Dungeness crab Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣋⣙⠷⣐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⢠⡴⠋⠁⠀⠀⣤⣔⣦⡀⠀⠀⠀⠈⠛⠿⣿⣯⡨⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⡼⠉⠀⢨⣇⠀⣄⠠⢋⣭⠟⠳⣴⠸⠖⠒⠢⢄⡀⠹⣧⣮⡁⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⣼⣇⣀⠔⢃⣌⡷⢾⡆⢸⣿⣿⣅⣙⢆⠀⠀⠠⠀⢁⡎⠙⢿⣧⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⡿⢁⣸⣿⣿⠧⢀⡴⠉⣿⣿⡗⠈⠀⣻⡦⠀⠈⠚⠃⣔⠈⠿⠇⠀⡔⠁
⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠳⣌⠓⢾⣿⣿⣿⣶⡏⢀⡴⠋⢙⣿⠄⢾⣿⣿⢄⠀⠀⠀⠀⠁⠀⠖⡸⠡
⠀⠀⠀⠀⠀⠀⠘⢻⣶⣤⡀⠙⡋⢹⣿⣿⣿⣥⠂⠈⠀⠂⢸⣿⠀⣸⢏⣧⠀⠃⠀⠀⠀⠀⠀⢾⠁⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⢦⣀⣻⡶⣍⣿⠟⠛⢣⡀⠀⠀⠜⠁⠀⠻⠹⠣⠀⠀⠀⠀⠐⣠⣶⠤⠚⠁⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣿⡷⠿⠿⣷⣦⠈⠀⠀⠀⠀⠇⠀⠆⠀⠀⠚⠉⣀⠀⠉⣉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢚⡩⠧⠎⠷⢶⣶⣤⣿⣿⣦⣤⣄⣀⣀⣀⣀⣀⣤⡴⢭⣥⣤⣀⠙⠛⠶⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⣠⠤⠁⠀⠀⠠⠛⠃⠛⠛⠛⠓⠛⠚⠛⠛⠳⠀⠰⠶⠾⡗⢀⠀⠈⠁⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠠⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡴⠋⠈⠻⣧⣤⣄⣤⣤⣤⣯⣴⣶⣶⣏⣀⠈⢙⣆⢠⠀⠀⠀⠠⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⢆⡀⠀⠀⠟⣿⣻⣾⣻⠟⠁⠈⡻⠉⠛⠃⡘⢽⠛⠁⠉⢉⠻⣯⣄⣀⡀⠀⢰⡞⠓⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢴⣤⠈⢀⠈⠁⠀⢶⣿⡷⡵⠁⠀⠀⠀⠀⣀⠀⠀⠂⠀⡴⡀⠀⠀⠀⠙⢮⡙⢿⡇⣀⣮⡕⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢲⡛⠀⠒⡿⢹⠁⠀⠀⠀⠀⠈⠻⣦⡄⣤⣶⠋⠀⠀⠀⠀⠀⠀⢃⣿⣏⠀⢈⠠⠞⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠙⠳⢈⠉⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⡉⠈⠁⠉⡀⠀⠀⠀⠀⠀⣰⡿⠟⠃⡽⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠦⣔⡛⠓⢢⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⢈⣴⠿⠴⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢲⣎⠉⠉⠉⢙⡿⣿⣷⣦⣄⠄⠀⠒⠀⢀⡠⢾⣯⣴⢂⣀⡀⠐⠒⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠴⠊⠀⠈⠉⠙⠛⠓⠚⢻⣟⣡⠌⠓⠒⠛⠛⠛⠂⠾⢽⣍⠉⠉⢀⣄⠀⠘⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠠⣾⣿⠀⠀⠀⠋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡽⠃⠀⠰⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘⠀⠠⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣔⠂⠄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠣⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠎⣠⣀⡠⠄⠀⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⠐⠠⢤⡤⠤⠓⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢘⣆
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠐⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠂⠈⠁⡹⠀⠄
⠀⠀⠀⠀⠀⢰⠀⠘⠆⠀⠀⠀⠈⠂⡤⠤⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠠⡀⠀⢀⡴⠂⠀⠀⠒
⠀⠀⠀⠀⠀⠰⠂⠀⠈⠛⣀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⢒⠖⠁
⠀⠀⠀⠀⠀⠀⢢⣀⡄⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠤⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⠄⠒⠐⠐⠃⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠼⠛⠉⠁⢀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠄⠋⠀⠀⠀⢠⠀⢰⠾⠛⠒⠶⠤⠶⠔⠺⢯⡍⠙⠒⠒⠐⠚⠤⠼⠃⠒⣀
⠀⠀⠀⠀⠀⠀⢤⠜⠁⠀⠀⠀⠀⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠠⠄⠴⠄


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⡟⡃⡀⣀⣀⠀⣊⡀⠈⢀⣉⠉⢠⣦⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠠⢾⣿⣶⣼⡿⠛⠀⠀⠀⠀⠀⣀⠉⠙⠻⢶⣤⡟⢻⣦⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⢚⢢⡀⠀⠸⡼⢻⣽⣭⣤⣶⠟⠛⣿⡏⠙⢿⣿⢷⣦⣤⣼⢯⣾⡟⠀⢀⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢏⣦⡀⡈⠳⣹⡟⠀⢸⣷⣦⠟⢳⣶⡾⠃⢀⣿⣿⣷⠟⠋⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⣀⣀⠠⢠⣼⣷⣑⠦⡀⠰⣤⣹⣿⣿⡂⣾⢛⡏⣤⣼⣿⡟⣠⣴⣿⣿⣯⣤⣤⣀⣀
⠀⠀⣠⡶⠞⠿⠟⠂⠉⣙⣻⢿⡳⣦⣓⢿⣿⣿⢿⢷⠾⣿⣷⣿⣿⣿⣳⣿⣷⣿⣿⣏⣉⠙⠛⠿⠿⢶⣤⡀
⠀⠘⠉⠀⠀⠀⣠⣴⣯⠟⠋⣻⡭⣻⣿⣿⣿⡷⠞⠻⠿⠿⠿⠿⣿⣿⢻⣿⣿⣟⡋⠙⢷⣶⣦⣄⠀⠀⠈⠙
⠀⠀⠀⠀⠀⣾⠛⠉⢀⣠⣮⠟⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⢿⣷⡄⠈⠉⢻⡆
⠀⠀⠀⠀⢸⠃⠀⢀⡾⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⡤⠶⢶⣶⠀⠐⣟⣛⠋⠛⢶⣤⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⢋⣴⡿⠋⣠⠴⠛⠁⠀⠀⠈⠉⠛⠳⢦⣍⣙⣻⣿⢦⠀⠀⢀⣀⠀⠀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣱⣯⣴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⣄⣀⣹⣷⡀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢀⡀⣤⣾⠇⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⢛⣻⣿⣉⣳⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢨⣮⣿⣧⣤⡶⠿⠟⠋⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⣀⠀⠀⣠⣶⣟⣉⠈⠉⠛⡿⠃⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⣤⣴⡾⠛⠛⣶⣤⣀⡢⠜⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣆⡀⠈⠀⠀⠀⢀⣾⠃⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠻⣟⢁⢀⣰⣾⣿⣿⠿⠿⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠻⢿⡉⠀⠀⠀⠀⠾⠛⠁⠀⠀⣀⡰⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⣀⣀⣈⠛⣻⣿⣿⣿⣿⣤⣤⣤⣶⣶⣀⣀⣠⣤⣤⣶⡤⡤⢶⣶⣶⣤⣴⣤⣯⡿⠿⠛⠛⢙⠄⠀⠀⢀⣀⣤⣶⠾⠛⠁
⠀⠀⢣⠀⠀⠀⠀⠀⠉⣥⣌⠙⠻⣿⣟⡛⠉⠙⠋⠙⠛⠛⢿⣿⣿⣿⠟⣋⣹⣧⣿⣮⣭⣭⣭⣉⠻⠿⠟⢁⣠⣴⠿⠖⠀⠀⠀⣉⣉⠉
⠀⠀⠘⠦⣄⣀⣀⣀⠀⠀⠙⠻⠶⢶⣿⣿⣿⣷⣶⣦⣤⣀⣀⠉⠉⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣴⣾⣿⡿⠷⠒⠀⠀⠀⠉⢉⡀
⠀⠀⠀⠀⠀⠉⠙⠛⠻⠏⣾⣷⣤⣠⣴⢭⣟⣿⣿⣿⣿⣿⣿⣿⣆⡀⠘⠿⠿⠿⠿⠿⠿⠟⠛⣡⣿⣿⣿⣶⠦⢸⣧⣤⣤⣤⣤⡾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢽⣿⣿⣿⣿⣿⡯⠀⣴⣶⣶⣶⣶⣶⣶⣟⣥⣿⣿⠭⠥⣄⣀⣉⠀⠀⠀⠀⠀⠘⣎⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣠⡤⠀⠐⠚⠛⣻⣿⠈⠿⠟⠙⠀⠉⠁⠀⠉⠀⠈⠙⠓⠛⠷⠦⣠⢄⣤⣤⣉⣫⣷⡄⠀⠀⠈⠳⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣾⠟⠚⠋⠁⢀⣼⣤⣤⠞⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢶⡆⠀⠀⢹⣿⡇⠀⠀⠀⠀⠸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⢀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⣼⠃⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢸⣿⡉⢳⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⣀⣀⡀⠀⣰⠇⣼⡟⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡝⠁⠀⠀⠀⠀⠀⠀⢻⣷⡄⣦⠀⠀⠀⠀⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢋⣤⣄⠀⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣰⣶⣾⣿⣆⡹⢿⣿⣆⢀⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣦⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠛⠋⠙⠛⢻⣿⣿⣷⣾⣿⣿⣿⡿⠿⠿⠿⠿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⢠⣤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢦⣤⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧⢹⣶⣶⡶⠂⠀⠀⠀⠀⠀⠀⠲⣶⣶⣾⢡⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠻⣿⣿⣿⠿⣫⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣮⡻⢿⣿⣿⡿⢣⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾⣿⣿⢆⣨⡵⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣭⣀⢺⣿⣿⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⠀⠀⠀⣿⣿⣧⣈⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢈⣠⣿⣿⡇⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣿⣿⣿⣿⠄⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀⢼⣿⣿⣿⡿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠙⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⢿⡟⠁⠀⠀⢠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠸⠿⠿⣷⣶⣤⣀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣠⣴⣶⡿⠿⠿⠐⠶⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣀⣀⣤⣬⣉⣉⣈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣈⣉⣩⣤⣄⣀⣀⠀⠀⠀⠈⠻⣦⣀
⠀⠀⠀⠀⠠⠞⠉⠀⠀⠀⢀⣄⠿⠿⠿⠿⠿⠿⠿⠿⣇⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣊⣉⣛⣏⢹⠿⠿⠿⠿⠿⢧⣄⠀⠀⠀⠈⠻⢧
⠀⠀⠀⠀⡜⠀⠀⠀⠀⢠⣾⠃⠀⠀⢀⣀⢠⣾⣿⣿⡿⠟⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠘⠭⣭⣿⣭⣤⢄⣀⠀⠀⠈⢩⣤⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠃⠀⠀⠀⠀⣾⠃⠀⠀⠀⢈⣿⣷⡌⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣴⣿⣏⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⠀⠀⢰⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠘⢿⣿⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⠟⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣮⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣮⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⢶⣤⠀⠀⠀⢠⣖⠶⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣔⣡⣾⡿⠟⠋⣉⣀⣀⣀⠀⠈⠙⢦⣳⣌⣀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢿⡀⣸⣤⡟⣩⠖⠋⠉⠉⠉⠉⠉⠉⠉⠓⢦⡝⢯⣠⡷⣀⡼⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢜⣖⡘⢟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡽⣟⣱⠽⠃⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣍⣳⣿⣷⣤⣀⣀⣀⣄⣠⣤⣀⣤⣶⣿⡿⣹⣩⣤⡼⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠶⣤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⢀⣔⡶⠶⠿⠿⠛⠛⠛⠛⠛⠻⠿⠿⠷⢤⣴⣄⠈⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⢘⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⢲⡒⢫⣟⣛⢓⡀⠀⠀⣐⢛⣛⣯⡓⣲⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠃⠀⠀⠀⣱⠈⣀⠴⠋⠀⠀⠀⠈⠳⢄⡈⢰⡁⠀⠀⠀⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢀⠇⠘⢢⣄⠤⢚⡥⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣝⠢⢄⣤⠚⠀⢇⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢸⠀⠀⢾⣗⠚⠁⠀⢀⣀⣠⠤⠤⠤⠤⠤⢄⣀⣀⠀⠀⠙⢒⣻⠆⠀⢸⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢇⢸⡀⠈⢇⠀⠀⢈⡧⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣏⠀⠀⢀⠇⠀⣸⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣳⡀⠈⢣⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠋⠀⣠⣃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⢤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠳⢼⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠴⠋⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⢒⡽⢁⣈⡉⠓⠬⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡮⠖⠋⣉⣀⡹⣝⠢⣀
⠀⠀⠀⠀⢀⠔⣫⠖⠁⢀⡠⠔⠚⠛⠛⠚⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠓⠚⠛⠛⠒⠤⣀⠀⠙⢮⡓⢄
⠀⠀⠀⠀⣷⠎⠀⠀⢠⠊⡷⠒⠒⣒⡶⠒⠒⠚⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠒⠒⠲⣖⡒⠒⠲⡏⢢⠀⠀⠈⢶⡇
⠀⠀⠀⢸⣸⠀⠀⢠⢃⡜⠀⠀⡏⠙⢢⡤⠴⠚⠁⠈⠓⠲⠤⠖⠒⠒⠦⠴⠒⠋⠀⠙⠲⠤⣤⠚⠉⡇⠀⠘⣄⢣⠀⠀⢸⣸
⠀⠀⠀⠀⢿⠀⠀⢸⡞⠀⠀⡞⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⡆⠀⠘⣾⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢱⣄⣀⡞⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣄⣀⣴⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠳⣄⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠃⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢴⣶⣿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣳⢶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡞⣵⠗⢆⣿⣭⣤⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⡭⡣⠤⣿⡲⣄
⠀⠀⠀⠀⠀⢠⣾⢋⣾⣿⠀⢀⣤⠜⠛⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢤⡀⠀⠈⢷⡌⢳⡄
⠀⠀⠀⠀⢠⣞⣿⣿⣿⢁⣴⡿⠁⠀⠀⠀⠀⠀⡀⢈⣇⣷⣴⣉⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣜⣿⣌⣿⡀
⢠⠀⠀⠀⣰⡿⢥⣌⠻⡿⠋⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣽⣾⣶⣷⣲⡦⣄⡀⠀⠙⢿⣿⠟⣻⠛⡇
⣿⡄⠀⢸⣿⣧⣿⣿⡠⠀⡰⣾⣿⣿⢿⣴⣿⣿⣟⡡⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⡷⣀⠀⡅⠀⢿⠘⣷⠀⣠
⢻⣇⠀⠈⢧⣿⣾⡟⢣⣜⣿⣿⣿⣽⣟⣿⣿⣿⣼⣿⢿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣛⣄⣷⠀⣸⣰⡏⣠⢿
⠘⣎⠻⣄⠀⠙⢻⠿⡿⣈⣻⣾⣯⢶⣿⡾⣿⣿⣿⣿⣿⣎⣿⡿⣿⣿⣟⣷⣿⣿⣿⣿⣏⣉⢏⣸⡿⢋⡜⢁⡟
⢤⡈⠦⣝⡒⢦⣀⠓⠚⠽⣿⣿⣷⣾⣧⣾⣿⣿⣿⣿⣿⣿⣿⣳⣩⣟⣿⣿⣿⣿⣿⣿⣯⣽⣼⢟⡵⢋⡴⠋⢀
⠘⣿⣦⣈⡛⠳⠿⢿⣽⣓⣚⠛⢿⣿⣿⣿⣿⣿⡿⠿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢟⣛⣽⡿⠖⣫⢥⣴⡞
⠀⠈⠙⠿⣯⣝⣳⠖⠒⠨⣭⠭⠭⠭⠽⡿⣿⣯⣼⣾⣿⣿⣿⣿⣾⣿⣿⡿⠛⠙⠛⢛⣭⠭⣖⣲⠯⠶⠛⠁
⠀⠀⠀⠀⠀⠉⠉⠙⠛⢛⣩⣾⣿⠿⠻⠛⣫⣽⣙⢻⣿⣿⣿⢿⣿⣏⣛⡛⠛⡛⣛⡓⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠤⠚⣩⡴⠚⢑⣾⣛⣩⣥⠼⠟⠋⠀⠉⠈⠚⢯⣏⣉⣛⣳⣿⡻⣌⣙⣶⣀
⠀⠀⠀⠀⠀⢠⣷⡶⠛⠉⠀⣐⢡⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣎⢷⡌⠉⠻⢬⣷⡀
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢀⠎⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⣷⠀⠀⠀⢿⣯
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⡿⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡁⠞⠀⠀⠀⠀⢀⣀⢀⣀⡀⠀⠀⠈⠂⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢀⣤⡶⣚⣛⣷⣀⣻⡮⢫⣿⣿⣆⣀⡀⠀⠘⠉⠣⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⢀⣿⡿⡋⠈⠉⣿⣿⣿⡅⣿⡬⡿⠓⣯⣽⣿⣧⣰⡀⠀⠀⠀⠀⠀⠀⠀⠍
⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⣹⣿⣾⣷⣶⣿⣿⣧⡿⣿⣟⢻⣦⣿⣿⣿⣿⣷⠁⠀⠀⠀⠀⢀⠔⠁
⠀⠐⠠⣀⣀⠀⠀⠀⠀⠁⠒⠛⢿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⠄⠓⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⣿⣿⣿⣿⣞⣯⣻⣿⣿⣿⣿⣿⡿⠁⠀⠂⠀⠸⠀⠀⠀⠐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢐⠒⠒⠈⢿⠿⣿⣿⣿⣿⣿⣿⡿⡿⠿⠐⠢⢤⠀⠂
⠀⠀⠀⠠⠊⠀⠀⠀⠀⡸⠈⠀⠉⠀⠀⠚⠀⠁⠉⠉⠉⠉⠉⠁⠐⠂⠀⠀⠉⠒⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠋⠉⠉⠉⠙⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠋⠉⠉⠉⠁⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⢀⣠⠴⠦⠦⠖⠢⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠖⠲⠴⠴⠦⣄⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡫⠀⠐⠀⠀⢰⠁⠀⠀⠀⠀⢀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢄⡀⠀⠀⠀⠀⠈⣆⠀⠠⠄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣹⡁⠀⠀⢀⣠⢸⣲⠒⣶⣚⣭⣵⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣯⣭⣓⡶⠒⣒⡿⣇⠀⠀⡀⠀⣟⠆
⠀⠀⠀⠀⠀⠀⠀⢡⣘⢻⣛⠷⣿⣿⣷⣿⣟⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣻⣿⣾⣿⣷⠞⣛⡛⣃⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠉⠉⠀⠱⣘⠛⣛⡩⠋⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠙⢍⣛⠛⣃⠎⢀⠉⠉⢱
⠀⠀⠀⠀⠀⠀⣼⡅⠀⠀⠀⠀⠀⢉⢮⠇⠀⠀⠀⠀⠀⠀⣀⠈⣉⡹⠷⠦⠤⠤⠴⠾⢏⣉⠁⣀⠀⠀⠀⠀⠀⠀⠸⡧⡈⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⢀⠸⣹⣇⣠⣶⢶⣾⢿⡫⠋⠀⠀⠀⠀⠀⢀⣀⢿⣿⣧⡀⠀⠀⠠⠄⠀⠀⠀⣼⣿⡿⢀⡀⠀⠀⠀⠀⠀⠙⢝⠿⣷⠶⣶⣄⣸⡇⠇⡀
