Boi Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣄⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⣴⠀⠀⣆⢸⡅⠀⣴⢾⠿⠛⠿⠻⡿⣿⢿⣿⣿⣿⠟⣛⣿⣯⣭⣬⠽⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣿⠀⣀⢏⠸⣇⣾⡷⣁⣤⣀⣄⣾⣱⠃⠈⠉⢉⣴⣿⣿⣿⣿⣿⣿⡟⠁⠉⠙⢿⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⢸⡆⢻⡀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢠⣿⣿⣿⣿⣿⢟⣛⣭⣤⣄⡀⠐⢄⠉⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢷⡈⢷⡹⣿⣿⣿⣿⣿⡿⠿⠛⠟⠛⠿⠿⣿⣿⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⠿⠶⡈⣿⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢀⡻⣮⣿⣮⣽⣿⡯⠿⠶⢾⣿⣿⣿⣶⣶⠈⠀⠉⠁⠀⠀⠈⠙⠛⠉⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⡁
⠙⠳⠨⣭⠛⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠿⠁⢣
⠀⠀⠀⠀⠸⢹⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠁⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠸⣿⣏⠀⠉⠉⠉⠉⠻⣯⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⣠⣾⠿⢿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣷⠀⢠⣤⠄⠀⠛⣫⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⢟⠁⠀⠀⣬⡿⢟⣁⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⢠⠆⣀⡀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⠏⠀⠀⠀⣤⣾⣿⣧⡄⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣞⠙⢳⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢀⣠⣼⣿⣿⣿⡏⠀⠀⠀⠀⠃⠀⠀⢀⣴⣮⠭⣄⠀⠀⡀⠀⢀⠀⢡⣿⣿⣇⡠⠁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣨⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡄⢀⡤⠀⠟⠀⠠⠀⠉⢀⡈⠁⠤⠊⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣤⣀⡀⠉⠋⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠉⠀⠀⠀⠀⢀⡤⠀⠀⠈⣦⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⠇⠈⠛⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⡆⢠⠤⠀⣴⠟⠐⢶⣤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡀⢸⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⣀⣀⣠⡄⠛⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⣌⡙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡠⠞⢁⣀⠤⠴⠒⠋⠉⠁⣨⡏⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣅⣀⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠸⠿⢧⣦⣤⣤⣤⣴⣶⣿⠟⢠⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⢉⡉⠉⠉⠀⣠⠆⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠐⠶⢤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣭⠤⠴⠶⠶⠖⠒⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⡇⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠻⢷⣦⣤⣤⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⣿⣿⠿⠋⠀⠀⠀⣀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⣀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⢀⣀⣤⣶⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠘⠿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣀⣤⣤⣤⣤⣶⣶⣿⣿⡿⠿⠛⠉⠁⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⣀⣠⣴⣶⣿⡿⠟⠃⠀⠀⠀⠀⠠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣦⣤⣤⣤⣤⣤⣶⣶⣶⣾⣿⡿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣀⣀⣤⠴⠖⠒⠛⠋⠉⠉⠉⠉⠉⠛⠓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣜⣦⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠺⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡍⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⢛⣻⣆⠀⠀⠀⠀⠀⠀⢿⠛⣷⠲⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡇⢻⡀⠀⠀⠀⠀⢀⡿⢀⡟⠛⠉⠁⠈⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠻⣆⡙⠶⢤⡤⠶⠋⣠⡞⠁⠀⢀⣀⣀⣤⠤⠖⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠷⠶⠶⠶⠦⠤⣤⣬⣿⡷⠶⠶⠶⠿⠷⠒⠚⠛⠉⠉⠁⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠲⠶⠶⠤⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠶⠶⠚⠛⠋⠉⠡⡆⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢀⣀⣤⠴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣄⣀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⠤⠶⠚⣿⠉⠁⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠈⠉⠉⣿⠉⠉⠙⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣿⠶⠚⠋⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠙⠛⠛⣿⠒⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠛⠛⠋⠉⠉⠀⠀⢸⠄⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠄⠒⠂⠤⢤⡴⠋⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠈⠹⢉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣅⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⢰⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠲⠖⢮⡀⠈⠙⠲⢄⠀⠀⠀⢀⠔⠁⠀⠀⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠞⠀⠀⡻⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⡄⠒⠳⡤⣀⣀⡀⠀⠀⠀⣀⡀⠀⣀⡠⢶⠀⠀⠱⠴⠚⢣⠀⠀⠀⢑⡤⠀⠀⠀⠀⠀⠀⠀⣀⣤⢀⣀⠽⠦⡀⠀⠀⠀⠀⠀⡾
⠀⠀⣠⠞⠁⠀⢸⠤⠄⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⡎⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠉⠛⠻⠛⠓⠒⠒⠋⠁⢉⡉⠀⠀⠀⡏⠳⣌⠀⠀⢀⠇
⠀⠀⢹⠋⠀⠀⢸⠠⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠇⠈⠳⣄⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⡇⠀⠈⢣⣀⠞⢄
⠀⠀⠓⠤⠲⢖⢽⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⢄⣀⣤⣆⣠⠎⠀⠀⠀⠈⢧⠀⢰⡀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠣⠤⠶⢹⡏⠀⠀⠱⡄
⠀⠀⠀⠀⠘⠋⠀⠳⣄⠀⠑⠒⠚⠀⠀⠀⠀⠀⠀⠀⣀⡥⠊⡸⠀⠀⠀⠀⢀⣬⠆⠀⠑⢄⡀⠙⠋⠁⠀⠀⠀⠀⠀⠀⢀⡠⡖⠁⠘⣄⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣢⣄⣀⣀⣀⣀⣠⣤⠔⠚⠑⠢⢄⡧⢿⠀⢀⣦⡸⠀⠀⠀⠀⣀⠬⢵⣦⣤⠤⢤⣤⠔⣲⠮⢥⡒⠓⢄⠀⠙⠦⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⣿⠉⠈⢁⡤⣇⡙⢖⠒⠒⠮⣷⣠⡷⠋⠀⠀⠀⠀⠀⢠⠃⠀⠐⡇⠉⠉⠋⠀⠀⣏⠀⠀⠙⢍⠙⣗⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⣰⠏⠀⢠⠮⠀⠈⢱⠀⠑⢦⡀⠈⠻⡁⠀⠀⠀⠀⠀⠀⢸⡀⠀⣴⠃⠀⠀⠀⠀⠀⠻⣄⣀⣀⢼⠀⡏⢡⠜⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⠋⠁⠀⠀⢸⠀⠀⠀⠀⠣⡀⠀⠱⡀⠠⠹⡀⠀⠀⠀⠀⠀⠀⢇⢻⡝⠀⠀⠀⠀⠀⠀⢄⠙⠻⡄⠀⡆⠹⢀⣰⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡟⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⢀⣴⠗⡐⠄⢡⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⢁⠎⠀⠀⠇⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠉⡹⠀⠀⡀⢸⠀⠀⠀⠀⠀⠀⠀⢼⣀⣀⠀⠀⠀⠀⢀⡠⠊⠀⠀⡜⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢥⣀⣁⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠍⠉⠉⠙⢰⣢⢀⡴⠤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢲⡶⠤⠤⠤⣤⠤⢤⠲⠿⣜⡢⠜⢁⠦⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢆⣠⠤⢄⡀⠀⠈⡷⢫⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢀⡟⠀⣸⠀⢐⠂⠈⠑⢭⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⢹⡤⠊⠀⡴⠡⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡉⠢⣄⡀⢀⡼⢧⡔⢁⡴⠻⡀⠀⠀⠀⢳⠤⠔⠃⠐⠤⣤⣤⣄⡤⡞⢄⠀⠀⠀⡎⠱⣖⡋⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢀⠖⠉⢇⠀⠈⣳⠆⠀⠹⠉⠀⠙⣧⣀⡠⠃⠀⠀⠀⢠⠂⠀⠀⠀⡜⠁⠀⠀⠀⠉⠢⠉⠒⡺⠀⠉⢙⠀⠰⠀⠀⠀⠀⡾⠀⢀⡀
⠀⠀⠀⠠⠀⡏⠀⠀⠀⠉⠈⠁⠀⠀⢸⠀⠀⠀⢳⡀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢫⠀⠀⠀⠀⢀⡞⠁⠈⡀⠙⠢
⠀⠀⡂⠀⠀⠹⣄⠀⢀⠀⡀⠀⠀⣠⠇⠀⠀⠀⠀⠙⠶⠶⠖⠚⠁⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠑⠦⠤⠴⠋⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠠⢀⠀⠈⠙⠈⠁⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣾⠛⠋⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠈⠙⢿⣿⢿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠰⠆⠀⢿⠇⠀⠀⢀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⡀⢸⠁⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠟⠀⠀⠀⠂⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡿⠟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡏⣧⠀⠀⡂⠀⠀⠒⠀⢀⣤⣠⡀⠀⠀⠀⠀⠁⠀⠈⣿⣸⠇⠀⠀⠦⠀⠘⣦⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡿⠀⠀⠁⠀⠀⠀⠀⠀⠛⠛⣿⡇⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠄⠀⠀⢠⡾⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠒⠋⠉⣿⣧⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⢶⣆⠀⠸⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠗⠀⠀⠀⢤⡄⠀⠠⠄⠀⠀⠉⢻⡀⠀⠀⠰⠦⠄⠀⠤⠀⠀⠾⣿⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⠸⣇⠀⢧⠀⣶⡀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣶⣦⣿⣦⣼⣠⣿⣷⡀⠸⣿⣶⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣸⣟⣿⣿⣦⣄⣿⣧⡿⣿⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣰⡿⣿⣿⣿⣿⣿⣿⣿⣷⠃⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣴⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡟⣷⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⢰⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠘⠋⠿⠿⣿⣿⣿⣿⣿⣿⡿⣿⠁⠘⠋⠁⠀⠺⢃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣾⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠁⠀⠐⣿⠀⠀⠀⠀⠀⠀⠸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢸⡄⠀⠀⠀⠀⠀⢻⡇⠉⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⡿⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⡶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣯⡭⠈⠛⠁⠈⠙⢷⣦⢦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⠉⠀⠀⠀⢀⣠⡶⢦⡴⠀⢙⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢿⣤⣤⣤⡄⠀⠐⠛⠋⢠⣎⠀⣤⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⢹⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣎⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠒⠂⠀⠀⠀⠀⠒⠓⠀⠀⠸⠏⣼⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣀⠀⠶⠆⠀⠀⠀⠀⠺⠆⠀⠀⠀⠠⣾⣿⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡷⠚⠿⣧⠋⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⠀⢀⠀⣬⠟⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⣇⠛⣁⡀⠀⠀⣠⣬⣄⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠉⠙⠧⡀⠀⠈⠉⠉⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠰⣿⣀⣤⣶⣶⣶⣾⣶⣤⠒⢀⡜⠃⢰⣄⡀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⢰⣶⣾⣿⠉⢿⣿⣿⣿⣿⣿⠷⣩⠀⠀⠀⠈⠻⠶⢀⡀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⡟⠀⠀⠈⠙⣿⡿⣷⣼⣿⣿⣴⡿⢿⡿⠁⠀⢀⢀⣤⣤⠶⠟⠛⣦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⣴⡶⢿⣯⠀⠈⠙⢦⡘⠁⠸⣿⣴⣷⡿⠛⠋⢁⣰⠎⢀⣴⡶⠛⠋⠀⠀⠀⠀⠀⠸⣦⣼⡇
⠀⠀⠀⠀⢀⡾⠟⠁⠀⠀⠉⠀⣠⣤⠼⠃⠀⣀⠈⢿⣿⡁⠀⣠⡞⣡⡾⣿⠏⠀⠀⠀⠀⢠⡄⠀⠀⠀⣿⣿⠃
⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⣨⣷⣄⠀⠀⠻⣦⠀⣿⡿⣶⡿⠛⣡⡾⠃⠀⠀⠀⠀⢀⣾⠁⠀⠀⢀⣼⣏
⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣴⠂⠻⠀⣿⠀⠀⠀⠀⠸⣿⠁⣿⣤⡾⠋⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣰⡿⠃⠙⣧
⠀⠀⠀⣿⡅⠀⠀⠀⠀⣾⠁⠀⠀⣀⠉⠀⠀⠀⠀⠀⢹⣇⠈⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡿⠋⠀⠀⠀⠻⣦
⠀⠀⠀⠹⣷⣄⣤⣤⣄⣉⢀⣠⡾⣋⠀⠀⠀⠀⠀⠀⣜⠻⣷⣍⠀⠀⠀⠀⠀⢠⡾⠟⢛⣉⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠋⠀⣿⠀⠀⠀⠀⠀⠀⢿⡄⣈⣿⠷⣤⣤⣶⠾⠋⠁⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⡀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⣶⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠁⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷⣄⠀⠀⣀⠀⠀⠀⣠⣤⣄⠀⠀⢀⣠⡼⠇⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠙⠻⣿⠛⢳⣤⣤⣤⠀⠉⠛⢷⡿⠋⢠⣴⣄⢀⣀⠀⠸⣦⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⠀⠀⠀⠸⠇⠀⠀⠀⢀⣀⣆⢀⡏⣧⠀⣿⣷⡀⠛⠰⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⡇⠀⠻⡄⠀⣄⠀⢰⠀⢠⠀⠀⠙⡇⠘⡇⣿⢠⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡇⠀⣾⠃⢠⡏⠀⠈⠀⠸⠀⢸⠀⡇⢠⡆⢸⣾⡏⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⣿⠇⠘⠇⢰⣇⣀⠸⠄⠀⢀⡇⠘⡇⠈⠛⢁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢀⣿⠀⡄⠠⣀⣿⣿⡆⢸⡄⢸⡇⠀⣷⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⡇⠀⣿⣿⣀⠇⢸⡇⢸⣷⠀⢸⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⢹⣿⠀⣸⡿⠀⣿⣦⢸⠁⣾⠋⡄⢸⠀⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢀⣼⡟⠀⣿⠀⣰⠉⡁⢸⣦⠛⠀⡇⢸⡇⢱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⡇⢰⡿⠀⣿⢻⣿⢺⣿⡆⠀⠀⢸⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣾⡇⣸⠇⢠⡏⢸⣿⢸⣿⠁⢀⠀⢠⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣿⣇⣀⢀⣿⣷⣿⡟⢠⡇⣠⡇⢀⣿⠏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠛⠛⠛⠛⢻⣿⣧⡘⠁⣿⣇⣨⣿⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⣾⠏⠉⠉⢿⣿⠋⠉⠉⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣦⠀⠀⠀⠀⢿⡆⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⣿⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠈⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣴⣿⠁⠀⠀⠀⠀⠀⠰⣿⡄⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠁⢸⡿⠀⠀⠀⠀⠀⠀⠀⣻⡿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⢨⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣼⣿⣤⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣷⣀⣴⣾⣿⡿⠿⠟⠀⠀⠀⠀⠀⣼⡿⠋⠙⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣦⣴⣿⣿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣾⣶⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣟⠒⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠳⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⡀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⡠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠻⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣸⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠦⠤⠤⢤⡀⠀⠀⠀⠀⢸⡦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡉⣇⠀⠀⠀⠀⣸⠇⡿⠒⠛⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠳⣌⠳⠤⠤⠞⣡⡾⠁⢀⣀⣠⡤⠴⠚⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠓⠒⠒⠒⠚⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⣀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣅⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠤⠖⡚⠉⠁⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⢩⡏⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣇⠀⢀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣀⣀⣸⣇⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠖⣿⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⠀⣀⣠⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠤⣤⣤⣼⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠴⠒⢻⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠁⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠉⠙⠛⠛⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠀⢀⠤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠊⢠⠔⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰⢶⠀⢀⣀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠛⠓⣄⠈⠳⠀⠀⠐⠂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠺⠷⠲⠆⠀⠀⠙⠛⠀⠀⠠⠀⢤⡀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⠀⠀⢀⡄⠀⠃⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⡀⠀⠀⠀⠀⠀⠀⠀⢘⣁⡀⢺⡄⠀⠀⠀⣠⣍⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⣀⠀⠀⠜⠛⠃⡼⠁⠀⠀⠀⠛⠛⠀⠿⠀⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠘⠦⠀⠀⠀⠙⢇⠀⠀⠀⠀⠀⠀⠚⠀⠀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁⠤⠤⠤⣀⡠⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠙⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠰⠆⠀⠉⠉⠙⠦⢤⣤⡄⠉⠁⠀⠀⠒⠶⠶⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠖⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⢀⡄⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⢀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⡔⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠘⠳⠤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠉⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡟⠁⠠⠤⠀⣀⣀⠀⠀⠉⠠⡀
⡀⠀⠀⠀⠀⠀⠶⡏⠁⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠰⠆⢠⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡇
⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀
⠀⠉⣄⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⡌⠀⠀⠀⠀⡤⠁
⠀⠀⠈⠓⢤⡄⠀⠀⠀⠀⠀⠱⣤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⡰⠃⠀⠀⠀⣤⠚
⠀⠀⠀⠀⠀⠀⠙⠃⢤⡄⠀⠀⠀⠛⢤⣄⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⡀⠠⠦⠀⢀⣀⣀⣀⣀⣀⠀⠀⠠⠴⠶⠈⠉⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠃⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠰⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠄⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠁⣦⠤⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡤⠄⠈⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢰
⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡆⠀⠀⠀⠀⠀⣶
⠀⠀⠀⢰⡷⠀⠀⠀⢾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠐⠛⣀
⠀⠀⠀⠀⠇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⠛⠀⣦
⠀⠀⠀⠀⡇⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠂⠀⠻⢿⣿⠂
⠀⠀⠀⠀⢄⡀⢀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠃⣀⠉
⠀⠀⠀⠀⠞⠋⠻⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⡤⠉⠀⠀⠀⠘⠷⠶⠤⠤
⠀⠀⠀⠀⠀⠉⠀⠈⠃⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⡆⠀⠁⠀⠀⠀⠙⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠦⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⡤⠀⠐⠒⢶⠉⢀⣤
⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠉⣀
⠀⠀⠸⣟⠀⠀⠀⠀⠀⠀⠀⠀⢘⡤
⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠙⠁⢙
⣀⣀⣀⣀⣁⣤⣤⣤⣤⣤⣴⣄⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠁⠈⠉⠉⠉⠉⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣴⡄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡀⠰⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣟⣿⣿⠿⢿⣿⣿⣿⣿⢿⣿⣷⣼⣿⣦⣄⠐⣦⣤⠤⢀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⣷⡾⠛⣿⡃⠀⠙⠋⣉⠀⢀⣠⣴⣿⣿⣿⣿⡟⠛⢿⣧⣾⠋⠁⢴⣶⣤⣀
⠀⠀⠀⠀⠀⡀⣼⡿⠋⠀⢠⡟⠀⢸⣿⡇⠀⣠⠞⠁⠰⣿⣿⢿⣿⡿⣿⠟⠁⠀⠘⣿⡇⠀⢀⡀⠹⣿⢹⣷⡀
⠀⠀⠀⠀⢰⣇⣿⠁⠀⠀⢸⡇⠀⠛⠉⠀⣼⠋⠀⠀⠀⠈⠁⠈⠁⠀⠁⠀⢠⣶⡷⠛⠀⠰⣦⡁⠘⠆⢸⣿
⠀⠀⠀⢀⣼⣷⡜⢷⣄⠀⠈⢡⣤⣤⡀⢰⠇⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡾⠟⠋⢠⡤⠴⢤⡀⠹⣶⡆⠿⠁
⠀⠀⠀⣿⣿⡏⢈⣦⠙⢦⠠⠘⢿⣿⡇⢸⠀⡇⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⣸⠷⠊⢹⣇
⠀⠀⠀⣿⣿⡷⢿⠇⠀⠀⠀⠀⠀⠛⠃⠈⠀⣧⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠈⠁⢠⡀⢸⡟
⠀⠀⠀⣿⣿⡇⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣆⠈⠋⣸⡇
⠀⠀⠀⢹⣿⡇⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⣿
⠀⠀⠀⠘⠟⣡⣴⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢴⣿
⢠⡖⠠⡄⠀⢉⡟⠀⠀⠀⠐⠻⠶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⠏
⢹⣴⣦⠘⣤⣿⠃⠀⠀⠀⠀⣀⣠⡄⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣶⡶⠿⠀⠀⠀⣀⠀⠀⠀⢀⣤⡀
⢸⠁⢸⣧⠁⠀⠀⠀⠀⢀⠼⣿⢿⣶⣶⣄⠀⢲⡄⠀⠀⠀⣀⠀⠀⢀⣀⣀⣴⠞⣷⠟⣫⡿⢦⡀⠋⣠⣽
