Needle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣒⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣽⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣱⣿⣣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡟⠹⠟⡽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡽⠑⠟⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⡾⠁⢀⡼⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡸⠁⠀⣰⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡼⠁⢠⡞⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣰⠃⠀⡼⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡼⠁⣠⠏⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣰⠃⢀⡞⠁⢀⡞
⠀⠀⠀⠀⠀⠔⠋⠀⠀⣼⠁⣰⠋⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⠀⢰⠃⢠⠎⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡼⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣰⠋⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⣰⢃⡞⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡴⠃⠀⢰⠃
⠀⠀⠀⠀⠀⠀⣰⢧⠞⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡼⠁⠀⠀⡏
⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎⠞⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠈⠳⠤⢤⡤⠤⠤⠒⠒⠛⠉⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠑⠤⢤⣤⠤⠤⠖⠒⠚⠉⠙⠓⠂
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠞


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⣶⡆⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⠿⢿⠿⣿⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⢡⢊⠭⠙⣎⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡘⣌⣀⣢⢎⣼⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣶⣾⣶⣾⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣿⣧⣴⡦⠈⠉⠈⠛⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣷⡿⠋⠉⠉⠻⣿⣿⠆⠀⢀⣰⣶⣦⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⡃⠀⠀⠀⠀⢸⣿⡶⠀⠰⣿⠂⠈⢽⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣶⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⣄⣀⣀⣠⣿⣿⠄⠀⠈⠹⠷⡶⠿⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣾⠋⠉⠙⢿⡷⠀⠀⠀⠀⣠⣴⣄⠀⣿⣿⡇⠀⣴⣄⡀⠿⠛⣿⡟⠻⠗⠀⠀⠀⠀⠀⡀⣾⣦⣿⣷⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⠀⠀⠀⣸⣟⠃⠀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢀⡀⠀⠀⢀⣴⣴⣶⣆⡀⠚⣿⡿⠋⠁⠈⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠿⣶⠾⡿⠉⠀⠰⣿⣿⣿⣿⠟⢉⣭⣭⣍⡙⢿⣿⣿⣿⡷⠄⠀⢾⡿⡋⣱⣿⡦⠿⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⣤⣄⣶⣆⣠⣄⠀⠀⠀⢀⣀⣼⣿⣿⠃⣼⢋⡥⠤⣌⠻⣆⢹⣿⣿⣄⣀⠀⠘⠿⢶⡾⠟⠁⠺⠿⣿⣦⣀⣀⣴⣿⠿
⠀⠀⠀⣿⡀⢈⡷⠀⢰⣶⣿⠟⠛⠛⠿⣷⣶⡄⠀⠸⡿⣿⣿⣿⠸⣇⢸⡀⠀⣸⠀⣿⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⠛⢿⡟⠛⠟
⠀⠀⠀⠈⠙⠋⠀⠀⣶⣿⠇⠀⠀⠀⠀⢹⣿⣦⠀⠀⢀⣼⣿⣿⣆⠻⢦⣍⣉⣥⡾⢃⣼⣿⣿⣄⠀⠀⠀⠀⢰⣷⣾⣿⣶⣿⠆
⠀⣀⣀⡀⠀⠀⠀⠀⢨⣿⣧⡀⠀⠀⢀⣼⣿⡅⠀⠀⠈⠿⠿⢿⣿⣷⣦⣭⣭⣥⣴⣿⣿⡿⢿⠟⠀⠀⠐⢿⣿⠟⠉⠀⠉⠻⣿⡟
⢸⡋⠉⣿⠀⠀⠀⠀⠈⠉⣿⡿⣿⡿⢿⡯⠉⢠⡞⠛⢧⠀⠀⢼⣿⡿⠿⣿⣿⡿⠿⣿⣿⠄⠀⠀⠀⠀⠺⢿⣿⠀⠀⠀⠀⠀⣿⡿⠇
⠈⠛⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠘⢧⣤⠞⠀⠀⠀⠈⠁⠀⠛⠛⠃⠀⠈⠀⠀⠀⠀⠀⠂⠐⢿⢿⣦⣄⣀⣠⣾⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠻⣿⠛⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⡆⠀⣀⣸⣿⣿⣿⣀⡀⠀⣼⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣿⣿⣿⠿⣛⣫⣭⣥⣶⣶⣮⣭⣭⣛⡻⢿⣿⣿⣷⣾⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣿⡿⣫⣴⣿⠿⠛⠋⠉⠉⠉⠉⠉⠛⠻⢿⣶⣌⡻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣠⣾⣿⡿⢫⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣮⠻⣿⣿⣦⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⡿⣱⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡹⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢱⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⢹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⡇⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⣿⣿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⡇⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⡹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⢿⣿⣷⣜⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠙⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣷⣝⠻⣿⣶⣤⣄⣀⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠿⣿⣿⣿⣶⣭⣝⣛⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠃⠀⠉⢹⣿⣿⣿⠉⠁⠈⢿⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠈⡧⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⣀⣴⢋⣿⠟⠉⠉⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⣠⣾⠿⢿⣶⡀⠀⠀⢰⣤⣼⣧⣴⣎⢀⡀⢹⣏⠀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⢀⣀⣤⣠⡄⠀⢸⠉⠈⣳⠀⠀⠀⠀⣠⣤⣤⡀
⠀⢻⣿⣄⣰⣿⢇⣘⣿⠟⠉⠉⠉⠉⠛⢿⣥⣽⢿⣄⡀⠀⣀⣴⣿⠁⠀⠀⢀⣸⡾⠋⠉⠉⠙⢿⡬⠓⢒⣡⣤⣄⡀⣸⡏⠀⠈⢿⠄
⠀⠀⠙⠛⠛⠉⣈⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣧⡌⠉⠻⠛⠻⣩⣶⣶⣶⣄⠠⣿⠀⠀⠀⠀⠀⠈⣿⢢⣿⠋⠉⠹⣿⡜⢷⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠈⢙⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡀⠀⠀⠀⢰⣿⡏⠀⢹⣿⠖⢻⣄⠀⠀⠀⢀⣰⣏⠙⣿⣦⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢻⣧⡀⠀⠀⠀⠀⢀⣰⣿⡅⠀⠀⠀⠀⠀⠻⢿⣶⡿⠟⠀⠈⠹⠟⢶⠲⠟⠓⠀⠀⠈⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠷⣶⣶⣶⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠴⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠷⢶⣦⣤⣤⣀⣀⣘⣛⣛⣛⣛⣀⣀⣀⣤⣤⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣞⠉⣓⢦⣀⠀⠀⠀⠈⠙⣿⣷⣶⣶⣤⣤⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠓⢿⡦⢽⡝⠢⣄⠀⠀⣿⢯⣙⡛⠿⠭⣭⣙⣛⣛⣛⣛⠿⣻⣭⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠙⠲⣿⡕⡒⡩⠭⠿⠷⠶⠶⠶⠶⠶⢖⣚⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀⠙⠢⣄⡀⠀⠀⠀⠀⠐⠒⠛⣛⣳⡽⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⢦⣄⡙⠻⣍⣩⣭⠽⠟⣛⣭⣷⡾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣋⡉⠉⠉⠁⢐⣂⡭⠭⠒⢋⣁⣤⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣖⣒⣀⣀⣩⣤⣤⣶⡶⢿⣿⣟⣀⢽⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠶⣭⣭⣍⣁⣀⠀⠀⠀⠉⠲⣌⡛⢷⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣯⣅⡒⠠⠬⠭⠭⠭⠉⠉⢉⣉⣈⡭⠶⢬⣙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡀⠙⠿⢿⣿⣿⣿⣿⣿⠛⠋⠉⠉⣀⣠⣴⡿⠋⣻⡶⢭⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠉⠓⠲⠤⢤⣬⣉⣉⣙⣛⢛⣛⣛⣛⣉⡩⠴⠚⢹⡇⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠲⠶⠦⣤⣤⣄⣀⣀⠀⠠⠤⠤⠶⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⢠⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠱⠰⠤⠄⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢄⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠒⠒⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⢎⣣⡀⠀⠀⠀⠀⠰⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠚⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢱⡇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢪⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠠⡐⠁⠀⠐⠲⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠈⠉⠀⠀⠀⠀⠀⠀⡰⠢⠤⠁
⠀⠀⠀⠀⠀⣬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠓⢍⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁
⠀⠀⢀⠂⠒⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠢⠀⠔⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠊⢷⡀⠀⠀⠀⠠⠀⠰⠊⠐⠠⠤⣀⡰⠃⠻⠞⠙⣄⠠⠔⠊⠁
⠀⠉⠚⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐
⠀⢀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠅⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠁⢃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⠀⠀⢀⢄⡀⢀⠁⠀⠀⠀⠄⠂⠐⢄
⠀⠀⠀⠑⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣈⠅⠀⠀⢀⢀⣀⣄⠀⠀⠀⠔⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⠐⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⢀⡞⡇⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⣼⡐⠦⣄⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱⣿⢣⣷⡎⠙⢰⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⢣⣿⠟⠁⠀⡿⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣛⣛⣛⣛⡛⠛⠛⠛⠛⣛⣛⡛⣼⠏⠊⠱⣄⡀⠀⡇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠖⣒⣒⣚⣛⣭⡭⠭⠖⢰⡟⠤⠀⠀⠈⠙⠶⣥⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠭⠭⠵⠶⢖⣒⣚⣫⢁⡿⣐⡋⠀⠀⠀⠀⠀⡆⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣒⣛⣛⣭⡭⠭⠥⠶⠒⣼⢡⠥⣒⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠖⣒⣒⣚⣛⣭⡭⢱⢃⣒⣫⠥⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⡭⡭⠭⠭⠶⣖⣒⣚⣩⡭⠵⣖⣋⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⣛⣛⣻⣩⣭⡭⠽⠶⣒⣚⣭⠽⡖⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⣒⣒⣒⡛⡻⠩⠥⠶⣒⣫⠥⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⠭⠭⠭⠭⠶⢰⢓⣚⣫⡭⠵⣖⣋⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣒⣛⣛⣩⡍⠠⠬⠶⣒⣚⣩⠭⡖⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠖⠒⠒⠛⢋⣭⠭⠶⠒⠋⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠲⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠳⢤⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠚⠁
⠀⠀⠀⠀⢀⣀⣀⣤⠴⠚⠁⠉⠛⠓⠒⠶⠦⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⠴⠖⠚⠉⠁
⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⡏⢹⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢤⡄⢠⡤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⣤⣤⡀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⢠⣤⣤⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠀⠀⣿⠀⠀⡇
⣰⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⠀⠀⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣤⠤⠤⡄⠀⠀⠀⣤⠤⠤⣄⠀⠀⠀⢠⠤⠤⣤⠀⠀⡇
⡇⠀⠀⠀⣾⠛⠛⣷⠀⠀⠀⢸⠀⠀⣿⣀⣀⡇⠀⠀⠀⣿⣀⣀⡿⠀⠀⠀⢸⣀⣀⣿⠀⠀⡇⠀⠀⠀⣾⠛⠛⡇
⡇⠀⠀⠀⠻⠤⠤⠟⠀⠀⠀⢸⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⡇⠀⠀⠀⠿⠤⠤⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣶⠒⠒⡆⠀⠀⠀⣴⠒⠒⣦⠀⠀⠀⢰⠒⠒⣶⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⣤⣤⡇⠀⠀⠀⢿⣤⣤⡟⠀⠀⠀⢸⣤⣤⡿⠀⠀⡇
⡇⠀⠀⠀⣴⠶⠶⣦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣴⠶⠶⡆
⡇⠀⠀⠀⢿⣀⣀⡿⠀⢀⡴⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠳⢦⡀⠀⢿⣀⣀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠴⣯⡤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡽⠦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⣤⣤⡄
⡇⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣥⣤⣤⣤⣤⣤⣤⣤⣧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⡇
⡇⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠋⠙⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⠿⠿⢿⡿⠿⠿⣉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠛⠓⠚⠛⠋⢝⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⢾⣇⠀⠀⠀⠀⠀⠀⢓⣿⠶⠶⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⣴⠾⣿⡆⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⠔⠂⠀⠀⠸⣇⣀⡀⠀⠀⠀⣀⣰⡿⠒⠀⠀⠲⣌⠛⢦⠀⠀⠀⠀⠀⢸⣧⣴⢟⣡⡾⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠙⠦⣤⣈⡓⠂⠄⠉⠉⠉⠉⠉⠉⠀⡀⠠⠤⠴⠒⠋⣀⣼⠀⠀⠀⠀⣠⡾⣿⣵⠿⠋⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠈⠉⠙⠛⠒⠒⠒⠒⠂⠀⠀⠀⠀⠤⠤⠒⢋⡡⢾⠀⢀⣔⣼⣋⣷⠞⢿⡄⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠤⠴⠒⠋⠉⣠⣿⣶⣿⣿⡾⠋⠀⠀⠈⢿⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⢊⢕⣵⣿⠟⠉⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠈⠻⡒⠒⠂⠀⠀⠀⠀⠀⠐⠢⠤⢤⣄⣤⡤⢚⣵⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢧⡀⠀⠈⠑⠄⡀⠀⠀⠀⠀⠒⠒⠒⠊⡩⢋⣴⣿⠷⠋⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣄⡙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣫⡴⠛⠉⠀⠠⠊⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠙⠛⠛⠂⠀⠀⠀⠀⣀⠖⣡⡾⢯⣄⣀⣀⣀⣠⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⣵⠟⠉⠩⠥⠴⠒⠛⠉⠀⠚⢹⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠈⠐⢄⠀⠐⢪⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠱⣄⠀⢀⠔⣩⠶⠀⠀⠙⠻⣍⡉⠉⠁⠀⠀⣀⣠⠴⢛⣿⠀⠀⣀⣴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⡮⢛⣵⡟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠒⠚⠉⠀⠀⢀⣠⣿⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⣪⡶⠋⠀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⡾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣙⣲⠶⠶⠚⠋⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡵⠟⣿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⡠⠟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⣿⠓⢬⡳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢾⡧⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠒⠦⠤⠤⣀⣠⠤⠞⠁⣾
⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢭⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠉⣁⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡛⠛⠛⠛⠛⠛⠛⢛⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⢀⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⢠⣶⣟⣿⣦⠀⠀⢀⣴⣛⡟⣶⡀⠀⠀⣴⠞⠛⢷⡄⠀⠀⣤⡼⢓⠿⣤⠀⠀⢠⣾⡿⢿⣦⠀⠀⢀⡲⠛⠛⢶⡀
⠀⠸⣟⡧⣟⣾⠀⠀⠘⣧⠚⡟⣼⠇⠀⠀⢿⣀⢀⣸⡇⠀⠀⠼⣍⡆⣳⠷⠀⠀⠸⣟⣤⣼⣽⠃⠀⠀⢧⠀⠀⣸⠃
⠀⠀⠉⠙⠉⠁⠀⠀⠀⠈⠉⠋⠁⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠙⠉⠋⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠉⠙
⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⢀⣠⢤⣤⡀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⣾⠁⠀⠈⣗⠀⠀⣻⠁⠀⠘⣟⠀⠀⣾⢥⠈⡢⢿⠀⠀⢸⣿⡋⣻⣿⠀⠀⢐⡏⠀⠀⢹⡂⠀⠀⡏⠀⠀⢹⡆
⠀⠀⠙⠲⠤⠞⠁⠀⠀⠉⢷⠶⡾⠉⠀⠀⠘⠷⢽⠴⠋⠀⠀⠈⠻⠿⠿⠛⠀⠀⠀⠓⠦⠴⠛⠀⠀⠀⠙⠦⠴⠞

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣶⢤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⢀⣤⠶⠾⠛⠛⠛⢉⣍⠩⣄⢦⡐⠎⠧⠰⠄⠻⠂⠾⠉⠇⠚⠿⠘⣧⡿⣨⣏⠻⣯⢉⣹⠛⠛⠷⢶⣤⣤⣀
⠀⠀⣾⡁⠀⠀⢀⣀⣀⢀⠛⠃⠈⠀⠀⣠⣴⣶⣿⣿⣿⡟⠛⠓⠒⠒⠒⠳⠶⢭⡉⠉⠁⠺⢇⣐⣠⡄⠀⠀⠉⠛⠿⣦⡀
⠀⠀⣿⣿⣦⣄⣀⡀⠛⠉⢳⢀⡄⢠⣀⠙⠛⠛⠿⠿⠿⢿⣶⣦⣤⣤⣤⣤⠤⠴⠛⢀⠀⠀⣤⣍⠁⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⢿⣿⣿⡝⡙⢹⠛⠲⠶⠦⣤⣤⣈⣁⣘⡛⠂⠰⠀⠀⠀⢲⣆⠐⢻⡄⠶⠲⡾⠉⠳⠀⠉⣉⣀⣠⣤⡴⢶⢿⣿⣿⡇
⠀⠀⠀⠙⣿⡟⠻⠾⣦⣤⣀⣀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⢀⣀⣸⣾⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣷⣤⣀⣈⠉⠉⠙⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠖⠒⠚⠛⠛⠛⣹⡿⠋
⠀⠀⠀⠀⠀⠀⣿⣓⠾⠭⢭⣟⣓⣶⣦⠤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣾⡏
⠀⠀⠀⠀⠀⢸⣿⣯⣭⣭⣲⣖⡲⠯⠭⠭⠿⣷⣶⣿⣿⣟⣿⣿⣯⢩⣭⣿⣯⣿⣿⡿⣿⣾⣿⣿⣯⣽⣿⣿⣗
⠀⠀⠀⠀⢰⣿⣿⠻⢯⣛⣛⡻⠿⠭⢭⣭⣗⣗⣒⣒⢒⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⢷⣶⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⠻⣌⡉⠚⣷⣿⣭⣭⣝⣛⣒⣒⣶⣿⣿⡿⠽⠿⠿⢿⠭⠿⣭⠻⣽⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⣿⡿⣷⣮⣭⣉⣉⡛⠿⢻⣿⣿⣻⣿⣿⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣚⣛⣿⣿⣿⣿⣿⣟⣿⡦
⠀⠀⠀⠀⠀⢠⣿⣿⣟⣓⠾⠭⣿⣿⣟⣻⣷⣶⣶⣯⣥⣥⣭⣭⣭⣽⣿⣯⣭⣭⣽⣿⣿⣻⣿⣿⣿⣿⣿⣿⡿⣏⠁
⠀⠀⠀⠀⠀⣿⠿⣭⣽⣽⣿⡿⣿⣳⣷⣒⣶⠿⠿⠿⠿⠯⠭⣽⡭⠭⠽⠿⠿⠿⠿⠿⢿⡿⣿⣿⣿⣿⣾⡿⢣⡟
⠀⠀⠀⠀⢀⣾⣷⣍⠓⣲⠬⠭⣙⣛⣶⣧⡬⠭⠭⠭⣭⣉⣉⣉⣉⣉⣻⣛⣋⣉⣉⢩⣭⣭⠿⠿⠟⣃⣠⣾⣏
⠀⠀⠀⠀⠘⣯⠻⣙⡳⢮⣽⣟⣒⣒⠲⠾⠿⠽⣿⣽⣷⣒⣒⣒⣒⠒⠒⠲⠒⢒⣚⣒⣯⣭⣶⣶⣯⣽⣿⣿⡟
⠀⠀⠀⠀⠀⢘⡿⣿⣭⣙⣛⠻⠿⠿⢿⣽⣿⣿⣷⣶⣶⣍⣡⣽⣿⣶⣶⡟⠛⠻⠛⢿⣷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣴⡟⣷⢬⣙⣺⠿⠿⠿⠷⣶⣶⣶⣾⣿⣭⡭⢭⣭⣭⣭⣭⣭⣭⣭⣿⣿⣿⣟⣛⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⢻⣭⣿⣓⣻⠿⣷⣶⣖⣒⠲⠶⠶⠯⠽⠭⠭⢭⣭⣭⣭⣭⣭⠭⠭⠽⠿⠿⠿⣿⣛⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣛⡻⢿⡿⣿⣶⣦⣭⣭⣭⣽⣛⣛⣛⣛⣓⣛⣓⣒⣒⣒⣶⣿⣟⣛⣛⣿⣯⣽⣿⣿⣛⣿⡃
⠀⠀⠀⠀⠀⢸⣟⡛⣭⣍⣛⣒⡢⠭⢭⢭⣍⣉⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⠻⣦⡀
⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣾⡿⠯⣭⣟⣺⣓⣒⣒⣶⢒⣀⠀⠶⠶⠆⠒⠆⡐⣒⣶⣺⣷⣾⣿⢿⣿⣿⣿⣿⡄⠈⢻⣄
⠀⠀⠀⠀⠀⢸⣿⠿⢿⣶⣶⡮⠭⣿⣿⣖⣷⣿⣿⡿⠿⠿⠻⡿⠿⣿⣿⡿⣿⣿⣿⣟⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⢸⣿⣝⣒⠒⠬⢭⣙⣒⣲⡶⠶⠶⠬⠿⠽⠿⠿⠿⠽⠿⠭⠥⠬⠴⢿⣟⣚⣖⣶⣭⠽⣿⣿⣿⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⢨⣿⢶⣬⣭⣍⣛⣓⣲⣾⣭⢭⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⢿⢥⣶⣶⣾⣿⣿⣿⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣀⣴⠿⣟⡳⣦⣭⣝⣛⣿⡿⣭⣭⣭⣍⣛⣛⣛⣛⣛⣛⣟⣛⣛⣛⣛⣛⣋⣉⣭⣭⣽⣷⣾⣿⣿⣿⣦⡀⠀⠀⣾⠁
⢀⣴⠾⠋⠀⠀⠀⠙⠻⢿⣷⣶⣬⣭⣭⣟⣛⣛⣓⣶⣾⠿⠟⠷⠿⠿⣿⢿⣿⣿⣛⣛⣛⣯⣽⣿⣿⠿⠿⠋⠉⠻⣶⣄⢸⡀
⢸⣿⣦⣀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣽⠛⣇⠀⠀⠀⠀⢀⡀
⠈⣿⣯⣿⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠃⠙⢷⣀⣴⠟⠛⠛⠳⢦⡀
⠀⠈⠻⣿⣧⣄⡀⠈⠙⠓⠶⠤⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠴⠶⠚⣿⣏⣿⣿⡿⠛⠀⠀⣠⡿⣧⣄⡀⠀⠀⣠⣿
⠀⠀⠀⠀⠉⠙⠻⠿⢶⣶⣤⣤⣀⣀⡈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⢀⣀⣠⣤⡿⠟⠛⠁⠀⣀⣴⠾⠋⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠿⠿⠿⠶⣶⣶⣶⣶⣶⣦⣤⣤⠶⠶⠶⠾⠿⠛⠋⠉⠁⠀⠀⠰⠶⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣄⣀⣄⣀⣀⣀⣄⣀⣀⣀⣠⣀⣀⣄⣀⣀⣄⣀⣀⣀⣀⣀⣠⣄⣀⣄⣤⣀⣠⣠⣄⣠⣤⣀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⣀⠀⠀⢀⣀⠀⠀⢀⠀⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡇⠀⠀⠀⠘⠋⠃⠀⠘⠤⠇⠀⠀⠽⠂⠀⠸⠈⠀⠀⠛⠀⠀⠸⠀⠀⠘⠻⠀⠀⠻⠄⠀⠀⢰⡇
⠀⠿⠤⣤⡤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⢀⣀⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⣿⡟⠛⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠠⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠄⠀⠠⠀⠀⠀⠸⠀⡄⠤⠤⠤⠄⠤⠤⠤⠤⢤⠀⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣀⣸⠗⠛⠛⠺⣧⣀⠀⠀⠀⠀⣀⣸⡷⢿⣿⢾⣇⣀⠀⠀⠀⠀⢀⣴⠋⠉⠉⠙⣦⡀⠀⠀⠀⠀⣀⣸⠷⢺⡗⠾⣇⣀⠀⠀⠀⠀⣀⣼⡻⢿⡽⢟⣧⣀
⠀⢀⣽⠛⣶⠛⠛⣶⠛⣯⡀⠀⠀⢀⣽⠏⣾⡿⢿⣷⠹⣯⡀⠀⠀⠀⣎⠀⢱⡶⢶⡎⠀⢹⠀⠀⠀⢀⣹⢓⠊⠀⠀⠑⠞⣏⡀⠀⠀⢀⣿⣎⣀⡼⢧⣀⣱⢿⡀
⠀⠈⣻⡀⠻⣤⣤⠟⢀⣟⠁⠀⠀⠈⣻⣾⣯⣦⣾⣽⣶⣟⠁⠀⠀⠀⢇⠀⡸⠷⠾⢇⠀⣸⠀⠀⠀⠈⣹⡤⢄⠀⠀⡠⣥⣏⠁⠀⠀⠈⣷⢏⠉⢳⡞⠉⡹⣾⠁
⠀⠀⠉⢹⣦⣼⣧⢴⡏⠉⠀⠀⠀⠀⠉⢹⡷⣦⣴⢾⡏⠉⠀⠀⠀⠀⠈⠻⣄⣀⣀⣠⠟⠁⠀⠀⠀⠀⠉⢹⡶⢼⡧⢶⡏⠉⠀⠀⠀⠀⠉⢳⣵⣞⣳⣮⡟⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀
⠀⠀⡀⣬⣾⣿⣿⣷⣥⢂⠀⠀⠀⠀⣀⣼⣛⣻⣟⣛⣧⣀⠀⠀⠀⠀⢠⡴⠟⢻⡟⠻⢶⡀⠀⠀⠀⠀⢰⣶⣴⣿⣿⣦⣶⡆⠀⠀⠀⠀⣰⡼⣟⡿⢿⣛⢷⣀
⠀⠐⣾⣿⠑⡤⢤⠊⣿⣷⠀⠀⠀⢀⣿⣸⣿⡿⢿⣿⣏⣿⣀⠀⠀⠐⣿⢷⡶⣻⣟⢶⡶⢿⡀⠀⠀⣀⣴⣿⠋⠀⠀⠙⣿⣦⣀⠀⠀⢐⣿⡼⢿⢗⡸⡿⢯⣿⡂
⠀⠠⢿⣿⡠⠓⠚⢄⣿⡿⠄⠀⠀⠈⣿⢹⣿⣷⣼⣿⡏⣿⠁⠀⠀⠈⣷⠀⣿⣽⣯⣾⢀⣿⠄⠀⠀⠙⠻⣿⣄⠀⠀⣠⣿⠟⠁⠀⠀⠨⣿⣳⣾⡮⠵⣷⣾⣿⠅
⠀⠀⠁⢛⢿⣿⣿⡿⡛⠌⠀⠀⠀⠀⠉⢻⣭⣽⣯⣭⡟⠉⠀⠀⠀⠀⠉⠿⣯⣤⣠⣽⡟⠃⠀⠀⠀⠀⠸⠿⠻⣿⣿⠟⠿⠇⠀⠀⠀⠀⠙⢷⣯⣷⣾⣭⡞⠋
⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣀⣸⣷⠿⠿⣾⣇⣀⠀⠀⠀⠀⣀⡼⠛⢻⡟⠛⢧⣀⠀⠀⠀⠀⣠⡾⠛⢻⡟⠛⢷⣀⠀⠀⠀⠀⣠⡶⢟⣩⣍⡻⢶⣄⠀⠀⠀⠀⣀⡼⣳⣯⣽⣟⢧⣀
⠀⢀⣽⡟⣨⠗⠺⣅⣹⣯⡀⠀⠀⢀⡿⠛⢦⡼⢧⡴⠛⢿⡀⠀⠀⢰⡿⠀⢀⡼⢧⡀⠈⢿⡂⠀⠀⢠⡟⡰⠁⢠⡄⠈⣎⢻⡄⠀⠀⢀⡿⣾⣟⡽⢯⣻⣷⢿⡀
⠀⠈⣻⣏⢙⡦⢴⡋⣸⣟⠁⠀⠀⠈⣷⣤⠞⢳⡞⠳⣤⣾⠁⠀⠀⠨⣷⣤⠞⠳⠞⠳⣤⣾⠅⠀⠀⠘⣧⡱⡔⠉⠉⢢⢎⣽⠃⠀⠀⠈⣳⢿⣯⣳⣞⣽⣟⣾⠁
⠀⠀⠉⢹⡿⣶⣶⢿⡏⠉⠀⠀⠀⠀⠉⢳⣤⣼⣧⣤⡞⠉⠀⠀⠀⠀⠙⢷⣤⣄⣠⣤⡾⠉⠀⠀⠀⠀⠙⠷⣮⣑⣋⣵⠾⠋⠀⠀⠀⠀⠉⢳⣽⣿⣻⢭⡞⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢰⣶⢟⣛⣛⡻⣶⡆⠀⠀⠀⠀⣀⣼⠿⢻⡟⠿⣧⣂⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⢰⣶⣿⠛⠛⣿⣶⡆⠀⠀⠀⠀⠰⣦⡶⢿⡿⢶⣴⠆
⠀⢈⣿⢱⢿⡷⢾⡿⡎⣿⡁⠀⠀⢐⣿⠧⢤⡼⢦⡤⠼⣿⡂⠀⠀⠠⣿⣿⣿⢟⡻⣿⣿⣿⡀⠀⠀⢈⣿⠛⠛⣤⣤⠛⠛⣿⡁⠀⠀⢀⣴⣏⡾⢋⡙⢷⣹⣧⡀
⠀⢈⣿⡸⣾⡷⢾⣳⢇⣿⡁⠀⠀⠨⣿⡄⢨⠷⠾⡅⢠⣿⠅⠀⠀⠈⣿⣿⣿⣮⣵⣿⣿⣿⠀⠀⠀⢈⣿⣤⣤⠛⠻⣤⣤⣿⡁⠀⠀⠈⠻⣏⢷⣌⣡⡾⣹⠟⠁
⠀⠀⠸⠿⣮⣭⣭⣵⠿⠇⠀⠀⠀⠀⠉⢻⣷⣤⣤⣾⡟⠍⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠀⠸⠿⣿⣤⣤⣿⠿⠇⠀⠀⠀⠀⠰⠟⠳⣾⣷⠞⠿⠆
⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣠⣼⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⣠⡾⠋⠉⠉⠙⢷⣄⠀⠀⠀⠀⢰⡶⠛⠋⠙⠛⣶⡆⠀⠀⠀⠀⣀⣸⡷⢿⡿⢿⣇⣀⠀⠀⠀⠀⣀⣼⢿⣛⣛⡿⣧⣀
⠀⢐⣿⣿⣿⠛⠛⣿⣿⣿⡂⠀⠀⢰⡟⠑⠢⡠⢤⠔⠊⢻⡆⠀⠀⢈⡿⠓⢲⡴⢦⡴⠏⢿⡁⠀⠀⢀⣽⣏⣀⡼⢧⣀⣹⣯⡀⠀⠀⢀⣿⢳⣯⡿⢿⣽⡞⣿⡀
⠀⠨⣿⣿⣿⣤⣤⣿⣿⣿⠅⠀⠀⠸⣧⡠⠔⠑⠊⠢⢄⣼⠇⠀⠀⢈⣷⠀⠈⢳⠞⠁⠀⣾⡁⠀⠀⠈⣻⣏⠉⢳⡞⠉⣹⣟⠁⠀⠀⠈⣿⡼⣟⣷⣾⣻⢧⣿⠁
⠀⠀⠙⢿⣿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠙⢷⣄⣀⣀⣤⡾⠋⠀⠀⠀⠀⠸⠷⣤⣜⣧⣤⠾⠇⠀⠀⠀⠀⠉⢹⣷⣾⣷⢾⡏⠉⠀⠀⠀⠀⠉⢻⣾⣭⣭⣷⡟⠉
⠀⠀⠀⠀⠈⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠁⠀⠀⠀⠀⢈⣭⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣶⠹⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣷⠀⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⡠⣾⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⡈⠛⢿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⣀⠙⢿⣿⣿⡟⣿⡽⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⡽⣿⣟⢿⣿⣿⡇⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣝⢿⣿⣿⣿⣿⣿⣷⣄⠙⢛⣬⠻⣿⣮⣝⣃⡸⠿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣾⣭⣿⣿⣿⣭⣽⣾⠿⠋⠀⠀⠉⠙⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢯⣫⡻⣷⢤⡀⡴⠋⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⡳⣕⣝⢮⡻⡻⣷⣄⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⣻⢽⡾⣯⡳⣝⢿⣮⡻⣝⢦⡀⠀⢈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣝⢿⢿⣾⣗⣯⣗⢿⣿⢷⣝⣮⡺⣷⣽⣷⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣟⢞⢮⡻⡳⣝⣟⣿⣺⢽⣺⣝⡺⢿⣟⡾⣷⣿⣯⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡻⣮⡻⡳⣝⢾⢮⡻⣿⣿⣿⡿⣷⣮⣗⡺⢭⣓⠿⢽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣍⢿⡯⣷⢽⡻⢮⡳⣕⢝⢮⡻⣷⣝⢷⣯⣟⡿⢿⣺⣭⣗⣲⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣝⢮⡳⡙⢾⣽⡿⣗⣯⣛⠷⣝⣿⡪⣷⢝⣮⣳⣯⣝⣺⠭⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣜⢮⣧⡑⣝⢦⡙⢯⣟⣿⣻⢽⣲⣭⣛⡺⢿⣿⣿⣷⣿⣮⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⢯⣳⣝⢿⣮⡳⢝⢮⡻⢿⣺⢽⣺⢿⣻⣯⣷⣶⣭⣝⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢯⣻⣿⣿⢿⣾⣝⡳⣽⣿⣦⡑⢽⡦⣙⢿⡺⢽⣒⠯⣟⣻⢯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⣻⡳⣝⢟⢿⣿⡯⣗⣯⣷⣭⣛⠾⣶⣝⠮⣷⣍⡳⣾⣭⣓⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⡳⣝⣝⢮⣫⡳⣝⢟⢷⣽⣻⢽⣿⢽⣶⣭⣝⡺⠿⣿⣦⣝⡾⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣝⡾⣯⡳⣝⢮⣮⡳⣵⣟⢾⣽⣺⢽⣚⠯⣗⣿⣿⣖⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣟⢿⣲⣝⡺⢷⣝⢮⡪⣷⣝⣿⡻⣝⡺⠭⣓⡾⢽⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣾⣿⡿⣷⡮⣝⡻⢮⣷⣽⣿⣿⣝⣿⣲⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⠀⠀⠙⠯⣗⡯⣗⣯⣷⣮⣭⣛⡻⠿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠈⠛⣷⡯⣟⡻⢽⣿⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠞⠁⠙⠳⣯⣽⣲⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⢤⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⢸⣧⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⣼⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢸⣿⢰⣿⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣄⢠⣴⣿⣟⢸⣿⢸⣿⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣿⢸⣿⠿⠿⠿⠿⢸⣿⣧⣴⠶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢿⣯⣸⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⣦⣴⣿⡇⣿⣿⢰⣶⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣇⣿⣿⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣉⣛⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣭⣍⡇⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⡍⣿⡇⣿⢙⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣼⡇⣿⢸⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡇⣿⢸⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠶⠾⠶⣷⢸⣿⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠻⣶⣿⣨⣿⠸⠿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠻⠿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣛⡛⣿⣾⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⣿⡇⣿⣶⣶⣶⣶⢸⣿⡟⠻⠶⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋⠁⠻⣿⣯⢹⣿⢸⣿⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠙⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢸⣿⠸⣿⢸⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢸⡏⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⢻⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣄⡤⠀⢠⠖⡄⠀⠠⣆⠀⠀⡶⠄⠀⢠⠀⠀⡶⠀⠀⣤⡀⠀⢠⠀⠀⠀⢸⡇
⠀⣟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡧⠤⠤⠤⠥⠤⠤⠤⠭⢤⠤⠬⠭⠤⠤⠥⠤⠤⠤⠤⠤⠤⡤⠤⠤⠥⠤⠬⠥⠤⠤⢼⠃
⠀⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⠀⠀⠐⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠇⠐⠀⠐⠀⠀⠂⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⠿⠂⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠂⠀⠂⠀⠐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠐⠀⠐⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇⢰⠒⠒⠒⠒⠒⠒⠒⠒⠂⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣀⠀⠀⢀⣀⠀⠀⢀⣃⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⠛⣳⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣠⠴⠚⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣋⡁⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⡴⠊⠁⠀⡤⠚⢻⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢞⣭⠞⠉⣠⠏⠀⢀⣤⠾⢩⢧⠀⠀⡴⠃⠀⠀⠀⠸⣄⡤⠊⠀⣀⡤⠤⢤⡇
⠀⠀⠀⠀⢀⡴⠋⠠⣿⣯⠴⠊⠁⣠⠴⠋⣠⠗⠁⡨⢧⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⡠⠔⠋⢹
⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⣠⠴⡫⢅⣤⠞⠁⡠⠚⠡⢩⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⢸⣁⣀⣠⣤⠶⠚⣉⠴⠑⡨⣚⠥⢒⣥⣴⢗⠕⡜⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠈⠁⠀⠈⢯⣥⠤⠒⢋⠡⢐⣨⡷⠟⠋⡡⣊⢔⡕⢀⢷⡀⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡓⠛⠛⠉⠁⠒⣈⡤⠞⠉⡠⡪⢊⡼⢋⢿⣄⣀⣀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣘⠫⠅⡊⠥⡠⢔⠅⡪⠔⣩⢗⠧⢊⣿⠀⢃⡷⠀⠀⠀⠙⠒⠚⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡔⠈⠊⠥⢚⣋⠥⢒⢞⣑⡵⡫⡽⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠅⠀⡰⠖⢉⡠⢞⡩⢞⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣖⡂⠨⠥⣐⣾⡭⢖⣫⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⠈⠛⠛⣉⣩⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢤⣬⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠴⠋⠉⠀⣰⠿⠿⢿⡿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠔⢁⣤⣶⣶⢌⠻⢷⣊⠈⠙⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⢠⢣⠀⡀⢀⡠⢂⣉⣤⣾⠻⣿⣿⣿⣿⣿⣿⣦⡝⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣇⣾⣿⣵⢋⠴⠋⠈⢿⡟⠆⠸⣿⣿⣿⣿⣿⣿⡛⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⡈⣱⠊⠀⣀⠀⢈⡇⠀⠀⢹⣿⣿⣿⣿⣿⡇⠏⢹⡀
⠀⠀⠀⢀⣠⡴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣋⠘⠛⠁⠙⠛⠛⠛⠀⣤⣾⣿⣿⣿⣿⣯⠇⢸⣽⠃
⠀⣠⡶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣟⣶⠶⠆⠀⠂⠀⠀⠉⠻⣿⣿⣟⣿⡏⠀⣾⡏
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⢻⣿⣄⣀⣀⡀⢀⣀⠀⠀⠀⠙⣿⣿⣿⡶⠁
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡔⠀⠀⠀⠀⠀⠀⠀⠻⢦⡽⡿⠋⠉⠁⠀⠀⠀⢀⠀⠀⠙⣷⠟⠁
⢈⠻⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠈⡀⡺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⢀⡀
⠈⠻⣦⣬⡉⠛⠛⠛⠛⠛⠛⠛⢉⠉⡤⠀⣾⠀⠈⠀⠀⠀⠀⠀⠀⠸⣤⢀⠀⡀⠀⠀⠀⠀⢶⠀⠀⠀⠈⣧⣄⡠⠒⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⣷⠄
⠀⠀⠈⠙⠲⠼⠄⠀⢀⣃⠈⠈⠄⠐⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢄⣠⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣄⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠦⣤⣤⡄⠀⠀⡀⠀⠀⠀⠙⢦⣤⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⠀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⡤⠤⠖⠲⢾⡉⠉⣩⠝⠛⠋⠙⠓⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠈⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⣠⠞⠁⠈⠳⣄
⠀⣀⡀⠀⠀⣰⠃⠀⠀⠀⠀⠈⠳⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢸⡷⣄⣠⠞⠁⠀⠀⠀⠀⠘⣧⠀⠀⢀⣀
⠀⠈⠉⠛⠲⠇⣼⠳⢤⡀⠀⠀⡰⠋⡇⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢰⠇⠈⢧⠀⠀⠀⣀⡤⠞⢧⠸⠒⠋⠉
⠀⠀⠀⠀⠀⣸⠃⠀⠀⠉⠳⡶⠁⠀⢳⠀⠀⠀⠈⠻⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡞⠀⠀⠈⢧⡴⠚⠉⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢰⠃⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⡠⠧⣀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢹⡀
⣀⣀⣀⡀⠐⡇⠀⣴⠤⣤⡎⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠉⠀⠀⠈⠉⠓⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⣇⣠⡤⠴⢦⠀⢸⡇⣀⣀⣀
⠀⠀⠉⠉⠉⠁⢰⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⡆⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⣳⡀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠀⠀⠀⠀⠀⠀⠀⢈⡟⠉⠀⠀⠀⠀⠀⠀⡼⢀⣀⣀⡀⠀⣧
⠀⠀⠀⠀⢀⣀⣸⡀⣾⠉⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠁⣿⠀⣏⣀⣀
⠀⠀⠈⠉⠉⠉⠀⠀⣿⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⣸⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⢀⡟

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣄⡀
⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⡇⠀⠀⠀⠀⣷⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶
⠀⠀⠀⠀⠛⠛⠛⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⢛⡃⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠠⠤⠧⠼⠀⠀⠤⠤⠤⠄⠀⠠⠼⠧⠄⠀⠠⠤⠤⠤⠀⠀⠴⠤⠮⠄⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠰⠲⠒⠲⠀⠀⠖⠾⠖⠂⠀⠰⠖⠒⠆⠀⠐⠶⠒⠒⠀⠀⠖⠲⠷⠂⠀⠐⠲⠒⠆
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣄⣠⣠⣠⣤⣠⣠
⠀⠀⠀⠀⠀⠠⠤⠤⣬⠥⢤⠄⠀⠀⢰⡆⠁⠀⠁⠈⠀⠀⠀⢉⡉⠀⣶⠀⠀⠀⠤⢤⢧⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠜⠓⠐⠂⠀⠀⠐⠀⠀⠀⠐⠀⠛⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣴⠏⠙⣧⡀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠀⣼⠟⠁⠀⠀⠈⠻⢦⣄⡀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⢿⡄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠀⡇⢸⢿⣄⠘⢷⣄⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣷⣇⠀⡇⠸⡄⢻⢷⣄⠙⠳⠶⣤⠾⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢹⡀⣿⣝⢷⣽⡀⢱⡈⢦⠹⣿⢦⣤⣀⣀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠁⢧⠈⡇⣿⣌⠑⠯⣻⢶⣷⣄⣳⣌⣳⣌⣉⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣧⡘⡆⠸⡀⢱⡸⣧⡙⠢⢄⣉⠛⠛⠿⠿⢿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠛⡆⢿⡛⢷⣄⢳⡀⠳⡙⢿⡓⠦⢌⣉⣙⣛⣻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠸⡄⢧⢸⣏⠲⢝⡻⢷⣄⣙⣦⡙⢶⣤⡤⣬⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⢧⠀⢧⠘⡆⢿⡓⠦⣉⠒⠮⢭⣍⣛⣛⣉⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣏⢳⣜⡄⠈⢆⠘⢎⢿⣦⣄⠉⠓⠒⠦⢤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⢝⡿⣦⣈⣣⡈⠳⣝⢷⣭⣓⣒⣶⠟⠁
⠀⠀⠀⠀⠀⢀⣴⠏⠀⠻⣦⣌⡓⠯⢽⣛⣛⣛⣛⣛⣩⡿⠁
⠀⠀⠀⠀⠀⢸⡇⠳⠀⠀⠈⠻⢯⣕⣒⠤⠤⣬⣥⡾⠋
⠀⠀⠀⠀⠀⠈⢷⣄⠠⠀⠀⢄⢀⣼⠟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣤⣶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠢⠄⣀⠀⠀⠀⠀⡎⠀⠑⠢⠄⣀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠉⠁⠸⠉⠒⠤⣀⡀⠀⠀⠉⣉⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢀⡀⠔⠗⠒⠒⠒⠒⢈⠏⠢⢀⠀⠀⠀⠀⢁⡜⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢫⠉⠁⠀⠀⡀⢾⠤⢀⢀⠀⡜⠣⡀⠀⠈⠐⠂⡴⡎⠀⠀⠈⢄
⠀⠀⠈⠒⠠⢀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⡱⢖⠂⠉⠀⠈⡄⠀⠀⢀⠃⠀⠈⠐⠀⣤⠊⠀⠘⡀⠀⠀⠈⠓⢄⡀⠀⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠈⠳⠂⠤⡄⠊⠁⢀⡠⠚⠀⠀⠱⣤⠄⠊⢩⠁⠀⡝⢄⠀⠀⣠⠎⠀⢇⠀⠀⠈⠦⡀⠀⠀⣀⠌⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢙⠛⠡⣄⢀⡀⡠⠎⠈⢢⡄⠴⠖⠲⢇⠀⡡⠊⠀⠣⠀⠀⠢⣀⢀⡠⠜⠎⠉⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠸⠀⠀⡇⠈⠍⠲⠤⣤⢊⠈⢦⢃⡘⡠⠊⠣⣀⡀⠀⢵⠖⠂⡍⠀⠀⠘⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⠆⠀⡘⠀⡰⠀⣠⣁⣸⠀⠬⠑⣿⣿⠔⣂⠉⡍⠀⠀⠇⠀⢀⠁⠀⠀⢀⠀⠀⠀⢸
⠀⠐⠂⠀⠀⠐⠚⠧⡒⠂⠛⢆⠘⢇⠉⠙⢄⠙⠄⠀⢱⡤⠊⡌⢡⡡⡔⠉⠉⢒⠴⠦⠄⣈⡀⠀⠀⠈⡄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢣⠀⢃⠀⠘⡀⢨⠴⠁⡈⠱⡆⣸⡀⣈⣦⡰⠁⠀⡠⠒⢉⡨⠛⠉⣐⠞⠢⢤⠄⣱⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠃⠈⠀⡠⠷⠁⣀⠀⠈⠲⠤⠄⣇⠀⠀⠱⣄⠎⠀⡰⠁⠀⣠⠊⠀⢀⠎⠁⠀⠀⠃⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢨⠴⠫⢄⡀⠀⠀⠑⡄⠇⠀⠀⠘⡀⠀⣀⠈⢣⡜⠀⠀⡐⠁⠀⡠⠁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠥⡀⠀⠀⠈⠢⡀⠀⢸⠤⠤⠤⢀⡗⠉⠀⠀⠀⠈⢆⡰⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠤⡀⠀⠈⠆⠀⠀⠀⠀⠀⠹⡤⠒⠉⠉⠉⠈⠳⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⡧⠐⠒⠒⠒⠠⣧⠄⠒⠒⠒⠒⠠⠌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠃⠀⠀⠀⠀⠀⢘⠀⠀⠀⣀⣀⣀⣀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠤⠐⠒⠀⠀⠐⠒⠖⠊⠁⠀⠀⠀⠀⠈⠉⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠇⠀⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣾⡆⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⢸⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣀⣀⣀⣸⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⡏⠉⠉⠉⢹⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣧⣤⣤⣤⣼⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⣀⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡃⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡀⠀⠀⠀⠀⠀⢸⣩⠏⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡯⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠉⠉⠁⠈⣉⠑⢢⣿⡘⡄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⢌⣻⡇⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠀⠣⠀⠠⢻⣿⢃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠇⢻⡄⠀⠀⠀⢀⢀⡜⠘⢯⢗⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠣⠀⠀⠀⠘⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠑⡖⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠏⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡘
⠀⠀⠀⠀⠀⢀⡀⠀⠠⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡔⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢀
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⣀⣠⠤⠄⠨⠈⠨⠉⠀⠀⠀⠀⠈⠉⠉⠁⠀⢠⠈⠀⠉⢻
⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢀⡟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⡀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡾⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢡⠀⠀⠀⠀⠀⠀⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣆⠀⠀⠀⠀⠀⠀⡼⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢇⠀⠀⠀⠀⢸⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡹⡄⠀⠀⠀⠀⢸⣇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⡄⠀⠀⠀⡾⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢹⡀⠀⠀⠀⢻⡟⢿⡀⠀⠀⠀⢀⣀⣤⣤⣄⠀⠀⠀⠀⣼⠁⡜⠁⠀⢀⡼⣷⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣶⢄⡀⠈⢳⣄⠻⣄⠀⢠⡿⠛⠉⠉⠈⠻⣆⠀⣰⢣⠞⠀⢀⣴⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣕⠠⡙⣦⠹⣤⡿⠁⠀⢀⣀⣀⣀⣘⣶⡣⠋⣀⣴⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣝⣷⣼⣷⠾⠛⠛⠉⠉⠉⠉⠉⠻⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣽⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣶⣤⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⡤⠊⢰⠏⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠐⡄⠙⠷⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡽⠀⠀⡟⠰⠁⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠘⣆⠘⣄⠀⠘⣧⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠃⠀⢸⣷⡇⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠘⣿⢛⡇⠀⢸⣆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⡀⠀⢸⡆⢧⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣧⠀⡇⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⢿⣈⣇⠀⠀⠀⠘⢿⣷⡀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⣿⣠⣧⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠈⢻⣍⢧⡀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⢠⡏⣼⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠪⢒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⠐⠒⠶⣶⣄⡀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⠀⠉⠻⢷⡂⠤⠒⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⢼⡇
⢀⡄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⢸⣦⡀⠀⠀⠀⠀⢀⡠⠛⠲⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⡈
⡟⠁⠀⠀⠘⡄⠀⠠⠄⠀⠀⠀⠀⣻⠀⠈⠳⡀⠀⢠⠔⠁⠀⠀⠀⠀⠙⢄⢀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠁
⢹⡄⠁⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢳⣤⣶⣷⣦⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠁
⠀⢹⡄⢰⣄⢤⣼⡄⠀⠀⠀⠀⠀⣸⣽⡄⠀⣀⢀⠆⡶⣄⡀⢐⡤⠀⣀⣀⣨⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠃⠀⠀⠀⠀⠀⣰⠄
⠀⠀⢻⣿⣷⡈⢿⣿⡆⠀⠀⠀⣠⡿⠻⠃⠚⠉⠉⠈⠐⠋⠻⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡴⠿⠟⢋⣡⣤⣶⠄⠀⢀⣠⡴⠞⠋
⠀⠀⠀⢿⣿⣷⡈⢻⣿⡄⠈⠈⠍⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠉⠉⠣⠛⡄⠀⠀⠀⠀⠀⠘⣧⣴⣾⣿⣿⣿⡿⠷⠛⠉
⠀⠀⠀⠈⢿⣿⣷⣼⠛⡇⢀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⢸⡉⠀⠀⠀⠠⡙⣶⣤⡀⠀⠀⠀⣀⣀⡀⣠⡴⠔⠋⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⢢⣼⠿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⣿⠈⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⣠⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡱⠂⠈⠛⢿⣿⣤⣌⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠚⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢆⠀⠙⠻⣿⣧⣄⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⣸⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⢀⢂⢀
⠀⠀⠀⠀⢺⢆⡀⠀⠀⠀⠀⣀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⠀⠈
⠀⠀⠀⠀⠀⠢⣍⣿⣷⣏⣍⣇⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣿⣇⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⠛⣿⡟⠛⠃⠀⢸
