Vaccination Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢟⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣯⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠹⣿⣿⣿⣿⣻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⣠⣽⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣟⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣾⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⡇⠀⢸⣿⠟⠿⠿⠿⠿⠿⠿⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣥⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡇⡀⠠⡀⠀⠤⢠⠄⢸⡇⠀⢸⣿⠀⠀⠄⠀⠠⡄⡤⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣸⡷⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⠛⠱⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠿⠿⠿⠿⠿⠛⠁⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠁



⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠉⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠿⠿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣀⠀⠀⣀⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⠛⠃⣀⣴⠟⠹⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠏⠻⣦⣀⠘⠛⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣥⡤⠶⠛⠉⣀⡴⠂⠘⣿⡄⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⣿⠃⠐⢦⣀⠉⠛⠶⢤⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠈⢉⡠⠖⠀⠘⣿⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢠⣿⠃⠀⠲⢤⡉⠁⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⢠⠖⠋⣀⠘⣿⡄⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⠃⣀⠙⠲⡄⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀⠀⠀⠀⠰⠛⣁⡄⠘⣿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣠⣿⠃⢠⣈⠙⠆⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠚⣁⡤⠆⠘⣿⣆⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⣰⣿⠃⠰⢤⣈⠓⠀⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠉⠀⠀⠀⢨⣿⠂⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠐⣿⡅⠀⠀⠀⠉⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⣀⣤⣾⣿⠃⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⣿⣷⣤⣀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶⡶⣿⡟⢿⡌⢿⣦⠀⠈⠻⠿⣶⣶⣶⣶⠿⠟⠁⠀⣴⡿⢩⡿⢻⣿⢶⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⢻⡌⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢡⡟⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠘⣿⣆⢻⡌⢻⣧⠀⣀⡀⠀⠀⢀⣀⠀⣼⡿⢡⡟⣰⣿⠃⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡈⠉⠛⠛⠿⣶⣦⠀⠀⠘⢿⣆⢻⡄⢻⣿⢛⣿⡇⢸⣿⡛⣿⡟⢠⡟⣰⡿⠃⠀⠀⣴⣶⠿⠟⠛⠉⢁⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡷⠀⠀⢀⡴⢻⣿⠀⠀⢠⣾⣿⡆⢻⣠⣵⡿⠋⠀⠀⠙⢿⣮⣄⡟⢰⣿⣷⡄⠀⠀⣿⡟⢦⡀⠀⠀⢾⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⡀⢀⣴⣯⣀⣸⣿⠀⠀⢿⣷⣶⡾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⣶⣾⡿⠀⠀⣿⡇⣀⣽⣦⡀⢀⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⢿⣿⠉⠛⠻⠿⣶⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⠿⠟⠛⠉⣿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣇⢠⡆⣰⡄⠉⣩⣍⠛⢛⣛⠛⠛⠛⠛⠛⠛⠛⡛⢛⣛⣛⣉⠉⡥⢤⡄⣶⡄⢾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢻⡾⢠⣧⣽⣸⡇⠈⢣⡏⠉⠓⣼⢸⢷⣸⡇⣼⢳⡈⢹⠁⢸⢸⡇⢀⡟⣿⠙⡿⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢷⣶⣬⣄⣈⡋⠓⠶⠋⠳⠤⠗⠿⠸⠀⠻⠧⠟⠛⠇⠸⠀⠘⠈⢛⣋⣠⣤⣶⡾⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⢀⣤⣄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⠒⠊⠉⠊⢀⠒⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀⠀⠠⠤⠓⠂⢒⣸⠓⠂⠀⠠⣭⣥⣤⠀⠀⠀⢄
⠀⠀⠀⠀⠀⡠⠒⠀⢈⣩⡭⠭⠤⢈⡁⠈⠛⢍⠭⠉⠷⢻⠁⠀⠒⠂⠀⠀⠀⠐⢒⠂⣭
⠀⠀⠀⠀⠀⠛⠂⠀⠠⠬⠬⠉⠭⠭⠤⠀⠐⣁⡄⠀⣨⢾⡀⠀⠀⠀⠀⣜⢜⢕⠥⠪⠞
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⢔⠔⠐⠀⠀⢺⣠⣆⠠⠀⣹⠷⠲⠤⠴⠤⠷⠚⢶⡕⠄
⠀⠀⠀⠀⠀⠀⠃⣤⣀⡀⠐⠐⠅⠔⡕⣑⢹⣛⡙⠂⣠⢸⠁⠀⠈⠉⠉⠉⠉⠁⡀⢀⠀⡄
⠀⠀⠀⠀⠀⢀⠜⠉⣁⡈⠉⠉⠋⠉⠁⢋⠓⠷⡀⡄⠀⠈⠀⠀⠀⠀⠀⠠⠪⠦⠨⠘⣀⡇
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠉⢉⢉⢉⠄⢀⠤⠂⣇⡂⠀⡂⠀⢨⡉⠀⢒⠒⡒⢂⡉⣉⢰⡇
⠀⠀⠀⠀⠀⡟⠢⠤⢀⡀⠀⠁⠀⠀⠐⠉⣉⠁⡇⣀⠀⡇⠀⢸⡈⡄⠽⣘⠀⢀⠀⡏⢻⡅
⠀⠀⠀⠀⠀⣷⣴⡆⡄⠄⣉⠉⠉⡉⢉⠡⠂⠀⡇⠀⠎⡇⠀⢸⣹⣶⡠⠭⠂⠨⠄⠒⠚⠄
⠀⠀⠀⠀⠀⠯⣈⠁⢆⠸⣀⠠⠀⠑⣌⢈⡁⠀⡇⠀⡀⠅⠀⢸⠀⡫⠑⣋⣉⡀⠀⣒⠀⡆
⠀⠀⠀⠀⠀⣾⠶⣭⡒⢒⠂⠤⠤⡤⠄⢒⠊⠁⡖⠒⢛⡀⠀⠀⠲⣦⠍⡀⠂⠀⠀⣀⠤⡇
⠀⠀⠀⠀⠀⢲⣅⣒⠀⠈⠝⠉⠀⠇⢸⡗⠀⡆⡏⠉⠉⢙⠀⠀⠀⠀⢉⣒⣒⣎⠩⠀⠀⡏⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⡙⠪⢕⡀⠸⠓⠂⠀⠀⠈⠀⣀⠃⡧⠀⠀⠈⠢⡙⢤⣐⠛⡬⣍⣸⣶⠟⣸⣒⢒⣒⡂⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠁⡀⠀⠀⡉⠐⠒⢒⠂⢀⡊⣠⠂⣗⣒⡂⠀⠀⠈⠑⠪⢥⣬⣬⣬⣴⡛⠠⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠘⢜⡦⣀⠨⠲⠃⠔⣤⢦⣴⠟⡾⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠪⢍⣊⠗⠒⠚⢉⢠⠚⣁⠀⠀⡀⠠⠄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠛⠉⠠⠤⠠⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠒⠒⠷⣿⡖⠐⠢⠀⠀⠀⢀⠀⢀⣀⣀⠀⠀⠀⠀⡠⠒⢒⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠈⠙⠲⣶⣶⣦⣀⢈⡿⠶⠴⠁⠀⠀⢻⣦⣤⣤⣄⣍⣑⠖⠀⣠⣤⡿⠋⠁⣠⡑⣀⢀⡀
⠀⠀⠀⠀⠀⡠⠊⠉⠉⠛⠲⢶⣿⣀⠀⠈⢻⣿⢿⠟⠁⠀⠀⠀⠀⠀⠘⠿⢿⠿⠟⠻⡇⠀⣸⡿⠋⢀⣼⣿⡟⠋⢉⣁⣌⠃
⠀⠀⠀⢀⠊⢀⣼⣿⣶⣤⣤⣾⣿⣿⣷⣶⠾⢱⢺⠀⠀⠀⠀⠀⠀⠀⣴⣤⣾⣦⣼⠘⠀⣠⣿⠇⠀⣸⣿⠏⠀⣰⣿⣿⣿⡀⠆
⠀⠀⠀⢘⠠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⡧⣼⡦⠤⣄⠀⠀⠀⠀⡖⠒⠀⠀⢺⠘⠶⠋⢰⠀⣰⣿⣿⣤⣾⣿⣿⣿⣟⣳⠈
⠀⠀⠀⢸⠈⣿⣿⣿⣿⣿⣿⣿⣿⣏⣁⠄⣰⠊⠀⠀⣠⠞⠀⠀⠀⠀⡆⠀⠁⠀⠙⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢡
⠀⠀⠀⢘⠈⠋⠹⢛⣿⣿⣿⣿⣿⢿⣿⣶⣯⣤⣴⡿⠁⠀⠀⠀⠀⠀⢷⣶⣶⣶⣾⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⡆
⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⡿⣳⣿⣿⣿⣿⣿⢻⠃⠀⠀⠀⠀⠀⠀⢾⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢱
⠀⠀⠀⠠⠀⢿⣿⣿⣿⡟⣿⠁⣿⣿⣿⣿⡟⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠚⠛⠛⠛⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠸⡆
⠀⠄⠀⠘⣣⣉⣿⣿⣿⣇⢻⡄⠻⡟⠋⣀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡘⡻⣿⣿⣿⣿⣿⣿⣯⣤⡈⡇
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⢾⣿⣿⣿⣿⣿⣿⣷⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠖⠋⠉⢉⣓⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⠂⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡇⠐⠣⣀⣘⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⡉⠙⠇⡈⢻⣿⣿⣿⣿
⠀⠀⠀⠀⢀⡴⠛⡆⣠⠁⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⢀⡔⣁⠤⠼⠒⠲⠤⣄
⠀⠀⠀⢀⣞⣀⣠⡇⠘⢦⣀⣠⠞⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢰⣀⠞⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⣠⠊⠉⢰⠃⠀⠀⠀⣀⠯⠂⠀⠀⠈⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⡴⠁⠀⠀⢀⡱⠖⠂⠉⠀⣀⠀⣀⡄⠀⠀⠈⢆⠀⠈⠀⡗⠂⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⡼⠳⣶⠖⠈⠀⠀⠀⣀⠔⠊⠁⠙⠳⠇⠀⣏⠛⠢⢦⠀⠀⠛⠀⠀⠐⡎
⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⣦⠐⠀⠀⠈⠣⣀⠀⠀⠀⢀⠇⠀⠀⠂⠀⢸⡎
⣀⣀⡠⢴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡌⠂⢄⣀⡼⠓⠢⣀⡸⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠈⡦⡀⠀⠨⡃⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⠀⠀⢠⠃⠈⠢⢄⠘⠤⣀⢀⡎⠈⠒⢄⡀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⠀⠀⡼⠀⠀⠀⠀⠑⠒⠈⠉⠀⠀⠀⠀⠈⠢⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡀
⠀⠀⠀⠀⠀⣤⡄⠀⢠⣄⠀⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⣢⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣵⣿⢿⣄
⠀⠀⠀⠀⠀⠙⣧⡤⢞⣛⣒⣚⣷⣤⣶⠆⠀⢀⠀⠀⠠⣿⣿⣿⡿⠄⠀⠀⠀⠀⢴⣄⢀⣞⣵⣿⠟⠁
⠀⠀⠀⠸⣷⠞⣵⣾⣿⣿⣀⣹⣿⣿⣿⣦⣤⠿⠗⠀⠀⠉⠟⠛⠀⠀⠀⠀⢀⡴⠋⠙⢷⣝⠟⠁
⠀⣶⣦⣠⢏⡾⢿⣧⣾⣿⣿⣿⣧⡿⠛⢻⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⢦⡀⠀⣹⢷⠄
⠀⠀⠀⢸⢸⣧⣼⣿⣁⣨⣿⣋⣿⣷⣤⣼⣿⡟⠃⠀⠀⠀⠀⠀⢀⡴⢋⠀⠳⡄⠀⣠⠞⠁
⠀⠀⠻⠻⣼⣿⣏⣽⡟⢻⣿⣿⡿⢻⣿⣏⣿⠷⣶⣦⠀⠀⢀⡴⢋⡀⠈⠳⠀⣠⠞⠁
⠀⠀⣠⣤⠿⣿⣿⡟⣻⣿⡏⠀⢹⣿⣿⣿⣧⡀⠀⠀⢀⡴⠋⢤⡀⠙⠂⣠⠞⠁⢠⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠙⠃⠀⣼⡿⢿⣿⣿⣿⣿⣿⠿⣏⠀⠉⠀⢀⡴⢋⣄⠻⣿⣾⣿⠟⠁⠀⠀⣿⢸⡇⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠀⢀⣾⠉⠉⠿⠆⠀⠿⡷⢀⡴⢋⣴⣝⢿⣿⣿⠟⡁⠀⠀⠀⠀⠈⠛⡟⠛⠋⠉⠉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⢀⡴⣯⣴⣦⣙⣿⣿⠟⠁⡜⡉⢆⠀⠀⠀⡠⠞⠁⠈⠉⠉⠁⠙⠲⡄
⠀⠐⢦⣰⣼⣡⡄⡀⠀⠀⠀⠀⠀⠀⡟⣵⣿⣿⣿⣿⠟⠁⠀⣸⢸⠁⠸⡄⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠤⣽⢾⡿⣾⣟⢻⡅⠀⠀⠀⠀⠀⢤⣝⢿⣿⣿⠟⠁⠀⠀⠀⢯⠄⠀⢀⡇⠀⡇⠐⠀⠒⠒⠒⠒⠒⠒⠒⠂⡇
⢐⣻⣿⣿⡿⣿⣿⠲⠀⠀⠀⣀⣵⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⢫⡀⠀⡇⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠈⠀⠛⡟⠻⠛⠌⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⢃⣀⣀⡙⡄⡇⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⣿⡇⡇⡇⢻⠀⢿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⣿⣿⡇⡇⡇⣾⠀⣾⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣼⣿⣿⡇⡇⡇⣿⣦⣿⣿⣿⣿⣿⣿⣿⡇⡇
⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡸⠿⠿⠿⢇⠇⢧⡙⠿⠿⠿⠿⠿⠿⠿⠟⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠠⠀⠭⠉⠉⠍⠩⠭⠭⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢹⣇⢰⡟⢠⡿⣷⠀⣼⠟⠛⠃⣾⠟⠛⠃⣿⠄⣿⣧⣸⡇⠀⣾⢿⡄⠛⣿⡟⠃⣿⢠⡾⠛⢻⣆⣿⣷⣄⣿
⠀⣻⣿⢀⣾⠛⢻⣧⠻⣷⣶⠆⠻⣶⣶⠆⣿⣂⣿⡈⢻⣇⣼⡟⠛⣿⡀⣿⡇⢀⣿⠈⢿⣦⣾⠃⣻⡇⢻⣿⡀
⣿⣿⠛⠛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠋⠉⠉⠉⠉⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠉⠉⠉⠉⠉⠉⠲⢄⡀⠀⠀⠀⠀⠀⣇⢠⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⡟⠋⠹⠬⡑⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⡀⢠⠀⢦⡀⢰⡄⢰⣄⠀⠀⠀⢣⠀⠀⠀⠀⠇⠀⣀⠀⠀⠀⠀⢣⠤⢤⠀⠈⡆⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠦⠼⠙⠚⠈⠉⠉⠉⠉⠉⢤⡉⠉⠀⢈⠀⠀⠀⠀⣠⡀⢿⡇⣀⠤⢶⠁⠀⢸⠃⠀⢰⠀⠀⢰⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠐⠉⠁⠀⠀⠀⠀⠀⣀⠀⠁⠀⢰⡼⢄⠀⠀⠸⡀⠈⠉⠉⣀⠄⠸⠀⢰⡇⠀⠀⠘⠀⠀⢸⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢾⣄⠀⠀⢾⣧⠀⠀⠀⠀⣿⡆⢀⡠⠚⠃⢨⠀⠀⠀⠑⡄⠉⠉⢀⡠⠀⢳⠆⠀⢸⠀⢸⠀⣠⢃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣈⡀⡗⠒⠚⣓⣊⡉⢉⣉⡉⠍⠀⢡⡰⠒⠁⠀⠀⠀⠀⠱⣈⠉⠁⢀⠴⠃⠀⢿⠈⣠⢆⡰⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠤⠤⠤⡤⠤⠔⠒⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⡲⠀⣠⠚⢳⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠶⢖⣶⣖⣲⡂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⠟⢁⣠⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⡏⠉⠈⠙⡀⡠⢖⡒⣀⡶⠴⠤⡀⠀⠀⠀⠀⡜⣨⠃⢒⠁⠞⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠖⣍⣭⣵⡀⣀⠷⣛⣩⣽⠋⠑⠢⠀⠀⡏⠑⠒⢴⡱⢁⢔⡥⠐⢦⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠐⣤⠁⠀⠈⠻⡓⢻⠁⠙⠯⠿⠖⠒⠉⠰⣅⡀⠀⣾⠔⡡⠊⠀⠀⠈⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠢⡀⠀⠀⠱⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⢤⣤⡖⠘⠦⣀⠀⠀⢀⣀⡤⠤⠖⠒⠉⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⣯⢕⣂⠀⠀⠉⠁⢙⡀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⢦⠙⠒⠒⢎⠉⠧⠄⠒⠒⠂⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⣸⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣎⡲⣴⢟⢦⣂⠄⠀⠀⣀⢤⣶⠃⣶⠀⡀⠀⠀⠀⢢⣴⣶⣾⣤⠂⣀
⠀⠀⠀⠀⢰⠷⠊⠃⠀⠀⠀⠀⠀⠀⠀⣀⣱⣾⣷⠐⢀⡫⣷⣃⠀⢰⣿⣿⣿⣍⣙⣿⣅⠀⢠⢨⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠠⠼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠖⣯⣿⣿⣷⣶⣶⣻⡀⠀⠸⣿⣿⣾⣝⣶⣿⡟⠀⠠⢀⠹⣿⡅⠁⠀⠈⠀⠁
⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠊⡝⠿⡿⠷⡛⠃⠀⠀⠀⠘⠻⠿⠿⠿⠋⠀⠀⠀⠈⢠⠋⠀⠐⡌⠂
⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡩⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠀⠀⢸⣾⣷⡆⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠁⠠⠂⣃⢁⡀⠁⢁
⠀⠀⠀⣤⢽⣿⣿⣯⣄⠀⠀⢐⣟⠂⠀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⢸⠛⣿⣿⣿⡇⠀⠠⠀⢠⡚⠡⢟⠈⠃⠀⡆
⠀⠀⠀⠀⠸⢹⣟⠇⠀⠀⠀⢀⣿⠇⠀⠀⠀⣾⡥⢿⣿⣭⡄⠀⠀⠀⠛⡶⠻⡿⠟⠃⠀⠀⠀⠀⢂⡀⠢⠖⢠⠙⠁
⠀⠀⠀⠀⠀⢸⣇⡀⠀⠀⠀⠆⣿⡄⠀⠀⠀⠸⠗⠾⠿⠟⠁⠀⠀⠀⡠⠋⠀⠹⠀⠀⠀⠀⠀⠂⠀⠄⠋⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⢸⡟⠃⠀⠀⠀⡊⣿⡆⠀⠀⠀⣂⣀⣀⣀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⢸⣧⠄⠀⠀⠀⡇⣿⡅⠀⠀⠸⡟⠛⠛⠿⠿⣟⠀⠀⢠⣶⣦⣖⣾⣷⣶⠀⠀⠀⢀⣀⣦⣴⣢⣶⠀⡀
⠀⠀⠀⠀⠀⢸⡋⠆⠀⠀⠀⠏⣯⡄⠀⠀⠰⠆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠉⠀⠈⠀⠀⠠⢤⣿⣿⡿⢏⢛⣎⡁
⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⣀⣦⣿⣇⣀⠀⠀⡄⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⢒⣯⣿⢕⡡⢁⣛⡒⠂
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠙⣿⠀⠉⠀⠀⣇⣀⣀⣀⣼⣿⠀⠀⠀⣄⣀⠀⣀⣁⡀⠀⠀⠈⢉⢿⣿⣮⣥⡣⣟⠉
⠀⠀⠀⠀⠠⠟⠓⠀⠀⠀⠀⠾⠛⠦⠀⠀⠀⠋⠛⠻⠟⠛⠋⠀⠀⠀⠛⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠃⠘⠀⠁

⣶⣾⠿⣶⠶⢶⣶⡶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣰⣆⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡿⠿⢿⣿⡿⣿⣦⡤⠀⠀⡴⣄⠀⢀⡽⣮⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣧⣀⣼⣿⣷⣿⣿⣇⠀⢀⡝⢮⡳⣏⠔⠁⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢻⣿⡏⠉⢹⠟⢁⠔⢁⠐⠀⠙⢮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⣿⠖⢁⠴⡉⠢⠀⠀⢀⡴⢋⣉⣁⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠙⢁⠔⠁⠄⠈⠀⢀⡴⠋⢀⣾⡿⣿⣁⣹⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠣⡀⠁⠀⢀⡴⢋⠀⠀⠉⠹⣷⣾⣿⡿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⡀⢀⡴⠋⠀⢻⣷⡒⠀⠈⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠓⠊⣉⠉⣴⣷⣤⣾⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣸⣿⣇⣀⣸⣿⣏⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⡻⠿⣿⣿⠿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⠀⠠⣄⣼⣤⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⡾⠛⠻⡆⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠚⠓⠂
⠀⠀⢺⢿⡿⠋⠀⠀⢿⣿⣿⠗⠀⠀⠀⠁⠀⠀⠁⠀⠀⡰⠋⠀⣭⠀⠀⢠⣶⣷⡆⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⣄⠀⣠⡶⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⠹⣿⡋⠀⢀⣤⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢋⣤⡶⢷⣄⠈⢻⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣁⣴⠟⠙⠷⠀⣹⡇⠀⣹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⣠⡾⠋⠻⠆⢀⣠⡾⠋⣠⡾⠋⠁⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢉⣴⠟⠙⠷⠀⢀⣴⠟⢉⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣡⡾⠛⠻⠆⠀⣠⡾⠋⣡⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣾⠋⠀⣾⠛⠷⠀⢀⣴⠟⢋⣴⠾⠋
⠀⠀⠀⠀⠀⣀⣄⡀⠀⢰⣿⣷⠀⢴⣷⠄⢸⣿⣿⡿⠀⠀⠀⠀⢹⣧⡀⠙⢷⣤⡶⠟⣡⡶⠟⠁
⠀⠀⠀⠀⠀⢻⣿⣿⡄⠀⠙⣿⠀⠘⡟⠀⢸⣿⠏⢰⣾⡆⢀⣶⠛⣉⣿⣦⡀⢉⣤⠾⠋
⠀⠀⠀⠀⢀⡀⠉⠙⢷⣄⣠⣼⣷⣶⣿⣶⣾⣧⣠⠟⠉⢀⡵⠛⠞⠋⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠿⢿⣦⣀⣼⣿⡿⠿⣿⣿⣏⣹⣿⣿⡿⢣⠔⠉⢀⣤⣤
⠀⢠⣤⣤⣄⠀⠀⣿⣿⣿⡀⠀⢸⣿⡟⠉⠈⠉⢠⣴⣿⡶⠿⠿⠟⠁
⠀⠘⠿⠿⠟⠻⢾⣿⣿⣿⣿⣾⣿⣿⣧⡀⠀⣀⣿⡟⢻⣿⠤⠴⢾⣿⠇
⢀⣶⣿⣶⣶⣦⣾⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣶⣶⣆
⠈⠻⠿⢟⣋⣁⣨⣿⣿⣿⣙⣿⣿⡟⠉⢹⣿⣧⣠⣿⣿⣇⡀⠉⠛⢿⡿⠏
⠀⠀⠐⢿⠿⠋⢀⣼⠿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⡿⠋⠈⠻⣿⣿⡆
⠀⠀⠀⢠⣾⣿⡿⠁⠀⢈⡿⠻⣿⣿⣿⡿⠿⡟⠉⠻⣦⣀⠀⠈⠉
⠀⠀⠀⠘⠛⠛⠁⢀⣴⣿⡇⣴⣇⢸⣿⡀⠀⢳⡀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠻⢛⣾⣿⣧⠀⢸⣿⡆⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠠⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠠⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠡⠖⠒⠒⠒⠼⠭⠁⠉⠐⠀⢀⠄⠂⠀⠰⠺⠝⠶⠆⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢄⠀⡐⠁⠐⠀⡠⠂⢀⠈⠀⠀⠀⠀⠀⢠⠊⠀⢅⠀⢠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢈⠂⠀⠀⠔⠀⠀⠀⠐⠲⠢⠄⠔⠖⠒⠆⠀⠢⡀⠀⡠⠒⠒⠂⠂⠂⠂⠂⢀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣱⣗⠦⣙⢤⡀⠀⡸⠄⠀⣀⠀⠀⠀⠀⠠⠀⡀⠆⢠⠁⠀⣀⠀⠀⠀⠀⢄⣀⡠⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠪⠀⠀⠀⠑⠚⠂⡆⠐⠂⠀⠠⢨⡍⠁⠀⠀⡇⠀⢸⠀⠀⡀⠠⠄⠩⢅⡁⠀⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡶⠡⠁⠀⠀⠀⠀⠀⠀⡇⠐⠀⢀⡀⠠⠄⠀⠈⠀⡇⠀⢈⠀⠀⠀⠠⡀⠢⠔⠰⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢐⠞⠔⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠛⠺⠀⠀⠀⠀⠀⠀⡇⢰⢸⠀⠀⠃⠀⡀⡅⠵⢈⠀⠀⠲
⠀⠀⠀⠀⣀⠀⠀⢀⠮⠰⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠣⢒⠭⠀⠀⠀⠀⠀⠀⡇⢸⢸⠀⠀⠀⠀⠁⠀⢀⡀⢀⡀⠬
⠀⠀⠀⠀⠐⠁⠢⢮⡒⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⠒⠀⠠⠤⠠⠄⠀⠀⠃⢺⢸⠀⠀⢄⡀⠀⠁⠀⠐⠒⣒⠤
