Fungal Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠈⠉⠀⠀⠉⠁⠂⠐⢤⡁⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠀⠁⠒⠲⠠⠀⣤⢀⠀⠀⡀⠀⢠⢀⠀⠀⢀⡀⢠⡀⣱⣵⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠈⢰⡶⡤⣤⣬⣀⣤⣀⠤⠀⠠⠄⠀⠤⠈⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢀⠺⠔⠒⠈⢀⠀⠈⠁⠊⠀⠀⠀⠀⠀⠱⠀⠀⢸⣄⡀⠉⠉⠻⢼⣿
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⣒⣱⡾⠋⠉⠉⢻⣿⣤⢂⣈⠀⠀⠀⠀⠀⣇⠀⢸⠁⠁⠐⣤⢤⣤⣼
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⣼⢰⡇⠀⠀⠀⠀⠀⢿⣭⠩⠭⠁⢿⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣴⠯⣸⣷⠀⠀⠀⠀⠀⠀⢹⡦⣈⡉⠙⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠘⢻⡟⣿⠀⠀⠒⠠⠤⢀⣸⣷⣆⡠⠌⠁⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠈⣠⣙⣿⠀⠀⠀⠀⠈⠻⡧⠈⠁⠀⠀⢀⠀⠁⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⢇⣀⠠⠄⠐⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⣀⠠⡄⠀⠀⠀⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⣾⠠⠄⠀⠀⡑⢦⡘⠘⣾⡀⠀⠀⠀⠐⣿⡇⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠃⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠈⠻⣶⣿⠃⠈⠀⠀⠆⣿⢻⠇⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢡⠞⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⢰⠂⡦⣷⣿⡌⠏⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠋⠀⠀⠀⠀⠀⠀⠀⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣞⢠⠁⣧⠈⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⠀⠂⡀⠀⡀⠀⡀⠀⣣⢈⡇⠄⠀⣀⢀⠀⠀⠀⠀⠀⡸⣇⢹⡌⡏⠇⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠙⢄⣴⢀⡥⡀⢧⠐⡓⣾⡃⣼⡇⢠⠀⠀⠀⠈⢀⠀⣤⠼⠱⡏⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠳⠃⠘⠃⠇⠻⠈⠻⠀⣅⠠⡄⢰⣾⣟⠛⣧⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠄⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣼⢿⡇⠀⢠⣀⠀⢀⠀⣠⣄⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠟⠛⠁⠺⡅⠀⠃⠂⠘⠻⢿⠀⡬⠉⠉⠟⢀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠸⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⡀⠃⠀⠀⠁⠌⠃⠀⢿⡋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⢰⠿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠁⠀⠀⠀⠑⠆⠀⠀⢸⣷⠀⣰⣼⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠈⠉⠁⡿⣿⣷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠘⡇⠈⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⡅⠀⠈⡟⠂⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠊⠀⠀⠈⠀⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⡆⢰⡄⢀
⠀⠀⠀⠀⠀⠀⢀⡄⢀⠰⡏⠂⡀⠀⢰⡆⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⠀⢀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠰⠈⢳⣼⠃
⠀⠀⠀⠀⠀⠀⠘⣷⡞⢀⠀⢠⡇⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠑⡦⢀⠀⠀⠀⠀⠀⢀⢀⠀⠸⣿⣀
⠀⠀⠀⠀⠀⠀⣤⣿⡟⠆⠀⠘⠡⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣰⣤⣤⣴⣥⢤⠤⠄⢀⡀⠈⡄⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠂⠀⠞⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣀⣤⣤⣴⣶⡿⣟⡿⢿⡛⠉⠀⠀⠐⣰⠃⠀⢣⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣏⡛⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⣸⢛⣿⠉⠀⣠⣴⣶⠶⢶⠒⢛⢻⣏⣿⣿⣿⡿⣊⡽⠗⠊⠁⠀⠀⠀⠀⠐⠁⠀⡀⠛⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠰⠀⠀⠀⠀⠀⠀⢀⣼⡟⢀⣠⣴⡟⠹⢿⣿⣷⣦⣳⣾⣿⣿⣿⣿⠛⣋⢥⣴⣮⣤⣀⣀⣂⡠⠈⠀⠀⠀⠀⠈⣧⡀⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡄⠰⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠧⣤⡶⠿⢿⢻⡿⢿⣿⠿⠿⠛⠋⠁⣈⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠋⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⡇⢰⡀⠀⠀⠀⣿⣿⣿⣿⣟⣃⠉⠀⠀⡀⠉⣁⣺⣋⣴⣶⣶⣶⢦⡼⣿⣿⣿⠿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣇⠈⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣄⣀⢢⣜⣿⡿⣿⣿⣿⣿⠅⠈⠁⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⡶⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⠀⢹⣼⠀⠀⠀⠀⠀⠰⡆⠀⠂⠀⠀⠀⡄⠀⠠⠀⢀⣸⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⢤⣄⠘⠐⡀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠄⢀⠀⣏⠀⠀⠀⠀⣀⡰⠌⠀⠙⠀⠀⠀⠀⢨⠀⣠⢸⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠈⠳⠄⠀⠈⠣⡀⠢⣀⢀⠀⠀⠈⠉⠁⠂⢀⡎⠀⣿⠀⢀⢂⣴⣿⡆⠀⠀⠀⠀⠀⠀⠘⣤⠞⢹⣿⠙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⢀⣶⡀⠀⢀⢃⣿⣤⠀⠀⠀⠀⠀⢈⡇⠄⢻⣇⡈⠘⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⣮⣉⠀⢻⡎⢸⣿⣿⣢⣠⣶⣶⡀⠁⡄⠀⠈⣿⠗⠀⠈⠿⠛⣻⣿⣷⣤⣤⣤⠤⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡠⠷⢾⣶⣿⠏⣿⣿⡿⢋⠹⠿⢮⣶⠀⠀⢀⡟⠃⡀⠄⠀⢠⡿⠉⣻⡎⠀⢄⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠿⣷⣿⣿⡇⢉⠻⠯⢁⣐⣲⣿⣧⣤⣈⡴⠊⠀⠀⢀⣾⠁⢘⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠦⢇⠀⢰⣿⣿⣿⣿⣿⣇⣿⣦⣰⣾⡿⠿⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣷⠶⠿⠿⠿⠟⢻⠶⠶⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢿⠻⣧⠀⠀⣀⣠⣀⡴⠀⠀⢰⡋⠘⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⡏⠰⠆⣤⣶⣾⣿⣿⣿⣿⣷⣾⣿⣿⣭⠛⢁⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠠⢈⣱⣶⣽⣿⡿⠟⠛⣿⣿⣿⣿⣶⣶⣾⣿⣶⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡉⢢⣿⣿⣿⡇⠀⠀⣿⣿⣿⣏⡻⠋⠙⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠁⠉⠀⠳⠟⢺⡇⠀⣰⣿⣿⣿⣦⡉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣶⣦⣄⡀⠀⠀⠀⠐⠀⠐⢈⠿⣷⢦⣽⣿⣷⠹⡏⠛⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢠⠀⡀⣠⣼⣿⣟⣿⣛⢋⣭⣿⡻⣦⠀⣤⠘⠠⣾⠵⣿⣾⣿⣿⣿⣿⣿⡗⠂⠀⢀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣾⡿⢿⣯⣿⣿⣾⣷⣿⣿⡿⠟⠉⢁⠀⠀⣤⠀⣤⣴⡿⢻⣛⣻⣩⣍⣿⣿⡷⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠋⠡⢼⣿⣿⣿⣿⣿⣿⣿⡿⢿⠟⠳⠂⠒⢀⠙⠀⠀⠀⣤⣿⣶⠿⣿⣿⣿⣍⡙⠟⠿⠿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠛⢾⣿⣿⣿⢻⣿⣻⡆⠀⠀⠀⠀⠈⠀⠀⠀⠾⡟⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⠐⠂⢹⣿⣿⣿⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⣿⣿⣿⠘⡿⠋⠔⠀⠀⠀⠠⣀⣶⣤⡘⠒⠚⣻⣿⣿⣿⡏⣿⣿⣿⣿⠀⠈⠁⣼⣿⣿⣽⣯⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⠀⠀⠀⠀⠀⣤⣤⣶⣿⣿⣿⣧⡀⠙⠁⠉⠉⣿⠁⣽⠏⠈⠀⠀⣠⣾⣿⣿⣿⠉⡤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠛⢫⣿⣿⣿⣿⣿⣿⡿⠆⢠⣤⡄⢨⠴⠋⢀⣤⣶⣾⣿⣿⣯⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠩⣌⣹⠛⢿⣿⡏⠿⠁⠀⣸⡿⠁⠀⠀⠀⠈⠙⠿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠸⡇⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣤⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢼⡷⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⢠⣾⣷⠁⠀⠀⠀⠀⠀⠀⠺⢟⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠐⠀⠀⠀⣾⢻⡏⡀⠀⢀⡺⠀⠀⠀⠀⠈⠙⠻⣿⠻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⢀⡀⠠⣄⢠⣿⣮⡶⠁⣀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀⠀⢀⣿⠀⣾⣇⣾⣏⢸⣿⣿⡇⣼⣿⠇⠀⠀⠀⠀⠀⠀⣤⢀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣄⢠⣿⣧⢸⡿⢿⠉⣸⣿⡿⢲⣿⣿⣤⡌⠀⠀⠀⠀⠀⠈⣻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣿⣿⣯⣿⣧⡸⣷⣿⣿⢷⣿⣿⣿⡷⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣤⣼⣿⡿⠛⠛⢿⢿⡏⠉⣿⡟⢻⣜⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠺⠿⠛⢋⣼⠿⢿⡏⢹⡏⠀⠙⡿⠛⠻⣿⠻⣿⣿⣿⣿⠋⢠⣾⠁⠀⢀⠁⢸⡇⢀⡿⣇⠀⠸⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠛⠁⠀⢨⡗⣾⡁⠀⠼⠓⣀⠀⢻⡆⠘⣿⡀⠁⢀⢻⠉⠀⠀⡾⢿⠛⠀⠈⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠛⢇⠀⠀⠀⢠⡜⢢⠉⠈⡆⠀⡼⠉⠀⢰⠇⣤⠋⢳⠀⠈⡤⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠘⡄⠨⢃⡈⠡⡆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠇⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣋⣵⠾⢋⣠⡾⢛⣩⡿⢳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠘⣿⣡⡴⠟⣩⣴⠟⣡⡾⠋⣀⣽⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⢀⣿⢋⣤⠾⢋⣴⠟⣩⣤⣾⣯⣥⡾⠋⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣽⠿⢷⣦⣤⣴⣿⡿⠛⣡⡾⠋⢁⣾⠋⠀⠀⠀⠙⣷⡾⠋⣨⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⣡⡶⠟⣁⣴⠟⢉⣴⠟⠉⣠⡶⠛⣿⠀⠀⠀⠀⠀⣼⣥⡾⠋⣠⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣴⠟⢉⣤⠾⠋⣡⡾⠋⢁⣤⠾⠋⣠⡾⠛⣷⣦⣤⣤⣾⠟⢁⣠⠾⣫⡴⠾⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠋⣠⡶⠛⣁⣴⣟⡉⣠⡴⠟⢁⣴⠞⠋⣠⡾⠋⣠⡼⠟⢁⣴⠟⢁⣼⠋⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣴⠟⢉⡴⠟⠉⠉⠉⠛⢿⣤⡴⠟⢁⣠⡾⢋⣠⡾⠿⢶⣾⠋⣡⡾⠛⣿⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⣠⣴⣿⠁⠀⠀⠀⠀⠀⠀⢹⣇⣴⠟⢁⣴⢿⡏⠀⠀⠀⢹⣿⢋⣠⡾⢛⣷⣄⣀⣀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣘⣧⠀⠀⠀⠀⠀⠀⠀⢸⡟⣡⡶⠛⢁⣼⣧⣀⣀⣀⣼⣷⠟⢉⣴⠟⣡⡾⣫⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣤⣀⣀⠀⠀⣠⣿⡿⠋⢀⣴⠟⢁⣼⠟⢉⣽⠟⢁⣴⣟⣡⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⣿⣿⣶⣶⣿⣷⣶⣿⣷⣶⣿⠷⠞⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣿⡟⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠋⣠⡶⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣤⣾⣿⠿⠟⠛⠷⢶⠆⠀⠀⠀⠀⠀⠀⠀⡀⠚⢪⡃⠒⣀⠐⣠⡀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠿⠿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⣄⣠⣬⡤⠄⠀⣀⠘⠻⣷⣦⠀⠀⠀⠀⠀⣀⣥⣤⣶⣷⣶⣿⣿⣿⡷⠀⠀⠀⡄⠀⡘⠛⠛⠛⢿⣶⠶⠟⠋⠛⠀⡆
⠀⠀⠀⢤⣤⣶⠈⣥⣶⣻⣛⢓⣿⣿⣍⢀⣀⡀⠀⠀⠈⠙⠛⠛⠛⠀⣹⢻⣯⠉⠀⠀⠀⠀⣷⡈⠁⠀⠀⠀⣼⣿⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠤⠿⢰⣿⣟⣿⠿⡼⣿⣿⣿⡍⠛⠁⠀⠀⠀⠀⠀⣒⣐⠚⠛⠀⠃⢀⡀⠀⠀⠀⠹⣿⣦⡞⠀⢻⢸⣿⡆⠸⣮⣴⣿⠇
⠀⠀⠀⠀⠀⢀⣸⣽⣿⣅⣤⣦⣹⣻⣿⡇⠀⠀⠀⢀⡰⠂⡰⢡⠏⠀⢀⣼⡤⠰⠅⠀⠀⠀⠀⠙⢿⣿⣾⣻⣾⡷⣿⣷⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢬⣿⢿⡿⠟⢿⡿⠛⠲⠀⠀⠀⠈⠙⣰⣿⣿⣶⣶⣿⣿⣿⡶⠾⠅⠀⠀⠀⠀⠈⠁⠉⠁⠉⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣭⣭⣉⠙⢿⣷⣦⡀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⢀⣤⢶⣲⣤⠀⠈⠻⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠿⡆⠈⣿⣿⢿⣆⠀⠀⠀⣠⠟⠉⠀⢄⠀⠀⠈⠻⣆⠀⠀⠀⢀⣿⡿⠾⠛⠁⠀⣀⣀⠙⠛⠿⠋⠁
⠀⠀⠀⠀⢰⣿⣿⣿⡿⣋⣀⣀⣈⣀⠹⣿⣿⡿⡆⠀⠀⡏⠀⢠⣤⣶⣷⣟⢚⠐⢹⠀⠀⠀⠘⣿⣷⡚⣻⣿⡏⠉⢿⣿⣿⣷⢂
⠀⠀⠀⠀⠈⠹⠟⣿⣿⣽⣿⠃⣿⠿⠿⠾⠇⠀⠀⠀⠀⠙⠲⠼⠾⠿⡼⠿⠷⠖⠋⠀⠀⠀⠀⠈⠛⠿⠿⢿⣿⠛⢺⡭⠹⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠠⠅⡈⠉⠉⠀⠁⠀⠒⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣼⡟⠀⠀⠳⣮⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣈⣧⣀⣶⣴⣧⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠙⠟⠁⣠⣤⣤⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠿⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠛⠛⠛⠛⠛⠉

⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⢤⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢛⡂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠐⠀⠀⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁⠀⠈⣼⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⡈⣻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⡀⠀⠀⠉⠉⡀⠘⣯⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣆⠀⠀⠀⠼⠃⠀⠀⠀⠀⠐⠲⣬⣷⡜⠃⠀⠸⠛⢿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡶⣴⢄⣀⣀⣀⣠⣤⣧⣶⢶⣿⣾⣶⣶⣦⣤⣴⣶⣧⡄⣠⣾⣿⠃⢀⣠⡈⣡⣴⣿⣿⣶⣿
⠀⠀⠀⠀⠛⠀⢠⢁⠀⣶⢷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠡⠋⠹⣿⣿⣿⡿⠛⠟⠛⠛⠛⢫⣛⣤⡴⠾⠿⠿⠛⠛⠋⠁
⠀⠀⡀⢠⠀⠀⣻⣿⣾⣷⣿⣿⠿⣻⣿⣿⣿⣿⣿⡟⠉⢀⠀⠀⠀⣿⣿⣿⣦⣤⣤⣴⡾⠖⠛⠉
⠀⠀⠐⢀⡘⣿⣿⣿⡿⠛⠋⠁⢠⣿⣿⣿⣿⠿⠟⠛⠛⠛⠂⠀⠀⠸⠟⣋⠉⠉⠉
⠀⠀⠀⠉⣿⣿⣿⣯⣀⡀⠀⠀⣾⣿⠿⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠘⣄
⠀⠀⠀⠈⠘⢿⡿⠿⢿⣷⣿⣿⡿⠟⠀⠈⠙⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸⡇
⠀⠀⠀⠁⠀⠀⠀⣀⠀⠀⠉⠁⠀⠀⠀⢰⡀⠘⠿⡆⠀⠀⡀⠀⠀⠀⢐⣸⠞⠁
⠀⠀⡀⠠⠀⠀⠚⠀⠉⠀⠀⠀⠀⠉⠀⢈⡀⣴⢻⣿⠀⠀⠈⠖⠚⠛⠹⣿⠀⠀⠀⠀⠀⠀⠀⢀⣄⣾⣆
⠀⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⠰⠾⠀⢈⡉⣀⣾⠻⣿⠂⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⣼⣯⣿⣿
⠀⠁⡈⠀⢀⣸⣦⣄⠀⠀⠀⢀⡀⠀⣾⡇⠍⡀⠂⠀⠄⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠰⠞⣳⣶⣍⢻⣿⣿
⠀⠀⠙⠃⢺⠻⠛⠛⠛⠺⢿⣿⣿⡿⠉⠀⠁⠀⠀⠀⠀⢣⡀⠈⠛⠲⢾⣿⠁⠀⢀⣻⣍⣨⣿⣽⣦⡽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣴⠄⠃⣀⠀⠈⠀⠘⣊⣷⡀⠀⠐⠊⠙⠄⠀⣰⣶⣉⢵⡼⣷⠿⠾⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠊⠻⣃⣀⡤⠀⠀⠀⣤⣿⣿⠁⢀⠌⢤⣚⢿⣦⣄⠙⡿⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣧⣰⣿⣦⣴⣆⣰⣿⣯⡉⠉⠙⠿⠑⠧⢥⡼⣿⣿⢂⣦⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⣻⣿⣿⠟⠛⠛⠋⠉⠀⠀⠀⠀⠠⣈⣐⣢⣿⣿⣿⣯⣿⠰⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠾⣦⠀⠀⠁⢠⣄⠀⠀⢀⢀⣠⣤⣤⣠⣴⣿⣰⣮⣅⣼⣿⣿⣿⣿⣾⠁⣤⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢰⠖⠐⠄⠀⣠⣛⣹⢼⠾⢿⣟⣻⣿⣟⢛⣹⡿⠏⠙⢋⠙⣏⢡⣾⠃⣤⣥⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠱⠀⠀⢲⠟⣩⣿⡟⣮⣄⢻⣦⠞⢹⠛⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡐⠂⠀⠀⠀⠈

