⣿⠀⣿⣿⣿⣧
⠀⢸⣿⣿⣿⢀⡇⢸⣿⣿⣿⣿⣿⢸⣧⢸⣿⣿⣿⣿⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⣿⡏⣹⠀⢸⣿⠀⣿⣿⣿⡟
⠀⢸⣿⣿⣿⢸⡇⢸⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠉⣇⣸⣿⣿⣿⣿⣿⠀⣿⡇⣿⠀⢸⣿⠀⣿⣿⣿⣿
⠀⢸⣿⣿⣿⢸⡇⣾⣿⣿⣿⣿⣿⠘⣿⣸⣿⣿⣿⣿⣿⣿⢸⣿⣿⠉⠉⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⢸⣿⣟⣿⠀⢸⣿⡀⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡜⠃⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⡏⢹⣿⠘⣿⣿⣿⣿⣿⡿⣿⣿⢸⣿⣿⠀⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣧⣶⣿⣿⣿⡇⢸⣿⠀⣿⣿⣿⣿⣿⡇⣿⣿⢸⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⢀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠀⢠⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠙⠛⠿⢿⣿⣿⣿⡇⢸⣿⢀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠀⢸⣿⣿⢠⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠛⠛⠛⠛⠿⢿⣿⣧⣿⣿⣿⣿⣿⠀⢸⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠙⠛⠓⠾⢿⣿⣾⣿⣿⣿⣿⣿⣿⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠙⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⣉⠤⠖⠊⠉⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣫⠵⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢖⡫⠕⠊⠁⠀⠀⠀⠀⠀⣀⠤⠖⠋⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢖⣫⠵⠚⠉⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⠥⠒⠉⠀⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⣠⠤⠖⣋⡭⠗⠋⠀⠀⠀⢀⡠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣨⠇
⠀⠀⢀⣼⠥⠒⠋⠁⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⡤⠖⠋
⠀⠀⢹⡁⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⣠⠴⠚⠁
⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⢀⡤⠖⠋⠁
⠀⠀⠀⠀⠹⡄⠀⠀⠀⢀⣠⠔⠊⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠙⠦⣤⠖⠋⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠸⠇⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣀⡴⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁


⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⡀⠀⠀⠀⠀⠀⡀
⣿⡏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣹⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡞⠀⢰⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣼⠃⡀⢸⣿⠃⠀⠀⣀⣰⣾⠀⠀⠀⠀⠐⢡⠎⠀⠀⠄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⠼⠁⠀⣶⣿⡇⢐⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠁⠀⠀⠀⣭⣭⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠐⣡⢀⣼⠏⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣞⠏⠎
