Jockey Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣿⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠙⣿⢛⡶⢶⡾⠛⢿⣵⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢄⣨⣷⡀⠛⢋⡵⠈⡨⢁⣾⡿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠀⠀⠀⢠⣰⡴⣄⢀⣀⣀⣀⡤⠉⢡⡴⢋⠂⢰⠃⣾⡟⠓⠾⠿⢷⣶⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⢀⠀⢉⣿⣿⣿⡧⠙⢉⣤⡶⠋⠀⠎⠀⠋⣰⡟⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡞⢻⠃⠀⢠⠂⠀⠀⠀⡀⠓⠠⢰⣽⣿⡿⠡⣴⣿⠉⠀⠀⡞⠀⢳⣎⡽⠁
⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⡿⠀⢸⡄⠀⠸⣄⠀⠀⢀⠘⢦⠀⠀⠀⠉⠉⠓⠍⢷⠀⠀⠀⠀⠀⠠⣿⡇
⠀⠀⠀⢀⣾⣿⣿⠿⣿⠟⠿⠁⠀⠀⠳⣄⣲⡈⠑⠀⣌⡇⠈⢿⣦⣄⣀⣀⠀⢠⠸⣰⠀⠁⠀⠀⠠⢼⡁
⠀⢀⡴⢿⠻⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣷⣌⠓⠀⢸⣿⠈⠉⠛⠛⠿⠾⠿⣦⣄⣲⠤⣤⣀⣀⡀⠀⠀⡀
⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⡇⠀⠀⠀⠀⠀⢀⡀⠀⠉⠻⢿⣿⣄⠈⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢧⣀⠀⠀⠀⠀⠙⠿⣤⣀⣀⣀⣈⣿⡷⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠤⣍⠛⠛⠿⠶⣦⡀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣶⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀
⣿⣿⢋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⡷
⠀⠀⠀⠀⠀⠀⢰⣴⣿⣤⣤⣤⣤⣀⡀⠀⠀⣴⣾⣷⣿⣀⠀⢄⡅⠂
⠀⠀⠀⠀⠀⣴⣿⣇⣗⣦⣾⣭⡛⢿⣿⣷⠀⣿⡿⠛⠿⣿⣆⢸⣿⣖⠞
⠀⠀⠀⠀⣰⣿⣿⡟⣿⣿⡽⣿⣿⣧⣌⠻⣿⣿⣇⣤⣦⣤⣦⠔⠁
⠀⠀⠀⢀⣷⣽⣫⣾⠿⠩⢿⣿⣿⣿⣿⣿⢹⣿⣿⡅⣦⣾⠁⠀⠤
⠀⠀⠀⢻⣿⡟⠉⠀⠀⠀⠀⢿⣿⣿⣿⣟⡼⣩⣿⣇⠹⢿⣶⠘⠋⠲⣿⣷⣄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠵⣻⣿⡀⢸⣿⠁⠀⠀⢻⣿⢋⣿⣼⣿⣧⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣖⣴⣿⣿⣿⣋⣿⣃⣀⠀⢸⡿⣮⣿⣿⣿⣿⣿⣿⠦⠄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣯⣽⣬⣿⣿⣿⣷⣿⣿⡟⠈⠻⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⣿⣿⡿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣾⣿⣿⡽⠋⠉⢹⣿⣿⢿⣿⣿⢟⠋⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⣾⡽⠛⠉⠁⠀⠀⠀⢸⣿⣿⣿⠟⠛⠁⠀⠀⠀⠀⡢⢊⠜
⠀⠀⠀⠀⠀⠀⠀⣯⡄⠀⠀⠙⢷⣄⠀⣤⡀⠀⣠⣿⣿⠟⠁⠀⠀⠀⠀⠀⡠⢚⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠛⠛⠙⠉⣸⣿⠿⠁⠀⠀⠀⠀⢀⠴⠊⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠁⠀⡀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢹⣯⠇⠀⠀⠀⠀⠁⠀⠀⠀⣾⡷⣄⠂⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠊⠀⠀⠀⠉⠒⠶⢶⣄⣤⣀⡀⠀⠀⢸⠁⠉⢽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⠋⠉⠁⠀⠀⠸⣦⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣬⣷⣄⣾⣿⠿⠻⣿⣿⡇⠀⠀⠀⣄⡀⢠⣄⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡏⠀⠀⠀⢹⣿⣿⠴⠚⠉⣣⣼⣷⢹⣦⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢿⣿⣿⣧⠀⠀⠀⣴⣟⡀⣀⣴⠾⠛⠉⣟⠀⢛⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⢹⣿⣿⣿⣇⠀⢴⡟⣫⡷⠋⠁⠀⠀⠀⡟⠈⠁⠘⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⢀⣼⠿⣿⣿⣿⣈⣽⠟⢩⠀⠀⠀⡠⡠⢀⡇⠀⣥⡠⢀⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡜⠁⠻⣾⠁⣤⠿⣿⠟⠁⠀⠀⠀⠁⠀⠈⠙⢁⣾⢣⠘⠦⠀⠸⣿⡇
⠀⢠⣸⡲⢤⣀⢀⡀⢤⣄⡀⠀⠀⠀⠀⠀⢀⡤⠀⠁⠈⢉⣀⠒⠒⠋⠉⠂⡘⢀⣿⣎⡠⠖⠁⡞⠐⠀⠀⠀⠀⠀⠀⠀⣠⣦⣙⢿⣄⣦⠀⠒⢻⢇⡱
⠀⠀⠻⣶⣤⣤⣤⣤⠀⠘⠛⠓⠒⠂⠀⣺⠋⠀⠀⠀⠀⢻⠋⠀⠀⠀⠀⢀⠃⣼⡟⠛⢶⣄⠘⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡟⠁⠊⢿⣦⡀⠈⠤⠘⢀
⠀⠀⠀⣬⣿⣿⣿⣶⣦⣤⣀⣠⣤⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣸⣿⠀⡀⠀⢠⠛⣦⡄⠀⠀⠀⠀⠀⠀⣡⣿⠏⠀⠀⠀⠀⠙⢿⣆⠸⠆⣾
⠀⠀⠀⠈⠙⠛⠻⠟⠻⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠿⠿⢙⠿⣦⣤⠎⠀⠈⠀⠈⠓⠄⢀⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢘⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⡾⢡⠃⠀⠀⠀⠀⠀⠀⠠⣄⠉⠙⠛⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⣷⠀⠀⢠⣿⣇⠈⠂⠀⠀⠀⠀⠐⠺⠇⠘⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⣎⣷⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⡄⠀⠀⢴⡇⠀⠀⢸⣿⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⣿⣿⣿⣿⣿⣷⣦⣴⣤⣤⣿⣇⠀⢻⣿⡅⠀⠀⠀⢱⣄⣀⣿⠃⢰⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡄⠈⢿⣿⠀⢀⠀⠀⡙⢿⣿⣠⡟⠀⠈⠲⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣿⡟⠁⠀⠘⣿⣿⣿⡿⠉⠙⠛⠿⠦⠌⣿⣷⠌⠳⣄⠈⠢⣻⣿⣷⣮⣷⣶⣤⡠⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠸⡟⠀⠀⠀⠀⡿⢛⡿⠁⠀⠀⠀⠀⠀⠀⠈⠛⠛⠲⠶⢯⣀⣀⠈⠢⡀⠉⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠂⠱⣄⠀⠀⠸⡀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⢶⣷⠀⠀⠀⣿⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠑⠢⡀⠉⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡿⢃⣀⠀⣰⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠑⡄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⢡⣿⣿⣿⡿⣃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡖⢰⠀⡇⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠼⣁⡘⠈⠉⠉⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⡇⠓⢐⡄⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣷⡇⢸⢾⡇⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⡆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠉⣠⣾⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⣐⣿⣿⣿⣿⠙⡱⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⢿⣷⣤⣀⠀⠀⣠⣴⣿⠿⠁⠀⠀⠐⢋⣻⣿⣿⡿⠿⣾⣧⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠄⠀⠀⠦⢤⣀⠀⢠⣶⣿⣿⣿⣷⣈⣙⠛⢛⣣⠐⡟⢿⣷⣤⠀⠒⡀⢐⣿⣿⠋⣀⣀⣀⣬⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣩⢟⠁⢄⡠⠀⠀⠀⢠⣤⣿⣎⠻⣿⣿⣍⠀⠀⠁⠀⢻⡿⣔⡀⠀⠙⢿⣿⣳⣿⠿⣻⠛⠉⠙⠉⠉⠀⠸⠿⠓
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠀⠰⣿⣿⣶⣮⣠⡆⠀⠀⣀⠀⢹⡏⠙⠒⠓⠚⢷⣤⡀⠀⠀⠀⠀⣄⠸⠄⠈⣿⣿⡋⡴⠃
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⢀⣴⡈⠈⢻⡟⠛⠀⣼⣿⣷⣈⢷⣄⡀⠀⠁⠀⠈⢻⣦⡀⢀⠠⠀⢺⠀⠀⣹⡧⡞
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁⠀⢻⣾⠋⢣⠈⢁⠀⠀⣿⣿⣿⣿⣶⣯⣍⣑⡀⠀⣶⣿⣿⣷⣿⣗⠄⠀⠀⢰⣿⣷⠁
⠀⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠸⣿⡁⣻⠀⠉⢀⣼⣿⣿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢰⣿⣿⡿
⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⣿⣿⡏⢀⠔⣿⣿⣿⠏⠀⠀⠀⠀⠈⠉⠉⣿⠿⣿⣿⣿⣿⡇⡘⠠⠿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⣾⣿⠟⠁⠀⢀⣴⣿⢛⣫⠄⠁⢰⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⢸⠇⠁⠀⠀⠀⠀⠉⠛⠻⢦⡀
⠀⠀⠀⠀⠙⠉⠀⠀⣶⣿⣿⠿⠋⠁⠀⣤⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣶
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠐⢺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣿⣧⣤⣀⣀⣀⠀⢀⠀⠀⠀⠀⠠⣦⣤⣋⣀⣀⣀⣀⡀⠀⢀⣀⣀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠋⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡟⡩⡿⠟⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⢐⣿⢻⣿⡁⣀⠀⠀⠀⠀⠀⠀⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⣄⣠⣟⣿⣿⢿⣛⣉⣀⡈⠃⠀⠀⠾⣷⣾⣿⡄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⠹⣿⣀⣿⣿⣿⣿⣿⣆⠈⣉⣹⠁⠀⣠⣶⣤⠈⠈⠉⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣦⣌⡛⠹⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣁⣴⣿⠏⠁⣯⠀⡄⠐⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⡈⢉⠁⠀⠙⠛⠉⠙⠻⣿⣿⡿⠻⣏⣿⣿⠃⢠⡟⣿⠾⠁⠀⠀⢈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠄⠀⠀⠀⢦⡀⠀⢠⣿⣿⣿⣿⣿⡿⠃⠀⠄⢿⣼⣿⠦⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠳⢀⠈⠈⣡⣴⣿⣿⣷⣿⠟⣩⣶⣿⠟⠁⠀⠀⡄⣾⣿⡿⠗⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⡷⢀⠴⠛⠋⡙⠀⠀⠀⢸⠃⣿⣿⣽⡶⢦⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠛⠏⢀⣴⠀⠀⠀⠀⡘⢀⣿⣿⣫⣟⡁⠁⠀⠀⢸⡀⠀⠀⠀⡿⢰⣿⣿⣿⠋⠛⠈⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⡀⠀⢸⡏⠀⠀⠀⣠⣧⠾⠯⣸⡏⠁⠑⠠⠀⠀⠀⡇⠀⠀⣸⡇⣸⣿⣿⣿⠀⠀⠚⠛⠋
⠀⠀⠀⣀⠀⠀⠀⣠⣾⣿⣿⣿⡇⠀⠀⠀⢰⠂⣾⣿⣿⣄⡀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⣿⣿⣿⣿
⢀⢀⣼⣿⣷⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⢰⡸⣷⣬⡙⣿⡏⣻⡃⠀⠀⠀⠀⠀⠀⠑⢢⣄⣉⣛⣛⣛⣻⣿⣿
⠈⣫⣿⣷⣶⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢷⣿⣿⣿⣷⡀⣬⡇⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⠟⠁⢸⡆⠀⠐⣼⣿⣿⣿⣿⣷⣿⣿⡇⢸⣧⠀⠀⠀⠄⠀⣼⣿⣿⠟⣠⣿⣿⣿⡇
⠀⠀⢈⣿⣿⠛⠉⠉⠉⠀⠀⠀⠀⢻⠀⠀⢻⣿⡇⠙⢿⣿⣿⣿⡇⠀⢿⡷⣿⢠⣀⣠⣿⣿⡇⠸⠋⢩⣿⣿
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢸⣿⠇⠀⠀⠈⠛⢿⡇⣶⠀⠐⠁⣿⣿⣿⣿⣈⢁⣠⣴⣿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⡆⠀⠀⠀⠀⠸⣷⠛⠃⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠷⠔⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣷⠀⠀⠀⠀⠀⢻⣟⠀⠀⠘⢒⠛⠛⠙⠋⠉⠙⠿⠿⣶⣷⣦⡈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠛⢿⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣇⣀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣀⠀⠀⠀⠀⠀⠀⢸⣿⣤⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡤⠄⠀⠀⠀⠹⣿⢳⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡇⠀⣾⣿⣆⢀⡘⠛⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠠⠈⠙⠿⠿⠛⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣦⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠊⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⣤⡄⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣄⡛⠂⣦⢀⠘⣿⠃⠀⠀⠀⢀⣀⣀⣠⣴⣾⢷⣿⣮⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢟⣀⠈⣉⠲⢶⢿⡇⠀⠀⠀⣾⣿⣿⣿⣿⠟⣘⣿⣿⣿⣷⣶⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⢠⣶⣶⣾⣄⣸⣿⡟⠉⢉⡤⢚⣿⣿⣿⣿⣿⣯⣷⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠐⣿⡟⠹⣿⠿⢾⠟⠉⠙⣳⣶⣿⣶⣿⡿⠿⠻⠿⣿⣿⣿⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⣰⡟⠀⠀⡿⠀⠀⠀⠀⢹⣮⣿⣿⣿⠏⠀⠀⠀⠀⠀⠉⠙⠳⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢛⣻⣧⣀⡀⢁⡄⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠀⠀⠀⢠⣦⣄⠀⢠⣴⣀⣿⣀⠀⠀⠀⠀⠀⠉⢹⣿⠉⠀⣻⠁⡈⠙⣶⣤⣱⣿⠃
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢿⣧⠀⢿⣦⣽⣿⣿⣦⡙⣿⣿⣷⣄⠀⠀⠉⠀⠒⠲⣿⡖⠋⠁⡀⠈⠳⣾⣿⣿⡿
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠷⠾⠋⠀⠀⢹⣛⣇⠹⣛⠃⢻⣿⡿⣿⣷⣦⣤⣀⣀⣀⣿⣇⠀⣾⡇⠀⠀⣸⢿⣿⠇
⠀⠀⠀⠀⣴⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠈⠫⣿⣶⣌⠃⢿⣿⡇⠀⠈⠙⠻⠿⠿⠿⢿⣿⣶⣦⣇⢠⣾⣻⣷⣀
⠀⠀⢀⣨⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣮⢭⣿⣶⣶⣄
⠀⣤⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⠀⠈⠙⣿⡇
⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡘⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣎⠑⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⣴⡾⠛⠁⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠈⢠⡄⠀⠀⠀⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡄⢀⣀⠀⢀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢸⣇⠈⢁⡀⢉⡝⠳⠃⠀⠀⣀⣤⣤⡄⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠻⣿⠃⢠⡷⠂⠀⠀⣤⣶⡿⠟⠋⠀⠙⣢⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠄⢈⡀⠚⠦⣤⣾⡿⠿⠉⠀⢀⠀⠀⠀⠈⣯⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣯⣭⡉⠉⠉⡇⠀⠉⠓⣶⡦⠀⣴⣿⠀⣐⡒⣦⣨⠶⠤⢤⡴⠁⣰⣿⣶⣀⣼⣯⣾⡋
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣴⣶⣶⣾⡿⠋⠁⠀⠀⠀⠀⢴⢇⠀⠀⠀⢘⠳⢾⡿⠋⣈⠉⠉⠉⠁⢠⡴⠋⠀⣰⠟⠀⠀⠀⠈⠻⣾⡇⠛
⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⠛⠁⠀⣿⡃⠀⠀⠀⠀⠀⠀⢜⠶⠦⠤⣄⡈⠐⡆⠵⠀⠁⠀⠀⠀⣀⡈⠁⣠⣾⠋
⠀⢹⣿⣿⣿⣿⢿⣿⢟⡿⠃⠀⠀⠀⢿⣿⣶⣤⣤⣤⠄⠀⠸⢀⡀⠀⠀⠀⠈⢁⠓⢀⣀⡀⠀⠀⠸⣿⣶⣿⠁
⠀⠀⠁⠛⠋⠁⠀⠉⠁⠀⠀⠀⠀⠀⠈⠻⢿⣟⠛⠧⢀⠀⢸⢘⠿⣶⣄⣀⠀⣿⠀⢻⠀⢀⠀⠠⠆⢹⡟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⠀⠀⣿⡇⠈⠻⢿⣿⣿⣄⣸⢧⡊⠀⢴⣤⡾⠣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⣿⠀⠀⠀⠀⠉⠙⠛⠚⢇⡀⠈⢻⡷⠤⠴⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣤⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣶⣇⢠⡄⠀⠲⣶⣆⣀⣀⣀⠰⣆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣧⣀⡀⠀⠙⠛⠛⠉⠁⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⢋⣙⣂
⠀⠀⠀⠀⠀⠀⣠⣾⢶⣶⣶⣤⡀⠀⠀⠀⠈⠉⠙⣹⠟⠿⣿⣶⣶⣶⣤
⠀⠀⠀⢀⣠⣾⡽⣣⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⣿⣦⡀⠈⢿⡿⠋⠁
⢀⣠⣴⣿⣿⢟⣾⢟⣼⣿⣿⣿⣿⣿⣿⣇⠀⢀⣀⣸⠟⠋⠀⣨
⢜⢷⣿⣿⠳⠿⠗⠉⠙⣾⣿⣿⣿⣿⣿⣿⣷⣿⣍⣩⣿⠟⠉⠁⠀⠀⠀⠀⠀⣀
⠀⠁⠚⠁⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣿⣿⣿⡟⣽⢿⣿⣦⡤⠸⣶⣶⣶⡾⢛⣽⣿⣿⣿⣿⣿⣷⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⣷⣬⣿⣿⢟⣼⣿⡌⢿⣿⣷⡘⡹⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣂⣬⣿⣿⣳⣿⣿⣿⣛⣳⣿⣿⡷⠀⣿⣿⣟⣠⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣶⣾⣿⣿⣿⣿⡿⣧⢸⣿⣿⣿⣿⣿⣿⠇⠘⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⡘⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⢹⣿⣿⣿⣿⣭⣾⣿⠯⣿⣿⣿⣿⠿⠋⠀⠀⠘⣿⣿⣿⣿⣿⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⣠⣽⣿⣾⣿⣿⣿⣿⣿⡴⠿⠿⠛⠂⠉⠉⠈⣿⣿⣿⡟⠀⠈⠛⢿⣿⣄⠀⠀⠀⠀⠈⠻⠿⢿⡿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠋⠉⠉⠀⠀⢹⡀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⠀⠀⣨⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⢃⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⠀⢀⣤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣄⠀⢀⠀⠀⠀⢸⣿⣀⣀⣀⡀⣀⠀⣴⣶⠖⠋⠉⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠫⢁⠀⠀⠈⠉⠉⠉⠉⠉⢻⣿⠋⠁⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤

⠀⠀⠀⠀⠀⠀⣀⠀⠄⠀⠀⡠⢄⠠⠀⠐
⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⠏⠀⣠⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⡀⠈⡙⠛⠻⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠏⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠉⠛⣻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠐⠛⠛⠛⠛⠿⡿⠻⠟⠋⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠁⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠈⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣶⣦⢀⣠⡀⠀⠀⠈⢻⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡎⠙⠛⠿⠟⠛⠛⠁⠀⠀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣷⣤⠀⠀⠀⠀⠀⠀⠀⠿⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⠿⠿⢿⣶⣤⣤⣀⣠⣴⣾⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣿⣿⣿⣷⣦⣄⣀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣾⣿⣿⣷⣶⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠉⠉⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣀⣤⣴⣿⣿⣿⣿⣿⡿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣼⡿⠛⠋⣩⣿⡿⠛⠉⠀⠀⠀⠀⠀⠉⠉⠛⠛⣿⡿⠛⠉⠙⠻⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⣰⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣶⠟⠋⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠺⠿⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄

⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣒⣂⣰⣿⣿⢻⠟⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣿⣿⣿⣿⡿⠓⠊⠀⠀⣀⣤⣶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡏⠀⢠⣠⣴⣾⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠙⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣠⣤⣤⣀⣵⣦⣾⣿⣿⣿⠿⣿⣿⣿⣿⣿⠟⠛⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢫⣿⣿⣿⣿⣿⣿⣿⣿⠃⠙⣿⣿⣿⣿⣿⣹⣿⣿⣴⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣴⣶⣶⣾⣿⣿⣿⢇⠟⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠐⢿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠏⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠙⠻⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣴⣾⣿⠿⠋⣰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠻⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠀⠸⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠉⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⢀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠻⣿⣤⣤⣤⣤⣶⡶⠟⠛⢁⣤⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡉⠀⠀⠀⠀⠀⢻⡟⠉⠉⠉⠁⠀⠀⠀⠘⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣽⣽⣽⣭⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣽⣿⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣡⡤⠀⠈⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠀⣀⣠⡖⠀⠀⠘⠧⢄⠄⠀⠀⠀⠈⠙⢿⣶⣤⣀
⠀⠀⠀⠀⠀⢀⠴⢛⣿⡿⠿⠿⠭⠥⡴⣿⣇⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⢻⣿⠟⠓
⠀⠀⠀⠘⣦⣀⠀⠉⠀⠀⠀⠀⠀⡜⠀⠈⠻⣿⣆⡴⠚⠳⠮⢅⠀⠀⠀⠈⠁⠀⢀⡴
⠀⠀⠿⢂⣯⠁⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⡔⠉
⠀⠀⠀⣼⠋⠀⠀⠀⣠⠀⠐⡴⢥⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⡇⣇⣠⡄⠀⣿⣷⠦⠄⠀⠀⡄
⠀⠀⢰⢿⣆⣀⣤⡾⠏⣀⠜⠁⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⢿⡈⢷⡈⠻⣿⡀⠻⣿⠀⢲⠲⣿⠇
⠀⢀⣟⠈⢛⠿⡋⠔⠊⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⡙⠢⠙⡇⠀⠀⠀⠸
⠀⠘⠿⣿⠟⠀⠀⠀⠀⠀⠀⣠⣶⣖⣋⠉⠀⠀⠀⠀⠀⡀⠀⢠⠄⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣍⠭⠭⠛⠒⠲⠶⡞⠁⠰⠋⠀⠀⠀⠰⠾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⣀⠀⠀⠈⠙⠒⠶⢤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⢿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⢀⣼⡇⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠃⣈⢡⡟⠻⣿⣿⠦⠏⣑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡛⢺⠧⠀⠋⠈⠁⢠⠀⠀⠈⠚⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢀⣠⣴⣿⣰⡆⠈⠁⠀⠀⠀⠀⠀⠱⣶⡗⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⣿⣿⠿⠿⣿⣿⡿⣿⣿⣶⣄⡀⠀⠀⠘⢲⣿⠻⣼⣿⡇⠐⠁⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠙⣯⠁⠀⠀⠀⢠⠏⠉⠀⠉⠉⠙⠳⣤⣀⡝⠂⣙⣦⠼⢿⡷⠀⠀⠀⠀⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⣠⣾⠟⡙⡷⣶⠃⠀⠀⠀⣠⠃⠀⠀⠀⠀⡀⠀⠀⢸⣿⣷⣮⣥⣀⣤⣄⣀⣀⡀⢀⣼⣿⠿⠿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⡿⠷⠊
⠀⠀⠀⢀⠀⠈⠩⡠⢵⣿⡅⣧⢤⢤⡼⢁⡀⠀⠀⠀⠐⠀⠀⠀⠀⢻⣿⣿⣿⡎⣽⠯⢢⡀⠀⠉⠙⢇⠀⠀⠀⠁⠉⠀⠈⠉⠀⠈⠉⠁
⠀⠠⣥⣉⣀⠄⡠⣿⠋⣻⣆⣞⠵⠋⢰⣿⡟⠃⢰⣾⡖⠀⠤⠄⠀⢸⣇⣿⣛⣃⣈⡼⠋⠑⡀⠀⠀⠀⠑⡤⠀⠀⢀
⠀⠀⠀⡀⠁⠈⠉⠛⠚⠛⠉⠁⠀⠀⠈⠛⣏⡀⠙⣿⡧⡀⠀⠀⠀⠆⣸⠏⠁⠉⠉⠀⠀⢀⠃⡀⠀⠀⠀⢁⡨⠖⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠎⢢⣤⠐⠕⠦⣤⣾⡾⠃⠀⠀⠀⠀⠀⠠⣂⠀⠠⠀⠀⠀⠀⠀⢀⠤⡀⢠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠈⣿⣶⠄⠀⠀⣻⣿⠤⠤⠤⠤⠤⡖⠋⠉⣉⠦⡀⠀⠀⠀⢀⢎⠁⠘⣂⠙⠿⠉⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠈⠑⠂⠐⠁⠀⠀⠀⠈⠂⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢤⣄⣸⡒⠂⠀⠀⠀⢄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⣻⡻⣿⠁⠀⠀⣠⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠱⣿⢄⠫⠦⣀⡴⣻⣿⣿⡿⠿⢿⣿⡄
⠀⠀⠀⣀⣠⣴⣶⣤⣀⠀⠀⠀⠀⡠⠴⢶⣦⣄⣀⣀⣀⣀⣀⣀⢌⠉⠁⣠⣴⢷⣿⣿⣿⠇
⠀⠀⠘⢿⣿⣿⠿⠿⢿⣷⣤⣤⡾⠃⠀⠈⠛⠿⡿⣩⠙⣥⡿⠋⣤⣴⣿⠿⠁⣰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⣦⣶⣾⣑⠆⠘⠿⠷⢺⣿⣿⣿⣿⣷⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣦⣿⣿⣿⣿⣿⡆⣦⣤⣸⣿⡿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡿⠃⣿⣿⠛⠣⠽⢿⣿⡿⣽⣦⣨⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⢀⣾⡟⠁⠀⠀⠀⠀⠀⣾⡿⠉⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠉⠛⠶⣤⣀⠀⠀⣸⠏⠀⠀⠀⢈⣿⠇
⠀⠀⠀⠀⠙⠁⠲⠀⠄⠀⠀⠀⠀⠀⠙⣷⠀⢀⣀⣀⣉⣶⠖⠋⠘⠳⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠂⢠⡀⢀⠀⠀⠒⠀⠘⠍⠰⣮⠉⠉⠋⠈⠀⠀⠀⣰⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠙⠙⠻⠿⠛⠟⠿⠾⠟⠓⠐⠚⠓⠐⠒⠂⠛⠐⠛⠿⠛⠋⠛⠙⠃⠘⠉

⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠻⡼⠀⠀⠀⠀⣸⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢰⠛⢦⠀⣠⠴⠚⠉⠉⠁⠀⠀⠀⠀⠉⠉⠓⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⡏⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⠻⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠋⠉⠲⢄⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⣿⠀⠀⠀⠀⠀⡴⠋⡇
⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⣄⠀⠀⣠⢦⡀⠀⢠⡄⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠞⠁⠀⢹
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⠤⠴⠁⣀⣁⠈⠁⠀⠉⠒⣁⠈⠂⢤⡤⠴⡞⠁⠀⠀⢀⡴⠃⣀⣀⣠⡼
⠀⠀⠀⠀⠀⢀⡇⠀⡄⢠⠀⠀⠀⠀⠀⠀⣀⠀⡸⠀⢠⠊⠁⡈⠑⡄⠀⠀⠀⠸⣴⠋⣠⡈⢣⣧⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⢈⠉⠀⠀⢸⠀⠺⠿⠂⡄⠀⠀⠀⠀⢷⠈⠛⠋⢸⣧
⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⠀⠔⠉⠁⠙⠦⠜⠀⠀⠀⠀⠑⠤⠤⠖⠁⠀⠀⠀⠀⢸⠉⠒⠀⠁⢸⠀⠀⠀⠀⣶⠒⠒⠒⠤⢤
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠨⠀⠀⠴⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⢸⡇⠀⠀⠀⠈⢳⡀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠳⢄⣀⡀⠀⠀⠀⠀⡋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠤⠤⠇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠹⡄⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⢸⠇⠀⠀⠀⠀⢠⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⡞⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠑⠲⠤⠤⣀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⣀⡤⠒⠁⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⣳⠃⠀⠀⠀⢀⠀⠀⠀⠀⠈⠢⣀⠀⢠⡏⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠇⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠈⠓⠋⣸⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠸⡄⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠉⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠑⡄⠀⠀⠀⠀⣰⠛⠦⣄⡀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠈⠢⣄⣀⠞⢧⠀⠀⠀⠉⠒⠶⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢀⡔⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣠⠞⠉⠉⠓⠦⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣦⢸⣧⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠙⠿⠛⠉⢈⠶⠒⡕⠊⢁⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣌⠁⠀⡤⠊⣠⢺⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⠀⢞⣽⠟⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⣨⠀⡼⠟⠋⠉⢈⣀⠉⠉⣲⣶⠀⠀⠀⠀⠀⠉⡢⠛⢄⠈⠻⣮⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⡞⣁⡄⠀⠀⠟⠉⠀⠀⡌⢠⡑⢄⠀⠀⠀⠀⢹⡄⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⢻⠃⠀⠀⠀⠻⠂⠀⣈⠙⡦⠈⣧⠀⠀⣰⣷⡑⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⡀⣠⠔⠂⠀⠉⠉⣀⠀⢸⠀⣴⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣄⠀⠃⠃⢲⣒⣀⣭⣭⡾⠀⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⡀⢀⠀⠻⠿⠿⠟⠁⣰⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⡑⠤⢀⠠⢔⣵⣿⣿⣿⣿⣟⣡⣴⣶⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⡟⠣⢅⡻⡿⢿⠿⢿⣿⣽⣿⣷⣿⣏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠓⠤⣦⠀⠈⢦⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡙⠀⢀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠻⣿⢻⢻⠋⠟⢳⢦⣄⡀⠀⠀⠀⠈⡅⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣄⠂⠄⠀⡜⡼⠸⠻⣦⣀⣴⠞⠓⢦⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⣾⠀⢠⢡⢡⠀⠃⢡⠹⠁⠘⢘⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢳⠆⠇⡎⠂⠀⣆⡾⠋⢂⣤⣴⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⡁⠘⠸⡸⠱⣆⡴⠋⣠⡴⡞⠟⢆⠇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠂⠃⢠⠀⢡⡥⣫⠴⠻⣯⠀⠀⠀⠘⠘⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣵⣴⡜⠀⣞⣴⠞⠋⠀⠀⠀⠘⢿⣄⢧⠁⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣷⡋⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣴⣮⣼⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠆⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⠀⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠆⠀⠀⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠉⠒⠢⣄⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣀⠖⠒⠉⠉⠉⠁⠒⠢⢌⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠔⠁⠈⢀⡀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣽⠢⠀⠑⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡇⠀⠀⡤⠀⠀⠤⠄⢀⣠⣾⡇⠀⠀⠀⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠻⣿⣶⡶⠿⠛⢻⡟⠀⠀⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠉⠻⠄⠀⠔⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⠤⠤⠤⡤⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⢠⠃
⡄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⡎
⠱⡀⠀⠀⠢⡀⠀⠀⠀⠀⠑⠀⠀⢄⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠘⢎⠀⠀⠈⠂⡤⢆⠀⠀⣀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠑⠦⡀⠄⠀⠀⡀⠀⠀⠁⠂⠤⣀⠈⠢⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠦⠤⡄⠀⠀⠀⠀⠀⠀⠈⠶⠥⣄⠀⠀⠀⠢⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠀⠀⠈⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡸⠁⣀⠀⠀⣀⠜⠀⠀⠈⠢⣀⠌⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⢠⠤⡀⢠⠙⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣬⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢩⠀⣠⠚⣣
⡇⠘⡁⠀⢷⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⠷⠁⡰⠉
⠃⡀⠀⠀⠠⠠⣤⣄⣡⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣴⡇⠀⠛⠀⠼⠁⠀⢀
⠀⢡⠀⠀⠀⠀⠀⠈⠁⢠⠄⠼⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠈⠡⠀⢀⣀⣠⡀⣠⡀⠀⠀⠀⠀⠀⠙⠶⣀⠈⠃⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠳⣀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠛⠀⠀⠋⠀⠀⠠⠟⠀⢹⠀⠀⠀⠀⠀⠀⠀⣹⠀⠘⠳⠄⠀⢀⡔⠦⢀
⠀⠀⠀⠈⠛⡦⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⣄⡀⠀⠀⠘⢦⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢰⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⢴⣆⢀⣀⠠⡤⣤⢤⡄⣀⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⢀⠌⠁⠀⠖⠋
⠀⠀⠀⠀⠀⠘⡇⠀⢹⡆⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠻⣄⠀⠀⠀⣰⠋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⢀⡤⠊⠁⠀⢀⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⠀⠀⣴⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠈⣦⠀⣀⣧⣠⠛⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⡆⠀⠀⣄⢠⡜⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⢨⡏⠀⠀⠀⠀⠀⠀⠐⠀⠈⠃⠀⠀⣀⡤⠞⣋⠀⠀⠀⢰⠀⠀⣿⡏⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡘⠁⠀⠀⠀⣤⣄⠤⡤⢤⣤⡤⠚⠛⠉⢀⣴⡿⠀⠀⠀⠘⠓⣤⣿⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣀⣤⡴⠾⢿⣿⠃⠀⠀⢠⣄⢀⡼⠋⠙⢶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⢋⣀⣀⣀⣼⠃⠀⠀⠀⢠⡟⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢣⠀⠀⠀⠀⠙⢿⣟⠉⠀⠀⣀⡤⠏⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠷⣀⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⠳⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡶⠛⠉⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⡏⠉⠉⠉⠉⠉⠀⠲⠶⠒⠦⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠶⡶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠉⠠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉⠉⠉⢛⠉⠉⠀⠘⠒⠒⣶⠶⠤⣤⣤⣤⡤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠘⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡆⠀⠀⠀⠀⠀⠀⢀⡦⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⡷⠀⠀⠀⠀⣀⡤⠋⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⠀⠀⢿⡀⢠⡄⠒⠉⠀⠀⠀⢀⢼⡇
