Hop Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⠀⠂⠠⠄⠀⠀⠀⠀⠀⠐⠀⠈⠁⠐⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢢⡀⠀⠀⠀⠀⠰⢀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠄⠐⠀⠀⠰⠾⠀⠀⠘⠀⠀⠰⡄⢀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢭⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⢀⣠⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢤⣀⣀⡀⠀⢀⠀⠀⠠⡄⠀⠀⠀⠠⠀⢀⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠐⠀⠀⠀⡀⠀⠀⢠⡼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣄⠀⠀⢀⠀⠉⠀⡰⠏⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠅⣘⠳⢦⣀⠀⣴⠟⠁⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠓⡶⠖⠚⡛⢤⣄⣦⡀⠉⠛⠁⠀⠀⠀⠀⠀⣨⠏⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⠼⡁⠀⠀⣨⠦⢄⡈⡇⠀⠀⠀⠀⠀⢠⡶⠞⠋⠴⠶⢦⣤⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣱⣄⣙⣶⠿⠃⣀⣴⠛⠛⡄⠀⠀⠀⠰⠈⢣⣀⡔⠉⠀⣰⡉⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠸⣴⠋⠁⢈⣏⠁⣠⠞⠁⢡⡄⡰⠀⠀⠀⠀⠐⣠⡞⠉⠉⠢⡴⠋⠀⣿⠈⠉⡀
⠀⠀⠀⠀⠀⠀⠂⠀⢠⣤⣏⠀⠉⣱⣀⡴⠞⠋⡅⠀⠀⠀⠀⠀⢘⣄⡀⢀⡴⠃⠤⣤⠿⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠰⣴⠞⠁⠈⡗⠚⠁⠀⣷⣠⡾⠀⠀⠀⠀⠀⠘⡾⠀⠙⡿⢄⡀⢠⠇⠀⢘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡣⣀⣀⣧⣀⣴⠟⣉⡽⠃⠀⠀⠀⠀⠀⠀⣤⡴⠊⠀⠀⣽⠏⠁⠐⡾⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢭⣨⣏⡶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⠳⣄⣰⢇⠀⣀⡞⢁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣾⠁⠀⣹⡛⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠁⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⢀⠇⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⡿⠋⢸⠀⢀⡄⠰⠀⠂⠀⡒⠀⢀⣀⣠⠄
⠀⠀⠀⠀⠀⢸⠀⢘⠐⠐⢉⠋⠀⠀⠀⠁⠒⠰⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡜⢹⡟⠤⡾⠀⡈⠂⢀⠀⣠⣀⣥⠴⠾⡟⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⢺⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⣇⢇⢙⣷⡇⣤⣴⣳⣾⠿⠿⣿⡛⠃⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣴⣶⣿⠐⠒⢾⣤⣀⠀⠀⠀⣀⠀⢀⣠⡴⢾⠉⠙⠉⠒⠀⠈⠃⢨⡿⢿⠿⠏⣡⣄⡀⠀⣀⣈⠰⠊
⠀⠀⠀⠀⠀⠀⠀⣺⣶⣾⠛⢋⣼⣧⣍⠀⣿⡶⢵⠄⠀⠘⢾⠯⢉⣀⣶⣖⣲⣦⣴⡒⠠⠶⣦⠼⢿⡟⠓⣀⠉⣍⢉⣈⢁⡀
⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⣾⠏⠉⢠⣿⡟⢻⠁⠀⢣⠀⠀⠀⠱⣌⠁⠔⢻⠿⣯⣲⣶⣶⣶⣿⣤⣾⣍⢩⠙⢿⡭⠷⡈⠎
⠀⠀⠀⠀⠀⠀⢧⣷⣿⡟⠃⣠⣴⣏⡁⠁⢸⡟⠲⣼⠀⠀⠀⠀⠈⠁⠒⢂⢔⣿⣿⣵⣿⣿⣽⡟⢃⡾⠟⣶⡾⢯⣀⠈⠆
⠀⠀⠀⠀⠀⠀⢸⠘⣿⣿⣾⡟⠇⠉⢩⣷⡿⣇⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⢾⣿⣿⣿⣿⡟⣿⣿⣿⠁⢠⡟⠁⠠⣿⠉⢛
⠀⠀⠀⠀⠀⠀⠈⢻⢿⣇⠋⢙⣦⣴⡟⠟⣰⡏⡙⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⡿⣿⡟⢿⣇⡽⣿⡍⣴⣿⡋⢉⣷⠏⣶⠼
⠀⠀⠀⠀⠀⠀⠀⠈⠞⣿⣶⠿⡌⢡⡿⢼⠧⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣷⣀⣵⡾⣿⣇⣻⣿⠿⠓⣤⡼⢏⢠⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⣩⠷⠫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⡧⢴⣛⣿⡿⢹⣷⣴⠿⣅⣾⣹⣃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠯⠉⠈⠀⠀⠉⠹⠿⠿⢾⠥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠈⠁⠈⠉⠲⢲⡀⠀⢰⠛⠾⡶⢶⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⠂⠀⢹⠀⡾⠀⠆⠀⢺⣆⣠⣤⣾⣧⣶⣿⣿⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⢤⡀⠀⣾⣇⣿⣰⣾⡆⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⣀⣉⣷⣼⢿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿⡷⢾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣠⠆⠘⠇⠁⢁⣾⡟⠋⠉⠉⠉⠉⠛⠳⣶⡒⠚⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠐⠒⢳⣾⠋⠀⠀⢀⣼⣿⣿⣷⣶⠤⣄⠀⠀⠀⠀⠀⠉⠓⢤⣀⢨⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⢀⡴⠋⠀⣀⣶⣶⣿⣿⡟⠉⠙⡳⠀⠈⢻⡓⠶⣄⠀⠀⠀⠀⠈⠳⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢀⡴⠋⣀⣤⣾⣿⣿⣿⣿⡿⣧⣀⠐⠀⠀⠀⢸⠷⢤⣌⣷⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⢊⣴⣿⣿⡿⣩⣿⣿⣏⣭⡾⣿⢻⢿⣶⣤⣄⢸⠳⣤⣀⠈⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠞⠁⣿⢏⣽⣾⣿⣿⣿⣿⡿⣷⠃⠀⠘⠙⠘⠉⠻⠇⢸⡏⠓⢦⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⣿⣿⠻⡿⢟⣃⣥⣾⢿⣦⡀⢀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣾⣿⣿⣿⣿⣿⢸⣿⣷⣄⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⣿⡿⢿⣿⣿⣿⣿⣿⠸⠛⢻⢿⣧⡐⠀⢠⣿⢻⣧⡀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡿⢿⣷⡿⣿⣿⣿⣯⢿⡆⡄⠀⠀⠈⢻⣦⣿⣾⠛⠛⠻⣦⣄⢤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣱⣿⣿⡇⣿⡿⠋⠇⠊⢻⡄⠈⠀⢀⡾⢿⣍⢻⣷⣀⠀⣿⠻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠛⠁⢀⣀⡴⠞⣿⣄⣴⠏⠀⢸⣇⢸⡟⠹⢷⡿⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣿⣣⣿⣷⣾⣿⣿⣿⡇⠃⠺⠛⠃⠀⠀⢸⣿⢻⡇⠀⠀⣧⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣿⣿⣼⣯⠿⠛⠻⣷⣀⠀⠀⠀⠀⢀⣾⢿⡎⠃⠀⢀⡏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣿⠋⣼⣿⣿⡦⠀⠀⠙⣿⡀⠀⢄⣴⠟⢿⣸⣿⡄⢠⡾⣧⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣯⢻⡧⠀⢀⣼⣿⣷⣴⡟⠃⠀⢀⣿⠙⠿⠋⢸⠃⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡍⣿⣅⣴⣿⡹⢿⡟⠉⠀⠀⣠⠞⣿⣷⣄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡽⢿⣿⡿⠃⣠⡾⣿⣄⣠⣾⠋⢀⡏⢻⠏⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⣿⣣⣾⡿⠀⣿⠻⠟⣷⠁⡾⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢧⡀⣼⡟⠀⣰⠋⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠱⠞⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢻⣿⣿⣿⣿⣥⣶⣷⡄⠀⠀⠀⣠⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠄⣀⣼⣿⣿⣿⣿⣿⣿⡦
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣀⡄⠸⣿⣿⣿⣿⡿⣿⢸⣿⣿⣿⣿⣿⠟⣋⣬⣍⣿⣿⣿⣿⣏⣀⡀
⠀⠀⠈⣿⣿⣿⣷⣿⣾⣦⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⡷⢼⣿⣿⡿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠈⠋⠛⢋⣵⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠙⠻⣿⣿⣮⡻⣿⣿⢻⣿⣿⣿⣯⠀⠀⠀⣾⣅⡀⢀⣀⢀⣤⡶⣶⣄⣀⠈⠉⠙⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢤⣴⣶⣿⣿⣿⣿⣿⣮⠻⣼⣿⣿⡿⠁⠀⠀⢀⣿⠉⠛⢺⣿⣯⣽⣷⢿⡿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠁
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⣶⠎⢩⡈⠀⣠⣤⣤⣾⠁⠀⠘⢿⣿⣿⢛⣷⣿⣿⣎⣿⣷⡦⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⢸⢠
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢹⡟⠋⠀⠀⢹⣆⢀⡀⠀⠛⠟⢿⣿⣷⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡟⡼⢸
⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠛⠁⣰⢶⣬⣤⣄⠀⢰⡿⣿⣿⣯⣄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠃⠁⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢞⣿⠻⣥⡀⢿⣿⣳⡿⣾⣟⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣽⡻⣿⣙⡇⢪⣿⠿⣷⣾⢻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⣿⡆⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣯⣭⣿⡿⢺⡏⠀⠈⠸⣶⣿⣵⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⠀⢈⣵⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⡟⠰⣶⣤⣰⣮⠹⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣮⡛⠰⠻⣟⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣦⠙⠿⠃⣴⣿⣹⣟⣫⣝⣆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠿⠇⠀⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠠⣾⣿⣟⣿⡿⣿⠦⠀⣷⡿⣏⣿⣵⣿⣏⠀⣼⣿⣿⡟⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⡷⢸⣴⣾⡗
⠀⠀⠀⠀⠀⠀⠰⣿⣾⣿⣼⣿⢻⡆⠀⣻⣝⣿⣿⣛⡟⡇⢈⣿⣧⣿⣹⣷⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢠⢋⣻⣭⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠞⠿⠋⠀⠀⠹⣿⣶⣿⡿⠃⠀⠀⠻⣯⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡌⢋⠌⠻⡟⠋⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⣿⣶⣿⣧⡀⠿⠛⣁⡤⢀⣤⣤⠄⠉⠻⠿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢇⢼⣾⡿⠋⢀⣼⡛⣧⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⣼⣿⣟⣿⡇⠀⢀⣿⣺⣿⣿⢳⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠉⠀⣠⡄⠀⠸⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⣵⣾⣿⣿⣿⠀⢿⣿⡟⣾⡿⣿⣏⣹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⣠⣿⠛⢷⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣷⣿⣻⣿⡿⠉⠀⣾⣿⣿⣿⣶⣮⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⣀⣴⡾⣯⠝⣷⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠋⠀⠀⠀⡷⢹⣭⣿⡿⢯⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢃⣴⢸⡟⡹⣶⣭⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⣠⠴⠞⠁⣀⣈⠿⣴⣙⠷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣉⢩⣀⣤⡢⡐⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⣀⣀⢀⣤⣄⣤⢄⣴⣶⣴⡾⠃⠀⠙⡻⣑⠬⡢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⢤⣸⡟⠋⢾⠿⠻⠿⠋⣿⣏⣸⡿⠌⣻⣥⣤⡀⢐⠭⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣌⢹⣯⡹⣿⣯⣽⣦⣴⠿⣷⡾⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠙⠻⠈⠉⠃⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣠⠾⠛⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⣛⣛⣛⣿⣤⠭⢭⢶⣫⡭⣭⣝⢦⣶⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⣿⣿⠶⡻⢽⣿⣽⠫⡶⠀⠉⡟⠈⠁⣿⢛⠈⣿⣛⡷⣯⡛⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡳⣡⠟⠁⠐⢘⣩⣸⡇⠁⠀⠀⠀⠃⠀⠀⠀⠀⢀⣟⢕⠯⡢⡙⢦⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣱⠏⠀⠀⠠⠪⣝⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠷⣕⢌⢊⠈⣷⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡟⠀⠀⠀⠀⠨⠺⠸⣇⠀⠀⠀⠀⠀⠀⠀⠠⢠⣿⠸⠀⠀⠀⠀⠀⢸⡏⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣸⣼⠃⠀⠀⠀⠀⠀⠈⠆⣻⡄⠀⠀⠀⠀⠀⠀⣠⢟⢻⣆⠀⠀⠀⠀⠀⠀⣷⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⡿⢘⡀⠀⠀⠀⣀⣰⣾⣏⡿⣦⡀⡄⢀⡀⣰⠏⡞⡦⠻⣦⡀⢀⠀⢀⡀⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣃⣭⣤⠶⢶⢛⠉⢹⠃⡇⡟⢨⢳⣄⠊⣰⡟⢻⠃⠇⠙⡏⣻⣶⠾⢤⣭⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡟⡅⠀⢸⠘⠈⠀⣾⠃⠁⠁⠁⠺⡹⣾⠋⣷⠀⠀⠀⠀⠀⠀⣿⠹⣷⢸⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠄⠀⠸⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠃⠀⢻⡄⠀⠀⠀⠀⡀⣿⠀⠉⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⣼⣿⢷⡀⠀⠀⠀⠀⠀⢠⡾⣿⣦⡀⠀⣸⢠⡏⡀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠄⣼⣿⠘⡇⢙⢶⣄⠀⢀⡴⠋⠚⠉⠎⢻⣦⣃⡿⣷⣁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢶⣼⣿⠐⠀⠁⠀⠀⠙⣷⠞⠁⠀⠀⠀⠀⣼⠋⠻⠋⢋⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠙⠸⣆⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⣰⡏⠀⠈⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠈⣷⡄⠤⣠⣟⡝⢧⡀⠰⢠⡾⢻⣷⡀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⢹⣶⣿⠀⠁⠀⢹⣶⡏⠀⣼⢿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⡉⠹⣆⠀⠀⣾⠁⢀⡾⢃⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣹⣧⣼⣿⣴⣿⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠛⠈⠁⠈

