Haircut Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⢶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣵⣖⠚⠉⠛⠻⣿⣟⡻⢿⡷⡄⠀⠀⠀⠀⠀⠀⠀⣿⢵⣧⠤⠞⠙⠛⠻⢿⣟⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠉⠉⠙⠳⣦⣄⢤⡙⢿⣦⠙⢹⡄⠀⠀⠀⠀⠀⠀⡟⠈⠛⠻⠿⣶⣤⡀⢄⠙⢿⣎⠳⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⢁⣀⠀⠀⠀⣼⠆⠙⢿⣿⡌⣿⣧⠀⡇⠀⠀⠀⠀⠀⢸⢀⣀⠀⠀⢠⣿⣊⣙⢾⣷⡈⣿⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⡘⠸⠇⠀⠀⠀⢸⣿⠿⠿⣿⣿⣹⣿⢠⡇⠀⠀⠀⠀⠀⡘⠹⠅⠀⠀⠀⣏⣩⡽⠿⣿⣷⢸⣿⢸⡇
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠘⠁⠞⣹⠟⣿⣿⡿⡼⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⠀⣼⡟⠡⠞⣹⢿⣿⣿⡏⣼⠁
⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢿⣿⣿⠃⠀⠀⠀⠀⠀⢩⡖⠶⢾⣿⣷⣸⣷⣄⠈⣇⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠄⠀⠈⠙⣖⠋⠁⠀⠀⠀⠀⠀⠀⣸⠟⢿⡷⣿⣿⣿⢿⡟⠀⠉⢳⡖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠒⠲⡖⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠘⠁⣛⣉⣵⠟⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠉⠉⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠒⢒⣒⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⠒⢒⣒⣲⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣵⣖⣚⠉⠙⠛⢿⣿⡛⢿⡷⣄⠀⠀⠀⠀⠀⠀⢸⣷⣷⣖⣒⠉⠉⠛⠿⣯⣟⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠉⠉⢻⣶⣄⢦⡙⢿⣧⡙⠹⡆⠀⠀⠀⠀⠀⡸⠁⠀⠈⠉⠻⣷⣤⡠⣌⢻⣷⡌⠞⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣀⡀⠀⠀⣼⣿⣿⣿⣿⡌⣿⣧⠀⡇⠀⠀⠀⠀⠀⣇⣀⣀⠀⠀⢴⣿⣿⣿⣿⣦⢻⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠸⠇⠀⠀⠀⢸⣿⠿⡛⣿⣿⣹⣿⢠⡇⠀⠀⠀⠀⠀⠉⠗⠀⠀⠀⢸⣿⠿⣛⢿⣿⣼⣿⡇⡟
⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠈⠁⢚⣱⣿⣿⣿⣿⡼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠾⠁⠘⣣⣿⣿⣿⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡷⣄⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢵⣤⣀⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣠⠆⠀⠈⠉⢟⠉⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢀⡄⠀⠀⠉⢻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒⡏⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⢞⠉⠁⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢤⡤⠤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣾⣿⣿⣾⣿⣿⣿⣦⣀⡉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⣻⣿⣿⣿⡜⠿⣿⣏⢿⠣⠿⣻⣿⡟⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠟⠁⠈⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⡇⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢄⣀⣀⣀⠤⠐⠲⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣹⠀⣏⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⣯⠣⠤⠀⠴⠃⠀⠈⠢⠄⠤⢚⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠰⡀⠀⠀⠀⣿⣧⠀⠀⢀⣤⣤⣤⣀⠀⢀⣾⣿⣧⠄⠀⠀⣠⣿⠛⠒⢰⠇
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠹⡄⠀⠈⣿⣿⣷⣿⠿⠿⠛⠿⠻⣿⣿⣿⣿⠁⠀⠀⣰⣿⠃⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠈⠙⢶⠦⣤⣿⣆⠀⢸⣿⣿⣿⣦⣰⣶⣦⣼⣿⣿⣿⣿⣄⠀⣰⡿⠃⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⢹⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣼⣿⠁⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠁⠀⠙⠻⠟⠿⠿⠿⠛⠿⠋⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣤⣶⠾⠿⢿⣿⣛⠓⠲⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣛⡛⠛⠶⠦⣄⡉⢳⡄⢀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⣿⢻⣿⡛⠿⢭⣉⠒⢦⡈⢳⡄⢿⡌⡇⢻⡽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⡟⣆⠙⢿⣕⡲⢬⡻⣆⣷⣹⣷⣿⣿⣿⣾⣷⣏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣳⣜⢮⣙⢦⣜⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⢿⣿⣿⣿⣿⣿⣿⡿⠟⡋⠁⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠉⠙⠛⠛⠋⠭⠒⠁⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠰⣿⣿⣿⣶⣤⣠⡾⣦⣤⣴⣾⣿⣿⡷⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠻⠿⠿⢿⡿⠟⣤⡜⠿⣿⠿⠿⠿⠃⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡀⠀⣀⠀⠀⠀⣸⣿⣿⡀⠀⠀⢀⡀⠀⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⣧⠀⠀⠉⠁⠉⠁⠀⢰⡿⣿⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢿⡒⠒⠒⠲⠲⠒⠒⣺⢳⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠘⢯⣭⣤⡀⣠⣬⡽⠟⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⣿⣹⣿⣿⣿⣿⣻⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣽⣿⣿⣿⣿⣿⣿⣼⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠠⣄⢤⡰⣄⢦⡰⣄⠠⣄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣐⣰⣦⣀⣒⠦⣄⣀⣀⣀⡰
⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⢿⣿⠿⠿⠿⢿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠗⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⡟⠛⠂
⠀⠀⠀⠀⠀⠀⠘⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠛⠛⠛⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠋⠉⠉⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⡰⣄⡀⡦⠤⣀⡤⠄⠀⢠
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣭⡻⠿⠗
⠀⠀⠀⠀⠀⠀⠹⡿⠋⠁⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠻⣿⣿⠿⠟⠛⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠉⠀⠀⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣷⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣷⣿⣿⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⣄⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢠⣶⣾⣿⣶⣶⣼⣿⣷⢶⣶⡎
⠀⠀⠀⠀⠀⠀⢿⣿⠟⠉⠉⠙⠛⠛⠛⠛⠿⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠿⣿⣿⡿⠛⠉⠋⠀⠀⠀⠀⠀⢸⡿⠟⠁⠀⠀⠈⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣄⢀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠻⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠛⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡷⠺⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠲⠓⡔⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⠉⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⣤⠄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣷⣤⣤⣤⣾⣷⣶⣶⣦⡀⠀⠀⠀⢠⣾⣿⣿⠁⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠸⣿⣿⣿⣦⡤⠤⣤⣾⣿⣿⣿⠀⠀⠀⠀⠀⢿⠿⣿⣶⣶⣶⣶⣾⣿⣿⣿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠋⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠙⠻⠛⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠒⠂⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢀⣄⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣾⣶⠋
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠁⠀⠀⠀⠉⣻⣦⡄⠀⠀⠀⠀⢠⣴⣶⣿⣿⣧⣀⣀⣀⣼⣿⣿⣶⣦⡀⠀⢀⣴⣶⣾⣿⣧⡀⠀⠀⣠⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⡀⠀⣀⣴⣿⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠙⠛⠋⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠃⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠦⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠳⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⡀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢴⠄⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⠈⠛⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣟⠀⠀⠀⢿⣿⠟⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣷⣤⣀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠒⠛⠛⠛⠃⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣴⣤⣶⢶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠗⠓⠛⠿⠿⠟⠋⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠃⢠⣄⡀
⠀⠀⠀⠀⠀⠠⠞⠁⠀⢀⢅⠀⣀⣀⣀⣙⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠓⣠⣴⡶⠞⠛⠃⠄
⠀⠀⠀⠀⠀⠀⣾⠻⠻⣿⠛⠛⢻⡾⣿⠏⢣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠒⠛⠫⢹⠷⠀⡀
⠀⠀⠀⠀⠀⠀⠹⣎⢁⢸⡆⠀⠀⠉⠡⣤⡜⠀⠀⠀⠀⠀⠀⠀⢀⣶⢸⠂⠀⠀⡀⢀⢀⡀⣾⡀⠀⣀⡤⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣤⣤⣤⡶⢷⣟⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⠆⡾⢹⡇⠿⣿⢰⡿⡼⢰⡟⠊⠘⣿⡁
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠾⢿⣯⣫⡽⢳⣿⣿⠀⠀⠀⠀⠀⠀⠀⣾⡇⠍⠀⠓⠙⣁⣀⠋⢈⠍⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⢐⡏⠀⠀⠀⠘⣿⣿⣿⣿⣿⡋⠀⡀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢷⡀⢰⡿⡈⢠⡶⡄⢰⠆⡄
⠀⠀⢚⢀⣴⣿⣇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣽⢿⡇⢾⣧⠇⣿⠔⠁
⠀⠀⢸⣿⣿⣿⣿⠀⠂⠠⠀⣼⣿⣿⣿⣿⣿⣿⡷⠀⠀⣤⠁⠀⠀⠀⠀⠀⠠⣋⡴⠏⠛⠈⠀⠀⠀⢸⡇
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠲⠷⠿⣿⣿⣿⣿⣿⠇⠀⠸⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣭⣴⣶⣶⣶⣶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣛⡶⠤⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⣭⡶⠶⠶⠶⠶⢶⣶⣤⣍⣙⠳⢦⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣛⠿⠿⠿⣿⣿⣶⣶⣮⣽⡻⢷⣦⡉⠳⣦⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣦⣭⣙⠻⢿⣿⣦⡈⡛⢦⡈⢳⡜⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠲⠄⠀⠀⠉⠛⠿⣿⣿⣦⡹⣿⣷⡹⣆⢉⠂⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠂⠀⠀⠀⠀⠀⢰⣌⠙⢿⣷⣿⣿⣷⢹⣧⠳⡄⡉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄⠹⣿⣿⣿⡌⣿⣇⢹⡜⡌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣿⡀⢿⣿⣿⠁⣿⣿⠀⣷⣿⡈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠧⢸⣿⡿⢸⣿⣿⠀⣿⣿⡇⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡿⢡⣿⣿⡿⣸⣿⡟⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣱⣿⣿⡿⣱⣿⠟⢡⣿⢏⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢋⣾⡿⠃⣰⠟⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⣱⣿⢫⡆⢈⣵⢢⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⣼⣿⢣⣿⢀⣾⠇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣿⣿⠀⢿⡏⢸⣿⣸⣿⠀⠁⠀⠛⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⣮⡛⠘⣿⣿⣿⣇⠐⢄⣙⣚⣵⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢻⣷⡘⣿⣦⣌⠻⣿⣿⣦⣔⡲⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡙⠷⡈⠙⠿⣷⣦⡙⢿⡟⠿⢿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⡛⢀⣜⣿⣷⣄⠈⢓⣮⡙⢿⡌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⣤⣬⡾⢏⣿⣿⣷⠉⠿⣹⢢⡑⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠁⠀⠀⠀⠀⣠⣶⣿⣿⠿⠋⠀⠈⠴⣿⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⣏⠁⠀⠀⠀⠀⠀⠀⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡵⠞⣀⢁⡤⣀⣤⡄⣠⣀⠄⡶⢣⣄⡄⠀⠀⣠⣤⢀⣄⡤⣠⣤⠞⠀⠀⣠⢋⣤⡤⠀⣤⢠⣞⡅⣠⢃⣤⣄⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠉⠙⠂⠙⠘⠊⠛⠛⠘⠃⠙⡿⠃⠀⠈⠃⠛⠙⠙⠃⠙⠙⠂⠀⢸⡻⠉⠊⠓⠛⠋⠉⠙⠉⠓⠙⠊⠙⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⣭⣵⣶⣶⣶⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣯⣽⣟⣛⡶⠄⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣋⣥⡷⠶⠶⠶⠶⣶⣶⣬⣍⣛⠳⢦⣄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣛⠿⠿⠿⣿⣿⣷⣶⣮⣽⡻⢷⣦⡉⠻⣆⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣶⣤⣍⣙⠻⢿⣿⣦⡈⡛⢦⡈⢳⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠲⠄⠀⠀⠉⠛⠿⣿⣷⣦⡹⣿⣷⡹⣆⢉⠂⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⢰⣌⠙⢿⣿⣿⣿⣷⢹⣧⠳⡄⡉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡀⠻⣿⣿⣿⡄⣿⣇⢻⡜⡌⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⣷⡀⢻⣿⣿⠁⣿⣿⠀⣷⣻⡈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠧⢸⣿⡟⢰⣿⣿⠀⣿⣿⠇⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡿⢡⣿⣿⡿⣸⣿⡿⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣱⣿⣿⡿⣱⣿⠟⢁⣾⢟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢋⣾⡿⠋⣰⠟⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣱⣿⢫⡄⢈⣵⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣼⣿⢣⣿⢀⣾⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢿⡏⢸⣿⣸⣿⠀⠁⠀⠛⡢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⣮⡻⠘⣿⣿⣿⣇⠀⢄⣚⣚⣡⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⢿⣧⠘⣿⣦⣈⠻⣿⣿⣦⣔⡺⣭⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⣦⡙⢷⡈⠛⢿⣷⣤⠙⢿⡿⠿⢿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⡛⢀⢜⣿⣷⣄⠈⢒⣮⡙⢷⡌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⠀⠀⣥⣭⣾⢟⣿⣿⣷⠈⠿⣹⢠⡃⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⣠⣴⣿⣿⠿⠋⠀⠈⠥⢿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣟⠉⠀⠀⠀⠀⠀⠀⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠞⣀⢁⡤⢀⣤⡄⣠⢀⠄⡴⢆⣄⡄⠀⠀⣠⣤⢀⣀⡤⢀⣤⠞⠀⠀⣠⠋⣠⣤⠀⣤⢀⣞⡁⣠⢂⣤⡄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠋⠙⠂⠛⠘⠃⠛⠛⠘⠃⠙⡿⠃⠀⠈⠃⠛⠙⠙⠃⠛⠙⠂⠀⢸⡻⠚⠊⠓⠛⠊⠙⠙⠘⠓⠙⠚⠙⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣶⣶⣶⣶⣶⡷⣿⣾⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣿⣷⣶⣶⣶⣾⣷⡾⠷⠾⠶⠾⠿⣾⡶⠿⡶⠿⠶⠶⢶

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠚⠛⠂⠀⠀⠀⠀⠀⡀⢈⣀⠉⡰⠀⠀⠀⠀⠀⢂⠤⠤⠤⠠⠦⠴⠀⠀⠀⠀⡀⠠⠌⠄⠙⢷⣄
⠀⠀⠘⠖⠖⠒⠲⠒⠢⠀⠀⠀⡀⠁⠀⠀⠀⢀⠀⠀⠀⡤⡉⠀⠀⠀⠀⠀⠀⠆⠀⠀⠸⡅⠀⠀⠁⠀⠫⠌
⠀⠀⠘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠰⡀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣀⣀⣀⣠⣤⢀⠂⠀⠀⠀⢀⡀⢉⡅⡉⡱⠀⠀⠀⠀⢀⡆⠀⠀⢸⣿⣧
⠀⠀⣾⠁⠀⠀⠀⠀⡄⠀⠀⠀⠀⡏⠀⠀⠀⠈⠐⠀⠀⠀⠀⠻⠋⠉⠉⠉⠉⡆⠀⠀⠀⢸⣤⠄⠤⠦⠂⠏
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠃⢀⡸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠅⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠓⠁⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢁⣠⣤⠀⠀⠀⠀⢀⡀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢣⠀⠀⠁⠀⠄⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⡇⠙⠀⠀⣴⣿⡶⠀⢠⢠⣴⣦⡇⠀⠀⠀⢠⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⣈⣳⠉⢡⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢡⡞⠀⢄⠀⠀⠀⠀⠆⢀⡄⠀⢀⠆⠀⠀⠀⢠⠴⠊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡀⠈⠀⠀⠀⢠⣄⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠰⣫⠥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣁⢀⡀⠃⠀⠐⢄⣀⣠⠁⠸⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⣲⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣪⠊⠀⠀⢱⠀⡤⠄⠒⠦⠖⢻⠚⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠈⠂⢄⠀⣀⣀⠀⢀⠄⣰⠀⣆⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⢰⠀⠀⠀⠀⠀⢡⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠈⣰⠋⢬⠐⢊⠙⣦⠶⣂⠐⠉⠖⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠀⠄⠀⠠⠀⣰⠇⠀⢈⡭⣤⠏⢀⠊⡍⣿⡷⠹⡆⡐⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⢀⡨⠖⠊⠀⢀⣄⡐⢻⢏⡜⣡⠎⠀⣿⣇⠀⠙⢮⡀⠬⠇⠂
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⡀⢀⠄⠈⡀⠔⢦⣀⡠⠔⠊⠁⠀⠀⢀⡔⠃⠀⠀⢽⢾⠊⠱⢂⠀⡟⢻⠀⢀⡒⠋⠃
⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠐⡀⣠⡊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠀⠀⠀⠀⡺⣼⡀⢸⣿⠃⠃⠘⠰⣿⠏⠀⠀⣿⠞⡆
⠀⠀⠀⠀⠀⢘⣿⡀⠀⠀⢀⠌⠁⠙⢦⡀⠀⠀⢀⣠⡤⠊⠁⠀⠀⠀⠀⠀⠀⠐⠜⡇⠀⠀⠀⠐⡁⠀⠀⠀⠀⢀⣏⠇⠅
⠀⠀⠀⠀⠀⠀⣿⡇⡠⠐⠁⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⣰⡷⠄⠀⠀⠀⠒⠀⠀⠀⡠⠢⠈⣧⡅
⠀⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠂⠐⠀⠀⠀⠀⠀⠀⠀⣴⣿⢸⡄⢺⠶⢖⠂⢒⡖⠀⢰⣤⠀⠻⡇
