Speedy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡶⠷⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠟⠛⠻⣶⡄
⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡟
⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡶⣾⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣠⡿⠁
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⢻⡈⠇⠀⣠⢟⡻⣿⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡿⠃
⠘⠛⠛⢿⣷⣦⣤⣀⡀⠀⣠⡴⢻⣧⣾⡇⠀⠀⢠⠟⢻⡿⡇⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠘⣿⣇
⢀⣠⣤⣤⣿⣿⣦⣉⣹⡿⠷⣤⡀⣿⣿⣤⣄⣀⡏⣴⣿⡁⠀⠀⠀⠈⢿⣦⣤⣶⣾⡿⠿⢶⣿⣿⣆
⠀⠀⠀⠀⠀⠈⠻⢿⣿⡋⠛⠛⢿⡻⣦⣬⡿⠉⣽⣛⣛⣛⠛⠛⠂⠀⠾⢋⡵⠋⠁⠀⠀⢠⡇⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠈⢷⣄⣀⢀⣠⣿⣿⢿⣿⣟⡛⠛⠛⠲⠾⣧⣄⡀⠀⣠⡿⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠀⣩⣿⣿⠿⠁⠈⠁⠀⠉⠛⠻⢶⣾⣤⣭⣥⣶⠿⠋⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠛⢻⣿⠷⣤⣶⣤⣤⣤⣤⣴⣶⡾⠟⠉⠙⠿⣧⣄⠀⠀⠀⠀⠀⢀⣠⣴⡾⠇
⠀⠀⢴⣦⡀⠀⣀⣤⣾⡿⠋⠘⢛⣹⣿⣤⣾⡿⣍⠙⢿⣌⠙⢷⣤⣤⣤⣤⣤⣤⣥⣴⣶⣶⡾⠿⠟⠋⠁
⠀⠀⠈⢳⣌⠻⣍⣈⣻⣿⣿⡟⠉⠀⠀⠀⠙⠻⢶⣤⣤⣿⣷⡄⠈⠛⢯⣍⠉⢉⣵⣿
⠀⠀⢠⡟⣠⠞⠛⠋⠉⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠷⣦⣤⣀⡀⢈⣿⠟⢿⠁
⠀⠀⣿⠞⠁⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠈⠉⠛⠻⢿⣄⢬⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣴⣶⣶⣶⣦⣤⣀⣀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠙⢦⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠉⠉⣍⠛⠛⣿⣿⣿⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣽⣆⣰⡟⠀⠙⠻⢿⣶⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡏⢹⠁⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣿⠀⣷⠸⣷⣶⣶⣶⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⣰⡾⠋⠉⠉⠉⠉⢉⣛⣁⣀⣏⣀⣀⣀⣀⣀⣀⡀⠀⠈⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⣼⣿⣴⣶⠶⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⢀⣠⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠉⠉⠉⠉⠛⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⡄⢤⡄⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣬⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡷⢂⣨⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡄⠠⣀⣠⣶⣄⡀⠀⢰⣿⣿⣿⣿⡿⢟⡱⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣾⣠⠤⠀⠉⣤⣶⣿⡿⠛⢺⡇⠀⣿⠿⣿⠟⢥⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡯⠐⣿⢋⣹⣿⣿⣿⣿⣿⢆⣼⢻⣿⣿⣿⣿⣿⣿⡟⣣⣴⣿⡿⠟⠉⠀⠀⢠⡇⠀⣿⡄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣿⣿⣿⣿⣿⢿⣿⡿⡇⢼⢣⣾⡟⣾⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⢀⠀⣿⣀⢀⣈⣵⣾⢿⣿⣿⣿⡷⠆⠄⠀⡾⠿⢉⣿⡿⠿⣿⣿⡟⠹⢷⡾⠃⠀⠀⠀⠀⡼⠁⠘⡇
⠀⠀⠀⠶⠶⠿⠿⠛⠻⡟⠛⣿⣿⣷⣾⣿⣷⣿⠟⠰⣿⣿⣿⣿⣾⣿⣿⣿⣿⣦⡌⣷⣾⠞⠁⠀⠀⣠⠜⠁⠀⠀⢀
⠀⠀⢀⣠⣤⣴⠶⠶⠚⢻⠉⠙⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣯⠛⠶⠿⠛⠉⠀⠀⠀⠀⢈
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠳⡀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣛⠻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⣿⣿⣿⠟⠻⠟⠻⣿⣿⣿⡟⢻⠂⣿⡟⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⢿⡍⣶⣿⣷⣾⣶⣿⣧⣬⣿⡟⣠⣿⣦⣿⣿⣿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⣼⣿⣻⣿⣶⣶⣽⣿⣿⣿⣿⣿⡿⣿⡇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣻⣿⣿⣯⡉⠻⣿⣏⣴⡿⢋⣼⣿⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠉⠉⠀⣸⣿⣇⠀⠀⠉⠉⠉⠉⠋⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⢀⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣿⣥⣶⣶⣶⣿⡿⢿⣿⣿⣄⡀⠀⡄
⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣥⢿⣿⠿⣿⣿⣦⡀⠀⠁
⣻⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠀⠈⠿⣿⣿⣦⡀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢡⣿⣿⣿⣿⡿⢻⣿⣿⣰⣿⡟⠀⠀⠀⠙⢿⣿⣿⣦⡀⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢸⣿⣿⣿⡿⢱⢸⣿⣿⢻⢻⡇⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣜⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⡀
⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠴⢶⣶⡦⠄⢻⡄⢻⣧⣿⣷⠀⠸⣿⣿⣃⣾⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣥⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣷⣦⡎⠱⠈⢿⣿⣿⣯⠺⣿⣟⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣶⣤⣄⡁⠐⠂⠀⠀⠀⠀⢀⠀⣠⣿
⣿⣇⠀⠀⣀⣿⣿⣿⣟⣛⣿⣿⣇⣼⣿⣀⣤⡐⢠⣿⣿⣿⡆⢿⣿⣿⠋⣀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣷⣶⣦⣤⣷⣶⣿⡿⠛
⣿⣿⣿⣿⣿⠁⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣤⠋⢿⣿⣷⢼⣿⣯⣄⣈⡑⠂⢀⠀⠀⠀⠈⠀⠚⠁⠀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⣿⣿⣿⣿⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣋⣵⣶⣿⣿⣿⠺⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀
⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⢆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣿⣯⠓
⣿⣿⣿⣿⣿⡿⣿⡝⢹⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⡯⢭⠛⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢊⣠⣴⣾⠥⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⠃⠴⣦⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠉⠤⠀⠀⠀⠀⠀⠠⡔⢄⠀⢀⡴⠛⠉⠢⣀⣀⣠⣤⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡴⠀⠀⠀⣀⠈⠀⠙⠾⢤⣤⠶⢚⣻⣉⣀⣤⣤⣷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⡴⠒⠒⠂⢀⡀⠀⠀⣠⠁⠀⠀⠀⠀⢂⠁⠀⠲⠿⠿⠿⢃⣴⠶⠎⠑⠙⡏⠉⣽⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠁⠀⠀⠀⠀⠀⠉⢫⣀⠉⢁⠠⠀⠀⠘⠛⠀⠀⠀⠀⠀⡠⠜⠀⠀⠀⠀⢰⣇⠶⠃⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠊⠀⠀⠀⢀⡞⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣰⠟⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠀⠤⠔⠚⠁⠀⠀⠈⠐⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠀⠀⢀⡀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣎⠉⢉⣀⣠⣼⢄⠀⠀⣿⠐⠉⠉⠙⠛⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠔⠊⢀⠴⠋⠛⠝⣡⠞⠉⠉⠈⠀⢢⡤⠤⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⢠⣞⣁⣠⡤⠖⠋⠁⠀⠀⠀⠀⠀⠈⣷⠀⢨
⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⣀⡜⢧⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢉⣡⡜⠡⠄⠒⠒⠀⠀⠂⠀⠒⠂⢤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠤⣀⣀⣀⣀⣀⣠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠇⠀⠀⢠⠤⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠀⡰⠃⠀⠀⠀⢀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⢛⠉⠉⠁⠀⠀⠈⠉⢋⠉⠠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⢈⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⠤⠬⢍⣉⡙⠛⠻⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣬⣉⡛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠚⠉⠀⠀⡾⠀⠀⠈⢈⣀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢀⠀⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⢠⠔⠂⠀⠄⠀⠀⠀⡀⠀⢸⠀⡀⠈⠀⠤⠤⠄⠐⠒⠒⠒⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⠒⠒⠒⠒⠒⠂⠄⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣤⣠⣾⣿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⣿⣿⣿⣿⣤⣀⡀⠀⠀⣀⣀⢀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⣿⣿⣇⣾⣿⣿⣿⣶⡞⡉⠀⠋⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢀⠐⠁
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠸⠏⢘⣿⣿⣿⣿⠟⠛⢿⣅⠀⠀⠀⠑⠀⡀
⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠘⠿⣏⢻⣷⣶⣦⠝⠻⠃⠂⠀⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣰⡤⠀⠀⠀⠀⠀⠈⣉⣀⠀⠀⠀⠀⠀⣀⡀⠠⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⢀⠦⠼⡥⠬⠿⣶⣖⡊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⡄⠀⠀⠠⠄⠀⠀⠄⠈⠂⠀⡐⠀⠀⠀⢀⠀⠐⠠⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠀⠠⠄⠠⠀⡀⠁⡀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀⠤⠤⠤⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣤⡀⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡐⠂⠙⠦⠛⠳⣦⣤⣶⣿⣿⣏⠹⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⣷⣷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠒⠈⠻⣿⠿⣿⣿⣿⣿⣿⠟⢺⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠊⠄⠀⠀⠀⠰⣿⠗⣀⣤⣹⣿⢿⣿⠆⡀⠁⠀⠀⣀⣴⣾⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠛⠁⠈⠁⠀⠀⢀⡀⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠤⢤⣠⣴⠚⠻⣆⣀⠀⠀⠀⠀⢹⡿⠃
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⣀⠀⠀⢀⠀⠀⢀⣴⣿⠏⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠒⠂⠀⠐⠠⠂⠀⠀⠀⠐⠀⠄⠀⠀⠠⠀⠈⠋⢁⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⢻⡟⠛⠷⣦⡀⠀⠀⠀⣠⡾⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠻⠂⠀⢀⡀⠀⠀⠟⠻⣦⠀⠀⣿⡁⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⢸⣇⣀⣀⠀⢀⣻⡆⠀⠘⢷⣤⣤⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠋⠀⠀⠀⠈⠛⠛⠛⠀⠙⣻⡇⠀⢀⣴⠾⠿⠿⢶⣄⠀⠀⠀⢠⣴⡶⠶⠿⠛⠛⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⣠⠀⠀⠀⠀⠀⠀⣤⣰⡟⠀⢰⡟⠁⠀⠀⠀⢀⢹⣟⠛⠿⢿⡇⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣰⣆⣀⣤⡾⠋⠀⠀⢸⡇⠀⠀⠀⠀⠛⠛⠛⠿⠶⠾⢿⣤⡀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⣷⠀⠐⠻⠷⠶⠶⣦⣤⣤⣤⣀⣼⡇⠀⣀⣀⣠⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠘⣿⠀⠉⠹⠿⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⣦⣄⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠀⠀⢀⣴⠟⠀⠀⣈⣽⠿⣦⣄⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⢀⣠⡾⠛⠁⠀⠈⣿⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⣀⣠⣴⠟⠋⠀⠀⠀⠀⠀⢸⣇⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⠁⢠⣴⠶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣬⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣴⡶⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⢀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠙⠲⡄
⠀⠀⠀⠀⠀⠀⢀⣜⢁⣴⣦⣿⣷⠀⠀⠀⠈⠻⣿⣿⠿⠿⠿⣿⡿⠿⢿⣿⣧⠀⠀⢹⡀
⠀⠀⠀⠀⠀⣴⠉⠉⠛⠿⣶⣭⡉⠀⠀⢀⡤⢚⠭⡅⠀⠀⣀⠭⢭⡓⠀⠀⠉⠳⣄⢸⠅
⠀⠀⠀⠀⠀⣿⣶⣦⢀⣾⣿⡟⠀⠀⡠⠋⢰⢁⣤⡇⠀⡔⢁⣄⢠⠇⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⣼⣿⡟⢿⣿⣿⠏⠀⠀⡴⠁⡀⠉⡾⠿⠀⢸⣀⣿⡿⠃⠀⠀⠀⠀⠀⢼⣿⣿⠖⠒⢦
⠀⠀⠀⠘⣽⣿⢧⣿⣿⠏⠀⠀⢰⠃⠚⡁⠀⠓⠄⠀⠀⠁⠀⠀⠀⠀⣠⢄⠀⠀⠀⠙⠁⠮⡀⢸
⠀⠀⠀⣰⣍⡉⡿⢿⡃⠀⠀⠀⢸⠀⠀⠐⠠⣀⠀⠀⠀⠀⢀⣀⠤⠊⣡⡎⠁⠀⠀⠀⠀⣨⡴⠋
⠀⠀⢠⠷⠦⣸⡀⡰⡇⠀⠀⠀⠘⣇⠀⠀⠀⠈⢫⣍⣉⣩⣤⣤⣶⣿⠟⠀⠀⠀⠀⢀⣼⣁
⠀⠀⢸⢷⣄⡼⠁⢼⡀⠀⠀⠀⣀⠼⠣⡀⠀⠐⠫⣉⣛⣛⣿⡿⠟⠁⠀⠀⠀⠀⣠⠊⠀⠈⠉⠓⠦⣄⡀
⠀⢰⢫⣷⡿⣷⡴⠃⢨⠛⠛⠉⠁⠀⠀⣿⢷⣤⣀⠀⠈⠉⠀⠀⠀⠀⢀⣠⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢄⡀
⢰⣿⣿⡟⠀⠙⢦⠔⠁⠀⠀⠀⠀⠀⠀⣿⢼⣧⣈⡉⠓⠒⠒⠒⠒⠋⢉⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢦
⠈⠛⠋⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⢀⣀⣿⣤⣷⣿⣽⣿⣿⣿⣿⣿⣿⡯⣇⣸⣧⠀⠀⡴⠒⠒⠒⠶⠶⠤⣤⣄⠃⣠⠷⢤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁⠸⣇⣷⣺⣿⢻⣿⣿⣛⣻⣿⡿⣽⣿⣙⣶⣾⠇⠀⠀⠀⠀⠀⠀⡸⢉⠉⠁⠀⢀⡇⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡼⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⠀⠀⠀⠀⠀⠀⠀⣳⣾⡇⢀⣤⠎⢠⡶⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣷⣯⣽⣿⣭⣯⣿⢷⣿⣿⣿⢧⡀⠀⠀⠀⠀⢠⣎⡀⠳⠟⠉⢀⡠⠚⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⢿⢻⣟⣻⣿⡾⣿⣿⣿⣻⣭⣿⣿⡷⠻⣄⠀⠀⠀⢸⣼⡿⣆⠀⢰⡿⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⣿⣿⣯⣿⣿⣮⣭⣿⣽⣽⣽⣿⣟⣲⠿⣮⣷⣽⡄⠀⠀⠸⣯⣻⢿⡖⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⣟⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣞⣟⣿⡄⠀⠀⠈⠙⠾⣧⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣾⣯⣭⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣭⣥⣧⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢀⣤⣶⡡⠔⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢈⣴⡿⡄⢠⣾⢟⣼⣿⣀⡤⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠴⠾⠟⣡⣾⠁⣿⣿⠿⠟⠛⢩⢰⣟⡀⣀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡀⣠⠞⢀⣤⣸⣿⠀⠈⠓⡂⠤⠿⣿⣿⣶⣏⣥
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣴⣤⣧⠈⣟⢸⡾⣿⠏⡝⠁⠀⠀⠀⠀⠀⢙⠻⡅⠙⢿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣶⣾⡟⠁⢿⣤⣾⣿⣿⣭⢹⣶⣤⡀⠀⠀⢀⣀⢨⠀⠘⠂⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠜⣿⣛⣭⣖⣯⣽⣇⢿⣿⣿⣶⡄⣾⠋⠸⠃⠀⠀⠀⣡⣀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡞⣷⣍⡻⣾⣿⣿⣿⡇⠹⠀⣎⠀⠠⠀⠈⠋⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣜⢿⣿⣿⡿⡿⣷⡇⠀⠸⣿⡢⣴⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣮⣿⣷⣽⢿⣿⡆⠀⠀⡀⠀⠉⠻⣿⣿⣯⣉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣍⡻⣿⡟⣷⠒⠈⠀⠀⠀⠀⠈⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣧⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠃⠘⠙⠂⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⢨⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠎⠀⠠⠄⠀⠀⠀⠀⠀⠀⢠⠂⣀⠄⠀⠀⡀
⠀⠀⠀⡌⠁⠀⠀⡄⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⠜⠋⠀⠐⠀⠀⠁
⠀⠀⠀⠀⡀⠄⠊⠒⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠄⠀⠄⠌⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠲⣍⠳
⢀⠊⢉⠉⢳⡀⡤⡄⠀⠀⠀⢠⠊⠁⢉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⠀⠀⠀⠀⠊⠁⠔
⠸⣀⣈⠤⠈⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⢊⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠄
⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠄⢀⣄⠀⠀⡀
⠀⠀⠀⠀⢄⠀⠀⠀⠀⢼⢀⠀⠀⠐⠄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠀⠀⠀⠉⠂⠀⠰
⢠⠂⠛⢦⠀⠉⠀⠀⠀⠈⠀⠉⠒⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠻⣄⣠⡾⡁⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⠀⠀⠀⠐⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠙⣷⠀⣀⡠⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⡽⠃⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠞⠋⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⢛⡛⢿⣦⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⡡⠒⠁⠀⠀⠀⠈⢻⡝
