Soy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢎⣷⣀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢏⠲⣧⠀⠀⠀⠀⠀⠀⠀⢰⠟⢉⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢧⠀⠀⠀⠀⠀⠀⠸⣄⠉⢀⣤⡄⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⣆⠀⠀⠀⠀⠀⠀⠀⡀⢻⡀⠀⢠⠀⠀⠀⠀⠠⡖⢀⣹⣆⣏⡜⣾⣡⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢆⡀⠀⠀⠀⠀⠀⠑⣮⣷⡠⡟⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⢀⢴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢀⡈⠂⣄⠀⠀⠀⠀⢸⠋⠉⣲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⠸⡿⠞⠛⠦⣷⡀⠀⠀⠀⢸⠆⠉⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠢⡀⠀⠀⠐⡄⠳⡄⠸⣷⣄⣀⢀⠄⡿⠇⠀⠀⠀⢸⡇⠀⠙⡇⠀⢀⡀⠄⠠⠤⠔⠒⠂⠤⠤⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠤⠀⠾⡄⠀⠇⠈⢧⡙⢿⣿⠟⠇⢂⠀⠀⢢⣸⣇⠀⠘⠃⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⠀⠀⢰⠀⠀⠀⠈⢆⢠⣴⣖⣚⡗⢦⡀⠻⣻⣶⠏⠀⠀⠀⠈⠉⠓⠒⠂⠀⠀⠀⠀⠀⠂⣉⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⣄⡄⠀⠉⠀⡿⠐⠁⠀⠀⠘⣽⠋⠉⠉⠉⢻⣧⠀⠈⠙⠆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠐⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠀⠀⠀⡀⠀⠀⢰⡙⢷⣶⠂⣤⣼⠻⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⢾⠁
⠀⠀⠀⠀⠠⠔⠰⠉⠢⡀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠈⢁⠀⠀⠀⠀⠻⣊⠙⠛⣋⣥⡤⣬⡢⡀⠀⠀⠀⠢⠄⣒⠂⠀⠀⠀⠀⠀⠀⠈⠈⠃⠴
⠀⢀⠄⠂⠀⠀⢀⡀⠀⠈⠐⠂⠔⠳⡀⠀⣀⠈⠀⠈⣁⠀⠀⠀⠀⠀⠀⠹⣞⣿⠁⠀⠀⢹⣾⡄⠀⠀⢠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠈⠲⢄⠐⠀⠀⠀⠀⠀⠀⠁⠀⢠⡭⠿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢞⣻⣅⣮⣾⠟⡇⠀⠀⠘⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⡉⠀⣈⡣⠤⠤⡀⠀⠀⠀⠀⠀⡠⠤⠌⡑⠊⠑⠢⠥⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠒⠀⡀⠐⠁⠀⠐⡠⠂⠘⠻⠏⣯⣀⡀⣀⡘⠁⠀⠑⠀⠘⢀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠠⠐⠒⢺⡗⠉⠉⠉⠻⡶⠓⠉⠙⡊⢙⢑⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢄⣀⡞⠻⣶⡒⠐⠂⠳⣆⣴⠶⢻⡀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢋⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠶⠶⠶⠖⠚⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⠭⣿⣛⢛⣛⣛⢛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠐⠢⡀⠀⠀⠀⠀⣀⠠⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⡀⠀⢀⡀⠀⠆⠰⠀⠀⠀⠀⣐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡠⠬⡁⠀⠀⠀⠃⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠓⠰⣀⡀⢣⡀⠅⢀⠠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂⢻⡍⠃⠃⠀⠋⠠⠤⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⠁⠠⠀⠐⣖⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠤⠒⠒⠒⠄⢀⡂⢠⠐⡴⡀⠀⢰⠂⠀⢀⠉⡀⠀⠀⠀⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠰⠀⠠⠀⣀⠀⠁⠑⡆⠊⠇⠀⠸⠀⠀⠀⡀⠀⢤⣄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠉⠳⠠⣅⣴⠀⠄⣤⣤⡀⠀⠀⠂⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠀⠠⠤⠄⠀⠠⠁⣌⠼⣧⣏⡀⣴⠚⣶⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⣤⡤⡦⠜⢀⠀⠞⠀⣀⣿⡇⠐⠹⡄⠈⠳⢄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣈⠀⠀⠀⣠⣾⠏⠀⠀⣶⡏⡇⠀⠀⢐⠂⠀⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠛⣩⠖⠀⠀⣀⡾⠀⣹⣿⠀⠀⠀⢣⡀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⢠⡶⠎⠀⠈⡟⠻⣄⠀⠀⠠⡟⢒⠂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠞⠁⢀⡴⢿⠐⡄⠈⡇⡀⠀⢙⡑⠿⣦⣤⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⠄⠀⠴⠟⠁⢰⡀⠈⠲⠴⠤⣀⠀⠦⡄⠀⠈⠂
⠀⠀⠀⣠⡶⢾⡟⠻⢿⠿⠿⡿⣛⣿⡻⡿⠖⣦⡀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠐⠀⠹⣀⠀⠈⠂⢀
⠀⠀⠈⠿⣓⣪⣜⣣⣓⣄⣒⣡⠰⣇⣧⣧⣤⡿⠃⠀⠀⠀⠀⠀⠀⠘⠄⡄⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠴⣾⠟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠛⢋⣭⡽⠟⢓⣶⠞⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡿⠋⠁⠀⢰⠋⠀⣠⠞⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠉⠀⢀⣀⣀⣀⠘⣤⠞⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⢀⣴⡾⠋⠀⠀⢀⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠁⠀⡞⠉⠀⠀⠀⣠⠟⠊⠀⠀⠂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠖⠋⠁⠁⠀⠀⠀⢸⡄⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠿⠟⠛⠋⢀⠴⠒⠊⠛⠲⢄⠀⠀⠳⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⣀⣀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢳⣠⠞⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢀⡾⢋⡴⠛⠉⠀⠈⠙⢦⠀⢸⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⣀⡴⠋⢠⠏⠀⠀⠀⠀⠀⠀⠈⡇⠸⡀⠀⢀⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⣿⠀⠀⡏⠀⠀⠀⠀⠀⢀⣀⣤⠷⠶⠛⠛⠋⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠙⢦⠀⡇⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⢀⣀⣤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠈⢧⡙⢢⡤⠞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣠⡴⠚⠉⠀⣀⣴⡦⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠚⠛⠉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠶⠶⠶⠶⠖⠛⠋⠉⠀⠀⢀⣠⡤⠶⠶⠶⠤⠤⣤⣤⣄⡀⠀⠈⠲⠦⠤⠴⠶⠚⠉
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠾⠛⠿⢶⠶⠦⠀⠀⠙⣆
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⣤⣤⣤⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⣀⣼⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢣⠀⠀⠠⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡜⠿⠉⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢳⡄⡜⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠜⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣧⣀⠀⠤⡈⡇⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠄⣸⠀⠀⠀⠀⠀⠠⠄⠄⠀⠒⠉⠁⠀⠀⠀⠀⠨⠻⡄⠀⠀⠇⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠄⠀⠀⢀⣤⡷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣀⡀⠀⠀⠀⠀⠀⠸⡒⢢⣤⢠⡤⠞⠉⠀⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢜⡴⠃⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠶⣄⠈⡇⠘⡁⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠗⠀⠀⠀⠀⠀⡠⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠈⡱⣼⡏⠉⠓⢦⣤
⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⢀⣀⡠⢚⡡⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢶⣄⣀⣤⠞⠁⣸⡇⠀⠀⠀⠘⢦⣤⢀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⢀⡟⠀⠀⣀⣀⣤⣡⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠈⢧⠀⠀⠀⠀⠀⠙⠀⠙⠀⠐⡖⠀⢂
⠀⠀⠀⠀⠀⠘⠉⠘⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠙⠲⠦⠤⠴⠛⠉⠀⠑⠤⣍⠀⣀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢻⡻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠿⣟⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⠇⠀⠀⢸⣿⠋⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠠⠀⢀⡀⢰⡙⢄⡀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⣿⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠁⠀⠀⠀⠉⠂⠱⣄⠀⠀⠀⠀⠀⠀⠈⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣋⠁⠀⢲⡌⠓⣾⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠁⠀⠀⢀⢃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡯⢄⡀⠀⠀⠀⡀⠤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠧⣢⡀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡲⡆⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠁⠀⠈⠀⠀⠐⡟⢛⣻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠄⡀⢀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠻⡀⠻⣄⠀⠀⣀⣀⣼⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠘⠉⠛⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠻⠿⠿⠿⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠤⣄⣒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠂⠐⠶⠤⠤⠤⠤⠄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠐⠒⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠐⠒⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠼⠀⠩⠁⠀⠀⠉⠁⠐⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠠⠦⣀⠄⡆⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⡋⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢠⠀⠠⠊⠤⡨⠂⠀⡔⢀⠱⠱⡜⢠⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠂⠅⢒⠀⠴⠄⠄⠀⠀⠀⠀⠀⠄⠂⠁⣿⠀⠐⠢⢅⠒⠄⠀⠣⠖⢀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠤⠀⠅⡨⠈⠂⠀⠊⠀⡨⠶⠃⠄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠁⠀⠂⡀⡇⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠂⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⡄⢸⠄⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⢄⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⡂⢈⠄⠀⠈⠖
⠀⠀⠀⠀⠀⠀⠄⠀⢢⠀⠉⢀⡀⠐⠂⠉⠀⢀⠉⡑⠀⠎⠉⣠⠜⠓⣆⠀⠂⠈⠁⣉⠀⠉⠛⡅⠀⣅⠀⠀⠈⠀⣠⡂⠀⠂
⠀⠀⠀⠀⠀⠣⣀⡀⠃⢈⠁⢉⠀⢀⠤⠄⠀⠁⠄⠢⠄⠠⠐⠆⠀⠘⢣⠔⠀⠖⠠⠈⠂⠠⡤⠤⠾⠀⠒⠑⡂⠈⠀⠑⣠⠤
⠀⠀⠀⠀⠀⡈⡀⠉⠁⠀⠔⠀⠀⠈⠀⠀⠀⠢⠠⠐⠀⠐⠀⢃⡘⠛⡀⠀⠔⠃⠀⠖⠀⠀⣀⠀⠀⠓⠀⠊⠑⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠈⠀⠰⡌⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣧⠀⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⢠⣦⣄⡀⠀⠀⢀⣀⡠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⢻⡏⠀⠀⠀⣠⡑⠀⡄⠀⠀⢀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡼⠃⠀⠀⢀⠉⡍⠆⠹⣶⣤⠃⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⡠⠞⠁⠀⠀⠀⠆⠀⠅⡰⠀⠈⢻⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠔⢉⡴⠀⢠⣥⠀⠀⠀⠀⠀⢪⠀⠤⣀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠆⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠈⠛⣆⡘⢁⣠⠖⠛⠃⠠⠄⠀⠀⠈⠑⠄⠀⠠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠛⠻⡇⠀⠀⢀⡤⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠜⠋⠀⠠⡀⠀⠀⢐⡧
⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣤⡴⠞⠁⠀⠀⠙⠄⠀⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠐⠂⠉⠀⠀⠀⠀⠀⠁⠀⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢆⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠖⠶⠖⠀⠁⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡿⢠⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣁⡀
⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢀⢙⣶⢇⠸⣀⡆⠀⠀⢀⠀⠀⠀⠈⠲⣄⣀⠀⠀⣀⣄⣀⢀⣀⡤⠞⠀⠀⠉⠉⠉⠁⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⠿⠿⠁⢀⡑⣝⠓⠳⠖⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢢⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⢸⠇⠉⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣿⠇⠀⠀⠀⠳⣄⣈⢉⣡⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⡤⢯⡀⠀⠀⠀⣹⡄⠀⠀⢠⡞⢹⡭⠂⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢬⠙⠒⠐⠚⣡⣈⠁⠀⠁⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡄⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢳⠀⠀⠀⠀⠀⡼⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢣⠀⠀⠀⡜⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⡆⣠⠎⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣴⣅⣀⡀⠤⠼⢿⠥⠤⣤⣤⣼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⣙⣯⡛⠒⠛⠉⠉⠁⠀⠘⣒⠤⣭⣉⡐⠠⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣁⡠⠔⠋⠁⠀⠉⠲⢤⣀⡀⠀⠀⢀⠀⠈⠉⠙⠓⠶⢦⣍⣮⠐⠢⢄⡀
⠀⠀⠀⠀⠀⣤⠤⠤⠤⠖⠛⠚⠉⠁⠀⠀⢀⣀⡀⣀⡴⠶⠋⠭⢍⢛⠶⠂⠒⠢⢤⣄⠀⠀⠈⠙⠓⢦⣄⣉⣒⣄
⠀⠀⠀⢀⣀⠬⠓⠦⣄⡀⠀⠀⢀⣀⣤⣶⡶⠞⡻⠉⠀⢀⣀⠤⠤⠿⡀⠺⠷⠆⠀⣱⠖⠋⠈⠑⠢⡄⠀⠉⠒⢽⡕⢄
⠀⠀⠈⠓⠦⠤⣀⣀⣈⣹⠟⠃⠉⠛⠫⡁⠀⠀⠘⢶⡶⠉⣴⣲⠆⠀⢈⡷⠒⠈⠉⠛⠢⡀⠀⠀⠀⠘⣆⡀⣀⠀⠈⠺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢄⣀⡀⢀⡟⠀⠀⠀⡠⠗⠒⠒⠒⢤⡴⢿⡀⢠⣤⣄⠀⢀⡟⠶⣦⡤⠞⠉⠙⠓⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠉⠀⠀⠀⢸⠁⣠⢤⣤⠀⠀⡓⠀⡹⠒⠂⠈⠛⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣉⣐⠢⠔⠁⢸⡀⠀⠀⢀⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠿⠿⠗⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣍⡑⢰⠋⠉⢳⠈⢣⡠⠃⠀⣯⠈⡗⠈⡏⣭⠃⠀⡸⢳⡀⠀⡟⢦⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠚⠈⠒⠐⠊⠀⠐⠃⠀⠐⠓⠒⠊⠐⠓⠒⠚⠐⠃⠈⠓⠀⠓⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⢤⡀⢀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⡀
⠀⠀⠀⠀⠱⢤⡀⠀⣠⠞⡀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⣟⣠⣕⠀⠀⠀⠀⡠⠄⠀⠀⡀⠀⠀⠒⠁
⠀⠀⠀⠈⢙⠀⠤⠀⠍⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠆⠀⠀⠈⠈⠈⠁⠀⠀⠂
⠀⠀⠀⢀⢆⡣⠔⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢀⠀⠆
⠀⠀⠀⠈⠀⡀⠄⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠠⠅⠀⡾⠀⠀⠀⠀⠀⠁⣀⠄⡀⠀⠀⠀⠀⠑⠠⠼⠁⣴
⠀⠀⠀⠀⠀⠨⠒⠐⠄⠶⠀⢡⠀⠀⠀⠀⣀⢀⡀⠀⢸⠁⠀⠀⠀⣾⠘⠁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⡀⠌⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣉⠈⠙⠃⠀⢸⠄⠀⠠⠈⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠂⠘⠀⠀⠀⣠⣈⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠈⢀⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⠀⡐⠲⠖⢂⠏⠀⢀⠀⠐⠺⠀⠀⠐⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⠘⡧⠈⠁⠀⠀⠈⢀⢀⡠⠐⠀⠀⠁⠀⡀⠘⡠⠂
⠀⠀⠀⠐⠂⠈⠑⠀⢈⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⣸⠎⠉⠀⠐⠁⠀⢀⢈⡁⠘⠁
⠀⠀⠀⠀⠠⠴⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠟⠛⠂
⠀⠀⠀⠀⠀⣀⣀⣄⣀⡀⠀⠀⠀⠀⠀⢄⡉⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢦⠵⠝⣴⣞⣿⣶⣵⣄⣀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠸⡌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠈⣉⠉⠛⠓⠛⠚⠛⡽
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⠤⠤⠀⡀⡄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠁
⠀⠀⣿⣿⠿⡿⣿⢿⠿⡿⣿⣿⡇⠀⠠⣛⣴⣺⡯⡿⢷⣦⣷⣮⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠘⣿⣷⣷⣿⣾⣷⣿⣿⡟⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⢙⡿⠁⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠉⠀⡤
⠀⠀⠀⠈⠛⠿⠿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠂⠈⠂⡀⠀⠀⠀⠒⠂⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢂⠤⠂⠀⠀⡌⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠄⢂⠉⢢⡀⠂⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢿⠐⠻⠀⠎⠄⡀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠈⠀⠀⠐⣆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⡔⢠⠀⣴⡀⠀⡎⠀⠠⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠀⠐⠀⢄⣀⠈⠂⠁⠃⠀⠀⠀⠀⠀⠐⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠈⠑⠒⠺⣀⣰⠶⠂⡀⡀⠀⠉⠒⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢈⡄⠀⡠⠃⢉⣿⠂⠸⡇⠺⢆⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠋⠉⠁⣰⡖⠀⠀⣽⢻⠀⠀⢡⠄⠀⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⡩⠁⠀⣀⣾⠛⣽⡆⠀⠀⢆⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠀⢀⡴⠞⠀⢸⡟⢧⡀⠀⠠⡟⠒⠀⠘⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⢠⡞⡇⠰⡀⢈⠄⠀⢩⠝⠳⠦⡤
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⡀⠀⠁⠀⠐⠋⠀⢱⠀⠈⠉⠓⢤⠀⠓⢄⠀⠡
⠀⠀⢰⣿⡙⢯⡍⡟⣋⡹⢙⣿⠉⠏⢹⣷⠄⠀⠀⠀⠀⠀⢇⠀⠀⠀⠈⠢⢀⠀⠁⠂
⠀⠀⠀⠁⠒⠷⠾⠶⠷⠶⠷⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⢸⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠐⠒⠤⠤⣀⠀⠀⢀⠞⠁⠀⠀⠀⢠⠀⠀⠄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⣀⡤⢆⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠺⣄⠀⠀⠀⠈⠙⢦⣇⠀⠀⠰⡇⢀⠴⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠰⠏⠁⠀⠀⣿⠛⠾⣝⠳⢦⣄⡀⢀⠀⠀⠈⢧⡀⠀⠀⠀⠂⢿⢀⣘⣄⣞⢕⡭⢖⡀⠀⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠀⠀⠀⠀⠉⠊⣍⡚⠲⠶⠤⣤⣉⠳⡄⠀⣀⣼⣽⡿⣂⢁⡏⠀⠈⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣔⣤⣠⣤⣤⣤⣈⣉⣉⣉⠠⠤⠬⠽⡿⠽⠟⠃⠘⠋⣿⣟⠁⢀⣀⣠⠞⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣖⠋⠉⠉⠉⠉⠙⠛⠂⠐⠶⠖⠛⠋⠀⠀⠀⠀⠀⠀⢀⠤⡝⠛⠛⢋⠉⠠⠀⣀⣀⣄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⡠⠄⣁⡾⣁⣆⣴⠛⠀⠐⠃⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢀⠔⠈⣥⣴⡄⢀⠶⣒⠳⠤⣀⣀⣀⣀⠤⠀⣠⠝⠺⠷⠿⠫⣚⠽⠋⠀⠀⠀⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡷⠊⠀⠀⠁⢸⠁⠀⠙⠿⠆⠈⠨⠭⠴⠖⠋⠉⠙⠛⠛⠛⠋⠁⠐⠀⠄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠈⠒⠄⠀⠐⠚⢦⡀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠁⠐⠐⠀⢄⠀⠀⡐⠋⢀⠂⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠓⡜⠁⠀⠲⢀⡀⠠⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⢰⡀⠀⠐⣠⠧⢂⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⣀⠀⢸⠁⣠⠇⠀⢸⡌⠐⠇⢀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⣀⣀⡈⠶⡄⠰⠟⠒⠒⡜⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠉⠄⠀⠀⡤⠓⡄⠀⠐⠁⠀⠃⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡶⠀⠀⠈⡄⠀⠀⠀⢠⠋⠈⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠄⠀⠀⢸⠀⠀⢀⣧⠎⢸⡇⠨⡄⠀⠊⠁⠀⠉⠑⠲⠆⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠀⢠⠀⠀⠀⠘⠀⠀⠸⠀⠒⡜⠀⠈⠀⢀⡄⠀⠀⠀⠢⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⣀⣀⣤⠤⠒⠀⠀⠀⢀⠀⡀⠀⠀⢤⣀⢁⠀⡼⠛⠁⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⣀⠉⠉⣇⣠⣦⠑⠉⢣⠀⠠⢀⡀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠓⠀⣿⠁⠛⢾⣧⠈⠘⢣⡎⠀⠀⣤⠊⢉⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠀⠀⡰⠃⠀⠀⢸⣟⠶⠀⠀⠁⣀⣄⠉⠙⠛⠒⠶⠀⠀⠄
⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⠀⢈⡝⠀⠀⠀⡙⢹⣿⡄⠀⠀⠸⠿⢦⣀⠀⠀⠀⢬⣴⡤⠄
⠀⠀⠀⢰⠃⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⣠⡜⠀⠀⠀⢀⣿⠀⣿⢸⣀⡀⠀⠀⠐⢬⡉⠙⠉⠋⠁
⠀⠀⠀⠘⣆⠻⡄⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠇⠀⠀⠤⠶⢛⡀⠀⠀⠀⣸⠃⠀⣿⠀⠀⠸⣦⡀⠀⠀⠳⣄
⠀⠀⠀⠀⠘⠢⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⣴⠂⠀⠊⠀⠀⢰⡎⠁⠀⠀⣤⠀⠀⠀⠈⠓⠠⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠛⠁⠀⠠⠁⠀⠠⣻⠃⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠚⠁⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⢧⠀⠀⠀⣼⠏⠀⠀⠀⠀⡴⣯⠅⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⢀⣥⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠓⠒⠚⠁⠀⠀⠀⠀⢸⠀⠁⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⢐⡥⠂⠈⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣠⣀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠀⠀⠤⠚⠁


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

⠀⠀⠀⠀⠀⠀⠀⢠⣤⣖⡂⠐⠒⠒⠦⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠉⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⡚⠉⠀⠀⠉⣐⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠐⠀⠠⢼⡀⠀⠀⠀⠀⣀⠤⠖⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡞⠛⠛⢧⡀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠂⠠⢄⡀⠐⠀⠤⠄⠀⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠀⠀⠁⡐⠄⡀⠀⠀⠄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⡀⠀⠀⠀⠀⠈⡆⠀⠐⠀⠀⠈⠢⡀⠀⡂⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠳⣄⠀⠀⠀⠀⠈⠑⠢⢄⣀⠀⠂⠆⠀⠘⠤⠀⠂⡔⢀⠃⠁⠘⢀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣶⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⢈⠢⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠳⡄⠀⠀⠀⠀⠀⢠⡅⠄⢸⣧⠀⠀⠀⠉⢲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠤⣄⡀⠐⠀⠈⠐⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⢸⠄⠀⠀⢧⢳⡀⠀⠌⠁⠀⠈⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⢀⠀⡟⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡳⡙⣄⠆⢀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠘⢆⡈⠀⣁⠞⠁⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⢮⠻⡦⣀⠢⠀⠀⢠⡄⠈⣀⡭⢑⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢳⡝⢮⡋⠕⠀⠉⠀⡜⠁⠀⠀⠀⠀⠑⢍⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⠀⠀⠀⠀⠀⢻⡄⠈⠓⠦⣄⠀⢡⠀⠀⠀⠀⠀⢀⠀⠡⠀⣉⠷⠖⠒⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠄⠄⠀⠀⠀⠀⡹⣄⠀⠀⠈⠑⠦⣁⠪⠀⠀⠂⠐⣠⠟⡰⠁⠀⠀⠀⠀⠀⠈⢗⠒⠶⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠒⣤⠀⠀⠀⠀⠈⢧⠀⠀⠀⡀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠓⠦⢥⣈⣉⣀⠀⠘⢄⠀⢀⠀⡀⠈⢀⡾⠈⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠐⠠⣸⠃⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠦⣭⣀⣂⣐⣲⡫⠴⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢊⣠⠟⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢀⠀⠀⠀⠀⡀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡀⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠄⠙⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⢀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠃⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⡀⠀⠁⠄⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⢀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⣀⣀⠀⢀⣈⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⢤⣤⣖⢒⡿⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠁⡰⠚⢋⡡⠞⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣀⡀⢇⡴⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⢀⣴⠏⠉⠈⣹⡏⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠛⠁⢰⠋⠀⠀⢀⡼⠛⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠖⠋⣁⣀⣀⠀⠀⠸⡀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⣀⡤⠖⠋⠉⠉⠁⢀⠔⠉⠀⠀⠀⠙⢆⢀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⢠⠞⣫⠖⠂⠀⠒⢄⠀⡎⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡾
⣠⠴⢃⡞⠁⠀⠀⠀⠀⠘⡆⢇⠀⢀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⣇⠀⢸⠀⠀⠀⠀⣀⡤⠴⠓⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠈⢧⠸⡀⠀⣠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⣀⠤⠖⠒⠒⠲⣄
⠀⠈⠓⢬⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⢀⡴⠋⠁⣤⠴⠚⠋⠀⠈
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠞⠉⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⠚⠉⠉⠀⠀⣠⠴⠒⣒⡒⠲⠶⠤⠤⣄⠀⠙⠒⠶⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠈⠓⠛⠛⠛⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⣀⣀⣀⣤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠠⠶⣫⡉⠉⠁⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠐⠀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣐⠀⠀⠐⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠉⠀⢀⠀⢈⡉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐⠸⣇⠀⠀⣀⡀⠠⠖⣋⣉⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠠⢀⠁⠐⠚⠀⠄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠑⠠⡀⠀⠐⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠻⣄⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠂⢨⠀⠀⠘⠀⠀⠈⢢⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢀⠘⠢⡀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠂⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⡰⠀⢸⡄⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠹⠄⠀⠀⠀⠀⠀⡆⠀⠀⢿⣆⠀⠂⢂⠭⠳⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠢⣄⠁⠀⠀⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠘⣎⢦⡀⡇⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠠⠂⠂⡼⠁⠀⠀⠀⠀⠀⡇⠐⠁⠀⠘⢢⡳⣔⡠⠀⡀⠀⠀⠀⡂⠉⠲⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠙⣎⢮⡡⠬⠀⠠⠴⠁⢠⠖⠁⠒⠉⠛⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠈⢧⠙⠦⣄⡀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢳⡉⠲⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠉⠳⢤⡈⢄⡀⠀⠈⠄⠂⢠⡄⡠⠃⠀⠀⠈⠁⠳⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢤⣀⠀⠀⠀⠀⠈⢆⠈⠀⠀⠀⠀⠀⠊⠳⣄⠀⠀⠀⠙⠲⣌⣐⠠⠤⠔⠋⠀⡅⠀⠀⠀⠀⡠⠀⢸⠠⠀⠉⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⡀⠘⣇⠀⠀⠀⠀⠈⢧⡀⠀⠁⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠉⠙⠛⠒⠲⠤⣌⡒⠄⢀⣀⣁⡴⢊⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠄⣱⠇⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠈⡉⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠠⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣁⠀⠀⠀⠀⠈⢛⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠙⠃⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⠶⠗⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣀
⠀⠀⠀⠜⠁⠀⠀⠙⢦⡀⠀⠀⢸⡀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠟⢃
⠀⠀⠀⠈⢤⡀⠀⠀⠘⣷⠀⠀⠀⠉⠐⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠉⠀⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡀⠀⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠄⠀⢄⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠈⠳⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠉⢿⣷⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⡀⠀⠀⠉⠻⠦⢀
⠀⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⣀⣈⡥⠞⢻⠏⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣶⣄⣀⢀⣠⠎⠁
⠀⠀⠀⠀⠠⠀⠀⠀⠉⠁⠉⣩⠞⢋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠇⡠⠀⠠⡴⠂⠀⠉⢠⠂⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⠠⣦
⠀⠀⠀⢀⡆⣠⡴⠞⠁⠐⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣠⣄⢹⠀⠙
⠀⠀⠀⢀⠀⠛⢀⡀⢀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣿⡏
⠀⠀⠀⠀⠀⠁⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠉⠀⠀⣀⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠀⠁

⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠤⠶⠶⢛⡽⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉⢁⣤⠶⠟⠉⣩⠿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡫⠟⠉⠀⠀⢰⠋⠀⢀⡴⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣀⣀⡀⠘⣄⡾⠋⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⣀⣴⡾⠉⠉⠉⠉⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠽⠋⠀⡼⠟⠋⠀⠀⠀⢀⡾⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠛⠋⠁⠀⠀⢸⠁⠀⠀⠀⠀⣠⠟⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠋⠁⢀⣀⣀⡀⠀⠀⠀⠸⡄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠚⠉⠁⢀⠴⠋⠉⠀⠉⠉⠳⣄⠀⠀⢙⣦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⢀⣠⡶⠟⠉⣀⣀⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⡴⠟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⣠⠟⢋⡴⠚⠉⠀⠀⠉⠓⣄⠀⢸⡃⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡟
⠀⣀⡼⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⡇⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⣾⠉⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣠⣵⠤⢶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⢿⡀⠀⢸⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠹⡆⠸⡄⠀⠀⢠⡞⡩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⢀⣠⠴⠶⠚⠛⠛⠲⢦⡀
⠀⠀⠙⢦⡉⣳⠞⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⠀⣀⣤⡾⠗⠀⠀⢹
⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣄⣀⣀⣀⣀⣀⣀⣤⡴⠶⠛⠛⠉⠉⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⣀⣀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⣉⣭⣭⣉⣉⣉⡙⠛⠛⠳⣄⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠙⠒⠒⠛⠛⠛⠋⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠦⢤⣤⣤⣤⣤⣤⠴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣁⡤⠄⠀⢀⣤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡠⠊⢉⢠⡄⣀⠔⠁⠀⠙⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣵⡶⠿⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⢠⣤⢴⡾⣿⠶⠚⠛⠶⣶⣤⣄⣀⣤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⡴⢋⠔⠁⠀⣰⠤⠿⠛⠛⠛⠛⠛⠓⠒⠛⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡴⠟⠡⠃⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠠⣊⠔⠁
⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⣉⣉⣂⣀⣀⣀⣉⣉⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠤⠤⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠁⢀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⡀⡠⠐⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢘⠀⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠳⠕⠠⠄⢀⡐⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢄⠠⢄⠀⢀⡠⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⣸⡇⠀⠠⠈⠆⡅⠢⠤⠄⠀⡠⠖⠄⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⣀⡞⡇⠀⠀⠀⠈⠀⠀⠀⠀⠸⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⠀⠀⢀⠸⠀⠆⢀⡤⢄⠀⡀⠐⣀⠀⢤⠃⠤⠄⢀⠀⡀⢀⠀⡀⠀⡀⡀⢠⠀⢀⠑⠂
⠀⠀⠀⠀⠀⠂⢈⡆⡄⠠⠤⠉⠈⠀⠴⠀⠃⠊⢀⡐⠈⣹⢄⠈⢐⠀⢄⠈⠉⠃⡠⢂⣀⡀⣀⢺⡀⠈
⠀⠀⠀⠀⠀⠒⠂⠀⢀⠐⠀⠎⠉⠀⣄⠁⠙⠀⠡⠇⢠⡄⠃⠀⠅⢁⡄⠀⠋⠉⠡⡄⠀⠇⠐⠂⠒⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠁⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢠⣿⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣉⣉⣉⣉⠛⠶⣼⡏⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣋⣴⠛⢉⣠⣤⣈⠙⣦⣿⣷⣀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⣉⣉⣉⣉⠛⠿⣄⣠⠿⠛⣉⣿⣍⣹⡿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⡴⠛⣉⣠⣤⣉⠙⢦⡾⢃⡴⠋⣹⣿⣼⣿⠛⢦⡘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣧⣾⣥⣾⣯⣤⣤⣽⣷⣼⣧⣾⣤⣾⣿⣿⣿⣿⣷⣬⣷⣼⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠘⣇⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⡿⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⢻⡀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⢸⠇⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⢤⣼⣇⠀⠀⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠀⠀⣿⠤⠶⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣀⣠⣆⣀⣠⣄⣤⣤⣄⣄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⣭⣽⣯⣭⠁⣀⣸⣇⣀⡀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠰⠶⣾⠷⠶⢾⡏⣿⣼⢩⡇⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢀⣼⡟⢲⣶⢸⡇⣿⢾⢨⡇⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠁⣻⣾⣁⠘⠷⠻⠞⠻⠇⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠐⠟⠉⠈⠉⠛⠛⠛⠛⠛⠛⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃





⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⢛⠛⣛⡿⠖⠛⢋⣉⡤⠄⢖⣊⣳⡂⠑⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⢸⠘⠷⠶⠖⠛⠉⠁⠀⣀⣀⣤⣬⣽⣦⠘⣄⣠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠛⠛⠛⠻⠷⢖⡤⡒⠋⠉⢉⣒⣂⣠⣬⣧⠘⣦⣤⣄⣬⡍⢻⡆
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠁⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣈⣛⣛⣛⠻⠿⠿⠿⠿⣛⣛⣛⣫⣭⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠁⠘⠲⠤⣤⣤⣀⣤⡔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⣧⠈⣉⡉⢁⠀⢀⡀⣀⣀⣀⣀⠤⠴⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠲⠶⢦⣤⣤⣤⣤⣤⣤⠴⠞⠋⠁⠀⠀⠀⠀⢿⡀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⢀⣠⣼⣇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⣦⣤⣤⣄⣀⠀⢀⣠⣤⠤⠀⠀⠤⠤⠤⣤⣤⣤⣤⣴⣾⣍⣀⣀⣻⡀⠉⠉⠉⠉⠉⡼⠋⠉⠈⠁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣭⡿⠛⠁⠀⢠⡰⣄⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⢿⡍⠉⠛⠛⠛⠲⡾⠷⠶⢤⣤⣤⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠟⠋⠁⠀⠀⠀⠀⠀⠱⡌⢳⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠻⣇⣤⡶⠶⠟⣾⠓⠚⠓⠲⢦⡀⠀⠈⠉⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠳⣄⠀⠀⢀⣤⠶⠚⣩⣶⡾⠛⠁⠀⠀⠁⠶⠞⠃⠀⠀⠀⢹⣦⣤⠞⠁⢸