⢸⡄⣿⠋⠀⣆⠀⠀⠀⠀⠀⠹⠦⠴⠏⠉⠋⠘⣿⠀⠀⠐⠃⠠⠞⢻⣏⣿⣥⡞⣱⠚⢁⣠⡤⣥⡸⠛⡟
⠈⣧⠘⣦⠀⣿⠀⠀⠀⢀⠄⡄⠀⠐⠀⡴⠀⠀⠙⠀⠀⠀⠀⠀⠀⢈⣽⠟⠙⡇⣯⢰⠋⣡⣴⡈⠷⡄⠇
⠀⢸⣆⠈⢇⢻⡄⠀⠀⠈⠚⠁⠈⠀⠈⠀⠀⣠⠖⠒⠒⢀⡀⠀⠀⣼⠃⠀⠀⢁⣏⣼⣠⣿⣿⡏⠀⣸⡆
⠀⠀⢻⣦⡀⠈⣷⡀⠀⠀⠀⠀⠀⠀⡀⠀⠾⠽⠷⠄⠀⣤⣭⡆⢠⡟⠀⠀⠀⠈⠻⣿⡿⠛⠙⠃⣴⠟
⠀⠀⠀⠉⠻⣧⡘⣧⠀⠀⠀⠀⣞⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⡶⠋⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠈⠁⢸⣧⠀⠀⠀⠀⠉⢀⢀⣠⣤⡶⠶⣶⣦⡀⠀⠈⢷⡀⠀⢸⣏⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⡀⠀⠀⠀⠈⠉⠁⣀⠀⠀⠀⢀⠉⠛⠂⠈⣿⣄⠀⠻⣆⠀⠀⠀⢻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⣿⢿⣦⣀⠀⠀⠀⠀⠰⠿⣿⣿⡟⠀⠀⠀⠀⠈⢿⣦⠀⠙⡆⠀⠀⠀⠈⢿⣽⡶⣄
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠹⣧⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⣼⠻⣧⣄⠀⠀⠀⠀⠀⢈⣿⣿⡘⡆
⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠙⠀⠀⠀⠙⠳⠶⠤⢴⡶⠶⠴⠶⠛⠁⢀⠄⠻⣄⠀⠉⠛⠷⢶⣶⣶⣿⣿⡿⠃⡿
⠀⠾⠿⢻⠟⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣌⠳⣄⠀⠀⠀⠀⢀⡴⢃⡼⢿⣦⡀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⢀
⠀⢤⣄⣀⣀⣀⣀⣤⣴⡾⢿⢿⣬⣙⣿⣷⡌⠂⠀⠀⠀⠊⣠⣞⣡⣾⠿⣿⣦⡀⠀⠀⠘⢿⣿⣷⣄⣀⣀⣴⣊
⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⣈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠟⠛⢉⣠⡤⠉⠛⠛⢶⡤⠤⠈⢻⣿⣿⣿⡛⠛⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣷⣶⣶⣶⣾⠿⠿⠛⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠻⣿⣿⡿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣤⠤⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠾⢸⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠰⡤⠣⡀⢀⠟⢄⠀⡸⢣⡀⠀⡸⢢⣀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢘⡧⠤⡆⠀⠀⢡⠀⠈⠁⠀⠠⠍⠀⠀⠈⠯⠀⠀⠈⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⡸⠤⣠⠆⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠄⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢷⠀⠀⠀⠀⠀⢀⠤⣀⠈⣉⣁⣢⡀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠐⠄⠀⠀⢀⣈⣱⡤⠀⠀⣀⣠⠞⠀⣠⠶⢒⡒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠉⠉⠉⠉⠒⠒⠠⠤⠤⠴⠲⢶⣛⣋⣉⣀⡴⠺⣁⣀⣹⡏⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠰⣟⠛⠋⢦⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣹⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠰⣷⣣⣦⡼⠓⠋⠉⠉⢹⠃⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠉⠉⠓⠒⠂⠤⢴⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⢀⡔⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢹⠀⠀⣀⡤⠞⢹⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⢿⡋⠉⠀⠀⠸⣥⡤⡤⠼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠀⠘⠒⠒⠒⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠔⠒⠒⠒⠒⠦⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠀⠉⠀⢒⡶⠉⠉⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⡀
⠀⠀⠀⠀⠀⣀⢤⠔⠋⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣀⠜⠣⣄⡀⠀⠀⠀⠀⠀⡠⠚⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢲⠀⠀⠈⣳⠀⠠⡤⠜⠁⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⣀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣀⣠⠞⠁⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⢱
⠀⠀⢀⡼⠀⠀⣍⠉⠉⠒⠠⠤⠔⠒⠂⠒⠊⢩⠃⠀⠀⠉⠀⢸⠀⠀⠀⠘⠡⣤⣤⠤⣀⡠⠤⠒⠛⠒⠊⠁⢹⢦⡀⠄⠀⠀⡎
⠀⠀⢫⠆⠀⠀⡏⡁⠀⠀⠀⠀⠀⠤⠀⠀⠀⡜⠀⠀⠀⠀⠀⡌⠓⢄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠘⠂⠀⠀⢸⠀⠙⢦⡀⣴⡁
⠀⠀⠎⡠⢤⢔⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢣⠀⢀⣀⢀⡜⠀⠀⠀⠱⡄⠀⡄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸⣄⣠⠞⡟⠁⠈⢢⡀
⠀⠀⠀⠀⠒⠁⠙⣄⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠈⣉⠜⢹⠀⠀⠀⠄⢀⣱⠀⠘⢆⠀⠑⠖⠉⠀⠀⠀⠀⠀⠀⢀⣤⠊⠘⣄⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠈⠑⢢⣄⣀⣀⣀⣀⣀⣤⠴⠚⠠⢄⣬⢾⠀⢀⣆⡼⠀⠀⠀⠀⡈⢰⣦⣤⠤⣤⣤⠤⣶⠮⡥⠞⠦⠀⠘⠦⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠈⣿⠉⠉⢉⡬⣍⠳⡒⠒⠠⣵⣠⡷⠋⠈⠀⠀⠀⠀⡌⠀⠀⣧⠉⠋⠋⠀⠀⡇⠀⠈⢫⠑⣗⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⣤⠋⠀⢰⠃⠀⠈⠇⠈⠢⡄⠀⠻⡁⠀⠀⠀⠀⠀⠀⡇⠀⣰⠇⠀⠀⠀⠀⡀⢳⣀⡠⠴⡇⢯⠩⠔⣻⠂
⠀⠀⠀⠀⠀⠀⠀⢇⣸⠏⠀⠀⠀⢹⠀⠀⠀⠈⠆⠀⠘⡄⠀⠘⡄⠀⠀⠀⠀⠀⠸⣜⣼⠀⠀⠀⠀⠀⠀⢄⠙⡣⠀⢰⢸⢠⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡿⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⢀⣴⠛⠈⠀⠲⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⡈⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡰⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠘⠣⡤⣤⣄⣀⣀⣤⢘⠀⠀⡰⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠓⠤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣕⢢⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠂⠀⠀⠀⢩⣂⠖⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠙⠉⠙⡖⠋⡏⠉⡑⠢⢅⡠⠉⠀⠀⠠⠐⡀⠀⠀⠀⠀⢑⠖⠋⠒⣄⢀⡴⠉⡳⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣏⠀⠀⠀⣸⠁⡰⠃⣠⡂⠀⠀⠙⢆⠀⢠⠔⠆⠀⠀⠀⠀⣠⡎⠀⠀⠀⡞⢅⣀⠜⠁⠀⠀⠙⢢
⠀⠀⠀⠀⠀⢀⠤⢾⠉⠑⢮⡙⠒⢶⠋⠛⢶⠚⠁⣰⠀⠀⠀⢘⠍⠀⠀⠈⡜⠋⠛⠉⠑⢝⠢⠤⣼⠁⠘⠷⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠏⠀⠈⠲⠔⠊⠀⠀⢠⠀⠀⠸⡕⠋⠁⠀⠀⠀⡜⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠀⢠⠋⠐⠘⠶⠄⠂
⠀⠀⠌⠀⠀⢆⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠙⠦⠤⠤⠴⠊⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠙⠦⣀⣀⠔⠁⠀⠀⠁⠀⠈⠖
⠀⠀⠁⠄⡀⠈⠑⠂⠃⠱⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⡾⠀⠀⡆⣿⠀⣠⣾⡿⠛⠙⠛⣻⢻⠟⣿⣿⡿⢛⣭⣿⣿⣦⣴⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢀⠀⣇⢀⣌⣃⢸⣴⣿⣎⣤⣦⣤⣴⣧⣏⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠏⠀⠀⠙⠻⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠸⡇⢻⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣿⣿⣿⣿⡿⣫⣭⣶⣶⣦⣄⠈⢦⡈⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣌⢷⣝⢿⣿⣿⣿⡿⢛⣋⣙⣉⣙⠛⠻⡿⡿⡿⠟⠟⢿⣿⣿⣿⡿⠟⠛⠳⣌⣿⣿⣿⣿⣿⣿⣿⣿
⣠⡀⠠⣝⣷⣿⣷⢿⣟⠛⢛⣫⣽⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢣
⠀⠉⠐⠒⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠃⠈⡆
⠀⠀⠀⠀⠊⢸⣿⡿⠫⠿⠿⠿⢿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠁⠀⣠⣤⣷
⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⠀⠀⡨⢻⣿⠟⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⣠⡾⠋⠉⢿
⠀⠀⠀⠀⠀⠀⢿⣿⣇⠠⡾⠊⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣎⠇⠀⠀⠘⢛⣭⣶⠂⠀⠀⠀⠀⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣴⡶⣤⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⢀⣼⣿⣿⣿⠃⠀⠀⠀⡞⠀⠀⠀⢀⡤⠆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠻⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢺⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠉⠓⠀⢰⡆⠀⠂⠀⠏⠛⠛⠊⣀⠄⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠛⠋⠛⠿⠻⠿⠿⠂⠀⠀⠀⠀⠀⠀⢿⣿⡿⠃⠐⠁⠈⠀⠀⢁⡀⠀⠐⢀⡈⠁⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢃⢻⠏⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⢠⠈⢁⡀⠀⣸⡇⢄⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⣶⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠈⠉⠀⢀⠻⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡜⣆⡛⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⣀⡤⠴⠖⠛⠉⣻⠋⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣮⣉⡀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠱⣶⣫⣍⣄⣀⣠⣤⣴⣾⠏⡀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠹⣆⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠉⠙⠛⠛⠛⠋⠁⣰⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠶⠟⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣾⣥⠤⠴⠶⠶⠶⠒⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⣿⠀⠀⠀⠀⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⠙⠿⣶⣤⣤⣄⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⠿⠟⠁⠀⠀⢀⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⣀⣀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⢀⣀⣤⣴⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣶⣾⣿⣿⠿⠟⠛⠉⢁⣠⣴⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⢁⣀⣀⣤⣴⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠖⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⣶⣶⣶⣶⣶⣾⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⠶⢂⣠⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣯⣴⣾⣿⣿⣿⣿⣏⣭⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡛⠿⣿⣭⡙⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⣿⣿⣟⢻⣿⣿⣿⢿⣍⠀⠙⠆⠈⢻⣏⠢⣄⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉⣠⣾⡿⡿⣻⡏⠀⠉⠉⠉⠳⣌⠁⠀⢀⠀⢆⢹⣷⣾⣿⣆⠀⠈⠙⢿⣶⣄
⠀⠀⠀⠀⠀⣠⣿⠟⠁⠀⣰⣿⣿⢁⠃⣿⣧⢘⠦⡀⠀⠀⢈⢷⡀⠠⡃⣌⣿⣿⡙⠷⠯⠻⣦⣀⠀⠙⢿⣷⡄
⠀⠀⠀⢀⣾⡟⠁⠀⣠⠾⣿⣿⡏⣾⢸⣿⣻⣿⣆⠙⢶⣄⢠⡈⣷⣵⣿⣿⣿⣿⣷⣶⡄⠀⠀⠙⢷⡄⠀⠙⢿⣦
⠀⠀⢠⣿⠏⠀⢀⡾⢋⡼⠛⣿⣿⣿⣾⡇⢻⡿⣟⢻⣾⣿⣷⣿⣾⣿⣿⠞⢻⡿⣧⠀⠀⠀⠁⠀⠀⠹⣦⠀⠈⢻⣷⡀
⠀⢠⣿⠋⠀⢠⡟⠁⠊⠀⠀⣿⣿⣿⣿⠷⣄⠁⣼⢏⣬⢿⣿⣿⢹⣿⣿⣼⣸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⢻⣷⡀
⢀⣿⠏⠀⢠⣿⡇⠀⠀⠀⠀⢸⢿⣿⣿⣶⠿⣷⠀⠀⠉⠀⡽⠁⠾⢻⣯⣼⣿⠳⠄⠀⡀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⢿⣧
⣼⡿⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⢨⡿⣏⢃⠀⢿⡀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠑⢀⠈⠢⡀⠀⠀⣿⣧⠀⠈⣿
⣿⠇⠀⢸⣿⠿⡇⠀⠀⠀⠀⠀⠈⠀⠃⠀⠳⢀⠁⠀⠀⠀⠀⢀⠞⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⣿⣿⡄⠀⢹
⣿⠀⠀⣿⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⣀⣠⣶⠋⡰⠀⠀⣻⣶⠾⢟⣽⠏⠀⠀⠀⠀⠀⠐⡀⠸⣽⡇⠀⢸
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣶⣾⣿⣿⣁⡴⠶⠚⠉⠁⠒⠉⢉⣼⠀⠀⠀⠀⠀⠀⠀⠱⠀⢸⡇⠀⢸
⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⡻⣿⣿⣿⡏⠁⠀⣀⣀⣠⣤⣶⣾⢟⣿⣄⣀⡀⠀⠀⠀⠀⡀⢂⢸⡇⠀⢸
⣿⡀⠀⢿⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣎⣿⣿⣷⣤⠘⢉⣿⡿⢫⣿⡷⠋⣁⡌⠁⠀⠉⠀⠢⢄⡈⠀⢺⡇⠀⢸
⣿⣇⠀⠸⣧⠀⠀⠑⢄⡀⠀⠀⠀⣀⡴⡞⣹⣿⣿⠸⡟⠇⣥⣀⣨⣭⣶⣿⡏⠀⣾⡏⠀⠀⢠⠇⠀⠐⠚⣿⣄⣿⠀⠀⣾
⠸⣿⡀⠀⢻⡄⠀⠀⠀⠈⠑⢢⣾⣿⡼⣱⡟⣾⣿⣷⣉⣻⣿⣿⡿⠁⣡⠟⠀⢀⣿⠃⠀⣰⠏⣀⣴⠊⣴⡟⣿⡇⠀⢰⣿
⠀⢻⣷⠀⠈⢿⡄⠀⠀⠀⠀⣼⣿⡏⣷⡿⠀⣿⣿⣿⡇⣽⣿⢿⠃⣼⠡⣆⠀⢸⣿⠀⣼⡟⣼⡿⠃⣼⣿⠀⢸⡇⢠⣿⠇
⠀⠀⢻⣧⡀⠀⢻⣤⠀⠀⢀⣿⣿⡇⣿⠃⣤⢿⣿⡿⢠⠡⢣⡏⠰⣿⠶⡻⢠⢸⣿⣿⣿⣴⡿⠁⢶⣿⠇⠀⠈⣷⣾⠏
⠀⠀⠀⠻⣷⡀⠀⠙⣦⡀⢼⣿⣿⣿⡏⣴⣿⠸⡿⠁⡾⢡⣿⡇⠀⠈⠉⠀⢸⡸⣿⣿⣿⣿⠃⢠⣿⡟⠀⣼⢰⣿⠏
⠀⠀⠀⠀⠙⣿⣦⠀⠈⠻⣿⣿⣿⣿⢃⣿⣿⡇⢡⡼⣡⣿⢿⡇⠀⢦⣤⡀⢸⡇⣿⣿⢻⣿⠀⣾⣿⠁⣸⣿⡾⠃
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣸⣿⣿⣿⡟⢸⣿⣤⣴⣴⠁⠉⠁⣻⡇⠀⠀⢹⡿⡌⣷⢻⣿⣼⣿⢸⣿⡏⢰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣿⣿⣿⠃⡇⣿⣿⣿⡿⢀⣴⣿⣿⠃⢀⣤⡾⠃⢸⣿⣾⣿⣿⣿⣿⡟⢀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡟⢠⠀⣿⣿⣿⣷⣿⣿⣿⣿⠠⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢤⣽⣿⣿⡙⢿⣿⣿⣿⠂⠀⢀⣤⡀⠈⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣷⣾⣿⡏⠀⠀⠁⢸⣇⣤⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⣟⣿⣿⠿⢿⣿⣿⣿⠿⠿⣿⣷⣾⣿⣦⣄⢰⣦⡤⢆⣠⠴⠶⡄
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⢻⡾⠟⣹⡇⠀⠈⢉⣁⠀⣀⣤⣾⣿⣿⣿⣿⠛⠛⣿⣶⠟⠁⠰⣶⣶⣤⡀
⠀⠀⠀⠀⢀⣠⣿⠏⠀⠀⣾⠁⢰⣿⠇⠀⣠⠟⠁⠘⢿⡿⢿⡿⢿⠿⠋⠀⠀⢹⡟⠀⢀⣀⠈⠿⠙⣿⡄
⠀⠀⠀⠀⢸⡼⣿⠀⠀⠀⡯⠀⠈⠁⠀⣼⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣼⡿⠋⠁⠀⠳⣦⡀⠳⢸⡿⠁
⠀⠀⠀⣰⣾⡷⣌⠳⡄⠀⠀⣶⣶⡆⢠⡇⣠⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠉⠀⠒⠒⠓⣦⢈⣿⡇⠛⠁
⠀⠀⠀⣿⣿⣇⣼⠇⠙⠆⠀⠙⢿⣿⠸⠇⡇⠀⠀⠀⠀⢀⣼⣿⡟⠁⠀⠀⠀⠀⠀⠴⠛⠁⢸⣷⡀
⠀⠀⠀⣿⣿⠛⠛⠀⣠⠀⠀⠀⠈⠁⠀⠀⢻⡄⠀⠀⢀⣾⣿⠏⠀⠀⠀⠀⠀⠀⢠⡀⠐⣦⢸⡇⠁
⠀⠀⠀⣿⣿⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠀⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢸⡇
⠀⠀⠀⢹⡿⠈⣉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⣾⠃
⠀⡀⢀⠈⠁⠿⡿⠁⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢉⣿
⢼⢡⡈⢣⣀⣾⠀⠀⠀⠀⠀⠈⠙⠛⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠈⠁⢀
⢸⠋⢻⡜⠋⠁⠀⠀⠀⠀⣶⣿⣯⣤⡀⠀⢤⡀⠀⠀⠀⠀⠀⠛⠛⠉⠉⣠⠤⣤⡶⢲⣤⡀⠴⠋⢳
⢸⠀⣾⠗⢠⡄⠀⠀⠀⠉⠸⣧⣉⣽⠛⠷⠀⣿⡀⠀⠠⠆⢀⡴⢾⣻⣿⣫⠞⣡⠞⠉⣈⣓⢠⡾⣿
⠘⣇⢻⡄⠸⡇⠀⠀⠀⢀⢀⠀⠀⠀⢠⠆⠀⠻⠀⠀⠀⠀⠀⠀⢙⣿⠟⢻⢸⡇⡴⠛⣩⡌⢷⡀⠿
⠀⢻⡄⠹⡄⣿⠀⠀⠀⠳⠛⠈⠀⠀⠁⠀⢀⡤⠶⠦⠀⠀⠀⠀⣾⠋⠀⠘⢠⢣⣇⣼⣿⡿⠀⢙⡆
⠀⠈⢿⣄⠀⢸⣇⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⡷⡄⠀⣨⣑⡆⢰⡇⠀⠀⠀⠹⢿⣿⠿⠛⠓⢠⡿⠁
⠀⠀⠀⠙⢿⣆⢻⡆⠀⠀⠀⢰⢚⡷⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⠃⠀⠀⢀⣤⡾⠁⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠉⠁⢿⡄⠀⠀⠀⠉⠀⡀⣀⣤⣶⠶⣴⣤⡀⠀⠘⣧⠀⠀⣿⡋⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠉⠉⢀⡀⠀⠀⢀⠉⠓⠀⠸⣷⡀⠘⢷⡀⠀⠀⠸⣧⣄
⠀⠀⠀⠀⠀⠀⢀⡴⢻⡿⢷⣄⡀⠀⠀⠀⠀⠿⠿⢿⠟⠀⠀⠀⠀⠘⣿⡄⠈⠳⠀⠀⠀⠀⢻⣭⣷⢄
⠀⠀⠀⠀⠀⢠⡞⠀⠀⢻⡄⠙⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⡟⠻⣦⣄⡀⠀⠀⠀⢀⣽⣿⡌⣇
⠀⠀⣀⣀⣴⠏⠀⠀⠀⢈⣃⠀⠀⠈⠛⠒⠶⠶⠶⠶⠶⠚⠉⠀⣰⠀⠳⣄⠀⠉⠛⠷⠶⠾⠿⠿⠿⠃⣿
⠀⠛⠛⡽⠃⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠈⢳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⠀⢀⡼⢁⣀⠀⠀⠀⠀⠻⣿⣧⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⣿⣶⣝⠢⡀⠀⠀⠀⢠⠎⣠⠟⣿⣷⡄⠀⠀⠀⠹⣿⣿⣄⠀⠀⠀⠀⢠
⠀⠑⠶⢦⣤⣴⡶⠿⠛⠛⠛⠿⠶⣽⣿⣧⣀⣠⣄⣠⣤⣾⣵⡾⠛⠹⣿⣷⣄⠀⠀⠘⢿⣿⣷⣤⣶⣾⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣤⣄⣉⣉⣉⣉⣉⣉⣩⣤⣴⡾⠃⠀⠀⠙⠻⠛⠀⠈⢿⣿⣿⣯⣭⣥

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⠟⣦⡴⠞⠋⠉⠉⠉⠉⠉⠙⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡈⠛⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢀⣀⣰⡀⠀⠀⣸⡆⠀⠀⢸⣆⠀⠀⠀⣷⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⠀⠀⠀⢻⠁⠑⠦⡴⠁⠘⢦⡤⠞⠚⠣⣀⡠⠋⠘⠲⠤⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠒⠺⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⡐⠳⢤⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠄⠠⠄⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣇⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠒⠒⠒⠢⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢠⡊⠀⠉⠉⠀⠉⠫⡉⢹⠀⠀⠀⠀⣀⡾⠁⠀⣀⣤⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡼⠷⠶⠤⣤⣀⡉⠉⠉⠁⠐⠒⢒⣛⣁⣤⣤⠶⠞⠋⢀⣠⠞⠉⢠⡖⡦⣬⣽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠛⠷⢤⣤⣤⡤⠶⠋⠙⣦⡴⣤⣾⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠉⠁⠹⡄⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣴⢃⣦⣰⠿⠞⠋⠉⠉⠉⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣠⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡤⣤⣀⣀⡀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠈⠉⠉⠑⠒⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢻⠀⠀⠀⠀⣠⡴⠋⡷⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣦⠴⠖⠛⠁⠀⠀⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⣽⡆⠀⠀⠀⠀⠀⠛⣟⣙⣛⣉⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⣂⣠⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⣿⣿⣿⣿⣯⣥⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⢯⡙⠻⣷⣮⡛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⣿⣿⣿⣿⡉⢿⠿⢿⡓⠦⡀⠈⠀⠈⢷⣝⢢⣀⠈⠛⢿⣦⣄
⠀⠀⠀⠀⠀⣠⣾⠟⠉⢀⣾⣿⢻⢳⣿⠀⡄⠀⠀⠉⢧⡀⠀⢂⠘⣼⣿⣿⣟⢷⣄⠀⠈⠻⣷⣄
⠀⠀⠀⢀⣴⡟⠁⠀⣰⣾⣿⣏⣦⣼⣿⣧⡌⠢⣄⡀⠢⢻⡤⣵⣴⣾⣿⣦⣈⠁⠉⠻⣦⡀⠈⠻⣷⡀
⠀⠀⢠⣾⠏⠀⣠⡞⣱⢿⣿⢻⣧⣿⢻⣿⡿⢦⣽⣿⣽⣮⣿⣿⣯⢿⣿⣟⠉⠀⠀⠀⠈⠻⣄⠀⠙⣿⣄
⠀⢠⣿⠃⠀⣴⠏⠐⠁⠀⣿⣾⣿⣿⣄⠃⢹⡞⣙⣿⣿⡿⣿⣿⣧⢆⡏⠙⠀⠀⠀⠀⠀⠀⠙⣧⡀⠈⢿⣆
⢀⣿⠇⠀⣼⣟⠀⠀⠀⠀⢻⣿⣿⣧⡾⢷⡈⠀⠉⠈⡟⠡⡿⣿⣩⣿⠷⡄⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠘⣿⡄
⣼⡟⠀⢰⣿⣿⠀⠀⠀⠀⠀⢸⢿⡙⡀⢺⡁⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⣿⣇⠀⢸⣷
⣿⠃⠀⣿⡿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⡉⠀⠀⠀⠀⡰⠃⠀⢻⠀⠀⠀⠀⡀⠀⠀⠀⠀⢂⠀⢿⣿⡀⠀⣿
⣿⠀⢰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⣀⣴⡾⠁⠆⢀⣸⠿⢟⣫⣽⠁⠀⠀⠀⠀⠀⠄⠘⢽⡇⠀⢹
⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣲⣶⣿⣿⣿⠴⠞⠋⠉⠀⠈⣁⣠⣾⠀⠀⠀⠀⠀⠀⠘⠀⢸⡇⠀⢸
⣿⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡻⣿⣿⡆⠀⣠⠤⣴⣶⢿⣿⣣⠟⠦⠤⠤⠀⠀⠀⠀⠠⣸⡇⠀⣸
⣿⡄⠀⣿⠀⠈⢀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣻⣿⡏⠆⠀⠼⣟⣡⣿⠋⢠⡶⠃⠀⢠⠀⠈⣑⣦⡀⣿⠁⠀⣿
⢻⣧⠀⠸⣇⠀⠀⠉⠢⢄⣀⣴⣞⡽⣼⢿⣿⣌⠧⢷⣿⣶⡿⠛⣻⠏⠀⣿⠇⠀⣠⠟⠀⡠⢂⣿⣷⡏⠀⢸⡿
⠈⣿⡆⠀⢻⡄⠀⠀⠀⠀⣸⣿⢿⣽⡏⢸⣿⣿⡗⣻⣿⠿⢁⡾⣁⠀⢰⣿⠀⣰⡟⣴⡿⢁⣾⠇⢹⡇⢠⣿⠃
⠀⠘⣿⡄⠀⠻⣄⠀⠀⠀⣿⣿⢸⡟⢀⢿⣿⣿⢁⢻⢫⡇⢸⣧⢽⠂⢸⣿⣶⡟⣼⡟⠡⣾⡟⠀⠘⣧⣾⠏
⠀⠀⠘⢿⣆⠀⠙⢦⡀⢼⣿⣿⣾⢃⣼⡌⣿⠃⡼⢡⣿⡃⠈⠁⠉⢸⡜⣿⣿⣿⡿⠀⣰⣿⠁⣰⢣⣿⠏
⠀⠀⠀⠈⢻⣧⡀⠈⠻⣿⣿⣿⡿⢸⣿⣷⠱⡼⣡⣿⣿⠃⠠⣤⣄⠀⡇⣿⣿⢹⡇⢰⣿⡏⢰⣿⡼⠃
⠀⠀⠀⠀⠀⠙⢿⣦⣼⣿⣿⣿⠇⣿⣷⣶⣶⠀⠉⢁⣿⠀⠀⠘⣿⢇⣿⢹⣿⣼⡇⣾⡿⢀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⢀⢸⣿⣿⣿⢠⣶⣿⡟⠀⠠⠾⠋⠘⣿⣿⣿⣿⣿⣿⣃⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢇⣼⠸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢿⣿⣮⣛⣿⣿⡟⠀⠠⠞⣦⠀⢿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⣛⠙⠿⠏⠀⠙⢿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⡿⠿⠋⠀⠀⠀⠀⢀⣀⠙⢋⡈⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣿⡿⠻⠷⠀⠀⠀⢀⣠⣶⡿⢋⡻⠋⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠉⠻⠿⠿⠛⠓⠀⠀⠈⠁⠀⢺⣇⠀⢾⠟⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠇⠘⠛⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠹⠟⢴⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢀⡀⠠⠶⠆⠀⠀⠀⠀⠸⠿⠂⠀⠀⠀⠀⣶⣿⡆⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠶⠿⣧⡜⠃⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣴⡶⠻⠇⠀⠀⠀⠀⣈⣉⣉⠀⠀⠀⠀⠀⠀⣼⡆⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠒⠺⢷⡀⠀⠀⠻⠿⠿⠃⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠃⠀⣴⡄⠙⠂⣀⣀⣀⣤⣄⡀⢠⡄⠀⢿⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣈⣛⣷⣶⢾⣿⣿⣿⣿⣿⣿⣇⣀⡜⠃⠀⠘⠻⣦⣀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠃⠀⠸⣿⣿⣿⣧⠀⢿⣿⣿⡿⣿⣿⣟⣱⡞⠀⠀⠀⠀⠀⠙⣃⣤⣄⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⠛⣿⡋⠀⢀⡀⠈⣿⡿⢛⣷⣾⣿⣿⣶⠿⠛⡿⠋⠀⠀⣀⣀⣴⡶⠟⠛⠋⠹⣷⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⣿⡄⠀⠈⠙⢦⡈⠁⠈⢿⣷⣿⣷⡿⠋⠉⠀⣠⠞⠁⣠⣶⠿⠋⠉⠀⠀⠀⠀⠀⠀⠸⣧⣼⡿
⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠉⠀⢀⣀⣀⠾⠏⠀⠀⠀⠻⣿⣟⠀⠀⢀⡞⢁⣴⢿⣿⠃⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣹⣧⡄⠀⠀⠸⣷⡄⠙⣿⣿⣤⣿⠟⠋⣰⠟⠁⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣸⣿
⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⢀⣤⠘⣿⢹⣷⠀⠀⠀⠈⠁⣼⣿⠀⣿⠁⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣾⠏⠹⣧⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⡿⠁⠀⠀⠘⠟⠀⠀⠀⠀⠀⠘⣷⡄⠹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣠⣾⠟⠁⠀⠀⢸⣇
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠿⠃⠀⢀⣴⠗⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⠿⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡾⠟⠛⠁⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠙⠿⣶⣾⣷⣶⣶⣤⣶⡾⢉⣦⠀⠀⠀⠀⠀⠀⢸⡇⠉⠿⣷⣄⠀⠀⠀⢀⣠⣶⠟⠁⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣸⡿⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⠿⠛⠻⠿⠿⠛⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⣦⡀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⢹⣷⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣀⣴⡆⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠻⣶⣶⡿⢷⣄⣀⣀⣉⠉⠛⠷⣦⣴⣾⠛⢁⣀⡀⠀⠉⠀⠀⣦⡀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⠀⢻⡇⠀⠙⠉⣿⡟⠁⠀⠀⠈⠋⠁⡀⠘⡛⣿⠀⣿⣦⡀⠸⣿⢀⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠻⡇⠀⢹⡇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠲⣾⡇⢸⡇⢻⠀⢻⣿⣿⣦⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⡇⠀⠈⣷⠀⢀⡆⠀⠾⠀⢸⡄⠀⡄⢸⡇⢸⠃⢸⣄⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⣾⡇⠀⣾⠃⢀⠀⠀⠀⠁⠀⠇⢸⡇⢸⡆⠘⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⢸⡟⠀⠙⠀⢼⣅⣠⠸⠇⠀⠀⢸⠁⠸⡇⠀⠉⢉⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⣾⠃⢠⠀⢠⢀⣿⡿⣇⠀⣇⠀⣼⠂⠀⣿⠀⢠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⢻⠀⢸⠀⢸⣿⣿⣀⡟⠀⣿⠀⣿⣦⠀⢹⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⣷⡎⠀⣾⡏⠈⣿⣧⢰⡇⢠⣿⢻⠀⢸⠇⢸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⣿⠇⢰⣿⠁⢀⠛⠛⠸⣧⢸⡏⢸⠀⢸⡇⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠀⢸⡿⠀⣿⣦⣷⣤⣿⣇⠀⠘⠀⢸⡇⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢃⡜⠀⣸⠃⠀⣿⢸⣿⠃⣿⣿⠃⠀⠀⠘⡅⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⢿⣏⠀⠟⠀⣼⡇⣸⣿⠀⣿⠃⢀⡘⠀⣸⣿⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣯⣤⣀⣨⣿⣿⣿⣿⠀⡏⢠⣾⠁⢸⣿⠏⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠛⠛⠛⠛⠛⢻⣿⣿⣀⠁⢸⣿⣀⣬⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⣿⠏⠉⠉⠻⣿⡿⠋⠉⠉⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣾⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣄⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡅⠉⠉⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣦⣄⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡏⠉⠉⠉⣹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣤⣾⣿⣿⡿⠻⠛⠁⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣶⣿⣿⡟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠀⠈⠉⠉⠉⠉⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠠⠄⠀⠉⠉⠉⠙⠿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠐⢠⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⠀⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠄⠀⠀⢀⣀⣀⠈⠁⣦⡘⠀⠀⠈⠠⣄