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠉⠙⠻⢷⣄⡀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠓⠾⣯⠔⣋⠀⠐⠂⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠐⡠⢀⡾⠋⠀⠀⠀⢀⢀⡀⠀⠑⢍⠛⠳⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠝⠃⠐⠆⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠃⢡⠏⠀⠀⠀⠀⠀⠈⠑⠈⠢⣀⠈⢆⠜⠁⢀⠀⣀⠀⡀⠠⠐⠈⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⡾⠀⡘⠀⠀⠈⠀⠀⠀⢀⣀⠤⠤⠐⠒⠈⠉⠉⠉⠙⢳⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠐⠉⠀⠀⠘⠀⢠⠁⠀⠀⣀⡤⠔⠊⠉⠀⠀⠀⠀⠀⣀⣁⠀⠀⠀⣸⢀⣴⠞⠁
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢾⠉⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠉⠛⠲⢶⣟⠋⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠷⠦⣀⣀⣀⡤⠚⠁⠐⠀⠀⢀⠈⠁⠀⠈⠓⢦⣀⣠⣤⣬⡓
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⣤⣤⣤⣤⢤⡖⢹⡄⠀⡜⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⣴⣿⣿⡁⣹⣿⡶⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡇⠀⠀⠀⠀⢋⡠⠒⠢⣀⣀⠀⠀⠐⠒⠀⠀⣀⡿⢿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠠⡄⠀⠀⠀⠀⢸⣤⠤⠤⠌⢁⣈⡏⠀⣀⢹⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣿⢦⡀⠀⠀⢸⡁⣠⠖⢁⣠⠄⣱⣦⣩⡏⠀⠀⠀⠀⠀⠀⠀⣠⢦⡀⠀⢠⠇⣽⢛⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠳⢦⡘⣇⢹⠈⠻⣦⠀⠢⠍⠁⠀⠀⠁⢰⢿⠈⢦⣄⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⢀⡞⢠⠇⡼⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢣⡾⠋⠉⠙⢶⡄⢻⣿⠈⡇⠀⠈⢷⡀⠢⠤⣠⡤⠴⠃⠈⠓⠦⣌⠙⢦⡀⠀⠀⠀⢠⡟⠀⢻⠋⠀⠁⠜⢡⣾
⠀⠀⠀⠀⢀⣤⣄⣤⠴⣋⡴⠋⠀⠀⠀⠀⠀⢻⣄⠏⢸⠃⠀⠀⠈⣷⠀⠀⡏⠀⠀⠀⠀⠀⠀⠸⣇⣀⡇⠀⠀⠀⢸⡇⠀⠀⠃⠀⠀⠐⢁⡟
⠀⠀⠀⠀⠈⠛⠒⠒⠛⠉⠀⠀⠀⠀⣀⣠⡤⠺⣧⡠⣿⠀⢀⡤⠖⢿⠀⠀⣟⡛⠒⠒⠄⠀⠀⢠⢟⣉⢳⠀⠀⠀⢸⡇⠀⠀⠀⢀⡠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠺⠭⢣⡿⠀⣿⡴⠋⠀⠀⠙⠖⡖⠛⢄⠀⠒⢆⠀⠀⣾⡘⠸⠛⠀⠀⢀⡟⠀⢰⠶⠚⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⡀⠀⠘⠳⠞⣹⠃⣸⣿⡄⠀⠉⠉⠺⡄⢆⠀⡈⠀⠀⠈⢄⠀⠈⠃⠈⠀⠀⢀⡞⠀⣴⠇
⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠈⠉⠛⠲⢤⡾⠁⢾⠟⣹⠁⠀⡴⠂⠀⠈⠉⠉⠀⠀⢂⠀⡈⢦⠀⠀⡼⢻⢀⡞⠀⣸⠃
⠀⠀⣴⠋⢀⠄⠀⠒⠀⠐⠀⠤⣄⣀⣀⣀⠠⠔⠁⢠⠏⠀⠘⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠈⠌⠳⣴⢃⣾⠋⠜⣰⠃
⠀⣼⠃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠓⠊⠀⢚⡡⠞⣹⠃
⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢨⡇⠀⠀⠀⠀⠀⠋⢀⡖⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠚⠦⠤⠤⠄⠀⠀⣀⣀⣤⡤⠤⣤⠤⠤⠶⠞⠁
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠙⣆⡀⠀⢀⣀⡠⠴⠒⠋⠉⠀⠀⠀⠀⠙⡄
⠸⡆⠀⠀⠀⠀⠀⢀⣴⠞⠋⣹⠀⢾⠋⠛⠒⠒⠚⢋⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠙⢀⡀⢀⣀⠴⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⠀⢤⣀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠒⠒⠒⠒⠲⠶⠤⣄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⣀⠀⠀⠀⠈⣁⠰⠖⠐⠚⠉⠉⠉⠈⠉⠀⠀⠒⠒⠒⠒⠀⠉⠛⣠⡶⠖⠛⠉⠉⠉⠉⠉⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣯⡥⠖⠀⡀⠀⠀⠀⣀⠰⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠖⠚⠛⠉⠉⡀⠋⣀⠴⠒⠶⠃⢀⡄⢊⡁⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠐⣠⣄⣀⣀⡤⣄⠾⠋⠁⠀⠀⢰⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⡆⢸⡆⢀⠴⠂⠀⠀⢀⣠⠞⢀⣀⠀⠈⣉⣥⣆⣀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⣿⣄⢻⡾⠅⠈⣀⠤⠄⠀⠀⠀⠀⠀⣸⣴⣿⠿⢛⣿⣿⣿⣷⣥⣤⣴⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠻⣌⠹⣄⢈⠿⢏⠑⠆⣠⡟⡰⠲⡜⢶⡶⠋⠀⠘⣿⣿⣷⣉⣹⣯⣿⣿⡿⠿⢿⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡈⢖⠉⠠⡀⠈⠳⣏⣾⡗⠀⠐⠉⡈⠁⠀⢀⣼⠂⠉⠛⠿⣿⠿⠟⠋⠀⠀⠈⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠣⡀⠉⣢⣴⠶⢶⣤⣥⠀⠲⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣤⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡟⠛⠉⠉⠉⠀⣸⠉⠉⠓⠛⠳⢶⣟⣈⠀⠲⠀⠀⢀⠀⠀⠀⠀⠀⠐⠺⡅
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠸⠃⠀⢠⠄⠐⠂⠐⠦⠄⠶⠄⠴⢾⣿⣶⣶⡶⠀⠲⠶⠶⣴⣶⣶⣶⣶⣶⠶⠶⠶⠶⠶⠶⢶⠶⣦⠂⠄⠠⠄⠠
⠀⠒⠒⠒⠒⠦⠼⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⣀⠏⠉⠍⠉⠀⠀⠀⠀⠀⠀⠀⠸⢏⠀⠀⠀⠀⠀⠛⠂⠘⠃⠘⠳⠶⢶⡴⢶⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠁⠀⠈⠙⠉⠙⣏⠀⡀⠀⢱⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⢀⣀⣀⠴⣧⠀⠡⠀⠈⢡⡙⠲⣤⣦⣄⣀⡀⠀⠀⠀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀⠤⠞⠀⠀⠀⠀⠀⢁⠀⠈⠉⠙⢛⣿⠛⠋⡃
⣀⣀⣀⣀⣠⡴⢦⣤⣀⣀⣀⣀⡀⠤⠛⠻⣿⣿⠀⠀⠀⠀⡀⠀⢆⠀⣆⠀⠀⠛⡿⠋⠀⠈⠣⠘⠛⢠⠶⠀⢀⣀⣀⣀⣀⡤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⠑⠂⠐⠃⠀⠈⠀⠘⢳⣖⠊⣀⠀⠠⠌⠀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⠤⠤⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠄⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠀⡀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠸⠃⡐⠁⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⠀⢀⠄⠀⣠⠊⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⣠⠃⠀⡠⠃⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡖⠁⢀⠄⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠤⠸⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⣻⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣽⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⣤⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠘⢾⠋⠀⠀⠀⠀⠀⠉⢹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢰⡀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢀⣄⣀⣴⣾⣿⣎⣵⣾⣿⣷⣄⠀⠠⣄⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢻⣧⡀⠀⣿⡟
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⣿⠀⣿⣿⠇⣤⣿⠇
⠀⠀⠀⠀⣼⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣾⣿⣼⣿⣿⣾⣮⡟
⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠈⠛⣽⣿⣿⣿⣿⣿⡿⢋⣙⡻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠟⣿⣿⠟⠁
⠀⠀⠀⠿⠇⠀⢿⣿⣿⣿⡇⠘⢿⣿⣿⣭⣙⣻⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡟⢡⡿⠿⣦⢀⣃⣾⠛⢿⣄
⠀⠀⠀⢀⣀⣀⣀⡈⠹⣿⣧⠲⣄⣉⡉⠋⠉⡁⢠⣶⣿⣿⣷⣦⣄⢀⣴⣾⣿⣿⣿⢃⢸⡇⠀⠈⣿⣿⣿⣦⣾⣿⣓⣲⣶⣄⡀
⠀⢀⠔⠁⠀⠀⠀⠙⢦⡈⢿⣧⡽⠥⠉⠋⠁⠀⣿⠏⠀⠀⠉⠛⢿⣿⠟⠛⠛⠋⠻⢁⣶⣿⣦⠘⢿⠟⠋⣱⣿⣵⣿⢿⣿⣿⡇
⠀⡘⠀⠀⠀⠀⠀⠀⠈⢧⠈⢿⣿⣦⣄⡀⠀⠀⣿⠀⢀⣤⠒⠀⠂⣙⣦⣀⣐⣢⣴⣿⣿⡗⠀⢀⣶⣾⣿⣿⡿⠟⣡⣾⣿⠟
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⢧⠘⣿⣿⣿⣿⡷⣦⣽⣷⠿⠿⢧⠀⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠛⠿⡿⣿⠿⠛⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣿⣿⣿⣸⣿⣿⣿⡔⣦⣘⣛⣿⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠘⢿⣧⣹⣟⣻⣿⣶⣿⣿⣿⡿⠀⠀⠀⠀⣿⠟⣛⣿⣿⣿⣿⣋⠉⣀⣿⣧⣤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠘⢷⣄⠙⠿⢿⣿⠉⠙⠉⠉⠉⠀⠀⠀⠀⡰⠴⠋⢀⣽⢿⡟⢻⣿⠆⠋⠛⠻⣥⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣤⣙⠛⣷⣶⢷⠀⠀⠀⡀⠀⠀⠀⠀⢠⣷⠶⡾⠟⠁⠸⣇⣠⠋⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠰⢶⣶⣿⣿⣿⣿⣿⡿⠋⠀⠈⢣⡀⠀⠻⣶⣤⣤⣤⣾⠗⠋⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠋⠀⣀⣤⠤⠈⠛⠶⠦⠤⢬⣭⣿⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⡀⠀⣉⠀⠀⠀⠀⠀⣀⣤⡉⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡶⠀⠀⠘⢿⣿⣷⣦⣤⠶⠶⠶⠶⣶⣦⣤⡐⢢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡶⠶⠶⢞⣻⣿⣿⣧⣤⡀⠀⠀⠙⠻⢿⣿⣿⣷⣶⣶⣴⣿⡛⠁⠉⠐⠂⠠⠦⢦⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣯⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣠⣄⣀⡉⠽⣿⣿⣿⣿⣿⣿⡿⠶⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠿⠻⠿⣿⠿⠛⠿⠟⠁⠀⠀⠀⠀⠉⠉⠉⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⣞⣛⣿⣿⣿⣿⣿⣛⣒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢹⡦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡇⠀⠙⢦
⠀⠀⠀⣶⢶⡶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠙⣮⣽⡖⠛⠛⢛⣿⣛⠛⠿⢧⣷⡀⠀⠈⡇
⠀⠀⢀⣿⣿⣿⣿⣷⣿⣿⣹⠇⠀⠀⢀⠔⡫⠖⢆⠀⠀⢀⡤⠶⢮⠁⠀⠀⠈⠙⢦⣀⡇
⠀⠀⢸⢹⣿⣿⣿⣿⣿⣿⠏⠀⠀⡴⠃⢰⢁⣤⡜⠀⡰⠃⣠⡀⡼⠀⠀⠀⠀⠀⢸⣿⣧
⠀⢀⣾⣬⣉⡙⠻⢟⡿⡏⠀⢀⡞⠀⠀⠉⡾⠿⠁⠠⡇⣸⣿⡟⠁⠀⠀⠀⠀⠀⢼⣿⣿⠷⠒⠲⡄
⠀⢸⣉⡉⠛⠛⠻⠷⣾⠀⠀⡼⠀⠚⠁⠀⠣⠄⠀⠀⠉⠀⠀⠀⠁⠀⣤⣄⠀⠀⠀⠙⠋⢰⣃⠀⣿
⠀⠙⢾⠺⠽⣶⣶⣤⡏⠀⠀⣇⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⢁⡎⠁⠀⠀⠀⠀⠀⣊⡴⠃
⠀⠀⢸⡀⢀⡴⠚⠉⠀⠀⠀⢹⡀⠀⠀⠀⠉⢖⠒⠒⠒⢊⣉⣡⣴⣾⡟⠀⠀⠀⠀⠀⢠⠏⠁
⣴⢊⠛⡶⠚⢦⠀⠀⠀⠀⠀⠀⣳⡄⠀⠀⠀⣜⠛⠿⠿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⡴⠋⠙⠓⠦⣤⣀
⣏⣁⣉⣧⣠⢟⠆⠀⠀⠀⢀⡴⠁⢸⣶⣄⡀⠈⠉⠒⠒⠚⠋⠉⠀⠀⠀⠀⢀⣤⠊⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⢞⣠⣀⣹⠀⠘⡦⢤⡤⠖⠋⠀⠀⠘⣿⣸⡟⠲⠦⣤⣀⣀⣀⣀⣤⠤⠶⣾⢫⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢈⠇⠀⡽⣶⡼⠁⢠⠀⠀⠀⠀⠀⠀⣿⠙⣿⣶⢤⣤⣄⣀⣀⣀⣠⣤⣴⡟⢳⡇⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⡰⠙⣆
⢸⠀⢠⠇⠙⠳⣴⡃⠀⠀⠀⠀⠀⢀⣿⣷⣿⣾⣿⣿⣾⣷⣶⣾⣿⣿⡿⣷⣾⢿⡀⢀⣿⠉⠉⠉⠉⠉⠙⠛⢲⠮⣤⠴⠋⠙⡶⢦⣀
⠸⠤⠞⠀⠀⠀⠀⠉⠓⠲⠶⠶⠚⠋⠙⣷⢻⣿⣿⣿⣿⣿⣿⣿⣿⢸⡯⣿⣿⣿⣽⣻⡇⠀⠀⠀⠀⠀⠀⠀⡞⢠⡀⠀⠀⢀⠇⠀⣙⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡻⢾⣻⣿⣿⣿⣯⡷⣿⣷⣿⢿⣟⣯⣾⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣅⣀⡴⣾⣶⣿⡿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣾⢷⣾⣿⣿⡿⠶⣿⣟⣻⣽⡶⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢀⣾⡿⣿⣭⣾⣿⡿⢩⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣿⡿⣿⣿⣿⣿⣿⣟⣟⣛⣻⣯⣿⣿⣿⣷⢚⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣿⢿⣿⣿⡾⠟⡠⠊⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢾⣿⣿⣿⢽⣻⣿⣽⣽⣿⣿⣯⣽⣿⣛⣿⢿⣿⣿⢞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⡿⠛⢉⡄⠊⠀⠀⠀⣸⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⢿⢏⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⣝⡻⡞⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠑⢆⠀⣀⠔⢡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣿⣟⣋⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣙⣋⣩⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣉⣥⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠈⠐⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠋⠀⠀⡤⠊⠀⠀⠠⠤⡐⠀⠾⠓⠒⠒⠈⠀⠀⢀
⠁⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈
⢤⠖⠀⠀⠀⢠⣀⠄⠀⠀⠀⠌⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⣄⠀⠀⠀⢶⠄
⠀⠀⠀⠀⠀⠋⠀⠀⠔⠛⠀⠀⠀⠀⠀⠢⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠱⡁⠀⠀⠈⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⡄⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢆⢤⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⡈⠄⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⡎⠂⠀⠀⠀⠀⠀⠀⢄⡇⡀
⠀⠀⠀⢆⠀⠁⣦⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⠇⠀⠀⠀⡇⠀⢀⠀⠈⠃⡃⠀⠀⠀⢀
⣄⠀⠐⢾⡔⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⢣⠃⠀⡠⠂⠁⠀⠀⠀⠀⠀⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⢀⡴⠀⠀⢄⡀⠀⠈⠀⣠⡆
⠀⠀⠘⠀⠘⠀⠀⠀⠀⣠⠀⠀⠀⠇⠀⠀⠀⢀⠀⠊⠙⠀⠀⡠⢺⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⠀⢀⣀⡊⠔⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠝⠋⠀⠀⠀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡠⠊⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠐⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠞⠋⠀⠀⠰⠀⠀⠀⠘⠄
⠐⠠⢀⣀⠀⠀⠀⠈⠉⠉⠁⠌⢠⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠀⠀⢀⡀⠀⠀⡀⠀⠀⡀⠈⠚
⠀⠀⠀⠀⠀⠠⠄⡐⢀⣀⡀⠀⠀⠑⠀⠀⢠⣧⠀⠀⠀⠀⠀⠐⠢⢤⣤⠀⡠⠋⠀⡠⠊⠇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠑⠹⠯⡁⠀⠀⠀⠀⠀⠈⠃⢀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠈⣢⣀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠈⠗⠀⠀⠀⠀⠀⠉⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⡀
⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣐⣛⣛⣉⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢾⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⣼⣇⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⣸⠀⠸⣧⣸⣦⡀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣠⠀⣿⣿⣿⡇⢰⡀⣷⡀⠀⠀⠀⠀⠀⠀⠀⣰⣏⣼⣿⢰⡄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⣷⣾⡟⣼⠁⣿⣿⣿⡧⣸⣴⠀⢀⣼⣿⠃⢠⡆⣿⣤⣿⣿⣿⣀⣾⡷⣸⣷⠀⣦⠀⠀⢀⠀⣼⣿⣿⣿⣿⣎⣿⣾⣿⣦⠄
⠀⠀⠀⠀⣄⠀⣴⣿⡇⣿⣿⣿⣿⣿⣤⣿⡿⣹⣿⣿⣿⣷⣿⣿⣿⣴⣿⣃⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣄⢷⣌⣷⣿⣿⣿⡟⣹⣿⣿⣿⣿⣿⣀⣦
⠀⠀⠀⠰⣿⣿⣿⢻⣷⣟⠹⡙⢿⣿⣿⣿⠰⣿⣿⠟⣿⣧⣿⣿⠙⢿⢻⣿⣿⡏⣿⠃⢿⠙⣿⡟⠠⣿⣿⢹⣿⡿⣿⣿⠻⣿⣿⠐⣿⣿⡿⠃⠟⢛⣿⣷⠏
⠀⠀⠀⠀⠀⠉⠉⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠉⠁⠈⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⢰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠀⢄
⠀⠀⠀⠀⠀⠀⣃⢣⠠⣀⠄⠠⡀⠀⡠⢔⣲⣦⣤⣭⣭⣭⢭⣴⣷⣌⠀⠀⠀⠨⠃⠰⠀⢄⣀
⠀⠀⠀⠀⠀⠘⠹⢸⢰⠋⢀⡐⡡⣫⣾⣿⣿⣿⣿⣿⣿⣯⣞⣾⣿⣿⣷⣄⣀⡀⠈⠀⠀⠀⠜⠁⢢
⠀⠀⠀⠀⠀⢠⠀⠸⣶⠐⠃⣔⠱⠷⠿⠻⣟⣛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢖⡒⠒⠚⠲⠤⣄⣀
⠀⠀⠀⠀⠀⣺⣆⠀⢧⠳⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⠭⣐⠀⠠⢀⠀⠀⠀⠀⠀⠠⠄⠉⠒⢄⠠⣀⠀⠉⠙⢦⣄
⠀⠀⠀⠀⠀⣸⡌⠙⠆⠀⠙⣦⢄⣀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣀⣙⣦⡀⠊⠐⡄⠀⠀⢀⣀⡘⣄⢀⣱⣌⣳⢰⢻⣿⢽⠆
⠀⠀⠀⠀⠨⡀⠘⢦⡀⠀⠀⢸⣆⡀⠉⢺⣿⣄⠢⡀⠀⠀⢠⡟⣽⣯⠁⠀⣨⠀⠙⣻⣭⣭⣥⣽⣤⣴⢾⡧⠸⠼⢶⡟
⠀⠀⠀⠀⢀⡩⢄⢸⣯⠲⣽⣾⠷⣿⡦⢸⣿⣿⣷⡐⣆⣀⠈⠷⠷⠶⠂⢠⠎⠀⢰⣿⣿⣿⣿⣿⣿⡏⣸⣷⢣⡿⠛⠃
⠀⠀⠀⠀⠁⢠⠚⢸⣸⡀⡟⠿⠸⣿⣿⡦⡿⢁⣼⣻⣆⠠⡙⠀⣒⣢⠜⢉⠈⢠⡘⠟⠛⠿⣻⠟⣋⣼⣯⣿⣯⣀
⠀⠀⠀⠀⡄⠨⠄⠀⡜⣷⣷⣌⣤⠘⠓⢲⣿⡿⢿⣿⡿⢷⣬⣽⣽⠶⠶⠶⠿⠤⢭⢭⣭⢠⣴⣶⣿⣿⣿⡇⢹⣿⣿⡄
⠀⠀⠀⠀⠁⠑⠀⠄⢳⣿⣿⠈⢪⣢⣹⣿⠋⠞⠿⢿⡇⠈⠿⠛⣿⣤⢻⣿⣿⣿⣋⠐⡏⠣⠧⠿⡟⣹⣿⡇⡘⠀⢉⡇
⠀⠀⠀⠀⠀⢸⠀⡄⠘⣿⣿⣠⣾⣽⣿⡏⠀⢀⣇⣼⠇⠀⠀⠀⣯⣶⣤⣭⣤⠖⠀⠐⠈⢲⣾⣿⣶⡿⠋⣠⣣⣤⡾
⠀⠀⠀⠀⠀⠈⠂⠓⠤⠻⠿⠿⠛⠛⢻⣷⣾⣷⡿⠋⢠⢢⣠⡞⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⠿⣿⣿⣶⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣤⣶⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⣠⠴⠒⡒⠒⢤⡀⠀⠀⠀⠀⡤⠼⠥⣄⠀⠀⠀⠀⠀⡟⣟⠀⠀⠀⠀⢀⡤⣒⣶⣿⠶⢝⣆
⠀⠀⢰⡥⠤⠤⠤⠤⠤⣽⠀⠀⠀⢺⡭⠭⠭⣭⠇⠀⠀⠀⢸⣓⠻⡄⠀⠀⢠⢫⣮⣅⡿⡿⣟⡯⡅
⠀⠀⠸⣏⣙⠾⠭⢞⣉⡿⠀⠀⠀⠀⡇⠀⠀⡞⠀⠀⠀⠀⢻⡸⢂⡇⠀⠀⢸⢸⢉⣏⠛⢽⣆⡇⡇
⠀⠀⠀⠓⠚⠢⣠⠞⠓⠃⠀⠀⠀⠀⣇⠀⢀⡇⠀⠀⠀⠀⣸⠀⠸⡇⠀⠀⠀⠣⣙⠭⠾⠶⣊⠜⠁
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⢀⡴⡟⣇⡟⡷⣄⠀⠀⠀⠀⠀⠀⡟⡇⠀⠀⠀⠀⠀⢀⡠⠤⠦⠤⡀⠀⠀⠀⠀⢀⡴⠋⠻⣷⡄
⠀⠀⠸⡍⠉⠉⠉⠉⣹⠀⠀⠀⠀⢸⠛⠛⠛⢳⢤⡀⠀⠀⢯⠤⡤⠤⡠⢬⠇⠀⠀⠀⡞⠀⠀⢀⡼⠁
⠀⠀⠀⣇⠀⠀⠀⠀⡇⠀⠀⢀⣀⣸⣄⣀⣀⣿⣊⡀⠀⠀⠘⡆⢇⢠⠇⡞⠀⠀⣠⣴⡹⠶⠒⠉
⠀⠀⠀⠘⠒⠒⠒⠚⠁⠀⠀⠀⠓⠶⠶⠶⠶⠶⠚⠁⠀⠀⠀⠙⠚⠚⠚⠁⠀⠀⠙⠚⠀⣠⣄⡀
⠀⠀⠀⠀⠀⣀⠴⡀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣄⠀⠀⢠⣞⣓⣞⣳⡄
⠀⠀⠀⡼⠭⠤⠤⠽⡄⠀⠀⢰⠋⠉⠛⠉⠙⠉⠉⢳⠀⠀⠀⠀⠀⣀⡞⠑⠺⡀⠀⠘⡟⢛⡻⢿⠃
⠀⠀⠀⣟⢲⡚⢲⠛⡇⠀⠀⣿⡯⢽⡯⠟⢭⣿⠽⣿⡇⠀⠀⢀⠴⣅⠈⠓⢼⠁⠀⠀⢸⡥⢒⡟
⠀⠀⠀⢸⡀⡇⡟⣸⠀⠀⠀⠙⠛⠿⠭⠭⠭⠽⠟⠛⠀⠀⠾⣙⣆⣈⣧⠴⠋⠀⠀⠀⠀⢟⣹⠁
⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⣴⢒⣟⣲⡀⠀⠀⣠⠒⢄⣠⠒⣄⠀⠀⠀⣾⢿⠆⣜⣳⡄⠀⠀⠀⠀⠀⣴⣽⡷⠦⡀
⠀⠀⠀⢀⡴⠋⠀⡯⢽⣻⠃⠀⢸⠁⡸⠈⡇⡹⠈⡇⠀⠀⣿⠸⣤⠏⠘⢦⠀⠀⢀⡤⣪⣾⠞⠀⢀⡗
⠀⠀⢰⠋⠀⣠⠞⣠⠞⠁⠀⠀⠈⢦⠃⡰⢧⠃⡴⠃⠀⢸⠁⠀⢹⠀⠀⢸⠀⠀⢗⡿⠚⠁⠀⡠⠊
⠀⠀⠀⠑⢞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⢸⣀⣀⣸⣀⣀⣸⠀⠀⠈⠳⠤⠴⠊

⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠇⠀⠀⠀⢂⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠈⠠⠂
⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠀⠀⠄⠀⠀⠀⠀⠘⡀⠀⠀⠄⠀⠀⠀⠀⠆⢀⠠⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣤⠀⠀⠀⠒⠂⠀⠀⠄⠀⠀⢠⠀⠀⠀⠀⠈⠀⠀⡄⠂⡄⠀⠀⠂⠀⠄⢦⡀⡀⠄⠄
⠀⠀⠀⠡⡳⠀⠀⠠⠐⠩⠀⠀⠀⠀⠀⢃⠀⡀⣀⡈⠎⠀⠀⠀⠈⠒⠜⠄⠤⠊⠀⠀⠀⠐⠆⢠⠃⠠⠂⠠


⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⠀⠁⠸⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⠄⠐⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠩⠤⠊⠄⠈⠀⠀⠀⠈⠂⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠔⠤⠀⠂⠁⠂⠀⠀⠀⠀⠀⠂⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⡀⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⣷⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⡀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⢠⣿⠁⠀⠀⢀⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⢰⠀⠀⠀⠀⢠⣿⣿⠁⠀⠀⢀⣾⠃⡀⠀⣴⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡏⠀⠀⢠⡇⠀⠀⠀⣰⣿⣿⠏⠀⢠⠀⣼⡟⢀⠃⣼⠟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠀⠀⣠⣿⠀⠀⢀⣼⣿⣿⡟⠀⠀⡄⢰⣿⡇⢸⠠⠏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⣿⣿⣿⠀⢀⣼⣿⠃⠀⣴⣿⣿⣿⡟⠀⠀⢰⡇⢸⣿⠇⣾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣼⣿⣿⣿⡇⢠⣾⣿⠏⢠⣾⣿⣿⣿⠏⠀⠀⠀⣾⠃⢸⠟⣰⢇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢰⠃⠀⠀⠀⣰⣿⠟⣿⣿⣇⡞⣿⡟⣴⣿⣿⣿⡿⠁⣰⢣⠆⣼⠟⠀⠈⣰⢏⡟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢺⠀⠀⠀⢠⣿⠃⠀⣿⠟⣿⢀⣿⣿⣿⠋⣼⡟⠀⢰⣷⣿⣾⠏⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⠀⣻⡄⠀⠀⣼⡇⠀⠀⡇⠀⡏⢸⣿⡿⠃⣼⣿⢡⢇⣿⣿⣿⠏⢀⣾⣿⠟⠁⠀⣠⡾⠀⠀⠀⢸⡇
⠀⠀⢸⠀⢿⣿⠀⠀⣿⠃⢰⣴⠇⠀⠀⢸⣿⠁⢰⣿⡏⣾⢸⣿⣿⣿⢠⣿⡿⠋⢀⣴⣿⡿⠃⠀⠀⠀⢸⡇
⠀⠀⢸⡄⠈⢿⡇⠀⣿⠀⣼⣿⠀⠀⠀⣼⡇⠀⣼⣿⢧⣿⢸⣿⣿⣿⣾⣿⠃⢠⣿⡫⠋⣠⡜⣡⠆⠀⢸⡇
⠀⠀⢸⣇⢠⣞⣿⠀⠁⢀⣿⡟⠀⠀⠀⣿⠀⠀⣿⢻⢸⣿⣸⣿⣿⣿⣿⡇⢠⡿⠁⣤⣾⡿⠛⠁⠀⠀⢸⡇
⠀⠀⠸⣿⣆⣿⣿⠀⠀⣸⣿⠃⠀⠀⠀⠉⠀⢀⣯⡏⣿⣿⢹⣿⡇⣿⣿⡇⢼⠁⣾⡿⡋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⣿⣿⣿⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⢱⣿⡟⣼⣿⡇⣿⣿⡇⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⣿⣿⢠⣾⠏⠀⠀⡆⠀⠀⠀⣰⣿⢇⠏⡿⣹⣿⣿⣾⣿⡿⠁⣼⡿⠋⠀⠀⠀⠀⣠⠀⠀⢸⡇
⠀⠀⠀⠀⠘⣿⡇⣼⠏⢠⠃⣴⡇⠀⠀⢠⣿⡏⢀⠀⣽⢿⣿⣿⠿⠋⡀⠞⠉⠀⠀⢀⣀⣴⡾⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠋⣼⠏⢠⠇⣰⣿⠃⠀⢀⣾⠏⢀⣾⡼⣿⣿⠟⠁⢀⡠⠔⠀⣠⣴⠾⠛⢡⡞⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢰⡏⢀⣾⣴⣿⠏⠀⢀⣾⠃⠀⣼⡇⣿⡿⠁⢀⣴⠋⠀⣠⡾⠋⠀⣰⣾⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡿⠀⣸⣿⣿⠏⠀⠀⡾⠁⠀⣰⣿⣧⢹⠇⢠⣿⡏⠀⠞⠁⠀⣠⣾⣿⣿⠃⣸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠘⠁⠀⣰⢿⣿⡏⣻⢠⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⠏⢀⢻⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⡴⢋⣾⣿⢳⣿⣾⣿⠃⠀⠀⢀⣾⡿⣭⣿⠏⠀⠎⢸⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡜⣡⣿⣿⣢⣿⣿⡿⠁⠀⠀⢠⣾⣿⣿⡿⠃⠀⠜⢀⡾⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢈⣼⣿⣿⣿⠿⠋⠁⠀⠀⠀⣰⣿⣿⠟⠋⠀⣠⣎⣤⣿⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡆⠀⢀⠞⣵⠿⠋⠁⠀⣀⡀⠀⣠⠾⠟⠋⢁⣀⣤⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣶⠀⠀⢂⠞⠁⠀⣠⣶⠿⠋⠛⢋⣠⣤⣶⣾⣿⣿⣿⣿⣿⣻⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⢀⠀⣠⠞⢉⠀⠀⡠⢚⣩⣭⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⡿⠃⠀⢀⡴⠁⠔⣡⡖⠁⠀⢀⣴⡿⠿⠿⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢀⣿⠋⢀⣤⣶⠟⠁⣠⣾⠋⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠜⢁⣴⡿⠟⠁⠠⠾⠋⠁⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠄⠀⠠⠤⠠⠄⠼⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠒⠂⠀⠀⠉⠉⠉⠉⠓⠒⣄⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠐⠂⠀⠀⠈⠘⡤⠶⠶⠬⠥⠀⠀⢀⣐⡂⠉⠩⠛⡇⠀⠀⠀⠀⠢⠘
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢛⣉⣉⣈⡉⠉⣉⠁⠨⠕⢲⠊⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠒⠲⠄⠤⠒⠒⠒⠒⠮⡁
⠀⠀⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢐⣀⣈⣉⣭⠉⠰⢒⠎⠀⠀⠀⠀⠴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣃⣀⣙⣛⣭⣍⣭⡕⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⣐⣒⣒⠲⠤⡜⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠐⠠⠄⣀⡀⠀⠀⠀⠈⠖⠒⠲⠰⠟⣍⣀⠤⠤⠒⠉⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⣛⠛⠛⠛⢡⠀⠀⠀⠀⠀⡒⠀⠀⢀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠐⠒⠠⠠⠤⠤⠤⠀⠀⡻⠶⠶⢶⣷⠊⠁⠀⠀⠀⠀⠀⠤⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡭⠭⠖⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⣿⠁⢠⠊⠙⣦⣏⡷⠀⣟⠹⡆⢰⠉⠙⡆⡴⠋⢻⡆⡞⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⡼⠦⠴⠃⠟⠳⠖⠿⠘⠷⠜⠦⠞⠅⠳⡤⠞⠇⠳⣤⠼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⣰⣊⡑⢢⠀⠀⠀⠀⠀⣀⢤⡤⠤⣄⣠⠤⡄
⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⣠⣖⣂⣁⣀⣲⣄⠀⠀⠀⠀⣞⣁⢀⣹⣌⢱⡄⠀⠀⠀⣈⠞⠀⡞⠺⣡⡞⢾⣽
⠀⠀⠀⠀⣢⣤⠤⢤⠤⡤⠤⢤⣤⣼⡀⠀⠀⠀⠀⢧⠒⠒⠒⠒⢲⠞⠀⠀⠀⠘⢧⠶⣖⣒⠶⢲⠋⠀⠀⠀⢸⡒⠼⠼⢄⠿⠼⠚⡏
⠀⠀⠀⠀⢱⣊⣉⡲⠮⠭⢷⠭⣙⣤⠃⠀⠀⠀⠀⠈⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢧⣎⢉⣱⡎⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠣⠤⠍⠮⡇⢙⡴⠵⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⣦⠙⡾⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢀⠔⢋⠤⡤⠤⠤⣉⢆⣴⠒⣆⠀⠀⠀⠀⣠⠔⠊⠻⡍⠙⢏⠐⠢⡀⠀⠀⡤⠒⠉⠉⠉⠙⠒⢄⠀⠀⠀⠀⠀⡴⠊⠉⠉⠲⣄
⠀⠀⠀⠀⠎⡞⢙⣚⣁⣡⢞⣕⣋⡀⣇⢸⠀⠀⠀⣜⠁⠀⠀⠀⠘⣄⣸⠦⠤⠜⠀⠘⡤⠤⠤⢤⠤⠤⠤⡞⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡆⡘⠨⠈⠁⠹⢉⠟⢲⢲⠉⠉⠀⠀⢠⠋⠣⡀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠸⡀⢇⠀⡇⢸⢠⠇⠀⠀⠀⠀⠇⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠰⡌⠳⡉⣉⣢⣀⡴⢁⡞⠀⠀⠀⠀⢸⠲⢄⣨⠗⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣⣘⡀⢃⣁⡜⠀⠀⠀⠀⣀⠇⣹⠦⠔⠋
⠀⠀⠀⠀⠀⠈⠓⠢⠤⠭⠔⠒⠉⠀⠀⠀⠀⠀⠸⡀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠯⣅⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢰⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡆⢠⣆⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢠⣴⡒⢒⣖⠲⠦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣧⣃⣀⣀⠀⠀⠀⠀⠀⠀⢩⠩⣇⡼⠼⡄⠀⠀⠀⠀⢀⠴⢄⡀⣠⠔⢄⡀
⠀⠀⠀⠀⢰⢛⡥⠞⠓⠦⠥⠭⡻⢄⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⣾⡻⣹⠀⠀⠀⠀⡈⠀⣍⠖⠒⢦⡀⠀⠀⡰⠋⠈⡆⠙⢇⢸⡀⠙⢦
⠀⠀⠀⠰⣏⡜⠀⠀⠀⠀⠀⠀⠘⣾⡄⠀⠀⣀⢀⣱⣄⣀⣀⣤⠞⡛⢩⡁⠀⠀⠀⡞⠁⠨⠁⠀⠀⠀⢹⠀⠀⡇⠀⢠⠁⠀⢸⠀⡧⠀⢸
⠀⠀⠀⢠⣋⣀⣀⣀⣀⣀⣀⣀⣀⡸⠀⠀⠀⠘⠢⢤⣤⣄⣀⣤⠤⠴⠊⠀⠀⠀⠀⡇⠀⢀⡇⠀⠀⠀⢸⠀⠀⠱⣄⠸⠀⣀⢮⡀⠇⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠴⡇⠀⠀⠀⢈⠀⠀⠀⠀⠑⠎⠁⠀⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢖⣿⡒⢢⠀⠀⠀⠀⠀⢸⢾⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⢱⢦⡀
⠀⠀⠀⠀⠀⣀⡠⠤⣐⣵⢿⡿⠊⠉⢣⠀⠀⠀⠀⡜⣈⢘⣄⠀⠀⠀⠀⠀⠀⣠⠊⠀⣹⣳⣲⡹⠀⠀⠀⡴⠲⡤⠒⠒⡤⠐⠲⡤⠤⢤
⠀⠀⠀⢰⡯⣒⡲⣩⣓⠵⠊⠀⠀⢀⡜⠀⠀⠀⠸⡩⢍⠉⠈⡆⠀⠀⠀⠀⡰⠗⠒⠉⠁⠀⣠⠇⠀⠀⢸⣄⣈⣁⠠⠄⣁⣀⢀⣁⣀⣠⡇
⠀⠀⠀⠺⣈⠞⠉⠁⠀⠀⠀⠀⡡⠊⠀⠀⠀⠀⠀⢱⢀⠇⠸⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⢮⣽⣚⡈⣉⣐⣒⢒⠮⣉⣳⢯⡽
⠀⠀⠀⠀⠘⢆⡀⠀⣀⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠸⡄⠀⠀⠑⠤⣄⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠢⠤⠒⠒⠠⠤⠁⠂⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⠤⠃⠀⠀⠀⠀⠈⠳⠴⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⡀⠀⠀⠀⠀⡠⠔⠂⠒⠈⠉⠀⠀⣀⢀⡀⣀⢀⣀⢀⡀
⠂⠀⠀⠀⠀⣀⠄⠄⠂⢀⣀⣨⠤⠥⡄⣐⣂⠂⢐⡐⠀⠄⠀⠀⠀⠀⢠⠀⠀⠀⠈
⠀⠀⠀⠠⡎⠀⠀⢠⠚⠉⢠⠔⣪⣥⣤⣤⠄⠀⠀⠀⠀⢢⠀⠈⠀⢀⣠⠲⠃⢈⠀⠨
⠀⠀⢠⡄⠘⠒⠀⠈⠀⠀⠈⠈⠁⠋⠲⠲⠌⠂⠀⠀⠐⠁⢐⠀⠈⠁⠀⢀⠔⠀⡸⠃
⠀⠀⠀⠉⠂⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠴⠂⠂⠁⢠⡢⠀⠁
⠀⠀⢀⠄⡌⠀⠈⠁⠉⠉⠉⠁⠒⠒⠒⠒⠛⠉⠉⠁⠀⠀⢀⠀⠠⠐⢁⡜
⠀⠀⠀⠂⠡⡀⠈⠠⠄⣀⣀⢀⡀⢀⣀⡀⠀⠀⠀⠈⠀⠐⠂⠀⠀⠄⣈⠃
⠀⠀⠀⠀⠀⠈⠂⠕⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⣀⠤⠀⠀⣀⡎⡁⠀⠀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣄⠀⠀⠀⠀⠀⠀⢀⣀⡠⠦⢴⠞⢋⣴⡁⠀⠀⡀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠑⢶⣤⡄⡠⠤⠤⠄⢒⡁⠀⡠⢆⢅⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣶⣚⣛⣓⠊⠴⢋⠆⡅⠀⠀⠀⡄⠂⣀⣤⠀⠀⠦⠀⠀⠖⠋⠐⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠁⠈⠹⣷⠶⠖⣷⠷⠊⡁⠀⠀⠀⠁⠀⢀⠘⠃⢀⡀⢷⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢰⣄⠀⠀⠉⠀⠐⠟⠒⢂⡺⢽⣻⡁⢀⠀⣀⠁⠀⠐⡃⠀⠠⡠⢀⠀⠂⠀⠀⠁⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠉⠀⠀⠂⢘⣾⣷⣶⡦⠝⣫⠁⠋⠐⢃⠤⢰⠁⢀⡄⢠⣦⣦⠈⠂⣈⠁⡀⠤⠀⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠀⠄⠀⡀⡀⢠⣾⣴⣶⠮⢿⣴⠃⠀⠀⡠⠁⠄⠌⠀⠈⠠⠄⠋⠉⡠⠔⠄⠀⢳⠠⠄⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠒⠠⠀⠃⠀⠀⢳⡭⠤⠄⢰⣺⡵⠁⠁⠀⡁⠠⠁⠈⠴⠐⠁⣰⠃⠀⠁⠒⠤⠀⠀⣇⠠⠄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠃⠒⢠⡀⠆⣭⠸⣿⡿⣛⣭⡿⣃⡶⠌⢀⠀⠈⠀⢁⠂⠀⠉⠉⠲⢄⠀⣀⣀⣈⠀⢀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣄⣰⣛⣿⠟⠀⠗⠉⢶⣿⣋⠙⠀⠚⠃⠀⠀⢠⠀⠰⣤⠢⠴⠦⠔⠊⠁⠀⢀⠀⠔⣸⠃⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠺⡷⠢⣤⣀⠀⠈⣩⡥⠤⠌⡀⠀⠠⡤⠤⠦⡤⠐⣬⣄⠀⢀⢄⣀⡀⣤⠠⢞⣃⠬⠒⠓⡡⠶⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢖⣚⣫⣭⠿⠶⠟⣛⣛⣋⡩⠭⠭⠭⠶⠶⠖⠒⠒⠒⠒⠲⠶⠶⠶⢾⣭⣽⣿⣿⣿
⠀⠀⠀⢀⣠⣴⠶⠛⠉⠥⠖⠚⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠭⠭⠭⠭⠽⣿⣿⣿⣿⡿
⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣬⣤⣭⣥⣤⣴⡿⠟⠋
⣾⡏⠀⠀⠀⠀⢀⣠⣤⣶⠾⠛⠉⠁⣀⣀⡤⠴⢒⣂⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠿⠛⠋⠁
⠙⢷⣦⣤⣴⣾⡿⠛⢉⣠⣤⠶⣚⣫⣭⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠉
⠀⠀⢮⣭⣙⣛⣛⣛⣻⣿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁
⠀⠀⠀⠙⢿⣿⣿⣿⣟⣻⠿⠯⠭⠭⠤⠤⠴⠶⠶⠒⠒⠂
⠀⠀⠀⠀⠀⠈⢻⣿⣶⣦⣭⣭⣭⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣟⣛⣛⡿⠷⠶⠶⠶⠖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣶⢮⣭⣭⣭⣛⣛⣓⣒⣒⣒⣂⣀⣀⣠⣤⣤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣶⣶⠶⠭⠭⢭⣭⣭⣭⠭⠥⠤⠤⠔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣩⣭⠭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⠭⠍⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠿⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡭⠭⠭⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠟⠭⠭⠭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣻⡴
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣭⣭⣿⣯⣽⣭⣿