⠀⠀⠀⠀⠀⢠⣶⡛⠋⠙⣇⠀⠉⠉⠛⠛⠛⠷⠶⢶⣤⣤⣤⣤⠼⡧⡙⡂⠜⣋⡤⠖⢿⣿⡟⢀⠔⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠙⠛⠛⠾⠿⠿⣿⣷⣶⣶⣤⣤⣤⣄⣀⣀⣤⣤⣷⣽⣿⠞⢉⣤⣴⠋⣹⡇⠼⠀⠀⠀⠀⣀⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢤⣈⠉⠉⠉⠛⠛⠛⠛⠻⠿⠶⠾⣷⣶⣾⣍⣷⡀⣀⣀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡼⠂⣰⢂⡿⠂⠀⠀⠀⠀⠀⡔⠒⠲⠦⣄⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢀⡴⠁⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠈⡷⠟⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⣤⡜⢀⠀⠈⠣⣄⣀⣀⣀⣀⡀⢀⣀⣴⠏⠀⣴⠃⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣙⣒⣒⣂⠀⠀⠀⠀⠈⠉⠉⢀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠲⠶⠶⠞⠛⠉⠀⠉⠉⠙⠓⠲⠶⠦⠤⠴⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⠀⠀⠏⠈⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠟⠿⠿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣠⠰⠒⠟⠻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠿⠿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⠿⢀⠠⠎⠠⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠁⠤⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢀⠠⠎⠠⠿⠿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⠉⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠋⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⡉⠉⣉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠟⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠉⠋⠛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠤⠰⠥⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠥⠐⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⣶⠀⣶⠀⣤⣭⣭⣭⣤⣤⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢉⣈⣤⠴⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠖⠛⠉⠁⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠛⣠⣶⣿⡿⠿⠿⠿⢿⣿⣶⣤⡀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⣉⣉⣉⠛⣠⣾⡿⢛⣡⣴⣶⣶⣶⣶⣬⡙⢿⣿⣆⠛⢋⣉⣉⡙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣴⠟⣋⣉⡉⢰⣿⡟⣠⣿⡿⢋⣭⣤⣬⣙⠻⣿⣆⢻⣿⣇⠉⣉⣉⡛⢷⡌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⣤⣬⣭⣬⣭⣤⣭⣩⣅⣩⣭⣥⣭⣭⣤⣭⣭⣭⣭⣭⣤⣭⣭⣤⣭⣭⣤⣭⣭⣥⣬⣉⣬⣭⣤⣤⠰⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢧⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⡟⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠸⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣾⠇⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠤⠀⣀⠀⣰⡧⠴⡗⠚⣻⣁⣸⣦⡤⠖
⠀⠀⠀⠀⠀⠀⠤⠴⣿⣚⡽⠛⠁⢸⠀⢠⡿⡆⢸⠀⢰⡇⢰⡇⢈⡽⠁⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⣠⣴⣟⣁⡤⢤⡤⢼⠁⢸⠃⢻⣼⠀⢸⡇⠀⠛⠋⢠⠀⠀⢀⠀⠀⢀⡼⣣⠖⠁
⠀⠀⠀⠀⠀⠈⠉⠉⣇⡤⠖⠋⠀⠻⠴⠺⠀⠀⠛⣤⠀⢠⡄⣷⠖⣾⡴⣦⠞⠀⢀⣞⢞⡵⠊
⠀⠀⠀⠀⠀⢀⡤⠾⡏⠀⠀⠀⠀⠀⢠⣀⣿⣠⠄⡟⠿⠿⢹⡏⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣤⣦⢶⠾⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⢀⣀⣤⣾⡜⠳⣻⠿⠂⠧⠖⠂⢀⣀⢀⣀⣤⣴⠶⣖⣉⣉⣉⠉⠒⢬⡁⠁
⠀⠀⠀⠀⠀⠀⠀⣷⣲⣾⡧⠛⠉⡏⠟⠈⠋⠀⠀⣀⡠⠊⠁⠀⠫⣴⢣⣀⠷⠈⡧⢄⠈⢳⠀⠀⠈⠳⢄⠀⠀⠐⠂
⠀⠀⠀⠀⠀⡰⠂⡏⠀⠸⠀⠀⠀⠁⢀⣠⣴⡖⠉⡱⠁⠀⠀⠀⠁⠙⢗⣭⠷⣴⡳⠌⡇⢸⠯⡉⢳⡀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠑⠞⠁⠀⠀⣀⡤⠖⠋⠁⡟⠋⡻⢤⡃⠀⡔⠒⠂⠉⢂⡜⠁⠂⠀⠙⣞⣠⠷⣄⡹⢸⡣⢚⣶⠤⡈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⡀⢀⡀⠄⠠⡈⠒⠌⠒⢦⣶⣆⡀⡌⠀⠀⠀⣀⣀⠈⢏⡲⠗⠧⣞⠔⣱⠋⡰⣁⠠⢠⠈⠲⡀
⠀⠀⠀⠀⠀⠀⣼⠡⣴⢤⠐⠢⣀⠢⡕⠂⢁⠠⠐⡧⠠⠠⡈⠛⢶⡃⠀⡎⠀⠀⠈⣤⠋⠀⠀⠀⠈⢾⠃⢜⠔⢁⠔⡦⢀⡦⢿
⠀⠀⠀⠀⠀⠀⡟⣆⠀⠸⡇⠦⠀⠂⠒⣈⠠⡄⠰⣄⠢⠀⣴⣦⣶⡌⠑⠧⣶⡄⢐⠁⢠⠄⠐⠒⢢⠈⢟⠳⣪⡭⠞⠋⠁⠀⡼
⠀⠀⠀⠀⠀⠀⡇⠀⠙⢦⡃⠘⠙⠣⠉⢮⢹⣈⠃⢈⣀⠴⢯⡁⡠⠛⠉⠙⢪⡑⠧⡀⢢⢠⢰⣄⣤⠤⠚⠋⡁⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠉⠳⣍⡀⡳⡜⠛⠛⡿⠉⠀⠀⠀⡸⢁⡴⠒⢢⠀⣇⣀⣨⠽⠚⠋⠁⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠉⠳⢤⣀⢸⠁⢠⠊⠑⡄⣇⣸⣠⡤⠜⠒⠉⠁⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠒⣚⠋⠉⠁⠀⠀⠒⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⠁⠁⢀⣀⡤⠴⠒⠋⠁⠀⢀⠀⣠⡀⣴⣠⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠙⠒⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⢠⠟⣷⠁⠿⠘⠃⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⡀⠀⠀⠀⡀
⣿⣯⢭⣟⡛⣛⣛⠛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⣠⡟⢡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋⠉⠛⠛⠋⠉⠉⠳⣄⠀⠀⣴⠏⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⣉⣤⠶
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠛⠛⠛⠋⠛⠶⠄⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⢁⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⢋⣡⡴⠟⠉
⠀⠀⠀⢀⣠⣤⢶⣿⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠀⠀⣰⠟⣰⡟⠉⠙⠛⢶⡄⠀⠀⠀⠀⠀⢀⣠⡶⠛⣉⣤⠾⠋⠁
⠀⢀⡴⠛⣁⣤⠾⣗⠶⠀⠀⣤⠶⠄⠀⠀⠀⠀⠿⠀⠀⢀⡾⢋⣼⠋⠀⠀⠀⠀⠀⡿⠳⣦⣀⣴⠞⢋⣡⡴⠛⠉
⣰⡏⣠⡾⢯⡜⠓⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣡⡿⠁⠀⠀⠀⣀⠀⢘⣻⡶⠟⣉⣴⠞⠋⠁
⣿⠀⣿⣹⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⠏⠀⠀⠀⠀⣀⣼⠷⢋⣥⡶⢻⡏⢸⡇
⢻⣦⡟⠷⠀⠀⠀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣠⡴⠞⣩⡴⠞⠋⠁⠀⢸⡇⢸⡇
⠈⣿⣿⣗⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣥⠶⠛⠉⠀⠀⠀⠀⠀⢈⣹⣿⠁
⠀⠘⣿⡙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⣿⠃
⠀⠀⠘⢷⡝⠶⠍⠛⠻⠶⣦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣶⠶⠟⠛⠉⠀⣼⠏
⠀⠀⠀⠈⠻⣾⠛⠦⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠙⠷⣝⡛⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠶⣦⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢦⣤⣤⣤⣤⣤⣤⠶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠴⣾⠋⣽⣹⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣷⠖⢋⣩⠖⠁⠀⠀⣀⠀⣾⣰⡇⡜⣿⢸⣇⠞⠀⠀⠀⡄
⠀⠀⠀⠀⠀⢀⣠⠖⡋⠁⣧⠴⠟⠒⠛⣻⠆⢰⡷⡟⡿⠑⠟⠀⠏⠀⡁⠀⢰⠇⣰⠇⠀⢀⣠
⠀⠀⠀⠀⣴⡯⠖⠋⠀⠀⣧⠀⠀⣠⠞⠁⠈⠙⠚⢠⠀⢀⢠⠀⣤⢸⣷⢛⣾⠞⢹⠀⠀⢋⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡴⠚⠁⠀⠀⠀⠀⠀⣸⠀⠿⠻⠞⣿⣼⠙⠋⣿⠀⠘⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣿⠀⠀⠀⣠⢤⡖⠀⠀⣿⣠⡴⠀⢸⡇⠛
⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠋⠁⠀⠀⠀⢀⣀⡠⢤⠔⠶⣒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠛⠉⠁⢢⡗⠀⠒⢠⠙⢆⠈⢲⠤⣀
⠀⠀⠀⠀⠀⠀⢀⣤⠖⠊⠁⠀⠀⣠⡄⡰⠒⠾⢭⡢⢤⡊⢀⣠⠤⠃⡠⠗⠒⠐⢸⠀⡰⢀⡬⠔⢊⣉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⣄⠎⠲⠃⠈⠁⠀⠀⣀⠓⠂⠉⠒⢤⣠⠊⢀⡀⠴⣒⠡⠶⠵⢍⠀⢎⢹⣂⠇⠀⣈⡆
⠀⠀⠀⠀⠀⠀⠀⡰⠒⠃⡄⠀⠀⠀⠀⠀⠀⠤⠏⠁⣨⡤⠒⠂⣏⢙⠤⣠⠊⢠⠂⡏⠌⡲⠑⣤⣈⡤⠖⠊⠁⡇
⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠐⠀⢀⣀⣶⣲⠃⠀⠀⡇⢀⣀⣰⣾⠄⠑⢩⠛⠲⣝⣓⣨⠴⠚⠉⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠉⠳⢤⣀⡔⡉⠀⠀⠀⣱⠼⠾⣔⠋⠀⠀⠈⢣⣀⣠⠷⠒⠋⠁⠀⠀⠀⠀⠀⠀⣀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠈⠙⠧⣄⣀⡞⠠⠀⠀⠈⣧⣠⠴⠒⠉⠁⠀⠀⠀⠈⠁⠀⢀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠀⠀⠀⠈⠉⢛⣒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣈⣁⣀⡤⠴⠚⠉⠁⠀⠀⠀⣀⡴⡴⠖⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠊⠀⠀⠀⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢆⣛⣛⡰⢦⡄⣀⣀⣀⡀⠀⢀⠈⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣿⢻⣿⣿⣿⣽⠿⢿⣍⠪⣿⣯⣅⣿⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣬⠻⣾⡿⠋⣩⣰⣔⣲⣆⠦⢜⢿⣿⣿⣻⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢷⡀⢀⣅⣿⡿⣋⣭⣭⣭⣤⣥⣭⣭⣭⡁⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡤⣑⠘⠟⣫⡾⠿⠿⠿⠟⠉⠉⠉⠹⢿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣌⣿⢰⣶⣶⣒⣒⣒⣒⣲⣶⣶⣶⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⢸⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⢀⠤⠔⠒⠒⠢⢄⡀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⣀⠄⠒⠋⠁⠀⠀⠀⠀⠀⠀⠈⠢⢄⡀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⡀⠀⠀⠑⠢⢄
⠀⠀⠀⠀⣀⠤⠒⠁⠀⠀⢀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠔⠒⠢⠤⠋⠙⣦⣀
⠀⡠⠔⠋⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⡄⠀⠀⠀⠊⠁⠑⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣀⠜⡠⠔⠒⠒⠠⠄⣶⠤⣑⠂⣄
⠸⣀⠀⠀⢠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠃⠀⠨⠀⠀⠀⡠⡔⠋⠀⠀⠂⠈⡹⠯⠶⡄⡀
⠀⠈⠑⢢⢄⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⢀⠤⠂⢡⠁⠀⡄⠀⠂⢇⡔⠉⠀⠀⠀⢋⠤⠒⠉⠠⠄⠀⢹⣌⡑⠢⣄
⠀⠀⠀⢸⠀⠉⠒⠤⡈⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡄⠤⠀⠀⠘⠢⡀⠀⠄⢀⣠⣦⡲⠀⡀⠂⠁⠀⠈⢀⡀⠄⠊⠁⣠⢹⠀⠀⠉⠂⢄
⠀⠀⠀⠈⠢⢄⡀⠀⠀⠑⠢⢌⠑⠢⢀⠀⠀⠀⠀⠀⢀⠤⠒⡋⠀⣰⠥⠴⢦⣤⣌⡂⢁⠈⠢⢄⢩⣴⡶⢶⠵⣄⠑⢀⣞⡉⠀⠂⠀⠀⠁⠀⡇⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠉⠒⠄⡉⠒⠀⠒⠊⠀⡰⠧⢭⣚⡥⢤⣥⣇⣀⣼⠚⠉⠉⠂⠆⠀⠀⢰⠁⠀⠀⠙⠉⠁⠙⢢⣀⣀⠀⠀⣀⢇⡠⢶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⠀⠀⠈⠑⠠⢀⠀⠀⠑⢤⢾⡁⠀⠀⠀⠀⠉⠛⢦⠀⢀⠀⢠⣄⠠⡀⢀⠠⠴⡁⠀⠀⠀⠈⠂⠈⡉⠄⠊⠁⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠉⠒⢄⡀⠉⠃⠀⠀⠀⠀⢀⡀⠜⣀⣀⢠⣌⡨⠏⣘⠁⡀⢠⠈⠁⠀⢀⡠⠒⠉⠀⠀⡀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠈⠑⠠⣀⠀⠀⠀⠁⠀⡎⠀⢀⠀⠀⢙⡄⠃⠀⠓⠁⣀⠤⠊⠁⠀⢀⡠⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠁⠢⢄⡀⠀⠃⠐⠀⠀⠀⠈⠃⢀⡠⠔⠈⠀⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠀⠈⠐⠤⣀⠀⠀⣀⠤⠂⠁⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠈⠁⠀⠀⣀⠤⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠠⠔⠊

⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣤⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⢠⣄⠀⠀⠙⢆⠀⠀⠁⠀⠀⠀⠀⠀⢠⢄⡀⠀⠀⢀⠎⠙⢦⡀⢀⠞⠁⠹⡀⠀⣠⠔⠋⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⣾⠗
⠀⠀⣄⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠓⠦⠌⠀⠀⠀⠙⠃⠀⠀⠀⠙⠊⠀⠀⠀⠀⡇⢀⡠⠖⠋⡇⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⣠⡾⠃
⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡖⠠⠤⢤⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⢀⣠⡀
⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠠⣇⡠⠔⠒⢻⠀⠀⠀⠀⠀⠀⠠⠖⠛⠉
⠀⠛⠻⠦⡀⠀⠀⠀⠀⢸⠀⠒⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣟⡻⡍⠧⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⣫⠵⣻⡟⣻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠤⠤⠤⣴⠀⠀⠀⠀⠁
⠀⠉⠉⠀⠀⢀⡤⠔⠒⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⢖⡫⢖⢉⡠⢟⢽⢧⢟⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠚⠉⣁⡴⢚⡭⠞⠛⣋⠝⣫⡞⢿⡾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⡀⡤⠤⠐⣫⠴⣊⡵⠚⠁⢀⡠⢞⡵⠋⠀⠹⡾⣵⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⡦
⠀⠀⠀⣠⠄⠒⠊⠁⠀⠀⠀⠀⠀⢸⣇⣠⠖⣋⠵⠊⠁⠀⣀⠴⣫⠞⠁⠀⠀⠀⠀⠠⡙⣆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⢀⣠⠞⣫⠴⠋⠁⠀⢀⣠⢞⡵⠊⠁⠀⠀⠀⠀⣀⣤⠴⢷⡜⣿⡆⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⢄⡀
⠀⠀⠀⠀⠀⠀⢉⠆⠀⣠⠖⣋⡴⣾⠁⠀⠀⢀⡴⣋⠴⠋⠀⢀⣠⡤⠶⠚⠋⠁⠀⠀⠸⣗⠃⠧⠀⠀⠉⠛⠲⠦⣤⣀⣀⣠⠴⠒⢦⠀⠀⠀⠀⠈⢓⡶
⠀⠀⠀⢀⡠⠚⠁⠰⣯⡤⠞⠁⠀⡏⠀⣠⠞⣡⣞⡥⠴⠒⠋⠉⠀⠀⠀⠀⠀⢀⠀⣤⢾⡿⠤⣤⡀⠀⠀⠀⢀⡠⠔⠋⠉⠀⠀⠀⠘⣇⠀⢀⣠⠖⠋
⠀⠀⠀⠉⠒⠢⢄⣀⠀⠀⠀⠀⢀⡷⢊⡵⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⢵⣺⣧⣯⠐⢯⠆⢁⣠⠜⠛⢉⡠⠶⠋⠀⠀⠀⠀⠀⣈⡷⠫⣄