⠀⠀⠀⠀⠀⠀⠈⠒⢍⡒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠠⡠⠂⠈⢸⠀⠀⠀⠀⠀⠈⠉⠉⠩⡉⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠐⠀⠀⠂⠐⠂⠀⠀⠒⠈⠈⠗⠤⣀⡐⠀⠒⠀⣊⢁⡑⠋⠘⢤⣀⠀⠀⢀⢄⠀⡨⠒⢠⠞⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠈⠁⠂⠐⠀⠀⠁⠊⠝⠒⠒⠴⠢⠖⠀⠀⠐⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣉⠻⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣄⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⡙⠻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣷⣦⣈⠻⢿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣷⣤⡉⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⡄⢀⣶⣦⡈⠛⢿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⣰⣿⣿⣿⣿⣷⣤⡉⠻⢿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠙⠻⣿⣿⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡀⠀⠀⠙⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢃⡐⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡿⢃⣼⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠋⣰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠋⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣾⣿⠟⠁⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⣠⣿⣋⣉⣉⣉⣉⣉⣉⣉⣉⣙⠏⣰⣿⣿⠋⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⢃⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⣸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⣾⣿⠏⠀⠀⢀⣶⣶⣦⠀⠀⣀⣼⣿⣷⣿⣿⣶⣿⣧⣀⠀⠀⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣴⣿⡿⠁⠀⠀⠀⠘⢿⣿⣿⣦⣾⣿⣿⣿⠛⠛⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⣰⣿⣿⣿⣿⣿⣿⣿⡟⢁⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢙⣛⣛⣻⣢⣾⣿⡿⢿⣿⣿⣿⣿⣧⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠛⠻⠿⠋⣰⣿⡿⠋⠀⠀⠀⠀⠀⣀⣄⡀⢀⣿⣿⣿⣻⣿⣿⣿⣟⣟⡀⢀⣹⣿⣿⣿⣿⣦⣤⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⠏⠀⠿⢷⣾⣿⠿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠀⢿⣿⣿⣿⣷⣤⣼⣿⣿⡟⠁⠈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡿⠋⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠙⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡏⠀⠀⠀⠀⢀⣹⣿⠛⠋⠉⠹⣿⡄⠀⠀⠀⠀⠙⠿⠿⠋
⠀⠀⠀⠀⠀⢠⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠒⠆⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡆⠀⠁⠈⠀⣀⠄⣀⠀⣠⣀⡋⡐⠀⢠⢊⠀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠀⠈⢰⠤⠐⠉⠉⠈⠁⠀⠀⠐⠢⠄⠁⠈⠓⠲⢂⡂⢊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⠈⠐⢄⠁⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠐⠂⠀⡴⠉⠀⠁⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⢠⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⡐⠎⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠠⠤⠀⠀⠀⠈⠨⠀⠀⠠⠄⠀⠹⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢡⡀⠀⡀
⠀⠀⠀⠐⢀⣀⠁⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⣱⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⡆⠀⠠
⠀⠀⠀⠈⠀⢸⠀⠉⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⣃
⠀⠀⠀⠀⠰⣽⠀⠀⢠⣙⠀⠀⠀⠀⠀⠀⠀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠂⠀⠀⠀⠀⡍⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢠⠐⠄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢥⠇⢀⡆⠂⠈
⠀⠀⠀⢤⠄⠀⢃⠀⠀⠀⠀⠀⠒⣀⠄⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠈
⠀⠀⠀⠀⠀⠀⢘⣆⠀⣁⠀⠀⠑⠃⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣸⡇⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⡤⣶⠋⢀
⠀⠀⠀⠀⠀⢀⡀⡠⠊⠻⣦⣀⡀⢲⣀⠀⠀⠀⠀⠀⢂⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣤⡞⠇⠀⠦
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢈⡾⢷⣄⠀⠀⢀⠀⣄⠂⡀⠀⠀⣤⡠⠀⠀⠀⠀⠀⠀⣈⣤⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⠻⠶⢻⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠛⠁⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠆⠀⡔⠋⠁⡽⠙⠛⠷⠒⠲⠖⠦⠨⠛⣏⠉⠀⢈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⠀⠘⠒⠂⠸⠀⠀⠀⠀⢀⠴⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠛⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢰⢀⣀⠀⢀⠀⣀⣀⡀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠒⠒⡄⢹⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⢸⠘⠒⠐⠒⠒⠒⠒⠂⠒⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠀⠀⠀⠀⠀⢸⢠⣀⣀⣀⣀⣀⣀⣀⣀⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠇⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠈⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢸⢸⣀⣀⣀⣀⣀⣀⣀⢀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡸⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠠⠠⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⣀⣀⣒⣒⣒⡒⣂⣀⣀⢀⠀⠀⠀⠀⠀⠒⠒⢒⡲⢁⡇⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢨⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠸⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠦⣤⣤⣤⣤⣤⣤⣤⣴⠴⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠁⠉⠉⠀⠀⠀⠈⠀⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⡉⠉⡉⠉⠉⠉⠁
⠀⠀⠀⡔⢀⠀⠠⢀⠀⢀⡀⠀⣀⠀⢀⡀⠀⡀⠂⢀⠀⣀⡀⠀⣀⣀⢀⡀⡀⡀⠀⢀⠀⢀⡀⠀⢀⠀⠀⡀⠠⠀⠀⣠⠈⢠⡤⣄⠀⠑⣄
⠀⠀⡼⠀⠦⠤⠤⠬⠤⠦⠡⠴⠌⠠⠄⠥⠴⠭⠥⠤⠥⠤⠬⠤⠬⠥⠬⠭⠴⠤⠤⠤⠅⠰⠨⠄⠀⠠⠀⠸⠄⠣⠀⠈⠓⠓⠲⠬⠦⠤⠊
⠀⠀⠳⠖⠲⠒⠒⠒⠶⠲⠖⠒⠒⠖⠒⠒⠒⠒⠒⠒⠒⠒⠖⠒⠒⠒⠒⠖⠲⠒⠒⠒⠲⠒⠒⠶⠒⠒⠒⠖⠶⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠓⢦⡀⠀⢠⠖⠃⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠈⣳⠀⠀⠀⠀⠀⠀⢸⠃⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣨⠇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⠛⢦⣀⣠⠴⠖⠋⣿⠀⠀⠀⢿⠓⠲⢤⡀⠀⣠⠞⢁⡴⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠳⡄⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠋⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠆⠀⠀⢀⡴⠚⠉⠙⣦⠀⠀⠀⠀⠀⠀⠠⠞⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠸⣄⣀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⠀⠀⠀⠀⣾⣠⠴⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢴⠋⠙⠳⣄⠀⠀⠈⠛⠓⠲⠶⠤⠴⠋⠉⢳⡀
⠀⢀⡞⠉⠙⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡤⠞⠀⠀⣀⣤⣤⣤⣤⣤⡄⠀⠀⢸⠇
⠀⠘⡇⠀⠀⢠⠤⠖⢶⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠙⠦⠴⠋
⠀⠀⠙⠦⠤⠞⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣞⠉⠉⢙⡆⠀⠀⠀⠀⠀⢠⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡴⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠹⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⢀⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣲⣄⠀⠈⠛⠉⠙⡆
⠀⠀⠀⠀⠀⢀⣀⣀⡞⠁⢀⡴⠋⠉⠓⠦⣤⣦⠀⠀⣇⣠⡤⠶⠛⠁⠈⡗⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⡇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡆⠀⣧⡀⠀⠀⠀⠀⠀⠳⣤⠴⠞⠁
⠀⠀⠀⠀⠀⠙⢦⣀⣀⡿⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⠶⡄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⣟⠀⣀⣨⠓⣅⣀⢀⣿⡬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⡀
⠀⠀⠀⠀⠀⣠⢤⠀⠀⡠⠋⠉⡤⢤⠀⠀⠀⠉⠸⢄⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⠀⠀⠀⢹⢴⡃⠀⠀⢀⡤⢄
⠀⠀⠀⠀⠀⠑⠺⢕⠊⢀⡀⠀⠁⠉⠀⠀⠯⡵⠀⠀⠛⡫⠖⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢎⠣⠴⢒⣋⡀⠉⠓⠦⢜⡴⠚
⠀⠀⠀⠀⠀⠀⠀⡜⠀⢯⠟⠀⠀⢯⣱⠀⠀⢠⠴⡄⠀⢧⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠘⠒⠁⢠⢤⡄⠘⢄
⠀⠀⠀⠀⣏⡱⢮⡁⠀⠀⠀⠀⠀⣀⣤⡀⠀⠈⠋⠀⠀⢈⠴⣍⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣙⣒⠚⠀⡤⡄⠀⠀⠀⠈⠉⠀⠀⠈⢒⣒⣉⡳
⠀⠀⠀⠀⠀⠀⠀⢣⠀⢞⣳⠀⠀⠑⠚⠀⠀⣴⣲⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡇⠀⠓⠁⠀⠯⡷⠀⠀⣞⠝⠀⡏⠀⠈
⠀⠀⠀⠀⠀⡴⢲⡯⠤⡀⠉⠀⠀⠰⣋⡗⠀⠈⠁⣠⠖⢽⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀⠀⢀⣄⡀⠀⠀⣀⣀⠀⠰⣇⣀⡀
⠀⠀⠀⠀⠀⠙⠊⠀⠀⢈⣆⣤⣀⣀⠀⢀⣀⣤⢺⠁⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡰⠋⠙⠦⣈⠓⠃⠀⠈⠓⣁⠴⠋⠲⣄⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡸⠀⠀⢈⣶⠉⠀⠀⣷⢱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡔⠒⠒⠒⢆⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠇⠀⠀⠀⠳⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⣔⠋⢳⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣅⡀⡇⠀⠀⡀⣸⠃⣇⠀⠀⠀⢸⢆⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠑⠒⠋⢉⠁⠀⠈⠉⠉⠒⠋⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢦⡀⢀⠞⠁⠀⠀⢼⣁⡽⠀⠀⢀⢤⣀⠀⠈⠳⡄⣀⡴⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠓⣌⠁⢀⣠⡀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠂⠀⠀⠈⣡⠖⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢸⣀⠇⠀⠀⠐⣍⢱⡄⠀⠀⠀⡠⢤⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠦⠤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠧⠜⠀⠀⠀⠧⠤⠖⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠊⠉⢳⡀⠀⠀⣀⠀⠀⠀⢀⡖⢛⡆⠀⠀⠀⢀⠀⠀⠀⠀⡎⠉⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠸⣉⣱⠀⠀⠀⠉⠉⠀⠀⠀⢴⢉⡵⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⣃⢤⡀⠈⠁⠀⠀⠀⢰⠚⣳⠀⠀⠈⠉⠀⢀⠤⢌⡳⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⠏⠀⠀⠑⢦⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡔⠋⠀⠀⠹⢤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣠⠒⠲⠤⢤⠀⡤⠤⠔⠒⢄⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⡜⠀⠀⠀⢀⡇⡇⠀⠀⠀⢸⡃⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣀⡹⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⢀⢠⣹⣴⣤⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠈⣽⣿⣿⣿⣿⣾⢃⡀⡀⡆⡀⢀⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣜⢹⢿⣿⣿⢿⡳⢮⣵⣿⣿⣷⣯⠄⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣟⠈⠙⠂⠘⢫⣿⣿⣿⣿⡿⠝⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠐⠋⠟⣟⠻⣱⡶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠘⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⢀⡀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠒⣼⣶⣶⣧⣖⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠰⢽⣿⣿⣿⣿⣧⠶⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⣀⣩⠿⡿⡿⡟⣭⢠⠄⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢢⣷⣾⣾⣼⡓⠃⣛⣽⣾⣾⣴⣃⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠮⣿⣿⣿⣿⣿⡥⠖⣻⣿⣿⣿⣿⣗⠆⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠠⡄⡈⣽⣻⢿⢿⣻⡅⡀⣽⢻⣿⣿⣿⣴⢢⣷⣿⣿⣼⣺⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⡀⠀⢸⢖⠣⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡸⠀⠠⠄⠘⠀⠈⠀⠀⠀⠀⠳⠈⢁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠁⢀⡀⠀⡀⠠⡇⢀⡀⢠⡀⣀⠀⠉⠄⢔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠀⠀⢠⡜⠁⠁⠉⠚⠛⠋⠉⠀⠀⠙⢦⡀⠀⠀⢐⠒
⠀⠀⠀⠀⠀⡀⠈⠠⠌⢁⡀⠢⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠄⠀⠀⠠⠤⠔⠂
⠀⠀⠀⠀⠀⠀⢀⠤⠂⢀⠀⢸⡇⠀⢠⠞⠁⠀⠀⠀⠉⠑⠀⠀⢸⣇⠀⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⣷⣀⡔⠒⠖⢤⡀⠀⡴⢈⣝⠰⣾⠃⠀⢀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⡟⠅⠀⢀⣽⡦⢁⣉⠘⢳⣦⠀⢼⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠂⡼⠀⢃⠀⠀⠘⡻⢞⠄⠘⡄⠀⠁⠠⡼⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⡞⢀⢑⠆⠈⠒⠫⠄⠀⠺⠓⣰⠎⡯⠀⢄⠀⠀⠀⠀⠀⣀⡠⠀⢀⡀
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠔⠀⠁⠈⢲⡀⠀⠒⠤⣄⣀⡴⠞⢠⠞⠁⠀⠀⠀⠀⢀⡔⠉⢠⠀⠉⠉⠠⣆
⠀⡆⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣤⠀⠈⠀⠁⣠⠞⠂⠀⠀⠀⠀⠀⣸⠀⠀⢾⡁⠀⠀⠀⡯
⠀⠘⡄⢳⣠⠎⠁⠀⠀⠀⠀⠀⠀⣀⡠⠚⢢⠙⠒⠛⠛⠁⠀⢰⡟⣀⠀⠀⡼⠋⠀⠀⠀⢱⣀⠀⡀⠁
⢀⡄⠀⡀⠙⠀⢰⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠱⠆⠚⠐⠀⠴⠁⢻⡀⠼⠁⠀⠀⢀⠆⠈⠁
⡤⠀⠀⠐⡆⠀⠈⡄⠀⠀⣿⠀⠀⠀⠀⠘⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⡸
⠑⠀⡸⠀⠛⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⠈⠧⣀⣀⠇
⠀⠀⠈⠢⡀⠀⠀⢰⠀⢸⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⡀⠀⢨⠉
⠀⠀⠀⠀⣦⠀⠀⠐⣇⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⡆⠀⠀⠀⢻⡀⠁⠀⡄
⠀⠀⠀⠀⣿⠀⠀⠀⢸⡏⠀⢸⡀⠀⠀⠀⠀⠂⠀⠀⠄⠇⠀⠀⠀⢸⡇⠀⢨⠇
⠀⠀⠀⠀⢸⢀⡀⠤⠜⢯⠄⢸⡇⠀⠀⠀⠀⠀⠀⠄⠀⡇⠀⠀⠀⠸⣇⢀⡟
⠀⠀⠀⠀⠈⠻⡄⠀⠀⠸⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡀⠤⣿⠊
⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⣠⠟⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠏⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠠⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⢠⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⢛⡏⠀⢲⠀⠁⠉⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠃⠀⠸⡄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠷⡄⠀⡸⠀⠀⠀⠀⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⠀⠀⠀⠀⡀⢰⠁⠀⣣⠔⠀⠀⠀⠐⠒⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⠒⠒⠂⢤⡀⢹⠇⠈⡗⢂⢁⠒⠉⠁⠻⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡀⠀⠈⠉⠉⠢⢠⠟⠀⠀⠀⠀⠘⠁
⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣙⠲⣤⣤⣀⡀⣉⠀⠤⠀⠀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⢿⠿⢿⠿⠻⠿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣛⣟⣛⣿⣿⣿⠿⠿⠿⠿⡻⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢰⡟⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡙⢻⡆⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣾⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠋⠩⠍⣉⡙⠛⠛⠿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⢉⣿⠋⠉⠀⠀⠀⠀⠀⠀⠃⠦⣀⠀⠀⠑⠢⢄⡀⠉⠙⠫⣀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣠⣤⣶⡤⠚⠉⠻⡇⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠈⠒⠤⣀⠀⠉⠲⢶⣦⣄⡀⠀⠀⠹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣾⣿⣿⠋⠀⠀⠠⠂⠙⣦⡀⠀⠈⠑⠢⠄⣀⠀⠀⠀⠀⠀⠈⠑⠂⢤⠀⠀⠉⠒⠤⠀⡈⠻⣧⠀⠀⠀⢻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢰⣿⡿⠁⠀⠀⢀⠆⠀⠀⢨⠿⢦⢄⡀⠀⠀⠈⠁⠒⠠⠄⣀⡀⠀⠀⠀⠈⠁⠒⠒⢀⣤⠼⠛⣻⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡿⠁⠀⢀⡴⠃⠀⢀⠔⠁⠀⠀⠉⠒⠭⣐⠒⠠⠤⠄⠀⣀⣀⣉⠉⠩⠭⠽⠿⠯⢁⣤⡴⣊⣩⠃⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⢀⡴⠊⠀⡠⠖⠁⠀⣠⡴⣁⣀⡀⠀⠀⠀⣠⠶⠦⣀⠲⠶⢦⡄⣤⣤⣤⣤⡤⠤⢦⠄⠀⠈⠉⠀⠉⣛⡆⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⢻⣿⡖⠒⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⢇⠰⢆⣬⡴⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢹⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠁⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⠀⠀⠀⢿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠃⠀⠀⣰⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠘⢿⣿⣿⣿
⡿⠛⠛⠉⠀⠀⢀⣴⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⡀⠀⠀⠙⠻⠿
⣿⣦⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣠⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠎⢌⣱⠎⠁⠒⠤⣄⡛⠿⠆⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡄⣠⠈⡜⠀⠙⠤⣀⠀⠀⠀⠈⠓⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⢧⠆⡐⠀⠀⠀⠀⠀⠉⠑⠢⢄⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⢐⠧⡌⠄⠀⡀⠀⠀⠀⠠⢤⠀⠀⢀⠀⠉⠲⠓⠄⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠎⡘⣈⡠⠞⠉⡇⠀⢀⣅⠤⠚⣷⠉⠑⢄⠀⡘⠄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠒⠚⢴⠛⠉⠀⠀⠐⠁⠀⠄⠙⡶⠎⠽⠃⠀⠀⠉⠹⡦⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠒⠛⡶⠚⠲⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠒⠒⣭⠶⠶⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⡤⠤⠳⠤⡤⢀⡀⠀⠳⠂⠉⠰⠞⠀⠀⠀⣠⣤⠮⡤⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠳⣆⡀⠀⠀⠀⠀⠈⠉⠒⠚⠀⠀⠀⠓⢒⠛⠛⠥⡆⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠐⠁⢰⡆⠀⢈⣉⠳⡦⣄⠀⠀⠀⡜⠎⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠛⠻⣤⣹⠃⠀⠣⠤⠜⡗⠈⡟⠓⠶⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⠃⡴⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠦⣤⣤⣴⢻⡇⠀⠀⣔⠀⢰⡇⠀⡟⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢈⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⠉⠙⠉⠉⠉⠘⡀⠀⠉⠀⡇⠀⣷⠀⠀⡇⠀⠀⠀⠀⠛⠉⠉⠉⠉⠈⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⡁⠀⣿⠀⠀⡇⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠓⠤⠤⠤⠤⠊⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠑⠤⠤⠤⠤⠚⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡟⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣀⣀⣤⣀⣀⣀⣄⣀⣧⣀⣠⣤⣤⣤⣤⣤⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢩⣴⣶⣒⣒⣒⣛⣭⠀⣿⠀⣥⣴⣶⣶⠶⣶⣾⡋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢄⠀⠀⠀⠀⠀⠀⠀⠜⣅⠢⡀⠀⠀⠀⠀⠀⠀⡡⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⠿⢿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣠⠤⠤⣤⣀⣀⡉⠉⠙⠛⠿⠿⣷⣶⣶⡤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡾⠀⡟⠀⠀⠀⠀⠉⠉⠙⠓⠒⠶⠤⣤⣀⣉⠉⠛⠚⠿⠿⢷⣶⣦⣤⣄⣀
⠀⠀⠀⣀⣠⠤⠖⠛⠉⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠶⠤⣤⣀⣈⡉⠉⠛⢿⢷
⣴⣾⣋⣉⠀⠀⢀⣠⠴⠒⢻⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡇⠀⡟⣸
⡇⠀⣀⡉⠉⢻⠉⠀⠀⠀⡿⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢸⢃⡇
⣇⠀⡏⢹⠀⢸⠀⠀⠀⢠⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡾⢸⠁
⣿⠀⣿⢸⠀⢸⠀⠀⠀⢸⠃⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢠⡇⡟
