Dozen Emojis & Symbols

⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠈⠀⠀⠀⠀⠷⠀⠈⠅⠨⠂⠬⠀⠛⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠁
⡠⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⡀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠐
⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣒⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣴⣶⠶⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⠿⣿⣿⠿⠋⠉⠀⠀⠉⠲⠂⠀⠀⣤⣄
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢳⡖⠀⢀⣤⡀⠀⠀⠀⠀⠀⣸⣿⣽⡄⠀⠀⠤
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠈⢻⣭⣭⠀⠀⣀⡀⠀⠀⠀⠸⣇⠀⣼⣯⣿⠀⠀⠀⠀⠒⠛⠻⣿⣧
⠀⠀⠀⠀⠈⠛⠿⢿⡿⠟⠉⠀⠀⠈⣿⣏⣈⡁⣀⡀⠀⠀⠀⠀⢹⡇⠀⣸⣿⣿⠀⠀⠀⠀⠟⠛⠛⢿⣿⣧⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⠙⢛⡄⠀⠀⠀⠀⣿⠞⡟⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⣧⠀⠀⠀⠀⠁⠀⠀⠘⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠻⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⣶⡿⠛⠛⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣤⣤⡶⠿⠿⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣤⣄⡀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣤⣤⣤⣤⡾⠿⠛⠛⠛⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⣠⣿⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣧⢋⡡⣯⠟⢾⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⡤⡤⢴⠏⠁
⠀⠀⠀⠀⠘⠻⠿⣿⣷⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⡿⣿⣿⣿⣟⣸⡶⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡟⡓⣾⢒⠚⢒⡲⠒⡞
⠀⠀⠀⠀⠀⠳⠆⡄⠀⡴⠦⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡅⣿⣹⣍⣽⣭⣹⠁
⠀⠀⠀⠀⢰⣶⣦⣤⣠⣤⣴⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠶⠾⣶⣷⣷⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⢀⡀⠀⠑⣆
⠀⠀⠀⠀⢸⣿⣛⣿⣿⣟⣟⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡿⡆⠀⠀⠀⠀⠉⢹⡦
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣷⡄⠀⠀⠀⠀⠀⢐⣀⣀⣀⣀⣀⠐⡀⠄⠀⠌⡐⢆⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠘⣷⡄
⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠑⠂⠂⠒⠁⠂⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⠀⢹⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠯⡁⠀⢀⠀⠀⠀⣿⠿⠿⠿⣤⢦⣼⣭⣥⣥⣤⣴⡼⠿⠿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢀⣿⠇
⠘⣿⣿⣿⡿⠿⡟⠉⠟⢤⠉⢐⣤⣴⣿⡟⠀⠀⠀⡇⠀⠂⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠁
⠀⠹⡋⡙⠃⢤⠀⣨⣤⣴⣿⣯⠿⠛⠉⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠄⠀⠤⠀⢸⡇⠒⠂⠀⣿
⠀⠀⢿⣧⣴⣾⣿⣿⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣻⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⣴⣾⣶⣶⣶⣶⣶⡀
⠀⠀⠈⠙⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠸⠟⠿⠿⠿⠿⠿⠟⠃



⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⣟⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⣤⣖⣄⠀⠀⠀⠀⠀⠈⢙⣿
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⣀⣸⡿⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⡰⣦⣤⣤⣖⣾⣸
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣽⣿⣿⣿⣷⣼⣿⣿⣷⠀⠀⠀⠀⠰⣾⣿⣿⣻⣿⣿⡌⢡⠀⠀⢹⣿⣿⡇
⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠛⠛⡿⣿⢩⣉⢻⡥⠘⠛⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠳⠺⠿⠎⠉⠛⠓⠾⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣶⣦⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠫⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠾⠉⠶⡄⠠⠆⠠⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠑⣠⠖⠐⣦⡇⣠⡄⠀⣴⠁⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⡁⠀⡄⠀⠀⠀⢄⠚⠟⠀⠀⠇⠀⠀⠀⣼⠁⠀⠀⠇⢀⢀⡀⣤⠤⠆⠀⠀⣀⠀⠤⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⢂⠀⠀⠸⠀⠀⠀⠀⢧⠀⢄⡀⠀⠀⠀⠀⠙⠒⠀⠐⠇⠈⠀⣈⢡⡀⣠⡞⠧⠉⠙⢂⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⣄⠀⠀⠀⠘⠢⠞⠻⠀⠉⣶⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠋⠉⠁⠠⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠂⠈⠴⠻⢇⠀⢰⣴⣤⡀⠀⠀⣞⡰⢢⠌
⠀⠀⡀⠠⢤⡶⠆⢘⠖⡇⠐⠶⣶⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⠤⠤⠂
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⣀⡀⠐⠊⠀⠀⠀⠀⠐⣄⠀⢀
⠀⠀⠀⠀⢀⠀⠰⠂⣌⡙⣟⢻⣿⣻⡆⡐⢠⠀⢀⠌⡱⠄⠀⠑⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠘⠀⠂⠀⠈⠉⠁⠀⠀⡈⠁⠀⠅⠘⠀⠈⠂⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⢀⠈⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢋⠂⠀⠆⣀⠿⠂⣹⣿⣷⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢀⠀⠀⠀⠉⠉⠩⠉⠀⠀⠉⠤⠀⠀⠀⠀⠀⠀⢖⢈⢱⢗⡼⢁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢷⣶⡃⢼⠡⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣀⢠⠃⠀⠀⠀⣠⠤⢄⡀⠀⢱⡀⢠⠖⠉⠀⠀⠈⠙⢦⡀⠀⠀⠀⢀⡠⠴⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠈⠙⣄⠀⠀⢸⠓⠒⠒⢻⠀⠀⡷⠁⠀⠀⣠⠤⠤⢤⠀⠱⡄⠀⡴⠉⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡶⠒⠉⠉⠉⠓⢴⠁⠀⠀⢀⡏⠓⠚⠙⡇⠀⢹⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠏⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⡠⠖⠋⠉⠙⠢⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣀⡠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣠⠎⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣍⠓⠒⢢
⠀⠀⢰⡉⠀⠶⢇⣀⣀⠀⣤⣔⡒⢚⢶⠒⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⣰⢏⣒⣒⣦⡤⠴⠒⢉⣁⠤⢲⠋
⠀⠀⠀⠉⠹⡓⠒⠢⠬⠭⣁⣀⠈⠙⠾⠯⢇⣀⣀⠀⠀⠀⣤⡤⠒⢲⡖⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡯⠞⠁⢀⣀⠤⠖⠊⠁⠀⠀⡞
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢹⠉⠑⢶⠢⠤⢬⣉⣉⠛⠁⠉⠉⠻⢽⣇⣀⠀⠀⠀⠀⢀⣀⡤⠔⠚⢉⣁⠤⠖⢺⠉⠈⡆⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠀⠀⡾⡀⠀⠀⠀⠀⠉⠙⡖⠒⢦⠤⢄⣀⣉⣉⣉⣉⣁⠤⠤⡶⠊⠉⠀⠀⠀⡾⡀⠀⢱⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠈⢇⡀⠀⠀⠀⣾⠀⢰⢁⣧⠀⠀⠀⠀⠀⠀⠇⠀⢸⡄⠀⠀⠀⠀⠈⡇⠀⠀⠀⡇⠀⠀⠀⠀⢠⠇⢣⠀⠀⡇⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠒⠚⠉⠸⡀⠀⠀⠀⠀⢸⠀⠀⡾⢱⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠒⠼⠦⠤⠼⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠦⠤⢼⣀⠠⠗⠚⢇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⢄⣀⣸⠀⠀⣀⡧⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⢛⡛⠿⠿⣿⣿⠋⠉⣴⠿⢿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⢡⣦⠀⠈⠏⠀⠰⣠⣶⡐⣿⣿⣿⣿⡟⢿⣿⣿⢿⣿⣿⢟⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⠏⠀⠁⠀⠀⠀⠀⣰⣤⢬⣿⣿⣿⣿⣿⣮⡻⣇⢨⡿⣫⣾⣿⣿⣿⣿⠿⢻⠇⠰⠆⠀⠀⣤⠀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⠚⢻⠶⣦⣈⢡⡾⣋⡿⠀⢹⣿⣉⣉⣉⣙⣛⠀⠀⢘⣛⣋⣉⣉⣹⡇⠀⣘⠂⠀⠈⠀⠀⠀⢰⠙⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⠂⣸⢒⡧⢿⣿⣗⣭⠾⣷⣌⣿⣿⠿⣿⣿⢫⡔⡳⣜⢿⣿⣿⣿⣿⣧⣤⣼⡀⠀⠀⠄⠀⢈⡅⠀⢨⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣼⢿⣏⠻⣿⣤⣾⡟⡸⣷⢸⡿⢃⣾⢟⣵⣿⣿⣳⣿⣷⣝⢿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣤⣿⣷⣤⣿⣿⣿
⣿⣿⣿⡟⠋⣈⢩⡻⣿⣼⣿⣷⣝⠷⠾⣴⣿⣿⡞⢡⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⠿⠟⠋⠁⠉⢻⣿⣿⣿
⣿⣿⣿⢰⣦⣿⣿⣷⣹⣿⣿⣿⣿⣷⣾⡟⠛⡉⠀⠪⠆⢿⣿⣿⡜⣿⠷⠻⢿⡿⣿⠿⠳⣿⣟⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡇⠀⠀⢻⣞⣿⣿⣿⠞⠧⠀⠈⠁⠉⠀⠰⠟⣾⣿⣯⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣀⣸⣇⣀⣀⣼⣿⣿⣿⣿⣦⠀⠆⠀⠈⠀⠀⠆⢠⣿⣿⣿⠟⣷⣾⣷⣦⣴⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⡟⠿⢻⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣾⣄⣾⠆⣠⣿⣿⡿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢻⣿⣿⣿⠋⣽⢸⡌⢿⣿⣿⣯⠿⣳⣿⣿⣿⣵⣤⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⠀⠀⠀⣿⣿⣷⠀⠔⣈⣅⡐⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⠈⣿⣿⣿⢠⡾⠟⠛⠿⣄⣿⣿⣿⣿⣿⠁⠀⠈⣿⣿⡿⠋⢿⣿⡿⢿⣿⡏⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠠⣿⣿⣿⠌⠀⠀⠄⠀⠉⣿⣿⣿⣿⢿⣵⣶⣾⣿⣟⠁⢴⡞⣿⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⠸⠧⠄⠀⡴⠿⣿⣿⠀⢠⠸⣿⣿⣿⣿⣿⣄⣀⣤⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⣽⣿⣀⣥⣾⣿⣿⣿⣿⣿⣿⣭⣭⣤⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠀⠐⠒⠒⠈⠉⠉⠉⠉⠉⠈⠉⢉⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡷⠆⠣⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⢳
⠀⠀⠀⢀⣤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⠇
⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠲⢼⡀
⠀⠀⢸⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⢷
⠀⠀⠈⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠘⡇
⠀⠀⠀⢸⣇⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣄⣠⡇⠀⢠⡄⠴⢄⡀
⠀⠀⠀⠀⠱⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠚⠳⡴⡏⠀⠀⠀⠙⣆⢀⣀
⠀⠀⠀⠀⠈⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⠾⠺⢷⣤⡾⡏⠀⠀⠀⠀⠃⠸⣆⣀⡀⢰⠟⠁⠈⠑
⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣤⡴⠆⠤⣰⣏⣴⡿⠁⠀⠀⢠⡗⠘⠃⠀⠀⠀⢀⡖⠉⠉⠉⠳⡏⠀⢀⠠⠄⠸⠦⣄⣀
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⡄⡀⠀⢀⡹⠉⠀⠀⠀⠘⣯⠑⠇⠀⠀⠀⣼⡥⠖⠛⠉⠳⣷⡞⠀⠀⢀⡠⠤⠗⠊⠉⠳⠶⢿⡎⠉
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠠⣴⠞⠋⠉⠉⠙⢿⡭⠉⠇⣀⠀⠀⠀⣤⡶⠴⠒⠦⣰⡄⡿⠁⠀⠀⠀⢀⡈⠷⠒⠋⠁⠀⠀⢀⡤⠔⠂⠉⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣲⠶⠚⢛⠏⠀⠀⠀⠀⢀⣘⡤⠒⠀⠺⣄⠀⠪⠝⠀⠀⠀⠀⠈⣳⣇⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⠀⠀⠀⠀⠀⢾⠋⠀⠀⠀⠀⠈⢅⣼⣀⡤⠤⠒⠂⠩⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⣸⡧⠤⠄⠚⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣠⣉⠑⠊⣸⠀⠀⠀⠀⠀⠖⢘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠧⣤⣠⡤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠉⠙⠒⠿⢄⠀⠀⠤⠀⠀⡊⠠⠤⠂⠀⠀⠀⠀⠀⣿⣅⠀⠀⠀⣀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣤⣸⣆⣀⠀⠐⠒⠋⠉⠀⢰⡆⠀⠀⠀⠀⠀⡀⠁⠺⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣠⡤⠬⠿⢿⣄⠀⠀⠀⠀⠀⢀⡾⣷⠦⣄⣀⠀⢶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣀⡀⢠⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⠙⠁⠐⠀⠀⠀⣀⠀⢀⠀⠀⠀⢀⡤⠆⢶⣄⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⠁⠓⣤⠃⠀⠀⠀⣿⣷⠞⠀⠀⠈⢻⣄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡦⠀⠀⣀⣠⣄⡀⠀⢀⡟⠀⠀⠈⢿⣿⣿⠗⠀⠘⢶⣾⣽⣷⣾⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⣠⣤⣄⢿⣿⣿⣿⠂⣼⡀⠠⢤⣄⣼⣿⡇⠀⠀⡀⠈⣿⣿⡿⠋⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡇⢛⣽⣿⣿⣤⣍⣉⡛⢸⣿⣿⡟⢰⠏⠀⠀⠀⠹⣿⣿⠟⠀⠀⠉⢿⣷⣿⣿⠿⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⡿⠀⣾⠀⠀⠐⢶⣴⣿⣿⡄⠀⠀⢄⣸⣿⣿⢋⣠⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠃⡸⠁⠀⠀⠀⠸⣧⢹⠏⠀⠀⠀⠈⣿⣿⣿⡿⠿⣡⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⢿⡿⣿⣿⣷⣶⣄⣀⣀⣠⣄⣀⠀⠀⢠⡷⠀⠀⠐⢶⡰⣿⣾⣦⠀⠀⠲⣤⣟⣸⣟⣥⣤⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⡞⠀⠀⠀⠀⠘⣧⢹⡟⠁⠀⠀⠀⠸⣿⣿⣿⣿⠻⠛⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⡇⠀⠀⠤⣄⣸⣿⣼⣇⡀⠀⣀⡀⢸⡟⣿⡿⢃⣠⡖⢸⡆
⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⣼⣿⣿⣿⣿⣿⡏⢠⡿⠁⠀⠀⠀⢸⡟⢻⡿⠁⠀⠀⠈⢻⡿⣿⣿⣿⣿⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠸⣄⠙⠻⠿⣿⣿⣿⢁⣶⣶⣶⣹⣿⣿⣿⣿⣿⠀⣼⠁⠀⠀⠀⢀⣿⡇⢘⠁⠀⠀⠀⠀⢸⣿⢸⣿⡿⠃⣈⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠱⡖⠒⠂⠠⠭⠍⠙⣛⠛⠷⠿⢿⣿⡿⠃⣼⠋⠄⡀⠀⠀⢸⣿⠗⠈⢳⣄⠀⠀⠀⣿⡟⣼⢿⠟⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠠⠤⠄⣀⡀⠀⠀⠉⠀⣶⣤⣶⠟⠀⠑⢶⠐⠒⠂⣭⣤⣤⣤⣈⣉⣀⣂⣀⠛⣋⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠛⠛⠁⠀⠀⠀⠀⠑⠶⠞⠟⠛⠋⠀⠈⠻⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⢀⡠⠔⢒⡒⠒⠒⠒⢒⣒⣒⣒⣶⠒⠒⠒⠀⠒⠀⠠⣤⣭⣭⣭⣭⣥⣤⣦⣬⣭⣭⣭⣭⣤⣤⣤⣤⣤⣶⣆⣶⣦⣄
⠀⠀⠀⠀⠀⠀⡞⢰⣿⣿⣿⠒⠀⠀⠈⠉⠉⠉⠁⠈⠉⠉⠀⠐⠒⠢⡀⠀⣴⡿⠂⠀⠀⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠑⢄⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⠀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢸⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⡆⣿⣿⣿⣿⣶⡌⠻⠛⢻⣵⣶⠀⠀⠀⠀⠀⠀⠀⠞⠀⡀⢻⣿⣄⠀⠀⠀⠀⠀⠀⢸⣷⣍⢻⠛⣿⣿⡇⢸⠀⠀⣼
⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⣿⡗⣀⣀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⣄⠀⠁⢸⣿⡿⠁⠀⠀⠀⠀⠀⢸⡿⢇⣸⣀⣿⣿⡇⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣷⢹⣿⣿⣿⠈⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠛⠁⣸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣷⠸⣿⡿⠿⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⣿⠃⠀⢿⣷⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠰⠃⣀⣀⣿
⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⢸⡗⢤⡄⠀⠀⠀⠈⣷⣦⡆⠀⠀⠀⠈⣗⢶⡆⠀⠀⠀⠘⣷⢴⡆⠀⠀⠀⠈⡷⣴⡆⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⢰⠟⠄⠀⠉⠙⣷⣷⣬⡅⠀⠀⠉⠻⣿⣿⣧⠀⠀⠈⠙⣿⣾⣥⡀⠀⠀⠐⢿⣾⣥⡄⠀⠀⠈⠻⣿⣧⣤⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⣿⡏⣿⠁⠀⠀⠀⠀⣿⢸⡏⠀⠀⠀⠀⣿⡎⣿⠀⠀⠀⠀⢸⡏⣿⠀⠀⠀⠀⠀⣿⡸⣿⠀⠀⠀⠀⢸⣿⣧⡀
⠀⠀⠀⠀⢸⡅⠂⠀⠀⣀⣈⣉⡁⡉⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⢤⣠⣟⡂⣉⠀⣀⣀⡀⡼⠃⢹⡄⠀⡀⠀⠀⠉⠑⠙⠀⠀⠀⠢⣼⡿⢟⣿
⠀⠀⠀⠀⠀⠘⢇⠉⠉⠙⢿⣿⣿⠻⡿⠛⠿⣿⣿⣿⡶⣾⣿⣿⣿⣿⣿⡿⠻⡿⣿⣿⣿⣿⣿⠿⣿⡾⠛⠿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠣⣀⣀⣾⡿⠃⠀⠹⣤⣀⣸⣿⠟⠀⠘⢧⣉⣀⣿⡿⠁⠀⠱⣄⣙⣿⣿⠏⠀⠘⣧⣀⣠⣿⡟⠀⠈⢧⣉⣻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠿⠟
⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣾⣿⣿⣿⣿⠿⠏⠉⠀⠀⣤⣼⣿
⠏⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠘⣿⣿⠿⠋⠉⠀⢀⣤⣼⣿⡿⠛⠁
⠆⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠼⠓⠠⠦⣤⣿⣿⡿⠟⠁
⠠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠄⢸⠟⠋



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣷⣶⣤⣀⠀⣀⣀⣠⣶⡄⣀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣾⣿⣇
⣷⡄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣥⣭⣭⣷⣶⡾⠻⠂⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠟⣻⣿⣿⣿⣿⣿⣯⣇
⠋⠄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⡶