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢀⡴⢋⣴⠋⠑⠤⣀⠀⠀⢀⡀⣀⡀⠠⢤⠞⡕⣐⠆⢽⣨⢟⢾⡰⠞⠉⠀⠀⠈⠉⠀⠀⠀⠀⢀⣠⡴⠚⢩⠁⠀⠈⠑⣄
⠀⠀⠀⠀⠀⡰⠃⠀⠀⢴⣋⡴⠋⠘⡆⠀⠀⠈⠑⠶⣭⠊⣉⠈⠉⡛⢛⠊⠡⣦⠘⢛⡐⡾⠁⠀⠀⠀⠀⠀⠀⣀⣤⡴⠚⠉⠀⠀⢀⡎⠀⣀⡤⠒⠉
⠀⠀⠀⠀⠘⠐⠀⠤⠄⡀⠉⠀⠀⠀⣳⠀⠀⠀⠀⠀⠀⠙⠳⢤⣃⣀⠦⠀⢴⣚⡌⢩⡾⠁⠀⠀⢀⣠⡤⠖⠛⠉⠙⢦⡀⠀⠀⠀⡼⠀⠛⣅
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣾⠟⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠮⣧⣀⢴⣟⣁⣤⠴⠚⠋⠁⠀⠀⣀⣴⣄⡼⠁⠀⠀⢰⠃⠀⢀⣈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠤⠤⢤⠀⢿⠦⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠉⠁⠀⠀⠀⠀⢀⣠⡾⠟⢩⠟⠁⠀⠀⢀⡏⢸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠑⠿⣣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣴⠿⠋⠀⣰⠋⠀⠀⠀⠀⣼⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣀⣤⠀⢿⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⡺⠟⠁⠀⠀⣼⠛⣆⠀⠀⠀⢰⠗⠒⠒⠛⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢸⡆⠀⠈⠛⢷⢤⡀⠀⠀⠀⠀⠀⠀⣧⣶⠟⠋⠀⠀⠀⢀⡾⠁⠀⠘⣆⠀⢀⡏
⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣧⠀⠀⠀⠀⠉⠻⢷⣦⣀⣠⣴⡾⡏⠁⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⢹⡄⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⣦
⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⢹⡄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⡇⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⣹⠇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠛⠁⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡼⠁⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢶⣄
⠀⠀⠀⠀⢠⣶⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⠞⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠉
⠀⠀⠀⠀⠈⠁⠀⣠⡾⠃⠀⠀⣰⠂⠀⢀⡀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⣿⠀⣰⠋⠀⣀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⣦⡀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠼⠟⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⢿⡔⣡⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠛⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠄⠀⠀⠀⠀⠀⢀⢠⣴⣶⣶⣶⢶⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣒⣒⣒⢒⠒⣒⣒⣒⣒⣒⣲⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠠⠤⠴⠒⠒⣚⢒⢒⣐⣐⡇⠀⠠⠤⠤⠤⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠉⠉⠉⠉⠀⠐⠒⠒⠒⠒⠒⠂⠠⠬⠭⠭⠭⠭⠍⠉⣹⣷⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠐⠒⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣉⣛⣟⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣮⣦⣤⣀⣀⣀⡀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣺⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⣿⣯⡢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣲⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣾⣿⣿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠙⣿⣿⣿⣿⠟⢉⣉⣈⣹⡿⢃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢰⣿⣿⣿⣤⣿⡇⢺⡇⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⡄⠘⣿⣿⠁⢠⣾⣿⠿⢋⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⣿⣿⣿⠀⣿⡇⢻⡇⢸⡇⢸⢿⡧⢾⡏⠴⣦⣽⣧⠈⣶⣾⣿⡿⠁⢰⣿⣿⡄⠘⣿⠀⠸⣟⢁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠻⣿⠇⣿⡃⢸⡇⠘⡇⠀⠊⠁⢸⡟⠳⠖⢸⣷⠀⠿⢿⡿⢧⣰⠿⠟⠛⠻⠀⠘⡧⣀⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡘⢿⣿⠿⠋⠁⠂⠀⠐⠒⠁⠈⠻⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣠⠶⠋⣠⢞⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡴⣋⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⡿⠋⣠⡶⡫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⠉⣤⢞⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡘⠷⣄⡀⡀⠀⠀⠀⣀⡴⠞⢁⡴⢚⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣀⣉⣛⣛⣛⣋⣉⡤⠶⣋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣉⠉⠉⠉⠉⢁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⣄⣀⠒⠒⠒⠠⠤⡤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠤⢀⣀⣤⣬⣤⣤⡤⢄⡀⠀⠠⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠈⠀⠀⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠐⠀⠣⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠉⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠂⠀⠀⠀⠀⠀⠤⠄⠀⠈⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢐⡸⠀⢈⠀⠐⠂⠀⠀⠉⠠⠒⠠⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢠⠤⡀⡐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢡⠊⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⢀⠔⣉⢉⡂⠀⠀⠄⠀⢣⢢⡙⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢙⡒⠤⠄⢀⣈⣀⠀⠀⠀⠀⠀⢀⡀⠄⢃⢸⠀⠘⠄⣑⠠⢄⠠⠀⣄⡖⢨⠀⢱⠈⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠐⢒⠒⠂⠀⠀⠩⠐⠂⠀⢠⢸⠀⠀⢀⡀⢱⠀⡆⠡⡀⢀⠜⠀⠈⣤⡴⣠⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⢸⠈⠀⠀⠈⠀⠡⠔⠁⠀⠀⠀⠀⠀⠎⠀⠀⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣀⣀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠸⠆⠔⠀⠢⣠⡄⠀⠀⠐⠀⢄⡀⠠⢠⠚⠀⠧⠐⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⠀⠘⠂⡄
⠀⠀⠀⠀⠀⢀⠠⠀⠘⢆⠀⠁⢈⣁⠘⠂⠉⠀⠀⡃⠈⠁⠠⠂⠀⠠⠔⠓⢧⠀⠘⠀⠤⠈⡁⠀⠓⠚⠅⠀⣀⠀⠂⠀⠀⢀⣜⠀⠠⠒
⠀⠀⠀⠀⠀⠘⢀⣀⠐⠃⠌⠉⢉⠈⠁⡴⠀⠀⠁⠀⠄⢦⠀⣤⠘⡆⠀⠘⠋⡵⠀⠴⠂⠄⠑⠀⢠⡤⠤⢪⠀⠐⠙⠒⠂⠃⠘⠂⡤⠔⡁
⠀⠀⠀⠀⠀⠀⠂⠀⠁⠄⣀⠀⠀⠀⠈⡀⠀⠀⠳⠄⠀⠖⠀⠠⠈⠁⡐⠳⡀⠁⡀⠂⠄⠠⠂⠀⠀⡀⠐⠀⠳⠀⠐⠁⠄⣈⡠⠉⠠⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠂⠀⠀⠀⠁⠀⠐⠁⠑⠤⠀⠐⠁⠀⠀⠀⠀⠀⠐⠁⠀⠄
⣶⣶⣿⣶⣶⡷⢾⣿⣿⣿⣿⣶⣶⣶⣾⣾⣾⣷⣾⣾⣷⣷⣿⣿⣾⣿⣶⣷⣾⣶⣾⣷⣾⣶⣿⣿⣾⣷⣾⣾⣿⣾⣷⣾⣿⣷⣾⠷⠶⠶⠿⠶⣶⡶⠶⠶⠶⠾⢶

⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢳⡉⠒⠢⡤⢄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠊⠀⠀⠀⠉⠈⠙⠔⡢⠜⠛⠛⠢⣒⠢⢤⡴⠂⠈⠁⠒⢞⠍⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣠⡀⠀⠠⠀⢠⠇⠐⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⡸⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢍⣀⣀⣀⠀⠄⠀⣀⠀⠔⠁⣈⣂⢀⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠒⠚⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠲⡀
⠀⠀⠀⠀⢀⡴⠒⠋⠉⠒⡄⠀⠀⠀⠘⡍⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⡏⠀⠀⠀⠀⢂⠇⠀⠀⠀⠀⠹⡄⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣨⠴⠃
⠀⠀⠀⠀⠑⠦⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠘⣄⠀⠌⠉⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⡄⠀⠀⠈⠳⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠶⠶⠶⠤⠤⣤⣤
⠀⠀⠀⠀⡠⠔⠤⢄⡀⠀⠀⢸⠀⡀⠀⢰⠁⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣠⠞
⠀⠀⠀⢸⠀⠀⠀⢀⠙⡆⠀⠈⠢⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢤⣀⣀⡀⣀⣀⣀⡀⠀⢀⣀⡠⠴⠊⠁
⠀⠀⠀⠀⠳⣄⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⣀⡀⢀⠀⣀⠀⣤⣀⠀⢀⣀⣤⡀⠀⣀⠀⠀⣄⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠋⢻⠸⣇⡟⠀⣿⢸⡇⠸⡏⠀⠀⢰⣿⠀⢸⣿⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⢸⠂⢿⡇⠀⣿⣼⡇⠀⣷⠶⠀⢸⢻⡇⢸⡿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣿⠀⣸⠀⢸⡇⠀⢹⠈⣧⠀⣿⠀⠀⣿⠀⣇⢸⡇⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢿⣄⡿⠀⢸⡇⠀⢸⣀⡟⠀⣿⣀⢀⡟⠛⣿⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠉⠀⠀⠈⠁⠀⠈⠉⠀⠀⠉⠉⠈⠁⠀⠘⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠴⠶⠦⣤⣦⣤⣤⣤⣀⣠⣿⣟⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⣯⣟⡻⢫⢖⠭⢭⡭⠾⢯⣶⠶⣲⡬⠿⢟⣿⣿⡁
⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⢻⢏⡴⢯⣭⠶⢿⡇⡏⡇⡯⣑⠾⣭⣥⣬⢍⢢⣹⣿⣵⢮⡳⡟⠛⠋⡿
⠀⠀⠀⠀⠀⠀⠈⢧⠀⢰⡎⡼⠿⣵⣪⡝⢿⡿⢧⣷⣷⠮⠿⣵⠖⠚⠋⠉⠉⠀⠀⠀⠹⡜⡆⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣸⢱⠃⠀⠀⠀⢉⡏⡔⡡⠚⠶⣭⡟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢱⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⣏⡀⠀⠀⠀⢸⢸⢱⠁⠀⠀⢸⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⡏⣇
⠀⠀⠀⠀⠀⠀⠀⢰⣧⣾⠈⠉⠙⠒⠲⡎⠸⢸⣀⠀⠀⠘⣿⢀⣀⣠⠤⠴⠖⠒⠋⠉⠁⠀⣷⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢠⢇⡟⠋⠈⠉⠁⠀⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠤⣄⣀⡀⠀⠀⠀⠀⡇⠀⠀⢀⣠⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢤⣧⠴⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⠀⠀⠀⠠⠀⠂⢄⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⢀⠠⠂⠁⠀⢀⡠⠖⠃⠂⢄⠀⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⡀⡀⣤⣦⣀⠀⡤⠒⠁⠀⠀⢀⠠⠚⠁⠂⠄⠀⠀⠀⠈⠓⠦⣀
⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠀⠀⠀⡄⣤⣤⢤⣀⠈⢁⡬⡝⣦⣤⣤⡀⠂⠁⠀⠀⢀⠠⠊⠉⠑⠀⡀⠀⠀⠀⠙⠢⢄⡀
⠀⠀⠀⢠⠔⠋⠁⠀⠀⢀⣤⣤⣤⣀⠈⠁⡬⡜⢶⣶⢧⣅⠈⠁⡀⡥⡤⢄⡀⠊⠀⠀⠀⢀⣠⡞⠉⣹⡤⠠⡀⠀⠀⠉⠓⢤
⠀⠀⠀⢸⠀⠀⠀⠀⠰⠋⠉⠛⢉⣽⣧⣄⣡⡄⠀⠁⠀⢈⢀⣺⠉⠀⠀⠀⠈⡆⠀⣠⡞⠅⣹⠭⢾⠿⢀⣴⠷⠉⠁⠀⠀⠸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠁⠉⠋⠁⡌⠗⠚⠲⢤⠋⠀⠈⢱⠒⠀⠨⠤⣕⢤⣌⣩⣞⣧⢀⣼⠗⠋⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠂⠀⠀⢀⡀⠇⣀⡄⠘⠠⢀⠀⠀⢢⠤⠙⢩⡄⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠆⠀⠈⠉⠁⡠⢒⣂⠳⢜⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠁⠉⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠠⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣭⣭⣭⣭⣭⣍⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢿⣿⣿⣿⣿⣿⠟⣫⣥⣶⣶⣮⣍⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⣭⣭⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⠂⢀⠀⣴⠟⠉⠉⠉⠉⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡶⣶⢶⠶⣶⡶⣶⣶⣶⣶⡶⢸⣿⣿⣿⣿⣇⣿⣿⠆⠹⠷⠟⠀⠀⠀⢀⣠⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⡖⣶⣶⣶⣶⣶⣶⢆⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⣠⠾⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⡇⠿⣫⣭⣭⣭⡛⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⢱⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⣠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢃⣛⣋⣭⣴⠶⠒⠶⣬⣝⣛⣫⣭⠶⠖⠶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡌⢿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣾⣿⣿⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠀⠀⠙⠻⠿⢿⣿⠿⠟⠛⠛⠛⠻⠿⣿⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠟⠋⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⢋⣁⣤⡴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⣉⣥⡴⠶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠞⣛⣩⣥⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⡶⣶⡶⣶⣾⣿⡯⠶⠶⠿⠿⠷⠶⠶⠶⠶⠶⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⡇⣿⠀⣿⠶⠶⠶⠶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⠿⠟⠛⢹⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⢸⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣄⣿⡴⠶⠶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⠶⠶⣾⣷⠟⠉⢀⣀⣠⣤⣄⣀⠈⠙⠳⣤⣤⠶⠶⠶⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⣤⠶⠚⠓⢶⡟⠁⢠⡶⠛⠉⢁⣀⡈⠉⠻⣦⡀⠘⢷⡶⠞⠛⠶⣦⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢠⡟⣡⡾⠛⢻⣿⠀⣰⡟⠀⣰⡞⠋⠉⠙⢷⣄⠈⣿⡄⠈⣿⡛⠛⢷⣌⢻⣄⣻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠛⠙⠋⠛⠛⠛⠛⠛⠋⠛⠛⠙⠛⠋⠛⠛⠛⠙⠛⠙⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢹⣿⣆