⠀⡠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠤⠤⠤⠤⠉⠡⠤⠤⠤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀
⠀⠢⠠⡤⠄⠠⠠⠠⠤⠄⠀⠠⠤⠄⠠⠀⠀⠠⠠⠀⠀⠀⠤⠠⠀⠠⠤⠄⠀⠀⠀⠠⠤⠤⠠⠀⠄⠤⡄⠠⠌
⠀⠀⠀⡆⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⠲⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠎⠀⠀⠀⠈⠇⠀⠀⠎⠀⠀⠀⠈⠇⠀⠀⠎⠀⠀⠀⠈⠇⠀⠀⠏⠀⠀⠀⠈⠇⠀⠀⡇
⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⠸⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⣸⢀⢀⠀⢸⡄⠀⠀⣸⠀⠀⢀⢸⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⡇⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⠀⢰⠒⠒⠒⢲⠀⠀⢀⡔⠒⢲⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢘⣠⡮⠤⠤⠤⢥⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠞⠀⠀⠀⠘⠆⠀⠀⠞⠀⠀⠀⠘⠆⠀⠀⠞⠀⠀⣀⡾⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣿⣿⣷⣶⣶⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠋⠛⠒⠐⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⣸⣀⣀⣀⣸⡀⠀⠀⠀⢸⠀⠀⠀⢰⣄⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠑⠀⠂⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⣿⢈⣽⡀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⢸
⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠍⠁⠀⠀⠁⣿⠿⠋⢡⡾⢋⣴⣦⣌⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣡⣴⣤⡙⢷⡄⢸⡀
⡖⠒⠒⠓⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠚⢇⠸⣿⣿⡿⢸⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⡇⢿⣿⣿⠇⠘⠓⠒⠚⠒
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠬⠥⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠬⠅⠂
⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠿⠇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢈⡑⣀⡀⠀⠀⠀⠀⣨⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⠒⠲⠲⠲⠄⠤⠤⠤⠤⠤⢤⣤⠤⠤⣿⣿
⠀⠀⠀⠀⠈⡋⠨⠍⣡⡴⠖⠺⠥⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⡀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⢰⠂⠀⠄⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡅⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠗⠤⠦⠴⠴⠶⠤⠤⠴⠤⣤⠤⠤⠤⠴⣿⡟
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠿⠿⠇⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣴⣿⡟⢛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠻⡿⠿⠿⠟⠻⡅⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⣀⠀⠄⠀⠔⠊⠱
⠀⠀⠀⠀⠀⠀⢸⣃⣀⢀⠀⢨⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⣦⣤⣤⣶⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡶



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢣⠉⠂⠤⢀⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡟⡗⢄⡀⠀⠀⠀⠀⢀⡠⣾⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠇⠹⡀⠈⠙⠓⠚⠋⠉⡼⠁⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⠒⠉⠀⠀⠀⠀⠀⢀⠌⠀⠀⠳⢄⡀⠀⢀⣀⣴⠁⠀⠀⠀⠑⢄⠀⠑⠤⣀⠀⢀⣀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⠑⡦⢤⣀⠀⠀⠀⠀⠀⠀⢀⣦⡈⠉⠁⣠⠋⠂⠀⠀⠀⠀⠀⠉⣒⡤⠴⢚⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⡄⠀⠉⠐⠤⠀⢀⡀⠈⠈⡀⠀⢡⠇⠀⠀⠀⣀⣤⠴⠚⠉⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⡠⠴⠊⠩⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠠⡀⢂⢟⡑⠂⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⠄⠀⣀⣇⠀⠀⠉⠱⡀⢀⣈⣀⢡⡄⠀⠀⠙⠛⠤⣀⡀⠀⢀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣀⡤⠞⠚⠋⠀⠀⠑⢆⠀⠀⡇⠈⠁⠀⠈⢧⠀⠀⡠⠂⠀⠀⠉⢓⡢⣥⣀⠀⢧
⠀⠀⠀⠀⠀⣀⡠⠶⠛⠛⠓⠒⠶⢤⣀⠀⠀⠀⠈⡆⠀⣷⠤⠶⠦⢤⡘⣆⠎⠀⢀⡤⠒⠉⠉⢀⣀⠬⠽⠾⠷⢀
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⢱⠰⠁⠀⠀⠀⠀⠈⠻⡄⢠⠏⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢸⠷⠒⠒⠒⠲⠤⢤⣀⢳⡞⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢠⡇⠤⠒⠒⠀⠀⠤⢤⣈⠙⢷⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⠤⠤⠤⠤⠤⢄⡀⠀⠀⠙⠺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡿⢿⣿⡿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣿⣿⣿⣿⣷⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣦⣾⣹⠟⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠈⢻⣿⣧⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡁⣹⣿⣼⡿⣦⣾⡟⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠸⣿⣭⣿⣿⣦⣴⣾⣿⡛⣿⡟
⠀⠀⠀⠀⣠⣤⡶⠶⣦⣤⣹⣿⣿⣾⡁⣹⣿⡿⠁⠀⠀⢀⣴⡿⠛⢻⡏⠙⢻⣷⣿⣿⣿⣉⣿⣿⢛⣿⣿⡟⠁
⠀⠀⢠⡾⢿⣧⠶⠶⠶⣌⠻⣯⡙⠛⠛⠛⠉⠀⠀⠀⠀⣾⠟⠳⣦⣼⣷⣴⠟⠉⣿⡌⠙⠛⠿⠿⠛⠛⠁
⠀⠀⣿⢁⡟⢹⡿⠛⢶⣼⣷⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣿⣧⣠⣿⠷⠶⣾⡇⠀⣠⣤⣤⣤⣤⣀⠀⠀⠀⣀⣀
⠀⠀⣿⡈⣧⠘⣷⣤⠾⢡⡟⢸⣇⣴⣿⣟⣿⣦⡀⠀⠀⠹⣯⡀⢠⡿⠙⢿⣄⣠⣿⣷⡿⢿⡅⠀⠈⣹⡿⣷⡀⡞⢷⡬⢷
⠀⠀⠘⢷⣌⣻⡿⠶⠶⢋⣰⡟⢹⣷⣾⠉⢹⣶⣿⠀⠀⠀⠘⠻⢿⣧⣤⣾⠿⠋⢹⡿⠀⠀⣻⠶⢾⡋⠀⢸⣿⠙⠾⠵⠋
⠀⠀⠀⠀⠙⠛⠷⠶⣾⣟⣁⡀⠈⢷⣾⣟⣿⡶⠃⠀⠀⠀⠀⣰⡾⢿⣦⠀⠀⠀⢸⣷⡴⠶⠿⣤⡼⠷⢶⣴⣿
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣧⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠹⢷⣾⠟⠀⠀⠀⠀⢻⣦⡀⠀⣾⠆⠀⣠⣿⢏⣤⣤⣦⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⣽⣿⣿⠇⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠙⠻⠿⠿⠿⠟⢫⣽⣿⠋⠁⠀⠀⠀⠉⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⣯⣶⡿⠟⠛⠛⡿⢷⣦⡀⠀⠀⠀⣰⡟⢻⣩⠿⣧⠀⠀⠀⠀⠀⠀⢠⣿⠋⠻⣷⣶⣶⣶⣶⣤⠀⢀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⡿⠙⣷⡖⠚⣾⢧⡄⠙⣿⡄⠀⠀⢻⣉⡿⣜⢒⡿⠀⠀⠀⠀⠀⠀⢼⡏⠀⠀⣾⣿⡟⠉⢻⣿⣿⠛⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢸⣃⣿⠟⢻⣦⢿⡶⢿⣷⠀⠀⠀⠛⠶⠾⢿⣷⡶⠿⠿⠶⣶⡄⢹⣧⠀⠀⢻⣿⣷⣤⣾⣿⠏⠀⠀⣿⠇
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠘⣿⡟⠻⣏⢻⡷⠿⣇⡼⠁⣸⡟⠀⠀⠀⠀⠀⣾⡟⠙⣷⣠⣄⣴⠟⠻⣿⣿⣦⡀⠀⠙⣿⠟⠛⠁⠀⢀⣼⠯
⠀⠀⠀⣠⣾⣿⣿⠋⠙⣿⣷⣝⣿⣄⠈⣿⠳⠒⠛⢷⣴⡟⠁⠀⠀⠀⠀⢰⣿⠀⢀⣿⠁⠈⣻⡄⠀⣿⡎⠙⠿⣦⣼⣿⣀⣀⣤⣶⠟⠋
⠀⠀⢰⣿⠁⠉⣿⣶⣶⡿⠛⢿⡎⠙⠻⢿⣶⣶⠿⠟⠉⠀⠀⠀⠀⠀⠀⠈⣿⡟⠋⠻⢶⡶⠟⠛⢻⣿⠃⠀⠀⠁⠉⠋⠛⠋⠉
⠀⠀⢸⣿⣷⠾⣿⣤⣿⣧⣤⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣷⣷⣦⣼⣿⣦⣄⣸⣇⣀⣴⡟⠃
⠀⠀⠘⢿⣇⠀⣸⣿⠁⠈⣿⡿⠃⠀⠀⠀⠀⠀⠀⣠⣾⣿⠛⠙⢿⡟⠉⢹⣿⣿⣏⡉⠛⠙⠙⠉⠀⢠⣴⢶⣄
⠀⠀⠀⠈⠙⠿⢿⣿⡷⣾⣋⣁⠀⠀⠀⠀⠀⠀⢐⣿⡿⠿⣷⣶⠿⠿⣶⣾⠟⠛⣿⣗⠀⠀⠀⠀⠀⠸⣷⣴⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣿⣿⣆⠀⠀⠀⠀⣺⣿⣅⣀⣼⡇⠀⠀⠈⣿⣦⣤⣿⣿⡀⣀⣾⣾⡿⠿⠿⣷⣤⠀⢠⠴⣶⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣍⣽⡿⣿⠀⠀⠀⠀⢘⣿⣿⠟⠛⢿⣦⣤⣾⡏⠉⢹⣿⣿⣼⡟⣡⡾⣿⠓⢶⣌⢻⣧⢾⡒⣭⠓⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⣿⣿⡷⠋⠀⠀⠀⠀⠀⠻⢿⣧⣤⣾⡏⠀⢹⣿⣶⣿⠟⢫⣿⠀⣏⣰⡟⢻⡶⢿⠋⣿⡍⠙⠛⠋⠉⠻⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⣿⣿⣿⡿⠟⠋⠀⠈⢿⣟⠻⣍⣙⣿⣤⠟⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣤⣍⣹⣧⣶⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢐⣈⣢⣄⣀⣔⣁⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡁⠀⠀⠀⠀⡁⢀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡀⠀⡀⠀⠀⢀⠀⡀⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⡄⢀⡀⡀⠀⣀⣀⠀⢀⢀⡀⢠⣨
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢂⠀⡀⠀⡀⢀⠀⢀⠀⡀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⡀⠅⠀⠀⡀⢸⢈⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠉⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⢀⣀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠠⠀⠀⠀⡄⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⢹⠃⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡧⠯⠠⠤⠤⠤⠠⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠣⠀⠘⠓⠠
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠰⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡂⠀⠀⠠
⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⠀⠈⡄⠀⠐⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⡌
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣬⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣉⠉⠉⣁⡾⡅⠀⠀⠀⠀⢀⣀⠤⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠⠞⢀⣯⡉⢉⠿⡄⠙⢤⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣄⣴⣃⣀⡜⡸⣏⠿⣄⡨⡿⠉⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠉⡏⢹⣙⣿⠮⣿⣀⣧⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡧⠞⡿⢼⡔⠁⣠⠝⠋⢙⡲⠿⠒⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠋⠑⣶⠃⠀⣧⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠑⢦⡟⢉⠉⢿⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⡜⠉⠉⠙⢾⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣒⠒⠒⠒⠒⠒⠒⠲⠶⠦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠚⠧⣬⣉⣉⡉⠙⠛⠛⠛⠛⣉⣉⣉⡽⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡿⢶⡶⢦⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⡶⢾⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡧⢸⡇⢸⠀⣿⠀⡇⢸⠁⣿⠀⡇⢸⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠾⢷⣼⣤⣿⣤⣧⣼⣥⣿⣤⣷⡾⠷⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢾⣷⣶⣶⣶⣶⣶⣶⣾⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣈⣉⠉⠉⠉⠉⣉⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢉⣛⠛⠛⠛⠛⠛⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠀⠀⠀⠀⠀⣠⡾⡛⢳⡄
⠀⠀⠀⠀⠀⠀⢀⣤⢤⣤⣤⣀⣀⠀⠀⠀⣿⠀⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠀⣿⡀⠀⠀⢠⡾⠻⢌⢁⣴⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⢈⡇⠯⠝⣧⠀⠀⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⡆⠀⣿⡀⠀⣨⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣶⣶⡾⠋⠀⠀⠙⠻⠿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⠿⠛⠁⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠫⠭⢻⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣾⣶⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⡇⠀⠀⠈⣿⣿⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣶⣾⣷⣧⣤⣤⣬⣟⣻⣛⣷⣖⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣿⠀⠀⠀⠀⠀⠀⣩⣽⡿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢀⣀⣤⡶⠟⢋⣹⣾⣷⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣵⡚⠻⠻⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⡇⠈⠉⠓⠿⢧⣂⠿⣻⣾⣿⣿⣿⣿⣺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣇⠀⠀⠀⠀⠀⢠⣿⣿⣶⣿⣿⣿⣯⢽⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠢⠶⠔⠄⢤⣄⣟⣾⣿⣿⣿⣿⣿⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⡇⠐⠢⠌⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⣤⣤⣶⠿⣛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⡻⢷⣶⢛⢾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣿⠉⠟⠳⠦⣣⣹⣿⣿⡿⢿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣿⠀⠦⠤⠤⠴⣽⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⣿⣀⡀⠀⠀⠀⠺⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⡁⣉⣹⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⠉⠉⣋⣻⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠓⠚⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣭⡛⢯⣿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣩⣛⢿⣿⣿⣿⡷⠜⣿⣾⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠺⣯⣾⣷⣆⣬⠟⠁⠀⠀⠈⠛⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡺⢣⣮⣴⣄⣻⠟⠁⠠⣭⠝⠂⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢻⣯⣝⣙⣻⡿⠋⠀⠀⠀⠀⠀⠶⢯⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣷⣮⣦⣜⣩⡿⠃⠀⠀⢀⣀⡀⠠⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⢛⡟⠋⠀⠹⣄⠀⠘⠲⠝⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⣿⠀⠘⣇⠀⠀⠀⠙⡷⡤⠴⠾⠛⠯⡀
⠀⠀⠀⠀⠀⠀⡴⠋⢶⣄⠀⠀⠘⣆⠀⠈⠓⠶⠤⣤⠞⠀⢰⠂⠀⠀⠘⣆
⠀⠀⠀⠀⠀⡼⠁⢈⢣⣻⣆⡀⠀⠈⠛⠶⠶⠶⠚⡟⠲⠚⠁⢄⡆⠀⡀⢹⡄
⠀⠀⠀⠀⢰⠃⠀⠸⣿⠇⠻⢷⢠⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⡇⣰⣿⠀⣧
⠀⠀⠀⠀⢸⣶⣶⣄⠘⢆⠀⣾⠟⠀⠀⠀⠀⠀⢠⡇⣇⠀⠀⠀⠉⣿⠇⠀⢸⡄
⠀⠀⠀⠀⠀⢹⡇⠈⠙⠲⢤⣸⡇⠀⠀⠀⠀⠀⠸⣤⡼⠀⠀⠀⠀⢿⡤⠖⢿⠁
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⣤⡀⠐⠆⠀⣶⠀⠀⡼⠀⠀⢸⢿⣿⠀⠀⠀⠀⣠⢔⡎⢙⣲⡀
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⢸⣷⣴⢄⣾⣉⣻⠙⣶⣾⠿⣛⣛⢿⡿⠶⠶⠾⠿⠿⠶⠶⠶⠮⠗⠋⢀⡾⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠈⣿⠂⢸⠉⢑⢈⢇⢻⠋⢠⠋⣨⣤⡉⢶⡀⠀⢀⠚⠛⠛⠛⠉⠉⠓⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⡗⢺⣤⡼⢢⠔⡸⠀⣼⠞⢣⠀⠈⠲⣛⡿⣿⣀⣀⣤⣤⣤⣤⡄⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠉⠓⠺⣟⠧⢼⣾⡱⣏⣽⣷⡦⣿⣿⣿⡿⣦⠀⠀⠀⠀⠈⠉⢻⣷⣍⡵⣿⡀⠄⣤⡤⣄⠐⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⣝⠿⣶⡬⣤⣍⣷⠶⠾⠿⢳⠦⣤⣤⣤⣤⣀⣹⣼⣻⠀⡓⣄⠀⠀⠀⠀⠈⠉⠁⠐⠒⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣤⣄⣀⣀⣀⡎⣿⠗⠸⡓⠓⣿⡇⠀⠀⠠⠔⡠⠈⠱⢆⠀⠀⣽⣿⡿⢀⠁⠋⢦
