Circus Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⣯⣄⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠉⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡏⠸⣇⠠⣕⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣿⣿⣿⣿⣧⣤⣿⣷⣿⣿⠇⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⣴⣿⣿⡏⣼⡏⣿⢻⡆⢷⡸⡆⣼⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡗⠀⢈⣼⣿⣿⡿⢠⣿⢸⣿⠘⡧⠈⣷⡙⣎⢧⡉⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠛⠀⢃⣴⣿⣿⣿⣿⢫⣿⡏⢸⠛⡀⢿⣭⠙⣷⡌⢷⡑⣄⠀⢃⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠃⠁⠀⣀⣴⣿⣿⣿⣿⡿⠁⣸⣿⠁⢸⠛⠁⢸⣆⡄⠘⢿⣆⠙⣮⡳⢮⡀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠠⡖⠀⠈⠁⠀⣀⣠⣾⣿⣿⣿⣿⣿⠟⠀⣰⣿⡏⠀⣿⣤⡇⠀⣿⣿⣆⠈⢿⣷⡈⠻⣦⡙⢦⣄⠀⠋⠀⠁⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣴⣻⣿⣿⣿⣿⣿⡿⢋⠀⣰⣿⣿⠁⠀⣏⣀⣻⠀⢸⣿⣿⠆⡀⠻⣿⣦⡈⠻⣶⣍⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⢈⣶⣿⣿⠇⠀⢰⣿⣿⣿⡀⠀⢿⣿⣷⣾⡀⠈⠻⣿⣦⡀⠙⠿⣦⣍⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢋⡀⢀⣴⣿⣿⣿⡟⠀⠀⣼⣿⣿⣿⡇⣐⠈⢿⣿⣿⣿⣦⣀⠈⢻⣿⣷⣤⣈⣙⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣫⠽⠛⣻⣵⣶⣿⣉⠁⠤⠈⢉⣿⣶⣶⣟⠹⠉⠉⣫⠶⣾⠻⠁⢒⡿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢛⠻⣿⣷⣿⣿⢿⠛⠙⠻⣿⣿⣿⡏⠀⠀⣿⣿⣿⠏⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠏⢹⣿⣿⡿⢠⣧⣞⣽⣿⣿⣿⣆⡑⣄⠀⣿⣿⣿⠁⠀⢀⣟⠋⡾⠀⠀⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣿⣿⡟⠀⠀⢸⠇⠀⠃⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⡿⠛⢹⡟⣿⡇⠀⠀⢸⠀⢨⠀⠀⣾⡏
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⢿⣄⣿⣿⣿⣿⣿⣿⢣⡆⡈⢿⣹⣇⣀⣀⣿⣶⣾⣤⣤⣿⣧⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠷⠾⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢻⣿⡝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣤⡾⠟⣡⣿⠋⣿⣌⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠘⣒⣛⣛⣉⣀⣀⣀⣉⣉⣀⣘⣛⣓⣀⣉⣛⣛⣒⣂⣐⣂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢠⣿⡿⠁⠀⣾⣿⡇⠀⣿⣿⠀⠉⣿⣿⡉⠉⢿⣿⡍⠉⢿⣧⡀
⠀⠀⠀⠀⠀⣤⡴⠚⠋⠀⠐⠛⠛⠁⠀⠘⠛⠛⠀⠀⠛⠛⠀⠀⠙⠛⠃⠀⠈⠛⠛⠂⠀⠙⠛⢦⣤
⠀⠀⠀⠀⠀⠀⠑⣴⡇⠀⢸⣷⣤⠎⠐⢢⣾⡆⠐⠳⣄⣠⠞⠂⣴⣷⡔⠂⠱⣤⣾⠀⠀⣿⣦⠊
⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣸⣿⣿⠀⠀⣼⣿⡇⢀⠊⣾⣿⠐⢀⣿⣿⡇⠀⠀⣿⣿⡆⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⣿⡟⠀⠀⣿⣿⡗⠀⣠⢿⡿⣄⠀⣿⡽⣧⠀⠀⢿⣿⡇⠀⠘⣿⣷
⠀⠀⠀⠀⠀⣰⣿⠇⠀⢸⣿⣿⠃⠠⢴⣾⣿⣧⡴⢋⣾⣿⠘⢷⣼⣿⣿⠀⠄⢸⣿⣿⡀⠀⢻⣿⣧
⠀⠀⠀⠀⣰⣿⡟⠀⠀⣾⣿⡍⠈⢀⣸⣿⠿⠋⢠⣾⣿⣿⡇⠄⠙⠻⣿⣆⡀⠈⣯⣿⢇⠀⠈⣿⣿⣧
⠀⠀⢀⣼⣿⣿⡁⠆⣰⣿⣿⠃⠀⠙⣏⠀⠀⣰⣿⣿⣿⣿⡇⠘⣦⠀⠀⣹⠏⠀⢹⣿⣿⡄⠰⠘⣿⣿⣷⡀
⢠⣴⣿⣹⣧⣥⣤⣴⣿⣿⣿⣤⣤⣤⣌⡻⢿⣿⣿⡿⠿⠛⠀⠀⠘⢿⠟⣡⣤⣤⣤⣿⣿⣷⣤⣤⣼⣷⣿⣽⣦
⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠂⠂⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⠀⠀⠀⠀⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠀⢀⡁
⠀⠀⠀⡈⠙⠉⠀⠙⠃⠈⠀⠀⠀⠀⠀⠀⠀⢰⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⢂⠜
⠀⠀⠀⠀⠦⡔⢦⣀⠦⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠆⢼⠀⢰⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢐⠒
⠀⢀⣀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⢇⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠰⠚⠂⠉⠐⠂
⠀⢾⠀⠠⠋⠀⠤⠘⢴⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠄⠄⠈⠉⠑⠢⠤⠖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂
⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠐⠀⠇⠐⠂⢰⢂⠀⠀⣀⠀⠀⠀⠀⢰⡄⠤⢺⡗⠠⢀⣴
⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⡾⠐⠁⠀⠒⣊⣤⢊⣀⣭⠃⠀⠀⠀⠀⠛⠃⠀⠈⠁⠀⠘⠉⢃
⣦⣤⡀⢀⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠈⠑⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠙⠿⠷⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣆⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠂⠒⠂
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣰⡦⡄⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠉⠀⠿⡀
⠀⠀⠀⠀⣀⣠⣤⢴⣗⣾⠟⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⣖⣦⠈⠓
⠀⠀⠀⠈⠿⠿⠿⠏⠈⠋⠁⠀⠈⠀⠀⠀⠀⠀⣠⡞⠛⠛⡟⣟⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡯⠀⠾
⠀⠀⠀⠀⠀⠀⢀⡄⠁⡠⠐⠀⠀⠀⠀⣀⡤⣪⣄⣤⡠⣄⠥⣨⡌⢣⠓⣤⣀⡀⠀⠀⠀⠂⡸⠉⠇⠀⡦
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠘⠃⠋⠉⠋⠙⠁⠃⠘⠀⠘⠉⠃⠀⠀⠀⠀⠇⣤⢶⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢤⡀⠀⠀⠀⠈⠉⠀⠀⠀⡠⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣾⡿⠉⠀⠀⡀⠀⠀⠉⢹⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⢀⡄⠘⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⢠⠀⠤⠀⡀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠩⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⢚
⠀⠀⠀⠀⢀⠀⡀⠀⡀⠀⠄⠀⠀⠀⠀⠈⠠⡂⠀⠀⠀⠀⠀⡾⡞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠸⠷⢾
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀⡀⢐⢀⠀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡨⠷⠼⢀⠁⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠡
⠀⢰⡄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠃⠀⠀⠀⣀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇
⠀⠈⠁⠀⣋⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠩⡄⠐⠂⠖⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢷
⠀⠀⠀⠐⡟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢈⡈⠀⡀⠀⠐⠀⠐⠀⠀⠀⣀⣀⠀⠀⠀⠀⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠇
⠀⠀⠀⢠⡿⡟⣄⠙⠀⣄⠛⢀⡆⠀⠐⠛⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠂⠘⠇⠀⠀⠀⢨⠅
⠀⠀⠀⢸⢳⣧⢻⠀⠀⠘⢻⡏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠋⠀⠀⠀⠉⠙⠂⠀⠀⠀⢹⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡇⠿⠆⠀⠀⠀⠀⠀⠀⢀⠋⣰⣄⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡐⢂
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠃⠛⠻⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⡙⢐⠹⣱
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡐⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠐⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠁⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠂⡄
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠄⠂⠀⠀⠰⠀⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⣤⠄⢀⠎
⠀⠀⠀⠀⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠔⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠠⠨⠴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠌⠀⠂⠀⠀⠐⠀⠉⠉⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢰⠺⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢀⠀⠀⠘
⠀⠀⠀⠀⢌⠀⠁⠀⠀⡔⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠈
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣰⡆⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁
⠀⠀⠀⠀⠈⡇⠀⠀⡀⠀⠀⢰⠀⠀⠀⣀⠤⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠸⢢⠂⠀⠀⠀⠀⠀⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⡿⠛⢿⡆⠀⢸⡇⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠘⣗⠋⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⣦⣼⣧⠤⠤⣶⣿⡇⠈⠀⠀⠀⠀⠈⠂
⠀⠀⠀⣆⠀⠐⠉⠀⠀⠀⠀⡷⠈⠀⠀⠀⠐⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣠⣽⣿⣿⣿⠀⣴⠛⠉⠁⠀⠠
⠀⠀⠀⣾⠣⡀⠀⠀⠀⠀⠈⡷⠎⠃⠀⠀⠀⠲⡅⠂⠀⠀⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠀⣤⢸⣿⠋⠉⢹⡏⠄⠛
⠀⠁⠒⠛⠂⠉⠂⠀⠀⢤⡀⣤⣄⠙⠂⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⡄⠀⡁⠀⢀⡟⠈⠙⠀⠀⢸⡇⠀⠀⠐⡖⠀⠠⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣺⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠂⠀⠉⠀⠀⠀⠀⠀⢸⡇⠦⠀⠀⠿⠀⠀⠀⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠐⠶⠶⠲⠶⠂⠀⠀⠀⠀⠀⠀⠤⠤⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⡜⡇⠙⢕⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⡡⠀⠀⡜⠀⢁⠀⠀⠑⢄⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢐⢢⠀⢀⠀⠀⠀⠀⡀⠔⠀⠀⠀⡜⠀⠀⠈⡄⠀⠀⠀⠉⠢⡀⠀⠉⠒⠤⣀⠀⢸⡩⠔
⠀⠀⠀⠀⣾⠃⠀⠀⠀⢠⢶⠁⠀⠀⠀⢀⣼⠠⠀⠀⠀⢱⡄⠀⠀⠀⠀⢈⠦⡀⠀⠀⠀⢘⠟⡇
⠀⠀⠀⠀⡇⠁⠒⠖⠊⠁⠘⠈⠀⠒⠒⠁⠘⠈⠐⢲⡆⠁⠈⠑⠂⠐⡖⠁⠀⡏⠐⠒⡎⠁⠀⡇
⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⡀⠀⠀⡆⠀⠀⡇⠀⢀⡟⡟⡀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⠀⠇⠀⠀⡇⠠⢫⠁⢡⠰⡀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠇⠀⠀⡇
⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠇⠀⢰⠀⠀⢀⡗⢁⠇⠀⠘⡀⠘⡇⠀⠀⡇⠀⠀⡇⠀⠀⢀⠀⠀⢰
⠀⠀⠀⠸⠀⠀⢸⠀⠀⢰⠀⠀⢸⠀⡠⠊⠀⡜⠀⠀⠀⢃⠀⠈⢆⠀⡇⠀⠀⡇⠀⠀⠘⠀⠀⠀⠇
⠀⠀⢀⠇⠀⠀⡎⠀⠀⡈⠀⠀⣴⠊⠀⠀⡰⠁⠀⠀⠀⠘⡄⠀⠀⠑⡇⠀⠀⡇⠀⠀⠀⡀⠀⠀⠸⡀
⠀⠀⡜⠀⠀⢠⠁⠀⠀⠇⠀⠀⠇⢣⠀⡠⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⡇⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢣
⠀⡔⠀⠀⢀⣎⣀⣀⠼⠀⠀⠸⠀⠠⠷⠁⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠧⠀⠀⠸⠀⠀⠀⠸⠄⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣦⣼⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠶⠛⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠜⠐⢾⣷⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠔⢀⣤⡀⡀⣸⣿⣿⣄⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠜⠟⠟⡿⣿⣿⣿⣿⡄⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠆⠌⡐⠀⠁⣿⢸⣏⣿⣿⣤⡾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⡡⠂⡰⠁⠞⠀⣿⠀⣿⡘⣿⣿⣷⣼⠷
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣤⠀⡈⢋⠜⢁⠸⠁⣴⠀⠀⣨⠀⢿⡇⢹⣿⡟⣿⣦⡀⣠⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⠒⢾⡇⠀⠉⣊⡰⠋⣀⡴⠁⢰⣿⢀⢰⣾⡀⢸⣿⡀⢻⣿⣎⢻⣿⣿⣄⠸⣷⠆⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡚⠁⢀⣜⡟⠁⢰⣿⣵⣿⣿⡿⡇⠈⣯⣡⠀⠻⣿⣦⡙⢿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⡀⠀⠀⢴⠿⠋⠀⢤⣏⠋⠀⣠⣿⣿⣿⣍⣥⣿⣷⣦⢸⣧⣧⠀⠹⣿⣷⣄⠙⢿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⢹⣶⠶⢿⣤⣤⣤⣾⣦⣀⣀⣴⡇⠀⢘⣿⠁⠀⡉⣧⣉⣉⣉⣥⣀⠈⢉⣹⣥⣤⣴⣶⣶⡟⠁
⠀⠀⠀⠀⠀⢘⡞⠀⢸⣿⠋⠉⢸⣿⡟⠋⢩⠟⣠⣾⣿⢧⢷⣷⠉⠙⠛⢿⣿⡟⢿⠟⣿⣿⠉⢻⣿⣿⣧
⠀⠀⠀⠀⠀⡶⠃⠀⣼⡏⠀⠀⣿⣿⠀⠀⣱⢟⣿⣿⣿⣇⢎⣿⢠⠀⠀⢸⡿⣇⠀⠀⢿⣿⡇⠀⢿⣿⣿⡄
⠀⠀⠀⠀⣼⣏⠀⣸⣿⠁⠀⣼⠿⣛⣵⡾⢃⣾⣿⣿⣿⣯⣿⣿⣸⠀⠀⠸⣿⣿⠀⠀⢸⣿⣿⣠⣸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠐⠉⠛⠛⠛⠳⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣿⣿⣷⣶⣾⣿⣿⠿⠛⠛⠻⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⠉⠉⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⡂⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣿⡿⣸⣏⢿⣿⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⠟⡼⢁⡟⢻⡈⢧⠻⣜⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢋⡴⢋⡾⠁⣼⠁⠈⣧⠈⢷⡙⢦⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⡴⠋⣠⡞⠁⣼⠃⠀⠀⠘⣧⠈⢳⣄⠙⢦⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⣋⣤⡞⠋⣀⡴⠋⢀⣼⠃⠀⠀⠀⠀⠘⣧⡀⠙⣦⣀⠙⢳⣤⣙⠳⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⣿⠉⠉⠉⣿⠉⠉⠉⢻⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠉⠉⣿⠉⠉⠉⣿⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢰⡆⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣾⣷⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢠⡇⢸⡆⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⣼⠁⠈⣧⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⢠⡏⠀⠀⢹⡄⠀⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⠀⣾⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⣿⠀⠀⠀⣧⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢸⣼⠃⠀⠀⠀⠀⠘⣧⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⣿⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣿⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢀⣀⣿⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣯⣀⡀⠀⣿⠀⠀⢸⡇
⣀⣠⣤⣤⣤⣤⣤⣤⣼⣧⣤⣴⣿⣾⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣷⣿⣦⣤⣼⣧⣤⣤⣤⣤⣤⣤⣄
⣿⠳⣄⠠⣤⠄⣠⠞⠉⠻⣦⡀⢦⠆⣠⡾⠉⠙⢦⡀⢦⡤⢀⡴⠋⠙⢧⡀⢴⡴⢀⡴⠋⠙⢷⣄⠰⡴⢀⣴⠟⠉⠳⣄⠠⣤⠄⣠⠞
⣿⣀⣈⣳⣤⡞⠁⠀⠀⣀⣈⣻⣦⣾⣋⣀⣀⣀⣀⣙⣦⣴⣋⣀⣀⣀⣀⣙⣦⣴⣋⣀⣀⣀⣀⣙⣷⣴⣟⣁⣀⠀⠀⠈⢳⣤⣞⣁⣀
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠴⠿⠃⠀⢠⣄⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡖⡹⢻⡳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣿⣼⣿⣏⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⢻⠏⠻⠁⣿⢹⣿⣿⣿⣿⣄⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉⠁⡰⣣⠏⡰⡄⠀⢹⢘⣿⣹⣿⣎⢿⣷⡄⠉⠛⠒⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⠛⠛⠉⠀⠀⠀⠀⣤⢞⣵⠏⠐⠹⠀⠄⢹⠘⣿⣇⢻⣿⣎⢻⣿⣷⣄⠀⠀⠀⠉⠙⠚⠲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢀⣴⠟⣡⡾⠃⣰⢃⠆⠀⠀⠬⠄⠙⢿⠀⢻⣿⣷⡝⢿⣿⣷⣄⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⣶⠟⡡⢪⠞⠁⢐⡁⠀⢊⣤⠖⢛⡚⢲⣤⡐⠠⠿⣿⣿⣦⠙⢿⣮⣻⣶⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣼⡾⢋⣡⢾⣴⣃⣠⣢⣿⣲⡾⠯⡬⢥⣿⡷⠾⠶⠾⢶⣶⣿⣿⣿⣷⣶⣍⣻⣷⡿⣟⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣤⣰⣶⠷⠾⣿⣿⠞⠛⣫⡿⣯⣉⣀⣠⣿⠟⢷⣤⣴⣾⣿⣷⣤⣤⣾⠿⢻⣧⣤⣬⣿⣿⣷⣤⣴⣿⣿⣍⣻⣿⢻⡴⣦⣄
⠀⠀⠀⠀⠀⠀⠘⣿⢷⣾⣷⣶⣿⣿⣿⠻⣿⡇⢿⣿⠉⡟⣱⡇⠀⠀⢩⣿⣿⣿⣿⠘⠸⠀⢸⡏⢿⣹⣿⣿⢻⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡌⣿⢹⡟⣿⠸⣿⡆⢿⡃⢸⣿⡃⡰⠋⡇⠁⠄⣸⣿⣿⣿⣿⣆⠀⠀⢸⣷⡈⠻⣿⠃⣸⣿⢿⣿⣸⣿⢸⡟⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠸⡇⣿⠀⢿⠁⢸⡇⠈⣿⠻⣆⡼⣷⣶⣶⣿⣿⣿⣿⣿⣿⣷⣤⣾⣇⢑⣠⣿⠀⢹⣇⣼⡟⣿⡏⢸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢱⣸⡆⡇⢹⠀⢸⡆⠈⡇⠀⢻⠀⢹⣷⣿⠏⣿⣿⣿⣿⣿⣿⣿⡟⠠⣿⣯⣾⡿⡟⢠⢸⠇⢹⢻⣿⡇⢹⣇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠚⠶⠾⠦⠦⠧⠦⠼⠤⠼⠿⠿⠤⠘⠛⠛⠛⠛⠛⠛⠃⠤⠼⠿⠿⢧⡷⠼⠼⠶⠞⠚⠛⠋⠉⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣦⣭⣦⣬⣬⣦⣵⣿⣽⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣒⣶⣿⣦⣤⣾⣷⣖⣊⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⠿⠿⢿⠻⡿⢫⡿⠿⢿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣽⡿⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠐⠛⠿⣿⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡠⣿⣯⣥⣴⣦⡄⠀⠀⠐⣄⢀⡴⠀⠀⠀⢤⣶⣦⣭⣿⡧⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣭⠉⣿⡶⠀⠀⢀⠀⢹⣾⠃⢀⠀⠀⠰⣾⣏⢩⣽⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⡿⢋⣾⣿⡄⣘⣶⣾⠁⠈⡟⠀⢸⣶⣞⡁⣼⣿⣆⠻⣿⣿⠧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠈⢿⣿⡇⠈⠛⢿⠀⠀⠁⠀⢸⠟⠊⠀⣿⣿⡟⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡇⠀⠈⢻⣿⣆⢠⠃⠀⠀⠀⠀⠀⢣⣀⣼⣿⠏⠀⠀⢿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢁⣿⣿⣴⣦⠀⠀⢻⣿⠇⠀⣤⣤⣠⣤⡄⠀⢻⣿⠏⠀⠀⣶⣼⣿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⢳⡈⣿⡀⠀⠀⠙⡟⠁⠀⠀⣸⡟⣠⠎⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠟⢹⣿⣿⡟⢰⠀⢻⣿⣷⣤⡀⠤⠦⠄⣠⣴⣿⣿⠇⢰⡀⣿⣿⣿⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣵⣿⠀⠘⣿⣿⣿⡀⠀⠀⠀⣨⣿⣿⡟⠀⢸⣷⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣼⣇⢁⠙⣿⣿⣶⣶⣾⣿⡿⠁⠁⣾⣌⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⢹⣿⢻⣿⣜⣶⣌⡻⣿⣿⡿⢋⣤⣞⣼⣿⢻⣿⠃⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠻⣿⣿⣟⡻⢿⣿⠿⢛⣿⣿⣿⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⢿⣆⢀⣴⠟⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣶⣦⣶⣶⣄⠀⢀⣴⣶⣦⣴⣾⣿⣿⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢀⣴⣶⣶⡀⠀⠀⣰⣶⣶⣄⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃⠀⣰⣿⠏⢻⣿⡇⠀⠀⣿⣿⠋⣿⣷⡀⠀⢻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠛⠉⠛⠀⠀⣿⣏⠎⢡⣽⡇⠀⠀⣿⣥⠉⢏⣿⡇⠀⠘⠋⠙⠻⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠁⠠⠂⠀⠀⠀⢻⣿⣇⣾⡿⠀⠀⠀⠸⣿⣆⣿⣿⠃⠀⠀⠀⢢⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⣥⠀⠀⠀⠀⠀⠉⠛⠋⣀⣤⣴⣤⣄⡉⠛⠋⠁⠀⠀⠀⠀⢀⡅⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⢣⠈⠀⢀⣶⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⢀⣶⡆⠀⠈⢠⠃⢠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡁⠀⠘⣿⣿⡄⠀⠘⢿⣿⣿⣿⣿⠟⠀⠀⣼⣿⡇⠀⠀⣁⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠹⣿⡿⣶⣄⣀⡈⠉⠉⢀⣀⣴⣾⣿⡿⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠙⢿⣮⡻⢿⣿⣿⣿⣿⠿⣫⣾⠟⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡈⠻⣿⣶⣦⣤⣤⣶⣿⡿⠃⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠈⠙⠛⠛⠛⠛⠉⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⠤⠤⠶⠖⠋