⢿⠀⣿⢸⠀⢸⡆⠀⠀⣾⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣼⢸⠇
⢸⠀⢸⢸⡄⢸⡇⠀⢀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⡇⡾
⢸⡄⠘⠾⠇⠸⡇⠀⢸⠃⠈⠙⠓⠲⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣼⣸⠇
⠘⡇⠰⣿⡇⠀⡇⠀⠈⠙⣒⡶⢤⣀⡀⠀⠈⠉⠙⠒⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⣏⡿
⠀⠙⠲⠦⢤⣤⣧⠴⠞⠋⠁⠀⠀⠀⢉⣽⠶⠦⣤⣀⡀⠀⠈⠉⠛⠲⠦⣤⣀⣀⠀⠀⢀⡏⠀⣼⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠯⣭⣙⡓⣶⣤⣄⣀⡀⠀⠉⠉⠛⠚⠁⢠⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢦⡀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠈⢹⡟⠓⠶⠤⣤⣀⣼⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣍⠳⠦⣄⣀⡀⠀⠀⠀⠀⢀⣠⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠬⣭⣭⣛⣛⣛⣭⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⡆⠀⠀⠀⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⣰⣿⡇⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⠀⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⣀⣤⣿⡿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣴⡏⠀⠀⠀⠀⠀⠀⢀⡴⠿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣧⣀⣼⣿⣿⣿⣿⣿⣿⣿⡟⠉⢻⣦⣀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣶⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠋⠀⢀⣿⣿⣿⣿⣿⣿⣦⣤⣤⣾⣿⣿⣿⠟⠛⠻⣿⣿⣿⣿⣤⣤⡤⠴⠿⡧
⠀⠀⠀⠀⠀⢸⣦⣄⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠞⠋⠉⠉⠉⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣄⣀⣀⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠛⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢿⣿⣿⣿⡿⠉⠉⠉⠉⠉⠉⠙⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⠖⠚⠛⠛⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣹⣿⣿⣿⠀⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣭⣭⡍⠁⠀⠀⠈⢻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠉⠻⣿⣿⣿⣿⣿⣟⡁⢉⣿⣿⣿⣿⣿⣿⣿⣷⣷⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⠞⠁⠀⠀⠀⠀⠀⠀⣸⠟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⠀⠸⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⡄⠙⢿⡟⠁⣴⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⠇⠀⢸⡇⠀⢿⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠠⠶⠶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⠉⠿⠶⢻⣯⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⡟⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⠛⠾⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠱⣋⠀⢸⡃⢈⣻⠋⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠁⢸⠀⠘⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⠀⠀⢸⠀⠀⢀⣠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣦⣼⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠾⠛⢻⠛⠻⢷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣷⣾⣶⣿⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣤⣸⣤⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡛⢻⠛⣻⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣼⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣹⣙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⢰⡾⢿⡆⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⣽⠆⠀⠀⠈⢻⡟⠁⠀⠀⠰⣿⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠈⠉⢿⡄⢀⣿⣀⣸⣇⣀⣿⡀⢀⡿⠉⠁⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣿⡀⠀⢷⣤⡾⠟⠛⠉⠉⢉⡉⠉⠙⠛⠿⢷⣤⡿⠀⢀⣿⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠟⠁⠀⣤⣶⣄⠀⠈⠁⠀⣠⡶⣦⡀⠉⠻⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢷⡆⠀⠛⣷⡟⠁⢠⡄⠈⢿⣴⠟⠀⠀⠀⠀⠹⢷⠾⠃⢀⡀⠈⠻⣾⠋⠀⢰⡾⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠛⠷⣾⠏⠀⢀⣀⡀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⣁⡀⠀⢹⣷⠾⠛⠻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡿⠀⠰⣿⣉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⣿⠄⠀⢿⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣦⣄⣀⣸⡇⠀⢀⠉⠉⠁⠀⠛⠀⠀⢰⡟⢻⡆⠀⠰⠆⠀⠈⠛⠋⠀⠀⢸⣇⣀⣀⣴⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠟⠉⠉⢹⡇⠀⠘⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⡤⠀⢸⡏⠉⠉⠿⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣧⠀⠀⣼⡟⣻⡆⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⡶⣦⡀⠀⠀⠀⣼⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣤⡶⢿⣆⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡾⠃⠀⠀⣰⡿⢶⣤⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡾⠃⠀⣠⡿⣧⡀⠀⢠⡄⠀⠀⢰⡾⢶⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣤⠀⠸⣷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⣀⠀⠀⠀⠘⠷⠟⠃⠀⠀⠖⠀⠀⣀⣴⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣿⠁⠀⣾⠛⢷⣦⣤⣀⣀⣀⣀⣀⣠⣤⣴⡾⠛⢷⠀⠈⣿⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⢀⣀⣾⠁⠈⣿⠉⢹⡏⠉⣿⠁⠘⣷⣀⡀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣿⠆⠀⠀⢠⣼⣧⡀⠀⠀⢰⣿⣹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠸⣷⣾⠇⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠒⠒⠒⠒⠒⠀⢄
⠀⠀⠀⠀⠀⡴⠊⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⢡
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠖⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠑⠢⠀⢀⣀⡀⠀⠠⠂⠄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠏⠉⠉⠉⠩⠽⠿⠿⠿⠿⣷⠀⠀⠂⠀⡠⠀⠀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠦⠂⠀⠀⢀⢂⠄⢎⣡⠀⠀⠀⠀⠀⢰⡠⠤⠔⠀⠀⠀⣿⠀⠤⠄⠊
⠀⠀⠀⠀⠀⡠⠔⠂⠉⠉⠉⠑⠢⡀⠀⠄⠀⠀⠰⠾⡀⢸⠣⠃⠀⠀⠀⠀⠨⡁⠀⠀⠀⠀⠀⣿⠀⠀⡠⠀⠂⠈⠀⠀⠈⠐⠢⡀
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠉⠀⠀⠀⠈⠄⠀⠀⠀⠀⠢⣙⡓⠂⠄⠀⠀⠀⠀⠠⡳⣴⠣⠄⠀⠀⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠂⠀⠀⡁⠀⠀⠉⠉⡡⠀⢄⠀⠀⠁⠈⠉⠉⠁⠉⠉⠍⡉⠉⠁⢸⠀⠂
⠀⠀⠀⠀⠀⠑⠢⠄⠀⠀⠀⢀⡀⢎⠀⠀⠀⠀⢐⠒⠐⠄⠀⠀⠀⠀⠀⠀⢈⠊⠀⠓⣤⠀⠀⢸⠀⠈⠂⠀⠀⢀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠆⠈⠀⠀⠀⣠⣄⣤⡀⠀⠀⢰⠀⠀⠈⡳⢄⠴⡓⠀⠀⢸⠀⣠⠊⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣀⣀⣀⣒⣠⣬⣦⣤⣴⣤⣤⣤⣬⣦⣶⣤⣥⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠤⠠⠄⠀⠄⢠⠈⡗⡺⣾⣽⣧⣗⣛⣗⠒⠚⠛⠛⢭⡷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⡤⠬⠤⡅⢭⠸⡅⢯⠿⣿⢹⡿⣤⠀⡤⠕⣀⠱⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠤⠤⠤⠤⠤⠍⠉⠁⠨⢭⣁⣉⣉⣩⠭⠽⠬⢤⢬⠭⣬⣥⣭⣿⣆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣤⣄⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣁⣀⣀⣀⣈⣁⣀⣀⣀⣀⣀⣀⣁⣠⣠⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣀⠴⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠂⠀⠀⠤⠚⠁⠀⠉⠐⠉⠀⠀⠀⠀⠐⠂⢙⠉⢥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⡀⠤⠞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠑⢆⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠈⠀⠀⢰⠀⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠈⠀⢀⠁⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢀⡀⣀⣀⣀⡀⠀⠀⢀⠜⠉⠣⠄⠠⠒⠉⠓⠦⣀⣀⠤⠆⠁⠠⠄⣈⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣼⠑⠆
⠀⠀⠀⠀⠀⠀⠸⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠸
⠀⠀⠀⠀⠀⡐⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⢃
⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡘⠁
⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠁⠲⠶⠀⠀⣀⣉⣤⡉⠀⠰⠶⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠊⠉⡙⠋⠀⠒⠒⢠⡄⠀⠀⠄⣤⠐⠒⠒⠂⠛⡍⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠸⠁⠲⣄⠀⠀⢀⡇⠀⠀⠀⣿⠀⠀⠀⣠⠔⢺⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⠃⠀⠀⠈⠓⡄⢨⡇⠀⠀⠀⣿⠀⣠⠊⠁⠀⠀⠲⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⢠⠉⠀⠀⠀⡠⠎⠀⢸⡇⠀⠀⠀⣽⠀⠈⢢⠀⠀⠀⠀⢡⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡆⠀⠀⠀⠀⠙⢄⠀⠀⡇⠀⠀⠀⠸⠀⠀⡰⠃⠀⠀⠀⠀⠦⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠳⣀⠇⠀⠀⠀⢀⢀⠎⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⢀⠴⠖⣨⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⣶⢀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠈⠓⠐⠋⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⣥
⠀⠀⣀⡤⠊⠀⠈⠑⠢⢄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⠤⠒⠁⠀⠣⢄⡀
⡤⠚⠁⠀⠀⠀⠀⠀⠀⠈⠙⡇⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣶⡟⠁⠀⠀⠀⠀⠀⠀⠉⠓
⠉⢰⠆⢀⡀⠀⠀⠀⠀⣀⡴⢄⠀⠀⣠⡞⠋⠉⠉⢛⢻⣶⡦⠤⢤⡤⠤⠴⠦⠤⠀⠀⠀⠀⠀⢸⠀⢀⡿⠀⡄⠀⠀⠀⠀⢀⠀⠰⡆
⠀⠈⠶⣟⠁⣰⠆⢀⣤⡀⣷⢀⣇⢠⡏⠀⠀⠀⠀⠀⠈⢹⡇⠀⠐⠇⠀⠀⡆⠀⠀⠀⠀⢀⠀⠀⠀⢨⡏⠀⠇⣀⡀⠠⣄⠈⣷⠶⠃
⠀⠀⠀⠈⠒⠉⠒⠂⠈⢻⡉⠃⢧⡘⣧⡀⠀⡀⠀⠀⠀⣼⡇⠀⠐⡆⠀⠀⡇⠀⠀⠀⡰⠺⣷⣦⠀⣼⠛⡀⢸⠁⠐⠀⠛⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⢸⣦⣬⣷⣶⣤⣤⣤⡾⢁⡦⠀⢠⡇⠀⠀⡇⠀⠀⠘⢤⡀⢘⣿⢠⡏⣭⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢻⣌⣤⣤⣤⢱⣿⠀⠠⢄⠨⣟⠀⢸⡇⠀⠀⡇⠈⠀⠀⠀⣹⣏⣹⣿⣀⡤⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡈⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⣿⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢁⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣿⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⠃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠈⠱⠆⠠⠄⠴⠋⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠈⠲⠄⠀⠠⠴⠋⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃⠀⠸⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠤⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠐⡇⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠆⠺⢿⣭⣭⣤⣤⣬⣽⡋⢸⡏⠘⢯⣭⣭⣭⣭⣽⡟⠿⠃⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠛⠃⠀⠀⠀⠀⠈⠉⠁⣼⠣⠘⠃⠀⠀⠀⠀⠀⠰⠟⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢭⠴⠆⢀⣀⡀⠰⢦⡁⠞⠓⢨⠴⠆⣀⣀⣀⡀⠰⠮⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠔⣠⡐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠐⠐⠉⠀⠈⡘⢠⠀⠀⠔⠈⠉⣦⠀⠀⡄⢃⠁⠀⠉⠂⠂⢤⣀⡀
⠀⠀⠸⢄⣉⠉⠛⠓⠒⠚⠋⠉⢉⠋⠁⠀⠀⠀⠀⢀⠜⠠⠃⠀⠈⣄⣀⣴⡿⠁⠀⠘⠄⠱⡀⠀⠀⠀⠀⠈⠉⡉⠉⠉⠛⠓⠒⠚⠉⢉⡠⠃
⠀⠀⠐⢍⡉⠉⠉⠀⠀⠀⠈⠉⠈⣁⣀⠀⠀⠀⠀⢬⠀⠂⡀⠀⡀⣸⢿⣿⡇⢀⡀⢀⡀⠀⡅⠀⠀⠀⠀⢀⣈⠈⠉⠀⠀⠀⠀⠉⠉⣉⡩⠆
⠀⠀⠀⠀⠠⡛⠉⠁⠀⠀⠉⠉⠁⠈⣠⣤⠀⠀⠀⠈⢳⣂⣬⣁⢤⢼⠀⣿⡧⣤⣈⡭⣐⡞⠁⠀⠀⠀⣤⣄⠁⠈⠉⠁⠀⠀⠈⠙⢛⠄
⠀⠀⠀⠀⠀⠀⠉⠐⠒⣖⠐⠂⠈⢁⣀⠠⠖⠂⣠⠀⣈⠀⢈⠁⢀⣻⠀⣷⣗⡀⠈⠁⠀⣁⠀⣄⠐⠒⠄⣀⡈⠁⠐⠂⡰⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠛⠐⠚⠋⠉⣁⣉⡉⢹⠀⣿⡏⢉⣉⣈⠉⠙⠓⠂⠛⠀⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢉⣹⠶⡃⠀⠀⣿⠅⢘⠶⢯⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠨⣿⠟⣋⡷⠀⣿⢶⣙⠻⣿⠅⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡈⠙⠫⠤⣀⣿⡤⠝⠋⢁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⡀⠉⠓⢤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢈⡛⣦⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠘⢧⣿⠃⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠉⠲⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⣨⡟⣷⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠻⡅⣿⠟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠻⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⡷⣄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⣿⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⢣⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣽⣆⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹⡏⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣫⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡤⢤⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢄⠀⠀⠀⠀⠈⡇⢸⠁⠀⠀⠀⠀⡠⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡞⠁⠀⣀⡴⠀⠀⠀⠀⢠⠇⠸⡄⠀⠀⠀⠀⢦⣀⠀⠈⢳⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢳⠀⠀⠀⠈⠳⠴⠯⣉⣧⠤⠖⠛⠋⠉⠓⠚⠋⠙⠛⠲⠤⣼⣀⠽⠦⠞⠁⠀⠀⠀⡞⠉⠉⠳⣄
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⡇⠀⠀⢀⣴⡋⠳⢤⣀⣀⠞⠁⠀⠀⡴⠋⠉⠙⡆⠀⠀⠀⠀⢰⠋⠉⠙⢦⠀⠀⠈⠳⣀⣀⡤⠞⣙⣦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠙⠦⠖⠋⠉⠉⠙⢦⣀⡼⠀⠀⠀⠀⠱⢤⣤⣾⡁⠀⠀⠀⠀⢈⣷⣤⡤⠞⠀⠀⠀⠀⢧⣀⡴⠋⠉⠉⠙⠲⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠷⠟⠀⠀⠀⠀⠻⠾⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⡀⠀⡾⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢿⠀⢀⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⡷⢤⡇⠀⢠⠏⠉⠙⣄⡤⣄⠀⠀⠀⡞⠉⠉⢳⠀⠀⠀⣠⢤⣠⠏⠉⠹⡄⠀⢸⡤⠾⡄⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢧⠀⢈⡷⠚⡇⠀⠘⢦⣀⡴⠋⠙⠃⠀⠀⠀⢦⣀⣀⡴⠀⠀⠀⠘⠋⠙⣆⣀⡴⠃⠀⢸⠓⢾⡁⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋⠀⠀⢷⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡾⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⠀⣖⡦⠀⠀⠀⠀⣴⣲⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃
⠀⠀⠀⠀⠀⡰⠒⠲⢄⣀⣀⣤⠞⠁⣱⠀⠀⠀⠀⡴⠋⠙⢻⡀⠀⠀⠀⠀⢀⡟⠋⠙⢦⠀⠀⠀⠀⡏⠈⠳⢤⣀⣀⡠⠖⠒⢆
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⣄⡴⠚⠉⠉⢣⡀⠀⠀⠱⣄⣀⣰⠃⠀⠀⠀⠀⠸⣄⣀⣠⠎⠀⠀⢀⠜⠉⠉⠓⢦⣠⠟⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⠀⠀⠀⠀⡟⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡼⠀⠀⠀⢀⡶⠒⢞⣁⡟⠓⠦⢤⣤⣠⠤⠤⣄⣤⡤⠴⠚⢹⣉⡷⠒⢶⡀⠀⠀⠀⢧⣀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠉⢑⠀⠀⠀⠀⠘⡆⢰⠃⠀⠀⠀⠀⡎⠉⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠀⠀⠀⠀⢀⡇⢸⡀⠀⠀⠀⠀⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⠖⠒⠋⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠆⠀⠀⣀⣀⠀⢀⣜⣫⣿⡂⢤⡀⠈⡩⠖⢶⡋⠉⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⣟⣩⣿⣽⣷⣶⠖⠁⠳⣵⣾⡶⢠⢦⡘⡇⠀⠀⠄⠀⢠⠶⢦⡄⠀⢀⣴⡏⢱⣦⣀
⠀⠀⠀⠁⡀⠀⢀⣀⣠⣀⠀⠀⠀⠀⣠⢠⠀⠀⠀⠀⠈⠻⠏⠛⠛⠃⠀⠀⢀⣨⣭⣽⡟⡾⠛⠉⢂⠀⢸⡀⠀⠀⠀⠀⠀⢸⣷⣿⣍⡿⢇⡳
⠀⠀⠀⠀⠀⠈⢣⣾⣿⣿⣶⣤⡀⠀⠈⠎⢇⠀⠀⢀⠀⢀⣤⣄⠀⠀⠀⠀⠇⢹⣿⡽⣷⣟⡱⢀⠀⢣⠘⣷⠄⠀⠀⠀⠁⣾⠿⣟⣷⣿⣯
⠀⠀⠀⠀⠀⠀⠀⢙⠛⢛⡻⣾⠁⠀⣀⠸⡌⢣⠀⠀⠀⡿⣿⣿⣿⣷⣶⡀⠘⠶⣑⣰⠆⠀⠀⠘⠀⣠⢓⡸⢂⠐⡲⢄⢰⣿⣷⡏⡫⢩⡃⢀⡀
⠀⠀⠰⠒⡂⢤⠤⣬⢭⠤⡁⠁⣄⡾⣽⡆⠹⢄⣱⡀⠀⠹⠿⣿⣷⢿⣿⣷⡄⠀⢉⠅⢐⢀⡀⡀⠀⠈⠛⠃⢸⡀⠀⠼⢼⠿⡻⠟⠛⢿⠃⠀⠁
⠀⠀⠀⢡⡁⣈⠾⣤⢃⢳⣾⠶⡿⡿⣷⣿⣯⠤⠀⢀⠶⠀⠀⠘⠢⠿⠻⠃⠀⠃⠂⠠⢘⣖⠇⢀⡤⠠⠤⡈⣧⠿⠶⠃⢀⡠⠀⠀⢀⢀
⠀⠀⠀⠀⢻⠑⢉⣭⠩⠋⢸⣴⣱⢻⣷⣿⡏⣰⡟⣿⣶⣤⡀⠀⢀⠀⠀⠠⡀⠉⠃⠀⠉⠁⠀⠘⣶⣖⠒⠃⡿⡇⡄⠀⠈⠁⠂⠈⠁⣢⠂
⠀⠀⠀⠆⣘⠚⠋⠁⠀⠀⢀⣻⢽⣷⣽⠋⠀⣡⣾⣿⣿⣿⡿⠀⣈⡁⠒⡲⢅⠈⢀⢈⡇⠀⣼⢹⡸⠏⠓⠀⠹⠉⠆⠀⠀⢠⡀⠀⠀⠈
⠀⠀⠀⠃⢨⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣼⣯⣿⡿⣿⣧⡄⠻⠿⡤⠒⠌⠀⠓⠉⢼⡈⣣⢴⣷⠂⠀⢧⠄⠀⠀⠘⠓⡈⠓⠀⠀⠀⡴⠀⠛
⢀⠀⠐⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⠻⡿⠻⠷⠂⠴⠤⠆⠀⠀⠈⣀⡄⠉⠉⢄⠏⠀⠀⠀⠀⠀⢀⡄⠋⠉⠉⠛⣂⠀⠁⠄
⡘⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠙⡷⠶⠀⠐⠀⠀⠀⠀⠀⢠⡌⠚⠠⣀⡀⠃⡄⢨⠉⢰⡿
⣤⣬⣃⠨⢧⡄⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠑⠲⠁⣀⣀⠀⠀⠀⠀⠀⠺⠆⠀⠠⠌⢿⣿⣀⣭⡆⢈⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠄⠠⠤⠄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢠⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠈⡦⢤⠄⡀⠠⠐⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠠⠠⠀⠀⠄⡺⠃⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⡀⠠⡀⠉⠒⢢⠖⢚⠉⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⡸⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠂⠴⠂⠁⠀⠃⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠀⠂⠊⠡⠈⠒⠂⠄
⠀⠀⠀⠀⡰⠄⠀⠑⠀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⠠⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠈⢄⠀⠀⠈⢀⠈⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⢀⠸⣄⠠⠁⠀⢱⠀⠑⢄⠀⠀⠀⠀⠁⠀⠁⠀⠀⡠⠈⠓⠀⠠⠴⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡻⡀⠸⠠⠀⠰⠓⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⠈⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠐⠠⡀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠠⠄⠀⢀⠄⠂⠈⣉⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠉⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⡀⠒⠂⢀⠀⠐⠀⠀⢤⡀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠀⠐⣀⠀⡈⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠨⢀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠆⠀⡀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠸⠀⠀⡀⠦⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠒⢄⠀⠸⠂⠊
⠀⠀⣰⡶⠶⠶⠶⠶⠶⠶⠶⢶⣆⠀⠀⢰⠀⠎⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⠀⡱
⠀⠀⡿⠐⠄⠀⠀⠀⠀⠀⡠⠀⢿⡀⠀⢸⠀⠑⠀⡆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⢀⠀⠀⡜⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⡇⠀⢸⠀⠀⣄⠈⠒⠂⢇⠀⠀⣠⡀⠀⠀⣀⡀⠼⠀⠥⠤⡤⠄⠠
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡰⣣⠀⠀⠀⢅⠒⠤⣀⠀⠀⠀⠠⠭⠋⠀⠀⣀⠄⠒⡡⠀⡀⠀⠀⣐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡴⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠮⠷⠤⠤⠤⠤⠥⠽⠤⠬⠴⠤⠥⠤⠤⠤⠭⠤⠭⠭⠤⠤⠧⠾⠥⠭⠬⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠥⠥⠤
⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⠿⢻⣗⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠇⠀⠀⠉⠀⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢸⠁⢿⡣⢰⣶⣶⡿⠁⠀⠀⣶⣖⣶⣀⣀⣀⣀⣷⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣡⡟⠹⠿⣷⣄⢨⡄⠀⣀⠚⠉⣽⠇⠀⠀⠀⣇⠄⠉⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢀⡠⣰⠃⠀⠁⣈⡽⠖⠒⠢⠽⢳⣀⠛⣠⡞⠁⠀⠀⠀⠀⣇⡩⠆⠀⠀⠀⠀⡇
⠀⠀⢀⡤⠚⢹⠇⢇⠴⡎⠉⠁⠀⠀⡣⠀⠀⣯⡧⠽⡟⢳⣄⠀⠀⠀⠀⣃⡕⠂⠀⠀⠀⠀⣧
⠀⢠⠏⠀⠀⠈⣻⠛⠀⡇⠀⠀⠀⠀⡆⠀⢘⣗⠛⠇⢀⡼⠈⠑⣤⠀⠀⠹⡞⠇⠀⠀⠀⠀⡇
⢰⡃⠀⢀⣀⣾⣇⣀⡼⠁⠐⠀⠀⠀⢁⠀⢸⣄⠒⠀⢨⡆⠀⣼⢿⠀⠀⠘⢿⠃⠀⠀⠀⠀⡇⠀⠀⣀⡀
⠀⠉⠉⠙⠛⠚⠉⢹⡇⠀⠄⢁⠀⠀⠈⡀⠈⠀⡝⣆⠈⠧⣤⣀⣺⠀⠀⡰⢄⡆⠀⠀⠀⠀⣇⡴⠋⠉⠉⠑⡆
⠀⠀⠀⠀⠀⠀⡤⢼⡇⠀⠈⢂⠂⠀⠀⢀⢸⠁⡏⠈⢦⡇⣸⠉⠛⡆⠀⡁⢎⡁⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠑⠢⠀⠀⢸⠀⡇⠀⠱⡔⠋⠀⠀⢱⡀⡣⡊⡁⠀⡠⠊⣠⡔⠲⣤⣠⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠧⡿⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠹⡄⣼⢇⠀⠀⠀⡼⠃⡅⡜⣃⣈⣓⡉⠀⡇⣸⠁⢸⠀⠸⡀
⠀⠚⠒⠒⠒⠒⠒⠓⠒⠒⠒⠒⠒⠒⢲⠋⠀⠀⠀⠹⡗⠒⠚⠒⠚⣧⣤⣧⣉⣿⠀⠰⣷⣶⣷⠗⠒⢺⠒⠒⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⠛⠛⠛⠓⠶⠶⣤⣄⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⡛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢞⣫⡿⠖⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠶⠶⠦⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠓⣾⠋⢰⡾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣤⠾⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠛⠓⠒⡆⠀⣿⠀⢸⠀⠀⢀⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠉⠉⠉⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠉⣙⣛⣛⡓⠒⠒⠒⠒⠒⠃⢀⡏⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⢻⠀⠀⠀⣿⠀⠀⣾⣿⣦
⠀⠀⠀⠀⠀⠀⣿⠀⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⢸⡇⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢸⠃⠀⣿⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⠇⠀⢰⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⢀⡏
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡟⠀⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠓⠒⠒⠚⠛⠛⠀⠀⢀⡟⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⢀⡟⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢿⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⢸⣇⡴⠇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡖⢲⠀⠀⠀⠀⣸⠃⠀⢀⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠩⠉⠉⢩⠉⠉⠉⠛⠛⠛⠛⠛⠀⠀⡿⠀⣠⡿⠉
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣿⣀⡴⠋⠈⠉⠉⠉⠛⡟⢛⡿⠛⠓⣷⡒⠶⠶⠶⠶⢶⣦⢤⣤⣤⣤⣤⣴⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠲⠶⠶⠶⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⣠⠖⠚⠁⢾⣇⡀⠀⠈⠙⠛⠓⠒⠒⠛⠁⠀⣠⡿⠀⠈⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣧⣀⣀⣀⠈⠻⠯⣿⣶⣶⣶⣶⣶⣒⣒⣾⡽⠟⠁⠀⠀⠀⠀⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⡤⠤⡤⢤⡤⠤⣤⢤⣤⣄⣈⣉⣉⣉⡉⠙⠛⠛⠛⠷⠾⠯⢭⣭⣥⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠋⡿⠓⡾⠓⣲⠓⡶⠧⢤⠧⢤⡯⣤⣽⣉⡟⢀⡖⢲⠖⢦⠤⢤⡀⠀⢩⠟⣇⠀⠀⠀⠀⣠⡴⢶⢞⡿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⣉⡽⠉⣹⠛⢲⠟⢲⠓⢲⠿⢤⡾⠤⣼⢥⣼⠁⡜⢉⡟⢒⠟⠒⡾⠁⡰⠋⣴⠏⠀⠀⣠⣞⣁⣠⣧⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⠤⠬⣉⣉⣉⣙⣛⣒⠛⠒⠛⠲⠾⠤⠴⠧⠴⠃⢸⣉⣽⣉⡟⠓⡾⢀⡜⣡⠞⠁⠀⠀⣰⠋⠀⠀⠀⠈⠉⠲⢄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠓⠒⠶⠶⠶⠯⠭⣭⣭⣉⣓⣒⣒⣒⠒⠒⠦⠤⠤⢭⣭⣉⣠⢋⡼⠃⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠓⠲⠶⠶⠶⠿⠟⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣴⣲⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣄⡀⢠⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⢸⡿⠀⠀⠀⢠⡇⠀⠀⠀⢸⣦⣤⣤⣀⡀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⣿⣿⡇⠀⠠⣤⣼⣿⠋⢹⣿⣿⡂⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠾⣿⣿⣿⣿⠿⠿⠿⠿⠟⠓⠲⠿⠿⢿⣶⣿⣿⣿⣧⣤⣶⣶⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠉⠋⠉⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠧⢾⠋⠿⣆⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⣀⠀⢸⣿⣧⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡟⠀⣀⠐⡄⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡇⠈⣿⡠⣻⣾⣷⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠐⠀⣼⡟⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⡀⣠⣿⡇⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣦⣶⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣵⣾⣿⣇⣿⣿⡄⠀⣾⣿⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢹⣿⢘⣿⣿⣴⣿⣿⢁⠽⡻⢿⢿⣿⡿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⡈⠛⠛⠉⠀⠀⠀⠱⠤⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⣧⠐⠁⠀⠀⠀⠐⠄⣗⡘⠉⠓⠒⠉⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣆⢿⡻⣤⣐⣤⠤⠀⠀⣽⣶⣦⣥⣀⣀⣤⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣯⠙⠌⠳⢬⣟⣁⠀⠀⢠⣿⣤⣦⡈⠉⠉⣉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠀⠀⢐⣽⣶⣤⣬⣵⣶⢟⢅⡠⣰⡴⣞⣻⢮⣿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣤⣶⣾⣿⣿⣿⣿⣿⣯⠺⣿⣶⢟⡽⠛⢉⡛⣿⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠳⣤⡄⢰⣆⣭⣩⢦⡿⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠧⠉⠘⠛⠁⠈⠓⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣄⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠉⠉⠉⠀⣹⠀⠀⠀⠀⢀⡗⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⢠⣾⠁⢠⣄⠀⣠⡤⢻⡄⠀⠀⠀⠀⠹⠿⠿⠟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠠⠤⢠⣤⠤⠤⠄⡄⠀⠀⠀⠈⢻⡀⠈⢉⢀⣉⠀⡾⠃⠀⠀⠀⢀⡤⠤⠒⠒⠒⠠⠄⡀
⠀⠀⠀⠀⢀⣠⠤⠒⠒⠒⠒⠲⠅⠀⠀⠀⠀⠀⣽⠀⠀⣉⠐⢸⠁⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣂⠁⠻⢿⣦⣤⡴⠃⠘⠢⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⢰⠁⠀⠀⠀⠀⢀⣠⡶⡛⠀⠈⠃⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣱⣄⡀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠘⠀⠀⠀⢀⡐⡏⢹⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⣠⠿⣄⠀⠀⠀⠀⠀⡞
⠀⠀⠀⣀⣤⡾⠭⣄⢰⣼⡇⠀⢀⡤⠤⡀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣋⣀⡤⠞⠓⢄⡀⢀⡞⠁
⠀⢰⠋⠁⠀⠀⠀⠀⡹⡿⠷⠔⠚⠁⠑⢣⡀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢻⠿⢿⣻⣿⣛⡛⣌
⠀⠈⢿⡶⣤⡀⡠⠪⣎⣀⠀⡀⠀⠀⣀⣰⠻⡄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⣹⠏⠉⠉⠈⠉⠈⢷
⠀⠀⠀⠙⢦⡁⣇⠔⠀⠀⠀⢹⣿⣿⣿⣿⣷⡈⠢⡀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠾⠒⠛⡁⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⣿⢻⣿⠋⡟⠛⠂⠙⠂⠒⠒⠒⠚⠑⢂⡠⠀⠀⣀⠤⣾⣇⣨⠭⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠻⠁⠀⠘⣄⠀⠀⠀⠀⠀⢠⣊⡀⢉⣪⡤⠾⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠈⠁⠀⠀⠀⠈⠙⠲⠤⠤⠤⠤⠴⠛⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⢰⣯⠀⠀⡘⢆⠀⠀⠀⠀⠀⣠⠀⠀⡀⡀⠀⠐⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⣿⣟⣡⢸⡵⡇⡠⢸⡏⠀⠀⠀⠀⠀⠘⣿⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⣩⠓⣿⡿⠿⢞⣿⣴⢇⣨⣭⡤⢤⠂⠆⠀⠀⢈⣿⣳⣾⣿⣿⣆⡀⠀⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⣾⣿⣇⣿⣱⣿⣷⡶⢞⣯⣄⣦⣤⡄⣉⠀⢀⣀⣴⣿⣿⣿⣟⢿⣿⡄⠀⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⣙⣿⣿⣿⡉⠙⣻⣯⠿⠿⢶⠶⡤⢼⣤⣀⣠⠛⣉⣽⠃⣸⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠁⣠⣿⣨⠿⣷⡿⢿⣷⣾⡟⠀⠀⢀⣨⣅⣀⠀⠐⠉⠄⠈⠃⣠⡞⠉⣏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡞⢩⢄⠉⠽⣪⣤⡾⢿⡟⣄⠠⠔⠻⢿⣿⢿⣿⣿⠚⠛⠉⠉⠙⣿⣤⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣿⠘⣿⠦⣶⠶⠚⣯⡀⠀⣾⠟⠁⢠⡄⠀⠀⢹⡌⢹⠟⠀⡀⣀⣠⠞⠛⢿⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⢹⣶⠟⠚⠛⠁⠋⢰⣿⡆⢤⣌⣧⠀⠀⠈⣿⡾⠷⠟⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠙⢿⡼⠟⢭⠓⠂⠐⠶⠋⢹⡿⢨⣿⣿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠈⠻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢣⠄⣧⡏⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡾⢟⠼⡃⠀⢶⠶⢀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠟⢀⢻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣀⠴⠒⢢⡀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⡓⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣎⣁⣀⣀⡀⠙⢄⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣰⢋⣩⣽⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡀⠀⠀⠉⠓⠮⡑⢤⣀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⠋⠴⢚⣷⡀⠀⠀⠀⠀⠀⠀⠘⢿⣈⣛⣰⡀⠀⢴⡄⢸⠖⠋⠀⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠹⣝⣿⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠛⠉⠲⠤⠴⠋⠀⣤⣠⠞⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠘⡟⣆⠀⠀⠀⠀⠀⠀⠀⡷⡀⠰⠤⠴⠀⠀⢀⡴⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡿⣼⡄⠀⠀⠀⠀⠀⠰⢇⣈⣓⣶⠴⠶⠒⠋⠀⡧⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣻⣙⣛⡩⣷⠈⠁⠀⠀⠀⠀⣠⠖⣻⠟⢱⠟⠳⣄⣠⠴⠚⣳⠈⢻⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠀⠀⣿⠀⣴⢲⣄⣠⠞⠁⠾⣅⡀⢸⣄⣴⡿⡅⠀⣰⠃⠀⢸⡉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣠⡏⠉⣿⠁⠀⠀⢠⠞⠁⢸⠈⢹⠀⠱⡾⠃⢀⣀⣈⡇⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⠀⣧⣴⣛⠉⠙⠒⠾⢧⡀⢸⠀⡟⢀⡼⠁⠀⠻⡄⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢀⣟⡭⠤⠀⠀⠀⠀⠀⠉⢿⡀⣧⠞⠀⠀⣠⠔⠃⠀⣀⠔⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣈⡯⠾⣉⣁⣀⣀⣀⣀⠀⠀⠙⢧⡀⢠⠞⠁⠀⠀⡴⠁⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠛⢉⡝⠀⠀⠈⠳⡄⡏⠀⠘⢦⣀⡠⠏⣹⠓⢤⡀⠀⡾⠁⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⢹⣄⡰⠃⠀⠀⠉⢺⡁⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠖⠒⠒⠒⠲⢄⣿⠀⠀⢀⡞⡼⠃⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⣸⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠓⠦⣀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⢄⣀⣀⠴⠁⠀⠀⠳

⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⣨⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣅
⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⣰⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢇⠀⠀⠀⠀⠀⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠋⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠓⠁⠀⠗⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠈⠒⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡄⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄⠀⠀⠀⡀⣀⡀⠠⡇⠀⢀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣁⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣁⣠⠀⠀⠠⠆⠒⠈⠁⠁⠈⢈⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠒⠒⠚⠉⠁⠀⠉⠙⠄⠤⠠⠤⠠⠦⠤⠴⠦⠤⠖⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠁⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣟⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣽⣯⣽⣿⣭⣿⣯⣯

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠘⠛⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢀⣴⣿⣿⣿⣿⣿⣿⡤⠤⠴⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡆⠚⠛⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠏⠉⠀⢀⣉⡘⠂⠀⣠⣖⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡍⠀⠀⠀⠀⠈⠛⠁⠀⠀⠛⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢽⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣼⣿⣿⡇⠻⢷⣤⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣭⣴⡆⠀⠀⠀⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⡐⢰⡟⡆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢿⡉⠻⡋⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⢁⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠚⠉⠀⠀⠈⠲⠌⠓⠄⠀⠀⠀⠀⠱⠀⢀⠀⠀⠀⠀⠀⠈⠀⢰⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢸⠀⠀⠰⡆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢣⡀⣼⠀⠀⠀⠇
⠀⠀⠀⠀⡰⠋⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⣸⣧⡖⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⢷⡏⠀⠀⠀⢰
⠀⠀⢀⡞⠁⠀⠀⠀⠀⢙⣦⡀⠀⠀⣀⢴⣿⣟⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⡞⠀⠀⠀⠀⢸
⠀⢀⠏⠀⠀⠀⠀⢀⠔⢯⣻⣷⣰⠞⠁⠈⠻⡿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⡸⠀⡇⠀⠀⠀⢸
⠀⠊⠀⠀⠀⠀⠀⠑⠦⡀⠉⠻⠏⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠤⠤⠆⠈⠀⠀⠀⣸⠀⣠⣇⠀⢻⡀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⢩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡏⠻⢧⠀⠁⠀⠀⠇
⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⣀⠀⠀⠀⠉⠒⢄⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⠀⠀⠈⠒⢼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠁

⠀⠀⠀⠀⠀⢀⣠⡴⣶⣶⣾⣶⣶⣦⣤⡀⠀⠀⠀⢀⣠⣤⣤⣠⡄⣠⠄⠀⠀⠀⢤⣤⡄⠀⠠⣤⡤⠀⠀⢤⣤⠄
⠀⠀⠀⣠⣾⣯⣷⣿⣿⣿⣿⣿⣿⣿⣾⣝⢷⣄⣠⡟⠁⢉⣽⡋⣸⣿⣿⣆⠀⣄⢺⣿⠀⠀⢀⠛⣗⠀⢀⣙⠻⣀⡀
⠀⠀⣰⢫⣾⡇⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣷⣽⡟⢧⣴⣿⠟⠁⠀⠈⠻⣿⡆⢿⡭⠥⣽⣿⣧⠶⠬⡇⢸⡥⠤⢼⡇
⠀⢰⣯⣿⣿⡇⠀⠀⣶⠆⠀⠀⣸⣿⡿⠟⠻⣿⡇⠀⠀⠀⠀⠀⢰⠆⠀⢹⡷⣾⠁⠀⠈⡇⡏⠉⠁⢡⢸⠉⠉⢹⡇
⢰⣾⣻⣿⣿⡇⠀⠀⢀⡀⠀⠘⠿⠋⠀⢀⣴⣿⣇⠀⣀⠀⠀⠀⢶⣶⡗⣠⢸⣿⢶⣶⡾⠿⠷⣶⣶⠾⠾⢶⣶⡾⠷⣤⡤
⠀⠸⡟⣿⣿⡇⠀⠀⢸⣿⡆⠀⠀⠀⠴⣿⣿⣿⣯⠉⣉⣹⡶⠾⢻⣾⡧⠃⣾⠇⠲⣿⡆⠀⠀⢷⣿⠀⠀⢸⣿⠇⢰⡆
⠀⠀⠻⣜⣿⣷⣶⣶⠞⠉⠀⢀⣤⡀⢀⣸⣿⣣⡟⢲⡝⠋⢀⣀⡇⠤⠄⢀⣷⣼⣋⠉⢙⣶⡏⡉⡈⠉⢟⣚⣓⣒⣿⣧
⠀⠀⠀⠙⢮⣛⢿⣿⣄⢀⣴⣿⣿⣿⡿⣻⡵⠋⠀⠈⢷⡘⠡⠽⣴⠚⠀⣸⣿⠀⠻⡶⠾⢿⣯⣦⣤⣂⣸⣥⡭⠏⣿⣿
⠠⣤⣤⣤⣤⣭⣿⣶⣭⣭⣭⣭⣽⣶⣿⣯⣤⣤⣤⣴⠟⢹⣦⣀⣈⣁⣴⠋⣿⠀⠀⠙⠀⠀⠀⠀⠈⣍⠉⠉⢤⣠⣏⡋
⠀⠉⠉⠉⣾⡷⠀⠁⢸⣿⠃⠉⢉⣩⣽⣿⡿⠟⠁⣿⠀⠀⣧⠈⣽⣿⣅⢸⡏⢳⣴⡾⠛⠀⠀⠀⠀⢹⡀⣼⠈⢷⠻⣿⠗
⠀⠀⠀⣶⠚⠛⢲⣴⠊⠉⢻⣶⡿⠿⠛⠉⠀⠀⢰⣇⡀⣠⢻⡼⢻⣿⠙⣿⠀⠀⠉⠉⣹⠖⠀⠀⠀⢸⣇⣿⠀⠈⢷⣿⠇
⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⢸⣿⠀⢴⠀⠀⠀⠀⣠⠽⠛⠃⠘⡇⣾⣿⣤⡇⠀⠀⢀⣼⠏⠀⢀⠀⠀⠀⣿⡏⠀⠀⠈⢿⡆
⢰⣶⣶⣿⣶⣶⣾⣿⡦⠤⢼⡏⠀⢸⡄⠀⠀⠀⠸⣆⠀⠀⠀⢹⣿⣿⡟⠀⠀⣠⣿⠏⠀⢰⣿⣇⡀⠀⣿⠀⠀⠀⡠⡞⠃
⠀⢡⢄⠀⠀⠀⠀⠀⠀⠀⣾⢣⠀⡇⢿⠀⠀⠰⣆⠻⣧⡀⠀⠀⢿⡿⠁⠀⣴⡿⠃⠈⠋⠉⠉⣹⠀⠀⢿⠀⣠⠞⣼⠃⠠
⠀⠈⣛⣀⢀⣀⡀⠀⠀⢀⡟⠸⡇⣿⠸⡇⠀⢠⠈⢣⡙⢷⣄⠀⣸⢁⣠⣞⡋⠀⠀⢀⠤⢀⣀⠉⠀⠀⢸⣦⠀⡸⣁⠴⠒
⠈⢯⢉⢛⣯⣭⠃⠀⠀⢀⡇⣧⢷⣿⠀⢻⡀⢀⠣⡠⠑⣴⣿⢷⣷⡟⠁⠉⡩⢝⣟⢉⡴⠋⠼⣛⠆⠀⢸⣿⠧⢭⣴⠶⠆
⢀⣘⣏⣙⣭⣃⣀⡀⣰⠿⣆⢹⡜⣿⡀⣤⣿⣌⢣⡙⠀⡿⠷⢦⣿⠀⠀⠀⡇⠐⢰⢻⢡⠋⣰⢡⢪⢀⣿⣷⡀⠸⢴⣶
⠘⠛⢿⣿⣿⣿⣿⣿⡟⠁⠹⣦⡷⠘⢣⣿⣿⣿⣆⢳⣠⡇⠀⢸⣟⠻⠿⣿⣀⣘⣃⠀⣿⠿⠛⠃⡼⢸⠁⠙⢻⡀⠀⠈⠁
⠀⠀⠀⢠⣿⣿⣿⡿⠀⢀⣠⣈⣿⡄⣸⣿⣿⣿⣿⣆⢹⣅⠀⠀⣿⣽⠉⠛⠛⠛⠿⠿⣿⣶⣀⢀⣡⠾⣶⣶⡈⠗⠀⣀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⢀⣈⣿⣟⣿⡻⣯⠀⢙⡏⠀⠀⢹⣿⣇⡀⠀⠀⠀⠀⠀⠀⠈⢉⣴⢀⣿⣿⣿⣿⣿⠿
⠀⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⢠⡿⠻⡚⠛⠊⣥⣿⣿⣸⠁⠀⠀⠈⣿⡟⠁⣠⣴⡆⠀⠀⠀⠀⣾⠇⢸⣿⣿⣯⢤⣤⡆
⠀⠶⠶⠶⠛⢻⣿⣦⣀⠀⠀⣿⠐⢲⣶⠿⠭⠿⠉⣟⢸⠤⠴⠶⠶⠾⠶⠤⠹⠷⣷⡆⠶⠶⠶⠷⠶⠖⠛⢛⠟⠃⣚⡃
⠀⠀⡇⠀⠀⢸⣿⡇⠉⡷⠦⣝⡲⣶⣾⡿⢥⣴⠆⠃⣾⠀⠀⠀⠀⠀⠀⠀⠀⡶⢭⢯⣯⡿⠿⠯⢽⣦⣤⣾⠀⡆⠉⠁
⠀⠀⡇⠀⠀⢸⠙⡇⠀⡇⡇⠨⢸⣿⣿⠆⠀⢿⣷⣴⠃⠀⠛⠛⠛⠛⠛⠻⠛⢇⢤⣆⣱⡑⢌⣻⡾⠀⠀⣿⢾⡿⠿⠁
⠀⠀⡇⠀⠀⠸⠀⡇⠀⠃⠀⠀⢸⣿⣿⣧⣶⣋⣹⣟⣷⣦⡄⠀⠀⠀⠀⠀⠀⠈⠦⣏⠁⠑⣠⠟⠁⠀⠀⣿⢸⡇⢰
⠀⠀⠇⠀⠀⠀⠰⡄⠀⠀⠀⠀⢸⣿⣿⠉⢻⠛⢻⡟⠉⢹⡆⠀⠀⠀⠀⠀⠀⠀⢐⡟⢲⠶⠚⢿⠇⠀⠀⣿⢸⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠀⠀⠀⢻⣿⠀⠈⠀⢸⡇⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠸⠀⢸⠀⠀⠀⠹⡘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠿⠀⠀⠀⠐⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠸⠀⠀⢸⠀⠀⠀⠀⡏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣭⣭⣭⣭⣭⣭⠉⠁⠀⠀⠀⠀⣨⣛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣤⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣿⣟⣛⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⡀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣯⣭⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠿⠿⠿⠿⠿⠿⠿⠷⣿⡇⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡇⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⠉⣭⡍⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡄⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠀⠶⠾⠿⠷⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣲⢾⠆⠀⠻⠃⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⡋⣽⣿⣥⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣯⣾⢟⠕⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡠⣪⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⡴⣫⣾⢟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⠟⠁⡠⣺⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢙⣿⣟⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡦⠙⠦⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠳⡏⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣌⢰⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿






⠀⠀⠀⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣰⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠲⡄⠀⢸⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠾⠃⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣒⣀⣒⣀⣀⣀⣠⣏⣀⣤⣤⡤⠴⠶⠖⢺