⠀⣀⣀⣀⣀⡀
⠀⠷⠾⠶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⢀⣤⢄⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⢶⡁⣸⣧⡌⡇⠐⣄⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠴⢚⣛⡛⡰⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣀⠀⢹⣶⡌⠀⠀⡄⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⡦⢡⣿⣃⠉⠁⠘⠃⢺⡿⡁⠀⠀⠀⠈⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠄⠶⡆⠀⠐⠒⣦⠿⣀⡈⠃⠘⢛⡏⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡤⢤⣧⡤⠴⠚⠟⠑⠿⠅⠀⠒⣻⣦⠁⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⣦⢀⡡⣤⣀⣠⣄⣠⠀⠀⠀⠐⣦⣤⠶⣤⠋⠉⢂⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣈⡇⠘⣉⠀⢉⠻⠏⢋⠱⡷⠲⡖⣎⠀⣘⠄⠘⢤⡀⠀⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠁⠉⠀⠀⣿⣤⣆⣠⣄⣨⢿⣟⠶⣴⡋⢩⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣀⣍⡀⢿⣇⡸⠁⠰⠴⣉⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣈⠰⠄⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡄⣠⠄⣀⠀⡄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⣇⠀⣸⠀⡇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠈⠁⠈⠉⠡⠈⠁⠀⠀⠀⠀⠄

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠁⠈⠉⠐⠂⢄⡀⠀⠀⠀⠈⠠⣼⠀⠀⡠⢾⠃⠶⠿⡳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠐⣤⣀⠈⠉⠲⢆⠀⠀⠀⠘⠄⠰⠁⢨⠀⢆⠀⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡄⠀⠀⠈⠳⢷⡂⠀⡑⣄⠀⠀⠘⣧⣀⣸⠀⠈⠑⠚⠋⠑⠂⠀⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡀⠀⠀⠀⠀⢙⣶⡀⢋⢃⠀⠀⠀⠙⡏⠀⠀⠀⢀⣀⣆⣜⠂⠈⠃⠀⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⢦⣤⠀⢀⣼⣿⣷⡟⢉⠃⠀⠀⢰⢃⡤⠒⠋⠀⠉⠙⠉⠛⠛⠶⢶⣄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡀⠀⠄⠤⠾⢹⡇⠏⠀⠈⠀⠀⠀⣼⠏⠀⠀⠀⠈⢢⣁⢀⣀⠀⠀⠀⠈⠉⣺⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡄⣀⡡⠼⠃⠘⢄⠀⠀⢀⣬⣹⠃⡀⠀⠀⠀⠃⠈⠈⠉⠁⠲⣦⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠘⢿⡿⡀⠀⠐⠂⣀⠠⠀⠁⠀⣀⣤⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣃⠀⠀⠀⠈⢧⡄⠀⠀⠉⠂⠈⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠀⠀⠀⢹⡝⡄⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠐⠂⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⣠⣴⣶⣆⡔⠋⠀⠀⠀⠀⢀⡜⠀⠱⣀⠀⠀⠀⢠⡻⣷⠀⠀⠀⠀⣀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣁⠀⣼⠙⣿⡄⠤⢤⣤⢀⠊⠀⠀⠀⠉⠃⠀⠀⠀⢡⠹⣷⠀⠀⠀⠤⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⢸⠁⠠⣭⡔⠉⠀⠐⠈⡻⢾⡷⣤⠀⠀⠀⠹⠀⠀⢠⡄⠀⢸⡇⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢦⠀⠘⠖⢠⡿⢀⠤⠀⠀⠄⠀⣈⠀⣿⡇⠀⠸⣿⠀⠀⠀⢡⡄⠀⠻⡄⠰⠀⢾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠘⣠⣀⣵⣼⡇⠀⠀⠂⠀⢠⡼⢧⠀⢸⡟⠀⡆⠀⡀⠀⠀⠀⠁⠶⠀⠀⢠⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠓⠋⠁⠀⠂⠀⠀⢀⣬⠀⣸⠷⣿⠇⢸⠃⢀⠷⣿⡀⠿⢲⡄⠙⣆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠘⠋⣹⠞⣁⡼⠋⠀⡏⠀⣆⡀⠀⠦⠄⠙⣧⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣣⠘⣡⡴⠋⠀⠀⡰⠇⠀⠹⡀⠈⠀⠀⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠁⢀⣤⡠⠐⠛⢧⠀⠀⠈⠓⠚⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠸⣿⠇⠀⠂⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟⢁⣴⡴⠃⠀⠀⢃⢠⡀⣀⣤⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡼⢠⣽⠋⠀⠀⠀⠀⡸⠻⣽⡟⠛⠲⡁⠠⣤⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢰⠃⣾⠁⠀⣀⣤⣀⣤⡇⢰⡋⠳⣄⡀⠀⠠⣄⠘⡶⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣱⡇⠀⠊⢁⣠⠿⡿⠷⣿⣷⣶⣶⣌⠱⡄⠀⡈⠳⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡟⠀⠏⣠⢟⠇⠐⠀⣠⡄⠘⡏⢣⢸⣷⡆⡀⠁⢶⢼⣉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁⣼⠂⠀⠀⠀⡄⠰⡅⢸⢦⣉⠧⣤⣤⣤⣀⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢂⣿⡆⠀⡀⠲⠀⠰⢀⡄⢸⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⢿⣧⠀⡁⠀⠀⢠⠆⡰⣊⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠹⣤⠁⠘⠀⡞⢐⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡰⣴⠧⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡟⣿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⣠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⢶⡿⡾⡿⡶⣶⡶⣦⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣡⠋⣹⠁⠘⢸⠀⡇⠃⠃⠀⣷⠁⠋⠎⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⣠⡴⣿⣄⠀⠀⠀⠁⠀⠀⣠⣿⣷⣄⡀⠈⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣤⣶⡟⠉⠹⠙⠹⢷⣦⣄⠀⣠⣾⡟⠌⠌⢎⠙⢛⣷⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢹⢟⣧⡀⠀⠀⢀⣠⢿⢹⢻⡟⠙⠛⣦⡀⠀⠀⢸⡏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⣸⠸⣧⣠⡾⡿⡇⠀⠈⠀⠁⠀⠀⡿⠟⢿⣦⡾⣻⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣀⣴⡿⡄⠸⠉⠁⠀⣿⣦⣄⠀⠀⣠⡾⣷⡀⠀⠀⢃⣿⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⠘⣼⢻⡀⢀⣠⣾⢿⠀⠛⡷⣞⠣⠁⢸⡟⡷⢶⠾⢸⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⠈⡟⠏⠃⠈⢸⣧⣀⠀⠘⠀⢀⣼⣇⠀⠈⠆⠸⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣯⣀⣠⣤⣤⡶⡿⠏⠋⣿⣶⣴⣿⠋⠙⠛⣲⣶⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠏⣿⣎⡀⠀⢀⣴⠋⠀⠋⢹⣷⣤⣾⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⢹⡁⠟⡷⠟⣿⣦⣀⣴⢿⠉⠃⣱⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣄⠀⣼⠛⢻⡟⠉⢸⡦⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠹⣄⣼⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡧⠾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠄⠀⠀⠀⠀⠀⠀⢸⣿⡷⢤⠀⠀⠀⠀⢀⡰⢾⢹⣿⣯⡍⢱⡾⢿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠁⠀⠀⠀⠀⠀⢀⣿⣿⣵⠯⣳⡀⠀⣠⣿⠃⣶⣦⡹⣾⠇⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡄⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣾⣷⣿⠀⣿⣿⣿⣿⢿⣿⣿⢰⣿⡿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣯⣿⣿⣿⡭⢠⣿⣿⠋⡹⣿⣿⢯⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡡⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣯⣿⣿⣟⣵⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣠⢀⣀⣈⣿⣿⣿⢻⣿⣿⣻⣶⣾⣿⡿⢋⢭⣾⣿⣿⡿⢋⣀⣤⣤⣶⣰⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣽⣿⣿⣿⣿⣿⡟⣉⣶⣾⣿⡿⠛⠉⢁⣴⣿⣿⣯⣿⢿⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣦⣦⡾⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠃⠈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣯⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡏⠘⠃⠙⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡱⠛⣿⣦⡀⠀⠋⡉⠀⠀⠀⢸⡇⠀⠀⠀⣠⣿⣯⢯⣤⠒⠢⠺⣦⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠀⣴⠋⠈⠉⣿⢷⡌⣳⡄⠀⠸⣧⠀⢀⣼⣿⣿⣇⠈⣿⡀⠀⠀⠘⠆⠀⠐⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡶⢾⡏⠀⢀⣴⢿⣤⣷⡿⠷⠒⢲⡞⠋⠀⢽⣿⠿⣿⣶⣿⠟⠦⣄⣀⡙⠂⡀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏⠀⣸⣷⣶⣿⣿⣾⡿⠃⠀⠀⠀⢸⣧⠀⠀⠀⢹⡄⠈⠻⣷⠀⠀⢹⡍⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡏⠻⢿⡇⣿⣿⣿⠁⠀⠀⣀⣶⠀⠙⣀⠀⠀⢸⡷⢀⠀⣿⣇⢀⠊⣠⡀⢰⡄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⣇⡴⢿⣿⣿⡇⢻⠶⣶⣾⡟⣿⡇⠀⠈⠃⣤⡸⠏⡾⠁⠘⣿⡇⠀⣿⠉⢀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⡋⢸⣷⣿⣿⣿⣧⣾⠀⠸⣇⠀⣠⣿⣆⣀⣼⡟⢁⡼⡃⢠⢀⣿⣄⠼⣿⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⣿⠿⣿⠹⢀⣴⣿⣿⠋⠉⢨⡏⠈⢁⡝⠀⢠⡟⢽⡟⣿⣧⣿⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠸⡄⢹⣟⠏⢹⣧⣀⡜⢁⣠⣿⡄⢠⡞⢀⠀⠈⠁⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢾⡻⢦⡞⠉⡟⠀⡾⠟⢀⡿⠋⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⢿⣾⢡⡞⠀⡴⢄⡠⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡿⠜⢧⡖⡁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠂⠈⠁


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠛⠋⠉⠉⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣄⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠋⠉⣠⣾⡿⠟⠉⣿⣿⣿⣿⣿⣏⠛⠿⢿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠀⠀⣠⣿⡿⠋⠀⠀⠀⢸⣿⡇⠈⠛⠿⣿⣦⣄⠀⠙⠻⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠈⠛⢿⣷⣄⠀⠀⠉⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠙⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠻⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠈⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠠⣿⡏⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⣿⡇⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⣀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣷⣤⣤⣄⣠⣤⣤⣶⣾⣿⠟⠋⣡⣿⠟⢹⣿⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⣿⡇⠙⢿⣷⣄⠉⠉⠛⠛⢿⣿⣦⣄⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣹⣿⠿⠛⠛⠛⠛⠉⠁⠀⢀⣾⡿⠋⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⠹⣿⣧⡀⠀⠀⠀⠈⠹⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⣾⣿⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⢹⣿⣧
⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠙⣿⣦⠀⠀⠀⠀⢿⣿⡆
⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠘⣿⣿⡀
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⢀⣾⣿⢿⣿⣿⣷⡀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢻⣿⡇
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢠⣾⡿⢋⣿⡿⠙⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⢸⣿⣿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⣾⡿⠁⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⣰⣿⣿⡅⠀⢻⣿⡿⣿⣧⣄⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠟⠁⢀⣾⡿⠃⠀⠀⠀⢈⣿⣿⣦⣀⠀⢀⣼⣿⠿⣿⣷⠀⠀⢻⣿⡈⠻⢿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⣿⣿⡀⠀⠀⣀⣀⣤⣤⣾⣿⠿⠋⠁⠀⢀⣾⡿⠁⠀⠀⠀⢠⣾⣿⠛⠻⣿⣷⣿⡿⠏⠀⢻⣿⡇⠀⠀⠻⣿⡄⠀⠙⠻⢿⣿⣶⣤⣄⠀⠀⠀⣿⣿
⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠸⣿⡟⠀⠀⠀⠸⣿⣷⠀⠀⠀⠹⣿⡄⠀⠀⠀⠉⠛⣿⣿⣿⣶⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⢻⣿⡟⠛⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢸⣿⣷⣄⠀⠀⠀⠀⠀⠹⠂⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡿⢛⣿⡿⣿⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⢸⣿⠃⢿⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⣿⣿⣿⠛⢿⣿⣦⣄⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⣠⣶⣿⡿⠋⠀⠀⠀⣾⣿⠀⠘⣿⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⣿⡿⢸⣿⡆⠀⠙⠻⣿⣿⣦⣤⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣤⣴⣿⡿⠟⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⠹⣿⣆⠀⠀⠀⠈⠀⠀⠀⠀⣾⣿⠇⠀⣿⡇⠀⠀⠀⣼⣿⡿⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠋⠀⠀⣿⡧⠀⠀⠀⠀⣿⡇⠀⠀⠀⠻⣿⣆⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⣿⡇⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣿⣿⣷⡄⠀⠀⢀⣼⣿⡏⠀⠀⠀⣿⠁⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇⠀⠀⢀⣿⡿⠋⢻⣿⣷⣶⣿⣿⣿⣿⡆⠀⠀⠉⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⢸⣿⡏⠉⠁⢸⣿⣿⣿⡄⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⣼⣿⡇⠀⠀⢸⣿⠀⠀⠀⣸⣿⠿⣿⣷⠀⠀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⣴⣿⣿⣧⠀⠀⢸⣿⠀⠀⢀⣿⣿⠀⢻⣿⣦⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣴⣿⡟⠹⣿⣆⠀⢸⣿⠀⠀⣸⣿⡇⠀⢸⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⢻⣿⡆⠈⠿⠀⢠⣿⡿⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⢠⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠤⠤⠤⠤⠤⣤⣀⣀⢸⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠇⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⠐⠂⣤⣀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢢⣄⢰⠀⠀⠀⠙⣧⡀⠀⠀⢀⣀⣤⠴⢛⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣅⣀⠀⠀⠀⠀⠀⠀⠒⠂⠉⠉⠻⣦⡀⠀⡀⠜⣷⣴⠞⠋⠉⣠⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠙⠛⠛⠶⢦⡀⠀⠀⠀⠀⠀⠈⢳⣤⡷⠟⢛⠛⣷⣶⣞⠉⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠬⢿⡍⠙⠲⠤⠤⠴⠃⠀⠀⠀⠀⣠⡶⠛⠉⠙⢦⣠⡿⠋⠉⠻⣷⣄⢠⡇⠀⠀⠀⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠐⠒⠒⠶⢿⣟⡛⠯⣟⢻⠀⠀⠀⠀⣴⠿⣆⠀⠀⠀⠈⠻⣧⢧⣀⣀⣠⠞⣡⣶⣥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣭⡶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢿⣦⣈⠻⣤⣀⠀⢰⠏⢀⣘⣛⣒⠒⣶⣶⡿⠋⠉⠛⠛⣿⣟⠀⠀⠀⠘⠢
⠀⠀⠀⠀⣄⣀⣤⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡔⠉⠀⠀⠀⢐⢹⣿⠿⠟⠛⠻⣶⣋⣉⣽⣛⣏⣛⣿⡄⠀⠀⠀⠀⣿⠻⣦⡀⠀⠀⠀⡆
⠀⠀⠀⠀⠈⠣⢄⡀⠀⠀⠀⠀⠀⠀⠠⠤⠴⠒⠞⠉⠁⣠⠀⠀⠀⠀⠀⣠⣿⡟⡄⠀⠀⠀⠙⢿⣭⣥⣤⣤⢬⡽⠿⢦⣀⠀⡰⠃⠀⠈⠙⣦⠄⢀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⣲⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢷⠀⣀⣠⡴⠟⠁⠀⢷⠀⠀⠀⠀⢀⣼⡿⢕⡊⠝⠻⣧⠀⠀⠀⢸⣆⡀⠀⠀⣸⠋⢷⡀⠀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⣀⣀⣀⣀⡀⠀⠀⢸⠃⠉⢈⡵⢿⡄⠀⠀⠀⠈⢣⡀⠤⠶⠛⠙⣷⠀⠀⠁⠀⢹⣇⠀⢠⡞⠉⠙⠶⠊⠀⠀⠀⣷⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣯⣤⡤⠤⡿⠀⢀⠞⠀⠀⠳⡀⠀⣀⣤⣾⡇⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⠇⠉⢅⠀⠀⠀⠀⣿⣄⠀⠀⠇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣸⢻⠀⠀⠀⠀⡘⠛⠉⠠⠉⢷⡀⠀⢀⣠⣾⡷⡀⠀⣰⡏⠀⢀⡾⠳⢤⣀⠼⠁⠈⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢂⣀⣠⠾⣇⠀⠀⠀⠀⢈⠟⠉⠉⠉⢸⡇⠙⠊⣿⠈⢐⡞⠃⠀⠀⠈⣆⠀⠀⠀⣸⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣴⡏⠉⠁⠀⠙⢆⣀⣀⡴⢿⠀⠀⠀⠐⣼⠀⠀⡸⠉⠃⠾⡻⣄⠀⠀⢀⡟⢦⡀⢠⣏⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⣼⠉⠁⠀⠈⢧⡀⠤⠚⡽⠢⠊⠀⠀⠀⠀⠑⢌⣱⠄⡸⡇⠀⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣓⠀⠀⢠⣾⡇⠀⠀⠀⠀⣰⡇⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠤⣾⠋⠈⡇⠀⠀⣠⠞⣣⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⠔⠱⠐⠻⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⣠⣄⣤⣠⣤⡀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣾⣿⣿⣷⣿⣿⣄⣠⠀⠐⠂⠠⠠⠀⠄
⠀⠀⠀⠀⠈⠁⠀⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⠋⢰⣯⣛⣿⣿⡿⣆⡀⠀⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠛⠛⠟⠿⠿⠛⠉⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢒⡦⣄⡂⢀⣠⠀⠀⠀⠀⠉⠛⢛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢧⣄⣸⠀⠂⡑⡺⣇⢀⣀⣀⠀⡠⠔⣯⡓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⢲⣶⠲⠺⣆⠐⠘⢿⣏⠂⢈⣭⣯⠀⠁⣼⣒⣚⣉⣉⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡆⠀⢘⣀⣸⠋⢢⣤⢼⡆⠀⠀⡉⣷⣾⡿⠁⠁⢰⠋⢁⡿⡇⠘⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⢹⡇⠈⢽⡄⠀⠀⠕⢙⣷⠄⣀⣼⡿⠖⢀⣤⣾⡙⠛⠀⡷⠋⢲⠒⠋
⠀⠀⠀⠀⠀⠀⢐⢒⣏⢋⠛⠓⢳⠀⠀⠀⣰⣟⠀⢾⡟⢻⣿⠃⣿⣿⠇⠀⣰⠷⠒⡛⡱⡲
⠀⠀⠀⠀⠀⠀⢨⢰⠈⢿⠀⠀⢀⣦⣤⣚⣿⣿⠟⠀⠀⠀⠁⢶⣿⣟⣃⣤⣯⠀⠀⡿⠀⡇⡄
⠀⠀⠀⠀⠀⠀⢀⠚⣤⣐⣿⡦⣭⣵⡾⠿⠚⠁⠀⠀⠀⠀⠀⠀⠉⠛⠟⢿⣯⡵⣾⣧⡼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⡶⢻⡇⠀⠸⠛⠶⣶⠶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⣉⣁⣼⠋⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠛⠋⠁⡢⣚⣯⣾⠃⠀⠀⡀⢠⣇⢦⠀⠀⠀⠀⢲⣄⠰⣄⠀⣉⠛⢷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡿⣿⣿⣿⣻⣿⡏⠀⡠⠘⣡⣿⣿⣮⡃⢣⡀⣄⢸⣿⣿⣶⣶⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣡⣾⡿⠻⣱⢿⣿⣰⣁⣵⣾⢻⡟⣿⣿⣿⣷⣤⣌⣸⣿⣿⣿⡿⣎⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢘⣽⣿⠇⢠⢣⣿⢻⣿⣿⡇⡟⢸⡇⢿⢻⣿⣿⣿⣿⣿⢿⡜⣿⢷⡹⣿⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠠⢫⡾⣿⠀⠀⡾⢁⡾⣿⣿⡇⠁⠸⠁⢼⠘⡟⢻⣹⣷⡹⡌⠇⢻⡜⣧⣿⣯⢿⣜⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠴⣫⣾⣿⠀⠀⠠⢋⣽⣿⣿⣧⠀⠀⠀⠘⢠⡇⣾⣿⣿⣷⣄⠀⠘⣧⢹⣿⣧⡁⠙⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣤⣾⣿⣿⣿⣀⣠⣾⣿⡏⣿⡏⣿⣧⡀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣷⣤⣹⣿⣿⣿⣿⣦⣈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡟⠁⠇⣿⣿⣿⡿⡇⣿⠀⣿⠀⣿⢿⣿⣦⣠⣿⡿⣿⢹⣿⣿⡿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠸⢳⣿⣿⡇⠀⠀⠀⡟⠀⡿⢸⡏⢻⣿⠹⠁⣿⢸⡇⣿⡇⣿⢹⣿⡻⡟⣿⢹⣿⡏⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣇⠀⠀⠋⣨⣿⣧⠀⠀⠀⠇⠀⠁⠈⢠⣿⣿⣧⠀⠀⠀⠇⣿⡇⣿⣾⣿⣧⠁⠹⠸⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣿⣄⣠⣾⠟⣿⣿⣧⠀⠀⠀⠀⠀⣰⣿⡿⣿⣿⣷⡀⠀⠀⢹⣧⣿⣿⣿⣿⣷⣀⠀⣧⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢻⣿⡟⠙⠀⡟⢹⣿⣷⡄⠀⣠⣾⡟⣿⠁⣿⢻⣿⣿⣦⡀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠘⢹⣿⡀⠀⠁⢸⢻⣿⣿⣾⠟⢹⡇⣿⠀⣿⠘⣿⢿⡿⣿⣿⡏⢿⢸⡇⣿⢹⣏⡿⢻⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⣿⢻⣧⡀⠀⠀⢩⣾⣿⣿⣆⠈⠀⠘⠀⡿⠀⡟⣸⣧⣿⣿⣿⡀⠀⠃⣿⣸⣿⣿⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⢸⠿⣷⣄⣴⡿⢻⢸⣿⣿⣷⡀⠀⠀⠃⢰⣧⣿⢿⡟⣿⣿⣿⣦⣰⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢹⣿⣇⠀⠘⢸⡇⢸⢿⣿⣷⡄⢀⣾⡟⢻⠀⡇⢿⢻⡟⣿⣿⡇⠈⣟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣾⢿⣿⣆⠀⠀⠀⠘⠘⣱⣿⣿⣿⣿⣧⡈⠀⠀⠘⢸⣷⣿⣿⣧⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⠉⢿⣷⡄⢀⣴⡾⠻⠘⠁⡟⣸⢿⣿⣶⣄⢀⡿⠟⣿⢻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⣿⢿⣿⣿⣷⣄⠀⠀⠁⣿⣼⣿⣿⣿⣿⣷⣀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣇⠟⠹⣿⢿⣷⣄⣼⣿⣿⣧⠛⣿⣻⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡾⠻⣷⣌⠙⢟⣫⡿⠛⢷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⡐⠛⠂⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⡕⣀
⠀⠀⠀⠀⠀⣠⡭⣀⡰⡃⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⡌⢙⣾⡷⣏⣔⠇
⠀⠀⠐⠒⠺⣿⢌⠀⠉⠲⠄⠀⠀⠀⠀⢀⣻⠟⢰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠪⠙⠻⣿⣫⠀⠀⠈
⠀⠀⠀⠐⠖⠃⠠⠀⠀⠀⠀⠀⠀⠀⡀⢁⠀⠀⠈⠀⠈⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣂⡑⡄⠠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠉⠻⡰⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢩⠖⠀⠊⠁
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠐⠠⠄⠀⠀⠀⠄⠂⠀⣠⣤⠶⠦⠄⡀⡆⠄⣀⢀⣠⡀⠀⣴⣆⠤⠒⠒⠒⢢⡀
⠀⠀⠀⠀⡠⠐⡐⢾⣿⣂⠁⠀⠀⠀⠀⠀⠐⡿⠕⠁⠀⢀⢀⣨⣽⠒⠒⢿⡟⠒⠊⢽⣶⢀⣠⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠈⠁⠥⢴⣖⠠⠄⠀⠀⠀⠀⠀⢠⡀⠢⠄⠆⢠⣼⡿⣷⡄⠛⡟⠣⣱⡌⡱⢠⢦⡔
⠀⠀⠀⠐⠀⠒⢀⡄⠁⠀⣀⣤⠀⠀⠤⠴⠖⠀⡐⢦⣎⡰⣄⡔⠣⢶⡔⡸⣥⠆⡽⡿⢁⠚⠛⡎
⠀⠀⠀⠀⠀⠀⠐⢧⠄⠀⡀⡀⠀⠀⠀⠀⠀⠀⢁⡜⠀⠷⠧⠛⠋⠀⠻⣕⠴⢷⣼⠁⠈⠈⠐⠐
⠀⠀⠁⠆⠀⠲⠂⠀⢲⢋⢠⣄⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠷⠃
⠀⠀⠀⠀⠁⡄⡀⠁⠀⡆⠼⠏⢁⠀⠀⠀⠀⣀⠤⠤⡄⢀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠤⠈⠤⠔⠉⠁⠀⠀⡀⠀⠂⠍⠀⠀⢠⢀⣈⠷⢤⠤⣦⣷⠊⠀⠀⠀⠀⠀⠀⠄⠀⢴⣤⣤⡴⣤⢷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠠⠀⢀⣀⡀⢤⢀⡏⠹⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⣈⡯⠉⠕⠶⠐⢍⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣍⡁⠀⠄⠈⠐⢧⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠉⠈⠋⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠙⢝⠲⢶⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠚⠒⠂⠲⢌⠢⡌⠑⣾⣯⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠡⠤⠴⠦⠤⢶⠦⠶⠤⠯⢥⣤⣬⣿⣄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠚⠒⠠⣄⣀⣀⠴⢒⡫⠷⣀⠠⢴⢿⣯⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠾⠛⠓⠒⠂⠐⠐⠚⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⡀⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣵⣝⣄⠀⢀⣠⣤⠶⠞⢛⠃⣹⢟⠢⣬⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠛⠛⢻⣯⡁⠈⠉⠙⡞⣾⠛⠁⠀⠀⠀⢰⢆⡍⠘⡦⣈⡳⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣽⣷⣤⠴⣦⠀⢹⠀⠀⠀⢸⡙⡇⠀⠀⠀⠀⠀⡜⣍⣷⢾⡿⣷⢤⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⣀⡤⢏⠀⠜⢳⣼⠀⠀⢀⣿⢁⣧⠀⠀⢀⠀⠀⢗⢌⣿⠀⢷⠪⢳⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠛⠉⠀⡼⢦⢴⢿⠇⠀⢀⡾⠛⠉⢉⡦⠶⢼⠁⠀⠈⣆⢛⠀⠀⣧⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⡴⣧⡀⢈⠎⣠⣴⣧⡤⡶⢚⢧⣀⡀⠀⢳⡀⠀⠘⠀⢢⡀⠃⣄⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡏⣜⠸⣿⡇⡸⠟⠋⢰⡿⣇⠀⠀⢀⣼⠟⢷⢶⠕⠀⠀⠀⠀⠙⢦⠘⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣶⣀⣤⣴⡶⠶⠶⠾⠧⢟⠳⠜⠉⠀⠀⠀⡎⠃⢉⡧⠔⠋⠁⣸⣆⠈⠆⠀⠀⠀⠀⠀⠀⠱⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢯⡽⣓⠻⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣶⡉⠀⠀⠀⣠⠋⠻⡎⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⢘⠻⠿⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢳⣄⡴⠚⣿⣄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢋⡇⣀⣠⠼⢻⢦⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣏⢈⡦⢾⡀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢉⡼⣇⠀⠈⢀⡿⢻⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠛⢋⣠⠞⢄⣰⣿⣄⡀⢦⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⠛⠁⠀⣰⡇⠁⠫⠙⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢹⣇⡴⠻⠁⣼⣦⡀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡴⢫⡟⢧⣀⣠⢷⡀⠛⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⠀⢸⠛⢊⠹⢦⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠷⠻⡀⡸⠷⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⡅⠀⣠⠟⢳⠤⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡛⠋⠉⠁⠉⠉⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠾⠁⠠⣀⣀⣦⣵⣴⣶⣶⡶⠁
⠀⠀⠀⠀⠀⢠⢎⣁⣴⠿⡻⠟⠛⠉⠙⠉⢛⢿⡷⣆⢰⡀
⠀⠀⢀⣀⣐⣁⢉⠉⠁⠈⠐⠀⠀⠀⠈⣀⣠⠞⠁⣸⡗⢿
⠀⠀⠀⠀⠀⠀⠀⢉⣑⣛⣓⣚⡲⠚⠚⠉⠁⣠⠟⢹⡇⠈⣷⡀⠀⠀⠀⡠⠀⠀⢀
⠀⠀⠀⠀⠀⢠⣚⢁⠾⠋⠉⣹⢦⣀⣦⢤⠾⠋⠀⢸⠀⠀⠀⠙⢦⣴⣿⡿⠿⣤⠤⣼⠛⠀⠀⡀
⠀⠀⠀⠀⢠⠞⠠⡿⠂⠄⠤⣿⠀⠘⣧⡜⠀⠀⠀⣸⠀⠀⠀⠀⠈⢸⠋⢁⣀⣼⡄⠈⠙⣏⠛⠈⡀
⠀⠀⢰⠂⢿⠄⢤⣧⠄⠐⠚⣏⣀⢀⣽⡇⣴⡚⣿⠻⢏⡀⠠⢠⣇⣿⡏⠈⢉⢹⣿⣿⣯⠙⡲⢾⣁⢀
⠀⠀⣾⢘⡻⡅⢤⣍⠀⠀⠀⣸⠃⢀⣿⡞⠙⠉⣩⠗⠋⠈⡧⠶⣋⣟⠃⣀⣦⣿⣿⠥⠈⢻⡿⠈⣛⠰
⠀⠀⠈⢹⡷⠒⠛⠋⡹⠉⢽⡧⠖⠁⣼⣠⣴⣞⠛⢆⠀⢠⣧⠄⠈⠏⠳⠿⠯⢍⣉⣲⢤⣶⣫⠽⠋⠃