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢐⣉⢰⡄⠀⠀⢀⣊⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠀⠀⠚⠋⠰⠁⠀⠀⠘⠋⠘⢃⣱
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠈⢻⡀⠀⠀⠀⠀⠠⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠈⠀⠀⠀⠀⢀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠐⠒⠂⣤⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠒⢤⣀⣀⡐⠆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡠⠆⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⣤⡀
⠀⠌⠀⠀⠀⠀⣀⠀⠐⠂⠈⢩⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠖⠀⣀⠀⠀⠀⠁⠂⡀
⡀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠆⢸⡇⠀⠀⠀⠉⠁⠀⠀⠸
⠁⡄⠀⠀⠀⠀⠡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠠⠀⠀⠀⢠⠄
⠀⠘⠦⣀⠀⠀⠀⠈⠢⣄⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⡜⠁⠀⢀⠤⠃
⠀⠀⠀⠈⠑⠒⢠⠀⠀⠈⠒⢄⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠊⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠠⠈⠁⠀⠤⠤⠤⠤⠄⠀⠈⠉⠀⠀⡀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⠄⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸⠄
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣤⢀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠰⠦⠀⠴⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⢸
⠀⠀⢿⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⣼⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠻⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⢡
⠀⠀⠀⡗⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠚⣀
⠀⠀⠀⡃⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⣉⣰⡆
⠀⠀⠀⡁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠠⠌⠙⠃⠄
⠀⠀⠀⠜⠓⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠤⠋⠀⠀⠐⠦⠤⠤
⠀⠀⠀⠀⠁⠀⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠙⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠁⠀⠀⠀⡀⠀⣀⣀⠖⠉
⠀⠀⣶⠀⠀⠀⠀⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠈
⠀⠀⢿⡀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠄⠀⠀⠀⠀⡁⠨
⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠉⠀⠋⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣴⣶⣶⣶⣶⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡟⠟⠙⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢻⣻⠀⠀⠸⢻⣿⣿⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠀⣀⠀⠀⢹⣿⠉⠋⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠐⠠⠀⠈⠀⠋⠀⠀⠸⠇⠀⠀⠀⠠⠄⠀⠉⠑⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⠀⠉⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂⠀⠀⢰⡏⠀⠀⢠⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⠀⠀⠀⣤⡦⠀⠉⠀⣠⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣴⣷⣾⡁⠀⠀⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⠄⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣶⡀⣿⠉⠉⠀⠀⠀⠀⢀⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡇⠀⠀⣠⠀⡀⠀⠉⠀⠀⢴⣶⣴⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⠀⠀⠀⠘⠃⠀⠀⠻⣾⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣼⠇⠐⠀⠉⠀⠀⠀⠀⠀⠀⠈⠋⠚⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⣄⠀⠀⠀⢀⣴⢏⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⡞⠃⢿⣿⡀⠀⠀⠀⠀⡄⠉⠁⠀⠀⠀⠀⠀⣴⣦⡀⠀⠸⣷⣾⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠸⠸⣇⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⡀⠀⢩⣿⣿⠀⢀⣼⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢩⡀⠀⠀⠀⠈⣿⡀⠀⣈⠋⠀⡀⠀⠀⠀⣿⡄⠀⠀⠀⠉⠈⠁⠈⠈⠉⠀⠀⠈⣹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⢹⣇⠀⢹⡀⠘⣿⡄⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣿⣿⣷⣶⣼⣿⣇⣸⣆⣴⣿⣿⡄⠀⢻⣿⣶⣆⠀⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣼⡿⣿⣿⣷⣄⠀⣼⣯⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣼⢿⣿⣿⣿⣿⣿⢷⣿⣿⣿⣱⡏⠈⢻⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠃⠀⠀⢸⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢻⣟⣿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣼⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⠀⠁⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢸⣿⡏⠀⢹⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⣿⡿⠛⠀⢠⣿⠉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢀⡿⠃⠀⠀⠀⠘⠉⠋⢿⣿⣿⣿⣿⣿⠿⠿⢹⣿⠀⠀⠀⠀⠀⠀⠈⠉⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠁⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠸⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⠈⠁⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢤⣴⣐⠠⠅⠈⠑⠀⠡⠦⡀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠀⠁⠀⠀⠀⢀⡀⠂⠀⡀⠀⠀⠀⠘⠄⠀⠀⠀⠀⡄⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠁⠀⠈⠀⢀⡤⠀⠀⠸⠀⠀⢀⣀⠁⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢀⠀⠆⠀⠀⠀⢠⣄⠀⠀⠈⠎⠀⠄⠰⣀⠔⠁⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⣄⣤⠀⣷⡄⠀⠈⠿⠀⠀⠀⢀⠃⠐⠠⠊⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⢀⠂⠀⠆⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⢈⣉⠀⡲⠁⡠⠺⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠐⢬⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢁⢰⠂⡠⠒⡒⢁⡐⠁⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⡤⠀⠀⠀⠀⠀⠉⠀⠀⠁⡿⠾⠇⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⡂⠀⠢⠄⠠⠤⠤⠤⢤⠀⠀⢠⣶⠶⠇⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣄⣀⣠⡜⠉⠀⠀⢀⠀⠱⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠄⣜⣉⡉⠉⢹⠐⠂⠺⡁⠀⡀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡴⣿⣭⠛⠻⡿⢧⠠⠃⠈⡩⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠈⠹⠤⠵⠛⠛⠣⠞⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠒⠀⠤⠤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡠⠀⠂⢄⠀⠀⠀⠀⢆⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡠⠊⠫⣱⡀⠀⠙⠡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⣦⠆⠉⡆⠀⡃⠤⠘⣂⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠁⢁⡤⠚⠀⢈⠀⠠⠘⠁⠀⢀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠔⠚⠉⠀⠀⠀⠀⠋⠐⡂⢀⡊⠥⠂

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⠛⠉⠙⠛⠛⠛⠛⠿⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⠠⠄⠒⠂⠀⠒⠂⠀⠒⠒⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠴⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠛⢃⡀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠱⣄⠀⠀⠀⣀⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡿⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠹⣿⣿⣿
⣿⣿⣿⣿⠃⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⣿⣿
⣿⣿⣿⣿⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣸⣿⣿
⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠉⠁⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡆⠀⢀⡀⠀⠀⠂⢀⡀⠐⠀⠀⣀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⡇⢀⣀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⣀⣠⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠟⠉⠦⣀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠚⢿⣿⣿⣿
⣿⡿⠛⠁⠀⠀⠀⠀⠐⠤⢀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠙⠻⣿
⣏⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⡿⠉⢠⡆⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣤⡀⠻⣶
⣿⣿⣶⡟⠀⢠⡏⠀⣼⣷⣬⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠋⣠⣶⣆⠀⢷⡀⠘⢷⣤⣾
⣿⣿⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⡋⠉⠉⠀⠀⠐⠒⠒⠂⠀⠀⠀⣠⡀⠀⠀⠐⠒⠒⠀⠀⠈⠉⠁⠀⢸⣷⣾⣿⣿⣿⣦⣼⣷⣤⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢤⣤⠐⠛⠛⠛⠛⠛⠛⠷⠂⣴⡖⠲⡖⠛⠛⠛⠛⠛⠲⢶⣤⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⢠⡴⠋⠋⠉⠉⠩⠭⠵⠦⠀⢾⡀⢠⠼⠛⠛⠛⠛⠛⠁⠀⣤⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣢⡁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣥⣤⣤⣤⣤⣤⣥⣴⣾⣿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣬⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⠤⠤⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠜⢿⡏⠀⠀⠀⠀⠀⢀⡇⠹⡖⠀⠐
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠒⠂⠠⠿⠤⠀⠀⠀⠀⠀⡼⠳⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣠⣖⢒⠸⠗⢲⡤⢦⠀⡴⢼⣿⢄⣿⣿⠻⠀⠀⡠⠤⠤
⠀⠀⠀⢀⣀⡠⠀⠀⣀⡀⢉⣭⣤⣤⣤⡀⠀⢃⠀⠃⠉⠉⠀⢸⠿⢉⡆⠀⠀⠴⠿
⠀⠀⠀⠸⠾⠛⠛⠀⠀⣤⣬⠫⠒⠚⠉⠋⠙⠉⠉⠀⠀⠀⣘⣟⡃⡈⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⣙⣻⣴⡯⠀⠀⡸⠈⠡⠄⠐
⠀⠀⠀⠴⠲⠖⠚⠋⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠃⠀⡰⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⠀⠀⠀⠀⠀⠀⠐⠺⠽⢦⣀⡜⠀⠀⠀⠀⠈⠄⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣿⠿⠋⣁⠀⠀⡀⠀⠀⠀⠀⠇⠀⢠⡄⠀⠀⠐⣁⠀⠀⣸⠐⠛⠒⢀
⠀⠀⠀⠀⠀⢀⠀⠀⡟⠋⠉⠹⢾⣿⣴⣶⣿⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠰⠚⠁⠀⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡉⠿⢿⣿⣋⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠙⠻⢶⣟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀⣴⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣴⣿⣿⡿⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣤⣶⣾⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⣶⡾⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢀⣴⣶⣶⣾⣿⣿⣿⣿⠟⠟⣿⣿
⠀⠀⠹⢿⣿⠟⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣾⣿⡏⣿⣿⣿⣿⣿⠏⠙⠁⠹⣿
⠀⠀⠀⠀⠹⣾⣿⣷⡄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⢙⣿⡏⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣦⣴⣾⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⣿⣿⣿⠸⣿⡇⢸⡟⠁
⠀⠀⠀⠀⠀⠠⡌⢶⡿⠛⠻⠃⠀⠀⠉⠘⢿⡜⡏⢻⡍⠁⢀⣿⣾⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⣿⣿⡍⣦⣿⣧⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⣿⣶⣾⡇⠀⢦⣀⠀⠈⢻⣿⣾⣿⡠⠄⢹⡏⢹⣇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠎⣸⣿⣟⣹⡿⣹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⠀⠀⠙⠳⣆⠀⢣⣹⣿⡇⠀⠈⣷⢸⣷⢿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣁⣼⣿⣿⣿⣿⣿⠋⣿
⠀⠀⡀⠀⠀⠀⠀⠀⠙⣿⢿⣿⡗⠀⠀⠀⠙⠻⣦⣿⠻⣧⠀⠀⣿⣞⣿⠙⠛⣿⣿⣿⣿⣷⠰⠿⣩⣿⣿⣿⣿⡿⣫⣤⣄⣿⠂
⠀⠀⠀⡠⠀⠀⠀⠀⠀⠘⣦⣽⣶⠗⠓⠒⠒⠒⠀⠙⣀⣾⣧⣀⣸⣿⣿⣷⠶⣿⣿⠯⣽⣿⣤⢰⣿⣿⣿⣿⣿⣿⣿⣷⣿⡏
⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠘⢺⣿⣴⣾⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⡟⢩⡄⠀⠀⢀⢀⣠⡄
⠀⠀⠀⠀⠀⠉⠋⣤⡀⠀⠀⠀⢻⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣼⣿⣿⣿⣇⠈⡀⠀⠀⠀⠉⣽⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⢠⣜⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡏⢸⣿⣿⣿⣿⣿⠀⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⡻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⡇⠈⠸⣿⣿⣿⣿⣿⣄⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣶⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠹⣿⣿⣾⣿⣿⠿⢿⣿⣿⢿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠐⠓⢿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠁⠀⠄⢠⠟⠁⣴⣶⢤⣀⣉⡈⢻⢹⣿⣿⣿⣿⣿⣿⣿⠿⠀⠸⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⢸⣷⣦⡄⠻⣿⣶⣄⣨⡟⢡⣤⣵⠉⠻⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣧⣴⣿⣿⣿⡇⠀⣼⣿⡿⠏⠀⠀⢻⣿⣆⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠘⠟⠀⠀⠀⠀⠀⣿⣿⣷⣾⣿⡟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠛⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠉⣾⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣯⠄⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⣴⣶⠋⣿⣿⣿⣿⣧⠀⠀⡀⠠⠴⠀⠠⠀⠀⢹⣿⣿⠭⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⢹⣿⣿⣿⣿⠈⠌⠹⠀⠀⠀⠊⢁⠀⢸⣿⣿⡖⠉⡀⠘⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⢸⣿⣿⣿⣿⣥⡄⠀⠀⠀⢀⠀⠁⠀⣼⣿⣿⠇⠀⢹⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⣠⠄⠉⢲⣦⡛⢻⣿⣿⣧⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⣼⣿⣷⣶⣾⣿⣿⡟⠀⣿⠻⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠉⠛⢰⣿⣿⣿⣿⣿⣿⣿⠃⠀⡿⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⣢⣾⣿⣿⡿⢿⣿⣿⣿⠂⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⢸⣿⣿⣿⡃⠀⠀⠀⠀⠹⠷⣀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⠈⣿⣿⣇⠁⠀⠀⠀⠀⠀⠀⠙⠧⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿⢿⣿⣿⡇⢹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡿⠃⢸⣿⣿⡁⠘⢿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠦⠄⠈⠉⠛⠛⠛⠛⠋⠉⠰⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⡀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢠⣤⢀⣀⣤⣴⠶⠛⠛⠛⠛⠛⢿⣿⠳⢦⣤⣀⣸⣇⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡀⠀⣀⣴⡿⠷⠛⠉⠀⠀⠀⠀⣠⡶⠶⠶⣆⡙⠀⠀⠀⠉⠻⠿⢧⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠏⠉⠈⠉⣠⠶⠛⠛⠷⣄⠀⣸⠃⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡟⠀⠀⠀⢠⠋⠀⢀⣤⣤⣿⠀⡇⠀⢀⣴⡟⠓⣿⡄⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢠⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣿⠀⠀⠀⠀⢸⠀⠀⣿⣿⣠⣿⠀⣇⠀⢸⣿⣧⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠟⠃
⠀⠀⢀⡶⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⠀⠀⠀⠘⢷⣀⢻⣿⡿⠋⠀⠙⢧⣼⣿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠋⠻⣿
⢀⡞⠛⡆⠀⢻⠙⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⢉⣻⣯⣤⠾⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⡇
⣰⡇⠀⢷⠀⠀⡇⠀⣶⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡶⠟⠁
⡏⢸⡆⠘⣇⠀⠰⠀⢸⡆⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠻⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⢣⠀⢷⠀⠘⠀⠀⠀⠈⣧⡴⠶⠦⣀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠘⣇⠀⠀⠀⢀⣀⣀⣰⡏⠀⣀⠤⠿⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠸⣇⠀⠰⠋⠉⠉⠉⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢿⣅⡀
⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣿⣶⣦⣤⣤⣤⣤⣤⣴⣶⠾⢻⣿⠁⠀⠀⠹⣿⠳⣦⡀
⠀⠀⠀⠀⠀⠙⠛⣷⡄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⢋⠛⠀⠀⢫⡍⠉⠉⠁⠀⠀⢀⣼⠇⠀⠀⠀⠀⢹⣇⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⡿⠀⠀⠀⠀⠻⢦⣤⣤⣤⠶⠟⠁⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢱⡀⠀⠀⣀⣤⢶⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⣧⣤⠞⠋⠀⢸⡇⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⡀⠀⣿⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠙⠁⠀⣠⡴⢿⣃⣠⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠈⠛⢶⣿⡄⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣤⠶⠟⠉⠀⠈⠋⠉⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢻⣄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠹⣆⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡴⠋⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⣠⠋⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣅⡜⠁⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣴⣶⡿⠀⠈⣦⢴⠟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⠶⠤⠤⠶⠶⠶⠶⠶⠶⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠈⣿⠀⠀⠀⠀⠿⠞⠁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⢤⣄⣀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢻⡇⠀⢀⣀⣀⣀⣀⣤⣤⣤⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢻⡿⠛⠛⠻⣷⡶⠶⠾⠇⠀⠀⠀⠘⠛⢻⡟⠛⠛⠉⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡾⠃⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⣀⣀⣀⣼⠏⠙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡖⠒⠒⠀⠉⠉⠉⠉⢩⣭⣽⠿⣦⣄⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠉⠉⠉⠙⢦⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠳⢦⣀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⠉⠙⣻⡆⠀⠀⠀⠀⠀⠀⠀⢸⣆⣀⣀⣀⣀⣠⣤⡤⠤⠶⠶⠶⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣉⠛⠛⠒⠠⠤⠤⠤⠴⠤⠶⠆⢘⣿⠁⠀⠀⠀⠀⠀⠀⠀⠸⢿⣀⣀⣀⣀⠀⣀⣀⣀⣀⣠⣤⡶⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣤⣼⣿⣿⣷⣶⣶⣿⣿⣿⣦⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠛⠋⠀⠉⠉⠀⠙⠛⠋⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁⡴⠶⠀⠀⠀⠀⠀⠀⠴⢦⡀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⣿⠟⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⡍⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡅⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠄⠀⠀⢀⡘⠁⠀⠀⠀⡠⠊⠀⠀⢠⠱⠄⡺⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⡀⠀⠈⠿⠛⠛⠃⢀⠠⠀⠊⠀⠀⢀⠤⠃⠀⠀⢚⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣢⡤⠤⣎⠁⠤⠀⠰⣶⠊⠁⣀⠄⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁⢿⣷⣼⣿⠀⠀⠀⠀⣿⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠔⢿⡀⠀⠀⠀⠀⠛⠈⠁⠀⠀⡔⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠓⠉⠀⠁⠀⣀⠨⠷⠤⢲⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣚⠥⡀⠔⠉⠁⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⣀⣀⣀⠀⠀⢀⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠏⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠉⠉⠍⠀⠙⡋⠉⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⠀⠀⠇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢷⣦⣠⡆⠀⠀⠀⢠⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠻⢯⣷⣿⠟⠀⠀⠀⠀⠸⣿⣷⣖⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠀⠀⡩⠊⠀⠀⠀⠀⠀⠀⠙⡛⠛⠉⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠒⠁
⡿⣿⠿⠿⡿⡻⢿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠴⠶⣤⣄⡀⠀⣠⡴⠖⠛⠛⠉⠉⢛⡟⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠜⠀⣀⣹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣆⣀⢀⡟⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣮⣍⣙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⡶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢠⡀⠀⠀⣿⣆⡀⠀⠀⠀⠈⠑⢤⣀⠀⠈⠻⣶⣄⠀⢸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣧⡀⠀⠸⣆⠙⠶⢤⣄⣀⣀⡀⠉⠳⣤⠀⠘⣿⣧⢸⡟⢷⣦⣤⣀⡀⠐⠢⡀⢦⡀⠀⢠⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⡏⠳⣄⢰⣌⠢⡄⠀⠀⠀⠈⠉⠙⠲⠈⢷⡄⣿⣿⣿⠃⢦⡙⢿⡉⠛⠳⢦⡙⣿⣷⡀⣸⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⣇⠀⠈⠻⢿⣍⣂⣠⣤⣄⣀⠀⠀⠀⠀⠈⣷⣿⠿⠃⠀⠀⠉⠂⠛⠆⠀⠀⠹⣿⣿⣧⣿⣿⣼⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠱⡹⡀⠠⠖⠛⠉⠉⠉⠀⠈⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠛⠛⠛⠛⠛⠲⢤⠀⢹⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠺⢧⣄⠀⠀⠀⠀⠻⡄⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡶⣄⠀⠀⢹⠀⠀⠘⠁⢠⣾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣦⠈⠙⠀⠀⠀⣿⣿⠿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢣⣠⡞⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⢸⣿⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⠩⡆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⡾⠋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢻⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠈⠛⢯⣛⣛⣛⣛⣛⡿⠟⠁⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⢾⠉⠳⢦⣄⡀⠀⠀⠀⣀⣤⠶⣿⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢠⣀⣿⠀⠀⠀⠈⠉⠛⠛⠛⠉⠀⠀⣧⠈⠙⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠘⢯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⠀⠀⢰⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⡞⠋⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡞⢠⡟⠛⠳⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡼⢡⡟⠀⠀⠀⠀⠀⠉⠛⠳⣤⣀
⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠐⠡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠳⡀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠄⠠⠋

⠀⢉⡹⠿⠟⠛⠛⠛⠁⠀⠀⠀⣠⣔⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⣿⣿⣿⣿⣿⣾⣿⣿⡇⠀⠀⠛⠒⠛⠛⠋⠻⠻
⣀⣿⠷⠶⠦⠤⠤⠤⠤⢄⢀⣀⡀⠉⠉⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⡿
⣿⠿⠟⠻⠶⠶⠶⠶⣦⡤⠄⠀⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⣶⣦⣀⣽⣿⣿⣿⣿⣿⠿⠁
⣶⣾⣷⣶⣶⣦⣤⣤⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣧⡘⢿⣻⣿⣿⣽⣿⣿⣿⡿⠿⠃
⣿⣿⠛⠛⠑⠛⠻⢬⣉⣉⡁⠀⠀⠈⠙⠃⠀⠒⠀⠀⠀⠸⣿⣿⣿⡆⢿⣿⣿⡿⠿⠿⠟⠷⠄
⣛⡛⢻⣷⣶⣶⣤⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠿⠶⢞⣻⡿⣾⣿⣶⡄
⣶⣭⣄⣈⣙⣛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⢿⣿⣿⣿⣿⡀⢙⣛⣛⣲⣂⣴⣂⣀⣀⣀⣀⣀⣀⡀⡀⢀⠀⣀
⠻⠟⠛⠛⠛⠛⠻⠿⠿⠿⣶⣦⣤⣤⣤⡄⠀⠀⠀⠀⢀⣿⡀⠀⣸⣿⡿⠋⠀⠀⢰⣤⣭⣭⣤⣤⣤⣀⣉⠩⣭⣤⣀⣀⢨⣥⣥
⣶⣶⣮⣤⣤⣴⠼⠶⠶⠦⠄⠤⠥⠀⠤⠚⠋⠙⠛⠋⢀⣹⣿⣿⣿⡿⠁⢀⡆⠀⠈⢿⢟⣛⣿⣯⣿⣷⣶⣾⣿⣿⣛⣿⣿⣿⣿
⣿⣿⣇⣠⡤⠤⠄⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⠿⢷⣾⣿⣇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣶⣶⡶⢛⠗⠚⢿⣿⣿⣿⡇⠀⠀⠀⠉⠉⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣷⠟⠃⠀⠀⠸⠿⠿⠀⠀⠀⠐⢶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢿⣽⣿⣷⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣀⣹⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣗⠙⠿⠟⠁⠀⠀⠀⣠⣾⣿⣿⣿⣶⡿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⣀⣠⣴⣶⣿⣿⣿⣷⡈⠉⠋⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠌⠉⠉⠉⠉⠘⠁⠉⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⠔⠂⠉⠉⠳⠐⠒⠒⠢⣄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠎⠀⠀⢀⡠⠄⡀⠀⠀⠀⠀⠈⢣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡀⢸⠀⠀⡜⠀⠀⠀⠘⠀⠀⠱⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠌⠻⠀⠀⡄⢀⠀⠀⠀⠀⠀⢀⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⠀⠈⠚⠀⢈⡵⢬⠥⠊⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⣄⡀⠀⡀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠦⢵⣜⠋⣀⣀⡠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠈⠉⠀⠀⠐⠍⣒⣐⡺⠏⠁⠉⠉⠐⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⡃⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⡄⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠣⢀⡀⠀⠀⠀⠶⢺⡀⠀⠀⠀⠘⡀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣨⡀⠀⠀⠀⠀⠐⠀⢤⡀⠢⣄⠀⡇⠀⣀⣀⣀⣹⣄⠀⠀⠀⠹⣄⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⠒⢄⣀⠀⠀⠀⠀⠀⡔⠉⡄⢡⠀⠀⢀⣀⡌⡀⠀⠀⠀⠀⡝⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢢⡄⢘⡀⠢⠄⣸⠆⠀⠀⠠⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⢉⡽⠍⠻⠀⠀⣠⠅⠂⠘⠐⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠂⠀⠿⡤⠂⡄⡀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠄⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠡⡀⠀⠀⡒⠀⠢⠀⠾⠰⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠁⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠂⠀⠀⠀⠘⠶⠯⠤⠤⠄⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⣎⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⠼⢷
⠀⠀⠀⠳⣤⠤⠬⠭⠭⠭⠭⠭⠭⢭⠭⠭⠽⠿⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠽⠿⠿⠿⠿⠿⠟⠛⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⣀⠀⠀⠈⠀⠀⠀⠀⠁⠉⠈⢸⣾⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠯⡂⠘⠋⣁⣠⠤⠤⠤⠤⣠⠞⠉⠉⠉⠁⠀⠘⣤⣾⣠⢞⡉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢡⢊⡀⠀⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡄⠀⠀⠀⠈⠛⠃⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣕⠀⠉⣠⠄⠀⠀⠀⠀⣀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡇⠀⠐⠃⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⣀⣩⠶⢄⣀⠜⠀⠀⢎⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⣆⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠸⡀⠈⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢡⠀⠀⠀⣠⠁⠈⢄⡄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠸⠀⠀⠐⠁⠑⡔⠉⠱⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⢐⠅⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢇⠀⠀⠃⠀⢰⠀⠀⠀⠀⠀⠀⠐⠂⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢤⡄⠀⠀⡆⠀⠀⣀⣠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⣾⣿⣿⣿⣿⡿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣿⣿⣿⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⠀⠀⠀⣀⣀⣠⡤⠔⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠥⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡯⢹⣠⣴⡋⣺⣲⣁⠜⢟⣴⡭⢠⡿⣿⣿⣿⣿⣧⠀⢣⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣾⣾⣿⡟⠉⡏⠀⡟⠀⠙⠳⠋⠛⠁⠈⣀⣾⠻⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠁⢉⡁⠈⠁⠸⡟⠀⠈⠀⠀⣏⡀⠀⢨⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠃⠀⠈⠉⠉⠙⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⣀⣀⣀⢼⣀⡀⠀⠈⠛⢻⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠈⠁⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣄⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢉⣁⠈⠀⢸⡿⠛⠛⠥⠄⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣠⠒⢄⠀⠀⠀⠀⡖⠒⢦⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⣿⢳⣾⣿⢂⡤⠶⣄⠳⠿⠟⠀⠿⠟⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⡜⠛⠀⠈⢀⠀⠀⠚⠁⠀⣀⠀⠀⠀⠋⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠴⢆⡀⠀⠉⠑⠢⠤⠤⠒⠉⢀⣠⠋⢉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⢶⠤⠤⢤⠖⠚⠉⣋⠹⠁⡸⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢟⣄⣀⣬⡷⣄⢎⠉⠧⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠉⠉⠉⠀⠈⢪⢳⠄⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢑⢣⠴⠒⠀⠈⢱⢠⢜⠏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠑⡞⢸⡀⠀⠀⠀⢸⠸⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢰⠃⠀⠉⠉⠉⠉⠉⠀⢹⣀⣄⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⢤⣀⣀⣤⠤⠖⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠈⣛⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⡀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡇⠀⠀⢸⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣽⣄⣀⣸⣀⣀⣮⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠉⠁⠉⡹⠉⠙⠋⠀⠀⢱
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣒⣒⣒⣋⣈⣓⣢⣴⣶⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⢒⣊⣉⣭⣭⣍⠙⠛⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣡⣴⣶⣿⣿⣿⣿⣿⣿⠿⠶⠄⠀⠀⢠⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⣤⣾⣛⣛⣻⣿⣿⣿⣿⣿⣟⣃⠀⠀⠀⢿⣿⣿⡿⠁⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⣤⣀⡉⠉⢉⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⣠⣼⣿⠏⢀⣸⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠐⠾⢿⣿⣿⣿⣿⣿⣿⣶⣶⣼⣿⣿⣶⣶⣾⡿⠛⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢺⠋⠉⠉⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠐⠒⠶⠿⢿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢮⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠖⠛⠿⠶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠶⠟⠛⠛⠛⠛⠳⠀⠀⢸⣿⣿⣿⡿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠘⣿⡇⠀⠀⠀⢠⢤⣶⣶⣤⡉⠻⣄⠀⠀⠀⠀⠀⣠⣐⣶⣦⣄⡀⠀⠀⠀⠈⣿⣿⠟⢠⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠱⣸⡇⠀⠀⠀⠛⠋⢿⠿⠟⠙⠀⢻⠀⠀⠀⠀⠸⠧⣿⣿⠟⠛⠋⠀⠀⠀⠀⠀⠋⢠⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢰⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠸⡇⠀⠀⠀⠀⠀⠀⠀⣴⠻⣧⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠂⠀⠒⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠙⠀⢶⡏⠉⠛⠛⠛⠛⠋⢉⠟⠀⠀⠟⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠻⣶⣦⠤⠴⠖⠒⠁⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⢆⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡰⠁⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣿⠈⢳⡀⠙⠢⢄⣀⣀⣀⣀⡤⠶⠋⠀⠀⢀⠞⠀⠀⡏⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⣿⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⢸⡇⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠀⠀⠀⠀⣿⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠳⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⢶⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⠀⠙⠻⣿⢶⣤⣀
⠤⠶⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣠⠞⠉⠳⣄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠙⠻⢶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡞⢻⠀⠀⠀⣹⣦⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⠏⠀⢸⠀⠀⢰⡟⠘⢧⠀⠀⠀⠀⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡼⠃⠀⠀⣼⣀⣠⣿⠀⠀⠈⢧⡀⠀⢠⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠋⠀⠀⢀⡜⠁⠉⠀⠈⣧⠀⠀⠀⠳⣄⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡞⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⠛⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡎⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠀⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⢀⠀⢡⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣤⣠⡀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⣇⡾⠀⡿⢚⡅⢪⠔⠁⠀⢺⡇⠈⠀⠀⠀⠀⠀⢰⠀⠀⡠⠂⠀⡰⡿⠻⡷⡄⠀⢢⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡾⠃⢀⠀⠀⠈⠀⠀⠀⢀⠀⠈⢹⠃⠀⠀⠀⠀⠀⢸⠀⠈⡠⢖⡊⠀⠀⠀⠀⠘⠦⢄⡑⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢧⠀⢀⠒⠄⠀⠀⠀⠀⠒⡀⠀⡘⠉⠂⠀⠀⠀⡠⢎⠊⠉⠀⠖⣢⠀⠀⠀⠀⢐⢺⠀⠉⠑⠿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠚⠀⠀⠀⠀⠈⠛⠀⠀⠃⡠⠀⠀⠀⡜⠀⠈⠦⣤⠀⠈⠋⠀⠀⠀⠀⠘⠋⠀⣠⡤⠃⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⡀⠀⢀⠀⠀⠀⡀⠀⠀⡘⠋⠀⠀⠀⠀⢩⠂⠀⡀⡸⡄⠀⠀⣀⠀⠀⠀⠀⠀⢀⠿⡀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡈⠉⢉⡀⠄⠊⠀⠀⠀⠀⠀⠶⠅⢀⠠⠜⠁⠘⠦⣀⠀⠁⠀⠁⣀⡴⠋⠀⠑⠤⢀⠨⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠈⠖⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⠽⠉⠉⠙⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠠⠖⠀⢙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡜⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⢹⠀⠀⠀⠀⠀⠀⡘⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⡆⠀⠀⠀⠀⢠⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢋⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠸⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⡘⡀⠠⡜⣷⠐⠒⠒⠒⠒⠒⠐⠒⠂⢲⡧⢓⠀⣸⠀⠀⠀⠠⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠉⠁⠀⠀⠈⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⡤⠄⢀⡼⣤⢀⡀⣀⠠⠜⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⡏⠉⠉⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢌⡀⢉⡁⢈⠁⣈⠥⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢬⡁⢉⡀⣌⠁⡩⠥⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠉⠉⠀⠀⡀⣄⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⡄⡄⠀⠉⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠐⣢⠦⠖⠉⠁⠈⠀⠢⢔⣖⠂⠛⠀⠀⠀⠀⠀⠀⠀⠻⠐⢒⡦⠖⠉⠀⠉⠐⠦⣴⡖⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠦⠤⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡶⣤⢤⣤⠄⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⣠⣶⣶⣿⣿⣖⢢⣶⣤⣉⣿⣦⣤⣀⠈⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠲⠔⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣫⠟⢣⢾⠽⣿⣿⣧⠴⠛⣿⣋⠁⠀⠐⠆⢠⣴⠈⠡⠀⠤⢀⠄⠁⠄⠀⠈⠱⡀
⠀⣤⣿⠀⠀⠀⡀⠠⠽⠟⠛⠉⠀⠈⢆⠘⣏⠓⠙⠁⣝⠛⡅⠀⠀⠀⠀⠀⠀⢬⣹⣤⣦⣞⢤⣉⠔⣁⠦⣡⢄⡠⠌⡀
⠀⠘⡇⠀⠐⠈⠀⠀⠀⠀⢄⠀⠲⡤⣀⠡⠹⡀⠀⠀⠻⠤⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⣿⣿⡷⣿⣾⣫⠿⡉⢊⡑⢀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢱⠹⣷⣶⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣽⢿⣽⠆⠐⣂⠙
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢳⣌⠙⠻⢮⡳⡌⣶⡀⠀⠀⠀⠀⢀⣶⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⡿⠋⡀⢠⠃
⠀⠉⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠳⢽⡀⢻⣿⣶⣿⢓⣦⠘⢀⠀⠀⠀⠀⣼⡟⠈⠿⠿⠛⠛⠛⠛⠛⠛⠿⣿⣋⠘⠁
⠀⠁⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠻⠿⠟⠋⠁⠀⠉⠉⠀⠀⠈⠋⢀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠆⠘⠁
⠀⠀⠀⡈⠲⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⡺⢆⠀⠀⠀⣷⣢⢄⣀⡀
⠀⠀⠀⠐⠬⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⣟⢛⡹⠷⠄⠀⠀⠒⠋⠉⠻⣱⠁⠀⠀⠀⢀⣀⠀⠤⡀
⠀⠀⢺⣿⣿⣷⣿⣿⣿⣿⣿⣶⣄⠳⣤⣤⡤⣤⡀⣀⣠⣀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⣀⠲⡖⠢⢰⣶⡀⡀⠀⠀⠉⢳⣦⣈⠑⠢⡀
⠀⠀⠈⠘⠿⠛⠛⠛⠏⠛⠉⠀⣩⠓⢤⣍⣙⣻⠿⢿⡿⣷⣿⢆⠾⠚⠲⠤⢤⠴⢦⠱⠌⡇⠌⣸⠙⢿⡧⠘⢛⠀⠈⠨⠛⠊⠻⣿⢾⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠈⣠⡶⢾⣿⡿⣿⠟⠂⠒⣂⠒⠋⠀⠀⠀⡀⡀⠀⠐⠚⣺⢃⢸⢛⡄⠀⠀⠀⠻⢳⡄⠱⣶⠷⠀⠿⢿⠏⢻
⠀⠀⠀⠀⠀⠀⠀⢀⠖⢁⡄⢐⣿⡿⣹⣿⡇⣾⣿⣿⡜⣿⣷⣾⠿⠷⢸⣿⣿⣹⣿⣿⣿⡗⠛⠈⠳⠀⠀⠀⠀⠀⢼⠀⢿⡿⠀⣾⠛⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠿⠃⠘⠻⠃⠻⠿⠟⠓⠿⠏⠾⠛⠟⠀⠛⠁⠈⠉⠋⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣠⣾⣷⠀⡴⠥⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠛⠉⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣾⣿⣿⣶⣶⣦⣤⣀⠀⠀⠀⠀⡴⠒⡶⠀⠀⠀⠀⠀⠀⠀⠚⢾
⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⢿⢿⣿⣿⣏⠉
⡀⠀⠀⠀⠄⠀⠀⠛⣿⡄⠀⠀⣿⣿⣿⣿⣿⣻⣷⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣶⣄⡀
⣯⠀⠀⠀⠀⠀⠀⠸⣿⣇⡀⠀⠸⡿⢿⣿⣻⡛⣇⣠⣟⣻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⣀⣤⣤⢴⣶
⠞⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠀⠀⠀⠀⠙⠛⢿⣿⣿⠟⠁⠀⠀⠀⢠⣤⣼⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⣟⣪
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠘⠛⠗⣦⡀⠀⢠⣤⡚⠋⠁⠀⠀⠀⠄⠀⠀⠀⣿⣏⣭⠏⠉⢛
⡎⠉⠈⠉⠉⠉⠒⠠⠔⠒⠒⠒⠒⠀⡈⠀⠀⠀⠀⢀⡇⠘⠋⠈⡿⠧⢀⡀⠀⡠⠦⠔⠃⠤⠤⠬⢽⠙⠀⠀⠉
⣄⡀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣈⡋⢤⠀⠢
⠉⠁⠀⠀⠀⠀⣀⢾⠀⠀⠀⠀⠀⣠⡈⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡴⠖⣋⣀⣾⣿⣫⣷⣷⡀
⠀⠀⠂⠀⠀⠘⠋⠉⠀⡀⠀⠐⠻⣿⣿⣷⣶⠒⠋⠒⠀⠀⠀⠀⠀⠀⠀⠾⣽⡿⠠⣼⠟⢹⠙⠀⡗⢀⡌⢧⡀
⢀⡄⠀⠀⠀⠀⠚⠁⠀⡀⠃⠀⣠⣿⣿⣿⣿⣿⣦⡀⠀⠄⠀⠀⢀⠙⠀⠂⠨⢷⣶⣌⠀⡀⠀⢀⣠⡋⠳⢴⠃
⡀⠉⠙⠀⢀⣰⠿⠖⢩⣤⣦⣴⣿⡿⠋⡤⢨⠻⢷⣿⣌⠤⠀⠀⡤⡦⣴⣾⢧⠴⢿⣿⣾⣧⣤⣮⢿⣀⠤⠄⠠
⠃⠀⠀⠘⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⢸⣇⡛⠀⠀⢿⣿⣄⠀⠀⠈⠉⠛⠀⠀⢀⣾⣷⣿⣿⣿⣿⣶⣄
⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠈⠻⢟⢣⡀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠛⠿⠿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢲⣶⠒⠀⠉⢹⣾⣰⣿⣿⣿⣿⡿⠁⢠⡾⣦⣀⠀⣾⣿⡀
⠀⠀⠀⢀⣼⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠠⢜⠟⡋⠉⠁⠀⠀⠈⠈⢱⣿⢻⣷⣾⣿⣿⣦
⣀⡀⠀⠀⠉⠀⢠⣤⣄⠀⠀⠴⢦⣤⣤⠀⠀⠀⣤⣷⣬⣿⠏⠐⠾⣷⢷⣠⠴⣾⣄⡀⢀⡀⠀⠈⠉⠛⠙⢟⢻
⣽⠛⠛⠀⠀⣀⣤⣼⠛⣿⠧⡆⠀⠉⠓⢢⡄⠀⠈⠙⠛⠁⠀⠀⠀⠰⣿⠆⠉⠉⠁⢠⡿⠃⠀⠀⠀⠀⠀⠀⠈
⢿⠯⠄⠠⣴⡟⠻⢯⢩⣥⣤⣄⠀⠀⠽⢽⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢶⣦⠀⠀⠀⠀⠀⣠
⣤⠄⠀⣿⣻⣿⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠐⠾⢶⣶⣶⣶⠀⠀⠛⠛⠿⠃⠀⠀⠀⠀⠉

⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠀⡀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠂⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠰⠀⢀⡀⠴⢂⣂⣀⡀⡤⣤⣤⢤⢀⣀⣒⠀⠧⢀⡀⠀⡈⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠈⠄⢀⠤⢐⡩⣄⠶⠟⠛⣈⡉⠉⢩⠩⡉⠉⠭⡉⠓⠛⠿⠶⣢⢍⡒⠤⡀⢀⠊⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⢀⠤⢊⣵⡮⢓⠩⡐⢀⠀⠅⡐⣰⠀⠈⣆⢁⢨⡄⠁⠀⢀⡔⠂⠀⠉⠚⢵⣦⣑⠤⡀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢝⡡⡮⠓⠁⠀⡅⠂⠁⠉⠀⣐⠂⠀⢡⡀⢟⡈⢛⠁⠀⡠⡁⠇⠀⠀⢀⡋⠂⡈⠚⢵⢌⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⡲⢋⡐⠅⠐⠀⠀⠀⢁⠠⡀⠡⠬⠅⠊⠁⠈⠁⠙⠒⠛⠼⠠⡇⣆⠀⠕⠁⣤⡑⠋⠀⠙⢷⣎⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢪⣞⠎⠀⡀⠉⠊⠱⠈⠀⡰⠞⠁⢀⠠⠐⠂⠀⠈⠀⠀⠉⠀⠐⠂⠄⡀⠉⠚⠀⣃⠊⣀⡀⠁⠉⢢⠱⣣⡕⡄
⠀⠀⠀⠀⠀⠀⠀⡠⢡⡳⠁⠈⠀⠐⢈⠂⢰⡌⠃⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣀⠁⢮⡀⠐⠈⠒⠂⢀⡜⢞⡌⢆
⠀⠀⠀⠀⠀⠀⡰⢡⡳⠁⠀⠈⠢⠀⣀⠠⠀⡄⠍⠉⡀⣠⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⠀⠉⠨⢀⠃⢡⡊⠄⢀⠁⠔⡈⢞⡌⠆
⠀⠀⠀⠀⠀⠀⢁⣿⠁⢪⠀⡁⠌⠀⠀⠁⡠⠇⠀⠀⠐⠮⠀⠀⠀⠀⣄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠷⠂⠀⠀⠸⠄⠈⠖⡀⠡⠳⠈⢩⠌⣿⡈⡀
⠀⠀⠀⠀⠀⠈⢸⡇⠰⠀⠁⠑⠠⠤⠀⠠⠁⢀⢀⠀⠀⠀⠀⠀⠀⠠⡀⢠⢆⡠⡆⢠⢀⠄⠀⠀⠀⠀⠀⡀⡀⠀⠆⠈⣥⠂⠁⠀⠀⠆⢸⡇⢀
⠀⠀⠀⠀⠀⠃⣿⠀⠄⠲⡤⢂⠰⠂⠀⠇⠀⠘⠘⡀⠀⠀⠀⠀⠰⠋⠇⢀⠸⠄⡟⣸⠀⠀⠀⠀⠀⠀⢀⠃⠁⠀⠰⠀⢨⡄⠂⠀⠈⣉⡀⣿⠸
⠀⠀⠀⠀⠀⢐⡖⠀⠠⠔⡠⠁⢄⠀⢰⠀⠀⠀⢁⠡⡀⢰⠀⠐⣝⠏⡅⣤⠃⠀⡏⠌⣾⡍⣠⠀⡇⠀⠌⡠⠀⠀⠀⡆⠀⡤⠠⢠⠀⠦⠀⣓⡂
⠀⠀⠀⠀⠀⠨⠧⠀⠀⠀⠈⠁⠈⠀⢸⠀⠀⠀⠈⠂⡁⠰⠀⣓⠤⢇⡂⠁⣀⠀⣇⢂⠣⢄⡄⢠⠆⢈⠔⠁⠀⠀⠀⡇⠀⠀⠀⠐⠠⠀⠁⠼⠇⠁
⠀⠀⠀⠀⠀⠈⣿⠀⠀⠤⠚⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⢃⠀⠀⠀⢠⠀⠀⠀⠀⠠⡀⠀⠀⡜⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠦⣀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠆⣟⡀⠀⠚⠒⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⠉⠉⠉⠉⠉⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠱⠤⠇⠀⢀⣳⠸
⠀⠀⠀⠀⠀⠀⠸⢧⠀⠀⠤⠙⠠⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⠢⢀⡀⠀⠀⢀⡀⠔⠂⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡤⠒⣄⠀⠀⡼⠏⠆
⠀⠀⠀⠀⠀⠀⢆⢫⠆⠀⠘⠂⠃⠀⡄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠒⠒⠁⠀⠰⡝⠠
