Hair Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⡀⠀⠁⠀⠀⠀⡀⠐⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢚⠀⣴⠟⢫⣹⡦⠤⡔⠀⠑⢬⣦⣄⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠄⢾⠁⣠⡞⠀⢹⠁⡀⠀⠀⠀⠀⠀⠀⠚⡀⠀⠙⣣⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃⠀⡞⠀⡟⠸⠁⡔⠀⠀⠳⢤⡴⠀⠁⡀⠀⠀⡠⠣⠀⠾⢿⡄⠀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⡠⠊⠀⡂⢀⣤⡃⠈⣇⠀⠁⢢⠀⠣⡀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⣶⡋⠁⣀⠇⢸⡅⠒⠀⠘⠖⠐
⠀⠀⠀⠀⠈⠀⠀⠀⢣⣝⠚⢟⡆⠙⠣⠤⠾⡳⢄⣀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⡹⠁⠀⠈⠄⡀⠐⣄⡀
⠀⠀⠈⠀⠀⠀⢰⠋⠀⠀⢡⠄⠁⠀⠀⠀⠀⠈⣀⡵⠞⠀⠠⣄⠂⠀⠀⠀⠀⢀⣠⡞⠁⢀⣀⣀⠀⡀⠀⠀⠙⠀⠐⠆⢤⣄
⠀⠀⡀⢀⣀⡴⠋⠈⢢⢠⠞⠀⠠⣦⠀⠀⣀⡜⠁⠀⠀⠘⠀⠀⠀⠀⠀⠘⠋⠉⠚⠲⠖⠀⠀⠉⠂⠄⠀⠀⠀⠀⣆⠀⠀⢹⡝⠄
⠀⠀⠚⠛⠉⠀⠀⠀⠈⠀⠀⠀⠀⠫⣤⣴⠟⠁⡄⠰⣤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⡀⠀⢀⠀⡴⠋⠀⢀⠀⠃⢸
⠀⠀⠀⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠄⠜⢇⠀⠀⠀⠀⢠⠀⢀⠀⠀⠀⠹⡄⠀⠀⢢⠀⠁⠀⠀⠀⢀⡀⠎⢀⠀⠂
⠀⠀⠉⠠⠀⢡⠤⠤⠀⡀⢆⠀⣀⠄⠀⠀⡀⣀⠀⠀⠀⢸⣶⡀⠀⢠⡻⣂⡜⠀⠀⠀⠀⠘⠦⠤⡴⠂⠀⢀⠀⠂⠀⠨⠔⠋⠀⣀⡄
⠀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⡀⠀⠋⢁⣀⡀⠀⣼⠃⠀⠀⠘⣾⠯⠀⠈⠀⠀⠀⡈⠀⠀⣃⠀⠀⠀⠀⠠⠀⠀⣴⡤⠊⠁
⠀⡻⡀⢻⡀⠀⠸⡀⠀⠀⣵⠀⠀⠀⠁⠀⢤⠸⣄⣣⠜⠃⠀⠀⠀⢰⠃⠀⠀⠀⠀⠐⠀⠑⠀⢸⣿⠀⠀⠀⠀⢠⡆⠘⠋⠄⠄
⠀⣠⠂⠀⠐⠀⠀⣽⠲⢤⣘⡿⠛⡃⠊⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠐⡆⠀⠀⠀⠀⠀⠻⣦⣀⣠⣴⡏⢠
⠀⢸⠀⠀⠀⣤⠀⣿⠑⠖⠠⢒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠁⠀⡈
⠀⠀⠐⠀⠀⠻⡄⢸⣆⡀⣀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⠀⢤
⠀⠀⠀⠀⠀⠀⢈⠉⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡠⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⣸⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃
⠀⠄



⠀⡀

⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢒⣈⣻⣦⣾⣿⣿⠷⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢋⠉⣡⣥⣿⣿⣿⣿⡳⠆⠠⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣴⣆⡄⣠⡿⠿⠋⠀⠀⠉⠳⣦⡄⠐⢜⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⡇⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢻⣤⣧⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⡟⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢰⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠰⡟⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢾⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⡻⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣟⠛⢌⠁⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⢷⣌⢿⣷⣾⣷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡛⣿⣿⣹⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣷⣦⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣷⣤⡀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠟⠋⣉⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠉⠙⠻⢦⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣯⣴⠶⠛⠋⠉⠴⠶⠶⠟⠛⠛⠛⣛⡻⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠛⠁⠀⠀⠀⣀⣠⣤⣴⣶⣶⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠿⠷⠦⠀⠀⠀⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠃⠀⢀⣤⣴⣾⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⠤⠄⠀⠀⢢⡈⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⠉⠀⠀⢀⣀⡤⠂⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠹⣆⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠀⠀⠀⣠⣴⣾⡿⠋⣀⣤⣾⡿⠿⣛⣯⣿⡿⠛⠉⠀⠀⠀⣀⣀⠀⠀⠀⠘⣷⣄⠀⢻⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣁⠄⠀⢀⣴⣿⣿⡿⢋⣴⣾⡿⠛⣩⣴⣿⠿⠋⢁⣠⣤⣴⡶⠿⠛⠉⠀⠀⢀⠀⠀⠘⣿⣷⣄⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⢀⣴⣿⢟⣿⢏⣴⣿⠟⣡⣴⣿⡿⣋⣥⣶⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢀⣾⣿⢋⣾⢃⣾⠟⣡⣾⣿⣿⣿⣿⠟⣻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠈⢷⡀⣿⡼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣾⣿⠃⣾⠇⣾⢋⣼⣿⣿⣿⡿⠋⠡⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⢣⢻⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⡿⢸⢣⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⢹⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⣾⡇⢠⣿⣿⣿⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠸⣷⣸⠃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢿⡇⣾⣿⣿⠋⠉⠉⠙⠛⠿⣶⣶⣤⡄⠀⠀⠀⠀⣤⣤⣶⡾⠟⠛⠉⠉⠉⠂⠀⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣧⣿⣿⡏⠀⡀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⣿⣿⣿⠀⠀⢙⣦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣖⠁⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠶⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡷⠖⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣉⣯⣹⣩⣯⣱⣍⣍⣏⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣭⣭⣽⣿⣿⣿⣭⣭⣭⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢈⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠙⠉⣩⣿⣟⠁⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⠋⠟⠁⠀⠀⠀⠀⠈⠳⣤⣄⠀⡀⠀⠀⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣠⡴⠞⡿⠿⣿⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢾⣦⣢⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⢀⣴⠞⠋⣠⠟⠀⠁⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⣄⡀⠀⠱⡀
⠀⠀⠄⠀⠀⠀⠀⠀⣀⣀⡀⢀⠀⠀⢀⣾⣶⣶⡟⠁⠀⣼⡏⠀⠀⠀⠀⠁⠈⠛⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⢃
⠀⠀⢀⣠⠴⠚⠉⠉⠉⠉⠉⠀⢠⣚⣻⣽⡿⠻⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠐
⠀⠈⣸⡇⠀⠀⠀⠀⢴⣾⣿⠟⠋⠟⠉⠁⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌
⠀⠀⠹⣧⠀⠀⠀⠀⠀⠉⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⢹⣆⣄⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⢆⡀⠀⠀⠀⠀⠀⢀⢸⣷⠀⠀⠀⡀⢀⠀⠀⠀⣀⣤⡤⠖⠁
⠀⠀⠀⡸⠛⠋⠉⠁⠀⠀⠈⠀⠐⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⠹⡄⠀⠀⠀⠀⢸⣿⢃⣴⣴⣿⡷⠷⠶⠚⠛⠉⠁
⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⢷⣄⠀⠀⠂⠰⠚⠁⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢸⡷⠋⠁
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⣬⣳⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡸⠁
⠀⣿⠀⠀⠀⠀⠀⠀⢀⠀⠴⠛⠋⠉⠉⠉⠉⠙⠛⠛⢿⣦⣤⣤⣤⡤⠆⠊⠀⠀⠀⠀⡠⠃
⠀⠸⡆⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠷⢶⣦⣴⣦⣤⣦⣾⡇
⠀⠀⢹⣠⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠙⣿⡟⢹
⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠁⢸
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⣹⣷⣿⣷⣶⣶⣶⣾⣿⠃⠀⠀⣾
⠀⣼⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⢤⣤⡤⠶⠾⣿⡿⠏⠀⠘⠀⠀⡿⠃
⠀⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠋⠁⠈⠀⠈⠀⢀⣴⠟⠏⠀⠀⠀⣀⠔⠋
⠀⣧⡸⠀⠀⠀⠀⠀⠀⠀⠒⠫⠉⠁⠀⠀⠀⠀⠀⢀⠴⠋⠏⠀⠀⠀⠀⣴⠃
⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠁
⠀⠘⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⣀⡤⡴⠶⠒⠋⠉
⠀⠀⠘⡄⠀⢀⠀⠀⠀⠀⠀⢀⣾⡾⢿⡟⠉⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⡆⣾⠀⠀⠀⠀⢀⠞⠁⠀⠘⣿⠔⡠⢀
⠀⠀⠀⢀⢾⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠀⠀⠀⠀⡟
⠀⠀⡈⠀⠀⠀⠀⠀⢀⠾⠁⠀⠀⠀⢸
⠀⢀⡇⡀⠀⣀⣤⠶⠊⢀⡔⠀⠀⠀⡎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠋⠉⠉⠉⠉⠳⠑⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣠⠴⠒⠛⠉⠀⠀⠀⢸⡆⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢀⡔⣠⡶⢋⡵⢊⣤⠖⠀⠀⠀⠀⢸⠇⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⣴⠟⣰⣿⢡⣿⠕⡠⠄⠀⠀⠀⠈⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡏⢰⣿⠃⣾⡏⡜⠁⠀⠀⠀⠀⠀⠀⠀⡰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⣷⠀⢻⠀⣿⣿⢰⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⢀⢀⣿⠃⠀⢸⣿⡇⢸⣿⣿⡇⠀⠀⢀⣠⡴⠞⠋⠁⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⣻⠵⠂⡀⡼⢻⡇⢸⡿⣿⣿⡄⠀⠙⠁⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡯⠤⠶⠚⠛⠁⠸⣷⣿⠇⢻⣿⣿⣄⠀⠐⠺⢿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⡀⠀⠀⠀⠀⣀⡼⠀⣿⡏⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡞⢁⣠⣤⣶⣶⠿⠋⠁⢘⢹⡇⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⣿⣿⣿⠋⣀⣀⣠⣴⡟⢸⠇⠀⡀⠀⣿⡟⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣿⣿⡿⠿⠿⠿⠟⠛⠉⠠⠋⠀⠴⠀⢠⣿⠇⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣤⣴⣶⠶⠦⠤⠤⠔⣊⣴⣿⠃⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡎⠀⣴⣿⠛⠁⠀⠀⠀⠀⠉⠉⣩⣴⡟⠀⢶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣿⡗⠀⠀⠀⢀⣤⣴⣶⡿⠟⠋⡀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣠⣮⣾⣿⠟⠋⠁⢀⣴⢆⣷⡀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠻⣿⣿⠏⠀⣀⣤⣶⣿⢟⣼⣿⡇⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣴⣾⠿⠛⠋⣡⣾⣿⡿⣱⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋⠀⢠⠎⣼⣿⣿⣟⣿⡿⠋⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⢀⡿⣸⣿⣿⣿⡿⠋⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣧⢻⣿⣿⢋⡄⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⢿⡏⣾⣧⡪⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⣿⣝⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⢛⣻⢿⣮⣍⣙⠓⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡫⢔⣤⣶⡟⣽⡲⢕⣨⣍⣛⠳⢮⣝⢮⣙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⡾⢋⣴⠏⠀⣿⡙⢦⣉⡙⠛⠻⠶⣮⣷⣿⣷⡝⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⡾⠋⢠⣿⠏⠀⠀⢳⣝⠦⣬⣙⡓⠶⢤⣄⠉⠛⢿⣿⣎⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⣿⠁⠀⣿⣿⠀⠀⠀⠀⠙⠻⢶⣮⣽⣛⠲⢮⣝⣦⡀⠙⣿⡎⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡘⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⡉⠻⣿⡄⠘⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠷⣄⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠘⣷⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⣸⡏⡼⣿⣿⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⣰⠟⡰⢱⣿⣿⣧⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⣴⠏⡴⢡⣿⣿⡝⢿⣧⡘⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⣼⠏⡰⢁⣿⣿⢡⡙⢦⣙⠻⠦⣍⡓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⡏⡄⠁⢸⡿⣟⢸⡉⠓⠻⠿⠷⣶⣬⣭⣛⡻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⡇⢸⡄⢸⡇⢿⠸⣯⣑⣒⣒⣂⣒⣀⣀⣈⣉⡁⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠳⡌⣷⠘⣇⢧⣙⠫⠭⣭⣤⣤⣤⣤⣌⣉⣙⠲⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠙⢿⡆⠘⣎⢧⠙⣿⣿⣭⢹⣿⣷⣜⣿⣧⢹⡒⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⣿⠘⣌⠪⡳⢌⣛⣋⣼⡯⢻⣿⣾⡏⡎⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⠀⡿⣷⡚⠶⠖⣚⡵⢁⡟⣿⣿⢳⣿⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⠄⠀⠀⠀⢸⡇⣸⣰⣝⠦⣍⣛⣉⣡⠴⢋⣼⣿⠏⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢿⡀⠀⠀⣠⠞⣰⣿⣅⠪⢷⣦⣤⣤⣤⣶⢿⣯⡅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⣭⣟⣫⣥⡾⢟⡻⢛⡷⡶⣤⡍⠉⠈⠁⢜⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠭⣙⣛⣉⡭⠔⢻⠀⣧⣿⢠⡋⠙⢿⣬⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⣆⣿⣌⠓⠛⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣙⡛⢛⣛⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣴⣾⣿⣿⣯⠀⠀⠀⠀⣀⣀⡉⣉⣛⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣺⣵⠞⠁⢩⡁⢿⠋⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣶⣝⡿⣦⣄
⠀⠀⠀⢀⣤⣶⣿⢿⣿⣿⣿⣿⠃⢠⡅⡄⠳⣌⠐⡀⠀⠀⠀⠀⠈⠻⣿⣍⠻⣧⠙⢿⣿⣿⣧⡀
⠀⠀⠠⠟⡽⢋⣴⠿⡿⠿⣻⣿⡀⠀⠑⣷⡀⡈⠑⠺⢦⣄⠀⠀⠀⠀⠙⣿⣆⠙⣧⠀⠙⢿⣿⣿⣄
⠀⠀⠀⠘⠁⣾⠃⢀⣠⠞⢻⣿⣧⡰⡀⠈⠻⣿⣾⣄⢠⣍⠻⢮⣦⢳⡄⠘⣿⡆⠈⠂⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠁⣿⣷⠟⠁⠀⠀⠻⣿⡻⣿⣦⣀⠀⠙⠻⢦⣝⣷⣄⢻⣧⢻⡄⠸⣿⠀⣄⠀⢲⡘⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠈⠁⠈⠙⠻⢷⣶⣤⣄⡈⠙⠻⣷⣿⣾⣷⠀⢿⡇⢸⣆⠘⣧⢹⣿⣿⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠀⠀⢠⡴⠖⣛⡉⠉⠉⠙⠻⢷⣄⠈⢿⣿⣿⡆⢸⣷⢸⣿⡀⣿⣾⡟⣿⣷⡁
⠀⠀⠀⠀⠀⠀⠀⠶⢶⣦⡀⠀⠀⠀⠰⠟⢿⡿⢻⠿⠋⠀⠀⠹⣧⠈⣿⣿⡇⢸⣿⣼⣿⡇⣿⣿⡇⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡆⣻⣿⡇⢸⡟⣿⢻⣷⡇⢹⣇⣿⣿⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢻⡇⣿⣿⠀⣼⠇⠟⠀⣿⢀⠈⣸⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⣾⣿⣿⡏⢠⡿⠀⠀⠀⠃⣾⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡏⢰⢻⡟⣿⡿⢀⣾⠃⡄⢀⡆⢰⣧⡞⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣷⣤⣤⠀⠀⠀⡾⠀⡿⠈⣽⣿⠇⣼⡏⣰⠃⣼⣧⣿⢏⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⢰⡇⠀⡷⢠⣿⡟⣼⡿⣰⣏⣼⣿⠟⣡⡾⢃⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣧⢸⣿⣱⡿⣱⣿⣾⠟⢁⡴⠏⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⢻⣾⣿⣿⣿⡿⠟⠁⠐⠋⠀⣠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣞⣿⣿⠟⠋⠀⠀⠀⢀⠄⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡼⣿⡿⢀⡤⠀⢀⡴⢃⡾⣳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠁⠀⠀⠀⠀⡼⣷⡹⣿⣷⣿⣴⡵⢋⣴⠟⣴⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⡇⠘⢷⣿⣿⣿⠟⣠⠞⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢰⡇⢀⠎⠻⣿⣿⡞⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣸⣶⡼⢻⣿⡾⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⣿⣿⡇⣾⣿⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃

⠀⠀⠀⠀⠀⠀⠄⠤⠠⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡠⠴⣶⣶⡶⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣾⠟⠛⠛⠿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⣴⣾⣯⡩⣿⣿⣟⠷⢶⣿⣿⣳⣮⣻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⣿⡯⠐⢚⣛⣿⠟⠀⠀⠀⠀⠈⠻⢿⣿⣿⣟⢿⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⣾⡿⣛⣥⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡍⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣾⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⢻⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⣧⠹⡟⣆
⠀⠀⠀⠀⠀⠀⠀⡼⣿⣿⢳⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣿⡆⢷⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠁⡟⡯⣿⣿⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢻⠇⠘⡇⣮⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⠀⠀⢹⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣇⡸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣿⣼⡇
⠀⠀⠀⠀⠀⢸⣇⣤⣿⣏⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠰⡀⢸⣿⡿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⡀⠻⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣴⣳⣼⣿⣷⡟
⠀⠀⠀⠀⠀⠀⣼⠈⠛⣷⣌⣢⣤⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢹⠀⡀⢈⣿⣿⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⡇⠙⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠸⣦⣱⣿⣿⣿⣧⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣿⣿⣤⣿⣿⣿⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠋⣴⣿⡿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣼⣿⣦⡹⣿⣿⣿⣷⡷
⠀⠀⠀⠀⠀⠀⠰⠚⢸⣿⣫⣾⣿⣿⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣇⣿⣿⣧⠹⣿⣿⣿⣟⠁
⠀⠀⠀⠀⣀⢤⣀⣤⣟⣽⣿⣿⢃⢟⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡸⣿⣿⣟⠄⢹⢛⣿⣮⠢
⠀⠀⠀⠌⣥⢚⢟⣿⣿⣟⡟⠃⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣅⠙⠿⣿⣦⣸⡟⡀⠈⢳⡀
⠀⠀⠜⣿⡟⠃⡸⣿⠟⠌⠀⢰⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢏⣻⣿⣿⣷⠶⢶⣿⣟⣷⣿⣤⠊⠁
⠀⠀⠈⡟⣠⠎⣠⢋⠀⠀⣠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡼⣫⣾⣷⣝⢿⣿⠻⠶⣶⣿⣿⠉⠈⢢
⠀⠀⢀⡿⣁⠜⣡⠎⣴⣿⡿⣍⢺⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣵⣿⣿⣿⡟⣷⣽⣀⣧⡈⢿⡃⠀⠀⠸
⠀⠀⢸⣷⢏⣾⢃⣾⣿⣿⣶⣌⣿⣧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣽⣿⣿⡿⡼⠡⣸⣿⢿⣿⣿⣦⣿⡄
⠀⠀⠘⣯⠈⣿⢿⣿⠿⢿⡿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢸⢸⣿⣿⣷⡀⠀⠘⠏⢪⢿⡌⠻⡙⢷⡄
⠀⠀⠀⠈⠀⢸⣿⣿⣷⡼⠃⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠸⡏⠛⣟⠂⠀⠀⠀⠹⣿⣦⣘⣦⣽⣄