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆
⠀⠀⠀⣄⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠦⠀⠀⢄⠈⠰⠤⠀⠒⢐⠂⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠸⡁⢀⠄⠀⢀⠀⠀⠀⢀⠐⠀⠀⠀⡐⠊⠀⠈⠸⡀⠀⠀⠀⠀⠐⠀⠈⡀⠄
⠰⡀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠈⠃⠀⠋⠀⡎⣉⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠑
⠀⠀⠐⠒⠒⠚⠁⠀⠀⠀⠀⠀⠊⢲⠂⢑⠤⠘⠁
⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⣀⠀⣀⡀⣀⡀⢀⡀⢀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⢰⣷⠘⣇⠁⢹⠀⣇⡀⡏⢹⢸⢹⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠼⠻⠴⣬⠇⢸⠀⡇⠀⢧⡼⠸⣼⠃⣧
⠀⠀⢠⡀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠌⠸⣧
⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠷⠴⠆⢁⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠻⠁⠁⠀⠉⠈⢲⡀⠀⠀⠀⠘⠟⠛⠲⢴⠿⣠⡒⠶⢅⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⢰⡃⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠦⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠐
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⡎⠱⣄⡀⢐⡀⠕⠒⢬⡧⠃⠀⠀⠀⠀⠀⠀⠀⢄⡐⠢⠀⣀
⠀⠀⢛⠀⠀⠀⠠⠈⠀⠈⠀⡀⡉⠀⠀⠀⠀⣤⠀⠐⠃⠐⠓⠒⢬⡀⢂⣤⠈⣠⠀⠀⠐⠀⠀⠀⡀⠄⠉
⠀⠀⠀⠈⠀⢁⠂⠀⠐⡀⠖⠃⠀⠀⠀⠀⠃⠠⠤⠦⣀⠐⢁⠝⢊⠀⠀⠈⠈⣀⣢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣁⣁⣀⡀⠠⠄⠐⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠒⠖⠒⢉⠙⠛⠉⠥⢤⣤⣤⣤⣤⣤⡤⠤⣤⣤⣤⣀⣉⣙⣛⡛⠛⠛⠓⠲⠦⠤⢄⣀
⠀⠀⢀⠤⠒⠀⡀⠄⠂⠠⠐⠊⡉⢍⣐⣀⣠⡤⠤⠶⠖⢂⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤⣉⣉⣙⣓⠒⠢⠤⢄⣀⠀⠉⠑
⢀⠔⠁⣠⠖⠁⠀⢈⣀⣤⣶⠿⢛⣋⣭⡤⠶⠶⠞⠛⠛⣉⣉⣀⣠⣄⣀⣀⣀⣀⡀⠀⠀⠈⠉⠉⠉⠙⠓⠒⠈⠉⠓⠶⢄
⡌⠀⠌⠁⢀⣤⢞⡿⣹⣥⡶⠟⢋⣡⣴⣶⠶⢖⣛⣫⠭⠉⠉⠀⠀⠀⠀⠀⠙⠛⠓⠒⠒⠒⠲⠶⠤⠤⣤⣀⣀
⠀⠀⢀⡶⢋⣸⠿⢋⠕⣫⡴⠞⠛⠉⢁⣀⠤⠔⣒⠒⠒⠒⠒⠒⠐⠀⠉⣉⣉⣉⣉⡉⠉⠉⠋⠐⠒⠀⠀⠀⠉⠙⠓⠦⣄
⠀⢀⣟⣴⠟⢁⢔⡵⢟⡩⡲⠼⢒⠩⠁⣀⣈⣤⣤⣤⣤⠤⠶⠶⠶⠶⠦⠉⠉⠉⠉⠉⠉⠉⠛⠓⠲⠤⢄⡀⠀⠀⠀⠀⠈
⠀⠈⡟⠁⣰⣻⢏⠔⠁⢀⣤⣶⡷⠞⢋⡉⠭⠴⠒⠒⠀⠀⠀⣉⣁⡀⠀⠀⠀⠀⠀⣤⣤⣄⣀⡀⠀⠀⠀⠉⠲⡄
⠀⠈⠃⠀⢱⠇⠀⢀⣴⣿⠟⢉⡤⣐⠄⡂⠤⠀⣀⣀⣀⣤⡤⠤⣤⡤⢤⡤⠤⠤⠤⠀⡀⠀⠀⠈⠑⠢⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⢠⡟⡐⣡⢖⡡⢊⣠⣤⡶⠞⢛⣩⡥⠴⠶⠒⠒⠒⠒⠒⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⡿⠀⠜⠁⣡⢶⠟⢋⣡⠴⠚⣛⠿⠟⠋⠉⢉⣁⣉⣉⣉⣀⣀⡀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⠁⡠⠊⠁⣢⡼⣆⣠⣤⣶⡾⣿⣿⠿⠿⠶⠦⠤⣄⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣡⠶⣫⡵⢟⣫⣭⣽⣶⣤⣤⣤⡤⠤⠤⠤⠄⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣠⠞⣵⡿⣫⢖⣵⣿⡿⢟⣯⠏⣩⠤⠶⢒⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡾⡏⢐⡵⢋⡕⡡⡺⠋⣠⣮⠴⠚⣛⣩⣵⣞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠡⠀⡾⠁⠎⠈⡐⣡⡞⡫⢂⣴⣾⡿⠋⢉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠀⡇⠀⠀⢠⡾⢻⢊⡴⡫⠋⣩⣴⠿⣫⠭⠉⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠸⡇⢆⡟⡐⣱⡿⢋⠕⣪⣴⣾⣯⡭⣭⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠁⠀⠠⢊⣾⢟⢟⢫⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡾⠐⢋⣴⠿⣋⣤⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣧⠞⡻⢛⣩⡥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠇⢸⣶⢟⣥⣶⡛⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣴⡿⢫⣥⡶⠚⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠃⣭⣤⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣥⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣵⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣸⣿⠀⠀⠀⡀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣿⣻⠀⠠⡄⢏⢧⠐⡀⠙⡀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⠁⠀⠠⠀⠀⣶⣿⣿⡇⢀⡀⠀⠀⠀⠣⠀⢸⡄⠈⢀⠶
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⢹⣿⣿⣿⣿⣇⠁⠀⠒⠒⠒⠒⠒⠛⠉⣁⡤⠒⠀⠈⠉⠉⢀⠐⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⢳⠒⠓⠂⠀⡀⠀⡸⢦⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⡀
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⠈⣦⣤⣄⠀⠀⢀⡏⣠⠈⢙⡳⢦⡤⣄⣀⣀⣀⣤⡤⠤⠆⠒⠃⡀
⠀⠀⠀⠀⠀⠀⡯⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣶⣤⣾⣧⡇⢸⠸⣡⣼⢇⡐⠂⠐⠀⠀⠄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅⠙⠿⠿⠿⣿⠻⠿⠿⣿⣿⣼⣿⣇⣀⡀⣠⢼⣋⣴⡡⠇
⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡀⠀⠀⠀⠠⡀⠀⠈⠛⠪⠿⠭⠬⠭⠭⠿⠖⠋⠁⠀⡇
⠀⠀⠀⠀⠀⠈⠓⠲⠶⠾⠿⠿⠿⠿⠿⠿⠟⠛⠁⠀⠉⠐⠒⠒⠂⠘⠑⠒⠀⠤⠤⠤⠤⠤⠤⠀⠤⠠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⣥⡶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡿⠦⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⣯⣅⠀⣾⣴⡄⢀⡀⠀⠀⠠⠴⢶⠆⠀⢀⠀⠙⠋⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠙⠛⠾⣧⠃⠀⠜⠀⣠⣀⡦⠀⣤⢡⡈⠻⢀⣤⠤⢹⡶⢦⠀⠁
⠀⠀⠀⠀⠀⢀⣀⣠⣤⢘⠈⠀⢀⠀⠍⢀⡙⢋⠀⡠⠀⣿⠏⠀⢠⠏⣤⣇⡀⣠⣯⣏⣩⡿⡟⠓⠓
⠀⠀⠀⠀⠀⣨⣧⠀⠈⢻⡇⠀⠊⠀⢀⡠⠂⠀⠀⠀⢀⠟⠀⠠⣏⣸⣿⡃⣱⣤⣸⣿⡟⠳⢬⢀⣢⣾⢷⠶
⠀⠀⠀⠀⠀⡿⠋⠀⠀⢠⣷⢀⣠⡴⠛⢥⣴⡆⠀⠀⣮⠴⢀⡚⣿⣿⣾⣧⡛⠹⠉⠻⢿⣤⣌⣻⣿⣿⣧⠳⡀⠀⣀
⠀⠀⠀⠀⠈⠁⠀⣠⣮⡞⢷⢀⡙⢿⣿⣶⣿⣷⣼⡟⠿⠆⠊⡁⢸⣿⣧⣿⣷⣤⡄⠀⠀⠙⢯⣉⢹⣿⢿⣷⠾⠆⠿
⠀⠀⠀⠀⣤⣡⡄⡋⢬⣿⣎⢧⢳⣀⡛⠙⣿⣿⣿⣼⡶⠀⠹⡜⢸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠹⣟⣯⡟⠁⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠘⠃⠀⠀⠸⣿⣿⡌⠣⠻⡇⠀⠘⢿⣯⣹⡯⠀⣦⣿⣿⣿⣿⡄⣿⣿⠀⠀⠀⠀⠀⢤⠼⣟⡅⠀⢠⡾⠆⠀⣰⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠁⠀⣠⠃⠀⠀⠈⢻⣿⣷⡖⣿⣿⣿⣿⣻⣇⢿⡿⠀⠀⠀⠀⠀⠸⢠⠏⠀⠰⢿⣧⡄⠀⠙⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⡆⠁⠀⠀⠀⠀⠈⣿⣿⣣⠿⣿⣿⣿⠙⢿⡌⣿⣆⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⢸⡇⠀⢰⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠋⡇⠸⣶⣄⡀⠀⠀⣿⠟⠉⠀⣨⠁⠻⣶⣤⣥⣌⣽⣦⣀⠀⢹⠏⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣸⣄⠈⢿⣷⠀⢠⣁⠈⣷⣾⡿⠇⠒⣾⠻⠉⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡏⠛⠻⠁⠀⠀⠉⠑⢿⣿⡇⠀⠀⢹⣦⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⣤⢾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢿⣇⠀⠀⠈⣿⠀⠀⠀⢠⣄⣼⢳⡶⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡄⠀⠀⠀⠘⣿⡄⠀⠀⠸⣧⡄⠀⢀⣿⠀⠀⢀⣿⣼⣷⠏⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⣿⣀⣀⡀⠻⣿⣦⡘⣿⣁⡠⢿⠟⠁⠀⠀⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠉⠁⠉⠛⠁⠈⠉⠁⠹⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡄⠀⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢰⣿⡇
⠀⠀⠀⠀⠀⠀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧⣸⣿⣧
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⣿⠄⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⣿⡿⠁⠀⠀⣸⣄⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢻⣿⣿⡟⠇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣠⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢠⡀⠹⣿⣿⣾⣿⣆⠀⠀⠀⢀⠀⠀⠀⣼⣇⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡟⣠⣾⣿⣿⣿⠏⠀⠀⠀⣄
⠀⠀⢀⡀⠀⠀⣸⡗⠀⠙⢿⣿⣿⣿⡆⠀⠀⠘⣧⠀⢸⣿⣿⣶⣤⣴⣿⣿⣿⠀⠀⢻⣦⡀⠀⠀⠙⢿⣿⣿⣿⠟⢰⣿⣿⣿⣿⣿⣆⡀⠀⣰⣿
⠀⠀⠀⢻⣷⣼⡟⠀⢸⣆⣨⣿⣿⣿⡟⠀⢀⣼⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣼⣿⣿⡆⠀⠀⠀⠙⠿⢣⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢠⡆⠀⠀⣿⡿⠀⠀⠈⢿⣿⣿⣿⣿⢃⣴⣿⡟⠁⢀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣾⣧⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣾⣿⣶⣾⣿⡇⠈⣷⡄⠀⠛⢿⡿⢡⣿⣿⣿⣧⣤⣼⡧⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠙⣶⡘⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀⢸⣇⢿⣿⣿⣿⣿⣿⢃⣶⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⢀⠈⠹⣿⣿⣿⣿⣿⠏⠀⠀⢸⣯⣿⣿⣿⣿⣿⡿⠁⣾⣿⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣧⣤⣾⣿⣿⣿⠏⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠙⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠴⠄⠀⠀⠀⠀⣀⡤⠄⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠒⠉⠉⠉⡩⢟⣿⣷⣶⣶⣤⣀⡠⣴⣖⡒⠋⠉⣉⣉⣉⡀⣀⠀⠀⠚⠋⠙⠢⠤⢤⣄⣀⡀
⠀⠀⠀⠀⢠⡴⠚⠉⢉⣠⣤⣤⣭⣌⠁⢾⢶⠻⠛⠛⠻⣿⣿⡭⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢤⡀⠛⣰⢤⠀⠀⠀⠀⠈⣃⣶⡀
⠀⠀⠀⠀⣿⠁⠀⢶⣾⣿⠟⠛⡻⣿⣷⡄⢼⠀⠀⠀⠈⣁⣸⡄⠀⢀⣤⣤⠀⣀⣴⣾⣿⠿⢛⡿⣿⣄⠈⣷⠢⠼⠃⠔⣠⣴⣶⣾⣿⠁
⠀⠀⠀⠀⠘⢧⣀⣸⣿⡇⠸⢿⠇⢹⣿⣷⠈⡑⠤⢀⣀⡠⠠⠀⠀⠀⠀⠀⠀⢈⣿⣿⡇⢰⣿⠆⢸⣿⣧⠀⠀⠃⢀⠈⢿⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣤⣶⣿⣏⡼⠣⣬⣤⣀⣀⣀⣀⣀⣤⠤⠤⠤⠤⠤⠽⠿⣿⣤⣍⣤⣾⣟⡿⠦⠤⣶⣶⣶⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠉⠀⠀⠀⠈⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣉⣁⣀⣀⡀⠀⠐⢶⣶⣶⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠩⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣤⣤⡦⣰⡄⠀⠀⢰⡦⢦⣰⣤⣤⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⣿⠀⣶⠓⣦⠀⠀⠀⠀⠀⣤⡀⢀⣠⡀⢠⣄⡀⣀⡀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⠀⠙⠒⣿⡄⠀⠀⣿⠀⠑⠾⠟⠛⠃⠚⠻⠟⠘⠛⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⢀⡿⠁⢈⠀⣯⠀⢰⠀⣤⢀⠀⡾⢀⣄⠀⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠟⠀⡇⠀⠋⠀⢿⢹⢠⡇⢸⡇⠀⡁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄⠑⠈⠁⠁⠁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢓⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠻⠂
⠀⠀⠀⠀⠀⠀⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⢀
⠀⢀⠤⠠⣤⣠⣇⡔⠇⠀⣀⠴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠤⠤⠤⣄⡀⠀⠀⢀⡠⠤⠀⠒⠂⠀⠀⠉⠀⣠⡴⢿⡀
⣤⣯⣤⡚⠀⠀⠈⡟⠉⠲⡀⠀⠀⢱⠀⠀⠀⣀⠤⢄⠀⠀⠀⢠⠖⠊⠉⠁⠀⠀⠀⠉⠘⠒⠢⣄⡙⣦⠀⡸⢤⣂⡀⠀⠀⠀⠀⣤⡼⢟⣴⣾⠇
⡇⠠⠀⠙⠶⢤⠴⠖⠒⠲⠿⣤⣤⣞⣳⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠒⠄⠀⠀⢀⡐⠀⡂⡀⠂⠘⠏⢹⡆⢰⠆⢤⣿⣋⠐⠒⢚⣩⡴⠟⠋
⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⢻⠶⠇⠀⠀⠀⠀⠀⡀⠀⠀⣸⣦⣤⣄⣄⣠⣄⣠⣦⣠⣦⣶⣶⣞⣿⠀⠀⠒⢄⠉⣁⠴⠟⠋⠁
⢸⠀⡆⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⣧⠀⠀⠨⣀⠀⠀⡇⠀⢾⢀⡙⠓⠛⠳⠶⠛⠷⠞⢛⣋⡭⠖⣻⠏⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠛⢆⡀
⠘⡄⠁⠀⠀⠀⠀⠀⡔⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢥⣭⡭⣭⣭⣭⣭⣭⠭⠴⠖⠋⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠸⠀⢀⠨⡸⡄
⠀⠉⠛⠛⠛⠓⠒⠺⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣈⠀⠀⠈⠇⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠠⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡘⠲⠂⠠⢀⠉⠛⢦⣄⠀⠀⠀⠀⢷⠀⡀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣀⣠⣴⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡀⠰⠞⠃⠀⠘⠀⠀⠀⠀⠀⠀⢀⡞⠱⠀⠀⠀⠀⠐⠢⡀⠈⠳⣄⠀⠀⠘⠇⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⢋⠉⢊⠈⢉⡏
⠀⠀⠀⠀⠀⢀⣘⠃⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀⡀⠀⠀⠀⠀⠈⠢⡀⠈⣆⠀⠀⠀⠃⠀⢀⡀⠸⠀⠀⠀⠀⠀⠠⡐⠠⠑⢨⠅
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢠⡟⠀⠀⠀⠀⠁⠀⠀⢀⣠⣶⠶⠛⠚⠋⠀⠀⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡏⠰⠀⠀⠀⢠⣀⣴⠾⠿⠋⠁⠀⠀⠀⠀⠀⢰⠀⠀⡈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠰⢀⠢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠂⠀⠀⠀⢀⡿⠀⡀⢈⣤⡶⠛⠋⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⢰⢬⠁
⠀⠀⠀⠀⠀⠀⠀⡆⢉⣻⣦⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⢸⣃⣴⠾⠛⠁⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠷⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⡠⠦⠴⢤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⡤⠶⠛⠿⢫⡥⠅⠀⠦⢤⡀⠀⠀⠀⠀⢀⣤⢠⣤⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⢀
⠀⠘⠀⢤⠀⠈⡇⠀⠀⠀⠀⠀⣀⠠⣤⡄⢠⠶⠤⠀⠀⠀⠀⠀⠀⣴⠋⠁⠉⠚⠀⠀⠀⠀⣀⠀⢀⣿⠀⠀⣠⣤⢴⣤⣤⣤⢴⡄⢠⠀⠀⠀⠈⢳⣿⣄⠼⠶⣤
⠀⡆⠘⡆⠀⠀⣿⠀⠀⠀⣠⣾⠟⢁⡽⠓⠉⠉⠛⠛⢴⣄⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⣀⣤⣞⣫⣽⠀⠀⠈⣽⡇⠉⢉⢻⡸⡇⣸⣤⠀⠀⠀⠀⠋⠀⠈
⠀⠃⣴⠃⠀⠀⣿⠀⠀⢰⠻⠏⡠⠊⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⣿⡀⠀⠁⣀⡠⢤⣶⠿⡿⣋⠁⣴⠼⠃⣤⠀⢸⣏⣤⣸⣄⣧⣷⣿⠃⠀⠀⠀⠀⠘
⠀⢸⠋⠆⠀⠀⡟⠀⠀⠘⢾⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠻⠿⡏⠉⢀⣬⠥⠌⣻⣯⣭⠶⡞⢻⠃⠀⠘⡆⠀⠉⠉⠉⠀⣼⠁⠀⠀⠀⠀⠀⢢⠤⠬⠉
⠀⢸⣠⡇⠀⢰⡇⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠀⠒⠒⠒⠁⠀⠀⠀⣳⣟⠦⠴⣤⣾⡿⠛⠋⠀⢀⡷⠋⠀⠀⠀⢷⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⡂⣀⠰⠊
⠀⠈⣇⠀⠀⣼⠁⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⣄⣀⣀⣉⣠⠤⠔⠚⠉⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠸⣆⣰⠇⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠐⠒⠐⠒⠋⠀⠀⠀⠀⡀⠔⠄
⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠿⠯⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢠⡄⠲⡀
⠀⠀⠀⢈⢧⠀⠀⠀⠠⡤⠤⠤⠤⢤⡄⣀⣀⣀⣀⣈⣐⣄⠀⠀⠀⠀⡤⢀⣤⣤⣴⠾⠿⠛⠉⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⣣⠀⠈⣢⡀
⠀⠀⠀⠘⡜⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢉⡀⠀⠙⡀⠀⠀⠔⠉⠝⠉⠁⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⢀⣀⣀⠤⠼⢧⡽⢤⣾⠿⠋
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠐⢦⠀⠀⠀⠀⣀⡴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠒⠀⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⣀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⢠
⠀⠐⠈⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⠀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣤⣤
⣿⣿⣴⣭⣦⣥⣽⣥⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣻⣮⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠿⣠⣤⣤⣶⣶⠈⠉⠉⠁⠀⠀⢠⡤⣤⣤⣤⣀⡀⢀⣾⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠋⠁⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⣹⣿⣿⣿⡿⠋⠀⠉⠙⠻⢿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣷⡄⢀⣀⣀⣘⠿⠟⠛⠛⣤⣤⣀⣀⠀⠿⠿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⢛⡟⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡙⠉⠉⠀⢻⣿⣿⣿⡀⠀⠀⠠⣿⣿⠯⠇⠀⠀⠀⠀⢨⣷⣦⣤⣀⡀⠀⠀⢀⣠⣶⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠺⢿⡿⢆⣀⣀⣤⣽⣿⡿⠶⠒⠚⠛⠋⠁⠀⠀⢀⣀⣀⡐⠛⣿⢿⣿⣿⡿⠛⠘⠻⣿⣿⣿⣿⣿⢿⣄⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣀⣉⣩⡗⠲⣄⠀⣼⡿⠁⠀⠀⠀⠐⢠⣬⡉⠒⢬⡙⢯⣅⠀⠀⠀⠀⠀⠙⠻⠟⠉⠁⠀⠀⢻⣧⣿
⠀⠀⡠⠤⠀⠀⠀⠀⠀⠠⠄⢐⠒⠒⠦⠤⢻⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢾⣷⡄⢙⡦⠈⠛⢓⠒⠄⠒⠿⣿⣦⡀⠀⠀⠀⠠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣘⡄⠀⠀⢀⡀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠠⠴⠀⠈⠉⠉⠀⠀⠀⠀⠀⠑⡄⠀⠀⡈⠛⠛⠀⣤⣴⣶⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢿⠁⠀⠀⠀⠀⠐⢤⣄⡀⠉⠓⠲⢤⣉⠒⢄⡀⠀⠀⠀⠀⠀⠀⠘⢆⠀⣂⣤⠀⠀⢻⣿⡿
⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⣦⣌⡻⢷⡀⣶⣯⣿⡄⢬⠻⣿⣉⣿⣶⢦⣌⡙⠲⣭⡳⢤⣀⣀⣀⣿⠀⠸⡆⢹⠘⡷⣿⡇⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠿⣶⣵⣿⡿⣿⣤⠸⣆⠈⠉⠛⠛⠺⠿⠿⣶⣄⡙⢦⡉⠙⠛⠉⠀⠀⠑⠈⢀⠟⢿⡀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠉⠛⠿⣿⠿⢯⠄⠈⠛⠦⠄⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⠀⠀⠀⢀⣴⣖⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⢾⣷⣶⡀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠒⠂⣸⡷⢂⡾⢛⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠳⠀⠠⠀⠀⠀⠀⠀⠐⠒⣛⠋⠁⠀⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠄⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⢩⠓
⠀⠀⠀⠀⢀⡤⠊⠁⠀⠠⢄⠘⢶⠂⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣫⣴⠾⠃⣤⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠀⢠⣾⡄
⠀⠀⠀⠀⢸⡀⡀⠀⠓⠒⡢⠤⠤⠾⠣⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⠾⡋⢻⡴⠏⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢿⡾⠇⠿⣤⣤⣦⠉
⠀⠀⠀⠀⢸⣿⠩⣀⡴⠊⠀⠀⠀⠀⠀⢮⣀⡀⠀⠀⠀⠀⣤⠂⣰⣟⢻⣅⡦⠟⠉⠀⢀⣰⡿⠃⠀⠀⠀⠀⠀⣆⠚⢧⣼⡇⠨⣶⡄⠒⠉⠉⠛
⠀⠀⠀⠀⠀⠙⠢⠿⠇⠀⠀⠘⠷⠜⠀⠈⣬⡃⠀⠀⠀⠀⠙⢀⣷⡜⠋⠁⢀⢀⣶⣿⡶⠛⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠹⣿⣯⡀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⣀⠀⡀⠀⠤⣤⣿⡇⠀⠀⠀⠀⠀⠀⠻⣀⣴⣷⣉⣴⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⡉⠛⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠶⠶⠆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡸⠂⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⠀⠀⢾⣁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⠒⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠼⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠦⠀⠀⠃⡀⠂⣨⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣼⡆⠀⠀⢾⠆⡠
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⠀⢀⣀⣐⢃⣠⣴⠾⢃⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⢿⡀⢻⣇⠀⠀⠰⢺⠃⣾⣷
⠀⠀⠀⠀⠀⣾⠁⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢶⣮⣽⣿⣿⣧⣬⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⡎⣿⡖⠏⠉⠓⠀⠀⢼⣯⣥⠂
⠀⠀⠀⠀⠀⢉⠀⠀⠠⣿⠇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣿⣏⣉⣙⣛⣭⣼⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠠⠟⠋
⠀⠀⠀⠀⠀⠈⠳⢀⣰⡙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⢠⡤⣤⡿⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣤⣠⣴⣿⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠠⡄⠀⡀
⠀⠀⠀⠀⠀⢠⣧⣤⣴⠶⢠⣾⣷⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠲⠦⠟⢿⣻⡆⠀⠀⠀⠀⠀⠀⠀⢠⡶⣾⣷⣶⡿⣿⣿⣿⣗⠂⡄⠐⡀⠁
⠀⠀⠀⠀⠀⠘⢩⠟⠳⠶⠷⣬⣭⣵⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⠁⠒⠂⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⢘⠿⣿⣹⣯⣷⣿⣛⣻⣿⣷⡇⠀⡟
⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⢈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠠⣴⣶⡖⢢⣾⡏⠀⠀⠀⠀⠀⠀⠀⠛⢶⣌⠙⢳⣤⣿⣿⠿⣋⣽⠃⣸⢳
⠀⠀⠀⠀⠀⠀⠀⣧⡐⢿⣿⣯⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⠬⠁⠼⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠀⣿⠸⣦⣾⣿⡇⠀⡿⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠈⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢷⠀⠉⠀⣉⠹⠟⠃⠘⠃⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠛⠶⠾⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⣶⣬⠓⣯⣾⡻⠛⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣇⣹⡧⠀⠀⢀⣼⣅⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢹⢻⡄
⠀⠀⠀⠀⠀⣠⠊⣿⣶⣶⡉⣭⢳⣾⣿⣖⣠⡇⠀⠀⠀⠀⠀⠀⠀⣿⡗⢩⣭⡧⠀⢠⣿⠧⣿⣯⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠟⣮⣤⡿
⠀⠀⠀⠀⠀⠛⢶⣿⣛⠻⣿⣶⣿⣤⢒⣽⡿⢋⣆⠀⠀⠀⠀⠀⠀⣶⠀⠀⢸⡇⠀⠈⡇⠀⠀⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠈⢃⠲⣼⠷⠆
⠀⠀⠀⠀⠀⠤⢂⠈⠙⠻⢿⣍⣙⠿⠼⢿⣤⡾⠛⠀⠀⠀⠀⠀⠀⣿⠀⣤⣼⡇⠀⠀⡟⢰⣶⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⡘⣟⣓⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠐⢠⣀⡠⠈⠙⠿⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠉⢹⣷⠀⠀⣷⠈⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢫⣽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⣸⣯⠀⠀⢹⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣦⡜⣷⣷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲⠤
⠀⠀⠀⠀⠀⠀⢻⣹⣿⣿⣿⢡⣷⢸⠗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠂⡀⠀⢶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣇⣸⡟⢸⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠁⠀⣡⠔⠃⣀⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⠶⠚⠛⠻⠿⣿⣿⣴⣶⡀
⠀⠀⠀⠀⠀⠐⣷⠿⠿⠿⠿⠿⢧⡾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠸⣧⣄⠐⠊⠁⢀⣴⣿⣿⣿⡟⣻⠀⠀⠀⠀⠀⠀⠰⢖⣀⣀⠐⠠⠔⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⢀⡐⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣟⡁⠤⣳⡿⢻⣿⣽⡿⠽⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢷⣦⣄⡀⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣿⣷⠾⣿⡿⠶⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡻⢶⣬⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⠼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠀⠈⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣖⣶⣶⣿⣿⣿⡿⢿⣿⣿⡷⢶⠶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠛⣿⡿⠋⢟⣽⣧⣶⣿⠿⡿⠇⢁⣾⣆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠛⠋⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠟⠢⣦⣤⣿⣿⡧⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠃⠀⠀⣀⠠⠊⠁⠻⣧⣿⢟⠙⡃⣤⣄⣾⣿⣿⣟⠅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⣐⡴⠥⠡⣤⣀⡀⠙⢯⡶⢶⣿⣿⣿⣿⣿⡟⠩⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⣴⠫⣹⣦⣤⣴⣶⣄⣩⠷⣄⠹⢿⣿⣿⣿⣿⣿⠫⢸⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⢀⡾⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢷⣁⢙⠿⣾⣿⡉⣲⣿⣯⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⣽⢿⠁⠀⢠⡾⠓⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠹⣿⣦⣀⠀⢹⡟⠻⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢻⡏⣿⢹⡉⣽⢹⣟⡦⣼⠿⢛⣄⢘⠞⣿⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢠⣿⣿⠿⣣⡇⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣿⣏⢨⣦⣿⣿⣿⣷⣿⣿⡟⣹⡏⠀⠁⡜⡿⠀⠀⠄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢍⣿⣿⠓⢠⡅⠁⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣵⣿⣿⣿⣿⣿⡿⣿⢿⣿⣽⠟⠀⠠⢎⡞⠀⠀⠀⠈⢷⣝⠻⣿⣿⠿⡿⠟⠙⣦⣿⡿⣪⣝⠈⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢻⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢻⣏⣠⣴⣷⢾⢶⢢⣄⠀⢀⠀⡙⣷⣷⣾⣶⣷⢼⡺⣟⡯⡮⠁⠁⢀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣵⣿⢿⡷⠿⠿⠿⢟⣹⣿⣯⣷⣿⢿⠿⡏⣧⣿⣿⣽⣿⡟⠀⢬⣻⣾⣿⣿⣿⣿⣿⣿⣎⢡⢀⡀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡿⢁⡤⠖⠒⠚⠛⢛⣿⣿⡿⢁⣿⣼⣿⣿⡿⡿⣿⣿⣯⣔⡙⢮⣿⣿⡿⣿⡿⠿⠟⠁⠀⡉⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠚⠁⠀⠀⠀⢀⣴⢿⣟⣧⡾⠟⣛⣽⣯⣢⣴⣶⣿⣷⣿⣿⣿⣷⣍⠛⠛⠂⠁⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠀⠀⠀⠀⢀⡴⣫⣷⣿⣿⣷⠶⠿⠿⣿⡿⣿⣿⣿⣿⡿⠏⠛⠛⠉⠀⣀⣤⣴⡶⢶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⡿⠟⠛⠛⠻⣏⢻⣿⣿⡿⡻⠇⠀⠀⠀⣀⣤⣶⢿⣿⣿⣿⢟⠺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣫⡾⠛⢉⣣⡀⠀⠀⠀⠀⢿⣷⡙⠳⣄⡀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠊⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⣋⣴⠷⣦⠁⠑⢄⡀⡀⠳⠄⠙⠻⣦⣬⢿⣿⣿⣏⣿⣿⣿⣿⣿⣿⡷⠬⠭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⡋⢑⣾⣿⣦⣀⣦⣿⣿⣄⠀⠀⠀⣰⠁⣾⣿⣿⣿⠻⣿⣿⣿⣿⣿⡍⢓⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠦⠞⣰⣿⣿⣿⣾⣿⣿⣿⣿⠿⠷⢄⢀⡌⣼⣿⣿⡿⣿⠙⣻⣿⣿⣿⣏⣠⢤⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡗⠀⠀⣀⡤⣺⣿⣿⣿⣿⣿⣿⡿⠛⠁⣄⢀⣤⠟⣽⣿⡿⠋⢰⡏⢠⣿⣿⣿⣉⢕⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢋⠞⠀⠖⠛⢃⠜⣫⣿⣿⣿⡿⡿⠛⣀⣤⣼⠃⠀⠀⣾⠿⠋⠀⣠⣟⣴⣿⠿⠝⣫⣷⠞⠁
⠀⠀⠀⠀⠀⠀⢠⡎⠀⢀⠤⠀⠁⠀⠀⠉⠊⠁⠀⣠⣴⣿⠟⡇⠀⠀⠀⠀⠀⣀⣴⣿⣿⣟⣯⡶⠟⠋
⠀⠀⠀⠀⠀⢠⣿⠀⠐⣠⠂⠀⠀⠀⠀⠀⢀⢄⡾⠋⠐⠁⠈⠀⠀⠀⠀⠀⠸⡿⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠉⢀⣾⠇⠔⢀⢀⣀⡴⠊⠠⢏⣀⡤⠂
⠀⠀⠀⢀⠃⠀⠀⢐⣡⣦⠞⠙⠉⠀⠀⠀⠀⠛⠋
⠀⠀⢀⣤⣴⠀⠰⠋⠁⢀⣀⡴⠊⠐⠋
⠀⠀⠀⠹⠃⣀⣤⣶⢾⠟⠁
⢀⠀⡰⠷⠛⠋⠉