⠀⠐⠋⣀⠀⠁⠙⣀⠀⠀⠀⠀⢀⡼⠶⠊⠙⠂⡀⠀⠀⠀⠀⡰⠃⢠⠃
⠀⠈⠲⠄⣀⣉⣉⣉⣉⣉⠉⣰⠛⠤⢀⡀⠀⠀⢡⡄⢀⣀⠐⠷⢀⡙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣠⣤⣤⣤⣬⣥⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠻⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⣀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡿⢦⣾⡄⢠⠾⢁⡐
⠀⠀⠀⠀⢰⡄⠈⠆⠈⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢈⣄⠀⠀⠀⠁⢠⠄⠁
⠀⢀⠀⠀⠀⠙⣄⠈⢀⠀⡀⢰⠀⠀⠀⡀⠀⢀⠀⠀⠀⠛⢥⡄⠈⠉⠓⠒⠲⡄
⠀⠈⢢⡀⠑⢆⡀⠆⠈⠴⠗⠈⢀⠀⠘⠀⡤⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠈⢢⣄⡙⠞⠁⢀⡠⠤⠛⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠀⡀⠀⠃
⠀⠀⢠⡄⠀⠉⢁⡀⠀⠘⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢂⠀⠂
⠀⠀⠀⠀⠀⠀⠈⢣⣀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⢤⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠚⠁⣼⠛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣀⣀⣀⣈⣧⠀⢡⡀⠀⠀⣀⣀⣀⣀⡠⠴⠶⠚⠋⠁⠀⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠃⡀⠀⠀⠀⠀⠀⠀⣀⠞⠉⠙⠛⠀⠉⠉⠉⠉⠙⠛⠿⣷⡀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢡⠀⠀⠀⣠⠶⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠈⠹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠲⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣃⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠋⠁⣠⣿⡆⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⢠⠀⠀⢷⡖⠒⠀⠶⠶⠶⠊⠉⠁⠀⠀⣠⣴⣿⣿⡇⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢸⠄⠀⠀⢲⣄⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⣦⠄⠀⠀⠀⠀⠀⠀⠀⠤⠒⠁⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⠁⢆⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠟⠁⠉⠁⠈⣿⡿⣿⠇⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠙⠻⢇⣀⠄⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠈⡍⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠉⣠⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣠⣾⡋⠀⠀⠀⠀⠀⠀⢀⣰⠞⠃
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋⠉⡛⢿⠓⠒⠒⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠲⠶⢶⣶⠶⠖⠋⢻⠉⠲⣤⠀⣃⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⢨⠷⡀⠀⣾⠀⠀⠀⠈⠋⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠢⢤⣤⡀⣀⣀⣀⣀⠀⣤⡴⠛⠁⠀⠀⠈⠀⢃⣴⠁⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠴⣦⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠈⢃⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠘⠃⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⡰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠤⠖⠛⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⢀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢇⠀⠀⠀⠀⣸⠘⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⣠⠾⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠳⣄⣀⡴⠉⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⣉⣠⠶⠛⢳⣄⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠉⢀⣴⠃⠀⢀⡾⠁⠀⠀⠀⠀⠀⡀⣀⣠⠤⠚⠛⠉⠁⠀⠀⠀⠹⣦⠀⠛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡏⠻⡀⠘⠿⠒⠲⠶⠒⠒⠒⠘⠋⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠘⠦⣀⠀⠀⠀⠀⢀⡟⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠈⠓⠶⠶⠶⠋⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠴⠤⠴⠴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠣⠴⠋⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣤⣄⡠⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⢘⣿⡄⡀
⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⢀⠀⠀⠀⠀⣀⣤⠀⢠⣀⠀⡄⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⢿⣿⠟⠇⠟⣦⣀
⠀⠀⠀⠀⠀⠀⢘⠛⠛⢰⣿⣿⣷⣦⡀⠉⠁⢠⣾⡛⠊⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡋⠀⠟⠃⠀⠀⢀⣸⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠁⣤⢼⣿⠟⠀⠉⠻⢿⣷⡤⠸⣿⣧⣾⠏⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⢀⣹⠶⡗⢫⡉⢿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣦⡀⠤⠐⠊⠉⠀⠠⢠⣥⣤⣤⠰⣞⡂⠀⠀⣀⣀⡀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠰⠃⠻⠟⢠⣾⠃⠈⣇⠈⠋⢻⣿⣧
⠀⠀⠀⠀⠀⠀⣭⡻⠟⠋⠸⣷⣌⠳⣄⠀⠀⢀⠎⠀⠈⢛⣶⡘⢿⣟⡀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠸⡀⠀⠈⠛⢿⣧
⠀⠀⠀⢆⣛⣴⠟⠀⠀⠀⠀⠘⢿⣧⠹⣆⢀⠸⠤⠖⠊⠁⠀⠀⢀⣿⣷⡄⠈⣯⣍⡙⠲⣄⠀⠀⠀⢀⠀⠀⠀⢀⣤⢤⣶⡟⡁⠀⠀⠀⠀⣧⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⣿⣤⣶⡇⠀⢀⣾⣀⣀⣀⣀⣘⣩⣭⣤⣶⣿⣿⣿⣷⡈⠃⠀⠀⠂⠀⠀⠔⢋⣤⡾⢋⣮⣤⣤⠶⢁⣼⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣠⡄⠀⠚⠩⢯⡘⢿⠿⠖⠒⣸⣿⣿⡷⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡘⢛⠁⠀⠛⠢⠰⣇⠀⠀⢀⡼⠹⣿⢟⣾⡀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠴⣾⠏⠀⠀⠈⠳⣤⠏⠀⠀⢵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⡀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠿⠙⠁⣴⣿⣿⢿⠿⠇⠙⠁⢀⣀⣤⡄⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣅⣀⣀⠐⠿⠂⠀⠀⣤⣌⠲⠾⠿⠟⠁⣰⢺⡆
⠀⠀⠀⠀⠀⣀⣀⣤⠀⠀⠀⠀⠀⠠⠤⢠⡶⠊⠉⠉⠁⢀⡠⠀⡠⣶⡌⢣⡀⠸⣿⣷⡿⠛⠻⠦⣀⠠
⢠⣤⣤⣶⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠈⢠⣮⣾⣿⣿⣿⣿⣐⡼⠋⠀⠀⠀⠀⠈⠉⠁
⠈⠿⠛⠛⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠸⣳⠋⣹⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⠉⡁⣀⢀⠠⢤⣄
⠀⠀⠀⢀⣦⠀⣠⣤⣤⡶⠛⠉⠙⠓⠶⣶⢦⣷⠀⠰⠞⠳⡄⠙⣻⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣀⠀⠠⣤⣤⠆⠀⠤⠼⠟
⢀⣀⣴⠋⣠⠞⠉⠉⠁⠀⠀⠀⠀⠀⠀⢹⡌⡏⠐⠀⠀⠀⠀⣰⠌⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠀⠘⣤⣴⣤⣴⡄⣀⠷⠆⢀⡄⠀⠀⠀⠀⡀
⠈⠽⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⢰⡆⠀⢀⣸⠟⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠘⠿⠯⠁⣴⠿⠛⠇⠀⠘⢡⣾⣌⡙⠓⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠠⠴⠾⠧⠦⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣴⣶⠖⠀⠀⠀⠀⠀⠚⠂⠀⠀⣿⡟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠒⠛⠓⠲⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⣰⣼⣿⡿⠂⢰⡾⠟⣀⣶⡽⠋⠀⠀⠀⠀⠑⣆⠀⠐⠆⠀⢻⣿⣄⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢾⣿⡿⠿⠛⠉⠀⠀⢋⠀⣸⡟⠁⢀⠀⠀⠀⠀⢰⢠⣿⣿⣷⡆⢻⣶⡿⠛⢧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠎⠀⢠⡏⠀⠀⣿⣄⡟⢛⣷⢸⣿⣿⣾⣏⠀⠀⠀⠳⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣧⣇⠀⠀⢀⣠⣤⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⣻⣷⣦⣀⣉⣡⣤⣤⣶⣾⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠺⣿⣿⠀⢠⣼⠟⠉⠀⠀⠁⠘⠻⠷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⣷⣤⡈⣿⠁⠀⢈⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣦⣤⡷⠒⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⢀⣰⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠛⠁⠀⠛⠋
⠀⠀⠀⠀⠀⢠⡴⠶⠛⠉⠀⢻⠹⣿⡟⡹⠋⠻⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⣿⠟⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢃⡆
⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡘⢡
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣛⡹⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣬⠑⠀⠀⣠⣿⣼⣇
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⢠⠿⠿⠁⠀⠀⡀⠀⠀⠀⢀⡡⠤⠀⠀⠀⠀⠐⣤⠇⠸⢿⠀⠂⠔⣿⣟⡠⠞⣷
⠀⠀⠀⠀⠀⠀⢸⡰⡆⠀⠀⠀⠀⢀⣶⡶⠀⠀⠀⠀⣳⡜⠀⠀⢘⡁⢠⡦⠀⡀⠀⠸⡆⡾⠀⠣⠈⠠⣼⣿⠟⠁⠀⣠⡇
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⡀⠀⠀⠚⡉⠀⠀⠤⡄⠀⡶⢰⡖⣠⡜⠃⠀⠁⠘⢇⣠⡴⠇⠽⢤⣀⠀⡇⠿⢃⣤⠶⠋⠁⣿⠀⢸⡀
⠀⠀⠠⠁⣿⣿⠟⢉⡁⢈⣯⡠⠰⡿⠇⠀⢀⣶⡇⢠⣧⣼⡅⠘⠀⣤⣼⡟⣴⣿⣁⣀⣀⣀⣤⡬⠷⠁⢴⠛⣵⠀⣀⠴⠀⢀⣎⢸⡄
⠀⠀⣇⢰⣏⡩⣶⢁⠀⡂⠀⡱⣤⠁⠸⠿⠤⠙⠁⡞⠁⠈⣧⠀⠀⠏⠙⠃⣾⣅⡀⣀⡀⣀⡄⡠⣾⣷⠈⢆⣿⠊⢉⠄⠀⢘⡁⢀⣙
⠀⠀⠸⡢⣿⣧⡜⢂⣉⠔⣤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠀⠀⢻⣯⣽⣯⣼⣧⠯⠻⢿⠇⠀⠀⠡⣔⠉⠀⠀⣫⡀⢀⣈⣇
⠀⠀⠀⠘⠻⢿⡿⠧⠁⠈⠔⠁⠀⠀⢀⣼⣤⣀⣀⠀⠀⠀⠀⡟⢃⠀⠀⢀⠀⠙⠶⣶⠶⠖⢒⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣯⣥⣤⣤⣭⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣏⠉⢄⠀⠀⢰⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠂⠘⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠌⡌⠐⠀⠆⠱⠈⠆⡆⠀⢘⠀⠠⠋⠰⠈⠀⠘⢠⠁⠚⠛⢠⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢭⡀⣿⣤⠀⢀⠀⠀⠴⠶⠄⠀⡀⠘⠃⢠⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠉⠛⣦⡄⠈⢠⣦⠾⠀⡆⢰⠝⠠⣶⣢⣿⣳⣀⡀
⠀⠀⠀⠀⢰⡒⠛⢲⡄⠀⠂⠁⢐⠀⠀⠀⢘⠏⠀⡼⣩⡿⢓⡚⣻⣶⣮⡁⠀⣀⣤⣄
⠀⠀⠀⠀⡿⠋⠀⢈⣧⣀⣠⠾⣡⣤⠀⠀⣎⡄⡰⣿⣿⣿⡼⠻⠛⢿⣄⡀⣾⣿⣯⠢⠀⢀
⠀⠀⠀⠈⡁⢠⡴⡞⢧⢈⠻⣷⣾⣿⣮⣟⠳⠄⢁⢸⣿⣿⣿⣴⡄⠀⠙⠯⣉⣻⣿⣷⠿⠸⠇
⠀⠀⠀⠰⡰⠨⠰⣿⣏⢯⢧⡌⠈⣿⣿⣿⡇⠀⢳⣹⣿⡿⣿⣿⠁⠀⠀⠀⠸⣿⠏⠀⠀⡀⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡻⠄⠈⠃⠀⠈⢻⣾⣃⣘⣾⣿⣿⣷⢹⣿⠀⠀⠀⠀⠻⢸⠂⢀⣾⠋⠀⢸⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⠀⠀⠀⠀⠀⣿⣿⣹⣿⣿⡿⢻⣎⣷⡄⠀⠀⠀⢠⠇⠀⠀⠉⣿⠀⢰⠻⡍
⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠘⢶⣄⠀⠀⣿⠟⠁⣨⠁⠳⣦⣭⣌⣿⣄⡀⠹⠏⠀⠀⠀⠀⡿⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣧⣀⣿⠇⠀⢤⣘⣷⣿⠃⠘⣟⠋⠉⠋⠙⠀⢀⠀⠀⠀⠀⣰⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠈⠀⠀⠀⠈⢻⣏⠀⠀⢻⡀⠀⠀⠀⢀⣏⡀⠀⠠⣞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣶⡄⠀⠀⣟⡀⠀⢸⡇⠀⢠⣟⣿⡜⠁⠀⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣀⣀⠛⢿⣆⣸⣇⡠⠿⠏⠁⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠀⠈⠁⠈⠉⠈⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡟⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⣀⣀⡾⠀⠟⣴⣿⣯⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⢻⣄⠀⢉⣡⠀⢀⡶⢸⠏⢀⡶⢀⡟⠻⣦
⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣶⣿⠇⣼⠟⠛⠛⡇⠀⠚⠁⡄⠀⢨⣴⠿⢃⣴⣿⠁
⠀⠀⠀⠀⠀⢸⣇⢠⣾⣟⣡⡏⣸⠏⢀⣴⣿⣷⣤⣴⣾⣿⣤⣤⣴⣾⣿⣟⣡⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣻⠟⠋⠉⠉⣀⠉⣀⠛⢁⣴⠀⣸⠃⢈⡤⢀⡉⠀⣡⠄⢈⠁⢀⡟⠀⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣤⣴⠞⠀⣿⠀⡟⠀⠰⠃⢰⡏⢠⡿⠃⢨⠁⣼⠟⠀⡍⠀⠞⠀⠘⣉⠀⠉⣿⡄
⠀⠀⠀⠀⣿⠙⠶⠟⠋⠀⣸⡇⠀⢿⣤⣴⣦⡔⠁⣦⣤⣶⠇⢰⣤⣴⣾⣧⣴⠎⣰⣿⠟⠻⢷⣿⠇
⠀⠀⠀⠀⠹⣷⣄⣀⣠⣾⣿⣧⣀⣀⣿⡍⡿⠀⢸⣿⢉⡏⠀⣿⠃⠘⠷⠟⠃⣰⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠙⠛⠛⠋⠘⣷⣤⣿⠇⣿⣄⣰⣿⣆⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠋⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⠀⠀⠀⠀⠀⣀⣴⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣻⠚⠇⠀⠀⢠⣾⣿⠿⠛⠉⢠⠷⡄⠴
⠀⠀⠀⠀⠀⠤⢒⣲⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⢤⣤⣤⣴⣟⢣⠀⣀⢀⣤⠟⠉⠢⢤⣾⣿⣤⣈⣢⣄⠀⢂
⠀⠀⠀⠠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⡿⢛⡿⠀⠉⠁⠈⠉⡷⠖⠒⡾⣆⣀⠀⣀⣀⣉⠳⠄⠈⠀⠀⠀⢀⣾⣿⠋⠀⠉⠉⠙⢿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠁⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣻⣿⣷⠄⠀⠺⠦⠀⠀⠀⣹⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⢠⣴⣤⣶⣶⡲⠄⠀⠾⠿⣥⠀⠀⢀⡀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠀⠀⠀⣠⠾⠛⠛⠛⠛⠻⢷⣤⣄⣀⠀⠀⠀⢠⣷⡀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣲⣾⣿⠟⠉⢀⣠⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣶⣾⣿⣷⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢉⣦⣤⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡗⠒⠶⣦⣌⠳
⠀⠀⠀⠀⣀⣤⣐⣢⠞⠁⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠉⠉⠁⠀⠀⡰⠋⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡓⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⣄⡀
⠀⠀⠀⠀⣀⣄⡀⠀⡀⣿⡆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿
⠀⠀⠀⠚⠛⠛⠟⠿⠛⠿⠛⠿⠟⠛⠛⠋⠝⠛⠛⠋⠉⠛⠙⠛⠁⠀⠀⠀⠀⠀⠀⠠⠤⣶⠄⢠⢄⠽⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⡠⠖⢚⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⢠⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⣀⣤⡀⠀⠈⢦⠈⢢⡀
⠀⠁⠀⠀⠀⢿⡉⢢⡀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠄⠀⠠⡶⠿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢠⣇⡼⠋⣰⠃⠀⠀⠀⢧⠀⢳
⠀⠀⠰⣟⠓⢮⣳⣄⠱⣼⡇⣠⠏⠀⠀⠀⠀⠀⠀⢀⡾⣫⡴⠆⠀⠀⠀⠀⠀⠀⠀⣀⡉⠻⣆⠀⠀⠀⠀⠀⣞⢉⣉⡙⢿⣄⠀⠀⢾⡥⠶⣦⠀⠀⢸
⠀⠀⠀⠈⠓⢄⡙⠁⣠⠟⠁⢳⡀⠀⠀⠀⠀⠀⠀⣸⠇⢁⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡦⠹⣧⠀⠀⠀⠀⠉⠁⠀⡝⠀⠈⢷⠀⠀⡤⠖⠋⠀⠀⢸
⠀⠀⠰⣏⣉⣈⡁⠰⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠙⢦⣄⡀⠀⠀⢀⣠⡴⠛⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠁⣤⣉⣱⡄⠀⠀⠏
⠀⠀⠀⠀⢠⠞⣡⣄⣀⣀⣄⠙⢦⡀⠀⠀⠀⠀⣠⣿⡀⢸⣇⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⣿⠀⢸⡷⣤⡀⠀⠀⠀⣰⠋⢠⡷⠒⠋⠁⠈⠉
⠀⠀⠀⠀⠘⠋⠀⠈⠉⠉⠈⠳⣄⠙⢦⡀⠀⠀⣧⣂⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢀⣀⣸⠇⠀⢠⡞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢦⠀⠈⣿⠀⠀⢿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠀⠀⣿⠁⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⢄⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣠⠞⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠈⢧⡈⠉⠙⠛⠛⢛⡿⠃⠀⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⡀⠈⠻⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⣄⣀⠉⠛⠛⠛⠋⠁⠀⣀⣠⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣀⣼⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⠀⢀⣠⠶⠛⠙⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠙⢦⡀⠀⣠⠏⣹⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⢻⡙⢦⣀⡴⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠲⠋⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠘⠛⠛⠛⣧⠀⢰⡆
⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢠⠄⣀⠀⡀⠀⡄⢠⠀⠲⠈⠁⢀⣿⡇⠀⢠⣤⣄⠈
⠀⠀⠈⣡⡤⢠⠄⠀⣴⡇⠀⠀⠀⠀⠀⢠⡄⠀⡤⡀⢄⠀⠁⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⠀⠹⠿⠟⠀⠀⢀⣄
⠀⠀⠀⠀⠰⠈⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠉⠁⠈⠉⠀⠁⠡⠶⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠐⡅⠀⢠⠃⠀⠀⠃⠈⢀⣴⣷⡀⠛⠛⢁⣀⣀⡀⠀⢀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠑⢸⠘⠀⠀⠐⠰⠾⣿⡿⠆⠀⠀⠀⠀⠈⠀⠈⠁⠀⠐⢛⡉⣉⢳⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣀⠀⠀⢠⣈⣠⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠖⢳⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠙⠀⠀⠘⠖⠃⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⢀⣠⣶⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢄⡒⠀⠀⠸⠀⠀⠀⠚⠁⠀⠛
⠀⠀⠀⠛⠛⠿⠇⠀⠛⠋⣤⡶⠀⡀⠀⠀⠈⠥⠀⠘⠂⠀⠀⢄⡐⠀⠀⠀⠸⡀⠀⠀⠀⡀⠀⠀⠛⠀⢀⢀⣀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠁⢀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠻⣿⣿⣿⣄⠀⠀⣶⡄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠸⠇⡌
⠀⠀⠀⠻⠿⢿⣿⡷⠾⠛⠀⠀⢨⣯⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⡎⠀⠀⠀⣌⠀⠤⠀⠀⢠⡄⣬⠀⠀⢠⡆⠅
⠀⠀⠀⠈⠁⠈⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣴⣦⣦⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣪⣵⣾⣿⣟⣭⣯⡹⣿⣿⣷⣦⣕⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢡⣾⣿⣿⣿⡿⢻⣿⣿⣷⣜⠻⣿⡇⢹⣿⣝⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣘⢛⣛⣻⣵⣾⣿⣿⣿⣿⣿⣿⣶⣥⣌⣩⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣿⣿⣿⠿⠋⠁⠀⠈⠙⠿⢿⣿⣿⣿⠟⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠿⠛⠛⠓⠂⠀⠀⠐⠚⠒⠒⠌⢻⣿⣦⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠿⣿⣿⡿⢠⡶⠶⠄⠹⠄⠀⠀⠴⠶⢦⡄⠀⢿⣿⡿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢠⣤⣿⣻⠇⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣿⣿⣹⡆⠀⠀⠀⠀⠛⠓⠈⠀⠠⡀⠀⠀⢰⣧⣿⣿⣏⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⣿⣿⣿⣿⣿⣧⠀⠀⢰⣾⣡⣄⣍⣝⣶⠽⠀⠀⣼⣹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⣠⠄⠉⠋⣽⣿⣿⡄⠀⠀⠻⠿⠻⡿⠟⠋⠀⠀⢀⣿⣿⣿⣍⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⣤⣤⣼⣿⣿⣿⣷⣄⠀⠀⠒⠖⠂⠀⠀⠀⣠⣾⣿⣿⣿⣿⣾⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣀⣤⡾⢻⣿⣿⣿⡟⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣛⣛⣙⢛⡛⣻⣿⣿⣿⢿⣿⠿⠛⠋⠉⠀⠘⣿⣿⣽⣷⣶⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⢫⡹⣿⣿⣿⠋⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡌⢿⣧⣼⣿⣿⠿⠟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢀⡀⣆⢧⢻⣿⣇⠀⠐⢝⢿⡄⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⢻⣿⣿⠇⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣦⣼⣿⣼⡞⡆⣿⣿⣿⣦⣤⡀⠙⢦⡀⠀⠀⠀⣰⡿⠃⠀⣀⣴⣿⡇⠁⠀⢀⣾⣇⡴⠉⢣
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡿⣿⣷⠹⡼⠻⣿⣿⢿⣿⣷⣦⣑⢄⢀⡾⣁⣤⣶⣿⠟⠋⠁⠀⢰⣀⡾⣿⣿⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠫⠀⠛⠛⠃⠁⢻⠻⣿⣷⡿⡟⠛⠛⠉⠀⠀⠀⠀⢠⣸⣽⣿⣿⠏⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠀⠀⠀⠀⠀⠀⠈⠃⠘⠃⠙⠛⠀⠀⠀⠀⠀⠚⠀⠀⠛⠻⠻⠋⠀⠛⠛⠁⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣉⡀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⠀⡀⠀⠘⣿⣀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣶⣛⣫⣥⣤⣶⡾⣡⣤⠤⠀⠁⠈⠀⢀⠀⠀⠀⠀⢀⡿⢹⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣤⠀⠘⠡⠭⠭⠵⠶⠶⠦⢤⣴⣟⣿⣶⣤⡀⢀⡴⠋⠀⠀⢀⣀⠉⡰⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠶⠄⣸⠷⠂⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠙⠻⠷⢏⠀⠀⠀⠀⠀⢠⠞⠀⠀⡴⠁⣲⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣿⠟⠛⠉⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⢠⡿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⢶⠻⠿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠙⠳⡄⠀⠀⠀⠀⣿⣿⠁⠳⡀
⠀⠀⠀⢀⣿⡇⠀⠀⠀⣠⡆⠐⢲⣾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡄⠣⣴⣶⡀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⣿⡟
⠀⠀⠀⡞⢿⣦⣤⣴⡾⠛⣠⡴⠋⠀⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠙⢦⣈⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠀⣻⠀⠀⠈⠙⣧
⠀⠀⣼⣇⣀⣼⢛⡥⠖⠋⠁⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠬⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠘
⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣠⠄⠀⠀⢀⣤⣾⡇⠀⠀⠀⠀⠀⠀⠘⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣂⣠⠴⠖⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠋⠁⣀⣤⠶⠛⠉⠀⠉⠀⠀⠀⠀⠀⢀⣀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣰⣾⣿⣿⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⢀⣀⠤⠤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢹⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢠⣾⡇⠀⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⠛⠻⠟⠛⠿⠿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⡥⠄⠀⠀⠀⠀⠀⢤⡀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠘⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⣾⣷⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡐⠀⠀⢀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⡤⢴⠆⠀⠀⠠⠂⠀⠀⡠⠀⠁⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢌⣀⣠⡔⢈⡀⠠⣤⠐⠈⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⢹⣧⣼⡷⠀⠀⠀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠠⠐⢻⡀⠀⠀⠀⠙⠈⠁⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⠁⠀⢀⠠⠀⠐⠛⠊⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠑⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⢉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠰⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣽⣷⠇⠀⠀⠀⣷⣿⣫⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⡻⠃⠀⠀⠀⠀⠘⠿⠿⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠀⠀⠀⢱⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⠙⡄⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣄⠐⢀⡜⠁⠄⠀⢀⠀⠀⡇⠀⠀⠀⠘⢆⠀⠀⠀⢸⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⣻⠟⡝⡟⢷⣶⣤⣀⣀⠞⠁⠀⢀⠀⠀⠈⠧⠀⠀⢈⣀⡀⠈⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠋⢀⡬⠐⠋⠘⠁⠁⠀⠐⠦⠙⢷⣦⣀⠀⠈⠙⠂⠀⡠⠚⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣡⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⣯⡣⡀⠒⠀⡼⠁⣰⠃⠀⠁⠀⠀⠀⢳⠐⠚
⠀⠀⠀⠀⠀⠀⠐⠟⡴⢡⣇⠛⠀⡀⠀⠀⠀⠐⠂⠀⠈⠀⢀⠀⠛⢸⣻⡷⣵⡀⠀⠱⡀⠹⣄⣀⣀⡤⠀⢠⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠹⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⠏⠀⠇⠻⠁⠀⠀⣹⢶⠤⢍⡭⠤⠖⠁⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⠦⢄⣀⣀⡠⠴⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⣀⠀⢰⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⠤⠤⠤⠤⠔⠚⢿⢄⡀⠀⠀⣀⢀⡠⠊⠁⠀⠀⠀⡇⠀⠀⢇
⠀⢠⠊⢉⣚⠉⢹⠄⠀⢀⣀⣠⠔⠋⡽⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠑⣾⡷⠏⠀⠀⠀⠀⠀⠀⣃⣀⠀⠈
⠀⢿⢠⠋⢀⢱⠈⠓⡄⠀⠞⠿⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠞⠁⠀⠀⠀⠀⠀⣀⣴⡉⠵⠮⣗⠒⣄
⠀⣰⠛⢻⠾⠊⡤⠞⠁⠀⠀⠀⠀⠘⠓⠋⢹⡏⠉⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣅⢎⣥⡼⢴⠃
⠀⠈⠋⢉⣓⣚⣆⡴⣦⣀⡠⠤⠤⠒⠒⢃⣸⠇⠉⠉⠁⣇⡀⠉⠙⠓⠒⠒⠦⠤⢄⣀⠉⢱⣌⡹⠯⣀⡼
⠀⠀⠀⠈⠓⣮⠿⠛⠉⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⡀
⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢘⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠋⠔⠋⠄⡰⠀⠀⠀⠂⠠⠀⠀⠂⠱⠘⠀⠶⠂⠋⠇⠃⡇⠱⠀⠀⠰⠂⠁⠀⠀⠴⠀⠆⠀⠸⠉⠱⠋⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠿⠿⣿⠿⠗⠀⠀⠀⠀⠉⠀⠀⢀⣴⣄⠀⠹⠇⢀⡀⠀⢰⣿⣿⣷
⣿⡀⠀⠀⣿⣿⣿⠄⠀⠆⠀⣴⡁⠀⠀⠀⣾⣿⣷⠄⠦⠀⣾⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠉
⣿⣷⡀⢴⣿⣿⣿⣧⠀⠀⣼⣿⣷⡀⠀⣠⣿⣿⣿⡄⠀⠰⣿⣿⣿⡷⠀⢀⣄⣀⠺⠿⢿⣿⠿⠟⠂
⠻⠋⠁⣼⡿⣿⣿⣷⠀⢀⣼⣿⣷⡀⠀⣹⣿⣿⣿⡏⠀⠼⠿⣿⠿⠿⠄⣹⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⢈
⠀⠠⠄⠀⠀⠙⢠⡄⠀⢉⣿⣿⣿⡉⠀⠛⠛⣿⠛⠛⠂⣄⢠⠘⠀⠀⠰⣿⣿⣿⠧⠀⠉⣀⠀⠐⢻⣤⡞
⣤⣤⣤⠄⠀⠀⣾⣿⡀⠚⠛⣿⠛⠛⠂⠀⠀⠉⠀⣸⡄⣏⢹⠀⣼⡆⠼⠿⣿⠟⠷⢀⣾⣿⣷⡀⠿⠛⣿
⠿⢿⢿⠀⠀⣼⣿⣿⣧⠀⠀⠉⠀⢦⡼⣆⠀⠃⢠⣿⣧⠈⠉⣸⣿⣿⡀⠀⡉⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿
⠀⠸⢸⠀⣰⣿⣿⣿⣿⣇⠀⠋⠀⣾⡿⠉⣀⠀⠛⢿⠿⠓⢰⣿⣿⣿⣷⠀⠉⠀⠰⢿⡿⣷⣿⡿⠆⠀⠏
⠀⠀⣰⡀⠛⠛⢻⡿⠛⠋⠀⠀⠀⠙⠶⠿⢫⣾⣿⣄⠀⢀⣿⣿⣿⣿⣿⣇⢰⡆⢰⣿⣿⣿⣿⣿⡆
⠀⣰⣿⣷⡀⠀⡀⡀⠀⠀⣴⣶⣶⡀⠀⠀⢿⣿⣿⣿⠆⠀⠀⠉⠻⠉⠁⠀⣾⣷⡀⠀⠀⠟⠀⠀⠀⠀⠈
⠀⣹⣿⣿⡉⢰⡍⣵⠀⢈⣿⣿⣿⡅⠀⣰⣿⣿⣿⣿⢧⠠⢶⣀⡴⠃⠀⠾⢿⡿⠷⠀⢀⡄⠀⣼⣶⣶⡀
⠴⣿⣿⣿⡿⠆⠉⠁⠀⢾⣿⣿⣿⡿⠄⣹⣿⣽⣿⣽⣯⠀⣾⢿⣷⣀⣁⠀⢀⡀⠀⢠⠀⠀⢠⣿⣿⣿⡄
⣰⣿⣿⣿⣷⣄⠀⠂⢀⣾⣿⣿⣿⣷⡄⠛⠛⢻⡟⠛⠛⠃⡄⠀⣿⣿⣿⣿⣿⠀⢠⣿⣧⡀⢻⣿⣿⣿⡟
⠀⠀⢻⠁⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⣄⠀⢄⠈⠁⢀⠀⢀⣿⡀⡟⡇⠀⢸⢸⠀⣠⣿⣿⡀⠟⠛⣿⠛⠛
⠀⣧⠀⠀⠂⣰⣶⣶⡀⠀⠃⠀⠀⠀⣸⣶⣏⠁⠀⣀⢀⣾⣿⣧⠀⠁⠀⠈⠈⠘⣻⣿⣿⡛⠀⠀⣉⠀⠤
⣼⣿⣆⠀⠀⣽⣿⣿⡏⠀⣶⣶⣶⣶⣿⠋⠙⠁⢠⣿⣎⠙⠟⠉⠁⢠⣾⣦⡀⠘⠛⢻⠛⠛⠂⣴⣿⣷⡀
⠻⡿⠟⠂⢾⣿⣿⣿⡿⠄⡟⡟⠛⢻⣿⠀⠀⠐⢻⣿⣟⠂⠀⠀⢰⣯⣿⣿⣷⡀⠀⡈⠀⠀⢠⣿⣿⣿⡅
⢠⡄⣀⠀⣼⣿⣿⣿⣿⣄⠧⠃⣤⠘⠟⠀⢀⣠⣿⣿⣿⣦⡀⠀⣴⢿⣷⢿⣿⣄⢰⡟⣳⠀⣹⣿⣿⣿⣏
⢾⣽⡅⠈⠁⠀⢹⠁⠀⠀⠀⣸⣿⡆⠀⠤⠀⣰⣿⣿⣿⣆⠀⠸⣿⡿⣿⣿⢿⡿⠀⠓⠃⡄⠉⠉⢿⠉⠉
⣤⣙⡟⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠘⠋⠉⣿⠉⠉⠃⣰⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣤⣿⣷⣾⣷⣶⣶⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣼⣿⡏⠀⠀⠀⠉⠉⠈⠙⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⠟⠋⠛⠛⠿⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠏⠋⠉⠈⠁⠀⠈⢻⣿⣿
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠘⣿⡇
⠀⠀⠀⠈⣜⠀⠀⠀⠁⡀⢀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠗⠎⠀⠄⠀⠁⠄⠠⠀⠀⠀⠀⠐
⠀⠀⠀⢠⣿⣦⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠤⠠⡉⠈⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀
⠀⠀⠀⣸⣿⣿⣿⣿⠗⠐⢿⣿⣷⠀⠀⠀⠂⢀⠄⠹⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡛⠃⠓
⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠈⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎
⠀⠀⠀⠈⠉⠉⢱⠀⠀⠀⠀⢈⠁⠀⢠⠀⠀⠀⠀⠀⠈⠩⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠦⠲⠀⠀⠀⠀⠸⠲⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢒⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠥⡄⠈⢴⠀⠀⠉⡙⠀⠀⠀⠀⠈⡤⠏⠘⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠠⢺⠇⠀⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠛⠀⠀⠀⠀⠀⢀⢤⡶⠶⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⡀⠀⠀⠀⠀⠀⠡⠈⠠⠀⠛⢆⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣦⠀⠀⠠⢠⠀⢀⣳⡆⠴⠮⢷⣎⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣄⣀⣤⣽⣄⡈⢏⠁⠀⠀⠀⠛⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣇⠈⣉⣝⠚⠻⠓⠺⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣿⣧⡀⠙⠿⣷⣦⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⢿⣿⣿⣶⣾⣿⣟⠁⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢶⡀⠀⠀⠀⢸⣾⣿⣿⣿⣿⣿⣿⡄⠒⠂⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣀⣤⡶⠋⠀⠈⣀⡀⢈⡅⠂⠈⡁⠀⠀⠀⠠⣸⣿⣿⣿⣿⣿⣿⣷⡀⠘⠃⢀⣀⡀⠉⣿⠃
⠀⠀⠀⢰⣿⠟⠀⠀⠀⠀⠉⠁⠈⠱⣦⠀⣿⣿⣦⣴⣾⣿⡿⠟⠟⠛⠁⠙⠋⠀⠀⠀⠉⠉⠉⠙⣿⡀
⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠈⣀⠐⠀⡀⠀⣹⡏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠉⢷⡀
⠀⠀⠀⠀⠻⠯⠄⠀⠀⠀⠀⠀⠓⠂⢀⢶⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣟⠻⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣈⠙⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⣦⡘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣰⣦⡒⠶⠶⠤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣓⣤⣤⣀⡀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢭⣉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠒⠀⠈⢑⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡟⠋⠉⠙⢦⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡼⠀⠀⠀⠀⠈⣇⠀⢀⣈⣋⠉⠀⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⡇⠀⠀⠀⢀⠀⢸⠔⠉⠀⠈⡇⠀⠀⠉⠐⢲⠖⠀⠀⠀⠀⠀⠀⢠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠧⡀⢿⠀⠀⢰⣏⣴⠃⠀⠀⠀⢀⠃⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢦⣝⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⢈⠘⢧⡀⠈⢁⣧⣦⠀⢀⠀⠚⠀⠀⢠⠉⠁⠿⠤⠤⠤⢄⡀⣀⣀⣈⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣏⠀⠂⠍⠃⠉⢇⣠⠞⠁⠀⠀⠀⠀⠋⠀⠀⠒⢶⣓⣲⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⡠⠃⡆⠀⠀⠤⣀⡀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠄⠀⢸⠃⠀⠈⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠂⠁⡀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⠠⣄⡀⠈⠁⠀⠀⢰⡀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠸⠈⢀⣀⣹⠒⠤⣀⠀⠘⠏⠁⠀⠀⠰⣶⠀⠀⠀⢢⢰⡢⣵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⡡⠆⡐⠀⠀⠛⣄⡠⢤⠑⠊⠀⠀⠐⣖⠒⠛⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠒⠤⡤⣤⠤⠤⠤⠤⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢴⣤⣆⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣀⠀⡘⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣴⣉⣬⣟⣷⡖⠂⠤⠤⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠓⠓⠒⠒⠒⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡇⠀⠀⠀⠀⠀⡇⠰⠖⠘⡸⢙⡤⢤⢐