⠀⠀⠀⠀⠀⠀⠀⡙⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠘⠈⠁⠍⣠⣤⠀⠀⠉⠟⠃⠀⣀⠀⠙⣜⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡸⠀⠂⠀⠀⠀⠀⠀⢀⠈⣀⠘⠉⠀⣴⡤⠀⠀⠘⠟⠂⠀⣈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣂⡀⣀⣀⣀⣀⣑⣂⣐⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠆⠘⠟⠀⠀⠀⠈⠁⠀⣴⡤⠀⠀⠈⠟⠋⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠄⢢⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⢀⠼⡋⠉⠳⢹⢰⠀⠀⠀⠀⠀⡠⠚⢉⠄⠀⡀⢀⠢⡀
⠀⢰⡉⠉⠁⡔⠀⠀⠀⠀⠀⣀⠀⠀⢠⢯⠏⠀⠀⠀⠀⠯⠀⠀⠀⠀⣸⠁⡴⠁⠔⠚⡁⠀⢧⢱
⠀⢸⡁⠰⣂⡼⠀⠃⠀⠂⠀⡡⠃⠀⠸⡀⠐⠈⠁⠀⠠⠐⠆⠀⠀⠀⡇⢰⠁⠄⠀⠀⠀⠈⢸⠈⡇
⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⢄⠀⠀⠀⠀⠁⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢊⠜

⠀⠀⢀⠤⣤⠀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠠⠠⠦
⠀⠀⡟⠚⠋⠤⣴⠖⠊⠀⠀⠀⠀⢀⡠⠤⠖⠋⢁⡔⡡⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠈⠐
⠀⠀⠈⠂⠉⠋⠁⠀⠤⠠⠀⠀⠐⠁⠀⣀⣤⠴⠋⣰⠃⠄⠀⠆⢠⡀⠀⠀⠀⠀⣀⠀⠐⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠰⠉⠀⢀⣀⠴⠃⠀⠀⠀⠀⢀⡇⠀⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡄⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠁⠀⠡⠄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠰⢀⠀⡀⠀⢀⣰⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡀⢆
⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⢁⡇⠀⠀⠀⠀⠀⠈⠂⠠⠄⠠⠌⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⡇⢨⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣲⣽⣾⣿⣏⠁⠀⠀⠀⠀⣉⠉⢙⡛⠷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢴⣪⡵⠛⠙⡛⠣⣿⠛⠀⠀⠀⠸⣿⣶⣿⣷⣿⣷⣶⣍⡻⣷⣦⡀
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⡟⠀⡐⢀⠻⣄⠻⢰⠀⠀⠀⠀⠈⠙⢿⣿⣟⠻⣿⡝⠻⣮⣿⣿⣦⡀
⠀⠀⠀⣠⢟⡽⠛⣩⣿⣿⣿⣿⣿⡁⠀⢃⢿⡀⠈⠳⢄⣧⡀⠀⠀⠀⠀⠀⠙⣿⣧⠘⢿⡄⠈⠻⣿⣿⣷⡄
⠀⠀⠀⠃⡾⢁⣾⠟⠙⠉⣩⣾⣿⣇⢀⠀⠘⢷⣤⣇⢠⠈⠙⠷⣦⡐⣄⢢⡀⠈⢿⣧⠀⠻⡀⠀⠙⣿⣿⣿⣆
⠀⠀⠀⠀⡇⢸⣧⢠⣴⠟⠁⠸⣟⣿⣦⣷⡀⠀⠙⠻⣿⣦⡘⣶⣄⠙⣾⣆⢳⡄⠘⣿⡇⠀⠀⠀⢀⠸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠘⣿⡿⠁⠀⠀⠀⠈⠻⣏⠻⣿⣶⣄⡀⠀⠉⠻⢮⣿⣷⣌⢿⡎⣿⡀⢸⣿⠀⢧⡀⠸⣆⢹⣿⣿⡿⡆
⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣛⣿⣿⣶⣦⣄⡈⠙⢿⣾⣿⣼⣧⠀⣿⡇⢸⣧⠀⣿⡌⣷⣿⣷⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠶⢦⡄⠀⠀⠀⠠⠶⠛⣋⣉⣁⡀⠀⢉⠙⠿⣦⡀⠹⣿⣿⣿⠀⢸⣧⢸⣿⡄⢹⣧⣿⢸⣿⣷⡡
⠀⠀⠀⠀⠀⠀⠀⠰⠞⢳⣦⡀⠀⠀⠀⠀⠟⠹⣿⡟⡿⠟⠋⠀⠀⠘⣷⡄⢹⣿⣿⠆⢸⣿⣼⣿⡇⣸⢿⣿⢸⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢘⣿⣿⠀⢸⡟⣿⠿⣷⡿⠘⣿⣸⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⢸⣿⡟⠀⣼⠃⡿⠀⣿⠇⠀⠃⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣸⣿⣾⣿⠇⢰⡟⠀⠀⠀⠟⢸⠇⣼⣿⣿⣿⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣷⣿⢿⣿⡟⢀⣿⠃⡀⠀⡄⢀⣿⣰⢟⣿⣿⡏⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⣼⠁⣼⠙⢏⣿⣿⠁⣼⡏⢠⠇⢰⡇⣼⣿⢋⣾⣿⣿⡧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠟⠃⠀⠀⢠⡏⠀⣿⠀⣼⣿⡇⣼⡿⢠⡟⢠⣿⣿⠟⣡⡾⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⣿⢀⣿⡟⣸⣿⢣⣿⣱⣿⡿⢋⣴⠟⢡⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⢸⣿⣧⣿⣣⣿⣿⠟⠋⣴⠟⠁⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠰⢸⣿⣿⣿⣿⡿⠟⠁⠀⠈⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢧⣿⣿⠟⠋⠀⠀⠀⠀⡠⢂⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢿⡄⡜⣿⡿⠀⣠⠆⠀⣠⠞⣡⡾⣱⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⡼⣷⡸⣽⣷⣾⣧⢎⡾⢃⣴⠏⣴⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⢠⡇⠘⣷⣽⣿⣿⡿⠋⣴⠟⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡜⠻⣿⣿⣷⠞⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⢱⣠⢈⣿⣿⣦⡾⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣾⣿⣿⢠⣿⣿⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣸⣿⣿⣷⣾⣿⡇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⢿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟

⠀⠀⠀⠀⠀⠀⢠⠀⡄⢠⠀⠄⠀⠀⠄⠀⠠⠀⠀⠀⠀⠠⠀⠀⠄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⢆⣸⡅⠀⠀⠈⠀⠀⠀⠀⢀⠄⢰⠊⠉⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠠⢀⠀⠤⠈⠀⠀⠀⠀⡱⠀⠀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣌⣀⣰⣾⠀⠀⠀⠚⢆⣀
⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠈⠀⠰⠀⠈⠀⡴⠁⠀⠀⠘⢄
⠀⠀⠀⠀⠀⢸⠀⢱⡀⠄⠆⠂⠀⠀⣠⠼⠗⢄⣰⠀⠀⠈⢗⢀
⠀⠀⠀⠀⠀⢸⠀⠀⠑⣄⣀⠄⠀⠀⠀⠇⠀⠀⣿⠀⠀⠀⠈⠆⠠⣀
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠇⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢀⠎⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⢀
⠀⠀⠀⠀⠀⠙⡶⠤⠤⠤⠄⠀⠀⠀⠈⢸⡆⠘⢎⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠙⢃⣐⣺⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢳⠤⠤⠴⠀⠀⠀⠀⠀⠀⠓⠤⣾⡤⠤⠵⠦⡀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡜⠀⢀⠀⠀⠀⠀⠀⠀⠈⢦⣜⠃⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⢧
⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢓⣶⣶⣶⠖⠒⠚⠓⢤⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢸⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⢸⣄⡁
⠀⠀⠀⠀⠀⣸⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠸⠈⠳⣌⡂
⠀⠀⠀⠀⠀⣿⣓⡉⠳⠶⢄⠀⠀⠀⠀⡀⠄⠒⠐⠂⠸⠤⠤⠐⠀⠂⠤⣈⠢⣀⠀⣀⠤⠛⠛⠛⠋⠛⠁
⠀⠀⠀⠀⠀⠈⠈⠉⠛⠛⠉⠀⠀⠀⠀⠑⠠⠄⠀⡀⣀⣀⣀⣀⣀⣀⡀⠤⠃⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣾⣶⣾⣷⣤⣿⣿⣶⣿⣿⣾⣿⣷⣶⣿⣿⣶⣿⣦⣤⣶⣷⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠛⠉⠉⠉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⣉⣩⣤⣤⣤⣬⣭⣛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠛⠉⣁⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠻⠿⢿⣿⣧⡙⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠟⠋⠁⠀⠀⣀⣤⡶⢶⣦⣀⠈⠛⢷⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⣾⣿⠟⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⠟⢁⡴⠺⢿⡝⢧⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢶⡀⠈⠀⠀⠹⣿⣿⠛⠻⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣘⠛⠻⣆⢻⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡿⠀⠀⠀⠀⠈⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣷⢀⡌⢸⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠛⣡⠾⢡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠋⢀⣴⢟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⣀⣤⡄⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢉⣴⣶⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠛⠛⠛⠛⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡤⠤⠴⠖⠛⢿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⡴⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⡀⠀⠀⠀⢸⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢠⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⢿⠙⠿⠟⢙⡟⠛⣟⠟⢻⡿⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣶⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠞⠄⠄⠀⠀⠤⠀⠀⠄⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠀⠆⢸⠇⠠⠶⠶⠀⠀⠐⠂⠀⠃⠀⠀⣉⣉⣇
⠀⠀⣠⣼⣽⡆⣤⡄⠁⡇⠠⠀⠙⣾⠂⡄⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠖⠒⠀⠀⠋⠉⠉⠉⠉⠉⠉⠃⠀⢀
⠀⠐⢿⣆⢠⣼⡿⠃⠀⡇⠀⡗⠀⢿⠀⠻⠀⢇⡄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠀⠀⠀⠀⠀⠀⣴⠄⠴⣄⠀⠀⠀⠀⢰⠀⢸
⠀⠀⠈⠛⠚⠉⠁⠀⠀⣇⢰⠇⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣁⣸⣇⠀⢀⠀⢰⠏⣩⡞⢿⣻⣷⠀⠀⠆⢸⠀⠐
⠀⠀⠀⠀⠃⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠃⠐⢦⠤⠀⠀⠃⢰⣤⣤⣴⣯⣦⣷⣴⣶⣾⣿⣿⣿⣷⣾⣷⠺⢄⡙⠁⢸⣿⠏⠀⠀⠀⣸⠁⢸
⠀⠀⠀⠀⡄⠀⠀⠀⠀⣷⠨⠀⠀⢸⠀⠀⠀⢸⣇⠀⠀⠀⡀⠀⠀⠂⠰⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢶⣯⠉⣁⣾⣷⡀⠀⠀⠘⣿⠀⢸
⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠘⠀⠀⣿⠀⠀⠀⣿⡏⣤⠘⠛⠛⢧⣄⣠⡀⠀⠀⠆⠨⠁⣭⠭⠭⠭⢡⢀⣿⣿⣿⣿⣿⣴⣿⠄⢺⣿⣿⣿⣿⡆⠀⠀⣿⠀⠈
⠀⠀⠀⠀⠿⠀⠀⠀⢰⣿⠀⠀⠀⢼⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆⠈⠛⠁⠀⠰⡇⠀⢤⠁⢠⡀⠀⣿⢈⣿⣾⣿⣿⣿⣄⣬⣾⣿⣿⣿⣿⣿⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣀⡤⢚⣶⣿⣿⣿⣷⠄⠀⠀⠸⠀⡄⠀⠀⠀⠄⠀⠀⣀⡀⠀⠁⣰⣿⣾⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠅⠁⠠⡇⠀⢸⡇
⠀⠀⠰⠶⢶⣶⣶⣶⣾⡟⠶⣓⣭⣯⡶⠛⢻⢿⣿⠁⠐⢘⠋⠒⠁⠀⠀⡄⣠⣠⣿⣿⡀⠱⠦⠈⠉⠙⠿⢃⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⢊⡁⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢡⡏⡉⠀⡖⢹⡿⡿⢾⣿⡄⠖⢫⡈⠈⠃⠺⣸⣿⣿⣷⣿⠛⠁⠀⠀⠀⠒⠀⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⡀⠀⣘⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡟⠸⡇⣿⠁⣹⢸⣿⠇⣖⡞⠉⠢⣄⠀⢡⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠙⠉⣿⣟⣿⣿⣿⡿⠙⠀⠂⠒⣛⡀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⣿⡆⠙⣿⣿⡗⣿⠧⣼⣢⣴⡆⠼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠙⣿⡉⣥⣀⠐⣶⠀⣀⣿⡇⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠇⠀⠀⢸⡟⠇⣽⣿⢽⠿⡞⣿⣻⣿⣷⢘⣿⣿⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢻⡄⢀⣿⣿⠃⢀⠉⠀⠠⠀⠉⠣⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠇⢰⣿⣿⡙⠀⣫⣼⣿⣿⣿⣿⣿⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠂⢹⣿⣷⡆⠒⠲⠲⣛⠛⢓⠒⠲⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣇⢄⠂⢹⡸⠙⢻⣼⡇⠀⠈⢛⣟⣽⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡈⣿⣟⣃⡀⠀⠀⠙⠁⠙⠀⢇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠚⠉⠀⣸⡧⢾⢿⣼⣇⠀⣳⠟⠕⢛⣣⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⠋⠛⠋⢿⣿⣷⣦⣤⢄⡀⠀⢻⡋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⠹⣀⣼⣿⣿⣿⣿⢿⣤⢖⣸⡟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢀⡴⠸⠆⠀⢀⡀⠹⢻⣿⢀⡶⠈⢊⢸⡇⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣠⣴⣿⣿⡟⠺⠋⠙⠁⢡⠞⠁⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⠋⣠⣖⣿⣿⣦⡁⠘⠩⢸⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣰⣾⠟⢿⡴⠶⠂⠀⠀⢀⠼⡫⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⣾⣯⣾⡿⣿⣿⠇⠀⠀⢸⡇⢁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐⠀⣄⡀⡀⣰⣦⣄⡀⠀⠀⠀⠠⠾⠋⠀⠀⠀⣤⣾⡻⣿⣿⠅⣼⣿⣿⠀⣲⣴⣾⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣳⣿⡹⣿⢿⢿⣿⣦⣤⣤⣤⣤⣴⣾⡟⣿⣿⣾⠃⣰⣿⣿⣿⣷⣿⣿⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣶⡿⣿⣻⣿⣉⢸⣿⣿⡿⢿⣿⡇⡿⢿⣿⣿⡿⠟⢹⣿⣿⣛⢿⣿⣿⣿⢿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⢿⠟⠛⠛⢹⡇⣤⢸⡷⠻⣦⣾⣿⣿⣧⣻⣿⣿⣣⣴⣿⣿⣿⣿⣧⡩⣉⣿⡿⢷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡼⠛⠁⣼⡇⠀⢘⣻⣿⡿⣿⣼⣿⡟⣿⣿⣿⣽⣿⣿⣿⣿⣿⠛⠁⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡠⠞⠋⠁⠀⠀⠀⢸⣰⣿⠾⠿⠿⠿⠿⠛⠉⢁⣁⡀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⣫⣾⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⠿⣿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠿⣿⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⡀⢠⠴⠄⠀⠸⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡠⠐⣒⡏⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣄⣀⡁⠁⠀⠁⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠠⡼⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠌⠘⣻⣯⣄⠀⠀⠀⠀⣠⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢁⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⡟⠀⢸⣿⣇⡀⢀⣠⠞⠁⠈⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠒⣆⣈⣤⣤⣀⡀⠀⠀⢀⡇⠀⠀⠿⠙⢿⡩⣷⠀⢀⠄⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣠⣿⣿⣿⣍⣻⣿⣿⣦⣴⠏⠀⠠⣤⠀⠀⡈⠁⢸⡀⠁⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⣿⡈⠉⠛⠛⠋⠉⠁⣿⡇⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠄⢼⡟⢀⣀⡀⠀⢀⣀⡀⢻⣧⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠻⠟⠀⠘⠻⠂⢸⢳⡄⢀⣸⠄⠀⠀⠀⠀⠀⠀⡿⣻⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⠀⣀⠀⠀⢸⠲⢺⣔⡂⠍⠀⠀⠀⠀⠀⠀⢃⡹⡆⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⣠⣉⣡⠰⠀⣼⣧⠘⢴⡄⠀⠈⠀⠀⠀⠀⠀⠀⢳⣦⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⠀⠀⠀⣠⢞⣿⣿⣷⣺⣇⡀⠀⠓⠒⠠⠤⠤⠠⣿⣛⣠⣤⣀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⡿⣿⣶⠟⠋⢠⣿⣿⢿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣷⠸⣿⠒⠀⢀⣀⡿⡏⠘⢸⡿⢿⣿⣿⡌⡿⠿⣶⣤⣀⡀⠀⠦⢤⣄⠀⢨⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠋⣿⢸⡄⠹⡄⢉⣠⠞⠱⠁⠀⢸⠇⠀⠘⠷⣧⣷⣾⡆⠻⣯⡿⠓⠶⠤⠬⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠡⠁⡏⠀⢹⠀⢷⠀⠙⠀⠀⠐⢀⠄⢀⠇⠀⠀⠀⠀⠀⠉⠻⣿⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣇⠀⠈⠇⠈⠃⠀⠀⢀⡴⠋⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⢿⡀⠠⣄⠀⠀⠴⠟⢋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠉⠑⠂⠒⠉⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠐
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡀⢰⠆⢰⣶⡆⠐⡶⠀⢰⠀⠀⡶⠀⢴⣴⡆⢰⡆⠀⣶⡆⠈⠀⢴⡆
⠀⠀⠀⠀⠀⣀⣀⣀⣀⠈⠉⠁⣉⣁⣈⠉⠁⢀⣉⣀⣈⡁⠀⠉⣀⣈⣈⣁⡈⠉⣀⣉⣁⣀⡀⢉⣁⣀⣀⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⣿⣿⣿⠀⠀⢸⣿⠿⠿⣿⡆⠀⣿⡿⠛⢻⣷⠀⣿⣿⠛⠛⠃⢸⣿⡟⠛⢿⣇
⠀⠀⠀⠀⠀⣿⣷⠀⣸⣿⠀⢀⣿⡟⣿⡄⠀⢸⣿⡀⣀⣿⡇⠀⣿⣿⣶⣿⣯⠀⣿⣿⠿⠿⠀⢸⣿⣧⣴⣾⠏
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⢸⣿⠁⣿⡇⠀⢸⣿⣿⣿⡟⠀⠀⣿⡿⠀⢹⣿⠆⣿⣿⣀⣀⡀⢸⣿⡏⢻⣿⡀
⠀⠀⠀⠀⠀⣿⣟⠛⢻⣿⠀⣼⣿⣿⣿⣿⠀⢸⣿⠀⢻⣿⣄⠀⣿⣿⣾⡿⠏⠀⣻⣿⣿⣛⣃⠘⠛⠃⠀⠛⠛
⠀⠀⠀⠀⠀⣿⣟⠀⣸⣿⠀⣿⡏⠁⢸⣿⡄⠸⠟⠂⠈⠛⢉⣠⣤⣤⣤⡀⣾⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠰⠟⠃⠀⠀⡠⠤⠤⠤⠜⠸⠇⠿⠅⢀⣨⡿⠃⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠠⠟⠛⠉⣁⣤⣤⣤⡶⠚⢛⠛⠛⠛⠛⠛⢛⣛⣿⡟⠉⣥⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⣠⠞⠹⣇⢈⣿⠇⢸⣏⣠⢀⡴⢺⡆⣸⡏⣸⡟⢸⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⡉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠉⣉⡁⠀⣿⢹⡿⣾⡇⣾⢣⣿⠷⠛⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣄⣤⣶⣿⣿⡿⢸⡟⣾⡇⠛⠛⠁⠾⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣿⠟⠋⠁⣸⠃⠾⠃⣀⣠⠴⠖⠚⠛⠛⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣤⠀⠀⢀⡴⠁⠀⠒⢫⣭⣤⣿⣶⣶⣶⣾⣿⣥⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⢸⣿⣤⡴⠋⠀⠀⠀⣀⣠⡤⠾⠛⠛⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠙⠛⣿⠿⠂
⠀⠀⠀⠀⠈⠉⠁⠀⣀⡤⠖⣛⡭⠶⠞⠋⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡇⠠⠄⡧⢄
⠀⠀⠀⠀⠀⠀⣠⡾⢋⣠⠾⣯⡛⠁⢴⣿⣯⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣹⣿⣿⣿⣿⣿⣿⣷⠀⣶⡠⠊
⠀⠀⠀⠀⠀⢰⣿⠀⠈⠀⣠⡟⠁⠡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣸⣏⣉⣉⣀⣏⡟⡄
⠀⠀⠀⠀⠀⠀⠛⠿⠟⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠁⣿⣿⣿⣿⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣿⣿⣿⡗⠞
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡘⡏⠀⠀⠙⠢⠬⢭⣍⣉⣉⣉⣉⣉⣛⣛⣛⡛⠛⠛⡛⢟⣴⣿⣿⡿⠟⠉⡇⠄
⠀⠀⠀⠀⠀⠀⢰⣤⠷⣄⡴⢦⣰⡄⠀⠀⣠⢾⣵⣧⣤⣼⣯⣭⣤⣀⣉⣹⣛⡁⠶⢻⠛⠋⠁⠈⠉⠉⠇⠄
⠀⠀⠀⠀⠀⠀⠸⠨⠿⠋⠽⠿⠹⠇⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣥⣼⣿⣿⡆⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⠀⢸⢻⣇⣿⣼⣟⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⠀⡇⠀⢠⠶⢤⣤⢰⡆⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⢠⠇⠀⠈⠓⠙⠋⠘⠃⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⡰⢣⣤⠉⠙⢛⣟⠉⠸⠓⠒⠠⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣭⣉⢙⠛⣿⠄⠀⠈⣿⠁⠉⠡⣤⣤⡀⠀⢀⠀⠀⠀⠀⠈⠙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣦⣤⣀⣀⠀⢸⣿⡀⠀⠀⠨⠿⠿⠿⠷⠙⠃⣀⣀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠾⠷⠶⠶⠶⠿⠛⠛⠛⠋⠉⠁

⠀⢂⡰⠀⠀⠀⠀⢸⣇⠀⠀⣩⢞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠘⣾⡿⠀⠀⠙⠋
⠀⠈⠁⠂⠀⠀⠰⣼⣿⠀⠀⢴⣯⠇⠀⠀⠀⡀⠀⠀⠀⠘⠛⠀⠀⠀⠾⣷⠄⠀⠀⠀⠙⡇⠀⠀⢀⣿
⠀⡴⣶⡄⠀⠀⠀⠻⡇⠀⠀⠀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣾⡇⠀⠀⢊⡴⡃⠀⠀⠀⠀⠃⠀⠀⢠⣿
⠀⢞⣵⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⡠⠁⡀⠀⠀⠀⢸⣇⠀⠀⣩⢞⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣫⢾⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠈⠨⠮⡈⠀⠀⢠⣸⣿⠀⠀⢴⣯⠇⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⣵⣻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⢿⡟⠀⠀⢀⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⡆
