Unintelligent Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⣾⠁⠀⠀⠀⢀⡀⢹⡆⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠉⠐⣇⢰⡞⢷⡄⣯⣽⠆⣷⠀⢼⠃⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀⡾⠀⢿⠈⠛⠛⠁⢀⣠⡀⢹⡆⠘⢷⡀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⢠⣾⣁⣀⠸⣇⠀⢶⣿⣭⡿⠁⣸⠃⣀⣈⣷⡄⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠉⠉⢻⡀⠙⢦⣄⣀⣀⣠⡾⠋⢈⡟⠉⠉⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⢀⡀⠈⢷⣄⠀⠈⠉⠉⠁⠀⣠⡾⠁⢀⡀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⢸⡇⠀⠀⠙⠳⠶⢶⡶⠶⠞⠋⠀⠀⢸⡇⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠛⠛⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⡾⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⡤⠀⣤⡤⠤⠀⣤⡤⠤⠀⣤⢤⣄⠀⠀⢠⡄⠠⢤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠁⠀⣿⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣇⠀⠀⣿⠛⠃⠀⣿⠛⠃⠀⣿⠛⠋⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣆⠀⣿⣀⣀⠀⣿⣀⣀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠈⠉⠉⠀⠉⠉⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣤⡄⠀⣤⠀⠀⣤⠀⠀⣤⠀⢀⣤⣤⡀⠀⣠⡀⠀⠀⣤⣤⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠰⣯⠀⠀⠀⣿⠀⠀⣿⡇⢠⣿⡄⢸⣿⠈⣿⠀⣿⠄⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⢷⡄⠀⣿⠀⢠⡟⣷⣸⢻⡇⠰⣿⠾⠋⠀⣿⡀⠀⠀⣿⠷⠆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⢨⡿⠀⣿⠀⢸⡇⢹⡟⢸⡇⠰⣿⠀⠀⠀⣿⣃⡀⠀⣿⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠁⠀⠉⠀⠈⠁⠈⠁⠈⠉⠈⠉⠀⠀⠀⠉⠉⠉⠀⠉⠉⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⣀⠀⣀⠀⢀⡀⠀⣀⣀⡀⠀⢀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠉⠁⠈⢹⡏⠉⠀⣿⠀⢸⡇⠀⣿⠉⣿⠀⢸⡇⠀⣿⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⣿⠴⠟⠀⢸⡇⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠘⠃⠀⠀⠙⠓⠛⠀⠀⠛⠀⠀⠀⠘⠃⠀⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⢀⣴⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⢠⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠁⠀⠀⠀⠀⠈⠙⢻⣿⡄
⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⠀⢀⣀⣤⣤⣀⣀⠀⠀⣿⡇
⠀⠀⠙⢿⣿⣿⠇⠀⠀⠀⠀⠀⢿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⣿⣯⣭⣿⣿⣿⣾⡿⠁
⠀⠀⠀⠀⠙⠉⠀⠀⢰⣿⣷⣆⠈⠋⠁⠀⠀⠀⣶⣿⣷⡀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⠉⠉⠉⠉⠉⣿⣿⣿⠇⠀⠀⢿⣿⣿⣿⣿⣯⣟⣛⣯⣿⣿⣿⣿⣿
⠀⠀⣀⣀⢀⠀⠀⠀⠘⠿⠟⠃⠀⢀⣀⡀⡀⠀⠛⠿⠟⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⢀⣿⣿⣿⣇⣀⣀⣀⣀⣀⢀⣀⣸⣿⣿⣿⡀⠀⢀⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⡀
⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⠋⠙⢦⠀⠀⠀⢀⣤⠀⠀⣠⡄⠀⠀⠀⠀⡼⠋⠙⣦
⠀⠀⡿⢿⢿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡿⢠⣿⣾⣿⣿⡇⠀⠀⠸⠿⠛⠛⠿⠏⠀⠀⠀⢸⣿⣾⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣿⣯⡿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⢾⣿⣹⡏
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢻⡿⢃⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠸⡌⢿⡿⢡
⠀⠀⠀⠀⠀⣠⣼⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣤⠳⣾⣷⠎
⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣄
⠀⠀⠀⣿⣿⣿⣿⣿⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⢸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠻⢿⣿⣆⡀⠀⠀⠀⠀⣿⣿⣿⣿⠶⠶⠶⠶⢾⣿⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠘⠿⠛⠃⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣶⣿⠿⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠶⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠿⠛⠛⠉⠉⠉⠉⠙⠛⠛⠿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⡀
⠀⠀⣴⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠇⠀⠀⠀⠀⠀⠀⠐⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿
⠀⠸⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠟⠛⠛⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠿⠿⢿⣿⣷⣤⡀
⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⡀
⠀⠀⠀⠀⣼⣿⣧⣤⣶⣶⣶⣶⡶⣶⣶⣦⣤⣀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⢹⣿⡟⠉⠀⠀⠈⠻⣷⠿⠛⠈⠉⠙⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣶⢶⣶⣦⣤⣤⣀⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⠀⠀⠀⠀⠛⠿⠿⠛⠙⠻⣷⣾⣿⠇
⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣶⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣀⣀⣀⣀⣴⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⢻⣿⣷⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⠉⠛⠿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣤⣀⣀⣀⣤⣾⣿⣿⠟⠉⠀⠀⠈⢻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⣠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠁
⢾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⣸⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⣤
⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⠟⠁
⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⣁⣴⣾⠋⠁
⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣵⣶⣶⡿⠛⠉
⠀⠀⠈⠻⢿⣿⣦⣄⣀⡀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠰⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠈⣓⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠉⠛⢋⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠃⠨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠖⠲⠂⠀⠶⠶⠆⠀⠐⠒⠒⠀⠀⠒⠒⠂⠀⠒⠒⠀⠁⠀⢠⡀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢣⡃⢀⣀⣀⠀⠀⣀⣀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⣈⣳⡟⠀⢀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⠀⣀⠀⠉⠟⠁⣠⡾⠀⢠⣤⢀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⣀⣨⣥⣤⡀⠉⠉⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣀⠀⠀⠀⢠⠀⣿⡏⢀⣴⣿⣯⢻⡏⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠛⠁⠭⠛⣿⢿⣇⠀⠀⠀⣠⣾⠿⣿⠟⠓⠦⠤⠀⠁⠀⠀⠐⠘⢻⣡⣯⣝⣿⣝⠸⣻⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⠒⠶⢶⡶⠂⣿⡿⠆⠀⢠⠟⠓⣾⠗⠺⣿⠗⠲⢲⡟⠀⠀⠀⠘⡏⣷⣿⣿⣟⢛⡗⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⡀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠊⠀⠀⠀⠀⢴⣾⠛⢿⡟⠿⢾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡏⠀⢿⡷⢒⣶⡬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠐⣾⣃⣬⠟⣲⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⡆⠀⠀⠀⠀⠁⠀⣀⣀⣀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣰⡞⠁⠀⠀⠀⣼⣿⣯⠰⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⣿⠀⠀⡤⠚⠳⠶⣿⠉⠭⠉⠘⠻⢄⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠸⣿⡎⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⣦⠀⠀⣀⣄⠀⠀⢀⡀⠀⡀⠀⠀⠳⡀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢨⡟⠀⠀⣟⠉⣈⣋⣁⠁⠈⠳⠀⠀⠀⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠔⡞⠀⠀⡼⠁⠈⠙⠉⠛⠃⠀⠀⠸⡆⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⣠⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠠⠄⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠸⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠰⣌⠀⠀⠀⠀⠀⠀⢀⣹⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠄⠀⠀⢠⣾⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣂⠀⠀⠀⠀⠀⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⣸⠂⠻⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⣿⡇⠈⠳⠶⠸⣷⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠺⢟⣿⣟⣷⠀⠀⣀⢠⡅⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⢿⣴⣄⡄⢀⣶⡠⠄⣨⢽⡛⢶⣄⡀
⠀⠀⠀⢀⣴⡾⠋⠁⠀⠀⠀⠀⠀⢹⣿⣻⣶⣇⣽⢸⡇⢀⢀⠀⡀⠀⡀⡀⢸⣦⢠⣷⣮⣃⠀⠑⠈⣒⠬⠑⡐⢭⣺⢽⡻⢿⣦⡀
⠀⠀⠀⠈⣭⣀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠹⢫⣿⢿⣾⡧⣿⢸⡄⣿⡀⢸⡇⢰⣤⣾⣿⣿⠗⡀⠄⠩⠅⢀⠄⠈⠀⠂⠉⠀⠱⣯⣷
⠀⠀⠀⠀⠈⠁⠙⠳⢶⣤⡀⠀⠀⠀⠀⠈⠈⠋⠟⠁⣴⢿⡗⠀⣿⣅⣼⣷⣴⣿⣿⠏⠃⣴⣺⠇⢔⣏⢀⡘⠖⢀⢆⣠⣤⡷⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣄⡀⠀⠀⠀⠀⠀⠋⠘⡏⣣⠱⢻⣿⣻⣿⣿⡏⡳⠿⣿⣷⣤⡴⢛⣭⣵⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⡉⠉⠛⢝⢛⣷⡙⣦⡾⠿⠞⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠦⠤⠤⢶⡶⠶⠶⠚⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⡀⠄⠀⠀⠀⠐⠔⠠⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⢶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆
⠀⠀⠀⠀⢀⣤⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⣸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣇
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿
⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠀⠀⠀⣿⠀⠀⠀⠀⢸⣇⠀⠀⠀⢠⣷⡀⠀⢀⡿⢶⡿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢟⠒⠒⠊⠁⠈⠳⠤⠞⠈⠓⠦⠤⠶⠋⠈⠓⠒⠚⠉⠀⠙⠛⠋⠁⠘⣷
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠉⠒⣄⠀⠀⠹⣧
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠋⠀⠀⠀⡴⠊⠉⠀⠉⠙⢆⠀⠀⠀⠀⠀⢰⠃⠀⢀⣤⡀⠀⠘⡆⠀⠀⠹⣧
⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⣀⡼⠀⠀⠀⢸⠁⠀⢠⣦⡄⠀⠸⡄⠀⠀⠀⠀⢸⠀⠀⠈⠛⠁⠀⢀⡇⠀⠀⠀⢻
⠀⠈⢿⣄⠀⠀⣀⣀⠀⠈⡏⠀⠀⠀⠀⢸⡀⠀⠈⠉⠀⠀⢰⠇⠀⠀⠀⠀⠈⠳⣄⣀⠀⣀⣠⠞⠀⠀⠀⠀⢸
⠀⠀⠀⠹⣶⠞⠉⡉⠙⠋⠁⠀⠀⠀⠀⠀⠳⢤⣀⣀⣀⡤⠋⣴⣤⣤⣤⣤⡶⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢿⣄⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣧⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠙⠛⠛⠛⢦⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡤⠏⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠋⠀⠀⠉⠙⠒⠲⠤⠤⠤⠤⠤⠴⠖⠚⠋⠉⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⣀⣀⣀⣀⣰⡚⠛⠛⠲⣦⣤⣄⣀⣀⣠⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠙⠉⠉⡇⠀⠀⠈⠙⡆⠀⠀⡟⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣀⣀⣀⣀⣿⣠⣤⠴⠚⠓⠶⠴⣧⣤⣤⣤⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠉⠉⠉⠛⠛⢛⣛⣛⠛⠉⠉⠉⠉⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡏⠉⢹⡇⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠘⣧⠀⣼⠃⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⢿⠀⣿⠀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠞⠛⠀⠛⠳⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠿⠿⣶⣦⣄⡀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⡄⢀⣤⡄⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⣹⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠀⠻⠃⠘⠋⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠙⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⡄⠀⠀⣾⣏⠀⠀⠀⠀⠀⠀⣀⣴⣦⠀⠀⠀⠈⣿⡆⠀⠰⣿⣏
⠀⠀⠀⠀⠀⠀⢀⣠⣦⣾⡟⠉⠀⠙⣿⡄⠀⣿⡿⠿⣶⣶⣶⣶⠿⠟⠋⠻⢷⣶⣤⣶⣿⠃⠀⠠⣿⠏
⠀⠀⠀⠀⢠⣾⠟⠋⠉⠁⢀⣤⣤⣶⡿⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⣰⣾⠟⠀⠀⠀⣴⣿⠋⠋⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⣼⣿⣵⡾⠿⣷⣄⠈⢿⣦⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⢿⠟⠁⠀⠀⠘⢿⣆⠈⢿⣧⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⢀⣻⣷⡀⠀⠀⠀⣿⣿⣦⣄⡀⠀⠀⣀⣠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠙⣿⣄⠐⠿⣿⡇⠉⠛⠻⠿⠿⠛⠋⠁⣿⡿⢷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⢿⣆⠀⠘⢿⣦⠀⠻⣷⣄⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠈⠉⠛⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠀⠈⢿⣧⠀⠈⢻⣧⡀⠈⢻⣷⡀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⢻⣷⡀⠀⢉⣀⣀⠀⠙⢿⣶⣾⠟⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⠀⣴⡿⠛⠻⣷⣄⡀⠉⠁⣀⣴⡿⠛⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠈⢻⣷⠀⢰⣿⠉⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⣠⣼⡟⠀⠸⣿⣄⡀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⢀⣾⠟⠉⢀⣴⠀⠈⠙⣿⣆⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢠⣿⠏⢀⣴⣾⠟⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣠⡀⢀⣴⣿⣿⣿⣿⣿⣷⣄⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣈⣁⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠾⠋⠙⣧⢰⡿⠿⠿⠿⠟⠋⠻⠿⣿⡇⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⣴⠷⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠞⠛⣧⠙⣧⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢷⣸⣷⡀⢀⣸⣷⣄⣀⣀⣤⣾⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣿⣿⣿⡌⢹⣇⠈⠉⠉⢀⣼⠉⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠈⢿⣿⡿⠆⠙⢷⣄⣴⠟⠁⠀⠀⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⠋⣴⠟⠷⣄⠙⢁⡴⠟⢳⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠏⠀⢀⣸⠇⢿⣀⠀⠀⢷⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⡾⠁⠀⣠⡟⣡⣶⠀⠙⢧⡀⠀⠙⡦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣰⠏⠀⠋⠁⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣤⣄⡀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣫⠖⠋⠉⠉⠙⠒⢄⠀⠀⠀⠀⡤⠒⠉⠉⠉⠙⠢⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡞⣭⠀⡀⠀⢳⠀⠀⡞⠀⢰⢫⣍⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠳⠬⠄⠁⠀⢸⠀⠀⡇⠀⠈⠦⠭⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣀⠤⠐⠊⠁⠀⠀⠈⠐⠂⠤⡀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠔⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡢⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢿⣿⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠔⠒



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣶⣿⣿⣶⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⡛⠿⣿⣿⠿⢛⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⠟⣥⢛⣿⣻⣷⡈⢁⣾⣟⣿⡛⣬⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⢋⡼⣯⢹⣟⣯⣵⡇⢸⢮⣼⣻⡏⣽⢣⡙⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣟⣿⣿⣿⣼⢯⡇⢸⡽⣧⢿⣻⣿⣻⣇⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⢷⣿⣷⢿⡯⣿⢻⡇⢸⡟⣿⢽⡿⣾⣿⡾⣇⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⣴⡛⣿⢿⣿⣾⣯⣿⡟⡅⢨⢻⣿⣽⣷⣿⡿⣷⠛⣦⢻⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣹⢾⣿⣾⡟⣯⣶⣙⣮⡇⢸⣼⣋⣶⣽⢺⣷⣿⣷⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠃⢿⣟⣷⣾⣟⣿⢿⣩⡿⡅⢨⢿⡍⡾⣿⣻⣷⣿⣳⡿⠘⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣷⣿⣜⢽⣿⣿⡅⢨⢿⣿⡿⣣⣿⣾⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⡹⣿⣟⢯⣝⣯⣿⡼⣿⠇⠘⣿⢧⣿⣽⣫⡿⣻⣻⢏⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡅⠿⣿⣟⣿⣻⡷⣿⣿⡃⢘⣽⣿⠼⣞⢿⣛⣿⠿⢸⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣶⣬⣄⣛⣓⣛⣪⣴⣦⣕⣛⣛⣛⣤⣵⣶⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⡿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣗⢸⣿⡟⢻⣿⣿⣷⡿⣿⣿⣼⡟⣿⢳⡟⣾⢿⡿⣿⡟⣿⣿⣿⣧⣿⣿⡟⢻⣿⡇⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⢸⣿⣿⣿⣿⣿⣧⣿⣿⣧⣿⣿⣿⣾⣷⣿⣾⣿⣿⣾⣯⣿⣿⣼⣿⣿⣿⣿⣿⡇⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣽⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠞⠲⠶⣤⡀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⢿⠀⠞⠀⣾⠋⠀⠀⠀⠀⠀⠈⠻⣆⠰⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣆⠀⢸⣇⡀⠀⠀⣀⣤⣦⠀⠀⢻⡆⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⢁⢀⣽⠇⢸⡏⠛⠛⠛⠉⠁⠈⠛⠛⣿⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣀⠀⢰⡟⠋⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠙⢷⡌⢻⡄⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣿⣆⠀⢀⣿⢷⣄⣀⣀⣠⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠙⣧⠈⢿⡄⠈⠉⠉⠁⣸⡏⠛⠳⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢻⣄⠘⢷⠀⠹⣦⡀⢠⡾⠋⠀⠀⠀⠀⠀⠉⠛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⠂⣠⠶⣤⡈⠻⠋⢀⣴⢶⣄⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⠇⠀⠈⢻⡆⢸⡏⠀⠀⢹⣆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡾⠋⠀⢀⣴⠟⢁⠘⠳⣦⠀⠀⠹⣦⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠁⠀⢠⡾⠁⣴⠟⠀⠀⠙⣧⡀⠀⠈⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⠀⠀⠀⠿




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡙⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣮⠙⠻⢿⣷⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣿⣷⣆⣄⣤⠌⣭⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⢣⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⣿⣿⣿⣿⡻⣿⣣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⣿⣼⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠋⠀⣼⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⠀⣀⣤⠖⠚⣉⣉⡉⢻⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⠀⣾⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⡿⣤⣿⣿⣿⢩⣴⣿⣟⣁⣠⣴⣿⣿⣿⣮⡟⣿⣷⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⢷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠓⠉⢉⣙⣣⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⠉⠋⣿⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⢶⣿⣿⣷⡌⢿⣿⣿⣿⣿⣿⡿⣿⣥⣿⣦⣀⢸⡇
⠀⠀⠀⢰⣿⣿⡿⣻⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣬⣿⣿⣿⣿⣷⡈⣿⡿⠟⠋⠁⠀⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠈⠻⣿⣿⣿⡿⢟⣫⣽⣭⡛⢿⣿⣿⣿⠿⠿⢿⣿⣿⣥⡟⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⣥⡀⠀⠘⠦⣿⠏⠙⠀⠀⠀⠈⠻⣿⣷⠀⠀⠀⠀⠀⠘⣿⣿⡟
⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡿⡿⣋⣉⠙⠿⠇⠀⠀⣤⡖⠀⠀⣀⣀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⢠⠬⣿⠟⣿⡟⣿⣷⣶⣼⠷⢾⣷⢾⠟⢿⠿⠃⠸⣏⡍⡦⠀⠀⠀⢀⣴⠋⡀
⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠘⠷⣾⡆⠘⠛⠉⠛⠛⣿⠀⠈⠛⠛⠋⠀⠀⠀⢱⡇⣾⠁⠀⠀⠀⢸⣿⠀⢸
⠀⠀⠀⢀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣂⡀⠀⠀⣤⡟⠀⠀⠠⡄⠀⠀⠀⠈⣼⠁⡏⠀⠀⠀⠀⢜⡻⠂⠛
⠀⠀⠀⣻⣗⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠛⠁⣸⣧⣟⠀⠀⣠⠛⡄⠀⠀⢀⡷⠞
⠀⠀⢀⡿⠟⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣼⠁⠉⢻⣷⣾⣅⣀⡈⠀⠀⣼
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠛⡒⢦⣥⣴⠶⠂⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⠃⠀⣀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣬⡽⠧⣤⠾⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣊⣭⣽⣿⡷⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣊⣵⣾⣿⣿⠋⢹⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣼⣿⣿⣿⣟⠁⠀⠈⠦⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⢟⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣬⣭⣥⣦⣄⣀⠀⠀⠀⠀⡇⢠⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⢸⠃⣿⣿⣿⣿⣿⣿⡆⢀⡤⠒⣿⣿⣷⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣻⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⢀⣼⣿⣿⠿⢿⠿⠻⣿⣿⣿⣿⣿⣿⣿⠃⠾⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⠿⣿⣿⣿⣷
⠀⠀⣠⠚⠛⢿⡁⠀⠀⠀⠀⠈⠙⠻⣿⡿⠟⢁⣠⣴⣾⣶⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠈⡻⠿⣿
⠀⢰⠁⢀⣀⣼⣿⠀⠀⠀⠀⠀⠀⢠⡿⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⣷
⠀⠀⠓⠤⢴⠿⠏⠀⠀⠀⠀⠀⠀⣾⣧⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠳⣀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡗⠁⣠⣤⠈⠙⣿⣿⠟⠁⠠⠶⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠒⠂⠀⠀⠈⡭⢖⡥⠄⠀⠻⢿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠈⠻⣿⣷⣄⠀⠀⣤⠋⠀⣠⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⢧⣘⢋⣷⣦⠘⢶⣾⣙⣯⣽⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠁⠀⢸⣿⠫⠛⣋⢠⣿⣇⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⢡⣀⠰⣒⠒⠶⡖⠂⠀⢸⣿⣿⣿⣿⣿⣿⣟⠉⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⠤⠤⠤⠄⠒⠁⠀⠀⠀⢈⢹⣆⠘⣧⠀⠸⢄⠀⣸⠟⢫⣿⣿⠿⠋⠙⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠁⠀⠀⡄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢣⠁⠈⠳⠀⣀⣋⣑⣰⣿⣿⡀⠀⠀⠀⢱⠀⠀⢰⠁⡄⠃⡀⡃⠀⡘⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢣⡀⠀⠀⠶⣤⣶⣾⣿⣿⣷⡄⠀⠀⠀⢣⠀⢸⢸⡁⢹⠗⢲⢀⠃⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢳⡀⠀⠀⠀⠀⣼⡿⠻⠿⣿⣆⠀⠀⠈⠆⠺⣝⠀⢸⠀⣾⠜⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠙⠒⠒⠛⠛⠛⠁⠀⣸⣿⣿⣧⡀⢀⡇⠀⢹⠀⢸⠀⣾⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠰⠊⠘⢿⢟⣿⠿⠿⣷⠠⣾⠀⣸⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⢳⠀⢻⡄⣿⠀⣿⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠰⣾⠁⣿⡆⣿⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣀⣴⣯⣴⣿⡗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠘⠛⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣾⣿⣻⣶⣤⣀⣠⡴⠞⣋⣭⣴⠶⠶⣭⣓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣽⠋⣠⣾⣿⣿⡷⠛⠋⠛⠻⢷⣕⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠋⠀⠀⠀⠀⠀⠹⣾⣿⣿⡿⣿⣿⣾⡿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠙⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢀⣀⠀⠀⠀⠻⣼⠏⣒⡢⠉⠡⣒⡈⣿⠃⢀⣀⣀⣀⢀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢻⣄⡉⠁⠀⠀⠉⣁⢉⣿⣿⡿⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠐
⠀⠀⠀⠀⠁⠀⠀⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣿⡱⣦⣶⣤⡾⢩⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⡐⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠳⡌⢉⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣌⡻⢋⣴⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣤⠾⠵⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⢹⣿⣿⣿⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠋⠁⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠟⠛⠉⣹⣿⣿⣿⠏⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣦⡀⣷⡀⠀⢠⣿⣿⣿⡟⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣶⠀⣴⠀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⣧⢠⣿⣿⣿⣿⠁⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢯⠄⠸⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠃⠈⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉

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





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⣣⠀⠁⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⣶⡍⢁⣶⣤⠀⠻⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣞⠘⠛⢁⣈⠻⠛⠀⠀⠀⠙⣿⠁⠀⠈⠏⠀⠀⠀⢀⡀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⢰⣇⠂⡀⠀⠀⠋⣀⠀⠙⣏⡠⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢚⠄⠀⠒⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⠟⢀⠅⠀⠀⠀⡉⠀⠀⢻⣿⣆⠀⢀⡠⠔⠂⠉⠄⠀⠀⠀⠀⠥⠀⡀⠀⠘⣿⣿⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⢀⡂⠀⠄⠀⠀⠁⠀⢀⣾⣿⣇⠀⠈⠀⠙⠀⠸⠃⠀⠀⡀⠀⢄⢀⠀⠀⠀⠉⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⣴⡖⣌⣤⡀⣄⡀⠀⠈⠉⢀⠀⠈⠀⠱⠄⠀⠀⠀⠉⠀⠀⢉⠀⠄⡀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣠⣶⣶⣤⣶⣾⣿⢯⣾⣿⠋⣰⣿⢿⣴⣤⡄⠀⠙⠆⠁⠈⠀⠀⠀⠀⠀⠀⠀⠘⠦⢐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠿⠿⠿⠛⠛⠉⢠⣾⣿⠇⠉⠉⢀⣾⡿⣟⣷⡆⡀⢀⣀⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠒⠈⠀⢸⣿⣍⠁⠀⣠⣾⣿⣿⣿⡟⠁⣸⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠄⠀⢰⣿⣿⣿⣿⠏⢠⣾⣿⡆⠘⣿⡇⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠃⠀⣿⣿⣿⣿⡟⠀⡆⢻⣿⣧⠀⣿⡇⠀⠀⠀⠀⠀⠐⡄⠆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⣿⣷⠀⠀⠀⠀⣠⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⣿⣿⣿⣿⡇⠀⠀⠀⢹⣿⠀⢹⠟⠀⠀⠐⠆⠁⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠠⡀⠘⣿⡄⠀⠀⣠⣶⣶⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠀⠀⡱⠀⠹⠏⣧⣾⡿⠋⠁⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠁⠘⣛⣛⣛⠛⠛⢯⣭⣴⣾⡄⠀⠀⠀⡙⠋⣀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⠀⠀⢤⣿⣿⣿⢀⠀⠈⠻⠿⣿⣿⡄⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠙⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣾⣷⡀⢹⡿⠁⢴⡶⢀⠀⢾⠇⠈⣉⡉⠓⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⣉⠀⠀⣡⣶⡄⠀⠈⠀⣤⠀⠈⠿⠟⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⣿⣿⣿⣿⣷⠀⠀⠀⢤⣤⣴⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⡀⠀⢠⣾⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⡄⡇⣰⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣼⣿⣿⠃
⢀⣤⣤⣤⣾⡟⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠸⣿⣿⠧⠤⣄
⠳⡿⣿⣿⣿⣇⠈⠉⣁⡐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⢀⣤⣶⠀⠀⠀⡀
⠀⠈⠑⢻⣿⡟⠀⢀⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⣀⠀⣿⣿⣿⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠹⠁⠀⠠⠤⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⣿⣿⣶⣤⣤⣤⠁
⠀⠀⠀⠀⠀⠣⣀⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠝⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡰⣦⡀⠀⠀⠀⣴⣦⢀⣀⣤⣴⣾⣿⣷⣄⣤⣴⣦⣤⣀⠀⠀⢾⣿⣿⣧⢀⣿⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⢶⣿⣿⣿⣿⠿⠿⠟⠛⠉⠈⠻⣿⣿⣿⣿⣷⣦⡀⠙⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠉⠛⢿⣿⣿⣿⡄⣿⡏⠀⠀⢀⡀⠀⠀⣀⡀⠀⢿⣿⣿⠿⢿⣿⣿⣷⣄⣼⣿⢿⣷⣽⣿⣿⣿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⠀⠀⠀⠉⠛⠟⠁⣿⣧⠀⢰⣿⡇⠀⠘⣿⣿⣦⢸⣿⣿⠀⠀⠈⠻⢿⣿⣯⠉⣠⣽⠟⠻⣿⣿⡄⠈⠙⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⡀⠀⠀⠀⠀⣿⣿⠀⣿⣿⠂⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⢻⣿⣇⣿⣿⠀⠀⢻⣿⡇⠀⠀⠀⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣷⣄⠀⠀⣿⣿⣄⣿⡏⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠈⣿⣧⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⣿⡿⣿⣿⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⢿⣇⠀⢀⣠⣴⣿⠟⠁⢻⣿⣧⡀⠀⠀⣿⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⡀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⠿⠛⠋⠁⠀⠀⠸⣿⣿⣷⣄⣀⣸⣀⣠⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣀⣀⣀⣾⣿⡟⠛⢿⣿⣦⣤⣤⣾⣿⡟⠁⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⢿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠙⠦


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣍⡻⣋⣝⢙⣟⣋⣿⣯⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⢜⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢈⠀⠀⠤⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠿⢀
⠘⠁⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⢀⣤⣄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⣈⠁⢀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡐⠠⠀⠀⠹⡀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⢷⡀⠀⠀⠀⠐⡄⠀⠀⠀⠢⡄
⠂⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⣦⠀⠸⠀⠀⡧⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢅
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣠⢞⡡⠂⠀⠀⠀⠙⢶⡄⠀⠀⠀⠀⠘⠀⠀⠀⠉⠀⠀⠙⠷⣄⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠛⠀⠀⠀⠁⠉⠀⠀⠀⠉⠒⢤⠤⠠⠀⠀⠀⠈⠉⠀⠠⠀⠀⠘⠃⠖
⢀⣤⠒⠀⠠⣤⣀⠀⠀⠀⠈⠋⢀⣀⡤⢾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⠀⠀⠠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠉⠀⡀
⡛⠀⠀⠀⠀⢈⣿⡇⠀⢠⡴⠛⠉⠀⠀⠀⡽⢡⠀⠐⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠱⠀⠀⠀⠈⠁
⣧⣀⣀⣀⣀⡿⠋⠀⣰⠋⠀⠀⠀⠀⠠⠊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠂⠀⠀⠀⠀⠄⢈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠐⠀⠀⠀⠰
⠈⠙⠛⠋⠁⢈⣠⡴⠗⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠈⠉⠀⢀⣠⣤⣤⢤⣄⡀⢀⠀⡀⠀⠀⠀⠀⠀⠑⠀⠀⠃⠐⠀⠀⠀⠀⠀⠀⢄⠀⠀⢀⠤
⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⢠⠀⢰⠋⠀⠀⠀⡀⠀⣄⠀⠈⠋⠀⠀⠀⠀⠈⢀⣄⠁⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄⠳
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⠌⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⡀⠀⠀⠀⠀⠀⡹⠀⠀⢀⠀⠀⠈⠀⠐⣄⡀⠀⠁⠀⠀⠀⢀⠀⠀⠐
⠀⠀⠀⠳⠤⠖⠂⠁⠂⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⣫⠂⠀⠀⠀⠈⠐⠠⠤⠤⠊⠀⠀⠀⠀⠀⠁⠀⢀⠐⠀⠀⠀⠀⠀⡄⠀⠀⠁
⠤⠤⠀⠀⠰⠀⢈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣐⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠄⠀⠀⠀⠀⠈⠊⠁⠢⢀⠀⠀⢄⣵⠀⠀⢰⠏⠀⠀⠀⣠⢀⠆⠀⠀⠀⠠⠀⠀⢀⡤⠂⠀⠀⠂⠀⠀⠀⢀⡀⠀⠂⠀⢰⠋⠉⠓⠀⠠⡀⠸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠓⠠⠀⡿⣡⠄⠀⠀⠠⠤⠶⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠄⢻⣄⠀⠀⠀⠐
⠄⠀⠀⠠⢀⠤⠀⠀⠀⠀⠀⠠⠀⠐⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⢀⠀⠈⠹⡆⠀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣇
⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡟⠀⠀⠀⠈⠙⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠘⢦⡀
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⢻⡄⠀⠀⠀⠐⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠙⠦⣄
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⣠⠀⢀⠓⠂⠀⠤⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠈⠁⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠰⡄⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡠⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀
⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠍
⠀⠀⢱⡀⠐⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⡆⠀⠐⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠟⠛⠂⠁⠀⠀⠀⠈⢦⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⣀⣤⣤⣤⣤⡤⠤⢤⡀⣧⠀⠀⣠⣯⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡴⢾⣿⠿⠟⠛⠋⠉⠀⠀⠀⢼⠁⠙⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⣼⣠⣄⣘⣿⣿⣿⣿⣿⡟⠻⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⣠⣤⢤⡤⠀⣠⣀⣻⡏⢭⣽⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⣠⣶⣿⣷⣶⡶⠾⢧⠀⠀⠀⠀⣶⠾⠟⠛⠋⠉⠉⠁⠀⢹⣟⣿⣿⣿⣿⡼⠀⠀⠀⠀⡀⠙⠄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⣿⡇⠀⠀⠀⢐⡆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠰⠶⠖⠓⠖
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣠⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠘⣿⣿⣿⡟⠻⠿⣿⣿⢽⣿⣿⣿⣷⡄⣀⡤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷
⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠈⣫⣾⣿⣿⣿⠿⠋⠁⠀⢀⡄⢀⣤⣤⣤⣄⠀⠀⠀⠀⣀⣀⠨⡀
⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⢸⣿⣿⣿⠟⠉⠀⠀⢀⣰⡾⢰⣿⣿⣿⣿⣿⡷⠀⠀⣿⣿⣿⣷⣾
⠀⠀⠀⠀⠀⢿⠟⠀⠀⠀⠘⣿⡿⠃⠀⠀⠀⠀⢼⣿⠀⠘⣿⣿⣿⣿⡿⠆⠀⢠⣾⣿⣿⣯⣉
⠀⠀⠀⠀⠀⠸⡖⢦⡀⠀⢰⡿⠁⠀⠀⢠⣶⠾⠋⠁⠀⠀⠈⠉⠉⠡⠀⠀⢀⣼⣿⡏⣭⠋⠘⡄
⠀⠀⠀⠀⠀⠀⣧⡠⢷⠀⠀⠁⠀⠀⠀⢸⡏⢠⣴⣆⢢⣀⠀⢀⣴⣤⣄⠀⠘⠋⠉⢸⠿⣧⣰⠃
⠀⠀⠀⠀⠀⠠⠶⠚⠋⠁⠀⠀⢤⣦⣶⣮⣴⣿⠛⠿⠀⢻⣿⣿⠿⠛⠋⠀⠀⢀⠠⠜⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣆⠀⠀⢸⡟⠛⠲⠒⠀⠁⣈⣤⣤⣴⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⡄⠀⢸⣷⣶⣶⣶⣿⣿⣿⣯⣥⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠸⠛⠛⠏⠉⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠳⠦⠤⣀⣠⣤⠴⠞⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⢛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣆⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢀⡾⢻⡆⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣆⣀⣀⠀⢀⣀⣀⣿⣁⣸⣿⣋⢸⡧⠤⠦⠄⠒⠒⠂⠈⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⡀⠀⠀⢀⣨⣇⡀⠀⠘⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠂⠤⠤⣤⣴⡿⣉⣉⡙⣷⣾⣋⣉⣉⣛⣦⣀⣀⣀⣀⣀⣀⣀⡀⠀⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡏⠀⣀⣀⣀⠈⡇⣉⣉⣉⠉⢹⣯⡉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⢦⣄⢀⣴⠟⠉⣷⡎⠀⠀⠀⢹⡟⠁⠀⠀⢡⣾⠋⠙⢦⡀⣀⣴⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⠔⠒⠠⡀⢙⡿⠁⣠⠞⠉⢏⠁⠉⠉⣸⣯⠉⠉⠉⣸⠋⠻⣦⠈⢻⡏⠀⡤⠒⠂⢄⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠁⠀⠀⠠⠀⣾⠁⢰⡟⢀⣠⡾⠟⠛⠛⠛⠛⠛⠛⠻⠷⣤⡀⢹⡇⠀⢿⡄⠠⠂⠀⠀⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣀⣵⠶⠶⠾⠾⢷⣤⣈⣿⠟⠁⠀⠀⠀⠠⠄⠀⠄⠀⠀⠀⠈⠙⣿⣁⣠⡼⠷⠿⠶⠶⣶⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣸⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢸⠁⠀⠛⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠈⠓⠤⠔⠊⠉⠙⠒⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡠⡤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉
⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⢠⡾⠉⠛⠷⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠛⠉⠹⣦
⠀⠀⣠⠶⠋⠉⣁⣀⣀⣈⠉⠛⠶⣄⠀⢠⡿⠁⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠹⣧
⠀⣼⠃⢠⡶⠋⠉⠀⠈⠉⠙⢶⣄⠈⢷⡾⠁⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠹⡆
⣸⠃⢰⡏⠀⠀⢀⡀⠀⠀⠀⠀⠙⣆⣼⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⡄
⡿⠀⣿⠀⠀⣴⠋⢹⡆⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠂⣠⡄⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣷
⣿⠀⢿⡀⠀⣿⣀⣾⡀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⢹⡆
⠸⣆⠈⢷⡀⠘⢧⣀⠉⠛⠚⠋⢹⡇⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢹⡄⠀⠀⠀⠀⠀⣿
⠀⠙⣦⡀⠙⢦⣀⠉⠛⠶⠶⠚⣿⠀⠀⠀⠀⠀⢰⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠈⠻⢦⡀⠙⠳⢦⣄⡀⣸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠉⣿⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⣷⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇⠀⠀⠀⠀⠀⢸⣇⣀⡴⠟⠋⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠛⠷⣄⠀⣿⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⠿⣆⠀⣴⡆⢺⡉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢹⡇⣿⠀⣴⡄⢀⡿⠷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠹⠾⠋⠹⢾⣇⢿⡀⠀⠀⠀⠀⠀⠐⢒⣶⠶⠶⠶⠶⠶⣖⠂⠀⠀⠀⠀⠀⠀⣸⢇⣿⠞⠉⠻⡾⠁⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⢷⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣰⠏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⢿⣿⠆⠀⢠⡇⠈⢻⣦⣤⡀⠀⡀⠀⡹⣦⠀⠀⢀⡞⡁⠀⡀⠀⣠⣤⣾⠋⠘⣷⠀⠀⢸⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⣿⢸⡿⣦⣠⡾⠁⣰⠏⠉⠉⠻⠞⠛⠿⠛⢻⡆⠀⣾⠛⠛⠟⠛⠾⠋⠉⠉⢷⡀⠹⣦⣠⡾⣿⢸⡇⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡼⠿⠛⠀⠀⠀⠀⠀⢿⣄⣀⣀⣠⣄⣀⣤⣤⡼⠇⠀⠿⣤⣤⣤⣀⣄⣀⣀⣀⣼⠇⠀⠀⠀⠀⠘⠿⠷⢤⠴⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⢀⠠⠤⢄⣀⠀⠀⠀⠀⠀⢀⡀⠀⣠⣀⣀⣀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢹⣂⣴⣷⢖⡟⡛⠾⣙⠲⣖⣶⡿⣛⣾⠿⠦⣼⡿⠤⠙⠋⠁⠀⠀⠈⠀⢐⡴⠒⢂⣀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⣶⡿⣫⡾⠁⣾⣀⣱⡀⢉⢴⣬⡉⠉⢁⡠⠖⠒⠒⠢⠦⣤⣀⣀⣀⣀⡠⠔⠃⣴⣾⠭⠟⢉⡀
⠀⠀⠀⠀⠀⠀⢀⡛⢿⣉⠈⢦⠟⠉⠛⣧⡄⠀⠀⢱⠞⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠳⠶⠤⢤⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠠⣈⢳⡆⢀⣶⡄⠈⢇⢀⡐⠁⠀⣠⣴⣿⣭⣭⣍⡙⠲⠦⣤⣤⠴⢾⠳⢓⡯⢷⣿⣭⣴⠏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣷⣿⣿⣻⣦⠔⠋⢀⡴⠚⣉⣥⣶⣿⢣⢒⠈⢦⣤⣤⣤⣦⣵⣾⠭⠴⣬⣿⣿⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠈⠿⣝⡲⣄⣝⣮⣉⣉⣁⣤⠴⣋⡴⠛⠁⠀⠀⢦⠔⢩⡵⡀⠙⠉⠉⠉⢉⣵⣞⣿⣿⣿⣿⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⠷⠶⣶⣿⣿⣿⡶⠟⠩⠴⠶⢤⣤⣴⣮⣂⣢⠜⣜⢄⠀⣀⣤⢟⡵⣫⡹⣾⣿⠏⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣱⡟⠓⠆⢶⢦⣤⣄⣀⣀⣠⣴⠯⠖⢚⣽⣟⡿⢯⠭⣟⣾⣷⣟⡹⣴⣿⣛⣹⡿⠟⠁⣠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣟⠿⡔⠦⡀⠈⠲⢬⢉⠉⢀⣠⣔⢾⣟⢋⣭⣽⡆⣠⣾⠟⣳⠿⢛⣋⣉⣭⣥⣤⣴⣾⣻⡟⣮⣻
⠀⠀⠀⠀⠀⠀⠀⠦⠀⢵⠀⠀⣶⣴⡞⡉⠙⠛⠳⣤⡉⢿⠔⠈⠋⠙⡿⢡⣾⡷⠚⠋⠛⠙⠒⣶⣾⢟⡗⠟⣻⡿⢹
⠀⠀⠀⠀⠀⠀⠀⠹⠀⠸⡄⠀⡎⢿⣿⣶⣿⣷⡾⣿⣿⣦⢫⠜⢠⢠⠀⣾⠿⡿⡓⣣⢚⡼⠞⣋⣭⢽⢿⠟⡻⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⡇⠀⡟⠆⡛⠛⠾⢾⣞⠽⠫⣿⡟⣰⢣⠃⣰⠏⣿⡿⡇⣰⠟⣠⡞⢻⡶⣶⣮⣔⣁⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢧⠐⣷⠤⠬⢭⣿⣦⢹⣞⡳⣼⣇⣧⡏⣰⡿⣥⢈⣶⡿⠋⣰⡿⣫⢚⠷⣾⡿⡋⢙⡿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡘⢦⣜⣿⣅⣚⣟⣛⣻⡵⠜⣛⡉⠥⠬⢬⣩⡟⣛⣋⣠⡶⠋⣈⣠⣍⡛⠰⠴⡞⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⠻⡙⠛⣻⠿⠶⣦⣽⣤⡶⠐⠒⠚⣼⣷⣯⣬⣥⡶⣟⠉⢺⣼⣯⣿⠻⢛⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⣳⣦⣬⣮⢤⢤⣄⠈⣟⠻⠰⡥⣀⢉⣽⠟⡞⠹⢹⢟⢺⡳⢦⣭⡾⣹⣷⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⠀⣶⣿⡄⣺⣶⠋⢀⠜⡠⠋⠨⠃⠜⡡⠁⠎⠼⠀⠛⣧⠀⡧⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠼⠾⢱⣿⢁⣄⣰⣛⡋⢳⡒⣂⣋⢉⣲⣤⣾⡂⠈⠩⡆⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣮⣿⠁⣵⣬⡙⠮⣝⢀⣀⣉⡉⠍⠚⠉⣻⠽⠇⣨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣈⣁⣟⠀⠀⢈⡼⢉⠔⣡⠄⡠⠈⣳⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠭⣴⠶⢋⡠⠋⡴⢁⡞⡰⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠁⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⢄
⠀⠀⠀⠒⠉⠘⠀⠀⠀⠰⢄⣀⠈⠀⠠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢠
⠃⠀⠁⠐⠤⢀⠁⠀⠀⢀⠠⠏⢀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⠀⠀⡄⠀⢠⠀⠀⠡⠜⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠃⢂⣴⡀⢄⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⢂⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠀⠈⠉⠉⠀⠒⠠⣤⢀⠀⠀⠐⡶⠣⣤⡄⢫⠀⠀⠀⡠
⠀⠀⠀⠀⠄⠀⠀⠄⠀⠠⠀⠀⠀⠈⠁⠑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠃⡶⠋⠈⠙⢷⠀⢈⠶⠂⠤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠒⠈⠁⠁⠶⡀⣸⠑⠁⠀⠀⠀⢆⣁⠀⢑
⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠈⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠂⠤⠎⣅⠈⠁⠈⠢⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡖⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣀⡀⠀⠀⠀⢀⣠⡖⠊⠀⠀⠠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣿⡿⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡠⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠓⡁⢀⡉⠂⢄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⠀⡀⠀⠄⣀⢀⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⢀⠈⠀⡸⠂⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠉⡉⠹⠿⠿⠋⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⢊⢀⡀⠀⠊⢃⡀⠔⠈⠆⠘⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢤⠀⠀⠠⣄⠀⠀⠀⠀⠀⢠⡀⠀⠀⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⣀⠈⢀⠀⠀⢧⠀⡁⠊⠳⡀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠘⠄⠀⠀⠘⡄⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠜⣛⠀⠀⠀⠈⣓⡤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⢙⠁⠀⠀⠈⢒⣀⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⢄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠎⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⣀⣀⣶⣓⣆⡀⠀⠀⠀⠀⠀⠀⢀⣴⡖⢂⣸⣆⣀⣀⢀⠀⠀⠀⠀⠀⢠⡾⢿⣶⣴⣾⣿⣿⣶⠀⠀⠀⠀⠀⢀⡴⡆⢀⣀⣀⠀⣠⣤
⠀⠀⡰⠛⠛⢿⣿⣿⣿⠿⠛⢦⠀⠀⢠⠴⠿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠶⠠⠄⠀⠀⠘⠁⢼⣿⠿⣿⣿⡍⠙⠀⠀⠀⠀⠀⢨⠄⣽⣿⣿⣿⣿⠁⠨⡅
⠀⠀⠁⠀⠀⠈⠉⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠟⠛⠛⠛⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠘⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⣤⣶⣶⣤⣀⣤⣶⣤⡀⠀⠀⠀⠀⠀⡶⢿⣆⣾⡗⣴⣿⠻⡗⠀⠀⠀⠀⠀⢠⢿⣄⣀⣤⣀⣀⡴⣤⠀⠀⠀⠀⠀⠀⣀⣠⣄⣠⣾⣁⣀⣀
⠀⠀⠈⠀⠀⠹⡿⠿⣿⠋⠀⠉⠀⠀⠀⠀⠀⠓⢸⣿⣿⣿⣿⣿⡆⠶⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠘⠉⠙⢿⣿⣿⣿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⣾⠿⣿⣷⣿⣿⣿⣿⡆⠀⠀⠀⠀⢤⠉⠉⣷⣶⣶⣶⠟⠛⠓⡀⠀⠀⠀⠀⢀⣀⣴⣆⣤⣦⣤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠻⢿⡄
⠀⠀⠀⠃⠐⠿⠿⠿⢿⡧⠈⠁⠀⠀⠀⠀⠀⠀⠸⠿⠿⡿⠿⠃⠀⠀⠁⠀⠀⠀⠀⠈⢹⡿⠿⠿⠿⢿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⠃


