Mythical creature Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⣀⠀⠀⡔⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠁
⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡡⠀⠠⠀⢄⡀⢤⡂⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⡖⠠⠖⠀⠈⠁
⠀⠐⠁⡀⠀⠀⠀⠐⠀⠀⠙⣴⠀⢃⣀⡈⠁⠀⣷⡀⡏⡀⠀⠀⠀⠀⠀⠀⢀⠀⡞
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⠿⠿⠇⠿⣻⣷⣤⠀⠉⠁⠛⡁⠀⡀⠀⠀⠀⠈⠀⡉⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠈⡿⠒⠦⠀⠀⡀⠀⠀⠀⠀⠈⠂⠀⠎⠡⠄⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣰⣤⠀⠁⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣿⠀⠀⠀⠓⠦⢀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢠⣾⠚⣡⣶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠑⠛⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢨⠇⢰⣿⡉⣿⣋⣿⣿⠶⠀⠂⠀⡀⠉⠀⠀⠀⠀⣠⡶⠀⢄⡀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢹⢸⣿⣟⣋⣻⣿⣿⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⡴⠋⠛⠾⠲⠀⠳⠤⠂⠠⢄⢀⠎
⠀⠀⠀⠀⠀⠰⣆⣼⣿⣿⣿⣏⣿⠀⠀⠀⢰⡞⠀⠐⠂⢄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠉⠿⡥⣝⢿⢿⢻⠄⠀⡀⠀⠃⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⢀⡀⠀⠨⠛⠀⠀⠁⠂⠸⣤⡄⠁⠀⠰⣄⡠
⠀⠀⠀⠀⠀⠱⠄⠶⠂
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢝⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⢌⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⠯⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠋⠈⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣾⣅⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠂⠀⠀⠀⠀⢀⣤⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠈⠛⠛⢿⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠀⠀⠀⠀⣠⣏⡜⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⡼⣛⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⡄⢠⣀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⢀⣠⡎⠀⠀⠀⢠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠶⠿⢷⣤⣄⡀⠀⠀⠈⣻⣿⣿⣿⣿⣾⣄⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣴⣿⣿⣿⣿⣿⣿⣿⣏⣴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣟⣟⣟⣻⣿⣿⣿⣿⣿⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣽⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⣛⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣯⣵⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⢿⣿⣿⢻⣿⣿⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⢿⣿⡟⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣦⣤⣤⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⢿⣻⠛⠿⡿⢿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣏⠛⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⢿⣿⠃⠀⢀⣸⣿⣷⡖⠋⡀⡀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠋⠹⣷⣄⣿⠀⢀⣼⣿⣿⣿⣟⠉⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⣀⣤⣿⣿⣿⢯⠀⢹⠀⠠⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠦⠤⣊⣉⢱⣿⣛⣿⣿⣧⣈⠇⢀⣼⣿⣿⣶⣶⣶⡶⢶⣶⣶⣴⣄⡠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⣛⣭⣿⣿⣿⣿⠟⠟⠀⣸⡿⠟⠛⠁⠀⠛⠉⠉⠉⠉⠀⠉⠉⠒⠦⠤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠾⢿⣯⠄⣀⣴⣾⣿⠿⠿⠛⣩⠄⠈⢠⠞⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦
⠀⠀⠀⠀⣠⡾⠛⠛⠠⠄⢀⣿⡿⢻⣋⠴⠀⡠⠊⢀⣾⣷⣯⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠄⠀⡄⢀⡀⣄⣤⠄⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⣠⠟⠉⠀⠀⠀⣠⣿⣿⡿⠗⡽⡡⢂⠈⣰⣎⣾⠿⠛⠛⠣⢦⣴⡾⠿⡿⣿⣿⣿⠿⠶⢾⣤⣶⣶⣿⣏⠀⠀⠀⠀⠀⠀⠀⣈⣤⣆⣤⠖
⠀⠀⣿⣾⡿⠓⠀⠀⠼⡟⣷⠹⠞⣼⢡⠟⣰⣿⡟⠣⠀⠀⠀⠀⠀⡁⢉⠰⣾⣿⡯⢿⣶⣤⣤⣿⣿⣿⣿⣿⣀⣀⣤⣄⣠⡤⠈⠛⠉⠉
⠀⠀⠉⠁⠀⠀⠀⠀⠀⣱⣿⡆⠀⢺⢻⢸⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠁⠀⠀⠄⠀⠀⡀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡁⠀⡇⣆⣿⣿⣏⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣤⡄⠀⠤⠒⠂⠀⣀⣠⣴⣾⠋⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⣿⣿⡉⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣡⣤⣤⣶⣶⣿⣿⣿⡿⠉⠁⢻⡆⠀⣠⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⢿⣿⣛⣚⣁⣹⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⡿⠃⠀⠀⠀⣿⡾⠟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⣿⣿⣿⣿⡆⠀⠀⠠⣸⡇⠬⣿⢫⣿⡿⠟⠋⠁⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⡏⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣿⣿⣿⡿⣿⣧⠀⡀⠀⣽⣷⡾⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠟⣴⠉⠈⡆⢸⣿⣷⡄⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⠀⠀⢀⣾⣿⣯⣀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠓⠂⠀⠀⠉⠀⠁⢻⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠀⠙⠿⣿⣿⣿⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠙⠀⠀⠀⠀⠈⠙⢳⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠖⠂⠀⠀⠀⠀⠀⠀⠙⣏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣳⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣦⣀⣀⡀⢀⣰⠛⠀⢀⡀⠀⠀⠀⠀⠀⣀⣴⣋⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣥⣉⠿⡿⢿⠋⠀⠉⠀⠉⠶⠂⢠⣼⣿⠋⠉⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠉⠁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⣀⡀⣀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⠋⠟⠻⠉⠇⠯⠄⠵⠼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡜⣀⠄⠀⠠⣤⠀⠀⢀⣠⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⣤⡄⢤⡄⢤⠄⡠⠤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⡗⢁⠞⣉⣀⣀⣀⣀⣀⠠⠤⠕⠺⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠋⠋⠓⠋⠓⠘⠚⠓⠒⠛⠘⠂⠀⠀⠀⠀⠀⢀⡴⢁⡴⠋⠀⣸⠟⠉⠁⠀⠀⠈⠉⠐⠶⣴⣇⡠⠞⠛⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⢋⡠⠛⢀⣴⣿⣛⣋⣀⠀⠀⠀⠀⠡⠀⢘⣯⣌⣁⠒⠒⠒⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢐⠛⠶⠋⠀⣰⡿⠿⠭⢭⣉⡑⣴⠏⣤⠄⣀⡴⣟⠒⠂⠭⣉⠛⠦⢤⣀⣀⠶⠒
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠋⢸⠶⠁⠀⣀⡤⠾⠥⢤⣐⡶⠞⠁⠲⣥⣴⡒⢛⡉⡑⠢⣌⠤⡀⡀⠉⠑⠒⠬⠵⢶⡶⠶⠚⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡀⠀⠐⠀⠀⠀⠀⠒⠾⢋⣠⠔⢂⠐⠒⢀⠑⠠⡄⢀⣀⣿⣷⣶⠂⠈⠑⠂⠑⠀⢩⣩⡤⠈⠐⠂⠤⣾⡷⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣷⣾⡀⢰⣾⡷⠒⠊⠀⠀⢀⠙⠓⡀⠈⢠⡴⠓⢄⠂⢀⠀⠀⠀⠀⠈⠑⠢⠤⠤⢤⠐⡞⠩⢉⠉⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠉⠀⠀⠀⠀⠉⠀⡀⠀⠁⠐⠈⠉⣚⠛⠚⠋⠀⠀⠀⢢⠂⢀⠀⠀⠀⠀⠀⢡⠀⠂⢈⠢⡐⠢⣀⠀⠀⡀⢅
⠀⠀⠀⠀⠀⠀⠀⡀⡈⢳⠄⠀⠀⢠⣔⠃⠉⢩⣼⡏⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠀⢸⠐⢀⡀⠈⠀⣀⠀⠁⡴⣷⠀⠀⠘⠓⠠⠩⣲⡀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⠂⠀⠐⠀⠀⡞⠈⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠃⠀⠀⠀⢠⠀⠀⠀⠙⢷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢁⣬⠗⠂⠀⡀⠀⠀⢠⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⠉⠀⠀⠀⠂⠀⠸⠀⠀⠳⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠀⡰⠋⠀⠀⠀⠚⢱⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⡤⠄⢠⠄⠀⠀⣰⠃⠀⠀⠀⠀⠙⠛⠓
⠀⠀⠀⠀⠠⠂⡁⠀⠀⠤⠊⠁⠐⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣰⠋⢀⠀⠀⠀⠀⣸⠋⠀⣦
⠀⠀⠀⠀⠒⠁⠀⠀⠀⣠⠄⡥⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠠⠀⡇⠀⠀⠀⠀⠠⠀⡧⡰⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⠔⠀⠀⠀⢷⠀⠀⠀⠀⠀⡂⢰⠁⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠋⠀⠠⠀⠀⠀⠀⠀⠀⣈⡷⣆⡀⠀⠀⠰⢀⠇⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠩⠀⠀⠀⠀⠀⠀⠠⠀⢀⡤⠖⣩⠀⠀⣑⠢⢄⡀⠈⠒⠤⣤
⠀⠀⠀⠠⢠⣈⡒⠀⠀⠀⠒⠀⠠⠀⠀⠀⢀⣀⡴⢋⠀⠀⠀⠀⢂⠀⢂⣠⠴⠊⠉⠀⠐⠃⢠⠐⠁⢠⠀⠉⠓⠦⣄⠄⠂⠀⠀⢤⡀
⠀⠀⠀⠀⢀⣈⣁⠄⠀⣀⡀⠀⠀⠀⠉⠙⢉⡽⠁⠀⠠⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡄⠀⠀⠹⠁⠀⠿⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠙⢢
⠀⠀⠀⠀⠀⠞⢳⡴⠈⠁⠈⠉⣀⡘⠃⠶⣾⠁⠁⠀⠀⠀⡀⢰⡟⠀⠙⠐⠂⠰⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠉⠀⠀⡏⠄⠀⠀⠄⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⡠⠀⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⡀⠀⠀⢿⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠒⠀⡀⠀⣀⠀⠔⠁⠀⠀⠲⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⢆⡀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠂⠀⠠⠈⠛⠾⢥⣄⡀⢀⣊⢀⣀⠀⢠⠀⠀⠀⠘⠁⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠂⠐⠀⠂⠀⡀⠀⠈⠉⠓⠾⢤⣦⡤⠀⠀⠀⠠⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣀⠀⠁⠀⢀⠀⠀⠀⠘⠀⠀⡈⠙⠳⠦⣤⠘⠃⠁⠠⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⡆⠀⠀⢀⠀⠀⠁⠀⠀⠐⠀⡀⠀⠁⠀⢀⠀⠀⠀⢁⠑⠄
⠀⠀⠀⠀⠀⠀⠀⢂⠄⢐⡀⢀⡴⠁⠸⠀⠀⠀⠀⠄⠀⠐⠀⠀⠘⡄⠓⣄⠀⠈⠁⢁⠐⠄
⠀⠀⠀⠀⠀⠐⠁⡀⠀⠘⡀⡼⠯⠁⢀⡀⠀⠀⣠⡆⠀⢸⠁⠀⠀⠇⠈⠿⡆⠈⠃⠀⠀⠘⠄
⠀⠀⠀⠀⠀⡄⠀⣁⠂⠀⠁⠧⠈⠀⠀⠉⢢⠜⠘⠀⠀⢨⡤⠚⠁⠀⠀⢂⠿⠘⠀⢂⠘⠀⢀
⠀⠀⠀⠀⠀⡀⠈⡇⢀⠆⡼⡄⠀⠀⡐⢰⡏⠃⠀⠀⠀⠙⠁⢀⠠⠂⠀⠀⣰⡄⢦⠀⠹⠀⠈
⠀⠀⠀⠀⠀⢡⠀⢠⠂⢄⢷⡙⢄⢀⠠⠄⣧⣀⠀⠀⡀⢀⡄⠀⣠⢁⠀⣨⢃⡟⣀⣡⠀⠠⠘
⠀⠀⠀⠀⠀⠸⠀⠈⠇⣈⠹⡎⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠃⠃⠀⠘⠈⠐⣩⠗⠀⠴⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢸⠉⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡐⠉⠒⠋
⠀⠀⠀⠀⠀⠀⠘⠀⠇⠀⠀⠀⠀⠀⠀⢠⠀⣀⠐⠂⠀⠀⢀⠀⠀⠀⠘⠁⠀⠀⠀⣠⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⠈⢠⠇⠀⠵⠲⠉⠀⠀⠓⣁⣶⠀⠂⠀⣶⡠⢀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⠀⠈⠲⣤⣀⠁⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠀⠀⠀⠀⢁⡄⣹⠁⠘⠀⠒⣆⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠸⠃⠁⠀⠄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠠

⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⢡⡿⠟⢳⡄⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⢀⣀⡀⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠺⠁⠀⠀⠀⡇⠀⠀⠀⠀⠓⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⠡⠄⠈⠁⠉⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⢐⡏⢀⠀⢄⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠊⠁⠐⠄⠀⠈⡋⠀⠇⠀⠃⠀⡀⠀⠀⠀⢀⡀⠋⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢭⣈⡁⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠀⠘⠂⠁⠀⠃⠀⠀⠀⠚⠀⠀⠀⠘
⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⠀⠰⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠆⠀⠲⠅⢢⡆⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡤⠀⠼⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡘⠁⠀⣿⠃⠸⠅⠀⠀⠀⠀⠆⣠⠂⠃⠈⠡⠄⡤⢤⣿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠄⠃⢹⢇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡘⠒⣠⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠐⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⠸⠄⢪⡀⢀⣀⡤⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⢜⣄⠀⠀⠀⠀⠠⠻⣀⡿⠛⠋⠉⠋⠙⠉⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⣀⢤⡄⠠
⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⣶⢟⣄⠀⠀⠀⠐⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢦⣀⡀⠀⠒⠃⣀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠓⠾⢡⡀⠀⠀⠀⠀⠉⠉⠙⢫⣥⡄⡀⠀⠀⠀⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠑⠀⠀⠀⠞⠣⠂⠀⣀
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠑⢴⡠⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣰⣷⡏⠹⡷⣦⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈
⠀⠀⠐⠴⠟⠂⠀⠀⢄⡀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⡶⢂⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⠀⠀⠀⠀⠐⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢒⣘⠯⡓⢈⣐⣻⣆⠀⠠⣀
⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠔⠀⠁⠀⠀⡀⠀⠀⠀⠀⠋⠻⠆⢈⠁⠀⣠⠤⠶⡀⠀⡀⠀⠍⠀⠀⠙⠀⠉⠀⠀⠀⠚⣯⡏⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⠘⣡⡄⠀⠀⠀⠀⠀⠐⢞⡂⣈⠖⠔⢂⠘⠈⢡⠀⠀⠀⠀⠀⠀⠀⠒⠸⠃⠠⡿⣇⠀⡀⢳⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⢀⠀⠀⡘⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠘⢄⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠏⠀⢐⣈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠀⠛⠀⠀⠛⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣾⠃⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢀⣿⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠙⣦⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣤⡾⠿⠛⠻⣷⡀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⠀⠀⠙⠻⣷⣤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⠛⠁⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠙⠿⣇⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣾⠏⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠂⠀⠀⠈⠛⣦⡀⢀⣆⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⢀⡴