⠀⠀⠀⠀⠀⠀⠻⣯⣅⡀⣀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠽⢝⠦⡨⡛⠂⠀⠀⠀⠀⠉⠻⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠡⠞⣽⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣴⠾⢿⣿⡿⣦⣄⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⡠⠋⠀⣠⣿⠉⣷⡌⣌⢳⣄⠈⠢⣀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⣠⠞⢁⣿⡄⢿⣷⠘⡆⠹⣆⢣⡈⢷⣄⠈⠻⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣀⣤⣴⡶⠞⢋⣡⣴⠟⠹⣧⠈⠁⠑⢿⡄⠹⣧⡣⠀⠈⠛⠶⠤⣍⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣾⠿⠋⣁⣴⣾⠟⠋⠀⠀⠀⠹⣆⠀⠀⠈⢿⡂⠙⣷⣄⠀⠀⠀⢀⣠⠔⠂⢙⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁⢠⣾⣿⡏⠁⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠈⢷⣄⠠⣙⠓⠒⢚⡉⠄⠀⢀⢀⣴⠿⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠀⣰⣿⣿⣿⣏⡉⠳⠄⠀⠀⠰⠞⢋⣹⣧⣠⣀⠀⠉⠓⠿⠿⣛⣥⠒⢀⣘⡵⣾⣿⣗⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣿⣿⣿⣿⠿⠿⠷⠀⠀⠀⠀⠞⢉⣿⡿⢷⣝⢿⣷⣶⣿⣿⣿⡛⣟⠻⣿⡿⣎⣿⣮⡁⠚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⣿⣿⡿⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢿⣿⣿⣷⣌⣧⠙⣷⡹⣎⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣿⡿⢡⣿⡀⠀⠀⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢻⣎⣿⠇⣿⡟⢳⡆⢿⣧⢻⡟⢻⡟⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⢀⣾⣿⣷⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢘⡼⠋⢠⡟⠘⢸⡇⣾⣿⡜⣿⡌⣿⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣴⢟⣻⣿⣿⣧⡘⢿⣟⣉⡻⠓⠀⠀⠀⠀⡠⢊⡴⠋⠁⣠⠞⢠⡄⢺⡇⣿⣿⡇⢸⡇⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡼⠃⢹⡿⢿⣿⣿⣷⡀⠈⠁⠀⠀⠀⢀⣴⣾⡷⠋⠁⣠⠞⢁⡰⠏⢠⣿⠀⣿⣿⣇⡼⢁⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡃⠀⣿⠇⣜⢿⣿⣿⣷⣤⣀⣤⣴⣾⣿⡿⠋⣀⣴⠟⣡⠴⢋⣀⣠⡿⢻⠀⣿⣿⡿⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⡄⠹⣦⣹⣿⣿⣿⣿⡿⢹⣿⠋⣠⡾⢋⡴⠛⠁⠴⣋⣥⠔⣠⡷⣾⡸⣿⣷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡿⡄⠘⣧⠐⠘⡿⢾⣿⣿⡿⢃⡾⢁⣼⠟⡵⠋⢀⡴⠛⢉⡴⣻⡟⠁⠀⠈⣷⡹⣿⣿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⠁⢹⡄⠈⢧⠀⠈⣀⣿⠟⢁⣾⡇⣾⢟⣼⠁⣴⠏⠀⣠⡟⠼⣿⠀⠀⢠⠞⠉⢹⣯⢹⡻⣎⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⠇⢠⣿⣿⡄⠈⣧⠀⠟⠁⣰⡿⣿⣧⣿⠘⡇⣸⡏⠀⢰⡿⠀⢰⡇⠀⠀⣍⠀⠀⠀⢻⡀⢳⡈⠳⣤⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣯⠛⠀⣿⠛⣿⡇⠀⢸⡆⣠⣾⣿⠃⣿⣿⣿⣇⢱⣿⡇⠀⣾⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠻⣦⡈⠻⣦⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⠀⢹⡇⣸⣴⠿⠋⡌⡟⢸⣿⣿⣿⣿⡌⣿⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠙⣿⢦⡈⠻⣦⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣤⡼⠟⠋⠁⢠⠞⡼⢁⣿⣿⣿⠸⣿⣧⣿⠀⠀⣿⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠈⠳⡙⠳⣌⡳⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣴⠟⠉⢀⣀⡤⠖⢁⡞⢡⣾⣿⣿⡟⣠⣿⣿⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡻⣦⡀⠙⢦⣄⠙⠻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠷⠛⢛⣉⣩⣤⡾⣫⣴⣿⡿⣱⣿⢇⡇⣿⣿⣿⠀⠀⣿⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣮⡻⣦⣀⠙⠻⢶⣤⣼⠁
⠀⠀⠀⠀⠀⠀⣠⠞⢉⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣿⡟⢸⠇⣿⡏⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⣿⣮⡙⠿⣿⠞⠻⡇
⠀⠀⠀⠀⠀⢠⣿⣾⠿⣛⣭⣥⡴⢾⠶⣶⣶⣛⣉⣴⣿⡿⠁⢏⣼⡿⢠⣿⠀⠀⡇⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢸⣿⢻⡌⢳⣄⡾⢃⡔
⠀⠀⠀⠀⣠⡞⢹⠏⣸⣿⣿⠏⡔⣵⠰⣿⡏⢻⣿⡿⠛⣀⣤⡿⣿⠃⣾⡏⠀⢀⡇⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣠⣶⡿⠁⠸⢣⡾⢏⡵⠋
⠀⠀⠀⠀⢿⡄⠸⣆⠸⣿⣿⡀⡇⣿⣄⠉⠱⣾⣟⣛⣛⣛⣭⡾⠁⡼⣿⠃⠀⣼⡷⠛⢧⣀⣀⣀⣀⡤⠴⠚⠉⠉⡱⠟⠉⠀⣠⢧⣿⠵⠋
⠀⠀⠀⠀⠀⠙⠳⣬⣑⠿⣿⣷⣵⡈⠳⠮⣥⡛⠛⣛⣻⡿⠋⢀⡞⣵⠃⠀⣰⣿⡇⠘⣷⣤⣈⣁⣤⣤⣤⣤⣤⣤⣤⠖⠒⣉⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣍⣛⠛⠷⠶⠾⠿⠶⠛⠉⣀⠴⣋⡾⠁⢀⣴⡟⣱⣧⠟⠋⠉⠉⠉⠁⠀⢠⣿⣿⠏⢹⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠳⣬⣙⡲⢤⣄⣀⣀⣀⣠⡤⠞⣋⣤⡶⠻⣿⠀⣿⡀⠀⠠⠒⠋⣉⣉⣩⡉⠽⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⣉⠋⢙⣿⣿⡶⠟⠋⠘⢧⡸⠘⠳⠆⣉⣭⠖⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣶⣴⣶⣶⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣸⣿⡿⣿⡿⢿⣿⣿⣿⣏⣉⡿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⡴⣽⡿⠃⠹⠁⠀⠉⠙⠋⠈⠉⠋⠀⠈⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⠎⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡎⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠀⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⢸⡇⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣸⣧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⡏⢸⣿⡁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠱⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢈⣿⡿⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⠁⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⡆⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣷⣄⡀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⣧⣿⠁⣦⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⣿⠙⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠞⠉⠀⠀⢿⣿⠀⣿⣷⡄⠀⠉⠻⣷⣦⣀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠃⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣤⡀⠀⠀⠈⠇⠀⠰⢿⣿⣶⡄⠀⠀⠙⡿⢿⣷⣾⣿⣿⣿⣿⠟⠀⠌⠀⠹⣿⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⠀⠀⠀⢰⡟⢿⣷⢎⠋⠠⠀⠠⠛⢿⣿⣿⣿⠃⢀⠀⢀⡄⢠⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠘⢠⣾⣿⣿⣻⣏⢷⡄⠀⠀⣾⠀⠈⠉⣹⠆⡏⠀⣼⠃⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠙⢮⣧⠀⠀⠹⠀⠀⠀⢿⡀⣷⠒⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⢠⡄⢻⣿⣿⡨⢣⡌⢻⡄⠀⢶⡄⠀⠀⢸⣇⠇⠀⣷⢰⣿⡟⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⠿⣟⠂⠙⢿⣇⠘⡇⠀⣿⠀⢸⢧⠀⢀⣿⣿⣄⣾⣸⠘⣿⠁⠋⣾⡇⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣼⢶⡈⠃⠀⠀⠉⠀⠋⠀⢸⠀⣾⠀⠀⣼⣿⣿⣿⣿⡉⠀⠉⣈⣰⠈⣧⠀⠀⣜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⢀⣀⠀⢠⡾⢀⡏⠘⣧⣿⣿⣿⣿⣿⡁⠀⠀⢻⣷⡇⠁⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣿⡇⠘⠃⣼⡇⣾⣿⢿⣿⡟⣿⣟⠁⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠘⠁⠀⠐⣿⣿⣿⡎⢸⣿⣿⣿⡿⠃⠀⢀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠸⣿⣿⡿⠃⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠄⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣭⣟⠲⢤⣤⣶⡿⠿⡇⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣭⣽⡓⠦⣤⣴⣾⠿⠷⠀⣤⣀
⠀⠀⠀⠀⠀⠀⢸⢹⠸⠋⠛⢿⣷⡄⠀⠀⠀⠉⠙⢿⡌⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⠟⠙⠻⣿⣦⠀⠀⠀⠈⠉⠻⣦⠻⠁
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠈⢻⣿⣄⡀⢀⣀⣠⣤⣭⣖⠒⠶⣦⡄⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠙⣿⣧⣀⠀⣀⣀⣤⣬⣷⡒⠲⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣓⠢⠀⢉⣟⠛⠉⡁⠀⠈⢻⣿⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠄⠈⣹⡛⠋⢉⠀⠀⠙⣿⡇⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⢸⡟⣧⡀⢸⣦⡀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠀⠀⣿⢳⣄⠈⣷⣄⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠈⢻⣷⣿⡿⣶⣤⣦⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠹⣷⣾⣿⢷⣦⣶⡴⠞
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢴⣿⡷⠈⣦⡀⣿⡏⡗⠀⢙⠎⣷⠀⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠠⣾⣿⠆⢱⣄⢸⣿⢹⠀⠈⡓⢹⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⢸⣷⠏⢀⣤⣶⠟⣰⠏⣰⡟⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⣿⡾⠁⣠⣴⡿⢃⡟⢡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣔⣿⣿⡏⠸⢡⡾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣾⣿⣿⠁⠏⣴⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠙⢿⣮⡻⣄⢻⡅⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⠙⢻⣷⣝⢦⡘⣯⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⣿⡇⠀⠈⣿⠋⣿⡆⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢸⣯⠀⠀⢹⡏⢹⣷⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠉⠛⢦⣔⠁⣠⣾⣿⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡈⠙⠳⣤⡋⢀⣼⣿⡂⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⢠⡞⣫⠊⠁⢺⣿⣷⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⡴⢛⡖⠉⠐⣿⣿⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠉⠳⡇⠀⠀⡀⠉⠻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡏⠛⢾⡀⠀⢀⠈⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢧⠀⢻⣆⠀⠓⢄⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⢻⡀⠘⣷⡀⠘⠢⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⣠⠞⠃⠀⠀⢀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢡⠴⠛⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉

⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀
⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣼⣿⣿⠁⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⢿⣿⣧⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⠉⠉⠉⢻⣿⡆
⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⢻⠏⠘⣿⣿⡆⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⢠⣿⣿⡇⠻⠟⠁⠏⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠠⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⠀⠀⠀⠘⠉⠷⠀⠀⠀⠀⠀⠀⣼⠷⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠈⠆⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠀⠀⠀⠀⢨⡏
⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠞⢳⣦
⠀⠀⠀⣠⣾⡿⠟⠛⠛⠻⠿⣷⡀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆
⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣄⡀⠀⠀⢀⡾⠟⠃
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣴⣿⣿⣿⡏⠘⢿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⢠⣿⠿⠿⠿⠿⠿⠿⢿⡄
⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⣸⣿⣿⡛⠇⠀⠈⠻⢿⣿⣿⣿⣿⣏⠉⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠟⣿⣿⣇⠀⠀⠀⠀⠀⠉⢻⣿⣿⡿⠃⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠘⢣⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⡿⣿⡉⠀⠀⠀⠀⠀⠀⣾⠿⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⣠⣾⣿⣿⠹⣿⣿⣿⣿⡟⣿⣿⣇⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟⠉⠉⣿⣿⡄⠀⠀⢀⣾⣿⣿⣿⠀⢹⣿⡟⣿⣿⣿⣿⡆
⠀⠀⢸⣿⣿⡏⠀⠻⣿⡇⠸⠁⢹⣿⡟⠀⠀⠀⣿⣿⡿⡏⢻⣿⠁⠀⠀⢸⣿⣇⠀⠀⠊⣿⣿⣿⠙⠀⠀⡟⠀⠘⣿⣿⣿⠁
⠀⠀⠀⢿⣿⡇⠀⠀⠘⠀⠀⢀⣿⡏⠃⠀⠀⠀⢹⣿⡇⠉⠀⠛⠀⠀⠀⢸⣿⠏⠀⠀⠀⠟⣿⣯⠀⠀⠀⠀⠀⠀⢹⣿⡿
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⣞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣾⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⢿⣟⣛⣛⣛⣛⠾⠿⣽⣂⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⣯⣭⣭⣭⣭⣿⣛⡷⣶⣍⡛⢦⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣭⣍⣙⡻⠿⣿⣶⣍⠻⢦⡙⢦⡹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠦⠄⠀⠉⠛⠿⣿⣷⣬⢻⣿⣆⢦⠙⠂⠳⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⣦⡙⢿⣿⣿⣿⡞⣷⡱⡄⡉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠹⣿⣿⡇⢹⣧⢹⡼⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⡄⣿⣿⡇⣾⣿⡄⣿⣿⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢸⡿⢡⣿⣿⢡⣿⡿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣡⣿⣿⢏⣾⡿⢡⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣳⣿⠋⣠⠟⣡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢏⣼⢟⡅⢈⡵⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⣾⣏⣾⠁⣾⢣⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⡘⣿⣾⣿⡈⠀⢈⢅⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⢷⣅⠙⣿⣿⣧⡈⢂⣾⣞⣣⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⣜⢿⡌⠻⣷⣦⡙⢿⣿⣶⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠲⠀⣹⣷⣌⡀⠢⢝⡻⣦⠹⣆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢩⣨⡾⢟⣿⣷⡜⠿⣻⢨⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠛⠀⠈⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠀⠀⠀⠀⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣻⡍⢛⣀⢀⣀⠀⡀⡀⣠⢄⡀⡀⠀⢀⣀⠀⣀⡀⣀⡰⠂⠀⢠⠆⣀⡀⢀⢀⣴⡃⡠⢀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠚⠹⠗⠋⠻⠚⠛⠓⠇⢛⡿⠀⠀⠋⠻⠚⠻⠛⠛⠇⠀⢰⡷⠛⠻⠒⠞⠚⠻⠙⠗⠻⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠡⣉⠒⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⡶⣚⠋⠁⢀⣀⠀⠀⠀⡀⠈⠓⠈⢻⡄⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣨⠞⢉⠔⡹⡡⠋⠀⠀⠀⢲⡅⠀⠁⠀⠀⢇⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡁⠀⠀⡘⠰⡇⣴⠀⠀⠀⠊⡇⢠⠀⠀⠀⠈⠀⣰⡔⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢉⢸⠁⠀⠀⡃⠀⢻⡈⠀⢀⡀⣄⠱⡘⣆⠀⠀⠐⠊⢁⣀⡀⠁⣽⠀⠀⠈⢠
⠀⠀⠀⠀⠀⠀⡤⠂⡟⠀⠀⠀⠈⡌⠀⠳⡄⠉⠓⢤⣽⣦⠈⣍⢸⠂⢀⠨⣓⠦⠔⠚⣉⡤⠶⢦⣿⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⡎⠀⠈⡀⠀⣠⠤⠚⠁⠀⡀⠸⠢⠈⢢⠀⠈⢢⣸⠏⡀⠀⠐⠢⢀⣉⠉⠁⠄⠀⢸⢿⠠⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⣸⠞⠁⢀⣀⡤⠞⠁⠀⠀⠇⠠⠀⠀⠘⢞⠄⠀⠀⠁⠀⠀⠈⠁⠁⠀⢀⢾⠀⠀⠀⠑⡇⣷⠁
⠀⠀⠀⠀⢀⠼⡖⢻⣿⡤⠤⠴⠂⠀⠀⠀⠀⢀⢇⠆⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⣹⡧⠐⠀⠘⡇⠹⣄
⠀⠀⠀⠀⢆⠀⡹⢸⡗⠋⠉⠉⠉⠉⢁⣀⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⠀⣳⠀⠀⠣⣀⠀⠉⠑⠒⢄
⠀⠀⠀⢀⡠⡿⢱⣏⠄⢀⡴⠟⢓⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⠈⠀⠐⠢⣌⠋⠁⠁
⠀⠀⡴⠁⠀⠀⠨⣾⡀⡊⠀⡠⠋⠁⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠱⠀⢦⣀⠀⠱⡌⠀⣦⠀⣀
⠀⠈⠁⠀⠀⠀⠌⠈⡛⢇⢠⠁⠀⡞⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⣦⠙⢶⣠⣀⠀⣾⠀⠘⡀
⠀⠀⠇⡀⠀⠀⠀⣀⡀⠀⠎⢀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⡄⠀⠻⣹⠀⡇⠀⡀
⠀⠀⡠⠶⠖⠛⣹⠃⠀⠀⠠⣈⠳⢤⡘⢦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⡏⠀⠀⠀⠀⡟⠓⢌⠀⠐⠄
⠀⠠⠀⠀⠀⠀⢇⡀⠀⠒⢤⠀⠀⠀⢹⢨⡏⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠘⡅⠂⠀⠀⠀⠹⣦⠀⢳⡴
⠀⠀⠑⠈⡆⠀⠀⠉⢳⠀⠀⡃⠀⠄⡵⡿⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡠⠛⠂⠂⠀⢠⡴⣹⡗⡿
⠀⣀⣀⡴⡟⠐⠀⠃⣸⢃⠴⠁⠠⠀⠼⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢴⣯⠞⠁⠀⢀⢠⠗⠉⣴⠃⠀⠁
⠀⠀⠈⡂⢰⠋⠀⠀⢰⡃⠀⠀⠀⡀⠣⣀⣈⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢠⠏⠁⢁⠀⠀⠜⠀⢀⡀⠣⡀⠤
⠀⠀⠀⠐⠨⠀⢀⡀⠈⠓⠀⡀⠘⡧⢀⠀⠈⠙⠄⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠸⠀⡐⠋⠁⢀⡴⠗⠛⠁⣄
⠀⠀⠀⠀⠀⡈⠘⠣⡴⢂⡴⠡⠊⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠛⢄⢰⡘⠄⠀⠀⠙⠢⠁⠠⡈
⠀⠀⠀⠀⠀⠦⣄⠀⠀⢯⠀⠀⠀⣄⠀⡊⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢌⡼⠁⠢⡘⡄⠀⠀⢘⡼⠁
⠀⠀⠀⠀⠀⠀⠸⠀⠠⠈⠛⠲⣄⢳⠂⢣⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠖⠉⠀⣀⣤⣾⠃⠀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡾⠁⠀⠀⠈⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢇⠀⠀⢎⡴⠛⠁⢀⠴⠋⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠀⡏⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⢿⠀⠀⠘⠁⠀⠀⠈
⠀⠄⠀⠀⠀⠀⠀⠀⡡⠄⠈⠣⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠘⡆
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠆⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠛⠛⠛⠛⠛⣉⣽⣿⣿⣿⣿⣿⣿⣿⣿⠋⡎⠀⠀⠉⠁⠀⠹⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠜⠁⠀⠀⠀⠀⠀⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢰⣿⣿⣿⡿⢡⠇
⠀⠀⠀⠀⠀⠀⢀⣼⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠸⠿⣿⣿⠇⠈⠣
⠀⠀⠀⠀⠐⠒⠋⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢹⡿⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⠇
⠈⠲⠶⣶⣾⣿⣿⣿⣿⣿⠿⠿⢛⣿⠟⠃⠸⣿⣇⠈⢟⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠏⣿⣿⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠙⢿⣆⠀⠀⠉⠻⠿⣿⣿⠿⠋⠀⣿⣿⣿⠀⠘⢿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⢀⠀⠀⠀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠟⢿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠻⣟⠁⠈⠙⠁⠀⠸⠟⠁⠀⠀⠀⠀⢻⣿⡏⠻⠿⣿⠙⣿⣿⡟⠀⠀⠀⠀⢸⣿⠋⠀⠀⠉⠉⠙⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠀⠀⠙⠀⣿⡿⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⢠⣿⣿⣇
⠀⠀⠀⠀⠀⣤⣤⣶⣾⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⢙⡗⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⠀⣾⣿⣿⣦⠀⠀⢰⣿⣿⣿⡿
⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠀⠘⢿⣿⣿⡇
⠀⠀⠀⠀⣿⡿⢿⠟⢿⣿⡿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠹⣿⣿⡀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣷⣄⡀⠀⠀⠀⠀⠀⢸⣿⠻⠃⠀⠀⠀⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠘⡿⣿⣿⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⢠⣿⣿⠟⠉⠛⠛⠛⢿⣿⣇⠀⠀⠀⠀⠙⢿⣷⠀⠀⠀⢰⣿⡿⠃⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠰⣿⡇⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⠿⣿⣿⡅⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣾⣿⣷⣄⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⢿⣿⣿⡿⠁⠀⠈⠻⣿⣿⣿⠄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⢠⣿⣿⡿⠿⠟⠙⢿⡇
⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠹⣿⠉⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⢻⠃⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⣾⣷⣿⣟⣷⣾⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⢀⣀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠙⠿⠿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠘⠋⠉⠉⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠋⠀⠐⠷⠶⠓⠀⠀⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠎⠀⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣯⠁⠀⠀⠀⠀⠀⠈⠀⢠⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⢣⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣰⣿⣿⣿⣿⣷⣶⣶⣶⡞⠋⠉⢸⢸⠁⢸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢸⣾⠀⢸⣿⡟⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠸⣿⠀⠈⣿⣿⡈⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢿⡆⠀⠸⣿⣧⠀⠙⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⡿⡀⠀⠘⣿⣧⡀⠀⠈⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣇⢳⠀⠀⠈⢿⣷⡀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⢸⠀⢃⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠈⡆⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⢀⣤⣶⣿⡭⠭⣭⣉⠙⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠁⠀⢀⠄⣴⣿⠟⠋⠀⠀⠀⠀⠉⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⡾⣻⢋⡴⣊⡭⠽⠿⣶⣄⠙⣦⣙⠲⠤⡀⡀⢀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⢣⡞⢰⣧⢋⢞⣡⠴⠚⠛⠓⠾⣧⢞⣟⠻⣖⡒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⢀⡔⠀⢠⢣⡏⠀⣿⣧⠾⣻⠕⠁⠀⠀⠀⠀⠘⢿⣎⢧⣈⠙⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡁⠳⠦⠖⢋⠀⡠⢣⡟⢠⡞⣫⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠳⣝⢦⡀⠉⠻⢝⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠀⠀⠀⠀⢀⠙⠦⠄⠚⣁⠔⢡⣿⢡⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢧⡙⢦⣄⠀⠑⡝⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠐⠤⠀⠀⠁⠒⠂⠈⡠⢀⣾⢇⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠈⠳⣝⢦⡙⢷⡀⠈⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣠⣾⡟⣼⣿⠛⠛⢷⣦⠀⠀⠀⠀⠠⡾⠟⠋⠉⠉⠉⠒⠤⠀⠀⠹⣎⠻⣦⡙⣆⢸⠁
⠀⠀⠀⠀⢠⠖⠲⣄⠀⠀⢠⠀⠀⠀⢀⡀⢀⣠⠾⢋⡿⣼⡏⢸⣤⣤⣤⣤⡀⠀⠀⠀⠀⢀⢥⣶⠾⢷⣶⡶⠂⠀⠀⠀⢹⣷⡈⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠈⢧⣀⠀⠀⠀⠀⠀⣠⣾⣿⠧⢬⣿⡟⠳⠟⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣆⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣈⣙⣛⣿⣿⣿⡿⢱⠏⢠⠞⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡀⢸⡿⡇⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⡽⢟⣫⡾⠋⣰⢏⡴⢋⡴⢻⡇⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⠋⠐⠁⣸⠁⡟⢸⡿
⠀⠀⠀⠈⠉⠉⠉⠀⣠⡴⠞⢋⡡⢖⡿⠋⢀⣼⣷⣫⠴⠋⠀⡠⣿⡀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⠞⢹⣴⡇⣸⠃
⠀⠀⠀⠀⠀⠲⠒⠋⠁⢠⠔⣩⠞⠉⣀⣴⠿⢟⡋⠕⣊⣴⠾⢋⠽⣷⠀⢀⣤⠬⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⡇⢸⡟⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⠞⣁⣴⣾⣭⠖⢉⣡⠴⠚⠉⢀⣠⡤⠚⡹⣧⠀⠠⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢠⢠⣟⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡜⢰⡷⠋⣹⠏⢡⢋⡶⠛⠛⠿⠿⠿⣛⣩⡴⠛⠁⠘⣧⠀⠙⠓⠀⠀⠀⠀⠀⠀⢀⠤⣶⠟⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢰⠁⣿⠀⠀⣿⠀⡏⡾⠁⣠⣴⣶⠿⠟⠛⠉⠀⣠⣤⣶⠟⣧⡀⠀⠀⣀⣀⠤⠂⠀⠀⣼⠋⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠹⣄⠈⣿⠀⣿⢇⣾⡿⠋⡠⠀⠀⣠⣴⣫⠵⠋⢁⡴⢈⡝⣩⢹⡇⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⣆⢀⠙⠳⣜⣧⣿⣸⡿⠁⡜⠁⢀⣼⠟⠋⣀⣤⠶⠋⣠⣾⢞⣿⢸⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣕⣀⠀⠈⠙⣿⡇⢠⡃⢀⡞⠁⣠⡾⠋⠁⡰⣼⡿⢡⠎⣾⢸⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⢿⡌⢧⢸⠀⢰⡟⠀⠀⣼⢡⣿⠁⡎⣼⣿⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣮⣿⡀⢸⡇⠀⠀⡇⡼⣿⣸⠀⡇⢿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣿⣦⡀⢳⣧⠹⣿⣆⢻⡘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣍⡓⠽⢿⣦⣕⣄⡉⠛⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠀⠉⠙⠛⠓⠂⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠉⠁⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⠴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠠⠤⠤⠤⢀⡀⠀⠘⡆⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠋⠁⠀⠀⠀⠀⠀⠀⠈⠑⢾⣧⣄⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢙⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠉⣀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⣠⠆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣒⠂⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⣠⡔⠊⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⢀⠖⠁⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⢇⣀⢀⣀⣀⢀⠀⠀⠀⠈⠑⠢⣦
⠀⠀⠀⠀⠀⣧⠐⠈⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠌⠀⠉⠀⠀⠈⠢⣀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⣀⠠⠈⠉⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠔⠂⠉⠁⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠠⠀⠁⠀⢀⡠⠐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠉⢀⣠⡤⠴⠒⠉⠀⠀⠀⠀⠀⣠⠤⠚⠁⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⠀⠀⠀⢀⠠⠔⠋⠈⠀⠀⠀⠀⠀⣀⠀⠄⠀⠊⠁⠀⠀⠀⢀⡠⠖⡛⠁
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡄⠀⠀⠀⠀⠈⢶⠁⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠔⠂⠉⠀⠀⠐⠀⠀⠀⠀⠀⡔⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠀⠀⠘⠄⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⠲⣀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⢀⡟
⠀⠀⢀⡀⠄⠂⠉⠀⠀⠀⠀⠀⠀⣀⡀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠢⡀⠀⠀⠀⠀⠐⢤⡄⠤⠊
⠀⡠⠋⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠀⡇⢀
⠀⢃⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⢋⣀⡠⠂⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⠁⠀⡄
⠀⠀⠐⡀⠀⠁⠀⠀⠀⠀⠠⠒⠉⠉⠁⠙⠉⠂⠈⠉⠈⠀⠀⢀⠆⠁⣠⠠⠄⠀⠲⠖⠉⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⡀⠊⠀⠀⠃
⠀⡀⠞⠀⠀⢀⠀⠀⠀⠀⡖⠀⠀⠣⠀⠀⠀⢀⠔⠁⠀⠀⠀⠘⠀⣇⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠛
⡆⠠⠔⠒⠂⠈⠀⠀⠀⠀⢂⠀⣀⡠⠃⠀⡠⠀⠀⢀⠀⠀⠀⠀⠀⠉⠣⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⡀⠀⠠⠊⠀⠀⢀⠀⠀⠂
⠁⣄⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣀⣀⠀⢀⡀⠠⠄⠊⠀⠀⠀⠀⠀⠀⠹⡆⠼⠆⠀⠀⠀⣇
⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠋⡄
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠠⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡴⠋⠁⠁⠀⠈⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⡐⠀⠀⢀⠀⠀⠘⡀⣀⠂⠀⠬⠋⡀⠐⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠃⣄⣇⡀⠀⠀⠀⢰⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡌⠀⠀⠀⠀⠀⠀⠒⠶⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢉⠃⠀⠲⠞⣃⣀⡀⠐⠁
⠀⠀⠀⠀⠲⠦⠚⡁⠠⠂⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠒⠐⠒⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣯⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣧⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠂⣨⣬⣭⣅⡘⢦⡄⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣎⣥⣬⡉⠳⣦⡉⠻⡓⣿⣼⣿⠇⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠉⣆⢹⣯⠂⠃⢸⡿⣿⡆⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⡀⣿⣾⡿⡄⠀⣼⣿⣿⠁⣼⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⣄⠀⠀⢤⡿⢿⣿⢡⣾⡻⣿⣧⣾⢿⣷⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠈⠙⣍⠻⣼⢻⣷⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⣠⡟⢠⣿⣷⣌⠳⣿⡙⠝⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣽⡹⡖⠀⠀⠀⠀⠀⠘⢿⣦⡙⢿⣽⣿⣇⠸⣿⡦⡉⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣽⣿⢸⣄⠀⣀⣠⠀⠀⠀⢹⣿⠆⣽⠟⠃⣼⠈⣷⠘⠈⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⠟⠛⠋⠁⠀⠀⠀⠀⠀⣠⢏⣡⠞⡏⠠⠚⢋⣼⠏⢀⣴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣿⣆⠑⣄⠻⣿⠁⢠⣿⣿⣾⡏⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠻⣿⡇⢈⣇⣿⣷⣬⠙⣿⣎⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⣴⢟⣵⣾⣿⣿⣿⡿⢃⣼⣿⡇⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠤⠖⠒⠋⠉⠀⠀⠀⠀⣈⢿⣿⣾⣿⢿⣿⣿⡏⢰⣟⣵⣏⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠾⠋⠀⢻⣿⠛⣿⡟⠙⢻⣦⡉⢯⣿⠗⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⢉⠏⠀⠀⠀⣤⡴⠋⣻⡇⠈⢻⣴⠟⠩⠖⠛⡷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠶⠶⠶⠶⠿⠿⠟⠁⠈⢻⣇⠐⣟⠠⠞⠋⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⣤⣀⣠⣀⣀⣀⣀⣀⣀⣤⢹⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⣦⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣘⣿⣬⣿⣾⣿⣷⣾⣟⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣹⣻⣿⣿⣿⣿⣿⣷⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢹⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢋⣽⣿⣿⣿⡿⢜⡇⠹⣝⢿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣫⠞⣤⣓⠁⠘⠁⠀⣁⣤⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⠀⠰⣤⠀⠀⠀⠀⢴⠖⠀⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡇⠀⠀⠀⠀⡇⠀⠀⠀⢰⣥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠿⡀⠀⠀⣀⣀⡀⠀⢠⠟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠃⠙⣄⠈⠛⠛⠁⣠⠋⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠛⠛⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠞⠀⠀⠀⠀⠙⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠻⢅⠀⠀⠀⠀⠀⠀⠀⠀⠈⠝⠒⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠤⠄⠒⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⣟⣳⡅⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⣄⡀⠀⠀⡀⢀⣇⣀⠩⡛⢤⢁⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡘⢇⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⣻⣷⡀⢀⡽⣼⣟⡹⠗⠟⠺⠟⣫⡭⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢳⣌⠇⠀⠀⠀⠐⠀⠀⠀⠀⢅
⠀⠀⠀⠀⢰⣿⣿⣷⠀⡀⢈⣿⠐⡒⣴⡟⠋⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⢤⣬⣅⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣶⣄⠐
⠀⠀⠀⠀⢸⣿⣿⣿⢈⣳⣫⡵⡀⣩⣏⠙⣣⣴⠟⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣡⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⣼⣟⠟⢻⣇⢿⣽⢿⣥⠽⠿⣿⠻⣷⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠁⡀⠀⠀⠀⠀⠀⠀⣀⠀⢰⠀⠀⠀⠀⠐⣣⡐
⠀⠀⠀⠀⠈⢻⡌⠘⠟⠟⢿⠯⠹⡦⢶⣿⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡌⠂⣄⠀⠀⠀⠈⠁⡆⢸⣦⡀⠀⢁⣨⠄
⠀⠀⠀⠀⠀⢈⣿⣄⠀⠀⢻⡙⠓⠀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣟⢀⠀⠂⠀⠀⠀⠀⡃⣼⠀⢿⡄⠀⡱⢀⡃
⠀⠀⠀⠀⣰⣿⣯⣽⡻⢶⣤⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠶⠤⠈⡀⣠⣾⣿⣿⠶⢹⠋⢰⡧⠾⠁⡀
⠀⠀⠀⣾⣿⣿⣧⡉⠓⢪⣅⣿⡄⢻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⣿⡷⢠⡼⠿⠕⠼⠋
⠀⠀⠀⣿⣿⣿⣿⣷⣄⢸⣿⣿⡇⢸⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠷⠴⠾⠐⠀⠀⠀⠁⠀⢒⣀⣥⠂⠀⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣄⢸⣿⣧⢸⣿⣿⣿⣿⣿⣿⣯⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣄⣠⣶⡛⠛⠾⠷⣿⣿⡍⣀⠂⠄⢐
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⡐⠄⠙⢿⣿⣿⣿⣦⣤⣈⠉⠈⠀⠐⠀⢶⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠁⠀⠉⠉⠉⠁⠈⠉⠉⠀⠈⠁⠈⠈⠁
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠠⠀⠔⣖⣠⡄⠠⡄⢠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡋⠉⢁⡄⠀⠈⠁⠈⣥⠀⢳⡀⡠⢄⠿⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠰⠁⢀⣀⡀⠰⠉⠁⢀⠆⠈⣘⡛⠢⠄⠠⣀⠾⠓⠀⠀⠋⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠾⠽⠁⡀⢰⢯⠆⢀⡀⠊⠠⡿⠉⠆⠀⢸⠗⠢⡀⣱⣄⠠⠃⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⠀⡀⠈⢀⡘⣤⡀⠁⡀⠀⣶⡄⣀⠈⡀⠀⠠⣀⠆⠈⠌⠈⠲⡑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣎⡄⢰⣅⠀⢸⣀⠣⠀⢀⣨⡁⠉⣉⠀⠀⢳⡾⠂⠀⠤⠀⠒⢆⠰⢧⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠁⢀⠘⣎⣀⠸⠭⢠⣄⡼⠀⠈⠉⠙⠤⢠⣄⠑⠄⠀⠈⡆⠰⣾⠰⡞⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡃⣀⡈⠫⠄⣸⠏⠀⣀⣀⠀⠀⠀⠀⣀⣈⠑⠚⢧⡈⠇⠀⡸⢀⠎⠴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠢⠈⠳⢇⡀⣼⠀⠀⠀⢩⡍⠀⠀⠀⠀⢉⡍⠀⠀⠀⡌⣀⢣⠀⢤⢠⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢣⡀⢲⡤⠼⡆⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠃⡹⢃⠑⡸⡴⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠑⠈⣄⣶⠶⡀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡜⢀⣀⡹⠋⠴⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠌⢭⡀⢀⠱⡄⠀⠀⠀⣀⢀⡀⠀⠀⠀⡰⣥⡴⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢠⣧⠈⢢⡀⠀⠀⠀⠀⠀⣠⡎⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠑⠒⠒⠒⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠒⠒⠒⠒⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠤⠤⠤⠤⠄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⠴⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠤⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⡞⠧⢤⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣤⡤⠤⠽⡀
⠀⠀⠀⠀⠀⣧⣀⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⢩⣭⠉⢡⡦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⡇
⠀⠀⠀⠀⢰⡇⠀⠉⠉⠛⠒⠒⠒⢲⠶⠶⠶⢶⠷⠶⠲⡗⠒⠒⠒⠚⠋⡏⠉⠉⠁⠀⠀⠀⢣
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⡞⠀⠀⠀⢷⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⡼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⠀⢸
⠀⠀⠀⠀⡗⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡆
⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢸⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠉⠛⠧⣶⣄⣠⢀⡀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠒⠛⠛⠋⠉⠀⠀⠀⠀⠉⠛⠷⠦⠤⠤⠭⠶⠖⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠀⢞⡹⣿⣿⣿⣿⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠘⠁⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⠷⠒⠒⠶⠄⠀⠠⠶⠒⠒⠀⣽⣿⣿⣿⣿⢿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⢿⠐⠛⠿⠳⠀⠀⠀⠾⠿⠟⠀⠯⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢟⣽⣿⣿⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⢿⣿⡇⠀⠀⠈⢂⠀⡐⠀⠀⠀⢸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢻⠿⡿⣄⠀⠑⣁⠀⣈⠊⠀⣠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⡦⣄⠀⠀⠀⡠⢾⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠉⠉⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢲⢯⣧⣀⠀⠀⠀⠀⠀⣀⣼⡽⡖⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠶⠟⣿⣍⠈⠣⡒⢌⠛⠫⠷⣶⠾⠟⢚⡡⢒⠍⠀⣩⣿⠳⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢀⠙⣿⣦⣀⠉⠙⠫⣹⣶⣙⣾⣏⠕⠉⠁⣠⣾⣿⠉⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠄⠻⠻⣷⣦⣤⣀⣀⣀⣀⣀⣤⣴⣾⠟⢟⠠⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣐⠡⢀⠋⠉⠿⠛⠿⠉⠙⠀⠀⣂⡥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠥⠤⠬⠤⠒⠚⠉⠁

⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣻⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠰⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⣠⠐⠀⠀⠀⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤
⠀⠀⠀⢠⡃⠀⠀⠠⢠⣾⠷⠂⠀⠀⠀⠀⢠⠂⠀⣀⠟⠀⠀⠀⠀⢀⣶⢻⠆⡀⢠⡀⠀⠀⠀⠀⠘⠻⠛⠀⠐⠀⠀⠀⢀⡀⡀
⠀⠀⠀⣼⡇⠀⠀⠰⠀⢉⢀⡀⠀⠀⠠⠀⠠⠁⠘⡏⠉⠋⠀⠀⠀⢸⠇⠀⠀⠀⠘⡏⠀⠐⠀⠀⠠⣾⠃⠀⠀⠀⠀⠀⢺⠁
⠀⠀⠠⢩⣆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⢉⠁⠀⠀⠀⠀⠘⠒⠀⠐⠂⠀⠀⠀⠀⠀⡤⠐⣿⣆⠀⠀⠀⠀⠨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣠⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣠⣾⡿⠻⠯⣝⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣴⢿⣿⡗⢌⡛⠶⣝⣾⣯⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣸⣿⢸⣌⢿⣶⢭⣒⢬⡙⢿⣷⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠀⠸⣹⢦⡙⠷⣬⣛⢿⣟⢿⣆⠹⣧⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠙⠳⣍⣓⠶⣬⡓⣿⡆⢻⡄⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⡤⠈⢹⣼⣿⣿⠃⣸⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠠⢄⠀⠀⠳⠦⠞⣻⡿⢃⡴⡣⣾⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⣼⠋⣰⢏⢔⣼⣿⣿⡌⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⣸⢣⠀⣾⡟⢎⣷⣝⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡇⢼⢸⣿⣼⣦⠍⡛⠳⠮⣽⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣻⡌⣾⡏⡷⡍⣻⣟⣛⣒⠲⠬⣈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠸⡇⡘⡧⡹⣿⢝⣿⣿⣷⡶⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠠⠀⢸⢃⣧⡿⣖⠮⢕⡻⠩⣿⣿⣽⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣦⣀⡤⣫⣾⠻⣌⡲⠭⠥⣒⣵⡿⣹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣿⠿⢟⣛⠷⢮⡍⠛⠻⣻⡗⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢉⢰⣼⡿⣼⠉⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣝⣷⣶⡶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⢩⠤⣠⣬⡛⠢⡀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⡟⣾⠀⠈⠉⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⢺⠏⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠌⠀⡸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡇⠙⣹⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠿⠟⠝⠟⢦⣄⠀⠀⠀⠀⠣⡜⠘⣇⢕⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢐⣦⡀⠑⠹⡇⠀⠀⠀⠀⡇⠰⢸⢕⡮⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠉⢻⣿⡄⢨⡀⢔⠀⢀⠘⡳⠀⠆⢹⡹⣹⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢄⠹⣦⣤⡘⢿⣧⡀⠛⢶⠟⠑⠀⠄⠀⠀⢊⣧⠐⣻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠳⣮⣿⠋⠙⢿⡇⠉⠒⠀⠠⠀⠀⢀⠀⠀⣿⡐⣈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠈⢻⡄⠀⢸⣿⣄⠀⠀⠀⢀⠴⠂⠀⠐⣿⡇⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⢳⠀⠀⣿⠋⠁⠈⠁⠀⠀⠀⠀⢘⣿⠀⠐⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⠻⣧⠀⠘⢧⠀⠀⠀⠀⠀⠀⢀⣾⠃⢀⣥⠏