⠀⠀⠀⠊⠼⠖⠁⠐⠒⠒⠉⠀⠀⠀⢇⡇⠈⠹⡄⢨⠷⠎⢹⣶⣦⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⣴⡿⣷⣀⠀⣈⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠻⢳⡀⠈⢿⠿⣷⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣴⡏⢷⡷⣀⡌⠹⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠈⣤⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⡱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠄⠤⠴⠊⠛⡊⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠉⠑⠀⠀⠈⠀⠀⢀⣀⠔⠉⠀⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠠⠛⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠂⠀⠐⢀⠆⠀⢀⣴⣊⡀⢠⠔⠈⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠤⡦⢤⣀⡐⠀⠀⠀⠄⠀⠀⢠⠄⠀⠠⠊⠉⠉⠁⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣩⣤⠞⠋⠛⡶⣿⣦⡈⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⣡⡞⠽⣷⠦⣄⡀⣹⠏⠀⠛⠷⣜⣇⠓⠀⠀⠀⠣⠀⠀⠠⠤⠄⠀⠒⠈⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⢙⡧⢤⡔⠁⠀⠈⠛⡿⢤⣀⡈⣸⠟⣷⢻⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⣴⡽⣶⠋⠐⣹⠧⢤⣀⣀⠜⢁⠀⢼⡿⠃⠰⡟⣘⣦⡆⣀⠤⡤⠴⢖⡀⠉⠁⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣤⣟⣻⣿⢵⣾⠿⠿⠿⢷⣄⡀⠘⣇⣀⣼⡿⠯⠙⠿⣟⠓⠒⠶⣄⠈⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣰⣿⣉⠀⠀⠙⣿⠷⢦⣕⡳⣿⣟⠻⣵⣾⠟⠀⠀⠂⠀⠘⢆⣀⠀⠈⠓⢦⠒⠁
⠀⠀⠀⠀⢀⣶⣄⣀⣠⠟⡻⢦⡞⠀⠀⠘⠛⠻⢮⡿⣆⣻⡿⠑⢦⡤⢤⣤⠀⠼⠋⠉⠉⠙⣟⢇
⠀⠀⠀⣠⠞⠁⠈⠹⠧⡀⠀⠈⠛⡶⠚⠋⠉⣽⠋⠀⣸⣿⣷⠀⢸⣇⠀⢃⢳⣤⣀⡤⢶⡏⢹⣎⡆
⠀⠀⢀⡿⠉⠉⠳⢶⣦⠞⠲⠄⡼⠁⠀⠀⢸⡏⠀⠀⠿⠙⢃⠀⢸⡇⠀⠘⡎⣆⠀⠀⠀⢡⣀⣿⣿
⠀⠀⠙⢟⡒⢒⡶⠛⢃⠀⠀⢺⣧⣀⠒⢁⡿⣦⣂⡴⡧⠃⠀⠢⣘⣿⣶⣤⣿⡽⢷⢶⡶⠖⠉⣿⠛
⠀⠀⠀⠀⠘⠿⠀⡀⣈⣳⣦⠔⠚⠩⢟⡟⠀⠨⠟⠁⠀⠀⠀⠀⠈⠉⠛⢿⣮⣀⣨⡇⠉⢳⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠶⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠃⠈⠓⠒⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠉⠁⠓⠠⣀⡀⠀⠀⠀⠁⢴⡇⠀⡠⠚⠇⠺⠟⡛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠈⠱⢦⣀⢀⠈⠓⢄⠀⠀⠀⠡⣀⠀⠀⡄⢓⡀⢀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠓⢯⡀⠈⢡⣀⠀⠀⠻⢤⣤⠁⠀⠈⠉⢁⠀⢁⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⣽⣧⢉⡌⡄⠀⠀⠈⡁⠀⠀⣀⠠⢤⣦⣮⣁⣄⡀⠀⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠞⠷⠂⢀⣴⣿⢿⠏⣀⠌⠀⠀⢠⣅⠔⠉⠁⢠⠀⠄⠀⠁⠈⠙⠳⠦⢄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠈⠐⢚⠁⣿⠘⡄⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⡁⠦⠠⣤⣄⣀⡀⢀⠀⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠐⠖⠚⠁⠀⠙⢄⡀⢀⣶⣾⠉⠢⢀⡀⠀⠀⠀⠀⠀⠀⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⡇⠀⠀⠁⠲⡌⠁⠀⠀⢴⠛⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡶⡄⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⢂⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡠⠚⠉⠀⠀⠀⢈⠇⢱⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠁⠲⠋⢙⡿⢿⡄⠀⠀⡀⠀⢠⠊⠀⠈⠳⢄⠀⠀⠘⠎⢿⡄⠀⠀⠀⠒⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⡺⠃⢸⡅⠸⠟⠉⠙⠿⣦⣧⡀⠀⠀⠀⢰⠀⠀⠀⠘⠀⢻⡄⠀⠀⠁⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢈⢁⠀⢧⡀⣹⡏⠀⡀⠀⢀⠈⠈⠋⣹⣦⠀⠰⣼⠀⠀⠘⠇⠀⠈⣧⠀⢀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡇⡠⠚⣀⣿⠀⠈⠡⠀⠀⣀⣞⠀⢉⣿⠀⠠⠉⡀⠀⠀⠈⠃⣤⠈⠀⠀⣶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠦⠵⠽⠛⠉⠠⠀⠀⠀⢀⠃⢈⣧⣼⡇⢀⡇⠀⣧⣶⠀⣤⣄⠈⠲⡄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠶⢻⡤⠎⢀⠾⠁⢸⠀⡜⢀⠉⡍⡁⠺⣆⠀⢿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡔⡄⢎⣠⠞⠁⠀⢀⡇⠀⠹⠁⠀⠀⠀⠘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠊⠅⣠⣀⠠⠾⢦⠀⠀⠘⠲⠐⠖⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠠⢿⡧⠀⠆⠀⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠋⣠⣠⠋⠀⠀⠂⢀⡀⢀⣤⣴⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⢰⠇⣼⠟⠁⠀⠀⠀⢸⠻⣽⡟⠛⠦⡁⠀⣨⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⡎⢰⠏⠀⣀⣤⣄⣤⡏⢠⡏⠳⣄⡀⠀⠠⣄⠘⡶⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣅⡎⢀⠊⢁⣠⡾⠿⠷⢿⣷⣶⡶⣌⠱⠄⠀⡈⠳⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⠁⠎⢀⢟⡇⠀⠀⡠⣄⠀⡇⢡⢼⣿⡆⠀⠑⠶⢾⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⣴⡇⠀⠀⠀⡤⠸⠆⢸⢶⣈⠗⠤⠤⢦⠤⠵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣾⡇⠀⣄⠚⠀⠐⢀⠆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠻⣷⠀⢀⢀⠈⢠⠃⣜⣥⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢈⠀⡏⣨⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⡾⠓⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣽⣿⣷⣞⣩⣿⣹⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⡄⠀⠀⠀⢠⣞⣿⣿⣿⣳⡄⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⢶⡲⣿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠁⠠
⠀⠠⣿⣾⠀⠀⢿⣿⢿⣿⡿⣿⣿⠀⠀⢠⣿⡷⠀⠀⠀⠀⠀⠀⣴⣿⣸⣿⣾⣷⣿⣇⣿⣦⠀⠀⠀⠀⠀⠀⠠⣀⢂⠀⢀⠀⢂⠀⠀⡀⠀⢀⠀⢀⢀⡤
⠀⠀⢻⣿⣆⠀⠺⣿⣿⣿⣿⣿⡿⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⡽⢿⣿⣿⣿⢧⣿⣿⡇⠀⠀⠀⠀⠀⠐⠪⣡⠄⠀⡠⠀⠀⠀⠀⢀⠀⠀⢦⡵⠒
⠀⠀⠈⢻⣿⣦⠀⠛⣾⣿⣷⠟⠀⢠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⣿⡿⣷⣾⢿⡿⣷⣾⢿⣿⠀⠀⠀⠀⠀⠀⠀⠉⣛⡜⡄⠀⠌⠀⠄⠉⠀⠀⡌⣞⡍
⠀⠀⠀⠀⠹⢿⣷⡄⠀⠉⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡿⢿⣿⣿⡿⢿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠧⡂⠀⠁⠀⠀⠀⣧⠚⠂
⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⡄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣟⣋⣿⡏⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⣄⣠⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⡰⣻⢟⠣⡐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⢀⣠⣤⣤⣄⠠⠃⠈⠂⣰⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⡜⠉⢺⠊⠹⡎⠁⢻⠈⠳⡀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢀⣠⣶⣟⣶⣄⠀⠀⢹⣿⡄
⠀⠀⢠⣾⡿⣿⣿⣿⣶⡀⢠⡾⣿⣿⣿⣻⣿⡆⠀⠀⠀⠀⠀⢰⣀⠤⢻⣀⠜⠱⢄⡜⠢⢄⡃⠀⠀⠀⠀⠀⠀⣿⣟⠀⢀⡼⣿⣿⡿⣿⣿⣧⠀⠀⣿⡷
⠀⠀⣾⣿⣿⣿⣿⣿⣾⠁⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⡇⢀⡎⠙⣄⣠⠊⠙⣄⢸⠁⠀⠀⠀⠀⠀⠄⢿⠋⠀⢸⣿⣟⣻⣿⣟⣿⣿⡀⠀⢻⠟⡀
⠀⠀⠹⢿⣿⡿⠿⠋⠁⠀⠀⠀⠉⠻⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠱⡏⢣⣴⠁⠁⣣⣰⠙⠞⠀⠀⠀⠀⠀⠀⠈⢄⠁⠀⠘⣿⠿⣿⣻⣿⢿⡿⠃⠀⢌⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⠀⡷⠴⡀⡸⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠘⠿⠷⠿⠿⠟⠃⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⠔⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠊⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠰⣝⢦⡀⠀⠀⠀⠀⡀⢀⣤⢶⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠸⡷⣝⡳⣄⡀⠸⡟⠋⠀⠀⠀⠐⠊⠉⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⢷⠀⠙⠳⢿⣷⣿⡀⠀⠀⢀⡀⠄⠂⠈⣻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘⣇⠠⣶⠛⠛⠉⠙⡟⢍⣁⢀⣀⣀⠠⠄⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⢸⣿⢳⡹⣆⢸⡀⠀⠂⡀⠃⠀⢫⠲⢄⠀⠐⠀⠄⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢹⡇⢠⡞⣽⢸⢷⡝⣾⣇⠀⠀⠀⢰⠀⠀⡆⠀⠑⢄⠀⠀⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣾⠶⠋⢀⡿⣿⠀⢻⡜⣿⣦⣤⡀⠀⠀⠀⠑⠀⠀⠈⠃⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⣀⣀⣼⣱⠇⠀⠀⢻⣜⣯⠉⠉⠻⢧⣤⣴⣤⣴⣄⣀⣀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⢀⡴⠛⢛⠁⠈⠙⠳⣄⠀⠀⢻⣈⡷⠚⠛⠟⠛⠳⣤⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⢀⣾⣀⣴⡏⠀⠀⣾⠀⠚⣧⠀⣼⠋⡁⠀⠐⢶⡶⠚⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⠄⠘⣧⣴⡞⠉⠳⣄⣿⠀⡿⠀⣷⣤⠀⠘⣇⡀⠀⢻⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣹⣴⡞⠁⠸⡇⠀⣾⢫⡇⠀⢷⠀⡿⠉⠓⣾⠋⠙⣷⠺⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣷⣤⠟⢻⢿⡟⠋⠀⠀⠈⢻⡷⣅⣀⣿⣦⣀⣼⠀⢻⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⢻⣀⣼⣯⠞⠀⠀⠀⠀⠀⠀⠛⢿⣭⣷⣄⠉⣿⠲⣾⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠮⣽⡿⢦⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⣾⠟⢻⠇⠀⠘⠛⠳⠶⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⣅⣤⡾⠃⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣏⣉⣐⣹⣪⣽⣾⠃⠀⠀⡄⣰⣧⢣⡀⠀⠀⠀⣷⣤⡐⠄⠠⠌⣙⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣻⣿⡿⢫⣿⡏⡀⡴⢈⣴⣿⣿⣷⣅⠱⠌⣆⢹⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣡⣾⣿⠙⣱⢯⣿⣤⣵⡾⢿⢹⡟⣿⣿⣿⣿⣶⣦⣼⣿⣿⣿⡻⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⢀⢜⣿⡇⠀⢃⡿⢫⣿⣿⡇⠏⢸⡇⢿⢸⡏⣿⣿⡿⣿⢻⠸⡏⣧⢹⣿⣿⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣡⠞⣽⡇⠀⠞⢡⢟⣽⣿⡇⠀⠈⠁⢻⢸⡇⣿⣿⣷⡘⠈⠇⢿⠸⣿⣿⢯⢻⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⢐⣡⣾⣿⡇⠀⢀⣥⣾⢿⣿⣿⡄⠀⠀⠀⢸⣇⣿⣿⣿⣿⣦⡀⠸⣧⣿⣿⣷⡀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⡿⢻⢻⣿⣧⣴⣿⢻⡏⣼⡇⣿⣿⣦⡀⠀⣼⣿⡏⣿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠈⡼⣿⣿⡇⠇⠸⠀⣿⠀⣿⢻⣿⣿⣾⢿⠙⡇⢹⢸⣿⠸⣿⣿⣿⣿⣿⡿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢡⠿⣿⣷⠀⠀⠀⡟⠀⠟⠸⠃⣼⣿⡌⠀⠇⢸⠘⣿⠀⣿⣹⣿⠹⠸⡇⣿⢻⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢿⡇⠀⢀⣴⣿⣿⣆⠀⠀⠃⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠀⣿⣆⣿⣿⣿⣧⡀⠁⢹⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠸⣿⣶⡟⠇⢸⢿⣿⣦⡀⠀⠀⣠⣾⣿⠏⣿⣿⣿⣦⡀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠟⣿⡄⠀⠘⢸⣿⣿⣷⣤⣾⢿⡇⣾⠀⣿⢸⣿⣿⣿⣶⣾⣿⡏⣿⢻⡟⣿⢻⣿⣍⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⣰⡿⣷⡀⠀⠈⠹⣻⣿⣿⡇⠈⠃⠸⠀⣿⠀⡿⣻⣏⣿⣿⣇⠁⠸⢸⡇⣿⣾⣅⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠃⡿⣷⡀⣀⣼⢿⡟⣿⣿⣦⠀⠀⠀⠇⢰⣇⣿⣿⡿⣿⣿⣦⡀⣸⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠃⠙⣿⣿⠙⠘⠁⡟⢸⡿⣷⣦⡀⠀⣼⡿⣿⠸⡇⣿⢿⣿⣿⣿⠋⢻⢏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣰⣿⣿⣆⠀⠀⠁⠸⠑⢫⣿⣷⣾⣿⣆⠘⠀⠀⠛⣼⣽⣿⣿⡄⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣇⠛⢻⣷⡄⠀⣠⣶⠿⠹⠏⡿⣻⣿⣷⣄⡀⢠⡿⠟⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⣿⣿⣿⣷⣄⠀⠀⠃⣿⣼⣿⣿⣿⣿⡇⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⠻⠹⣿⢿⣷⣄⣸⣿⣿⣏⠟⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣾⠿⣷⣍⠛⠿⣻⡿⠻⣦⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠋⠁

⣿⣿⣦⣀⠀⢙⡦⢧⣛⠓⢻⡴⠨⢙⡾⠿⣧⣿⣿⣿⢿⡏⣻⣯⢭⡝⠀⠀⠿⠃⠴⠈⢻⡆⠙⠇⠙⠻⣧⣶⡰⠟⠈⣺⠏⠀⣙
⣿⣿⣿⣿⣿⣷⣤⢜⡛⠁⠀⠛⠢⠀⡿⠺⢿⠵⠿⣿⣿⠻⡻⣭⣟⠦⡆⠉⢸⡇⣀⢂⠢⠀⠀⠀⠀⠀⢹⡏⠀⠐⠆⣈⢀⡤⠻
⣿⣿⣿⣿⣿⣍⣿⣶⢸⣿⢤⡤⠀⠠⠁⢰⣦⣟⠛⠿⠟⠺⠻⠾⡮⠘⠀⣴⠾⢿⣽⢾⠄⠀⢤⠀⠀⢠⣀⠃⠀⠂⠀⣸⣄⣤⡿
⣿⣿⣥⣤⣴⠋⣸⣏⣴⡶⢆⣀⡄⠀⣠⣜⠂⠈⠁⠀⠀⠀⠀⠀⠀⡆⠀⢈⡄⢰⠧⠇⢉⡐⠂⠉⠰⣀⣀⠈⠀⠀⢚⡯⠩⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣭⣵⣶⠿⢫⡉⡶⠖⠂⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠉⡰⠧⢄⡄⢻⠁⠈⠀⠈⠀⠀⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡇⠁⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢒⣧⣿⠀⠀⠀⠴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣡⣼⣧⣿⡟⠀⠀⡀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⠀⢀⠀⠀⠿⣏⡛⡷⠾⣷⣳⡃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⡀⠀⠀⠀⠡⣄⡀⠀⠀⠀⠠⠀⠀⠀⣄⠊⠉⠀⡷⠈⣠⡏⠁⢠⠀⠀⠀⣠⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣛⢛⣋⣪⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠄⠈⢐⣀⣠⡘⠋⠸⠁⠀⠀⠀⠀⠀⢈⣿⠝