⠀⠀⠀⠀⠠⡀⠀⠀⠸⣦⡀⠄⠀⠀⠈⠣⠀⠀⠀⠀⠹⣿⣅⠀⠀⠀⠀⠈⢿⣴⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠙⢦⡰⣄⠀⠂⠀⣤⡀⠀⠀⠈⠙⢧⡀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠘⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠹⢆⠀⠀⠀⠛⠾⣟⠷⣤⣌⣙⣦⡀⠀⠀⠀⠻⣦⠀⠀⠀⢸⡀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⢠⠈⠳⣄⠉⠛⠛⣷⠶⢶⣤⣤⣿⠃⠀⠀⢨⣇⠀⠀⣠⡟⠁⠀⠀⠀⣠⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠉⠓⢾⣄⠀⠈⢷⡄⠀⣿⡀⠀⠙⠻⣿⣦⡆⠀⠈⢿⣤⣾⠋⠀⠀⠀⢀⣼⣁⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡀⠀⠀⠀⠉⠳⡄⠀⠉⠲⣤⡙⢤⠀⠀⠘⢇⢻⡄⠀⢸⡿⠀⠀⠀⣀⣴⣿⠿⠛⠉⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⠀⠹⣆⠀⠐⠛⠛⠻⢷⣦⡀⢸⡀⢿⡄⢸⣷⠀⣷⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⣀⣈⠑⠿⠷⣦⣄⡀⠀⠀⠈⠛⢿⣧⠸⣷⠈⣿⣄⣿⣿⠟⠷⣤⣀⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡻⠛⠶⣄⡉⠻⣍⠳⢶⣤⣀⠀⠹⣆⣻⡇⠈⢻⣿⣇⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⣀⠀⠈⠀⠀⡠⠌⡙⣷⣮⡑⠂⢤⣙⠿⣷⣽⣿⣷⡀⢀⠻⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠟⠛⠛⠛⠛⠛⠛⢛⠻⠿⣷⣤⣈⣳⣦⣍⣿⣿⣷⣌⣷⣿⣿⣶⣦⣤⣤⣶⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⡤⠖⠊⠉⠉⠉⠙⠛⠛⠶⣶⢶⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠉⠛⠋⠉⠉⢁⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠠⠄⠒⠉⠀⠀⢒⣠⣴⣿⣿⣿⣿⡿⠟⢛⠿⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠤⠶⠛⣉⣁⣴⣿⣿⣿⣿⣶⡞⠋⠀⠀⣀⣤⣄⣟⠀⣿⡗⢀⡀⠀⠀⠀⠀⠤⠤⠜⠓⠂
⠀⠀⠀⠀⠀⠀⣠⠶⡞⠛⠙⠛⠁⠀⠀⠀⠀⠛⣉⣽⣿⣿⣿⣿⣿⣿⠁⠀⢆⣴⣿⣶⣿⣧⠀⣿⠀⠈⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⢀⣤⡤⠐⣂⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣷⡟⠀⠀⠘⠀⣴⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠔⠂⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢻⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠘⣿⣇⠀⠈⡁⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢻⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣙⣿⣿⣿⣿⣿⣿⠶⠆⠀⣄
⠀⠀⠀⠀⠀⠀⠐⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠿⠛⠃
⠀⠀⠀⠀⠀⡠⠊⢁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣠⠖⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡔⠁⠠⠚⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠈⠀⠀⠀⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠛⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⡟⠉⠘⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠉⠀⠀⢠⠟⠠⠋⢠⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⠟⠙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⡀⢻⡿⠿⡟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣀⢷⣾⣿⣇⢠⣾⣷⣅⠀⣀⣀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⠾⣿⣢⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣛⣿⣿⣿⣿⣿⣿⣋⣿⣿⠿⠿⣿⣿⣿⣿⣻⡏⠀⣰⣿⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣶⣷⣾⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠿⣿⣿⣿⠀⠿⣽⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠐⣿⣿⣧⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡿⢋⡷⠊⠽⢿⣿⣿⣿⡿⡟⣹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠸⠛⠿⣿⣿⣿⣿⣿⣿⢟⣿⢟⡟⠿⢿⣿⠗⠀⢻⣶⡿⢁⠀⠀⢪⣿⣿⣿⣾⡻⢟⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣀⠀⡿⠁⠙⠷⠦⣼⣿⣟⣿⣿⢿⣩⣽⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⢔⣣⠞⠁⠀⠀⠀⢰⣿⣿⣿⣿⣯⣶⣤⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢁⣤⣷⠿⠁⠀⠀⠀⠀⢴⣿⠟⢏⡉⠀⢽⡟⠀⠉
⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢟⣉⣡⣶⣶⣦⣤⣶⣿⡟⠁⠀⠀⠀⠉⠉
⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠿⢿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣀⡀⢤⣠⣶⣿⣿⣿⡿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⠛⠋⠘⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠻⣿⡾⡇⠀⠀⠀⣈⣛⡙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⢹⣶⣄⣤⣶⣾⣿⣷⣶⣶⣿⣿⣿⣻⣿⣟⣯⡳⣲⣤⣤⣤⡀
⠀⠀⢀⣀⣤⣾⣿⣿⣿⣷⡿⢷⣿⣿⣷⣽⣿⣯⣥⣉⡙⠻⣿⣷⣟⣿⢿⣿⡿⢳⡄⡀
⠠⠿⠿⢟⣿⡿⣿⣿⠟⢋⠀⠉⠉⠛⠿⠿⣿⣷⣭⡷⣿⠀⠀⠙⠛⠿⡷⡛⠚⡌⠁
⠀⠐⠒⠛⠛⠋⠉⠀⠀⠀⠀⠀⠘⠓⠒⠂⠈⠻⢿⡟⡏⠀⠀⠀⠀⠀⠀⠁⠈⠁


⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⣿⠿⠷⠶⠿⠯⣟⡷⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡼⣯⢿⡿⠑⠄⠁⠀⠀⠀⠀⠉⠳⣶⠞⣦⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠀⠐⠁⣂⡄⠐⢤⠀⠀⠀⠀⠀⠈⢳⡚⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠚⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠁⡉⠀⠀⠀⠀⠀⠋⢻⣤⠜⠲⢤⠀⠀⠀⣠⡆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⠁⠀⠀⠀⠀⠀⡆⣼⣶⡎⠁⢹⠃⠀⠀⠈⡠⡴⠚⣽⠧⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⡰⠁⠀⠀⠀⠀⠀⠈⠁⣾⣤⣿⠀⠘⠀⠀⡠⢊⣼⠁⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⣀⣀⠀⠀⠀⢠⠀⠀⠀⠀⣤⠌⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⡺⠇⠀⠞⣠⠞⠁⣬⠻⠶⢤⠶⢒⣲⠖⡜
⠀⠀⠀⠀⡢⠀⢀⣫⠦⢒⢶⣿⣿⣠⣤⡈⠀⠄⠀⠀⠀⠠⠔⣰⡇⢄⡀⠀⠀⠀⠀⠐⠓⠀⢻⣻⡦⣀⠞⠃⠀⠀⠀⠀⠀⣿⠀⠀⠉⣀
⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠛⠃⠀⠚⢛⠿⣶⣄⣀⢠⠠⢀⠔⠋⠒⠉⠐⠚⠛⠃⠀⠀⠀⠀⠸⡿⠟⠁⣤⠄⠀⠀⠀⠀⢨⡎⠛⠒⣿⠋⠁⠵
⠀⠀⠀⠐⠀⠀⠠⠄⠰⡆⠀⠀⣠⠀⠦⣤⣶⠟⠛⢿⣤⢣⠂⡀⠀⠀⠀⠀⠀⠀⠀⡠⠌⢠⡄⢷⡀⠈⠁⠀⠀⣠⡦⠀⠀⠀⠀⣔⢻⣍⡀
⠀⠀⠀⠀⠀⢁⠀⢰⣠⣵⣄⣠⢀⣤⡾⠊⣠⣤⣤⡀⠛⢬⠈⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠶⣄⢀⡀⠡⠀⢀⣤⠄⢈⢉⠁⠤⢉⡟⠓⠄
⠀⠀⢠⡐⠀⠢⣪⡿⢿⣿⣿⣥⠟⠉⠀⣼⠛⣠⠿⠷⠀⢢⠈⠁⠀⠀⠀⠀⠀⣴⠄⠂⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⡀⠈⢁⠀⣈⡀⠠⢔⢚⣧⠀⡃
⠀⠀⠀⠑⢀⡞⠁⠞⠁⠒⣿⡟⠀⡀⠄⠑⢾⣟⣆⠰⢷⡀⠀⠀⠀⠀⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠠⠑⠲⣤⣉⠁⣀⠀⠀⡈⠑⢦⡃
⠀⠀⡀⠐⠀⠀⠐⢂⣀⡴⢀⡄⠀⢣⡢⣄⠀⣊⣥⠴⠾⠿⣦⣤⣦⡙⠩⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠠⠉⡀⠝⠶⣤⠾⠗⠀⣠⣤⢽⡄
⠀⠘⠏⠀⠀⠐⢠⢾⣷⢃⠾⣡⠀⣸⣳⡿⣛⣛⡵⢶⣯⣠⣶⣿⡻⣿⣆⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡅⠐⠂⢠⡄⣗⢉⢧⡀⣽⠾⣽⡀⢿
⠀⠀⠄⠀⠐⠂⠈⢰⣷⣱⣼⣵⠄⡁⣿⣿⠟⡛⡷⣾⢿⡿⠋⣿⠿⢻⣿⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠔⠁⠃⢙⡐⠀⠉⣚⢆⢹⡆⡀⠸⢷⡸⡄
⠀⠀⠀⠐⠐⠠⠒⢿⣿⠾⣿⠋⠀⣿⢫⡅⣼⠀⠹⠇⡘⠉⠉⠁⠋⠈⠂⠚⣛⣿⣷⣦⣄⣀⠀⠐⠚⣀⠀⢏⡴⠂⠀⠢⡈⠁⠀⢘⢨⢸⢍⣷⠟⠳⠓⣟
⠀⠀⠀⡀⠀⠀⣠⠘⣿⢄⡇⡀⠀⠙⠳⠺⠉⠣⡐⠀⣠⣶⠶⠾⣿⢿⣷⣶⣔⠀⠻⣶⣿⣿⣟⢶⣦⣤⣙⣦⡤⡼⣶⠶⠾⠧⢴⣭⣀⠉⠘⣿⣈⣷⡀⢸
⠀⠀⠀⠀⢀⠀⠴⢀⡘⢾⠇⠀⠀⠠⣄⢩⠂⠀⢠⣞⡿⣶⠟⢉⡛⢻⣾⣻⣷⢟⣦⡱⡼⣭⢹⣙⣍⣽⡷⠖⠋⠉⠉⠉⡁⢀⣄⣈⣹⡳⣦⣈⡛⡄⢣⠈⠁
⠀⠀⠀⠀⠀⠀⠠⡂⢹⣇⢠⠀⣧⣤⠛⡃⠀⢰⣿⣿⢫⡆⣴⢻⢿⣷⣏⣟⣿⠆⠩⢷⠳⣾⠉⠀⠀⠀⠀⠀⠀⢠⡦⡎⠉⡁⠀⠁⠀⡅⢾⢳⣏⡑
⠀⠀⠀⠘⢄⢩⠄⢁⣸⡹⣄⠀⣪⢻⠷⢠⢈⢸⣿⡛⠈⠀⠻⡾⠶⣋⢾⣟⣿⠙⣰⠋⠻⢿⣷⣄⠀⠀⠀⢀⣼⣟⣿⢀⠑⠀⠀⢣⣀⠀⡹⣆⣏⢳⣇
⠀⠀⠀⢀⠀⠀⠀⠀⠛⠀⠀⢳⣿⣿⡷⣤⠀⠸⣿⣿⣶⣶⠷⠈⠉⡁⢹⣿⣏⣹⠏⠀⠀⠀⠈⠉⠛⣒⣀⣾⣿⣾⡿⠠⠁⠄⠀⠐⠀⠀⡞⣿⣹⢸⣇
⠀⠀⢀⠈⠈⠰⠂⠀⠀⢠⣶⠘⣋⣿⣿⣌⠀⠄⠹⣷⣦⣤⡤⠁⠐⣠⣬⢟⣤⠏⠀⠀⠀⠀⢀⣴⣾⡿⣽⣿⣿⡿⠉⠄⠀⢀⢀⠀⠀⡚⣇⣿⣪⣼⡏
⠀⠀⠈⠁⠀⠀⠀⠀⠀⡟⠛⠘⠛⢿⣿⣧⡝⠂⡀⠙⢿⣯⣤⣴⡷⢛⡥⠚⠁⠠⠀⠀⠐⣴⣿⣿⡳⣻⢧⣿⠟⠁⠨⠘⠌⠌⠀⡤⢻⢁⣾⢯⢇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡜⠃⣀⡂⠙⢿⣿⣱⠖⡀⠀⠙⣻⢟⣴⠻⠁⠂⠀⠀⠀⢀⣼⣿⣳⣯⣾⡿⠟⣁⠠⠨⠠⢀⢂⡀⢛⠟⣲⣿⡿⠋⡤⠚⡸
⠀⠀⠀⠀⠀⠀⠔⠂⠀⠘⢣⣎⢅⠐⠙⠀⢹⣿⣯⡘⣉⡴⣣⠋⠁⠀⠀⠄⠀⢠⡶⢿⡿⡟⡛⠋⠉⠄⡀⠈⠈⡤⣤⢌⣉⠓⣯⣾⣿⡿⠅⣠⢴⡀⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣆⢄⠸⣧⠤⢀⡶⣠⣾⣿⣿⣿⠞⠜⠁⠀⠀⠀⣀⣴⡟⢉⣁⡀⠀⣀⠀⣄⣴⣴⠈⣺⠛⠲⠚⣨⣥⣾⠟⢡⠉⣦⠔⣁⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠎⠳⣙⠷⣶⣾⣵⡯⠾⠛⠁⠀⡀⠀⣠⣴⣾⣇⣀⣛⣋⢉⢳⠔⠋⠷⢞⢃⣈⣛⣥⣴⣾⠿⢋⣻⠄⣤⣈⣵⡼⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣼⡓⢬⡑⠒⠂⠲⠛⣄⡚⠉⣀⠿⢿⠿⢿⠿⣿⣿⣿⠟⠻⠟⣛⣿⣿⡿⠟⣉⡀⠰⠆⠈⠉⠀⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣷⣀⣤⣾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡏⣹⠟⢁⣼⣶⠶⢾⣧⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠽⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣜⠿⠷⠒⡿⠋⠙⢳⣎⠉⠛⣯⣀⡀⠀⠀⠁⠈⠀⠄⠀⢆⣀⣀⠤⣄⣒⣤⣷⢾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡴⠾⢁⣠⣃⠉⠻⣷⡄⠀⣫⡤⢀⡼⠁⠀⣀⠀⠊⢩⣴⣾⣿⣿⡿⠿⠁⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡿⣿⠟⣿⣿⡀⣸⠀⠀⢸⣷⡁⢂⠔⠁⢀⣶⣿⣿⡿⠛⠋⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢆⣀⡠⢄⣤⠟⠋⠁⣶⡿⠉⢀⠃⠀⠀⣽⠟⡥⠏⣠⣾⣿⣿⣿⣟⣈⣀⣀⣀⣀⣀⠘⠿⢤⡀
⠀⠀⡿⠐⣀⡟⣩⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⣀⣹⡭⠿⠋⠀⠀⢈⣿⣿⣦⢸⣿⣿⣿⣿⣿⠿⠛⠋⠉⠁⠀⠈⠈⠉⡿⠉⠙
⠀⠀⠉⣓⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠊⠉⠈⠁⠀⠀⠀⢾⣾⣿⣟⠙⡟⢻⣿⣿⣿⣿⡤⢀⡀⠀⠀⠀⠀⠀⡌
⠀⣗⣘⠚⠆⠓⠦⣄⠀⠀⠀⠀⠀⣴⠗⠀⠀⠀⠀⠀⢀⣀⡴⠚⠛⠛⠻⠟⣉⣵⣾⣿⣿⡤⢅⡀⠀⠀⠈⠓⠦⣄⣐⣇
⠀⠈⠉⠉⡋⢺⠳⡈⠑⢤⣄⡔⠉⠀⠀⠀⠀⢠⠖⠋⣉⢛⣰⣾⣷⡿⢿⣿⣿⡿⣿⣶⠀⠁⠀⠈⠢⡀⠀⢠⠆⠉⠉⠉⠃
⠀⠀⠀⢀⡷⠜⠀⠈⢶⣴⠋⠀⠀⠀⠀⠀⠔⢣⡾⠟⠉⠀⠁⠀⠘⡁⠈⠀⠀⠁⠀⠈⠂⠠⡀⠀⠀⢁⢐⡏
⠀⠀⠀⠀⠀⠀⠀⣠⣸⣿⣄⠀⠀⠀⠀⢀⠾⠋⢀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠹⡆
⠀⠀⠀⠀⠀⠀⢀⣸⣷⡿⠍⠓⠂⠀⠀⠀⠀⠀⢿⡯⢟⣿⢿⣿⣷⣶⣤⣶⡀⣀⣤⣤⣴⣂⡲⢜⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣟⣴⣾⡆⠀⠀⠀⠀⠀⠀⠛⠀⠀⡈⠀⠀⡈⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣾⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡻⣾⣧⠀⢰⡀⠀⠀⠀⠀⢠⣦⣽⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⡧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⣵⢿⣇⡧⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣾⡉⢻⣟⡿⢿⣣⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡸⠉⠙⠛⠷⣜⡓⣄⣹⠏⠛⠛⠛⠋⠁⠀⠈⠀⠙⣿⡋⢹⡈⢸⡏⡏⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⠄⠰⠦⢤⣄⣣⠸⢿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⣏⢨⡇⠐⣧⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢀⣀⣀⠀⠀⠀⠉⢆⢈⣦⡈⠁⣀⣀⠀⠈⠉⠐⣄⣸⣉⣩⠄⡀⣸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⣞⡽⠛⠚⢲⠦⢀⠀⠈⡄⢿⠟⢀⣭⣟⠉⠛⣷⡀⠀⢿⠄⠁⣠⢣⣐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡏⠀⠀⠀⠛⠒⣦⡥⠟⠀⣴⣾⠋⠁⠀⠉⠉⠛⠳⡆⠘⣷⣾⢣⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢱⠄⠀⠀⢀⣼⣦⣴⠏⢰⣿⣷⣤⠄⢀⠀⣀⠀⢠⠇⢸⡿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⠀⠉⠡⢿⣿⣿⣿⡾⠟⠀⢨⡟⠛⠿⠿⢵⠶⢋⡴⠋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠓⠒⠚⠋⢁⠤⠀⢄⣚⡷⠶⠶⠶⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⠴⠷⢶⣾⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣶⣶⣶⠶⣶⢖⠚⠂
⠀⠀⠀⠀⠀⠀⣈⠀⠀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⠏⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠁⡠⠲⡇⣀⠦⠄⠀⠀⠀⠀⠀⡤⣶⣒⠤⣄⣀⢀⣀⡤⢒⣿⡷⠿⠲⠦⣀
⠀⠀⠀⠀⠀⢠⣴⠀⠀⢀⢙⠀⠀⠀⠀⠀⠀⡨⠺⠋⠿⠻⠶⣯⣽⠁⣴⣧⠧⠤⣤⣤⡀⠈⢣
⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣾⡁⢰⠂⠀⠈⢿⣌⢄⠀⢀⠀⠐⠦⣵⠷⠤⡀
⠀⠀⠀⠀⠀⠀⠠⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⣿⠿⠇⠀⠀⠀⠀⠈⠻⢧⡉⢺⣄⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠢⠒⠀⠀⠀⠀⠀⠀⡷⠄⠙⠷⠶⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢇⢦⠀⡇⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⢲⠤⠀⠀⠀⡈⢂⠀⠀⠀⠀⢠⣟⠿⡿⡿⢿⡿⣿⣦⠀⠻⣿⢷⣦⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⣆⣦⣤⣾⠇⠀⠀⠀⡄⠀⣾⣿⣿⣿⣿⠿⣿⣾⣿⠄⠀⠸⡆⠹⣧⠀⠀⠀⠟⠻⡧⠀⠀⠀⢀⣤⣤⣤⣤⣄⠒⠁
⠀⠀⢐⣿⡿⠛⠁⠀⢀⠀⠀⠃⠀⠸⣿⣿⣿⣯⣶⣿⣿⡿⠀⠀⠀⢳⡈⣻⡧⡀⢼⠀⢶⠀⠀⠀⣠⣿⣏⣉⣭⠏⠉⠁
⠀⠀⠘⣿⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠙⢥⠔⣼⡿⠀⡴⡴⠖⠾⡏⠀⢈⡵⠋
⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣄⠉⠁⠒⠸⠦⢮⣉⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⢀⢀⣀⡀⠀⠀⠀⢴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⣀⣀⣀⠍⠑⠲⠶⠤⠂
⠀⠀⠀⠀⠠⠶⣿⡷⢰⡿⠻⢫⣞⠁⠀⢹⣿⠁⠢⠀⠈⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠋
⠀⠀⠀⠁⢑⡆⣸⠁⣾⠃⠂⣿⣿⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢴⢶⣦⣄⡀⡀
⠀⠀⠀⣰⣾⠒⣥⣾⡿⠟⣸⣿⠏⠀⠀⠘⠂⠈⠂⠀⠀⠀⠀⠀⠀⣟⠛⠻⣟⠛⠛⠫⠰⡄⠀⠀⠀⠀⣄⣀
⠀⠀⠀⡟⠁⠀⢀⣀⣤⣴⣿⡿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠄⠈⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣥⢄⡀
⠀⠀⠀⡇⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⠦⠄⠀⠴⠀⠀⠰⡏⣷⡘⣷⡆
⠀⠀⠀⠱⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡉⠫⠠⠐
⠀⠀⠀⠀⠐⢀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡏⠀⠐
⠀⠀⠀⠀⢀⡸⣀⠀⣍⡓⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠄⠀⠀⠀⠀⡀⠀⠀⢰⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠙⠀⣿⡙⣫⡍⢩⡍⠹⠉⣟⢹⠟⣭⡟⡍⢹⠉⣿⣿⢹⣶⡏⠍⢻⠟⢭⡟⡉⢹⡍⣩⡏⣿⢹⡏⡍⣻⠻⢽⠛⢽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣼⣴⣧⣽⣧⣿⣧⣼⣧⣼⣤⣿⣼⣦⣹⣧⣦⣼⣤⣽⣷⣘⣻⣧⣷⣼⣤⣿⣥⣦⣼⣧⣽⣧⣭⣼⣧⣧⣽⣌⣿⣝⣼⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⡴⠶⡄
⠀⠀⠀⠀⠀⣰⣗⡈⠁⠀⠀⠀⠀⠀⠀⠘⠳⠄⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉
⠀⠀⠀⠀⢀⣭⣌⣁⠀⠀⠀⠀⠀⠲⠖⠀⢰⣆⣸⣇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠘⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠻⣏⠋⣀⣀⠀⠀⠀⠀⣀⠀⠠⣿⣿⣿⣾⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⠀⡹
⠀⠀⠀⠀⠀⠀⠈⢹⣿⠉⠀⠀⠀⠉⠁⠀⠀⠉⢻⣿⡟⢿⣿⡿⣆⣀⡤⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠙⡿⠛⠡⡏⢻⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠀⠀⠐⠿⠟

⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⠀⢰⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⠶⢾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⣴⣶⣆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⣸⣧⣠⣸⣿⣦⣀⠀⠀⠀⠀⠀⢈⣻⣿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⡿⠈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⣿⢿⠉⣽⢿⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣤⠿⠃
⠀⠀⠀⠀⠀⠸⠃⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠀⠘⠚⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠭⠃

⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠠⠾⠿⣿⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠛⣆
⠀⠀⠀⠀⠀⠀⠐⣿⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠈⠁⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣀⠋
⠀⠀⠀⠀⠀⠀⣀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣦⣤⣤⣤⣤⡀⠀⠀⠀⠀⠙⢠⣴⣷⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⡠
⠀⠀⠀⠀⠀⢰⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠻⡟⠛⢿⣿⡈⠇⠀⠀⠀⠀⠀⢾⠛⡿⠛⢻⡿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣼⠀⡆
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡇⠀⠌⠀⠅⠀⠀⠀⠀⠀⠀⠈⠀⣰⠀⠠⠁⡸⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠰⠂⠀⠀⠀⠠⣶⡟⠃⠀⠀⠀⠀⠀⢸⡇⢰⣾⣿⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⠈