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠁⣣⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⡴⠙⣡⠆
⠀⠀⠀⠀⠀⠀⡠⠀⠀⡁⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⣰⠧⣾⠋⠀⠀⣠⡤⢤⣤⣤⠤⡀
⠀⠀⠀⠀⠀⣰⠁⡀⢊⣤⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⡁⣰⡏⠀⠀⠀⣿⣻⣿⣻⣟⢶⣷
⠀⠀⠀⠀⢰⣏⣩⣍⠛⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⡀⠀⠁⣽⡿⠀⠀⠀⠀⡇⠀⣙⣶⡉⢠⡟
⠀⠀⠀⠀⠘⠯⣿⣯⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢸⣿⡆⠀⠟⠀⠀⠀⠀⠀⡇⠀⠀⠙⡟⠁⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢠⠏⢈⣥⠎⠀⠀⠀⠀⠀⠀⡇⢀⠀⡐⣷⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠠⠀⠀⠀⠀⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⡅⠀⠰⡁⢤⢂⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣶⡂⠀⢞⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⢡⠀⠾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡾⠛⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⢛⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⡀⠙⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠢⢐⢿⣻⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⣿⣦⣼⢡⢏⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡄⢳⣯⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣾⣯⠁⠹⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⡟
⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⢘⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣃⣀⣀⣾⡁⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠈⠁⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⡇⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡅⢩⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⢀⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⡌⠉⢻⡀⠀⠀⠀⡠⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⢣⡔⢺⡇⠀⠀⠸⠧⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣿⠈⢡⠀⠀⡜⠃⠾⠁
⠀⡿⣀⢈⡆⢠⡁⢀⠃
⠀⡇⠈⠀⠀⠀⠁⠙⠀⣾⣿⣽⢷⣦⠀⣠⣤⡤⣤⣄⠀⢀⣀⡀⣀⡀
⠀⣇⡀⠀⠀⠀⠀⠈⡇⠋⠀⠀⠀⢱⠀⡿⠛⠛⠛⢻⠀⢸⠿⠷⠯⠿⠆⢰⣷⣞⣶⣶⠄⠀⣤⣤⢤⣤⡄
⠀⡏⠉⠁⠐⠒⠒⠒⠓⠄⠤⠤⠤⢼⠀⣇⣀⣀⡀⢸⠀⣷⠀⠀⠀⠀⠄⢸⠁⠀⠀⠈⡆⠀⡟⠉⠉⠙⢣
⠀⠲⠦⠤⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠛⠛⠂⠀⠐⠶⠶⠠⠄⢀⣠⣇⣠⡅⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠉⠉⢰⠲⠖⠒⠂⠀⠤⠤⡤⠤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠆⠆⠀⠀⠀⠀⠀⠀⡇⠀⢹⡁⠈⢡⠀⠈⣻⢛⠋⠁⠀⠒⠶⠶⠀⠷⠠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡜⡘⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠈⡄⠀⠀⡎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢠⢡⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⢱⠀⠀⢰⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⡆⠀⠀⡎⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠎⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠸⡀⠀⠸⢡