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠀⠀⠀⠀⠘⠐⠻⠈⠒⠀⠱⢎⣀⢒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⠀⠀⠀⠀⣰⣀⡀⢠⡴⡴⢤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⣽⠀⠀⠀⠀⠩⠸⠿⣿⡖⣲⡀⠿⡗⠳⠗
⠀⠀⠀⠀⠀⠀⣴⣯⣿⠿⣷⠆⠀⠀⠀⢴⣶⢶⡂⣟⣸⣶⣶⡶⠲⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠐⣿⣯⣿⢿⣿⣾⠄⠀⠀⢸⡿⣢⣴⣷⣶⣶⣶⡔⠤⠖⠀⠀⠀⠀⡼⠀⠀⠀⠔⠀⡰
⠀⠀⠀⠀⠀⣼⣹⣹⣿⣿⢻⣿⣦⠀⠀⠐⣪⣼⣿⣿⠟⢻⣿⡇⠄⡀⠀⠤⠀⠈⠑⠀⢀⣴⠃⢀
⠀⠀⠀⠀⢠⣟⣻⣿⣿⢿⣿⢿⡾⢀⠀⠈⠁⠐⣿⠁⢀⡈⢻⣇⠈⡇⢀⡰⢄⣠⣦⣾⢟⠋⢀⠄
⠀⠀⠀⠀⡷⣿⣿⣿⡏⠈⣹⣗⣿⠟⠢⣤⡀⠈⡿⡬⠿⠧⣿⡇⠀⠃⠀⠀⠈⠀⠁⢸⠁⣆⠊⣀⡀
⠀⠀⠀⠀⢛⠻⠁⢘⠟⢱⡏⠛⠀⡟⢰⡄⢀⠠⡴⣦⣾⡧⠾⠭⣦⠀⠀⠀⠀⠀⠀⠈⡇⡃⠀⠈⣤⠰⡃
⠀⠀⠀⠀⢷⡤⠀⣰⠀⠈⠀⢀⣿⡟⣾⠁⣸⡇⠠⠾⠿⠷⠶⠶⠶⠶⠺⠁⠀⠂⡤⠈⠁⠀⣿⠀⢻⠞⣧
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡀⠀⠘⠺⢝⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣛⣭⣽⣶⡶⢾⡥⠤⠤⠤⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡹⠊⠛⠉⠉⠀⠤⣄⠀⠀⠀⠀⠀⠈⣧⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠈⠀⠀⠀⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠀⠀⢠⢆⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠔⠂⢀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠞⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠘⠈⠀⢀⡠⠖⡆⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⠄⢀⣠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢔⠀⠀⠀⠀⠰⡒⠊⠉⣀⠄⡇⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠹⣌⡉⢅⡾⠀⠀⠀⠀⣠⠏⠀⠀⠠⢐⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠉⠉⠀⠀⠀⣠⠔⢁⣀⠴⢳⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⢴⠒⠋⡍⡉⠀⠤⠒⠊
⠀⠀⠀⠀⡀⠈⠤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠐⢊⣠⠀⠑⠚⠁⢡
⠀⠀⠀⠀⠠⠀⠀⢁⣀⡠⠖⠀⠤⠤⠔⠚⠤⠐⠒⠉⡜⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡴⠤⢍⡑⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠄⢄⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⣇⣀⠀⠈⢑⠖⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢂⡭⢷⡀⢀⣀⣀⣈⡑⠲⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⢸⢀⠇⠉⠉⠈⠁⠈
⠀⠈⠂⠀⡀⠀⠀⡴⢢⠜⠁⠀⠀⠉⠉⠀⠀⠀⠈⠐⠢⢤⣠⠀⠉⠉⠒⡖⠀⠀⡠⣇⠜
⠈⠢⠠⡄⠈⠑⠲⣄⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⣸⢅⠀⠥⠒⠉
⠀⠀⠀⠪⡲⢄⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⣤⣤⣤⣤⣬⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⠞⠀⠀⠀⠀⠀⠠⣤⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡿⠃⠐⣦⣤⣶⡿⠟⠛⠉⠛⠛⠻⢷⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠛⠋⠙⠻⢿⣿⣿⣿⠟⢿⡃⠂⠀⠀⠀⣀⠀⣀⢰⣱⣿⡃
⠀⠀⠀⠀⠀⣠⣤⣶⣶⣿⣷⣦⣀⠀⢀⣀⣀⣀⣀⣀⣰⣷⣶⣤⡄⠀⢙⠻⣿⣶⡾⠋⠢⢄⡀⠀⣠⢭⣵⠻⠡⠍⡋
⠀⠀⠀⢀⣴⣿⣿⢿⣿⣿⡿⢿⡿⠛⠉⠀⠀⠀⠀⡀⠙⠛⣻⣻⣿⣶⣿⡇⢉⠁⠀⠻⡂⢀⣼⡞⠥⡢⢇⢰⢂⢀⡇
⠀⠀⣠⣾⣿⣿⠇⠽⠛⠉⢠⠏⡄⠀⠀⠀⠀⠀⡰⠃⢠⣾⡟⡹⠿⣿⡟⠀⡘⠀⠀⠀⠀⠼⠟⠉⠈⣿⡿⠻⣿⠉⣁
⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⣠⠄⠀⠰⡈⠙⠿⠓⠀⠹⣷⡀⢹⣄⠀⠘⠈⠁⣢⣶⡖⢹⣇⠀⠛⠦⠋
⠀⠀⣨⡿⠟⠁⠀⣀⠤⠤⠋⠆⠃⠀⠠⢚⣥⢶⣶⣦⣍⣂⠀⠀⠀⢀⣘⣿⣾⡿⠶⢤⣠⣾⡏⠿⣲⣾⠋
⠀⠀⠀⠀⠀⠀⠀⡇⢀⣠⣤⣤⠤⠖⠒⠋⠀⠀⠀⠀⠉⠉⠉⠉⠋⠉⠉⠉⠀⠀⠀⠀⠘⠿⠂⠊⠉
⠀⠀⠀⠀⠀⠀⣰⣤⠋⠀⣿⡇
⠀⠀⠀⣠⡤⠴⢧⢇⣀⣰⣿⠇
⠀⠀⠰⠿⠇⠒⠁⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠿⠿⣏⣀⣠⠖⠛⠛⠛⢀⡠⣤⣶⣶⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠐⠘⠿⢿⣇⡠⣶⣶⡆⡀
⠀⠀⠀⠀⣠⣤⣶⠿⠛⠂⠀⠀⠀⢬⡷⣀⡜⠋⠁⠠⣶⣩⣾⣗⢿⠟⠉⠉⠙⠃⢤⣶⡿⢻⡇⠠⠶⣠⣶⣥⣄⠈⠉⢀⠟⠉⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢿⣻⣿⡿⠀⢠⣾⣿⣿⣟⡯⠄⠀⠀⠀⠀⠀⠀⢀⣄⢜⣡⠖⠋⠛⠻⠾⡟⣷⢤⠟
⠀⠀⠀⠀⠀⠀⡠⠖⡀⠀⢀⣠⣾⣟⠟⠓⠒⢦⡄⠈⠉⣉⠙⣏⠒⠦⡀⠀⠀⠀⠀⠌⠉⠀⠀⠀⠀⠀⠀⢠⠋⠀⠂
⠀⠀⠀⢀⡠⢊⠐⠉⠀⠘⢉⠈⠁⠀⠀⠀⠀⠀⠉⠄⠀⠀⠁⠜⠄⠀⠀⠉⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠅⠀⠀⠤⡮⠅⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣆⡀⢀⢈⠂⠀⠄⠠⠤⠀⠀⣤⣤⢀⣤⣤⣤⠬⣄⣤⡤⠤⠤⠤⠤⠀⠄
⠀⠀⠈⠉⠛⠛⠛⠛⠉⠉⠉⠉⠉⠙⠉⠋⠉⠙⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣦⣤⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠹⣯⣩⠟⡀⠀⠀⠀⠀⡀⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠰⡀⠀⣅⡰⢀⡠⡷⠐⠚⠛⠋⢉⡀⠙⠟⣦⣄
⠀⠀⠀⢀⢀⣀⣀⣀⠀⠀⠀⣀⠄⠀⢀⡴⠤⠀⠠⣠⠊⠀⢀⣷⡄⠘⢾⣌⢙⣒⡬⠟⠛⠿⠾⠷⠔⠴⠾⣟
⠀⠀⠀⠀⢿⡉⠻⠏⢛⡻⠟⣵⡿⠟⠁⠀⠀⠀⠀⣸⣦⠀⠈⢳⠛⠁⡤⣽⡯⠁⣄⡀⠀⠀⠀⣴⠀⣠⡄⠈⡧
⠀⠀⠀⠀⠻⣭⡛⠛⢉⣤⠞⡩⡜⠀⠀⠀⠀⠀⠐⠃⠉⢣⡀⠀⠂⠀⠙⠠⠝⠋⠀⠀⠐⠤⠘⣏⢣⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⣿⠁⡇⡇⠀⠀⠀⠀⠀⢁⡶⠖⠉⢁⡤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠉⠢⠰⣶⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠘⠃⠀⠀⠀⠀⢠⣙⣰⠆⠸⡙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢙⠀⠀⠀⠈⢂⣉⡠
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡆⢸⣦⡀⠀⠀⡠⠊⠑⠷⣌⡄⠀⢣⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠡⠼⠃⠉⠂⠈⠀⠀⠀⠀⠀⠁⠐⠒⠒⠷⢄⣀⢢⡀⠀⠀⠀⠲⣯⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠢⣀⠀⠀⠑⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠐⡧⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⠀⠈⠀⡑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢷⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⡄⢉⣢⣈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠈


⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣼⣷⣬⣿⣶⣥⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡄⠀⠀⣀⣤⠿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣿⣿⡿⠃⢠⣾⣿⣿⣾⣿⣿⣏⡀⠀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢀⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢈⢻⣿
⠀⠀⠀⢠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠛
⠀⠀⠀⠀⠘⠛⠉⠁⢸⣿⡿⢿⣿⣿⠻⠿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡛⠛⠻⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⣹⡟⢰⣿⣿⡟⠁⠈⢿⡇⠀⠀⢈⡟⠀⠀⠀⠈⣻⠿⣦
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢿⣇⢸⡇⢻⡄⠀⠀⠈⣷⠀⠰⠞⠁⠀⠀⣤⠼⠋⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠙⢾⣧⡄⠻⣄⡀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠁⠀⠀⠈⠁

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

⠀⠀⠀⡀
⠀⠀⠀⠋⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠐⠄⠃⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠭⢹⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⢰⠄⠀⢀⠀⠀⠠⢠⠀⠀⠀⢀⠀⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠐⠀⠀⠀⠀⠠⠉⠀⡀⠀⡄⠀⡀⠘⠀⠀⠘⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡈⡴⢶⡎⣀⠀⠠⠄⠀⠔⠁⠑⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡙⡚⠣⡀⠀⠀⠀⠀⠢⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠲⠀⣨⠡⡠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢴⣟⣳⠡⠠⠤⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⢀⠀⠀⣠⠐⠫⠁⠃⡄⠀⢠⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠠⢔⡀⠄⠀⠀⠤⠀⠌⢢⣀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠐⢨⢇⠜⡕⠂⠀⠀⣀⠀⠀⠈⠑⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⡱⠒⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢿⣿⣿⡲⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣷⠿⣿⣿⣿⣏⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠘⣿⣿⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣤⣤⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠟⠛⡛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣹⡟⣛⠻⣿⣷⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⣿⣶⣶⣤⣠⣠⣴⣼⣿⣿⣿⣿⡏⠉⠙⠛⣛⠛⠉⢉⠈⠉⢿⣿⣿⣿⣿⡿⢹⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣁⡈⢙⣿⣿⣿⣿⣿⣿⣇⣀⣙⠉⣭⣭⡍⣛⠁⠀⠀⢿⣿⡏⢿⣿⡘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣷⣦⣀⠈⣿⣿⡈⣿⣷⠹⡿⠁
⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠛⢻⡟⠻⡿⠿⠿⠿⠿⠿⠛⠿⠿⠿⠿⠿⣿⣿⣓⢺⡧
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⠤⣿⠁⠀⢈⣏⠉
⠀⠀⠀⠀⠀⠀⠀⢶⡶⠿⠿⢭⣿⢿⣟⠛⠻⠿⣯⡿⠿⠿⠟⠉⠉⠙⠛⠻⠿⣿⣿⣟⡻⡟⢿⣤⣄⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠶⠋⠋⠀⠈⠁⠀⠀⠀⠀⠠⠿⠤⠒⠂⠐⠀⠀⠀⠀⠠⠍⠛⠫⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⣀⡀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢸⣿⠀⠉⢁⠀⠛⡹⢾⠍⠀⠀⠀⢀⣀⣀⡠⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⢶⣿⡆⠀⢹⣤⠈⠁⠀⢀⣀⣴⣾⡿⠟⠋⠻⡌⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠙⠂⢀⣿⠁⠀⣠⣴⣿⡿⠟⠁⠀⠀⠀⠀⠘⣿⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⠤⠤⡤⠀⠠⣄⣀⡀⠀⠐⠉⣠⡖⢀⣀⡉⠛⠻⠟⠉⠀⠀⣠⠆⢀⣤⣀⡀⠹⣿⣄⣠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠶⠀⠀⠀⡏⠀⠀⠀⠈⣿⡟⣀⣾⣿⠇⣠⣬⣭⣷⡶⠋⠑⣰⠞⠓⢒⣾⠋⠙⠶⠞⣷⡟⣿
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣴⣶⣿⡿⠿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢲⠧⠄⠀⠀⠀⢈⠘⠻⣿⠋⢠⠀⠀⠀⠀⠀⠠⡾⠃⠀⢠⣾⠃⠀⠀⠀⠀⠈⠻⣿⠄
⠀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠠⣿⣃⠀⠀⠀⠀⠀⠀⠀⢮⠳⠒⠲⠤⣄⡀⠉⡖⠼⠂⠀⠀⠀⠀⠀⣤⡀⠁⢀⣴⠟⠁
⠀⢘⣿⣿⣿⣿⣿⣿⣿⡟⣽⠏⠀⠀⠀⠀⢿⣿⣷⣤⣤⣤⣤⠄⠀⢸⢀⡀⠀⠀⠀⠀⠈⢁⠘⠂⣀⣀⠀⠀⠀⠸⣿⣶⣿⡇
⠀⠀⠈⠒⠛⠋⠁⠐⠋⠉⠀⠀⠀⠀⠀⠀⠈⠻⢿⣟⠻⠿⠅⡀⠀⢸⠘⢿⣶⣄⡀⠀⠀⣿⠀⠸⡇⠀⠀⠀⣀⠔⢹⡟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢀⠀⠀⠈⢸⣷⠚⠽⣿⣿⣶⣿⣀⠀⡧⣠⠏⢀⣠⣀⣼⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣿⣿⡄⠀⠀⠉⠙⠿⠿⠿⣇⣿⠂⠀⠙⣿⣏⣀⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡀⠈⠋⠉⠉⠉⠉⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢀⣀⣉⢀⡀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠈⢀⡀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣧⣸⠆⠀⠈⠻⢿⣤⣤⣤⣤⡘⠷⠀⠀⠀⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⣀⣤⠶⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠤⠤⢀⢀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡔
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠄⡀⠂⠀⠄⠀⢀⠀⢘⡀⠀⠀⠀⡠⠀⠀⠨⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡀⡋⠁⠀⠈⠂⠀⠀⢠⠃⠀⠀⠉⠀⢤⠈⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢄⠀⠀⠀⢐⢯⠀⠀⠀⠀⠖⠀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣾⣧⣜⢦⠀⠀⣞⠀⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠈⠙⢦⣠⠻⣧⣾⠇⠀⠀⠀⠀⡀⠀⠠⠂⢀⠀⠀⠀⠺
⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⠀⢰⣧⣼⡦⢿⣦⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⠁
⠀⠹⢄⠀⠀⠀⡄⠀⢳⠀⠀⠀⠀⣠⡆⠀⣨⣍⣭⠖⣩⡿⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠶⠁⠀⠀⠐⠁
⠀⠀⠀⠛⠢⡈⢧⠀⠘⠀⠀⠀⠀⠉⢃⠀⠙⠛⢿⣶⣿⡇⠀⠀⠀⢀⠔⣙⣀⡚⣀⢠⠀⠀⢀⣠⡦⣉⣠⠊
⠠⡄⠀⢀⠻⣤⠿⠗⠀⠘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⡷⠀⣠⡴⠟⠛⠛⠙⡁⠉⣠⣦⣴⣿⢿⠔⣠⠃
⠀⠀⠀⠀⠄⠻⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄⣦⡀⠸⢿⠟⠢⢀⠀⠐⠗⠀⠻⠛⠋⣹⣿⡿⠀⢀⡿⠤⠄
⠀⠀⠀⠄⠈⠂⢒⣇⠀⠀⠀⠠⠀⣼⣷⠄⠘⠿⠟⢹⣶⣿⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡟⢐⠀⠈⠁
⠀⠀⠀⠀⠐⠌⣻⣿⣦⡀⣠⢤⡾⠋⠀⠀⠀⣶⣤⣷⡟⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠹⣍⡻⠦⠈⠁⠐⠒⠀⠀⠈⠉⠉⣶⣤⣿⠀⠀⠀⣀⠀⠀⠠⠐⢀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⣶⡿⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠛⢠⣤⣀⠀⠀⠙⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠁⠀⢸⡟⡤⠄⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠣⣀⣀⡈⢒⠀⠐⠀⠀⠀⡀⠀⠊
⠠⠒⠂⠀⣰⠏⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⡾⠀⢀⣄⣶⣤⣀⣦⢀⡾⠁⡌⠸⠛⠉⠠⠘⠋⠀⠀⠄⠘
⠀⠀⠀⠐⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢈⣻⡿⠍⢻⣿⡊⠀⠀⠀⠊⠁⠠⢄⠀⡠⠒⠀⠉⠛
⠀⠀⠀⢇⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡁⣨⡴⠚⠍⠉⢐⠦⣄⠈⠑⢦⡀⠀⠤⠒⠉⠉⠑⠒
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠍⠀⢾⣦⡷⠻⣾⡿⠀⠀⠀⠁⠀⠢⣤⣞⡁
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠷⠲⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢬⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣼⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡉⠆⠀⠀⠀⠀⠀⣤⢾⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡏⣻⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⠻⠿⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣀⣀⣸⣿⣿⣿⣿⣿⣿⡀⠀⠀⡀⠼⠻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⢴⣾⣿⣿⣧⣽⣾⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣶⡎⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⣀⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠾⠟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡀⣀⣰⣿⣿⣿⣾⣏⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⠄⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⢿⣿⣿⠟⣛⣿⣏⣿⣿⡿⢿⣶⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣹⣿⠿⠟⣿⡇
⠀⠀⠀⢀⣴⣿⣿⡟⠁⢹⡟⢿⣿⡿⣸⣿⣿⣽⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠻⠤⠿⠿⣿⣶⣶⣿⡇
⠀⠀⠀⣸⣿⣿⡟⠀⠀⠘⣿⣶⣶⣾⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣧
⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠀⠹⣧⡀
⠀⠀⠀⠸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣯⣵⡄⠀⠀⠀⠀⠙⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠙⠛⠹⣿⣿⣇⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢠⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣿⣍⡛⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄⠙⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣦⡀⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂

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

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠿⠯⢠⣾⣧⡀⠐⠀⠀⣾⠁⢨⡷⠀⣴⢒⣦⠀⢉⡀⢰⢿⠂⣰⡿⣛⠻⣇⠀⡞⠀⠈⢳⠋⠀⠙⡆
⠀⠀⠀⣰⣿⡟⣿⣷⡀⠀⠀⠹⢤⣀⡀⠀⢻⣌⣉⢰⠫⣽⠂⠀⠀⠹⣷⣭⣾⡏⠀⠳⡀⠀⢺⠀⠀⡸⠃
⠀⠀⣴⣿⣷⡅⣥⣿⣿⣄⠀⣶⠆⢨⠟⣉⠉⠉⡉⠙⢳⡀⠀⠃⠀⠐⢶⠭⠶⠆⠀⠀⠙⢦⣸⣠⠞⠁
⠀⠈⢉⣽⡿⠇⡿⣿⣏⠉⠀⡀⠀⠘⠦⠿⠀⠀⣉⣀⣰⣧⣄⣀⣀⡘⠻⠖⠿⠃⢀⣤⡀⠀⠙⣡⡶⣤⡀