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣭⢆⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⢇⡤⢀⡔⠀⠀⠀⠀⠀⠠⠾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢺⡵⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢶⠀⠘⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡏⠀⠀⠀⠀⢹⡷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⡶⠀⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠰⡟⢻⡶⠚
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠀⢾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⠿⡛⠁⣄⡄⠀⠀⠀⢰⢶⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢇⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⣿⡶⠤⢌⡮⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢺⣾⠃⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣠⣀⡀⠘⡧⠈⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⡓⠛⢹⣼⠻⢿⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠠⡁⣨⣿⡿⢻⢿⠿⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡄⠀⠀⠀⠀⠀⠀⠰⡎⠉⢷⢸⣿⠋⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠡⣖⠒⠈⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⢖⣻⡿⠿⢻⡞⠿⢿⣟⡲⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠐⢁⡴⠻⠋⠀⢀⡟⢻⡀⠀⠙⢟⢦⡉⠂⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⣀⣀⣀⣴⠋⠀⠀⢀⣰⡏⠀⠀⢻⣦⡀⠀⠀⠙⣦⣀⣀⣀⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠋⠉⢉⡽⢁⣀⡤⠖⠋⠘⡇⠀⠀⠈⠆⠹⡶⢤⣀⡈⢯⡉⠻⣝⢿⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠄⠀⠀⠛⣿⡏⠁⠀⠀⠀⡸⢷⠀⡀⠀⠀⠀⠹⠈⣏⢹⡋⠛⠀⠈⢢⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⢡⠟⣧⠀⠀⠀⠀⠀⠀⠈⠘⡇⠀⠀⠀⠀⠁⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢈⣿⡄⠀⠀⠀⠀⠁⣰⠏⠳⣄⠀⠀⠀⠀⠀⢠⣷⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣠⡤⠖⢻⢩⣧⠀⠀⢀⣠⠞⠁⠀⠀⠜⢳⢦⡀⠀⠀⣼⠁⠙⢳⣤⣄⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠋⠀⠀⠀⠈⡜⢸⣧⣴⡋⠀⠀⠀⠀⠀⠀⠎⣸⢙⣦⣼⡇⠀⠀⠀⢣⢹⡙⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟⠀⠀⠀⠀⠐⢀⡟⠈⠈⣧⠀⠀⠀⠀⠀⠀⠰⠃⣼⠁⠁⢻⡀⠀⠀⠀⠀⠇⢻⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡞⠀⠀⠠⠋⣷⡀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠘⣳⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⡴⣯⠀⠀⠀⠀⠰⢁⡟⢦⡀⢀⡴⢿⡀⠀⠀⠀⠀⡏⣹⢦⡀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣠⠞⠁⢹⡄⠀⠀⠀⢁⡞⠁⠀⠙⠋⠀⣸⢷⡀⠀⠀⠀⢠⠇⢸⡷⣄⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⡀⠀⠘⢻⣆⠀⢠⡟⠀⠀⠀⠀⠀⠀⠃⡞⢻⡄⠀⣰⠏⠀⠀⠁⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠊⣼⠳⠏⢹⣄⠀⠀⠀⠀⠀⠘⣠⠏⠹⢞⣇⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣰⠃⠀⠀⠘⢹⢦⡀⠀⠀⢀⡴⠋⠀⠀⠘⡹⣆⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠻⣆⠀⠀⠀⠃⡼⠙⢦⡴⢋⣧⠀⠀⠀⠐⣱⠟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⣾⠁⠀⠀⠀⢀⠞⣷⡀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠉⢧⣀⠀⠀⣠⡾⠉⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠾⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣷⣄⣀⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⢛⣹⡿⠟⣿⣿⣿⣟⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⢀⣴⠟⠋⠀⠀⢹⡇⠈⠛⢿⣦⡈⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⣠⠟⠁⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⠻⣦⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⡾⠋⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⡞⠁⠀⠀⠀⠀⢀⣼⠟⢸⡟⢿⣄⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢸⣧⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⠀⠀⠸⣿⠀⠀⠈⢻⣶⣤⣀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⣰⣿⠁⠀⠀⠀⢀⣀⣤⡾⣿⡿⣿⡇⠀⠀⠀⠀⡿⠀⠀⠀⢸⣿⢿⣿⡻⠷⣶⣤⣄⡀⢸⡿
⠀⠀⠀⠈⢙⣿⠿⠿⠿⠛⠋⢁⣴⠟⠁⣿⡇⠀⠀⠀⠀⡇⠀⠀⠀⢸⣿⠀⠙⢷⣄⠀⠈⠙⢻⣿⡇
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⣠⡾⠋⠀⢠⣿⡇⠀⠀⠀⠀⡇⠀⠀⠀⣼⡏⠀⠀⠈⠻⣧⡀⠀⠈⢿⣇
⠀⠀⠀⣾⠇⠀⠀⠀⢀⡴⠋⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⡇⠀⠀⢀⣿⣇⠀⠀⠀⠀⠙⢷⡀⠀⠘⣿⡄
⠀⠀⢸⡿⠀⠀⠀⡰⠋⠀⠀⠀⠀⣼⣿⣿⢿⡆⠀⠀⠀⠃⠀⠀⣾⣿⣿⣆⠀⠀⠀⠀⠈⢷⠀⠀⢻⣧
⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣼⠏⠈⢿⣆⠀⠀⠀⢀⣼⣟⠀⢿⣿⣷⣀⠀⠀⠀⠈⠇⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠀⠀⣀⣤⡾⠛⠁⣼⠏⠀⠀⢨⣿⣷⣄⣀⣾⠟⣿⡆⠈⢿⡌⠻⢷⣦⣀⡀⠀⠀⢸⣿
⠀⠀⢿⣷⣶⣶⣾⠿⠛⠉⠀⠀⣼⠋⠀⠀⢠⣿⠃⠈⢻⡿⠁⠀⢸⣧⠀⠈⢻⡄⠀⠉⠛⢿⣷⣦⣸⣿
⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢠⡾⠁⠀⠀⣴⡿⠃⠀⠀⢸⣷⠀⠀⠘⣿⠀⠀⠀⠻⣆⠀⠀⢸⣿⠉⠻⠟
⠀⠀⠀⠀⠀⣿⠃⠀⠀⠰⠋⠀⠀⣠⣾⣿⣇⠀⠀⠀⠸⣿⠀⠀⠀⣿⣆⠀⠀⠀⠙⠆⠀⢸⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⣴⠟⢹⡟⣿⡀⠀⠀⠀⡿⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡆⠀⢀⣠⣾⠟⠁⠀⣿⠇⢹⣇⠀⠀⠀⡇⠀⠀⢸⣿⢻⡇⠙⠻⣷⣤⣀⣼⡟
⠀⠀⠀⠀⠀⢹⣷⣶⡿⠋⣿⠀⠀⠀⣿⠀⠀⢻⣆⠀⠀⠀⠀⢀⣿⠇⢸⡇⠀⠀⣿⡟⠻⠿⠃
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣿⡄⠀⠀⣿⠀⠀⢀⣿⣦⡀⠀⢀⣾⡏⠀⢸⡇⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⠀⠀⣾⡏⢹⣿⠶⠿⣿⣷⡀⠈⠁⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⣼⡟⠀⢸⡟⠀⠀⣿⢿⣧⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀⣰⣿⣷⠀⢸⡇⠀⢸⡿⠈⣿⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠁⢹⣧⠘⠇⢀⣿⠇⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏

⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⠨⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠐⠉⠁⠀⠀⠠⢀⠑⡖⠄⠀⠠⢰⣀⠉⢂⠀⠀⠀⡄⢀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⢀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈⠀⢭⠀⠀⠁⠈⡁⢧⠀⠄⠀⠤⠈⠑⠢⠃⠀⠀⠀⠠⢤⢄⡀⠀⡀
⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⢂⡘⠂⠀⠀⠀⠂⡸⠂⠀⠀⠀⠀⢘⣄⠉⠁⠀⠀⡜⠀⠄⣀⠽⣦⣂⢤⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢴⠁⡀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠾⢅⠩⠁⡠⠠⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⢠⢀⠄⠀⠀⠀⠀⠀⠀⠀⢶⣆⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠠⠄⠂⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⣰⡄
⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠲⡀⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠓⠀⠀⠀⠀⠀⢠⠆⠀⢶⠰⠚⠤⠀⠀⡠⠐⢀
⠀⠀⡀⢀⣀⠐⠀⠀⠀⠀⠀⢈⣬⠃⠀⠀⢠⡀⣁⠈⠀⠀⠀⠀⠂⣶⠆⠈⠑⡢⣄⡂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡷⢋⠙⠉⠀⢀⡬⠁⠈⠋⠽⡛⡲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡿⢁⣠⠤⠂⠀⠈⠑⠒⠀⠀⠀⠐⠬⣩⣽⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣧⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡻⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⣿⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⡄⡘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟⢱⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣠⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣣⠋⠀⠀⠄⠀⠀⠀⠀⠀⢰⠀⠀⢠⠀⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⢋⡆⢀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⢧⣿⣇⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⣼⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⡿⢾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣵⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢀⡴⢾⣿⡿⠿⠛⣉⢁⣼⣵⣴⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⣿⣿⣿⣿⡗⣸⣿⣿⣿⣿⣿⣿⣿⣷⢀⡀⠀⢀⣼⠃⣽⢋⠇⠀⠰⡏⣸⣿⡟⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣸⢿⣿⡟⠸⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⡿⣿⣿⣟⣾⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠜⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣷⣿⣷⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡎⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⢹⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⣎⢿⣿⣿⣿⡿⠿⣿⣟⠻⣿⣧⣶⡄⠀⠀⠀⠀⢾⢟⣫⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠈⠸⢿⣿⣿⣿⣿⣹⢿⣿⣿⣿⣿⣿⡛⠛⠋⠥⣤⡤⣀⠀⠰⣶⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢀⡝⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣁⡛⠋⠠⠬⠐⣣⡐⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣀⢘⣓⣀⡈⠱⠶⠀⠶⢿⣿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⢈⡀⡱⢿⣿⣿⣿⣇⣻⣿⡿⣝⣿⡿⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠛⠋⠉⠛⠻⠿⠿⠿⠶⠖⠀⢀⣀⣈⣤⣥⣴⣼⣿⣿⡿⢿⡾⡿⢽⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⣿⢿⣿⣿⡷⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣭⣭⣍⣉⠛⠷⢦⣤⣤⣤⣼⣧⣀⣀⣀⣤⠴⠋⢉⣩⠭⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠿⠷⠶⣶⡶⡟⠋⠀⠂⠙⠿⣿⣶⣶⠿⠿⠷⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡫⠐⡩⠩⠭⠭⣿⣧⠁⠀⠀⠀⠀⠀⠀⣾⣯⠥⠬⠥⢉⠢⢝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠁⠀⠀⠀⠀⣰⡟⢻⣆⠀⠀⠀⠀⠀⣸⠟⢻⣆⠀⠀⠀⠀⠈⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⣼⠏⠀⠀⠙⠦⣀⠀⣠⠾⠋⠀⠀⠹⣦⡀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⡴⣿⡍⠄⠀⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠠⢩⣷⢤⡀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠓⠉⣰⢏⠷⣌⠀⠀⠀⢀⡾⠿⢷⡀⠀⠀⠀⣡⢿⠻⣆⠉⠚⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢀⡏⠌⠀⠈⠑⢦⠾⠋⠀⠀⠀⠙⠶⡔⠋⠁⠀⢁⢻⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣼⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⣆⡀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡏⠒⣄⠀⠀⣠⡟⢻⣄⠀⠀⠀⣠⡟⢷⡄⠀⠀⡠⠃⢹⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣱⢺⠋⠀⠀⠈⠓⣶⠚⠁⠀⠀⠙⡗⡎⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣴⢿⡈⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠡⢰⡿⣦⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢧⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⢀⡾⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⢳⡀⠀⢠⡟⠀⢻⡄⠀⢠⡞⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠲⠻⡄⠀⢰⠛⠖⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣈⣿⣿⣿⣿⣶⣿⣷⡀⠀⠀⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢼⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣗⣠⡼⣿⣿⣿⣿⣿⡿⠃
⠀⠀⢠⣤⣤⣄⣠⣀⡀⣸⣿⣿⣶⣤⠘⣿⣿⣿⣯⣿⢘⣷⣿⣿⣿⠋⢡⣾⣷⣾⣿⣿⣿⣶⣦⡄
⠀⠀⠸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⡙⠿⢿⣿⢟⡽⠛⠛⠋⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠆
⠀⠀⠈⠻⢿⣿⣿⣝⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⡟⠀⠀⠀⠀⣀⣀⡀⠈⠻⠿⠿⠿
⠀⠀⣀⣠⣴⣾⣿⣿⣷⣝⢿⣾⣿⣿⡟⠁⠀⠀⣿⠟⢶⢴⣷⡞⣿⣭⣿⣖⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣯⣤⡝⢛⠋⠁⣀⣀⣴⠟⠀⠠⣾⣿⣿⢿⣵⣿⣾⢿⣷⣄⠀⠀⠀⠀⠀⠀⢀⢰⢣⢠⢃
⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⠂⠈⣳⠞⠋⠀⠹⣦⠀⠀⠈⠻⠿⣿⣿⣭⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠈⣾⢸⢸⢸
⠀⠀⠀⠀⠉⠙⠛⠿⡿⠛⠁⣰⣶⣩⣤⡀⠀⡾⣿⣿⣏⣀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠙⠘⠺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⢟⣟⢫⣄⠸⣿⣻⡿⣾⣟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⣠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣿⡱⣿⡹⠀⣿⠿⣷⣷⣻⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⡧⢼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣮⣴⡿⣋⡿⠀⠈⠘⢷⣿⣽⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡿⢃⣺⣷⣦⣷⡈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⠀⢙⣯⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣶⣶⣷⣄⠛⢠⢿⣷⣿⡽⣶⡷⠀⢠⣄⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠛⡀⢿⣟⠁
⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⣿⣿⠁⠘⡿⣭⣾⣿⣿⡷⠠⣾⡿⣿⣽⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛⢰⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣻⣿⠿⠀⠀⣿⣞⣻⣷⡗⠛⠐⣿⣿⣿⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡜⠿⠧⢧⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠘⠋⠉⢡⣤⣄⣤⡄⠸⣿⠿⠉⠀⢀⣀⠈⠻⢿⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⣦⣴⡷⠀⣠⣞⢅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣻⣟⣷⣿⠆⠀⢸⣻⣶⣿⣯⣤⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠙⠋⢀⡀⠉⢳⡍
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣫⣽⣿⣿⡆⢰⣿⡿⣯⣿⣾⡟⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⣴⡾⣧⣴⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣿⣿⣷⡟⠁⢰⣿⣿⣾⣧⣍⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢀⣴⣿⡹⢷⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠋⠀⠀⢸⠟⣯⣽⡿⢯⣾⡇⠁⠀⠀⠀⠀⠀⠀⣀⡴⢋⣤⢼⠟⢣⣮⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠴⠞⠁⣀⡸⢧⣤⣽⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣤⣲⢕⠮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣀⣤⣄⣤⣦⣴⣆⣶⣿⣾⠟⠁⠀⠩⣪⢕⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠼⣏⢁⣿⣋⣙⣋⣸⡯⠬⣿⣂⢽⣶⡶⠀⠝⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⠾⣷⡜⠿⣶⠿⢷⠟⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠉⠀⠀⠉⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣶⣾⣿⣶⣶⣦⣄⣠⣴⣶⣶⣿⣶⣶⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⣿⡇⠀⠀⠀⢈⣽⣿⣿⣯⡁⠀⠀⠀⢸⣿⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠹⣿⣄⠀⢀⣾⡟⠁⠈⢻⣷⡀⠀⣠⣿⠏⠀⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠈⠻⣷⣾⣿⡀⠀⠀⢀⣿⣷⣾⠟⠁⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠿⠿⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠻⣷⣄⣠⣾⠟⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣄⣀⣀⣀⣠⣽⣿⣿⣯⣄⣀⣀⣀⣠⣤⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠁⠀⠉⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣤⡀⠀⠀⢀⣤⣶⣦⣄⠀⠀⢰⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠉⢻⣿⠀⢠⣿⡏⠁⠉⢿⣧⠀⢸⣿⠀⠈⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣾⡿⠀⢸⣿⡇⠀⠀⠈⠁⠀⢸⣿⣤⣤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡛⠉⠀⠀⢸⣿⡇⢀⣴⣶⣤⠀⢸⣿⠛⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠙⢿⣦⡀⠀⠘⣿⣇⡀⣉⣿⡟⠀⢸⣿⠀⠀⣠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠙⠿⠀⠀⠈⠛⠿⠿⠋⠀⠀⠸⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣿⣿⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠷⢶⣿⣯⢶⡶⣽⣿⡶⠾⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣏⣁⣤⣶⣿⣟⠁⠀⠀⠈⣻⣿⣷⣤⣈⣹⡿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⢿⣶⣕⣪⣶⠿⣿⣿⣿⣿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡽⠏⠜⣸⡏⡿⠏⡇⠸⠉⣿⣿⠀⠇⢻⣿⣿⢸⡅⠁⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡇⣀⣠⣾⣿⣧⠀⠘⠀⠀⢀⣿⣿⡀⠀⠈⢹⡿⣼⣿⣷⣄⣙⢺⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⣿⢿⠛⡿⣎⠦⣢⣴⡿⢿⡿⢿⣦⣔⠼⣳⢿⣿⣿⣿⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢸⠘⣇⣿⠘⠀⠁⢻⣿⠹⠀⠇⠸⠃⢸⠀⡏⣿⡟⠈⠎⣿⣿⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⣿⣜⣀⡤⣠⣾⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⣷⣄⠄⣸⢣⣿⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣶⣿⣿⣿⡿⠿⣦⣀⢤⡤⣀⣴⠻⠉⣿⣿⢻⣶⣿⣿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡻⡏⢷⡹⠁⠃⠈⢸⣿⢷⣾⣿⡇⠀⢠⣿⣿⡿⢹⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣸⣷⣄⠄⣰⡏⠘⠸⠀⠃⢻⣆⠸⣫⣾⣇⣮⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⣿⣿⡟⠻⢷⡀⠀⠀⢀⡾⡟⢻⣿⢿⠟⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣃⢀⣾⡿⣎⣱⣿⣷⡀⣼⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣟⡟⢻⣹⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣖⠒⠒⠚⠛⠛⠓⠒⠲⠦⣤⣍⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⣀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣀⣤⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡓⠀⠀⠀⠀⠀⠀⠀⠠⠤⠼⠓⢾⡀⠀⠀⠈⢻⣄⣠⡤⠖⠛⣉⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠙⣦⣀⣬⡾⠿⣷⣤⣠⠞⠋⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣖⠒⠦⣄⣀⣀⡽⠀⠀⠀⠀⢀⣤⠞⠛⠧⣄⣠⡾⠋⠻⣶⣄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠈⠁⠀⠀⠀⠤⣶⣶⠲⢖⣶⢻⠁⠀⠀⠀⣼⢯⡀⠀⠀⠈⠻⣦⢧⠀⠀⣀⠔⣡⣔⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣙⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣦⡙⠾⣄⡀⠀⣸⠃⠀⠿⣒⠲⠶⣶⣿⠟⠙⠻⠷⣿⡋⠁⠀⠉⠂
⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠊⠁⠀⠀⠈⢹⣿⠾⠟⠛⢿⣴⣋⣩⣟⣻⣛⣿⡅⠀⠀⠀⠀⣿⠳⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠛⠉⢁⡆⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠹⢿⣭⣤⣤⡤⣽⠟⠦⣀⠀⡰⠁⠀⠈⢳⡦⠄⡸⣧
⠀⠀⠀⠀⠀⠀⠀⢉⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡇⣀⣠⡴⠟⠁⠀⢻⠀⠀⠀⠀⣠⣿⡏⠀⠈⠙⢷⡀⠀⠀⢸⣦⡀⠀⢠⠟⠙⢆⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣤⣀⣤⠀⠀⠀⡜⠀⠀⣴⠞⣧⠀⠀⠀⠀⢡⡄⠔⠛⠁⢸⡇⠀⠀⠀⠈⣧⣀⣠⠋⠀⠉⠓⡁⠀⠀⠸⢦⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠖⠒⠇⣀⡜⠁⠀⠈⢢⣀⣤⠴⢿⡄⠀⠀⠀⠈⣷⠀⠀⠀⢰⠇⠀⢠⡀⠀⠀⢠⠷⢤⣀⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠆⠀⠀⠀⣴⠁⠀⠀⠀⠳⣀⣠⠤⠞⣿⠣⣀⣴⡯⠀⣠⡏⠙⠒⢶⠋⠀⠀⠈⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠦⠶⠚⢻⡀⠀⠀⢀⣴⠁⠀⠀⠀⡿⠀⠁⡼⠄⢀⣧⠀⠀⠀⢸⣧⠀⠀⢠⠇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡚⣿⠀⠀⠀⠀⢹⠶⠖⠋⢹⡀⠀⢀⡴⣇⠀⠔⠀⠀⠉⢌⠳⢄⠀⢸⠏⠳⣄⠘⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⠀⠀⠀⢠⡾⠀⠀⠀⠀⢳⠂⠁⣰⠃⠁⠀⠀⠀⠀⠀⠑⠛⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠃⢀⡴⠋⡇⠀⠀⢀⣰⢟⠧⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠁⢠⢿⠀⢴⡋⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢬⠀⢀⣀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣊⣩⠖⠛⠳⣜⢿⣿⡛⣐⣌⢢⡀⠀⠀⠀⠀⣠⡄⣉⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣝⠋⠀⢠⡶⣿⠟⠉⠀⠀⠀⡉⠩⠛⢤⠐⠂⢩⠄⠄⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣁⠈⡹⠋⣱⣾⡿⢦⣀⠐⢾⣿⠿⠿⠿⢼⣯⡀⠀⡔⢁⣠⣗⢐⣦⠀⢀⣠⡀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⠋⡽⠁⠪⠽⡟⢀⣴⠿⠛⠋⠠⡤⠤⣤⣦⣘⡰⣔⣰⢫⣱⡻⢩⡻⠾⣯⡍⢛⣻⣵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡶⢡⡴⠿⡿⠉⣯⣼⡿⠦⣄⣀⠀⠀⠀⡵⠋⠉⢩⡙⣼⡄⠙⣋⠁⢿⣦⠙⢿⠿⢿⣿⡻⣇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢓⠖⠲⣾⠁⣰⣿⢋⣴⡾⠋⠉⠉⣩⡿⠁⠀⣰⡿⠂⡀⣿⡄⠘⡄⠈⢋⣀⡘⡄⠀⠉⠛⠌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢣⢶⠏⠴⠞⡿⢛⣿⠃⠁⠘⠃⠀⠰⠚⠻⡇⠀⢰⡟⠁⢠⣅⠻⠿⠃⠸⣦⣀⠙⢻⣿⣶⣤⢤⣶⡶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢁⣾⣿⢋⣽⣡⢽⣷⣶⠖⡿⠀⠀⠀⠀⣠⡇⠀⠋⢀⡴⠋⣿⡟⢶⡶⢶⣾⣿⠳⣼⣇⠈⢻⣸⣿⣷⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠇⠻⡟⣹⠀⢣⠴⢶⣶⠹⣿⢧⣶⡾⠇⠀⠀⣿⠃⣸⣦⢸⣿⣯⡀⢿⢻⠿⣦⣼⣮⡟⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠇⢿⠀⠊⠀⣨⣧⠀⠀⡼⠋⠀⣶⡀⠀⣿⢴⠃⡟⣸⣿⣿⣿⣀⣿⣄⠹⢇⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠺⣴⠖⠉⣰⣿⠀⠠⢀⣠⠊⢸⣿⣆⣡⣷⣄⢡⣿⡾⣯⣟⠛⠏⠉⣷⣼⡉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡄⠈⣉⠴⣤⠊⢉⣿⡆⠘⣱⠛⡿⡿⠁⠉⠻⣻⣟⢿⣷⣄⣸⡅⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠏⠷⢋⡇⠘⢿⡄⠘⡩⢷⡾⣿⡤⠊⠀⠀⠀⠀⠈⠉⠓⠿⠹⣿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠉⣧⡖⠡⣷⡿⠠⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠛⠃⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⣠⡾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⣾⠻⢷⡾⢋⣤⡽⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡟⣉⣭⣥⣬⣭⡿⢾⣟⠛⡫⡔⡇⣿⣶⣹⣿⣿⣿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣿⠟⣉⣷⣿⢻⣏⢸⢹⠀⠀⡇⡇⠃⠋⢹⠹⣿⣿⡭⣛⢷⣍⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣝⣵⠏⠀⠀⢋⣭⡆⣿⠘⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣱⣟⠪⡳⡝⢷⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣽⠇⠀⠀⠀⠪⢝⡏⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢸⢳⡕⠌⢮⠊⣷⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡟⠀⠀⠀⠀⠠⠚⢕⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡗⠗⠀⠀⠀⠀⠀⢸⡇⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣹⣾⡇⡀⠀⠀⠀⠀⠀⠹⠞⣷⠀⠀⠀⠀⠀⠀⠀⢂⡾⢛⣷⡀⠀⠀⠀⠀⠀⠘⣧⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⣿⠸⠁⠀⠀⠀⠀⣀⣶⣾⣻⢳⣄⠀⢀⠀⡠⢠⣾⡁⣿⠘⣷⣄⠀⠀⠀⢀⠀⣿⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣋⣡⣥⣴⠶⡛⡏⡿⣿⢹⢱⢹⢷⡘⠖⢠⣿⡾⢻⡟⡘⣮⣛⣶⣧⣽⣥⣃⣘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡿⢩⢩⢸⠘⠈⡇⢹⣇⡏⡞⠘⠸⢸⣻⣴⡟⣿⠁⠆⠀⠁⠉⠚⢸⣷⣷⣾⢿⡏⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠈⢸⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢃⡟⠇⢿⡆⠀⠀⠀⠀⠀⠀⣿⠘⠟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⠀⢈⠀⠀⠀⣾⣿⡄⢀⠀⠀⠀⠀⠀⠀⣸⣷⣄⢀⠀⠀⡴⢸⡇⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢀⠀⣼⣿⣿⠛⢦⡀⠀⠀⡀⢀⡼⣫⡿⢙⢷⣝⢴⢃⣿⣷⠆⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⢊⣾⡿⠃⠋⠉⠢⠙⢦⡀⣰⠏⠔⠙⠁⠘⢙⣿⢷⣼⣯⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⣿⢿⡇⠀⠀⠀⠀⠀⠈⣿⠹⠁⠀⠀⠀⠀⣼⠏⠺⢿⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠈⠀⠻⣆⠀⠀⠀⠀⣼⣿⡀⠀⠀⢀⠞⣴⣿⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠢⢸⣷⡄⢇⣾⣿⣟⢷⣜⠒⣡⣾⢳⣿⣇⠐⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣾⡷⣿⣿⣏⠁⠀⠁⢹⡷⡿⠋⣼⠷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠁⢻⣄⢀⢀⣿⠁⢀⣼⢏⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣆⣻⣦⣼⣿⣴⣿⣧⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠈⠛⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠉⠀⠀⠀⠀⠉⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⡀⠀⠀⢀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣴⣶⣦⣄⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⣶⣾⣿⣿⣿⡿⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡖⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣿⣿⣿⣿⣿⣻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣤⣄⣀⣀⣀⣀⡀
