Shtick Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣒⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠁⠀⠀⠀⠄⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠠⠈⠁⠈⣀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡆⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠠⠊⠀⠈⠀⡂⠀⠃⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⠁⠐⠪⠔⠂⠀⠁⠀⢄⢀⠀⠀⠁⢐⣤⣘⠶⣤⣖⠋⠉⠙⢢⠔⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣧⣤⣤⣠⣔⣉⡀⠈⠒⠄⣀⡐⡉⠀⠀⠠⡁⡠⠄⠒⢲⣼⣯⡦⣔⡁⣀⣀⣀⡉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣐⠈⠻⢿⣿⣿⠟⠉⢀⣠⣶⣿⣿⣿⣿⣮⣭⣉⣉⣭⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⣛⡛⠿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⢷⣦⣬⣉⣉⣉⣡⣤⣶⣿⣿⣷⣶⣬⣉⣛⠛⠋⣉⣤⣶⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣤⣉⠛⠻⠿⠟⠛⢉⣀⣠⣄⣀⣉⠛⠿⠿⠿⠿⢛⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣶⣶⣾⣿⠿⠛⠛⢛⣿⣿⣶⡶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣠⣾⣿⣿⣿⢶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣿⣿⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣱⣿⣿⣿⢟⣵⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⠟⠠⡀⠰⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣳⣿⣿⣿⣳⣿⣿⣿⣿⣿⡿⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⣀⣿⣿⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠙⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⠛⢋⣭⣴⣶⣿⡆⠀⠀⢀⣀⣠⣤⣤⣴⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⢿⣛⣯⣽⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠊⠉⠙⠁⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⠀⠉⠓⠶⠶⠶⠶⢤⣤⣴⠂
⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠒⠀⠀⠀⠀⠐⠀⠸⣧⠈
⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠦⡴⠶⢦⣶⣤⡤⠦⢤⡴⠟
⠀⠀⠀⠀⠈⠠⣍⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠳⢍⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣀⡀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣻⡶⠷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⣻⡏⠙⣿⣙⢉⡿⠋⢹⣿⡉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⡌⢳⡄⠀⠀⠀⠀⠀⠀⢠⠁⠀⢀⣿⡟⠙⠦⠟⠻⣴⠃⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠛⢙⣶⡄⠀⠀⠀⠀⢸⠀⢠⣼⣿⡃⠀⠀⠀⠀⠀⠀⢘⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡀⠙⣦⠀⠀⠀⢸⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢈⣿⡎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠳⣄⠀⣼⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⡇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⢷⣽⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠁⠀⠘⣿⡗⠀⠀⠀⠀⠀⠀⠀⠸⠳⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢈⡀⣷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⣤⣤⣤⡤⣤⣤⡄⢸⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠦⠴⠖⠒⠑⠀⠀⠀⠀⠀⠈⣾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⢶⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣯⢸⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⡇⠘⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⢸⡃⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣠⡇⢸⣧⣤⣤⣄⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠉⠀⠈⣧⠸⠇⠈⠁⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣤⣀⠒⠾⠅⠿⣤⣄⡐⢲⡯
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠄⠙⠛⠛⠛⠛⠛⠻⢶⣦⣄⣤⡤⠤⠞⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡰⠖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠛⠻⣷⣤⣤⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⡴⠶⠺⣷⠈⡇⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣰⡏⠀⢀⣴⠋⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠟⠛⠛⠋⠀⠀⠀⠀⢀⡴⠟⠛⠃⠀⠀⠀⠀⠀⠛⢶⡆⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡷⠀⠀⢀⣾⣿⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⢿⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⢀⡾⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿
⠀⠀⠀⢀⣴⠏⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣶⠄⠀⠀⠀⣠⠟
⠀⠀⢠⡞⠁⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⣀⡾⠋
⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠞⠋
⣶⣅⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣶⣦⣤⣤⣤⣤⣤⣶⡶⠞⠛⠋
⠀⠉⠛⠛⠶⠶⠶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠻⠿⠷⢶⣤⣤⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠒⠺⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠛⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣰⣇⡀⠀⠘⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⠀⠀⠀⠉⠛⠷⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠈⠛⠋⠉⠀⢀⣿⠃⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣤⣤⣤⠞⠛⢿⣇⠀⠀⠀⠀⠀⠀⢘⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣶⣶⣤⣤⣴⠟⠉⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣴⠚⠛⠛⠛⠛⠻⣦
⣀⣀⣀⣀⣀⣻⣷⣶⣶⣶⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣛⣛⣛⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣤⣤⣀⠘⢷⡄⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣙⡗⠘⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⠉⠛⠛⠶⣮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡾⠿⠷⠦⠀⠀⠀⠀⠀⠀⡴⠾⠿⢷⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣤⣤⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⣤⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢻⡟⠁⠀⠀⠀⠈⠙⣿⡿⢿⣿⠋⠁⠀⠀⠀⠈⢻⡟⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢸⣇⠀⠺⠶⠶⠖⢀⣿⠃⠘⣿⡀⠲⠶⠶⠃⠀⣸⡇⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣈⠻⣦⣄⣀⣀⣤⡾⠃⣤⡄⠘⢷⣤⣀⣀⣠⣴⠟⣁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢮⣍⠉⠉⢡⡀⠀⠀⠀⠀⣀⡈⠉⠉⣩⡴⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⠛⠛⠛⠉⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⣶⣶⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠉⠉⠉⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⣧⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠘⣷⣄⠀⠀⠀⢀⣠⡿⠁⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠟⢷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠙⠛⣷⡿⠛⠉⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠻⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣿⢡⡿⢷⣤⡍⠙⠻⢶⡄⢀⣠⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣀⣠⡶⠟⠛⢩⣤⡾⢿⡎⣿⡀
⠀⠀⠀⠀⠀⠀⠸⣧⡘⢛⣛⡋⠁⠀⠀⢨⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠛⢷⣄⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⡇⠀⠀⠈⢙⣛⡛⢃⣼⠇
⠀⠀⠀⠀⢀⣤⡴⠿⢛⣛⣭⣥⠀⢀⣠⣾⣿⣿⣶⣶⣶⣤⣤⣄⣀⣰⡏⠀⠀⠀⠀⠀⠀⢹⣆⣀⣠⣤⣤⣶⣶⣶⣾⣿⣷⣄⡀⠀⣬⣍⣛⡛⠻⢦⣤⡀
⠀⠀⠀⠀⠸⠷⠶⢛⣿⠟⣫⡷⠟⠋⠉⠛⠻⢿⣶⣦⣤⣉⠉⠛⠛⠿⣧⣀⣀⣀⣀⣀⣀⣼⠿⠛⠛⢉⣉⣤⣴⣶⡿⠟⠛⠉⠙⠻⢾⣝⠻⣿⡛⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠸⠷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣛⠿⢿⣶⣶⣮⣍⠛⠛⠛⣛⣭⣥⣶⣶⡿⢿⣿⠛⠉⠠⢤⣀⣀⠀⠀⠀⠀⠉⠻⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠛⠛⢶⣭⣯⣴⣶⣾⠿⠿⣛⣻⣽⡿⠛⠛⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⠇⠀⠀⠀⢻⣿⠟⠩⠥⠤⠤⠤⠍⠻⣿⡟⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢀⣴⡟⢁⡀⠀⠀⠤⠤⠤⠀⠒⠘⠻⣦⡀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⠾⠋⠉⠀⠉⠀⠀⠀⠀⠤⠤⠤⠀⢀⣀⣈⡛⣷⣦⣼⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠂⠉⠉⠉⠙⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⡠⠂⠀⠀⠀⠒⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠗⠀⣠⠎⠀⠀⠀⠀⠈⢀⣴⣄⢠⣶⡄⠀⠀⠀⠀⠘⢦⠀⠀⢠⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⢒⡏⠀⠀⠀⠀⠀⠀⠸⣿⡟⢸⣿⠇⠀⠀⠀⠀⠀⠈⣷⣀⡈⣀⠴
⠀⠀⠀⠀⠀⠀⠀⡰⠛⠇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣡⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠹⢤⡤⠞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⡀⠀⠀⢸⣇⠉⢀⠇
⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⡞⠈⣉⣁
⠀⠀⠀⠀⠀⠀⠀⣇⠰⠁⢰⠃⢳⡀⠀⠀⠙⠲⠤⠤⠤⠤⠔⠒⠉⠀⠀⠀⠀⢀⡼⡇⠘⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠋⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠳⣄⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣀⣀⣀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢰⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢸⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠐⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣡⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠫⣿⣦⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠈⠘⢿⣆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠠⣿⡾⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠗
⠉⠉⠉⠉⠉⠉⠉⠉⠳⢍⡉⠉⠉⠁⠻⣿⡇⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⠶⠿⣤⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⡈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠊⠉⢉⡱⠃
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡁⢀⡀⡀⠠⠤⣠⣀⠐⠤⠤⠤⠌⠉⠉⠡⡄
⠀⠀⠀⣦⣙⣓⣂⡀⠆⠀⠀⠀⠀⢠⠁⠀⡀⡀⢀⣀⡀⠀⣤⡄⣀⠠⠤⠀⠄⡤⠀⠀⣧
⠀⠀⣴⣧⣶⣶⣶⣶⠗⡄⠀⠀⠀⡎⠀⠀⠀⠈⠉⢀⢉⠠⢀⠀⠀⣀⣠⠀⠀⢤⣀⡀⠘⣆
⠀⠀⢿⣿⣿⣿⣿⣿⡿⠇⠀⢀⠞⠀⢀⣅⢩⠁⡉⡉⣁⡁⠀⢈⣭⣁⣡⢈⡀⡀⠤⣀⠀⠈⢣⠀⡀
⠀⠀⠤⠿⠿⠿⠛⠛⠁⠘⠯⣍⣃⣀⣂⣀⣀⡀⠒⠒⠒⠒⠒⠒⠒⠀⢐⣀⣈⣁⣈⣉⣀⡉⠠⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠤⠐⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣤⣡⠯⠛⢿⡟⠛⠷⣶⣾⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠳⠻⠷⠶⣤⢶⠦⢤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣤⣄⣴⣦⣄⢀⡀⢸⣃⣀⢀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣠⣤⣄⣤⣤⣤⣈⣤⠶⠶⡶⠾⠒⠶⢤⣤⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣁⠉⡉⣉⠉⠉⠙⠡⣦⣀⣦⣤⣤⣴⣄⡁⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⢀⣦⡾⠛⠛⠛⠿⠷⠞⠈⠉⠉⠁⠀⠀⠉⠀⠘⠻⠶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⡿⠏⠀⠀⣠⠤⢠⣤⣀⣘⣆⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠇⠀⠀⠀⠀⣌⠉⠙⠙⠛⠻⣶⣦⣶⣦⣴⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣈⣤⣠⣾⡀⠀⣤⣠⣓⣴⠿⠛⠋⠙⠉⠈⠀⠈⠉⠉⠉⠙⢿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢⣤⣼⣠⣀⣼⡆⣤⣦⣤⣦⠤⡤⣤⣴⣆⡆
⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣷⣤⣿⣄⡀⣀⠄⠀⠀⠈⢉⣉⣩⠿⢿⣾⣿⠟⠳⠛⠉⠛⠻⠆
⠀⠀⠀⠀⠀⠀⢠⠈⠉⠀⠀⠀⠁⠈⠙⠛⠷⣶⣞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠐
⠀⠀⠀⠀⢠⣔⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠓⠶⣦⣴⣤⣴⣤⣤⣿⠃⢠⣤⣄⡀⡆
⠀⠀⠀⠀⠺⠿⢷⣶⣦⣷⣤⣶⣼⣇⣰⣤⣤⣤⣦⣀⣄⣠⣤⣀⣄⣈⣩⣍⣥⣤⣴⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⢀⣠⠖⠛⠉⠙⠒⠦⣄⡀⠀⠀⣀⡀
⠀⠀⠀⠠⣏⣀⣀⡀⠀⠀⠀⠀⠀⠉⠲⢤⡿⣈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠒⠲⠤⣄⡀⣏⠙⠦⡙⢦⡀⠀⠐⠛⠂⠀⠀⠀⢀⣈⣀⣀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢭⣁⡀⠀⠈⠢⣽⠶⠒⠦⣄⡀⣠⠞⠉⠁⠀⠈⠙⣆
⢀⡔⠋⠀⠀⠈⠉⠓⠒⠦⠤⣄⣀⡇⠀⡤⠤⠤⢍⣑⡒⠤⣏⠀⠀⠀⠀⠙⢧⣴⡆⠀⠀⠀⠀⡿
⠻⠤⢤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠉⠉⣺⣄⠀⠀⠀⠰⠎⣷⠀⠀⠀⠀⣰⠃⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⣠⠞⠁⠈⠑⠒⢤⣀⣠⠟⠶⠶⠒⠛⠓⠚⠋⠉⠉⠁⢰⠏⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣠⣄⡀⠀⠀⡼⠁⠀⣠⣶⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢍⢂⢸⡇⠀⣸⢁⠷⡴⠋⣳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠽⠿⣗⣒⡪⠭⠔⠛⠻⢿⠫⡭⣍⣙⣾⣁⡰⠋⢓⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⣀⡠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠦⠀⠘⢯⣠⠎⢙⠶⢮⣀⣠⠟⠓⠲⢤⣤⣤⣤⣤⣤⠤⠤⠟⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⠓⠯⢄⣼⣁⡀⠀⠀⠀⣼⡉⠉⠉⠲⠖⠒⠒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣦⡤⠒⠊⠛⣒⢢⡀⠜⠁⠀⠀⠀⠀⠱⡖⣆⠀⠀⢲⣄⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢿⠀⠀⠀⠀⠁⠀⣠⡀⢠⡀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⠲⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠙⢄⠈⠓⡶⠶⠶⠒⠋⠉⢧⡤⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣙⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⣫⠛⣷⡶⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠭⣽⠄⠀⠈⢶⠒⠒⠒⠒⠚⠉⠉⠹⣖⠩⠗⢞⡽⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡴⠂⢀⡀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠈⠉⠑⠦⣀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠉⠀⠙⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⡐⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⡷⣸⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⢰⡾⡀⠀⠀⠀⠀⠀⠀⢠⠇⡟⡄⠘⣆⢀⣶⠀⠀⠀⠀⠀⠀⡨⢷⠈⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠁⠠⠀⡸⠀⠀⠀⠀⠀⠀⢸⢸⡇⣧⠀⠸⡜⡘⡆⠀⠀⠀⠀⠐⢆⠠⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⡠⠎⠀⠀⠀⠈⡏⢦⠀⢸⠀⠓⠋⠀⠀⡇⢱⢹⡄⠀⠀⠀⠀⠀⠑⣀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⢀⡀⢧⠀⠀⠀⠀⠳⠎⢠⡇⡤⡄⠀⠀⠀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣸⠈⡟⡌⢆⠀⠀⠀⠀⠀⡸⢸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⣠⠤⣀⠀⡇⠀⢻⡼⠁⢹⣸⠀⠀⠀⠀⠀⠳⠞⠀⣧⢀⣠⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣤⡀
⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⣆⡇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⣧⢯⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⡀
⠀⠀⣰⣿⣯⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠘⠁⠀⠀⠀⠀⠀⠀⣰⠃⢠⠄⠀⠀⠀⢸⣸⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢷⡄⢠⠀⣿⣦⣾⠂⠀⠀⠀⠈⣿⡆⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀
⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⣿⣣⣿⣿⣿⣾⡇⠀⠀⠀⢸⡞⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢿⣆⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣁⣠⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠈⣧⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⠀⠀⠀⣴⣃⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡚⣽⠒⠋⠉⠳⢄⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣀⡴⠚⠉⠉⠉⣏⡚⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣽⡤⠤⠤⠶⠶⣹⠷⠒⠹⢿⣿⠿⡿⡿⢿⠿⠷⠾⣛⠛⠲⠦⠶⠤⢤⣭⣀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⠋⠁⢀⣠⠔⠁⡏⠀⠃⡇⠈⠳⡀⡀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠭⢿⢣⣠⣴⠟⠁⠀⢰⠃⡠⠀⢧⠀⠀⠈⢪⡓⡀⡜⣪⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠐⡺⠟⠁⠀⠀⠀⣼⢴⡥⠄⣸⠀⠀⠀⠀⠙⢾⡖⠊⡸⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣇⠈⣁⡱⣽⠀⠀⠀⠀⠀⠀⠉⠓⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡄⢈⠿⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠛⠋⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⠏⠀⠃⠀⠀⢈⣦⡄⠀⠀⠀⠀⠀⢀⡾⣿⡿⣦⠀⠀⠀⢈⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣯⠀⠀⠀⠂⠀⠻⣿⠟⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠁⠉⠉⠙⡽⠝
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⢳⣿⣿⠀⣤⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⡀⣀⡀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠻⠿⠟⠿⠻⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣹⡽


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣀⣸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣜⣋⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⡻⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⢠⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠿⠇⣠⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠿⢿⣿⣿⣿⢿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠰⠆⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠾⣿⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⢳⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣙⣋⣭⣴⣦⣅⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣩⣴⣦⣍⣙⣋⡀
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠛⣋⣍⡛⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⡛⠁⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣷⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⠉
⠀⠀⠀⠀⠀⠀⣀⡹⣶⠉⠡⡾⢃⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⡙⢷⠌⠡⡾⢋⣀
⠀⠀⠀⠀⢀⣾⡿⣷⣤⠀⢠⣴⡿⣿⣆⠀⠀⠀⠀⠀⣀⡀⢬⣭⡅⢀⣀⠀⠀⠀⠀⠀⣴⣿⢿⣦⠄⠀⣴⡿⣿⣧
⠀⠀⠀⠀⣼⡿⠁⣿⡏⣸⡆⢿⡇⠸⣿⡆⠀⠀⠀⣠⡝⢿⠆⠀⢶⠟⣡⣄⠀⠀⠀⢸⣿⠃⢸⡟⣸⣇⢻⡇⠈⣿⡇
⠀⠀⠀⠀⣿⡇⠀⣿⣄⠻⠏⣼⡇⠀⢿⡇⠀⢀⣾⣿⣿⣦⡆⠀⢦⣼⣿⣿⣷⡀⠀⢸⡏⠀⢸⣇⠻⠟⣸⣧⠀⢻⡇
⠀⠀⠀⠀⣿⠇⠀⣛⡛⣃⠚⡛⡃⠀⢸⡇⠀⣾⣿⠃⢸⣿⢁⣿⠸⣿⡇⠹⣿⣧⠀⢸⡇⠀⢛⢛⣓⣚⣛⠛⠀⢸⡇
⠀⠀⠀⠀⠠⠀⠀⣿⣿⡿⣿⣿⣿⠀⢠⠄⠀⣿⡏⠀⣸⣏⢸⣿⠇⣻⡇⠀⢻⣿⠀⠰⠄⠀⣿⣿⡿⢿⣿⣿⠀⠠⠄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⣿⡇⠀⣿⣿⣦⣉⣾⣿⡧⠀⢸⣿⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠛⠃⠀⣶⣶⣶⣶⣶⣶⣶⠀⠘⠛⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⣿⣿⣿⠀⠀⠀⠀⠁⠀⠀⣿⣿⣿⠉⣿⣿⣿⠀⠀⠈⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣩⣭⡄⣬⣭⡁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣩⣥⡅⢨⣭⣍
⠀⠀⠀⠀⠀⠀⢈⣛⣛⣁⣙⣛⣋⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⢀⣀⣙⣛⣃⣘⣛⣋
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⢿⣿⣿⣿⣇⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣆⠀⠀⠀⠈⢉⣉⣉⠀⣍⣉⡉⠀⠀⠀⠀⣰⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⡀⠀⠀⠿⠿⠿⠀⠿⠿⠷⠀⠀⢀⣴⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠋⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿


⠲⡶⢀⣶⡄⢰⠀⢰⡖⠂⣶⡀⡆⢲⡖⠀⢰⣆⢰⡆⢠⣶⡀⣶⡄⡆⢠⣶⡀⣴⠖⠂⣴⠒⢠⣦⢰⣆⢰⡖⢠⣦⣰⡔⢶
⣸⣇⣸⣛⣇⣸⣤⣸⣯⣅⣏⣹⣇⣸⣇⣀⣸⣹⣏⣇⣼⣛⣇⣿⣹⣇⣼⣛⣧⣳⣬⣇⣿⣭⣸⣙⣏⣿⣸⣭⣈⣏⣻⣁⣸
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⠃⠀⣤⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣧⠀⠀⠀⠀⣿⣤⠤⣄⠀⠀⢀⣤⡀⠀⢀⣤⣴⡇⠀⢠⣿⠀⣾⣡⡼⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡆⠀⠀⢸⣿⣡⠆⠘⣧⣴⣿⡀⡿⢀⡾⢡⣿⠁⢀⣾⠇⠀⣿⠁⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⣾⠉⣟⠛⢻⡏⢾⣀⡿⠛⢹⣇⡞⢸⣀⡞⣿⠀⠀⠈⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⣷⢀⡿⠀⠻⠶⠛⠀⠈⠋⠁⠀⠀⠉⠀⠈⠉⠘⠷⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⠇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣀⣤⡤⠶⠒⢛⣋⣩⡍⠉⠉⠉⠍⠉⠉⠉⢩⠗⠋⡭⠛⠛⠛⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠉⠀⢀⣀⡀⠀⠀⠀⢀⣀⣤⣀⣠⠒⡏⢹⠛⣟⢦⡈⢢⡀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣰⠋⢉⡉⠉⠓⢲⣯⠔⢶⡞⠉⣦⡑⢌⠲⣌⠳⡙⣄⢹⡀⠀⣀⠀⠀⣧⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣀⣤⣅⠀⠀⠀⢀⣠⠾⠾⠤⣾⢣⡘⢇⠙⢌⠳⣌⢣⡹⣌⢦⠳⠞⠉⠓⡚⢁⠛⢄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠐⠃⠀⣰⠋⠉⠀⢘⡾⠳⢖⡫⠭⠤⠴⠊⣉⣉⣧⢳⡈⢣⠀⠙⢎⢧⠉⠙⣆⠳⡀⠀⠯⣀⡁⠀⠈⢣⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣀⣀⣸⣠⠴⣻⠿⠶⣒⣋⣠⠴⠋⣉⡭⠤⠤⠜⣧⢹⠀⢙⡷⠛⠦⡓⢄⠸⣗⢭⡢⢄⠀⢹⠀⠀⢀⢧⢀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠐⣇⡔⣋⡥⠤⠤⣴⡾⠕⢒⣊⡵⠒⠒⢒⣫⠥⢖⡒⠂⢁⣹⡆⢧⠈⠓⢦⠀⠈⢢⡱⡌⢆⠙⣦⡱⡌⠑⢒⡡⠊⠘⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠊⠥⠒⠒⣫⡥⠖⠛⠦⣄⠀⣠⠞⣡⠖⣫⠥⠒⢊⡭⠂⣹⡌⢧⡀⠘⠢⠒⠚⣳⣝⡢⣥⣤⡾⠞⠛⠛⠒⠒⢤⡇⠠⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡩⢤⡟⠁⠀⠀⠀⠀⠈⢳⡏⠐⣅⡀⠓⠤⢤⣈⡳⣄⢣⠘⢦⣉⣓⣒⣈⡭⢶⠎⣩⠋⣼⠀⠀⠀⠄⠀⠀⠀⢳⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠸⡏⢷⣜⠀⣠⠞⠛⢦⡀⠀⠈⣿⠦⣄⣉⡉⠓⢦⡀⠈⢧⠹⡗⠤⠏⢉⢏⣠⠬⠥⢖⣡⠾⠖⠓⠒⢦⡀⠀⠀⢀⡾⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢦⣇⠀⠀⠘⡇⠀⠀⣿⠈⣀⠒⠙⢳⣄⡑⠒⠚⠒⢿⣄⡰⣫⠏⡴⠚⠉⢹⠁⠀⠀⠀⠀⠀⠙⣆⡠⠚⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠷⣤⣀⡇⠀⠀⣿⠀⣒⠛⠛⠋⣊⠽⢛⢝⡛⢦⣉⠙⣁⣾⠴⠒⠒⠺⡄⠀⠀⠀⠁⠀⠀⢻⢀⣠⠞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠈⠙⠳⢦⣴⣏⠒⠚⠿⠭⠭⢒⡪⢟⡭⠒⠉⠉⠉⡿⠁⠀⠠⠀⠀⢹⡀⠂⠀⠀⣀⣤⠾⠛⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⣶⣭⣥⣖⣉⣀⣀⠀⠀⢰⡃⢀⣀⣀⣀⣀⣨⣧⡴⠞⠛⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠠⠄⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠉⠁⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢀⡜⠛⣿⢿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠁⠀⢠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢀⣴⠋⠘⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠂⠀⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⢀⣤⣶⣶⣶⣷⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣼⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡿⣿⡿⣻⣿⣦⣄⣀⣀⣀⣀⣀⣤⣶⣿⠏⠀⠀⠀⠀⠙⢿⣿⣿⡿⠿⢿⣿⣖⣀⣀⣤⣼⣿⣿⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢼⢸⡇⣿⣧⠑⣽⣦⣈⣉⣙⣛⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⣿⡛⠟⣉⣽⣻⣿⡏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠹⣏⠻⣶⣦⣭⣭⣽⡿⢟⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⣷⡻⠛⢋⡽⠋⢠⡾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠇⠀⠈⠀⠀⠈⣉⣉⣥⠴⣊⣽⡿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠃⠘⢷⡄⠉⠀⣰⡟⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⢩⣤⣾⡿⠋⠀⠻⣿⣿⣧⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠙⠛⠻⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢹⣿⣿⣆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⡄⠀⠀⠀⠀⡇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠇⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⢀⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⣸⡇⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣠⠟⠀⣸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⢰⡏⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⡿⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢠⣿⠃⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣾⣿⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠿⠃⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠘⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠘⢿⣿⣧⠻⣿⠿⠆


⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠈⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣤⣤⣼⣿⣦⣤⣄⠀⠀⠀⢠⣿⣿⣿⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⢛⠰⠆⠀⠀⠀⣀⣬⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠛⠿⠛⠛⡟⠁
⠀⠀⠀⠀⠀⠀⣴⣾⣷⣦⠈⠀⠀⠀⠀⠀⣵⡾⣿⠛⠉⠻⠇⠀⠀⠀⠀⠘⣣⣾⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⣿⣴⢢⣶⣶⠜⡆⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⠀⠀⠀⣴⣿⣦⣠⣤⡀
⠀⠀⠀⠀⠀⢸⣿⠀⠻⣿⣿⠟⠀⠀⠀⠀⠀⠀⢘⢿⢟⣀⡀⠀⠀⠀⠀⠉⢿⣿⡿⢿⣿⡗
⠀⠀⠀⠀⠀⠀⣁⠀⣤⣼⣧⣤⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡀⠀⠀⠀⠰⠲⠀⣠⣤⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠘⠾⠿⠿⠿⠿⠀⠀⠀⠀⠀⠸⠿⠻⠿⠿⠿⠇⠀⠀⠀⠀⠈⠁⠿⠿⠿⠿⠇


⠀⠀⠀⠀⠀⠀⣴⣿⣶⠀⢾⠷⠀⠀⠀⠀⣀⠤⣄⣠⣤⣀⣀⠀⠀⠀⠀⣤⣦⣤⣶⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠘⣿⠋⠈⢀⢀⣷⡄⠀⠀⣇⠀⣨⣟⣛⣿⣏⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⢠⣦⣿⣴⡄⣀⣀⠏⠁⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⡿⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠻⠟⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⡿⡿⠿⡟⢛⣋⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣤⣤⣴⣶⣶⣤⣤⣴⣶⣾⠿⠿⠛⣛⣋⣩⣭⣤⣴⣶⣿⡿⠿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⠿⣿⢿⢛⡏⡟⡇⣿⣿⣿⣿⣶⣶⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⡏⢹⢸⠀⡇⢸⢰⡇⡏⡇⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⡿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⢾⢸⠀⡇⢸⠸⡆⡇⡇⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣾⢸⠀⡇⢸⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡏⢸⠀⡇⢸⠸⣤⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠀⡇⢸⢸⣿⡅⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠀⡇⣼⢸⢻⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⣾⢸⡇⡟⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⡇⣿⢸⡇⡇⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣨⣿⡇⣇⣿⣸⣇⣧⣸⣀⣇⡇⣿⣧⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣋⠭⢭⣉⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣯⣤⠤⠬⣝⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣾⣷⣾⣿⣿⣿⣥⣴⠚⣉⣉⣭⣉⣍⣉⣙⠒⠢⢬⡙⠿⣷⣶⣬⣷⣼⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⣋⡭⠥⠒⠒⠒⠒⠢⠤⢍⣉⠛⢿⣷⣤⠤⠤⠤⠤⢬⣉⡓⢦⣬⣷⡿⢛⣉⣉⣉⣛⠻⢿⣦⣄
⠀⠀⠀⠀⠀⢠⣿⠟⣵⠚⣡⠖⣊⣉⠭⠭⠭⣉⡓⠒⠤⣉⠲⣌⢻⣷⡖⠒⠒⠒⠦⣍⣾⡟⣡⠞⣩⠤⠴⠤⢬⡙⢦⡙⣿⣆
⠀⠀⠀⠀⣠⣼⣿⣴⣧⣾⣵⣾⣷⣶⣶⣶⣶⣶⣿⣷⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣶⣾⣿⣶⣿⣼⣧⣶⣿⣿⣦⣽⣤⣷⣼⣿⣄⡀
⠀⠀⠀⠀⣿⣟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠈⣿⡆⠛⠃⠀⠛⠛⠀⠘⠛⠀⠛⠛⠀⠘⠛⠀⠛⠃⠀⠘⠛⠃⠘⠛⠃⠀⠛⠛⠀⠘⠛⠀⠘⠛⠃⠘⠛⠃⠛⢃⣿⡇
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⡿⠋
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠛⠛⠛⠿⠿⠿⠿⠿⢿⣶⣾⡿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⣿⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠃
⠀⢰
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠛⣿⣿⣿⣿⡿⣿⣿⣿⡟⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠉⠛⣻⣛⣽⡋⠀⠈⣿⡟⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⢀⣘⠛⣉⡀⠀⠀⠈⠉⠀⣠⣤⣤⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣼⡿⠛⠋⠙⠳⣀⠀⠀⠈⢁⡀⢤⣄⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣥⣾⠗⠘⣶⣿⣀⠀⠙⠋⠀⠐⠈⠁⢻⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡟⠁⠉⠀⠈⠉⣋⣩⣀⣠⡀⠀⠀⠀⠀⣸⣅⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠀⢀⡀⠀⠸⠟⠻⠟⠁⠀⠀⡀⢤⡠⣿⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣦⠈⠉⠂⠀⣀⣤⣄⣤⠤⣤⣉⠘⣇⢹⡇⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣛⣿⣿⣿⡇⠠⣾⣿⣏⣀⣀⣀⣤⣬⣿⠓⠗⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⡆⠹⣟⠛⠉⠉⠉⠀⠉⠁⠀⠀⢀⣷⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡄⠙⠀⠚⠋⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⣙⠿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣿⣿⣿⣿⣿⣿⡉⠛⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⡠⠞⠹⣿⣯⢹⡆⣿⠸⠏⣷
⠀⠀⠀⠀⠀⢀⣾⣯⡀⣿⣿⠟⠛⢛⡏⢩⡿⠤⠀⠈⠿⣿⣧⣴⣶⣴⣦⠶⠋⠀⠀⠀⠁⠘⠀⠁⢹⣿⣿⠋⠉⠉⠁⠈⠉⢉⣉
⠀⠀⠀⠀⢀⣼⣯⠈⢿⣿⡏⠀⠀⠀⠁⡼⡃⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢸⡿⠁⠀⠂⠀⠀⠀⠠⠷
⠀⠀⠀⣠⣾⠃⠀⢀⢼⣿⡇⢠⡀⣀⣠⠀⠁⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⣿⠀⠀⠀⡟⠁
⠀⠀⠘⣇⣤⠀⠀⠈⣾⣿⢼⣿⣷⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⣤⣶⣞⠁⡀
⠀⠀⣈⣥⠾⣦⣴⡿⠻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⠟⢠⣽⣆
⣾⣿⠦⣴⣶⠾⠛⠓⢤⣿⣿⣿⣿⡿⡄⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⣀⣀⣠⣠⣦⣼⣿⣯⣀⣶⣿⡿
⢹⣿⣾⣝⣶⣦⡄⠀⠀⢈⠟⠻⣿⠁⠈⠀⠑⢄⠀⠀⠀⢀⠀⠀⠀⣠⡀⠀⠐⡄⢪⣾⡟⣼⣿⣿⡿⠹⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠙⠿⣿⣿⣿⣿⣿⣶⠃⠀⠀⠈⠀⠀⠀⠀⡈⠁⠒⠀⠈⠉⠶⢦⣿⣿⠷⠾⠛⠙⣿⡟⠛⠿⠿⠃⠀⠙⠻⠿⠿⠟⠉
⠀⠀⠀⠈⠙⠛⠿⣿⣟⢀⡄⠀⠀⠀⠀⠀⢠⠹⠄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢸⣧⣀⣠⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣶⣤⣤⣶⣬⣆⣀⣀⣀⠀⠀⠀⠐⡤⠂⠀⠀⠀⢀⡈⠛⠛⣭⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣇⠉⠉⠁⠐⠒⠒⠘⠀⠂⠐⠈⠉⠁⠀⠀⠀⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⣿⣇⠙⠁⠀⠀⠀⠀⠀⠀⢈⣤⣤⣠⡴⠆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣯⠀⢀⡀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠠⢀⡆⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄⣣⣼⠀⠀⠀⠀⢸⣿⣿⣿⣿⠢⡉⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡧⠁⣹⣷⠀⠀⢀⢨⣿⣿⣿⣿⣷⡉⠂⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡶⢻⡟⠀⠀⢸⡎⠀⠀⠀⢻⣿⣿⡆⠀⠀⠶⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⡄⣿⠀⠀⠀⠇⠀⠀⠀⠘⣿⣟⢿⣶⡁⡦⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⣿⣇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢹⠇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡍⢙⠆⠀⡀⠀⠀⠀⠀⠸⣿⡄⠁⠀⠀⠘⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡟⠑⠂⠀⠀⠀⠀⠀⠀⢻⡷⠀⠀⠀⠀⠐⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡦⠀⠀⠀⠀⠀⠈⠓⠘⠒⢀⣀⣤⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⡿⠃⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣻⣿⣿⣟⣀⣀⣀⣀⣀
⣿⣿⢋⣉⡿⠟⠿⠿⠿⠟⠻⠟⠻⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣤⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣦⡀
⠀⠀⣰⣿⣿⣟⢿⢽⣙⢿⣿⣧⠀⠀⠀⣼⣿⠏⢁⣴⠈⢻⣿⣿⡄⠀⠀⢠⣿⣿⡽⠛⠛⠻⡝⣿⣧⠀⠀⠀⣼⣿⢟⣿⡛⣻⣟⢿⣿⡆
⠀⠀⣿⣿⡷⣿⢻⢿⣽⣿⣿⣿⠀⠀⢰⣿⣿⡄⠀⠋⠀⠀⣿⣿⡇⠀⠀⣾⣿⣿⣿⣟⣿⡛⣷⣿⣿⡆⠀⠀⣿⣿⣹⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠹⣿⣿⣍⡿⢺⣩⣾⣿⡟⠀⠀⠀⢿⣿⣿⡆⠀⢠⣬⣿⣿⠇⠀⠀⠸⣿⣿⡿⠭⣯⠤⣿⣿⡿⠀⠀⠀⢻⣿⣬⣭⣍⣭⣬⣽⣿⠏
⠀⠀⠀⠙⠻⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠛⠿⣷⣷⣿⡿⠟⠁⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠟⠃

⠀⠀⠀⠀⣠⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣤⣄
⠀⠀⢠⣾⡿⢛⢛⠛⢿⢛⣿⣆⠀⠀⠀⣴⣿⣫⣭⣭⣭⣭⣻⣷⡀⠀⠀⢀⣾⡿⠿⠿⠻⠿⠿⣿⣆⠀⠀⠀⣰⣿⣿⣻⣿⣿⡿⣿⣷⡄
⠀⠀⣾⣿⡇⣿⣿⣿⢰⢰⣿⣿⠀⠀⢰⣿⣿⢾⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣷⣶⡶⡲⣶⣦⣿⣿⡄⠀⠀⣿⣿⣿⣿⣿⣺⡇⣿⣿⣷
⠀⠀⢻⣿⡇⣿⣭⡿⢨⢼⣿⡿⠀⠀⠈⣿⡿⠶⢶⣶⣶⠶⢿⣿⡇⠀⠀⢹⣿⡿⣿⣮⢫⣾⡿⣿⣿⠁⠀⠀⣿⣿⣟⠛⢛⠛⠃⣿⣿⡏
⠀⠀⠀⠻⢿⣶⣶⣶⣿⣿⠟⠁⠀⠀⠀⠘⠷⣶⣾⣿⣷⣶⡶⠋⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠈⠻⣿⣷⣶⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀
⠀⠀⢀⣴⣿⣿⣿⡿⢿⣿⣷⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⠛⣿⣦⠀⠀⠀⢀⣴⣿⣿⡿⠿⢿⣿⣷⡄⠀⠀⠀⢠⡾⣿⡿⠿⢿⣿⣿⣦⡀
⠀⠀⣼⣿⣿⣿⣿⠃⣨⣿⣿⣿⠀⠀⢠⣿⣿⣿⣿⠟⠁⣰⣿⣿⡇⠀⠀⣸⣿⣿⣿⡟⢀⣀⣹⣿⣿⡀⠀⠀⣿⣧⣄⠀⢠⣴⣴⣶⣿⣧
⠀⠀⢿⣿⣿⣛⠁⣼⣿⣿⣿⣿⠀⠀⠘⣿⣿⡟⣉⢀⣰⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⣿⣾⣿⣿⣿⣿⠃⠀⠀⣿⣿⠧⠤⠿⣿⣿⣿⣿⡟
⠀⠀⠈⢿⣿⣯⣴⣿⣿⣿⡿⠃⠀⠀⠀⠙⢿⣿⣦⣿⣿⣿⣿⠟⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠙⢿⣦⣦⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⢀⣴⣿⣿⠿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣄
⠀⠀⣼⣿⣿⣿⣿⣿⣿⠟⣿⣷⠀⠀⠀⣿⣿⣿⡿⠂⢻⣿⣿⣿⡆⠀⠀⢸⣿⣿⢏⡬⠫⣌⢿⣿⣷⠀⠀⠀⣾⣿⣟⢩⢖⣀⡙⣿⣿⣇
⠀⠀⣿⣿⡿⢿⠛⡏⠐⠀⣿⣿⠀⠀⠸⣿⣿⣵⣶⣦⣶⣶⣿⣿⡇⠀⠀⢻⣿⣧⡸⡛⢘⠻⣠⣿⣿⠇⠀⠀⣿⣿⡢⢦⣌⢿⡿⣸⣿⣿
⠀⠀⠘⣿⣷⣾⣶⣷⣾⣶⣿⠏⠀⠀⠀⠻⣟⣘⣿⡃⣿⡟⢸⡿⠃⠀⠀⠘⢿⣿⣧⣼⣿⣤⣿⣿⠟⠀⠀⠀⠹⣿⣿⣶⣍⣡⣾⣿⣿⠃
⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣏⠀⣹⣏⣛⣛⣛⣛⣻⣙⣏⣛⣻⣛⣛⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣛⣻⣿⣟⣛⣟⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣬⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢋⡿⣟⠲⣄⠀⡈⠢⣄⠀⠀⠀⠀⠀⠀⠈⣉⣽⢿⣰⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⣀⠤⣩⡇⢿⡘⡜⢦⠈⠢⣈⠳⢄⠀⠘⣻⡶⢫⣵⣿⣿⠟⠵⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⡶⠟⣫⡵⠞⠁⢸⡄⠀⢻⡌⢿⡀⠈⢹⣖⡛⢲⣿⣰⣿⣿⠟⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣴⣿⣋⡀⠀⠀⢀⣻⡄⠀⠳⣄⣉⣒⠈⣹⡷⢫⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⣿⣿⣷⡮⠀⠀⠩⢲⣿⣶⣦⣤⣭⣿⣿⣿⢃⣾⣿⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⢡⣿⣆⠀⣀⣀⠀⠀⠀⠀⢸⢃⣽⣿⠃⣼⡟⣿⣿⡜⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣿⣿⣎⠻⠍⠉⠀⢀⣰⡦⢺⡿⠁⣸⣿⢱⣿⣇⠇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⣿⣜⢿⣿⣦⣀⣤⣴⡿⢋⣴⠏⠀⢰⣿⢿⠘⣿⣏⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⢻⠘⢷⣿⣿⣿⢹⢏⣴⣻⠃⠀⢠⣿⡷⠚⢣⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡟⢰⡄⢣⡀⢨⡿⣡⣏⣾⣳⠃⠀⢠⣿⣿⠁⢠⠞⢹⡟⡳⣕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡗⢀⡿⣿⠀⠇⢀⣴⢿⣿⣿⠃⠀⢀⣾⡟⢸⠀⠀⠀⠀⣷⠘⣎⠳⣍⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⣿⣠⠾⢋⣘⣸⣿⠃⠀⢀⣾⣿⠁⡼⠀⠀⠀⠀⢸⣧⡈⢳⢬⡳⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⠟⣉⡤⠖⠋⠉⣹⢇⠄⢀⣾⡿⠻⠷⣧⣀⣀⣠⣤⣾⣿⣷⣄⠑⢌⠚⢦⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⣓⠟⠁⠀⠀⠀⣰⢃⠎⣰⣿⡿⠁⠀⠈⠲⣝⢿⣿⣿⣿⣿⣿⣾⡳⣤⣙⣶⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠞⠁⠀⠀⠀⠀⣰⢏⡞⣰⣿⡿⠁⠀⠀⠀⠀⠙⣦⠹⣿⣿⣿⣿⡿⣟⢦⡹⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡁⡀⠀⠀⠀⠀⣰⢏⡞⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⡆⢿⣿⣿⡟⠁⢻⣟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⠀⠀⠀⠀⣰⢏⣞⣼⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣗⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣰⢏⣾⣾⢏⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⣴⢯⡟⣾⢃⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠈⠁⠈⠉



⣿⢿⠿⡿⢿⢟⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣛⣛⣻⣿⣛⣛⣻⣟⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠖⠶⢦⣴⠶⠾⣷⣤⠶⠶⣶⠛⠉⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⣀⡀⠀⠀⠛⠀⠀⠉⠀⠀⠀⠈⠃⠀⠀⠀⢨⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⣛⣿⣶⣤⣤⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⣠⡴⠶⠟⠛⠛⠋⠉⠉⠉⠛⠛⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⡶⠿⠟⢛⣿⡟⣉⣉⣉⢻⣯⡉⠉⠙⠛⠿⣷⣤⡀⠀⠀⢸⣇⠀⠀⠻⣤⣤⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⠿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣴⡿⠛⠉⠁⠀⠀⠀⢸⡧⢠⣤⣨⣿⣿⣽⡇⠀⠀⠀⠀⠈⠙⣿⣆⠀⠛⢩⡇⠀⠀⠀⠉⠻⣏⠉⠉⠉⠉⢩⣿⠟⠁
⠀⠀⠀⠀⠀⣰⡾⠋⠉⠀⠀⠉⠳⣦⠀⠀⠀⠀⠀⠈⠷⣤⣀⣀⣀⣤⠿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣧⣠⠞⠁⠀⠀⠀⠀⠀⠹⣇⠀⠀⣰⡟⠁
⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠙⣉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢠⡿
⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⡖⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⢀⣼⣿⣶⣦⡀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠻⣄⠀⠀⠚⠉⠀⠈⢿⣷⠀⠀⠀⠀⠀⠈⣇⢸⡁
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⣿⠀⣷⡀⢹⡆⠀⢀⣄⣀⣀⣾⡟⠀⠀⠀⠀⢀⣴⠟⠋⠛⢷⣆
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣀⠀⠀⣀⣤⣾⣷⣶⢾⣿⡟⠛⠋⠁⠀⠀⠀⠀⢠⡟⠀⡿⡇⠀⣿⠀⠈⠛⢻⣿⠋⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⠛⠉⠀⠀⠀⠀⢹⣧⠀⣴⠒⠶⠶⠶⠶⠋⢀⣼⢢⡇⠀⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⣿⠀⣀
⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢷⡀⢦⣄⣀⣀⣴⠾⢅⣸⠃⣸⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣠⣿⣷⣿⣿
⡀⠹⣧⡀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠙⢦⣉⠩⣭⣦⡴⠞⣁⣴⣯⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⠿⠛⣹⡿⠁
⠻⢷⣬⣹⣶⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠉⠛⠶⠶⠶⠟⢫⡟⠈⢻⣆⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠛⠛⣷⣴⣾⠟⠁
⠀⢶⣬⡽⢿⣿⠿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢻⡆⠀⢰⣿⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣤⣄⣀⣀⣀⣀⣀⣠⣿⣿⠟⠲⢦⣄⣠⡴⠶⢾⣧⡴⠖⠀⠀⠘⣿⣀⣸⣿⣤⣤⣤⣶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⣹⡿⣿⠀⠀⣀⣬⡏⢀⠀⢠⣿⣀⡀⠀⠀⠀⣿⠋⠉⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣿⡀⠈⠁⢈⣷⣬⣭⣿⠋⢉⡿⠿⢶⣾⡟⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠘⣷⡀⣠⡾⢻⣤⢻⡉⡖⣻⠃⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠘⠛⠋⠀⠀⢻⣸⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢃⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣸⣇⣀⣤⣤⣤⣤⣤⣤⣶⣶⣶⠾⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⣿⠋⠉⠉⠉⠉⠉⠉⠉⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠶⠦⣤⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⡧⠴⠶⢶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⣰⣀⣠⣤⡌⠙⡆⠀⠀⠀⠀⢠⠟⢠⣤⣀⣀⣇⣀⣼⠇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣽⣭⣥⣤⣤⣤⣤⣼⣷⣿⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣷⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⣋⣭⣥⣤⣤⣬⣭⣙⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⣡⣶⣿⣿⠿⢛⣛⣛⣛⠻⣿⣿⣶⣌⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢃⡾⢋⣼⣿⡏⣼⣿⠿⢿⣿⣷⢸⣿⣷⡌⢷⡌⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢁⡿⢠⣿⣿⣿⡇⣙⣡⣶⣶⣌⣛⢸⣿⣿⣿⣆⢻⡌⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣼⠃⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⡈⣷⢸⣿⣷
⠀⠀⢀⣴⣶⣶⡆⣿⣿⡇⣿⠀⣿⣿⣿⣿⡿⠧⢙⠛⠛⣋⠼⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢰⣶⣾⣶⡄
⠀⠀⢸⠿⢋⣍⡃⢸⣿⣇⢸⡆⢿⣿⡿⢩⣶⣷⣌⠛⠛⣡⣾⣶⡌⢿⣿⣿⣿⡏⣸⣿⡟⢈⣭⡛⠿⡇
⠀⠀⢴⣶⣿⣿⣿⡌⣿⣿⡄⢻⣬⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⡿⢡⣿⣿⢁⣿⣿⣿⣶⡦
⢀⣤⣆⢻⣿⣿⣿⡗⠈⢿⣿⣆⠻⣿⣧⠹⢿⣿⣿⣿⣿⣿⣿⡿⠟⣰⣿⠟⣠⣿⡿⠃⠺⣿⣿⣿⡟⣰⣦
⣾⣿⣿⣷⣬⣉⣥⣶⣿⣌⠻⣿⣷⣬⡙⠿⣶⣶⣶⣶⣶⣶⣶⣶⠿⢋⣡⣾⣿⠟⣡⣿⣶⣬⣉⣥⣶⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠈⠛⢿⣿⣷⣶⣬⣭⣍⣩⣭⣥⣴⣾⣿⡿⠛⠁⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⣤
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢀⣠⣶⣿⣿⣿⡟⡡⠉⣛⣋⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢰⣇⠻⠿⠿⠿⠂
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⢸⡏⣴⣶⣶⣶⠄
⠀⢸⣿⣿⣿⣿⣿⣿⢹⣿⣿⡇⢸⣿⣿⣿⣿⣿⢸⡗⣠⣭⣭⣥⡀
⠀⢸⣿⣿⣿⣿⣟⢰⣶⢘⣿⡇⣼⣿⣿⣿⣿⣿⢸⡧⢙⣛⣛⣛⠁
⠀⠸⣿⣿⣿⣿⣿⣶⣴⣾⣿⡇⠈⠉⠉⠉⠉⠉⢸⣇⠻⠿⠿⠿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⢰⣿⣿⣆⣶⠉⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠉⠘⣻⣟⣻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⠛⠛⢩⣤⣤⣤⣤⣤⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡿⠻⠛⠉⠉⠛⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠛⠂⠀⡀⠴⠆⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣿⣇
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡁⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠙⠦⣤⡤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡄⠀⢀⣀⣀⣀⠀⣤⣄⡀
⠀⠈⠉⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠀⠀⠙⣟⠁⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢰⣿⡆⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⣿⡇⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⢀⠀⠀⢼⣿⣿⣿⣿⡄⢸⣿⣇⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠁⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⡏⠀⠀⠷⣄⣀⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠈⣿⡟⠀⢸⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⠃⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⣀⢀⣴⣶⣶⣄⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠀⢀⣿⣿⢸⣿⣿⣿⣿⠀⢀⣿⡿⠋⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠿⠿⣿⠿⠂⠙⠛⠛⠁⠀⣸⣿⢁⡤⠶⣄⠀⢈⡭⢭⣍⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⡇⢸⡀⠀⣸⠗⣯⠀⠀⣹⢀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿⣿⡄⠉⠛⠉⠀⠈⠛⠛⠁⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⢀⣀⡤⢾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠈⠉⠉⣩⣥⣤⣄⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠛⢛⣿⠃⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⢀⣤⣤⡀⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠘⢿⣿⣷⣶⣤⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⣿⠆⠀⢸⣿⠿⠿⣿⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠉⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠾⠋⠀⠀⠈⠻⠶⠾⠟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⡄⠀⣠⣾⣿⣿⣷⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠸⣤⠴⠋