⠀⠠⠾⢿⣿⡆⣾⣿⡿⠧⠀⡁⢀⠀⠀⣀⣶⣀⠙⣿⣿⣿⣿⣿⡿⠟⠀⠄⠀⠀⠈⠛⠃⠀⠸⣕⡿⣫⡇
⠀⠀⣠⣿⣯⡃⢋⣿⣷⡀⢸⣎⣹⡁⠐⣺⣿⣗⠂⢸⣿⣿⡟⠁⠀⢠⡄⢶⢦⣤⠀⠸⣿⠟⠀⢉⣉⠉
⠀⠐⢛⠛⠛⣷⡟⠛⠛⠛⠈⠑⣋⡁⠀⠀⠛⠀⠀⢸⣿⣿⣧⠀⢸⡗⢠⣶⣦⠐⣷⠀⠀⠀⣰⢿⣿⡷
⠀⠀⣉⠠⡄⢉⠁⠀⣠⡀⠠⣾⣿⣿⣆⠀⠀⠙⠀⢸⡿⡟⡿⣧⠘⢫⡄⣩⣡⣬⠋⠀⢼⣿⣇⣠⡿⠃
⠀⢠⣭⡷⢳⣯⣄⠘⠿⠏⠀⠘⢿⣄⣸⣿⣯⠄⢀⣿⡁⠀⠀⣽⣧⠀⠀⠉⠁⠀⠀⠄⠀⠀⠉⣉⠀⢠
⠀⠀⠴⢋⡏⠳⠀⠀⢀⠀⠀⢀⣀⣉⣉⣉⣤⣶⣿⡿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣦⠀⠀⢀⣼⣿⣦
⠀⠀⠀⢠⣤⡄⠘⠁⠘⢷⣾⣿⣿⣿⠿⠻⣻⣿⡃⠀⢐⣻⣿⣿⡿⠛⠉⢀⣿⠏⣿⡆⢠⣾⣯⣩⣿⣷⡀
⠀⢰⣿⡟⣙⢻⣷⠄⠀⢸⣿⣿⣿⣧⣀⢀⢸⣿⣷⣦⣷⣿⡿⠋⠀⠘⠃⢸⡿⠀⣿⠇⢀⣼⣿⢙⣿⣧⡀
⠀⠀⢽⣧⣭⣾⡏⠀⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣁⡤⢤⣀⠀⣰⡞⠁⣴⠟⠀⠀⣼⣟⠛⣿⣇
⠀⢰⣶⣭⢀⣭⣶⡆⠀⢰⣿⣿⣿⣿⠋⠉⠀⣠⣀⠀⡼⢃⢮⣷⡂⢧⠈⡀⠀⢡⡀⠀⠈⠉⠉⡿⠉⠉⠁
⠀⠈⠻⠿⠿⠿⠛⠀⢠⣿⢣⣿⡟⠁⠖⠀⠀⠛⠋⠀⠸⣆⡛⢟⣂⠏⠈⠁⣰⣿⣿⡄⢀⡄⠀⠠⡞⢳⠄
⠀⠀⠠⠀⠀⠒⣠⣴⡿⠃⣼⡟⠀⣰⠚⣶⣴⣾⣶⡀⠀⠀⣍⢉⠀⠀⠀⠐⣿⣧⣾⡟⠂⠀⢀⡀⢉⠉
⠀⢠⡶⣲⢶⣰⡿⠋⣠⣾⠟⢁⠀⠹⣾⣿⣿⣿⡿⠀⠐⢶⡿⢻⡷⠂⠀⠰⣿⣧⣾⡿⠂⠐⠶⢟⡻⠶⠂
⠀⠸⣞⣿⣳⠇⠀⠀⡿⠋⠐⣏⣱⠀⠘⢿⡿⠋⠀⠆⠐⠛⡿⢿⠛⠃⠀⠰⠛⢧⡾⠿⠂⠐⢋⡶⢾⠙
⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⢘⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⢀⠀⠀⠀⠀⢀⣤⠀⢀⣀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⢿⣿⠿⠇⡟⣦⣀
⠀⠀⠀⠀⠀⠀⢘⠛⠛⢰⣿⣿⣷⣤⠀⠉⠉⢠⣼⡛⠎⡀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠟⠋⠀⠀⢀⣹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠉⣀⢼⣿⠟⠀⠙⠻⢿⣦⡤⢸⣿⣧⣾⠏⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⣹⠶⡇⢫⠉⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⣡⣾⣿⣦⣀⠠⠔⠒⠉⠁⠠⢨⣭⣤⣤⢴⡾⡆⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠰⠂⠿⠟⢠⣾⠃⠈⣇⠈⠋⢻⣿⣧
⠀⠀⠀⠀⠀⠀⣨⣻⠟⠏⠺⣷⣌⢲⡀⠀⠀⠀⠌⠀⠈⠛⣦⡘⢿⣶⡀⠉⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠸⡀⠀⠈⠻⢿
⠀⠀⠀⢄⢒⣴⠞⠁⠀⠀⠀⠙⣿⣆⠹⣆⠀⡸⡤⠔⠚⠉⠀⠀⢀⣿⣧⡄⠈⣭⣙⡛⠶⣄⠀⠀⠀⠀⠂⠀⠀⠀⣠⣤⣴⡿⡁⠀⠀⠀⠀⣧
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⣿⣤⣷⡆⠀⢀⣶⡀⠀⠀⣀⣘⣋⣩⣥⣴⣿⣿⣿⣦⡈⠆⠀⠀⠂⠀⠀⠔⢋⣡⡾⠛⣾⣀⣠⡴⢂⣴⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣠⡀⠀⠐⢫⣭⠱⣿⠿⠧⠀⢰⣿⣿⡧⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠘⠟⠁⠀⠒⢦⠨⣇⠀⠀⢀⡴⠛⣿⢟⣵⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣤⣶⡇⠀⠀⠈⢧⣀⠜⠀⠀⢥⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠙⠃⣤⣾⣿⣿⣿⡎⠻⠀⠀⠀⣤⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⡀⠀⠀⠠⣿⠍⠀⠀⣄⡁⠤⡴⡿⠟⠉⣀⢴⡆
⠀⠀⠀⠀⠀⠀⣀⣀⠀⠠⠤⠤⠀⢀⡤⢀⣶⠘⠋⠉⠉⠀⢀⠀⢠⢦⡉⢉⠀⠲⣶⣶⡾⠻⢿⣄⡀⢀
⢀⣀⣠⣴⣾⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠘⢁⣜⣷⣿⣿⣾⣷⣐⣼⠛⠀⠀⠀⠀⠈⠃
⠘⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢦⠟⣻⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣤⠀⢀⣀⣠⣴⠞⠛⠳⠶⢤⣤⣆⣧⠀⢠⡶⢯⡀⠹⢿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⢠⣤⣀⡄⠀⢀⡰⡷
⠀⢀⣠⠞⢁⡴⠛⠛⠉⠀⠀⠀⠀⠀⠀⢻⡈⡟⠠⠀⠀⠀⠀⣠⡉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠹⣀⣠⣉⣉⡀⠀⢦⡄⠀⡀
⠘⣻⣥⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⡆⠀⠀⣰⡶⠿⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠉⠠⠰⢿⣟⠋⣨⣷⠿⡇⠀⠘⢃⣴⡄⠲⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⢁⣠⣼⣏⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⣠⣤⡤⠀⠀⠉⠀⠀⠴⠄⠀⠈⣿⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠿⠦⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠄⢠⣴⡶⠠⣴⣻⠟⠀⠀⠀⠀⠠⡀⠀⠠⡆⠀⢿⣷⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣴⣶⣿⣿⠿⠛⠁⠀⠚⠁⢰⣿⠋⠁⠀⠀⠀⠀⢠⢠⣿⣶⣦⡀⢶⣬⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣤⣄⣜⠁⢀⡞⠀⠀⣢⡀⡾⠾⣿⢹⣿⣷⣼⡟⠁⠀⠈⢦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⢀⠀⡀⡀⡠⢸⠠⡂⢠⠠⡀⡄⡂⠄⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢖⠂⣆⠗⠁⠇⠀⠣⠸⠀⠚⠘⠀⠃⠃⠁⠀⠀⡏⢀⠇⠄⠐⢰⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠤⠄⠤⢄⠈⠁⠀⠈⢁⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠎⠉⠀⠁⠈⠀⢀⣀⠀⠈⠑⠒⠖⠢⠤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠔⠉⠁⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠠⡉⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠐⠢⠭⢥⡝⠀⠀⠁⠀⣀⠖⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠄
⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠷⣮⡍⠁⠈⢣⠀⢀⣀⠆⠁⠀⠈⠙⠔⠆⠤⠤⠦⠀⠀⢀⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⢸⢸⢀⠆⠋⠁⠀⠀⠀⢹⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢇⠀⢁⠏⡼⠀⠀⠀⠀⠀⠠⡘⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃⢣⠀⠀⠀⠀⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠈⢢⣀⠜⠀⠀⠶⠀⠀⠀⡀⡄⢂⠀⠀⠀⠰⡇⠀⠀⠀⠠⠦⠀⠀⠣⣤⢠⡀⡰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⣦⣀⠀⠀⠀⠀⠀⢀⣠⠞⠓⢤⣀⠀⠐⠐⠀⠀⠀⠀⢀⡠⠔⠁⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⣿⠐⠒⢲⣙⠷⡀⠈⠑⠀⠓⠋⠁⠀⠒⠒⠂⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⡿⠃⠀⠸⠄⢠⣆⠀⠀⠀⣀⠀⣀⠐⠤⣦⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⠀⣈⠀⠛⠂⢸⠀⠉⠉⠂⠴⠗⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⢀⠄⠉⠁⠰⠦⢼⡀⢀⢀⠀⡀⡀⢄⡒⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠃⠴⠻⠦⠐⠛⠒⠀⠀⢘⠁⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡃⠀⠐⠀⠀⢀⠀⡀⠐⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡁⠀⢈⢩⡀⠀⠀⡀⣄⡄⠀⢈⡁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡢⡄⢀⣴⡆⣶⣄⠀⡤⣲⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣌⢳⡡⢼⣟⡓⢻⣿⠤⣱⢋⡜⠀⢀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣀⡀⠹⣷⣦⡀⠈⢿⢿⡛⡧⠹⣼⢦⣙⡮⢨⡾⠡⡟⣻⢿⠏⢀⣴⠿⠋⣀⣤⡤
⠀⠀⠀⠀⠀⠀⣤⣙⣿⣷⡤⠜⠛⣲⢼⣶⣽⢰⢂⣶⡝⢨⠐⣴⣆⠲⢸⣵⣾⠻⣍⣁⠒⢾⠿⣷⣶⠆
⠀⠀⠀⠀⠀⣴⣾⠋⣱⣞⣀⣼⣻⣷⣬⣿⡮⢨⣟⢞⡓⠄⠔⣻⢞⣯⠨⣼⣯⣵⣌⡱⠦⣌⢿⣆⠙⢿⣆
⠀⢀⡄⢲⣜⠋⢀⣼⠋⠉⠅⠀⠉⠉⠉⠁⠈⠁⠁⠉⠁⠀⠀⠉⠁⠁⠉⠈⠉⠙⠋⠡⠠⢀⠈⠻⢄⠀⢉⣶⠂⣄
⣠⣬⣌⣿⣿⠊⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠈⢺⣿⣏⣬⣤
⠈⠙⢿⠿⣻⠠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡦⢸⡿⢾⠟⠉
⠀⠐⠈⢶⢟⣀⡌⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠨⣄⡘⢷⠎
⠀⠀⠀⠀⣉⣿⡁⠀⠀⠀⠀⠃⠈⠀⢣⠢⢂⠔⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠈⠐⠂⢀⠀⠀⠀⠀⠀⣹⣏⡁
⠀⠀⢐⠾⣦⠉⢓⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢐⡋⢥⡾⢖⠂
⢀⣤⣾⢖⣿⠒⠟⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⠗⢺⣗⢾⣢⣄
⠉⢛⢉⢿⣿⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⢼⣿⢏⢙⠋
⠈⠈⠃⠘⢏⡀⠈⢲⣄⠈⠀⠐⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠠⠀⠂⡀⢰⠈⠀⠀⡙⠄⠋
⠀⠀⠀⠀⠈⢿⣶⡈⢻⣎⠛⢶⣻⣿⠾⣷⠮⠱⣗⣽⣃⠤⢄⣻⣕⡷⠩⢿⣿⣾⣗⣢⡴⠎⣴⡟⢀⣴⡟
⠀⠀⠀⠀⠀⠀⠾⢛⣷⣾⠦⠌⢉⡳⢿⣿⢾⢨⡙⣾⢇⠰⢀⢿⡞⣩⢸⢾⣿⣋⠟⠛⣀⣼⣭⡶⢛⡍
⠀⠀⠀⠀⠀⠀⠐⠛⠉⢡⣶⡾⠋⠀⣽⣴⣋⣦⢠⢶⡽⢞⡦⠴⢦⠠⣎⣳⣼⡅⠙⢶⣤⡌⠛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉⢉⠟⣱⠫⢼⣗⣋⣸⣾⠬⢳⡙⢏⠉⠙⠀⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⡣⠇⠘⢿⡇⣿⠟⠀⠧⣻⣼
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣇⣴⣾⣿⣿⣿⣤⣀⣀⠀⠀⢀⣶⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⡁⠀⠀⠉⠻⣿⣿⣇⢰⡇⣿⣿⣿⣿⣿⢿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠉⣉⢻⣦⠀⠀⠀⠈⣿⣷⡿⢳⢿⣿⣿⣿⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡝⠃⠀⠀⠀⠸⣿⣵⣿⣻⡻⣿⡿⠃⠀⠀⣀⣀⣤⣤⣾⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠗⢾⣇⠀⠤⣀⠀⣈⣿⣶⣭⡛⢿⠃⠒⠅⠀⠛⠿⢿⣿⣿⣿⣿⣯⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠼⠪⠯⣄⠀⢀⣼⣿⣿⡿⣩⣿⣶⠅⢀⡠⠀⠘⠂⠐⢂⠈⠙⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⡀⢀⣼⣿⣿⣿⡿⠁⠘⣿⣿⠶⣭⣤⣶⣶⣶⣿⣦⠓⢆⡀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⣀⣼⣿⡷⠄⠀⠀⠀⢸⣿⣟⣛⡻⠿⠱⣖⣦⣾⣿⣷⣹⣿⣿⣿⣿⣿⣿⠶⠚⠛⠁⠀⠙⢻⡀
⠀⠀⠀⠀⣾⡟⠁⠀⠃⠀⠀⠀⠀⠀⢹⣿⣿⣷⡀⠀⢀⣴⡻⠿⣿⡿⠽⠏⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢹⡷⠄⠀⠀⠄⢀⣀⠀⢺⣸⡈⠙⡟⣼⣺⡅⠻⣿⣿⣿⠂⠈⠻⠆⠐⣠⣶⣦⣦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠓⠻⠇⢇⠀⢑⣾⣭⣴⣾⡿⠛⠁⠀⠀⠀⠄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⣾⣱⢿⣿⣿⣿⣻⠎⠀⠈⠁⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣴⣄⠀⠈⠙⠻⠿⣿⡇⠀⠀⠁⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠐⣄⠀⢀⢐⣈⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣔⠨⣝⣻⣿⣿⣿⣿⣿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡖⠙⢀⠦⠙⣿⣿⣿⣿⣿⣟⠂⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠁⠀⠀⠀⠈⠈⠁⢼⡟⢹⣿⣿⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠘⠙⠀⠀⢹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣏
⠀⣀⡀⢀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⣀⣉⡍⠀⠀⠀⠀⡀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⢀⣼⡆⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⢡⡟⠻⣿⣿⠦⠏⣐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠛⢸⠧⠀⠋⠈⠁⢠⠀⠀⠈⠚⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡷⢀⣠⣴⣿⣠⡆⠈⠁⠀⠀⠀⠀⠀⠱⣶⡗⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣿⣿⣿⠿⠿⣿⣿⡿⣿⣿⣶⣄⡀⠀⠀⠘⢲⣿⠻⣽⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣆⠙⣯⠉⠀⠀⠀⢨⠏⠉⠀⠉⠉⠙⠲⣤⣀⡟⠂⣙⣦⡜⢿⡷⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⣠⣾⡟⡙⡷⣶⠃⠀⠀⠀⣠⠃⠀⠀⠀⠀⡀⠀⠀⢸⣿⣷⣾⣥⣀⣠⣄⢀⣀⠀⢀⣼⣿⡿⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠷⠊
⠀⠀⠀⢀⠀⠈⠩⡠⢵⣿⡅⣧⢤⢤⡴⢁⡀⠀⠀⠀⠐⠀⠀⠀⠀⢻⣿⣿⣿⡍⣽⠯⢢⡀⠀⠉⠙⢇⠀⠀⠀⠁⠉⠀⠈⠉⠀⠈⠉⠁
⠀⠠⣥⣉⣀⠄⡤⣿⠋⣻⡄⣞⠵⠋⢰⣿⡟⠃⣴⣾⡖⠠⠤⠄⠀⢨⣇⣿⣛⣃⣈⡼⠋⠑⡀⠀⠀⠀⠑⡤⠀⠀⢀
⠀⠀⠀⣀⠁⠈⠉⠛⠒⠛⠉⠁⠀⠀⠈⠛⣏⡀⠙⣿⡧⠀⠀⠀⠀⠆⣸⠏⠁⠉⠉⠀⠀⢀⠣⡀⠀⠀⠀⢃⡨⠖⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠙⠎⢢⣄⠐⠑⠤⣤⣾⡾⠃⠀⠀⠀⠀⠀⠠⣂⠀⠡⠀⠀⠀⠀⠀⢀⠠⡀⢠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⠄⠀⠀⣻⣿⠤⠤⠤⠤⠤⡖⠋⠉⣉⠦⡀⠀⠀⠀⢀⢎⠁⠘⣂⠉⠿⠉⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠈⠑⠂⠔⠁⠀⠁⠀⠈⠂⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠻⢀⠘⡀⠃⠀⠶⠧⠊⠃⢡⡴⠀⠶
⠀⠡⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⠁⠀⠠⣤⢠⣆⣃⠦⢯⢡⠑⣄⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠉⠓⠁⠈⠀⠀⠘⠑⠀⡄⠉⠉⢠⠀⠊⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠩⠁⠀⠒⠐⠊
⠀⠀⢠⡀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡀⠡⠀⠀⠀⠂⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⢤⡶⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠤⡨⠂⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡤⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠉⠉⠉⠉⠉⠰⠆⣀⠀⢀⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⠰⠶⠀⠀⠀⠐⠶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠁⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠈⠃⠀⠀⠀⠀⠈⠁⠐⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠉⠁⠒⢀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠒⠛⡀⠀⠐⠤⡀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠸⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠘⢦⣀⠀⠈⠳⢄⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠤⠤⢱⣄⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⢡⡌⡄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠰⠀⠀⠀⠀⢰⡆⠀⢰⠀⡄
⠀⠈⠀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢰⣆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣠⠸⣏⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢠⡷
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⢙⠀⠏⠢⡈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⣘⠃⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⢀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⠀⠐⠛⠋⠀⠀⠀⢠⡀⠀⠀⢀⡴⠋⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠘⠠⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⠑⠂⠉⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⢀⠾⣠
⠀⠀⢃⠔⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠈⠑⠒⣤⠄⠤⠐⠒⠲⣋⢣⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⠎⢠⡉
⠀⠀⠀⠠⢃⠀⠀⠀⠀⠀⠈⠓⠤⢀⡀⢠⠞⠁⠀⠀⠀⠀⠀⠐⠦⢄⡀⠀⡄⠀⠀⣩⡌⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢠⡄
⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠄⠀⢠⣤⠀⠀⠀⡀⠀⠀⠉⠚⠛⣤⠊⣡⢂⠈⢣⡀⠀⠀⠀⠀⠀⣀⡤⠒⠁⠀⢰⠇⠁
⠀⠀⠀⠀⢀⡘⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠄⣀⡴⠁⠀⠀⠀⠀⠀⠀⠀⣹⠀⠠⡀⠉⠒⠒⠒⠒⠺⡉⢀⣔⣀⠔⠀⠆⠙⢤⠆
⠀⠀⠀⠐⠈⠀⠀⠙⠦⣀⠀⠀⠀⢀⡠⠜⠃⡠⠀⡠⠒⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠱⡀⠀⠀⠀⠀⠀⠉⠈⠀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠈⡀⠔⢁⠄⠊⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣄
⢀⡄⠀⠀⠀⠀⠄⠀⢀⡔⢳⠤⠘⢋⡠⠚⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠱⡄⠙⠦⡀⠀⠀⠀⢀⡀⠀⠀⠠⠊⢀⠙⢄⡀
⠶⠀⠀⠠⡀⠀⠀⠘⠇⠀⠀⡶⠜⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠈⠦⡀⠈⠃⠤⠀⡜⠳⡀⠠⠃⠀⠋⠁⠤⠃
⠀⠀⠀⠀⠈⠀⠀⠀⠛⠓⢤⡅⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⠳⢄⡀⠀⡇⠀⡜⠁⠠⠂⠈⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠙⠃⡀⠶⠄⠔⠈⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⢬⡥⠤⠒⠂⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣿⠉⠀⠒⠒⠨⠤⠤⠄⢀⡀⠀⠀⣤⠤⠤⠤⠀⢀⣀⣀⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣷⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠺⠆⠀⠀⠀⠀⠀⢰⣄⠤⣤⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⠰⠖⠶⠊⣄⠀⠀⠀⡌⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠈⢣⡄⠀⠁⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⣦⠀⠀⠀⠀⠈⡄⢿⣄⠀⠀⢀⡜⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠳⣤⣤⣤⠞⠀⢸⠈⠑⠒⠉⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⣀⡿⣦⣤⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠀⠀⠈⠂
