Luffy from One Piece Emojis & Symbols

⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣦⣧⣄⡀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠈⠀⢠⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣷⣦⣄⡀⠀⢇
⠀⠀⠀⠀⠀⢀⢦⣴⣿⣿⣿⠿⡟⠛⠉⠁⠀⠀⢀⣀⣤⣤⣤⣤⣴⣷⣴⣦⣤⣤⣄⣀⠀⠈⠉⠉⠛⠢⢼⡀
⠀⠀⠀⠀⢀⣼⣿⡿⠛⢉⠀⠘⠿⣳⣥⣾⣶⠃⠈⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣶⣄⠀⢀⡀⠀⠈⠂⢄
⠀⠀⢀⣴⡿⢛⣥⣶⣌⠁⢀⣠⣾⡟⣿⣿⡟⠀⠀⠘⣿⣿⣿⣿⡇⢻⣿⡌⢻⣿⣿⣿⣿⣿⣿⣿⣷⣅⠀⠀⠀⠀⠑⢄
⢀⡴⢛⣵⣶⣾⣿⢟⣥⣶⣿⣿⡟⠀⢻⣿⠃⠀⠀⠀⠘⣿⣿⣿⡇⠀⠹⣧⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢣
⢋⢔⣌⡙⠻⠿⣿⣿⣿⣿⣿⣿⠃⠀⠘⣿⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠹⣆⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠁⠻⠟⠟⠐⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠙⠀⢀⣿⣿⣿⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⣿⣿⣿⣿⢿⣿⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⢀⣀⣤⣶⣥⣄⣹⣿⠈⢿⣿⣿⣿⣿⣷⣤⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⠸⣿⠁⠀⣨⠿⠛⠻⢶⣶⠄⠀⠀⠀⠀⢠⠖⢉⡟⠉⢀⣀⠀⠙⣿⠀⢸⣿⣿⣿⣿⣻⡿⠃⠀⣰
⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⢿⣄⡜⠁⢀⣤⣄⠀⠙⣆⠀⠀⠀⠀⡟⠀⠈⠀⠀⠻⠿⠃⠀⠐⠀⢸⣿⣿⣿⣿⣿⡿⠂⡴⠁
⠀⠀⣰⣿⣿⠿⣿⣿⣿⠀⠈⠉⠇⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⢤⣀⣀⣀⣀⠴⠀⠀⣾⣿⣿⣿⣿⢟⡡⠚
⠀⠀⠛⠙⣷⢸⠻⢿⣿⡇⠀⠀⠀⢤⣀⣀⣀⡠⠄⠀⠀⠀⠀⠸⢿⡆⠀⠠⢬⣉⣭⡤⢿⠁⢀⣿⣿⣇⣨⠟⠁
⠀⢀⢄⠀⣿⡄⠀⣻⢿⣷⡀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠂⠚⠁⠀⠀⠀⠀⢈⡄⠀⠀⣼⣿⠟⠁
⠲⢼⣣⣴⣿⠹⣦⣘⢾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⠖⠋⠀⠀⣠⡟⠁
⠻⣿⣿⣻⣿⣿⣼⡟⢳⠮⣍⣻⣆⠀⠀⠀⠙⠓⠒⠛⠛⠛⠛⠛⢛⣉⡉⠄⠀⠀⠀⢀⣴⣇⡀
⠻⠶⠤⠭⠽⠲⠯⠤⠿⠤⠼⡿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢻⡷⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠻⣭⣿⡏⠙⢷⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡿⢻⣽⠻⣽⠿⣿⣿⣧⣿⣦⣿
⠀⠀⠀⠀⠀⣠⣴⢖⢉⣿⣟⡷⢯⣿⣿⠀⠀⠈⠙⠻⢿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣻⡟⢻⣯⣼⣿⣿⣿⢿
⠀⠀⣠⣴⣿⠋⠳⣾⣿⣇⣘⢶⣇⣀⠨⡄⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠿⣘⢷⠤
⢠⡾⠻⣏⣠⣻⣤⣿⣿⣿⣧⣴⣆⣠⣷⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠘⡿⠃⢻⣿⠛⢿⣿⣿⣿⡿⣿⢾⠙⣤⠻
⣿⣤⣴⠈⣹⣁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠏⠀⠀⢀⠼⣿⢒⠾⢾⡛⣿⡿⢇⡠⠻⣈⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠉⠉⠉⠉⠋⠉⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠟⠛⠉⠉⣀⠐⠒⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⡆⠀⢤⠀⠙⠛⠉⠛⠛⠻⢶⡀
⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠙⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠁⢤⡀⠀⠀⠀⢠⠉⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢰⡏⠀⣰⠃
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠙⣠⡆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠉⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⡄⠀⠀⣀⣠⣄⣤⣰⣦⣴⣦⣤⣴⣄⣀⣠⣄⠀⠀⠀⠀⢿⠀⠀⢰⠁⠀⠀⠀⣤⣀⣀⣀⣀⠀⡀⣀⡴⠃⠀⠀⡇
⡇⠀⠀⠀⠀⢱⣤⣾⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⢤⣾⣦⣴⣦⣤⠖⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣀⠀⠀⠀⡇
⡇⠀⠀⠀⠠⣾⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⠀⠀⠀⡇
⣇⠀⠀⠀⠀⢻⣿⣿⣀⣀⡀⣤⣤⡴⠒⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⣛⣛⡒⠒⠀⢤⣤⡤⠀⠀⢸⠏⠀⠀⠀⠃
⡏⠀⠀⠀⠀⠘⢿⣿⣿⠟⢣⠋⠀⠀⠀⣀⠉⠉⣨⣽⠛⠀⣀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣛⣋⡀⠀⠀⢰⠀⢈⠉⢿
⡇⠀⠀⠀⠀⠀⠀⠈⠟⢀⣾⠂⠀⠀⢸⠃⠀⠀⢿⣿⠤⣤⠼⠀⠀⢸⠆⠀⢰⡆⠀⢸⣿⣇⠀⢘⠂⣼⡀⢻⠃⣼⡀
⡇⠀⠀⠀⠀⠀⠀⠀⣶⡌⠁⡀⠀⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢸⠗⠠⠇⠉⠃⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢧⡴⠀⡶⠀⠀⠀⠀⢇
⠃⠀⠀⠀⠀⠀⠀⣠⣷⠀⢸⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⢀⣠⣦⣬⣷⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⢠⡞⢀⠞⠁⢸⠇⠀⠀⠀⠀⠈⢁
⠀⠛⠤⠤⠀⠚⠃⠀⠀⡀⢸⠀⠀⠛⢃⠄⠀⠀⠀⠀⢾⠿⠛⠉⣀⡉⢉⡉⠙⠛⠿⠆⠀⠀⠀⣰⡏⠀⣿⡄⢠⠟⠀⠘⠢⣄⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡌⣷⢠⣦⠈⠀⠁⠒⣀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⣠⡞⠋⣠⡜⠛⠋⢀⠀⠀⠀⠀⠀⠉⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡟⠙⠛⠁⠀⠠⠛⠳⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠰⣁⠀⠀⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⠟⣠⣿⡆⠁⠀⠀⠀⠈⠑⠲⢦⡀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢈⡷⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠙⠛⠦⡀⠀⠀⢀⡤⠂⠘⢿⡀⠀⠀⠀⠀⢠⣾⣁⠀⠀⢀⡜⠁⠘⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠁⡀⠉⠰⠶⠋⠀⠀⠀⠘⣧⡀⠀⠀⢀⠺⣿⠟⠒⠶⠞⠁⣴⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⠧⣀⢀⡜⢠⠏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡿⠁⣾⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠃⠀⣿⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⣷⠰⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠸⠀⠠⠆⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣴⠆⠁⠀⠀⠀⠀⢠⡀⠀⠁⢤⡀⠀⠀⠀⠀⠀⠀⠸⠃⣀⡀⠀⢻⠀⠀⠀⠀⠀⠸⠧⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠎⠀⠀⠀⢨⣁⠀⠀⠀⠀⠀⣀⣤⠚⣉⠙⢢⣄⡀⠀⠀⠀⣴⠃⠀⣿⠂⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠘⡏⠀⠀⠀⠀⠀⠙⠋⠀⠀⢴⡾⠋⢁⡴⠉⠓⢄⠙⢿⡁⠲⠶⡿⠀⢸⠁⠀⠀⠀⠀⣠⡜⠁