⠀⠀⠀⢰⡟⢠⡻⠿⠿⡱⠚⢲⠋⠀⠀⠀⠀⠀⣴⠲⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠙⠖⣦⠀⠀⠀⠀⠀⠙⡶⠓⢎⠿⠿⢟⡄⢻⡆
⢰⡿⠀⣧⡇⢰⢩⡅⠉⠀⠀⢀⡇⠀⠀⠀⡠⢾⠋⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠙⡷⢄⠀⠀⠀⢸⡁⠀⠀⠈⠈⡍⡆⢸⣸⠀⢿
⡞⡇⠀⣿⡗⡌⠻⡻⣆⠀⠀⢠⡓⡄⣀⡎⠏⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⣀⢠⢚⡄⠀⠀⣴⢿⠟⢡⢺⣿⠀⢸
⣷⣧⠀⠘⣷⣸⡄⠙⣿⣿⣶⣾⣿⣿⠩⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠈⠉⣯⣿⣷⣶⣾⣯⠏⢠⣇⣾⠇⠀⣼
⢹⡆⢅⠀⠙⡿⣟⢦⡈⠻⣻⠿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢹⣿⠿⣟⠟⢁⡴⣻⢿⠃⠀⡠⣱
⠐⡿⣌⣆⠀⠈⠛⠷⣿⣦⣀⠉⠲⡞⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢳⠖⠉⣀⣴⣿⠾⠛⠁⠀⣰⣡⣿
⠀⠘⢿⣿⢦⡀⠀⠀⠈⠻⣦⣵⣦⣧⠸⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣾⠇⣼⣴⣮⢷⠟⠁⠀⠀⢀⠴⣿⡿⠃
⠀⠀⠀⠳⣷⣬⣵⣒⠢⠤⣈⡛⢾⣮⣧⡙⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢋⣼⣥⡷⢛⣁⠤⠔⣒⣊⣥⣾⠞
⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣶⣭⣷⣿⠿⣷⣌⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣼⣾⠟⣡⣮⠿⣯⣾⣽⣶⡾⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠈⢙⣛⣿⣶⣿⢧⡙⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣿⡿⢋⡼⣿⣶⡟⣛⡋⠁⢀⣀⢀
⠀⠀⠀⠀⠀⡔⣻⣏⡷⠶⠿⠿⣿⣶⢿⣦⣭⣝⡻⣆⠻⣿⣿⣾⡀⠀⠀⠀⠀⠀⢠⠀⠀⢠⣴⣿⣿⠟⣰⢟⣪⣡⣴⣷⣾⣷⠿⠷⠶⢶⣹⣟⢢
⠀⠀⠀⢀⡜⡰⠋⠀⠀⢀⠴⣭⣉⣈⣉⠛⠻⢿⣿⣿⣳⡙⣿⣿⣷⣤⣔⠀⠀⠀⠀⣧⣴⣿⣿⣿⢋⣞⣿⣿⡿⢿⠛⣉⣉⣍⣽⠦⡀⠀⠀⠙⢄⢣⡀
⠀⠀⠀⣾⣿⠁⠀⠀⢠⢋⣼⡟⠛⠛⠛⠛⠿⠿⣽⣦⣭⣷⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣾⣭⣴⣯⠾⠿⠛⠛⠛⠛⢻⣷⡘⡄⠀⠀⠈⢷⣷
⠀⠀⠀⣭⡇⠀⠀⠀⣇⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠛⠢⠬⠭⠭⠿⠿⠿⠿⠭⠭⠥⠔⠓⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣸⠀⠀⠀⢸⣩
⠀⠀⠀⠹⣣⠀⠀⢸⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⡇⠀⠀⣼⠏
⠀⠀⠀⠀⠘⠃⠀⠀⠻⣟⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣫⠟⠀⠀⠘⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣾⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣴⠒⠛⣙⣓⣦⡀⠰⣯⠥⣤⢤⡉⠙⡶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⣴⡾⠋⠀⠀⢳⠀⡈⣉⣭⠽⠀⠀⠈⠑⠦⣄⠙⠎⠀⠀⠈⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡙⣄⠀⠀⣰⠋⣻⣷⡶⠊⢀⣼⣪⢞⣁⢠⣀⣀⡀⡀⠀⣀⡀⠙⢶⣦⣤⣶⡿⠷⠛⣦⠀⠀⠀⠀⢀⡴⢫⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣌⣧⡀⣿⣇⣿⣿⣽⣶⢟⣫⡥⠾⢆⡉⠉⢩⡉⢉⡋⠁⠿⢦⣀⣉⣿⡯⣷⠀⢠⣟⡇⠀⣠⢴⢋⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠙⣷⣿⣿⣄⢀⣤⡟⣁⠲⠀⠈⠃⠀⠀⠀⠀⠀⠀⢀⠆⠙⢫⣀⠀⢠⣶⣦⣾⡵⠋⣱⠿⠋
⠀⠀⠀⢀⣤⠤⣤⣤⣤⣤⣽⣦⣾⡟⠿⣿⣿⣿⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⣻⣿⡿⠛⢻⣷⣶⣞⣁⣀⣀⣀⣀⣀
⠀⠀⡜⣹⠶⠶⠾⠟⠳⠤⢤⣾⣏⣵⡶⠲⠶⣧⡉⠓⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⢀⠀⢀⣤⣶⠶⠟⠋⢉⣷⣤⣤⠼⣟⣢⣤⣬⡗⢦
⠀⠀⣿⠁⠀⢀⡤⠖⢲⣶⡋⢙⣿⣍⠙⠛⠿⠿⣷⣄⣑⡒⠦⡄⠿⠀⠀⠇⣀⠤⠚⡡⠴⡏⠛⠋⣁⠠⠽⣙⣭⡛⠒⠶⢤⣀⠀⠀⠹⣾
⠀⠀⢻⠀⠀⣞⣰⠒⠋⠉⢉⣩⣤⣬⠛⢻⡟⠛⠻⣿⠷⣿⣳⣄⣀⢀⣐⠂⣡⢖⡻⠴⠲⠛⡛⢿⣙⡿⠿⠯⢭⡉⠓⢬⣶⣨⣷
⠀⠀⠀⠀⠀⠘⠇⠀⠀⣰⢿⠀⣠⣾⣿⡿⠛⠒⠾⠿⠟⠋⠉⠉⠓⠛⠛⠒⠋⠉⠑⠶⠷⠾⠞⠛⢿⣷⣤⢄⡲⠘⣆⠀⠀⠙⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣯⡞⠀⢹⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁






⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣤⣴⣶⠂⠀⠒⢿⣶⣆
⠀⠀⠀⣇⠀⣿⣟⠉⣀⣀⣀⣀⠀⢉⣿⠀⣼
⠀⢠⡀⠘⢧⣙⣿⣿⣿⣿⣿⣿⣿⣿⣏⡴⢃⣼
⠀⠀⠉⠓⣒⣛⣻⣿⣿⣿⣿⣿⣿⡿⠿⠖⠊⠁
⠀⠀⡴⠋⠉⠉⠉⣩⣽⠟⠛⠿⣿⡍⠉⠙⠳⠄
⠀⠀⠀⠀⠀⠀⢾⡅⠀⠀⠀⠀⢀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⢀⣠⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣻⣥⠐⢧⠘⠂⠄
⠀⠀⠀⠀⠀⠀⢀⣤⠀⣘⣴⣶⢦⣤⣴⣾⢿⣿⣿⣿⣿⣿⠟⠙⠿⠋⠫⠝⠋⠻⠿⠟⠛⠛⠏⣿⣿⣦⣘⣻⣶⣈
⠀⠀⠀⠀⢀⣴⣭⣏⣀⣈⢙⡧⡞⢯⣿⡿⠇⠿⠛⠉⠉⢉⡠⠐⢄⠘⢈⡆⠀⢄⠠⢴⣶⡿⠿⢟⣻⣿⣿⣿⣟⠾⡇
⠀⠀⠀⢰⣿⢋⠺⣿⠿⠿⠯⠿⠛⣷⣶⣶⣄⡀⠀⠔⠃⠺⣷⠒⣹⣸⡮⡗⠀⢸⠾⣏⣤⣴⣶⣿⣿⣽⣿⡿⠗⣾⣿⣦
⠀⠀⠀⠀⡟⢻⡏⠀⠀⣀⣴⣶⣆⣀⣙⣻⣯⢉⠀⠀⠉⠉⠋⣻⡍⢉⠦⠖⠤⠈⡤⠘⢫⣝⣟⣩⣶⣻⣿⣷⠀⡟⠫⣷⡶⡀
⠀⠀⠀⠀⢧⣭⣟⣠⣾⣿⢿⣟⠻⢿⣿⣿⡟⠠⠀⠊⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⣿⣦⡿⠿⠀⣿⠀⣸⡇⠹⡄
⠀⠀⠀⠀⠸⣻⣽⣿⣿⣿⣿⣿⢿⣾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠪⠚⠳⠴⢆⠀⠀⢹⡇⠘⡿⣧
⠀⠀⠀⠀⠘⣾⣿⣿⠉⢿⠾⠿⠟⢿⡖⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⠁⢐⠑⠀⠀⠀⠀⣷⢿⠅⠈⡀
⠀⠀⠀⠀⠁⣷⣅⠁⠀⠀⠀⠒⠁⠀⠀⢉⠀⠉⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣸⡀⠂⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢠
⠀⠀⠀⠀⠀⠉⣽⠇⠈⠢⠀⠀⠀⠀⠀⠀⢄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢘⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠋⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡤⠀⣄⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣍⡇⠀⢀⠀⠈⣫⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀⡨⠟⠋⠉⠁⠈⠈⠉⠂⡁⠀⢀
⠀⠀⠀⡠⠂⠉⠉⠲⢤⣀⢙⣴⠎⠉⠳⣤⢋⢀⣠⠖⠉⠉⠐⠦⡀⠀⠀⠀⢄⠀⠀⢤⡞⠓⢀⠀⠀⠂⠀⠀⠂⢀⣀⠀⠙⢯⡐⠀⢀
⠀⠀⠀⠀⠀⠐⠦⠠⢄⣈⣽⠃⠀⠀⠀⢹⣯⣉⡀⠤⠴⠖⠀⠀⠁⠀⠀⠀⡀⠁⠐⠂⠹⢷⡟⠀⠀⠀⠀⠀⠀⠀⠙⢶⡶⠞⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠧⣀⣀⣠⡼⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⢨⡁⠹⠿⢶⣄⡀⢀⣴⠾⠟⢓⣚⠀⠓⠀⠉
⠀⠀⠀⠀⠆⠀⠰⡖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠖⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠰⠆⠉⠉⠉⠀⠈⠁⠺⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⠄⠀⠀⠀⠀⣀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⠈⠀⢀⠀⠀⡀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠠⠦⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⡄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠐⢶⡖⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠂⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⣆⠀⡘⢁⡀
⠀⠀⠀⠀⠀⠸⠸⣷⡀⠀⠀⠀⣰⡈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠒⠶⣾⡀⠊⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠿⠛⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠠⣿⡀⠀⡸⠋⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠖⠀⠉⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠂⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠖⠈⠀⠀⢭⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡄⠠⠿⠋
⠀⠀⠀⠀⢰⣆⣤⢀⠀⢠⠿⡅
⠀⠀⠀⠀⠈⠉⠛⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢄⢠⡄⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣧⠤⠈⠁⠀⠘⠋⡁⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠞⠛⢀⠀⡀⠚⠛⠏⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢆⢁⠀⠀⠐⠀⠐⠂⠠⡄⠀⠲⠺⡄
⠀⠀⠀⠀⠀⠀⠀⢈⣋⡀⠂⠀⠀⠀⠉⠀⠀⠠⣘⡁⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⠀⢀⣿⡎⠀⠀⠀⠀⠀⠀⠀⢘⣿⣧⠄
⠀⠀⠀⠀⠀⠒⠤⠄⣿⡁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⡘⢷⣄⣀⠀⠀⠀⢀⣴⡾⠋⣠⠀⠆⠈
⠀⠀⠀⠈⠂⠶⠐⢦⡭⠿⢦⣄⠀⠀⢀⣤⣶⣿⣧⡴⠂⠰⠒⠁⠀⠀⠀⠀⠀⠒⠀⢃⠀⠛⡀⠛⠿⠷⠦⠶⠿⠟⠃⠹⠍⣀⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠊⠁⢀⠤⠈⠛⠛⠒⠙⠛⠋⠳⡄⠀⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⣿⣿⣿⣿⠿⠾⠿⡯⣿⣿⣿⠙⠋⣿⠿⠇⠀⠀⣀⢤⣤⣤⣴⣤⡶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⠙⠉⠀⡀⠀⠀⠰⣿⣷⣿⡏⠀⠀⠈⠙⡭⠆⠀⢸⠛⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣾⣿⡉⠛⠁⠀⠀⠀⢀⠂⢀⣠⣼⢇⣴⢀⡰⠛⠉⠁⢰⡄⠀⠀⠀⠀⠀⠚⠓⠿⣂⡄⠈
⠀⠀⠀⠀⠀⠀⠀⣤⢾⠟⠒⠛⠛⠓⢪⣿⣿⣿⡿⣡⣿⢷⡶⠶⣶⡶⠥⠴⢞⠛⠛⠿⠉⠁⣠⠤⠤⠞⠛⢓⣒⣒⢲⣄
⠀⠀⠀⠀⠀⣴⢽⣧⡜⠀⠀⠀⠀⠀⣾⣿⣿⣟⣡⢛⣫⣤⣭⣥⣼⣥⣤⠴⠿⠒⠒⠺⠯⠛⠉⠀⠀⠈⠉⠉⠉⠀⢸⣿⡄
⠀⠀⠀⠀⡾⠁⢸⣿⠀⠀⠀⠀⠀⢀⡟⢉⢘⣿⣿⡟⠛⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣿⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠃⠸⣶⠁⠈⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣸⣷⡄⢻⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡿⡇⠙⠢⢉⣢⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢷⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⢧

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡍⣠⣀⡀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⢀⣠⡤⠭⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣱⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣜⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠉⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠠⠀⡀⠖⠱⠄
⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣠⣄⠀⠠⠀⠀⠀⠐⠛⠃⠀⠄⡀⠀⠀⠀⠀⠁⠀⣤⣤⠄
⠀⠀⠀⢀⠁⠷⠀⣀⡀⠐⠀⠀⠀⡠⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡔⠀⠀⠀⡠⠊
⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠂⠀⣀⠤⠒⢢
⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠉⠠⢤⡞⠁
⠀⠀⠀⠀⢠⠄⠠⠤⠤⠤⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠗⠒⠒⢓⠳
⠀⠀⠀⠀⠀⠙⠶⠖⢂⣀⣀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢿⡐⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠊⠥⢤⣴⠤⢺⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠔⠛⠁⠀⠈⠐⠂⠀⠠⠄⠀⠐⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠻⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⠦⠶⣴⢀⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡄⢘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⣄⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣆⣿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⡸⠳⠛⢯⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡏⠀⠈⣿⡆⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠙⠿⠒⢀⡀⢀⠀⡀⢀⠠⠠⢤⢀⣀⣀⣀⠀⠀⠀⠏⠀⠀⢀⣿⡇⠀⠀⢀⣡⡇
⠀⠀⠀⠀⠀⢷⠃⠀⠀⣷⠀⠀⢀⡼⠄⢃⣈⠀⠀⠤⠤⠤⠶⠀⠴⠶⣿⣧⣭⣤⡳⠶⣷⣶⣤⣾⡿⠁⠀⠀⠈⢹
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⣤⣴⠛⠀⣰⠎⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠸⣿⡋⠻⣿⣷⣍⢻⣿⠟⠁⠀⠀⠀⢠⡞⠀⠀⠀⠀⡀
⠀⠸⡆⠀⠀⠈⣦⠶⠀⠀⠈⠙⠂⣴⠃⠀⠀⠀⠀⣿⠀⠀⠀⡆⠀⠀⢀⣿⡃⠀⠈⣿⣿⠛⡇⠀⠀⢀⣼⣷⡟⠀⠀⠀⢀⡾
⠀⠀⢳⡴⢄⠀⠈⢦⠈⢠⡤⠀⢠⡿⢆⣀⢀⡀⣀⣼⡷⣄⠀⡇⢀⣀⣼⣷⡁⠀⢀⣿⣿⡆⢿⣤⡎⣠⣿⠋⠀⠀⢀⣜⡽⠁
⠀⠀⠀⠑⢆⠐⠄⠀⠉⠃⠇⠀⠸⠁⠀⠈⠉⠀⡼⠉⠀⠀⠀⠇⠉⠉⠙⢿⡛⠛⠛⣿⣿⢇⣼⠿⠛⠋⠁⠀⢀⠔⣼⡏⠁
⠀⠀⠀⠀⠀⠳⢤⣹⡇⠐⢾⣐⡆⡆⠀⠀⠀⠀⠳⣄⠀⠀⠀⡆⠀⠀⠀⡸⠁⠀⢀⣿⣿⣾⣿⡄⣤⣴⠒⠒⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠂⡙⣄⠀⠀⠀⠀⠀⠛⠒⡄⡅⠀⠀⢾⣅⡀⢀⣾⣿⣿⠿⠿⠛⠋⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠊⠀⠉⣻⣮⡛⢶⣖⠀⠀⠀⢈⡅⠁⠀⠀⠀⠈⣹⣿⣿⣿⣧⠤⠤⣀
⠀⠀⠀⠀⠀⠀⣠⣅⡘⠉⠉⠉⠉⣀⡀⡬⣦⡙⢦⣄⣀⣺⡄⠆⠀⠀⣀⣤⣿⣿⡉⠛⠛⠻⠶⠾⣿⠶⡄
⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⢰⣁⠀⠀⠐⠋⠙⠿⠶⢒⣉⣁⣀⣰⣿⠿⠿⠟⠻⠿⣍⣐⣄⠀⠀⠘⣷⠛
⠀⠀⠀⠀⠀⠰⣞⡀⠀⠀⠀⣈⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠇⠀⠀⠘⣧⡇
⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠟⠀⠀⣠⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠁⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠶⠀⠀⢨⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢡⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠁⠀⡀⠀⠀⠀⠈⠉⠀⠀⠈⠑⠀⠀⠀⠀⠀⠙⠒⠀⢻⠂⠀⠸⢠⡀
⠀⠀⠀⠀⠀⡒⠠⠀⠈⡍⠹⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⣹⡏⠀⠠⠖⢠⠇
⠀⠀⠀⠘⠀⠘⣆⡠⢀⠑⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⢀⡔⠀⢀
⠀⠀⠀⠐⠂⠀⠈⠢⢤⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠋⠠⢀⠉
⠀⠀⠀⠀⠀⣀⠀⢀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⣂⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠈⠀⠤⠵⠆⠀⠴⠶⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣵⣖⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠠⠔⠚⠋⢱⣄⣀⠀⠀⠀⠀⣀⣠⣤⢾⡑⠀⠀⠉⠉⠛⠓⠦⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠰⠊⠀⠀⠀⡶⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠙⠓⠞⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⠆⠀⠀⠀⠸⠁⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡼⠛⠰⣊⠉⢹⣄⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⢠⡇⡀⣿⣦⣤⣼⠿⠽⠤⠤⠴⠦⢤⣀⡀⠈⣆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢘⠉⠀⣿⣄⠀⣾⠀⠀⢀⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠐⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠆⠀⣷⡀⠘⣯⣤⣿⡖⠟⠤⠷⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣲⡤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡔⠈⠳⣤⣈⣿⡯⢉⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠉⠉⠄⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⣤⡀⢌⠙⣿⣇⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠺⣇⣠⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠨⡙⠶⣥⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢾⣻⣿⣧⣤⣴⣾⡿⠋⠀⠀⣀⡿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠀⠈⠳⢿⣶⣤⣽⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢐⣿⣟⣛⠛⠉⠁⠀⢮⠴⠛⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠀⠀⠀⠀⠀⠀⡈⠙⠛⣿⡽⠀⠰⠀⠀⠰⠂⡁⣷⣾⣿⡟⠙⠛⠶⠶⠶⠟⠛⠉⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠜⣶⠲⠶⠚⠛⠳⣅⠊⠀⣀⣢⣾⣿⣿⡟⠉⠙⠳⠶⠦⠤⠴⠴⠷⣦⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⡏⠀⠉⠙⠳⠶⢤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⠄⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠰⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡰⠟⠃⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠛⠻⠿⢿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢹⣿⠛⠛⠛⠛⠂⠀⠀⠀⠉⠿⣆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⣀⣀⡀⡀⠀⢀⣀⣀⣠⣴⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⡀⠢⢿⡿⠟⠛⠁⠀⠀⠙⠛⠿⠭⠄⢀⣲⣤⣠⡀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⣤⣾⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠿⣿⣷⡟⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⢂⢲⣭⡖⠂⠂⢮⣶⠊⣄⠀⠀⠈⠉⠁⠀⣿⣾⠀⢀
⠀⠀⠀⢰⡀⠀⢸⣠⣧⣼⡇⠀⠠⢀⠰⠀⠁⢨⣌⠙⠀⠀⠀⠈⣯⣤⠀⠈⠰⣤⢤⡤⠄⣿⡏⠀⢸
⠀⠀⠀⠀⢇⠀⠀⠻⢿⡟⣤⣤⠣⡈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠏⢿⡿⠋⠀⣰⠟
⠀⠀⠀⢀⠈⠲⣴⡀⡀⠈⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠏⢀⣤⠾⠁⠀⡔
⠀⠀⠀⠈⠳⣄⡈⠛⠷⠷⠴⢶⢦⡀⣄⡀⠀⢤⣠⡀⠀⠀⠀⠀⣠⣤⣴⣦⣄⣀⡼⠿⡛⢡⣠⠾⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⠶⠶⠬⣿⡿⠿⣿⣿⡿⠟⠀⠀⠀⠀⠉⢉⠛⠫⠟⠛⠟⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⣤⡶⠶⠦⠾⠿⠷⡄⠀⠀⠀⠀⠀⠀⣰⠛⠛⠓⠀⠉⠲⠦⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠢⠷⠟⠒⠤⠤⣀⣀⢀⡈⠋⠙⠛⠺⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

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

⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⣈⠀⠈⠘⣿⣿⡇⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⢉⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⡌⢈⡉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⢻⣿⣿⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓⠚⠋⢸⣿