⠀⠀⠀⠀⢀⠀⢀⢄⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠓⠀⡄
⠀⠀⠀⠀⠘⡵⠨⠌⡶⠐⠒⠤⢄⡀⠐⠈⣁⡀⣂⠄⠀⠐⢆
⠀⠀⠀⠀⠸⢀⡔⠊⡇⠀⠀⠀⠀⠈⠀⠀⢒⠂⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠎⠈⠀⠇⢀⠀⣀⠤⠂⠁⠀⠀⢠⠙⡄⠈⠆⢠⢀⠠⠄⢀⣀⠀⠀⣠⢶⠽⡬⢭⣒⡲
⠀⠀⠀⠀⡄⠠⠄⠀⡠⠞⢯⠁⢦⠀⠀⢀⡀⠀⠁⠘⢆⠀⢀⠈⣏⠒⢄⠈⠉⠙⠢⡄⠀⠈⠙⠒⠒⠁
⠀⠀⠀⢸⠰⠀⡖⠀⠀⠀⠈⡇⠀⡠⠖⠉⠀⠀⠀⠀⠀⠑⠀⠳⠘⠆⠊⠀⠘⡆⠀⠈⠂
⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠁⠁⢠⠀⠀⠀⠀⠀⠀⠀⢰⢠⠀⣄⢒⡄⠀⠀⠃⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⢠⠀⠀⠉⠁⠀⠀⠀⠀⡏⡎⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⣀⠈⡆⣆⠀⠒⠂⠀⠀⡼⢸⠁⢀⣀⡤⠔⠋⠀⠢⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⣅⠃⠈⠑⠒⠀⠠⠞⠓⠀⠀⠁⠠⠤⠤⠤⠤⠖⠈⠓⡂⠀⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇⠀⢸⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣁⠀⢸⣀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠀⢀⠀⠐⢠⡄⠀⠀⠀⠠⣸⣀⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⣸⣠⠄⠀⠀⠀⢀⡄⢂⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠱⠘⠀⠀⠀⠠⠄⣻⠄⠑⠆⣲⢒⡐⠂⠀⠐⠛⡝⠒⠀⠈⠒⣀⢀⠰⢀⠄⡠⠎⠀⠐⠊⡍⠓⠀⠀⠀⢂⡚⢖⠀⠎⠡⣞⠠⠄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠂⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⢈⡁⠆⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠀⠀⠈⠁⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠘⢘⡠⢣⢜⡳⠜⠀⠁⠁⠈⠀⠀⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠄⡹⠃⡭⠀⠀⠀⢀⠐⠑⠠⠍⠂⠑⢦⡀⠀⠀⠨⢐⠟⡠⠀⠀⠛⠀⢀⠀⠀⢠⠀⠀⠀⠀⡀⠠⠄
⠀⢀⠀⢿⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠀⠀⠐⠄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠀⠀⠀⠀⠂⣼⠆
⠀⠀⠂⠠⡄⠀⠀⡀⠀⠀⠠⡇⠀⠆⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠆⢨⠅⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠤⠢⠀⢠⠀⠀⠐⠀⢰⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⡠⠀⠔⠤⠤⠄⠀⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠒⠀⣷⠀⠀⠀⠑⣄⠀⣀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⣀⠌⠀⠀⠀⣴⠀⠔⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠒⣀⠋⠰⠀⠀⠙⠛⠀⢣⠆⠀⠀⠀⡄⠀⠀⠐⡞⠂⠛⠊⠀⠀⠒⠘⣀⠐⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⡄
⠀⠀⠀⠀⢤⠘⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠘⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣠⠁
⠀⠀⠀⠀⠀⡀⠊⢠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠤⠑⢂⠄
⠀⠀⠀⠀⠀⢀⠐⠆⢀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠘⠢⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡰⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠔⠛
⠀⠀⢠⡄⠀⠀⠀⠉⡈⠂⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠐⢃⡈⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠤⠄
⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠄⠄⠡⠄⠤⠀⠀⠠⠀⠀⠂⠀⠀⠀⡈⢀⣀⠀⠄⢂⠀⠀⠀⠀⣀⡰⠤⡀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠃⠀⠀⠀⠀⠂⠀⠄⠀⠀⠤⠠⠤⠀⠠
⠀⠀⠀⠐⠈⢀⠂⠀⠀⠀⠀⠘⢣⣽⡀⣁⢒⢡⡤⠭⠒⢠⠊⠉⠀⠀⠀⢉⣇⣀⠟⠀⠀⠀⠘⡙⡄⠐⠭⠤⡌⠐⣀⢀⣼⣔⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠐⠋⣕⠉⠉⠀⡃⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠘⡉⢢⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⢲⢺⠀⡉⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡗⢒⠢⠔⠂⠀⠀⠀⠠⡄
⠀⠀⠀⢈⣰⠀⠀⠀⠀⠂⠒⢋⠁⡈⠐⠀⠀⠀⠀⠀⠀⠀⠗⠔⠃⠠⠄⠀⡀⠀⠀⠃⠘⠀⠃⠌⠀⠀⠀⠀⠀⠀⠈⠐⢉⠀⡉⠒⠀⠂⠀⠀⠁⣦⠀⠆
⠀⠀⠀⠈⡃⡄⠀⠀⠀⠈⠄⢨⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⡁⠠⠁⠀⠀⠀⢁⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠠⠀⠀⠐⠀⠀⠀⠄⢰⡈⠣⢀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠓⡄⠀⠀⠀⠀⠐⠂⢲⠀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠁⠀⠀⢀⠞⠁⠃⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣉⣀⡐⠀⠐⠒⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢟⣛⣩⣭⣭⣛⣓⡀⠀⠀⠀⢮⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⠵⣾⢿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⢿⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡼⠁⠈⠀⠀⠀⠀⠐⠒⠲⠄⠀⠀⠀⠀⠀⠀⠸⣷⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠠⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠁⠀⠀⢀⠤⡀⠀⠔⡡⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡤⣄⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣧⠃⠀⠈⢀⠡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠁⠀⢰⠚⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⡇⠀⠀⠀⠀⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢙⠂⠤⠞⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣏⠀⠀⠀⠀⠀⠈⣏⠉⠁⠀⣠⠔⣻⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠘⢇⡉⣉⡠⣺⠏⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡠⠚⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⣌⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⡀⠀⠀⠀⣀⣠⡴⢾⠁⠒⠒⠉⣁⡠⠟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢋⡀⢀⡞⢰⠒⠊⠉⠁
⠀⠀⠀⠀⠀⡠⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠒⠒⠒⠉⣀⣤⡆⠀⠉⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡴⠂⠤⠤⠤⠤⠖⠚⠦⠔⠒⠋⠁⡼⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠈⠙⠲⠤⣀⡉⠑⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠲⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣦⠤⣄⢀⣀⣠⠋⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⡭⢷⡄⠀⢀⣀⣀⣀⣈⠑⠲⠦⢤⣤⣀⣀⡀⠀⣀⡁⠀⠀⠀⠀⠀⢸⠀⡟⠈⠀⠀⠈⠉⠉⠉
⠀⠀⠒⠄⠀⠀⠀⠀⠀⡰⢮⡠⠚⠁⠀⠀⠛⠉⠉⠉⠁⠈⠉⠁⠢⢤⣀⡀⢀⠉⠉⠉⠉⠴⡦⠀⠀⠀⢠⠋⡰⠃
⠘⢌⠢⡀⠈⠑⠢⢄⡘⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⣸⠣⠤⠔⣊⡭⠛⠁
⠀⠈⠱⣈⠢⡀⠀⠀⠈⢳⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠏⠙⠋⠉⠁
⠀⠀⠀⠈⠑⢌⠳⢄⢀⡞

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

⠀⠀⠀⠀⠀⠀⣤⡴⠆⢀⠀⠀⠀⢠⣴⡶⢂⡀⠀⠀⣀⣤⠖⣠⣤⡴⢀⡀⠀⠀⠀⠀⠀⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤
⠀⠀⣀⣠⣤⣤⣽⣷⣾⡟⣷⣤⣤⣼⣿⣷⣿⣿⣤⣤⣹⣿⣴⣿⣿⣶⡿⢿⣄⣀⣀⣀⣀⣹⣽⣶⣿⢿⡄⠀⠀⣀⣤⣤⣿⣿⣶⣿⣆
⠐⠛⠉⠙⣿⠿⠿⢿⣿⡿⠛⣿⠿⠿⣿⣿⢋⣽⠿⠿⠿⣿⣿⣿⣿⣿⣇⠉⠉⣩⢿⡿⠿⠿⣿⣿⣇⡀⠐⠛⠛⠻⣿⡿⣿⣿⣟
⠀⠀⠀⠀⠙⣆⠀⠀⠃⠁⠀⠈⠦⠤⠼⠬⠞⠁⠀⠀⠠⠇⠘⠧⠄⠈⠳⠤⠊⠁⠏⠀⠀⠀⠀⠘⠄⠑⠀⠀⠀⠀⠹⣦⠀⠘⢊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠟⢙⣿⣿⣿⣿⡿⠛⠋⣹⣿⣿⣿⠿⠟⢙⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⣿⣿
⣿⣿⣿⡿⠿⠷⠄⠘⠉⠀⠈⠟⠛⠃⠀⠈⠁⠈⠻⠯⠀⠀⠋⠀⠀⠈⠀⠈⠻⣿⣿⠿⡿⠄⠀⠋⠉⠈⢻⣿⣿⡿⠿⠯⠀⠀⠊⠉⢻
⣝⣁⣠⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⢰⣷⣎⣉⣁⡀⠀⠀⠀⠀⢀⣶⣤
⣿⣿⣿⣿⣄⠻⣿⢓⣠⢸⣿⣤⠚⠿⠆⠂⠁⣰⣿⡿⠿⢰⣄⠐⢖⠂⠀⠿⣣⣔⢡⣾⣿⣷⣶⣆⢲⡌⢿⣿⣿⣿⡀⠚⢖⠆⡠⣸⣿
⣿⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣦⣼⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣦⣽⣿⣿⣿⣿⣿⣦⣾⣿⣷⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣙⣳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡙⠿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣤⣴⣿⣷⣆⠈⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻⠟⢻⣿⠿⣿⣿⣿⣿⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣠⠴⠛⠃⣤⣾⣿⣿⣿⣬⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣧⠄⢀⣴⣿⣅⢀⣿⣿⣍⡻⡟⣳⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⢆⢀⡷⣼⣿⣿⡿⠟⢁⡰⠎⣩⣿⣿⠏⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⢿⣿⣿⡟⠉⣃⢲⡛⠛⠛⠻⢆⣈⣤⣤⠴⠋⣹⠇⠀⠀⢀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⢀⢉⡼⠋⠁⢠⣾⣿⣯⠝⢶⡤⡴⠾⠿⠟⢽⢳⣾⣥⣤⣤⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢶⠞⠛⣿⡀⢀⡀⣀⣾⣿⣿⣿⣶⣤⣤⣤⣷⣶⣶⡘⣟⣿⢿⣿⠿⢿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣁⢀⣤⠂⠀⢀⣨⣷⣨⣿⣿⣿⣽⣿⣿⣿⡇⠀⢹⣿⡟⠿⡿⢧⣿⣶⣤⣴⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣒⠉⠙⠓⠲⠧⢬⣁⠛⠉⣫⣼⣄⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠛⢹⢿⣻⣽⠟⠋⢱⣾⠿⢿⣦⡽⠟⠛⠿⣿⡷⣶⡄⢠⡼⢋⢀⡀⠀⣠⣶⢶⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂⣀⠠⣦⠰⠟⠉⠀⣀⣠⣾⣿⣄⣼⣿⣖⠶⢤⣤⡬⢿⣶⠊⠁⠙⠿⣿⣿⣶⣟⣁⣾⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⡆⣉⢩⣅⠀⠀⠘⣧⡀⣠⣾⠟⠛⠋⣠⣦⠀⠛⢿⣦⣆⡀⠀⠀⠙⠿⣷⣠⣤⣀⠈⠉⣉⠉⣭⣁⣴⣶⣾⠇
⠀⠀⠀⠀⠀⡸⢸⣿⣿⣦⡆⠀⠙⠣⣾⠟⣿⡷⢀⣷⣷⡟⠾⠁⣀⣴⣿⠧⣿⣷⡀⠀⠙⣿⣿⠂⠀⠀⡖⠉⢻⡟⣿⣿⣶⣿⣶⡏⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠐⣷⣾⡿⢣⡟⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠀⠀⢻⣿⣥⣿⠒⢿⣿⡇⠀⠀⠈⢿⡿⣿⣇⠰⢲⠞⠀⠀⠙⢿⣇⡁⠀⠀⠀⠀⠀⣠⡄⢰⣞
⠀⠀⠀⠀⢀⡿⠙⢱⡟⠀⣴⣶⣶⣶⠀⣠⣴⠿⠋⢀⡰⣷⣾⣿⣿⠿⢿⠦⢾⣿⣷⣶⠀⢀⠈⠙⠻⢿⣿⣛⢿⡷⢦⣄⣠⣈⢉⠓⢦⣤⣤⣾⢣⢿⣼⣿⡇
⠀⠀⠀⠀⢸⠃⠀⡾⢸⣿⡘⣿⣿⣿⡟⢉⠀⠀⣰⣿⣷⡸⣿⣷⣿⣤⣴⣄⣴⣿⣟⣻⣾⣿⡆⢰⣶⣾⣯⠉⠳⣄⠸⠟⠛⠛⢾⣿⣤⣦⣤⣿⣿⡜⠛⡿⠤⠴⠂
⠀⠀⠀⠀⢸⡆⢸⡇⠀⣿⣧⠹⣿⣿⣿⡘⢷⣿⠟⢛⣉⢷⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠙⣿⠇⣾⣿⣿⡟⢀⡼⠋⣀⣀⠀⠀⠀⣽⢻⡽⠿⣿⡍⠀⢸⠷⠿⠂
⠀⠀⠀⠀⢸⣇⠈⢷⡖⠛⡿⠃⠉⣀⣀⣰⣞⢳⣼⣿⣿⠿⠿⢿⣿⣿⡇⠀⢀⣤⣤⣄⢸⠟⠀⠿⢿⣿⣧⣾⣵⣿⠟⠻⣿⡆⠀⣷⢰⣇⣴⡟⣠⢆⡇
⠀⠀⢀⡴⠾⣷⡀⠈⢿⡾⣧⣴⣿⣿⠋⠉⣿⣿⣿⣿⡇⠀⠀⠸⡏⢿⡇⣰⠿⠋⠉⢻⡟⠶⠦⠶⣦⣿⣿⣿⣿⠀⠀⠀⢹⣿⢠⡷⠖⢛⣫⣾⣡⣾⣀⡀
⠀⠀⠈⢰⣿⣿⠧⡀⢨⢷⣿⣿⠷⠛⠀⠀⣸⣿⣿⡏⠤⠀⠀⠀⣿⡘⣹⣿⠀⠀⠀⣼⠀⠁⠀⠀⣜⢹⣿⠟⣿⠄⠰⠆⣼⣾⣯⠶⢒⣻⣿⣟⣥⣤⢀⡉⠁⣀
⠀⠀⠀⠀⠉⠃⠀⢻⣿⣼⣟⢣⣀⠐⣠⢤⣿⣿⣿⣿⠔⣄⠶⢰⣿⠇⢻⣿⠣⡜⠃⠻⡇⠶⠀⠀⠉⣻⣟⢦⡀⠶⣤⠴⠂⣠⢿⣾⡟⠋⢠⡅⠈⡿⠏⠛⠃
⠀⠀⠀⠀⠀⣼⢻⡏⠙⢯⠙⠿⢿⣯⣤⡞⠁⣉⣿⡀⠀⠀⠑⠉⠉⠀⣸⡈⣳⣶⣶⡾⠿⢆⠰⡶⠚⢿⣿⣿⣿⣶⣦⣀⠶⠋⣀⠬⠛⢁⡾⠶
⠀⠀⠀⠀⢰⡇⡀⠃⣀⢈⣷⣀⣀⠈⢩⡙⠚⠻⠿⣷⣤⣀⣀⣀⣄⣰⣿⣿⣯⢿⣿⣇⢀⡜⠉⢱⣄⣼⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣸⣧⣄⠀⠈⡄
⠀⠀⠀⠀⣸⢟⣡⠋⣹⣿⣿⣿⡿⢣⢸⡏⡷⣶⠀⠀⠉⠉⠻⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠀⠀⡧
⠀⠀⠀⢀⡾⢹⡇⠀⢿⢻⠝⡏⠀⠀⢸⡇⡇⣿⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⣤⢸⠆⢶⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⣠⡄⣲⠴⢷⠀⣸⡿⢥⣠⠔⠋
⠀⠀⠀⢘⣷⠟⢳⡀⢀⡟⣰⠀⠀⠀⠘⠃⠗⠿⢸⡏⢸⡇⢸⡇⢰⡆⡆⠀⣿⠀⠀⢼⠐⡆⣷⠁⣷⠛⣤⣿⢸⣿⣿⠷⠟⠃⠈⢠⡏⠀⢠⡗
⠀⠀⠀⢸⡏⠀⢸⣿⣼⠁⠈⠉⠛⠲⠦⠤⣤⣀⣀⡀⠈⠁⠈⠃⠈⠃⡇⠀⠙⠀⠀⠚⠚⠃⠛⠀⠉⠀⠉⠀⠀⠀⣀⣀⠤⠤⠒⠉⢷⣠⠟
⠀⠀⠀⠈⢿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢴⡦⠶⠶⠖⠙⠃⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⢛⣛⣛⡉⣉⠙⣶⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⢩⣖⣦⣶⣆⢸⣿⠃⢹⡇⠸⢷⣿⠀⣿⠉⠉⠉⠛⣲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⢸⡟⠁⠟⢻⠈⠋⠳⠈⠁⠒⢚⣁⣀⣯⣀⣀⣀⣘⠛⢭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠷⡶⠀⠀⣻⣄⣡⣤⡤⠴⠶⠚⠛⠛⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠙⠋⠀⣀⡴⠃⠀⢀⢄
⠀⠀⠀⠀⠀⠀⠠⠀⡇⢨⡄⠀⣼⣡⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠴⡆⠀⠀⠀⣹⣶⢶⡖⢲⢶⡶⡾⢯⣹⡇⠀⠀⣸⡎
⠀⠀⠀⠀⠀⠀⢸⡇⣇⢸⡀⠀⠉⠁⠀⢠⣤⡤⢤⠤⣤⢲⣶⢿⡁⢹⣌⣿⠀⠀⠀⠿⠋⠈⢷⠇⠈⠻⠃⠀⠙⠃⢸⡇⣿⡇⢠⣤⡄
⠀⠀⣴⢒⠊⠉⠙⠋⠛⠙⢓⠒⢦⡀⠀⢸⣿⡇⢸⣧⣽⠸⣿⠾⡇⠀⠋⠈⠀⠀⠀⠀⢸⡇⠀⢰⣢⠀⠀⣴⣦⠀⢸⣧⠻⠇⠈⠛⠉
⠀⠀⡧⢀⣀⢉⣭⡉⣩⣅⢠⣤⡀⢻⠀⠈⠈⠀⠈⠀⡁⠀⠁⠀⠐⣦⣀⢿⣠⢿⠀⢰⢟⣧⠀⣰⠿⡀⠀⢸⣻⣦⠈⠿⠀⢠⣦⣀⡀⠻⠃
⠀⠀⢷⡟⢻⣿⣸⣇⣟⣻⡿⢹⡇⡿⠀⠀⠀⠀⣶⣾⡿⣰⠀⠀⠀⠹⡉⣂⡡⠼⠀⣯⣵⡿⠀⣿⠃⢷⡀⣾⣭⣼⠇⠀⠺⣼⢿⣻⠅
⠀⠀⣸⣗⠀⣤⠙⠿⣃⠉⢠⣌⣻⠧⣄⢠⡖⠒⢻⣿⡏⢉⣳⡄⠀⡶⣄⢠⢺⠂⠀⢀⡀⣀⠀⢙⡓⠛⠁⠀⠀⠀⠀⢀⣀⡙⠋⠁⣀⢀
⠀⢸⠁⠈⠉⠓⠒⠚⣉⠉⠉⠉⠀⠀⠘⡇⢳⠚⠹⣿⡏⠉⣹⠁⣤⣷⣿⣿⣾⣧⠀⡸⠖⢶⣉⡳⣆⡵⠀⡶⠉⢲⣤⠟⠁⢸⡆⠀⠻⠟
⠀⢸⣴⢶⣄⡞⣷⣼⢻⡆⣾⢻⢠⡟⣦⣿⢸⡀⠀⣿⡇⠀⡟⠀⢸⣏⠋⠋⣭⡏⠀⠁⠈⠁⢀⣉⡉⠀⠀⣷⡀⠀⠉⠀⠀⣼⠃⣶⣤⡤
⠀⠀⣏⠈⠿⠃⠻⡿⠈⠷⣏⠈⢿⡃⣸⡅⠀⠣⢀⠛⠓⠚⠁⠀⠈⣿⣀⣴⣉⡇⠀⠀⠀⣰⠋⠀⠙⣦⠀⠈⠙⠲⢤⣠⡼⠃⠀⠻⠟⠁
⠀⠀⢿⠺⡧⢮⡥⣬⢤⡴⠶⢋⣛⣬⠾⠃⠀⣼⢁⢹⣠⡖⢒⡄⠀⠈⠉⠉⠉⠀⢀⡴⠒⣿⠀⠀⠀⣿⠞⠉⣶⠀⠀⣽⠁⠀⢀⡤⠖⠂⢤
⠀⣠⡀⠉⠓⠒⠒⠒⠒⠛⠉⠉⠀⢠⣤⡀⢠⡬⠽⡿⠻⣿⠏⣤⢰⣿⣤⣶⠀⣤⠞⠳⣄⢿⠀⠀⢀⣟⣠⠤⣿⡁⠀⠙⢦⠀⡞
⠲⡋⠹⡃⢤⢤⣶⣶⡾⣹⠀⡤⣄⢨⣿⡁⠸⡇⠀⣇⢠⡧⠀⢼⢻⣿⣿⡅⣸⠃⠀⠀⢹⣮⠧⣴⠻⡿⠁⠀⠈⣿⠀⠀⣼⠀⡇⠀⠀⠀⢠
⠘⠛⠉⠁⠘⠚⠉⢁⡀⠀⣠⢿⡅⠺⠾⠇⠀⣇⠀⣿⢸⠇⣀⣿⣸⣿⣸⠇⢻⡀⠀⢀⣾⡏⠀⢸⣿⡇⠀⠀⢀⡿⠀⠴⠃⠀⣸⠶⠴⠒⠋
⢸⠖⢦⠀⠀⣠⠚⣵⠿⠀⢑⣛⣁⡔⢰⠀⠀⠈⠉⠉⠉⠉⠁⣀⣀⠀⠀⠀⠀⠙⢶⠋⣿⡄⠀⣸⡿⠙⣦⡴⢛⣀⣀⡀⠀⢠⡇⠀⣸⣷⣶
⠉⠻⠁⠐⢊⣲⠗⠁⣴⠰⣿⣌⠿⠃⠘⢡⣾⠀⠀⠀⢀⡴⠺⣅⡽⢦⡀⠀⠀⠀⠀⠀⠈⢳⣤⠟⠠⠀⡄⡴⠋⠀⠀⠈⢢⠈⢇⠈⠛⡽⠋
⠘⠃⣠⡤⣤⣁⣀⣀⡈⠙⠋⠀⠀⢀⠴⢭⠭⠵⢦⠀⢯⢌⣓⣉⣫⣭⡽⠀⣔⠀⠀⠡⣠⠀⡅⢀⠃⠜⠐⡇⠀⠀⠀⠀⣸⠇⢸⡆⢠⣤
⠀⠀⣿⠀⠖⠀⢀⣠⡾⣷⣀⠀⠀⡾⢉⢚⣩⢉⡹⣄⢸⢸⣻⢹⢸⣿⠃⠀⣾⡆⠀⠀⠈⢳⡁⢀⠎⠀⠀⢻⣄⣀⣀⡴⠋⢀⡼⠀⠙⠛⠁
⠀⠀⢻⣆⣤⡾⠛⢋⣠⠾⣹⠃⠈⣶⣦⡭⡶⣤⡖⡟⠀⠉⠛⠛⠛⠁⡔⠀⠉⠙⠒⢄⡀⠀⣿⠀⠀⠀⢠⠞⠋⠁⠀⠀⣶⡄⠀⣶⣤⠄
⠀⠀⠀⠉⣷⣤⠴⢛⡵⠚⢡⣄⡄⠹⢾⣷⣷⣿⠼⠁⡠⣠⣶⣄⠀⢺⡄⠀⠀⠀⡀⣠⡗⠀⣿⡀⠀⠀⠸⡀⠠⣿⣖⠀⣀⣠⡆⠘⠁
⠀⠀⠀⠘⠧⠔⠚⠉⠀⠀⢮⢿⠗⠀⠀⠀⠀⠀⠀⠀⣱⢺⡟⡞⠁⠀⠘⡏⡏⣿⣏⣿⠃⠐⠁⠀⠀⠀⢀⡇⠀⠁⠀⣠⣯⡀⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣁⣀⣀⣀⣷⣿⣟⣛⣛⣀⣀⣀⣀⣀⣠⣏⣀⣀⣀⣀⣈⣀⣈⣃⣀⣀⣀⣀
⣿⣿⠛⡻⢛⡛⠛⣟⢛⡿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⡠⠖⢚⣉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⢠⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⣀⣤⡄⠀⠈⢦⠈⢢⡀
⠀⠁⠀⠀⠀⢿⡉⢢⡀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠄⠀⠠⡶⠿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢠⣇⡼⠋⣰⠇⠀⠀⠀⣧⠀⢳
⠀⠀⠰⣏⠓⢮⣳⣄⠱⣼⡇⣠⠏⠀⠀⠀⠀⠀⠀⢀⡾⣫⡴⠆⠀⠀⠀⠀⠀⠀⠀⣀⡉⠻⣆⠀⠀⠀⠀⠀⣞⢉⣈⡙⢿⣄⠀⠀⢾⡥⠶⣦⠀⠀⢸
⠀⠀⠀⠈⠳⢄⡙⠁⣰⠟⠁⢳⡀⠀⠀⠀⠀⠀⠀⣸⠇⢁⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡦⠹⣧⠀⠀⠀⠀⠉⠁⠀⡝⠀⠈⢷⠀⠀⡤⠖⠋⠀⠀⢸
⠀⠀⠰⣏⣉⣉⡁⢰⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠙⢦⣄⡀⠀⠀⢀⣠⡴⠛⡇⠀⢻⡆⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠀⣤⣉⣱⡄⠀⠀⠏
⠀⠀⠀⠀⢠⠞⣡⣄⣀⣀⣄⠙⢦⡀⠀⠀⠀⠀⣠⣿⡀⢸⣇⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⣿⠀⢸⡷⣤⡀⠀⠀⠀⣰⠋⢠⡷⠒⠋⠁⠈⠉