⠀⠀⠀⠀⠀⢠⡤⠀⠀⠠⢤⣴⠃⠀⠀⠀⠀⠀⢰⠀⠀⢰⠀⠀⠤⠋⠀⠀⠀⠀⠓⡄⠙⡄⠀⡃⠀⣸⠇⢄⠀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣄⣸⣆⠀⢦⡀⠀⠀⠀⢀⡠⠿⠾⢇⣠⢁⡜⠁⠀⠈⠁⣶⡶⠃
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣰⡾⠛⠉⢛⠠⢂⠀⠹⣄⡀⠠⠛⠧⠄⠤⠼⢿⣾⡁⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠁⣴⠆⣀⡈⠀⢹⣶⡦⠈⢁⡄⠲⠆⠀⠀⠀⠀⠈⠙⣶⡖⠋
⠀⠀⠀⠀⠀⢸⡦⠠⠤⠤⢎⣀⠀⠀⠀⣀⠁⢰⠀⡴⠛⠻⠗⣦⠙⢧⠈⠙⢻⡗⠀⠀⠀⠀⠀⢀⣠⠉
⠀⠀⠀⠀⠀⠈⠁⣠⠰⠖⠈⠉⠰⢄⠀⢹⡶⠂⢰⠀⠀⢀⣾⡁⠀⢸⡇⠀⠀⠂⠀⠹⡷⠀⠒⠈⠁
⠀⠀⠀⠀⠀⣆⠺⠋⢀⡴⠛⣦⠀⢸⠀⣾⠀⠀⠻⠀⢠⣿⠋⢻⡄⢾⣧⠀⠀⢰⠀⠀⢠⡀
⠀⠀⠀⠀⠀⣿⡆⠀⠸⡄⠀⠙⡾⠁⠀⣿⢀⣾⠇⠐⠃⠘⠀⠿⠀⢸⠃⠀⠀⠀⢀⠀⠀⠰⡀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠈⠡⠄⠰⢧⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢠⡖⠀⠃⣸⠀⠀⠀⠀⠈⣀⢀⡀⠻⡄⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⢿⠀⠀⠀⠀⠸⢀⠟⠀⣿⠁⠀⠀⠀⠀⠀⢹⠋⠀⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠃⠀⠀⠀⢠⡟⠀⠀⠁⣤⠀⠀⠀⠀⠀⢸⠄⠀⠀⠈⠠⢀⡀⠀⢘⣦
⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⢀⣼⠇⠀⡀⠀⣹⠇⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠋⠀⣾⠁⠀⣿⢆⡀⠀⠀⠸⣿
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣇⠀⠀⠀⠀⠐⢄⣰⣷⣶⠋⠀⠀⣿⡆⠁⠀⠀⠀⠈⢡
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠛⠀⠈⠣⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢀⣿⠀⠀⠀⠀⣀⣠⣶⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⠓⠒⠂⠀⠁⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⣦⠀⠤⠤⠀⠀⠀⢀⣸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⡆
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠖⠒⠗⠀⠀⠀⠀⠈⢿⠁⠈
⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠆⠀⠀⢀⡀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠄⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⢨⣿
⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⠂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠉⠀⠀⠈⠁⠀⠰⢄⡀
⠀⠀⢀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⣠⣤⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡀⠀⠀⠀⠀⠀⠈⠻
⠀⠀⠈⠀⠀⠈⢷⣦⣤⣀⣀⣤⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣦⣀⠀⠀⠀⢀⣴