⠀⠀⠀⠀⠀⠀⣠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⡾⠛⠻⣶⣿⡀⢈⣾⠛⢻⡆⠀⠀⠀⠀⡴⠾⣿⣷⣦⣶⣿⡷⣦⠀⠀⠀⠀⠀⠐⠞⣷⣼⣏⣼⣿⠛⠇⠀⠀⠀⠀⠴⠛⠛⠻⣿⣿⣿⣿⡿⠋⠉⠓
⠀⠀⢡⡀⠀⣿⣿⣿⣿⡇⠀⣼⠀⠀⠀⠀⠚⠀⠀⠈⠿⠛⠻⠃⠀⠀⠆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠉⠹⡇
⠀⠀⠀⠀⠀⠉⠙⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⢀⣤⣶⣶⣴⣿⣦⣶⣤⡀⠀⠀⠀⠀⠀⡼⢷⣀⣤⣤⣄⣠⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⣀⡀
⠀⠀⠈⠀⢀⣿⣿⣿⣿⡏⠀⠉⠀⠀⠀⠀⠀⠖⢸⣿⣿⣿⣿⡇⠀⠶⠀⠀⠀⠀⠀⠀⣐⠛⣿⣿⣿⡟⢳⡀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⢻⡇
⠀⠀⠀⠀⢸⠏⠀⠀⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢡⣾⣿⣿⣿⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⢘⠇
⠀⠀⠀⠀⠠⠄⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⢩⠀⠂⡆⡀⡄⠈⠻⣄⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣡⡖⣲⢸⠈⢰⣅⣧⠁⠀⡄⠘⣆⢠⠔⠋⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⢰⢠⠛⡎⢸⣿⣿⣿⣷⣄⡧⡄⢻⣧⡀⠀⠀⠀⠀⠠⢀⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣘⠘⢀⣱⣾⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣄⡄⢢⣴⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢲⡿⠛⠋⠘⠳⠿⣿⣿⣿⣶⡹⣿⣿⣷⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣛⡉⣉⣀⣝⣛⠾⠟⠛⣿⣿⣎⡻⢿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⠛⠛⣏⡓⠊⠽⡄⠀⢛⣿⢯⣹⣷⢼⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⠀⠀⠀⠀⣠⣺⣍⠉⠉⠀⠀⠀⠹⠆⢰⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⢷⣴⣴⡾⠁⠀⠈⠙⠲⢄⠀⠠⣐⠞⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠢⢌⣀⣀⣀⣀⣀⡤⠂⠀⢰⡜⠀⣴⠛⠑⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣽⣷⣦⣤⣤⣭⣥⡤⠊⣀⡼⠏⠀⠀⣿⠀⠀⢻⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣼⣿⡟⣻⣿⣿⣿⡶⠛⣻⠋⠀⠀⠀⢸⣿⠀⠀⠀⣿⣇
⠀⠀⠀⠀⢀⡴⣄⠀⣠⠞⠁⣺⣿⠀⠏⠁⠀⢹⢟⡛⠻⡏⠀⠀⠀⠀⣿⡟⠀⠀⠀⢻⠘⡆
⠀⠀⠀⠀⠘⢧⡈⠓⠋⠳⣴⣿⣿⡶⢤⣄⣀⣸⡴⠀⠀⠁⠀⠀⠀⣸⣿⡇⠀⠀⠀⣾⢶⠋
⠀⠀⠀⠀⡀⠚⣡⠖⣦⡀⢸⠟⠉⡇⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⣻⣿⣦⣤⣀⣀⣿⢸⡆
⠀⠀⠀⠸⠤⡞⢁⣼⢁⡇⡏⠀⣸⠃⠀⠀⠀⣤⣶⡇⠀⠀⠀⠀⣼⠿⠉⣀⠀⠀⠀⣨⡇⣿
⠀⠀⠀⠀⠀⠓⠋⢻⠞⠑⠃⠀⠈⠙⠒⠛⣿⣿⣿⣷⣄⣠⣠⣼⣷⣤⣾⣿⠁⣾⣧⣯⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠒⠲⠦⣄⣯⠀⠈⣿⠀⠈⠉⠉⠉⣿⠿⠷⠚⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣄⢀⠀⠀⢈⢻⡟⢳⡟⠉⠉⠙⠲⣶⣿⣀⣀⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠻⢿⣿⣿⣿⣿⢿⣿⣿⣷⣶⣤⣀⣀⣤⣵⣌⣷⡶⠚⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠛⠓⠂⠀⠈⠳⢿⣿⣿⣿⣿⢿⣋⠕⠹⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣿⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠠⠤⠤⠠
⠀⠀⠀⠘⢷⣛⠓⣂⠔⠀⠀⢀⡼⣿⠀⠀⢐⠀⠀⠔⠉⠛⠁⠀⠁⠀⡠⠀⠀⠀⢾⡀⠐⢀⢂⣀⡔⢰⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡼⠟⠐⠁⠀⠀⠘⠦⠤⠂⠐⠢⠀⠀⠐⠊⠀⠀⠀⠀⠈⠉⠓⠊⠉
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢢⡑⠂⠀⠀⠧⠥⠤⢤⠂
⠀⠀⠀⠹⠶⠄⠀⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣨⠇⠀⡄⠀⠀⠀⠀⢠⣴⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠧⡤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⣨⣿⣀⠀⠀⠀⠈⠂⠄⢀⣈⠭⠋⠀⠀⠠⠤⣤⣤⣾⠟⠁⠀⠀⠀⠀⠀⠈⠐⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠈⠑⡀⡀⣦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠠⣄⣀
⠀⠀⠀⢦⣤⠤⠴⠆⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠁⠀⠈⠀⠀⢸⠀⠀⠂⠀⠐⡊⠀⠔⠀⠈⠻⢦⡀⣀⣠⠾⠃⠀⠘⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠰⠀⠀⠀⢀⡠⠄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣴⠖⠁
⠀⠀⢠⠂⢠⣷⠤⣀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⢀⡤⠀⢤⠀⠁⡐⠒⠲⢦⠀⠀⠀⠀⠀⠙⠛⠀⠁
⠀⠀⠘⢶⣿⣁⣠⣻⠇⣄⢰⣿⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡀⠀⠀⡀⠘⡀⠀⣀⡾⠀⠀⠤⣤⣴⣶⣶⣶⣶⣶⢀⣀
⠀⠀⠀⠀⠀⠙⠛⠋⠀⣈⡛⠋⠀⠀⠀⠈⠓⢄⣀⣀⠄⠀⠀⠀⠀⢙⠚⠋⠀⢀⠈⣈⠉⠁⠀⢶⣾⣿⣿⣼⣿⣿⣿⣿⣿⡿⠄
⠀⠀⠀⣀⣀⣴⡇⠀⠒⠛⠃⠀⢧⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣾⣅⠀⢐⠀⠀⠀⡌⡇⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠶⠊⠀⠐⠒⠛⠁⠀⠀⠋⠉⠉⠉⠉⠉⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠳⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠀⠉⠉⠐⠺⣍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠚⠉⠉⠉⠁⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠂⠀⢷⠀⠀⠀⢀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⡀⠀⠈⠀⠀⠀⠀⠘⠇⠸⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⣤⠀⠀⠀⠀⠁⠀⠀⢁⣀⣤⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠘⣆⠀⠀⠤⠄⠐⠈⠉⠑⠜⠀⠀⢀⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠖⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⢲⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠆⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡇⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⡤⣀⠀⠀⠀⠀⠀⠇⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢰⠀⢠⢣⠀⠀⡆
⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⠈⡀⠀⢃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠀⡆⠀⠃⠀⠈⢣
⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢠⠁⠀⢰⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣻⣤⣤⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣞⣁⣀⣀⣀⣀⣳⣤⣶⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠖⠒⠈⠀⠒⠛⢃⡀⠀⠀⠀⠀⣠⣴⠶⠔⣇
⠀⠀⠀⢀⡶⠦⣦⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡀⠀⠠⠄⠤⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⢀⠘⡄⢀
⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣇⣃⣀⣧⣤⠁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠖⠚⠛⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠡⠉⠠⠀⠀⠀⠀⠀⠀⣄⠀⠘⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠘⣃⣤⣠⣤⣄⠀⠐⠲⠎⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⣤
⠀⠀⠀⢀⠀⠀⠠⠠⠀⠀⠀⠀⠀⣴⢀⠖⠉⢀⠀⠀⢡⠈⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⡤⣄⠀⠀⠀⠀⠈⠉⠋⠉
⠀⠀⠈⠈⠀⢀⠀⠀⠀⢀⡀⠀⠀⣸⢛⡁⠶⠈⠀⠰⠆⢤⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣤⠟
⠀⠀⣤⠴⠂⠛⣡⡴⠦⣭⡁⢰⣿⠇⠀⢹⡄⠀⣠⠀⣴⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠠⣾⠏⠀⠀⠘⠿⠟⠻⣄⣀⣀⣈⠇⠀⠀⠀⡄⠀⠀⠀⢰⡞⢁⡴⢲⡄⠀⠀⠀⠺⠖⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣤⠴⢮⣻⣼⡏
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣈⠃⠀⠀⠈⠁⠑⠀⣀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠛⠀⠠⠀⠐⠀⠒
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡂⠄⠨⠀⠠⠅⠸⠀⠀⠓⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣭⣭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠵
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⢠⠀⠐⠀⠀⠀⣀⣠⡤⠠⠀⠀⠀⠆⠰⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠶⠂⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡚⡒⠀⠀⠀⡀⠉⠵⠆⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠈⠁⣈⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠐⠆⠀⢨⣍⡥⠀⠁⠀⢁⠀⠒⠀⠄⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⢀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡤⠤⠀⠈⢙⠉⠀⠋⠀⠀⠤⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠻⠚⠴⢇
⠀⠀⠀⠀⠐⠶⠲⠶⠶⠶⠒⠂⠀⠀⠀⠰⠞⠀⠀⠀⠀⠴⠦⠄⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⢀
⠃⠓⠢⣤⠀⢰⣶⣀⠀⠉⢉⣴⠒⠀⠀⠀⠀⠀⡀⠂⡔⠀⠀⠀⠀⠀⠀⢀⠀⡔⢄⠀⠀⢤⣀⣀⠀⠀⠀⠀⢠⠾⠿⠋⠀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠰⠀⠁⠀⠀⠀⠀⠰⠶⠀⠐⠈⠔⠂⠀⠀⠀⢀⡀⠹⠋⠠⠄⠀⠀⠀⠀⢠⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣾⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣄⠙⣿⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠙⡎⠉⢹⠀⠀⠀⠀⠀⠀⠀⣠⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠄⠀⠀⠀⠀⠀⣓⣀⢾⣤⣶⣥⣴⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢢⠀⠀⠠⢾⠟⠁⡿⣆⣮⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⡄⠀⢀⡀⠰⠁⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡠⠐⠁⠀⠀⠀⠀⣿⣿⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⣽⢟⣽⣹⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠟⠻⣿⡇⢍⡙⣯⢤⣴⣾⡯⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⡴⣠⢿⡿⣌⠈⠉⠉⢀⡟⡽⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⢱⢯⠟⠀⠀⠀⠀⠀⡾⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣿⠏⠀⠀⠀⠀⢠⠶⠿⢻⠂⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡻⣕⢄⠀⠀⠀⠀⣃⠀⠄⢀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡻⡾⢂⠀⠀⠿⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⢀⡈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠄⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠲⣤⣀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⢠⣠⣄⠀⠀⡴⠚⠙⠋⠉⠉⠙⠲⣄⠀⠀⠀⣾⠶⣿⠀⠀⠀⠘⣿⡗⢦⠀⠰⢿⡿⠇⠀⠀⣠⢶⣿⡟
⠀⠀⠀⠀⠈⠛⢿⣶⣼⣧⠞⢦⡴⠓⢦⡀⠀⠈⠳⠒⠋⣡⠞⠁⠀⠀⠀⢀⣼⠿⡄⠱⣀⣸⣇⣀⠀⡴⢡⠞⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠹⣔⣿⠀⢴⠄⠀⠀⢳⡜⡄⠐⠀⡴⠁⠀⠀⠀⠀⣴⡟⠁⠀⣇⣤⣷⣼⣇⣀⣉⠃⢸⠀⠀⠙⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⡆⢶⡿⠷⠀⢿⣧⡄⢸⠁⠀⠀⠀⠀⠀⠸⣧⠀⠀⣾⡇⠀⠘⠋⠀⢸⡗⢸⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢄⠄⢂⣴⠢⣾⣿⠃⣿⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⢳⢶⡄⢠⠶⡞⢦⢾⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢿⠟⠋⢀⡠⢿⣽⣦⣸⣄⠀⠀⠀⠀⠀⠀⢸⣧⡖⢿⣈⢉⠀⠄⠉⠁⣸⠟⠰⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⢁⣷⣶⡾⠋⠀⢀⣯⡙⠹⡏⠓⠀⠀⠀⠀⠀⠸⠋⠀⠀⠳⡔⢦⡥⢊⣱⣇⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⣀⡔⠊⠁⣿⡀⠀⠀⢠⡾⠁⠈⠑⢣⣀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⡟⠂⠤⠊⢁⡟⢣⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢦⣸⠱⡄⡰⢹⣧⠶⠛⠋⠉⠉⠁⠀⠀⠀⠀⣠⠖⠙⢆⠈⠳⣄⠀⡠⠊⢀⡔⠋⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠑⠂⠈⠋⣀⠒⠊⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⡶⠶⢀⡤⠞⢛⣲⡶⠀⠹⣄⡀⢚⣻⠟⠳⣄⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢀⠠⠤⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢙⣶⡋⠀⢸⡉⠀⣸⠛⠛⠿⣿⢯⡹⡄⠀⣨⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⣬⠰⠲⢨⠖⠀⠰⠀⠀⠀⠀⠀⠀⠰⠋⠀⠙⠀⢸⣿⣰⠧⢴⣦⠄⠘⢀⣅⣿⣔⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠔⠿⣸⡻⠸⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠙⠀⠛⠛⠀⠀⠻⢷⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠛⠉⢉⡀⠋⠕⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣆⠰⣄⣐⣾⠃⣸⠇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢈⢰⣧⡈⠃⠀⣶⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠘⣦⡈⠉⠉⢹⠃
⠀⠀⠀⠀⠀⠀⣨⣇⠀⠀⢊⣼⢀⠀⢀⣸⠁⣏⠄⡀⠀⡰⠂⠀⠀⠀⢀⣀⣤⠶⣏⡁⠀⠀⢸⡿⠲⠶⠏
⠀⠀⠀⠀⠀⠀⢹⡏⠀⠈⠀⢠⣎⠕⠛⡟⠀⠀⠑⠂⠐⣿⠆⠀⠀⢰⠋⠀⠀⠀⠀⠹⣄⠀⣼⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠠⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢹⣄⡟⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠈⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡟⡛⢻⣟⣻⢿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡿⠿⠿⠉⠋⠉⠻⠿⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⡔⠒⠲⠆⡀⠀⠶⠶⠒⠂⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠷⣾⡿⠋⠀⠸⠧⠆⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣇⠉⠷⠿⣿⣠⠤⠀⠄⠀⠀⠀⠀⢹⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢰⣿⣗⠀⠀⠀⠀⠀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣆⢨⠛⣱⣶⣶⣦⣲⠀⠁⡟⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣿⣸⣿⣇⠈⠁⠈⠋⠐⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠙⠀⠀⠀⠀⠀⣀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡶⠾⠀⠀⠚⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡘⣿⣿⡇⠀⠀⠀⠀⡼⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢿⡇⠀⢹⡇⠀⠀⠀⠀⠃⣼⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠾⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣤⣶⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⣀⢀⣊⣀⠀⢅⠀⣸⠀⡀⠀⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠆⢲⣤⣾⠃⣔⣰⠘⠱⣄⡀⠈⠣⡀⠄⠀⢀⠀⡌⡠⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⣏⠎⡞⠏⠀⠀⡷⠎⠀⠘⢿⠆⠀⠀⠂⠙⠀⠀⠊⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡤⠥⠛⣖⠂⠠⢀⠀⠈⠁⠇⠀⠀⠠⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⡆⠀⡖⠁⠀⠀⡨⠃⠀⠀⠄⠀⢰⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⠃⣰⠇⣀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠑⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⡀⠈⡕⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠘⣄⣧⠖⠂⠀⠀⠀⠀⠈⢁⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⣠⣤⡼⠀⠀⠀⠀⠀⠀⠀⡄⢈⡄⠥⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠚⠟⠠⠀⠉⠛⠀⠀⠀⠶⠶⠆⠀⠁⠀⠀⡀⠠⠄⠂⠉⠀⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠚⠓⠒⠂⠀⠀⠀⠀⠀⠐⠀⣀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠂⣄⡀⠀⠀⠀⠀⠀⡆⠀⡶⠖⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣀⣶⠏⣀⢆⡁⠠⡀⠀⠀⠀⠀⠘⠁⠀⢇⠀⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⡄⢷⡶⠊⢁⠤⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡖
⠀⠀⠀⠀⠀⡀⠀⠀⠀⡎⢀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⣏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠜⠂⠲⠿⠓⠶⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠔⠛⣈⡗⠐⢢⠒⡚⢦
⠀⠀⠀⠀⠀⡀⢒⣿⢤⡅⡀⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠐⢆⠉⠣⡤⢄⠁⢸⠋
⠀⠀⠀⠀⠀⠀⡎⠀⣸⡁⢀⣄⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠈⢿⣀⠀⢡⠀⣁⠸⢂⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠶⠚⠓⠀⠀⠉⠉⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡋⠁⠀⠀⠀⠀⠙⠳⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠻⣿⣷⡀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⣠⠖⠋⠉⠉⠓⠤⠀⠀⠀⢀⢿⣦⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠑⠀⠀⠈⠈⠻⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡞⠻⣿⣦⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠘⣷⣾⣿⡿⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠈⢙⠙⠁⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠂⠁⠀⠀⡴⠛⠳⠀⠀⠀⠙⠲⠤⠤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣷⡒⠢⠤⢀⣀⡀⠀⠀⠀⠀⣀⣀⠤⠔⣪⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣦⣄⣀⣉⣉⣉⣉⣉⣀⣤⣴⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠶⠶⢤⣌⣻⡒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠦⠤⠤⣤⣄⣀⠀⠀⠉⠛⠄⠀⠙⢦⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣊⣉⣀⣀⡀⠀⠀⠀⠀⠀⠘⠂⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣯⣀⣀⠀⠀⠀⠀⠀⠞⠂⠉⠉⣉⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡜⠁⠩⠿⣥⣀⠀⠀⠀⠀⠀⣈⣉⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢲⡖⣤⡠⠤⠶⠒⠒⠲⢦⣘⣤⠞⠉⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠈⠙⢦⡈⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠸⣅⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⠀⠀⠀⠀⠀⣤⣀⢳⡀
⠀⠀⠀⠀⠀⠐⢧⣀⠀⠀⠉⠲⡖⠛⠉⠻⡴⠆⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⠀⠀⣰⠥⠤⠤⠤⡀⠀⠀⠀⠀⢠⡘⡎⠙⠃
⠀⠀⠀⠀⠀⠀⢠⡞⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠀⠀⣒⡤⡀⠀⠀⠀⠈⠁⠴⠚⠛⠃⡸⠀⠀⣰⡆⢸⠏⠛
⠀⠀⠀⠀⠀⠀⠸⣍⠵⠖⠤⣤⣤⣤⣤⡤⠤⠴⢶⣒⠛⠋⠉⠁⠀⠀⠈⠀⢀⡤⠚⠙⠒⠒⠒⠋⠉⠁⠈⠀⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⠳⠴⠋⠀⠀⠀⠀⠈⠉⠲⡄⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡜⠀⢀⣞⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠌⠉⠉⠁⠀⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢰⢸⠇
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⡇⠀⠀⠘⢸
⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠸⢷⣶⡶⠋⢸
⠀⠀⠀⠀⠁⣰⠒⠲⡄⠀⠀⠀⠀⠀⢀⡞⠁⣀⣀⣀⣀⡟⣻⠀⠀⢸
⠀⠀⠀⠀⠀⢹⠀⠀⢹⡤⠤⠤⠴⠒⠋⢉⠟⡻⠛⠙⠿⣧⡿⢶⣆⣸
⠀⠀⠀⠀⠀⠀⢳⠖⢻⡀⠀⠀⠠⠤⢤⣏⣼⠁⠀⠀⠀⠀⢠⠀⢸⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠳⡄⢧⠀⠀⠉⣁⡤⠿⡌⠉⠓⠀⠀⠈⠁⢠⠞⢀⣧⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠷⠚⠋⠁⠀⠀⠙⠲⢤⣀⣀⠤⢞⣁⣴⢋⡼⠙⠚⠁⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣍⣉⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⢰⣿⡟⣿⣧⡀⠀⠀⢀⣼⣿⢻⣿⡄⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠈⠙⠻⣿⣷⣄⠀⠀⢠⣿⡟⠀⠈⢿⣷⡄⢠⣾⡿⠁⠀⢻⣿⡄⠀⠀⣠⣾⣿⠟⠋⠁⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣶⡆⠀⠙⣿⣷⣄⣾⣿⠁⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⠈⣿⣷⣠⣾⣿⠋⠀⢰⣶⣄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⣼⣿⠏⠀⠀⠀⠈⢻⣿⣿⠃⠀⠀⠀⢰⣿⠟⠻⣿⡆⠀⠀⠀⠘⣿⣿⡟⠁⠀⠀⠀⠹⣿⣧⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣾⡿⠃⠀⢸⣿⠆⠀⠀⠙⠃⠀⠀⠀⠀⠸⣿⣷⣾⣿⠇⠀⠀⠀⠀⠘⠋⠀⠀⠰⣿⡇⠀⠘⢿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢀⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢸⣿⡋⠙⣿⣷⠀⢸⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⡇⠀⣾⣿⠋⢙⣿⡇
⠀⠀⠀⠀⠀⠈⠿⣿⣿⡿⠃⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠁⠀⠀⠘⢿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠻⢿⣿⡿⠿⠃⠀⠀⠀⠀⠘⠿⢿⣿⠿⠟⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⢿⣿⣍⡉⠉⠉⢉⣩⣿⡟⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠋⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠈⠻⣿⣷⣦⣤⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⠟⠁⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⠀⠀⣼⣿⠏⠀⣀⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣤⣀⠀⠹⣿⣧⠀⠀⣠⣤⣄⣀⡀
⠀⠀⠀⣠⣾⡿⠿⠛⠛⠋⠀⣼⣿⣷⣾⣿⠿⠛⣿⣿⠁⠀⢀⣴⣿⡆⠀⠀⢰⣿⣦⡀⠀⠈⣿⣿⠛⠿⣿⣷⣾⣿⣧⠀⠙⠛⠛⠿⢿⣷⣄
⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠙⠛⠉⠁⠀⠀⢀⣿⡟⠀⣠⣾⡿⣿⣧⠀⠀⣼⣿⢿⣷⣄⠀⢻⣿⠀⠀⠀⠈⠉⠛⠋⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣾⡿⠋⠀⢹⣿⡆⢰⣿⡏⠀⠙⢿⣷⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠋⠀⠀⠀⠈⢿⣷⣾⡿⠁⠀⠀⠀⠙⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣻⣿⣻⣻⣟⣟⣟⣿⣻⣛⣻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⠿⠟⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠒⠛⠛⠛⠓⠲⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣤⠴⠶⠖⠛⠛⠛⠛⠛⠛⠛⠓⠶⠶⢤⣄⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⡤⠶⠖⠚⠛⠛⣋⣉⣉⣛⠛⠛⠒⠶⠦⣼⣏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣩⣥⣴⡶⢖⣻⣿⡏⠉⠉⠉⠉⠉⠉⣿⣿⣗⠶⣶⢤⣭⣹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⡇⠈⠁⡀⢈⡗⠿⢟⣫⣄⠀⠀⠀⠀⠀⢀⣬⣛⠾⠗⣏⠀⡀⠈⠀⣿⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠁⠀⠁⠀⠈⠙⢿⡁⣴⠋⠀⠘⣧⠀⠀⠀⢠⡟⠀⠈⢳⡄⣹⠟⠉⠀⠀⠁⠀⢸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠈⠀⠀⢠⡶⠒⢶⣼⢣⡇⣠⠶⡄⢹⠀⠀⠀⢸⠃⣠⣄⠈⣧⢻⣴⠖⠲⣦⠀⠀⠈⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⣼⠃⣿⠀⠀⣠⠿⠀⠻⢯⣄⣿⠟⣠⣤⣤⡘⢷⣇⣼⠷⠃⠸⢧⡀⠀⢸⡇⢳⣦⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣦⣀⣏⠀⢤⡀⠀⠀⢀⡞⠁⠀⠀⠙⣆⠀⠀⠀⣠⠆⢈⣇⣠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣦⡀⢻⣄⠀⠘⣧⠀⠀⠀⣠⡟⠀⢀⣼⠃⣠⡞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢿⣛⢦⣌⣛⠒⢚⣋⣤⢞⣻⠇⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣯⠛⠛⠛⠛⠛⠛⠛⢫⡟⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡘⢷⡛⠙⢷⠏⠙⣷⠟⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣙⠛⠓⠛⢛⣡⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠈⠀⠈⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠤⠔⠒⠋⠉⠉⠓⢲⠏⠀⠀⠀⠀⢀⡴⠷⠿⠦⣤⠴⠃⠀⠙⣆⣀⣀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⡤⠊⠀⠀⠀⠀⠀⠀⠀⣀⣾⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣏⠀⠀⠀⠀⠉⠀⠈⡇⠀⢀⡴⠀⠀⢀⡴⠓⠒⠒⠠⠤⠄
⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⢀⠗⠂⠄⠀⣿⣆⢀⡆⠀⠀⠀⢀⣷⠞⠛⠉⢳⣖⡭⠠⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣼⣧⡀⠀⢀⣠⡏⠀⠀⠀⢠⡼⠛⠛⢧⡀⠀⠀⠈⢷⣀⣀⣀⣼⣧⣤⣀⢀⠀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠊⠉⠉⢢⣀⣘⣿⡷⠞⡽⠁⠀⠀⠀⣾⠀⠀⠀⠀⠙⣆⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠻⠷⠶⠶⠛⠁⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⣄⣀⡠⠤⠒⠒⠒⠒⠒⠲⢾⡀⠀⠀⡴⠋⠉⠑⠄
⠀⠤⠤⠤⠤⣄⡀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⢠⠁⠀⠀⠀⡄
⢣⡄⠀⠀⠀⠈⠙⢦⡀⠸⡄⠀⠀⠀⢰⠃⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⢀⣤⠤⢤⣄⠀⠀⠀⠀⠀⣸⡇⡆⠀⠀⠸⢧⣄⡀⠠⠄⣄
⠀⠉⠐⠢⣄⠀⠀⠀⠹⡄⠈⠢⣄⣀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢰⠉⠀⠀⠀⠈⠙⠶⢤⣤⡴⠛⠀⠙⣄⡀
⠀⠀⠀⣠⠏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠉⠁⠄⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⣇⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣀⣀⣀⣤⡴
⠀⢀⡎⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠁⠀⠘⢦⣀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠈⠣⣄⣀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠁⠀⠄⠀⠐⢀⠀⠀⡀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢁⣀⠄⠀⠀⠀⠀⠀⠀⣛⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣾⣿⣶⣾⣿⣿⣿⠶⠀⠀⠈⡘⣻⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣍⣻⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⣿⠃⠙⠈⠳⣍⠉⠙⠛⠛⢛⣽⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣹⣤⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡛⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣯⣽⣿⣿⣯⣿⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣀⣭⣿⣿⣿⣷⣬⡆⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣠⣴⣿⣶⣶⣶⣤⡄⠀⠀⠀⠀⠈⠉⠁⠀⠙⡿⠿⣿⡇⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣸⣤⣴⣶⣶⣾⣷⣶⣤⣹⣿⡿⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⢸⣯⢸⣿⠏⢿⣛⣛⣻⣿⣿⣿⡆⠺⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠈⠛⣾⠛⣿⢿⡟⢹⣿⣿⠁⠀⢸⣿⣏⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠈⠐⠈⢁⡿⠛⡏⠀⠀⣿⣿⣏⡙⠛⠋⣸⣿⣿⣿⣿⡷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣻⣿⣿⣦⠉⠛⠻⢿⣛⣿⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣏⡀⠁⠀⠀⠀⠀⣹⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠄⡀⠀⠀⢠⡞⠁⠀⠀⠀⠀⣸⣿⡟⣷⡄⠀⢀⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⠘⠀⠀⣿⠷⣶⣤⣀⣠⣴⣿⣿⣴⣿⡇⣶⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⣿⡶⠄⠈⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠸⢀⠀⣼⣏⠠⠤⠤⠤⠤⣤⣤⣄⣈⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡼⢿⣭⡟⠁⠀⠂⢤⣤⣤⣤⣄⣠⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡄⢀⣄⠁⠀⠀⠀⠀⢐⠃⠀⠙⠳⣤⢀⣀⠀⠀⣼⣍⢿⣿⣿⣿⣿⣿⠁⢻⣇
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉⠛⠀⢤⣤⣴⣶⣾⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⣾⠈⠄⠀⠀⠉⣺⣿⣿⣿⣿⠇⠀⢈⡏⢷⣄
⠀⠀⠀⠀⠀⠙⠛⠳⠶⠾⢷⣿⣄⣈⣿⣿⢿⣿⠇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠙⠿⣷⣶⣶⣿⣿⣿⣿⣿⠇⠀⠀⠰⠇⠀⠙⠦⢤⡀
⠀⠀⠀⠀⠲⢦⣤⣄⣀⣠⣇⡀⣈⡉⢹⡇⣸⣿⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠁⠀⠀⢰⡟⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢀⣀⡀⠀⠉⢉⡉⣉⣙⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡀⠀⠚⣿⡿⠛⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠙⠛⢷⡄
⠀⠀⠀⠀⣈⠉⠛⠿⣿⣿⣛⠋⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⣸⡏⢀⣦⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠂
⠀⠀⠀⠀⠈⠛⠲⠾⠿⠿⢿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⢠⠿⢧⣶⣤⡤⢐⣿⠏⠁⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠿⠟⣠⡾⢁⡄⠀⣠⡾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⣿⡟⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⣾⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⣩⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣾⡿⠟⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣩⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⣡⡾⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⠿⠿⠿⠻⢿⠟⠙⠿⢿⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣛⣛⣻⣟⣿⣿⣿⣟⣛⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠿⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢋⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣶⣿⣿⣿⠿⠟⣛⣫⣭⣭⣥⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢛⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣬⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢟⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣍⣛⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣾⣿⣿⣿⣿⠿⣛⣩⣙⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣛⣉⣙⡻⢿⣿⣿⣷⣶⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣦⢻⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣋⣿⣎⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡘⣿⡟⢿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣝⠻⠿⠿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠻⠿⠿⢛⣵⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢻⣿⣿⣿⣿⣿⠋⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⣿⣿⣿⣿⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⢻⣿⣿⣿⣿⣧⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡛⢣⣿⣿⣿⣿⡿⣡⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡙⢿⣿⣿⣿⣷⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣱⣿⣿⣿⣿⢟⣴⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⣙⢿⣿⣿⣿⣷⣭⣛⠿⠿⣿⣿⠿⠿⢟⣋⣵⣿⣿⣿⡿⢟⣵⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠛⠿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⠛⠉⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠶⠖⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠶⠶⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⣠⠀⢀⡄⠀⢘⡃⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⢘⠁⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⣿⣤⣼⡇⠀⢸⡇⠀⠻⣯⣬⡀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣭⣌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢿⠀⠸⡇⠀⢸⡇⠀⢤⣤⣼⠇⠀⠀⠀⠀⠀⢸⠇⠀⢤⣤⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⢠⣤⣤⣤⠀⠀⣤⠀⣠⠀⠀⣠⣤⣤⡄⠀⢨⡅⠀⣴⡶⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⢦⡄⠀⠀⢸⡏⠀⠀⠘⣿⠀⣸⡇⠀⣻⣦⡼⠏⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠴⠶⠾⠋⠀⠀⠸⠇⠀⠀⠀⠻⠶⠟⠀⠀⠺⠀⠀⠀⠀⠸⠇⠀⠻⠶⠞⠃⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣼⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣼⢳⡄⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢀⣠⣿⣀⣷⣄⣀⣀⣀⣤⡾⠟⠉⠀⠀⠀⣼⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠀⠀⠀⠀⠀⣰⣏⡴⠞⠛⠋⠉⠉⠉⠉⠉⠉⠛⠋⠀⠀⠀⠀⠀⢰⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣴⡾⠻⣿⠿⢶⣿⠀⠀⠀⠀⣠⡴⠞⠛⢿⡿⠛⣷⡄⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣄⣼⠏⠀⠉⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⢻⡗⠶⠶⠶⠖⠛⠉⠀⣄⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⡏⠀⠠⠄⠀⣀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠙⠛⠂⢀⡄⠀⢠⣤⣿⣄⡀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠉⠉⢹⣇⠀⠀⢀⡀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠘⠃⢀⡀⠀⠀⣤⣄⣼⠃⠀⠉⠂⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠋⠻⣦⣄⣈⣁⣀⣀⣤⡶⠋⠙⢷⣦⣄⣀⣀⣀⣰⣦⣤⠾⠛⠛⢦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠛⠉⠁⠀⠀⠀⠀⠈⠉⠙⠋⠉⠉⠁⠹⡆⠀⠀⠀⠁⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣤⣤⣤⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⢀⡀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⢙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⣿⢿⣿⣿⣿⣶⡄⠀⠀⠀⠛⣿⣦⡀⠀⠀⠀⣠⠶⢋⣠⣀⣀⣤⣦⡲⣤⣾⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠙⣯⣿⣿⣿⣯⣶⣶⡂⠀⠀⠉⢿⣿⣆⣠⠞⡁⠀⠊⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⠿⠟⠡⣦⣭⣽⣷⣶⣽⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⢹⣾⣿⣿⣿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⢿⣿⡿⢻⣭⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⢿⡿⠁⠀⠀⠀⠈⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣿⣯⣴⠿⠿⠿⠛⠛⠋⠁⠀⠀⠀⠀⢀⣠⣦⡹⣟⣿⣿⡟⠀⠀⠀⠀⠀⠰⠰⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠒⠀⠀⠀⠀⢀⡴⠏⢁⣠⠴⠶⠶⠖⠒⠶⠶⢤⣤⣀⠀⢠⣻⣿⣿⣿⣾⡿⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠷⠄⢻⡀⠀⠀⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠙⠶⣝⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠺
⠀⠀⠀⠀⠀⠀⠸⠿⠷⠟⠃⠀⠀⠹⠇⠀⠀⠀⠆⣀⣴⡾⠿⠟⠛⠓⢦⠀⠀⠀⠈⢻⡿⠛⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⣿⣦⡀⠀⠛⣩⣴⣠⣤⣤⣄⣄⡦⠀⠀⢀⠏⠰⠋⠀⣿⠀⠀⠀⠀⠰⠶⠰⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⢿⣿⡀⠀⠿⣿⡏⠸⢿⣻⠗⠀⠀⠀⠀⠀⢤⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢮⡽⠀⠀⠀⠀⠘⠛⠋⠁⠀⠀⠀⢀⡄⠀⠾⠐⣰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠒⢶⣤⡀⠀⠀⠀⠀⠀⡜⢁⣴⡤⣂⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⠀⢀⣤⡄⣠⠟⠈⠢⠀⠀⠀⠀⠀⠉⠙⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣴⣞⣀⣤⣤⣤⣤⣡⠀⠄⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣍⣄⣠⣴⣾⣿⡿⠃⠀⠀⠀⢰⣼⡏⠘⣿⠿⣷⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣯⣿⡿⡍⠸⠛⠿⠁⠀⠀⠀⠀⣿⡏⠀⠀⠸⣇⠉⠙⢿⣿⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠿⣿⣿⡿⢀⣀⠠⢂⡀⠀⠀⠀⠀⣸⣿⡏⠀⠀⣴⠋⠀⠀⠀⠈⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⣤⣴⡿⠁⠀⠀⢠⣷⣿⣿⣧⡴⠟⠁⢐⣮⣶⣶⣶⣵⣝⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣅⠀⠀⢀⣠⣿⡿⠿⠛⠁⠀⠀⠀⢨⣿⣿⠉⠉⠛⠛⠙⠛⠛⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣤⣶⣿⡿⠛⠁⢀⣤⡀⠠⡀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡿⠟⠛⠉⠉⣿⣿⡟⢉⡄⠀⣰⠏⠀⠙⢶⣿⣦⢳⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠉⠁⠀⠀⠀⠀⠀⢹⣿⣷⣿⣧⣼⠃⠀⠀⠀⠀⠙⢿⣿⡷⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢋⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⣽⣽⡇⠂⠀⠀⠀⠀⠀⢠⡿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⠤⠀⠀⠀⠠⣄⣀⠀⠀⡁⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢤⡀⠀⠀⠀⠄⠠⠀⠀⢠⠀⠀⠀⠀⠀⣀⠠⡶⠆⠀⢆⠿⡁⡅⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠠⠇⠀⠀⠀⠀⠀⡠⠆⠀⠈⠁⣀⠀⠀⠐⠒⠹⠁
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⡀⠀⠀⠀⣄⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡄⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⢀⠶⣭⠙⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⡎⠁⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⠀⠀⠴







⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠶⠏⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠓⠘⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠈
⠀⠀⠀⡠⡖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣖⠀⠐⣷⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠
⠐⠀⠐⣴⠔⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣮⣯⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠉⠤⠼⠂⠀⠀⠀⢤⣤⠂⠀⠀⣰⡿⠀⠠⠼
⠀⠀⠀⠈⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⣀⣄⠀⠀⣀⠀⡀⠀⠀⣀⣀⣄⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣴⡓⢀⣲⣀⣄⠀⠀⠀⠀⡀⠀⢀⣀
⣿⡯⣟⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⢀⠄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡈⠉⢁⠀⠏⢠⣋⣀⣀⣀⣤⣶⠊
⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⠛⢻⣕⣶⣢⣧⣶⣷⣶⣶⣿⣿⡿⠋⠉⠉⠁⡀
⠀⢨⡉⠓⠒⣲⣌⣀⣔⣄⣁⠀⡄⠀⠀⠀⣠⣬⢤⣍⣛⣃⣬⣽⣷⣿⣿⣻⣩⣙⣻⣟⣿⡿⠁⠀⠀⠈⠉
⠀⠈⣯⣫⣭⡶⠟⢓⣺⡟⣏⠀⠀⠒⠶⠾⠋⠉⠳⣬⣭⣯⡭⢴⣶⣶⣾⣿⣽⡯⢷⡿⠋⠀⠀⢀⠄⡘
⠀⠀⢛⣫⣟⣽⣟⣑⢎⣽⣮⢳⠂⠤⠤⠤⠦⣤⠀⠀⢨⣿⣿⣿⣿⢷⣶⣾⣽⣿⣋⣤⣄⣤⣶⣅⡠⠓
⠀⠀⠀⠉⠉⢙⣻⣖⣻⣯⢺⢆⠀⠀⠀⢀⣀⣌⡇⠀⢸⣿⡟⠁⣤⡀⠨⣙⣫⣽⣷⢮⡉⠍⠛⢦⣄
⠀⢰⣄⣛⣲⣤⣤⣷⠳⣗⡷⡘⣟⡝⣏⣽⣛⢟⠇⠀⠈⠛⠛⠮⣭⣤⣭⡶⣶⣄⠁⠠⠉⣰⠀⢸⡯⠛⠴⢤
⠀⠈⢮⡋⢓⡙⠘⢻⣧⠈⣿⢿⣓⣋⣩⢥⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⣿⢷⣿⣧⡴⠮⣄⣀⣀⣤⣱⡖
⠀⢀⣄⠈⠀⠈⢻⠧⣿⣷⣌⡳⠭⡭⢶⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣿⣛⣻⡵⣋⣝⣪⣉⡉⠙⠛⣃
⠀⠘⢧⣄⠀⢊⣍⣙⣾⣿⣿⣻⣿⣿⣿⣿⣾⣶⠊⠁⠀⠀⠀⠀⠠⢀⣧⣬⣉⣫⣽⣿⣶⣿⠿⠛⠛⠿⠥⠞⠉⣀
⠀⠀⠦⠜⢉⠀⣿⣟⡭⣿⣿⡿⠿⣿⣿⣍⣿⣿⣿⣷⣦⣤⡄⣠⠀⣿⠿⠷⣷⡞⣶⣾⢷⡾⣒⣚⡳⢄⡀⠀⢸⠃
⠀⠀⠉⠙⣶⣿⣻⢏⠞⠛⢙⠯⣁⣲⣬⢭⣿⣻⡿⠛⠛⠋⠀⣿⠂⠙⡟⠷⣿⣟⠿⠿⣿⡿⠭⢍⢻⣤⣥⠄⠁
⠀⠀⠀⣨⠝⠋⠁⠀⠀⠊⠉⠳⠖⢶⣽⣷⣽⣽⡇⠀⠀⠀⠀⣾⣄⠀⣇⣤⣼⡳⣭⣽⠏⠀⠁⠀⣧⣨⣥⠛⠛⠁
⠀⠀⠀⠹⠰⠄⠈⠀⡄⠐⠀⠀⢀⣁⣉⠿⠿⠾⣧⣀⣀⣠⣴⣿⣥⣾⠿⠋⠉⠑⠦⠜⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣴⣿⣿⢋⠀⠀⠀⢴⠂⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢹⣿⠿⠿⣀⠀⠐⢦⠇⠀⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣎⣤⣤⢰⡆⠺⣏⡀
⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣧⡬⢾⠶⠆⢸⣿⢻⣷⣄⡀
⠀⠀⠀⠀⣄⣈⡆⠁⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣯⣤⣴⣏⣿⠎⢟⢿⣷⣦⣀⠀⣀
⠀⠀⠀⠀⠀⠈⢣⣄⣢⠀⠀⢀⣴⡿⢋⣹⡟⣹⠻⡿⢿⣿⣍⢵⡿⣷⢀⣩⣿⣿⣿⠷⠟
⠀⠀⠀⠀⠀⠀⠀⢝⣽⡽⣦⣯⡥⣠⣭⣤⡘⠻⣿⣤⢟⠛⠛⠄⢼⢯⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠿⢷⣼⣟⣳⠽⠟⢹⣿⣦⠙⢿⠖⣺⣟⠒⡠⠃⠀⠙⢷⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⢹⣿⡁⠀⢣⠀⠻⣿⣷⠐⠠⠉⢄⠌⣿⠃⠀⠀⢨⡇⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢸⣷⠽⢴⠆⠀⠀⣽⣿⡽⠤⠐⠁⠀⣀⡠⣶⠀⠠⠼⣃⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⡏⠀⠈⠁⠑⠠⠁⠉⠂⠀⢠⡶⡎⡑⠂⡼⠛⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣌⡀⠴⠀⠈⠀⠀⡁⠀⡐⠀⢀⠀⠀⠀⡃⢈⡄⠀⠰⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⣸⣄⣀⢠⣴⡟⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡿⠉⠀⠈⢿⣿⣯⠁⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢘⠁⠀⠀⠀⠈⢛⡓⠛⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠈⠀⠀⠀⠀⠸⡿⡮⠜⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⢸⣤⠠⠜⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠍⡁⡀⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣦⡀⠈⠀⢢⠁⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⠿⡆⠀⠴⢦⣤⣤⣄⣀⡀⠀⣀⡤⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⠰⢤⡀⠀⠀⢀⣈⣉⠉⠻⢿⣦⠷⠿⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣕⣄⠴⠴⠶⠆⠀⠈⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣨⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣅⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠈⡿⠁⣩⣿⠀⡟⢻⠀⣷⣸⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢼⣧⠄⢧⠾⠀⡇⢸⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣬⣭⣬⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣾⠿⠿⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠿⢾⡁⣀⠻⢶⣬⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠶⠒⠉⠐⢍⡁⢀⡆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⢀⡈⢓⡊⠀⠀⠻⢟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣄⡀⣀⣀⣠⠜⠉⣠⡜⠆⡄⠀⣯⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠙⡿⠭⣭⠤⠴⠚⠁⡇⠀⡇⣠⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠦⠤⣶⣤⣴⣤⣾⡇⠀⡇⢸⠲⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣖⠺⠿⠿⡿⠿⢿⡇⠀⣿⡜⢀⣤⣽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⣈⠁⠒⠒⠒⠉⢀⡘⣹⢣⠋⠀⠈⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣈⠉⠉⠀⢀⡰⢫⣾⠏⢸⠀⠀⠀⢻⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠿⡛⠂⠋⣩⠖⢉⡏⠀⢼⠀⠀⠀⠘⣧⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⣷⢀⡞⣁⠀⣼⠀⠀⣾⠀⠀⠀⠀⢹⡀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⣿⠉⠉⢳⡇⠀⠀⢿⠀⠀⠀⠀⢸⡇⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠃⠀⠀⠈⠁⠀⠀⣼⠀⠀⠀⠀⢸⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢤⢼⡇⣀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢥⣤⣤⣤⣤⣀⣀⣀⣠⣴⡧⣴⣾⠿⠿⢻⠻⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢺⡇⠀⠀⣀⠀⠀⠀⠀⣼⠀⢠⡆⠀⣆⢸⡶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⡇⠀⠀⠈⠛⠶⡆⠀⠙⠶⠊⠙⠚⠉⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷
⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠛⠛⠲⢶⣂⣤⡤⠾⠷⠶⠶⣶⣶⡤⣄⣀⣀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⢾⡁⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠊⣉⡛⠙⠛⠓⠿⣄
⠀⠀⠀⠀⠀⠀⠸⢍⣓⠒⠒⠒⠒⢺⡿⠦⣤⣠⣤⣤⣤⣶⣶⣚⠫⠭⢯⣛⣒⣒⣚⣋⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠒⠒⠒⠊⠉⠉⠁


⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣯⣽⣽⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣶⣾⣿⣷⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣶⣿⣿⣿⠟⠻⠃
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣻⣟⠋⠏⠇
⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠶⣤⠄
⠀⠀⣿⣿⣷⣶⣶⣾⣿⣿⠋⠀⠀⠀⠀⠀⠉⠀⢀⡀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠘⣿⣿⣿⡏⠫⡍⠛⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⢃⢀⣎⡁⡹⡤
⠀⠀⠀⠈⢿⣿⣧⣤⠳⠄⠀⠀⠀⠀⠀⠀⠀⠈⡙⠚⠁⠐⠢⢼⠿⠀⣤⣤⣤⡠⠈⣠⣤⣏⣿⣇
⠀⠀⠀⠀⠀⠙⣿⡟⠀⠐⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⢔⠧⠤⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠙⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢰⠇⡀⠀⠀⠀⠀⠀⢀⡿⡆⠱⠒⠒⠒⠉⠉⠂
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠢⣀⠀⠀⠀⢡⠘⡷⣀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠑⠢⣠⣌⢆⠻⣮⣓⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⠛⠶⣌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡴⡀⢀⠀⡔⠆⠈⠁⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢱⠞⣶⡙⠀⠀⠀⣀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠁⠀⠄⡀⠀⡰⢄⠀⠁⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⠀⠀⠀⠰⠇⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣀⣤⣆⠀⠀⠀⠘⠁⠀⠀⠀⠀⢀⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠛⠁⠀⠈⠉⠉⠐⠊⠉⠉⢀⠖⠊⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣿⡀⠀⠀⣰⣿⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⠿⠿⠇⠀⠀⠈⠙⠛⠛⠛⠛⠋

⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠦⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⣤⣄⡀
⠶⠆⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡌⠑⣀⣀⣀⠀⣴⣶⣿⡿⠙⣻⡏⣬⣿⡾⣿⣷⣶⣄⣀⣀⣀⣀
⣠⣶⣾⣿⣿⣿⣿⣿⣿⡿⠛⠻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡀⠀⢢⡀⠀⠀⠀⠈⢿⣆⠘⣿⣿⣿⠉⠙⣻⣤⢾⡋⠀⠈⣿⣄⢀⣿⣍⣿⣟⣻⣿⣿⡇
⡿⠿⠋⣉⠁⠀⠀⠀⠀⠁⡄⠀⠿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢿⣄⠀⠡⠀⠀⠀⠀⠀⢻⣆⠘⡏⢻⣶⡴⢿⣥⣿⣿⣷⠾⢿⠏⢙⣿⣡⣽⣛⣿⡟⢿⡇
⣀⣠⣴⣀⣨⣿⣶⡄⠀⢠⣄⠀⠀⠀⠀⠘⢯⡻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⢻⡆⠸⣟⠛⣷⣾⢻⣤⣀⣿⡶⢿⡗⢻⠋⢙⣏⠉⣻⠿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⢀⣤⠀⠀⠀⠀⢈⣿⣿⣿⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢿⡄⠛⠻⣟⣻⣼⡿⣿⣻⣷⣾⡿⠻⡿⠻⡿⠛⣿⣷⢼⣿
⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡁⠀⠀⠀⠀⠚⠁⠈⠛⣉⠙⠃⠈⠁⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠸⣇⢸⣦⡿⣿⡉⢷⣾⠛⣿⣸⣷⣴⣿⡶⣿⣆⣰⣇⠈⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⢄⠝⠁⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⠈⠛⣧⣈⣷⡞⠛⣦⣿⣿⣿⡇⢹⣧⡈⣋⠈⢻⠛⢿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡇⠸⣷⠿⣿⡇⢹⣿⠋⣿⣿⣿⣿⣾⣿⣿⡿⣷⢾⣷⣬
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⠀⢀⣀⠀⠀⠀⠀⠟⢀⣽⣦⣤⣷⡞⣻⣾⣿⣿⣿⣿⣿⣿⣿⣷⣹⣦⣙⡋
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣀⣀⣀⣰⢿⡿⠛⠿⠟⠙⠿⠋⠻⠿⠛⣿⠀⠀⠀⢈⡁⣉⣧⣤⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣻⣝⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⣿⣬⡙⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢼
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠘⢿⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡈⠀⠀⠈⠁⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠘⢿⣿⡆⠀⠀⠄⢐⣿⡇⠀⠄⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⠿⣿⡆⠀⢠⡜⣿⡄⠀⠀⠈⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡟⠋⠾⠿⡀⠀⠈⠉⠙⠻⢷⡀⠀⠀⠀⢰⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⠟⣡⣾⠁⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⡄⠀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢿⣿⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣤⡕⠤⠿⠋⠀⠀⠀⠠⠤⠤⠤⡿⢿⣿⣿⣿
⠀⢸⣿⣿⡿⣷⠀⢹⡍⠀⠀⠀⠀⠀⠀⠀⣄⠙⣆⠀⠀⠘⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣄⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⣈⢻⣿
⠀⢸⣿⣿⣿⠿⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣿⣦⣤⣀⣿⠟⢋⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠛⣩⣴⠞⠁⠀⠻⣷⡀⠀⢀⠀⠀⡀⠀⠹⢿⣿
⠀⢸⣿⣿⠟⣯⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠀⠈⠛⠤⣤⣄⠀⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⣾⠿⠟⠈⠉⠁⠀⠀⠀⠀⣿⡇⠀⠸⢇⠀⠻⣦⡔⣿⡟
⠀⠈⣿⣿⣠⡝⣾⣿⣯⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠙⢷⣤⣤⣤⡴⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠘⢷⡄⢹⡇⣼⢧
⠀⢸⣿⣿⠟⠁⠸⣟⣷⣙⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠻⠿⠟⠀⡿⢩⣦⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠀⠀⠀⠀⠈⢳⡞⠳⢿⣿⢧
⣠⡿⠋⠀⠀⡀⠀⠻⢜⣿⣼⣱⠂⠀⠀⠁⠀⠀⣾⣿⣿⠆⠀⠲⣤⡀⠀⣀⡀⠀⠀⠀⣴⣿⠀⡅⠸⣿⠀⠀⣤⡠⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⢠⣿⡿⣷
⠇⠀⠀⠀⠀⢷⣀⠀⠈⢻⣯⠀⠀⠈⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠈⠛⣻⣿⣿⠀⠀⢀⣿⠈⠀⢠⠀⠻⡆⠀⠸⣿⣾⢿⣧⠀⢀⠀⠀⠀⠀⠀⢀⡿⢶⣿⣿⣿⡟
⡃⠀⠀⠀⠀⢸⣿⣿⣦⣜⠿⣆⡄⠀⡐⠀⠀⠀⠀⠈⠻⡇⠀⣰⢠⣿⡟⠁⠙⣷⣶⡾⠇⠀⠀⠘⡄⢰⡆⠀⢀⡹⡿⠈⢿⣶⡾⠀⠀⠀⠀⠀⡼⢁⡼⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠰⣼⣦⠀⠙⠀⠸⣧⣄⠁⠀⣠⣦⡄⠀⠀⠁⠀⢿⢸⣿⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⡀⠀⠻⣦⠀⠙⣆⠀⠀⠘⠀⢀⠀⠀⠀⣦⠐⢚⣿⣿⣿⣿⠯
⢣⡀⢰⣦⠀⠀⢹⣿⣷⣀⠆⠀⠻⠞⢤⠘⠻⠟⠙⠀⠀⠀⠀⢸⡞⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⢋⣰⣏⣼⣯⣿⣿⣿
⠘⣧⠀⠻⣆⠀⠀⠹⠏⣿⣿⡡⡀⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢸⣧⢉⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⠀⠀⢀⡜⡠⢥⣿⢶⡗⣶⢿⣿⣿
⠀⢹⣧⡀⠙⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⢩⣿⡟⠶⣄⡀⠘⠘⣿⡘⢌⠁⠀⠀⢸⣇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣄⣀⠄⢀⣴⠁⠀⢱⣀⣦⣸⣷⣶⠶⣦⣽
⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣀⣀⣿⣿⣶⣥⡟⢶⣦⣿⡄⣀⠀⠀⠀⣸⠏⣀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣤⣛⣡⡴⣿⠺⡟⢙⡉⣅⣹⣦⣧⣸
⠀⠀⣿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠛⢿⣿⣿⣮⣽⣯⣴⡛⠛⠿⢦⣿⣁⣤⣦⣤⣀⣀⣰⠀⠀⡀⢀⣀⡔⣩⣿⣿⣝⣿⣿⣿⡷⢾⡟⢿⠛⠿⢿⡟
⠀⠀⣿⣿⣿⡎⠙⠻⠷⣶⣤⠀⠀⠀⠀⠀⠀⠀⣀⣤⠜⣿⣽⣏⣻⡿⢻⣿⡿⢷⣭⣭⣴⣄⣢⣉⣭⣝⠛⠛⣧⠜⣦⠀⡉⠛⠹⣯⣻⣿⣿⣿⣿⡧⣾⣧⣾⣾⣿
⠀⠀⣿⣿⣿⣿⡀⢠⡄⠈⠉⠁⠂⠀⠒⢀⡛⣉⣛⣭⠾⣙⣿⡛⢿⣟⢿⡿⣷⣿⣭⣼⣟⣹⡿⣿⡿⣿⣾⣏⣸⠇⣼⡇⢻⣳⣤⣙⣿⣿⣿⣿⣿⣷⣼⣇⣸⣍⣻
⠀⠀⣿⣿⣞⣿⡟⢿⠀⠀⠀⠀⠀⣀⣶⢟⣵⣟⣽⣾⣿⠻⣎⢻⣌⢿⡜⣯⢼⣋⣿⣉⣿⣉⣿⣿⡟⢋⣋⠛⠛⢷⣏⡶⠾⢃⣿⢾⠣⣦⣌⠙⠻⣿⣿⣯⣽⡋⢛
⠀⣰⡿⣿⣹⣿⠁⠀⠀⠀⢀⣴⣿⣿⣶⣿⣷⣿⡟⣿⣿⣿⣿⣷⣽⣖⣯⠽⣞⡟⢻⠏⣯⣼⡧⣼⣅⣿⠻⣷⣧⡴⠿⢷⣶⣾⣛⡾⢷⡿⣼⣷⣦⣽⣿⣿⣿⣿⢿
⢰⣿⣿⢿⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⠟⠉⠀⠈⠻⢿⣏⣿⣋⣿⣿⣿⣿⣿⣿⣿⣶⣟⣋⣼⣤⣾⣄⣽⣿⣿⣿⡇⢀⠀⠀⠙⠻⢷⣾⣓⣾⣏⣿⣿⢿⡝⠻⣿⣾
⣿⣿⣿⣿⣿⣃⣐⣿⡿⣳⣿⣽⠿⣿⡀⠀⠀⠀⠀⠀⠉⠻⣧⣿⣨⣿⣿⣿⣍⣿⢹⣿⣿⣩⣀⣹⣿⣿⣿⣿⣟⣧⠈⣷⣤⡀⠀⠀⠿⣿⣷⣾⣹⣏⡿⣽⣧⡘⣿
⣿⣯⣿⣿⣿⡿⣿⣿⣿⣿⣿⣽⣟⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣧⣾⢤⡿⣼⢿⣿⣿⣟⣛⣸⣿⣿⣿⣿⣿⣀⣸⣿⣿⣷⣶⣤⣀⠈⠛⠿⣿⣿⣾⣻⣿⣾
⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣿⢺⡷⣿⣿⣿⣟⣹⣿⣿⡿⠋⠉⠉⠛⠻⢿⣿⣿⣿⣿⣶⣤⡈⣿⢿⣿⣿⣿
⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⡀⠀⠉⠻⠿⣿⣿⣏⠿⠛⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠙⠿⣿⡶⠟⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣶⣶⣤⣼⣿⡿⠋⠀⠀⠀⢀⣠⡾⠃⠀⠰⣿⣿⣶⣶⣶⣶⣶⣶⣈⣿⣷⠀⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢿⣼⣧⣽⣿⣿⣶⣶⣷⣾⡿⠏⠀⠀⠀⠀⠈⠉⠁⠀⠀⢸⣿⣿⣿⣿⡿⠃⣿