⠀⠀⠀⢀⣀⣠⣤⣤⣀⣠⣀⡀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠤⡀⠀⢀⣀⢀⣤⣴⣤⡀
⠀⠀⠀⠿⠿⠟⠛⢉⣽⣿⡿⠿⠆⢀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣠⣀⠐⠛⢳⡖⠋⠀⡴⠁⣽⣿⣦⣤⣼⠀⢤⠛⢋⠛⢻⣼⣽⣆
⠀⠀⠀⣷⠲⣀⣠⠐⠋⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⠋⢩⣉⣠⣾⣿⠿⠀⠸⣷⣾⣿⣿⠁⠀⠀⠀⣤⠒⠷⣞⢏⢀⣹⣟
⠀⠀⠀⢿⣿⠋⠁⠀⠀⠠⠾⠟⠁⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣾⣿⡿⠃⠀⠀⠀⠀⠈⠛⢳⣎⣰⣦⡀⠀⠁⠀⣸⣶⣿⣾⠛⠈
⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠛⠁⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢠⢢⠀⣹⠅⠀⠀⠀⢀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⢶⣤⡀⠀⠈⠀⠀⠀⠉⠉
⠀⠀⠈⠋⠀⠨⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠠⠤⠀⠀⠀⣀⣠⡤⣦⣤⣴⣤⣤⣄⡠⠀⠀⠀⠀⠈⠉⠉⢹⠋⠉⠁⠀⠀⣀⣀⣠⣤⣀
⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠙⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢥⣶⣿⣿⣿⣿⣿⣿⣻⡶⣤⡄⢀⠀⠀⠀⠘⠀⠀⠀⠀⢼⣷⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⡟⠻⠉⠛⣆⠀⠀⣻⢉⣿⠀⠀⠀⣠⠚⠉⠉⢹⣶⡀⠀⠈⠳⣼⡟⠛⣛⠿⠿⣿⡿⣃⠀⠀⠈⠀⠀⠄⢰⡆⠀⠀⠀⠙⠻⣿⡟⣽⡿⠇
⠀⠀⠘⣿⣿⣇⢀⠄⠀⠋⠀⠀⣿⣨⣿⡆⠀⢸⠇⠀⠀⠀⠈⢻⣷⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⠖⠁⠀⠀⠀⠈⠀⠸⠞⠁⠀⠀⠀⠀⠀⠈⠽⠏
⠀⠀⠀⠈⠛⣿⡞⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⢆⠀⠀⠀⢐⣼⠟⠀⠀⠀⠀⢸⣿⡇⣾⣿⠃⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⢀⣴⡶⠂⠒⠒⠲⣦⡀
⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠈⠻⢿⣿⠏⠀⠀⠀⡳⠀⡀⢺⡟⠀⠀⠀⠀⠀⢘⣟⣧⣿⡿⠀⠀⠀⣿⣿⣋⠙⣻⡆⠀⠀⠀⠀⠸⢿⣷⣶⣶⣶⣾⣿⡇
⠀⠀⠀⣠⣤⣴⣶⣶⣶⣶⣦⡀⠀⣀⣀⣀⡀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠿⣿⣿⢿⣿⡆⠀⠀⠻⢿⣿⣿⠿⠃⠀⠀⣰⣶⣶⣦⡀⢘⢻⡎⠛⠉
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣟⢡⣉⣛⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⠂⠀⠀⠀⠀⠀⠀⢼⡇⠀⠀⠀⢸⣿⣿⣿⡿⣷⠀⣼⠁
⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⡿⠃⠀⡽⠀⢻⡄⠀⠀⠉⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠛⢻⣸⠒⠋⢀⣿⡀
⠀⠀⠀⠀⠀⢉⢻⣟⣿⠛⠁⠀⠀⡇⠀⠀⡇⠀⡀⠀⠀⠀⡇⢀⡄⠀⠀⢘⣿⣰⢆⣄⡀⠀⠀⠀⠀⠲⡷⢰⡟⠲⠂⠀⠀⠀⣾⣻⡇⠀⠘⢹⡇
⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⠀⠀⠀⠳⢤⠼⠃⠀⠃⠀⠀⠀⡇⢸⣇⠀⡼⠛⢞⡟⠒⠉⠠⠀⠀⠀⠀⠀⠇⠈⣣⠀⠀⠀⠀⠀⠛⠿⠃⠀⠀⠘⠁
⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⣠⣤⠤⢄⠀⠀⠀⠀⠀⠀⣇⣸⢹⠀⢠⣗⣞⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢶⣿⡆
⠀⠀⠘⢻⣿⠉⠀⠀⠀⠀⠀⢸⣿⣇⣀⠀⡇⢀⡄⠀⠀⠀⣿⠧⠞⠀⠀⢿⣟⣧⣉⢻⡕⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢼⡟⠁⡀⢀⣬⡃
⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠘⠛⠻⠟⠛⠃⠈⠣⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠒⠶⠶⠶⠾⠷⠀⠀⠐⠀⠀⠁⠀⠀⠐⠃⠒⠛⠛⠛⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⡈⠉⠉⠉⠉⠉⠳⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠁⢠⣄⠄⠒⠰⠂⣀⠒⠶⠏⠑⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄⠀⠐⠛⢋⡐⠁⠒⢀⡆⠀⠀⢠⣀⣙⡋⠚⠒⢶⣿⣤⠑⠄
⠀⠀⠀⠀⠀⢀⠀⠶⡖⠁⠀⠀⠀⠀⡴⠂⠊⠀⠀⠈⠉⠀⠀⠙⠁⠳⠀⠈⠙⠿⠃⢀⣀⣀⣉⠀⣄⣼⣷⣄
⠀⠀⠀⠀⠈⠀⠐⠊⠀⠀⠀⠀⡰⠏⠀⠖⡀⠀⠀⠀⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⠏⣀⣉⠛⠋⣉⣿⣶⡄
⠀⡴⣡⡒⣀⣦⡰⣦⢄⣀⣴⠃⣁⠠⣀⣼⡁⠀⢀⢀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠛⠫⠿⠈⠹⠟⠉⠂⠈⠓⢢⣄⡀
⠂⠐⠉⡁⠈⠙⢧⡉⠀⣀⣈⣋⠹⣿⣿⠋⢙⣿⣟⣛⡿⠿⣦⣰⣿⣤⣄⣀⢠⠀⢀⡀⠈⠀⠀⠄⠀⢂⡀⠀⢦⡀⡆⢤⡀⢀⠈⣉⠳⣤⡀
⠁⠀⠀⠀⠈⠁⠀⢺⣗⠋⠉⠻⠂⠈⢀⠀⠘⠹⠋⠉⣤⡄⢈⠠⣶⣼⠯⠉⣾⣿⣿⣳⣶⣶⠶⣶⣄⣨⣅⣀⣈⣇⡙⠎⢻⡜⢿⣿⢿⣜⣿⣦⣄
⠈⠁⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠋⠶⠪⠀⢀⡤⢤⠾⠟⠀⠐⢿⠀⠼⣯⣿⠗⣏⢿⡿⣶⣿⣿⣿⣿⣿⣿⡻⣿⣷⠾⣿⣴⣯⣛⣍⢻⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⣿⣿⠂⠘⠀⠁⠉⠁⠀⠘⠈⠛⠉⠁⠞⠻⠆⠰⠀⠾⠙⠛⠛⠛⢿⢯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢸⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⠆⠀⠀⠀⠀⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣿⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠙⠀⠛⠀⢀⡅⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢸⡿⣷⡁⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠈⢳⣌⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⢶⡖⢦⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠠⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢹⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣸⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣈⣙⣛⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠲⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣬⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠠⠔⠔⠖⢊⣾⣿⣿⣿⣿⣿⣿⠟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣦⣌⠒⠂⠐⠀⠚⢿⡟⠙⢉⡉⠉⠀⢀⣿⣷⡄
⠀⠀⠀⢀⣠⣤⠶⠶⠶⣶⣦⣤⣀⠀⠀⠀⠀⠐⠀⠀⠙⣿⣷⣄⠀⠀⠀⢼⡇⠀⠋⠀⠀⣠⣿⣿⣿⣿⣦
⠀⢀⣴⡟⠋⠀⣀⠀⡀⡀⢈⣹⣿⣷⡄⠀⠀⣴⢾⣿⣿⠻⠟⠻⠀⠀⠀⣸⡇⠈⠐⠂⠼⢿⣿⠿⠿⠿⠿
⠀⢸⣿⣿⣿⣯⣭⠭⣭⣭⠿⠛⠿⣿⣧⣄⠀⠈⠉⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡿
⠀⠀⢿⣿⣿⣿⣿⣉⣹⣿⡀⢀⡀⠸⠿⣿⣷⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡟⡀⠀⠀⠀⠀⣾⠁
⠀⠀⣀⣤⣈⠉⣹⠋⣿⡿⣷⣾⣧⣴⣴⣿⣿⠀⠀⢠⠀⢻⡀⠀⠰⠀⣼⡇⠀⢘⠀⠐⣽⠏
⠀⣎⣡⣞⡻⢠⣷⠚⢿⣷⢿⣿⡟⠿⠿⠟⠁⠀⠀⠀⢂⠈⣧⠀⣀⡀⣻⣧⠃⠃⢀⡾⠁
⢀⢉⣩⣙⣿⣏⡹⣿⡿⣟⣸⣟⣀⣠⡆⢠⣤⠀⠀⠀⣀⢀⣿⣦⣿⡌⣿⣿⣜⣵⣿⠃
⢸⣼⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⣷⡶⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⣿⢿⠃
⠈⠛⠛⠛⠛⠛⠛⠚⠟⠛⠛⠛⠛⠻⠿⠻⠿⠃⠀⠀⠀⠀⠿⣿⡿⣿⠿⠟⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⢉⡙⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠄⠉⡓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠒⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠂⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣀⡠⣄⣸⣣⡀⠀⣀⡴⣀⣠⢆⣀⣠⠁⣄⢤⣠⢦⣀⡤⣀⣄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠞⠛⠉⠉⠉⡉⠉⠙⡗⠀⠈⠈⠁⠄⠀⢠⠋⠁⠀⣍⣉⡩⠭⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠒⠒⠒⢲⠃⣀⠀⣤⠀⣀⠈⠸⡉⠉⠉⠀⢀⠠⠐⠉⠁⠉⠕⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠁⠉⠀⠉⠁⢹⠒⠁⠀⠀⢀⡁⠈⠐⠀⠀⠀⠒⠰⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢸⡀⠀⠰⢋⣩⣏⣉⠻⢻⠖⠲⢶⡤⢤⣜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⡷⠀⢨⠋⠀⠀⠈⠙⠲⣄⠀⠸⡇⠅⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢰⠃⢠⢎⢆⠀⣀⣤⠔⠲⢺⡄⠀⡷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢀⡔⢠⠃⠀⡆⢠⢴⣿⠁⠈⣧⢲⡞⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠎⣰⠃⠀⠀⠸⢍⣎⣸⡄⠀⢸⢿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⢸⠃⠀⠸⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡜⠀⠀⠀⠀⠀⠆⠹⡄⠀⠀⠀⠀⠀⡾⠀⠀⠀⣟⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣁⣀⣀⣀⣀⣀⣀⣴⠃⠀⠀⠀⠀⠀⠧⠴⠦⠤⠷⠤⢄⡠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠿⠛⠛⠛⢿⡟⠿⠿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠆
⠀⠀⠀⠀⠀⠀⠀⠠⡶⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠉⢻⣆⠀⠀⠀⠀⠘⢷⣄⣀⣠⡾⠋⠛⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠻⣧⡀⠀⠀⣸⡏⠀⠀⣠⡴⠶⣤⡉⠉⠉⠀⢀⣾⠋⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡋⠀⠀⠈⠛⠛⠛⠉⠀⠀⣼⡏⠀⠀⠈⢻⡆⠀⠀⠘⣿⡀⠀⠀⣨⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠉⠻⣦⠀⢀⣀⣀⣠⣤⣤⣤⣽⣧⣤⣤⣤⣿⣧⣄⣀⣀⡈⠙⠛⠛⠋⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣃⣤⣶⣿⠿⢿⡛⠉⣿⠉⠁⣿⠀⠀⢻⠁⠀⠀⡿⠉⠉⣹⠛⠛⢷⣶⣤⣄⡀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠋⠁⠻⣄⠀⠓⠀⠘⠀⠀⠈⠀⠀⠈⠀⠀⠀⠁⠀⠀⠋⠀⠠⠟⠀⣠⡿⠻⢷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣖⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢉⣀⢀⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣿⣀⡀⢠⡆⠀⠀⣿⠃⠀⠀⠀⠀⣿⡇⠀⠰⣄⢀⣳⣤⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣿⣿⣿⠀⠀⠀⠀⡄⢹⣿⠿⠿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⢹⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠄⡞⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢰⢰⠁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢠⠃⡄⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⡟⠀⠁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡀⠀⠀⠀⠀⡜⠁⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⣶⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠀⠀⠒⠂⠠⠌⠋⠙⠫⠙⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠋⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠷⣤⡀
⠀⠀⠀⠀⠀⢀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢶⣶⣤⣴⣄⣄
⠀⢀⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⠀⠀⠐⢿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣿⣿⣿⣿⣇⠀⠤⠸⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣯⣿⣿⣿⣿⡀⠀⠀⠿⡿⠇⢠⣤⣶⠀⠀⠀⠀⠀⠀⢀⣀⡺⣿⣿⣿⠯⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⠉⠙⠙⠿⠃⢀⣀⢤⣤⣤⣾⣿⠿⠁⠀⠀⠀⣠⣰⠋⢙⣿⣽⣿⠟⣀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣾⠛⠉⣽⣿⣿⠁⠀⠀⠀⠀⣠⣌⣻⡟⠲⠆⠉⠈⠻⠟⢯⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⣹⣿⡇⠀⠀⠀⠀⣉⣙⠋⠳⠀⠀⢠⠀⠀⠀⡉⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⠀⠀⠀⠀⠙⠛⡠⠦⠀⠀⠈⢳⡄⢾⡿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣤⣄⢁⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⢾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢋⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⡆⠀⠀⠀⠀⠀⠈⠐⣶⡄⠀⠀⠀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣿⣷⠀⠀⢀⡀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡧⣿⣿⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⢻⣿⣧⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢹⣿⣿⡆⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⣿⣿⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠶⡰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⢺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡟⡑⢿⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢧⣼⠟⣻⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⣆⢹⠆⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡡⢚⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠲⠶⠦⣤⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⣀⣤⣤⣶⣶⡶⡶⣶⠶⠶⡶⠤⣤⣄⣈⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⡟⠋⠉⢀⣰⡁⡅⢳⣹⠀⡄⠱⣦⡄⡴⣋⡝⢳⣦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡋⠀⠬⠄⠀⣀⣵⣿⣴⡄⣿⣿⣧⣶⢿⣯⣾⣿⣋⠀⠜⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣅⣀⣸⠯⢝⡿⣷⢭⣻⣿⢳⡯⣯⡟⠿⣋⡟⢉⣙⣿⡛⠣⠶⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣍⣹⣾⡟⠉⠷⣶⣶⡻⣿⢳⢋⡀⡴⡟⣡⣿⠟⢿⣿⣶⣾⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣷⣿⣷⡝⠘⠼⠎⢹⣇⣰⣿⣿⣶⠷⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠀⠀⠀⢸⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠾⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢒⡩⠝⠛⠛⠛⠲⣄
⠀⠀⠀⠀⠀⣀⣠⣤⣄⣠⣴⣴⣟⣿⣿⣿⣿⡟⣿⢿⣿⣆⠀⠀⠀⠀⠀⣀⠤⠄⠂⣀⡄⡒⠙⡡⠒⠁⠀⠀⠄⠤⠤⡀⠈⠃
⠀⠀⢠⣶⣿⣿⣾⣯⣾⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣹⡟⠀⠀⠀⢠⠎⠁⠀⠈⠁⠛⠁⡠⠊⠀⠶⠀⠀⠀⠀⠀⡀⠘⠂
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡶⠀⣀⡀⡀⠀⠀⠂
⠀⠀⠸⣿⣿⢿⣿⣿⣿⣿⣻⣿⣷⣬⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠈⢅⠀⢤⠀⠀⠀⠀⠔⢆⠀⠙⠏⠀⠄⠀⠀⠂
⠀⠀⠀⠈⠻⠿⣿⣿⢿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣤⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠉⠉⠙⢳⣦⣤⠀⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣽⣟⣻⢿⣿⣾⣿⣿⣷⣶⠖⢫⣵⣤⡀⠀⠀⠀⠀⠀⠀⠼⡚⠑⠃⡖⢤⡀⠉⠀⠀⠀⠝⣭⠿⠟⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣶⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱⠠⠀⡀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣷⣿⣽⣭⡿⠟⠋⠀⠀⠀⠀⠀⠀⣦⠀⠀⠁⠢⢀⠀⠀⠀⠀⠀⠁⢉⡙⠉⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⡄⠅⠈⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⣼⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⢠⠆⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⡀⠀⠠⢀⠀⠀⠰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢼⣿⢿⣿⣿⣿⣿⣷⡧⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣸⡇⢰⡀⢠⠀⠀⠁⣝⣄⣠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠎⠃⠿⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠨⣛⣾⣷⣿⣶⡀⣰⠟⠉⠀⡄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⡟⠀⢤⠤⠤⠀⡀⢰⣶⣶⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⢿⡶⠶⢯⠀⠙⠀⢀⡐⠂⠀⣽⠛⠛⢾⣫⡟⠓⠙⠛⠷⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣟⡩⠛⣀⡀⠀⣀⣀⡀⠀⠈⢿⣷⢀⠁⠀⠀⠀⠀⠁⠦⠠⢀⣀⠀⠹⢿⡦⣄
⠀⠀⠀⠀⣠⢔⣿⣿⡟⣿⡛⠊⢰⡿⠿⠛⠈⠁⠋⠁⣠⣾⠛⠀⠈⠁⠀⠒⠁⠂⠀⠀⠀⢀⣀⡁⠀⠠⠛⡿⣷⡀
⠀⢀⡴⠟⠀⡿⠛⠋⠁⡟⡭⠓⠦⠀⠀⣀⠀⠀⠀⠈⠉⠁⠳⡆⠀⠘⠛⠀⠀⢀⣤⡀⠀⠀⠀⠁⠀⠀⠸⣤⣼⣿⡆
⢠⣾⣦⠰⡆⣠⡶⠒⠛⢩⣄⠀⡄⠀⠀⠛⠳⠶⠒⠀⠀⠈⠁⢀⠠⠠⡄⠀⡴⠉⠻⠇⠀⠀⢀⠀⡀⠄⠀⢚⣤⡝⣷⣶⠶⠦⣤⣀
⣼⡛⣿⢄⡀⠀⣤⣠⢦⡰⣉⠀⠀⡄⠀⠀⡀⠀⠀⠀⢰⠈⠤⠉⠁⠐⢠⡄⠀⠀⠀⠃⢈⠀⠈⠦⠱⢀⠜⠈⢻⣿⡿⠹⠂⡀⠈⠉⡓⠦⣄
⣿⣧⠘⠫⢅⡙⠚⠡⢺⠾⠡⢶⠦⠄⠀⣠⣀⣄⣄⣆⣀⣠⣤⣀⣄⣀⡉⣁⣀⣀⣰⣖⣥⣀⣀⣤⣀⣤⣴⣾⣿⠟⠁⠀⠀⠋⠀⠲⢄⣀⠈⠹⣦⡀
⢹⣿⡏⠀⢀⣉⣽⡿⢶⣶⣶⣶⣷⣿⣶⣼⣤⣼⣿⣿⣿⢿⣿⠿⡿⡿⠿⠿⠿⠛⠛⠛⠛⠛⠛⣛⣭⣿⣿⣿⣷⡲⣶⠆⠀⠀⠀⠀⠀⠉⠀⢠⡀⢻⡄
⠀⢻⣷⣴⣿⡟⠥⠒⡉⠁⠉⠉⠉⠉⠙⠛⠛⡉⣀⣿⠛⠃⠀⠈⠀⠐⣄⣀⣤⣤⣤⣴⣴⣾⣿⠿⠛⠛⠛⠛⣿⣿⣿⠿⢷⣶⣶⣦⣤⣄⡀⠀⠀⠛⢿
⠀⠀⠈⠻⣿⣿⣿⣎⣘⣡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⡷⠀⠠⠀⠀⠀⢸⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢀⠛⠋⠁⠀⠀⠀⠉⠙⠻⢿⣷⣶⣴⣆⢸
⠀⠀⠀⠀⠀⠉⠛⠿⡿⠿⣿⣷⣾⣷⣾⣾⣾⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⢷⣴⣴⣶⣤⣶⣶⣤⣦⣤⣀⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣛⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣯⣷⠈⠀⠃⡐⠄⠚⠛⠋⠉⠉⠉⠙⠉⠉⠛⢻⡟⠀⠀⠀⠀⠀⠀⠀⣠⣦⣀⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢀⣶⡿⣿⡿⠿⠿⣻⠟⠻⣿⣿⣿⣮⣻⣿⣿⣯⣿⠲⣤⡇⠀⠀⠀⢺⡿⠿⠃⠀⠀⢀⣠⣶⣾⣿⣿⡿⡿⢷⣾⣷⣾⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⢀⣶⣻⢋⡑⠋⢚⣋⣀⣩⣼⣿⣾⣿⣿⣿⣿⣿⣿⣧⢿⣸⠋⠀⠀⠀⠀⢠⣷⣴⠀⣠⣴⣿⣿⣿⣯⣿⣧⣤⣀⡈⠀⠈⢀⠹⣷⣤⡀
⢀⣿⣿⣿⣿⣿⣿⣿⡿⠿⢻⡿⠉⠉⠉⠉⠉⠉⠉⣿⣿⡸⡏⠀⠀⠀⠀⠀⠀⢈⣿⣶⣿⣿⠟⠋⠉⠉⠉⣉⠝⠛⠿⠿⢶⣦⣄⣾⣭⣿⣦⡀
⠈⢿⣿⣿⡿⠛⠉⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣠⠁⠀⠀⠀⠀⠀⠀⠘⢻⣿⡅⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀
⠀⠀⠻⢿⣶⣀⣀⡀⠀⠀⣠⡀⡀⠀⠀⠀⠀⢠⡦⢀⣿⣿⡅⢣⡄⠀⠀⠀⠀⠀⢱⡄⠙⢿⣷⣶⣶⡿⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠉⠉⠛⠿⠛⢻⡇⡇⠀⠀⠀⠀⠀⠀⢨⣏⢹⣿⠈⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠙⢻⣿⠁⠀⢀⠊⠀⢰⣆⡀⢀⠀⠀⢀⠀⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣿⡄⠄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣼⠇⠀⠀⠁⠀⠀⠟⠻⠿⢿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⣼⡟⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣧⠀⠀⠀⠀⠀⠀⠀⠠⣸⣷⣿⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠇⠀⠀⠀⡄⠀⠐⣷
⠀⠀⠀⠀⢀⣀⠀⣀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣿⣶⣛⣇⣀
⠀⠀⠀⠈⣶⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣎⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢀⣦⣰⡀⠀⠸⣿⣿⣿⣤⣷⣦⣀⡀
⠀⠀⠀⠀⠈⢬⣿⣷⣧⣤⣤⣀⣠⣦⣤⣀⣦⣤⣦⣴⣿⣿⣿⣿⣤⡿⢤⡠⣤⣀⠀⠈⣠⡶⢟⣻⣿⣶⣿⣿⣄⢤⣼⠻⣿⠛⢿⣍⡍⠁
⠀⠀⠀⠀⠤⡚⠡⡉⠛⠻⠛⠛⢿⣷⣿⣿⣿⣿⡿⠿⣝⠟⢿⠛⢻⠇⣸⠆⡩⠸⠿⠙⢉⠃⢮⣤⣈⡻⡔⣄⠟⡋⣤⠐⠢⣤⠀⠈⠁⠁⢠⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⡰⠛⠉⠉⡿⠆⠽⠿⠾⠿⠟⠒⠉⣦⠓⠛⡧⣾⣿⣿⣿⣷⢿⣿⣿⣿⢿⣷⠛⠉⠀⣁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢼⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠆⣠⣼⡟⠀⣈⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣥⣼⣿⣿⠃⣀⡿⣦⣬⣽⠿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣧⠐⠈⣧⣿⣿⣿⣿⡿⣿⣿⠿⠋⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠹⣿⣿⣿⣿⣿⡿⣿⣿⣦⣄⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡏⣀⣼⣿⣿⣿⣿⣤⠀⡨⢿⣿⣿⣿⣷⣬⣿⣿⣿⣧⣘⣻⠟⠳⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢉⡀⠀⠀⠘⢿⣿⡍⠁⣨⣿⣿⣿⣿⣿⡿⠶⣦⣤⣽⣧⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢾⣿⣿⡿⠛⠉⠀⠈⠙⠛⠛⠢⠀⠙⢿⣦⡙⢦⡀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⠁⣤⢿⣿⣿⣿⣿⠿⠿⣿⣿⠿⠃⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢻⣦⣄⡀
⠀⠀⠀⠀⢀⠀⢨⣿⣿⣿⣿⣿⡿⠏⠀⠼⡿⠿⠿⠋⠁⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠰⣁⠆⠀⠈⠀⠀⠀⠈⠻⣁
⠀⠀⡀⢀⡄⠀⢸⣿⣿⣿⣿⣅⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠃⢀⣀⠃⠈⣱⣿⣄⠀⢀
⠀⢈⡍⠉⢳⣤⣼⣿⣿⠟⠋⠀⠀⢠⡀⠀⠀⠀⠂⠀⠀⣴⠀⠏⡀⠀⠀⠈⠀⣾⠀⣇⠀⠇⢀⣼⠆⣠⣿⣤⣾⣿⣿⡇⠀⢀⡟⠀⠀⡠⠀⠀⠈⡆
⠀⡸⢻⠃⠈⠉⢉⠁⢫⠀⠄⠀⠀⠈⠀⠀⠀⠀⡀⢠⠈⠀⠋⣠⣧⠀⢠⣶⣠⣿⣦⢿⣆⣰⣿⣿⣿⣿⣿⣿⣿⣿⡷⢿⠁⢈⢀⡤⠚⠀⠀⢀
⠀⠇⠀⡠⠁⠠⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⡰⠐⢆⣞⢳⣄⡹⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣋⣻⣤⣄⣈⣁⡤⢊⣀⣤⡀⠀⢀
⠀⠀⠀⠀⢢⡀⠀⠀⠈⢄⣀⡄⠀⠀⠀⣤⣠⣜⣶⣿⣿⣷⣿⣿⣾⣿⢾⣿⣿⡟⠛⠛⣽⣿⣽⡿⣍⡿⣿⡿⣿⣷⣿⣿⣿⣿⣷⡾⠿⠋⠀⠀⢠⠎
⠀⠀⢀⠀⠀⠰⢶⣦⣤⣤⣹⣿⣿⣷⣾⣿⣿⣿⣿⡿⠿⠛⠛⣿⠀⣭⠀⡇⠻⠀⠀⠁⣾⣿⣿⣏⠺⢯⡹⢿⣦⠙⠛⠛⠻⢿⡿⠷⠆⠀⣀⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠃⠀⠀⠀⠀⣿⠟⠈⠹⠋⠀⠀⠈⠀⠒⣀⠀⠀⠀⠀⠙⢂⣤⣬⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠋⠀⠶⠶⠤⣤⣤⣤⣽⣿⣶⣶⣶⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠸⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠆⠠⡄⢸⠃⢰⢦⡄⢷⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣿⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢼⣤⠀⠀⠄⠀⡆⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠢⠀⠀⠹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⡀⠀⠀⠀⠀⠀⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢸⣦⠈⠃⠀⠀⠰⣁⠀⠀⠀⢀⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠷⣄⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠀⠀⠀⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢄⣀⣀⠀⠀⠀⠀⠀⠐⠦⠴⠆⠠⠴⠀⠄⠠⠞⠛⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣹⣷⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣀⣤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠈⠀⠀⠂⠀⠐⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀⠀⠀⠀⠹⠿⠿⡟⠓⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⢨⣇⠤⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠃⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⣾⡄⠀⠀⠀⠀⠀⠁⠘⠋⠰⠃⣀
⠀⠀⠀⠀⠀⢤⣤⠀⠀⠀⠀⠀⠁⠀⠀⢙⠀⠀⠀⠀⠀⠀⠀⣸⣆⣴⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⡿⠀⢲⠈⠹
⠀⠀⠀⠀⠀⠀⠉⠀⢠⡾⣿⡿⢷⠌⢀⣼⠁⠀⠀⠀⢸⢠⣼⣿⣿⡿⠟⠛⢻⠿⡁⠀⠀⠀⠀⠀⠀⠀⣁⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⢧⣾⣶⣬⣿⢦⠀⣀⣤⣿⢸⣿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⡇⠀⢅
⠀⠀⠀⠀⠀⠀⠀⠴⠀⣴⣿⡿⠟⣻⣴⠂⠀⠉⠁⠟⠛⣾⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠻⣷⣤⠄
⠀⠀⠀⠀⠀⠀⠉⡀⠠⠿⠶⠛⠿⣿⣿⣳⠆⣾⡆⢠⠀⠈⠁⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⢂⡀⢾⣧⣤⣤⡟⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠿⠂⠀⠀⣤⣀⡈⡁⠀⢀⣿⡃⡀⢀⡀⠀⠀⢠⢇⣸⣤⣰⣶⣦⡿⠘⠿⢿⡒⠀⠉⠉⠀⢠⡄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠌⡭⠈⣴⣷⠖⢈⡛⠻⠾⢷⡆⠞⠋⠙⠟⠉⠉⠩⠉⠁⣄⢼⣿⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠉⠀⠭⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠁⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠰⠋⠀⣰⣶⣒⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣾⣧⣂⠄⡀⠀⠁⣀⠀⢾⡆⢸⣿⢷⢽⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⢿⣖⠴⠆⣤⡹⡏⠀⠣⣸⠿⠍⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⢊⠱⡟⠠⣤⣵⡇⣼⣠⠜⡓⠚⠁⣂⠡⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡰⠟⠛⣬⢙⢏⠃⡄⣽⣿⢰⡟⡡⢆⣠⣀⠺⠙⠉
⠀⠀⠀⠀⠀⠀⣤⣠⣸⡋⠴⠶⢦⣷⣦⣳⢲⡈⢿⣿⣾⣣⡖⠁⠉⢀⠀⡠⠦⠤⡤
⠀⠀⠀⠀⢤⣷⠀⢿⣧⣠⠂⠬⠭⢟⡋⡙⢿⡇⣴⢮⡱⣏⣠⠞⠡⡽⠋
⠀⠀⢀⣶⣦⣌⣷⣿⡿⡿⡦⠶⠀⢻⡏⠊⠘⢿⣿⣿⡷⠸⢟⣊⠔
⠀⠀⠟⠋⣻⡷⣾⣯⣽⠟⠀⠄⠲⠄⢣⢑⣴⣼⢉⣿⡧⣟⢀⡺⢮⣁⣀⠄
⠀⠀⠀⠀⢶⡿⠿⢿⡿⡴⠒⠀⠀⠀⠀⢹⠟⢿⣸⣦⠙⠚⢋⣠⠆⠘⠁
⠀⠀⠀⠀⢲⠿⣶⢫⣿⣦⠃⠀⠀⠀⢈⢇⣿⣸⣿⢻⣇⣣⡹⣻⡂⣠⣆
⠀⠀⠀⠘⣾⣸⣿⣼⣿⢻⣋⠁⠀⠀⠈⠣⣸⣷⣯⠀⢻⡿⣷⣯⣎⠙⠃
⠀⠀⠀⣴⡿⡿⣷⣽⣿⣿⣯⢀⣀⣠⡞⠩⡷⢿⣿⡕⣾⣿⠟⢏⣽⠇
⠀⢠⡾⠿⣿⣿⣿⣿⣿⣿⠿⠓⠀⠈⠉⢛⣿⣶⣿⣿⢸⣿⠘⠌⣽⣀⣠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢴⣍⡏⣿⣿⡿⣻⡆⠘⣸⠀⢿⡏⠡⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢡⢲⠇⣿⡟⣽⣤⢸⢱⡆⠁⣷⣙⣧⡄⣀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣿⡾⡖⡿⡃⢹⢿⠄⢾⡇⢜⢹⣿⢿⣷⣿⣼⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠈⣸⡿⣶⣧⡿⣈⢀⣻⣼⡏⡟⡞⣿⣿⢟⡍⢿⢿⡀
⠀⠀⠀⢀⢄⣀⡀⢀⠤⣤⣄⡴⣿⣇⣿⣿⢱⣿⣦⣘⢿⣷⣿⣇⣿⡟⣾⣷⣾⣾⡧
⠀⠀⠀⠀⢾⡎⠄⡴⢂⡘⠳⢿⢟⠻⡳⣷⣿⢸⣿⣿⢨⢟⡿⢿⣿⣿⣿⣿⣿⣿⠷⠓⠒⠶⢶⣶⢷⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄
⠀⠀⠀⠀⠸⣿⡞⢰⠏⠀⠀⣸⢨⡇⣰⢸⣏⠙⢿⣿⣾⡾⣬⡘⣿⢻⠹⣿⣿⠋⢠⠄⢀⠆⠀⠈⡏⢃⠉⠙⡆⠀⠀⠀⠀⠀⡀⠀⡔
⠀⠀⠀⠐⠀⠉⢅⣽⠀⣰⡾⠟⠚⠛⢳⠏⠱⣦⣴⣆⢛⣿⢻⣿⣝⣷⣿⣿⣿⡄⣼⢀⡆⠀⠀⣼⠃⣼⢸⣧⣿⢀⣶⣤⣴⡯⠈⣶⣼
⠀⠀⠀⠀⢀⣴⡶⣿⠗⢯⡶⠛⠃⣄⠁⢆⠠⠿⠛⠽⡟⠟⠢⠹⣿⣻⠹⢏⡭⡝⠛⠁⣥⡾⠿⢉⣾⣿⣿⣿⣋⢙⣓⠑⢤⢻⡇
⠀⠀⠀⠀⠀⠛⠲⠁⡄⢼⠀⡄⠀⠈⣧⣸⣿⡿⠓⠲⠲⠤⡆⢤⣤⣁⢚⡻⠆⠀⠉⠁⠻⡂⣬⠙⣽⣝⣉⣝⠛⠓⠛⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢆⠷⣼⣦⡘⢶⡶⠿⣦⢸⡿⣧⡴⠞⠓⠛⠛⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠛⠙⠀⠘⠛⠞⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⢀⠸⡀⠀⠸⠘⡀⣀⢤⠀⠀⠀⠀⠀⠀⠀⢠⣝⡾⢷⣫⣤⣀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠄⠀⣀⡀⠀⠀⠈⠁⠀⠀⣴⣑⣁⠈⠀⠈⢣⣄⠙⡄⠀⢘⡿⣷⡾⢿⡛⠯⠝⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡀⠀⢠⠀⡈⡄⢰⢀⣀⣄⢺⣷⢷⡆⠁⠠⠐⡀⡂⡏⠀⠉⠉⠉⠉⠋⣿⡟⢻⠀⠀⠈⢹⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠰⢃⢀⡠⠐⠉⠀⠀⠀⢀⠤⠿⠚⠂⠑⡃⣚⣿⡠⢄⡅⠀⠀⠀⠀⠀⠀⣿⡧⢸⠀⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⡶⠤⠀⠀⠀⠠⠄⠀⠀⣀⣀⢀⡤⠖⠉⠀⠀⠉⠑⠫⢧⣤⡀⠀⠀⠠⣻⡅⢸⠀⠀⠀⠚⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⢐⠀⠀⡎⠠⠂⢀⠀⠀⠀⠀⠀⠀⢀⣰⣷⣶⠀⠀⠀⠀⠀⠀⠀⣤⣽⡷⡤⠀⠀⢹⡇⢸⠀⠨⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠂⡋⠠⠁⠀⠐⠀⠀⠀⠀⠐⠂⡰⠁⠛⠻⠀⣾⢒⠶⡲⢦⠀⠿⠿⠙⣽⡀⠀⣸⣇⢸⠀⠀⢠⠒⠟⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢠⣆⣀⣈⠁⠊⣁⣀⣀⣀⡀⣰⠃⠀⠀⠈⠀⢸⣯⣶⣡⣾⠀⠐⠖⠂⠙⣿⣿⣿⣿⣿⣷⣃⣾⣷⠾⣷⣷⢦⡀
⠀⠀⠀⠀⠐⠀⠀⢀⡠⢒⠘⠊⠉⠉⠭⢙⣳⣧⣍⠀⠀⠈⠉⠻⠀⢰⠖⠒⢿⢳⣶⣴⠖⣾⡗⠒⡂⠑⠒⡛⠋⣛⡛⣛⠛⠓⢈⣕⡺⣿⣿⣞⢧
⠀⠀⠄⠀⠀⠀⠀⠀⣛⠙⠙⠚⠓⠺⢶⠋⠁⠀⢹⢀⠀⠀⠀⠀⠀⢸⠁⠆⢸⣿⢹⡿⣛⢽⠉⡞⢹⠀⠀⡏⠉⣿⡏⣹⠉⠉⠉⠉⠁⠻⢯⡁⠣⡀
⠀⠀⠀⠴⠤⣤⣤⣴⣟⣰⠀⡀⠀⠀⠰⠀⠉⠁⢸⣸⣤⣄⠴⠴⠀⣸⣦⣀⠾⠿⠾⠧⢥⣼⣶⣀⣲⣀⣺⣷⣴⣿⡷⠿⠿⠶⠶⠶⠗⠒⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠀⠁⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⢀⠀⠀⠀⠀⡠⠐
⠀⠀⠀⡀⠀⢀⣿⣄⣀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣐⠶⠤⠤⡀⠀⠈⠁⠐⠄⠀⡀
⠛⠓⠣⠼⢀⣹⣿⡆⠀⠀⣿⣻⡧⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠦⠄⠈⣏⠉⠉⠉⠙⠓⠀⠀⣴⠀⠀⡄⠐⠄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠀⠉⠙⠓⠲⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠠⠀⠨⠀⠀⢤⡀⠙⢮⣷⣿⠀⠀⣿⠀⣆
⠀⢀⡀⠠⠄⡄⠀⠀⠀⢄⠀⠀⠄⢐⠠⠀⡀⠀⠀⠉⠉⠋⠿⣷⣶⣤⣄⡀⠀⢀⠀⠀⠀⠀⠄⠀⠀⣷⡀⠀⢻⣦⠈⢿⡇⠀⣸⣿⠀⣿⠐⢒⠒⠚⠓
⢀⠀⡀⠄⠀⠄⠀⠀⠀⢀⡉⠒⢄⡀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣿⣷⢾⡇⠀⠀⢸⡂⠀⠀⢹⣷⠀⠈⣿⡇⢸⡏⢠⣿⡏⣸⡏⠀⠀⠂⢁
⠡⠐⠂⡂⠂⠅⠏⣠⡒⢆⠀⢀⠄⠉⠢⣄⠀⠈⠀⣀⣠⠤⠶⠒⠛⠉⠉⠀⠀⠀⣇⠀⠀⢸⡇⠀⠀⠘⣿⡇⠀⢹⣿⢸⠟⠋⠙⠲⠟⣀⣠⡀⠀⠈
⠁⠐⠠⢅⠁⢎⠐⠢⠁⠬⠡⠃⢴⣃⣤⠼⠗⠛⠉⠁⠐⠀⡀⠒⠀⡀⠀⠠⣴⣆⠘⡄⠀⠀⢻⠀⠀⠀⢾⣧⠰⠈⠉⢻⡆⠀⠀⠈⠺⠿⠋⠀⠀⠠
⠸⢁⡇⠀⠐⠂⡠⠚⣀⣤⠷⠚⠋⠡⠀⠀⠀⠀⠄⠀⠀⠉⠀⠀⢴⡁⠐⠖⠒⠶⢤⣌⣦⡀⠈⠦⠀⠀⠺⣿⣥⠄⠀⠀⠠⠀⠀⠤⠀⡀⠀⠈⠀⣀⠐
⢤⢉⠹⢐⣪⡴⠞⠋⠁⠠⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠌⠁⠐⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠀⠀⠉⠅⠀⠂⠠⠖⠀⠀⠀⠀⠋⠁⠀⠀⠀⠁
⣠⡵⠞⠋⠁⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⡀⠀⠹⠶⠷⠿⠶⠄⠀⢀⠀⠀⠀⠁⠀⠤⠀⠀⢀⡔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠴⡶⢖⢲⠛⠛⠋⢉⡍⢍⡝⡻⠛⡳⠒⣶⠶⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢶⡚⠫⣍⠳⢌⠢⡙⢌⠀⠇⠀⡇⠸⠀⠜⠔⣡⡞⠡⢊⡡⠖⣋⠤⠞⢉⠹⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠲⢌⡑⡂⠍⡒⠅⠢⣈⣤⣬⣄⣡⠞⠛⠲⠞⠉⠉⠳⠴⠶⣭⣴⠶⡶⣄⠒⣩⡷⣤⠉⡓⣦⣀
⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠳⡄⠁⠀⣴⣾⣿⣿⣿⣿⣿⡴⢶⢶⠶⣶⠖⣶⡶⠒⢿⠾⠶⢷⣿⣰⡿⠧⣿⡟⠉⠁⠈⠳⢦⡀
⠀⠀⠀⣠⠞⠹⢧⡀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠶⣾⣿⣿⣿⣿⣿⣿⡿⠶⠶⡶⣶⢶⡾⠁⠀⠀⠀⢱⡄⢸⣿⢹⡇⠈⣿⡇⠀⠀⠀⢀⣠⠿⢦⡀
⠀⠀⡼⠃⠀⠀⠀⠁⠀⠀⢄⣠⣤⠞⠁⠀⣀⣷⣿⣿⣿⣿⣿⣿⣿⣈⣓⣙⣙⣈⣻⠇⠀⠀⠀⠀⠀⣧⢸⣿⢸⡇⢰⣿⡇⠀⠀⠀⠈⢡⠀⢈⣷⡄
⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣟⣏⣙⣁⣛⣘⣃⣸⣤⣤⣤⣀⣀⣠⣯⣸⣿⣼⣅⣈⣿⣇⣀⣀⣀⣀⠀⠀⠈⠀⠻⡄
⣸⣷⣶⣶⣶⣦⣤⣤⣀⣀⣀⣠⣤⠤⠤⠶⠶⣟⣛⣿⡿⠿⡟⢛⣛⣯⠿⠟⠛⣋⣭⡤⠶⢚⣫⡭⠖⠋⠁⣠⡴⠛⠉⢀⣩⠟⠉⡽⠉⣟⠛⣟⠲⣶⢿
⣿⠢⠬⢭⣅⣉⢉⣙⠛⠻⠻⡿⣿⣶⣦⡶⠟⢉⣁⣀⣤⠾⠛⠉⠁⣀⣤⠶⠛⠋⢁⣤⠞⠋⠁⠄⣠⡴⢛⠁⠀⢀⡴⠋⠀⠀⡾⢁⢰⣗⢀⣹⣄⣈⣻
⣏⠑⠮⣄⡠⡉⠙⠓⢦⣴⣠⣠⣴⢛⡩⢀⣤⣞⡋⠉⠉⣀⣤⣶⣻⣥⣤⣤⣤⣞⠋⠀⠀⠀⣠⡾⠋⠀⢁⡀⢠⡟⣁⣤⠶⠟⣩⠉⠉⠉⢉⣉⣁⡤⢾
⢻⢳⢤⣤⣉⣳⣷⣿⣛⣫⣯⣥⣬⣩⣿⣿⣟⣉⣹⣻⣿⣽⣽⣭⣭⡁⠹⠛⠟⢻⢧⣀⣠⡾⠋⠀⠀⠀⡚⣵⠿⠭⠕⠒⠉⠀⠀⠀⠀⠠⢝⣛⣓⣟⣿
⠈⢿⣭⣼⣻⣿⣿⡿⣿⠷⣿⣷⣾⣷⣿⣿⣿⣿⡻⣿⣿⣿⠿⣿⣯⣟⡿⣷⢶⣤⣈⣻⣿⡟⣽⢳⣠⢄⣼⣟⣿⣟⣷⣖⣶⣦⣀⣤⣤⣀⣀⡈⠉⣨⠏
⠀⠈⢷⠦⣠⡠⠍⢿⣟⣷⣾⣅⣇⢉⢻⣷⡿⡻⣿⢤⢿⡿⣶⣿⣻⣿⣿⣿⣯⣽⡟⣛⣮⣏⡙⠾⢯⣿⣿⣽⢧⡿⣿⢻⣭⡟⣽⣛⢷⢶⡭⠽⣽⠏
⠀⠀⠀⠳⣄⠹⣯⠭⠇⡹⢤⢿⠿⣿⣦⣻⣶⣻⣫⢿⣷⣟⡻⣯⣿⣟⣯⣟⠿⡽⣷⢿⣆⡿⣓⣾⢧⣸⡟⣛⣧⠹⡷⣟⣲⡧⠯⣝⡟⣼⣭⡿⠁
⠀⠀⠀⠀⠈⠳⢿⡅⣢⢍⡛⣿⡁⣹⣿⡇⣿⣿⣟⣏⢿⣾⣷⣛⣟⣳⣝⣯⣾⡟⣫⣻⣞⢿⣭⢿⣦⣯⣿⣧⣞⣷⢛⡟⢾⣓⢦⣈⣳⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣴⣐⢢⣟⡙⣿⣿⣿⣿⣼⣿⣭⡑⢿⣚⣿⣴⣿⡷⢿⢯⣿⣫⣿⣿⡛⣾⡷⣾⣩⣿⣝⣿⣟⣿⣟⣽⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣥⣼⣿⣿⣿⡟⡷⢮⣉⠟⣾⡿⡻⣼⣗⡿⢳⣿⣿⣮⣽⣛⣷⣿⣿⣯⣴⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠿⢶⣟⣚⣇⣿⣯⣍⣽⣷⣿⣿⣿⣿⣭⣭⠾⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠹⣿⣿⣷⣼⣾⣼⣾⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠀⢀⡀⠱⠜⣾⣿⣿⣿⣿⣿⣿⣿⡟⡟
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⢀⠀⠀⠙⠪⢢⣎⣿⣿⣿⣿⣿⣿⣿⣏⠵⠓
⠀⠀⠀⠀⠀⠀⢤⣿⣩⣴⣷⣿⡷⢄⣿⠅⣀⠇⢽⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⢘⠿⣿⣿⣿⣿⣤⣤⡁⢠⣐⢄⣸⣿⣿⣽⣽⣯⡇⠐⣄⠉⠀⣠⡄⢐⡤⠛⠛⠁⡂
⠀⠀⠀⠀⠀⠀⠄⠓⠿⢻⢯⣉⣹⠟⠿⡟⢻⢾⣿⣿⣿⣿⣿⣿⣄⢀⣈⣠⠄⣈⠉⠁⠀⣱⣸⡔⡇
⠀⠀⠀⠀⠀⠰⣤⣤⣀⣀⣾⣿⣿⣿⣄⣉⠘⣽⣿⣿⣿⣾⢻⣿⣿⡉⣍⠉⠉⡉⢩⣾⣺⣴⢿⣻⣶⣷⡀
⠀⠀⠀⠀⠀⢢⣸⣿⣿⡿⣿⣿⣿⡌⣿⣿⣴⣿⣿⣽⣿⣿⣿⣿⣿⣧⢀⣿⣿⣷⣷⣿⢿⣻⣿⣟⣿⢽⣿
⠀⠀⠀⠀⡀⠘⣽⣿⣿⣿⣿⣿⡏⠙⢿⣿⣽⣿⡟⡛⢿⠿⠛⠟⠛⠛⠐⠿⣿⣿⣿⣿⣯⢻⣞⣿⣾⣿⢿⣃⡄
⠀⠀⠀⣌⢱⣼⣾⣿⣿⣿⠿⡿⠛⡠⠿⠿⠿⠛⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠙⠁
⠀⠀⠀⠉⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠒⢠
⠀⠀⠀⠘⠢⠐⣄⠀⠀⠀⠴⠢⡄⠀⢀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⣀⡀⢠⣤⠀⠀⣀⠀⠀⣀⣤⣄
⢰⣤⣤⠴⠶⠤⠾⠶⠾⠀⠀⠀⠒⠂⠛⠓⠰⠿⠧⠴⠷⠾⠦⠤⠤⠴⠆⠀⣤⠤⢤⡄⠁⠀⠀⢰⣋⣁⡀⠉⡿⠔⠢⡄⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠋⠙⠁⠛⠋⠁⠠⠾⠿⠿⠿⠶⠤⠤⠤⠧⠄⠘⠛
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠉⠀⠀⠠⣴⣞⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⡀
⠄⡀⠀⡀⢠⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⡉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠁⠁⠁⠈⠉⠉⠁⠉
⡆⣄⡷⠀⡼⣄⠀⢀⡀⠀⣸⠡⢛⣀⣧⣻⣽⣇⣀⣀⣀⣀⣀⠀⣀⣀⣴⡷⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡆⠏⡟⢡⣟⣾⣾⢀⣇⣆⣟⠰⠮⡉⠁⣀⠀⠀⠀⠂⠀⠀⠀⢠⡇⠀⠘⣻⢿⣯⡆⠆⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠈⠘⠀⡄
⠟⢰⣰⡧⣩⡎⠙⠞⠒⠈⠀⢀⡠⠔⠈⠀⠀⠀⢀⠐⢀⡠⣤⡚⠀⠀⠈⠻⢿⣷⢣⢸⡌⠆⢠⠀⢹⠀⡅⢀⠀⢀⠀⠰⢀
⣟⣸⡿⠛⠊⠁⠀⢀⣠⡴⠚⢉⠀⠄⠀⠀⠀⠄⠁⢀⠞⡤⠺⠁⠀⠀⠀⢀⣤⣶⢟⡋⢿⢀⢈⡄⡄⣀⢣⡈⠀⣸⠃⠀⣾⠈⡀
⠁⠀⠀⠀⢀⡤⠴⠋⢁⡀⠢⠀⠀⡠⣂⠄⠀⠀⡰⠋⠘⡽⠀⠈⢠⠀⠀⠀⢈⡿⣸⡅⣻⣈⡌⢰⡟⢋⠀⡇⠁⠇⠀⠂⢸⢸⣷
⢀⣠⠴⠚⠁⠀⢀⡸⠁⠀⠀⣐⣄⠣⠀⠀⠀⡜⢡⢠⣼⠃⠀⠀⣾⠀⠀⠀⢈⣸⣛⣛⣋⢿⣃⡘⣧⠄⠀⣷⢠⠀⡼⠀⡖⢨⡾
⠉⠀⠀⣠⣶⠙⠋⠀⢀⢠⣳⠹⠌⠀⠀⢠⠌⠄⣠⡻⠇⠀⠀⠐⠇⠀⠀⠀⠩⠑⠛⠋⢉⢴⡇⣧⢸⣧⠀⠁⡿⢡⣻⠀⡿⡘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⣦⡀⠀⠀⠀⠀⢠⣾⣿⣿
⠀⠀⣤⣤⣴⡄⢀⡀⠀⠀⣀⣀⠀⠀⢠⣤⣿⣧⣤⣤⠀⡀⠀⣤⣤⣴⣾⣷⣄⣤⣤⣴⠦⣠⣾⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣶⣤⢖⣤⣤⣡⣸⣾
⠛⢻⢻⢛⠛⠃⡛⠛⠓⢛⠛⠻⠛⠛⠛⠛⠛⠃⠉⠁⠀⠈⠀⠈⠉⠙⠉⠉⠛⡿⠟⠛⠛⠉⠉⠀⠀⢰⠆⠀⠀⠀⠸⡭⠿⣿⣿⠾⣿⣿⣿⠿⣿
⢠⠁⠀⡀⠀⠢⠈⠀⢈⠀⠀⠐⠂⠀⠀⠀⠄⠀⠀⠀⠒⠂⢀⡀⠈⠁⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠲⢤⡀⠉⠒⠤⣀⠒⠤
⡘⢰⠀⡧⠘⡀⠀⢄⠀⠑⠄⡀⠀⠢⠈⠁⠢⢀⠀⠢⢄⠀⠀⢀⠀⠂⠀⠤⠄⠀⣀⠤⠂⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠐⠦⢀⠙⠲⠤⣀⠙⠲
⠁⢼⠀⢰⠀⢠⠀⠀⢂⠀⠀⠙⢦⡀⠀⠁⠀⡀⠙⠢⡀⠙⠢⣀⠁⠦⣀⠀⠈⠉⠀⠀⠚⠀⠀⠐⠂⠀⠠⠔⠒⠈⣀⡠⠤⠂⢀⠁⠲⣄⠈⠑⢤
⠀⣼⠀⢸⡄⠀⠇⠀⠀⠳⡄⠀⠀⠹⠄⠀⠀⠈⠢⠄⠈⠲⠄⠈⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠐⠖⠆⠀⠠⢠⠒⠈⠁⢄⠈⠙⢦⡀
⠀⠸⠀⠀⠧⠀⠘⠀⠀⠀⠡⠬⠍⠀⠀⠀⡀⠤⠈⠀⠀⢀⡐⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠶⠆⠀⠀⡤⠤⠀⠳⠀⡀⠃
⠠⠀⠐⠀⠀⠂⠀⣀⠀⠠⡄⠀⠢⠀⠒⠀⠁⠀⠀⡀⠀⠀⠉⢀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠉⠀⠈⠛⠒⠀⠀⠠⠤⠀⠀⠲⠒⢀⠉⢀⠤
⢶⠀⠘⠀⠉⠁⠀⠀⡀⠀⠠⠤⠄⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⡐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠄⠀⠐⢀⣤⠄⠀⠈⠁⣀⡤
⠀⠀⢀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣄⣀⡄⠀⠀⢀⣠⠄⠀⠀⠈⠀⠰
⠠⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⡀⠀⠈⠀⣠⡦
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠒⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⣀⣀⠀⠀⠀⠀⣠⡴⠆
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⡠
⠀⠀⠀⠀⠉⠙⠒⠤⠀⠀⠀⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠊⠁⠀⠀⣀⠀⠀⠈
⢾⣶⡄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠰⠓⢤⠄⡀⠀⠀⠀⠀⠀⠄⠠⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠉⠉⠉
⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠉⠁
⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠚⠂⠀⠀⠀⠀⠀⠰⠗⠂⠂⠒⠀⠀⠀⠀⠚⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⢤⡤⠤⠖⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⢀⣤⠀⠀⠀⠈⣿⡟⠁
⠀⠀⠀⠈⠉⠁⠄⠀⠈⠙⠛⠓⠀⠀⠀⠀⠀⢀⡴⠾⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⠀⠀⠄⠒⠂⠀⠀⠀⠘⠉⠉⠉