⠀⠀⠀⠀⠀⢀⣴⡞
⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠄
⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⢸⣿⡟⣧⠀⠀⠀⢀⠀⠀⠀⠀⢰⣿⠀⠀⢀⠀⢸⡇⠀⢸⠀⢱⢰⣿⣇
⠀⠀⠀⠀⠀⡜⠉⢠⣸⣇⠀⢀⣼⣇⠀⠀⢀⣿⣿⡀⠀⡘⢦⣈⣿⡄⢸⣧⣾⣿⣯⠻⡄⠀⠀⢱
⠀⠀⠀⠀⠈⠀⠀⠸⣿⠙⠳⡾⠉⣿⣦⣴⠟⢻⡿⢻⣦⡇⢸⡟⢹⢇⡞⢿⣯⡾⠋⠀⢿⣾⠀⠚⠀⢠⡀
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠰⡀⢻⡇⠉⠀⠘⠇⠀⠀⠁⡞⠀⠀⠊⠀⠊⠀⠀⠀⠀⠈⠟⠁⠀⠀⢼⠁
⠀⠀⠀⠈⠧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠴⠶⠶⠶⠶⠶⢤⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣛⣛⣋⣩⣍⣉⣙⣛⠻⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣤⣾⣿⣿⣷⣿⣿⣴⣶⣶⣶⣤⣤⡤⢟⣿⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠉⠀⠀⠈⠉⠒⠛⠋⠉⠛⠛⠛⢛⣡⠄⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⢿⣛⣛⣛⣛⣛⣛⣛⣛⣉⣭⣴⣶⣿⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⠤⠤⠤⠤⣤⣤⠤⠤⠶⠖⠛⠉⠁⢰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠐⠢⠬⢭⣭⣭⣭⣭⣭⣭⣿⠿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣽⣛⣓⠒⠶⠶⠒⠒⠒⠂⣀⡀⠐⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢑⠒⠶⠶⠶⠶⠶⠶⠾⢟⣻⣧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣉⣉⣉⣉⣉⣉⣡⠤⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢭⣛⣛⣛⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠦⠭⠭⠭⠥⠄⠀⠀⠀⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡿⢿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⣒⠒⠂⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠒⠒⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡑⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣓⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⠛⠛⢻⠟⠭⣫⡙⣿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠶⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠌⠋⠛⠛⠛⠛⡛⠁⠀⠀⠈⢻⢸⡟⠛⣷⠶⠶⠾⠿⠿⠷⠶⠶⠶⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠈⠉⠉⠁⠀⠀⢠⡆⠀⠀⠀⠀⠀⢷⣶⣶⣶⣿⣿⠀⠰⣤⣿⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠂⡤⠤⠤⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⣼⡷⠤⢐⣶⣾⣾⣏⠛⠛⣉⣁⣼⣿⣆⣿⢛⣦⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⢆⣿⠀⠀⠀⠀⠀⢸⣧
⠀⢀⡀⠀⠀⠼⠄⡀⠀⠀⠀⠀⡌⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠭⠾⠼⠧⢽⣿⣿⣯⢻⣿⠙⣿⣿⣿⣿⣿⣷⣇⠀⢀⢸⣾⣿⡆⠀⠀⠀⠀⢸⣿
⢀⡏⠀⠀⠀⠀⢰⣿⣄⣀⡀⠌⣀⣀⣀⣈⣿⣿⣿⠟⣿⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⢸⣿⠀⢰⣿⣿⣿⣿⣿⡏⢀⠔⠛⠉⠙⠃⢢⡀⠀⠀⢸⣿
⢸⠀⠀⠀⠀⠀⢸⣿⣿⠿⠁⣰⣿⣿⣿⣿⣿⣿⡇⠸⠿⠤⠶⠶⠶⢶⣶⣤⣤⣾⡿⠈⠀⣠⣿⣿⣿⣿⣿⣿⡇⡎⠀⠀⠀⠀⠀⣾⡗⠀⠀⢸⣿
⢸⠀⠀⠀⠀⠀⢸⣿⣿⡿⠷⠾⠯⢽⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣻⣿⣿⣿⣿⢿⠶⡇⠀⠀⠀⠀⢠⣿⣿⢀⡀⢸⢿
⣼⠀⠀⠀⠀⠀⣿⣿⣿⡷⢸⣶⣶⣶⣾⣦⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣤⣶⣏⣉⣉⣉⣉⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄
⢿⣀⣀⣀⣀⣤⣿⣿⠿⢧⣾⣿⡿⠿⠿⠿⢿⣿⡆⢂⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⡿⠿⠿⠟⠁
⢸⣿⣿⣿⣿⣿⣿⡿⠀⠈⠁⠀⠈⠉⠉⠉⠉⠀⠂⠀⠄⣀⣀⣀⣀⣀⢴⣿⣿⣿⠿⠒⠒⠒⠒⠒⠒⠒⠒⢪⢙⣇⡀⣀⣀⣀⣼⣿⡇
⠀⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠤⠤⠤⠤⠈⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⡀⠀⠀⠀⣠⠤⡾⠿⠻⠿⠧⠤⠤⢤⣀⣀⡖
⢱⣖⠋⠻⣅⠐⠈⠀⢠⠔⠒⠒⠒⠦⠤⠤⣉⣉⠓⠲⠤⢬⣑⡒⠢⠄⣀
⢸⡉⠳⣄⡈⠓⠦⣄⣀⠀⠀⠀⠈⠉⠒⠒⠒⠬⠍⣛⠒⠦⣤⣉⡙⠒⠦⣄⣀⠀⠀⠀⡀
⠀⠑⢦⣀⠉⠓⠦⣄⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠀⠈⠉⠓⠲⠤⣌⡙⠒⠤⣀⠁
⠠⣀⠎⠉⠙⠒⠤⢔⡉⠛⠲⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣉⠒⢤⡀⠀⠀⠠
⠀⠈⢷⠒⠤⢀⡀⠀⠀⠉⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣈⡐⠒⠥⠤⣒⣂⡈⠙⣦⠈⠁⠸⡄
⠀⠀⢈⠑⢦⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠲⠦⠤⠭⠭⠥⠤⠤⠤⢳⡀
⠀⠀⡀⢠⡆⠉⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠠⠤⢄⣀⣀⡀⠀⠀⠀⢂⣠⠾⠁
⠀⠀⠱⣧⣀⠁⠂⠄⠀⠀⠀⠉⠉⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⢤⣄⣀⣉⣉⣙⣏⣉⠄
⠀⠀⠀⠘⢯⡙⠓⠶⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣄⣀⣀⣀⣀⣀⣉⣭⠽⠋
⠀⠀⢈⠀⠀⠉⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⣀⣀⣀⣈⣉⣉⣩⡧⠄
⠀⠀⢺⡀⠘⣦⡃⠀⠈⠉⠛⠓⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣭⢽⡋
⠀⠀⠀⠙⢦⣘⡝⠓⠶⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠉⠉⠉⢉⣡⠴⣺⣃⠀⣴
⠀⠀⠀⠀⠀⠈⠙⢷⡶⠤⢤⣄⣀⣀⢠⡄⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠚⢛⣉⡤⠞⠁⢈⣽⣇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣠⠴⠋⣠⣯⣀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⡈⠉⠉⠉⠉⠉⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠈⢁⡴⠚⠁⠀⣨⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢯⣛⣛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢚⡥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠛⢻⡛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠊⢀⡀⠂⠠⠴⢛⡷⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⢾⣧⣄⣀⡀⠀⠀⢀⣀⣀⡀⠀⠀⠁⠀⠰⠚⢉⣠⡴⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⠛⠉⠉⠀⠀⠀⠠⠄⠲⠶⠛⠋⢁⣴⣣⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣄⣀⣀⡰⠄⠀⠀⠀⢀⡀⣠⡭⠛⠁⣈⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢉⡹⠋⠉⠁⠀⠀⠀⠀⠉⢀⠠⠖⠛⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢶⣉⡉⠁⠀⠀⡀⠀⠀⠠⠖⠛⢧⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠉⠁⠀⠀⠀⠁⠀⠀⢀⣠⣤⣴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠛⠒⡀⠀⠀⠀⢈⣩⣥⠖⠋⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣹⣗⡒⠠⠁⠀⠀⠀⢀⣠⣬⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢭⣈⠉⠀⠀⠐⣀⣐⠣⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠀⠈⠙⠓⠲⠦⢼⡧⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡌⠷⠀⠀⠀⠀⣀⢳⡖⠚⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠄⠈⠁⠀⠐⣶⠂⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠭⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⣑⠶⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣯⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⣲⢤⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡋⢀⡴⣾⣽⣻⡄⠀⠀⠀⠈⠻⣿⣿⡿⠿⠿⠟⣛⡿⠿⠿⣿⣿⡄⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢠⠎⠉⠛⠿⣿⣶⣭⡋⠀⠀⠀⢀⡤⢚⡭⡍⠁⠀⠀⢈⣩⣭⡓⠀⠀⠈⠙⢦⣄⢸⡇
⠀⠀⠀⠀⠀⣸⣶⣤⡀⢀⡶⡿⡾⠁⠀⠀⡰⠋⢰⠃⣀⣹⠀⢀⠞⠁⡀⢀⡇⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⢠⣿⡿⣿⣿⣞⡿⡽⠁⠀⢀⡞⠁⠀⠺⣼⣿⠃⠀⡏⢠⡿⣿⠞⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⡤⣄
⠀⠀⠀⢰⢿⣿⣿⢠⣿⣿⡟⠀⠀⠀⡞⢀⡴⠀⠀⣇⠈⠀⠀⠓⠚⠛⠓⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⢡⠖⠈⣧
⠀⠀⠀⢈⡷⢾⣧⢿⣿⠟⠀⠀⠀⢰⡇⠀⠐⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠜⢹⠦⠀⠀⠀⠀⠀⢉⡗⢀⠏
⠀⠀⢀⡞⠒⢤⡼⠛⢻⡀⠀⠀⠀⠸⡇⠀⠀⠈⠓⢤⣀⣀⣀⣀⣀⠤⠔⠊⣁⣤⡞⠀⠀⠀⠀⠀⢀⡴⠒⠉
⠀⠀⣸⠓⠒⢤⣇⡠⢻⠃⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢳⣤⣤⣤⣤⣶⣶⣿⣿⠟⠀⠀⠀⠀⠀⢀⡾⣤⣀⡀
⠀⠀⣹⣻⣦⣴⠃⠰⣇⡀⠀⠀⠀⣀⡤⠛⢦⡀⠀⠈⠫⣍⣛⣛⣻⣿⡿⠟⠁⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠉⠓⠦⣄⡀
⠀⣴⣳⣿⡿⣷⣦⠞⠁⡹⠓⠛⠉⠁⠀⠀⢸⣿⢶⣄⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀
⣼⣷⣿⡿⠁⠘⣧⣠⠔⠁⠀⠀⠀⠀⠀⠀⢸⣗⣽⣇⠉⠛⠲⠶⠦⠤⠴⠶⠒⠋⢹⣱⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠻⠿⠟⠁⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠘⣯⠈⣯⣟⣳⣶⣶⠤⠤⢤⠶⣶⣶⢿⠉⣿⡆⠀⠀⣀⣠⣤⣤⣤⣄⣀⣀⠀⢀⡰⠁⢹⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠤⠤⠤⠴⠖⠛⢿⣛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⢿⣻⣷⠄⣴⡟⠀⠀⠀⠀⠀⠈⠉⢹⡟⠶⠶⠋⠀⢹⣆⢏⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣿⣧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⣴⠄⠀⠀⠀⠀⠀⠀⠀⣾⣡⣦⡀⠀⠀⣾⣇⣝⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣮⣿⣿⣟⠷⣾⣭⣯⣿⣿⣯⢿⣯⣻⣿⣶⡶⠀⠀⠀⠀⠀⠀⠀⢠⠼⣿⣿⣤⡼⠿⠯⣿⡿⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⡿⣷⢾⣶⡿⠶⠿⣟⣻⣯⣷⣞⣷⣄⠀⠀⠀⠀⠀⣰⣧⣀⠙⠛⠁⠀⣀⡔⠉⢁⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣏⣮⣿⣾⣽⣻⣿⣻⡟⣟⣟⣻⣿⡽⣾⣿⣻⣧⣝⣦⠀⠀⠀⠀⢿⣮⡛⣧⠀⠀⣼⡿⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣿⣿⣿⣿⢟⣿⣿⣿⣽⣿⣿⣽⣿⣿⣾⣻⢿⣿⣿⣿⡹⣧⠀⠀⠀⠸⣏⢿⢾⣧⠎⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣽⣻⠟⣵⠟⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠛⠳⢿⣟⡿⡽⣆⠀⠀⠀⠈⠳⢧⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣿⣒⣛⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠛⣹⣀⣀⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣷⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⢠⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⠀⠀⠀⢸⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠁⠀⠀⠀⡰⠀⠀⠀⠀⠀⢀⣾⣿⠇⠀⠀⠀⠀⣼⡿⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⢀⠃⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⣼⡿⠁⡀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⣠⡞⠀⠀⠀⠀⣴⣿⣿⡿⠁⠀⢀⠀⣸⣿⡃⣸⠁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⠀⢀⣴⣿⠁⠀⠀⣠⣾⣿⣿⡿⠁⠀⢠⡄⢰⣿⣿⠃⡿⢰⠋
⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠁⠀⣰⣿⣿⠃⠀⣠⣾⣿⣿⣿⡿⢁⠀⠀⣸⡇⢸⣿⡿⢀⠳
⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡏⢀⣼⣿⣿⠃⣠⣾⣿⣿⣿⣿⠋⢀⠂⠀⢠⣿⠁⢨⡿⢁⡿⣈
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣼⣿⡿⠙⡿⣿⠇⣼⢡⣿⢇⣴⣿⡿⣿⣿⠟⠁⣰⢋⡞⣠⣾⠋⠀⠈⣠⣟⣼⠃
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢰⣿⠏⠀⢰⡇⣿⢸⠇⣸⣿⣾⣿⠋⣴⣿⠏⡌⣰⣧⣞⣴⡿⠁⠀⣠⣾⣿⡞⠁
⠀⠀⠀⡄⠀⣿⣆⠀⠀⠀⣿⡏⠀⠀⢸⠃⠋⢸⠀⣿⣿⣿⠁⣸⣿⡏⣼⢠⣿⣿⣿⡿⠁⣠⣾⣿⡟⠉⠀⢀⣤⡾
⠀⠀⠀⡇⠀⢿⣿⡄⠀⠀⣿⠁⢠⡇⣾⠀⠀⠈⢀⣿⡟⠀⢠⣿⣿⢰⡏⢸⣿⣿⣿⠁⣼⣿⡿⠋⢀⣠⣶⣿⡟⠁⠠
⠀⠀⠀⣧⠀⡘⣿⣷⠀⠀⡿⠀⣸⣿⡟⠀⠀⠀⢸⣿⠀⠀⣼⣿⡇⣾⣇⢸⣿⣿⡿⣸⣿⡿⠁⢀⣾⣿⠟⠋⣠⠶⣡⠔⠁
⠀⠀⠀⣿⡄⠱⣍⢿⡆⠀⡇⢀⣿⣿⠇⠀⠀⠀⢸⡇⠀⠀⣿⢩⢣⣿⣿⢸⣿⡟⣿⣿⣿⠁⢠⡿⠋⣠⣴⣿⡷⠞⠁
⠀⠀⠀⢿⣿⡄⣿⣿⣷⠀⠃⣸⣿⡿⠀⡀⠀⠀⢸⠀⠀⢸⡇⡾⣸⣿⣿⢸⣿⣧⢹⣿⣿⠀⣿⢀⣼⣿⢟⠁
⠀⠀⠀⠘⣿⣿⣿⣿⡿⠀⢠⣿⡿⠁⠀⡇⠀⠀⠸⠀⢸⣼⣸⢣⣿⣿⢇⣾⣿⡇⣼⣿⣿⠀⠁⣼⣿⡿⠋
⠀⠀⠀⠀⠘⣿⢿⣿⡇⢠⣿⡿⠁⠀⢀⡇⠀⠀⠀⠀⣾⣿⢇⡞⢹⢟⣾⣿⣟⣴⣿⣿⠟⢀⣴⣿⠏⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠸⣿⣿⢁⣿⠟⢀⡞⢀⣼⡇⠀⠀⠀⣸⣿⠏⢌⢀⢫⣾⣿⣿⣿⢿⠿⠋⣠⠞⠋⠀⠀⠀⠀⣀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠘⠇⣼⠏⢀⡞⢀⣼⣿⠁⠀⠀⣰⣿⠋⠀⣼⣠⣟⣵⣿⠿⠋⠀⢀⣈⠄⠀⢀⣤⣴⠾⠛⣩⡟
⠀⠀⠀⠀⠀⠀⡈⣾⠃⢀⡾⣠⣾⣿⠇⠀⠀⣼⡿⠁⠀⣼⡇⣏⣿⡟⠁⢀⣠⠞⠋⠀⣠⡾⠋⠉⢠⣴⣾⡟
⠀⠀⠀⠀⠀⠠⢰⠏⠀⣼⣿⣿⣿⠋⠀⠀⣼⠟⠀⠀⣼⣿⣇⢻⡟⠀⣠⣾⡏⠀⣠⠞⠉⠀⣠⣶⣿⣿⡿⠀⣾
⠀⠀⠀⠀⠀⠀⡎⠀⢠⣿⣿⡟⠁⠀⠀⢼⠋⠀⠀⣰⣿⡿⢻⢸⠃⣰⣿⡿⠀⠀⠀⠀⢀⣼⣿⣟⣾⡿⠁⢠⢻
⠀⠀⠀⠀⠀⢠⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⢀⡼⠋⣾⡷⣄⣾⣰⣿⣿⠃⠀⠀⠀⢠⣿⣿⢻⣾⡿⠁⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢠⠏⣰⣿⡟⢇⣿⣿⣿⡿⠁⠀⠀⠀⣰⣿⣿⣿⣿⠏⠀⢠⠊⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠰⢣⣾⣿⣿⣿⣾⡿⠟⠋⠀⠀⠀⢀⣾⣿⣽⡿⠛⠁⢀⡴⠁⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣶⠀⠀⠀⡴⢋⣽⣿⠿⠋⠁⠀⠀⠀⠀⢀⣴⣿⠿⠟⠉⢀⣠⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠠⢸⣇⢰⡇⠀⢀⠞⣰⠟⠋⠀⢀⣀⣤⡶⣮⡤⠞⠋⣁⣀⣤⣴⣾⣿⣿⣿⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⡟⠀⠀⢀⠞⠁⠀⣠⡶⠟⠋⠁⠀⣀⡴⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⠁
⠀⠀⠀⠀⠀⠀⢠⣡⣿⡟⠀⠐⠀⣠⠄⢀⡾⢋⡠⠆⠀⠀⢊⣥⣶⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⢀⣠⡾⠁⠐⣡⡴⠋⠀⠀⣠⣴⠿⠛⠋⠉⠉⠁
⠀⠀⠀⠀⢰⡿⠋⢀⣤⣶⣿⠋⠀⣠⣾⠟⠁⠀⣠⠾⠋⠁
⠀⠀⠀⢀⠟⢠⣾⠿⠟⠋⠀⠐⠚⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢠⣶⣿⠯⠷⠦⠤⢤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡿⠻⣶⣮⣽⣿⣷⣶⣶⣭⠍⠛⠛⠿⠷⣶⣤⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⡈⠉⠛⠿⠿⣿⣿⣷⣶⣶⣤⣤⣤⣀⣈⣉⣿⡦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡀⠀⠀⠀⠀⠲⢤⣤⣬⣉⣉⣉⣉⣉⣉⣻⣿⣿⣷⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣄⡀⠀⠀⠀⣤⣤⣤⣭⣭⣭⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣉⠀⠀⠀⣻⣿⣿⣷⣾⣿⣿⣿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣉⠁⠀⠀⣹⣿⣿⣿⣿⡷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣉⡁⠀⠀⢤⣭⣭⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡍⠀⠀⣈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣗⣂⠀⠈⠭⣭⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣥⣤⡄⠘⢻⣿⣿⣤⣄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠙⠿⢿⣥⠀⠸⢿⣿⣿⣆⡈⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⣿⣗⡘⢿⣿⣿⣿⣤⡁⠀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠂⠀⢰⣦⠈⠀⠀⠸⠿⣦⠉⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠐⠀⠀⠙⠋⠀⡀⠀⢴⡄⠉⣧⢸⣿⣿⣿⠁⠸⣿⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠈⠀⠀⠀⢈⡀⢀⡙⠀⣿⣾⣿⣿⠏⠁⠀⠀⡀⠈⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⢀⠀⠀⢤⣤⣿⣿⣷⢰⣿⣿⡿⣣⣀⠀⠀⠀⠡⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠄⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢀⣴⣿⡇⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⢠⣾⣿⡟⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⢿⠀⢠⣿⣿⡿⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⠏⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣤⡀⠀⠀⢸⣿⣿⣿⣇⠀⠀⠈⣿⡀⣸⠏⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⢸⣿⣿⣿⣿⣦⠀⠀⠸⡇⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠹⣷⡈⢿⡆⠈⣿⣿⣿⣿⣿⣷⡀⠀⠁⠀⠀⠀⠀⣾⠏⠀⠀⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠈⠁⠈⠿⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⣤⠀⠀⠏⠀⠀⠀⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢀⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⣾⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠀⠀⠙⣿⣦⠀⠸⣿⣿⣿⣿⣿⡇⠀⣿⣿⣧⠀⠀⠀⠸⢏⣼⠟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄⠀⠀⢻⣿⡇⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡆⠀⢀⣀⠸⠁⠀⠀⣠⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣧⠀⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠁⢰⣿⠃⠀⢠⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣤⠀⠀⣿⣿⣿⣷⣿⣿⣿⠀⠀⠹⣿⣿⣿⣧⣠⣾⣿⡄⢰⠟⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡙⣷⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⠀⠴⢋⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠘⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡇⠒⠫⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⢹⣿⣿⡇⠙⢿⣿⡀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⠀⠀⠉⠙⠀⠀⠀⢀⣿⡿⠃⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠘⠋⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣦⣀⠀⠀⠀⠀⣀⣴⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠒⠒⠒⠒⡶⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⠉⠀⠀⢀⡤⠤⣤⡀⠀⠹⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢤⣠⠖⠋⠁⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠙⣆⠀⠹⣄⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⡾⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣤⠤⠞⠁⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢻⠀⠀⠀⠀⣀⡤⠖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⣇⠀⣧⡤⠖⣋⡥⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⣁⠴⠚⣹⣚⡥⢶⡋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢀⡴⠁⠀⣀⣠⢼⣦⡀⠳⣄⣀⣠⡤⠤⠶⢲⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⠉⠉⣉⡥⠚⠉⠀⠉⠉⠙⡇⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⠤⠤⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠉⠉⠙⠳⣄⠀⠀⠀⠐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⠋⠉⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⢀⣀⡀⠀⠀⠄⠀⠙⠀⠀⠀⠀⠀⠠⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⣠⡴⠚⠉⠀⠀⠀⠀⣠⡤⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⡀⢹⡆⠀⠀⠀⡾⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡴⠋⠁⠀⠀⠀⢀⣠⠖⠋⠁⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢁⡼⠁⠐⢁⡜⠀⠀⠀⢀⡤⠊⢀⠂
⠀⠀⠀⠀⠀⠀⠜⠁⢀⡴⠋⠀⠀⠀⠀⠀⣴⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢀⡴⠋⠀⢀⠤⠖⠉⣀⠴⠁⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁⢀⡠⠞⠋⠀⠀⠈⠀⣀⠤⠚⠁⠀⢀⠀⠐⢠⡞
⠀⠀⠀⠀⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠖⠚⠁⠀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠉⠀⢀⣠⠶⠊⢀⣤⠖⠋⠄
⠀⠀⢀⠞⠁⠀⠀⠀⠳⡄⠀⠀⠐⠶⠲⠶⠖⠒⠶⠒⠊⠉⠁⢀⣀⣤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢁⣤⠶⠊⠉⠀⢀⡜
⠀⠀⢸⡀⠀⠀⠀⠀⠄⠀⠉⣀⣒⣀⣀⣀⣀⣀⡤⡤⠤⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠂⠉⠁⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠻⢆⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⢀⣀⣀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠛⠉⠁⠀⠛⠲⢄⡰
⠀⠀⠀⠀⠀⠉⠛⠛⠒⠶⠶⠶⠶⠒⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⣠⠴⠋⢷
⠀⠀⠀⠀⠀⠉⠑⠖⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣤⠔⠒⠈⠁⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠖⠒⠒⠒⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢉⣁⣠⠤⠒⢂⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠶⠚⢛⣉⠩⠤⠒⠂⢁⣀⠴⠛⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠶⠊⠉⠀⠀⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠄⠐⠒⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠤⠤⠀⠐⠒⠒⠈⠉⠉⠀⠀⠀⢀⣀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡒⠂⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠉⠉⠉⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡴⠛⠁⠈⠉⠓⠶⠤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠤⢤⣤⣤⣤⠴⠶⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠒⠙⢢⣄⡀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣠
⠀⠀⠀⠀⠀⢠⠊⢀⡾⠃⠈⠓⠤⢄⡈⠉⠒⠒⠲⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣀⡀⣀⣀⡴⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠳⣼⡁⠀⠀⠀⠀⠀⠁⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢹⡋⠳⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⣤⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠠⠤⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠲⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣻⣶⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣙⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠶⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡀⢒⣠⣤⡶⠏⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⢀⣀⡀⠀⠈⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⣀⠽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠐⠛⠉⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠿⢦⣄
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⣀⣀⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠐⠒⠂⠀⠀⠀⠀⠠⠴⢞⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠂⠀⠀⠀⠀⢀⣠⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⠈⡀⠀⠀⣙⠇
⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣴⠋
⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠓⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠂⣮⣟⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠃⠀⠀⠈⠁⠀⠀⠈⠁⠹⣿⠿⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠙⠃⠀⠀⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠀⣲⣶⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠰⠾⠷⠶⠚⠋⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠖⠀⠀⣳⠁⠀⠀⠀⠀⠀⣀⣤⣶⣤⣄⡀⠠⣤⠠⣤
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⣴⢳⡆⠀⠈⠀⠒⠂⠀⠠⠦⠀⢸⣟⣿⣿⣿⣿⣯⣦⠘⡆⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⣿⣿⣿⣿⣿⣿⢷⡶⣅⣼⡄⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⣷⣿⣿⣿⣿⠟⡴⢀⡜⠉⢳⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣮⣐⣓⣃⣀⠈⠀⣜⠀⠘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⡀⠐⣖⡘⢇⣥⠀⡇⠀⠀⢲⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢩⡱⠉⠐⠘⠗⠤⣡⡼⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⢱⢄⣀⡘⢱⠒⠂⠀⢀⣀⢨⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠃⠀⠑⠒⠇⠀⠀⠀⢐⠁⠸⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⠶⠶⠶⢶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠋⣉⣁⣤⣤⣤⣶⣶⣶⣦⣤⣤⣈⡉⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⣉⣤⣴⡾⠿⠛⣋⣉⣭⣥⣤⣤⣤⣤⣬⣍⡙⢿⣦⡀⠈⠃
⠀⠀⠀⠀⠀⠀⣠⡿⢋⣴⣾⣿⣿⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣧⠀⢀
⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣴⣿⢿⡏⢀⡾
⠀⠀⠀⠀⠀⠈⠀⠀⠻⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠟⣛⣋⣭⣵⣶⣿⡿⢛⣵⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠰⣄⠀⠈⠻⣯⣝⣛⠻⠿⠿⠿⣿⣿⠿⠿⠿⠛⢋⣡⣴⣿⣿⠟⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣸⣻⣿⣿⣷⣶⣶⣤⣤⣤⣶⣶⣾⣿⣿⣿⡟⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁⠀⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡤⢤⣿⣿⣿⣿⣿⠥⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⣿⣿⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⠟⠓⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳

⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⡀⢀⡁⠀⠀⠀⠀⠀⢀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠴⠲⢤⣀⡐⠒⠂⠀⡀⠀⠄⠒
⠀⠀⠀⠑⣂⣾⢛⣻⠗⠶⣽⢳⣖⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠎⣰⣌⣩⢥⡓⢒⡦⣟⣩⠇⢀
⠀⠀⢰⣿⢻⣿⠿⠿⢿⣿⣿⣿⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⢳⣶⣩⠵⠲⢿⣿⣶⣾⣷⣆⣻⠇
⠀⠀⠀⣁⠩⢵⣶⡆⠈⠁⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢤⣄⣀⣀⣂⣤⠀⠀⠀⣠⣠⣤⡤
⠀⠀⠀⠉⠉⠉⠛⠛⠛⠒⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠉⠁⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⢤⣖⠚⢿⣤⠿⣿⣶⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠤⣞⠃⢿⠿⡇⢡⣜⠉⢀⣼⣿⠜⠥⣆⠰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⣉⡀⠴⠰⠀⠀⣐⣚⣙⣀⣐⣔⣀⣀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⠶⣄
⠀⢠⣶⡖⠒⠒⠚⠛⠉⢉⣁⣠⠼⣀⠈⠛⠶⢤⡀
⠀⢸⠉⠻⢶⣶⣉⡉⢩⢍⠑⠀⡀⠈⠑⠦⣄⣸⡇
⠀⠸⣄⡀⠀⠈⠉⣿⢄⡹⠝⢀⣨⣵⣶⡶⠟⣿⠁
⠀⠀⠘⡏⠓⠲⣤⠿⠶⠾⡟⠛⣿⠁⠈⢧⣀⣽⡦⡤⢤⣤⣄
⠀⠀⠀⠸⡄⠀⢻⡄⠀⠀⠳⣄⣹⠶⠞⠋⠉⠀⡐⢀⣾⣿⣿
⠀⠀⠀⠀⠙⢦⡀⠻⠦⠴⠞⠉⠀⠨⢀⠀⠀⢀⠀⢸⠛⢿⠛⢶⣶⣄
⠀⠀⢀⣤⣶⣾⣿⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠈⠀⢸⣴⣧⣸⣿⢿⣿
⠀⣰⣿⣿⣿⣿⣻⣿⣿⣿⣿⡽⠁⠀⠀⠀⠀⠀⠀⠈⠻⣁⠙⢟⢿⠟⠉⡉⡳⣦⣀⣤⣶⣶⣦⣄
⠀⣿⣿⡟⣿⣿⣧⣿⣿⣿⣿⣷⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢸⠀⠓⢠⠘⠻⢿⣿⣿⣿⡆
⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠦⣄⡉⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⣉⡡⠊⠻⣿⣿⣿⣀⡀
⠀⠈⠿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠉⠛⠲⢤⣤⣿⡿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠉⠉⢉⣳
⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⢄⣀⡠⠆⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣿⣿⣿⣿⣿⡿⢿⡧⠤⣀⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠉⠙⠒⠛⠛⠿⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠔⠒⠈⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠵⢶⣒⣠⣤⣄⣀⡀⠀⢀⡠⠀⠀⡥⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣉⠠⠄⠞⢼⣿⣯⢿⣷⡶⢰⢡⣶⣿⡢⠷⠋⠡⠀⣀⠉⢐⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⣥⠈⠁⡁⠂⠲⢤⣥⣟⠀⠰⠜⠛⠋⢀⡄⠀⢀⣾⢼⠃⠈
⠀⠀⠀⠀⠀⠀⢰⠻⠀⠀⠀⣿⣽⢗⣾⢿⠅⡀⠀⠀⠉⠉⠁⠶⣴⣾⢆⣾⡇⠀⡟⠁⣬⠆
⠀⠀⠀⠀⠠⠤⠈⠁⠀⠀⠀⠀⠀⠈⢡⠔⠈⠀⠀⠀⠀⣀⢐⠄⣽⣻⡎⢿⡀⠾⠤⠚⠁
⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠀⠀⠌⠂⠈⠀⠀⠀⣰⣻⡭⢂⠈⢋⠉⠚⠁
⠀⠀⠀⢻⣷⠄⠀⠉⠙⠛⠒⠠⠌⠀⠀⠀⠀⠒⢰⢱⡾⢀⣴⠋⠁
⠀⠀⠀⠀⠈⠑⠠⢀⡀⠀⢀⣤⣀⡀⠨⠀⣀⢰⣃⠸⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⠿⣿⠟⠛⢻⡟⠛⠻⣿⠿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡿⠛⠉⢧⠀⠀⢸⡄⠀⠘⠃⠀⢠⡇⠀⠀⡼⠉⠛⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⢻⡄⠀⠈⠀⠀⠘⠃⠀⠀⠀⠀⠘⠁⠀⠀⠁⠀⢠⠞⠀⠙⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁⠙⠆⠀⠀⠙⠀⠀⠀⠣⡊⢰⠉⡆⢳⢈⠵⡉⡆⠀⠀⠀⠋⠀⠀⠰⠋⠈⠻⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠋⠳⣄⠀⠀⠀⠀⡮⣶⠉⡂⠀⠈⠀⠉⠀⠀⠉⠁⠉⠀⡇⢼⣵⣉⠆⠀⠀⠀⣠⠞⠙⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣤⡀⠀⠈⢡⡄⡠⢄⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡀⢀⣈⠁⠀⢀⣤⢿⣿⣿⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠁⠀⠀⠘⠜⠳⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠣⠜⠀⠀⠈⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⣾⣿⣿⠉⠓⠶⠤⢠⡄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣤⠠⢄⠤⠶⠚⠉⣿⣿⣷
⠀⠀⠀⠀⢰⣿⣿⣧⣀⡀⠀⠀⠙⠋⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣝⣣⣬⣤⡤⢀⣀⣼⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣷⣦⣀⣤⣤⣴⣶⣶⣿⠿⠿⠛⠛⠋⠉⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⠶⠶⠶⠶⢈⠕⣏⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠉⠀⠀⢈⡝⢊⡱⣉⡃⠶⠶⠶⠶⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⡟⠉⠁⠀⠀⠀⠔⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠠⢤⡠⢄⠀⠀⠈⠉⢻⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣿⣿⣀⡤⠶⠒⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠐⠃⠐⠂⠒⠶⢤⣀⣿⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⠀⠀⠀⠀⠰⠀⠸⠷⠥⠏⠟⠿⠔⠸⠯⠇⠀⠆⠀⠀⠀⠀⣠⣶⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠃⠀⢀⡤⠴⣶⡶⢾⣿⣿⣿⣿⣿⠿⠛⠛⠛⢻⣟⣻⡿⢶⣶⣦⣄⠀⠀⠼⠃⣀⣀⣀
⠀⠀⠀⢀⣀⡀⠆⠀⢸⣷⣶⣤⣤⣭⣭⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⣶⡿⠿⢿⣛⣫⡆⠀⠀⠀⠤⣤⡉⠛⡆
⠀⠀⠀⡏⢡⣾⡛⠀⠈⢻⣿⠿⠿⠿⠿⠭⠭⢭⣭⣭⣭⣭⣠⣤⣿⣋⣉⣉⣩⣭⣵⡶⢀⣀⣤⠤⠾⡛⠀⢤⡀
⠀⠒⡀⠀⠀⢈⡉⠛⠛⠛⢶⣶⣶⣒⣒⡒⠀⠀⠀⠀⠠⣤⣤⣤⣤⣶⣶⣶⣿⣿⣿⡷⠀⠀⠀⠀⢀⣹⡇⠐⠟
⠀⠀⠀⠀⣀⠻⢤⣤⣤⣀⣘⣿⣿⣿⠿⠍⠉⠩⠤⠤⠤⠤⠬⠿⢿⣿⣿⣿⣿⣿⡿⠤⠤⠶⠖⠚⠋⠉
⠀⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⠻⣿⣯⣭⣍⣉⣑⠶⠶⠤⠤⠤⠤⣬⣭⣵⣶⣾⡟⢁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣴⠒⣤⡀⠛⣻⣿⣷⠒⢒⣲⣶⣤⣤⣤⣤⣶⣶⣿⣿⣻⣯⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣁⡉⠀⠸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⣭⣉⣛⣿⣿⣿⣷⣖⠛⠛⠋⠀⠈⠡⠶⠿⠞⣛⡋⠀⢀⣉⣷⠘⡦⠀⠀⠀⠀⠰⡿⠉⠞⠁⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠿⣿⣿⣯⢭⠝⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣏⡉⣐⣶⠀⠀⠀⠀⠀⠀⠙⠂⣴⠏⣠⠄⢈⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡗⠂⠈⠿⠿⢿⣿⣒⣶⣶⣖⠛⠛⡻⠿⠿⣟⣛⣿⣿⡁⠀⢀⣀⡀⠀⠀⠀⠘⠁⡴⠋⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠂⠀⠀⠬⠿⠿⠿⠷⣶⣶⣶⣾⣋⣙⡛⠟⠯⡿⣀⠒⠚⠋⢠⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⣤⣀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠉⢙⠛⠻⠿⢷⣶⣾⣿⣭⠿⣦⣤⠶⠟⠀⠀⠀⠉⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠘⠉⠀⣹⣆⣤⠤⠴⠖⠀⠀⠀⠀⢤⡘⠦⠤⢤⠀⠰⠾⠿⢿⡿⢿⡧⠀⠀⠪⠅⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⣤⠀⠀⠉⣥⠶⠖⠛⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⣷⣿⡏⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣿⡀⠀⢀⣄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡶⠀⠰⠰⣇⣼⣿⣋⠀⠸⠀⠀⠦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠉⠉⠀⠁