⠀⠀⠀⠀⠀⠀⢀⡾⠁⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⢻⣆
⠀⠀⠀⠀⠀⢀⡾⠁⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠹⣆
⠀⠀⠀⠀⢀⣾⠁⠀⠀⠈⣿⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⢻⣦
⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢹⡎⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢁⣾⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠈⣷⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡍⢠⡅⠀⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⣠⣴⣿⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣿⡿⠋⢻⣥⢻⣷⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣾⡾⢰⡌⣿⣿⣿⣯⡀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢻⣟⣿⡟⠋⠘⣧⣾⣿⠛⡟⢸⣏⠙⠑⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣧⣻⣿⠏⠀⠀⠀⢉⣻⣟⠚⠃⡎⣿⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠋⡿⣰⠏⠉⠀⠀⠀⠀⢸⣻⡇⢀⡜⠀⢸⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⡋⠡⢀⣤⣿⡿⠀⠀⠀⠀⠀⠀⣰⣿⠁⡞⠀⠀⣸⣇⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⡡⠟⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⢀⣾⡿⠃⠐⠀⠀⠀⣹⢻⡄⢌⠲⢤⡀⠀⠀⠀⠀⠀⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢡⠈⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠐⣾⣿⡟⠀⠀⠆⠀⠀⠐⡿⠀⢳⣦⡀⠀⠉⠐⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⢖⣟⣿⡄⢺⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠈⠀⠀⠀⢀⠃⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠉⠁⠀⠀⠑⡝⠳⣲⣾⠁⠀⠀⠀⠀⠀⠀⡠⢡⣿⠇⠀⢠⠀⠀⠀⠀⡞⠀⠀⠀⣠⠀⠀⠤⠄⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⢤⠶⠉⣯⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⠁⠁⠀⠀⠀⠀⠀⠀⠈⣰⡿⠋⠀⠀⠀⠳⢠⠀⠈⠆⠀⠀⠀⠘⡗⣆⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠱⣄⠘⢧⣄⣀⣤⣤⠶⣶⡿⠋⠁⠀⠀⠀⠀⠀⠠⣶⣾⠟⠉⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠻⠿⠿⢿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⣇⠀⠀⣿⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢯⣅⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣈⢀⡀⣙⡟⡇⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣦⣤⣤⣤⣤⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠠⢐⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣻⣛⣛⣻⣫⣛⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⣭⣿⣯⣿⣭⣭⣯⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡠⠉⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡠⠁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠐⠒⠒⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠠⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣈⡀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣔⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣈⣄⡀⠀⠀⠀⡰⢖⠯⡐⠀⡏⠈⠉⠒⢠⠀⢠⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⣉⢻⣙⠫⠯⠝⡿⣿⣻⠟⠢⡑⠆⠀⣇⠀⠀⠀⠀⡰⠁⠀⠉⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡀⡐⠋⢨⢁⠄⢳⡈⣉⣽⠟⢡⣿⣽⣦⡠⢨⡇⢰⠀⠁⡸⠀⡔⠁⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠷⢆⠀⠠⡞⢉⣲⣶⡽⢁⠌⠀⠀⠁⢀⠴⣿⣖⢛⠻⣟⣾⡆⣸⣄⣴⠃⡜⡀⠀⢀⡔⠁
⠀⠀⠀⠀⠰⡈⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠤⡷⢾⣿⣀⣛⣷⡾⢄⡞⡋⢨⡈⠓⡈⣤⣙⢯⡇⣿⢸⢃⣎⣤⡩⠷⠻⠒⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢍⣩⣿⠀⣼⣃⢷⡈⣑⡿⠿⢿⣛⣿⡧⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⡟⣐⣰⡧⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠐⠈⢹⠛⠛⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣶⠤⠖⠒⠒⠉⡰⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠑⠢⢤⣀⣀⣀⣀⣠⣴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⠘⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠃⠀⠘⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠘⣆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠘⢆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠈⢧⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⢳⣸⡇⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⣧⠤⠖⠋⠁

⠀⠀⠀⠀⠤⡤⢀⡀⠀⠀⠴⠶⢤⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠰⢤⡤⣀⣀⢀⠃⠒⠒⢦⢈⠗⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠘⢩⠏⡧⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠠⣵⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠃⠀⠀⣠⠞⠀⠑⢬⡂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠆⠀⡹⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⡀⠉⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⣴⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⢀⣿⠗⠉⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⡄⣀⠀⠀⣠⠴⢥⣌⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸
⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣆⣀⡀⣸⠵⠓⠲⣄⣹⢾⠈⠉⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡿⠃⠀⠀⠀⡽⣽⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠨⢤⡍⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀⣀⠀⠠⠀⠂⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⣷⣿⣧⠀⣀⠀⢷⠃⠀⠀⠀⠀⡀⠄⠀⠉⡁⠀⢃⣬⡁⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠉⠁⠀⠉⠀⠁⠀⠀⠀⢀⠈⠀⡄⠀⠀⢰⣦⡼⡿⠛⠉⠀⢄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢰⣶⣴⠿⠛⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠒⠠⠃⠒⠦⠿⠿⠿⠿⠛⢷⣝⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠛⣉⣦⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣀⣤⠶⠛⠉⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⣁⣠⡴⠞⠋⠉⠀⠀⢀⣀⣤⠶⡞⠋⢉⣹⠆
⠀⠀⠀⠀⠀⠀⢰⡶⢤⣤⣤⣄⠀⠀⣀⣤⠶⠛⣁⡤⠶⠛⠉⠀⠀⢀⣀⣤⠶⠚⠋⣁⣸⡤⠷⠛⠋⠁
⠀⠀⠀⢠⡾⢋⢻⢥⣂⣰⡄⢸⣷⣛⣩⡴⠖⠋⠉⠀⣀⣠⡤⠶⠛⠋⣁⣤⡴⠞⠋⠉
⠀⠀⣰⠏⣤⡍⢊⠀⣉⡙⠙⡄⢸⡏⠁⢀⣠⡤⠶⠛⢉⣡⣤⠶⠚⠋⠉
⠀⠈⢻⡆⢔⣩⠖⢢⡄⠰⣌⣡⡼⠷⠚⢋⣡⣤⠶⠚⠋⠁
⠀⠀⠈⣷⠰⢍⣡⡼⠶⠛⣉⣡⡴⠶⠛⠉⠁
⠀⠀⠀⠈⢻⣏⣥⣶⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠁⠀⠀⠠⡀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠁
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢
⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠐⠂⠈⠁⠈⡏⠛⠒⠤⢀⣀
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠴⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠉⠉⠒⠢⠄⡀
⠀⠀⠀⠀⢀⣙⠦⣤⣖⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⢀⠠⠐⠀⠉⠀⠀⠀⠉⠑⠢⢄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠠⢄⣀⠀⣀⠔⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠛⠁⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⣄⠀⠀⠀⠀⠀⠈⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠠⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠁⠒
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠧⣄⡀⡀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣼⣷⡄⣀⡀⢀⣶⠶⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣿⣟⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠉⡬⠈⡟⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠋⠓⠦⢤⣄⣀⣀⣀⠀⣀⡤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠓⠤⠤⠤⠤⠔⠒⠒⠒⠋⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⡀⠀⠀⠀⠀⡷⠀⠀⠀⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⣃⠠⠔⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠈⠀⠀⠀⠀⠀⠀⡒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣷⣴⡁⠔⠐⠈⠑⢤⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⠀⢀⣼⢳⣒⡭⢛⡷⢺⣿⡟⢸⢿⠃⠀⠀⠀⠠⠐⠁⠀⠂⢀⡠⠄⠤⡀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠀⠀⠀⠀⠀⡴⣽⣺⣎⢻⣭⠏⠑⠤⣠⣇⠿⠿⠶⠤⠤⣀⠁⠀⡀⣄⡤⠊⠀⠀⠀⠐⡀⢠⠀⠈⡀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠐⠐⠉⠁⠀⠈⠓⠳⣽⡗⠣⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⣈⣧⡖⠉⠠⠁⠀⠀⡀⠐⠂⣁⡀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⡠⠒⠁⡀⠐⠀⠂⠀⢀⣄⢠⠬⡁⠂⢇⣀⠀⠀⠀⠀⠀⠠⣠⠊⣠⣤⣤⣈⠃⠤⡆⠁⠀⠀⠀⠀⠙⠀⠀⠇
⠀⠀⠀⠁⠀⠀⡄⠁⠀⠀⠈⠁⣾⠆⠀⢧⠘⠚⣀⠇⠀⢀⣀⠉⠂⢀⡀⠀⠀⠃⠀⠈⠀⠀⠉⠀⠀⠈⡄
⠀⠀⠀⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠓⠊⠀⠀⠀⠈⠂⠰⣆⠀⠀⠀⠀⠀⠶⣛⠀⢠
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠂⡀⠀⠸⡱⠀⠀⠀⢀⠀⠀⠀⡤⠒⠒⠦⡀⠔⢢⠀⠀⠈⠢⣐⠲⠶⠾⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠣⠞⡵⠀⡸⠈⠁⠀⠀⠀⠀⠀⠙⠈⣁⠄⠂⠁⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⠠⠴⠀⢉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊
⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉
⠀⠀⠀⠀⠀⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢷⣄⠀⠀⠀⢀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣡⠊⢻⡛⡆⣊⠀⡀⠀⣀⠔⠒⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡷⠁⠀⢀⢇⠴⠀⠉⠑⠊⠠⣤⡀⢀⢤⠽⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠛⢰⡇⠀⡠⠃⢸⣀⠀⠀⠈⠑⠢⣄⠙⡓⠦⣄⣈⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡇⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠂⢰⠁⢀⠌⢧⠉⠐⠂⠤⠠⣀⢣⡑⣀⣈⣯⠵⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠙⢧⠀⠀⠀⠀⠀⣰⣁⠤⠂⠒⠢⡀⠀⡏⢰⠃⠀⠘⠆⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⣂⠀⠀⠀⡐⢹⠋⠀⠀⠀⠀⣦⡼⡱⠇⠀⠀⠀⣠⡄⣀⠀⠠⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠂⠀⠀⠀⠀⢹⠀⠀⢠⡷⠇⢬⠀⠀⠀⢸⠇⠀⢻⠂⠀⠀⠰⠁⠸⡀⠈⠁⠂⠌⡄
⠀⠀⣀⠀⠀⠀⠀⡴⠿⠿⢳⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⡸⢿⠀⢀⠀⠀⣠⡟⠀⡅⠈⢧⠀⢠⠃⠀⠀⢩⡀⠀⠀⢀⡇
⡰⠛⠋⠻⡄⠀⢸⡆⠀⠀⢸⠁⠀⢰⠃⠀⠀⠀⠀⠀⠀⠠⣾⠀⠀⢑⠈⠿⢷⣔⡤⠏⠀⢀⠃⠀⠀⠣⣈⠀⠀⠀⢀⢯⠑⠮⢨⠀⠀⠀⠀⢠⢤
⣯⠀⠀⢠⣷⠀⠸⡤⠀⡀⡿⠀⠀⢸⠃⠀⠀⠀⠀⠀⡀⢠⠆⠀⠀⠘⡆⠀⠸⠀⠀⠀⠀⠨⠀⠀⠀⠀⠈⠑⠢⢄⡘⠀⠡⡀⡌⠀⠀⠀⠀⡞⡒
⠘⢶⣤⣵⡼⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠄⣠⠀⠠⠊⢷⡟⠀⠀⠀⠀⡁⢈⡠⠖⠉⠉⠁⢀⢣⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠳⠀⠀⠀⠀⠀⡄⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠄⠀⠀⡈⠀⠸⠃⠀⠀⠀⢸⡁⡎⠣⠀⠀⠀⠀⣴⠂⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠁⠄⡀