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⡀⠤⠤⡀⠀⠀⢀⡀⠀⠀⠀⠀⡰⠁⢷⡀⠌⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⢀⣠⣦⠵⠒⠀⢢⡀⢠⣄⣩⡽⠁⢳⡿⢿⣿⣾⠇⡄⣁
⠀⠀⠀⠀⢀⠠⠀⠨⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⢣⢬⣬⣽⠃⣾⡟⠀⠀⣼⣿⣾⣴⣈⣩⣶⣤⣀⠀⡀
⠀⠀⡤⣊⠐⣖⡂⠙⠂⠀⠀⡖⠨⠆⠀⠀⠀⠀⠀⠈⢯⣽⢇⣶⢻⣿⠶⠙⠫⠟⢿⡟⠙⠻⣿⠟⠻⠇
⠀⠀⢧⠉⡘⡎⢉⠀⠀⠀⠀⡀⠀⠂⠔⠈⠀⠐⠀⠀⠘⡧⠿⢿⡞⠁⣀⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⡶
⠀⠀⠈⢆⠁⣽⡌⠀⠀⢀⠠⡀⠀⠄⠓⠐⠂⠔⠁⠈⠀⠹⠀⢰⡆⠀⠀⠁⠀⢤⢄⠀⣿⣶⣄⡀⡼⢁⡀
⠀⠀⠀⠈⠆⢟⣷⠠⠀⠀⠀⣈⠁⠀⠀⠀⠀⠄⢉⡀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⡀⠺⣿⣿⣿⣿⠿⣿⣆⡀
⠀⠀⠀⠀⠘⣤⢿⣧⠀⠢⠄⠤⠀⠀⠀⢀⠀⠀⠠⠄⠁⠂⠈⢦⣀⠀⠂⠄⢠⣤⣾⡅⣀⡞⣽⠿⢿⣷⣿⣿⠷
⠀⠀⠀⠀⠀⠘⡜⣿⣆⠀⠀⢐⢃⡀⠀⠀⠀⡀⠓⢄⠠⠀⠀⠘⣟⡳⣦⣤⣸⣟⣹⣿⠏⣼⠳⣶⣾⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠘⡌⣿⡄⠀⠀⠂⠀⠉⠈⡁⠀⠀⠄⠀⠀⠀⢀⡹⣠⠏⠛⠻⠻⢻⡿⣼⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡘⣷⡀⠀⠀⠀⠀⠈⠀⠁⠀⠀⣠⠀⠀⢈⣀⠡⠈⠉⠉⢈⡁⣹⣏⡍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⣧⠀⠀⠀⠀⠤⠰⠀⠀⠈⠀⠀⠰⢀⣩⣄⣠⠄⣤⡈⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢧⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠛⡯⠙⡛⠹⢗⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠢⠀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠈⠨⠋⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠈⠒⣦⣤⣤⣤⣀⣀⡠⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠓⢄⡀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠈⠉⠢⡈⠙⠳⠒⠚⠉⠀⠀⠈⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢨⠀⠀⣰⠀⠀⡀⢀⠀⠰⡀⠈⣦⠆⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡟⠀⢠⠀⠀⢸⡇⠀⠇⠀⠰⡡⠊⣤⣄⢳⢀⡏⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⡇⠀⡇⣠⣴⡟⠀⠤⢤⡀⠈⠀⠀⠀⠀⣸⢞⠇⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣧⣿⣿⠀⠐⠒⠦⠄⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠉⠁⢙⣟⡏⠀⠀⠒⠤⡀⠀⣠⠄⢠⣿⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣿⡟⠒⠤⠤⡀⢠⣾⣧⢄⡸⣿⡆⠀⠀⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⢀⣀⣀⠤⠖⠋⢀⣠⠟⠋⢿⣐⠠⠀⠀⣾⡿⡇⢸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣼⡀⠀⠀⠀⠀⠀⠀⢰⠀⢠⡇
⠀⠈⠓⠒⠒⠒⠊⠉⠀⠀⠀⣼⠮⢭⣥⣠⣿⡵⢃⣼⣿⣿⣿⣿⡛⢶⡦⢾⣤⣤⣤⣤⡜⠸⣿⠇⠀⠀⠀⠀⠀⠀⡏⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠯⠷⠌⣿⡶⠿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣠⠈⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡄⠀⣼⠁⠀⣿⣿⣿⣿⣿⣿⠙⠿⠿⡇⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣴⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢀⡟⠀⠀⢹⣿⣿⠿⠿⢿⡄⠀⠀⡇⠀⠀⠀⢀⣀⣼⣀⣀⣤⣴⠟⠛⠉⠉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠦⢀⡾⠁⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⢰⠁⠀⠀⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢰⢹⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠃⢈⣰⠟⠁⠀⠀⠀⠀⢸⣧⠀⠀⢠⠀⠀⠃⠀⠀⢸⣿⡿⠋⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠧⣤⣤⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢸⢠⠁⠀⠀⢀⣼⠋⠀⠀⠀⢋⠹⣿⡆⠀⠀⠀⡄⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⡀⠀⠀⣾⠃⠀⠀⠀⠀⠸⡄⢿⣧⠀⠀⠀⡇⠀⢰⡏⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠤⠀⠈⢷⠈⢀⠄⢻⣶⡄⠀⠀⠀⢀⠁⢸⡿⠀⢀⡀⢱⠀⠈⢇⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠀⠀⢨⡇⠀⠀⢸⣿⣿⠀⠀⠀⠈⠓⠚⡇⠀⠠⠄⠀⢧⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣄⠀⠀⣸⢿⣷⣶⣾⡿⠁⠀⠀⠀⠀⠀⠀⣧⣠⣤⣤⣤⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁

⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣾⣿⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⣿⣧⠐⢠⠀⠀⠀⠊⠉⠉⠀⡀
⠀⠀⠹⣿⣿⣿⡷⠀⠀⠄⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⢶⡾⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⣿⣿⠂⠀⢀⡔⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⠦⠀⠀⠀⠼⣿⡆⣠⣿⢉⢓⣀⠀⠀⠀⠀⠀⢠⣴⡦⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠒⠋⠀⠀⠀⠀⣸⣧⢋⢇⣼⣿⠇⠀⡀⠠⠤⠶⠛⠉⠀⠀⠀⠀⠁⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡿⣿⣿⣿⣾⡏⠁⠀⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡈⢻⡿⣿⣿⠃⠀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿⣇⣹⣾⣿⣿⡀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⢀⡴⢒⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⠀⢘⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⢠⣶⣄⣀⣠⣀⡀⠀⠀⠀⠑⠾⠋
⠀⠀⠀⡇⠂⡒⢤⣄⣀⣠⡀⠀⠀⠈⠙⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠁
⠀⠀⠀⢿⣤⣮⣿⣿⣿⣿⡷⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⠂
⠀⠀⠀⢸⡉⠉⢉⠻⠋⠁⠀⢀⣤⢤⣖⣋⠉⠀⠀⠀⠀⣀⢐⠀⢠⣾⡿⠄⢠⣾⣏⣀⣀⢀⢲⢩⣵⡗
⠀⠀⠘⢾⣷⣄⠘⢿⣷⣤⣾⣿⠿⠿⣿⠋⠀⠀⠀⣀⡐⠉⠸⢀⣄⣿⣿⣗⡟⠘⣿⠁⣹⡿⠛⢷⣆
⠀⠀⠀⣺⣷⣿⣿⡺⢿⣿⣿⠋⢾⣿⣿⣦⣴⡂⠀⣿⣛⠛⠂⠀⣿⣟⣏⡙⣿⣽⣿⣿⣴⣧⣤⢶⡌
⠀⠀⢀⣡⡼⣿⣿⣿⣾⣿⡿⢿⣾⣿⣟⣿⠋⠀⠀⠻⢿⣷⣌⣾⣿⣿⣿⣿⣋⣙⣿⡮⣤⣯⡶⠻⠁
⠀⠀⠀⠀⠀⠀⠈⠛⣿⠷⠟⠛⠓⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⣠⠴⠶⠦⣤⡤⠖⠒⢄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠘⣁⡄⠀⠘⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⡎⠀⠀⡀⠘⠢⣄⠀⠈⠣
⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⡐⢄⡂⢄⢢⡘⣦
⠀⠀⠀⣿⣾⢏⡠⠀⣠⠔⠡⢾⣿⣡⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣿⢿⣖⠈⡂⣤⡘⢮⠃
⠀⠀⠀⠸⣯⠞⢠⣞⠁⢊⣴⣿⣿⣯⡻⠛⠀⠀⠀⠀⠀⠀⢹⡿⠿⣿⣿⣿⣷⡌⠳⣝⢦⡱⡄⠁
⠀⠀⠀⠀⠿⣶⡿⢑⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠹⣾⣿⣿⣿⣿⢝⢦⡙⣷⡝⡆
⠀⠀⠀⠀⠀⠙⠇⣼⠇⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡡⣵⡈⠿
⠀⠀⠀⠀⠀⠀⠀⠙⢲⣟⣿⣿⣿⣿⣿⠁⣤⠀⠀⣀⡂⠀⠀⡇⠈⣿⣿⡿⣿⣾⣿⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⣼⣿⡟⠐⣿⡀⠀⣒⡆⠀⠀⣤⠄⡘⢿⣿⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⣰⡇⠀⠯⠄⠀⢀⣿⣦⠀⠈⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⣀⣤⠤⠀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢙⠘⠿⣯⠁⠀⠀⠀⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈⡄⢠⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠘⢆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢣⠈⣷⠀⠀⠀⡀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠌

⠀⠀⢀⡤⠒⠉⠉⠉⠑⠢⣄
⠀⡴⠁⠀⠀⠀⠀⢀⠀⠀⠈⢣
⡜⠀⠀⢐⠞⠓⣤⠖⠛⢦⠀⠀⠇
⡇⠐⡀⢸⠀⠀⡻⢣⡤⠃⠀⠀⢸
⢣⠀⠳⣄⡙⠋⠀⠀⢀⡠⠆⠀⡘
⠈⢦⡀⠀⠉⠉⠉⠉⠀⠀⠀⡠⠃
⠀⠀⠙⠢⢤⣀⣀⣀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⠈⠲⣄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠸⡀⠑⠂⡀⣀⡀⠀⢠⣾⠀⢸⡀⠀⠀⡐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠈⠻⣷⡀⠈
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠑⢂⢟⣟⠻⣥⡎⠋⠁⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⡔⡡⠔⠀⠙⠛⠤
⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠈⠲⢤⡄⠀⢠⠢⠎⠨⠧⠘⣣⣤⣤⣶⠤⠞⠀⢡⠤⠶⢋⡟⠉⠑⢮⡞⣄
⠀⠀⠀⠀⣴⡉⢷⠂⠈⠉⠇⠀⠀⠀⠀⠉⠒⢾⠃⢀⡤⢀⡞⠁⠀⠀⠙⡆⠀⢀⣘⡄⠑⡏⠀⠀⠀⢈⡗⠏
⠀⠀⠀⢠⠧⡑⠢⣉⡋⠀⡀⠀⠀⠀⠀⠀⠀⣿⠀⢈⢤⠈⠳⣤⣤⣤⠞⠁⠰⢉⠋⣑⣄⡙⢤⣀⣠⣾⠓⠞
⠀⠀⠀⠈⡗⢬⡖⠚⣁⡤⠟⠂⠲⡀⠀⠀⠀⠸⡀⠠⡠⠙⣀⠄⠂⠰⠀⠀⠈⢮⡀⡐⠈⣏⠉⠀⠘⠘⠉
⠀⠀⠀⠀⠈⠉⣽⠫⠀⠈⠀⣀⡤⠇⢀⡀⣀⡀⢱⣀⠀⠀⠈⢁⣰⠃⠀⠀⠀⢠⠓⣄⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠙⢦⡴⠶⣏⠉⢣⡀⢸⠘⢣⠠⠖⠈⢹⣿⡂⠻⠏⢉⠿⢙⣳⡏⣠⠋⠑⠖⠃
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⢦⡀⠙⢾⣄⠻⣀⠆⣆⣈⠀⠅⠀⠀⠀⠀⢨⢀⠟⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠈⠉⡹⠃⠈⠁⠐⢎⠳⡊⣉⠟⠁⠈
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⠃⠀⠀⠀⠀⠸⡆⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢻⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠘⠇⠀⠘⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠦⠴⠇⣀⡀⡀⠀⠄⠠⠤⠖⠀⡂⢉⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢰⠂⠐⠄⠀⠀⠀⠀⠀⠘⢃⣀⠇⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠁⠀⢈⠖⠒⠲⠶⢶⡆⠀⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠶⡾⠇⠀⠀⠀⢸⣧⣤⣀⣼
⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣄⠀⠀⠀⠀⠀⠀⣇⣀⡧⠀⠀⠀⠀⢀⡗⢀⡇⠀⠀⠀⠐⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠈⠙⠖⠐⠒⠾⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠉⠓⠒⠲⠒⠖⠋⠁⠀⠀⠙⣧
⠀⠀⠀⠀⠀⢧⠀⠤⠤⢀⣀⠲⠶⠖⠒⢲⠒⠒⠒⠂⡁⠀⠀⠀⡓⠒⠒⠒⠲⠦⢤⣄⣀⡀⠀⠀⠀⠀⣀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢄⠀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠛⠒⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⢻⢹⡏⣿⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⢸⠺⡇⢿⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⣙⣈⣉⣁⣫⣘⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢧⢯⡇⡟⣿⢻⣹⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⡽⣣⢏⡾⣸⡇⣿⠘⣧⢻⡝⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣫⠕⣫⠞⣡⡿⢡⣿⠁⣿⡆⠹⣧⠹⣦⣙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣩⠴⣋⣵⠟⣡⣾⡿⠁⣼⡿⠀⣿⣷⠀⠹⣷⡈⠻⣷⣌⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⢛⣩⡴⠞⣋⣵⡿⠛⣡⣾⣿⠟⠀⣾⣿⠇⠀⢹⣿⣆⠀⠙⣿⣦⡈⠛⢿⣦⣌⡛⠿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠫⠉⢅⠲⠾⢟⠉⢴⣾⣿⣋⣤⣾⣿⡿⠃⢀⣼⣿⡿⠀⠀⠸⣿⣿⡄⠀⠈⢻⣿⣦⣀⠉⠻⢿⣷⣦⣍⡛⠿⠿⡷⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⡇⠠⣤⡀⢤⣤⠒⠁⠐⠄⡉⠉⢉⡭⠵⣈⡩⠶⠭⣄⡉⣉⡤⠤⠟⠉⢻⣶⠀⣭⣤⠒⣾⡏⢳⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣿⡇⢸⣿⠀⠀⡀⢠⠀⠀⠁⠐⢲⠀⠉⠐⠒⠂⠀⠄⢂⡀⠀⠀⣿⣿⠀⢿⣿⠀⣿⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧⠀⣿⡇⢸⣿⠀⠐⠀⡜⢀⡄⠀⠀⠈⠀⢀⠀⠀⠡⠀⢠⡀⠳⡈⡇⣿⣿⠀⢸⣿⠀⢻⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢸⣿⠀⣿⡇⢸⣿⠀⠀⠐⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠡⡗⣿⣿⠀⢸⣿⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣸⡇⠀⣿⡇⢸⣿⠀⢙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠙⣿⣿⠀⢸⣿⠀⢸⣇⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠿⢧⣤⣿⣀⡈⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⣿⣿⣀⣼⣿⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠰⡆⢁⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠜⠋⠁⣠⣄⣧⠤⠀⢄⠁⠈⣉⢐⣂⡞⠰⣴⣾⣷⣶⣶⡀⡉⠂
⠀⠀⠀⠀⠀⠀⡀⠀⡂⠁⣀⡴⠶⠛⠉⠁⠀⠀⠀⠀⢇⢘⣉⣻⡟⢠⣶⠏⠀⠉⣷⣧⣞⣨⢛⢂⣤⣀⡀
⠀⠀⢀⠤⠂⣥⠤⠐⡎⠉⠀⢀⠤⢰⡀⠀⠀⠀⠀⠀⠈⣞⢻⡟⢀⡼⣿⣦⡄⠾⢿⡿⣽⡿⠻⣿⣿⣿⣿⡦
⠀⠀⢇⠛⠂⣇⠺⠂⠀⠀⠀⠈⠁⠀⠀⡠⠐⠀⢀⠀⠀⠘⡟⣥⣿⣥⠟⠁⠀⠀⠁⠀⠑⠁⠀⠈⠛⠀⠀⢡
⠀⠀⠘⡆⠳⢸⡄⠆⠀⠀⠀⠐⠀⢀⠀⣅⢀⡀⣁⠆⠠⠄⢻⡀⠈⡍⠀⠙⠳⠀⠀⠀⠀⠐⣤⣀⠀⠀⢠⠯⠀⢀
⠀⠀⠀⠘⡄⣻⣷⡀⠀⠀⠂⠉⢠⠊⠁⠀⠀⠀⠑⠤⠀⠀⠄⢆⠀⠱⠀⢀⠀⠀⠀⠙⠂⠀⣿⣿⣷⣦⣯⣴⣷⣄
⠀⠀⠀⠀⠸⡜⢺⣧⠁⢀⠀⠀⡂⢀⠀⠀⠀⠀⠁⠐⠂⠈⠀⠈⣄⠀⠀⢈⠀⠀⠀⠀⣠⠂⠈⣻⢿⣿⣿⣏⣻⣿⣶⡠
⠀⠀⠀⠀⠀⠹⡝⣿⣗⢀⠉⠈⢣⠀⠀⠀⠠⠀⠀⣈⠁⠀⠀⠀⡘⣷⣄⡀⠈⠀⢶⣿⣿⣴⡴⢣⣟⣉⣹⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠱⡘⣿⡄⠀⡀⢂⡰⠂⡄⢤⠄⠂⠀⠱⠈⠀⠀⠀⢹⡦⠹⣿⣶⣿⣦⣾⡿⢡⣿⣨⣹⣿⡫⠊
⠀⠀⠀⠀⠀⠀⠀⠱⡸⣷⡀⠀⠁⠄⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⡆⢣⠘⡄⢄⠉⠩⠿⢧⣿⡻⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⢴⠌⠀⠀⠀⠀⢴⢀⣞⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠸⣇⠀⠀⠀⠀⠐⠘⠀⠈⠀⠀⠀⢐⣃⣤⣶⣤⡖⠀⣶⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⢝⠀⣩⠁⠛⡻⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠢⠄⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⠍⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀⠀⢀⠤⠔⠒⠚⠛⠃⠒⠢⠤⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠘⠿⢿⣿⡻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠒⠦⢤⣀
⠀⠀⠀⠁⠈⡃⠀⠀⠀⠀⠀⠀⢙⣀⡀⢰⣄⠀⠀⠀⠀⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡂
⠀⠀⠀⠀⡀⠁⠀⢄⠀⠀⣰⣿⣸⠏⠁⣰⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⡀
⠀⠀⠀⠀⠆⠀⠀⢀⣳⣤⡍⠉⢠⢆⣴⣿⣿⣷⡄⢠⣾⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢷
⠀⠀⠀⠀⠀⠛⠻⠟⠟⠋⠀⠀⠉⠀⠀⠀⠀⠘⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠶⠾⠯⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠃⠀⠀⠀⠀⣰⣾⣄⣀⠀⠀⢀⣴⣶⣿⣧⠴⠀⠀⠀⠀⠀⢰⣿⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⣴⣿⣿⣾⠿⣾⣶⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⡀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠉⠉⠉⠛⠛⠋⠉⠉⠙⠻⣶⡀⣤⡀⢸⠅⢱⡀⠀⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡏⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢹⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠘⣟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠻⠻⠀⣠⣶⡴⠃⡀⠀⣤⣦⣤⢤⡤⠀⠀⠁⠀⠀⠶⠶⢦⣉⣡⣠⠼⡿⢿⠿⠷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠼⠛⠉⠹⣟⣷⣰⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣦⡈⠛⠏⠒⢦⢾⣗⢹⣦⠀⠠⠈⠑⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⡙⣀⠈⠃⣄⢆⣈⠏⡿⠇⠀⢰⡄⠀⠈⠉⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠂⠀⠀⠚⠚⣿⠀⢀⠀⢀⣾⠁⣤⠀⣠⣿⠟⠲⠂⠀⠒⠂⠀⠈⠉⠀⠀⠈⠉⠉⠉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡄⠢⡤⠛⣠⠇⠀⣾⡇⢀⣿⠀⠀⣆⣼⠇⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠸⣦⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠣⣐⣃⣤⠛⠆⣸⣿⠷⠀⠁⠀⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠄⠀⠀⠀⠀⠀⠀⢠⢝⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠉⠳⣂⣴⣿⣿⣇⡗⣤⠀⢀⡶⠉⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⡖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢷⣆⡻⠙⣉⠁⠈⠃⡀⣀⣤⠃⢺⡀⠀⠀⠀⠀⠀⠀⠀⢾⣷⠀⠀⠀⠀⠀⣼⠃⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣼⣍⡀⣦⣸⣿⣿⣏⢀⡀⡉⣉⡉⠀⠀⠄⠐⠒⠁⠀⡄⠀⠀⠀⢸⠉⠛⢸⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⢻⢿⣿⣿⡟⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣴⠀⠠⠀⠈⢆⠀⠀⢀⣴⣜⡻⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⡟⠀⠀⠠⣀⠘⢳⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡐⠀⠀⠀⠑
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠂⠀⢠⣀⡀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡤⠋⠀⠀⣔⢂⡀⣨
⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠀⠈⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠈⠉⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠈
⠀⠀⠀⠀⡠⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠛⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠰⠆⠴⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠿⠀⣦⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣠⢾⠿⡟⣦⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⣻⠉⠀⠍⡽⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣆⠀⠀⠀⡰⡽⢁⣿⢸⡇⢿⡄⠀⠀⢀⡟⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡟⣎⢦⠤⣪⡾⠁⣼⣷⣿⣷⡘⢿⣦⣤⡿⣹⡏⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⡏⠾⠗⣋⣥⣾⢋⣠⢞⡽⠡⠧⠹⣷⡀⠻⣿⣻⣿⣧⣼⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢚⡇⣮⠑⣰⡀⢂⣽⠘⣿⠉⠁⠀⠄⡊⠀⠀⠀⠉⢻⠇⣦⡀⢀⣎⠉⣽⢹⣗
⠀⠀⠀⠀⠀⠀⠀⣾⢁⡿⠀⣿⠁⢸⡟⠀⣯⣷⣶⣦⣤⣤⣬⣬⣴⣶⣽⡀⢸⡇⠀⣿⡀⢻⡌⢿⡀
⠀⠀⠀⠀⠀⢀⣾⠇⣼⠃⣸⡿⠀⣼⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣷⠀⢻⣧⠘⣷⡘⣷⣄
⠀⠀⠀⠠⠶⠿⠁⠼⠇⠠⠿⠇⠠⠿⠡⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠄⠿⠆⠸⠿⠦⣨⠷⠬⡿⠷⢤

⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣴⠇⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣻⣏⣾⣧⣀⠉⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⠛⡟⢯⡛⡛⢏⠂⠀⠀⠀⠀⠀⠀⠀⠘⠙⣾⣷⡋⣿⠀⠀⠁⠀⠠⡰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⣶⣨⣆⡓⠄⠸⠀⠀⢠⠀⠀⠀⠀⢀⢰⣶⣏⣿⡷⣶⡤⣄⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⣼⣿⣿⡿⢩⡇⠀⠀⠀⠀⠘⠆⠀⠀⠀⢸⣆⢻⣿⡿⠁⣾⣧⡹⢧⡀⠀⠠⠀⠠⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⣿⡇⢙⡿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⢿⣿⣿⣶⣿⠙⣿⣿⡷⠀⠀⠀⠀⠈⢹⢷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢨⣿⣿⣿⣷⣤⣴⡼⠉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢲⣶⣤⣿⣿⣏⠀⠈⢻⣷⡀⠀⠀⠀⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡀⠈⠀⣿⣿⣿⣿⡟⣅⣾⣿⡇⠀⠀⢀⣀⣄⠀⠈⡘⡆⠀⡸⠞⢿⣿⣽⣿⣿⡇⠀⠀⠈⠁⠀⠀⠀⠚⠻⣷
⠀⠀⠀⠀⢠⣀⡄⠀⠠⠀⠀⠉⠀⠀⠠⠟⠉⠀⠡⢂⣽⣿⣿⡇⠀⠀⢸⣉⠉⠄⠀⠸⡀⢀⠃⡘⠚⠉⢩⢿⡟⣷⠀⠀⠘⢿⠄⠀⠀⠀⣠⡍
⠀⠀⠀⠀⠈⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣤⣿⣿⠉⣿⣷⠀⠀⠀⠇⠀⣶⡆⠀⣅⢨⠀⣵⠈⢰⢘⡭⡇⡟⠀⠀⡉⠈⢻⠂⠀⠀⠁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠞⢩⡿⠿⡿⡟⠀⠹⣷⣄⠀⠀⡀⠀⠀⢠⡀⠹⡇⠀⠹⠼⣒⡉⣈⠇⡇⠀⠀⣉⣹⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣴⡀⠱⣄⠱⣾⣦⠀⠘⠻⣴⣄⣁⠈⠀⣾⠳⠀⣿⡄⡀⢸⠖⢹⠉⢸⠁⠀⠀⢹⣿⢿⠒⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⠈⠳⠿⣿⡇⠐⠂⠘⢿⣿⣿⠒⠻⡿⣶⢿⣿⣴⣾⠀⠘⠴⠇⠀⢸⡀⠉⠇⢈⠀⠉⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢻⣷⣿⣿⠷⣄⡀⠈⣿⡀⠀⠀⠀⣷⡄⡀⠀⢱⡜⣾⠛⠛⠃⠀⠀⠀⢸⣦⡼⠏⢠⠀⢈⡀⢠⣷⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢘⠁⢹⡟⠀⠈⠙⣦⢨⣷⠀⠀⠀⢹⡻⣷⡄⠘⣿⣏⢰⡀⠀⠀⠀⠀⠈⢿⣿⣦⣶⡤⠾⠀⠈⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠊⠀⡞⠀⠀⠀⠀⣘⡃⠏⠀⠀⠀⠀⣿⣻⣷⠀⠉⢉⡜⣿⡌⠁⠀⠈⠀⠈⣿⣿⠁⡀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣡⡤⠖⠚⠉⠀⠀⠀⠀⠀⢸⠃⣷⠈⠐⠛⠚⠋⠻⢁⣀⡴⢿⡿⠃⣯⣦⣬⠉⣷⣄⠸⣽⣿⣇⣀⡀⣿⣧⣶⣷⡆
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠶⠿⠿⠛⠻⠿⠏⠀⣿⣿⣿⣷⣧⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀

⣠⣤⣤⣤⣤⣀⣀⣄⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣄⣠⣤⣤⣀⣀⣄⣀⣀⣀⣀⣤⣤⣤⣀⣠⣤⣤⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣠⣀⣀⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠁⢂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠑⢄⡀⠀⠀⠀⢃⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠁⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡠⠠⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡈⠀⠀⠀⠀⠢⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠁⠀⠀⠀⠀⠀⢀⠁⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠱⠀⠀⠀⠀⠀⠈⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠄⠠⠀⠀⠀⠌⠀⠀⠀⣤⠲⣦⠀⠀⠀⠀⠀⣠⡶⢢⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⡐⠀⠀⠀⠀⣿⣿⣿⠂⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⡐⠀⠀⠰⠀⢃⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠊⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠁⠀⠀⢰⣄⠀⠀⡀⠀⠈⢰⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⢀⠔⠃⠠⠄⢀⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⡀⠀⢀⠄⠈⠉⢀⠀⠀⠀⠂⠈⠐⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⠀⢔⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⠃⠉⠄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠋⠀⠀⠐⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠑⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠁⠸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠀⠢⠘⠀⠑⠀⠒⠀⠀⠀⠈⢀⠀⠀⠀⠇⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢠⡇⠀⠂⠀⢀⠂⠒⠀⠚⠇⠈⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⢣⡀⠀⠀⠀⠀⣀⠀⠈⠆⠀⠀⠊⠀⠀⡄⠀⢀⠄⠀⠤⠂⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⢄⠀⠀⠳⣄⢰⠀⠀⠲⡄⠀⢦⠀⢰⠀⠀⢀⢶⢰⠀⠀⠐⡄⠀⡄⠀⠀⠰⡄⠀⠀⠀⠀⠀⢰⡄
⠀⠀⣎⢧⡀⢤⣌⣻⠄⠀⠀⠹⡄⠈⢧⠘⣆⠀⢸⠀⢯⠃⠰⡀⢹⠀⠹⡄⠀⠀⠹⡄⢸⡆⠀⠒⠁⢱
⠀⠀⡇⠑⠹⠆⠱⠈⠑⠄⠀⠀⡽⡄⢠⡆⠘⣆⠸⠀⠀⠱⡀⠳⡌⠠⠳⡙⠄⠀⢀⠔⠋⠸⡀⠀⠀⢸
⢡⠀⠰⡀⠀⢀⠐⠀⠀⠀⠀⢸⢇⠱⣼⠹⣄⠘⠀⠀⠀⠀⠡⢀⠘⡄⠀⠱⢀⠀⠁⠀⠀⠀⠁⠀⠀⠘
⠈⢧⠀⠁⠀⢸⢇⠀⠀⠀⠀⠘⠈⠂⣿⠀⠈⠣⡀⠀⠀⠀⠀⢈⡇⠰⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢀⠀⠳⠄⣷⡈⠄⠉⠒⠂⠀⡿⠄⠀⣿⠀⠀⠀⠈⠳⢄⠀⠀⠊⠸⠸⣆⠀⠀⠻⠀⠀⠀⣀⡠⣶⠀⢠⠎
⠈⢣⡀⠀⠰⡈⠢⣀⠀⠀⠀⢣⠀⠀⢿⠀⠀⡀⠀⠀⠀⠳⡀⠀⠁⠀⡿⠀⠀⠃⠀⠀⠀⠀⡠⢃⡴⠋
⠀⠀⠳⣄⠀⠁⡀⠀⠁⢀⠀⠰⡇⠠⢸⠀⠀⠁⠀⠆⠀⠀⢣⠀⠀⡠⠃⠀⠄⡀⠀⠀⠀⠪⠔⠋⠀⣠⠆
⠀⠑⢤⡈⠓⠄⠙⠿⢈⡁⠀⠀⠳⡀⠀⢆⠄⠀⠀⠀⠀⠀⠘⠀⠈⠀⢀⣀⣀⠬⠟⠁⣀⣀⣠⠴⠚⠁
⠀⠀⠀⠉⠓⠦⠤⠤⠂⡴⠁⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠙⢦⡀⣉⡁⠀⠀⣀⣀⠄
⠀⠉⠑⠒⠲⠶⠒⠊⡼⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠙⠒⠶⠒⠂
⠐⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⡇⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⡄⠑⠲⠤⠤⠤
⠀⠀⠀⠀⠀⡴⢰⡄⠀⠄⠀⢤⣴⣦⡄⠀⠀⡀⠀⠈⠃⠀⢀⠀⠀⢠⣴⣤⡤⠀⠐⠀⠀⡇⣄
⠀⢀⣠⠴⠊⠀⢈⢧⠀⠀⠀⢈⠻⠿⢾⡂⠈⠀⠀⠀⠀⠀⠀⠁⢸⡷⠿⠛⠀⠀⠀⠀⣸⡇⠈⠲⣄⡀
⠀⠁⠀⠀⠀⢠⠇⣸⠀⠀⠀⠀⠁⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⡀⣏⢱⡀⠀⠀⠈⠁
⠀⠀⠀⣀⠴⠃⢠⠇⠈⣆⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⡼⠁⢹⡀⠳⣄⡀
⠀⠂⠉⠀⠀⣠⠋⠀⣧⠘⡆⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠈⠀⠀⣸⠁⡼⠀⠳⡀⠀⠉⠒
⠀⠀⢀⡠⠚⠁⣰⠃⠘⣆⠀⠀⠊⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠘⠄⠀⠠⠀⠀⠀⣸⠃⠸⡄⠙⠦⡀
⠀⠀⠁⠀⣠⠞⡥⠂⠀⠘⢴⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⡶⠃⠀⠐⢜⢦⡀⠈
⠀⡀⠴⠊⢁⡞⠀⠀⡆⠀⢸⡀⢤⠂⠁⠈⠓⡛⢶⣦⣴⡶⢛⠞⠁⠠⢚⡤⢠⡇⠀⢰⠀⠈⢳⠈⠓⠤
⠀⣠⠖⠁⣸⠀⣠⠆⣗⡃⢸⣧⠈⠭⠐⠀⠀⠈⠒⢿⡿⠀⠁⠀⠀⡂⠡⢀⣽⠇⢨⣽⠠⡀⠈⡇⠈⠲⣄
⠘⠁⠀⣠⡧⠞⢁⡴⣻⠀⢸⠚⣧⡀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⢀⣾⠙⡇⠘⡟⢦⡙⠦⣝⡄⠀⠈
⠀⠀⠉⠁⣠⠔⠋⣰⠃⠀⡼⢠⢇⠹⡗⠒⠛⠉⠁⠈⠁⠉⠉⠛⠒⣺⠋⢸⡀⣧⠀⠙⣄⠙⠢⣄⠉⠁⠂
⠀⢀⠴⠋⠀⠀⡰⠁⠀⢰⢇⠞⡾⡆⢘⢦⠀⠀⠀⠀⠀⠀⠀⢀⡴⡃⢸⢿⢳⡸⡄⠀⠘⣆⠀⠀⠙⠢⡀
⠀⠁⠀⠀⠀⠠⠁⠀⠀⠸⠘⠸⠇⠇⠸⠨⢙⠦⣄⣀⣀⣠⠴⠋⡆⠇⠘⢸⠀⠇⠇⠀⠀⠘⠄

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠀⠀⢠⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣠⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⡄
⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢻⣿⣿⣿⡇⠋⢠⣿⣿⣿⡇⣝⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⡆⠀⠀⠀⢀⣧⣿⠇⠀⠀⠀⣿⢧
⣿⣿⣿⣿⣿⣿⣷⣴⣶⣶⢸⣿⣿⠋⠀⢰⣿⣿⣿⡿⠃⣿⣿⣿⣿⡟⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⡇⠀⠀⣴⣿⣿⠁⠀⠀⠀⣠⣼⠈⠀⠀⢰⣶⡆
⣿⣿⣿⣿⣿⡿⠿⢸⡟⠉⢸⣿⡇⢰⢋⢈⣉⠹⣿⣷⣴⣿⣿⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⣠⣶⣧⠀⢸⡿⣱⢿⠿⣷⡄⠀⠙⠋⠀⠀⢠⢸⡅
⣿⣿⣿⣿⣿⣧⣠⣾⣷⣶⣿⣿⣧⡘⢾⡟⡿⢀⣿⣿⣿⣿⣿⣿⣷⣄⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠈⠉⠁⠀⠘⣿⣝⠸⡄⣸⡇⠀⠀⠀⠀⠀⠈⠻⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠒⣆⣶⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⡙⣟⠉⠀⠀⢀⠀⠀⣀⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣃⣿⣿⣿⡇⠀⣸⣿⣠⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⡄⠀⠻⢿⣧⠟⠀⠀⣸⣿⡿⠀⡿⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣁⣀⣠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⣤⢤⡤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⢀⣀⠀⠀⡀⡀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠁⢤⣤⣝⣚⡐⢂⡐⢀⡒⢈⠀⠤⠀⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠣⣬⡑⠺⡦⢼⡉⢤⠈⢡⡌⠁⠦⠀⠒⢪⣉⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠲⣤⣽⣉⣿⣟⠓⡀⠶⠂⡀⠿⢀⡘⠃⣤⣥⠾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⡛⢷⣬⣯⣭⣿⡋⠛⣀⡘⠛⣠⡈⠉⣤⢤⡞⢒⣉⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣽⠛⢳⡶⠿⣭⣥⣤⠉⢡⣤⠉⠡⡶⠀⠸⠟⣏⡥⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⡐⢮⣿⡟⣳⣾⠷⠆⠀⠶⠆⠀⠶⠆⣀⠚⢛⣯⣵⢶⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣅⡚⢬⣾⣏⣻⣿⡛⠓⣀⡘⠛⣀⡈⠛⣠⣬⡵⢷⣹⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠆⣹⠛⣷⡾⢿⣭⣥⣤⠉⢡⣤⠉⠁⣶⠀⠰⢟⣎⡅⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⢯⣽⣟⣳⣶⠿⠆⠀⢶⠆⠀⠶⠆⣀⠚⢓⣯⡡⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⡺⢬⣿⣏⣹⣿⡛⠛⣀⡘⠛⣀⡈⠛⣠⣬⡵⢷⣂⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣤⣀⣶⢄⣹⠻⣶⣾⣿⣬⣤⣄⠉⢡⣤⠉⠡⣤⠀⠰⢛⣏⣥⣴⣀⣤⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣾⡏⡇⣾⣿⣽⣿⣷⣾⣿⠂⠀⠲⠂⠀⠲⠂⠀⠚⠋⣽⣿⣷⡾⢋⣷⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⣷⣿⣿⡿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⡏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣿⠀⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣺⣿⣿⣿⣿⣿⣿⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢻⣿⣿⣿⣿⣿⡏⢀⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⣇⢻⡀⠻⣿⣿⣿⣿⣿⣿⣿⠿⢿⢾⣿⣿⣿⣿⡟⢁⡞⣸⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡘⣆⠙⣄⠙⢿⣿⣿⣿⣿⣿⣶⣦⣿⣿⣿⡿⠏⣠⠎⣰⠇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⢦⡀⠙⢦⣝⠻⠿⢿⣽⣯⣼⣿⠿⣿⡵⠊⢁⡴⠋⢦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣙⠦⣀⡀⠉⠑⠒⡖⢲⠒⠂⠉⢁⣠⠴⣋⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣳⣮⣍⣛⠛⠛⠃⠘⠛⠛⣛⣩⡴⣞⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⢻⡿⠟⠙⢿⡟⠻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣿⣿⠀⣿⠒⠒⠠⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠋⠛⠻⠿⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⠏⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣠⣀⢀⡀⡀⠀⠀⠀⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣿⠿⢳⣞⣡⣤⣶⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠛⠩⠿⠿⠿⠿⠟⣛⣛⠛⠿⠿⠿⠿⠍⠙⠛⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣻⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣵⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠓⠲⢤⡀⠀⠀⢀⣠⠤⠴⠒⠒⠒⠒⠒⠲⠤⣄⡀⠀⠀⢀⣠⡤⠤⠶⠦⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣈⡷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠒⠒⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢸⠃⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⣠⡀⠀⠀⠀⠀⠙⠦⣄⠀⠀⢈⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠺⡁⠙⠦⣄⣀⠀⠀⠀⣀⣩⡗⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⢧⡀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠀⠳⣄
⠀⠙⢆⠀⠈⠉⠉⠙⠉⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⠤⢤⣀⣀⣀⣀⡤⠔⠋⠁⠀⠉⠙⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠈⠙⠲⠤⢤⣀⣀⣀⣀⡤⠤⠖⠚⠉⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⡤⠄⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠙⡏⢸⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢻⣸⡇⠀⠀⠀⠀⠀⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡏⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⢂⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⡏⢦⠀⠀⢀⣤⣤⣾⠂⠀⢀⣤⣴⡄⠂⠑⣌⣠⣤⡀⠀⠀⢠⠊⡗⣄
⠀⠀⠀⠀⠀⠀⡠⠒⠊⢠⣷⠈⠳⣦⣸⣿⢙⠿⡟⡻⠏⠹⢟⢋⣿⣏⣿⣿⣿⣧⣤⣶⡇⣴⣧⠈⣶⣤⣄⡀⠠⠄
⠀⠀⠀⠀⠸⣟⢿⡖⣷⡛⣞⢱⣖⣾⠛⠉⠉⠀⠘⠁⢨⣤⢶⣽⠀⣠⡀⣀⡤⢸⣭⣁⣴⣹⣿⣿⣿⣿⣿⣿⢂⣄
⠀⠀⠠⠀⠀⡇⣾⠀⣿⠈⡇⢹⠀⣳⠐⠀⣠⠀⠀⠀⠌⠉⠀⠸⣷⣤⠄⢤⠤⠤⣴⡧⣤⢼⡿⢿⡏⣿⠿⡷⠾⠿⠇
⠀⠀⠀⠆⠘⠇⠉⠀⣴⠀⡇⢸⡀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⡀⠀⠀⠀⠀⢘⠀⢩⠀⡃⢰⠄⢘⠀⣇⠀⡤
⠀⠀⠐⠀⢸⡀⣾⠀⠿⠀⠇⠸⡇⢠⡃⠤⠆⠀⠇⠀⠀⠀⠀⠀⠇⡅⠄⠀⠀⠀⠣⠀⡿⠀⠇⠸⠂⣼⠀⣿
⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣀⣄⠤⢤⡤⠼⠒⢒⡀⠀⡀⢀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠐⠒⠭⠉⠀⠀⠀⢠⡄⠃⠀⠂⠀⠈⠡⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⣀⣀⣄⣯⣼⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢉⡉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡒⣉⠀⡆⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠉⣥⡤⢴⢶⣤⡤⠤⠴⠶⠿⠟⠛⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡈⠁⠢⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡡⠅⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠠⠟⠳⠦⠀⠀⢀⡇⠀⠰⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠨⠅⠨⠿⠿⠓⠐⠀⠀⠀⠀⠈⢧⡄⢍⡀⠂⠀⠉⠁⠀⠀⡻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠑⠊⠇⠱⠀⢀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠠⠀⡤⢊⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢣⠀⠀⣠⣤⠀⠀⠀⠀⠀⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣸⠀⠀⣿⠗⠀⠀⠂⣀⣀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢅⢀⠀⠈⠀⠀⠀⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⡟⢛⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⣈⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣍⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠃


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠒⠒⠂⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⡀⠀⠀⡠⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡜⢠⠂⠰⠁⠀⠀⠀⠀⠀⢀⠔⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠐⢉⡏⢉⠓⠂⠀⠀⠀⠀⢢⡋⢢⠄⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢈⡁⡌⠉⠒⠀⠀⠀⠀⠀⠸⡏⠡⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⡀⠈⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⡠⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣈⠑⠂⠄⣀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠒⢉⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠁⠢⣤⠄⡀⢈⣉⣉⣉⣉⠁⢠⠀⠄⠂⢁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠬⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⣎⡈⠃⠀⠀⠀⢀⠀⢀⡤⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⣀⠉⠑⠒⠒⠒⠒⠈⢁⡠⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⠁⠀⠐⠀⠀⠉⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠒⠀⠠⠤⠤⠄⠐⠊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣀⠀⢀⡠⠔⣒⠶⢤
⠻⡉⢑⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠛⠒⠈⠛⠋⢀⢔⡛⠉⣉⡭⠶⠶⠒⢒⠖
⠀⠘⢦⣂⣀⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣶⣶⠴⠤⣀⣈⠀⢂⣍⡉⣳⠯⢅⡐⠀⢀⣀⠸⠦⣤⣀
⠀⠀⠀⠀⠀⠙⡄⠱⡄⠀⠀⠀⠀⠀⢀⣴⢯⡄⣇⠈⢱⠀⠃⠀⠀⠀⠙⣱⠊⡦⢹⠀⠐⠢⢍⡲⢤⡼⠷⢄
⠀⠀⠀⠀⠀⠀⡽⠀⣹⠀⠀⠀⠀⠀⠀⣠⠏⠀⢈⠟⣥⡀⠀⡴⣶⣦⠀⠘⠦⠀⠋⠰⡀⠠⣄⠈⠁⠈⠳⡄
⠀⠀⠀⢀⡤⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⠟⢲⠊⢈⣄⣛⣁⠀⠛⠛⠃⠀⠀⡄⣄⠀⠢⡑⠤⣈⠓⠤⣈⢷⡻⠄
⠀⠀⢀⡞⠁⢠⠏⠀⠀⢀⣠⠶⠞⠛⠓⢲⠃⡤⢼⠿⣿⠟⠃⠀⠐⣧⢀⣀⡏⢸⡆⠀⡇⠀⠈⠃⠀⠈⠣⢳⡀
⠀⠀⢸⡁⠀⣏⠀⠀⣴⠟⠁⠀⠀⠀⠀⣨⢾⡀⡈⣳⠛⠲⠦⠶⠚⣡⣿⢻⠇⡸⠀⡞⠀⠀⢆⠀⠀⢦⠀⣬⣧
⠀⠀⠈⢧⡀⠹⠶⣾⠁⠀⠀⠀⠀⠀⠀⠉⢠⠿⣄⢏⡰⠀⣠⢸⠏⢰⠃⡏⡜⠁⡀⠳⠀⠀⠈⠳⡄⢸⡆⢹⠈
⠀⠀⠀⠀⠙⠲⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡈⠈⠁⢉⠈⠁⡀⡏⠀⢿⠁⠀⢇⠀⠀⡄⠀⠀⣷⢸⣷⡾
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⡀⠀⠀⠱⡼⢧⠀⠠⠎⠀⢠⠁⠡⠀⠘⠆⠀⠈⣧⠀⣷⠀⠀⣿⢸⠋⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣦⣤⣀⡁⡸⢇⠀⡆⢰⡇⠀⠀⠀⢣⠀⣄⠀⠃⢀⣏⠀⡼⢻⠏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⠈⢺⠳⡀⢣⡀⠀⢀⠞⠀⣼⠀⢀⡞⠹⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⡼⠛⠁⠀⠀⣴⠛⡇⠀⠀⠙⢾⢱⢤⣯⣴⣾⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⢠⡞⠁⠀⠀⢀⡼⠃⠀⡇⠀⠀⠀⢀⡏⣾⠛⠻⠿⢛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣸⣇⠀⠀⣰⠋⠀⠀⠀⣇⠀⠀⠀⢸⠀⣹⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡟⢸⡀⠀⡇⠀⠀⠀⠀⢻⡀⠀⠀⢸⡀⡇⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢾⠀⠐⡇⢸⠁⠀⣇⠀⣀⠀⠀⠘⡇⠀⠀⠈⣧⡇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠙⠚⢧⠈⠉⡉⠉⠹⡄⠀⣇⣀⡀⢠⣿⠀⠄⠼⠧⠶⠦⡄
⠀⠀⢀⠀⠀⠀⠘⣧⣤⣤⣄⣤⠾⢦⠤⠧⠞⠴⠃⣰⠋⠀⠀⠀⠈⠳⣄⡀⢀⡄⣠⡿
⠀⠀⠊⠃⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠴⣤⢦⡴⠴⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⣴⡶⠶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⠿⣮⣵⠱⠿⢿⡚⠻⣅⢉⢳⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠽⢟⠻⣉⢒⣖⣀⣌⣃⡸⡿⢾⠟⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⡶⢆⣙⣦⠉⠉⠙⠉⠉⠁⣸⣠⣄⣴⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⢭⣏⢻⡍⢵⠆⣀⡀⠀⠀⢠⣿⣿⣟⡿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡗⣦⡈⣐⡺⣙⠫⠝⢧⡔⡈⠉⠉⠉⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣟⡵⣏⡳⢿⡻⠬⢷⠴⣖⣷⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡳⣭⣛⣶⠥⣮⡉⣖⣒⣶⡢⢦⣾⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣱⢽⣵⣪⣟⣛⣿⢟⡽⣯⢴⡯⣸⣿⠈⠏⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠨⢙⡒⠯⣶⢶⣚⣫⢗⠛⣺⠧⢋⣿⡆⣺⢊⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠈⣯⠙⢒⡙⠉⡠⢗⣐⠏⡴⢟⣞⠛⣿⠦⠖⡎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡢⠝⠲⣂⣸⣏⣭⠔⣋⡴⣚⠝⣽⢿⣟⢻⡕⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢟⣉⣠⢒⣸⠦⠶⠋⣁⡸⢵⡎⡼⡾⣿⣿⡻⣷⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡬⠴⠣⠤⡥⠒⠣⡭⢖⣫⡞⡴⣣⢼⣿⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣔⡗⣓⠿⢟⣩⣖⡾⠃⢞⣩⡦⠋⠻⡿⣷⡻⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣶⣬⣔⣛⠲⣦⠬⠭⠭⢖⣞⡽⢛⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠏⠈⠳⠷⢹⣆⠸⣭⣭⠵⣶⣪⡴⣻⣿⣿⣿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣤⣤⣱⣏⠐⠚⡉⠛⠻⠬⡠⢼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⡂⣠⣶⣫⡞⠁⠀⠀⣽⡶⠢⣯⠂⠤⢤⡀⢲⡹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣰⢭⣙⢀⣤⡴⢻⠁⠀⠘⢷⣤⣤⡈⢄⡼⣝⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣙⠂⠈⣿⣿⣯⣛⠢⢄⡀⠘⣿⠀⠙⠀⠁⢸⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠳⡌⠙⠲⠇⠀⢿⡄⢀⡴⠀⠗⢧⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⢸⣷⠀⢿⡂⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⠀⠀⠀⠀⠀⠀⠈⣿⡄⠈⠥⡀⡲⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⡆⠀⠀⠀⠀⠀⠀⠛⠉⢒⢀⣇⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠳⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠁⣀⣀⣠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠸⠊⠀⠀⠀⠀⠀⠐⠟⠠⠌⣹⣻⣿⡯⡾⠻⠄⠉⠁⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣨⣖⣶⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠁⠠⣾⣿⠃⢹⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⢨⡄⠀⡐⠢⠀⢨⠀⢛⣀⣠⣾⡟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠈⠁⣒⠒⢠⣆⠀⠀⣾⠋⣽⣿⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⡧⠁
⠀⠀⠀⠀⠀⢀⡡⠈⢉⢿⡽⠀⢀⣿⣶⣿⠈⠀⠀⠀⠀⠀⠀⠈⠂⢀⣠⡶⠊⠀⠰⣆
⠀⠀⠀⠀⠤⠥⠐⠲⢤⣷⡧⠀⣸⣿⣟⣿⡆⠀⠀⠀⠀⠀⠀⣀⣤⣾⠋⠀⠀⠀⠀⢢⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠰⡒⣉⢐⠿⠟⠁⣠⠟⠉⠀⡇⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠁⠀⠀⠀⠀⢠⣾⠟⣆
⠙⠒⠀⠁⠈⠙⡓⢶⡶⠒⠈⠀⠀⠀⠀⠐⣶⣾⣾⣿⣿⣿⣿⡿⡌⠀⠀⠀⠀⠀⠀⠀⢸⡔⠀⠀⠀⠀⠀⠀⢀⠉⢃⠀⠀⠀⠀⠀⠀⣹⡏⠀⢠
⠀⠀⠀⠀⠀⠰⠀⢿⠃⠀⠀⠀⠀⠀⠀⠀⡉⠃⣻⣿⣿⣷⣿⠻⠿⡀⠀⠀⠀⠀⢠⠀⠈⡗⠄⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡏
⠀⠀⠀⠀⠀⠈⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⢁⣄⠋⣿⣿⣿⣿⡇⠀⠠⠀⠀⠀⠀⠀⠀⠀⢹⣂⠀⠀⠀⠀⡄⠃⢇⠀⠀⠀⠀⢀⠔⡞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⣿⣹⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠈⢻⣿⣿⡀⢄⠀⠀⠀⡼⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠈⠀⢢⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⣻⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢯⣷⠈⠀⠀⡼⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡏⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢠⠂⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢻⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⡠
⠀⠿⠦⡤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⡀⠀⠀⠀⠐⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢧⠄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣷⠊⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⢧⠁⠀⢤⣱⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠉⠉⠉⠀⠀⠁⠀⠀⠀⢻⡀⠂⠀⠀⠉⠁⠠⠀⠀⠀⠀⣡⠣⠒⠀⠀⠀⠀⠀⠀⡼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠁⠐⠐⠈⠉⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣶⣶⣶⣿⣿⣿⣿⣭⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡛⠻⠿⠛⠛⠛⠉⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⡏⡏⣿⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡽⢃⡾⢠⣷⠸⣧⠻⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⣶⠟⢡⣾⠃⢸⣿⠀⢻⣧⠘⢷⣍⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣡⣾⣿⠋⢠⣾⡟⠀⣸⣿⡄⠘⣿⣧⠈⠻⣷⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣠⣾⣿⠟⠁⢠⣿⣿⠁⠀⣿⣿⡇⠀⢹⣿⣧⠀⠙⢿⣷⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⣠⣾⣿⡿⠋⠀⢠⣿⣿⠇⠀⢀⣿⣿⣧⠀⠀⣿⣿⣷⡀⠈⠻⣿⣷⣄⠈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢀⣴⣿⣿⣿⠟⠁⠀⣠⣿⣿⡟⠀⠀⢸⣿⣿⣿⠀⠀⠸⣿⣿⣷⡀⠀⠘⢿⣿⣷⣤⡀⠙⠲⣄⡀
⠀⠀⠀⠀⠀⣠⠴⠋⠀⢀⣴⣿⣿⣿⡿⠃⠀⠀⣰⣿⣿⣿⠁⠀⠀⣸⣿⣿⣿⡄⠀⠀⢻⣿⣿⣷⡀⠀⠀⠹⣿⣿⣿⣦⡀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠸⣅⠀⢀⡴⣿⣿⣿⣿⠋⠀⠀⠀⣰⣿⣿⣿⡏⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣷⡀⠀⠀⠈⢻⣿⣿⣿⣆⠀⢀⣠⠃
⠀⠀⠀⠀⠀⠈⠉⠉⣾⣮⣿⠛⠙⢦⣀⣀⡼⠻⣿⣿⡿⣆⠀⠀⢀⠿⣿⣿⣿⢧⡀⠀⢀⡿⢿⣿⡿⣳⢄⣀⡠⠎⠙⣻⣯⣶⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⠀⠰⣶⣿⣿⡆⠀⠀⠈⣭⣽⣽⡇⠈⠉⠀⣶⣭⣩⡍⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⡿⣇⣀⣀⣀⠿⣿⣿⡇⠀⠀⢀⣿⣿⣿⡇⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄⠀⠀⢹⣿⣿⣧⠀⠀⠀⣻⡵⣟⣯⣽⣿⣭⣝⡳⣿⡃⠀⠀⢸⣿⣿⣿⠀⠀⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇⠀⠀⢸⣿⣿⣿⠀⠀⣴⢯⣾⣿⣿⣿⠸⣿⣿⣿⣯⣻⡄⠀⢸⣿⣿⡟⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⣿⣿⣿⡀⢰⣏⣿⣿⣿⣿⠃⠀⢻⣿⣿⣿⣧⣿⠀⣾⣿⣿⡇⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⢿⣿⣿⡇⢸⡿⣿⣿⡿⠃⠀⠀⠀⠙⢿⣿⣿⣿⠀⣿⣿⣿⠃⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⢸⣿⣿⡇⢸⡧⡽⣅⠀⠀⠀⠀⠀⠀⠀⡭⡍⣿⢀⣿⣿⣿⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠀⠀⠈⣿⣿⣿⠈⣇⣿⣿⣦⡀⠀⠀⠀⣀⣼⣿⡇⣿⢸⣿⣿⡇⠀⠀⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠿⠿⠿⠟⠁⠀⠀⠀⠙⠿⠿⠇⠏⠘⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⣿⣿⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⢸⣿⡇⠀⢀⣶
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣿⣿⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⡇⠀⣸⣿
⠀⠀⠀⠀⣄⠀⠀⣿⡇⠀⣿⣿⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢸⣿⡇⠀⣿⡇⠀⢀⡄
⠀⠀⠀⠀⢻⣧⠀⣿⣿⠀⣿⣿⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⡀⣸⣿⡇⣸⣿⠇⢠⣿⠃
⠀⠀⠀⠀⠀⢿⣧⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣤⣾⡿⠀⠘⢿⣷⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣿⡿⠁⠀⠀⠀⠀⠹⣿⣷⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⢿⠿⠟

⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠠⣀⠄⠐⠀⠈⠉⠐⠂⠤⡀
⠀⠠⠆⠀⠀⠀⠀⠀⠀⢄⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠂⠐⠐⠂⠠⠤
⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⡂⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢛⠋
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢤
⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠈⠂⠔⠀⠀⠀⠀⠀⠀⢠⢤⡁⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⡀⠀⠑⠀⠀⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠃⠂⢀⡀⠀⠊⡁⠀⠀⠀⠀⢠⠋⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠀⠂⠀⠀⠀⢀⡏⠀⢠⢀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⡄⠀⠀⠀⠈⠐⠀⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⣀⠄⠀⠂⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠄⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠠⠄⠰⡎⠀⠀⠀⠀⡘⠀⠀⡂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠃⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠂⠀⢀⠀⠀⠘⡀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⢠⠀⡆⠀⣀⢀⡲⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠘⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⡆⠀⡆⢦⠶⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁⠀⢠⡈⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠃⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠱⡀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈⠦⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠹⠟⠋⠉⠉⠙⠒⠒⠋⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣶⣴⣴⣶⣯⡥⠤⠶⣦⣄⣀⠀⠒⠺⠿⣝⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡁⠀⢠⣼⣿⡹⣴⣦⠄⢸⣿⣍⠙⠓⠂⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠊⠁⠛⠛⠀⣤⣇⢻⡃⢿⣁⣴⣿⡿⣯⡛⢦⠀⠈⠛⢷⡶⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠁⠀⢠⣄⣀⣿⡏⢧⡳⡈⠿⣯⡙⠳⣌⠑⠈⠳⡀⠀⠀⠹⣆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⢻⢺⡟⣿⡿⣷⠈⢣⡈⠀⠈⠻⣄⠈⢧⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⡼⢸⡇⠹⡇⠹⡄⠀⠑⠄⠀⠀⠹⡆⠀⠀⠀⠀⠀⢢⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢤⣼⡇⠀⠘⠀⠹⣦⡀⠀⠀⠀⠀⠙⢄⡀⠀⠀⣀⠀⢹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⡀⠀⠀⢀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⣀⣸⠳⡄⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡏⠋⠀⢹⡄⠀⠀⣰⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⣧⠀⣰⠃⢹⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⣨⡄⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠇⠀⠀⢸⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣦⡄⠀⠀⠉⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠃⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡀⠀⠀⣀⠀⠀⣄⣾⣿⣇⣀⣀⡀⣰⣿⣿⡇⠀⠀⠀⠀⠀⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣠⣾⣿⣧⣼⣿⣟⣻⠀⠀⠀⠀⣀⣈⣽⣷⣤⣀⣄⣠⣶⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠛⠋⠙⠣⣤⣿⠟⠁⠀⠉⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣷⠞⠛⠛⢇⠹⠇⠰⠛⢛⠛⠢⣻⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡙⢿⣿⣦⣴⢾⣟⢿⡇⢀⣾⣿⢽⣶⡠⢿⡿⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⢻⣜⣿⠏⠋⠀⣉⣴⣶⠒⢸⡃⠚⠉⠀⠘⡇⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠞⠯⡅⠀⢀⣾⣿⣿⣿⣷⣾⣧⢀⣀⠀⢀⡧⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣠⣷⣰⣿⣿⣿⠿⠿⣿⠿⢋⣠⠄⠁⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣧⠸⢩⣿⣿⣛⣛⠿⠋⠀⠀⣰⠟⠃⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣿⣿⣧⣄⠉⠛⠉⠁⠀⢀⣤⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⢻⣽⣿⣿⣿⣿⣷⣄⣀⣀⣠⣾⢏⡼⠋⣛⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡠⠚⠹⣾⢟⡿⣫⠿⡟⣿⣿⡴⠋⢀⡔⠁⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣧⣈⡀⠋⢐⣯⠚⣁⠞⠁⠀⣰⠛⣦⠤⣀⣤⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣿⣿⣿⣿⡷⠊⠀⠈⠁⠀⠰⠈⠀⣀⠚⢿⣿⣿⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢻⡅⠀⠀⠀⠀⠀⠀⠀⠜⠁⣀⣾⢯⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣶⣶⣶⠶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡳⠀⠀⠇⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠂⠀⠀⠁⠐⢀⠉⠃⠀⠉⣉⡀⠀⠀⠀⠐⠈⠠⢶⡷⠆⠀⠀⠉⠁⠉⠉⠏⠉⠛⠳⣶⣄⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡎⡀
⠀⠀⠀⠁⠀⢀⡤⠒⠩⠀⠀⠀⠒⠀⠀⡀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⡅
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡄⠀⠀⠀⠈⢮⣐⢄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠸⠀⡀⠘⠁⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡆⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⢠⡤⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠖⢻⠇⠀⠀⠀⠐⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠐⠋⡁
⠀⠀⠀⠀⠤⣾⠁⠀⠀⠀⢀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⡇⠀⠀⢠⠀⠀⠈⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠘⣓
⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⣘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠠⠋⠀⠀⠁⠀⠀⠀⠈⢹⡄⠀⠀⠀⠀⣉
⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⢸⣹⡄⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠂⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢉⠁
⠀⠀⠀⢠⠼⠀⠀⠀⠀⢀⣸⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠀⠀⠀⠐⣀⣀⡀⠀⠀⢀⡞⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠄⠰⡇⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠆⣤⣶⠟⠋⠉⠉⡙⠛⣦⡞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⢈⣭⣶⣶⣦⣤⣄⠁⠀⠀⠀⠀⢀⡇⠀⠀⣀⢀⣰⠏⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⢨⣄⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢧⠀⠠⠴⠀⠐⠏⠋⣈⡠⠟⠋⠀⠀⠀⠀⣼⠀⠀⢐⣭⠞⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠷⣼⣶⡄⠀⣀⢸⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡇⣠⣶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣾⡇⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⢻⡿⣽⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⡟⢇⢏⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣅⣅⠙⢃⠰⡄⡄⠀⠀⡀⢠⣾⡏⠀⣁⣼⠿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠶⠀⣒⣀⠡⢘⣄⣠⠁⢃⣙⡀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠚⠋⠁⠈⠛⠲⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⡄⢠⡶⢀⡄
⠀⠀⠀⠀⠀⠀⣤⠅⠀⠀⢀⠛⠃⠀⠀⠀⢸⠧⣿⣇⡞⢀
⠀⠀⠀⠀⠀⠘⢁⡀⠘⢶⡾⢦⠀⠀⠀⠀⣼⢀⡇⢹⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⡀⠀⠈⠀⢀⣹⣿⡄⠀⠀⠀⠀⠻⠏⠓⢾⢆⣴⡁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⢠⠀⠀⢠⠆⠀⠉⢟⠀⠀⠀⠀⠀⠘⠃⠀⠀⠙⢻⣿⠇⠀⠀⠀⠀⢀⡤⠊⠀⢀⡤
⠀⠀⠀⢀⠀⠀⣼⠃⡀⠀⡍⠀⠀⠀⠀⠀⠀⠠⠀⠈⠈⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣾⠃
⠀⠀⠀⠈⠀⠱⢼⣀⡜⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⢀⠜⠀⢀⢰⡟⠃
⠀⠀⠀⢠⡿⢀⣮⡼⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠾⣿⣿⠃⠀⠐⠄⠛⠁⠀⠀⠀⣀⠄⠀⢀⠄
⠀⠀⠀⢸⢃⣾⣿⣧⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣝⠁⢥⡀⠀⠀⠀⢀⠀⠈⣛⣷⣾⣯⡄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣼⣿⣴⡏⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠋⠀⠀⠀⠀⢈⡿⠿⢿⣿⣿⡛⠓⠂
⠀⠀⠀⢸⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠐⢉⠻⠶⠇⠉⠋⠉⠉⠀⠀⠀⠒⠈
⠀⠀⠀⠸⢻⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⢐⠁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢸⡆⠀⡇⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⢀⢿⡏⠀⠀⠀⠀⠀⠀⠐⢄⡀⠠⡠⣄⢠⡄⡟⠀⠀⠀⠀⠀⠀⢀⠠⢤⢀⣀⠀⣀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⢀⠀⠀⠙⢷⣴⣿⡛⠓⠃⠀⠀⠀⠀⠀⠀⠹⠷⢮⡁⠠⠙⠿⠟⢢⣄
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⡁⣦⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢀⡇⠀⠈⡁⠄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⢠⣾⣷⡇⠀⠀⠀⠀⠀⠀⢢⡜⢿⣿⣾⡉⠐⢛⡐⠛
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠈⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈⠠⣈⠀⠄⠀⠗⢁⠄⡀
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠶⢄⠀⠀⠀⠀⠀⢀⣴⠰⢦
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⡄⠚⠀⠀⠀⠀⠀⠀⠰⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⢦⣿⠆⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣷⡄⠀⠀⠀⠀⠀⣠⣶⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢨⣾⠈⣷⣤⠀⠀⠀⠀⢋⢸⣿⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢹⢧⡁⢸⣿⣷⣿⠏⠀⠀⠀⠘⢿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠙⣿⣿⣿⣿⣆⠀⠀⠀⢄⣻⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⡷⢾⣿⣿⣿⠁⠀⠀⠀⠀⣸⡘⢿⡋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠄⣟⣰⣿⠒⠀⠀⠀⠀⢹⣷⣼⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠰⡧⢾⣷⡧⠀⠀⠀⢀⣼⡗⢰⠌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⠀⠀⢴⣾⣿⣗⡀⠀⣀⢠⣿⣿⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⣮⡋⣿⣏⠶⠀⠈⢸⣿⣿⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⢰⠟⡝⠛⣿⠀⠀⠀⣾⣽⣿⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢷⣦⡧⠀⣿⠰⡟⠀⣿⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣿⡇⢸⣿⡇⠀⣿⠄⣦⢠⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠛⣇⠀⣿⠋⠀⢹⠀⣍⢸⡇⣺⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣀⡇⠀⠺⠁⠀⠸⠀⠀⡀⠸⢦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⣰⡇⣤⣴⣂⣀⡄⢠⢸⣷⣦⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⠁⠀⢨⠅⠀⠀⠈⠶⠿⠯⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⢉⣿⢿⡗⠄⢾⣟⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣿⣿⢀⠀⠃⢀⠸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣯⡠⠇⠀⠯⣠⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣇⠘⡟⠃⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣟⠿⣿⣍⣉⣿⣯⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠸⣿⣿⣿⣻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣲⣦⣄⡀⠀⠀⠀⠀⠘⢷⣼⠃⠀⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠃⠀⠀⡠⠉⠘⣿⡇⠈⡇⠐⠒⠒⠲⠾⣿⡐⠀⠘⢿⣿⣿⣿⣿⠋
⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⢹⠃⢀⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠘⣿⣿⢿⡇
⠀⠀⠀⠿⠟⠁⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠁⠀⢀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠘⣤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠤⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⡀⠀⠀⠑⠘⠊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉


⣿⣿⢛⢿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢚⡛⠛⠻⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢀⠙⠁⠰⠶⠀⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠛⠁⣴⠄⠠⡶⠀⠛⠙⠒⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡦⠀⢠⡄⠀⠀⠶⠆⠀⣶⡾⠉⠉⠛⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣤⣤⣬⠵⠖⣊⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣶⣶⡾⠿⢟⠛⠉⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠙⠳⡄⠀⠀⠀⠈⠁⠈⠉⠳⡄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢰⠀⠀⠀⠀⠀⢹⡀⠀⡆⠀⠀⠀⠀⠀⢹⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⣘⡇⠀⠀⠀⢸⡄⠀⣆⠀⠀⠈⡇⠀⡁⢀⢠⣇⠀⠀⡌⠀⠀⡄
⠀⠀⠀⠀⠀⢉⠭⣭⡣⡀⠀⠀⠀⡇⠈⠉⡁⠀⠀⡇⡎⠀⠀⢀⡉⠐⠀⡇⠀⢠⡠⢒⠲⣆
⠀⠀⠀⠀⡇⡀⠀⢜⣷⠈⠀⠀⠀⡇⠐⠄⠃⠀⠀⠷⠰⠀⠀⠀⣁⠋⠀⢧⠀⢸⡀⠁⠀⠘⠂
⠀⠀⠀⠀⣇⢃⢀⣶⣿⠀⣠⠀⠀⡇⠀⣀⠿⢀⣤⣶⣶⣶⣶⣄⠛⠤⢤⣼⠃⢤⡁⠀⠀⠈
⠀⠀⠀⠀⠘⣦⢈⣝⣇⠎⢀⠤⠤⢿⣏⠀⠀⣾⣿⣤⣿⣿⣿⣿⣧⠀⢀⠤⢦⣄⢳⠀⡀
⠀⠀⢀⣀⣀⣈⣷⠌⠁⡀⠀⠀⢀⠀⠈⢳⡜⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⡀⠘⠂⠀⣧⣤⣄⡀
⠀⢪⣍⠉⡻⠍⣿⠀⠀⣇⠀⠀⠃⠀⣀⠀⠙⣿⣿⣿⣿⣿⣿⡿⠃⠀⡀⠀⠁⠀⠀⠀⢷⢲⣟⣷
⡇⣿⣿⣷⠁⣾⣿⣷⠶⡿⣄⠀⠀⠀⠘⣿⡆⠌⠛⠿⣿⣯⠽⠪⢴⣾⠇⠀⠀⠀⡤⣤⠌⠙⠿⠹⠶⠶⢦
⢻⡽⣿⣿⡇⣿⣿⣿⠀⣹⣜⢷⣄⠀⠀⠈⠻⣶⣤⣤⣀⣀⣤⣴⡟⠁⠀⠀⡀⠀⢀⣥⣭⡉⢂⣴⣿⣿⠆
⠀⠹⣎⢿⣷⢹⣿⣿⡆⣿⣿⣷⣌⠛⠶⣤⡀⠈⠻⢭⣉⣉⡍⠉⠀⠠⠒⢰⠇⡀⣼⣿⣿⣿⢰⣿⣿⠏
⠀⠀⠘⢧⢻⠘⣿⣿⡏⣿⣿⣿⡟⢷⣄⠈⠻⣦⣀⠀⠀⠀⢀⡠⠀⠀⢠⣧⠀⡇⣿⣿⣿⡏⣸⣿⡟
⠀⠀⠀⠈⣧⠀⣿⣿⡇⣿⣿⣿⡇⣼⣿⣦⡀⠀⠉⠙⠛⠛⠉⠀⠀⣠⣾⣿⠂⡇⣿⣿⣿⠃⣿⣿⠡
⠀⠀⠀⠀⢸⠐⢿⡿⠀⢹⣿⣿⣿⢿⣿⣿⣿⠳⣶⣶⣶⣿⠒⠲⣿⣿⣿⣿⠀⣿⣿⣿⡿⢠⣿⡟⡄
⠀⠀⠀⠀⠈⠷⡤⠴⡆⣾⣿⣿⢧⢸⣿⣿⣿⡀⣿⣿⣿⣿⡄⠈⣿⣿⣿⣿⠀⣿⣿⣿⡇⢈⣻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣛⣛⣼⢸⣿⣿⣿⠇⢹⣿⣿⣿⣿⠀⢿⣿⣿⡟⢸⠛⠿⠿⠡⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠹⣦⡛⠟⣋⠐⢦⡙⣿⣿⠟⠶⣌⡛⠛⠃⠘⠛⠶⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢴⢰⡂
⠀⠀⠀⠀⠀⠀⢦⡤⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠧⠀⡷⠂⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⠀⣀⠐⢄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡠⠀⠀⠜⠃⠀⠛⠀⠈⠢⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠠⠀⠁⠀⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⡀⠐⡈⠤⠀⠀⡠⠒⠉⠙⠳⣄⠀⣴⠄⠠⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⡀
⠀⠀⣀⠀⠀⠀⡀⠤⠙⠂⠑⠀⠀⢀⠀⡞⠁⠀⠀⠀⠀⠈⢧⠤⠀⡇⠀⠀⠀⠀⠐⡄⠂⡠⠖⠁⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⡆⠰
⠀⠀⠈⠙⢢⡈⠁⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⡇⣠⠁⠀⠀⠠⠀⣄⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠈⠐⡀⠁
⠀⠀⠀⠀⠀⢱⡀⠀⡠⠀⢁⠀⠃⠊⠐⡇⠀⠀⠀⠀⠀⠀⠀⠙⠒⣁⠀⠀⠀⢀⡀⠰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢉⠥⠐⠄⠀⠀⠂⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⣠⡀⠇⠀⡀⠘⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠾⠀⠀⠔⠈⠀⢷⠀⠀⢠⢀⡀⣄⠀⠸⡄⡀⠀⠀⠄⠅⡈⠀⠀⡴⠀⠀⠀⠀⠀⡸⠡⡄⢱
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠁⣠⠀⠀⠀⡀⢸⡅⠀⠘⠀⠑⠀⠀⠀⣏⣃⡁⢀⠀⠀⠎⠀⠈⠁⠨⠀⠀⠀⠰⡅⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⠁⠃⠀⢂⣀⠨⠇⠀⠀⠀⠀⠀⠀⢀⡤⠤⡄⠂⠛⠠⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠉⠂⠀⠄⠀⠲⡀⠀⠀⠀⠀⠀⢸⠁⠀⡄⢀⡀⡌⠀⠀⠀⠀⠀⠀⢀⠎⢈⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡈⠀⠀⠀⠀⡌⢧⠀⢀⠀⠀⠀⡞⠠⠁⠁⠈⢨⠁⠀⠀⠀⠀⠀⢀⠈⠠⠈⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠰⣀⡁⠘⣆⠀⠀⠀⣰⠃⠀⢀⠀⠀⢸⠀⠀⠀⠀⢀⡴⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢴⡇⠀⠤⠈⢢⠄⡴⠃⠄⠀⠀⠀⠀⢸⡀⠀⢀⠄⠋⠀⠈⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠄⢷⠀⠀⢀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠘⠁⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢐⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠞⠀⠀⠠⡇⠖