⠀⠀⠀⠀⠀⠀⢠⣾⠿⠲⠢⢄⡼⣶⣤⣀⣀⣀⣴⢻⡗⡀⢹⡆⠀⠀⠑⢶⣆⣀⡀⠀⣋⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⣾⠁⡀⠀⠀⠀⣑⢟⣿⣿⡿⠫⠄⣺⠃⣷⠀⢹⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⡏⠈⠀⠀⣀⡀⠈⠙⣕⢮⣿⣆⡁⠋⣰⡇⠀⡘
⠀⠀⠀⠀⠀⠀⠘⠀⢀⢼⡍⠙⠳⣤⣉⢧⡈⠘⡷⠾⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡀⢻⡿⠂⠀⠀⠀⠉⠛⠒⠒⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁
⠀⠀⠀⠀⠀⠀⢀⡔⡠⠁
⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⣶⠶⣶⣤⣠⡾⠿⠻⠿⢶⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⠻⠿⠏⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄
⠀⠀⠀⠀⠀⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠟⠛⠁⠀⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⢼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⢘⣿⠀⠀⠀⠀⢀⣤⣦⣄⣀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⢀⣀⣀⠀⠀⠀⠀⠈⢻⣧
⢀⣾⠃⠀⠀⠀⠀⣾⡏⠀⠙⣿⣉⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣽⠛⠋⠻⣷⠀⠀⠀⠀⠀⣿
⣾⠃⠀⠀⠀⠀⠀⢻⣧⣀⠀⠈⠉⠉⠀⠉⠛⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠛⠛⠁⠀⢀⣰⡿⠀⠀⠀⠀⠀⢹⣦
⠻⣧⡀⠀⠀⠀⠀⠀⠉⠛⣿⡄⠀⠀⠀⠀⠀⠀⢠⣄⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠀⢸⡟
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠸⣧
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⢶⣶⡶⣶⢶⣶⡶⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠘⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠙⠷⣶⠾⠛⠀⠀⠀⢀⣠⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠈⠉⢿⡆⠀⠀⠀⠀⠀⠀⡀⠀⠉⠻⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠘⢷⣦⣶⠿⢶⣶⠿⠛⢷⣶⠾⠷⣿⡇⠀⠀⠀⠀⠀⠀⣾⠿⣶⣶⣶⣤⣤⡶⢶⣤⣤⡶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠈⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠻⠷⣶⣶⣶⡾⠟⠛⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⠉⠛⠿⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢻⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⡏⠉⠛⢷⣦
⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠃⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⣸⡿⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠉⠻⣦⡀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⢸⡏⠀⠀⠀⣴⡟⠀⠀⠀⠉⠛⢷⣦⣄⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠸⠇⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠘⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠀⡀⢴⡶⢴⠶⡰⣶⠄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣀⣶⢉⡗⢥⡽⠟⠙⡆⠐⣇⢲⣇⣨⡄⠀⣄⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣞⡁⢨⢿⣇⡇⢈⡘⣧⢧⣠⡍⣹⣯⣹⡗⢸⣿⡿⠹⣾⢧⠶⣄⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⣽⡉⠈⣵⢲⠄⡻⣁⣹⢵⣷⢪⡶⣿⢻⣍⡠⢉⣽⢾⠀⣼⣭⡞⣸⢯⣿⣷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡝⣶⣌⡿⠂⠩⣴⠡⣧⠈⣿⢼⣄⣝⡞⣼⢡⣿⡗⠋⣴⣷⣾⣶⣿⣶⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢞⢻⣿⣿⣾⣦⡤⠲⣗⡻⣶⠳⣤⠻⡏⠽⣿⣋⢿⢁⠍⠟⠾⢛⢯⣹⣿⡟⢮⣼⠃⣝⣳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣫⣀⠏⢹⣿⣯⠘⢃⡈⣿⡿⣆⢼⠓⣍⣼⣿⣿⢻⠚⣤⠔⢶⡟⢳⣏⣴⣿⡾⣷⣷⠞⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⢋⠹⠈⢃⡸⠻⣞⡿⢿⡹⣳⣏⡙⠶⠿⣿⡿⠐⣄⢤⠇⣾⣾⣷⣿⣿⣿⣿⠑⣬⠿⠟⡁⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡔⠉⢻⠓⣾⡤⠎⡀⢳⣴⡡⣅⣄⣋⠀⢄⣄⠘⣶⣿⣰⣦⣿⣾⣿⣿⢿⡟⣾⠿⣿⣟⠃⣼⣾⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⠴⡿⢾⣷⣶⢴⣠⡚⠉⢿⢿⣯⠿⠓⠈⠁⠀⠀⠀⠀⠛⢿⣿⣿⣿⣦⣿⣽⣾⣷⣿⣷⣬⣿⣥⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⡟⠋⠙⠧⢖⣮⣙⣏⠻⢟⡹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣾⡥⢼⣷⣿⣾⢷⣾⣿⣟⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡍⢿⣿⣿⡟⣾⠛⡤⣽⣯⣳⣾⣧⠴⠶⢶⣤⠀⠀⠀⣴⡶⠶⠦⣌⣽⣶⣿⣿⣿⡇⢸⣿⢲⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢾⣿⣿⡞⣻⣿⣶⣧⣾⣽⡟⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣰⣿⣾⣿⣿⣿⣿⣻⣯⣭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⢿⠛⣻⡿⢻⣟⣿⣯⠛⠃⠘⠻⢿⠟⠀⠀⠀⠀⠀⠻⡿⠛⠃⠀⠁⣸⣷⡼⣿⣽⣽⡷⡮⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢸⣿⠶⣯⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⣩⠿⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠏⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡇⠀⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣼⣷⡤⠄⠀⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠀⠐⠒⠺⣿⡟⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣀⢀⣿⣿⡀⣀⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠛⠛⠛⢿⣿⡞⠛⠑⠛⠛⠛⠛⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⡅⠉⠉⠉⠉⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣀⠀⣿⣿⣧⣤⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉⠛⠛⠛⠋⠁⢹⣿⡏⠉⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣼⣿⣧⣤⠤⢤⣤⡴⠶⠶⠄⠀⠶⠖⠒⠒⠛⠛⠛⢻⣿⣿⡋⠒⠒⠒⠶⠶⠖⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠘⣿⣸⣇⣀⣀⣀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠤⢤⣤⣤⣶⠦⠄⢸⣿⣿⠴⠶⠶⠛⠛⠓⠀⠈⠛⠉⠉⠉⠉⠉⠉⠁⠀⣿⡉⣿⡉⠉⠉⠛⠋
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠹⠇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⠶⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠟⠁⠀⢉⠩⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⠀⠀⠀⠀⠙⢦⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡹⠀⠢⢄⠀⢠⠤⠀⠈⣧⢀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠘⠷⠂⠀⠀⠲⠗⡀⣿⢸⡜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⣀⡀⠀⢠⢇⣿⣼⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡔⣄⣉⡁⣠⠏⣼⣿⣿⠇⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⣷⢿⡿⣵⢋⣾⣿⣿⡿⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣰⡟⢸⡾⣡⣿⣿⣿⡿⠁⣼⢹⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⣋⣴⠟⣡⠟⣰⣿⣿⣿⣿⠁⢠⡏⢸⣷⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⠾⠋⠁⣴⠏⢰⣿⣿⣿⣿⠇⡄⣼⢰⠀⣿⡸⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣴⢋⠀⠀⠀⣼⡏⠀⣿⣿⣿⣿⣿⢀⣧⡇⢸⣇⠸⣧⠹⣄⠀⠀⠀⠀⢠⢸
⠀⠀⠀⠀⠀⠀⢸⣧⠋⠀⠀⣸⣿⠁⢰⣿⣿⣿⣿⣿⠸⣿⣇⠈⣿⣦⡘⢷⣬⣙⠒⠒⣒⣡⠏
⠀⠀⠀⠀⠀⠀⠀⢏⠀⡆⠀⢻⣿⡄⠀⣿⣿⣿⣿⣿⡄⢻⣿⡄⠈⠿⣿⣶⣮⣍⣉⣭⣭⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣄⠘⢿⣷⡀⠘⣿⣝⠿⣿⣿⣄⠙⠿⣦⣀⠈⠉⠉⠉⢉⣉⣠⣤⣄⡀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠮⣿⣿⣦⡈⠻⣷⣄⡉⠛⠓⠢⠀⠀⣀⣠⣴⡾⠟⠋⠉⠉⠉⠙⢷⡄⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⡌⠛⠿⠿⢶⣶⠾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⢠⠖⣻⣶⣦⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣵⣶⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣯⠛⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣎⡗⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⢀⣿⣟⣵⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⠃⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣬⡦⢀⣸⣿⣿⣿⣹⣿⣿⡟⠋⠙⠛⠛⠉⠻⠷⠶⠿⣿⣿⣀⣻⣿⣿⡿⣦⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡟⢠⡟⢩⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⢿⣹⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣻⣿⣿⣧⣿⣷⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡌⣿⣴⣿⣿⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⢿⡿⢿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢻⣻⣿⣗⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢰⣇⣹⣿⣿⡟⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢹⣼⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠤⣿⣿⣿⣿⣿⣷⡋⠴⠚⠉⠁⠲⢶⡄⠀⠀⠀⠀⢠⡶⠖⠐⠚⠂⠄⠀⠈⡐⠏⡋⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣾⣿⣿⣿⣿⣿⣿⠇⢶⣶⣶⣶⣤⣦⡄⠀⠀⠀⠀⣤⣤⣤⣤⣴⣶⡆⠀⠀⠈⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⠈⠙⠉⠁⠃⠀⠀⠀⠘⠛⠈⠉⠋⠀⠉⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⣿⣿⣿⣿⣿⣿⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢿⣿⣿⣿⣿⠇⣾⣿⣷⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡱⢆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣸⣿⣿⣿⣿⣆⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⣿⡌⠟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⡀⠀⠀⠀⠀⣀⣴⣶⣦⣀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣇⢹⣾⡀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢡⠜⣿⢛⣿⣿⡏⢸⣿⣿⣿⣷⡄⠀⠀⠀⠙⠣⣈⡁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢻⣷⡄⠸
⠀⠀⠀⠀⠀⠀⠀⢫⡘⡄⡷⠀⢻⣼⣿⣿⣇⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡠⢳⣿⣿⣆⢣⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣆⠈⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢁⣿⡏⠁⢠⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⠛⠓⠒⠒⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣮⡟⠃⢰⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢈⡄⢠⠈⠙⢻⡏⠻⣿⣿⢣⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⢰⣿⠁⢼⣷⣷⣾⣿⠆⢹⣿⣏⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣾
⠀⠀⠀⠀⠀⠀⢀⠇⢀⣌⣿⣿⣾⣿⡤⣿⡿⢸⣿⣿⢿⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⣏⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠈⡲⢰⣿⣿⠻⣧⡿⣷⠋⠀⠈⣿⣿⣽⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠃⣿⡟⢻⡟⣿⠁
⠀⠀⠀⠀⠀⠀⠸⠁⢸⣿⡇⠀⢿⡄⠹⠂⠀⠀⠈⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡙⠃⠏⢹⢨⢻⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢠⠊⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢖⣥⣶⣾⣿⣶⣦⣌⠹⣦⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢫⣾⣿⠟⠛⠛⠛⠻⣿⣆⢹⡇⢹⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⢣⣿⡟⠁⠀⢀⡤⠤⠄⠘⣿⣿⠃⠈⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣫⣴⣿⠟⡄⠀⢰⡟⣠⠒⠀⠀⠹⣧⣤⣤⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⢋⣡⡜⠁⢠⣿⢱⢃⣀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠵⠒⠛⠉⣁⣠⣴⡿⣣⠏⠈⠀⠀⢀⡀⠀⠻⢿⠿⣿⡟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣴⣶⣾⡿⠟⠛⣉⠥⢚⡏⠀⠀⠛⠛⠛⠂⠀⠀⠈⣸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢫⣿⣿⡿⠁⠀⢠⣾⣧⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⣿⠃⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⣿⣧⠀⠀⢸⣿⣿⡄⢠⣤⣤⣀⠀⠀⠀⠀⢠⣿⣿⣿⠃⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠛⢿⣿⣷⣦⣄⡙⠿⣿⡜⠻⠛⠁⠀⠀⣀⣴⣿⠹⣿⣿⡀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡆⠀⠈⢿⣿⣿⣿⣷⠈⣿⣦⣤⠀⠀⠈⠁⠘⣿⣧⠙⢿⣷⡀⢻⣷⡀
⠀⠀⠀⠀⠀⣠⡶⠟⠛⣉⣁⣀⣠⣤⣿⣿⣿⣿⢟⣠⣿⣿⡟⠀⠀⠀⠀⠀⠈⢻⣷⣄⠈⢻⣆⠹⣿⣆
⠀⠀⠀⠀⢸⡏⢠⣾⠿⠟⠛⠛⠛⣛⣋⣭⣵⡾⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣰⣿⠀⣿⣿
⠀⠀⠀⠀⠘⢇⢸⣿⡀⠀⡖⠀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣩⠴⠾⣛⣡⡾⠟⠁
⠀⠀⠀⠀⠀⠈⠈⠻⣷⡘⡇⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠇⢸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⣀⠀⠀⠀⠀⢀⡀⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠢⠠⠊⠀⠀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⠒⠄⠓⠰⠋⠉⠛⠙⠛⠉⠙⠛⠒⠮⠉⠙⠚⠲⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠄⠀⠀⠀⠱⣄⣀⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀⠀⠀⠄⠀⠀⠀⠀⠀⠄⠘⠲⠔⠊⠓⠘⠦⢤⠀⠀⠀⠀⠤⠬⠤⢤
⠀⠀⠀⠀⠀⠀⡒⠒⢦⠈⡍⠀⠀⣐⡖⠂⠀⢀⡀⠰⣄⣀⣀⠐⠄⠒⠲⣄⣀⠐⢚⠲⠤⠰⠤⣄⡀⠀⠈⠉⠙⠆
⠀⠀⠀⠀⠀⠤⠽⣦⠀⠀⠙⣀⡀⠙⠲⠤⠴⡌⠉⠀⣀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⡈⠁⠀⠀⠀⣘⠂
⠀⠀⠀⠀⠰⣜⠁⠀⢀⡀⠠⡀⠀⠀⠀⢠⠄⠀⠔⠋⠉⠀⠀⠀⠀⠀⠀⠠⠤⢤⡀⠀⠈⠁⠀⠈⠉⠙⠀⠀⠨⠧⢤⡀
⠀⠀⠠⣘⠋⠀⠀⢀⠀⠀⡀⠁⠀⠀⠃⣰⠀⠀⠀⢠⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠈⠀⠈⠒⣄⣄⠀⢀⡀⠰⡆⠠⢮⡉⢙⣄
⠀⠀⠀⢈⠽⢤⠀⠈⠙⠈⠀⠳⠄⣉⠋⠃⠀⠀⠈⠁⣤⡄⠀⠀⠀⠀⠀⢀⣀⡈⠙⠀⠀⠀⠃⢍⡉⠁⠙⠒⠀⠤⣤⠉⢀⠈⡀
⠀⠀⢄⠎⠁⠀⠀⠀⠀⠀⠈⠀⠎⠀⡑⡆⠀⠀⠀⣀⡉⠁⠀⠀⠀⠀⠀⠈⠛⠁⣀⠀⠀⠀⡔⡁⠈⡄⠀⠀⠁⠀⠀⠀⠈⠉⠉⠓
⠀⠀⠘⠦⠄⠀⠀⠀⠀⠀⠀⠰⠀⠠⢾⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠿⠠⠀⡀⠀⠀⠀⠀⡀⢀⠐⢦⡄
⠀⠀⠀⠳⣄⣀⡘⠁⠀⠀⡄⠀⠣⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣰⠓⠂⠀⠂⠀⠈⠈⠉⠀⠁
⠀⠀⠀⢦⡀⠀⠠⣀⠀⡀⠈⠉⠒⢀⡸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠦⡄⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢈⠍⠀⠀⠀⠙⠶⠘⠦⠤⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠄⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠈⠓⠶⠂⠀⠀⠀⠠⠄⢀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⡀⠀⠀⠤⠀⠠⣄⣈⠃
⠀⠀⠀⠀⠀⠮⣄⠄⠀⢄⡀⠐⠆⠈⠁⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠒⠚⠒⠒⠋⠙⠳⣤⣀⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⠤⠤⠤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣮⣷⣦⣴⣧⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡷
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⢱⠎⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠺⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⠄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠶⣍⣃⡀
⠀⠀⠀⠀⢐⡶⠀⡀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⣆⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢀⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡀⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠸⣷⠈
⠀⠀⠀⠀⡀⣠⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣤⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢠⡆⠈⠑⡀
⠀⠀⠀⠘⠀⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡍⠓⠃⢀⡀⠀⠀⠀⠀⠀⠀⢠⠄⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣠⠽⢇⠄
⢄⣠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⣀⣾⣿⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⣠⡴
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡆⠉⠀⠀⠀⠑⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣿⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡙⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤
⣿⣿⣇⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⣠⣶⣶⡶⢤⡀
⠀⠀⠀⠀⢀⣰⡴⢷⠿⠉⠛⠏⠽⠿⣿⣿⡶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣵⡿⣿⣿⣿⣿⣄⠙⠂
⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⡋⣀⠻⠉⣀⣾⠿⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠟⠀⠉⠉⠛⠻⣿⡄⠀⠀⡄
⠀⣠⣾⣿⣿⣿⣿⣿⡿⢿⢰⣮⡀⣾⣿⠷⠤⢾⣿⣲⣿⣥⣦⡄⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⢸⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢁⣼⠟⣿⡇⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⢸⣧
⢰⣿⣿⣿⣿⡛⠛⠋⣿⣸⡝⠛⣿⣴⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣸⣿⠄
⠈⣿⣿⢏⡽⢿⣿⣶⣿⣿⣷⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⣀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣠⣿⣿⠃
⠀⣿⣿⣾⢡⠏⠀⠈⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣾⣿⣿⢟⡁
⠀⠘⣿⣿⣿⡆⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣶⣶⣞⣛⡭⠾⠿⠿⢿⣷
⠀⠀⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠈⣿⣿⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡹⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⣦⣸⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢦⠰⣶⠀⠀⡐⢦⡀⠀⢤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣤⡐⣠⠙⠻⢦⡐⠌⢻⣎⢣⠹⣧⢣⢻⣆⢱⡀⡘⢠⣮⠘⡱⠆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣛⡛⠲⢀⣼⢡⠀⠀⡀⢝⢦⠀⢻⠆⠀⢹⠀⢠⡿⢀⢁⡇⣤⢇⠸⢁⡾⠋⢞⣭⣶⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣋⣍⣉⣓⣈⣁⠛⠠⣄⡙⣎⣾⠇⠞⠘⢀⡟⡰⠀⡇⠸⠸⢠⢇⡎⠀⡜⢀⣠⠏⣁⣀⣈⣷⣮⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡷⢒⡖⢡⢾⡵⠶⠦⢨⢙⣿⠩⢳⣄⠀⠿⡟⢠⠆⣰⠏⣰⠃⢸⢁⡄⠀⠛⠈⠑⣂⠠⠍⠥⠉⠛⠶⣦⣌⠩⠙⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠘⡛⢋⢡⡈⡿⢡⡁⣦⠄⠋⣼⢦⢯⠙⡆⠀⣡⠀⡾⢁⠀⣯⢀⡿⣨⡶⠟⠛⢛⠋⠩⠒⠛⠛⠛⠶⣶⣤⠉⠻⢦⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣿⣷⢼⠊⠃⡈⠓⠀⣏⢚⣜⠒⢳⡀⣰⠞⠀⡌⢀⠟⡰⠋⠁⣀⣀⣤⡤⠀⠀⠒⠒⠲⢤⣤⣄⠹⣷⡔⡄⢀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⣝⣙⡔⣛⣿⣧⠀⠀⠈⢀⠉⢡⠀⠁⠃⣸⠰⠋⠀⠈⢁⡴⠟⢋⣉⡤⠒⠉⠔⠒⠲⢤⣄⡈⢻⡆⣸⣇⠁⢸⡤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢮⡳⣽⣄⣀⡻⠄⣀⠀⡎⠀⢸⣀⡴⠒⠓⣲⡞⠁⣠⠋⡠⠞⢋⣡⠔⠠⠤⢄⠀⣨⢀⣈⡷⢨⡟⣹⢏⡆⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣠⠓⠉⠉⠉⠻⢄⣀⡀⠞⡁⣼⠉⢹⡆⡼⠋⠀⠊⡀⡌⠴⠰⢟⣠⠖⣲⢆⡜⣠⠋⣺⠟⣠⣟⣴⢃⡞⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⢊⡴⢤⣄⡴⠦⢭⡁⠀⣤⠈⣤⠊⣗⠀⣥⢀⣾⡇⢸⡟⠉⡩⢣⡀⣉⡀⠈⢁⠞⡵⢞⣿⣿⣣⠿⠉
⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⢹⠛⠐⢴⠋⢠⠂⠀⠃⢀⠩⠤⡟⢰⣷⡀⠹⠀⢿⠁⣸⣇⣤⢗⣨⣤⢏⣡⣴⣫⣌⣠⡤⠌⠁⠁
⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⢛⣤⡤⠿⠤⠼⣄⠒⠈⢡⣶⡀⠁⠈⠿⠿⠆⠀⠈⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠘⠻⣿⣟⣿⡏⢰⠆⢀⣾⡏⠀⣆⠘⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠘⠀⠘⠛⠃⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⢟⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⡒⢦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣏⠁⠀⠀⠛⠂⠀⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣶⣄⣴⣿⡿⣛⣭⣽⣧⣄⡀
⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⣀⠴⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠊⠉⠉⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣛⣥⣤⣶⣶⣶⣬⣿⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠈⠻⢿⣿⣿⣿⣮⣭⣭⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠖
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⠹⠛⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣶⡘⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣛⣛⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣄⣀⣀⣀⠠⣤⣤⡀⠀⢢⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣶⣮⣹⣄⡘⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣷⣿⣿⣤⣄⣀⡀⠀⠀⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠶⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠛⠛⠛⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⣴⣾⡇⢸⣷⣦⡀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⣼⡏⢹⣧⣼⣿⠹⣿⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠻⢷⣤⣴⡿⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢻⣧⠈⠀⠁⠀⣰⡿⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡀⠺⠆⢀⣨⣿⡀⠾⠇⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠙⢿⡆⢠⣾⠟⠁⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣴⡾⠿⠛⢙⣿⣶⡶⠟⠋⠛⣷⣀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠸⠇⠸⡿⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⢸⣿⠀⠶⠆⣸⡏⠀⠀⠀⠀⣴⡿⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣄⣀⣀⣀⣤⣶⠟⠋⠀⠀⠀⢀⣴⡶⠿⠟⢻⣷⣶⣶⠟⠁⠀⠀⠀⣠⡿⠀⠺⠂⣸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⣸⣟⠠⣶⠀⣾⡏⠀⠀⠀⠀⣴⡿⠛⠿⣧⣴⣶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠾⠟⢿⣦⣤⣶⠟⠀⠀⠀⠀⣸⡿⠀⠾⠂⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢠⣦⠀⣾⠏⠁⠀⠀⢀⣴⡾⠿⢿⣥⣤⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⠿⢷⣤⣤⣼⠟⠀⠀⠀⠀⣸⡏⠠⡶⠀⣿⡏⠀⠀⠀⢀⣴⠿⠛⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢠⡄⢀⣾⠋⠉⠀⠀⢀⣴⡶⠾⣿⣅⣤⣤⣾⠟⠀⠀⠀⠀⣸⡟⠐⠿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⠿⣧⣌⣡⣼⠟⠀⠀⠀⠀⣾⡏⢠⡦⠈⣿⠋⠀⠀⠀⢀⣴⠿⠟⠋⢁⣴⣶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⣠⡀⢠⣿⠋⠉⠁⠀⢀⣤⣶⣾⣿⣀⣠⣤⣾⠟⠀⠀⠀⠀⣼⡏⠰⠶⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⢿⣦⣈⣁⣼⠏⠀⠀⠀⠀⣿⠏⣠⡄⢹⣿⠋⠉⠀⠀⢀⣴⡾⠿⠛⢿⣶⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣉⠁⢰⡿⠛⠛⠁⠀⠀⠀⠀⠀⠻⣧⡈⢀⣾⠏⠀⠀⠀⠀⣿⡇⢰⡦⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠛⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⢾⣿⣀⣀⣴⣶⠾⠟⣷⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢠⡄⠉⠛⣿⠃⣤⡄⢰⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⣶⠿⠿⣷⣤⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣶⣲⣶⣶⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⣿⣿⣿⣿⣿⡿⣛⡛⣉⠉⢙⣛⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠻⣏⠀⢸⠏⠉⢻⣿⣿⣿⠀⢀⣀⣤⣤⣽⣶⣝⢿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡀⠈⠀⠀⠀⣠⡾⣿⣿⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣏⣀⣀⠀⠀⢾⠆⠀⠀⠀⠀⣿⣿⣿⡿⡻⠟⠋⠉⠉⠉⠛⠻⣿⣿⣿⣎⠙⢿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠋⠀⢀⠀⠀⡀⠀⠙⣿⣿⣿⠟⡩⠊⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⡀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⡏⠀⠀⣷⣄⢀⣼⡿⠛⠚⠓⠶⣶⡆⠀⠀⠀⢀⣴⠶⠶⣿⣿⣿⣿⣷⠀⣿⡇
⠀⠀⠀⠀⠀⣠⠔⠒⠀⠀⠈⣿⣿⣿⣿⣿⣶⣾⣿⣿⡿⣫⣤⣶⣾⢷⣦⠈⠀⠀⠀⠀⢀⣤⣦⣤⣿⣿⣿⣿⠟⢌⣿⡇
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣿⣿⣿⢹⣿⣿⣿⣿⡏⠐⠿⣿⣿⣯⠁⠘⠆⠀⠀⠀⣼⣟⠃⢹⣿⣿⣿⣯⣵⣾⣾⣿⠁
⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠷⢿⡇⠀⠀⠹⢮⣁⣀⠀⠀⠀⠀⠀⠉⠍⢠⡼⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⢸⣿⣿⡶⣶⡾⠿⣿⣿⣿⣿⣿⣿⣦⡓⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⡿⣿⣻⣽⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠙⢯⣉⠉⢁⣤⣾⣿⣿⣿⣿⡀⠀⠘⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠳⢦⣤⣀⡀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣀⣈⣽⣿⣿⡿⣻⡿⣿⣿⣿⣿⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡿⠃⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣾⣫⣾⣿⠟⢡⠇⠀⠀⠀⠀⢸⣿⣦⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠛⠛⠉⠀⢠⠏⠀⠀⠀⠀⠀⠀⠻⠑⡒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⠤⢴⡒⠒⠂⠀⠀⠈⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠤⠐⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⠀⠈⠉⠒⠐⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠀⢀⣀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠇⣴⠀⠀⠀⠂⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠆⠀⢺⠈⠉⠀⠠⠀⠀⠀⠀⡶⠠⠚⠀⠀⠀⠸⣄⠀⠀⠣⡀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠜⠂⠀⠀⠀⠘⡀⠀⠀⢄⢠⣬⣤⣀⠀⠀⠀⣀⡀⠈⠹⠀⠀⠈⣇⠀⢧⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣧⣀⠀⠉⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⡉⠳⣤⠇
⠀⠀⠀⠀⠀⠀⠠⠒⠢⠀⠀⠀⠂⠀⠀⠰⡈⠉⠛⠛⠙⡄⠀⠀⢠⣶⡾⡳⣤⠀⢀⠀⠁⠀⠃
⠀⠀⠀⠀⠠⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢰⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⠀⠀⠷⠀⠀⠆⠀⠒⢀⠀⠠⠠⠀⠀⠀⡈⠈⠀⠐⢠⠃
⠀⠀⠀⠀⠀⠙⠦⠀⠀⢨⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠐⠄⠀⠂⠀⢀⡴⠃⠀⠀⠀⠘⠄⠠
⠀⠀⠀⠀⠀⠀⠑⢀⠀⠂⠑⠄⠀⢠⣤⡐⠁⠠⡀⠀⠀⠀⠈⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠘⠁⢱⡀⢀⠈⣠⡤⠴⠚⢡⠀⠀⠀⠀⠀⠀⠂⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⢦⡀⠀⠀⢧⠀⠲⢛⢷⠑⠀⡌⠳⢄⠀⠠⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢹⠀⠀⢸⠀⠀⠀⢠⡆⢰⡁⠀⠈⠁⠊⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠦⡀⠈⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠚⠁⠀⠀⠀⠒⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠋⠀⢴⣶⠆⢀⢠⣦⡆⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠤⠇⢀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠋⠀⠀⡀⠀⠈⠐⠃⡠⠂⠀⠀⠀⡯⣁
⠀⠀⠀⠀⠀⠀⠀⢠⢊⢀⠘⠀⠀⢀⢀⠉⠐⠀⠐⠊⠀⠀⠀⠀⠀⢘⣰⣦
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢡⠠⠄⠐⠒⠈⠊⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠆⠀⠠⣔⣃⠒⠻⢀⣀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢡⠀⠀⠀⠀⠄⠀⠀⣠⠃⠈⣂⣼⠞⠁⡺⣶⡔⡄⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⠔⠂⠁⠠⠔⢛⣃⣊⠔⠀⣿⡀⠙⢦⡀⠇
⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠑⠤⠒⠉⠀⠀⡠⠂⠀⠀⢸⣻⠁⣥⡄⠋⠇⣤⡍⠈⢀⣀
⠀⠀⠀⠀⢹⡇⠀⠀⠀⠚⠢⡀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠈⢻⠀⠋⠀⠀⠀⠉⠁⠀⣿⠜
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠑⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⠀⠀⠀⠂⠀⢀⠐⠙⣧
⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⡵⠘⠛⠆⣠⡏⠀⠷⢀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⣀⠈⠁⢴⡆⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠸⠇⠀⠀⠋⠀⣠⡄⠀⠐⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢦⠆⠀⠀⣚⡻⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠟⣹⠟⠉⠁⠁⠀⠐⠠⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⡠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣐⠊⠨⠀⠫⣦⢀⣀⣀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⠀⠀⠀⠀⠀⠀⠀⢀⡐⠃⠀⠀⠀⠀⠋⠁⢈⠵⣿⣶⣄
⠀⠀⠀⠀⠀⠀⣑⡄⠀⠀⠀⠁⠂⠠⢤⣀⣀⠀⠀⠀⠀⠀⠀⣃⡤⠀⠀⠀⠀⠀⠀⣬⠣⠿⠀⠀⡀⠀⡀⠀⠀⢈⣴⣶⣮⢷
⢀⣦⠀⠀⢴⠩⢿⣇⠂⠀⠀⠀⡀⠀⠀⠈⠙⠛⠤⣀⡀⠀⠀⣏⣀⣀⣀⡀⠤⠒⠋⠉⠚⢫⢤⠤⠟⢻⡻⠶⣤⠈⣁⣽⠹⢼
⡆⠈⠲⣀⢷⡄⠀⠈⠆⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢌⡺⠭⢈⡠⠊
⠈⠂⠀⠈⠀⠀⠐⠂⠀⣠⣤⣤⡤⢄⣀⣀⣀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠶⠤⠤⣴⣶⣶⣀⣤⣤⣴
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⣀⣤⣴⣶⣶⣤⣤⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⣀⢰⣆⠀⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣀⢸⣿⡝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⢃⣾⣿⣿⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣶⣷⣿⣿⣿⡟⣸⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣽⡻⣟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⢀⣀⣀⣀⣈⠻⣿⣿⡆⠀⢹⣿⣶⡌⢿
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢀⠀⠈⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠈⠉⠉⠉⠛⠻⠟⠘⣿⣿⢀⣿⣿⣿⢇⡟
⠀⠀⠀⠀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠘⣧⡀⠙⠿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⣶⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣄⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡙⠛⠿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⡀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠉⠛⠒⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣄⠈⠛⢦⡀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢦⣄⠀⢀⡾⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⠛⠛⠻⠿⣿⣿⣿⣿⣦⡀⠈⠳⣼⣆⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣷⣄⠈⢿⣿⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⠈⣿⣿⣦⢄⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠋⠀⠘⠈⠙⠀⠙⠈⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣒⣲⣦⣟⣯⡶⡿⣿⣿⢷⣌⠻⣷⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⣊⣏⣽⣿⣮⣿⣿⣿⣿⣿⣟⣿⣟⣷⣿⡣⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢋⣭⣿⣿⣿⢿⣿⣿⣿⣿⣿⣻⣟⣽⣽⣷⣼⡏⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢉⣿⣿⣾⣿⣿⣿⣿⠿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢷⣿⣭⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⢵⣿⣿⣽⣿⣿⣿⣿⣟⣳⣄⠀⠚⠋⢭⣭⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⣿⡝⣿⣿⣿⣿⣿⠇⠉⠀⠀⠀⠈⠁⠀⠀⣠⣿⢣⣿⣼⣿⣿⣟⠿⣿⢷⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⡞⢛⣴⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠠⠋⢏⣾⣿⣿⣿⣿⣿⣀⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠋⣠⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⢠⣤⣤⣀⠀⠀⠀⣼⢦⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣷⡟⢿⢃⣿⡿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⣈⣿⣀⣽⡿⣿⣿⣾⣿⣿⣿⣿⣿⣧⣄⣤⣤⣶⠉⠀⢧⣸⣻⡍⠀⠙⢿⡛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡽⣿⣿⡿⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠑⠁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠰⣡⣾⣿⢿⣿⣿⣷⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣴⣛⣷⣏⣠⣿⣿⣫⣾⣿⣿⣻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⡫⣀⣿⣿⢀⣿⣿⣿⣷⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⢧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢿⣿⢾⣿⣿⣽⣿⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣿⣿⣿⣿⣿⡿⢋⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣼⠟⣿⡟⣿⡟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣴⠟⡾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣟⣿⣟⣛⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠛⠿⢳⠞⠻⠟⠉⠙⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡔⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡜⡰⢀⠆⡘⠀⠸⠀⠀⡄⠸⠀⢀⢷⡀⢢⡐⠀⠙⢦⠁⠀⢣⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠊⢠⠃⡀⠀⠀⣰⡇⡀⡠⠋⠀⢣⡀⡙⠢⠀⠀⠁⠀⠀⠀⢣⢳
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢠⣷⣄⡴⠿⠿⠋⠀⠀⠀⠀⠱⣝⢄⠀⠀⠀⠀⠀⠀⠘⣾⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢀⡴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡱⡄⠀⠀⠀⠀⠀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⠄⠀⠀⠀⠀⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠆⠀⠀⠈⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠀⠀⡹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠱⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⡀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣷⣀⢀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡼⠀⣠⡀⢀⡄⡀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣇⢐⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣼⣿⠴⠾⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⢀⡜⠁⠀⠀⣠⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠙⠉⠀⠀⠀⡴⠁⠀⠀⠀⠁⢀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣵⣶⡶⠂⠤⣄⣠⡼⠁⣠⠞⢁⠀⠀⡈⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣿⡟⠀⠀⡀⠀⢉⣴⡖⠁⠊⠉⣀⠀⢉⢀⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⣠⣾⢃⣠⡾⢹⡇⠀⠈⠉⠉⠀⠈⠛⢿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣷⠿⢻⣿⣿⠋⠀⠈⢀⣀⠀⠀⠀⣠⠖⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣿⡙⣅⡀⠸⠘⣿⢠⡎⠀⠻⠿⠗⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡌⢳⡘⢿⣷⣄⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣦⠃⢸⢿⣿⣵⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⣴⣡⣾⠟⠁⢸⡟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⣿⠀⢿⡀⠀⣸⡿⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⢀⣽⣟⠧⣄⢵⣏⠁⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠋⠀⠈⢣⢹⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢰⠀⠀⣼⠉⢀⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠆⠁⠀⢼⣏⠀⠾⡆⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⠳⠀⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢫⣾⠿⣛⣯⣽⣟⣛⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣿⡏⡞⣩⣴⣶⣶⣮⣭⣲⢍⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡜⢷⡙⢧⠁⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢛⠛⠛⠃⠈⣁⠀⠉⠉⠉⠉⠻⢿⢿⣿⣿⣿⣦⡘⢧
⠀⠀⠀⠀⠀⠀⠀⡰⢋⣴⣾⡿⢛⡵⣢⡞⠁⠀⠀⠀⠀⠀⠀⠀⠳⡹⣿⣿⣿⣿⣜⣧
⠀⠀⠀⠀⠀⠀⠀⣴⡿⣻⣟⣴⢏⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣾⡆
⠀⠀⠀⠀⠀⠀⢰⡿⣱⣿⣿⢃⣾⣟⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠘⡇⣿⣿⡏⣾⣿⠃⠀⠈⠙⠷⣦⡄⠀⠀⢠⣴⠾⠋⠁⠀⣿⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⢱⣿⣿⢀⣴⢶⣶⡦⡄⠁⠀⠀⠈⢠⢴⣶⡶⣦⣸⣿⣿⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢸⣿⣧⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠈⣿⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠄⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⡼⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⡀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢤⡀⠀⠀⠀⠀⠀⡠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠙⠦⠤⠤⠴⠋⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣭⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⢔⣿⠿⠿⠿⠛⠛⠛⠉⠑⠒
⠀⠀⠀⠀⠀⡠⣰⡟⠏⠀⣠⢖⣫⣤⣄⣤⣀⣙⠛⠷⣮⣽⣲⣤⡀
⠀⠀⠀⠀⢰⠃⣿⢨⠀⠀⣿⣿⠵⣛⣩⣵⣿⣭⣽⣛⠶⣝⢿⣿⡿⡆
⠀⠀⠀⢀⣸⡆⠻⢎⠀⠀⠈⠣⣾⣿⠋⠀⠛⠻⠿⡟⣿⣦⡑⠙⣷⡙⣦⢀
⠀⠀⠀⢸⣯⣈⡁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠘⠟⣿⣷⡘⣧⢈⢻⣷⣄
⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠇⢹⠈⢷⡙⢿⣷⣄
⠀⢾⡇⢀⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣿⠀⣄⢻⣦⡹⢿⣷⣄
⠀⠈⠁⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⠘⣹⣦⡙⢿⣦⣝⢿⣿⣄⡀
⠠⠶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⡄⣷⣽⣿⣦⣝⢿⣷⣬⡻⢿⣦⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢸⡏⣽⣟⢷⣬⣙⠷⣮⡛⢿⣷⣽⡻⣆
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡹⣿⣷⣌⣿⣿⣿⣷⣦⣬⡛⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡕⠹⣮⡻⢿⣿⣿⠟⡻⠿⢿⣷⡈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣬⣙⠷⣦⣤⡤⠤⠄⣠⢠⠹⡄⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠿⠿⠶⠖⢲⠶⢿⡵⢟⡄⢧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣂⡠⣴⣶⣶⣖⠶⠖⣊⡙⢋⣵⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣼⡿⠀⣿⣟⣿⣾⣿⢟⣭⠾⢛⡵⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠘⠻⢫⡿⠁⠉⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠁⢦⢦⣀⠀⣙⣀⣀⣀⡤⠖⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢷⣭⣛⠶⠶⠶⣒⡂⢀⣴⡇⠀⠹⣶⣤⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠛⠛⠋⠭⣶⢿⣿⡇⠀⠀⠪⣛⠒⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠞⢀⢂⠘⣎⣮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡈⠉⠀⢠⣾⣿⠇⣹⣿⣷⡌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⣿⡾⣛⢉⣥⠞⣰⢟⠞⠉⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣴⡿⢋⣾⣷⣫⢡⡾⠃⢀⡀⠀⠀⡿⣸⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⣿⡟⣵⣷⠏⡴⣴⠫⠂⠀⠸⢃⡿⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡽⣧⣿⢿⡼⡇⠃⠀⠀⠀⠀⠞⣡⡟⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⢦⡙⠆⢻⣆⠀⠀⠀⠀⠤⠞⠋⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠻⣆⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠜⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠚⠁

⠀⠀⠀⢺⠀⠉⠒⠒⠒⠒⠤⠼⣷⣤⠏⠀⠉⠀⣠⣞⣁⣀⡀
⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣠⠞⠁⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀
⠀⠀⠀⠀⠀⠈⠓⠤⣤⠤⠶⠖⠒⠶⣄⠀⡿⠀⠀⠀⠀⠀⢠⣠⣀⣸⡇
⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⡿⡰⠳⢤⣤⣤⠴⠞⠉⠀⢀⡟
⠀⠀⠀⠀⠀⣿⠞⠛⠛⠳⠶⣶⠖⢻⡿⣅⢀⡼⠴⠤⠤⢤⡤⣶⠟
⠀⠀⠀⠀⠀⢹⡀⠀⠀⢠⠊⣏⣀⣴⠷⠾⠷⢤⣀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⣈⡷⠞⠛⠉⠉⠉⢸⠷⠒⢦⠀⠀⠉⠳⡴⠛⠁
⠀⠀⠀⣠⠾⠛⠒⠦⣄⠀⠀⠀⠸⡄⠀⠸⡇⠀⠀⠀⢹
⠀⠀⢸⠃⠀⠀⠀⠀⣼⠛⠲⣄⣀⣘⣆⠘⣧⠀⠀⠀⠸⠠⣖⠛⠓⢦
⠀⠀⠸⡄⠀⠀⠀⣐⣾⢇⣼⠀⠀⢀⡎⣧⠘⢿⣄⠀⢸⣀⠏⠀⠀⠀⡇
⠀⠀⠀⠈⠒⠶⠾⢛⡵⠋⡏⠀⢠⡾⢔⡟⠳⢿⣍⡳⢬⣧⣄⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠻⣶⠿⠶⢾⣅⡀⠀⠀⠀⠙⢦⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢸⡇⠀⢀⣤⡧⠙⢦⡀⠀⠀⢸⠇
⠀⠀⠀⠀⣠⠴⠶⠾⠷⣄⠈⠳⣷⣿⡽⠃⠀⠀⢻⡄⢀⡞
⠀⠀⠀⣼⣡⠌⠀⠀⠀⠈⠳⣄⢠⠏⠀⠀⠀⠀⢸⡟⠋⠓⢦⡄
⠀⠀⠀⢹⡏⠙⢦⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡼⠃⢀⣠⠤⢽⡄
⠀⠀⠀⠀⠹⡄⠀⠳⣄⠀⠀⢸⡇⠀⠀⠀⢀⡜⠁⣠⠏⠀⠀⢠⠇
⠀⠀⠀⠀⣠⠞⠓⢦⠈⠙⠒⠚⢷⡀⠀⣰⠋⠀⢀⡏⠀⠀⣠⠎
⣀⣀⣀⣸⣻⣀⣤⣾⣋⣒⣢⣄⣈⣻⣦⣿⣤⣤⣊⣀⣠⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⣿⣿⢿⠿⣿⣿⣿⣧⡀⠀⠀⠀⢀⢠⣿⣿⣿⣾⣿⣷⣆⡀⠀⠀⠀⣴⣿⣿⣿⣿⡿⠿⠆
⠀⠀⠀⠀⢀⡀⠀⠙⠻⣿⠀⠀⠀⠀⢸⣿⡟⠃⠈⠀⠁⠙⣿⣿⡧⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⣿⡗⠀⠀⠀⣿⣿⣿⣿⡟⠋⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠉⠁⠀⠈⠀⢻⣿⠆⠀⠀⠀⢻⣿⣿⣿⡿⠗⢀⣀⠈⠀⠀⠀⠀⢻⡟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⣿⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⣿⣷⣤⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡦
⠀⠀⠀⠀⣤⣤⣶⣴⣤⡄⠀⠀⠀⠀⠀⠀⢄⣠⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣤⡆⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⠂
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣲⣾⡿⠟⠛⠛⢻⣧⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⡏⠁⠐
⠀⠀⢻⣿⣿⡟⡿⡿⠀⠹⡇⠀⠀⠀⠈⠀⣿⠠⠤⠀⠤⠄⠗⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⣿⣿⠃⠀⠀⢀⣤⣶⣾⣿⣿⡃
⠀⠀⠀⠘⠋⠛⠃⠑⠋⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰⣤⡀⠀⠀⠀⠘⠋⣀⡀⠀⣀⢹⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⢀⠀⢿⣿⡉⠀⠀⠀⠀⠀⠈⠁⠈⠉⠁⠀⠀⠀⣿⣿⣿⡿⠫⠛⢿⣿⡄
⠀⠀⠀⠀⠙⣤⣠⣀⡴⠂⠀⠀⠀⠀⠀⠀⢸⣿⠂⠙⠉⠀⢸⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠟⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⡀⢀⠀⠀⠀⠀⠀⣁⣰⣶⣶⣴⣦⡁⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠅
⠀⠀⠀⠀⠉⣹⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠀⠀⢀⣿⠋⠉⠉⠉⠻⣷⡀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣦⡄
⠀⠀⠀⡤⠀⠈⠉⠉⠉⠙⣿⠁⠀⠀⠀⠘⠃⠀⠙⠛⠛⢿⣿⣿⣿⣿⠆⠀⠀⢸⣇⢀⡀⠀⡠⠄⣹⡇⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠈⡿⠻⡿⢻⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠻⠁
⠀⠀⠀⠂⢀⣠⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣰⡇⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠿⠍
⠀⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⠖⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣿⣽⣯⣮⣽⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣽⣿⣽⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢸⠀⠀⠀⢂⠠⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠀⢠⠄⠀⠂⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠣⡑⡀⠀⠀⢀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢁⠔⠁⠀⠀⠀⢰⠀⠀⠁⠀⠀⢄⠐⡀⠀⠀⠄⠱⡈⠄⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠈⠄⠱⡈⠄⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣇⠀⠰⠶⠆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠠⠀⠀⠀⡀⠱⠈⠄⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠡⣤⣤⣀⢀⢀⠀⠀⠀⠀⡆⠰⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠇⢀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠉⠉⠀⠘⠀⠂⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⢠⠀⠀⠀⣰⠃⠐⢰⠂⡐⠀⠇⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠇⡀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⢀⠆⠉⠐⠀⠰⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠜⡐⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠋⠀⢠⠀⡠⢂⡔⠀⠀⢀⠀⠀⠀⠘⠀⠀⠈⢐⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡴⢡⠊⣀⠄⠐⢃⡔⢁⠔⢀⠄⣡⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠚⡰⢃⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⡏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡈⠉⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣭⣯⣝⠓⠶⣤⣶⠿⢿⣿⠄⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠟⠙⠻⣿⣷⡄⠀⠀⠀⠀⠉⠛⣷⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⣀⣀⣬⣷⡒⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠙⡛⠛⠛⠋⠉⠉⠙⣿⣿⠀⢀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⢰⡿⣆⠀⠸⣆⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠙⢷⣦⣹⣷⣦⣀⢡⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⣤⣄⠹⣄⠈⢻⣯⢿⡍⠛⢿⢻⣅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⠛⠛⢀⢸⣦⣸⠟⠸⢁⣠⡴⢸⡿⢀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠐⠀⠀⣿⡏⣀⣴⣿⡟⢡⠟⣡⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣿⡿⢿⡀⠻⣴⠟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⡏⠈⠻⣿⣦⡑⣄⠻⣄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣾⠁⣿⣇⠀⠀⢸⡟⠈⣿⡄⠨⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠈⠙⠳⣤⡋⢀⣴⣿⣗⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⡾⠛⡵⠋⠀⣿⣿⣦⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠷⣼⠁⠀⠀⠈⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⣧⠀⠸⣧⠀⠘⢆⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⠃⣠⡿⠃⠀⠀⠘⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⡴⣾⣿⣟⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣹⣿⣦⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⣿⣿⡗⡆
⠀⠀⠀⠀⠀⣸⠱⠏⠀⠀⢰⣾⠄⠀⠀⠀⠀⠀⢸⣿⡏⠁⠀⠀⢻⣧⠀⠀⠀⠀⠀⠘⢧⠸⠋⠙⢻⢣⣇
⠀⠀⠀⠀⠀⣿⣄⠂⠀⠀⢀⣿⡆⠀⠀⠀⠀⣤⣾⣧⢷⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠰⣿⣩⡦⣘⣣⢾⣿⡷⠀⠀⠀⠀⠘⢿⣿⣶⡃⠶⢢⣿⠏⠀⠀⠀⠀⠀⠀⠀⣄⠖⡠⢸⠏⢸⡇
⠀⠀⠀⢀⡴⣯⢱⡆⢀⣅⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⣯⣀⣾⡇
⠀⠀⠀⠘⢧⣬⡟⠀⢸⡟⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⡇
⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠈⢸⠛⠁
⠀⠀⠀⠀⠀⢀⣤⠶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢷⣺⣯⣶⣄
⠀⠀⠀⠀⢠⣿⠛⣾⠿⠟⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣧⢱⣿
⠀⠀⠀⢠⣾⢿⠗⠝⠛⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⡄⡀⡀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⠏
⠀⠀⠀⠘⠿⢿⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢍⡁⠀⠉
⠀⠀⠀⠀⠀⠊⠀⠀⠂⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢢⡄
⠀⠀⠀⠀⠀⣠⠞⠋⠙⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠿⠟⠉⢻⣿
⠀⠀⠀⠀⣠⣿⠀⣠⠔⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⣿⣽⢧⡷⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⢠⠻⠃⠠⠟⠻⣝⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⢠⣾⣿⣇
⠀⠀⠀⠈⣿⠱⢾⣦⠑⣣⣿⣄⠀⠀⠀⠀⠀⢠⣿⣠⠂⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠞⠟⣿⡆
⠀⠀⠀⠘⣟⢣⡞⠋⠀⣿⣤⠟⠀⠀⠀⠀⠀⣸⣌⠻⡀⢤⠄⡰⡿⠿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣴⣶⣾⠋⠉⠁
⠀⠀⠀⢰⣿⡿⠋⠀⠀⠘⢻⣷⠀⠀⠀⠀⢸⡏⢁⣤⡯⠐⠊⢷⣉⢻⡿⠀⠀⠀⠀⠀⠀⠀⢿⠋⣋⢸⠄
⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠁⠻⠟⠁⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⣠⣤⣤⣄⡀⠀⠀⣠⣤⣤⣀⠀⠀⣀⣤⣤⣀⠀⠀⣀⣤⣤⣤⡀⠀⢀⣀⣤⣤⣀⠀⠀⣀⣤⣤⣄⡀⠀⢀⣤⣤⣄⡀
⠀⣾⣿⠿⠿⢿⣿⡄⣾⣿⣿⠿⢿⡇⢸⡿⠛⠛⠿⣧⢰⡿⢿⣿⡿⢿⡄⣿⠛⠛⠛⠻⡇⢸⠋⠛⠟⠛⢻⢰⡟⠛⠟⠛⢻
⠀⣿⠃⠀⠀⠀⣿⠇⠛⠀⠀⠀⠀⠃⠸⠂⠀⠀⠀⠟⠈⠃⠀⠀⠀⠘⠁⠛⠀⠀⠀⠀⠇⠘⠀⠀⠀⠀⠸⠘⠃⠀⠀⠀⠘
⠀⠈⠁⣀⣀⠀⠁⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣠⣤⣤⣄⠀⠀⣠⣄⣀
⠀⣼⣿⠿⠿⣷⡄⢰⡿⠿⣿⠿⢷⠀⣾⠉⠉⠉⢻⡄⢰⡿⣿⣿⣿⣧⠀⣼⠛⠛⠛⢷⠀⣼⡏⠙⠋⠹⣧⠀⣿⣿⠿⢿⣦
⠀⠿⠀⠀⠀⠘⡇⠸⠀⠀⠀⠀⠸⠀⠟⠀⠀⠀⠸⠃⢸⠀⠀⠀⠉⠿⠀⠟⠀⠀⠀⠸⠀⢻⡆⠀⠀⢀⡿⠀⡏⠀⠀⠀⢸
⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢁⣀⡀⠈⠁⠀⠀⠀⣀⣀
⠀⢰⡿⠛⠻⣿⡄⢠⣾⡿⠿⢿⣷⠀⣼⣿⡿⣿⣷⠀⣼⣿⡿⣿⣷⡀⣼⣿⣿⣿⣶⠀⣠⣾⣿⣿⣿⣷⣄⠀⣼⣿⢿⣿⣷
⠀⣿⠁⠀⠀⠘⡷⢸⡏⠀⠀⠀⣻⠀⣿⠉⠀⠈⢻⠀⡟⠀⠀⠀⢹⠃⢿⠈⠛⠛⢿⠃⣿⣿⡿⠿⠿⣿⣿⠀⡿⠁⠀⠙⢿
⠀⠘⠆⠀⠀⠰⠃⠀⠇⠀⠀⠀⠿⠀⠘⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠻⣧⠀⠀⠀⣸⠟⠀⠿⠀⠀⠀⠞
⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⢠⣿⣿⣿⣿⣷⡄⠀⠀⠀⢸⣿⣿⢿⣿⡆⠀⠀⠀⣾⣿⣿⣿⠿⣆⠀⠀⠀⣰⣿⠛⠻⣿⣦⠀⠀⠀⣰⣿⣿⣿⣿⣷
⠀⣾⡟⠛⠛⠛⢻⣿⡄⠀⠀⣿⡇⠀⠀⠈⣿⠀⠀⢾⣿⡿⠛⠁⠀⢻⠀⠀⢀⣿⡇⠀⠀⠈⣿⡇⠀⢸⣿⡿⠉⠛⠻⣿⣧
⠀⣿⣷⠀⠀⢀⣾⡿⠃⠀⠀⣿⡇⠀⠀⢀⡿⠀⠀⣿⣿⣧⠀⠀⢀⣿⠄⠀⢸⣿⣧⠀⠀⢠⣿⠇⢀⣼⣿⣇⠀⠀⢀⣿⣿
⠀⢿⣿⣇⠀⠸⣿⠁⠀⠀⠀⣻⣧⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠸⣿⡀⠀⠈⣿⣿⠀⠀⠻⣏⠀⠀⣿⣿⣿⠀⠀⢸⣿⣿
⠀⣸⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠽⣿⠏⠀⠀⠺⣿⡏
⢀⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣀⠀⢻⠁⠀⠀⢀⣠⣤⣄⠀⠀⠁⠀⠀⣀⣀⡈
⠈⢿⣿⠃⠀⣀⣀⠀⠀⠀⢻⣿⣿⠟⠀⣠⣶⣿⣿⣶⣶⣶⡀⠀⣿⣿⣿⣿⣷⡄⠀⣰⣿⣿⣿⣿⣷⠀⠀⣴⣿⣿⣿⣿⣷
⠀⠀⠁⣰⣿⣿⣿⣿⣆⠀⠈⠛⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠘⠃⠀⠀⢻⣿⡇⣰⣿⡿⠛⠉⠀⣿⣆⠀⢿⡟⠀⠀⠀⠁
⠀⠀⢠⣿⡟⠛⠛⠻⣿⡆⠀⢠⣾⣿⣿⣷⡄⠀⠀⠼⠁⠀⢀⣀⡀⠀⠀⠀⡝⠀⢻⣿⣷⠀⠀⢠⣿⣿⠀⠀⣇
⠀⠀⠘⣿⣇⠀⠀⢰⣿⠃⢀⣿⣿⠿⠟⠹⣷⠀⠀⢠⣤⣴⣿⣿⣿⣧⣤⡄⠀⠀⢸⣿⣿⠃⠀⢸⣿⡃⠀⠀⠹⣿⣯⣽⠆
⠀⠀⠀⣸⣿⠀⠀⢿⣇⠀⠸⣿⡇⠀⠀⢀⣿⠇⠀⠘⠿⠿⠙⠛⠛⢿⠿⠃⠀⠀⠘⣿⣇⠀⠀⠀⠉⠓⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠲⢦⡀⠢⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣿⣧⠀⠈⣧⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢰⠀⡟⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⣀⣀⣠⣤⣤⣶⣾⣿⡿⢣⡟⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⣴⣾⣿⣿⣿⣿⣿⡿⠿⠟⣋⣴⡿⠁⠀⠀⢹⡏⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⡿⠟⠋⠉⠁⠀⠀⣠⣴⣾⣿⠟⣤⠴⠖⠒⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠉⠀⠀⠀⣀⣤⣶⣿⣿⡿⠛⠁⠀⠀⣀⣠⣄⡀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⢀⣴⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⣿⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⣿⣿⣿⣿⣿⡿⢀⣿⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿⣿⣿⣿⡇⢸⣿⣧⠀⠀⠀⠀⣴⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⣿⣿⣿⣿⣿⡇⠸⣿⣎⠣⡀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣧⠀⢻⣿⣦⡈⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧⡀⠙⢿⣿⣦⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣷⣄⠀⠉⠛⢿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠙⠻⣿⣿⣿⣿⣷⣦⣀⠀⠀⠉⠛⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⢠⠂⡆⠀⠉⠛⠻⢿⣿⣿⣷⣦⣀⠀⢤⡀⠈⠙⠿⣿⣦⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢸⠀⣿⣄⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣤⡙⢢⡀⣄⠈⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⢻⣿⣷⣄⠀⠀⠀⠀⣠⣤⣈⣛⣿⣿⣄⠱⠸⣷⡀⠘⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣌⢻⣿⣭⣛⠲⠤⠀⠀⠈⠙⠻⣿⣿⣿⡄⠀⣿⣧⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⠙⢿⣿⣿⣶⣤⣀⠀⠀⠀⠘⣿⣿⠁⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⡄⢀⠀⠟⠁⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢁⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠉⠉⠁⠀⠀⠈⠉⠉⠛⠻⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⣠⡾⠁⠀⣴⠃⢀⠀⣠⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⢀⣼⣿⠁⢠⣾⡇⢀⣾⠀⣿⣦⡀⠀⠙⠶⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⡴⠀⠀⢠⣿⣿⡇⢀⣿⣿⠃⣾⣿⠀⣿⣿⣿⣶⣤⣄⣀⡈⠉⠉⠉⠉⠉⠉⠙⠛⢿⣇
⠀⠀⠀⠀⠀⠀⣼⣿⢃⡾⠁⠀⣰⣿⣿⣿⡇⣾⢿⣿⣦⣿⣿⣇⠹⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀⠙⠆
⠀⠀⠀⠀⠀⢰⣿⢃⣾⠁⠀⢰⣿⣿⣿⢸⣧⣿⠈⢿⣿⣿⠻⣿⣦⠙⢿⣿⣿⣷⣦⣄⣉⠉⠉⠁⠀⠈⠉⠙⠻⡄
⠀⠀⠀⠀⠀⢸⡟⣼⠇⠀⢠⣿⣿⡟⢿⠀⠙⢿⣿⢾⣿⣿⣧⠈⠛⠷⣤⣽⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡢⢄⡀⠐
⠀⠀⠀⠀⠀⢸⣇⣿⡄⠀⣼⣿⣿⡇⠈⠁⣀⣠⣌⣀⠈⠛⠉⠁⠀⠀⠛⠉⣀⣭⣿⣿⣶⣭⡙⠛⠿⣿⣿⣶⣝⢦
⠀⠀⠀⠀⠀⠈⣿⣿⠃⠀⣿⣿⣿⡇⠀⠺⠿⢹⣿⠏⠙⠀⠀⠀⠀⠀⠀⠊⠹⣿⡏⠽⠟⠛⢿⣷⣄⠈⠙⢿⣿⡎⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣼⡇⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣷⡄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢳⡄⢻⣿⡄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣜⣿⣷⠹⣷⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣷⠈⣿⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⢀⢋⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠙⠦⡀⠀⠀⠀⠂⠀⠈⠙⠀⠀⠀⠀⠀⠀⣸⣿⣣⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣴⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⢦⡀⠀⠀⠀⠀⠀⣠⠞⣿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠙⠲⠶⠶⠖⠋⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠖⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⡖⠀⠀⠀⠀⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⡈⠸⢃⠀⠀⠀⡀⢂⠄⢀⠌⠉⠙⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡿⠓⠨⠔⢋⣁⣂⡁⣨⡅⡇⢠⠃⠀⠀⠘⣯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢼⣧⡀⡆⣾⡿⠒⠉⢉⣿⣷⠋⠁⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⢀⣶⢸⠏⢱⣿⢻⣿⣿⠘⠛⠳⠋⢀⠈⠐⠀⠀⠠⣟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢟⠀⠊⣸⣯⢸⣟⣿⠀⠀⠀⠀⠀⠉⠁⠀⠀⢻⢸⢧⡰
⠀⠀⠀⠀⠀⠀⠀⠠⡴⠛⣿⡄⢸⣤⠞⠉⢀⠿⢋⣿⠆⠀⠀⠀⠀⠀⠀⠀⢀⡽⣾⢶⣿
⠀⠀⠀⠀⠀⠀⢌⢸⡇⣰⠛⣿⡆⢹⣶⢪⣵⣾⢋⠕⠁⠀⠀⠀⠀⠈⢀⣰⡿⣿⠘⢞⢍⠰
⠀⠀⠀⠀⠀⠀⠈⢧⢁⠣⠜⠛⢥⠋⢫⢡⠙⣿⣦⡁⠀⠀⠈⠢⢤⡀⠀⠛⠀⡽⠚⠛⢯⢳⡄⡀
⠀⠀⠀⠀⠀⠀⡠⠞⣾⣿⡀⢀⣾⣤⢟⡮⠖⢹⡿⢛⠀⠀⠀⠀⠸⠀⠀⠀⠀⠻⣄⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠘⠀⣇⡏⡟⠁⠙⡄⢿⣻⡄⠀⠹⢄⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡦⠂⣌⢸⣿⠀⡆⡀⡜⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠠⡣⢻⡟⠁⠀⣷⣧⡛⢷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠌⠀⣿⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⣣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠇⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⢸⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠇⠀⢀⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣿⣿⣿⠏⠀⢀⣼⡿⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣉⣭⣽⣿⣿⣿⣿⣿⠟⠁⢀⣠⡾⣿⢣⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢀⣠⣴⡿⢋⣼⣷⣿⡟⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⣀⣠⣴⣶⡿⠛⢉⣴⣿⣿⣿⠟⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⣛⣉⣠⣤⣴⣶⣶⡿⠿⠟⠛⠉⢁⣠⣶⣿⣿⣿⡿⢋⣴⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣫⣶⣿⣿⠟⠀⣀⣤⣴⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣴⡿⠛⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣤⣶⠞⠛⣁⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠁⠀⠀⣀⣠⣾⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⢋⣠⠶⣻⣿⣿⣿⣿⣿⣿⣿⠿⢛⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠹⡿⢿⣿⣟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⡴⢋⣵⣾⣿⣿⣿⣿⣿⡿⠋⢡⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏⢠⣾⣿⣿⣿⣿⣿⣿⠏⠀⣰⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⣿⣿⣿⣿⠏⣿⡏⠀⢠⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣿⣿⣿⣿⡿⠀⣿⡇⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⣠⣶⣦⣴⣦⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣇⠀⢻⣧⠀⠀⢿⣿⣿⣿⣿⢀⡤⠀⢀⠀⠀⠙⠿⠿⠿⠟⠁⠀⢀⠄⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⠈⢿⣧⡀⠈⠻⣿⣿⣿⣿⡇⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⡠⠋⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⠹⣿⣆⡀⠙⠿⣿⣿⣷⠀⠀⠀⠈⠑⠢⠤⠤⠔⠋⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠈⠻⢿⣦⡀⠈⠻⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠉⠂⠀⠀⠙⠿⣷⣤⡀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⢦⡀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣄⡈⠳⢦⣄⡀⠀⠀⠀⢀⡉⠛⠿⣶⣤⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣦⣌⣉⠛⠶⣤⣀⠙⠻⣶⣤⣉⠻⢷⣦⡀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣶⣭⣳⣦⡈⢿⣿⣷⣦⡉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣄⣀⣈⠉⠛⢿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⡄⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⡙⢿⣿⣿⣿⠇⣸⣿⣿⣿⡇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡿⠟⣁⣼⣿⣿⣿⠟⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣡⣴⣾⣿⣿⣿⣯⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢤⣄⣀⢀⡀⠀⠀⠀⠀⢀⡀⣀⢀⡠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡿⢾⣩⣏⡿⠿⣩⠷⡀⠀⠀⢠⣾⢻⣿⢿⣱⣍⡿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣫⣏⢞⣻⣋⡹⢠⣿⣷⡷⣤⢤⣼⣿⣷⣂⣉⣹⣟⡖⣽⣻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡗⣾⠿⢽⣿⣟⣿⣒⣿⣿⣾⣿⣿⣿⣒⣿⣛⡿⣭⡽⣶⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⣟⣿⣻⢿⣿⠿⢋⡤⠽⠿⠿⠃⠤⡙⠿⣿⡟⣟⢷⡏⣻⠟⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢶⣵⣿⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣿⣾⠲⣿⡻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡝⢿⡿⣟⣟⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣀⣻⡿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢦⠼⣿⡿⣿⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢻⠿⣿⠧⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡖⢃⣟⣿⠋⡇⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣶⣹⡘⢲⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣍⣿⣫⣾⣿⡟⡩⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣴⢿⡳⡌⣿⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡐⢻⡛⣻⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⣼⠗⣟⣋⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣨⠅⢘⣯⡌⢁⣵⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢈⡬⢁⣽⡃⠬