⠀⠀⠀⠀⠀⠀⠈⠎⢯⢄⠀⠀⠀⢄⡜⠗⣀⠀⠐⠄⡀⠀⠀⢀⣔⠁⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⢀⠤⠂⠀⢀⣄⡀⠀⠀⠀⠀⣠⡝⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢆⢻⣦⠀⠒⢘⢃⡮⠏⣦⠀⠀⠈⠂⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠒⠁⠀⣀⣴⢉⠫⣵⣗⠄⠀⣴⡟⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡙⣵⣄⠈⠁⣀⠐⠁⠚⠓⠄⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣧⡀⣸⠏⢚⢢⠑⡬⠋⣠⣮⢋⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡫⣶⣄⠉⠾⣀⠾⢃⢠⡗⠃⠀⠤⢤⢀⣀⣀⣀⠀⠠⡄⡦⠀⠈⠁⠑⡈⠟⣠⠞⠀⣠⣾⢝⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡑⢷⡦⣀⠉⠀⠌⠀⠀⡀⠂⢀⠈⠁⣠⢠⠀⠀⠀⢠⢀⢶⠌⠶⠗⠀⣀⢴⡾⢊⠕⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⡓⠯⣖⡤⣀⠀⠈⠓⠒⠁⠦⠴⠦⠃⠀⠘⠒⠋⠉⣀⣤⣶⠝⢊⠡⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠡⢉⡛⠻⠵⠶⣦⣖⣶⣴⣰⣶⠶⠦⠟⢛⡉⠍⠒⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣐⣂⣀⣀⣀⣒⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⠙⢯⡛⢿⣿⡛⠿⣿⣟⠿⣿⣮⡙⠻⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠉⠑⠀⠈⠉⠁⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⣠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣄⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢰⠋⡙⢻⣿⡇⠀⠈⠁⢀⣀⣀⡀⠈⠀⠀⠀⠀⠀⠀⠈⢁⣀⡀⡀⠀⠀⠁⠀⢻⠟⣋⠙⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢢⡿⠀⠀⠠⡎⠡⣯⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⠈⠣⠀⠀⠀⢘⡔⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⣧⠀⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⣿⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠆⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⢖⠒⠒⠒⠒⠒⠒⡲⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠉⠐⠒⠒⠈⠉⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⢄⣀⣀⡠⠔⠚⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⢀⣠⠤⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⢠⡟⠁⠁⠀⢰⣤⠀⠀⢻⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⠼⢯⠀⠀⠀⢡⡀⠻⣧
⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⡟⠀⠀⠈⠆⠀⠀⠀⢣⠀⢿⡇
⠀⠀⠀⠀⠀⠔⠊⠀⣠⣤⣤⣤⣤⠀⠀⠀⠀⠀⠛⠀⠻⠂⠀⠈⠀⠀⢠⣴⡌⡆⣸⡇
⠀⠀⠀⠔⠁⠀⢠⠛⠅⠀⠀⠀⠉⠓⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡟⢰⣿⡇
⠀⠀⠀⠈⢄⠀⠁⠀⢄⠀⣠⠖⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⢸⣿⠁⠘⣿⠃
⠀⠀⠀⠀⠈⠂⠀⠈⠛⠻⠥⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⣹⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⢀⣵⣄⠀⠀⠀⠉⠉⠀⠀⠀⠀⠘⠃⠀⢶⣾⠟⢰⣾⠟⢻⣿⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣽⢿⡷⣄⠀⠀⢀⣀⣄⠀⠀⣰⠀⢠⠀⠉⢀⠼⠋⠠⣿⣿⠁⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⠲⣻⣿⣿⡿⢶⣿⡠⠀⠀⣴⠯⢐⣨⣾⣿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⠢⢄⣀⡟⠀⢀⡼⢿⣿⡿⠟⠛⠋⠉⠉⢀⣽
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣤⣿⠔⠊⠁⠀⢀⡴⠖⣶⡿⠋⣿
⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠸⡇⢀⡤⠐⠚⠛⠛⠋⢅⠀⢰⣿
⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣧⢸⡇⠀⢀⣀⣠⣤⠀⠈⠘⠋
⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢸⣾⡇⠀⢨⠉⠙⢦⢮⡑⠂⠛⠲⠄⣀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⡇⠀⠈⠲⣤⣒⣋⡉⠀⠀⠀⠀⠈⠑⠦⡀
⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡗⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠒⠤⣀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⠝⣻⠿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠿⡻⢿⣦⣄⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⠻⣿⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠘⠻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⡆⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⠿⠷⠀⠀⠀⠰⠧⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⠀⡀⡀⠀⠀⡄⠀⡀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠀⠂⠠⠀⢀⠀⣤⠀⠀⠀⠄⠠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠁⠀⠀⠀⠁⠀⠁⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠈⢣⡀⠀⢻⠀⠀⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠙⠳⢤⡀⠀⠱⠀⠘⠀⠀⠀⠈⠻⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⣠⣤⣤⣄⠀⠀⠀⠈⢻⣿⣷
⠀⢀⣀⢠⡷⣤⠾⡀⠀⠀⢀⣤⣤⠀⠘⡀⠀⠥⠀⠀⠀⠀⣶⣶⣶⣦⠀⠀⢠⣾⣿⣿⣿⣿⣷⡀⠀⠀⠈⣿⣿⡇
⠀⣀⣻⠉⠁⠀⠀⠛⠟⠀⠈⢿⣿⣷⡀⠁⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⠃⠀⠀⣰⣿⣿⠁
⢀⣹⠆⠀⠀⢀⣤⣤⣄⡀⠀⠀⢻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠙⠿⢿⡿⠿⠃⢀⣀⣼⣿⡿⠃
⠈⣉⡁⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣦⣤⣴⣶⣾⣿⣿⣿⣿⡟⠁
⣸⣿⡿⠀⣿⣿⣿⣿⣿⣿⡟⣀⣀⣨⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡇⠀⠈⢻⢿⣿⠿⣻⣾⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢻⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠉⠙⠛⠙⢿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣧
⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠹⣇
⠀⢰⣿⣿⣿⣿⣿⢛⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢻⡆
⠀⠀⠙⢿⣿⣿⣟⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢻⡄
⠀⠀⠀⠀⠙⢿⣽⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢿
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⣿⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠻⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢞⣛⣿⣭⣭⣍⣭⣟⣛⠷⣦⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣩⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣭⣝⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣃⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢣⣿⣿⣇⣻⣿⣇⢿⣿⣿⣿⠹⣿⣿⡿⣿⣿⣿⣿⣿⣿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢸⠟⠻⣿⣿⠟⡼⠎⠿⣟⣵⣷⣟⣛⠹⣿⡿⠿⣛⢹⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⠃⠐⠙⣻⣼⠿⢶⣾⣿⣿⣿⣿⣧⣵⣶⣾⣿⣿⣘⢟⡣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡒⠆⠀⠈⣷⣶⣶⣾⣿⣿⣿⣿⣿⣤⣤⣿⣿⡿⣛⣚⣣⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⢤⢜⡻⢿⣿⣿⣭⣭⣍⣿⣿⣿⢛⣫⡜⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠿⢶⠎⠹⠿⠉⢩⠶⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢯⣾⢫⣾⠸⣿⢹⡎⢿⡎⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣾⣿⣧⢿⡖⡏⢹⢏⣾⣿⡜⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣾⣿⢸⣿⡊⣇⠁⢣⣿⣿⢹⣷⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡳⣢⣵⢸⣿⣿⡀⢠⣿⣿⣿⢨⣋⢥⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠻⣷⡰⣶⣶⣨⣭⣭⣩⣸⠟⠶⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡸⠿⢹⣿⣧⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣉⠸⠿⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣝⣣⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠆⠐⠤⠄⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠤⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⡊⠁⠀⠈⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠄⠀⠀⣐⣦⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠰⢠⣄⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠘⠆⠀⠀⠑⠉⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠷⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠘⣆⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⡂⠀⠀⣠⠒⣒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠲⠿⢄⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡩⠛⠉⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⠃⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠰⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡆⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢰⡀⣀⣀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢤⣤⣀⠀⠈⠉⢩⣇⠀⢡
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⡈⠁⠡⡱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠘⡄⠀⠀⠘⣽⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡧⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢎⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣔⠏⠀⠀⠀⣣⡴⠒⠒⣷⣄⠚
⠀⠀⠀⠀⠀⠀⢀⡿⡷⠤⠤⠔⠀⠒⠺⠀⠀⠀⠀⠀⠉⠺⣢⡤⣊⡡⠌⣉⣲⠕⠁⠀⠀⠀⢰⠁⢀⣶⠖⠛⠺⡆⢃
⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⡠⠆⠤⣀⠼⠀⠀⠀⠀⠀⢸⣢⡜⢁⡀⠀⠀⢹⠈
⠀⠀⠀⠀⠀⢠⠃⢸⠃⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠈⠢⠔⠒⠋⠀⠀⠀⠀⠀⠀⢀⠉⠨⣵⡀⢀⡠⠞⢀⡆
⠀⠀⠀⠀⠀⠈⠒⠺⣄⣀⣆⠀⠀⣀⣀⡠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠋⠉⢛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠉⠁⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠘⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣄⡀⠈⣽⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠶⠦⣄⠀⠀⢀⠴⠿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢖⣉⠀⣠⠛⠉⠀⠈⠑⢄
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠑⡔⠁⠀⠀⠀⠈⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠃⢏⡠⠟⢡⠂⠉⠢⡀⠀⠈⡆
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢠⠋⠀⠀⠀⠘⢦⠀⠙⣄⡤⠆
⠀⠀⠀⠀⠀⠘⠒⠀⠀⠄⠐⠒⠦⠤⠤⠤⠤⠤⠀⠀⠠⠄⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠤⠤⠤⠤⠴⠒⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⣀⢀⠄⠀⠀⠀⠀⠀⠁⠀⠀⣤⠀⢠⣀
⠀⠀⠀⠀⢀⠀⠠⡄⣤⡜⡆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠱⣿⠋⠀⠀⠀⠀⢀⡀⠀⠀⢰⡇⢰⣿⡏
⠀⠀⣤⣾⣮⠇⠀⠙⢺⣿⡁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣶⣿⣦⠀⠀⠀⠀⣨⡅⠀⠀⢀⡇⠀⣿⡇
⠀⠀⠀⣹⣿⠀⠀⣶⣾⣿⣷⣶⠀⠀⠀⠀⠻⣧⣏⠀⣸⣿⣿⣿⡂⠀⠀⠸⣿⣿⣷⣶⣶⣶⣶⣶⣶⡖
⠀⠀⠈⣻⠇⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠛⢿⣿⣿⣿⣿⡿⠿⠿⡇⠀⠀⠀⣿⡟⠉⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠀⠀⠀⠛⠛⠛⠛⠛⠀⠀⠀⠀⠈⠛⢹⡤⠈⠁⠀⠀⠈⠀⠀⠀⡿⡇⠀⠙⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡄
⠀⠀⢸⣷⣿⣿⣿⣗⢺⣳⠀⢀⣴⣿⣿⣿⣿⣿⡆⠀⠀⠻⣽⣧⡀⠀⠀⠀⠀⠀⢶⣦⡞⠀⣀⠀⣿⡇
⠀⠀⢘⣛⠹⡴⣯⡇⠀⠀⠠⡀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠖⣾⣿⣿⣾⣟⠋⠉⠁
⠀⠀⢸⣿⣿⣿⣿⡇⠰⣿⣷⠃⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠘⠟⢻⣁⠙⠂
⠀⠀⢾⣿⣿⢿⣿⡇⢀⣨⣿⣅⢸⡿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢸⣿⣿⣿⣿⣿⠾⣽⣧⣌⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⡄⠀⠀⠀⣀
⠀⠀⠈⢹⣿⠋⣿⣿⡆⠈⣿⡏⣿⣿⢹⣿⣿⠉⠁⠀⠀⠀⠀⠀⣬⣷⣦⡀⠙⣾⣷⡀⠤⠬⣿⡀
⠀⠀⠀⢸⣿⠀⢻⣿⠃⠀⣿⡇⣿⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠳⣿⡟⠀⠀⣿⡿⠇⠀⢀⣿⡧
⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⡇⠀⠀⣿⡇⠀⠀⠈⣯

⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠖⠈⠉⠉⠉⠉⠉⠉⠁⠰⠦⢤⡀
⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⠉⠉⠉⠁⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⢸⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠘⣿⡿⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⠲⠶⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⢉⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢾⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠖⠉
⠀⢾⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠁⢰⠖⠴⠶⠄⠘⠋⠀⣠⡞⣀⡴⠞⠁⢀⣠⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾
⠀⠸⠀⠀⠀⢀⣀⡀⣠⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⣧⠀⣀⣤⣄⠀⠀⠛⠛⠛⠋⠀⠚⠛⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤
⠀⠀⣧⠀⣠⡎⠀⠈⢿⡀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠴⠶⠶⠯⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠷⣄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢹⡆⠈⢷⠀⠀⠈⢧⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⠀⣠⣴⡀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⠀⠀⠈⠢⡄⢰⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⢹⡀⠈⣷⠀⠀⠘⣧⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣼⣷⡀⠀⠀⠀⡜⠁⢸⡀⡇⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⢻⡀⠘⣇⠀⠀⠹⡆⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠄⠛⠋⠀⠀⠞⠀⠀⠀⠀⢿⣿⡿⠀⠀⠀⠀⠧⠤⢘⣋⡛⢦⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠈⢧⠀⠘⣄⠀⠀⢸⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢊⡀⠌⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⡰⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠈⣦⠀⢹⡄⠀⠀⢻⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠐⠋⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠸⣇⠀⢱⡀⠀⠈⢧⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠻⠟⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⢿⡀⠀⠈⣧⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠘⢧⠀⠀⠘⡆⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠠⠤⠤⠤⣤⡤⠤⠔⠺⣿⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠘⣧⠀⠀⠸⡆⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠉⠁⠒⠀⢤⣼⣛⣢⣈⠷⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠹⡆⠀⠀⢻⡄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⣠⠴⠳⢦⡀⠒⣶⠟⠋⠉⠤⠤⠒⠓⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠶⣿⡄⠀⠀⢳⡀⢰⡄⠀⠀⠀⠀⠀⣀⣰⠓⠚⢁⠋⢀⠞⠁⠀⠀⠀⠹⢠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡂⢸⢿⡀⠀⠈⣷⠀⠻⡀⠀⠀⠀⠀⠀⢸⡇⠀⣼⠔⠁⠀⠀⠀⠀⠀⠀⢼⡿⣄⠀⠀⠀⠀⠀⢀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡏⠀⢳⡀⠀⠈⣇⠀⢇⡀⠀⠀⣀⣀⠈⣧⠜⠁⠀⠀⠀⠀⠀⠀⢀⡼⠀⢇⠙⢦⡀⠀⠀⢰⣁⡰⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⣷⠀⠀⠸⣆⠈⣥⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢸⠀⠈⠛⢦⣄⣀⣀⣠⠞⠃⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⣇⠀⠀⠻⡄⠈⡆⠀⠀⢀⠔⠀⠈⠣⣄⠀⠀⢀⠞⠁⠀⠀⠀⠈⣶⢠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠸⣆⠀⠀⢳⡄⠹⣀⡔⠂⠀⠀⠀⢀⠜⢳⠖⠋⠀⠀⠀⠀⠀⠀⠹⡎⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠹⡄⠀⠈⣷⡀⠉⠀⠀⠀⠀⠔⠛⠀⠈⠄⠀⠀⠀⠀⠀⠀⡠⠎⠙⣦⡙⠷⣤⣀⣀⡀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢛⠢⡤⠈⠀⠀⢀⡠⠊⠀⠀⠀⠷⠀⠀⠁⠂⠀⠀⠐⠂⠀⠀⠔⠈⠙⠒⢤⣬⠥⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣥⡬⠀⠀⠀⣠⣖⠈⢣⠀⠀⠀⢠⠀⠈⠒⠠⠤⠤⠤⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣥⡖⢠⡄⠠⠀⠹⡄⠈⡆⠀⠀⠘⠢⣤⡄⠀⢀⣀⣀⡀⣀⠀⢀⡀⢠⣤⢤⡴⢶⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠛⠚⣧⠀⠀⢹⡄⠸⡆⠀⠀⢠⡄⠀⠀⠀⠉⠉⠉⢿⣈⠀⢁⡴⠃⣠⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠘⡆⠀⠀⢷⠀⠻⡀⢀⡼⠄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⣠⠚⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠹⡆⠀⠀⢧⠀⢣⡄⢰⠆⢠⡄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠻⡀⠀⠘⣇⠈⢷⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢳⡀⠀⠙⡄⠈⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⣧⠀⠀⢹⡄⠸⡏⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⣧⠀⠀⢿⡀⠸⠗⠒⠖⠒⠒⠀⠀⢾⡇⠀⠉⠉⠉⠁⢀⣠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠸⣇⠀⠀⢷⠀⢻⡖⠒⠀⠠⣤⠤⠎⠙⡏⠉⠉⠉⠉⠉⠁
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠹⡄⠀⠘⢇⠀⢣⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⢻⡄⠀⠘⣄⠈⣆⠀⠀⠇⠀⠀⠀⢰⠀⠀⠀⠀⠀⡅
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢳⠀⠀⢹⡄⠸⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡇
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢷⠀⠀⣱⡀⣸⣇⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⢸⠷⠊⠉⠀⢀⣯⡀⠀⠀⠀⠀⠶⠀⠀⠉⠉⢹
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⣠⠖⠋⠉⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣇⢀⣀⣀⣨⡄
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⣿⣧⠀⠀⠀⣹
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠂⠈⠛⡄⠀⠀⠀⠀⢀⣀⡸⣶⡈⠁⠀⠀⠀⠀⢀⣌
⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠐⠀⣀⣀⣀⣸⡄⠀⡰⠞⠉⠻⠇⣿⠃⠶⠖⠚⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠈⠻⣀⣀⣀⡀⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⠉⠉⠁⠐⠒⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠂⠉⠉⠉⠉⠀⠒⠒⠒⠲⢦⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡇⠀⠠⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⢘⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢈⡇⡠⢤⣶⡤⠄⠀⠀⠤⢰⣦⡤⡀⢸⡀⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢹⣤⠇⠑⢿⣿⡷⠀⠀⠀⠸⣿⣿⠗⠁⠸⣃⡇⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢼⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠛⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠦⠄⠀⢀⠠⣤⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⡄⠀⠀⠀⢀⠇⠈⠳⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⠀⠈⠒⠒⠒⠉⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⡀
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠘⣄⡴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢦⣠⠋⠀⠀⠙⠆
⠀⠀⠀⠀⠀⢠⠁⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⡄⠀⠀⠘⡆
⠀⠀⠀⠀⣠⠃⠀⠀⡰⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⢆⠀⠀⠘⣄
⠀⠀⠀⢰⠆⠀⢀⡜⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠈⢣⡀⠀⠸⡆
⠀⠀⢀⠇⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠱⡀⠀⠘⡀
⠀⢀⠏⠀⠀⠸⢄⡀⠀⠀⠀⠀⠀⠀⢣⣀⣀⡀⠀⠀⢀⣀⣀⣀⠠⠤⢞⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⠹
⢠⣾⡄⠀⠀⡔⠀⠛⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠀⠀⠀⠀⠛⠐⢣⠀⠀⢠⡷
⠓⣿⣱⢸⡆⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢠⢰⡇⣞⣿
⠀⠈⠙⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠈⠋⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⣀⡀⣀⢀⡀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠉⢩⠉⠉⠁⠀⠈⠙⡋⠉⠉⠁⢹⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡃⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠒⠒⠳⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠔⠚⠉⠉⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠚⠂⠤⢀
⠀⠀⠀⠀⠀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠶⠆⠀⢀⣤⣤⣀⣀⣠⡜
⣤⣤⣤⣤⣤⣼⣶⣶⣶⣶⣦⣤⣬⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⠀⠐⠀⡀⠀⠀⠀⠂⠀⠀⠀⣀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠈⡀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠜⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠄⠒⠒⠚⠁⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠠⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠣⠀⠀⠐⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠯⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⡳⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⡂⠀⣀⣔⣐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠚⠳⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠘⠉⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⠁⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡅⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡇⠀⠀⠀⠀⠀⠀⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠣⢴⡖⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⢀⠂⠊⢅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⢢⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⡈⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠈⣤⠔⠐⣧⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⠤⠄⠀⠐⠒⠇⠀⠀⠀⠀⠙⢷⡤⢎⡠⢌⣁⡖⠋⠀⠀⠀⡜⠀⣰⠖⠛⠳⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠡⡔⠒⠠⡤⠃⠀⠀⠀⠀⣹⣶⡁⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢀⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠈⠉⠀⠀⠀⠀⠀⠀⢇⣀⠶⣇⣠⠴⠅⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡖⠗⠀⠀⠐⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣧⡀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠦⢄⠀⢀⠠⠾⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡪⠀⡠⠊⡁⠀⠈⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠁⠂⠀⠀⠀⠀⠑⠒⠤⡀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⡥⠚⠁⢆⠈⠠⡄⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠨⠤⠤⡌⠀⠀⠀⠀⣳⠀⠼⠖⠋