⠀⢰⣶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣿⡇⣶⣶⣤⣤⣤⣤⣤⣄⡐⠶⠶⣶⣾⣿⠷
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣟⢶⣦⣤⣄⡀
⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢤⣍⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠙⠿⣿⣿⣿⠋⠀⠀⠈⢿⣿⡎⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢀⣀⡀⠸⣿⡇⠀⠈⠙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣽⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢿⣿⡿⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣞⣿⣫⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⡿⢸⣿⣿⣿⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣿⡇⣸⣿⣿⣿⣿⠘⣿⡇⠀⠀⠀⠀⠀⠀⠘⠿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠿⠃
⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣤⠀⠀⠀⠀⠀⢸⣿⠁⢹⣿⣿⣿⣿⠀⣿⡧⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠁⠀⢻⣿⡏⣿⣿⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣸⣿⣿⣿⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠉⢸⣿⡏⣿⣿⠈⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣟⣿⣿⠈⠁⠀⠀⠀⠀⠀⠈⠁⣿⣿⢹⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣆⡀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠁⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢂⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣍⢻⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣤⣿⢹⣟⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣸⡏⡾⢸⢿⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢋⡾⣡⠏⣼⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡴⣫⡴⣫⣾⣛⣛⠷⣤⡲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡲⢾⣉⡞⣵⢫⡞⠋⢩⠾⠿⣿⢮⣻⣯⠍⣙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣈⠳⣬⣙⠒⢸⡇⣧⡻⢮⣽⢳⡎⠃⣼⣛⠿⢧⣛⠩⠶⢋⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣿⣿⣤⣿⣮⣉⣻⣿⣟⣥⣞⣩⣿⣻⣶⣼⣷⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣦⣤⣀⣉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣀⣠⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢳⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠰⠦⣀⣀⢀⣾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢿⣷⣤⣤⡛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⢀⡀⠉⢹⠟⢿⣿⣦⣼⣏⣉⡿⠛⢿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣴⡄⠈⠁⠀⢾⣀⠀⢏⡉⠛⢿⣿⣶⣦⣀⠀⠈⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠿⠿⠁⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠃⠈⠙⠻⠷⢾⣧⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠗⠓⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣙⠿⠶⠶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⡉⠐⠶⠒⠒⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⠶⣶⠶⠶⠶⠶⠖⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⣨⡀⠲⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡄⠐⠒⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠙⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⠦⠀⠱⢶⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠐⣆⠀⠳⡄⠻⡆⢰⠀⠀⠀⠀⠀⠀⠀⢹
⠀⡄⠀⠛⠁⠀⠀⠀⠇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠰⡟
⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠘⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇
⠀⠀⠀⠸⠦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⢀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠃⠀⣀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠠⡍⠱⢄⠀⠀⠋⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠾⠀⠘
⠀⠀⠘⢧⣤⣴⡆⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⣀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠁⠀⠉⠉⠉⠀⠉⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣴⣦⣶⣶⣶⣶⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⣶⣾⣩⢩⡵⢾⠓⠛⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⣯⡼⠛⣩⡽⠝⠁⢶⣌⠳⣌⡠⢻⣤⣿⡳⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠘⢿⣣⣞⡴⡿⠚⠓⠳⠶⠒⠻⠆⠀⣻⣍⣭⣍⣙⣿⣯⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠋⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⠈⠉⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣠⢀⣤⠀⢈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⣀⣛⣁⣀⣤⡟⠅⠠⠜⣻⢿⣦⣀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⠿⠋
⠀⠀⠠⢤⢤⠉⢩⠕⠀⡀⣶⡆⠐⢰⠛⠓⣂⠈⣿⢿⣶⣶⣶⣤⣿⣿⣿⣿⣋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡇⠀⣰⡷⡟⡄⠀⠜⠛⠢⢙⣾⠛⢿⣟⡛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠋⠀⠈⠐⡄⡴⠂⣤⠀⠙⠳⣄⠉⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠀⠈⢳⡌⠳⢄⡈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁



⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠒⠀⠀⢐⣒⡒⠲⠦⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣋⣥⣶⣶⣖⡃⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣓⡶⢤⣀⠀⠀⠀⠀⠀⣠⠤⣚⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣚⣽⣾⣿⣿⡿⠿⠟⠛⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⡶⣮⣝⢦⣀⣠⣬⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣾⣿⡿⠛⠉⣀⠀⠀⢀⡀⠀⠐⠒⠲⠶⢤⢠⣶⡆⠠⠄⠔⠒⠂⠀⠀⠈⠙⠻⣿⣷⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣸⢹⣿⠟⠁⠀⠀⠐⠛⢡⡔⠁⣀⣀⠀⢀⣠⡰⣾⣶⠤⡄⠂⠀⠀⠠⢔⠉⠀⡢⢐⣬⠾⠛⡉⣼
⠀⠀⠀⠀⠀⠀⠀⢿⣎⢩⠀⠀⣀⠁⡀⠀⠘⠇⠘⢉⣥⠗⠄⣽⡟⠙⣾⣄⡉⣰⡆⠀⠈⠿⣦⣄⣉⠿⠖⠀⢈⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣏⠑⢄⠀⠀⠀⠈⠀⠀⢤⣤⠟⠁⠀⢄⠉⠀⠀⣈⡙⡀⠘⣛⣀⡀⠰⠉⠀⠀⠀⣠⣴⡟⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁⣀⣠⣤⣶⣿⣿⢿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠶⠶⠶⠶⠶⠒⠾⢿⣿⣿⣿⣿⣿⡿⠋⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⣻⣿⣿⣿⣿⠏⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠿⠟⣂⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣭⣿⡿⢛⣿⠞⠋⠀⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠈⢰⠀⢄⣀⡀⠀⠀⠀⠀⠀⠈⣛⣉⣯⣷⡞⣫⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠋⠉⠉⠉⠉⠙⠛⣿⣿⡿⢟⣫⣾⣿⣿⣿⡿⠁⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠉⠒⠲⠶⠶⠶⢶⣶⣾⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣾⣧⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⠛⠛⠋⠁⠀⠀⠀⠀⠀⢔⣪⣾⣿⣯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⠭⣭⣭⡽⠿⠿⠿⠿⠛⠿⠿⠛⠛⠋⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣭⣭⣽⣷⣽⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣶⣶⣿⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠉⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣏
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣶⣶⣦
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠋⠉⠁
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⡄
⠀⢀⣠⣼⣿⣿⣿⣿⣿⠿⠿⠇
⢴⣿⣿⣿⣿⣿⡿⠿⠉
⢸⣿⣿⠟⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⢽⣿⣿⡿⢻⣟⡝⢿⣿⣿⣿⠀⠀⠀⣿⣿⡿⠋⢡⡌⠙⣿⣿⣿⡇⠀⠀⢸⣿⣿⢫⠭⠭⠭⢽⢻⣿⣿⠀⠀⠀⣿⣿⡿⣟⡿⠿⣟⡿⣿⣿⡇
⠀⠀⢺⣿⣿⢾⡷⣧⣟⣿⣽⣿⣿⠀⠀⠀⣿⣿⡇⠀⠹⠃⠀⠸⣿⣿⡇⠀⠀⢸⣿⣿⢼⢶⢶⣶⢾⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⢻⣿⣿⣝⢷⡏⣾⢛⣼⣿⣿⠀⠀⠀⣿⣿⣿⣆⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⣿⢽⣿⣿⣯⣸⢸⣿⣿⠀⠀⠀⣿⣿⣟⠛⠛⠛⠛⢛⣿⣿⡇
⠀⠀⢹⣿⣿⣿⣷⣶⣷⣿⣿⣿⣿⠀⠀⠠⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢹⣿⣿⣩⣭⣭⡍⡇⢸⣿⣿⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⡇⣿⣿⡇⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⢹⣿⣿⠀⠀⠀⣿⣿⣿⢱⣿⣿⣷⢹⣿⣿⡇
⠀⠀⢸⣿⣿⢻⣿⣿⡇⡏⢸⣿⣿⠀⠀⠀⣿⣿⡟⠿⠿⠿⠿⠇⣿⣿⡇⠀⠀⢸⣿⣿⣿⡟⣥⡙⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⢺⣿⣯⣽⢸⣿⣿⡇
⠀⠀⢸⣿⣿⠻⠿⠿⠇⣷⣾⣿⣿⠀⠀⠀⣿⣿⠛⠛⣿⣿⡟⠛⢻⣿⡇⠀⠀⢸⣿⣿⣿⣷⣭⣶⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⡈⠉⠍⠉⣸⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⢾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠀⠀⠐⣿⣿⣿⣿⣿⣿⡿⠻⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡿⠀⣩⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⣼⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠆⢀⠈⣻⣿⣿⠀⠀⠀⣿⣿⣀⠀⠀⣀⣤⣼⣿⣿⡇
⠀⠀⢸⣿⣿⡿⠛⢁⣾⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⢟⠁⠀⣼⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣟⣀⣰⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣃⣴⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣷⣌⣣⣿⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣇⣀⣀⣽⣿⣿⣿⣿⡇
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⣿⣿⣿⣿⣏⢙⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠿⢿⠿⢿⣿⣿⣿⠀⠀⠀⣿⣿⣿⡿⠟⡛⠿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⡿⡟⢹⣿⣿⠀⠀⠀⣿⣿⡿⠿⠥⢤⠽⠿⣿⣿⡇⠀⠀⢸⣿⣿⢏⣾⡉⣹⣞⢿⣿⣿⠀⠀⠀⣿⣿⡏⠆⠚⣵⣮⡈⣿⣿⡇
⠀⠀⢼⣿⣟⠉⡏⢹⠀⡇⢸⣿⣿⠀⠀⠀⣿⣿⠿⣿⣿⢼⣿⡷⣿⣿⡇⠀⠀⢸⣿⣿⣮⠫⣠⡬⢫⣾⣿⣿⠀⠀⠀⣿⣿⣇⡻⢷⣌⠿⣣⣿⣿⡇
⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣀⣿⣏⣘⣿⣃⣹⣿⡇⠀⠀⢸⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣷⣥⣾⣿⣿⣿⡇
⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃

⠀⠀⠀⠀⢀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠠⠁⢀⢀⠀⡁⠀⠀⠀⠀⠀⢅⡂⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠡⠀⠀⠀⠀⠀⠚⣒⡲⠂⠁⡀
⠀⠀⠀⠀⠀⠀⠶⠀⢀⣀⣀⠂⠄⠀⠀⠀⠀⠈⠲⠄⠐⠀⢐⡂⠂⠀⠀⠀⠥⠗⠖⠂⡀⠀⠀⠀⠀⠀⠁⠨⠄⠂⢀⠃
⠀⠃⠉⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠁⣤⠠⠂⠀⠀⠀⠀⠀⢎⣀
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠀⠀⠀⠀⠀⠨⠸⠇
⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠰⠇⠆⠸⠄⠤⠀⠀⠀⠀⠀⠐⠐⠂⠀⠐⠂⠐⠀⠀⠀⠀⠰⠀⠒⠀⠀⠐⠀⠂⠀⠀⠰⠀⠒⠀⠀⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠀⠀⠀⠀⠆⢎⠠⠖⠀⠴⠰⠀⠀⠀⠀⠀⢐⠈⠀⠑⡀
⠀⠀⠀⠀⠐⠈⠀⠠⠁⠀⠉⠀⠀⠀⠀⠀⠣⠀⢈⠁⠐⠀⠀⠀⠀⠘⡀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⢁⡀⠐
⠀⠀⠀⠀⠐⠀⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠢⢀⢀⠠
⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠐⠖⣠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠎⠑⠂⠈
⠀⠀⠁⡉⠀⠀⠀⡰⠂⠀⢂⣀⠢⠀⠀⠀⠐⢅⠌⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠐⠀⠈⠂⠄⠊⠀⠀⠀⠀⠁⠀⠀⠀⠠⠆⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠈⠁⠀⠀⠀⠁⠈⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁
⠀⠀⠀⡀⠄⠀⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀
⠀⠠⠁⠀⠠⠀⠀⠀⠀⠊⠀⠀⠀⢈⠀⠀⠀⠀⠂⠀⠀⠀⢐⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠃⠀⠀⠐⠉⠀⣀⠨⣏⡉⠁
⠀⠀⠁⡰⢦⠀⠀⠀⠀⠀⠀⢰⠂⡁⠀⠀⠰⠀⠀⠀⣤⢀⡀⠀⠩⠀⠀⠀⠀⠀⠠⠦⠀⠀⠌⠀⠀⠈⠂⠀⡄⠀⠠⠂
⠀⠀⠐⠀⠀⠁⠀⠪⠐⠀⠂⠀⠀⢠⠀⠀⢈⡀⠈⠁⠐⠈⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠂⠄⡄⠀⠀⠀⠀⠀⠀⢀⠐⡀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠠⢆⠄⠀⠄⠀⠀⠀⠀⢠⠰⠀⡆
⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠈⠂⠃⠁⠀⠀⠀⠀⠀⠐⠁⠀⠩⠌⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⠤⣤⡄⠀⠀⢀⢤⠤⣤⠀⠀⠀⠀⢀⢀⡈⠀⢠⡀⢀⠀⠀⠀⠀⢀⡌⠀⢠⡀⢀⠀⠀⠀⠀⠀⡤⢠⡄⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠋⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣰⣿⣿⠟⢻⣷⠀⠀⠀⠀⣼⡟⠻⣿⣿⣇⣠⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⢀⣤⣾⡿⠟⠋⠉⠉⠉⠉⠉⠛⠻⣿⣿⡏⠀⠀⢻⡇⠀⠀⢸⡟⠀⠀⢹⣿⣿⠟⠛⠋⠉⠉⠉⠉⠙⠻⢿⣷⣤⡀
⠀⠀⣠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠸⣧⠀⠀⢸⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⣠⣿⣶⣶⣿⣅⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣼⡟⠁⠀⠀⠈⢻⣧⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⣠⣾⠿⠿⠿⢿⣿⣿⡆⠀⠀⠀⠀⠈⠻⢷⣦⣤⡾⠟⠁⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠿⣿⣆⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠈⠙⠿⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣄⣀⠀⠿⠋⠁⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⢸⣿
⠀⣿⣇⣠⣤⣤⣿⠃⠀⠀⠀⠀⣰⣷⣠⡶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢶⣄⣼⣆⠀⠀⠀⠀⠈⣿⣤⣤⣄⣸⣿
⣰⣿⠛⠉⠉⠉⢻⣷⠀⠀⠀⢠⣿⡿⠋⢀⣀⣤⣶⣶⠿⠿⠿⠿⠿⠿⠿⠿⣶⣶⣤⣄⡀⠉⢿⣿⡆⠀⠀⠀⣼⡟⠉⠉⠉⠹⣿
⣿⣧⠀⠀⠀⠀⠀⣿⠃⠀⠀⣸⣿⣷⣴⡿⠛⠉⠀⠀⠐⠲⢖⣛⣛⡒⠖⠂⠀⠀⠉⠛⢿⣶⣾⣿⣧⠀⠀⠀⣿⡀⠀⠀⠀⠀⣸
⠘⣿⣦⣀⣀⣠⣾⠟⠀⠀⢀⣿⣿⣿⠋⣀⣤⣤⣤⣀⡀⠀⠉⠉⠉⠉⠀⢀⣀⣤⣤⣄⣀⠈⣿⣿⣿⡀⠀⠀⠻⣷⣄⣀⣀⣴⣿
⠀⠈⠉⢻⣿⡏⠀⠀⠀⠀⣸⣿⣿⡇⠈⠉⠉⠉⡙⠻⣿⣶⣤⠂⠘⢦⣶⣿⠿⢋⠉⠉⠉⠁⢸⣿⣿⣇⠀⠀⠀⠀⢹⣿⡟⠉⠁
⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣿⠙⣿⣷⠀⠀⠶⠿⠻⢿⣿⣿⣷⠀⠀⣾⣿⣿⡿⠿⠿⠶⠀⠀⣾⣿⠏⢿⡄⠀⠀⠀⠸⣿⡇
⠀⠀⠀⢸⣿⡅⠀⠀⠀⢸⡇⢠⣿⠛⠁⠀⠀⠈⠙⠛⠉⠀⣿⠀⠀⣿⠀⠉⠛⠋⠁⠀⠀⠈⠙⣿⣄⢸⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⣷⣿⡇⠀⠀⠀⠀⠀⠀⢀⡴⠶⣿⠀⠀⣽⠶⢦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⢿⣇⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⣰⣿⢀⣴⠇⠀⠀⠸⣦⡀⣿⣆⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠸⣿⡄⠀⠀⢸⣿⣿⣿⣿⣦⣄⠐⠾⠛⠉⠓⠹⣇⡀⢀⣰⠏⠚⠋⠛⠷⠂⣠⣴⣾⣿⣿⣿⡇⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⢻⣷⣤⡴⠾⢿⣿⣿⣿⡉⠙⠧⠀⡀⠀⠀⠀⠈⠻⠟⠃⠀⠀⠀⢀⠀⠰⠛⠉⣿⣿⣿⡿⠷⢶⣤⣾⡟
⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠈⣿⣿⡇⠀⠀⢰⣇⣠⣴⠶⠶⠶⠶⠶⠶⣦⣄⣸⡆⠀⠀⢸⣿⣿⠁⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⢀⣾⣿⣿⡀⠀⢼⡟⠉⢀⣤⣶⣶⣶⣶⣤⡀⠉⢻⣧⠀⢀⣿⣿⣿⡀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⣽⣿⣶⣶⣶⠿⠟⠀⢻⣷⡀⠘⠷⠖⠻⠛⠉⠉⠉⠉⠛⠟⠳⠾⠃⢀⣾⡟⠀⠻⠿⣶⣤⣶⣿⣿⡀
⠀⠀⠀⣠⣶⣾⣟⡃⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⢈⣻⣷⣶⣄
⠀⠀⠀⢻⣿⣿⠿⠿⠿⣷⣶⣶⣤⣤⠀⠀⠈⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠁⠀⠀⣤⣤⣶⣶⣾⡿⠿⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡏⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⢹⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⠻⢿⣶⣤⣤⣤⣤⣶⡿⠟⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⣀⣀⣀⠀⠈⠉⠉⠉⠉⠁⠀⣀⣀⣀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣶⣾⣿⠿⠿⠿⠿⣿⣧⡀⠀⠀⠀⠀⢀⣼⣿⠿⠿⠿⠿⣿⣷⣶⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⣠⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡧
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠉⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣺⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠕⠊⠁⣀⠤⠒⠉⣀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠭⠒⢉⡠⠤⠒⣉⠤⠔⣂⠥⠄⠺⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⡪⠕⢊⡠⠄⣂⠥⢔⣂⠥⠐⣊⣁⣤⣴⠦⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠝⠊⣁⣴⠮⢙⡂⠥⠒⠈⣁⡀⠤⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠮⢛⣩⡴⣞⠫⠕⠂⢉⡡⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⣺⣿⣽⢿⠅⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣏⣿⢧⣿⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡷⡽⣟⢯⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣳⢯⣟⡾⡽⢤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠛⠉⠉⣉⣉⣤⣤⣤⣤⣤⣤⣧⣏⣿⣾⣽⠁⢀⣀⣀⣀⣀⣀⣀⠉⠉⠉⠉⠛⠓⢲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣉⣠⡴⣞⣿⢽⣷⣶⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡦⢤⣯⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠂⠭⣙⡛⠛⠾⠿⠿⠶⠶⣿⣿⣷⣶⣿⣭⣟⣶⣝⣳⣽⣿⣿⣿⣿⣿⠭⠿⠷⠾⠟⠛⣛⡋⠥⢐⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠬⠅⠀⠀⠉⠉⠐⠒⠒⠂⠠⠶⠤⠤⠤⠤⠀⠀⠠⠆⠀⠀⠠⠆⠀⠀⠐⠒⠒⠈⠉⠁⠀⠀⠨⢽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡒⠒⣂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠔⢉⠄⠒⡒⡀⠉⠪⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡬⠐⠈⠀⡜⠁⠈⢣⠁⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠹⡀⣼⣀⠈⠁⠀⡤⢀⠌⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢕⠢⢈⣉⣀⡉⠄⡪⠊⠀⠀⠀⠀⠀⠀⢀⡀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡒⠤⠤⣀⣀⣀⠀⠀⠀⠉⠒⠂⠐⠒⠉⠀⠀⣀⣀⠀⠤⠄⢀⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⠤⠤⢭⣭⣭⣭⣭⣭⣭⣭⡥⠤⠤⠶⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠉⠙⠳⣄⣀⠈⢳⠤⣤⠴⠖⠒⠋⠉⠉⣩⣵⣦⣄⣀⣀⠉⠉⠉⢹⠷⡖⢲⠦⢄⡴⣄
⠀⠀⠀⠀⠀⠀⡠⠼⠦⣀⠀⠈⠁⠱⡼⡄⠘⡤⡄⠀⠀⠀⣰⠟⣾⢻⡏⠻⣟⠛⢦⡀⡸⢰⢓⠏⣷⡜⣱⣏⢳⡦⣄
⠀⠀⠀⠀⠀⠀⠳⢄⣀⠘⢤⣀⡀⣄⠙⢏⠀⠀⠸⡀⠀⢰⡟⢸⣧⣿⣽⠶⣿⠛⢿⡅⡇⡏⡼⢰⡟⡰⢃⢜⡥⠋⣀⡙⢦⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣀⣀⠀⣉⠳⢝⢮⠑⢽⣆⡑⢤⣸⣧⠎⢠⢿⣇⡀⠙⡦⠺⣽⢸⢰⢣⣟⡜⡵⣣⣪⣴⣮⡷⣟⡭⠔⢿⠁
⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⢟⣛⣦⡙⣌⠳⣷⡀⠠⢭⡳⡝⣿⣦⠜⢠⣇⠉⢲⣁⣸⡿⣇⣇⣾⢎⢞⣾⢿⡿⣻⠵⠊⠁⣠⠤⣌⢧
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣇⢿⣬⣝⠈⢾⢦⠹⡳⣄⡀⣹⢷⣼⣈⠷⠿⠓⠛⠛⠿⢾⣿⣿⣿⢯⢫⣯⢗⢽⠾⠄⣾⠑⠛⠃⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣷⣮⡞⢣⢣⣹⣶⣲⡷⢷⣬⣷⡿⠗⠊⡟⠑⠋⠀⠀⠀⠀⠀⠀⠈⢫⡉⠛⠻⢟⡁⡞⢀⡴⠋⡛⢉⠗⠲⣜⡿
⠀⠀⠀⠀⠀⠀⠀⢻⠙⢧⣸⣾⣿⠳⢤⡙⢆⡼⠁⠀⠀⠀⠳⢤⣀⣀⣀⠀⣀⣀⣀⡤⠞⠁⠀⠀⠀⠙⢦⡇⡠⠞⢉⣿⣀⣠⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠙⠿⣄⠑⠤⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣾⣀⣸⠤⣏⡤⠊⠁⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠉⠒⠧⢮⣘⣟⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⣴⡾⢟⣉⡤⠗⠊⠁⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠙⠢⣄⡀⠀⠀⠀⠀⠉⠙⠒⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠚⠛⠋⠉⠁⠀⠀⠀⠀⣀⡤⠚⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠉⠓⠢⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠒⠤⣀⡀⠀⠀⠈⠉⠙⠓⠒⠒⠲⠤⠤⠤⠤⠤⠤⠖⠒⠒⠚⠋⠉⠁⠀⠀⢀⣀⢤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠉⠓⠒⠤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠒⠚⠉⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣍⠓⠒⠲⠶⠶⠶⠶⠶⠒⠒⠋⢉⡡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠦⠤⠤⠖⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣭⣽⣶⡿⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢠⣄⢠⡄⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⣼⣿⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠀⣀⣀⣠⣀⡀⠀⠀⣼⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠉⠉⣩⣀⣤⣾⣿⢿⡋⠁⠈⠁⠀⠀⠉⠀⠈⢹⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⣿⣿⠋⠩⠀⠀⠀⠀⢴⣤⣿⣥⣔⡶⡄⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠈⣷⡀⠀⠀⠀⠐⣴⣿⣿⣿⣿⣿⣿⣞⡆⠀⠀⠀⢠⡄⠀⠲⠿⠿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣠⢫⣢⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⢿⣛⢿⣵⡀⠠⢤⠙⢿⣿⣿⣿⠿⢿⢏⡀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣿⡿⡆⠀⠀⠀⠈⡻⠿⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠒⠦⠀⠀⠤⠀⠨⣵⡭⢿⣟⠷⠆⠀⠀⠀⠀⠉⠀⠀⢦⠀⠀⠀⠀⠛⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⣤⣄⡉⠀⠐⠁⠀⣀⣀⠀⠀⡀⠀⠀⠀⠘⡂⠰⠆⠀⠀⠀⠀⢴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⣿⣿⢁⠀⣤⣛⠀⠉⢹⣧⠘⠃⠀⠀⠙⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣓⣯⣼⣿⣿⡇⢲⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⢿⣿⣧⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⠛⠉⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣃⣿⣿⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠁⠀⠀⠀⠀⠀⠘⢿⣿⡷⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⣨⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⠉⢾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣻⣿⡏⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⣀⣀⣄⣉⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠛⠃⢀⣾⣿⡀⠀⣿⡇⠀⠀⣿⣟⣛⠃⢸⣿⣦⢸⣿⠘⠛⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣼⣧⣼⣷⠀⣿⣇⣀⡀⣿⣏⣉⡁⢸⡇⠹⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠀⠙⠁⠉⠉⠉⠁⠉⠉⠉⠁⠈⠁⠀⠈⠉⠀⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡤⠶⠶⠶⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢿⡉⢷⡄⢘⡷⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⢧⠀⣷⠸⣇⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⠤⢤⣄⡀⠙⠻⣧⡾⠛⠉⠉⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠙⠶⣄⡈⢙⡦⠀⠀⠀⢠⡀⣿⠰⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠈⣿⢻⡇⠀⠀⠀⢸⡇⣿⠀⠀⠈⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⡇⠀⣶⠀⢸⡷⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⣸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢻⡆⠀⠀⠀⠀⣧⠀⣿⠀⣿⠀⠀⠀⠀⢰⡟⠙⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠃
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⣤⣼⣇⡀⠀⠀⠀⣿⠀⣿⠀⣿⠀⠀⠀⢀⣸⣧⣤⣿⣀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠈⠹⣆⠀⠀⠙⠛⠛⠛⠃⠀⠀⣰⠏⠁⠀⠀⠀⠉⢳⡄⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠐⠏⣀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣁⡼⣧⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠞⠋⠉⠉⠉⢹⠋⠙⣿⠉⠉⠛⢦⣄⣀
⠀⠀⢀⣀⣰⢷⣀⣀⠀⠀⠙⣿⠁⠈⡿⠋⠀⠀⣀⣠⣿⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠘⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⡶⢶⣄⡀⠀⠉⠳⣄
⠀⠀⠈⢻⡅⠀⣽⠛⠀⠀⠀⠿⠛⠛⠿⠀⠀⠈⠻⣯⠀⢹⡟⠁⠀⠀⠀⠀⠀⠀⣠⡶⠿⠶⠀⠀⠀⣠⡴⢶⣤⡴⠂⠀⣿⡉⣠⣄⠉⣿⠄⠀⠀⠙⣦
⠀⠀⠀⠸⠟⠛⠻⠄⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠰⠟⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⣀⣽⠂⠀⠀⠀⠀⠀⢿⣥⡈⢁⣠⡿⠂⠀⠰⢦⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠉⠙⢦⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣻⣦⠀⠀⠀⠀⣏⠁⠀⢸⣟⡷⠀⠀⢈⡉⠛⠛⢉⠡⣄⠀⠀⠀⢸
⢀⣠⣼⠿⣤⣀⡀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣴⠿⣦⣀⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⣳⠄⠀⠉⠀⠀⣼⠋⠁⠀⠀⢈⠀⣿⠀⠀⠚⢻
⠈⢻⣆⣀⣾⠋⠀⠀⠀⠸⣇⠀⠀⠀⠀⣸⠇⠀⠀⠀⠙⣧⣀⣰⡏⠁⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠙⠶⣴⠟⢷⡴⠛⣁⠒⠠⠄⠖⣠⠟⠀⠀⠀⣸
⠀⠘⠋⠉⠛⠂⠀⠀⢀⣀⣽⣷⠦⠴⣾⣏⣀⡀⠀⠀⠘⠛⠉⠙⠇⠀⠀⠀⠀⠀⠀⣈⠳⣤⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠛⠉⠀⢠⠴⣾⠃
⠀⠀⠀⠀⠀⣴⠞⠛⠋⠉⠀⠙⠶⠶⠋⠀⠉⠙⠛⠳⣄⠀⠀⠀⠘⣦⡀⠀⠀⢠⡾⠃⠀⠀⠉⠉⢷⣄⣀⣀⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠰⠆⢸⡇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢹⡆⠰⠆⠀⠈⠻⣄⣴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠚⠻⢦⣄⣀⣠⡶⠀⠀⢸⡆
⠀⠒⠐⠂⢸⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠐⠂⠒⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣆⠀⠀⠀⣿
⣤⣦⣼⣧⣾⣦⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣾⣶⣼⣷⣼⣧⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣴⣷⣦⣤⣤⣿⣦⣦
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣽⣷⠶⣦⣄⣰⣄⡀
⠀⠀⠀⠀⠀⠀⠉⠉⣙⣿⣷⣶⣦⣭⣁⠀⠈⠙⠯⠙⢦⡀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠽⠿⣿⣿⣿⣶⣤⣀⠀⠀⠙⣶⡿⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠿⢷⣦⣀
⠀⠀⠀⠀⠀⣀⣠⣤⠿⠏⠉⠉⠉⠙⠛⠛⠿⠛⢿⣷⣤⡀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄
⠀⠀⠀⠀⣠⢝⣛⣡⣤⣶⣶⣶⣶⣶⣶⣶⣤⣄⠀⠈⢻⣿⣦⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠼⠛⠉⡽⠟⠋⠟⠉⣚⣯⣭⣭⣟⣿⣿⣿⣦⡀⠙⣿⣷⣬⣷⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⣀⣀⣿⡟⠛⠛⠛⠛⠂⠀⠀⠉⠉⢉⣿⣿⠿⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠙⠛⠿⣷⣦⣀⢻⣿⡀
⠀⠀⠀⢀⡼⢟⣋⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡄
⠀⠀⣰⠿⠛⢻⠟⠛⣿⠿⠛⠻⠿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣤⣄⣀⣀
⠀⣀⣴⣾⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣷⣦⣀
⣰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢀⡤⠶⠶⣶⣄⡀⠀⠀⠀⠀⢀⣠⣶⠶⠶⢤⡀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⣿⠀⠀⠠⣤⣶⣾⣿⣿⣿⠃⠘⣿⣿⣿⣷⣶⣤⠄⠀⠀⣿⣿⡿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠙⠿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⡿⠁⣼⣿⣿⠀⠀⠀⠀⠀⠠⠼⠛⠏⠀⠀⠸⠛⠧⠄⠀⠀⠀⠀⠀⣿⣿⣧⠈⢿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⠿⠋
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⣧⠀⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⣼⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣤⣹⣿⣿⣷⠀⣀⣀⣤⠟⠉⠁⠀⠀⠈⠉⠻⣤⣀⣀⠀⣾⣿⣿⣏⣤⡀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣴⢿⣼⠀⣿⣿⣿⣿⡀⠈⠛⢛⣷⣾⣷⣀⣀⣾⣷⣾⡛⠛⠁⢀⣿⣿⣿⣿⠀⣧⡿⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣅⠸⡄⠿⢀⣿⣿⣿⣿⣧⣶⡿⠿⣿⣟⣛⣩⣍⣛⣻⣿⠿⢿⣶⣼⣿⣿⣿⣿⠀⠿⢀⠇⣨⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣿⠶⠛⠶⠟⢻⣿⣿⣿⣿⣿⠀⠰⣦⣬⣉⣉⣉⣉⣥⣴⠆⠀⣿⣿⣿⣿⣿⡏⠳⠶⠟⠶⣿⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡝⡆⣀⡤⠶⠛⣿⣿⣿⣿⣿⣆⠀⠉⠀⢸⣿⣿⡇⠀⠉⠀⣰⣿⣿⣿⣿⣿⠛⠲⢦⣀⢰⢫⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡿⣅⠀⢀⡤⠞⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⡟⠳⢦⣀⠀⣩⢿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣽⣿⡋⢀⣴⠞⠉⢙⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡋⠉⠳⣦⡀⢙⣿⣯⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣎⣷⣟⡁⠀⣴⠟⠁⣰⡏⠉⣿⢿⣿⣿⡿⢿⠉⢹⡆⠈⠻⣆⠀⢈⣻⣾⣱⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣯⣽⣾⣯⣄⢠⡟⠀⢸⡇⠀⢻⡟⠀⢸⡇⠀⢻⡄⣠⣼⣷⣫⣽⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣷⣮⣿⣛⡶⣼⣥⣤⣸⣇⣤⣬⣧⣶⣛⣿⣵⣾⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣶⣷⣿⣿⣶⣶⣿⣿⣿⡿⠿⠿⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣿⣯⣽⣿⣿⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣁⣍⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣾⣤⣦⣤⡀
⠀⠀⠀⠀⣠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⢻⣿⣿⣿⣿⠁⠈⠈⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠛⣿⣿⡟⠐⢤⡀⣤⠤⠂⠀⠛⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠘⣿⡇⠀⠘⠀⠹⠂⠀⠀⠀⠈⢹⡿⣻⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⠤⠂⠙⡄⠀⠀⠀⣸⣱⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⢀⠾⠶⢻⣕⣶⠖⠀⠀⠀⣠⣿⡿⠛⠋⠁
⠀⠀⠀⠀⡠⠆⢙⡿⠶⣼⣿⡷⠃⠀⣠⣾⣿⣿⣿⣶⣶⣶⠒⠂
⠀⠀⢀⡴⠓⣲⣾⣷⣤⠌⣿⣥⣴⢎⣙⠋⣿⢿⡿⠟⢁⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢸⣰⣿⣿⣿⣾⡟⠀⡧⠈⠉⠀⠙⣷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠊⡹⠀⣠⠊⠇
⠀⢠⣾⣿⡿⣿⠿⠿⠁⠀⡼⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⠃⣴⠁⡎⠀⡠⢲⠂
⣴⣿⡿⠋⠙⢤⡀⠀⠤⠰⣇⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠸⣷⠇⡘⣠⠎⡰⠃
⠉⠉⠠⠀⠀⢸⠷⣀⣀⣤⣿⠀⠀⠀⠐⣦⣄⠀⠈⠉⠒⠒⠀⠀⠘⣧⠀⠀⠀⠰⠀⡀⢄⠀⢠⠃⠀⢀⠀⠛⢁⠌⣀⡠
⠀⠀⠀⢢⡀⠘⠀⠀⡉⠺⢿⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢁⠱⡄⠑⠎⠀⣰⣄⠒⠀⠀⢉⡩⠖⠋

⠀⠀⠀⠸⡄⢰⠀⠀⢨⣁⣾⡆⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⡟⣿⡀⠀⠀⠀⢆⠸⡀⠀⠀⢹⡿⠀⠀⣰⠃
⠀⠀⠀⠀⡇⢸⠀⡄⢄⡈⠀⢧⣀⣀⡀⠤⢞⡤⠀⠀⠀⠀⠀⢀⡼⠀⠘⣧⠀⠀⠔⠛⣲⢧⠀⢀⠘⠁⢀⡜⠁
⠀⠀⠀⠀⡇⠸⠀⠘⠂⠀⠀⢸⠧⠤⠤⠀⠁⠀⠐⠂⢐⠶⠾⣿⠃⠀⠀⠘⡇⠀⠀⠀⠀⠄⠀⢈⣷⡖⠉
⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢀⣾⣷⠶⠂⠀⠀⠀⠀⠀⠀⠠⠚⣿⡀⠀⠀⠀⠘⢦⣤⣠⣬⣷⡽⠟⠉
⠀⠀⠀⠀⠀⠈⠳⠤⣀⡴⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠑⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠙⠲⢄⡀
⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠛⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠈⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠉⡐⠂⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣬⣁⠒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠿⠛⡏⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⡿⣟⣋⣭⣭⣯⣭⣽⣯⡛⣯⣤⣼⡆⠀⠀⢀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣷⣿⣿⣿⣿⣶⣿⣿⣿⢿⣿⠟⠋⠁⠀⠀⠀⠉⠈⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣤⣤⣤⣶⣶⣶⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣴⣶⣶⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠳⡄⠀⠀⢠⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⠀⠰⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣧⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠑⠲⣄⠀⠀⠊⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠄⢸⣀⣀⣆⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⠉⠀⠀⠈⢹⠀⠀⠀⠛⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠘⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢀⢤⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⢓⡄
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⡤
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢰⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⡔⠁⢀⡴⠋⠀⠀⠀⠀⢀⣠⠴⠚⢉⣠⠵
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠁⣀⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡴⠋⢀⡴⠋⠀⠀⢀⣠⠴⠊⣁⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢀⢜⡴⠋⢀⡔⠋⠀⣀⠤⠚⣉⡤⠒⠋⠁
⠀⠀⠀⠀⠀⢈⣇⣀⡀⢀⠤⠄⠰⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠙⡿⠋⣠⠞⠁⣀⠴⢊⡡⠔⠊⠁
⠀⠀⠀⠀⣰⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢤⠊⢱⡶⠊⣤⠖⠉
⠀⠀⠀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠘⠦⢤⡀
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢦
⣞⠉⠙⠡⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠤⠊⢰⠃
⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠚⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣿⡧⠀⠀⠀⣠⣶⡄⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠀⢹⣿⣿⣿⠟⠙⢿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣶⡍⠀⠀⠀⠀⠙⢛⣧⡌⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⠉⢶⣿⣿⣿⣿⣦⡤⠚⢷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣧⠀⣤⣤⢴⡄⣸⣿⣿⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠸⣿⠛⠛⡛⣿⠇⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⣰⣿⠏⠘⠻⣿⣿⠿⣾⣷⡿⠛⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⠏⠀⣼⣧⣄⡈⠀⣈⠁⠉⣀⣤⠀⠘⣿⣿⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠁⠀⠉⠁⠀⢸⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⡇⠀⠀⣿⣿⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⠈⣛⠿⡟⣿⣿⣿⣿⣿⣿⢃⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⠀⢈⣿⣭⣿⣿⣽⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿
⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⢸⣿⣿⣿⣄⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠄⠀⣐⣂⣀⣀⣀⣠⣤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠚⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣛⣚⠀⡄⣇⣩⣉⣩⠩⠅⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠶⠧⠦⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⡡⢭⣍⠂⠘⠒⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⡇⠛⠀⠠⠥⠀⢄⣍⡐⡠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⣄⠄⠘⡇⠀⢤⡤⡄⢴⡄⠈⡆⠄⡈⡴⡂⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣍⢀⡘⡙⠀⠄⠓⠟⠁⠓⢄⠻⢋⠤⠣⡀⠀⣀⣱⠈⢑⣤⡉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢀⠭⢄⠁⠂⡀⠉⣐⡂⣮⠤⠶⠞⣠⣀⡀⠃⠉⢉⣀⡄⢍⡸⢼⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡐⠏⡣⡁⠬⢧⡀⠀⣀⡀⠋⢐⠁⣈⡢⠊⠬⠂⠛⠶⡄⡘⢂⠂⣇⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡛⢷⣾⣤⣃⣀⣚⠂⠊⠀⠀⡀⠴⡄⢾⢗⣀⣀⣄⣠⣬⣥⡶⠾⢛⣭⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠳⢦⣭⣉⣙⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣉⣥⡤⠶⠛⠉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⢉⠉⠉⠛⣿⠟⣿⡿⠟⠿⣿⠿⠿⢿⣬⠻⣷⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠿⣛⣛⣛⠿⣾⣯⣼⢯⣼⢟⡟⠖⣾⠿⣶⣽⣧⣽⡇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢛⣉⣙⠻⢾⡷⡿⡿⣻⢿⢻⢣⠇⠹⣟⣉⣉⡙⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣉⣙⢿⡬⡇⣼⡳⣛⣾⣺⣿⡿⣧⡞⣉⣉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢸⣧⣹⣟⢿⣮⣻⡷⠿⢃⣿⢰⡟⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⣮⣝⣧⣿⢏⣴⠞⣛⣡⡾⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠿⠿⠾⠿⠾⠟⠛⠋