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡌⠛⡹⣿⡿⣅⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠸⢇⠛⢡⣦
⠀⠀⠀⠀⠀⠀⠀⡔⢹⡶⣿⣤⠙⢷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣰⠊⣤⣽⢲⡏⠢
⠀⠀⠀⠀⠀⠀⠀⢐⣼⣾⣽⣿⣶⡿⢋⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠘⢻⣖⣿⣯⣷⣧⡂
⠀⠀⠀⠀⠀⠀⡞⡿⡫⣟⣽⡀⣧⣷⣏⣉⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣉⣽⣾⣽⢀⡉⣻⢝⢿⢳
⠀⠀⠀⠀⠀⠀⢙⢿⡿⢏⡙⢿⣿⣫⡗⡟⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⢿⠺⣝⣿⡿⢋⡙⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢹⠞⣁⣷⣽⣦⡟⡹⡾⡋⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⢱⢏⢺⣤⡯⣾⣉⠳⡀
⠀⠀⠀⠀⠀⠀⠈⠿⡻⣼⠺⢧⡿⢗⣳⣍⡟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣻⣱⣟⡺⢷⡴⠟⣗⢟⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣾⡍⡙⣻⡷⣺⣟⣚⢛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⣣⣻⡓⢮⡛⢋⢉⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⢷⣷⡝⢏⡞⣍⡳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⢳⡱⢫⣾⡾⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣧⡜⠲⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⠆⢣⣼⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣽⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⣿⡿⠁⣭⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠁⠀⢙⣉⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠔⡺⠒⢂⣶⡟⠘⠢⣄⡛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠐⠏⣿⡴⠀⠀⢀⢯⣿⣤⠠⡶⢻⣿⣿⡎⣦⡀
⠀⠀⠀⠠⣾⣯⡜⠋⠉⣀⠀⠀⢀⣶⣿⣿⡇⢀⠀⠈⠊⢻⡏⠀⣠⡜⣿⣿⣿⢸⣿⡀
⠀⠀⠀⡴⣿⣯⠞⠋⠉⠙⠛⠛⠋⠉⠀⠉⠻⡗⠀⠀⠐⢸⣿⣾⣿⣿⣿⣿⡿⠋⣿⣾⣆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣿⢿⠛⠟⡏⢸⣷⡇⢣⣸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠄⠡⠃⠌⠀⡉⣾⠉⠙⢫⣇⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠈⠀⣼⣨⣿⠀⠀⢀⢹⡅⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣿⠀⠀⠀⣀⡼⢘⣩⣾⣿⡆⠀⠀⠛⣷⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢠⣯⠁⠀⣠⡿⢻⢋⡵⠋⣩⢷⣤⣀⠀⣿⣧⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⠸⣿⣷⠀⣿⠟⣷⡿⢁⡜⠁⠀⠀⠉⠁⢸⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠀⠀⣿⣆⠀⢴⣾⡿⢥⣾⣷⣤⣤⣶⠀⠀⢻⣿⣿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⠀⠀⠀⠈⢿⣆⣩⠟⣡⠎⠀⠀⠀⠀⠹⠃⠀⢸⣟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣾⡷⢆⡀⠀⠈⠀⢠⣾⠋⠀⠀⠀⠀⠀⠰⠀⠇⠀⢻⠘⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⡇⠀⠈⢻⣿⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠸⡇⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣇⡿⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⡀⠀⡇⠘⣿⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣦⠀⣠⣿⣿⣶⣶⣦⣴⣼⣿⣿⣷⣿⣿⣶⣦⣶⠀⢷⣶⣿⣥⣠⣶⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⡿⠟⠛⠛⠛⠛⠛⠋⠉⠉⠙⠛⠛⠛⠟⣀⠈⠙⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠴⠶⣄⡀⣹⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣇⠀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠛⠛⠒⠖⠛⠛⠛⠛⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠙⠛⠋⠉⠉⠉⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣄⢸⡆
⠀⠀⠀⠀⠀⠀⢰⡇⢸⣀⠀⠀⠀⠀⣼⠁⠶⠖⠒⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠖⠒⠶⠜⣇⠀⠀⠀⠀⣀⣿⠈⣧
⠀⠀⠀⠀⠀⢀⡟⡴⠋⢹⠀⠀⠀⢀⡟⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⢿⠀⠀⠀⢸⡏⠙⢷⡹⡆
⠀⠀⠀⠀⠀⡾⢹⡇⠀⢻⠀⠀⠀⢸⡇⠀⠀⠻⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠿⠀⠀⢸⡇⠀⠀⢸⡟⠃⠈⡇⢹⡄
⠀⠀⠀⠀⣼⠃⠸⡇⠀⢸⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⢰⡇⠀⢷⡀
⠀⠀⠀⢰⡇⠀⠀⠹⣄⢸⡇⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠃⣠⡞⠀⠀⠘⣧
⠀⠀⠀⣾⠀⠀⠀⠀⠈⠻⣇⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣾⠞⠋⠀⠀⠀⠀⢹⡄
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢻⡌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⢡⡏⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣷⠀⠀⠙⠳⢦⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠉⠀⠀⡾⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡘⣧⠀⠀⠀⢸⠉⠙⠛⠲⠶⠖⠛⠋⠁⢸⠀⠀⠀⣼⢃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠙⣦⡀⣶⡀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⣰⠄⣠⠟
⠀⠀⠀⠀⠀⠈⠻⠿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⠀⣀⣠⣤⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣤⣄⣀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠛⠛⠉⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠉⠙⠛⠳⠶⢤⣤⣀⡀
⠀⠀⢀⣠⠴⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⣀⣀⣤⣤⠴⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠦⣄⡀
⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠉⠙⠓⠒⠶⠶⠶⠒⠛⠋⠁⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⣠⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣴⣦⣤⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠰⣿⣿⡿⣿⣿⠿⠿⣿⣇⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⢸⣿⡿⠙⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠻⠏⠀⠈⠁⠀⠀⠛⠁⠀⠀⠀⠀⢸⣿⣧⠙⠻⣿⠉⣿⣿⠁⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠈⢀⣿⡏⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⢠⣿⣿⣷
⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠰⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠹⣿⣿⡇
⠀⠀⠀⠀⣿⡟⠛⠉⠛⠿⠛⡿⠁⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠻⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠀⠘⣿⣿⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢿⣿⣿⡇⠀⠀⠸⣿⣿⣿
⠀⠀⠀⢀⣿⣿⡿⢿⡿⣿⣿⣿⣧⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠘⠹⠿⠃⠀⠀⠀⠘⠛⠏
⠀⠀⠀⢸⣿⠏⠁⠀⠀⠀⠈⣿⣿⠆⠀⠀⠀⠀⠈⠋⠁⠀⠀⠘⠛⠋
⠀⠀⠀⠈⠻⠆⠀⠀⠀⠀⠀⡿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣶⠂⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⡀
⠀⠀⠀⢀⣴⣿⣿⣿⣶⣶⣦⣶⣶⣄⠀⠀⠀⠀⣸⡟⠉⠀⠀⠈⠉⣿⣷⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⡟⠉⠉⠻⣿⣿⣿⡇⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⣸⣿⣿⣿⡿⠟⢿⣇
⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⠀⠘⣿⡟⠁⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣷⣿⣿⣿⣿⣾⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠛⡛⡟⠋⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⢿⣄⠉⢄⣿⣧⣤⠀⠘⠻⠻⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠙⠇⢨⣦⣿⣿⣿⣿⣧⡀⢠⣤⣼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⢹⢸⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡾⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡯⢀⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣾⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣛⡽⢿⣿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⣾⡭⢤⠀⠒⠂⠴⠦⣀⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣰⠏⢁⣠⡤⠀⠀⠀⠀⢀⠀⠈⢷⣦⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣴⠋⠁⠀⠀⠀⠀⠀⠀⠙⡄⠀⣿⠁⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢺⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⣯⠀⢸⡀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠁⠘⡇⠸⡇⠂⠠⠄⠀⠀⠀⠤⠀⠀⠄⢸⡟⠆⠈⢳⣌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣮⡵⠟⠋⠐⠀⢙⡄⡳⠾⠛⠦⠀⠀⠀⠐⣛⠶⠒⢘⡧⠈⠢⣄⠀⠉⠑⠒⠒⠲⠲⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠈⣏⠉⠙⠚⠒⢛⠛⠶⣦⣈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⣿⠜⢴⣴⡇⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⢷⣄⡀⠈⢷⣤⣌⠿⣶⢷⡀⠀⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡄⠈⢻⣇⠀⠀⢧⠠⠋⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠙⠳⣄⣀⠈⠻⣷⣴⣀⢳⡀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢹⡆⠀⣹⣦⢰⠹⡆⠈⠘⣦⡀⠀⠀⠀⠀⠀⠀⠈⠳⣵⣍⠳⣆⠈⢷⡀⡀⠹⡌⠪⣛⢷⣤⣀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡸⠃⡰⠘⣿⢿⡇⢷⠀⠀⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⡀⢷⠘⣆⢱⣄⡈⠙⠺⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⠀⡠⢊⣴⠞⠁⢠⠇⢸⣷⡜⠀⡠⠓⠁⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⡀⠀⠐⡾⠀⢈⢾⣟⣬⠉⠉⠳⣝⡙⢟⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠋⠀⣠⢚⣱⠋⠁⢀⡴⠋⣠⢺⣿⠇⣼⠁⠀⣠⠂⠀⠀⠀⠀⠀⠀⣠⠎⠀⢼⣧⡤⠞⢡⡇⠀⢿⣇⠨⣧⡀⠀⢸⣷⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡞⢁⣠⠖⣋⡴⢚⣡⡶⣿⢰⡯⢠⠞⠁⠀⠀⠀⠀⢀⣴⠿⡥⠒⠚⠉⠀⠀⠀⣼⠁⠀⢺⣧⠀⢮⠟⠆⠘⣿⡇⠀⣺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⡜⣡⢞⠏⢚⣿⠏⠀⠁⢾⣿⠃⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠈⠣⡀⠀⠀⢠⡇⠀⠀⣸⣿⠀⠀⣼⠀⠀⣿⡇⢠⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣾⡟⠀⠁⢀⣾⠋⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡼⢀⡃⠀⠀⡞⠀⠀⠀⢣⠁⠀⠀⣿⠀⠀⡎⢷⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠤⠊⡾⠁⢀⠔⢠⠄⠁⠽⢧⡀⠀⠀⠀⠀⢀⠉⠓⠒⠊⢳⣼⠃⠀⣼⠁⠀⠀⢂⠌⠀⠀⠠⡏⠘⡈⢧⠈⠡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠐⡁⠀⡼⠔⣱⠀⠀⠀⠀⠙⠲⠄⠤⠔⠋⠀⠀⠀⠀⣼⡇⠀⢸⠃⠀⠀⠀⡌⠀⠀⣠⢸⠀⠀⠘⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣷⡀⢃⣠⣯⠀⠿⡀⢦⡀⢐⣠⡀⠀⠀⠀⠀⠀⠀⢀⡔⣿⠀⠀⢼⠀⠀⠀⡸⠁⠀⠀⣽⡆⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢹⡍⢮⡨⠻⣦⡘⠿⢤⣬⣓⣿⣿⣷⣄⠀⠀⢀⠔⠉⣼⡇⠀⣆⢸⡀⠀⢠⠁⠀⠈⠀⢸⡄⠀⠀⠀⠀⡆⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣡⠎⠀⠀⢳⣼⣜⡝⠷⠄⡀⠉⠉⠻⣮⡛⣆⡔⠁⡠⠊⡅⢸⢦⡘⢌⢷⡄⠈⢆⠀⠀⠀⠈⠓⠦⣤⡤⠞⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣩⠞⠁⠀⠀⠀⠈⣿⣿⠀⢁⠀⠈⠑⠦⣄⡇⠉⣼⡅⠜⠁⢸⠃⣸⠀⠈⠐⢿⣿⡶⣦⡑⢄⡈⢦⠈⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣰⠋⠀⠀⠀⠀⠀⠀⠱⣟⢠⣸⢧⡀⠀⢀⡼⢿⣆⣾⠈⠓⠤⡘⣿⡗⠀⠀⢰⠀⠁⣹⡀⠉⠲⢝⡒⣦⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠠⠀⠀⠀⣠⡿⠃⠀⢻⡏⠿⠻⠿⠖⢺⡏⣸⠀⠀⠀⠀⠈⠑⠒⠒⠋⠀⠀⣸⠇⠀⠀⠀⠁⣮⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⠀⠀⠀⠙⠊⠉⠀⠂⠀⠀⡇⡄⠀⠀⢀⣜⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⢀⣔⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢰⡷⠧⠶⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣄
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣼⣿⣿⡿⠋⠉⠉⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⣸⣿⣿⣿⡟⠂⠐⠐⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣃⠀⢀⠹⣿⡟
⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⠀⣼⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢦⠾⠗⠔⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣦⡾⠁⠀⠀⠀⢵⣉⡆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠡⠇⠘⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⡽⣇⠈⠀⠀⠁⢠⡽⡜⡀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⡿⠉⠀⠈⢰⣿⣿⣿⡗
⠀⠀⠀⠀⠀⢰⠇⣈⣽⡆⠀⠀⠠⠟⣟⠞⢱⠀⠀⠀⠀⠀⠀⠨⢚⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⠮
⠀⠀⠀⠀⠀⢌⣬⠘⠉⠁⠀⠀⠀⠈⠙⣧⣰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣤⣠⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⣿⣿⣿⣶⣤⣤⣤⣤⣶⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣦⣤⣤⣤⣴⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠿⠿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⡿⠟⠋

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

⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣄
⠀⠀⠀⢀⣾⣿⡟⠉⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⢾⣿⣿⡏⠈⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠁⠀⢹⠟
⠀⠀⠀⠀⠻⠿⠗⠀⠐⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣰⠖⠂⢀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣾⡿⡿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⢿⠃⠄⠠⠈⣆⡆⠀⠀⠀⠀⠀⣼⣿⣿⡧⠁⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠜⣙⣧⠀⠀⠰⣯⠖⡄⠀⠀⠀⠀⠸⣿⣿⣇⠀⢀⣿⣿⡿
⠀⠀⠀⢀⣵⣍⠀⠀⠀⠀⣈⣦⡆⠀⠀⠀⠀⠀⠀⢻⣿⣦⣼⡿⠋⠁⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠚⠿⣿⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣶⣦⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡦⡀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀
⠀⠀⣴⣾⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⢸⣿⡿⠋⠀⠻⣧⠀⠀⠀⠀⢠⡜⠋⠙⠛⠠⣷⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⢿⣧⠀⠀⠀⠀⣠⣴⣶⣴⣶⣄
⠀⣿⣿⣿⠈⠁⠙⣿⣿⣿⣇⠀⠀⠀⣾⡟⠀⠀⠀⠀⣿⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠻⠟⠀⠀⠈⠷⠀⠀⠀⢠⣿⡿⠿⠿⠿⢿⣧
⠀⠿⣿⡟⠀⠀⠀⢸⡿⣿⠏⠀⠀⠀⠘⢧⡀⠀⢀⡼⠃⠀⠀⠀⠘⢿⣿⣦⠀⠀⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⢈⣿
⠀⠀⢹⣧⡀⠀⠀⣬⠟⠿⠀⠀⠀⠀⠠⣿⣷⡀⠈⠀⠀⠀⠀⠀⢰⣾⠛⠛⠀⠀⠛⠛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠙⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀
⠀⠀⢀⣠⣤⣤⣦⣄⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣴⣿⣧⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣄
⠀⢰⣿⣿⠿⠛⠻⣿⣿⣦⠀⠀⠀⢠⣿⣿⣿⡿⠋⠙⣿⣦⠀⠀⠀⢸⡿⠟⠛⠛⠿⢿⡆⠀⠀⠀⠀⢀⣾⣿⢿⣿⠉⢻⡆⠀⠀⠀⠀⢀⣠⣀⡀
⢠⣾⣿⡇⠀⠀⠀⠸⣿⣿⣷⠀⠀⠸⢿⣿⠋⠀⠀⢰⣿⠟⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⣏⠁⠀⠀⠀⠃⠀⠀⢀⣴⣿⣿⣿⣿⣦
⣮⣿⣿⡀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣷⠀⢠⣾⣿⣦⠀⠀⠀⢸⣷⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢾⣿⣟⠁⠀⠀⠀⠀⠀⠀⠸⣿⡿⢻⣿⣿⣿⡆
⠙⢻⣿⣷⣆⠀⢀⣴⣿⣿⣿⠃⠀⠀⣨⣿⣿⠆⢴⡿⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠈⢿⣿⡦⠀⢰⠀⠀⠀⠀⠀⣽⠃⠀⠉⢻⣿
⠀⢻⣿⣿⠟⠀⠘⠿⣿⣿⣿⠀⠀⠀⣻⡟⠁⠀⢈⣿⡟⠀⠀⠀⠀⠘⣿⠇⠀⠀⢸⣿⠁⠀⠀⠀⠀⠈⠻⣿⡗⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠃
⠀⠀⡿⠉⠀⠀⠀⠀⠉⠙⠁⠀⠀⠀⠹⣟⠀⠀⠈⠿⠂⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠈⠀⠈
⠀⠀⠀⠀⣄⣤⣀⣄⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣦⡄⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⣄
⠀⢀⣠⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢠⣿⣿⠟⠁⠻⣿⡀⠀⠀⠀⠀⣿⠟⠙⠛⠻⢿⡆⠀⠀⠀⠀⢰⣿⡿⠿⠿⣿⠀⠀⠀⠀⠀⣤⣶⣶⣶⣦⣄
⠀⢸⣿⣿⡇⡉⠘⢿⣿⣿⣷⠀⠀⠀⢸⣿⠁⠀⠀⠀⣿⡇⠀⠀⠀⣴⣿⠀⠀⠀⠀⢨⣿⡄⠀⠀⠀⠛⠿⠁⠀⠀⠹⠂⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣆
⠀⠸⣿⣿⠀⠀⠀⢒⣿⣿⣿⠀⠀⠀⠀⢻⣄⠀⠀⣰⠟⠀⠀⠀⢀⣿⣿⣷⡄⠀⣠⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⣹⣿
⠀⠀⠹⣿⣄⠀⠀⢠⡿⠿⠇⠀⠀⠀⠀⢾⣿⣄⠀⠁⠀⠀⠀⠀⠘⣿⡟⠉⠁⠀⠙⠛⢿⣿⠀⠀⠀⠀⠀⠀⠐⢿⣦⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠸⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⣠⣤⣤⣶⣤⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⣰⣶⣾⣷⣶⡄
⠀⠀⣾⣿⠿⠛⠛⣿⣿⣷⡀⠀⠀⠀⣾⣿⣿⣿⠿⠉⢿⣷⠀⠀⠀⢀⣿⠿⠿⠿⠿⢿⣧⠀⠀⠀⢀⣼⣿⣿⣿⠟⠻⣿⠀⠀⠀⠀⠀⣀⣠⣀
⠀⣶⣿⡟⠀⠀⠀⠈⣿⣿⣿⡄⠀⠀⢿⣿⡟⠁⠀⠀⣿⡿⠁⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠛⠀⠀⠀⢠⣿⣿⣿⣿⣷⡆
⢠⣽⣿⣁⠀⠀⠀⠀⢘⣿⣿⡄⠀⠀⢾⣿⣿⡆⠀⣼⣿⣧⡀⠀⠀⠸⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿
⠈⢻⣿⣿⣦⠀⠀⣠⣿⣿⣿⠋⠀⠀⢀⣿⣿⡧⢠⣿⣿⡿⠀⠀⠀⠀⣿⣇⠀⠀⠀⣾⡿⠀⠀⠀⠀⣹⣿⣿⡄⢠⣆⠀⠀⠀⠀⠸⡿⠀⠀⠙⣿⡏
⠀⠻⣿⣿⡟⠃⠈⠻⢿⣿⣿⠇⠀⠀⢸⣿⠋⠀⠀⣿⡿⠁⠀⠀⠀⠀⣿⡟⠀⠀⠘⣿⡇⠀⠀⠀⠀⠙⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘
⠀⠀⢻⠉⠁⠀⠀⠀⠈⠉⠋⠀⠀⠀⠘⣿⠃⠀⠀⢻⠇⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⠤⣄⠀⠉⠻⠶⠒⠀⠉⠀⠉⠱⠇⠒⠒⠚⠁⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⠀⠀⠀⠀⠀⠈⠓⢶⡶⠤⠐⠒⠚⠉⠴⠦⠄⠀⣀⣀⠀⠀⠀⠀⠘⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡲⠶⠒⠉⠀⠁⣰⠃⠀⠀⠀⠀⢀⢀⣄⠉⠂⠀⠀⠀⢀⡀⠤⠄⠀⣀⣀⣤⠤⠶⠛⠛⠋⠀⠙⠛⠓⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢀⠀⠀⢀⠐⣸⠃⢀⠤⠐⠂⠀⠀⠀⠉⠉⠓⠒⠶⢒⣒⡶⠶⠿⠛⠛⠛⠁⠚⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢺⠕⠀⠀⠀⠀⠀⢀⡼⠀⢿⠀⠀⠀⡀⠄⠀⠀⠀⠤⣀⢀⡤⠖⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠀⠀⠀⠀⠀⠈⠉⠂
⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⣥⠔⠊⠁⠒⠄⠀⠠⠀⠀⠲⡀⠀⠀⠀⣀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠑⠂⡀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⢀⠀⠆⠀⠀⢀⣤⣤⣶⡶⠒⠿⣏⢄⠑⡀⠀⢀⡀⠀⠈⠁⠂⠀⠈⠢⠄⢀⣈⠁⢨⡀⢠⡉⠂⠀⠀⢀⠀⠀⠀⠢⠀⣀⣐⣭⡁
⠀⠀⠀⠗⠀⠃⠀⢠⠋⠀⠚⠋⠀⠀⠀⠀⠑⠀⠈⠢⡀⠈⠒⠶⣤⣀⡀⠀⠀⠀⠀⠈⠳⢭⡁⠤⣈⢬⠶⠦⠤⠓⠒⠛⠒⠊
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⡀⠀⣤⠀⠀⠀⠠⠼⢲⠤⠬⣶⡤⠤⠤⠽⣿⠿⣿⣷⣦⣤⣤⣖⣛⡶⣄⠀⠀⠀⠀⢀⠐