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠈⠉⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢈⠁⠀⠀⠀⢀⠀⠀⠸
⠀⠀⠀⢀⠀⠀⢀⢀⠠⡇⠀⠀⠀⠠⠀⠠⡀⢠⠃⡄⠀⠂⡀⠀⠂⠔⠀⠀⠀⢄⠀⡄⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠛⠉⠛⠛⠛⠛⠛⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⠀⣾⠛⢿⣿⣿⣶⣤⣤⣤⣀⠀⠀⠀⠀⠀⠸⣿⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⢿⠀⠟⠀⠀⠀⠙⠛⠫⣤⣤⠈⠉⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠛⠲⣤⣤⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠤⣶⣤⣄⢀⡀⢹⣦⠀⠀⠀⠀⠀⠹⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠉⠁⠀⠀⠀⠀⠀⣶⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠠⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠀⠀⠀⠀⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣿⡄⠀⠀⣄⠀⠀⠀⠀⣀⠀⠀⢲⠆⠙⠀⠀⠀⠀⠖⠀⠀⠀⣀⠀⠀⠐⢢⠀⠀⠀⣀⢀⣿⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣿⠁⠀⠀⠉⠀⠀⠘⢿⣿⣿⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠴⡿⠟⠁⠀⠀⠀⢘⠀⠀⠀⠀⠁⠀⠠⢿⡿⠛⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢰⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣠⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠻⠷⠶⠖⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠒⠲⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⢿⠇⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡶⠶⣦⣤⠄⢀⠀⠀⢀⣀⠀⣠⠤⣶⡶⠖⠛⣿⡏⢻⠆⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⢰⣿⠁⠀⠀⠀⠀⠀⢀⣿⡁⠉⠀⠀⠀⠀⠀⣀⣾⡏⣷⡏⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠏⣸⡆⠀⠀⠀⠴⠋⠀⠙⣀⠀⠀⠀⠠⠎⢹⣏⡹⠇⠀⠀⠀⠀⠀⠀⢠⡾⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⣿⡇⠀⠀⢰⡟⢡⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡴⢧⣼⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣿⠀⠀⠀⠈⢣⡜⢣⡄⠀⠀⠻⠀⠀⠀⠀⢀⣴⠟⣱⠎⠁⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⣿⠀⠀⠀⠀⠈⢳⡌⠙⢦⣤⣀⠀⢠⣤⡴⠛⣥⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠙⢦⣄⠉⠙⠛⠋⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⣀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢀⣾⠋⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠒⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⢡⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⢷⣶⣤⡄⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢸⣟⠙⣦⡤⣤⣄⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⣠⡿⠟⠁⠀⠀⢠⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⢰⣿⣀⣴⡇⣸⠄⣼⠋⣿⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠉⠉⠉⠻⣇⣀⣿⡀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠷⠶⢦⣄⣀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⠁⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣠⣤⣄⠀⠈⠙⢷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠾⠃⠀⠀⠀⠉⠻⢦⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠛⠢⣤⡄⠀⠀⠀⠀⠀⠀⢤⣤⡿⢿⠆⢼⣇⠀⠀⠀⠀⠀⢀⡼⠛⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣆⣀⣠⣴⠟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⡄⣀⣀⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠀⠈⠙⢛⣛⡁⠠⣴⡦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⣿⠀⠈⠉⠉⠉⠉⠉⠉⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⢿⡿⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠟⠈⡿⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣸⠁⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠛⠛⠛⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣿⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣤⣤⣲⣶⣶⣶⣤⣄⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣤⡴⠶⠈⣉⣉⣀⣴⣦⡀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣏⣉⡉⠁⠉⢉⣉⣻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡤⠶⠈⠉⠉⠉⠁⠶⠿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⣀⡀⣀⣀⣀⣀⡀⢀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠉⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣴⠇⠀⠀⠀⢀⢲⠀⠀⠀⠀⠀⠀⠀⣉⡐⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠘⡇⠀⠀⣾⣧⡀⠀⠀⠀⠀⠀⠆⠉⠙⠒⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠀⠀⠸⣷⡦⠼⠿⠟⠶⠿⠶⠖⠂⠀⠀⣥⡀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢰⠀⢀⠀⠀⠀⠐⠶⣄⠠⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡘⣧⣈⢺⢋⢗⣠⣷⠀⠐⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢾⡿⠃⠈⠉⠉⠛⢿⣿⡷⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠀⠀⠘⣿⣃⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠄⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠈⢶⣦⡀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⡀⠘⣀⠀⠀⠀⠀⠸⣿⠛⡿⢻⠇⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⣿⣗⣷⡀⠈⠀⠈⠀⢈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠋⠀⣤⠀⡀⠀⢠⠀⢀⡄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠐⠀⠀⠃⠿⠀⠀⠀⠈⠀⢠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣐⣠⣴⣦⣹⠃⠀⠀⠀⢀⡄⢡⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠛⢧⡀⠀⠂⢸⣧⣴⣿⣿⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⣶⣄⠀⠀⠀⠀⣤⣤⣏⠸⣿⣿⣿⣿⣿⡿⠈⠁⣴⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⢸⣿⡇⠀⠀⠀⣴⣿⣿⣿⣵⣍⠛⠟⠻⢿⠆⠀⡼⢃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣠⣤⢸⣿⣏⠀⠀⠀⠘⠉⢻⣿⣿⣿⣿⣶⣶⣶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠄⣤⠶⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠈⣿⣿⣧⢀⠀⠀⠀⠈⠉⠻⠛⠛⡟⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⣴⣿⣿⣿⣶⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠿⠙⠻⢟⡠⣤⣀⡀⠀⠀⠀⠀⠁⠐⠀⠀⠀⢀⣀⢰⡔⠋⣡⣾⣿⡿⠟⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢌⣘⠉⠑⠃⠶⠆⣾⠦⣾⠆⠟⠊⢋⡀⠔⠚⠁⢻⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠞⠆⠲⢠⠄⠀⠙⠠⠖⠓⠈⠁⠀⠀⠀⠄⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⢀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠨⡳⡏⠁⠸⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣦⡀⢹⣦⡄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢌⠳⢤⣴⣮⡐⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡿⢿⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠢⠼⣛⠓⠙⣿⣿⣦⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣀⠀⢴⣿⢿⡹⣷⣤⠿
⠀⠀⠀⠠⣄⣀⠀⣀⠤⠴⠀⠄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠀⠀⣄⣴⣿⡸⡄
⠀⠀⠀⣠⣶⣿⣿⡁⠈⠀⠰⠀⠀⠒⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⢐⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣠⠀⠀⡘⠙⡏⡇⢻
⠀⠀⢀⣵⣾⣿⣿⣿⡄⠀⠀⠀⢠⣾⠃⠀⣠⣴⣿⣿⣿⣿⡿⠟⠛⢦⡀⠀⠉⠉⠄⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⢈⠀⠘⠛⠋
⠀⠀⠘⠿⣿⣿⣧⡐⠄⢀⠀⠀⢉⣹⣶⣿⣿⣿⡿⢟⠫⠅⠀⣤⠀⠀⠈⠁⠀⠈⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣫⠕⠀⠀⡍
⠀⠀⠀⠀⠀⠉⠉⠉⠈⢳⣾⣿⣿⣿⣿⣿⠟⠅⣂⡡⢔⠛⠉⠻⠂⠀⠀⡀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⡴⠂⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⡈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⢀⣴⡏⠑⢆⣀⣀⣀⣀⣀⡼⠁⠀⢰⠇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠂⠬⡏⠉⢀⡀⣀⡈⡏⠉⠉⢩⠇⢀⡔⠃⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠤⠒⠊⠁⠠⠀⠀⠀⠀⠈⠁⠀⠸⣷⣧⣤⠤⣾⠴⠋⠀⠀⣴⠋⣼⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡿⠛⡁⠠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⣀⡤⠞⠁⠀⠀⢈⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠋⠀⣠⠄⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢀⡾⣇⠀⠀⣠⢾⡁⠀⠀⣀⡤⠚⠁⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠋⠁⠀⠀⣰⠁⡀⠔⠁⡼⠉⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⡼⠦⠚⠁⠘⡉⠛⠋⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠉⠉⠀⠀⠀⢀⠂⠉⠁⠀⠀⢰⠀⢀⠔⠀⠀⢴⡶⠄⠀⠃⢀⢈⡓⠀⠀⠀⠀⠟⠆⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠀⡠⠁⠀⠀⠀⠂⢠⡇⣀⠉⠀⠘⡭⠛⠶⠖⠀⢦⡜⡉⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⡤⢤⣤⣶⣶⣤⣼⡇⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣀⠅⠀⠀⠀⠀⡞⢠⠇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⡴⠛⠛⠛⠳⣞⣉⡠⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢿⢠⠀⠀⠀⢰⡇⢸⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠛⢟⠈⠳⡄⠀⢀⠖⠉⠉⠉⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⡏⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠼⢣⡀⠈⠉⠶⣙⠂⠂⠀⠈⢠⣴⠃⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⣼⢠⡇⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠻⡖⠒⠚⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡄⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡎⢈⡶⢧⡀⠑⢄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⡇⠀⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢃⡎⣠⠔⠻⣄⠀⢙⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⢸⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⡎⢻⠃⠀⠀⠈⠳⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⣽⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢲⠀⠀⠀⢩⣟⣿⠿⣿⠟⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠀⠂⠀⠂⠂⠁⠂⠃⠀⠀⠐⠐⠐⠀⠀⠀⠀⠀⠀⠁⠀⠁⠃⠄⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠕⠉⠀⠊⠀⠀⠀⠉⠑⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠁⠀⠀⢀⣀⠄⢀⠤⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⣠⢚⠍⠐⠈⠁⠀⣘⡄⡠⢆⠄⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠀⠀⠁⠀⣀⠀⠀⠀⠀⣀⠏⢁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⣢⠁⠀⠛⠀⠀⠀⠀⠛⠁⠠⡤⠤⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢈⣦⣤⣀⣀⢀⣀⠀⠄⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⡩⠉⢇⡿⡋⠃⠀⣠⡘⠱⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠠⠃⠀⡇⠀⠙⠀⠸⠙⠇⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⣠⠛⠆⢀⡇⣠⠀⠀⠀⠀⢠⣼⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⣠⠁⠀⣼⡿⣠⠃⠀⠀⠀⠀⠘⡻⡀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠃⠠⠪⢈⡔⠀⠀⠀⠀⠀⠀⠀⢃⠈⠒⠄⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⣁⠿⡦⢄⣀⠀⠀⠀⠀⢀⣸⠀⠀⠐⠁⢄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠁⠈⢀⡅⠀⠉⠉⠉⢉⣉⠁⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡀⠀⠀⡘⣏⠀⢀⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠈⠹⠉⠀⠀⢙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣀⣘⠀⠀⠀⠀⠘⡃⢀⡭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⢦⣘⣆⠀⠀⠀⢰⠄⠁⠒⠝⠶⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⠀⠁⠀⠑⡄⠀⠸⡄⠀⢄⣈⣀⣀⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣦⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠌⠀⠀⠀⠀⣴⣿⡿⠟⠋⠁⠀⠀⠀⠈⠙⢿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⣴⣾⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡆
⠀⠀⠀⠀⠠⠐⠿⢟⣁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⡀
⠰⣦⣤⣶⣶⣶⣿⣿⣿⠀⠀⠀⠀⡐⣶⣖⠀⠀⠀⣴⡆⠀⠻⣿⣄⠀⠀⠀⠀⢀⣠⣴⣿⣿⠿⠿⢛⡫⠃
⠀⠈⠛⣿⣿⣿⣿⣿⡿⠿⣇⠀⠘⣿⣿⣿⡵⣤⠀⠹⣿⢠⣼⣟⣿⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠈⢻⡟⣩⣽⣿⣿⣿⣿⣦⠹⣿⣿⡆⠀⠀⠀⠁⠸⡿⠟⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⢀⣴⠟⠋⠙⠿⣿⣿⣿⣿⣿⣧⣿⡛⠃⠀⠀⠀⠀⠖⣀⣤⣶⣾⣿⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠁⠈⢿⠉⠉⠀⠀⠀⠀⠰⣿⣹⣿⡿⠋⠛⠿⠷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣀⣠⣄⠀⠀⢀⣀⠀⣀⣾⢿⣿⣿⠆⠙⠐⠾⠿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣟⣿⣇⣠⢾⡇⠾⣿⣿⣶⣿⢿⡠⠄⠀⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⡟⢿⣿⣿⣾⠇⠀⠉⠉⢹⣿⣶⣷⣤⣶⠂⢹⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡌⢿⣿⣿⠀⠀⠀⠀⠈⠻⠿⠿⢿⡟⣰⣶⡷⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣎⢿⡏⠀⠀⠀⠀⠀⠀⣀⢾⡿⠖⠉⢀⣴⣿⣿⣧⡳⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠏⠛⠿⣿⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣥⣿⡄⠀⠀⠀⠀⠀⣴⡿⢾⣿⣿⣿⣿⠀⠀⠀⠈⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠻⢿⣷⡀⠀⠀⠀⣰⣦⣔⣿⢿⣿⡿⠃⠀⠀⠀⠀⠀⣠⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⣻⣷⣾⡄⡀⢹⣿⣿⣿⣷⡿⠶⣤⣄⡀⠀⡀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣴⣶⣿⣿⠿⠿⠿⢿⣷⣾⣿⣿⣿⣿⡇⢀⣇⠀⢩⣵⢣⢨⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠠⠹⠏⠉⠉⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢀⣾⣿⡀⠈⠛⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣿⣾⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣋⣤⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣵⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣽⣿⣿⣿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠉⠉⠉⠉⠈⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣀⠠⡀⠀⠀⠀⠀⠀⣁⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠛⠁⠀⠀⠉⠙⠶⠼⠀⠑⢆⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡴⠋⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⡀⠀⠣⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⢆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢨⠀⠀⠀⠀⠀⢸⣷⠆⠀⠀⠀⢸⣷⠄⠀⠀⠀⠀⠹⠆⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣱⠀⠀⠀⠀⠀⠀⠀⢰⣇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢠⣁⠐⠠⠄⠑⢊⡁⣀⡀⠀⠀⠀⠀⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠈⢿⡁⠀⢠⣤⣤⣤⡿⠃⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠉⠰⠖⠛⠋⠁⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢤⡤⢤⣤⠀⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠈⠀⠀⠀⠊⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠢⡀⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⢀⣀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠒⠉⠂⢄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠨⠁⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠐⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡄⠖⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⢄⠀⠂⢠⠐⠈⠁⠀⠠⠄⠂⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢠⠀⠀⠀⢈⠁⠀⠄
⠀⠃⠐⠂⠀⠃⠀⠀⠐⠂⠈⠁⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⢀⠐⠀⠒⠈
⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠳⢄⡀⠀⠀⠀⠀⡴⠚⠛⡆⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢹⠂⠀⠀⠠⡆⠀⠀⡇⠀⠀⠀⣀⠀⣀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠛⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⣇⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡏⠈⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡄⣀⣸⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠑⠶⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⢤⡤⠄⣀⡀⠴⠀⠀⠀⠧⣄⣀⣀⣀⣀⣀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢀⣸⡀⠀⠀⠀⠀⢈⣀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠍⠈⠀⡇⠀⠀⠀⠀⢠⠈⠈⢡⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢎⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡜⠀⠀⠀⠀⠀⠙⠲⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⣀⠀⠀⠀⠈⠉⠀⠀⠊⠉⡴⠀⠀⡴⠓⠀⠀⠀⠀⠒⠒⠉⠀⢀⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣛⣒⣒⣒⣒⣒⣋⣁⣀⣀⣛⣤⣤⣤⣤⣤⣤⣤⣔⣊⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠿⢻⠷⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡤⢶⠾⢾⣆⠀⠀⠀⢀⣴⣿⣿⣿⠃⠈⠃⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢠⠴⢾⠿⠾⠋⠀⢀⣤⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡳⠠⠤⢦⣤⣶⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⠀⠀⢺⣿⣌⣻⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⡄⢸⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠘⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡋⠀⢿⣿⣿⣿⣿⣿⣿⣷⣤⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⢀⣠⣤⣤⣴⣶⣶⣶⣦⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⢠⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣴⣾⠿⣯⠿⢿⣿⣿⣿⡟⢛⢿⡿⠻⢤⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⢀⣿⣷⠎⠈⠁⠀⣼⣿⣿⠋⢁⣟⣉⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⢿⣿⣿⣿⣿⣿⠀⠠⠙⠃⠀⠀⠀⢠⣾⣿⣿⣿⡁⢚⣵⣾⠞⠁⠙⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡇⠀⠀⠁⠀⠀⠀⢀⣿⣿⣿⣿⣿⣾⣿⡗⠁⠀⠀⠀⠀⠉⠪⣽⣿⣤⡤⠴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣄⠀⢀⡠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣛⣈⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣟⣥⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⡸⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠉⠀⠀⠓⢠⣈⡤⢶⣿⣿⣿⢋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠚⢉⣀⠈⣻⡯⠰⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡑⢺⣿⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣯⣴⣾⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠛⠿⠿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡠⠤⠤⠤⠤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⣠⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⣠⠞⠡⠠⠤⡜⠢⣀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⡀⠈⠉⠉⠉⠁⠀⡌⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⡇⠀⠀⣀⠒⠒⠀⡇⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠀⣷⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠐⠒⠒⠒⠒⠒⣉⠀⠐⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡖⣴⠀⢸⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣈⠁⡇⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠰⣰⠀⢸⠁⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⣛⡂⠀⢡⠒⡞⠀⠀⡞⠀⠀⠀⢸
⠄⠀⠀⠀⠀⠤⠄⠀⠄⠄⣤⡌⣤⡀⠀⠘⠓⠃⠀⡄⢠⠇⠀⢀⡇⠀⠀⠀⠸⡄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠁⠄⠀⠀⠀⢠⢇⡸⠀⠀⡸⠀⠀⠀⠀⠀⣇