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠇
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣵⣾⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣯⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣿⣿⣿⡿⣣⣤⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢟⣼⣿⣿⢏⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠔⠣⡠⠀⠢⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠹⠄⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡄⠀⠀⠀⠀⠠⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⡄⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠁⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠀⠄⡀
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⡄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠂⠀⢀⠁⠃
⠀⠐⠤⠀⡀⠐⠀⠀⠤⠂⠀⠀⠀⢠⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠢⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠢⠀⠀⢠⠀⠈⢃⠉⠀⠀⠀⠀⠄⠈⠄⠀⠀⠀⠀⠀⢥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡡⠎⠀⠐⠀⠌
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⡠⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠨⠄⠀⠀⠒⢀⢀⠀⡀⠀⠠⢐⠂⢠⠀⠀⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠀⠈⠁⠀⠀⠀⠈⠪⠀⠠⠀⢀⠀⠀⠀⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠀⣄⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠂⠀⠀⠀⠀⠀⣈⠔
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢟⡿⠿⣿⡿⠛⣵⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡛⡱⠆⠘⠃⠀⠴⠦⣿⣿⣿⣿⣻⣿⠿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣴⣤⣀⠀⣠⡾⣏⢻⣿⠿⠿⠿⠟⠂⠻⠿⠿⠿⣿⠋⠹⠀⠉⠀⠘⠂⡾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⢘⡯⣿⣶⣿⣿⣵⣎⣿⡿⣿⣿⣿⡶⣜⣿⣿⣿⣿⣤⣼⠀⠀⠀⠀⡸⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⡞⣧⡹⣷⢾⣟⣼⣾⡟⣱⣿⣿⣿⣧⣿⣿⣽⣿⣿⣿⣿⣷⣤⣤⣴⢿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣏⣤⣶⣟⣿⣿⣿⣾⣭⣾⠿⠏⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⠉⠋⠀⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣴⠀⠘⣷⣿⣿⣿⠿⠀⠙⠙⠁⠽⢿⣿⣏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⣤⣿⣀⣤⣿⣿⣿⣿⡄⠂⡀⢀⡐⢀⣾⣿⡿⢻⣿⣷⣦⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⢹⣿⣿⣿⢿⢿⢿⣿⣶⣿⣾⣧⣾⣿⣋⢠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢰⣿⣿⠁⢘⣘⠁⢩⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⢸⣿⣿⡴⠛⠛⢳⣼⣿⣿⣿⡏⠀⣹⣿⠟⠹⣿⢿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣤⡀⠀⢠⣼⣿⠛⢻⣿⣿⣿⣿⠀⠛⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⠈⠉⠈⠉⢸⣿⣀⣵⣿⣿⣿⣿⣿⣛⣉⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⢉⢷
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⠔⢁⡌
⠀⠀⠀⠀⡠⠃⠀⠀⠀⡠⠃⡰⢧⡈⠂⢒⣈⣁⣀⣀⠀⠉⠐⠂⠄⡀
⠀⠀⠀⡔⠀⠀⠀⠀⡔⠁⣜⠳⢤⣉⣉⠉⣀⣀⣀⠈⠉⠓⠢⢄⡀⠀⠑⠢⠄⣀
⠀⠀⠬⠄⠤⠤⢀⡜⢀⠞⢌⠓⠤⠄⠤⠤⢤⣄⣈⠙⠲⢤⡀⠀⠉⠓⢦⣄⠀⠀⠈⠁⠀⠒⠂⠀⠀⠀⠄⠀⠀⠐⠒⠀⠀⠈⠉⠁⠀⠀⣀⠨⠝⡆
⠀⠀⠆⠀⠀⠀⠀⢡⡌⢳⡈⠐⠒⠒⠲⠤⣄⡀⠉⠳⢦⡀⠉⠳⡢⢄⠀⠀⠉⠋⠰⠄⣀⠀⠤⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠒⠂⠉⠁⢀⠠⠐⡡⠁
⠀⠀⠈⠀⠀⠒⠐⢎⣱⠀⠙⠣⡐⠤⣄⡀⠀⠙⢦⡀⠀⠙⢦⡀⠈⠐⠌⡐⠢⠄⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠉⢉⣉⠀⢺⡀
⠀⠀⠀⠀⢀⠌⠀⠀⠙⡀⠀⠀⠑⠄⠈⠙⢦⡀⠀⠙⢦⣀⠀⠙⠲⣄⡀⠈⠁⠒⠤⢀⡉⠀⠒⠂⠀⠀⠒⠂⠀⠀⠐⠒⠂⠈⠉⠁⢀⡠⠔⡸⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢢⠀⠀⠀⠈⢆⠀⠈⢋⢢⠀⠀⠡⡑⢄⡀⠀⠙⠳⢤⡀⣀⠀⠀⠀⠉⠀⠒⠒⠀⠀⠤⠤⠤⠤⠤⠤⠔⢈⣁⡠⠎⡀
⠀⠀⠀⠀⡄⠀⡄⠀⡄⠀⢂⠀⠀⠀⠀⢣⠀⠀⠣⠑⢄⠀⠈⠢⡈⠒⠄⡀⠀⠈⠀⠊⠥⢀⠀⠒⠀⠤⠤⠤⠤⠤⠀⠤⠤⠤⠤⠤⠌⢳⠐⠀⡧
⠀⠀⠀⠀⡄⠀⢁⠀⠰⡀⠀⢂⠀⠀⠀⠀⠱⡀⠀⠡⡀⠑⠄⡀⠈⠒⢄⡀⠁⠒⠠⠄⣀⡀⠀⠀⠈⠉⠀⠒⠒⠒⠀⠀⠤⣤⡀⠒⣩⠚⢈⠔⠁
⠀⠀⠀⠀⠁⠀⠐⡀⠀⠱⡀⠀⠣⠀⠀⠀⠀⠘⢄⠀⠐⢄⠀⠈⠢⢀⠀⠈⠑⠠⢀⡀⠀⠀⠉⠈⠀⠐⠒⠂⠀⠠⠤⠤⠤⠤⠬⢽⡤⢽⡇
⠀⠀⠀⠀⠘⡀⠀⢡⠀⠀⠑⠄⠀⠑⠄⠀⠀⠀⠀⠣⡀⠀⠑⢄⠀⠀⠉⠂⠄⣀⠀⠈⠑⠂⠠⠄⣀⢀⣀⠀⠀⣀⡀⠠⠄⠀⡢⠝⣋⠕⠁
⠀⠀⠀⠀⠀⠡⡀⠀⢂⠀⠀⠈⠢⡀⠈⠢⡀⠀⠀⠀⠈⠢⡀⠀⠑⠤⡀⠀⠀⠀⠉⠐⠂⠤⢀⣀⠀⠀⠉⠉⠒⠤⢀⡠⠐⠪⠔⠊
⠀⠀⠀⠀⠀⠀⠱⡀⠀⢂⠀⠀⠀⠈⠢⠀⠈⠢⡀⠀⠀⠀⠈⠂⢄⠀⠈⠐⠄⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠤⠄⠈⣣⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠣⡀⠀⠀⠀⠐⢦⡀⠈⠒⢄⠀⠀⠀⠀⠉⠂⠄⡀⠈⠉⠋⠅⣒⠂⠀⠠⠤⠤⠤⠀⠀⢒⣠⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠑⢄⠀⠀⠀⠈⠫⡢⠀⠀⠈⠂⠄⡀⠀⠀⠀⠈⠑⠢⠄⣀⠀⠉⠒⠤⣤⡤⠀⢒⠈⠉⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠢⡀⠀⠀⠀⠈⠢⡁⠢⢄⠀⠀⠑⠂⠤⣀⠀⠀⠀⠀⠉⠐⠠⠄⡈⠑⢄⠠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠈⠢⢀⠀⠀⠀⠈⠢⣀⠈⠑⠢⣄⢀⡀⠀⠉⠐⠂⠠⠤⠀⠀⠀⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠑⢄⡀⠀⠀⠀⠑⠤⡀⠀⠑⠢⡁⠂⠒⠂⠀⠀⠴⠶⠆⢥⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠈⠒⠄⡀⠀⠀⠈⠐⠄⡀⠈⠓⢆⠀⠀⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠈⠑⠢⠄⣀⠀⠈⠑⠠⣈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⠀⠈⠁⠒⠀⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠿⣿⢿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢟⠋⠉⠉⠁⠀⠸⠉⠉⠓⠢⢍⡛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠳⡈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠻⣑⢤⡀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⢣⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡳⣄⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠨⡄⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⠝⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡠⠋⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣤⡀⠠⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡵⢊⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⣷⣖⣠⠤⠤⠤⠼⠤⠄⣒⣩⣥⣶⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠟⠛⠛⠛⠻⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣸⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢃⡤⣄⢀⣣⣃⠀⠠⠟⠁⠀⠀⠀⠀⠀⠀⠻⡆⠀⣿⡇⠘⠃⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⣸⠶⠛⠉⠉⠭⣭⡳⣄⠀⢠⡖⠲⡦⣄⣠⠴⠒⠒⠛⢿⣴⣛⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣠⠤⢶⡾⠃⠀⠉⣻⡷⣾⡁⠀⠀⠀⠀⠀⠈⠛⠧⡀⢠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠀⣾⠀⢀⣠⠖⢻⣽⡶⠞⠓⠒⠛⠛⠻⠿⢯⣉⠉⠳⣄⠀⠀⠀⠀⠀⢢⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠯⣶⠞⠋⢀⣀⣤⡤⠤⠤⢤⣤⣀⠀⠈⠙⠦⣿⣿⣦⣄⡀⠀⢸⠂⡇⠐⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⣠⠞⠁⣠⡾⠟⠉⠀⠀⠀⠀⠚⠀⠈⠙⠲⣄⠀⠈⠳⡄⠀⠙⠳⣦⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⠃⣠⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠙⣆⠀⢆⠘⣆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⢣⠏⠀⠀⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠘⣆⠘⡆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡇⡾⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⡀⢻⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⠶⠀⠀⠀⠀⠀⠈⡶⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇⢸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⣇⠀⠀⠀⠀⠀⠀⠀⣱⠶⠃⠀⠀⠀⠀⠀⠀⠀⠒⣿⠀⠀⡇⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣇⢸⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠃⡟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣆⢳⡀⠀⠀⠀⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡞⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢦⡙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⡽⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠢⣍⡒⠤⣀⡀⠶⠀⠀⠀⢀⣠⠖⠋⠀⢀⣾⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⣀⠉⠛⠒⠲⠶⠒⠒⠛⠉⠀⣀⣤⡖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠈⠙⠓⠶⠶⠶⠶⠶⠖⠚⠋⠙⣦⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⡎⣽⠹⠏⣹⠊⠆⠀⠀⠀⠀⢠⢎⡿⢹⠉⣷⠙⡆⠀⠀⠀⠀⢰⢫⡏⠽⢉⡗⠱⠀⠀⠀⠀⠀⡰⣹⠏⠟⢩⡎⢳
⠀⠀⠀⠀⠀⠀⢸⠃⠁⠀⠀⣀⠀⡆⠀⠀⠀⠀⢸⠀⠁⠀⠀⠁⠀⡇⠀⠀⠀⠀⡌⠈⢀⡀⣀⠀⢰⠀⠀⠀⠀⠀⡇⠈⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⣸⡄⢰⡂⠀⣀⠀⣇⡀⠀⠀⢀⣸⠀⣰⠄⢰⣀⢀⣃⠀⠀⠀⢀⣃⠀⡁⠀⢀⡁⢸⣀⠀⠀⠀⣀⡇⢀⡘⠈⢃⠀⣼⡀
⠀⠀⠀⠀⠀⠸⠬⡇⠀⠁⢀⡈⢀⠧⠜⠀⠀⠣⢼⡀⢨⠛⢢⠀⢸⠥⠗⠀⠐⠧⢽⠀⣀⣀⣀⠀⣸⠤⠃⠀⠘⠤⢇⢠⠔⠒⠢⡀⡯⠼
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠈⣀⠜⠀⠀⠀⠀⠀⠀⠣⡈⠢⠚⣠⠎⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠘⢌⠊⠉⢛⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡏⡾⠐⠇⢼⠉⠃⠀⠀⠀⠀⢠⠣⡟⠸⠁⡿⠉⡆⠀⠀⠀⠀⢸⢱⠇⠺⠠⡏⠙⠀⠀⠀⠀⠀⡜⢼⠇⠏⢰⠏⢹
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⣀⠀⠀⡀⠀⡇⠀⠀⠀⠀⡄⠀⠠⠂⠢⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣸⡆⣤⡆⠐⣥⠀⣧⣀⠀⠀⣀⣼⠀⢤⠀⠠⣄⢠⣥⡀⠀⠀⣠⣥⢠⢤⠀⠤⡄⢸⣄⡀⠀⢀⣠⡇⢠⡄⠀⣠⡄⣾⣀
⠀⠀⠀⠀⠀⠙⠒⡇⢰⣾⣿⣦⢀⠗⠊⠀⠀⠑⠺⡄⢙⡤⣤⡁⢸⠒⠋⠀⠈⠓⢺⠀⠀⣶⠀⠀⡼⠒⠁⠀⠈⠒⢧⠀⠀⡠⠄⠀⡗⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢄⣉⣉⡡⠊⠀⠀⠀⠀⠀⠀⠑⢌⣉⣉⡵⠃⠀⠀⠀⠀⠀⠀⠳⣀⢈⣀⠜⠁⠀⠀⠀⠀⠀⠈⠢⣀⢀⣠⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⠒⠒⠂⠉⠉⠉⠉⠉⠑⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⣠⣤⡶⣲⠶⠻⢟⠟⠛⠒⢲⠶⠦⣤⣀⡀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⡤⡾⢛⢹⣴⣿⠶⠿⠻⠿⠛⠿⠻⠳⠻⡿⢾⣵⣬⣝⡛⠲⢤⣀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⡞⡙⣤⣷⠿⠋⠁⡀⣤⢰⡆⢰⣶⡄⢠⣦⠀⣠⡄⢀⡀⠊⠙⠫⢖⣬⣋⡳⢦⣤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⡏⣨⣶⡟⢉⣥⡄⢶⡌⠟⠉⠀⠀⢸⣿⡇⠀⠁⠀⠉⠁⠘⠏⢴⡗⢀⣤⡈⠙⠿⣯⣭⢛⣶⣦⣄⣑⣤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⣴⠻⣠⣾⠏⠁⣴⡹⣧⢻⣆⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣤⡄⠀⠽⢗⣴⣚⡻⣮⡝⡻⣦⡠
⠀⠀⠀⠀⡠⠃⢀⡞⢀⣾⠟⢁⣀⠻⠎⠀⠸⣧⠹⡧⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠉⠰⠿⢀⡀⠉⠿⢶⣛⡻⣦⡟⢷⣆⠒
⠀⠀⠀⡐⠁⢠⠏⢤⡿⠃⠀⣮⡙⠀⠀⠀⠀⠀⠡⠘⣆⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢡⡶⠀⠙⢿⣯⣃⠙⣎⣿⣆⣐⡀
⠀⠀⡐⠀⣰⠏⣄⡞⠁⣰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠁⡘⣆⠀⠀⠀⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣆⠀⠘⢿⠁⠈⢏⠹⣷⡤
⠀⠠⠀⢰⠿⠐⣾⠅⣀⠉⠻⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠟⠋⢡⣤⠀⠀⢂⠀⠘⣧⢸⣧⠄
⠀⡄⠀⡞⢂⣻⠇⢀⣙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠈⠄⠀⠸⣧⢻⡇⠀⠅
⢰⠀⢸⣏⣾⡛⢀⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠸⠀⠀⢳⢿⡿⠂
⡈⠀⣬⢋⣿⠃⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⡆⠀⢸⣱⣿⣭⡄
⡇⠀⣯⢪⣿⠂⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⠀⠁⠀⢸⢕⣿⡂
⡇⢰⡇⠐⣿⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⡛⠋⠀⠀⠀⢸⢺⣿⣒⡀
⠀⠐⢧⠪⣷⠀⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠆⠀⣞⢽⣿⠒
⢰⠀⠷⢌⣻⣆⠈⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⡸⠀⢀⠗⣹⡿
⠈⡆⠘⣗⠨⢧⡀⠀⡶⠆⢀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣈⠋⠀⢠⠁⠀⣘⡩⣾⣯⡭⠩
⠀⢡⠀⢹⡬⢹⣧⡀⠰⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣶⣄⡈⠛⠀⠀⡌⠀⢠⠇⣺⡟⠉⠁⠄
⠀⠘⡄⠀⢻⣆⠻⣧⡀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏⠀⠀⡜⠀⢠⢛⣭⣟⣃⠈
⠀⠀⠈⢆⠸⢳⡌⠹⣯⡄⠀⠺⠓⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⢀⠈⠀⡰⢋⣿⡟⠓⠒
⠀⠀⠀⠈⠣⣞⡻⣶⡞⢷⡢⡀⠈⠋⣴⡆⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⠘⠷⠀⠀⠀⡠⠋⢀⡜⢭⣵⡟⠒⠒⠂
⠀⠀⠀⠀⠀⠙⣧⡟⢷⣔⣻⣷⡀⠀⠀⠰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢴⣆⠛⠃⠀⠀⣠⠞⠀⣰⢓⣰⡾⠯⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣎⣹⢶⡀⠛⠦⣀⠀⠈⠁⠾⢂⣶⠀⣀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⠀⣤⡙⠿⠟⠈⠁⠀⣀⠔⠊⠀⣠⠾⢉⣰⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣅⠋⢦⣄⠈⠑⠢⢤⣀⠀⠁⠘⠛⠸⡷⠘⣿⣷⢰⣆⢸⡆⠙⠃⠉⠁⠀⣀⡤⠒⠊⠁⢀⣠⠾⢍⣽⠟⠓⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⢻⣧⡨⢱⠢⣄⡀⠀⠉⠐⠢⠤⠄⢀⣀⣈⣁⣀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠀⣀⣤⡶⣯⣥⡶⠿⠡⠬⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠽⠷⢧⣡⢉⢉⠱⡰⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢤⡔⠶⢙⣩⣤⡜⠛⠋⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⢀⣙⣛⡳⣶⡤⣬⣌⣏⣫⡹⡙⢝⠏⡫⡙⠬⢑⣬⣦⠶⠽⠛⠛⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⠖⠚⠛⠛⠛⣛⡛⠛⢛⣛⣛⣛⣛⡡⠬⠌⠉⠁

⠀⠀⠀⠀⠀⣀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢄
⠀⠀⡇⠀⠈⡵⢄⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢁⠠⡈⠁⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⠀⠙⣡⠀⣉⠋⠀⠀⠀⠀⠀⡆
⠀⠀⡇⠀⢰⠀⠈⣾⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠊⠊⠀⠁⠐⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠙⢇⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠚⡗⠛⢻⡋⣃⡠⠤⠒⠚⠛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠐⠠⠀⡀⢨⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠀⠂⢄⡀⠀⠀⠇
⠀⠀⡇⣀⠀⠛⠓⠋⠁⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣤⣦⣴⡎⠀⠀⠀⡇⠀⠀⠀⠀⢀⠰⡁⠀⢀⣀⠀⢀⠠⠤⢋⠦⡀⠅
⠀⠀⡇⠈⠲⢄⠈⠲⠦⠴⠶⠖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠄⠂⠁⠀⠀⡆
⠀⠀⡇⠀⠀⠠⣉⡒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⡃
⠀⠀⣇⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠃⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⡆⠀⠀⠀⠀⠋⠁⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢈⠙⠀⠀⠁⠈⡆
⠀⠀⡇⠀⠀⠀⠀⠂⠀⠑⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢢⠈⠛⡓⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠈⠓⠀⠀⠀⠀⡄
⠀⠀⡇⠀⠀⠀⠀⡠⠠⠔⠢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢸⣤⡤⢿⣤⠇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⣨⡹⠂⠀⡇
⠀⠀⡇⠀⠀⢠⠞⠀⠀⣀⡀⠆⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⡀⠀⠉⠀⠀⠐⠒⠲⢦⡀⡇⠀⠀⠀⠀⠀⠀⣐⠓⠮⠟⠓⠐⠋⣭⠄⠀⠀⡇
⠀⠀⡇⠀⠀⢸⣈⣶⣾⣿⣿⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⣀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣤⣀⣀⣼⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠈⢿⣿⣿⣿⡿⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠢⡀⠈⠓⠶⠶⠛⠁⡇⠀⠀⠀⠀⠀⠀⠘⢿⠿⠿⠿⠿⣿⠃⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠉⠛⠁⠐⠊⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠙⠂⠈⠁⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⠀⠀⠀⠀⢉⣒⡒⠉⠁⠀⠀⠀⠀⡃
⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠉
⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀
⠀⠀⡖⠒⢒⣦⡄⠄⢖⡒⠲⠶⠒⠒⠒⠒⠀⠀⠀⠀⠐⠒⠒⠒⠒⠂⠨⠒⠚⠓⠀⠒⠒⠐⡆⠀⠀⠀⠐⠒⠒⠚⠋⣠⢶⡒⢊⢛⠓⠒⠒⠒⡆
⠀⠀⡇⠀⠙⠛⠓⠒⠢⠁⢙⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢲⣀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣼⢁⣀⠀⢃⣤⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡸⠻⠤⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠟⠚⠋⠀⠱⣄⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠍⠀⠀⠀⠀⠐⡦⠅⠀⠀⠀⠀⠀⠀⡀⠀⢀⣤⡀⠀⠈⠁⠉⢳⡄⠅
⠀⠀⡇⠀⠀⠀⠀⠀⢦⣤⣠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⠦⠤⠤⠴⠶⠛⠁⡆⠀⠀⠀⠀⠀⢶⣁⠀⠘⠦⠤⠴⠶⠶⠒⠋⠀⡇
⠀⠀⡇⠀⠀⠀⠀⢀⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⡑⠲⢤⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠤⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣤⣤⢤⡴⡄⣬⣤⢭⣧⢤⢤⣤⣤⡤⣤
⠀⠀⠢⠖⠡⠀⠀⢠⠀⠀⠐⠈⠂⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠉⠩⠉⠉⠨⠍⠛⠛⠋⠙⠉⠛⠙⠋⠙⠙⡟
⠀⡁⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⣄⡀⠀⠀⠀⠀⡇⡇⠘⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂⠀⠘⠀⠀⡧
⡟⢀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⡇⢀⣈⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠀⠀⣿
⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠁⠀⡏⠀⠀⠀⣿⢓⣶⣦⣐⣖⣀⣶⣆⣶⣶⣾⣆⣶⣶⣳⣶⣶⣷
⠰⡔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠉⠈⠉⠈⠀⠀⠁
⠀⠙⢌⠢⠈⠀⠀⠀⠀⠀⠄⡀⢌⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠉⠒⠤⢭⣠⣤⠬⠅⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠉⡍⠉⢡⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠃⠀⢠⡟⠃⠀⠀⠀⠀⠀⠀⠀⢔⡢⠖⠊⢹⡄⠉⠛⠷⣠
⠀⠀⠀⠀⠀⠀⢀⡆⠈⠈⢰⠄⠈⠉⣏⣉⠁⠀⢰⡂⠁⠠⡀⠀⠀⠠⡞⠛⠓⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⢺⠋⠀⠀⠀⢾⡗⠀⠀⠀⠘⢶
⠀⠀⠀⠀⠀⠀⠘⢲⡀⠖⣠⠔⠉⠀⠠⠀⠀⢉⠂⡝⢶⢀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢈⠀⠀⣀⡀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⡼⠁⢠⣄⠀⠀⠀⢀⡀⠑⠘⡀⠁⠀⠀⠀⠀⣷⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡈⠁⠀⠀⠀⠘⠁⠉⠋⠁⠀⠉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠤⠀⠉⠳⣦⡞⠋⠀⠠⠀⠁⠀⠀⠀⠀⠀⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣜⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠫⢄⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠎⣐⡅⣀⡀⣀⡠⢦⣔⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠠⠀⠀⠄⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠿⡟⠛⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠒⠒⠂⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠠⠙⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠄⠛⠳⢶⣄⠀⠀⠀⠀⠀⢸⣻⠞⠁⠀⠀⠀⠀⢀⠀⠤⠀⠁⠼⠤⠥⢈⣀⣤⣀⡀
⠸⠧⠥⠴⠦⠤⠄⠄⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⠀⠀⠀⠀⠙⢦⣀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠰⠾⠿⠾⠿⠿⠿⠷⠶⠶⠶⠶⠶⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⡄⠀⠀⠀⠀⠀⢻⡇⢀⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠘⣇⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠏⢈⠁⠁⠀⡀⢀⠀⠀⠀⠙⢏⠁
⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠻⣿⣶⠄⠀⠀⢠⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡅⠀⠈⠂⠀⢀⢸
⠀⠀⠈⠐⠄⠂⠂⠈⠤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡧⠐⠾⢿⡗⢦⡴⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⡀⠀⠀⢀⡀⠀⣀⠀⠠⢸⠈
⠀⠀⠀⡀⠀⠀⠀⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⣰⠀⠠⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠈⠈⠀⢸⠸
⠀⠠⢁⠄⠀⠀⠀⠐⠀⠑⡄⠀⠀⠀⠀⠀⢀⡎⠀⠀⢀⣔⠁⠀⠼⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠠⠄⢀⠠⡀⠀⠀⡀
⠀⢃⠇⠀⠀⠀⠀⠀⠀⠱⢡⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⡏⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠂⠀⡞⢹⡏⠙⠂⠈⠀⠀⡀⡆
⠀⡀⠃⠀⠀⠀⠀⠀⠀⢈⢨⠀⠀⠀⠀⠀⣿⠀⢀⡀⠀⠇⠉⣠⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠇⡈⠡⠀⠀⠀⠀⠀⠀⠇
⢠⣗⣐⣒⣆⣓⡒⠚⠒⠒⠚⠆⠀⠀⠀⠀⠈⠠⣈⣀⣠⣤⣼⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠚⠛⠋⠙⠛⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⣀⠀⠀⠰⠆⠀⠀⢀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠈⠁⠀⢸⡇⠀⠀⠋⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⢀⣴⠟⠁⠀⠀⠀⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣠⠈⠁⠀⠀⠀⠀⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠁⠀⠀⢰⡆⠀⠀⠈⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣤⡴⠞⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠛⠉⠉⠀⣀⢀⡀⣀⠀⡉⠉⠛⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⠁⢠⡀⠆⠘⠀⠁⠈⡘⣃⠁⠈⠀⠃⠰⢀⡄⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢄⠙⠈⠓⡄⡄⠀⠀⠀⣇⣜⠁⠀⠀⠀⢠⠚⠀⠋⡠⠀⠙⢿⣷⣄
⠀⠀⠀⠀⣴⣿⠟⠀⣄⠑⠀⠀⠀⠀⠇⠇⠀⠀⣤⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠊⣠⠀⠻⣿⣦
⠀⠀⠀⣸⣿⠏⠀⠲⢦⢀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠖⠀⠹⣿⣇
⠀⠀⢠⣿⡟⠀⣈⠁⠀⢸⠻⡸⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠈⣁⠀⢻⣿⡄
⠀⠀⣾⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣧
⠀⠀⣿⣿⠀⢘⣂⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣐⡃⠀⣿⣿
⠀⠀⣿⣿⠀⢨⠍⠀⠛⢀⣀⣠⣤⣤⡶⠶⠟⠛⠛⠻⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠩⡅⠀⣿⣿
⠀⠀⢿⣿⡄⠀⠂⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣿⡿
⠀⠀⠸⣿⣧⠀⢉⡀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⣰⡆⠀⢀⡉⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣆⠀⠴⠚⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠓⠦⠀⣰⣿⡏
⠀⠀⠀⠀⠻⣿⣦⠀⠋⡠⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢄⠙⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠊⣠⠀⡤⠁⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠐⢥⠀⣄⠑⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠘⠁⠆⢠⠀⡀⢀⢨⡅⡀⢀⠀⡄⠰⠈⠃⢀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⠀⠉⠈⠁⠉⠀⣀⣀⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉

⣟⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⢰⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠄
⠀⠀⠈⠛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⡁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠂⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠘⠂


⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⡆⠀⡀⢠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⡌⡀⢠⠁⠀⢀⠀⠀⠀⠀⠀⢁⠁⠈⠁⢨⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠖⠀⠳⢝⠂
⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠐⠀⠠⠐⠀⠀⠀⠀⠀⠘⡤⢠⠠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⠤⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔
⠀⠀⠀⣠⡅⠀⣀⢀⠀⠀⠀⠀⠀⢀⡥⠀⠄⢉⠀⠀⠀⠀⠀⠀⠀⠄⡑⠀⠀⡀⠂⠀⠀⠀⠀⡄⠨⠐⠀⡀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢
⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⠈⠀⠂⠀⠀⠀⠀⠀⠤⠐⠀⠀⠄
⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠁⠀⠀⠀⠀⠀⠀⠉⠲⠀⠁
⣀⣠⣄⣄⣄⣤⣄⣠⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⢛⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣹⡓⠀⢀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠀⠀⢸⠟⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⢰⠈⠀⠭⠅⠀⠀⠭⠀⠅⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡄⠁⢨⡙⠀⠀⣩⠄⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⡦⠐⠏⠀⠶⠆⠀⠆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠠⣤⠄⠾⠀⠴⠦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠀⢸⣿⣶⠾⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⢀⠀⣿⣶⣾⢏⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡀⠀⠀⠀⠀⡄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠁⠀⠀⡠⣀⣁⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡷⠨⠁⠀⠀⠍⠚⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⠌⢐⣒⠢⠩⠚⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠀⠀⠀⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣷⡄⠀⢤⣤⣠⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠛⣁⣀⣸⣀⡙⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⣩⣷⡾⠟⠛⠛⠛⠛⠛⠿⢶⣿⣙⠛⠿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⣤⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⡄⠀⠀⠈⠙⢢⣦⡄⠀⠤⠤⠄⢤⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠉⠀⠀⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠻⠷⣶⣶⣶⣶⡶⠿⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠁⣀⣠⠤⠴⠒⠒⠚⠛⠛⠛⠓⠒⠒⠲⠤⢤⣀⡀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣀⡤⠒⢋⣁⡤⠴⠒⠚⠋⠉⠉⠉⢉⡉⠉⠛⠒⠲⠤⣄⣉⠓⠢⣄⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣀⠴⠊⣁⠴⠚⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡉⠳⢄⡀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠟⠀⣠⠞⢁⡴⠊⠁⣶⠀⠀⡆⠀⠀⠀⠀⠀⢸⡀⡸⣷⣁⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠙⠲⣄⠙⢦⡀⠈⢻⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠁⢠⠞⢁⡴⠋⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⣧⡼⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣄⠙⢦⠀⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣿⠏⠀⡰⠃⣰⠋⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠳⡀⠳⡄⠈⢿⣿⣿⣿
⣿⣿⣿⠋⠀⡼⠁⡼⢁⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⡄⠙⡄⠀⢻⣿⣿
⣿⣿⠏⠀⡼⢀⡞⠀⢸⡇⢰⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠘⣆⠸⡄⠀⢿⣿
⣿⡟⠀⣸⠁⡼⠀⠀⠸⠇⠀⠣⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⡴⠀⠠⠤⠄⠀⠀⠸⡄⢹⡀⠘⣿
⣿⠁⢀⡇⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠳⠊⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣇⠀⢹
⣿⠀⢸⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⠀⠸
⡇⠀⣾⠀⡇⠀⢠⡆⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⡇⢸⡇
⡇⠀⣿⠀⡇⠀⠀⠑⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⡇⢘⡇
⡇⠀⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⢸⡇
⣿⠀⢸⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠀⢰
⣿⡄⠈⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡜⠀⡏⠀⣸
⣿⣧⠀⢹⡀⢳⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡇⠀⣰⠃⣸⠁⢠⣿
⣿⣿⣆⠀⢳⡀⢣⠀⠀⢙⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⢠⠃⢰⠃⢀⣾⣿
⣿⣿⣿⣄⠀⢳⡀⢳⡀⠻⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠃⢀⣼⣿⣿
⣿⣿⣿⣿⣆⠀⠱⣄⠙⣄⠀⠀⠀⠀⠀⠀⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⠂⠀⠀⠀⢀⡜⢁⡴⠁⢀⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣧⡀⠈⢦⡈⠳⣄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⢀⡴⠋⣠⠎⠀⣠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢦⡈⠓⢤⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠔⠋⣠⠞⠁⣠⣾⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⠳⢄⡉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠸⡇⢹⠆⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⠴⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠉⠓⠦⣄⣉⠙⠒⠦⠤⣤⣄⣀⣉⣀⣀⣠⡤⠤⠖⠚⢉⣁⡤⠖⠋⠁⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠈⠉⠓⠒⠦⠤⠤⠤⣤⡤⠤⠤⠤⠖⠒⠋⠉⠁⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⢀⣀⣤⣤⣤⣶⠶⠶⠶⠶⠶⣶⣶⣤⣤⣤⣀⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣠⣴⣾⣿⣿⣿⣿⣷⠶⠶⠶⠶⠶⠦⠤⢤⣭⣤⣭⣿⡿⢷⢶⣌⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣾⣿⣿⡿⠛⢋⣉⣤⣤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣌⣉⠛⢿⣿⣿⣷⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢈⣷⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⢿⣷⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡖⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣾⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⢀⣤⣀⣂⣂⣀⣂⠀⠰⠦⠀⠒⠀⣀⠐⠀⢠⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢙⡆⠀⠀⠀⠙⢿⣿⣿⣯⣯⡌⠄⣠⣴⣿⡿⠋⠀⠀⠸⡿⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡾⠁⠀⠀⠀⠀⠀⠋⠻⣿⣿⣦⣾⠿⠋⠉⠀⠀⠀⠀⠀⢳⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠈⠀⠀⠀⠀⠀⠀⠀⢀⣏⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡗⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠘⢿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡛⢳⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⢀⣟⡑⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢨⠀⠀⠀⠠⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣟⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠘⠀⢀⠄⠀⠀⠀⠀⡠⠐⠋⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣶⠀⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠳⢄⡀⠀⠀⠀⡆⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⣸⡿⣠⣴⠃⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠘⠦⣄⠀⠀⣮⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠻⣿⣿⣿⠀⠠⣄⣀⣠⣼⣄⠀⠀⠀⠀⠀⠀⠀⠚⠃⣠⣾⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣤⣴⣾⢟⡻⠓⠠⠉⠒⠀⠀⠀⠤⡀⠈⣻⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⣿⣿⣿⣿⣿⣷⣤⡀⠆⠀⠀⠀⠀⠀⠀⣰⣿⣷⣿⣿⣿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢈⣿⣉⠛⠿⠿⡛⢛⠿⠷⠶⢶⡶⠾⣿⠟⠛⠛⠙⣫⣽⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡻⠶⠰⠿⡐⠍⠁⠒⠒⠒⠄⠀⠤⠀⠀⠀⠐⠀⠂⠈⠉⠀⣿⠿⠦⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡙⠢⣄⠀⠈⠐⠲⠄⢀⣤⡀⣀⣀⠀⠀⠤⠤⠄⠠⠄⠐⠀⡀⣠⠖⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠄⠘⠃⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠁⠈⠉⠀⠈

⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣉⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣄⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⡿⢿⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⢸⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⡎⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⠀⠉⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⠀⠀⡉⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⣶⣤⣄⣀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠋⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣿⣿⣿⣿⣿⠁⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣿⣿⠁⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣼⣷⣤⣄⡀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣀⠤⠄⠒⣂⣀⣀⣐⡒⠠⠤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⠔⡫⠔⠂⠉⠀⠀⠀⠀⠀⠀⠉⠐⠬⣑⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⢋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⢳⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠇⡎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⡆⢇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢦⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⠞⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠳⣜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡡⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠈⠓⢬⣓⠦⢤⣀⣀⣀⣀⣀⣀⡤⠔⣊⡵⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠈⠉⠒⠲⠮⠭⠭⠴⠖⠚⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⢿⢛⠿⢿⣿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣯⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣀⣀⣀⣀⣀⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠉⠉⣿⡏⠉⠁⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣀⣠⣤⣤⣶⣿⣷⣤⣤⣤⣀⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⣴⣾⣿⡿⣿⡛⠿⠿⠿⠿⠿⠟⣻⠿⣿⣿⣶⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋⢁⣴⣿⡿⡿⠟⠂⠉⠀⠐⠀⠁⠸⠀⠁⠐⠀⠈⠁⠋⠟⡿⣿⣷⣄⠉⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢀⣴⣿⣟⠟⡈⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠙⢟⣿⣷⣄⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡝⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠷⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢆⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠐⠦⢄⢼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠠⣪⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣗⢄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⢔⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣦⢜⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢘⢤⣪⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣶⣦⣤⣂⣠⣡⣬⢤⣥⣀⣲⣬⣶⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠒⠒⠋⠉⠉⠉⠉⠉⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⣀⡤⠴⠒⢚⣉⣉⣉⣉⣉⣉⣉⡙⠒⠲⠤⣄⡈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢁⡠⠖⢋⡡⠔⠚⠉⠉⠀⠀⡄⠀⠤⣄⠀⠈⠉⠙⠒⠤⣉⠓⠦⣀⠉⠲⣄
⠀⠀⠀⠀⠀⢀⡴⠋⣀⠔⢉⡴⠊⡅⠀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢠⡄⠉⠢⣌⠑⢤⡈⠳⣄
⠀⠀⠀⠀⣠⠋⢀⠜⢁⠔⠁⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠑⢄⠙⢆⠈⠣⡀
⠀⠀⠀⡼⠁⡰⠃⡴⠁⠀⠀⠀⠀⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠱⣄⠳⡀⠙⡄
⠀⠀⡜⠀⡼⢁⠞⢠⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠈⢆⠹⡄⠘⡄
⠀⡼⠀⡼⢁⠎⠀⢸⠀⣯⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢆⠹⡄⠸⡄
⢠⠃⢠⠃⡞⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠘⡆⢳⠀⢳
⡼⠀⡞⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢳⠘⡆⠸
⡇⠀⡇⢸⠀⢀⠄⢦⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⣀⣀⣤⠤⠤⠶⠒⠒⠻⠖⠉⠀⠐⠦⠀⢸⠀⡇
⡇⠀⡇⢸⠀⠈⢃⠟⠛⠧⠼⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢄⡼⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⢘⠀⡇
⡇⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⡇
⢫⠀⢧⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠇⢰
⠘⡄⠘⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠃⡼⠀⡜
⠀⢳⠀⢱⡈⢇⠀⠀⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠗⢀⠏⣰⠃⢰⠃
⠀⠀⢣⠀⠳⡈⢦⠀⠇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⠃⢠⠃
⠀⠀⠀⢳⡀⠱⣄⠳⡄⠀⠀⠀⠀⠠⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠂⠀⠀⠀⡴⢃⡴⠁⣰⠃
⠀⠀⠀⠀⠙⣄⠈⠢⡈⠢⣀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⢀⡠⠊⡠⠊⢀⡜⠁
⠀⠀⠀⠀⠀⠈⠳⣄⠈⠢⣌⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢤⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⢋⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠑⠢⣌⡑⠲⠤⣄⣀⡀⠀⠀⠣⠘⠀⠀⣀⣀⡤⠴⠒⣉⡤⠖⠉⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡉⠑⠒⠤⠤⣍⣉⣉⣉⣉⣉⣉⡥⠤⠔⠒⠉⢁⡠⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣀⣀⣀⣀⣀⣀⠤⠤⠔⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠿⠛⠛⠋⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⣰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⢠⣷⣶⣤⣤⠀⠀⠀⣀⣀⣠⡀⠀⠀⢠⣾⡿⠿⠿⠟⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⡿⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⡀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠢⠀⠀⠀⡀⠀⠀⠀⢀⠀⢀⣴⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⠉⠙⡹⢋⠏⣩⢉⠉⢩⡆⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠀⣾⠖⢱⣧⣬⣜⣱⣧⡟⢸⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣽⢰⣿⣿⢻⡏⢰⣟⡔⢹⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣻⠀⣠⢆⣆⣧⣌⠈⠀⢸⡧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⣺⣿⣿⣿⡇⠸⣆⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣧⣿⣿⣏⡉⠩⠁⠹⠆⢸⡅⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣠⡽⢯⡥⠤⢤⡤⢤⠀⠠⠈⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣤⣄⣀⣤⢤⠤⢤⡤⡤⠤⠬⡇⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣼⠭⠈⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠻⠟⠻⢾⡇⠀⠀⠀⠀⣰⣯⣼⠤⢼⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣷⠀⠀⠀⢠⣿⣿⣸⠀⢘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣶⣶⣼⠟⠀⠀⠀⠸⠧⣿⣽⣉⡬⠴⣿⣿⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣀⡀⣀⣀⣀⣾⣿⣀⣀⣀⣠⣿⣁⣀⡀⣀
⠀⠀⠀⠀⠀⠠⠿⠿⠿⠛⠿⠿⠻⣿⡿⢿⣿⡿⠿⠻⠿⠛⠿⢿⡟⢿⣿⠿⢿⡟⠛⠛⢻⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⢿⣿⣼⣿⡇⠀⠀⠀⠀⠀⠸⣇⠈⢿⡀⣼⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣧⣠⠤⠀⣠⣉⡭⢿⠀⠄⠤⢠⠤⡤⠤⣼⠶⡼⠧⠭⠤⠤⠤⣤⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠊⠉⠈⠁⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣀⡤⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⣤⣀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠀⠀⢈⠆⠀⢄⢀⠀⠠⡊⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡄⢊⠡⠀⠀⠀⠃⠀⠀⠄⠑⠠⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠒⡁⠔⢉⣠⡤⠖⠒⠒⡒⠒⠲⢤⣄⠈⠂⠈⠂⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡴⠋⠁⡆⠀⠀⡄⠱⠄⠀⠀⠌⠙⢦⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⠏⡀⠀⠀⠇⠸⠀⠁⠈⠀⠀⠇⠀⢀⠀⠹⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⠀⢹⢰⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠁⠹⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⣠⡀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⡀⢠⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠁⠘⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡆⠀⡈⠢⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⠀⢠⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⣆⠁⠀⠀⢨⠁⠀⢀⠀⠀⠰⡂⠀⠀⠈⣠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠳⣄⡐⠂⠀⠀⢛⡃⠀⠀⠰⢂⣠⠞⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢠⡈⠙⠲⠦⠤⠨⠄⠤⠴⠖⠋⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠤⠤⠀⠀⠂⠈⠀⠑⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠒⣶⠒⠦⣤⣀⠀⠀⢰⢀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⣀⡔⠾⣀⠀⠉⠀⠀⠀⠙⢷⣄⣈⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢤⣀⣬⣿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⠹⣿⠂⠀⠀⡱⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠀⠀⣸⣿⣿⣿⣿⡌⠉⠀⠀⠀⠀⠀⠀⠀⢡⢴⣿⣶⠐⠃⠱⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣠⣠⣶⣿⢏⣿⣿⣿⡆⢀⡀⠀⢀⠠⠤⠾⣞⠁⠀⠀⢿⡅⠆⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡤⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣾⡓⠀⠀⠀⠀⠀⢫⠀⠀⠉⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⣀⣠⣿⣖⣻⣿⣾⣿⣟⣿⣿⣿⣿⣿⣶⡦⠀⠀⠀⠘⠄⠀⢢⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀⠸⣶⠀⢸⣦⡾⣿⣤⣔⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠛⠛⠋⠀⠈⠉⠉⠉⣉⣉⣭⣽⣿⣬⣽⠙⠓⠒⠒⠛⠛⠁⠀⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡤⠀⢀⣠⣆⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠂⡤⠀⠘⠀⠊⠈⠉⠛⢻⣿⣤⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⡀⠀⠀⠀⢀⡷⠋
⠀⠀⠀⠀⠀⡘⢀⣧⣤⡤⠀⠀⢀⣠⣴⣿⠿⢋⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠿⠆⠀⡰
⠀⠀⠀⠀⠰⠶⠤⠴⢴⣤⠤⠤⠤⠤⠤⠤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠀⠀⠀⠀⢀⣦⣴⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠁⠀⠀⠀⣠⠼⠛⠋⠉⠉⠁⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠲⡠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭
⠀⠀⢂⢸⡌⢎⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⢀⠂⠈⠀⠁
⠀⠀⠈⢧⢶⣴⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠒⠰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠
⠀⠀⠀⠀⠺⠆⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣆⣆⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠘
⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⡐⡀⠀⠁⠀⠀⠀⠀⠀⠀⣀⢀⣄⢀⣀⡀⡆
⠀⠀⠀⠀⠀⠈⠿⡿⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⡸⣳⠷⠶⠶⢷⡷⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⢀⡢⠄⠃⠀⠀⠀⡼⢩⠋⠀⠀⠀⠈⢣⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠈⢸⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠁⠨⣶⠿⡍⠆⠀⠀⠀⠀⠀⠀⠰⠐⠲⠿⠟⠀⠀⠘⠿⡄⠀⠀⠀⠀⢠⡷
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢰⠾⣆⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠐⠰⢻⠀⠀⠀⠀⠀⢱⡄
⠀⠀⢀⡞⠋⢉⠟⢳⡀⠠⠄⠀⠀⢀⡔⢯⠉⠉⠲⡀⠀⠀⠀⠀⠀⡄⢀⡻⠀⠀⠀⠃⠠⠐⠛
⠀⠀⣬⠀⠠⠾⠦⠸⠇⣭⡟⠀⠀⢸⠀⠘⢷⠀⠀⡇⠀⠀⠀⠀⠀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠃
⠀⠀⠘⢄⣀⣀⣠⠆⠀⠴⠀⠀⠀⠈⠳⣄⣀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣤⣠⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠄⠀⠀⠀⠀⢈⠛⠛⠛⢲⣦
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⡆⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⢃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠐⠠⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠑

⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⣠⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠄⠀⠀⠀⠀⠠⢁⣀⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⠂⠨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏
⠀⠀⠀⣠⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠠⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀
⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢒⠂⣄⠀⠀⠀⠀⠀⠀⢠⣸⠛⠛⢷⣖⡀⠀⠀⠀⠀⠀⢀⣑⠂⠀⠂⠀⢀
⠀⠀⠀⠘⠲⠂⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠲⠈⠛⠊⠠⠀⠀⠀⠀⢰⣾⡃⠀⠀⢀⣽⠧⠀⠀⠀⠀⠠⠂⠈⠋⠀⠀⢀
⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠉⠙⢠⠀⡈⠉⠁⠀⠀⠀⠀⠑⠒⡂⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠈⠗⠁⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠭⣭⣙⣻⠃⠀⠀⠀⠀⠔⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢈⠕⠀⠀⠀⠐
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢲⡗⠀⠙⠛⠃⠀⠀⠀⠀⠛⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀⢤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣷⠆⡁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣥⣄⣀⡒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠉⠉⠀⠈⠉⠛⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⠃⠀⠀⡄⠀⠘⢻⣿⡒⢦⡄⠀⠸⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⣼⣠⠀⣠⣮⣾⣱⣧⣾⡄⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃⠈⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣿⣄⠀⣿⠉⡟⠈⢻⣿⣿⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣶⣽⠀⡇⠀⢸⣿⣿⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⣤⣴⣮⣭⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⢁⣉⣛⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠒⠊⠉⠉⠈⠉⠉⠉⠉⠉⠐⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⣀⡠⠘⠁⠀⠀⠀⣀⠤⠶⡋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⢦⣄⡀⠀⠀⠀⠙⢷⣤⡀
⠀⠀⠀⠀⠀⣤⠋⠀⠀⠀⠀⠐⣉⠤⠂⠁⠀⠀⠁⠀⢰⠀⡆⠀⠀⠀⠀⠀⠀⠙⢻⣷⡆⡀⠀⠀⠙⢿⣦
⠀⠀⠀⢀⠖⠁⠀⠀⡀⠀⣠⠊⠄⠐⢄⢢⠀⠀⠀⠀⢨⡄⡇⠀⠀⠀⠀⢀⠞⠀⠀⠈⠻⣮⡢⡀⠀⠀⠙⣷⡀
⠀⠀⠠⠋⠀⠀⢀⡼⢡⠞⠀⠁⠀⠀⠈⠳⣧⠀⠀⠀⠈⠛⠃⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⡈⠳⣿⣶⡀⠀⠸⣷⡀
⠀⢠⠆⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢀⣀⠀⠹⣿⣿⣟⣄⠸⣷
⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⠛⠦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⣡⠾⠛⠉⠀⠀⠹⣿⣿⣿⡆⢻⣇
⠘⠃⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡷⢘⣿
⠆⠀⠀⠀⡄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⠀⣿
⠀⠀⠀⢤⡗⠠⠀⣔⣒⢂⢈⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⣹⠇⠀⠀⠀⠀⠀⠀⠀⠰⠷⠶⠿⢷⠀⢸⣿⡇⣿⡄⣸
⠀⠀⠀⢫⢻⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⣿⣇⣹⠁⣿
⢠⠀⠀⢸⣾⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣾⣿⠸⡃⠀⣿
⠀⠆⠀⠈⣿⢫⡀⠀⢠⣔⡋⢞⡟⠀⠀⠀⠀⠀⠀⠀⢀⣟⠁⠢⢄⠀⠀⠀⠀⠀⠿⣿⡧⢶⡄⠀⣸⣿⡗⡛⠁⣰⡏
⠀⠀⢀⠀⢹⣇⢿⣤⠄⢩⠔⠁⠀⠀⠀⢀⡀⠀⠀⠀⢸⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠋⠀⣰⣯⠟⡈⠁⣰⡟
⠀⠀⠀⠠⠀⣿⣯⠹⣃⣄⠀⠀⠀⠀⠠⡿⡅⠀⠀⣤⠠⣷⠆⠀⠒⠂⠀⢦⡀⠀⠀⠀⠀⢀⡼⡧⢃⠞⠀⢰⡟
⠀⠀⠀⠐⢷⠘⣿⣷⣬⣿⣷⣄⠀⠐⠟⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠑⠄⠀⣈⣴⣿⠟⣵⠏⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠌⠛⣿⣷⣮⣻⢦⣄⡁⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⢀⣠⡴⣾⠽⢋⣵⠞⠁⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠈⠳⠿⣿⣿⣿⣶⣭⣘⡉⠃⠀⠂⠄⠠⣤⣥⣤⣴⣶⣿⡿⠕⣋⣤⣶⠿⢁⣠⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣉⣩⣁⣤⣥⣴⣶⡿⢿⢋⡡⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠨⠾⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣟⡴⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣤⣀⣤⡀⠀⠀⠀⠀⢀⣤⡄⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⢛⣟⠆⠀⠀⢠⣽⣯⡀⢀⣿⣇⠀⠀⠀⠀⢀⣬⣯⣬⣿⣥⠀⠀⠀⢠⣤⣿⣅⣻⢳
⠀⠀⠀⠀⠀⠀⣹⣏⠀⢀⡀⣠⢾⣿⠀⠀⠀⣿⣿⣿⣿⣾⣿⣿⣧⣴⠄⠀⠸⣿⣿⣿⣿⡿⠂⠀⢠⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣟⣻⣮⠋⠉⣼⣿⣇⠀⠀⠋⣿⣿⠘⢹⣿⣿⡝⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⠈⢸⣿⣿⣿⡇⠛
⠀⠀⠀⠀⠀⠀⣿⠋⣻⡇⠀⠀⠉⣿⠁⠀⠀⠀⣿⣿⠀⢈⡟⣿⠀⠀⠀⠀⠀⣸⡏⣿⢹⡇⠀⠀⠀⢸⡟⣿⡟⢷⣄
⠀⠀⠀⠀⠘⠛⠛⠁⠙⠀⠀⠀⠀⠋⠀⠀⠀⠀⠃⠙⠀⠈⠃⠉⠀⠀⠀⠀⠀⠙⠀⠘⠉⠃⠀⠀⠀⠈⠁⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣄⠘⠿⢁⣀⣀⠀⠀⠀⠀⠀⠀⣠⣄⠻⠃⠀⠀⠀⠀⣤⡻⢃⡀⠀⠀⠀⣀⡀⠀⠘⠿⠃⠀⢀⣀
⠀⠀⠀⠀⢀⢀⣹⣟⢁⣀⠘⣿⡉⠑⠀⠀⠀⠀⢠⣝⣋⣀⡿⣇⠀⠀⢨⣟⠃⢛⣏⠇⠀⠀⠛⣏⡀⠀⠀⠀⢀⣻⠛
⠀⠀⠀⠀⠐⣾⣿⠋⢫⡽⠾⣿⡇⠀⠀⠀⠀⢀⣿⣿⣽⣿⡇⠃⠀⠀⣿⣿⣦⣾⣿⠀⠀⠀⠀⢿⣿⢶⣤⠶⣿⡿
⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⣴⣿⣷⠀⠀⠀⠐⠋⠚⣿⣿⠀⠀⠀⠀⠀⣿⣧⠙⣹⣿⡆⠀⠀⠀⠘⣿⡆⢀⣾⣿⡇
⠀⠀⠀⢀⣴⡿⠈⣿⠀⠀⣿⠈⠻⠶⠆⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣿⢻⡇⠙⡿⢷⡄⠀⠀⠀⠸⣿⠀⣿⠋⠃
⠀⠀⠀⠙⠁⠀⠀⠛⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠋⠘⠃⠈⠃⠀⠁⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠰⣿⠂⣴⡄⠀⠀⠀⠀⠀⠀⠀⠠⣶⣄⠀⠀⠀⠸⣿⠀⠀⢴⣶⠀⠀⠀⠀⠰⣿⠆⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣾⡂⠀⠀⠀⠀⢠⣶⢠⣾⡞⠀⠀⣰⣿⣿⣷⣠⣶⡷⢆⡀⠀⢠⣾⣿⣷⡀⠀⠀⣰⣾⣶⡀
⠀⠀⠀⠀⣿⣿⣿⣇⣿⣿⢷⠀⠀⠀⢰⣾⣭⣾⣿⡇⠀⠀⣿⣿⣿⣿⡟⣯⣴⡛⠁⠀⢸⢿⣿⣿⡇⣤⢰⢿⣿⣿⢷
⠀⠀⠀⠀⠀⣿⣿⡻⣿⣿⠂⠀⠀⠀⣿⣿⣤⡴⣿⠧⠀⠀⠀⣿⣿⡇⠸⣿⣿⠇⠀⠀⠀⢸⣿⡇⠙⣿⠋⠸⣿⣿⠂
⠀⠀⠀⠀⠀⣿⢸⡧⡇⣿⠀⠀⢠⣤⣿⡇⢾⠁⢿⠀⠀⠀⠀⣿⢸⡇⠀⡿⢹⠀⠀⠀⠀⢼⠹⡇⠠⡿⠀⠸⡇⣿
⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⣹⣟⡀⠀⠀⠀⠀⠀⢘⣿⡃⠀⠀⠀⠀⠀⠀⣘⣟⡀⠀⢻⡗⠀⢀⣻⣏⡀⢀⣻⣃
⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⣼⣿⣿⣿⣼⣿⣿⣦⡿⣿⣿⣷⣿⣿⣿⣆
⠀⠀⠀⠀⠛⣿⣿⠻⣧⣯⠀⣶⡜⣹⣿⣷⠛⠀⠀⠀⡞⠃⣿⣿⡝⢣⣿⣿⡝⠃⣿⣿⠸⣻⣿⣿⡻⢤⣿⡀⣶
⠀⠀⠀⠀⠀⣿⢿⡆⢸⣿⢻⣿⠀⢹⡿⡏⠀⠀⠀⠀⡇⠀⣿⢿⡇⠈⣿⣿⠁⢠⡿⣿⠀⢹⡟⣿⠀⠀⣿⠛⣿⡇⣻
⠀⠀⠀⠀⠀⠛⠘⠃⠘⠛⠀⠛⠀⠘⠃⠃⠀⠀⠀⠀⠃⠀⠛⠘⠃⠀⠋⠙⠀⠘⠃⠛⠀⠘⠃⠛⠀⠘⠛⠀⠛⠀⠛