⠀⠀⠀⠀⠀⢀⣀⡠⠤⠴⠟⠛⠛⠉⠉⠉⠉⢹⢹⣏⠀⠀⢸⣾⣾⡟⠛⠛⠓⠲⠮⠶⠴⠾⣧⣀⣈⣿⠓⠿⠴⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠲⢀⠀⠃⠄⠻⣼⣿⠀⢠⡏⢉⣯⠗⠀⠀⠀⠀⠀⣀⣀⣤⡤⠾⠟⠛⠓⠶⠤⣄⡀⠀⠀⠀⠤⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠈⠉⠛⣉⠩⠤⠀⠤⠒⠒⠉⠉⠀⠀⠀⣀⣀⡠⠤⢤⣒⣒⣮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⣀⣀⠠⠤⠤⣐⣒⣪⡭⠥⠖⠒⠚⠉⠉⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢄⣒⣢⡭⠥⠶⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣷⣦⣤⣴⣾⣿⣿⣿⣿⣷⣦⣤⣴⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠛⠉⠉⠉⠉⠉⠉⠛⠻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠉⢀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣄⡀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⢀⣶⣶⣶⣿⣿⠃⢀⣴⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣦⠀⠙⣿⣷⣶⣶⣦
⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⠃⢀⣾⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⡇⠀⣸⣿
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣧⠀⢹⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠀⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⢻⣿⣿⣷⣄⡀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠏⠀⣼⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⢻⣿⣦⡀⠙⢿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠟⠁⢀⣼⣿⠟⠛⠛⠋⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣦⣀⠈⠙⠛⠿⠿⠿⠿⠛⠋⠁⣠⣴⣿⣿⡇⠀⢀⣤⣄⠀⢀⣸⣿⣿⣿⣀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣴⣶⣿⣿⣿⣿⣿⣷⠐⣿⣿⣿⡿⠿⠛⠛⠛⠛⠻⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠟⠉⠈⠉⢻⣿⣿⣿⣿⡟⠉⠁⠙⠻⡿⠟⠁⢠⣿⠟⠉⣠⣴⣾⣿⣿⣷⣶⣤⡈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⠀⠀⠀⠀⠀⢰⣿⣿⣿⠏⢠⣾⣿⡿⠛⠋⠉⠛⠻⣿⣿⣦⠈⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡏⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠘⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⢿⣿⣧⡀⠀⠀⠀⠀⠀⣰⣿⣿⠇⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢿⣧⡈⠻⣿⣿⣶⣦⣤⣶⣾⣿⡿⠃⣠⣿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣈⠉⠛⠛⠛⠛⠋⢁⣠⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⠟⠿⣿⣶⣶⣾⡿⠟⢿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠸⣿⣿⡟⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠅⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⣀⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡀⢆⠀⠀⠀⢿⣿⣷⡀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡙⣿⣿⣎⢣⡀⠀⠈⠻⣿⡷⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣾⡻⡿⣌⠳⣌⠻⣿⣆⠙⢦⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢫⠟⢻⣿⣿⡿⢿⣌⠻⣿⣦⣈⢿⣷⣄⠙⠳⣤⣤⣠⣴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠫⡇⠀⠀⣿⣿⣿⣿⣶⣝⢷⣮⣻⣿⣷⣿⣿⠿⡶⢤⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢳⡀⠀⢷⡹⣿⣿⣿⣽⣦⣙⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠹⣦⡀⠙⠻⢿⣿⣷⣯⣽⣿⣇⣀⣤⠶⠚⠋⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⠻⣦⣄⡀⠈⠉⠉⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡙⠻⠿⠶⢶⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠒⠒⠉⠀⠀⠀⠀⠠⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠛⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⠄⠴⠄⠐⠂⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣠⠴⣋⡉⢉⡇⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣠⠞⣡⢾⣿⡷⠋⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡴⠞⠁⣸⠿⢿⡃⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⣠⠶⠚⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⢯⣀⡴⠊⠁⠀⠀⠙⠒⠒⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢚⣡⠖⠋⠉⠙⠲⠦⣄⣤⠴⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡥⠖⠋⠀⠀⠀⠀⠀⢀⡾⠉⠙⣆⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣞⡭⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⣠⠟⠀⠀⠀⠀⠳⣄⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⡽⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉
⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁
⠀⠀⢀⣠⠾⠛⠉
⠀⠐⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⠻⢷⣦⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⢠⣿⠋⠀⠀⡏⢻⠀⠀⠙⣿⡄⠀⠀⠀⠀⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠾
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡏⢰⡞⢚⠃⠘⡓⣲⠀⢹⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣿⡀⠉⠉⡇⢸⠉⠉⢀⣿⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⢿⣦⣀⣉⣉⣀⣴⡿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⢸⡿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣾⡇
⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇
⠀⠀⢸⣿⣀⣀⣸⣇⣠⣤⣤⣄⣼⣇⣀⣀⣿⣄⣤⣤⣤⣄⣄⣼⣧⣠⣤⣄⣄⣼⣧⣤⣤⣤⣤⣤⣤⣿⣄⣤⣸⣧⣠⣤⣤⣄⣼⣧⣠⣄⣿⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⢀⣔⣶⣢⡄⠀⠀⣤⣖⣲⣄⠀⠀⣠⢶⣶⢦⠀⠀⢀⠔⣶⠢⡀⠀⠀⡠⡶⣖⣄⠀⠀⣠⠖⡒⢢⡀⠀⢀⠔⣒⡢⡄⠀⠀⡴⡖⡲⢄
⠀⠀⠸⡿⣤⢿⡏⠀⠈⣷⡧⢿⡿⠀⠀⢿⣼⣼⣼⠃⠀⠸⡛⣤⢛⠇⠀⠀⢟⣄⡼⡻⠀⠀⢣⠚⠜⣡⠃⠀⠸⡘⠯⢃⡏⠀⠈⣗⠢⠓⡾
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⢠⣾⠿⣷⡄⠀⢀⣶⣿⣻⣦⠀⠀⣴⢿⣿⢶⡀⠀⢠⡞⣿⢓⣆⠀⢀⣶⣿⢿⣦⠀⠀⣰⢎⣟⢲⡀⠀⢠⣞⣿⡳⣆⠀⢀⡶⣏⡹⢦
⠀⠀⠸⣾⣦⣿⠏⠀⠈⢷⣟⣻⡿⠀⠀⢻⣼⣾⣼⠃⠀⠸⣏⣷⣉⠟⠀⠘⢿⣧⣾⡿⠀⠀⢳⣈⣚⡴⠃⠀⠸⣜⣳⣳⠏⠀⠈⢷⣱⣋⡾
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉
⠀⠀⢠⣮⠿⣵⡆⠀⠀⣴⣯⣿⣦⠀⠀⣴⢿⢿⢷⡄⠀⢠⣊⠿⣑⡆⠀⠀⣶⠟⢯⣦⠀⠀⡴⢋⢏⡳⡀⠀⢠⢎⣭⡕⡆⠀⠀⠔⠏⡝⢢
⠀⠀⠘⢟⣶⡻⠇⠀⠀⠿⣗⣿⠟⠀⠀⠹⣼⣾⡼⠁⠀⠘⢍⣶⣩⠇⠀⠀⢿⣦⣜⠟⠀⠀⠱⣉⣚⡴⠁⠀⠘⢌⣓⡡⠇⠀⠀⠯⣁⣉⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠚⣫⣿⣿⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡏⠞⠀⠀⠁⠂⠁⠀⠀⠺⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⡇⠀⠀⠀⠒⡀⠀⠈⡀⠀⠈⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠿⠿⠩⢽⣿⣿⣦⡁⠀⠀⠀⠔⣒⠾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣰⠀⠈⠀⠀⠙⠛⠱⣴⡤⠀⢒⣻⣟⡳⣼⠁⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⡠⣄⣀⣲⣺⠟⣉⡓⡥⢄⢨⠱⢜⣷⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⣿⢯⡛⠙⢟⣩⡄⢠⠉⢋⠢⢄⣠⣦⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡍⠀⠀⠀⠀⠀⠈⠿⣷⣴⡤⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⣴⣶⡾⠃⢀⡔⠀⠀⠀⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣾⣿⣷⣤⣶⡛⠀⠀⠀⠀⠀⠀⠀⣹⣿⣶⡄
⠀⠔⢀⢰⡖⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣷⡿⢿⣿⣿⣷⣶⣶⣶⣶⡿⣿⣟⣷⣿⣿⢷⣤
⠀⢱⠈⣠⠌⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣿⡿⣿⢿⣿⣷⣿⣿⡿⣿⣟⣽⣿⣿⣿⣿⣿⣿⣿⢽⡆
⢀⡠⠛⠃⢴⠄⢄⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣞⣿⣽⣾⣯⣿⣟⣿⢛⣿⣾⣿⣿⣻⣿⣽⣾⣿⠄
⠀⠈⠀⠤⣀⠀⠀⠰⠈⠆⠀⠀⠀⠀⣴⣿⣞⣿⣟⣿⣿⣿⣿⣷⣿⢻⣿⣿⣿⣿⣿⣿⡿⣿⣻⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠲⡀⠀⠀⠢⡀⠀⢰⣿⣮⣿⣽⣹⣿⣿⣷⣯⣻⣿⢿⣟⣾⡷⣿⣿⡻⣿⣻⣿⣭⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠲⣿⣽⣿⡿⣿⣾⣿⣿⣾⣿⣿⡿⢿⠹⣿⣷⣿⣟⣷⡿⣿⣛⣿⢛⣭⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⣼⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⠿⣛⣻⣛⣿⣾⣟⡻⣯⣻⣷⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣤⡴⠿⠿⠯⠿⣿⣿⣿⣿⣎⣿⣿⢿⡿⣿⣿⣿⣿⣿⡿⣯⣽⣿⣻⠟⣯⣿
⠀⠀⢀⡀⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠚⠿⣿⣾⣿⣿⣿⡿⣿⣿⣯⣿⣿⣿⣿⣽⣻⣾⣟⣿⡟⣻
⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⢐⠢⠤⠲⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠁⠛⣿⣩⣿⣿⡿⠿⠟⠛⠉
⠧⡀⠈⠁⠀⠀⠀⠀⠀⠠⡀⠀⡀⢁⠀⠀⠀⠠⠀⠒⠠⠤⢀⣀⣀⡀⠀⠀⣀⠀⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠱⡀⠀⠀⠀⠀⢀⠴⡶⢄⠀⠀⠀⠀⠀⡰⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣇⠀⠀⠀⢠⠋⢰⣷⠈⢣⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀⠀⠀⣶⡄⠘⡿⠀⣴⠆⠀⠀⣸⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢙⡙⢦⠀⠸⣷⢸⡟⢰⡿⠀⣠⠞⣉⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡳⣄⢹⣆⢃⣾⢁⡴⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⠿⠿⢿⣿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡶⠲⣦⢶⢺⣿⣤⠆⠐⢦⣽⣿⣶⢶⡶⠲⣶⣶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠿⠈⠃⠈⠁⠀⠀⠀⠀⣠⢺⣿⣵⠏⠈⢷⣽⣿⢢⡀⠀⠀⠀⠀⠉⠀⠊⠸⠿⢦
⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⣽⡛⢿⣡⣇⢀⣧⣹⡟⣻⣝⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⡋⠀⣰⠋⡔⢠⣿⣿⣾⣿⢧⡐⡌⢷⡀⠈⡛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠷⠊⠀⢰⡋⡸⠀⢿⠀⠙⠏⠁⢹⠇⠸⡈⣳⠀⠈⠲⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠀⠀⠀⠀⣼⡿⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠹⣿⡆⠀⠀⠀⠨⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠍⠁⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣻⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢀⣉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣁⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇




⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣄⣀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣰⠿⠯⣏⠀⢈⡯⠿⢷⡀⠀⠀⠀⠀⣴⡿⢿⣏⠁⠀⠀⠈⣹⡿⢿⣦
⠀⠀⠀⣰⣿⣿⣿⡿⠿⢿⣿⣿⣿⡆⠀⠀⢰⣿⣀⣠⡟⠉⠙⣧⣀⣸⣷⠀⠀⢠⣾⠏⠀⠈⢿⣆⣠⣀⣴⡟⠀⠀⠻⣷
⠀⠀⠀⢺⣿⣿⣿⣇⠀⣬⣿⣿⣿⡇⠀⠀⢸⣷⠉⠙⣧⣀⣠⡟⠉⢳⣿⠀⠀⣼⡏⠀⠀⢀⣾⣿⠿⠿⣻⣧⠀⠀⠀⢻⣇
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠹⣦⣴⡏⠀⠉⣗⣤⡾⠁⠀⠀⣿⡿⠿⠿⢿⣿⣇⠀⢀⣽⣿⠿⠿⠿⢿⣷
⠀⠀⠀⠀⠈⠛⠿⢿⣿⡿⠿⠋⠀⠀⣄⣰⣄⣠⠈⠙⠻⠲⠚⠛⠉⠀⠀⠀⠀⢹⣧⠀⠀⠀⢻⣷⣿⣿⣿⡋⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡿⠛⠛⢿⣷⠂⠀⠀⠀⢀⣠⣠⣠⣀⡀⠀⠻⣧⡀⣠⡿⠁⠀⠀⠘⣷⡀⣠⣾⠟
⠀⠀⠀⠀⠀⣠⣴⠶⠶⣦⣄⠀⠙⣿⣇⢀⠀⣸⣿⠃⠀⣀⣾⠿⣿⡋⢹⡿⢿⣶⡀⠈⠻⢿⣧⣤⣠⣠⣤⣼⡿⠛⠁
⠀⠀⠀⠀⣸⡟⢱⠒⠲⡎⢻⡆⠈⠉⡿⢿⡿⢿⠉⠁⢐⣾⡿⣤⣾⠟⠿⣧⡼⢿⣷⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢻⣇⡸⠤⠤⢇⣼⡇⠀⠀⠀⠀⢀⠀⠀⠀⠐⣿⣤⣼⣧⣀⣠⣿⣦⣼⣿⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠙⠷⠶⡶⠿⠋⠀⠀⠀⣤⣷⣾⣾⣧⠆⠀⠻⣿⣄⣼⠛⢻⣧⣸⣿⠃⠀⠀⠀⢐⣾⡿⠋⠋⠉⠙⢽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢀⣹⣿⠁⠀⠀⢹⣿⡁⠀⠘⠹⠻⢿⠿⠿⠛⠀⠀⠀⠀⠀⣺⣿⡏⠀⠀⠀⠀⠀⣹⣿⡗
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄⠀⠞⣿⣄⣀⣠⣾⡷⠂⠀⣀⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡢⠀⠀⠀⣰⣿⣿⠃
⠀⠀⠀⠀⣾⣿⣿⡿⠿⣿⣿⣿⣆⠈⠁⠟⠻⠙⠋⠁⠀⢤⣿⣿⣻⡿⣤⠄⠀⠀⣠⣶⣶⣦⡄⠘⠿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠐⣿⣿⣿⣧⣤⣾⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠐⢾⣾⡁⢀⣷⣿⠖⠀⢸⣿⡏⠉⣿⣿⠀⠀⠀⠁⠉⠉
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠟⢀⣠⣴⡶⣶⢦⣦⣄⠀⠋⣻⢾⡿⢾⠙⠃⠀⠘⢿⣿⣾⣿⠟⠀⠀⣀⣄⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⢠⣿⠿⡁⠀⡇⠀⣨⠿⣿⡄⠀⠀⠁⣀⣤⣤⣤⣄⡀⠀⠁⠀⢀⣰⡟⢿⣿⣬⣿⡿⢷⣆
⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⣃⣀⣈⡶⠛⣾⣁⣀⣸⣷⠀⠀⣴⣿⣿⠿⠿⣿⣿⣦⠀⠀⣲⣿⣿⠫⠊⠉⠑⠽⣿⣿⣆
⠀⠀⠀⠀⣼⡿⡛⢛⢿⣦⠀⣿⡇⠀⣨⠷⡶⠿⡀⠀⣸⣟⠀⢰⣿⣿⢱⡋⢙⢎⣿⣿⡄⠀⢺⣦⣿⣁⠀⠀⠀⢜⡷⣼⡗
⠀⠀⠀⠀⣿⣇⠄⠠⣸⣿⠀⠘⢿⣾⡁⠀⡇⠀⣈⣶⠿⠃⠀⠀⢿⣿⣷⣭⣭⣾⣿⡿⠀⠀⠘⢻⣏⣷⣯⠦⡵⣏⣹⡟
⠀⠀⠀⠀⠘⠻⠿⠿⠟⠁⠀⠀⠈⠘⠛⠻⠿⠟⠏⠋⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠋⠀⠀⠀⠀⠀⠋⠟⠿⠶⠿⠻⠁

⣠⡄⠒⠒⠒⠀⠘⠋⠉⠀⠉⠉⠉⠉⣭⣍⣙⣛⠛⢛⣋⣩⡭⠉⠉⠉⠉⠉⠉⠙⠁⠐⠒⠒⠒⢤⡄
⠑⠢⠭⠭⠍⠉⣻⣀⣀⣐⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⡂⣀⣨⡏⠉⠭⠭⠥⠔⠃
⠀⠀⡷⣦⠤⢤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⣤⠶⠶⠶⢾⠤⠤⢀⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠑⢄⣀⠤⠠⣄
⠀⠀⣗⠒⠒⠀⠤⠤⠤⠤⠤⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⠀⠠⠤⠤⠤⠄⠐⠒⣒⣘⣹⠀⠀⠀⠀⢀⠜⣡⠒⡄⡸
⠀⠀⡇⠈⠉⠉⠉⠙⠃⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠘⠉⠉⠉⠁⠀⠀⢸⠀⠀⠀⢠⠎⠀⠙⠺⣤⣃⣀⠀⢀⣀⣀⡀⠤⠔
⠀⠀⣧⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠤⢼⠀⠀⢠⠃⠀⠀⠀⡠⠃
⠀⠀⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⠀⢠⠃⠀⠀⠀⡰⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠃⠀⠀⠀⡜⠁
⠀⠀⣏⣉⣀⣐⣒⠒⠒⠒⠒⠒⠂⠀⠀⠠⠤⠤⠤⠤⠤⠤⠄⠀⠀⠐⠒⠒⠒⢒⣒⣀⣉⢉⣹⠃⠀⠀⢀⠎
⠀⠀⡧⠤⠤⠤⠤⠄⠉⣩⣍⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⠉⠠⠤⠤⠤⢴⠃⠀⠀⢠⠋
⠀⠀⣇⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠃⠀⠀⡰⠃
⠀⠀⡇⠤⠤⠤⠉⣩⣉⣉⣈⣉⣛⣁⣀⣀⣀⣀⣂⣀⣀⣀⣀⣙⣋⣉⣉⣉⡍⠡⠤⢤⠃⠀⠀⡴⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣼
⠀⠀⣇⡉⠉⠉⠉⠙⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠋⢡⠃⠀⢀⣎⣬
⠀⠀⣥⣀⣈⣉⣉⡉⠉⠛⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠘⠋⣉⠏⠀⢠⡏⠤⠴
⡴⣘⣓⡤⠤⠄⢀⣀⣈⣉⣉⣉⣉⣉⣉⡁⠉⠉⠉⠉⣁⣈⣉⣉⣉⣉⣉⣁⠏⠀⣰⠥⠤⢤⣒⣓⢦
⠑⠒⠠⠤⠬⠍⠉⢉⣌⣉⣉⣛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣙⠛⢉⠏⠀⡴⠩⠭⠥⠤⠐⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏

⠀⠀⠀⣠⠴⠖⠒⠂⣠⣄⠀⠀⠀⠀⢀⡤⠶⠒⠒⠲⢤⡀⠀⠀⠀⢀⣠⠤⢤⡤⡤⢄⡀⠀⠀⠀⠀⣠⠴⠒⢒⡲⠦⣄⠀⠀⠀⠀⠀⢀⠤⠆⣶⠦⣄
⠀⠀⢰⣁⡤⠴⠚⠉⠁⢀⡇⠀⠀⢠⡗⠒⠒⠒⠒⠒⢒⣿⠀⠀⠀⣿⡀⠀⠘⡐⠀⠀⢹⡆⠀⠀⠀⡷⣦⣝⣃⣀⣠⡴⡇⠀⠀⠀⠠⣟⠓⢲⡟⠒⣨⡇
⠀⠀⠘⢝⡓⠲⠶⠖⢚⡻⠁⠀⠀⠘⢏⡓⠒⠒⠒⠚⣉⠜⠀⠀⠀⠳⢌⣙⣛⣓⣒⡨⠝⠀⠀⠀⠀⠳⣄⣀⣀⣀⠤⠔⠃⠀⠀⠀⠀⠳⠅⠛⠻⠿⠋