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠖⠒⠒⠛⠛⠛⠒⠒⠤⢤⣄⡀
⠀⠀⠀⠀⠀⢀⠄⠊⠁⡄⣴⣇⣶⣶⣷⣾⣿⣄⣶⣶⣠⡄⣉⠑⠀⢀⡀
⠀⠀⠀⠀⠐⣡⣀⣾⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢀⠈⠲⣄
⠀⠀⠀⠀⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣇⡈⠓⢤⡀
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠙⠦⣄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠈⠑⠢⠤⠤⠤⠄⢀
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣦⣶⣿⣷⠀⢀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣏⣾⡎⣿⣿⣿⡿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⡇⣉⣿⢾⣿⠋⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣴⣿⠿⢻⣿⣿⣿⣿⡇⠘⠇⣸⣿⣤⠀⠈⠻⠻⠋⣿⣿⡿⢿⠿⡿
⠀⠀⠀⠀⠀⠺⠏⣱⠠⢿⣿⡟⠐⠡⣶⢤⣴⣿⣿⣇⡀⠀⠀⠀⠀⠙⠋⡀⠘⡶⠄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⠃⠉⢠⣤⣾⣿⣿⣿⣿⣿⣵⣤⣄⣤⣠⣤⡃⠀⠈
⠀⠀⠀⠀⠀⢠⢸⡇⠀⣼⢻⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣀⣠
⠀⠀⠀⠀⠀⠸⣼⠇⠀⡿⠀⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣩⣛⣻⣿⠉⠁
⠀⠀⠀⠀⠀⢠⠏⢀⠞⠁⢠⠔⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠘⢛⢃⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠒⢒⠓⠓⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠈⢉⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠂⠀⠐⠚⠘⠘⣤⡀⠀⠀⠄⢀⣀⠀⠀⠀⠀⢰⠶⢀⣠⡴⠄⠁⠀⠀⣍⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⡤⠆⠲⡄⠀⣌⡓⠀⠤⠔⠒⠒⠉⠉⠉⠀⠈⠉⠉⢉⠙⢒⠤⢤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠄⡀⠀⢀⠀⣀⣀⣷⠀⠁⠀⠀⠀⠀⠀⢀⡀⠄⠤⠶⢶⡟⠛⠁⢀⣠⣧⡬⢄⣀⠈⠁⠒⠆⠀⠀⠄⣀
⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀⠀⠀⢰⠀⠀⠀⠀⠨⠁⢀⡾⠀⠀⠀⠀⢀⠀⣈⣒⡤⢓⣂⠄⢐⡀⠄⢈⡀⠠⠥⠀⠠⢉⡉⠥⠄⠌⠐⣀⡠⠽⠧
⠀⠀⠀⠀⠀⡍⠀⠀⠐⠊⣳⢸⠀⠈⡀⠈⠀⠀⢈⢉⠉⢰⠒⣶⣌⣘⠳⡖⡭⠐⠚⠭⠤⣍⡉⠚⢯⠥⢞⡩⠤⣀⠈⠠⣄⣲⣤⠶⠍⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠀⠀⢀⣈⣁⣸⣄⣌⠂⠤⠤⠤⠤⠤⠆⠹⠒⠀⠀⢆⠻⡇⢈⠁⠀⠈⠁⠀⠈⠑⠒⠠⠔⠠⠀⠐⠀⠈⠁⠤⣐⠤⣐⡢⢀⣀⣵⠠⠔
⠀⠀⠀⠀⠈⠁⠒⠒⢺⠙⠉⠐⠒⢜⠂⠀⠀⠀⠀⢀⠁⢾⣣⠄⠀⠰⣒⡟⢛⠾⠛⠴⠖⠢⠾⢷⣶⣦⣴⣄⡀⠇⠴⠖⢒⣉⡁⠥⠔⠒⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠥⠬⠼⠦⠤⣀⣀⣀⣠⡴⠋⠉⠉⠋⠉⣕⠖⠅⠿⠁⠀⠀⢀⠀⠀⠈⠈⠉⠓⠑⠠⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠉⠀⠀⠈⡍⠁⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠐⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⢛⡯⠝⠋⠉⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠿⠁⠸⣆⣀⣴⣾⣓⠂⠀⠈⢻⣿⣿⢣⢶⡙⣿⣿⣿⣿⣟⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