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣧⣒⣯⣭⣭⣭⣭⣩⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠒
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⠏⠘⢧⡘⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⣿⠃⠀⠀⠀⣀⣀⡀⡜⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣀⠀⡤⢠⣤⠄⠰⠾⠂⣐⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣴⣿⣿⠏⠀⠀⠀⣾⣿⣿⡿⠁⠀⠀⠂⠀⠀⡰⠀⠀⢀⣈⣑⣤⡇⠈⠉⠀⠺⠂⣀⠩⠆
⠀⠲⠶⠶⠶⢶⣶⣿⣿⣿⠟⠒⠒⠒⠒⠚⠛⠛⠓⠛⠛⠛⠋⠉⢩⠋⠉⠉⠁⠈⢅⣀⡀⠀⢀⠀⠲⠂⠉⠀⠃
⠀⠀⠀⢀⣴⣿⣿⡿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⣐⣪⡟⢲⠦⠍⠂
⠀⢀⣠⣿⣿⣿⣋⣀⣽⣄⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠖⡾⠒⠒⠋⠉⠉⢩⠻⡁⢀⡤⠖⠂
⠀⢨⣿⣿⠟⠉⠁⣠⣼⣷⣶⣶⣄⠀⠀⠀⠀⠀⠀⡴⠛⠛⣳⠁⠀⠀⠀⠀⠀⣸⣴⠟
⠀⢸⣿⠏⠀⢀⣾⣿⠿⣿⠿⢿⣿⣷⣄⣀⣀⣀⣀⣓⣾⣿⡥⠤⠤⠶⠒⠒⠛⢿⠛⣆
⠀⢺⣿⠶⠶⢾⣿⠛⠛⢻⡛⠛⠙⣿⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣣⡞
⠀⠈⢿⡀⠀⠘⣿⠀⠀⠸⡇⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠞⢾⠋⠁⢇
⠀⢀⣈⣷⣄⣠⣬⣷⣦⣤⣷⠤⢴⡿⠷⠖⠒⠚⠛⠋⠉⠉⠉⠀⢀⣤⣤⣄⠀⠀⣠⠏
⠀⠈⠉⠉⠉⠛⠦⢤⣤⣤⣿⡶⠋⠁⠀⠀⠀⠀⢰⣾⣿⡷⠀⠀⣸⣿⣿⡿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣀⣠⣤⣤⠤⠼⠿⠛⠟⠛⠉⠉⠀⠀⠁
⠀⠰⠶⠶⢲⣿⣿⣟⠛⠋⠉⣿⠉⠁
⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⣠⡌
⠀⠀⠀⠀⠻⣿⡟⠋⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣿⣷⣶⣿⠋⢠⣶⣷⣶⣶⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣁⣾⣿⣿⣿⣿⣿⣿⣋⣶⣿⣿⠏⠁⠀⠒⠲⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢱⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠹⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⡀⠀⠈⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣈⡉⠛⠻⠿⣿⣿⠀⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⣉⣉⠉⠛⠻⠿⣿⣿⣿⣶⣦⣤⡀⠀⣮⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣶⣶⣤⣌⣉⠛⠻⠿⠁⢀⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⢠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣄⣈⡙⠛⠿⠿⣿⣿⣿⣶⡄⠀⣰⣿⣿⣿⣿⣿⣍⠙
⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣶⣦⣤⣌⡉⠛⠃⠀⣾⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄⣉⡉⠛⠻⠿⣿⣿⣷⠀⢰⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣈⣹⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣷⣦⣤⣀⠁⠀⢸⣿⣿⣿⣿⣿⠛⠋
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣉⠙⠛⠿⣿⣿⠃⠘⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣿⣯⣩⣏⣉⡉⠉⠛⠛⠻⠿⣿⣿⣿⣷⣶⣤⣀⠀⠄⣼⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣈⡉⠛⠛⠻⠋⠀⢠⣿⣿⣿⣿⡟⠛⠃
⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣠⣍⡉⠉⠙⠛⠿⢿⣿⣿⣿⣶⣦⠀⠀⣾⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣈⡉⠉⠃⠀⠀⣿⣿⣿⣿⣧⣈⠁
⠀⢀⣶⣸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⡀⠀⠉⠉⠙⠛⠻⠿⣿⣿⣿⡿⠀⠀⢀⣿⣿⣿⡿⠿⠿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⡆
⠀⠘⢿⡿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠙⠋
⠀⠀⠀⠙⠻⢿⣝⢿⣿⣿⣿⣿⠞⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠙⠲⠽⢿⣿⣿⣷⣄⣈⠙⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣷⣦⣍⣙⣛⣻⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠛⠙⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⢿⣿⣿⣿⠇⣴⠾⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⢀⢰⣿⣿⡄⠁⢀⣿⡀⠙⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣶⣿⣿⣿⣿⣄⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠹⣿⣟⣦⡉⠸⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠙⣿⠻⢿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣴⣦⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⣀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⢀⡀⣤⣶⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⣶⠓⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠈⠉⠉⠉⠉⠲⢠⣄⡀
⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣀⢀⠤⣀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢸⣦⣀⡀⠀⠀⡆⢻⡿⢂⡙⠃⠀⠀⢀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⡏⠉⠈⠻⢆⡀⠈⠁⠈⠋⠉⠀⠀⢠⡞⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⡇⢀⣀⠠⢾⡃⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⢨⡿⠦⢀
⠀⠀⠀⠀⠀⢃⣘⡏⠂⠴⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠈⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢻⣄⣄⡴⠀⠀⠀⢀⡠⠶⠦⠶⢾⣟⠁⠀⠀⠀⠀⠀⠙⢿⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⢀⡴⠟⠡⡄⠀⠀⠀⠀⠘⠷⢦⡀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢰⠃⠀⠀⠀⠈⠑⢦⣀⣤⠆⠀⠀⠈⠐⠠⣀⠀⠀⠀⠀⠀⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡊⠀⠀⡀⠀⣠⠀⠀⢹⡄⠀⠀⢀⡀⠀⠀⠈⠓⣦⣀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢿⠄⠀⠋⠀⠀⠰⣶⡏⠀⠀⢤⠋⠁⠀⢀⣀⣠⠄⠉⠛⢦⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⠀⢸⡀⠀⠀⠀⠀⠑⢀⠀⠀⣦⡀⠀⣠⠀⠁⠀⠀⠀⠀⠈⠳⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠉⠁⠀⠀⠀⠀⠈⢷⠸⡁⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⣹⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠱⣄⠀⠀⠷⠴⣦⣄⡀⠀⠀⣀⠛⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⡷⠠⡀⠀⠀⠈⠉⠛⠛⢧⣄⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡀⠀⠈⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠀⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠐⣿⠀⠀⠀⠙⣦⡀⠀⠀⠀⣤⠀⠀⠠⠿⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠙⠁⢄⣀⡠⠦⡈⠳⣤⠄⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠂⣀⠁⠀⠀⡀⠀⠀⠀⡀⠈⠀⢄⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣄⠐⣄⡀⠀⠄⠀⠀⠀⠈⠙⠲⠢⣬⠒⣤⠀⠀⠁⠒⠀⠈⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠒⢴⣶⡖⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣀⣤⣤⣤⣀⣀⣀⣀⣀⣤⣶⣿⣿⣟⣛⣿⣷⣦⣤⣤⡀⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⢻⡜⠄⠈⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⠀⠈⠁⠀⠙⣿⣯⠀⠀⠀⠀⠀⠀⠈⠳⣬⣙⠓⠀⠀⠀⠀⠀⠀⠀⢱⠈⠐⢿⠁⠀⠉⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠙⠲⢤⣄⡒⠢⣤⣤⠼⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣛⣡⣤⣤⣄⠀⠶⠶⠶⠛⠀⠸⣆⠀⠀⠈⠙⠳⣤⣀⣄⡀⠀⠀⣤⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠸⡅⠉⡟⠛⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢿⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠘⢇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠘⡀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⢃⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠩⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⣠⡀⠉⠓⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡤⠤⠤⠤⠤⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⡠⠒⡄⠀⠀⡴⠁⠀⠀⣠⠔⠠⠚⠁⠈⠓⠤⣀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀
⣰⠊⠉⢈⡽⠅⠴⠃⡠⠊⠀⠶⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠈⠑⡄⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⡏⠀⠀⠳⣄⣀⡠⠊⠁⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⢷⠀⠂⠀⠀⠀⠐⠀⠀⠀⢀⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠜⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠈⠢⣄⣀⣀⣀⣀⣀⣀⠔⢁⣴⢁⡤⠤⢼⠀⠀⠀⠀⣠⠞⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⣀⣠⣔⡩⠖⠉⠈⠁⠀⠀⠀⠣⠤⠒⠯⢤⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⣀⣥⠀⠀⠀⢰⠁⠈⠉⠒⠒⠤⠤⠤⢤⣤⣤⠤⠤⢤⠤⠤⠤⡄⠀⠀⠀⠀⡎⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠣⡀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡀⠀⠀⠀⢱⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡤⢤⣀⠤⠜⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡎⠸⡀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⠁⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⢀⠤⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⢀⣀⡀⡜⠀⠀⡧⢄⣠⣄⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠈⠉⠁⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣭⣭⣭⣛⡻⠷⣦⣄
⠀⠀⠰⣄⡀⠀⣀⡀⠠⠤⠤⣴⣶⣾⣿⣿⣿⣿⣯⣭⣿⣛⣿⢿⣻⢷⣤⣽⣿⣷⣄
⠀⠀⠀⠈⠛⢷⣶⣬⣭⣽⣿⣿⣿⣿⣿⣿⣷⣺⣦⡴⠶⠦⢌⣁⣈⠙⢮⡛⣿⣿⣿⡳⣄⡀
⠀⠀⠀⠀⠙⢦⣌⣙⠛⠿⠿⠿⠿⠿⠛⠛⣉⡭⠶⠒⣛⣛⣳⣿⣿⣿⡷⠿⣿⣿⣿⣷⣹⣿
⠀⠀⠀⠀⠀⠐⠾⠿⣿⠿⠿⣟⣻⣽⡾⢛⣡⡴⢞⣫⣭⡭⠠⣤⡰⣬⣭⣅⣙⣦⡉⠻⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠒⣻⣿⣿⡿⢋⣴⣿⣯⣼⣭⣿⣿⣿⣷⣬⡛⠶⣭⣭⠛⠛⠛⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠟⣻⣿⢏⣴⣿⣿⠟⢉⣥⡾⠟⢋⣭⡽⠟⠛⠿⠿⠃⠀⠀⠰⢤⠾⣷⠒⠻⠦⡀
⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⢯⣾⣿⠟⠁⣰⣿⠟⠀⣴⣿⡍⠀⣴⠞⠂⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣀⣈⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠟⣿⣿⣿⠏⠀⣼⣿⡟⠀⢸⣿⡿⠀⣼⠃⣠⣤⡀⠀⠀⠀⠀⠀⢿⣁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠛⢰⣿⣿⡟⠀⣸⣿⣿⡇⠀⢸⣿⣧⠀⣏⣾⣿⣿⣿⡄⣠⢄⣠⡴⠿⠿⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⣿⣱⠀⣿⢸⣿⡇⢰⡌⣿⣿⣆⢿⣿⣿⣿⣿⣿⡏⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⣿⠀⡇⢸⣿⣇⢸⣷⣹⣿⣿⣾⣿⠋⠀⠀⠙⠇⣿⣏⣲⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠇⢸⣿⢻⡌⣿⣷⣿⣧⠹⣿⡄⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠈⢿⡎⢿⣽⣿⢿⣿⣧⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢷⡈⢿⣿⣆⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠻⣿⣧⡈⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡀⠀⠉⠛⠳⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠘⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠢⡄⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠡⢄⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⢆⣀⠀⠀⣀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠸⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢁⠀⣀⡉⠀⠉⢆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠢⢀⡀⠔
⠀⠀⠑⠂⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠰⢀
⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡀⠀⠀⢀⠁⠰⠁⡐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠆⠀⠴
⠁⡀⠔⡂⠀⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣠⠀⠀⠀⡈
⠀⠂⠂⠈⠐⠐⠀⣟⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡀⠹⠀⠁⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⣰⠃⢱⢄⡀⠠⠀⠀⠀⠀⠀⠠⠀⠀⣠⢴⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⢨⡀⠑⠀⠠⣇⢨⠂⠀⠊⠁⠡⢸⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠄⢼⣀⡧⠤⠤⠤⠀⠀⠀⠶⠥⠤⣆⣸⡖⠒⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠅⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⡁⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡤⢠⣀⢀⠀⣀⣠⠤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠰⠀⠀⠀⠆⠀⠀⠂⠐⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠜⠣⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠸⠓⠺⠇⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠛⠀⠀⢺⣿⠀⠀⠘⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣀⣄⡀⠀⠀⢠⣿⣿⣆⠀⠀⠀⣠⣄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠙⠛⠀⠀⣰⡿⡿⢿⢻⣦⠀⠀⠛⠛⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣀⣄⡀⠀⠀⢀⣜⡿⢱⡇⢸⡆⢿⡣⡀⠀⠀⠀⣠⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠹⠻⠁⠀⡠⣪⡾⢁⣾⡇⢸⣷⠈⢿⣝⢄⠀⠈⠟⠿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣿⠂⠀⢀⡀⠀⠀⢀⢞⣴⡟⠁⣼⣿⠀⠘⣿⣇⠈⢿⣦⡳⣄⠀⠀⠀⡀⠀⠐⣾⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠟⣀⠔⣱⣿⠏⠀⣰⣿⣿⠀⠀⣿⣿⡄⠀⠻⣿⣎⠳⣄⠘⠛⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡠⢊⣵⣾⠟⠁⠀⣰⣿⣿⡇⠈⠀⢻⣿⣷⡀⠀⠙⢿⣷⣬⡓⢄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣩⣶⣿⠟⠁⠀⠀⣰⣿⣿⣿⠁⠀⠀⢸⣿⣿⣷⡀⠀⠀⠙⢿⣿⣦⣝⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣟⣕⡂⠀⣠⣾⣿⣿⣿⣿⡬⠭⠭⢤⣿⣿⣿⣿⣆⠀⠀⢐⢋⣻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣍⠻⢯⣝⣛⣛⣛⡫⠟⣫⣾⣿⣶⣶⣿⣿⣝⠿⣟⡻⠿⢟⢛⣫⠿⣫⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣽⣷⣤⣄⣉⣉⣥⣴⣿⣿⠙⣿⣿⣿⣿⢸⣿⣷⣬⣍⣉⣉⣩⣤⣾⣏⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣇⣿⣟⣀⣹⣿⣿⣁⣸⣿⣿⣀⣿⠿⠿⣏⣈⣿⣿⣇⣉⣿⣿⣏⣀⣿⣿⣸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⣲⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣯⣽⣻⣿⣭⣷⣾⣿⡷⠶⠿⠿⣿⣿⣿⣿⣲⣶⣿⣿⣿⣿⣿⣿⣷⣾⣿⣠⣇⣠⢸⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣽⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣽⢾⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣽⣿⡿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣟⣿⣿⠿⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣯⣿⡿⣿⣿⣿⣿⣿⣿⢿⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣷⣾⣝⣸⣿⣟⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣽⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣿⣻⢿⣻⣾⢾⣿⡟⣿⣯⡿⢟⣾⣿⣿⣿⣿⡏
⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠿⠛⠉⠈⠿⠿⠛⠉⢿⣿⣿⠟⣟⣿⣿⡾⣯⡾⣿⡿⣹⣟⣿⣫⣿⣿⡟
⣀⣼⣿⣿⠋⠉⠀⠁⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣯⣾⣿⣿⢯⢾⣿⡿⣿⡻⢟⣾⠃
⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡽⣿⣿⣭⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠣⣿⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⡎⠀⠀⠀⠀⣀⡤⠀⠘⡇⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠴⠀⣀⡸⠀⠀⣠⠖⠋⡁⠠⠆⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⢀⡶⠄⠀⠚⢀⡀⠘⠛⠃⠀⠈⡀⠴⠀⣡⠤⣤⣄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠌⠇⠀⢾⠁⠀⠾⠀⠀⢀⣄⠀⠠⡄⠀⠁⠀⣀⠧⣊⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢀⠉⠀⠀⢁⣦⠀⠀⠀⠈⠁⠀⠂⠂⠐⢏⡂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢈⣡⡌⠀⠐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⡤⡶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⢠⡄⠀⠀⠀⠀⢰⠛⠛⠳⢶⠄⠀⠀⠀⢻⣄⠛⢾⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠤⠀⣀⠄⠀⠀⠀⠺⠷⠁⢀⣤⣠⠂⠀⠀⠀⠀⠀⠉⠉⠁⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡾⠀⠂⡞⣵⠢⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⢀⣬⡄⠘⢁⡙⠀⢨⡋⠁
⠀⠀⠀⠀⣀⡀⢀⡀⢠⡞⠁⣄⠁⢤⡈⠉⠀⢀⠈⠁
⠀⠀⠀⢀⡍⠭⠉⠉⠉⣃⠈⠋⠀⠀⠶⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠰⣏⣪⢳⠀⠀⠀⠀⠀⣠⣤⡶⠶⠆⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠃⡄
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠹⣦⣌⡭⠓⣄⠀⠀⠀⠀⢨⠀⠀⣀⡄⠀⠀⠀⠈⡒⢏⡀⢀⡁⠤⢀⣀⠀⢀⡴⡁⠴⢀⡀⠓⠀⠠⠤
⠀⠀⠀⠀⠀⠉⠀⢠⢖⣓⠦⠔⠀⠀⠀⠀⠋⠀⠀⠀⢴⡾⠀⠈⠋⠁⠀⠀⠀⠀⠉⠛⠱⠎⠠⡀⠛⠛⠀⠈⣄⠛⠀⠈⠁⠛⠦⠤⣔⣚⡀
⠀⠀⠀⣤⠀⠀⠰⠆⠫⠕⣀⡤⠀⠀⠀⢤⡁⠃⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡶⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⢠⡄⢀⡄⠀⣀⡴⠚⢁⠀⠖⠀⠀⠀⠀⠀⢛⠁⠀⢀⡀⠻⠁⠀⠂⠀⡄⠒⠒⢀⡙⠆⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠘⠃⠀⢁⠰⠆⠉⠀⠀⠀⣠⠾⠷⠄⠈⢀⡀⢛⠁⢀⣵⠄⠀⠄⠰⠚
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠞⠁⠀⠀⠖⠂⠉⠁⠀⠲⠟⠁⢰⢩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠛⠁⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠈⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⢤⠄⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣦⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣏⠉⠹⠉⠀⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠘⠂⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡏⣴⣷⣶⣦⢰⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⡏⣝⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠙⢻⣿⣶⣶⣿⠏⠋⠉⠀⣹⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠒⠒⠀⠈⠉⠉⠀⠀⠀⠴⠦⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠛⠋⠉⠀⠄⠀⡄⠀⠀⠀⠈⡀⠉⡀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⣦⣄⡀⠀⠸⡁⠤⡤⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠐⠸⠿⠿⠉⠀⢀⠜⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠠⠀⠀⠀⠀⠀⠀⢸⠦⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠀⠈⡄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠉⠈⠓⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢐⢈⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⢀⡀⢰⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠈⠁⠐⡁⢀⣀⣄⣀⠀⠀⠀⠀⠔⠠⠊⠀⠈⠴⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠁⠀⡀⠄⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⡁⡀⠀⠁⠀⠁⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠴⠀⠀⠀⢠⠄⢴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠒⠈⠁⠂⠁⠀⠜⠁⠀⠀⠈⠂⣴⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢸⣣⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠉⠉⠁⠘⠂⠲⣄⠂⠠⠀⠀⠀⠀⠀⠤⢠⣔⣈⠠⠐⠁⠀⠀⠀⠀⢀⠀⡩
⠀⠀⠀⠀⠀⣀⠤⠔⡊⠀⣖⠋⠀⠐⠀⠀⠐⠒⠁⠙⠚⠧⢠⡴⠄⠚⠑⠈⠀⠀⠀⠀⠀⠀⡀⠀⠰⡶⣡⠔⠁
⠀⠀⡠⠆⠉⣀⠐⡏⠑⠀⠀⠀⠀⠈⠀⢐⡒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣠⠾⠊⠁
⠀⠀⠁⠀⠀⠉⠀⠀⠐⠒⠀⠈⠁⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢎⣴⠏⠀⠰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠔⢀⡀⣀⢴⢾⣠⣤⡔⠀⠀⠀⠀⠀⣀⡀⢀⠄⢀⡠⠒⠚⠋⡏⠚⣫⡤⠀⡜⠉⠉⠺⢆
⠀⠀⠀⠀⠀⣀⡴⢒⠅⠊⠀⠈⠀⠀⠀⠀⠉⠘⠀⠀⠚⠿⠛⠁⠈⠁⠰⣄⣀⠴⠔⠋⠀⡄⠘⢤⠿⣱⠢⡀⢱⡦⡆
⠀⠀⠀⡠⢚⠥⠊⡡⠀⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁⠀⠈⠙⠛⠃
⢀⢔⡠⢒⣡⠔⠁⢀⠴⠊
⠳⠈⣴⠋⣀⠤⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢿⣓⠲⠄⢂⠀⠹⢷⡟⠁⢀⠀⠴⢒⣻⠃⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣓⡖⠂⠀⠉⠉⠐⠄⠳⣄⠈⠁⡴⠋⠔⠋⠉⠀⠀⠒⣞⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠍⡉⠁⠀⠀⠀⠀⠀⠀⠸⣆⡾⠁⠀⠀⠀⠀⠀⠀⠉⡉⠭⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠂⠐⠿⣿⣿⣷⣶⣦⣄⠠⣀⡀⢻⠃⣀⡠⢀⣤⣶⣾⣿⣿⣿⠗⠀⠐⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣛⡉⠁⠀⠠⣹⣿⣿⠋⣁⣒⠶⣬⡳⣽⣷⣼⢰⣿⡵⣫⡴⠖⣂⠉⢻⣿⣿⡁⠀⠀⠀⣉⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠛⠁⠀⣀⠄⠀⣁⡿⣡⣾⣿⠿⠓⠀⠁⠈⠀⠀⠀⠈⠀⠁⠀⠻⢿⡿⢦⡹⣦⡁⠀⢄⡀⠀⠉⠻⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⢠⣄⣨⡥⢊⣵⡿⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣷⣍⠢⣭⣁⣤⠈⠙⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠁⢀⠀⠀⠀⠀⢲⡿⣻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣗⠂⠀⠀⠀⢄⠀⠙⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠂⠀⠀⠀⢋⠞⣯⠀⠀⠠⣶⣦⡀⡄⠀⠀⠀⠀⠔⣰⣶⠦⠀⠀⢸⡟⢎⠃⠀⠀⠈⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢀⣀⣀⣠⣤⠖⠋⣰⢿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⢷⡌⠓⢦⣤⣀⣀⣀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠛⣉⡍⠀⠀⢀⡾⠁⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠙⣆⠀⠀⠀⢉⡛⠿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⡾⠁⠀⠀⠀⠈⠀⣠⣿⡿⠁⢀⠂⠀⣀⢀⡀⠀⠀⣀⢀⡄⠈⢂⠀⠹⣿⣷⡀⠈⠃⠀⠀⠀⠹⣦⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠈⣠⡟⠀⢀⡴⣺⠎⢀⣴⠋⡼⠁⣸⡎⠀⠀⠈⠛⠻⣦⡾⠛⠋⠀⠀⠘⣧⡀⢹⡌⠳⣄⠈⢿⡲⣄⠀⠘⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⡎⡾⠁⠴⠋⠁⣰⠃⣰⡿⢻⡀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⣴⠿⣷⠀⢳⠀⠈⠳⠄⢻⡌⣇⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡇⠄⠀⠀⠐⢁⣴⡿⠁⠘⣻⠀⠤⠶⠚⠉⠙⠲⠤⠄⢸⡟⠄⠹⣷⣄⠑⡀⠀⠀⡄⣷⢿⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⢀⠀⡜⢠⡞⠁⠇⠀⢠⡿⣷⣄⣀⠀⠀⠀⠀⣀⣴⡿⣧⠀⠀⠃⠙⣦⠈⡀⢀⢠⢩⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣠⠁⣾⠁⠀⠀⠀⣼⠁⢸⠿⣿⣷⣦⣾⣿⡿⢻⠃⢸⡆⠀⠀⠀⢸⡆⠃⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⢿⡄⢠⡆⠀⠛⠀⢸⡀⢹⡌⢻⠋⣸⠃⣸⠀⠘⠃⠀⣠⠀⢸⢇⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⠁⠈⢧⠀⠀⢸⡄⢷⡌⠿⠈⠘⠏⣰⠏⣴⠀⠀⢀⡏⠀⠡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠻⢦⣟⠀⠀⠀⠀⣿⠼⠋⢀⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠋


⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠔⠈⠉⠉⠐⠤⠐⠒⠒⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠁⠀⠀⠀⠀⠀⠀⢉⡉⠈⠩⠀⡂⠤⣀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠐⠄
⠀⠀⠀⠀⠀⢀⠄⠁⠀⠎⠀⠀⠠⠄⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠐⠂⠐⠉⡂⠀⠐⠁⠑⢠⡤⠄⠚⢄⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⡀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠓⠄⠀⠠⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⢈⣶⣶⣧⣰⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⡰
⠀⠀⠀⠀⢀⠣⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠀⠀⠀⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠈⢡
⠀⠀⠀⠀⠈⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠢⡀⠀⠀⠀⠈⠐⠤⠤⠤⠐⠁⠀⠀⢀⠄⠊⠀⠀⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⣀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢠⡀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠢⠐⠉⠢⢀⠀⠀⠐⠀⠤⠤⠄⠐⠈⠀⠀⢀⠔⠁⠀⡠⠁⠀⡠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠢⡀⠀⠈⠀⠂⠀⠄⠀⠀⠠⠄⠐⠈⠀⢀⠤⠊⠀⡀⠀⠀⠀⠀⠀⡄⠘⣀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⣀⠀⠀⠄⢀⣀⡀⠀⠀⠀⣀⠀⠄⠐⠀⡀⠄⠂⠀⠀⠀⢠⡄⢆⠀⠀⠀⠘⢁⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⠀⠠⠤⠤⠄⠀⠒⠒⠉⠁⠀⠀⠀⡀⠄⠊⠁⠀⠄⠂⠀⠀⠀⠉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⣠⡀⢀⠈⠐⢀⡀⢀⠀⠂⠁⠀⣀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠈⢀⠠⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠸⠁⠉⢸⠀⠀⢻⣁⣹⠟⠀⠀⡗⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠒⠈⣠⣔⣤⠈⠐⠂⠠⠀⠏⡄⠀⠀⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠙⠦⠞⠀⠀⠀⠰⠀⠀⠀⠀⠀⡠⠚
⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⡄⠀⠀⠀⢠⠀⠀⠀⣠⢖⣤⠀⠀⠀⠀⠀⢸⠀⠒⠁
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠀⠙⠦⠚⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠀⢀⡴⢀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⠀⠀⠀⣀⠤⣤⠀⠀⠀⠀⠀⠀⡀
⢀⠔⠀⠈⠐⠌⡐⠤⣀⣸⠧⠤⢤⣤⣤⡀⢀⣀⣀⢙⣭⣝⣀⡀⠀⣰⢀⣄⡤⠠⢄
⠃⣤⠀⠀⠀⠀⠈⠠⠄⠁⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃
⠀⠠⠞⢠⡄⠀⡀⠀⠀⠀⡀
⠀⠀⠐⠛⠒⠁⠀⠚⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣰⠀⢀⣀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠒⠒⠒⠒⢺⠁⠀⠀⡶⠒⠒⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠀⠀⠀⠀⣬⠀⠀⠀⢧⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠀⠆⠁⠀⠉⠢⡀⠀⠀⣨⠀⠀⠀⠀⡇⠀⠀⠀⢠⠄⠀⠀⠀⢌⠀⠀⢀⡐⠉⠀⠈⠁
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⠀⠀⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢴⣾⣿⡏⡗⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠏⠟⠧⠇⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⡿⡿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⣾⡷⣳⢣⡇⣷⢻⡻⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢔⡪⢗⡽⢋⣼⠃⣾⠃⣿⡈⢷⡜⢷⣝⠢⣄⡀
⠀⠀⠀⠀⠀⢀⣀⡤⠖⣪⡥⢞⣭⠾⢋⣴⡿⠃⣼⡿⠀⣿⣧⠀⢻⣦⡙⠳⣮⣙⠳⣦⣤⣀
⠀⠀⠀⠀⠈⠉⢁⠚⣻⠁⠾⠿⠡⡶⠿⠟⠀⣼⣿⠃⠀⠸⡿⣆⠀⠙⢿⣦⡈⠙⠿⠶⢭⡙⠟⡟⠒
⠀⠀⠀⠀⠀⠀⢸⢸⡇⢸⣶⢰⣆⠀⠀⠀⠀⠀⢀⡁⠀⢉⠐⠀⠐⠚⠁⠀⣿⠀⣾⡇⢹⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠈⣧⢸⣇⢸⡟⠀⠀⠈⡠⠀⠀⠂⠀⠀⠀⠀⢀⠰⡐⠀⣿⠀⣿⡇⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡄⣿⢸⣿⢸⡇⢀⣀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⢂⣿⠀⣿⡇⢸⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠳⠧⣸⣿⠘⣿⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠘⣿⠀⣿⣇⣸⠧⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡁⠈⡃⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠁⠀⠁⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣘⢻⣟⢛⡡⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣝⢦⡤⢤⡴⣯⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠇⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣶⣦⣴⣦⣴⣶⣤⣶⣤⣶⣦⣴⣦⣤⣶⣤⣶⣤⣴⣦⣴⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠆⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣺⣿⣚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠆⠀⠆⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣹⣟⣉⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣛⠀⠤⠠⠤⢈⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡾⣿⣽⣿⣽⡗⠆
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠀⠀⢀⣸⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢃⠻⠿⠿⠟⡡
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣍⠉⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣇⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⡴⢂⠐⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣧⠐⢀⣀⠀⠀⠙⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⢐⣭⠉⠸⠗⣈⣿⣶⠠⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣿⣷⣆⠙⠖⠀⣹⠃⠰⢁⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠉⠻⠿⣿⠦⠄⣀⡉⢠⡂⣤⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢗⣫⢹⠢⡬⣷⡍⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠗⣿⣿⡼⣿⡄⡺⢭⢭⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⢿⣿⢯⡽⣿⣿⡛⡯⣳⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠁⢬⣿⣿⣿⢗⠷⣿⠻⠖⣿⠽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣮⢁⣿⣼⣿⡝⣻⣯⣼⠿⡷⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠛⠿⣮⡟⣕⠌⢹⡿⣤⣤⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣜⣆⠻⣿⣝⣷⣾⣯⢿⢷⢋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡸⡤⠌⣿⡌⠉⢻⣬⣼⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡝⣲⣎⡻⣄⠀⢹⠙⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡰⠮⣆⠑⣛⣧⣀⡅⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢿⡴⢭⢥⣖⢺⡓⠟⢿⣿⣧⣀⠙⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣄⢀⠀⠀⠀⢓⣸⠂⠀⢻⣿⣮⡳⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠘⠁⠀⠀⠀⢿⡿⣟⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⠒⠐⠉⠈⠛⠀⠀⠀⠀⠈⢻⡙⣯⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡄⠀⠀⠀⠀⠀⠁⠘⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⡆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠉⠀⠀⠈⠓⠦⠤⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⢉⡭⠼⢤⡀⠀⠀⠀⢀⣠⣀⠀⠰⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡴⠁⠀⣄⠀⢸⣠⣄⣰⠁⣀⠀⠹⡄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢣⡀⠀⠈⢡⣤⣤⢤⣤⡀⠁⠀⣰⠃⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢁⠆⠰⢈⡻⢌⠻⠃⢠⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⡤⡆⠀⢠⡏⠀⠀⢼⣁⣈⡆⠀⠀⢧⠀⠀⠀⣈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠘⣆⠀⢄⠈⢛⡋⠀⡀⠀⡼⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢶⠀⠀⠈⠓⢬⡉⠙⠋⣩⠵⠊⠀⠀⠀⠀⢠⠇⡰⠂⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣰⣀⠀⠀⠈⠢⠜⠀⠀⠀⢀⡜⣆⡴⠋⠈⠳⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠈⠓⠦⣄⡀⢀⣠⠤⠖⠋⡇⠁⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠤⣳⡀⠀⠀⠀⠙⠋⠀⠀⠀⣰⢃⣠⠤⣄⡀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠈⢳⠀⠀⠀⢹⠉⠀⠀⢰⠏⠀⠀⠀⠀⠙⡆⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠲⣄⠈⡇⠀⠀⢸⠀⠀⠀⡞⠀⣠⠔⠀⠀⢠⠇⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠳⣇⡀⠀⢸⠀⣀⣀⣃⠞⠁⠀⠀⡠⠏⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⠉⠉⠁⢠⠞⠉⠉⠉⣻⠋⠁⠀⠉⢦⠀⠚⠛⠓⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣅⠀⣅⣀⠸⡀⡄⠀⡀⢻⡀⣀⢀⡆⡸⠀⠀⡆⢈⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠉⠀⠀⠀⠀⠈⠉⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⢀⣤⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⢷⣦⡀⣿⣦⣤⡾⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡋⠀⠀⠀⠀⠛⣿⠿⠋⢡⣴⠾⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠳⣦⡏⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⣄⡀⠀⠀⠀⠙⠦⣀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⣤⣤⣀⣈⠉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠋⠹⢟⣿⣿⡿⠛⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠏⡀⠀⠀⠟⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣴⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢀⣤⣤⡄⠐⠀⠈⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⠀⠈⠛⠻⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠛⠉⠠⡠⢂⣶⣶⣦⡀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣥⣄⠈⠀⠀⠀⢸⣿⣿⣿⣇⣴⣶⣦⡀⠀⠘⢄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⣿⣿⣿⣧⠀⠀⠀⠀⠻⣿⣿⠏⢻⣿⣿⣿⡄⠀⠈⡍⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⠁⢸⣿⣿⡿⢫⡀⠀⠠⠂⠀⢀⡀⠀⠀⢹⠿⠿⠃⠀⠀⠠⢀⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠉⠉⠀⣾⣿⠀⠀⢀⣾⣿⣿⣦⠀⠘⠀⠀⢠⣾⣿⣿⣯⡀⢠
⠀⠀⠀⠀⠀⠀⠰⠋⠉⠁⠀⣀⣤⣤⣀⠠⣾⣿⣿⠀⠀⠈⣿⣿⣿⣿⠁⠀⣦⠀⠈⢿⣿⣿⣿⣷⢨
⠀⠀⠀⠀⠄⠰⠀⠹⢄⠀⣾⣿⣿⣿⣿⠆⣿⠿⠃⠀⡴⠀⠹⢿⡿⠟⠀⠈⣿⣷⣄⠀⠙⠻⠟⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣟⠿⣿⠟⠋⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠒⣄⣰⣾⣿⣦⣀⣀⡀⠀⣠⣤⣤⡒⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠙⠟⠁⠀⢼⣿⣿⣿⣷⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣤⡈⠀⠀⠸⣿⣿⣿⡿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆⠀⠀⠈⠙⠋⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠗⣤⣤⣄⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡸⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣴⣿⣿⠘⠻⠛⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣿⣿⣿⠀⠀⣠⣤⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢻⠇⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⣾⡀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠁⢄⢠⣿⣿⡦⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⣶⣾⣿⣿⡧⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠙⠿⣿⣇⡀⠰⢴⠆⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⢶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠘⠃⠊⠁⠀⠀⠈⠀⠐⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣔⠀⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠓⠢⢤⡀⠀⠀⢠⡐⢧⡀⢠⠐⠀⢠⢦⠀⠀⡀⠀⢀⣤⣀⣀⣀⣘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠀⠒⠶⣂⣀⡉⣶⡉⢨⡓⠄⠢⣤⣙⠚⠛⡉⠛⡦⡟⢋⡙⢻⠻⠛⢩⠍⠙⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢀⠘⠿⠷⠶⠄⠹⣧⣌⣻⣦⡍⠁⢒⣻⡃⠈⢱⡆⣀⠐⠀⢀⠁⢈⠂⠀⠉⣠⠾⣍⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣁⣘⡋⠀⠀⠠⢤⣀⠀⠕⠂⠛⠻⠯⠂⠰⠿⠈⢐⠒⠞⠁⠀⠈⠁⠂⢀⠀⢠⣾⠙⢦⠈⠻⣦⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣛⠉⠉⠛⢡⣤⣀⣀⣰⣍⠓⠠⠀⠀⠀⠀⢀⠉⠀⠉⡁⠀⠀⠈⠠⠔⠀⡁⠀⢿⣍⠁⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣶⣦⣤⣤⡌⣙⠀⠀⣤⣄⠀⠀⠀⠀⢤⡾⠿⠏⠋⠉⠀⠀⠀⣠⠾⠾⢦⣽⣷⣄⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⡶⠞⠉⡙⠋⠉⢐⣠⠤⠈⠁⠦⠀⠉⠁⠀⠀⡸⠿⢷⣦⣄⡀⠀⣀⠀⠀⢡⣶⣶⣦⣽⡿⢿⣿⣦⣄⣀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣽⣿⡶⠛⣻⠛⠑⡾⠟⠓⢶⡀⠀⠀⠹⣿⣆⡙⢷⡈⠿⢶⣍⡻⣿⣿⣄⠀⠀⣿⣿⣿⣿⣿⣿⣦⣽⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣽⣟⢉⡩⠛⠲⠿⠷⠀⠴⣄⠙⠒⠠⠀⠙⢷⣄⠀⠀⢚⣲⣿⣗⡛⠻⣿⣷⣴⣿⣿⣿⣿⣿⣿⡿⢻⠕⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠱⠿⠷⠦⠶⡆⠘⠒⢂⠀⠀⠰⠂⠀⠀⠀⠀⠈⠀⠸⠿⢷⣮⣽⣦⡸⠏⢻⣿⣿⣿⠿⣿⠽⠊⠁⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠶⠛⠓⠲⢶⣶⣦⣦⣤⣵⣷⠎⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠛⠿⣽⣷⣤⣹⣷⣾⣷⠄⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⢀⢀⣠⣶⣶⣶⡦⢤⣤⣌⡛⠏⣛⢛⠛⢿⡏⠘⡁⠀⣠⡀⠁⠀⠐⡦⠀⠀⠀⠀⠁⠙⠢⠄⠀⣾⣯⡉⣛⣿⣿⣧⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣈⣉⣉⣭⣿⣿⣿⣿⡽⠻⢷⡤⣤⡁⠑⠢⣈⢿⡁⠈⠉⠑⠒⠒⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠻⣿⡿⠇
⠀⠀⠀⠀⠀⠁⢉⣿⣿⠿⢿⣿⣿⡏⠻⠀⡙⡞⣾⡻⣶⡓⠈⣰⠏⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠠⠖⣶⣿⡿⢿⢓⠝⠋⠴⠁⠀⠀⠀⠀⠈⣋⠡⠽⣿⣦⠟⠂⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⢦⣀⡁
⠀⠀⢰⣿⣿⠋⠋⠐⠛⠠⠈⠀⠒⠀⠀⠀⠀⠀⠘⠃⠀⠁⢻⡆⠆⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠃
⠀⠀⠸⠟⡋⢠⠊⠀⠊⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠐⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠀⠀⠈⢿⣦⡐
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠞⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⢭⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⡁⠙⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣻⡷⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠘⢦⣤⣤⡾⠷⣀⣈⣿⠂⠔⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠟⣔⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣀⣤⡄⠀⠀⠀⠀⠀⠈⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢸⡿⣾⢿⡝⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠈⠉⡴⠤⣤⠚⠀⠊⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⣍⣉⢺⡍⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠾⠁⠀⢹⡤⠀⠀⠈⠂⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⢉⢙⣷⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⣏⣩⣩⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣙⠗
⣀⣀⣀⢀⣀⡀⡀⣀⠀⣀⣀⢀⢀⣀⢀⣀⡀⢀⣀⠀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⠀⠀⢀⠀⣀⣀⡀⡀⢀⠀⣀⡀⢀⡀⠀⣀⡀⠀⠀⡀
⣿⠉⠉⣿⢛⠛⢛⢛⡛⢻⡛⣻⡛⠛⢻⠟⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣻⣿⣿⣛⣛⣛⣛⣛⣻

⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠒⠶⠀⠒⠉⠉⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠰⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⣀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠊⢹⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢠⡀
⠀⠀⠀⢀⠀⢻⡀⠜⢀⣴⠇⣠⣾⣧⣀⣼⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠓⠤⣤⡤⠦⠶⠞⠛⠛⠉⢛⣿⡀⠀⠀⠀⠀⠠⣄⣤⣤⣄⣠⡞⠁⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣄⠀⠀⠀⠀⣿⢿⢿⣿⣯⠀⠀⠀⠀⠀⢀⣾⡿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⢼⠁⠀⠈⠉⠛⠳⠀⠀⢀⣤⣿⣿⣧⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠉⠝⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠄⠘⠇⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠀⠀⠉⠚⢆⠇⠀⠀⠀⣴⠋⠀⠀⠀⣀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⠛⠛⠙⠛⠓⠈⠁⠙⠙⠛⠛⠀⠘⠘⠛⠛⠛⠀⠐⠛⠛⠙⠓⠛⠛⠛⠙⠛⠒⠂⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣦⣄⣀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀⣀⣀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠎⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠻⣿⣿⣿⣿⣿⣿⡄⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠇⠘⣿⣿⠀⠀⠀⠀⠈⠛⢿⣿⡛⠿⣿⣷⡀⠈⠛⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⢈⣿⠀⠀⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟⠀⠠⣶⡿⠟⠀⠀⠀⠀⠿⠿⠟⠁⠀⠶⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠂⠥⡀⠀⠀⠀⠀⠀⠈⡠⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢘⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠑⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢄
⠀⠀⡀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⢂⣤⠀⠀⢀⡘⢀⡂⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆
⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠈⠀⠈⢠⡀⢠⡄⠁⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔
⠀⠀⠉⠂⠀⠊⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠌⠀⠀⠀⠈⠀⡀⡁⠀⠀⠀⠀⠲⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠈⢂⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⣀⠀⠄⡔⠁⠀⠀⠀⠀⠀⠀⡌⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⣆⠀⠀⠀⠀⠐⠀⠰⠄⠀⡠⠂
⠃⠀⠀⠬⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⡅⠀⠀⠀⠐⠂
⠀⣊⡀⠐⠤⣀⡀⢀⡧⢀⡀⠀⠀⠀⢃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⢀⠀⠐⡧⠀⠀⠐⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠄⠀⠐⠠⡸⠢⡀⠀⠀⢀⠀⠠⠀⠀⠠⠒⠂⠀⠀⠀⣀⠞⣄⠀⠀⠄⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠁⠀⡏⢲⠤⢈⠈⠠⢀⠀⡀⠀⢁⡠⠴⡞⢻⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡇⢸⠀⠀⠀⠈⠟⠐⠏⡀⠠⠀⠀⡇⢸⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠚⠛⡧⢼⠒⠒⠒⠒⠲⠂⠒⠘⠃⠒⠘⡧⢼⡏⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠨⡀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣄⣀⣀⣀⣀⣀⢀⣀⡠⡤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠠⠀⣩⠁⠉⠀⢹⠀⠀⠰⠄⠤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧⣴⣦⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣴⣶⣴⣤⣤⣤⣤⣤⣤⣴⣦⣴⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢶⣏⡤⠖⠒⢦⣀⠤⠤⢄⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠨⠀⠀⠀⠈⠁⠀⠰⢎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣀⡤⠶⣶⣃⡠⠖⠻⠷⠖⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠂⠂⡉⠳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠩⠀⠐⠀⠀⠀⢠⠀⠀⡄⠀⢢⡀⠙⢿⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⡁⠐⠀⡠⠁⠀⠀⠀⠀⠘⠀⠀⠠⡀⠀⠱⣄⠀⠑⢦⡙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⣠⠴⠖⠋⠁⣀⠔⠊⠀⡠⠊⠀⠀⡜⠀⠀⠀⡐⠀⠀⠀⢳⡀⠀⠈⠣⡀⠀⠉⠢⣀⠈⠙⠳⠦⣄
⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⠁⠀⠀⠾⠀⠀⠀⢺⠀⠠⠤⠤⢦⠤⠤⠤⠤⡷⠦⠖⠀⠛⠖⠒⠒⠚⢳⠚⠛⠉⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⡡⠀⠀⠀⢸⠀⠀⠀⠀⣸⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠙⢦⣤⠴⣿⣄⣀⣤⢧⡀⠀⢀⡾⣧⡀⠀⣠⢿⣄⠀⠀⣠⢿⣄⠀⢀⣼⠻⣄⣀⡴⠋⡳⢶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡦⠌⢙⠀⠀⠉⣉⡍⠀⠈⠉⠉⠁⠀⠈⠉⠉⠁⠀⠉⠛⡏⠀⠀⢈⠁⠀⠀⠃⢸⡇
⠀⠀⠀⠀⠀⠀⢀⡇⢰⠗⠀⢸⠀⠀⠀⡇⡠⠀⢠⠀⠀⠀⢰⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠸⠀⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⣿⠀⠀⡁⠀⠀⠀⠇⠀⠀⢘⠀⠀⠀⡅⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢸⠀⠀⠀⡏⠀⡀⠨⠀⠀⠀⠀⣿⡀⠄⡆⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⡆⠈⡇
⠀⠀⠀⠀⠀⠀⡾⠀⡤⡀⠀⡘⠀⠀⠀⡧⠊⠀⠐⠀⠀⠀⢰⣿⣇⠈⢦⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⢣⠀⢿
⠀⠀⠀⠀⠀⢠⠇⠀⡆⠀⠂⡆⠀⠀⢨⠀⠀⠀⢸⠀⠀⠀⣸⣿⣿⡄⠀⠣⡀⠀⢰⠀⠀⠀⡆⠀⠀⠘⠀⠸⡆
⠀⠀⠀⠀⠀⡾⠀⠸⢡⠔⠡⠀⠀⠀⢘⢀⠄⠀⢸⠀⠀⢀⣿⣿⣿⣧⠀⠀⠈⠢⠘⡀⠀⠀⢱⠀⠀⠀⡄⠀⢻⡄
⠀⠀⠀⠀⣼⠃⢀⡟⠃⠊⡸⠀⠀⠀⡞⠁⡀⠀⠤⠀⢀⣾⣿⣿⣿⣿⣆⠀⡐⠁⠀⡧⠀⠀⠘⡀⠀⠀⢸⡀⢈⣧
⠀⠀⠀⠀⠙⠢⢮⣅⣊⣀⡃⠀⠀⢀⡃⠀⠀⠀⡌⢀⡞⠀⠀⠀⠀⠀⠙⣶⣀⣀⣀⣹⣀⣀⣀⣥⣤⠤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⣠⡾⠟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⡟⠳⠄⠀⠀⢿⣿⣿⣿⣿⠇⠻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠉⠒⡗⠊⠁⠀⠀⠀⣤⣶⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠙
⠀⢰⡏⣰⠿⢷⡄⠀⢠⣾⣿⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣾⣿⠆⢠⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⢀⣀⣀⣸⠇
⠀⠈⢣⠁⠈⣀⠏⠀⠀⠉⠉⠀⠀⠀⢠⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠉⠒⠐⠃⠀⠀⠀⢀⣤⣤⣤⡄⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⢀⣈⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡟⠿⠿⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⡇⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠸⠟⠠⠂⠀⠸⠛⠻⠀⠀⠀⣠⣶⡆⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡇⠀⠀⠀⠀⣴⣶⣤⣤⣤⡄
⢀⡀⠀⠀⠀⠀⢀⣈⢀⣀⡀⠀⢠⡿⠛⢻⠀⠀⣴⣶⣾⢀⣤⣤⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠛⢿⣧⠀⠀⠀⠠⣾⣿⠇⢿⣿⠟⠀⠀⠀⠀⣿⣭⣭⣭⣽⠁
⢸⣷⣤⡀⠀⠀⠙⠋⠈⠛⠁⠀⣾⠁⠀⣼⠀⠀⠉⠛⠁⠘⠛⠛⠁⠀⢰⣿⣿⣿⠀⠀⠀⠀⣿⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⠛⢻
⠈⡇⠉⠳⠀⠀⢀⠀⠀⠀⠀⠀⣯⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⣀⣀⢠⡏⠀⠀⠀⠀⠀⢰⠃⠀⠀⡶⠆⠀⠀⠀⢀⣤⣤⣿⠀⣀⣀⣸
⠀⣧⠀⠀⠀⠀⡸⠀⠀⠀⢀⣤⣿⣿⠏⠀⠀⠀⠀⠀⢰⣶⣶⣶⣷⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡼⠀⠀⠘⠀⡼⠀⠀⠀⠹⢿⣿⠟⠸⣿⣿⡿
⣿⡿⠀⢀⣠⣀⠇⠀⢀⣴⣿⡿⢻⠁⠀⠀⠀⠀⠀⠀⠘⣷⣶⣶⣿⡄⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⢠⣶⣶⠇⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠉⠁⠀⠈⠛⠛⠀⢀⣾⡟⠁⢀⣸⣦⣀⡀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣇⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠈⠙⠉⠀⠀⢰⠟⣠⣤⣄⠀⠀⠀⢘⠛⠛⠀⠀⢀⡄⠀⠈
⠀⢀⣀⠀⠀⠀⠀⢸⡟⠀⢰⡿⠻⡟⠻⣷⡄⠀⠀⣠⣶⣾⡀⣾⣿⣿⠀⠀⠀⠀⢸⠉⠉⡇⠀⠀⠀⢀⣠⣤⡄⠀⠸⡀⠏⠠⢸⠇⠀⣀⣸⢀⣼⠀⠀⣼⠃
⠀⣼⠻⢿⣶⡤⠀⠸⣇⠀⢻⡀⠀⡇⠀⢹⡇⠀⠀⠙⠿⠛⠁⠉⠋⠁⠀⠀⠀⣀⣨⢀⣤⣧⠀⠀⠀⢸⣿⠶⢧⠀⠀⠈⠒⠐⠋⠀⠀⠙⠋⠈⠉⠀⣰⠏
⣸⠻⠿⣷⣾⠃⠀⠀⠘⢦⣀⠑⠀⣧⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠉⠁⠀⠀⠀⣀⡆⣠⣼⡀⠀⠀⣿⢆⠐⠀⠀⠀⠀⠀⠀⢠⡟
⠃⠀⠀⢠⠏⠀⠀⠀⠀⠀⠈⠉⠉⢻⠁⠀⠀⢠⣶⣾⠿⢿⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣦⣄⣾⠁⠀⢠⣾⣶
⢸⣿⣷⡟⠀⠀⠀⠀⢀⣴⣶⡄⠀⢸⠀⠀⠀⠘⣇⠀⠀⠈⣧⠀⠀⠀⠀⣾⠉⣃⠀⠀⠀⠘⣷⣄⠀⠀⢀⣄⣤⣤⣤⣤⡄⠀⠀⠀⠘⢿⣿⣿⣿⠇⠀⠀⠀⠙⠛
⠀⠉⠉⠀⠀⠀⠀⠀⠘⣿⠿⠃⢀⡼⠀⠀⠀⠀⢹⡀⠀⢀⣸⡆⠀⠀⠀⠅⢠⡟⠀⠀⠀⠀⡇⠈⠀⠀⢸⡿⠿⠿⠿⠿⣧⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⣤⣤⣤
⣴⣦⣤⣤⣤⣄⡀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⣤⣶⣧⠸⣿⣿⠇⠀⠀⢀⣼⡿⠁⠀⠀⣠⣶⣧⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢿⠶⢿
⣿⠛⠛⠻⠿⢿⡇⠀⠀⠀⢀⣄⠀⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⢠⣿⠋⢃⣀⠀⠀⠈⠋⠁⠀⠀⠀⠀⣏⠉⠉⠉⠉⢿⠀⠀⠀⠸⡿⣶⣄⠀⠀⣤⣼⢠⣶⡆
⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣿⠁⣴⠟⡛⢿⡆⠀⠀⠀⠀⠀⣀⡀⣿⠀⠀⣀⣀⣸⡄⠀⠀⠀⢳⠀⠈⠃⠀⠉⠉⠀⠉
⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⢿⣿⣦⡀⠀⠀⠀⠀⢹⠳⣆⠀⠀⠹⡄⠛⠄⢃⢀⡇⠀⠀⠀⢰⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⣇
⠀⢀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣿⠀⠉⠻⣧⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠈⠓⠒⠚⡏⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠙⠛⠛⠋⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⣠⣶⣶⣤
⠰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠀⠀⠀⣴⣾⡇⠀⠀⠀⠀⠀⣴⣦⠀⢡⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⠻⣿⣿⣿
⠀⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠙⠃⠖⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣤⡄⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⡿⣶⣦⣤⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⡴⠋⢹⠀⠀⠀⠀⠀⡿⠿⢷⠆⠀⠀⠀⠀⢠
⢿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⡿⣷⣶⣭⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣴⣶⡿⠀⠀⠀⢸⢁⣴⠏⠀⠀⠀⠀⠀⠃⠀⢸⠀⠀⠀⠀⠀⠸
⣠⡇⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢀⣤⣄⣸⠃⠀⠀⣹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠀⢀⣤⣶⣿⠟⠁⠀⠀⠀⠰⣿⡿⢰⣶⣾⠀⠀⠀⢠⣤⣆
⠛⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⣴⣿⣶⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠿⠋⣁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⡼⠛⡇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⢠⡾⢻⠻⣷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⣧⡾⠁⠀⠀⠀⢸⡆⠀⠀⠀⣄⡀⠀⠀⠀⠀⢀⣤⣤⣤⣼⡇⢀⣤⣶⣶⣼⡇⠀⠀⠀⠀⢇⠘⠀⠘⠀⣿⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡿⢿⣦
⣶⣶⣶⣾⣿⡷⢷⣾⣾⡿⣾⣷⣶⣷⣿⣿⣿⣿⣾⣶⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣶⣾⣷⣾⣿⣿⣿⣾⡶⠶⢶⠾⠿⣾⣶⠶⠾⠿⡿⠾⠿⠾⢿

⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠠⡄⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠉⠉⠐⠄⠀⠀⢀⣀⣀⣀⣀⣤⣄
⠀⠀⠀⠀⠀⣞⠁⠀⠀⢱⡄⠠⡎⢀⠊⠢⡀⠀⠀⣀⠀⢀⠃⠀⠀⠀⠀⣤⣘⣒⠩⠭⠍⠀⠲⣖⠐⡏
⠀⠀⠀⣴⠒⣎⠲⠤⣀⡐⣣⣢⢞⣦⠘⣧⠈⠲⡎⣴⣿⣾⠀⠀⠀⠀⠀⠈⡏⢸⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠈⠉⠁⠀⠀⠈⢮⣊⡵⣿⣽⡷⡄⠀⢠⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⣧⠸⠀⠀⠄⠀⠀⣘⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢮⢞⢡⡾⣟⣤⠀⠱⣄⠣⠀⠀⠀⠀⠀⠀⢀⡽⠤⠷⠄⠛⢋⣉⠭⢤⣔⣃⡀
⠀⠀⢀⠀⠀⠀⡀⠀⢴⡦⠀⠀⠑⣥⢞⣫⡾⣣⡀⠘⠀⢣⡀⠀⠀⢀⣀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠄⠃⠀⠀⠀⢀⠴⠐⠂⠂⠉⡂⠈⢯⣕⣾⣯⣷⡀⠀⠈⢏⠐⠒⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿
⠣⠀⢀⠀⠀⠀⠐⠢⣄⠀⠀⢠⡃⠒⡘⢻⣫⣵⡾⢯⠀⣛⠄⠀⣀⠪⣆⠼⠂⢤⣠⣄⠰⠾⡟⠻⠿⠟⠛⠋
⠀⠀⠈⠚⠢⡀⠀⢀⣜⠿⡅⡍⠙⠼⠁⠒⢻⣷⣾⠋⠁⠀⡠⠔⠣⡻⡀⠩⠄⢈⠛⣿⣧⡀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢨⠓⠂⠒⢴⠀⣿⣿⢋⠥⠒⠁⠀⢀⣀⣅⡱⢰⡈⣷⠀⠘⡿⠁⠀⠀⠀⠀⣀⣐⡄
⠀⠀⠀⠀⡠⠔⠂⠀⠉⠀⠁⠀⠀⠀⢨⣀⠟⠋⠁⢀⣠⠔⠋⠁⠀⢀⣀⠈⠁⠀⠀⢠⡅⡇⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⡜⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠉⠀⣀⠤⠒⠥⣀⠀⠀⢀⣾⠟⠿⣽⣆⠀⠀⠈⠉⠀⠀⠀⠀⢰⣤⡀⠛
⠀⠀⠀⡀⢀⡄⠀⡇⠀⠙⠀⠀⠀⠘⢦⣍⣀⠀⠀⠀⡀⠑⠀⠈⢿⡗⢠⣿⡜⠁⠢⡀⠀⠀⠀⠀⠀⣾⣿⡿
⠀⠀⠀⢣⠸⣿⠀⠈⠒⠊⠀⠀⢀⠀⣀⣨⣿⣿⣿⡶⢤⣄⡀⠀⠀⠉⠛⠙⠂⠀⠃⠴⠑⢄⠀⠀⢀⠛⠋⠁
⠀⠀⠀⠘⠦⡙⠁⠀⠀⠀⠀⠐⠁⠀⣿⣿⠿⣿⣿⣀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢵⣄⠊⠀⣀
⠀⠀⠀⠀⠀⠈⠁⠒⠒⠂⢁⠤⠐⠊⢁⡀⠀⠘⢿⣿⣿⣿⣿⣧⣤⣴⠖⠈⠁⠈⠠⢄⠀⠀⡒⠒⠵⢆⠀⠱⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⡌⠀⠀⠈⣆⠈⢿⣿⣿⣿⠟⠋⡤⠒⠒⠂⠀⣶⡶⠁⠶⣅⠀⠀⢀⠇⠀⠀⠘⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠐⠄⣀⡀⠍⠀⠈⢿⠛⢁⣀⣀⡈⢆⡎⠀⠀⠉⠀⠀⠐⠤⠈⡉⠁⠀⠀⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⢀⠤⠄⠀⠀⢂⠁⠒⠒⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⢑⣁⣤⣴⡾⠀⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣟⣛⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣹⣛⣝⣛⣟⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠠⠄⠂⠀⠀⠀⠐⠂⠀⠄⡀
⠀⣾⣿⣷⣠⣀⣠⢤⡀⠀⠀⠀⠀⠈⡑⣄
⠀⢻⣿⣿⣿⣧⣀⣤⣼⠁⠀⠀⠀⠀⠈⠺⠴⣄
⠀⢸⡿⣿⣿⣿⡟⢻⡿⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠒⠄⢀⡀
⠀⠈⠛⠇⠈⠉⢹⣿⡿⠂⠀⠀⢀⠂⠀⢀⠀⠀⠀⣄⡀⢦⡉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢋⠁⢠⡌⠁⠈⠠⠪⣦⠂⠀⠈⠙⠉⠉⠉⣴⠦⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢮⡙⡋⠃⣆⢀⣥⣿⣇⠀⠀⠀⠀⠀⣀⡄⡈⡄⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣶⣿⣿⣿⣿⣿⣿⠌⡁⠀⠀⢠⣦⢀⡀⠐⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣯⣿⡑⠅⡀⠀⠃⠀⠀⠔⠈⠀⠀⠀⠀⠱⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣽⣿⣿⣷⣄⠀⠦⠠⠄⢠⠠⠀⠀⠀⠀⢀⠀⠘⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠉⣿⠋⠳⢷⣶⣤⡀⠀⢠⠀⠀⣦⣬⠈⠀⠀⠘⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠱⠔⢊⣤⠤⢾⣿⣿⠿⠟⢲⣶⣿⣿⠲⣿⣶⣤⠁⣙⡀⢹⡙⠳⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠅⠊⠀⠀⠀⠀⠉⠑⠻⣿⣿⠀⠀⠉⠀⠀⠀⠙⢅⠳⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡑⠂⠐⠀⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⡴⢚⣉⠉⠲⣄
⣼⢰⠋⠉⠣⠀⠈⠳⣤⣀⠀⢀⣠⣤⠶⠶⠶⠲⠶⠶⠶⣤⣄⣀
⡇⡎⠀⠀⠀⠳⢄⣀⣨⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⢀⣠⡤⠖⠚⠛⠉⠉⠙⠛⠶⣤⡀
⣇⠇⠀⠀⠀⠀⡠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠶⠟⠒⢋⣩⡤⠴⠶⠒⠒⠛⠓⠲⣤⡀⠈⢿⡄
⢻⢠⠀⠀⢠⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣄⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⣿
⢸⣼⡆⠀⡟⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣸⣀⡀
⠀⡟⣷⣀⡇⡜⢉⡇⠀⠀⢀⣶⠀⠀⠀⠀⢀⠖⠢⢄⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠏⠉⠛⢷⣄
⠀⡇⢃⡹⠀⣇⢿⡇⠀⠀⢸⢻⠀⠀⠀⠀⡌⢠⣘⣾⡇⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠙⣷⣀
⠀⠹⡌⠣⣀⢈⣟⣀⡈⠁⢸⢸⡄⠀⠀⠀⠙⠬⠿⠛⠓⠒⠀⠀⠀⢀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠈⢿⢷⡄
⠀⠀⢻⢦⣌⡟⠁⡀⠉⠀⠘⡄⠳⣄⠀⠀⠀⠀⠀⠐⣲⣤⠀⠀⠀⠈⢷⠀⠘⣇⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⡄
⠀⠀⠀⠑⣼⠉⠈⠉⠃⠀⠀⠙⣆⠈⣳⣤⣠⣤⠶⠋⠁⠘⠀⠀⢀⣴⠛⣧⠀⠈⠳⠶⠒⢉⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⠹⡄
⠀⠀⠀⠀⣿⠒⠒⠲⠀⠀⠀⡴⠋⠉⠉⠉⠀⠀⣀⣤⣦⡤⠴⠚⠉⠀⠀⠙⠷⣦⣤⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⠳⢬⣶⡤⢤⢤
⠀⠀⠀⠀⣿⠤⠤⢄⠀⠀⣼⠧⠤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢸⡈
⠀⠀⠀⠀⢿⣀⣀⣀⠀⢰⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠸⡇⣀⡈⠀⢸⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⣿⠉⠉⠁⢸⠀⠘⣧⡀⠀⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⢸⡉⠉⠀⢸⡀⠀⠈⣷⣤⡈⠳⣄⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠈⡏⠉⠂⠘⡇⠀⠀⢹⡄⠉⠓⠛⠓⠶⠤⡤⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠶⠶⠒⣿⠉⢧⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣏⠈⠁⠀⡇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⢠⣏⣉⡁⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⡰⠓⠂⠄⠀⡼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⡗⠀⠀⠀⠀⠀⠀⠀⣟⣁⡐⢂⣀⡼⠃⡾⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠻⠖⠻⠦⠼⠃⠀⠀⠀⠀⡿⠄⡠⡀⣀⠀⠀⢻⠀⡟⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⠉⠉⠀⠀⣧⠛⢂⠄⠄⠀⠀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠦⢴⣤⠵⠖⠋⢠⣇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣃⠴⡄⠀⠀⠀⣀⡼⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡀⠉⠙⠁⠉⢀⡴⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠄⠀⠰⢀⠀⠁⢭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡀⢤⡤⢠⠀⢰⠂⠸⠂⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢳⡀⠙⠒⠂⣴⣿⠋⠀⢀⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⣧⠀⠀⠘⢯⠁⠀⠠⠈⠀⠀⢇⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠳⡴⣄⡞⠓⢃⠐⣐⡇⠀⣸⣷⠆
⠀⠀⠀⠀⢠⣀⠀⠐⠤⠤⣴⣶⣄⠀⠀⠘⠳⠆⠀⠸⠇⠀⠀⠀⠀⠉⠉⠙⠛⠛⣟⠈⠑⠀⠄⣀
⠀⠀⠀⠀⢈⠉⠻⠖⣒⣋⠡⠙⠠⠁⠠⠀⠀⠀⠰⠄⠒⠚⠓⠊⠀⠀⠂⠰⡄⠀⣀⠠⢴⡦⡄⠀⠌⠧⠄
⠀⠀⢀⣶⡷⠔⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠚⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠐⠲⢄⡀⠸⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⡠⠒⠉⠁⣨⣇⣀⣀
⠀⠀⠀⠀⡠⠒⠂⠁⠈⠓⠀⠀⢀⣉⣦⣻⣍⣉⣉⣉⡉⠀⠀⠀⠀⠤⠤⠴⠧⠤⠤⢎⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠻⢤⡤⠤⠤⣴⠿⠛⠂⠒⠒⠒⠒⠾⢷⡖⠒⠒⠲⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣇⡀
⠀⣠⠖⠋⠉⠁⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⠔⠈⠀⠁⠢⡀⠀⡠⠀⠀⠠⡀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲
⡜⠁⠀⠀⠀⠀⢠⠔⠒⠓⠢⣴⠃⠀⠀⠀⡎⠀⠀⠀⠀⠀⠰⡎⠀⠀⠀⠀⠘⡄⠀⠀⠀⢳⡎⠁⠀⠉⠢⡀
⢑⠄⠀⠀⠀⢠⠁⠀⠠⡀⢀⡇⠀⠀⠀⠀⡇⠀⠀⣚⣧⠀⠀⡇⣽⡇⠀⠀⠀⡇⠀⠀⠀⠈⠳⢤⡔⠀⠀⢡⠀⠀⠀⠀⠙
⣾⠀⠀⠀⠀⠘⡀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠱⡀⠀⠛⣋⡠⠴⠧⠽⢥⣀⡀⡰⠁⠀⠀⠀⠀⠀⠀⠙⢦⣀⠆
⠘⠦⣀⣠⠀⠀⢱⠞⠁⠀⠀⢀⡀⠀⠀⠤⢤⡵⠊⠉⠀⣠⠂⠀⠐⡄⠀⠉⠳⣤⠤⠄⠤⠤⢄⣀⡀⠀⠙⢦⠀⠀⡦⠐⠊
⠀⠀⠀⠸⡀⣰⠋⠀⡠⠔⠈⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠘⠠⠤⠤⠃⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⠓⠦⡈⢳⡔⠃
⠀⠀⠀⠀⣼⠁⡠⠊⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣳⡀
⠀⠀⠀⢰⡇⡜⠀⠀⠀⠰⠻⡅⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣸⢠⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠃⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⢻⡍⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠈⠉⠒⠒⠒⠒⠒⠋⠉⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⡟⠢⢤⣀⣀⡀⠀⠀⢀⣀⣠⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⢩⠉⣉⣹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⠒⠀⠉⢀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠏⣉⣁⠼⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠰⢟⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⣀⡴⠒⡞⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣄⡀⠘⡆⠡⠈⠲⣄⣠⣤⣄⣀⡼⠁⠀⠀⣇⠀⣀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣇⠀⠀⣤⢤⣿⠴⠂⢸⡶⣦⠀⠀⡊⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡆⠀⠉⢉⠿⠤⠤⢿⡀⠀⡴⠒⠒⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠜⠀⠀⡴⠋⠀⠀⠀⠀⠉⠙⠲⣄⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠤⠶⠏⠀⠀⠀⠀⠀⠽⠿⠿⠿⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠄⠛⠀⢺⣦⣾⡆⠚⠀⢸⣧⣼⡃⠘⠃⢰⣷⣴⡇⠐⠃⢠⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⢠⣿⣿⣿⠁⠀⠀⢸⣿⣿⡇⠀⠀⠘⣿⣿⣿⡄⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠸⢿⣶⡆⠀⠀⠀⠀⢀⣤⣾⡿⠃⠀⠀⠰⢿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠸⣿⣿⣿⣄⠀⠀⠘⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⡂
⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣠⣤⣾⣿⣿⣋⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⢿⡿⢻⡿⢻⣿⠻⣿⠻⣿⠿⣿⠿⣿⡟⣿⠟⣿⡟⣿⡟⣿⡿⢿⡿⢿⡿⢻⣿⢻⣿⠻⣿⠻⣿⢿⣿⠿⣿⠟⣿⠟⣿⡟⢿⡟⢿⡿⢿⣿
⠀⠀⠀⠀⠀⡆⠁⠀⠁⠀⠁⢰⡇⠀⠁⠀⠉⠀⢈⣧⠈⠀⠈⠀⠈⠀⣿⠀⠀⢁⡈⠁⠀⣿⠀⠁⠀⠁⠀⠁⣼⡉⠀⠉⠀⠈⠀⣸⡄⠈⠀⠈⠀⠈⢱
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⢸⣿⣄⣴⣺⣇⣦⢰⣿⡇⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⢣⣿⣿⡜⣷⣭⡓⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⢸⣿⣿⣷⠀⠀⠀⢸⣿⣿⣿⠀⠀⣴⣾⣿⡿⢋⣾⣿⣿⣷⡙⢿⣿⣷⣦⠀⠀⣿⣿⣿⡆⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⣼⣿⣇
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠙⠋⢀⣾⣿⣿⣿⣿⣷⡀⠙⠋⠀⠀⢰⣿⣿⣿⣧⠀⠀⢠⣿⣿⣿⣿⠀⠀⢀⣿⣿⣿
⠀⠀⠀⢰⣿⣿⠉⡇⠀⢰⠇⣿⣿⠃⣇⠀⢠⡇⣿⣿⡇⣧⠀⣴⠁⠐⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⣦⠀⡾⢸⣿⡿⢸⡀⠀⣸⠸⣿⣿⠸⡆⠀⢸⠙⣿⣿⡇
⠀⠀⠀⣾⣿⡟⠀⢿⠀⣼⠀⢹⣿⠀⢹⠀⣸⠀⢸⣿⠀⢸⣾⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣷⡀⠀⣿⡇⠈⣇⢀⡏⠀⣿⡇⠀⣧⠀⡟⠀⢿⣿⣿⡀
⠀⣀⣼⣿⣿⣇⣀⣸⣆⣇⣀⣘⣇⣀⣘⣇⣏⣀⣈⣏⣀⣸⣿⣿⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⣿⣿⣇⣀⣹⣀⣀⣻⣸⣃⣀⣸⣃⣀⣸⣰⣇⣀⣸⣿⣿⣧⡀
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⣤⣼⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⠟⣽⡏⢹⣿⠻⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⣀⣤⣶⡾⣟⣿⡿⠟⢋⣴⠟⠁⢰⣿⠁⠈⣿⡆⠈⠻⣦⡙⠻⢿⣿⣻⢿⣶⣤⣀
⢀⣀⣤⣶⠿⠛⣉⣴⣾⠟⠉⢀⣴⡿⠃⠀⢀⣿⡇⠀⠀⢸⣿⡀⠀⠙⢿⣦⡀⠉⠻⣷⣦⣉⠛⠿⣶⣤⣀
⣿⡟⠛⠛⠛⢿⣿⠛⠛⠛⠛⣿⡟⠛⠛⠛⢻⣿⠛⠛⠛⠛⣿⡟⠛⠛⠛⢻⣿⠛⠛⠛⠛⣿⡿⠛⠛⠛⢻
⢻⣧⡀⠀⢀⣼⣿⣄⠀⠀⣠⣿⣧⡀⠀⢀⣼⣿⣄⠀⠀⣠⣿⣧⡀⠀⢀⣼⣿⣄⠀⠀⣠⣿⣧⡀⠀⢀⣼
⠀⠙⢻⣿⠟⠋⠈⠛⢿⡿⠛⠁⠙⠻⣿⠟⠋⠈⠛⢿⡿⠛⠁⠙⠻⣿⠟⠋⠈⠛⢿⡿⠛⠁⠙⠻⣿⡟⠋
⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⣸⡿⢿⣇⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⢀⣿⠃⠘⣿⡀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⡇⠀⠀⣼⡏⠀⠀⢹⣧⠀⠀⢸⣧⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⠃⠀⣼⡟⠀⠀⠀⠀⢻⣧⠀⠘⣿⡀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⠀⣼⡟⢀⣾⠟⠀⠀⠀⠀⠀⠀⠻⣷⡀⢻⣧⠀⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⢸⡇⠀⣰⣿⣵⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣮⣿⣆⠀⢸⡇⠀⠀⠀⠀⣿⡇
⠠⣶⣾⣿⣶⣶⣶⣶⣾⣷⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣾⣷⣶⣶⣶⣶⣿⣷⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠿⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠉⠁⠂⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⠟⠋⠩⠐⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⢛⠩⠁⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣶⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡶⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡆⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠘⠻⠟⠋
⠀⠀⠀⠀⢀⣾⣿⠟⠀⢀⣸⣦⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟
⠀⠀⠀⠀⢸⣿⡏⠀⢠⣿⠟⡏⠉⠙⢿⣷⡀
⠀⠀⠀⠀⠸⣿⡇⠀⢸⡇⠀⣷⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠹⣿⣄⠀⠙⠀⢸⡀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⣟⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣼⡷⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⢰⣾⣿⡿⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⢀⣶⣿⣿⠇⠀⠀⠀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠙⠋⠁