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡜⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡇⠀⠀⠀⠀⠶⠀⠀⠇⡆
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣧⠀⠀⠀⠀⠘⠆⠀⠸⠰⠆
⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⠃⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠸⡀⠀⢃⠲
⠀⠀⠀⠀⠀⢰⠁⠀⠀⡎⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠃⠀⠘⠁⠇
⠀⠀⠀⠀⢀⠇⠀⠀⡸⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⡄⠀⠙⡘⡀
⠀⠀⠀⠀⣌⣀⣀⣠⣁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠄⠿⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢃⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣄⣀⢡⢡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠔⡀⠀⠀⣠⠐⠉⠁⣀⢀⡀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⡈⠀⠀⠀⠄⠀⠀⠤⠀⠈⠘⠉⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⡹⠝⠂⠀⣁⢢
⠀⠀⠀⠀⠀⠀⢊⡁⠀⢀⡤⠊⣀⠔⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⣁⠆⠀⠀⢀⣠⠈⡀⠐⠀⠁
⠀⠀⠀⡠⣀⡀⠧⢈⡡⠝⢤⡖⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠀⠀⠁⣤⣒⠋⠉⠈⢉⡠⠀⠀⣈
⠀⠀⠰⠀⠟⠡⠞⠂⠱⡤⠄⣡⠀⠈⠀⠀⠀⠀⠀⠀⠂⡡⠄⠀⠀⠀⠀⠀⠐⠀⠀⣠⡄⠂⠐⠛⠉⠀⢊⣧⠾⠚⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡯⢃⡤⣀⠔⠊⠁⠀⠀⠀⣠⠴⠦⠀⠁⠀⢀⣠⠔⣞⡭⠦⡉⠂⠀⠀⡢⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠉⠁⠴⠉⠊⠀⢠⡀⠠⠂⠐⣈⡤⠀⢀⣠⠴⠚⡽⠂⠉⡄⠀⠀⠀⠀⠀⠊⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢬⢻⢰⠀⠐⢾⢤⠀⠀⠄⠒⠉⠀⣠⠴⢊⣡⣴⠾⠋⠁⡠⠀⠰⠂⠄⠀⠀⠀⠀⡀⠀⠀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⠲⠶⠾⠇⢠⣄⢀⡷⠀⠀⠀⢀⣨⣔⡾⠋⠉⠀⠀⡀⠀⠂⠀⠀⠀⠀⠔⠀⠀⢀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣚⣉⡡⣶⡀⢸⣏⠊⠉⡟⠗⢀⣠⡶⣻⠟⠉⠀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⢠⠏⠘⠁⠤⠀⠚⣗⣀⣈⣼⣭⢔⣚⠁⠡⢊⡡⠀⠀⠀⠀⡚⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⠀⠞⢛⡀⢈⡫⠀⢸⢉⣯⣗⠉⢰⡜⠖⢿⠄⠖⠀⢀⡠⠆
⠀⠀⠀⠀⠀⠀⢀⠚⢰⣧⣈⠳⡀⢻⡿⠉⡡⠠⢸⢍⡦⠀⠁⠸⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠑⠲⠞⠭⠴⠖⠛⠁⠀⡳⢄⣆⠚⣥⣰⠟⡾⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣈⡥⠒⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠮⠄⠱⠾⠛⠈⠀⠀⠀⠀⠀⠠⠀⠈⠀⢀⡡⠖⠋⠁⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣪⣐⡀⣠⠴⠚⠁⠀⠀⣀⢴⠂⠀⠊⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠆⠀⠀⠀⣠⠶⠛⢲⢄⣏⠁⠘⠉⠀⢀⣀⡴⣶⠯⠞⠋⠀⠐⠀⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⠌⠙⢂⠼⢉⡱⡍⣦⣀⢀⣀⡭⣦⡉⠄⠀⠀⠀⡀⠀⢀⠠⠊⠁