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣴⣒⣖⣒⣶⣒⣒⣖⡒⣲⠖⠒⡶⠖⢤⠤⠭⢭⠭⢭⣍⠉⠉⣩⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡾⢁⠾⢤⠾⢤⡾⠥⣤⠯⢬⡽⠭⢭⠿⠭⣽⢯⣉⣽⣋⡲⣾⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢠⠟⡠⠛⢲⠛⢲⠞⠒⡴⠃⢢⠞⠤⡴⠧⢤⠾⠥⣤⠮⣤⢞⡗⠁⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⢠⠟⠐⠿⠿⠿⠿⠿⠿⠿⠷⠾⠟⠖⠾⢷⣶⣿⣶⣺⣗⣲⡽⠋⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⡿⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠋⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⣤⠴⠞⠋⠉
⠀⠀⠀⠀⠙⠛⠛⠛⠒⠒⠲⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣄⣀⣀⣀⣀⣿⣤⠴⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠰⠄⡈⠗⠈⠁⠐⣁⡠⠀⠀⢴⠋⠀⠀⠈⠑⠢⢄⡀⢀⡀⠆⠒⠒⠒⠶⣀⠀⠀⠀⣠⠺⠚⠰⠷⢄⠄⠀⠀⠘⡀⠢⠵⠀⠆⣀⠔⠀⠀⠀⡄⡈⢁⡱⠀⢠
⠀⡤⠤⠁⠀⠀⠀⠀⠈⠤⠆⠀⠘⠖⠖⠂⠦⣀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢀⡸⠀⠐⢸⡍⠀⠀⠂⠀⠻⠄⠀⠀⡐⠃⠀⠀⠀⠀⠀⡣⠀⠀⠀⢀⢗⠈⠃⡇
⠀⢀⡰⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠈⠉⠀⠀⠉⠉⠁⠁⠀⠀⠈⢫⣄⢀⣀⡰⠛⠈⠀⠀⢑⠅⠀⠀⠀⠀⠀⡤⠈⠀⠀⠈⠀⢤⡄⠀⠰⠆
⠀⠀⠀⠞⠀⣀⠀⠢⠄⠀⠀⢠⠀⠒⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠎⠈⠱⠂⠀⠀⠀⢀⠀⠀⠍⠈⠐⢀⠀⠀⠀⠄⠸⠰⢀⡄⠦⠉⡁⠀⠀⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠘⠠⡈⠂⠀⠀⠀⠀⢰⠄⠀⠂⠈⠒⠀⠀⠀⢀⠀⠀⣐⠆⠞⠠⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠟
⠀⠀⠀⡔⠒⢢⠀⠀⠀⠀⠀⠀⠢⠀⠈⡀⠀⡈⠀⠀⠠⠐⠄⠠⠂⠀⠆⠀⠁⠄⠀⠀⠉⠀⠀⣩⠄⣀⣀⠓⠠⣠⠀⠀⣀⣀⡀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠠⠐
⠀⠀⠰⠀⠠⠀⠀⠀⠴⠀⠠⠁⢰⢐⠈⠡⠀⠀⠀⠈⠈⠀⠈⠀⠀⠠⡄⠄⠁⠀⠀⠀⠀⠀⠀⠀⠘⠃⠉⠀⠀⠱⢵⣄⡈⠉⠛⠲⣤
⠀⠀⠂⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⣀⣀⡀⠀⠤⠤⠐⠉⠉⠢⡄⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠑⡥⠀⠀⠀⠂⡀⠀⠔⠂
⠀⢘⠀⠀⢠⠃⠀⠰⠉⠈⠱⠀⠀⠀⠀⠀⠒⠀⠀⡄⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⡸⠀⠀⣠⠮⠃⠘⣵⠀⠈⡗⠀⠀⠀⠀⠀⠀⠁⠀⢁⠂
⠀⠈⢲⠐⠁⠀⠀⡄⠀⠀⠌⠀⠀⠀⢀⠀⣀⠀⢂⠈⠀⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⡀⠠⠄⠖⠁⠀⢬⠋⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠈⠈⠄⣅⢆⠂
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⢀⠀⠈⢀⠀⠀⠀⠆⢠⠀⠈⠑⠒⠰⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠰⠜⠛⠋⠀⠰⠀⠀⢸⠀⠂⠀⠘⡄
⠀⠀⠂⠀⠀⠀⠀⡄⠀⡸⠀⠀⠀⠈⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⡀⢀⢀⣀⣀⣤⣴⣶⣾⣶⣶⣤⡀⠀⠀⠀⠋⠀⠀⠀⠰⠓⠁⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣤⠀⠀⡗⠀⡀⠀⠀⠀⠀⠀⣴⣾⡿⠟⠟⣛⠻⠻⡟⠟⠉⠹⠿⣿⣏⠉⡹⢿⣿⣿⡆⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡄⣀⠀⠢⣀⣄⡀⢁⠠⠆⠀⠠⣼⡏⢸⢰⡄⠀⠈⠰⢧⠙⡆⠹⡿⠷⠂⢹⣯⣥⣾⡟⠻⣿⠀⠀⠀⠀⠀⠀⠀⡀⢡⠀⡄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡁⠀⣋⠀⢀⡀⠀⢰⣿⡇⠀⣸⣇⣸⣤⣷⣴⣷⡴⢾⡿⠶⢾⡟⢛⠛⠙⡆⢁⢻⡇⠀⠀⢀⠄⠡⠂⠀⠄⠐⠌⠁⢄⡰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠉⠋⠁⢀⡀⠀⠀⠐⣿⠟⡛⢿⠉⡝⢿⠀⣦⣶⣶⠸⣇⠘⠛⣣⡈⠠⣆⠁⢸⣾⡷⠀⠀⠸⡀⠀⢀⠈⡀⠉⡡⢀⠀⢀⡀
⠀⠀⡴⠒⠒⠲⣄⠀⠀⠀⠀⠀⠈⠁⠀⠴⠀⠀⠀⠀⠀⠀⡹⣧⠡⣜⠃⠰⣄⠡⡙⠋⢹⣧⢹⡄⠙⠛⢣⢠⣈⣀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢄⠀⢠⠈⠢⠬⠤⠃
⠀⢼⠀⠀⠀⠀⢈⡄⠀⢀⠀⠴⠀⠀⢤⠀⢠⠀⢀⡀⠀⠀⠀⠨⠛⠶⢾⠶⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠋⠁⠀⣀⠀⡔⠆⠀⠀⠀⠑⠂
⠀⠘⢦⣀⣀⣀⠜⠀⠀⠀⠀⠀⠀⠓⢤⠦⢤⡔⢁⡀⠀⢔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡬⣃⡟⢠⣌⢀⠀⠀⠀⠀⠀⡀
⠀⠀⣀⠀⠀⠀⣀⠤⠤⢄⡀⠘⠀⠀⠹⢄⣠⠏⠀⣀⠀⢀⡄⠀⡀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⢖⣄⣸⠶⠛⠉⠉⠀⠀⠀⠠⡀⠐
⠀⢘⠁⢸⠆⡼⠁⠀⠀⠀⠘⡆⠀⠒⠀⡐⠀⠦⠀⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠁⡰⠋⠙⡆⠀⠀⠀⠐⠢⠼⠛⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠑⠀⠀⠁⢀
⠀⠈⠉⠉⠀⢇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⡎⠀⠈⢡⠀⠀⠀⠀⠈⠢⠔⠁⠀⠀⣠⢠⣤⣤⣤⠲⣾⣩⠻⡍⢣⠀⠉⠀⠀⡀⠀⠀⡠⠀⢀
⠀⠀⠄⠀⠀⠈⠓⠤⡤⠤⠊⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⠀⠀⠀⠑⠄⠔⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠐⣩⡆⣿⡞⠷⠍⠛⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠘⠀⢀⡀⠀⡀⠀⠀⠤⠀⠀⡀⠀⠀⢀⡀⢀⡀⠀⠀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠄⠀⠈⠀⢈⠀⠀⠀⠀⠀⠀⠁⠀⠀⡠⢀⠂⢒⡠⡀
⠀⠀⣐⠂⢘⠀⡀⡎⢃⢩⡩⠤⠼⢄⡠⢀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡀⠄⠀⠀⠀⠀⠀⡀⠀⠂⠀⠀⠀⠀⢀⡀⢀⠄⠲⠒⠠⡀⢀⠀⢀⠎⠂⠱⠀⠀⠼⢈⠆
⠀⠆⢨⠖⠉⠉⠈⠁⠈⠀⠀⠈⠉⠀⠓⢼⠀⠀⠀⣠⠴⠟⠉⠋⢹⡆⠀⠀⠀⣠⠖⠑⢠⠅⠂⠀⠀⠀⠁⠀⠀⢨⡠⠀⠀⠀⠐⣩⠀⠀⢨⢰⠀⡄⠀⡄⠀⠰⠈
⠀⡌⢸⡀⠀⠀⠀⠀⢀⠀⠀⢀⠀⡀⢫⡉⠀⢠⠞⠁⣠⠤⠤⠖⠛⠀⣀⡠⠔⠁⠀⢀⡜⠀⠀⠀⠒⠀⠀⠀⠐⠨⢴⡄⠀⠀⢄⠈⠀⠀⠀⠢⡑⠄⢀⠀⢔⠡⠂
⠀⠀⣃⠜⡘⠤⠤⠄⢂⠠⡌⣤⠗⠈⠆⠀⠀⢯⡼⠊⠁⠀⠀⠀⢰⡋⠉⣀⣀⠤⠒⠉⠀⠀⠀⠠⠆⣀⠀⠀⠀⠀⢠⠂⠋⠣⠁⠀⠀⠀⢀⡀⠀⠁⠀⠈⠁
⠀⠀⠀⠈⠀⠆⠖⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣿⡛⣛⢻⣾⣗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠛⠋⠛⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⢖⡶⣄⣠⢾⠝⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠈⠁⠇⠘⠤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣆⠀⢨⣀⡄⢀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⢄⠀⠀⠒⣿⣿⠏⠀⠠⠀⠠⠂⠿⣿⢷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⡈⠁⣱⠠⢉⡄⢀⠀⠀⠀⠐⠀⠐⠈⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡴⠖⠂⡆⠂⠀⠀⠐⠉⢠⡁⠀⢸⡇⠈⢿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠸⠙⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠡⣀⡴⢖⡠⠖⠲⢶⠦⠈⠆⠀⡘⢻⡿⢀⡴⠋⠀⠀⠀⣀⠀⣤⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⣦⣠⣴⣾⣯⣥⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠱⡏⠀⠐⠒⠤⠀⣿⣾⠃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠤⠼⠿⠛⠉⠉⠉⠉⠉⠉⣷⣄⠀⢀⡀⢀⣄⡎⢧⠀⠁⢀⠀⠀⡀⠘⢻⡇⠀⠀⠀⠀⢀⣀⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢟⣧⣵⠸⡆⠐⡇⠠⠄⡀⠀⠸⠃⠀⠈⠉⠑⢏⠡⡀⠀⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡩⠍⠉⠡⠞⠛⢻⣄⡇⠀⠑⠀⠀⠈⢰⣶⡶⢤⣀⠀⠀⢸⠇⡵⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠱⡀⠀⠀⠀⠀⣾⣧⣤⠠⠀⠀⠀⠀⣾⣿⠃⠀⠈⠛⠲⣾⢠⣧⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠑⠀⠀⠀⠜⠛⠛⠁⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠁⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠑⠒⠄⢀⠐⢂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠨⢤⡠⠀⠀⠀⠀⠀⠀⠘⠂⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠚⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠘⠛⠛⠶⠄⠀⠀⠀⠀⠀⠀⠱⠢⠀⠀⠀⠀⠀⠀⠀⠀⢉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣤⠀⠀⠄⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⡠⠴⢦⡀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⣤⠀⠠⠤⣦⢸⣆⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣰⠀⢸⣿⠀⠀⠐⠂
⠀⠀⠘⡢⠤⠤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠒⠂⠐⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠴⠋⢸⠀⠀⣎
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠦⠤⢤⠔⠚⠁⠀⠀⠀⠀⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣤⠤⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠔⠀⡲⠊
⠀⠀⠀⣗⣀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠈⠳⣄
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠡⢄⠀⠑⠲⡤⠤⣤⠤⢠⠔⠖⠚⢙⣫⠀⠀⠀⠀⠀⠈⠳⣄
⠀⡞⠁⠱⣄⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⢀⠊⠀⠀⠀⣡⠀⠀⡖⠁⠀⠀⢞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⢀⠡⠤⣄⠚⢀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠑⠢⡀⠀⠙⣠⠎⢀⣠⠄⠶⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢇⡀⢀⣈⠆⢸⡟⠃⠀⡀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠿⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⢧
⢨⡉⠉⠙⡄⢸⡇⠀⡘⠉⢧⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠠⠆⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠈⢧
⠰⡉⠙⡏⠀⢸⡁⠎⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠸
⠀⠈⠉⡇⠀⢸⡇⠀⠀⠀⣰⠃⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠰⢄⣠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠇⠀⢸⠁⠀⢀⡴⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠁⠀⢀⣀⣀⡀⢇⡘⠦⡀⠀⠀⠀⢀⣀⣤⠔⠋⠀⠀⠀⠀⠀⡸
⠀⠀⠀⢧⣀⡼⠀⠘⠋⠑⠦⣀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠠⠄⢀⣀⣀⣀⣀⠈⢽⠀⠀⠙⢏⠉⠀⠀⠀⣂⠀⠀⠀⠀⢀⠜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠈⠄⠀⠀⠀⠨⡄⠀⣠⠔⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⣀⠀⠀⠠⠋⠀⠀⠀⠀⠀⢰⠀⠀⣀⠠⠖⠈⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡆⣤⣤⣤⣤⠖⢀⠄⠀⠀⢠⠏⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠀⠀⠀⢀⡠⠂⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠉⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠧⠤⠤⠤⢤⠤⠄⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠢⠤⠤⢤⡤⠤⠤⠤⠤⠤⢤⡤⠄⠀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠄⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠸⠆⠀⢀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠤⣤⣤⠀⠀⠶⠖⠛⠋⠙⠥⠄⠤⠄⠖⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣢⣤⣤⣤⣤⣤⣚⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡶⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡏⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⢧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣹⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⣏⡀
⠀⠀⠀⠀⠀⢀⡼⢋⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡙⢦⡀
⠀⠀⠀⢀⡴⠋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⢦⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⢰⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡯⢿⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⢀⠀⠈⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢙⣟⠁⠛⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢶⠄⣿⣥⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢠⠃⠀⠘⢻⣿⣷⣶⣦⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⠇⠀⠀⢀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠀⣌⠹⣿⠛⠐⢿⣿⡏⢹⣿⣿⡿⢿⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⣾⡀⠀⠀⠀⠑⡀⢠⠀⠀⢀⡀⣯⣀⠁⡐⠒⠂⣵⡆⢀⠹⠟⠁⠘⢿⠿⠃⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡃⠀⠀⠀⠀⠙⠀⠀⢀⠈⠁⢀⣀⡀⠍⠀⠀⡿⣿⣿⡇⢒⠀⣤⢸⣶⣶⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⣶⣦⣤⡆⠀⠄⢸⠟⠛⣧⠀⠀⠀⡏⣾⢳⣭⢘⡂⠈⠘⠓⠚⠛⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠾⠁⠉⠁⠟⠛⠋⠀⢰⢧⠀⠀⢹⠀⠠⠀⠁⠹⠏⠃⢸⠇⡟⣴⣦⣤⣴⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⣀⠀⠀⠀⠉⠁⠀⢀⠀⢀⣀⠀⡀⠀⠀⢠⡀⠞⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠒⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠐⡱⣀⠀⠀⣀⠀⠀⠀⣤⡔⠀⠀⠀⠈⠉⠀⠀⢀⣀⡀⠀⠀⣀⢀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠁⠁⠀⠀⢰⠀⠀⠀⡇⠁⢀⠀⠀⠀⠀⠀⠀⠌⠛⠒⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⡀⠀⠀⠀⠀⠈⠈⠒
⠀⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠘⠟⠿⠿⠟⠛⠟⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⠻⠗⠂⠀⠄⠢⠰⠀⠀⠀⠇⠂⠀⡀⠀⠀⠠⢤⣤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠙⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⡻⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⣄⠀⠀⠀⠀⠀⣀⣀⠀⣠⠀⠀⠀⢀⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠁⠀⠀⠀⠀⠈⢉⡹⠿⠟⠀⠀⠀⠀⠘⠏⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡆⠀⠀⢴⣦⠀⠀⠀⣀⣰⠈⠀⠀⠀⢀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢊⠉⣀⣤⡄⠀⢀⡀⠀⠚⡛⠁⠀⠈⢻⣾⣧⠀⠀⠰⢻⠃⠀⠀⠀⠀⠀⣶⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣢⣤⣽⣿⣶⣿⣠⠱⣦⡆⠀⢿⣿⠀⠀⢿⡏⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⣦⣾⣿⣿⣿⣿⣿⣷⣷⣎⡟⠻⢶⣿⣿⣶⡶⢠⡄⣠⠀⢿⠇⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⠀⠀⠀⠰⠳⢦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⡛⣾⣏⡓⡸⣦⡍⢀⣀⡀⠀⠰⣄⣠⠀⠦⠾⠷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠛⠀⠀⠰⢿⠇⠀⠉⠛⠃⠀⢀⠀⠀⠀⠀⠀⠀⢒⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠸⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡴⠀⠀⠀⠀⠘⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠞⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠠⡞⠋⠀⠈⣻⠿⣿⣿⣿⣿⣿⣿⡿⠿⣏⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢸⡏⠉⠉⢹⣄⠀⢙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡾⠍⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠁⡒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣼⣿⣷⠀⢸⠡⢀⠀⠐⢈⡇⠀⠀⠀⠀⠀⠀⠀⣆⣀⣀⣄⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣷⡁⣿⣿⢀⣄⢺⡔⣇⢀⠄⠉⡓⠀⠀⠀⠀⠀⠀⠀⠸⠟⠿⢿⣶⣽⣿⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠂⠋⠉⣙⢣⣿⣿⣸⡿⠘⣿⣆⡼⠠⣀⣉⡀⡀⢀⠀⠀⠀⠀⠀⠈⣲⡇⣶⡼⣿⣿⣧⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡄⣿⢇⣴⣶⣬⣅⣬⡴⠏⠉⡄⠈⢀⡑⠒⠆⠀⠀⠀⢨⣿⡾⠿⠿⢶⣿⣿⡎⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠻⢿⡖⣿⢸⡿⠿⢿⣭⣡⢦⣿⣿⣛⠋⠉⠑⢠⠀⢀⠙⠚⣿⠿⢿⣟⡛⢷⡽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣤⣶⡷⣟⡘⢧⡀⠌⣠⡿⠟⠛⠉⢻⡉⠹⠓⠲⠶⢦⡤⠴⢁⡤⠴⠀⢀⡾⣧⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⢀⣠⣾⡈⠞⠷⠂⠥⢣⣎⣷⡿⡏⠁⠀⠀⣀⣀⣧⣀⠀⠀⠀⠀⠠⠀⠀⠃⣀⣶⠋⠐⠇⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⣴⡟⠉⡀⠈⢙⡫⢊⣄⣴⡿⢿⡦⠀⢀⡠⠼⠿⣿⣿⠿⣯⣷⣿⠶⠚⠚⠊⠉⠉⢻⣧⣤⣄⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠉⣽⣐⣁⣈⢉⣠⣴⡟⠃⠁⢀⣸⠶⠖⠁⠀⠀⠀⠈⠻⣆⠻⢟⡽⠄⠀⠀⠀⠀⣀⡴⢿⡾⠾⠏⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⢹⣏⢨⣜⣈⣤⣼⠗⠢⠀⢸⣟⠈⠀⠀⢿⠀⠀⠀⠀⢿⡄⠺⢁⣀⣤⣤⣤⣞⠁⠀⠀⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣿⢿⡋⠉⠉⠀⠀⠀⢰⡿⣿⡅⢲⣦⣼⡇⠀⠀⠀⢸⡷⠖⠛⠉⠁⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠉⠿⣠⡾⠛⢆⠉⠒⠀⠐⠒⠉⠀⣿⡟⠠⣽⣿⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢃⡀⡻⢸⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡟⣸⠆⠈⢋⣀⣀⡀⠀⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⢩⣴⠉⣰⠀⠈⠛⠉⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⡿⠁⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠤⠄⣀⠀⠀⢀⡤⠴⠖⣚⠛⠋⠩⠍⣁⠐⠒⠠⢀⡀⠀⢀⠤⠤⠀⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⡀⣽⣎⡀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠈⠑⠢⢀⠀⠙⠠⢀⡀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⡐⠀⠀⢀⠔⠂⡠⠋⠀⡈⢳⡉⠓⠀⠄⡀⠀⠀⠀⠀⠈⠙⠢⢄⡀⠉⠒⠠⢀⠉⠉⢄⡢⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠜⠀⠀⢀⠁⠀⠙⣦⣀⠀⠀⠈⠐⠠⠄⣀⠀⠀⠀⠀⠁⠒⠠⠄⠈⣀⣢⠼⡇⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠌⠀⠀⡠⠃⠀⢠⠊⠀⠉⠫⢔⠂⠤⢀⣀⠀⠈⠁⠐⠒⣒⢒⣶⠶⢊⣉⡤⠐⡃
⠀⠀⠀⠀⠀⠰⠀⢀⠔⠁⢀⠄⠊⢀⣤⢌⣁⠀⠀⠀⢀⡤⠤⣀⠶⠤⣤⢈⣥⣠⣀⡤⠤⡄⠐⠊⠉⠈⠛⣣
⠀⠀⠀⠀⠀⠀⠀⠈⢙⠛⡒⠂⠈⠁⠀⠀⠀⠀⠀⠉⠙⡀⢒⡸⠶⠀⠌⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⢋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢰⠀⠀⠀⡀
⠀⠀⠀⠀⡸⠁⠀⢠⠁⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢆⠀⠀⢄
⢀⡀⠤⠒⠁⠀⡠⠃⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠈⢄⠀⠈⠣⢄⡀
⠣⢀⣀⣀⠠⠒⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⠢⢄⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠠⢁⣂⠖⠀⠠⢠⡐⠳⠆⠀⢀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⠄⢤⠡⠁⠉⠂⠄⡀⠀⠀⠁⠲⠌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡰⢲⠢⠁⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⡝⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠠⢑⠣⠃⡠⢄⠀⠀⠔⠒⣄⡠⢄⠀⠉⢉⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⢁⡦⠖⠈⠀⠘⠀⠀⢃⣁⠜⡇⠀⠑⠠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠤⠶⡤⠧⠄⠀⠀⠀⠀⠈⡁⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠰⠶⣵⠶⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⢤⢤⡤⠤⠑⠤⠄⣀⠀⠘⠂⠉⠰⠂⠀⣀⣠⣤⠦⠤⢀⡠⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠒⠤⣀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⣍⠍⠓⠆⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠀⡉⠒⠦⣄⡀⠀⠀⠀⠀⣜⠎⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⠤⢤⣄⢰⡇⠀⡖⠂⣽⠱⣦⢄⣘⠮⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠈⣿⠁⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⡮⢱⢜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠦⣤⣴⣞⣏⠀⠀⡆⠀⡞⠀⢸⠀⢸⠆⠀⠀⠀⠀⠀⣀⢁⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠉⠉⠉⠁⠈⡄⠈⠁⠀⠀⢸⠀⢸⠀⠀⠀⠀⠉⠉⠉⠉⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⡀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⢆⠀⠀⠀⢀⠃⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠆⠀⠀⠀⠀⠀⠐⠃⠀⠸⠀⠀⠢⠤⠤⠤⠤⠤⠤⠄⠤⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⢤⣤⣤⡄⣬⢀⣤⣤⣤⢤⣤⣄⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠨⠷⠾⠟⠯⠭⠴⠄⡿⠠⠶⠾⠟⠛⠲⠾⠅⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢗⡠⠄⠀⠀⠠⢄⡪⠳⢕⡠⠀⣀⣀⠀⢤⡲⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣭⣀⣀⣸⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢋⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⠇⠀⠀⠀⠀⠀⠀⣼⡏⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⡁⣷⠀⠀⠀⠀⢀⣠⣤⣼⣿⢤⣤⣀⡀⠀⠀⠀⣿⠳⣜⠦⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢱⠟⢠⣿⡄⣠⡴⠞⠋⠉⠀⠀⠀⠘⠃⠈⠉⠓⢦⣀⡇⣀⣬⡷⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣾⣯⣤⣻⡇⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⢲⡄⠀⠹⠿⣏⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣷⡙⠇⣿⠀⠀⠀⠘⢷⣀⣠⠾⣧
⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠏⠀⠀⠀⠀⢸⡟⠛⠾⠋
⠀⠀⠀⠀⠀⠀⢀⡈⣷⣤⣀⢰⣷⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⣿⢸⠀⣉⣭⣿⠗⠀⢀⡾⢋⣠⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠯⠤⠴⣶⡟⢻
⠀⠀⠀⠀⠀⠀⠹⣧⡾⠛⠉⠀⠘⢧⡀⠸⣧⣈⣩⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⢶⣀⣟⡶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠈⠁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠈⢿⣇⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣲⡶⠋⣷⣤⡀⠀⠀⢰⣦⡴⠚⠁⠀⠀⠀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣯⣄⣀⣿⠈⠉⠉⠉⢹⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡴⠭⠙⢻⡆⠀⠀⠀⢸⣗⣠⠿⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⢸⣯⣵⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⢋⣿⣿⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⡴⢋⣴⠟⣡⡾⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠻⣿⣷⣟⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⣀⡴⠋⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣤⣤⣤⣤⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⡴⠋⢀⣴⠟⣡⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠒⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣌⠳⣴⠟⣡⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢷⡿⢷⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣉⡉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢨⣭⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡟⢹⡇⠀⠀⠀⠀⠀⠀⣿⢘⣛⣃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣿⢨⣭⡉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣤⣤⣤⣤⣾⠃⠀⠀⠀⠀⣾⠀⠀⢹⡆⠀⠀⠀⠀⠀⣿⠰⠶⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠁⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⢿⡀⠀⣸⠇⠀⠀⠀⠀⠀⣿⠐⠒⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣀⣀⡀⢸⡇⠀⠀⣴⠟⠀⠀⠙⢷⡀⠀⠀⠀⠀⣿⠈⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣄⣠⡿⣰⠟⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢽⠉⠉⠉⠉⢹⡇⢸⡇⠀⠀⣿⠀⠀⣶⠶⢾⡇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⣿⡀⡀
⠀⠀⠀⠀⠀⢀⣤⡀⠈⠁⠀⠻⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣻⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⣷⠀⢸⡇⠀⠀⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣿
⠀⠀⠀⠀⠀⣾⠁⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠻⠦⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⣿⠀⠀⡟⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⣄⣀⣀⣀⣸⡇⢸⡇⠀⠀⣿⠀⠀⠷⠶⢾⡇⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⣉⣉⣩⣍⣉⣤⡿⠀⠀⠀⠻⣤⣀⣀⣠⡾⠃⠀⠀⠀⠀⣿⣛⣛⣛⣛⣛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣏⣈⡷⢤⡞⠉⠉⢳⡀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠳⣄⣀⠼⠁⠀⠀⠀⠀⢧⣀⡀⠀⠀⠀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠽⠒⠲⢤⡀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢻⠀⠀⢰⠇⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⢀⡞⠀⢀⡟⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⠞⠁⠈⠙⠒⠚⠉⠀⠀⡼⠁⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣀⡤⠾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⣏⠀⠀⠀⠀⠀⠀⠀⢀⠟⠦⣄⣀⣀⣠⠞⠁⣀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠙⣦⠀⠀⠀⠀⠀⡼⠀⠀⢀⣀⠀⠀⠀⢸⣉⡷⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⣠⠞⠉⠛⠧⣄⠀⢯⣨⠇⠀⠀⠀⠉⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣯⣤⣤⣤⣀⣨⠟⠲⢤⡀⠀⠀⡤⢤⡀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⣿⠀⠀⠀⠈⢻⠀⠀⠀⠙⡆⠈⠳⠾⠁⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣀⣿⣀⣀⣀⣠⡾⣤⣀⣀⣰⣃⣀⣀⣀⣀⡀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠈⠉⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠸⠀⢸⠀⡡⡇⢸⠀⡞⡄⠠⢔⠀⠀⠠⠀⠀⠣⡀⢸⠀⡧⡇⡎⠈⡂⡶⠁⡇
⣶⣶⡶⣶⣶⣶⣾⣶⣾⣶⣶⣷⣶⣶⣶⣶⣾⣶⣾⣶⣾⣶⣶⣶⣷⣶⣶⣶⣶⣾⣾⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠢⣀⠠⠤⠤⠤⠖⣉⡀⠀⠀⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⡁⠀⠀⠱⡀⠀⠀⠀⠀⠀⠈⠓⢄⠀⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠁⠀⢢⡴⠂⠤⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢷⣄⠀⢸⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠆
⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⡇⠈⠉⠁⠈⡆⠑⠢⢤⠠⠤⣄⣀⣀⣠⠤⠄⢤⠤⠃
⠀⠀⠀⠀⠀⠀⢎⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⡀⠀⠀⠀⡄⠀⠀⠀⢀⣘⠤⢼⣤
⠀⠀⠀⠀⠀⠀⠘⣇⡄⠀⣰⢀⢰⠀⣀⡀⠀⠀⣇⣤⡖⠒⠈⠙⢲⢺⡁⠀⡄⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁⠉⠀⠲⡤⡎⠓⠜⠁⠈⢧⡀⠈⣀⢏⠀⠙⠦⠤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠀⠈⠉⠀⠀⠙⢁⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣕⡒⢶⢤⠤⡤⠴⡒⡍⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠙⠣⣤⡓⠐⣇⢧⠀⢸⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠈⠑⡤⠛⠚⠓⠁⠚⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠳⢄⡀⠀⠀⠀⠀⠀⢰⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡏⠒⠲⠤⠤⠰⡾⠋⠀⠀⠑⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⡀⠀⠀⠀⠀⢸⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⢸⢲
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢸⠀⠀⠀⠀⢀⠆⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠈⢷⠤⠴⢚⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠀⠀⠀⠀⠸⢄⣀⡀⣀⣀⣰⣀⡠⠤⢤⠒⠊⠉⠀⠀⡖⠒⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⡀⠀⠀⠀⠀⡠⠚⠫⡀⠀⠄⣬⣀⢀⣼⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⢀⠀⠀⡠⠊⠘⢠⣀⡈⢒⡺⠄⣀⠠⠾⣦⡀⠀⠀⢀⠇⠀⠸⢀⠀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⡆⠀⡅⠀⠀⠀⢇⠈⢉⡷⠀⠈⠂⠐⠁⠙⡄⠐⠁⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠰⡀⢣⠀⠀⠀⠀⠉⠉⠀⡇⠀⠀⠀⢀⡰⠁⢡⠀⠀⠀⠀⠀⡘⠀⠃
⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠐⠁⠈⠢⢄⡀⠀⠀⢀⡠⠋⠁⠒⠈⠁⠀⠀⠀⠑⠤⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢠⢲⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⢀⠹⡆⢀⠀⠸⣆⠱⣾⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠈⠀⠀⠀⢡⣾⣇⡀⠻⠑⠈⠋⠿⠇⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⢀⣠⣄⣤⣴⣶⣿⠗⠒⠀⢤⡀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠙⣿⡿⠛⠁⠀⢠⠀⠜⣻⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⣠⡴⠆⠰⢀⣿⡇⡇⠀⠘⡄⢧
⠀⠀⠀⠀⠀⠀⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠂⠘⣡⡎⢀⢼⣿⣷⣿⠀⡇⣧⢸
⠀⠀⠀⠐⢆⡴⠲⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⠀⠐⠘⢿⡀⢉⣾⣿⣿⣿⡇⠙⠳⢸
⠀⠀⠀⢐⡬⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⡄⠀⠄⣼⠇⣼⣿⣿⠇⢿⠇⠀⠀⠈
⠀⠀⠀⠘⡇⠀⠀⣸⣿⣿⣿⡷⣤⣄⠰⠄⠐⠒⠀⠀⡀⠀⠀⠀⠑⠄⠀⠠⢠⣿⣿⢿⣿⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠿⢻⣏⣩⠀⠀⣟⠀⠀⠀⠀⠀⠸⢋⠄⠀⠄⠀⠀⠀⠀⣀⢵⣿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠘⠀⣶⣶⣶⠶⡶⣄⣈⣈⣴⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⢳⡀⠃⢉⠙⣡⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠁⠂⢘⣛⣫⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⡇⠀⠀⠀⠀⠀⠀⢈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⢸⣷⡆⠀⠀⠀⠀⠀⣹⣿⠟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠒⠚⠀⣻⣿⣋⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⣿⣿⣥⣇⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⢰⡄⢀⣠⣼⣿⣿⣿⣭⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢢⠀⠀⠙⣷⡵⣦⣀⡀⣸⣿⣿⣿⣿⣿⣿⡿⠟⠛⠻⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡆⠀⢧⣠⡖⣿⣿⡏⢿⡇⢻⣿⢃⣿⣷⢿⣿⣷⡆⢀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⣿⠇⢸⡀⣿⣾⡷⠞⠛⣾⣿⠿⣦⡈⢿⡇⢼⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⢨⠆⢿⣿⠁⠀⢰⠟⠁⠀⠀⠛⢿⡇⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠳⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠈⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⠀⣽⣿⣾⠟⣿⣆⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠈⠉⠛⠿⣧⣈⠙⠻⢿⡝⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠸⣆⡀⢀⣠⠶⠛⠛⣳⣾⠷⢤⡀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⡾⠃⠀⠀⠙⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠆⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⣹⠿⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠈⠉⠓⢦⣄⠀⠐⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡞⠛⠶⣤⣀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⣩⣿⠶⡍⠛⣆⠀⠀⠀⠈⠁⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠁⣸⣇⣀⣠⣿⣤⣤⣤⣤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⣴⠟⠉⠙⠋⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⢀⣼⠏
⠀⠀⠀⠀⠀⠀⢀⣾⠃⢠⡾⠃
⠀⠀⠀⠀⠀⢠⣾⣁⣠⡿⠁
⢠⣴⢶⣶⣄⠈⠉⠉⠉
⠈⠛⠛⠛⢁⣾⢿⡄
⠀⠀⠀⠀⠸⣇⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣄⣄⣤⣄⣤⠀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡋⠀⠀⣀⡀⠀⢀⣀⡀⡀⠀⠀⠀⠀⣧⣿⣿⣿⡟⠀⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣗⣠⠞⣻⣿⢿⣿⢠⠶⠴⠋⠀⠀⠀⠀⠘⣿⡿⢿⠥⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠀⠀⠳⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠸⡇⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢀⠀⠀⠀⠠⣵⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢣⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣄⣀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⡄⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⣾⣿⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⣤⡿⡇⠙

⠀⠀⠀⠀⠀⠀⠀⠀⡞⢉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣘⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⣤⣶⣤⣶⣧⣴⣦⣴⣦⣤⣶⣤⣶⣤⣵⣦⣴⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⢻⣿⣾⣿⣾⣿⣶⣿⣷⡟⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠐⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠿⠿⠿⠿⠿⠿⠿⠿⠷⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡾⠿⠿⠿⠿⠿⠿⠿⣶⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⢀⣴⡾⠿⠿⠿⠿⠿⠿⠿⠗⠀⢸⣿⠀⠀⠀⠀⣰⡿⠋⠀⠙⠿⣉⠻⣿⣿⣿⣿⠉⠉⠉⠉⠉⢹⣿⠁⠀⠺⠿⠿⠿⠿⠿⠿⠿⢷⣦⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣿⡇⠀⠀⠀⠀⠻⣷⣄⠀⢸⣿⠀⠀⠀⠀⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠘⢿⣶⣾⠿⠿⠿⠿⠿⠿⢿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠙⠻⠿⠟⠀⠀⠀⢸⣿⠁⣀⣠⣤⣶⡾⠿⢨⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⢀⡀⠀⠀⣿⡇⠀⠛⣋⣩⣤⣶⣦⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠘⢿⣦⣸⣿⠀⠀⠺⠛⠋⠉⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠻⣿⣧⣤⣤⣤⣤⣤⣄⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣴⣤⣤⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣤⣤
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⡀⠀⠀⠀⠀⠀⣀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠁⠈⠀⠰⠀⢅⡠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢃⣤⢁⢀⠀⠀⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠘⠓⠉⠁⡈⢀⠃⣀⠶⠚⠃⠀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⡀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⡈⠕⠊
⠀⠀⠀⠀⠐⠀⠀⡀⠄⠢⠌⠉⠀⠈⠙⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⣠⠄⠒⠁
⠀⠀⠀⠀⠄⡰⠊⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠢⡔⠀⠠
⠀⠀⠀⠀⠌⠀⠠⡀⠀⠀⠀⣄⣀⣀⡜⣂⠀⠀⠀⠂⠐⠄⠀⠀⡀⠀⠀⠠
⠀⠀⠀⡌⠀⡄⠀⠀⠀⠀⠀⢻⠀⣮⢧⣉⡠⠀⠀⠂⠀⠘⡬⡄⠠⠀⡀
⠀⠀⢀⣲⠀⠓⠒⠈⠀⠀⠀⠈⠳⠤⢎⡤⢀⣀⣀⠐⠀⠁⠃
⠀⠀⠀⡄⠀⠀⡀⡀⠀⢀⠀⠀⡠⠂⠚⢀⡄⠂⠉⠀⠠⠘⡇⠀⢀
⠀⠀⠀⢱⡀⠀⠐⠄⡀⢀⠀⠀⠗⢠⡜⠁⢀⠤⠒⠀⠀⡰⠁⠀⠈
⠀⠀⠀⠀⠑⡹⠂⠀⠀⠈⠀⠀⠀⠃⡠⠊⣡⠄⠀⠀⡰⠁⠀⠀⠁
⠀⠀⠀⢀⡀⠉⠢⣀⠀⠀⠀⠘⡥⠊⠀⠈⠀⢀⡠⠊⠁⠀⠆
⠀⠀⠀⠀⠈⠒⠀⠈⠉⠓⠤⠤⠤⠤⠤⠴⠊⠁⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡀⠀⠀⢀⡀
⣿⣿⡿⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⢠⠀⢠⠤⡀⠀⠀⢈⠀⠉⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⢄⠀⢸⠀⠀⠸⠀⠀⠀⠸⠀⠎⠀⠐⢢⠀⠀⡔⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣢⠘⠖⠂⠉⠉⠉⠁⠃⠘⠒⠰⢥⡁⢀⠔⠁⣀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⢢⠀⡠⠖⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠢⠈⠁⠰⡉⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⡀⠛⠀⡀⢀⠈⠀⢱⠀⠀⠀⠀⠀⠀⡨⠢⣤⠄⠀⠀⠀⠈⠱⡨⠥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠄⡔⠁⠁⠀⠀⠈⠂⠴⠋⠑⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣈⡠⠄⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠡⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠊⠉⠑⣏⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠅⠄⢲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡰⠒⡄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡌⠀⠠⠒⠢⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠘⢃⣉⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠉⡇⠀⢓⠀⢠⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠒⠐⠊⠉⡄⠈⡇⢀⡀
⠀⠀⠀⠀⠀⡏⠁⠒⢂⣁⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⡆⠀⠀⠆⠀⠀⠐⠒⠦⣠⠃⢰⠃⠀⠍
⠀⠀⠀⠀⠀⠑⡠⠂⠁⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠤⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⢐⠈⠑⡄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠂⠀⠀⡼⠉⠑⢦⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⣤⠀⠀⠀⠀⠉⠲⠁⠀⠀⠀⠀⠀⠀⠀⢰⠔⠛⠀⠀⠀⠀⢀⠼⡣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠒⢄⠀⠂⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠔⠀⠀⢶⠃⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣀⠄⠀⡠⡱⠢⢄⢀⡀⠀⠀⠀⠀⠀⠠⠀⡀⠀⡀⡤⠚⠑⡈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠁⢀⠆⢨⠉⠒⠒⣶⠒⠒⠒⠆⠹⡉⠀⡨⠂⠀⢐⢀⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⡣⠀⠠⠠⠆⠀⠸⡀⣀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣖⣺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣒⣟
⠀⠀⠀⠀⠀⠀⣿⢉⣉⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⣿⠐⠒⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⣿⢨⣭⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⠐⠒⠒⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠿⠿⠿⢻
⠀⠀⠀⠀⠀⠀⣿⠨⡭⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢭⠅⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⣿⠐⠒⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠒⠂⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⠤⠤⢤⠟
⠀⠀⠀⠀⠀⠀⣿⠨⠭⠤⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠭⠥⠄⢸⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⠲⣄
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⢰⣒⣛⣛⣛⣛⣛⣛⣲⡆⠀⠀⠀⠀⠀⣖⣚⣛⣛⣛⣛⣛⣓⣶⠀⠀⠀⠀⠀⣿⣿⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠙⠋⠁⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣈⣀⣉⣁⣀⣿⣿
⠀⠀⠀⠀⠀⠀⣤⣧⣼⣧⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣤⣧⣼⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠄⣀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣀⢄⠀⠀⠈⢀⠠⠐⠈⠀⠀⠈⠀⠉⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⠈⡗⠉⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠢⠧⣀⣀⠜⠲⣤⠖⠛⠓⢦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢇⠀⠀⠀⢨⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⢵⣊⠈⢉⠛⠑⠒⢤⡀⠀⠀⡀⡠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠰⠀⠀⠀⠀⠇⠐⠐⠁
⠀⠀⠀⠀⠀⢤⢠⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢀⠠⠀⠠⠔⠁
⠀⠀⠀⠀⢀⠚⠘⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠀⠀⠀⢀⠗⡀⠀⠀⠈⠃⠀⠒⢦⡔⠒⣶⣶⣾⣿⠿⠿⣿⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠋⢸⡔⠁⠀⠑⠀⠀⠀⢀⠀⠀⠈⢿⣿⣿⡟⠉⠉⠀⠀⠜⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢰⠀⠄⠀⠀⠐⠀⠀⠀⠢⡀⠀⠸⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠔⠓⠒⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢀⣀⡀⠀⠀⠈⠂⡤⠀⣀⣀⡀⠀⠀⠀⠐⠈⠀⠀⠀⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠤⢱⠄⠁⠀⠀⠀⠀⠀⠂⠈⠩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣆⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠉⡀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⢂⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠉⠀⠈⠢⡀⠀⠤⠂⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣤⡀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠉⠂⠀⠀⠀⢀⣀⣀⣤⣄⣀⣀⣬⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢠⡴⠿⠟⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠉⠉⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣈⠁
⣶⣶⠶⠶⠶⢶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣦⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⠀⠀⡎⠀⠁⢘⠀⠀⠀⠡⠀⠔⠁⡤⢄⠀⠀⡔⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢂⠀⠱⡀⢣⠤⠞⠚⠒⠒⠲⠃⠸⠤⢤⣣⠊⠀⡰⠁⠀⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠢⡀⠉⣪⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢌⠑⠊⣠⠚⠉⡡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠱⠞⠁⠀⠠⠂⠐⠄⠀⠀⠀⠀⠀⠀⢰⢄⢀⠸⠀⠀⠈⠑⢮⢐⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⢉⡝⠀⠀⠉⠀⠡⠀⡠⠛⠄⠀⠀⠀⠀⠀⠁⠠⠋⠀⠀⠀⠀⢀⡈⠁⠈⢣
⠴⠂⣀⣀⠀⡀⠀⠀⠀⠠⠂⠼⣐⡎⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠄⠤⠠⡼⣀⡀⠀⠀⠀⠀⠀⣠⠴⠖
⢹⠁⠀⣿⠍⠐⠶⣤⠀⡇⢀⣀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⢥⡼⣀⠤⣀⠔⠋⢀⣴⡍
⢨⠀⠀⠀⠀⠘⠀⠈⡳⣎⡁⣠⠇⠀⢀⣄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠿⠗⠁⠀⢹⠀⡴⠁⢸
⠀⡀⠀⣦⠀⠀⣮⠟⠁⢣⠤⣽⠀⠰⠁⠀⢹⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠤⠠⡀⠠⡶⡺⠋⠙⠺⢿⡅⠀⢨
⢀⠀⠀⡿⣀⡴⠋⣰⠲⠧⠤⠾⠂⠀⠑⠀⠋⠁⠂⠀⠨⠀⠀⠈⠀⠀⠀⠀⢀⠖⠁⠁⡄⠀⠀⣀⣀⣀⠀⠇⢀⡟⡓⢲⠰⢅⠸⡣⡄⢸
⢀⣄⣠⡿⠋⣀⠔⠻⡀⢀⠔⠒⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠀⠀⠀⠀⠀⠢⠀⠔⠀⠀⠀⠀⠀⠈⠈⠀⢼⣀⠔⣁⣀⠈⠑⢀⠈⢺⣄
⠒⠚⠓⠒⠊⠁⠀⠀⠙⠁⣀⠠⠜⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⠤⢬⡉⠀⢸⠂⠀⠀⠀⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠊⠀⠘⣄⠀⠀⠀⠀⠰⠧⣀⠄⠀⠀⠀⠀⠀⠀⠀⢄⢀⣦⠀⠠⠀⠀⠀⢠⣏⠀⠀⠉⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⢃⡀⠀⢁⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⡰⠀⠀⢀⡴⠉⠪⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⠇⠈⢢⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⣠⡀⠻⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠜⠀⡔⢱⠋⠉⠇⢰⢤⣀⣀⢀⡀⠀⣀⠀⣤⠴⠖⠫⡀⠈⠆⠀⠈⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠜⠀⠸⡀⠀⠈⢩⠉⠉⡸⠀⠘⢆⠐⠚⠁⠀⠰⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢒⣋⣭⡭⣍⡓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠰⣍⡀⠀⠀⣀⡼⢀⠌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠀⠀⠭⠭⠭⠑⠒⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⠦⢤⣀⣀⣀⣠⠤⠔⠚⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠯⢭⣙⠲⣄
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡀⠈⠳⡌⢣
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠔⢋⡩⢝⣦⠀⢹⡘
⠀⠀⠀⠀⠀⠀⢸⡈⠳⢤⣀⣀⡀⠀⠀⣀⣠⠤⠒⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⣡⢔⣿⡾⢋⠝⠀⢘⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠑⠢⡄⠀⠉⠉⠉⠀⣀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠤⢤⣀⣠⠖⢁⣤⣮⡿⠛⣡⠔⢧⣀⣀⠼⠚
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠈⢁⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⠉⠓⣌⢳⣾⡿⠟⢁⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠈⡇⢄⡀⡇⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣸⠔⠚⠉⠉⠉⠓⢄⠈⢧⢻⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠗⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⠔⡿⠀⠀⠀⠀⣠⡴⠎⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠠⣉⠒⠒⠦⠤⠤⠖⠒⢋⡡⠚⠁⠀⣠⠴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠉⠁⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡮⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠙⢦⠀⠀⠀⢠⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡤⠚⠙⢢⠀⠀⠀⣧⠀⠀⣨⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⡈⠆⢀⡼⠀⠀⢀⣇⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⠞⠓⠚⠡⠀⠀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢊⡴⠊
⠀⠀⠀⠀⠀⢀⡠⢖⡽⠊⠁
⠀⠀⠀⣠⠖⣩⠖⠁
⢀⡴⣪⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠪⣁⡝⠀⠰⡉⢱⠀⠀⢰⠒⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠣⣶⣇⡠⠤⠆⢙⡏⠊⢉⠋⣧⣉⠑⠖⣶⢉⡀⠰⡒⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⢄⣧⠀⣠⠔⠚⢏⠓⠀⠀⠀⠀⠀⠀⣸⠯⢭⡀⢀⡽⢅⠈⢉⣴⣉⡥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠍⣡⢄⠀⠈⠀⢀⡤⠤⣄⠀⠀⠙⡆⡷⠂⢠⢇⡶⠃⠀⠤⢫⣴⡜⠀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⡆⣠⠞⠁⠉⠮⢤⣸⣀⠀⠀⠛⠠⡈⣎⠀⠀⠈⠵⠽⢁⡤⠚⢯⠂⠀⡠⠛⡇⡙⠐⣣⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠒⣌⠣⠄⠀⠀⠀⠀⣙⣀⠀⠀⠀⠀⠩⠂⡡⠀⠠⢄⠀⠈⡇⢰⠤⠃⢊⢊⠎⠔⢌⢌⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⠃⠀⠀⢀⠤⡄⠀⠉⠀⠀⠀⠀⠁⠀⠓⣤⠊⣞⠀⢀⡬⣿⢃⠔⣱⢥⡪⢀⠒⡊⢤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⡀⠁⠀⠀⢌⢀⣑⡀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠒⠃⠠⠜⢀⢨⠄⠠⠂⠀⡃⢑⡌⡰⠠⣷⡠⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣩⠩⣴⠤⠀⠀⠈⠁⠨⠇⠀⠀⠀⡅⠈⡅⠀⡔⢈⠙⢢⠤⡨⠈⠭⠪⡫⣈⠗⠑⠡⡪⡪⠠⠖⠢⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡂⠽⢰⠒⠦⠄⠀⠀⠁⠀⠀⠀⠙⠾⠽⠀⣻⡶⠗⠃⠺⣂⣉⢊⡉⣎⠕⠀⡬⡙⢘⣚⠔⠧⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⣴⣤⠀⠘⠦⠛⣟⠗⠀⢠⠦⠔⣢⡄⠀⠀⠐⢙⡠⢐⠀⢓⢪⢄⡧⣰⠕⡡⠪⣐⡗⡩⢕⡱⠒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠹⡄⢀⠀⠀⠀⠁⠀⠈⠦⢦⡿⢖⡫⡢⠆⠯⡸⢼⢒⡋⢑⠑⡪⢔⡩⢜⢕⠈⢉⡗⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠛⣡⢴⢆⠀⢀⣠⠀⠒⢭⡲⣵⠞⡳⢖⡳⢖⣡⡺⢙⠄⣉⠗⠡⢎⢕⣌⡗⠙⡤⣕⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠮⠛⢮⡳⣖⠒⢡⠊⠥⢒⡋⣠⠉⢒⣩⢄⢹⠠⢚⠭⢋⢔⡠⣈⣦⣀⣵⣊⢤⣄⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⡠⣽⡓⡂⠂⢼⣉⡀⡗⠾⣋⢥⡎⢈⣺⡃⢠⠸⣸⣂⣤⣤⣩⣧⠏⠩⣦⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡸⠀⡠⠿⢡⡶⣂⠫⢉⡠⠕⡪⠕⡩⣋⠥⢒⢝⣀⡸⡧⠤⡔⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠜⠈⠒⢎⠱⣂⠴⡀⡵⠢⢔⡪⠔⢪⣔⠋⡞⠋⠉⠦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠘⠅⢩⠃⠳⡚⠛⠃⠈⠁⠀⡵⡈⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠃⠈⠉⠋⠁⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⣀⠀⢀⠀⢀⡠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠔⠒⠭⢒⠢⠔⡒⠤⣒⠎⢒⡪⠥⢒⡩⠔⠢⠅⠒⠬⠓⠪⠕⠂⠩⠒⠉⠔⠊⠁⠒⠉⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡶⠶⠖⠂⠀⠀⠀⠀⠀⠀⠀⠔⠒⠖⠀⠀⠀⠀⠙⠻⠷⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠋⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢛⣷⣦⡀
⠀⠀⠀⠀⣤⣾⡛⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⣀⣀⣀⣠⣀⣀⣀⡄⢀⣀⠀⢀⣉⡭⢿⡆
⠀⠀⠀⠀⣿⡏⠝⠉⠻⠿⠒⠶⠶⡾⠽⡷⠶⠆⠛⠚⠳⠛⠂⠂⡛⠛⠺⠛⠛⠉⠜⠛⠛⠉⠉⢹⠇⠀⠀⠃
⠀⠀⠀⠀⣿⠅⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠈⠐⠊⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡂
⠀⠀⠀⠀⣿⡇⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⢰⡀⠀⡷⠀⠀⠀⠀⠀⣀⣀⠠⠄⠂⠉⢹⡀
⠀⠀⠀⠀⢻⡅⠠⣤⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⢃⢀⣠⡤⠂⠋⠉⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⣀⣦⡄⠙⠋⠁⠉⠈⠀⠀⠀⠀⠀⢀⣀⢤⣀⣶⠟⢛⠃
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡄⠀⠀⠀⡀⠋⠉⣹⠃⣀⠀⠀⠀⢀⣀⣠⣴⡴⠶⠴⠋⠋⠉
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⠀⢀⡀⠀⠀⠀⣤⣌⡉⠉⠉⠀⠁⠙⢿⠉⠀⠀⠀⠁⠠⢄⠠⡄⠀⠀⣄
⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⡴⠀⠈⠇⠠⡈⠛⠛⢀⢿⠀⠀⠁⠀⠀⠈⠠⠀⠀⠀⠀⠀⣴⠀⠀⠘⢁⣸⡇⢰⣦
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠀⠁⠀⢷⠛⠀⠀⠈⠈⢀⡟⠀⠀⠎⢱⣦⠀⠀⠀⠘⠐⣤⣾⣏⣀⣲⠦⡄⠿⢥⣼⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣷⣦⡄⠀⠀⡀⠈⠐⠞⠁⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⢹⣿⣯⣥⢢⠈⠛⢄⡙⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃⣿⡟⠀⠀⠀⠛⢁⣠⣄⠀⠀⠀⠳⠷⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⢿⣆⠀⢰⣬⣥
⠀⠀⠀⠀⠀⠀⠀⠈⢩⣵⣿⣣⡄⡄⠀⠀⠈⠈⢉⡁⠁⣐⢴⡀⠀⠀⠀⣀⠤⠲⠾⠛⢿⠏⠹⣟⣁⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠐⠒⠛⠻⣿⣿⣷⠄⠀⣀⠀⠀⠀⠀⠙⢻⣿⠃⣘⣽⣋⠣⠀⠀⠀⠀⠘⠢⠀⢯⡉⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡦⠀⡙⢩⡄⠀⠀⢀⣼⣿⣶⣿⣶⠿⠂⠀⠀⠀⠀⠀⠂⠀⠀⣥⠰⠗⢼⣷⡇
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣄⣄⣈⣤⣥⣬⣴⣿⣍⣤⣦⠤⣴⢿⣿⣿⠗⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⢈⡿⡁
⠀⠀⠀⠀⠀⠀⠯⠈⠻⢿⡟⠉⢙⣦⠁⢿⣿⣿⣿⣻⣭⠋⠰⠤⢘⠒⠈⠛⠀⠀⠀⠀⠀⠀⣄⠀⣿⡆⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡁⡜⠀⠉⠈⠉⠀⠀⠉⢩⡆⢀⠀⠀⠄⠀⡀⠀⢺⣆⠀⠀⣶⠽⢤⠛⠈⠀⠀⠄⠇
⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠀⠄⠀⠀⣸⢇⡖⠀⠀⠀⠀⠀⡴⠏⣻⡆⠀⠀⠀⢘⡇⢠⠀⣠⡶⠆
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠄⢠⣿⠤⠀⡠⢾⡇⠼⣿⠀⠀⠀⠀⠀⠁⣶⣿⠿⢀⠀⢀⣿⣰⠀⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠟⠁⠀⢀⠅⠀⠀⢰⣿⠀⠀⠀⠀⠀⠈⠀⢻⠀⠈⠀⢠⣽⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠂⠀⠘⠛⠃⠀⠀⢀⣀⠤⠤⠾⠗⠒⠚⠆⠧⠒⠒⠂⠊⠁


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

⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡇⣛⢸⢈⡇⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠀⠀⠀⠀⠀⠀⡄⠤⠀⠀⢀⠈⠀⠁⠉⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢤⠁⠀⡀⠁⠀⠀⡀⠦⠠⢄⣀⠀⠀⠀⠀⠀⠀⠃⠠⡄⠈
⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠃⠀⠀⠀⠖⠀⠉
⠀⠀⠸⡿⠗⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⢀⠀⢀⡈⠁⠀⠴⠶⠀⠰⠆⠀⠀⠀⠜⠓⠂
⠀⢠⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⢀⠀⠀⠢⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠐⠫⠅⠀⠰⠃⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⢴⡄⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠃⠀⡄⠀⠀⠀⠐⠀⠀⢠⣀⣠
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠓⠲⠶⠤⡤⠴⠶⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆
⠀⠀⠀⣿⠉⠹⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⣿⠦⠞⠁⡷⢦⡀⢀⣤⣄⠀⣄⣀⢠⣤⣠⡀⢀⣀⠀⠀⣠⡀⢀⡤⣄⠀⡀⢀⡐⢻⠛⡄⢀⡀⠀⢀⣀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⡇⠈⡇⣏⣀⣿⣀⡏⠀⢸⢹⠁⡇⣾⣸⣇⡸⣅⣠⢸⣛⣋⠼⣇⣸⣅⢸⠀⡇⢯⣀⡤⣏⣸⣅⢸
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠁⠈⠁⠀⠀⠈⠁⠀⠈⠁⠀⠈⠁⠀⠀⠀⠁⠀⠉⠀⠈⠁⠀⠈
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣾⣶⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠻⠿⠀⢰⣾⣿⣷⣶⠀⠈⠠⢀⠀⢰⠿⠱⣦⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠹⣿⡏⠉⠀⠀⣀⠀⢀⠈⢹⣻⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡄⠘⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⣿⡿⠟⠉⢿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠟⠛⠁⠻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠋⠁⣀⡀⠀⠀⠀⠀⠙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⢀⡀⠀⠀⣐⠈⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⣰⡆⠀⠀⢀⣌⠀⢨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠃⠀⠘⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⠈⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⡀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠗⠀⠀⢾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⠀⠀⢠⣶⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠀⠀⠀⢀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣿⣇⠀⠀⠈⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⠀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠹⠿⣷⣄⣾⠿⠂⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⣤⢰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠀⣼⡆⣹⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢠⡀⠀⣀
⠀⢠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠁⢘⣿⣿⣿⣿⣿⣿⣿⡟⠿⢶⣿⠃⠀⣿
⣠⣿⣿⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠋⠉⠉⠀⠘⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⠀⠀⠻⠇

⠀⠀⠀⠀⠀⢀⣤⡶⣒⣒⣒⣒⣒⣲⣶⣒⣒⣒⣒⣒⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⠒⠒⠒⢶⣦
⠀⠀⠀⠀⠀⣾⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⣿⣹
⠀⠀⠀⠀⠀⡏⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⢲⡆⣿⢸⡆
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⢸⡇
⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸⠃
⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢸
⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣼
⠀⠀⠀⠀⢘⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡿⣿
⠀⠀⠀⠀⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿
⠀⠀⠀⠀⠀⣿⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡇⣿
⠀⠀⠀⠀⠀⢹⠀⠻⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠚⠃⠀⣧⡏
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠤⠤⠤⠤⠄⡆⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣚⣛⣛⣛⣉⣉⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢤⣤⣀
⠀⠄⠀⠀⠀⠀⠀⠀⣰⢏⡭⣭⢭⡭⢭⡭⢭⠭⢭⠭⢭⡭⢭⡭⢭⡉⢩⠭⣽⠿⡿⢿⠻⣆⠀⠀⠀⣾⣡⠜⠋⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣰⠏⡾⢭⠯⢽⠭⢽⠭⢽⠭⢽⡭⢽⡯⠭⡯⠭⡇⠘⣟⣻⣛⣻⣚⣧⠹⣆⠀⠀⠹⣅⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⣰⣏⣈⣓⣛⣒⣛⣒⣚⣒⣚⣒⣒⣓⣚⣓⣒⣓⣒⣛⣀⣱⣤⣷⣤⣷⣤⣇⣹⣆⠀⠀⠘⢧⡀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠻⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠟⠀⠀⠀⠀⠛⠶⠴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣾⠉⡇⣿⠀⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠱⢌⠀⠀⡇⢠⡇⡿⠠⣧⢠⡤⠠⠶⠛⠛⠛⠓⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡾⢷⡀⠀⠢⢀⣇⠈⣷⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⣿⣿⣷⠆⠙⣆⠀⠻⠇⠀⠈⠁⠀⠀⠀⠀⠀⣄⣴⣠⠴⣶⢦⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡸⠂⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣽⢦⣶⡶⡒⠊⠁⢠⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⣿⣿⠓⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⢿⣿⡿⠛⠁⠀⣀⣀⣵⣾⡟⡂⠠⠒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢹⣿⡿⠃⣿⣏⡁⢀⣀⠀⠂⣌⢰⡜⢿⣿⣿⡟⠟⠃⠀⢠⣴⣿⣿⣿⣿⠟⠉⠀⠛⠓⢁
⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⢗⣾⢿⡱⣆⠘⢿⣟⡼⣻⣇⣸⣿⣿⣣⣾⣿⠇⠃⠀⠀⢠⣿⣿⡿⠟⢻⣿⣿⣧⣶⣦⣻⣟⠁
⠀⠀⠀⠀⠀⠀⣰⣿⣿⡋⠈⠳⢈⢀⠘⡿⡆⢸⣟⣛⣿⣾⣿⣿⣿⣿⣏⠀⠀⠀⢁⣸⣿⠟⠀⠀⠀⠈⣿⣿⠋⠛⢉⠻⠟⠁
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⡀⣄⣖⣠⣐⠞⠀⣼⣿⣿⣿⠾⠄⠀⢻⣿⣿⣦⣀⣴⣿⣾⣃⡀⢤⡄⠀⠀⣿⡇⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⣿⣹⣿⣿⡓⣤⣷⣮⣿⠗⣰⣿⡟⢻⢽⡂⠦⠠⣎⠿⣿⣯⣿⣿⣿⣿⠟⠃⠀⠉⢦⡸⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣍⠺⠿⠿⠛⡩⠴⠞⠉⠀⠀⠀⠀⠈⠠⠿⠈⠀⠀⠘⣿⣿⣿⠋⠀⠀⠀⠀⠀⠹⣿⠇
⠀⠀⠀⠀⠀⣿⣿⡏⣻⣿⠿⡏⢞⡧⠀⠀⠀⠐⠀⠀⠀⢸⡆⢟⢡⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢠⣙⣿⡄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣟⠈⢭⠁⣿⡀⠀⠀⠀⠀⠀⠀⢸⠁⠈⠛⠑⠁⢸⣿⣿⡄⠀⠐⢸⠷⣶⣼⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣿⣗⣿⣿⣄⢮⣔⡎⢹⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠈⢠⣼⢻⣿⣷⡄⠀⣦⡈⠁⠏⢉⡝
⠀⠀⠀⠀⠀⢻⣿⣟⣿⣿⣇⠈⢁⠀⡸⣄⠀⠉⠀⠀⠀⣁⡩⠴⠖⠚⠉⢀⣿⣿⣿⣿⣦⣼⣿⣶⣾⣿⣇
⠀⠀⠀⠀⠀⢸⣿⣯⣿⣻⡿⡂⠀⠁⠀⠁⠀⠁⠀⣠⢚⠁⠀⠀⠀⢠⠒⣼⣿⣿⣿⠿⣿⠹⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⣿⣿⣺⣧⢿⡙⠆⠂⠀⠀⠀⠀⢚⠵⠋⠀⠐⢀⠀⠈⣦⣿⣿⣿⢻⣇⠘⠳⢞⡋⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⡻⣿⣿⣿⣿⠆⠀⠀⢀⡔⠃⠀⠀⠀⠀⡠⠐⣸⣿⡿⣿⣗⠈⣿⢦⣌⣉⠁⠈⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣆⢀⠔⠋⠂⠂⡀⠀⠀⠀⣰⣾⣿⠿⣼⣿⡅⠀⣿⠎⡿⢯⡙⢾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣯⡶⣉⡀⠜⠣⡁⠿⡩⠡⠚⣿⣿⢻⡶⣿⡄⠙⢷⡿⣷⠄⡀⣻⣶⣿⡿⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣝⣧⢫⡂⠀⢸⡀⢠⠰⠃⣾⠇⡸⢲⡿⢿⣦⣉⠛⠛⣏⣛⣿⣿⣿⠋⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠉⣿⣿⣾⠲⣤⡀⡀⠀⣇⡎⠀⣰⣿⣤⣇⢨⣿⣦⣿⣿⠷⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣢⡌⢠⡄⢙⣧⣀⣿⣳⣿⣘⠛⣷⣯⡛⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆⢢⡇⢸⠻⣿⣧⣿⣻⣿⠃⠀⢈⣲⡈⢹⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡄⢻⣾⣿⣿⣿⣶⣟⣧⣲⣿⣿⣷⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣄⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠋⠍⠁⠀⠀⠀⣈⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⠿⠉⠀⠀⠆⠀⠀⠀⠀⠀⢱⡀⠈⠛⢻⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣼⠿⠋⠁⠀⠉⠀⠘⠳⢄⡀⠀⢀⠊⠀⠈⢹⣯⣍⣙⣿⣿⣆