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

⠀⠀⠀⢰⡶⣦⠠⠀⠀⡄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⢀⢠
⠀⠀⠀⢸⣇⣿⣴⣤⢀⣶⣾⣗⡶⣴⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⠀⠀⡸⠀⢸⠘
⠀⠀⠀⢸⡇⢸⣇⣿⣸⣷⣿⣻⣇⣿⡀⠀⠀⠀⣠⠞⠁⠀⢀⠀⠤⠄⠀⠈⠙⠹⠅⢠⠀⠀⠀⠀⡇⠀⠀⡇⠀⠈
⠀⠀⠀⠈⠁⠀⠉⠉⠈⠉⠈⠁⠉⠈⠀⡠⠀⢀⠏⠀⠀⠀⢉⠜⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠥⣤⡀⠀⠀⠠⠀⢸⠃⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠧⣤⡑⠝⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠰⣆⠀⠀⠀⠀⡄⠘⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠐⠒⡤⠀⡡⢈⡉⠙⢏⡤⠜⠁⠀⠀⠀⠀⢦⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠁⠀⠀⠀⠁⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠊⠐⣪⠤⣀⠠⠁⠐⢠⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⢂⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠈⠈⠁⢂⠔⠚⠁⠀⠀⠀⢠⠐⠀⠐⠀⡰⠆⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡟⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⢀⡔⠁⠀⠀⢰⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡐⠸⣤⡠⢥⣤⡿⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⡰⣶⡊⢉⡄⠐⠂⠩⠃⢹⡒⠶⠟⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠨⠅⠑⣀⣸⠧⠀⠉⠀⠱⢄⡀⠀⠬⡕⠩⠙⡂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⠐⢀⠀⠈⠴⠀⠰⣆⠀⢐⣉⣄⡀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣶
⠀⠀⠀⠀⠐⠀⠀⠀⠩⣬⣤⠙⠒⠶⠀⡲⠂⠩⡄⠒⠷⠦⠈⠁⣦⡠⠚⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⠲⠿⠤⠦⡮⠤⠦⠵⣦⣤⠋⠁⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⠀⠀⠃⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠤⠀⣀⠀⠀⠀⠀⢀⡀⠊⠯⠉⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠇⠀⠀⠁⠀⠨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⣤⡀
⠀⠀⠀⠀⡤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡠⠬⢍⣔⠵⢀⠊⢴⡈⠅⠠⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠂⠀⠘⠄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣀⢠⣀⠜⠫⠛⠁⠀⣨⡗⠁⡙⡉⠒⠤⣶⡍⠓⣀⡀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢀⠒⢀⡤⠧⠦⣮⠁⢉⡏⢀⣀⣋⢄⢈⠢⣑⠶⡼⢀⠋⢩⠀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠰⡀⠀⠀⠀⢸⠃⠀⠀⡠⣮⠀⡞⢋⠓⣄⡌⠀⢸⣌⢲⡂⢒⡡⠂⡎⠀⠀⣅⡫⠂⠈⠠⠊⠑⣀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣿⠀⠀⡜⠀⠀⣸⠁⢰⠂⡉⠀⠀⡁⠀⠉⡋⣱⠄⠀⣩⡀⠀⣀⣀⢠⣶⠊⠔⠃⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢢⠀⠀⢹⣄⠨⡇⣰⠛⠉⢆⢂⠠⠐⠢⠰⢆⠖⡉⢹⢹⡜⡲⢧⣀⡀⢉⠔⠊⢐⠈⣿⡶⠿⠩⠢⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢣⠀⠘⣿⣨⣵⣏⠀⢠⡟⢹⠛⠲⡤⠀⡇⢀⠞⢣⢸⢄⠀⢸⣭⣉⣊⣀⡀⠴⠀⠈⢸⠑⡂⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⣙⣲⣬⢥⣄⠑⠀⠰⢄⣼⣾⢂⠨⠘⣃⣀⠠⠨⢈⣛⡞⣐⡨⠀⠀⠖⠄⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠆⠀⠈⢷⣆⡎⠈⣄⠁⠬⣡⣄⣠⠉⠈⠳⣄⣦⡦⠛⠉⠍⢉⢘⠀⠉⡔⠣⣀⢈⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠆⠀⠈⢋⠻⣶⣄⢙⠖⠋⣀⣈⡄⠔⠀⣥⠦⢤⣄⡐⢆⠲⠆⢂⠜⡀⢀⣴⠟⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠱⡄⠙⠻⢶⣬⣁⣩⢀⠁⢊⠲⠇⠀⢒⡑⣠⣋⣩⣥⡶⠟⡋⢐⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⠈⢦⡀⠀⠀⠉⠙⠛⠻⠿⠿⠷⠿⠿⠿⠛⠛⠉⡑⠂⠅⣪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠉⠀⠈⠂⠀⠀⠀⠀⣂⡤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⣟⡳⠦⣤⣄⣀⣀⣤⣀⣀⣤⡴⢞⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⢳⠀⡄⠀⠀⠀⠻⣿⣷⣶⣬⣭⣭⣭⣭⣴⣶⣾⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡜⡆⢻⠹⡄⠀⠀⠀⠈⠑⠪⠭⣭⣭⣭⣭⡭⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⡄⢧⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⡴⣍⠉⠑⠤⡀
⠀⠀⠀⠀⠙⢭⣛⠶⣤⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠲⣝⢦⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣕⠈⢢⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠈⠉⠑⠦⣴⠶⣄⡤⠒⢦⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⣇⡀⣀⡤⠤⢈⠿⠗⣓⡔⠀⣚⠣⡄⡈⣹⣙⡞⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠡⢀⣉⠀⢰⡀⠔⠘⡞⠍⡁⠈⠾⡛⡳⠠⢅⠀⢨⠚⢐⠮⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠈⡟⠁⠉⢛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢹⡟⠛⠛⠛⠛⠛⠛⠛⠛⢆⠀⠀⠂⠇⠒⠀⡸⠛⠛⠛⠛⠛⠛⠛⠛⢻⡏⠁
⠀⢀⠀⡠⢄⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠐⢀⠈⠑⠒⠁⠀⠑⠚⠁⡠⠂⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡠⢄⠀⡀
⣀⠇⣈⡇⠈⢢⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠤⠤⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⡔⠁⢘⣁⠸
⢣⠤⠤⠀⠀⣀⣉⠉⠒⠺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠒⠉⣉⣀⠀⠀⠤⠤
⠑⠒⠒⠒⠊⠀⠀⠈⠉⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠉⠁⠀⠀⠑⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⡏⢹⡏⢹⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠢⠤⡇⢸⡇⢸⠤⠔⠒⠒⠢⢄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⢤⣄⣀⣠⠤⠖⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⡀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣧
⡇⠀⠀⠘⡄⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠦⠤⣀⣀
⢸⠀⠀⠀⡇⠀⠀⢸⡀⠀⠀⢸⠀⠀⠀⢀⡤⠖⠊⠉⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠳⡄⠀⠀⠉⠓⠢
⠈⡄⠀⠀⢹⠀⠀⢸⡇⠀⠀⢸⠀⠀⠰⣅⠀⠀⠀⢰⠃⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⢾⣿⣿⠆⠀⠀⠀⢹⠀⠀⠀⠀⣠
⠀⠇⠀⠀⢸⡀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠈⠓⠲⠤⣾⠀⠀⠀⠀⠈⠉⠉⢀⣀⣀⣀⠤⠤⣉⠁⠀⠀⠀⠀⢸⡷⠒⠒⠋⠁
⠀⢸⠀⠀⠀⡇⠀⠈⡇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⣠⠞⠉⢀⣀⡀⣀⡀⣬⣹⡄⠀⠀⠀⢸⠇
⠀⠀⢧⠀⠀⢇⠀⠀⡇⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠠⠾⠻⠚⣛⣋⡥⠬⠿⠿⠓⠿⠷⣄⡀⢠⠟
⠀⠀⠈⢿⣐⣋⡒⢉⡉⠭⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠛⢦⡀
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣄⣀⣀⣀⣀⣤⡤⠶⠚⠉⠀⠀⠀⠙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇
⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⡟⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⢠⡇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⢸⡇⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢿⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠳⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⡟⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⣦⠞⠛⢷⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠈⣿⠦⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠂⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⣷⣶⣦⠀⠀⠀⠀⠀⣤⣿⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⣀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠼⠀⠀⠀⠙⠿⠿⠿⠿⠟⠛⠿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠿⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⣀⣤⣴⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⢠⣠⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⢸⣿⣿⣾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢸⣿⡿⠋⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣧⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢹⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠻⣿⠟⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⠀⢿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠈⠻⣿⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⣦⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠁⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⢔⠋⠈⠀⠀⡄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡉⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠨⠄⠀⠀⠀⠙⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠠⠀⠀⠍⠀⠒⠂⠀⡆⠈⠄⠀⠀⠀⠘⠀⠀⠠⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠠⢀⡀⠒⠂⠀⠀⠒⠆⠐⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⠛⠀⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠘⣤⡈⠱⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⣄⠀⠀⠀⢸⠏⢂⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡘
⠀⠀⠐⣀⠀⠀⠀⠀⢉⠀⠀⢀⠀⠸⡄⠀⠀⠃⠀⢀⡀⠀⠈⠁⠁⠀⠀⠰⡀⠑⠀⠀⠀⠀⠀⠀⠀⠳⣄⠠⠀⠒⠈⠀⠀⠀⠀⠐
⠀⠀⠀⠈⠀⠀⠀⡀⠀⢀⢀⡤⠇⠀⠀⠈⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⠤⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠂⠀⠀⠀⢄⠀⢄⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣶⣶⣿
⠀⠀⠠⠂⠈⡀⠀⠤⠀⠀⠁⠀⡄⠘⢄⣀⣀⣄⠀⠁⢀⣄⠀⠀⠀⠀⠀⠙⡀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢀⠨⠁⠀⠓⢌⠁
⠀⠀⡀⠀⠀⠁⠀⠀⠀⡄⠠⢃⣤⣤⣬⣏⠁⢹⡤⠀⠀⢉⣥⠄⠠⢀⠀⠀⣡⠾⠃⠙⠦⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡀
⠀⠀⠛⠾⡇⠀⢀⢠⣶⣿⡤⠻⣯⣥⠔⠾⢶⣶⡦⢀⣀⣮⣻⣶⠶⠚⠢⡀⢀⠳⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠢⡀
⢃⠀⠀⠀⠂⠀⠈⠙⢻⡁⣀⣴⢿⣿⣷⣶⡾⣿⡄⠀⢹⠿⣿⠟⢣⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠑⠀⠀⠀⠀⠀⠀⠈⠉⡃⣼⣷⣶⣶⣶⣦⣤⡆⣠⡬⣿⣿⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⣿⣄⣠⣴⣭⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠰⠀⣀
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣷⣶⣾⣶⣶⣶⣶⣶⣾⣿⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣶⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⠏⠉⠙⢻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⠃⠉⠀⠘⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣂⠀⠀⠀⢀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⡗⠶⠖⣿⣬⣿⡄⠀⠀⠀⠀⠀⠀⣤⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⠿⠃⠐⠒⠒⠀⠿⣿⣷⣤⣀⣀⠀⠀⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣤⣴⣶⣾⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⡀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠟⠉⠁⠀⠀⠀⣿⣿⣿⡆⢰⣿⡏⠉⠛⠻⠿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠉⢙⣋⣛⡛⠻⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⣿⠃⠀⠀⠀⣀⣀⣀⠈⠉
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣦⣄⠀⢤⠀⠀⠀⠀⠀⠘⢋⣤⣶⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠚⠏⢿⣿⠏⠟⠚⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠂⠐⠚⠛⠒⠂⢲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷

⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠈⠲⠿⣉⡑⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⢬⣉⡛⠲⠶⢤⣄⡀⠀⠀⠀⠀⠀⡀⠄⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠦⢤⣈⡉⠒⠶⠂⠁⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠒⣛⣋⣉⣀⡙⢿⣦⣀⠀⠀⠀⠀⠀⢀⡜⠀⢬⣭⣭⣛⣒⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⢡⡖⠋⠉⠁⠀⠀⠀⠀⠀⠈⠛⠿⠶⡶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠉⢳⠈⣿⠆
⠀⠀⠀⠀⢀⡤⠴⠶⢤⣄⣀⠀⠘⠳⠾⠷⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠧⠾⠋⠀⠀⠀⣀⡤⠤⠤⢤⣀
⠀⠀⠀⡔⢡⡶⠛⠉⠙⠫⠉⡷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⡇⢾⠋⠩⠂⠉⠉⠲⣌⢳⡀
⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡄⣧
⠀⠀⠈⢧⡹⣆⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⣸⢃⡟
⠀⠀⠀⠈⠳⣌⠦⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡴⢋⡾⠁
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣫⡴⠋
⠀⠀⠀⠀⠀⠀⠈⣧⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠟⠛⠉⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠦⠤⢄⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⣀⣀⣀⣠⣤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⢂⣀⣀⠤⠄⠀⠚⠛⠳⠭⠥⠤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠐⠒⠂⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠔⠀⠀⢀⠀⠀⠀⠆⠀⠀⢀⢠⣰⣂⣀⣀⣀⠩⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠌⠀⠀⠀⠀⠁⣃⠴⠚⠒⠊⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠊⠀⠀⠀⠀⠀⡀⠆⡀⠄⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠁⠀⠀⠀⠀⠀⡠⢣⠠⠒⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⢰⢞⣁⠠⠐⠈⠁⠀⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⢀⠠⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣁⠀⠄⠀⠀⠄⡀⠀⠈⢄⠀⡀⠤⠀⠒⠀⠀⠀⠀⠀⠒⠀⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠠⠄⠊⠀⠀⠀⠀⠀⠀⢒⠘⠀⠐⡀⠀⣁⣴⣁⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠀⢈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠑⢌⡈⢀⡼⢩⠤⠄⡈⠉⠉⠙⠒⠶⢤⠀⠀⠀⠀⠀⠀⡈⠢⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠠⠊⢠⠋⠀⡡⠁⠐⠀⠀⠀⠘⡄⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠈⠀⠈⠄⠱
⠀⠀⠀⠀⠀⠠⠁⠀⡀⠀⠀⠐⠀⠀⢃⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⠃⢠⡆⠀⠀⠈⠕⢄⠀⣴⣥⡀⠀⠀⢰⠃⠉⠉⠉⠉⠑⠢⠐⠈⣦⡀⠀⠡
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠠⡅⠀⡐⠀⠀⠀⠀⠈⠰⢞⠌⠪⠙⠗⠒⠃⠀⠀⠀⠀⠀⠀⡌⠀⠀⢸⠂⠀⠀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠡⡀⠐⣡⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⠘⠧⠐⠉⠀⠀⠀⠠⣀⢸⣠⣁⡀⢨⣀⣀⣀⣁⣉⣀⠀⠀⣸⠛⣦⠀⣸⡄⡀⢀⣡
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠈⠂⠀⠀⠀⡀⠀⠀⠀⠠⢀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠙⢸⠁⠀⠈⢉⡉⠉⠋⠋⠛⠛⠿⠳⠛⢰⣿⡿⠿⢠⢏⣸⡱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠠⢨⡟⠀⡰⠣⣱⢟⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣠⣶⣿⣤⣤⣤⡶⡶⠶⠖⡚⠤⠊⢠⡾⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⣠⠠⠀⠁⠀⣈⠐⠋⠻⠅⠀⠀⠀⠀⠀⠄⠀⠂⡉⡤⠐⢋⠜⠁⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠔⠈⠱⢄⠀⠀⠀⠈⠓⠒⠰⠂⠤⠠⠆⠰⠂⠊⢉⠀⠀⡰⠋⠊⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠠⠖⠈⠁⠀⠀⠠⠀⠓⠢⢄⡀⠠⠀⠀⠄⠈⠀⠀⠂⠀⠄⣠⠞⠀⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠃⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠈⢙⠃⠦⢤⠆⡤⢢⠖⠟⡡⠂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢀⠤⠐⢂⠉⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠈⠀⠁⠈⠐⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⣄⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣯⣍⣈⠙⠻⠳⣄⠀⣀⣀⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠿⠿⢿⣿⢦⣄⠈⠻⡉⠉⠀⠀⠀⠈⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣩⣥⣶⣶⣶⣦⣤⣀⠙⢷⣄⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣫⡬⠥⠿⠟⠛⠻⠿⣷⣬⡿⠷⠗⠒⠒⠒⠲⠶⠶⣤⣄⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣛⣭⣤⣶⣶⣶⣶⣶⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠛⠉⠛⠉⢉⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠶⠶⠶⠶⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠠⠶⢦⣄⣀⡀⣀⣀⡴⠶⠤⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣀⣀⣀⣀⣀⣀⣠⡞⣹⣿⠀⠀⠚⠛⣻⠿⠁⢹⢿⡛⠛⠂⠀⢸⣿⠹⣦⣀⣀⣀⣀⣀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⢻⡄⢿⣿⣄⠀⠀⠀⣠⡄⠀⠀⣄⠀⠀⠀⢀⣾⣿⠃⣿⠉⠙⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣇⠲⣿⣿⡇⠰⢶⣅⣤⠀⣠⣌⣷⠶⠀⣿⣿⡗⢢⣯⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⢹⡘⢣⣿⣿⣷⣴⠾⠿⣿⣭⣽⡿⠿⢶⣼⣿⣿⣇⠛⡸⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠋⢉⡩⢿⣿⣿⣧⠀⠛⠲⣶⡶⠚⠃⣸⣿⣿⣿⢭⣉⠉⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣍⣠⠞⢻⣿⣿⣷⣶⣾⣿⣿⣶⣾⣿⣿⣿⠛⢤⣈⣽⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣟⣦⡞⠁⣰⠋⡻⢿⣿⣿⣿⣿⢻⡋⢧⡀⠑⣤⢟⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣾⢥⣰⡃⢸⠉⢻⠁⢻⠀⣧⣠⢷⣾⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⣾⣾⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠛⠋⠉⠁




⣿⢿⠿⡿⡿⡟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣻⣛⣻⣿⣿⣛⣻⣿⣿⣿⣿⢻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⣄⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡀⠀⢀⣑⠆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⠃⠀⠙⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⣀⠀⠀⠀⢀⠠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠉⠉⠉⠀⠀⢀⠌
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢤⠤⠤⠂⠁
⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠈⠁⠁⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠘⢿⣿⠃⠉⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠄⠒⠈⠁⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠈⠉⠀⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⢀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣭⣝⣁⣀⣀⣀⣀⣈⣒⣒⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠸⠀⠀⠀⠀⡔⠥⠀⠱⣦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠑⡄⢨⠀⠀⠀⠀⢾⣄⣄⠀⢉⡇⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⣠⠶⠶⣦⠀⠀⠰⠀⠣⡀⠀⠀⠀⡠⠃⡠⡿⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠐⡺⠀⠻⡋⠀⡠⠃⠀⠀⠀⠉⠉⠀⡠⢊⠞⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⣀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠱⡣⡀⠈⠉⠀⠀⠀⠀⠀⠀⡠⢊⠔⢁⠔⡡⠊⠁⠉⠢⡀⠀⠀⠀⡰⠃⠀⣠⠞⠉⡠⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠒⢽⣄⠀⠙⢼⣦⡀⠀⠀⠀⠀⠀⠜⡔⠁⡴⠁⠘⠀⢰⣐⡄⠀⡆⠀⠀⡰⠁⠀⡼⠃⡠⠊⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⢤⢶⡆⡦⠑⢤⡀⠈⠻⣦⡀⠀⠀⣈⠌⢠⠊⠀⠀⣀⠵⠺⠛⣀⠜⠀⠀⢀⠃⠀⣼⡡⠊⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠺⢮⣍⣀⣀⠀⠈⢲⣄⠈⢻⣆⣠⣯⢠⣃⣠⡪⠋⢁⡠⠔⠈⠀⠀⠀⠀⢨⠀⠀⠉⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣤⠄⢀⣀⣈⡙⠻⢷⢴⠳⡀⢹⣾⡇⡶⣮⠊⡠⠶⠭⠤⣁⣒⡂⠤⢀⣀⠎⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⠴⠚⠉⠁⠀⠈⣉⢐⣭⡳⣜⣞⣿⣷⣼⠇⣼⣅⣀⣀⣀⠀⠈⠉⠙⢲⠃⢠⡺⢇
⠀⠀⠀⠀⠀⠀⢠⠖⡡⠖⠉⠀⠀⣀⡤⠖⣋⡉⠉⠀⠄⠉⢿⣎⣿⣿⣿⡏⠠⠦⠀⠉⠉⡙⠛⠶⡴⠁⠠⡳⠉⠲⣝⢆
⠀⠀⠀⠀⠀⢠⢁⡞⠀⠀⣠⠖⠋⠁⠀⠐⠀⠁⠀⠀⠀⠠⠀⢻⡸⢿⢿⡁⠀⠄⠀⠀⠀⠐⢂⠜⠀⠀⡑⠳⢦⡀⠈⣆⢡
⠀⠀⠀⠀⠀⠸⡘⣇⢠⠞⠁⠀⠘⠑⠀⠀⠐⠀⠀⠀⠀⠄⠀⠙⠻⠼⣼⠉⠀⠠⠀⠀⠠⠀⠉⡀⠀⠺⠀⠠⡀⠱⡄⣸⢸
⠀⠀⠀⠀⠀⠀⢷⣜⠻⣄⠤⠀⠅⠐⠂⠀⠂⢀⣀⠀⠀⢀⠀⠀⠀⠀⠈⠁⢀⠀⠂⠰⠆⠀⠀⠡⢀⠀⠠⠄⠀⡠⢟⣵⡇
⠀⠀⠀⠀⠀⠀⠸⣎⣗⣬⡓⠦⣄⡀⠀⠀⢀⠀⠀⢀⡐⠈⠀⠢⡀⠀⠀⢂⠀⠁⠙⠀⠄⢀⠀⠀⢀⣀⠤⢒⣩⣶⡿⣾
⠀⠀⠀⠀⠀⠀⠀⢻⡯⣿⢿⢳⣶⢭⣝⣒⠦⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠴⠶⢒⣋⢥⢴⣺⢿⣴⡿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣵⣺⣟⣾⣷⣟⡯⣟⢷⣖⣶⡶⣶⣶⣶⡭⣥⣴⣶⣆⡶⣾⣿⣿⢷⣿⣾⣿⣾⣯⣯⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣻⣿⡛⠿⣿⣿⣿⣙⠛⡓⣿⡏⠉⠙⢟⠀⠈⠛⠛⠋⠙⣿⠟⠋⣉⣭⡴⠿⠛⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣞⣳⡈⠀⡀⠙⠻⢷⣇⣀⡿⠀⢀⣸⠦⢄⢠⠀⢀⢠⠋⣸⠾⠛⠀⠊⠡⣀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣿⣄⠀⠀⠁⢸⡏⠉⠀⠀⡞⠡⢠⢂⠀⠋⠉⠉⠻⡃⠀⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣿⣦⣤⣨⣗⣶⠔⠚⢡⠈⠂⠁⣠⠎⠁⣠⣴⡥⣴⡶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡇⠀⡀⠀⠉⠐⢹⠀⠀⢠⣟⣿⣋⣯⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣬⠩⠙⣛⣛⠒⠒⠛⢓⣒⠛⠉⢩⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠐⠒⠒⠒⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄
⠀⠀⠀⠀⠀⣴⣦⣤⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠠⢯⣉⣭⠀⠀⠀⣴⠇⠀⠀⠀⠀⠀⠛⣋⣥⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⣘⠻⣿⡆
⠀⠀⠀⠀⠀⠏⢩⣴⡶⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣬⣿⣯⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠈⣻⣛⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⣀⣀⣘⣿⠗⠀⠁
⠀⠀⠀⠀⢀⣠⣴⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣆
⠀⠀⠀⢠⡿⢹⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠙⠷⠶⠶
⠀⠀⠀⠈⠁⢸⣿⣿⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢻⣿
⠀⢀⣤⣤⣤⣿⠏⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢸⣿⡀
⠀⠀⠉⠉⠉⠁⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠿⠧
⠀⠀⠀⠀⢰⣦⣙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⢿⡿⠂⠀⠀⢀⣄
⠀⠀⠀⠀⠀⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⢿⣤⣴⢿⣿⣿⣿⢶⣄⣾⠃
⠀⠀⠀⢀⣠⣿⣿⣷⡄⠀⠀⠀⠀⠀⠰⠟⠛⠋⢸⣿⣿⣷⣸⡗⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣿⣿⡇⠀⠉⠁
⠀⠀⠀⠛⢋⣿⣿⡟⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡿⣿⡇⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣧
⠀⠀⠀⢠⣾⠏⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢸⣿
⠀⠀⠀⣾⡏⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⣿⡇
⠀⠀⠀⠛⠁⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠙⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⡻⠿⢛⠿⢿⠿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⠒⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⣴⠦⠖⠒⠒⠤⢤⠀⠤⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀
⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠔⠀⢠⢄⠥⣽⠖⠋⠀⠀⠀⣀⠄⠒⠀⣷⢀⣴⠖⠈⠣⠰⠢⢤⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠴⢞⡁⣴⠢⠰⢲⣾⣁⠒⠂⠐⠁⠌⠀⣀⠐⠀⡈⠡⠡⠀⢀⠀⢀⠀⠨⢷⣀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⢀⡏⠀⠀⠀⠀⠀⣠⢺⠀⣤⡬⣤⡎⠳⡱⢑⠨⠌⠋⢻⣦⠃⠠⠃⠀⠀⠀⢥⡁⠂⠡⡈⢒⡐⠠⣐⢩⡈⠱⡄⠀⠀⠀⠀⠀⢳
⠀⢸⠱⠀⠀⠀⠀⢰⠃⠲⡼⢁⢀⣤⠹⣕⠬⡂⢕⠁⠂⣶⣏⣤⣴⡤⡪⢛⠑⠀⠂⣴⣾⢷⠉⣫⢮⣨⡯⣁⠚⠉⡄⠀⠀⠀⠀⢸⡇
⠀⠸⡆⠀⠀⠀⠀⢠⠀⡣⢀⡲⠬⠉⣺⠄⠠⣯⡙⠓⡆⡎⡀⠨⢖⠀⣠⢡⠎⠦⣠⠱⢼⢮⠆⣈⣿⢀⡁⢫⢡⠀⠇⠀⠀⠀⠀⣸⠃
⠀⠀⠙⣆⠀⠀⠀⠀⢣⡐⢖⢻⣒⣞⠍⡀⢲⡐⡦⣰⠿⢑⠐⡔⠡⡚⠀⠀⠀⠔⠁⡼⠶⡎⢞⠂⠉⠀⠳⣤⢠⠜⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠈⠳⢤⡀⠀⠀⠙⠪⣄⠪⠁⠀⠁⠉⢀⢸⠃⢉⡆⢄⢧⡞⢈⠰⠀⠑⠌⢓⠁⠊⢡⣐⢙⡓⣤⡧⠗⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠁⠒⠤⢤⣬⡁⠶⠤⢬⢑⡁⢂⡸⠢⠉⣂⣩⡀⠄⠅⢄⣀⡯⠷⠚⠉⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢤⣤⣀⣉⡉⠙⠀⠒⠛⠒⠚⠑⠒⠒⠒⠛⢋⣈⣁⣀⣤⠤⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⢐⠈⠆⡀⣀⢀⣀⣀⠀⢀⡤⡀⢀⠀⣀⠆⠀⢢⠓⡆⣠⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢄⣤⠄⠀⠳⡀⠀⠀⠐⠒⠂⠓⠀⠊⠐⠛⠂⠐⠓⠒⠒⠚⠒⠒⠀⠚⠒⠈⠓⠒⠚⠐⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠒⠂⡁⠁⡘⠋⠁⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⢤⠀⢀⠤⠤⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣷⣟⠀⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠦⠠⠄⠂⠻⠏⠁⠀⠤⠘
⠀⠀⠀⠀⡼⡀⠀⠀⠀⠀⠀⣠⠄⢒⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠂⠄⠀⠈⠔⠀⠀⠀⠀⠀⠀⢈⢷⡀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⢋⡆
⠀⠀⠀⡀⠀⠀⠤⠤⠤⠠⡆⡄⠀⠀⠀⠀⠀⠈⠘⠧⣀⠀⠈⠁⠀⡒⠀⠀⡀⠀⠠⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⡢
⢐⡒⠒⠀⠈⠂⣀⣉⡉⡉⢋⡙⡠⢠⣷⣶⣲⣲⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⢸⠀⣿⣶⣶⣧⣀⡀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠐⠒⡒⣒⡒
⠀⠀⠀⢰⣞⠃⠈⠁⠀⠀⠠⠞⠒⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⢹⣷⣏⣁⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣶⠖⠀⠀⠀⡶⠉⠢⡄
⠀⠃⠂⢸⣿⠀⡖⠒⡶⢀⢀⢀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⣈⣙⢿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠻⠏⡄
⠀⢸⣬⢤⣳⣶⣿⣻⣻⡓⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣯⣿⣽⣿⣿⣧⣀⡀⠀⠀⠀⢀⣀⣀⡤⠖⠛⠛⠛⠿⢋⠐⣀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠍⠉⠭⠭⠭⠽⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠽⠟⠲⠛⠛⠙⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣋⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠶⠶⠶⣶⡶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⢶⣶⣶⡾⠿⠿⣦⣶⠿⠻⢿⣶⡿⠻⢿⣭⣭⣉⣉⣩⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡅⠀⠀⣿⠃⠀⠀⢰⣿⠀⠀⣾⣏⣉⣉⣉⣉⠁
⠀⠀⠀⠀⠀⠠⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣻⣇⠀⠀⢿⡇⠀⠀⣸⡇⠀⢠⣿⣋⣉⣉⣉⣹⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡄⠀⢸⣷⠀⠀⣿⡇⠀⢸⣿⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⡿⠀⠀⣿⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣾⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣿⡇⠀⢰⣿⡀⠀⠀⣼⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠉⠁⠀⠀⣿⡇⠀⢸⣿⠻⣷⣴⡿⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⡿⠁⠀⠀⣠⣴⡾⠟⠻⣷⡀⠀⢿⡆⠈⢻⣧⠀⠀⣰⡿⠁⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢰⡿⠁⠀⢠⣾⠟⠁⠀⣠⣴⣿⡇⠀⠸⣷⠀⠀⢻⣇⣴⣿⡿⠿⠛⠛⠛⠛⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣠⣾⡇⠀⠀⣿⠃⠀⢀⣾⠏⠁⣼⡇⠀⢀⣿⠀⠀⢸⣿⠋⠁⢀⣀⣤⣤⣤⣀⡀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢰⡿⢿⡇⠀⠀⣿⠀⠀⠸⣿⠀⣰⡿⠀⠀⣼⡟⠀⠀⣼⣧⣴⠿⠛⠋⠉⠉⠉⠙⠻⢷⣄⠀⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠸⣿⡀⠀⠻⣷⣄⣠⣿⡿⠋⠀⢀⣼⠟⠀⠀⣰⡿⠋⠁⣀⣤⣶⠾⠿⣷⡄⠀⢸⣿⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⡿⠶⠶⠿⠷⠶⠿⠿⠶⠶⠾⠿⠿⠿⠶⠶⠶⠿⠿⠶⠶⠾⠿⠷⠶⠾⠿⠿⠶⠶⠶⠿⠷⠶⠿⠿⠶⠶⠶⠶⠾⠿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠆⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⣎⢱⡀⡠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⣀⣁⠀⠀⠐⡉⠀⠀⠈⠁⠒⢤⣀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠷⢄⠤⢃⣀⡄⠀⠀⠀⠀⠀⢑⠠⢀⠀⢀⠖⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣩⠁⠀⢸⡈⢧⡀⠀⢀⡰⠀⠐⠃⢀⠕⠁⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠃⠀⠀⠀⡇⢇⡱⢦⡀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢀⡤⠒
⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠘⣴⣍⠃⠀⠀⠀⠀⠸⡈⠓⢆⡈⠓⠠⢀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⢀⠃⣅⠀⠀⠀⠀⠀⠀⢣⡈⠢⡀⠀⠀⠀⠀⢱⠀⠀⠉⠒⢧⣠⠁⠑⠒⠂⠤⠄⠠⠤⠗
⢀⠄⠒⠉⣀⠃⠸⠒⠒⠀⠀⠠⠎⠈⠢⢈⠒⢄⠀⠀⠀⢣⠀⠀⠀⡧⠤⠭⡁⠐⠂⠂⠀⠀⠀⠀⢀⠄⢀⠂⣄
⢷⣀⡠⠂⠦⢄⠦⢀⡀⠀⠀⠀⠀⠀⠀⠘⠱⢄⠉⠢⣀⠀⢣⡀⡐⣠⠤⠀⠟⠀⠀⠀⣀⠀⡀⠀⠘⢠⣾⣿⣿⡀⠀⡀
⠀⠉⠀⠀⠀⠀⠀⠀⠉⢲⢄⠀⠄⠀⠀⠀⠀⢀⠉⠢⡀⠑⠤⡑⢶⡈⡄⠀⠀⠀⣠⠚⣤⠟⠙⢆⠀⠻⠿⠿⠛⠜⠋⡜⢴⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢺⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠘⡒⢄⠈⠑⠦⣉⠒⠂⠚⠒⠋⠀⠀⠀⠈⣆⠔⡄⠀⡀⢆⠐⠷⣿⢆
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠁⠀⠀⠀⠀⡀⠀⡔⢷⢯⠞⡤⠑⠢⡀⠀⠙⠲⢤⣀⠀⢰⠂⠒⢲⣹⣂⢡⠤⠂⠀⠑⢶⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢎⡹⢽⠁⠀⠀⠉⠓⢄⠀⠀⠈⠉⢚⢆⠠⠋⠀⠸⣑⣄⡠⠢⢐⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⡒⠀⢄⣀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠑⢢⣀⡐⠉⣑⢠⡀⠀⢁⠩⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠉⠑⠶⠄⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠋⠘⢤⠠⠙⢶⣶⣆⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⡘⠀⢀⡏⠉⠐⠒⠀⠤⠄⢀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠋⡭⠣⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠜⡋⠖⢀⣒⠶⠆⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣦⡀⠀⠀⠀⠀⡲⠞⡛⠶⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠋⠙⠋⠐⠻⠾⠫⠆