⠀⠀⠀⠀⠘⠋⠀⠈⠉⠉⠈⠳⣄⠙⢦⡀⠀⠀⣧⣂⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢀⣀⣸⠇⠀⢠⡞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢦⠀⠈⣿⠀⠀⢿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠀⠀⣿⠁⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⣄⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣠⠞⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠈⢧⡈⠉⠙⠛⠛⢛⡿⠃⠀⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠢⡀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠙⢦⣀⡀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⣄⣀⠉⠛⠛⠛⠋⠁⠀⣀⣠⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢀⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⡆⢸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⣀⣼⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡷⠀⠀⢀⣠⠶⠛⠙⠃⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠙⢦⡀⠀⣠⠟⣽⠛⠳⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⢻⡙⢦⣀⡴⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠲⠋⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡉⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⣀⡉⣒⡶⢤⣍⡙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣦⡀⠈⢦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣿⣯⣉⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠠⡹⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣾⣿⣿⣿⣿⣿⣿⣧⣿⣭⢫⣞⣿⣿⠐⣷⠻⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠸⢸⡟⠛⠛⠛⠋⠙⠁⠉⠋⠿⢻⣾⣿⣿⣼⠘⢢⡟⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⡆⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣯⣂⠛⢹⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢿⣴⣯⢟⠆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣿⣿⣯⣐⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡇⠀⣷⣤⣤⠀⠀⠀⠰⠿⣿⣯⣭⣍⠑⠀⠘⣿⣿⡷⣭⣋⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠹⠾⠦⠀⠀⠀⠀⠀⠁⠀⠈⠙⠃⠀⠀⠹⣿⣿⣦⠌⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡳⣄⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⣻⣿⣿⣮⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⣀⢀⣼⣿⣿⣿⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⠛⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⣿⣿⣿⡿⢹⢸⠹⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡇⠰⢸⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⣿⣿⣿⣿⡃⠀⠈⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡲⢤⣀⣀⣀⠠⠒⠉⠀⠀⢠⣿⣿⣿⣿⡇⠀⠁⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡀⠸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⡃⠀⠣⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣧⡰⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⢦⣤⡂
⠀⠀⠀⠀⠀⡀⠀⡀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠆⠀⠶⢮⣭⠀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠊⠢⠙⢿⣷⡖
⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠛⢀⠀⠀⠹⣆⠀⠀⠀⠀⢠⣿⣿⡿⣿⣧⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠈⠘⡉⣀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠹⠀⠀⠀⢀⠃⣿⣿⠃⠘⢿⣷⠀⠀⣀⣀⢿⢿⡀
⠀⠀⠀⡀⠹⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠀⠀⢀⠂⢠⣿⣡⢖⡒⢚⠻⡈⠋⠉⠀⠀⠂⠃
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢲⠏⠀⠀⠀⠀⢝⠉⠑⠋⠀⠀⠀⠈
⠜⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⢠⡄⠀⣼⠁⠀⠀⠀⢀⣀⡀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣿⣿⣿⡿⣿⡿⠿⣿⣿⣿⠿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣾⣶⣾⣶⣾⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠿⠶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣥⣴⣶⣶⣶⣶⣦⣤⣉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠙⠛⠛⠛⠛⠛⠟⠻⠿⣿⠂⠀⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠈⠉⠁⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⡞⢙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠊⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⡄⠀⠀⠀⠀⠀⠠⢬⣤⡄⠀⠀⠀⠀⠀⠀⣴⡆⢰⡤⢧⣦
⠀⠀⠀⣀⣢⡷⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠛⠤⢀⡠⣦⡶⡀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⣴⣿⠧⠘⠀⠐⠛
⠀⠀⠀⠐⠋⠀⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠌⣁⠠⡉⠁⢴⣤⣤⣤⡀⠀⣠⣤⣶⣾⠟⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡁⠂⢁⠂⠄⢀⠀⠀⠀⠀⢀⠀⠀⠠⡊⠄⢈⠀⠊⠀⠀⠀⠐⢀⡘⠙⠛⠣⠄⠻⠋⠁⢡⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠂⠀⢀⠤⣀⠨⣀⠄⠆⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠂⠀⠀⠀⠐⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠃⠀⠖⢂⠢
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⢀⠠⠐⠻⣀⠇
⠀⠀⠀⠀⠀⠀⡔⣀⣡⠉⠑⠐⠀⠀⠀⠀⠐⠒⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠮⡄⢨
⠀⠀⠀⠀⠀⠀⠃⠘⠁⠑⢀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⢣⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠂⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⠏⠀⣠⣤⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀⡀⠈⠙⠛⠻⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠋⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠈⠛⣛⣿⣿⣿⣿⣿⣿⡿⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠐⠛⠛⠛⠛⠿⡿⠻⠟⠋⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⠁⠀⠉⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀⠈⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣶⣦⢀⣠⡀⠀⠀⠈⢻⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡎⠉⠛⠿⠟⠛⠛⠁⠀⠀⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠿⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠋⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠙⣿⡇⠀⠀⣼⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣸⡿⠀⠀⣼⡟⠀⠀⡀⠀⠀⠀⢠⣾⠂⠀⠀⢀⣶⠆⠀⠁⢀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠷⠶⣾⡿⠀⣠⡾⠛⣿⠂⢠⣿⡗⢩⣿⠀⣾⡧⠊⣿⠂⢻⣿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⠇⢰⡟⢁⣾⡇⢀⣾⣯⢤⣾⠏⣼⣿⢤⣼⠟⢀⣿⠃⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿⣄⠈⠛⠁⠘⠛⣽⡏⠙⠋⠁⢰⡿⠙⠛⠉⠀⠀⠛⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠃⠀⠀⠀⣴⠞⣻⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣴⡆⢀⣴⡷⠀⠀⢰⣄⠀⠀⣶⠄⠀⠀⠀⠀⠀⠐⠛⠟⠁
⠀⠀⠀⠀⠀⢠⣶⠿⣿⡟⠉⠙⢦⠀⠁⢀⣤⠉⠉⠉⠛⠛⢻⣾⢿⣶⣿⣯⣤⣀⣠⣼⠟⠀⣼⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠹⠅⣰⣿⣀⣀⣤⠏⢀⣠⣌⠉⢰⣛⡆⠀⠀⣼⠏⢠⡿⢁⣭⣭⡍⢉⣡⣤⣴⡟⠀⠀⢠⣴⠶⣤⠀⢀⣴⠆⢠⣶
⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⢹⡄⢹⡿⠁⢀⣼⢛⡿⠀⣸⡏⠀⣾⡷⢋⣿⠋⣴⠟⠉⢠⡟⠀⠀⣴⡿⢁⣼⠇⢀⣾⠏⣠⣿⠃
⠀⠀⠀⠀⠀⢠⣿⡃⢀⣀⣤⠟⠁⣿⣧⠶⠋⢰⣿⣥⢾⣿⠀⢸⡟⠁⢼⣇⡤⢿⣤⡶⠻⣧⣴⠞⢿⡥⠊⣿⡤⢺⣿⠔⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣴⡟⠀⠐⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠶⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣾⣯⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣬⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⡾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⠖⠟⠳⢲⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⢰⣶⣶⡆⢱⣶⣶⡆⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠾⢽⢯⠷⢀⢀⠆
⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⡷⠀⠀⢸⣿⣿⡇⢸⣿⣿⠁⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢰⣿⢾⣿⣄
⠀⠹⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⣸⣿⣧⣤⣄⠀⠀⠐⢿⠿⠀⠿⠿⠁
⠀⠀⠀⠉⠉⠉⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠀⠀⠿⠟⠛⠉⠛⠛⠃⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⡁
⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⣚⣷⡴⠀⠀⢦⣴⣗⠚⠀⠋⣤⠀⠀⠀⠀⠀⢈⡉⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⡻⢟⡀⠀⠀⠈⠁⢖⣿⣶⣶⣾⣳⠋⠁⠀⠈⢃⡄⠀⠀⠀⠀⣈⡠⠄⠀⠸⠿⠆⠀⣀⠂⠀⡀⠀⠺⠿⠂
⠀⢻⣿⣸⣿⣿⣇⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣤⠄⠀⢀⣈⣁⡤⠖⠀⠀⠀⠀⠀⠈⠉⠹⣷
⠀⠈⠿⠋⠀⠀⠙⠻⠋⠀⠀⠈⣿⠛⠛⢻⠏⠀⢠⣦⠀⠀⢡⡆⠀⠈⣡⣀⡠⠵⠚⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣾⣶⣶⣧⣄⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠀⠎⠸⠂⠀⠈⠈⠁⠀⠂⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣠⣶⣿⠓⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣾⣿⣿⠟⠃⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠐⠻⠿⠿⠀⠀⣤⣄⣻⣿⣿⣟⣠⣠⠀⠀⠀⢀⣴⣄⡘⠇
⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠂⠀⠀⠙⢿⣿⣷⣆⣹⡿⠋⠀⠀⣴⣿⣤⣬⣿⡆
⠀⠰⠿⠿⠿⠿⢿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠋⠈⠿⢿⠷⠀⠀⠿⠿⠀⠸⠿⠇
⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⢀⣬⠁⠱⠆⠀⠀⠀⠀⠀⠀⠀⣔⣰⣶⣠⣾⣧⢀⣤⠀⠀⣴⡟⠛⠉⠉⠙⢳⣆⠀⠀⢀⣴⣾⣦⡀
⠀⠀⠘⢆⡛⣡⡿⡿⠳⠇⢀⠀⠠⠀⠀⢀⣹⣷⣽⣿⣿⣿⣿⠃⠀⢘⣿⠀⠀⠀⠀⠀⠀⣯⠀⢠⣿⡿⠟⢻⣿⡆
⠀⠘⠛⣤⠢⣤⠷⢆⠈⣘⠋⠀⢇⠀⠀⠸⢿⣿⣿⠿⠿⠟⠃⠀⠀⠀⢿⣤⣀⡀⢀⡀⣰⡟⠀⢺⣿⣭⢀⠰⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠘⠁⠙⠁⠀⠀⠑⠂⠤⠤⠤⠤⠤⠄⠂⠀⠀⠀⠀⠀⠉⠙⡿⡿⠛⠉⠀⠀⠈⠻⢿⣿⣿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠒⠚⠛⠉⠉⠉⠉⠉⠛⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉⠳⣄⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠛⢦⣄⡀⠀⠀⢀⡏⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⢿⡀⠀⠀⠀⠀⠈⠉⠛⠶⠼⠇⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠚⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⡞⠛⠾⣦⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣀⡼⠟⠛⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠳⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠙⠿⠀⠀⢠⡶⠛⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠚⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⢷⣄⣤⡄⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⠀⠹⠄⠀⠀⠀⠀⠀⢀⣄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡿
⠀⢠⣦⣤⣶⡀⠀⠀⠈⢧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣸⠃⠀⠀⢠⣶⣤⣶⡄
⢴⡾⣯⢻⡏⣷⢻⠄⠀⠈⣧⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⠏⠀⠠⡟⣾⣽⡿⣽⢷
⣴⢿⡍⠀⠀⠉⡏⠀⠀⠀⠈⢧⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣰⠋⠀⠀⠀⢻⠉⠀⠀⠱⠿
⠈⠛⢦⣀⠀⠀⠛⢦⡀⠀⠀⠈⠳⣽⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣛⣛⣛⣛⣛⣛⣛⣛⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡼⠃⠀⠀⢀⡴⠟⠀⢀⣀⡴⠛
⠀⠀⠀⠈⠳⡀⠀⠀⠙⠳⣄⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠉⠉⠉⠉⠉⠉⢩⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡵⠋⠀⠀⣠⠞⠋⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠳⣤⡀⠀⠉⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⣤⡤⠖⠋⠀⠀⠀⠀⠀⠀⢀⣤⣾⡷⠋⠀⢀⣤⠞⠁⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠈⠙⠳⢤⣀⠈⠙⠻⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠁⣀⡤⠞⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠀⠈⠙⠛⠿⠶⢶⣶⠤⠤⠤⠤⣴⡶⠶⠿⠛⠋⠉⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠉⠳⡶⣄⡀⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⢀⣠⣶⠞⠉⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⠀⣰⠃⠀⠙⠳⠤⠞⠛⢦⡀⣀⣴⠛⠳⣤⠞⠋⠀⠘⣆⠀⠀⠀⣀⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣦⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣠⣤⡄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢀⣴⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡀⢠⡄⠀⠀⠀⠀⠀⠀⢠⣾⠏⡜⣦⠛⢽⣉⣷⣤⠀⠄
⠀⠀⣠⣴⡾⠿⣶⣶⣡⠞⢭⡧⠀⠃⠀⠀⠀⠀⠀⣴⢯⣵⡄⣽⠘⢷⣄⣘⡁⠀⢡⣐
⠀⣴⣿⣿⣿⣇⡀⢘⣻⢷⣶⣦⣠⣦⡀⠀⢀⣀⠀⠙⠊⠁⠘⣿⡆⠀⠙⡿⣏⡙⢿⢾⣦⡒⠀⠀⠀⡀
⣰⢿⣿⠯⠟⠻⣿⣶⣬⣛⠿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣆⣤⣄⣹⣿⣄⠀⠁⠙⠿⣼⣾⣿⢃⣷⠀⠀⠘⠿⣷⣦
⠙⠛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⣉⣿⣿⣿⣿⣿⣿⡟⠉⠋⢹⣿⣿⡀⠀⠀⢂⠀⣠⣾⢿⣧
⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⡟⣷⣿⠿⠛⢿⣿⣿⣿⠃⠀⠀⠈⢿⣿⣷⣤⠀⠛⠻⣏⣉⡲⣿⢀
⠀⠀⠀⠀⠀⢀⠘⣡⠞⠛⠙⠋⠉⠉⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠻⠜⠃
⠀⠀⠀⠀⡀⢈⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣧⠀⠀⠀⠀⢸⡵⠧⡘⠇⠀⠰⡆
⠀⠀⢀⡴⠙⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠘⢧⣤⡄⠀⠀⠁⠈⠳⡀⠀⠛⠼
⠀⠀⠀⠀⣰⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣀⠁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠢⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠉⠁⠹⠖⠂⢤⡀⠀⠀⠀⠀⠀⠀⠠⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⢦⣀⡤⠞⠉⠱⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠰⡎⢀⡠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⠖⠳⠎⠀⢠⠈⠑⠲⠦⢤⣤⡤⢀⠀⢤⡤⢤⡄⠀⠐⣠⡿⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠐⣇⡠⡏⠀⠀⢸⠃⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤⣤⡤⠾⣴⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠉⠁⠀⢀⡜⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠙⢧⡀⠀⠒⠒⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠉⢰
⡀⠀⠀⣠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡐⠰⠆
⠀⠀⢠⡼⠆⠀⠀⡀⠀⠀⠀⢀⡀⠂⠀⠠⠤⣀⡀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠄⢠⡇⠀⠀⠈⠉⠀⠀⢀
⠈⠁⢠⡜⡀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠲⠲⢤⣌⣉⠒⠠⣄⣀⡙⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⣀⠠⠤⠒⣊⣤⠖⠁⠙⢦⣀⠀⠀⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⣌⣉⠓⠚⠛⡿⠛⠛⠛⣿⠟⠛⠛⠉⢀⣀⣀⣀⠠⠤⠔⠒⠊⠉⠀⠀⠀⠀⠀⠈⠁⠘⠧⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡙⠀⠈⠙⠉⠁⠴⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠒⠒⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠡⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⢀⣀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠏⠀⢰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⣱⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠲⠶⠶⠶⠺⠟⠋⠁⠄⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠑⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⢀⡁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⢨
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