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

⠀⠀⠀⠀⠀⢀⡤⠂⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⡐⠉⢣⡀
⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠈⢦⢀⡀⠀⠀⠀⠀⡆⠀⠀⠱⡀
⠀⣠⠄⠀⢶⠀⠀⠀⠀⠀⠀⢰⠀⠈⠆⠀⠀⢰⠀⠀⠀⠀⢳
⠐⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠀⠀⡌⠀⣀⡀⠀⠀⠧⢀
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⢠⠃⠀⠀⡜⠀⢦⣉⣀⢠⡀⠀⡤⠤⠺⣤⣤⣄
⢰⠀⠀⠀⠀⠈⠢⣀⣀⠔⠁⠀⠀⣰⢗⣿⡿⠁⠃⠶⢡⣀⡑⠂⡇⠘⡿⣏⠃
⠈⢂⠀⠀⠀⠀⠀⡿⢧⣀⣀⠠⠊⠀⣽⡟⡇⠀⠀⠀⢿⣿⡿⠀⣁⠀⣷⠈⠁
⠀⠀⠑⢄⣀⡠⠎⠀⠸⠏⠀⠀⠀⠀⠉⠀⠸⡀⠹⡶⠶⠿⠾⢽⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠘⡗⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣝⣲⣶⣟⡡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠙
⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠤⠞
⠀⠀⠀⠀⠀⠈⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠼⢮⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠘⢤⠞⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⢀⠼⢄⠀⠀⠀⠀⠀⠀⠙⠢⣤⠎
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣀⠤⠖⠋⠁⠀⠀⠀⠀⠸⣄⠼
⠀⠀⠀⠀⠀⠀⡇⠈⣱⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⢄
⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⣰⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢎⡁⠀⠀⠀⠀⠀⢀⣈⡽⠃⠀⠀⠀⠀⠀⡆⠄⠐⠒⠒⠐⠒⠒⠠⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⠀⣀⠠⠔⠲⠶⠀⣀⠶⠄⠀⠀⠀⠀⢤⣤⣄⣀⠀⠊⠀⠀⠀⠤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡘⠁⣀⠌⠻⠁⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠎⠀⠀⠉⡹⠛⠓⢦⡀⠀⠈⣧⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⢰⡎⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠂⢀⣧⢀⣀⢸⣧
⠀⠀⠀⠀⠀⢀⣿⡶⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣦⣤⣾⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⣿⡿⡏⠀⠀⢠⣶⣼⣿⣿⣿⣯⣼⣿⣿⣿⣿⣿⡷⠖⠂⠀⠀⠀⠀⠀⢠⣿⣿⢻⣿⣼⣿⣷⢆⡄⡄
⠀⠀⠀⠀⠀⠀⣿⣾⣧⠀⣰⣿⣿⣿⠟⠛⠛⠛⠛⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⣿⣿⣿⣟⣾⣷⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣸⣿⣿⠋⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⢸⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠌⠳⡀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⢸⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⣀⣀⡴⠁⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⡿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⢀⠀⢄⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣐⣾⠏⠀⠀⠀⢰⣶⠀⠀⢀⠀⠈⠪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⢠⠘⠋⠛⠳⣤⣀⠀⠀⢸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢸⣿⣿⡿⠖⠋⠙⠛⢶⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠳⠄⠀⠀⠀⠀⠀⠀⢠⣾⡏⠀⠀⠀⠀⢠⣼⠿⠋⠀⣠⣴⠖⢢⣄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⡔⡉⠑⡆⠀⠀⠺⣿⣗⣀⣀⡉⣉⣉⣁⣤⠚⡉⠛⡁⠀⠀⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣫⣽⠁⠀⠀⠀⡑⠁⣠⠃⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⢷⣄⣉⣠⠃⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⢀⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣦⣀⠌⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⣀⡀
⠀⠀⠀⠀⠀⣠⡞⠚⢿⣿⣿⣿⣁⣰⣷⣆⣤⣶⣤⣤⣤⣼⢱⣦⣤⣤⣤⣦⣴⣶⣶⣾⣷⣾⣟⡛⠿⣥⠈⢡⣤
⠀⠀⠀⠀⢀⠉⢀⡠⠚⠋⢉⡽⠟⠛⠉⠉⠉⠉⠀⠀⠀⣀⣂⡭⠿⠛⠛⠉⠉⠉⠀⠀⠀⠈⠉⠙⠳⣌⣷⣬⣄⠄
⠀⠀⠀⠀⠈⠉⠛⠦⠴⢴⣏⣀⠀⠀⢀⣠⠄⠂⠈⠉⠉⠉⠛⠳⡶⠄⠐⠲⠶⢋⠉⠙⢻⡦⠤⠤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠒⠒⠖⠂⠀⠐⠒⠢⠤⠤⠤⠄⠐⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⢤⣀⠀⠀⠀⠀⠀⣠⠴⡒⠈⢛⡋⢐⠒⠦⢄⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠙⢦⠀⠀⠘⠁⠶⠶⠤⢤⡉⠁⠢⠈⣢⠍⠢⡀⠀⠀⠀⠠⠤⠄⢄⣀
⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠑⠦⣀⠒⠈⠲⣄⠠⠘⠀⠀⠁⠀⡠⠐⠈⢉⢓⡲⣤⣍⡢⡀
⠀⠀⠀⡼⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠸⡄⢤⢦⡀⠳⢤⡈⠀⠑⠂⠀⠀⠀⠀⣷⠒⠋⠉⠉⠉⠉⠉⠙⠛⠦⣄⡀
⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⠒⠋⠉⠀⠀⠀⠀⠈⢄⠣⢜⠃⢤⠚⠒⢤⣀⠀⠀⠀⠂⢀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠈⠙⠒⠶⠤⠤⠶⠒⠊⠉⠀⠀⢀⡀⠤⠤⠤⣄⡀⠀⠀⠀⠈⠣⡀⠳⢤⡉⠁⣠⠐⠀⠀⠀⣦⢌⠓⠶⠤⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠊⠉⠁⠀⠀⠁⠀⠐⠙⠆⠀⠀⠀⠀⠈⠢⣄⣉⠒⠦⠤⠌⣍⠢⣉⠒⢭⠒⢤⠬⠉⠳⣄⣀⣀⣀⡀⠤⠖⡫⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠶⠛⠋⠉⠓⢤⡈⠐⠤⠀⣀⡀⠀⠠⠀⠀⢀⠀⢒⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠙⠁⠀⣩⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣤⣤⡤⠤⠒⠋⣁⣤⠖⠒⠒⠒⠒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠄⡠⠤⠴⢋⢁⡠⠤⠐⠊⠉⠙⠁⠀⠆⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⢀⣠⠖⠉⢁⣀⣀⠠⠀⠀⠀⠂⣀⣬⠵⠖⢒⠛⠿⠅⡈⠳⠤⠾⠀⠀⢳⡄⠀⠀⠀⢀⡔⡉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⢠⠋⠔⠚⢋⣁⡀⠤⠀⣤⠀⠀⢀⣀⡀⠠⣄⠀⣀⠆⠀⠥⠤⠀⢀⠀⠀⡜⠀⠀⠀⠀⡞⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡏⠠⠖⢰⣊⣁⠩⠤⠴⠚⠊⣉⠃⠁⠐⠚⠁⠤⠠⠄⠀⠠⡐⠂⠣⢀⠞⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⢻⡉⠋⠈⣈⣀⣠⠴⠔⠊⠉⠀⠀⠀⠠⠄⠀⠀⠀⠃⠀⠐⠀⣣⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⣀⡀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠑⢄⠀⣀⣀⣀⠤⢀⠈⠀⣀⡤⠚⠁⠀⠁⠒⠒⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠼⠤⠖⠒⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠿⣦⣄⣁⣠⠄⠀⣾⣦⣠⣤⣴⣶⣶⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣡⣤⠤⠤⣬⣿⣿⣿⡒⠀⠈⡉⣉⣁⣀⡀⠀⠙⠧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣉⡅⠀⠀⣀⣤⣿⣿⣿⠅⠀⠈⠉⠁⠀⠉⠙⠳⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⡿⣿⣿⡆⠀⢀⡓⢶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣤⣿⣿⣿⣿⠄⠀⠉⠁⠰⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣥⡄⠀⠀⠀⠀⠹⣿⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⠟⠋⠉⠛⠛⠛⠛⠻⣆⠀⢻⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⢻⠇⣠⣶⣶⣶⣶⣶⣶⣦⢸⡆⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⡏⠸⣾⡿⠛⠻⠿⠿⠿⠿⣿⣿⡇⢸⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀⠻⡀⠀⠀⣀⣀⠀⠀⢿⡿⠀⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣄⠶⣤⣄⣠⣤⣴⠟⠁⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣭⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⠾⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁

⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠰⠠⡀⢤⠄
⠀⠀⠀⠀⠀⢹⡿⠿⠄⠀⠀⠀⠀⣴⠀⠀⡀⠀⠀⠀⠀⣏⠉⠁⠔⠈⢁⠉⠀⠀⣶⣾⣦⠀⢢
⠀⠀⠀⠀⠀⢘⣅⡈⠀⠀⠀⠀⠀⠈⠀⣀⡥⠀⢀⢂⣠⣯⣀⡀⢣⣄⣀⡂⢀⠀⠈⠁⢉⣄
⠀⠀⠀⠀⠀⣸⣿⣿⣦⡠⣄⠀⠀⢐⣿⡿⠟⢷⠿⣿⣿⡃⠉⠿⣀⠀⣸⠏⠉⠃⠀⢒⣻⣿⣷⡄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠛⠀⠀⢠⢮⡿⠃⠀⠀⠀⣿⠃⠀⢰⣾⣷⣶⠿⣀⡂⢰⣆⣨⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣻⣿⣿⡇⠆⠀⠘⢷⣆⠤⠀⢀⣰⠟⠡⡖⠙⠟⠋⠀⠈⠋⠷⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡆⢰⠀⣿⣿⠀⠀⠉⢻⣄⠀⢩⠆⠀⠀⠀⠀⠀⢀⣿⣿⣷⣌⡻⣿⠟
⠀⠀⠀⢸⠟⢿⣿⣿⣿⣼⣿⣿⣾⣿⣿⠀⣰⠀⠸⣿⣿⡗⢲⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣍⠙
⠀⠀⠀⠸⣟⣶⣶⣤⣈⡇⢈⣿⣿⣿⡏⠀⣿⠀⠀⣿⠛⠀⢘⣿⣿⢸⡞⣿⣿⣿⡿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⣿⣿⣿⠙⢿⡇⠀⡿⣿⣿⡀⠀⣿⣇⠀⣿⣀⡀⠸⣿⣿⣿⣿⣿⣿⣿⣧⠟⠻⠁⣿⣿⠃
⠀⠀⠀⠀⠛⢻⣿⡶⠘⠀⠀⢰⣾⣿⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⣿⣿⡘⣿⣿⣇⠀⠀⠀⡟⠿
⠀⠀⠀⠀⠀⠀⡿⠿⠀⠀⠀⠀⠘⠛⠛⠋⠻⣿⡟⠻⠿⠿⠿⣿⠛⠟⠙⠃⠈⠿⠟⢻⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡆⢰⡆⠀⠀⠀⢠⡜⢳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⡀⠀⠀⠀⣷⡄⠀⠀⠀⠀⢸⠇⢸⠀⠀⠀⠀⠀⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⡀⡐⡁⠀⠀⠀⣿⣿⣦⡀⠀⠀⢸⠀⠛⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣿⡇⠙⢿⣷⣤⣬⣀⣀⣴⣿⣄⡀⢀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠈⠙⠛⠛⠋⠁⠙⠛⠿⠛⠛⣿⡟⠀⢠⡤⠤⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢷⣶⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠿⢧⣀⢘⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠛⣦⣞⠁⢀⣀⣀⡀⢈⡿⢻⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⢉⣀⣠⡾⠷⠤⢤⣄⣾⣻⢧⡀⠀⢠⡇⠀⠸⣧⣼⠟⣭⠿⠛⠛⠙⣿⡇⠀⢿⣀⡀
⠀⠀⣀⣀⡄⠀⠀⠀⠉⠛⣿⡟⠁⠀⣀⣀⠀⠈⠙⣿⠅⠙⣆⣸⠀⠀⠀⢹⣿⣄⢹⣤⡤⠀⠀⡿⡷⠒⠺⣄
⠀⠀⣽⣀⡡⠀⠀⠀⠀⠀⣿⣿⢴⡾⠛⠛⠳⡄⠀⠘⣷⣀⠈⠛⠀⢰⡄⠀⢻⡉⠉⠉⠀⣀⡼⣡⣿⣤⣤⡟⠛⠛⠓
⠀⠀⠀⠀⠁⠀⠀⠀⣠⣾⣿⣿⣿⡀⠀⠀⠀⣻⠀⠀⣿⣿⣦⡀⠀⡾⢷⣀⣈⣿⠶⠶⠟⠋⣼⠁⠈
⠀⠀⠀⢀⣠⡤⠶⠿⠿⠿⣿⣿⣿⡳⢦⡤⠔⠃⠀⣰⣿⣿⡌⢳⡶⠇⠈⠛⠉⠁⢀⢠⣀⣀⣘⣦⡀⠀⠀⠠⠁⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠿⢿⣿⣀⣀⣀⣠⣴⣿⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠜⠳⡘⢦⡀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡴⠃⠀⠀⠟⠛⠛⠋⠉⠀⠉⢿⡄⠀⠀⣠⠖⠋⠉⠓⠦⡆⠀⠙⠦⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣷⢀⡾⠁⣠⠖⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡞⠃⠐⠆⠀⠀⠀⠀⢹⡾⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠗⠲⡼⠀⠀⠀⠀⠀⢸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣉⣉⠓⢦⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠈⠛⠋⠉⠉⢻⡄⢱⠀⠀⠀⠀⠀⠐⢯⣥⣤⣍⡙⢦⡀
⠀⠀⠀⣰⠋⣠⠤⠤⣄⠙⡆⠀⠀⠀⠀⠀⢀⣠⠤⠚⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠧
⠀⠀⠐⠧⡼⠁⠀⣠⠞⢀⡇⠀⠀⠀⠀⠀⠸⡁⢰⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠞⢀⠇
⠀⠀⠀⠀⠀⢰⠛⢡⡴⠋⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣠⠤⠴⠖⠋
⠀⠀⠀⠀⠀⠈⢧⠘⣦⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⣄⠀⠀⠀⠀⠀⠀⢀⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠈⠷⠞⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⣈⢷⡞
⠀⠀⠀⠀⠀⠀⠀⣾⠉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⠗
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⡤⠶⠒⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⣀⢀⡀⣀⣀⡀⠀⢀⣀⣀⡀⣀⢀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣀⡀⠧⢠⡟⠉⠉⠙⢷⠏⠉⠉⠛⣇⠰⠇⣀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⣑⡄⠀⢸⡇⡤⠂⠠⣬⡤⠀⠠⢄⣹⠀⠀⢄⣽⠲⢦⣿⡄
⠀⠀⠀⠀⢀⣠⡇⠀⠲⠅⠀⠀⢀⡼⢿⠀⠀⠁⢨⠀⠀⠀⢸⠷⣄⠀⠀⣻⣀⣠⣤⣿⠷⠶⣤⡀
⠀⠀⠀⣴⠋⠙⣻⠃⢀⠈⠙⠀⡟⢀⣠⡷⠶⠶⠿⠷⠶⠶⣧⣀⠸⡇⠘⠁⣿⠟⠉⢀⡀⠀⠀⠻⡄
⠀⠀⢰⡇⢀⣰⠷⠶⠶⠷⢤⣀⣿⠋⠁⠀⠀⠠⠀⠠⠀⠀⠀⠙⢿⣇⣠⠴⠿⠶⠶⢾⣷⠀⠀⢀⣿⡄
⠀⠀⠀⣷⠋⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⢷⣴⣼⠏⢷⡀
⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⣧
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠈
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣋⣸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠉⠉⠉⠉⢣
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠙⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣤⣤⣤⠤⠴⠶⠚⠁
⠀⠀⠀⠀⢠⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠰⠀⠀⠔⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢸⠇⠀⠀⠀⢀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⡿⠀⠀⠀⠀⣸⠃⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⢸⡇⠀⠀⠀⢀⡟⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⢀⣸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡀
⠀⢰⣇⣤⠀⠀⡀⢸⢃⡾⠋⠁⠘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠈⠙⣆
⢠⡞⠉⠙⣧⡾⠳⣼⡾⡇⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿
⡾⠀⠀⠀⠀⠀⠀⠈⢷⣿⡄⠀⠀⠀⠀⠀⣏⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⣰⠇
⣷⠀⠀⢰⡿⣿⠆⠀⣸⠇⢙⣦⣤⡀⢀⡀⣙⣆⠀⢀⣾⣁⢀⡀⣠⣤⣾⠃
⠸⣆⠀⢨⣧⡟⠳⠶⠋⢰⡏⠀⠈⠙⠋⠛⠉⢹⡆⢸⡋⠙⠋⠙⠋⠀⠈⢷
⣀⣈⣛⣋⣉⣀⣀⣀⣀⣈⣳⣤⣴⣒⣒⣒⣒⣋⣀⣈⣓⣒⣒⣒⣲⣦⣴⣛⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠓⠦⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡤⢤⣀⣀
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠀⠁⠉⠳⡄
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⡄
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀
⠀⠀⠀⣏⢀⣤⣠⣤⣤⣠⣤⣤⣆⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⡀⠀⠀⡸⠁⠀⠀⠀⡀
⠀⠀⠠⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠙⠶⠤⠼⠋⠧⣄⣀⣤⠞⠁⠈⢹⠁⠀⢀⡇
⠀⠀⠀⢙⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣸⠀⢠⠋
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢣⣤⣤⣄⣀
⠀⠀⠀⢠⣿⢠⠓⢦⠠⢀⠀⠒⣆⣀⣠⠐⠂⢲⣄⠀⢀⣀⠤⢤⡤⠔⠢⠐⠆⢸⡇⠄⢸⠁⠀⠈⠉⠳⣦
⠀⠀⠀⢸⣿⢸⠀⢈⠀⠀⠀⠀⠘⠿⠋⢰⡄⠘⠿⠿⠋⠀⠀⠀⡆⠀⠀⠀⠀⢸⡇⢀⡿⠒⠲⢦⡀⠀⠈⠳⣄
⠀⠀⠀⢸⣿⢸⠀⠘⢄⡀⠀⠀⠀⢀⡴⠃⠳⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⢸⡇⣸⠁⠀⠀⠀⠙⢦⡀⠀⠘⢧⡀
⠀⠀⠀⡚⣿⠀⠀⠀⠀⠈⠁⠁⠉⠁⠀⠀⠀⠈⠓⠀⠀⠔⠚⠁⠀⢲⣦⠀⢸⢸⡇⢿⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠳⡄
⠀⠀⡠⠋⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢸⠈⣧⠈⠳⡄⠀⠀⠀⠀⠀⠹⣆⠀⠀⠹⣆
⠀⣸⠁⠀⡇⠀⢠⡾⠋⠙⢲⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠁⠀⠀⠀⠈⠀⣿⠀⠀⢧⡄⠀⠀⠀⠀⠀⠘⣦⠀⠀⠙⣆
⠀⢻⡀⠀⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡆⣿⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢹⡄
⠀⠘⢦⢰⡇⡆⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠃⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⣇
⠀⠀⠀⢰⡇⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠛⠛⠁⠀⠀⠈⠑⠢⣄⢠⢸⡇⠀⣀⣴⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡿
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣸⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢸⡇
⠀⠀⠀⣸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏⠀⠀⠂⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⢀⣤⠶⠶⣤⣤⣤⣤⡴⠞⠁⠀⢀⣾⠁
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢄⠀⠀⠀⠀⠀⢀⡜⢡⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⢀⡴⠛⠁⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣤⣶⠃⠀⠑⠲⠤⠤⠖⠋⣀⡴⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⣀⣴⠟
⣼⠁⠀⠀⠀⠀⠈⠑⠂⠤⠄⢀⡀⢀⣀⣀⣀⣀⣉⠉⠁⣀⠀⠀⠤⠄⠒⠊⠉⠀⠀⠀⠀⠀⢹⡆⠀⢀⣠⣴⠾⠋
⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠓⠛⠛⠉
⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠉⠻⣷⡶⢦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⢦⣬⣭⣭⣍⣭⣍⣭⣭⣭⣭⣤⠾⠿⠟⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢛⣿⣿⣿⣾⡆⢠⣄⣀⡀⣀⠀⣀⣀⣤⣤⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣁⣩⣿⣿⣿⣾⣭⣯⣽⡿⠛⠋⠉⠉⡙⠛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠉⠈⢉⡛⣿⣿⣿⡟⠀⠀⠀⠀⠘⠀⣾⠀⠀⢀⡀⠶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢹⠛⠟⠋⠰⠀⠀⠀⠀⣦⣤⠿⣧⡀⣸⣿⣴⣶⣬⡙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢊⡉⠳⣦⡀⠀⠀⢸⡇⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣯⣭⣦⣽⡯⠙⢙⣿⣿⣿⣿⣜⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣀⣌⠻⢦⣄⢸⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⢱⣦⣶⣿⣿⣿⣿⣿⣶⣶⣶⣝⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⣿⣿⣰⣤⣤⡉⣻⠁⢀⡴⠖⠉⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣛⢿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣿⣿⣿⣿⠿⠧⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡿⢛⣵⣦⠀⠀⠀⠀⠀⠀⢀⠀⡤⠤⠄⠲⠒⠿⠿⠿⣿⣿⡟⠋⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⢋⠀⣾⣿⡿⠀⠀⡀⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡿⢻⣿⣿⡿⠻⣿⣿⣿⠿⣿⢟⣾⣿⣿⣿⡷⠀⠀⠀⠀⠀⣴⡇⠀⣠⣤⣤⣀⣀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣼⣡⣾⡿⠋⠀⠀⠀⠙⣿⣾⣧⣾⣿⣿⠟⠉⠀⠀⠀⢀⣤⣾⠋⣴⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣧⡘⠀⠀⠊⠁
⠀⠀⠀⠀⠀⣠⣼⡿⠛⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢾⡿⠀⠀⢿⣿⣿⡿⢿⡿⠛⠀⠀⣰⣿⣿⣿⡇⢡⠀⠀⣤⠀⢰⡀
⠀⠀⠀⠀⢠⣘⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⢀⠀⣀⣤⡶⠇⠀⠀⠈⠁⠹⠄⠈⠁⠀⠀⣰⣿⣿⠉⠉⠁⢸⠀⠀⢹⣗⠻⡷⠆
⠀⠀⠀⠰⣋⣘⠇⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠇⠀⡀⢠⡟⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠋⠃⣼⠧⣤⠞
⠀⠀⠀⢸⣿⡿⢾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀⢠⡿⠋⠀⡤⠚⠻⢿⡋⠀⢤⣴⡦⢾⡟⠀⠀⠀⠀⢀⢀⢠⡄⡀
⠀⠀⠀⠘⠿⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢰⣿⣷⡀⢸⣿⣿⣯⣴⠶⠟⠃⠁⢀⣨⣤⣄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡄⠉⠈⢿⣿⣿⣿⢁⡡⠐⢶⣿⣿⣻⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⠿⣇⠀⠀⠘⠋⠛⠛⠋⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢶⣤⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⡟⠿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣀⣘⣻⣿⡀⠀⠀⠀⠀⠑⠀⠃⠰⠀⠀⠆⠄⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋⠛⠉⠝⣶⣿⣿⡿⣿⠇⠙⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣶⠀⠴⣶⣶⣏⠉⠛⠛⠶⣦⣤⣤⣀⣀⣀⣀⣤⠤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⣉⣙⣻⣶⣿⣿⣿⠟⣀⠀⠀⣀⣸⠛⠋⠁⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⣾⣿⡿⠷⣼⠿⣿⢿⣏⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣀⣤⠴⠒⠉⢉⣤⠴⠛⠉⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⢻⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡍⠣⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡜⠻⢦⡤⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠦⡀⢮⡉⢉⠙⠓⠚⠛⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⠢⠐⠂⠀⠒⠄⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠘⡆
⠀⠀⠀⣀⣀⣘⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⢢⡀⠀⠀⠀⢠⠁⠀⠀⠀⠘⡄
⢀⠔⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⠀⠀⠀⡜⠀⠀⠀⠀⠀⠘⡄
⡎⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⢸⠀⠀⢀⡃⣀⡀⠠⠤⠤⠀⠚⠒⡄
⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⡜⠀⠐⣍⠀⣀⢠⡤⠤⢀⠶⠲⡚⢶⢒⡤⣄
⢃⠀⠀⠀⠀⠀⠈⠢⣀⠀⣀⡴⠋⠀⠀⠀⣠⠤⣲⣿⠏⢠⠃⣦⠸⣀⣘⠰⠆⢰⠀⢻⣾⣝⠷
⠘⡄⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⣠⠞⠡⢪⣿⢿⠀⠈⠀⠀⢸⣿⣿⣧⠀⠈⠀⠸⡆⠙⠓
⠀⠘⢦⡀⠀⠀⠀⢠⠆⠅⣱⡶⠂⠉⠀⠀⠀⠻⠃⠘⡄⠀⢤⣀⣈⣛⣛⣡⢤⡖⠁⢐⡇
⠀⠀⠀⠉⠒⢤⣖⠁⠀⠀⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢏⡘⠛⠓⣓⡸⠁⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣄
⠀⠀⠀⠀⠀⠘⡚⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠬⣙⣿⣟⣫⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠤⠴
⠀⠀⠀⠀⠀⠀⠸⠀⢠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣟⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢡⡏⠀⣳⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠁⠚⠁⠀⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠸⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⡴⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠘⢆⣀⡼
⠀⠀⠀⠀⠀⠀⠀⡔⠉⠑⣦⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠑⣤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠉⠁⠀⠀⠀⠀⠀⠉⠉⢐⣺⠁⠀⠀⠀⠀⠀⠘⡀⣀⠀⠤⠤⠤⠄⠠⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠦⠤⠤⠤⠤⠴⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠀⠀⢀⣀⣀⠀⠀⠀⠄⠼⠃
⣿⣿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠉⠀⠒⠒⠦⠠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢸⠁⠀⢀⠤⠐⠒⠊⠁⠀⠀⠀⠀⡔⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠒⠁⠀⡰⠁⠀⠀⠀⢀⠀⠀⠀⠀⡴⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⢀⠎⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⡄⠀⠀⠜⠀⠀⠀⠀⠈⠀⢀⠖⠉⢁⠀⠈⢀⠎⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠂⠒⠁⢀⠞⠁⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠛⠤⠀⠀⠁⣠⠊⠐⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⣠⠤⠐⠒⠊⠀⠀⠀⡀⠀⠀⠀⠘⠁⠢⠀⠀⠀⠀⡠⠒⠈⠁⡀⠀⢀⡼⠀⠀⠀⠀⠐⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⡠⠊⠀⠀⢀⠔⠁⠀⠀⠀⠀⡀⠀⠀⢈⡤⠀⠠⡄⠀⠀⡠⠄⠀⠈⠀⠀⠠⠀⠈⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡀
⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⣀⠤⠒⠁⢀⠀⠀⠀⣀⠜⡀⠀⠀⠦⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⣀⠀⠀⠀⠈⠢⡀
⠀⠀⢰⠁⠀⢀⡰⠃⠀⡠⠀⣠⠁⠀⠀⡔⠁⢀⡀⠀⠈⠁⠐⢊⠑⠄⠈⢀⣀⡠⠴⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢄⠀⠱⡀
⠀⠀⡆⠀⠰⠁⠀⡠⠎⡰⠋⠀⡠⠶⡊⠀⠀⠀⡠⠄⠀⠀⠀⢀⠔⠂⠉⠁⠀⠀⠀⢃⣀⣀⠤⠤⠤⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠃
⠀⢠⠁⢀⠆⠀⠘⠀⢀⡀⠤⠊⠀⠀⠀⡠⠔⠊⠀⠂⡰⠀⢡⠃⠀⠀⢀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠀⡸⠁
⠀⠸⡀⡜⠀⠀⠁⠀⡏⡀⠀⠐⠀⢀⢎⡀⠈⠂⡠⠊⠉⢠⠂⣠⠴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⠤⢤⣀⡀⠑⣄⠀⡀⠠⠄⢒⡡⠊
⠀⠀⢣⡱⡀⠀⡔⠋⠀⠈⠂⠀⣀⠜⠀⢀⣠⠎⠑⢀⡴⠁⡜⠁⠀⠀⠀⢀⣀⡤⠴⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢮⡆⠀⠐⠈⠁
⠀⠀⠀⠱⣁⠘⠀⢀⣀⡀⣠⠎⠀⣀⠴⠁⢀⡠⠖⠁⢀⡞⠀⢀⡀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸
⠀⠀⠀⠀⠈⠓⠦⠥⢀⣈⣀⣒⡊⠤⠤⠒⠉⠀⠀⠀⣎⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⡀⠀⠈⠛⠋⣉⣁⡤⠤⠖⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣀⠤⠴⠶⠤⢤⣄⡀⠀⠀⠀⠀⣀⡀⣀⡀⠄⠃
⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠈
⠀⠀⠀⠰⡄⠀⢀⣀⠀⠀⠀⠀⢀⠎⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⡄
⠀⠀⠀⠀⠈⠁⠀⡈⠳⠠⠤⠤⠮⠄⡀⠈⠳⣦⣤⠀⠀⢀⣤⠴⠎⠁⠀⠀⠀⢹⡄⢀⠀⠀⠂⠐⠤⠤⠐⠊⠁
⠀⠀⠀⠀⡐⠊⠀⠀⣀⣀⠀⠤⠄⠒⠁⡆⠠⡥⠜⠀⠀⠸⣅⣺⠄⡡⠀⠀⠒⠤⠤⢄⣀⡈⠐⠤⡀
⠀⠀⠀⠀⠹⣄⠀⠀⠅⠀⠀⢲⠀⠀⠀⠘⠳⠶⠖⠀⠀⠐⠶⠶⠚⠁⠀⠀⠀⠀⢰⠀⠈⢷⡀⠀⢸⡆
⠀⠀⠀⠀⠰⠉⠛⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠙⠂⠺⡀
⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⡀⠤⢤⡀⠀⣠⠔⠚⠳⣄⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⣀⣀⣀⣀⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄
⢀⠀⠧⠠⠄⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⣿⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉⠉⠉⠁
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠄⠐⠓⠂⠤⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠙⠓⠒⠲⠤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⡀⢀⢀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⢀⠐⢆⣁⡠⠦⠘⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢖⣀⣈⣳
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣴⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣶⣄⡀
⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠒⠒⣦⣀⠀⢀⡖⠉⠉⠉⠙⢿⡻⣦⡀
⠀⠀⠀⠀⡴⠋⣠⠴⠒⠒⠒⠒⠶⢤⣄⠀⠀⠀⠀⠀⠀⢀⠇⠈⠉⠉⠀⠀⠀⠀⠀⠀⡹⣆⠻⡟⢾⡛⠙⠛⠓⢦⣀⡀
⠀⠀⠀⡼⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣽⣆⠹⡄⠹⡄⠀⠀⠀⣏⠙⠳⢦⡀
⠀⠀⢸⣷⠷⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢳⠀⠀⠀⠀⠀⢻⠀⠀⠀⠹⣦
⠀⠀⣿⠃⠀⠀⣿⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⣇⠸⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠳⣄
⠀⢸⠙⢄⣀⣴⣿⣿⣿⣿⣿⣿⣷⠀⠀⠱⡃⠆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⣶⠏⠀⠘⣿⠀⠀⡄⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⢹⢷⡀
⠀⡎⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⣸⠁⠠⠀⠀⢀⣎⠀⠀⠀⠀⢳⠘⠀⢻⡀
⠀⣧⠀⠀⠀⣿⣿⣿⣿⣿⡿⢿⣿⠀⠀⠀⠀⠀⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢰⡏⢠⣧⠤⠤⣤⠞⠉⠉⠳⣤⠤⠀⢸⠀⠃⠈⡇
⠀⢸⡀⠀⠀⠹⣿⣿⣿⣿⣧⣾⢃⣄⣄⡀⢀⡄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⣼⣡⠟⠁⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢸⠀⠀⠀⡇
⠀⠈⢷⡀⠀⠀⡈⠛⠿⠟⠋⠁⠉⣿⠿⢣⣾⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢸⠀⢀⢠⡇
⠀⠀⠘⣷⢄⡪⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⡞⠠⠃⡾
⠀⠀⠀⠈⢷⡍⠒⠦⠤⠤⠴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠊⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠈⣠⡴⠋⢀⣤⠤⢤⣄⡀⠀⠀⠀⠀⣀⣠⡤⠚⠀⠀⢀⣠⠞
⠀⠁⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠑⠠⠀⠀⠀⠠⢀⠤⠷⠤⠒⠤⣀⣀⣤⠾⠋⢀⡔⠉⠀⡀⠀⠀⠉⠙⠛⠿⠭⠭⠀⠀⠠⣊⡵⠛⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣀⣀⣀⣀⣠⣤⡴⠶⠛⠉⠀⠀⠀⡞⠀⣠⠿⠶⠦⢤⣔⣂⣀⡠⠤⠴⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⠲⢤⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠲⢇⡠⠼⠿⢿⣧⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠛⠛⢷⣶⣶⡆⠾⡦⡶⠎⡿⠛⠁⣀⢄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠙⠛⠇⠴⣦⡧⢌⠧⠔⠊⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡘⢳⣬⡴⠛⠀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣔⣴⣿⠁⡴⢫⠀⡝⠄⣹⡦⢣⠄⠀⠀⠀⠀⠀⠀⠠⢠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⢖⡆⡉⠂⠀⣀⣀⣀⡀⠈⡋⠸⣛⣢⡇⣘⢼⡀⣤⣽⠁⠬⠄⠀⠀⠐⣶⣶⡅⠄⠁⠀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⢿⣿⣏⠲⠺⠀⢠⠏⣻⢥⣿⣿⣷⣴⣿⢦⠀⠈⢳⠊⠱⡻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠦⣑⣀⣞⣠⠟⣛⠛⠛⢛⡉⠙⣄⣱⣤⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⣛⢿⣿⣆⣀⣾⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣺⢽⣶⣽⣺⢽⣿⣿⣿⣿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣾⣽⣿⣷⣮⣻⣿⣿⡿⠁⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣴⠊⠻⠋⠛⠉⠙⠛⠻⢿⡃⠲⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡧⢞⡀⠀⠀⠀⠀⠀⠀⠀⠩⡂⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣼⡿⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠁⣈⠐⡀⠀⡀⠀⠀⠄⢀⠀⠀⢀⠈
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⠂⠃⢙⠀⠀⠀⠒⠂⢸⠀⢠⡄⢠⠄⠀⠀⢠⡃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡆⠀⠨⡀⠀⠂⠀⡀⢍⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠘⡀⠄⠀⢸⠦⠾⠆⠀⠒⡀⠐⠀⡆⠀⠑⠄
⠀⠀⢀⠀⠐⠀⠀⠀⢠⣿⣧⠀⠀⠀⠄⠐⠀⠁⠈⢢⠀⠀⠀⡄⠀⠀⣰⡀⠆⠀⢨⡄⠀⣠⡄⠇⠀⠀⢈⠀⢐⠀⠀⠀⠘⡄⠀⣷⣀⠤⠲⡄
⠀⠀⢨⢒⠤⠄⠄⢒⠃⠀⠈⠙⣇⠈⠀⠀⠀⠠⠶⢿⣄⢀⠀⠈⠁⠚⠉⠁⢸⠀⡀⠀⠈⠁⠀⢠⠀⠀⡰⠀⠨⠇⠰⠀⠀⠁⠀⠟⠀⠀⠀⠘⡆
⠀⠀⡆⠀⠈⠀⣠⠉⠀⠀⠀⠠⠈⢦⡀⠀⠀⠀⠀⠀⠙⢾⡆⠀⠀⠀⠀⠀⢰⠀⠁⠈⠂⠀⣠⡾⠆⠾⢿⠈⠀⠀⠀⠀⣀⡄⠀⢆⡈⠀⠁⣀⠁
⠀⢄⣃⠀⠀⠘⠛⣀⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢻⢲⠠⠀⠀⣴⣇⠀⠀⣀⣤⠚⠉⠀⠀⠀⠈⣆⠀⠀⡀⠘⠉⠀⠀⢸⠉⠁⠁⠇
⠀⠀⠉⠀⠀⠀⠀⠈⢦⡀⠀⠀⢠⢾⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠙⠒⢾⠁⠈⠋⠉⢹⡇⠀⠀⠀⡀⠀⠀⠘⡶⠋⠀⢀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⠋⠀⠀⠀⠑⢦⡤⠒⠀⠀⠀⢹⠁⡠⠈⢀⠈⡦⠀⠀⠀⠀⢇⠀⠈⠀⠀⡀⡄⢴⡞⠢⡰⠈⣈⠀⠀⡇
⠀⠀⠀⠘⠀⠀⠁⠀⠀⡃⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡦⢀⡀⢁⣠⠅⠴⠀⠀⠀⠎⡆⠀⠂⠁⠁⠀⢼⠁⠀⠈⠑⠦⢄⡼⠷⠦⠐⠁⡈
⠀⠘⢄⡀⠀⠀⠀⠀⣰⣷⣤⠀⠀⠀⣠⠀⠀⠤⠰⠔⠚⠘⣓⡀⣀⣤⡋⠀⠀⠀⡁⠀⠀⢈⠆⠀⠀⠀⠀⠀⢡⠀⠀⠀⢀⣴⠀⠀⠀⡄⠀⠱⡀
⠀⠀⠀⠱⠤⣀⠠⠞⠯⠁⠀⠉⠀⠘⠉⢧⠀⠀⠀⠀⠀⣠⠏⠀⠈⠇⠉⠛⠫⠇⡇⡰⢰⠃⠀⠀⠀⠀⠀⠀⠈⣧⡤⠘⡁⠙⡄⠈⠆⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠁⠈⠀⠀⢼⡟⠁⠀⠀⠀⠸⡀⠀⠀⠰⡀⠀⠀⠀⠀⢀⡘⠀⠀⠈⠁⠈⠉⠂⠀⠘⠇⠃⠈⠀⠸⡄⠀⢠⡈⠉
⡆⠀⠀⠀⠀⠃⠈⠀⠀⠀⠛⠢⣀⡠⠂⠀⢃⠀⠀⠀⢀⠀⠙⡄⢀⠂⠈⠀⠀⠴⣶⠿⣅⢀⠀⠀⠀⠀⠁⠐⠀⢾⣁⠀⣀⡠⣤⠆⠀⠀⢠
⢣⡀⠀⠀⠀⢠⠐⠂⠰⡖⠀⠀⠀⢠⣤⡇⠈⣆⠀⠀⠀⢴⡶⠁⠀⢀⠀⠀⠀⡀⢡⢠⠈⡃⠄⠀⠀⠀⠀⠀⠀⠈⡏⠁⠀⠀⣿⠏⠄⠀⠠⡆
⠀⠁⡈⠀⠀⠘⠀⠸⠀⢹⢖⠆⡄⡀⢀⡃⠀⠈⢿⡇⠌⠎⢿⡀⠀⠀⠀⢀⢾⣇⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⠶⠾⣯⠀⠀⠀⠀⢠
⠀⠀⠡⠶⠦⠶⢶⡞⠛⠛⠮⡀⠁⠀⠀⠁⠁⣁⣼⠃⠀⠀⠀⠙⢆⣠⡾⠀⠀⠀⠑⠀⠘⢻⡷⠤⠀⠀⡀⠀⠙⣿⠃⠀⠀⠀⠆⡁⠀⠉⢙⠃
⠀⡀⠀⠀⠀⠀⠈⢇⠀⠀⠄⠈⠉⢻⠟⠁⠀⠀⠸⡀⠀⠀⣄⡀⠄⠀⡇⠠⠤⠤⠃⡀⢀⠎⣀⡀⠛⠀⠁⠀⢶⣞⠀⠀⠎⠀⠀⠂⠀⠀⢸
⠀⡇⠀⢀⠀⠂⠀⢲⠀⠐⢄⠀⠀⠨⠀⠰⠀⠀⠀⠉⠑⠠⢀⣀⣠⠼⠷⣤⣄⡀⠀⠀⠉⢛⠟⣧⠆⠂⠒⠁⣿⠟⢳⠆⠀⠤⠀⢀⠀⢢
⠀⢣⠀⠀⠀⠐⠀⠀⣣⠀⠀⠑⠠⡀⡀⠒⠒⠒⢢⡀⠀⠀⠀⣸⠁⠀⠀⠘⢏⡀⠀⠀⢀⠋⠀⠅⠀⠀⠒⢶⠃⠄⠌⠂⢤⠀⡀⠀⠀⠨
⠀⠀⠈⠄⠀⠀⠀⠀⠈⠁⠀⠀⢵⡈⠷⠄⠀⠀⠀⠀⠁⠂⠐⠉⠑⠀⠀⠀⠁⠈⠀⠀⠈⠆⠀⠠⢄⠀⠀⠹⠌⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣄⡀⠈⠉⠉⠛⠛⠛⠋⠉⠉⠁⣀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠃⠀⢀⣴⣿⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⢻⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠒⢿⣿⢻⡿⢷⣄⠀⢀⣴⠿⣿⢻⣿⠷⢺⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⢻⣿⠟⠋⣉⣤⣌⡉⠛⢿⣿⠃⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆⠀⠈⠃⠀⢸⣿⣿⣿⣿⠀⠀⠉⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⡀⢸⣷⣄⡘⠿⢿⡿⠏⢀⣴⣿⠀⣰⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⡿⢷⡀⢻⣿⡿⢷⣶⣶⠾⣿⣿⠃⣰⠿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠙⢿⣶⣶⣶⣾⠿⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣙⢷⣄⡉⠉⠉⣁⣤⢞⣁⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣮⣉⣛⣉⣩⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠟⠋⠀⠀⠀⠈⠛⠿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣻⡛⡛⢻⡟⣻⢿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠦⡜⢲⠀⠀⠀⢀⡀⠀⠀⠠⡄⠈⠣⡀⠘⡄⠈⠀⠓⠊⠁⡀⠠⠀⠀⠠⡀⣀⠀⠀⣼⢗⠞⢕⠟⠁⠹⣼⠀⢁⣀⢧
⠀⠁⠀⠐⢂⣀⠀⠀⠀⠰⠆⠀⠀⡀⠈⠢⡘⠀⠀⠀⠖⢨⠅⠖⠈⠁⠠⡄⠰⠶⠸⢈⣇⡸⡘⠀⠀⠁⠀⡄⠀⠁⠠
⠀⠈⢿⣿⣿⣿⠁⠀⠀⠀⠄⠁⠀⠐⠃⠀⠈⠻⠆⠀⠁⠄⠐⠀⠀⠀⠀⠁⠀⠀⠀⠙⠖⠊⠃⠀⠀⣄⡤⢀⡁⠀⡄
⠀⠀⠀⢿⣿⠃⠀⢀⣧⠀⠀⠀⢀⠠⠀⠀⠐⣀⠀⠀⠁⠀⠀⠀⢴⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣸⢋⡉⢛⣓⣓⠿
⠀⠀⢴⡎⠁⠀⠀⠘⠋⣀⣀⣀⣀⣀⡀⠃⠀⡄⠀⠀⠀⠀⠸⢄⠀⠐⠃⠀⠤⠀⣶⠄⢬⠿⢆⡹⠧⠿⠧⣬⣩⡕⠿
⠎⠀⠀⣔⢕⣚⠶⡀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠴⠚⢡⡆⢰⡇⢠⡀⠀⠀⠀⠐⠛⠀⠀⠀⠛⢋⠈⠛
⠀⠄⠀⢋⢜⠈⡎⡆⠘⣿⣿⣿⣿⣿⡇⠨⠂⠉⣁⡂⠀⠀⢠⠄⠀⢠⡆⢀⠈⢁⠘⠃⠀⠀⠁⠀⠰⠆⠈⠱⠓⣴⡔
⢀⡂⠛⢀⢋⠸⠳⡁⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠀⠀⠀⠰⡇⠈⠁⢺⠇⠙⠀⣤⠀⠀⢀⠀⣠⡀⠀⠀⠀⠁
⠘⢌⣭⠫⡆⣤⠞⠘⠀⠋⠉⠉⠙⠋⣁⠀⠀⢄⡄⠀⠀⠀⠀⠀⠀⠸⠀⡄⠀⣶⠀⠛⠀⣀⠀⠀⠈⣥⠀⠀⢂⠀⠳
⠀⠀⠁⠀⠈⢀⠀⢠⣆⡃⢠⡄⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⡀⣠⢠⠄⠀⠁⠀⠀⠀⡤⠐⠀⠆⠀⠀⠀⠀⠀⠈⠐⠂
⠀⠀⠀⢰⣆⣠⣶⣿⣿⣿⣿⣿⣾⣥⠆⠀⠀⣶⠄⠀⠀⠤⠯⠑⡖⢙⡖⠀⠀⢀⠀⠀⡀⠈⠌⠠
⠀⠀⠔⠀⣹⣿⣿⣿⣷⡿⣿⣿⣿⣿⢀⡁⠀⠘⠀⠀⠐⠑⡎⣹⣉⠍⠅⢠⣦⠀⠀⠀⠀⠀⠀⠰⠂⢱⠄⡀
⠀⠀⠀⠰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⠃⠀⠀⠀⠀⠀⠈⠨⠁⠘⠀⢀⡀⣀⡀⠠⠆⠘⢄⠐⠀⠀⠘⠒⢉⡁⠄⡀
⡀⠀⠀⣀⠀⠈⠛⠿⠿⠿⠿⠿⠛⠀⠀⣤⡀⣤⠶⣆⠀⠀⠂⠀⢀⡮⠳⠔⠛⠺⠀⠀⠀⢀⠀⠀⣀⠀⢨⠰⠈⠁
⠀⠀⠄⠉⠀⠀⠰⠀⠀⠀⡄⠀⡆⠀⠸⡯⣚⣇⣸⢮⢞⣠⣆⠀⢀⢸⢧⢢⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡀
⠀⠀⢻⣿⣿⡿⠀⢀⡀⠀⢀⠀⠀⠀⠶⠀⠛⡘⠓⠀⠈⠓⠀⠀⠈⢗⢆⡩⡆⢀⡀⠀⠁⠀⠒⠀⡀⠀⠀⠀⢠⡀⠡
⠂⠀⠀⢻⡿⠙⠛⠟⠀⠀⠸⢷⣄⠀⠀⠙⠏⠁⠀⠛⠀⠀⠀⠀⠛⠈⠈⠉⠠⣿⡿⣠⣾⡧⠀⠀⡀⠀⠀⠀⠄⠃⠐
⢰⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡀⠀⠀⢀⡀⠀⠠⠆⠀⠜⠃⠀⣠⣄⢟⣽⣿⣧⠀⣤⠈⠁⠂⠐⡀
⡿⠧⠠⠠⠸⠂⠀⠀⠀⠀⠀⠀⠀⡌⢡⡀⠀⠀⢈⠁⠠⠁⠜⠂⢤⠀⡄⢿⢿⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡂⢐⡀⠀⠀⢀⠀⠀⠦⠀⠀⠶⠂⠀⠆⠰⠈⠀⠁⠠⠁⠀⠀⠀⠀⡠⠁⠈⠿⣿⣭⡄⠀⢠⡄
⣁⡽⢼⣠⠀⠛⠃⠰⠀⠄⠀⠀⠒⠀⢈⢃⠚⠀⢠⡀⠄⠀⠠⠁⡠⢁⠠⠀⠀⡠⢉⠀⠀⠈⡃⠀⢀⠀⢀⠀⢠⠀⡳
⡼⢤⠼⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠁⠀⡘⢃⠀⡷⠀⣀⢀⡀⠀⠈⠏⠀⠀⠐⠈⠀⡴⠾⠍⡻⢏⢩⡢⠀⣀⡍⢨
⠀⠁⡁

⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⠿⠟⠻⠟⠿⠿⠿⠻⠿⠿⣿⠟⠿⠿⠿⠿⠿⠿⠟⢟⠻⠿⠿⠿⠿⠿⠿⠛⠿⠿⠟⠛⠿⢿
⣿⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⡀⠐⡄⢀⠀⠀⡆⠀⢄⠀⠤⠀⠠⢄⣴⣾⢃⠔⠀⠀⠀⠙⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸
⣿⠀⠀⠀⡀⠀⠘⣄⢀⡄⠀⠀⠀⡇⠀⣿⢈⠀⢸⣿⠀⣸⡄⠀⣼⣴⣿⣿⣿⠏⠘⠀⠀⠀⠁⠀⡀⠀⣄⠀⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⢸
⣿⡄⠀⠾⠃⠀⠀⠸⣇⢹⣄⠁⢠⣷⠀⣿⣾⡆⠸⢋⠀⣿⡧⣼⣿⣿⡿⣻⡿⠀⣀⣴⣦⣴⠞⠉⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⣿⡷⣿⢦⣼⣿⢷⣿⣿⣷⣷⣼⣿⣿⣿⣿⣿⣫⣠⣿⣶⣾⠿⠛⣿⡡⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠀⠀⠾⠗⠀⢸
⣿⠁⠈⠲⣦⣤⣀⡁⠘⣷⣽⡈⣿⢁⢻⣽⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣥⣒⣡⠄⠀⢈⣃⡤⠀⠀⠈⠀⢀⣀⠀⠀⠀⠁⠀⠀⢸
⣿⡀⠀⢰⠆⠉⢻⣿⡳⣎⢻⣷⣿⣸⡌⣿⣿⡅⢿⣿⣿⢻⣿⡿⣱⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠀⣤⣼
⣿⡿⣷⣶⣦⣤⣀⠹⣷⣼⡄⡏⣿⣯⣷⣿⣿⣇⣼⣿⣿⣿⣿⣿⣿⣿⡽⢾⣿⣿⣿⠟⠉⢐⣾⣯⣤⣄⣀⣰⣀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸
⣿⣃⣈⠛⣿⣿⢮⣿⣿⣿⣧⡙⣿⣿⣿⣿⣿⣿⡿⠋⠭⠠⠌⢉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣫⣵⣖⣞⣉⠉⠉⠁⠀⠀⠈⠘⠛⠀⠀⠀⠀⠀⢀⣸
⣿⡟⢿⣟⡻⢿⣿⣿⣿⣿⣿⡷⠟⠛⠛⠛⠉⢠⡇⢀⣠⣤⣾⣧⡄⠈⢻⣿⣿⣿⣿⣿⣿⣫⡕⠀⠀⠀⠃⠀⠀⢀⢠⠄⠀⢀⠀⣬⠀⠄⠀⠘⢻
⣿⣦⣤⣝⣯⡓⣿⣿⣿⣿⡟⢀⠀⠀⠀⠀⠀⣿⢠⣿⠋⠀⠀⠀⢹⣦⡸⡷⣻⣿⣿⣿⣿⡿⠃⠀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⣿⡁⠉⠻⣿⣿⡿⣿⣿⣿⡷⠿⣿⡚⣿⠀⠀⢣⢿⣿⣄⣀⣀⣴⣾⣿⣼⣿⣝⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣁⡀⠰⠂⠀⠀⠀⣠⠀⠂⠀⠀⢸
⣿⣷⣶⣶⣽⣿⣿⣿⣿⣟⠔⡠⠴⣻⣿⡀⠀⠀⣾⣿⣿⠿⢿⢻⠋⣿⣿⣥⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠁⠀⠀⠀⠀⠀⢼
⣿⡇⠘⠿⠷⣿⣿⣿⣿⣿⢞⣴⣾⣀⣿⣇⣠⠾⠉⢹⣗⢎⣠⣶⣟⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠄⠀⠀⠀⠀⠀⠀⣾
⣿⡇⣠⡶⠿⠿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⡷⠖⠀⠀⠘⣿⣭⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠐⠀⠺⠇⠀⠀⢺
⣿⠅⠀⠀⢠⣤⠀⠈⣙⣿⣾⣿⣽⣿⡏⣸⣇⡀⠀⠀⠀⣿⣿⡿⢟⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⠟⣿⡠⠄⠀⢠⣿⣍⡳⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠰⠂⠀⠀⠀⠀⠀⠀⢸
⣿⡀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣸⣿⠟⢁⡼⠨⠓⠊⠉⠉⠁⠀⢀⠀⢀⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠛⠀⠀⠀⢸
⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠰⡿⢀⡀⢀⠀⣄⢰⢲⡇⣷⢸⡘⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠠⠀⠐⠀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠘⡇⢳⠘⡇⢣⢸⡟⣇⠹⡄⢇⢷⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⠀⠀⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣿⣼⣧⣷⣼⣦⣷⣾⣦⣷⣾⣾⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣬⣤⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡫⠉⠉⠉⠉⠓⢄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⡀⢀⡏⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠋⠀⠀⠀⠀⠀⡴⠚⠉⠓⠲⠶⠶⠛⠉⠀⠀⠉⢢⠀⣠⠞⠉⠙⣿⡇⠀⠀⠀⠈⠒⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠛⣿⠁⠀⠀⠀⢠⠖⠛⣇⣀⣀⣀⣀⠀⣠⠔⠒⠲⠦⣤⣴⠾⠋⠀⠀⠀⣿⠣⡀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣀⣿⡀⠀⠀⠀⢸⣴⢞⠋⠉⠈⠉⣩⠿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠙⢗⠢⣠⠖⠋⢳⡀⠞⠉⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠋⠁⠀⠈⠈⠙⠒⣤⠟⠑⠁⠀⠀⢀⡞⠑⠀⢻⣀⠀⠀⠀⠀⠀⠀⣀⠔⢋⠐⡲⢤⣄⣠⠜⠁⠀⠀⢸⠃⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⣴⣶⣯⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡾⠰⠀⠀⠀⠈⡙⠛⠒⠒⠚⠻⣤⣰⡇⠁⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠈⠙⠻⠿⣿⣧⡀⠀⠀⢿⡀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⢀⣠⣯⡴⠞⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠈⠉⠛⠒⠾⠷⣤⣄⣤⣾⣷⡀⠀⠀⠀⢈⣇⣀⣤⡀⠀⠀⠀⢈⡃⣉⣓⣶⡶⠶⠞⠛⠉⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠒⠶⠾⠿⠷⠶⠚⠓⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣟⣛⡻⠿⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⠖⠚⠛






⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣾⣴⣿⣶⣿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⡏⠉⠉⠉⢉⣭⣍⢉⡉⠉⠉⣉⣭⣍⠉⢉⡉⠉⢉⣉⠉⢉⣭⣭⡉⠉⠉⠉⢙⠃
⡇⠀⠀⠀⠀⣥⠼⠚⠇⠀⠀⣯⣤⠀⠀⣜⣷⠀⠿⣤⡄⢸⣥⠄
⡇⠀⠀⠀⠘⠃⠀⠘⠷⠶⠂⠿⠶⠖⠾⠉⠙⠇⠶⠶⠋⠺⠷⠶⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠
⡇⠀⡄⠀⣦⠀⣄⠲⠄⢰⡶⠖⠀⠀⠀⢶⣀⡄⢀⣤⣄⢠⠀⣀⢠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⢢⡀
⡇⠀⣧⣀⡿⢤⣙⣳⡆⢸⡗⠀⠀⠀⠀⠈⣿⠀⣿⣀⡿⢸⣄⡏⢸⣷⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⠀⠀⢀⠟
⡇⠀⠈⠉⠁⠈⠉⠉⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠉⠀⠈⠀⠙⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⢈
⡇⠀⠀⠀⠀⠀⢸⣟⣷⠀⣟⣻⠆⢀⣾⣆⠀⣶⠀⣿⣆⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠠⠜⠀⠀⠀⠀⠈⠳⡄
⡇⠀⠀⠀⠀⠀⢸⣏⡿⠒⠟⠷⠤⠼⠛⢻⡄⠯⠸⡇⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡘
⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠄⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠀⠘⢷⠖⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠤⠙⠛⠉⠁⠀⠀⣘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⠀⠀⡠⠆⠀⠀⣀⡈⠃⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣴⡦⢸⣿⡇⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⣙⠃⠀⠅⠖⢠⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠈⢧⡄⣠⠖⠳⢦⡼⠛⢛⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠉⢉⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡏⠁⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠶⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠄⠒⠒⠀⠉⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠄⠚⠓⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⣸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢘⠀⠀⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡆⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⢀⠀⢠⠀⠀⠀⣤⢠⠀⣀⠀⠀⡀⢠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠄⢀⣄⠀⠀⠀⡤⢀⡄⠠⠄⠀⠀⠀⡤⢀⡀
⠀⠈⠂⠈⠀⠐⠂⠃⠘⠀⠉⠀⠀⠐⠂⠘⠀⠛⠀⠘⠃⠀⠃⠀⠀⠈⠉⠀⠃⠀⠀⠀⠆⠀⠂⠉⠙⠀⠌⠀⠛⠈⠉⠒⠀⠑⠄⠈⠓⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠖⠚⢉⡟⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢀⡜⠁⠀⢀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⡀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣯⡀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣹⣦
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⢄⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠈⡆
⠀⠀⠀⠀⠈⢢⡁⠀⠀⠀⠀⠈⠳⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢱
⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⣠⣤⠽⠶⠟⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠀⠉⠙⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠈⠻⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢠⠞⠋⠀⠀⠀⠔⠂⠉⠉⠉⠉⠉⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⢀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠔⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠒⠒⠀⣷⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠠⠊⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⣄⣀⣀⣀⠀⠐⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⣀⣤⡴⠞⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣄⡉⠉⠙⠛⠶⠶⠶⠶⠶⠶⠶⠶⠞⠉⠀⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⢤⣀⡀⠢⠤⢤⡄⠠⠄⠒⠂⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡒⠚⠙⠶⡚⠛⠦⣄⠀⠀⠀⠀⠉⠓⠶⠤⠴⠶⠚⠋⠛⠋⠉⠀⠀⠐⠤⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠠⣇⣴⠶⣄⠙⢦⡄⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⠀⠀⣀⣀⣀⠀⠀⠄⢠⠂
⠀⠀⠀⠀⠀⠀⢀⠴⢦⠙⢄⡤⢬⠁⠀⠉⡷⠀⠧⢀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⡄⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡽⠂⡎⠉⠀⠠⠎⢱⡟⠓⠚⢦⠣⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⣻⡀
⠀⠀⠀⠀⠀⠀⢀⢤⡂⢠⠞⠁⠀⣈⠒⠦⣌⠳⡄⠀⠀⠀⢁⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡴⡆⣸⣶⠇⡀⡀⢀⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠈⣦⠁⠀⠣⣠⠜⢀⣉⣀⡈⠲⢬⡁⠂⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⣿⢻⠃⠀⠘⣧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠲⠤⠆⠰⡉⠀⠈⠙⣇⡀⠙⢲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠇⠈⠀⡠⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⠀⡌⠀⠒⠒⠉⢻⡆⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠀⠀⠐⣊⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⠒⠤⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠒⠛⠋⠁


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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⡇⠘⠛⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠙⠒⠚⢃⡼⠇⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠉⠁⠀⠀⠀⠀⠀⠀⠻⣤⣼⠀⠀⠀⠀⢠⡶⢆⠀⠀⠀⢀⡴⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠲⣄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠶⠶⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠙⠋⠀⠀⠀⠛⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⣀⡠⠶⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣦⣀⠀⠀⠀⠀⢾⡁⠀⣳⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀⠀⠀⠙⠚⠁⠀⠀⠸⣇⡀⣹⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢸⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⢾⣭⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠸⣇⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⢀⣤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢿⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠀⠲⢤⣄⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠘⠶⣤⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⣿⠟⣶⣤⣀⣿⠀⣰⡾⠛⠛⠛⠳⢦⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⠛⠛⠛⢉⣽⠇⣸⠏⠠⢤⣀⠀⠀⠀⢱⡄⢸⡇⠀⣴⠞⢛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠤⠤⠀⠤⠐⠛⠁⠀⣿⠀⠀⢸⣿⡟⣷⣤⣼⠇⣾⠀⣾⠁⡼⠋⣍⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠁⠀⠀⠀⠐⠒⠒⠒⠒⠲⣤⡀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠘⣇⠀⠀⠙⠛⠋⣠⡟⢠⠟⠀⠻⣄⠻⠆⠜⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠉⠓⠲⠒⠉⠉⠉⠰⠶⠶⠚⠉⢠⡟⠀⠀⠀⠈⠑⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣀⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠉⠉⠉⠛⢦⡀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⣰⠟
⠀⢀⠀⠀⢀⡄⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣠⠚⠁
⢠⡎⠀⢰⠏⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠋⠁
⠀⠁⠀⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢻⡌⠛⠳⠶⠦⠀⣤⣤⠄⢤⡶⠶⠖⠛⠋
⠀⠀⠀⡇⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣶⠀⣽
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠈⣧⠀⢀⠟⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠈⠳⣾⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡍⠉⠉⣷⠄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⣠⠾⠛⣻⡷⠶⠾⣧⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠋⠀⠛⠁⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⠋
⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠋⠓⠒⢦⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠄
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⣷⡀⠀⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⠶⣤⣀⣀⣀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⡀⠉⠉⠉⠀⠹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⠰⣆⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢰⡟⠀⠀⠀⡟⠉⠒⠒⠒⠢⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠙⣆⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⣼⠁⠀⠀⣸⠃⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠸⣇⠀⠀⠀⠀⠈⢷⡀⠀⢸⡇⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠈⢷⠀⣼⠃⠀⠀⢸⣇⣀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠉⠉⠙⠛⠒⠤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢸⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠘⡄⠙⠻⠶⣦⣄⡀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⡀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟⢿⣿⣿⡏⠁⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⣾⣿⣿⣧⠒⢺⣦⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢿⣿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢀⣿⣿⣭⣿⡇⢨⡅⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡴⠦⣾⣿⣿⣿⣿⣿⠷⠾⢀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣨⣿⣿⣿⣿⣧⣤⣶⠟⠁⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣂⡰⣟⠉⠺⣿⡿⠻⠿⡇⠃⢀⣈⣹⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣽⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣱⣦⣬⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠋⠉⠙⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠟⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⣿⡿⢯⠛⣿⣿⣿⣿⡏⠶⠀⠶⠀⠀⢀⡀⢤⡀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠃⡀⠉⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣌⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡒⢠⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⡉⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣯⣵⣾⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡉⠉⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣿⣿⢩⣽⣾⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠛⠋⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢻⣿⠶⠋⢹⣷⡀⠀⠀⠀⠀⠀⠀⣿⡏⠛⡵⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣧⡛⢷⣬⣥⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠐⢟⠀⠂⠈⠉⢷⡀⠀⠀⠀⠀⣠⡿⠇⠀⠁⠓⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣽⣿⣿⣿⣿⣿⡿⢿⠿⡄⠀⠀⠀⠀⠈⠀⠙⠀⠀⠀⠉⠄⠀⠀⢈⡿⠆
⠀⠀⠀⢀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣝⣩⡉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢀⠐⠳⣴⡿
⠀⠀⢰⣿⣿⣿⡿⠟⠻⠟⠛⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠰⠀⠸⠄⠀⠀⠀⠀⠀⠀⠙⢿⣇⣶⠿⠆
⠀⠀⠸⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣀⣀⠀⠀⠀⢀⣠⣤⡾⢟⣠⠇
⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠙⣿⠏⠜⣻⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠛⠛⠛⠉⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢲⣄⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠙⢿⣿⣿⠃⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⠀⢻⡏⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠘⠀⠘⠃⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠙⢿⣿⣿⣿⣟⠛⠛⣿⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⢿⣿⣿⠏⠀⠀⠘⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠰⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⢳⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡃⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣧⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣿⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢀⡀⢀⣠⡾⢛⣿⣿⠿⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢣⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠴⠯⠭⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣄⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡟⢲⣾⣾⣶⡄⣶⣶⣶⡆⣶⣶⣶⣶⢶⣦⢰⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡏⠉⠁⢸⣿⣁⣿⢷⣿⡇⣿⣧⠁⢠⣿⠇⠸⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⢸⣿⢿⡇⠀⣿⠶⢿⡇⠀⣼⠇⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣠⡇⢀⣿⣼⣿⣶⣿⣿⣿⣷⣾⣯⣀⡀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠮⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠤⠭⠭⠍⠁⠀⠀⠀⠀⠀⠀⠁⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡀⢀⣠⣀⡀⠀⠀⠀⠀⣠⣶⡆⢄⠲⡄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠘⣴⠀⠿⣿⠏⣆⠀⢀⡀⠙⠛⠁⢸⡇⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⣄⣀⣀⣠⠿⠚⠊⠳⣤⣤⣴⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢹⣿⠿⠛⠁⣴⣰⣆⡄⠈⠻⠿⠿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣆⠀⠀⠀⡠⠀⠘⠋⠙⠃⠀⠀⣤⡀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢼⣧⡤⢤⣤⠄⠠⠤⠶⣿⠃⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⣿⣶⠀⠀⠈⠀⠀⢸⣿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⢏⠻⠶⢶⢶⢶⠞⠋⠌⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠆⠉⠐⠀⠐⠐⠈⠀⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠐⠂⠐⠞⠋