⠀⠀⠀⢲⣿⣿⣶⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⢉⣒⣙⡛⢷⣿⣅⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠛⠋⠅
⠀⠀⠀⢾⠻⡟⠛⣿⢿⡟⠻⠷⠀⠀⠀⠀⠰⣿⣏⣉⣛⣛⣻⣿⠆⠀⠀⠀⠀⢼⡦⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠓⠉⠀⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠈⠉⠐⠄⢠⠀⠀⢀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠤⣼⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⡄⠀⡀
⠀⠀⠀⠀⠀⢀⠟⣿⣿⠧⣀⠀⠀⠀⠀⠀⣼⡟⣦⣦⣄⣀⣼⣛⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣏⠀⣿⡿⣄
⠀⠀⠀⠀⢀⣾⣿⡿⢿⡆⠘⠀⠀⠀⠀⠤⠿⠃⣹⣿⠻⠿⢿⡿⠳⡄⠀⠘⠋⣰⣾⠿⣿⣿⣿⣿⣷⣿⣗⡀
⠀⠀⠀⠠⠟⢱⠛⠁⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⢏⠀⠀⢸⠀⠐⠇⠀⠀⠀⢯⡸⢰⣏⡘⢯⠙⣻⡋⠙⠃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⠀⠘⠂

⠀⠀⠀⠀⠀⢰⣿⣆⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⠀⢀⣴⣷
⠀⠀⠀⡀⢠⣾⠟⠋⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠐⠹⠆⠀⠀⠀⠀⠘⣿⣇⣰⣏⢠⣿⡿
⠀⠀⠀⠀⣴⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣶⣲⣶⣤⣬⣧⡈⡀⠀⠀⠀⠀⠀⠈⠛⣿⡟⠟⠋⠁
⠀⠀⠀⠀⢿⣿⠁⣀⡀⢠⠀⠀⠀⠀⠀⢠⣾⣿⡞⣿⠉⠛⢿⡟⠻⠀⠀⠀⠀⠀⠀⠀⠚⢳⣦⣤⣄
⠀⠀⠀⠀⠈⠻⣧⣿⡽⠸⠷⠂⠀⠀⠀⠈⠉⠀⡄⠸⠄⠀⠈⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣠⣤⣶⣶⣶⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⣷⣷⣤⣄⣀⠀⠀⢠⡀⠠⣜⣶⡀⠀⠞⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣰⣿⣿⣭⣤⣴⣶⣶⣿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⢹⣝⠿⣿⣿⣿⣶⣤⣿⣶⣞⣷⣤⡆⠀⠐⠲⠶⣶⢄⠄⠀⠀⠀⢠⣾⣿⣿⣿⠋⡛⣋⡉⠉⣉⣑⣦⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠈⠒⢾⣿⣿⣿⣿⣿⣿⣿⣟⣿⠀⠀⠀⠀⢠⣴⠚⠁⠀⠀⠀⠀⣿⣿⡦⠃⣄⣉⣁⠈⠛⠀⣉⠀⠉⠠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠿⣿⣿⣿⣿⠏⣿⣤⠄⠀⠀⠉⠛⡷⠐⠒⠆⠀⣴⣿⣍⡍⠀⢦⢐⣉⣋⠻⢍⠉⠑⠢⣄⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡘⠛⣿⣯⡜⡟⢉⣿⠂⠂⣠⠞⣷⣀⣠⣴⣾⣟⣿⡏⢠⡷⠒⠤⢁⠉⢁⡠⡀⣠⣄⢉⣻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣯⡙⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣿⣻⣯⣷⣾⡇⠀⠀⠁⢼⢁⣻⣿⣟⣹⣿⣟⣿⣎⣿⣶⣶⣄⣦⣠⡿⢾⣿⡙⠻⢿⣿⡄⠹⢿⣿⣿⣿⣿⣿⣧⡁⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⣝⠟⢣⣤⣆⢓⠍⣉⣴⡟⣿⣿⣶⣺⠿⣿⣿⣿⣽⡟⣿⠙⢿⠀⠙⠁⠀⠀⠈⠁⠀⠈⠛⢿⣿⣿⣿⣿⣷⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⠇⢛⠂⠀⠋⠹⠉⠀⠁⠀⠸⠏⢻⡩⣧⣵⡿⠟⠀⠀⠈⠉⠲⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠟⠻⣿⣿⣿⣿⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⠀⢠⡄⠀⢀⡀⣠⠀⠀⠤⣤⣀⣾⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⢙⢷⠠⣄⠀⠉⠠⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠘⢀⠁⢀⡀⠞⠺⠛⣄⢀⣠⠇⡿⣝⢿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⢦⠐⡤⣱⡐⠲⢲⣟⠋⢻⡟⠻⠁⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⢰⣙⣄⡀⣠⠗⠀⣴⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠒⣽⠷⠶⢾⣇⡾⢿⣤⣈⣁⡃⢰⠞⠛⠛⢿⠇⠀⠀⠀⢀⢠⠀⠀⠘⠁⠁⠀⠀⠺⢍⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⣶⠁⠀⠀⠘⠋⠀⢈⡹⣾⠏⠀⠘⡄⣠⣀⠸⠀⠀⠀⠀⠀⠻⡗⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣾⠂⠀⠀⠀⡀⡠⢔⡯⠚⠁⠀⠀⠀⠙⠙⢯⡀⠀⢰⡆⠀⠀⠀⠰⡈⠑⣤⡄⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠆⡀⢠⣀⣦⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣾⠁⠀⠀⠀⠀⠘⠆⠻⢧⣤⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡏⣜⣠⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣰⣶⡦⣶⣾⡷⣯⣱⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⡯⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣿⡄⠀⠀⠀⢠⣤⣤⣄⣄⠸⠁
⠀⠀⠀⠀⠀⠀⠻⡟⠗⠛⠁⠀⣿⣿⡾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⢶⠶⡇⠀⠀⠀⠉⠀⠉⠍⠩
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃⠸⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣤⣶⢶⣶⣶⣦⣶⣶⣶⣿⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠟⠛⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⡿⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡰⢦⣿⣫⣿⣿⣿⣿⣿⣿⣶⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠂⠠⣤⡀⢀⣠⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣾⠿⠷⠛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣽⢿⡳⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⢺⠁
⠀⠀⢠⣤⣼⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⠀⣀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠀⢸⠁⠀⠰⠆
⠀⠀⠈⠻⠛⢿⣿⣿⣇⢰⣀⠀⣶⣶⣀⡀⡌⢿⣧⣷⣼⣿⣾⣿⣿⣿⣷⠿⠟⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⢐⣀⠈⣀⣠⣀⣀⣀
⠀⠀⠀⠀⠀⠈⠉⠛⣿⠿⣿⣤⣌⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⣿⣿⠿⠄⡶⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠉⠉⠙⠛⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡏⠀⠀⢸⡀⠀⠀⢀⣵⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣗⠤⡅⠀⢸⠀⣠⣦⣿⠟⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⡿⣟⠙⢿⣤⡀⠐⣼⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠙⣿⢿⣿⡟⠀⢸⠘⢻⣿⠟⢿⠻⢯⣷⡄⣭⣹⣿⣿⣟⣛⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠿⠀⠀⢸⠀⢸⣻⠀⢸⠀⠀⢹⠉⠙⠉⠙⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⣆⠀⣈⠁⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠀⠀⠀⠀⠀⠉⢱⡆⠈⠀⠔
⠀⠀⠀⠀⠀⣀⢀⡉⡄⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⣤⠖⠋⠉⠙⠒⡀⠛⠃
⠀⠀⠀⠀⠀⠈⠭⣧⡊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢀⡼⠁⠀⠀⢀⣾⡷⠂
⠀⠀⠀⠀⢀⣴⡜⠁⡄⠀⢀⣠⡀⠀⠓⣆⠀⠀⠐⣁⠌⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣴⠛⠁⡅⠠⣏⠀⠈⠉⠑⠲⢴⡌⢣⣤⣴⡿⠀⠀⠀⠀⣠⠏
⠀⠀⣰⠏⠀⠀⠉⢺⡘⠓⠀⠀⠀⠀⠘⠀⠈⠁⠈⠁⠀⠀⢀⣴⣯⠖
⠀⠀⠈⢣⡀⠀⠀⠀⠳⣦⠀⠀⠀⠀⠀⠰⠀⠀⣧⣤⣀⣬⠟⠁
⠀⠀⠀⠘⠈⠐⠢⢄⠀⠈⠙⠶⠤⣤⣤⣶⢠⡼⠿⡿⠛⠙⢉⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠀⢀⡀⢀⡀⢀⡀⢀⡀⠀⠀⡀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠸⠀⣼⣧⢜⠡⠊⡆⠈⠁⠴⢪⠈⠀⢀⣴⣩⣠⠔⠀⠉⣄⢀⠐⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⡚⠋⡜⠂⠁⠀⠃⢀⣀⠴⢂⠂⠡⠭⠛⠀⠀⠉⠀⠐⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠂⠀⠀⡇
⠀⢸⡛⣠⠃⢀⡄⣀⠀⠁⠀⠒⡃⢠⠆⠁⠀⠀⠀⠀⠀⠠⠒⠒⠒⠒⠤⠎⠀⠂⠐⠂⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⡇
⠀⠸⠊⠁⠠⠊⡞⠕⡀⠈⠉⠀⣀⠀⣤⢘⣦⣧⣾⣾⣼⣇⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⢸⠀⣀⢃⠚⠀⠀⠐⢶⡼⠟⠋⠉⠉⠀⠀⠀⠀⠀⠙⠻⣶⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⢐⠀⠈⠀⠙⠁⠀⠠⣬⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⠀⠀⠀⠶⡗⢀⠈⠓⠂⢤⠄⠐⠆⠠⠀⡇
⠀⢈⡜⠁⠠⡂⠀⢰⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣄⠞⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠠⢃⡄⡄⠁⠀⣩⣿⠇⠀⠰⠂⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠻⣦⠔⠀⠁⠀⡀⠀⠀⠀⠀⠄⠀⠁
⠀⢀⠘⡀⠀⠀⠀⠖⣻⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠠⠀⠈⠛⠒⢀⠚⣩⢷⣅⣀⣀⣆⣶⢠⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⠀⠀⢠⠄⢀⡤⠀⢦⠄⠠⠄⠀⠆⠀⠀⠀⠇
⠀⢠⡤⣶⣀⠀⣨⠐⢣⡏⠌⠁⠀⠉⠛⠺⣅⣶⡄⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠈⠱⠁⠫⡇⠇⠉⣄⠁⠀⠀⠀⠀⠀⠀⠈⠛⣷⢀⠀⠀⢀⡀⣺⠄
⠀⢠⠐⡄⣀⣿⡌⢢⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⢄⣔⣿⠋⠁⢸⠶⣃⣀⢈⣦⠢⠀⠀⠉⣀⠀⠈⠁⠀⠀⠀⠆
⠀⢸⠙⡷⠟⣹⢿⡅⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⣠⣥⡏⣾⠙⢁⡞⠁⢋⠞⠛⠡⠘⡇⠀⠁⡈⣡⠆⠀⠀⠀⠀⠀⠄
⠀⠸⠀⠁⠘⠃⢩⠃⢠⠄⠀⠀⠀⠀⠀⠀⠀⣸⡏⠉⡇⠻⠀⢸⢣⠏⠀⢠⠎⠀⢰⡏⠄⠈⠛⠓⠂⠉⠠⠢⠋⠁⠆
⠀⠀⠀⠀⠀⡀⠰⠇⠀⠀⢀⡿⣴⠂⠀⠀⢠⣽⠋⡃⠓⠤⠑⠞⠻⢧⣴⠿⣤⣴⡟⢠⣏⣤⣄⠀⣠⠶⣰⠂⠠⠀⡄
⠀⠸⠀⠀⠀⡴⠊⠀⠀⢀⡴⠁⢸⡃⠀⠀⣾⠛⠀⠸⢷⠎⠛⢲⡄⠀⠀⣤⡂⠙⣴⣯⣴⡿⠛⠂⠥⢦⡶⠂⠀⠀⠂
⠀⢠⡀⠀⢹⠐⠀⢰⡎⠉⢦⠀⠘⣓⠠⠀⣶⡆⠐⢒⠀⠀⢠⠶⣭⡤⢈⠁⢄⠀⠠⠀⠁⠀⠀⠂⠀⣭⢥⠃⠀⠀⠇
⠀⢘⠀⠈⠍⠄⠰⢾⡁⠀⠀⢡⡀⠟⠂⠀⠹⣃⠀⠀⠀⠀⢀⡀⠈⠀⠀⠡⠀⠀⠀⠀⠐⠂⢀⠴⠊⠁⠀⠁⣠⢀⡄
⠀⠀⠀⠀⣶⣦⣠⣌⡀⠀⠀⠀⠇⠺⢄⡀⠀⠹⣗⣤⡀⢁⢸⣿⣾⣶⡿⣆⠀⡠⠀⠸⠃⠰⠶⠒⠋⠹⣿⣧⣄⣁⡄
⠀⢸⠴⢺⠇⣼⣿⠘⣇⡀⣀⠀⠀⠀⠈⡿⠀⠀⠈⠙⣷⡜⢄⠙⠟⠛⠃⠁⢀⠀⢠⣘⣟⡒⠒⠒⠖⢻⣿⣿⣿⣿⡇
⠀⠸⠂⢀⡉⠛⡛⢒⡭⠟⠉⠓⢄⢀⡘⠀⠀⠐⠀⠍⣠⣟⣤⠐⠂⠀⢀⣾⣮⣤⣄⢉⣾⠍⠁⠀⠀⠀⠀⠙⠿⣯⡅
⠀⢰⠀⠀⠀⠈⠉⠁⠀⠀⠀⠇⢀⠜⠁⠀⠀⢰⠎⠀⠉⠈⠛⣤⠂⣼⠏⠛⠛⠙⢻⡞⢹⢻⣾⠗⣶⣶⣶⣦⣶⠟⡁
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠶⠀⠀⠀⠀⡞⠂⠀⠤⣄⣠⡽⡗⠀⣴⠀⠀⠀⠀⠈⢲⡉⠘⠁⠈⡯⢉⣄⣠⡄
⠀⠐⠀⠀⠀⢠⡂⠀⢀⠀⡴⠂⠲⣄⠀⠀⠀⠉⣣⠀⠀⠀⠉⠀⡀⠀⢹⠇⣤⡀⠀⠀⠘⠛⢄⠀⠿⣤⢧⡀⢣⡀
⠀⠐⠀⠈⣞⣾⣧⢰⡯⠊⠀⠀⠈⠉⣣⣄⠀⠀⠙⢦⡀⠆⣀⡀⠀⠀⠀⠀⠈⠃⣱⢦⡀⣤⡼⣆⠀⠉⠀⠁
⠀⢨⠀⠚⢿⣿⡱⠟⠀⠀⠀⠀⠀⠀⠀⠉⣆⣀⣠⣼⣇⠀⠈⠁⡆⠀⠀⠢⠐⠉⠁⠈⢙⡟⢡⡘⣇⠐⠂⠀⢀
⠀⠨⠀⠀⠀⠈⢃⠀⠀⢂⠀⠀⠀⠀⠀⢀⣿⠋⣿⡉⢻⡄⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⠈⣇⣸⣧⣼⣟⡦⡔⠉⠀⡃
⠀⠘⠀⠀⠀⠀⠀⠆⢀⢾⣸⣀⡤⠶⠊⠙⣯⣤⣿⣧⣼⠇⠰⠧⠀⠀⠀⠀⠀⠀⣶⠈⢹⠗⠉⠉⠁⠀⠀⠀⠀⠀⠇
⠀⠨⠀⠀⠀⠀⠀⠈⡸⣾⣿⣯⡼⠋⠀⠀⠀⠀⠀⠠⡞⠁⠀⠀⠀⠀⠀⠀⠈⣷⣿⣷⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠘⠀⠂⠀⠀⠀⠀⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠂⠀⠈⠛⠛⠃⠀⠀⠒⠀⠀⠀⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠁⠀⠀⠀⠀⢀⠔⠀⠀⡤⠊⠁⠀⠠⠾⣖⣋⠉⠀⠀⠠⠂⠄⠸⡄
⠀⠀⠀⠀⠀⢄⡀⢠⣄⠲⢶⣶⡀⠁⠰⣶⠁⠔⢖⡋⠀⢶⣀⡀⠒⠒⠂⠒⠻⠫⣙⡃⠋⠠⡀⠒⠲⣄⠰⠒⠒⠒⠒⢲⡆⡀⡒⡆
⠀⠀⠀⠀⠀⠀⠑⠤⠀⠢⢀⣙⢣⠀⠀⠉⢧⠀⠈⠱⡄⠀⠀⠈⠁⠀⠀⠀⠀⠁⠏⡆⠀⠀⠀⠀⠉⠀⠉⠁⢠⠀⠀⠀⠐⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠂⠀⣀⡀⠈⠈⠂⠀⠀⠀⠈⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⢀⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⠀⢀⠱⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠀⠑⠄⠘⠡⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡶⠀⡂⠀⠙⠒⣠⡀⠈⠢⢈⡆⠀⠀⢀⣀⠀⠀⠀⠄⠀⢀⣀⡀⡀⠀⢠⡀⠀⠀⠀⢿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠙⠒⠒⠒⠒⠚⠛⠚⠛⠛⠀⠀⣿⣿⣧⠀⠀⠐⠚⠃⠤⠉⠛⣠⠝⡳⠆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⡀⠀⠠⠁⠀⠀⠀⠐⠊⠀⠔⠁⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠤⠀⠀⠐⠋⠀⠀⠀⣠⣿⡃⠀⠀⠘⢿⡟⠿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠀⠙⠿⠿⣦⣤⣠⡄⠀⠀⠤⠀⠰⠟⣿⣿⠃⠀⠀⣤⠾⠁⠀⠙⣿⠿⣿⣶⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢻⡇⠀⠀⠀⣀⣠⣾⣿⣿⠀⠀⠀⠁⠀⠀⠀⢠⡀⠀⠀⠹⣧⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠤⠀⠀⠀⠀⠀⠠⢤⠤⢶⣿⣅⠀⢠⣤⣿⡾⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⣿⣶⠏
⠀⠀⠀⠀⠀⠀⣀⢔⣻⣤⣤⠄⢀⣀⣴⠶⠾⠿⠿⠿⠶⢿⣿⡿⠀⢀⠀⠀⢀⣿⠏⠀⠀⠀⡀⠀⠀⠀⢀⣭⣿⣿⣴⣸⣿⢣⡶
⠀⠀⠀⠀⠀⠀⢁⣾⠛⢡⡴⠾⠟⠛⠁⠀⠀⠀⡀⠀⣴⣿⣿⣀⣾⣏⠀⠴⣿⡏⠀⠀⠀⢸⣿⣤⠀⣼⣿⡋⢀⣿⠏⣿⣿⠏⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣶⡿⠤⠶⣤⣤⣶⣦⡤⠀⢁⣼⠿⢸⠟⠛⠘⣿⡀⣶⡿⠀⠀⠀⠀⠈⠛⠻⠀⢿⣿⡀⢸⣿⠰⣿⡋
⠀⠀⠀⠀⠀⠸⣿⣟⠀⠀⠀⡴⠛⠛⠋⠁⠀⣀⣿⣿⣆⢠⡀⠃⢠⡈⠁⠈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠻⣧⡀⢀⣴⠂⠀⠀⣠⠶⣶⠟⠁⠸⡏⣿⣷⠂⣼⣿⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡙⣿⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢻⣇⣰⡿⣾⣿⣶⣏⣠⣤⡞⢙⣿⣫⠞⠋⡁⠴⠉⣴⢞⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⢣⠟⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢋⣵⡾⢋⡝⢂⣴⣉⣐⣥⣴⠓⠁⢀⡼⠃⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡇⠠⠟⢉⡜⣡⣴⣿⡿⢋⣿⣿⠉⠀⠀⣠⣴⣶⡿⠟⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢶⣀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠈⢳⣆⣸⣿⣾⠿⠟⠁⠶⣬⣽⠿⣤⡰⢾⠯⠉⠛⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢴⡟⠳⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠃⠀⢀⣾⢂⣈⣡⠴⠊⣡⠆⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⢠⣿⠿⠟⠛⠁⠀⠀⠟⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠃⢶⡆⢠⠄⣴⠆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠘⠓⠲⡜⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢆⡁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⡉⢉⣤⡈⣳⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡀⠞⡣⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠁⢿⣧⣼⡿⢀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡅⣷⡀⠀⠀⠀⠀⠀⠈⠑⡄⠀⠟⢿⣿⣇⣸⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⣤⣿⣅⠀⠀⠄⠀⠀⠀⠀⠑⠄⢸⣼⡿⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣛⠃⠀⠀⠙⠿⠻⣆⠀⠀⢀⠀⠀⢀⢀⣤⠿⠏⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢣⣤⣄⠈⢶⣤⡄⠀⠘⢿⣆⠀⠀⠚⠐⠛⠻⡈⠡⠀⠀⣾⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡈⠓⠈⢏⣸⣇⡀⠀⠿⣷⣴⣀⢰⣶⠐⠁⠀⢀⡞⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⠿⠃⠀⠀⠈⡁⣾⣿⠦⠀⠛⠉⣨⠟⢀⠟⢀⠴⠟⠀⡟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠂⡿⠂⠀⠀⢰⣿⠀⣿⡟⣠⡆⠀⠘⠁⢠⣤⠒⢁⡴⣧⡿⣀⡀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣶⠃⠀⠀⠀⣸⣿⣴⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⢪⡵⠊⠚⠛⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢹⡇⠀⠸⣇⠻⣬⣛⣿⣿⢳⡄⣖⡀⣀⣡⠤⠛⣧⠦⢀⣻⣦⠀⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢹⣼⣧⠀⡄⢸⣆⣩⣿⣿⣿⣿⡿⣿⣿⠛⢁⣠⠖⢁⣰⠯⠛⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⣿⣿⣦⢿⣄⣿⣿⣿⣿⠟⣡⣾⣿⣿⣶⣿⣧⡴⠞⠋⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⢿⣷⡿⢸⣿⣼⣿⡟⠋⣼⣯⣿⣿⠟⠛⠋⠀⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠿⣾⣿⣧⣿⣿⣿⡟⠁⣾⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣿⣿⣏⢷⣼⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣏⣻⣿⣿⣼⣿⣿⠟⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⠈⠁⣀⠈⢠⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣙⢿⣷⣦⡀⣠⡄⢘⠂⣀⣶⣌⣇⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢴⣦⣽⣧⣐⣫⣤⣌⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠉




⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣟⣛⣛⣛⣛⣋⣛⣛⣃⣈⣙⣛⣟⣛⡉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣬⣭⣭⣭⣭⣭⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣍⣭⣭⣉⣭⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⡴⢶⡽⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠄⢀⡀⠀⠀⠀⢀⣠⣠⣀⡀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⢇⡄
⠀⠀⠀⠀⠨⠵⢞⣳⡟⠻⠂⠀⠀⠀⣠⣀⣤⡀⠀⠀⠀⠈⢢⣄⣠⡄⠃⠀⠀⠀⣿⣿⡟⠁⠀⣀⣾⣿⢋⠈⠀⠀⠀⠀⠀⠀⠐⣯⠁⡄