⢷⡄⠛⠿⢯⡼⠟⠁⠀⠀⢠⣿⣿⢸⣼⢃⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⣾⡁⠀⠀⡀⠀⠀⠀⠀⠀⣼⣿⣿⣾⣁⢾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⣁⣀⣤⣼⣷⣶⣦⣰⣿⣿⣿⣿⣿⡟⠁⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⡏⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣫⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣰⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⡿⠿⠻⠟
⠀⠀⠀⠀⢀⡀⠘⡳⢶⠀⠰⣀⣀⡴⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⢀⣴⣿⣿⡆⣀⡀
⠀⠀⠀⠀⡘⣻⡖⠛⡟⣶⡀⠈⢹⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠙⣿⠷⠶⠟⣿⠅⠀⡿⠀⠀⠀⠀⠀⠈⠿⢿⠟⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠙⢷⠒⢻⣯⣤⡴⢡⣤⡮⠀⠀⡀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣷⣦⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⠀⠀⠀⠐⠿⣟⣛⠁⠀⠀⠀⣤⣾⣿⣷⣿⡿⠿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣀⠤⠀⠀⠈⢿⣿⣧⠀⠀⠀⠸⠏⠉⠈⠋⠙⢿⠿⠟⠿⠻⠎⠻⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠋⠁⠀⠀⠀⠀⠀⠛⠛⠧
⠀⠀⠀⠀⠀⠀⠈⠉

⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠛⠷⠄⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣴⠞⠋⠉⠁⠀⠀⠀⠉⠉⠛⠷⣦⡀⠀⠀⠀⠾⠛⠁⠀⠀⠀⠀⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣠⡶⠛⠋⠉⠛⠛⢶⣤⡀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⣸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠛⠷⣤⣄⣀⣀⣀⣀⣈⣷⣴⠾⠋⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣿⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⣿⠀⠈⢻⣄⠀⠀⢀⣠⡶⢶⣄⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⣿⠀⠀⠀⢀⣀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠏⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⣿⠃
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣰⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⣦⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⠶⠛⠁⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⣿⡀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠸⣧⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⠀⢀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⠶⣦⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡤⠀⠠⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⠈⠀⠀⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠤⠄⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢡⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣠⠄⠀⠀⣀⣠⣠⣤⡶⠊⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⣴⣤⣤⣄⣤⣶⣶⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠉⠉⠉⠉⠉⠭⠭⠩⠭⠍⠍⣉⠉⣉⠐⣯⣮⣉⣙⣿⣾⠆⢉⠻⣎⠓⡻⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠊⠉⣉⣉⣉⣩⣥⣶⣤⠌⠙⠿⠛⠛⠿⠒⠈⠀⠈⠉⠉
⠀⠀⠀⠑⠒⠒⠒⠀⠀⠀⠸⠍⢀⣀⣈⡉⠉⠉⠁⠉⠙⠉⠀⠉⠀⠀⠐⠒⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠀⠹⠒⠉⠉⠑⠔⠒⡲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠚⠒⠒⠒⠲⢄⡀⠀⢀⡠⢘⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡞⠉⠉⠉⠓⣆⡟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠼⡤⠀⠀⠀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢼⣤⢀⣀⠀⠀⣗⣒⡄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣾⣿⣛⣻⠷⣾⣛⣛⣭⣿⣷⡾⡧⣤⣴⢾⣛⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⠀⠀⢘⡇⠉⠉⠛⠻⠿⣿⠿⠛⠉⠁⣿⠻⠷⢿⠍⠛⠋⣉⣸⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣶⣾⠿⠿⠛⢻⡇⠷⢀⡀⢠⠄⢸⠀⡀⠀⠀⣹⣤⣤⣼⣶⠟⣛⡥⠖⠛⡇⠀⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢈⣽⣿⣷⢶⣿⣷⣾⣷⣷⣌⣁⣼⡦⠶⢚⣯⠿⠁⣁⡤⠞⠋⠁⢀⠰⠆⡿⢹⡽⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⣿⣮⣯⠐⡆⡭⢙⡛⣿⠉⣻⠾⠗⣏⣡⡷⠞⠋⠁⢀⠄⠠⠅⠈⣀⣴⠗⣫⢶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣧⢼⠿⡟⠷⣷⣬⡄⣿⣁⣴⣶⠞⠛⢉⠀⡆⣂⠀⠈⢁⣠⣶⡿⢋⡵⣊⠁⣘⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⣤⠴⡞⠛⡇⣿⠴⠛⣧⣌⣠⠋⡛⡟⣿⠁⢸⠀⠘⠘⠀⢏⣸⠴⣚⡽⠛⣡⠞⡉⠂⢐⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⣉⠑⣢⠴⠛⢳⡿⠔⠊⣀⠤⠈⣙⠷⣾⢻⣄⣸⡇⢀⡤⠖⣫⠴⠚⣁⡴⠛⠑⡠⢊⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⡁⠂⣠⠴⠋⣁⠤⠂⢁⠠⠒⢉⠠⠐⡉⠄⣂⠽⣸⡏⠝⣛⡥⠖⠋⢀⡤⠚⡁⠀⠀⢊⡴⠋⡠⠃
⠀⠀⠀⠀⢤⢀⠀⣀⡴⠚⢁⠁⠂⣤⠴⠋⡡⠔⠈⣀⠄⠊⡡⠔⠊⠠⠀⢀⣔⣚⣧⠴⠟⠛⠉⠁⣀⡤⠞⢉⠀⢀⠔⢁⡴⠿⠖⠋
⠀⠀⠀⠀⠘⣧⣞⣁⣈⣒⣢⣔⣋⣀⣔⣈⣀⣤⣊⣠⣴⣥⡤⠴⠶⠖⠚⠋⠉⠁⠀⠀⢀⣠⠴⠚⠉⠰⠀⡠⢀⣤⠞⡛⠠
⠀⠀⠀⠀⠀⠙⢢⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠛⠉⠀⠒⠀⠀⡠⢊⣠⢟⡹⠊
⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠶⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⡠⠐⢁⡴⠿⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠈⠉⠉⠉⠀⠀⡔⠉⢻⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⣠⣴⠏⢔⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⡀⠈⢧⣶⣾⠎⠀⠀⠀⠀⠀⠐⠈⠠⠔⣲⡞⠑⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡶⣼⣿⡷⠶⠬⣄⣀⡠⣴⣬⣴⢊⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⡿⠿⡒⠾⠛⠙⠋⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⢔⡲⠈⠐⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⣥⣾⡷⠛⠁⠀⠀⠀⠀⣠⣭⡒⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠙⠲⢿⡛⠻⢥⡀⠀⠀⠀⢀⣤⣮⡥⢋⡴⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢐⣠⣭⣿⣶⣤⣀⠀⠀⠀⠈⠙⠲⢽⣦⣴⣾⠿⣛⣥⣺⢵⣚⣭⣶⡆
⠀⠀⠀⠀⠀⠀⣀⣠⢤⢲⣻⠿⠟⠋⠉⠘⢦⡉⠛⢦⣄⡀⠀⠀⠀⠈⣉⢲⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠈⠐⠠⢉⡛⠋⠁⠀⠀⠀⠀⢀⣠⣤⣿⣷⡶⣞⣻⣷⡆⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⢄⡀⠀⠀⣊⠕⡢⢄⡠⢶⣯⡥⢞⣛⣭⣶⣿⣿⣿⣽⡇⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⠛⢷⣶⣍⡂⠀⠉⠓⢶⣾⣿⣿⣿⣿⣿⡿⢿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠑⢤⡀⠨⡛⢿⣷⣤⣀⢸⣿⡿⢿⣻⣭⣷⣿⡿⣟⣯⣷⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠈⢹⣾⣭⣶⣿⣿⠿⣛⣭⣷⣿⡿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠢⣀⠈⡳⢦⣸⡿⢟⣛⣭⢶⣻⣿⣿⣿⡿⣟⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⡂⠙⠿⣟⣭⣾⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠑⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢰⠀⠀⠀⠀⠀⠀⠀⠱⣄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⡀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢠⠃⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⡀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠁⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣀⡀⠠⡯⠣⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⠀⠀⠀⢸⠀⠀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣼⣏⣩⣖⢦⣤⣄⣀⣀⠀⢈⡇⠀⠀⠁⡀⢀⣆⣩⣤⡶⠃
⠀⠀⠀⢀⣀⣊⣛⣿⣷⣴⣶⣶⣦⣼⣿⣿⣿⣿⣓⡻⠶⠾⠶⠾⢒⡚⢳⣿⣿⣭⣉⣙⠝⢁
⠀⠒⠀⠀⠿⡉⠹⠿⠶⠟⡉⠽⢯⣭⠋⡐⠿⠿⠧⠽⠿⠟⢀⣽⣿⡋⠙⣛⡂⠘⠛⠯⠐⠛⠀⠀⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠀⠖⠐⠛⠛⠧⠄⠀⠛⠂⠰⠦⠉⠉⠡⠄⠀⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣈⣶⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠇⠀⢏⡉⠛⠂⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠑⢄⠀⠀⣸⠀⠀⠈⢳⣄⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡳⠀⠀⠀⠑⢀⡟⠀⠀⠀⣸⠛⠣⡀⠀⡿⠀⠒⡶⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠄⡀⠀⣴⠛⠡⢤⠐⡄⠀⠀⠈⠓⡀⢀⡴