⠂⠀⢀⡇⠀⠀⠀⣇⠀⠐⠒⠂⠀⣸⠀⠀⠀⢸⡀⠢⠤⠤⠔⢀⠇⠀⠀⠀⣇⠀⠀⠛⠀⠀⣸⠀⠀⠀⢸⡀
⠤⠴⠊⠀⠀⠀⠀⠈⠓⠤⠤⠤⠞⠁⠀⠀⠀⠀⠑⠦⠤⠠⠴⠊⠀⠀⠀⠀⠈⠳⠤⠤⠤⠞⠁⠀⠀⠀⠀⠑⠦

⠶⢦⡀⠀⠀⠀⠀⠀⣠⡤⠤⢤⣄⠀⠀⠀⠀⠀⢀⡴⠶⠶⢦⡀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⢀⡴
⠠⠀⠙⡆⠀⠀⠀⡞⢁⠀⠀⠀⡈⢳⠀⠀⠀⢰⠋⠴⠀⠀⠦⠙⡄⠀⠀⠀⡞⠁⠛⠛⠛⠈⢳⠀⠀⠀⢰⠋
⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠤⠀⠀⢸⠀⠀⠀⢰⠀⢢⣀⣀⡜⠀⡇⠀⠀⠀⡇⠈⠀⣀⠀⠁⢸⠀⠀⠀⢸
⣀⣠⠜⠁⠀⠀⠀⠙⢤⣀⣀⣀⡤⠋⠀⠀⠀⠈⠣⣄⣀⣀⣠⠜⠁⠀⠀⠀⠙⢤⣀⡀⣀⡤⠋⠀⠀⠀⠈⠣⣄
⡀
⣼⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠳⡀⠀⠀⢍⡷⠋⡀⠀⢀⠙⢾⡩⠀⠀⢠⠞⠁⠀⠀⠈⠳⡀⠀⠀⠀⡴⠋⠇⠩⠀⠙⢆⠀⠀⠀⢀⠞⠁
⣀⠀⠂⡇⠀⠀⠀⡇⠀⠁⣤⠈⠀⢸⠀⠀⠀⢸⠀⡁⠀⠀⢈⠀⡇⠀⠀⠀⡇⠈⢀⣀⡀⠁⢸⠀⠀⠀⢸⠐
⠀⢀⡰⠃⠀⠀⠀⠳⣀⠀⠉⠀⣀⠞⠀⠀⠀⠘⢆⡈⠉⠈⢁⡰⠃⠀⠀⠀⠳⣀⠀⠀⠀⣀⠞⠀⠀⠀⠘⢆⡀
⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠉⠑⢦⠀⠀⠀⠀⢠⠖⣿⣿⣿⠲⡄⠀⠀⠀⠀⡴⠋⠉⠉⠙⢦⠀⠀⠀⠀⣠⠖⠋⠉⠙⠲⣄⠀⠀⠀⠀⡴⠊
⠘⠀⠀⡇⠀⠀⠀⡇⢠⠀⠀⠀⡄⢸⠀⠀⠀⢸⠀⡉⠀⠀⢉⠀⡇⠀⠀⠀⡇⠐⠀⠀⠀⠂⢸⠀⠀⠀⢸
⠀⠀⢠⠇⠀⠀⠀⢧⠀⠀⠒⠀⠀⡼⠀⠀⠀⠸⡄⠙⠒⠒⠋⢀⠇⠀⠀⠀⢧⠀⠀⠉⠀⠀⡼⠀⠀⠀⠸⡄
⠔⠚⠁⠀⠀⠀⠀⠀⠑⠒⠤⠒⠊⠀⠀⠀⠀⠀⠈⠒⠢⠔⠚⠁⠀⠀⠀⠀⠀⠑⠒⠤⠒⠊⠀⠀⠀⠀⠀⠈⠓
⢢
⠒⠢⣄⠀⠀⠀⠀⢀⡤⠒⠲⢲⢤⡀⠀⠀⠀⠀⣠⠔⠒⠒⠢⣄⠀⠀⠀⠤⣀⡤⠒⠒⠒⢤⣀⠤⠀⠀⠀⣠⠔
⠀⠀⡈⡆⠀⠀⠀⡎⠠⠀⠑⠈⠄⢱⠀⠀⠀⠰⠁⠄⠀⠀⠠⠈⡆⠀⠀⠑⡏⠀⠆⠀⠰⠀⢹⠊⠀⠀⢰⢁
⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣟⣻⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⡿⠿⠿⠿⠿⠿⠿⣷⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠿⠛⠉⠉⠀⠀⠀⡀⠀⠠⠄⠀⢀⠀⠀⠈⠉⠙⠛⠿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠋⠁⠀⡀⠄⠒⠀⠀⠀⠀⠠⣤⠀⠠⠄⢦⡄⠀⠈⠀⠃⠠⠀⠀⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⠙⢿⣶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⢀⠄⠈⠀⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⠀⢀⣀⣋⣀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⢤⠀⠙⢿⣷⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠄⠀⠀⠀⠀⠀⢸⡀⠀⠀⣇⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠄⠀⠙⣿⣦
⠀⠀⠀⢠⣾⡟⠁⠠⠀⠀⠀⠀⠀⠀⠀⠼⠇⠀⠠⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⢿⣷⡀
⠀⠀⢠⣿⠏⠀⠦⠀⢠⣤⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠆⠀⢻⣷⡀
⠀⢀⣿⡏⠀⡠⠀⠀⢸⡏⠀⣸⡃⠀⢸⡇⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠀⢻⣷
⠀⣼⡟⠀⢠⠀⠀⠀⢸⣇⠀⢹⡇⠀⢸⠇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⡀⠈⣿⣇
⢰⣿⠃⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠚⠛⠀⠀⠀⢀⠀⢸⣿
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⣿⡇⠀⠀⠀⣠⠂⠘⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠃⠀⢿
⣿⡇⠀⠆⠀⠙⠶⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⣬⡀⠀⠀⠶⠀⢹
⣿⡇⠀⢀⠀⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣕⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⡄⠀⣿
⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠐⣿⣿⣿⡟⠁⢀⠀⠀⣿
⠸⣿⡄⠀⠀⠀⠀⠀⢀⡀⢀⡐⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣦⣤⣴⣶⣾⣿⣿⣿⣦⣈⠀⢸⣿
⠀⢻⣷⠀⠙⡀⠀⠀⠸⣆⣠⠃⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⡏
⠀⠈⢿⣇⠀⠐⠀⠀⣴⠉⠙⣷⣰⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⢀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣦⠀⠖⠀⠈⠓⠚⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠈⢿⣧⡀⠐⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠛⠛⠛⠉⠙⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠻⣷⣄⠀⠲⠀⠀⠀⠈⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡈⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠘⣿⡧
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠃⢀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⠠⣿⣿⣿⣿⣿⣿⣿⢠⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠁⢾⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⡇⠈⢻⡆⠀⠀⠀⠀⠀⠘⠂⠈⠁⢠⠀⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣼⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠓⠀⠚⠁⠀⠀⠀⢀⠀⠤⠐⠈⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣧⠁⠉⠀⠘⠂⠀⠉⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋⠁⣼⣿⣿⣿⢿⣿⣿⡏⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⣿⡿⣷⣶⣶⣶⣶⣾⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢸⣿⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣏⣻⣿⣦⣈⣏⣙⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣾⣿⣿⣿⣑⣾⣿⣛⣿⣁⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣀⣈⠐⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠉⠉⠙⠻⢷⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⡟⠀⠀⠀⠀⠀⢤⣤⣤⣄⣀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡘⠀⠀⠀⡰⢻⣆⣌⢛⣦⠀⠘⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⢰⣧⣶⣠⣦⣷⣿⣼⣿⣿⣿⠀⢈⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣿⣦⠀⢸⡏⠸⡏⠀⢹⣿⣿⡇⠐⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⡇⠀⡇⠀⢸⣿⣿⣷⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣅⣤⣡⣤⣬⣭⣭⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠋⠻⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠷⠒⠒⠛⠛⠛⠛⠿⠦⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠚⠉⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠠⠄⢀⡀⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠴⠊⠀⠀⠀⠀⣠⠔⠚⡛⠥⠐⢀⡀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⢐⠵⠁⠀⠀⠀⡀⠐⢉⠤⠂⠁⠐⠂⠀⠀⠀⠀⡶⠀⡷⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣀⡀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⢰⠊⠀⠀⠀⠀⠀⣠⡖⠡⠀⠐⣄⢳⡀⠀⠀⠀⠀⢠⡄⣿⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠙⢷⣝⢦⡀⠀⠀⠈⢿⣆
⠀⠀⠀⡰⠁⠀⠀⠀⣴⢃⡴⠃⠈⠀⠀⠀⠈⠳⣳⡀⠀⠀⠀⠘⠓⠛⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡙⢶⣿⣦⡀⠄⠈⢿⣆
⠀⠀⡴⠀⠀⠀⢀⡾⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠔⠈⠀⠈⢻⣿⣿⣶⣆⠀⢿⣆
⠀⢠⠃⠀⠀⠀⠞⠠⠀⠀⠒⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣡⣴⠿⠛⠀⠀⢻⣿⣿⣿⣶⠘⣿⡄
⠀⡄⠀⠀⠀⡘⢀⠁⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⣠⠖⠉⠀⠘⠋⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡄⢻⣧
⡀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠸⣿
⠃⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⣦⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⠀⣿⡇⢿⡇⠐⣿
⠀⠀⠀⠐⣼⡇⠰⠀⠬⠍⠉⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠛⠛⠿⢀⡀⣿⣟⢸⣷⡀⣿
⠀⠀⠀⠘⡎⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⡿⣌⠧⠀⣿
⢠⠀⠀⠀⣧⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⣸⣿⡇⢟⠁⢀⣿
⠈⡤⠀⠀⢹⡿⢣⠀⠀⢀⣄⢖⣫⢱⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠠⣄⠀⠀⠀⠀⠀⠀⣾⣿⣶⣤⣦⡀⠀⢠⣿⡻⠼⠞⠀⣸⡏
⠀⠐⢀⠀⠘⣷⡉⣧⡀⡈⠻⣋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠉⠛⠶⠙⠀⢠⡿⡷⢡⠚⠀⣸⡟
⠀⠀⠀⢂⠀⢻⣷⡹⢿⡇⠀⠋⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⢘⣧⡀⠀⢀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⣏⡟⣡⠎⠀⣰⡟⠁
⠀⠀⠀⠀⡣⡈⣿⣿⣌⠢⣾⣦⠀⠀⠀⢀⣤⠏⠗⠀⠀⠘⠃⠈⠛⠉⠀⠀⠀⠈⠀⠻⣄⠀⠀⠠⠄⣠⣾⣯⢏⣴⠁⠀⣰⠟
⠀⠀⠀⠀⠘⢷⠈⢿⣿⣷⣝⠿⣷⣄⠀⠈⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠂⢀⣤⣾⣯⠟⣡⠟⠁⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢂⠉⢿⣿⣿⣮⡝⢷⣦⢄⡀⡀⠀⠀⠀⠘⠂⠰⡆⡀⠀⠠⠀⡀⣠⡴⣞⠿⠟⢋⣴⡟⠉⠀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⢻⣿⣿⣿⣷⣮⣘⣤⠉⡀⠐⠒⠂⠤⣤⣤⣶⣶⣶⣿⣿⠿⢞⣋⣴⣶⡿⠗⣀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣭⣍⣉⣁⣀⣠⣤⣤⣶⣶⣿⢿⢛⣉⢤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣥⡴⠞⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠉⠙⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣀⡤⠖⠛⠉⠉⠀⠀⣄⢀⡀⠀⠉⠉⠛⠲⢤⣀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠚⠁⢀⢀⠀⠀⠀⠀⠀⠟⠰⠧⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⠀⠛⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢦⡀⠙⢷⡄
⠀⠀⠀⠀⣰⡿⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢿⡄
⠀⠀⠀⢠⡿⠁⢠⠏⢠⢠⣶⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢻⡄⠈⣿⡄
⠀⠀⠀⣾⠃⢀⡟⠀⠈⠀⠉⠁⠀⠀⠀⠀⠘⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢷⠀⠸⣷
⠀⠀⢰⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⡄
⠀⠀⢸⡇⠀⣾⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡇⠀⢹⡇
⠀⠀⢸⡇⠀⢻⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠾⠃⡇⠀⢸⡇
⠀⠀⠸⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠀⠀⠀⠀⠀⢰⡇⠀⣿⠃
⠀⠀⠀⢿⡆⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠘⣿⡀⠘⣧⠀⢘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡇⠀⣼⠃⢀⣿⠃
⠀⠀⠀⠀⠘⣷⡀⠘⢧⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢿⣄⠈⠳⣄⠀⠀⠀⣲⡖⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢀⣰⠟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⢦⣀⠙⠁⠀⠀⠀⠀⢼⣧⡀⠀⠀⠀⠀⠈⠟⠀⣀⡴⠛⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠉⠛⠶⢤⣄⣀⡀⠈⠉⠀⠀⢀⣀⣠⡤⠶⠛⠉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉

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

⠖⠀⠀⢀⡇⠀⠀⠀⠈⣇⠀⠀⠰⠤⠆⠀⠀⣸⠁⠀⠀⠀⢹⡀⠲⣄⣀⣀⡠⠖⢀⡏⠀⠀⠀⠈⣇⠀⠀⠀⠿⠀⠀⠀⣸⠁⠀⠀⠀⢸
⠀⢀⣠⠞⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⠳⣄⡀
⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠶⠼⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠷⠾⠷⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠷
⢀⡀⠈⠳⡀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⠀⠙⣆⠀⠀⠀⠀⢀⠞⢡⡄⠀⠀⢠⡈⢳⡀⠀⠀⠀⠀⣰⠋⠿⡿⠿⠿⠾⠙⣆⠀⠀⠀⠀⢀⠞⠁⢀
⠌⠁⠀⠀⡧⠀⠀⠀⢠⡇⠐⠃⠀⠀⠀⠘⠂⢸⡄⠀⠀⠀⣼⠀⢠⠀⠀⠀⠀⡄⠀⣧⠀⠀⠀⢠⡇⠠⠄⠀⠀⠀⠠⠄⢸⡄⠀⠀⠀⣸⠀⠀⠈
⠀⠀⠀⢠⠇⠀⠀⠀⠀⢧⠀⠀⠈⠛⠁⠀⠀⡼⠀⠀⠀⠀⠸⡄⠈⠳⠤⠤⠞⠁⢠⠇⠀⠀⠀⠀⢧⠀⠀⠀⠶⠀⠀⠀⡼⠀⠀⠀⠀⠸⡄
⣀⣠⠴⠋⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠙⠦⣄
⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠛⠒⢤⡀⠀⠀⠀⢠⠤⡄⣠⠔⠚⠛⠓⠲⣄⢠⠤⡀⠀⠀⠀⢀⡴⠒⠛⠛⠒⢦⡀⠀⠀⠀⠀⠀⠀⣠⢶⢺⢟⠓⠢⣄⠀⠀⠀⠀⠀⠀⢀⡤⠒
⠀⠀⠀⠹⡄⠀⠀⠘⠢⡟⠁⣠⡄⠀⢠⣄⠈⢿⠔⠃⠀⠀⢠⠏⢀⠀⠀⠀⠀⡀⠹⡄⠀⠀⠀⠀⡾⠁⣘⠢⠞⠁⣀⠈⢧⠀⠀⠀⠀⢠⠋
⣀⠀⠐⠀⡇⠀⠀⠀⠰⡇⠀⠀⠀⣤⠀⠀⠀⢸⠆⠀⠀⠀⢿⠀⣉⠀⠀⠀⠀⣉⠀⡷⠀⠀⠀⠰⡇⠀⠁⢀⠀⡀⠈⠀⢸⠆⠀⠀⠀⢼⠀⠂
⠁⠀⠀⣰⠃⠀⠀⠀⠀⢳⡀⠀⠀⠉⠀⠀⢀⡞⠀⠀⠀⠀⠘⣆⠈⠑⠒⠒⠚⠁⣰⠃⠀⠀⠀⠀⢳⡀⠀⠈⠉⠁⠀⢀⡞⠀⠀⠀⠀⠘⣆
⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠦

⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠉⠉⠳⣄⠀⠀⠀⠀⠀⢀⡴⣾⣿⣿⣿⣷⢦⡀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠉⠉⠳⣄⠀⠀⠀⠀⠀⢀⡄⠛⠉⠉⠙⠛⢤⡀⠀⠀⠀⠀⠀⣠⠞⠋
⠰⠆⠀⠘⡆⠀⠀⠀⠀⡞⠀⠉⠉⠉⠙⠉⠀⢻⠀⠀⠀⠀⣰⠁⠘⠓⠀⠀⠚⠃⠘⡆⠀⠀⠀⠀⡟⢀⡄⠀⠀⠀⢠⡀⢳⠀⠀⠀⠀⢸⠃⠀⠰
⠁⠀⠀⠀⡇⠀⠀⠀⠘⡇⠈⠁⠀⣀⠀⠈⠁⢸⠃⠀⠀⠀⢻⠀⠸⣄⠀⠀⣠⠇⠀⡟⠀⠀⠀⠘⡇⠀⠀⢠⠤⠄⠀⠀⢸⠃⠀⠀⠀⢹
⠀⠀⢀⡼⠁⠀⠀⠀⠀⠹⣄⠀⠀⠉⠀⠀⣠⠏⠀⠀⠀⠀⠈⢧⡀⠈⠉⠉⠁⢀⡼⠁⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠈⢧⡀
⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠖⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓
⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠙⢦⠀⠀⠀⠀⠀⢠⠞⠁⠠⣏⢃⡏⠳⡄⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠙⢦⠀⠀⠀⢰⡉⣧⠞⠁⠀⠀⠀⠈⠳⣼⢉⡇⠀⠀⠀⡴⠋
⠀⠀⢠⠈⡇⠀⠀⠀⢀⡇⠀⠒⠀⠈⠁⠒⠀⢹⡀⠀⠀⠀⣼⠀⠴⠀⠀⠀⠀⠦⠀⣇⠀⠀⠀⢈⡇⠀⠛⠃⠀⠘⠓⠀⢹⡁⠀⠀⠀⣸⠁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠳⠶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⢉⣠⣤⠶⠶⠞⢻⡿⠛⠶⠶⢦⣤⣈⠙⠳⣦⣄
⠀⠀⠀⠀⢀⡴⠟⢁⣤⠾⣯⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⢹⡿⢦⣄⠙⢷⣄
⠀⠀⠀⣴⠟⢁⣴⠟⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢷⣄⠙⢷⡀
⠀⢀⣾⠃⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠈⢻⡄
⠀⣼⠃⣰⠟⠻⠆⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⣷⠀⢻⡄
⢸⡏⢠⡟⠀⠀⠀⠀⠀⠀⠻⣇⡈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡀⠀⠸⣇⠈⣷
⣿⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⣻⡶⠶⣤⣠⣤⠶⠞⢛⣋⣥⣴⠾⠃⠀⠀⣿⠀⢹⡄
⣿⠀⣿⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠈⣿⡴⠶⠛⠋⠉⠀⠀⠀⠀⢠⣤⣾⡆⢸⡇
⣿⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣠⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⠇
⢹⡆⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠈⣷⠀⢻⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣿⠶⢦⣤⡀
⠀⠘⣧⠈⢻⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠶⢦⣌⠻⣦
⠀⠀⠈⢷⡀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢹⡆⢹
⠀⠀⠀⠀⠻⣦⡈⠻⢦⣄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⣸⠇⣸
⠀⠀⠀⠀⠀⠈⠙⢶⣄⡉⠛⠶⣦⣤⣀⣀⣸⣇⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣿⠶⠞⢋⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣀⣉⡉⠉⠉⢉⣉⣉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⢶⣶⣶⢶⣶⡶⢶⣶⡶⢖⡶⢲⡶⣶⡶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⠩⢗⠽⣫⠖⠵⡫⢞⡽⣫⠔⣩⠞⣩⠞⣩⡴⢛⡡⠞⣹⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢆⡙⠲⠤⣤⣀⣀⡈⠀⠁⠘⠁⠚⠡⠊⢠⠞⢁⠴⠋⢐⣋⣥⠴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡉⠀⡄⢀⡉⣉⠙⠛⠛⠛⠒⠲⠶⠶⠶⠖⠛⢋⢉⡁⠠⠘⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠳⠶⠦⣭⣬⣇⣀⣃⣀⣚⣀⣰⣀⣘⣣⣭⣤⡴⢶⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠰⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠒⠉⠙⠲⢤⣀⣀⣠⣤⠶⠶⠶⠶⣤⣄⠐⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠙⠯⣅⡀⠀⠀⠀⠀⠀⠙⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠈⠙⠛⠛⠛⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣄⡴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⢉⠔⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⠞⠁⠀⣠⠖⠒⢦⡀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⡞⣁⡤⢶⠶⠋⠀⠀⠀⠙⠳⢦⣀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠗⢈⡛⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠶⠟⠋⡡⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡈⠃⢴⠄⢈⡉⢉⠛⠛⠛⠛⠛⠛⡛⢉⡉⠡⡄⠲⠈⣁⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣬⣤⣉⣀⣁⣘⣃⣙⣀⣁⣀⣥⣤⡴⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠶⠆⢴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠦⣤⡆⠀⡐
⠀⠀⠀⣀⠴⢊⠉⠁⠀⠀⠈⠑⠒⠤⡀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣤⡤⠤⡀⡀⠈⠙⢦⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⢀⡼⢃⠤⠄⠶⠒⠂⡀⠆⢀⠀⠀⢸⣆⠀⠀⠀⠀⠸⡀⠐⠉⠀⠁⠐⠋⠀⠛⢆⠀⡎⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠸⠀⠑⠤⡀⠀⠀⠀⢙
⠀⢸⢰⠁⠀⠀⡀⠀⠀⠀⠐⠀⡑⠢⠄⢹⡄⠀⠀⠀⣀⡁⠀⠉⠀⠀⠀⠈⠉⠀⠀⣀⡀⠀⠀⠀⢺⠂⠀⢠⠊⠖⠋⠀⠀⠉⠉⠓⠎⢲⠄⠀⡨
⠀⢸⠌⠀⠀⠀⢀⠀⠀⠁⠀⠀⢨⢤⠲⣸⠇⠀⠀⢸⠀⡍⠀⠀⠀⠀⠀⡀⢀⠀⠀⡅⠛⠀⠀⠀⠈⠑⠦⢸⡆⡀⠀⠀⠀⠀⡀⠀⠀⠈⡆⢤⠁
⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣬⠜⠰⢹⡀⠀⠀⠀⠑⣦⠀⠀⡀⠀⠀⠈⠁⠀⢀⡗⠁⠀⠀⠀⠀⠀⡄⡾⠈⠁⠀⡀⠀⠐⠁⠀⠀⠀⢻⡘
⠐⠀⠐⠀⡆⠀⠀⠀⠀⠀⠀⢤⠐⢆⢇⠀⢱⠀⠀⠀⠀⢰⡀⠀⠑⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠘⠘⠂⠀⠀⠀⠢⠀⠇⡜⠀⠀⠆⠀⠀⠀⠀⠀⠑⢄⡀⠉⠉⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠒⠒⠒⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠠⠄
⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⢕⠢⡀⠀⠀⠀⠀⠀⠀⣤⢾⣭⣵⠦⠀⠀⠀⠀⢄
⢀⠎⢄⠁⠀⠀⠀⠀⢀⣤⠊⢀⠀⠀⠢⡘⡀⠀⠀⠠⠀⡀⠲⣀⣐⣤⣈⣦⣄⣢⡤⠄⠘⡄⢣⠹⡄⠀⠀⠀⠀⠪⠚⠉⠉⠀⠀⠀⢰⡀⡆⠀⢧
⣸⠀⠈⠀⠀⠀⠲⠀⠊⣀⣀⡀⠑⠀⠀⠀⠀⠀⠀⠀⡳⠗⠀⠈⠉⠁⠄⠀⠄⠀⢡⠀⠀⣿⠈⠀⡇⠀⠀⠀⠐⠆⡁⠀⠁⠐⠂⠀⠈⠇⠇⠀⡜
⢻⠀⠀⠀⠀⠀⢛⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠂⠀⠀⠈⠃⠀⠀⠀⠀⠧⠤⣽⠀⠀⡇⠀⠀⢠⠚⠦⡔⠂⠀⠀⠉⠀⡘⢠⠔⢄⠸
⠈⣆⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡂⠀⠀⠁⠀⠀⠀⢀⠂⠀⢠⠀⠀⠀⠀⠀⠀⠀⣚⡠⠃⠀⠸⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠀⠀⠘⣇⠌⠐
⠀⠘⠀⡠⠀⠀⠠⠐⠂⠀⠀⡰⢀⠆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢀⠃⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⢀⡇⠈⠁
⢦⣬⢄⠇⡜⠀⠀⠀⠀⠀⠀⡠⢺⣠⣘⢮⠶⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠐⡇⢀⠐⢄⣀⡀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⣠⠎
⠈⠉⠁⠉⠀⠀⠈⠐⠐⠂⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠈⠀⠀⠉⠛⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠙⢢⣀⣠⠀⠊
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣾⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠏⠀⠀⢠⣶⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠸⠋⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠛⠛⠛⠛⠛⠷⣦⡀⠀⠀⠀⠀⠛⠀⠀⠛⠀⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣼⣏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠒⠶⣾⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣿⣷⣾⣿⣿⣾⣭⣍⠛⣦⠰⣧⣤⣼⠇⢾⣁⣀⣴⣾⣿⣿⣿⣿⣿⣷⣄⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⡄⢀⡞⣽⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⣀⣿⣄⣀⣀⣈⠙⠻⢿⣿⣿⣿⣿⣿⣿⡟⣧⠀⢿⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⡿⠀⡾⠀⣿⣿⣿⣿⣿⡟⣉⣤⠶⣛⣫⣭⣭⣿⣿⣿⣿⣿⣷⣶⣿⡟⠻⣿⣿⣿⡇⢸⠄⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣇⠀⢿⣿⡿⢋⡽⠛⣩⣶⣿⠿⠛⠛⢉⣉⠉⠉⠛⠻⠿⣿⣿⣿⣷⣄⠉⠛⠧⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠻⣤⠞⠁⣰⠋⣠⣾⡿⢋⡅⠀⠀⠀⢸⣿⠀⠀⠀⠀⠖⠈⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣾⡿⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⡿⢡⡄⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⡄⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀⠀⠀⠀⠀⠀⠀⠴⠶⠀⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢿⣸⣿⡄⠛⠃⠀⠀⠀⠀⠀⠀⢨⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢗⣿⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢿⡀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡾⢸⣿⠁⢠⡏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⣇⠀⠹⣿⣿⣧⠀⡤⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⢡⣿⠏⢀⡾⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠙⣦⠀⠹⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣠⠞⣱⣿⠋⢠⠞⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢳⡀⠈⢻⣿⣿⣷⣤⣘⠃⠀⠀⠀⣿⠄⠀⠀⣀⣤⠞⣫⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢻⣿⣿⣿⣿⣶⣶⣶⣶⠶⣖⣛⣭⣴⡿⢿⣯⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⠋⠛⠻⠿⢿⣿⣿⠿⠿⠟⠛⠉⠀⠈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠶⠶⢶⠶⠶⠶⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠓⣀⣀⣀⣀⣨⣄⣀⣀⣀⡀⠋⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠓⣀⣤⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⢦⣄⡐⠋⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⠀⠈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠈⢡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣀⡀⠏⠀⢀⣤⢶⡖⠶⣄⠀⠀⠈⠻⣦⡉⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠆⣴⣿⣿⣿⣷⣴⠟⠁⢸⡟⠀⠈⠳⣄⠀⠀⠈⠳⣄⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢠⡟⠙⢀⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠘⢿⡙⠃⠀⠀⠀⠘⢷⡀⠀⢀⣹⣦⠘⠙⣧
⠀⠀⠀⠀⠀⢀⡟⠀⢀⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠐⠛⠁⠈⣻⠟⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⣽⠖⠋⠁⠘⣧⠀⠸⣇
⠀⠀⠀⠀⠀⣼⣧⠀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢠⡟⠁⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⢸⡆⢠⣿⡄
⠀⠀⠀⠀⢀⡏⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⡞⠁⠀⠀⢀⡴⠋⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠈⣇
⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣠⡟⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿
⠀⠀⠀⠀⢸⡷⠂⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠋⠁⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠲⣿
⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠉⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠶⠄⣴⠛⠉⠙⢶⡀⠴⠒⠀⠀⠀⠀⡿⠀⢠⡏
⠀⠀⠀⠀⠀⢹⡏⠀⢻⣿⣿⣿⣿⣿⡀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⠃⠈⣿⠁
⠀⠀⠀⠀⠀⠈⢷⠀⠈⢿⣿⣿⣿⣿⣷⣄⡀⠀⠀⢉⣹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠛⠁⠘⠳⠶⠶⠋⠀⠙⠓⠀⠀⢠⠏⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠘⢧⣤⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢷⠀⠀⠀⠀⣰⠏⢠⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⣀⡌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣄⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⣠⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣤⡖⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣤⠴⠶⠛⠉⠐⣦⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⡦⠀⠈⠉⠉⢉⠉⠉⠉⠀⠀⣆⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠶⠾⠶⠖⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠿⠿⠿⢿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠿⠛⠉⠀⠀⠀⣿⢸⠀⠀⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠞⠁⠀⠲⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠖⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠤⠤⠀⠀⠀⣠⣤⣤⣤⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⣎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠎⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⢀⡔⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣤⣀⡀⠀⠘⠃⠀⣀⣠⣴⣾⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁



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

⠀⠀⠀⠀⢀⠶⠀⠀⢀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⡀⠀⠀⡰⠆⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⠶
⠀⠀⠀⠀⠂⠈⠀⠈⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠐⠀⠁⠀⠁⠁⠀⠀⠀⠀⠀⠁⠘⠀⠀⠃⠀⠀⠁⠈⠀⠀⠂⠈⠀⠈
⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⢀⣤⣠⣄⠀⠀⠀⠀⢠⣠⡄⠀⠀⠀⢰⣦⣄⣄⠀⠀⠀⢀⣤⣄⣤⡄
⠀⠀⠀⠀⠀⠸⠯⠿⠀⠀⠀⠀⠀⠿⠹⠇⠀⠀⠀⠘⠻⠧⠏⠀⠀⠀⠀⠿⠹⠇⠀⠀⠀⠈⠻⠉⠿⠀⠀⠀⠘⠀⠿⠓⠛

⠀⠀⠀⠀⡠⣲⠯⠲⣦⡀⠀⠠⣒⠯⠓⢦⡀⠀⢀⢖⡺⠕⢶⣆⠀⢀⣴⡺⠝⠶⣄⠀⠀⣴⣲⠽⠲⣆⠀⠀⡤⣶⠭⠲⣦⡀
⠀⠀⠀⠀⠏⠀⠀⠀⠘⠃⠀⠛⠀⠀⠀⠈⠋⠀⠸⠁⠀⠀⠀⠛⠀⠘⠃⠀⠀⠀⠹⠀⠀⠋⠀⠀⠀⠑⠃⠀⠋⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⢀⡄⣠⠀⠀⠀⠀⠀⣄⢠⠀⠀⠀⠀⢀⣄⢠⣄⠀⠀⠀⠀⢠⢠⡄⠀⠀⠀⢰⣤⡀⣄⠀⠀⠀⢀⣤⡄⣤⡄
⠀⠀⠀⠀⠀⠸⠇⠿⠀⠀⠀⠀⠀⠧⠸⠇⠀⠀⠀⠘⠿⠸⠏⠀⠀⠀⠀⠿⠸⠇⠀⠀⠀⠈⠻⠆⠿⠀⠀⠀⠘⠁⠇⠟⠛
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢰⣿⣿⠷⠀⠀⠀⢀⣾⣿⠷⠄⠀⠀⠀⣾⣿⠿⠆⠀⠀⠀⣼⣿⡿⠦⠀⠀⠀⣰⣿⡿⠧⠀⠀⠀⠀⠀⠀⠘⢿⣿⡆
⠀⠀⠀⠀⠈⠐⠀⠐⠀⠀⠀⠈⠀⠂⠐⠀⠀⠀⠠⠁⠂⠀⠂⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠁⠐⠀⠂⠀⠀⠀⠀⠐⠀⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢄⣐⡀⠀⠀⠀⠀⢄⣐⡀
⠀⠀⠀⠀⠀⠀⠯⠯⠀⠀⠀⠀⠀⠿⠹⠀⠀⠀⠀⠀⠯⠩⠇⠀⠀⠀⠀⠸⠽⠅⠀⠀⠀⠀⠰⠻⠇⠀⠀⠀⠠⠈⠟⠯⠤

⠀⠀⠀⠀⣴⣦⣴⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢀⣄⠀⠀⠀⠀⠀⠀⣀⠀⣤⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⢠⣤⡀⣠⣄
⠀⠀⠀⢠⣾⣥⣬⡟⠁⠀⠀⠀⠀⠀⣀⠀⣴⡌⢰⣶⠀⣀⠀⠀⠀⠀⠀⣤⣼⣷⣬⣯⠄⠀⠀⠀⠀⠀⠈⢻⣿⣄⣠⣿⡞⠁⠀⠀⠀⠀⠀⣨⣟⠁⠙⣿⡄
⠀⠀⠀⢸⣷⡟⣿⡇⠀⠀⠀⠀⠀⠀⠙⠷⣿⣦⢲⣷⡞⠋⠀⠀⠀⠀⢺⣿⣾⣍⣩⣿⣽⡅⠀⠀⠀⠀⠀⢸⣿⠈⠁⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣄⣻⣟⣿
⠀⠀⠀⠀⢨⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠸⠿⠃⠀⠀⠀⠀⠀⠘⠿⠻⠿⠻⠟⠻⠃⠀⠀⠀⠀⠀⠸⠛⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠿⠗⠈⠱⠻⠗


⠀⠀⠀⠀⣠⣤⢠⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⡄⣠⣤⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⢀⣤⡀
⠀⠀⠀⢀⣽⣏⣤⣿⣥⠀⠀⠀⠀⠀⠀⣈⣿⡁⣙⡟⣁⠀⠀⠀⠀⠀⠀⣿⠇⣹⣟⡘⢿⠃⠀⠀⠀⠀⠀⠸⣿⡟⠰⣿⡷⠀⠀⠀⠀⠀⠀⠈⣻⣟⠘⢿⠟
⠀⠀⠠⣿⣹⣿⣸⣿⣿⡇⠀⠀⠀⠀⠀⣿⡿⠉⠉⢿⡏⠀⠀⠀⠀⠀⢸⡿⣿⣋⠉⣱⢿⣷⠀⠀⠀⠀⣶⣿⣷⠔⡒⢼⣷⣦⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠈⠻⠿⠸⠿⠟⠁⠀⠀⠀⠀⠀⠾⠛⠖⠾⠛⠅⠀⠀⠀⠀⠀⠘⠛⠈⠉⠉⠁⠛⠋⠀⠀⠀⠀⠛⠉⠋⠀⠀⠈⠉⠋⠀⠀⠀⠀⠘⠻⠷⠞⠋⠘⠛⠁


⠀⠀⠀⠀⣾⣿⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡶⢶⣄⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄
⠀⠀⠀⢀⣼⣯⡜⢿⠃⠀⠀⠀⠀⠀⠀⠀⣀⣽⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣟⣳⣤⠟⠀⠀⠀⠀⠀⠀⢴⣬⣭⣙⣡⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⢯⣄
⠀⠀⠀⢸⣿⣻⣷⣻⡇⠀⠀⠀⠀⠀⠀⣀⠋⠋⠙⠙⣀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⢾⣯⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⢸⣏⠉⢤⣿⡆
⠀⠀⠀⠘⣿⣿⡾⠙⠃⠀⠀⠀⠀⠀⠀⠙⠡⡀⢀⠌⠋⠀⠀⠀⠀⠀⠀⠙⢿⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠰⣶⢠⣦⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢠⣶⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡦
⠀⠀⠀⣰⣾⣿⣶⣷⡆⠀⠀⠀⠀⠀⢠⣽⣧⡀⢀⣹⣯⡀⠀⠀⠀⠀⠀⣨⣿⡍⢿⡿⣤⠀⠀⠀⠀⠀⠀⣠⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣷⣆
⠀⠀⠀⣿⣿⣿⣛⢿⠇⠀⠀⠀⠀⠀⢸⣿⣿⢷⡾⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⡧⣿⣿⡦⠀⠀⠀⠀⠀⠸⣿⣭⣟⠢⢶⣠⡀⠀⠀⠀⠀⠀⠀⢘⣿⣟⣿
⠀⠀⠀⠻⠿⠿⠪⠿⠆⠀⠀⠀⠀⠀⠈⠛⠛⠈⠀⠛⠛⠁⠀⠀⠀⠀⠀⠚⠛⠓⠚⠋⠃⠀⠀⠀⠀⠀⠶⠿⠇⠛⠀⠚⠛⠓⠀⠀⠀⠀⠀⠀⢿⣿⣿⡯


⠀⠀⠀⢀⡀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠐⣤⡂⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣤
⠀⠀⠀⣿⡿⢿⠿⢀⣹⠛⠀⠀⠀⠀⢀⡙⣿⡃⣈⣛⢣⡄⠀⠀⠀⠘⢦⣽⣛⠀⠠⣛⣿⡴⠋⠀⠀⠀⠀⣿⠀⢠⡄⣭⠀⠀⠀⠀⠀⠀⢀⣶⣏⠀⣀⣼⣷⡄
⠀⠀⠰⣷⣶⣾⣿⡟⠋⠀⠀⠀⠀⠀⢸⣿⣿⡿⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⢷⡼⢿⣿⠀⠀⠀⠀⠀⢰⣿⢀⠚⠂⣿⠀⠀⠀⠀⠀⠀⢸⣷⢿⠉⣼⠟⠝⠁
⠀⠀⠀⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠁⠈⠒⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠉⠉⠁⠉⠀⠉⠉⠉⠉⠉⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿
⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⡈⣳⠶⠚⠛⠿⢷⣤⠀⡀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠇⡻⣿⡴⠛⠙⢿⡟⣿⣗⣧⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡟⢧⠀⣀⠄⠀⠈⠹⢦⡀⠀⠀⠀⠀⠀⠀⠣⢗⠙⢶⣀⣠⢈⠁⠛⣹⣿⡿⠀⠀⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠓⠀⠳⣄⠀⠀⠂⠀⢀⣱⡴⣶⣄⣀⢀⣀⠀⠀⠳⡈⢭⡀⠀⠀⠘⠷⠛⣥⡎⠙⡷⢎⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠙⠓⢤⡈⢀⣙⡛⠛⣦⡴⢭⣿⣶⡢⠎⠀⣿⡘⣍⠃⠀⠀⠀⠀⠈⠀⣴⠻⣆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣭⣤⣭⣡⣄⣴⠦⠀⠸⠶⣚⢣⣌⠧⠘⡇⣸⣦⠤⠤⣒⣾⠋⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⣤⡘⣦⣧⣬⣁⣠⠖⡚⠛⣿⣿⣷⡴⣿⣿⣶⠞⠋⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣧⣿⢟⣿⣿⢻⣿⠿⣷⠾⣯⣝⠳⠯⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠫⢹⡿⢸⣿⡟⢳⡈⣷⠓⠶⠷⣎⠈⣿⣵⣏⠀⠀⢀⡤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⣟⣷⢤⣵⣌⠓⢽⣛⣟⡛⠁⠰⢿⡆⠀⣼⡅⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⣶⣭⠳⠷⣦⠉⣉⣉⣙⠛⠻⠷⡄⠀⠙⢿⢿⣡⡄⠁
⠀⠀⠀⠀⠀⢀⣤⣶⢶⡀⠀⠀⠀⠋⠀⠀⠀⢺⣿⣿⣷⣚⢛⣋⣡⣿⣿⣷⣤⡀⠲⣄⢀⠀⢿⢿⡇
⠀⠀⠀⠀⡀⠀⣱⣿⡟⢻⡄⠀⠀⡄⠀⢰⣾⣿⣿⣿⣿⣿⣟⠉⠛⠷⠿⣯⣿⣧⠀⢘⡇⢷⣌⠿⡃
⠀⠀⠀⠰⠃⣼⡟⠁⠈⠀⡇⠀⠀⠹⣆⠘⢛⣿⠿⠋⢸⡇⠉⢳⡀⠐⣄⠀⠳⠉⡆⢻⡱⠆⢿⠀⠀⠀⠶
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠞⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⡸⢁⡜⡀⢷⠀⢸⡆⠀⠀⣇⢸⡇⠁⢸⣶⣦⣿⠛
⠀⠀⠀⠀⠘⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣢⡾⠉⠀⣼⣷⢂⣾⠃⠈⠁⣎⣴⣿⡀⢠⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⢀⢴⠿⠛⠉⠠⣔⣻⣷⣿⠟⠀⠀⠀⣼⠻⣿⢧⣵⠋⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠳⣄⠈⠀⠀⠐⢲⣾⣦⣾⠞⡡⢔⢋⠉⣁⣭⡥⠄⠂⡠⠚⠋⠀⢧⡈⢧⡽⠂⠠⠉⠁⡠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣀⣩⣤⠁⣮⠔⣩⣾⡿⠛⠁⣤⣰⢋⡴⠁⠀⠀⠈⢷⡌⢀⠀⠀⠀⠀⠀⢀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠇⢰⣿⡜⣱⡏⠀⠰⢶⣥⢇⠸⣠⠀⠀⠀⠀⠀⠙⠈⠑⢒⡛⠀⢈⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⣿⠂⠀⠀⢸⣿⣷⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⢀⡀⠀⢘⠟⠛⠒⠈
⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⢷⡄⣬⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠨⠿⠶⠗⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿
⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠉⣀⡴⠐⡖⠮⣄⡀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠶⡿⠛⢀⣀⣉⣀⡘⠫⡓⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠊⡁⠀⠀⠀⣩⠶⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠙⢦⣤⠞⠉⠁⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡀⠀⠀⠈⠉⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠑⢄⠀⠀⠀⠀⢂⠄⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⣦⣌⡁⣀⡉⢄⣶⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉⠁⠀⠀⠑⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⡿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡿⠟⠋⠉⠄⠘⠀⣬⣠⣧⢠⣅⠈⠉⠛⠻⡿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⠟⢉⠘⣂⠀⣀⠀⠀⠀⢸⡿⣿⢠⡿⠀⠀⠀⠀⢠⡝⠙⠿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⢟⠁⠂⠀⠀⣿⠀⣿⠀⠀⠀⢸⣿⣿⣤⣤⠀⠀⠀⠀⢸⡇⠀⠀⠉⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⣻⡆⣿⠀⠀⠀⠈⠉⣿⠈⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠸⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠐⠖⣶⢠⡖⢦⡀⠉⠁⠉⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣾⠉⣷⡞⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⠁⠀⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣔⠋⠀⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣯⣻⣽⣿⣿⠃⡈⠀⠀⠿⠙⠇⠼⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠟⠃⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠠⢀⣤⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⣶⠄⢿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠠⠼⣿⢀⣿⡄⠒⠲⠶⠶⠶⠶⠶⠶⠶⠾⣿⣿⣷⣶⡶⠶⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⣷⡄⢿⣿⣿⡗
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠠⠠⣞⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⠿⠃⢺⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡆⠂⠈⠉⢀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠈⣿⣿⣿⠇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⠈⣀⠀⢿⣤⣤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣀⠀⢀⣹⣿⣿⡟
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⢁⡤⡎⠉⢹⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡉⠠⣤⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⡠⠎⠁⠉⠀⠀⢰⣶⣶⡶⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠿⠿⠟⠀⠀⠀⠀⠀⢢⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣆⠁⡀⠀⠀⠀⣠⠞⠀⠀⠀⢀⣾⣁⣿⠀⠀⠀⢀⡥⠈⣷⠀⠀⠀⠀⣱⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⡐⠀⡤⠻⠿⠇⠀⠀⠸⣿⠀⢹⡇⠀⠀⠘⠓⠐⠫⡄⢀⣴⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⡀⠐⠀⠄⢀⡙⢲⠛⡁⠀⠄⠐⢀⣡⣴⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣶⣶⣶⣶⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣩⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠀⠀⠀⣤⡴⠶⠔⠿⠃⡀⢠⢠⡄⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣀⡀⠀⢠⡄⠀⢦⣠⣆⠀⠀⠰⠁⠀⠀⠶⠋⢁⡀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠸⠟⡃⠄⢰⣶⠿⢈⣉⣭⣤⣤⣦⣴⣤⣿⣧⣀⡺⠻⣦⣤⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡈⠁⢀⣠⣴⣾⠛⠛⣻⣿⣿⣿⣿⠛⠛⠛⣻⣿⣿⣦⣄⡀⢠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠶⠀⠈⣁⣿⣿⣿⣿⣧⣄⡨⣽⣿⣿⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⣠⡦⢀⣶⣿⣿⣿⣿⣿⣹⣿⡇⠋⠈⠛⠛⠋⠙⠛⠻⣿⣿⣼⣿⣿⣿⣿⣿⣷⣠⣬⠁⠀⠄⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠿⢀⣾⣿⣿⣿⣿⣿⣿⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢹⣿⣿⣿⣿⣿⣿⣿⡄⢘⡁⠀⠀⢾⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠘⠀⠀⠺⢦⣾⣿⣿⣿⣿⣿⣿⣿⢸⣟⠠⣠⣤⣤⣤⣤⣤⣤⣤⣼⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣞⣁⡀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡄⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠀⠙⢿⣿⣿⣿⣿⡿⣽⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠁⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡻⠇⠈⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣷⡄⠀⠙⢿⠿⠉⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⡎⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡹⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣙⣿⣿⠀⠀⠈⠀⠀⢠⣿⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⢠⠆⠰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠛⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣿⣿⣿⣷⠀⠀⠀⠠⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠟⠁⠀⠀⠀⠀⠈⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⣟⣈⡁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣞⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⢋⡈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠛⢶⠄⠘⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇⠀⠀⢀⣴⣦⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⢋⡄⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠐⢤⡄⠈⢻⣿⣿⣿⣿⣿⢿⣿⣧⣀⣴⣿⣿⡿⢿⣦⣠⣿⣿⢿⣿⣿⣿⣿⣿⡿⠋⢘⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡄⠙⠿⣿⣿⣿⣾⠟⠿⢧⣶⣿⣿⣷⠒⠿⠿⢿⣿⣾⣿⣿⣿⡿⠋⣄⡀⢊⡙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠟⢴⣌⣻⢿⣿⣤⣈⣿⣿⣿⣿⣿⣭⣥⣤⣿⣿⣿⡿⠟⠉⠠⣶⡈⡁⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⢨⡿⠻⢂⣤⠉⡙⢻⢻⠿⠿⡿⡿⠟⠛⣛⠉⢡⡄⢀⠛⠀⠂⢀⣄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⢰⣶⡄⠻⠆⠛⢃⣈⡁⠀⠀⣿⣿⠶⣿⠁⠈⠁⣿⠁⠛⠀⠙⠋
⠀⠀⠀⠀⠀⠐⠀⢱⡜⢂⣀⣀⣠⣄⠱⣆⠀⣄⠀⣰⠃⠀⡙⠁⠾⠂⠠⠚⠛⠋⢰⣿⠀⠁⢴⡷⠀⠀⠀⠀⠀⠄
⠀⠀⢀⠀⠀⠀⢀⣡⣼⣿⣿⣿⡏⣩⣦⣦⣶⣿⣿⡟⠀⠢⠉⠀⠀⠀⠀⠀⡀⠀⠀⠁⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠠⠶⠶⠆⠀⣀⡀⢠⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢇⣠⣤⢠⣬⡈⢿⣏⣸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢣⣿⣿⣿⢻⣿⣿⣎⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣶⣿⣿⣬⣿⣷⡆⣘⣸⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡙⢿⣿⣹⢛⣉⡑⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⢀⣠⡀⣿⣿⣇⠉⢹⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣿⣿⡟⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⡏⢸⣿⣷⢸⣿⣷⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⣿⡇⢸⣿⣿⢸⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣷⣾⣿⣿⣾⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣻⣿⣿⣿⣿⣿⣿⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇





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

⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠠⠆⠀⠛
⠀⠀⠀⠠⢀⡃⠈⠱⣄⢀⠀⡀⠁⢀⡀⠀⠀⠫⣀⢄⠀⡑⠤
⠀⠀⠀⠀⠉⠙⠀⠀⢬⠀⠣⣅⠑⠈⣧⠐⣇⢱⠈⣹⡎⡗⢀⡀
⠀⠀⠀⠀⢸⣄⡑⢲⡀⠒⢲⡘⡎⠀⢳⣬⣼⣿⣿⣿⣿⣿⣣⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠆⢭⡓⢦⣳⠈⠻⣦⣸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠢⡀⢀⠉⢻⠛⢷⣇⣼⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠂⠀⠠
⠀⠀⠀⠉⠄⠁⠂⠘⢾⣤⠈⠒⣼⣿⣿⣿⣿⡇⠉⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢉⠀⢂⡴⡀⡀⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⠀⠀⠀⠙⣿⣦⣾⣿⣿⣿⣿⣿⣿⣆⠁⡻⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣅⣈⠞⡟⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⢆⡈⠈⠀⠰⢤⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⠠⢤⠉⠻⣿⣧⣀⠀⡀⠀⠀⠀⢰⣿⣿⣿⠞⢠⠖⠁⠁
⠀⠀⠀⠘⠘⢆⠉⠢⠈⡳⣼⣿⣿⣿⣿⠃⠰⡄⢽⣿⣷⣤⡀⠀⠈⠛⢿⣷⣷⣄⡀⠀⠿⣿⣿⡿⠖⡃⢀⠀⣠⠌⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠈⠉⠛⢦⣼⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠙⢿⢿⣧⣀⣀⣀⣈⠙⠻⢿⣿⣾⣿⣿⣶⠦⣀⣀⣰⠁⠀⠀⠀⠀⢂⠑⠂
⠀⠀⠀⠡⠃⢆⡀⢀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣯⣍⠥⠄⢻⣿⣿⣿⣿⡯⠔⠈⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠨⠗⢿⣿⣿⣿⣿⣿⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠳⠂⠈⢿⣿⣿⣿⡔⠒⠃⢀⠀⠂⡠⢂⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⣿⣿⣯⡴⠶⠤⠤⠖⠋⠁⠁⠈⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹⣿⣿⣿⣧⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣿⡇⠀⡄⠀⠀⠀⠐⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣆⠀⠈⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⣿⣿⣿⡇⣣⠄⠈⠐⡡⠊⠃⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⢤⡀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⡿⠁⢀⣰⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣿⣿⣿⣿⠟⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠉⠉⣄⡴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⠿⣿⣿⣿⣦⡶⠯⡤⠴⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⣿⣿⣿⣿⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡜⣿⣿⣀⡤⠤⡶
⠀⠀⠀⣠⣤⣤⣽⣿⣿⣿⣿⣿⣿⣧⣦⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣻⣿⣏⣸⠚⠁⠀⠀⢂
⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢉⣿⣿⣿⣦⣄⣢⡀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣥⣄⣀⡐⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠟⠋⠉⠉⠉⠉⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠃⠀⠀⡀⠀⠐⢶⣶⡆⠤⣄⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⢄⠀⢀⢸⣹⡇⣶⣬⣇⠀⢨⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⢂⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀⠀⣿⠛⣿⠛⢻⣿⣿⡏⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣻⠀⣇⠀⢸⣿⣿⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⢀⡋⣀⣐⣛⣟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⠻⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠤⠐⠶⠖⠾⠷⠷⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄
⠀⠀⠀⠀⠀⢀⢤⡶⠂⠀⠀⠀⢀⠤⠒⡉⠍⠄⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣧⣤⣀⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⢀⠀⠀⠀⠀⠀⠸⠀⢿⠀⠀⠀⠀⠀⠀⡀⠀⠛⢿⣇⢄⡀⠀⠀⠙⢿⣄
⠀⠀⠀⢠⠟⠀⠀⠀⣠⠀⡠⢊⠐⠀⠈⢦⡳⡀⠀⠀⠀⠀⣆⣸⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⢷⣕⣦⠀⠀⠈⢻⣦
⠀⠀⢀⠋⠀⠀⠀⣴⢃⠞⠁⠀⠀⠀⠀⠀⠑⡷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣄⡀⠈⢻⣧
⠀⠰⠏⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⣤⠀⠘⣿⣿⣿⣷⡈⢿⣆
⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⡴⠟⠋⠀⠀⠀⠘⣿⣿⣿⡷⠸⣿
⠈⠃⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠀⣻
⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⠁⢸
⠀⠀⠀⠄⣿⠀⠄⢠⣒⡒⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿⢷⠀⠀⣿⡇⢹⣇⢨
⠀⠀⠀⢻⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣟⡘⠇⢸
⢠⠀⠀⠈⣮⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣸⣿⠷⡏⠀⢸
⠀⡆⠀⠀⢿⠟⡄⠀⠀⣤⡲⡫⢵⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠒⠄⠀⠀⠀⠀⠀⠰⢿⣷⣦⢦⡄⠀⢠⣿⡿⠾⠋⢀⣾
⠀⠈⠀⠀⢹⣧⢹⣆⣀⠙⣫⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠒⠙⠁⣠⡟⡟⢡⠋⢀⣾⠃
⠀⠀⠀⢀⠀⣻⣦⡛⡗⢀⠉⠀⠀⠀⠀⢀⣾⡃⠀⠀⣀⠀⣷⣤⠀⠠⠄⢀⢠⡀⠀⠀⠀⠀⠀⠀⣴⣏⠟⡴⠃⠀⣼⠋
⠀⠀⠀⠀⣦⠘⢿⣷⣌⢻⣷⣄⠂⠀⠠⡾⢉⠁⠀⠀⠉⠀⠉⠁⠀⠀⠀⡀⠀⠙⢆⠀⠀⢚⣤⣿⡷⣣⡾⠀⢀⡜⠁
⠀⠀⠀⠀⠈⠃⢈⠻⣿⣷⣍⡻⣷⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠈⠀⠀⠀⠀⣠⣼⣿⠿⣫⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣼⣿⣿⣿⣧⣝⡻⠳⠲⠤⠀⣀⡀⠀⠀⠅⣀⣀⣀⣤⣴⣖⣭⠮⣋⣡⣾⠛⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢻⣿⣿⣿⣿⣾⣯⣤⣄⣀⡀⠀⠽⠛⠛⠛⣛⣛⣋⣩⣶⣾⠿⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡵⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣶⣪⠃⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡉⢁⣀⣀⣼⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠭⣅⢀⣴⠾⣧⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡜⢡⣴⡄⠛⢸⡟⢀⡾⠀⠀⢩⢿⣿⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡷⢾⡟⠀⠀⣠⠝⠏⠓⠲⣾⡗⠚⠀⢹⣦⡀⣠⠖⢉⣠⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⠛⣿⣟⣷⣌⠁⣠⡴⡆⠀⠀⠙⣦⣶⣻⠟⣷⡏⠀⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⢾⣿⢱⣿⣏⣹⠋⠶⠽⠞⠁⠀⠀⠀⠉⢧⠀⣀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡈⣋⣛⣣⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⢿⡏⠃⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠻⣟⢋⡟⢠⡮⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⡿⡀⠈⠳⢦⣤⣀⣀⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠃⢸⣸⠇⠀⠁⠀⠀⠀⠀⠀⠀⢠⡶⢀⣻⢯⣤⠿⢇⠁⠀⠀⢀⠀⡠⢻⣦⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⠦⡀⠀⠀⠀⠀⠀⠀⠈⢙⠟⣫⣻⣶⡖⣠⣶⡿⠿⠀⠐⠣⢾⡛⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣦⡈⠀⠀⠀⠀⠀⠀⠀⢸⡀⠛⢛⣁⡉⠟⣇⠤⠤⠶⠖⠚⠛⢉⣩⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠋⠑⠶⠉⠙⠒⠿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠙⠻⢯⣹⠿⠃⠛⠛⠀⠀⠀⠀⠀⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢀⡄⠀⠇⠀⠀⠀⠀⠘⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠿⠞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠘⠞⠀⠀⣶⣆⡴⠾⠗⠉⠈⠀⡅⠀⠀⠀⠈⠣⡀⠀⠀⠀⣠⣤⠀⢱⣤⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⢴⠟⢿⠀⠀⠀⢠⡄⢠⠃⢀⠃⠀⢴⠶⢷⠛⠀⠈⠉⠉⢿⡟⣃⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⣡⠃⠀⠀⠀⠀⠀⠂⢈⢣⠀⠀⠸⢻⣧⣿⣙⣛⡤⠀⠀⠀⠀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣧⡀⠻⠀⢠⠀⠀⡀⠈⠀⣣⠞⠁⢀⣼⡀⠠⡀⠀⡷⢸⠠⢀⣠⠞⠋⢀⡟⢉⣽⣋⠳⡆⠀⠘⠿⢇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠳⣄⣀⠀⠀⢀⣣⣴⡾⡁⢀⣴⣿⣿⣿⣶⡄⣷⣾⡏⠸⣇⣶⠄⢤⣶⣿⣟⣻⣿⣿⣥⠤⡴⠚⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣀⠈⢩⣛⡋⠯⢿⣽⣡⣾⣿⣿⣿⣿⣿⣬⠧⣿⡿⡱⠁⠘⠿⣟⠉⡺⢿⣤⣀⣄⣄⡀⣾⢺⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⣰⠿⡷⣹⠿⣧⣼⠟⠋⠉⠉⠀⠉⣈⠁⣌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠓⠊⠁⠀⢸⣏⠀⠀⠀⠀⣴⢋⣀⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣷⣌⣋⡽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣈⠓⣠⠏⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣝⣧⡶⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠁

⠀⠀⠀⢀⣀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⡿⠿⠿⠟⠛⠛⠛⠉⠁⠀⠒⠶⠆⠲⠄⢤⡄⠀⠀⡀
⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣏⣀⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⣤
⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣧⠀⠀⠉⠉⠙⠙⠟⠁⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢉⡹⠖⢛
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣴⣶⣶⣦⣤⣶⣶⣦⣤⣤⣄⣤⣄⣠⡴⠤⠶⠶⠖⠐⠒⠃⠒⠀⠐⠂⠀⢉⣁⣠⣴⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣯⣽⣿⣭⣤⣍⣁⣀⣁⣀⣀⣈⣀⣀⣄⣤⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⠿⠟⠛⢹⣿
⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⢿⢻⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⣿⠿⠿⠿⠛⠛⠙⠋⠉⠉⠉⠉⠁⠀⠀⢀⣀⣀⣤⣴⣿⡿⠁
⠀⠈⣿⣿⣿⣿⣧⠀⠉⠉⠉⠉⠋⠉⠛⠛⠿⠾⠿⠿⠿⠷⠾⠿⠷⠿⠿⠷⠿⠄⠄⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣿⣿⠟⠉⢃
⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣿⣧⣤⢀⣿⡇
⠀⠸⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢠⣿⣿⣿⡿⣏
⠀⠀⣿⣿⣧⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⣿⣿⡟⠁⣀⡀
⠀⠠⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣶⣻⠇
⠀⢰⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡶⢶⠦⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣾⡟
⠀⠈⣿⡏⣤⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⡇
⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⡀⠀⠀⠀⡴⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣽⡏
⠀⢠⣿⣿⣅⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⠉⠃
⠀⢸⣿⣿⠟⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⡂⣄
⠀⢸⣯⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢟⣽⠃
⠀⢸⣿⣏⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⣟
⠀⠸⣿⣇⢲⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⣼⣿
⠀⢠⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣝⠃
⠀⠘⣿⣿⡿⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿
⠀⠀⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿
⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⠁⠀⠀⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣧
⠀⢺⣿⣿⠏⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠛⠛⠃⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣋⣽⠃
⠀⠀⣿⣿⠞⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠋
⠀⢸⣿⣧⡾⣿⠀⠀⠂⠀⠀⠀⠀⣠⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣁⠀⢀
⠀⢸⣿⠋⣱⣿⡆⠰⠀⠀⠀⣠⣾⠟⠛⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡆
⠀⢸⣿⡿⣻⣿⠁⠀⠀⢀⣾⣟⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁
⠀⣼⣿⣿⣿⣿⠀⡀⢀⣾⣿⣿⠛⠇⠺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠈⢿⣿⣿⣟⡟
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢻⣷⣤⣤⣀⡀⣸⣿⣿⣿⣿
⢀⣼⣿⡟⢉⣿⣿⣿⣿⣿⣿⣯⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣤⠴⠾⢿⣯⣿⢿⣿⣿⣤⣴⢆⡀⢭⣬⣷⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣹⣿⣻⣿⣿⣾⣿⣿⣯⣯⣿⣿⣿⣷⣿⣭⣮⣽⣽⣿⢶⣾⡿⣾⡿⠿⠫⠛⢦⡙⢿⣻⣿⣾⣿⣾⣿⣿⣿⣿
⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠍⢹⣿⠛⣿⣿⠛⠟⠙⢋⠋⠑⠈⢈⣡⣡⣤⣾⡷⠶⠚⠛⢛⣉⣥⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣽⣿⣿⣿⣿⣿⣿⣿⡟⠚⣿⡿⡟⠛⣟⠛⠋⣙⣛⣁⣤⣡⣬⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠟⠛⠋⠋⢉⣵⣾⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣛⣀⣬⣤⣴⠶⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣀⣀⣍⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠁⠀⠀⢈⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⠤⣖⣲⡭⢿⣛⣋⣁⣀⣠⡶⠦⣤⠀⠈⢙⣁⣀⣀⣀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⣰⠀⢸⣶⡭⠷⢚⣻⠿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠓⠿⣄⡀⠸⣏⢆⠀⠉⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⡴⠞⠉⢀⡠⠤⣒⣺⣭⡭⢭⣭⣙⣒⠤⣄⠀⠙⠳⣜⢯⣣⡀⠀⠈⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢁⡴⠋⠀⡠⣖⡽⠞⡋⠁⠀⠀⡆⠀⢀⡆⢭⡛⢦⣑⢄⠀⠈⢿⣷⡵⡄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡠⣪⠞⡉⠀⠀⠛⠀⠀⠀⠀⠀⣨⡕⠾⠃⠀⠙⢦⡳⡀⠀⢻⡽⣞⢆⠀⣼⠰⡆
⠀⠀⠀⠀⠀⠀⢠⡏⠀⡜⡴⢃⣺⣿⠂⠀⠀⠀⠀⠀⠀⣼⢫⠇⠀⠀⠀⢠⠌⢳⡙⡄⠀⢷⠘⢮⡾⠃
⠀⠀⠀⠀⠀⢀⡟⠀⡜⡼⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⣰⣷⠏⠀⠀⠀⠀⠀⠀⠀⢷⢱⠀⠸⡇
⠀⠀⠀⠀⠀⣸⠃⢰⣿⡇⠀⠀⠀⠀⠀⠀⣠⠦⢤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⡇
⠀⠀⠀⠀⠀⢹⠀⠘⣿⡇⠚⠁⠀⠀⠀⠘⢧⡴⠞⠛⠻⡀⠀⠀⠀⠀⠀⠀⠘⠃⣸⢣⠃⢀⡇⢰
⠀⠀⠀⠀⠀⠘⣇⠀⢳⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢠⡏⡜⠀⣼⠁⠏
⠀⠀⠀⠀⠘⠄⢹⡄⠀⢿⣧⠀⠤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣀⡾⢣⣞⡜⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢳⡄⠈⠻⣷⣄⠀⠀⣴⣶⠆⠀⠀⠀⠀⢀⠀⠀⠀⢉⡵⣫⠎⠀⣰⠃
⠀⠀⠀⠀⠀⠀⣠⠾⣝⣦⡀⠈⠻⣷⣤⣛⡛⠀⠀⠋⠀⠀⢈⣁⡤⢖⡫⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠻⠴⠞⠉⠙⢦⣄⠈⠙⠻⠭⠭⠽⣛⣛⣛⠭⠕⠚⠉⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠦⣤⣀⣀⣀⣀⣀⣤⣤⣶⡟⠉⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣶⣿⠆⣿⣿⣿⣿⡏⠉⠁⠉⠁⠈⠀⠉⠉⠉⠁⠉⠁⠉⢹⣷⣶⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⢿⢧⣿⣟⣛⡿⣧⣤⡤⠤⠤⠤⠤⠤⠴⠤⣤⣄⣠⣤⣼⣿⣿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡇⢸⠷⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠿⠙⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠰⡈⠋⠀⠀⠀⠀⠀⢶⣾⠡⠃⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢄⠀⠀⠀⠀⢀⡾⠁⠁⠀⢀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⣾⣿⠀⠀⠀⠀⠀⢄⠀⢀⡮⠊⠀⠀⠀⠈⠷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⢿⣷⠀⠀⠀⠀⡀⠈⢸⠀⠑⠀⠀⠀⠀⠀⡾⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠈⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠸⡀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠣⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠂⠀⠈⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⡇⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠀⠀⠑⢾⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⡇⣸⣦⢤⣤⡤⠀⠀⠀⠀⠀⠠⠧⣴⣃⠐⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣾⣾⡞⣿⣯⣽⣷⡟⠛⠳⠒⠒⠒⠒⠲⠖⠒⠛⠋⠑⠛⢻⣷⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠿⣿⢆⣿⣿⣿⣿⣇⡀⠀⠀⡀⢀⠀⠀⢀⡀⣀⣀⣀⣀⣸⣿⠿⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠉⠉⠲⡄⠀⢀⡴⠋⠉⠉⠳⣄⠀⠀⡴⠊⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⣠⠞⠀⠘⢆⠀⠀⠀⢀⠞⠀⠘⢆⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠛⠋⠙⠾⣟⠛⠛⠉⠀⠈⠙⠛⢛⡿⠞⠉⠛⠋⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⠀⠀⠀⠀⠀⣤⠀⠀⡇⠀⠀⠀⠀⡇⠀⢠⡄⠀⠀⠀⠀⢰⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠧⣤⠼⣤⣤⣤⣤⣤⠿⣤⡴⢧⣤⣤⣤⣤⡷⣤⡼⢧⣤⣤⣤⣤⡼⣤⡤⠇






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

⠀⠀⠀⠀⠀⠀⠀⣠⣴⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⠀⣿⣧⡉⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣯⣉⠀⠀⠠⡅⠀⠀⢿⡿⠁⠀⠀⠀⠀⠀⠀⢿⣿⠁⢰⣾⣶⣶⣦⢸⣿⣿⡇⡃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠮⠸⠀⢠⣾⣷⣄⠀⠀⠀⠀⠀⢀⣼⣧⡄⠘⠿⠿⠿⠋⢀⣹⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠘⠻⣦⣨⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣾⣿⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⣠⣤⣬⣿⣿⣿⡯⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⣤⣤⣤⣤⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠃⠀⠿⠟⠉⠉⠉⠿⠗⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⠁⠁⠀⠈⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡄⠙⠋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣯⣿⡶⠀⣤⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠋⢘⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⣿⣿⣷⣌⣿⣟⣿⣽⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣿⣶⣶⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣯⣿⣯⣽⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠋⠛⠋⠋⠛⠋⠛⠛⠉⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠁⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠒⠀⠀⠀⠛⠛⠛⠒⠒⠒⠲⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠒⠒⠶⣶⣦⣶⣤⠤⠤⣤⣀⣀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠐⣺⠉⣹⠆⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡉⠉⢲⡞⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⣹⠧⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠹⡿⡃⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢠⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⠀⠡⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡀⣰⠏⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢻⡀⠀⠀⢄⢠⡤⠀⠀⡁⠀⠤⠔⣲⣿⡿⠁⠀⠀⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⣧⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢌⡴⠋⠀⠀⠀⢰⠁⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⢀⡿⠀⠀⠀⠀⠀⠻⡄⠀⠐⣴⠋⠀⠀⠀⠀⠀⣦⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠨⡛⣻⠄⠀⠀⠀⠀⠀⠀⢸⢐⢸⡇⠀⠀⠀⠀⠀⠀⢈⡟⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠙⣦⠀⠀⠀⠀⠀⡠⠂⠈⠀⠳⣄⠀⠀⠀⠀⠀⣠⠛⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡾⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⣧⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣸⠃⠀⢀⠔⠁⠀⠀⠀⠰⡄⠀⠀⠀⠙⢦⡀⠀⠸⡆⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡟⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠹⣆⠀⢻⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠘⣆⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢦⢹⡆⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠖⠚⠉⠀⢸⣸⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢢⡡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⣣⣏⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢴⠂⢹⡅⠚⠛⠭⠥⠶⠖⠒⠒⠲⠶⠶⠦⠭⠿⠛⠓⢛⠇⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠙⠻⠋⢁⣀⠂⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⣶⣒⣫⡽⠿⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠂⠀⠰⠦⠔⠒⠒⠒⠒⠒⠋⠉⠉