⠤⠠⠤⣴⣦⣤⣤⣤⣤⠤⠼⢴⣦⣤⣤⣀⣀⡀⠈⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠉⠉⠁⠛⠒⠄⠙⠁⠉⠺⠁⠖⠂⢴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡡⣤⡤⠶⠒⠒⠂⠀⠀⠀⢀⡤⠰⠆⠤⠄⠀⢼⠛⣹⠀⣀⠤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠟⡥⠊⣁⠤⠐⠒⠈⠉⠉⠉⠐⠊⠀⠀⠀⠀⠀⠀⠀⢿⣽⡔⠡⠲⠀⠸
⠀⠀⠀⠀⠀⠀⠀⢀⣎⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⢀⡶⠀⠀⠀⠀⠀⠙⠀⣀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠀⠀⠐⠢⡀⠀⣴⡶⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠩⠶⠀⠀⠈⢱⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠨⠧⢄⡠⠔⠓⠤⣤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⠀⠈⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⢀⠀⠈⢂⣰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠀⠀⠐⠀⠀⠀⠀⠀⢤⠚⠋⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⠀⠤⠄⠒⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡄⠀⠃⢀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡇⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢈⣿⣿⣿⣿⣿⣇⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡾⢷⠀⢀⡠⢒⣒⡲⠞⣡⡀⠛⠏⢺⣿⣿⣿⣿⣿⡟⠛⠿⠓⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠿⣯⣽⠶⡿⠀⣿⠛⢷⡌⢉⣿⣧⠀⡴⠝⢿⣿⡿⠟⣓⣂⡰⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠤⣾⡏⣤⠀⠀⣇⠀⢀⣘⡃⢀⡀⠘⠣⣴⡃⢘⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠛⠀⠤⣄⣼⡀⢈⣟⠀⠈⣛⡁⣉⡍⠠⠤⢾⠈⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢰⠇⠉⠉⠀⠉⠉⠀⠀⠁⠄⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⡠⣆⣀⠀⠈⠑⠛⢿⠓⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠥⠤⢤⣀⡀⠀⢹⣌⠙⠳⢿⣿⣶⣦⣦⣾⣠⣄⣰⣴⣀⣴⣶⣶⣶⣾⡯⠽⡶⠖⠒⠂⠀⠒⠒⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠙⠓⠒⠒⠒⠒⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠐⠂⠐⠀⠀⠂⠂⠂⠀⠃⠂⠁⠀⠀⠐⠀⠐⠀⠀⠀⠀⠈⠀⠁⠘⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠉⠉⠛⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⠚⠉⠀⠀⣠⣼⠀⠖⢶⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠗⠀⠀⠀⣀⣜⠁⠈⢿⠀⢈⣆⠀⠀⠀⠈⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡖⠆⢠⠾⠥⠤⠇⠀⠈⠑⠣⠼⡆⠀⡀⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⣎⠀⢰⠇⠰⡟⠛⡆⠀⠀⠀⣾⣿⡿⣄⠑⣄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣖⢢⣸⢹⡀⠀⠀⠀⠈⠛⢲⠛⠀⠀⠉⠈⠁⠴⠲⠆⠉⠉⠁⠉⢳⡾⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣃⠘⠂⠙⠸⡇⠀⠀⠀⠀⠀⠀⠳⣦⡀⠀⠰⡖⠒⠲⠂⡶⠂⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠁⠀⠀⠀⠙⢩⠞⠁⠀⠀⠀⠀⠈⠙⢦⣄⠈⠁⠀⠈⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀⣀⣀⣠⣬⡟⠻⢿⣟⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢱⠀⠀⠀⢀⡴⠋⠁⠀⠹⣷⠤⣀⣀⣀⡬⠟⠛⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⣦⣴⣶⠋⠀⢀⡀⠀⠀⠀⠉⠓⠛⠉⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⠈⢿⣧⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢀⣀⣠⢿⣦⠋⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⢶⠀⣀⢤⣬⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⢺⣫⠔⠋⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⡄⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢻⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡹⠛⠐⠒⢲⠒⡖⠛⠉⢋⠉⠙⡏⠹⡄⠀⡼⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠑⣤⡇⠀⣿⠋⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠇⢃⣻⡆⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠘⡄⢿⡝⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏⣇⠀⠀⠀⠀⠀⢱⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢤⠤⣠⣠⣤⣴⠃⠹⣤⣤⣤⣤⠤⠤⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢸⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡾⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠤⡀⠴⡇⠀⠀⠀⠀⠀⠀⣧⠤⠴⠚⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⢯⡙⠦⠤⠤⡆⠀⠀⠀⠀⠀⠀⠙⡒⠤⠞⣩⡽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠚⠙⢍⣢⠥⠴⠋⢀⣴⠞⡧⠀⠀⠀⠀⠀⠀⢸⣗⡌⢪⠐⠯⣔⡹⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⡢⢤⣤⠤⢔⠒⣒⣋⣉⣉⠇⠀⠀⠀⠀⠀⠀⠈⠓⠻⠶⣦⣄⡀⠀⣀⣠⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⣿⠟⠿⣿⣦⡀⠉⠛⢿⣦⠘⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢣⡿⠘⠃⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠘⠃⠀⠀⠀⢻⡄⢠⡀⠀⠀⠀⠀⠀⣴⡀⢀⠀⢸⣆⣻⣦⣻⡄
⠀⠀⠀⠀⠀⠀⠀⡟⠛⠀⠀⠀⠀⠀⣧⡀⢻⡌⢷⣄⠀⠀⠀⢀⠘⣷⣜⣷⣄⣹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢰⡦⠀⠸⣷⣀⠻⣦⣿⣧⣄⠀⢸⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠰⣆⣴⣾⣿⠀⠀⠈⠻⣿⠞⠻⣿⣿⣿⣮⣿⣿⣮⣙⢿⣿⡟⠿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠻⠿⠛⠋⠀⠈⠁⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠋⠀⠀⠀⠀⠀⠀⣠⠔⣶⣤⡀⠀⠀⠀⢠⠄⢀⡤⣶⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢾⡰⣄⠀⠀⠀⠀⠀⠀⠀⣀⣉⠉⠀⠀⠀⠀⢾⠀⠀⠀⠘⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣍⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠃⠀⠈⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠠⠶⠦⠤⠤⢤⣶⠆⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠊⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠉⠙⠓⠶⠤⠤⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢬⣁⡒⠲⠦⠤⠤⣤⠠⠤⠖⠚⢋⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠲⠶⠶⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣴⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠀⠀⠀⠛⠃⠀⠀⠀⠉⠁⠀⠀⠸⠛⠀⠀⠀⢴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢠⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⣄⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠀⢀⣀⣄⠐⠛⡏⠀⠳⣤⣁⠀⡇⠀⢿⠀⡠⢛⠀⠀⣶⠀⡀⠀⠀⠀⠀⠀⠀⠈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣄⣨⣄⠀⢳⠀⠠⣤⡼⠰⠇⠇⠸⢠⠓⢾⠀⡜⢹⣰⠁⡴⠲⡆⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢯⠀⠙⢦⠘⡄⠈⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠸⢃⡄⠹⡆⠁⡠⠂⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⣴⠛⢢⡈⠳⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⣡⠞⢳⡤⠋⣀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣠⠤⠀⠑⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡤⠚⢁⡤⣄⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠞⠙⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠚⠳⠤⠞
⠀⠀⠀⠀⠈⠛⠓⠶⠦⢤⣄⣀⡀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠊⠉⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⠀⠐⠂
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠶⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠸⠻⠀⠀⠀⠀⢠⡔⠺⡅⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⡀⠀⠀⠀⠳⠂⠀⠀⠀⠀⠀⠀⠀⣠⠤⠬⠦⣄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⢀⡼⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⢃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠧⣄⠀⠀⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⢺⠗⠀⠀⠀⠀⠠⠞⠁⢀⡴⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡉⠢⣀⠈⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢙⡶⢋⡔⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠻⣀⡼⢤⡈⠓⠀⠀⠀⠀⠀⠹⠟
⠀⠀⠀⠀⠀⠀⠐⣾⠆⠀⠀⠀⠀⠀⠋⡠⠋⡰⠋⠈⠇⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠄⢸⠁⢀⣄⠈⠣⠀⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢧⡠⠀⠀⠳⡄⠁⠀⠀⠀⠀⡆⢰⡄⢰⠘⢻⠁⠘⣦⠶⠀⠳⢤⡼⠃⠀⠀⠀⠀⠀⠀⠹⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠃⠀⠀⠒⠀⠀⡇⢸⠘⣼⠀⠀⣇⠀⠹⠤⠖⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠸⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⣾⡆⠀⠀⢀⣤⡀⠀⠀⢤⣦⠀⠀⠀⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠶⠞⠛⠛⠛⠛⠛⠻⠶⢶⣤⣄⡀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⡇⢹⣇⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠸⠟⠁⠀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠋⣿⣀⡀
⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣤
⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⣿⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡞⢹⡇⠀⠀⠀⠀⣴⠛⣇⠀⠀⠀⠀⡼⠹⣇⠀⠀⠀⢀⡾⢧⣤⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣤⠦⣄⡀⠀⠘⢷⣄⣀⣤⠾⠋⠀⠈⢷⣀⢀⣠⡾⠁⠀⠹⢦⣀⣤⠞⠁⠀⠙⢦⣠⡴⠟⠀⠀⣸⠇⠰⣿⣹⠆⠀⡀⠀⠀⢀⣾
⠙⠒⠛⠃⠀⠀⣾⠁⠀⠀⠀⠐⢦⣄⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⣤⡀⠀⢸⣇⣠⣤⣼⡏
⠀⣤⣶⡶⠀⠀⣿⠀⠀⠀⠚⠶⠶⢮⣽⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣶⠶⠆⠀⠀⣇⠀⠘⠳⠟⠀⣾⠉⠀⠀⢹⡇
⠀⠛⠋⠀⠀⣴⠏⠀⠀⠀⠀⣴⠶⠖⠛⣯⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠙⢿⠳⠶⢦⣤⣄⠀⠀⠹⣆⠀⢀⣠⡾⢧⠄⠀⣀⣼⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣏⠀⠀⣰⠏⠀⠀⠠⠞⠋⢉⠻⡦⠀⠀⠀⣾⠁⠀⠀⡞⠁⠀⠀⠀⠉⠛⠛⣡⣤⣴⣶⠿⠛⠁
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⡟⠀⠀⠹⡄⠀⠀⠀⠐⠂⠙⠂⠀⠀⠀⠀⠹⣆⠀⠀⢹⡄⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⢈⡧⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⣞⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⢸⡅⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⡼⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⢷⣤⣤⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣵⣴⣶⣾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠻⣿⡛⠻⠶⠶⠶⣶⣶⢶⣶⠒⠛⠛⠉⠉⠙⣷⡍⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠈⢿⣄⣀⣠⡾⠋⠀⠀⠙⠳⣤⣀⣀⣤⠾⠋⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⢠⣶⠚⠃⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠘⠳⣶
⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠛⠛⣿⠿⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠾⣿⠛⠓⠀⠀⠀⢀⣿⠉
⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⢸⡇⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⢸⡆⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⡀⢸⡇⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡧⠀⠀⢸⡇⢀⣀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡾⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⠶⠶⢤⣤⣤⣠⡾⠋⠻⢦⣤⣤⣤⠴⠶⠻⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⢻⡇⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠳⠶⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⠟⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣶⣤⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠟⠃
⣿⣿⣿⡿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⢿⣯⣗⣻⣦⣤⢴⣟⡛⣽⠿⣿⣿⣿⣿⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⢿⣿⣿⣭⠴⣿⢯⡽⣃⣧⣿⢡⣽⡿⣛⡴⢋⡤⠆⢈⠛⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠞⠛⠙⠫⠎⡵⢋⣾⡿⢋⣴⣫⡼⢹⣿⠾⠋⡴⣊⡵⠊⠀⠀⠀⠈⢧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠻⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠐⠋⠟⣰⠞⣽⠟⣬⢾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢸⣿⣿⣿⣸⣷
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣷⠀⠀⠀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣤⣦⣤⣄⡀⠀⢰⣝⡄⠀⠀⠀⠀⣀⣤⣤⡐⠹⠿⣿⡷⠿⣟⡿⠁⠀⠀⠀⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⠛⠋⠉⠁⢠⣿⣿⣿⡄⠀⣠⣾⣿⣿⣿⣿⣷⣤⡀⠀⠀⠈⠑⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⣠⣾⣆⣀⢸⣿⣿⡿⢀⣴⣿⣿⣿⣿⢿⣟⠫⠈⠙⠮⠵⠤⢠⡀⠀⠀⢀⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⢠⢢⣼⣭⠇⠀⠀⠀⣀⣴⣿⣿⣿⣿⡟⠸⠋⣠⣿⣷⣿⣿⣿⠗⣾⣌⠃⠀⠀⠀⠀⠀⣼⠏⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣀⣤⠁⠺⠀⠀⠀⠀⠘⠛⠀⢀⣴⠟⢹⣿⡿⢿⣦⣿⠁⠀⣴⣿⣿⣿⣿⡿⢁⣠⡔⠃⠀⠀⠀⠀⠀⡼⠁⠀⢀⣼⣇⠹⣿⣿⣿⠂
⠀⠀⠀⠀⢠⣿⡟⢻⠀⠀⠀⠀⠀⠀⠀⠀⠚⣋⠠⠾⠛⡶⢿⣿⡇⠀⢰⣿⣿⡿⡿⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⣤⠷⠷⠆
⠀⠀⠀⠀⣿⣿⡧⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠉⠁⢀⣾⠏⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⣠⣴⠄
⠀⠀⠀⠀⢿⠃⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠇⠀⠀⠀⡾⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢻⡾⠋⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣄⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣤⣤⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠛⠦⡙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⣴⣧⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⣼⠿⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠠⠾⣿⣿⡁⠀⢀⣤⣿⣉⠙⣿⣿⣿⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠑⠂⣙⡟⠋⠉⢰⣿⣿⠟⣁⣴⣾⣿⣷⣤⠄⣀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⢀⣾⡿⠁⢀⣿⠶⠀⣀⣀⡨⡵⠖⣉⢠⣟⣿⠻⢿⣿⣾⠟⠁⡠⠀⠀⣼⣇⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣴⠛⠉⠁⢶⣏⡁⠀⠚⠋⠉⣄⣵⣤⣿⢿⣿⠃⠀⠀⠉⠉⢈⣁⠀⠀⡰⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢸⡇⠀⠀⠀⠉⢿⣿⣾⣿⢿⠟⠛⠛⠛⠚⢻⣿⣷⡀⣠⠴⠚⡡⠚⠀⣤⣹⣿⠋⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⠀⡀⠀⠀⠙⠇⠆⠀⠀⠀⢀⣀⣼⣿⣿⡿⠛⠛⢦⣞⣠⣿⣾⣿⡿⠃⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣷⠀⠀⠀⠈⠛⠻⠟⠻⣿⠟⠛⠉⠀⠀⠀⢀⣼⣿⣿⣿⣿⠟⠁⣠⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠁⣀⣼⣿⣽⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⢻⣟⣉⣀⣠⠾⠿⠿⠿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⢶⠶⠶⠾⠿⠟⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢇⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠛⠓⠶⠊⠁⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠀⠀⠀⣴⠀⣠⣀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡈⠁⠀⠀⠀⠀⣠⠞⢳⣠⠁⠘⡄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⢀⣤⠾⣅⠀⠀⢻⡀⣀⢽⡄⠀⠀⠀⠀⢸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡞⠛⠀⣰⠿⠄⠶⠞⠀⠀⠀⠉⠙⠂⢿⡀⠠⡀⠀⢸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⠀⣼⠁⢰⡿⠛⢷⠀⠀⠀⠀⣴⣿⢿⢳⡄⠙⣄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⢤⡀⡎⢳⠀⠀⠀⠀⠀⠐⠿⢤⡴⠻⠀⠈⠛⠀⠛⢀⡤⣤⡄⠛⠛⠛⠊⠛⢦⣼⣦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠹⣆⠹⡇⢸⡇⠀⠀⠀⠀⠀⠀⠈⢧⣘⡀⠀⠀⠀⣄⣀⣀⣀⣀⣀⡄⠀⠀⠀⣸⣧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠲⡄⠀⠀⠀⠈⣇⣠⠀⣤⠀⠀⠀⠀⠀⠙⠿⣄⠀⠀⠙⢅⠉⠉⢡⠞⠃⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠐⠂⠀⠀⠈⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⣀⠀⢀⡀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⢤⣿⠛⠻⢿⣏⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠻⣷⣤⣀⣀⣀⣀⣩⡽⠟⠛⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣇⢀⣤⣤⠞⠁⠀⠀⠀⠀⠀⠈⠓⠺⠭⠿⠛⠁⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⠙⢿⣿⡄⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠘⠀⠀⠻⣷⡀⣸⣃⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⣀⣤⡤⠶⠻⣷⣋⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⣿⣀⡤⢶⡶⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⡎⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣡⠖⠉⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⣇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣠⣼⠁⠀⢸⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠜⠁⠀⠀⠁⢻⠉⡏⠀⠀⠀⢦⡀⠀⡇⠀⢻⡀⣀⠼⠁⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠙⢦⣇⠀⢸⣟⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢦⣀⠀⠀⠀⠀⠀⠀⢹⠸⡀⣿⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠈⡆⢸⣿⠳⠤⠖⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡾⣧⠀⠀⠀⠀⠀⠀⠸⡀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⢣⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢳⠒⠶⠶⠶⣶⠾⠁⠈⠳⠶⢶⠶⠶⠖⠒⠚⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠘⣇⣀⣀⣠⠼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠯⡉⠉⢉⣉⡄⠀⠀⠀⠀⠀⠀⠀⢻⣍⡁⢀⣠⠟⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣟⢩⡫⠑⣆⠉⠉⣉⣩⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠨⣉⠥⢲⣏⠍⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⠁⠈⠲⠼⠖⠒⠋⠁⢀⣾⣟⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⣯⢵⡈⠦⠉⠓⠾⠥⠚⠓⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢍⡲⢶⣶⠒⣒⣚⣛⣉⣉⣉⣭⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠷⣶⣀⣤⠀⢀⣀⣤⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠥⠔⠚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣤⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣶⣿⣿⣿⣿⣿⣷⣅⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⣿⣿⠁⠀⠈⣿⣿⣿⠟⠉⠀⠙⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠺⡝⠉⢄⠘⠀⠻⠛⠁⠀⠀⠀⠀⠀⣿⣿⡇⠀⡌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠮⠿⠁⠀⣦⠉⠖⠐⡤⢄
⠀⠀⠀⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣱⠈⠀⠀⠀⠀⡇⠀⠓⣰⠀⣋⡆
⠀⠀⠀⠀⢀⡼⠡⣻⡿⠶⠦⠤⣄⡀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⠉⠁⡄
⠀⠀⠀⡠⠋⠀⠾⠁⠀⠀⠀⠀⠀⠉⠲⣿⠷⢄⣀⡠⠤⠋⠉⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢸⠃⠀⠀⠀⠀⢀⣀⣀⣤⣠⡀
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠀⠀⠀⡄⠀⢀⣠⠴⡏⠉⢄⠀⠀⠀⢀⣅⣤⡀
⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠼⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠒⠲⣆⠴⠋⠀⠀⠘⡄⠈⡄⠀⠀⠀⢠⡤⠜
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⢀⠇⠀⢀⡤⠚⠁⠀⠀⠀⠀⢀⣧⠰⠧⠀⠀⠀⡊
⠀⢸⠀⠀⠀⠀⠀⠀⣄⣆⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⣼⡤⠚⠁⠀⠀⠀⠀⠀⠀⡔⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠈⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠚⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⣿⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⢹⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡿
⠀⠀⠘⣯⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⡦⢤⣀⠀⠀⣠⠞⠁⣸⠃
⠀⠀⠀⢸⣆⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠠⠤⠤⣷⠀⠈⠙⠻⠤⣤⡴⠏
⠀⠀⠀⠀⣿⣧⠑⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠉⣉⡉⠁⠀⠉⢳⡀
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣧
⠀⠀⠀⠀⢸⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣸⡄
⠀⠀⠀⠀⢸⡇⠀⠘⠢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡇
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⡇⢹⠄
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠙⠛⠾⣦⡀
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠛⠒⠒⠒⠼⢧⡀
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣶⣶⣶⣤⣄⣸⡄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣀⠤⢂⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⢿⡛⣿⡆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣈⣉⣉⣉⣉⣉⣁⣁⣉⣉⣉⣉⣉⣉⣁⣈⣁⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⢠⡟⣠⠴⠚⠉⠉⠉⠉⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣽⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠹⡆
⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⣤⣴⢻⣦⣤⡄⠹⣄⠀⠀⠀⢿
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗⢰⣷⡾⠿⠉⠁⠘⠃⠀⢹⠶⣿⡆⠀⠀⠈⡇
⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⢰⡆⢠⠃⡿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⠀⠀⡇
⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⢸⣷⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠸⢙⠀⣸
⠀⠀⠀⠀⢱⣿⡄⠀⠀⠀⢀⡞⠁⠀⠀⣶⠄⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⣼⣡⠏
⠀⠀⠀⠀⠈⣧⠻⡄⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠘⢧⡀⠘⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣌⣣⡀⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠚⠁⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣶⡤⣀⣀⠀⠀⣀⣀⡤⠚⢿⡄⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠹⣦⣀⡉⠉⠉⠀⠀⢀⣠⡿⠁⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠚⠋⠁⠀⠀⠀⠀⢠⡀⠀⢀⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠸⣟⡒⠤⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠉⠉⠛⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣿⣦⡤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⡾⠿⢻⢢⣀⡟
⠀⠀⠀⠀⠀⠀⢿⠀⠶⡆⢸⠃⠈⠙⠛⠻⠿⠞⠛⠛⠿⠿⠿⠿⠟⠛⠛⠋⠉⠀⠀⠀⠸⣟⣸⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⣱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⡄⠀⠀⠀⠀⠀⠀⠀⠛⠓⠂⠐⣶⠒⠚⠋⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣀⣤⡤⣤⣄⡀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⣠⠴⠚⠛⠛⠛⢦⣼⠋⠀⠀⠀⠈⠙⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣦⣤⡀⠀⣠⣰⠿⢶⣶⡶⣶⣖⣋⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠓⠉⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠹⢒⠉⠍⠉⠓⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢀⢖⠀⠀⣀⠀⠀⠀⣶⠀⠀⠀⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⡼⠈⠓⠶⠶⠗⠛⠿⠥⣶⡃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣸⠁⠈⣀⠀⠀⠀⠀⣀⠁⠀⣅⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠀⠘⠿⠀⠀⠀⠀⠿⠇⠀⠸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣫⡽⢳⣤⣀⣀⣀⣀⣤⢲⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⠞⠁⡆⢸⠑⠬⣤⣠⡥⠿⠀⡆⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢷⣯⡀⢠⡇⡆⠀⠀⠘⠃⣄⣰⡇⣧⢀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⠁⢹⣾⣸⠇⠀⠀⠀⠀⢁⢀⣿⣿⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⠟⢿⡏⠀⠀⠀⠀⠀⠀⠁⠘⡿⣆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢿⠃⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡷⡿⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⣤⡼⢀⣷⣄⣀⠀⠀⠀⠀⠀⣀⡤⣶⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢻⠂⠉⠉⠉⣉⡉⠉⠉⠐⠁⠀⢭⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣷⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣤⡼⠉⣷⠠⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⣶⢟⠀⡟⢵⣾⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣀⣸⡶⣷⣉⣀⣉⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠤⣀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⢈⡀⠉⠓⠈⠁⠀⠙⠘⠋⠐⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⢥⡶⠚⠋⣉⣁⠤⢤⠀⣀⠖⠈⠁⠒⠀⠐⢅⢰⣀⠔⡃⠐⡄
⠀⠀⠀⠀⠀⠀⠈⡕⣁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠻⠃⠈⠉⢠⠁
⠀⠀⠀⠀⠀⠀⣼⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⡀⠀⢠⣄⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠷⠤⠄⠐⠣⠤⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡇⠀⠀⠀⠀⠀⠈⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠇⠀⠀⠀⠀⢀⠴⢃⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢀⡀⣀⡀⠠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢀⡔⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⣟⣿⣷⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣛⣿⣽⣷⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⠁⠀⠀⣀⣀⣠⡤⠒⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠨⡧⣴⡋⣾⠳⡅⠘⢣⠆⠀⣼⠻⡿⣿⣼⣦⠀⠁⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢧⢠⣶⢀⠀⠀⡇⠀⢀⠂⠁⠀⠐⢠⣔⠀⣾⣖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⢄⡰⠀⢈⡏⠀⣹⣋⣉⡅⠠⠄⡏⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⣤⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣔⡒⠒⠚⠒⠲⡖⠺⢽⣶⣤⣤⣀⣸⢀⡀⣀⣀⢀⡀⣀⣠⣤⣴⣚⣆⠤⠤⠄⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠺⠤⠠⠄⠠⠐⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⣛⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⠄⠀⠈⠃⠀⣀⡆⠀⠀⠀⠤⠀⠠⠀⠀⠀⢀⣴⣶⣶⠤⣄⠀⠀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠄⠁⠛⠿⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣄⣸⡆⡸⠋⠉⡩⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢐⢎⡀⠈⠃⠀⠀⠀⠀⠠⠀⠀⠐⠤⠉⢿⣿⠿⣻⠃⠀⠐⣆⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣌⠛⠖⠊⠀⠀⡀⢉⡷⠈⢠⣾⣶⠀⣴⠀⠀⠀⣠⠞⠛⠻⡷⡌⣏⣴⣌⠙⢿⠁⣙⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠜⠛⡃⠀⠀⠀⠘⣦⣀⠘⣢⣾⣿⣿⡭⠁⠀⠀⣾⡏⡀⠀⢀⣖⠀⣿⣤⡴⠙⣤⡻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⠃⠀⠀⣰⣿⣷⠥⢄⢠⣹⣡⠻⣽⣧⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⡁⠈⠉⠁⠀⠀⠀⣸⣿⣿⡟⢈⣿⣽⣿⠀⠀⠰⢿⡿⣿⠟⠚⣿⣿⠆⡀⣶⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣧⢀⡀⠀⠀⠀⠀⢹⣿⣿⣧⣾⣿⣿⣿⣇⠀⢰⣿⢟⡟⠀⣼⣿⠃⠀⢡⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣯⡿⠿⠿⠓⠀⠐⠘⣿⣿⣿⡿⣿⣿⣿⣿⡄⣿⣿⡿⣃⣾⣿⣿⣷⣧⣀⣾⡆⣈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡇⡀⠀⠀⠀⠀⠆⢹⠿⣻⣿⡇⠈⢻⣿⣿⣧⢿⣙⡇⢉⣉⣝⣹⣅⡉⠈⣉⣁⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠒⠈⠛⠳⠾⠿⠿⠦⡄⣾⣿⣿⡟⠁⠀⠸⣿⣿⡟⠿⢿⣯⣯⣭⣽⣯⣍⣉⣩⣭⣉⣹⣇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⣿⣿⠟⠓⢶⠛⠁⠻⢿⡿⠿⠁⠉⠈⠁⠤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⣿⡟⠀⣠⣦⣤⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠈⠛⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⣿⣿⣿⡇⠈⠉⠁⠀⣀⠀⠀⠀⢿⣿⡟⠛⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠈⠈⢿⣿⡿⠇⠀⠈⠁⠊⢀⡀⠀⠀⠘⠛⠀⠠⣿⠃⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠄⡄⠀⣜⠙⣧⠀⠠⡤⠀⠀⠋⠀⠀⠀⠀⠀⠀⠺⣿⠀⠄⠇
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⢪⡠⠊⠉⠹⠁⡀⠈⠰⣁⡀⠀⠀⠰⠀⢀⠀⠀⠀⠘⠀⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔
⠀⠀⠀⠀⠀⠀⠓⠀⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⢀⣀⡼⠴⢥⡔⠀⠐⠢⡀⠀⠀⢀⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⢶⠀⠳⢄⣂⠀⠀⠀⠀⠀⢀⡤⢠⠖⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠑⢤⠁⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠉⠒⠒⢖⡊⠁⠈⢺⠀⠀⠀⠀⠀⠆⠀⠸⠀⠀⠀⠀⢸⠳⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⠀⠀⠉⡗⡀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⢀⠎⠃⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠓⠉⠒⢤⡀⢱⠀⠀⠀⠇⠀⡌⠀⠀⣠⡎⠑⡦⠄⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠀⠀⢰⠤⠊⠀⠀⠈⠀⣹⠀⣃⡠⢈⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠋⠒⠀⣞⠢⣀⡠⠄⠐⠊⠁⠀⠀⢠⣊⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠉⠐⠒⢤⡀⠈⡛⣴⠂⠀⠀⠀⠂⠀⠀⠀⠒⠂⠀⠉
⠀⠀⠀⠀⠀⠀⣠⡇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠉⠉⠉⠉⢻⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣒⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣇⢹⣋⣛⣛⣙⣫⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⣠⣄⠀⣲⠤⢂⠀⣀⣀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠟⠁⠂⠀⠀⠀⠀⣀⣲⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠰⠀⠀⠜⠻⠃⠉⠈⡙⡙⠁⠀⠀⠷⠒⠙⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⡉⠁⠀⠀⠀⠉⠁⠀⠒⠀⠀⠀⠠⠄⡄⠀⠀⠈⠑⡀⠀⠀⠀⠈⢐⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠴⠀⠀⢠⠎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠇⠄⠀⠐⠀⡠⠃⢤⢄⡀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡖⢀⣃⠀⠈⠘⠁⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠦⠀⠀⡀⠀⠀⠐⠠⠤⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⡈⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠀⠀⠁⠐⠀⠀⠃⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⠔⠁⠀⠀⠀⠠⡴⢆⣈⠆⠀⢀⠤⡄⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠣⠉⡘⡄⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢢⠦⠀⠀⣄⡀⠀⢀⠆⠀⠀⠀⠀⠀⢤⡴⠤⠀⠀⠀⠀⠀⠀⠐⠀⠐⢤⡤⡀⠤⢀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠐⠁⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠊⠈⠀⠁⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠛⠀⠀⠀⠀⢀⠒⠐⠒⠊⠘⠀⠀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣄⣀⣠⡶⠦⢤⣤⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⢀⣀⠀⠉⠁⠀⠀⠀⠈⠁⠙⠉⠸⣷⣢⡄
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢔⡶⠚⠋⢉⣉⣀⣀⣀⡀⣀⠜⠊⠁⠒⠂⠀⠘⣅⣰⡃⡠⠒⠈⢲
⠀⠀⠀⠀⠀⠀⠀⢋⠔⢁⠤⠒⠉⠁⠀⠀⠀⠀⢰⠁⠀⠀⢀⡀⠀⠀⠀⠘⢾⡟⠰⠛⠁⢸
⠀⠀⠀⠀⠀⠀⢀⣮⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⠢⡀⠞⢁⣀⡀⠀⠀⠀⡀⠰⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡡⠬⣄⡋⠀⢉⣦⠀⠼⣿⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠒⠒⠛⣇⠀⠀⢠⠃⠀⢋⢆⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠀⠐⠂⠂⠒⠛⠒⠒⠋⠀⠙⠛⠑⠀⠀⠘⠻⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢡⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⡏⠘⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠁⠀⠀⠀⡠⠓⡀⢰⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢰⠀⠀⠀⡰⢇⠀⢈⡶⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠈⢦⠤⠊⠀⠀⢹⠁⠀⢀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠘⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢈⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠹⡄⠸⠸⠀⠀⢃⠀⠀⠀⠀⠀⢀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⠲⠶⠂⠀⠘⡀⠀⠀⣠⣤⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣠⣾⣿⣿⣿⣿⡿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣿⣿⣿⣿⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠁⠀⠀⢀⡀⣀⣀⡤⠐⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢹⣄⣠⡞⢡⡷⣞⠉⠺⢗⣴⡏⢈⡻⣿⣿⣿⣿⣯⡀⠹⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣝⣿⡿⠋⢟⠁⢹⠇⠉⠳⡤⠛⠻⠣⢿⢁⣼⠿⣯⡩⠖⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⣯⡋⣛⠗⢺⠿⠄⣾⠆⠀⢾⠆⠰⢶⠄⠀⢻⠿⢴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠉⢹⠚⡷⠒⠿⠤⠄⠺⠒⠒⠿⡉⠉⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠠⢾⣤⣧⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⠒⠀⠋⠀⠲⡖⠺⢽⣲⣦⢤⣀⡀⢸⠀⠀⢀⣀⠀⡀⣀⣀⣠⣤⠤⢴⣓⣀⣀⠠⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠁⠀⠉⠀⠀⠉⠋⠛⠋⠙⠉⠉⠙⠚⠛⠓⠓⠳⠿⠿⠽⠯⠭⣭⠅⠀⠀⠤⠤⠤⠖⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠰⠶⠀⠉⠉⠉⠉⠉⠉⠉⠑⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠐⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⡀⠠⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠂⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠸⠃⢀⠀⢀⣀⠤⠄⣻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠀⠤⠤⢈⣻⠀⠀⠐⠄⠡⠀⠀⠀⢃
⠀⠀⠀⠀⠰⠆⠀⣾⠖⠋⠀⠀⠀⢁⡤⠀⠚⠓⠠⢄⠀⠀⠀⠀⠀⢀⡔⢊⣀⠚⠣⣄⠘⠓⠤⣀⠀⠀⠀⠀⢸
⠀⠀⢀⣤⠷⢄⣸⠇⠀⠀⠀⠀⣴⠋⢠⣾⣟⣡⣤⠈⣦⠀⠀⠀⠀⣏⣰⣿⣿⣴⣇⣈⣣⠀⠀⠈⢻⡆⠀⠀⢸⡀
⠀⠀⡟⠀⠀⢤⣽⠀⠀⠀⠀⠰⡶⠖⠛⠛⠉⠉⠉⠙⠛⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣧⠀⠀⢸⡄
⠀⠀⠻⣤⣀⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢈⠇
⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⢦⣄⡀
⠀⠀⠀⠘⣆⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣀⠀⠀⠀⠀⠀⢀⣀⡤⡶⠁⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠹⠇
⠀⠀⠀⠀⠹⣆⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠷⣍⠑⠒⠚⠒⠒⠈⠉⣰⠇⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤
⠀⠀⠀⠀⠀⠈⠳⣤⣀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣤⣤⣤⠴⠛⠁⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⡒⠀⠲⡄⠀⠀⡴⠊⠀⣠⠞⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠻⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⣀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢘⣀⡊⠀⠀⠛⠥⠚⠁⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⢤⣤⣤⣤⣤⣤⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠉⠉⠳⠄⠀⠀⣤⠒⠁⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠲⣶⠒⠋⠀⠀⠀⠉⠓⣶⠖⠖⠒⠢⢤⣤⣄⣀⡀⠀⠀⠀⠀⣀⣀⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠳⢦⣤⣤⣤⡤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡉⠉⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠳⠶⠶⠲⠞⠉⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠶⠶⠦⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⣠⣤⣿⠶⠶⠶⠶⠶⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡌⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠑⣄⢾⠶⠶⠶⠶⠶⠶⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣟
⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠈⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⢀⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⢈⣤⠀⠸⠀⢠⠀⠀⢿⣀⢙⡄⠀⠀⠀⠀⢠⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠉⠛⠷⣄⡼⣆⡠⠞⠉⠋⠁⠀⠀⠀⠀⠺⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢄⣀⡠⠀⠀⠀⢸⡇⠀⠀⠀⠹⡄⠀⠀⢀⣀⣀⣠⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠉⠁⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⢀⣀⣀⢸⡇⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠀⠀⠀⠀⠀⠀⢻⣀⣀⡀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⣠⠴⠷⠮⢬⣁⠀⠀⠉⠉⠉⠉⠉⠉⢩⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⣿⠉⠀⢈⣁⣠⣤⣤⠄⠀⢹⣄
⢠⡞⠁⠀⠀⠀⠀⠉⠓⡄⠀⠀⠀⠀⠀⠀⠻⠤⣄⣹⡆⠀⠀⠀⠀⠀⠀⣧⠴⠚⠉⠉⠀⠀⠀⠀⠀⠈⠉⠲⣄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠂⠀⠀⠀⠤⢤⡴⠋⠉⣇⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠛⠒⠒⠲⠶⠶⠶⠶⠶⠶⠷⠤⠤⠤⠤⠤⠴⢤⣤⠼⠋⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⠟
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣾⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⢀⡀⠀⡀⠀⣷⣶⣶⣶⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣧⡀⠀⠀⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡷⠯⠤⠤
⠀⠀⠀⠀⣀⣀⣤⣶⣿⣿⣿⠟⠉⠉⠉⠈⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⣿⣿⣷⠶⠶⠒⠒⠒
⠀⠀⠀⠀⢘⣿⡿⠿⣻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣗⣐⣒⣀⡀
⠀⠀⠀⢠⣼⣶⣤⣶⡿⠁⠀⢀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⢿⣿⣿⠿⠿⠖⠦
⠀⠀⠀⠨⢿⣿⣿⣿⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⢿⠿⠭⠭⠄
⠀⠀⠀⠀⢹⣿⣿⣿⡀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣭⣧⣤⣶⣿⣿⡭⠭⠁
⠀⠀⠀⣀⣼⣿⣿⡋⠀⢀⢸⣿⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣖⡒⣒⡂
⠀⠀⢞⣿⣿⣿⣿⣿⣦⣼⣾⣿⠗⢉⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣛⣓⠂⠁
⠀⠀⠁⢿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠉⠉
⠀⠀⢠⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠟⣹⡿⢿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣾⣿⣀⠀⠀⠀⢠
⠀⠀⠀⠺⠋⠀⠈⠹⣿⣿⣿⣶⣾⣿⣶⣦⣄⣾⣦⣸⣿⣿⣿⣷⣀⣰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠉⠉⠁⠉⠉⠁⠀⠀⠉⠉⠁

⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⣨⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⣴⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⠏
⠀⠀⣠⣿⣿⣿⣿⠁⠀⠀⣠⠀⠀⠀⠀⣹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣿⣧⣶⡶⠋
⠀⠀⠀⠉⣿⣿⣿⡄⠀⠀⠈⠑⠲⠶⠟⢻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⣿⣿⣿⡇
⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣼⣿⣿⠁
⠀⠀⠀⣼⠋⠀⠙⢷⡀⠀⠀⠀⠀⠀⣿⠁⠈⠙⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⡿⠀⠀⠀⠈⢷⠀⠀⠀⠠⠾⡷⠀⠀⠀⠀⠙⠿⣶⣴⣤⣄⣠⡄⠀⠀⢀⣀⣼⣿⣿⡏
⠀⠀⠚⠛⣀⠀⠀⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⢠⡿⢿⣿⣿⣿⠃
⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⢀⣴⣾⣿⣷⡀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶
⠀⠛⠋⠙⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣶⣤⣀⡀
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠺⡿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢿⣿⣿⣿⡍⠙⢦
⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠜⣡⣿⣿⣿⠃⠀⠀⠀⠀⠀⠶⠋⠁⠀⠀⢈⣿⣿⣿⣿⠀⠈⠛
⠀⠀⠀⠀⢹⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⣿⡿
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⣿⣥⣄⣀⡀
⠀⠀⠀⠠⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠺⠷⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⠀⠀⠀⠈⠉⠉⢻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣤⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⠀⠀⠀⠻⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣆⠀⣰⣿⣞⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄⠀⠈⠈⣿⣿⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠖⠐⠈⠀⠀⠘⠃⠀⢶⣾⣾⡇⢀⣀⣘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠂⠀⠀⠀⠒⠂⣀⡀⠀⠋⢳⡎⠀⠀⠘⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠠⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠒⢲⣤⢦⠤⠀⠾⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠲⣏⠛⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢿⡿⠿⠻⠿⢿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⠄⠈⣀⣀⡀⣀⣀⡀⠄⠀⠀⣧⠀⠀⠀⠀⠀⠀⡀
⠀⠞⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⢠⣶⣬⣤⠀⠀⢠⠋⠀⠀⠘⣿⠿⡹⠹⠦⣇⣃⣽⠁⠀⠀⠄⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣤⠴⠦⠲⠶⠦⣤⡉⠙⣿⠆⢀⠂⠀⠀⢀⡞⠀⠀⠀⠀⠀⡼⠛⠁⠀⠀⠀⠇
⠀⠀⠦⢀⣀⡀⣿⢿⡿⠿⢿⣿⠿⠃⠀⡟⢛⠃⠀⠠⠄⣑⣡⣤⡁⠀⠀⠨⡇
⠀⠀⠀⢹⠛⠋⠀⠸⠁⠀⠈⢻⠀⡄⠤⠿⣯⣀⣀⣨⣿⣿⣿⣿⠀⠀⠀⢰⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⢀⡶⠆⠐⢿⣿⣿⣿⣿⣿⡛⣻⣤⣄⣬⣿⣿⣿⣿⠿⠛⠉⠰⠦
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠀⠘⠐⠲⠿⣶⠀⠰⠶⠎⠙⠛⠁⠻⢿⡿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⣄⣀⡄⠀⣤⣀⣤⡀⣀⣀⣠⣀⡄⢀⣠⡀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢠⣤⣤⡄⣤⣤⡤⣤⡀⣤⣤⢤⡄⣤⣤⣤⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠤⠀⡤⠤⠤⠤⠤⠤⠦⠤⣤⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠦⠤⠄⠤⠤⠤⠤⠠⠤⠶⠶⠤⠄⠤⠤⢤⠤⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠶⠶⠶⠲⠴⠶⠆⠶⠦⠰⠄⠶⠶⠶⠴⠶⠶⠶⠆⢸⠀⠀⠀⠀⠀⠀⠀⡀⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠴⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠛⠓⠒⠒⠶⠶⠦⠀⠀⢤⠤⠤⠤⠤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠊⢦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠑⠵⡠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠠⠂⢄⡁
⠀⣀⠄⢤⢄⠀⠀⠀⠀⠀⠳⢄⡴⡀⠀⠀⣀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢶⡄
⠠⣭⠒⣌⢳⡱⡀⢀⠂⢠⠀⠀⠑⢌⢣⣀⣬⢄⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡆
⠀⠉⠁⠘⡄⢇⢃⠎⠀⠀⠛⢦⡀⠀⠙⡋⢈⡯⣤⣖⠀⡀⠰⠘⣄⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢣⠀⢹⣸⠎⠀⠀⠀⠀⠀⠉⠲⡌⠵⠅⠀⠀⢀⠈⠁⠀⠀⠁⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢣⠀⡏⠀⠀⠰⢄⣀⣀⡀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢶⣁⣀⠀⢀⠂
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