⠀⠀⡠⠖⢒⣛⣛⣛⠒⠲⡄⠀⠀⠀⣾⣿⡿⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡦⢐⣷⡆
⠀⠀⡓⠶⠤⠭⠭⢭⢤⠔⣻⠀⠀⠈⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⡂⠀⠐⠹⠆⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⢏⠒⠦⠤⠤⠤⠀⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠗⠷⠚⠛⠒⣄⠀⠀⠀⠀⠀⠀⠈⠲⡕⠄⠀⠀⠀⢸⣿⣶⣿⡿
⠀⠀⠀⠁⠒⠀⠂⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠸⡇⢠⠤⠔⠒⢲⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠸⡜⢋⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢹⠀⠀⠀⠀⠀⠀⢻⡈⠉⠁⠀⠈⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠄
⠀⠀⠀⠀⠀⠰⡶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣀⣰⢶⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠖⠚⠁⠀⠀⠀⠀⠀⠈⠻⣏⣀⡇
⢀⠤⠚⠆⠀⠀⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⢠⠑⠁⠀⠈⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠰⡒
⠀⠀⠄⠀⢃⠀⠀⠀⠀⠀⠈⠉⠛⢦⣘⣷⠀⡀⣠⣛⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠰⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⢾⣿⣿⣿⣦⠀⠀⠀⠘
⢂⠉⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂⠁⠀⠀⠀⠀⠀⠀⠀⢷⣺⣿⣿⣿⡿
⠀⢣⠄⢀⠈⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠛⠛
⠀⠀⠳⡤⢤⠊⡧⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⢀⠀⠰⠷⣶⣶⡶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡝⢶⡆⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠊⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡉⠢⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠑⠒⠀⠉
⠀⢰⠀⠀⠀⠀⠀⠤⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⢀⠀⠼⠂⣈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡾⠑⠒⠓⠒⠬⣉⠹
⠀⠀⠓⠀⠀⠀⠀⡀⠙⢄⡠⠀⠀⠀⠀⠀⠀⡔⠁⢠⠀⠈⡀⢀⢠⠀⢑⡬⠂⠀⠀⠀⠀⠀⠀⠁⣀⣠⣶⣒⣓⣓⣶⣀
⠀⠀⠀⠀⢀⡴⠛⠷⡤⠊⠀⠀⠀⠀⠀⢠⠎⠠⠀⠀⠠⠄⠀⢠⡤⠒⠁⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠢⡀⠀⠒⢀⡤⠚⠁⠀⠀⠀⠀⢀⣸⡍⢆⠀⠀⠀⠘⣿⣿⣉⣉⡉⠛⠛⣿⣿⡇⠀⠀⠀⢠⣤⡴⢦
⠀⠀⠘⢄⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡒⢈⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⢧⣠⠏
⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣁⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣐⠄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠤⠀⠀⠀⠀⠀⠄⢈⠀⠀⣀
⠀⠀⠀⠀⠀⠐⠊⢠⡔⠉⠀⠀⠀⠀⠀⠀⠈⢻⣽⣯⡉⠀⠎⠀⢀⠀⠀⠀⠀⠀⠶⠤⣀⡋⢀⡄⠀⠀⠀⠀⠀⠀⠤⣄⠒⠀⣱⠂
⠀⠀⠀⢈⠥⠒⠒⠦⠿⣶⠶⠀⣀⣀⡠⢔⠴⠯⡀⠑⠂⠴⠤⠔⠁⠀⠀⠀⠀⠀⠀⡠⠤⠨⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣦⡀
⠀⠀⡴⠉⠁⢠⠈⡀⠀⠇⠀⠁⡉⠉⠁⠨⢠⠂⠈⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠔⠁⠑⠒⠀⣄⠀⠀⠀⠀⠀⠀⢀⠰⠦⠈⠛⢯⠂
⠀⠀⠂⠀⠀⠁⠀⡏⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡀⠀⠀⠀⠀⠀⠁⡀⠐⠀⣤⡤⢈⡁⠀⠀⠓⠀⠐⠋⠁⣀⠀⠈⡻⠉
⠀⠀⠀⠠⠁⠀⠀⢣⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠸⠋⠓⠒⠀⢐⡒⠴⠛⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⢔⠉⡐⠀⢈⡦⠒⡂⠀⠀⠀⢰⢆⡴⠂⢶⡍⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠀⠘⠀⠈⠀⠠⠉⠠⠤⠐⠁⠀⠑⠂⠾⠤⢾⢈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢘⠄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠁
⠀⠀⠀⡀⠀⠀⣀⢠⠀⠱⣰⡦⠂⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠐⠠⠤⣀⣀
⠀⠀⠀⠄⠀⠈⢤⡈⢢⣐⠋⠀⠀⠀⠀⠀⠘⣆⠀⠀⠋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠚⠉
⠀⠈⠀⠀⠀⠀⡀⢁⣄⣻⠀⠀⠀⠀⠀⠀⠀⡷⠐⠐⠃⡇⡀⠀⠀⠀⠀⠀⠠⠜⢋⡁⠒⣳⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠳⠢⡀
⠀⠀⠀⠀⡀⠊⠀⠀⠈⠙⠄⠀⠀⠀⠀⢀⠼⠋⠁⡘⠈⠃⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠈⠀⠀⠀⠡⣤⠶⠤⠶⠶⣚⡂⠀⡄⡁⠙
⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠠⠯⠔⠂⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⡀⠁⠀⣀⣼⡐
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠦⠁⠀⣀⢎⣠⠟⠁⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⡉⡛⢿⣙⣠⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣠⠶⠛⠉⠀⠙⢀⣁⣒⣤⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣽⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣿⠀⢠⠾⠋⠨⠛⣤⠀⠀⠀⠁⣸⠛⢻⢿⢿⣎⢥⢴⣶⣶⣦⣤⣀⠈⠛⢿⣿⢛⡟⠂⠔⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢰⣿⣉⣴⣿⠀⠀⢀⣤⠛⣷⠀⢻⡀⠛⣶⣿⣫⣬⠘⢾⡌⠉⠀⠀⠈⠙⢻⠒⣤⣽⣿⣿⡆⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⣿⡇⢿⣿⣿⣤⠴⣫⣀⣶⣼⠻⢮⠇⢈⣽⣿⣿⣷⡀⣨⡷⡀⠀⠀⠀⠓⢀⢤⡟⠈⠻⣿⣿⡄⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡬⣿⣿⠇⣄⣽⣿⣿⣿⢆⣈⣰⣦⠙⣿⣿⣿⠗⠎⠀⢨⣿⣧⡤⠀⠐⠮⡿⣴⠃⠈⠿⡧⠀⢀⡾⢘⠇
⠀⡤⠀⠀⠀⠀⢹⣧⡘⠷⢶⣾⣿⣿⣿⣧⣶⣿⠛⢀⣴⠟⠛⢿⣿⣄⣴⣿⣿⣿⡿⠀⠦⠀⠈⢫⠙⡋⠀⡰⢂⣾⢣⠈
⠀⢳⣾⣦⣤⣄⠀⠈⠻⡖⢶⣿⣿⣿⣿⣟⠛⠧⠸⠟⠁⡄⢠⣿⣿⠛⠙⣿⢿⣟⠛⢷⡀⠀⠀⠀⠀⠀⠀⣠⠾⠡⠎⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠹⣿⣿⣿⡷⠤⢠⣦⣀⣿⣿⣿⣿⣿⣞⠀⠀⠀⠀⠀⢸⣿⠘⠀⣾⢻⣼⡿⠀⠈⠀⠀⠀⠀⠁⢠⣄⠘⢂⣠⡔⠒⢈⣩⠄⠀⠀⠀⠁
⠀⠀⠀⠈⠛⠾⠿⣶⣤⣼⣿⠻⣶⣿⣛⠙⠇⢗⡤⠀⠀⠀⠛⢠⡄⠀⢸⠂⠙⡿⢶⡀⠀⠀⠀⢈⣡⢶⢿⡟⡋⠉⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⡟⠻⠿⠿⣿⣿⣦⡋⠁⠁⠀⠀⡀⣸⡇⠀⠾⠀⠀⠀⠁⠈⣀⣴⡾⠯⠄⡍⣌
⠀⠀⠀⠀⢠⠴⢋⣩⣤⡞⢡⣴⣦⣤⣄⣹⣿⣿⣄⣀⣠⠤⠉⠀⠀⠀⠄⢀⣀⣀⣠⣞⣋⣀⣴⡖⠚⠓⠛⢶⣌⠒⣄⡀
⠀⠀⠀⣴⠟⠚⠋⠁⠀⠀⣠⠰⠙⠉⠉⠛⢛⣏⣯⣻⠽⣿⣿⢿⣭⣿⣽⢿⣿⣾⣿⡀⠉⣩⣉⣛⣷⢆⠀⠀⠈⠓⠪⣱⣄
⠀⠀⠀⡷⠀⠀⠀⢀⣤⠾⠋⠛⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠓⠌⠉⠉⠛⢧⠱⠄⠀⠀⠀⠀⢬⣃
⠀⠀⠀⠀⠀⢀⢤⡿⠉⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣠⠿⠟⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠰⠍⠉⠉⠋⠙⠭⡆⠀⠀⡠⠤⠤⣤⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⣰⠂⠀⠰⠂⠀⠀⠀⠀⠋⠁
⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢲⡐⣀⣿⠀⠀⠈⢹⠁⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠄⠁⠀⣠⢀⡀⠀⠀⠈⠀⠁⠁⠀⠀⠀⢸
⠀⢀⡴⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠈⠉⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠠⠆⠀⠀⠀⢸⠀⠀⠀⠁⠀⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠰⠃⠀⠀⠐⠀⠀⠀⠠⣂⠄⠀⠀⢰⠀⠀⠃⠀⠀⠰⠘⡄⠀⠀⣗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⡠⠄⠈⠁⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠹⠁⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⡀⠀⠀⠀⠀⢀⡠⠒⠉⣠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⡀⠀⠁⠀⠌⠂⠀⠀⠀⣀⠔⠉⢀⡴⠊⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠄⠂
⠀⠀⠀⠄⠈⠀⠠⠀⠀⠀⠀⢀⠤⠊⢀⡠⠞⠁⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⣀⡠⡠⠁
⠤⠈⠀⡀⠄⠀⠀⠀⠀⡠⠖⠁⣀⠔⠋⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠕⠂⠀⠀⠀⠒⠀⠀⠀⣴⣾⣧
⠘⠁⠀⠀⠀⠀⣀⠔⠊⢀⡤⠚⠁⠀⠀⠂⠀⠀⢀⠔⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠀⠸⣿⣿⡡
⠀⠀⠀⠀⡠⠞⣡⢀⠖⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠰⠋⠁⠀⣠⠺⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢈⠀⣀⠔⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⣀⡀⡄⠀⣀⠈
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⣤⠤⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠄⠀⠄⠀⠀⠀⠀⡠⠀⣒⡀⢒⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣁⢰⣿⠙⠋⠉⠉⠈⠁⠓⠀⣋⣁⣀⡀⠉⢀⣵⠉⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡤⣾⣿⣶⣧⠜⠛⠉⠉⠁⠀⠈⠉⠉⠛⠛⠿⣿⣦⣄⣈⣿⡿⣷⡀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠘⡷⡄⠀⠀⠀⠀⢀⡟⢻⣿⣿⣿⣿⠿⠋⠀⣀⣠⣴⠦⣤⣠⣤⣴⣧⣤⣀⡀⠈⠙⠻⢿⣧⣰⣿⣿⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢞⢢⡄⠀⠀⠀⢿⡼⢛⣧⣿⣤⣄⣴⣿⣿⠛⠛⣻⣿⡇⠘⠻⣿⣿⡿⣿⣷⣶⣿⣿⢻⣮⣿⠃⠀⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢽⠦⣄⠀⠈⠑⢿⣿⢿⣿⡟⠁⢹⣿⣤⣧⣿⠟⣧⣤⣴⣿⠏⠀⢸⣿⣿⣿⣧⡿⠋⠀⣀⣴⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠼⣷⢤⣄⠀⠙⠻⢿⣇⠀⣼⣿⣿⣿⡇⢠⣿⣿⣿⣧⡀⣴⣾⣿⣿⡿⠋⢀⣠⣾⣽⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⣠⡭⣧⣌⠳⢦⣄⠀⢿⣿⣿⣿⣿⣿⣯⣙⣫⣽⣿⣿⣽⣿⣿⣿⡿⢃⣴⣿⣿⣿⣿⣿⡿⣿⣿⣷⣦⣤⣄⡀
⠀⠀⠀⣀⣠⣶⢿⣿⣿⡦⠐⠛⠉⠷⢯⣛⡻⣦⣍⠙⡞⣿⣿⣿⣿⣿⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠿⢿⣿⣽⣿⣿⣿⣋⣀⡀⠉⠉⠛⠿⠿⠗⠿⣶⣄
⠀⠀⣼⡿⠛⠉⠉⠉⠁⠀⠀⣠⢴⣶⡿⣱⡧⣬⣝⠳⣦⣴⣟⣿⣿⣳⣦⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⡾⠻⣿⣏⣤⣄⣀⠀⠀⠀⠀⠀⠉⠳
⠀⠸⠋⠀⠀⠀⠀⠀⣠⣴⣾⣷⠿⠋⠀⣉⣩⡖⢤⣬⣭⣽⣿⡽⠋⠉⠉⠛⠉⠉⠉⠉⠉⠻⠿⣏⣉⣈⣻⣿⣿⣓⣤⣄⠈⠻⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠋⠁⠀⣠⣬⣵⠟⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠹⣿⣿⣷⣄⠀⠀⠈⠹⣿⡄
⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢠⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣦⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⡶⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠁
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⣤⡀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⢸⣧⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣿⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⢀⣼⠗
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⣶⡶⠟⠁
⠀⠀⠀⠀⢠⣀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠐⠛⣿⣿⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢹⡿⠋⠁⠀⣠⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡏⠙⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢘⣷⡀⠀⠀⠀⠀⠙⠷⣦⡀⠀⠀⠀⠀⠀⢀⣿⣿⡍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠦⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣄⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢻⡉⠉⢹⠻⣦⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠉⠁⠀⣱⠀⠀⠀⢰⠁⠀⠱
⠀⠀⠀⠀⠀⢀⡠⠔⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠈⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⡠⠔⠚⠋⠉⠉⠳⠖⠄⡀
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠐⠦⣄⡀⠀⠀⣰⠏⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⡾⠀⠀⠈⢂⣀
⠀⠀⠀⠀⠀⠠⢤⣤⣀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣼⣥⣤⣤⡶⠉⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠂⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠛⣷⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠤⠀⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣥⣤⣤⣼⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⡟⠋⠉⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⣿⠏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⣿⣦⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡟⠛⢿⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣿⣧⣤⣼⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠐⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠘⠂⠇⠀⠀⠈⠈⡃⠀⠀⠀⠃⠀⠀⠀⢀⠄⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠀⠄⠂⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣤
⠀⠀⠴⠒⠒⠒⠒⠲⠒⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⢱⡆⠀⠀⠀⠀⠀⠄
⠀⠀⡇⢀⠖⠉⠁⠉⠢⡀⢰⠉⠉⡆⡊⠀⠙⠄⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠞⠃⠀⢀⠀⠀⠀⢀
⠀⠀⡇⡌⠀⠀⠀⠀⠀⠱⠈⡲⣒⠁⢙⡶⡊⠀⠀⠀⠀⠀⠀⠀⣀⠀⣃⠀⠀⠠⢀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠛⠄
⠀⠀⡇⠡⡀⠀⠀⠀⠀⡔⢸⠀⠀⡇⣇⠀⢈⠀⠀⠀⠀⠀⠀⠀⡅⠀⠬⠀⠀⢀⣀⣀⡀⠀⠠⠀⠰⠀⠀⠀⠀⠁⠤⠀⠄
⠀⠀⣇⠀⠑⠢⡤⠄⠊⠀⠀⠉⠉⠀⠈⠉⠁⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⠀⠤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⠌⡆⠀⠀⠀⡇⠘⠀⠀⠀⠀⠅⡀⠀⠀⠀⠀⡈⠂⠀⠰⠰⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠁⠀⠀⢮⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡾⢩⠏⢿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢤⢀⠠⢄⡀
⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠄⠀⠀⠀⡅⡆⠀⠹⢍⡘⢹⠀⢀⠔⠛⠋⠲⡄⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠀⠀⢻⡀⢀⢈⡬⢷⡾⠀⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⠀⢠⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠢⡉⠓⠛⠛⠉⠀⠀⠀⠓⠤⠤⠖⠁⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⢢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⢄⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠒⠀⠘⠀⠀⠀⠀⠸⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠄⠀⠀⠀⠀⡀⠀⠀⠡⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠘⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⡀⠀⠐⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⣀⠀⠀⣀⠤⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡘⠀⢀⠞⠳
⠀⠀⠀⠘⠀⠙⢦⡀⡇⠀⠀⠱⡀⠀⠀⠀⠀⡰⠉⢀⣴⣦⣄⠑⡎⢀⣴⣶⣤⠀⠈⢆⠀⠀⠀⠀⢠⠎⠀⠀⠀⣩⠔⠁⠀⠀⡇
⠀⠀⠀⢡⠀⠀⠀⠙⠃⠀⠀⠀⠘⡀⠀⠀⠀⡁⠀⣿⣿⠙⣿⡆⡇⢸⣿⡋⣻⡇⠀⢸⠀⠀⠀⠀⡈⠀⠀⠀⠀⠁⠀⠀⠀⡜
⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠱⣀⠘⢿⣿⠟⣡⠗⢌⡻⣿⠿⢁⡴⠃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠙⠢⠄⣀⡀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠁⣒⡲⠈⠧⠤⠤⠏⠉⠏⠠⠄⣀⠀⠀⠀⢀⠎⠀⢀⠤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠙⠢⣄⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠑⢤⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⢎⠀⠀⠀⢀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⣈⠑⠢⠤⣀⠇⠀⠀⠀⠀⠀⣀⣠⡤⠤⢴⠒⡎⢋⣹⠼⡟⠀⠀⠀⠀⠀⠈⠦⠔⠊⣡⠤⡄⠈⠢⡀
⠀⠀⠀⠀⠀⡴⠁⢀⠜⠀⠉⣒⣦⣄⠀⠀⠀⠀⠀⠀⠙⢗⠷⠺⠔⡖⢻⠙⣠⠞⠀⠀⠀⠀⠀⠀⢠⡴⠖⢧⠀⠀⠈⢆⠀⠱⡀
⠀⠀⠀⠀⡜⠀⢠⠎⠀⢀⠔⠁⡤⢉⣃⠀⠀⠀⠀⠀⠀⠀⠙⠲⠧⠵⠜⠊⠁⠀⠀⠀⠀⠀⠀⣀⠤⠐⠲⡀⠱⡀⠀⠈⢆⣀⠅
⠀⠀⠀⠀⠈⠒⠁⠀⢀⠎⠀⡜⠀⠀⡰⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠱⠀⠀⠀⢱⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠜⠀⠀⢠⠁⢠⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠀⠀⠀⢡⠀⢃⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠋⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣠⣠⣴⡶⣾⡅⠻⠆⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠊⠑⢛⠷⣠⠻⣛⡵⠟⠙⢧⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡄⠈⠀⠀⠐⡏⢸⣶⡟⠟⠠⡀⠀⠈⣷
⠀⠀⠀⠀⠀⠤⠈⠁⠀⠀⠀⠄⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣠⠜⠋⠀⠀⠀⠃⠀⠐⠀⢸⡇
⠀⠀⡠⠀⠀⠄⠁⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡋⠅⡊⢉⠂⡀⠀⠀⠁⡀⠀⠈⣧⡀
⠀⢈⡀⠀⠀⠀⠀⣷⣄⠈⠁⢰⡸⡃⠀⠀⠀⠀⢀⣀⣴⣾⣷⠷⠒⠋⠁⠈⠓⠤⣦⠀⠀⠐⡀⠀⣿
⠀⢀⠍⠢⣐⠋⠛⠛⣿⣷⣶⣾⡟⠙⢶⣶⣶⣞⣿⣿⣯⣍⣀⠀⠀⠀⠀⠀⠀⠀⡜⠢⡀⠀⠀⠀⡿