⠀⠀⠀⠀⠀⠀⠁⢻⠀⠀⠀⠀⠀⠰⣿⠿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡇⠠⠔
⠀⠀⠀⠀⠀⠀⠀⡎⠰⠃⠀⠀⠀⠈⢾⣿⣽⡞⡗⠀⠀⠀⠀⢲⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⢀⠀⠀⠀⠀⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠉⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⣤⠀⢘⣦⠾⠁⠀⠠⠐⣦⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⢠⣤⢞⠀⠀⢀⠈⡄⢠⠀⣀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠔⠋⠀⠊⠀⠀⠀⠀⠀⠀⠅⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡟⠋⠉⠁⠀⠀⠈⠀⢑⡺⡍⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⡇⠀⠀⠀⠀⠀⠈⠰⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⢀⡄⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣨⠁⣉⠀⠀⠀⣶⡶⠀⠀⠀⡇⠀⡿⠋⠁⠀⠀⠀⠀⠀⠀⡀⢉⠀⠀⠀⠀⠆
⠀⠄⠀⠀⠀⠀⣀⠀⡦⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⡃⡿⣃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣿⠀⠀⠀⠾⡷⠄
⠀⠂⠀⠀⠀⠀⠈⢰⡖⠚⠀⠀⠸⠿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠉⠳⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣾⠆⠄
⠀⠂⠀⠀⠀⠀⠀⠘⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉
⠀⡂
⠀⠁
⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⠀⠀⢀⣰⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢾⣱⡎⠀⣡⡿⣿⠿⠛⠃⠀⢀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠿⠟⠿⠋⠀⠳⠋⢀⡤⠐⠒⣻⣿⣍⡃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢦⡤⠤⠤⠤⣦⣐⢩⡞⢠⣆⣿⣥⣴⣾⣿⣿⣟⠉⣉⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣟⣿⣶⣶⣶⣤⣌⠁⠂⠾⠿⠿⣿⣿⣿⣿⣛⡛⠋⠉⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⡈⠁⣼⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣸⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣣⣽⣿⣿⣿⣙⣷⠀⠀⠀⢀⡄⠀⠀⢠⣷⣦⠉⢿⡿⢿⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣈⣁⣦⣽⠵⣟⠿⣯⢿⡆⠀⠈⠉⠁⠀⠀⣼⡟⠿⢹⠟⣰⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⡟⠈⠻⡗⣊⠿⢷⣶⣾⡟⠀⠀⠀⠀⠀⢰⣯⣤⡴⡟⣤⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣦⡀⠓⣆⠿⠆⢀⣿⡉⠀⠀⠀⡠⠀⢰⣿⠿⠿⠳⠿⢟⡻⠁⠈⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠸⡇⢀⣷⣿⣿⡀⠀⠀⠀⢀⣤⣿⣟⢀⣠⣶⠈⠛⠁⠁⠀⠰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠘⣿⣻⣿⣿⠟⠀⠀⣀⣣⣤⣾⣿⣿⣿⡈⣿⡩⠀⠿⠉⠉⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡯⠁⠀⣠⣬⣿⣿⣿⣿⣿⡿⠟⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⡹⢿⣿⣿⡿⢄⣉⢴⣿⣿⣿⣿⣿⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡘⣠⣿⣿⣿⠋⢀⣹⣿⣿⣿⠰⣿⣿⣿⣿⡙⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢃⣾⣿⣿⣿⣿⣿⣯⣟⣹⡟⠀⠀⠀⠀⠀⠀⠐⢂⡄⠀⠠⠀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣟⢩⣯⣼⠟⡿⢿⣣⣴⣶⣦⠔⢛⣯⡦⠀⠀⠀⠀⡐⢡⣾⣿⣾⣧⣔⡘⣁⢤⡀
⠀⠀⠀⢅⠀⠀⠀⠀⣠⠀⣠⣿⣿⣿⣿⠿⢫⣷⣿⣿⣿⢿⣿⣼⣿⣧⠘⢀⠀⢰⢳⡞⠋⠜⠁⠀⠀⠀⢻⡇⠁
⠀⠀⠀⠀⠙⠓⠧⡤⠤⠚⣿⢿⣿⣿⣿⢶⣽⣿⣿⣿⠇⠀⠻⠿⢿⣿⣔⣞⣇⢸⣿⠀⣴⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⣹⣿⣿⣿⣯⣿⣿⣿⣿⡀⠀⠁⠠⢈⡿⣿⣿⣿⣾⣿⣷⣷
⠀⠀⠀⠀⠀⠀⢰⠞⣵⠄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠄⠻⠷⠯⢛⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢘⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠣⣀⠁⠀⢀⠀⠀⠀⠘⢾⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡹⠈⠀⠛⠋⢹⣿⣿⣿⣿⣿⣿⣦⡄⠀⠈⠀⢀⡀⠀⠁⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠻⣿⣿⣮⣻⣿⣿⣿⡁⠀⠀⠉⠀⢸⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣦⣤⣴⣿⣿⡿⢋⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⣤⣀⠠⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡿⢿⣿⣿⠶⣾⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⡶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⠺⠀⢠⠄⢹⠧⠱⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠧⠰⠉⠀⡀⠨⠦⡈⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢔⣤⠀⠀⠢⠀⣈⠶⠀⡟⠂⠀⠀⢀⠴⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⡁⠀⠀⠀⠳⡆⢠⡏⠀⠀⠐⠆⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠠⢯⠀⢩⡖⠊⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧⠀⠁⣀⠀⡱⢆⢹⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⣤⠔⠓⢠⠒⠠⠈⠘⠀⠉⠙⢉⡦⣀⠀⠀⠀⠀⠀⠈⠓⡶⢤⣨⡞⢳⠖⠼⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠏⣵⢖⣀⡄⠐⠀⠀⢉⣤⡀⠘⣌⠃⠀⠀⠀⠀⠀⠀⠘⠂⠉⠉⣁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠠⡞⢷⠀⠀⠉⠒⠧⣦⡼⠁⠻⣏⣱⠄⠀⠀⣀⣤⣤⡀⠀⣀⠴⠁⠠⠤⡀⠈⣳⠃⠀⠀⢀⡰⡀⠂⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢺⣟⠉⠉⠁⠐⠼⡧⠀⡀⠈⠑⢤⠞⣹⠓⣉⠁⢹⣄⣌⠀⠄⣀⣀⣈⢇⣘⠔⠀⠈⠘⣠⠃⠀⣀⣀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⢣⣀⣀⣀⠀⣹⠀⠀⠁⠀⠀⠙⣾⠀⠀⠀⠢⣀⣸⣄⠀⠀⣠⣾⣿⣿⣿⣷⣾⣶⣛⣤⡫⠏⣧⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⠄⠀⠈⢨⡆⠀⢀⠀⠀⠀⠈⢳⡁⡀⠀⢀⠡⢨⣳⣮⣤⣤⣭⣭⣄⣈⡛⠟⠉⢿⠁⡀⢡⡀⠿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠉⠚⠓⢲⠏⣤⡄⠀⠀⠙⠂⠀⣀⠹⣄⠀⠜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠾⠂⢀⠀⠀⡀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢧⠀⢀⣀⠀⢱⠉⠁⠀⠉⠲⡇⠀⠈⠣⡘⣆⣰⣾⣿⣻⣿⣿⢿⡟⣿⢻⠟⢙⡃⡴⠒⣀⠀⣤⠤⣤⣧⠋
⠀⠀⠀⠀⣤⠤⣾⠁⢝⣦⠔⢮⠱⡏⠀⠀⠈⢣⠤⠄⠤⠄⢠⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣸⣿⣿⠜⡷⠀⢀⣍⣀⣉⡐⠡⢴⡥⠊
⠀⠀⢀⡤⠿⢀⠬⢀⠈⠙⠀⠚⢱⡧⠀⠀⠀⡧⠀⠀⠠⣤⠄⣁⠀⠀⠀⠈⣿⣿⣿⣿⣿⢿⡟⣿⡃⠀⠀⡰⠃⠀⠃⠈⠉⠒⠐⠃
⠀⠀⣿⡖⢀⠤⠀⡀⠐⠀⠀⠂⣿⢱⣷⠀⢸⡀⢾⡗⠀⠀⢢⠀⠉⠢⠀⠀⢸⣿⣿⣿⣿⣾⣣⣿⣷⣤⣈⣀⠀⢠
⠀⠀⠘⡴⣤⠄⣤⠤⣀⠉⣤⡀⠈⢦⡴⠊⠀⠁⢆⠉⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⡟⢸⣿⠋⠉⠛⢿⣷⢹⢄⠁⠂⠈⢂
⠀⠀⠀⠙⢤⡦⢌⢂⣉⣀⣡⡄⣙⣤⣷⣤⡤⠔⠚⠓⢤⡀⠀⡀⠱⢄⠀⠀⣾⣿⣵⣿⠁⠀⠀⠀⠀⣿⣫⣐⠀⠉⠢
⠀⠀⠀⠀⠘⠠⠐⠋⠁⠈⣠⣿⣿⡿⠛⠁⠀⠀⡀⠠⠄⠠⠭⠀⡒⠢⣴⣤⣿⣽⢞⠜⠀⠀⠀⠀⠀⢸⣷⣏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣩⠏⣰⣦⡄⢀⣆⠀⡀⢠⣄⠀⠀⠈⠀⠀⣼⣿⡞⠁⠀⢀⠃⠀⠀⢀⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣏⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣿⣇⠹⡄⢀⡎⠀⠀⠀⣼⢿⣿⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢛⡇⣀⠀⣢⠀⠀⠞⠀⠀⠀⠀⠀⠐⠀⠀⠈⢿⡟⢳⡈⡟⠀⠀⠀⡘⠙⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡘⠛⠁⠀⠀⡈⠀⠀⠀⠀⢀⡄⠠⠀⠀⢠⡎⢧⠞⡙⠀⠀⠀⡔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⣿⠀⢀⠀⠈⡅⢀⢀⡀⠀⢠⠀⠀⠑⠠⣾⣋⣠⢶⡇⠀⠀⠀⣷⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⣡⣄⡁⠀⠔⠋⠀⡌⠀⠀⣈⠀⠠⢤⡾⠁⣸⠁⠞⡅⠀⠀⠀⡧⠄⡙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⡙⠋⠉⠃⠀⠀⠰⢄⠀⠀⠀⠁⣠⣾⠁⠀⠉⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠈⠓⠦⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠠⣶⣶⡄⠀⠀⢴⣶⢶⡴⠁⠈⠀⠀⠀⠀⠀⢡⠀⠀⠀⠋⠀⠀⢀⠀⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⣀⣈⠉⢀⣠⠶⣄⠀⡎⣷⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⣴⡏⠉⠉⠙⠉⠀⠀⠈⠉⣠⠿⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆⢈⡰⠖⠚⠉⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡞⢿⣧⠀⠀⠀⠀⠀⣀⠞⣉⡀⠀⠙⢢⣠⠔⠒⠤⠤⢤⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠘⠃⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠉⠀⠀⣀⣠⠞⠀⢀⡥⣤⠤⠤⠐⠒⠒⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠰⠁⠒⠂⠀⠀⣀⡜⢁⡀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠯⠛⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣉⣿⣉⣗⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣤⠸⠉⠁⠀⠙⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠻⠁⢸⠂⠀⠀⠘⣸⠜⡄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢀⠀⠀⠴⢲⡋⠁⠀⠈⠉⢹⠙⠀⠤⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⢀⣿⢡⠐⠀⠄⠀⣾⣁⠀⣀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡅⠈⠀⠀⠀⢉⠙⣿⠓⢪⠒⠁⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠳⡀⠀⠀⠀⠀⢙⠯⠀⠀⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢔⡌⠇⠀⠀⠀⠈⠤⢂⡴⠁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠈⠀⠇⠀⠀⠀⠀⠀⣤⠀⡌⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠀⠊⠳⠀⠀⠀⠀⠀⠀⠘⠒⠒⣀⡂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠀⠂⠈⠣⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣂⡠⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣖⣀⠀⠠⠄⡨⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠄⠠⠠⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂
⠀⠀⠀⠀⠀⠀⠘⠁⠈⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⣿⣿⣿⣟⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⡿⠿⠻⠷⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠿⠀⣀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡄⠀⠀⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢤⣀⡹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠈⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⡆⠀⠀⠀⡄⠀⣦⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⣿⣧⣼⡆⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣹⡆⠀⠀⠸⠛⠁⠐⢼⣆⢀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⡌⡐⠀⡀⠐⠻⣉⣠⠀⠀⠐⢠⣇⣻⣴⣿⣿⣷⣦⡄⡀
⠀⠀⠀⠀⠀⠄⠠⢧⡀⡠⣰⣦⣈⣻⢷⣄⠀⠀⠙⢳⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⢠⡾⠓⣣⣃⠰⣧⠿⣾⣏⣻⢿⣿⣿⣿⣦⣄⡀⠈⠁⠀⠀⠈⠘⠻⣿⣿⣦
⠀⠀⢙⡷⠚⠛⠸⣿⣿⣏⣿⣿⣿⢿⢿⣻⠿⠯⢭⡅⠀⠲⠀⠀⠀⠀⠀⠀⠻⣿⣗
⠀⠀⠰⣆⣀⣀⣤⣷⣯⣭⣼⣷⣭⣽⣿⡿⢳⣤⣾⡗⠂⡀⠀⢀⠀⡆⠀⠀⠀⢹⣿⣆⢠
⠀⠀⠀⠀⠉⣭⣭⣿⣿⣿⣿⠿⡉⠨⣍⠀⠉⠸⠅⠀⠀⠀⠀⠈⠀⠀⢀⠁⠀⠈⠿⣿⡄⠂⠀⠀⠀⠲⣿⣷⡶⠂
⠀⠀⠀⠀⠀⠈⣽⣿⣮⣿⣇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣻⣿⣄⡀⠀⠀⠻⣿⣬⠀⠀⠀⠈⠉⠉⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠟⠷⠏⠋⠐⠀⠀⠀⠈⠛⢿⣶⣤⣄⣠⣴⣶⣶⣦⡀⠀⠀⣢⡶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣿⣒⣀⣿⡿⠿⠓⢢⣴⣶⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⠿⠟⠉⣯⣄⣀⠀⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡁⢠⣴⣿⣿⣿⣿⠃⣤⣼⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠛⠛⠛⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢛⠁⢊⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⢾⣿⣋⣀⣀⠴⠾⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⢁⣀⠀⠻⣯⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⡾⢇⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢠⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣍⠛⠂⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠐⠶⢦⣄⡠⣤⠀⠀⠀⠀⠀⠈⠀⠈⠻⣧⠰⠚⣓⠲⢥⣀⡢⢲⣤⣠⡀⢀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⡎⢡⡀⢀⡀⢀⠈⠃⣠⡰⢟⣿⡶⢏⠢⡥⠶⡍⣿⢮⣿⣿⣤⠒⢠
⠀⠀⠀⠀⠙⢦⣔⠲⣤⣿⣈⡉⠉⠉⠀⠑⣠⣷⣳⣾⣿⣶⣦⣶⣷⡎⣿⣿⣿⣿⣿⣿⣶⢿⡤⡄
⠀⠀⠀⠀⠀⠈⠛⣷⣌⣻⣷⠵⡢⡂⠢⠁⠘⢙⣹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌
⠀⠀⠀⠀⠀⠀⠱⣽⣿⣿⣿⡟⣓⠹⠌⣷⣈⠼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⢦⠙⣿⣿⣿⣷⣯⣨⡾⣿⣟⡻⠷⣦⣜⣿⡿⠿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠂⠱⣾⣿⣿⣿⣿⣷⣤⠀⠈⠙⢦⣌⠉⠻⣧⣄⢩⠿⠝⠯⠙⠛⠛⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣤⣭⠻⣿⣷⡄⠠⣤⡛⠄⢹⢶⠂⠈⣿⢦⣤⠀⠛⠛⢿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠙⠿⠿⣿⣿⣿⣿⣷⣾⣿⣂⠘⡁⢒⣀⡀⠈⢙⠤⢤⣄⣤⣀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⡀⡀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠛⡱⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣁⣤⡾⠃⢠⡄⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣟⣫⣭⣿⣍⠶⠀⠀⠀⢸⣿⣿⣷⣄⠀⠀⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⠋⠹⣶⡤⠈⠁⠀⠈⠙⠻⣿⣿⣶⣿⣿⠃⠐⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠃⠁⠘⣤⠁⠕⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣧⣤⣶⣿⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣫⡾⠁⠀⠀⢀⡄⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⠻⣾⣿⣯⣽⣯⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠂⠀⠉⢀⠐⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⠋⠁⠈⠙⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⠄⠀⣠⣴⣶⣿⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣿⣿⢿⣿⣧⠀⠀⣀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡔⣼⣿⣯⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⠻⣿⣦⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠁⠘⢿⣭⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣯⢣⡀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠻⠿⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⣤⣭⣍⠙⢷⣬⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣵⣶⢹⡇⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠷⣭⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⣰⣿⠋⢸⣧⣴⣾⠟⠛⢛⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡦⣤⣤⣤⣤⣶⠿⠋⠀⣠⣾⡿⠁⢀⣼⣿⠟⢁⣠⣾⣿⡿⠛⠉⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣝⢿⡶⣤⣤⣀⣤⣤⣴⡾⠟⢉⣠⣾⡿⠋⣡⣴⣿⣿⠟⠋⠀⣠⡤⠤⣄⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡳⠯⣭⣭⣭⣥⠴⠾⠛⠋⣁⣴⣾⡿⠟⠉⠀⠀⠀⢸⡏⢰⣤⡿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣤⣤⣴⣶⠾⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠿⣦⣬⣴⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠥⠤⣌⠹⡦⣤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡜⢷⡖⡶⢸⡆
⠀⠀⠀⠀⠀⠀⡴⢋⡔⢁⣠⣂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠈⡧⠠⠧⣧⣼⣧⣄
⠀⠀⠀⠀⠀⣸⠃⡞⣠⠋⡰⠋⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠖⠚⠒⠲⠦⠤⢀⠀⢧⡀⠈⠢⣀⣀⡼⠁⢠⡖⢋⣁⡤⣰⠇