⠀⠈⠉⠀⠀⠀⠀⠆⠀⠀⠀⠈⣉⡀⠀⠀⢋⠿⡃⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡐⠁⠀⠀⠀⠀⢹⡇
⠀⢸⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢿⡇⠀⠀⡡⢚⡅⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠊⠩⢎⡁⠀⠀⢠⣸⣧
⠀⢸⣿⠀⠀⡐⢓⡜⠂⠀⠀⠀⢸⣷⠀⠀⠔⣫⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⡀⠀⠀⠀⠈⢿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠃⠀⠀⠸⣼⣿⠀⠀⠺⠷⠃⠀⠀⠀⡄⠀⠀⠀⠈⣉⡀⠀⠀⢚⡿⠂⠀⠀⠀⠈⡇
⠀⠈⠋⠀⠀⠀⠴⣷⠄⠀⠀⠀⠹⡟⠀⠀⠀⣤⠀⠀⠀⢀⠆⠀⠀⠀⠀⣿⡇⠀⠀⣩⢞⡁
⢀⠀⣿⡇⠀⠀⢚⡵⠂⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⠬⡠⠄⠀⠀⠀⢸⣇⠀⠀⠴⣫⡄
⠀⡀⢿⡇⠀⠀⣩⢞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠑⠄⠀⠀⠸⣼⣿⠀⠀⠺⡷⠃⠀⠀⠀⡄
⠀⢠⣸⡇⠀⠀⢴⣯⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⣠⣦⡀⠀⠀⠀⠻⡇⠀⠀⠀⣤⠀⠀⠀⠀⠆
⠀⠈⢿⡇⠀⠀⠀⣁⠀⠀⠀⠐⠂⠀⠀⠀⠀⣤⡄⠀⠀⡴⣫⡄⠀⠀⠀⠀⠃⠀⠀⠀⣿⠀⠀⠀⢌⠠⠄
⠀⠀⠈⠇⠀⠀⠀⣿⠀⠀⠀⠘⠁⠀⠀⠐⠀⢻⣇⠀⠀⢞⡵⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠐⠑⠄
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠔⢣⡜⠂⠀⠀⢀⢸⣿⠀⠀⣫⣞⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠇⠀⠀⠈⣾⣿⠀⠀⠈⠋⠀⠀⠀⠀⠆⠀⠀⠀⠀⣤⡄⠀⠀⡵⣫⡅
⠀⠀⠄⠀⠀⠀⠈⣉⡀⠀⠀⠾⣿⠆⠀⠀⠀⠘⡇⠀⠀⠀⣿⠀⠀⠀⢠⠃⠀⠀⠀⠀⢻⡇⠀⠀⢞⡵⠆
⠀⠰⠂⠀⠀⠀⠀⣿⡇⠀⠀⢚⡵⡇⠀⠀⠀⠀⠁⠀⠀⢠⣿⠀⠀⠔⠓⡜⠂⠀⠀⠂⢸⣧⠀⠀⣫⣞⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⠛⠉⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⢁⣀⣀⡀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡶⠞⠂⠈⢁⡤⠀⠘⣿⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠚⣰⠀⢀⠀⢀⠠⡉⠐⠋⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢓⡶⢽⡲⡶⠑⠘⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣴⣶⣶⣤⣤⡀⠀⠘⢦⡓⠀⠚⢁⠀⠀⠀⠀⠀⢀⡷⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠋⠉⠀⠀⠉⠙⠛⠻⣷⣦⡀⠙⢦⡁⠁⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⠴⠆⠀⠀⠤⣄⠀⠀⢿⣿⣿⢀⡤⠟⠲⢤⣤⠴⠚⠁⠀⠀⠀⠀⡄⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢴⢀⠀⠀⣄⠈⠀⠀⠸⣿⣿⢯⡤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣓⠄⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠾⣡⢤⠀⠁⠀⠀⠀⠀⠙⠙⣷⢍⣻⠅⡉⠉⠒⠲⠤⣄⣀⣀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠘⠿⣖⣩⠿⠂⠀⠀⠀⣼⡛⠛⠁⠉⠓⠶⠔⠲⣞⠒⠶⢴⡇⠈⠉⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⢋⠲⣦⣤⡴⠒⠒⠻⢤⣉⡽⠃⠀⠀⠀⢠⠔⠚⣦⡀⠈⠉⠛⠒⠲⠦⠤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⡀⠙⠒⠀⠒⠈⣀⠄⠀⠪⠭⣓⢦⡄⠀⠉⠀⠈⠀⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠙⠧⣄⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀⠀⠀⠀⠀⠉⠒⠢⠤⠤
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠱⡐⠒⠢⠤⠤⠤⠤⠶⠒⠋⠁⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠺⢧⣄⡀⠀⠀⠈⠑⠦⢤⣀⡀⠀⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠈⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣠⣤⡀⠀⢤⣤⣤⠄⠠⣤⣤⣄⠀⢀⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠘⣿⣛⣷⠀⢸⣯⣽⣧⠀⢸⡷⠆⠀⢸⣿⣶⡟⠀⢿⣭⡉⠀⣿⣃⣀⣿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡶⢿⣦⠀⠀⣼⣿⣧⠀⠀⣿⠉⣿⡀⢸⣷⣶⠿⠁⠸⠷⠶⠇⠸⠇⠸⠷⠀⢶⣼⠟⢀⣿⠙⢻⡇⢰⡟⠁⢹⡇⢀⣿⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣛⣻⡇⠀⠿⠉⠙⠃⠀⠉⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡁⠀⠙⠁⠈⠻⠶⠟⠁⣸⡟⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠁⠀⣀⣠⠤⠴⠖⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠶⠦⢤⣀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⢀⡀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣷⣿⣿⡟⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⢀⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠁⠀⠀⠑⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠊⠀⠀⠈⠁⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⢘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠈⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠫⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠝⠁⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠑⢄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠨⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠈⠢⠄⠈⢠⠀⠀⡠⠗⠁⠀⠀⠀⢐⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠕⠀⠠⠪⠂⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣀⡀⣀⡠⠬⠭⣥⡄⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠠⠼⠓⡉⢠⡆⠀⠀⠀⠙⣴⠟⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⣀⣄⣦⠸⡧⠈⣻⠂⠀⠀⢀⣇⢐⠆⠈⠀⢠⣴⠘⢧⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⠾⠟⠁⢀⡠⡄⠀⣿⣝⢿⠀⠛⠁⣠⡶⠶⠶⠟⠋⠀⣀⢰⡸⡇⠹⡄⠘⠁⠀⣠⣴⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠠⣿⡘⣗⢷⠀⠋⢉⡤⠐⣫⡾⠋⠀⠀⠀⠀⠀⠐⣇⡈⠳⠛⠀⢀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠸⠃⠉⢀⣠⠔⣉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠠⢄⠉⠢⣑⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⢴⣾⠟⢋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡙⠳⣿⡷⢉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠉⣈⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⣩⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣽⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠉⠉⠛⠛⠛⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡧⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⢿⣿⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠧⢽⣦⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣦⠀⢴⠶⠒⠀⠸⠿⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣀⠒⠒⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣄⢀⡇⠈⠀⠈⠑⠂⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠬⣽⠇⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⣌⡔⢸⢿⣷⣲⠐⠀⠀⠀⠀⠀⡸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠙⠀⡇⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⢠⡾⠁⢿⣿⣿⠉⠁⠀⠀⣠⠞⢁⠏⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠰⣀⣀⠀⠔⠋⠀⠀⠀⠀⠀⠀⠘⡇⠀⠘⢿⡿⣷⣶⠶⣟⡡⠐⠀⠠⠂⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣠⣥⣴⣶⣾⣷⣶⣦⡀⠀⢀⡼⠃⠀⠀⠈⠁⠈⢳⠊⢻⠀⣠⠎⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣾⠿⣿⣷⣿⣦⣿⣿⡿⢿⣶⠟⠁⠀⠘⣷⠀⠀⠀⠀⠀⠘⣦⠠⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⡀⠀⢸⣿⠀⠀⠈⠉⠉⠀⠀⠀⢸⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⠀⢻⡟⢀⣀⠀⠀⠀⠀⠀⠀⠘⣿⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠬⠌⠂⠀⠙⠂⠤⠀⠚⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣷⣆⡠⠌⡒⠀⠂⠀⠀⠀⠀⠀⢸⡙⢷⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡀⠀⠲⠴⠊⢀⠀⢠⣧⠁⢺⣊⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⢿⣈⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠁⠐⢿⠀⠛⠉⢀⣾⣿⣿⣈⣹⡇⠀⠀⢯⣀⠀⠀⠀⠀⠠⢠⣿⣧⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣀⡀⠀⣀⣴⠏⣿⣿⣿⣿⡽⣷⣄⣀⠀⠉⠉⠉⠈⠓⠐⠚⠿⠤⠶⠶⢦⣤⣀⠀⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢻⣟⢿⠿⠿⠟⠁⢀⣿⢻⡟⣿⣿⠻⣿⣿⡿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠘⣿⡟⠒⠒⢀⣀⣽⢿⠀⠃⢸⡿⢿⣿⣿⣷⠨⡟⠿⣷⣦⣄⣀⠀⠰⠶⢦⣤⠀⠀⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡾⠻⣿⢸⡆⠘⣧⠈⠉⣡⡾⢣⠏⠀⠀⣾⡇⠀⠈⢿⣾⣴⣧⣷⣮⡿⣿⣿⡟⠲⢤⣤⣨⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⡍⢸⡇⠀⣿⠀⣿⡀⠘⠧⠈⠉⠠⠋⢀⠀⢠⠃⡇⠀⠀⠀⠀⠙⠻⣷⣿⢡⠋⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠘⠀⢸⠀⠀⠹⡄⠈⠇⠀⠀⠀⠀⢀⡴⠃⢀⠏⠀⠇⠀⡄⠀⠀⠀⠀⠈⠻⣾⣵⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢸⡆⠀⠀⠁⠀⠀⠀⢀⣤⡴⠋⢀⠠⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠈⣷⡀⠑⠲⢤⡀⠈⠉⣁⡤⠈⠁⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠁⠉⠁⠀⠀⠀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⣠⡴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡇⠀⡠⢤⠀⢺⢄⡀⢠⠖⠂⠀⡔⢰⠈⠧⣄⠀⣀⢀⠈⠆⠀⡴⢤⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⠒⠚⢄⢀⠀⡸⠀⡄⢸⠀⡌⢠⡇⣼⠉⢁⠎⢀⠃⣼⠀⠀⠁⠀⡘⠁⡐⢀⠎⢀⡄⣠⠋⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠶⠐⢀⡘⠘⠐⠓⠊⠀⠈⠉⠉⠁⠀⠀⠉⠁⠀⠀⠉⠀⠑⠒⠊⠉⠀⠀⠓⠛⠄⠚⠶⣁⠼⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡷⠶⣷⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣇⣀⡀⣴⡆
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠁⠀⠀⠀⠈⠙⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡖⠤⢀⣤⣶⣶⣾⣿⠷⠶⢾⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⣠⢾⣿⣋⡁⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣶⡀⠀⠀⠀⠀⠀⠀⡿⠸⠇⠘⣸⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⣿⢸⡟⠛⠛⠃⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⠀⠀⠀⠀⠀⣇⢀⠀⠀⠁⢞⣿⣿⠟
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠙⠎⢧⠀⠀⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠻⣿⣿⣿⣶⣾⣿⠋
⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠜⡄⠀⢻⣿⣿⣿⠿⣿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠀⠈⢿⣿⣷⠀⠀⠉⠻⣷⣄⠀⣼⣿⠃⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠈⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠐⡀⠀⠘⢿⣿⣆⠀⠀⠀⣾⣻⣿⣿⠋⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠈⢆⠀⠈⠻⣿⣿⣶⣶⣿⣿⡿⠻⣦⡀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⡀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠙⠆⠐⢻⣿⣿⣿⠟⠋⠀⠀⠘⢷⡄⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⣀⠀⠹⢿⣧⡀⠀⢀⣀⣤⣾⣿⣿⠟⠻⣿⣄⠀⠀⣠⢖⡈⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠿⠿⣿⠿⠿⠛⠛⠋⠉⠙⢶⣤⣼⣿⣷⣄⢁⣿⣿⠟⣱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣀⣀⣹⣿⣿⣿⣿⣿⡿⣷⣿⠟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⢿⣿⡿⠿⣿⡜⡇⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠒⢄⠀⡰⠁⢀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⠻⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠜⠰⠣⠃⢆⡜⠜⢱⠁⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⣰⣿⢟⣵⡿⢏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢀⡔⠁⣄⠀⡠⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣷⣄⠀⠀⠼⣟⣵⣿⠏⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠞⠐⢖⡁⡜⢀⡀⠀⣀⢠⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣟⣿⣿⠏⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠋⠀⠀⠰⠓⠁⠎⠸⠱⠁⠏⠸⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢠⣤⣄⠀⠀⢀⢀⡀⢀⡤⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠄
⠀⠀⠀⢰⠟⢋⣩⠀⠀⡇⠀⡱⢮⠀⢨⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢋⣡
⠀⠀⠀⢰⡾⠟⢛⠀⠀⠀⢀⠀⠀⠡⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢰⡾⠟⢋
⠀⠀⠀⢨⣴⡾⢿⠀⠀⠀⠀⠂⢠⡀⠀⠀⠀⠢⠀⠀⠀⠌⠀⠀⠀⢠⠁⠈⠀⠀⠀⣚⣿⠀⠀⢸⣴⡾⠟
⠀⠀⠀⢘⣡⣴⠿⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠊⠉⠱⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⣒⣾⠀⠀⢘⣡⣴⠿
⠀⠀⠀⠸⣋⣥⣾⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠤⣾⠀⠀⠸⢋⣥⣶
⠀⠀⠀⠸⠛⣉⣴⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢰⠁⠀⠀⠀⠀⠠⠭⣽⠀⠀⠸⠟⣉⣴
⠀⠀⠀⠁⠈⠉⠀⠁⠀⠀⠀⢠⣾⠀⠀⠀⠀⠀⣀⣀⣀⣀⣐⣂⣩⠭⠀⠀⠀⠀⠀⠉⠭⠀⠀⠁⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⢀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠤⢰⡾⠀⠀⠀⠀⠀⢷⠀⠄⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⢠⠀⣦⠀⠀⠀⠀⠀⠀⠀⣾⡘⠀⠀⠀⠀⠀⢢⡞⠀⠀⠀⠀⠀⢀⠔⢋⣉⡙⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⠆⠀⠀⠈⢼⣷⠄⠀⠀⠀⠄⣿⠇⠁⠀⠀⠀⠀⣠⠀⣾⣉⡷⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢶⣄⠀⣤⠎⠡⠀⠀⠀⠀⠶⠁⠘⠦⡈⡉⢡⠴⠁⠨⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⣀⣼⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⢠⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⡟⠀⠐⠀⠀⠀⠀⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡻⢿⣧⠀⣸⣿⣿⡄⠄⠀⠀⣠⣿⠃⠀⢀⣀⣠⣴⣦⠀⠀⠀⠀⠀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⣇⠁⣿⡿⣿⡭⢀⣠⡿⣁⡠⢲⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣻⣷⣾⣿⣿⡟⠀⠀⠈⣿⣿⡏⠙⠻⣷⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣾⣷⣶⣤⣄⣈⢿⣷⣿⣧⣿⡯⣤⣀⡀⠀⣿⣿⡇⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣠⣄⣤⣧⣀⣠⣄⣤⣤⣤⣤⣴⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⢷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⡉⣙⣿⣿⣿⣿⣿⣿⣭⣭⣽⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠙⡻⢫⣴⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣯⣻⣿⣿⣿⠟⠿⡿⣿⠿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣦⣤⣴⣇⠸⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣿⡿⣿⣿⣿⣿⣶⣶⣮⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⠿⢻⣿⣿⣿⣿⣿⢏⣿⣿⣟⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⠙⠟⠺⠾⠿⠇⠈⠉⣟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣍⣸⣿⣿⣿⣿⣿⡟⡃⠒⠛⠁⢶⠿⢿⣿⣯⣿⣿⣿⣿⣿⣿⣶⣶⡶⠶⠿⢲⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⠛⠛⠉⠀⣟⠀⣿⡄⠠⠠⠾⠽⠛⠛⠏⣿⡏⠩⡏⠁⠘⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠋
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣼⠀⢀⠀⠀⠀⠀⢠⡆⣿⣇⠀⠁⠀⢸⠀⠀⠀⣮⣀⠀⠀⣇⠀⠀⡀⠰
⠀⠀⠲⠦⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⠇⠀⠙⠀⠙⠀⠀⠀⠀⠘⠁⠈⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣟⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⠴⣼⣿⣿⣿⡷⢶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠠⣿⣿⣿⣟⠁⠀⠈⠙⠛⠻⠋⠉⠑⠛⠓⠒⠒⠶⠶⠆⠀⠀⣠
⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⠀⠀⠀⠀⢤⡄⣀⣒⠒⠶⠆⣿⣿⣿⣿⠁⠀⣀⣠⣶⣄⠀⠀⠀⠀⠀⠀⣀⠀⡼⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠚⠛⣯⡄⠀⠀⡀⠀⢈⢳⡇⠀⠀⡀⠀⠀⣻⣿⣿⡿⢀⣈⣿⣿⣿⠿⠷⢦⣄⡀⠀⣧⣿⢠⡇⠀⠀⡘
⠀⠀⠀⣾⡆⣶⣼⣿⣿⠠⠄⠀⠁⠀⠰⠆⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⣿⣿⠿⠿⣿⡶⣤⣹⣿⣦⡛⣿⣿⣄⠄⢠⢁⣆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣤⣼⠆⠀⠠⠤⠀⠀⣿⣿⣿⣿⣿⣿⡤⠀⠀⠀⣰⠟⢻⣿⠃⡋⢻⣿⣿⣦⣧⣼⡿
⠀⠀⠀⣿⣿⣿⡟⢿⢻⢀⠠⣶⣿⣿⠉⠀⠀⠀⠀⠐⠀⣿⣿⣿⣿⡿⣻⡄⠀⠀⣼⣿⢴⡿⡟⠀⣲⡏⠘⣿⣿⣿⣿⣷
⠀⠀⠀⢿⣿⣿⣧⢾⣿⣿⣷⣿⣿⢿⢿⠉⠒⠲⠒⠒⠐⣿⣾⠋⢻⣧⣻⣯⣤⣾⣻⣿⣾⡟⠁⢀⣞⠉⢠⠋⠛⠛⠻⢿⠇
⠀⠀⠀⠀⣿⣿⣿⣾⣾⣿⣟⣛⣛⡥⡀⢸⠀⠀⠒⠲⢴⣿⡿⣧⣾⡿⣿⣿⣿⣿⠿⠿⠿⠃⠀⣾⠉⠀⠀⠀⠈
⠀⠀⠀⣤⣿⣿⣿⣿⡟⢻⣿⣿⣿⠛⠿⠀⠀⢀⣀⡀⢸⣿⡋⢻⣿⣾⣾⣿⣿⣿⣷⣂⣷⡇⣼⣹⠀⠀⡀
⠀⠀⠀⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⠀⣤⣸⣇⣿⣿⣷⣼⣿⡿⠺⠿⡿⢻⡿⠿⠿⠿⠿⢇⣼⣁⣿⡄⠀⠃⠀⢀⡀
⠀⠀⠀⢿⡿⠿⣿⣿⠟⠉⠉⠛⢿⡆⠀⠈⠙⠛⠿⢿⣿⣷⣿⣿⣷⣯⣀⠇⠀⠀⠀⠀⠀⠉⠛⠿⠿⠦⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⢀⣀⢤⣄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⢴⠇⠀⢠⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠒⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢸⠀⠀⠀⠒⠐⠀⠀⠄⠄⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠤⠀⡸⠀⠀⠀⠀⠀⠀⢀⣄⢠⠆⠀⠀⠀⠀⠀⢘⠀⠀⠀⡚⠀⠀⠀⠄
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⡄⠀⡇⠀⠀⠀⠀⢀⠔⠙⠁⠞⠰⠧⣦⠀⠀⠀⡨⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⡠⠋⠀⡀⠀⠀⠀⠀⢂⡄⠀⢠⣧⣄⣀⢀⡇⠂⠀⠐⠀⢠⠀⡀
⠀⠀⢀⠀⠠⠄⠀⣀⣀⠀⠀⠀⢰⠀⢃⠀⠀⠀⡇⡰⠊⠉⠀⠈⠛⢤⢨⠃⡄⠘⠦⣅⡀⢦⡇
⠀⠀⠁⠀⠀⠜⠉⠀⠀⠙⡄⠀⢸⠀⢸⠀⠀⢐⠱⡁⠀⠀⠀⠀⠀⠀⡿⠄⢇⠲⠀⢨⢁⡇⠀⠈⠁⠀⠀⠀⠀⠀⡇⠀⢉
⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠀⠸⠀⢸⠀⠀⠀⠻⠁⣖⣒⡒⣒⣢⠌⠋⠀⠀⠣⡑⣇⠞⠀⠀⠁⠁⠀⠀⠀⠀⢠⡃⠀⡄
⠀⠠⠀⣦⣼⡀⠀⠀⠀⢀⠆⠀⠀⡆⢸⠀⠀⠀⠀⠑⢮⠁⠀⠀⡠⠊⠀⠀⠀⣠⢉⢌⡢⢢⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⢀⠀⡁⢄⣀⡤⡎⠀⠀⠀⠿⣿⠤⡀⠀⢀⢀⡄⠁⠀⠈⠢⠄⡀⠀⢞⣁⣽⠓⢶⠋⠀⠈⠉⠛⠒⠤⡴⢉⡁⠈
⠀⠀⠀⠀⠀⠀⣼⠀⢀⠇⢀⣀⠀⢷⠠⣏⢾⣶⣁⠀⠀⠚⠀⠄⠀⠀⠀⢢⡴⠁⢘⠄⠛⠉⠓⠢⢄⡀⠀⠀⣆⢁⣀⠈⠁
⠒⠀⠀⢀⢓⣋⠠⠬⠥⠤⠤⠈⠕⠣⠀⠙⠳⡌⠛⡓⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡰⡮⠃⠸⡂
⠉⠉⠠⡵⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠎⡀⡉⢷⠀⠀⠀⠀⠀⢠⡗⣤⢠⡜⠀⠀⠁⠀⠆⠀⠀⠀⢰⠀⠀⠀⠐⢹⡶
⠤⠤⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣸⠒⠒⠒⢺⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⢄⠀⠀⢠⠊
⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢅⡀⠠⠤⢸⡀⠀⠈⠀⠀⡸⠀⠁⠂⠄⣀⠀⠀⠀⠀⠀⠀⠴⠢⠄⢀⡠⠃⠠
⠀⢀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢈⠀⠀⠀⡸⠀⠀⠁⠁⠄⢾⠀⠀⠐⠀⠀⠉⠒⠦⢀⡀⠀⠀⠀⠘⠁
⡀⠈⢦⠙⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠉⠀⠀⡀⠀⠀⠹⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠙⣉⡁⠆
⠀⠀⠈⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠂⠀⠀⢀⠀⠀⠐⠂⠀⠀⡠⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠌⠀⠀⢀⠄⠀⠀⣿⠀⠀⠀⡦
⠰⠀⠀⠀⠈⠁⠱⠲⣀⠉⠉⢉⡼⢹⠀⠀⠀⠀⣨⠀⠀⠀⡻⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠉⡁⢀⡀⣘⠁⢀⣀⠤⢼⠀⠀⠀⡇⠇⠀⠀⢸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣈⡶⠉⢁⠸⠀⢹⡍⡀⠁⠀⡆⠀⠀⠀⠇⢸⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠒⠒⠚⠉⠙⢉⠁⡴⠊⠄⠸⠀⠆⠊⠀⠈⡂⠀⠃⠀⠀⢸⠂⠈⡇⠀⠀⢃⠀⠀⠀⠀⠀⠁⠀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠙⠤⣀⣀⠀⠀⣀⡠⠞⣠⠊⠉⠉⠁⢻⠀⢀⡼⠚⠛⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠘⠛⠛⠛⠋⠀⠘⠧⠤⠤⠦⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣀⣤⡶⠶⢶⣶⣶⣶⠶⠦⣄⣄⣀⣀⣀⠄⠒⠐⠂⣄
⠀⠀⠀⣀⣠⣤⣶⣾⣿⡿⠿⠿⣯⡴⠿⠿⠿⠷⢶⣤⡈⠙⣷⣆⠀⢻⣷⣸⣄⣀⣀⣠⣴⣿⣧⣀⡀
⢠⣴⣿⡿⠟⠛⠉⠁⣀⠀⠰⠚⠁⢴⣄⠐⣦⡀⣠⡙⣿⣆⠘⣿⡆⠿⢿⣿⣿⠠⠤⠧⢼⣿⣿⠻⣿⣷⣤
⢸⣿⡇⠀⠀⠀⠀⠀⠁⠐⢩⠾⣦⢐⣹⣦⡘⣿⣿⣧⡘⣿⠰⣿⠇⣠⣾⣿⣿⠀⣶⣶⢸⣿⡇⠀⠀⣿⣿
⢸⣿⡇⡠⠀⠀⢠⠂⠀⢂⠀⢼⣿⣦⡈⠉⠙⠻⢿⡧⠷⠋⠐⠙⠛⢭⣿⣧⣏⠀⠛⠋⠈⠸⠀⠀⠀⣿⣿
⠸⣿⡇⠁⠀⢀⠃⠱⢰⠃⠀⠘⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⠃⠀⣀⣤⢠⣾⡇⠀⠀⣿⡿
⠀⣿⡇⠀⠀⠙⡄⠀⣾⠀⠀⢸⣿⠃⠶⠶⠒⣲⡀⠀⠀⠔⠶⠶⠶⡌⢿⣿⣼⡇⣿⣿⢸⠏⠁⠀⢀⢿⡇
⠀⣿⢸⠀⠀⢀⡜⡶⢻⠀⠀⢸⡏⠀⡠⢶⡗⠯⠃⠀⠀⠰⢶⡽⢎⠛⠀⣿⠟⠃⠉⠀⠀⣇⡀⠀⢸⢸⡇
⠀⢹⡏⠀⠀⠈⢤⡜⠸⡀⠈⣿⠇⠀⠀⠈⠁⠀⠀⠄⠀⠀⠀⠉⠁⠀⢀⢦⣞⠡⣴⣶⣾⣿⡇⠀⡸⣾⠁
⠀⠸⣧⠀⠀⠀⠸⡇⠀⡇⠀⠱⣇⠀⠀⠀⢀⣀⣤⣤⣤⣴⣀⠀⠀⠀⠀⡟⠁⡾⠿⠿⠛⡇⠀⠀⣧⡿
⠀⠀⢿⣷⠀⠀⠀⡇⠠⠃⠀⠈⢿⣷⣤⢖⣩⣭⣥⢶⣾⣭⣬⣙⠦⣠⣼⡟⠀⠀⠀⠀⣀⣧⡆⢸⣿⠇
⠀⠀⠸⣿⣇⠀⠃⢧⠀⢠⠀⠐⣾⣿⣾⣿⣿⡉⢹⣶⣿⡉⣹⣿⣿⣿⣟⣿⣀⢰⣿⣿⣿⡟⣧⣿⡿
⠀⠀⠀⢿⣿⡄⠈⠃⢦⢸⠀⠀⢼⣧⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡇⣷⠾⠭⡥⢤⣷⣾⣼⠇
⠀⠀⠀⠈⢿⣳⠀⠀⠀⣨⡄⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣼⣿⣤⠤⢧⣼⣿⣿⡟
⠀⠀⠀⠀⠘⣿⣧⠀⠀⢹⠀⡂⠨⡏⠀⠈⣷⠀⡂⢸⠂⢐⠀⡇⢐⡦⣖⠀⡂⢸⡀⠀⢈⢛⡝
⠀⠀⠀⠀⠀⠘⢿⣳⡀⠻⠤⠴⠖⡴⠾⠤⠶⠤⠷⢶⠤⠶⢴⡷⠶⠦⡶⠶⠦⠼⠃⣰⣳⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⡧⠄⠘⡇⠠⠄⢸⠀⠿⢀⡇⠨⡤⠆⠀⠀⢀⣴⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠛⠒⠛⠛⠚⣷⣾⣿⣶⠛⠓⠛⠃⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣄⣀⣀⠀⠛⠋⢁⣀⣀⣤⣴⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⣿⣿⡿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠂⠒⠶⠶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⡏⠀⢸⣿⣿⣿⣿⡿⢹⣿⣿⣦⢀⣀⣠⣤⣶⣦⣤⣶⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⢸⣿⣿⢽⡿⢀⣈⡻⣿⣿⣿⣿⢿⡛⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠿⠀⢸⣿⣧⣀⣙⣻⣛⢳⡎⠙⣿⡏⠀⢠⣴⣾⣿⣿⣿⣿⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠧⠀⣿⠇⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠋⠙⣴⣿⠃⠒⠋⠨⣽⣻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡿⠋⠠⠀⢲⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠘⠛⠀⡠⠔⠁⣴⣷⣶⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⡿⠀⠀⠀⡠⠊⠁⠀⠀⣿⣿⣿⣿⣿⡿⡿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢀⠔⠁⠀⠀⣀⣠⣶⣿⣿⣿⣿⣷⡄⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠊⠀⢀⣴⣿⣿⢿⣿⣿⣿⣿⡜⠿⣿⣦⡀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⣴⠃⢀⣴⠿⠋⠡⣾⠘⣧⣿⣿⣿⣿⡄⣈⠙⢳⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⠉⢿⡿⠁⠀⠖⣴⣿⣦⣝⣿⢻⣿⣏⣿⡏⢀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⣸⣷⣄⣠⣾⣿⣿⣿⣿⣿⣷⣽⣿⣿⠀⣿⣿⠛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠸⠀⢀⣧⣼⣿⠀⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⡆⣼⠀⠀⠀⣰⠞⣿⣿⠛⠟⠃⠸⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣿⠀⢀⠀⠀⠸⠋⡿⠀⢀⣷⢶⣶⠄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣤⢸⡇⠀⠀⠀⣠⣶⣿⠇⢾⡀⣔⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣧⠀⠀⠘⣿⠟⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠿⠿⠿⠿⠸⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⡀⠠⢄⡀
⠀⠀⠀⠀⠀⠐⠒⠲⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢨⠋⠀⠀⠀⠈⠆⠀⠠⠚⠁⠀⠀⡄
⠀⠀⠀⠐⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⠄⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⡄⠈⠀⠀⠀⠀⣀⠂
⠀⠀⠀⠀⠀⣑⣾⣿⠿⠗⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣿⡄⡀⢠⣤⠔⠊⠁
⠀⠀⠀⠀⢠⡀⠀⠀⠀⠲⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⣸⣧⠀⣾⠵⠆
⠀⠀⠀⠀⠈⠣⣤⠀⣀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠈⢡⣾⠇⢀⡶
⠀⠀⠀⠀⠀⡰⣏⣄⢹⡿⠋⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢀⡼⠁⣼⠃
⠀⠀⠀⠀⠀⠉⠉⢻⠈⢀⠴⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠟⠉⣈⡷⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠽⢁⠀⠀⠀⠻⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡇⠺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠈⡗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠴⠿⠉⣶⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⢠⣴⣧⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠟⢻⡿⠛⠘⠁⠀⠀⠀⠀⠀⠸⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢂⣸⣿⣿⡇⠀⠀⠀⠀⠙⠛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣼⠟⢋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠁⢀⡏⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⠁⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠈⠓⠁⠈⢿⣶⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠶⠶⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡈⢐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢐⠂⢈⡠⣌⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢅⠘⠿⠿⣦⣕⡆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⣬⡐⠤⣀⠈⠻⢶⢿⡖⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⢸⣿⣄⠀⠁⠐⠠⠠⠔⠹⣟⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠺⣿⡄⠀⠀⠀⠀⠀⠀⠀⠑⠲⣽⢷⣦⡖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⢀⡀⠤⢐⣢⣍⣤⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣄⣀⣠⡤⠵⠚⠛⠛⠠⣭⣤⣶⣾⣿⣿⠿⠛⠋⠉⠉⠋⠙⠖⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠉⠉⠉⠀⠀⢀⣠⣉⣠⣤⣼⣿⣿⡟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢦⣤⡾⠶⠶⠶⠿⠛⠛⠛⢫⡀⠀⣺⣿⣿⠆⠀⠀⣀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡼⠉⠁⠀⠀⣼⠏⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⢉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⣸⣿⣠⠊⠀⠀⠀⣠⣴⡿⢳⡄⣼⡿⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣼⢁⠀⠀⢀⣤⡤⢼⣿⠀⣴⣿⢁⣿⡏⠀⣿⣿⠟⢉⣿⠀⢰⣿⠟⣠⠞⠀⡿⠷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⠞⠋⢻⣇⢠⣿⡏⠀⣿⣿⠀⣸⠿⢻⣿⠁⢀⣿⡿⣦⣼⡏⢀⣿⣿⠋⠁⣠⠞⠀⢰⣿⠇⡠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣠⡿⠁⢸⣿⡇⢠⣿⡏⡠⠁⠀⣾⣿⣠⠚⣿⣇⡼⠋⠀⠈⠿⣿⠤⠞⠁⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠞⣸⠃⠀⠸⣿⠷⠋⠿⠋⠀⠀⠀⠙⠋⠁⠀⠉⠁⢀⣀⠀⠀⠀⢀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠉⠁⠀⣴⣿⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⣸⣿⣯⠔⠁⠀⠀⣴⣾⠷⣆⠀⢠⣿⡖⠒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⣀⠴⣿⣿⠅⣴⡦⠀⣸⣿⡇⠀⣿⠀⣼⣿⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡅⢸⣿⡟⢸⣿⠁⣠⣿⣟⢷⣠⡧⠈⣿⡟⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢸⣿⡇⣼⣿⠇⣾⣿⠜⠁⠻⢿⡶⠟⠀⢸⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⣼⡿⠃⠻⠟⠀⠀⠁⠀⠀⠀⠀⠀⠐⠊⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣤⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⠿⢿⣟⡙⠿⡿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣛⣻⣿⣿⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⢀⡜⠁⠀⠀⣠⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠙⠉⠀⠀⠀⡴⠁⠀⠀⠀⠁⢀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⣵⣶⡶⠂⠤⣄⣠⡼⠁⣠⠞⢁⠀⠀⡈⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⣿⡟⠀⠀⡀⠀⢉⣴⡖⠁⠊⠉⣀⠀⢉⢀⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠀⣠⣾⢃⣠⡾⢹⡇⠀⠈⠉⠉⠀⠈⠛⢿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡿⣷⠿⢻⣿⣿⠋⠀⠈⢀⣀⠀⠀⠀⣠⠖⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⣿⡙⣅⡀⠸⠘⣿⢠⡎⠀⠻⠿⠗⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡌⢳⡘⢿⣷⣄⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⣦⠃⢸⢿⣿⣵⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡴⠋⣴⣡⣾⠟⠁⢸⡟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⣿⠀⢿⡀⠀⣸⡿⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠐⣿⡄⢀⣽⣟⠧⣄⢵⣏⠁⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡿⠋⠀⠈⢣⢹⣧⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⢰⠀⠀⣼⠉⢀⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠆⠁⠀⢼⣏⠀⠾⡆⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠉⠳⠀⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠈⠉⠙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢼⠁⠀⠀⢹⡄⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠋⠉⠒⠲⢶⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠈⢷⡄⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣛⡻⢿⣿⣿⣿⣿⣦⣄⠀⠀⠀⣴⣿⣿⣷⡀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⠻⣿⣿⣿⣿⣷⣄⠀⠘⣿⣿⣿⣷⡀⢀⣾⣧⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢯⣿⣿⣿⠟⠉⠀⠉⠻⣿⣿⣿⣿⣿⣷⣌⢿⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡿⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡎⢿⣿⣿⣻⣿⣇⠉⠉⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣸⣿⢧⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣟⢿⣿⡜⣿⣿⣧⢿⣿⡄⠀⠀⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣿⡿⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣮⢻⣿⡽⣿⣿⡼⣿⣷⠀⠀⠀⢻⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣽⣿⣿⣿⣷⢿⣿⡆⠀⠀⠀⠹⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣿⣿⠃⣿⣿⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠹⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣝⢿⣿⣿⣿⣄⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⣠⡖⠋⡉⠑⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⣿⡽⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣧⠀⠀⠀⠀⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⢿⣿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣧⢿⣿⣷⢿⣿⣷⡀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⣴⣿⣿⣿⣧⣿⡾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠸⣿⡞⣿⣿⣸⣿⣿⣿⡆⢀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⣿⣿⣿⡼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡽⣿⣯⣿⣿⣿⡇⠀⠈⠙⠓⠚⠛⠉⠀⠀⣿⣿⣿⣿⢻⣿⣿⡽⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣿⣯⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣮⣿⣿⣷⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠁⢸⠃⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⡟⣠⢠⡇⠢⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡏⢼⣆⣄⠀⠀⠚⣀
⠀⠀⠀⠀⠀⠀⠀⠁⣠⠜⠛⠀⠉⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠂⠎⠙⠟⣸⣷⠶⢖⠷⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⢠⠄⠀⠀⠒⠃⠀⠀⣈⣟⣷⠃⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⢨⡙⠂⡖⠀⠠⠄⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⣘⢹⡗⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡇⠀⢀⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠀⡀⣱⣞⡵⣳⡌⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⢺⣾⡷⠺⠁⠀⠛⡸⠟⡈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠐⠒⠀⠐⠀⠀⠀⠁⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⡴⣾⣿⣟⢦⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⣹⣿⣦⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⣿⣿⡗⡆
⠀⠀⠀⠀⠀⣸⠱⠏⠀⠀⢰⣾⠄⠀⠀⠀⠀⠀⢸⣿⡏⠁⠀⠀⢻⣧⠀⠀⠀⠀⠀⠘⢧⠸⠋⠙⢻⢣⣇
⠀⠀⠀⠀⠀⣿⣄⠂⠀⠀⢀⣿⡆⠀⠀⠀⠀⣤⣾⣧⢷⡄⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄
⠀⠀⠀⠀⠰⣿⣩⡦⣘⣣⢾⣿⡷⠀⠀⠀⠀⠘⢿⣿⣶⡃⠶⢢⣿⠏⠀⠀⠀⠀⠀⠀⠀⣄⠖⡠⢸⠏⢸⡇
⠀⠀⠀⢀⡴⣯⢱⡆⢀⣅⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⣯⣀⣾⡇
⠀⠀⠀⠘⢧⣬⡟⠀⢸⡟⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⡇
⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠈⢸⠛⠁
⠀⠀⠀⠀⠀⢀⣤⠶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⢷⣺⣯⣶⣄
⠀⠀⠀⠀⢠⣿⠛⣾⠿⠟⣹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢻⣧⢱⣿
⠀⠀⠀⢠⣾⢿⠗⠝⠛⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⡄⡀⡀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⠏
⠀⠀⠀⠘⠿⢿⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⢍⡁⠀⠉
⠀⠀⠀⠀⠀⠊⠀⠀⠂⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⢢⡄
⠀⠀⠀⠀⠀⣠⠞⠋⠙⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠿⠟⠉⢻⣿
⠀⠀⠀⠀⣠⣿⠀⣠⠔⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⣿⣽⢧⡷⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⢠⠻⠃⠠⠟⠻⣝⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣒⢠⣾⣿⣇
⠀⠀⠀⠈⣿⠱⢾⣦⠑⣣⣿⣄⠀⠀⠀⠀⠀⢠⣿⣠⠂⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠞⠟⣿⡆
⠀⠀⠀⠘⣟⢣⡞⠋⠀⣿⣤⠟⠀⠀⠀⠀⠀⣸⣌⠻⡀⢤⠄⡰⡿⠿⣆⠀⠀⠀⠀⠀⠀⠀⣾⣴⣶⣾⠋⠉⠁
⠀⠀⠀⢰⣿⡿⠋⠀⠀⠘⢻⣷⠀⠀⠀⠀⢸⡏⢁⣤⡯⠐⠊⢷⣉⢻⡿⠀⠀⠀⠀⠀⠀⠀⢿⠋⣋⢸⠄
⠀⠀⠀⠀⠙⠗⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠁⠻⠟⠁⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣤⣦⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⢀⠁⣸⣿⣿⠇⠀⢀⣶⣿⡛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠨⣿⠿⠃⢀⣰⣿⡻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣀⣀⣀⠀⠐⠄⣠⣶⡟⡭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣣⠾⢿⣯⣎⢰⡏⠉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠛⣄⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢴⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣦⡀⠀⠀⠀⠀⠀⢀⡄⠚⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⠉⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⡀⣠⣟⡿⠶⠶⣤⣀⣀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠐⢦⡤⠀⠰⡿⣿⣿⠆⠀⠀⠀⠀⠀⣾⣴⣿⣿⣿⣓⣀⣨⣍⣛⢿⣗⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢰⣦⡀⠸⢿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠘⢿⡋⠙⣿⣿⣆⢻⣿⣿⣶⡀⠀⠀⠀⠀⠀⢰⠀⢨⣿⢿⡏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢛⣿⣿⣿⣿⣿⡻⡄⠀⠀⠀⠀⠈⠀⠘⠻⣪⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⢸⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢾⡇⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣸⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⢠⣜⣻⣿⣿⠀⠀⠀⠀⠉⠐⣥⣀⡀⠆⠀⣹⣯⠙⣿⡿⢋⣿⣟⣽⣧⣿⡿⠀⠀⠀⠀⣃⠀⢠⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⢸⣿⣷⣆⠘⠀⠀⠀⠀⠀⠠⠜⠁⠁⠀⠀⠘⢿⣦⠈⢿⢸⠌⣿⣼⢃⣿⠀⠀⠀⠀⠀⣷⡀⢸⣿⣿⣇⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠀⠀⠀⠀⢀⠒⣀⠀⠀⠀⠀⠀⠘⣳⠀⠀⠀⡼⢿⣿⣾⠟⠀⠀⠀⠀⠀⣿⡇⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠄⢻⣿⣿⠀⠀⠀⠀⢺⣗⠚⢧⣄⠀⢀⣠⣼⡉⠀⠘⣿⣧⣞⢋⠁⠀⠀⠀⠀⠀⠀⣿⠃⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢀⣶⣿⡟⢻⠀⠀⠀⠀⢘⡻⣿⠿⠟⠀⢾⣟⠟⠃⠀⢸⣿⣿⣷⡿⠀⠀⠀⠀⠀⠀⠀⡧⠀⢠⣿⣿⡟⢹
⠀⠀⠀⠀⠀⠀⠀⢈⠈⠁⠀⢻⣿⣿⣿⡇⠀⠀⠀⢀⡛⠁⠀⠀⣾⡿⠗⠀⠀⢀⣼⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠉⢹⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⡿⠁⠀⠀⠀⠈⢧⣶⠀⣤⣭⢴⣦⣶⣿⠿⢿⣿⣿⡿⠄⠀⠀⠀⠀⠀⠀⠘⠂⠒⠲⢿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠀⢰⣿⣿⣷⣤⣤⣶⣤⣀⠀⠀⠀⠀⠀⠀⠼⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠁⠀⠸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣶⣤⣀⡀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣦⡀⠀⠀⠀⠀⣀⣤⣴⣆⠈⠀⠠⣤⣶⣤⣤⡿⠟⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⣦⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣜⢿⣿⣼⡇⢠⣴⣶⣿⣿⣿⣿⣿⣷⣮⣤⣈⣹⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢹⣾⢟⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠉⠀⠙⢿⠗⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠹⢟⣵⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠛⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣆
⠀⠀⠀⠀⠀⠘⣿⡧⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⡾⠋⣴⣶⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢴⣶⡘⢷⣄⠀⠀⠀⠀⢀⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠰⡽⠇⣼⡟⢠⣿⡇⠀⢠⣶⠄⣤⠀⣤⡄⢤⣤⠀⠀⣾⣧⠸⡦⡈⢿⡷⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠾⠿⠀⠀⢸⡇⢰⣟⠀⢽⡇⠀⣷⠀⠀⢹⡿⠆⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠀⠉⠉⠀⠉

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

⠰⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢒⠒⠒⠒⠒⠒⠒⠒⠒
⠰⠀⠀⠁⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠄⠀⢀⠄⢠⡀⠀⠨⢂⠈⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⠀⠀⠈⠁
⢸⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠐⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶
⢘⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⢿⣻⣿⢿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣿⢼⣿⣿⣿⣿⣿⣿⣿⣿⡧⣧⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡟⢛⡌⠻⢟⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⠈
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣿⠟⢰⠘⢃⡔⢳⢌⠻⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁⠀⠀⠩⣿⣿⡟⠀⢸⠀⣼⣇⠻⠈⠈⢿⣿⣟⡝⠁⠈⠻⢿⣿⣿⣿⣿⡀
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⡀⠀⠀⠸⡟⠄⠀⢄⡿⡯⢙⢷⡅⡈⢈⢻⡟⠀⠀⠀⣰⣿⣿⣿⣿⣿
⢸⠂⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄⠀⣠⢋⡠⢀⣎⣓⣨⣉⣊⣻⡤⣢⠙⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⢠
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠀⢥⠃⣿⣿⣿⣿⣿⣿⠘⡤⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡀
⢸⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠘⡿⢿⠿⣿⣿⡿⡫⠸⠃⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣭⣭⣭⣭⣭⣭⣭⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⠀⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢸⠀⠀⠀⠀⠀⡀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢀
⢸⠀⠀⠀⣸⣾⣛⠳⡀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠄⡉⣤⣴⡆
⢸⠀⠀⠀⣹⣿⣿⣿⢧⣤⣄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡀⡀⠸⢗⢼⣿⠋
⢸⠀⠀⣀⠌⠹⠿⠿⠘⠧⠼⢧⣽⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠠⠿⡉⠀⢀⠿⠋
⢸⠀⠺⠧⡄⢣⠀⠀⠐⢫⣭⣀⡀⣸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⣿⣿⡟⠀⠶⣯⡅⠂⠀⠰⠋⠀⠞⠳
⢸⠀⠤⠶⣠⣤⡤⠀⠀⠤⠈⡭⢅⠰⠻⣿⣿⣿⣟⡿⠡⣿⣿⢙⡇⢸⣷⣾⣿⣿⣿⣿⣿⡟⠁⠀⠛⠃⠀⠀⢨⣤⣶⣦⣴⠲
⢸⠀⠐⠃⠋⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⣿⣿⡿⡿⢣⣾⣿⣿⠙⠃⣿⣿⣿⣇⢿⣿⣿⡟⠀⠀⠀⠸⠣⠀⠀⠐⠁⠛⠚⠛⠂
⢸⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠋⠀⠀⠄⠀⠻⣿⢡⣿⣿⣿⣿⠀⢀⣿⣿⣿⣿⣯⢽⠟⠀⠄⠀⠀⠀⠂⠀⡐⢤⣄
⢸⠀⠀⠀⠀⠀⣾⡶⠀⠀⣠⠆⣀⡁⠲⠀⠀⠀⢻⣿⣿⣿⣿⣦⣹⣿⣿⣿⣿⠟⠀⠀⠁⠄⠸⣆⠀⡀⢸⠃⢰⣿⡐
⢸⠀⠀⠀⠀⠀⠛⠀⠠⠄⢹⣦⡎⠁⠆⡀⠂⠀⠈⠉⠻⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠰⢀⠀⠉⠠⣟⡷⢰⡄⠈⠙⠙
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠛⠀⠀⠐⠀⠀⢠⠀⠈⢻⣿⣿⠟⢁⠀⢢⠀⢀⠁⠀⠀⠀⠈⢣⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⣡⣾⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠿⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠼⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡂⡀⠠⠴⠂⠀⠸⠇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣒⡃⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⢄⣀⡀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⠐⡞⠀⠀⠀⠀⠀⠀⣀⠔⠁⡀⢀⠘⠸⣧⠄⠀⠀⠀⠀⡠⢻⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⡟⠀⠘⣿⣇⣀⣀⣠⠞⠁⠈⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠒⣂⣠⣤⣤⣤⣀⠀⠀⢀⡇⠀⠀⠟⠙⢻⡩⢶⠀⠠⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣴⣿⣿⣿⣮⣽⣿⣿⣦⡴⠋⠀⠐⣤⠀⠀⠈⠀⠸⡄⠁⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⣿⡀⠉⠉⠉⠉⠉⠀⣿⡇⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢺⡟⢠⣄⡀⠀⢀⣀⠀⢻⡧⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠚⠋⠀⠈⠛⠂⢸⠃⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢀⢀⡀⠀⠀⠸⠊⢒⠘⠐⠀⠀⠀⠀⠀⠀⠀⠃⡼⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠤⠉⠤⠔⢀⣼⣧⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⢱⢦⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣄⠀⠀⠀⣰⠊⣿⣿⣾⡸⣧⣀⠀⠙⠒⠐⠀⠀⠀⠿⢋⡤⠤⣤⣀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢿⡻⠿⠿⠟⠁⢠⣿⣿⢿⣷⢻⣿⣻⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣷⠘⣯⠐⣐⠀⢠⡟⡟⠈⢸⡟⠿⣿⡿⡀⠽⠛⢷⣦⣤⣀⠀⠒⠲⢤⣀⣠⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⣾⠋⣿⠸⡆⠘⣆⢠⡴⠏⠐⠁⠀⡸⠃⠀⠈⠻⢧⣶⣺⠆⠙⠧⡟⠉⠓⠲⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⡇⠀⢸⠀⢳⡀⠈⠀⠀⠀⢠⠄⢀⠁⠘⠀⠀⠀⠀⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡧⠀⠈⠃⠀⠀⠀⢀⣠⠖⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⢻⡄⠀⠤⣀⠀⠒⠋⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⠀⠉⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠸⣷⣴⢄⡀⠀⠀⢹⣷⡀⠀⡾⡀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣀⠀⢙⣿⣦⣤⢳⡀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢿⣿⣷⣷⣶⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⣧⢀⣿⠉⠻⢿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢹⣾⡇⠀⠀⠀⠙⠹⣇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿⡁⡘⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⣇⡇⠀⠀⠀⠀⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄⢿⡃⠀⠀⠀⠀⠀⠈⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠙⣄⣸⡟⠙⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢨⡿⣀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣾⢣⡟⢦⡀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠉⣾⡇⠀⠙⢆⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠘⢷⡄⠀⠀⢀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠂⠀⠈⣿⣶⣤⠀⢣⡀⠈⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠳⣦⡀⢸⣿⣴⣂⡀⠑⢮⡟⠁⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⡱⣄⡀⠀⠈⠙⢿⣿⣿⡟⣀⠂⠈⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢠⡏⠀⠀⠙⠷⣤⡀⢀⣿⣿⠀⠘⢧⣤⣀⣀⣀⣠⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢬⣌⡷⢶⣄⡀⠀⠘⣿⣿⡟⣠⡆⢸⣿⣿⣿⣭⣭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⡿⢿⣿⣿⣿⣿⠿⣱⠟⡰⠟⠀⠻⣟⠿⡍⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡠⠛⠉⣿⣿⣯⡞⠁⠸⠁⠀⠀⠀⠈⠓⠒⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⣿⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠠⣤⣀⣉⡙⠛⠛⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⣌⣿⡋⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣭⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠿⠿⠿⠿⠿⠿⢭⣽⣓⠂⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣾⣛⣛⣛⣛⣛⡿⢶⣦⣍⡛⠶⣄⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣭⣙⣛⡛⠻⠿⣿⣿⣶⣝⠻⢷⣌⠻⣄⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠉⠛⠿⣿⣷⣦⣍⠻⣿⣷⡤⡝⠳⡌⢧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⢉⠻⢿⣷⣼⣿⣿⡹⣆⢢⡀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠀⠀⠀⠸⣷⣄⠙⣿⣿⣿⡇⢿⣆⢳⡰⡰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣦⠸⣿⣿⡇⢸⣿⡄⣷⣷⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡿⡄⣿⣿⠇⣿⣿⡇⣿⣿⡇⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⡟⣸⣿⣿⢣⣿⡿⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣼⣿⣿⢫⣿⡿⢁⣾⢏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⣵⡿⠋⣠⠟⢡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢫⣾⢟⡔⢈⣵⢢⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢡⣿⣏⣾⠃⣾⡇⡟⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠘⢿⡘⣿⣦⣿⡇⠁⠀⣩⣰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠘⣯⣁⠹⣿⣿⣷⡀⢒⡬⠥⢞⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢦⠻⣷⡙⢿⣷⣌⡛⢿⣿⣶⣽⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡤⠈⣷⣮⡓⠀⢌⡛⢿⣦⠹⣯⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢜⣼⡿⣿⣦⡠⣺⢷⠙⡧⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠙⢛⣭⣾⣿⡿⠃⠐⢚⣼⡇⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠈⠀⠀⠀⠀⠀⠰⡟⠋⠉⠁⠀⠀⠀⠴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠞⣄⣡⠤⡠⣴⢀⣄⡤⡰⢓⣄⡤⠀⢀⣤⡤⣠⣤⢄⣤⡞⠁⠀⣠⢋⣤⡤⡠⡆⣰⣫⣠⢌⣤⣆⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠙⠊⠁⠙⠈⠉⠃⠛⠈⠽⠃⠀⠈⠈⠃⠉⠙⠈⠉⠃⠀⢠⡻⠉⠉⠋⠙⠉⠉⠉⠙⠈⠋⠉⠉⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣤⣄⣀
⠀⠀⠀⠻⣝⠮⣝⡲⢤⣀
⠀⠀⠀⠀⠈⠳⣄⠙⠲⢬⣙⠦⣄
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠓⠦⣝⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠲⢭⡲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠉⠓⠿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠐⠂⠀⠈⠙⡦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠈⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⡇⠀⠉⠙⠒⠦⠤⠤⠤⠴⠒⠋⢀⡠⠖⠚⠛⠓⠲⣄⠐⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡤⠤⠒⠛⢤⡀⠀⠀⠀⠀⠀⠀⠀⡐⢀⡞⠀⠀⠀⠀⠀⠀⠈⢳⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⠛⢷⣤⡀⠀⠡⠘⣇⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠉⠻⣷⢤⡀⠈⠳⢤⣀⣀⣀⠴⠚⠁⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡓⠦⣤⣀⣀⣀⣠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣠⠴⠒⠛⠓⠲⢤⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⠀⡼⠁⠀⠀⠀⠀⠀⠀⠱⡄⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡗⡀⢷⡀⠀⠀⠀⠀⠀⠀⣰⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠑⠦⣄⣀⣀⡤⠖⠁⢐⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠀⠀⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠟⠟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣛⠻⠻⣿⣿⣿⣿⠀⠀⠀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⣿⣿⣿⢰⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣾⣿⣿⣿⠀⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⣿⣿⢜⠮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠼⢹⣿⣿⣿⠀⠀⠀⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⡇
⣿⣿⣿⠈⠀⠈⠪⣻⢿⡿⠿⠿⣿⠿⡻⠟⢁⠀⣿⣿⣿⣿⠀⠀⢸⠈⢄⠈⠢⣀⡀⢀⣀⣀⠀⣀⠴⠊⡀⠃⡇
⣿⣿⣿⡠⡀⣬⠆⠀⠉⠉⠉⠉⠉⠁⠀⠴⡅⢠⢻⣿⣿⣿⠀⠀⠈⢄⢰⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣖⢠⠃
⣿⣿⣿⣿⡵⢄⠠⠜⠾⠄⠰⠀⡴⠞⡤⢄⣠⢳⣿⣿⣿⣿⠀⠀⠀⠈⣦⡠⠴⡱⡗⡄⠐⠀⠶⠊⠆⢄⣰⠃
⣿⣿⣿⣿⡹⠁⢀⣈⠺⣟⠀⢀⡿⢛⣀⠀⠙⠞⣿⣿⣿⣿⠀⠀⠀⢠⠏⢄⣀⣙⢿⣇⠤⣔⣿⣋⣨⡠⠘⣇
⣿⣿⣿⣿⡃⠀⠀⢇⠀⠈⢳⡊⠀⢀⡄⠀⠀⢃⣿⣿⣿⣿⠀⠀⠀⢸⠁⠀⠈⣄⠀⠙⣷⠋⠀⢀⡗⠀⠀⣿
⣿⣿⣿⣿⡳⠶⣪⡩⡋⢷⡿⣿⠞⣹⢩⣔⠶⢦⣿⣿⣿⣿⠀⠀⠀⠘⠶⠖⠂⢼⠰⣶⠾⢷⡞⣻⠙⠺⠷⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣮⣖⢌⡡⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣙⡡⠚⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠻⠥⠵⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣘⢤⣬⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠘⠛⠁⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠁⢂⣠⡔⠀⠀⠻⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⡿⢷⡤⢀⣀⡴⠂⠀⠉⠓⠠⢄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⢵⣶⣿⠥⠤⠀⠀⢄⠀⠀⠈⠉⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣽⠉⠀⠀⠺⠂⠆⡄⠈⠀⢁⠀⠀⠀⠐⣀⢀⠐⠁⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⢉⠐⠀⠀⠀⠀⢤⠀⠁⠀⠀⠀⠈⠢⠄⠀⠀⠙⢄⠀⠀⣴⢳⣀
⠀⠀⠀⠀⠀⢠⠖⡙⠀⠤⢀⠀⠐⡳⠀⢧⠀⠀⠀⠀⢀⡀⠈⠢⡀⠀⠉⠲⡤⡄⢩⣆
⠀⢀⣀⠀⠀⠁⠀⠀⠐⠂⠀⠀⠉⠛⢦⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⢃⠘⠄⢻⡆
⠀⠈⡿⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠖⣮⠐⣤⠼⡧
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠚⢻⡟⠁⠀⠁⡼⠁
⠀⢸⠀⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⡄⠀⠀⠀⢠⠟⠀⡀⡠⠂
⠀⣾⠀⡥⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠻⢄⡀⠀⢰⠃⣠⠖⡹⣧
⢀⣿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠱⠤⠁⠀⡀⢘⠃⢹⣆⡀
⢸⡇⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠄⠁⠀⠀⢛⡀⠘⠳⡈⠁⠀⠈⡁⠈⠉⠙⢦
⣼⡇⠰⠀⡆⣠⠤⠤⠤⠤⠄⠀⠀⡀⠤⠴⠖⢩⠀⠀⣀⣀⠴⠛⢧⣀⡰⠚⠀⠉⠉⠀⠀⠀⠀⠀⠳⣦⡀
⠀⢅⣀⢀⡌⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠉⠁⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡀
⠀⠀⠸⠋⢲⡷⢿⣿⣿⣿⣿⣿⣿⣿⠟⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣆⡀
⠀⠀⠀⢟⡰⢰⣻⣿⣿⣿⣿⣿⠏⣴⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠹⣟⣿⣧⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⡟⢲⢠⣤⣄⡀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿⣶⣶⣯⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢸⣿⡏⠛⠛⠁⠛⠛⠛⠛⠛⠛⠛⠛⢳⡆⠛⠛⠛⠛⠋⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⢸⣿⡇⢀⡾⢧⣄⠀⠀⠀⠀⠀⠀⠀⠈⠃⡆⠀⠀⠀⠀⠀⣀⣀⣀⣿⡏⢻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⢸⣿⡇⠸⠧⠒⠛⠓⠒⣶⣶⣶⣶⣶⣶⣄⣻⣲⠖⠿⢯⣿⣟⣿⣛⣿⠀⠀⠙⢿⣿⣿⣷⣀⡴⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⢩⢿⡇⣷⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⡿⠆
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⢸⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠛⠉⢀⣠⠖
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢑⠂⠉⠉⠉⠹⣃⣹⣿⣿⣿⣿⣿⣿⣿⣏⣿⠀⠀⠀⠀⠈⣿⠿⠛⠉⣱⣤⣴⣾⠟⠁
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⠿⠟⠛⣉⡠⢿⣶⣶⣠⣤⠚⠁⡄⢀⣼⣿⠿⠏
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⣿⣇⡠⡄⠒⠈⠋⠘⠄⠒⠈⠀⠍⢀⠰⡟⠉⠉⠁⡌⡀
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣶⣞⣿⠀⠀⠀⠀⠀⢐⢸⢇⡐⠈⠆⠂⠈⢀⡠⠀⢀⣀⣤⣾⡷⢇⣒⣂⠠⠚
⠀⠀⠀⠀⠀⠉⠉⠀⠀⣀⡠⠽⠟⠛⠟⠀⠀⠀⠀⠀⠀⢂⣁⣌⣹⡟⣓⣦⡀⠀⢀⠠⠄⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⣰⡏⠁⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠁⠈⠉⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣤⣄⣈⡉⠃⠀⠀⠀⠀⠐⠂⠀⠀⠈⠉⢀⣀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⢿⣿⣷⣶⣾⣿⣿⣿⡿⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⢂⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠡⠀⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢶⣄⡀
⠀⠀⠀⠀⠠⠐⠚⠋⠓⠂⠀⠀⠀⠀⠀⠀⠀⡀⠂⢉⠀⠀⠀⡲⡀⠀⠀⠀⠀⠀⠠⠆⢀⡦⣤⣴⣴⢊⣶⣤⡾⡀⠀⠀⠀⠀⠀⡠⢦⡴⠤⠄⠀⠛⢿⣦⡀
⠀⠀⠀⣰⣦⣦⣀⣤⣠⣄⣄⣄⡀⠀⠀⠀⠀⠁⠖⠚⠉⠓⠛⠁⠈⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠀⣸⠉⡜⠂⠠⠤⠀⠡⡞⣙⠇
⠀⠀⠀⣡⡉⠈⠉⠉⠈⠉⠈⠀⠃⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⠀⠀⢀⣤⠋⠀⠀⠀⠀⢹⡕⠀⠀⠀⠀⠀⠀⠘⢫
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⠒⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⣀
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠸⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⡄⣈⡆⠀⠀⠀⠀⠀⠀⠀⠉⣉⡁⠘⢉⢣⡀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠠⢾⡿⣷⡄
⠀⠀⠀⡆⡟⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⣌⠗⠋⠈⠛⠋⠛⠏⣹⠀⠀⠀⠀⠀⠀⣾⣷⠶⠛⠷⠗⠚⠷⣇⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⢐⠏⡁⢿⡇
⠀⠀⠀⢿⠂⠀⠀⠀⠀⠀⠀⢵⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⢸⡿⠟⠓⠚⠛⠋⠁⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

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


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

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⣀⡀
⠀⠀⠐⠁⠉⢄⠀⠀⢰⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⣤⣿⣭⠁⠀⠀⠀⣀⣡
⠀⠀⠘⠦⠠⠓⠀⠀⠺⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠁⠀⠀⡄⠀⠀⠀⠀⢹⣝⠋⠀⠀⠀⠀⠉⠃⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠈⠉⢱⣶⡾⠁⠀⠀⠀⠀⢢⣉⠀⠀⠀⠀⠀⠆⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⡷⠆⠰⢾⡄⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠠⠄⢳⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣥⠒⠒⣬⡞⠀⠀⠀⠀⢠⠭⡄⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣷⡆⠸⢛⠀⠀⠀⠀⠀⣴⣶⢠⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣧⡄⠰⢞⠀⠀⠀⠀⠀⠋⣿⣏⠀⠀⠀⠀⠀⠐⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⢀⣿⣿⣿⡟⢛⡆⠀⠀⠀⠀⣇⡄⢰⠾⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡏⡅⢠⣾⠀⠀⠀⠀⠀⣠⣿⣿⠀⠀⠀⠀⠀⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⡴⠏⠁⢨⠴⢦⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠇⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣞⠻⣿⣿⡏⡀⠀⠀⠀⠁⡇⠀⠀⢸⠈⠀⠀⠀⠀⠸⠆⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠷⢆⣠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠐⠈⠉⠀⠀⠀⢀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⢀⡀⠀⠀⠀⣷⠆⠤⠃
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⢀⢊⢠
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠈⠦
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠂⠀⠀⠐⠂⠡⠤⠀⠣⠐⠀⠀⠀⠀⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠕⢾⣿⣿⡿⣫⡿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⢰⣤⣄⣤⣴⣾⣿⣿⣿⣿⢿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡍⠙⠛⠛⠋⣈⠉⠉⠉⠉⣹⢿⣿⡿⠃⠀⢻⣿⡿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠉⠁⢰⠶⠶⣦⢶⣮⣿⣯⡟⠈⢻⣿⣦⣄⣈⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠀⠀⠟⣿⣿⣿⣿⡇⠀⠀⣿⡿⢟⣉⡵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠸⡟⠛⠋⠍⠛⠲⠆⡤⠴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠶⠏⠀⠀⠀⢋⡀⣠⠀⠀⠐⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣀⡀⠄⠀⠀⠀⠀⠉⢀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠁⠢⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠉⠉⠉⠑⠀⠐⠒⠲⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢤⠴⠒⠶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡶⠧⠤⢤⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠀⠀⠂⠰⠆⠄⠄⠀⠠⣀⡀⠤
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡀⢻⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⡦⠀⠀⠁⢰⡆⣀
⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠴⠶⠶⠖⢢⡍⠁⢰⠁⠀⠀⠀⠀⠀⠀⢀⡤⣿⠐⣸⢁⣄⠀⠀⠀⢀⠀⡇⠶⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⠉⠈⠉⠹⡧⠀⠀⠀⣇⠒⠂⠈⠀⢸⠃⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢸⡇⠀⠀⠀⠀⡶⠉⠀⢀⠀⠀⠀⠀⠀⠘⢠⠄⠀⠀⣽⣦⣄⣀⠀⣠⠃⠀⠀⠀⠀⠀⠘⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⠀⠀⠀⠀⠀⣠⠆⠙⠓⠂⠉⠯⣤⡀⠹⠂⡀⠀⠶⣜⣁⠀⢦⣸⠆⠀⠀⡀
⠀⠀⡠⠋⠀⠀⢀⠾⠊⠋⠈⠣⡄⠀⠀⢸⠀⠀⣇⠀⠀⠀⠤⠰⠁⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠉⢦⡆⠈⠳⡎⠰⠅⠀⠂⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠊⠄⢠⠋⠀⠀⠀⠀⠀⠰⠀⠀⠘⡇⠀⣿⠀⠀⠀⠦⣼⠀⣀⠀⠀⠤⠀⣀⣀⢬⡹⠀⠸⢌⢫⡄⡾⢠⠓⠀⠤⠀⡀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠴
⠀⠀⠆⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡇⠀⢣⠀⠀⠀⠀⠡⠀⢳⣤⠀⠀⠲⢶⠏⡘⠀⠀⠀⠈⢦⡙⣤⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡬⠀⢀⡆
⠀⠀⠀⠀⣦⣼⡄⠀⠀⠀⠀⠀⡸⠃⠀⠀⡇⠀⣺⠀⠀⠀⠀⠀⠙⠢⡄⠐⠂⠂⡠⠖⠀⠀⠀⠀⢀⡌⡁⢬⡳⠖⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠙
⠀⠀⠀⠀⠁⠀⠉⠠⡀⠀⣀⡴⠁⠀⠀⠀⡑⣿⣫⣄⠀⠀⠀⠀⠀⡀⠏⠀⠀⠘⢦⡀⠀⠀⠀⢸⠋⢘⡿⠢⣴⡜⠁⠀⠉⠙⠛⠶⠤⣀⣠⠊⠉⠁⠐
⠀⠀⠀⠀⡈⠁⠈⡀⡏⠉⠁⢠⠀⠀⠀⠀⣧⡀⢩⡄⣰⢄⣴⠋⠉⠃⢀⣄⡰⠀⠀⠉⠉⢃⡀⢈⡟⠉⠡⣠⣼⢓⡤⢄⡀⠀⠀⠀⠀⠀⡁⠀⠉⠀⠐⠆
⣀⡐⠀⠁⣀⠀⣀⠔⠓⠰⠆⠉⠉⠑⠶⡄⣙⠙⠡⢶⣅⠛⢷⣶⡄⠀⠀⠅⠠⠀⠀⠀⠀⠁⠸⠋⠀⠀⡜⠁⠀⠀⠀⠀⠀⠋⠰⠂⡄⠈⠣⣤⠲⠶⠀⠐
⣉⣈⣁⣀⡀⣀⣡⠤⠤⠖⠘⠓⠒⠒⠒⠢⡙⢄⡀⠈⠘⡣⣀⣩⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡰⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡷⠊⠀⠻⡑⠉
⠀⠀⠀⠈⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠻⢯⡄⢀⠸⢇⠀⠀⠀⠀⠀⠀⠀⣸⢣⣤⢠⡴⠁⠀⠀⠀⠁⠴⠀⠀⠀⠀⠀⡮⠀⠀⠀⠀⢐⢹⡷
⣀⣀⣄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢸⠤⠤⠤⠤⢾⠀⠀⠀⠀⠀⠀⠀⡇⢺⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠁⠀⠠⣇⡀⠀⠀⠀⠠⠘⠁
⠀⠀⡌⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡯⠄⠀⠀⠀⣺⠀⠀⠀⠘⠀⠀⠀⡃⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠈⠀⠀⣰⠆
⣀⠀⣐⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡆⢳⠒⠒⠀⠉⢹⠂⠀⠤⠤⠤⠀⠤⠁⠀⠀⠈⠉⠒⠦⣄⡀⠀⠀⠀⠀⠀⠘⠁⠈⢠⠤⠚⠁⠀⠘⠃
⠀⠠⡴⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠤⠀⣀⢀⠀⡈⠠⠉⠉⠈⠁⠂⢉⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢠⣄⠀⣀⠀⠀⠀⠀⠀⠒⠁
⣀⠀⠸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⡇⠀⠀⢀⡀⠀⠀⠈⡏⠀⠠⣀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⣡⣈⡐⠇⡀
⠉⠀⠀⠙⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠤⠀⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠢⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠔
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠹⡄⣀⣀⣀⢀⡀⣀⠠⠤⠤⣤⠖⠂⠁⠠⠄⠀⢻⠀⠀⠀⢸⣷⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡄⠂⠁
⠀⠁⠀⠀⠀⠁⠀⠀⢩⠙⢤⣀⡀⠀⣠⠞⠁⡇⠀⢀⣀⠀⠀⠾⠀⠀⠀⢰⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠈⣄⠀⠀⢸⠀⢀⣀⣀⡏⠀⢀⣀⡠⢤⡇⠀⠀⠀⢺⠀⡇⠀⠀⠀⢛⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠞⠻⠀⠈⠁⠉⡟⠛⠈⠁⠀⢀⡇⠀⠀⠀⢸⠀⢰⠀⠀⠀⢸⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣤⠤⠤⢶⠒⠒⠚⢉⠭⡀⠀⠀⠀⣇⣠⠋⠁⠂⡄⠀⠙⠀⠀⠀⠀⣏⠀⠸⡆⠀⠀⠘⡇⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠠⠀⠁
⠈⠀⠀⠀⠀⠀⠑⠄⠀⣏⠀⠘⠀⠁⠀⠁⠁⠀⠀⢀⠇⠀⣿⢀⣀⣠⣰⡇⠀⠀⢿⡀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣈⣉⣀⣀⠤⠖⢁⣤⠎⠀⠀⠀⠀⢸⡇⠀⢠⡞⠉⠁⠀⠈⣳⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠛⠛⠋⠉⠉⠉⠁⠀⠸⠶⠒⢤⠒⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠋⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣻⣛⣛⣟⣻⣻⣿⣿⠛⣿⣟⠛⢛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⣠⠔⠋⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠀⠀⢀⡤⠊⠁⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⢀⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⡷⠋⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠖⢉⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠴⠶⠞⠋⢸⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢁⡤⠶⠒⠲⣤⡀⠀⢸⡶⡏⠈⡇
⠀⠀⠀⠀⠀⠀⢠⡟⠀⣰⠋⠀⠀⠀⠀⠀⢿⠀⡟⠀⡗⠚⢣⡀
⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⠀⠀⠀⠀⢀⡟⢸⠃⠀⡟⠀⣈⣉⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠘⣧⠈⠣⡀⠀⢀⣀⡴⠋⣰⠏⠀⢰⢠⡞⠉⠉⠙⢷⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⣉⣉⣡⠴⠛⠁⠀⠀⡎⢸⡇⠀⠀⠀⠀⢷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢠⡇⠘⣧⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠙⣦⡀⠀⣀⡾⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⠙⠛⠉⢀⣾⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣛⣲⣒⣛⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡋⡥⣽⣷⣿⣿⡟⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠐⣾⣿⡿⠿⠿⣍⣁⠂⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⢆⡼⠛⠉⠁⠀⠀⠀⠙⢷⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣟⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⡎⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⣄⣧⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢻⡿⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢸⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⠸⡻⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣜⠿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣯⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣟⠹⣿⣍⣣⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⣳⣼⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⣿⣟⣍⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢍⢿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣇⡙⢶⣄⣠⡤⠴⠷⠦⢤⣀⣴⠶⣣⢻⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢠⡈⢙⠿⠿⠛⠉⠈⠛⠻⠿⡋⠋⣀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡟⣿⡦⠀⠒⡀⠄⠉⠤⠀⠒⠀⢾⡿⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⡧⠸⣿⠆⠐⠂⠀⠌⠁⠄⡐⠂⠠⣾⡟⢌⣿⣿⣿⣿
⠀⠀⠀⠀⠈⣷⠈⢠⡀⠀⠀⠉⠭⠁⠀⠀⣠⡌⠡⣟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡠⠉⣤⡀⠀⠀⠩⠍⠁⠀⠀⣠⠈⢠⢼⣿⣿⣿⣿
⠀⠀⠀⠠⣾⡇⢰⢀⠉⠓⣴⡞⠀⢳⡄⠈⢉⠀⡄⢺⣦⠄⠀⠀⢸⣿⣿⣿⡛⢵⠂⡄⢈⠉⠀⣴⠃⠘⢦⠄⠋⢁⢰⠀⣶⢛⣿⣿⣿
⠀⠀⠀⠀⠘⠻⣆⠀⢜⠀⠀⣤⣤⣄⠀⠐⡅⢀⣶⠟⠁⠀⠀⠀⢸⣿⣿⣿⣿⣦⣕⢆⠀⠕⠀⢀⣤⣤⠄⠀⢘⠄⠀⢞⣵⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⢳⡄⠇⠀⠶⣰⠆⠀⠟⣠⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣮⡣⠘⠃⠀⢖⡰⠆⠀⠇⢐⣵⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⡖⡴⣿⠦⠾⣴⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣛⣻⣻⣜⠒⣾⣷⡶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⠘⢿⣿⡿⢂⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣭⣯⣭⠉⣽⣭⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣩⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣑⡠⡡⣒⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣹⣿⣿⠋⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠟⠏⠀⠀⠀⠀⠀⣹⣿⣿⠋⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠟⠏
⠉⠀⠁⠀⠀⠀⠀⠀⠴⣾⣿⣯⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠴⣾⣿⣯⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⣀⣴⣄⡀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⣀⣴⣄⡀
⠀⠀⢠⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠙⣿⠿⠁⠀⠀⠀⠀⢠⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠙⡿⠿⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⣶⣦⠄⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⣶⣦⠄⠀⠀⠀⠀⠀⢠⣶⡀
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⢘⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⣤⣶⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠘⠋⠙⠀⠀⠀⠀⠀⣤⣴⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠘⠋
⢀⡀⠀⠀⠀⠿⠿⠏⠀⠀⠀⠀⠀⢹⣿⣿⡖⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠿⠿⠏⠀⠀⠀⠀⠀⢹⣿⣿⠖⠀⠀⠀⠀⠀⠀⣠
⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⢘⣿⣿⡶⠀⠀⠀⠀⣸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⢸⣿⣿⡶
⠀⠁⠀⠀⠀⠀⠀⠀⢐⣿⣶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋
⠀⠀⢀⣄⠀⠀⠀⠀⠛⠻⠟⠛⠀⠀⠀⢀⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠛⠻⠟⠛⠁⠀⠀⢀⣀⣶⡄
⠀⠲⣾⣿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢶⣾⣇⡀⠀⠀⠀⠲⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢶⣾⣧⡀
⠀⠀⠿⠋⠙⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠻⡏⠁⠀⠀⠀⠀⠿⠋⠙⠀⠀⠀⣠⣶⡀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠛⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡏⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡏⠀⠀⠀⠀⢀⣤
⣀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⢤⣴⣄⡀⠀⠀⠀⢀⣀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⢤⣴⣄⡄
⣹⣿⣿⠋⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠟⠏⠀⠀⠀⠀⠀⣹⣿⣿⠋⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠿⠏
⠉⠀⠁⠀⠀⠀⠀⠀⠰⣾⣿⣯⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠴⣾⣿⣯⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣹⣿⣿⠇⠀⠀⠀⠀⣀⣴⣄⡀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣸⣿⣿⠇⠀⠀⠀⠀⣀⣴⣄⡀
⠀⠀⢠⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠙⣿⠿⠁⠀⠀⠀⠀⢠⣿⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠙⣿⠿⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⣶⣦⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠠⣶⣦⠀⠀⠀⠀⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⠀⢀
⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⢘⣿⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⣤⣶⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠘⠋⠙⠀⠀⠀⠀⠀⣤⣶⣆⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠘⠋
⢀⡀⠀⠀⠀⠿⠿⠋⠁⠀⠀⠀⠀⢹⣿⣿⡖⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠿⠿⠏⠁⠀⠀⠀⠀⢹⣿⣿⡖⠀⠀⠀⠀⠀⠀⣤
⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⢸⣿⣿⡶⠀⠀⠀⠀⣹⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⢸⣿⣿⡶
⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⢰⣿⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠋
⠀⠀⢀⣄⠀⠀⠀⠀⠛⠻⠟⠛⠁⠀⠀⢀⣀⣶⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠛⠻⠟⠛⠁⠀⠀⢀⣀⣶⡄
⠀⠲⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢶⣾⣧⡀⠀⠀⠀⠲⣾⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⢶⣾⣧⡀
⠀⠀⠿⠋⠉⠀⠀⢀⣠⣶⣀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠛⠻⡏⠁⠀⠀⠀⠀⠿⠋⠙⠀⠀⢀⣠⣶⣀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠛⠻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡏⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡏⠀⠀⠀⠀⢀⣤
⣀⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡏⠀⠀⠀⠀⢤⣴⣤⡄⠀⠀⠀⢀⣄⣴⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⢤⣴⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠛⠉⠀⢀⠀⣀⡤⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣤⣴⠞⠋⠁⠀⢀⣤⢞⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣋⣵⡾⠟⠋⡉⠁⠀⠀⣠⣴⡶⠿⠛⠛⠛⣛⣻⡟⠛⠛⠛⠉⠀⢀⠄⢀⣴⢟⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⣾⠟⠁⠀⠀⠀⠀⢀⣴⡿⠋⣁⡤⠀⠀⠈⠉⠉⠀⢀⡠⠀⠀⠀⡰⠋⣠⡟⣡⡞
⠀⠀⠀⠀⠀⢀⣾⠏⠀⢠⡟⠁⠀⡀⠀⠀⠀⣠⡿⠉⣠⠞⠁⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⡼⠁⣼⠏⣾⡟
⠀⠀⠀⠀⢠⣿⠃⠀⢠⡟⠀⢀⡜⠁⠀⠀⣰⠏⢠⠞⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡼⠁⣼⠏⣸⢸⣇
⠀⠀⠀⠀⣾⠃⠈⠀⣼⠁⠀⡞⠀⠀⠀⠀⠏⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣸⡏⠀⡟⠘⣿⡀
⠀⠀⠀⢸⡿⠀⠇⠀⡏⠀⣸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⠀⠀⡇⠀⢿⣧
⠀⠀⠀⢸⡇⠐⠀⠀⡀⠀⡇⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠃⠀⠀⠃⠀⠸⣿⡄
⠀⠀⠀⢸⣧⠀⣀⠀⡇⠀⠁⠀⢠⡿⠀⠀⢀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠈⣿⡀⢹⡀⢻⡀⠀⠀⢸⡇⡌⢠⡏⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣰⠇⠀⠀⠀⠀⠀⡇⠀⣿⣿⡇
⠀⠀⠀⠀⠸⣷⠈⣧⠘⣧⠠⡆⢸⡇⡇⣾⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣼⠏⠀⠀⠀⠀⠀⠀⠁⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠹⣷⡘⣧⠘⣧⣿⡘⣿⣧⡏⠀⢠⡟⠀⠀⠀⣠⠀⠀⣠⠞⣡⣴⠟⠁⠀⠀⠀⠀⠀⢀⡆⠀⣼⢏⡇⣿⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣜⣷⡘⢿⣷⡹⣿⣇⠀⣾⠁⢀⣴⠟⠁⢠⣾⣵⡾⠛⠁⠀⠀⠀⠀⠀⠀⢠⡾⠀⣸⠟⢸⢱⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣿⣿⣿⣿⣼⣇⣴⢟⣡⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣼⠋⠀⠀⣾⢿⡇
⠀⠀⠀⠀⠀⠀⢠⡴⠶⠶⢿⣿⣿⣥⡭⠙⣿⢿⣿⣿⣭⣤⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠞⠁⠀⢀⣼⠏⣸⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠾⢫⣿⣇⣾⠛⠇⣈⣿⡿⠿⣶⣤⣤⣤⣤⣤⣤⠤⠀⠐⢉⠀⠀⠀⣠⠞⢁⣴⡿⠁⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⢿⣦⣿⣿⢿⣷⣶⣤⣤⣄⣀⣀⣀⣤⣤⣶⠾⢋⣡⠴⠋⣁⣴⠟⠋⡰⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣹⠿⣿⣾⣿⠳⢦⣭⣉⣉⣉⠉⠉⣀⣀⡀⠁⠀⠀⠛⠉⠀⠀⠊⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠙⠿⣦⣔⡭⢍⣉⣙⡉⠉⠉⢀⣀⣀⡤⠴⠊⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣬⣭⣍⣭⣤⣤⣴⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠒⠲⠶⠶⠶⠦⣤⣰⣿⣀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⣀⣤⡴⠶⢶⡞⠉⠁⠀⣀⣀⣀⣀⣤⠶⠶⠶⢤⣿⠫⢿⣧⣄⡤⢴⣴⠟⠁
⠰⠶⠶⠾⣟⠉⠀⠀⠀⠸⣷⠀⠀⠘⠻⠟⢉⣩⡤⠶⠶⠀⣸⣿⣶⣶⣿⣟⣿⠿⢙⡇
⠀⠀⠀⠀⣿⠀⠀⠀⣄⠀⢻⡾⢷⣤⣀⡈⠋⠁⠀⠀⠀⢠⣿⣧⣿⣹⣿⠟⠁⠀⠋⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⣿⠀⠀⢰⣯⣿⡼⠃⠀⠈⠉⠉⠛⠶⣦⣄⠴⣿⣿⣸⣿⠟⠃⣶⣀⡀⠀⢀⣀⣶⡿⠟⣟⣀⣠⡼⠥⠝⠛⠻⠿⣷⣦⣄
⠀⠀⠀⣠⡴⣶⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠟⣩⡀⠈⢯⣿⣿⣶⣿⡟⣉⣴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢷⣦⡀
⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⣿⣟⣀⠘⣿⣇⠀⢠⣬⣿⡿⢁⣾⡿⢃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣾⣯⠿⠛⢻⣧⠙⢿⣳⣄⣉⣉⣴⡾⢟⣷⣿⣄⡄⠀⠀⠀⠀⠀⢠⣿⡄⡦⣀⣴⣯⣿⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣹⠟⠉⠀⣠⠴⢿⣿⡆⢀⣉⠛⣿⣇⣤⣴⣿⣿⣿⡟⢠⣿⣾⣰⣾⣷⡾⡿⣿⣷⣿⣿⣿⡎⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠁⠀⠀⠐⠆⣠⣾⣽⣿⠙⣿⣿⣿⣿⠟⢠⡟⠁⢼⠆⢸⠿⣿⡏⠙⠀⠀⠀⣠⣿⣿⣟⣽⡷⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢹⣿⡧⢾⡄⠀⠀⠈⠁⠀⣉⡾⢠⣏⠀⠀⣼⠏⠁⠀⢠⣄⣠⣿⠿⢿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⡼⠙⣚⢿⠿⢷⣾⣿⣆⠘⠁⠀⠀⠀⡀⠀⡤⠀⢿⣿⡛⠋⠀⢈⣉⡡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣯⡀⠀⠀⠀⠀⢀⣤⠞⠋⠈⠉⠀⠀⠛⣷⣿⣿⣿⠿⣏⠀⣠⡤⢀⣾⡇⣸⢇⣾⣻⠇⠀⢠⣤⣤⠍⠁
⠀⠀⠀⠀⠀⢀⣠⣤⣴⠛⠁⠀⠈⠻⣆⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⡾⠉⠻⠿⢿⡿⣿⣿⣷⡿⢫⣼⣿⡿⠿⠋⠀⠀⢨⡿⡇
⠀⠀⠐⠷⠚⠋⠁⠀⢹⣇⠀⠀⠀⠀⣿⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠶⣶⡀⠙⠶⢾⠛⠋⠛⠛⠋⠀⠀⠀⠀⢀⣸⣁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢠⣦⢿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⠁⠀⠀⠀⠀⠉⠻⣦⣈⣧⡴⢶⡀⠀⠀⠀⣶⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣤⠟⠋⠁⠀⠀⠀⠀⠀⢀⣠⡿⠈⠛⠛⠛⠛⠓⠶⠶⣤⣈⣿⣿⡁⠀⠙⢦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⣠⡆⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠒⠛⠉⠉⣉⣛⣿⠀⠀⠀⠈
⠀⠀⠀⣠⣤⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠰⣦⣤⣤⣤⡴⠶⠶⠖⠛⠓⠲⠶⠯⣭⣛⢳⡆
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴⣶⣶⠶⠶⠶⠶⠶⠶⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉

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

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

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

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⡀⢀⣀⣄⠀⢀⢀⡀⢀⡀⠀⠀⣀⣀⢀⡀⣀⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⢾⢦⠔⠘⠀⠾⡟⠛⠿⠛⣦⠀⠀⠀⠀⠀⠀⠞⠻⠿⠟⣻⡧⠙⠶⢦⣦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢺⡆⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣸⡟⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⡩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠈⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⠄⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⢠⠖⢉⣭⠉⠯⣭⣖⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣿⣧⡈⠀⠁⠀⣽⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⡀⣀⣀⣀⣀⠀⠀⣿⣿⣧⠀⣠⣷⡞⠓⠉⠉⠱⣶⣿⢱⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⠧⡅⢹⠀⡏⣿⠀⠀⣿⣿⣿⣶⡷⢟⣿⣉⣩⣭⣭⣯⣩⣚⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠛⠃⠘⠀⠛⠋⠀⠀⣿⣿⡟⠀⠙⣿⢟⣻⣿⣿⡟⠛⠻⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⢸⣿⠟⠋⠀⢰⣾⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⠈⢀⣠⣾⡇⠘⠛⢸⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣤⣄⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠲⠶⠿⠿⠿⠿⠿⠷⠶⠞⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⠀⣾⣿⠀⢸⣿⣶⡄⢰⣶⣶⡄⣶⡶⠶⢰⣶⣶⡆⢠⣶⢶⡆⢰⣶⣾⡗⢠⣿⢿⡆⢸⣿⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡿⠀⣿⣿⡇⢸⣯⣿⠇⢸⣯⣿⠇⣿⣧⡀⢸⣿⣼⡇⠘⢿⣎⠋⢸⣿⣽⡇⢸⣿⢸⡇⢸⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿⢀⣿⣽⡇⢸⡟⣿⡇⢸⡟⣿⡇⣿⡏⠁⢸⣿⢹⣗⢀⣌⢿⣧⢸⣿⣻⣇⢸⣿⢸⡇⢸⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⢸⡿⢹⡇⢸⠗⠿⠇⠸⠿⠿⠃⠿⠿⠷⠸⠿⠸⠿⠈⠿⠿⠏⠠⠿⠿⠇⠘⢿⣾⠇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⣀⣀⣀⣤⣤⡤⠤⠤⣶⣶⡶⠶⠶⢶⠶⠶⢶⣶⡶⢶⠶⢶⠦⠤⠤⠤⣤⣤⣄⣀⣀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⢠⣴⣿⡿⠃⠀⣠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⠸⠟⠁⣠⡄⢸⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡀⡰⣿⣷⣶⣿⣿⣧⣼⣾⣿⣦⠀⠀⠀⣤⢠⢤⡀⢤⡄⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣷⣶⣯⣭⣉⣉⡩⡭⣭⣤⣾⠇⠀⠀⣿⢘⣺⠇⣼⠁⣟⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠘⣟⡷⠤⠤⠤⠼⢿⣿⡞⡏⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⡟⣿⣿⠀⠀⢄⣀⢙⣯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠈⠺⢽⣗⣄⣉⡭⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣰⣦⣀⢀⣀⣀⠀⢀⡀⠒⠻⢿⣷⣶⠿⠛⢛⣁⡀⠀⣀⣀⣀⣀⣴⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⠙⢻⣿⢟⣽⡷⢋⣴⣾⣿⣿⣷⣶⡀⠘⠃⢰⣷⣿⣿⣿⣷⣎⠙⢷⣮⡻⣿⠟⠉⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡄⢣⣾⣿⠃⠈⢉⣩⣝⣛⠿⢻⣷⡀⢠⣿⠿⢟⣛⣩⣌⠀⠀⠘⣿⣷⡙⣰⡦⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠁⣾⣿⡁⠀⣐⣚⣿⣿⣿⣿⣦⣹⣷⣿⢣⣾⣿⣿⣿⣿⣶⣂⠀⢘⣿⣧⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡖⣼⣿⠋⠀⣸⠿⣯⣭⣛⣿⡿⣏⣁⣉⣉⣈⡹⢿⣿⣫⣭⣿⠻⣇⠀⠹⣿⣇⢲⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⣿⠏⢈⡭⠅⣀⠔⣡⠄⠻⢌⠉⡟⡞⢉⡉⠋⠛⣁⡉⢧⢯⠉⡱⡇⢠⣜⡢⣀⠲⢭⡁⠹⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢃⣴⠏⠀⢈⣵⣿⣿⠀⠿⢯⣾⡇⠃⠘⡏⠀⠀⣿⠁⠞⢸⣷⡽⠿⢀⣿⣿⣮⡁⠀⢻⣶⡙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢡⣿⡏⠀⠀⢸⣿⣿⣿⣿⣣⣿⣿⣷⡀⠀⠻⣴⡶⠋⠀⣠⣾⣿⣿⢸⣿⣿⣿⣿⡁⠀⠀⢻⣿⡈⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣡⣿⣿⡃⠀⠀⠸⢿⣿⣿⣿⢸⣿⣿⠿⠛⠀⢤⡈⢡⡄⠈⠙⢿⣧⣿⠹⣿⣿⣿⠟⠇⠀⠀⢸⣿⣷⣄⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣧⠀⠀⠀⢀⡿⠿⢋⡘⣿⣃⣤⣤⡴⠿⠃⠘⠷⠦⢤⣤⣘⡿⢀⡙⠿⢯⣀⠀⠀⢀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡦⠀⠀⠈⣴⣾⣿⣷⠘⣿⣥⠀⣦⣶⣶⣾⣶⣼⠀⣬⣿⢁⣿⣿⣷⣦⠈⠀⠀⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⣸⣿⣿⣿⣿⣄⡟⣿⣦⡏⠀⠸⠁⠈⢹⣼⡟⢿⡈⣿⣿⣿⣿⣆⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀⡿⠸⣿⣿⣿⣿⣿⡏⡿⠀⠀⠀⠀⠀⠈⣿⢹⣿⣧⣿⣿⠿⠃⢹⣠⣾⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡙⠿⣿⣇⠀⠀⠈⠻⣿⡟⣇⡇⠀⠀⠀⠀⠀⠀⠹⣼⣹⣿⠟⠁⠀⠀⣸⣿⠟⣃⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣌⠻⣶⡄⠀⠀⢹⡇⣿⣷⠀⠀⠀⠀⠀⢀⡷⣯⣿⡏⠀⠀⢴⣾⠏⣰⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣆⢹⣿⡀⠀⢸⢇⡟⠉⡀⠀⠀⠀⠀⡈⠑⢻⠹⡇⠀⢘⣿⠇⣼⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣝⠛⠄⠈⠘⢷⡀⠟⠶⢾⡶⠞⠛⢠⡟⠀⠀⠐⢛⣣⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣧⡀⠀⠈⢦⣅⠳⣦⣤⣤⣴⢇⣡⠞⠀⠀⣀⡴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⢿⣷⣌⣿⣯⣵⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡙⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠤⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠚⠉⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⠁⠀⠀⣈⣁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⢿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⡤⣦⡀
⠀⠀⠛⠁⠀⠈⢷⡀⠀⢠⣾⣄⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠈⢷⣶⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⢀⣿⣷⣦⣄⡀
⠀⠀⢀⣴⣧⡀⠀⠀⢻⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣄⡀
⠀⣴⣿⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠹⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⡀⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠙⣦⣀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣂⠒⠁
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁⠀⠀⠘⠿⠇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⣩⣶⣿⡆
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠉⠀⠰⠾⠿⣿⡿⠉⡁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢹⣿⡀
⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⣿⠃
⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⣤⠶⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⣠⣶⣿⣿⡿⠃⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⠀⢀⣴⣿⣿⣿⣿⠟⠀⠀⠀⡐⠄⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢰⠈⢂⠀⠀⣠⣾⡟⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⣾⣧⠈⠻⣿⣿⠟⠀⠀⠀⠀⠀⢰⡃⠀⠀⢣⣼⣿⡟⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⢀⣴⣿⣿⣿⣿⣧⡀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣇⢀⣴⣿⣿⢏⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠗⠄⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⠟⠁⠈⠑⢄⡀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠈⠠⣤⣀⠀⠀⠀⣾⣿⣿⡿⣇⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣜⢿⠟⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⠉⠛⢳⠀⠀⠀⠈⠳⣄
⠀⠀⢠⡞⠁⠀⠀⠀⠀⢸⣿⣀⣨⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣀⣴⣿⠿⣥⣿⣷⣦⣀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠙⠃⠈⠻⠿⠿⠿⠷⣦⣄⡀⠀⠈⠙⠒⠲⠤⠼⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⣠⠞⠋⠉⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠉⠻⣦⡀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡾⠁⠀⠀⢀⣼⡇⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣷⣶⣦⣀⠀⠀⠀⢿⣦⡀⠀⠀⠈⣷⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣯⡀⣀⣴⣿⠟⠀⡤⢴⣾⣿⣿⣿⡿⡿⢿⢻⡿⣿⣿⣿⣷⣶⢦⡄⠙⠿⣶⣤⣴⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠋⠀⠀⠀⡇⢸⠈⡇⢿⢸⠀⡇⢸⠘⡇⡇⢸⠀⡇⢻⢸⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠃⠘⠀⠇⠠⠸⠀⠇⢸⠀⠇⡇⠸⠀⠇⠐⠘⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢷⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⣧⠀⠀⠀⠀⠀⠀⣸⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠘⣇⠀⠀⠀⠀⣰⠇⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠸⣆⠀⠀⢠⠏⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠹⡄⢠⡏⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢻⡟⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⡾⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣼⠁⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣠⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠛⠋⢰⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡾⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡴⣿⡁⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣾⠁⠀⢀⡴⠋⠀⠈⠻⣆⠀⠀⢙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠛⢦⠟⠁⠀⠀⠀⠀⠈⢳⡴⠋⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢀⣤⢞⣩⠤⠤⢤⣀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣀⣤⣤⣬⣛⢦⡀
⠀⠀⠀⠀⢠⡞⡵⠋⠀⠀⠀⠀⠈⢷⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⡞⠁⠀⠀⠀⠉⢳⡝⣦
⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⣼⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡆⠀⠀⠀⠀⠀⠀⣿⢸⡆
⠀⠀⠀⠀⠸⡆⢳⣄⠀⠀⠀⠀⢠⡇⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣷⠀⠀⠀⠀⠀⣠⠏⣼
⠀⠀⠀⠀⠀⠹⣦⡙⠳⢤⣤⠴⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠳⣤⣤⡤⠞⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠴⠖⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠻⠿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡾⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⢡⡆⠀⠓⠾⣻⣿⣿⣯⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣱⣿⣻⣿⡿⠆⠊⢱⡏⣿⣧
⠀⠀⠀⠀⠀⠀⣿⡏⢸⡇⠀⠀⠀⠻⡞⢿⣿⣇⠈⠛⠷⣶⡶⣶⠶⠶⠶⢶⠶⠶⣶⢶⣶⠾⠛⠁⣰⣿⡿⣻⠾⠁⠀⠀⢸⡇⣼⣿
⠀⠀⠀⠀⠀⢀⣿⡇⢸⡷⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠠⡿⢠⡦⣤⡶⢤⢆⡿⠈⠄⢀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⢸⡇⢹⣿
⠀⠀⠀⠀⠀⢸⣿⡇⢸⣧⢀⠀⠀⠀⠀⠀⠈⢿⡣⠄⠐⢻⠛⣿⢧⠑⠀⢱⠛⡞⢳⠟⡶⡶⢠⣿⡟⠁⠀⠀⠀⠀⠀⣀⣼⡇⢸⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣇⢸⣿⣎⣑⢆⣰⠀⠀⣒⠈⢧⡐⠀⠨⡰⢃⠖⠀⠀⠀⠀⠹⡐⢃⠀⢀⢢⡿⢁⠀⠀⠀⣀⠐⢛⣹⣿⡇⢼⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣟⣾⣿⣿⠿⠾⠛⠓⠶⠟⣤⣊⢻⣄⢔⠔⢻⣷⣄⠀⠀⣠⣾⠟⣆⢣⣠⡟⣁⢤⡤⠶⠶⠛⠻⠿⣿⣿⣷⣸⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣏⣁⠋⡂⠁⢀⠨⣣⢝⣆⠀⠀⠀⢋⠁⠈⠀⠋⠀⠈⣷⣯⡄⠀⠀⠀⠊⠐⢉⣻⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠛⠍⠱⠁⠆⣠⠖⠃⠘⢿⣄⢠⡀⢨⣥⣤⡆⢤⣀⣸⡿⠁⠘⢷⣄⠠⠀⠀⠰⡛⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣸⣿⣿⣿⠟⠋⠀⠀⣁⣀⣠⡴⠞⠁⠀⠀⣴⠟⠉⠀⠀⠂⠀⠀⠀⠀⠀⠉⠹⣆⠀⠀⠉⠓⢦⣄⣀⣀⠀⠀⠉⠻⣿⣿⣿⣧
⠀⠀⠀⢠⣿⣿⢛⣥⠶⠾⠛⠛⠋⠉⠁⣀⠀⠀⠀⣜⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⣇⠀⠀⠀⣀⠉⠉⠙⠛⠻⠶⠶⣤⡹⣿⣿⡆
⠀⠀⠀⣼⡿⠷⠋⠀⢀⣀⠤⠄⢒⣂⣤⣤⣤⣬⣭⣭⣄⡂⠨⡁⠐⡦⠀⠂⠠⡀⠀⠄⢘⣢⣭⣭⣯⣭⣬⣥⣘⡒⠁⠄⣀⡀⠀⠙⠛⢿⣷
⠀⠀⢰⡟⠤⢀⣴⠞⠋⠁⣠⠾⠛⠉⠉⢉⣛⣿⣿⣿⣿⣿⣷⡜⡾⠿⠀⢀⣐⢦⢣⣾⣿⣿⣿⣿⣿⡛⠉⠉⠛⡛⠷⣦⡈⠙⠳⣦⡀⠀⢻⡆
⠀⠀⣼⣥⡾⢛⠁⠄⣑⣚⢖⣲⣦⣉⠘⠻⢿⣿⣜⣋⣿⣿⣿⡇⡟⠰⣶⣌⠙⠪⢸⣿⣿⣿⣙⣣⣾⡿⠟⠃⢀⣤⣒⣾⣛⣀⠂⡈⠻⢷⣬⣿
⠀⢠⣿⠀⡤⢁⡴⠚⠭⠽⠿⣯⣝⠯⣷⣄⠀⠙⠻⣿⡛⠛⢿⣷⠀⠀⠠⠉⠀⠀⣾⡿⠟⠛⣿⡿⠋⠀⣠⣾⠯⣻⣽⠿⠯⠭⠛⢦⡁⢐⢆⣿⡀
⠀⢸⡇⡌⢱⠏⡴⠋⠀⡀⢂⡀⠈⠳⣜⢷⣷⣄⠀⠘⢷⣄⠀⢻⣇⡀⠀⠀⠀⢸⡿⠁⢠⡾⠋⠀⢀⣾⣿⢫⡞⠉⢀⠑⠀⠆⠉⢦⡙⣏⢾⢻⡇
⠀⢸⡇⣷⣿⠈⠀⠰⠀⢤⡁⠐⠀⠀⠹⣮⣿⠙⢷⣄⠀⢻⡖⢸⣿⠁⠀⠂⠀⣿⡇⢀⡾⠁⣀⣴⠟⣹⣧⡏⠀⠀⠁⠀⠀⠀⠄⠘⠃⣹⠈⢸⡇
⠀⢸⡇⠃⢹⣇⠀⠀⠀⠀⢰⣤⣅⡈⢀⣿⣿⢸⠂⠹⣧⡘⠀⠸⠏⠃⠀⠀⠃⠿⠇⢸⡇⣴⠟⠡⠀⣿⣿⠃⢔⣡⣴⠂⠀⠀⠀⠀⣰⡏⠂⢸⡇
⠀⣾⡇⣴⠀⠻⠇⠀⠀⢉⠈⠉⠙⠿⣶⣜⣿⢏⣠⠀⠘⣷⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⣾⡏⠀⢀⠱⣟⣯⣴⡟⠋⠉⠀⠀⠈⠁⠀⠟⠁⠀⢸⡇
⠀⣿⢡⡟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣌⠀⠁⣿⠃⠀⡟⠀⠀⠀⠀⢸⠀⢸⣿⠀⠀⣨⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿
⠀⣿⣼⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⡿⠁⠀⠇⠀⠀⠀⠀⠨⠀⠸⣧⠈⣼⡟⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠉⡈⣿
⠀⣿⡇⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⡀⣿⡇⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠠⠀⠠⣤⣄⡀⡀⠀⠀⠀⠀⠀⡇⣿
⠀⣿⡇⠘⡷⠀⠀⠀⠐⠢⠀⠀⠤⠀⠀⠀⠀⠈⢿⣿⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢀⣤⠖⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠸⡇⣿