⠀⠀⠀⠀⠐⠲⡆⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡃⣡⢾⢄⡷⣿⢿⡿⠋⠀⠀⢻⡄⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⣻⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⣦⣹⣖⣫⡴⠂⡈⠁⠀⠀⠀⢈⣇
⠀⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠀⠉⠛⢻⣿⣷⣿⢿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⡿⣛⣵⣴⣦⣀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⣿⠿⠟⣷⣿⣿⣿⣿⣦⣀⡀⠀⠀⢽⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⠅⠒⠄⣶⣍⡻⢿⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⣿⣿⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⢼⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣧⠀⠀⠐⡟⢡⡿⣻⣿⣿⣿⣿⣿⠶⣄⣀⡲⢿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠆⠀⢘⢆⡾⣱⣿⣿⣿⠻⣿⢿⠑⠈⠹⣙⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣏⡿⠃⡭⡸⣿⣿⡿⠋⠙⣿⡝⠧⡄⣰⡀⣻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⣧⡞⣹⡇⢠⡶⠐⠻⣿⣿⣧⣶⢹⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣹⣿⣿⣿⠟⠁⢹⣿⠜⣸⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡟⠉⠀⠘⣸⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠸⠯⠀⠀⠀⠀⠿⠼⠻⠿⠿⠿⡿⠿⠿⠿⡿⡿⠿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⢶⡺⠟⠻⠋⢁⣀⣥⣨⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠟⠋⠁⠈⠑⠂⣁⣠⣴⣾⣿⡿⠟⠉⢰⠾⠋⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⡁⡨⠤⢒⣀⠤⠀⠀⣒⣿⣿⣿⠿⢋⡡⢀⠀⠀⠁⠴⣋⣥⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣭⠒⠉⠀⠀⠀⠒⠉⠉⠀⠀⣠⣿⣿⠟⡋⢁⡴⠟⣋⠤⠀⡀⢀⣴⣿⢿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠭⡪⣝⢦⣤⠴⠚⠁⣠⣴⣶⡿⠟⢉⠴⠋⠀⠀⠔⠋⣀⣴⣿⣿⠟⠋⠒⠉⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢈⠈⠳⣿⡄⢀⣤⡾⡛⠭⣀⡤⠖⠋⠁⣠⣶⣴⣿⠿⣛⣽⣯⠔⣊⣤⠞⣋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢃⣀⣐⣈⣄⣤⣽⠿⣿⣟⠉⠐⠉⣀⡴⢂⣴⣾⢿⣛⣭⠶⠛⠙⠉⡴⠋⠟⣁⣼⣾⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡜⢋⠉⡅⣠⡿⠉⠒⠄⠈⢻⣷⡶⢋⣵⠾⠛⠩⠒⡹⠋⢀⠔⠂⣠⣴⣊⣴⡿⠛⠵⢛⣆
⠀⠀⠀⠀⠀⠀⣴⡟⡩⢿⣷⡈⢮⣼⣟⣉⣀⣴⣆⣤⣥⣽⡟⢋⣴⡤⠔⠂⢀⣠⠾⢳⢂⣴⣿⠟⣫⠕⣀⠔⢒⣫⡽
⠀⠀⠀⠀⠀⣾⡝⠀⠁⠚⢻⣷⣿⠿⡫⢁⣾⣿⣿⠿⣅⠹⣽⣏⠀⠀⠐⠊⢁⣴⣴⡷⣟⠋⠠⢊⣵⣞⣥⡾⠟⠉
⠀⠀⠀⠀⠀⣿⠀⢴⣤⣾⣿⢫⠀⠀⠀⣼⣿⣟⠁⠀⠀⠀⠿⣿⡴⢖⣫⡵⠟⣋⠵⢞⣩⣠⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠻⠶⠿⠿⠿⢿⡀⠁⠜⣾⣟⣁⣀⣀⣤⣷⣒⣀⣿⠶⠛⠉⣚⠉⢀⣚⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣠⣿⣿⣟⠛⠟⠛⠻⡯⡹⣿⡤⠒⠋⠁⣺⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠙⠳⢮⣐⠄⠀⣿⣇⣤⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⡀⡤⠄⢀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠐⠂⠊⡆⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⡴⠁⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣧⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢇⠀⠀⠀⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠉⠀⠀⠀⠠⠦⠠⠾⠀⠀⠀⠈⠐⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠉⠙⠂⡀⠀⣄⠀⣀⡀⣀⠠⠤⠤⠄⠀⠀⠀⠀⠈⢸⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠛⠐⠀⠀⠀⠀⠐⠒⠄⠀⠀⠂⠀⢀⡌⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⡀⠔⠁⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠨⣃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⣀⣉⣿⠀⠀⢰⡁⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠒⠃⠀⠀⠀⠐⠋⠙⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⢨⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⢀⠀⠀⠠⡵⠖⠀⢃⠀⠀⢸⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⢶⣤⣤⣤⣤⠾⠀⠐⠿⢻⢛⣄⠠⠖⠀⢾⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠚⠀⠀⠀⠒⠒⠒⠋⠀⠙⠶⠒⠒⠒⠀⠀⠀⠐⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣴⠀⠀⠀⠀⠀⠀⣀⡤⠂
⠀⠀⠀⠀⠄⡄⠀⠠⢠⠀⠀⠀⠀⠌⠹⣄⠀⠀⢀⡴⠚⠉⡀⠂⠒⠄⠀⠀⠀⠀⣔⡄
⠀⠀⠀⢀⠛⢧⠀⠨⢻⡄⠀⠀⠀⠓⠾⣿⠖⠚⠁⡄⠀⡐⢌⠀⠀⠁⠀⠀⠀⢠⠛⢧
⠀⠀⠀⠘⢤⡿⠷⡃⠸⣿⡤⢀⠀⠤⠾⠁⠀⠀⠀⢹⠴⣟⠊⠐⢁⠀⠁⡢⠰⡑⢤⣾⠷
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣠⣖⠐⣶⣶⡁⠀⠀⠀⡀⠐⠘⣎⠀⠉⣀⠄⠀⠀⡄⠀⠀⢘⢂⢀⣴⡄
⠀⠀⠀⠀⠀⠀⢀⣀⣾⠋⠁⠑⠿⢛⢿⠇⠆⠀⣠⡀⠀⠁⢹⡟⠋⠘⠤⣿⣽⠿⠀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⢀⡤⠞⢩⠩⠝⡄⠀⠀⠀⠒⢇⢀⠤⠀⠸⢖⠈⠉⠛⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⣠⠖⠋⠀⠐⢇⠀⣓⣿⡆⢀⡄⠀⡼⣾⠤⠤⠄⠘⠀⠀⠀⠀⠀⠀⠀⢶⡎⠀⠀⠀⠀⠀⣠⠀⠀⠵⠠
⠸⣬⣀⡀⠀⡈⠀⠊⠛⠉⠉⢅⠢⢤⣷⡿⣀⢴⠁⢠⣤⣦⠀⠀⠀⠀⡈⠐⣻⡀⡀⢰⠀⢠⣤⣧⡄⠠⠃
⠀⠀⠉⠙⠳⢶⡍⠒⠀⠀⣢⣀⠀⠁⠀⠀⠐⠤⠙⠛⠿⠁⡠⠐⠀⢤⣥⢈⣭⣿⠿⠦⠽⠿⠛⢋⣡⣴⣦
⠀⠀⠀⠀⠀⠀⠢⣬⣬⣿⡿⠿⠶⢥⣄⣐⡀⠄⢀⠀⠀⠀⠑⠀⠀⠀⠃⡝⠛⠒⠂⣀⠀⠒⠛⠉⢉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠉⠉⠛⠶⣦⣥⣀⣀⠠⠄⡀⠀⠈⠒⠒⠒⢛⢢⡞⠈⠡⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⢀⣠⣶⡷⠄⠀⠀⠀⠈⠉⠙⠛⠷⢶⢯⣅⣲⠤⢐⣂⠀⠙⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⣬⣅⡂⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠟⠻⣷⣶⣤⣄⡀⠀⠀⠀⣀⣠⣴⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠈⠻⡏⠙⢿⣆⢀⡴⠋⠁⠀⠈⠉⠛⠻⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢹⡇⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⡾⠛⠛⠛⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣤⡴⠾⠋⠉⠉⠻⢦⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠀⠀⠈⠙⣄
⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣤⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⢷⣄
⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠉⠙⠛⠿⠶⣤⡀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣿
⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣁⣀⣀⡀⠀⣀⣴⣿⠃⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⠦⣤⣄
⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⢻⣿⠁⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⢠⣿⣧⣄
⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠁⠀⡹
⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣤⣤⣀⣀⣠⣤⡶⠟⠋⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⠀⢹⣿⡏⠁⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⡿⣿⣷⣿⣿⣿⣄⡀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⡿⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠀⢸
⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠀⢰⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⣿
⢹⣿⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⣧⡈⠙⢿⣿⣶⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⣴⠿⠻⠿⠁⠀⠀⣿
⠈⢿⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⠃⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣍⣷⠶⠀⠀⠀⠀⢠⣿
⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢻⣿⣿⣿⡌⠻⣿⡙⠛⠛⠿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⢿⡟⣿⡉⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢿⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡰⠶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠒⠒⠓⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠓⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣾⠏⠀⠀⠀⠠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡉⠃⠀⠉⠿⠁⢀⡞⠉⠉⣹⡅⠛⢐⣧⡿⢋⡿⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⢯⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡽⡆
⠀⠀⠀⢸⠀⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠋⠀⡇
⠀⠀⠀⢸⠀⠀⠀⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡏
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⣇⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡇
⠀⠀⠀⢸⠀⣠⠞⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⣄⠀⡇
⠀⠀⠀⠸⣖⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢰⠛⠛⠛⠛⠛⠛⠛⠛⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠒⠒⠒⠒⠒⠒⠒⠒⣷⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠶⠞⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀

⡿⠉⠩⠝⠉⣉⠉⠉⠉⢉⠉⠩⠭⠭⠭⠭⠉⠉⠛⠉⠉⠙⠋⢉⠉⠉⠉⠉⠉⠉⠋⠛⠻⠟⠉⠉⠉⠙⠭⠭⠩⠿⢿⡭⠿⠟⠛⠛⠻⠿⣿⠿⠹⠏⠭⠭⢽
⣿⠯⠭⠤⠀⠀⠀⠀⠀⠀⡘⠃⠀⠐⠂⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠤⠀⠀⠀⣀⣀⠈⠀⠀⠀⠒⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒
⣧⡄⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤⠀⠀⠀⠙⠉⠛⠃⢐⣒⡒⢾⣷⣤⣤⣄⣀⣀⣀⣀⠐⠛⠛⠶⠄⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠀⠀⠀⠤