⠀⡀⠄⣠⠄⣀⡀⠀⡀⠀⣀⠀⡀
⢦⡈⠲⢄⡉⠒⢭⡑⠂⠤⢀⠋⠥⣂⠢⠤⣐⡂⠤⢀⣀
⠀⠙⢦⡀⠙⠢⣀⠈⠓⢦⣀⠉⠲⢤⣌⡁⠂⠬⣉⠓⠲⠭⣝⡲⠤⠤⣄⡀
⢳⡀⠀⠙⣦⠀⠈⠳⣄⠀⠈⠳⢶⣀⠈⠙⠳⢤⣀⡉⠑⠶⢤⣍⠙⠒⠦⣭⣙⠒⠦⠤⣄⣀
⠈⢻⣤⠀⠈⢳⡄⠀⠈⠻⣤⠀⠀⠉⠳⣤⡀⠀⠈⠑⠶⣤⣀⠈⠙⠲⢦⣀⠉⠙⠲⢦⣄⡉⠛⠲⢦⣄⡀
⠀⠀⢷⡀⠀⠀⠙⣤⠀⠀⠈⢣⣄⠀⠀⠈⠻⣦⣄⠀⠀⠈⠙⠷⣤⡀⠀⠉⠛⠶⣄⡀⠈⠙⠲⣦⣀⠈⠉⠛⠶⣄⣀
⡇⠀⠈⢿⡆⠀⠀⠙⣷⡀⠀⠀⠙⠷⡀⠀⠀⠀⠙⢷⣦⠀⠀⠀⠈⠛⢷⣤⡀⠀⠈⠛⠷⣤⣀⠀⠉⠳⢦⣤⣄⡈⠙⠳⢦
⣿⣶⠶⠾⠿⣶⣶⣶⣿⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣾⣿⣿⣿⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡴⣶⣶⣶⣶⣲⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠻⢿⣿⣟⣿⣿⡟⠀⠀⣤⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⢦⣶⣿⢾⣧⡘⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣯⣨⣿⣿⣆⢿⣯⣛⢜⣀⣦⠀⡀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⠁⠀⠀⠀⠀⠀⣠⣌⣙⢻⣿⣿⣿⣿⣿⣌⠿⢿⡾⠿⠁⠀⠁⠀⠀⠀⠀⣠⡾⣿⠄
⠀⠀⠀⠀⠀⠀⠀⣈⣁⣀⣀⣤⣤⣤⡶⠾⠟⠉⠉⠉⠳⣽⣿⣿⣿⣧⣿⣿⣦⣦⣀⠀⠀⠀⢀⣈⣤⣾⣿⣧⣿⡆
⠀⠀⠀⠀⠀⡴⠋⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⡛⠃⢏⣹⣅
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⣦⡄⢀⢰⣴⣶⡀⠀⠀⣠⣾⣿⣿⣾⣿⣿⣿⣿⣻⣿⣌⣿⠋⣿⣷⡿⣗⡓⣏⣛⢻
⠀⠀⠀⠀⠀⠀⠉⠠⢀⣤⣿⣿⣧⣤⣿⣿⣿⣤⣤⣴⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣨⠃⠀⢻⣿⣻⣻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢿⣹⣿⣿⠟⠿⢿⣿⣿⣰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⠂⠀⠈⢱⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⣀⡀⠙⡿⠋⣀⣀⡈⣻⣿⣟⢿⣇⣿⣿⣿⡟⠛⠿⣿⣿⣿⣿⣷⠈⠀⠀⠸⡿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⢫⡛⢿⡟⠙⢛⢿⡟⢿⣾⣿⣿⣻⣧⢿⡟⠀⠀⠈⣭⢽⣿⡄⠀⣀⡖⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠙⢾⣧⣿⣻⣿⡏⢀⣸⣋⠀⢙⣶⣿⣷⣶⡀⠀⠹⡯⣥⣶⠙⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢄⠉⠉⠉⠻⣼⢡⣙⣿⣿⣿⠿⠟⠀⠀⠀⣀⣀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⢻⡏⠻⠍⠽⢁⠀⠀⣈⡤⠤⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⢹⣿⠠⢝⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⡴⠁⠉⠠⠀⠆⠺⠟⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡼⠁⠀⠋⢠⡄⠐⠀⠃⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⡼⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣼⠀⣠⠖⡋⡽⡫⢻⣿⢻⡻⡿⡓⢖⡲⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠤⠀⢸⣧⢞⡕⢁⣞⡼⠥⢼⣿⠉⠓⠷⣙⡆⠙⣎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⣾⡿⣀⣤⣀⠈⠙⠒⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢰⣿⡟⣆⠀⠀⠀⠀⠀⢠⣿⠟⡹⠟⠨⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠵⠶⣤⡉⠀⢸⠀⠀⠀⠀⢀⣾⡏⠀⠁⠘⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠄⠒⡠⣸⣿⣄⠘⡇⠀⠀⢀⣾⣿⢀⣠⣤⣶⣶⣿⣭⣓⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⠊⡴⢡⡟⠛⣷⡇⠀⢀⣾⣿⡿⢹⣃⢸⣷⠷⣝⣟⢮⡳⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⡟⠒⠷⠛⠀⣼⣿⡟⠓⠋⠁⢸⣿⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⢰⣿⣿⣇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⢸⣿⣿⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⢇⠈⢣⠘⠁⣿⣿⣿⣇⠀⠀⣿⢸⣿⣿⣧⠀⢠⣿⣿⣧⠀⠀⠶⠀⠋⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡳⡈⡇⢤⠻⡇⡟⢿⣄⢰⣿⡸⣿⡟⣿⡆⢻⢿⣿⡿⠂⢀⠀⡀⢀⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⢴⣌⣢⣻⡴⣆⠻⢦⢹⣦⠹⢃⡌⢧⣏⡟⢻⡏⠀⢌⣠⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠙⢧⣁⡿⠛⣿⠀⠇⠳⢸⠻⣄⣨⠟⠛⠛⠋⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠋

⡂⠙⠿⠧⠤⠀⠤⢠⣀⢹⠇⢀⠡⠀⡈⠑⠀⠐⠀⢀⠈⠱⣧⠀⠄
⠀⠔⢤⠀⠜⠳⡄⠈⣿⠢⠂⠈⠑⠁⣄⠀⠀⠀⠀⠉⢄⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢸⣧⣨⢤⠐⠊⠙⣆⢫⡴⢂⡀⠀⡀⠈⢿⡇⠀⠀⠀⠀⠠
⠨⠋⠀⢸⡇⡐⢾⡟⠀⠀⠀⡉⠢⣌⡐⠊⠁
⠀⠀⠀⠘⠃⠘⠀⠈⠀⠀⠀⠀⠀⣿⡇⠂⠀⠀⠀⠌
⠀⠐⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣀⠠⣠⡾⠿⡚⢿⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢷⡄⢴⡆⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⢀⠀⠘⡏⠀⢰⣿⠂⠀⣛⣿⠀⢀⣿
⠀⢸⡇⣤⡄⢽⠀⠀⣾⡿⢿⣇⣬⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⢐⠀⠀⠚⡟⠒⡟⢫⡟⣤⠊⣏⡄⢀⣀⣀
⠀⠀⠀⠘⠃⠀⠀⠀⠉⠁⠈⠁⠀⠉⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠁⠀⠁⢘⣿⣿⣿⣧⠈⠉⠉⠉
⡒⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣐⢄⡢⣀⢄⡠⣀⣁⣰⠦⠴⠒⠛⠋⡸⠀⠀⠀⠀⠁⠀⡇⢸⡇⣿⢸
⠭⣓⡪⢭⣒⠦⢄⡀⠀⠀⠀⠀⠀⠀⢈⣳⡽⠶⠓⠋⠉⠀⠀⠀⠀⠠⠘⠁⢡⡀⠀⠀⠀⠀⠀⡇⢸⢠⡇⣿⠄⡄⡠
⠭⣒⡫⢵⣒⠭⣓⡪⠅⣂⣠⠴⠒⠋⠉⠀⠀⠀⠀⠒⢷⣆⠀⠀⠠⡤⠀⠁⠠⡿⠀⠀⠀⠀⠀⠁⣉⣿⠋⠙⣘⠟⠣⢀
⣯⢕⣚⡭⢖⣫⠵⠒⠋⠅⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠋⢉⠿⡀⢠⡿⢡⡀
⢞⣫⠵⠚⠉⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠐⠞⠃⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠃
⣿⣷⣶⣶⠶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⢶

⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⣤⠀⠃⠀⠻⢅⡶⠀⠀⠀⠀⠀⢀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠃⠀⠀⠀⠀⠀⠀⠠⡂
⠀⠀⠀⠙⢧⠀⣠⠤⡀⣐⣄⡄⠀⠀⢴⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠁
⠀⠀⠀⠐⢿⣯⢉⠒⢠⠄⢤⡧⠀⠀⢘⠄⠀⢠⢄⢬⠀⠀⠀⠀⠀⠀⠀⠛⢀⠀⠀⠀⠀⠀⠀⣀⢢⠀⣾⠁
⠀⠀⠀⠀⠀⠙⢎⠀⠃⠂⣯⠕⣋⠄⠨⣆⠀⠀⡺⣄⢢⠄⣀⠋⠁⠠⠀⣤⡒⠀⠀⢰⡇⠈⠁⠉⢰⣄⣉
⠀⠀⠀⠁⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⡈⡅⣿⠘⡐⣻⣂⠀⠀⠀⠘⢿⣂⣀⢼⡆⠀⠀⣀⣼⠋
⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠈⠁
⠀⠀⠀⣀⢤⣇⣾⡶⠄⢀⡀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣉⠁⠃⠁⠔⠀⣴⣶⡀⠀⠀⠀⠠⠀⠀⢦⣆⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⢠⡀
⠀⢠⣬⡇⠀⣰⣧⡔⣲⣨⣶⡄⠀⠀⠀⠀⠀⢨⡿⢿⣷⡠⣄⠀⠀⠀⠀⡌⡠⢀⢄⠀⢦⠀⠀⠀⡄⠀⢱
⠀⠈⢹⡅⠀⠉⢸⡸⢿⣻⣏⣡⡖⠀⠀⠠⢤⡗⠚⣲⣤⠱⠿⠷⠀⠀⠀⡇⣠⠐⠀⠁⠈⣷⢁⡀⠑⡀⢸⠃
⠀⠐⠞⢃⠀⠀⠙⢧⣸⢁⢛⣃⣠⠀⠀⠀⠈⠛⠄⠈⢋⣾⡟⢰⡿⠀⠀⠑⡘⠄⢀⡀⢢⣼⣄⠷⡖⢘⠑⠄
⠀⠀⠀⠀⠁⠀⠀⠠⠀⠉⠉⠅⠀⠀⠀⠀⠀⠠⠀⠈⠉⠒⠒⠂⠃⠀⠀⠀⠲⠧⠠⣁⢈⢈⢀⡤⠖⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⢣⢳⡀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣅⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⢢⠣⣀⠿⡂⠀⠀⠄⡰⠀⠀⡀⠎⠀⠀⠁⠀⠀⠀⠚⡄⢸⢀⠆
⠀⠀⠀⠀⠀⠀⠐⢢⡓⠏⠒⠀⣈⠁⠟⠉⠁⠉⠢⠀⠀⣀⠆⠐⠐⠂⠕⠲⢎⠑⠤
⠀⠀⠀⠀⠀⢐⠤⠒⠒⠒⡲⣤⢂⣿⣀⠀⠀⠀⠀⡁⢀⠁⠀⠀⠀⠀⢸⣀⢢⠄⣠⢣⡄
⠀⠀⠀⠀⡔⠁⠀⠊⠀⢀⠀⠉⠉⠉⠀⢌⢑⡴⠉⠀⡠⠛⠉⠀⠑⣎⠁⠀⠈⠺⢉⡔⠉⠧⢲⢀⠀⠀⠀⠀⠀⠀⡀⡀⠤⢀⠄
⠀⠀⠀⠀⠁⠌⠀⠀⢀⠁⠀⠀⠀⠁⠀⢰⠁⠀⢠⠈⠐⠆⠀⠀⠁⣀⡍⢂⡠⠊⠁⠹⢄⠤⣜⣉⠠⠤⢐⡀⠨⠅⠒⢀⣉⠀⠀⡀
⠀⠀⠀⢀⣦⡀⠀⢀⣨⢀⠀⠀⣀⣠⠀⣀⣦⡀⠀⠀⢠⣴⢀⣩⣬⡴⢃⣨⡅⠀⢠⠇⣊⠥⠴⢖⣚⢊⡀⠨⠭⠀⠁⠐⠒⠀⠉
⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠠⠄⠀⠲⡖⠀⠈⠁
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠠⠄⠈⠡⠤⠀⠄⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡰
⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⡀⠀⣀⡀⠀⠀⠀⠀⠠⠀⠁⠐⠒⠒⠘⠋⠉⠁⠒⡚
⠀⠀⠀⠀⠀⠀⢀⡾⢻⣍⠹⣇⠀⢀⠀⣀⠀⠠⠤⠠⠀⠀⠒⠂⠀⠀⠀⢩⢸
⠀⠀⠀⠀⠀⠀⠘⣯⡈⡛⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣢⣬⣷⠶⠛⢾⠿⢶⣤⡒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣡⣾⣿⣾⣿⡷⡀⣸⣧⡀⠎⣽⢷⡝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢫⣼⢇⠋⠙⢉⣠⠷⠿⠬⢳⣮⡻⠧⢿⠸⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡲⣛⡵⡞⠉⢭⢀⣴⣟⣽⣶⣶⣆⠐⠂⠌⣙⠿⡉⣧⣿⡿⢿⡷⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⣾⣾⡍⡆⢱⠀⣿⡿⠛⠁⣾⠿⢿⣿⠿⣷⣦⠘⡷⢘⢿⢿⡿⠏⢹⡎⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣾⠛⠿⠟⠁⣸⣼⣋⣡⣀⠡⡘⢇⣼⣷⣤⣿⡿⡟⢃⣜⣙⣯⠻⢖⠀⢳⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⡇⠈⢐⡶⢛⣉⣤⣬⣭⣙⣛⡻⣦⣍⣙⣫⣭⣥⣶⠿⢋⣉⣁⣤⠬⢿⣦⣻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣁⣼⡾⠻⢫⣶⠟⢿⣿⠏⡉⡍⢙⠿⠚⢿⣿⣶⢶⡿⠿⠛⠋⢏⠻⣻⣷⣾⣼⠙⠛⠓⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠄⣠⣤⢟⣽⢿⣷⠇⣸⣴⣯⣵⣬⣶⣶⣾⠞⣫⢿⠭⢭⣻⡟⣿⢺⠟⠶⠶⣷⣶⣷⣾⣿⠿⢿⢻⣿⣿⣿⡦⣬⣧
⠀⠀⠀⠀⠀⠀⢠⠎⢩⢣⡟⢁⣀⡴⠿⣟⡽⠟⠋⠩⣉⡛⢌⠩⠡⠀⠀⠀⠀⠀⠆⡠⡀⢂⡤⢒⠉⠁⠉⡛⠲⢦⣦⣽⣿⣿⣴⣩⣝⡷⡀
⠀⠀⠀⠀⠀⠀⠸⡀⠸⢻⣊⡩⠗⣂⠉⠅⢀⣠⡤⢶⣦⠽⢦⠀⠀⡆⠀⠀⠀⣀⢠⡑⣴⣿⠖⠦⠀⡒⠂⢀⡹⠶⠬⠉⠉⣙⢛⡡⣍⢷⠆
⠀⠀⠀⠀⠀⠀⠀⢳⣄⡪⠙⠿⠷⣦⣄⡀⢘⡫⠊⡿⠛⠛⠛⡆⠆⠀⠀⡀⠀⠸⡾⠁⣿⢿⢿⣷⣒⡀⠀⠀⠀⠀⣠⡌⢉⢅⡶⠼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠭⡑⠢⢕⣛⣛⢛⠛⠷⣧⣆⣐⡇⠀⠀⠀⡇⢠⢀⡇⢰⣿⠀⢀⠱⠌⠀⠤⠚⠋⢉⣠⡴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠚⠛⠳⠤⣘⣋⡃⠀⠆⣰⠇⢸⠸⠀⢂⣿⣉⠉⠁⠠⠀⡲⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠄⠀⡐⠜⠀⢸⠀⠀⢊⠎⢧⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡈⠀⢠⠆⠀⠀⢸⡄⠀⠀⠡⠨⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡼⠀⠀⠀⢠⠳⠀⠀⠄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠘⠀⠃⠐⠈⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢶⠀⣹⡇⠀⠀⠀⠀⠀⠀⠇⢀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⢆⠀⢻⡄⠀⠀⠀⡀⠀⡄⢸⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣂⠹⣷⠱⡀⠀⡃⢀⣣⢇⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣵⣽⣾⣷⠾⠛⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⢰⣦⣄⠀⣶⣴⣆⣰⣷⠀⣤⣾⡆⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⢿⣷⣼⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣿⠁⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠘⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣽⣿⠿⣿⣿⣿⣿⡿⣿⢿⣻⡟⡯⣿⣿⣛⣿⣿⣻⣿⣿⡷⣻⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣻⣿⠿⡿⣿⡻⢟⡟⣪⣿⣿⣿⣿⣾⣟⣇⣷⣟⣶⣾⢾⣿⡿⢻⣿⣯⠿⣾⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⠿⠿⠛⠋⠋⠙⠳⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣟⣯⠵⠛⠙⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⣽⣶⣔⡦⢴⣶⣦⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣷⣦⠔⠒⢶⠖⣍⠉⠙⢤⠀⢜⣿⣿⣌⠟⠛⠻⣿⣿⣿⣿⣿⡿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢹⡏⠀⠀⠀⢺⣿⡒⠋⠊⠀⠙⠋⣵⠉⠀⠀⢸⣿⡇⡄⢀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡦⠄⠀⣈⢣⠀⠀⠀⣿⣿⠃⠀⣀⣤⣿⡇⣿⡇⠀⠀⣸⣿⠇⢻⣿⣿⣿⠿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣡⡇⠀⠧⠀⠁⠛⠉⠙⡍⠉⠉⢩⠉⠉⣹⡟⠁⣴⡆⠈⡧⢸⣸⣿⣿⠁⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣖⠚⠀⠀⠀⠀⠀⠇⠀⠀⠼⠀⣠⣿⣷⣶⣿⣷⣾⠻⢉⠸⠿⠯⠀⠚⠋⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠷⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣫⣉⠛⢛⡟⠛⠛⠛⠤⠀⣀⠀⠀⠈⣧⠀⠀⠻⣦⠘⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡄⠀⠀⠀⠀⣏⠀⠀⡿⠋⠉⠀⠀⠈⠳⡐⠀⠀⠀⠀⣈⢁⠀⠀⢼⣧⠀⠀⢻⣧⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠙⣦⡀⠈⢷⡗⠀⢸⣿⡆⠀⠘⣿⡇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣂⠀⠀⠀⠀⠀⠀⠃⣰⠀⠀⠀⠀⠄⠀⠀⠀⣿⡄⠀⠘⣿⣆⠈⣿⡄⠸⣿⣷⠀⠀⣿⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠒⠛⠀⠀⠀⠀⠒⢿⠀⠀⠀⢀⡶⠀⠀⠀⢸⣿⡀⠀⢻⣿⡀⢸⡇⠐⣿⣿⠀⠀⣿⣿⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡶⠤⠀⢀⠀⠀⠀⠸⡄⠀⠀⠈⣧⠀⠀⠀⠸⣿⡇⠀⠸⣿⡇⢸⠃⠀⢸⣿⠀⠀⣿⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠳⡀⠀⠀⢻⠀⠀⠀⠀⣽⡇⠀⠀⣿⠇⡜⠀⠀⢸⡟⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⣇⠀⠀⠀⢸⡇⠀⠀⠿⠋⠒⠒⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠘⠓⠀⠁


⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⠿⠿⣿⡿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⢀⣴⠾⡻⠛⢍⢻⢹⣟⣶⣶⢶⡄
⠀⠀⠀⠀⣾⠔⣉⣲⣶⣾⣿⢾⣿⣿⠷⠚⠋⠈⠁⠐⠒⠶⣿⡶⡄
⠀⠀⠀⠀⢻⣿⣽⣿⣿⣿⣷⢀⣿⡿⣛⣒⣠⣤⡤⣤⣶⣶⣾⣿⣿⠎⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⣀⠙⠋⠉⠉⠙⢿⣿⡿⠞⣋⣠⣴⣾⢾⣷⣥⡶⠭⠞⣽⣧⣤⣶⣶⣷⣯⠉⠻⣦
⠀⠀⣰⣟⠛⣯⣏⣋⣳⣶⣼⣿⣿⡀⢔⣾⣿⡇⢠⣿⣿⣿⣖⢋⣻⣿⡏⠀⣿⣿⣿⣶⢀⠽
⠀⠀⠘⠿⢿⣮⣿⡟⠛⣿⣿⡿⣿⣷⣿⣯⣼⣄⣼⣿⣽⣷⣽⣿⣟⣻⡷⢠⣿⡭⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⣡⠾⢿⣿⣿⡟⢻⣿⣿⡟⢈⣿⣿⡏⢹⣿⣿⣿⣿⡈⣻⣶⡶⠶⠒⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⠺⣿⡀⣾⣿⣿⣇⢸⣿⣿⠇⣾⣿⣿⠵⠿⣿⣿⣿⡿⡗⠉⠈⠻⡖⠒⠂⠀⠀⠀⠀⠈⠳⢄⣀⡤
⠀⠀⠀⠀⠠⠁⠠⡿⠿⠿⣿⣿⣿⣽⣿⢏⣼⣿⣿⢖⣲⡖⣿⠿⠒⡞⠉⠳⠀⠀⠀⠀⠀⠀⠀⣠⣴⣖⠒⠾⣿⣿⡿⢻⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠾⠿⣿⡿⢿⠏⠆⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣦⠈⣿⣷⠿⠟⠛⠋
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠄⡀⠀⠁⠀⠀⠀⠘⣿⡏⠀⠀⠀⠠⢄⠈⡆⠀⠀⢳⡄⠀⠤⢚⣉⣛⠟⣿⣟⡹⣧⣿⣾⢸⣳⢆⠀⣀
⠀⠀⠀⠀⢹⣶⣄⡀⠀⠀⠀⠑⣾⡀⠀⢀⠀⠀⠹⣧⠀⠀⣠⠤⠦⠿⠿⠤⣀⣾⢃⣤⣾⣿⡟⢋⣴⡭⠛⣿⣿⣿⣵⡾⡷⠾⠷⠃
⠀⠀⠀⠀⠸⠛⠉⣉⡀⠀⠀⢀⡿⢁⡴⣡⣶⣶⣶⣭⠀⠠⠁⠀⢀⠠⠴⢶⡟⢁⢻⣿⣿⣿⣿⣮⣴⠇⠀⣨⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⣄⠞⠹⢡⣠⣿⣿⣿⣿⣿⣄⠃⠀⠀⠁⠀⠀⠀⡷⠘⠌⢿⣿⣿⣿⢟⣉⠄⣸⣿⣿⣿⠟⣋⣥⣄
⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⣿⣀⠀⠀⢿⡛⢸⣿⣿⣿⣿⡀⠀⠀⣴⣶⠤⠚⠁⠀⠀⠀⢀⣀⣤⣿⠀⠀⣿⣿⣦⣴⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠘⡇⢸⣿⡿⣻⣿⣷⣆⢬⠑⠻⠿⣿⣿⣿⡇⠀⠀⢠⢶⠀⢤⣤⣤⣤⣤⣄⡀⠐⢿⠀⠀⠈⠓⠉⢉⠀⢀⠉
⠀⠀⠀⠀⠀⡄⠐⠛⠙⠊⠻⣿⣿⣿⣄⢳⠀⢠⣦⡈⢸⡇⣄⠀⠀⠸⢢⣦⣿⠏⣦⢻⣿⣏⠀⠀⠁⢠⣤⣄⡘⣦⣅⣸⡆
⠀⠀⠀⠀⠀⠸⣄⠀⠙⢷⡄⡿⠻⣿⣿⡆⠀⠀⠹⠇⠘⢰⡿⠢⣦⣶⣾⠋⠁⠀⡿⠨⠉⢹⠛⣛⢻⣻⡋⠙⢻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠒⠋⠉⠁⠀⠀⠀⠀⠀⠛⠇⠀⠀⠈⠁⠀⠀⠰⡇⣷⠑⢤⠭⠿⠯⠶⢦⡀⣼⠿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⢀⠀⠀⢀⠀⠀⡀⢀⣀⡦⠂⠀⠀⠀⠀⠀⢂⠀⠛⠉⠃⠀⠀⠀⠀⠀⠀⢋⠁⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡚⠦⠶⣿⠋⡿⠩⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠋⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠁⠀⢹⣿⣿⣿⣿⡿⠷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣇⡀⣠⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⠋⠉⠹⣿⣿⣿⣿⠟⠉⠙⢿⣦
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⠀⢀⣿⣿⣿⠟⠉⠻⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣦⣤⣼⣿⣿⣿⣿⣄⠀⢀⣼⣿⣧
⠀⠀⠀⠀⠘⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣦⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣧⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⢠⣿⣄⣼⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⡉⣹⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠹⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⠀⠀⠀⢀⣶⠀⣤⠀⢀⣼⡟⠀⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣿⣧⡀⠀⣾⠁⠀⣠⡾⠃⣠⣿⣧⣾⣿⣇⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣿⣿⣾⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢉⣠⣤⣤⠶⢶⡶⢦⣤⣄⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣯⣾⡶⢻⠆⠸⣷⢾⣿⣿⢃⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⡀⠀⠀⠈⠲⢭⣛⣛⣻⠀⠀⣿⠭⠽⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⠸⣆⠀⠀⠀⠀⠀⠀⡿⠀⢸⠇⠀⠀⢀⣠⠴⠤⡀
⠀⠀⠀⠀⠀⠀⢀⠞⢋⣉⣀⣀⣀⠀⠘⠳⡄⠀⠀⠀⢰⠇⠀⡼⠀⣴⠞⠋⠁⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠘⢦⣛⠿⢿⠇⣿⣿⡿⢆⡽⠀⠀⢀⡏⠀⢠⠇⠘⣇⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⡟⠒⠋⠉⠀⠀⠀⡞⠀⠀⡞⠀⠀⠈⠓⣦⢤⣀⣀⣀⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⠀⠀⠀⣸⠁⠀⣸⠁⠀⠀⠀⣰⠋⢸⠃⠀⠀⣀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⢧⠀⠀⠀⢀⡇⠀⢀⡇⠀⠀⠀⡴⠁⢠⠇⡴⠚⡋⠁⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠒⢦⡀⠘⡄⠈⢣⡀⠀⢸⠀⠀⢸⠁⠀⠀⡼⠁⢠⠏⠀⠹⢜⣯⡟⣲⣦⡈⢳
⠀⠀⠀⠀⠀⣠⠾⠁⠀⠀⠉⢳⡜⣆⠀⠹⣄⣾⠀⠀⣸⠀⠀⢸⠃⢀⡏⠀⠀⠀⣠⠏⢠⠟⠿⠵⠊
⠀⠀⠀⠀⠀⠻⢤⣀⣀⣠⠤⠚⠁⠘⣆⠀⠸⣿⠀⣠⣿⡄⠀⣿⣠⢸⠁⢀⡤⠚⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⢧⡀⠀⠀⠸⡆⠀⢿⣗⢞⣿⣧⢸⣷⣷⢸⡶⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠙⠳⠶⢦⣿⡡⠜⣿⡙⠜⢹⢾⡟⡣⢾⣇⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠿⠿⠛⠸⣧⠂⣸⠗⠒⠋⠀⢻⣌⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⣴⣿⣿⣷⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣐⣾⣷⡆⢀⠀⠀⠀⠖⢙⡀⠀⠈⠃⠁⠀⡴⠶⠀⠀⠀⢰⠶⠶⣤⠀⣀
⠀⠀⠀⠻⠿⠛⠟⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⢀⣬⣿⣿⣷⣢⡔⠂⠀⠀⣿⣾⡞⢧⡄⠀⣄⠙⠆⠀⠀⠀⠈⢁⡄⠉⠈⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⢠⣚⢄⣹⡇⠸⢄⠀⠀⠀⠀⠀⠀⠀⠴⠇⣰⣋⠀⠀⠀⠀⠀⣿⠋⠀⠰⣷⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠀⠀⠀⠀⢺⣹⢺⡛⣇⡐⢘⣤⡄⠀⠀⠀⢰⡕⠐⢲⠀⠹⠇⠀⠀⠀⠀⠀⠀⣰⠂⡌⠛⠁
⠀⠀⠀⠈⣛⡦⣀⣤⣄⠙⢿⠛⠙⠀⠀⠀⠀⠈⢕⢺⢯⡝⢩⣽⣿⡃⠀⠀⠀⠀⠀⠀⢹⣦⠀⠀⠀⠀⢀⠀⠀⠀⣨⣶⠉⠆
⠀⠀⠀⠹⣿⣷⢾⣿⡾⠀⠀⠀⠀⠀⢠⡄⠀⠀⣟⡻⢶⣿⣿⠋⠙⠁⠀⠀⣤⡄⠀⠀⠀⠙⠀⠀⠀⢠⠁
⠀⠀⠀⠀⢸⣿⢾⡋⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢫⣼⣿⠉⠛⠀⠀⠀⣠⡶⠿⣷⣶⣄⠀⠀⠀⠀⠤⠀⢰⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠈⠙⠛⠁⠀⢺⣿⢻⠀⠀⠀⠀⠀⠀⠀⠔⢿⠆⠀⠀⠀⠀⣷⣷⠾⣾⣿⣿⠀⠀⠀⠘⠁⣆⣿⠀⠀⠀⠀⠐⠾⢣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⣧⣿⣿⡿⠟⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠲⣴⣿⣟⡷
⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠈⣩⡟⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠉⠰⠃
⠀⠀⢠⣿⣻⣿⣿⣿⡆⠀⠀⠀⣰⢾⣿⣧⡀⠀⠰⠟⠀⠛⠀⠀⠀⠀⠀⢀⡤⣤⡀⠉⠇⠀⠀⠀⠀⠋⠀⠀⡀
⠀⠀⠘⠿⢟⢿⣿⣿⠇⠀⠀⠀⡉⠀⠈⠙⢻⡆⠀⠀⠀⠀⠠⠀⠀⠀⠀⣬⡾⣿⠀⢠⣄⠀⠀⠀⠘⠻⠿⠐⠁⠀⠀⣠⣴⡆
⠀⠀⠀⠀⠛⠻⠛⠋⠀⠀⠀⠸⡿⣄⣀⣤⣼⠇⠀⠀⢀⣠⠐⣀⠀⠀⠀⠻⠅⣠⢸⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠐⠛

⠀⠄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠃⠓⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠂⠀⣒⣒⠲⣂⠀⠓⣀⣒⡂⠐⣖⣚⡀⠛⣂⣙⣂⡛⣆⠐⢂⣘⣳⠀⣲⣘⠃⠠⠓⠈⢻⠂⠛⢂⡼⠏⢻⡧⠛
⠂⢀⡒⠀⠀⢹⡆⠀⠘⢳⠀⠀⠉⣷⠆⠀⠈⠳⡀⠀⠒⣃⡀⠚⠃⡀⠘⠁⢠⡘⠃⠀⣬⡋⠀⢀⣍⠀⠀⣍
⣀⠘⠉⠀⠀⡈⠉⠀⠀⢘⣳⣀⠀⠉⢉⣥⡀⠈⠙⢀⣄⣉⠉⠀⢈⣹⡁⠀⠈⠍⠀⠀⠉⣁⣀⠀⠈⢀⣴⣤
⡛⢓⠄⠦⠀⢹⡓⠤⠒⠀⠍⢉⡁⠀⠈⣁⡈⠀⣀⠭⠉⢉⠁⡀⠈⢙⠋⡄⠈⠙⠣⠀⠀⢙⣙⡂⠄⠠⠉⠉⡄
⣿⣿⣾⣷⣶⣾⣿⣷⣾⣾⣾⣿⣿⣿⣿⣿⣷⣶⣿⣷⣿⣾⣿⣿⣷⣾⣿⣷⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣾⣿
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉⠩⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠚⠋⠑⡒⢐⣀⡀⠀⠀⡀⠀⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⡾⠸⠟⠛⢓⠶⢤⣄⡑⠛⢖⠄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⡿⣿⠫⠑⠌⠂⢄⡀⠀⠁⢃⠀⠀⠀⠀⠀⠀⠈⠙⠺⢤⣁⠢⡐⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠃⢘⡀⢰⢬⣓⣴⠿⠷⣄⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⢳⣜⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣦⣸⠵⠋⡉⡄⠠⢰⠐⠀⠅⠐⠈⢱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⡪⢋⠑⡈⢆⢱⠸⠰⣠⢖⠋⠉⠙⢿⢇⡀⣰⠋⠒⠒⠂⢀⠄⠀⢒⣚⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⣉⠏⠠⡈⠣⣝⣌⢺⣠⣾⠅⠘⡀⠀⠀⠀⠐⠈⢃⠀⢠⠀⢀⢵⣎⡭⣍⣀⠙⢻⣧
⠀⠀⠀⠀⠀⠀⡀⠀⠉⢒⣐⠖⠓⠂⠉⢱⠾⠁⡀⠼⠀⠤⠠⠤⠤⠀⢀⠱⡀⣰⠁⠀⡀⠉⠻⢽⣾⡄⡻
⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠯⠉⠉⠀⠀⡆⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠡⣧⡅⠀⠀⢀⠀⠀⠀⠙⢗⠇
⠀⠀⠀⠀⠈⠳⡢⡀⠀⠀⠐⢄⡀⠀⠀⠻⣄⠠⠀⣀⣀⣀⣀⣀⣀⡀⢤⡵⢻⠇⠁⣶⠇⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠳⡈⢂⠐⢀⠸⡙⠓⠶⡄⠈⢷⡄⣹⢄⣀⡀⢺⡇⠔⠁⠀⠁⣀⣼⡟⡉⠒⠶⠆⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠄⠢⣵⡀⠀⠈⢦⡜⡀⠸⠀⠀⠀⠀⢿⠀⢀⡤⠚⣩⠿⠛⠒⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡚⠩⠉⠉⠲⢼⠀⣁⠆⠀⠀⠀⠀⠘⡿⠉⣰⠿⠁⠀⠀⠘⢉⡴⠟⠀⠀⡠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠉⠶⠀⠉⠉⠑⣀⡀⠸⠤⢸⡗⢀⡀⠀⣐⢲⣷⣴⣣⣰⡒⠂⠐⠈⠉⠁⠶⣥⠈⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠐⢶⡄⠂⠀⠀⠀⠀⢐⠪⠋⠁⠀⠀⠠⡤⠟⢒⠀⢶⢶⣟⡯⢄⡀⠀⠀⢉⠀⠐⠠⠄⠀⠀⠁⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣁⠀⠀⠀⠀⠀⡀⠡⢁⠠⠀⢥⠀⡀⠈⡂⢌⠑⠢⢄⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠓⠀⢠⠀⡀⠀⢁⠐⠀⠈⠂⠀⠀⠄⠷⠀⠀⠩⣃⣈⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠁⠀⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠒⢛⢻⠒⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⢻⣹⣥⣀⣠⢄⠀⠈⢩⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⠏⠁⠢⠋⠙⠶⢀
⠀⠀⠀⠀⠀⢠⣶⡴⠿⠛⠛⠻⢿⣿⣾⣾⣤⣀⣹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⠑⠓⠂⠒⢗⡆⡄⡀⠠⢠⡀
⠀⠀⠀⣰⣾⣿⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⡀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠋⠺⣶⣠⣿⣦⡀
⠀⠀⠀⡻⢏⠛⠀⠀⠀⠐⣿⣿⣷⣶⣿⣿⠓⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣺⣄⠀⠀⠀⢀⣠⡀⣀⣀⣀⣀⣀⢀⡀⠉⠁
⠀⠀⠀⠀⠉⠀⣴⢲⠀⢲⠮⣿⣿⣿⣿⣿⡇⠀⢀⣀⠾⣽⡇⠀⠀⠀⠀⠀⢛⠶⠀⡀⠀⠀⡘⢿⡇⣾⠋⣩⣭⠁⠀⠀⠀⠄
⠀⠀⢰⠿⢰⡇⢴⡟⣠⠈⢻⡙⡛⢿⢿⣿⣛⠀⢯⠘⣣⣼⡗⠀⡀⠀⠀⠀⡠⢀⠄⢁⠅⠀⠆⠙⣱⣥⡀⠁⣿⠀⢀⠒⠄⠁⢂
⠀⣈⢻⡃⣾⡴⠺⢦⣻⣿⣬⣽⢧⣠⣦⣤⣿⣶⣻⣷⣧⣿⣇⡞⠁⠀⠠⠌⡀⠐⢀⡥⠐⡪⠄⠂⠀⠈⠋⠛⠃⠀⢀⣤⣴⠳⠈⠠⠂
⠀⠾⠋⠉⠻⠋⠄⠈⢻⡿⣞⢷⡏⢙⣷⣯⣻⣿⣟⠁⠙⠛⠿⠛⠃⠀⣒⠤⠄⣰⠄⠈⠈⡻⡦⢘⣫⠖⠙⠳⢀⢒⢁⠀⠈⣵⣀
⠀⠀⠀⠀⠈⠁⠀⠀⠼⠿⠟⠋⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠋⠈⠀⠈⠁⠀⠉⠓⠂



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⠚⠿⣻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⡄
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣧⣤⣼⣿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠟⣿⣶⠍⡀⠀⠤⣤⡀
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣈⠭⠤⣀⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣈⣿⣶⣿⣿⣾⣿⣦⣤⣿⣿⡆
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠈⠀⢽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠛⢿⣿⣯⠀⠀⠉⢻⣿⣿⣿⣛⠻⠛⠁
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⡋⢹⣧⠀⠼⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣾⣿⡿⣿⣷⣿⣿⣿⣿⣿⡟⣻⣿⠆
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⣱⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠛⣿⡟⢙⣿⣿⣉⡗⣿⣿⣿⣿⣿⣷⡆
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⣠⣤⣿⣾⣿⣶⣇⣸⡟⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣿⣿⢩⣿⣟⣽⣿⣻⣟⣿⣯⣽⣿⢷⡿⠃
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣽⣿⣯⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠛⠋⠓⠋⠙⠛⠛⠀⠉⠉
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠚⠛⠷⠖⠚⣿⠞⣷⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣶⣶⣾⣿⣷⡿⠋⠁
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⠀⢤⣴⣷⣾⡟⠹⣿⣿⣿⡿⢡⣴⡄
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⣿⣿⡀⠀⠀⠀⠀⠙⢿⣿⠁⠀⡙⢻⣿⡅⢻⣿⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠰⠈⣀⠐⢠⡄
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣹⡟⡻⣿⠀⠀⠀⠀⠀⣼⢿⡆⢠⣥⠘⣯⠀⣰⣿⡉⠀⠀⠀⠀⠀⠀⠾⢡⠞⣁⠞⠠⣦⢘⠵⢢⡄
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⣿⢡⣋⣿⡀⠀⠀⣠⣴⣶⣿⣶⣾⣿⣦⣿⡿⠿⢿⣷⣦⣄⠀⠀⠀⣤⣶⣮⣽⠷⣤⡶⣾⣹⣟⣧
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⣪⣿⣻⣇⠀⠀⠀⠙⠛⠩⠿⠟⠻⢟⠛⢿⠶⠾⠟⠃⠀⠀⠀⠀⠸⣿⣿⣽⡟⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡿⢬⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠺⣿⣤⣦⣽⣾⣯⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣫⣿⣿⡛⢻⠶⣽⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⢿⣏⠀⣀⠀⢿⣿⣠⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⣤⠿⢶⣉⡀⠀⢹⣿⢿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠁
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠀⠾⠉⠿⠶⠿⠍⠯⠟⠋⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⣠⣤⣤⡿⠇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇
⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⢀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⠛⠁⠀⠀⠀⢠⡬⠢
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠛⠉⠁⠀⢀⡀⠀⠀⠀⢠⣆⠀⠀⠀⣀⠀⠉⠒⠐⠀⠠⠄⠀⡀
⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⣀⠀⠀⢀⡀⠀⠘⠁⠀⠀⠀⠈⠁⠀⢀⠀⠉⠀⠀⠰⠆⠀⠀⠀⠀⠀⠈⠑⣆
⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠁⠀⠀⠘⠁⠀⠀⠀⠀⠃⠀⢀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢓⡲⠋
⠀⠀⠀⠀⠀⠉⠛⢿⣟⣶⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⢤⠤⠔⠒⠐⠚⡉⠵⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠿⢶⣤⣍⣷⣌⣁⣀⣨⡏⢀⣤⡶⠶⠗⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡏⠉⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣽⣥⣀⣀⣠⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⣷⣤⣆⡤⠼⠲⡅⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢆⢰⠀⣿⡟⠀⠀⠀⠀⢣⠀⢀⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢑⢌⣿⣀⣿⠁⠀⠀⠀⠀⠈⠇⢸⢁⡎⡹⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⢸⣧⢹⣿⠀⠀⠀⠀⠀⠀⠀⣼⡸⢀⡧⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡟⣼⣿⡟⡇⢀⠀⠘⠀⠀⠀⢻⡗⣹⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢁⡇⣿⣋⡗⠸⠀⠄⠀⠀⠀⣼⣡⠇⣼⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⣧⡇⢸⠀⡇⠀⡆⣿⣿⣾⣋⠥⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⣿⣷⣼⣆⣧⣷⡿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠒⠒⢒⣒⣒⣒⠲⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⢋⠉⠀⠀⠀⠀⠀⠀⠰⡏⠀⠀⠈⢙⡄⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠁⠀⠀⣎⣉⣩⠗⠀⠀⠀⠀⠀⠀⠈⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡫⠴⠚⠛⠒⢦⡀⠀⠉⠀⠀⣀⡤⠤⠤⢤⣤⣀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣄⡀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡎⠀⠀⠀⠀⠀⣠⠇⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⡞⠁⠀⠀⠀⠀⠈⢦⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠑⠦⠶⠒⠒⠋⠁⠀⠀⠀⣏⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠹⣄
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀⣀⠀⠀⠀⠈⠒⠒⠉⠉⠉⠀⠀⠀⠀⣠⣄⣀⠈⠲⠶⠶⠖⠚⠁⠀⠀⠀⡠⠤⣌⢧
⠀⠀⠀⠀⡾⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⢸⡾⡆
⠀⠀⠀⠀⡇⠀⠀⠸⡅⠀⠀⠀⠀⠀⢀⣠⡤⠚⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⠀⠀⠀⢀⡴⠲⢤⡀⠀⠙⠉⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠉⠂⠀⠰⠒⠊⢉⣠⠴⢖⡋⠉⠙⢯⠀⢹⠀⣯⢸⠀⡜⢀⠎⠀⡠⠋⣩⠛⢒⣦⣈⠳⣄⣀⡷⠀⠀⠀⠀⡧
⠀⠀⠀⠀⢹⡀⠀⡤⠴⢤⠀⠀⣠⡖⠋⠒⠤⣀⡉⠲⢄⡈⢣⠸⡆⢸⢸⢰⢣⠏⣠⢞⡴⢋⡤⠚⣉⡠⠜⠳⣄⠀⠀⠀⡠⢄⢠⠇
⠀⠀⠀⠀⠀⠙⢆⠃⠀⠜⠁⡼⠡⠬⢍⣛⣒⠦⣍⣙⠒⠽⣦⣱⡧⠴⠶⠾⠧⢼⣟⣵⠾⠵⢒⣋⡥⡤⠖⣒⣙⣇⠀⡎⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⢷⠒⠒⠲⠒⠒⠛⢛⣛⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⣛⣛⣋⠭⠭⠥⠤⢠⡟⢀⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣻⣮⣉⣉⠩⠭⠶⢚⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣦⣤⠶⠶⠾⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠓⠋⠉

⠀⠙⠿⠀⠀⠰⢠⣄⠹⠀⡀⠀⠈⠁⠀⠀⠀⠀⠁⣷
⢀⡒⠂⠀⠙⣆⠀⡿⠐⠀⠀⠁⠦⣀⡀⠀⠀⠀⠀⠋
⠸⡷⠊⣄⠀⢀⣧⠐⠀⠠⠀⠀⠀⡻⠃
⠈⠀⠀⠯⠀⠈⠁⠀⠀⠀⠀⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⢤⠻⠂⢻⣦⠀⢀
⠀⢀⠀⠀⢀⠀⠀⢰⣟⣷⠐⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⣼⡇⢀⣬⣲⠀⢸⡇
⠀⠹⠒⡶⠚⠀⠀⢿⠉⠿⠶⠶⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠸⠀⠨⠱⣏⣴⣰⠄⠤⠤⠶
⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠌⠁⠀⠀⠀⠀⠀⢰⡖⡟⣻
⣞⠴⣑⠠⢀⡀⠀⠀⠀⠀⠀⠀⠑⢮⣪⡵⠵⠚⠚⠉⠉⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢸⠃⡇⡏
⡮⣝⡢⢍⡲⢌⡂⠄⣀⡠⠤⠒⠊⠁⠀⠀⠀⠰⣄⠀⠀⢀⡄⠀⠀⠵⠀⠀⠀⠀⠀⢘⡺⠃⠓⡺⠏
⡭⣗⡯⢕⣪⠵⠒⠩⠁⠀⠀⠀⠀⠀⠰⠆⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⢈⠐⠀⢰⠟⣤
⣫⠵⠚⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠄⠐⠤⠀⠴⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⡌⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢰⠄⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⡊⠀⠈⡆⠀⠀⠀⠀⠀⢀⠀⢤⠈⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⠢⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⣀⣄⣀⢤⠔⠄
⠀⠀⠀⠀⠇⠀⠀⠐⢠⡤⢐⠂⠉⠀⠀⠀⠒⠀⠀⠁⠂⠄⠀⠀⣀⡀⣠⠔⠈⠔⠠⠠⡓⠄⠈⢄⡠⠄⠀⣀⢄⣠⠊⠒⠦⠠⠀⠀⠓⠊⠀⠀⠉
⠀⠀⠀⠀⢠⠀⠐⠀⢨⠑⠢⠌⡁⠒⠀⠄⠀⠀⠀⠀⠀⠄⠂⠈⠤⠿⠃⠀⠁⠸⠀⠀⠈⠢⠀⠀⠑⠀⠲⣁⣀⣠⣤⣤⣤⡤⠖⣶⢒⡲⣖⣲⣒⠒⠐⠄
⠀⠀⠀⠀⠀⠒⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠚⠀⠖⠈⠀⠀⠀⠀⠀⠈⠢⣀⠂⠀⠀⣀⡠⠔⣂⡥⠙⡊⠕⠋⡁⠔⠊⢁⠠⠐⡠⠒⡁⠔⡡⠔⠉⠩⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠤⠀⠘⠁⠀⠐⠳⠆⢉⡅⠊⢁⠠⢰⣍⣠⠔⢁⡠⠐⠈⡠⠐⢁⠄⠊⡠⠊⠀⢀⠔⠁⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠬⢑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢺⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢰⠃⠀⢰⠆⣁⡠⠒⢁⠠⠂⠁⠔⠁⠀⡠⠂⠁⡠⠈⢀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠑⡀⠁⠂⠄⡀⠀⠀⡀⠀⠈⠉⣨⠄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⣠⣰⠀⠈⢉⡘⡧⢤⡊⠁⡀⠔⠁⠀⡠⠊⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠂⠒⠒⡈⠁⢀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠠⠑⠋⠈⡁⠠⠀⠂⣉⣙⠈⠱⠒⡰⠣⡄⣠⠂⡠⠔
⠀⠀⠀⠀⠀⠈⠛⠓⠒⠒⠒⠃⠀⠀⢀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠈⢉⡁⠠⠂⡰⠀⠀⠀⠀⠀⢀⠠⠒⠉⢄⡒⠢⣌⣉⠒⠩⢴⡒⢬⠉⡑⢄⡈⣑⠒⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢀⡧⠀⠒⡿⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠀⠠⠒⠈⠀⠀⠐⠀⣠⢊⡀⠎⡉⠒⠠⢍⣉⠒⠒⠅⣓⠢⠬⠢⣒⠆⢅⠙⠠⡁
⠀⠀⠀⠀⠀⠘⡗⠀⠁⠀⠀⡇⢀⣠⣿⠀⢀⠤⠐⠀⠁⠀⠄⠂⠁⠀⠠⠀⠀⠀⡠⢒⢊⠣⣈⠦⡑⡂⠚⢷⡤⢀⡈⠓⠢⢤⡁⠒⢄⠈⠣⢄⠈⠓
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠰⡏⢁⡠⠚⠀⠁⠀⠠⠂⠁⠀⠀⠀⠀⠀⠀⢀⣤⣊⣀⠛⠩⣉⠲⠤⡀⡉⠂⠤⣍⠢⢉⡐⠢⠄⢈⡃⠂⢌⡐⠤⡁⠒⢀
⠀⠀⠀⠀⢀⡠⡇⠀⡁⣀⠤⠊⠁⠀⢀⠠⠂⠀⠀⠀⢀⣀⡄⠂⠀⠰⠐⢈⣘⠆⠁⡑⠦⠄⢉⡒⢄⢉⠒⠤⢈⠐⠢⠬⡉⠒⢢⡈⠒⠠⢈⠑⠢⢕⡂
⠀⠀⠀⠀⠀⢀⣡⠆⠙⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢤⠌⠀⠀⣠⠟⢁⡘⠉⡀⠱⡄⣈⠑⠂⣤⠜⣠⢄⠉⠲⢄⡙⠢⠀⡈⠢⢄⡉⠢⢀⡀⠉⠠⢀⡉
⠀⠀⠀⠀⠴⠛⠁⠀⠀⡀⠈⠐⠀⠀⠀⠀⠀⠀⠘⠀⢰⣤⠞⠃⢾⡁⠈⠢⣌⠒⢄⢀⠙⠂⢄⠀⠘⢄⠑⠢⠤⡈⠢⣄⠉⠢⡀⠈⠉⠂⢌⠠⢄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡞⠀⡎⠉⠦⡀⠑⠠⡀⠑⠆⣀⠙⠢⡀⠙⢤⡀⠉⠒⠢⣈⠒⠤⠉⠒⣌⡒⠤⡀⠀⠈⠢⣉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠒⠀⠉⠒⠀⠘⠂⠀⠀⠑⠀⠈⠐⠀⠈⠂⠀⠀⠈⠀⠀⠂⠀⠀⠁⠀⠉⠂⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣙⠛⢗⣤⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⡶⠿⣷⣿⠉⠷⢿⡞⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠃⡻⠃⣤⣈⣉⣠⣶⣄⣴⣮⣰⣯⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣮⣶⡤⠿⢞⣛⣋⣭⣭⣭⣭⣉⣉⠛⠿⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⣉⣴⣶⠟⠛⣋⣩⣍⣀⣠⠀⠉⣻⣷⠄⠈⣿⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⢰⡟⠁⣈⣻⣿⣿⣿⣿⡿⠿⣛⣩⠞⣋⣠⣾⢟⡻⠿⣟⠿⣶⡖⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣼⣿⣾⣄⠀⠀⠉⠃⠉⠋⠉⠙⢻⣿⣿⠉⠉⠈⠀⣿⢡⢾⢳⣿⡟⣤⡑⢮⣹⢯⡿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣽⣿⣧⣾⣷⡤⠀⠀⠁⠀⠀⠀⠈⣿⣿⠄⠀⠀⠀⠀⡾⡷⣏⢣⣴⣞⡷⡄⢿⣻⡷⣾⡃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⢚⠓⠾⠿⠋⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠰⠿⠓⠚⠋⣻⣥⣀⣠⡄⣻⣳⠿
⠀⠀⠀⠀⠀⢎⡵⢶⣾⡿⣷⣼⠾⢷⣷⣶⣶⣦⣄⠀⡇⠀⢠⣿⣿⣿⠀⠀⠔⠀⠤⠀⣴⠾⣹⣿⣿⣿⠷⠧⢿⡛⡆
⠀⠀⠀⠀⠀⢾⣳⢼⣿⡳⣾⣧⠀⢰⣿⣿⣿⣿⢿⣼⢇⣴⣿⣿⡿⠁⠠⠂⠀⠀⣠⡞⢙⣶⢿⣿⠫⠻⠙⣾⡎⣟⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠥⠼⠟⠉⠀⠀⡟⢾⣟⣷⣿⠟⠛⠿⠟⣿⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠨⠙⢄⣠⡸⠿⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡘⡄⠀⠉⠀⠈⣧⡀⣀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠙⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣶⠿⠿⠳⠀⠀⢠⣮⣹⣷⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣤⣴⣤⠀⠀⠿⢻⡟⡉⡹⢏⣀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⣀⡆⣀⡿⣷⣼⣿⡄⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠿⢋⠟⠁⠀⠘⣿⣿⡟⣿⣷⢿⣹⡄⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⢆⠀⠀⠀⢻⣼⣾⡿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡂⠈⠈⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣫⣵⣿⣿⣿⣿⣿⣿⣶⣭⣝⡛⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡤⠚⠉⢀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠤⠤⣬⣱⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢟⣥⣾⣿⠿⠿⠿⢛⣛⣛⣛⣛⣛⣛⣛⣛⡳⠎⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣼⠖⠛⠉⣭⠤⠤⠄⢀⠒⢲⠀⠐⠒⠒⢒⣂⣤⠴⠋⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⣭⣵⣶⣖⣛⣻⣻⡿⣿⡿⣿⣭⣭⣭⡭⠿⢿⣛⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠒⠒⢸⠀⣟⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣭⣭⣭⣭⣭⣽⡇⣿⢱⣯⣭⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⡎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣙⣛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿







⠀⠀⠀⠀⠂

⡏⠉⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠛⠛⠉⠉⠉⠙⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠚⠈⠁⠐⠒⢄
⡇⠀⠀⠀⢠⢤⣦⣄⠂⠀⣴⠾⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠂⣀⢀⠀⠤⡴⣶⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠾⠛⢹⠂⠀⠁⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⣼⡷
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢠⣯
⡇⠀⠀⠀⠀⠀⠀⢰⡆⠀⢹⠀⠁⠀⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡄⠀⠱⠀⠀⠤⣿⡗⠀⢾⠀⡼
⡇⠀⠀⠀⠀⠀⣧⠸⠀⠀⢸⡆⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢀⡄⢷⠀⠀⣧⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠘⠗
⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠁⠀⢸⡆⣀⠀⠀⠀⠀⠀⠀⠢⡐⢎⣧⢠⣿⡆⣠⠆⠀⠀⠀⢀⣀⡀
⡇⠀⠀⠀⠀⠀⠳⢿⣀⣠⣤⡀⣀⣼⣿⠟⠀⠀⡀⠀⠀⠀⠀⠘⠌⢿⡌⢯⣹⠟⠀⠀⣀⣤⣿⣿⣿⣷⢶⣤⣀⠀⠀⠀⠈⣶⡷⢶⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⢁⣈⡁⠀⠀⠘⠃⠀⠀⠀⡀⠀⠘⠿⠟⠛⠓⠀⣠⣾⣿⡟⢿⣿⡿⠟⢃⣉⡙⠆⠀⠀⠀⠈⠀⡜⠋
⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⠶⠋⣭⠉⠀⠀⠀⠀⢀⢆⣀⣀⣊⣠⡀⠀⠀⠀⢰⠿⣿⣿⣀⡈⣥⣴⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠙⢻⣶⡄⠀⠀⠘⢹⡿⣿⠆⠂⠀⠀⢀⣠⣥⠈⠉⠉⠉⠀⠀⠀⠘⠛⠿⠟⢿⣿⠟⢃⡞⠁
⡇⠀⠀⠀⢀⡌⠀⠀⠀⠀⠀⢠⠀⣷⡄⠀⠀⠀⠀⠉⠁⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢨⠁⠀⠀⠀⢠⣀⣶⠖⢶⣶⣶
⡟⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠷⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠸⡆⠈⠛⠙⠉⠙⠛⠀⢸⡏
⡇⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⣙⠟⢰⣄⠀⠁⠀⠀⠀⢀⠀⢸⠀⢈⣷⠶⣦⣤⣀
⡇⠀⠐⠷⢾⡎⠁⠀⠠⣄⣀⣀⣀⣀⣀⣠⡤⠤⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠷⠦⣿⡁⠈⠀⠀⠀⠛⠀⠀⠀⠈⡯⠀⣭⠉⠉
⡇⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠉⠉⡏⠁⠀⠀⠀⠀⠀⠀⣀⠈⣽⠏⣿⢿⠷⠆⠀⠀⠀⠈⡗⢉⣉⡥⠀⣤⠀⢠⡀⠀⠀⠀⣇⠀⡇⢀⢠⡤
⡇⠀⠀⠀⢰⡇⠀⠀⠐⠀⠸⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⡸⠀⠉⢲⡥⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠸⣿⣿⠀⠀⡀⢻⣤⡷⠘⣤⠁
⡇⠀⠀⠀⠘⠁⠀⢄⠀⠻⡀⠀⠀⡀⠀⠀⢰⡀⠀⠀⠲⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⢞⣿⠚⠛⡀⠃⠋
⡇⠀⠀⠀⠀⠀⠀⢀⣤⣴⡓⣶⣶⣿⣿⠿⡿⠋⠀⠀⢸⠀⠀⠀⢸⠁⡄⠀⠀⠀⠀⠐⢶⡄⠀⢀⣤⡤⠀⠀⠀⠀⠀⠈⠁⠀⠉
⡇⠀⠀⠀⠀⠀⠀⠈⠛⠙⠛⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢾⠀⠀⠀⣠⣄⠀⠤⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⠺⡖
⡇⠀⠀⢰⣐⡒⠶⢶⣶⠶⠶⣶⣤⣤⠀⡀⠀⠀⠀⠀⢸⠆⠀⠀⢸⡧⡄⠀⠀⢽⡀⠀⢸⠀⠀⠸⠀⠀⠀⢀⠀⠀⠀⠀⠀⣦⠄⠁⠐
⡇⠀⠀⠀⠉⠙⠛⠻⢿⠆⢠⡾⣿⣻⣿⣿⡿⠇⠀⣧⠀⡌⣾⠀⢸⣷⡇⠀⠀⢸⡇⠀⠚⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⣾
⡇⠀⠀⠀⢀⣠⣤⣤⣿⠀⢸⣧⠀⠀⠀⠀⠀⠀⢀⡛⠀⠛⠃⠀⠘⠻⠃⠀⠀⣼⠃⠀⠀⠀⠀⠠⢃⣠⠄⠢⠌⠀⠀⠀⠀⣿
⡇⠀⢶⠶⠿⣿⠆⠘⣿⠀⠸⠟⠀⠀⠀⣀⣤⣶⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠖⠰⠞⠒⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠈⠃⠈⡿⠀⠠⡄⠀⠀⢐⣩⣄⣙⠛⣿⠛⣿⣷⡄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢀⡀⠀⠀⢀⡇⠀⠀⡀
⡇⠀⠀⠀⠀⡀⢀⢀⡇⠀⠀⣰⢀⠀⠀⠉⠛⠻⠿⠧⢶⣯⣿⣵⣄⠀⠀⠀⠀⠙⠛⠿⠁⠀⠶⠾⠓⠚⠁⠛⠉⠀⠀⣬⣿⠇⢠⠀⠃
⡇⠀⠀⣷⡶⣿⡌⡼⠁⣤⠀⢸⠿⠆⠀⠰⢦⡄⠐⠀⠋⠉⠈⠁⠀⠀⠀⢀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⠀⠘⢱⠀⣴
⡏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣤⠀⠀⠁⠀⡶⠶⡆⠀⠈⠛⠚⠿⠏⠁⠀⠀⠀⠀⠐⠛⠛⠁⠀⠀⠀⠀⠄
⡇⠀⠀⠀⢀⡀⠀⣀⣷⣠⡀⠤⠖⢯⢦⡀⠈⠙⠟⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣤⣠⣄⣠⣴⡆
⡇⠀⠀⡘⢿⣟⠻⢿⡙⠇⠾⠗⣾⢻⡇⢃⣠⣤⡀⠈⠛⠃⠈⠋⠈⠻⢶⡀⠀⣠⣾⠏⠙⠀⠀⠀⠀⡄⠀⠀⠀⣤⣤⣤⠀⠉⠈⠉
⡇⠀⠀⠦⣼⣿⣿⡉⠁⠀⠀⠀⡸⠈⠀⡉⢀⣿⠈⣦⡀⠀⠀⠀⢠⣤⠀⠀⠀⣽⡇⠀⠀⢀⠀⠀⢈⡳⠖⠄⠀⠈⢉⠁⠀⠀⠀⠀⠂
⡇⠀⠀⠀⣼⣿⢀⡶⢿⣴⡶⠿⠟⠚⠿⠷⢻⠟⡿⠢⠃⠀⠀⠀⠈⠀⢠⡄⠀⢿⡇⢰⣿⡭⠀⠀⣸⠁⠀⠀⠀⠀⠘⣶⣷⣾⡣⣴⠆
⡃⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠒⠠⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡄⠟⠤⠺⠟⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠿⠻⠿⠋⠁
⡇
⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⢰⠀⠀⣿⠀⢰⠇⠀⣰⠂⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠙⢦⠀⠹⡀⠀⠘⡇⠀⡏⠀⡼⠀⢠⠃⠀⢀⡞⠁⠀⠀⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⣄⠀⠙⢦⡀⠳⡀⠘⠀⠀⠃⠀⠃⠀⠃⢀⠃⠀⡠⠋⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⡀⠉⠢⢄⠙⠦⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠔⠁⢀⠴⠊⠀⠀⠀⠀⣀⣤⠴⠾⣭⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡽⠖⠒⠲⣤⣀⡈⠁⠀⢠⡴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠤⠔⠚⠩⠋⢀⡀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠹⠞⠙⢆⢸⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠖⠂⠀⢼⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⣦⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣐⣶⣲⣆⣾⣾⣿⣦⣼⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⢼⣆⠀⠀⠀⠀⠐⢰⠑⢺⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣇⣀⣀⣀⣼⣿⣫⣿⣿⣿⣿⣿⣿⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾⣿⣶⣶⣤⣄⣠⣯⡴⣯⠷⡶⢶⠗⡶⠲⡶⢦⡤⣤⣤⣦⣼⡀⢸⣯⣿⡽⠷⠿⣟⣛⣛⣿⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⡟⣿⢩⠇⣿⢁⡏⢠⡏⣼⢁⡇⣸⠁⡾⢠⠏⠴⠡⠞⣱⠯⢛⣫⠥⠖⣂⣩⡭⠤⠔⠒⣒⣒⣒⣒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⣸⢰⡇⣸⠁⠾⠀⣋⣨⡤⢤⠲⡖⢦⠒⡺⠖⣉⠔⢊⡩⠔⣚⣩⠥⠒⢒⣉⣉⡭⠭⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⣉⣴⢲⡙⣏⠳⡜⢆⠳⠈⡠⢊⡠⢊⡠⠚⣩⠔⢋⣡⠴⠚⣋⡩⠤⠶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠂⠷⠘⡆⢱⡘⢠⠎⡠⢋⡴⢋⡴⢋⡤⠚⢉⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠉⠀⠉