⠀⡎⠀⢠⣬⣷⣶⣿⡟⢒⣲⠾⣟⠿⣀⠙⢿⡿⣟⠛⡛⠿⣿⣿⣕⢢⡀⠀⠀⢠⠁⠀⠙⡄⠀⢰⠇
⠂⠀⢀⠞⢡⣲⣿⣿⠁⣬⣉⠳⣾⢷⡟⣇⠀⠙⢮⡻⣦⡑⢢⠰⣿⢳⡇⠀⠀⠀⠀⠀⠀⠈⢠⡿
⠀⠀⠀⠀⡹⢋⢻⣿⣷⣴⡦⢽⡮⠈⠁⠀⠀⠀⠀⠑⣬⠱⡀⠀⢹⡈⡇⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠁⠀⡞⢟⡏⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡇⠀⠀⣿⢹⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠘⢀⠀⠱⡤⣱⡀⣈⠳⠦⣤⣤⣄⡀⣀⣀⣀⡀⡟⣿⠀⠴⠁⠏⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠈⠒⠤⠀⢉⣯⣭⣰⣿⠧⣤⣭⣿⣿⡟⠁⡁⣿⢈⣀⣀⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡿⠿⠉⠉⠉⠛⢛⢿⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢣⠾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⣿⣿⣿⣆⣠⣾⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣼⣷⣦⣠⡿
⠀⠀⠀⠈⠹⠿⠛⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣄⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢹⣿⣿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣠⠀⠀⠀⠀⢀⡾⠂⠀⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⡀⠀⣴⣾⣿⣿⣧⢶⣀⣠⣴⠟⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠀⠀⠀⠻⣆⡘⢿⣿⣿⣿⣿⣿⣯⣅⡀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⠀⠀⠀⠀⠈⠛⠛⠛⠉⣻⣿⣿⣿⣿⣿⡷⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣶⠄⣴⣶⣆⢈⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢹⣿⠁
⠀⠀⠀⠀⢠⣿⠀⣻⣿⣯⣾⡟⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⢸⡿⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⢠⣤⣤⡄
⠀⠀⠀⠀⠀⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒⠟⠀⠀⢤⣄⣀⠀⣀⣻⣿⣟⣀⢾⣿⣿⠟⢐⡟
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⢻⣿⣿⡿⣿⣿⣿⣶⣾⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⣿⣿⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠟⠉⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠈⠻⣷⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⡿⠀⠙⣿⣟
⠀⠀⠀⠀⢠⡼⠇⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠀⠀⠙⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠲⢿⣒⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣞⠉⠉⣡⠼⠟⠓⠀⠀⠀⠈⢳⣀⣠⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠈⠓⠾⣁⣀⣀⠀⠀⠻⢿⣛⡋⠉⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢳⡀⢀⣠⣤⠤⠤⠶⠚⠋⠀⠀⠀⠀⠀⠉⠓⠲⣄⠀⢀⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢹⠀⠉⡽⠁⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢸⡉⠉⣰⠃⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⢺⡀⣰⡧⠖⠚⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠙⠒⠾⣇⢀⡟⠀⠈⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡏⣇⠀⠘⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⢀⡏⡿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢻⠙⣆⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⡞⢹⠃⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠷⡄⠳⣏⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡽⠃⡴⢷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⠓⢦⡽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣥⠖⢛⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢚⣒⣚⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⣒⡚⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣯⡥⠽⠤⠬⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠯⠤⠼⠥⣭⡟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡇⠀⠀⡴⢿⣉⣉⣙⣷⣦⣤⣀⣀⣀⣠⣤⣖⣿⣉⣉⣹⠟⡄⠀⠀⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢸⣧⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣿⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⢽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠵⠋⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⢻⠿⠋⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⡠⠶⠖⠀⠖⠀⠆⠀⠆⠀⠛⠒⠠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⣀⣈⠁⠀⢠⠂⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣀⠀⢐⠀⠀⠁⠀⠀⠄⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⡯⠖⠀⠀⢹⠁
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠉⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⢒⠒⠐⠂⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠓⠋⠉
⠀⠀⠀⠀⠠⡀⢰⠀⠀⡆⡆⠘⠀⠀⡇⠀⠀⡏⠀⠀⡔⢸⣽⢲⡄⢁⣈⠀⠀⢀⡀⠐⠃⠀⢀⠉⠀⠀⠘⡁⠀⠒⠒⢀⡀
⠀⠀⠀⠀⠀⠁⠈⢀⠟⠛⢻⡁⠀⢠⡇⠄⢠⠃⠀⠸⣚⣚⣗⣰⡇⢀⠀⠀⠀⠈⠀⠀⠀⣰⠛⢿⡄⣴⠀⠀⠀⠀⣄⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⡇⠀⠈⠁⠀⠀⠙⢋⠏⠃⠄⢀⠁⠀⡀⢠⠀⠀⠀⠘⠳⠾⠃⢸⠀⠀⠀⢰⡟⠊⠃⠀⢠
⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⣿⠀⠘⠓⠀⠀⢸⢠⡖⠀⠸⠀⢰⠀⢸⠀⢀⣇⢀⠀⠀⢰⠛⠉⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠈
⠀⠀⠊⠀⠀⠀⠀⠈⠆⠀⢰⠏⣀⠯⠀⠀⠀⠈⢸⡇⠀⠀⠀⢸⠀⠈⠀⠐⠶⢺⡀⢀⡀⠧⠔⠰⠶⠆⠀⠀⠐⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢒⡋⣀⢹⠀⠀⡄⠀⠀⠠⠁⠆⡏⠀⢸⠀⠀⢠⡆⠀⠀⠀⠈⢱⢸⠀⠀⢆⠉⠀⠀⠀⢰⡇⠀⠁
⠀⠀⠛⠒⠋⠁⠀⠀⠀⠸⡝⠹⢸⡀⠀⠳⠀⠀⡄⠀⠀⠁⠴⠞⠀⠤⢹⠀⠘⠌⠀⠀⠈⢘⠀⠀⢸
⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠑⠃⠀⢡⣤⣤⡄⢰⠁⠀⠀⠀⢠⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠇⠀⠀⠀⣀
⠐⠒⠒⠒⠒⠒⠒⠀⠒⠒⠓⠒⢢⣤⣤⣤⡄⢸⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⡄⠀⢀⡃⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠘⠀⠀⠂⠀⠀⠏⠀⠀⠀⠀⠀⠋⠀⠀⠄⠀⠃⠀⠀⠳⠐

⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡒⠢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡿⠛⠉⠀⠀⠀⠀⠀⠈⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠻⡦⣀⠀⠀⠀⠀⠀⠠⢦⣤⠊⠀⠀⠀⣠⣤⡦⣮⣤⣀⠀⠀⠈⢳⡶⠞⠀⠀⠀⠀⠀⢀⠔⡟
⠀⠀⠀⠳⣌⢳⡀⠀⠀⠀⠸⣰⡷⠚⠋⣈⡀⠉⢉⠂⠀⠋⢉⣿⠍⣷⣾⣧⡴⠀⠀⠀⢀⣼⣫⠜⠁
⠢⣀⠀⠀⠈⠫⣙⣲⣄⣀⣀⣸⠇⠀⠀⠈⠁⠆⢈⠁⠀⠀⠸⠝⠓⠚⠊⣟⣁⡀⣠⣒⣋⠝⠁⠀⠀⣠⠔
⠀⠈⠙⠶⣤⣄⡀⠉⡉⠓⠛⠟⢶⠄⠂⠒⠀⡊⢳⡩⠀⠀⠀⡲⠡⢒⣶⠾⠟⠓⢋⣉⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠓⠻⠷⠶⣶⣮⣅⣠⢅⣤⣤⡉⢠⣰⣏⣖⣼⣾⣿⡶⠿⠛⠓⠋⠉⠉
⠀⠀⠀⠀⢀⣤⣴⡶⠶⠛⠓⢿⣷⣶⣶⣿⣷⣦⣬⣁⣤⣬⣿⣿⣧⣶⣿⢿⡏⠛⠓⠛⠷⣶⣄⡀
⠀⠀⢀⠴⠋⠉⠀⠀⣠⣶⠿⠋⠙⠛⠛⠟⠋⠉⠉⠛⠋⠉⠉⠛⠛⠛⠋⠉⠙⢿⣶⡄⠀⠀⠈⠛⠦
⠀⠀⠀⠀⠀⠀⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢆

⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⣶⡾⠿⠿⠷⠆⠀⠀⢀⡀⢀⣠⣤⣴⣶⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⣿⣥⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡶⣦⣷⢛⣡⣶⣶⣶⡯⣮⣅⡀⣤⣢⣝⠿⢿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⣀⣹⣿⣧⠀⠀⠀⣀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣿⣛⠻⣿⣿⣼⣿⣿⣶⣿⣿⠛⠛⣻⣿⣿⡿⣽⡿⣿⣿⣿⣿⣷⣤⠀⡠⠄⣾⣀
⠀⠀⠀⠀⢺⣿⡋⡹⣿⣄⢀⣼⣧⡜⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⢃⣄⣼⣿⣿⣏⣴⣿⣿⣜⣿⣿⣿⣟⣫⣦⠉⣼⠿⢹⣷⡆
⠀⠀⠀⠀⠈⢿⣧⢳⡙⠿⣾⣿⣿⣿⣿⡿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣊⣿⣝⣿⣿⣿⣧⣿⠆⣼⡿⠁
⠀⠀⠀⠀⠀⣈⣻⣷⣝⣀⣿⣿⣿⡾⣿⣧⣿⣿⣾⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⠐⣨⢿⣿⣿⣟⣠⣼⢟⡁
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⢿⣿⣿⣿⡰⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣼⣿⣧⢹⣏⢿⣯⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣯⣯⣽⠋⣙⡻⢿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⡿⣟⣿⣌⣽⣿⣭⣿⣥⣿⣭⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣿⣶⣿⡿⢶⣶⣝⠷⣍⠙⢿⣿⣷⡈⢋⣙⣿⣟⣩⣋⣰⣿⣿⣿⡿⢋⡀⣻⣧⣘⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⣿⣿⣿⡽⢿⣿⡿⠟⠀⠀⠘⢿⣿⣷⣿⣴⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣯⣿⣿⡿⠃⠀⠀⠉⠁⠀⠀⠀⠀⠈⠙⠻⠿⡟⢿⠇
⠀⠀⠀⢀⣿⡿⣻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠛⠃⠈⠙⠛⠻⠿⠿⠟⠛⠋⠉⠁⠛⠛⠛⠋⠁
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣤⣶⡶⠶⣤⡀⠀⠀⢒⣛⣛⣛⡻⣿⣶⣄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣻⣿⣷⣶⣶⠶⠀⠀⠀⠀⠈⠙⠻⣿⣿⣳⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢿⢟⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⣾⣿⣿⡏⣿⣿⣿⠿⠋⠁⠀⢰⣤⣀⣄⣤⣤⣀⣀⣀⡶⠀⠀⠀⠉⠛⠿⢷⢺⣿⣿⣿⡄⠀⠀⠀⠀⣠⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠰⣿⣿⣿⣻⣭⣤⣄⠀⣠⣴⣿⣿⣽⣿⣿⣧⣿⣿⣿⣿⣿⣿⣦⣄⢀⣠⣤⣶⣮⣿⣿⡿⠁⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡈⢛⣿⣿⣿⣿⠯⣾⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⠈⢀⣤⣾⡿⠋
⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣀⡈⠻⣷⣬⣝⡛⠿⡟⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⠿⣛⣻⣿⡖⠿⠟⠉⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡾⠛⠛⠛⠛⠋⠙⠻⠿⠿⠷⣶⣮⣹⣿⣿⡧⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣛⣫⣭⣶⣿⢿⡿⠿⠟⠛⠻⠿⠟⠛⠛⣴⡄
⠀⠀⠛⠀⠀⠀⠀⠀⣀⣠⡄⣤⣤⣤⣤⣦⣬⣍⣛⣛⡿⠿⢿⣿⣟⣻⡿⣿⣹⣿⣿⠸⣿⣿⣿⣽⡿⣛⣵⣾⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⣰⠿⠿⠛⠉⠈⠉⠉⠀⠈⣉⣉⡙⣛⣻⡿⠋⠰⠋⣙⢿⣿⣿⣿⣿⣻⣷⠾⣻⣹⣿⣻⣻⣟⡛⣻⣟⠉⢉⡀⠀⠀⠉⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⣠⣶⣶⡆⢿⣿⠛⠿⢿⣿⡧⠿⠆⠙⠃⠩⠿⠟⠹⠿⠇⠘⠉⠻⠿⣿⡿⠿⠏⠛⣯⣾⣿⣿⣿⢶⣄⠀⠀⠈⠉⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡻⢿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢟⣯⣿⣿⡄⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡠⠲⢦⠀⠀⠀⠀⣀⣤⣴⠶⠶⠶⠄⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠄⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣷⠚⠻⣧⠀⢀⣼⡿⢯⡤⠂⠀⠀⠀⠀⠀⢰⡀⠋⠈⠠⠀⠀⠀⠂⠀⠤⠀⠈⠑⢠⡄⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⢹⡄⠸⣿⣤⣀⡀⣰⡆⠀⡄⠀⠀⡶⣵⡬⢄⢄⣤⢢⣀⢀⠤⣠⣤⣤⣠⣿⣾⡆⠀⠀⡀⣀⠤⡶⣉
⠀⠀⠀⢀⣴⣶⣶⣤⣈⠙⠡⠅⠀⠀⡉⠛⠛⠛⠛⠛⠁⠀⠀⠉⠙⠃⠙⠋⠛⠚⠛⠋⠚⠛⠃⠛⠉⠋⠉⠀⠀⠘⠈⠁⡖⢛⡃
⠀⠀⠀⠘⢧⣬⣉⣉⣻⣻⠶⢀⠄⠈⠀⠀⠰⣄⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⡴⠀⢀⣠⡤⠾⢿⣦⠔⣾⣠⠁
⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠂⠐⠈⠀⠀⠈⡙⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⡀⣺⠀⣰⠞⢩⠃⢀⣨⠃⠤⠂⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢌⣀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⠊⣉⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡿⢄⠀⠀⣸⠏⠀⠀⠨⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠐⠀⠐⠒⢈⣉⡉⠙⠷⢤⣄⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⢲⣷⡟⢂⢀⠖
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡠⡤⣄⣀⠈⣀⣀⣐⠛⢋⡁⠀⠀⠀⠀⡀⢀⡞⠁⣀⠾⠁⠀⠐⡉⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⣠⡾⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠌⠙⠋⠁⠀⠈⠈⠥⣈⠉⠁⠀⠀⠀⠀⠀⠘⠿⣡⠖⠁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠋⠉⠉⠈⠉⠛⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢏⢈⠮⠁⣦⣀
⠀⠀⠀⠀⢀⠄⣶⠀⠀⠀⠀⠀⠣⢢⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢶⡆⠀⠀⢰⠀⠀⠀⠀⠀⠀⡄⡀⠀⡀⠠⢰⡆⠁⢄⢊⠽⣢
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⡄⠢⠄⠀⠀⠈⠀⠀⠙⠟⠒⠚⠿⠂⠀⠀⠀⡦⠘⠃⠉⠠⠼⢄⠀⠀⠀⠿⡸⠂⠔
⠀⠀⠀⡀⠀⣾⠀⠀⠀⠐⡀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⢿⣀⣤⣄⣠⠔⠀⠀⣠⣤⣀⡀⠀⠀⠀⠠⣀⣶⣀⣠⡠⣌⣀⠶⠃⠀⠀⠀⡠⡄
⠀⠀⠀⡇⠀⠘⠀⠀⠀⡀⠇⣠⡀⠀⣄⡈⠲⢠⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠴⡏⠀⠀⢙⡄⠀⠀⠀⠈⠿⠃⠀⢰⣞⠧⠃⢀⣐⠤⠂⠈⠋⠔⣢⡀
⠀⠀⠀⣄⠰⠁⢠⠇⠈⢹⠇⠻⠈⠢⣈⣏⣀⡌⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠱⠄⣀⠔⠅⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢁⡴⢀⣀⠀⢀⡿⠁
⠀⠀⠀⠈⢄⣀⣹⠦⢀⡀⠀⠀⠘⣆⢉⣼⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠳⣴⢠⣦⠐⠚
⠀⠀⠀⠀⠀⢩⣱⠧⠤⡇⣀⠀⠠⠻⡆⢀⣿⡀⠀⠀⠀⣴⠺⠛⢿⣿⣿⣶⣄⠀⠀⠀⢀⣠⣴⢶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠛⡀⠈⢇⠀⠀⠀⠀⡯⠈⠂⠹⡄⠀⠀⠀⠀⡀⢀⣀⣡⡴⠟⠀⣠⡴⣿⣗⣽⣷⣿⡿⠏⠀⠀⠀⠀⢀⣤⠀⠀⠉⠁⢈⡀⣀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠰⠠⠀⠹⢶⡖⠐⣿⡋⠀⠀⠀⠃⠀⠀⠚⠟⠺⠛⠛⠁⣀⡀⠀⢿⣿⣇⡾⠿⠟⠋⠀⠀⠀⠀⠐⠐⠛⣻⣷⡶⠖⠛⠙⢋⣿⣿⡆⢨
⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⠸⣏⡀⢨⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⠖⠚⣳⠁⢀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⢀⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢧⠀⠀⢷⠀⠀⠀⠀⠀⢀⠀⠔⢩⣀⡐⡪⠋⠀⡿⠓⢦⠀⠻⡓⠓⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡿⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣹⠀⠀⠀⠀⠀⠀⠠⠚⠿⠉⠁⠀⠀⠀⣄⠀⠄⢧⠀⠻⡦⣰⠀⠉⢢
⠀⠀⠀⠀⠀⠈⠀⠁⠀⣀⠀⠠⣿⠤⠼⠗⢀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⠀⣾⠀⠀⠈⠙⠶⠦⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠴⣄⣈⠦⠴⠤⠚⠉⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠾⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⠤⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠈⢷⡀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⣦⣈⣷⣶⡾⣃⢶
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠙⠷⠿⣥⡖⠀⠀⣠⠴⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢛⠩⠍⠩⠥⠀⠩⠞⡇
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠱⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠐⠀⠀⠀⣉⢹
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡆⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠀⠀⠄⠀⠀⢰⢸
⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢬⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⠴⠤⢾⡶⢦⣿⣷⡷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⢀⡀⠀⠸⣻
⠀⠀⠀⠁⢠⡇⡘⠀⠀⢀⣨⣔⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⠶⠋⠁⠀⠀⠀⠈⠀⠀⠀⠈⠳⠾⠢⡀⠀⠀⠀⠀⠀⠀⢀⡎⠉⢀⢀⠉⠳⢾⣿⠀⠀⠀⠀⡀
⠀⠀⡶⠀⢸⣿⣿⠞⠉⣅⣈⠙⡁⠀⠀⠀⠀⢀⠤⢤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣼⠇⠀⠀⠈⠀⠤⢼⡇⠀⠀⠀⠀⡇
⠀⠀⣿⣤⠘⣿⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⡄⠀⠀⢿⡀⠠⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡁
⠀⠀⢣⠀⢰⣿⠁⠀⠀⠀⠤⢼⡇⠀⢀⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡄⢸⣇⢀⡀⠀⠐⢀⡟⠀⠀⠀⠀⢁⡇
⡀⠀⠈⢧⠀⣿⣅⠠⠄⠀⢀⢸⡆⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⣿⠀⠀⠀⣛⡿⠁⠀⠀⠀⠘⢻
⡇⠀⠀⠘⣧⣾⣿⣄⡂⠀⡀⢠⣿⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣯⣿⣷⣭⣶⣿⠁⠀⠀⠀⠀⢠⠏
⡇⠀⠀⠀⠘⣆⠹⣿⣿⣷⣦⣿⡿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣟⠛⠛⠛⣫⠃⠀⣠⠀⣰⠟
⠸⣤⠄⠀⠀⠈⢧⡙⢿⣭⣭⣽⣾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣭⣿⡿⢁⡀⠀⣭⡟⠋
⠀⠹⡄⠀⠀⠀⠀⠳⣄⠹⢿⣟⠿⢿⣿⡦⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠟⢉⠔⠉⢠⡴⠏