⠀⠀⠀⠀⠀⢠⡄⢠⣶⡆⠒⡀⠀⠀⠀⠀⠀⠠⠀⢲⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⣿⣷⠘⢂
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣁⡐⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⣺⢿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠛⠈⠛⠃⠙⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠋⠘⠛⠛⠘


⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⠒⢠⣶⡦⢀⡀⠀⠀⠀⠀⠀⠀⣤⢰⣾⣦⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣃⠚⠀⠀⠀⠀⠀⠀⠐⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⢿⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠘⠛⠃⠛⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠙⠛⠛⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠲⢄
⠀⠀⠀⠔⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠰⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⠲⠤⠴⠒⠀⠀⠀⢄⣀⣀⠤⠜
⠀⢣⡀⠀⢀⣀⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠉⠉⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣤⠤⠒⠒⠒⠒⠺⢟⠛⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠲
⠀⠀⠀⠀⠀⠀⠀⡗⠈⠁⠋⢶⠆⠀⠀⠀⠾⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⡦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠥⠐⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢀⣀⣀⣀⣀⡠⠚⠁⠀⠀⡌⠀⠀⡠⠂⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠈⠉⠁⠀⠀⠀⠀⡠⠊⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⢤⡤⠤⠒⠉⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡷⢤⣀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣿⠀⠈⠉⠂⢎⡛⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣿⠀⠀⠀⠀⠀⠉⠓⠪⠽⣛⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣁⣀⣀⣀⣀⣿⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠬⠿⢟⣚⡿⠭⠭⢭⣖⣒⣒⣂⣐⣀⣀⡬⠤⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⢈⠈⠉⠄⠁⢤⣀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡀⠀⡀⢄⣀⣀⡀⡄⠒⠶⠀⠈⣊⠙⠛⠓⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠚⠈⠀⠋⠀⠀⠀⠀⠀⠀⢔⣌⠙⠳⠻⣆⣈⣠⢈⠀⠀⡀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢀⢶⠗⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠉⠒⠀⠉⠳⠲⠿⠓⠂⠀⠁⠚⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠮⡏⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠄⠀⣀⣀⣀⡂⠠⢀⠀⠀⠀⠀⠀⠀⠈⠑⠆⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠔⠠⠀⠑⠀⠀⠀⠀⠀⣀⡀⠐⠚⠁⠈⠤⠀⠉⠀⠀⠀⠂⣄⠀⢀⠀⠹⡃⠹⡄⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠠⡳⠈⡠⠠⠓⠀⠊⠀⠀⢠⠀⠀⢀⠄⠘⠰⢋⠔⣀⠄⠌⢤⠡⣄⠩⠒⢦⣄⣠⡇⣀⢛⡆⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠠⢀⣌⠄⠀⠀⠊⠀⣀⠈⠀⠀⠀⢁⠎⠁⠀⠀⠀⠀⠀⢀⠀⡀⠈⣿⢠⡿⢨⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠂⠄⠀⠀⠀⠀⢡⠀⠀⠀⠀⡌⠀⠤⢂⠴⠚⢀⠀⢸⢠⡁⢿⡄⢈⢁⣹⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⢲⡄⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠈⠀⡀⠀⠁⠁⠀⢠⠀⠀⡴⠄⣂⠈⢁⣼⠁⣸⣾⡾⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠡⡙⠂⠀⡀⢠⠀⠀⠀⠀⠘⠈⠀⠀⠐⡑⠁⡆⠀⡰⢠⡜⠃⠀⠈⠉⠀⠁⢠⠇⢀⠇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠵⣪⣔⠉⠀⠀⠀⠀⠀⠄⢸⡧⣢⠊⠀⠀⠀⠀⠀⠀⠄⣼⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⢀⠀⠀⠀⠀⠲⢶⠶⠶⠐⠒⢐⡊⢍⠃⠑⡀⠀⠰⠄⠀⣄⣾⠃⡏⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠀⠁⠀⠈⠋⠉⠀⠀⠠⠀⠈⠁⠘⠁⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠒⢶⡀⠀⣀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡐⠅⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⣿⣶⡖⠀⢀⣔⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣤⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⡀⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡟⣉⡻⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣻⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣲⣾⠿⢿⣛⣿⣿⣯⣿⣯⡭⡭⣍⣽⡿⣿⣯⣽⢿⣿⡿⣟⠿⠿⣷⣶⣦⣤⣀
⠀⠀⢀⡤⢖⣯⣷⣛⣩⣥⣶⣿⢿⣿⣿⣿⣯⣿⣿⣹⣾⡿⣗⣶⢖⢾⣻⣷⣛⣿⣵⡵⣶⣶⣮⣭⡛⠾⣽⡲⢄⡀
⠀⡴⣫⣾⢛⣽⣿⣿⣵⣿⡯⣺⣿⣿⣿⣿⡿⣿⣷⢿⡟⡟⣿⢿⡞⣟⣽⣵⢞⣯⣽⣿⣺⣿⣿⣿⣽⢢⣌⠛⣷⣝⢦
⡼⣱⣿⡱⡿⢿⡶⣿⣿⣿⣟⣽⣿⣿⣿⣯⣿⣿⣿⡜⣺⣿⣿⡞⣿⣻⡿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⢽⣣⣈⢿⣎
⡇⢿⣇⠓⠻⣿⣿⣶⣽⣿⢿⣿⣿⣿⣮⣛⣿⣿⣆⣿⣿⣟⣿⣞⣯⣯⢻⡜⣿⡿⢿⣿⣟⣿⡿⣿⣿⣯⣾⣿⣿⣸⡿
⠹⡜⢿⣄⠀⠻⠿⣿⢿⠿⣯⣿⡿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢗⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠿⣫⡿⢣
⠀⠈⠢⣝⠷⣤⣀⠻⣿⣭⣷⣾⣿⣿⣾⣿⣿⣿⣿⣟⢭⣝⣿⣿⣿⣿⣽⣿⣷⣿⣿⣿⣻⣯⣿⡿⠟⣋⣤⠾⣫⠔⠁
⠀⠀⠀⠈⠙⠲⢽⣛⠶⣬⣍⣉⡉⠛⢿⣷⣾⣽⣿⣿⣛⣥⣿⢿⣿⣿⣿⣿⣽⣿⠿⢛⣛⣡⣤⠶⣛⡯⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠭⠭⣛⣓⣺⡿⠿⠶⠦⠤⠤⠤⠭⠥⠶⠶⢶⣖⣛⣛⠭⠭⠓⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠙⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠈⣧⠖⠛⠛⠓⠢
⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠰⠐⠀⠀⠀⢰⠁
⣏⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢚⠇⠀⢠⠃⢀⣠⡀⠀⠀⢀
⠹⡄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢔⡥⠞⠉⠀⠉⠒⠚⠁
⠀⠘⢦⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣤⡴⠋⣠⠴⠲⠄
⠀⠀⠈⠳⣄⠀⠈⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠴⣏⠀⠸⠂⠂⠀⠞
⠀⠀⠀⢀⣘⣦⠀⠈⢧⡴⠲⣄⣞⠉⡷⠶⢤⡞⠁⣹⠶⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠐⢂⡼
⠀⢠⠞⢉⡕⠒⠓⢄⠀⢻⡖⢻⡍⠳⡅⣢⣀⣳⠖⢋⠑⢾⠀⣸⣆⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⣀⡠⠤⠒⠊⠉
⠀⠸⢤⡈⠢⣴⡚⠛⢷⡤⠗⠋⠙⣄⣘⡄⢀⠽⠖⠧⣴⡮⠭⣍⠓⠟⠱⣄⣤⣀⠀⡴⢃⡠⠖⠋⠁
⠀⠀⢸⡇⠀⠀⠉⠒⠰⢏⣙⠉⡏⠁⠈⠹⡇⠀⢀⣴⠋⠐⠀⡠⠗⠚⠒⡅⠀⠈⣟⡴⢯⣀
⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⣽⣉⣉⡀⠙⣏⠉⠝⢂⠀⢂⠟⠒⠚⡏⢱⠀⠘⡆
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠒⠓⠒⠓⠒⠒⠒⠛⠉⡤⠞⠁
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣤⣀⣀⣀⠀⢀⣀⣀⣤⠴⠚⠉
⣿⣿⡷⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⢩⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠰⠞⠃⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⢶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⣀⣀⣤⡶⠄⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢀⡀⠀⠈⠛⠻⠿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢀⣾⣧⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣰⣿⣿⠏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠻⠶⣤⣤⣤⣤⣤⣤⣤⣴⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⣰⣿⢋⣿⡏⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣀⣾⡿⠃⢸⣿⠃⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣤⣾⠟⠋⠀⠀⢸⣿⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠟⠋⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠒⠀⠀⠉⠁⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⢠⠀⠃⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠁⠀⢀⠁⠀⠀⠀⠑⢄⠀⡐⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⢈⠂⠀⠄⠂⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⡔⠉⠉⠀⠀⠀⠠⠚⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠁⢀⡤⠄⠀⠀⠀⠀⠀⠠⠂⠀⠊⠀⠀⠀⠰⢶⣄⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠄
⠀⢀⠀⠐⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡈⠀⠀⠐⠀⣀⠀⠀⢒⠁⠊⡀⠀⠀⠀⠀⠀⠈⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⢄
⠀⠁⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⡠⢦⠀⠹⢀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠙⠓⠀⠀⠀⠀⠐⠂
⠀⠀⢇⣀⠔⠀⠀⠀⠀⠀⠀⠀⠔⠁⠈⠁⠈⠀⢠⠀⠂⠁⣂⡤⠔⠒⠂⠀⠠⠄⠀⠀⠠⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⡀⠀⡠⠋⠀⠈⠀⠀⠀⠀⠀⡠⠂⠀⢂⠊⡡⢀⡈⢙⡉⠐⠠⢄⡁⠊⠻⣔⡂⠓⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠆
⠀⢀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⣀⣀⠔⠉⠀⠀⢀⢂⠀⣰⣾⠋⠴⠋⢔⡡⠔⠢⢭⡀⢙⢦⡙⢆⢳⡈⡐⢄⠀⠀⠀⠀⠀⠠⢀⠐
⠀⠐⡀⠀⠀⠀⠀⠀⡠⠤⠤⠊⠀⠀⠀⠀⠀⠀⡄⣚⠐⡓⡁⢐⢀⡆⠁⠀⠀⠀⠀⠙⠆⠁⡙⠄⠀⡇⠉⠢⠁
⠀⠀⢇⠠⠄⠀⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠆⡀⠀⢡⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡄⢨⠸⠀⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠁⢠⠜⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠇⡈⠀⠀⠀⠀⠂⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠀⡂⠀⠀⠀⠀⢸⠇⠼⠸⡆⠖⠸⠄⢄⠀⠀⠀⠀⠀⠀⠀⢈⠰⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⡰⠀⠈⠐
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠁⠂⠀⠀⠀⢀⢀⢸⠀⡇⠀⢴⠀⠢⠡⡀⠀⠀⠀⠀⠀⠈⢶⠀⡀⢣⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⢀⠧⠀⠀⠑⠀⠤⠀⠀⠀⠀⠀⠀⠘⠘⠎⢧⠹⡀⢌⢧⡀⠐⠔⠄⠀⠀⢀⢡⠆⠎⢠⠆⡌⠀⠌⠐⠀⠀⠀⠈⠂⠤⢀⡀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠘⠀⠀⡈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠦⣘⠣⡀⠈⠶⠱⠆⡐⠈⢀⠀⠥⠂⢈⡄⠈⠸⠁⢊⠈⠀⢀⢤⣄⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢢⡀⢛⠢⠙⢀⣀⡚⠨⠤⠞⠵⠞⣠⠂⢊⠔⠁⠀⢀⠁⠈⠛⢻⠀⠀⠀⠠⠁⠛⠛⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠙⠒⠒⠈⣊⡤⠚⡋⢀⣈⠅⠂⠁⠀⠀⠀⢸⠀⠀⠀⢦⡀⠀⡠⠃⠀⠀⢠⠂
⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠄⠀⠀⠀⠈⠉⠉⢁⠀⠤⠀⠀⠠⢄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢄⠀⠑⠠⠀⠀⠀⠠⠐
⠀⠢⠅⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢠⠖⠂⠒⠢⡀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡑⠀⠀⠀⠀⠀⠇⠀⠀⠸⢄⠀⠀⠀⠀⠈⢀⠠⠤⠄⡀⠀⠀⠀⠉⠀⠀⢠
⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠃⠀⠀⠀⠀⢠⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠰
⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠤⠊
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠉⠙⠒⠂⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠚⠛⢳⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠻⠐⠀⠐⠸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢻⡟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢡⣀⣠⡍⠀⣤⣀⣀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡷⠀⠈⠉⠉⠀⠘⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣄⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠡⠤⠀⠀⠤⢹⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡧⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠛⠛⠃⣾⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡟⠀⠀⠀⠹⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠏⣀⠈⠀⢀⣈⡩⠿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠒⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢸⣃⠠⠤⡀⣬⣠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠐⠼⠃⡀⠊⢰⣷⡌⡰⠀⢀⡀
⢀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠟⠉⠁⠁⢰⢈⠆⡴⠓⡆⢜⠩⠀⠁⠀⣆⠠⡀
⠀⠀⠀⡈⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠐⠁⠄⠎⠀⠚⢣⣨⠀⠄⠀⢰⣀⡰⢄
⠶⠞⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣀⠀⣀⣦⠀⠃⠃⠞⡀⠄⠺⢃⠔⢤⡓⠄⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⡮⢊⠀⠔⠂⠄⢀⡌⡀⠀⠦⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣀⢤⡞⠁⢀⠀⠁⠀⠁⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠇⠸⠀⡴⠁⠀⠀⠀⣴⣦⢚⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠘⠋⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢁⠃⠀⠀⠀⠂⠳⠀⠰⠃⢈⣆⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡆⠘⢯⠀⡁⠀⠀⠀⡇⠠⢀⠀⢠⠳⡷⣄⠈⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠉⠁⠠⡇⢀⠆⡼⠀⠀⠀⢈⠆⠔⠉⠂⠈⡟⣀⠀⠆⠰⠈⡛⠻⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⢀⠐⠚⢩⠂⠀⠐⢱⠛⠚⡁⠊⡐⠁⠀⡂⠰⡇⠂⢠⢄⠤⢯⡠⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⡸⠃⠀⢀⠼⡄⡄⠀⡃⠀⠁⠀⡖⠡⢦⠘⠀⠀⠏⠎⡇⠉⠐⠄⣘⡁⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⡀⠀⣁⠆⠐⠉⠀⠍⢀⡐⢧⠰⡀⠀⠀⠘⠺⠀⠀⡜⠠⡄⠈⢶⡄⠸⠑⠐⠡⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⢰⣏⡼⣨⢂⠀⠀⠀⣱⡈⡇⠡⠀⠀⡠⠘⡐⠞⡁⢠⠹⡀⢰⢥⠘⢠⠐⠀⡋⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣾⠤⠊⠐⡉⠄⠀⠀⠈⠽⡇⡃⠅⠀⠈⠀⣆⠉⡞⠀⣠⣜⣀⣎⡤⠗⡄⢚⢸⠄⠈⠂⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⡄⢠⣡⡤⣤⠂⠀⢢⠀⠀⠀⡴⠡⠁⠐⠠⠠⢀⡿⠃⣷⢀⣐⡀⠀⡀⡠⠊⠀⠲⣄⣯⡰⠜⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠘⢄⢲⠰⡇⠀⢀⢰⠆⠐⠀⠤⠀⢀⡋⠉⢁⡈⠀⠠⠟⢧⡞⠁⠁⣡⣛⠃⠀⠄⠨⣟⠏⣆⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⢈⠓⡹⠀⠚⠦⣀⡴⡲⠶⡒⢇⠚⠛⠂⢶⣈⠣⠀⠀⣀⡴⠊⠁⠩⠀⠀⠀⠓⢤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⡳⢬⣀⡠⠶⢦⡈⠀⢀⣘⠃⠄⣀⡤⢈⠀⠈⠁⠀⠀⠀⠇⠀⠀⢈⣃⣠⣴⠾⢛⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠉⠛⠿⠶⣶⣤⣌⣁⣀⡀⡀⣇⢀⢀⢁⣐⣦⣤⣌⣤⣴⡶⠿⠛⠋⢑⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⢐⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠁⠐⢂⠨⠀⠈⣂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣈⡑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠴⠊⠁
⣿⢿⢿⠿⣿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣻⣿⣟⣟⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣭⣭⣛⣛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⢹⡟⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣶⠀⠀⢰⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣷⠄⠀⠄⢀⣀⠀⠀⠀⠀⠸⢯⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⠋⠉⠉⠁⠀⠀⠀⠀⠛⠙⠓⢶⣤⣄⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢟⣳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⡵⠄⠀⡄⢠⣶⠾⠿⠀⠀⠀⠀⠀⠀⢠⣯⣡⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣀⠀⢲⡄⢠⠀⠀⠀⠀⠀⢀⡀⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⣿⣷⠆⠀⠰⠆⠀⠀⠀⠀⢀⢖⣥⣶⣶⣶⣮⡂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⣠⣶⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢃⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣾⠀⠀⢠⠴⠶⢤⣾⠟⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⢡⣾⣿⣆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠁⠀⢀⠠⠋⠀⠀⠀⠀⠀⠀⠀⢠⢙⣛⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣄⠀⠀⠀⠀⠚⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⢀⠂⢀⡄⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠘⠃⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⢶⡆⠀⠀⠘⠿⠋⠉⠁⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣸⣷⠀⠀⠀⠀⠀⠈⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⢿⣧⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣦⣼⣿⣶⣤⣿⠄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣿⣿⡆⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣯⣰⣿⣿⣷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⠏⠁⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠛⢻⡿⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠈⠋⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⣶⠶⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠰⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣶⣾⣦⣄⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠐⡒⠂⡂⠄⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠤⠖⠦⡄
⠀⠀⠀⠀⠀⣁⢘⠁⠀⡁⠀⠀⠀⠀⢐⣤⣠⠶⣤⠀⠀⠀⠀⣉⢓⠣⠄⠘⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡚⢀⡀⣀⠜⡆
⠀⠀⠀⠀⠠⠆⠠⠆⢂⣁⠀⠀⠀⠀⠈⠉⠀⠲⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⡀⠀⠀⡀⠶⠀⠀⠀⣀⠐⡁⡁⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠠⠄⠰⠀⠀⢠⠀⠀⠀⠀⠀⠀⡠⠜⠁⠀⠀⠀⠁⠉⠀⠁⢸⢙⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠤⠀⠀⠂⠁
⠀⠀⠀⠀⠈⠓⡞⠠⠀⡈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⢸⢀⠀⠀⠀⠀⠑⠆⠰⠀⠀⠀⠀⠀⠄⠀⢨⠀⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠠⠎⠒⠢⣀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⢔⠈⠁⡂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⡴
⠀⠀⠀⠀⠀⡁⠂⢘⠀⠉⠀⠀⠀⠀⡀⠁⠈⡀⠀⠀⠀⠀⠀⣃⠀⠀⣘⠀⠀⠀⠀⠈⠀⠀⠀⠈⠠⠀⠀⠀⢎⢰⡀⢀⢓
⠀⠀⠀⠀⠀⠡⠀⠀⠠⠀⠀⠀⠀⠠⠐⠄⠀⠐⠀⠀⠀⠀⠀⠈⠎⠹⠁⠀⠀⠀⠀⠸⠀⠀⠀⠴⠀⠀⠀⠀⡄⠃⢠⣈⣌⡅

⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⡤⠤⠤⠤⠤⣄⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢌⡠⠄⠀⠀⠀⠀⠀⠈⠰⠤⠑⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⡀⠄⠀⠄⠦⣠⠀⠀⠀⠀⣟⣟⣟⣻⣳
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈⠀⠀⠀⠀⠀⢶⣳⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⡥⣄⡴⣄⡤⠄⠀⠀⠀⠤⠠⠤⢭⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠅⠀⠀⡀⠀⠀⠀⠀⡀⢸⣁⢂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠀⡨⠅⠉⠉⠉⠁⠀⠀⠀⠁⠀⡹⢝⠝⠃
⠀⠀⠀⠀⠀⣀⡃⠂⣤⡀⠀⠀⠀⠀⢈⣉⠂⠈⠂⠀⠀⠀⠀⠑⠈⠉⠀⠀⠀⠀⠀⠀⠀⠂⠤⠠⠄⠀⠀⠀⣰⡉⡈⣰⣀⡀
⠀⠀⠀⠀⠀⠑⠷⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠒⠒⠒⠒⠂⠀⠀⠀⠠⠴⠳⠚⠞⠆⠀⠀⠀⠀⠰⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣠⠀⠀⠀⠀⠀⢀⡤⠒⠀⠐⠒⠦⢼⡆⠄⠴⠶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄
⠀⢶⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⠀⠀⠀⠀⡏⢷⠤⠖⢊⣅⠀⠀⠀⢀⣠⠞⠉⠙⠁⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠉⢿⣆⢦⣴⣶⣷⣶⣦⣂⡶⢠⡄⠀⠀⠈⠻⣷⣄⡀⠀⠈⠓⠄⠐⢈⡉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⢰⣏⡸⠛⠋⠙⣛⠻⠷⣾⣿⣿⣃⡀⠀⠀⠈⠻⣍⠐⠂⠀⢤⡼⠉⠙⢿⡆⠀⠐⠂⠀⠀⣤⣤⣤⣤⣤⡴⠿⠂
⠀⠀⠀⣾⠈⠻⣦⡀⠀⠀⠉⠒⢮⡛⢿⣿⣿⣆⠀⠀⠀⠀⠈⠐⠀⠸⡇⠀⣀⣠⡟⠀⠀⠀⠀⠀⠉⠉⠀⢀⠔
⠀⠀⡘⡛⠂⠀⠈⠻⣆⠀⠀⠀⠀⠙⢦⠙⣿⣿⣄⣀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⡀⢀⠀⠀⠀⠀⠀⠀⠃
⠀⢠⠁⠀⠀⠀⣀⡴⢻⡿⣆⠀⠀⠀⠈⢷⠘⣿⣷⣁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡍⠀⠘⠓⢶⡄
⠀⣾⣄⡤⣾⡟⠉⠀⠀⣿⢹⡄⠀⠀⠀⠀⢷⠘⣿⣿⡀⠀⠀⠀⠀⠀⣀⠞⠘⠿⢿⣿⡏⢠⣀⣀⣺⠋⠁
⢮⣿⣿⢿⣿⠀⠀⠀⠀⣿⡆⣿⠀⠀⠀⠀⠈⡄⢹⡏⠓⠀⠀⠀⠀⣿⣴⢆⠀⢀⡀⣮⠁⣼⠋⠉⠉⢦
⠀⠉⠁⠘⠋⠀⠀⠀⠀⣿⡇⢹⡆⠀⠀⠀⠀⠉⠈⣷⡀⠀⠀⠀⠀⠀⠀⠑⠀⠋⡠⠉⠉⠉⠒⢄⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡞⠇⠀⡈⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⡈⠀⠀⢠⠁⠀⠀⠀⠀⠀⠈⠂⠄⡀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠐⠀⠉⠐⠂⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⢀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢢
⢀⣴⣾⣭⣒⠤⣀⠀⣾⠏⠐⠂⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⣠⠟⠉⠁⠉⡙⠒⡤⣄⠀⠀⠀⠀⠀⠀⠁⠺⣓⣄
⠘⢿⣿⡿⣿⣿⣦⣝⣿⣄⡄⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠁⣤⡸⣿⣮⣻⣶⣤⣤⣄⡀⠀⠀⠀⠑⠅⠠⣀
⠀⠈⠻⣿⣎⢿⣿⣿⡟⣧⡄⢈⡥⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠈⠀⠀⠰⠀⠠⡄⢰⡞⣿⣷⣿⡌⠻⣿⣿⣷⣄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠘⢿⣷⣝⠯⣴⡉⣵⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠐⠆⠀⠲⣀⣀⣈⣿⣿⣿⣼⣻⠀⢿⣿⣿⣿⣷⣄⠀⢀
⠀⠀⠀⠀⠈⢻⡟⠲⢱⣉⣀⠈⠻⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⢠⡄⠁⢙⣫⣿⣍⡟⣷⣸⠀⠸⣿⣿⣿⣿⣿⣦⡼
⠀⠀⠀⠀⠀⠈⠑⢬⠮⣭⣿⡇⠀⠀⠈⠙⠛⠻⠿⠾⠶⠶⠿⠾⠶⢾⣷⡀⠀⠠⠀⠀⢰⣏⣽⣯⣯⠇⡇⠀⠀⠉⠻⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠹⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠘⠿⠿⠸⠀⠸⡀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠈⠀⠂⣤⠀⣤⣕⣤⣀⣀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⣦⡾⢦⡈⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡚⣤⡗⣿⣿⣷⣦⠙⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠺⠿⣿⣿⡄⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠹⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠙⠛⠷⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣶⣆
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶
⣿⣿⣥⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢰⠞⠓⢦⡀
⠀⠀⠘⣆⠀⠀⠉⠢⣀
⠀⠀⠀⠈⠳⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⢀
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠣⡀⠀⠀⠀⠀⢠⠖⢢⣤⣤⣴⡇⠄⣸⣉⣹⠟⡉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠙⢦⣼⠋⢳⣼⠦⣰⠁⢆⡈⣟⡾⠁⡄⠈⣣⠴⠚⠭⣿⠓⢦⡤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣛⡷⣄⡀⠙⢷⠋⣻⣦⠈⠳⡏⠉⢉⡷⠚⠲⣾⠗⠋⠉⢢⡨⠷⠾⡀⠀⠀⢹⣍⡉⠙⢲⡄
⠀⠀⠀⠀⠀⠀⢰⡏⠀⣞⠀⣠⠤⠿⣦⡤⠟⠁⣀⣻⠦⠞⢦⡏⠐⠐⣰⣃⠈⣁⣤⣯⡀⠀⠄⢸⠖⢛⠛⣄⡹⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠘⠷⢤⡌⠙⠓⠲⠤⠄⣽⣙⣉⢹⡅⠀⠀⠀⠳⡶⠊⢁⡀⠉⣧⣐⣂⣸⣀⣴⡥⠬⠴⠚⠉⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠦⠤⠤⣤⡤⠤⠤⠶⠒⠋⠉




⣿⣿⠟⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠓⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠒⠀⠆⠀⠀⠀⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠘⠤⢀⠄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠀⠀⠀⠀⢆⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠖⠣⢤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠨⠤⢄⡉⠢⢄⠀⠀⡼⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⣠⠖⢩⢁⡠⢤⡀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢿⣦⠀⠡⡜
⠀⠀⠀⠀⠀⡜⠁⠀⠼⠩⠄⠀⠀⠙⡄⠰⠢⠽⠀⠀⡀⠀⠀⠀⠀⢰⣿⣷⠀⢀⣴⣶⣮⣻⣷⣔
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠇⣤⣤⣤⣌⣤⠄⠀⢀⠀⣾⣿⡟⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡏⠀⠺⣿⣉⠉⠉⠉⠉⠒⠒⠒⠾⠿⠯⢿⣯⣱⡀⠏⠈⠛⡿⠁⣿⢻⣿⣿⡿⢻⣿⡿⠀⠀⠀⣠⣴⣤⡢⡀
⠀⠀⠀⠀⡇⠀⠀⠀⡹⠛⠶⠤⣀⣀⣀⣀⣤⡤⠤⢤⠖⠫⣇⠴⣦⣞⣀⣰⣿⣦⡈⢁⣠⣾⣿⣷⣶⣶⣿⣿⣿⣿⣿⡜⠄
⠀⠀⠀⠀⢷⠀⠀⠀⠒⠔⠒⠤⢄⣀⠀⠀⠀⠀⡰⠃⠀⡴⠁⢨⣿⣿⡛⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⡄
⠀⠀⠀⠀⢸⠀⠀⠀⢠⠃⠀⠀⠀⠀⠈⠁⠀⠀⡁⠀⠘⠀⠀⢸⣿⣟⠷⠷⢿⣿⣿⣿⡟⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⡄
⠀⠀⠀⠀⢸⠀⠀⡠⠋⠀⠀⠀⡀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠀⠈⡻⣿⣷⣶⣿⣿⣿⢟⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⡄
⠀⠀⠀⢀⣏⠤⠊⠁⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⡎⡀⠀⠀⢀⠊⠙⠛⠟⠛⠋⠁⠀⠢⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣷⡱⡀
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢸⠄⠡⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣼⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠠⠔⠁⢷⢺⣗⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣿⡏⠉⠻⣦⣤⣀⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⡸⠉⠛⠿⣿⠿⠿⠿⣟⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣟⡕⠁⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣷⣤⣴⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣠⣤⣠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⢃⠁⠄⡀⠀⠀⠀⠈⠙⠯⣟⡶⣥⣒⠠⣀
⠀⠀⠀⠀⠑⢄⠈⠒⠄⣀⣀⣀⣠⣤⡽⢿⣭⣻⣷⣮⣔⡠⢤⣀⣀⡀
⠀⠀⠀⠀⣠⡤⠧⡀⠐⠚⠻⣄⠀⠀⠀⠀⠀⠈⠙⠳⢭⣋⠗⡲⢭⣓⠿⢳⠶⣤⣀
⠀⠀⠀⡎⠋⠀⠀⠀⠀⠀⠀⠘⣧⣤⣴⣶⠖⣶⣶⣲⡶⠮⣿⣶⣥⣊⠝⡲⠦⣄⡛⠷⢀⡀
⠀⠀⠀⡗⢦⣠⣴⣧⣸⣄⣀⣠⡞⣎⠙⣷⢿⢿⣿⣽⣾⢾⣯⣽⣷⣿⣿⣶⣌⢐⠠⢦⡔⠠⠅⡐⠠⠀⡀
⠀⠀⠀⢡⠀⠉⠙⠛⠿⠶⠿⢦⣏⣧⣮⣮⣷⣷⣿⣷⣯⣩⣿⡿⠿⢿⡿⠻⠍⠓⠫⣒⠨⣐⠠⠀⠈⠀⠆⠇
⠀⠀⠀⠈⡍⠐⠠⠄⣀⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣀⠠⣴⣮⡏⠈⠐⠨⣐⣬⡇
⠀⠀⠀⠀⠘⣏⠩⡳⡦⠀⠩⡴⠶⣆⠐⠒⠒⣒⣶⣒⠐⠒⠒⣰⡞⠿⢦⠤⡺⡡⡞
⠀⠀⠀⠀⠀⠈⢮⣾⣮⣶⡾⠏⣂⢴⣦⣭⣟⠫⠉⣳⣬⠀⢙⢕⣪⣲⣿⣷⣶⠞
⠀⠀⠀⠀⠀⠀⠀⠑⢿⠐⠚⠂⠕⡪⠩⠩⢵⠫⠿⠿⣿⡛⣿⡛⠝⠂⢈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣖⡂⠀⠤⠤⠤⠤⠤⠤⠤⠀⠐⡒⣲⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠚⠛⠋⠁⠀⠐⠛⠚⠒⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⣀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣠⣤⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡅⠹⣿⣿⣿⡆⠀⣤⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢁⣤⡀⢰⣿⣿⣿⣿⠏⢨⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠙⣿⡿⢀⣾⣿⣿⣷⣤⣈⡉⠛⠛⠿⠿⠟⠛⢋⣁⣤⣾⣿⣿⣷⡀⢻⣿⡟⠁⣴⣿⣿⡇
⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢿⣿⣿⣿⣆⠈⠁⣼⣿⣿⣿⣿⣿⣿⣿⡇⢰⣶⣶⡆⢸⣿⣿⣿⣿⣿⣿⣿⣧⠈⠋⣠⣾⣿⣿⡿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠈⢿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⡿⠃⣴⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿
⠀⠘⢿⣿⣿⣿⡟⠿⣿⣿⣿⠿⣷⡈⢻⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⡟⢀⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⡿⢁⣼⣿⠿⠛⠋⣁⣤⣴⣿⣿⡿⠃
⠀⠀⠈⠻⣿⣿⣿⣦⣈⠙⢿⠀⣿⣷⡄⠙⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣇⠘⢿⣿⣿⡿⠃⣸⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⠋⠀⢈⣁⣤⣴⣶⣄⢹⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠛⣿⣿⣿⣷⣦⡀⠻⢿⣿⣦⡈⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣤⠉⠻⣿⣷⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⢶⣶⡶⢶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⣾⣧⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠈⠛

⠀⠀⠀⠐⢿⡶⠗⢀⡾⣦⠀⢰⣷⠀⠀⣾⠟⠓⢠⣷⣆⢰⡆⠚⢷⡿⠇⠀⢀⣴⠟⠻⠂⢰⡟⢳⡆⢠⡶⠛⢷⡄⢶⡄⣴⣦⢰⡷⠘⢿⡾⠗⢰⡆⠀⣾⡆
⠀⠀⠀⠀⢸⡷⠀⣾⣧⣿⡆⢸⣿⠀⠀⣿⡛⠓⢸⣏⠹⣾⡇⠀⢸⡇⠀⠀⢸⣧⠀⢿⡆⢸⡷⢾⡇⢸⣇⠀⣸⡇⠘⣷⡿⢻⣾⠇⠀⢸⡇⠀⢸⡟⠛⢿⡇
⣤⣤⣤⣤⣤⣧⣴⣿⣤⣬⣿⣼⣿⣿⣧⣿⣿⣿⣬⣯⣤⣽⣧⣤⣼⣧⣤⣤⣤⣽⣿⣿⣧⣼⣧⣼⣿⣤⣽⣾⣯⣤⣤⣿⣧⣼⣿⣤⣤⣼⣧⣤⣼⣧⣤⣼⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠐⠊⠙⢊⢙⠪⠁⠒⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⣾⣄⣂⡼⠄⠀⠀⠀⠠⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠇
⠀⠀⠀⠀⠀⠀⠰⣁⠀⣠⡶⠞⠛⠋⢈⣉⠁⠒⡲⠥⢶⠄⠀⣀⡷⠀⠀⠀⠀⠀⠀⢰⡇⡘⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠙⡻⠶⢋⣉⣉⣉⣉⣉⣉⣩⠶⠾⣟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠇⣎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢲⣤⣈⠉⠉⢉⣤⣴⠒⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⢰⣍⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⠙⠉⠁⠀⣉⣉⣁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡴⢿⣷⠁⠀⠀⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠰⠖⠛⠉⠉⠁⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⣰⠃⠹⣇⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⢀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠞⢹⠗⠲⢾⣯⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡰⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠈⢳⣌⡑⠲⠤⣀⠀⠀⠀⠀⠀⡼⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢠⠀⢢⠙⠦⣀⠈⠉⠓⠢⣤⠞⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣰⡦⢼⡆⠀⠈⠳⢄⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢨⡆⢰⠟⠀⠀⠀⠀⠙⠦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠆⠀⠀⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠞⢲⢶⡦⠯⣶⠾⠿⢳⠚⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⠹⡧⡜⠿⠗⠚⠻⢴⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢇⠱⡀⠀⠀⣀⠎⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠀⣸⠓⠊⢳⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢹⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣖⠊⠉⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠭⠬⠉⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠖⢾⣏⣙⣙⣋⣉⣉⣩⣋⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠒⠒⢠⡀⢁⠁⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⣄⠔⠋⠀⢁⠕⠢⠤⡀⠀⠀⠀⠀⣁⠐⠐⢦⡄⠤⠠⠀⣀
⠀⠀⠀⢀⠤⠖⠚⠉⠁⠀⠀⠀⠐⠁⠈⠀⢀⠂⠀⠀⠀⠐⠈⠁⠂⠂⠀⠈⠠⢄⠀⠀⠀⢄⠀⠈⠙⠂⠤⡀
⠀⠀⢐⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠐⢄⠀⠨⠒⠂⠀⡄⠈⠐⢀⢀⠠⠄⠒⠂⠀⠀⠈⠛⠩⡀⠀⠀⠀⠀⠈⡆
⠀⠀⠈⠓⢄⡀⠀⠀⠔⠉⠑⠁⠁⠀⠀⠔⡁⠠⠀⢘⡀⠀⠠⠐⠨⠀⠀⠀⡀⠈⠀⠀⠢⠀⠀⠀⠀⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠈⠑⠓⠦⣤⣤⣄⣀⠀⠀⠀⠀⠀⠚⠀⠀⠐⠀⠀⠀⠁⠀⠪⠀⣀⣀⣀⣠⣤⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣍⡉⠉⠙⠛⠛⠛⠛⠒⠓⠒⠖⠛⠒⠛⠉⠉⣉⡽⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⢤⣤⣤⡬⠯⠤⠤⠤⠴⠒⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣭⣽⣿⣿⣯⣿⣽⣯⣿

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦
⠀⠀⠀⠀⠀⢾⣮⣧⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠛⡟
⠀⠀⠀⠀⠀⠙⠛⣿⣿⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣻⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢘⣿⡿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣿⣿⣇⠈⣻⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣧⠙⣿⡛⢠⣿⣿⣿⠀⠀⠀⠀⠀⣱⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⢻⣾⣿⣷⣿⣿⣿⣿⣇⣀⣠⣄⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣽⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣥⠀⠈⠻⠿⠿⢿⡟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠉⢿⣿⣿⣿⣿⣷⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢰⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⠀⠀⠀⠀⠀⠀⢠⣤⣤⣞⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠄⠀⠀⠀⠀⠉⠛⠻⠿⠆⠀⠀⠀⠸⠿⠿⠷⠦⠤⠤