⠀⠀⠀⠀⠀⢀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⢀⢔⠆⡻⠁⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀
⠀⠀⠀⠠⠊⠄⠂⠀⠐⡌⢂⠀⠀⠀⠀⠀⠳⢣⢝⡤⣒⡤⡩⣪⣂⠀⠀⠀⠀⠀⠀⡂⠄⠀⠈⢠⠠
⠀⠀⠀⠀⠈⢀⠀⠀⠀⠸⢈⠀⠀⠀⠀⠀⢏⠇⣨⡫⠪⣎⣶⠡⢹⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠘⢌⠢⠄⠠⠔⠁⠎⠀⠀⠀⠀⢪⣪⣦⢉⣚⡡⢡⣆⡾⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠈⠂⠓⠶⢊⡲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠿⢷⣦⡀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⣠⠖⢋⣉⣉⡙⠲⣄
⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠠⠀⠀⠀⢰⠋⣴⡾⠟⠻⢦⡙⣿⡄⠀⠀⠀⢀⣾⡿⠉⠀⠀⠈⠻⣿⣆⠀⠀⠀⣼⢁⡾⣫⠖⠲⢍⢷⡘⣇
⠀⠀⠀⠀⠀⠸⡏⠀⢈⣇⡀⠀⠀⠀⠀⠀⢸⡟⢰⠃⠀⢨⠇⣿⡇⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⡇⢸⡱⡇⠀⠀⢸⢸⠇⡿
⠀⠀⠀⠀⠀⠈⠛⠳⠻⠑⠀⢀⠀⠀⠀⠀⠸⣿⡘⢷⣤⣤⡾⠟⣠⠇⠀⠀⠘⣿⣧⡀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠱⡈⢧⡙⠮⠬⠷⣋⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣤⣤⣴⠾⠋⠀⠀⠀⠀⠈⠻⢿⣿⡿⡿⠛⠁⠀⠀⠀⠀⠀⠑⠄⠉⠛⠛⠛⠉

⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⠛⠻⢷⣦⡀⠀⠀⠀⠀⣠⡲⣝⣛⣛⣦⣢⢄
⠀⠀⠀⢠⣿⣿⡽⠛⠻⢿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢡⡖⣫⠭⢍⠲⡘⣷⡀⠀⠀⢰⢯⡿⡯⣒⣒⣟⣟⣝⢆
⠀⠀⠀⣿⣿⢿⠀⠀⠈⣮⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣿⢸⠀⠀⠀⡃⡗⢸⡇⠀⠀⢬⢺⣸⣸⡥⣼⣹⢹⢸⣺
⠀⠀⠀⢘⣿⣿⣗⣶⣖⣵⣿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠹⣌⠢⢔⡮⠞⣡⡿⠀⠀⠀⠸⣷⣻⣿⠯⠝⣿⣿⣟⡎
⠀⠀⠀⠀⠙⠻⡿⠿⢿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⠈⠛⠶⠶⠛⠉⠀⠀⠀⠀⠀⠈⠓⠯⠟⡯⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⡱⠊⠉⠐⢝⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣭⣶⣾⣷⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗⡀⠀⠀⠈⡆⣾⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣾⡏⠀⠈⢹⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣝⡢⠤⢚⣔⣿⠧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⢿⣷⣾⠟⡣⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠁⠀⠈⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠩⠖⠊⠑⢊⣉⣙⣛⠤⢤⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢁⣀⡀⠀⢀⠒⠶⠮⠭⠉⠉⠛⠒⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠟⠄⠁⠀⠀⠀⠀⠈⠓⠠⠄⣀⡀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠙⠀⠀⣀⡀⠀⠀⠀⠨⢝⣶⣶⣦⠄⠙⢦⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠐⠋⠀⠀⠀⢀⡀⠀⠀⠙⠛⡋⠀⠀⡜⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⡤⠤⠒⠈⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠐⠁⢠⠞⢀⠴⠃
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⠂⣡⠞⠁
⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⠰⡄⠀⠉⡀⡤⠀⡀
⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠠⠖⠋⠁⠀⠀⠉⠛⠓⠒⠓⠂⣸⠋⠀⢈⠁⠘⠁⠀⠀⠀⠀⡄⠀⠄⠁⠢⡀
⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢇⠀⠀⠀⠠⠴⠒⣄⠀⠀⠈⠢⠁⢀⠠⠔⠋
⠀⠀⠀⠘⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠐⠒⠉⠀⠀⠀⠈⠳⣀⠀⠀⠰⣆⠌⠀⠀⡔⠉⠈⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠓⢄⡀⠀⡠⠊⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠋⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡓⠀⠠⠄⣀⣀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠇⠀⠀⠀⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠊⠀⠘⠀⠀⠈⠁⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠁
⠀⠀⠀⠀⠀⣠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠈⠚⠀⡩⠊
⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⣠⠄⠴⡉⠉⢣⠀⡝⠛⢲⡀
⠀⠀⠀⢀⡤⢧⡀⣠⣿⣶⣿⣿⣿⣿⣿⣷⣤⡤⠤⣄
⠀⠀⢰⡋⠀⢀⣾⣿⣿⠿⠿⣿⠛⠛⢯⠀⠙⡆⣀⣠⠃
⠀⠀⢠⠟⠒⣿⣿⣿⡇⠀⠀⠈⡆⠀⣼⡤⠞⠫⡀⠀⢱
⠀⠀⠱⡄⠀⣿⣿⣿⠗⠤⣀⣠⠞⠉⠀⣇⠀⣠⠗⢴⠋
⠀⠀⠀⠙⠂⢿⣿⡿⠀⡀⠀⠑⠤⠤⠜⠁⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠉⢆
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠉⠙⠂⠀⠀⢄⢤⣀⠴⢂⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢠⠈⢁⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⣀⠤⠤⣀⣠⠖⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠰⠀⣠⠀⡄⢸⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠊⠝⠉⠙⠶⠤⣤⣤⠤⠚⣦⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠁⠀⠀⠀⠀⠈⠲⠤⣠⣀⣀⣀⣀⠴⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⢰⠓⠦⡤⠶⠦⣄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠊⠀⠀⠀⠀⠀⠀⠉⠉⠀⢩⠀⢸⡉⠑⠦⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣄⣀⣀⣀⠀⠀⠈⠙⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⠇
⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⠒⠒⠒⠒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠋⠁
⠀⠀⠀⠀⡠⠊⠡⠀⠀⠀⠄⠀⠀⠒⠀⠠⡈⢳⡀
⠀⠀⠀⠀⣇⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⠀⢀⣠⡇
⠀⠀⠀⠀⣧⠉⠒⠲⠦⡤⠤⠄⠔⢶⠒⢋⣽⡞⣽⡄
⠀⠀⠀⠀⣿⣙⣓⣚⣺⣦⣤⠤⢶⡧⢾⠛⠒⢏⡹⡀
⠀⠀⠀⠀⢼⠀⣉⢹⠀⢸⠉⠉⡉⠚⠐⢅⠠⢃⡵⢇
⠀⠀⠀⠠⡛⢦⣄⣙⣤⣼⣉⣉⣥⡤⡴⠖⠚⣹⣤⡏
⠀⠀⠀⠀⠸⡖⠢⠤⢤⣌⣀⣀⣠⣤⣏⡴⢾⠉⢠⠃
⠀⠀⠀⠀⠀⠱⢄⣁⠀⠀⠉⠁⠀⠠⠂⣀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠀⠀⠀⠀⢀⡀⠀⡏⠈⠣⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠙⡆⠀⠀⠀⢨⢷⡀⠰⢈⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⡟⠀⠀⠀⠘⣕⠙⠢⡀⠁⠪⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡀⠀⠀⠀⠀⠀⠱⡨⢀⠌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠊⣣⠀⣴⠀⠀⠀⠀⠘⢤⡀⠀⠂⠑⣄⠀⠀⠀⠀⢠⠜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠌⡠⢁⡿⢸⠹⣄⢀⡀⠀⠀⠀⠙⣬⠀⢑⠙⡆⠀⠀⡔⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢂⠅⡢⠰⡇⠀⠑⠬⣢⠹⡳⣄⠀⠀⠸⡆⠐⠂⢻⠀⠀⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠐⠁⣄⠀⢷⡀⠀⠀⠈⠀⠇⡈⣷⠀⡸⠂⡷⠀⢹⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⢸⡨⠀⡄⠣⡀⠑⢄⠀⢀⡬⠁⠀⣾⡼⠃⡐⢨⠀⡵⠀⢸⡇⣷
⠀⠀⠀⠀⠀⠘⡈⠑⢢⡀⠀⠀⢣⠀⠸⡄⠈⠢⡈⢳⡞⢀⠞⠀⠍⢀⠞⠀⠎⢰⠃⠀⡸⡁⢸⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠂⠀⣙⡀⠀⠀⠑⣄⠐⡄⠀⠐⡀⢠⠃⠀⣇⠔⠁⠀⠌⢠⠏⣠⡞⠃⠀⠀⡇⠀⠀⠀⣠⢷
⠀⠀⠀⠀⠀⠀⠈⠹⢖⠤⣁⠀⠀⠀⠘⡆⠈⠀⠀⢱⡎⠀⠀⠋⠀⠀⡜⢠⠿⠚⠉⣀⡤⠀⢰⠇⠀⢀⠌⣁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⢦⠀⠀⢿⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠁⢀⠄⠒⠉⡔⠀⢰⠏⠀⠀⠧⢊⡁⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢦⡀⢳⢀⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⢀⠎⢀⡴⠃⢀⡀⢀⡔⢉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⡀⠰⡈⡟⢁⠌⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⡞⢡⣴⡏⢰⠋⠈⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠃⠀⠆⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠀⣷⠟⠲⣇⡟⡰⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡀⠇⠸⠀⠟⠱⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢴⠀⠱⣄⡃⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠔⡡⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⢀⠀⠀⠀⠈⠀⠀⠤⠀⠀⢀⣀⣀⣀⡀⠀⠔⠈⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠈⠀⠈⠐⠠⠐⠨⠀⡂⠄⡀⠄⠠⠀⡠⢀⠐⠀⠠⢐⠀⢀⠀⡠⢔⠢⢀⡐⢀⠔⡢

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⡖⢺⠛⢻⠛⢛⡗⢲⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢶⡋⢻⡀⢷⠈⣇⠘⠂⠸⠀⠘⠁⡾⠀⠟⢠⠏⢹⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡋⠳⡄⠛⠀⠓⠀⢀⢀⢀⠀⠀⠀⡀⢀⢀⠀⠀⠛⠀⠋⣰⠋⢙⡶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢧⡀⠙⠀⠉⡀⢀⠀⠀⠀⠘⠘⠀⠀⠀⠐⠂⠘⠀⠀⠀⠀⠀⠁⠐⠋⣀⡼⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠦⡄⠙⠀⠀⠀⠋⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠼⠀⠀⠘⠁⣠⠖⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡉⠓⠦⠀⡄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄⡄⠀⠴⠋⣉⣧
⠀⠀⠀⠀⠀⠀⠀⣸⣁⣉⠃⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡾⠋⠁⠁⠀⠀⠉⣉⣈⣧
⠀⠀⠀⠀⠀⠀⢠⣯⣬⡍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣩⠞⠋⠀⠀⠀⠀⠀⠀⠉⢩⣥⣽⡆
⠀⠀⠀⠀⠀⠀⣼⠤⠤⠤⠀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢴⣞⠉⢀⡴⠋⠁⠀⠀⠀⠀⠀⢦⡆⡆⠀⠤⠤⠤⣧
⠀⠀⠀⠀⠀⠀⡿⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠈⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠒⠒⢿
⠀⠀⠀⠀⠀⠀⡟⠙⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⢻
⠀⠀⠀⠀⠀⠀⣿⠋⠉⡀⠀⡀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⣀⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⡆⠀⣀⠉⠙⣿
⠀⠀⠀⠀⠀⠀⢸⡋⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⣉⠉⢙⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠋⣁⣤⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠙⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣏⣡⡤⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⢨⡇⡇⡇⠀⠀⢤⣉⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⠇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⢶⣾⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠉⠁⠀⢀⣀⣀⣈⣉⠁⠀⠈⠋⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⢀⣤⣶⣿⣿⣿⡿⣿⣿⣿⣷⣦⣄⠀⠀⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢀⣾⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣷⣄⠀⠘⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⣰⣿⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⣿⣆⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⡸⠿⣿⣿⣿⣿⣿⠋⣤⡉⠻⠿⠿⠿⣿⠿⠿⣿⣷⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠆⣸⣿⣿⣿⣿⣿⣿⡆⢉⣤⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠆⢹⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢿⣿⣟⣡⣿⣿⣁⣾⣿⣿⣿⣿⣿⣍⣹⣿⣿⠃⠀⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠙⢿⣿⣿⣿⣿⣿⡉⣿⣿⣿⣿⣿⣿⠟⠁⠀⣤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠉⠛⠿⠿⣿⣿⣿⠿⠿⠛⠉⠀⠀⠰⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠦⠀⣀⠀⠀⠀⠀⠀⣀⠀⣤⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠀⠛⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠰⠀⠀⠀⢀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⣩⣴⣦⠠⠖⠞⠈⠀⢠⣤⣾⠋⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⣴⣿⣿⣿⣿⣄⠤⢀⣄⣶⠿⠦⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⣼⣿⡿⢟⣿⣿⣿⣿⣦⠊⡏⠀⠀⠀⠀⣰⠈⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣰⣿⡿⡏⠟⠈⠉⠉⠙⠛⢽⠷⣄⠀⢀⠀⢐⡾⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⢰⣿⡿⣱⠋⠀⠀⠀⠀⠀⠀⠈⢻⠙⢿⣦⣄⢛⡀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⢰⡆⡄⢰⠏⣠⡼⣿⣂⣹⡀⠀⠀⠀⠀⠀⠀⠀⣺⠁⠀⣼⢻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠘⠈⠀⠀⠀⠀⠀⠎⣸⣿⣯⣤⣿⣿⢇⡀⠀⠀⠀⠀⠀⣸⠆⢀⣾⣛⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠖⠂⣀⣼⠿⠳⠱⠿⣿⣿⣷⣿⠂⠶⠆⠾⠛⢀⡤⣮⣼⡿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⢀⣀⣀⡀⠀⡠⠖⠋⠁⡀⠄⠈⠋⠃⣸⣿⠉⠃⠀⠀⠀⠈⠸⢗⡷⡿⠛⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠋⠉⢁⠴⠊⣀⠴⠃⡁⢀⣤⠀⣠⠞⠛⠬⡁⠀⢄⣠⣤⣾⣿⣳⡴⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠘⠈⠀⠒⠋⠀⠈⢀⠜⠁⣀⣀⣠⣬⣿⣾⠿⢿⠝⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠈⢀⣠⣴⣾⡻⡇⠠⠴⣋⣾⣾⣿⣿⠿⠛⠉⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢑⣣⣾⣟⡻⣿⣿⣿⣄⠙⢦⣙⢿⡟⠉⠀⠀⠀⠀⠀⣻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠞⢫⣵⣾⣿⢯⠻⢿⣿⣿⣦⣽⠁⠈⢸⡀⠀⠀⠀⢀⣲⡃
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠠⠐⣠⣾⣿⣿⣿⣄⡀⢀⣙⣿⢿⠇⠀⢐⠸⣦⠀⠀⠀⣴⡟⠠⠁
⠀⠀⠀⠀⠀⠠⡿⠃⡠⠔⣺⣿⣿⣿⣿⡿⠋⢉⡿⠟⢫⠎⠀⠀⠈⣸⠃⠀⠀⡲⠏⠐⠁
⠀⠀⠀⠀⠀⡰⠁⢀⠠⠊⠉⠛⠿⠋⢉⣠⣾⠿⠀⠀⠀⠐⠀⠀⠐⢁⠠⠞⠉
⠀⠀⠀⠀⢼⠇⢐⡅⠀⠀⠀⠀⠀⣴⠏⠙⠁
⠀⠀⠀⡀⠀⠀⠞⢀⡤⠤⠒⠋⠐⣥⠖⠁
⠀⠀⣀⡤⠀⠴⠋⠉⣀⠄⠴⠂
⠀⠀⠹⢃⣠⡴⠾⠋
⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⢰⡿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢠⢿⡅⠀⠀⠀⠀⠀⠀⠀⠸⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢸⢻⡆⠀⣼⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇⠀⣿⠈⣇⠀⠀⠀⠀⠀⢀⠀⠀⢹⡄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣇
⠀⠀⠀⠀⠀⠀⣸⠃⢸⠃⠀⢹⠀⢻⡀⠀⠀⠀⠀⠀⢷⡄⠀⢳⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⠇⠀⣼⠀⠀⢸⡇⠈⣇⠀⠀⠀⠀⠀⢸⢹⡀⠈⢧⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⡞⠀⠀⡏⠀⠀⠀⣇⠀⢸⡄⠀⠀⠀⠀⣸⠈⡇⠀⠈⣇⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⡿⡇⠀⢸⡆⢻⡀⠀⠀⠀⠀⠀⠀⠀⡏⣿
⠀⠀⠀⠀⢰⠇⠀⠀⡇⠀⠀⠀⣿⠀⠀⣧⠀⠀⠀⠀⡟⠀⡇⠀⠀⠸⡆⠀⠀⢻⡀⠀⠀⠀⠀⢰⠃⡇⠀⠀⣧⠈⣇⠀⠀⠀⠀⠀⠀⠀⣇⢹⡀
⠀⠀⠀⠀⢸⠀⠀⠀⢿⣤⣀⣴⠏⠀⠀⢻⢠⣇⠀⠀⡇⠀⣇⠀⠀⢀⣇⠀⠀⠀⣇⠀⠀⠀⠀⡟⠀⡇⠀⠀⢹⠀⢸⡄⠀⠀⠀⠀⠀⠀⢻⠈⣧
⠀⠀⣄⠀⢸⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⡿⢸⢹⡄⠀⣧⠀⢻⡶⠶⠾⡟⠀⠀⠀⡿⢠⡄⠀⢠⡇⠀⣧⣀⣀⡼⠀⠀⣧⠀⠀⡄⠀⠀⠀⠈⣇⠘⢧
⠀⠀⣿⡇⠸⡆⠀⢀⠀⠀⠀⠀⠀⠀⢠⠇⠘⡆⢳⡀⠸⡄⠀⠙⠀⠀⠁⠀⠀⢀⡇⣿⡇⠀⠸⡇⠀⠘⠿⠋⠁⠀⠀⢹⠀⠀⢹⣆⠀⠀⠀⢹⡄⠈⢳⡄
⠀⢀⡇⣿⠀⢧⠀⡾⣆⠀⠀⠀⠀⢀⡟⠀⠀⢻⡈⢧⠀⠹⣆⠀⠀⠀⠀⠀⠀⣼⠁⢷⢻⡀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡟⡆⠀⠀⠀⢷⠀⠀⠙⣆
⡀⣸⠁⣿⠀⠸⣶⡇⢸⡄⠀⠀⠀⡾⠀⠀⠀⠀⣧⠈⣧⠀⠘⢦⠀⠀⠀⠀⢠⡇⠀⢸⡆⢳⠀⢹⡀⠀⠀⠀⣶⠷⣦⣸⠀⠀⠸⡇⢹⡀⠀⠀⠘⡆⠀⠀⠘
⡇⡏⠀⣿⠀⠀⢹⡇⠀⡇⠀⠀⢰⡇⠀⠀⠀⠀⣿⠀⢸⡄⠀⠈⢷⡀⠀⠀⢸⣇⠀⣸⡇⠘⡇⠀⢷⠀⠀⠀⡇⠀⢸⡏⠀⠀⢸⡇⠈⡇⠀⠀⠀⢻
⣿⣷⠀⢻⣄⠀⠀⠀⢠⡇⠀⠀⠸⣧⠀⠀⠀⣠⣿⠀⢘⡇⠀⠀⠘⣧⠀⠀⠘⣿⠛⠉⠁⢠⡇⠀⠈⢧⠀⠀⢳⡀⠀⠀⠀⠀⣸⠁⠀⣷⠀⠀⠀⢸⡆
⠹⣿⡄⠀⠻⣷⠶⠾⣿⠃⠀⠀⠀⢻⣿⠷⠿⠻⠃⠀⢸⠃⠀⠀⠀⣿⡆⠀⠀⠈⠀⠀⠀⢸⠁⠀⠀⠈⢧⠀⠀⠻⢶⣤⣴⡾⠋⠀⠀⡟⠀⠀⠀⣸⡇
⠀⠈⠻⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣦⣤⣤⣴⣿⡇⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⣀⣼⣿⡆⠀⠀⠀⠀⠀⠀⠀⣼⠻⠿⠛⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⠀⠈⠉⠉⠉⣿⠃⠀⠀⠀⠀⠀⠀⠘⢿⠛⠛⠋⠁⠈⣇⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠤⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢈⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⠁⠀⠀⠀⠀⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣭⣋⣿⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⣀⡀⠀⡀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⡀⠈⢁⠀⢛⡝⠾⠁⠀⠀⣀⣤⣤⣤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠿⠀⢸⡴⠀⠀⢀⣀⣤⣿⠿⠛⠉⠙⢆⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⠈⢁⠀⠿⣄⣠⣶⣿⡿⠋⠁⠀⠀⠀⠀⠈⣯⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⣉⠉⠉⠑⡆⠀⠐⠒⢤⣤⡄⢁⣴⡏⠁⠠⢤⡉⢙⣥⣀⣀⣴⠋⢠⣾⢦⠀⣸⣟⣦⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣴⡿⠛⠋⠉⠉⠁⠀⣀⡇⠀⠀⠀⠈⠟⣤⣼⡿⠃⠘⠛⠚⠛⠋⢀⣤⠎⠀⢠⡿⠉⠉⠉⠑⠿⣿⡿⠰
⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⠟⠉⢹⣿⠃⠀⠀⠀⠀⠀⠀⡸⣍⣁⣀⠀⠀⠢⢤⢙⡄⠘⠂⠀⠀⠀⠀⠙⠁⠀⣰⡿⠁⠀⠀⠀⠀⠀⠈⠋
⠀⠲⣿⣿⣿⣿⣿⣿⣿⢿⡿⠃⠀⠀⢸⣿⣦⣀⡂⠀⠀⠀⠀⢸⠀⠀⠈⠉⠓⠂⠙⠨⡅⠀⠀⠀⠀⠈⢻⣦⣤⣿⠋
⠀⠈⠛⠿⠿⠟⠉⠽⠗⠊⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣏⠀⠀⢨⢸⣶⣄⡀⠀⠀⢸⡇⢠⡖⠈⠀⠀⢀⡈⢿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⡈⠁⠀⠘⢰⡮⠟⣿⣶⣦⣼⡇⠀⣱⢀⠔⠀⣉⢀⣼⢗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⠀⠀⠉⠛⠿⠿⣷⣿⣻⠄⠈⠻⣿⣋⠈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠙⠛⠛⠋⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠶⢀⣈⣁⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣤⡷⠀⠈⠛⢿⣦⣤⣤⣤⡚⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣤⣀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢠⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣀⠘⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⡎⠀⢱⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠃⠀⠀⠈⠉⠁⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠈⠇⠀⠀⠀⠀⢀⠊⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⠔⠄⠀⢀⠔⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠸⣒⢶⠌⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠌⠁⠀⠀⠁⠀⠘
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡢⠀⠐⠐⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠢⠐⠀⠉⠀⠂⢄⠀⠠⠀⠀⠀⢀⠄⠀⠈⠁⠃⠀⠀⠀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣊⣀⣀⣀⣀⣀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⡴⣫⣭⣥⣭⣬⣥⣬⣥⣬⣍⢧⠀⡰⠋⠀⠀⠀⠉⢦⠀⡰⠋⠀⣀⠀⠙⢦⠀⣰⠟⠋⠉⠉⠻⣦⠀⡴⠋⠁⠀⠀⠙⢦⠀⡴⠋⠀⠀⠀⠙⢆
⠀⠀⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠀⡇⠀⢀⣴⡆⠀⠈⣶⡇⠀⣾⣟⣷⡀⢈⡆⣿⠀⠀⠶⠊⠁⣻⢶⡇⠀⠀⠶⠤⠤⢸⣶⡃⠀⠀⠶⠂⠀⢸⡆
⠀⠀⠻⣿⣿⣿⣿⣶⣤⣿⣿⣿⣿⠟⠀⠣⡀⣸⣿⣧⢀⡼⠀⠳⣀⠈⠉⠁⢈⡼⠀⠻⣦⠀⠀⠐⣰⠟⠀⠳⣤⡤⡀⣤⣤⠞⠀⢣⣠⣤⣶⣤⣄⠜
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⣙⣛⣋⣉⠀⠀⠀⠈⠙⠒⠛⠉⠀⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠈⠙⢛⣋⣁⣀⣀⣀⣈⣙⣛⡋⠁
⠀⠀⠀⣠⣶⡿⠿⠿⣶⣤⡀⠀⠀⠀⢠⠔⠋⡁⢠⡄⣈⠙⠲⣄⠀⠀⠀⠀⣠⠖⢋⠉⡉⢉⡓⢦⡀⠀⠀⠀⣴⣿⣿⢿⠛⣟⠛⠛⣻⠛⡿⣿⣿⣦
⠀⠀⣾⡟⣵⣿⣿⣿⣮⢻⣷⠀⠀⣰⠃⢀⠂⠀⠀⢠⠀⠐⣀⠘⣆⠀⠀⡼⠁⠰⠈⠀⠀⢈⠤⠂⠙⣆⠀⢰⣿⣉⣧⣼⡶⠟⠚⠓⠻⢶⣧⣼⣋⣿⡆
⠀⠀⣿⣥⣿⣯⣤⣽⣿⣭⣿⠆⠀⡏⠀⠀⠀⠀⣾⣿⠀⠀⠠⠄⢸⠀⢸⡇⠀⢁⣀⡀⠀⣋⣀⢩⡄⢹⠀⢸⣿⣿⠟⢉⣀⣠⣤⣤⣄⣈⡉⠻⣿⣿⡇
⠀⠀⠉⠉⠉⣉⣉⣉⠉⠉⠉⠁⠀⣇⣀⣀⣀⣀⣘⣃⣀⣀⣀⣀⣸⠀⢸⣧⣶⣿⣿⣿⣾⣿⣿⣷⣦⣼⠀⠸⣿⣿⣾⣿⣿⣿⣿⣾⣿⣿⣿⣷⣿⣿⠇
⠀⠀⢠⡴⠋⠉⠉⠉⠙⢦⡀⠀⠀⢀⣠⡤⠤⢤⣄⠀⠀⠀⠀⢀⣠⡤⣤⣄⡀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣠⣤⣤⣤
⠀⢠⡏⠀⠀⠀⣀⠀⣀⡄⢻⠀⢠⠏⠁⠀⠀⠀⠈⢳⡄⠀⣴⠫⠐⠀⠀⠠⠙⣦⠀⢰⣿⠟⣉⣤⣤⣤⣌⠙⢿⡇⠀⢰⣿⣿⣿⠟⠁⠀⢀
⠀⠸⡇⠀⠀⠘⠛⠁⠀⠀⣸⠁⡟⠀⠀⠰⣶⠖⠀⠀⣧⢸⡇⠀⣠⡐⢢⣄⠩⢸⡆⢸⢃⣾⣿⡿⠟⠛⣿⣷⣌⡿⠀⢸⣿⣿⡏⠀⠀⠀⠀⠑⢄⡀
⠀⠀⠻⣄⠀⢀⠀⠀⢀⣠⠋⠀⠓⠒⠒⠒⠒⠒⠒⠒⠋⠘⠛⠛⠛⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠀⠀⠀⠛⠛⠛⠃⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠙⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠲⣤⡀⠀⠀⣠⠶⠛⠛⠛⠳⣦⡀⠀⠀⣠⡶⠛⠛⠛⢲⣄⠀⠀⠀⣠⠶⠛⠛⠛⠶⣤⡀
⠀⠀⢀⡴⠶⠶⡶⢶⠶⠶⠶⠶⢤⡀⠀⣼⠃⠠⣀⠀⠀⠀⠈⢳⡀⣼⠁⠀⠳⡀⠀⠀⡈⢷⡀⣼⠃⠀⠀⠀⠀⠀⠈⣷⠀⣼⠃⠀⢀⠀⠀⠀⠈⢷
⠀⠀⣿⣴⣿⣿⣿⣿⣿⣷⣎⢁⡨⡇⠀⣿⠀⠀⠈⠻⠂⠀⠀⢸⠇⣗⠀⠀⠀⠻⠂⠀⡂⢸⠇⣇⠀⠀⠀⠛⠦⣄⠀⣸⠀⣿⠀⠀⠀⠻⠂⠀⠀⣸⠇
⠀⠀⣿⣿⣿⣿⡿⠛⠛⣿⣿⣀⣀⡇⠀⠘⣦⡀⠀⠀⠀⠀⣠⠟⠀⠹⣄⠰⣶⣶⣶⡦⣠⠟⠀⠹⣦⡀⠀⠀⠀⢀⣰⠏⠀⠹⣆⣠⣤⣤⣤⡤⣠⠟
⠀⠀⠙⠛⠛⠛⠛⠓⠒⠛⠛⠛⠛⠃⠀⠀⠀⠙⠓⠖⠒⠛⠁⠀⠀⠀⠈⠙⠲⠶⠖⠚⠁⠀⠀⠀⠈⠙⠒⠶⠒⠋⠁⠀⠀⠀⠈⠙⠛⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⢄⠈⢷⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⣸⠃⢻⡀⠀⢀⣾⠀⠀⢹⡆⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⣸⠇⢰⠀⠀⠀⠀⠀⢠⠈⢻⡄⠀⡄⠀⢠⣿⠀⠈⣷⠀⢸⣷⡀⠀⣸⡇⠀⠀⠀⢠⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢀⠀⣿⠀⠘⣧⠀⠀⠀⠀⣼⠀⠀⣿⠀⣧⠀⠀⢿⣷⣶⡿⠀⠸⣿⣿⣿⡿⠀⣧⠀⠀⣾⠀⠀⠀⣿⠀⣇
⠀⠀⠀⠀⢸⡄⠀⢀⣾⠇⢾⡀⣿⡄⠀⢸⣧⠀⠀⣼⣇⠀⣠⡿⠀⢻⣧⠀⠀⢻⣿⡇⠀⠀⢻⣿⣿⠁⠀⠹⣆⠀⢹⣷⣤⣶⡏⠀⢹⡄
⠀⠀⢀⣴⡿⠁⠀⣾⡟⠀⠈⣷⠘⣿⣷⣿⠇⡄⠀⢿⣿⣿⣿⠁⠀⢀⣿⠀⠀⠈⣿⣧⡀⢀⣾⣿⣿⡀⠀⠀⣹⡆⠀⢻⣿⣿⡀⠀⢀⣿
⠀⢀⣾⡟⠀⠀⠀⣿⣷⣄⣤⡟⠀⠘⣿⡟⠀⢳⡀⠈⢻⣿⣿⣄⣠⣼⣿⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⡿⠁⠀⠀⣿⣿⣷⣶⣿⣿
⠀⠈⣿⣇⠀⠀⢀⣿⣿⣿⣟⠀⠀⢀⣿⡀⠀⢘⣇⠀⠀⢹⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣷⣶⣿⣿⣿⣿⣿⣦⣤⣾⣿⣷⣶⣾⣿⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠿⠿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠿⠛⠉⠀⠀⠀⣀⡀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠛⠿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠀⠀⡀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢀⡀⠀⠙⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠻⣷⡀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠿⠃⠀⠀⠀⢀⣾⠟⠀⠀⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⢀⡀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠏⠀⠀⠀⠀⢀⡀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠙⣿⣷⡄
⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⣴⣿⡟⠁⠀⢠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠆⠀⠈⢻⣿⣆
⠀⠀⣰⣿⡿⠁⠀⠀⠀⠈⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠁⠀⠀⠀⠈⢿⣿⣆
⠀⢀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀
⠀⣸⣿⡟⠀⠀⢶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⡶⠀⠀⢻⣿⣇
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣷⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⣿⣿
⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣶⣭⡻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣿⡇⠀⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿
⠈⠛⠛⠛⠛⠛⠛⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠈⠛⠿⠿⢿⠿⠟⠁⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣤⣤⣀⣀⣀⣀⣤⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⡿⠿⠟⠋⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠤⠀⠀⠀⠀⠄⠄⠄⠀⠆⠄⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠂⢀⠀⠄⠀⠂⠂

⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠉⠁
⡀⠀⠀⠀⠀⠀⣀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⠛⠋
⣇⠀⠀⠀⠠⠾⢿⢛⣩⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠋
⣿⣿⠖⠀⠀⠀⠸⡿⢻⡟⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⣿⣿⠟⠁
⣷⣦⣤⣀⡀⠀⠘⠒⢻⣧⣿⣤⡞⠋⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⡿⠋
⣿⣿⣿⣿⣿⣏⣀⣀⡙⣿⣿⣿⣄⣤⣿⣥⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡟⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣢⣼⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣴⣤⣤⣤⣤⣤⣤⣄⣤⣾⣦⣤⣄⣀⣴⣿⣿⣿⣤⣀⣿⣀⣀⣰⣾⣿⣂⣀⣀⣤⣶⣾⣦⣰⣾⣇⣸⡃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢙⣻⡽⣦⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⣠⠾⠷⣴⣯⣙⠻⠾⠁⠀⠀⠀⢀⠽⠿⠟⠉⠁⠀⠶⠶⢬⡉⠙⠛⠷⣄⠀⢸
⠀⠀⠀⠀⠀⢸⣇⣀⠀⠀⣹⡿⣿⠆⠀⠀⣠⠞⢡⠊⠙⡆⠀⢀⠔⠋⠉⣧⠀⠀⠀⠀⠘⣷⣼

⠀⠀⠀⠀⠀⣼⣿⣿⣷⣸⣿⣿⠃⠀⠀⡴⠁⠠⡇⣴⣷⠁⢠⠃⣰⣶⡰⠃⠀⠀⠀⠀⠀⣽⣿⣦⣀⣀
⠀⠀⠀⠀⣼⣿⣿⠉⣽⣿⣷⠃⠀⠀⡼⢁⡤⠀⢸⠉⠃⠀⠹⠤⠿⠾⠁⠀⠀⠀⠀⠀⠀⠻⣿⡟⢉⠍⠹⡄
⠀⠀⠀⠀⢳⣽⣟⣼⣿⡿⠁⠀⠀⢰⠃⠈⢂⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⣦⠀⠀⠀⠉⠀⢛⡄⢠⠇
⠀⠀⠀⣰⠧⢤⣹⠟⢿⡁⠀⠀⠀⢸⡀⠀⠀⠑⠤⣀⡀⠀⠀⢀⣀⠤⠔⠊⣠⡾⠀⠀⠀⠀⠀⢀⡬⠖⠋
⠀⠀⢀⠗⠒⢤⣧⣀⢖⠇⠀⠀⠀⠀⢣⡀⠀⠀⠀⠘⣤⣭⣭⣤⣤⣴⣶⣿⡟⠁⠀⠀⠀⠀⢀⣾⣀⡀
⠀⠀⢸⣟⣦⣠⠟⠀⣎⠀⠀⠀⠀⢀⡤⠳⣄⠀⠀⠘⢍⣙⣛⠛⣿⣿⠿⠋⠀⠀⠀⠀⠀⡰⠋⠀⠀⠉⠛⠲⢤⡀
⠀⢠⢿⣿⣿⢿⣦⠴⠁⢹⠓⠒⠋⠁⠀⠀⣿⡳⣦⣀⠀⠀⠉⠉⠁⠀⠀⠀⠀⢀⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀
⢠⣿⣾⣿⠋⠈⢧⣀⠔⠁⠀⠀⠀⠀⠀⠀⢸⣯⣿⣌⠙⠓⠲⠶⠶⠶⠶⠒⠋⢹⣗⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠘⠿⠟⠃⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣯⣟⣳⣶⣶⣶⣶⣶⣶⣶⣻⠉⣿⡄⠀⠀⣠⠤⠤⠤⣤⣤⣀⣀⢀⡠⠂⢈⣦⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠖⠛⠉⣿⢟⡿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣯⣷⣤⣼⠇⠀⠀⠀⠀⠀⠀⠈⡝⠓⠒⠋⠀⠈⡆⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⡀⠘⣷⣿⢿⣿⣿⣿⣏⣿⣿⣯⣿⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠸⣇⣶⡄⠀⠀⣰⠁⣀⡰⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣞⣿⡿⡿⣿⣿⡿⣿⢷⣿⣿⣭⡷⣿⡇⠀⠀⠀⠀⠀⠀⢀⡔⢻⣾⡧⠴⠛⠁⣸⡿⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢿⡿⣿⣿⣿⣿⣻⣿⣛⣻⣯⣽⣷⣟⣯⡿⣆⠀⠀⠀⠀⢀⡿⢦⣄⠉⠁⢀⣤⠞⠁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡻⣾⣿⣿⣷⣿⡯⣿⣿⣿⣭⣽⠷⣾⣿⣿⣿⣻⣜⣧⠀⠀⠀⠈⣿⣧⣽⣆⢀⡼⠟⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣟⣿⣿⣿⣯⡽⠿⠷⠿⠶⠾⠶⠿⠯⣭⣿⣟⢿⢷⡾⣧⠀⠀⠀⠻⣴⡹⢿⡏⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣫⣵⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⣿⣽⣇⠀⠀⠀⠀⠙⠺⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⣿⣿⣯⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣤⣾⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⠁⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠟⠁⠀⠈⠚⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠈⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡠⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠥⠒⠥⠀⠀⠀⠁⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠁⢀⠁⠄⡀⠔⢹
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠐⠁⠀⠀⠒⠦⠛⡊⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡈⡈⡀⠊⢀⠄⠀⢰⡴⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⢀⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠈⡀⠐⠀⠀⠀⠊


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣾⣿⣿⡿⠿⠿⠿⠟⠛⠛⠛⠛⠻⠿⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠛⠋⠁⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⢟⣣⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠛⠁⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣛⣫⣭⣶⣿⣿⣿⣿⠿⣫⣵⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣄⣀⠴⠾⠿⠿⠿⠿⠿⣟⣛⣛⣛⣭⣭⣵⣶⣾⣿⣿⣿⣿⣿⠿⣛⣭⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣛⣭⣵⣾⣿⣿⣿⣿⡿⠟⠋⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠿⠿⠟⣛⣛⣫⣭⣵⣶⣶⣿⣿⣿⣿⡿⠿⢛⣛⣭⠴⣒⣥⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⣒⠒⠲⠶⠿⠿⢿⠿⢿⠿⠿⠿⠿⠿⣛⣛⣫⣭⣭⣶⡶⠿⣛⣭⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⠿⢛⣫⣵⣶⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⢛⣛⣛⣩⣭⣶⣾⣿⣿⡿⠿⠿⢛⣋⠉⣡⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣤⣶⣶⣶⣶⡿⣟⣯⣷⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⢿⣿⣿⣛⣿⣭⣿⣾⣿⡿⠿⣟⢋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣷⣶⣾⣿⣿⣭⣭⣽⣿⠶⠶⣚⣯⣷⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢿⣿⣿⣯⣿⣶⣶⣿⣿⣿⢿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣷⣶⣶⣿⣿⣿⠿⢶⣖⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⣚⣛⣿⣽⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢛⣛⣻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⣤⢴⣿⣿⣿⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢚⣛⡻⠛⣟⢛⣿⣿⣿⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡊⠁⣤⣤⣶⡋⠃⢠⠟⠉⠹⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⣭⣿⣿⠻⣿⣿⡿⢃⣸⣀⣾⣿⡜⢪⣟⡿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠹⣦⣈⠁⣘⣿⣿⣿⣽⣟⣛⡻⠿⡷⣟⢟⣶⡾⠿⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠈⠈⣹⣿⣿⣿⡿⠁⠀⠨⠍⣛⠿⠂⣮⠎⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⣻⣿⣿⠟⢁⠀⢨⡴⣂⢭⣖⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣶⣶⣾⣶⠛⠲⠋⠉⠁⠀⠈⠙⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠂⠉⠀⢠⣿⣿⠿⠟⠋⠈⠻⡻⠟⠿⢷⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠼⠉⠀⠀⠀⠀⠀⠀⠀⠲⣦⣸⠀⠀⠀⠷⣄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⣀⣰⠶⠖⠚⠂⠀⠺⠤⠤⣤⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠯⠍⠓⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣀⠀⠠⠀⡠⡀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⡄⢀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠸⠀⠇⠀⠄⠀⠁⠀⠀⠠⠀⠠⠀⠙⠇⠠⠀⠺⠠⠄⠸⠀⠑
⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢯⠟
⠀⠀⠀⠠⡶⡦⠰⠶

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⣀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠐⠂⠀⠀⠨⠭⠭⠍⠁⣀⠀⠐⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢉⠠⠀⠀⠀⠀⠀⣀⣀⣀⣀⣄⣀⠀⠀⠈⠁⠒⠀⠄⠉⡒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⡉⠔⠈⠀⢀⣤⡴⢶⣟⣫⡯⠭⠴⠯⠾⠭⠭⣽⣗⣶⡦⣤⣀⣀⠀⠁⠐⠠⢁⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠔⢉⠀⠈⠀⣤⡴⢟⡿⠟⡚⠉⠁⢀⠀⠀⣿⡀⠀⠰⠀⠀⣰⡌⠉⠓⠺⢽⣳⢦⣄⠀⠀⠈⠂⢍⠂⢄
⠀⠀⠀⠀⢀⠔⢁⠔⠁⠀⣠⡽⣫⠖⠋⡀⠈⢿⣄⠀⠀⠀⡀⡙⢃⡀⠀⠀⠀⠿⠀⠀⠀⠀⢀⣬⠙⠮⣟⡦⣀⠀⠀⠑⢤⡑⢄
⠀⠀⠀⡠⠃⠐⠁⢀⣀⡼⣫⠞⢡⣀⠀⠁⠀⢨⢡⣄⠤⡀⠉⠛⠓⠂⠀⠂⠿⠖⡰⠀⠀⠀⠿⠃⠀⠠⠀⠙⢮⡳⡄⠀⠀⠙⢦⡑⠄
⠀⠀⡰⠡⠂⠀⠠⡵⣿⠞⠁⣀⠀⠙⣃⠀⣀⡀⡀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠏⠑⠣⠂⠀⠀⢀⣴⠆⠙⢮⣳⡀⠀⠈⠻⣌⢂
⠀⠠⢡⠀⠀⢀⡽⡿⢃⣀⠀⠀⠀⠀⠈⠙⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⢀⢄⠙⠁⠀⠠⠀⠑⣝⣄⠀⠀⠙⣧⠡⡀
⢀⠂⠀⠀⠀⣾⣿⠃⡀⠉⠛⠀⠶⠆⡰⢆⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠈⠉⠀⠀⠀⢀⣴⡶⠈⢿⢦⠀⠀⠘⣷⠱
⠀⢀⠀⠀⢼⣿⡇⣀⣀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠲⢖⡂⠈⠁⠀⡠⠘⣦⣷⡀⠀⢘⣧⢡
⡀⠈⠀⠠⣾⢹⠀⠉⠉⠃⠘⠽⠓⠜⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡘⣎⣧⠀⠀⢽
⡇⠂⠀⢐⣿⣾⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠯⠦⠦⠘⠛⠉⠁⢻⢹⡀⠀⢸⡇
⠁⠃⠀⠠⣿⢹⠀⠶⠶⠆⢶⠤⣖⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡖⡃⡳⢘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢸⣏⡇⠀⢰⡇
⠀⠀⠀⠐⢻⣸⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠟⠻⠿⣸⠿⠤⠤⠬⣤⣔⣀⡀⠀⠀⠐⠋⠛⠷⠕⠶⠶⢶⠆⢸⢾⠁⠀⢸⣇
⠀⠀⠀⠈⢹⡟⡆⢠⣤⠶⠀⠻⠺⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠁⠉⠑⠓⠶⠥⢤⣤⣠⣄⠀⠀⡾⡾⠀⠀⣸⣽
⠀⠀⠀⠀⠨⣿⢱⡀⠠⠀⠀⠀⡤⢤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠎⠶⠦⣤⣄⡀⠀⣸⢵⠃⠀⢐⡿⡎
⠀⠀⠐⡀⠀⠙⣏⢧⠰⠶⠓⠀⠉⠈⠉⠀⢀⣤⣤⣤⡤⠤⣤⣤⡤⣤⣤⢤⣤⣤⣤⣤⣄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣴⣷⡟⠀⣰⣿⣿⠃
⠀⠀⠠⠀⠀⠀⠙⣞⣦⠠⠂⠀⡀⠰⢉⠆⢸⠀⠀⠀⠀⠀⡤⠤⢠⡤⣄⠠⢤⡀⣤⡄⣥⢌⢛⠘⠿⡓⠛⠂⠀⠈⠓⠀⣼⣿⠏⠀⢠⣿⣿⠋
⠀⠀⠀⠀⢁⠀⠀⠘⢿⣧⠰⠟⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢙⣳⢸⣇⣿⠐⣺⡅⠀⣇⢙⣳⢨⠀⠀⠀⠀⠙⢷⡄⢠⣾⡽⠃⠀⠀⣼⢯⠋
⠀⠀⠀⠀⠠⣳⡄⠀⠈⠻⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⣈⠉⣀⠀⠁⡈⠉⢸⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⢀⡴⣷⠃
⠀⠀⠀⠀⠀⠙⢿⣤⠀⠀⠐⢄⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⡶⡏⠶⡇⢰⢸⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣴⢟⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠻⡷⣄⠀⠀⠈⠢⣀⠀⠈⠓⠒⠀⠀⠀⠀⠀⠀⠠⠀⠩⠤⠩⠁⠄⠀⠘⠃⠀⠀⢀⠤⠊⠀⠀⢀⣤⣿⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠉⠐⠀⠄⡀⠀⠀⠀⠀⠀⣊⣠⣷⡀⠀⠀⠀⠀⢀⡀⠀⠊⠁⠀⠀⢀⣰⣿⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣤⣀⠀⠀⠀⠀⠉⠀⠀⠐⠒⠘⠛⠛⠀⠒⠂⠀⠈⠁⠀⠀⢀⣀⣤⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠫⠿⣓⣶⣦⣤⣠⣀⣀⣀⣀⣠⣀⢀⣀⣀⣰⣠⣤⣴⣾⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠰⣦⣾⣿⣤⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠴⠛⠋⠛⠇⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠗⠒⠒⠚⠛⠛⠛⠛⠛⠒⠲⠶⢼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣄⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⢋⡥⣤⡉⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⡿⠀⠈⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⣇⢷⣤⡴⢃⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠙⠶⠤⠶⠛⠁⠀⠀⠀⡴⠖⠒⠒⠻⠉⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣧⣂⣀⣂⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⢰⠫⣟⣛⡳⠶⠶⣶⣤⣤⣤⣭⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠈⠣⣕⡻⢿⣿⣗⣚⣛⣳⢒⡿⠀⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁⠈⠉⠉⡛⠒⠛⢻⡋⠀⠰⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡷⠀⠀⠀⢀⣴⣟⠋⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⣿⣿⣿⣷⠆⠀⢸⡶⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⡿⣿⡿⠁⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢷⣄⡀⠘⠋⠀⠀⢀⣴⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡤⠖⠋⠁⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠖⠒⠒⠒⠓⠒⠒⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⡗⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠄⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣿⢿⣿⣿⣿⣿⣿⣿⠏⠀⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣀⠀⠀⣻⣿⣷⣶⣿⣿⣿⡿⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣿⣿⣿⣿⣷⡿⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠏⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠏⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⠿⠿⠿⠿⠓