⠀⠀⠹⣦⣈⣄⣀⠀⠈⠻⢽⣯⣅⣲⣤⣿⣦⢄⡦⣤⣵⢀⠀⠀⠀⠀⣄⠀⠆⣠⡀⠠⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⠄⣹⣯⡥⢤⣴⣁⣠⠶⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠻⣅⠀⢀⡈⢀⣈⣉⣽⡛⠛⡛⢷⣼⣿⣷⣯⣶⠗⠢⡠⣼⡾⢁⢸⣿⣿⠉⠙⣆⢀⠐⡠⣴⣤⣶⡯⢛⣫⣿⣏⣠⣬⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⡶⣧⡾⠁⡐⠂⠠⢹⡋⠀⠀⠈⢢⣷⣻⣼⠟⣩⣾⡟⣛⠋⠉⠁⠀⢠⡄⠀⠀⠀⠀⠀⣀⣼⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣁⢌⡁⠁⠀⢒⢢⣌⣳⡹⣿⣿⣅⠣⠅⠉⠄⠛⠅⠀⠀⢀⣴⣿⣿⢏⣾⣿⠟⠛⠛⠛⠋⠙⠻⡋⠛⠻⠶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢀⡐⠀⣠⡼⠛⠛⠛⠛⠛⠛⠿⠿⠿⢿⡿⣟⣿⣿⣾⣿⣾⣿⣿⣾⣧⣶⣿⣾⡿⢣⣾⣿⣷⣿⡶⠦⢦⣤⣶⣷⣟⠐⠠
⠀⠀⠀⠀⠀⠀⢾⣷⣿⠟⠀⠀⣀⣼⣯⣄⣄⢀⡀⣴⣦⣧⣿⣷⣮⣟⣻⣛⣿⣻⣛⣛⣛⣛⣭⣶⣿⣻⣇⣀⣀⣀⣠⣦⣼⡏⠢⡙⠷⣄⠀⠂
⠀⠀⠀⠀⠠⣃⣠⣿⠃⠀⠀⣰⠋⢸⡟⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠉⠉⠉⠙⢷⡆⠀⡀⠘⣷⠶⠆
⠀⠀⠀⠀⣵⣿⠟⠁⠀⠀⢠⣟⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣴⠀⠹⡆⠠
⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠈⣿⡯⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⡇⠀⠀⠹⣦⣥
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⢻⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⡇⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢻⣿⣅⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠋⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠛⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣄⣶⡐⠨⠁⠈⠑⠄⠁⡤⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⡒⠃⠀⠀⠀⠀⠀⢂⠀⡀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢠⢈⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⡀⠀⠈⠂⠀⠐⠀⠀⡠⠄⠀⠈⡆⠀⠀⢀⠈⠈⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⣀⠠⠪⠀⠀⠀⣠⡁⠀⠀⠈⢑⠀⡀⠠⡇⢀⠔⠁⠱⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⣐⡄⢰⣦⠀⠀⠘⣿⠀⠀⠀⢀⠢⠀⠆⠘⠁⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠈⠿⠃⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠃⠐⠀⢠⠊⢀⠰⠁⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢠⠀⢉⠠⣥⠒⢁⠌⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠃⢠⠀⠀⢀⣀⠀⠤⠄⠒⠂⠉⠄⣦⣤⡆⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡕⠂⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⠈⠉⠉⠉⡙⠀⠀⠘⠋⠉⢹⠤⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠄⠀⢸⠷⠿⠿⢿⠀⡀⢀⠄⠂⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⢠⣷⢶⣤⣤⣼⡆⡀⠌⠳⠤⢔⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢲⠙⠿⢿⠠⢍⣯⣽⣌⢁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠈⠋⠉⠉⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠉⠁⠚⠊⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠒⠀⠁⢢⣀⡀⠀⠀⢡⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡔⠁⠑⣼⠄⠀⠈⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⡶⠃⢀⡇⢀⠥⠀⠊⣡⠤⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⢀⣠⠶⠋⠀⠰⠀⡀⠊⠁⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠉⠀⠀⠀⠀⠀⠁⠈⠈⠀⠉⠄⠊
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⡥⣄⠀⠀⢄⠀⢀⣠⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠁⠀⠀⢦⠀⠀⠱⠂⠀⢀⣀⣀⣨⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠱⠶⠖⠁⠀⠙⠛⠁⠀⢀⣀⠁⠊⡠⠂⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⣸⡇⠈⠀⠀⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠉⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣼⡇⠀⠀⠐⠒⠉⡄⠀⠸⣠⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠸⠿⠇⠀⠀⠀⠀⠀⠸⠆⠀⠛⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠀⠁⠀⠀⠀⠡⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠛⠃⠀⠀⠀⠈⠀⠀⣀⡜⠁⣀⠆⠀⠀⠀⠀⠀⠘⣄⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⠆⠘⠀⠀⠀⠀⠀⢄⡀⠀⠌⠀⣶⡄⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡃⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⠤⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉⠹⠆⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠆⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣿⣿⣷⣶⣤⣶⣶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⠇⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢀⡼⢩⠟⣹⣿⣿⠛⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⢀⠏⢀⣿⣿⣿⡆⢠⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠈⠀⠈⠁⡟⢸⡇⣼⡟⠃⠇⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢨⡇⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⢿⠁⣿⠁⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣤⣠⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢻⡿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⣿⣿⡇⠀⢰⡇
⠀⠀⠀⢸⣷⣶⣶⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠅⠀⠀⠀⣿⣿⣷⠀⠸⠇⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣤⣾⠀⠀⣶⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡏
⠀⠀⠀⠀⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡞⠂⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡷⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡇
⠀⠀⠀⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣴⣧⠀⣿⡇⠀⠀⠀⠀⣿⣇⡀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣷⣿⢰⣾⣷⡆⠀⠀⢀⣘⣿⣧⣇⡀
⠀⠀⠀⠀⢿⣿⣿⠃⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⢠⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⢸⣿⢈⡇⠀⠀⠘⣿⠂⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠘⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠈⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⣀⣤⣄⣀
⠀⠀⠀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢹⣶⣾⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢀⡾⠋⠀⠀⠙⢷⡄⠀⠀⠀⠀⣀⣙⣛⣛⣛⣛⣛⣋⣠⠀⠀⠀⠈⠻⠟⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⠁⠀⠀⠀⠀⠘⣷
⠀⠀⠀⣿⣿⢶⣶⣶⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠉⠉⠉⠉⠋
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢹⡄⠀⠀⠀⢀⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠻⠲⠒⠓⠓⠚
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠙⢶⣤⣾⡷⠏⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⢚⣛⣾⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⠀⣾⠉⠙⠻⣿⣿⠟⠋⠉⣷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⡆⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠦⣄⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⣀⣴⡾⠃
⠀⠀⠀⠿⠿⠛⠛⠛⠁⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡏⢹⢀⡴⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⢾⡇⢸⣻⠇⠀⣇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⢤⣀⣸⣵⠃⢸⡇⣜⣻⢀⡾⠁⣿⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠚⣉⣿⣍⣙⣶⣄⠀⠀⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⣾⠁⣠⠿⡛⠲⣾⡙⣆
⠀⠀⠀⠀⠀⢴⣿⣒⣋⣉⣁⡀⠀⠉⢧⠀⠳⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⡴⠁⢰⠛⢦⣬⠿⣾
⠀⠀⣀⣶⠋⠉⢉⣹⣤⣄⣈⣭⡽⠶⣌⠳⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣷⠖⠓⠲⠶⣽⡄⠈
⢠⢞⡥⠞⠋⠋⠉⠀⠀⠀⠀⠀⠙⢦⡀⠙⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡇⠀⠀⠀⣰⠃⠙⣶
⠿⠋⠀⠀⠀⠀⣀⣠⢤⡴⠶⠶⠶⠦⠬⠷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠑⣾⠷⠤⠤⡾⠁⠀⠀⣸
⠀⠀⠀⠀⣰⠊⢁⣠⠞⠓⢒⣒⣶⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢐⣽⣿⣿⣄⠤⣈⡿⠃⠀⠀⣰⠛⠒⠒⠚⡏
⠀⢀⡴⠚⣉⡿⠋⠀⣰⣞⣉⣁⣀⡤⢴⡃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢞⣽⣿⣷⣼⡻⠿⠟⢨⡧⠋⠀⠀⠀⣠⡇⠀⠀⠀⢰⠃
⠀⣏⣤⠞⠋⠀⠀⡜⠁⢸⠁⠀⠀⠀⡴⠙⢾⣁⣰⠃⢠⠄⢀⡄⠀⡄⢸⡙⠿⠛⢈⡗⢢⣶⡋⠀⣀⠴⢚⣉⡞⠙⠢⣤⣴⠏
⠀⡇⢸⠀⠀⠀⢠⣧⢴⡇⠀⠀⠀⢰⠇⠀⠀⠉⣿⠚⠿⠦⠾⠧⠴⠧⠬⠿⠶⠒⢻⣿⠉⠙⣣⣾⠵⠚⣉⡿⠁⠀⢀⠞
⠀⢧⢸⠀⠀⠀⠸⡀⢰⡇⠀⠀⠀⠘⣦⠤⠖⠚⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⣀⡴⠏⢀⣠⠴⠋
⠀⠈⠻⡄⠀⠀⠀⠱⡞⢧⡀⠀⠀⠀⢷⠀⠀⠀⢀⡿⠒⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣙⣦⣀⠀⠈⠳⠦⡖⠋⠀⠀⠀⠀⠀⣤⣭⣭⣭⣽⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠘⠢⢤⣀⣀⣀⣀⣀⣀⡤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣾⡇⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢧⢿⠟⢬⠋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣸⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⠆⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢾⣥⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠘⠛⢻⡟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣨⣿⡿⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠳⣶⣎⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢯⠁⣾⡷⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢙⣢⣄⣽⢀⡀⣀⢴⠒⣎⠏⢙⣫⠛⠂⢏⢿⡖⡻⡗⡶⠢⣤⣤⠼⢦⣿⣴⠀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠸⣿⣄⠀⠻⣌⣽⣿⣿⣤⣮⡷⠛⠛⠙⠿⣿⣵⠠⣶⣿⠽⠧⣤⠀⠉⠟⣳⣦⠏⠁⣴⢿⠃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠹⣧⡴⠀⢈⣯⣉⡟⣿⣿⡛⠁⠀⠀⠀⢈⡆⠀⡟⠃⠀⠀⣐⣿⣿⣆⠙⣣⣤⢶⡧⠈⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡷⠄⠀⠀⠈⣧⣿⣿⣛⣿⣾⠻⠻⢻⣦⣤⠀⠀⠀⢡⠠⠁⠀⢀⠛⢁⡿⠇⠀⢏⣘⡻⠋⠀⢀⣀⣴⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢧⡠⡄⡀⠈⣷⠶⠉⠀⠀⠀⠀⠐⠟⠿⣷⣢⣤⣤⣴⠞⠛⠛⠋⠀⠀⠀⣾⢧⢘⣿⣾⠏⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠻⠿⠻⢇⡀⠀⠈⠁⠠⡄⢀⠀⠀⢀⣥⣾⡆⠀⠀⡀⠴⠐⣉⣁⣉⣤⢞⠉⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣶⣦⣼⠷⠾⠿⠛⠛⠛⠓⢶⣴⣷⠄⢹⡇⢈⣤⣢⠌⣀⣴⠋⠉⢉⣙⠐⠾⠒⠊⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⠛⠁⠀⡀⡀⠀⢤⣴⠾⢦⡬⠛⠋⠉⠛⠊⠓⠒⠉⠑⠚⠻⠷⠆⠆⠘⠏⠐⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⢀⣹⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢴⣆⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠛⠛⠀⠀⣀⡤⠤⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣟⠀⠄⡤⣴⡶⠛⠉⠀⠈⣁⣤⠤⠀⠀⠛⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠤⠐⠂⠀⠹⣽⣆⠀⠚⢙⣦⠀⠀⠀⠓⢿⠀⠀⠀⠀⣀⣀⣠⡤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⡹⠛⠻⣷⡺⣇⠀⠀⠀⠻⣷⡀⠀⠻⣷⡀⠀⠈⢸⠀⠀⠀⢸⠏⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠀⠀⠙⢮⣳⡄⠀⠀⢸⡻⣄⣀⣜⣷⣀⣰⣾⠀⡠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⣣
⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⣶⡶⢦⣵⣿⣿⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡷⣻⣤⣄⣠⣤⡄⣀⢀⣀⡀⠀⢀⡀⠀⠀⢸⡗⣶⣴⣿⠿⢻⡆
⠀⠀⠀⠀⠑⠤⣀⡀⠀⠉⠉⣷⣴⠶⣬⣻⣟⠛⠉⠛⠿⢻⣿⣿⠁⠉⠳⣒⠖⣦⣍⠙⠟⣼⠻⠋⣿⣷⠛⣻⣷⢻⣿⣿⣟⠿⣿⣶⠿⣿⡷⠏⠽⣇⣀⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣮⠁⠀⠀⠁⠀⠁⠁⠛⣿⣿⠀⠀⠐⠀⠁⠉⠈⠓⢠⣿⠀⠀⠚⣿⠀⠙⣿⠀⠀⠚⣿⠀⠙⣿⠀⢻⣿⡄⠐⣯⠉⠉⠙⠃
⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠐⣼⣦⡑⣤⣤⡠⠂⠀⠀⢆⢻⣿⣿⠀⠀⠐⠤⠂⣲⡼⠋⡈⣿⠀⠀⣼⣿⠀⢼⣿⢀⣠⣽⡿⣀⣼⣿⢀⣿⣟⠀⣘⣷⠴⠶⣶⠃
⠀⠀⠀⠠⠊⠁⠀⠀⠀⣤⡶⠏⠈⠩⣴⣾⣻⣶⣶⣾⣿⣿⣿⣿⣦⣒⣞⣙⣋⣭⣶⣼⣷⣝⣾⣿⡿⣿⣷⡿⠯⠿⢿⡿⠛⠿⠟⠛⠛⢻⡟⣧⣽⣇⡀⢠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠉⠋⢡⢞⡿⠛⠛⢿⢟⡿⠿⡿⣿⡿⣿⣿⠿⣿⣟⠛⠛⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⠙⠻⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⣀⠀⠀⣠⢖⣵⠛⠁⠀⢀⣼⡟⠀⢀⣽⡏⠀⠈⢻⠀⠀⠙⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡦⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⡷⠾⡛⠘⠃⠀⠀⣠⡾⠋⠀⡠⣾⠏⠀⠀⡐⣸⠀⠀⠀⠘⢷⡤⣄⣀⣀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⣾⣾⠃⠀⡲⣶⣁⠀⠀⠀⠓⠿⣀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⡀⠀⠀⠉⠙⠳⢦⣀⡒⠀⢈⡭⠁⠀⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⣠⣀⣀⣴⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠤⠴⠂⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠈⠁⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠤⣾⡄
⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⣗⢀⣘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠒⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⣠⠟⠒⠊⣤⠀⢄⠀⠀⠀⣶⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⣻⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣫⠀⠠⢀⡟⠳⣞⠀⠀⠀⠀⡀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡇⢠⣾⠀⠀⢰⠄
⠀⠀⠀⠀⠀⡤⠀⠀⠀⡤⠀⢀⠀⠀⠀⠀⠰⠒⠀⠀⢀⣼⣜⠁⢀⠀⡀⠀⣡⣭⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠈⡆⢺⠁⠀⠀⡄⢹⣀⠄⣹⠀⠉⢸⡇⣠⡼⠁
⠀⠐⢒⡞⠉⠁⠀⠀⠀⠈⠉⢷⠒⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢱⠄⠓⠃⠾⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠄⠀⠀⠀⠢⡄⡮⢻⡀⠀⢸⠇⠱⣦⠖⠁
⠀⠈⠈⠰⣄⠀⠀⢀⠀⠀⢠⢂⠕⠆⠀⠀⠀⠀⠀⠀⠈⢧⠘⢇⠠⠀⢀⡼⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⣺⠀⠀⠀⠓⠒⣇⣠⢻⣯⣠⣿⣄⠨⠇
⠀⠀⠀⠀⠀⠽⠈⠉⠁⠀⠄⠘⠀⠀⠀⢰⡇⠀⡆⠀⠀⠀⠀⠀⠐⠀⠘⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡼⡇⠀⠀⣿⣤⠀⠀⠀⠘⠣⠺⠈⠉⠩⠂⠒
⠀⠀⠀⠀⠀⠉⠒⠀⠀⠁⠀⠀⠀⠀⣀⠸⢃⡀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡖⠁⠀⠀⠃⢄⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠃⠀⠀⣢⠀⠀⠀⠀⠀⣸⣂⣄⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠎⠀⢈⠈⠀⠀⠰⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡃⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⢛⡏⣤⠅⣄⡄⣄⣐⡀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⣧⣄⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢷⠘⠀⠁⠙⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠈⠁⠉⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢗⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠇⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⠀⠀⠀⠈⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠜⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣠⣴⣧⣄⡠⠄⠊⢀⣠⣾⣿⣦⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠈⠉⠈⠁⠀⠀⢀⣴⣾⣿⣿⠿⠛⠉⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠘⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠈⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⣸⣿⣶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿
⠀⠀⠀⣴⣾⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⠉⠉⠛⠓⠒⠶⠦⠤⠤⠤⠤⠤⠴⠖⠒⠊⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠈⣿⡟⠀⠀⢠⣶⣿⣿⡿⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⢧⣀⣀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢀⡀⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⡒⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⡷⡤⢄⣀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡄⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠃⠉⠛⣵⢀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⢢⠀⠁⠀⠔⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡁⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠈⠄⠈⠁⠀⠀⣀⠤⠀⠀⠀⠀⠀⠰⢂⠀⠀⠄⠀⡀⠄⠃
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠡⠤⠘⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠒⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠝⠀⠆⠀⠀⠀⠀⠐⠂⡆⠀⠀⠀⠀⠀⠀⠐⠠⠄⢄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠧⠀⠄⠀⠐⠒⠉⠀⠁⠈⠁⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⢤⠠⠀⡈⠀⠀⠓⠄
⠀⠀⠀⠀⢀⠀⢠⡄⠘⠰⠇⠐⢃⠰⠈⠁⠀⠀⠀⠀⠀⢀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⠀⠀⠈⢀⡞⠀⠠⠃
⠀⠀⠀⠀⠋⠁⠀⠀⢀⠂⠀⠀⠈⠀⢀⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠘⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡴⠁⠀⢀⠁⠀⠀⠐⡰⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠈⠗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡘⠂⠰⠅⠀⠀⠀⠀⠜⠀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠈⠸⢧⡀⠀⠀⠀⠆⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣥⡇⠀⠀⠀⠀⠀⣙⠒⠤⣾⠄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠗⠲⠀⠐⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⢀⡀⠀⠀⠀⠈⠀⣥⠻⠀⠖⠂
⠀⠀⠀⠀⠀⠀⡀⠂⠈⡀⠖⠐⠃⢀⠤⠂⢀⡎⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠂⠂⢉⡨⠄⢽⠁⠀⠀⠀⢀⡞⠁⠑⠄⡀⠀⠀⡀⡄
⠀⠀⠀⠀⠠⡄⠘⢄⠐⠀⡀⠀⠀⠈⢆⣀⠞⠁⢠⠇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠈⠀⠀⠀⣾⠆⠀⠀⠀⠾⠉⠀⠦⢀⡈⠁⠈⠉
⠀⠀⠀⠀⠀⠱⣄⠀⠁⠐⡃⠀⠀⠀⠈⡤⠀⠈⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣿⠀⠄⢠⠌⠀⠀⠁⠀⣀⠈⠂
⠀⠀⠀⠀⠀⢄⣀⢉⣀⡈⢵⡀⠀⠀⡠⠄⣀⠒⠈⠀⠀⠀⠀⠀⠀⡄⠀⠞⠘⡇⠀⠀⠀⠈⠓⠊⠉⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠊⠰⠀⠀⠃⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠈⠆⠀⠈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⣸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠐⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢸⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣌⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣷⡶⠆⠀⠀⠀⠀⠀⣀⣠⣤⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⢀⣼⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠈⠁⠚⠛⠛⠓
⠀⠀⠀⠀⠀⠘⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠴⣆⠀⠀⠀⠁⣠⣶⡀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⢀⣶⣄⢀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣶⣤⣾⣿⠋⠀⠀⠀⠀⠀⠀⠸⣿⣷⣶⣶⣿⡟⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡿⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢐⡅⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠊⢀⣼⣿⣛⣻⣋⣭⣭⠍⠀⠀⠀⠀⠀⠿⢶⣭⣙⠻⣻⡋⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣞⣀⣴⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⣶⣌⣷⣤⡘⢆⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⢸⡆⠀⠀⣧⣄⠀⢠⡏⠉⣹⣿⠿⣋⣠⣴⡶⠖⠛⠟⠿⠟⠿⠿⠿⠶⠶⢦⣤⣀⠈⠻⣿⣿⡿⠚⢓⡄⠀⠀⢨⡜
⠀⠀⠀⠀⠀⠈⢧⢦⠀⣷⢸⡀⣿⣷⣿⡟⣁⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣌⠛⣦⣠⣼⡿⢀⡄⣸⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠸⣏⢤⡘⣧⡳⠈⣿⡿⣧⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠗⣿⣿⡿⣱⢛⣶⠇⠀⢠⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣎⢳⡝⠿⣶⣠⠿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣷⡞⣻⣿⣮⡷⠛⢉⡠⢎⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣽⣓⠦⣭⣿⣿⣿⣿⣿⣦⣄⡀⢀⢀⢀⣤⠀⣠⣄⣠⡀⣠⣴⣾⣿⣿⣿⣿⢋⣭⠥⣄⣐⣠⣽⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢙⡛⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠵⢿⣛⠛⣛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⣯⠼⠟⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠾⠭⣗⣦⣕⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠀⠀⣰⠚⣷⣷⣿⣭⣿⣿⡿⠿⠿⠿⣿⡿⡿⠿⣿⣷⣶⣶⣶⣷⣯⣀⡀⢠⣄⠈⠙⠻⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠁⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠙⠿⣿⣿⣷⡄⠀⠀⠀⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⢀⡏⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⢽⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠋⣉⣷⣖⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠘⠟⠀⢬⣾⡋⠁⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠔⣰⠇⣰⠒⠛⠀⠀⠽⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⡖⠀⠀⠀⠈⠉⠠⡄⠀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠎⡻⠁⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⠻⣾⣳⡄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢰⣷⣖⠃⠀⠰⡿⣏⣊⠀⠀⠌⢻⡧⠨⢻⣷⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣴⠎⢻⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢽⣿⢰⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠶⠓⡦⠀⠀⠀⠀⠀⠀⠀⠉⣺⡙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢢⢰⡇⠀⠀⠉⠀⠀⠀⠀⢠⡖⣦⢔⣿⡟⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⣼⣇⣤⠀⠀⠀⡀⠀⠀⠀⡠⠎⡸⠀⠀⠈⢑⡄⠀⠄⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣽⣿⡆⠀⠰⠟⢛⠂⠉⢀⡔⠁⠀⠤⠚⠉⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⢛⣻⡟⠁⠀⠀⠠⠇⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠏⡻⠁⠀⠀⠀⠼⠀⠀⠀⠀⠀⢀⡆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⣶⢏⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣵⡾⠟⠀⠀⠐⠛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⣿⣿⡘⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠸⠟⠉⠀⠀⠀⠀⣀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⢸⠃⠀⠀⠠⡀⠰⡀⠀⠀⢓⣶⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣄⠀⠀⠀⠁⠀⠀⠀⠂⢈⣙⠻⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣠⡜⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢠⠟⢠⠀⠀⠀⠉⠙⢯⣀⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠄⡀⢁⣄⠲⣽⣷⠦⣿⢳⣔⠏
⠀⠀⠀⠀⠉⠉⠙⠛⠋⠙⠋⠉⠉⠋⠐⠀⠀⣀⣇⠀⠀⠀⠀⠀⡙⣦⠙⢷⠀⠹⡷⢻⠆⠐⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⠀⠀⠀⠀⠁⣈⣠⣬⣇⣀⠀⠀⣧⢦⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣷⡿⣟⣿⣿⣄⠰⣻⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠹⣯⠙⢿⣿⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣴⠀⢰⠂⢰⠀⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢀⠇⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣸⠀⠀⠀⠀⠀⠀⢰

⠀⠀⠀⢸⣿⡆⠀⠀⠈⣹⡇⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠀⠀⠀⡄⠀⢸⡇⠀⡀⠀⢿⣿⠂⠀⣴⣶⣶⣶⣦⣴⣶⣶⣶⡄
⠀⠀⠀⣼⣿⣧⡀⠀⣾⣿⣿⣆⠀⠘⣿⣿⣿⣿⡇⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⣶⠀⠀⡇⠀⢸⡇⢰⡇⠀⢸⡟⠀⠀⡿⠉⠙⡟⠉⠙⡟⠉⠙⡇
⠀⠀⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⠀⠀⢘⣿⡀⠀⠀⠀⠀⠀⢾⣿⠀⠀⣿⡇⠀⣿⠀⠀⡇⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠻⠦⠶⠿⠦⠾⠷⠦⠾⠃
⠀⠀⠘⣿⣿⣿⠃⠀⣿⣿⣿⣿⠀⠀⢿⣿⣿⡿⠀⠀⢸⣿⡇⠀⢸⣿⡆⠀⢺⣿⠀⠀⣿⡇⠀⣿⠀⢠⣷⠀⢸⡇⢸⡇⠀⢸⡇⠀⠀⠐⠀⠐⠒⠒⠒⠒
⠀⠀⠀⢻⣿⡿⠀⠀⢿⣿⣿⡟⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⠇⠀⢸⣿⠇⠀⢸⣿⠀⠀⣿⠇⠀⣿⠀⠐⣿⠀⢸⡇⠸⡇⠀⠀⡇⠀⠀⠚⠂⠚⠛⠛⠛⠒⠒⠒⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣾⣯⡀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⠀⠀⢀⣿⣿⡟⠃⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣿⡟⡇⠀⠀⢠⣤⡹⢿⠃⠀⢰⣿⣦⣤⣤⣴⣾⣷⣶⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡄⠀⠀⠀⠀⣬⣬⡉⠀⠀⠀⣿⣿⣿⣿⠿⢿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⡇⠀⠀⠼⠿⠇⠀⠃⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣷⠀⠘⢠⣶⣿⣿⣷⣤⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⡏
⠀⠀⢸⣿⣿⡗⠀⢁⠰⠶⠀⡇⠈⠀⠀⠈⠻⢿⡿⣿⣿⣿⣿⣿⠀⠀⣼⣇⠀⠀⣼⣷⠀⠈⡏⠀⢠⣿⡈⠄⠀⢸⣿⣿⣿⣿⣿⠀⠀⣀⣀⣀⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠐⠿⠿⠇⠀⠸⠦⠤⠾⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠁⠀⠸⠿⠿⠃⠘⠿⠿⠃⠀⠁⠀⠻⠿⠗⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⢿⠿⠿⠿⠷⡿⠿⠿⢿⡇
⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⢀⣄⣄⣄⣤⣠⣀⣠⡀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇
⠀⠀⢹⣟⢸⠠⠀⠆⠀⠀⠀⣿⣿⣧⣤⣼⣿⢿⡃⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣴⡿⢻⡟⡝⠙⢻⣷⡀⠀⠀⠀⠀⣼⣿⣄⠀⠀⠀⠀⠿⠶⠶⠶⠶⠷⠶⠶⠾⠇
⠀⠀⢸⡇⢸⢐⠀⡇⠀⠀⢸⡏⠈⡉⠀⠉⠁⢹⡇⠀⠀⠀⣿⣿⣄⠀⠀⠀⢸⣿⢠⡿⡱⠀⢀⡿⢻⡷⠀⠀⠀⢀⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣄⡀
⠀⠀⢸⡇⢸⢈⠀⡇⠀⠀⠘⡅⠀⠁⠀⠀⠀⢨⡇⠀⠀⡆⢡⣠⡌⢰⠀⠀⠘⣿⣿⡱⠁⢀⣾⣣⣿⡏⠀⠀⢰⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣞⣩⣶⠟⠋⡲
⠀⠀⢸⡇⢸⠘⠀⠇⠀⠀⠀⡟⠈⠉⣿⣿⣿⣿⠁⠀⠀⣇⠈⠉⠀⣾⠀⠀⢀⠈⢻⣿⣶⣾⣿⣿⠋⣀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡿⠛⠁⢀⣤⡇
⠀⠀⠈⠃⠘⠈⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠛⠛⠒⠛⠋⠀⠀⠈⠛⠛⠓⠒⠒⠚⠛⠛⠋⠀⠀⠈⠛⠛⠛⠛⠛⠁⠀⠀⠀⣀⠴⠚⠳⠦⣴⡋⣸⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢷⣦⣄⠀⣀⡴⠛⠁
⠀⠀⢈⣩⠿⠛⠻⢯⣉⠀⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣾⣿⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⢠⡟⠁⠀⠀⠀⠀⠈⢧⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢰⣿⢿⢏⣤⡄⠀⠀⠀⢸⣿⣿⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣷⣶⡆
⠀⢾⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣰⠋⢋⣼⣿⣧⡀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠘⣧⡀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣿⣿⣦⣤⣄⣀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠔⠁⠀⠘⣿⣿⣿⠆⠀⠀⣿⣿⣿⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠙⠲⠦⠶⠚⠋⠀⠀⠀⠀⠻⠿⠛⠤⠬⠿⠀⠀⠀⠀⠘⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠿⠿⠿⠃⠀⠀⠀⠙⠻⠿⠿⠿⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡍⣿⠛⠿⠿⠿⠻⠟⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠙⢿⣿⣿⣷
⢀⣿⡟⣹⣿⣿⣿⣿⣿⠿⠿⠛⠻⠿⠿⢿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⣸⣿⣰⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢹⣿⣷
⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠶⣄
⢻⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠉⠙⠂⠈⠋⢹⡇
⠘⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⠀⠀⠀⣠⣄⡀
⠀⠘⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⣾⠉⠈⣻⡄
⠀⠀⠘⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠟⠀⠀⠀⢠⡄⠀⠀⢸⡇⢀⡏⠀⢀⣿⣀
⠀⠀⠀⣼⢋⣩⣉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⢸⣿⠋⠉⠱⡟⠀⢹⠷⣦
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⣤⣤⣤⠴⠶⢾⠋⠀⠀⠀⠀⠀⣸⣿⠀⠶⢾⣇⣀⡟⠀⣼⠁
⠀⠀⠀⠘⠷⢦⢾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⣈⡆⠀⠀⠀⠀⢠⡟⠹⣦⡀⠀⠉⠉⠙⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠈⠉⠀⠀⠀⢀⣴⠟⠀⠀⣸⠇⠀⠀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠛⣶⣦⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣦⣤⣀⣀⣀⣀⣀⣀⣠⣤⠴⢞⣻⡟⠁⠀⢄⡀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠀⢿⠉⠉⠉⠛⠻⠿⠟⠛⠋⠁⠀⠀⠀⠈⠻⣷⣤⣄⣀⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⣤⣼⠷⣶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡟⠀⠀⢠⡾⠁⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠻⣄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢹⡆⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧
⠀⠀⠀⠀⠀⠀⠙⠶⣆⠀⢀⠀⠀⣦⡾⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠛⠛⠉⠙⠛⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠶⠶⠋⠀⠀⠀⠀⠸⣧⣠⣤⡴⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⠒⣶⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⣴⠶⠶⠛⠛⠉⠀⠀⠀⢀⡟⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⣼⣅⡀⠀⠹⣧⡜⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠈⠷⢦⣍⣑⠂⠤⢨⡟⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⣏⣹⣌⡟⠛⠛⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣍⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡯⠤⠤⠧⠤⠤⠤⠤⠤⠤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠶⣤⣄⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⠰⠖⠒⠂⠀⠀⠀⠀⢠⣔⠂⠀⠒⢯⣿⣿⠃⠈⠙⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣻⠯⣶⣿⠿⠟⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠹⡄⠀⣠⠖⠁⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠙⢦⡄⢼⡁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⠈⠓⠋⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣦⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⢀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⣀⣼⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠚⠀⣻⠀⣼⠡⣴⣉⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⠶⠄⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠉⠛⠳⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⣾⠁⠁⠀⠀⠀⠀⢻⣷⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⢠⡄⠀⠀⡄⠀⠀⠀⠀⠀⢼⡷⣦⠀⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠰⣿⣿⡤⢤⣀⡀⠀⠀⠀⣤⡴⡆⠀⠀⠀⠀⠀⠀⣿⠇⠀⡀
⠀⠀⠀⠀⠀⠈⣟⠆⠀⠹⡀⠀⠀⠀⠀⢸⣧⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢄⠀⠈⢻⣆⠀⠀⣿⣼⠃⠀⠀⠀⠀⣠⣾⠏⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠦⣀⠀⠀⠐⣿⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠸⠏⢠⣿⣿⣿⣆⣀⣠⣴⡿⠟⠁⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠚⢧⡉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣟⣿⠿⠋⠉⠉⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠤⣀⡀⢀⠀⠄⠈⠒⠘⠛⢶⣄⠀⠀⠀⠀⢦⡀⠀⠀⢀⡀⠀⠀⣠⣴⣶⡿⠋⠉⠉⠀⠠⢄⡄⠀⢀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢉⡙⠛⠓⠒⠂⠉⢉⡿⣿⣶⣶⣦⣾⡿⣿⣶⣿⣷⣾⣿⣿⣿⡿⠉⠑⠒⠚⠒⠒⠚⣭⡉⢀⡀
⠀⠀⠀⣤⣤⣤⣤⣤⡀⠀⣈⣥⠤⠤⠛⠶⠶⠶⠖⠒⠂⠐⢻⣿⣼⢿⣿⣿⣶⣿⣻⣶⣿⣿⡿⣿⣿⣿⣒⡴⠶⠦⠤⠤⢤⠆⠄⢀⡉⠁⠀⠀⣲⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡠⠀⠀⠈⣧⢀⣠⢴⠊⠀⠀⠀⠀⣸⠖⠉⠀⠁⠀⠉⠉⠁⠈⠠⣄⠀⠀⠀⠉⠑⢠⣤⣤⣼⠀⠀⠲⣤⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠄⠀⠀⡔⠉⠀⡼⠀⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠀⢧⡀⠀⢠⡈⠀⠰⠆⠁
⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⢀⠜⠋⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠙⠃⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⣦⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠀⣤⠶⠤⣤⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⣥⡾⣋⡤⠚⠁⠀⠈⠛⠲⢬⣙⣻⣿⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢀⡀⠀⠀⢈⣼⡷⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣤⣿⣆⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣴⣿⣉⣤⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⡴⠛⠷⣾⡆⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣥⡽⠟⢳⣤⣀⡤⠶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⣿⣿⣿⣦⠀⠋⠀⠀⣰⠏⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠘⢯⣤⣴⣿⣿⡟⠛⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣈⣙⣻⣅⠀⠀⠀⠚⠁⣀⣀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣂⠲⣦⣼⠿⠿⠿⠿⢿⣿⣿⣶⣶⢿⣿⣻⣿⣿⣿⠿⣿⣿⠏⢉⣤⡄⠀⠀⠴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠡⢄⣀⣀⣀⠉⠳⠦⢿⣿⣶⣦⣤⣄⣀⡉⠛⢡⣾⣿⣿⣿⣿⣿⣿⣯⣶⣾⡿⠅⠀⠀⠒⢋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⠶⠿⠿⣿⣿⣿⣿⣿⣦⡘⠛⠛⠟⠿⢟⣛⣽⣿⣿⠷⠂⣷⠤⠤⠴⠾⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⢿⣿⡿⣿⠿⠛⠛⠛⠛⠛⠻⣿⣿⣛⠒⠒⠂⠤⣤⡀⠀⠘⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠖⠋⢠⣦⣤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⣬⠉⠛⣻⡇⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠛⠀⠀⢿⡟⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⢠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⢐⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣽⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⢿⣷⣤⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣯⡀⠀⠈⠙⠛⠛⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠛⠿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠛⠻⠿⢿⣿⡟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⢀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣶⣶⣤⡀
⠀⠀⠀⣯⢺⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠘⣎⢿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠙⡆⠀⠀⠀⠐⠋⢹⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠘⣮⣿⢫⣿⣿⣦⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡃⠀⠀⠀⢸⣷⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢮⡙⢿⣿⣿⣷⣄⠀⠀⠀⢀⣧⣤⣤⣤⣤⣤⣧⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠻⣿⣿⣿⣦⡀⠀⢰⣿⡀⢀⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡜⠁⢺⣿⣿⡄⠘⣛⣣⣬⣭⣭⣭⣭⣄⣀⠀⢸⣿⣿⠀⠀⠀⠀⣠⡾⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⢀⣀⣤⣶⣦⡀⠀⠀⠀⠻⣄⠈⢛⣩⣶⣿⠿⠛⠋⠉⠉⠉⠉⠛⠻⢿⣷⣌⡻⠀⠀⠀⣰⡟⢀⡄⠈⢙⣿⣿⣿⡿⠃⠀⣠⣶⣦⣄⡀
⢾⣧⡤⢿⣿⣿⡟⡆⠀⠀⠀⠈⣱⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⣰⡟⢠⣾⠁⢠⣾⣿⣿⡿⠀⠀⠀⡇⠀⠈⠙⢿⣦
⢺⣿⣷⠀⣿⣿⣷⢹⡄⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣟⣠⣿⠃⢠⣿⣿⣿⡿⠁⠀⠀⡜⣰⡀⠀⠀⠀⢹⠂
⠈⢿⣿⣧⢻⣿⣿⣧⢻⡀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣿⠟⢤⣿⣿⣿⡿⠁⠀⢀⣼⣿⡟⢡⠆⣠⠎⠁
⠀⠘⣿⣿⣇⢻⣿⣿⣯⢷⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢀⣾⣿⣿⡿⠁⠀⣠⣾⡿⠋⣰⣿⡿⠁
⠀⠀⠸⣿⣿⡆⠿⣿⣿⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⣿⣿⠙⠺⣿⣿⡿⢁⣤⡾⠋⠎⢀⡼⣿⡿⠀⠀⠀⢀⣠⡾
⠀⠀⠀⠹⣿⣿⣷⠹⣿⣿⣿⣷⠀⠀⠀⣠⣴⣾⡿⢿⣛⡿⠭⠭⠭⠭⠭⠽⣟⣿⢿⣿⣧⣄⠀⣼⡿⢡⣾⣿⣦⣀⠀⠞⢱⣿⠁⠀⢠⠂⢿⣿⠇
⠀⠀⠀⠀⢻⣿⣿⣆⢻⣿⣿⣿⣧⣴⣿⣿⡯⠗⠋⠁⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⠉⠺⢻⣿⣿⣿⢡⣿⣿⠛⣿⣿⣷⣤⣼⣿⠀⣰⠟⣡⣿⡿
⠀⠀⠀⠀⠀⢿⣿⠈⣦⡹⣿⣿⣿⣿⡿⠋⢀⡠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⡀⢹⣿⣿⣿⣿⠧⢄⣿⣿⣿⣿⣿⠃⣴⠃⣴⣿⡟
⠀⠀⠀⠀⠀⠘⣿⣦⠛⠛⣾⣿⣿⠏⢀⠔⠉⠀⠀⢀⣄⣀⢸⣧⣀⡤⢖⣀⠀⠀⠀⠀⠈⢠⣿⣿⣿⣧⢀⣼⣿⣿⣿⣿⠃⠚⢅⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⣸⣿⣿⡀⠁⠀⠀⠀⠀⣀⡤⠟⠋⢻⣥⠶⣈⣭⠉⢀⠀⠀⢀⣼⣿⣿⣿⣿⣸⣿⣿⣿⣿⠇⠀⠀⠀⣰⣹
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⣻⣿⣧⡳⣄⠀⠀⠐⠿⢥⣭⣶⣒⣻⡧⠞⣋⡤⠖⠁⣀⣴⣿⣿⣿⣿⣿⣏⣝⣻⣿⡿⣵⡆⠀⠀⡴⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠙⣿⣿⣷⣬⡑⣢⣄⣀⣀⠀⠀⠉⠉⣉⣉⣡⣤⣶⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⡿⠛⣽⣽⡇⢀⣼⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠉⠀⠀⠀⠑⢦⡙⠻⢟⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣬⡛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⣉⣠⣽⣿⠟⠀⠀⠀⠀⠀⠀⢰⢸⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣯⣿⣿⣿⣿⣿⣥⣴⣶⣶⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡜⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠇⠀⠀⠀⠀⠀⣠⣾⣾⣿⡟⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⢶⣤⣤⣤⣿⣾⣿⡿⢋⣤⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⠾⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢡⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢤⡒⣫⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⣀⣽⣿⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⣾⣿⣿⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠒⠤⡀⠀⠂⠀⠤⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢠⣿⠿⠃⢰⣃⡀⠤⠄⠒⠒⠒⠒⠶⠾⠤⢤⣁⡐⠬⠁⢠⠂⠀⠈⢢⠁
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⢵⠄⠀⠀⠀⢡
⠀⠎⠀⠈⠐⠢⢄⡀⠀⠀⠀⠧⣄⣩⠗⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠒⠢⡀
⠀⠀⠈⠁⠒⠦⣄⠈⠲⠄⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⡤⠋⠀⠈⠆⢰
⠀⠀⠀⠀⠀⠀⠀⠙⠎⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⠿⠛⠻⢿⣿⣿⠟⠉⠁⠒⡶⢦⠀⠸⢸
⠀⠀⠀⠀⡤⠐⠒⠘⠦⠶⢤⣀⣼⡀⠀⠀⢀⣴⣿⣿⠁⣤⡉⣿⣿⣿⣐⠿⢂⣼⢟⡥⠤⣀⣀⡴⠁⢸⠀⠀⡜
⠀⠀⠀⠀⡀⠀⡀⠘⠃⣀⠀⠀⠀⣸⠃⣠⡿⠿⣿⣿⣦⣭⣵⠿⠿⢿⡛⠻⠝⠚⠉⠀⠀⠀⡠⠀⢠⠏⠀⠀⠁
⠀⠀⠀⠀⠰⠀⠆⠀⠀⠀⠩⠉⠩⠒⠋⠀⠈⠉⠒⠒⠒⠂⠉⠉⣉⠤⠔⠒⠚⠋⠓⠒⠒⡘⢧⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⠄⣢⡀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⣀⣀⣀⠀⠀⠸⡞
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠐⡙⠻⡿⠯⣵⣽⣶⠶⠿⠋⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠁⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠐⠲⠶⢄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⢀⣐⠀⠁⠀⠀⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢠⠀⠀⠁⣀⠠⠤⠤⠤⠤⠀⠤⠤⠤⠄⢀⣀⠠⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠍⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠴⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢰⡆⠉⠐⠒⠦⣄⠀⠀⠀⠀⠀⠀⣄⣴⣚⠋⠉⠁⠄⢈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡈⠀⠈⠉⠩⠗⠓⠒⠒⠚⠛⠻⢈⠁⢀⠀⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⣸⣿⣿⣿⣇⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣀⡀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠈⢻⣿⡿⠛⣿⣿⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠉⢀⡀⢿⠟⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⣀⣴⣾⡇⣿⠃⠀⣼⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣿⣿⣿⣄⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠅⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇⠀⣈⣉⣙⠛⢷⣌⢿⣿⣿⣿⣿⣿⣿⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣷⣦⡉⠀⢻⣿⣿⣿⣿⡿⢸⡇⠀⠀⣿⣉⠛⣿⣿⣿⣷⣶⣿⣿⣽⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⡿⢟⣽⠇⠀⢸⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣿⣿⡇⢸⣏⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠈⠉⠛⠛⠛⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠋⢠⣾⣿⣷⣾⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⣿⣿⣿⣶⣿⣿⣿⠿⠿⠃⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⡟⠛⢻⣏⣨⣿⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡗⠶⠀⠐⠞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣃⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣯⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⣴⠾⠛⣉⣀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣦⠘⣷⡙⠿⣿⣿⠿⠟⣋⣙⣛⡛⢿⣿⣿⣿⡿⢋⣴⠟⢋⣤⣶⣿⣿⣿⡆⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡌⠻⠷⡶⠶⠞⠛⣉⣉⡙⠛⠷⣦⣭⣥⡶⠟⢁⣴⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣤⣤⣤⣴⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⢀⣤⣶⣴⣾⣿⠟⠋⠛⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠔⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟
⠀⢻⣿⠃⠀⠀⠀⠀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠸⡏⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣤⣿⣷⣶⣶⣶⣶⣶⣾⣷⡛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⠟⠋⢻⡅⠀⠈⣹⣟⣻⣶⣤⡀⠀⠘⠂⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢀⣾⣏⣉⠄⠀⠀⠀⠀⠀⠹⣶⠾⠛⠉⠉⠉⠙⠻⢆⠀⠀⠀⠀⠉⠉⠩⠿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠂⢠
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡀⠀⢀⣴⣿⠏⠙⢳⣄⡀⠀⠀⢀⣀⠀⣷⣶⣶⣤⣤⣠⣤⡄
⠀⠀⠀⠀⠀⠀⢳⠄⠀⠘⡇⠀⡈⢸⡇⠀⢀⣿⡿⠿⠿⠿⠛⠛⠭⠚⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢓⠀⠹⣜⣇⠀⣸⡏⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡔⠀⠈⢧⡀⡈⠻⣧⣽⣓⡿⢬⣷⠤⠥⠘⠃⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⢀⣻⣤⡴⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣯⣴⣽⣿⣿⠑⠀⠀⠀⠀⠀⠀⡠⠤⠈⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⢠⣿⣿⡆
⠀⠀⠀⠀⠀⠠⣀⣠⣤⡴⣇⠀⠙⠻⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⠿⢷⢀⣾⠻⣿⠃
⠀⠀⠀⠀⢀⡼⠉⠀⠀⢀⡜⢳⣄⢀⡚⣯⢀⡀⠀⠀⢀⣤⠆⠀⣨⡟⠀⠀⢀⠒⠀⠂⠀⡀⠀⠀⠀⢀⣿⠒⠀⠉⠙⢤⡏
⠀⠀⠀⠀⠘⠀⠀⢀⡶⠋⠙⢦⡙⢻⡇⠉⠢⢽⠗⠞⠞⢉⡴⢛⣠⡼⠆⠂⠁⠀⠀⠀⢈⣑⠀⠀⢀⢼⣏⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⢲⠋⠀⠀⠀⠀⠙⠺⡇⠀⠀⢿⠀⣀⡴⠋⢴⡿⠁⠀⠀⠀⠀⠀⠀⣄⣿⢙⠀⠀⢨⣾⣿⣶⡀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⣠⡾⢛⠀⠀⣾⣿⠛⠀⠀⠀⠀⠀⣀⣿⢉⠉⢀⡴⠛⠋⠹⣿⣿⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠠⠀⠉⢺⠿⣿⣷⡠⡰⣶⣤⣴⡟⢣⢐⣒⠋⠁⠀⢀⣵⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⡞⠒⢦⣤⠈⣁⡸⢛⡏⢩⣗⠛⣛⠛⠻⢿⣿⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣄⠈⡑⢬⠁⠙⣧⡀⠙⠲⠤⣄⡀⠀⠄⠀⠀⠀⠀⢠⣄⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣻⠇⠈⢓⡾⠈⠙⠲⢤⡁⢢⠍⠙⠛⠂⠛⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣄⣰⠏⠀⢀⣼⠇⠀⠀⠀⠀⠉⠛⠳⢦⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣤⣄⣀⣀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⢠⣿⠛⢦⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡀⠀⠀⠈⣿⡀⢸⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢀⣴⣶⣶⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⠹⣧⠀⠀⠀⢿⡇⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⣿⡄⠀⠀⢸⣷⠀⢹⠀⠀⠀⢀⣿⣿⡇⠀⠉⢹⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣹⣷⠀⢠⣾⣿⡖⠘⣿⡇⠀⢸⣿⣿⡇⠀⠀⡞⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡉⣹⠿⢇⠈⣿⣿⣷⠆⢿⣿⠀⢸⣿⣿⠃⠀⢰⠇⠀⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠸⡀⣿⣿⣿⡂⠸⣿⡇⢸⣿⡿⠲⠤⡜⠀⡼⢁⣼⣿⠏⠙⠻⣿⣿⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣷⣾⣿⣿⡇⠀⢸⣇⣿⣿⡇⠀⣾⠃⡼⢡⣾⣿⡟⠀⠀⠀⢈⡿⠛⠋⠉
⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⠟⠋⠉⠉⠉⠙⠻⣿⣿⣷⡀⠸⣿⣿⣿⡯⢉⣿⡼⣡⣿⣿⣿⣄⡀⠀⢠⠎
⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⣿⣿⣿⠙⢺⡿⣵⣿⣿⣿⣯⣀⣼⣷⠋⠀⠀⢀⠀⠀⢐
⠀⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⠁⠀⣼⣿⣿⣿⣿⣟⣵⣿⣿⣷⡶⣶⡆⢸⣷⣴⡟
⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⢹⣿⡀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⡇⣿⡇⢸⠛⠛⡇
⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢰⣿⣿⣿⣷⡼⢤⣤⠧⣄⠠
⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣽⣿⣿⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢸⣿⣿⣿⣿⠀⣸⣿⠀⢸
⢻⣿⣿⣦⣀⡀⠀⠀⢀⣀⣤⣶⢿⣿⣿⡿⠛⠉⠉⢙⣿⣿⣾⣿⣿⣿⣿⣿⣀⢐⣗⣀⣽⠀⢸⣿⣿⣿⣿⠁⣿⣿⠃⢹
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠋⠉⠻⣦⣤⠞⠋⠀⢈⣹⢿⣿⣿⣿⣿⣮⣽⣿⣿⣿⠀⢸⣿⣿⣿⣿⠀⣿⣿⡇⢹
⠀⠀⠉⠛⠛⣻⣿⡿⠋⠉⠉⠻⣦⣴⠞⠋⠉⢙⣷⣶⡞⠉⠀⢀⣉⣿⣿⣿⠫⣭⣿⣯⣿⠀⢸⣿⣿⣿⣿⠀⣿⣿⡇⢸
⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⠶⠋⠉⠙⣷⣶⠞⠋⠁⢀⣙⣷⣾⣿⣿⣿⡿⢿⠀⢿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⣿⣿⡇⢸
⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣶⣤⠖⠋⠁⠉⣻⣶⣾⣿⣿⣿⣿⠿⠛⠉⠀⢸⡇⢸⣿⣿⣿⠀⠘⠻⠿⠿⠟⠓⠛⠛⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣀⡤⠶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣷⣾⣧⣤⣴⣾⣹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣿⣿⡏⣻⣣⣿⣿⠁⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠾⠿⣿⣿⣷⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡫⡾⣿⣿⣿⣷⡿⠟⣋⣭⣶⠾⢿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠟⢋⣥⣶⣿⣿⣅⠀⡴⠺⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠿⣋⣤⣾⣿⢿⣿⣿⣿⡿⠀⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢟⣵⣾⢟⣿⠟⠁⠀⠙⠛⠋⠀⠀⠈⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⣿⣿⠸⠃⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠋⠀⠀⣶⣿⡂⠀⠠⡀⠀⠀⠀⠀⠁⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⣹⠀⠀⠀⠁⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢰⣷⠃⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⠈⢿⣷⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⣄⡙⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣶⣍⡻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣦⡙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣮⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠒⠒⠒⠒⠒⠂⠒⠆⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠄⠀⢀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⡔⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡚⠀⣀⣀⢘⢲⡀⠐⠀⠀⠀⢰⠒⢢⠤⠄⠀⠀⠠⠄⠀⠀⠀⠀⢃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡔⠃⠊⠙⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠘⢀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠂⠘⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠖⠲⠖⠒⠒⠒⠂⠒⠂⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⡠⠒⠂⠐⠒⢤⠀⠀⡴⠂⠑⢢⠀⠖⠉⠙⢆⢸⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠇
⠀⠀⠀⡇⡘⠀⠀⠀⠀⠀⠀⢱⡀⢣⡀⢀⡼⠀⢦⣀⣀⠎⢸⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⡃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⣠⠤⢤⠄
⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⠆⠠⢄⠀⡴⠒⠢⡄⢸⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠧⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⡇⠱⡀⠀⠀⠀⠀⠀⣰⠁⣇⠀⠀⣸⠀⣄⠀⢀⡄⢸⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠰⠄⠀⠀⠠⠄⠄⠤⠀⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡇⠀⠈⠢⠤⠀⠤⠞⠁⠀⠈⠉⠈⠁⠀⠈⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⡀⠀⢀⣄
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠳⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠉⠒⢄⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⣀⣀⣀⡀⠠⠊⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣅⠼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠴⠊⢠⠘⠅⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠀⡀⠤⢀
⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠄⡀⠀⠀⠀⠈⠀⠄⠀⠀⠤⠂⠀⠘⡆⠀⢀⡔⠚⠒⠒⠢⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⠀⠀⠀⢨⣀⠠⠀⠠⠐⠨⠀⠐⡇⠀⡎⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⢀⡠⠀⠀⠀⠀⠑⢤⡐⣀⢶⠡⡈⠎⠀⠀⠡⡀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠔⠉⠑⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠈⠛⠂⠑⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⢐⠀⠀⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡦⠀⠀⠠⠄⠀⢠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠢⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠊⠀⠁⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⣷⠀⠀⢰⠋⠉⠙⡄⢠⠖⠒⠒⠒⠒⠒⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⣿⣿⣿⡆⠀⠘⠛⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢸⣿⠀⠀⠘⢄⣀⡠⠃⢸⣀⣀⣀⣀⣀⣀⡇⠀⡶⠶⠶⠶⠶⠶⢶⠄⠘⢷⣶⡿⠟⠛⠁⣴⣿⣿⣷⡀⠀⠈⠻⣦⡀⢀⡼⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣬⣽⡄⠀⢠⣾⣿⣷⡄⢸⣛⣿⣻⣟⣿⣛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⢸⣿⣿⣿⣿⣷⠀⣠⢤⣬⡿⣿⣁⡤⣄⠀⠀⠀⣿
⠀⠀⠀⠸⣿⣿⡇⠀⠹⣿⣿⡿⠃⠸⣭⣿⣽⣯⣿⣭⠇⠀⠶⠶⠶⠶⠶⠶⠶⠆⠀⠈⠿⠿⠃⠀⠘⠿⠿⠿⠿⠛⠀⠳⠶⠋⠀⠀⠙⠷⠞⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠠⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠸⣿⣿⣆⠀⢀⡏⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⣠⣾⣶⣾⣦⡄⠀⠀⢸⢿
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⡄⠹⣿⣿⣆⠈⣧⡀⢀⡼⠀⣶⣶⣶⣶⣶⣶⣆⡀⣿⣿⣿⡟⠀⢸⣿⣿⣿⣿⣿⡇⣤⠀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⣀⣀
⠀⠀⠀⣼⣧⣿⠀⠀⢰⡆⢽⡇⠀⠹⣯⠁⠀⠈⣿⣏⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⡇⠀⠘⣿⣿⣿⣿⣿⠃⡟⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⡟
⠀⠀⠀⠉⠈⠉⠀⠀⠸⠇⢿⠇⠀⠀⠹⣧⠀⠀⢿⡟⠀⠀⠛⠛⠛⠛⠛⠛⠛⠁⠈⠉⠉⠇⠀⠀⠀⠙⠛⠋⠀⢈⣃⡀⠹⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡀⠀⠀⠀⠀⢀⡀⠀⣠⣤⡀⠀⢀⣠⡆⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⡀
⠀⠀⠀⢀⣀⣿⣿⣧⣀⠀⠀⠀⢠⣀⣀⡅⠀⠀⠀⠀⢰⣤⠀⠸⣿⠃⠀⣸⣿⡇⠀⢠⣴⠋⠀⠀⠈⢳⣤⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⡞⠀⠀⠀⠀⠘⡆⠀⠀⢸⠛⠛⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠞⠛⠣⠀⠸⣿⡄⠀⠀⠀⣸⡿⠀⠀⣿⡿⠿⠀⠀⡴⠶⠆⠀⠶⠦⡄⠸⣿⡿⠃
⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⢰⣶⣶⣶⠀⠀⠙⢿⣷⣒⣾⠟⠁⠀⠀⣿⡇⠀⢠⠊⠉⠑⢄⢀⠔⠉⠉⠢
⠀⠀⠀⣇⣀⣀⣀⣀⣀⡇⠀⠀⢺⣀⣀⡇⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⢸⣿⣿⣿⠀⠀⠀⢀⣀⣯⣀⡀⠀⠀⠀⣿⡇⠀⠸⡀⠀⠀⡝⠙⣇⠀⠀⣠⠃⡀
⠀⠀⠀⠈⣡⣤⡄⠀⠉⠀⣤⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠀⠈⠛⠛⠛⠀⠀⠀⠈⠉⠉⠉⠁⣤⠀⠀⠈⠀⠀⠀⠈⠉⠁⠀⢀⣈⠉⠉⠀⢠⡟
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢹⠀⠀⠀⠀⣠⣶⣷⣶⣄⠀⠀⠀⢰⣿⠛⣿⠛⢻⡇⠀⠀⠀⣀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢸⣿⡂⠀⠀⠀⡝
⠀⠀⠀⣠⣿⣿⣿⣄⠀⠀⢸⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣦⡀⠸⣿⠛⣿⠛⢻⡇⠀⣴⠿⣹⢟⣳⣆⠀⣿⠀⠀⠀⢀⣴⣷⣾⣦⡀⠀⢸⣿⡇⠀⠀⢠⣶⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠀⣶⣿⣷⠆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠛⣿⠛⢻⡇⠰⣿⢿⢿⣳⡿⣿⠀⣿⠀⣶⠀⣾⣿⣿⣿⣿⣿⠀⢸⣿⠇⠀⠀⢸⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⢿⣿⣿⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣿⣿⣿⣿⡇⠀⠙⢯⣶⣽⠾⠋⠀⣿⠀⣿⡀⢿⣿⣿⣿⣿⡿⠀⢸⣿⠁⠀⠀⠸⣿⠃
⠀⠀⠀⠉⢉⣋⢉⡁⠀⠈⠉⠉⠀⠀⠈⠉⠉⠉⠉⢉⣁⣀⣀⣀⣛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠉⠉⠉⠉⣤⣄⠀⠉⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠔⠛⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠄⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠸⠿⢿⣿⣿⣿⠿⠇⠀⣰⣶⣄⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⢸⣿
⠀⠀⢀⣴⣶⣶⣶⣤⡀⠀⠈⢳⣄⢀⡴⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣤⠤⠾⠿⠿⠿⠤⢤⡀⢻⣿⣿⣦⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⣸⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣷⢀⡴⠲⣽⣿⡵⠲⣄⠀⢈⣉⣉⣉⢉⡉⢉⣉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠈⣟⠉⢻⣷⣄⡀⠈⡉⠉⠀⠀⠀⠀⠀⣿⣿
⠀⠘⣿⣿⣿⣿⣿⣿⣿⠈⢦⣤⠞⠘⢦⣤⠏⠀⠙⡟⠙⡏⢹⡏⠘⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠹⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⣿⣿⠄
⠀⠀⠈⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⡇⠀⡇⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠚⠛⠁⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠉
⠀⠀⢰⠛⣿⠒⣶⠒⣾⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⠛⠛⠛⠛⢻⣿⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠚⠿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠂⠐⠒⠒⠒⠒⠒⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠈⠁⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣩⣿⣿⣿⣿⠀⠘⢲⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⡇⣿⣿⣿⣿⣿⠀⠀⠈⡇⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢈⠇⣿⣿⣿⣿⣿⠀⠀⠀⣇⢿⡿
⣿⣏⠛⠿⣿⣿⣟⣩⢍⣙⣛⣛⠛⠃⠈⠀⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣷⣶⣤⠤⠶⣶⡲⢄⠀⢀⣀⣤⠆
⣿⣿⣿⣷⣶⣤⣌⡙⠀⠀⠉⠛⠷⠀⠀⢰⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣿⣥⣴⣾⣿⣿⡴⠾⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⠀⠀⢠⣾⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣶⣤⣀⠀⢹⣿⣿⣿⠀⠰⡿⠿⠛⠁⢻⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣸⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⢀⠋⡀⠀⠀⠀⠀⢰⠀⢹⡀⡀⠀⠀⣈⡁⠠⠴⡲⣂⠕⢀⠄⠀⠀⠀⠀⠀⣀⠴⣅⠙⡦⣀⣄⢀⣈⣽⡷⠀⢀⣀⣐⣲⣶⠶⠀⠀⠀⠚⣡⣶⣶⣦⣇⣄
⠀⢀⢸⢰⡇⠀⠀⠀⠀⠸⣦⣸⡇⢱⠀⢲⡜⠁⠈⠈⠒⠖⠶⠦⠀⠐⠀⠀⠀⠀⠘⠄⠔⠈⠙⣷⠿⢛⠋⠁⢀⣴⣟⣫⡼⠭⠴⠤⡤⣀⠀⠀⠋⠉⠀⣀⣃⠘
⠀⢸⢸⣿⣿⠀⠀⣄⡀⠀⢻⣿⡇⡜⠀⠈⠿⠲⣦⠤⠄⢤⢠⣤⣤⡥⣤⣤⠀⠀⠀⠘⠀⠉⠛⠖⠂⠀⠀⠀⠘⠿⢯⣤⣴⣭⣭⡾⠞⠁⠀⠀⠀⠀⣒⣥⠚⠋
⠀⠈⢆⢻⣿⣠⠖⠲⠷⣦⣼⡟⡴⢃⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⡀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⡘⢙⣛⣿⡀⠀⢰⣿⣟⢛⣳⠀⠀⠀⢀⡄⢖⡄⠀⠀⠀⠀⠀⠖⢋⣩⡤⢠⣤⣴⣦⣬⣍⣒⣂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣐⡬⣷⣆⠂
⠀⠀⠀⠄⠐⡘⢻⡀⢸⣟⡒⠀⠀⠀⠀⠀⡅⠠⣛⠢⣠⠀⠀⠐⠁⡴⣟⡿⡀⢘⠋⠤⠄⠈⢩⡥⠛⠛⠃⠀⠀⠀⠀⠀⢰⡷⠶⠾⠗⠒⠉⠀⠀⠉⠳⣦⢦
⠀⠀⠈⠂⠡⠊⣒⣓⣒⠑⢍⠒⠀⠀⠀⠀⠹⠂⠙⠊⠋⠀⠀⠘⠘⢟⣿⣷⣿⢾⣮⣵⣶⣾⠧⠀⠀⠈⠉⠀⠀⠀⠀⢸⡇⢦⡀⠀⣠⣤⣤⣀⡀⠠⠂⣸⠇⡔
⠀⠀⠀⠀⠀⠀⣒⢒⣒⠀⠀⠀⠀⡀⣀⣤⣠⣀⣀⡀⠀⠀⠀⠀⠙⠉⠐⠠⠍⢠⠿⠏⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⣾⣿⣿⣿⢿⣿⣾⣍⠅⢊⡀
⠀⠀⠀⠀⠀⠀⠐⡶⠂⠀⠀⠀⢨⠾⡉⠉⠙⠻⢿⣽⣟⣰⣆⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣤⡶⢿⢛⣫⣿⠄⠀⠐⠉⠈⠻⢿⣿⢿⣿⣿⣿⣟⠉⠉⠀⠐
⠀⠀⠀⠀⠀⠀⠠⢻⢧⡀⠀⠀⠈⠛⠿⢶⣴⠶⠦⠾⠿⠿⢿⣶⣿⣽⣾⣥⣦⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠐⠈⠀⠔⠉⠉⠙⠛⠋⠉⠢⡀⠈⠢
⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉
⣶⡾⣒⣷⣷⣾⣶⣶⣷⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣶⣾⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣥⣷⣶⣶⣶⣶⣦⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⢉⣉⣽⣀⣀⠀⣀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠉⢹⠒⠒⠠⠤⠤⣤⡤⠤⠤⠤⠤⠤⠒⠒⠊⠉⠉⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⣀⣠⣤⡀⠀⠀⠀⠠⠞⠒⠈⠁⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡠⠖⠒⠄⠀⠀⠆⠀⠀⠀⠀⠘⢟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢂⠈⢹⠉⠀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⡌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⢀⠇⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢻⣶⣦⣤⣄⣀⣤⠤⠔⠒⠉⠁⠀⠀⠀⠀⠀⡼⠀⠀⡜⠀⠀⠀⡠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⢿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⢀⠗⠢⠀⡘⠀⣰⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠓⠤⠥⠼⠒⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢸⢦⠀⠛⠁⢾⠉⣀⡠
⠀⠀⠀⠀⠀⢠⠒⡄⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠊⠁⠀⠀⠀⠀⠀⢈⡼⠀⠀⠀⠀⠙⠋⢲
⠀⢠⠔⠒⠠⣼⡄⠧⢄⠀⠀⣸⣀⣀⣀⣀⣀⣀⠀⠀⠀⡔⣤⠤⠬⢭⣟⠻⠤⠤⠞⢥⣄⣀⠀⠀⠀⠀⠀⠀⣠⠔⢋⣤⠊⠑⡂⠉⠉⠈⠉
⠀⠀⠑⢂⣦⠀⠀⠀⠀⣉⡀⣠⠤⠤⠤⠤⠤⠤⠬⢉⣿⡟⠁⠀⠀⡸⠙⢅⡰⢆⡰⢻⠀⠀⢹⢧⠤⠔⢒⡉⠄⠊⠁⠀⠉⡹⠃
⠀⠀⡞⢁⡠⠔⢲⠀⠖⢱⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⢇⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⠀⠀⣸⣹⠃⠉⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠈⠁⠀⠀⠘⠤⠜⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠖⡞⠀⠀⠀⠀⠀⠀⢠⡤⠔⠁⠀⠀⠀⠀⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠄⣀⠠⠤⠤⠒⠋⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⠉⢐⣶⠀⠒⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠭⠁⣒⡒⠠⢄⣀⣾⣋⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⡴⠒⢢⣀⠤⠒⠊⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣿⢤⠊⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣠⠎
⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣄⠤⣄
⡏⠹⡷⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢈⡇
⡇⠀⢇⢣⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⡀⠀⠀⠀⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠡⣞
⡇⠀⢸⠘⡀⠀⠀⢀⣠⠤⠒⠚⣉⡩⢇⠀⠀⠀⠀⠀⠈⠳⠤⣀⣀⣀⣀⣀⣀⣀⠤⠔⠊⠁⠀⠀⢸
⠃⠀⠀⡆⢧⢴⢎⣍⣀⠴⠒⠉⠀⠀⠈⠓⠤⠤⠴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⢸⠀⠀⡇⢸⡾⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠴⣄⡀⢀⣠⣄⣀⣀⡠⠔⠁
⠈⣆⠀⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠸
⠀⠘⢦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠙⠓⠂⠉⠉⠁⠒⢒⣶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣉⣉⣒⣒⣒⣒⣒⣊⣉⣉⠠⠤⠤⠼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣶⢷⡶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢿⢶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⣼⡏⣵⠃⢀⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣃⠨⣏⢻⡌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣏⣼⠏⢠⠎⢀⡔⢁⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⡆⠸⡄⢻⡎⣷
⠀⠀⠀⠀⠀⠀⣰⣿⣧⠤⠀⣸⠖⠋⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⡈⣶⡁⣀⠻⣾⠆
⠀⠀⠀⠀⠀⢠⣿⣧⡄⠀⢳⡌⠛⢛⣿⠋⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣭⠙⣩⠶⠈⠓⣿⡃
⠀⠀⠀⠀⠀⣾⡿⠧⡄⠉⠛⠀⢸⣿⣿⡆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣠⠖⠂⠀⠀⠀⢺⣿⣿⠉⠙⠀⢀⣻⣿⡇
⠀⠀⠀⠀⠀⢻⣻⡄⠂⡁⢠⣤⣿⣿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠘⢿⣿⣷⠀⠀⠈⢩⣿⡅
⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⢿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⢳⣾⣯⠋
⠀⠀⠀⠀⠀⠀⠘⢿⣧⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠗⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣴⣾⣿⣷⣤⢀⣤⡾⢷⣄⠀⢸⣷⣿⠀⠀⣴⡿⢶⣤⡀⣤⣿⣿⢷⡦⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣉⣛⡟⣿⣿⣏⡀⠈⢿⣦⡾⣿⣿⠷⣴⡟⠀⢀⣽⣿⡿⢋⢛⣙⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣿⣭⣿⣿⣷⣬⣿⡁⡼⠀⠂⠿⣤⣾⣿⣿⣭⣿⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⢀⣺⠉⠉⠙⠻⣿⣿⡟⣁⠁⠀⢀⢿⣿⣿⠟⠋⠉⠹⣇⣀⠀⣤⣀⡀
⠀⠀⠀⢀⣴⣻⠟⠛⠛⠛⠻⠷⣾⣿⣖⡀⡀⠀⢸⣿⣾⣥⡆⣀⣸⣿⣿⠃⠀⢀⣀⣾⣿⣷⠿⠟⠛⠓⠛⠿⣗⣦
⠀⠀⣠⣾⠋⣡⣤⣐⣶⣶⣖⣲⣾⣅⡉⠙⠻⣶⣦⡿⣛⠛⢛⡎⠛⠻⢿⣴⡾⠟⠋⢉⣼⣶⣒⣲⣶⣶⣂⣤⣈⠹⣷⣄
⠀⠀⠙⢁⣺⡿⠋⠉⠁⠈⠛⠻⣿⠷⣦⣵⣦⣈⡿⣰⣯⠀⢸⠀⠀⠀⡌⣋⣡⣶⣮⣶⢾⡿⠛⠋⠁⠉⠉⠙⣿⣓⣌⠋
⠀⠠⣞⡿⠋⠀⠀⠀⠀⡀⣠⣤⣾⣄⣀⣈⡉⠻⡇⡟⠏⣇⢸⠀⡄⠀⡟⣿⠛⢉⣁⣀⣤⣧⣤⡄⢀⠀⠀⠀⠈⠻⢯⢳
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠾⠛⠛⣿⠛⠛⠛⠛⢻⡇⠀⠀⠈⢸⠀⠀⠀⠀⣹⠟⠛⠛⠛⠛⣿⠛⠻⢷⣿⣦
⠀⠀⠀⠀⠀⣠⣼⡟⠁⠀⠀⢈⣷⣤⣤⣤⡴⢾⣇⢀⡀⠀⢘⠀⠀⠀⠀⣿⠶⢦⣤⣤⣤⣿⡀⠀⠀⠈⢿⣦⡄
⠀⠀⠀⠀⠰⠏⠈⠀⠀⢠⣞⣿⣿⠉⠁⠀⠀⠀⢻⣿⣿⣿⡿⠿⢿⣿⣾⡏⠀⠀⠀⠈⠉⣿⣿⣳⡀⠀⠀⠁⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⢩⡇⠀⠀⠀⠀⠀⢸⣿⣿⣧⣤⣀⣰⣿⣿⠇⠀⠀⠀⠀⠀⣿⠉⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⡇⠀⠀⠀⠀⠀⢨⣿⣿⡟⠉⠉⢹⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⣿⣿⡿⠛⠛⢿⣿⡟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⠿⠶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡶⢦⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢼⣿⣧⣀⣄⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣟⠒⠶⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⡴⠁⣿⠃⠘⣿⡆⢠⣿⠁⠙⣷⠙⢆⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⣀⣼⣧⣠⣤⣿⣿⣾⣷⣤⣀⣿⣆⣸⠀⠀⠀⠀⠀⠀⠈⢆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣯⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