⠀⠀⠀⠀⠀⢀⣠⠤⢤⣤⣀
⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢰⣷⡆
⠀⠀⠀⢸⣇⣴⣶⣦⡀⠀⠀⠈⣿⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠻⣿⣿⡿⠃⠀⠀⠀⢿⣿⡇⠀⢠⣤⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⣇⡀⠀⠀⠀⠀⠀⠀⣟⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠛⠁⠀⢸⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠛⢦⣘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠀⠀⢸⠀⣀⣀⠀⠀⠀⠘⢻⡏⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⢸⠋⢙⣿⠀⠀⠀⢠⣼⣷⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡼⠋⠀⠀⠀⠘⢻⡋⠉⡇⠀⠀⠀⢀⣴⣾⣿⠀⠀⠀⢀⣴⣾⣿⡇⠀⠀⠀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠘⠃⠀⠁⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⡄⠀⠠⠀⠀⡐⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⠄⠀⠀⢀⠀⠈⠄⠀⠐⠀⠀⠀⠀⢸⠀⢀⠀⢀⠂⠀⠠⠀⠀⠐⠀⠀⡀⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠠⡀⠀⠂⠀⠈⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠄⠀⡠⠊⠀⠀⠀⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠑⠀⠀⠀⢀⠀⠠⡀⠀⠀⠀⣀⢠⠀⡤⢠⠄⠀⠠⢄⠀⠀⠀⢀⠄⠀⢀⠄⠀⠀⢀⠄⠈⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠐⠠⢀⠀⠀⠁⠠⠀⠀⠈⠢⡀⠑⠀⠀⠀⠘⠃⠃⠈⣀⣁⣀⣀⣀⣀⢀⠑⠄⠐⠁⠀⠔⠁⠀⠀⠀⠁⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠑⠠⡀⠈⠀⠀⠀⠰⠖⢛⡉⢀⣭⣭⣥⣬⣭⣽⡀⠳⠖⠀⠀⠀⢀⠄⠈⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠄⡀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠸⢛⣉⡁⢠⣬⣤⣤⣤⣤⣯⠀⡛⢏⠀⠠⠂⠀⠀⠀⠁⠀⠠⠀⠂⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠂⠀⠀⠈⠀⠠⠀⠀⣛⣋⣩⠀⢠⣤⣤⠤⣤⣤⣭⠐⡙⢃⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣩⣤⠀⢴⡶⠶⠶⢶⣦⣬⢀⣙⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣥⡤⠀⢴⣶⠶⠶⢶⣶⣦⢈⣙⡃⠀⠀⠀⠀⠀⠀⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠍⠀⠀⣥⣤⠦⠀⠶⠶⠶⠶⠶⢶⡄⢨⣍⠄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⢀⣤⠦⠶⠀⠶⠶⠶⠶⠶⠶⠆⢠⡭⠀⠀⠈⡀⠀⠀⠀⠄⢀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠈⢄⣆⢷⢸⡄⣧⢸⡆⢀⠀⠀⣀⠄⠀⠀⠂⠄⡁⠀⠀⠄⡀⠀⠁⠐⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⡠⠐⠈⡠⠂⡀⠀⠙⢫⠃⠛⠘⠣⢟⡄⢐⠊⠀⠀⠀⠀⡁⠀⠀⠑⠠⢀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠠⠀⠀⢀⠔⠁⡀⠈⠀⠀⠀⠘⠀⢲⣖⠀⢸⣥⡌⢀⠀⠀⠈⢂⠈⠀⡀⠐⠄⠀⠈⠀⠄⡀⠀⠀⠁⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡠⠂⠁⠀⠀⠀⠔⠀⡔⠀⠀⠑⡀⢀⡣⠐⠋⠀⠀⠂⠀⠢⠀⠑⡀⠈⠂⠀⠀⠂⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠔⠊⠀⢀⠔⠀⢀⠈⠀⡐⠀⠠⠀⠐⢉⡉⡗⠈⡁⠀⠀⠀⠄⠀⠡⠀⠈⢀⠀⠀⠂⠀⠀⠑⠠⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⢀⠔⠁⠀⡀⠁⠀⡜⠀⠀⠀⠀⠘⣿⠇⠃⠀⢁⠀⠀⠂⠈⠄⠀⠐⡀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠃⠛⢀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡇⢀⣸⠀⠀⠀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣹⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢻⣿⣿⣿⡉⢻⣿⠙⠦⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣋⠁⠀⠘⣿⡿⣿⣿⢿⡟⠀⠀⠴⠓⠽⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠂⠠⡄⠀⠀⠀⠋⠤⠖⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠡⡑⠀⢸⡀⠐⡄⢀⠀⠄⡆⠰⡰⢀⠁⡠⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠢⡈⠀⢈⠂⡐⢌⢜⢄⢳⣀⣿⣼⣴⣥⣇⣅⣁⠆⡔⢡⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡑⢜⢄⣩⡾⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣅⣞⢔⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡈⢱⡟⠁⢀⣖⡽⠿⣿⣿⣿⣿⡿⣿⣿⡉⠙⢻⣅⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠐⣰⠏⢀⣞⣿⣷⣷⢞⣙⣭⣭⣟⠻⣿⣿⣿⣦⠀⠹⣆⠡⠤⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠬⠄⠀⠀⢠⡟⢀⣿⣿⣯⡋⢽⣧⣿⣿⣿⣿⣿⡎⣿⣾⣿⣧⠀⣿⣆⠉⠡⠄
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠤⢼⣧⣼⣿⣿⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡆⣿⣿⠤⠄⠀⠉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢼⣿⣿⣿⣿⣿⣯⠻⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣿⡿⠀⠀⠠⠄
⠀⠀⠀⢀⠀⠄⠒⠊⠉⡀⠀⡺⣿⣿⣿⣿⣿⣿⣿⣬⣻⣛⣛⣿⣿⣿⣿⠿⠛⢿⣿⣿⠇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠔⠀⠐⠙⣿⣿⣿⣿⣿⣋⡛⠻⠟⠿⠟⠛⠋⠀⢀⣠⣾⣿⠯⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣽⡟⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠿⠋⢡
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣒⠠⠭⢒⣀⠀⣀⣀⣀⣐⡀⠠⠄⠒⡺⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠐⠒⠒⠒⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠴⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠉⢁⡠⠂⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⣨⠏⠀⠈⠁⢀⡄⠊⣀⠴⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠄⠐⠀⡈⠀⠀⠐⡋⠁⠀⠀⠀⠂⣁⡤⠚⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠤⠠⣤⠖⠉⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡀⠄⠀⠀⠀⠀⡀⠄⢀⣠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣶⣶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢯⡻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⢛⣋⣭⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⡛⣛⣉⣭⡴⠾⣛⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣉⣭⣽⠿⠟⢛⣻⣿⠟⣫⡿⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠙⠳⠶⣾⣟⣋⣭⣶⣟⣭⡾⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠀⠠⠀⠠⠀⠀⠀⡠⠔⠒⡒⠢⣄⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡔⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠘⠀⠀⠄⠀⠐⢤⡀⠀⠀⠀⣠⠖⡀⠁⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⢐⣄⢦⡀⠀⠀⠤⠀⠀⣨⠂⣯⢺⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢀⡀⡄⡔⡣⠟⠰⠊⢂⣤⣭⣤⣈⠁⢛⡒⠃⠀⠐⢄⢸⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠈⠉⠁⠀⠀⠀⢠⣿⣿⣿⣇⣹⣦⠀⢀⡀⠀⠀⠀⢸⠏⠉⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠹⡇⠀⠀⠀⢀⣀⡤⠊⠘⣿⣿⣿⣿⣿⡿⠀⠀⠈⣉⠲⠀⢸⢰⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠑⢷⡀⢀⢞⣭⣄⣀⡀⠀⠈⠛⠿⠟⠋⢀⣠⣴⣾⠋⠌⢠⠗⢫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⣄⢆⠈⢻⣿⣿⣿⣿⡗⠚⠛⢻⣿⣿⡿⠁⠀⢠⠯⠴⢏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠲⠜⢢⡀⠀⠑⠝⠿⠛⠷⡶⢶⣿⣿⢏⠃⠀⣠⠋⠀⠀⠀⠀⠉⠐⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠄⣦⠝⢦⡀⠈⢈⠂⡀⠈⠀⣈⢟⠆⠀⡴⠁⠀⢀⡦⠄⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡀⣀⠠⠄⠀⠠⠀⡈⢢⡀⠑⠤⠥⠤⠴⠋⠀⡼⠁⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠑⢄
⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⢀⠜⡡⠊⡠⠢⠙⠦⣤⣤⣤⣤⠤⠞⠁⢤⣤⠤⠒⠃⠉⠈⠉⠉⠀⠀⠀⠈⠉⠛⠢⠧
⠀⠀⠀⠀⡴⢋⡔⠀⠀⠀⠀⠀⠀⢁⠜⠀⠀⠁⠀⠀⠀⠀⢀⢂⣐⡖⠂⠁
⠀⠀⠀⠘⢧⡎⡴⢀⡀⡀⠀⠀⠀⠀⠀⠀⡀⢐⠠⢠⠰⢰⠬⠆⠈
⠀⠀⠀⠀⠘⠤⠴⡏⠀⣼⠃⠇⣰⠀⡀⡗⠀⡾⠬⠜⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⠤⠖⠙⠶⠞⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠠⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⡿⡧⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⣿⠋⢻⠖⡿⢿⠳⡷⠻⠷⡿⠻⣛⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠠⠁⡔⠁⡄⠀⡇⠀⢣⠈⡶⡛⠀⢁⢈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠘⡀⠐⡌⠺⡀⠀⡇⠀⡠⠋⡔⠀⠇⢰⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣔⣠⡎⢀⢀⣀⣀⣡⢀⡈⣆⢈⢢⣧⣊⣀⣌⣀⣸⣀⣀⣄⣀⣡⣀⣑⣄
⠀⠀⠀⠀⠀⠀⠀⠎⠁⢸⠈⠁⢸⠆⠹⠬⠤⣬⣤⣥⡥⢣⢤⢤⣤⡥⠨⠞⠅⢻⠈⠀⢻⠈⠈⢣
⠀⠀⠀⠀⠀⠀⡜⠀⢀⠇⠀⠀⡾⠲⠬⡤⢥⣿⣼⣷⣣⣠⣼⣼⣬⢧⡥⠬⠐⠒⢇⠀⠀⢣⠀⠀⢣
⠀⠀⠀⠀⠀⡰⠀⠀⡼⠀⠀⢰⠃⠀⢠⠁⡸⠀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠈⢆⠀⠀⢣⠀⠀⢣
⠀⠀⠀⠀⡰⠁⠀⢠⠃⠀⠀⡎⠀⠀⡜⢀⠃⢰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠀⠀⠘⡄⠀⠀⢣⠀⠀⠱⡀
⠀⠀⠀⣴⣁⠀⢀⡏⠀⠀⡸⠀⠀⢠⠃⡞⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢃⠀⢃⠀⠀⠹⡄⠀⠀⢣⠀⣀⡸⠄
⠀⠀⠀⠀⠈⠉⠙⠓⠒⠴⠧⠤⠤⢮⡼⠀⢀⠇⠁⠀⠀⠀⠀⠀⠨⡀⠀⠘⡤⠤⠦⠤⠴⠚⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠈⣟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢀⣄
⠀⠀⠀⠀⣽⠞⠀⠀⠀⣆⡄⠈⠀⠀⠀⣀⡇⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠛⠃⠀⠀⠀⢦
⠀⠀⠀⡞⣹⠾⢷⣆⠀⠴⠃⠀⠀⠀⠘⠛⠁⠀⠀⠺⠿⠃⠀⠾⠿⠃⠀⠾⠟⠀⠀⠀⠀⢠⣤⠄⠀⠰⠿⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⣫
⠀⠀⠈⢇⠙⠏⢨⡟⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡖⣱⡜⣱⣼⣰⠀⠀⠣
⠀⠀⠀⠀⡅⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣀⠀⠘⠃⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠱⠏⠡⠏⠁
⠀⠀⠀⠀⠑⠋⠀⠀⠀⣠⣤⢀⠀⠀⠀⠀⢰⠒⠀⠀⠀⢻⣧⠄⠸⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⣀⠀⣀⡄⠀⢀⡀⢠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⣁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠾⠏⠀⠛⠁⠀⠟⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⠟⠁⠀⠈⠉⢀⠀⠀⠀⢸⣽⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠾⠏⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⣤⡄⣀⣤⠀⠈⠆⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤
⠀⠀⠀⠀⣶⠗⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢠⣤⣠⢀⡤⡄⠀⠀⣝⠳⡀⠀⠀⠀⠘⠋⠀⠛⠋⠀⠀⠐⡀⠀⠀⠀⠀⠀⠿⠿⠿⢿⠀⠀⢠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⠋⠛⠛⣸⠒⠃⠀⠀⠈⠻⠄⠀⠀⠀⣠⠤⣤⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⢸
⠀⢀⢯⠇⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠀⠀⠀⠀⠀⢷⡆⠈⣿⠠⠄⠀⠀⠀⠀⠀⠀⣴⣶⠇⢠⣶⡾⢰⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠰⠿⠋⠀⠀⢀⣤⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⣤⠀⣤⠠⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣤⠀⣀⣄⠈⣭⡆⠸⠟⠀⠈⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢈⠀⠰⢏⡴⠀⠀⠐⠛⡇⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠛⠋⠀⠛⠁⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠾⠟⠀⠀⠀⠀⢿⠟⠀⠀⠀⠛⠀⠀⠀⠈⠋⡇⠀⠀⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⡧⠔⡅⠀⠀⠀⣿⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⡀⠘⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⠀⠀⠖⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠌⠉⠚⠿⢗⣶⠀⠀⢰⡏⠸⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠰⠶⠶⠶⡦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⡶⠶⠀⠀⠀⠀⣴⡶⠞⢀⣀⣀⠀⠀⠈⠀⠀⠘⠧⠄⠂
⠀⠀⠀⠀⢀⠀⠀⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⢠⣶⠆
⠀⠀⠀⢀⣯⠀⠀⣤⠞⡀⠀⠀⣠⣤⠀⢀⣠⠀⠀⣀⠀⠀⠀⠀⢰⡟⡿⠇⠀⠀⣷⡿⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⣀⡀⠀⠀⠀⢀
⠀⠀⠀⠚⠉⠀⠀⡴⢛⡅⠀⠀⢉⠁⠀⠛⠉⠀⠺⡛⠀⠀⠀⠀⠈⠻⡦⠀⠀⠀⡇⠀⠀⠀⣴⡶⠀⢘⡇⠀⠸⠿⠛⠃⠀⠀⠐⠚⠗⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠉⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠶⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠉⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣋⣀⡀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⠉⠉⠙⠓⢦⣄⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦⣤⡀⢸⡧⣤⠄⣀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⠛⢿⣱⣟⡀⠀⠙⢦⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣄⡀⠀⠀⣠⠋⠀⠈⠑⠦⣾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠈⠙⢲⡔⠁⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢦⣀⡀⠀⢏⡀⠀⣀⠴⣳⡄⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠉⠓⠦⣍⡉⠀⣠⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠦⣄⡀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠉⠓⠦⢄⣀⣀⡜⣇⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠢⣄⡀⠀⠀⠀⢙⠁⠁⠘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢮⠏⠙⠒⡶⠋⠀⠀⠀⠀⠀⠠⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⠏⠀⠀⣼⠁⠀⠀⠀⠐⠀⡀⠠⣖⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⢤⣀⣼⠟⠻⢶⡄⠀⠒⠚⢳⠀⠹⡆⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢢⡴⠟⠋⠀⠀⠀⢿⡄⠀⠀⠀⢣⠀⠙⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⢿⣄⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⢿⡄

⠃⢄⢉⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡉⣠
⠃⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠇
⠀⠀⠀⠈⢻⣦⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣠⣴⡿⠁
⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣰⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⢀⠀⠀⠀⣴⡃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠙⠁⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⠀⠀⢙⣶⡆⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⢀⣀⣤⡀⠁⠐⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠁⠀⠀⠈⠉⠛⢁⡀⠀⠙⠒⠄⢈⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⣬⠿⠃⠒⠀⠃⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠈⠓⡶⢿⡏⠀⠀⠀⠀⠀⠀⢀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠁⠀⢀⣀⢀⡀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣦⣤⣤⣴⣶⣿⡟⢡⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠀⣟⠉⠀⡸⠃⠀⣠
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠈⠁⠀⣴⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠠⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣺⠔⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⣠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠀⠀⠀⢠⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⡀
⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠳⠄⢀⢀⠸⢘⠌⠀⠁⠇⠀⠀⠄⠸⠈⠁⠇⠀⠀⠀⠀⠹⣿⣿⣦
⠄⠀⠀⢀⣼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠙⠻⣧⡀
⡀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡆
⡄⠊⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⣠⢀⠀⣀⡀⣠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣀⠸⠛⠀⡀⣀⣉⠋⠀⠀⠀⠀⣝⠂⠐⠄
⠀⠀⠀⠀⠀⠀⠠⠊⢀⠀⠀⠀⠀⡸⡀⠈⠀⠀⢀⡀⠀⠀⠀⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠖⠀⠦⡄⢀⢔⠁⢄⠘⠃⠀⠊⡀⠀⠑⠄⠀⠀⢈⡄⠑⡀
⠀⠀⠀⠀⠀⠨⠔⡀⠀⠀⢦⠀⠠⠀⠀⠀⠀⠀⠠⡰⠀⢨⠄⠀⠀⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠉⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢠⡀⠀⡀⠀⠀⠀⢀⠀⠀⡠⠐⠂⠀⠀⠸⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢤⣤⠄⠠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠈⠾⢤⣤⣤⣤⡴⠢⡐⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠟⠇⠀⠠⠄⠸⠀⠀⠈⣏⠁⠀⠀⡁⠀⠈⢈⠃⠐⠂⠀⠀⠦⡀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠄⠘⢠⠄⠀⠀⠑⢶⠊⠀⢲⠔⠊⠀⠀⠔⠁⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠀⠌⠈
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠰⡄⠀⠈⠂⠠⠤⠠⠐⠀⠀⠀⢀⢠⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠂⠀⠀⠀⠊⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡄⣶⠘⠶⠀⠀⠀⠀⠀⡀⠀⢀⠞⠈⠁⠀⡐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠛⣀⠒⠁⠁⣴⠔⠁⠀⠀⡰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⣦⠈⠓⠀⠁⡀⠒⠊⡰⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⢄⡀⠉⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡠⠂⠐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠑⠄⡀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠐⠞⠈⠀⠢⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠒⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠑⠢⡀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠑⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠳⠀⠀⢄⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠄⠀⠀⠄⠀⠀⠔⡨⢀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⡀⠀⠀⢀⠒⠁⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠰⠠⠄⠊⠀⠀⠀⠐⠂⠤⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣅⢸⠀⠀⠄⠺⡀⢠⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠈