Feathered Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⣠⣤⢀⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⣰⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠈⢸⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠀⠀⠊⠀⠀⡀⠰⠁⠀⠀⠀⠐⠂⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠓⠀⠀⠀⢀⣴⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠂⢰⠇⠎⠀⢀⣢⠛⠁⠁⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢢⢣⡇⠁⠀⠈⣠⣤⣾⣿⣿⠤⠘⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢾⡞⠉⢡⣤⣶⠥⠽⠷⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣶⣼⠿⠾⢿⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠉


⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠠⣄⡀
⠀⢦⠀⠀⠙⠣⡀⢱⣄⡀
⠀⠈⠈⠲⣴⡄⠀⠈⠗⠇⠀⢀
⠀⠀⠀⠆⠙⢳⣦⠀⡀⠀⠀⠘⠇
⠀⠀⠀⠀⠂⠒⠻⢶⣿⣀⠀⠀⠀⠸⣧⡀⠀⠄
⠀⠀⠀⠀⣀⣀⣀⡀⢼⣿⡆⠀⠀⠀⠈⣧⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠙⠋⠉⠛⠚⠛⠅⠀⠀⠀⠀⠉⠀⠀⠀⠀⢈⢣⡀
⠀⠀⠀⠀⠐⠤⠀⡀⠀⢈⣳⣶⣆⠐⡀⠀⠀⠀⠀⠀⠈⡌⠱⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢽⣟⣾⠀⠀⠀⠀⠀⠀⠀⠃⠀⢧⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⠀⡀⠀⠀⠀⠀⠠⠀⠀⠸⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠿⣷⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠒⢳⡻⣶⠀⣽⠀⠀⠀⠀⠀⠀⠀⠠⠀⢂⣾⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠊⠳⣿⣤⣠⡀⡀⠀⠀⠀⠀⠀⡎⠀⢹⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢮⣻⣿⣧⡇⠀⠀⢀⠀⢸⠀⠀⡞⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣇⠀⠀⣸⠀⡞⠀⢀⠀⡿⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣍⡀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣾⣣⢠⡇⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣶⣤⣴⠶⣶⠖⠻⠿⢯⣻⢷⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣝⡛⠀⠀⠀⠀⠐⠛⢉⣳⣯⣿⣧⡀⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⢁⠠⣬⣛⡲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠷⠒⠐⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡚⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣰⣿⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⢿⠿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⢯⣦⣆⣏⠀⡸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⣿⣿⠗⠁⠐⢹⠴⢶⠇
⠀⠀⠀⠀⠀⠀⠙⣶⢤⣸⣿⣧⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢻⣿⠁⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⣿⡆⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢹⣿⡤⠀⠀⠀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠻⣷⡄⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⣴⣷⠇⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠾⣿⣿⣿⣿⠀⠀⣰⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⣤⣀⠉⢻⣿⡀⠀⠙⠁⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠿⠻⢆⣱⣄⠀⠀⢠⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣦⣶⠻⢿⣿⣶⣶⡀⠀⡀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠷⠄⠛⠻⠿⠿⣿⣿⣾⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠶⠚⠋⠉

⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣶⠒⢄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⣧⣀⣀⣀⡀⢸⣦⠼⣇⠀⠀⠀⠀⠀⣎⢹⣇⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⢋⠙⣛⣉⠛⠛⣿⠀⢼⡄⠀⠀⠀⠈⢿⣿⣿⠳⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣾⣿⣿⣿⠆
⠀⠀⠀⠀⢸⣶⡀⠈⠉⠀⣽⠁⢜⢇⠀⠀⠀⠀⢿⣝⣿⡔⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣼⣿⣿⣿⣿⠁
⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⣹⣧⡃⣼⠀⠀⠀⠀⣼⠿⣿⣟⣽⠀⠀⠀⢠⢿⣿⣿⡯⠷⢚⣻⢿⣇⠀⠀⠀⣿⣯⣿⣿⣽
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠈⡿⠁⠞⡇⠀⠀⠀⣿⣷⣾⣿⣿⡇⠀⠀⣸⣿⣿⡿⠖⣛⣭⢶⣿⠟⠀⠀⢸⣿⣿⡟⣛⣏
⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢨⡿⠀⠸⡇⠀⠀⠀⣿⣽⣿⣿⣯⣿⠀⠀⣿⣟⢿⣷⣞⠽⣚⣩⡴⡇⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⠄⠙⠒⣀⠲⢷⢃⠔⠇⠀⠀⠀⠘⣿⣉⣹⣿⢟⠀⠀⣿⣮⣿⣯⡷⣛⣉⣠⣼⠃⠀⠀⢸⣿⣿⣿⣟⣿
⠀⠀⠀⠀⠀⠘⣷⣶⣵⣤⣨⡟⠀⢘⠀⠀⠀⠀⠀⣿⣻⣿⡿⣾⠀⠀⢹⣷⣽⣯⠖⣛⣯⣵⡿⠀⠀⠀⢸⠻⣿⠾⣛⣿
⠀⠀⠀⠀⠀⠀⢻⣻⣯⣌⠹⢽⢀⠈⠀⠀⠀⠀⠀⣟⣿⢿⣷⣿⠀⠀⠘⠿⣿⡿⣟⣿⣽⣿⠇⠀⠀⠀⢸⣿⣿⢿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠈⠄⠀⠀⠀⠀⢿⣟⣻⣿⣽⠁⠀⡀⢔⢮⣻⣿⣿⣿⠟⠀⠀⠀⠀⢸⣿⣿⣟⣭⣿⠂
⠀⠀⠀⠀⠀⠀⢈⠲⢬⣙⣿⡀⣄⠀⠀⠀⠀⠀⠀⢸⣛⣿⣿⠟⠀⠀⠘⢦⣑⠯⣿⠟⢋⠔⠀⠀⠀⠀⢸⣿⣿⣿⣿⠅⡀
⠀⠀⠀⠀⠀⠀⠀⢁⠒⢺⣏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠩⡏⡟⡇⠀⠀⠀⠀⠛⢻⢿⣋⠅⠘⠁⠀⠀⠀⠀⠹⢟⣻⣧⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⠴⠤⠄⠀⠀⠀⠀⠀⠈⡽⢳⡻⠀⠀⠀⠀⢀⣠⣤⣾⡖⠚⠁⠀⠀⠀⠀⠀⠀⣈⣿⡷⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⠟⣤
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣈⡄⢹⠋⠀⠟⠇
⠀⠀⠀⠀⠀⠀⠀⠺⡟⠻⢽⡾⠁⣄⠀⠀⠀⠀⠀⠀⢸⠟⠙⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢸⡂⢸⡀⠀⠀⠞⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠘⠋⠀⠀⠀⠀⠀⠀⣾⠆⠀⣼⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⡦⣽⢲⡅⠀⡄⣰⢃
⠀⠀⠀⠀⠀⠀⠈⠄⢀⣂⠀⢿⠀⢀⣿⠀⠀⠀⠀⢀⡻⣦⠀⢸⡂⢀⡶⠀⠀⠀⠀⠀⠀⠀⡅⠙⢸⣇⣨⡾⢑⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡀⢺⠀⡼⠃⠀⠀⠀⠀⠈⠻⣎⠳⣼⡖⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠟⠁⡴⠋⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠈⠀⢸⠀⠀⠀⠀⠀⠀⢌⠳⣬⡇⠀⢈⡇⠀⠀⠀⠀⠀⢀⠀⠀⣻⠤⠊⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⢡⣝⢲⡓⣾⠀⠠⢲⠀⠀⠀⠀⠀⠀⠀⠀⠬⣧⠀⣸⡇⠀⠀⠀⠀⠀⠀⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢳⡋⠓⠤⣿⠀⠠⠞⠀⠀⠀⠀⠀⣠⣀⠀⠀⣷⣴⣿⡇⠀⠀⠀⠀⠀⡄⠀⠐⡇⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⣿⠘⡠⠀⠀⠀⠀⠀⠀⠰⠈⠑⠂⣿⢛⣾⡇⠀⠀⠀⠀⠀⡃⠀⢸⠇⣀⠄⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡆⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣿⣾⣿⠃⠀⠀⠀⠀⠀⢷⣄⣸⠤⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠀⠀⠀⠀⠀⠀⠀⠀⠌⠒⠄⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠻⣿⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠂⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠉
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⡁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠈⣙⠿⣿⣿⣿⣿⣿⣿⣿⣇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢫⡷⠀⠀⠀⠀⠀⠠⠞⠂⠉
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⠇⠀⡀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣼⠀⠀⠈⠀⡔⠀⠀⠀⠠⠀⠀⠐
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠁⢹⠇⠀⠀⡌⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⠀⡜⠀⠀⠐⠈⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⢀⡏⢰⠀⠀⠀⠀⠃⠀⠀⠀⣀⡴⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⠇⠀⠃⠀⠀⢐⠀⠀⠀⠐⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⣾⠀⡀⠀⠀⠀⠉⠀⠀⠀⢀⠔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢠⣧⠀⠐⠄⠀⠀⠀⠀⠀⠔⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⠉⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⡇⠀⠙⠆⠀⠀⡆⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠰⡝⢆⠀⠀⠀⠀⣶⠀⠀⠀⠰⠁⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠀⠀⠀⠇⠈⠳⣄⠀⠀⢸⡄⠀⠀⠀⠠⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇⠀⠀⢄⠐⠀⠀⠀⠀⠀⠈⣇⠀⠀⢠⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⡁⢄⢀⠀⠀⠀⠀⢹⡀⠀⠘⠀⠀⠄⠀⠀⢄
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡤⠀⠀⠀⠹⡀⠙⠳⠦⡀⠀⠀⢣⠀⠀⠀⠀⣀⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠸⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠈⢇⠀⠀⡜⠁⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠱⡀⠄⠀⠠⠄⠀⠘⣆⠀⡠⡊⠀⠀⠑
⠀⠀⠀⠀⠀⡤⠴⠦⢤⡀⠈⣿⣿⣿⠿⠛⠋⠉⠙⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠒⠂⠠⠤⠼⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡿⠟⠑⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠐⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠰⠋

⡆
⣧
⠈
⠀⢇⠑
⠀⠈⢧⠀⠁⠂⠤⣄⡀
⠀⠀⢀⠱⢄⡀⠀⠀⠙⠦⣲⣄⡀
⠀⠀⠈⢠⠀⠓⢆⡀⠀⠀⠙⡈⠝⠃
⠀⠀⠀⠀⠙⠲⠌⠙⣶⣤⠀⣹⠀⠀⠀⠘⢦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣄⠀⠀⢀⠀⠁⠀⠈⠉⠓⠄⠀⡦⡄
⠀⠀⠀⠀⠀⠀⠢⡤⠀⠁⠀⠀⠉⠳⢶⣄⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⡀⠐⣀⢀⣀⠘⠻⣷⣦⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⡀⠀⠁⠄⡈⠛⠻⠿⢷⣶⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠁⠀⠀⠀⠀⠉⠀⠈⠙⠷⣶⠤⣀⠀⠀⠀⠀⠁⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣵⣦⡀⠀⠀⠀⠈⠑⠁⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠛⢿⣦⣄⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠶⣶⡶⢖⠒⠈⠉⠀⠀⠀⠀⠀⠚⢻⣷⣳⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠂⠀⠀⠀⠀⠐⠚⠛⠈⠙⢷⣦⡀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠥⢤⣄⣀⡀⠀⠀⠀⠀⠀⠈⠿⣦⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⢄⣀⠀⠀⠀⠀⠰⠿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠐⢚⣷⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⠥⠄⠀⢁⡉⠹⣌⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠯⠤⠍⠿⠿⠿⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣏⣷⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣾⣿⣿⣷⡀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⠘⣇⣰⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⢀⣏⣿⣧⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠸⣿⣿⣿⣿⣿⡇⢘⣿⣿⣿⣿⡏⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣌⠿⣿⣿⣿⠁⡼⣽⣿⣿⣿⢶⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣌⠻⣿⣀⣴⣿⣿⣾⣿⢸⡇⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣍⡻⠿⣯⢿⣿⣿⣿⡿⣼⠁⢸⣿⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣯⣻⣿⣾⣿⣿⣿⣷⢻⠂⣿⣿⣿⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⣿⣿⣿⣿⣿⣿⣿⣟⣰⣿⣿⡟⣸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣍⠻⢿⣿⣿⣿⣯⣿⣿⢳⣿⣳⢣⣿⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣭⣛⡿⣿⡿⣇⢻⣿⣿⢸⡇⢀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠉⠛⠻⢿⣮⡻⣿⣿⢸⡁⢸⣯⣿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣒⠂⠀⠤⠄⣁⣙⢿⣜⡇⢸⢹⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣬⣀⣀⣈⠻⢜⠻⣦⣸⡇⢸⡇⠀⠀⢸⡇⢀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⣿⠿⢿⣿⣿⣮⡓⢧⣸⡗⠀⠀⢸⡇⣼⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣍⣙⠛⠚⠿⡿⠿⣦⣙⢷⡀⠀⣾⣿⣿⣿⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣝⣿⣿⣿⣯⣭⣽⣿⣿⣷⣍⠀⢹⣿⣿⣿⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⢍⡉⠛⠛⢛⣻⣯⣽⣷⣮⡛⢿⣿⣿⢺⡇⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢶⣤⣀⣀⠀⠀⠛⠛⣋⡛⢳⣙⠿⣽⡷⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⣍⣉⣉⣭⠤⠒⠒⠺⠿⠷⣌⠳⣇⢠⡇⣤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠆⣀⠀⠉⠉⠉⠉⡉⠓⢄⡸⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡠⣼⠿⠿⣦⡙⢇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠙⠆⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠄⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⢠⠠⣤⠀⠀⣤⡤⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠈⠛⠁⣤⣶⣦⣴⣶⡄⠘⠃⢀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠻⣶⠀⠰⠀⠀⠀⠠⠀⠀⠀⢰⠟⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠂⠀⠀⠰
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⡶⠾⠟⠋⢶⡆⠀⣤⠀⠹⡟⠈⠛⠳⢶⣶⣿⣀⡀⣠⡄
⠀⠀⠀⠠⠀⠀⠀⢀⠀⢤⣤⣦⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠛⠛⢶⣤⡀
⠀⠀⠀⠀⣠⡶⠆⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠚⢻⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣁
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⢹
⠀⠀⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⢸⠀⠀⠀⠀⠀⢠⠀⠸⠿⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⣸
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⡆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢿⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣿⡷⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣿⡇⠀⣻⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢸⣿⣧⣠⠛⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢀⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⡿⣿⡇⠈⢷
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣸⣿⠀⢻⣷⡀⠸⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⣿⣿⠀⢸⣿⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⣀⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢀⢸⣿⠀⠀⣿⣿⠀⠘⣇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠘
⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⣀⡾⢸⡿⠀⠀⣿⣿⣆⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣿⠇⣼⡇⠀⠀⢹⣿⣿⡆⠀⣴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠇⠀⣸⡏⠀⣿⠁⠀⠀⢸⣿⣿⣷⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⢸⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⢠⣾⣿⠁⢰⣿⠀⠀⠀⠸⣿⣏⢻⣧⠀⠀⠇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⢠⡞⠁⢀⣿⣿⠃⠀⣿⣿⠀⠀⠀⠀⠻⣿⡈⣿⣷⠀⠰⣧⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠄
⠀⠀⠀⢸⡀⠀⠇⠀⠀⠀⠀⠀⠛⠁⢀⣾⣿⠏⠀⢠⣿⣿⠀⠀⠀⠀⠀⢸⣧⠸⣿⣆⠀⠘⣧⠀⠀⠀⠀⠀⢰
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣠⠀⠀⢠⣿⣿⡟⠀⠀⣾⡏⣿⠀⠀⠀⠀⠀⠀⢻⡆⢹⣿⣆⠀⠉⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠘⣤⠀⠀⠀⠀⢀⡛⠀⠁⣿⣿⣿⠀⣀⠀⣿⡇⠉⠀⠀⠀⠀⠀⠀⠘⣷⡀⢿⣿⣷
⠀⠀⠀⠀⠙⣦⠀⠀⠀⠸⠃⠀⣰⣿⣿⣿⣴⡟⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠈⣿⣿⡄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠈⢣⡀⠀⢀⠀⢀⣿⣿⣿⣿⡿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠘⣿⣧⡀⠀⠇
⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⣠⠈⢻⡏⠉⠉⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡈⣹⡿⢀⣀⠀⢀⣄⠐
⠀⠀⠀⠀⠀⠀⠃⠀⠙⣿⣿⣶⣤⣽⣦⣤⣶⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢿⣶⠿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⡄⢀⡘⠁⠀⠀⠈⠉⠉⠉⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠟⠀⠀⣀⣤⣤⣤⣤⣤⣀⠓⠄⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣤⠞⠃⠈⠻⣶⡀
⠀⠀⠀⠀⠀⠀⡆⢀⣾⠏⠀⠀⠀⠀⠀⢻⡆⠈⠳⡀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⢈⠀⠀⢀⡼⠃⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠆⠀⡀⠀⠀⠀⠀⣿⠀⠀⠈⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⡇⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⡀⠀⠠⠀⠰⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠉⠉⣀⣤⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠆⠀⠴⠉⠉⠁⠰⠶⠄⠀⠀⠀⠀⠀⠀⠰⠆⠀⠁⠀⠈⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠉⠈⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣧⣴⣤⣤⣴⣦⣶⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⢤⣀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⣠⡖⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠉⠙⢦⣬⣉⣛⣓⠓⢠⡀⠀⠀⡚⣒⣘⣡⠶⠛⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠓⠶⠀⠀⠊⠉⠉⠋⠙⢧⠞⠙⠉⠉⠁⠈⠑⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡄⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡀⠀⢀⠞⢉⡑⣴⣜⣧⠐⡀⠢⡀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢦⡀⠨⠢⢼⣤⣨⣴⢇⠀⣼⣦⣍⣠⠇⠔⠠⠠⢻⠃
⠀⠀⠀⠀⠀⠀⣀⣀⣠⢤⣴⣈⣷⣀⠀⠉⠶⣀⢁⠀⠉⠀⠈⠞⢏⡉⣈⡁⢀⡁⣴⠣⢺⣅⣄⠠⠠⠒⢂⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢒⠪⠽⣻⣙⠣⠔⠒⠼⠭⣭⣍⣉⡁⠀⣀⣭⣅⣬⣭⣴⣦⡗⢋⣡⣠⠤⠔⠊⠁
⠀⠀⠀⠀⢀⠄⠂⣀⠤⠶⠇⠀⠀⠈⠉⠑⠒⠂⠤⢤⡰⠿⠙⠎⠅⠙⢉⡠⠔⠾⠛⠋⠉⠁⠀⢲⣄⣁⠀⠐⠂⢀
⠀⠀⠀⠀⠀⢈⡠⠥⢚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠚⠀⠀⠒⠁⠀⠀⠀⠀⠀⠈⢦⠄⣈⡉⠂⠀⡀
⠀⠀⠀⠠⠀⡤⢒⣩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⣈⡁⠀⠁
⠀⠀⠀⠀⢁⢰⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠁
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠒⡆⠠⠤⠀⡀⠀⠄⢀⠀⠀⠀⠀⠂⠀⡀⠤⠔⠒⢉⡽⢁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⢙⡦⡀⠀⠉⠈⠙⠤⠤⡤⠴⠊⠁⠀⠀⡠⠴⠙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣬⣴⡦⣐⣦⠤⠤⢀⣀⠀⡀⣀⣠⣤⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃⠿⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠯⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⣤⠤⣤⠤⣤⣤⠤⣤⢤⣖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⠈⢳⣿⢃⠎⠀⢀⠀⠑⡄
⠀⣤⡤⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢧⠀⠳⠿⠃⢸⠁⢸⠀⠈⠶⠃⢀⡇⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢤⡄
⠈⢇⠀⠀⠀⠀⠀⠁⠐⠒⠒⠩⠵⠾⢷⣶⣤⣴⣫⣦⠄⢑⣶⣤⠶⠛⠰⠍⠛⠛⠒⠉⠉⠀⠀⠀⠀⠀⡇
⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀
⠘⢄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠤⠀⠀⠀⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⢀⣸⣻⣿⣟⢒⣤⣴⠒⣿⣏⣹⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣈⣉⣀⣀⣀⣀⣀⣉⣀⢈⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠺⠿⠿⠟⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢍⡉⠉⠉⠁⠠⠀⠀⠀⠈⠉⠛⠒⠦⠤⢤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢠⠤⠀⠂⠠⣄⠉⠠⡌⡠⣤⡤⠤⡐⠒⣉⡽⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⣴⣶⣤⠈⡀⠀⣿⡑⣁⣤⡄⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢧⠀⣾⣿⡯⠄⡅⢠⠇⠋⠻⡿⢏⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠈⠳⣤⣀⡡⠔⠁⠈⠀⠓⠤⠴⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠆⠆⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠠⡄⠀⠀⠀⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⡌⡙⠤⣴⡴⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⠆⠈⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡆⢠⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⢻⣽⣯⠟⣶⣲⣶⣶⡶⡐⠶⠛⠉⣿⠙⢤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡋⣹⠀⢗⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⣰⡁⢪⢤⣆⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⣀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠄⣁⠀⠀⠀⠀⣀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠉⠒⠤⣀⠀⣤⠀⠒⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠆⢸⡄⠄⡠⠔⠊⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣢⠀⠀⠀⠉⠃⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡀⠒⠀⠆⠀⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣠⣶⠿⠛⠋⠉⠛⠻⣶⣄⠀⠀⢀⣶⡾⠛⠋⠉⠙⠻⢿⣧⣈⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠈⢻⣆⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⠿⠻⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠞⠉⣿⡆⠀⠀⠀⠀⡄⢲⣦⠀⢠⣿⣿⣧⠀⢠⡐⣶⡄⠀⠀⠀⢀⣿⡷⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⡴⠋⢀⠔⠙⢿⣦⡀⠀⠀⠿⣿⢟⣤⠟⠋⠉⠛⢧⣜⠿⠿⠃⠀⢀⣠⣾⠿⡀⠈⢧⣀
⠀⠀⠀⣀⣀⡷⠄⣏⠉⠛⠷⢤⣎⣀⠀⠀⠙⠻⠿⠶⠶⠾⠿⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⠛⠁⠀⠑⡄⢈⣻⡵⠶⣆⣤
⠀⠀⠀⠘⡇⠉⠁⠒⠬⢍⣐⠒⠠⢄⣉⠒⠤⣀⠀⠀⠀⠀⠀⡗⢆⠀⠀⢠⠞⠇⠀⠀⠀⠀⠀⠀⡠⠤⠒⣋⡉⠤⠔⣚⣩⠷⣿⣤
⠀⠀⢸⣛⡳⠶⠒⠈⠙⢢⠀⠉⠓⠦⢄⡉⠲⢌⡙⢦⡀⠀⠀⠘⠢⠉⠎⠡⠊⠀⠀⠀⣠⠔⣂⡥⠖⢂⣉⠤⠖⠒⡝⠉⠉⠙⠳⢼⡀
⢀⠒⠉⠁⠀⠀⠀⠀⢀⠼⠦⢄⣀⠀⠀⠈⠓⢤⡙⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⢀⠔⣋⠴⢊⣡⠴⠚⢉⣀⡤⠖⢋⣱⣄⠀⠀⠀⠀⠈⠑⠂
⡈⠀⠀⢀⣀⠤⠀⠚⠻⡳⠦⣤⣈⠙⠓⠦⣤⣀⠈⠓⢷⡄⠙⢦⠀⠀⠀⢀⣔⡥⠊⡡⠶⣉⣠⡴⠞⢋⣡⡴⠞⠋⡡⠊⠑⠂⠤⣀⡀
⡗⠊⠉⠉⠀⠀⠀⠀⢀⠇⠀⠀⠈⠙⠓⠦⢤⣈⣙⠓⠶⢬⣦⣈⣧⡀⣠⠟⣁⢐⣩⠽⢦⠈⢣⡴⠚⠋⠁⠀⠀⠀⠱⣀⠀⠀⠀⠀⠉⠑⠂
⡇⠀⢀⣀⢠⡤⢴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣦⢸⡛⠒⠶⠶⠚⢿⠉⡤⠖⠋⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣱⠶⢤⣀⡀
⠸⡉⠉⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⡧⠤⣀⣀⡤⢼⠀⣷⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢠⣾⣧⡀⠀⠀⠉⠙⣲
⠀⠑⣤⣤⠴⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡧⣤⣄⣀⣤⣼⠀⢸⠀⠀⣤⡀⠀⠀⢸⣿⣿⣇⠀⢸⣿⣀⣽⡲⣤⣄⡰⠃
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⠀⢸⡆⢀⣿⣿⣦⡀⠀⣿⣯⣿⠇⠀⠙⠛⠋⠀⡇⢸⡇
⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢸⡇⢸⡇⠸⠿⠋⠛⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⡇⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⢛⣨⠇
⠀⠀⠀⠀⠳⠦⣭⣙⣛⠶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠒⠶⠶⠶⠒⡇⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢚⣫⡥⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢷⡲⠬⣭⣛⡓⠶⠤⣤⣀⡀⢸⠀⣷⠲⠤⠤⠤⠴⡇⠀⣿⠀⣀⣠⡤⠶⢚⣩⣥⠖⢋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣍⡲⢤⡀⠉⠙⠒⠲⠬⣭⣉⡀⣿⠀⠀⠀⠀⠀⡇⠀⢛⣋⣭⠴⠒⣋⡥⢞⣡⣖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠉⢏⣻⣿⣿⣶⣦⣤⣤⣀⣈⣉⣛⣦⣄⣀⣠⣤⣿⣚⣩⣥⣴⣦⣿⡷⣾⣉⠎⠉⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠒⠋⡯⠤⠤⣙⣿⡔⠉⢙⡏⠉⠉⠉⠉⠉⠉⠉⠉⣟⠉⠙⣼⣟⡡⠤⠬⡏⠓⠢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣄⣠⡜⠀⠑⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠖⠁⠘⣤⣀⡤⠃
⣿⡿⠿⣿⣿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣧⣽⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠻⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⠀⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢠⣿⣿⡇⠸⠛⠉⣉⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠈⠻⣿⡿⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣗⣀⣴⣶⡿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢋⣭⣷⡄⠀⠀⠀⠀⠀⠀⣴⣿⣿⣉⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⠟⠋⢸⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⠋⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⣀⣠⣾⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄⡀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡿⠟⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⡾⢋⣾⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⡰⠁⠇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⢀⣾⠝⠀⠀⠀⠀⠀⠀⠀⣀⣿⢟⣵⣿⣿⠗⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⣼⡷⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣽⢿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⡷⠈⠀⠔⠿⠶⠖⠀⠀⠀⠀⠀⢀⣧⣿⢫⣿⣿⡿⠁⠀⠠⠀⠀⠀⠀⢀⠠⠈⠇⡸⠁⢠⠀⠀⠀⠀⠀⠀⠀⣿⣜⣿⡿⣾⣿⡟
⠀⠀⠀⠀⠀⢸⣿⢣⠂⠤⠤⣤⠄⠀⠀⠀⠀⠀⠀⡘⣿⡏⣾⣿⠟⣡⠀⠀⠀⠀⠀⠀⠀⠀⠍⠒⠀⡀⠀⠀⣀⡄⠀⠀⠀⡀⢶⡸⣿⡿⣱⣿⣿⠅
⠀⠀⠀⠀⠀⣾⢋⣏⡤⠶⢒⡂⠀⠀⠀⠀⠀⠀⠐⢾⣿⡇⣿⣿⣿⢟⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣉⠉⠀⠀⠀⠀⣿⣾⣷⢏⣽⣿⠟⡉
⠀⠀⠀⠀⠀⠸⣼⣡⣦⣬⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠞⠛⠀⠀⠀⠀⠀⠻⣿⣣⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠉⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢄⠎⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣠⠇⣿⠄
⠀⠀⠀⠀⠀⠀⠀⣠⢠⣿⡿⣫⣾⣻⡵⠂⠀⠀⠀⠀⠀⠀⠀⠌⢠⠏⢈⡄⠀⠀⠀⠀⠀⠀⢠⣤⣿⢿⣿⣾⠁⠀⠀⠀⠀⠀⠀⣌⢷⠟⣰⣟⡀
⠀⠀⠀⠀⠀⣠⢸⣿⣿⢋⣾⣿⣿⠗⠀⠀⠀⠀⠀⢠⠀⠀⠀⡠⠋⠀⠘⠀⠀⠀⠀⠀⢀⣄⣼⣿⢏⣾⣿⡿⠀⠀⠀⠀⢠⢰⣾⡿⢫⣾⡛⠓⠋
⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⡷⠶⠀⠀⠀⠀⠀⠀⣸⠀⢀⠞⠁⣐⣋⡽⠀⠀⠀⠀⢀⣾⣻⣿⢯⣾⡟⢋⠀⠀⠀⠀⠀⢿⣧⠟⣱⣟⣒⣒⣉
⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣤⠄⠀⠀⠀⠀⠀⠀⠇⡴⢁⣠⣄⡩⠄⠀⠀⠀⠀⠀⠈⢿⣿⢣⣿⣿⣿⠍⠀⠀⠀⠀⢠⣻⢇⣾⣿⣦⠴⠖⠃
⠀⠀⠀⠀⠀⣰⠛⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣼⡇⣾⠿⠿⠅⠀⠀⠀⠀⠀⠀⠻⠸⣿⣶⠶⠂
⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠋⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣀⢀⠀⠀⠀⠴⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣷⣿⣭⡯⡤⠀⠐⠚⠁⢀⠻⢁⢂⠀⠀⠄
⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣷⣶⣦⡖⣡⣴⣠⡏⣆⡎⡼⣿⠀⢀⠰
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣾⣿⣼⡱⣿⣵⣾⣆⡇⠀⠈⠰⡀
⠀⢀⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣧⡆
⠴⠟⠋⣠⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⢠⣇⣠⡆⡄
⠀⠀⣼⠟⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡟
⠀⠈⠁⠀⣰⡿⠋⠋⣽⣿⠿⣿⣿⣿⣿⡿⣿⣿⢻⣿⢻⣿⢸⡏⠿⠙⡿⢹⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠋⠀⠀⣸⡿⠃⣸⣿⠋⣾⡟⢱⣿⠇⢺⡏⠈⠋⠀⠁⠀⠀⠀⠀⠁⠙⠹⢻⡿⣿⢿⡿⣿⠿
⠀⠀⠀⠀⠀⠀⢰⡟⠁⢰⣿⠃⢸⡿⠁⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠘⠃⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠃⠀⠘
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⢖⣿⠶⠋⠻⠿⣿⣽⣿⣭⣭⣤⣀
⠀⠀⠀⠀⠀⡰⠁⣩⣶⣬⣥⣾⣅⣀⡙⢿⣝⠿⣿⣾⣅⣀
⠀⠀⠀⠀⣰⠁⠀⠨⣭⣀⡀⢤⣥⣉⡃⣸⣷⣖⠛⢯⣿⣾⡛⠳⠦⡀
⠀⠀⠀⠀⡇⠀⠠⣿⡿⣿⠿⣧⡽⣿⣿⣿⠿⣿⣷⣾⡻⣝⣟⢿⣯
⠀⠀⠀⠀⢷⠀⢸⣿⠶⠿⢷⣶⡷⢿⣿⣷⣦⠙⣷⣟⢿⡎⣿⢦⣉⠳⣦⣄
⠀⠀⠀⠀⠸⡆⢸⣿⣿⣿⡆⣿⠿⠿⢿⣷⣶⣿⣿⠹⣷⣶⣿⣧⡹⣷⣦⣬⠕
⠀⠀⠀⠀⠀⢷⠸⣟⡭⠶⣺⣿⣟⠛⣿⣿⡜⣿⣿⣷⠈⣟⢿⣿⣶⡌⢻⣾⣄
⠀⠀⠀⠀⠀⣸⠀⣏⣁⣶⡿⣿⢿⣷⣮⣛⣱⣿⣦⣤⣴⣟⠣⠙⣿⣝⡆⢻⣿⣦⣄
⠀⠀⠀⠀⠠⡟⣡⠌⣥⣥⡀⠙⢸⣿⣿⠋⣿⢷⣌⢻⣿⣯⡁⣷⠹⣿⡭⡀⢿⣿⣽⡃
⠀⠀⠀⠀⠈⠁⠀⠀⠙⠉⠁⠀⠀⢉⣛⣸⣷⣸⣿⣷⡘⠘⠃⣼⣧⢹⣿⡂⢸⣿⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠀⣿⣿⣿⣶⣾⣟⠉⢷⣄⣁⣾⠇⢃⠉⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠟⠘⣾⣦⠙⠙⢦⢡⡟⣻⣟⢿⣆⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢺⢧⡀⠈⢣⡸⣷⢻⡖⢤⣈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣷⣌⡁⢻⣄⢹⡿⣦⣙⡀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠹⣯⡩⡙⣦⠓⢜⣯⠉⣢⡙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠻⣦⣈⢧⠘⡏⣷⣴⣤⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠽⢣⡳⣄⠛⠟⢿⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣭⣥⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣷⣶⣿⣿⠿⣶⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣦⣀⠀⠀⠀⠀⠐⢋⡭⢷⠀⠀⠐⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⡹⢃⠻⠂⠀⠀⠀⠀⣀⣀⣸⣀⠀⢠⣴⣴⠞⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠈⠁⠀⠀⠀⠀⠀⠈⠙⢻⣿⣶⣶⠟⢂⣜⣶⡴⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⣀⣀⡆⠀⠀⠀⠀⡴⣶⢿⢿⣿⣿⡿⠀⠂⠀⠀⠶⠦⡄⠀⠀⠀⣄⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣅⣨⣿⡏⠀⠀⠀⣤⣤⣀⣸⢺⣿⣿⣶⡀⠀⠀⠀⠀⢀⣀⡀⢀⣾⣿⣐⣨⣿⣇
⠀⠀⠀⠀⠀⠀⢲⠀⠀⢿⣿⣿⣿⡇⡄⠀⢠⣶⣴⣤⣼⢸⣿⣿⠀⠀⠈⠀⠀⣤⣤⠄⠀⠘⢿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⡼⠛⢯⣠⣆⠘⢿⣿⣿⠃⡋⠀⠘⠛⠛⠛⠛⢸⣿⣿⠀⠀⠀⡀⠀⠁⠀⡴⣧⠀⠀⠉⡛⠉⠉⣴⡴⣆
⠀⠀⠀⢀⡟⠶⣅⣁⣿⣀⣀⡹⠇⢠⠟⠳⠘⠓⠚⠒⠖⠘⠿⡷⠒⠒⢺⠗⠒⠀⣴⣀⣁⠀⠀⠀⠙⢦⠚⠛⡂⣐⣧
⠀⠀⣠⣶⣷⣴⠋⠉⠙⣤⠀⠀⠀⢨⠗⢺⠀⢴⣶⣶⠀⠀⢀⡀⠀⣀⣤⣤⡀⠀⠉⢿⡋⣀⡀⠶⠶⢾⠶⠶⣿⣟⠉⢅
⠀⠀⣿⣿⣿⠋⠙⠛⠛⡿⠒⠒⠐⢿⡶⣾⠀⠀⣿⠇⠀⠀⣾⣯⡿⣿⣿⣿⠁⠀⠾⣿⣿⡟⠈⠀⠀⣸⠀⢸⣿⣷⣶⡾
⠀⠀⠸⣿⣿⣆⡀⠀⠀⢧⣄⣀⣠⣼⣿⣞⠀⠀⠉⠁⠀⢹⣿⣯⡏⠉⠉⠉⠀⠀⢀⣿⣿⣿⣶⠀⠀⠞⠀⠈⢛⣿⡿
⠀⠀⠀⢽⣿⣿⡿⣿⠀⠀⠻⣿⣿⣿⣿⡿⠀⠀⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠀⠀⣻⣝⣿⡿⠀⠀⣆⠀⢠⣾⣿⠇
⠀⠀⠀⠈⡈⣿⠦⣞⠀⢈⠀⠈⠛⣟⣫⠅⠀⠀⠸⠿⠿⠿⢹⣏⣿⣿⣿⡟⠀⠀⠀⠛⢟⣿⠀⠀⠀⠈⠻⣿⡋⢛
⠀⠀⠀⠀⠉⡧⣤⡏⠀⠠⠀⠰⠺⣅⣨⡇⠀⠀⢠⣤⣤⣠⢸⡗⣀⣀⣀⡀⠀⠀⠀⢲⡶⢿⠄⠀⠀⢰⡶⡿⣷⣶
⠀⠀⠀⠀⢹⣧⣤⠗⠆⠐⠀⠰⢾⡋⢹⠀⠀⠀⠠⣿⣿⣿⣼⡇⣿⣿⣿⡇⠀⠀⠀⢸⣦⣄⡄⠀⠀⢸⣇⣠⣼⣗
⠀⠀⠀⠀⠸⣅⣨⠷⠂⠐⠀⠤⣔⠉⠓⠀⠀⠀⠀⠛⠛⠛⠓⠓⠛⠛⠛⠁⠀⠀⠀⠀⠫⣿⠁⠀⠀⠈⠙⣿⠅⠉
⠀⠀⠀⠀⠀⢉⣹⠴⠄⠀⢀⣤⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⠀⠀⠀⡟⠟⢷⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠾⠄⠀⠀⢧⣴⡼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⣿⢯⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⣷⠘⣼⣿⣾⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢳⡄⢹⢀⣿⣿⣿⠟⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣷⡌⠳⣄⠹⡔⡇⣿⡏⢀⡼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⡟⡎⠀⠸⡆⣩⣿⣿⡶⣫⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠄⣇⠀⠀⢱⢹⣿⠟⠋⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠈⠀⠀⢸⣟⡫⠖⠁⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠁⠀⢠⠀⣼⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢸⡈⠀⠀⠈⠆⢸⡀⡇⡴⠃⡀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣇⠠⠀⠀⢣⠈⣧⡿⠋⡁⠠⢀⣀⣠⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⠸⡟⡿⡆⢃⠀⢀⡵⣛⣀⡤⠶⠒⠉⠁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠁⠀⠹⣼⣿⠈⣶⣾⣻⡿⠝⠀⠀⠀⠀⣠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡆⠀⠀⢹⣿⡿⠿⠋⠀⠀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠀⢀⣾⣯⣴⣖⣚⠉⠉⠉⠐⣚⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⣶⡀⢤⡀⠀⢸⣟⣼⡿⠿⠯⠟⠛⠉⠙⠛⢛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⢂⠁⠘⣄⣼⠟⠀⠀⠒⠒⠀⠀⠈⠈⢉⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⡈⠙⢿⣄⠈⢢⡶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⠤⠦⠀⠀⠙⢶⣦⣻⡾⠛⠋⠉⢠⣤⣀⣀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣧⠀⣸⣿⣯⠤⠀⠐⠀⠀⠀⠐⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⡀⢀⣿⣾⣿⡧⠔⠢⠤⠔⠲⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣷⠿⠯⠉⠉⠉⠁⠚⠛⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⠟⠛⠋⠉⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⠁⠀⣠⡀⠀⠀⠀⢀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠟⠁⠀⠀⢀⣴⡟⠁⠀⠀⣴⣿⠟⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠛⠁⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⠉⠁⠀⠀⠹⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏⠀⠀⠀⠀⠀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣴⣾⠙⢿⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠛⢻⡇⠀⠀⠀⠀⠀⢀⡿⣿⣦⡀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⢼⡇⠀⠀⠀⠀⢀⣿⡇⢨⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠘⠃⠀⠀⠀⠀⣼⠿⣃⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⣦⣈⣁⣤⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠇⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢠⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⣾⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⣸⣿⣟⢁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⣆⢸⡿⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⣼⠀⠀⠀⣿⣿⣿⠁⢀⣿⡏⠀⠀⠀⠀⢀⣴⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⣿⠀⠀⠀⣿⣿⡏⠀⣾⡟⠀⠀⠀⣠⣾⡿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⣿⠇⠀⣿⡇⠀⢀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡟⠀⢠⣿⡇⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠈⣷⠀⠀⠀⠀⣿⠁⠀⢸⣿⡇⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⡿⠀⠀⠀⠀⢻⠃⠀⢸⣿⠇⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⣿⣿⡟⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⢠⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⣿⣿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇
⠀⠀⠀⣸⣿⣿⠀⠀⠀⠈⢣⣄⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⣹⣿⡇
⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⣿⣿⠀⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⢰⣿⣿
⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠹⣷⡆⠀⠀⢠⣿⡏⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⢀⣿⣿⠏
⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠿⣷⡀⠀⣾⣿⠇⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿
⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⢀⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠇
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⣰⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡀
⠀⣿⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⢠⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠃⢘⣿⣿⡇
⢰⣿⣿⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢸⣿⡇⠀⠀⢀⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⢿⣿⣷
⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⠃⢀⣴⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⢠⣿⣿⠁⠀⣾⣿⣶⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⢿⣀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⢸⣿⡟⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄⢸⣿⡇⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠻⣧⣼⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢿⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠸⣿⣿⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⣴⡿⠏⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⣴⠛⠁⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⣀
⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⣼⣿⠂
⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⣠⣾⣿⣿⡄
⠀⠀⣿⣿⠁⠹⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⠀⢠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⡆⠀⠘⢿⣷⡄⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⣿⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠉⢰⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠈⠻⣷⣄⠀⠀⠀⢸⣿⡇⠀⠘⣿⣿⠀⠀⢀⣾⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠿⠋⠀⣠⣿⣿⣿
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠈⠋⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠁⠀⠀⡄⠙⣿⣿⠏
⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⣿⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⠋⠀⠀⠀⠀⠀⠀⣴⣿⡿
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣿⡇⠀⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⣤⡀⠘⣿⣧⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠛⣷⣤⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁
⠀⠀⠀⠈⣿⣿⣆⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣤⣤⡀
⠀⠀⠀⠀⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠿⠛⢻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠟⠁⠀⣠⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀⠀⢂⠀⠀⠀⢸⣿⡇⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⡀⠀⢿⣆⠀⠘⣿⡇⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠋
⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣷⣀⠀⠻⣷⣴⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⢿⣿⣿⣦⡀
⠀⣴⣿⣿⣿⡛⠉⠉⠛⠛⠻⠿⠀⠀⠈⠻⣿⣿⠀⠀⠀⣿⣷⠀⠀⠀⠘⠛⠛⠁⣀⣤⣴⣾⣿⣿⣿⡧
⠀⠛⠿⢿⣿⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣿⣿⠀⠀⠀⠀⢀⣴⣾⣿⣿⣟⣉⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣷⣤⡀⠀⢸⣿⣇⠀⠀⣿⣿⠀⠀⠀⠀⠛⠛⠉⠙⠛⠿⠿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠿⠋⠉⠛⠁⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣾⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣤⣶⣶⣤⣄⠀⠀⣿⣿⠀⠀⢻⣿⣆⣠⣾⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠿⢿⣿⣿⣷⣶⣿⣿⠀⠀⠀⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠘⡿⠟⠁⠀⠀⠀⠀⠈⠛⠿⣿⣿⡄⠀⠀⢻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏

⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣿⣷⢦⣤⣤⣄⡀
⠀⠀⠀⠀⡄⠸⠿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣷⣶⢄
⠀⠀⠀⢸⡇⠀⠰⢿⣿⣿⣿⠷⠟⣿⣿⣭⣿⣿⣿⡿⣿⣿⣦⣀
⠀⠀⠀⠀⠱⣄⠀⠿⠿⠿⣿⣗⡛⢿⣦⣄⣉⡹⢿⣿⡶⣬⣭⡳⣵⣢⡀
⠀⠀⠀⠀⠀⠈⠳⢄⢰⣾⣻⣟⣛⣛⣹⣟⠻⠙⠂⣻⣷⣶⣾⡓⢮⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣞⡛⣿⣟⡙⣿⣻⣟⠻⣾⣿⣿⣿⣦⡙⠳⣟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⢦⣤⣾⣏⠙⠿⣶⣟⢯⡙⠶⣽⡋⠳⢌⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢱⣷⣿⠾⡿⣦⣮⡘⠉⠀⣿⣀⠺⣯⡙⢾⣿⣦⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠂⢻⠿⣿⡶⣟⠿⣿⡛⣿⣷⡉⠛⠙⢷⠈⠻⡝⣎⢿⡧
⠀⠀⠀⠀⠀⠀⠀⡦⢤⡤⠖⢁⣀⢸⣿⣶⣥⣿⣶⣤⣍⣾⠉⠻⣄⠀⠘⣷⡄⠹⣎⢯⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣶⣄⣶⣿⡿⢿⣯⣙⢚⣿⣹⣿⣿⣾⣷⡤⠹⣆⠒⣿⢹⣆⠹⣌⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡝⢿⣿⣮⡃⣹⣟⠍⣿⣿⣦⠹⡆⣸⠀⠻⣦⢹⡜⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢿⣾⣻⣿⡾⣾⣿⣿⠞⢿⣿⣿⡟⢦⣙⣐⣿⣏⠀⢤⣹⡆⢻⡜⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠽⡿⢿⣯⣟⠻⣿⣷⢾⣿⠿⣧⡀⠈⢻⣄⠙⣦⠀⡙⣷⢮⣿⡘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠻⠼⠏⣗⡘⣿⣮⡀⠙⣆⠘⣧⠀⠘⣧⣌⢻⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢼⡇⣿⣦⡘⣷⠈⢷⡠⠸⣧⡓⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠸⠆⢹⣆⠸⣷⡄⢿⣷⡅⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠣⡈⢿⡞⢧⠿⡄⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⠃⠀⠀⠁⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣤⣦⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢫⡶⢶⣶⡄⣻⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣵⢋⣤⣾⣿⣿⠋⠀⢻⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⣴⣾⣿⣿⡿⠆⠀⢸⡟⠀⠈⡇⢀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⡷⠆⠀⣸⡇⠀⠀⠻⣸⡀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣾⣿⡇⠀⠀⣿⡖⠀⠀⣿⠉⠁⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠇⠀⠀⣿⡏⠀⠀⣿⠀⠀⠐⠀⢀⣀⡰⡿⣿⣷⡆⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡉⣿⣿⣆⠀⢰⣿⣇⠀⠀⣿⡀⠀⠀⠀⢀⠛⠛⠉⠉⠩⢴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣯⡄⢸⣿⠏⠀⠀⣿⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⡿⠀⢸⣿⡇⠀⠀⣿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡞⣿⣿⣷⡇⢸⣿⠇⠀⠀⣿⠀⠀⠀⢸⠙⢶⡄⠀⠐⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠌⠻⠂⠀⠀⠙⠀⠀⠀⠸⠦⠈⠋⠀⠆⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡇⠆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣍⡇⢠⣜⡠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠃⠀⠀⢀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡿⠀⢀⢔⠕⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣼⣶⣿⣿⣶⣤⠘⢷⣦⠴⢧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠈⠂⢸⣿⣟⢻⣿⣻⠟⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡀⠀⡇⠀⠀⠀⢸⢿⣿⣷⣌⣰⢶⠝⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⢀⡈⣶⠀⠀⠀⠀⠘⣮⣤⣋⣛⣉⣶⡀⠈⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠿⠉⠁⠘⠀⠀⠀⠀⠀⠘⣯⡹⡉⢍⠀⠳⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠳⣟⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠙⣿⠷⠔⠋⠰⣝⠂⠀⠀⠀⠀⠀⠹⡾⡀
⠀⠀⠀⠀⠀⠐⢄⠀⣿⣈⡀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣎⠠⠼⢿⡀⢻⡴⣆⠀⠀⠀⠀⠀⣾⠋
⠀⠀⠀⠀⣀⡀⠀⠉⣿⣿⡧⠀⠀⠒⠀⠞⢾⣿⠛⣿⠏⠀⠀⢄⡙⠿⢇⡟⠀⠀⠀⠃⠘⣇⣤⢜⣂
⠀⠀⢀⠼⠫⠁⠟⠶⠽⡯⠁⠀⠀⠀⠀⠀⢸⡿⠴⢹⠀⠀⠀⠄⠙⢦⠀⠁⠰⠧⠿⠚⢹⠟⡠⠚⠛⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠐⢤⠙⢏⠛⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢢⢃⡈⢢⣤⣹⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠐⢄⠑⣌⢒⠹⠀⣯⡑⢝⠊⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡈⢘⠇⡀⣆⠓⠌⢃⢈⣷⣿⠂⠙⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠆⠏⠈⣳⠁⢀⣸⢶⡘⣷⠣⡡⡑⠕⣌⠺⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⠄⡤⠄⢹⠜⡀⣷⢡⢱⡀⡌⠀⠈⢢⠈⢣⡉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠐⢃⠰⠧⠀⢶⡏⡀⢃⣿⡆⢳⠳⠠⡀⢣⠀⠱⣈⠙⣞⡝⡄
⠀⠀⠀⠀⠠⠴⠛⠁⠀⠠⠔⠁⢀⡎⢰⡆⡁⣇⣿⢹⡇⠸⣇⠀⢱⡀⢣⡁⢸⡇⡸⣿⣹⡀
⠀⠀⠀⠀⠀⠀⢀⠐⠊⢠⣄⡉⠀⢹⢀⠀⣿⣿⣿⢸⣄⣸⢿⣆⠀⣇⣸⡧⡼⡙⡝⣜⢆⠸⡄
⠀⠀⠀⢸⠒⠦⠥⠠⡤⠚⢘⡀⣠⣦⣼⢻⡟⣇⢸⡎⠈⢧⠈⠏⢏⢯⠡⠱⡄⠹⡌⠈⢎⢦⠘⣆
⠀⠀⢰⠙⡖⢀⣔⣁⣆⡄⠈⣹⣝⢇⡿⢈⢀⣿⢻⠃⠀⠒⠘⠠⡈⣎⢇⠡⠹⡀⡘⡜⢆⠈⢧⠘⢆
⠀⠀⣟⡰⠁⡆⠸⠁⠏⡄⢸⠈⠰⠘⡇⣿⣿⣿⠼⠀⡄⢈⠀⠁⢣⠸⣌⢆⠑⢡⠐⡘⡘⡆⠈⢆⠘⣆
⠀⠀⠨⡷⠺⠛⢳⡎⡛⠢⢸⣦⣵⣾⣿⡿⡏⢻⠄⣀⡇⢹⡀⠘⠀⡀⠘⡌⠄⠀⠀⢱⠀⠸⡄⠈⣆⠘⡆
⠀⠀⠀⢳⡀⢤⢌⣄⣵⡁⣾⢿⠛⢇⠙⠄⢰⢸⠀⣷⡀⢸⡇⠀⢷⣽⡀⢹⠈⡄⠄⡄⢣⠀⢱⡀⠘⡆⠘⣆
⠀⠀⠀⠀⠉⡆⠀⢊⠜⡁⠟⠀⡆⢸⠀⡇⡈⢸⢰⠋⠿⣘⠽⡄⠘⣿⡱⡀⢣⠀⠘⣼⡌⢻⡄⢣⠀⠸⡄⠹⡄
⠀⠀⠀⠀⠀⠈⠻⢥⠞⠁⡇⢠⠃⡼⢰⠃⡇⣸⠏⠀⠀⠀⠀⠈⠋⠀⠻⡄⠀⢳⡄⠹⡁⠈⡇⠈⡇⠀⢳⡆⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠜⠀⡼⢠⠃⣼⢰⠷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠜⢿⡄⢳⣄⠸⣆⠘⣴⠀⣷⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣼⢃⣼⡷⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠜⢻⡀⢻⡆⢹⡇⠸⡇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠞⠻⡀⢳⠀⢻⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠣⣨⠣⣬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠎
⠀⠀⠀⠀⠀⠡⠐⠠⢀⣀⡄⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠬⠀⠠⠐⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡕⠁
⠀⠀⠀⠀⠀⠀⠱⡀⠀⣨⣴⠶⢤⣄⡀⠀⠀⣀⣤⠴⢶⣤⡄⢄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡢⠊
⠀⠀⠀⠀⢀⣀⣀⢀⣿⠋⠀⠀⠀⠈⠻⡄⣼⠋⠀⠀⠀⠈⠹⣇⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⠜
⠀⠀⠀⠀⠻⠋⢛⢿⡇⠀⠀⠰⣼⡇⠀⣿⣇⠀⣮⣿⠀⠀⠀⣿⡛⠋⠿⠂⠀⠀⠀⣠⠤⡀⠀⠀⠀⠀⠔⡵⠁
⠀⠀⠀⠀⣀⡴⠃⠌⠻⣄⡀⠀⠀⣀⠼⠛⠻⢦⣀⠀⠀⣀⣴⢏⠙⣄⢀⠀⠀⠀⠀⡇⠀⠈⢢⡀⡠⡢⠊
⠀⠀⢠⠊⡝⢀⠊⠀⠀⠈⠙⠛⠛⢡⡀⠀⠀⣠⠙⠛⠛⠉⠀⠀⠂⠈⢆⠈⠉⠁⡨⠷⡀⠀⠀⠙⢾
⠀⡰⠁⡜⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡆⠀⠀⡇⠀⠈⠢⡀⠀⠀⠁
⡰⠁⢰⠁⡄⠀⠀⠀⠀⠀⠀⠀⠒⠂⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠈⣦⡀⠀⠈⠐⠄
⡇⠀⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠇⠝⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠇⢀⢾⡀⠈⠐⢄⡀
⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⢱⢦⣀⠜
⢃⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠂⠈⠉
⠈⢾⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠈⢧⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢠⠇
⠀⠀⠈⢣⡱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⠋
⠀⠀⠀⠀⠉⣪⣗⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⣮⣞⠁
⠀⠀⠀⠀⠸⣀⠰⡞⣛⢿⡗⠶⠶⠶⠶⠶⠶⠶⠶⠶⢺⡟⠛⣳⠆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠣⠤⠌⠑⠒⠁⠀⠀⠀⠀⠀⠈⠒⠊⠣⠤⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣶⣿⣿⣿⣿⣷⡶⠆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⡿⢟⢍⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⣪⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⡻⢛⣵⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⢟⢕⣭⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⡫⣪⣷⣿⣿⣿⣿⣿⣿⣿⣛⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢧⣿⣿⣿⣿⣿⣿⢟⢕⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣼⣿⣿⣿⡿⣻⢕⣵⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣫⢞⣵⣿⣿⣿⣿⣿⣿⣿⣭⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢟⣩⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢸⣿⣿⣿⣿⢟⣵⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣹⣿⣿⢿⣷⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣣⡾⣡⣾⣿⣿⣿⣿⣿⣭⣥⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢯⡾⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⣵⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢿⡾⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⠞⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⢟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠃
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡿⠞⠃
⠀⠀⠀⠀⣴⡟⠁⠈⠉⠙⠉⠉⠁
⠀⠀⢀⣾⠋
⠀⣠⡟⠁
⡰⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡐⣠⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣟⣫⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢼⣘⠋⠁⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡚⣀⡅⣾⣿⠄⠀⠀⠀⠀⠄⠒⠒⠀⢄
⠀⠀⠀⠀⠀⠀⡳⣿⣴⢞⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⠟⠉⠙⠻⡻⡈⣵⠀⠀⢀⠆⠀⣰⣦⡀⠀⠲⡀
⠀⠀⠀⠀⠀⢀⢍⡻⣒⣫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠔⣥⣦⡡⠀⢃⣿⠀⠀⢸⠀⠀⣿⡀⢹⡄⠀⣧
⠀⠀⠀⠀⠀⢸⡦⣽⣵⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣽⠀⡘⠛⠛⢃⠀⢩⣇⠀⠀⠈⡆⠀⠸⣧⡀⣿⠀⡇⢐
⠀⠀⠀⠀⠀⣾⢿⣿⢖⡲⠇⠀⠀⠀⡄⠀⢀⣤⡄⠀⠀⣄⠀⠀⠀⠀⣻⣅⠀⠀⠀⠈⠁⢀⡟⠟⠀⠀⠀⠘⡄⠀⠹⣷⡏
⠀⠀⠀⠀⠀⡛⠲⣿⠷⠚⠃⠀⠀⢸⡁⡀⢸⣿⡿⠀⠘⡝⡆⠀⠀⠀⠹⡻⡢⠀⢀⠠⡠⠚⠠⠀⠀⠀⡀⢇⠰⡄⠀⢹⠃⠀⠀⣠
⠀⠀⠀⠀⠀⣯⣜⡃⠀⠀⠀⠀⠀⠘⣇⠃⠈⣿⠃⠀⠄⠁⣿⠀⠀⠀⣤⠹⣆⠉⢠⣞⠕⣡⠂⠀⠀⠀⠙⠂⠱⡝⠄⡸
⠀⠀⠀⠀⠀⠛⠲⣀⢔⣬⡆⠀⠀⠀⠹⣆⠀⠻⠀⢐⠆⢠⡏⠀⠀⠀⣿⣇⣿⡧⣏⡴⠞⠁⠀⠀⠀⠀⠀⠁⠀⢹⣄⡅
⠀⠀⠀⠀⠀⣶⣦⣿⠯⣚⡅⠀⠀⠀⠀⠙⢧⠀⡄⠂⢠⠏⠀⠀⠀⢰⣽⣿⣿⡿⣫⡶⠀⠀⠀⠀⠀⢀⢸⠀⠀⢠⠏⠀⣀⣠
⠀⠀⠀⠀⠀⣾⣭⢸⡭⠵⣮⠀⠀⠀⡦⡀⠘⡆⠁⣰⢫⡄⠀⠀⠀⢸⣿⣿⣿⢟⣫⠁⠀⠀⠀⠀⠀⣾⣾⡄⠀⠛⠛⠛⣹⠇
⠀⠀⠀⠀⠀⢹⣺⠸⢀⣠⡷⠀⠀⠀⢷⠀⠁⠈⣴⡷⠋⠀⠀⠀⠀⣸⣿⣿⡯⠟⠃⠀⠀⠀⠀⠀⠀⣿⣧⠃⢂⣨⡴⠞⠁
⠀⠀⠀⠀⠀⠘⣛⡿⢋⢥⢮⠀⠀⠀⠘⢷⣄⠀⢻⠇⠀⠁⠀⠀⠀⠸⡟⠿⢀⣠⠆⠀⠀⠀⠀⠀⠀⠸⡏⡾⢿⠃
⠀⠀⠀⠀⠀⠀⣟⢿⢅⣥⠞⠀⠀⠀⠀⠘⠯⡛⠾⢠⠴⠀⠀⠀⠀⠀⢛⠪⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢈⣻⠋⣠⠔⠀⠀⠀⠀⠀⠈⠀⢢⢇⣀⠀⠀⠀⠀⠀⠈⠣⠐⠀⠀⠀⠀⠀⠀⠀⠀⢤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣼⡇⠀⠀⠀⠸⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠸⠿⠉⠁⠀⠀⠀⠀⠘⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇
⠀⠀⢠⣿⠁⠀⠀⠀⠀⠙⠻⢿⣷⣶⣶⣶⣶⣿⠿⠋⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿
⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⡄⠀⠀⠀⠀⣼⣿⠟⠀⠀⠀⢀⣸⣿⣿⣄⣀
⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⣰⠇⠀⠀⠀⢸⣿⠛⠁⠀⠀⠀⠀⣤⣤⣀⠀⠀⢻⣿⣿⡇⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣋⠁
⠀⠀⠀⠀⠘⠟⠢⡀⠀⠀⠀⠻⣇⠀⠀⠀⣰⡏⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠈⠟⠻⣧⡀⢸⣿⣿⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠈⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⡿⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠁⠀⠀⠀⠀⠘⠿⣷⡀
⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⣀⣽⣶⣦⣀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⢿⡏⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠙⠛⠟⠂⠠⠀⠀⠀⠈⠉⠉⣹⢿⣷⡆⢀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⣸⣷⡀⠀⠀⠀⣼⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⡿⡏⠁
⠀⠀⠀⠐⠿⣿⡇⠀⠀⠀⠸⣿⣿⣦⣀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢀⣿⡟⠂⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠙⢿⣿⣶⡶⣶⠿⠿⡿⡹⠃
⠀⠀⠀⠀⠈⠹⠷⠀⠀⠀⠀⠈⠁⠉⠉⠙⠛⠳⠶⠾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢠⣾⡿⠿⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣁⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣌⠉⠒⢄⣀⠀⢀⡁⠠⠂⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠠⠄⢰⡄⠄⣀⠤⠒⠉⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠈⠉⠻⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠌⠤⠀⠄⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠀⣠⣶⡿⠿⠛⠻⠷⣦⣄⡀⠀⠀⠀⣤⣶⠿⠛⠛⠻⢿⣶⣔⠐⣢⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣦⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⣠⠟⢿⣇⠀⠀⠀⠀⠀⣀⣄⠀⠀⢸⣿⣿⡆⠀⠀⣠⣄⠀⠀⠀⠀⠀⣼⣿⡉⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⢀⠜⣿⣄⠀⠀⠀⢸⣦⣿⡇⣠⣿⠿⠿⣷⡄⢸⣶⣿⡇⠀⠀⠀⣰⣿⠃⠹⣄
⠀⠀⠀⠀⠀⣄⠀⣿⠲⢾⣟⠀⣠⠃⠀⠈⠻⢷⣦⣤⣤⣭⣽⡾⠉⠀⠀⠀⠀⠙⢷⣯⣭⣤⣤⣶⡾⠟⠉⢢⠀⠘⢷⢤⣀⡀⢀
⠀⠀⠀⠰⡖⠿⢥⣟⣢⠤⢤⣉⣙⠒⠦⢤⣀⠀⠈⠉⠉⠉⠁⢱⣄⠀⠀⠀⠀⣠⡆⠉⠉⠉⠉⠁⠀⠀⢀⣀⡵⠴⠚⣋⣩⠷⣿⣠⡀
⠀⠀⢠⣤⣷⣀⣀⠠⠬⣉⠒⠠⢌⡉⠑⠢⢌⡑⠢⣄⠀⠀⠀⠈⢄⠑⢄⡠⠊⡰⠀⠀⠀⠀⢀⡠⠔⣊⡩⠤⠒⢊⣉⢭⠴⠒⠯⣼⢻⡇
⠀⣀⠸⠖⠊⠉⠀⠀⠀⢨⡆⠀⠀⠈⠙⠲⢄⡉⠳⣌⠳⣄⠀⠀⠀⠉⠋⠃⠉⠀⠀⢀⡴⢚⡡⠖⢋⣡⠴⠒⠋⣁⡤⢾⠀⠀⠀⠀⠉⠓⠤⣀
⢰⠁⠀⠀⠀⠀⠀⣀⣴⣎⡉⠙⠲⢤⣄⡀⠀⠈⠓⢬⣳⡈⠳⣄⠀⠀⠀⠀⠀⢀⠔⣡⠖⢉⠤⠚⠉⣀⡤⠖⠋⣁⡤⠞⣳⣤⣀⠀⠀⠀⠀⠀⠹
⣄⢀⡠⠤⠒⠊⠁⠀⠈⢫⠉⠓⠲⢤⣀⡉⠙⠲⢤⣄⣈⠻⢆⠈⢧⠀⠀⢀⣴⠗⢊⡠⢆⠙⢦⡖⠋⣁⡤⠖⠋⠁⠀⢸⠁⠀⠀⠉⠒⠤⣄⡀
⡏⠀⠀⠀⠀⠀⠀⣀⡤⠋⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣉⡙⠲⣷⣖⡓⠀⠞⣡⣴⡥⠖⢋⣳⣠⡽⠏⠁⠀⠀⠀⠀⠀⠘⠢⣄⡀⠀⠀⠀⠀⠈⠙
⢣⣠⡤⠴⠔⠒⠛⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⣿⠈⠉⠉⠉⠁⣿⠰⡏⠉⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠓⠲⠤⣄⣀⣰
⠘⡆⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠒⠒⠲⠖⠒⢿⠀⣷⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⢰⣿⠿⢄⡀⠀⠀⠀⢨⠇
⠀⠈⠲⢶⡛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⠒⠶⠶⠴⠶⢾⠀⣿⠀⠀⣷⣄⠀⠀⠸⣿⡿⣷⡄⠸⣿⣦⣶⡏⣷⠦⡔⠃
⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢸⠀⢸⡄⢸⣿⣿⣷⣄⠀⢻⣷⠿⠃⠀⠈⠉⠉⠀⢹⠀⣷
⠀⠀⠀⠸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⢸⠀⢸⡇⠘⠟⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿
⠀⠀⠀⠀⡇⠿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣛⣥⠿
⠀⠀⠀⠀⠛⠲⠦⣭⣙⣛⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠋⠓⠒⠒⠒⠚⡇⢸⡇⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⣛⣩⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠻⣗⠲⠬⣍⣙⡓⠶⠤⣤⣀⣀⣿⢸⡗⠒⠲⠴⠖⠺⡇⠈⣇⢀⣀⣤⠴⠞⣋⡭⣴⠞⢉⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣮⣑⠦⣄⡀⠉⠙⠓⠶⠤⣍⣁⢘⡇⠀⠀⠀⠀⠀⡇⠀⢛⣉⡥⠶⠚⣉⡥⢚⣥⣶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⢹⣉⣶⢿⣷⣶⣤⣤⣤⣀⣈⣉⣳⣤⣄⣀⣠⣴⣷⣪⣭⣤⣴⣶⣯⠷⣮⣉⠎⠁⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠖⠊⣹⠤⠠⢌⣻⣧⠋⠉⣻⠉⠉⠉⠉⠉⠉⠉⠉⠙⡏⠉⠑⣼⣗⡡⠄⠤⣏⠑⠢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⣀⡠⠋⠘⠲⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠖⠃⠹⢄⣀⡤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠚⠉⠉⠉⠉⠋⠉⠉⢽⣿⣿⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⢀⣀⣀⣰⣻⣿⣷⣷⣦⣄⡀⠀⠙⠛⠲⠯⡿⣿⣛⡲⣤
⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⡞⠉⠠⠾⣿⣿⣶⣾⣿⣿⣿⡿⠛⢳⣤⡀⠀⠠⣌⠙⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣟⠀⠀⣸⣿⣭⣿⣻⣿⣿⣭⣻⣧⡙⢷⣤⠙⣾⣷⣾⣷⣦⢀⠙⢯⣔⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢄⢻⡀⠀⠀⢀⣑⣤⣼⣿⣭⣝⣛⣿⣿⣟⠿⣷⣚⣿⣿⣿⣿⣷⣷⡜⢯⣿⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡉⠀⠀⠀⠉⢉⡤⢴⣤⣬⣭⣭⣿⣿⣷⣬⣻⣿⣟⠻⣿⣿⢿⣷⣦⡙⠿⣿⣟⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠠⡴⠖⢥⣤⣜⢿⣦⣤⣄⣉⠻⣿⣿⣍⢳⣽⣟⣦⢻⣿⣤⡀⠀⠈⢳⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢔⣢⣄⣂⣉⣾⠟⢮⣍⠹⢿⡌⢿⣿⣿⣿⣿⡿⢾⣯⡻⣿⡄⠀⠀⢻⣮⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣰⣏⡝⠛⣯⣉⢻⠶⣤⣹⣿⡷⢷⣿⣳⣄⠉⢯⣿⣮⢿⣷⣹⣿⣷⡄⠀⠹⣷⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠋⠋⠛⠿⢊⣿⣷⠶⣤⡾⣍⡙⠦⣽⢿⣿⣷⢼⣷⣽⡌⣿⡿⣟⣿⣿⣤⣀⠙⣿⣷⣵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⢺⣿⣛⣛⣛⣛⣙⣛⣿⣧⡈⠻⣿⢿⡄⢹⣶⣶⡟⣿⡿⣏⣷⣼⣿⣿⡿⣿⣷⣄⠙⣿⡻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣶⣾⡿⣿⣯⣙⠻⣿⣿⣛⢻⡐⢿⣶⠻⣟⢷⣄⣻⡝⣿⣽⣿⡞⣷⣴⣿⣿⡦⢼⣿⣿⡆⠘⢿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⡾⣽⣮⣿⣟⣃⣷⣬⣉⣋⣿⠿⣶⡿⣿⣿⣸⣷⣷⣟⢿⣏⢸⡇⠘⣿⣿⣿⡄⠸⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣿⣿⣿⣿⣟⡻⣯⡉⠿⠿⣿⣣⡀⢹⣧⡈⢻⣷⡽⡎⣿⡜⣟⣶⣳⡄⣿⡏⣿⢧⠘⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠙⢿⣮⣙⣿⣷⣶⣾⠿⠟⠛⢿⣄⡹⣿⠞⣮⣇⣿⡛⣷⢷⣟⠹⣿⠼⣆⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠈⠛⠛⠿⣻⣷⣨⣿⠓⢹⣿⣟⠹⣏⠻⣟⣸⣿⢻⡋⠈⢿⡸⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠙⠙⢷⠻⠟⠹⣿⣿⣿⣿⣿⣿⣿⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠰⡆⠿⠿⠿⠿⠻⠛⠿⠘⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡿⣆⢿⡾⣿⡷⠒⢾⡶⠷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡼⡌⢇⣧⢇⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⣸⣿⠀⠨⣇⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣿⡆⠀⡟⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⠀⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢻⡄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠳⣄⠀⠀⠀⢿⠦⠤⠤⠒⠒⠒⠒⠶⠤⠒⢒⣺⠃⠀⠀⠀⡠⠊⠉⠳⣄
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠈⢳⠀⠀⠈⡷⣄⠀⠀⠀⠀⠀⠀⠀⣴⢻⡇⠀⠀⢠⠋⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⢠⡏⣀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⡇⢺⠷⢤⠀⠀⢀⣰⣾⡅⢸⡇⠀⠀⢸⠀⠀⠀⠀⠘⢶⣄⠹⡄
⠀⠀⠀⢀⡆⠍⠀⠐⠆⠐⣏⣠⠄⠸⢆⠀⠀⡇⠈⠉⢁⠱⣀⠞⡍⣉⠀⢻⠀⠀⢀⠞⡀⢀⠀⠳⠤⠜⠀⠉⠱⡄
⠀⠀⠀⡼⠀⠀⠆⠀⠀⠀⠀⢀⡀⠀⠘⠦⣼⢀⠀⠀⠀⠀⠉⠀⠀⠀⡀⡄⢇⣠⠋⠀⠑⠈⠋⠀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⢰⠃⠀⡄⠀⣀⢼⠁⠀⠈⠀⠀⠀⠀⡇⠀⠘⠀⢄⠀⠀⠠⡄⠠⠁⠇⠘⠀⠀⠀⠀⢀⠂⠀⠀⠠⠄⠐⠀⡀⢧
⠀⠀⡏⢠⠛⡇⠀⠀⢸⢰⠆⠁⢀⠃⠀⠀⡆⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⣘⡀⠘⠀⠀⠁⠀⠀⠚⠈⢁⠘⡆
⠀⢰⣠⠏⡆⠀⠀⠀⠈⢰⢠⠀⠼⠀⠀⠀⡧⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠀⢐⠈⠒⠀⢀⠀⣀⠀⠃⠀⡀⠀⠀⠈⠀⢻
⠀⠘⠁⠀⠀⡃⢀⢃⠀⠉⢸⠀⢳⠀⡄⡆⢱⠄⢀⠀⢠⠀⠀⠀⠀⠁⠁⡄⡾⢠⠀⢆⠈⡆⣹⠀⣈⠀⠡⡀⠀⠀⠀⠰⡄
⠀⡇⠀⢸⠀⠈⠁⡌⢠⢃⡞⠋⠈⠂⠀⠈⢹⢀⡘⡇⠘⠄⠀⠀⠀⡴⠀⣼⡷⠞⠈⠈⠃⠀⠈⡆⢸⠀⠀⠈⠀⠀⠀⠀⢧
⢠⠀⠀⠀⠀⠀⢀⠇⡜⠉⠀⠀⠀⠀⠀⠀⠘⡄⠀⢈⠀⠰⠄⠃⠐⠃⠨⢰⠀⠀⠀⠀⠀⠀⠀⠈⠉⡄⢠⠀⠀⠀⠀⠀⢸
⢸⠀⢠⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⠁⠀⠀⠀⠀⠃⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠘
⢰⠀⠘⠀⠈⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠲⠀⠀⢠⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⠀⠘
⢈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠄⠀⠀⠐⡀⠀⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⡆⠀⡇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢈⠊⣧⠠⣁⠰⡄⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⡁
⡇⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢠⠆⠺⠬⠤⠾⠷⠧⠦⠇⠼⠼⠤⠄⣀⡤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⢧⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠀⠀⢀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡰⡿⣀⣀⣸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢯⢩⠁⡉⠉⠉⠉⠉⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣘⣀⠀⠀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀⡀⠘⣈⣠⠀⠀⠀⢀⡄⢂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠁⠒⢲⢉⠟⠒⠒⠛⠛⠋⠉⠉⠉⠛⡑⠉⠉⠉⠙⡩⠽⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠠⠀⠀⢺⣼⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠶⠤⢬⠤⠤⣤⣤⣤⣤⣤⣤⣬⣦⡧⠤⠤⣤⢌⣷⣀⣄⣠⣄⣈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠤⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡝⢆⡱⠿⣼⠁⠠
⠀⠀⠀⠀⠀⠀⠀⣧⣼⣼⡰⣟⠵⠛⢠⣦⡖
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⡀⠐⢀⣰⡯⢈⡿⣂
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠊⢀⠔⢈⡿⠟⠉⢔⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⣿⣴⣇⠔⠉⠀⢁⣴⢟⠟⣠
⠀⠀⠀⠀⠀⠀⠈⠷⢿⣏⡿⠀⢀⡴⢋⠀⠈⠠⡍⠀⡄
⠀⠀⠀⠀⠀⠀⠠⢶⣾⣿⡇⣰⣿⠔⢁⡴⣢⣾⠇⣰⣟⡄
⠀⠀⠀⠀⠀⠀⠰⣖⡻⣿⣿⠻⣡⠞⣡⣾⡿⢁⡜⣿⠇⣲⡄
⠀⠀⠀⠀⠀⠀⠀⡏⣁⢿⣯⠟⠁⣠⡿⢋⡴⠃⢁⣴⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣾⢿⣿⠀⢰⣯⣶⠏⠀⣠⣾⡟⢠⡽⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠹⣾⢸⣆⡿⣹⡟⠀⣴⣿⠏⠃⣽⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠘⣦⣿⠀⢿⣁⡟⠀⣸⣿⠃⣠⠜⡻⢟⡿⢋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣆⠸⣿⣧⢠⡿⢡⡾⠋⢀⡵⠞⣁⣾⢟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣹⣄⢿⣿⡈⣵⠟⢀⡴⢋⡠⣺⡾⢃⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠏⡰⢋⡔⣣⡾⠋⡠⣾⠋⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣡⣞⣿⣄⠁⠊⣿⠋⢠⢮⠟⣠⠞⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡶⢿⣿⣿⡀⣾⠃⢠⡣⢃⣴⡏⢀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠄⠹⣿⡟⢠⠷⢡⠚⣿⠀⠙⠃⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡶⢸⣧⣿⣶⠃⣾⠛⣠⡄⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣰⢿⣿⣿⡇⣸⡇⢠⡍⢰⣿⣟⠁⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠹⣿⣿⣻⣧⣿⠀⡰⣰⣿⠏⠘⢂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⢿⡿⣿⣀⢱⣿⡟⣼⣆⠈⢱⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠐⢾⣧⣿⣿⢱⡏⠻⣶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⡙⣿⡇⣷⡘⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣈⣀⣀
⣿⠿⡿⢿⠿⠿⠿⠻⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠿⠿⣿⣿⣿⠿⠿⠿⡿⠿⠻⠿⢿

⠀⠀⠀⠀⠀⠀⣠⣴⣤⣤⣤⣤⣤⢤⣤⣤⣤⣄
⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣈⣿⡿⣼⣵⣾⣿⣿⢓⣶⣦⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⡻⢻⣷⣿⠝⢿⣟⣿⣿⣷⣦⣀
⠀⠀⠀⢺⣿⣿⣿⡿⣿⣿⣿⣿⣹⡟⢻⣛⢟⣷⣰⡟⣿⡿⢿⢿⣿⣕⣄
⠀⠀⠀⢸⣿⣿⣿⡿⣿⢿⣿⣿⣿⣗⣄⣿⣿⣟⣻⣮⣿⢻⣿⣿⣿⣯⣿⣷⣀
⠀⠀⠀⠀⠻⣿⣿⣾⡿⣸⣿⣿⣿⠷⣾⣿⡃⣿⣿⣛⣿⣷⣍⢿⣿⣷⣿⣹⣿⣧⡀
⠀⠀⠀⠀⠀⢾⣻⣿⣥⡏⣿⡿⣿⡻⣯⡏⠳⣝⡠⡙⣷⣭⡛⠷⢦⣌⣼⣯⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠸⣿⡿⣿⣾⢿⣷⡹⣷⣺⠻⢦⣀⠑⢧⡸⣦⡟⣷⣾⣝⢿⣺⣿⣿⣿⣻⡆
⠀⠀⠀⠀⠀⠀⠸⡿⣿⣿⢿⣿⡻⣿⣿⣾⣀⣩⠛⢶⣭⣿⢿⢯⡙⢏⡿⣿⠷⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡗⠿⣾⢿⡏⢹⢿⠛⣷⣄⠠⡄⠹⢦⡙⠈⣿⣯⡻⣿⣽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣻⣿⣿⣝⣲⣯⡸⣿⣦⡈⠂⡗⠹⢦⡀⠀⠁⢹⢦⡘⠯⢿⢯⡐⢿⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣟⣱⣿⠠⡙⢦⡀⠀⡎⢻⣦⡀⠈⠇⢙⠦⣈⠃⣏⣓⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠙⢿⠿⣿⣦⠉⠢⠙⢦⡄⠀⠘⢿⣦⡀⠼⠐⠘⠳⣄⠙⢿⣿⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢻⣗⢄⡈⠀⡙⠳⣄⠈⣟⢿⣦⡀⠂⡇⢀⢻⠦⣬⣥⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢻⣿⣦⡀⠀⢈⠳⣮⡀⢟⢿⣦⡀⠈⢿⡇⢻⣆⡨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣾⣗⢄⡀⢸⡙⣶⣈⠟⣿⣶⣄⠃⢹⢯⠗⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣟⣦⡀⠘⠈⠳⣬⡋⢿⣷⣤⡘⠃⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣤⡈⠪⠙⢶⣝⣿⣿⣦⣌⠻⣷⣆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣲⣄⡀⠜⣛⣾⣗⡋⠻⢮⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⡲⣄⡁⢻⡛⢷⣦⣬⣾⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢭⣳⣤⣉⠻⡟⠟⢺⡷⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⡶⠾⡷⣄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄⠀⢠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⣀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡏⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣿⣿⣷⣾⣯⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣏⢁⠈⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⡟⠛⠋⠐⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣾⣿⣿⡛⣾⣿⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡈⢿⣿⠟⠃⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣯⣠⣴⣖⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⡦⠄⠀⣤⣴⣿⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⣲⣿⣾⣷⣿⣿⣟⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⣿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣿⣾⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡯⣿⣏⣿⣽⡏⢉⢁⣀⢀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⣿⣿⣷⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣷⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⡿⢻⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣯⣿⣿⣿⣿⣿⣿⣧⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠛⠿⠿⠿⠿⠿⠿⠛⠻⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⣶⡀⣿⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇⣸⡆
⠀⠀⠀⠀⡟⢷⣇⢠⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⢨⣷⢏⡇
⠀⠀⣶⡀⣿⠈⢻⡜⣆⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡏⣼⠃⢸⢇⣴⡆
⠀⠀⢹⡝⠿⣆⣆⠻⣿⣆⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⣾⡿⢁⣆⡿⠛⣽⠁
⠀⠀⠀⢷⡀⠹⣿⣦⡙⢿⣦⡀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⣿⠟⣠⣿⡟⠁⣰⠇
⠀⠀⠠⣌⣳⣄⣌⠻⣷⣄⠙⢿⣄⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⢠⣾⠟⢁⣴⡿⢋⢄⣴⣃⡤
⠀⠀⠀⠻⣍⠙⠻⣷⣌⡻⣷⣄⠙⢷⣄⠀⠈⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⣠⣾⠟⢁⣴⡿⣋⣴⡿⠛⢉⡿⠁
⠀⠀⠀⠀⠘⠳⣝⢮⣝⠿⣾⣿⣷⣤⣉⣳⣦⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⣠⣶⣋⣤⣶⣿⣿⡾⣛⣥⢞⡵⠋
⠀⠀⠀⠀⠈⢿⡛⢛⡻⠿⣷⣿⣿⣿⣿⢭⣉⡀⢰⣦⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢠⣶⠀⣈⣭⣽⣿⣿⣿⣷⡿⠿⡛⢛⣻⠏
⠀⠀⠀⠀⠀⠀⢽⣷⠿⠷⣶⣾⣽⣿⣿⣿⠷⢦⠘⣇⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢀⡇⣠⠶⢿⣿⣿⣿⣽⣶⠶⠿⢷⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠩⣷⣷⣶⣦⣤⣼⣿⢾⣿⣾⣿⣭⠁⠙⠿⣿⣷⣆⠀⠀⢠⣶⣿⡿⠛⠁⢩⣽⣿⣾⣿⣾⣿⣤⣤⣶⣶⣷⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⢶⣶⢶⣶⣿⡿⣿⣯⣿⣿⠀⠀⠀⠀⠀⡿⠀⠀⠸⡇⠀⠀⠀⠀⢼⣿⣯⣿⡿⣿⣷⣶⣶⡶⢶⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⣿⠿⣟⣿⣿⡓⠛⣠⡶⣶⣿⣿⠃⠀⠀⠀⠻⣿⣷⡶⣦⡘⡛⣻⣿⣟⡿⢿⣟⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡷⣟⣭⠾⣿⡷⣻⡿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⣿⡳⣿⡿⣯⣙⡷⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠿⠟⢻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠛⠾⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣛⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⢸⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢸⢇⡿⠋⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠜⣿⠀⡞⠘⢠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⢰⡇⠪⠈⢟⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⣼⢠⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠁⠀⠀⡿⢸⣿⢟⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⡇⢘⣵⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⡇⠸⡿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⡇⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⡇⠈⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⡇⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠣⠓⢘⠀⠂⠀⠀⠘⠃⠀⠀⠀⠀⣠⠀⡆⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⡁⠁⠀⢠⣪⣈⠀⠀⠀⠀⠘⠦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣹⡆⢀⠀⠠⠀⠀⠀⠀⠄⠀⠈⠙⠉⠈⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠙⠁⠀⠐⠁⠀⠈⠙⢍⡆⠀⠴⠃⠠⡠⠡⠁⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠘⠄⠀⠀⠀⣐⡀⠀⠀⠀⠒⢼⣉⠇⡀⣠⣌⡀⠀⠀⣀⠆⠐⠆⠀⢀⣀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⢠⣴⣦⠀⠴⠺⣇⡒⠴⡂⠙⠋⠃⠀⢀⢄⠸⠄⠼⣤⠠⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠠⡀⠀⠎⠙⠂⡀⠈⠁⢿⠁⠀⣻⠊⠀⠁⠀⠀⡂⠀⠀⣤⣀⡠⠄⠈⠉⠁⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠉⠀⠀⣌⠄⠐⠀⠈⠳⡀⣿⠔⠺⣄⠀⠀⠀⠀⡀⠙⠋⠁⠰⠂⠃⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⡀⠀⠄⢀⡆⡹⠲⠷⣀⡿⢥⣀⠠⠎⠁⠠⣄⠀⠈⠰⠀⡤⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠀⠙⣑⣾⡁⠁⠀⠀⣀⣹⠀⣀⠰⣾⣷⠆⠀⢢⠶⡄⠀⣠⠁⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠛⢡⠔⠳⣄⣰⡆⠀⢹⠀⠅⠈⡆⠀⢐⠢⢿⠴⠒⠋⠑⡄⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⢿⡇⠀⠀⠀⠀⠓⠛⠳⣤⣿⠘⠶⣴⡥⠦⣤⡼⠛⠳⠤⠐⠐⠁⠈⠩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠛⠀⠀⠀⠀⢠⠖⠀⠀⠙⣿⣷⣾⠃⠀⡀⠈⢓⠀⠀⠀⢰⣶⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠘⣿⣿⡄⠐⠻⢷⠦⠌⡑⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠤⠴⡛⢛⠻⣦⡄⢀⣀⣀⣀⣀⣀⣐⣀⣀⠀⠀⠀⣈⠡⣤⡰⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠾⣯⣥⣴⣿⠃⠀⠀⠀⠀⠀⠒⠒⠒⣒⣂⣈⣉⣉⣉⠀⡂⣀⣀⡀
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠉⠉⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⠉⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣴⡿⠋⠀⠉⠉⠛⠷⠀⠀⢠⠒⠉⠀⠀⠈⠛⢷⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣤⣼⡟⠀⠀⠀⢀⣀⡀⠀⢡⢠⠃⠀⡀⣄⡀⠀⠀⠈⣿⣤⣴⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢈⠝⢻⡇⠀⠀⠀⠻⣾⠇⠀⢸⣹⠀⠀⠷⡿⠇⠀⠀⠀⣿⣏⠁⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠠⠊⢿⣦⡀⠀⠀⠀⠀⠐⠃⠈⠑⢤⡀⠀⠀⠀⣀⣼⢏⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡱⠁⡰⠁⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⢀⠛⠻⠟⠛⠋⠁⠀⠡⡀⠹⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⡰⠁⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠹⡀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠊⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡇
⠀⠀⠀⠀⠀⠀⢣⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢀⠇
⠀⠀⠀⠀⠀⠀⠘⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠙⣝⣴⢷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⡿⣞⣩⠉⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠙⠒⠒⠘⠓⠉⠉⠋⠉⠉⠉⠉⠉⠉⠙⠉⠑⠚⠁⠒⠚⠋⠓⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⢀⡴⠖⠛⠛⠶⢦⡀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⢻⣆⢰⡟⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⢻⡄⠈⢛⡟⢻⡟⠁⣠⡟⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢰⡇⢸⡇⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠞⠀⠀⠀⠀⢻⣴⣧⣼⣦⡟⠀⠀⠀⠀⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⢦⣄⡀⠀⠀⣹⠶⠶⠶⢶⣏⠀⠀⢀⣠⡴⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠶⣤⣌⡛⢶⣄⠈⠹⣟⣻⠛⠁⣠⡶⢛⣡⣤⠶⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⠶⠶⢤⣬⣙⡳⣽⣷⡀⠈⠁⢀⣾⣯⢞⣋⣥⡤⠶⠶⠶⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠶⠶⠶⠾⢿⣿⣿⣿⣆⣰⣿⣿⣿⡯⠷⠶⠶⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢁⡶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⢶⡉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣆⡀⢀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢰⢆⡆⡖⢢⢆⢦⡄⡆⠀⡆⢰⠒⣤⠒⡀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠀⠁⠁⠉⠈⠈⠁⠁⠀⠁⠈⠉⠀⠁⠁⠁

⠀⢸⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣼
⠀⠀⢿⣆⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣼⠏
⠀⠠⣌⢿⣧⡹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢣⣾⠏⣠
⠀⢤⣝⣷⣝⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⣿⣿⣿⣣⠄
⠀⠀⢽⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡥
⠀⠐⢶⣿⣿⣿⣿⣿⣿⣿⣭⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣽⣿⣿⣿⣿⣿⣿⣥⠶
⠀⠠⣤⣿⣿⣿⣿⣿⣿⡯⣿⣿⣏⣀⣹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡃⣘⣿⣿⡯⢽⣿⣿⣿⣿⣿⣯⣤⠄
⠀⠀⠴⠾⠛⣋⣭⣵⣶⡶⣾⢻⡿⣳⢞⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⣰⣿⡿⣯⣶⣾⣿⠻⣶⣶⣿⣿⣿⣿⣿⠧⠤
⠀⠀⠀⠀⢚⣿⡿⢟⣵⣾⣷⡾⣸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢸⣿⡟⣷⣞⣿⣿⣻⣶⣿⡿⣿⡿⣿⣟⠂
⠀⠀⠀⠀⠈⠉⢚⣩⠾⣛⣡⣾⣿⣭⢿⣿⣿⡿⣿⡇⠀⠀⠀⠀⣿⠿⣬⣿⣿⢿⣵⣿⣿⣿⣿⣿⣟⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠐⢋⢿⡿⡿⣻⣯⠞⢫⣾⣿⣿⣷⡄⠀⠀⣰⣿⣤⣿⣮⣛⣾⣿⣿⣿⣿⢯⠻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠇⡵⢞⢮⢞⣿⣿⣷⣼⠀⢘⣵⣿⣟⣿⣿⣿⣿⡿⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⢾⠹⠄⢸⣿⡿⠿⠟⠻⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠳⢤
⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡖⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣴⡦⡀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠙⠂⢸⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠙⢛⠀⢠⢠⡀⠀⠀⠀⠀⠀⠀⠰⠀⠈⡐⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⢀⠄⠁⡼⠁⠀⠀⠀⠀⠀⠀⣶⠀⠀⠄⢩⠎⠄⠂
⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢠⠞⠀⡀⠀⠀⠀⠀⠀⠄⢿⠠⠀⡴⠃⠀⠀⡀
⠀⠀⠀⠀⢲⠀⠉⠁⢄⠀⡄⠈⠀⠀⡀⠀⠀⠀⠀⠀⡖⣫⡙⠀⡸⣫⠞⡡⠋⠀⠀⠀⠀⠀⠀⠈⢦⠐⡐⠆
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠃⢠⠄⠒⠁⠀⠀⠀⠀⠀⢷⠰⠡⢀⣰⡣⢞⠵⠃⠀⠀⠀⠀⠀⠀⠂⠈⠳⣄⠀⠀⠀⢠⡁
⠀⠀⠀⠀⠰⣄⠈⠢⡀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢸⣫⠖⣡⠖⠃⠀⠀⠀⠀⠘⠣⡀⠄⠀⠀⢀⠀⢀⠞⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠁⠊⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⢧⡀⠈⢾⡵⢪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠀⢈⡴⠋⠄⠄
⠀⠀⠀⠀⠀⠢⢄⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣸⡎⣀⠂⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⠁⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠑⠤⡏⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⣳⠃⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⡀⠠
⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⣠⠖⠀⠁⠀⠀⠀⠀⠀⠀⠏⠠⣄⠠⣿⡤⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⢀⠔⠂⠀⠀⣀
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⡀
⠀⠀⠀⣸⠀⠀⠀⠀⠂⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⢱
⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⡼
⡇⠀⠀⢷⠀⠀⠀⠀⠂⢀⠄⠀⠠⠀⠀⠰⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⡇⠀⠀⢸⠁⡀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢂
⡇⠀⠀⠸⡄⠀⠀⡘⠀⠀⠀⠂⠀⠀⠀⠀⢱⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠰⡭
⡇⠀⠀⠀⣇⠀⢰⠁⠀⠀⡜⠀⠀⢀⣀⣀⣸⣄⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠃⠀⠀⠀⠘⣄⡏⠀⠀⢠⣥⡶⠟⠋⠉⠉⠉⠙⠃⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠂⣸
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠘⡃⠀⢠⠀⠀⠀⢀⣰⠆⠀⡼⠻⣷⣄⠀⠀⠀⠀⣀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⢀⡴⠉⠳⡤⣀⠀⢱⠂⠙⠄⣀⠠⠣⠃⡠⠒⡁⠀⢼⣿⠶⠒⠀⠐⠀⡀⠀⢀⡾
⠀⠀⠀⢸⢀⠀⠠⡇⠿⣶⣦⠀⡀⠀⠀⡀⠀⢰⣧⠸⢇⠔⠁⢁⠀⠀⠀⠀⠀⠘⡴⠞⠁⠀⠀⠀⢀
⠀⠀⢀⣸⡌⠓⢰⠀⠑⣪⡟⡏⠻⠂⠾⠃⠟⣛⠯⣶⠁⠀⠀⢸⠉⢠⡀⠀⠀⣠⢇⠀⠀⢀⠀⠄⠤⠀⠀⣀
⠀⢀⠎⠀⠉⠒⢺⠀⠀⣿⣷⡀⠉⣰⣶⣿⣅⢀⣴⣿⠀⠀⠀⡜⠈⠋⣁⡠⠞⢱⠈⡇⠀⡀⠀⠊⠀⠠⠈⠡⡰⢐⠂
⠀⢾⠀⠀⠀⠀⢸⢀⣄⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇⠀⠀⢠⠟⠉⠉⠉⠀⠀⠀⠐⡘⡆⢰⠴⡄⣤⣤⣄⢄⠀⠈
⠀⠈⠑⢲⡋⠀⠈⣆⠉⢹⡊⣿⣿⣿⣿⡇⢰⡞⠦⠾⠂⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢰⠂⠤⠰⠠⠆⠰⠀⠄
⠀⠀⠀⡔⠉⠉⣉⠜⠓⠈⠁⣿⠻⢿⣿⠃⢸⣷⡧⠔⠊⠀⠈⠓⢞⠉⠀⠀⠀⠀⠠⢁⡏⠀⣀⠠⠀⠀⠀⠀⠄⠀⢰
⠀⠀⠀⣩⣛⡽⠃⠀⠀⠀⠀⠀⠀⠀⠈⣹⡿⠁⠀⠀⠀⠀⠀⠉⠉⢳⠀⠀⠀⠀⠀⠸⣄⡘⠀⠐⠅⠀⠄⠂
⠠⠶⢿⣿⣿⠁⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⡔⠒⠦⣍⣉⠉⠒⢠⠀⠁
⠀⠀⠀⠁⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠛⠢⣆⣀⡀⠀⠀⠀⠀⣰⣿⡟⠛⠛⠛⠁⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠚⠉

⠀⠠⣆⠀⠀⠀⠀⠠⣿⡧⢠⡷⠀⡶⢤⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⢤⢴⠂⠾⣤⠠⣿⡇⠀⠀⠀⠀⣐⠆
⠀⠀⠈⠀⠄⠀⠁⠠⠉⢻⣵⣛⣾⠇⠘⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠃⠰⣿⣘⣮⡿⠋⠄⠈⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⠟⢻⣦⣤⡌⣿⣿⣇⡀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣿⣿⠍⣤⣴⣾⠺⠃⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢻⣦⠈⠻⡿⢶⣿⣦⠐⠀⠀⠁⢠⣾⣷⢪⡾⠋⣰⡟⠊⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣮⣛⡻⣿⠁⠀⠀⠈⣿⢯⣓⣱⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⡿⠷⠁⠀⠀⠀⠳⢿⠿⠛⠃⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡠⠄⠀⠀⠐⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⢀⣴⣴⣶⠆⠐⠺⠗⠄⠠⠤⣈⣿⣿⡶⠂⠐⠀⠀⠀⠠⠀⠀⠀⠀⠈⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡸⢼⣿⣤⣿⣤⠧⠤⠖⡤⡄⢠⣌⣹⠗⣴⣬⣿⢿⠷⠄⠄⣄⡠⠀⠀⠀⠀⢀⣠⠷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡶⣾⣚⡏⢷⡿⠿⠂⠰⠶⣿⡟⠋⠚⣿⣶⣼⠿⠇⣠⡤⣀⡀⠁⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡎⢿⢋⣿⡜⢍⠢⠀⠁⡀⢻⣯⠙⣷⣶⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠦⢀⠠⢸⡿⡄⣿⢷⡄⢈⣆⣰⣀⣿⣶⡍⢛⢯⠈⠁⠀⠀⠀⠀⠀⢀⠀⠂⠉⠁
⠀⠠⠀⠀⠀⠀⠰⠦⠒⡘⢦⢾⡁⣷⡻⣧⠀⠈⡿⣯⠙⣟⠟⠒⣀⡀⠀⠀⠀⠀⠀⠀⠂
⠀⠠⠓⢼⠬⢹⡾⢿⣿⣖⠶⠧⠸⠿⠁⠀⢀⣾⡻⡎⠳⠸⠓⠆
⠀⠀⠴⢄⡚⣭⣼⣿⣟⣋⣰⠃⠆⠀⠀⣠⣿⣟⠧
⠀⠀⠀⠈⣶⠻⢟⡉⠻⠉⢃⣴⣶⣷⣶⣿⠙⠇
⠀⠀⠀⢀⠼⠻⣈⣡⣼⣿⣿⣿⣧⢸⠁⠙
⠀⠀⠀⠀⡖⣾⡇⡏⢽⠟⠁⠻
⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⣿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⣷⠘⣼⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢳⡄⢹⢠⣿⣿⣿⠟⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣷⡌⠳⣄⠹⡔⡗⣿⡏⢀⡴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢁⡟⡏⠀⠸⡆⣙⣷⣿⡶⣫⠜⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠇⣇⠀⠀⢱⢹⣿⠟⠋⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⢠⠀⠈⠀⠀⢸⣞⡯⠖⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⠁⠀⢠⠀⣼⠀⠀⠀⢠⠟⠉⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢸⡈⠀⠀⠈⠄⢸⡈⡇⡰⠃⡀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣇⠠⠀⠀⢣⠈⣧⡿⠋⡁⠠⠀⣀⣠⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⢸⣟⡿⡆⢃⠀⠀⡵⣛⣀⡤⠶⠒⠍⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠁⠀⠹⣾⣿⠈⣶⣾⣻⡯⠝⠀⠀⠀⠀⣠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡆⠀⠀⢹⣿⡿⠿⠋⠁⠀⠀⠀⠀⢀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣀⣾⣯⣴⣶⣚⠉⠉⠉⢐⣚⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⣶⡀⢤⡀⠀⢸⣟⣼⡿⠿⠿⠿⡿⠭⣽⣿⣟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⢂⠃⠘⣆⣼⠟⠁⠀⠒⠒⠚⠃⣀⡀⣉⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣀⡈⠹⢿⣆⠈⢣⡶⠛⠁⠀⠀⠀⠀⠀⠠⠦⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠦⠤⠦⠀⠀⠙⢷⣦⣻⡾⠛⠉⠉⢠⣤⣀⣀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⣧⠀⣸⣿⣯⠤⠀⠒⠀⠀⠀⠐⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⢀⣿⣾⣿⠷⠔⠶⠤⠀⠲⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⣷⠿⠯⠉⠉⠉⠀⠚⠛⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠉
⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣠⣭⡇
⠀⠀⢠⠀⠀⢰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢾⡿⠋⢡
⠀⠀⢀⠳⣀⠤⠤⠤⣂⠬⠄⣂⡄⠀⠀⠀⠀⣛⣛⢛⣛⡛⠀⠀⠀⠀⠘⠠⠀⠀⣀⠀⠀⠠⠘⠀⠀⠀⢰⣦⣤⣤⣀⡄⡀
⠀⠈⢄⣒⡪⠭⠭⢙⠵⠊⠁⠀⠀⠀⠀⠀⣾⡿⠿⡿⠿⣿⡆⠀⠀⠀⠀⠀⠀⢠⣾⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢺⣷⢀⡄
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠋⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣙⣺⣹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⡏⣿⡭⢿⠺
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⠀⠀⠰⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠈⠋⠉⠁⠉⠀⣿⡖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣴⣿⣷⡦⢸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣷⡆⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣯⣿⣿⡇⠛⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣨⣵⣒⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠉⠉⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢩⡄
⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠿⡶⠿⠷⠶⡿⢶⠾⢾⠄⠀⠀⠀⠀⠀⠀⠀⣤⣤⣠⣙⣛⢛⣉⣤⣤
⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠒⢳⠒⡗⠒⠗⢺⠒⡺⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣿⣿⣿⣫⣭⣿
⠐⠀⠀⠐⣿⣿⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢍⣫⣛⣹⡹⣋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⠿⠟
⣤⣤⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣽⣿⡿⠿⣤⣿⣭⣝⣛⠓⣘⣿⣝⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠼⣽⣿⣿⣿⣿⣿⣿⣽⡿⢿⣯⣝⡳⣮⣭⣙⡛⢿⣮⣛⠷⣿⣛⣿⡿⣿⣿⣦⣄
⠀⠀⠀⠀⠙⠙⣷⣿⣵⣟⣷⣽⣿⢶⣬⣙⣟⣼⣿⣗⢮⣄⠙⢷⣤⡀⠻⠏⣿⣌⣝⠻⣝⠷⣄
⠀⠀⠀⠀⠀⠀⠈⣿⢻⣯⡙⣿⣿⣷⣬⣙⣿⣭⣧⠻⣿⣿⠷⠌⣿⣙⠛⣿⣿⣿⣮⢁⠘⢷⡉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣽⡿⣿⡿⣧⢩⢹⡟⢧⣀⡀⠳⣤⣴⡟⠉⠳⣶⣾⠟⠘⢷⡀⠀⠹⣅⠙⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⣿⣾⣽⣿⡷⣦⣳⠀⠹⣮⡂⢹⣦⠘⣶⣤⢹⣿⣷⡄⠀⠙⢧⡢⠘⢶⢀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣷⡈⠛⢮⣟⠳⢤⣽⣵⡿⣿⣄⡩⢿⢂⣿⣷⠙⣌⠀⠈⢳⡀⠈⢧⡄⠌⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢋⡿⣿⢿⣦⡘⠍⠻⡾⣷⣦⣾⣻⣿⣟⢶⠿⢇⣼⡇⠈⠳⡄⠀⠙⣆⠄⢳⡄⠈⢳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣽⡎⣿⡛⠳⢄⡃⠨⣻⡗⢸⡧⠙⢻⡶⠾⠟⣧⡀⠀⠙⣆⠀⠈⢧⡀⢳⡐⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣷⣌⣿⡄⠀⢻⡟⠳⠶⣿⠁⠀⠌⢿⡀⠀⠈⢳⡄⠀⠈⢷⡄⠀⠳⡀⢷⡀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⣿⠁⢦⡀⢳⡀⠀⠌⢶⣅⠀⢸⣿⣦⠀⠀⠙⣆⠀⠀⠹⣄⠀⢱⡌⢷⠀⠀⢷⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⢷⡘⣿⢦⡐⠈⢹⡅⠀⣿⠘⢷⡀⠀⠈⢷⡀⠀⠘⢧⠀⢳⡈⢧⡀⠈⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢿⡀⢉⢷⣤⣈⣰⣿⡀⠂⠱⣄⠀⠀⠳⣄⠀⠈⢧⠀⢳⡘⣯⠰⠸⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠋⢯⣹⡏⠛⢦⡀⠘⢦⡀⠀⠙⣆⠀⠘⢧⠈⢷⢹⡆⠀⢻⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢟⢿⠛⣧⡀⡀⠳⣄⠀⠳⡄⠀⠈⢦⡀⠘⣧⠘⣧⢻⠑⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠳⣌⠁⠘⢷⡀⠘⢦⠀⠀⠳⡄⠈⣆⠘⡎⢧⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢳⡬⢄⠳⡄⠈⠳⡄⠀⠙⣆⠘⣆⠹⡘⣆⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢹⢶⣤⡙⣦⠀⠘⢦⡀⠘⣧⠘⣆⢣⣹⡀⣿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠻⣷⣌⢣⡀⠈⢳⡀⠸⡆⠹⡍⢷⣇⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠻⡇⠳⡄⠈⢧⠀⢹⡄⢹⡜⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⡷⠀⢸⡆⠘⡇⠈⣧⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠁⠀⣨⠃⢀⡇⢀⡧⠘⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠛⠀⠾⠅⠐⠈

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⢀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠱⡍⠒⠤⣀⠀⣄⠠⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠀⡎⠀⡠⠔⠊⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠙⡦⠀⠀⠙⣃⣄⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣐⠀⢂⠠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣴⡿⠋⠉⠉⠉⠛⢷⡄⠀⣰⠟⠋⠉⠉⠉⠛⢷⡔⠊
⠀⠀⠀⠀⠀⣠⣶⣶⣤⣼⡏⠀⠀⠀⢀⣀⡀⠀⢿⣴⡏⠀⡀⣄⡀⠀⠀⠈⣿⣤⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠙⠃⢀⠝⢻⡇⠀⠀⠀⠻⣾⠟⠀⣼⣿⡇⠀⠷⡿⠇⠀⠀⠀⣿⣏⠁⠀⠛⠁
⠀⠀⠀⠀⠀⢀⣴⠋⠠⠊⢻⣦⡀⠀⠀⠀⣀⡴⠛⠙⠻⢤⡀⠀⠀⠀⣀⣼⠯⠀⠳⣄⡀
⠀⠀⠀⣠⠞⡱⠁⡰⠁⠀⠀⠈⠛⠛⠛⠛⢫⡀⠀⠀⠀⣀⠛⠛⠛⠛⠋⠁⠀⠱⡀⠹⡝⢦⡀
⠀⠀⡔⠁⡰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢍⠢⡀⠈⠝⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠹⡄⠙⣄
⠀⡞⠀⢰⠃⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⢱⠀⠘⣆
⣸⠀⢀⡏⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠢⠤⠄⡐⠁⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣇⠀⠸⡄
⡇⠀⣸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⠠⠋⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⡇⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠐⠂⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠘⡇⠀⡇
⢣⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢠⠇
⠘⡄⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡇⡜
⠀⠙⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⡼⠁
⠀⠀⠘⣆⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢠⠟
⠀⠀⠀⠈⢦⡙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⠏
⠀⠀⠀⠀⠀⠙⢮⣓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⣡⠞⠁
⠀⠀⠀⠀⠀⢠⠊⠙⣍⣿⢷⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⠿⣞⣩⠉⢱⡄
⠀⠀⠀⠀⠀⠈⠓⠊⢹⠒⠒⢸⢷⡉⢉⠏⠉⠉⠉⠉⠉⠉⠹⡁⢁⠾⡇⠒⠺⡏⠑⠊

⢠⣿⣆
⢸⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀
⣿⣿⣿⣿⡿⠿⠿⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣄
⣿⣿⣿⣏⠀⠀⠀⠀⠈⠷⢄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠈⠑⢦⡀
⢸⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠑⡄
⠸⣿⣿⣿⡿⠓⠲⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠘⣿⣿⠿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⢿⣿⡄⠀⠀⠀⠀⠀⡄⠀⣠⣤⣄⣴⣿⣿⡇⢀⣠⣤⣤
⠀⠈⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡛⠀⠀⠀⠀⢠⠁⢰⣿⣿⣿⣿⣿⢿⣷⣿⣿⣿⣿⣠⣴⣶⣦
⠀⠀⠈⢿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⠛⠉⠇⠀⠀⠸⠛⠛⠿⣻⣿⣿⣿⣿⣿⣶⣶⡆
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⣿⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣷⣶⣶⣶⣆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠲⠂⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣻⡀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⣇⠀⠻⠟⠁⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣾⠃⠀⢀⠴⡄⠀⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠟⠁⠀⣠⣿⣠⠇⠀⠀⠀⠈⠓⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⢲⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⣀⣠⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⣀⣀⣠⠴⠾⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡴⣞⣫⣭⡦⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣴⣾⠟⢡⣾⣵⣿⣿⣷⣾⡿⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢠⢫⣿⠏⢰⣿⣟⣿⠞⣵⣟⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⡆⣿⡿⠀⣿⣿⡞⣿⡷⡿⣿⡿⡾⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⡏⣿⣧⢰⡏⣿⡇⠀⣿⣿⣷⣿⣿⣷⣿⠟⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠐⡇⣿⣿⣟⣿⢹⡇⢸⣿⣵⢿⣻⣾⠟⣁⣴⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⡇⠀⢳⢸⣏⡿⣿⣾⣧⣿⣿⣻⣾⣿⣷⣿⢟⡵⣯⣩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⢿⡆⢸⡆⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣺⣿⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠘⡆⢿⡆⣧⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣞⠃⣵⢸⡇⣸⢳⣿⣿⠿⠽⠯⢵⣉⡁⠘⠒⠒⠒⠛⢋⣡
⠀⠀⠀⠀⠀⠀⠀⢠⡏⣷⢻⠀⣼⣼⣷⣷⡿⣿⣿⡿⠿⣗⠶⠶⣾⣭⣽⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⡌⡟⢻⣿⣿⡷⠿⠾⢿⣿⣿⣖⣻⠗⠲⠶⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠒⠀⣜⣮⣧⢱⣾⣯⣴⡶⠶⠶⢶⣶⣶⣿⡿⠏
⠀⠀⠀⠀⠀⠀⢀⡀⠸⣿⣿⣾⠟⣯⡿⠟⠛⢛⡻⠟⠋⢁⣙⠔
⠀⠀⠀⠀⠀⠀⠠⡨⠀⣾⣿⣯⡾⠛⣻⡆⠀⠘⠒
⠀⠀⠀⠀⠀⠀⠀⠠⢒⠸⣿⠝⢛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⠀⠘⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⢀⣤⠂⢚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡟⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡀⠀⠀⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠞⠋⣿⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡁⠀⠀⠀⣀⣤⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢻⡇⢀⣴⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣴⣿⣿⠿⣿⠻⣿⣿
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⠿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣾⣿⣿⣿⣥⠾⢿⣀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠉⠙⠻⣿⡋⠀⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⣿⣟⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠙⠻⣷⢲⡄⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠀⢨⣭⣟⠉⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⢠⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣦⣀⣀⣈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣯⣽⣯⣹⠋⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⣾⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣿⠀⠹⣿⠇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⢦⣌⠈⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣴⣿⣶⡇⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⣠⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⠋⢀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢷⣤⢶⠶⠒⢚⣩⣉⡫⠍⠭⣋⣩⣙⠒⠲⢶⢦⣴⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣶⠿⠷⣶⣄⢬⣴⣝⢂⢚⣵⣬⢄⣴⡶⠿⢷⣦⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠕⠉⢈⡀⠈⢧⢽⢸⢯⢿⢸⢥⠏⠀⣈⠈⠑⢽⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⢶⢋⣉⡑⡜⡾⣿⣿⣿⡿⡞⡔⣉⣉⢳⠆⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡏⡀⠀⢠⡸⣜⣛⣡⠇⢹⣿⣿⣿⣿⠁⢧⣙⣛⡼⣠⠀⠀⡈⣿⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⡟⣡⠀⠘⠛⠢⠛⠁⢠⢠⡟⠀⠘⣧⢠⠀⠙⠣⠚⠛⠀⢠⡙⡿⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣾⡳⡝⠽⡇⡶⠀⢀⡀⢰⣦⣾⣿⣇⠀⢀⣿⣿⣦⣶⠀⣀⠀⠰⡆⡿⠝⡵⣻⣆
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⣄⡘⢧⣤⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣤⠟⣀⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣘⠛⢻⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠛⢛⣠⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣼⡿⠃⠀⠀⠀⠀⠀⠀⣄⣴⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠽⢟⡡⠀⠀⠀⠀⠠⣠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣴⣷⡾⠁⣀⢀⠴⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣽⣯⡧⠒⡋⢁⡴⠚⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢐⣾⣛⣽⣿⠟⠉⣠⠾⠃⠁⠀⠐⠀⠀⣀⠀⠀⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣀⣶⡾⠟⠛⠛⢶⡾⠵⢈⣀⣥⠴⠶⠻⠿⢛⠍⠀⢀⣀⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠾⣩⣵⣢⣤⣶⣚⡼⠟⠛⠉⠋⠁⠀⠈⣐⣠⡤⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣛⣷⣷⣿⢽⠏⢉⣉⣽⣏⣠⣠⣤⣤⡶⠴⣿⡿⠶⠶⠒⠒⠒⠲⠖⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⣿⠿⣻⣿⣿⢿⣿⣁⡾⠁⠘⠁⠀⠀⠀⠚⠁⠀⢀⡠⢄⠠⠔⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣛⣷⣿⣿⢿⣿⣿⠗⣿⣏⣄⣀⣄⣀⣀⣀⣤⣤⣆⣀⣂⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⣯⣻⣿⡯⣿⣷⣦⡙⣿⣍⣀⣉⣀⣂⠈⡉⠉⠀⠀⠈⠁⠈⠉⠉⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣣⣿⣽⣿⣛⣿⣯⡆⢻⣿⣯⣭⣙⡛⡋⠉⠓⠛⠶⠶⣶⣶⡦⠦⢤⣤⡤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣟⢻⣗⣀⣻⢿⣿⡿⣿⣯⡽⠿⢾⣭⣙⠻⠛⠦⣄⣀⠈⠙⠛⠛⢷⣶⡂⠑⠛⠛⠟⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣾⣟⣿⣿⣦⢻⡿⠎⣿⠙⢷⣬⡀⠜⠷⢔⡀⠨⣿⣿⣷⢶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣋⣿⣿⣧⡙⣯⣼⠯⠟⢿⣆⣊⣉⡉⠳⣄⠈⠑⢶⡄⠀⠉⠃⠀⠈⠉⠉⠁
⠀⠀⠀⠀⢤⣤⣴⣶⣿⣿⣿⣿⡿⢋⣿⣿⡈⢿⣟⣃⣼⢘⠻⢮⣑⠲⠽⢿⣍⠃⠘⢷⣤⠈⢿⣿⣧⣄
⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⢦⣿⡿⣯⠹⣷⢿⡄⣿⣍⣛⣦⣀⣀⠈⠻⣌⠠⢌⢳⡄⠀⠹⡧⠀⠀⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⢸⣷⡾⣯⣿⡄⠙⠛⢦⡈⠓⠻⢧⡈⠉⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⢻⣿⠻⠿⢿⣮⡿⡿⢦⡀⠈⣷⣀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠻⣧⠀⣀⠘⢷⣄⡸⣧⣲⠌⣿⡳⣆⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣯⠻⣷⣔⢝⠂⢻⡔⠸⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠘⢿⣷⣕⠻⢿⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢣⠀⠿⠿⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⢸⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠜⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡍⣆⡠⠋⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢸⡞⡇⣠⠞⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⠙⡄⢠⠊⡆⠀⠀⠀⠀⠀⠀⠀⡤⢶⠎⠀⡴⠁⣱⢻⠁⡠⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡟⠋⢣⡸⠛⢳⣰⣏⡇⠀⠀⠀⣼⡲⠃⣠⠊⡠⠊⣸⣾⡟⣋⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣜⡅⠘⣦⢀⠸⡇⢀⢸⠃⠀⣇⡤⡀⡸⡽⢁⠜⡡⢊⣤⢞⣻⣷⡯⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⠈⢧⡰⣜⠺⢏⡹⢏⠿⣚⡶⣧⣽⣶⡟⣠⣧⣪⣾⣿⡿⠕⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣷⣿⣷⡯⠗⠚⣓⡉⠑⠛⠚⠽⣷⠺⣿⡟⢻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡬⢽⣷⣒⠴⠶⣦⣄⣸⣿⣿⠟⡡⡄⠀⠀⣇⢵⡀⠀⠀⠀⠀⠑⡤⠃⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣷⣾⣷⣷⣾⣿⢿⠃⠀⢧⠾⡀⠀⠸⣄⢇⣀⣤⠀⠀⠀⢱⡰⣷⢀⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⢻⠉⢹⠀⠀⠘⠴⠣⡶⣶⠀⠨⣭⣀⠀⠀⠀⢸⠉⢹⡙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣬⡆⠀⠾⣫⡄⠉⠁⠀⠀⠀⠀⠀⠀⠀⣮⠙⡿⣤⠤⠾⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⣿⢄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⢳⣇⣈⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣟⣿⡷⡤⣀⠀⠀⠀⠀⠀⣀⡤⠊⡸⠈⠫⣸⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠁⠑⣄⠉⠉⠈⠉⠉⠈⢆⢰⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⡴⢞⣫⠻⡟⠻⣍⡉⠒⠲⢤⣀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⣀⡤⠖⠒⢉⣩⠟⢻⠟⣝⡳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢶⡀⠙⠦⣀⡈⠁⠒⠲⠬⢽⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⡯⠥⠖⠒⠊⢁⣠⠴⠋⢀⡶⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠑⣄⡀⠀⠉⠉⣫⢶⡞⠉⠉⠉⠳⣄⢸⡇⣠⠞⠉⠉⠉⢳⡶⣝⠉⠉⠀⢀⣤⠋⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣅⢀⠽⠶⣤⡋⠀⠀⠑⠦⣤⣀⣀⣨⣿⣿⣅⣀⣀⣤⠴⠊⠀⠀⢹⣤⠶⠯⡀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠿⣖⡾⢿⠟⠋⠉⢩⠟⣿⣿⠻⡍⠉⠙⠻⡿⢷⣞⠿⠤⠤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠁⠀⢾⡀⢀⡴⢻⢾⢻⡟⡷⡟⢦⡀⢀⡷⠀⠈⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣞⢑⣦⣠⠜⠛⢠⣌⡏⠘⠃⢹⣥⡄⠛⠫⣄⣴⡊⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣌⠗⢤⡖⢡⠏⠀⠀⠀⠀⠹⡈⢲⡤⠺⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠣⠘⠁⠀⠀⠀⠀⠀⠀⠈⠃⠜⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠶⢳⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⠙⣆⠈⢧⠊⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⣿⠀⠸⡏⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡆⠀⡇⠀⠀⣿⠁⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⠁⣼⡴⠚⠛⣿⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⠊⠉⢳⠀⣠⣟⠕⠚⢇⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣼⣴⠿⠧⢤⠀⠸⣤⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢳⠀⣠⡟⠁⠀⣀⣨⡧⠖⢻⠖⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⣿⣗⣛⡯⡭⢨⣤⠴⢺⣓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⠉⠙⠛⠛⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡰⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠦⣀⠀⠀⠀⠘⠀⡀⠀⠀⠀⠀⠀⣤⠄⠀⠀⠁⠀⠀⢠⡚⠀⠀⣀⠤⢒⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠈⠑⠢⡟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣘⣃⠁⡋⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⡀⠀⣠⣴⠿⠿⠷⢶⣤⣀⣀⣀⣀⣠⣴⠶⠿⠿⢶⣦⡄⠑⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢨⣿⠏⠁⠀⠀⠀⠀⠈⢻⣆⢀⣾⠋⠀⠀⠀⠀⠀⠈⢻⣯⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢿⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⠋⢻⣇⠀⠀⠀⢀⡐⣦⡀⢀⣿⣿⣇⠀⣠⢲⣦⠀⠀⠀⢀⣿⢧⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡞⠁⡰⠋⠻⣦⣀⠀⠘⠿⢿⣣⠞⠋⠉⠛⢦⣙⡿⠟⠀⣀⣠⡾⠯⡀⠙⣤⡀
⠀⠀⠀⠀⠀⠠⣤⢽⡢⠷⣈⣙⠒⠶⠤⣀⡀⠈⠛⠛⠿⠟⠛⣇⠀⠀⠀⠀⢀⡟⠻⠿⠟⠛⠋⠀⠀⢑⣀⡸⠿⢒⣦⡆⡀
⠀⠀⠀⠀⠀⣠⣹⡀⢈⣉⣑⠂⠭⣉⠒⠢⢍⡑⠦⣀⠀⠀⠀⠩⡑⢄⢀⠔⡩⠀⠀⠀⠀⢀⡠⢔⣊⠩⠄⢒⣊⡭⠴⣫⡷⣶
⠀⠀⠀⠀⣀⠿⠜⠋⠁⠀⠀⢱⠀⠀⠈⠑⠠⣌⠳⢌⠳⢄⠀⠀⠈⠒⠓⠈⠀⠀⢀⠔⣪⠵⢚⣩⠤⠒⢈⣡⢼⠀⠀⠀⠉⠛⠤⣀
⠀⠀⠀⠘⠀⠀⠀⠀⢀⣠⣴⣉⡉⠓⠦⢄⣀⠀⠉⠢⢵⣄⠳⣄⠀⠀⠀⠀⣠⢖⠵⢊⡠⠚⢉⣠⠴⠚⣉⡤⠞⡳⠤⣀⠀⠀⠀⠀⢱
⠀⠀⠀⡧⠤⠔⠚⠉⠀⠀⠈⡆⠉⠛⠲⠤⣌⣉⠓⠦⢤⣙⢢⡈⢧⠀⢀⡶⢛⡡⢐⣍⠙⢞⣉⡤⠖⠋⠁⠀⢸⡀⠀⠀⠉⠓⠢⢄⣀
⠀⠀⠀⡇⠀⠀⠀⣀⣠⠔⠊⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢬⡉⡿⠯⠭⠭⠴⡿⢚⡭⠼⢷⠚⠃⠀⠀⠀⠀⠀⠀⠙⢢⣄⣀⠀⠀⠀⢨
⠀⠀⠀⢹⠚⠉⠉⠀⢈⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⣀⣀⣀⣀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣴⣏⠀⠀⠉⠑⢲⡎
⠀⠀⠀⠀⠣⣤⡤⣖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣷⢤⣀⣀⣠⣼⠀⡇⠀⢠⣄⠀⠀⢻⣿⣿⡀⠸⣿⣀⣽⢢⣤⣀⠞
⠀⠀⠀⠀⠀⠀⣇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⢸⠀⣿⠀⣿⣿⣧⡀⠘⣿⣾⠟⠀⠙⠛⠋⠘⡇⣿
⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⠀⠀⠀⠀⢸⠀⢿⠐⠻⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⢹⠸⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢖⣋⡥⠞