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⢀⢜⢧⠀⠀⠀⠀⠀⠀⡠⢆⠀⠲⡄⠀⠀⠀⠀⠀⡴⡡⠒⠒⠪⠣⠀⠀⠀⠀⠀⠀⢠⢮⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⠕⠁⢸⠇⠀⠀⠀⠀⢸⡀⡨⢶⠁⢸⠀⠀⠀⠀⠀⡗⡀⠀⠀⢀⢆⠂⠀⠀⠀⠀⠀⡇⠀⠙⠦⣄⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡊⠁⢀⣠⠎⠀⠀⠀⠀⠀⠀⠙⠤⠤⠗⠁⠀⠀⠀⠀⠀⠙⠾⣦⣐⡭⠊⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⡼⠋
⠀⠀⠀⠀⠀⠀⣀⡠⠄⡀⠈⠉⠉⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢤⡀
⠀⠀⠀⢀⣴⣊⠁⠀⠀⢈⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⢀⢿⠀⠀⠀⢀⡠⠒⠉⠀⠈⠃⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⣠⡚⠁⠀⠀⢀⠇
⠀⢀⡔⠁⠀⠈⢳⢀⡴⠋⠀⠀⠀⣠⠊⠙⢵⣄⢀⠴⣫⠞⠀⠀⣰⣟⡳⣦⠀⠀⡠⠂⠀⠀⢠⠞⠁⠀⠀⠀⣠⢞⠕⠁⠀⠀⠀⣠⠊⠀⠑⢤⡀⡠⠊
⠀⢸⠀⠀⢀⣠⠞⠋⠀⠀⠀⢀⡜⠁⠀⢀⡠⢟⡵⠊⠁⠀⠀⣜⣿⣺⣽⣽⡧⠚⠁⠀⠀⢠⣇⡀⣀⡀⢔⡫⠚⠁⠀⠀⠀⠀⢰⡃⠀⠀⢀⠔⠋
⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⢦⣤⠖⠉⠊⠁⠀⠀⠀⠀⠀⠙⠾⠷⠛⠉⠀⠀⠀⠀⠀⠀⠓⠁⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⡤⣔⣒⣒⠢⣄
⠀⠀⠀⡠⠴⠤⠶⠤⠶⠦⣄⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠂⣎⠀⠀⠀⢱⠈⡇⠀⠀⠀⠀⢠⢶⣒⡉⠉⣏⢉⣀⢶⡄
⠀⠀⠀⠳⠤⠄⣀⣀⣠⠶⠞⠀⠀⠀⠀⢯⠲⠤⣤⣤⠤⢶⠃⠀⠀⠀⠀⠘⡧⠬⣭⣃⡭⠥⢊⠇⠀⠀⠀⠀⠘⠦⠤⠤⠥⢤⣤⣤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠐⠚⠁⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠐⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⠄⠠⠄⠐⠒⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⡆⠀⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠛⡺⠀⠂⠠⠤⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠴⠖⠂⠀⠀⠠⠤⠄⠋⠆⠀⠔⠘⠀⠀⡁
⠀⠀⠈⠀⠤⡉⠂⢄⣀⠀⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡥⠀⠚⠐⢻⠿⠷⢐⠐⠂⠀⠀⢄⡀
⠀⠀⠀⠁⠀⠑⢄⠀⠈⠑⠀⣀⡀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡐⠁⠀⢦⠈⠃⠀⠀⠄⠂⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠐⠀⠀⠁⠀⠈⠉⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⡔⠀⠀⠀⠀⠂⠀⠀⠀⠈⡄⠂⠀⢺
⠀⠀⠀⠀⠈⠆⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠒⠒⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠆
⠀⠀⠀⠇⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠢⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⡂⠀⠀⠤⠄⣡⣶⡀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠤⠖⠀⢙⣋⣧⡀⠓⠂⠀⢐⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⡘⠉⠀⠑⢄⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⢦⠀⠀⠀⠱⡀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠘⠀⠂⠠
⠀⠀⠀⠀⠀⠈⠀⠠⣄⠀⠀⣼⠀⡀⠊⠀⡈⠔⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠤⡱⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠳⢀⠈⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠤⠐⠀⠘⠁⣉⠄⠨⠿⠈
⠀⠀⠀⠀⠀⠀⠂⢀⠋⠀⢀⡄⠀⠀⠁⢠⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠐⠁⠀⡘⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡜⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠂⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠀⣀⠘⡄⠀⡆⢀
⠀⠀⠀⣤⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠘⣶⡊⠀⠀⠀⠈⠓⣷⣾⠕⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⣤⠔⠁⠀⠀⠀⢦⣤⠖
⠀⠀⠀⠻⠇⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠊⣿⠓⠄⠀⠀⠀⠱⣿⣿⢧⠁⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠺⣿⠦⠀⠀⠀⠀⠺⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⣀⣈⢑⣄⠀⣰⣽⡩⠭⣍⡐⣄⠀⠀⠀⠀⠀⠀⠐⢤⣀⣇⣼⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⣶⠏⠓⠐
⠀⠀⠀⠐⠉⠁⣼⡿⠟⠻⠿⠋⠑⡄⠀⠀⠈⠂⠀⠀⠀⢀⠀⠚⠛⠛⠟⠻⡒⠀⠀⠀⠀⠀⠀⠈⠀⠀⣼⣿⣿⡿⢶⡤⢄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠠⠔⠒⠛⠩⠏⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⠴⠖⠒⠒⠲⣄⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠻⢄⣀⣀⣀⡠⠞⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢚⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡆⢭⣉⣁⣒⣒⣒⣒⠒⣒⣒⡒⠩⢍⣒⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡯⠭⠬⠭⠭⠭⠥⠤⣴⣒⡶⠾⠓⣋⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠉⠉⠉⠉⣭⣭⣥⠴⠶⣶⣛⣻⠯⠕⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠿⠿⠿⠟⠛⠓⠚⠋⢉⣉⣠⠤⠔⠚⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢒⣒⣒⣒⣒⣊⣉⢭⠥⠤⠒⠒⠈⣁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠙⠛⣛⣛⣛⣲⡖⠒⠿⠭⠭⠭⠭⠄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠭⠤⠤⠤⠤⠤⠴⢶⣶⣒⡺⣶⠶⠭⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡉⠉⠉⠉⢉⣉⣉⣀⣀⠤⠤⢒⣪⢽⡦⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⢬⣍⣉⣉⣉⡉⠩⠤⠤⠶⠒⠋⣉⡤⢺⣦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢸⣥⣒⡂⠤⠤⠤⠤⠤⣤⢔⣒⡭⠤⣒⡽⠃⠈⡆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠈⠉⠙⠛⠛⠛⠦⠬⠭⠒⠒⠚⠉⠁⠀⠀⢠⡟⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣕⣢⠤⣤⣄⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⡤⢴⣺⠽⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠨⠭⠭⠭⠭⠭⠝⠒⠒⠋⠉⠀⠀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠶⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⣀⠔⠒⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⠀⠀⠀⠀⣰⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⣤⣴⣿⣿⣿⣿⡀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⣠⣦⣄⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣦
⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣤
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣆⣠⣤⣿⣿⣿⣄⣀⣴⣿⣦
⢀⣤⣤⣤⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣾⣿⣆
⠉⠻⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢠⣤⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧⣄
⢀⣤⣾⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡏⠉
⠀⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⠃
⠀⠀⠀⠀⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⠻⠿⠋⠀⣿⣿⣿⡄⠀⠀⢠⣶⣶⣤⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠈⣿⣿⣿⣿⡿⠿⠟⢿⣿⣿⣿⣿⠀⣠⣾⣷⣄⣠⣴⣿⣿⣿⣿⣿⣷⣿⣿⣿⡏⠀⠙⠛⠁⠀⠀⣿⣿⣿⠁⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠹⠿⠿⠛⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣾⣷⣄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⣼⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⡇⠰⠁⠀⠀⠀⠀⠀⠀⠀⠉⢟⣿⣿⣿⣿⣷⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣤⣤⣶⣿⣿⣿⣿⣿⡿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠟⠉⠉⠹⣿⣿⣿⠇⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⡠⢚⡹⢯⠓⢄⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠌⢀⣿⡇⠘⣧⡀⠱⡀⠀⠀⠀⠀⢺⠃
⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠈⠟⢄⠀⠀⢸⠀⢈⣿⠇⠨⢻⣇⠀⣧⠀⠀⠀⡐⡿⠀⠀⠀⠀⠀⣠⡤
⠀⠀⠀⠀⠀⠙⢀⢀⠀⠀⠀⠀⠈⠻⡁⠠⣼⠀⢈⣺⣿⠐⣞⡁⠀⣏⠠⢶⡦⠖⠁⠀⠀⠀⣠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢇⠠⠀⠀⠀⠀⠀⠐⠼⣧⡈⣩⣥⢀⣬⣥⣼⡧⠖⠉⠀⠀⠀⣀⢀⡦⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣅⠠⣤⡄⠀⣶⣮⣿⡟⢻⡿⠛⢿⣡⠦⣯⣈⣿⣵⣫⠽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢁⡴⣺⢿⣧⣘⢧⣤⣿⡽⣖⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⠼⢫⡞⣯⣸⣺⣸⡟⣯⠪⡓⢬⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⠛⢁⣣⠎⠐⡏⠙⠙⠁⠇⠈⠣⣴⢆⠈⠣⣕⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣫⠞⠁⢠⢎⠏⠀⠀⠘⠀⠀⠸⠀⠀⠀⠈⢣⡠⡀⠈⠳⢯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠟⠁⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⢂⠀⠀⠉⠢⣦
⠀⠀⠀⠀⠀⠀⠀⢠⡼⠁⠀⠀⠀⢺⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⠔⠋⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠁⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⠁⠀⠀⠁⠢⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠶⠲⠤⣀⠸⣄⣀⣀⣀⣀⣀⡀⠤⡏⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠂⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢄⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠠⠀⣀⣑⣅⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠻⡀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣠⠴⠒⠋⣉⠝⢓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣤⣠⠤⠖⠁⠀⢠⠏⠀⢀⠄⢊⠤⠊⠀⢠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠫⢍⡉⠈⠉⠒⢄⡀⠀⠀⠀⠀⢸⠤⠎⣀⣐⡁⠀⠄⢊⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠁⠂⠌⡑⠢⢄⡀⠙⢆⠀⠀⠀⠀⠑⠒⠶⠶⠶⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠐⠠⠀⢀⡀⢁⣢⡬⠑⠊⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠉⠻⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣀⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣶⣶⣶⣶⣶⣶⣿⡇⠀⣿⠀⢸⠁⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣉⣹⣋⣉⣿⣿⡟⠛⣿⠛⢻⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣉⣹⣏⣉⣿⣿⡷⠶⣿⠶⢾⠶⠶⣿⣧⣤⣤⣤⣤⣤⣤⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣉⣹⣉⣉⣿⣿⣇⣀⣿⣀⣸⣀⣀⣿⣿⡯⠤⢽⠥⢼⡧⠤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣹⣉⣹⡏⣉⣿⡟⠛⠛⣛⠛⣛⠛⠛⠛⣿⡗⠒⣺⠒⢺⡗⠒⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠉⠀⠉⠀⠀⠀⣿⣯⣭⣽⣍⣽⣯⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⣤⡄⣤⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠛⠛⠛⠛⠛⠛⠁⠀⠈⠉⠁⠉⠁⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀
⢀⣨⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢰⣶
⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣽⣿⣥⡀⠀⠀⡀⣸⢿
⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣆⠸⡟⢹⢸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣤⣤⣿⣿⣿⣿⣿⢾⡷⢾⣸⣤⣾⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣧⣸⣿⠀⢹⡇⢀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⣾⣧⠋⠉⠉⠛
⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣻⣿⣿⣤⡤⠼⠿⠿⠿⠃
⠉⢻⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢏⣿⣿⣻⣿⣿⣿⠋
⠀⢸⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣼⣿⣇⣿⣿⣿⣿⡀
⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⢁⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⡶⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣦⠀⠀⠀⠀⠀⠙⠓⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡅⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⢻⣄⠀⠀⠢⣤⣀⣀⠛⠻⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⣸⠐⡇⠀⠀⠀⠀⡠⠊
⠀⠀⠈⠛⢷⣆⡀⠀⣿⣿⣷⣦⣌⠙⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⢩⡄⠀⡠⠊⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⢿⣦⣌⣿⣯⣉⠂⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣻⣾⣄⡀⠀⠀⠀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⢏⢿⣿⡉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣱⡏⢸⣙⣦⣺⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣼⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⡤⣛⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠶⠔⠒⠋⠀⢀⣰⠃⠀⠀⠀⠰⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⡀⠀⠀⣠⣾⡇
⣼⣤⢀⡀⢀⡀⡀⠀⠀⠐⠊⠁⠀⠀⠀⣀⣤⠾⠛⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠁⠲⠦⠴⣱⣶⣿⣿⡿⠃
⢿⣿⣌⠉⠀⠤⠀⠀⠀⢀⣀⣠⠴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣐⡒⠖⣻⡯⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢹⠽⠟⠋
⠀⠉⠛⠿⠶⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣀⣀
⠀⠀⠀⢀⣶⠾⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⢟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁
⠀⠀⠀⡀⠀⠤⠶⠂⠐⠀⠀⠀⠀⠀⠀⠀⠘⢦⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣾⣴⠎
⠀⠀⢸⣧⡤⡴⠒⡶⡀⠀⡄⠀⠀⠀⠀⠀⠀⢻⡨⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⢸⣯⣧⡬⡤⠤⠿⡄⢧⠀⠀⠀⠀⠀⠀⠈⣷⣿⡄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⣿⡅⣿⠀⠀⠀⠀⠀⠀⢸⣯⣭⠉⠁⠀⠀⠀⠀⠐⠐
⠀⠀⠘⣷⣟⠀⡀⢀⠀⣤⣐⠀⠀⠀⠀⠀⠀⠀⠸⡾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⡇⢿⠀⠀⠀⠀⠀⠀⠸⣿⣿⠄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠹⣿⡴⡿⠿⡣⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢿⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⠷⣾⡄⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠙⢶⠖⠛⠛⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣰⣾⣿⠇⠀⠀⠀⠀⠀⠀⠈⢿⡽⠄⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣀⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠷⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⣀⣀⣀⢀⣀⣀⣀⠤⠔⠒⠉⠀⢰⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠈⠙⠒⠤⠄⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⡙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⢠⠋⢫⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⡇⠀⠈⠑⠢⢄⣀⡀⠀⠀⠀⢀⡔⠁⠀⠀⢣
⠀⠀⠀⠀⢀⣀⣀⠤⠔⠊⠁⠀⠀⠀⠀⢀⠔⠁⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⠀⠀⠀⠀⠀⠉⠉⠉⡹⠻⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⢹⠑⠢⣄⠀⠀⠀⢀⣀⡤⠒⠁⠀⠀⠀⠀⢀⢷⠤⠤⠤⠄⠒⠊⠁⢸⠀⠑⠤⣀⣀⠀⠀⠀⢠⠞⠀⠀⢣⠀⠀⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠙⡾⢍⡉⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠳⡀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠈⠉⢉⡝⢣⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠈⠑⠤⣀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠈⠓⠤⣀⣀⠤⠔⠋⠀⠀⠀⢀⠷⡒⠀⠒⠂⠉⡇⠙⠦⢤⡀⣀⡠⠋⠀⠀⢣⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⣀⣩⠱⡆
⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠋⠑⠢⣄⠀⢀⣀⠔⠋⠀⠱⡀⠀⠀⣀⡇⠀⠀⠀⢀⠎⢱⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⣈⡳⢦⠖⠊⠉⠀⢀⠎
⠀⠀⠀⡰⠁⠀⠀⠀⡸⠀⠀⠀⠀⠀⡇⠀⠀⠀⢹⠣⢄⡀⠀⢀⡴⠻⡉⠉⢸⠉⠒⢒⣶⠁⠀⠀⠳⣄⠀⠀⠀⣀⠨⢷⠖⠊⠉⠀⢀⠎⠀⠀⠀⠀⡌
⠀⢀⡔⠁⠀⠀⠀⡴⠁⠀⠀⠀⠀⣰⠃⠀⠀⠀⣸⠀⠀⠈⡟⠥⣀⡤⠻⡈⢩⠑⡲⢏⠈⠳⣄⣀⡤⠤⡿⠋⠉⠀⠀⡌⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⡇
⢤⠎⠀⠀⠀⣠⣊⣀⣀⣀⣀⣀⣴⣁⣀⣀⣀⣰⢃⣀⡀⣰⠃⢀⡜⠑⠢⣵⣬⣞⠤⠔⢟⠊⢹⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⡇
⠀⠳⡄⠀⠀⠀⠑⡄⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⢄⠀⠀⠉⡇⠉⢹⢉⠥⣻⢿⢟⠒⠒⡼⠦⢈⣆⡀⠀⠱⡀⠀⠀⠀⢱⡀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢱
⠀⠀⠘⢆⠀⠀⠀⠈⡆⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⠀⠀⢀⣧⡔⠋⠳⣴⠃⣘⢀⣱⣼⠀⢠⠊⠁⠉⠉⡳⠟⠢⠤⢀⣀⢳⡀⠀⠀⠀⠱⡄⠀⠀⠀⠀⢣
⠀⠀⠀⠈⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸⡠⠒⠉⠀⠉⠳⣤⣋⣀⣙⡃⣀⠬⠶⡇⠀⠀⢠⠊⠀⠀⠀⠀⠀⡠⠜⠋⠑⠒⠢⠤⢈⣦⡀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⢰⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⣀⠴⠊⠉⠉⠑⢤⡀⠀⢠⠃⠀⠀⠈⡏⠀⠀⠀⠈⢦⡀⡇⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠉⠉⠒⠂⢬
⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⠀⠀⣠⠼⠯⢄⡀⠀⠀⠀⠀⠀⠱⣠⣃⣀⣀⣀⠀⢃⠀⣀⠤⠐⠒⠹⢇⠀⠀⠀⡸⠁⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⣠⠔⠊⠁
⠀⠀⠀⠀⢸⠀⠀⠀⣇⡠⠒⠉⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⢠⠏⠀⠀⠀⠈⠙⢺⠋⠀⠀⠀⠀⠀⠀⠱⣄⠀⡇⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⢀⠇⣀⠤⠚⠉⠉⠑⠢⢄⠀⠀⠀⠀⠀⠀⠑⡄⢠⠃⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⢀⣀⠠⠤⡌⣢⡆⠀⠀⠀⡜⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠾⠮⣄⡀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⢸⠷⠔⠒⠒⠒⠒⠤⢄⡀⡇⣠⠔⠊⠁⠀⠀⠀⠀⠀⠈⢦⡀⢰⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠘⢆⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣆⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⢣⢀⠎⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠤⠖⠒⠒⠒⠒⠒⠠⠿⣄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⢈⠟⠒⠉⠁⠀⠀⠀⠉⠑⠒⢤⡀⠈⡄⢠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡟⠁⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⠀⠠⠤⠤⢈⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣎⡠⠤⠀⠒⠒⠒⠒⠠⠤⣀⡀⠀⠀⠀⢰⠀⠀⠀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⢄⠸⢀⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢀⣠⡏⢹⣄⡀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠼⠯⣿⠸⠦⡄⢠⠴⠇⣿⠽⠧⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠛⠛⣿⣀⣀⣛⣛⣀⣀⣿⠛⠛⠛⡟
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⢻⠀⢀⣀⣉⠉⠉⣉⣉⠉⠉⣉⣀⡀⠀⡟⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⢹⠀⢸⠀⢸⡁⣿⠀⢸⡇⢸⡇⠀⣿⢈⡇⠀⡇⠀⡏⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⠧⠼⠀⢸⠀⢈⣉⣉⠀⠀⣉⣉⠀⠀⣉⣉⡁⠀⡇⠀⠧⠼⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡶⢶⠀⢸⠀⢸⠉⣿⠀⢸⡏⢹⡇⠀⣿⠉⡇⠀⡇⠀⡶⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⠀⢸⠀⠈⠉⠉⠀⠈⠉⠉⠁⠀⠉⠉⠁⠀⡇⠀⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣉⣉⠀⢸⠀⠸⡿⠿⠿⠿⢯⡽⠿⠿⠿⢽⠇⠀⡇⠀⣉⣉⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡏⢹⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⡇⠀⡏⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠷⠾⠀⢸⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⡇⠀⠷⠾⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠾⠤⠤⠧⠤⠤⠤⠼⠧⠤⠤⠤⠼⠤⠤⠷⠤⠤⠤⠤⠼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠀⠀⠀⠀⣠⣶⠖⢦⣄⠀⠀⠉⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡀⠀⣞⡿⢳⡾⠟⢃⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡰⣧⡀⠐⠛⢳⡶⠛⡣⠈⠆⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⠈⢿⣿⣿⣯⣶⣀⣿⣺⣷⣿⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣄⠙⢿⣿⣿⣿⣿⣿⠿⠋⠁⣀⣀⣠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣿⣿⢿⣿⣿⡶⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠾⠿⣿⣶⢿⣿⣿⡿⣿⣿⣷⣦⣶⡿⠿⠿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣰⣿⣿⣿⣴⣿⣯⡿⢷⣶⣶⣶⣤⡀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⠀⢀⣿⠋⢿⡍⠉⠿⡥⠗⠀⠀⠀⠈⠙⣷⡀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⣠⠟⠋⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠈⢿⡄
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠖⠊⢹⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⣶⣶⣤⡀⠀⠸⢀⠀⠇⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿⣿⣿⣿⣶⣶⣮⣄⣲⣲⣷⣴⣶⣶⣾⣿⣿⣿⣗⣦⣸⣿⣿⣿⡆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡇⣿⣿⣿⣿⣿⣿⢷⣟⣿⣟⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⡇⣿⢿⣿⠛⠛⠛⠷⠿⠿⠿⠿⠛⠛⠻⣿⣿⡿⣿⣿⣿⣿⢻⢇⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⠏⡀⣿⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠀⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⡿⣧⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢤⡄⢤⣤⡤⢸⢣⣭⠀⠀⠉⠐⠒⠐⠐⠐⠚⠂⠀⣿⣽⣿⣛⣷⣿⣿⣿⠀⠦⡀
⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣍⣉⣍⣉⣉⣉⣁⣀⣀⣠⣁⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠭⠯⠽⠽⠿⣿⢿⣿⣿⣿⣿⣿⣻⢿⡿⠿

⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠘⠓⣶⣿⣿⣷⣼⣠⣤⡀⠀⠀⠀⠀⠀⠐⠒⠐⡆⡇⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⡴⠶⣿⣿⣿⣿⣷⣟⡴⢧⡀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣖⣠⠖⠚⠉⠀⠀⠀⠀⢀⣀⣨⠀⣰⠃⠀⠀⠀⢀⣀⡠⠤⠀⠒⠒⠒
⠀⠐⢁⣤⢾⣿⣿⣿⣿⣯⣀⠀⠙⣆⠀⣠⠋⠉⣿⣿⣿⠿⣟⠛⠑⢦⣀⠀⠀⡔⠊⠹⢋⣧⣦⣧⣀⣠⣔⣊⣁⡀
⠀⢀⡞⠁⢸⡟⣿⠟⠿⡇⠹⣇⠀⠀⠠⠋⠀⠀⠉⢩⠏⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⢰⣿⣿⣿⣿⠿⡍
⠀⡞⠀⠀⢸⠇⣿⠀⠀⡇⠀⠸⡆⠀⠀⠀⠰⠖⠊⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠟⢻⠀⠘⢄
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠋⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⢰⡄
⠀⠀⠀⠐⢀⠀⣀⣤⣄⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣠⣤⡄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⡇⠀⠀⣼⠑⡀
⠀⠀⠋⠲⡘⣴⣿⣿⣿⡇⡌⡰⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡥⠀⡇⠀⠀⡇⠀⣼
⠀⠀⠀⠀⣘⣷⣽⣿⣿⣥⣗⣁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢮⣿⣿⣿⡥⠞⠉⠉⠉⠀⠁⠀⠀⠀⠀⣞⠉⠦⣷⣾⣧⣧⠜⠉⣦
⠀⢀⡴⠋⢸⠙⢻⠟⠻⡟⢹⠉⠳⡀⠀⠀⠀⠀⠀⢀⡴⢒⣿⣿⣿⣿⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡞⣿⣿⣿⡟⣶⠞⠁
⠀⡌⠀⠀⢸⠀⠁⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢀⠖⠁⠀⡿⠈⠋⠉⠁⡇⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠻⣿⡿⠁⢸
⠀⠁⠀⠀⠈⡆⠀⠀⠀⠀⢨⠀⠀⠈⠆⠀⠈⠁⠀⠀⡞⠀⠀⠀⠀⠀⢡⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠖⠛⠛⠛⠛⠛⠛⠛⠛⠟⠲⠶⠦⣤⣀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡘⠦⠀⠀⠀⠀⠲⡿⢿⠿⠿⠷⠀⠀⠀⠀⠀⢀⣈⣷⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠛⠿⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠻⢻⣿⠀⢻⣤⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡦⣤⣤⣄⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣶⣾⠟⣡⠖⣿⡽⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣽⣷⣿⣿⣟⣿⣿⣿⣿⣿⣿⢿⣿⣿⠿⢋⡤⠞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣍⢽⣿⣿⣟⢟⠿⣭⣟⣯⣿⢫⣙⣻⣿⣷⣶⡏⠀⠀⠀⣀⣾⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢾⣵⣿⣿⣿⣿⣾⣷⣯⣝⣭⡻⢿⣻⣿⣿⣿⠃⠀⣠⠞⡿⠁⠈⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣟⣽⣿⣿⡿⠟⣩⣷⣾⣷⣶⣭⣭⣽⠀⣰⠃⠀⢷⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢟⣾⣿⣽⣿⠞⣫⣔⣛⣻⣭⣿⢯⣿⣿⣿⣿⣷⠏⠀⠀⠀⠑⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣦⢭⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣻⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⢹⣷⣖⣶⣶⣷⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⣸⣿⣿⣿⣿⣿⣽⣿⣿⣽⣿⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢰⣿⣿⣌⡙⠻⠿⠿⠷⠿⣿⠾⠿⠿⠿⠿⠿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣉⡙⡛⢒⠶⠶⠀⠦⠶⠤⠔⠒⠛⠋⠉⠉⠉⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⠬⠤⣥⣤⣄⣀⣀⣀⣀⣠⣤⡤⠶⠶⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠉⠁⠀⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢷⠀⠛⢢⡴⢅⡀⠀⠀⠀⠀⣜⠉⠀⢣⠀⠀⠀⠀⣎⠀⠀⠂⠉⠉⣢⡎⠉⠒⠒⠀⢰⠋
⠀⠀⠀⠀⠴⠊⠉⡑⠂⠀⠀⠀⠀⠈⠄⠤⢈⡀⠀⠀⠀⢿⠀⠀⠀⣧⠀⠀⠀⡎⠉⠉⠁⠀⡝⠓⠃⠄⠐⠒⢠⡞
⠀⠀⠀⠔⠉⠣⡀⢀⠤⠀⢀⠀⠀⠀⠠⣀⡤⠉⠀⠀⠀⠻⠢⠘⠓⠁⠀⢀⠔⡇⠀⠀⠀⠀⢱⠦⠤⠤⠤⠤⣾⠃
⠀⠀⠈⠀⠀⢀⡔⠁⠀⠀⢸⠀⠀⠀⠀⠜⠱⡀⢈⠀⠀⠀⠀⠀⠀⠀⠒⠁⠠⢇⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣯⠇
⠀⠀⠘⡀⠀⡜⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠤⠤⠄⢼⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠈⠂⢆⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣀⡀⠀⠐⡛⠂⠠⠤⠤⠐⣳⠈⠑⢆
⠀⠀⠀⠀⢀⡀⠙⠒⠀⠀⠐⠋⠓⠒⠊⣁⠀⠀⡠⠄⠠⢄⡀⠉⢶⠀⠀⡔⠉⣀⢀⡈⢢⠀⠈⠉⠁⠀⠀⠉⠁⠔⠒⠁
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⡐⠡⠀⡎⠀⡆⢰⠄⢱⠀⠈⣦⠀⢄⠐⠇⠸⠁⡰⠀⠀⠄⠀⠐⠀⡄
⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠐⠁⠁⠀⠣⡀⠃⠈⢁⡜⠀⠀⠀⠣⠀⠐⠠⠤⠒⠁⠀⠀⠁⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠆⠈⢆⠀⠀⠀⡰⢀⠆⠀⠀⠀⠈⢀⣀⠁⢀⣀⣀⡔⢀⣀⣀⣤⣤⣤⣤⣄⣀⡃⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⡄⠈⢆⠀⡀⠁⠎⠀⠀⠀⢀⠞⠁⠀⠀⠀⢸⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⢦⠁⡜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⢠⠤⢄⠀⠀⠀⠀⠀⡔⠉⠉⠑⡄⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠼⢶⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇⠣⣀⠰⠀⠀⠀⠀⠀⢇⡀⠀⣠⠇⠀⠀⠀⢸⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠈⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢘⣇⠠⠄⠒⠋⠉⠒⠢⢄⠀⠉⢉⠕⠚⠒⡄⠀⢸⠀⢀⠃
⠀⠀⠀⠀⠀⢀⠤⠒⠃⣌⠁⠀⠰⠀⠀⠀⠀⠀⠀⠉⠉⠯⢭⠋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⢄⣀⣠⠇⠀⢸⣀⣸
⠀⠀⠀⠀⠐⡡⠊⠁⠠⢸⠀⠰⢀⠨⠑⢦⡀⠀⠀⢄⠀⠀⠸⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⢸
⠀⠀⠀⡌⣰⠁⠀⢀⠀⡜⠀⠅⢪⠀⢀⠀⡇⠀⢀⣠⢴⢙⡞⡧⢤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⡅⢧⣀⠔⣡⠞⠀⠀⠈⠢⠬⠤⠚⠀⡰⠁⠘⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠑⠤⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠤⢄⣀⣀⣠⣤⣴⣤⣂⣀⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠏
⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⡚⠉⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢛⡭⠖⠉⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣋⠴⢚⡩⠀⠀⠀⠀⠀⣀⣤⣶⠾⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠐⣈⡤⠚⠁⠀⠊⢁⡤⢖⡫⠝⠈⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁⠀⣀⡴⠚⠁⠀⠀⣀⡴⣞⠵⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⡵⠊⣀⠴⠚⢁⠀⠀⣀⣴⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣤⡀⠈⢧⡀
⠀⠀⠀⠀⣠⠴⢋⠵⠊⠁⠔⠊⠁⠔⠊⣡⣴⣿⠟⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⡟⠀⠀⠈⣿⡄⠈⠳⣄