⡿⢿⠿⠿⠿⡿⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠄⠶⠀⠀⠈⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⣾⠀⠀⠀⢀⣠⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⣄
⣴⡗⠀⠀⠈⠃⠀⡠⡖⠀⠀⡏⠀⠀⠰⢦⠎⢢⠀⠀⠈⣁⠀⠀⠀⡑⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⠇⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠠⠂⠁⢶⠀⠀⠀⣲⣿⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠉⠀⠀⠀⠀⠀⢀⢴⡜⠃⢰⡂⠀⠀⠀⠀⠈⠀⠀⢰⣿⠃⠐⠇⠀⡠⠠⠀⠀⠀⠀⡀⠀⠄
⣆⠀⠀⠀⠀⠈⠛⠶⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠠⠄⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡌⠀⣧⣀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⢀⠠⢀⡀⠀⠀⠀⠊⠀⠀⠀⠀⢀
⣘⣣⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⠠⠀⢐
⠸⣾⠍⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠐⢀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠁⠀⠀⠨⠀⠈
⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⣠⠀⠀⠀⠂⠀⠀⠀⢀⡀⠀⠈⢠⡀⠀⠨⣀⠀⠒⠀⠀⠀⠀⠀⠀⢀⠀⡀
⡄⠃⠚⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠉⠀⡇⠀⢱⣄⡀⠈⠐⢸⢀⠰⠀⠠⠀⠀⠿⠟⠠⠠
⠐⠘⠀⢀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠇⠀⠈⣡⡠⢀⡀⠀⠈⢄⠸⠠⠙⠢⡀⡼⠀⠀⠀⠀⠄⠰⣅⠀⠀⠈⠃
⠀⠀⠢⣿⣀⣀⡁⠉⠀⢀⠀⠀⠀⠀⠀⠂⠜⠉⠈⠀⠀⠠⠀⠁⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃
⠀⠀⠀⡇⠀⠀⠉⡆⠀⠈⠑⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⠠⠄⣀⠀⠀⠀⠓⡀⠀⠐
⠀⡎⠀⠈⠲⠤⣀⣇⣀⣀⠀⠀⠀⢢⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠐⠄⠀⠀⠀⢠⠠⠴⡖
⠘⠀⠀⠀⠀⠀⢸⡁⠀⠈⡄⠀⠀⠀⢤⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢹⠊⠁
⢀⣀⣄⣨⣤⡄⠀⠙⠒⡾⠦⣄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠜⠃⠀⠈⠛⠧⠂⠀⠘⢇⡀⢸⠀⠀⠀⢀⠐⠀⠀⢀⡄⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢺⡇
⡠⡀⠀⣀⡝⠀⠀⢄⠀⠀⠈⢹⡒⡄⠀⠀⢀⣤⣄⣀⡀⠀⠀⠀⠠⠄⠀⠄⢀⣀⠀⠀⠀⡀⠀⠂⠀⠀⠈
⠀⠀⠂⠈⠀⠀⠀⠘⠉⠀⠀⢀⠙⠧⠄⠀⠀⠉⠉⠙⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢈⢐⡂⠀⠀⠃⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠙⢦
⠀⠀⣀⠴⠶⠶⢶⠀⠀⢠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣁⡀⠄⡀⠀⠴⠶⠾⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠽⠛⢻⠇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠂⠀⠰⡀⠀⠀⠀⠈⠙⠁⠀⠀⠀⣼⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⣤⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠸⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⣦⠀⠀⣳⠀⠁⠀⣼⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠈⠠⠀⠀⠀⠀⢠⣦⠀⠐⢀⠀⢀⠀⣿⣷⠀⢰⠀⢐⠇
⠀⠀⠀⠀⠀⢧⡀⡅⠀⠀⢻⡀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢢⠀⠀⢸⡄⠸⠀⠐⠂⠀⠄⠀⠀⠀⠀⠀⠈⠀⣄⠈⠀⠀⠁⠙⣿⠀⠈⠠⠏
⠀⠀⠀⠀⠀⢸⣅⠁⠀⠀⠘⠃⠀⢸⡇⠀⡀⠀⠐⠀⠀⠀⠀⢄⠁⡈⢶⠀⡸⣷⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⢠⣽⠂⠀⠀⠀⠀⠀⢘⣿⣾⠆⠀⢀⠀⠀⠀⠀⠀⠣⡵⢸⡇⢿⡟⣧⡿⠀⠀⠀⢀⣴⣾⣶⣤⣀⡀⠀⠀⠀⠀⠀⠠⣀⣴⣄
⠀⠀⠀⠀⠀⠈⠉⠻⠖⠚⠷⠀⠾⠪⠟⠉⠀⠀⣮⡄⠀⠀⠀⠀⠀⢀⣁⣿⣀⣶⡋⠀⢀⣠⣶⡿⢿⣿⣾⢿⣝⠻⣷⣦⠀⠀⠀⠘⠻⠋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⣿⡿⣿⠇⠸⣿⠁⣁⣤⣴⣦⣬⠁⠀⠀⠀⣀⠀⣘⡀
⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⡎⢉⣀⣿⡀⠘⠀⠀⠀⠀⠨⠴⠤⣤⣤⠦⠆⠀⠀⠀⢀⣘⣛⣧⣛⣴⠄⠐⢲⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠈⠙⡽⠿⠀⠀⠀⠁⠸⠏⣿⠃⠈⠀⠀⠀⢠⣤⡴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠻⠻⡉⠁⡖
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠠⠀⠹⠤⠀⠀⠀⠀⠀⠈⡀⠀⠀⣀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⣠⣤⣴⡖⠊⢻⣿⡶⠀⠀⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣾⣧⡿⠇⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣃⠀⠉⠙⠋⠉⠉⠋⠀⣾⡁
⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢈⡹⠏⢠⣂⡀⠛⠀⠀⠀⢀⡀⠀⣆⠀⢸⣿⠲⣶⣤⣀
⠀⠐⢦⢤⣗⠛⠀⠀⢠⣀⠀⠀⠀⠀⣀⣀⣀⣀⠠⠔⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⢶⠦⣿⡁⠚⠀⠀⠀⠘⠇⠀⠈⠀⢸⡇⠀⣍⠉⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠙⠋⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⣻⡟⢻⣿⣿⣦⠀⢀⠀⠀⠘⡇⠈⢉⣡⠀⣤⠀⠀⣀⠀⠀⠀⢸⡇⠀⡇⢀⢀⣤
⠀⠀⠀⠀⣼⠃⠀⠀⠄⠀⢠⠄⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⠙⢠⣙⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣾⣷⡆⠀⠀⠈⣷⢰⡇⠺⣄⠁
⠀⠀⠀⠀⡟⠀⠀⠀⠀⡆⠁⠀⠀⡀⠀⠀⢀⡄⠀⠀⠀⠤⠉⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠗⠄⢀⡆⠺⠟⠀⠐⠟
⠀⠀⠀⠀⠀⠀⠈⠁⠀⣈⡆⣀⣀⣴⣶⣦⣤⡿⠆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⣤⣤⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠛⠃⠘⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠾⣿⠿⡦⠛⠿⠛⠉⠉⠉⠁⠀⢠⠀⠓⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠘⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢰⡀⠀⠀⢸⡇⠀⠀⠐⠻⡇⠀⠋⢰⠃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⠙⠹⡃⠛⠀⡠
⠀⠀⠐⠧⠉⠉⠒⢾⣿⠛⠙⢻⣶⣶⣂⡀⢄⡀⠀⠀⠀⠈⠃⢀⠀⢸⣟⣷⠀⠀⠀⣷⠀⠀⣸⠀⠀⠈⢠⡆⠀⢰⠀⠀⠀⠀⠀⢸⡇⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠉⠉⠐⠿⠀⢰⠟⠛⠂⠘⠛⠚⠛⠀⠀⡇⠐⣀⠿⠀⢨⣿⡟⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⡾⠀⠀⣀⠢⡈⠀⠀⠀⢸⡇⠀⠀⢀⠁
⠀⠀⠀⠀⣀⣤⠴⢾⣿⠀⢸⣳⠀⠀⠀⠀⠀⠀⠀⣀⣁⠀⠁⠀⠀⠀⠙⠃⠀⠀⣴⣯⣀⠀⢀⢀⠀⢀⣤⣾⡋⠥⠃⠀⠀⠀⠀⢸⠃⠀⠀⠸
⠀⠰⠖⠛⠻⠟⠀⠘⣿⠀⢸⠉⠀⠀⠀⢠⠤⠖⢿⠻⣿⣷⣤⡀⠀⠡⠀⠀⠀⠀⠀⠉⠉⠁⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠀⠈⠧⠀⠀⡀⠀⠀⠀⢠⣤⣤⣄⠀⡾⠈⣻⠌⣢⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⣀⣀⣀⣤⣴⣶⠶⠀⠀⢠⡟⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⡀⢠⡆⠀⠀⢠⡀⠀⠀⠀⠉⠉⠻⠘⠐⣶⣶⢾⣤⠦⠀⠀⠀⠀⠈⠉⠸⠃⠀⠐⠿⠛⠋⠉⠀⠉⠀⠀⠀⢸⣿⡇⠀⡄⠉⠀⡆
⠀⠀⢰⣶⠞⢥⠁⡼⠀⣰⠀⠠⠟⠃⠀⠀⠂⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠠⣄⢐⡀⢉⣤⠄⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠸⠛⠀⠀⠙⠃⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣤⡘⠀⡀⠀⠀⡾⠿⠇⠀⠀⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⢀⠀⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⣀⠀⠐⢊⠒⢄⠀⠀⠛⠃⢠⣟⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠴⠴⢶⠴⠶⠗
⠀⠀⠀⠚⢻⠁⠰⢆⠛⡇⠴⠗⢰⠎⡆⠈⠀⣤⣤⠀⠈⠙⠁⠀⠉⠈⠙⠷⡄⠀⢀⣾⠟⠉⠁⠀⠀⠀⢠⡄⠀⠀⠀⣤⣶⡶⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣾⡷⣶⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⢰⡆⠈⢦⠀⠀⠀⠀⢠⣦⠀⠀⠀⢸⣿⠀⠀⠀⡤⠀⠀⣈⠟⠚⠀⠀⠀⠐⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠈⡇⢀⡤⠶⢠⠴⠾⠟⠒⠶⠶⠀⡿⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⡟⠀⣿⣧⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⣳⣿⣽⣣⡷⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠘⠱⠂⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⣉⡙⠋⠁⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣽⠶⢖⣋⣚⣚⣓⣀⡀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣶⣦⣤⣀⠀⠀⣦⣀⡄⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣶⣿⣭⣙⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⣿⣶⣌⣛⢿⣮⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣻⠿⣿⣿⣿⣿⡟⠁⠀⠀⢘⣿⣿⣿⣿⣿⣷⣝⢿⣷⡥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡿⠐⠒⠳⣼⣿⣿⣿⣿⣿⣿⣝⣿⣷⡽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢤⢄⡀⡀⠀⣹⠃⠀⠀⠀⢹⣿⣯⣿⣿⣿⢿⢿⢿⢻⣿⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⢾⣿⣟⠻⠞⠁⠀⠀⠀⢰⠏⠈⠉⠉⠙⠻⢿⣷⣧⣶⡿⡻
⠀⠀⠀⠀⠀⠀⠀⡷⠖⣉⣤⣶⣶⣾⣿⣿⣻⣷⣾⣿⣦⣀⣀⣘⣠⣷⣌⣀⣤⠀⠀⠀⠀⠀⠀⠈⠈⠉
⠀⠀⠀⠀⠀⠀⢠⣐⣻⣿⣿⣿⣿⡏⠈⠁⣿⠿⠿⠛⠛⡿⠛⠙⠋⢹⠟⠋⠍⠃
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⡇⠀⠀⡄⠀⠀⠀⠘⠷⠀⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢠⣧⠀⡀⠈⣴⡿⠶⣤⢾⡝⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⠿⠻⣿⡟⠁⠀⢸⡧⢠⣆⡤⢤⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀⠀⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠰⡀⠀⢸⡇⠸⣶⣿⣿⣿⣯⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⣾⠀⢎⣴⡷⡆⢹⣿⡀⠈⢉⡇⠀⠀⢹⣟⠛⠛⠛⠋⠉⠉⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣀⠀⡟⢷⡀⡟⠀⠈⡎⢸⡇⠤⣾⣃⣀⣀⣘⣻⣀⠀⠀⠀⠀⠀⠀⠠⣮⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠆⠀⠀⢻⠃⠀⠀⣿⡇⠀⠀⠀⣩⣥⢆⡼⠍⠉⠉⠉⣏⡠⠔⠒⠀⠀⢂⠰⢷⠣⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡄⠀⠀⢀⡈⣷⠀⠀⣿⡇⣀⣄⠴⠋⣿⠀⡟⠂⠀⠀⣠⣿⠏⠀⠀⠀⠀⠘⠛⢪⠧⢁⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⠺⠤⢖⠀⢚⣡⡿⢄⢀⣿⡧⠀⠀⠀⠀⣾⣶⣃⠀⠀⠀⢹⣏⡀⠄⢠⡀⣤⣰⢆⡈⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠱⢄⣀⡙⠶⡯⣸⠷⡆⢺⡿⠁⠀⠀⡄⢰⣿⡿⠟⢧⠀⠀⣼⠋⣻⡦⡶⣷⡟⢃⡐⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⢉⣳⣶⣊⣻⡾⢸⡿⣿⣷⠀⣡⢾⣍⣿⢻⠃⣠⠀⠀⢰⡇⣐⣻⣤⣾⡟⠩⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠉⠪⣯⡍⢾⣾⡿⣿⣵⠞⣡⣄⣎⢿⣷⡀⠀⠀⠀⢟⣿⣿⣥⣞⣿⢘⣿⣿⡟⡛⠟⠳⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣦⢹⢷⣾⣿⣿⣿⣱⡿⢩⣿⠿⣾⠛⡇⣾⡤⢻⣿⣾⣿⣿⣥⣿⣟⡽⠁⠋⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡁⣿⣿⢸⣿⣿⣿⣿⣿⣹⣯⡾⣶⣯⣽⢲⡗⢹⡆⣻⣿⣿⣿⣿⢹⣾⡁⢁⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⣿⡉⠘⢿⣿⣿⣿⣿⡏⡟⡷⢻⣿⡟⡎⢠⣿⣿⣿⡟⠈⣿⢻⡇⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢄⡀⠀⢠⣄⠀⢀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⠄⠂⠈⠉⠉⠀⢀⡈⠁⠁⠀⡀⠀⠁⠀⠰⠔⢀⠀⠀⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠒⠁⠀⢀⠀⠀⠀⠀⠡⠜⠀⡖⠀⣀⠀⠀⠀⠁⢤⡇⢰⠄⠀⠛⠂⢠⢄⠒⢌⡑⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣦⠀⠠⠒⠀⠐⠂⢀⡱⠀⠀⢠⣤⣄⡀⠠⠄⠴⠦⠤⠤⠄⠤⠤⠶⠞⠀⠀⠀⣀⡀⣙⡮⢀⣝⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠟⠂⢠⡂⠀⢀⠀⠚⢙⡠⠽⠒⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢄⡶⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣌⢠⡴⠀⠶⠀⡦⠌⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⢀⡀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠤⠼⡮
⠀⠀⠀⠀⠀⠀⠀⣾⡁⠌⡂⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡢⢄⣄⢀⣄⡞⣠⣾⡋⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⢟⡥⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠠⠐⣛⡻⢿⡟⠉⠈⠉⠁⠈⡏⠴⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠂
⠀⠀⠀⠀⠀⠀⡾⠋⢀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠉⠉⠉⠀⠈⣑⠞⢧⠀⠀⠀⠀⠀⣛⣷⡦⠄⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠚⠉⠉⠉⠀⠸⡇⠀⠀⠀⠀⢻⠧⠐⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠛⠀⠤⠤⠄⠤⠤⠤⠄⠤⠤⠤⠤⠐⠓⠚⠛⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠤⠒⠊⠁⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⠀⠀⣠⠀⢀⣠⢤⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡾⠃⠀⠀⠋⣰⠟⠙⢧⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⢷⣦⡤⠔⠃⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠇⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⣀⣀⣤⣔⣢⣄⣄⣄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣛⢏⣉⣟⣻⡯⣶⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡙⠛⠿⠈⣯⣫⣌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣷⣤⣴⡾⠇⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣂⠀⣴⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣸⣇⣂⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⠀⢀⣤⣽⣿⣿⡍⡁⠀⠀⠀⢠⡄⢠⣿⠋⡉⡀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⠀⠈⣿⣿⣿⣿⣇⡽⠀⠰⣿⣌⠉⠙⣿⣿⡟⣋⣄
⠀⠀⢠⠔⢡⡄⠀⠀⠀⠀⠀⠒⠀⠐⠂⠰⢿⣿⣿⣿⣶⣶⣛⣿⣿⣿⣿⣿⠂⠀⢹⣿⢛⠛⣿⣿⢿⠟⡛⢒⣃⣀⡄
⠀⣠⣿⣗⣊⢡⣤⠐⠒⠒⠒⠒⠲⠰⠒⠲⠾⠿⠿⠿⠾⠿⠿⠾⠿⠟⠻⠿⠿⠿⠿⠟⠘⠀⠿⠿⠿⠇⠤⠏⠍⠀⠂⠐
⠀⠀⠀⢉⠁⠀⠈⠁⢁⠀⠀⣀⡀⠀⣀⠀⢒⢀⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⢲⣍⡙⢆⣭⠂⢲⣝⢻⢼⣳⣾⢈⡱⡇⢁⢧⢲⡀⢀⡔⠊⠒⠾⣘⠈⢀⣴⠐⢺⠠⣆⠴⢂⣰⣔⣂⠀⢁⡀⣖⠢⢀⡀
⠀⠀⢼⡇⡇⣜⡴⢠⡉⣻⠀⣸⣹⢡⡞⢹⡇⢪⡽⢻⢷⣆⠑⢰⢪⠷⠿⠸⠏⣣⣇⡁⠔⣿⣌⡿⠀⡦⢄⣠⢼⠁⢻⠟⠉
⠀⠀⢺⡇⣷⢾⡏⡤⣽⣿⢨⡷⣽⢒⣷⢸⣧⣬⡴⢼⡔⣼⣍⡦⣼⣞⠃⣶⠊⡘⠎⠹⣧⣩⣮⠁⣨⢛⣉⡁⢀⢠⠞⠒⡂
⠀⠀⠼⡧⢿⢸⣝⡇⡀⣿⠀⢛⢼⠆⣿⠸⡇⢿⡏⣸⠣⢅⣮⢘⣽⣿⡃⣻⠘⢛⡦⠠⣿⢴⢿⡟⣻⣫⣾⣴⢀⣫⢚⡠
⠀⠀⠚⠓⠂⠼⢧⣤⣥⣏⣈⡋⣽⠓⠛⠰⡇⠽⠴⣽⠄⢌⠧⣧⠀⣽⡅⣭⠘⣭⠂⢲⣿⠈⢰⡧⣾⡤⣿⣭⢨⣏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠚⠓⠒⠶⠼⠤⢤⣀⣿⣀⠀⠀⢴⠁⠈⠀⠐⣿⠀⠲⠿⣾⠤⠏⣀⣰⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠴⠤⣠⣯⣀⣀⡀⢻⠀⠀⠀⢸⡅⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⢰⣾⠆⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠂⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠈⠀⠀⠀⠀⠀⣺⣿⠄⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⢰⣶⠀⠀⠀⠐⠆⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⣤⠀⠀⠀⠉⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⢀⠀⠀⠀⠤⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢖⠁⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⢀⣶⣆
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠒⠒⠒⠲⠢⠤⠌⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠙⠁
⠀⠀⠛⠀⠀⠀⠃⢀⠀⠄⠁⠀⠀⠀⠈⠀⠀⠀⠀⠠⡀⢈⠁⠂⠤⢤⣤⠤⠥⠤⠒⠒⠂⡒⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⡀⠑⠤⣀⡀⣂⣤⠤⣼⣄⣌⣤⢂⡠⠄⠀⠀⠄⢊⠕⠀⡀⠀⠈
⠀⠀⠀⠠⣤⣆⠀⠀⠀⠀⠑⢄⣀⠈⣡⣶⠶⢶⣾⠃⠀⠀⢀⢹⣿⣄⡀⠠⠤⠐⠊⢁⡠⠎⣠⡴⠁
⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠠⠬⣙⠋⢭⣭⣽⠃⠀⠀⠀⠘⡽⡍⠳⣤⣉⢈⡩⠁⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠤⢀⠀⠀⠀⠀⡟⠛⠓⠊⠒⠂⠀⠉⠀⠀⠒⠀⠄⢀⠀⠀⠀⠀⠀⢀⡀⠤⠂⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠈⠁⠠⠆⠀⢀⠀⠀⠀⡀⠠⠆⠀⠘⠃⠀⠀⠈⠋⠀⠇⠀⠀⠀⡄⠀⢀⡀
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠒⡵⠖⠀⠐⠥⠆⠀⠀⠀⠀⠀⠀⢀⠧⡀⠀⠁⠐⠂⠀⠈⠁⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⢶⡆⠀⢈⠄⠀⠑⠀⠀⠀⠀⠀⠀⣱⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠰⠀⠀⠀⠀⠈⠈⠁⠀⠠⠄⢡⠄⠀⠀⠀⠀⠀⠀⠉
⠀⠀⣠⣦⡄⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠒⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆
⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠤
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⠁
⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢠⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠓⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠸⠷⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⢰⡿⠛⠶⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣄⡀⠀⠀⠀⢀⣀⣀⠀⠤⠤⣤⡀⠀⢠⠖⠁⣀⡉⠒⠀⠀⠀⢠⣴⣶⣄
⠀⠀⠀⠀⢈⣤⠀⣴⠀⢠⠄⢙⠀⠒⠐⠄⠀⠂⠀⠀⠀⣠⣾⣯⣾⣿⣿⣿⣦⣀⡀⠀⣿⣿⠀⢤⣤⠿⡟⠀⠄⢈⣡⣍⣡⣄⠀⠡⡀⢸⣿⡈⠹⣷
⠀⠀⠀⠀⠈⢁⣀⡝⠀⣤⡠⢇⢤⡈⠂⠐⠇⠈⠀⠀⠀⣿⣿⣾⡿⣿⣿⣿⣴⣿⠿⣶⣄⠈⠠⢤⣀⣷⡀⠈⢰⠙⣣⡄⢿⠟⣀⠀⠐⠀⢹⣷⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠦⣸⣌⡷⣚⠦⡆⠰⠀⠀⠀⠙⢿⣿⣿⢛⣿⠏⣡⣬⡛⢿⣿⣿⣦⡀⠉⠉⠁⠀⠈⣿⡀⠀⠀⣠⡛⣶⡇⠀⠀⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡘⢣⣾⡤⣖⣂⣝⡛⠒⠀⠀⢠⣆⠀⠀⠙⠻⠿⠿⣿⣷⣶⣿⡷⣿⣿⣿⣿⠀⠀⠀⠀⣤⣼⣶⠆⣀⣟⠈⣿⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣋⣉⠓⣱⠧⢿⡛⠽⡇⢰⠆⠀⢿⡿⢿⡏⠐⢒⡤⣤⡈⠙⠿⣿⣿⣿⠿⠃⠀⠀⠀⠃⣿⠉⠀⢀⡅⢉⡉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠲⣿⣤⣤⣿⡕⠃⠉⠀⠀⠈⠳⠦⣤⣐⣂⣰⠖⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⣾⠁⠈⠉⣿⡇⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢦⡄⠀⠐⢦⠈⠛⠙⢈⠻⠊⠀⠀⠀⠀⠀⣤⣤⣍⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣯⡄⣴⣿⣧⠙⣷⠀⠀⠀⢰⣿⢿⡇
⠀⠀⠀⠀⠳⠤⣀⠑⣄⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣴⡾⢧⠿⣿⢻⣿⡄⠀⠀⠀⠀⠀⢀⠀⢰⡄⠀⠀⠀⠀⠀⢡⡘⣄⣿⣿⣿⣴⣿⡇⠀⠀⣾⣯⣿⠇
⠀⠀⠀⠀⣑⡂⠬⡷⢬⠳⡄⠰⣷⣦⣀⠀⠀⠀⢸⣏⢀⣬⣤⢰⣄⣽⡇⠀⠀⠀⠀⠂⢸⡀⢼⠁⠀⠀⠀⠀⠀⠀⠐⡌⣼⣿⣿⡟⣿⣧⠀⠀⢿⣿⠏
⠀⠀⠀⠀⠈⠑⠿⣾⡻⣧⣌⠒⣀⡉⠁⠀⠀⠀⠀⢿⣯⣵⠿⠈⢛⣉⠀⠀⠀⠀⠀⠀⠈⣇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢿⣿⣿⣿⣿⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣈⣙⡻⣍⣻⠷⢶⣦⡽⣖⡲⠄⠀⢈⣥⣾⣷⢸⣟⣿⣷⡄⠀⠀⠀⠀⡸⢦⡅⢀⠈⡀⠀⠀⠀⠀⠀⠈⠢⡙⠿⠿⢃
⠀⠀⠀⠀⢰⣿⠉⡸⠓⣻⣷⢞⣩⡍⢿⣿⣽⣯⡶⠀⠘⣿⡕⢋⣄⣹⣽⣿⡄⠀⠀⠄⠑⠂⠀⢷⠀⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠁⠸⢃⣾⠉⣉⡛⠳⣜⣧⢹⣽⣧⣽⡇⠀⠙⠿⠶⠟⠈⠛⠉⠀⠀⠀⠀⠀⠈⠀⣼⢀⣿⣶⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⡀⠈⢁⡞⢀⠙⣤⣿⣽⡸⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠙⢀⠀⠀⠀⠀⠀⡀⣠⣾⢿⢾⣽⡇⠀⠄
⠀⠀⠀⠀⣼⣿⢿⡆⠘⣷⠛⡿⡟⠋⣸⠛⠀⠈⠉⠁⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⢀⣇⠇⠀⠆⠀⠀⠀⠀⠀⠰⣽⣷⢟⣻⡿⣿⠃⠈
⠀⠀⠀⠀⢸⣧⢸⡏⠀⣴⢾⠆⡷⠀⠉⠀⢀⠈⢡⣀⡀⠔⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠠⠐⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⢻⣏⣼⣿⣿⡇⠠
⠀⠀⠀⠀⠀⣿⣼⡇⠀⣿⠆⠀⠀⠀⢠⡦⠉⠙⢀⡄⢸⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⣶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠉⡽⢿⡗⣄
⠀⠀⠀⠀⠀⠿⠛⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣠⠚⠁⢠⠙⢨⠃⡅⠀⠀⠀⠀⠀⠀⢀⡙⣤⠂⣌⡀⠁
⠀⠀⠀⠀⠀⠀⣠⠶⠟⠓⢦⡀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠰⠷⡈⠁⠀⡄⠀⠀⣤⠔⠁⠀⠀⠀⠸⠀⠀⣷⠄⠀⠀⠀⠀⢀⠈⠠⠐⢊⣈⡄⠁
⠀⠀⠀⠀⠀⢰⠏⠀⠠⠤⠀⠉⢢⡀⠈⠃⠀⠀⠀⠀⠀⠰⡖⠓⡆⠀⢱⠀⠀⠀⠀⠁⠀⠀⠋⡀⠀⠈⠉⡄⠀⢀⣾⡿⢛⡷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡟⠐⠀⠀⢀⡀⠀⠀⡇⠀⣀⢤⣤⣤⣤⢦⣤⡈⠆⣀⣤⣸⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⡿⢂⠀⠀⠋⠀⠑⢲⠇⢸⣉⠩⠉⠛⠀⠀⠛⣵⠀⠀⠈⠉⠳⡄⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⡼⢃⡄⢠⣤
⠀⠀⠀⠀⠀⡆⢸⠀⡀⠰⡦⢀⠎⠀⠸⣭⣆⠿⣿⣿⢶⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠶⠊⠛⢶⣄⠀⠙⠛⠶⣾⡿⠵⠦⣖⡛⢛⣋⠉
⠀⠀⠀⠀⠀⣷⠘⠀⡇⠠⠄⣸⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⠀⠀⣀⣀⢀⣤⣄⢀⣤⣶⡆⢀⣀⠀⠀⠙⠛⠒⠛⠛⢉⣠⡶⠚⠡⣿⣻⣽⣦
⠀⠀⠀⠀⠀⣿⠀⢠⠃⠐⢠⡇⠀⣾⢻⣧⣞⢻⣆⣤⡀⠀⢀⣤⣿⡿⢣⡏⣽⠏⢉⣿⠰⣿⠛⢿⡄⠀⠀⠀⣠⠴⠋⠁⣠⣄⠀⣤⡛⠁⠉⣰⣾⡆
⠀⠀⠀⠀⠀⣾⠀⢘⠀⢀⠞⠀⠀⣽⣿⣍⣙⠋⢿⣉⣧⠀⢻⡏⠻⣶⡟⣤⣿⣿⣿⠿⣇⣿⣤⣾⠃⠀⢀⡴⠁⢀⡴⠞⠉⢡⡄⠉⠁⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠳⠾⠖⠋⠀⠀⠀⠛⣽⣟⠙⢷⣶⠙⣏⠀⠨⣷⣴⠟⠹⠷⣾⣏⡹⢦⣿⠎⠉⠁⠀⢠⠞⠁⠀⢻⡀⠀⠀⠀⢡⣤⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡷⣦⠁⠀⠀⠀⠠⠾⠷⠚⠛⠒⠚⠛⠀⠀⠀⠀⠀⠘⢧⠘⠇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡶⡄⠀⠀⠀⠀⠀⠰⢋⣉⢉⣤⣤⡀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⢃⡐⠶⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⣾⣿⠎⠙⢿⣧⣴⡀⠀⢰⣶⡤⢻⣟⢼⣇⣸⣯⣤⠀⠀⠙⢢⣄⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠸⣿⣙⣤⢢⣄⠛⠋⠀⠀⠀⠉⢰⡟⠻⣯⣹⣿⣿⡟⠃⠀⠉⠀⠈⣙⠶⠦⣬⣦⣼⣤⡜⠀⠀⠀⣦⠀⠀⢀⠜⠀⣠⠄
⠀⠀⠀⠀⠀⠉⢭⣤⣯⡿⠀⣴⣿⡦⠀⠀⠈⠿⣶⡿⣿⣷⡾⠏⠀⠀⠀⠀⡎⢀⣤⡤⢴⢾⡿⠟⠓⠒⢐⡾⠧⠴⣮⡁⠐⢶⣥⠤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⣾⠙⣯⣵⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠤⠊⠀⠀⠀⠁⠀⠀⢀⡠⠒⠂⠀⠀⠀⢀⠈⠐⠤⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀⠀⠀⠀⠀⢀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠂⠠⠈⠈⡄⠀⢤⠀⠀⢀⠀⡀⠀⠀⠐⠀⠀⡈⠀⠀⠀⠀⢀⠨⠶⠥⠄⠈⠂⠑⠀⡀⢀⠄⢈⠀⠴
⠀⠀⢈⠁⠃⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠈⠀⠁⠢⠀⠁⠁⠁⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠐⣀⢀⠂⠂⠰⢑⡁⢐⡴
⠀⠀⠈⠀⠁⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡒⠀⡀⠀⠀⠐⠐⠐⠀⠸⠀⣀⠀⠀⠀⠀⠀⠂⠐⢂⢀⠒⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠆⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠤⢄⡀⠀⠂⠀⠂⠢⠀⠲⢦⡀⠐⡥⢄
⠀⠈⠀⡄⢀⠀⡀⠀⠀⠘⢃⢂⠀⠈⢀⣄⠀⠀⣀⠀⠀⠀⢀⣁⠓⠈⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢥⡤⡄⠀⢰⠄⢀⠀⠀⠛⠀⠑⠊
⠀⠀⡀⠓⠀⠄⠠⠼⠀⠀⠈⠀⠀⢀⠐⠀⠁⢨⠙⡞⡛⠀⠈⠈⠀⡂⢠⠁⠂⠀⠊⠁⠀⠛⡏⠏⠭⠉⠁⠀⠁⠀⠀⠐⠐⠀⢀
⠀⠀⠀⠠⠖⠐⠦⠀⠀⠀⠀⠀⠀⢐⠴⠀⢠⠀⠑⠊⠁⠤⠁⢊⡀⠀⡸⠀⠀⠐⠀⠀⠠⠀⠀⠀⢀⣀⢠⡀⠀⠀⠀⠒⠀⠠⠄⡀
⠀⡎⠉⢣⠀⠇⠀⠀⠀⠥⠀⠀⡀⠈⠶⠀⠪⡨⠀⠂⠤⠐⠁⠃⢀⣈⠀⠀⠀⠀⠁⠈⠀⠀⢼⡥⣞⠛⡘⠁⠂⠈⠂⠴⡆⠠⠀⠂
⠀⢄⠂⡴⠀⠀⠀⢀⠀⠀⠀⠀⠘⠃⠀⠆⠀⠀⠀⠀⣤⠀⠀⠀⠠⠌⠀⠸⠂⠀⣤⠀⠂⢴⡀⡠⠔⠊⠤⠄⠀⠐⠍⠀⠐⠄⠀⡀⢀
⠀⠀⣀⠀⠀⠀⠐⠆⠀⠀⠀⠉⠁⠀⠀⠀⠤⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⠀⠂⢀⠈⢀⠈⠀⡀⡀⠀⠀⠀⢀⡐⢿⠂⣀⠀⠀⠉⢙
⠀⠀⣏⡂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠄⠀⠀⠈⠀⠀⠠⠔⠀⠘⠀⠀⢃⠀⠠⠀⠀⢀⠐⠐⠀⠀⠀⠀⢹⠀⠀⠀⠁⠀⠑⠁
⠀⢀⠀⠀⠐⠀⠀⠠⠀⠀⡄⣤⢠⠀⡀⠂⠀⠀⠀⡀⠄⢶⣲⣤⠂⠀⠀⠀⠀⠄⠀⠙⡀⠀⠀⠈⠈⠀⡰⠁⠀⠈⠀⠀⠀⠠
⠐⠪⢬⠐⠲⠀⠀⠀⠀⠀⠀⡆⠀⠐⠠⠂⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠴⣂⠀⠀⡀⢠⣒⡆⢀⠀⠀⠁⠀⠀⠠⠂⠄
⠀⠤⡀⢃⠂⠀⠀⠀⠀⠀⠀⠀⠁⠓⠒⠀⠀⠄⠀⢰⣖⠀⠀⠀⠀⠀⠀⠸⠁⠐⠈⠁⠀⠀⡁⡀⠁⡒⢀⠁⠂⠘⠄⠀⠃⠀⠰⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠂⠀⠐⠈⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡋⢻⡙⠳⣤⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣼⡟⠛⢛⡴⠞⠱⣴⡿⢵⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡗⠉⢠⣤⡿⢻⣿⣧⣀⡀⢝⢋⣶⣿⡿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠇⣤⡞⣃⠀⠹⣭⣏⣹⣿⠜⣠⢻⡻⡤⠿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣏⣀⡔⣋⣥⡤⣭⣭⣷⣮⣭⡽⣷⣾⣫⣭⣤⡞⡷⣿⣦⢀
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣾⢋⣹⣟⣩⣙⣧⣶⣴⡯⣬⣿⣽⣯⣭⣤⣥⣽⣿⣿⢧⡤⣤⣝⣆
⠀⠀⠀⠀⢰⠊⣫⢏⣠⣴⣻⠿⠋⠉⢝⡋⠉⠀⠀⠀⠁⠀⠐⣠⠖⠉⠉⠛⠢⣮⣿⣿⣬⢽⣧
⠀⠀⠀⠀⠘⣌⢹⢷⣮⣀⠀⠠⡶⢿⢿⢷⡀⠀⢀⠀⠀⡅⣸⣿⣥⡀⠀⠐⠛⠃⠠⢞⣥⣶⠟
⠀⠀⠀⠀⠀⠈⠛⠢⢐⠩⠛⠚⣒⠷⣤⣠⡇⠀⢸⠂⠆⠇⣿⠀⣙⣁⠭⢐⠂⣭⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠙⠒⠹⠁⠀⠼⠀⠀⢠⣿⡧⠤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠸⡆⠀⠑⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠘⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡀⢹⣄⠀⠀⠀⠀⡄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣽⣳⣤⣤⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠄⠁⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠶⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠠⠤⠤⠔⠆⠑⠒⠶⠒⠂⠶⠶⠦⠤⢤⢤⡀⠀⠀⠀⠈⢀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢃⠀⠀⠀⣀⣐⣲⣶⡦⠴⠦⠤⣀⣴⣤⠤⠤⠈⠒⠂⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠚⠦⠈⠩⡌⠁⠉⡇⠀⠀⠀⠏⠋⠉⠁⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠁⢉⠃⠈⠀⠐⢠⠂⠆⠂⠒⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠄⠀⠀⠤⠀⠀⠀⠁⠀⠀⠀⠡⠀⡀⠀⡀⠀⠀⡀⠀⠀⠀⢀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠐⠀⠀⠐⠄⢠⢌⡀⠀⠀⠀⠆⠠⠤⠢⢤⣈⡁⠀⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣴⣶⡆⠂⠐⢴⡄⠅⠀⠀⠠⠄⢻⠋⠁⠓⣤⣉⡙⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠠⢔⠑⠀⠀⠘⠉⠃⠀⠀⠀⠶⢈⠀⠀⠘⣛⠉⠂⡀⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠐⠀⠀⠊⠈⠁⠀⠀⢀⠀⠠⠀⠈⠡⠀⠀⠅⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢠⠆⠀⠔⣦⠉⠥⡌⠒⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠀⢐⣝⡉⠀⠘⡝⠓⠄⢑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠀⠀⠉⠁⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡮⠀⠠⡌⢸⡀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠉⠕⢧⡖⠁⠀⠀⠀⠀⠀⠀⠀⣠⣔⠂⠉⣉⣁⣠⣈⡙⠲⠠⢄
⠀⠀⠀⠀⠀⠀⠀⢸⢠⠊⡀⠀⠀⠀⠛⠀⠒⠒⠒⠒⡰⠋⢁⣠⣾⣿⣿⣿⣿⣿⣿⣿⢻⣆⡙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢠⡿⣿⣿⣿⣿⣿⣿⠛⠻⠛⢘⠿⢿⢦⡈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡇⠀⠁⠀⠀⠀⠀⠀⠀⡘⠁⢠⣿⣿⠁⠐⠛⠻⢿⡇⠀⠀⠀⠀⠀⣘⡀⠈⠋⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣾⣿⣿⣧⡥⠤⢄⠀⢡⠀⠀⢀⣀⠠⡈⠓⠀⢑⢆⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠀⠀⠀⠀⠀⠀⢰⡎⣠⠿⠏⠀⢀⠀⡀⢀⣃⣀⠧⣤⡅⡯⡤⢷⡀⠰⣌⢊⢢⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⡇⠀⠀⠀⢸⠎⠀⠀⠀⠈⢀⠀⠀⡆⠹⠐⠳⠄⠀⠰⠎⠛⠈⢹⣯⣭⣿⣿⣦⠴⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢣⠀⢰⠀⡇⠀⠀⠠⣤⠤⣾⡞⠚⢆⡁⢀⠀⠀⠀⠀⢠⠔⢚⡁⠐⠰⠀⠈⠷⣄⠀⠀⣰⠇⢀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⡇⠇⠈⠑⠀⢸⠷⣨⢇⠀⠄⣱⠊⠳⡀⠀⢠⠃⢰⠏⠀⠀⠀⠀⠀⠀⠈⠑⢣⡀⠋⢃⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠸⡌⠒⢶⠒⢻⣿⣿⡄⡀⠀⡏⠀⣦⣑⠀⢰⡆⠄⠀⠀⠀⠀⠀⠐⡤⠀⢀⡜⠀⠀⠀⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠺⣄⠈⢆⢰⠘⢿⣧⣇⣀⠙⡀⡽⠛⠛⠻⠗⠒⢦⡀⠈⠢⡀⣰⡃⣀⣽⣠⡦⣀⡰⣄⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⡻⠿⠋⠉⢠⡨⢧⣺⠁⠀⠀⠀⢀⡄⠀⡟⠋⠁⠉⠉⠉⠉⠁⠀⡀⣀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠰⠃⢰⠉⠱⡄⠟⣣⡀⠀⠀⢾⣷⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⡌⠀⢺⠀⠐⡄⢠⠟⢳⠦⠔⠃⠘⠻⠢⣄⣀⣠⣤⣤⣶⢶⣭⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢫⠀⡴⠁⢀⠯⣅⠀⠀⠀⠀⠀⠀⢸⡭⣏⠼⡁⢯⣜⣧⡤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠺⢄⠼⠢⠄⣀⢀⡀⠴⠋⠈⠉⠁⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⣀⣠⣤⠤⠶⢶⠶⠦⢤⣤⣀⡀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢠⢴⠟⠍⡁⡀⢹⠀⠀⠘⡇⡀⡀⢄⢪⠻⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⢦⡄⢳⡀⢳⣃⠎⡜⠰⠀⢸⡇⠀⠀⡇⢂⠰⡈⣦⡵⢋⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⢦⡓⢦⣈⣙⠓⠛⠾⢾⡇⠀⠀⣗⣚⣋⣩⠥⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡈⠉⠉⠙⠛⣻⠀⠀⠀⡏⠉⢀⣠⠴⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣠⡤⠶⠶⡶⠒⢲⡶⠦⣤⣄⡀⠙⣆⠀⠀⠀⡟⠀⠀⢰⣇⡴⠋⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣿⡕⡐⡀⡀⡇⠀⢸⡇⢀⢀⢈⢻⡆⣸⠀⠀⣸⠃⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣝⣓⠳⠷⡇⠀⠘⡶⠾⠶⢛⣩⠴⠃⠀⢠⡏⠀⠀⢠⡇⣀⣀⣤⣤⡤⢀⣀⣀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠤⣄⡀⠉⠉⠉⣧⠀⠀⣿⠛⠉⠉⠀⠀⠀⢀⡞⠀⠀⠀⡾⠻⣭⣀⣀⢲⠀⠀⡯⠈⡉⣙⣦⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⠦⣀⠀⢹⡀⠀⢹⡄⠀⠀⠀⠀⠀⡾⠀⠀⠀⣰⠛⠲⠤⣌⣉⡿⠀⠀⣟⣋⣉⣉⡥⠖⠃
⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣇⠀⠈⢧⠀⠀⠀⠀⣸⠃⠀⠀⢠⡏⠀⠀⠀⠀⢠⠇⠀⢰⡏⠉⠁
⠀⠀⠀⠀⠀⠀⢿⠀⣴⢖⠛⢻⠉⢹⢋⣉⣻⠆⢨⠃⢹⡄⠀⠘⣇⠀⠀⢠⡇⠀⠀⠀⡾⠀⠀⠀⠀⢀⡞⠀⠀⣸⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠳⠬⣭⣙⣻⠀⢸⣭⠭⠴⠖⠃⠀⠀⢷⠀⠀⠘⣆⠀⡾⠀⠀⠀⢰⠇⠀⠀⠀⢀⡞⢀⡴⠚⠉⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢧⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠘⣦⡇⠀⠀⠀⣾⠀⠀⠀⢠⠞⢰⠋⣀⣀⣀⣀⣀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⢧⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⣆⠀⠀⠀⡇⠀⢀⡴⠋⠀⠘⢦⣛⠦⣾⠃⢈⣛⣓⣦⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⠳⡄⠀⠀⠀⠀⠀⢹⡄⠀⠀⠹⡄⠀⠀⡇⣠⠟⠀⠀⠀⣠⠏⠉⢙⠏⠀⡼⠷⠽⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠙⢦⡀⠀⠀⠀⠀⢷⠀⠀⠀⢹⡀⠀⡟⠁⠀⠀⢀⡼⠁⠀⣰⠋⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠙⣦⠀⠀⠀⠸⡇⠀⠀⠀⢧⠀⢧⠀⠀⣠⠏⠀⣠⠞⠁⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠈⢳⡀⠀⠀⣷⠀⠀⠀⢸⡄⠸⡄⣴⠃⢀⡼⠃⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠻⡄⠀⢹⠀⠀⠀⠈⡇⠀⢻⡁⢠⠟⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢹⡄⣾⠀⠀⠀⢰⠇⠀⢸⣇⡟⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⢀⣸⠇⢻⣄⠀⢀⣾⣄⣤⠞⠘⣧⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣦⢄⠈⠠⠟⣹⠯⠫⠀⠈⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠃⠠⠿⠀⠉⠠⠀⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⡀⠈⠃⢈⠀⠆⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⠈
⠀⠀⠀⠀⠀⠀⡆⠡⠐⠀⢴⠆⠀⠁⠀⢦⡀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⣀⠢⡀⠊⢦⡍⠆⠀⠐
⠀⠀⠀⠀⠀⠀⢻⠐⡀⠀⠀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠘⢆⠀⠀⠨⡠⠈⣳⣙⠆⠀⢊⡀⢂⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠹⠆⠈⠿⡆⡀⠱⣊⣴⢛⣦⠁⠒⠀⡁
⠀⠀⠀⠀⠀⢀⠀⠀⠠⠈⠠⠀⠀⠁⠀⢄⡀⠀⠀⠀⠀⠂⠑⠌⠰⠄⠀⢌⠉⣣⣈⠈⣀⡀⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣁⠀⠀⠀⠀⠀⠙⣁⡔⠀⡀⠀⠐⡀⠃⣈⠁⢀⡛⠀⠂⠲⠖⠐⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠉⠄⠀⠀⠈⠒⠲⠀⡰⠡⠳⠆⠰⠄⢤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠃⠀⠂⠠⠀⠉⠀⠀⠐⠄⠈⠀⠁⠀⠀⠀⠄⢠⠂⠀⠀⠀⠆⠄
⠀⠀⠀⠀⠂⠀⠅⠀⢴⠀⠄⠀⠀⠀⠀⠀⢀⠀⠴⠀⠀⠀⠀⠈⠑⢴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠰⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⠀⢈⠠⠀⠀⠘⢁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠤⠄⠀⠂⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠁⠀⠆⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠁⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠠⠈⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠇⠀⠀⠠⠠⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠩⡈⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢡⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⢰⠆⠚⢠⡄⠀⠀⢀⣀⣠⣶⠶⠲⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣰⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢙⠁
⠀⠘⢠⠓⠀⠂⠀⠀⠿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⡤⠈⠀⠀⠰⠋⠀⠀⢀⣄⢀⡀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠘⠁⢠⡀⣤⠀⠒⠁⠈⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣴⡖⠀⠀⡰⠆⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠄⡀⠀⣿⡄⠀⠁⠀⠀⠀⠀⠠⣠⣦⡀⠀⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠙⠋⠀⠀⠰⢂⠀⠀⠀⠀⠈⠁⢄⠀⠀⠀⠀⠀⠀⠈⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠡⠙⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠁⠀⠀⠈⠀⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠈⠀⠀⠀⠲⠄⡀⠀⠀⠐⠀⠟⠀⠐⠠⠠⠀⠀⠀⠀⠀⠀⠈⢽⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠚⠁⠀⢀⡀⠀⣐⠁⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣰⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠄⠈⠀⣀⠀⠈⠀⠀⠀⡀⠶⣳⡋⠀⣀⠞⠁⠀⠀⠀⠂⠀⠀⠂⠁⡀⠉⠬⠋⠀⠃⣀⠤⠀⠀⠀⠀⣠⡾⠀⢟⡷⠧
⠀⠀⠀⡰⠃⠀⠀⢀⣀⠤⠤⠤⠀⠀⠀⠀⢀⣸⠄⠀⠀⠀⠊⣙⣧⠝⠓⠊⠡⡀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣻⣧⢰⡌⠷⢶
⠀⠀⡐⠃⠀⢀⠀⠈⠄⠘⠀⠀⠀⣠⠀⠀⠈⠁⠀⠀⠀⢠⠈⠁⢠⡤⠂⠠⢐⡏⠀⡀⠀⠀⠀⣰⡀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⢄⠀⠈⠉⢠⣸⡗⣶⠒⠠
⠀⠀⡗⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⢨⠀⢀⠀⠀⠰⠀⠀⠈⠆⣠⠃⠀⢰⡂⠀⠀⢈⠀⠀⠀⡼⠥⠋⠑⠀⠀⢀⡀⠀⠀⠀⠀⠀⠆⠀⠀⠉⠋⢿⢿⠟⡄⠃
⠀⠘⡟⠀⠀⠄⠀⠀⠀⠀⠴⠆⠀⠀⠀⠁⠀⠀⠂⠀⡀⠴⠄⡁⠀⢰⡇⢱⣄⠀⠋⠀⠀⢀⠃⠀⡢⠀⠀⠀⠙⠃⠠⠀⠀⢀⠔⠀⠀⠀⠀⠀⠘⠉⠃⠁
⠀⢠⣣⠀⠀⠀⠀⠒⠒⠂⠀⠀⢀⣀⠀⡰⠀⠀⢻⢩⠘⣴⣋⠴⢅⣿⡁⢀⣹⠀⠀⠀⠀⠈⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⢰⡄⠰⠄⡀
⠀⢨⡇⠀⢈⣿⠆⠀⠀⠀⠀⠠⠌⠁⠀⠙⠖⠀⠸⡄⠾⠀⡁⢠⡾⠻⣄⡺⡌⠀⠀⠀⠀⠀⣄⠀⠘⢹⣇⠀⠀⠀⠀⠀⠆⠀⢡⠀⠄⠀⠈⠀⢡⡀⢀⠃
⠀⠀⠁⠀⣸⡉⠀⠄⠀⠀⠀⠀⢆⠀⢈⠀⡀⠄⠀⠹⣄⠀⢱⠏⠐⡃⠙⡼⠀⠀⣖⡐⣲⣾⣴⠁⠀⠘⡇⠀⡄⠂⠀⠈⢤⣀⠀⠀⢀⠀⢠⡀⠈⠁⠙⠈
⠀⢢⣶⡆⠋⢾⠶⣄⡀⠀⠀⠀⠚⠴⠁⠀⠀⠀⣀⠀⠙⢶⡟⠀⣤⠰⠂⠀⠀⠀⠀⠜⢹⢛⡤⠗⠀⠀⠀⠀⠉⡀⣀⠀⠀⠉⠉⢉⣁⣀⣹⡦⠖
⠀⠀⠈⠁⠀⠰⠟⠋⠁⠀⠀⠀⠀⣐⡒⠤⡖⠀⠀⠀⠀⠀⠙⠢⠄⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢎⡵⡻⠁⠀⢀⡞⠋⠉⠈⠹⡇
⡀⠀⠀⠀⠀⣤⠘⠀⢀⡠⠔⠚⢒⣨⠥⠲⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠋⠀⠀⠀⠀⢸⡃⠀⠀⣀⡼⠁⠠⣄⡀
⠸⠗⡄⢐⡟⠁⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠑⠀⠀⠀⡀⠀⠀⠠⠀⠀⢠⠀⠀⠀⠀⣮⣀⠉⠀⠰⣾⠁⠀⠀⠀⢠⣴⡦⢼⣧⠶⠛⠉⠀⠀⡒⠸⡆
⢰⣾⣵⡊⣀⡔⠀⠀⠀⠒⢈⡀⠀⠄⠀⠈⡄⠀⠀⠀⠀⠈⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠐⣶⠃⠀⠀⠀⠀⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠐⠟
⠈⠾⣬⡷⠋⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠤⣀⣀⠴⠛⠢⠀⠀⠀⠀⠀⠀⢄⡙⠀⠀⠀⣀⣤⣿⡤⠶⠞⠁⠀⢀⣈⣲⠀⡀
⠀⠀⠈⡇⠀⠀⠀⠀⡌⠐⢀⡈⠈⠀⠀⠁⠊⠀⠀⠀⠀⠀⠀⡀⠠⡄⠀⠀⢀⠠⠄⠀⡆⣠⠘⠃⠀⠀⠀⢸⡏⠁⠈⣿⠀⠀⠀⡀⢹⣿⡿⠚⠏⢹⣦⡀
⠀⠀⢀⡅⠀⢀⠀⠀⠆⠠⢠⠀⠌⠁⠁⠀⠀⠀⠀⠂⠀⠀⠰⠄⠵⠀⢀⠀⢈⡀⠀⠀⠠⡀⠤⠈⠀⠀⠀⢸⣇⣠⠴⠋⠀⠀⠀⠉⠊⣵⠀⠀⠀⠀⠉⠛
⠀⠀⢸⡟⠀⠘⢆⠀⠀⠙⠀⢀⣀⠀⠀⠀⣀⣠⡤⠤⠤⠄⠶⠶⠀⠀⠈⠀⠀⠉⣁⣀⣀⡈⠀⠁⠀⢠⡞⢻⡏⠀⠀⠀⡈⣷⠀⠀⠀⢷⣠⡄⠀⣠⢾
⠀⠀⠘⠃⠀⠀⠈⢦⡀⠀⠀⠘⠼⠀⠈⠉⠈⠁⠀⠀⠠⠀⠁⠀⠀⠀⣠⡤⠴⢿⡿⠿⠟⠋⠀⠀⢀⣼⠷⠋⠀⠀⠀⠈⠉⠳⡀⠀⠀⠀⠉⠳⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⣶⣾⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⠶⢶⣶⡶⢶⡶⠶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⡿⠿⠻⠋⠙⠙⣛⠻⠻⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠻⣉⣥⣴⣶⣶⣿⣿⣿⣿⣿⣷⣦⣤⣠⣍⠻⠷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣡⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣭⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣽⠻⡟⢻⡋⣛⠱⣿⣿⣿⣿⣿⣿⣿⣿⣷⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣭⣟⠠⣽⢄⡁⠃⣐⣧⣽⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣯⣿⣷⣿⣽⣿⠇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⣄⣀⡀⠀⠉⠛⠉⠙⠛⢿⣿⣿⡏⠸⢰⣿⣿⣿⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⣠⣮⣶⣷⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢱⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⣡⣤⣀⡀⠠⡀
⠀⠀⡠⣴⣾⡿⠋⣉⣩⣤⣭⣥⣤⣤⣭⣭⣛⢧⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⠀⢀⣀⠀⢀⠴⠶⠿⣿⠿⠿⠿⠿⠿⣷⣽⢦⡀
⠀⢨⡾⣫⣥⣶⣾⡿⢿⡿⣿⡿⣟⣻⣿⣿⣿⠟⠀⠀⠀⠀⢸⠐⢸⣿⠀⠀⠀⣿⣷⣾⣿⣶⣶⣷⣾⣶⣭⣿⣽⣖⣗⢶⠮⣝⡄
⠀⠋⣴⣿⣿⣿⣭⣷⣶⣶⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣅
⠀⠈⠛⠛⠛⠋⠁⠀⠀⠈⠇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⡆⠀⠀⠐⠿⣿⡿⠿⠛⠛⢿⣝⣿⢙⣩⣁⣹⣿⣿⣿⣿⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣡⢸⠿⠻⠿⠿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣜⣻⣛⢿⣿⣿⣷⣄⡀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡂⢹⡀⠀⠀⠀⠀⠀⣦⠀⢰⠿⣿⣟⣟⣿⠛⣿⣿⣿⣭⣙⠀⠀⣿⠀⢀⠀⠀⠀⠀⢀⣟
⠀⠀⣤⣀⠀⠀⠀⠀⢐⠂⡀⠘⣇⡀⢹⡄⠀⠀⠀⢐⣯⠂⢸⣿⣿⣿⡟⢻⣇⠛⠿⠟⠛⠋⠀⢀⣿⢠⡌⠀⠀⠀⣦⣼
⠀⠀⢃⣀⠾⠀⠀⠀⠨⠄⠐⠀⠙⣿⣄⢷⠀⠀⠀⠰⢼⠚⢸⡇⠟⠿⡇⢸⡟⠀⢀⡄⠀⠀⠀⣼⡏⢀⠃⠀⢠⣤⣼⡉
⠀⠀⠈⢀⣀⣸⠃⠀⠠⣆⣀⠒⠀⠈⢻⡆⠢⡀⠀⠐⢺⠚⢻⡇⠀⠀⡇⢸⢣⠀⢬⣗⠀⠀⣸⡟⠃⡘⠀⣀⡈⡟⠉⠁⠀⡀
⠀⠀⠀⠈⠛⠙⢀⣴⠀⠀⣠⠀⢃⠀⠀⢷⠈⠻⢧⡘⢹⡤⢸⡇⢰⠀⡇⣸⣼⠀⣲⡃⠀⣰⡟⢀⢰⠃⠀⠉⡿⠻⠂⠀⠂⡀
⠀⠀⠀⠀⠀⠒⣥⠊⠀⣄⢐⡄⠿⠃⠀⣌⠷⠳⡄⢳⣼⣦⣼⠇⢺⣶⢷⡟⢟⣀⢙⠋⣻⠏⠀⢀⠁⠀⣷⣤⡗⣠⠘⡃⠐⠑
⠀⠀⣀⠙⠀⠙⣷⣁⡸⢧⡌⢋⠠⢶⣤⢾⡿⠛⣿⢺⣿⣿⣿⣜⢸⣿⣝⠁⣶⣈⣴⣾⢏⡜⣀⡏⢰⠈⢻⡇⣀⢳⡡⠢⣠⡄
⠀⠀⠉⠁⢼⣠⡄⢼⣽⠛⢃⠁⢲⡆⠀⣾⡇⢸⡿⠞⠿⣿⣿⣿⣾⡿⢋⣼⣹⣿⣿⢋⡾⠁⣹⣅⡰⣁⡄⠳⠓⡆⢆⠂
⠀⠀⠀⠀⠀⠉⣴⠀⡀⠸⠘⡶⠘⣋⣿⣿⡇⣸⣧⣀⢀⡿⠋⣻⣟⣴⣿⣿⣿⣿⠇⠈⡁⣰⣿⣿⣿⣿⣼⣆⣸⢃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠁⣰⣾⣴⣰⣾⣿⣿⣿⣟⣿⣏⠙⢿⣷⣠⡟⠁⠀⣡⣿⣿⣿⠀⠀⢠⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦⠁⠖
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⡏⣿⣧⡴⢶⡿⣿⣿⡿⣤⣿⡿⠿⣿⣿⣿⡿⠛⠛⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠻⠯⣽⠟⠻⠿⠟⠛⠢⠴⡿⠄⠳⠤⣄⠁⠈⠉⠀⠀⠉⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠐⠒⡒⠒⠒⠂⠄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠠⠒⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⡄⠀⠀⠀⠠⣀⡠⠂⠀⠀⢠⠂⠈⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠘⠀⠀⠀⠀⠀⠔⠒⢢⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠒⠲⠌⠉⣁⡈⠁⠀⠀⠀⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠐⠁⠉⣄⣀⠤⡄⠂⠂⠉⠀⠀⠀⠀⠀⠀⠀⠗⢄⠀⠀⠀⠉⢉⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠑⣀⠠⠐⠋⠁⢀⠔⠁⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢈⡐⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⢁⠀⣀⠄⠊⠁⢀⠜⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠀⠰⠀⠀⠀⠄⠀⠰⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠙⠒⠀⠀⠀⠐⡆⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠀⠉⠀⠀⠀⠀⣀⢀⣠⣤⣿⣿⣷⣶⣶⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣿⡿⣫⣿⣶⣤⢠⣄⢺⣷⣿⣿⠻⡟⠿⠻⠯⠛⠉⠙⣀⠉⠙⢻⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣧⡤⣚⢽⣿⣿⡿⠟⠱⠘⣞⣧⠹⠉⠋⠁⣴⣴⣦⡸⣷⣤⣾⣿⣷⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣟⣭⡿⣿⣿⣷⣄⣰⣿⣇⣿⣾⣿⣿⡜⣿⣿⣿⣷⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⢯⡿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⠈⠁⠉⢁⣀⣀⣄⣀⣀⠀⠀⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠙⠁⠁⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣸⠿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠙⠋⠁⠀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡟⢿⣿⣿⣿⣿⣿⣿⣿⡝⠟⠿⠉⠁⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⡛⣿⠻⠟⠟⣹⠿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⠉⠛⠛⠋⠈⠁⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡵⠞⠀⠑⠂⠈⣀⣽⡤⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣶⣶⣶⣿⣿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣽⠿⠿⣿⡗⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣭⣿⣿⠿⠏⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠛⠛⠂⠉⠓⠚⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡿⣍⣀⡒⠂⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠧⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣓⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣴⣒⣒⡂⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢙⣦⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢘⢈⠉⠩⠭⠀⠀⠀⠀⠀⠠⢀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠐⠮⣭⠉⠩⠁⠀⠀⠀⢨⣷⣿⣿⣿⣿⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢀⢙⣶⣶⣶⣈⣵⣶⣣⣼⣿⠛⢛⣉⣥⣠⣿⡟⢁⠬⣿⡶⠟⣷⠶⢶⠾⣶⣾⡿⢋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣘⣫⣽⣿⣿⣿⣯⡟⠻⠋⠸⠁⠈⠁⠀⠀⡀⢀⣦⠀⣨⡀⣤⠽⢿⡿⣷⠿⠛⢀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣧⣽⣶⣟⣾⣿⣵⣿⠛⠀⠀⠀⠀⣄⠀⠀⠀⢰⠇⣾⣿⣶⣿⣿⡧⠔⠋⢁⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡴⡿⣧⣿⣿⣿⣿⣿⠋⠀⠀⡄⠀⢠⣿⣶⢠⣾⣿⡴⠟⠙⡋⠁⠀⠶⠂⠋⢉⣸⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣵⣿⣿⣿⣿⣿⣿⡇⠀⠀⢡⢿⣆⢸⣿⠿⠛⣭⣵⣂⠛⠟⠛⠁⠀⠀⠠⢈⣴⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⣿⡇⢸⡇⢺⣼⡿⢯⣽⡓⠶⠌⠉⠁⠀⠀⠐⠶⣦⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣇⢸⣿⡾⠫⢞⡓⠈⠁⠀⠀⠀⠀⠀⢄⡈⢀⣠⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠟⢣⡏⢼⡶⠀⠀⠀⠀⠈⠳⣤⣀⡀⢈⣹⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠓⠦⣄⠐⣀⢀⣤⣠⡤⠩⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠃

⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣄⣤⣤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠐⠒⠮⣉⡽⢁⠜⣰⠋⠒⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⢞⠀⡞⠀⠉⣆⠀⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⢀⠤⠒⠀⠉⢙⠗⢖⠋⠁⠀⠀⠀⠀⠢⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠂⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘⡄⡰⡀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡖⠁⠀⠀⠉⡰⠖⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢷⠻⣕⣲⠔⠓⠒⠉⠉⠉⠉⠉⠘⠁⠀⠀⠒⠂⠤⠤⠤⠄⠀⣀⣠⡀⠀⠀⠁⠀⡄⠀⠳⡶⠶⡏⠈⣶⠓⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢲⣼⡦⢬⣷⣀⢰⠀⢤⣀⣀⡀⠀⣀⣠⣤⣤⣤⠤⠴⣖⠈⠉⠀⠀⠀⠀⠀⠀⢠⣔⠁⠀⢠⠛⡋⠙⣏⠁⡿⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡏⢲⠤⡀⠀⠀⠉⠙⢦⡈⠁⠉⠉⠡⠉⠐⠀⠉⠉⠒⢤⣈⠁⠐⠂⠀⠀⣀⡀⠸⠊⣸⠦⣾⠓⡋⢻⡉⢹⣦⣼⣉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠳⡈⠲⣄⠀⠀⠀⠈⠳⢤⡀⠠⠀⠀⠀⢀⠀⠄⠀⠈⠛⢿⡃⠀⠀⠀⣠⠖⠛⠛⠒⣿⡂⠙⠳⢾⣉⣤⣼⣉⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠀⠀⠐⣄⠀⡙⢲⠤⣄⠀⠀⠑⣦⣄⣀⠀⠀⠈⡀⠀⠄⠀⠂⠙⢦⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡧⣀⣯⠉⡤⡀
⠀⠀⠀⠀⠀⢀⡤⠶⠒⠂⢾⡖⠓⠒⠒⢾⡀⢈⠂⠐⢈⠉⠒⢤⡈⠳⡉⠓⠦⣤⢀⡤⢤⣬⣠⣐⡠⣳⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠦⢿⠄⡟⢢⡆
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠈⠻⡗⠤⢄⣄⠨⠻⣷⠶⠶⠶⣛⡯⠭⠤⣼⣾⡇⠤⠤⣀⠀⠀⢀⡞⠁⠀⠈⠓⣏⣹
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡊⠀⢀⠂⡁⠀⠈⢳⣄⣉⠁⠀⠈⢧⠀⡜⠁⠀⠀⠀⠀⠿⠄⠀⠀⠀⢱⣴⠟
⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⢧⡙⢦⣄⡀⠀⠀⠀⢀⠻⣟⠛⠳⠶⠞⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠓⠤⣌⠳⣌⠛⢶⣀⣠⠂⢁⠈⣆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣤⠤⣤⠴⠛⠁⠀⠀⠀⠙⠲⠼⣆⠀⠉⠙⠛⠶⠖⠞⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⣶⡖⣲⣶⣶⠲⢶⣤⡤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠒⢉⣿⣿⣿⡟⠿⠿⣿⣿⣵⣿⣿⣿⣿⣷⣾⡿⢿⣿⣶⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣴⠾⠟⣻⣿⣷⣾⣿⠋⣹⣿⣿⣶⡶⡿⠿⠿⠡⣤⣽⣿⣿⣧⣴⣿⣿⣟⣁⣰⣿⣿⣶⣤⣀
⠀⠀⠀⢀⣼⣿⡴⠶⠶⢛⣛⣛⣫⣭⣉⣭⣤⣴⣶⡲⣶⣶⣶⣶⣶⣶⣶⣦⣶⣭⣍⣙⣛⣛⠻⠿⠿⣯⣙⣛⠳⣤
⠀⠀⠀⡾⢫⣵⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣤⣈⠻⠟⣛⣋⠛⠛⢑⡿⠿⠿⣿⣿⣿⣿⣿⣷⣦⣬⣝⠻⢿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⡍⠉⠉⠀⣿⣿⣿⡇⠀⠀⢉⣉⣉⣛⣛⣛⡻⠿⣿⣿⣿⣿⣿⣷⣍⠃
⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢟⣩⡀⣿⣿⣿⠁⣾⣶⣦⣄⡺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⡰⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠿⠿⠿⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁⣠⣶⣭⣭⣭⣥⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠿⣿⣿⣿⣧⠀⢀⣠⣤⣶⢖⣿⣟⣿⣿⣷⣲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣶⣿⣿⣶⠍⠛⣻⣶⣿⠛⠻⣿⣿⡿⠟⣿⣿⣽⣟⣿⣍⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⣴⣯⣽⣿⣿⢿⣿⣿⣿⣶⣿⠛⣿⠟⣿⣿⣛⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⢰⣿⠻⣿⣿⣁⣼⣿⣧⣼⣿⣿⣿⣿⡿⠻⢿⣿⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡇⠘⣿⣤⣽⣿⠿⣿⣿⣿⣿⣅⠀⢛⣿⣿⣿⣿⣿⣦⣬⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⠀⠈⠻⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⡋⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠙⢛⠿⢯⣁⣸⣿⣿⣿⣿⣿⣿⣿⣄⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣿⣿⣶⡍⠛⠷⠦⠤⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣼⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢰⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣿⣿⣿⣿⣿⣿⢿⡏⢸⣿⣿⣿⣿⣿⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠾⠿⠟⠋⠀⠀⠻⢿⣿⣿⣿⡿⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⣑⣤⡄⠀⠤⠤⠄⣀⡀
⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⢀⣀⠤⢤⢤⣤⣤⣉⣁⠖⠲⠤⢄⡈⠑⠢⢄
⠀⠀⠀⣠⠴⠋⠀⢀⡠⠔⠈⠉⠀⠀⠀⠀⠀⢉⠄⠋⠀⠀⢀⠞⠁⠉⠙⠒⠽⢦⣀⡀
⠀⠀⠸⠥⠶⠿⠭⢭⠤⢤⠤⠄⠀⠄⠒⠂⠉⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⡰⠁⡠⠃⠀⡆⢰⠀⠀⢀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⣀⣀⢀⣀⡀⣉⣉⡉⢹
⠀⠀⠀⠀⠀⠰⠁⡐⠀⠀⢠⠀⢸⠀⡰⠁⠀⠀⢀⣾⠀⠀⣀⡤⠖⠋⠁⢠⠀⠀⠀⠁⣀⠴⠋
⠀⠀⠀⠀⢀⠃⡰⠀⠀⠀⠘⠀⢸⡴⠁⠀⠰⠀⠘⠀⠊⠙⠀⣀⠀⠘⣦⠘⡇⠀⣀⠀⢵⡀⢃
⠀⠀⠀⢠⠜⠚⠧⠀⡐⠀⠁⢀⡞⠀⠀⠀⠀⡄⠀⠈⠀⠐⠒⠒⠀⠈⢉⡠⠒⠉⠀⠀⠀⠹⡞⢄
⠀⠀⠀⠼⢶⠒⠺⡀⢣⠀⢰⠈⣀⠀⠄⠀⠄⡇⠸⢦⠾⠀⣠⣤⣤⠠⠙⢤⡀⠀⠀⢀⠀⠀⣘⣮⠤⡀
⠀⡄⠀⠀⢸⠀⠀⣷⠟⢀⡨⡆⠈⠁⠀⢰⣾⣿⣄⠀⣀⣼⣿⣿⡾⠃⢀⡦⣤⠋⠀⣴⣉⠀⢀⠜⠀⠀⡀
⡆⡇⠀⠀⢸⠀⠀⡇⢀⣮⢠⣾⡀⠘⡆⠸⣿⣿⡟⠀⠸⡿⠛⠉⣀⠤⠞⠋⢀⢴⡇⠈⡇⠀⢸⠀⠀⠀⡇
⡇⠂⠀⠀⠘⠀⠀⠇⠀⣿⣿⣿⣧⠀⢰⠀⡸⠋⠀⠀⠀⠘⠳⡎⠁⢀⠀⣴⣿⣿⡇⠀⠃⠀⠈⠀⠀⠀⠁
⢃⠠⠀⠀⠀⡆⠀⠘⠀⠘⠿⣿⣿⠆⠚⢋⡁⠀⣰⣿⣷⡀⠀⣉⠛⠛⣾⣿⡿⠛⠀⠘⠀⢀⠇⠀⠀⠀⢀
⠈⢷⡼⣦⣤⣿⣀⢀⣇⡀⠀⠈⠉⠀⠀⠙⠁⢄⣿⡿⣿⣇⠄⠙⠁⠀⠈⠉⠀⠀⣠⣇⢀⣼⣦⣤⣞⣾⠟
⠀⠀⠙⠿⣛⣿⠿⢿⠿⣯⣀⣐⣶⣷⣶⣄⠀⢼⡿⠃⢿⣿⠄⢀⣴⣾⣷⣖⢀⣤⠿⢿⠷⣿⣟⡻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣼⠀⠈⠃⠀⠈⠀⠀⠀⣠⡀⠀⠀⠈⠀⠀⠏⢸⠐⠉⢸⠹
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⢀⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠠⡄⠀⢸⣸⠠⠈⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠹⣻⢱⠀⣀⣎⠡⠀⠀⠀⡀⡗⠜⠁⠀⠀⡄⢠⣀⡄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢸⢸⢹⠘⢧⠐⠔⣠⡆⡦⠀⠀⡰⢠⡇⢸⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈⢈⢸⢸⢈⢸⢃⡇⡇⡆⢄⡐⠀⠀⢧⡼⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱⠈⠁⠀⠉⠘⢸⠀⠀⡇⢰⠀⠀⠃⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠄⠀⠀⠀⢇⠼⠀⠀⡄⢸⠀⠀⠀⠐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⡇⢰
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣧⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢼⠀⡶⠆⠀⠀⠀⠀⠀⠀⣀⡈⢁⣈⢁⠈⠁⣀⠀⠀⠀⠀⠀⡇⠠⠶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉
⠀⠀⢸⢠⣀⣠⠀⠈⠀⠀⠀⠀⣿⠉⠉⠉⠈⠈⠀⣿⠀⠀⠀⠀⠀⡇⣤⣀⡄⠀⠀⠀⠀⠀⠀⢠⠂⠀⠈⢢
⠀⠀⣼⣀⣉⣁⢰⣶⡆⠀⠀⠀⠿⠒⠂⠒⠂⠐⠂⠿⠀⠀⠀⠀⢀⣇⣈⣉⡀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠟


⠀⠀⠀⠀⢠⠶⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠲⠀⠖⡆⠀⠀⠀⠀⠀⠀⡠⠖⡒⠤⡀
⠀⠀⠀⠀⢸⣄⣀⡼⠀⠀⠀⠀⡖⠉⠉⡩⡉⠉⢱⡄⠀⠀⠀⠀⢰⣴⠀⣀⡇⠀⠀⠀⠀⠀⡜⠴⠜⠃⢶⢸
⠀⠀⣠⡶⠋⠀⠁⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠊⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⠀⠰⣔⠇⠘⢓⠜
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⣶⡎⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⢂⣠⣤⢂
⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⣀⡽⠀⠀⠀⠀⠀⠀⠈⠀⢻⢿⠃⠘
⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠉⠘⠂


⠀⠀⠀⣤⠈⠛⢫⣄⠀⠀⠀⠀⠀⢀⡨⠄⢤⠂⠀⠀⠀⠀⠀⠀⠀⠀⢙⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡐⠠⠄⠃⠎⠀⠀⠀⠀⠀⠀⢀⣂⣈⡄⠀⠀⠀⠀⠀⠀⠀⣄⣿⡀
⠀⠀⠷⣶⠾⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠋⡒⠉⠁⠀⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⣠⣿⣄⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⡐⠀⢀⠀⠀⠀⠀⠁⠀⠵⠆⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⡀⢀⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠢⡈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⠤⠂⠀⢄⢅⡀⠀⠂⠁⠂⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠄⡀⢀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠒⠒⠀⠨⣀⠼⠀⠀⠀⠈⠈⢀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠔⠀⠂⠀⠀⠀⠀⡆⠀⠒⠀⠢⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠂⢀⣀⠒⠁⠀⠀⠀⠠⠤⠤⠾⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠀⢀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⢀⠄⠄⠀⠀⠀⠐⢂⠴⠀⠠⡀⠀⠚⡛⣛⠛⠉⡉⠉⠀⠀⠄⢀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠘⣤⠀⠈⠘⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠲⢇⢠⣦⡄
⠀⠀⠀⠀⢄⢠⠀⠀⠀⠀⢲⡄⣀⣁⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⡀⠀⣀⠀⢀⡀⠀⠀⢦⡘⢁⡤
⠀⠀⠀⠈⠙⢚⠀⠐⠀⠀⠀⠀⣿⣹⠇⠈⣿⠀⣸⠏⠈⣷⠀⣿⠀⠀⢠⡏⠀⢻⡄⣼⠁⢈⡁⠀⠘⣆⣼⠃⠀⠀⡀⠉⠁⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢙⡇⢀⣿⠀⢻⣄⣀⡿⠀⣿⣀⣀⠸⣇⣀⣼⠃⢿⣀⣈⡿⠁⠀⣸⠇⠀⠀⠠⠁⠠⡁⠀⠤⠆
⠀⠀⠀⠀⢀⠀⠠⠀⠄⠀⠄⠀⠉⠉⠀⠈⠉⠁⠀⠉⠉⠀⠀⠉⠉⠉⠀⠉⠉⠁⠀⠈⠉⠉⠁⠀⠀⠉⢀⠀⡐⠀⠐⢄⢀⡂⠂
⠀⠀⠀⠀⠀⠀⠓⠀⠈⠀⠀⠀⡁⢈⠆⠀⡰⠢⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠆
⠀⠀⠀⠀⠀⠠⠒⠤⠒⠀⠀⠈⠂⠁⠀⠈⠐⠁⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠀⠀⠐⠀⠀⠁⠒⠈⢀⠀⡘⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⡀⠀⣖⠁⡐⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠑⠄⠀⠀⠀⠈⠁⠀⠀⠠⠰⠀⠀⡐⠒⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠰⠂⠛⠂⠀⠀⠘⠃⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣗⠀⠃⠀⠃⠀⠆⠐⡵⠀⡔⠀⠠⠀⠀⢲⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢀⠀⠀⠄⠀⠁⠐⢀⠐⠈⠁⠀⠀⠀⠆⠑⠁⠐⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠼⡄⠀⠀⠠⠟⠂⠐⠢⠄⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢴⠂⢮⡉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⣹⠉⠉⠙⠲⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢄⠀⣀⡴⡁⠀⠀⢢⡀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠈⠉⠁⠀⢇⡀⠀⣰⠃⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠉⠉⠁⠀⡄⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⢠⠀⠀⣠⠇⠀⠀⠀⠀⢀⣀⡀⠀⠹⠦⠞⠉⣷
⠀⠀⠀⠀⢀⣠⠴⠛⠁⠈⢹⡆⠀⠀⠙⠒⠚⠁⠀⠀⠀⠠⠚⠉⢩⠇⠀⠀⠀⠀⠀⢻⡄
⠀⢀⡴⠚⣉⠄⠀⠐⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠶⠋⠀⠀⠀⠀⢰⠀⠘⣇
⣠⡟⠓⢋⠁⠤⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢻⡄
⣿⠀⠘⡄⠀⠀⢰⡀⠈⠙⠳⢦⣄⡀⠀⠀⠙⠓⠒⠚⠁⠀⠀⠀⠀⡆⠀⠀⡧⠀⠀⠀⠘⣷
⢿⡄⠀⠙⣄⠀⠀⢧⠀⠀⠀⢸⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠚⠁⠀⠀⠀⠀⠸⣇
⠈⢷⡄⠀⠈⢷⡀⠘⢧⠀⠀⢸⡆⠀⠀⢨⠿⣦⣄⠀⠀⠀⠀⠀⢠⠴⢤⡀⠀⠀⣠⠞⠳⣄⢻⡄
⠀⠀⠙⢷⣄⠀⠙⠷⣌⢷⡀⠀⢻⡄⠀⢸⡇⠀⠙⣷⣦⣀⠀⠀⠀⠀⠀⢻⠀⠀⢃⠀⠀⡿⠈⣷⡀
⠀⠀⠀⠀⠉⠛⠶⢦⣌⡓⢿⣦⣀⠻⣆⠸⡇⠀⣼⠇⢀⣹⠿⢦⣄⡀⠔⠋⠀⠀⠀⠉⠉⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠁⠁⠘⠻⠚⢿⣅⣀⣀⣨⡽⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠈⠻⣍⣀⣀⣀⣤⡤⠈⠙⠷⣤⡀⠀⢠⠂⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⠳⣤⣄⣤⣤⣤⣤⣬⠿⣆⢸⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠑⢄⡀⠀⠀⠀⢀⡿⠀⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠙⣆⠀⠀⠀⠈⠉⠒⠒⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠃⡄⠀⠀⠀⢀⡴⠁⠀⢰⠀⢷⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⡇⠀⠀⣠⠞⠀⠀⠀⣼⠀⠈⠻⣦⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣴⠏⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀⡾⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⢠⡿⠁⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⣰⣟⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣼⠋⠹⣦⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⣰⠟⠁⠀⠀⠈⠻⣦⡀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠻⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠋⠉⠃⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⠴⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠾⢭⡀⠀⠈⠢⠱⡀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠡⠤⠀⠀⠈⠒⢄⠀⠑⡌⢈⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠃⠀⠒⠂⠂⠐⠢⠦⢤⣤⣈⣼⡇⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠤⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⣠⡴⠂⠈⢉⢛⡶⢠⣄⣀⠀⢀⠀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢶⠶⠊⠉⠁⠀⠀⠀⠁⠞⢠⠋⠀⢀⡀⣉⢁⡀⠀⢀⠤⠀⠀⠀⣀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⡠⠞⠁⠈⠀⠀⠄⠁⠀⢀⣀⠤⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠀⠘⢀⡎⢠⠀⠀⠀⠐⠀⠤⠄⣤⢞⡉⢉⡄⠀⠐⠒⠀⠙⠲⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⣸⠁⢸⠀⠂⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠘⣧⠀⠀⠀⢠⣿⠀⠈⠧⣤⡀⠀⠀⠀⢀⣀⡀⡄⠀⠀⣀⠀⠀⢀⣀⡤⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⢻⣇⠀⠸⣸⡿⠀⢰⠀⠀⠉⣡⠉⠉⠱⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠘⣿⣇⠀⢿⣷⠀⢸⠀⠀⢠⡇⠀⠀⢠⠂⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⡞⣄⢸⣿⡄⣸⠀⣠⡟⠀⢀⢄⡌⡘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠈⢿⣦⡹⣆⡻⠱⣯⠼⠟⠀⢀⣡⡾⢊⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⠛⡛⠁⠘⢵⠎⠠⢴⠿⠿⠟⠋⠑⠊⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⣠⡤⢶⢶⠚⠛⣿⢟⢶⢶⣤⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⡟⡕⡱⢡⢹⡆⠀⢻⠈⢎⢣⢣⡟⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠘⠦⣍⣓⠳⠧⢾⡇⠀⢸⣚⣚⣯⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠊⠉⠙⢦⠀⠀⠀⠀⠀⠈⠉⠉⢹⠀⠀⣼⠁⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⡟⠀⠀⡏⠀⠀⣀⡤⠖⠋⠉⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⡤⠤⢤⡤⢤⣤⣄⣀⠈⢦⠀⠀⠀⠀⣸⠁⠀⢸⠃⢠⠏⠁⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⢬⣛⡺⢿⠀⢸⠧⠶⠿⣃⠞⠀⠀⠀⢰⠃⠀⢀⡏⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⢸⠛⠋⠉⠁⠀⠀⠀⢠⠏⠀⠀⡼⠁⠀⠀⠀⠙⢲⡤⣄⣀⣀⣀⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡄⠀⠀⠀⠀⠀⠀⡞⠀⠀⢰⠃⠀⠀⠀⠀⢀⡞⠀⡼⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢧⠀⠀⠀⠀⠀⢸⠃⠀⠀⡟⠀⠀⠀⠀⢠⠏⠀⣸⠃⣠⠤⠒⠋⠁⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⡀⠈⢧⡀⠀⠀⠀⡟⠀⠀⢸⠃⠀⠀⠀⣰⠋⠀⣰⠃⠐⡇⢶⡶⢤⢄⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠉⠳⢤⡀⢳⡀⠀⠳⡄⠀⢰⡇⠀⠀⢸⠀⠀⠀⢰⠇⠀⣰⠃⠀⠀⠈⠲⢽⠏⢠⣿⣳⡄⢸
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠉⣦⢳⡄⠀⠙⣆⢸⠀⠀⠀⢸⠀⠀⢀⡏⠀⢀⡏⠀⠀⠀⢀⡴⠋⢠⠞⠛⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣀⣤⠴⠚⠁⠀⢻⡀⠀⠈⢿⠀⠀⠀⢸⡄⠀⢸⠁⠀⢸⠁⠀⣀⠴⠋⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⣄⠀⠀⠀⠀⢳⠀⠀⠸⡆⠀⠀⠈⣧⠀⡿⠀⠀⢸⣤⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠈⠳⠤⠤⣤⡈⣇⠀⠀⢳⠀⠀⠀⠸⡆⡇⠀⠀⢸⡁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⢀⡽⢹⡀⠀⠈⣧⣀⣀⣰⠃⢻⠀⠀⢈⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⢧⣀⣠⠟⠈⠉⠀⠀⠈⠳⠤⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠲⣾⣴⡶⠀⠀⠀⠀⠀⠀⠀⢀⣤⠒
⢺⡟⡿⠿⡷⠀⠀⠀⠀⠀⠠⠞⠛⠒⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⣾⣿⣟⣤⣰⣾⣦⠛⣶⣿⣿⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣚⣳⣶⣶⣦⣾⣿⣿⣷⣶⡶⠀⣠⠀⢶⣶⣾⣾⣿⣶⣆⣀⣀⣤⣠⣄⡀⣀⣤⡀⢰⣷⣶⣶⣂⡀
⠻⣿⠿⠿⣿⠿⠿⠳⠾⠿⠿⠤⠁⠀⠀⠀⠀⠸⠆⠀⠀⠉⠉⠉⠉⠉⠿⠏⠁⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠉⠛⠉⠏⠋⡉⠁⠉⠉⠉⠉⡉⡍⡛⢛⠋
⡋⠤⠚⣁⠤⠖⠊⠉⣀⠴⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠠⠉⠈⢉⡉⠀⠒⢀⠀⠀⠀⠀⠀⠀⠠⠐⠊⠡⠀⢈⠀⠈⠀⠂⠠⢀⠀⡀⢆
⣠⠔⠋⢁⡠⠴⠒⢉⡠⠔⠂⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠠⢄⡀⠀⠠⣄⠀⠐⠀⢀⠀⠀⢀⠄⠂⠀⠠⠄⠈⠀⠀⠀⠀⠀⠄⠁⠀⠐⠀⢀⠀⢸⠀⠇⠸
⢀⣠⠔⠉⠀⡠⠀⠉⠀⠠⠄⣀⡀⠑⠒⠤⠄⡀⠁⠐⠀⠀⠀⠐⠂⠀⠉⠋⠀⠀⢀⡤⠀⢀⠠⠊⠁⡀⠔⠁⠀⠀⠊⠀⠀⣠⠞⠀⠀⠀⠌⠀⢀⠄⠀⡀⠀⢷
⠋⠁⢀⠔⠋⠀⡀⠚⠀⢤⠀⡀⠉⠠⠶⡶⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠖⠉⢀⡠⠊⠁⣀⠄⠁⠀⠀⢀⠞⠀⠀⠀⣠⠋⠀⠀⠐⠀⢀⡇⠀⣸
⠀⠐⠍⠀⣠⠎⠀⣀⣀⠀⠁⢀⣄⡄⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢉⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠂⡀⠂⠀⠀⠀⡄⠀⣼⠀⠀⡿
⠀⣀⠐⠢⠀⠠⣄⡈⠉⢀⡀⠀⠐⠀⠠⣤⡖⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠃⠀⠀⠀⠁⠀⠚⠀⠤⠀⢀⠀⠀⡐⠀⠀⠄⠀⠠⠀⠀⣀⠀⢀
⠣⠈⠁⠀⠐⠦⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⡀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⣀⠀⠁⠀⠘⠃⠀⠰⠖⠀⡤⠀⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠒⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⠀⠈⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡣⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡔⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡜⠘⠰⢀⣠⣴⣶⣿⡿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣷⣶⣄⠃⠀⣀⠀⣠
⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⣠⣾⢿⣿⣿⣿⣯⣄⣼⣿⡟⠛⠻⣿⣾⣿⣿⣿⣿⣬⣿⣿⣷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢠⣿⣿⣿⡿⣿⣿⣽⣿⣿⣿⡇⠀⠀⠘⣿⣿⡿⣿⣿⣿⣿⣿⠏⣰⠆
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠑⡈⠻⣿⣿⣷⣶⣾⣟⢙⣿⣿⡇⠀⠀⢠⢿⣿⣷⣾⣿⡷⠟⠋⠔⠀⠀⣀⡤⠀⢠
⠀⠀⠀⠀⠀⣀⣀⡈⠑⢄⠀⢸⠀⠤⠉⠉⠛⠛⠛⠋⠉⠙⠁⠀⠀⠸⡀⠀⠀⠀⠀⡺⠂⣬⠔⡁⠈⠉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠚⠚⠢⠀⠀⠀⠀⠀⠀⢀⣡⡀⣠⠀⡀⠀⠁⠀⠀⡠⠀⢀⠠⠛⠚⠲⠶⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠲⡒⠀⢁⠢⣼⠀⠠⠀⡄⠈⠀⠃⠑⡛⠆⠀⢸⠅⢊⠄⠐⠲⠒⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡔⢁⣤⣼⣆⠀⠀⠀⠙⠛⠣⠀⠰⠀⠀⠀⠀⢠⠀⠀⡰⠛⠛⠁⠀⠀⢠⣾⣤⣄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⢇⣿⣿⣿⠏⢀⠖⠉⠉⠒⣄⠁⠀⢀⠠⠠⠤⣘⠀⠀⢁⡔⠉⠉⠐⢄⠘⣿⣿⣿⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠆⠀⢀⣾⣿⣿⠀⣴⠁⠀⠀⠀⠈⢣⠀⢸⣿⣷⡄⠀⢈⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣘⣿⡿⠋⠀⢠⠀⠀⠀⠀⠀⡸⠀⠈⠻⣿⣇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⡤⡶⡒⢻⡏⠉⢹⡟⡝⢟⢷⢶⣄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣾⢏⠎⡴⢱⢱⡇⠀⠘⡿⡘⡎⢦⢳⣽⢃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⠻⠼⣥⣧⣼⡇⠀⠀⣧⠷⠿⣚⣫⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠓⢦⡀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢺⡇⠀⠀⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⡇⠀⠀⠀⠀⢀⡤⠖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣼⠀⠀⣠⠶⠚⠉⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⣴⣒⡒⣲⡖⠒⡶⠶⠦⣄⠀⢳⡀⠀⠀⠀⠀⡾⠀⠀⢠⡏⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⣙⣛⣺⠀⢀⡷⠶⠿⢛⣡⠞⠀⠀⠀⠀⣼⠁⠀⠀⡼⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠀⢸⡏⠉⠉⠉⠀⠀⠀⠀⠀⣸⠃⠀⠀⣸⠃⠀⠀⠀⠀⠙⠲⣤⣄⣀⣀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⡼⠁⢠⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢹⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⡼⠁⠀⡾⠀⠀⣀⡤⠴⠚⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢳⡀⠀⠀⠀⠀⢰⡇⠀⠀⢠⡇⠀⠀⠀⠀⢠⡞⠀⠀⡼⠁⢰⠋⣁⣀⡀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠈⢧⠀⠀⠳⡄⠀⠀⠀⣾⠀⠀⠀⢸⠀⠀⠀⠀⢠⡟⠀⠀⡼⠁⠀⠘⢦⡻⣭⣹⠓⢲⣤⡀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠳⢤⡈⢧⠀⠀⠙⣦⠀⠀⡏⠀⠀⠀⣼⠀⠀⠀⢀⡟⠀⠀⣼⠁⠀⠀⠀⠀⠙⣳⠃⢀⣟⡶⠿⢂⡼
⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣹⠈⢷⡀⠀⠈⢧⣰⡇⠀⠀⠀⢸⠀⠀⠀⣼⠁⠀⢰⡇⠀⠀⠀⠀⣠⠞⠁⢠⠞⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠳⢤⣄⣀⣀⣀⣤⠴⠚⠁⠀⠈⢧⠀⠀⠀⢻⡇⠀⠀⠀⢸⡄⠀⢠⡏⠀⠀⢸⠀⠀⢀⣤⠞⠁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢹⡄⠀⠀⠀⠀⠈⣧⠀⠀⠀⣇⠀⠀⠀⠈⣧⠀⢸⠃⠀⠀⢸⣀⡴⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠙⢦⣀⣀⣀⡀⠸⡆⠀⠀⢹⡀⠀⠀⠀⠸⡆⢸⠀⠀⠀⢸⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡀⠀⠀⠈⠉⠙⣧⢷⠀⠀⠀⢳⡀⠀⠀⢀⡗⢸⡄⠀⠀⢸⡇⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⠤⠤⠶⠃⠸⡆⠀⠀⢨⡟⠲⠶⠚⠁⠈⢧⡀⠀⣸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠴⠞⠁⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⣀⡀⠰⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠱⣆⠴⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣰⣾⡿⠀⠀⠀⠀⢨⡆⣄⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣤⣶⣦⣠⡤⠾⣧⣿⣟⠉⠻⢧⠀⠀⠀⣶⢸⣷⣿⠁⣸⣿⣅⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠙⠿⠈⠫⣄⠀⠙⣿⣿⡀⢀⡀⠀⢴⡆⣻⣿⠛⣽⢢⡿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠘⣿⣷⣸⡏⠀⢸⠃⢀⣏⠀⡢⠿⣷⢶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⢰⡄⠈⠁⠀⢀⠀⠀⣤⠼⣿⣿⡇⠀⢹⡗⠈⠉⠈⠀⢸⣿⡿⠉⠉⡉⢉⣷⡷⣦
⠀⠀⠀⡠⠖⠛⠛⠛⠉⠀⠓⠀⠀⠘⣿⣼⣿⣇⠀⢸⠇⠀⠀⠀⠀⡜⠁⠀⠀⠀⢀⡾⠃⠀⠈⠓
⠀⠀⠀⠶⣤⡤⠄⠄⡀⠀⠀⢀⠈⠰⣼⣿⣿⣿⠀⢘⡀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁
⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠈⠉⠶⣝⣻⣿⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡌⢻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠳⠶⣥⣌⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡇⠄⠀⠀⠀⠀⠠⠚⡀
⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⣙⣿⣿⣶⣶⡄⠀⠰⢷⣿⣿⣩⡖⠛⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⣷⣶⠶⠚⠋⠉⠁⠀⠈⠻⢿⣿⣶⣤⣀⠈⢹⠉⠙⢷
⠀⠀⠀⠀⠀⠸⣇⠠⠈⠉⠉⢉⠀⠀⠀⠀⠀⠀⠀⢀⡠⣴⣞⣻⡿⢿⠆⢺⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠙⠷⠶⣦⣤⣀⣀⣀⡀⠀⠀⢐⠊⣿⡶⠉⣘⣿⣤⡷⠾⠈⣿⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣰⣿⠛⠃⠀⢴⠛⠋⠁⣠⠞⠀⡟⢧⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠁⠀⠀⢠⠋⠀⢀⡌⠁⠀⢸⠃⠈⡆⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⣾⠀⠀⠀⠘⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠃


⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠸⠿⠟⠀⠀⢰⣶⠀⣰⠚⣿⣭⡁⢳⣿⠋⠀⠘⠇⠀⠀⠈⠙⠛⠉⠀⠙⠛⠋⠀⠙⠓⠋⠁⠈⠛⠚⠉⠈⠑⠒⠀⡀
⠀⣴⣤⣴⣶⡦⢤⠀⣿⣤⣿⣤⣥⣬⣭⣤⠀⠀⠀⠀⠀⠎⠈⢳⡞⠁⢱⡞⠁⠹⣶⠁⠈⣶⠁⠈⣷⠏⠀⢳⠋⠀⢹⡀⠀⠀⠀⠀⠀⠀⠺
⢀⡈⢵⠌⡋⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠈⢧⠀⢸⢧⠀⢠⢿⡀⢀⢿⡄⠀⡾⣇⠀⡸⣇⠀⡸⢷⡀⣠⠀⣶⣦⣀⣶
⠛⠟⠀⠺⠿⠀⠀⠻⡼⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⢀⣄⠈⢁⡁⠀⠀⠱⠄⠉⡌⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣁⡀⠀⢻⣿⣿⣿
⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠐⣦⣷⣷⣾⣷⠲⣞⠀⠀⠀⠀⢀⣀⡀⠀⠁⠠⠀⠀⠋⡙⠛⠉
⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣧⠀⠀⠀⠀⠚⠢⠀⢀⠀⠈⢑⣾⣿⣟⠋⣋⣿⣿⣿⣿⡥⠖⠀⣾⣿⣷⠀⢸⠀⠶⢐⣇⣇⣐⠈
⣛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⠽⠿⠽⠀⠀⣀⣠⣤⠖⠀⢠⣄⠰⢼⣿⣿⣿⡴⣿⣿⣇⣨⣿⣧⣀⡀⠈⠉⠁⠀⠈⠀⠀⢉⡙⠉⢛⣀
⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⢐⡀⡊⠉⠁⠀⢙⠀⠐⠊⢀⠽⢷⣿⣿⣤⣽⣿⣾⣿⢻⣇⠀⠀⠀⠀⠀⠀⠠⠀⠀⣸⣇⠀⠈⠩
⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⣸⠃⢈⣶⡶⢾⣉⠀⠀⠄⠂⠐⠊⢻⣧⣼⣿⣿⣾⠿⡃⠈⠀⠀⠄⢀⣀⣤⣤⣤⠀⠐⠦⠄⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠙⢧⠀⣉⠀⠀⠀⣈⠀⠀⠉⠄⠀⠐⠃⠀⠈⠏⠃⠘⠄⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⣸⠁⡨⠷⠚⠛⢥⠀⠀⠀⣠⡄⠈⠀⠠⠀⣤⣦⡄⠀⠀⠀⣴⡶⣾⣿⠿⠿⠛⠋⠁⠸⠿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠟⠲⣧⠀⠲⣀⣀⣠⡴⠀⠀⠀⠨⠀⠘⠀⠀⠀⠻⠿⠃⠀⠀⠊⠉⣈⣉⣁⡀⠀⢀⡀⠘⠳⣶⠤⣿⣿
⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢸⠇⠴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣴⡄⠀⠀⠀⠀⠉⠙⠋⠁⠉⠁⠀⠰⠶⠀⠶⠈⣁
⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⣀⡉⠀⠰⠤⠀⠀⠀⣘⣃⢀⠀⠤⠀⠠⣶⡶⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⣀⡄⣄⠠⠀⠀⠂⠂⠘⢿
⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠈⠛⠘⠿⠿⠺⠗⢠⣴⣄⠈⣿⣿⡿⣧⣶⠆⠀⡉⠉⠀⣿⣿⣿⣿⣷⠀⠀⠀⠰⡀⠂⠀⡃
⠀⠀⠒⠒⠀⡍⢳⣧⣻⢽⠀⠀⠀⠀⣀⣄⠻⠃⠈⢹⣷⡄⠛⠛⠋⠉⠿⠀⠘⠃⢠⠀⠻⣿⣷⡿⠛⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠦⡈
⠀⣾⠀⠀⠀⠀⡾⠃⡵⡞⠀⠠⠀⣀⡿⡿⠃⣠⡰⣾⣿⣵⣶⣦⠄⠀⠠⢴⣲⡀⠀⠠⡦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠃⢀
⠂⠹⠂⠀⠀⢰⡱⣿⣻⠋⡄⠠⢀⣛⠁⣾⠆⣽⣧⡋⠘⢿⠓⢮⠑⠀⠀⠋⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⢠
⠄⠀⠀⠀⠀⠀⠓⠛⠁⠀⠀⠀⠈⠁⠈⠛⠀⢸⣿⣧⠀⢀⡀⣀⣠⡤⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⡀⠀⠾⢶⡄⠀⠀⠀⠀⢠⡴⣶⣿⣽⣦⡄⡀⢻⣿⣷⣽⡯⠽⠋⠀⠀⣀⠀⠀⠀⠠⠀⠄⠀⠁⠀⢠⣶⡄⠀⠀⠄⣁⣨⣀⡀⠄⠈⠁
⠠⡇⠰⠶⠀⣄⠰⠆⠀⢠⣿⣿⠏⢻⣿⣿⣧⡀⠈⢻⣿⢿⣶⡾⣶⠀⠈⠙⠀⠀⠀⣀⠀⢠⣦⡄⠀⠈⠙⠁⠀⣰⡿⣿⣿⣿⢻⣶⡀
⢠⡇⢴⡤⠀⢿⠀⣦⠀⠀⠻⣿⣿⣶⣾⣽⡇⠀⢀⣿⣿⣾⣿⣷⠿⣷⣦⠀⠈⠀⠀⠈⠁⠈⡟⠁⠀⠿⠂⠈⢠⣿⣷⣿⣿⣿⣿⣿⣇⠈
⠈⠃⣀⣀⡀⢀⣀⣘⠀⠀⠀⠈⠉⠻⢋⠁⠀⣀⠀⢉⡛⠻⠿⣿⣶⡆⢰⣾⣶⣄⠛⣶⡦⢾⣿⡧⠰⣿⡆⠐⠂⢿⣿⣻⣿⣿⣯⣿⠇⠒
⣴⣿⢶⣆⣙⣻⢶⡨⡕⡲⡦⠀⠀⠄⠀⢠⡀⠈⠃⠈⡉⠛⠀⠀⠈⠀⠀⠙⠋⠀⠘⠃⠀⣠⣯⠀⠰⠆⠰⡶⠀⠈⡙⠿⢿⠿⠟⡉⠀⢀⣤
⣪⣿⣼⣟⡡⠪⠿⠄⢅⣧⣤⡽⠀⣀⣀⡿⠁⢰⠄⣴⠏⠐⠞⠛⢲⣤⠖⠛⠓⢦⡶⠛⠓⣎⡵⠚⠓⠲⣤⠖⠓⠂⠀⠁⠘⠀⠁⠀⠀⣿⢿
⢠⣧⣄⠀⠠⠴⠶⠀⠀⠀⢀⣄⣼⣿⣿⣿⠀⢠⡄⢠⡴⠀⠀⠀⢠⣟⠀⠀⢀⣿⡀⠀⠀⣸⡃⠀⠀⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⡋⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣀⠀⠀⢶⣿⣷⡆⠀⠉⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⣿⣏⣀⣴⣿⣿⣿⣷⣾⣿⠿⠻⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⢰⡿⠟⠀⠀⠀⠉⠉⠀⣠⣄⠀⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟⠁⠉⣿⣿⣿⣦⣴⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠙⠛⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢻⣿⣯⣀⣸⣿⣿⣶⣾⣿⡿⠛⠉⣻⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⠸⢿⠇⠀⠀⠉⠁⠀⢠⣴⣶⡆⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⠀⠀⠀⠀⠈⠉⢁⣀⣠⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⣀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⣬⠀⠀⠀⣤⠀⠀⠀⠛⠟
⠀⠀⠀⠀⠙⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠈⠀⠐⠰⢦⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠪⠷⣤⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠁⠀⠰⠶⠀⠠⠼⠛⠛⠫⠽⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠈⠉⠛⠛⠿⢦⣄
⠀⠀⠀⠀⠀⠀⢀⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⢀⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⣤⣄⣤⡄⢀
⠀⠀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣎⠀⠀⠨⣿⣿⣿⣿⣿
⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⣾⣿⣿⣿⡀⠀⡀⢹⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣦
⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣻⣿⣿⣿⣧⠀⠀⠀⣿⣿⡇⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠻⠿⢿⣿⠄⠀⠀⢈⣉⣀⣸⣿⣿⠂⠀⠀⠀⠀⠀⠰⠾⢷⣞⣻⣿⡿⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⡠⣶⠿⢼⣿⣿⡛⠋⠁⠀⠀⠀⣀⠾⣷⣦⣀⠘⠛⠻⣿⣴⠾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⢀⠻⢿⣿⡇⠀⠀⠀⠀⠸⠿⣷⢿⡀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣷⠀⠀⠀⠀⠰⣷⠆⢀⠁⠀⠀⢷⣄⠀⣴⣦⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⢛⣿⣿⡇⠀⠀⠀⠀⠀⠘⠉⠉⢀⣀⠀⠙⠀⠻⠏⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣤⣽⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣹⣿⣿⡆⠀⠀⠀⠀⠀⠀⣄⣀⡀⠀⠁⠈⠰⣽⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠐⠿⠿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡿⢿⣿⡀⠀⠀⠤⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠗⣿⣿⣧⠀⠂⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⢹⣿⣿⡄⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣿⣿⣧⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣹⣿⣿⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡼⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢒⡠⢾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡁⢘⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡞⣁⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢣⣼⠷⣻⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣆⠙⡶⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⡃⠖⣻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⢾⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠓⠢⢄⢀⣀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⡱⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠊⠀⢀⡠⠂⠈⠀⠀⠀⠀⠈⢂
⠀⠀⠀⢀⠀⠤⠐⠈⠀⠀⠀⠀⢈⣀⣀⣀⣀⡠⠴⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠘⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠐⠉⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⡀⢀⠠⠀⠀⣤⡂⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⡀⠀⠀⠀⠀⡅⢰⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠂⠓⠂⠀⠀⠀⠀⠀⠃⠠⠃⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠃⠀⠂⠐⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠅⠄⠀⡀⠈⣄
⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠠⢄⠠⠀⠀⡀⠀⡈⠀⡀⢀⠚
⠀⠀⠀⠐⠀⠀⠀⢀⡀⠤⠄⠠⠒⠠⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠈⠉⠐⠂⠀⠒⠀⠀⠁
⠀⠄⠀⡠⠔⠐⠉⠀⣀⠀⠤⡖⠒⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⡀
⠀⡰⠁⠀⢐⣶⡌⠁⠀⠀⠀⠹⠅⠐⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁
⠀⠄⠀⠄⠐⠂⠀⠐⠒⠀⠀⠀⠁⠀⠀⠐⠉⠐⠈⠂⠀⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠⠀⠀⠐⠈⠉⠉⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡀⠀⠀⠀⡈⠁⠄⠁⠀⠀⠒⠂⠀⠀⠀⠁⠀⢣⠈⠠⢖⣉⡠⠤⠀⠀⡀⠠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠙⡀⠀⠀⠄⢀⡀⠀⠀⠀⠉⠁⠀⠈⠀⠀⠁
⠀⠐⢆⠀⠈⠰⢀⠀⠰⡃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⢀
⠀⠀⠀⠑⣀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⣀⣀⡐⠄⡀
⠀⠀⠀⠀⠈⠄⢀⢁⠠⠀⡄⠀⠀⠀⠀⠈⠀⠈⠉⠂⡀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⡀⢀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⡆⠈⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠐⠐⠄⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠐⠂⢀⠀⠀⠀⠆⢀⡀⢂
⠀⠀⠀⠀⠈⡡⠀⡁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠲⠄⣻⡀⢡⣷⠧⣊⠰⡀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⢣⡉⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⡾⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡟⢉⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣤⣴⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣦⣤⣼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠉⢉⣁⠒⠬⡑⣄⠱⠘⠀⢡⢈⡤⢋⠍⡛⠿⢛⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠅⠐⢒⡂⠨⠍⠀⣿⣷⣾⠟⠾⠛⠷⢿⣯⠤⣐⣥⣔⣨⣀⣀⣐⡘⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠶⠤⣴⣦⣡⣔⣿⡇⡔⠣⠐⠠⢼⣿⣿⣯⡭⠭⠿⠷⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡉⠀⠀⠈⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣴⣶⣴⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢉⠀⠀⡄⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠀⠀⠐⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠊⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠠⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠇⣰⠃⢀⠀⢀⡶⠀⠀⠀⡇⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡄⠘⣦⠏⢠⡆⢀⣞⡇⠀⢠⣶⣏⡼⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣟⠀⣾⡇⡼⢸⢁⡄⣜⣁⣿⡡⢞⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣌⣏⢻⣦⡿⢳⡇⣾⡾⣿⡟⣿⠏⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠻⠀⠳⠁⠈⠀⠛⠁⠋⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⣀⣀⣀⠀⠀⠀⠀⢤⣀⡀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⣬⡤⠠
⣶⠶⠶⠆⠀⠀⠐⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉⠀⠈⠉⠉⠉⠭⠅⠈⠉⠩⠍⠁⠠⠈⠀⠀⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⠤⠼⠇⠐⠀⠐⠐⠓⠒⠒⠀⠀⠀⠀⠒⠒⠀⠄⠀⠠⡄⣀⠀⠠⠄⠒⠂⠀⠚⠛⠒⠂⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⠛⣛⡋⠀⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠒⠒⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠺⣿⠋⠀⣿⣿⣧⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⡀⠀⠀⠀⠀⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⢻⡛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠉⢇⣀⣀⣸⠁⠀⠀⠈⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠴⡚⠉⢉⡠⠇⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣀⣠⠃⠀⠀⠀⢀⣤⡀⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⢷⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣀⡼⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠒⠲⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣀⣸⡇⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢠⡾⠁⠈⠁⠀⠀⠀⢀⣁⣀⣀⣀⣀⣀⣈⣷⣄⣀⣀⣼⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠞⠛⣉⠉⠻⣍⠙⣍⠹⠉⢹⠉⢹⢹⠋⡽⠛⣻⠟⢶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⠰⣟⠒⠤⣄⡑⢦⡈⢧⡈⢣⣱⣬⠤⠴⠶⣾⣤⣞⣡⠔⢁⠤⠞⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣬⣙⡲⢦⣽⣶⣿⣦⣿⣭⠴⠶⠶⠄⠀⠀⠀⠈⠉⠻⣧⠔⣚⣉⣈⣹⣦⡠⠤⣄⡀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⣤⣾⣿⣓⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠉⠐⠶⠶⠾⣧⠀⠈⢳⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠯⠭⠄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠿⠿⢭⣙⡒⢿⣄⣀⡾⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣭⣭⣍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠶⣄⣀⠉⣻⠏⠉⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣶⡖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣦⣤⣤⣉⡙⠋⠀⠀⠀⢀⣠⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣻⣿⣛⣒⣒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⠙⠛⠿⠿⢿⠿⣿⣿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣛⠒⣒⡒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⣤⣶⣦⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣛⣛⣛⣓⣚⣛⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣭⣤⠤⠦⠐⢒⣀⣐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⣭⣭⣭⣭⣭⣭⣭⣭⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣩⣭⣭⣥⣤⣤⣄⣀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣷⠀⠀⡰⡆
⠀⠀⠀⠀⠀⠙⡒⠬⣏⣿⣿⠿⡟⠚⠋⠉⠥⣤⣤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠹⣆⡜⢸⣠
⠀⠀⠀⠀⠀⠀⠐⡀⠀⣿⣿⠉⡿⢿⠛⢫⣭⠒⢂⣿⣏⣉⣉⠉⠉⠥⠤⠤⠤⠀⠀⠇⠳⣿⠀⠈⡙
⠀⠀⠀⠀⢠⣤⣴⣷⣾⣿⣿⣜⠁⠘⠿⠏⢻⡉⡸⢹⡇⣿⠤⡶⠒⠒⠒⠒⠒⠚⠃⠸⠀⠙⣷⣗⣇⣴⠖
⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠙⠧⠀⠁⠉⠼⠶⠒⠛⠉⠉⠉⠹⠶⠔⠃⠀⠀⠙⠹⠷⠖⣺⣟⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠒⠲⠤⣄⣀⣠⠄
⣀⣀⣀⣀⣀⢀⣀⣀⠀⣀⣀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠐⢠⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠐⠀⠠⢀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄
⠀⠐⠐⠀⠘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠗⠀⢠⠇
⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢦
⠀⠀⠀⠁⠀⠄⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠈⠁⢀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡎⢱⠀⠀⢀⣠⡄⠀⣀⡠⢀⡀⣀⠀⠀⢀⠀⠀⣀⣀⡀⠘⡂
⠀⠀⠀⢲⠀⠘⠀⠀⠈⠀⠀⠌⠀⠀⠀⠈⣠⠼⢁⣀⡇⠀⠕⠀⣄⢸⠠⢁⡀⢟⠀⠈⡇⠀⠀⠦⠀⠃⠁
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁⠀⣀⣈⢀⣁⣀⣈⣀⣁⣀⣀⣈⢀⣉⣁⣀⡇⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠉⠈⠉⠈⠉⠉⠉⠈⠉⠉⢉⠉⠉⠉⠁⠀⠀⡀⠃⠈⠀⠘⠀⠈⠈
⠀⠐⠀⠀⠀⠠⠀⠀⠀⡄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠄⠡⠤⠤⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠄⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤
⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠁⢠⡟⢁⠀⠀⠀⠐⠁⣁⢀⡠⠃
⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠂⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⡄⢠⠈⠈⠁⠀⠀⠀⠀⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⡀⢠⡄⠀⠀⠀⠒⠒⡄⠀⠀⠀⠀⡀⠑⠀⠀⠠⠘⠀⠀⠑⠁⠀⠀⠀⡀
⠀⢨⠀⠀⢀⣤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⣒⣓⣒⣢⠀⠀⠀⠠⠒⠀⠢⠄
⠀⠀⠀⠀⠰⡀⠊⡉⠀⠀⠡⠀⠠⠈⠀⠀⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⡀⠒⠄⠀⠈⠁⠀⠀⠀⡤⡄⠠⡤⠀⣠⢤⡄⢠⡄⠀⢀⡤⢤⡀⢠⡤⣤⠀⢤⠀⣤⠀⠀⠀⠀⠄⠐⠀⡀
⠀⠀⠀⠈⠠⠀⡆⠐⠀⠀⡷⣇⠀⡇⢰⡇⠀⣿⢸⡇⠀⢸⡃⢘⡇⣿⠀⢴⡄⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠁⠐⠁⠀⠀⠆⠀⠀⠀⠓⠋⠐⠓⠀⠛⠚⠁⠘⠓⠒⠈⠓⠚⠁⠘⠒⠛⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠀⠀⡀⢀⣤⡤⡄⠀⠀⠀⢸
⠀⢀⡀⠀⠀⠀⡁⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠘⠐⠀⠀⠀⠀⠀⠼⠥⠬⢛⠡⠁⠀⠀⠠⠂⠀⡁
⠀⠉⠁⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠒⠉⠠⠒
⠀⠄⢂⠀⠤⡀⠀⢀⡐⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠈⠀⡀⠀⡀⠀⠀⠀⠀⠈⠁
⠀⠀⡀⠀⢀⠀⠀⠀⠔⠠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠌
⠀⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⡰⠀⠈⡁⢀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠂⠀⢀⠀⡀⠠
⠀⠀⠀⠀⢀⡔⠀⠀⡀⠀⠀⠀⠈⡦⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠄⠀⠈⠀⡀⠀⠀⠀⠀⠶⠀⢀⠄⠀⠂⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣫⣷⣿⣿⣿⣿⣿⣾⣝⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⣿⣿⣿⣿
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣸⣿⣿⣿
⠀⠀⠀⠀⠛⠒⠒⠒⠒⢒⡒⠒⠒⡒⠒⠚⠙⠒⠃⠀⠀⠀⠀⣿⣿⣿⣿⣧⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣶⣭⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠤⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣛⣛⣻⣵⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⣤⣀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠛⠋⠀⠀⠀⠠⡄⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠠⠄⢀⠤⠒⠒⠓⠒⠦⣄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⡠⠞⠁⡀⠢⣸⢠⣰⢀⠀⣹⡄⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⣾⣥⣦⣿⣿⣷⣿⣿⣿⣿⣿⣿⠿⣆⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⣠⣴⣭⣽⣿⣿⢿⣿⣿⣿⣿⡏⢙⣿⣖⣭⡽⠶⠖⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣀⣤⠾⠋⠁⢿⣆⡙⠛⠋⠛⣠⣼⠇⠙⢿⣿⣭⣭⠥⠤⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠉⠉⢛⣿⣷⡀⠈⠛⠧⠖⠖⠺⠕⠋⠀⣠⣶⣟⠉⠉⠀⠀⢀⡄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡦⢠⡾⠟⠋⢙⣿⠞⢦⣄⣀⣠⣴⣤⣶⡂⢋⡹⠛⢷⣄⠀⠀⠘⠇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠉⠀⠀⡠⢿⢯⡀⠀⣼⣿⠋⠀⠉⣉⠿⢯⡀⠀⠀⠉⠂⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠊⠀⠈⢹⣿⠗⠒⠒⠊⠁⠀⠀⠁⠀⠀⢤⠀⠀⠀⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡀⠀⠀⠐⠲⠀⠀⠀⠀⠸⠙⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠈⠓⠶⠊⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠤⠤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠉⠉⠙⡻⢻⣿⠛⠛⠚⠛⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠊⢀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡀⢸⡇⠀⠀⠀⣉⡾⣿⠀⠀⢀⡖⡀⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠡⡇⠀⠀⣼⠃⠀⠀⠐⠋⠁⢻⣇⠀⡾⠉⢠⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠁⠀⢠⡟⠀⠀⠀⠀⠀⠀⠊⠻⣆⠃⡏⠈⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣙⣷⡅⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣦⣤⣶⠶⠾⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁⠀⠀⣀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣤⣾⣿⣷⣤⣿⣿⡋⢹⡄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠻⠃⠀⠀⠛⠃⠀⢰⣦⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢻⣿⣀⣿⣿⡟⢻⣿⣉⣿⣿⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⣤⠀⠾⠇⠀⢠⣤⠀⢶⠆⠀⠀⠀⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣷⣾⣿⡿⠻⣿⣷⣿⣿⣿⣏⣈⣿⡿⡃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡇⠉⠀⠀⣤⡀⠀⠁⠀⠀⠀⠿⠂⢀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⣹⣇⣸⣷⣶⡿⣿⣿⣀⣿⡿⠿⣿⡄⠂⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⣠⣿⠗⠘⠿⠀⠉⢁⡀⠀⠾⠇⠀⣠⡀⢛⣯⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣶⣶⣶⡶⠾⠏⢻⡿⠿⣿⣤⣾⣿⡿⠿⣷⠶⠏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⠉⠉⢉⣉⣠⣧⠀⣠⡀⠛⠃⠀⠀⣀⡀⢉⣅⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⠻⣿⣿⠀⠀⠀⠀⠑⠖⡻⢿⢹⣏⠷⢴⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡅⢰⠀⠀⣸⣿⣿⣿⣷⣋⡥⣀⠆⠀⢂⡙⣥⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⢱⣿⣿⡄⠀⠀⢀⠀⡀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣼⠀⠀⠹⣿⣿⣿⢿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⠹⣿⠀⢠⣬⣥⣥⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢰⠀⢹⣿⠛⠚⠛⢻⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠀⣿⡆⠈⣿⣿⣿⡇⠀⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⢸⡆⢸⣿⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠉⠉⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⣷⣾⣿⣷⣶⣶⣾⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠀⠀⠀⠀⣀⢀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⣿⡟⣿⢹⣻⡿⣶⢏⣿⣯⢿⢹⡇⣿⡹⢸⡿⡎⢻⣟⠿⢳⡧⣿⡇⡾⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢹⣷⡿⠈⢷⠀⣿⠘⣿⠀⢸⣿⡇⣿⡃⢸⡇⡛⠸⢁⠆⡿⢀⣼⢱⡇⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠶⠈⠀⠋⠀⠏⡼⠸⠇⠷⠹⠂⠀⢘⣤⠤⠈⡈⢀⠏⠇⠾⠁⡰⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠀⠐⠀⢀⡘⠢⠀⠊⣐⣡⣤⣑⡺⠊⠀⠀⢑⣅⠀⣀⠀⡄⡾⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢠⡠⠞⠁⠐⣙⡧⠋⢹⠏⣂⣀⡤⠠⠄⠀⠘⣬⡒⢈⠀⡁⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢁⠎⡠⠒⠲⢬⡆⠀⠈⠘⢍⠉⣠⠂⢠⠙⣦⡐⠙⢮⡳⡘⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠦⣎⠁⠠⣀⣤⣧⠶⢣⣶⣾⡿⠋⢀⣯⡀⢐⣿⡇⠀⠱⡅⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣦⣄⠀⠛⣿⣿⣶⢤⣳⡆⢂⣄⣾⣿⡀⢸⣇⡀⠀⠀⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⢿⣷⣤⠿⣿⣿⣷⡀⢹⠋⠑⠻⠟⠁⢿⣿⣿⠀⡴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢍⠀⠀⠘⠻⣼⠟⠗⠈⢂⢁⣀⢀⠄⠈⠈⠻⠾⠃⢠⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠴⠚⠺⣙⠂⠃⡉⠈⡀⠂⠠⡿⢟⠁⠀⠠⡄⠘⡄⢀⡜⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣖⢣⠈⣹⡉⠨⠠⠈⣉⠀⢰⡄⠚⠛⢁⣴⣈⡢⢣⠎⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⣾⡁⠠⡀⠀⠂⠀⠀⢀⣛⠶⠈⠁⠀⣏⣉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢨⠃⠀⢀⠖⠀⠀⠀⡸⠁⢀⠴⢖⠜⢹⠁⠈⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡁⠘⢾⡼⡯⢜⡹⢓⠆⠀⣉⣴⠀⠊⠀⠀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⢰⠘⠿⡅⡘⠓⠊⠁⠘⠹⠀⢀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠐⠂⠀⠈⠰

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

⠀⢀⡠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⠀⠂⠠⠄
⠀⢶⡒⠤⠀⡀⠀⠉⠓⠧⣄⣀⠀⠄⠀⠀⠀⠀⠀⢠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠔⠊⠀⡀⠀⠀⠀⠀⠀⠀⠉⠠
⠀⠀⠙⠦⣀⣁⠢⢄⡀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠠⢳⣀⣠⠃⠀⠀⠀⠀⠀⠠⠐⠀⠉⠀⠀⠀⠀⠀⢀⠠⠄⠊⠀⠀⠀⢀⠢⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⢙⠷⣶⣭⣒⣠⣄⣀⣀⡤⠀⠀⠀⢦⡤⢽⡏⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠰⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣤⣄⢀⣤⣐⢀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠆⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣏⢿⣿⠁⠁⣿⢰⡀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠁⠀⢸⡇⠈⣿⢷⡄⡂⠈
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⡈⠉⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⣀⣠⡀⡀⣀⣀⡀⡀⣀⢀⢀⣀⡀⣀⣀⣀⣀⡀⡀⢀⣀⠀⡄⡀⣀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠻⠿⠧⠀⠧⠟⠷⠣⠻⠟⠿⠂⠻⠿⠼⠽⠟⠟⠻⠜⠼⠿⠇⠸⠽⠤⠇⠿⠿⠼⠷⠀⠨⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠤⢀⣀⣠⢤⠒⠀⣒⣠⠀⠓⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⡐⢞⢕⣭⣹⣷⣮⣀⠠⠙⡟⣺⣿⣿⡶⡖⠂⠦⠑⡄⠀⠀⢀⠤⠄⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠑⠄⠀⠀⢰⣤⠀⠀⠀⠀⡄
⠀⠒⠠⠉⢈⣿⡼⠿⠓⠒⠚⠡⢾⡿⠿⠏⠥⠅⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠝⠒⠶⠠⠤⠀⠀⠀⠀⢯⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣤⡶⣤⠀⠀⠀⢊⠀⠀⠀⠀⢰⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⠸⣶⣶⠶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣴⡖⣾⣏⠁⢹⡴⠟⠁⠀⠀⠀⠀⠁⠒⠒⣂⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠠⠎⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠐⠀⠉⠁⠀⠀⠙⠤⠭⠓⠀⠄⡀
⠀⠀⠀⠀⠀⣠⠂⠁⠀⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠑⢄
⠀⠀⠀⠀⠊⠀⠑⠀⠀⠀⠀⠠⡀⠀⢈⠆⠀⠀⠘⠀⠘⠀⠀⠀⡱⡀
⠀⠀⠠⢣⠀⠀⠀⣀⣀⠤⠤⠠⠄⠄⠤⠀⠄⢠⠄⡤⠀⣀⠀⠀⠢⠨⡄
⠀⢀⠃⢁⠄⠈⠛⠠⠄⡑⠀⠠⡁⣀⢀⣠⡠⡡⢊⠠⢂⠁⠬⠒⠢⡀⢠
⠀⠈⠄⢈⡀⣀⣀⢀⣐⣀⣥⠞⠉⠉⠀⠀⠉⠉⢲⣬⡆⠀⠀⠈⢀⡑⡘
⠀⠀⠀⠐⠅⡒⢤⣰⣡⣡⣻⡄⠀⠀⠀⠀⠀⠀⣸⣏⣍⣕⡤⢒⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠀⠀⠀⠀⠀⠰⠛⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣠⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢶⣛⣭⣿⣳⣶⣪⣳⢶⢞⣻⣲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⣠⣎⠀⠀⢸⢗⡿⣽⠴⠿⣭⣗⣏⢯⡷⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠒⠒⠂⠂⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣾⣯⣾⣼⣧⣀⣀⣼⡻⠿⣾⣷⣶⣿⣿⣿⡿⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⣿⣿⣿⡿⠿⢞⠻⡛⢛⠻⡙⡏⡛⢛⠟⡻⠻⢿⠿⣿⣿⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠋⠉⠁⠒⠤⢑⣮⣲⣧⣧⢧⢧⣯⣊⣴⡊⠥⠐⢊⣉⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠠⠭⠭⣭⣿⣷⡟⠉⠀⠀⠀⠀⠀⠉⣿⣿⣧⡬⠤⠤⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢠⣿⣿⣯⣵⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡃⠀⠀⠀⠀⠀⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠙⠳⠦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢦⡀
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢸⠇⠀⠀⠀⠀⠠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⢀⣠⠶⠛⠋⠉⠉⠉⠛⠶⣄
⠀⠀⠘⠶⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠖⠛⠉⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⢿⠉⠉⠁⠀⠈⣧⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⡴⠞⠋⠉⠙⡆⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⡤⠞⠳⣽⣦⣀⠀⢠⠏⠀⠀⠂⠁⡴⠋⠀⠀⡴⠀⠀⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠀⠀⠀⢀⡞⠀⠰⡄⢸⢁⡴⢂⠀⠀⢀⡟
⠀⠀⠀⠀⢀⡴⠋⠀⢀⣠⡤⠴⠶⠒⠲⠦⠤⣄⠀⠀⠘⢦⠀⠀⡞⠀⢀⡀⢳⡼⠾⣴⣵⠂⢀⡼⠁
⠀⠀⠀⢠⡟⠀⢠⠞⠉⠀⢆⠀⢸⠀⠀⢀⡄⠀⠙⢦⡀⠈⢧⢸⡃⠀⠀⣉⣿⡀⠀⠀⠉⠛⠺⠷⣶⠟⢳⡄
⠀⠀⠀⣿⠀⢠⠏⠀⠀⠀⠈⣦⣼⣀⣠⠋⢀⡠⠄⠀⢣⠀⢸⠈⣇⠀⠀⠁⡰⠛⣦⡀⠀⠀⢀⡜⠁⠀⢀⡇
⠀⠀⠀⣿⠀⢸⡀⠀⢈⣉⣹⠃⠀⠈⢳⠖⠉⠀⠀⠀⡼⠀⣸⠀⠘⠆⠀⢀⠀⠈⣠⠿⣦⡀⡾⠀⠀⠀⡼⠁
⠀⠀⠀⠘⢧⣀⠳⢤⣅⣀⡞⠀⠀⠀⡟⠛⠓⠂⠀⡰⠃⣰⠷⣤⣀⠀⣀⣀⣤⠞⠁⠀⠈⠳⣷⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠉⠛⡷⠛⠋⠛⠲⢤⡀⡗⠲⠶⠒⢋⣠⠞⠁⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠂⠀⠀⠉⣿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣀⣠⣤⠟