⠀⠀⠀⠀⠀⠀⡠⠒⠐⠶⣄⠀⢠⡏⠈⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢸⠩⠃⠀⠀⠀⠀⠀⢘⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⠒⠶⠪⣤⡀
⠀⠀⠀⠀⠀⣼⠀⠀⠀⢠⠇⠀⢰⠴⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠘⡇⣇⡀⢀⠀⠀⠀⣞⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣹⠋⠀⠀⠀⡦⡌
⠀⠀⠀⠀⠀⢻⣦⣠⣿⠎⠀⠀⢸⠃⠀⠀⠀⠀⠀⢐⣿⠁⠀⠀⠀⠀⠱⠹⣾⡤⠃⠀⣲⠇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣠⠰⡄⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⣴⣇⠀⠀⢠⢀⠋⣽⠀⠀⠀⠀⠀⠀⠱⠌⠓⠁⠐⠊⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣖⡸⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣉⠴⠄⢀⡠⠤⣸⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⢀⣖⠉⠉⠁⡤⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⢙⡶⢘⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⢠⠈⠈⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⣇⢀⣎⠀⠀⠀⠀⠠⡆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢃⡔⠋⠀⠉⠛⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣖⠂⠂⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢻⢸⠇⠀⠄⠀⠀⢘⡇⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡿⠀⠀⠠⠄⢀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠠⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⣟⣇⠇⠀⠀⠄⢲⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣠⣯⠀⠈⢠⣠⡼⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡅⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢧⠀⣽⠀⡠⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠁⠙⠺⠯⠬⠛⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠠⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠀⠊⠀⠀⢀⡇⠀⠀⠀⠀⠀⢠⠞⢠⡼⠟⠳⠲⡄⠀⢐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢤⣀⡼⠀⠀⠀⠀⠀⠀⠟⣰⠋⠀⠀⠀⠀⡿⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠜⠂⣻⡆⠠⠀⢀⡾⠁⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⠁⣁⣀⠻⢵⠠⡡⠞⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⡽⡿⠉⠈⠉⠉⣷⡀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠁⠀⢟⡃⠄⠀⠀⣮⡝⠁⢦⠞⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣚⣥⣤⣥⣤⣖⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⡰⠁⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⠜⠀⠀⠀⢀⠎⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠁⢠⠊⠀⠀⠀⡠⠃⠀⠀⠀⢨⠓⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠑⠧⡀⠀⠀⡔⠁⠀⠀⠀⡰⠁⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⣁⣀⣀⣈⠦⣌⠀⠀⠀⢀⠜⡀⢐⠈⠀⡠⠄⠀⡜⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⡇⠀⠀⠀⠀⠀⠀⠑⠢⣠⠊⠀⠉⠀⠀⢄⠊⢀⠊⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡍⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠸⠁⡠⠃⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⣶⠁⠀⠀⠉⠉⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣔⠁⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠎
⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣠⡤⠖⣚⣋⣉⣉⣉⣁⣀⣀⣤⣤⣬⣭⠭⠭⠭⠭⠥⠤⠤⠤⠭⠭⠭⠭⠭⠽⣷⣶⣤⣀⡀
⠀⠐⣏⠙⢛⣻⣿⣿⡿⠿⠿⠿⠿⠿⠭⣭⣭⣭⣭⣉⣉⣉⣉⣉⣉⣭⣭⣭⣭⣭⠥⠤⠤⠤⠤⣤⣬⡉⢧
⠀⠀⠙⣆⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠟
⠀⠀⠀⠈⠲⠿⠶⠒⠒⠒⠶⠚⣶⣄⠒⢒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣲⣶⡶⢄⡤⠶⢶⠻⠋
⠀⠀⠀⠀⢀⠀⣨⠿⢟⣿⠿⠷⢿⡿⠯⣭⣭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢭⣭⣭⣭⣭⠝⠋⠀⠂⢸
⠀⠀⠀⠀⡾⡜⢡⠖⠉⠀⠀⠀⢸⣷⠃⣀⡬⠓⢦⠀⠀⠀⠀⠀⣠⠔⢋⣩⠽⠋⠁⢠⣤⢀⠀⢸⡷⣄
⠀⠀⠀⢰⢇⣷⠃⠀⠠⠴⠶⣦⢼⠿⠋⠀⠀⠀⠈⢳⡀⠀⠀⣴⣧⠴⠋⠁⠀⠀⠀⠈⠁⠘⠃⠘⡇⠀⠙⢆
⠀⠀⠀⣸⢸⡇⠀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠴⠚⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣸⠃
⠀⠀⠀⡏⢸⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠚⠂⠀⠀⢸⣠⠾⠃
⠀⠀⠀⡇⣿⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢥⡤⣴⡄⠀⠀⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⢴⠦⢼⣿⡆
⠀⠀⢸⡇⡏⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣹⣍⡉⠙⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⣠⣿⣤⣼⢸⠁
⠀⠀⠀⢧⢧⡀⠀⠱⣄⠀⠀⠀⠀⠀⢀⣤⣾⣿⣯⣿⠯⣙⠓⠒⠚⠛⠛⣉⣭⣿⣟⣛⠻⢧⣤⣄⣸⣿
⠀⠀⠀⠈⢷⠬⣝⣛⣛⣓⣒⣒⣒⣒⡿⣺⣥⣄⡏⠈⠳⠈⠉⠓⣖⣚⠉⢁⡼⣍⣛⣿⢻⡽⢯⣉⡏⡧
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠉⠉⠉⠉⢸⣿⠁⠘⢿⣲⣠⣶⣤⣴⣚⣁⣘⣞⣉⣀⣉⣩⣯⠬⠤⠚⡹
⠀⠀⠀⠀⠀⠈⠙⠶⢤⣤⣀⣀⣀⣘⣋⣀⣀⣀⣀⣉⣀⣀⡀⠉⠉⠉⠀⠀⣀⣀⣀⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠉⠁⠀⠀⠀⠀⠀⠈⠠⢅⣒⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠉⠘⠛⠽⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢶⣤
⠀⠉⠐⠒⠲⠀⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠉
⠀⠀⡄⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⡞⠉
⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠁⠀⠀⠈⢳⠀⠀⠀⠀⡌
⠀⡔⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠮⣵⣄⣀⠤⠔⠚⠉⠀⠀⠀⠀⢀⠔⠴⣠⠋⠀⠀⠀⢰
⠀⠈⠂⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⡴⠃⠀⠀⠀⠀⡏
⠀⠀⠀⠈⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠔⠈⠀⠀⢀⡞⠁⠀⠀⠀⠀⡸
⠀⠀⠀⠀⢹⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠊⠁⠀⠀⠀⢠⠞⠛⡀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⣇⠔⠉⠂⠀⠀⠀⠀⢠⠏⠀⠀⠱⡄⠀⢀⡏
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠑⠠⢀⡀⣀⡤⠚⡅⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢳⠀⡼
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠈⠀⡇⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⣳⠃
⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡰⠁⠀⠀⠀⠀⣠⠴⠋⠁⠐⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⣼⠀⠀⡴⠁⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠐⠉⠐⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⣿⢀⠜⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠝⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣿⣡⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠄⠀⠀⠀⠀⠀⠠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⡂⢀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⡿⠿⠿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣽⢹⣿⢻⢻⣿⣶⣶⣶⢶⢦⢤⢤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣾⣿⣼⣿⣿⣿⣟⣿⣿⣸⢸⢿⣿⣿⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⢳⡿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠶⢦⣤⣀⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣀⣭⠿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠶⠦⠤⣤⣤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⡤⠶⠛⠉⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⡴⠖⠓⢶⡄⠈⠉⣉⣉⡉⠙⠛⠛⠛⠒⠒⠒⠶⠶⢿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣾⠀⣞⠓⠶⠃⢀⡾⢉⣀⠙⢧⢰⠛⣦⠀⢠⠶⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠲⣌⠙⢦⡀⣼⠁⡟⢹⠀⣸⠘⣇⢸⣴⠋⣼⠃⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣶⠶⣬⠗⢀⡇⣿⠀⢧⠞⢀⡟⠀⢸⡆⢀⡞⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠶⠖⠋⠀⠘⢦⣀⣠⡾⠁⠀⢸⡃⣾⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⣀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⠈⠛⠃⠀⢀⣀⡀⢸⡇⢀⣀⢀⡴⣦⢸⡻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⡼⢹⡇⢰⠋⠀⣿⣰⠟⡇⢸⠋⣷⠀⢰⠟⡇⣼⢹⡇⢸⣧⢸⢹⣏⡇⢸⠈⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠁⣼⠀⣿⠀⠀⠈⠁⠸⣇⣼⠀⣿⠀⢸⠀⠙⠁⢼⡀⣸⠛⠾⠃⠀⠀⠸⠶⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣀⡿⠀⠀⠀⠀⠀⠉⠁⢰⡇⠀⣸⠀⠀⠀⠈⠙⢻⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣠⠟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠶⠶⠤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠲⠶⠦⠤⣤⣤⣀⣼⡴⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣤⣤⣤⣤⡤⠴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣤⣤⡴⣄⠐⠒⠒⠀⠦⢄⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠰⠟⢻⣿⣾⣿⣦⣼⣿⣿⠀⠀⠀⢸⣿⣿⣿⠟⠋⢉⣶⣿⡟
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡶⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠐⢶⣄⡀⠀⠀⠀⠀⠈⠉⠡⠾⢿⣿⡿⣿⠀⠀⣀⡼⠛⠛⠻⠚⠻⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠙⣯⠙⣿⣷⣿⣧⠀⣿⡿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣟⣁⣀⣠⣤⣴⣶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠙⣫⣿⣿⣦⣿⣄⣿⡇⠀⠀⠀⠀⠀⠀⠸⣻⣿⣷⣰⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠈⢷⡀⠻⣯⠻⢿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣭⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣌⠙⠿⠋⢠⣶⣶⣶⠖⠀⣼⠃⠀⠀⠀⠀⠀⣻⣦⣈⣷⣬⡿⠷⠾⠿⠟⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣹⠁⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣷⣀⡄⣿⣙⡿⠃⢀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠋⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⠀⠈⠉⠻⠿⠛⠉⢽⠀⢈⡀⠀⠞⠁
⠀⠀⠀⠀⢀⡴⢿⣿⣿⡃⢉⠀⠀⠀⢠⣠⠟⢻⣿⣿⣿⡟⠃⣤⣔⠀⠐⠀⠀⠘⢦⣽⣇⣤
⠀⠀⢰⣾⠋⣠⣿⣿⢛⣷⠋⠀⠀⠀⢹⣄⣀⡾⠋⣿⣿⣿⠶⣼⣿⣌⣿⣆⠀⠀⣘⠛⠉⠁
⠀⠀⢸⣿⣿⠟⣿⣿⣿⠃⠀⠀⠀⠀⠀⠻⣿⣇⣰⣿⣿⣿⣷⡌⣿⡆⢹⣿⠀⠀⣿⠃⠀⠀⠀⠀⣰⣿⣷⣶⣦⣤⣴⣶⡚⠛⠂
⠀⠀⠘⣿⡏⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⣻⣿⣯⣻⣿⡀⢠⣥⠚⠛⢿⣷⣶⣶⡆⢾⣿⠟⠀⣿⣿⠿⢿⣿⣿⣿⣦⣀⠀⠀⠀⢠⣶⢀⡀
⠀⠀⠀⠹⣿⣿⣿⣾⡇⠠⠶⠦⠀⠀⠤⠠⡄⠀⣦⡄⣻⣿⣿⣿⠟⠉⠿⣿⣿⣿⣟⣉⣤⣶⡤⠚⣟⠀⢠⣿⠃⠀⣾⣿⡿⣿⣿⡁⠉⠙⢇⠀⣟⠻⣿⣶⡄
⠀⠀⠀⠀⢿⣿⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣿⣿⣿⡉⠲⣾⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⣾⡇⠰⠏⠀⢸⣿⠋⠀⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠙⠛⠂
⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⢹⣿⣿⣿⣿⠃⢀⡿⢃⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠺⠋⠀⢠⡿⠛⣿⣿⡿⣆
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢻⣿⣿⠛⣶⠟⣠⣟⠀⢹⡇⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢸⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠛⣿⣿⠓⣾⠋⣸⣷⣿⠀⢸⣿⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣸⣿⢠⣅⣾⣿⡿⠋⠀⠘⣿⣇⠀⠠⢀⣶⣿⠖⠒⠢⢤⣀⠀⠀⠀⠀⠀⠐⠁⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠃⢼⣿⠏⠀⠀⠀⠀⢸⣿⣆⣴⣿⣿⣧⡀⠀⠄⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣾⣿⣿⢾⣿⣿⣿⣿⣥⣄⣤⣤⣾⡟⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣶⣶⣶⣿⣿⣿⣶⣤⠀⠀⠀⠀⢿⠀⠀⡠⠶⠷⢟⣿⣿⣾⣿⣻⣏⣹⡉⠛⣏⡙⢻⣟⡛⠻⠿⡿⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠋⢉⣩⣽⠟⠋⠁⠀⢀⠀⠐⠓⠦⢼⣧⠞⠀⠀⠀⣾⣿⣿⣿⣿⣟⡿⢿⣿⠂⠉⠀⠀⠈⠙⠶⠄⠶⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⣴⠀⠀⣰⣿⣤⣶⠀⠀⠀⢿⡀⠀⠀⢉⣿⠘⣿⣿⠙⢿⣷⣀⠻⡄