⠀⠀⠀⢯⡁⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⣷⣤⣤⠾⣻⠇⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠳⣄⠀⠀⠀⣀⣴⡿⠟⠉⠀⠀⠀⠀⠀⣠⠤⠖⠶⢤⡤⠶⠦⢤⡴⢛⡟⢿⣞⣋⠁⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠈⠳⣤⢞⠕⠋⠀⠀⠀⠀⠀⠀⠀⣼⠓⠒⠲⠤⢬⣷⠀⠀⣿⠀⢰⡇⠀⣿⠀⣩⡶⠶⠶⣦⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⣀⣠⡤⢾⡷⣴⠒⢻⣷⣴⣷⣞⡃⣼⠁⠈⢢⣤⠿⡆⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣰⣿⠑⣄⠈⢳⡄⢸⠁⣧⡴⠋⢻⠀⠈⣪⡽⠿⠿⣟⣉⣡⢴⡧⠶⠶⣤⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⢹⡹⣄⡈⠳⣸⡇⢘⣧⡘⢷⣤⡾⠀⢰⡟⠶⠴⠤⣼⣧⠀⣸⡀⠀⠀⢈⣷⠀⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠈⠳⣬⣽⡿⠋⣠⢾⠈⠙⣍⢳⣠⡶⠾⣿⣛⣛⣫⠵⠋⠀⠻⣍⣛⣛⣻⠇⠀⠀⢀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠹⣌⠓⠦⣬⣾⠹⣗⢦⡈⠻⣇⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠉⡿⠲⠚⠉⠹⡎⠳⣽⣤⡾⠂⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⢸⡘⢦⣄⣀⣼⠇⠀⠀⠀⠀⠀⢠⠏⠀⠀⢹⡆⠀⠀⠀⢀⢴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠙⠲⠶⠾⠋⠀⢀⣀⣀⠀⠀⢺⣧⡤⠴⣾⠇⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⡴⠉⢹⠉⢳⡄⠀⠉⠛⠉⠁⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⢰⣧⡀⣸⣀⣾⠇⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠻⠿⠯⠽⠋⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⢀⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠒⠁⠠⠀⠀⠒⢢⠄⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠄⠂⠀⠀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡐⠠⢄⡀⠀⠢⡀⠀⢊⠊⠙⢆
⠀⠀⠀⠀⠀⠈⢈⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠐⠨⣐⠲⠈⠡⠁⠀⢐⡪⡅
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠰⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠄⠐⢹⡂⠀⠄⠰⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠂⠀⠀⠀⠁⠄⢁⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠁⠈⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠀⠐⢀⡠⠚⠁⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠤⠴⠂⡀⠀⠀⡐⠂⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠔⠀⢈⠃⢀⡀⠀⠀⠀⠄⠁⠄
⠀⠀⠀⢰⠀⠀⠀⣀⠄⢢⠔⠄⠀⠒⠒⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⢠⠠⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠀⠀⠠
⠀⠀⠀⠈⠀⠠⠀⠀⣀⠈⠂⡰⠒⢇⠀⢸⠀⠃⠀⠀⠀⠀⠀⠠⠀⡑⠁⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⢀⠆⠀⠀⠔⠁⠂⡂⢀⠀⠢⡀⠉⠜⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠰⡁⡈⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠰⠁⠫⠝⠀⠀⠎⠁
⠀⠀⠀⠀⠀⠑⠄⠀⠁⠀⠐⠈⠂⢈⠀⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⢀⠔⠁⠐⠀⠀⠀⠀⠀⠀⠁⠅⡀⠀⠀⠄⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠠⣤⣀⡀⠀⢀⠀⢀⣼⢶⡚⠳⠀⢀⣀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠾⢶⣦⣽⠀⠈⠉⠙⠁⠂⠉⠁⢰⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠠⠐⢢⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠰⠀⠀⣸⢳⡖⠀⠀⠉⠠⠤⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⣠⠞⣁⠀⠀⠀⠀⠀⢠⠃⠀⠀⡿⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⣀⡤⠚⠁⠉⠙⠛⠁⠀⠀⢀⡎⠀⢠⠴⠃⠀⠉⠓⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢠⡀⠓⠋⠟⠀⠀⠠⣀⠄⠀⠀⣸⠃⠀⠀⠀⠰⠋⠘
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀⠀⠠⢟⡧⣀⣀⣀⠰⡄⠀⠀⠀⠀⠒⣼⠶⠴⠚⠈⣳⢄⡀⡄⣀⡠⠂
⠀⠀⠀⠀⠀⠀⠨⠃⠀⠀⠀⠀⡿⠇⠀⠀⠱⣈⣹⠋⠈⠙⠄⢰⣶⣄⣀⣅⠀⣀⣠⣴⣧⠀⠀⠈⠁
⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⠀⠀⠐⠤⣄⢶⡾⠛⠛⠀⠀⠀⠈⣟⠉⠉⢹⣛⣛⣯⡅⠀⢢⠁⣠⠄
⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣌⢻⣋⠙⠻⠿⢿⡄⠂⠈⠚⠁⠀⠀⡀
⠀⠀⠀⠤⠤⠂⠀⠀⠀⡀⠐⢢⣄⠀⣀⠀⠀⠀⠹⣀⠀⢠⠞⣿⢽⣯⣷⡄⠀⠀⢉⡄⠀⢠⣄⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠹⠀⠈⠀⠒⠒⢂⠀⠉⠁⠈⠀⠀⠀⡳⠳⠾⠛⠋⠀⠉⠀⠈⠀⠀⠈⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠀⡠⣇⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⢠⡔⠊⠀⠀⡀
⠀⠀⢀⠠⠂⠀⠀⠀⠹⠿⠄⠀⠀⠀⠀⠀⠀⠜⠀⠀⡁⠁⠀⠀⠀⣠⡴⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠁⠀⠠⡀
⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠔⠉⠀⢠⣀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣠⡷⠾⠛⢹⡄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢇⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠈⠙⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣤⣀⣀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⢀⣀⣠⣴⣾⠁⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣟⣾⡿⢭⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⡭⠭⢿⣿⣿⣿⠄⠀⠀⢿⠀⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡿⢭⣿⣿⣟⣛⣒⣖⣒⣒⣚⣛⣛⣿⣭⣭⢽⣷⣿⠀⢀⣠⠴⣞⡡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣯⢽⣿⣟⣓⣒⣳⣶⣶⣶⣶⣶⣾⣓⣶⣻⣿⡯⣏⡵⢞⣩⡴⠒⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢯⣿⣻⣞⣶⣿⠿⠿⠭⠭⠭⠿⠿⠿⠷⣻⣴⣾⡿⢞⠋⠁⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣲⣿⠿⢭⠭⣭⣭⣭⡭⠋⡭⠷⡿⢟⣋⣷⣿⣽⠄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣗⣲⣿⡯⢭⣽⣿⣿⣿⣿⣿⣿⣿⣯⡭⠽⢿⣿⣿⣿⠅⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣳⡿⣯⣽⣿⣟⣛⣒⣒⣒⣒⣲⣟⣛⣿⣯⡽⢿⣟⣿⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢿⣩⣵⡾⢞⣂⣶⣾⣿⣿⣿⣷⣶⣚⣒⣿⣿⡭⢿⣟⡂⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠿⣻⣽⣒⣶⣿⠿⢿⣭⠭⣭⡽⠿⢿⣷⣚⣛⣿⡽⣿⡀⠀⢠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠛⠉⠶⣽⣟⣲⡿⢿⠭⢭⣭⣭⣭⣭⣭⠭⠭⠽⢿⣶⣺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⠀⣠⠴⢿⣟⣳⡿⠿⣭⣿⣿⣿⣿⣛⣛⣿⣿⣿⣭⠽⢿⣿⣻⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⣧⣄⡛⠻⠿⢯⣽⣿⣟⣛⣛⣛⣛⣛⣛⣻⣟⣻⣿⡽⠿⠟⠛⣀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣬⣭⣭⣭⣥⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⡟⠛⡏⡒⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣉⡁⠀⣁⣹⣾⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠓⢒⡒⠒⢒⠂⠂⡀⢒⡒⠳⢄⣳⡤⢏⣿⣟⠒⣐⠒⢒⡒⢒⡂⠒⢻⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠛⠀⠘⠃⠘⠃⠀⠂⠐⢛⡋⠂⠘⠂⠛⠀⠘⠃⠘⠂⠀⢸⣀⣘
⠀⠀⠀⠀⠀⠀⠛⢩⠍⠈⡭⠉⢩⡄⠩⠅⢩⠉⢩⠉⠩⠉⣭⠉⡭⠉⢩⡄⢩⡭⠉⢹⠉⢹
⠀⠀⠀⠀⠀⠀⣄⡈⡉⠀⠉⠀⠀⠁⠠⠀⠠⠠⠀⠀⠈⠀⠉⠀⠀⠀⠈⠀⠀⠠⠤⣼⣶⣾
⠀⠀⠀⠀⠀⠀⠂⠀⠯⠀⠺⠄⠨⠅⠐⠇⠸⠅⠰⠄⠨⠀⠿⠀⠯⠀⠸⠂⠐⠇⠀⣠⣀⣸
⠀⠀⠀⠀⠀⠀⠙⠉⣥⠉⠡⡌⢠⠍⢠⡌⢩⡍⢩⡉⣩⠈⡌⠉⣬⠉⢩⡍⢩⡍⠁⢯⠉⢹
⠀⠀⠀⠀⠀⠀⢀⣀⠉⢁⠈⠁⠈⠁⠈⠁⠀⠁⠈⠁⠭⠤⠭⠤⠤⠤⠤⠤⠤⠤⠤⣿⣶⣽
⠀⠀⠀⠀⠀⠀⠈⠀⠔⡂⢐⡂⢒⡂⠀⠰⣄⣐⢂⣀⣄⣀⡆⠀⣒⠀⣶⡂⢰⣒⠀⢹⠀⣸
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠼⡇⠂⠐⠀⠀⠀⢡⠀⠠⠄⠈⠁⠄⠁⠀⣼⢤⢼
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠁⠉⠉⠉⠉⠉⠉⠉⠙⠐⠒⠐⠒⠒⠒⠒⠚⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠶⠛⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣦⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⢸⡇⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠘⢷⣄⣀⠀⠀⠀⢀⣀⣠⣴⣿⣦⡀⠀⠀⠈⠙⠻⠶⢶⣤⣤⣤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⣰⠛⢿⡉⠉⠉⠉⠉⠀⣸⠃⠈⠛⢶⣤⣀⣀⣀⣠⠴⠚⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣰⠃⠀⠀⠙⣦⣤⣤⠤⢶⡿⣤⡀⠀⢀⣠⡽⠟⣿⠁⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠒⠲⠶⣾⣯⣄⣀⣀⣠⡞⠁⠀⠀⠀⣴⠃⠙⣄⠀⣼⠁⣈⣿⣿⠛⠁⠀⠀⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠉⠉⠻⣟⠛⠓⠲⢾⡧⠤⢤⣌⣢⡷⠚⠉⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠹⣆⠀⠀⠈⣷⠀⣠⠖⠉⣏⠙⠛⢶⣿⣦⣤⣄⣀⣹⣆⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⣿⡀⠀⢀⣽⠿⢧⣤⣄⡘⡆⣰⠋⠀⠀⠀⠀⣨⠟⠉⠉⠉⣹⠟⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⢸⣧⡶⠋⠀⠀⠀⠀⠈⠙⢿⡇⠀⠀⠀⣠⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⠟⠛⠛⠶⠶⢦⣤⣀⡀⠀⠀⢷⡀⠀⡴⠃⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣜⣧⡼⠁⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠛⠛⠛⠛⠳⠶⢶⣤⣄⣀⠀⠀⠀⠈⠻⣇⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣄⠀⢻⡄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠾⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡷⢦⠤⣄⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠖⠊⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠁⡈⡇⠀⢠⠏⠀⠀⠀⠀⠀⠂⠈⠉⠉⠀⢀⡠⢴⠾⠀⠀⠙⠢⠄⠤⠤⠤⠤⢤⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁⢀⡏⡠⠤⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⡇⠀⠀⣄⣀⣠⠖⠋⠀⣀⣠⠴⠚⠃⠀⠀⠀⠀⠑⠢⠤⣤⠒⠁
⠀⠛⠳⢤⣄⡀⠀⠀⠀⠈⢷⢸⣇⠀⢠⡞⠛⢄⠤⠤⠤⠤⠀⠀⠀⠀⠀⡀⠀⠀⡀⠔⠈⡏
⠀⠀⠀⠀⠐⡌⠉⠛⢷⣶⣾⣿⣿⣷⢟⣀⣀⡤⠵⣤⠠⠄⠚⠀⠰⡄⢀⡨⠛⠉
⠀⠀⠀⠀⢀⣸⠶⣾⣿⣿⣿⣿⣿⠛⠋⠻⠊⢤⢖⣈⣗⣲⡤⣓⠒⣉⠁⠀⢠⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⡀
⠀⠀⢠⠔⠋⠀⢸⣿⣿⣿⣿⣿⠛⣆⡄⠀⡧⠈⠘⣽⠛⡍⠐⠁⣠⠉⠉⠀⠘⠓⠒⠀⠀⠠⠬⡦⣄⣀⡀⠑⢤⡀
⠀⠔⠁⠀⠀⠀⠀⠙⠛⠋⠁⡏⠀⡸⢷⡀⠀⠀⢱⡀⠀⠈⠁⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣸⣇⠁⠀⠀⠀⠀⡀⠘⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⣠⣞⡠⠤⠂⠁⢸⠁⠁⠀⠀⠀⠀⢹⡴⠂⠀⠀⢺⡃⠀⠀⠀⠀⠀⠀⠒⣚⠷⢄⡀⠀⠀⠀⡀
⠀⠀⠒⠉⠉⠉⠉⠑⠂⠀⢘⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠇⣠⠀⠀⠉⠉⠉⠳⣖⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢀⡔⣁⠀⠀⠀⠀⠀⠀⠈⢣⠀⢀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠞⠋⠁⠀⠀⠀⠀⠀⠀⠁⠒⠳⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣾⢃⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⡀⣀⣤⡴⠶⢚⡿⢁⡾⢋⣭⠛⠛⠛⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠁⢀⣤⣤⣤⡀⠈⠉⠙⠻⣄⠀⢀⣾⠃⠚⢡⡾⠁⠀⠀⠀⠀⠀⠈⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣀⠈⠛⠛⠛⠁⢀⣀⣠⡴⠋⠀⣼⠃⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣿⣿⡿⢿⣿⠀⠀⣼⠃⠀⣰⠏⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⣶⠶⠶⠶⣶⣚⣿⣿⠀⣼⠃⢀⣼⠋⣠⠶⠛⠉⠉⠙⠛⢶⣄⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⢯⣭⣭⣿⣭⣭⣭⢶⣿⣼⠇⢀⡾⢁⡾⠃⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⢸⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⡲⠶⠶⠶⢶⣚⣻⣿⠏⣠⡟⠁⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⢯⣭⣭⣿⣯⣭⡭⣿⠏⣰⠏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⢀⡿⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣛⣶⠶⠶⠶⢶⣾⠏⣼⡋⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣾⠷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠷⣯⣭⣽⣿⣯⣽⣟⡾⠛⠻⣦⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠳⣦⣄⣀⣀⣀⣀⣠⣿⣿⣡⣤⠾⠋⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⣀⣀⣤⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⡖⡗⠚⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠚⠒⠓⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⡇⡇⠀⠀⠠⣏⣉⣹⣹⠆⠀⢀⡴⣲⣲⡄⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇⠀⣠⠤⠤⠤⠤⠤⠤⡀⠸⡸⣄⡸⡹⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⡇⡜⠓⠀⠀⠀⠀⠀⠀⣥⠀⠈⠙⠊⠀⠀⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡟⠁⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡾⠿⣦⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡶⣗⡊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠿⠾⠟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢧⠯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⠴⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤
⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⢶⠚⣽⡞⡿⡿⡙⣬⡟⢱⠖⠒⣲⠶⢲⠒⡶⡖⢺⠋⢩⢣
⠀⠀⠀⠀⠀⠹⡙⢍⠿⠯⣉⠉⠢⡀⠀⠈⢑⠷⢧⣽⡛⡟⢻⣀⠵⢧⠀⡼⠁⠀⡸⢰⠛⢀⠎⠀⣸⢸
⠀⠀⠀⠀⠀⠀⢇⠈⢧⠀⠀⠉⢲⣼⣀⣔⠁⠀⢠⠇⠈⡗⠉⢣⠀⢀⣵⢧⡀⢰⣱⠃⡠⠊⠀⢠⢃⠎
⠀⠀⠀⠀⠀⠀⢸⠀⢸⡄⠀⠀⣨⡞⠉⠉⠻⢧⡞⣀⣀⣇⣠⠬⣷⠛⠁⢀⣙⣾⣧⠊⠀⠀⡰⢃⠎
⠀⠀⠀⠀⠀⠀⡎⠀⣼⢃⡠⠞⠛⠛⠻⢶⣄⡸⠋⠀⠀⢹⠀⣀⣈⣆⡴⠁⠀⠀⠙⠢⡀⡜⠀⣸⠆
⠀⠀⠀⠀⠀⢠⠃⣰⠿⠯⠤⠤⣄⠀⠀⠀⢹⠓⠉⠉⠻⢾⡞⠁⠀⠙⣇⢀⠴⣒⡭⠽⠟⠳⣄⣿
⠀⠀⠀⠀⣀⣿⣾⣋⠉⠉⠉⠉⠒⢵⣄⠀⡎⣀⣀⣀⡀⠀⡇⠀⠀⣀⡸⣏⡞⠁⠀⠀⡠⠾⠋⠛⢄
⠀⠀⠀⠈⠀⠀⠙⠢⡉⠲⢄⠀⠀⠀⠙⡾⠋⠁⠀⠀⠈⠑⡗⠿⠉⠀⠀⠘⣆⠀⢀⡞⠀⠀⢀⡠⠊⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠳⡀⠀⢰⡧⠔⠒⠒⠒⠢⢄⢸⠀⠀⠀⡀⠀⠘⣆⢸⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢱⢀⡏⣀⠤⠴⠒⠤⢄⡀⢹⠀⣴⠿⠋⢉⣉⣛⣾⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⣿⢞⣁⣤⣤⣄⣀⠀⠈⢪⣧⠏⡠⠊⠉⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠋⠉⠉⠀⠀⠁⠀⠉⠲⣄⢿⡜⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠂⢀⣾⡆⠀⠀⠐⣲⣔⢒⡦
⠀⠀⠀⢀⡠⠖⡲⠤⠭⠠⠔⠀⣤⠖⠁⠀⠀⠀⠀⠀⠠⣀⠀⠀⠉⠀⠀⠈⠲⣤⠀⠢⠄⠭⠤⢖⠲⠄⡀
⠀⣠⢎⡩⠂⠁⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢀⣤⠀⢘⢠⡄⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠈⠐⢍⡱⣄
⢜⠔⠁⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⢀⠀⠀⠄⠀⠁⠠⠒⠄⣀⠠⠈⠀⠀⠀⠀⢠⠀⢀⡀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⡠⡊⠗⠒⠉⠀⠗⡇⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠺⠀⠉⠂⠺⢑⢄
⠀⠀⠀⣠⢋⠌⠀⠀⠀⢀⡀⡒⢷⠀⠀⠀⠑⠀⠄⠀⠀⠀⠀⠄⠐⠀⠀⠀⡀⡾⢒⢄⡀⠀⠀⠀⠡⡝⢄
⠀⠀⡠⢡⠃⠀⠀⢀⠒⠵⠉⠀⠈⠛⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠛⠁⠀⠉⠮⡒⡀⠀⠀⠘⡌⢄
⠀⠀⢅⠃⠀⠀⠠⢁⠃⠀⠀⠀⠀⠀⠔⠝⠢⠧⢄⣀⡖⢲⣀⠠⠐⠗⠫⠢⠀⠀⠀⠀⠀⠘⡈⠄⠀⠀⠘⡨⡀
⠀⠀⠈⠀⠀⢀⣓⡆⠀⠀⠀⠀⠀⠎⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠱⠀⠀⠀⠀⠀⠐⣚⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⡺⠀⠀⠀⠀⠀⣸⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣇⠀⠀⠀⠀⠀⢗⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