⢀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡶⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡠⣏⣬⡇⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣟⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⢀⠛⣉⣿⣉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⠀⠀⡀⠲⠐⠂⠦⣽⡛⣿⡗⠛⢾⢛⡌⠀⢀⡄⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢹⢀⡤⢤⡤⣤⣀⠀⠛⠛⢧⣤⡋⢫⣿⠇⠀⢿⡖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡠⣴⣾⣷⣶⢧⡤⠀⠳⠀⢰⣄⣸⡕⣆⡀⠀⠀⢀⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠒⢳⠞⣛⣀⣿⣀⢀⡧⣴⡯⣍⡄⣅⣈⡋⢙⣿⣛⣿⣿⣌⠑
⠀⠀⠀⠀⠀⠀⢀⡶⠛⠁⠀⠞⢃⣸⣿⡈⡓⠲⡶⠀⠑⠞⣻⢿⡘⣿⠥⣀⠟⠛⣷⣄
⠀⠀⠀⠀⠀⠂⠀⣀⠀⠀⠀⠰⣼⣇⣻⠏⢷⡄⠃⠀⠀⠀⠙⠐⠋⠻⠀⠈⠀⠈⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠟⣃⠘⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣢⠰⣄⣀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢸⠟⣣⠛
⠀⠀⠀⠛⢅⠀⣧⣴⣶⢶⠉⠈⠉⠀⠀⠀⠀⠀⠀⠄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠘⠛⡻⠂⢀⡀
⠀⠀⠀⢀⣼⠛⠟⠉⠹⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠤⢶⠚⢳⣄⠁⠈⠑⠄
⠀⠀⠀⠀⠉⡶⠴⡦⠤⠻⣿⠂⠀⠀⠀⠀⠀⠀⠀⠁⡤⠀⠠⠄⠈⠁⠀⠀⠀⠀⠈⢇⠀⡼⠳⡇⠸⣆
⠀⠀⠀⠀⠘⠣⣄⠧⣄⢀⡟⠀⠀⠀⠀⠀⠐⠢⢤⠤⣤⡀⡤⠁⠈⠁⠀⠀⠀⠀⠸⣾⠴⠛⠋⠀⠀⠙⡀⠐
⠀⠀⠀⠀⠀⠀⠁⠦⠿⢤⢽⡟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠤⠞⠋⠉⠈⠓⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠈⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣑⣴⠀⢀⠌⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⡈⢦⡜⠀⠀⠨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠴⠲⠠⠄⡀⠀⠀⢢⠀⠈⠉⢣⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠁⠠⢤⣀⢱⠀⠀⠉⢄⠀⠀⠱⡀⠉⠉⢃⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠁⡠⠝⢓⡾⢀⠸⠀⠀⠀⠈⠉⠀⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠲⢠⣈⣠⠔⡳⠄⠀⢀⠀⠀⠐⠀⠈⠒⠀⠀⠤⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠐⠈⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠨⠐⠈⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠀⠈⣁⣢⣀⡀⠪⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⡄⡒⠒⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠂⠁⠀⠀⡀⠊⠁⠀⠀⠀⠉⠲⡀⠠⠀⠀⠀⠀⠀⠀⠄⢀⠍⠁⠀⠀⠀⠑⡀⠀⠑⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠓⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠢⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠑⡄
⠀⠀⠀⠀⠀⢀⠀⠐⠁⡈⠀⠀⠐⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠃⠀⠈⣄⠀⠀⠀⠀⢀⡂⠀⠐⢄⠀⣈⡆
⠀⠀⠀⠀⠀⠈⢢⠤⢀⠅⠀⠀⠀⠠⠚⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠃⠄⠄⠂⠀⠙⠐⠀⠤⠊⠐⠄⠠⠄⠁
⠀⠀⠀⠀⠀⠀⠆⠀⢌⡀⢀⡀⠤⠀⠀⠈⡄⠀⠀⠀⠀⠤⠊⠢⣀⠀⠀⠡⡀⠀⡄⠐⣄⠀⢀⠦⡀⠀⠀⠠⠀⠀⢃⠀⠀⠄
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠈⠁⠀⠀⠀⠄⠀⠁⠀⠆⠈⠈⠀⢊⠐⠰⠁⠀⠀⠑⠲⠉⢄⡀⠐⠁⠐⠊
⠀⠀⠀⠀⠠⣀⡀⣀⢼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⠂⠄⣐⠀⠀⠀⠙⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡀⠔⠹⠀⠐⠂⠁⠀⠀⡘⠂⠤⠓⢄⡀⠈⠀⠀⠀⠀⠀⠀⠒⠁⠄⠈⡉⠐⠠⡤⢀⠠⠃⠁
⠀⠀⠀⠀⠀⠀⠐⠀⢀⠲⠁⠀⠀⠀⠀⠅⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠒⠴⠀⠌
⠀⠀⠀⠀⠀⠀⠃⢲⠁⠀⠡⠀⠀⠀⢀⡶⣀⠄⠀⢃⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠰⢦⠀⠂⠀⠀⠀⠀⠀⢀⠦⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⢀⠠⢄⠠⠐⢆⡀⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠐⠉⠈⠲⡄⣠⠤⠄⠒⠒⠒⠒⠒⠒⠒⠢⠤⣀⠀⡀⡠⠔⠒⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢀⣀⡄⠠⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠠⠀⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡴⢫⠞⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⡄⡜⢠⠃⠜⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⣠⠀⠀⠀⢠⢆⠀⠀⠀⠈⢂⠘⣇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡇⡈⠀⡄⠀⢀⡠⣴⠟⠀⠀⢀⣠⡾⠃⠀⣀⠔⠁⠈⢆⠀⠀⠀⠀⠂⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⢁⣞⠴⠪⠵⠯⠤⠔⠒⠪⠵⠭⠔⠒⠋⠀⠀⠀⠀⠈⠣⡀⠀⠀⠈⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠟⠁⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠌⠢⡀⢧⡁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡏⠀⠀⠀⢠⣀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣠⠀⠈⠙⢻⠗⣄
⠀⠀⠀⠀⠀⠠⠂⠀⠸⡀⢰⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠀⢸⠀⡸⠀⢀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠚⡄⠀⠀⠀⠻⣿⣿⠟⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢿⣿⡿⠃⠀⠀⠀⡟⠊⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠠⠀⠠⡹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠂⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠁⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠄⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⢒⡟⠙⠭⣉⣉⠭⡋⢻⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⢠⠊⠀⡞⠀⠀⠀⠀⠀⠁⠀⠀⢣⠈⢢⠀⠀⠖⠙⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠢⠈⠀⠀⠀⣰⠉⠂⡴⠉⠒⠢⠤⠔⠓⠤⠔⠒⠊⢦⠊⢣⢀⠕⠀⠈⠉⠉⠒⠊⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢖⠉⠉⣁⢢⠐⡇⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⡏⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠠⠁⠀⠀⠃⠉⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⢣⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢀⡼⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡄⡱⠀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠉⠀⠀⠉⢒⠶⠤⠤⣤⣀⣀⣀⣀⡠⠤⠤⢤⠚⠉⠉⡀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠚⠁⠀⠀⠀⠀⠉⠒⠤⠔⠀⠀⠀⠀⠈⠁⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢻⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⣇⣴⢾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⢿⣶⣖⠲⠶⠶⣷⣿⣄⠀⢀⡄⠀⠀⠀⠀⠀⠀⢰⣤⣄⣻⡀⡿⣠⡤⢶⡇⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠶⠶⣬⣾⡟⠁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣹⠀⠀⠀⠀⠈⠙⣷⣸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣄⡉⢻⣷⢋⣠⡾⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡤⠿⡗⣛⡥⢾⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⣿⠉⠛⠻⢷⣦⣴⠶⠤⣼⣏⣷⠀⠀⢀⡞⠀⠀⠀⠰⣭⣿⣏⣹⠉⣵⠞⢻⡟⠀⠀⠀⠀⠀⠀⠀⠘⣶⡒⠲⢶⣄⣿⠏⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢰⡇⠈⠙⣿⡄⢰⡶⠶⣬⣿⣷⠀⠀⠈⠙⠻⣦⣿⡇⠀⠀⠀⠀⠻⣦⣀⣙⣿⣥⣴⣟⡥⡖⠀⠀⠀⠀⠀⠀⠀⡈⠻⣶⡶⠾⣿⢚⣯⡴⣾
⠀⠀⠀⢀⣀⣀⣸⡇⠀⠀⢸⣇⢸⡇⠀⠀⠉⠙⢷⣤⣀⣀⠀⢹⣿⠁⠀⠀⢀⠀⠰⣶⠿⢷⣼⣦⠞⠁⣰⠃⠀⢀⠀⠀⠀⠀⠀⠹⣍⠉⠙⠲⣿⠟⠁⣰⡏
⠀⠀⠈⠉⠙⠓⠾⣧⠀⠀⣸⠻⣾⣇⠀⠀⢠⣀⣀⣹⡏⠉⠛⠿⣿⣆⢀⣴⡏⠀⠀⠘⢳⣤⣤⡿⣿⠉⣡⡶⠚⡿⠀⠀⠀⠀⡀⠀⢈⡙⠒⠚⣿⣺⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣆⡀⣿⠀⠈⢿⡄⠀⢸⠀⠉⠙⠻⢶⣄⠀⠈⢻⣏⡿⠀⠀⠀⠀⠈⢿⣍⠉⠛⣷⣏⣤⠾⠁⣀⣤⡆⠀⠙⣯⡉⠙⠳⣼⡟⢁⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠿⣿⡄⠀⢸⡿⠶⢼⡀⠀⠀⡀⠀⠙⣷⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠉⣛⡛⠉⢻⣦⣠⠟⠁⣼⠃⡀⠀⠈⠙⠛⢛⣿⠋⠉⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⡇⠀⠘⣇⠀⠰⡏⠙⠒⠿⣤⣞⠘⣇⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣤⣼⢿⣶⡋⠁⠀⠹⣍⠉⠛⠶⣼⡟⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠷⠶⣿⡀⠀⠙⢦⣀⡇⠀⠀⠀⠀⠙⢷⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠈⠛⠒⣺⣿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠷⠒⠲⠾⢷⣀⡀⣀⣨⣇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣭⣭⣤⡘⢆⠀⠀⠀⢀⡾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠉⠁⠀⠈⠉⠻⢿⣿⣿⣿⢿⣄⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⡔⠄⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠘⠠⠡⠐⠰⠂⠀⠐⠠⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠿⡏⢠⡇⠐⠇⠀⢀⠀⣂⡤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⠀⠘⣘⢈⣷⣤⣤⣶⡶⠿⢿⣟⠀⡠
⠀⠀⠀⠀⠀⢀⣴⣶⣾⢿⡈⡙⢷⡤⡀⠀⢠⣀⣤⠶⢛⡈⡁⠉⠀⢈⡀⢟⣿⣟⠱⠦⣄⠠⠤⠘
⠀⠀⠀⠀⠀⡼⣿⣿⣴⠞⢻⣦⣼⡏⠹⡀⠀⠉⢁⠒⢛⣷⣿⣿⣄⣬⣽⣉⣹⠩⠶⣴⣢⡒⡴⠂
⠀⠀⠀⠀⠈⣼⣿⡟⢁⣠⣞⠙⢀⡷⢠⡅⠀⠀⠈⠐⠤⢁⢼⣿⣿⣿⣿⡿⢫⡿⣧⡽⣇⠉⠄
⠀⠀⠀⠀⠀⣇⣿⣷⣿⠋⠉⣽⡾⣧⣐⠁⠀⠀⠀⠀⠀⠰⣿⣿⣿⡿⢻⣿⡟⢀⣟⣁⣸⡏⢋
⠀⠀⠀⠀⠀⠈⡿⣧⣠⡷⠞⣍⣰⠃⠉⠀⠀⠀⠀⠀⠀⠰⣿⠛⢏⣿⣾⢿⣦⣿⠋⢸⠏⣹⠋
⠀⠀⠀⠀⠀⠀⠈⠹⣯⣵⣺⡥⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣗⣏⡽⣿⣾⢷⣅⡾⣋⣗⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠉⠉⠀⠈⠉⠷⠿⠯⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣙⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣎⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⣀⣤⡴⠶⢦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⣿⣿⣿⣶⢿⢿⡿⡟⣿⣿⣿⣿⡿⢷⣦⣬⣝⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣨⣾⣿⡟⡁⠘⠸⡇⡇⡇⢹⣿⣝⢮⡀⠈⠛⢶⣟⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣾⣿⣿⣿⠃⢠⠀⡀⢠⡇⠁⡇⢸⣿⢻⢧⠹⡄⠀⠈⢿⣆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠕⡽⡿⣹⡏⠀⠈⠀⠁⢸⠃⢠⠁⡈⣿⣇⢯⢧⢱⡀⠀⠘⣿⣏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠎⠜⡼⡱⣻⡇⢰⠐⡀⡄⡼⠀⠘⠀⣹⣿⣿⡯⢮⠣⡑⢄⡀⠘⢦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⡞⡕⡵⣻⣿⣄⢇⢁⠁⡇⠀⠁⣼⡿⣿⡙⡿⣶⣭⣂⣁⣉⣑⣈⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡞⡜⠈⣠⢞⠏⢹⠻⣄⠀⠀⢀⣾⡇⣇⠃⣧⢻⢸⢻⣿⡟⠻⡟⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣸⠐⣠⣶⡟⠛⠀⠈⠆⣿⢳⣄⣾⡼⡇⠀⠀⢸⡼⡘⡄⣿⠇⠀⣿⣿⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣴⠚⢹⢹⣇⡄⠀⠀⡀⣿⢸⢹⡻⢱⣧⠠⠀⠈⢧⠁⠁⢸⡄⠃⢹⢧⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⡏⡞⣿⡀⠀⡆⠀⡟⡌⠆⣱⠿⡿⣦⡑⠄⠘⢆⠀⢸⣇⠀⢸⠘⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⢱⢱⣿⣷⣄⠀⠀⡇⠀⣴⠿⠀⡇⢻⢹⡷⣦⣬⣃⠘⣿⣦⣀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡌⡼⠇⢹⢹⣿⣄⠃⣼⡿⠀⠄⠀⢸⢸⢣⢇⢿⡿⣿⡇⡿⡟⢿⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⣰⢃⠃⢸⡈⠘⣼⣷⡁⢿⡄⠀⠀⢸⢸⢸⢈⡾⢹⡿⣰⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠘⡖⠀⠊⣆⠀⣿⣿⡆⠈⠹⣦⡀⠘⠈⣠⣾⡇⠸⢱⠃⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⢄⢰⣿⡞⣷⡀⠄⠻⠹⡷⡔⢩⣿⣿⡄⡌⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠳⠅⢹⢷⢹⡟⠲⣄⣸⢿⠀⣸⢁⣿⢻⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⠁⢀⢸⠃⠀⣿⠻⢼⠸⠁⣰⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣽⢣⣀⠟⢠⢯⠴⡮⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢰⠋⠀⠰⣶⡞⢠⡠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⣤⣾⣵⣤⠟⠀⠀⠀⠝⣷⣼⡇⠀⠀⢳⠂