⠀⠀⠀⠀⣠⢾⡁⡠⠶⠤⠴⠒⡳⢶⡀⠀⠈⠉⢙⡆⠀⠀⣿⣥⠾⢿⢿⣿⣿⡄
⠀⠀⠀⢠⡣⢣⡍⠀⠀⢀⠜⠐⠒⢸⡄⠀⢀⡴⠊⠀⠀⢸⣿⣿⣦⣶⣾⣿⣿⣆⠈⠐
⠀⠀⠀⣾⢡⠎⠀⣀⠀⠀⠀⣠⡤⢄⢻⡆⢸⡇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣧⣀⣾⣷⠋
⠀⠀⢰⣷⢸⠀⠀⠀⣤⠀⠀⢁⡤⠚⠀⢻⣼⡇⢰⣶⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⡿⢾⣳⣄
⠀⠀⢸⣿⡙⡧⠁⠀⠘⠓⢤⡀⠀⠀⠈⠁⠙⣿⣿⣿⣀⣤⣄⠀⢀⣿⣿⣿⣿⣿⣴⣿⣿⣿⠄
⠀⠀⠀⢿⣇⣇⠀⢀⠅⠀⢀⠻⠽⠶⣞⡶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠈⢿⣿⡷⢧⣔⠐⡀⠀⢀⡴⣟⡙⡋⢻⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⢿⡿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠈⢻⣿⣷⣿⣿⡿⡞⠋⣔⣿⣷⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⢀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⠋⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠹⣧⣿⡇⡆⣿⣿⣿⣿⣿⣿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⡿⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⠁⣿⣿⣿⣿⣿⣿⡿⢷⡐⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠤⢲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡗⣿⣿⣿⣿⣿⣿⣿⣿⣌⡻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣫⣿⣿⣿⣿⣿⣥⣿⣿⣿⣎⢻⣿⣷⡀
⠀⠀⠀⠀⢠⣾⠞⠉⠻⠏⠩⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣜⢿⣿⣷⡙⣿⣧
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⣰⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⡿⢿⣿⢻⣿⣿⣿⣿⣿⣿⣷⡿⣯⢷⡌⢻⡄
⠀⠀⢠⡆⠐⠀⠀⠀⡀⣴⣿⣿⣿⡀⢰⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣯⣿⣿⣿⣷⣦⣄⡀⠙⢿⣿⣿⣧⠸⣿⣿⡴⡅
⠀⠀⠘⢂⠀⢸⣧⣾⣿⣿⣿⡟⠉⠁⢸⣿⣿⣿⣿⣳⣿⣿⣴⡏⠀⣿⣿⡿⠻⣿⠛⢯⣙⢳⣤⡙⣿⣿⣷⣼⣿⢷⡜
⠀⢀⠂⠈⠈⠉⢻⡿⣿⣿⣿⣧⠀⢀⣿⣿⣟⣿⣾⠙⣿⡟⠻⣿⡄⠛⠋⠀⠀⡿⠀⠀⠉⢷⣽⣷⣌⠙⣿⡅⢻⣏⠇⢶
⠀⢸⠀⢀⠀⢀⣿⣇⣿⣿⣿⣿⢧⡘⣿⣿⣿⣿⣿⣀⣯⣴⣶⣾⠇⠀⠀⠀⠈⣇⠀⠀⠀⠀⢻⣿⣿⡄⠛⣿⠆⣿⣆⠘⡃
⠀⢸⡀⠹⣖⣿⣿⣧⣾⣿⣿⣿⠈⢃⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣷⠀⠀⠀⠀⡾⠀⠀⠀⠀⠠⢑⣿⣿⠆⢺⣷⠸⠿⠁
⠀⠀⠉⠉⠛⠿⠍⢰⣟⣿⣿⡿⢦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⢰⣿⣄⣤⣴⣿⣷⣢⣥⣴⣾⣿⣿⣿⡿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠯⠸⢿⣿⣿⠟⠋⠙⠛⠿⠿⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⡴⠒⠛⠋⠉⠉⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣥⣤⣤⣤⣤⣤⣤⣤⣤⠀⢹⡆
⠀⠀⠀⣼⣧⠀⢸⠋⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡇
⠀⠀⠀⢻⢻⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⢸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⢸⡏⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠘⡇⣇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⡇⢿⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⢿⢸⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡧⠀⡇
⠀⠀⠀⠀⢸⡘⡇⠈⣧⣤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠛⠉⢉⣉⣉⣉⣉⣉⣀⣀⣇
⠀⠀⠀⠀⠈⣇⣧⣤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠛⠛⠛⠋⠉⠉⠉⠉⠉⢉⣉⣁⣂⣀⣀⡀⠘⣆
⠀⠀⠀⠀⠀⢿⠹⡄⠀⢀⡤⠤⡤⠄⢤⠒⡖⠒⡖⠘⣏⢹⣉⣙⣍⣉⣅⠨⣦⠽⡤⢧⠼⡄⠘⣆
⠀⠀⠀⠀⠀⠘⣆⢳⡀⠈⢧⠄⠺⡖⠚⡖⠙⡉⢹⣉⣹⣄⣵⡤⠴⡤⠼⡆⠲⡒⢳⠚⡗⠙⡄⠘⣇
⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠈⢷⠒⠳⠀⠙⡉⠹⡀⢣⠀⢱⣀⣣⡀⢵⡤⠼⡤⢷⠐⢳⠚⡂⠹⡀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⠘⣷⠒⢳⠒⠹⡉⢻⣉⣇⣀⣣⣀⣧⣠⣥⠤⠵⠴⠷⠚⠓⠛⠋⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠘⠒⠚⠛⠋⠉⠉⠉⠉⣭⡤⠤⠤⠴⠶⠒⠒⢲⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⣀⡀⢀⣀⣀⡀⠠⢷⡀⠀⠀⠀⠀⠀⠀⠀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠒⠒⢚⣒⣂⣈⣉⣤⡤⢤⣴⣶⣶⣾⠯⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡷⠤⣤⣤⣶⣖⣒⣚⣻⣿⣭⠭⠭⠽⠿⠓⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠄⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡤⠤⢤⡀⠀⠀⣀⡀⣤⡀⣀⣀⣠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡧⠤⢷⣀⠉⠁⠀⠠⠄⠀⠀⠀⡎⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⢠⢈⡗⢶⠛⠉⠒⠲⣤⣰⠃⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠤⠤⠊⠀⠸⡄⠀⠒⢀⠏⢹⠁⢀⡤⠾⣇⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠑⢄⡙⠶⠚⠉⠑⠒⠉⠀⠈⢛⣡⠔⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡴⢇⡀⠀⠀⠀⠀⠨⠓⢲⡤⡤⢴⠖⠲⠏⡉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢒⡏⠦⢈⠓⠢⣀⠔⢩⠀⡄⠉⠀⠏⡔⠁⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠛⠁⠲⡌⠀⠀⠀⠀⢸⢀⠃⠀⠀⡐⢁⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡸⠢⠄⢀⠠⠆⡎⠀⠀⠀⠇⠈⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠔⠁⠀⠀⠀⢰⠘⠀⠀⠀⠀⢠⠔⠉⠂⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡂⠀⠀⡠⠔⠣⢄⠀⠀⢀⠌⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠂⢤⡕⡀⠀⢀⠗⠐⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢠⠃⠀⠀⢸⠓⠛⠡⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠸⠀⠘⠂⠀⠀⠀⠀⠆⠒⠉⠘⠄⠀⡎⠀⠀⠀⢑⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⠢⠤⢀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⢤⣀⡀⠀⠈⠀⠀⠀⠘⡄⠀⡠⠂⠀⠣⡀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠀⠄⠀⠠⠍⠒⠒⠂⠀⢻⠀⠀⠈⠉⠒⠒⠂⠤⣴⠁⠌⠀⠀⠀⠀⠈⡀⠰⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠤⠄⠠⠄⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠑⠤⠤⠤⠤⠤⠐⠊⠁⠀⢣⠀⠀⠀⠀⡸⠁⠎⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠊⠁⡈⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣠⣿⠃⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠛⠛⠛⠛⠛⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠿⣦⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⠀⠀⣽⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⡾⠟⠛⠷⠟⠋⠀⠀⠀⠀⣠⣴⠾⠛⠀⠀⠀⠀⠀⠀⠘⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢻⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣾⡏⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⢀⣴⠟⠛⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣷⣤⡀⠀⣾⠃⠀⠀⠀⠀⠈⠻⣦⣴⡶⠶⣦⣀⣀⣴⠶⢶⣦⣀⣠⣴⠶⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣅⣼⡇⠀⢻⣇⠀⠀⠀⠀⢀⣴⠿⢿⣦⠀⠈⠉⠉⠁⠀⠀⠈⠙⣉⡁⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠙⢷⣄⢀⣴⠟⠁⠀⠀⠙⢷⣄⠛⠀⠀⠀⠀⠀⣿⣛⣿⠄⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣅⠀⠀⠀⠀⠀⠈⣿⠀⢠⣾⢻⣦⠀⠈⠛⠋⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠈⠻⣦⣀⠀⠀⢀⣼⠿⣧⠈⠻⠿⠋⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠀⠀⠀⠀⣿⠛⠻⠟⠛⠁⢀⣿⠀⠀⠀⠶⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⣀⣀⣠⣿⣀⣀⣀⣀⣤⣾⣏⣀⣠⣤⣴⠾⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠡⠔⠒⠊⠉⠉⠙⠒⠒⠤⣤⣀⡀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⣠⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢀⡞⡛⠉⠛⠋⠉⣉⣨⣏⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣿⡿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠀⠀⠀⢀⣾⣯⡀⠈⠉⠀⣀⠤⡀⠀⠰⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠉⡅⠄⠀⠀⠀⠀⠸⣿⣷⠺⠧⠀⠀⠀⠶⠶⠀⠀⢸⠻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⡀⢹⠏⡐⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣼⣿⣿⢻⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠁⠀⠘⠂⡞⠀⠀⠀⠀⠀⠀⢠⣿⡀⢀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣾⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢸⠀⠀⠀⡜⢀⠀⠀⠀⠀⠀⢠⣾⠻⢷⡈⣯⡿⠆⠀⠀⠀⣀⢠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠉⠁⠀⢀⠊⠀⠀⠀⠀⠀⢘⠻⢶⣿⣧⠀⠀⠀⣀⣴⠞⠁⢼⣏⣏⣏⠉⢹⡟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠛⢿⣿⠛⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠖⠊⡩⠜⠆⡀⠔⠀⠀⠀⠀⢽⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠛⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⣤⣾⡷⠒⠁⠀⠀⠀⠀⠀⠀⠈⢿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⣠⡾⠛⠁⠠⠄⠈⠁⡀⠐⠒⠓⠂⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠤⠊⠀⡠⠟⠁⠀⢀⣠⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣹⣿⢿⣿⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢇⠀⠀⣔⡊⠉⠀⠂⠈⣀⡠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡤⠞⠉⠁⠂⠠⠔⠉⡁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣰⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠈⠉⠛⠒⠆⠀⠀⠐⢶⡇⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⠤⢶⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢠⠂⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣤⣤⠀⣀⣀⣀⣀⣀⣤⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠐⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠠⢀⡔⢁⡤⠤⣤⠄⣀⡀⠀⠂⢄⠀⠀⡀⠤⠄⢀⢀⢠⣀⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⡀⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠉⠡⠀⠀⠈⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠐⠌⠀⠀⠀⠐⠀⠀⠂⣀⣀⣀⡀⡀⣀⣀⡀⠤⡐⠀⠈⠂⠠⠤⠤⠤⠤⡤⠤⠠⢄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣢⣀⠠⡄⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠋⠊⠈⠊⢑⣀⢐⠉⢁⠐⠒⠀⠠⠤⠀⠀⠀⡂⠀⢸⠀⠀⠒⠂⠀⠤⠀⠑⠒⠊⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⡴⠡⠁⠀⠀⠀⠀⠀⢸⠀⣀⠀⠀⠁⠘⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⠜⠄⠀⠀⠀⠀⠀⠀⠀⢸⠆⢀⣄⡴⠀⠈⠀⠀⠀⠠⡇⠀⢰⠀⠀⠀⠈⠢⠌⢁⠬⠁⠀⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⡠⠀⡠⠁⠸⠀⠀⠀⠀⠈⡇⠀⠘⠀⠀⠀⠀⠄⠃⠈⠀⠈⠀⠠⡇
⠀⠀⠀⠀⠒⠄⡀⢀⠂⠱⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠨⢀⡂⠀⠀⠀⠀⠀⠀⠈⡁⢀⢸⠀⠀⠀⠀⠀⠤⠠⠄⠀⣒⠚
⠀⠀⠀⠀⠀⠀⠨⠑⠬⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠂⠀⠀⠀⠀⠉⢙⠘⠀⠀⠐⠂⠤⠬⡥⡤⠤⠰⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⡐⢀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠀⠈⠀⠀⠈⠀⠁⠈⠀⠀⠀⠁⠒⠤⡤⢌⡡⢨⠤⠐⠈⡀⠀⠳⠤⢀⡀⠀⠊⠂⠐⠁⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠉⠉⠑⠉⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠠⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⢀⠃⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⠚⡆⠀⠀⠄⠓⠁⠑⠀⠐⠀⠠
⠀⠀⠄⠀⠀⠑⠀⠀⠀⠀⠀⡇⠀⠀⠂⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢀⠆⠀⠀⠑⠒⠊⠀⠀⠀⢀⣦⠀⢢⢀⠢⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⢀⠎⠀⠺⠇⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⡤⣄⡀⢀
⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⣁⠰⠄
⠀⠀⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠆⠀⠀⠀⠰⠇
⠀⠀⠈⠉⠈⠉⠉⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠊⠔⠁⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠂⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠁⡄⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠀⠀⠈⠈⡀⠀⠀⠀⠠⠼
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠠⠘⠀⡃⠈⡀⠄⠀⡀⠁⠐⠀⠀⠠⠀⠀⢃⠀⠀⠀⡄⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠠⠄⠄⠈⠠⠃⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠊⠑⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠄⡰⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠢⣀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢨⠃⠀⠀⢀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⢐⠆⠀⠀⠀⡀⢂⠀⠀⠀⣮⣁⣠⣤⠆⠀⠀⠀⡆
⠀⠀⠀⠀⡄⢘⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠈⠀⠀⠀⠈⢀⠀⠀⠀⠄⠀⠀⠀⠄⠁⠀⠀⠶⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠒⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⡉⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠆⠐⠚⠛⠓⠀⠀⠀⢈⡉⠉⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠁⠀⠄⠠⠂⠈⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⡀⠀⢀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⢀⠴⡁
⠀⠀⠀⢦⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠐⠀⠀⢠⠀⠀⠀⠤⠴⠂⠀⠀⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠠⠤⠀⠀⠄⠀⠀⠀⠈⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠛⠋⢻⣿⣷⠀⠀⠀⠀⠁⠀⢤⠄⠀⠉⠀⠀⠀⠀⠀⡿⠛⠉⠙⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⣠⣶⣶⣶⣶⣶⣾⡄⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠤⠀⠀⣰⣿⡿⠛⠋⠁⠹⣿⣿⡆⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣼⣷⡄⢠⡿⠋⠀⠀⠀⠀⠀⠈⠛⣿⠀⠀⠀⠸⣿⡏⠉⠉⣹⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡕⠀⠐⡌⣻⣿⡇⢸⣿⣶⡀⠀⠀⠀⠀⢠⣶⣿⠀⠀⠀⠀⢙⡿⠶⠾⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠉⠀⢸⣦⣼⠁⠸⣿⡃⠀⠻⣿⣷⣄⠀⢀⣠⣿⣿⠟⠀⠀⠈⢾⠉⠑⠀⠀⠊⠙⢧⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠃⢀⡠⠈⠛⢹⠄⢰⠏⠻⠀⠀⠀⠀⠀⡀⠱⠀⠀⠀⠀⠀⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⠄⠌⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⣼⠀⠀⠀⠀⡆⠀⠀⣁⡀⢘⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠐⠀⠀⠀⠸⠗⠀⠀⠐⠐⡇⠀⠀⢻⠃⢸⠀⠀⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⣶⣶⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠄⠂⠒⠤⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⡦⠤⠤⠤⠤⠤⠤⠼⠀⢘⡰⠂⠀⠀⠀⡀⣀⠀⠀⠀⠸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠰⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠘⠀⠀⠈⠁⠉⠁⠀⠀⠃⠦
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡤⠤⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⢱⠀⠀⠘⠀⠀⠀⠀⠂⠀⡇⠰⠀⠘⠀⠀⠀⠀⠈⠀⠀⠐⠀⠐⠀⠀⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠛⠒⠒⠒⠒⠁⠘⠒⠒⠒⠒⠃⠙⠛⠛⠛⠋⠀⠈⠛⠛⠛⠛⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠄⠀⠁⠐⣤⡀⠀⠐⡆⠀⢰⣧⠠⠤⠄⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⢰⣶⣶⠚⠀⠀⠁⠀⠸⣿⣆⢀⠂⠠⠈⢃⡀⠀⠀⠀⠀⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⡀⢰⣿⣿⣀⠀⡠⠋⠀⡀⠀⣿⣿⠀⡀⠰⡇⠈⣅⠀⠀⠀⢦⣄⣤⣿⡿⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠢⣄⢘⣿⣿⣿⣿⡟⢹⠃⢀⡶⠀⠀⣿⣿⡄⠀⠊⣿⠀⠸⠀⠀⠀⣼⣿⣿⣯⠀⢠⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⢰⣿⣿⣿⣿⣿⡿⠅⣸⠀⡆⢶⠀⠀⣿⣿⡌⠀⠀⣿⠀⢠⠠⡆⠸⣿⣿⣿⣷⣠⣿⡟⢐⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡏⠈⠉⢟⠋⢻⣿⣿⣷⡾⠁⠘⡇⢸⡆⢸⣿⣟⡟⠃⠈⣷⠀⠼⢠⡇⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠓⠀⠀⠀⣠⠄⠸⠷⣌⡇⠀⠀⣁⣸⡄⣼⣿⣿⡇⠃⠀⢿⡀⠘⢺⠁⠀⡿⠟⠻⣿⣿⣿⣷⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣄⡴⠶⠞⠁⠲⢲⣀⠈⠳⣤⡀⠰⣾⡇⢹⣿⣯⣇⠀⠀⡿⠀⠀⠃⠀⡘⠁⠀⣹⣿⣁⣉⢛⣻⢿⣾⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⣦⣀⠑⠀⣯⣇⣼⣿⣿⣿⣥⣀⡇⠀⠀⡇⠠⠁⢀⡴⠋⠁⠀⠙⠟⠛⠛⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⡗⢿⣿⡟⢯⡟⢿⣽⣷⡀⠀⡇⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⡾⠧⣀⠀⠀⡠⣠⡄⠀⣸⣿⣇⣬⠅⠻⢿⣥⣿⢟⣯⢟⡀⣷⡏⠀⢠⠀⠀⠀⠰⠃⠀⠀⢀⠀⠀⠛⣷⣄
⠀⠀⠀⠀⠀⢰⠂⠀⠸⣁⠀⠋⠀⠀⠀⠐⠆⠀⠀⠀⢀⣾⣿⡏⠠⠀⣹⣲⣿⣿⡒⡳⣾⣿⣾⡇⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⠁⢄⡠⢾⣿⣇
⠀⠀⠀⠀⠀⠀⢠⠆⠀⡀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⢺⣿⣿⠃⡠⢾⡟⠉⠉⣉⠛⠿⢄⠙⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡙⣧
⠀⠀⠀⠀⡀⠀⠀⠀⠔⠻⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⣿⣿⣿⠊⠀⠌⣷⡇⡟⠶⣶⡣⠄⠡⠈⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣼⢿⣿
⠀⠀⠖⠀⠡⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⣀⠙⢶⣤⠐⢬⣅⡀⠀⠀⣸⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣟⡫⣧
⠀⢐⠃⠀⠀⠀⠁⠀⠀⠆⠀⠀⠐⠀⠀⠀⡆⠀⠀⣾⣿⣵⣿⡤⣿⣯⠀⠠⣿⠀⠀⣸⣿⣼⢶⣽⣿⡗⢄⠀⠀⠐⠀⣀⠀⠀⠀⡇⠀⠈⠈⢰⠹⡇
⠀⠒⠰⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠁⠀⣸⠗⢿⡧⠘⢰⠏⠀⠀⠀⣿⠀⠒⢈⣿⡿⠀⠈⠛⠀⠀⠀⢀⡀⢀⣼⠂⠀⠘⠣⠀⠀⠀⣾⠁⠻⡄
⠀⠂⠀⡄⠀⠀⢀⠀⠀⠀⠀⢀⣀⣠⠴⠃⣠⠴⢡⣴⡿⠏⡤⠊⠀⠄⢠⠀⣿⠀⠀⠀⣄⡙⢄⣸⣧⣤⡗⢐⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⣠⡟⠆⠀⡿
⠀⡉⠀⠀⠀⠀⡆⠀⠀⣠⠖⠁⠀⠀⠀⠀⣴⠞⠋⠁⠐⠊⡁⠀⠀⠀⡿⠀⢛⠀⠀⢠⣿⡿⣾⣿⣿⣌⡛⠿⣄⢰⡀⠀⠀⠠⣶⠀⠀⢸⡿⢃⠀⣹⠇
⠀⠀⠀⠇⠀⠀⢹⡀⠀⠀⠀⠀⢀⣤⣾⠛⣉⣀⡁⣀⣰⣯⡊⣠⠴⢶⠎⡉⡛⠁⠑⠾⣗⠙⠃⣈⣷⡙⠣⠁⠋⠻⣷⣄⠀⠀⠀⢨⣘⠿⣯⣧⣽⣿
⠀⠀⠀⠈⢄⠀⣸⠃⠀⣠⣶⣿⡿⡟⡽⡟⡏⢹⢿⡿⣿⣍⠻⠇⣠⣤⡇⠀⠦⠀⢤⣀⣿⡷⠊⠁⢨⣿⣶⠈⢱⢴⢮⣍⣷⣄⠀⠀⠈⣻⣿⣿⣿⡿
⠀⠀⠀⠀⠘⡗⠁⢀⣴⢿⣿⡟⠁⠀⠠⡏⠇⠀⠀⠐⡘⡮⡁⢤⡩⣤⣼⠀⠁⠀⣌⣀⣿⡥⢒⣥⣾⠏⡞⢠⠀⠀⠀⡜⠁⠋⠻⣷⣾⣿⢟⣽⣿⠁
⠀⠀⠀⠀⠀⠹⣦⣾⣳⣿⣿⠁⠀⠆⠄⠈⠊⢄⠀⢶⣲⣑⣿⣆⠳⡉⠙⠀⡅⢐⢋⣿⡟⣰⣿⡿⠋⣠⠁⠎⡃⡄⡀⣅⠀⠀⡧⢹⣭⣽⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠄⠀⡴⣤⠀⠠⠘⣦⣀⣽⣿⣿⣿⣇⢥⠀⠀⢇⠨⣸⡿⣼⣟⣛⣥⣾⣯⠈⡰⡹⠃⢇⢅⠀⢀⡟⢸⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣍⣿⣿⡆⡄⢷⡘⢗⡀⠄⢀⠈⢻⣟⣿⣿⣿⣠⡄⠀⡀⢠⣿⣿⣿⣿⡿⠿⠟⢁⣸⡟⠁⢠⡾⡀⠀⣾⣧⣘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⡌⠿⣟⢿⣿⣷⣦⣆⣠⢬⣙⠿⢿⣿⡇⢀⣲⠺⣿⣿⣿⣿⣿⠷⠛⣫⡏⣠⣠⠞⠁⡁⣰⣿⢳⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣟⡳⢮⣽⣿⣿⣿⣿⣿⣿⣾⣿⣷⣶⣴⣶⣿⡿⣿⣿⣿⣷⡾⠷⠾⠟⡉⣠⢠⣷⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣬⣍⣛⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⢁⠀⣠⢂⣔⣴⣧⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣝⡛⣿⣿⣿⣿⣿⣿⣿⠟⠘⣿⣿⠻⢿⣿⣿⣿⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠿⠿⠛⠉⠀⠀⠈⠛⠷⠿⠿⠿⠿⠿⠛⠛⠉