⠀⠀⠀⠀⢰⡇⢀⡧⠛⣲⡇⠀⠀⠀⠈⡆⠀⠀⣠⣾⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠲⡄⠈⠁⠀⠐⠿⠞⠉⢀⣴⠿⠄
⠀⠀⠀⠀⠈⡿⢋⠴⡿⠘⣆⠀⢸⠀⠀⠇⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣄⠀⠀⠀⠀⢀⣴⣋⡴⠖⣰⠃
⠀⠀⠀⠀⢸⠁⠺⢀⡄⠀⠹⡴⠟⠀⡞⡰⠋⠀⠀⠀⠀⠀⠀⠆⢰⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⣧⠀⠀⠀⠀⠉⠉⠀⢰⡇
⠀⠀⠀⠀⣸⠁⡤⠋⡠⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⡠⢀⣤⠄⠁⠓⠔⣴⣄⡀⠀⠀⠀⠀⠀⠘⣏⠳⣄⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⢸⡄⠃⠀⡇⡞⠀⠀⠀⠀⡏⠀⠀⠀⠁⠀⠀⣰⢏⣤⣶⣶⣿⣶⣦⣝⢿⣆⠀⠀⠀⠀⠀⢿⡀⠘⣇⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⣇⠀⠀⠗⠃⠀⠀⠀⢀⡇⠀⠀⠀⠀⠐⠺⢷⡟⠻⣿⡻⠷⢠⡿⠙⣷⣽⠯⠀⠀⠀⠀⢸⡇⠀⢹⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⢠⡞⢿⠶⠂⠀⠀⠀⠀⠘⢿⣳⠭⠟⠛⠋⣤⣾⣿⠇⠀⠀⠀⠀⠀⣸⠃⠀⠈⣇⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠈⣿⠗⣀⣠⡴⣣⠟⠁⠘⣇⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠸⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡏⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡀⢀⠤⠤⡤⣐⣠⣼⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣛⣛⡛⠛⢛⣛⣛⣩⡵⡀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⡢⢯⣭⣽⣭⣯⡽⠽⢖⣻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣌⠙⠛⠒⠒⠒⢶⣶⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠮⣽⣶⣀⣀⣒⣾⣛⣿⡿⠿⠛⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣶⣶⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⢍⡱⠋⠉⠀⠀⠉⠒⢰⠆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⠻⠿⠓⢟⣀⠬⠀⠈⡇⠀⠀⠀⠀⠀⠀⠈⣳⡜
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⢿⣠⣤⡀⢼⣇⠀⢀⣞⠀⢲⣾⣿⣶⠛⠃⠀⣤⡙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⣧⣶⡿⠊⣡⣤⣤⡐⢀⣀⡤⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣏⢹⠉⠁⣞⣵⣷⣤⡎⣿⣿⣿⣿⣿⠯⠃⠉
⠀⠀⠀⠠⠤⠤⡄⢸⠀⠀⣆⡖⠀⠁⠃⣸⠘⣿⢻⡋⣰⡏⠻⠛⡌⣽⣿⣿⣿⡅⠎
⠀⠀⠀⠀⠈⠂⠀⢸⣇⠀⣡⠁⠀⠀⠂⠘⠆⢻⣆⣿⣿⣧⢀⣰⢧⣻⣿⠯⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡖⠛⠗⠀⠀⠀⠀⠐⠂⠹⣦⣽⣿⣮⡿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢰⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠓⠀⡤⠊⠁⠀⠀⠀⠀⣄⣾⣿⣿⣿⠒⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢿⡧⠄⠀⠤⠀⠀⠀⠀⠀⠘⢛⣿⠛⣿⣽⣿⡿⠘⢹⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣤⣠⣼⠷⠶⠓⠒⠀⠀⠀⠀⠀⢀⡟⠙⢯⣜⣥⣿⢥⣞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠂⠈⠉⠙⠙⠆
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠐⢀⠸⡇⠈⠑⢄⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠈⠳⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂
⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⡈⠃⠀⠀⠤⠀⠀⠈⠀⢀⠀⠱⡀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡄
⠀⠀⠀⢀⠐⠂⠐⠂⡃⢸⠀⠀⠈⠀⠀⢀⢀⡀⡀⠀⠀⠙⠢⣽⣀⡌⠀⣀⠶⡤⢤⣀⠀⠀⠀⣶⢰⠟
⠀⠀⠀⠀⡠⠐⠀⠀⠁⢨⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣣⠏⠀⡄⠀⠀⠀⠀⠤⣴⣯⣀⢀⡀⠀⢀⠶⡃
⠀⠀⠀⠀⠀⣠⠖⣃⠀⠀⢀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣯⣭⣭⠛⠀⠁
⠀⠀⠀⠀⠀⠨⠁⠙⡂⠀⠈⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠛⢿⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠂⠀⠈⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠢⣀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠆

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣷
⠀⠀⠐⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡟
⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣇⡇
⠀⠀⠀⢻⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣿⡇
⠀⠀⠉⠈⣇⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢦⢀⡞⠿
⠀⠀⠀⠀⢻⣇⠰⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⡏
⠀⠀⠀⠐⢦⡹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠩
⠀⠀⠀⠀⠀⠙⢟⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠱⡍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢼⣧⡄⠐⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠀⣀⠀⠀⠀⣸⣷⣿⠀⡄⠀⡼⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠂⠸⣴⡏⠙⢀⠀⠉⠁⠙⣠⠀⢰⠁⡤
⠀⠀⠀⠀⠀⠀⠀⠘⠀⣴⡀⠀⠲⠒⢈⠻⠋⠛⠐⠛⢧⢸⣆⠚⠀⠀⠃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢳⣀⠀⠀⠈⠀⣶⡱⠀⣀⠸⢘⣭⠳⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡚⣡⠞⠠⣌⠋⠈⣽⠏⢈⠀⢠⠟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⣰⠃⢀⣀⠀⠅⠀⠘⠀⡞⠀⠙⠀⠀⢠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡏⠀⠀⠰⡆⢃⠃⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡼⠃⡟⠀⠀⠀⠀⢹⣦⡐⠉⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⢸⡇⢸⠃⠂⠀⠀⠀⢨⡾⠉⢢⣀⡀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠐⠀⠀⢀⣴⡟⠁⣘⠀⠀⣄⠀⠀⢸⣯⡶⠀⠀⢷⡄⠀⠀⢀⡀⠀⢀⡹⣦⣄
⠀⠀⠀⠀⠀⣔⡠⢔⣉⠀⠀⢀⣽⠀⠀⠘⣆⠀⠸⣏⠀⠀⠀⠀⠈⢢⡿⠮⠁⠀⠀⠀⠆⠈⢳⡀
⠀⠀⠀⠠⠀⠈⡛⣴⠋⠀⠠⠃⠈⠀⠀⠀⢹⡀⠀⢻⣀⡠⠂⠀⠀⠀⢱⣄⠠⣈⡀⠒⡲⠖⢻⡷⣴⣦⣀
⠀⢀⠀⠋⠀⣀⠜⣱⣧⠐⠁⠀⠀⠀⠀⣰⣾⡀⠀⠀⠙⢦⠀⠀⠀⠀⠸⠮⢛⡿⠯⣉⠉⣭⠀⠀⠀⠀⢙⡄
⠀⠈⠙⠛⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢺⠟⢟⠀⠤⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢀⡀⠒⠒⠄
⠀⠀⠀⠀⠀⠀⢀⠈⠂⠈⠆⠀⠀⠀⢀⠅⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡟⡉⠉⠂⢰⠤⠐⠓
⠀⠀⠀⠀⠀⠀⠈⠁⠸⢄⠀⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠈⠃⠀⠠⠈⢄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠁⢀⠀⠀⠀⠠⡀
⠀⠉⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡄⠀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡠⠀⠤⠶⠈⠊⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⡆⢀⡀⠑⠠⠐⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢦⠧⢄⠉⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣦⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣾⣿⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣾⠟⠀⠀⠀⠀⠹⣷⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠋⢠⣿⣿⣿⣿⣧⡈⠛⠛⠛⠿⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⢿⣏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⡿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣅⠐⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣠⣴⠾⠷⣦⣄⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠿⠛⣉⣀⡙⠻⢿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠾⠛⠉⠀⠀⠀⠀⠉⠛⠷⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⡈⣉⣤⣶⣿⣿⣿⣿⣷⣦⣌⡉⣀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣸⢇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⣴⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠢⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⢡⡾⢃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⠀⠈⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠛⣡⡶⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡂⠀⠈⠛⠻⠿⢶⣶⣶⣶⣶⣶⡿⠿⠟⣋⣥⠴⢛⣩⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣬⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠦⡀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠉⢠⣄⣀⠀⠀⠀⠀⢱⣼⣷⣾⠄⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⡀⢤⣚⣿⢿⡟⠀⠀⠀⠈⠀⠀⠀⠑⠄⠀⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠑⠤⣄⡀⠀⠀⠀⠀⠀⢈⡄⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠔⠊⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣠⣿⣿⣻⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣤⡠⡀⠀⠀⠂⠄⠐⠰⢄⣀⢀⡠⠀⠠⠉⠀⠐⠀⠀⢹⢿⣿⡿⠄
⠀⠀⠀⠀⠀⠸⠅⠀⠀⢀⠄⠉⠀⠀⢰⠹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠏⠃⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁⠀⠀⠀⠸⡺⣄⠀⡀⠀⠀⢃⠀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⠀⠤⠄⣀⠈⠻⢿⣾⣷⡤⠂⠁⠀⠀⠸⠁⠀⣤⣀⠀⢀⡀
⠀⠀⠀⠈⠀⠀⠀⡠⢮⡿⠁⡀⠀⠠⠬⠿⢻⣿⣿⢷⡦⠤⠖⠚⠁⠙⢷⡏⠿⠛⠀⠀⡁⠀⠐
⠀⠀⠀⠾⠁⠉⠀⠀⠈⠁⠀⠀⡄⢀⣨⠶⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠜⠃⢨⠀⣠⠀⠀⠂⠀⠀⠀⠀⠀⠀⣰⠌⠀⢀⡏⢓⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠠⣿⡃⠀⠀⠑⠒⠀⠒⠂⠁⠀⠀⠀⠩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠢
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢛⣩⢿⣿⣿⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⡿⠊⠙⠛⠋⠁⠻⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⡃⠚⢣⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⣿⡔⠌⠁⠀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢾⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⠋⠀⠀⠀⣤⣾⣷⡈⠃⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢯⡀⠀⠀⠀⣤⣾⣿⣿⣿⣷⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣉⡱⠃⠘⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢸⠟⢫⡅⠀⢰⣿⣿⣿⣿⣿⣿⣿⡄⠀⢀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡄⣰⠃⠀⠀⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢻⣃⣠⠄⠀⠘⢧⠀⠸⣿⣿⣿⣿⣿⣿⣿⣧⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠚⣿⡟⢺⣷⠀⢾⣿⡄⠈⠙⠻⠿⠟⠋⠉⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠛⢋⡄⢸⠿⡀⣤⣴⠂⠀⡀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣸⣟⣇⡸⣆⣹⠿⠿⠓⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⠁⠂⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⣦⣤⣿⠀⠀⡈⠀⣼⠧⣤⣄⠤⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢨⡧⢠⡄⢰⡧⠀⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢠⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⡄⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠁⢠⠀⡀⢠⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠈⠠⠃⠈⠑⠘⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡶⠀⠀⣀⣤⠆⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⡻⠻⠆⠀⠀⠀⠀⠀⠀⠴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⢀⠀⠀⢀⣠⣆⣠⠀⠀⠃⠀⢀⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⡿⠁⠀⢠⡋⠉⠹⢃⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠶⠆⠂⠀⠺⢿⢇⡀⡀⠀⡸⠿⠛⢀⡿⢶⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⢻⡃⢠⠇⠀⢠⠂⠀⠠⠀⠀⠀⣴⣿⠟⠲⣄⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⠏⠀⠀⡇⠀⢠⣿⠟⠻⠀⠀⢹⠆⠀⠀⠶⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⢠⣄⠀⠘⢀⡄⠀⠃⠀⠉⠉⠀⡴⠞⠀⠀⠀⠀⢠⡆⠈⢢⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠈⠀⠀⠀⠞⢷⡛⠱⠄⠀⠀⡴⠀⠀⠀⠀⠀⠀⠐⠿⠃⣀⣸⡟⠦
⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡄⢲⡆⠀⢠⣷⢀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⢸⡋⠁
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⡎⠑⠇⠀⢘⠉⠘⡇⢢⣿⡶⠊⠀⠀⠀⠀⠐⠂⠀⠀⠁
⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠃⠀⠠⡘⢾⡄⠀⣏⡼⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⣷⡄⡟⢀⣶⣇⠀⠀⠀⠲⢀⣄⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⡀⢠⡄⢱⠙⡄⢸⣿⢋⡀⣘⣀⠀⠘⠛⠛⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣾⡇⢸⡄⠁⠀⡇⢈⠷⠛⡟⢃⣠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡇⢸⡇⠀⢀⣤⣾⣶⡷⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡀⠀⠈⡇⣸⣷⣷⡿⠟⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⢳⡏⣿⠥⣶⣶⣽⣿⡴⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠠⣄⡀⠀⠀⠀⣸⠇⣿⣠⣨⣿⣿⣿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠈⠙⢦⡀⠀⣿⡔⡿⠃⠛⠛⢿⣼⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢿⡆⢿⡿⡇⠀⢀⣀⡀⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣿⠾⠃⠀⠁⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣶⡧
⠀⠀⠀⠈⠛⢶⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠤⠴⠿⣶⡀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠤⢠⣭⣥⡄⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣈⠉⠙⢿⣆⠀⣿⡿⠁⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣭⡗⣩⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠟⠛⠉⠁⢀⣾⣧⡙⣯⣙⠛⠻⠿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠈⠻⣟⣛⡁⠀⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣏⡀⠀⠉⣿⣭⣤⣀⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣯⣤⠄⠀⠈⢿⣦⣶⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡿⠿⣽⣖⠄⠀⠀⠀⠹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠟⠁⠀⠀⠈⠙⠶⣶⡄⠀⠀⠘⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣸⡯⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢀⣿⡗⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣾⣿⡇⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢰⣿⣿⡧⠀⠀⠀⠀⢿⡄
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡟⠀⣽⣿⣿⡧⠀⠀⠀⠀⠘⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠙⠿⣿⣏⡛⠙⠋⠋⠙⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⢴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠙⠛⠛⠛⠛⠛⣉⣿⡿⠟