⣿⣁⡀⡀⠀⠀⠀⠀⠐⠒⢒⣚⣒⣒⣒⣂⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠊⣁⣀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠐⠒⠒⠀⠒⣚⡢⠿⠿⠋⠉⠁⠚⢛
⡟⠉⠀⠉⠉⠉⠉⠋⠙⠉⠛⠍⠉⠉⣉⡀⢀⠸⢂⠐⠒⠒⠒⠶⠂⠉⠉⠉⠉⠉⠉⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠉⠛⠀⠈⠛⠓⠂⠀⠈⠛
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣬⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣥⣴⣤⣤⣄⣀⣤⣤⣤⣤⣤⣄⣀⣤⣄⣀⣀⣀⣄⣀⣀⣠
⡄⡨⠁⠀⠀⠀⠀⠈⠀⠄⠉⠉⠀⠀⠀⠲⣤⣒⣂⣂⡀⠒⠂⣀⣉⠉⠉⠀⣹⣿⠓⠒⠒⠀⠚⠛⠿⠿⠟⠋⠙⠁⠚⠓⠚⠷⠶⠂⠐⠚⠒⠃⠀⠈⠈⠙⠛
⠀⠀⠀⠀⣀⡀⠠⠤⠤⠠⠔⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢿⡟⠁⠀⠐⠒⠒⠒⠀⠘⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠠⠁⠀⠀⠢
⠀⠀⠠⣤⣭⣽⠶⠒⠒⠚⠒⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⢀⣼⣿⣉⡁⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠶⠀⠀⡀⠀⠀⠀⣬⡷⠶⠒⠒
⠉⠉⠉⠁⢉⣉⠉⠙⠂⠀⠀⢀⣉⣀⣀⢀⢂⣑⣒⢃⡐⠀⠀⠀⠤⠀⠤⠤⣾⣿⡶⠖⠂⠀⠀⠴⢶⠶⠶⢶⡲⠆⠆⠀⠐⠛⠒⠒⠂⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠐⠀⠑⠉⠉⠩⠉⠠⣉⠭⢤⠭⡄⠩⠭⠭⠉⠉⠁⠛⠛⠚⠉⠀⠀⢻⡿⠋⠛⠁⠈⠛⠑⠋⠁⠀⠘⠙⠉⠉⠉⠁⠀⠀⡀
⣦⠤⠴⠶⠤⠴⠴⠶⠶⢶⠾⠿⠶⠶⠶⣦⣴⢦⣶⡶⣦⠴⠶⠶⠦⣤⠤⠶⠾⣷⣶⣶⣤⣶⣤⣤⣤⣴⣶⠶⠶⠶⣶⢶⣶⣦⡴⠴⠶⠶⠶⠶⠶⠾⠷⠷⠶
⣇⠀⠀⢀⡀⣀⣀⣀⡀⠀⠀⠀⠀⠂⡀⣈⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠃
⣿⠤⠤⠈⠉⠈⠧⠍⠛⠿⠿⠶⠤⠤⠧⠎⠉⠁⠁⠉⠃⠛⠻⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣀⢉⣈⡐⢲⠾⣅⣈⣨⣯⡉⠂⠀⠐⠒⣛
⡿⠶⠀⠀⠀⠀⡀⠀⠀⠉⠉⠁⠀⢠⠄⠀⠀⠀⠐⠒⢒⣂⣴⣦⡤⠐⠒⠒⠒⠒⠀⠘⠃⣀⣀⣄⠀⠀⠌⢩⠉⠁⢀⣀⣉⣙⣉⣈⣉⣙⡟⠻⠒⠒⠊⠉⢋
⣧⣤⣀⡀⠉⠉⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⢉⣉⡉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣁⣂⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠃
⣟⠃⠀⠀⠠⠦⣀⢀⠠⠰⠢⠀⠬⠥⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⣉⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣀⣀⣀⣀⢀⠀⢀⣠⣠⣄⣉⠁⠁⣀⢀⣐
⠛⠒⡾⠛⠛⠓⠾⠶⠖⠶⣶⠶⠿⠿⠿⠛⠿⠛⠿⠓⠿⠻⠒⠓⠛⠛⠛⠛⢿⡿⠛⠛⠛⠓⠛⠛⠛⠳⠖⠒⠶⠿⠷⠿⠿⠿⠛⠛⠻⠿⠿⠿⣿⠷⡿⠳⠟
⠤⢤⣤⠄⢀⠠⡄⣄⣠⣤⣌⣀⣈⢈⣉⣀⣠⣤⣤⡤⠄⠠⠤⠤⠤⠤⠀⠠⣼⣷⣀⣀⣀⡀⡠⣤⡀⠀⢠⣀⣀⣀⣀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠁
⣈⣉⣀⣠⠤⠤⠄⠀⠀⠀⠀⢈⠁⠠⠀⠤⠤⠰⠤⠀⠀⠀⠀⠀⠀⠤⠤⢤⣽⣿⣥⣤⣀⣀⣀⠚⠷⠖⠲⠗⠒⠐⠠⡤⠤⠄⠀⠀⠀⣀⠀⣀⣀⣀⢀⣀⣀
⣛⣳⠶⠦⠤⠤⠤⠤⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠭⠙⠃⠓⠖⠀⠀⠀⢨⡿⠂⠀⡀⠀⠀⠀⣀⣂⡒⠒⠠⠖⠀⠀⠀⠀⠐⢳⠶⠤⠤⠀⠉⠁⠀⠀⠚
⠉⠁⠀⠐⠒⠒⠀⠦⠤⠀⠀⠠⠄⠠⠀⠠⠀⠀⡀⠀⠀⠀⠀⠂⠐⠒⠚⢻⣿⡏⠉⣀⣀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠐⠠⠂
⠀⣀⠀⠀⠀⠐⢂⡐⠂⠀⣀⡀⠀⠬⠤⠶⠤⠍⠡⠉⠠⠤⠄⠐⠂⠒⠒⠀⣺⣿⡄⢀⡄⣀⣤⣤⣦⣤⣤⣀⣄⢠⣤⠖⢶⠦⠤⠠⠤⢤⠄⠀⠀⣀⣰⣐⣂
⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣛⠻⠟⠛⠟⠛⠛⠛⠛⡟⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠛⠿⠛⡋⠛⠛⠛⠛⠛⠛⠉⠛⠛⠋⠉⠉⠉⠛⠉⠛⠙
⣿⠀⠀⠀⠀⡁⠠⣤⢬⣭⣁⢀⣉⣉⣩⣭⡬⠀⢠⡄⢤⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⢀⠀⣄⣀⠀⢠⠄⠀⠀⠤⠤⠀⠀⠀⢀⣤
⣯⠌⠀⠀⠈⠉⠉⠋⠉⠈⠉⢉⠉⠉⠀⠀⠀⢀⣂⡀⣀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠤⠤⠄⠀⠀⠀⠀⠤⠤⠄⠐⠀⠀⠉⠈⣧⠔⠲⠶⠶⣶⣀⣀⣀⡀⠤⡬