⠀⣿⡇⣃⠁⠀⠀⠀⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠰⠃⠀⠀⢠⠀⠀⠐⠀⣄⣀⠀⠀⠂⢿⣴⡟⠁⠀⠀⠈⡀⢀⡈⠁⡀⠀⠀⢀⠛⢠⢸⣿
⠀⣿⡇⢣⡀⠐⠀⡀⠀⢀⠈⠠⢀⠀⠀⠀⠲⣤⣸⡿⠀⠀⠀⢉⡍⠀⠀⠀⠀⢰⡋⠄⠀⣀⢸⣿⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⡔⠖⠀⠀⣼⢸⣿
⠀⣿⡇⠘⣧⡀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⢸⣿⡇⠐⠀⠀⠈⠀⠀⠀⠀⠀⠠⠁⠀⠀⠍⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⢸⣿
⠀⣿⡇⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡦⠀⢠⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢸⣿
⠀⣿⡇⣄⢠⣄⠙⢷⡔⠂⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠁⠀⠀⠲⠂⣀⣀⠸⠄⠀⠀⠀⠀⠄⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⡾⠋⣠⠔⣧⢸⣿
⠀⢸⡇⣿⢀⠙⣷⡀⠙⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠠⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⢁⣼⠋⠀⢿⢸⡟
⠀⢸⣧⠃⡌⣀⠈⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⠃⠀⠀⠀⠀⣠⠀⠁⠈⢠⡞⢁⡁⢀⡼⣾⡇
⠀⠀⣿⠀⣤⠘⡃⠂⠹⣆⠣⠀⠀⠀⠀⠀⠀⠈⣿⣦⠀⠀⠀⠛⢿⣿⣿⣿⣿⡿⠋⠁⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⢸⣣⡟⠀⢸⡅⠪⠀⣿⠁
⠀⠀⢻⡆⢠⠊⡉⢆⠀⢹⡇⠀⢀⠄⣄⣀⣀⠀⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣾⣿⡯⠀⠄⣁⡀⣀⣀⠄⢰⡏⠀⠠⠆⠀⠀⢸⣿
⠀⠀⢸⣧⠀⠐⠀⠀⠄⠀⠘⣦⡁⠀⠀⢤⠤⠆⢻⡟⣿⣿⣷⣤⣄⣀⣀⣀⣀⣀⣤⣾⣿⣿⣿⣇⢰⠤⢤⠄⠀⠉⣰⠛⠁⠜⠀⠀⠃⠀⣾⡇
⠀⠀⠀⢿⡇⠀⠀⠀⠘⠣⠀⠸⣷⠢⠄⠂⠱⢠⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣿⣿⡀⠀⠁⠄⠀⣼⠃⠁⡀⢂⡁⢨⠀⢸⡿
⠀⠀⠀⠘⣷⡀⠀⣉⠙⠓⢦⠀⠜⢷⣦⣀⢻⣰⣿⢻⣧⡙⢿⣛⢻⡿⠿⠿⠿⡟⣻⡽⢋⣾⡟⣿⡄⠰⢂⣠⡾⠁⠠⠀⠒⠉⢁⠀⢀⣾⠃
⠀⠀⠀⠀⠹⣷⡈⢘⣃⡀⠈⠓⠀⠈⢣⠘⢷⠈⣿⡇⣿⣷⠀⠀⠈⠗⠘⠂⠻⠉⠀⢠⣾⣿⢹⣿⠁⣴⠋⠰⡁⠀⠀⠂⠀⣀⡄⢁⣾⠏
⠀⠀⠀⠀⠀⠹⣷⡄⠉⠁⠀⣠⠇⠀⠈⢇⠈⡇⢼⡇⢹⣿⣇⢱⠀⠀⠁⠠⠀⡰⠆⣾⣿⡇⢸⣿⢠⡏⢠⠃⠀⠀⣀⡀⠀⠉⢡⣾⠏
⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⢀⠀⠀⠀⠀⠈⢧⡸⣿⡇⠈⣿⣿⡄⠁⠀⠀⢠⠈⠀⢰⣿⡿⠐⡅⣿⠈⣠⠃⠀⢀⠐⠀⡀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⡈⢷⣀⠀⠀⠀⠈⢳⣸⡇⠡⠼⣿⣿⣄⡈⠶⠾⢁⣴⣿⣿⠃⡰⢰⡏⡜⠁⠀⠄⠊⠀⣤⠁⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡌⠻⣄⠀⠀⠀⠀⢳⡎⠀⠰⠌⠛⠻⠿⠿⠿⢿⠿⠛⠁⠀⢡⡟⡘⠀⢀⠠⠔⢀⡜⢡⣾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣧⣀⣶⣀⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⢸⣿⣯⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣿⠙⠿⠿⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣶⣿⡁⣰⡎⢹⡿⢶⡖⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠶⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢯⣤⡿⡇⠛⠋⠙⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢿⣻⣧⡀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡟⢻⣿⣿⣿⣿⡿⢿⣭⣿⣿⣶⣶⣶⣿⣷⠆⠀⠀⠀⠀⣀⣠⣴⣿⡿⠁⢠⣿⡟⠹⣷⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠉⢻⣿⣿⣿⣿⣿⡿⢿⣧⣘⠉⠁⠀⠀⠀⢶⣿⣿⣿⣿⡟⠁⠀⠈⣿⣷⡴⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠉⠉⠀⣀⠄⠀⠀⠀⠙⣻⠿⠋⠀⠀⠀⠀⢹⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠉⠙⢿⣿⣧⠼⠟⠁⠈⠃⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠈⣗⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡐⠒⠿⢯⡁⠀⠀⠁⠚⠁⡰⠂⠀⠀⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢻⣶⣆⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⣸⣿⡟⢠⣿⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡷⠟⠛⣿⣿⣿⣤⣄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢹⣟⠃⢀⣾⣿⣿⣿⣿⡟⠛⠃⠀⢴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢀⣼⣿⡿⠙⢿⣿⣿⡟⠿⠇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢨⣿⣿⡿⠋⠀⠀⠈⢿⣿⣿⡶⠀⠒⣾⠟⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠿⠟⠀⠻⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡆⠉⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠛⠛⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⢀⡀⠀⢹⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣀⠀⠀⠀⠀⣰⣿⣿⡿⠀⢨⡇⠀⠈⠻⢿⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠉⣶⣶⠀⢠⣿⣿⣿⣧⠀⢹⡃⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⣤⣤⣯⣤⣴⣄⣿⣿⣶⣾⣿⣿⣿⣿⣷⣾⣿⣶⣶⣾⣷⣤⣤⣀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⢿⣿⡿⠟⠛⠻⣿⣿⣿⣿⣿⡛⠛⠛⣛⠛⠛⠿⠿⠿⡿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⡇⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⢀⣿⣷⣦⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⣸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢰⡇⠈⢻⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡾⠀⠀⠀⢻⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠉⢻⣿⠃⠀⠀⣠⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠋⠀⣴⡿⠉⠻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⣰⣿⡁⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⢠⣿⣿⡅⣀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⢀⠀⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⡄⢹⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣇⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢻⠛⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣄⠀⢈⠙⢿⣶⣾⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⢀⣠⣿⠿⣽⣿⣝⠛⠛⢛⡉⠉⠉⠋⠉⠉⠛⠻⠶⠾⣿⣿⣾⠿⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⡿⠿⠛⠉⠛⢻⣿⣶⣿⠛⢣⠀⢠⣶⣄⡀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣹⣿⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣧⠀⠘⠻⣿⣿⣶⣤⣀⣀⠀⠀⣀⣤⣶⣿⣿⣦⣠⣾⣇⡌⢁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⠘⢿⣿⣿⣿⣿⣿⣿⠟⢻⡟⡀⢻⡍⠋⠉⠿⠿⠿⣶⣦⡑⠆
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣶⣾⣿⣿⣿⣿⣴⣿⠤⣴⣿⡯⠀⣓⠈⠀⣠⠀⠀⠀⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣶⣿⣿⣿⣟⢷⣤⡿⠃⠀⠈
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠉⠛⠛⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣴⣦⣴⣶⣶⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡏⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡟⣼⠇⣿⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣰⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⡿⠻⠿⣿⣿⡿⢿⠟⣱⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣿⢣⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣠⣿⣯⣿⣟⣋⠙⣿⠁⠚⠁⠈⣿⡿⢉⣉⣉⡽⢉⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢸⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠙⠿⣍⣀⣀⣿⣿⡇⠙⠀⠀⠀⠀⠀⣰⠭⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠸⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠼⠿⠋⣿⣿⣿⣿⣿⣿⣿⡸⡀
⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⢹⡇⠁
⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⢸⡇⠈
⠀⠀⠀⠀⠀⠀⣸⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⡞
⠀⠀⠀⠀⠀⢠⠟⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠰⢾⣿⣿⣿⣿⠿⠋⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠀⠀⠀⠤⠤⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠇⢹⣿⡟⠿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⢠⡟⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠿⠛⠛⠋⠙⣿⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⣥⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠘⠷⣬⠝⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠛⠛⠛⠉⠀⠀⠙⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠒⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢇⡠⠐⣶⡄⠀⠀⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⡿⠋⠀⠀⡸⠛⣷⣶⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠀⠀⠀⠰⠟⡹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⡰⠁⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⣀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣤⠚⠀⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡊⡉⠣⡀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠐⠋⠛⠉⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠃⢿⡀⢡⡀⠀⣤⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣇⠀⢰⢹⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀⠀⣿⣗⣼⣯⠄⢇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⡀⠀⠀⠀⢈⣷⡿⢰⠆⠃
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣠⣨⠽⣷⣶⣤⣀⠀⠀⠈⠂⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠁⠃
⠀⠀⠀⠀⠀⣠⠋⠀⠀⣰⠋⠀⠀⠀⢹⠉⠛⠻⠷⢦⣤⣀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⠃⠀⠀⢰⡏⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠙⠃⠄⠁⠀⠠⣄⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⢢
⠀⠀⠀⠀⣾⠀⡄⠀⣿⠁⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⡀⠀⠛⠦⢾⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢳⠟⠢⡀
⠀⠀⠀⢠⡇⢰⡇⡇⣿⡄⠐⠶⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⢦⡀⢠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡶⠝⠓⠒⠊⠉⠓⠚⠁
⠀⠀⠀⢸⣷⢸⣿⡇⣿⣯⠑⢺⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠟⠛⠉⠁
⠀⠀⠀⢸⣿⡎⣿⣷⠿⠀⠀⠀⠉⠻⠷
⠀⠀⠀⠘⠉⣿⣿⡌
⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣦⣦⣶⣶⣶⣶⣶⣶⣴⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠒⠢⣄⢀⣀⣀⣤⡤⢶⣶⡖⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠾⣧⣔⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣿⣶⣦⣤⣀⣀⣀⣸⠇⠀⢀⡽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣮⣰⢀⡀⠀⠀⠀⠀⠙⠉⠉⠀⠀⠀⠀⠉⠙⣿⡿⣧⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⢶⣶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⣦⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⡘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣖⡈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢃⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⡀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣖⡀⠙⣆⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣖⡀⠈⢧⠀⠀⠀⠀⣠⡿⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⣂⠈⢳⡀⠀⣴⡟⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣔⣂⠀⠳⣾⠏⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⣂⠀⠙⣆⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡷⣂⠀⠘⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠹⣷⣂⣀⣨⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡧⢦⡄⢹⡍⢀⡴⢬⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠈⠿⡭⢿⠏⠹⡜⢯⡿⠃⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢠⠋⠀⠀⠙⡄⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣀⣰⠃⠀⠀⠀⠀⠘⣆⣀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣀⡴⡞⠉⢩⠇⠀⠀⠀⠀⠀⠀⠸⡏⠙⢳⢦⣀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣠⠞⠁⢀⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢸⡆⠈⠳⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞⠁⠀⠀⢸⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠈⣇⠀⠀⠈⠳⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢁⡞⠁⠀⠀⠀⢀⡟⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⡀⠀⠀⠀⠈⢳⡈⠳⢤⡀
⠀⠀⠀⠀⠀⢀⡤⢖⣛⣛⡛⠉⠀⢠⠏⠀⠀⠀⠀⠀⡼⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢧⠀⠀⠀⠀⠀⠙⡄⠀⠉⢛⣛⣓⡲⢤⡀
⠀⠀⠀⠀⡴⣫⡾⠋⠉⠉⠙⣧⠠⡏⠀⠀⠀⠀⠀⡼⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢧⠀⠀⠀⠀⠀⢹⡄⣴⠋⠉⠉⠙⢷⣝⢦
⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⢸⠆⡇⠀⠀⠀⣀⡼⠁⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⢷⣀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠹⣯⣧
⠀⠀⠀⣯⣿⠀⠀⠀⠀⠀⢀⡾⣸⡃⢀⡴⣫⣥⣤⣤⣀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢀⣤⣤⣬⣝⢦⡀⢘⣇⢷⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢿⢻⣄⠀⠀⢀⣠⠞⣹⣅⣹⣫⡾⠋⠀⠀⠀⠹⣆⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢰⠏⠀⠀⠀⠙⢷⣝⢏⣨⣏⠳⣄⡀⠀⠀⣠⡿⡿
⠀⠀⢀⡜⣳⢭⣟⣛⣭⠵⠋⠁⢸⣷⡟⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡀⠀⠀⠀⠀⠀⢻⣿⡇⠈⠑⠮⣽⣛⣛⡭⣞⢣⡀
⢀⡴⢋⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣼⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡌⢧⡀⠀⠀⠀⠀⢸⣟⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⡙⢦
⠻⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢿⣄⣀⣀⣴⢞⡱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡳⢦⣀⣀⣠⡾⣳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣮⣭⣵⣖⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣲⣮⣭⣵⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠖⠀⡀⠐⢂⡀⠉⠏⠉⠃⠀⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠒⠄⠀⠀⠀⡠⠄⠁⠘⠁⢀⠚⠡⠂⠊⠀⠀⢠⠘⠀⢡⠀⠀⠀⠙⠲⢄
⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡌⠁⠁⠀⠀⠠⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⠀⠀⢲⣄
⠀⠀⣴⢷⣦⣤⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⢀⠌⠀⢣
⠀⠀⠃⢤⡌⠙⠻⢶⣼⣄⠀⠀⠀⠀⠀⠀⣠⠄⠃⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠘⡆
⠀⠀⠀⠀⡀⠉⠂⠄⣈⠛⠷⢶⣆⠀⠀⢻⡇⠄⠀⠀⢰⡀⠘⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⡰⠀⠀⠀⠀⠈⠉⠣⠄⡉⢧⠀⢸⣷⡈⠆⠀⠀⢃⠀⠘⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⡀⠀⠀⢹
⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢄⠘⢇⢸⡞⡖⢸⡀⠀⠀⣇⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡠⠂⠉⠉⠙⠂⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢺⡈⣇⢹⣼⠀⠄⠀⠸⢤⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠱⡅⠨⢇⠀⠘⣜⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠶⣤⣸⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⢨⠠⠤⠉⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠇⠀⠀⡎⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⢀⠀⠀⠀⠀⢨⠀⢀⠆⠀⠸⢡⠢⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠳⠀⠀⠀⢘⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⣇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠖⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⣲⠶⠒⠀⠈⠁⠀⠀⠀⣀⠴⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⢀⡠⠖⠁⠒⠊⠁⠀⠀⠀⠀⠀⠀⠔⠈⠁⠀⡸⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⠒⠒⠒⠀⠀⠒⠊
⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣀⠔⡊⠁⠀⠀⠐⠒⠒⠂⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⡏
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢄⡜
⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡉⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⢀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣀⣀⠤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡐⢿⣿⣿⣿⣿⣿⣿⡿⢂⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣧⡙⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢋⣼⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢸⣿⣿⡟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⡇⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠉⠉⠉⠉⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⡄⢠⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣮⣥⣬⣵⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉




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