⠀⠀⠀⠈⠻⢿⣦⣀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⢂⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡐⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣤⡶⠋⣡⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⣄⠈⠛⢷⣄⠰⢤⡀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠋⡵⠟⢁⣴⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⡀⠙⠻⢦⡙⢷⣤⡀⠀⢀⠈⠋⠀⠀⣀⣰⣖⣡⣾⡿⢫⠔⠋⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠈⠙⠳⢬⣉⠓⠀⣠⣿⣦⠀⠀⠀⠉⠋⠀⠀⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠤⣍⡐⠂⢀⣼⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢉⡓⠢⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⣀⠲⠤⢠⣾⣿⣿⣿⠿⠛⠃⣿⣿⣿⣷⣄⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠬⢁⣴⣿⣿⡿⠛⣁⣴⣾⣶⣌⡉⠿⣿⣿⣦⡀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠠⠌⣡⣾⣿⠟⢉⣴⣾⠟⠋⠀⠈⠛⢿⣶⣌⠙⢿⣿⣆⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣀⣾⠟⢋⣤⣾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣌⠛⢷⡀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠰⢋⣥⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣙⠂⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣶⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⠀⠙⠻⣿⣇
⠀⠀⠀⠀⠀⠐⠐⠢⠞⠔⠿⠫⠖⠖⡷⠾⠾⠖⡷⠺⠶⠕⠷⠶⢚⠶⡷⠲⠞⡵⠳⢞⠖⠣⢺⠖⠷⢺⠖⠷⠢⠾⠟⠶⠲⠂⠀⠐


⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢇
⠀⢠⣤⣤⣤⣤⣤⣼⣿⡇⢹⣤⣤⣤⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⢿⡇⠀⢧⣤⣤⣤⣤⣤⡀⠀⢠⣄⣀⣀⣀⣀⡤⣿⣇⠘⢄⣀⣀⣀⣀⣤
⠀⠀⠈⠳⢶⣤⣄⣀⢻⠇⢀⣀⣤⣴⠾⠋⠀⠀⠀⠀⠙⢦⣄⣀⠀⠈⠃⠀⠀⢀⣀⡴⠞⠁⠀⠀⠀⠈⠳⢦⣤⣀⣤⣽⣿⣤⣄⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠙⡟⠛⣡⣦⡙⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠉⠁⡀⡀⠉⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡯⢫⣿⣿⣯⠉⠛⠉
⠀⠀⠀⠀⠀⡸⢀⣼⣿⢿⣷⣄⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⣬⡷⣧⣄⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⣼⣿⠿⣿⣆⠘⡀
⠀⠀⠀⠀⢠⣷⠟⠋⠀⠀⠈⠛⢶⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⡴⠛⠁⠀⠀⠙⠳⣼⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⠁⠀⠈⠙⢷⣥
⣀⣀⣀⣀⣚⣁⣒⣒⣒⣒⣀⣀⣀⣙⣂⣀⣀⣀⣀⣀⣀⣒⣋⣒⣒⣒⣒⣒⣒⣒⣚⣓⣀⣀⣀⣀⣀⣀⣀⣚⣓⣒⣒⣒⣒⣒⣒⣒⣛⣂⣀⣀

⠀⠀⠀⣀⣀⡀⠛⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣖⣾⣶⣶
⣴⣾⣿⣟⣟⠛⢿⠛⠟⠶⡦⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⡀⣴⣶⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⢶⡍⢑⣠⣤⣤⣴⣾⣿⠟
⠈⠻⣿⣿⣏⣙⣙⣉⠹⠊⠛⠒⣦⠀⠭⣐⡂⠤⡄⢀⠀⣿⡿⡿⠥⣬⢯⠙⠹⡿⣤⣄⢀⡀⣀⣤⣤⣶⣺⠍⠡⠴⣒⣮⠁⢀⡀⢘⢠⣤⣴⡾⠛⠁
⠀⠀⠀⠙⣿⣿⡼⡥⣠⢟⠀⠈⢹⢏⠑⣒⣒⡽⢯⣿⣿⣤⣿⣟⣻⠏⠀⢀⣽⠘⢓⠙⣂⠈⡯⠿⣟⣟⣛⡋⢯⣖⣚⣳⣖⣚⣳⣶⢾⣿⠁
⠀⠀⠀⠀⣿⡿⠿⠒⠓⠚⠒⠒⠒⠚⠐⠒⠒⠒⠓⠓⠫⠟⠋⠩⠝⡀⠀⡙⠟⠙⠊⠙⣀⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⡀
⠀⠀⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠈⠀⡀⢿⠗⠀⠐⠊⠄⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢹⣀⠔⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠙⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠘⠃⠀⠤⠂⠀⢀⠀⢀⡀⠚⡀⠀⠁⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠁⠀⠠⡀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⠀⠂⠀⠀⠀⠒⠀⠸⡀⠄⠁⠀⠀⠀⠈⠀⠀⡀⢀⠀⠀⠀⠠⡆⠀⠀⡀⠊
⠀⠀⠀⠠⠄⠂⠀⠀⣂⠂⠀⠁⠁⢀⠤⠄⠀⠂⣀⠄⠀⠢⢄⠀⡴⡄⠀⠀⠀⠉⡉⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⠀⠨⠃⠀⠀⠂⠀⠒⠁⠀⠀⣂⠀⠀⠀⠀⠔⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⡆
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠈⠛⢿⣯⣭⣉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢛⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠀⠀⠀⠀⠈⢻⣿⣿⠀⢀⣾⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⠙⣿⣠⣿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢋⣿⢳⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⡿⠟⠛⠉⠁⠀⠀⢠⣾⣿⠀⠹⣿⣿⣿⣿⣷⣶⣤⣤⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⣦⣀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⣠⣾⣿⣿⣿⣿⣿⠟⠉⠀⠀⠙⠷⣤⡀⠀⠀⠀⠙⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⣴⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠈⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣀⠀⠙⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⣴⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠊⠀⢴⣾⣿⣿⠀⠘⠈⠙⠓⠀⠀⠒⠀⠀⠀⠐⠁⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠋⣼⣿⡿⠀⡀
⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠘⠀⠀⠀⠄⠠⠀⠀⠀⢀
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠄⠀⠀⠀⠀⠀⠀⠀⠋⡟⠁⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣾
⢀⠔⠈⠁⠈⠁⣀⠀⠀⠀⠀⢠⣖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠲⢀⣾⣏
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠤⠀⡴⠊⠉⠉⠐⢄⠀⠀⠀⠁⢄⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣀⡀⢀⣤⣿⣿⣟⡡
⠁⠉⠁⠉⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠦⣄⡠⠲⠒⠖⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣋⣀
⡀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠈⠑⣤⢄⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡻⣿⣿⣷
⠙⠂⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠴⠊⠁⠀⠀⢀⣿⡟⠃⠀⠃⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠠⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠙⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣷⡀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⡀
⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣄⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣹⠁
⠀⠀⠀⠀⠈⢆⠀⠉⠁⠀⢄⠵⠶⣦⠀⠀⠀⠀⠀⢀⠀⠀⢽⣿⣿⡇⠀⣾⠇⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⢀⣀⢀⡀⠉⠉⢻⣿⣿⣧⢠⣿⣷⣾⣿⡶⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠙⠻⣿⣿⡿⠿⠿⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠳⡀⠀⣰⠁⠀⠠⡄
⠀⡄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⡀⠐⠃⠀⠁⠈
⠀⠐⢄⢿⡄⡀⠀⠀⠀⠀⠕⠀⠄⠀⠀⠀⠀⠀⠀⠒⢂⣩⡴⠋⡉⠀⠙⢷⡀⠈⠀⠀⠐⠒⠒⠄⠠⠄⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠆
⠀⠀⠀⠀⠙⠁⠀⡎⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠋⠅⡔⠁⠀⠋⠀⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⣴⣶⣤⣬⡄⠀⠀⠀⠀⠀⠀⠀⡀
⡠⠖⠀⠀⠀⠀⠀⠀⠁⠰⣀⠀⢀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⠀⠀⠀⠈⠙⠙⠓⠚⠠⢄⢀
⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠉⠉⠒⠆⠒⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⣦⢠⠀⢀⣴⣿
⠀⠀⠀⠀⠇⠑⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢣⡀⣄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⢺⣧⣠⣿⣿⣿⣿⠏
⠒⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢿⣿⣄⡀⠀⠸⠁⠀⠀⢀⣤⣞⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣟⠋⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠁⠀⢀⣀⠀⠀⠀⠈⢻⣿⣷⣀⢠⠀⠀⣤⣿⣿⠋⢀⠀⠠⠠⠀⠠⠄⠀⡸⠉⠐⠙⠧⣐⠀⠤⠤
⠲⢀⠈⠁⠀⠁⠀⠀⠠⠀⠀⠀⢀⠂⠁⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⠈⠻⣿⣿⠛⠃⢸⣿⣿⡡⠊⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠒⠒
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠄⠀⢈⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣾⣷⣶⣦⣉⡀⠃⢌⣿⣟⡡⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠎⠀⠀⠀⠐⠚⠉⠉⠉⠻⠛⠛⠛⢻⡿⢿⠁⣠⡆⢫⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠈⠡⠀⢷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣤⣦⣤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠢⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢏⡜⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠂⠈⠙⡆⠈⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢠⡼⣄⣀⡂⠀⠀⠀⣇⠀⠀⠠⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣡⠤⠶⠚⣟⠁⠀⠀⢠⠝⠀⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠾⠵⠒⠶⣅⣀⢸⡆⡀⠀⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠁⠈⢣⡀⠀⢰⠀⠈⠉⠑⢢⡤⠂⠀⠀⠀⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢦⡤⠀⠀⢀⠀⡀⠈⠳⡀⠀⠀⢸⡆⠀⠀⢁⢀⠀⠀⠇⠀⠀⠀⡔⡠⠀⠀⠉⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣠⠀⠀⠀⠀⣀⣀⡀⠐⠄⠀⠀⠙⠀⠐⢌⡴⢂⠀⠀⠀⠀⣐⣎⠋⠀⠉⠂⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠈⠢⠀⠀⠸⡄⠀⠐⠀⠀⠈⢢⠹⢦⣾⠠⠂⠀⠀⠠⠀⠐⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠁⢀⡊⠀⠀⡄⠤⠆⠀⣤⡀⠁⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢡⢀⠀⢠⡀⠀⠀⠀⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁⠠⠶⠂⠒⡻⢆⠀⠤⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⣷⡋⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⡇⠐⠚⠞⠁⠀⣈⣀⠤⠤⠬⢶⡀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⠆⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⢀⣀⡤⠂⠉⠁⠀⡀⠠⠀⠐⠳⠆⣀⠀⠀⡇⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠈⢀⠄⡜⠱⠶⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄⡄⠤⣠⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠖⠉⢶⠞⠁⠀⠀⠀⠀⠀⠀⠡⡀⠀⠙⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢨⠦⣴⡟⠠⠄
⠀⠀⠀⠀⢀⠀⠂⢚⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⠆⠀⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣁⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⡈⣤⠀⣤⣄⡂⠀⠀⠀⠀⠀⢀⢤⠔⠳⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⠒⠒⠀⠀⠉⠉⠉⠉⠀⠐⠒⠒⠠⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢤⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⢶⣤⡙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣤⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡌⠢⡀
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠢⡀
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠰⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⡿
⠀⠠⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠗
⠀⠀⢋⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠌⠰⣏
⠀⠀⠀⠐⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡱⠁
⠀⠀⠀⠀⠙⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠑⠽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡘⠛⠓⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⡰⠶⠚⠛⡀
⠀⠀⠀⠀⠀⢀⠁⠀⠀⡾⠛⠻⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⣷⠀⠀⠀⠃
⠀⠀⠀⠀⠀⢸⣷⣤⣸⠇⢀⡀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⢸⡀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠔⠈⠻⡄⠀⠻⣟⡳⢄⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢾⡿⠋⠀⣰⠟⠐⠒⠓⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠀⣉⣲⣬⣓⣒⠶⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠐⢪⠛⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠧⠀⠀⠈⠙⠻⠿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⡿⠟⠋⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠱⠄⠀⠀⠀⢠⡾⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢥⠀⠀⠀⠀⠀⣰⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣷⣤⣀⠀⠀⢀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠳⡄⠀⠀⠈⣁⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣷⡀⠠⣿⠶⠶⠶⠦⢤⣤⣤⠤⠤⠶⠶⢶⣧⠀⢀⣸⣶⣾⣿⣿⡿⠇⠉⠘⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⣦⣷⣮⣽⣛⠿⢿⣿⣷⣿⣶⣤⣤⣤⣤⣄⣀⣠⣠⣤⣤⣴⣿⣶⣿⣿⠿⠟⣛⣭⣶⣇⡀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠰⣤⣀⠀⡤⣿⣿⡿⠛⠋⠉⠈⠓⠂⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣋⣩⣭⠵⠒⠚⠋⠉⠁⠙⠿⠧⣄⡀⠀⣠⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣴⣃⠈⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣦⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⡤⠴⠔⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣙⣛⣛⣛⣛⣛⣟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣤⣤⣶⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⢳⠀⢦⣄⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢹⣶⣿⣿⠟⠻⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣾⣿⣿⡇⠀⠀⢸⣿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠈⠁⠀⢀⣾⣿⣿⠟⠛⠛⢷
⠀⠀⠀⠀⠰⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀⣠⣴⣿⣿⣿⣧⣤⣤⣄⡸
⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠉⢿⣷
⠀⠀⠀⠀⠀⢾⣦⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠁⠀⠀⢤⣄⣀⣠⣿⡿
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡆⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠛⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢰⣶⢀⡿⠉⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⡄⠙⠋⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣠⣿⣿⣿⣿⠇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠻⣿⣿⣿⣷⡘⢷⣄⡀⠀⢀⣿⣿⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠹⣿⣿⣿⣷⡄⠙⠿⠿⠿⠿⠛⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠙⣿⣿⣿⣷⡀⠀⣤⡶⠄⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠘⢿⣿⣿⣷⠀⠘⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠹⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠘⢿⣿⣯⡙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠙⢿⣿⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠉⠛⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡄⢀⢢⡄⣤⣻⣾⣿⣹⡏
⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠸⠿⠿⠿⠿⠾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢠⣿⠛⠙⠃⠀⠀⠀⠀⠸⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣤⣤⣤⣄⣀⣀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡟
⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⠏
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⢺⣿⡟⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠸⣿⡀⠀⠀⠀⠀⠉⠉⠉⢹⣿⣿⣿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢀⢠⣰⠶⢖⠐⢯⢢⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡿⡣⠢⠄⣏⣱⢺⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠼⠀⠄⢀⣾⣸⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣁⡄⢀⡄⠀⠀⠀⠀⠀⠀⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠐⠀⠀⠋⠈⡾⠈⢳⠀⠀⠀⠀⠀⠀⠀⠤⠤⢀⠴⠖⠚⠲⡆⠀⠀⠀⠀⠀⢠⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⣷⠀⠸⡆⠀⠀⠀⠀⠀⠀⢃⡖⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⣀⣼⠃⠀⠀⠀⠀⠀⢄⡴⠀⢉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡚⠅⠀⠈⠈⠀⣶⡀⢱⢰⣝⣟⣹⣾⠉⠀⠀⠀⠀⣀⠤⢄⣄⡠⢖⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣼⣷⣢⢰⡆⡰⠀⣿⠀⠈⠀⡠⣬⣁⠍⠀⠀⠀⣠⡎⣼⡷⠰⢟⣴⣛⡢⢎⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠐⠃⠈⠻⢷⣬⣜⣿⣆⠀⣦⠄⢅⠐⠘⠂⢠⣾⣿⠑⠏⠛⡦⠘⣹⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠊⠀⢀⢀⣀⡀⠀⠘⠁⢿⣿⣿⠀⡇⠐⡀⠅⢁⡶⠟⠋⠹⣤⣯⠀⠥⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠠⠔⢆⠀⠁⠐⠀⠀⠀⠉⢣⠀⠀⡀⢀⣼⢻⠉⠉⣀⣫⣶⣶⠞⢉⢁⡠⢅⡀⠅⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠠⣠⠂⠀⢠⣾⡇⢄⠨⠿⡃⣼⣿⣷⢯⡿⠍⢫⠔⠚⡰⣗⡦⠀⣤⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡆⠀⣰⠀⠀⢸⡄⠐⣶⠾⠛⠟⠻⠯⢇⠀⠠⠰⢦⣄⢻⡽⢮⠒⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⡰⠋⠀⣠⣿⢐⡄⣿⠠⢀⡞⠻⡄⠠⠀⠀⠘⠦⠊⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⣰⣟⡿⢿⠀⠁⢀⠜⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣸⢡⡇⠃⠘⠟⠀⠊⠛
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣴⡇⢀⣯⠟⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⠀⣘⠕⠠⢀⠈⢠⠂⡘⣸⠀⢸⣅
⠀⠀⠀⠀⠀⢸⠓⠈⠉⠲⡄⠌⣾⡿⡟⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠙⣳⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡇⠀⠘⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇

⣽⣿⡟⠁⠀⠀⠀⠀⣤⣠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠟⠋⠀⠀⠀⠀⣽⣿⡟⠁⠀⠀⠀⠀⣤⣠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠚⢿⠷⠀⠀⠀⢀⣠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⠷⠀⠀⠀⢀⣠⣆⡀
⠀⠀⢰⣶⣧⡀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠿⠀⠀⠀⠀⢶⣿⡶⠀⠀⠀⠀⢰⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠿⠀⠀⠀⠀⢶⣿⡶
⠀⠀⠛⠙⠏⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠛⠛⠏⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠉⠁
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⣿⣿⠂⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢶⣶⣄⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⢶⣶⣄⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉
⣤⣀⠀⠀⠘⠛⠁⠀⠀⠀⠀⢹⠿⠏⠀⠀⠀⠀⣀⣴⠀⠀⠀⠀⢠⣤⣀⠀⠀⠘⠛⠁⠀⠀⠀⠀⢹⠿⠏⠀⠀⠀⠀⣀⣴⡀
⠻⠏⠀⠀⠀⠀⠀⢠⣄⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⡿⠋⠀⠀⠀⠼⠻⠏⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡿⡿⠋
⠀⠀⢀⠀⠀⠀⠠⠿⡿⠷⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠿⡿⠷⠀⠀⠀⠀⣠
⠀⢶⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠂⠀⠀⠀⢤⣾⣆⠀⠀⠀⢶⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠂⠀⠀⠀⢤⣾⣆
⠀⠘⠋⠉⠁⠀⣀⣴⣆⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠻⠋⠀⠀⠀⠘⠋⠉⠁⠀⣀⣴⣆⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠻⠋
⠀⢀⠀⠀⠀⠀⠸⠛⠟⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠛⠟⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⢀
⢶⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠇⠀⠀⠀⠘⣿⡷⠂⠀⠀⠐⢶⣿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠧⠀⠀⠀⠘⣿⡷⠂
⠛⠙⠃⠀⠀⠀⠀⣤⣿⣞⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠙⠃⠀⠀⠀⠀⣤⣿⣞⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⢀⣀⡄⠀⠀⠀⠙⠉⠀⠀⠀⢸⣾⣷⠆⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠙⠉⠀⠀⠀⢐⣾⣷⠂⠀⠀⠀⣀⣤⣀
⠀⠀⣼⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⠿⠏⠀⠀⠀⠀⣼⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⠿⠏
⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡗⠀⠀⠀⠀⢠⣴⣧⡄⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⣾⡗⠀⠀⠀⠀⢠⣴⣧⡀⠀⠀⠀⠀⣤
⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠀⠀⠀⠀⠀⠿⢿⠁⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⢹⣿⠗⠀⠀⠀⠀⢶⣿⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠗⠀⠀⠀⠀⢶⣿⡤⠀⠀⠀⠀⠀⢀
⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⣶⣿⡦⠀⠀⠀⢸⣿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⣶⣿⡤
⠈⠁⠀⠀⠀⠀⠀⣼⣷⣇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⣼⣷⣇⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋
⠀⣀⣴⣄⡀⠀⠈⠉⠋⠉⠀⠀⠀⢤⣾⣤⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣀⣴⣄⡀⠀⠈⠉⠋⠉⠀⠀⠀⢤⣾⣤⠀⠀⠀⠀⠀⣠⡀
⠀⢹⡿⠿⠅⠀⠀⢀⡀⠀⠀⠀⠀⠸⠛⠛⠀⠀⠀⠀⣽⣿⡷⠀⠀⠀⢹⡿⠿⠅⠀⠀⢀⡀⠀⠀⠀⠀⠺⠛⠛⠀⠀⠀⠀⢽⣿⡷
⠀⠀⠀⠀⠀⠀⢲⣿⣷⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣷⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣴⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⣿⡗⠀⠀⠀⢠⣤⣄⡀⠀⠀⢀⣀⣴⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⣿⡗⠀⠀⠀⢠⣤⣄⡀

⠀⣀⠤⠼⠏⠒⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠔⢐⣀⠂⠀⠀⠒⠂⢈⡤⢉⡀⠀⠈⠄
⠎⠁⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠁⠀⠂⠀⠀⠠⠀⢀⠋⠉⠀⠀⠀⠀⠀⠀⠙⢣⡀⠁⣁⠀⠀⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀
⠁⢄⠀⠀⠀⠀⠀⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠂
⠀⣠⠀⠢⠂⠤⠤⠤⠴⠒⠁⠁⡀⣀⣀⠀⢀⠀⠄⠀⣀⠀⢐⠀⠀⠀⣀⠀⠀⠀⠄⠀⠀⠈⠑⠠⠀⠀⠀⠈
⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠈⡄⠀⠄⠲⠀⣉⡀⠘⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⡼⠀⣀⠀⠀⠉⠐⠢
⠀⠀⠀⠀⢀⠒⠀⠀⠀⠀⠀⠀⠘⠠⠅⠀⠉⠄⢼⠀⠀⠐⠀⠀⠀⠠⣀⡀⠀⠀⠀⠀⠑⠀⡈⠀⠀⠠⠄
⠣⡄⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⢂⠀⠘⠂⢠⠁⠌⠀⢀⠀⠀⠆⠀⠒⠀⢈⠄⣤⠀⠀⡐⠀⠀⠁⠐⠄⡄⢀
⠀⠀⡟⠀⣠⠕⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠁⠈⠀⠈⠀⠀⠀⠂⠤⡄⠀⠠⠄⠀⢰
⢠⠔⢧⠀⠀⡶⠀⢤⢀⠀⠂⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠁⠠⠀⡄
⠀⠀⠸⣄⠀⠖⠀⠸⡄⠀⠀⠀⠀⠈⠲⢤⢰⣤⡄⠀⠀⠀⢀⣤⡄⣠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⡄⠈⠁
⠀⠀⠀⠜⠲⢆⣄⠤⠓⠶⣦⣸⢁⠤⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡷⠀⠠⠀⠀⠀⠀⠀⠟⠃⠁
⠁⣢⠘⠀⠀⠀⠸⡄⠀⠀⠈⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⢻⡵⠈⡂⠀⠀⠀⠀⣀⠙⠀⠂
⡎⠁⠣⣀⠀⠀⢀⠨⡦⢠⡄⠅⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠠⠾⠀⠀⡸⠲⢀⠀⠀⠀⣀⡀⠠
⠁⠀⠀⢈⣍⠁⠀⠀⠘⡇⠀⠂⠀⠘⢢⢳⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⢶⡟⢠⡇⠰⠊⠀⠀⠈⠚⠲⠺⡁
⡀⠀⠀⢨⠉⢠⠀⠀⠀⡇⠀⠀⠂⠀⠀⡼⣧⠑⠄⠀⠀⠀⠀⠀⠂⠀⠘⠁⠐⠁⠀⠆⠀⠀⠀⠀⠀⠀⠐
⠣⣄⠀⠀⢘⠁⠀⢀⡴⠁⠀⠀⠄⠀⠀⠀⣿⠺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⢀⣈⢟⡒⠜⠂⠒⠋⠀⠲⣄⠀⠀⠀⠀⡀⢛⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⠔⠃⠤⠤
⠁⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠀⠀⠀⠀⠀⠠⠲⠉⢡⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠨⠖⢀⠎⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⢰⡇
⠀⠀⠀⠀⠀⡐⠁⡐⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⡠⠴⠤⠄⣜⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢣⣄
⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠈⡆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘
⠀⠀⠁⠠⠄⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠁⠀⠀⠀⠀⠀⠀⡔
⠐⣾⣷⠀⠀⠰⠇⠀⠀⠀⠀⠰⢠⡀⠀⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠈⠁⠀⢠⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⢲⠒⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⣘⣀⣁⣀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⢀
⠀⠀⣀⣠⣿⣆⡀⠀⠀⠀⠀⢀⡰⠣⡠⠀⠀⠀⠀⣀⠀⠂⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⣀⣀
⠀⠀⠀⠈⢿⠁⠀⠀⠀⠀⠀⠀⢠⠂⠣⠀⠀⠀⠀⠈⣃⠤⣄⠁⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢈⡟⢯
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠂⢘⠀⠄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⡄⣀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠇⠀⠛⢀⡉⠀⠀⠀⠂⠀⢤⠄⠈⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠈⢳⡄⣚⠃
⠀⠀⠀⠈⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠐⠂⠉⠃
⠀⠀⠀⠀⠠⠄⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠂⠺⠠⠀⠃⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⢠⢠⡾⢤⠤⠀⠀⠠⣶⣿⣶⣀⣠⠔⠁⠀⠀⠄⠈⠃⡔
⠀⠀⣠⡀⠡⠂⣴⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⣙⠶⣻⡀⠀⠀⠀⠟⠙⠳⠀⠀⠀⠀⠀⠀⡤⠐⠨⡄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡄⠀⠂⠀⠀⠀⠁⠀⡀⣀⠘⡀⠀⠀⠀⠀⠀⢀⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣷⣤⡄
⠀⠐⣾⣿⡆⠀⠀⠀⠀⠀⠀⠉⠀⠀⠄⠁⠀⠀⠀⠉⣹⠿⣶⡀⠀⠀⠀⠰⠶⠴⠂⠀⠀⠀⠀⠀⢙⣿⣿⣇
⠀⠀⠋⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠘⠁⠀⠉
⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⣶⣼⠧⠖⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠙⠳⠄⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⢀⡀⠁⠠⠀⠀⠀⡀⡀⠸⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠤⠀⠀⠀⠠⡖⠂⠡⡶
⠀⠠⠂⠄⠀⠤⠄⠀⠀⠀⠀⠈⢓⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠇⠊⠲⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠚⠉⠍⠉⠉⠉⠙⡓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⣀⣠⣀⣄⣀⣲⣄⣀⠀⠀⠀⢹⠀⠀⠀⠲⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⢶⣚⡩⠭⠟⠚⠛⠉⠹⠟⠀⠈⠁⠀⠀⠀⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣽⣿⠿⠛⠉⠀⢀⣀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣀⠤⠛⠉⠀⠄⠀⣴⣾⣿⡿⠿⣿⣿⣧⠀⠀⠀⢀⡀⠀⣨⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⡀⠀⠐⢤⠿⣿⣿⣶⣶⠾⠿⠟⠁⠀⣀⣴⡿⠃⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⡄⠀⠀⠈⠈⠀⠀⠀⠀⠀⣠⣤⣴⣔⠂⠐⠟⠰⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣷⡀⠀⠀⣀⣤⣤⣷⣴⣷⣶⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠀⠈⠻⣯⡀⠀⠛⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⢻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣵⣷⣮⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣤
⠀⠀⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡏⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⠓
⠀⠀⠀⠀⠀⢹⢿⡿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⢑⣶⣿⣶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣤⣔⠁
⠀⠀⠀⠉⠉⠉⠉⢹⣿⣿⣿⡋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢰⡀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠿⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡟⠀⠑⠄⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣷⣄⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠃⠈⠰⢧⡤⢣⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣋⠝⢃⠂⡀⠀⣀⣀⢴⡠⠀⠠
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠤⠤⠀⠁⠀⠈⣴⢤⡀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠀⡠⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠱⡛⠣⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣉⣤⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠊⠁
⠀⠀⠀⠀⢀⠈⠉⠀⣬⠧⠀⠀⢤⡄⠀⠀⣠⡄⠀⢀⡀⠀⡀⠀⢤⡄⡄
⠀⠀⠀⠀⠀⠀⠔⠂⠉⢷⡀⠢⠾⠷⠫⠞⠿⠅⣤⢛⠉⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⣄⣠⠴⠈⡈⠀⠄⠲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⡀⠀⢀⣶
⠀⠀⠀⡡⣄⠀⢀⡤⠀⠀⠀⠀⠀⣀⣠⠈⠀⣠⣤⣼⠂⠙⠃⠀⣰⡾⠁
⠀⠠⡎⢠⣾⡶⠟⠀⠀⠀⣠⣾⣿⣿⠃⠀⢀⡼⣿⡟⠉⣷⣦⡚⠉
⠀⢀⠀⠀⠈⠀⠀⠀⠀⢀⣿⣿⢿⡇⠀⢰⣾⢁⣿⣅⣀⣠⣿⡇
⠀⠈⠀⣤⣤⣤⣤⣢⣾⣿⡿⠃⢼⠁⢠⠘⣿⢸⣿⡟⠉⠀⠉⠓⠺⠦⠀⠂
⠀⠀⠀⠀⠉⠓⠚⠻⠟⣿⠇⠀⢸⡆⢸⣿⢸⠀⣿⡄⠀⠈⠳⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠇⣸⡇⠸⠿⢠⡆⣿⣿⣄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠀⣿⣿⡎⠀⢸⡇⣿⣿⢿⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⡿⠋⠀⠀⢸⣇⣿⠋⢘⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠈⠀⠀⠀⠀⢃⣼⠏⠀⣾⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⣍⣠⡾⠟⣵⠀⠲⡘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⠀⠀⠀⢸⣧⣀⡛⠠⣶⡀⠀⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠈⣿⡆⠀⢸⡏⢈⣿⣧⣤⣏⢀⡛⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠉⢿⡿⢁⣿⠇⠀⠈⢻⣿⢷⣴⣟⠉⣿⠃⢠⡦⢶