⣇⠀⠀⠛⠉⠀⠤⠀⠀⠀⠀⠉⠁⠀⠀⠲⠴⠖⠒⠒⠛⣛⣉⡁⠀⠀⠀⠀⠀⠀⠲⠶⠶⠛⣓⠀⠀⠀⠀⠁⠀⠀⠲⠤⠐⠒⠒⣒⣒⠒⠒⠀⠀⠀⠀⠀⠒
⣿⠍⠉⠁⣀⡀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠀⠀⢒⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⢤⡄⠠⠤⠄⠀⠀⠈⢙
⣷⣀⣈⣠⣀⣀⣉⣁⣀⣀⣈⢀⣉⣭⣁⣠⣀⣶⣤⣀⣂⣄⣀⣀⣀⣀⣀⣄⣘⣳⣴⣦⣤⣀⣀⣀⣒⣒⣂⣐⣛⣿⣒⣲⣶⣀⣢⣰⣶⣖⣶⣶⣤⣄⣠⣀⣰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡶⠶⢶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠛⠛⠭⠥⠸⣿⠀⠀⠀⠀⠉⢻⠻⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠉⠛⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠘⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⠀⠀⠀⠀⠔⠀⠀⠤⠤⠤⡄⢤⢀⡀⣾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⠀⠀⠀⠀⢠⡟⠉⠻⡷⠃⣰⠀⠀⠀⠀⠀⣀⡠⠤⠤⠄⢠⢉⣈⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⢠⡿⠁⠰⠀⠀⢀⢻⠷⠆⣀⢪⠄⢁⠀⠖⠒⠲⠶⢆⣄⡉⠳⡈⢻⣶⡀⠀⠀⠀⣠⢴⢾⣿⡀
⠀⠀⠀⠀⢀⡿⠃⠀⠃⠀⠀⠟⠳⠃⠀⢹⠃⣶⠋⠄⠀⠀⢐⡀⠀⠀⠊⢦⢹⡄⣼⣿⢶⠀⠁⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⢸⠁⠀⠀⡜⢠⠏⢸⢠⠀⢠⡧⠝⠐⢲⢘⣿⠀⡇⢰⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠈⣼⠸⠀⣠⣧⣼⡄⢇⠘⣴⠋⠀⠀⠀⡌⢸⡜⢰⠃⠈⣿⡄⠀⠀⢸⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣿⠀⣀⣀⠀⠀⢋⠺⠏⠙⢇⠁⡠⣄⠳⠬⠀⣩⡀⠜⢀⣾⠋⡻⠁⣿⣿⢷⠀⠠⢼⠠⣤⣼⡇
⠀⠀⠀⠀⠀⢻⡟⠉⠁⠀⠀⠈⠳⣞⢦⣈⠷⣌⡒⠕⠒⡒⣶⡤⢀⠝⢁⠜⠀⠀⡇⢸⠀⠀⡰⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⢀⠰⢿⡻⡳⢤⣉⠓⠲⠦⢬⡤⠴⠒⠉⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⣠⣴⠯⠑⠀⠀⠈⠓⠛⢯⡑⠦⣘⣶⣤⢖⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠀⠀⠀⠂⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⠁⠀⠀⠀⠀⠐⢤⡤⠄⡀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠿⢷⣦⣄⣀⣀⣀⠀⢘⡇⠀⠀⠀⣀⣠⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁

⣀⣀⡀⡀⠀⠀⠀⡀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⠀⢀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠒⢒⢶⣖⡂⠀⠀⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⢦⠀⠀⠀⠀⠀⡠⢤⡠⣄⡀⠀⠠⣄⠀⠀⡀⠀⠀⢀⣤⡀⠀⠀⡀
⠀⠀⠀⢸⠀⠂⠀⠀⠀⠆⠀⢠⢤⡶⢾⣿⣶⠖⠂⠀⠀⣸⠀⠀⠀⠀⣸⣧⡀⢹⢿⠿⣄⠀⠙⠀⢀⠹⡆⠀⠈⢹⠇⢠⣸⣿⡆
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢐⠚⠊⡭⠝⠀⠴⠄⠀⠀⠸⠀⠀⠀⠀⠹⠿⠅⠜⠀⢴⠏⠀⠀⣰⣾⣶⣿⡄⠀⣈⣀⣼⣿⣏⡡
⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⣀
⠀⠀⠀⠀⠀⠠⠺⠁⠀⠠⠄⢈⠀⢅⠄⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠀⠠⢤⣀⣀⣀⣀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⢰⠇⠀⠀⠐⢈⡁⢀⠙⠘⠃⡀⠀⠀⢠⣤⣤⠄⠀⢀⠤⡄⢰⣶⠂⠀⠀⠍⠉⠉⠀⠀⣈⣉⣉⣉⣉⣉⣑⡒⢂⣀⣛⣒⣺
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠺⠿⠀⠂⠀⠀⡇⠀⠀⢠⠤⠀⠀⠀⠀⠈⢉⣉⣉⠉⠀⠤⠤⣄⠀⠀⠀⠈⠉⠉⣉⢙⠛⡿⠿⠿⠟⠩⠤⠽
⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢨⠤⠄⠀⢤⠄⠀⠉⠉⠉⠩⠤⠤⠬⣿⣉⣙⣻⣛⡛⠃⠀⠈⠀⠉⠁⠈⠠⠍⠁
⠀⠀⠀⠀⠀⠀⠙⠒⠤⣀⡀⠤⠴⠊⠀⠀⠀⠀⠈⠁⠈⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