⠀⠀⠀⠀⠀⠀⠈⠉⠓⢲⡬⢭⣛⣲⠦⢤⣀⣀⠀⠀⠀⠀⡇⢸⣈⠉⠉⠉⢹⡆⢸⡄⠀⠀⠀⣀⣠⠴⢖⣋⢭⡶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⢬⡉⠓⠲⠭⣭⣛⡒⠶⠇⢸⠉⠉⠉⠉⢹⡇⢸⡧⠶⢚⣫⠭⢒⡿⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢳⠾⣷⣦⣤⣀⣀⠉⠙⠒⠾⣄⡀⠀⠀⣠⣧⠴⢒⣋⣭⣴⣾⡵⢚⠟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⢻⣋⢍⡻⣿⠙⠛⢻⠛⠛⠛⠛⠛⠛⠛⢻⠟⠛⢍⣿⢋⠍⣹⡟⠤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⡼⠋⠳⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠹⢄⣀⡤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⢨⣿⢷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⡆⢢⣸⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢟⢠⣿⣿⣿⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡙⠳⣄⠘⣦⢸⣸⣿⡟⠉⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣿⣟⠄⠙⣆⠘⣾⣿⣿⣡⠞⣫⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣿⢸⠀⠀⠘⡆⣾⣿⣿⡥⠊⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠈⢸⠄⠀⠀⣇⣿⡟⣁⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢸⠀⢸⡇⠀⡀⠀⢀⣟⠵⠊⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢨⠃⠀⠘⠄⢸⡇⣰⠀⣀⠞⠁⠀⠀⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⡈⠀⠀⠀⡡⠘⡇⣿⣠⣋⠠⠀⠈⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⡿⣇⠠⡀⠀⠐⡄⢻⡿⠛⠠⠐⠄⣒⣄⢴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠻⡝⠿⡆⢡⠀⠀⡼⢛⣁⣤⠖⠒⠊⠘⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠻⢰⡴⠀⣧⣾⣿⠟⠠⠀⠀⠀⠀⠀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢰⣿⣶⠿⠛⠁⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡆⠀⢀⣾⣏⣤⣶⠖⠛⠚⠋⠉⠍⣛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⢀⠀⠀⠀⣷⣇⣴⣿⣿⣿⣭⠽⠿⠗⠾⠿⠿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⠂⠉⢦⢀⣿⠟⠉⠀⠂⠀⠐⠒⠶⠦⠔⢒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠻⣶⡄⠀⠷⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⡀⣠⣤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠉⠙⠶⣄⣌⣳⣁⣴⠿⠁⠐⠒⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠠⣄⠀⠘⣿⡾⠛⠉⠀⠉⠉⠉⠙⠛⠳⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣄⠀⠀⠀⣟⣠⣾⡿⠟⢋⠁⣀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⣴⣿⣿⣿⣭⣭⣉⣉⣉⣈⣉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠿⠛⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⣀⣿⠀⣆⠀⠘⡄⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⠀⢧⠀⠈⡇⠀⢹⠁⠀⣰⠁⣠⠤⠤⢄⡀⢀⡇⠀⠀⢰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣤⢀⡈⢣⠈⢳⡀⢱⠀⢸⠀⡔⠁⡞⢀⡀⣤⠀⠙⣾⠀⠀⠀⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠻⠉⢹⠀⡸⠀⠀⡷⢺⠆⢺⠶⠁⠀⡇⢈⠋⢙⠞⢀⡇⠀⠀⣸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⢳⣈⣄⡇⠀⠀⣏⣡⠖⢁⣴⠀⠀⢹⣸⢀⡎⡤⢺⠁⠀⣰⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⡡⢃⣧⠤⠤⠽⢯⣉⡭⠿⠶⠤⣿⡹⡝⢿⣅⣸⡀⣰⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠉⠙⠯⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠽⠃⠀⣨⡟⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠉⠛⠻⠿⠿⠿⠛⠛⠋⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⡇⢀⡀⠀⠀⢀⣤⡄⠀⠀⠀⠀⢈⡿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠿⠃⣼⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠓⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣿⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠛⠉⠁⠀⢠⠇⠀⠑⢦⡀⠀⠀⠀⣀⡤⠞⠁⠀⢹⠀⠀⠉⠉⠙⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⣩⠗⠒⠯⣅⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⠖⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠘⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢦⡀
⠀⠀⠀⠀⠀⢀⡀⠀⡰⠀⠂⠀⠀⠀⠀⠀⠀⠘⠋⢴⡤⠈⢂⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠑⠀⠇⠀⡤⠀⠀⠀⠀⠀⠀⢇⠀⠙⢋⢀⠄⡄⠀⠀⠀⠀⠀⠐⠀⢉⢀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠠⠊⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⠈⡴⠁⠀⠀⠀⠀⠀⣿⠀⠀⣠⠎
⠀⠀⠀⢀⠓⠦⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⡠⠠⠀⢠⢞⡤⣰⠀⠀⠀⠀⠀⠸⡀⠘⣁⠀⠀⠂
⠀⠀⠀⠘⠢⡀⠀⠀⠈⢀⡀⡴⠀⠀⠀⠀⠘⡔⠕⡀⢡⣋⢞⡵⠀⠀⠀⠀⠀⠄⠙⢤⡀⠀⠀⢀
⠀⠀⠀⠠⣄⠑⢆⠀⡄⠀⠀⠀⠀⠀⠀⠀⠠⠳⣀⢸⣫⢟⡡⠂⠀⠀⠀⠀⠲⡀⠄⠀⠁⡀⢀⠜
⠀⠀⠀⠀⠈⠁⠈⠐⠀⣠⠖⠀⠀⠀⠀⠀⠘⢆⠈⢲⠕⠉⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⠀⣴⠋⠂
⠀⠀⠀⠀⠀⠢⡀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢺⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠐
⠀⠀⠀⠀⠀⠀⠈⢩⠀⡀⠀⠀⠀⠀⠀⠀⠀⡖⡂⢈⣾⣀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠠⣰⠐
⠀⠀⠀⠀⠀⠀⠀⠘⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣟⠁⡠⠈⠀⠀⠀⠀⠀⠀⠈⢦⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠙⢦⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠛⠀⠀⡀⠈⢻⣆⡀⠀⢀⣠⠾⠇⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠟⠁⠀⠀⡀⠀⠠⠄⠙⢳⣶⠋⢡⡄⠀⠀⠀⠀⠀⠀⢀⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⢀⡀⠂⣤⣴⣶⢏⣡⠤⠀⠶⣴⡿⠦⣀⣴⡾⠟⠹⣿⣆⡰⠶⢀⠲⢀⣱⣶⡟⠻⡆⠀⠁⠀⠀⠐⠀⣀⣴⣶⠛
⠀⣤⠀⢠⣶⣿⠿⣧⡜⢫⣀⠀⣀⣨⣷⣾⣿⢿⣷⣶⣦⣈⡟⢿⣦⡛⣶⣿⠿⣏⣤⣶⣷⣶⣶⣤⣄⣴⣾⣿⠿⣯⣧⣤⣄
⢾⣽⣿⢿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣟⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠠⣆⡄⢀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⢉⠐⡀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⢸⣷⡇
⠀⠀⠀⢠⣼⣿⠅⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠘⡗⡁⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠹⣼⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢾⡏⠀⠀⠀⠀⠀⢼⡟⠁⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃
⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢹⡆⡀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠀⠈⢶⡀⠀⠀⠀⠀⠀⠉⢶⡀⠀⠀⠀⠀⠀⠘⣿⡦⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠠⣸⣇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡀⢧⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣿⠂⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⢼⡗
⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠘⠀⠒⠂⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠲⢤⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠐⠀⠀⠀⠀⠀⣆⠓⠀⠀⠀⢀⣠⠴⢊⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠙⠲⠤⣀⣸⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠂⢸⠂⠂⠴⠊⠉⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡆⠀⠀⠀⠈⠃⣢⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣁⣠⣔⡲⠒⠀⠸⠀⠠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⣠⣶⡿⠟⠛⠛⠛⠛⠿⣶⣤⠠⠤⠤⢀⣴⡾⠟⠛⠋⠛⠛⠿⣿⣮⡀⣩⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣦⣼⣿⠃⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢻⣧⣿⠇⠀⠀⡠⣤⣄⠀⠀⠀⠀⠀⢹⣿⣴⣶⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠈⢉⡿⢻⣿⠀⠀⠀⠀⠀⢸⣷⣿⣿⠀⠀⣸⣿⣿⡀⠀⠸⣷⣾⣿⠆⠀⠀⠀⠀⢸⣿⡛⠉⠁⠈⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣸⣿⣆⠀⠀⠀⠀⠀⠙⠛⠁⠀⢠⣿⣿⣿⣧⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⣾⡟⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠁⢀⡴⠁⠘⢿⣷⣄⡀⠀⠀⠀⢀⣀⡴⠛⠉⠀⠈⠙⠳⣤⣀⠀⠀⠀⢀⣀⣴⣿⠟⢆⠀⠈⢷⣤⡀
⠀⠀⠀⠀⠀⣠⠖⢉⡞⠁⢀⠎⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⢻⡀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⢣⡀⠈⢳⡉⠲⣄
⠀⠀⠀⢀⡼⠃⢀⡞⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⢦⡀⠀⠀⣠⠞⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢳⡀⠈⢦⡀
⠀⠀⢠⠞⠀⢀⡾⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡙⠢⢞⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢷⡀⠀⠳⡄
⠀⢠⠏⠀⠀⣼⠁⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⣧⠀⠀⠹⡄
⢀⡏⠀⠀⢸⠇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠓⢄⠀⠀⠀⠀⢀⠔⠈⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠸⡆⠀⠀⢹
⣸⠀⠀⢀⡟⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣉⣺⠉⠐⣆⡀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢿
⡏⠀⠀⢸⠃⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢉⠽⠤⠼⡉⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡆
⡇⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣀⡤⠋⠀⠀⠀⠀⠉⠦⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣇
⢇⠀⠀⡏⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿
⢸⡀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⢀
⠀⢧⠘⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⠀⡼
⠀⠘⣆⣷⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⣰⠃
⠀⠀⠘⢿⡄⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣸⡷⠃
⠀⠀⠀⠈⢷⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⡟⠁
⠀⠀⠀⠀⠈⢷⡀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣠⠟
⠀⠀⠀⠀⠀⠀⠻⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢟⠾⢽⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣞⣭⠶⢛⠝⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢀⣸⢶⠟⠋⠻⢟⣿⠛⡿⠿⣷⡶⠶⠶⣶⠶⠶⠶⠶⠶⠶⣾⡿⠿⡟⢻⣟⠟⠋⠛⢶⢮⣀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣼⠊⠉⠙⢢⡟⣏⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢈⡟⣧⠚⠉⠉⢻⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣿⣿⣿⡆⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣵⣷⣾⣷⡝⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⠋⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡔⣬⣭⢆⣤⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⠿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢿⣎⢫⣾⠿⠿⣿⣿⡜⣿⣿⣿⣿⠿⠏⣾⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⠻⣿⣿⡸⢟⣫⣶⣷⡕⢉⢡⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⢸⣿⣿⣿⠀⢹⣿⣿⣿⣿⠿⠋⠁⠈⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⢸⣿⣿⣿⠀⠀⠉⠛⠛⠁⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⢠⣤⣤⣴⣿⣿⣦⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠉⠛




⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣻⣿⣛⣿⣻

⠀⠀⠀⣠⣴⣶⣶⣿⣿⣶⣴⠦⣤⣀
⠀⠀⣸⢿⣯⣽⣿⣿⣿⣝⢿⣛⣋⣿⣿⣷⣦⣄⡀
⠀⠀⠘⢴⣷⣾⣿⣻⣶⡲⢿⣿⣻⡿⣿⣿⡟⣻⣿⣶⣄⡀
⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣾⣷⣌⣻⡿⠳⢿⡻⣇⡛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⣿⣧⠙⣧⡀⠙⢶⡷⡖⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣭⣸⣿⣧⠘⣿⢦⡀⠑⣬⢧⣤⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣻⡽⣷⢹⣴⡟⢶⡉⠳⣝⢦⡈⠳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⢿⡿⣿⣿⠳⣄⠙⢦⠈⠳⡻⣧⠘⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⣾⢧⡀⡑⢦⡈⢪⣳⣆⡙⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣦⣝⢦⡤⡙⢮⡙⢿⣾⠌⢣⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠿⢷⣿⣤⡠⡘⢦⡙⢷⣄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠻⣦⣤⠙⢦⡙⢷⣝⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣶⣼⡢⣿⣶⡻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠈⢿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠹⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠺⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣀⣀⠀⠈⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡞⣩⡍⢳⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⢺⡀⠀⠳⢥⡤⠟⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠁⠀⢣⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡧⣼⣿⡧⡉⠁⢩⡅
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡙⢰⣿⠟⢹⡜⠀⡲⠁
⠀⠀⠀⠀⠀⠀⡿⢟⣵⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠿⢋⠔⠿⠒⠉⠀⣀⠠⠄⠐⠒⠒⡄
⠀⠀⠀⠀⠀⠀⠙⣟⠵⡉⢢⠣⡀⠀⣀⠀⠀⠀⠀⢀⡌⢀⠔⠁⠀⣠⠔⠂⠉⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡑⢄⠑⡌⢾⡇⠀⠀⠀⠀⠰⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠣⡈⠢⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡜⢄⠑⢌⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠱⡀⠢⡑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢌⢦⠔⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠑⠤⡀⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠄⠠⠾⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠒⠈⠁⠀⠀⠉⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠟⠻⠉⠉⡿⢻⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠙⠃

⣿⣿⣿⣿⠿⠛⠉⠉⠁⠈⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠉⠙⢿⣏⠿⣿⣿⣿⣿⠟⠋⠀⠈⠻⢿⣿⣿⣿⡿⠿
⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠢⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⣸
⣿⣿⠄⠄⠀⠀⠀⠤⠀⠀⠄⠀⠠⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣀⠀⠀⠀⣀⠀⢀⡀⠀⣀⠀⢀⣀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠐⠀⠐⠂⠀⠒⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿
⣿⣿⣿⣿⣿⠋⠁⠂⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣶⣤⢀⡀⠀⡀⢀⣀⢤⣶⣿⣿⣿⣿⣿⠟⠑⠂⠀⠂⠀⠂⠙⣿⣿⣿⣿
⣿⣿⣿⡿⠋⠀⠀⢈⠐⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠃⠀⠀⡀⠀⡀⠀⠈⢿⣿⣿⣿⣿⠋⠀⠀⠰⠐⠒⠢⠀⠀⠘⢿⣿⣿
⣿⣿⡿⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠘⠀⠀⠰⠀⠀⠀⠻⣿⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣿
⣿⣿⠀⠀⠀⠀⠈⠓⢴⡤⠠⠉⠀⠀⠀⠀⠟⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⢿
⣿⣿⠀⠀⠀⠀⠀⠀⢀⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠓⠐⠆⠘⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣾
⣿⣿⠀⠀⠀⠠⢀⠠⠒⠒⢄⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣐⢆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⢠⠐⠐⢄⠀⠂⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡿⠁⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣧⡀⠀⠀⠀⠀⠀⣀⣄⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣤⣶⣶⣦⣄⡀⠀⢀⣀⣠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣬⣍⡻⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⠉⠛⠷⣮⣝⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠙⠻⣦⣙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠰⣄⠀⠀⠈⠙⢷⣌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠇⠀⠀⠀⠻⣆⠀⠀⠀⠀⠙⢷⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣿⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠀⠀⢸⡇⠀⠀⠀⠀⣶⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⢸⡇⠀⠈⡇⠀⠀⡇⠀⣿⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠘⣿⠀⠀⠀⠀⠀⢳⠀⠋⢠⡆⠀⢹⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡆⢸⣇⠀⢿⡄⠀⠀⠀⠀⢸⠀⠀⣼⠇⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡀⠸⣇⠀⢿⣄⠘⣧⠀⠀⠀⠀⣸⠀⣰⡟⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⡇⠀⢿⡆⠀⠻⣦⠈⠀⠀⠀⢀⣿⠠⠏⠀⢠⠇⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣷⠀⠈⢻⣆⠀⠈⠃⠀⠀⠀⢸⡏⠀⠀⣰⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠙⣷⡀⠀⠙⣧⡀⠀⠀⡄⠀⣿⡇⢀⡾⠋⠀⣀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣷⡀⠀⠈⠻⣦⠀⠈⣷⠀⠀⠀⢸⡿⠀⡾⠁⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡟⢷⣤⡀⠀⠙⢧⠀⠀⠀⠀⢀⣿⠁⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⢁⣀⠉⠻⣦⡀⠀⠃⠀⠀⢀⣾⠏⠀⠴⠛⠉⠀⠀⣀⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⣿⠈⢉⡀⠖⡈⢻⡄⠀⠀⠀⣼⡏⠀⣀⣤⡴⠶⠟⠛⠋⢩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠀⢸⡿⣦⡁⠄⢻⠀⠀⣼⠏⠰⠛⢉⠠⠀⣤⡄⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡻⣆⢸⡇⠘⢿⡖⠈⢆⣼⠏⢀⣀⣠⣤⠄⠀⠉⠀⢠⡿⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢿⣬⡇⢀⡈⣿⢀⣿⣿⠾⠛⠉⢩⡟⠀⠀⠀⣴⢏⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⡇⠈⢳⣼⣧⢸⡿⢛⡁⠀⠀⣰⠟⠁⣀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⡇⢀⠀⠹⣿⣿⣷⢟⣡⠴⠾⠿⠞⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⡇⢸⡇⠀⣿⠋⠉⢁⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠻⣼⣇⢠⡟⢠⡾⢋⣠⠀⠀⢀⣾⠃⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣌⠻⣾⠃⢩⡶⠛⠁⣀⣴⣟⣵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢸⣴⡟⠁⢚⣻⣿⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⢹⣿⢻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣟⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⣿⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⢶⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⢿⠀⠀⠀⠀⠀⠙⢷⣽⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢁⣶⣿⠀⠀⠀⠀⠀⢤⡈⢷⣭⣛⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣯⠀⠀⠀⠀⠀⠀⠹⣄⠙⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣿⣿⣿⣿⢿⣿⠡