⠀⠀⠀⠀⣀⡠⠚⢷⡀⠀⠀⠀⠉⠛⢿⢶⡀⠀⠀⠈⠀⣽⣇⠀⠀⠘⢧⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⢤⣾⣿⠀⠀⠀⠀⠲⢦⣆⠀⠀⠀⠉⠓⠀⠀⠀⠀⣸⢻⢣⣀⠀⠀⠀⠀⣤⠀⠛⠂⢀⣾⠃
⠀⠀⠀⠀⠙⢧⣀⡀⠀⠀⠀⠙⠛⠶⡄⠀⠀⠀⠀⢰⠆⠀⠈⡽⢿⣳⠄⠀⠀⠈⠛⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠙⠳⠆⠀⠀⠁⠀⠀⠀⠀⠠⣤⡴⠋⠀⠀⢠⠀⠼⢃⣽⡇⠀⠑⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⣸⠀⠀⠈⠁⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⣴⠀⠀⢠⠂⠀⠀⠀⠀⢠⣿⡇⠀⠀⣤⠀⠀⢀⡆⠀⠈⠋⢁
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠤⡺⠋⠠⠵⠋⠀⠀⠀⠀⠀⠼⠿⣇⣀⠀⣿⡆⠀⢸⣿⠀⢀⡀⠈⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢇⣤⣼⣿⠀⢸⣧⢠⣿
⠀⠀⠀⠀⣀⣀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣄⣿⡿⣺⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⡀⣀⣀⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠚⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠭⣌⣁⣀⠀⠀⠀⠉⢢⣀⡄⠀⠀⠀⢀⡴⠃⠀⢁⡀⠙⠒⠦⡄⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⡎⠀⠀⠀⠀⠉⠃⠀⠀⠈⣦⠀⣠⠝⠗⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⣍⠀⠀⠀⣄⣀⠀⠀⠀⠀⠀⣹⠀⠀⢀⡇⡒⠀⠀⠐⡶⠀⠀⠀⠀⢻⡎⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠈⣡⢶⠀⠀⠀⠀⡇⠀⢀⠇⠀⠉⠀⠀⠘⠯⡄⠀⠀⠀⡼⡇⡀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠐⠒⠉⠀⠀⢀⡴⠁⠀⠸⣆⠀⠀⠀⠀⠀⠴⠋⠀⢀⡜⢱⠁⠁⠀⠀⢰⡶⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⢴⣆⣀⡀⣀⣤⠀⠂⠁⠀⠀⠀⠀⠘⠢⣤⠤⠀⠀⢘⡞⠓⠉⠀⠹⡆⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠉⠀⠉⠉⠈⠷⡀⠀⠀⠀⠀⠀⡴⠃⠈⠁⠉⠉⠉⠈⢣⡀⠀⠀⠙⢤⣤⡆⣀⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢠⣰⡇⠀⠀⢸⠀⠀⠀⠀⣠⠇⠀⠀⠀⢀⡇⠀⠀⠀⢳⡀⠀⢀⠊⠉⠉⠀⢀⡬⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣣⣰⡄⠀⠀⠀⠙⣇⠀⠀⠈⡇⠀⠀⢀⣇⠀⠀⠀⠀⠈⠉⢳⣀⠀⠀⣧⠀⡜⠀⠀⠀⡆⠚⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠙⠁⠈⠓⠦⠄⢀⡟⠂⠀⠀⠇⠀⠀⡞⠈⠑⢲⡖⠂⢒⠶⡟⠁⠀⢀⡟⢸⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⣤⠴⠶⠶⠀⠰⡏⠀⠀⠀⠀⠀⠀⢀⢴⡁⠀⠀⢀⠔⠀⠀⣸⠁⠀⠀⠀⠀⠀⢸⠀⠹⣤⠴⠊⠀⢸⣄⠀⠀⠀⣴⠋⠀⢀
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢻⠀⢙⠶⠐⠁⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠈⢓⡎⠀⠀⠀⠀⣇⡈⠉⠉⠹⡌⢱⠞⠁
⠀⠀⠀⠀⢸⠳⢤⡘⣦⣤⣀⣀⡽⠀⠀⠀⠀⠀⠀⠈⢱⠛⠂⠀⢀⡴⠊⠀⠘⢧⠀⠀⠀⠀⠀⠀⢸⠇⠀⣠⠖⠈⠀⠀⠀⠀⠀⠉⡿
⠀⠀⠀⠀⡿⠀⠀⠈⡽⠀⢉⡟⠀⠀⠀⠀⠀⠀⣀⡤⠎⠀⠀⢰⠏⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⡰⠃⠀⢸⣅⡀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⢰⠧⣄⣀⡼⠁⠀⡾⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⢸⠷⢤⣀⣰⠖⡿⠀⠀⠀⠀⣸⠀⠀⠀⢸⠀⠉⠉⡟⡏⠉⠉⠉⣷⣀
⣄⠒⠂⠁⠀⠀⢸⠀⠀⠀⣇⠀⠀⠀⣀⣾⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡿⠀⣇⠀⠀⠀⠀⣏⠀⢀⣠⣃⠀⢀⠞⠀⠙⢤⣠⠔⠁⠘⣆
⠈⠙⠒⠒⠒⠖⠋⠀⠀⠀⠈⠻⣆⠋⠁⠀⢻⡄⠀⠀⠀⠀⠀⣸⣇⣀⣰⠃⠀⠸⢤⣀⣤⠴⢻⣏⠁⠀⠉⢉⡏⠀⠀⠀⠀⠈⠙⣦⠞⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⡇⠀⠀⣔⡒⠚⠁⠀⠀⣸⠀⠀⠀⠈⢳⡀⠀⠀⡇⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠊⢿⡀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠘⣧⣀⡠⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢧⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣀⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣠⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣴⣿⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠹⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⣋⣭⣍⠀⠀⠈⢿⣆⠀⠻⣶⡟⣄⣴⣾⣄⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣠⣀⣤⣴⣷⣿⣿⣿⣿⣿⣿⡌⠑⠦⠀⠀⠀⠀⠀⠘⠿⣮⡁⠈⠉⠙⢦⣤⣴⣿⠿⠷⣶⢤⡀⣀⡼⠛⠛⢷⠄
⠀⠀⠀⠙⠋⠳⠈⠻⠿⢉⣤⣿⣿⣷⠀⠀⠀⠀⢀⣠⣤⣴⡆⣄⡀⠁⠐⠂⢠⣌⣀⡁⠀⠀⣠⡶⣿⣿⠙⠄⠀⠀⠀⠙⠓⠤⠴⠶⠶⣤⡀
⠀⠀⠀⠀⠀⢴⣶⠿⣆⠈⠻⢿⡏⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡊⠛⠷⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⢀⣾⠀⠀⠘⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠙⠷⣶⣶⣿⣿⣿⣿⢛⠽⢿⣷⣾⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠋⠀⠀⠀⠀⣾⣿⡀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⠈⠁⠎⠀⠼⢻⣿⠿⢻⡃⠀⠀⠀⠀⠠⣴⣿⣿⡙⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠰⡚⠀⠈
⠀⠀⠀⣶⣤⣄⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠶⠀⠀⡴⠃⠀⣤⣀⣠⣄⠀⠀⠀⠀⠀⠙⢫⣷⣤⣄⠀⠀⣿⣿⣿⣿⡞⠉⠈⠁
⠀⠀⠀⢿⡿⠿⠷⣿⣶⣦⣀⣄⠀⠀⠀⣼⣧⠀⠀⠀⠈⠀⠠⢖⣼⣿⣿⢯⡿⠷⡀⠀⠀⠀⠀⠀⠉⠙⠀⠀⣴⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢄⠀⣿⣿⣿⣿⣿⣷⣾⣦⣿⣿⣆⣾⣦⠀⠀⠀⠀⡈⠙⠻⠧⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⠆⣀
⠀⠀⠀⢤⣤⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡴⣄⠀⣾⣿⣴⣆⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣶
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠙⠿⢿⣿⡿⠿⣿⣿⣿⡿⢿⣿⣿⣦⣿⣿⣿⣿⣴⣦⠀⢠⣄⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣯⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠁⠈⠙⠛⠛⠛⠶⠿⠿⠅⠀⠀⠰⢷⣴⠿⠟⠋⠙⠛⠻⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢋⣠⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣠⡶⢿⡞⣫⡽⢦⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣥⣷⣾⡾⣿⡿⠉⣵⡿⣿⣯⣿⣿⣿⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣻⠟⠉⠜⣻⣹⠓⠋⠁⠀⠀⠀⠈⠃⢻⢿⣿⣝⢿⡻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠏⠀⠀⠚⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⡷⠝⠅⢻⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⡿⠀⠀⠀⠐⠹⣼⡆⠀⠀⠀⠀⠀⠀⣼⡟⠃⠀⠀⠀⠘⡏⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠗⢻⡇⠆⠀⠀⠀⢈⣽⢿⣄⠀⡀⠀⢀⡼⢳⣻⣄⠀⠀⠀⠀⣿⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣘⣡⣤⣶⡞⣻⢿⡿⢙⢧⡹⢈⣾⡿⡿⠱⣽⣷⣶⣶⣝⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⠋⠐⠀⠛⠃⣿⠙⠘⠈⠛⣿⡟⣿⡀⠀⠁⠁⠙⣿⢷⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠈⠀⢸⣇⠀⠀⢀⡄⡟⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢀⣾⣿⡟⣦⡀⠀⠀⣴⢿⠿⣷⣄⠞⣼⣷⢢⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⣾⡟⠈⠀⠀⠹⣦⣾⠁⠈⠀⢁⡿⣳⡿⢿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠛⢧⡀⠀⠀⢀⣿⠀⠀⠀⠄⣼⠃⠉⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡂⣿⣄⣢⣿⢿⢷⣄⣣⡞⣿⣧⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡻⠿⡇⠀⢀⡿⠟⣠⠟⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣹⣆⣼⣇⣼⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠋⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣇⣸⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⡺⠟⣩⣴⠛⠉⣹⣏⠉⠻⣦⣍⠻⢗⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⠁⠀⠀⡴⠃⠘⣦⠀⠈⠈⠳⣄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠶⠶⠒⡾⠃⠀⣀⡴⠚⡇⠀⠀⠸⠓⣦⣀⠀⠈⢷⠲⢶⠶⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡁⠀⠠⠾⣷⠖⠋⠁⠀⢀⣷⠀⠀⠀⠁⠘⠏⢻⠲⡾⠷⠄⠑⢌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢜⣿⠀⠀⠀⠀⠈⡼⣆⠀⠀⠀⠀⠀⠀⠃⣿⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢊⣿⠀⠀⠀⠀⠈⢠⡞⢦⡀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⣀⡤⠖⡻⣹⡄⠀⠀⣀⡴⠋⠀⠀⡝⣦⣄⠀⠀⢠⡏⠙⠲⣤⣀⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣾⠋⠁⠀⠀⢁⠏⣷⣤⡞⠁⠀⠀⠀⠀⠀⠃⡞⢳⣤⣾⠀⠀⠀⠱⠸⡙⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⣸⠃⠁⢻⡀⠀⠀⠀⠀⠀⠘⢁⡟⠈⠘⣇⠀⠀⠀⠀⠃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡴⠃⠀⠀⠏⣻⣄⠀⠀⠀⠀⣠⠏⠀⠀⠀⢻⣦⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣠⠞⡇⠀⠀⠀⢠⢃⡞⠳⣄⣠⠞⢳⡀⠀⠀⠀⠸⢸⠷⣄⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣼⠁⠀⢹⡄⠀⠀⢀⡞⠁⠀⠈⠁⠀⡟⣳⡀⠀⠀⢠⠏⠀⡟⣧⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢹⡀⠀⠈⡿⣆⣠⢿⡀⠀⠀⠀⠀⠀⢠⢃⡿⣄⣰⠋⠀⠀⠃⡏⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⣰⠃⠉⠀⣷⡄⠀⠀⠀⠀⣠⠞⠀⠉⡽⣆⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣰⣏⠀⠀⠀⠸⢹⠷⣄⣠⠾⡇⠀⠀⠀⡰⣹⣦⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣆⠀⠀⢠⠟⠀⠘⠃⠘⣿⡄⠀⠀⣰⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡟⢦⠀⠀⠀⠈⡴⢻⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡴⠋⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠁⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠲⡶⣤⡀⢸⣏⠀⠠⡀⡘⠀⠀⢻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠅⠀⠀⠉⢻⡎⣷⠀⠀⢹⠃⠀⢀⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⠈⡆⠀⠳⠾⠃⢠⠿⠀⣠⢾⣋⣠⣤⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡁⠀⠈⠙⣄⡀⠀⢀⡿⠀⠀⠙⠋⠉⠀⠉⠀⠈⠉⣭⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⢨⣓⣦⣸⣧⣤⡤⢤⡠⢤⠀⠂⠀⢀⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣀⣄⣁⣸⠋⣷⠈⢆⠀⠂⠀⠁⠀⣀⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡿⣯⣹⣄⣀⡀⠈⢩⣩⣥⣌⣻⣧⣄⣀⣠⣤⣼⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣡⣀⣬⡇⠈⠻⡝⣶⠋⠈⠁⣈⣿⠁⠉⠉⢉⣡⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠟⡩⢿⠀⠚⢉⡟⠲⣿⠻⡄⠀⠀⣈⣹⣆⣠⣾⠋⠁⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⢠⣿⢿⠁⠚⢶⡖⠋⡿⣤⢼⣧⣇⣠⣾⠷⣯⣭⣥⣬⣷⠀⠀⠀⣠⠖⠛⠛⡷⢤⡀
⠀⠀⠀⠀⠀⢸⣿⣘⣓⡞⠉⠿⡤⢴⣿⡼⠋⢠⡿⠁⣰⢿⠀⢸⣷⢮⣳⡀⠐⣇⠀⠀⣰⠇⠀⢻⡄
⠀⠀⠀⠀⠀⠈⠿⢽⣍⣷⡶⠾⠷⠛⠋⠀⠀⢸⣠⠾⣧⡀⢠⡎⠀⠃⣿⣷⠀⠈⠛⠛⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⣾⢻⡉⠙⠈⣳⠶⢻⣿⠀⠀⡞⠹⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣸⠿⣶⠴⠟⡶⣿⠁⠀⠀⠙⠲⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⡗⢾⣷⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠈⠂⠠⢄⡀⠀⠀⠀⠀⠀⠀⢀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠣⣠⡟⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠉⠛⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⡇⠀⠎⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢻⡌⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣼⠁⠀⡿⠁⠀⠀⠀⠀⣀⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠉⠉⠛⠘⢃⡠⢀⠀⠀⡞⠁⠀⠀⠈⠀⠉⠯⢐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢗⣾⣇⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠙⢄⠄⠀⠀⠀⠀⠁⠀⠀⠀⢱⠃
⠀⠀⠀⠀⠀⡠⠀⠀⢀⢤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧⠀⠈⠢⡀⠠⠀⠀⠀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠣⢀⣀⣸⠀⡄⠀⠀⠀⡀⠐⢉⠊⠏⠛⠉⠁⠀⠀⡞⠓⠢⢄⣐⡂⠀⡿⠟⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢉⢆⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣰⣤⣤⣶⣶⠛⠉⠀⠄⣀⢀⠀⠀⠉⠒⠛
⠀⠀⠀⡀⠐⠀⠀⠀⠀⠈⢦⡀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⠔⠁⠈⠂⠄⣀⡀
⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠉⠉⠒⠀⠁⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠈⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