⠀⠀⠀⠀⠀⠀⠈⣁⣤⡿⠋⠀⠀⣰⡇⠀⢸⣿⣿⣿⡟⠀⠀⠀⢸⣇⢰⣿⣿⠇⠀⢸⡿⠱⣿⣿⢹⡇⢻⡀
⠀⠀⠀⠀⠀⠰⣿⣿⡟⠀⠀⢀⣶⣿⡷⠴⠿⠟⠛⠁⠀⠀⠀⠀⠈⣃⣸⣿⡟⠀⣷⣿⡇⠀⠀⣍⡿⣧⠈⣧
⠀⠀⠀⠀⠀⢠⢻⡟⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠏⠀⠀⠁⢸⡁⠀⠹⢿⠷⢾⣿⣿⡀
⠀⠀⠀⠀⠀⠸⡟⢀⡀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣾⠿⣿⠀⠀⠀⣷⢼⡇⠀⠀⠠⣰⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⡇⠈⣿⠀⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣾⣿⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⣿⣿⣿⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣴⡏⠀⠈⢻⣿⣿⠀⠀⠀⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⠟⠀⢰⣴⡿⣿⣿⡇⠈⠳⡆⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡿⠋⠀⠀⠈⠹⡇⠹⢿⣧⠀⠀⠉⢄⣬⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⠾⠋⠀⠀⠀⢰⣦⣾⠃⠀⠈⣿⡄⠀⠀⠀⠙⢤⣀⢘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⢻⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡴⠂⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⡆⠀⠀⠀⠀⠀⢠⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠸⡄⠀⠀⠀⢠⠋⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢱⠀⢀⡴⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠏⠀⠀⠀⠀⣼⣤⣊⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠗⠿⣿⣶⣒⣋⣉⣩⣝⣋⣙⣛⣿⣟⠓⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⡿⠗⢦⡀⠀⠀⠀⠀⠐⠒⠂⠬⠭⢗⣲⣤⣔⡚⠓⠶⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢋⣀⡤⠖⠋⠁⠀⠀⠀⠉⠷⣦⣄⣀⢀⣀⣀⡀⠀⠀⠀⠈⠉⠛⠳⢶⣬⣧⡈⠑⠢⣄⡀
⠀⠀⠠⣤⠤⠤⠤⠖⠚⠛⠋⠉⠀⠀⠀⠀⣀⣀⣀⣀⣴⠶⠟⠓⠪⢭⣙⡷⠒⠒⠒⠲⢴⡤⡀⠀⠀⠈⠉⠛⠷⣤⣠⣍⣢⢄
⠀⠀⣀⡬⠗⠦⣄⡀⠀⠀⠀⠀⠀⣀⣴⣴⡶⠗⣻⠋⠀⠀⠀⣀⡤⠤⠽⣄⠲⠿⣶⠄⠀⢹⡶⠒⠉⠉⠒⠦⡀⠀⠈⠑⠮⣷⡕⣄
⠺⠭⣥⣀⡀⠀⠀⠋⣻⠶⠒⠚⠛⠾⡋⠁⠀⠀⠹⣴⣤⠔⢉⡤⢤⣤⠀⠈⢳⡶⠖⠒⠒⠻⢧⡀⠀⠀⠀⠀⠘⣆⢀⠀⠀⠀⠙⢮⣧
⠀⠀⠀⠈⠉⠉⠋⠉⣇⣀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⢹⡠⠬⠭⢍⣀⢀⣴⡏⠀⠀⠀⠀⠀⠀⢹⣦⣤⠀⠀⣠⠟⠳⠮⢭⣄⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠭⣶⣦⠤⠞⠀⠀⠀⢀⠎⠁⠀⠀⠀⠀⠈⢫⠀⠱⢤⣉⡯⠽⠤⣴⡊⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠰⠯⠽⠃⣀⢄⡼⠀⡴⠉⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠾⠍⠒⠉⠀⠀⢧⡀⠀⣀⣤⣤⢀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣙⠀⡖⠉⠉⢲⠈⠻⣀⠟⠁⠘⣏⣉⣷⠀⢹⣉⡍⠃⠀⢨⠻⡄⠀⠈⡟⢦⡈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠽⠃⠳⠤⠤⠞⠀⠀⠿⠄⠀⠠⠧⠤⠼⠂⠼⠤⠥⠖⠠⠏⠉⠹⠤⠠⠧⠀⠙⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠐⠐⠀⠀⠀⠀⠂⠐⠀⠐⠀⠂⠀⠀⠀⠀⠀⠈⠐
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⢚⣂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠞⣛⣩⣥⠴⠶⠚⠛⠉⠉⣀⣠⣤⣤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⢛⣋⣩⣤⠶⠞⠛⣉⣁⣤⣤⣤⣶⡿⢿⣛⣻⣭⡽⠶⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡾⣡⢞⣫⣍⢹⡶⣤⣠⡤⠶⢞⣋⣩⣥⠶⠞⠛⣉⣉⣤⣴⡶⠿⢿⣛⣛⣭⡽⠷⠞⠛⠉⠉⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡀⠀⣀⣄⣼⠿⢿⣷⣿⣼⣿⣿⣾⢿⣯⣷⠖⠛⣉⣉⣤⣤⣶⠾⣟⣛⣭⡽⠶⠚⠛⠉⠉⠁⠀⠀⣀⣀⣤⣤⠶⠶⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠈⢠⣞⣭⣾⣿⣿⣷⣿⣿⡿⣯⡻⣏⢷⡹⡏⢿⣟⣿⣿⠿⠟⠛⠛⠉⠁⢀⣀⣀⣤⡤⠶⠶⠚⠛⠋⠉⠁
⠀⠀⠀⢰⣿⣿⡿⠿⢛⣋⣩⣿⣿⣿⣎⣷⢻⠈⣿⣷⢸⣏⣁⣀⣠⣤⠴⠶⠟⠛⠋⡉⠁
⠀⠀⠀⣼⣿⣵⣶⣿⣿⣿⡿⠷⣿⣿⡟⣿⢸⣼⣯⣇⣼⣿⠉⠁
⠀⠀⢰⣿⣿⣿⣛⣩⣥⣴⣶⣾⣿⡿⣹⢇⣾⣽⣿⣿⡟⢯⣷⡄
⠀⠀⢾⣼⠃⠉⠉⠘⣿⣷⣿⣿⣿⣹⢻⣿⡿⣿⣿⡟⣧⠘⠿⠃
⠀⠀⠈⠁⠀⠀⠀⠀⣿⣿⢹⣿⡧⡏⣾⢸⣿⣿⣿⣿⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣟⡿⣧⣿⣧⢿⣿⡿⣿⡏⣿⢉⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣼⣷⣿⣿⣹⣧⡻⣷⣿⢷⡹⣿⠿⢻⡇⡿
⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⣿⣿⣿⣯⣷⣜⣿⣿⣳⣼⢷⡌⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⢻⣿⣿⣿⣿⣿⣞⣿⣿⢿⣧⣿⠀⠙⢿⡝⣦
⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠘⣷⣿⣿⡙⢧⣽⣿⣿⣿⣷⡟⠀⠀⠀⠷⠟
⠀⠀⠀⠀⢀⠀⢿⢻⡄⠀⠀⣿⣿⡇⣿⡷⢾⢿⡟⠉⢿⣷⡀
⠀⠀⠀⠀⠈⠣⠈⢷⣿⡄⠀⣿⣿⣇⣿⡷⢏⡼⠃⠀⠈⢻⣝⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⢿⡙⣿⡿⢳⠏⠀⠀⠀⠀⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣫⣾⣟⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⣳⢿⣿⣿⣿⣷⣶⣟⣳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣽⣾⣿⠛⠁⠀⢻⣿⣄⠉⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠁⣿⣿⠀⠀⠀⠀⠹⣟⢦⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠉⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡆⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠌⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢌⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠀⢀⢊⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠘⠀⠀⠠⠣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠈⠒⠒⠠⡀⠀⠀⠀⠀⡆⡥⠐⡲⡱⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⡠⢈⢂⣀⣴⠩⢰⠀⠔⡔⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢀⣴⣶⣶⢶⡫⣿⣾⣿⣿⣿⣦⣌⣌⠄⠂⢁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠤⣌⣄⣞⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠋⣁⠠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠙⠛⠻⠿⣿⣿⣿⣏⣡⠊⠄⠂⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠉⡏⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⡇⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠇⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠶⠶⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⡶⢶⣦⣤⣄⡀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠙⢿⡖⠦⣄⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠀⠀⣀⡤⠖⠛⠙⢻⣆⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠈⢳⣿
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢀⡴⣚⣡⡤⠴⠶⠦⢤⣿⡆⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢠⡿
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢠⠿⠛⠉⠀⠀⠀⠀⠀⢀⣿⠁⠀⠈⠻⣶⣤⣀⡀⠀⠀⣀⣠⣿⠟⠁
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠁⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣀⣀⣀⣀⣤⡶⠟⠉⠀⣠⡶⢾⡿⠻⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁⠀⠀⢠⡾⠋⢀⡞⢀⡴⠋⠉⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢸⢁⡞⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⡾⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⡶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⢒⣛⣛⣶⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⣋⣉⣭⣤⡶⠾⠟⠛⠋⢉⣙⣯⡳⣝⢦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢲⠞⢁⣴⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠲⠤⠤⠼⠳⠽⠟⠛⠦⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⢉⣀⣀⡴⠃⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⡪⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠿⣄⠀⠀⣠⣖⠋⠀⠉⢑⣲⣾⣆⠀⢀⡀⠤⠐⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠊⠉⠀⢀⡠⠝⢳
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠉⠛⢲⡏⠉⠛⠳⣾⡿⠟⠋⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⠉⠁⢀⣀⡤⠴⠒⠊⠉⠁⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⢸⠃⠀⠀⠀⣿⠤⣄⣀⣀⠀⠉⠁⠒⠀⠠⠤⠐⠒⠉⠁⢀⣀⣤⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣝⣲⢶⡾⣶⣦⣄⠀⣿⠀⠀⠈⠉⠛⠛⠶⠶⠦⠤⠤⠴⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⢒⣋⣭⢴⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠻⣇⣤⢄⢹⣷⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠖⢚⣋⣭⢴⠶⠟⠛⠉⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⡄⠀⡏⠰⠜⢸⣿⡯⢭⣙⡓⠲⠤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡶⠶⠞⢛⣋⣥⣶⠾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⢸⡿⡇⡇⠉⠙⠛⠶⢶⣤⣭⣉⣉⣉⣩⣭⣤⡶⠾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠠⢄⢸⡿⡇⡇⠁⠒⠠⡀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠘⠀⠘⢮⣊⣼⠟⠃⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡉⠒⠠⢄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣈⡉⠒⠠⢄⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠶⠖⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⢀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡤⣤⣤⣤⣤⠔⠉⠀⠀⠀⠀⠈⠛⣦⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⢈⡁⠙⣯⠀⠀⠀⠀⠀⠀⠀⠁⣸⡿⢍⠉⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣠⣀⡀⠠⠁⢻⡗⡦⣤⣤⣠⣤⡤⢞⣿⠷⣤⣀⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠂⠈⠛⠳⠶⣤⣍⡓⠾⢶⠷⠶⠞⠛⠁⠀⠀⠈⠙⠛⠶⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠈⠛⠻⢷⣶⣖⡀⢀⠈⠚⠷⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠉⠙⠦⠤⣌⠁⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠻⠿⠿⠿⠿⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠻⠿⠿⠿⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣠⠀⠀⠀⢀⣠⣤⣤⡀⠀⢠⣤⡄⠀⠀⢰⡴⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡞⠉⠉⠻⠀⠀⢀⡞⠁⠀⠈⠙⢦⠀⠙⢦⡀⣠⠞⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⢧⣀⣀⣀⠀⠀⢸⠁⠀⠀⠀⠀⠸⡇⠀⠀⢹⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠉⠉⠉⢻⡄⢸⡀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣤⠀⠀⠀⣠⠇⠈⠷⣄⣀⠀⣀⡼⠃⠀⠀⣸⣇⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢛⡙⠛⠛⢁⡀⠀⣀⠈⢉⣉⠉⠀⣀⡀⠈⢉⣉⣅⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡟⢦⡴⢻⡇⠀⢹⠁⠈⡏⠀⠀⢸⣇⠴⢏⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⠀⣸⣇⠀⣼⡄⢠⣧⣠⣷⢈⣧⠀⠈⣧⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⣩⢛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠴⠶⢯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⠲⢦⡄⠀⢀⣽⣄⣄⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⡤⠼⢧⣄⡰⠖⣴⠟⠚⠛⠛⢦⡄⠙⣷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡏⠀⠀⢀⡼⠿⠻⠿⣶⠦⡤⣤⠞⠛⠛⠛⢦⡀⠀⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠀⠀⠙⠛⠚⠿⣄⣀⣀⣠⡾⠃⠰⢧⣸⣦⣀⣀⡼⠃⠀⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⠶⣏⠈⢻⣖⣚⣛⣛⣛⢩⣉⣭⣭⣿⣷⠶⠖⠛⠛⣷⣎⢧⡀
⠀⠀⠀⠀⠀⣿⢡⣶⣶⣾⡉⠷⣔⠊⢍⡚⠻⣿⣿⣿⣿⡎⢿⣿⠟⢁⣠⣴⠶⠚⠉⠛⢻⣧⣳⡀
⠀⠀⠀⠀⠀⢻⣆⢿⡗⠀⠙⢦⡀⠙⣄⠙⣄⠹⣽⣽⣶⣷⡞⢿⣷⠏⠀⣀⣀⠤⠀⠂⣺⣿⡷⠻⣄
⠀⠀⠀⠀⠀⢸⣿⡌⣧⠀⠀⠀⠻⣄⠘⡆⠸⣦⠿⠿⠿⠿⢟⣘⣛⣃⠈⣭⣤⡴⢲⡶⠶⢾⣿⣶⡙⣦
⠀⠀⠀⠀⠀⢸⣿⣷⡘⢒⣂⣠⣤⡭⢤⣶⣶⠶⠖⡻⠟⠛⣩⣌⡙⢿⣷⡹⠁⣇⠀⢿⡟⠛⠛⣻⣿⣜⢧⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣧⢹⣿⡉⠐⣝⡦⠈⠀⠐⠮⠁⠀⣀⠠⡅⠀⣠⣌⠛⢄⠸⢦⡈⠻⣿⣛⣉⣉⣽⡮⠿⡀
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⢻⣿⣦⡀⠀⣼⠃⠄⢂⣀⠘⠽⢲⣴⠦⠀⢁⣶⣛⣒⣦⣭⣭⣿⣿⣷⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣼⣿⣿⣿⣦⣬⣥⣤⣽⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