⠀⠀⡄⠀⠀⠀⠊⢀⠥⠚⠰⠄⠀⠀⠀⠀⠀⠀⢠⣠⣤⣀⠀⠐⠲⡶⠾⠿⠿⠛⢻⠋⠡⢤⣿⠛⣨⠙⠒⠀⠈
⠀⠐⠣⡀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⢩⠉⠻⠀⣁⠀⠀⠀⠀⢰⠃⠀⠀⢰⠏⠂⠙⠄⠀⠀⢀⡄
⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠓⠖⠙⠻⠤⣟⠠⠐⠁⠀⠀⠀⣨⡀⠐⠶⣶⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠉⠩⠔⠒⠀⠀⠀⠀⠀⠀⠧⣚⣠⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⢠⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⢂⠠⡈⠦⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠄⠀⠀⠀⠀⠠⡮⠉⠀⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⣾⠂⠈⠢⡀⠀⠀⠀⠀⡀⠀⢀⠀⢀⡾⠂⠀⠀⢠⠤⠖⠊⠁
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠼⣇⠈⠀⡏⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣠
⠀⠀⠀⠀⠀⠇⠀⠠⡀⠂⢀⡤⠤⡤⢤⣤⣜⣧⣤⣰⣷⡾⠛⠧⠀⠤⠤⣤⡀⠀⠁⠀⣀⡔⠁⠈
⠀⠀⠀⠀⠀⠈⢄⡀⠀⠚⢁⠄⠀⠀⢀⡤⢔⣛⣻⠵⠿⠚⠓⢀⠀⠈⠄⠈⠍⠉⠉⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⡟⠲⠬⡴⠞⠋⠁⡀⠀⠀⠉⠃⠀⠀⡰⠀⠈⠢⣵⣄⠀⣀⢤⣀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢣⡄⠄⠈⠀⠀⠉⠳⠄⠀⠀⣠⠤⠴⠓⠂⠀⠀⠀⠀⠁⠉⠑⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠈⠉⠚⠢⢀⠢⣦⠏⠀⣀⠼⠁⠀⣁⣀⡀⣄⡤⠐⢠⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠠⡌⠀⡆⠀⡃⣤⢞⠡⠀⠀⠙⠉⢉⠇⡈⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠁⠀⡼⠁⡼⠀⡰⠀⠀⡠⠁⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢦⢰⡇⠀⣠⠚⢁⡴⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⣼⠀⢑⡼⠀⢂⠅⠀⠃⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡻⠧⣸⠇⠀⠉⣠⠀⠋⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⢧⡴⠋⠙⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢠⠊⠀⠀⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡠⠔⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⡠⡉⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠙⠁⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠄⠀⠇⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⡴⢻⠀⢸⠓⣄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⢸⠁⠀⠸⡇⠈⠓⠛⠀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠙⢺⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠑⢤⠀⢀⠴⠃⠀⠀⣸⠀⠀⠀⠀⠀⠀⣠⠤⠔⠛⢦⣀⡤⠒⠛⢆⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠸⠀⠸⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⢰⡃⠀⠀⢠⠇⠀⠀⠀⢀⠞⠒⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⣀⣀⣠⠴⠊⠁⠀⠀⠀⢠⠖⠒⠉⢑⠤⠴⠊⠀⠀⠀⢀⡜⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⡏⠀⠀⠀⠀⡔⠉⠉⢳⡴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⢈⡷⠒⠋⠀⠀⢀⣀⡼⠁⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⣀⠀⠀⡼⠀⠀⠀⠀⡴⠁⠀⠉⣶⠋⠉⠀⠀⢀⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⡼⠋⠉⠀⠀⣀⡤⢴⡃⢀⣀⡠⠇⠀⠀⠀⢀⡏⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢧⣀⣀⡰⠃⠀⠀⠀⢰⠃⠀⠀⢹⠋⠀⠀⠀⢀⠴⠒⠋⢀⡠⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢰⠋⠁⠀⠀⢀⠴⠶⢏⢀⡠⠤⠎⠀⠀⠀⠀⣾⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⣠⠤⠴⠛⢦⣀⡤⠋⠀⠀⠀⠀⡇⠀⠀⠀⡏⠀⠀⠀⢀⡔⠚⠉⠈⡷⠖⠚⠁
⠀⠀⠀⠀⠀⢰⠁⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡞⠀⠀⠀⢀⡸⣄⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠳⠤⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠙⠦⡶⠋⠉⠀⣸⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢀⣠⠤⢧⣀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠐⠒⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⡈⠁⡿⠀⠀⢀⠀⢀⡠⠒⡡⠄⢀⡬⠉⠉⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⣿⠥⢒⣡⡼⠶⠁⠄⠊⠀⡞⡁⡾⠀⢰⠏⠀⠀⠀⢸⡗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠞⣰⣯⠀⠔⡅⢁⣴⠶⢚⠷⠶⣴⣿⣇⡧⠴⠃⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⣿⣿⣮⢰⢠⣿⣿⣏⡀⢀⣠⣾⡟⣿⠃⠀⠀⠀⠀⠀⠀⢸⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡔⠀⣰⣧⢸⡟⠈⣾⡞⢩⣿⣿⣗⠒⠛⠛⠚⠉⠀⠉⠉⠑⠀⠀⠠⢼⣏⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣾⣿⠿⠈⠔⠀⣽⡈⢸⡟⢿⣾⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠰⣧⢹⣯⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡏⠁⣿⠀⣠⡴⠛⠉⣾⡇⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷⣴⣍⣻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣽⠋⠈⣿⣷⡄⠻⣿⠃⢀⣀⠤⢋⣣⣾⠟⡁⠀⠀⠀⠀⠀⢀⡀⢀⠀⢠⣿⡙⢷⠙⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⣯⢀⣾⠉⣿⣿⠀⢹⣿⡏⢴⣦⣿⠋⣴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠄⣽⢟⢿⣿⠸⢃⠱⡅⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⡸⣇⡴⢿⣋⣴⠛⢣⡝⣾⠛⣿⣧⣌⠢⡀⠀⠀⠓⠦⣠⡀⣀⠴⠿⠣⢚⡧⠾⠿⢷⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣷⣾⠱⠊⢁⣷⢀⣼⠷⢺⣤⠿⣿⣿⣷⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⢿⣇⠀⠀⠀⠙⢾⠇⠈
⠀⠀⠀⠀⠀⠀⠀⡰⠋⡡⣸⢿⣿⡦⢠⡾⢻⣿⣡⡞⠉⠀⢸⠏⢡⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢧⡻⡘⡏⠀⠀⠱⡌⢿⣜⠷⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠐⣤⡇⢘⣿⣃⢸⡆⡀⣠⠟⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⡰⠊⢀⡆⡸⢵⡞⠉⠀⢸⣷⡇⡻⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠎⠀⠹⣇⠀⡄⢸⣿⡇⡧⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠠⠀⠄⠀⠀⠀⠦⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂⠁⠈⠇⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⣴⡋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠒⠿⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢀⡀⠀⠐⠦⣄⣀⠀⠀⠀⣀⠀⢀
⠀⢀⠀⡠⠚⠒⣶⡄⠀⠀⠠⣘⣊⡩⠀⠐⠀⣈⠢⡀⠀⠀⠀⠀⠠⠤⡀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢨⠎⠀⠀⢠⠈⠸⢗⠐⠛⠉⠉⠉⠩⠈⠀⢙⢧⣈⢦⣀⠀⠀⢀⠤⢼⠧⠀⠠⠦⠔⠀⠤⠐⠒⠦⣄⡀⠀⠀⢀⠅
⠀⡆⢀⣉⠀⠈⠲⢄⡄⠢⢀⢀⠀⠀⠀⠳⡄⠑⢦⣍⡛⠫⣉⣀⣉⣠⡮⠴⠂⢀⠉⣛⣉⡛⣒⡾⠶⠤⠤⠬⢍⣀⢤⣀
⠀⠀⠀⠉⠀⠀⠀⢸⠇⡐⠀⢿⠑⠶⠶⢶⣬⣆⡀⠘⠿⣶⡴⠉⠁⠀⠠⡤⠬⡿⠿⠵⢄⠀⠢⠉⠓⠦⠒⠂⠉
⠀⠀⠀⠀⠀⠀⠀⡏⠠⠁⣤⢈⢗⢋⡀⢆⠻⠿⣿⣗⠢⢌⠹⠓⢶⢛⠟⠲⠶⠾⣄⠒⠢⣕⣄⠀⣑⢀⠀⠀⠈⢉⣤⣀⢀⣀⠀⠀⠀⠀⠀⠔⠐
⠀⠀⠀⠀⠀⠀⠘⡄⡆⠀⢾⠆⠱⣄⠑⠊⡶⠶⣌⠻⠷⣬⡕⣂⠀⠀⠀⠀⠀⠠⠄⠙⠆⡈⠙⠙⠻⣟⣿⣷⣀⡀⢀⡀⠀⠈⠑⠲⢤⣀⡀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠙⠐⠿⣮⣧⡀⠈⠙⠉⠀⠀⠈⢦⠻⢯⡛⠷⠍⠀⠀⠀⡰⠉⠀⠉⠒⠾⡶⢦⣤⠾⣫⣭⣍⠉⠙⠙⠲⠶⠤⠄⠀⠀⠉⠁
⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠟⠛⠿⢟⠛⠛⠁⣀⢈⠢⡈⠛⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⣄⠤⠀⠐⠀⠛⠻⢄⠀⡄⡀⠀⠀⠀⠐
⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠚⢂⣀⡀⠙⠉⠁⠀⠀⠀⣈⠢⡀⠙⠳⣖⠦⣦⡤⠄⣀⣐⡸⠷⣌⠑⠲⣶⡐⠢⢄⢀⣤⡇⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠓⠲⢯⡉⠠⡍⡉⢩⣭⣭⡀⠀⠁⢳⡀⠀⢀⠉⠁⠺⢵⣤⣌⣉⣉⡊⠱⣦⡀⠉⠒⠂⠐⠲⣇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⢦⠘⠻⠀⠀⠉⠃⡄⣿⣾⣗⢤⣤⠄⠀⠊⠙⠉⠉⠉⠣⠈⠙⢢⠀⠉⠀⠒⢄⣀⣘⣦⣎⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠄⠀⠀⡘⠀⠈⢣⡀⠀⠀⠙⠤⢾⣿⣷⣤⣁⠀⠀⠀⠠⠒⠀⠀⠇⠀⠈⡗⠀⠡⠌⠶⠤⠤⠤⠄⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠉⠒⠀⠀⠀⠀⠈⠀⠀⠈⠉⢻⠐⠢⠄⠃⠀⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣤⣶⠾⠿⢿⣿⣛⠓⠲⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣛⡛⠛⠶⠦⣄⡉⢳⡄⢀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣿⢻⣿⡛⠿⢭⣉⠒⢦⡈⢳⡄⢿⡌⡇⢻⡽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⡟⣆⠙⢿⣕⡲⢬⡻⣆⣷⣹⣷⣿⣿⣿⣾⣷⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣳⣜⢮⣙⢦⣜⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⡋⠁⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠉⠙⠛⠛⠋⠭⠒⠁⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠰⣿⣿⣿⣶⣤⣠⡾⣦⣤⣴⣾⣿⣿⡷⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠻⠿⠿⢿⡿⠟⣤⡜⠿⣿⠿⠿⠿⠃⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⣀⠀⠀⠀⣸⣿⣿⡀⠀⠀⢀⡀⠀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣧⠀⠀⠉⠁⠉⠁⠀⢰⡿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢿⡒⠒⠒⠲⠲⠒⠒⣺⢳⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠘⢯⣭⣤⡀⣠⣬⡽⠟⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⣿⣹⣿⣿⣿⣿⣻⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣽⣿⣿⣿⣿⣿⣿⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣛⣿⣧⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⣡⠀⣴⣾⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡇⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⣷⣶⣿⡿⠿⢛⣛⣿⣿⣿⣛⣥⣿⣿⣿⣿⣿⡆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣟⣡⣶⢾⣿⣿⣿⠿⢟⣹⣿⣯⠙⠛⠙⠛⣦⡀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⣭⠶⠋⠁⠀⠀⠀⠈⢷⡝⠻⣶⣷⣄⣘⣿⣷⣽⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢡⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠙⠿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⣿⣟⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⡿⢻⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠉⠛⠛⠻⠷⠶⣦⠀⠀⠀⠀⣴⠶⠿⠛⠛⠉⠉⠙⠀⣿⣿⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⣠⡖⢻⣷⢢⠀⠀⠀⠀⠀⠀⠀⣚⣟⡛⠶⠄⠀⢠⣿⠋⣸⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣇⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢠⠻⢻⣧⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣴⣿⡇⠀⢻⡎⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣽⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⡄⠀⡿⣸⣇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⡇⣿⠀⢿⡍⣿⡄
⠀⠀⠀⠀⠀⠀⢀⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣾⣟⣡⣿⡇⣿⣶⠘⣷⢹⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⣿⣿⣿⣿⣿⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣼⣿⣿⠀⣿⣿⡇⣹⢯⡄⠰
⠀⠀⠀⠀⠀⠀⠸⡄⠸⣿⣿⡆⡿⢿⣿⣿⣿⣷⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣴⣿⡿⣧⢹⡌⢿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣇⠁⠨⣿⣿⢣⡟⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣡⡘⣿⣿⣱⣿⢸⡇⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⣧⣰⢹⣇⢻⣦⡀⢹⣷⡏⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢡⣿⣧⠹⣿⣿⣿⣾⡇⢹⣯⢻⡆
⠀⠀⠀⠀⠀⢀⡼⣿⢻⡆⣿⣆⢽⣷⣼⠋⣼⣿⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠘⣿⣷⣝⣿⣿⣿⢁⠸⢧⠀⢿
⠀⠀⠀⠀⠀⢚⣿⢣⣾⣷⣼⣿⣮⣿⣿⣦⣹⣿⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣟⣷⢸⣿⣿⣿⣿⣿⣿⣼⠀⡏⡦⢸⡇
⠀⠀⠀⣠⢾⡏⠸⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣴⢻⡄⣿⡇
⠀⠀⣰⠋⢈⣧⡾⢟⣼⣿⡟⣼⣿⣿⣿⣿⣟⣼⢟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⣿⢸⡇⣿⡈⣿⡟⣿⣿⣿⣿⣿⣾⣷
⠀⢰⠁⡠⠋⠀⢀⣾⣿⡿⣸⣯⣿⣿⣿⡿⢋⣵⡿⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠸⣯⢷⣌⣷⡘⣿⣿⣿⣿⣿⣿⡝⢿⡄
⠀⠇⠐⠁⠀⣰⣿⣿⡿⣧⣿⣿⣿⣿⣿⣿⣿⠟⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⢷⣜⢿⣿⡾⣷⢉⡿⠹⣿⣿⣿⣇⠈⢷
⠀⠀⠀⠀⠰⢻⡟⡇⠃⣿⣿⡟⣱⣿⣫⣿⣿⣼⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣌⡙⠻⣿⣿⣷⢸⠇⣿⢹⣿⣿⠀⠈⠣
⠀⠀⠀⠀⠀⣸⠀⠁⠀⠸⢯⣾⣿⣿⣿⣿⣿⣟⣧⢡⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢿⣼⣷⡝⢿⣿⣷⣦⣙⢿⣿⣠⢻⣿⡿⣏⢣
⠀⠀⠀⠀⢠⣿⣦⠀⠀⣰⣿⣿⣸⣿⣿⣿⣿⣿⡎⣷⡿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⢸⣿⢸⣿⣮⢻⣷⣜⢿⣿⣿⢻⣼⣿⣿⢡⠀⢣
⠀⠀⠀⠀⠋⢈⠻⣦⡰⣿⢱⡇⣿⡿⣿⣿⣿⣿⣤⠙⣿⠹⡄⠀⠀⠀⠀⠀⠘⣿⡿⠀⢸⡟⣼⣿⣿⣧⣿⣿⣧⢻⡿⡄⢿⣿⣿⠀⣧⠀⢃
⠀⠀⠀⠀⠀⠘⠀⢸⠿⣇⠸⣧⣿⠀⢸⣿⣿⣿⣿⠀⠸⠂⢻⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⡇⣿⣿⡏⣿⡏⢿⣿⣿⢻⣿⣎⢿⣿⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠛⣠⣿⣇⣼⣾⣿⣿⣿⡿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⠃⢸⢠⠟⢀⣿⣿⠁⣿⡇⣸⣿⣿⣼⠉⣿⣶⣿⡆⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⣿⣿⣿⣼⣿⣿⢋⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⡟⠀⣿⢁⡇⢿⣿⡟⣄⣿⡏⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠝⣹⣿⡿⠃⢨⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣰⡏⣼⠀⣾⢹⣧⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠠⣿⣷⣿⠀⡏⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠹⣿⣿⢳⡇⠀⠀⠙⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠹⡇⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⡐⠂⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⣡⣶⡶⠶⣶⢦⣄⡙⢷⡌⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣥⣤⣬⡉⠻⣷⣦⡈⠹⡝⢲⣧⣿⣿⡏⠊⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠊⠁⠀⠈⠙⢧⠈⣿⡟⠢⢹⠀⣿⢿⣯⣿⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⢸⣧⣼⣿⠂⠘⠀⣿⣿⣿⡿⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡀⠀⠀⠀⠔⠂⣼⢿⣿⣏⢦⠀⣼⣿⣿⡿⠃⣴⣿⣿⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡖⢶⠀⠀⠐⢳⡞⠛⢿⣿⠨⣾⣟⡜⣿⡷⣾⢿⣹⣿⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣯⡙⢿⣼⢯⣿⣿⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⠇⢸⣿⣷⣝⠻⣝⢿⠿⣿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡟⢷⣄⡀⢀⡀⠀⠀⠀⠀⢰⣿⡟⠀⣿⣟⢿⣮⣳⡌⢻⣷⣌⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣴⣿⣞⢷⡠⠄⠈⠀⠀⠀⠀⠈⠛⢿⣶⣌⠛⢿⣿⣿⡇⢀⠻⣿⡓⢆⠹⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢻⣶⣿⣿⠘⣧⣀⣀⣀⣤⠔⠀⠀⠀⠀⣿⡾⠀⣼⡿⠋⠁⣼⠂⣹⢧⠈⡆⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡿⠿⠟⠛⠉⠁⠀⠈⠉⠀⠀⠀⢀⡾⢋⣡⡴⣿⠁⣠⠾⠟⣠⣿⠇⠰⣠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣿⣿⡄⢻⣀⠡⣶⣿⠟⠁⣠⣾⣿⣻⢿⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢾⣿⣿⣄⠘⢧⡘⣿⣄⠀⢿⣷⣿⢿⣿⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢹⣿⣿⠀⣼⣷⣿⣟⣿⣦⡈⢿⣿⣄⡉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣾⠛⣵⣾⣿⣿⣿⣿⣿⡿⢀⣼⣿⣿⢃⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣏⣠⣤⡤⠴⠶⠛⠃⠉⠁⠀⠀⠀⠀⢧⣿⣿⣾⣷⣽⢿⣷⣿⡿⠋⣰⣿⣿⣿⠁⢾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣀⣠⣴⡮⠀⢹⣿⣿⣿⣿⣿⡿⠿⣷⣄⠙⠻⣟⣿⣷⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣤⡶⣾⠿⠟⠋⠉⠀⠀⢀⣿⣿⣄⠹⣿⣄⠀⣼⣿⡿⠀⣹⣿⢛⡥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⡀⠀⠀⣴⣶⠋⠀⣹⣿⠀⢈⣿⣾⡟⢡⠚⠟⠉⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠶⠶⣶⡶⡶⠿⠿⠿⠋⠀⠈⠻⣷⠀⢸⣿⠁⣴⠛⠉⠀⠀⠰⡄⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠙⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠟⠛⠷⠷⠶⠶⠶⠶⠿⠛⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣀⣒⣒⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⠿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣿⠿⠟⠛⠻⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⠿⢻⣿⠇
⠀⠀⠀⠀⠋⠀⣤⠀⠀⠀⣽⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⡤⠀⠀⢠⣤⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠿⠁⠀⠠⠄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡃⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠐⠀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⣛⣭⣭⣟⡿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠋⠁⠀⠀⠈⠙⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠿⠿⠿⠿⠛⣿⣇
⠀⠀⠀⠘⣿⣿⠿⠿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⢀⠀⢸⡏
⠀⠀⠀⠀⢻⠏⠃⠀⠈⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡁⠈⠁⠀⠀⠈⠁⢈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠐⠂⠈⠁
⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣄⠀⠀⣠⣾⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⠉⠀⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠁⠉⠀⠉⢷
⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢃⣀⠀⠀⢀⣘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⡀⠀⠀⢀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠉⠀⠀⠈⠁⠸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⠈⠢⡀⠀⡀⠀⢸⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣟⡻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡭⠶⢟⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣟⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠁⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣁⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⣦⣤⣾⣿⣿⣿⣶⣦⣌⣩⣿⣿⣶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠟⢹⣿⣿⣿⣿⡿⠻⠿⡟⡿⡻⣿⡛⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠈⣿⠏⠀⣼⠀⠀⠀⠀⠀⠀⠘⡇⠀⢻⡟⢿⣿⣿⣿⣿⣧⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⣿⠀⠀⡿⠀⠀⠀⠀⠀⠀⡀⢻⡄⢠⣻⣌⠻⣿⣿⣿⣿⣆⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⣿⣿⣶⣿⣼⣦⣴⣴⣷⣼⣿⣿⣿⣿⣿⣿⣦⠙⢿⣿⣿⣿⣶⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣧⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠳⣄⢻⣿⡿⠛⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠙⠛⠛⠿⢿⣿⠿⠿⠿⠟⠛⢿⠿⠛⠋⠉⠁⠀⠘⢷⣿⣷⣄⠀⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠀⠀⠀⠀⣿⣿⣿⣦⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⠈⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣄⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡉⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⣸⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣸⣿⣿⣧⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣥⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