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢹⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠣⣌⣛⠻⢿⣿⠟⢳⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⣿⣿⢳⡖
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⣁⠌⠙⢁⠀⠀⠀⠀⠀⠀⢄⢨⡙⠿⣿⣿⡄⠋⢀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢏⣠
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⣿⣿⣡⡴⢚⡯⠀⠀⠀⠀⠀⡀⠈⠓⠿⢿⣿⣿⡗⢀⠒⠛⠀⠀⠀⠀⠀⠀⣦⣿⣿⣿⣿⣿⣿⣯⡅
⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⢸⣿⡿⠋⣊⣿⠇⠀⠀⠀⠀⠀⣈⣲⣶⣮⣽⣭⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣗⠉
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢸⡯⠐⠉⢀⡥⠂⠀⠀⠀⠀⠀⠠⢤⣭⣯⣿⣿⣿⣿⠀⠰⠁⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠂⢀⣤⡄⠀⠀⠀⠀⠀⠀⠰⣬⣿⣻⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣟⡠⠀⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⢿⣷⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⡿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⣶⣠⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣻⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⡗⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣽⣿⣿⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣻⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⣿⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⣿⣡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣇⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⠁
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⡀⠌
⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⣼⡀⠀⠀⠀⠀⠀⣀⡤⠌⠐
⠀⠀⠀⠀⢹⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣿⠀⠀⡀⠀⠀⠀⠁⡀⠀⠘
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⡄⠀⠀⠇⢀⡌⠀⠀⠀⠔⡡
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢻⡇⠀⠀⢀⡞⠀⠀⡀⠐⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠀⠈⠃⠀⠀⡜⠀⠀⢀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢀⣠⣪⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⠀⠀⠀⠀⠀⢰⡇⠸⡄⠀⠀⠀⡌⠀⠀⠀⢀⡴⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣼⠇⠀⠑⠀⠀⢀⠃⠀⠀⠐⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⡿⠰⡀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣸⣧⠀⠘⢄⠀⠀⠈⠀⠀⠠⠊⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣿⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡆⠀⠙⠄⠀⠀⢠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢬⡳⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡘⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢯⡄⠀⠀⠘⠇⠘⢦⡀⠀⠀⠘⡆⠀⠀⠀⠀⡠⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠃⠀⠀⡀⠂⠀⠀⠉⠀⠀⠀⢷⠀⠀⠀⢰⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠸⡆⠀⠠⡇⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⡀⠀⠀⠈⣦⠑⢴⣤⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣾⠃⠀⠀⠀⠘⣆⠀⠠⠉⠓⠀⠀⠈⣧⠀⠀⠀⣠⠞⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠘⣆⠀⠰⠃⡠⠄⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⠖⠀⠀⠀⠀⠀⠀⠹⡆⢠⡈
⠀⠀⠀⠀⠀⠔⠛⠛⠛⠶⣄⠈⣿⣿⠿⠛⠉⠁⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠉⠀⠒⠒⠺⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⡈⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠙⣆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠈⠀⠙⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣿⢻⣿⣿⣿⣿⣻⣿⣯⣶⣀⣤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃⠀⠉⠀⠿⠟⠻⢛⠿⢟⠻⠿⠿⣿⣛⣛⡛⠻⠟⠋⠁
⠀⠀⠀⠀⠀⠀⢠⡈⠀⡀⣄⢀⣤⣦⣶⣀⣀⣈⠛⠿⣟⣫⣟⣫⣉⡻⢧⣤⡆⡀
⠀⠀⠀⠀⠀⢠⠀⠀⢄⣳⣿⣿⣿⣿⣿⣟⣿⢦⣤⣤⣈⣻⡙⢿⣿⢿⣧⣌⠻⣦⡀
⠀⠀⠀⠀⠀⠊⠀⠒⣿⣿⣿⣿⣿⣷⣶⡟⣷⡸⣟⢿⣿⢯⠿⡷⣤⣌⡳⢟⡻⣮⡻⣄
⠀⠀⠀⠀⠀⠘⢄⠀⠈⣻⣿⣿⣿⣿⣿⣿⣾⢿⣯⠰⢽⢿⣅⠻⣜⣟⢿⣦⣜⢊⣷⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠬⡹⣿⡶⣿⢷⣾⣿⣿⡷⠘⢿⣿⠆⢿⣷⣄⡻⢿⣧⠋⢻⣿⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢻⣿⣻⣿⣿⣿⣿⣷⣦⣉⣀⣿⣿⣿⣿⣦⡙⢳⡈⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢻⣯⣿⣿⣟⣿⣿⣿⠊⣿⣿⡻⢿⣿⣷⡛⣿⣷⣄⠁⢹⠻⣿⣶⣄⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣟⣶⣾⣿⣿⣻⣌⣻⣽⣿⡝⣿⣿⣿⣾⣧⣝⢸⣷⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⣭⣀⠈⠻⠾⣹⡿⣿⣿⢺⣿⣏⣿⣿⡿⣿⣷⣭⣥⣼⣿⣿⣿⣿⣿⣦⣽⡿⣿⣯⣒⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢦⣘⢷⣿⣼⣿⣿⣿⣿⢿⣿⣈⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢈⡙⠋⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠓⢌⡻⣿⢿⠿⣟⢉⣿⣿⣶⣿⣿⣻⣿⣿⣿⣿⣿⣾⡆⠀⠀⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⣶⣿⣿⣿⣿⣿⣝⣿⡿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⢹⣿⣷⣹⣿⣧⠘⠋⣸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡟⣿⣿⣻⣿⣧⡴⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⡿⣦⣿⣿⣿⡹⣿⣹⣿⣿⣿⣿⠿⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡿⣷⠘⠟⣏⠹⠏⢹⡿⣿⣿⣦⣔⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢌⠀⠹⣏⢿⡽⣟⠻⢿⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠻⠌⢻⣭⣯⣤⡁⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⣿⡻⣿⣿⣷⣶⣦⣀⠃⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⢿⠻⣿⡍⠛⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒


⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣶⣆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⣄⠀⠀⠀⠀⠀⢠⣾⣿⣯⣤⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠈⣷⡀⠀⠀⠀⠀⣿⣿⣿⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⠤⣄⢀⡤⢤⡀⠀⣿⢀⡟⢀⡿⢻⡆⠀⠀⠈⣿⣟⡋⠙⠛⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣼⡁⢹⡏⠀⢹⡏⠀⢬⣧⣾⡏⠀⠀⣸⢣⡟⢻⡄⠀⢀⣿⣿⣯⣙⡉⠉
⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⣤⣿⠞⢻⠀⢸⡇⠀⣼⠀⠀⣼⠟⣼⠇⠀⠀⣠⠟⣴⢛⡷⠀⠛⣿⡿⠿⠏⠉⠈⠅
⠀⠀⠀⠀⠀⢸⠈⠳⣄⠀⠀⠀⠀⣨⣗⠻⡄⠘⣇⠀⣇⣀⣿⣀⣠⡟⢠⡟⠀⢀⡴⠋⠀⣴⡿⠷⣄⠐⢿⡿⠒⡖
⠀⠀⠀⠀⠀⠈⡗⢤⡀⠙⢦⣄⢀⣿⡙⣦⢹⣤⡾⣿⣻⣾⣽⣬⣽⣿⣿⣷⡞⣋⣠⡴⠞⠋⢃⣴⣿⣶⣿⣿⣶⣿⣶
⠀⠀⠀⠀⠀⠀⢹⡦⢌⡀⠀⠈⠙⢧⣝⢮⣟⣵⢟⣋⠁⢿⣷⡄⣀⣘⠻⠿⢿⣍⣁⣤⠴⠚⢉⣠⣿⣯⣩⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢳⡠⠤⣀⣀⣀⠀⢨⢽⣿⠁⠸⣾⣧⣈⣿⠏⠿⠟⠁⡞⠀⠈⡷⠖⠒⠚⣉⣀⣼⣿⣿⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣖⣂⠀⢈⣉⣻⡀⢷⣴⡧⠈⠋⠛⠞⠀⠀⠀⠀⠙⣶⠾⠿⠿⣿⣽⣭⡵⠛⢻⣿⡿⠓⠒⠒⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠛⢶⣾⣛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣿⣿⠿⠛⠋⠀⠀⠚⣿⠟⠒⠲⡖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡏⠓⢦⣄⣀⣀⣀⣠⡴⣺⡿⠛⠛⠛⠁⠀⠀⠀⠀⠀⢰⣿⣿⣗⣒⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⣏⣁⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠞⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣽⣭⣅⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⣿⡟⠛⠓⠒⠒⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⠟⠒⠖⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣤⠶⢟⠋⠁⠁⠈⠙⠻⣶⣿⣟⠛⠻⢶⠶⣤⣀
⠀⠀⠀⢀⡾⠫⠀⠀⠀⠀⠀⠀⠠⣬⣅⣠⠙⡛⣻⡷⣤⣄⣀⡉⠛⢲⣦⣤⣄⣀
⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢦⣀⠁⡙⠶⣬⡉⠉⠙⠿⣿⣿⣿⣿⣭⣉⣹⡀
⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠹⠟⣶⣅⣙⠛⣿⣯⣍⣉⣉⢿⣟⡛⣿⣯⡙⠲⣄
⠀⠀⠀⠀⠙⠷⢦⣤⡀⠀⠀⠀⠀⣿⣌⡙⢲⣬⣉⣉⣣⣽⣯⣙⠻⠟⠉⣿⣿⣶⣬⠽⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⢠⣝⠻⣿⣷⣬⣿⣿⣿⣭⣉⣉⣻⣿⣿⡿⠿⢯⡟⠷⣌⢻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠚⣵⢮⣽⢿⣯⣽⣿⢾⣿⣿⣿⢍⣿⣿⣷⢶⡿⣶⣿⣴⣿⣿⣟⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡈⠳⢮⣵⣿⣿⣿⣷⣦⣭⣟⣻⣿⣿⣿⣷⠾⣿⣿⢿⣟⠹⢿⣾⡉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠺⣶⣾⣶⢭⣙⡷⢧⣍⣽⣿⠛⢿⡷⣄⢹⢿⡘⣷⣆⡻⢿⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠾⣿⣿⣟⢷⣿⣷⣿⡎⠻⣟⢷⣜⡄⢈⠷⡜⠻⢿⣮⣻⠍⢿⣿⣦⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠰⣶⡿⣿⣭⣤⣷⣍⣛⣟⣀⣿⣷⣟⢿⣼⠀⠹⣦⠚⢿⣿⣿⣏⢁⠻⣷⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣮⣹⣧⣽⣿⢧⡌⠹⢿⣿⡿⣿⣿⣭⣟⠀⠀⢋⢷⣠⢽⣿⡿⣼⣧⠼⢷⣜⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣬⣿⣷⣶⣿⣇⠙⢷⣎⢻⡷⣦⡛⢿⣿⢷⣦⡈⠂⢿⣶⣹⣇⠈⢻⣴⣌⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣴⣿⣿⣿⢳⣾⣿⣷⣶⣿⣿⣧⣹⠿⣦⢻⡾⣿⣟⢶⣄⡙⠏⢿⣅⠱⣿⣿⣿⣿⣟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣿⣿⣿⣯⣟⣷⣷⣾⣿⣿⣿⣿⡿⢶⣦⡾⣯⠺⡷⣽⣯⡟⢶⣿⢋⠳⣌⡻⢿⣿⣽⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡻⣏⡛⠛⠻⣿⣟⠻⠾⣿⣿⣷⣶⣖⢻⣯⣻⣿⣯⣟⢮⢟⢣⡙⣧⠀⠎⠻⣎⣿⣿⢿⣮⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢭⡙⠷⣶⡌⣿⣿⣶⡖⠛⢻⣿⣯⣿⣿⣿⣻⡟⢿⣿⢦⣤⣡⣿⣶⣄⠀⠈⠻⣿⣯⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣾⣿⣿⣿⣿⠿⠮⣹⣿⣿⣯⣉⠻⣶⣄⣙⣿⣏⣾⣗⣀⣉⡳⣦⣀⠈⠓⢏⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⣙⢿⣿⣿⣿⣷⣭⣿⣿⣶⣤⣘⠙⢿⣿⣿⣿⣯⣭⡡⢮⡍⠲⣄⠈⠳⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠦⠟⠛⠿⣿⣿⣯⣽⠿⢿⣿⣿⣮⠿⢌⡙⠛⠿⣿⢷⣾⣓⠈⠳⣦⡈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠠⢄⣙⣛⣃⣠⡿⢿⣶⡌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣌⣙⠈⠀⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡙⠟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠁

⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠺⣿⣿⣄⠀⠀⠀⠀⠀⢀⣵⠖⠛⠛⢤⡀⠀⠀⠀⠀⠉⠛⠛⠿⢻⠿⣿⣿⣷⠀⠀⠐⠿⠟⠟⠛⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠈⠻⣟⠀⠀⠀⡴⣴⡿⠋⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡀
⠀⠀⠀⢀⣾⠀⠀⠀⠈⠀⠀⢰⣷⡯⠀⠀⠀⠀⠀⢠⡿⠏⠀⠀⠀⢀⣤⣄⣀⢴⣤⡦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠚⠛⠛⠉⠛⠉⠙⠛⠿⠶⢾⣤⣀⣣⡀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⣼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⢾⣆⢀⡀⠀⠀⠘
⠀⠀⠀⠀⠘⠻⣦⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢹⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠛⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠁⠙⢶⣦⣤⣤⣤⠖⠀⠀⠀⠈⢿⣿⣿⡿⠀⠀⠀⠀⣀⣴⣾⣷⣿⣿⣿⣿⣿⠶⠀⠀⢠⠀⠀⠀⠉⠉⠛⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⢠⣾⣿⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠂⠀⠀⡠⠔⠒⠒⠀⠀⠀⠰⢻⣧
⠀⠀⠀⠀⠀⠀⣀⣼⠟⠛⠉⠀⠀⢀⣰⣴⠶⠾⣶⠦⢄⠀⠀⠀⠋⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⢀⣼⡇⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⢘⣶⣿⠁⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠿⠀⠀⠀⠀⠀⠀⢀⣾⠛⠀⠀⠀⠀⠄⣼⠇
⠀⠀⠀⠠⣼⡿⠃⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⣠⣯⣷⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣠⣾⠗
⠀⠀⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠾⠷⠿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⠀⠐⠛⠋⠁
⠀⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠜⢉⡙⢳⠉⠀⢀⣀⠐⠀⢄⠀⠁⠲⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢤⡀⠈⣄⡀⠉⠉⠂⠃⡁⠀⠀⠂⢀⠘⠀⣸⡀⠈⠰⠀⡀⠃⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢳⠀⠀⠒⠈⠉⠀⠈⠀⠀⡀⡔⠠⡌⠳⢀⣀⣠⢴⠁⠈⠉⠱⣔⢀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⣀⣄⢠⣀⣂⠌⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⢠⠂⣉⠆⡀⠀⠈⢿⠈⠉⠗⠀⡜⠈⠐⠤⢤⠎⢉⣍⠓⠄
⠀⠀⠐⠃⠉⠀⠈⠉⠙⠦⡘⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠈⠀⠀⠀⠀⠈⠲⠤⠤⠒⠀⠀⠀⠀⠴⠄⠷⠌⠓⠛⠒⢤⡀
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠣
⡆⠀⠀⠀⠀⠀⠀⠀⢠⡆⢰⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠁⠉
⠘⢄⠀⠀⠀⠀⠀⠒⠠⠤⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠤⠤⠀⠂⠁
⠀⠀⠙⠲⠤⠤⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠐⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠓⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠲⢂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠃⠀⠀⠂⠀⡇
⠀⠀⠀⠄⠀⠰⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰
⠀⠀⠀⡀
⣀⣀⣀⣈⣀⣀⣀⣀⣀⣲⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣒⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠉⣢⠀⠀⢀⠀⠊⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠒⠀⠤⠚⢹⠀⠀⡇⠐⠀⠀⠀⠒⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠦⡤⠀⠤⢤⠊⠠⢄⠈⠒⠄⠤⠀⣀⣠⡮⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⣰⡆⠀⠀⣸⡇⠁⢞⢀⢸⡄⠀⠀⢀⠿⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠘⡗⠂⠒⢹⠇⣴⣘⠳⡍⢷⠄⠤⠋⠜⠀⠤⠄⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⣀⡠⠌⢥⡔⠋⡼⠃⢹⡀⠀⠈⢄⠀⣈⠀⠀⠀⡀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠻⠧⣄⠀⠘⢦⠀⠀⢸⢧⢀⡠⠄⣍⠀⠑⢄⠀⠈⠑⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣿⠀⠀⠠⠓⠈⠓⠠⠼⠋⠀⣸⠀⠏⠀⠐⢸⠣⠀⠈⠑⢌⠳⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⣄⡠⣅⢦⠀⢸⢻⡐⠔⠁⠀⡘⠃⡄⠀⠀⠀⣳⠠⠀⠀⣠⠄⠰⢸⡄⠀⠈⢷⠹⡄⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠘⢘⣈⠥⠈⡀⡀⠘⡆⠇⠲⣄⡄⠀⠀⠧⠐⠐⠒⢱⠀⠢⠐⠡⠤⣀⢐⡇⢀⡰⣿⠀⢻⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠍⢐⡠⠧⠒⠡⡀⢹⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⡈⠘⠲⠄⠴⠂⠑⠚⠓⠉⠀⣿⠀⠈⠇⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠒⠉⣅⠀⠀⠀⣧⠑⣾⢸⠀⠀⠀⢠⠀⠈⡆⡎⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⡠⠀⠀⠈⢯⠑⠦⠴⣿⠈⢣⠘⡆⠀⠀⢀⠇⠀⣿⡇⢰⠇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⠀⢸⡀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡝⢁⡀⠈⠒⠈⠑⣦⣠⠇⠀⠈⢆⣧⠀⠀⡸⠀⠀⣿⡇⣸⠀⠀⢰⠀⠀⣼⠀⠀⠀⠀⠀⣿⠀⠘⣇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠖⢁⡠⠀⢀⠔⡥⠊⠛⠒⠦⣀⠤⣿⠀⢰⠃⠀⢰⣿⡇⣿⠀⠀⢸⠀⠀⣿⠀⠀⢠⡇⠀⣿⠀⠀⠹⡄⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢠⠎⠀⣠⠞⢸⠅⠀⣀⣀⠠⢬⡂⡈⡄⢸⠀⠀⣸⠸⡇⠘⢦⡀⢾⠀⠀⣿⠀⡇⢸⡇⠀⢻⡀⠀⠀⠙⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡟⠠⡏⠈⢀⡾⢀⡀⠀⢀⠀⠐⣿⠑⣳⠛⠀⠀⡿⠀⢷⠀⠀⠉⢹⡀⠀⣿⠀⠀⠈⣧⠀⠀⠳⣠⠎⠐⠂⠠⠂
⠀⠀⠀⠀⣰⠉⠁⠒⠀⣸⠃⠀⢰⠇⠀⢸⠁⢠⠃⠀⠛⠛⠋⠈⠢⠟⠀⠀⠀⡇⠀⠸⡆⠀⠸⢀⡇⠀⡿⠀⠀⠀⠘⣦⠀⠀⠉⠰⡇
⠀⠀⠀⠀⣿⠀⠀⠀⣰⠃⠀⠀⡟⠐⠤⢘⡇⣼⠀⠀⠀⡀⠀⡄⠀⡆⠀⡀⠀⠀⠀⠀⡇⠀⠀⡞⠀⢠⡇⠀⠀⢠⠀⠘⢧⡀⠀⠀⢻⡀
⠀⠀⠀⠀⢿⠀⠀⠀⡏⠀⠀⢀⡇⠀⠀⠀⠙⠃⠀⠀⢰⠇⠸⠃⠀⠹⡄⢧⡀⠀⠀⠀⠀⠀⣸⠁⠀⠼⠁⠀⠀⠀⠁⠀⠈⢳⡀⠀⠀⠳⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠷⠀⢀⠞⠁⣀⠤⠀⡴⠂⠀⣰⠏⠀⠀⣠⠞⢉⠁⠀⠳⡄⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢴⡁⠀⠈⠁⠀⢀⡀⠰⠞⣡⠖⠊⠉⠀⠀⢀⡆⠀⠀⠘⠆⠀⠀⠀⠀⠳⠔⠋⠉⠉⠉⠁⠀⠀⠉⠒⠦⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