⣿⣿⣿⢿⣿⠶⣾⣿⣿⣾⣿⡷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣶⣶⣦⣶⣦⣾⣯⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡴⠖⠛⠋⠉⠙⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⡾⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⢻⠛⠛⠛⡟⠋⠉⠻⠛⠉⠙⣿⡇
⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⡇⢹⠀⢸⠈⣹⠂⡧⡎⢹⠤⢰⠉⡧⣽⡇
⠀⣸⠃⠀⠀⠀⠀⠀⢀⣬⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⡇⠈⠀⣸⠘⠓⢀⣇⠙⠊⢀⡈⠓⠁⣾⡇
⠀⢹⡄⠀⠀⠀⠀⠀⡸⠛⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣧⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣼⠟⠡⠀⠐⠀⠠⠉⠻⣆⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣾⠁⠂⠀⠀⠀⡔⠀⠀⠂⠘⣇
⠀⠀⠀⠀⠙⠳⢦⣤⣄⣀⣠⣤⡴⠞⠋⠀⠀⠀⠀⣿⠐⠀⠀⠀⠚⠁⠀⠀⠐⠂⣿⠇⠀⠀⢀⡴⠚⠋⠉⠉⠉⠉⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢠⡀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀
⠀⠀⠀⠀⠀⠀⡴⠚⠛⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣌⡀⠀⠀⣈⣠⣾⠋⠀⠀⣨⣿⣿⣿⣷⣦⠀⢤⡤⠀⣠⣶⣶⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠳⢴⣿⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⢟⣩⣤⣼⣧⣤⣝⡻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠟⠋⣡⣿⠟⡋⠉⠉⠉⠉⢙⠿⣿⣍⠻⢿⠇
⠀⠀⠀⢠⣾⠟⠋⠁⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠇⠀⠀⠀⠀⢀⣸⡛⣅⠀⠉⠀⠀⢀⡴⠋⠀⣘⣿⣧
⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⠁⠘⣿⡇
⠀⠀⣿⠇⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⠁⠐⠒⠒⠚⠁⠀⠀⠀⠈⠁⣿⡇
⠀⠀⢿⡇⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣸⡏⠀⣴⣶⣶⢰⡶⣶⡄⣶⢶⣦⣴⡶⣶⠀⠀⠘⣿⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⡿
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣿⣾⣿⢸⣿⣼⡇⡗⣿⣿⣟⣿⢺⠀⠀⠀⠘⢿⣦⣀⠂⠀⢠⠀⠀⢣⣠⣾⠟⠁
⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⣤⣴⡾⠋⠀⠀⠀⠿⠿⠿⠸⠶⠾⠃⠿⠾⠟⠻⠷⠿⠀⠀⠀⢀⡼⠋⠻⠿⢶⣶⣶⡾⠿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠁⠀⠀⣀⣤⡀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠘⢿⣿⡿⠟⠁⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣦⠀⠀⠁⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⣾⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢿⣿⣿⠃⠀⠀⠘⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡽⠿⣿⣿⣿⣿⣷⣶⣿⣿⠏⠀⠀⠀⠀⠹⣿⣿⣶⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⢀⠀⠀⠀⠀⠀⠀⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠠⣤⡄⠀⠀⢤⣤⠀⠈⠙⠛⠿⢿⣧⡀⠀⣿⡧⠀⠀⠀⠀⢾⣿⠀⢀⣼⡿⠿⠛⠋⠁⠀⣤⡤⠀⠀⢠⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠈⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⢀⠔⠃⠈⠳⣄⠀⠀⠀⠒⠒⠈⢁⠀⠀⠀⠀⠀⠞⠁⠀⠀⠉⠀⠀⠀⠀⠀⠂⠉⠀⠉⣠⠀⠀⠀⠀⠀⠉⠡⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠦⠀⣄⡀⠀⠀⠀⢀⠴⠂⠀⢤
⠀⠀⠀⠀⠀⠀⠠⠞⠁⠠⠉⠢⠀⠀⠀⠝⠲⠰⠤⠤⠄⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠃⠀⠀⠀⠀⠃⠀⠀⠤⣤⠤⠦⣤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠀⠀⠐⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠐⠋⠁⠀⠉⢕⠀⠀⠀⠰⠋⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠙⠃⠙⠁⠀⠀⠀⠀⠄⠤⠀⠤⠤⠀⠀⠀⠙⠓⠗⠘⠋⠋⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠄⢠⡀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠊⠉⠙⠘⠉⢭⠀⠀⠀⠊⠀⠀⠠⠈⠂⠀⠀⠀⠞⠉⠁⠉⠑⠄⠀⠀⠔⠋⠁⠙⠳⠄⠀⠀⠀⢀⠤⠄⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⣤⣄⠀⢀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣠⣰⣶⡄⣥⢻⣬⣿⣿⠆⠿⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⣿⣿⠐⢿⣿⣿⣿⡋⢸⣿⡆⠆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠠⣶⡎⢻⣶⣿⣿⣿⣿⣿⣿⣫⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠹⠿⣿⣿⠟⢻⠙⣿⣿⠷⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠁⠛⠀⠀⠀⢻⠻⠆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣀⣀⣀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⡿⠟⠛⠛⠛⠛⠛⠛⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⢁⡀⠀⠀⠀⢰⡆⠀⠀⠛⢛⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠹⠄⠀⠀⠘⠃⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠈⠛⠃⠀⢀⣤⣴⣤⣄⠀⠀⠀⣠⣤⡄⠘⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⣹⣿⠀⢀⣼⢿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠘⠛⠛⠀⠀⠀⠀⣀⣴⡿⠃⣰⣿⣧⣼⣿⡄⠀⠀⠛⠛⠃⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣯⣶⡄⠉⠉⢉⣿⡏⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠠⠞⠀⠀⠀⢸⡇⠀⠀⠀⠳⠄⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣀⡀⠀⠀⠈⠁⠀⠀⢀⣀⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣷⣾⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠐⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢂⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠀⠀⠀⡠⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⡁⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⣶⠆⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠈⣿⡇⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡴⠛⠁⣀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡠⢀⠐⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠟⠂⠀⠰⠟⠶⠲⠾⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠠⡀⢄
⠀⠀⠀⠀⠀⠜⡠⠂⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠣
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⢰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠠
⠀⠀⠀⠄⡐⠀⠀⢀⡀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢂
⠀⠀⠀⠐⠀⠀⠀⢸⡇⠀⣠⣟⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠘⣿⠀⠀⠀⠂⠀⡀
⠀⡀⠀⠁⠀⠀⠀⢸⡇⠀⢸⡿⢿⣿⣎⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⠞⠁
⢀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠈⠿⠄⠙⠿⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣤⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢦⣤⣄⡀⠀⠀⠀⠀⢸⣿
⡇⠀⠀⠀⢠⡄⠐⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣤⣀⠀⢈⣿⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀
⡇⠀⠀⠀⣿⡀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣿⣷⣦⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣠⠇
⡇⠀⠀⠀⢈⡝⠂⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠻⣿⣿⠟⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠙⣶
⠇⠀⠀⠀⠈⠃⠀⠚⠁⠀⠀⠀⠀⠀⠀⠠⠄⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠤⠼⠋
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠
⠈⠀⠀⠀⠀⠀⠀⣠⠄⠀⢤⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠃⠀⡀⠀⠀⠀⢿⣦⣀⡘⠀⠀⠁⠂⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠠⠀⠀⢰⠃⠀⠈⢻⡆⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⣿⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠃⠀⠈⠓⠀⠀⠛⠁⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⡈
⠀⠀⠀⠀⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰
⠀⠀⠀⠀⠀⢢⠑⠀⠀⠀⠀⠀⠀⠀⣼⡟⣴⠶⠶⠶⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡔
⠀⠀⠀⠀⠀⠀⠑⠀⠠⡀⠀⠀⠀⢸⣿⠃⢀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⢦⣄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠈⠢⡀⠀⢼⡿⠀⢈⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣰⣏⡀⢈⡉⠀⠀⠀⠀⠀⠀⠀⣰⣦⠀⢀⣽⠀⠀⠀⠀⠀⠐⢁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢄⠀⠀⢠⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠛⠐⠋⠁⠀⠀⡠⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢂⡁⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠀⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⡿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣤⣤⣤⡄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠛⠛⠲⠶⣤⣀
⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠤⠄⡀⠀⠀⡀⣀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠙⢦⣄⠀⠸⣿⣿⣿⣄⡀
⠀⣰⠏⠀⠀⠈⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠂⠀⠈⢻⣷⣄⠉⠿⣿⣿⣿⣆
⢠⡏⠀⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⣿⣿⡀
⣾⠀⠠⢄⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠈⠞⢑⠀⠸⡇⠙⠿⠟⣿⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⣲⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⣿⣿⣿⡀
⣿⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⠀⠀⢀⠠⠄⠀⢠⡇⠀⠀⣠⣾⣿⣿⣿⡇
⢸⡆⠀⠓⠒⠂⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⣸⠁⣰⣿⣿⣿⠿⣿⣿⣷
⠀⢻⡄⠀⠀⠀⠠⣄⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⠀⠀⠀⣰⠇⠸⣿⣿⠋⠀⠀⠹⣿⣿⣆
⠀⠀⠹⣄⠀⠘⠓⠲⠀⠀⣀⠀⠀⠀⢀⡀⡀⠀⠀⠁⠈⠡⢀⡼⠃⠀⠀⢻⣿⣇⠀⠀⠀⠙⢿⣿⣷⣦⡀
⠀⠀⠀⠈⠳⣤⡀⠀⠀⠨⠬⠑⠀⠀⠐⠪⠍⠀⠀⠀⣀⡴⠛⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠈⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⡀⠀⢀⣀⣀⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⢀⣹⣿⣧⠀⠀⠀⠀⠀⠀⠀⢉⣿⠿
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⢰⣿⣿⣿⣿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠿⠿⠿⢿⣿⡀
⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠤⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠄⠀⠀⠄⠀⣾⢿⠇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠻⠦⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠠⠶⠋⠀⠀⣀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠉⠛⠛⠛⢛⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡏⢁⠀⠀⢀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⢛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠤⠀⠀⠀⡤⡾⠁⠀⠀⠀⣠⣴⣶⣶⣦⣄⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⠿⠟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠛⠿⠟⠿⠿⣿⣧
⠀⠀⠀⠀⠿⡇⠀⢀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠄⠀⠠⠀⠸⠟⡿
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠘⠲⠀⠀⠀⠐⠂⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠓⠂⠀⠀⠒⠛⠁⠀⠀⢀⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠁⠁⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠛⠛⠻⠿⠿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠒⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠚⠛⠉⠉⠉⠉⠉⠙⠛⠒⠲⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠋⠁⣀⣠⠤⠴⠒⠒⠒⠛⠛⠛⠓⠒⠒⠲⠤⢤⣀⡀⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⣀⡤⠒⢋⣁⡤⠴⠒⠚⠋⠉⠉⠉⢉⡉⠉⠛⠒⠲⠤⣄⣉⠓⠢⢄⡀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠴⠊⣁⠤⠚⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡉⠳⢄⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠞⢁⡴⠚⠁⣶⠀⠀⡆⠀⠀⠀⠀⠀⢸⡀⠀⣄⣁⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠉⠲⣄⠙⢦⡀⠙⢦⡀
⠀⠀⠀⠀⢀⡜⠁⢠⠞⢁⡴⠋⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⣄⠙⢦⠀⠙⣄
⠀⠀⠀⢠⠏⠀⡰⠃⣰⠋⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠳⡀⠳⡄⠈⢧
⠀⠀⢠⠋⠀⡼⠁⡼⢁⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣄⠙⡄⠀⢣
⠀⢠⠏⠀⡼⢀⡞⠀⢸⡇⢰⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠘⣆⠸⡄⠈⢧
⠀⡞⠀⣸⠁⡼⠀⠀⠸⠇⠀⠣⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠸⡄⢹⡀⠘⡆
⢸⠃⢀⡇⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣇⠀⢻
⣼⠀⢸⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⣤⡄⠀⠀⠀⠀⠸⡆⢸⠀⠸
⡇⠀⣾⠀⡇⠀⢠⡆⢱⡄⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⣀⣀⣠⡤⠤⠶⠖⠒⠛⠋⠉⠙⠚⠁⠀⠀⠚⠇⠀⠀⡇⢸⡇
⡇⠀⣿⠀⡇⠀⠀⠑⡼⠛⠳⠤⠿⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣄⣠⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠀⠀⡇⢸⡇
⡇⠀⢿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⢸⡇
⢻⠀⢸⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⠀⢰
⠸⡄⠈⡇⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡜⠀⡏⠀⣼
⠀⢧⠀⢹⡀⢳⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢈⡇⠀⣰⠃⣸⠁⢠⠇
⠀⠈⣆⠀⢳⡀⢣⠀⠀⢙⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⢠⠃⢰⠃⢀⡞
⠀⠀⠘⣆⠀⢳⡀⢳⡀⠻⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⣰⠃⢀⡞
⠀⠀⠀⠘⢆⠀⠱⣄⠙⣄⠀⠀⠀⠀⠀⠀⠤⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⠂⠀⠀⠀⢀⡜⢁⡴⠁⢀⡞
⠀⠀⠀⠀⠈⢣⡀⠈⢦⡈⠳⣄⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⢀⡴⠋⣠⠎⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢦⡈⠓⢤⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠔⠋⣠⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠳⢄⡉⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠸⡇⢹⠆⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⠴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠉⠓⠦⣄⣉⠙⠒⠦⠤⣤⣄⣀⣉⣀⣀⣤⡤⠤⠖⠚⢉⣁⡤⠖⠋⠁⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠈⠉⠓⠒⠦⠤⠤⠤⣤⡤⠤⠤⠤⠖⠒⠋⠉⠁⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠤⣤⣄⣀⣀⣀⣀⣤⡤⠤⠴⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠉⠀⢀⣀⣤⡴⠶⠶⠶⠶⠖⠶⠶⠶⢤⣤⣀⡀⠀⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠁⢀⣠⡶⠟⠋⠉⠀⠀⠀⠀⢠⡄⢠⣤⠀⠀⠀⠀⠉⠙⠳⢶⣄⡀⠈⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠰⢿⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣠⡾⠋⠀⠀⠀⢸⠇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠙⢷⣄⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣦⡀⠈⢻⣧
⠀⠀⠀⠀⠀⣾⡟⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢻⣧
⠀⠀⠀⠀⣼⡟⠀⢀⡿⠁⣶⠀⣿⣷⠀⠀⠀⠀⠀⠀⣶⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⠀⠈⣷⡀⠀⢿⣇
⠀⠀⠀⢰⣿⠃⠀⣼⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⣧⠀⠘⣿⡄
⠀⠀⠀⣼⡟⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢿⡇
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⠐⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡇⠀⢸⡿
⠀⠀⠀⢹⣷⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⡄⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠘⣿⡄⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢰⣿⠁
⠀⠀⠀⠀⠹⣷⡀⠀⢿⡄⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⢠⡟⠀⢀⣿⠇
⠀⠀⠀⠀⠀⢻⣷⡀⠈⢻⡄⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⡟⠁⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠻⣦⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠈⠳⣄⡀⠀⠀⣭⣿⠇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⣯⡉⠀⠀⠀⢀⣴⠟⠁⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⠻⢦⣄⠀⠋⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠚⠃⠀⣠⡴⠟⠁⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⠉⠀⣀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⣀⠀⠉⠉⠛⠛⠳⠶⠶⠶⠶⠛⠛⠛⠉⠁⠀⣀⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠈⠉

⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⢀⡴⠋⠀⠀⠂⠀⠈⠳⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡖⠋⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⡞⠀⠀⠀⠀⢀⠎⠀⠀⠘⡆⠀⠀⡞⠀⠀⠀⠀⠀⢀⡠⠀⠘⡆⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠋⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⢀⡇⠀⠘⡇⠀⠀⠀⠀⠓⠤⣀⠀⢀⡇
⠀⠀⠀⠹⣌⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠈⠓⠤⣤⣤⡤⠴⠋⠀⠀⠀⠀⠀⠈⠳⠤⣤⣤⡤⠴⠋⠀⠀⠀⠀⠀⠈⠳⠤⣤⣤⡤⠶⠋⠀⠀⠀⠀⠀⠈⠳⠤⣤⣤⡤⠖⠋
⠀⠀⠀⠀⠀⢀⣠⣤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣄⣀
⠀⠀⠀⢀⡴⠋⠀⠀⠂⠀⠈⠳⣄⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠙⢆⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⠞⠉⠀⠀⠂⠀⠈⠳⣄
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠣⡀⠀⠀⠀⡇⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠀⠀⡇⠀⠀⠀⢠⠃⠀⠀⠀⢀⡇⠀⠘⡇⠀⢀⠤⠒⠃⠀⠀⠀⢀⡇
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠑⠀⢀⡼⠁⠀⠀⠘⣆⠀⠀⠀⠇⠀⠀⠀⡴⠃⠀⠀⠹⣄⠀⠀⠁⠀⠀⠀⢀⡼⠁⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠈⠳⠦⣤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠈⠓⠦⢤⣤⡤⠴⠋⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠈⠳⠤⣤⣤⡤⠖⠋
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀
⠀⠀⠀⢀⡴⠋⠁⠀⠂⠀⠈⠳⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡔⠋⠀⠀⡀⠀⠉⠳⣄
⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡞⠀⠀⢄⡀⠀⠀⠀⠀⠘⡆⠀⠀⡞⠀⠀⠀⢢⠀⠀⠀⠀⠸⡆⠀⠀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠸⡆
⠀⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠃⠀⠀⠀⢀⡇⠀⠘⡇⠀⠀⠀⠀⠁⠀⠀⠀⢀⡇
⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠈⠳⠤⣤⣤⡤⠴⠋⠀⠀⠀⠀⠀⠈⠳⠤⣤⣤⡤⠴⠋⠀⠀⠀⠀⠀⠈⠳⠤⣤⣤⡤⠴⠋⠀⠀⠀⠀⠀⠉⠲⠤⣤⣤⡤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣲⡖⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⠀⠀⠀⠀⠀⣿⣂⠘⠁⡰⠷⣤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠟⠊⠀⠀⠀⠀⠀⠀⠻⣐⠒⢐⣗⠠⠿⠅⢰⡇⠀⠀⠀⠀⠀⠀⡔⠉⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣡⣒⣁⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠤⠅⠒⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠥⢬⠁⠀⠁⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠈⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⡀⠀⣀⠀⢀⠀⠀⠀⠀⣀⠦⢒⣂⣠⣤⣤⣀⣒⠠⢇⡀⠀⠀⠀⠀⠀⠀⡯⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣩⠖⠋⠉⠀⠀⢰⠀⠀⠈⠉⠓⢮⡑⢄⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣡⠞⠁⠀⠙⠀⠀⠀⠘⠀⠀⠀⠘⠁⠀⠈⢢⡱⡀
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠈⠐⠂⡜⣴⠁⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢠⠄⢱⡜⡄⠒⠈⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⢰⠃⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢳⢠⠀⠀⠀⠀⠀⢀⠔⡣⠎⣴⣉⠙⡦⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢸⠀⣀⡀⠀⠀⠀⠀⠈⠓⢤⠖⠁⠀⠀⠀⠀⠀⣀⡀⢸⠌⠀⠀⠀⠀⠀⡒⢰⠥⣰⡀⢸⠉⠓⢲
⠀⠀⠀⣸⣷⣶⣶⣶⣶⣶⣶⣶⣷⠀⠀⠀⢰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠦⢼⢀⡇⢀⠋⢐⠗⡼
⠀⠀⢠⣉⣀⣀⣶⣶⣶⣶⣎⣁⣉⡆⠀⠀⠀⡈⣆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡎⡈⠀⠀⠀⠀⠀⠈⠲⣤⣁⣍⡱⣓⠔⠁
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠒⠉⠑⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡐⠉⠑⠂⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⡳⣄⠀⠖⠀⠀⠀⢰⠀⠀⠀⠐⠆⢀⡴⡫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠢⢝⠲⠤⣤⣀⣈⣀⣀⡤⠴⣚⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡯⠓⡽⡂⠀⠀⠀⠀⢀⠋⠁⠒⠂⠀⠀⠒⠂⠉⢃⠀⠀⠀⠀⠀⠀⠀⡠⡺⡉⢱⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣎⠵⣳⣚⠳⡲⠽⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠸⠅⠸⣗⠀⠇⢣⣬⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⣄⠀⡈⢞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⣖⠒⢒⣶⣒⠀⠀⠀⠀⠀⠀⠀⢀⢖⡭⠚⠂⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠣⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡆⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠃⠐⣶⡂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠶⠶⠴⠭⠯⠯

⠀⠀⠀⠀⢀⣤⡶⠶⠶⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⣦⣄⡀
⠀⠀⠀⣴⠟⠁⠀⠀⠂⠀⠈⠙⣷⡀⠀⠀⠀⢠⣾⠋⠁⠀⠐⠀⠀⠈⠻⣦⠀⠀⠀⠀⣰⡟⠉⠀⠀⠂⠀⠈⠙⣷⡄⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠉⢻⣦
⠀⠀⢸⡟⠀⠀⠀⠀⡴⠁⠀⠀⠘⣷⠀⠀⠀⣾⠃⠀⠀⠀⢀⣀⠄⠀⠀⢹⡇⠀⠀⢰⡟⠀⠃⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⢸⣇⠈⠀⠀⠀⠁⠀⠀⠀⢠⡿⠀⠀⠀⣿⡄⠁⠀⠀⠈⠀⠀⠀⠀⣸⡇⠀⠀⢸⣇⠈⠀⠀⠀⠉⠉⠁⠀⢀⣿⠀⠀⠀⣿⡄⠁⠀⠀⠈⠑⠤⠀⠀⣸⡇
⠀⠀⠀⢻⣆⡁⠀⠀⡀⠀⠀⣠⡾⠃⠀⠀⠀⠘⣷⣌⠀⠀⢀⠀⠀⠀⣰⡟⠀⠀⠀⠀⢻⣆⠁⠀⠀⡀⠀⠀⢀⣾⠃⠀⠀⠀⠘⣷⣌⠀⠀⢀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠙⠻⠶⣦⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠈⠙⠷⢶⣤⣤⠶⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⡶⠟⠋

⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⢠⡾⠛⢉⠀⡄⠈⠉⠻⣦⡀⠀⠀⠀⠀⣴⠟⠋⠁⢠⠀⠉⠛⢷⡄⠀⠀⠀⠀⣠⡾⠛⠉⠀⡄⠈⠙⠻⣦⠀⠀⠀⠀⢀⣴⠟⠋⠁⠠⠀⠉⠛⢷⡄
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢠⡿⠁⠆⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⢸⣇⠐⠀⠀⠀⢢⠀⠀⠀⠀⣿⠀⠀⠀⣿⡀⠂⠀⠀⢰⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠐⠀⠀⢠⠇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠂⠀⡠⠖⠀⠀⠀⠀⢸⡇
⠀⠀⠈⢿⣄⠂⠀⠀⠀⠁⠀⠀⣼⠏⠀⠀⠀⠹⣧⡘⠀⠀⠘⠀⠀⠀⢠⡿⠁⠀⠀⠈⢿⡄⠃⠀⠃⠀⠀⠀⠀⣼⠏⠀⠀⠀⠹⣧⡘⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⠻⢷⣤⣄⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠙⠿⣦⣄⣈⣀⣤⡶⠟⠁⠀⠀⠀⠀⠈⠻⢶⣤⣀⣀⣠⣴⠾⠋⠀⠀⠀⠀⠀⠙⠷⣦⣄⣈⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⢀⣴⠿⠛⠉⡉⠉⠛⢷⣄⠀⠀⠀⠀⠀⣠⡾⠛⠋⢉⠉⠙⠻⣦⡀⠀⠀⠀⠀⢀⣴⠿⠛⠉⡉⠉⠛⢷⡄⠀⠀⠀⠀⠀⣠⡾⠟⠉⢉⠉⠙⠻⣦⡀
⠀⠀⢀⣾⠁⠄⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⣸⡟⠠⢀⠀⠈⠀⠀⠀⠈⢿⡀⠀⠀⢠⣿⠃⠄⠀⡄⠀⠀⠀⠀⢹⣆⠀⠀⠀⣼⡟⠀⠀⠀⢰⠀⠀⠀⠈⣿⡀
⠀⠀⢸⡇⠀⠀⠤⠤⠄⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠄⠀⠉⠲⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠠⠀⠀⠈⠇⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠄⠀⠀⠸⠀⠀⠀⠀⢸⡇
⠀⠀⠘⣿⡀⠆⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢻⣇⠠⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠘⣷⡀⠄⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⢻⣇⠰⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠘⢿⣦⣀⣀⠂⣀⣠⡾⠋⠀⠀⠀⠀⠀⠹⢷⣄⣀⡐⣀⣀⣴⠿⠃⠀⠀⠀⠀⠘⢿⣦⣀⣀⠀⣀⣠⣶⠟⠀⠀⠀⠀⠀⠻⢷⣄⣀⣐⣀⣀⣴⠿⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠒⠒⠶⠶⠶⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣤⠶⠛⠛⠛⠛⠛⠲⢶⣤⡉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡃⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⡼⠉⠛⠳⣦⠀⡏⢨⡇⢠⡶⠛⠉⠉⠉⠛⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠙⠉⠀⠀⠀⠀⣸⣇⣇⣀⣇⡟⠛⠲⠶⢤⣄⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⣉⣉⣭⣽⣶⣦⣄⠀⠈⠛⢦⡀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣤⣤⠶⢞⣫⠶⢚⣋⣭⣤⣤⣤⣤⣀⠉⠻⣦⣀⠀⢳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣠⡾⢋⣴⠞⠉⠁⢠⠤⡄⠀⠀⠉⠛⢶⣄⠙⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠋⡴⠋⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠘⢷⡈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢠⡿⢡⡞⠁⠀⡠⠖⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡈⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣾⠁⡞⠀⠀⠀⠑⠦⣄⡈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡇⢸⡇⣠⣀⠀⠀⠀⠀⠉⢿⡉⣹⡀⠀⠀⠀⠸⣭⠇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠸⡇⢸⡇⠓⠋⠀⠀⠀⠀⠀⠀⠉⢧⠹⣄⠀⠀⠀⠀⠀⣼⠃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⢷⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⣳⠀⠀⠀⣼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠈⢣⡘⢷⣀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠈⠁⠀⣠⡾⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⡀⠉⠛⠶⣤⣄⣀⣀⣙⣁⣠⣤⡴⠞⠉⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⢙⣆⠀⠀⠀⠀⠉⠉⠉⣀⣀⣤⡤⢶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡴⠋⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⢢⣼⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⡠⠄
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⣡⡤⠈
⢿⣿⣿⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢛⣲⣦⣄⣀⡾⠛⠋
⠀⡿⣿⣿⣿⣵⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡦⣴⣾⣿⣿⠿⣴⣯⣾⠟⢿⣶⠈⣝⡦
⣿⣿⣿⣿⣿⣿⣿⣴⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⡀⠀⠀⠀⢀⣴⡿⢋⣥⠚⠋⢛⣛⠿⡘⠟⣿⢻⣿⣸⣿⠀⠈⣠
⠙⠛⠿⣿⣿⣿⣿⣿⣿⣴⣻⣶⣶⠚⠛⠛⠛⢓⣶⣿⡏⣭⣿⣛⠀⢀⢙⣿⡏⣰⡟⢀⣴⡿⣟⣿⣻⣦⡆⣻⣿⣿⡉⠁⢀⣤⠇
⣤⣤⣤⣬⣭⣿⣿⣿⣿⣿⣿⣉⣫⠿⠋⠉⢉⣿⣿⣯⣿⡏⠀⠀⢨⣧⡿⣹⣯⣿⡄⣾⡟⢰⡿⣯⣽⣿⣸⣿⣿⣿⡅⠀⠉
⣿⣿⠋⠁⢀⢉⣿⣿⣿⣿⣿⣿⣷⣤⡚⠉⠨⣿⣿⣿⣿⣷⣄⣺⣾⣟⣵⠟⣼⡿⣷⣽⣧⠈⠁⠀⠙⡛⠻⠟⣿⣿⡇
⣧⣿⠀⢠⡟⣿⠉⠈⠉⠙⠻⢿⣾⣿⡿⣧⡀⢿⣿⣇⣿⡿⣿⠿⠿⠛⣡⡾⢻⡆⢹⣧⡽⣷⣀⣠⠚⠉⢀⣴⡿⠿
⡿⠃⠀⢸⡇⢸⡄⠀⠀⠀⣀⣤⣬⣽⣿⣿⣿⣾⣿⣿⣿⣷⣷⣶⣶⡿⣏⣤⣾⡇⢰⣿⣧⣿⣿⣁⣤⣾⣿⣿⠀⠠
⠀⠀⠀⢸⣇⣿⠀⠀⢀⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣯⣼⣿⣥⣬⣴⠟⣩⣿⣿⣿⣟⠁⣼⠸⣷⡀⠀⠂⠀⠀⠰⠖
⠀⣀⣤⣿⡿⠃⠀⠀⠈⢿⣿⣶⠟⠛⣿⡿⣿⣿⣿⠿⢛⣫⠴⣿⠛⠉⠉⢉⣩⣴⣿⣿⣿⣿⣿⣿⣷⠈⠴⣤⡿⠀⠀⠀⠀⠀⠀⡄
⡟⢻⣿⡟⠁⣬⣁⣤⠚⠀⠙⠿⣶⡶⢏⡠⠄⢀⣠⡴⠟⠁⠀⢻⡄⢠⣤⣾⠟⢉⣠⣴⣶⣶⣦⣤⠙⢻⣤⠋⠀⠀⠀⠀⠀⠀⠀⠉
⣴⡿⠋⣠⣾⣿⠟⣁⣤⣤⣴⣦⠶⣒⣯⠶⠛⠛⠉⢀⣴⣶⣄⠀⢰⣿⣿⠏⣰⣿⠟⢻⣿⣿⣿⣿⣇⢰⣿⣷⣄⣀⡀⠀⢀⣀⠠⠐⠂
⢋⣤⡾⠋⣨⣷⣿⡿⠟⠉⢁⣴⡿⠋⠀⠀⠀⠀⠀⣿⠋⠛⢿⣧⢸⣿⣿⠀⣿⣷⢆⣰⣿⣿⣿⢏⣿⡿⠛⣭⡝⠛⠛⠛⠁⡠
⠿⢋⣴⣿⣿⡿⠋⠀⠀⠀⣾⡟⠀⡴⠛⠛⠷⣦⠀⣿⡀⠀⣸⣿⠹⣿⣿⡀⢚⣿⣾⣿⡿⣿⣟⣼⣿⠀⠀⠀⢰⠀⠀⠀⣼⠃
⣴⠿⣿⡿⠋⢀⣴⠎⠀⠀⢻⡇⢸⡅⠛⡇⠀⣿⡇⠈⠛⢉⣿⣿⠀⠹⣿⣷⣸⣿⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣴⡟
⣫⡾⠋⢀⣤⣾⠃⠀⠀⠀⠈⢷⡈⠛⠛⠁⣰⡿⠃⠀⣀⣾⣿⣿⣄⠀⠀⠉⠛⣷⣬⣉⡉⠛⠫⣿⣉⠿⣷⣶⣶⣶⣾⡿⢦⣄
⢋⣠⣴⣿⠟⢁⠀⠀⠀⠀⠀⠈⢻⣷⣶⣿⣟⠁⣠⣶⣿⣿⠟⢁⣸⣖⣤⣀⠀⠛⠿⢿⣿⣿⣶⣄⠙⠳⣦⢩⣿⣿⣿⠑⠦⠙⠆
⣿⣿⣿⣡⣶⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⠟⣡⣾⣿⣿⠟⢁⣴⣿⣿⣿⣿⠿⢷⣶⢶⣦⡈⠻⣯⣿⣿⡄⠹⠀⣿⣿⠃⢠⠀⡇⠀⡆
⢋⣤⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⠟⣁⣼⣿⣿⣿⡿⠿⠁⠀⢸⡇⠀⠸⡇⠀⠈⣿⣽⣿⠀⣴⡿⠃⡸⠇⡠⠃⡠
⣿⣿⣿⣿⡿⣿⣵⣾⣿⣿⣿⡿⢋⣴⢟⣽⡿⢃⣴⡿⠿⠿⠋⠅⠀⠂⠀⠀⢸⡇⠀⠀⠀⠀⣠⣿⣿⡟⠸⣯⣛⣯⠕⠊⡀⠔⠁
⣿⠿⠻⣝⣿⠿⠙⢻⣿⠟⢁⣴⣿⣷⣾⠟⢰⣿⠋⢐⣁⣀⣈⠀⠀⠀⠀⠀⠘⣿⣄⠀⢀⣾⣿⠿⠋⠖⣠⣴⣿⣷⣶⣈⠁
⠀⠀⠀⠉⠀⠀⣠⡞⠁⢀⣾⣿⣿⣿⣟⠀⣿⠃⣴⣿⣿⣿⣯⣝⣻⣦⡀⠀⠀⠈⢿⣷⣼⣿⡏⠀⠀⠸⣟⠁⡀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⢀⣴⠏⠀⡀⣾⣿⣿⣿⣿⡇⠀⣿⢠⣿⠋⠉⠀⠉⢩⣿⣿⣿⣦⡀⠀⠀⠙⠻⣿⣿⣄⣀⠀⠙⠆⠀⠀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿
⠀⠀⠀⢀⡾⠁⢀⣾⠇⣿⣿⣿⣿⣿⣧⡑⠉⢸⣿⣄⣀⣀⠀⢜⣹⣿⣿⣿⣿⡌⢷⣄⠀⠹⣿⣿⣿⣷⣦⣤⣶⣿⣿⡟⣁⣀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿
⣀⣴⣾⠟⠀⢠⣿⡏⣶⡙⣿⣿⣿⣿⢿⣿⣿⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⠀⣿⣷⡀⠺⣿⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⠏⠀⣠⣿⡿⢰⣿⣷⡟⢿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢹⣿⣥⣿⣿⣿⡆⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠈⠙⢻⣿⣿
⣿⠋⠀⣰⣿⣿⠃⣸⣿⣿⡿⣆⠙⢿⣿⣿⣷⣍⡛⠿⡿⠿⠿⠛⠋⠁⠀⢀⣼⣿⣿⣿⢻⣿⣿⠀⠀⢀⣤⣭⣿⣿⣿⠿⢿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿
⠋⠀⢠⣿⣿⠟⢀⣿⣿⣿⡄⠈⠳⣄⡈⠙⠻⢿⣿⣿⣶⣶⣶⣶⣶⣶⣾⡿⣻⣿⠟⢡⣿⣿⡿⢠⡾⢛⣉⣉⡛⠻⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉
⠀⣠⣿⣿⡿⠀⢸⣿⣿⣿⠀⠀⠀⠈⠉⠒⠤⠤⣀⣉⣉⣛⣛⣛⣯⣿⣷⣿⡿⢁⣴⢟⣿⡿⠁⣯⡾⢿⣿⣿⣿⣆⠀⠙⣿⣷⣆
⢰⡿⢿⣿⡇⢀⣼⣿⣿⣿⠓⠀⠀⠀⠀⠀⠒⠒⠋⠯⠭⣭⣭⣽⣿⣿⠟⣫⣿⠟⣵⠟⠋⠀⠀⠙⠁⠀⣿⣿⣿⡿⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡿⠃⣾⣿⢁⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣾⣿⡿⠛⠁⣼⣿⡏⣾⠏⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⠀⣼⣿⣿⣿⣷⣄⣀⣀⣠⣤⣶⣿⣿
⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠿⠛⣫⣴⣾⡆⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⢀⠄⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⠸⣿⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡇⠘⢿⣿⣿⣆⡀⢀⣀⣤⣶⣿⣿⣿⠟⠁⣿⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢀⢿⣿⢠⡇⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣷⠀⣀⠙⠿⢿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢠⣿⠀⠈⢿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣤⣿⣿⣿⣿⠆⠀⢻⣿⣿⡇⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⢷⡄⠀⣈⣉⣡⣤⠶⠚⠁⠀⠀⠘⣿⣷⣀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡀⠀⠸⣿⣿⣷⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣷⣄⡀⠈⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⡆⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⡛⡿⢿⣷⣶⣶⣶⣶⣿⣿⣿⣷⣶⣦⣀⠈⢻⣿⣿⣿⣿⣶⣦⣬⣉⡛⠛⠿⠿⠟
⣿⣿⣿⣿⣿⣷⠀⠀⢹⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠲⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⢬⣿⡛⠳⢿⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠛⠛⠀⠈⠁⠀⠀⠀⠀⠻⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⣀⠤⠔⠒⠒⠀⠐⠒⠒⠠⢄⣀⠀⠀⠀⣴⡏⠤⠄⠒⠂⠀⠂⠤⠄⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⠀⠀⠀⠀⠀⠀⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠘⢁⣤⠴⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠙⠶⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢀⣀⠈⠳⡀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣤⣼⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⣼⣿⣿⣄⡄⢳⠀⠀⠀⠀⢰⡇⢰⣾⣿⣇⣀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢻⣄⡀
⠀⠀⣼⠁⢸⡇⠀⠀⠘⠉⠉⠛⠻⢤⡀⠀⠀⠻⣿⣿⡿⠃⣸⠀⠀⠀⠀⢸⡇⠹⣿⣿⣿⠏⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣺⠉⣷
⠀⠀⢹⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⢄⣀⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠻⣄⣀⣉⣀⣠⣤⠖⠋⠉⠉⠛⠒⠀⠀⠀⣿⢀⣿
⠀⠀⠀⠙⠶⣿⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣄⣰⠟⠉⠱⡄⠀⠀⠉⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠾⠁
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣤⣤⣴⣟⣻⣄⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⣼⠐⢻⣶⣈⢉⢩⡏⢹⡆⢀⣠⡤⠖⢳⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣷⣶⣿⣼⣼⣼⣿⣾⣼⣗⣺⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⢀⣤⠶⠚⠋⣉⣀⠤⠤⠤⠤⠤⢄⣈⡉⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠾⠋⠁⠀⠊⠉⠀⠀⠀⠂⠀⠀⠀⢀⠆⠈⠉⠒⠄⠈⠛⢦⡀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡤⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡀⠀⠀⠀⠀⠲⣄⠙⢶⡾⢧⣄⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠋⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⠀⠀⠚⠈⢢⡈⠻⣤⠿⠻⣷⣿⡿⣆
⠀⠀⠀⠀⣠⡾⢋⣾⠁⡰⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢹⣆⠀⢹⣿⢿⣿⡄
⠀⠀⣠⠞⠁⠀⣼⠃⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡱⠀⢿⣄⣠⡿⠈⠉⠙⢦⡀
⢀⡾⠃⣴⠛⠛⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⣿⠷⢶⣄⠀⠀⠀⠻
⣾⠁⠘⣃⣀⣴⡋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⢹
⠹⢦⣀⢠⡟⠁⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⣴⠟⠛⣿⠋⠀⠀⣠⡴
⠀⠀⠉⠛⢻⣤⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣉⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢿⣀⣀⣼⠷⠞⠛⠉
⠀⠀⠀⠀⢿⡀⠀⣿⠰⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣼⠋⠙⣧
⠀⠀⠀⠀⠈⠛⠛⢿⡀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠠⢄⢠⠃⣰⠿⠦⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠱⡄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡰⠁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡠⠊⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠑⠢⣄⣀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⡤⠒⠁⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣦⣄⣀⡙⠀⠒⠂⠶⠆⠳⠀⠚⢉⣀⣤⣶⠛⠁⠸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠉⢻⡟⠓⠶⠶⠶⠶⠛⠛⠋⠉⠀⣿⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠘⠳⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⢶⣶⣶⣶⣶⣶⢖⣶⡶⢖⡶⢲⡶⣶⡶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠋⠫⢕⠝⣫⠖⠷⡫⢞⡽⣫⠔⣩⠞⣩⠞⣉⡴⢛⣡⠞⣹⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢂⡙⠲⠤⣤⣀⣀⡈⠀⠁⠘⠁⠚⠡⠊⢡⠞⢁⠴⠋⢐⣋⣡⠴⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡛⠀⡄⢀⡉⣉⠙⠛⠛⠓⠲⠶⠶⠶⠶⠶⠚⢛⢉⡉⠠⠘⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠛⠲⠶⢤⣭⣬⣃⣀⣃⣀⣚⣀⣲⣀⣘⣡⣭⣤⡤⠶⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠰⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⠒⠉⠛⠲⢤⣀⣀⣠⣤⠶⠶⠶⠶⣤⣄⠐⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠙⢯⣅⠀⠀⠀⠀⠀⠀⠙⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠈⠙⠓⠛⠛⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣄⡤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢋⠔⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣡⠔⠁⠀⢠⠴⠒⢦⡀⠀⠀⠀⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⡾⣁⣤⢴⡶⠋⠀⠀⠀⠙⠲⣤⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢀⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⢦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠶⠈⠳⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠟⠛⡡⢄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⡘⠃⣤⠄⢈⡉⠛⠛⠛⠛⠛⠛⠛⡛⢋⡉⢁⡄⠲⠈⢁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣌⣀⣙⣀⣃⣘⣃⣙⣀⣉⣀⣁⣤⣤⡶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠄⠒⠒⠒⠀⠉⠁⠐⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠁⠀⡀⠀⠠⡀⠂⠲⠀⠀⠶⠀⢐⡒⠀⢠⠄⠀⣀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⡀⠄⠲⠡⠀⠃⠈⠀⠀⠀⢰⡆⢀⡶⢶⣦⠀⠀⠈⠉⠐⠫⠌⣐⠤⣀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⡠⠐⠈⢛⡀⢀⣤⠀⠀⠀⠀⠀⠀⢸⡇⠈⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⢀⣈⠐⢟⠢⢄⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⠴⡀⠔⠁⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⡼⠧⢤⡿⣴⡶⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠣⣝⢦⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡠⠊⢀⠴⡥⠊⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠀⠀⠀⠀⠀⠑⣌⢦⡀⠑⣄
⠀⠀⠀⠀⢀⠜⠀⡠⢤⠈⠀⠀⠀⠀⠀⢤⡉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠈⠣⡑⡄⠈⢦
⠀⠀⠀⢠⠏⠀⣴⡀⣀⠀⣠⣤⡀⠀⠀⠈⢿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠘⠌⢆⠀⢣
⠀⠀⢠⠃⠀⡄⠈⠀⣿⢰⠃⠀⢹⡄⠀⠀⠘⣿⣿⣏⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣹⠇⠈⠋⢂⠀⢣
⠀⢠⠃⠀⠌⡘⠀⠀⣿⢸⡄⠀⣼⠀⠀⠀⠀⠘⠴⠻⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⠄⠀⡆⠀⠀⢃
⠀⡎⠀⠈⠱⠀⠀⠈⠉⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠐⡠⠀⠈⠄
⢀⠁⠀⢦⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢸⠀⢀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠸
⠀⠀⠀⠀⠀⣼⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡖⠀⠀⠒
⠀⠀⠘⠃⠀⠻⠷⠆⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⣶⠄⠀⡤
⢀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟
⢸⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠈⠙⠀⠰
⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠆
⠀⠂⠀⠐⠣⠀⠀⠀⣠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠰⠀⠀⢠⠃
⠀⠈⡄⠀⢀⠄⠀⠀⢻⣶⡋⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡇⠀⠀⢀⢀⠀⠀⠆
⠀⠀⠘⠀⠀⢀⣀⢰⡏⠙⣷⠀⢀⣤⢤⢶⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠥⢼⡧⠖⠀⢀⠃⠀⠌
⠀⠀⠀⠐⡀⠈⠁⠀⠙⠚⠁⢀⣟⡾⣯⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠛⠃⢀⠌
⠀⠀⠀⠀⠈⢆⠀⠁⠀⠄⣠⣿⠿⠒⠛⠁⣠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠈⠐⠁⢠⠊
⠀⠀⠀⠀⠀⠀⠣⡀⠈⠼⠛⡁⠀⠀⠀⠀⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣟⣛⡋⠀⠀⠀⠀⢀⠔⠡⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠠⠐⢄⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡗⠀⠀⡀⠀⠡⠊⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠈⠁⠀⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠛⢷⡄⠀⠀⠀⠀⠀⠀⣀⠴⠛⢠⡔⠀⠐⠀⢀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠀⠀⠁⠀⡤⢀⡀⠀⠀⠀⠘⢷⣤⡴⠃⠀⠀⠀⢀⠀⠄⠀⠠⠄⠀⠁⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⣀⠀⠀⠀⠐⠀⠁⠄⠀⠀⠿⠀⠐⠀⠤⠙⠀⠀⠁⠀⢀⠠⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢠⣤⠀⠀⠀⠀⣀⣠⣼⡃⠀⠀⠀⠀⠀⠀⠀⠰⢆⣀⠙⠻⢋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠺⣇⠿⢀⡄
⠀⢠⣶⣿⡟⠋⠁⠻⢶⣿⡶⠟⠀⠀⠀⠀⠀⠀⠀⠸⢭⣿⡽⠆⠯⣼⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢶⡀⢠⣾⣷⣦⡀
⠀⢸⣾⣿⣷⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠙⢆⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠚⠃⠛⠾⣿⡟⠁
⠀⠈⢹⣿⡏⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⠀⠀⠀⢰⡟⣿
⠀⠀⢸⢹⠃⠀⠀⠀⢸⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠃⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠹⡇⠀⠀⡼⠁⢹
⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⣤⠘⠟⠛⠃⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠛⢡⡄⢸⣷⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢱⡎⠀⠀⢀⠀⢐⢐⠃
⠀⢀⡾⣿⡟⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣾⣿⡇⠀⢀⣼⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⡶⠚⠀⠸⢧⣿⣅
⠀⠈⠛⣻⣷⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠀⠀⣠⣊⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠀⢸⣿⡏⠣
⠀⠀⢀⡿⢻⡇⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡇⠀⠈⠛⠻⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⢸⣿⡇
⠀⠀⠸⠃⠘⠇⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠇⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠸⠹⠃
⠀⠀⠀⠀⢀⣒⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⣆⣉⡉⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠙⠚⢓⣖⣱⠀⠀⠀⣤⠙⠶⠛⠀⠀⠀⠀⠀⠀⢨⣿⣤⣴⣶⢠⣶⡤⢤⣽⣅
⠀⠀⠀⢿⣿⠉⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠰⣻⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠈⠻⠒⠋⠀⠀⢻⣿
⠀⠀⢠⣿⣿⡄⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣷⠀⢠⣾⣿⡅⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⡿⡿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢹⡇⠛⡿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⡇⠀⠀⠀⠀⠀⢸⠃⣿
⠀⠀⠀⠁⠁⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠃⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡟⠋⠛⢳⣆⣀⣀⣤⣤⣤⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣤⣤⣤⣀⣀⡶⠛⠋⠛⣧
⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠒⠚⠒⢲⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠛⠛⠃⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⣼⠃⢀⣤⡀⠀⠀⣄⠀⡄⢠⣤⣄⠈⢁⡀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⡇⠀⢸⡀⡷⠀⠀⣿⠀⡇⢸⠀⣿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣸⠛⣧⠀⠀⣿⠻⡇⢸⠿⡏⠀⠐⠿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣧⠀⢻⣤⡿⠀⠀⣿⠀⡇⢸⠀⣿⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⢀⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⠀⢀⣀⠀⢶⠆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠋⠀⠀⠀⠀⠀⠤⠤⣤⣤⠶⠿⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⣠⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠘⢧⣤⣤⡾⠋⠉⠉⠙⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠛⠛⠉⠉⠉⠻⣦⣤⣴⠟





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