⠀⠀⠀⠀⠀⠀⠆⣠⣾⣿⣿⣰⣧⠐⡞⢁⣾⡿⠀⠀⠀⠀⠙⣿⣯⣉⣿⠟⢠⣶⢀⣴⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⣠⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⡾⢁⣴⡄⣀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⢿⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣼⡃⣀⠚
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡈⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣋⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⡇⠀⠘⣿⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣮⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣶⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢙⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠉⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣏⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⠀⡆⠙⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡘⣦⠘⢿⣿⣳⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣿⣿⣿⣌⢳⣄⠈⠻⣿⣯⣟⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢰⣿⣿⣿⣿⣧⡙⢷⣄⠹⣿⣿⣷⣦⡉⠻⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢡⣿⣿⣿⣿⣿⣿⣷⡌⢻⣧⡘⣿⣿⣿⣿⣶⣄⡉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⣴⣿⣿⡿⣿⣿⣿⣿⣿⣿⣼⣿⣷⣽⣿⣿⣷⠈⠙⢿⣦⡈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣿⣿⡿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⣿⣷⡄⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣴⣿⡿⠛⠁⢠⣿⠇⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⢹⣿⡄⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢋⡠⠞⡛⢋⣸⠇⠀⠀⢸⣿⠀⠀⠘⣿⣿⣿⣿⣿⠁⢿⣿⣿⣿⡟⠀⠀⣸⡏⢨⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⡠⠔⠃⢀⡞⢀⡎⣿⠀⠀⠀⢸⣿⠀⠀⠀⢻⣿⣿⣿⣿⠀⢸⣿⣿⣿⠃⠀⢠⡿⠁⣼⣿⡟
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⢠⠏⠀⢸⡇⣼⠀⠀⠀⢸⣿⠀⠀⣾⠀⣿⡿⡟⣿⠀⢸⣿⢿⡟⠀⠀⣼⠃⣸⣿⡿⠁⠀⣠
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢰⠏⠀⠀⣼⠀⣼⠀⠀⢀⣾⡟⠀⠀⢻⠀⣿⠧⡧⣾⠟⠛⠁⣞⠁⢀⡾⠁⢠⡿⠋⠀⢀⡴⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢀⠏⠀⠀⢀⡿⠀⠘⡄⠀⣼⣟⢀⠀⠀⠸⣼⠃⠀⠀⠀⠀⠀⠀⠉⡶⠋⠁⢠⡏⠀⢀⡴⠋
⠀⠀⠁⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢀⡿⠀⠀⢀⣾⠃⣠⠞⢷⣴⣿⡟⠈⠙⢤⣀⡿⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠊⣀⡴⠋
⠀⢀⡄⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⣸⡇⡀⣠⣾⠃⠘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⡾⠋
⠀⠘⠃⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠛⠟⠁⠘⠿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣴⣿⠏
⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⣿⡿⠁
⠆⢀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⡟
⠠⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⠁
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠰⡒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⣽⣾⣷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡄⠛⠍⢙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠁⡀⣰⢀⠁⠁⠘⢧⠴⢦⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⠟⣄⠠⠁⢠⠀⣠⠃⠀⠙⣴⣯⣾⣷⣡⣴⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⣽⠷⠾⠓⠠⠼⣋⣁⣁⠠⠔⠉⠹⢿⠛⠭⡛⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡉⢀⠄⣀⡀⠀⣻⣗⣀⡈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠱⡐⠋⠉⠠⠆⠉⠿⢿⣃⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢿⣀⠀⠈⣁⣠⡶⣶⣶⣏⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢻⡽⠿⠋⠻⣽⣟⣽⣠⢠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠲⣤⣶⠘⠟⣧⣷⣚⠁⠈⠁
⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡿⣭⣀⡀⠀⠀⠈⢻⡄⠆⢹⡟⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⠄⣈⡁⠒⢦⣆⣻⠈⠀⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡈⠛⣷⣰⣿⣷⠶⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣈⣽⣿⡽⠐⡏⠉⠃⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡻⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⣿⠃⢹⡄⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⢇⣾⠏⠀⠀⣇⠀⠀⣰⡿⠋⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡇⣼⣋⠑⠤⡀⢻⣤⠾⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⣹⢏⡭⢒⣦⢿⣛⠉⢂⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠿⠛⠻⣿⣿⣿⣀⣴⠟⢋⡥⠨⣄⠈⡆⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠿⠿⠛⢷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⢒⡂⢺⡄⣨⣠⡟⠁
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⡿⣿⣷⣬⣍⣰⣤⣾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠉⠈⢿⣿⣿⣿⣿⣿⡛⠛⢿⣿⡉⡆⠘⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣌⣻⣧⡔⠀⡅⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣮⣸⣛⣿⣷⣄⣀⣀⣘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠉⣻⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⣿⣿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⢿⣷⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢘⡿⡏⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⡇
⡀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠂⣀⣤⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣆⠀⠀⠀⣀
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠼⠻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠟
⠀⠀⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⣠⣦⣀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣆⡀⠀⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠈⡿⠻⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⢲⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠐⠿⠀⠀⠀⢺⣿⡂
⡀⠀⠀⢒⠷⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⣠⡀⠀⠀⢙⠿⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⣠⡀⠀⠀⠠⣾⡤⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⠰⠿⡟⠁⠀⠀⢀⣠⡀⠀⠀⢠⣷⡄⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠸⠿⡏
⠙⠃⠀⠀⠀⠉⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠈⠉⠁⠀⠀⠀⡄
⠀⣤⠀⠀⠀⠀⠀⠀⠀⢸⡿⣏⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢺⠿⡍⠀⠀⣄
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠇
⡀⠀⠠⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠄⠀⠀⣀⣤⡀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡦⠄⠀⠀⣀
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠻⠁⠀⠀⠀⠀⠀⠀⠀⠐⠿⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠟
⠀⠀⠀⢰⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⡶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣤⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠹⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠽⠛⠧⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠛⠃⠀⠀⢲⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢺⣿⡍
⡄⠀⠀⠐⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢐⠿⡏
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⣆⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣠⡄
⣰⡀⠀⠀⠠⣾⡶⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠰⠛⡟⠁⠀⠀⢀⣠⡀⠀⠀⢠⣿⡖⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠼⠻⡏
⠙⠃⠀⠀⠀⠁⠉⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠈⠁⠁⠀⠀⢀⣄
⣀⣤⣀⣀⣀⣀⣀⣀⣀⣘⣿⣏⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣦⣀⣀⣀⣀⣀⣀⣀⣹⣿⣅⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣻⣦⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⡟⠷⣖⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣷⠀⢸⠇⠀⠀⢀⣠⣴⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⣿⣆⣀⡀⢀⣴⣿⣿⣤⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣷⣾⣟⠋⣴⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣾⣿⣿⣷⣾⡯⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⢻⣿⣿⣿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢻⣿⣿⣿⡀⠀⢿⣯⡙⢿⣿⣿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣁⣿⣿⣿⣿⣿⣤⣀⠙⢷⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢻⣤⡀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣏⠀⢹⠃⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⡟⣿⣿⣿⣿⡿⢻⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⡀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠈⠄⠀⠀⡞⡀⠄⠒⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠠⣀⡠⠀⠀⠀⠀⠀⢀⠞⠉⠀⢀⠠⠄⠒⠠⣴⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢀⠀⠀⠄⣰⠏⢀⡔⠉⠀⠀⡀⠔⠁⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠔⠁⢀⣪⣾⢏⣴⡟⠀⠀⠀⠈⠀⢠⢆⠖⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⠀⢀⠄⢀⡼⢋⠊⢸⡿⢁⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⢀⢎⣴⠏⠀⠀⠀⠸⠁⠘⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠠⠄⠀⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡇⣾⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢄⠀⠳⣄⠀⠀⠀⠀⠀⠀⢸⡇⢱⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⠏⠇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⣶⣴⣽⣶⣦⣀⠀⠀⠈⢃⠀⢾⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢡⠀⠀⢸⡁⠄⠀⠀⠀⠀⢀⠀⠀⠠⢀⣠⣤⣾⠟⠋⢁⣴⠉⠙⠫⡉⠋⠐⠦⣀⠳⣌⢐⢂
⠀⠀⠀⢠⡄⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢃⠐⢄⠀⠀⠄⠠⠙⠷⠒⠫⠋⠠⠋⠀⢠⠊⡵⠑⡀⠀⢱⡤⡄⠀⠀⠱⡈⠚⡏⢱⠀⢀⡀
⠀⠀⠀⢸⠈⡆⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠉⢄⠈⠀⠀⠀⠀⠄⠈⠀⠀⠁⠀⠀⠋⠀⠇⠀⠃⡀⠀⢠⡈⠀⠀⠀⠱⣀⠁⠘⡎⠀⠁
⠀⠀⠀⣧⣄⠹⢀⠀⠠⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠢⣤⣀⡀⢀⣠⡄⠀⠀⠀⢀⠀⠀⠤⠀⡰⠀⠀⠁⢀⠞⠁⠇⡤⣼⡆⢀⠁⡀⣿
⠀⠀⠀⣇⢘⠄⣏⠄⢀⡆⢀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠀⠀⠁⠒⠀⢀⠀⡀⠀⠀⠀⠀⢀⣼⠀⠀⠁⠀⢿⢀⡅⡼⠁⠂⠁⠐⣇⠀⡷⢨⣀⠉⢸
⠀⠀⠀⣿⠈⢠⠻⠈⠀⠀⠀⠂⡀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠦⡔⠈⠀⠀⠀⠀⠀⠸⣸⠏⠀⠀⠀⠀⠀⢸⠀⠁⢸⠇⢀⣼
⠀⠀⠀⢿⡄⣸⢀⡖⠈⠀⢠⠈⠁⡀⠀⠀⠀⠀⠀⠀⠐⠒⠄⡻⡄⠀⠠⠀⠀⠀⠀⠁⢦⠀⠀⠀⠀⠀⠀⡟⠄⠁⠀⡀⢂⠐⠀⡇⠀⠸⠀⢸⡏
⠀⠀⠀⠸⣟⣯⠀⠁⠀⣇⡨⣬⣁⠸⠀⡄⠀⠀⠀⠀⢀⣠⣽⣧⣽⡖⠿⣆⢀⡰⠀⠀⠘⠀⠀⠀⠄⡀⠀⠅⠐⡈⠁⠠⢀⢀⠄⣱⠀⠀⡆⣸⠇
⠀⠀⠀⠀⢻⢳⠀⢀⡼⠋⡰⣿⠝⠀⣆⠁⢠⠀⠀⠈⡟⠼⡟⠁⠀⠀⠀⠀⢹⣷⠞⠀⠚⠀⠀⠀⠀⠁⢄⣦⣿⡎⠐⠀⠀⠀⣰⡟⣇⠀⣧⡟
⠀⠀⠀⠀⠈⢿⢇⡿⠀⡼⢸⢃⠀⠄⡇⠜⠈⠀⣄⡈⡧⠀⠀⢠⠀⠀⠀⠀⢀⢏⡔⠀⠊⠀⠘⠀⢀⠓⢾⣧⠾⣝⡳⡥⢂⡔⠁⠀⢻⡆⡿⠁
⠀⠀⠀⠀⠀⠈⢿⡇⠀⡇⠘⠀⠉⠀⠛⠁⢀⠞⢉⢰⠃⢀⡀⠈⠄⠀⠤⠀⠀⠀⠀⠀⠀⠐⠉⠹⠦⡙⢦⡿⣿⣯⣿⣠⠏⠀⠀⠀⣸⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠣⠀⠘⠄⠀⠀⠀⠀⠂⠁⡠⠃⠃⡠⠜⠐⠂⠂⠌⠃⠈⠂⡀⠀⠀⠀⠀⠀⠆⠰⠋⠈⠀⢈⣙⡋⢠⢤⡀⡀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⢢⠀⢀⣀⠀⠈⠀⡐⡔⠀⠀⢀⠀⣀⢀⡘⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡜⣼⠀⢱⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠁⠒⠉⠉⠁⠀⣠⠞⠈⢀⢀⢤⠀⠲⢁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⠈⣧⢰⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⡄⠀⢀⣴⠃⢣⡈⢷⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠁⠀⢈⢂⣸⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢚⠙⢥⠉⣁⠶⠊⠷⠀⠁⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠘⠒⠾⠮⢤⣀⣀⣀⠀⢀⣀⣀⣀⡠⠤⠴⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠔⠋⢹⠀⠀⠀⠀⠀⠀⠀⡜⠁⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠰⠒⠛⢦⠀⠀⠀⠀⠀⢸⣷⠀⢸⠀⠀⠀⢀⡤⡴⣶⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣤⠤⣷⡀⠀⠀⣸⠷⢄⡀⠀⠀⣸⠿⠆⠼⢀⣀⣀⣉⣀⣀⣀⣁⠈⠉⠙⢲⣶⠖⣾⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣶⣌⠙⢦⣈⠁⠀⠀⣻⣄⠰⣃⣤⣶⠿⠿⠿⣿⢛⠋⠉⠉⠉⢿⣶⣴⣯⣤⣾⣿⠿⠛⢛⣹⣿⣤⣄
⠀⠀⠀⠀⠀⠈⠙⢿⣿⡏⠀⣩⣗⣄⠀⠉⣩⠾⠿⣿⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⢨⣟⠁⠀⠉⢡⡴⢒⢞⣿⣿⣿⠿⣿⡆
⠀⠀⠀⠀⣠⡤⠒⡶⣟⠓⠮⠿⣘⡸⢠⡞⠇⠠⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠿⡷⢀⡴⠻⣿⣻⣼⣿⠿⢁⣶⠟⢻⣶⣄
⠀⠀⠀⢸⣿⣶⣦⣈⠙⠳⢶⡶⠤⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣌⡛⠒⠉⣠⡞⠁⠙⢛⣡⣶⢟⣡⣼⣿⣿⢷⣀
⠀⠀⠀⠈⠙⠻⢧⣭⠀⣠⠀⢹⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⣰⠽⣷⡄⢀⣼⠟⣙⢦⣿⣿⣼⣃⣀⣽⡇
⠀⠀⠀⠀⠀⢀⡴⡿⢿⡙⠛⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⠙⠦⠥⠽⣻⡀⠀⠉⣿⡿⣛⣵⣭⣤⣝⡇
⠀⠀⠀⠀⠀⠸⡿⣖⣀⠹⣴⡇⣿⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⣦⠀⠄⠫⡹⣧⡾⢿⣿⣋⠁⠈⢻⠞
⠀⠀⠀⠀⠀⠀⠙⣿⢷⡀⢉⡉⠛⣿⣷⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠚⠃⢹⣿⡇⠨⣽⡽⠦⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠸⠿⢦⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⣣⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡛⠻⣷⣤⣄⣀⣠⡄⠀⠀⠂⠀⠀⢀⣀⣠⣴⡿⠛⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠙⣿⣿⣿⣿⣿⣋⣉⠛⠛⠛⠛⠛⠛⠋⠉⣈⣤⡶⠞⠋⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣴⣾⣿⣿⣿⣋⡟⠛⠛⠛⠛⠛⠒⠿⠿⠛⠋⠉⠀⠀⣀⣀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣷⣤⣤⢄⡀⡀⢀⢀⠀⠀⣠⣀⣠⣾⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⢀⡀⠀⢀⢀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⢀⣀
⣿⣿⡟⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