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠒⠂⢢⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡠⠒⠊⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠔⢋⡠⢖⠉⡿⠃⠀⠀⠀⠀⠀⠀⠑⠄⣀⠔⠀⠀⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡀⠀⡜⣁⣀⡀⠀⠀⢠⣒⠒⢄⠀⠀⠀⠀⠀⠈⢫⡀⢍⠲⣄
⠀⠀⠀⠀⠀⡠⠴⠒⠄⠀⠀⢀⣰⣦⠴⠖⠲⠧⡌⠀⢰⡏⣁⣈⣉⢳⠋⠉⠉⢉⣛⡳⣤⣀⣀⣴⣄⣀⠷⠤⣤⡈⢦⠀⠀⢠⠊⠙⠆⢤
⠀⠀⠀⠀⠘⣄⠀⢀⠆⢀⠞⠉⠀⠀⠀⠀⠀⠀⠘⡦⠚⠉⠁⠀⠈⠉⠓⢤⡜⠉⠉⠈⢻⡀⠀⠈⢀⠼⠓⠒⠚⠓⠚⠧⣀⠀⢆⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢀⣀⡀⢳⠀⠀⠀⠀⠳⣄⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣋⣉⠁⢸⡆⠛⠇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⢬⣤⣀⣀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡔⠁⠀⠀⠑⣼⠔⠂⠒⡄⠀⠀⢸⡄⠀⠀⠀⠀⠀⣀⣀⣠⡶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠈⠉⠉⠉⠛⠛⠛⠓⠒⠒⠲⡀⠀⠀⢀⡀⡯⢀⡀⠀⢸⠖⠒⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⡰⠳⡀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⢀⣄⣀⠖⠲⡀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠈⠉⠉⠀⠀⠈⠉⠉⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⠖⠲⣀⣠⡀
⢀⠷⣀⡈⡦⠀⠑⡄⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢀⣀⣀⡀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢠⠎⠀⢴⢁⣀⠾
⢧⣀⣀⡁⠑⠀⠀⠉⠉⠒⠲⠼⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠧⠒⠒⠉⠉⠀⠀⠊⢈⣀⣀
⠸⣀⣀⡠⡠⠤⠞⠉⠑⠒⠒⠲⠤⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠤⠖⠒⠒⠊⠉⠳⠤⠄⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡟⡟⣻⢻⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠛⠒⠾⠇⡇⢻⠸⠧⠒⠛⠉⠉⠙⠓⢦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣟⣻⣤⣤⣤⣤⣤⣤⣤⣤⣤⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣦⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢟⣻⣟⣛⣛⣛⣿⣿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡇⣿⣿⣿⡿⠿⠿⠘⠛⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣄⣀⣀⣀⣀⣠⣤⠴⢟⣴⣿⣿⣏⠀⠀⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡷⢏⠉⢉⣍⠉⠀⢀⠀⠈⠻⣿⣿⠟⠀⢸⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠜⠀⠘⠛⢁⠀⢿⠗⠀⡔⢻⠃⠀⢀⠌⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⡶⠠⠝⠁⢀⡀⠀⠷⠃⠀⢀⠎⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⢟⠶⠶⠶⡷⠁⢸⠁⠀⢠⠋⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⡻⣿⣿⣶⣷⣶⣮⣤⣶⣿⠀⠔⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣼⣿⣶⣮⣝⣛⣿⣻⣛⣛⡉⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠔⢰⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡆⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡔⠁⠀⣾⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠎⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣄⡆⠀⠀⠀⢀⣛⡻⢿⣿⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⣠⣶⣿⣿⣿⣿⣷⣾⢤⣯⣭⣭⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠱⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠰⣄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷
⠀⠈⠻⣶⣤⣄⡻⣝⢿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀
⠀⠀⠀⠉⠻⣯⡻⠎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣽⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣣⣄⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣛⣓⣒⣒⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⣤⠤⢤⡀⠀⠀⣰⢏⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⣹
⠀⠀⠀⠀⠀⢀⡴⠶⠶⠶⠿⣝⠀⠀⠀⠀⠀⠙⢶⡼⣣⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣋⣥⠖⠋⠁
⠀⠀⢀⣠⠴⣾⣿⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⣠⠞⣴⠋⠙⠓⢦⡀⠀⠀⠀⣀⡴⢞⣫⡴⠚⠉
⢀⡾⣋⡴⣞⠻⠃⠀⣴⠛⠂⠀⠀⠘⠃⠀⣴⢫⡾⠁⠀⠀⠀⣼⣙⣳⠴⣛⡥⠖⠋⠁
⣿⢰⣿⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⠏⠀⠀⠀⣸⡦⢞⣫⣴⠚⣯
⢿⣼⡻⠀⠀⢠⡤⣄⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⣠⠴⣚⡭⠖⠋⠁⢸⣀⡿
⠘⣿⣿⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣀⣀⡀⠈⠵⠛⠉⠀⠀⠀⢀⣠⣿⠃
⠀⠘⣧⣯⣟⠶⢤⣄⣀⡀⠀⠀⠀⠈⠁⠈⠛⠀⠀⢀⣀⣠⡴⠶⠛⣱⠏
⠀⠀⠈⢳⣖⠦⠀⠈⠉⠉⠛⠛⠛⠓⠒⠚⠛⠛⠛⠉⠉⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠙⢿⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣀⣀⡀⠀⠀⠀⣀⣀⣠⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠮⢭⣭⣭⣭⣭⣭⡭⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣠⣤⣤⡀⠀⠀⠀⢀⡾⢃⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⣻
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣴⡟⠷⠀⠈⠉⠁⠀⠀⠙⢦⣀⣰⠟⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢋⣡⡴⠞⠉
⠀⠀⠀⠀⠀⠀⢀⣿⣁⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⢀⣽⢋⣼⢯⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⣉⣤⠶⠋⠁
⠀⠀⢀⣤⠶⠛⣻⢓⡖⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠀⣠⡞⣡⡾⠁⠀⠀⠉⢷⣤⡀⠀⠀⣀⣤⠾⢋⣡⡴⠟⠉
⢀⡴⢋⣠⠶⣿⡙⠎⠁⠀⢸⠋⠉⠀⠀⠀⠀⠉⠀⠀⣴⢏⣴⠏⠀⠀⠀⠀⠀⣼⣍⣹⡷⠞⣋⣴⠞⠋⠁
⣾⠁⣾⢭⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡾⠃⠀⠀⠀⠐⣧⣤⠶⢛⣡⡶⠻⣏
⣿⣀⣿⣛⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠰⠟⠀⠀⠀⣀⡴⠞⣋⡴⠞⠋⠁⣿⠀⣿
⠸⣿⣷⡍⠀⠀⠀⠋⠉⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢛⣭⠶⠛⠉⠀⠀⠀⠀⢻⣴⡏
⠀⠹⣟⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢦⣄⠀⠈⠋⠁⠀⠀⠀⠀⠀⣀⣴⢾⡿
⠀⠀⠻⣦⠾⣽⠛⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣀⣤⣴⠶⠟⠋⢡⡾⠁
⠀⠀⠀⠙⢷⡲⠶⠄⠀⠈⠉⠙⠛⠛⠛⠷⠶⠶⠶⠶⠾⠛⠛⠛⠋⠉⠁⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠈⠻⣦⡲⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣍⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠶⣤⣤⣤⣀⣀⣀⣠⣤⣤⡴⢶⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⢦⣤⣭⣭⣭⣥⣤⠶⠶⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⠀⠀⢀⣀⣀⣀⣀⣈⣛⣃⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡙⠛⠛⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⣿⣿⣿⡿⢡⣾⣧⠰⣦⡘⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠁⣾⣷⡎⣿⣿⠇⣾⣿⣿⣇⢻⣷⡌⢿⣯⣭⣭⣭⣭⣭⣭⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧⣙⣋⣥⣿⣿⠀⣿⣿⣿⣿⢸⣿⣿⡘⣿⣿⠟⠛⠛⠛⠛⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡿⢋⣉⡙⢿⣿⣿⣿⣿⠀⣿⣿⣿⡿⢸⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠰⣿⡇⢿⣿⡗⣸⣿⣿⣿⣿⡇⢿⣿⣿⡇⣾⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣾⡿⠛⠻⣿⣿⣷⢸⣿⣿⢠⣿⣿⣿⡇⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⣸⣿⣿⡆⣿⡇⣼⣿⣿⡟⠐⠛⠛⠛⠛⠛⠛⠛⠛⢛⡛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣬⣴⣿⡿⢋⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠿⠋⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⢠⠄⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠢⣼⡄⠀⠀⢐⠀⠀⠀⣦⠀⠀⠀⠁⠀⠀⢰⠗⠈⠉⢰⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⠀⠀⠻⣄⣀⣠⠋⠉⠢⠴⠀⠀⠀⠊⠉⠒⢶⠀⠀⠀⠰⠋⠀⠀⣤⠈⠉⠀⠠⠞⠀⢀⡧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠀⠈⠛⣶⠟⠁⠉⠳⣤⡄⠀⠀⣠⡆⣀⣤⡀⠀⠘⠓⡶⠖⠳⠄⣰⡀⠀⠀⢠⣠⠊⠀⠀⣹⡀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡀⢸⣄⠙⠲⡶⠖⠉⠳⢤⡤⢶⠙⠙⢶⣾⡏⡏⠁⠀⠠⡀⢀⣴⠋⢾⡀⠀⠀⡛⢶⠴⠛⠏⢢⣤⡴⠟⢿⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢰⣧⡈⢯⠀⠀⣠⠀⠀⠀⣿⠀⠀⠀⣀⠀⠀⠀⡜⠒⡆⠀⠀⢠⡇⠀⠈⠻⠶⢴⡁⠀⠀⠀⠀⠀⠀⠀⢀⡌⠙⢒⠞⢁⣾⡄
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⣀⠀⠿⡄⠀⠀⣬⣷⣤⠖⠙⠦⣤⠞⠛⠋⡍⠲⠶⠋⢳⣤⣤⣖⠀⠀⢸⣦⡤⠜⠀⠀⠈⣢⣾⠿⢛⣡⣴⣿⣿⣷
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣶⣤⣤⣉⡀⠀⠉⠄⠀⢠⣿⣦⣀⡠⠻⢷⡤⠶⠿⣤⡄⠉⠓⠒⢻⠀⠀⠘⠂⠀⠀⠁⠀⠀⠘⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆
⠀⠀⢀⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣦
⠀⠀⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⢏⣿
⠀⠀⠀⢿⣷⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢟⣥⣿⡟
⠀⠀⠀⠈⠻⣿⣿⣮⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣮⣭⣛⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣀⣠⡤⠴⠒⠋⠁⠀⡀⠔⠊
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠙⠛⠿⢿⠟⠛⠳⣶⣶⣯⣭⣭⣽⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣭⣭⣭⣵⣶⡏⠉⢀⠖⠀⠀⠖⠊⠁⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠸⣄⣀⣀⡀⠀⠀⠀⠀⠀⠸⣄⣀⣀⡈⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠉⠈⢦⣄⣠⡤⠀⠀⠀⠀⠐⠶⠶⠚
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁

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

⠀⢀⣞⡉⠉⠓⠦⡀
⠀⠈⢦⡛⠷⣦⣄⠈⠢⡀
⠀⠀⠀⠉⠑⠲⢍⡳⢦⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⠄⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠠⠘⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⢒⢶⠚⢦⡤⠞⠋⠳⣤⡤⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⣇⡀⢀⡠⠠⠤⣀⠷⡕⠈⣲⡔⠐⣂⠑⡤⢄⠈⣩⢉⡖⠋⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠯⢅⡈⠙⠁⠈⠂⢀⠼⠤⣖⢛⣛⣁⣉⣶⠷⢲⠤⢈⠃⠈⢈⠖⠢⢤⡭⠷⢤⡀
⠀⠀⠀⠀⠀⠀⡤⠷⠤⠶⠶⠾⠾⠷⠶⠿⠷⠶⠖⠁⠀⠈⠳⡶⠂⠙⢶⠶⠶⠷⠾⠿⠿⠿⠷⠶⠶⠦⠾⡄
⠀⠀⠀⠀⠀⠈⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡇⠀⠀⠠⠀⡇⠄⠀⠀⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠏
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⠌⠡⣀⣀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀⠀⠉⠁⠀⠀⠈⠉⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠀⠀⠐⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⣀⣠⢽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢥⣄⡀
⠀⠀⠀⠀⡴⠊⢀⠤⠖⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠤⢄⠈⠲⡄
⠀⠀⣀⡜⠀⣰⠁⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢱⡀⠘⣄⡀
⢠⠞⠁⡀⠀⢸⠀⠀⠀⠀⠀⠹⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠼⠁⠀⠀⠀⠀⢸⠀⠀⡀⠙⢦
⢇⣠⢎⡟⢦⣈⡇⠀⠀⠀⠀⠀⠀⠈⠻⠤⠤⠤⠤⠤⢤⡤⣤⡤⣤⠤⠤⠤⠤⠤⠼⠋⠀⠀⠀⠀⠀⠀⠀⣏⣠⠞⣏⢦⣀
⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠲⠤⣼⠀⡇⡇⢸⡤⠴⠒⠒⠒⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⣀⣀⣀⣀⣀⣀⣀⣀⡇⣇⣀⣀⣀⣀⣀⣀⣀⣀⡷
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣽⣿⢸⢸⢸⢸⣟⠛⣷⣄⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣴⣶⠶⠶⠶⠶⢶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣻⣿⢸⢸⢸⢸⣿⠀⠈⠛⣛⣛⣛⣛⣛⣉⣉⣉⣁⣀⣀⡀⠀⠀⢀⣀⣀⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⢸⢸⢸⣿⣀⣠⡿⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⢸⢸⢸⢸⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⢸⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⢸⠀⣿⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣴⣿⣾⣾⣾⣤⣿⣴⠾⣛⣛⡭⢍⣛⣛⠻⣶⣄⣀⣀⣠⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣠⡾⢋⣥⣶⡾⢛⡩⠤⠒⣒⣒⣒⣒⠬⢝⡻⣾⣥⠖⢒⣒⠒⠬⣹⣾⠟⣛⡩⠭⠭⠭⠭⠭⣙⡻⢶⣤⡀
⠀⠀⠀⠀⢰⡿⢰⢣⣿⢃⠞⣡⢒⣩⠭⢒⣒⣒⣨⣍⣓⣌⣢⣻⣯⣭⣭⣍⣿⣟⣰⣊⡵⢚⣩⠭⠭⠭⢍⡒⠬⡳⣝⢿⣆
⠀⠀⢰⣶⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠷⠿⠿⠿⠿⢶⣶
⠀⠀⠀⠹⣷⣜⠛⠀⠛⠛⠀⠚⠃⠀⠒⠓⠀⠐⠛⠂⠘⠛⠃⠀⠛⠋⠀⠘⠛⠁⠘⠛⠀⠘⠛⠀⠚⠓⠀⠐⠒⠀⠶⠶⣠⣾⠏
⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣋⣉⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣉⣉⣉⣉⣉⣉⣉⣹⡏
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠒⠂⠉⠉⠉⠉⠑⠲⢤⡀
⠀⠀⢀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⢀⡇
⡇⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⣖⠞⠊⠉
⠑⠦⠤⠤⠒⠺⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣈⡇
⠀⠀⠀⠀⠀⠀⢸⢀⡠⢤⠶⠒⠉⠉⠉⠀⣀⡐⠒⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠘⠟⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⣀⣀⣀⡠⠔⠊⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠑⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⢤⡤⠔⠒⠉⠀⠀⠀⠀⠀⢠⠂⡠⢤⣄⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡷⢄⣀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠠⣒⣂⡤⠀⠀⠀⠀⠐⠂⢤⣄⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡇⠀⠈⠙⠒⢎⡀⠙⠦⣌⡀⠒⠒⣖⣖⠒⠒⡲⠠⡐⠢⠄⠖⠀⠉⠃⠘⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠉⠓⠦⢄⣉⣙⠒⠦⠤⣄⣈⡀⠄⠀⠂⠀⠐⠂⠀⢀⠀⠈⠂⢙
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⠤⠤⠒⠒⠉⠳⠭⠿⠓⠲⠤⠤⣭⣉⣉⣙⣛⡒⠒⠒⢒⣚⣋⣉⣁
⣤⣦⣤⣤⣦⣤⣤⣼⣿⣿⣷⣾⣿⣿⣿⣿⣭⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠛⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢰⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣼⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣷⣄⣰⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋⠉⠻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⣤⡤⠀⠀⠻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈⠂⠀⠄⠀⠐⠷⠀⠀⠰⠄⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣤⠤⠼⠛⠁⠀⠀⠀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣯⡗⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⠀⠀⠀⠀⠀⣴⣦⣹⣧⣴⠋⣻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⢿⣧⡀⠀⠀⠀⠈⠙⣿⣿⢻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⣿⣿⣄⠀⢀⣤⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⢿⣿⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡷⣾⣿⣿⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣤⣼⣿⣧⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣁⣹⣿⣏⣉⣿⣿⣧⡀
⠀⠀⠀⣤⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠀⠀⡀
⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠈⠋⠘⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠋⠛⠁⠀⠀⠈⠻⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠁⠘⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠠⠤⠄⠀⠀⢀⡀⢀⣀⣀⣀⣀⠀⠀⢠⣤⠐⠃⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠸⡇⠀⠀⠀⠙⠀⠀⠀⠀⣠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣘⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇






⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⠂⠘⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⡿⢿⣿⡿⣿⠿⣿⡿⣏⠹⢿⢿⣿⣿⠿⠿⢿⡿⠛⣿⣿⣿⡿⠛⠛⢻⠿⢿⣿⣿⠿⠿⠿⣿⡿⠻⠿⠿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣿⣶⣾⣶⣶⣦⣿⣶⣦⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⣴⠰⠐⠀⠀⠀⠀⠀⢀⣀⠤⠖⠒⠚⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠙⠀⠀⠄⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠁⠀⢤⡀⠀⣠⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⡠⡘⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠈⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠠⡀⠈⢋⠀⠅⠰⠟⠀⢦⡀⠤⠒⡀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⢀⠁⠀⠀⠀⢠⠀⠀⡵⠐⢔⢂⢱⠇⡌⠐⠠⢱⢶⣤⣀⡀⠐⠀⠀⠀⠀⢡
⠀⠀⠀⢈⠀⠀⢠⢠⢀⠀⠋⠉⠃⡞⠑⠡⠀⠁⡸⠄⠠⠤⠀⠉⠉⠁⠤⠄⠀⠀⠀⡆
⠀⠀⠀⠘⡀⠀⠀⡶⣀⢂⡤⠀⠙⠓⢂⠲⣖⣀⡀⠀⣻⠀⠠⠁⠠⠚⠁⡀⠀⠀⢠⠁
⠀⠀⠀⠀⠱⡀⠀⠀⠈⠀⡀⠊⠓⠈⠁⠀⠤⠉⢉⢛⣓⠀⢀⣀⠈⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠙⢆⠀⠀⠙⠄⢀⠀⠀⠁⡀⠀⣄⡈⠁⠀⠀⠸⠃⠞⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⣀⠀⠈⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⢶⡶⣦⣤⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⢸⡇⡷⠆⠀⠀⠙⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⢸⡇⣿⠃⠀⢀⣰⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⢸⡇⣿⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣹⡇⣧⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡼⣣⣷⢹⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣵⢋⡴⠋⠈⢧⡹⣌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⢮⡳⠮⣝⣲⠦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⣶⣶⣾⡿⣿⣿⢶⣯⣭⣓⡾⣝⡻⢿⣿⣷⣶⣶⡶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⢀⡴⢛⡵⣋⡵⢛⡥⢞⣫⣵⡾⠿⢿⣶⣝⢦⡙⢮⡻⣟⢿⣦⡀⠀⠈⠉⠛⠲⢤⣀
⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⣤⣠⠞⡴⢋⣼⣫⠴⢟⣛⣭⠵⠖⣚⣋⣭⠭⠿⣧⡙⣆⢳⡜⣮⢳⣹⣄⡤⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠈⠛⠾⣷⣿⣭⣶⣻⣭⣵⣖⣻⣭⡭⢶⣖⣚⣫⣽⣧⢸⣆⣿⣸⣮⠷⠛⠁⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣶⣦⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⡤⣴⣶⡶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠤⢤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⠒⠛⠉

⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠉⠉⠁⠒⠒⠀⠉⠉⠉⠉⠑⠢⢤⡀
⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡧
⡆⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⢲⡖⡶⠒⠊
⠳⣄⣀⣀⡠⠶⢩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣤⠄⠒⠒⠒⠀⠉⠩⠉⠉⠉⢧
⠀⠀⠀⠀⠀⠀⢸⠆⠉⠀⠉⣴⡆⠀⠀⠀⠐⠿⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠓⠒⠒⠊⠉⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⡀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⡉⠉⠀⠀⠀⠀⠀⢀⠤⠰⠀⠒⠀⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡟⠲⢤⣄⡀⢀⠰⣇⡀⠀⣀⣀⣈⠒⠒⡒⠀⠐⠂⠀⣀⠀⠉⠳⠶⢦⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠⡇⠀⠀⠈⠁⠲⣄⣈⠛⠷⢤⣌⡀⠘⠙⠀⠀⠜⠁⠀⠀⠀⠉⠁⠀⠈⠀⠉⢙⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠐⣂⣤⢤⣍⡑⠒⠢⠤⠤⣈⣀⣈⣀⡀⠈⠡⠄⠀⣀⣈⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣊⡀⣀⣀⣀⣠⠼⠧⠤⠔⠒⠒⠂⠈⢻⠁⠀⠙⠒⠚⠋⠉⠒⠒⠲⠤⠤⠤⢤⣤⣉⣉⣉⠥⣤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡄⠂⢰⡶⠦⣤⣤⣤⣤⠔⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠁⠀⠀⡌⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⣠⡄⣒⣂⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⠤⢀⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣵⣶⣶⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠙⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣾⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠘⠻⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣠⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣧⣶⣤
⠀⠀⠀⠀⠀⠿⣿⠀⠀⢰⣿⡧⠀⠀⠀⣀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣾⣿⣷⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠉⠁⣠⣴⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣄⠰⣌⠻⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣼⣿⣿⠿⢛⣥⠆⠀⣀⣠⣤⣴⣶
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣶⠦⣭⣭⣝⣛⣛⣛⣛⣛⣛⣛⣛⣩⣭⣭⣴⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣷⣦⣄⡀⠉⠉⣉⣉⣭⣭⣿⣿⣿⣿⣿⣻⣿⣻⣯⣿⣿⣿⣿⣿⣿⣿⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣾⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⠀⠀⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⠈⠛⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⢠⣿⣾⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣦⣄⠀⠀⠀⢴⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⣦⣀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢠⣴⠾⠛⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣾⠋⠉⠛⢛⡁⠀⠀⠀⠘⠷⠶⠟⠛⠻⢶⣄⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⡥⠀⠀⢀⡛⠂⠀⠀⠀⠖⡳⠀⠀⠀⠀⠀⠩⠿⠉⢉⠉⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡿⠁⠀⠀⠀⣠⠄⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡾⠛⠋⠉⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⣠⠀⠀⠀⠘⠒⠀⠀⠀⠣⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⣴⡟⡥⠌⠀⠀⣠⠄⠀⠀⠀⢠⠤⠀⠘⠁⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠷⣶⡄
⠀⠀⠀⠀⠀⠀⣿⢁⡀⠀⠀⠀⠀⠀⠀⠀⢰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⠃⠀⠀⢸⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣿
⠀⠀⠀⠀⢠⣴⡿⠚⠂⠀⠀⢾⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠳⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠘⣿
⠀⠀⠀⠀⣸⣧⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠀⣿⣤⡀
⠀⢀⣴⠟⢩⡿⠀⠙⡊⠀⠀⠀⠰⣯⠄⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠰⣏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡟⠈⠻⣦
⣰⡟⠁⢰⣿⢁⡀⠀⢧⡀⠀⠀⣀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⣷⠀⠀⣿
⣿⣧⡀⠈⢿⣌⠉⠀⠀⠀⠀⠘⠳⠔⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⣸⠏
⣿⠘⢷⣄⡀⠙⠻⣦⣄⡀⠀⠶⠀⠀⢰⠟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⡦⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣴⡿⠋⢀⣴⠏
⢹⡆⠀⠈⠛⠶⣤⡀⠉⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋⠀⠀⢠⣶⠶⠿⠋⢀⣴⠟⠁⠀⢸
⠘⣧⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠛⠶⠶⠿⠿⣷⣤⣤⣤⣤⣈⣁⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⠶⠶⠶⠟⢀⣠⡴⠞⠋⠁⠀⠀⢀⣿
⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢤⣤⣄⣀⣉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⣀⣠⣤⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣄⠀⠀⠈⠙⠛⠛⠶⠶⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠛⠋⠁⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠞⠛⠉