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⡯⠀⠀⠀⠀⠀⠈⢿⡏⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠟⢛⣛⣁⣀⣀⠀⠀⠀⠉⠉⠘⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣿⣿⣿⡒⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⡀⠀⠤⠀⢰⠂⠀⠀⣀⢀⡀⠀⢀⠀⠀⠀⢸⡏⢻⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠁⣀⠀⠀⣿⡀⠏⢢⡿⣿⡟⢠⡎⠀⠀⠀⠘⣷⢹⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⡇⠈⣿⡆⢇⢹⠀⠀⣿⣿⠀⠀⠀⢸⡇⠨⠀⠀⠀⠀⠀⣿⠠⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣷⣄⢹⣇⣾⡄⡅⡀⢹⡏⠀⢸⣷⣿⠁⢠⡀⠀⠀⠀⠀⢻⡗⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢿⡸⣿⣿⣻⣧⡀⣷⢸⠇⠀⣼⣿⠇⠀⣼⣧⣤⠄⢰⡄⠚⠓⠼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣼⠷⠛⠛⠉⢻⣿⣿⣾⡰⢠⣿⣿⣶⣦⣿⣿⣿⣴⣀⣤⣐⣴⣦⡀⠘⠿⡃⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠙⠻⠿⡿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠋⠛⠉⠉⠁⠀⠀⠀⠉⠉⠙⠟⣿⣿⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⠸⣿⠉⠁⣀⡤⠴⠶⠶⣶⠶⡶⠖⠲⠦⠤⠤⠴⠃⠈⠂⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠘⢦⡨⢛⣩⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⠐⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢀⡈⠉⠘⡿⠿⣛⣋⣻⣍⣉⡋⢉⡉⠙⣛⣻⣛⡛⠛⠃⠲⠂⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠉⢁⣶⣾⣿⣿⣿⣿⣿⠿⠿⠷⢶⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣠⣈⠻⢿⣍⣍⡀⠀⡀⠆⠁⠈⠀⠀⠀⠈⠈⠉⠑⣢⣦⣠⡴⠏
⠀⠀⠀⠀⠤⠤⠶⣶⣾⣿⣿⣤⣤⣤⣤⣤⣼⣇⣤⣤⣤⣤⣴⠿⠶⠞⢛⣟⣓⣛⣛⣻⣟⣀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣻⣟⣿⣿⣿⣿⣷⣶⣶⣶⣷⣿⣾⣿⠿⠿⠿⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠙⠛⠿⠿⠿⠿⠟⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠶⢀⡠⢠⠤⠐⠀⠀⠀⠀⠀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠘⠇⠀⠈⠛⠒⠚⠡⠶⠤⠤⠤⠄⠀⢈⠀⠀⠀⠱⠆⠀⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⠀⠀⠀⠁⠀⠀⠀⡀⠒⠒⢒⣒⡉⠉⠀⠀⠀⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠣⡂⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣐⡤⣴⣾⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⣆⡀⠀⠀⠀⠀⠀⠈⠭⣐⣰⣽⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⢹⠿⡛⠋⠋⢀⠀⢀⣭⡅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⣶⣴⡤⠀⠀⢸⣻⠅⠀⠀⠠⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⣏⣉⣄⠤⢠⡽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢀⢀⠀⠀⠠⣉⣙⣛⣏⣄⠘⡛⠀⠀⠠⠀⠀⠀⠀⠀⠠⠄⠀⠈⠀⠁⠐⠀⢱⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠀⠀⠀⠀⠈⠄⠈⠂⠀⡁⠇⢌⣻⣛⣷⠌⢀⣊⠧⡀⠈⠂⠠⡀⠀⠀⢀⣀⡠⢀⡀⠀⠀⠀⠀⣈⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠒⠰⠁⠀⡀⠏⠈⠈⠀⢈⣛⠡⠉⠀⣘⣿⣿⠀⠈⠀⡠⠀⡅⡁⠀⠀⠀⠂⠀⡱⠀⠄⢀⡄⢂⠀⠀⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠂⠀⠦⡂⢁⠀⠄⡀⠀⠀⡷⠦⣀⣀⢬⣿⡯⠡⡀⠀⠁⠀⡈⠣⠀⠀⠀⠠⠄⠁⠀⠀⣘⠈⠁⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠂⢀⠐⠕⡂⠠⠀⢠⡀⠂⠀⡀⠁⣻⡏⠁⣜⢫⢾⣀⢥⢊⡀⢪⠁⠀⠀⠀⠀⠈⠢⠀⠀⢤⢋⣈⣁⠔⠏⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡄⠀⣠⠋⠠⠀⠘⠱⠂⠜⡴⢦⣍⡍⠹⣿⣒⡰⢝⡊⠄⣸⣄⠀⠁⠀⢁⡂⠠⠀⠀⠣⡀⠀⡞⠨⠦⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠉⠀⠐⠀⠰⠀⠀⠀⠀⢁⠀⠙⠿⣣⠈⠻⣶⣟⣦⣥⠀⡅⢠⠀⠀⠀⠈⠀⠀⢐⣄⠃⡆⠘⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠀⠘⠻⠇⠀⠀⠀⢀⠀⡆⢀⠄⠁⠈⠁⠠⠭⠿⣿⠏⡉⡴⠃⠀⠀⠀⠄⠊⠙⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠀⠀⠑⠈⠀⠁⠂⠀⠀⠀⠀⢀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀⠀⠉⠻⢿⣷⣄
⠀⠀⠀⠀⣠⣾⡿⠋⠀⢀⣤⣾⣿⡟⠋⢹⣿⣆⣀⣿⣿⣀⣀⣿⡟⠙⠻⣿⣷⣤⡀⠀⠙⢿⣷⣄
⠀⠀⠀⣴⣿⠏⠀⢀⣴⣿⠟⠉⢻⣿⣶⣿⠿⠿⠛⠛⠛⠛⠿⠿⣿⣶⣾⣿⠋⠻⣿⣦⡀⠀⠹⣿⣦
⠀⢀⣾⡿⠃⠀⣠⣿⡿⢿⣷⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣴⣿⢿⣿⣄⠀⠘⢿⣷⡀
⠀⣼⣿⠃⠀⣰⣿⣯⣀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⣹⣿⣆⠀⠈⣿⣧
⢰⣿⡇⠀⢰⣿⡟⠛⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘⣿⣿⠿⢻⣿⡆⠀⠸⣿
⣾⣿⠀⠀⣿⣿⣤⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠁⠀⠸⣿⣆⣀⣿⣷⠀⠀⣿
⣿⣿⠀⠀⣿⡟⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣾⡿⠋⠁⠀⠀⠀⠀⣿⣿⠛⢻⣿⠀⠀⣿
⣿⣿⠀⠀⣿⣧⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⣼⣿⠀⠀⣿
⢿⣿⠀⠀⣿⣿⡋⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢛⣿⣿⠀⠀⣿
⠸⣿⡆⠀⠸⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠇⠀⢸⣿
⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠈⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⢻⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⣰⣿⠟
⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇⠀⠀⠀⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣀⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠁

⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡰⠀⠀⠦⢔⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠞⠀⠘⡄
⠀⢰⠃⠀⠀⢀⡾⢡⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡘⣧⠀⠀⠀⢹⡄
⠀⣿⠀⠀⠀⠘⠷⢋⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠳⠟⠀⠀⠀⢸⡇
⠀⠹⣆⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢀⡼⠁
⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠀⣠⣾⠟⠋⠀⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⣼⡿⠋⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠰⡜⢿⣧⡀⠀⠀⠀⠀⠀⠀⢀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⣿⣇⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠘⡎⢿⣧⠀⠀⠀⠀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢹⣿⡀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⣠⣾⠟⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠻⣷⣾⣟⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⣾⡿⠀⣼⡿⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠈⣿⡆⠀⠉⠛⠛⠉⠀⠀⠀⢀⡼⡇⠀⠀⠀⢰⣿⠇⣼⡿⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⢠⠏⠀⢧⠀⠀⠀⢸⣿⣾⡿⠁⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣿⡀⠀⠀⠹⣿⡀⠀⠀⠀⠀⣰⠃⠀⠀⠘⡆⠀⠀⠀⠉⠉⠀⠀⠀⡄⣼⠃⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣷⡀⠀⠀⣿⡇⠀⠀⠀⢰⠇⠀⠀⠀⠀⢹⡀⠀⠀⢀⠀⠀⠀⢸⡇⣿⠀⠀⠀⢀⠞⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⡘⢿⣦⣤⣿⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣇⠀⢀⠜⠀⠀⠀⠀⢣⡹⢦⣤⠴⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⠓⠮⠭⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢸⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⡀⠸⡆⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠉⢂⠱⡀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⠀⠀⠀⠀⠑⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⢿⠢⠞⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠐⢂⡿
⠀⢶⣶⡌⢷⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⡾⣡⣾⠗
⠀⠀⣿⠙⠟⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠛⢋⡿
⠀⠀⠘⢧⣀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣀⣠⡾⠁
⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⢀⢄⡀⠀⠀⣤⣶⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠘⠉⠀⣰⠁⠉⠳⠤⣄⣩⢤⡀⠘⠋⠁⠀⢠⡄⠐⠈⠁⠀⠉⠀⠀⠉⠂⣀⠀⠀⠀⡎⢠⠊⠀⣡⠼⠓⢄⠀⠀⠻⣠⠗
⠀⠀⠐⢾⣿⣿⠀⣴⣦⠄⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⢢⠀⠀⠀⠀⠈⠑⠢⠄⠸⠀⣀⠄⠈⠙⠀⠀⠀⡇⠼⠚⣩⢒⣶⡲⡈⠳⡀⠀⠉
⠀⠀⠀⠈⠋⣤⣀⠘⠉⠀⠀⠀⡇⠀⠀⣠⡤⠦⠤⠤⠖⠒⣙⠃⠀⠀⠀⡀⠀⠀⠀⠀⢴⠎⠀⡠⢒⣒⡢⠀⠀⠃⢰⠀⢇⢇⣀⢟⠇⠀⠱⡀⢒⠢
⠀⠀⠀⠀⠈⠿⠿⠀⠀⠀⠀⠀⠈⠩⠭⠗⠁⢀⣰⡆⠀⠀⢣⡕⠒⠒⠒⠤⡑⠀⠀⠰⠁⠀⢠⠎⠁⣀⠈⠳⡀⠀⠸⣧⠈⠓⠒⠋⠀⢠⡄⠱⡇⣧⠁
⠀⠀⢀⣀⣀⠀⠀⢀⡔⡆⢀⡄⢀⣤⠀⠀⠀⢀⣉⠀⢸⠀⡎⠱⡄⠀⠀⠀⠹⠀⠸⠇⠀⠀⠇⡴⢋⣠⡙⡆⢱⡀⠀⣡⠀⠶⠀⠙⠂⢀⡠⠖⢣⠘⡆
⠀⣵⢿⠟⣉⡉⢆⠉⢹⡿⢷⠿⢋⡀⡤⠴⠪⠈⢎⠆⠈⠀⢃⠀⠘⢆⠀⠐⡰⠀⠀⠱⠀⡇⡆⢇⠛⠿⠃⡸⠀⡇⠈⣆⠳⠒⠒⢒⣈⡥⢴⣾⠉⠣⣼
⠀⢷⡄⢈⣿⣿⡿⠀⠈⠀⣀⠊⠁⠀⠀⠀⠀⠀⣸⡴⠀⠀⠀⠑⠤⠤⠵⢾⡅⠀⠀⠀⣤⠈⢱⡀⠑⠒⠊⠁⠀⢸⠀⠱⡈⡏⠉⢧⣤⣴⡿⡙⡄
⠀⠈⠓⠾⠟⠋⣠⠖⠂⠉⢙⣆⡠⠤⠖⠒⠉⢩⣖⡀⠀⢀⡠⠤⣀⠀⢀⡠⠤⣀⠂⠀⣀⠀⠀⠱⡀⠀⠀⠀⠀⠀⡆⠀⣕⠆⠀⢸⠙⠞⠾⢡⢶
⠀⠀⣠⣤⠶⠭⠜⢷⡎⠉⠀⠀⠀⠴⠀⠀⠀⠈⠛⣁⠜⠁⠀⠀⠀⠑⠁⠀⠀⠀⠱⠊⠀⢉⣭⢢⠑⡄⠀⠀⠀⠀⢰⠀⠀⠁⠀⠀⠟⢶⢧⢸
⠀⠀⠀⠀⠀⠀⢠⠟⡓⡄⠀⠀⢴⣾⡇⠐⠀⡰⢊⠀⡴⠎⠁⢤⢤⡀⡴⢳⠀⡞⠁⢀⠄⠀⠀⢠⠀⠘⢆⣄⢳⡀⣾⡆⠀⠀⠀⠀⠀⢀⠀⠻
⠀⣀⠄⡶⢠⢂⠎⠀⡇⠘⢆⠐⠠⣉⠀⠀⠀⡇⠛⠀⠑⠒⢤⢘⡼⠁⠃⠘⡇⠻⣆⠈⠯⠥⠀⡰⠀⠀⠀⠻⢮⡝⠾⠁⠀⠰⣄⣀⣠⠟⢄⣀⢤⡶⠂
⠀⣁⠸⠃⢀⠎⠀⠀⡇⠀⡈⢧⠀⠈⠀⠀⠀⠈⢢⠀⠐⠐⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠃⠀⢀⠶⡀⠈⠁⠀⠀⠀⠀⡡⠀⣀⠤⠤⣄⠈⠧⣀
⠠⠤⣣⢀⠎⠌⠀⠀⠃⠘⠋⠀⠁⠀⠀⠀⠀⠀⣈⠳⢄⡀⢀⡠⢀⣀⠀⢿⡠⠎⠀⠀⠀⠀⠀⠀⠀⠙⣤⠋⠀⠀⠀⠀⣠⠔⠁⢰⠁⠀⠀⡬⡇⠀⡎
⠐⠮⠁⠎⠀⠤⡀⠀⠀⠀⢠⠄⢈⠧⢴⡖⠒⠒⠭⣄⠀⠈⠁⠀⠀⠀⡈⠀⠀⠀⢀⣴⠖⠛⠉⣉⡑⢦⡈⣀⠤⠤⢤⠀⠀⢈⡇⠀⠣⢄⣠⠕⠁⣀⣣
⠀⣴⢾⡧⢈⣵⣞⠀⠀⠀⠀⡴⠣⠦⠕⠁⠀⡖⢤⠀⢱⡀⠀⠀⠀⠾⣙⠆⠀⡴⠁⠀⢠⠤⡄⠑⢚⡤⡻⠁⢀⡔⡰⠀⠀⠋⠉⠙⣤⠖⠢⣀⡎⠀⠈
⠠⡷⡗⢍⢦⠻⢻⠆⠀⠀⢸⠀⠀⢀⠄⢠⡄⠙⠤⠃⠀⡇⠀⣔⡆⠀⠙⠀⢸⠁⣔⣢⠈⢓⡡⢖⡩⠊⠀⠀⣨⠾⡀⠈⠉⠒⠀⠀⠁⠀⠀⠙⠀⢀⡀
⠀⠻⣜⠂⠙⣳⠞⠀⠀⠀⢸⠀⠐⢧⡶⠟⠁⡠⠄⢰⣄⡇⢤⠀⠀⠀⠀⠀⠸⣀⣨⠔⣚⡭⠖⠉⠀⢀⣠⢎⣠⠜⠀⠀⠑⠀⠀⣶⠀⠀⠺⠟⠀⠻⠛
⠀⠑⠂⠈⡝⢋⡒⢦⡀⠀⠈⠣⡔⠢⣀⠀⢼⣤⡴⢯⡟⠀⠀⢀⡜⣦⠀⢁⡜⠙⠃⠉⠁⠀⣀⠤⠚⢁⡽⠀⠀⠀⠈⠳⡄⠀⠀⠐⠒⠒⠠⠄
⠀⠀⠀⡈⡄⠉⢛⣤⡈⢦⡀⠀⠈⠒⠶⠤⢤⡤⡒⠉⠀⠀⠀⠀⠘⠃⠀⠮⣀⣀⣤⠔⢒⡟⠈⠓⠚⠉⠠⠀⠀⠀⠀⢀⡤⠔⠒⠋⢹⠓⠲⢤⡀
⠀⢀⣀⠙⢌⠢⡉⠘⠡⣾⣭⠦⡀⠀⠀⠀⠈⠙⠞⠀⠀⠀⠀⠀⢰⣦⣤⠆⠀⠰⠈⡉⠉⠀⡀⠀⠐⠀⠀⢣⠀⠀⡰⠋⠀⡘⠀⢐⠁⠀⠀⠀⠙⡄
⠀⢹⠿⠀⠀⠁⠈⠒⡄⠈⠹⣲⣏⠢⡀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠺⣧⡼⡦⠀⠀⠀⢰⠀⠀⠱⠀⠀⣰⣤⠀⠀⢰⠁⢀⡠⠔⠀⢜⣀⠔⡄⣠⢄⢼
⠀⠀⢀⡀⠀⠀⢀⡉⠒⠲⣄⠘⠻⣧⡈⠢⡀⠀⠀⠀⠀⡴⠴⢤⡀⠈⠀⠀⠀⣀⠤⡀⠃⠀⠀⠀⠀⠙⠋⠀⡀⢸⠴⠞⠀⡀⠀⠀⠀⠀⠘⠁⠀⢸
⠀⠜⢁⣠⣴⡒⠲⢬⡂⠀⢨⠳⡀⢬⠽⢃⡼⠀⠀⠀⠀⠘⢲⠚⢁⡠⢤⠚⣏⣘⡄⢧⠀⢀⠠⠰⠀⠊⢱⠀⢰⠘⣤⠖⢋⠜⠀⢀⡎⠉⠲⠖⠂⡞
⠀⢠⢿⣝⡛⠻⣿⣿⣿⡀⡀⠑⢌⠒⡲⢭⠶⠶⠲⠖⢲⣺⠉⢹⠅⣗⢛⣉⣩⣀⡧⠼⠀⢫⠀⠀⠀⠜⡙⡀⠀⠱⠘⢦⡞⠀⠰⡍⠀⠀⠀⣀⠞⢠⠂
⠀⢸⣮⣙⡿⣿⣷⣿⣿⡇⢱⠀⠀⠀⢣⠬⡦⠾⡒⣻⡩⠟⠖⠺⠉⠉⢠⠄⠒⠄⢀⡀⣀⠈⣦⠒⠉⠁⠐⠁⠀⠀⠀⠀⠈⠑⠒⠓⠒⠒⠋⠡⠒⠁
⠀⠈⠻⣭⣽⣛⣿⣿⠟⠀⠎⠀⠀⢃⠘⠤⠞⠚⢉⠴⢚⡆⠰⠖⡇⠀⠀⠉⣁⡀⠣⠤⠚⠀⢸⡀⠀⠀⠁⠀⠀⢴⣷⡆⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠉⠉⠉⠀⠐⠀⠀⠀⠀⠀⠀⠘⠶⠝⠈⠉⠀⠤⠶⠉⠁⠀⠀⠘⠂⠞⠀⠀⠦⠵⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠷⠸⠁

⠀⠀⠀⣠⡤⠤⠤⠤⣤⣄⣀⡀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠘⣧⠶⠒⠒⠲⠶⠦⣬⣉⡛⠶⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣼⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠙⢦⡀
⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠙⢿⣿⣿⣿⣿⠿⠿⢿⠿⠿⣿⣿⣿⣧⠀⠀⠀⣷
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⣠⢔⣛⡛⠃⠀⠀⠐⣛⣛⡳⠄⠀⠉⠛⠷⣄⠀⣼
⠀⠀⠀⠀⠀⢹⠋⠙⢲⡶⠞⠋⠀⠀⠀⣠⠞⢡⠋⠀⢹⠀⠀⡠⠊⠉⠈⣷⠀⠀⠀⠀⠀⠹⣷⣟
⠀⠀⠀⠀⠀⢸⠒⢲⡾⠀⠀⠀⠀⢀⡼⠁⠀⢟⣰⣿⠏⠀⡜⠁⣴⣷⣰⠃⠀⠀⠀⠀⠀⢀⣿⣿⣦⣀⣀⡀
⠀⠀⠀⠀⠀⡟⠀⡾⠀⠀⠀⠀⠀⡞⠀⣠⠀⢠⡋⠙⠄⠀⠧⠼⠿⠾⠁⡀⠀⠀⠀⠀⠀⠘⢿⣿⠟⢉⠌⠙⡆
⠀⠀⠀⠀⢸⠁⣸⠃⠀⠀⠀⠀⢸⠁⠊⢃⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣤⡀⠀⠀⠀⠋⠀⠿⡄⢀⡇
⠀⠀⠀⢀⡏⢀⡟⠀⠀⠀⠀⠀⢸⠀⠀⠀⠳⢄⡀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢁⣠⠇⠁⠀⠀⠀⠀⠀⣈⡥⠞
⠀⠀⠀⣸⠀⣸⠇⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⣏⣓⠒⠚⣉⣉⣠⣤⣶⣿⠏⠀⠀⠀⠀⠀⢀⡼⠁
⢰⠚⠙⢳⠖⠛⣄⠀⠀⠀⠀⠀⠀⢘⣦⠀⠀⠀⠠⣜⠛⡿⠿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⣠⠞⠉⠛⠲⠦⣄⡀
⣞⢉⣉⢻⡀⣸⢸⠀⠀⠀⠀⠀⣠⠏⠀⣳⣤⡀⠀⠈⠙⠒⠲⠖⠛⠋⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⢸⢉⠈⠙⣿⠁⢾⣤⣀⣀⡤⠞⠁⠀⠀⢽⣾⢹⡷⢦⣤⣀⣀⠀⠀⠀⢀⣀⣠⢤⡶⣛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠈⡟⠉⢹⣯⣄⡼⠀⢨⠀⠀⠀⠀⠀⠀⢸⡿⢺⡷⣤⣀⣈⠉⠉⠉⠉⠉⢁⣀⣸⡯⢧⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀
⢸⠃⠀⣼⠁⢿⣅⣠⠎⠀⠀⠀⠀⠀⠀⠸⣷⣤⣿⣾⣯⣭⣿⣿⣿⣿⣿⣭⣽⡹⣇⣸⣿⡀⠀⢠⡖⠒⠒⠒⠒⠲⠶⢦⣤⣤⡊⠀⣰⠓⢤⡴⠦⣄⡀
⡿⠀⢠⡟⠀⠀⠈⠛⠦⣤⣀⣀⣀⣀⡤⠞⢿⡶⣿⡿⣿⢛⣿⢿⣿⣿⣛⡟⣿⣸⣾⣿⡾⣷⠆⠛⠃⠀⠀⠀⠀⠀⠀⠀⣰⡏⠙⠋⠁⠀⠈⣿⢱⡮⣿
⣧⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠘⣷⣿⣿⣿⣿⣻⢿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⠓⡟⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣷⡆⠀⣄⣼⢣⣿⣵⡟
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣟⣿⡿⠻⢷⡾⢿⡿⢿⣿⣻⣯⣷⢿⣿⡃⠀⠀⠀⠀⠀⠀⠀⡴⠛⢿⣿⡷⠞⠛⠉⣹⡿⠃⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢿⣿⣿⣿⣟⣿⣿⣿⣿⣛⣛⣫⣭⣷⢾⣟⣽⡻⣄⠀⠀⠀⠀⠀⣾⠶⣤⡈⠉⠀⢀⣤⠞⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣫⣷⣿⣿⣶⡾⢯⣭⣹⣿⣿⣿⣿⣯⣿⣷⣿⣽⣿⢿⣹⣦⠀⠀⠀⠀⣿⢮⣏⢷⡀⢀⡼⠟⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣗⣿⣿⣿⣿⢿⣛⣷⣾⣯⣿⣭⣭⣽⣷⣿⣛⣾⡿⣿⣿⣟⡻⣇⠀⠀⠀⠸⣷⡹⠿⣷⠋⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡹⡟⢟⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢮⣿⠷⢹⣆⠀⠀⠀⠈⠙⠷⣿⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣬⣿⣛⣋⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠛⣋⣽⣀⣀⣀⡀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣴⣶⡞⠂⠀⣀⣤⠴⠤⡄⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣤⣶⠁⢀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠘⢬⡉⠉⢷⡶⠞⠁⣀⠠⣀⣄⣲⣯⣿⠿⠷⢦⡄⠀⢀⣠⣶⣶⠀⢀⣀⠀⣀⠄⠈⠩⠷⠶⢫⣿⠿⠤⣆
⠀⠀⠀⠀⠐⠶⠿⠟⠋⠀⠁⠀⠀⠀⢸⡾⣥⣶⡄⠀⠀⢈⣶⣰⣿⣿⡝⠋⠁⠀⠀⠀⠀⠛⠻⠛⠁⣽⠀⢘⣧⣼⣿⣊⣷⣆⠀⢠⡨⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣚⢯⣾⠿⠃⠀⢰⢿⣿⣿⣧⣿⡚⠀⠀⠀⠀⠀⠀⠀⢀⢶⡖⠥⠞⠋⠀⠈⠙⠛⢯⡹⠓⠞⠁
⠀⠀⠀⠀⠀⠀⢠⠖⣋⠄⠀⣀⠤⣚⡿⠝⠉⠉⠙⠷⡆⠀⠀⠀⡀⢻⡌⠛⠢⢀⡀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠘⢀
⠀⠀⠀⢀⡠⠔⡁⠈⠀⠀⠈⢡⠌⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠲⠳⡀⠀⠀⠉⠀⠘⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠤⣯⡅⢀⡠⢀⣀⣀⣀⣀⣀⢀⣀⣀⣢⣀⠀⣀⣈⠂⠀⠀⠠⠤⠀⠀⢠⣤⣄⢠⣤⢤⣤⡤⣅⣤⣤⠤⠤⠤⠤⣤⠀⠄
⠄⠄⠀⠉⠓⠓⠛⠛⠛⠒⠒⠐⠙⠛⠓⠛⠛⠋⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠋⠉⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠒⠀⠁⠈⠈⣁⢀⣀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠀⠂⠤⣀
⠀⠀⢀⠤⠒⠉⠀⠀⠠⠀⠒⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠀⠄⣀⠀⠀⠀⠑⢦
⠀⠰⠁⠀⠀⠀⠀⢀⠠⠄⡒⠠⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠁
⠀⠰⡀⠀⢀⠔⢈⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠂⠤⢀⡀⠀⠀⠀⠀⡠⠃
⠀⠀⠉⠢⣌⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣢⡴⠕⠊
⠀⠀⠀⠀⠘⠬⡙⠳⢦⣤⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠄⠒⠉⢃⠄
⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⢍⣙⣢⠮⠭⠭⠀⣀⣀⣀⣀⣈⣉⡉⠀⠁⠀⠂⠀⣀⣠⠤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⡀⠁⠡⣍⠙⡓⠛⠚⠲⠶⠴⠦⠀⠉⡛⠒⡒⠪⠭⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠐⠶⠤⠉⣉⢁⣀⠀⠀⠀⠀⠀⠉⠉⠉⣁⣤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡖⠭⠭⠭⠛⠛⠛⠛⠋⠭⠉⠁⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢶⣚⡀⣀⣀⣀⣀⣠⠶⠷⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⠤⠄⠀⠈⠀⠈⣉⡶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⠄⠠⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠍⠁⢌⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