⠣⠀⠀⠈⠲⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⣽⠁⢄⠀⠈⠙⢃⢠⣤⣴⣟⠋⠶⠖⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣾⠀⠀⠙⠢⡀⠀⢩⣯⠻⣿⣿⣶⡆⠠⢑⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⣿⢠⠀⠀⠀⠈⠱⣾⢀⠷⠈⠻⣿⡧⠀⠀⢹⡦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠠⠐⢤⢆⡜⢤⡀⠀⠀⠀⠀⣬⣿⢸⡆⠀⠀⠑⢄⠀⣾⠁⠀⠐⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣀⣀⠀⠂⣿⠃⣼⡁⡀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠛⢿⠁⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠢⢄⡀⢰⡿⠀⠓⢈⣡⣗⣤⣿⢏⢬⡀⠀⢀⡀⠀⠀⠀⠀⠑⢄⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⢤⣀⣤⠀⢭⡼⠾⠑⠍⢉⢜⣷⠒⡄⣠⡄⠀⠀⠀⠀⠉⠢⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣄⡙⢷⣤⣿⡿⣟⡑⢶⡶⢾⣭⣀⠁⢡⠇⢢⡄⣰⠄⠀⡄⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢡⠀⠈⠙⠿⢾⣿⣄⠀⠉⠳⠶⣄⣻⡟⠈⠉⢢⡄⠁⠉⣿⠀⣼⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⠀⠀⠀⠀⠆⠙⣯⠀⠀⠀⠀⠀⠉⠳⠢⢄⣿⠀⠈⠐⣜⠃⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠀⣼⣷⣠⠀⠀⠀⠈⡎⣿⢀⡖⠁⠀⠀⠀⠀⠀⠀⠉⢲⢶⣀⠘⢗⠢⢽⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡘⠸⠎⣰⣻⣿⠹⢆⡆⢠⡞⢰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣿⣿⣁⠀⠈⠒⠠⣀⠀⢀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠔⠀⠨⢻⣿⣽⣏⡋⣶⢿⡟⡡⣿⣿⡇⠀⠘⡆⠈⡀⠀⠀⢀⠀⠠⣴⢀⡝⣿⢿⣆⡄⣳⡀⠀⠉⠛⢅⡀
⠀⠀⠀⠀⠀⠃⠐⠒⠒⠻⡿⣿⡿⢝⣿⡿⠁⣸⣿⣿⣇⣿⡀⣇⢸⣿⢠⠆⡈⠀⠀⣿⣸⠀⣾⣿⡟⠃⡧⢠⢀⣦⠀⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⢀⢀⣀⢳⡾⢀⣓⣈⣭⠥⣽⣯⣟⣻⢻⡏⠉⢱⠈⠋⠈⣀⣋⣠⣼⣽⣧⣼⣯⣭⣡⣤⡅⠐⣻⢱⠀⡸⠀⢀⡝
⠀⠀⠀⠀⠀⠀⡿⢿⢿⣿⣿⣽⣿⣄⣀⣠⣌⡛⡈⣝⡤⢶⣶⣿⠿⢛⣩⣭⣤⡭⠽⢿⡿⠿⣿⣿⣿⣿⡀⠀⠀⠀⠢⠰⣴⡿⠁
⠀⠀⠀⠀⠀⣸⠇⠰⣚⠞⠈⠙⢹⣿⣿⣿⣯⡤⡾⢿⡖⣋⠣⠞⠛⠙⠁⠠⠤⠤⠤⠤⣤⣷⣿⣾⠇⣿⣿⡄⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⢿⠀⢒⠃⠈⢥⣀⠒⠠⠽⠛⣽⣯⣿⣿⢿⠿⠂⠀⠀⠀⠀⠒⣀⣀⣀⣀⣒⣻⣿⢏⣾⡉⡽⠓⠀⠀⠂
⠀⠀⠀⠀⠀⢸⡄⠀⠉⣀⠒⠢⠄⠤⠄⢨⢏⣉⢺⠙⣦⣀⠈⠀⠀⠀⠀⠁⠤⠤⣤⣭⣿⡿⣵⡟⢣⠋
⠀⠀⠀⠀⠀⡸⣿⡍⢐⡀⠤⢍⣁⠐⠂⣚⢀⠈⣩⡾⢷⡤⠄⠒⣂⠀⠤⠐⢒⣠⣶⣾⢟⣾⢇⡘⠁
⠀⠀⠀⠀⠀⠷⠞⡛⠘⣿⣷⣒⣲⣒⣀⠩⣇⣀⡴⠾⠛⠓⠛⢟⣣⣴⣶⣭⠴⢶⣿⣟⣾⢁⣮⡄⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠉⠈⠉⠉⠙⠁⠈⠉⠀⠀⠀⠉⠉⠉⠉⠛⠛⠓⠂⠀⠙⠻⠿⠷⠾⠖⠋⠙⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠠⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠲⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠉⠑⠒⢤⡀
⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈⣖⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠲⢄⣀⠀⠀⢀⡠⠤⠤⠄⢀⣀⡔⠲⡎⠁⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡉⠓⢺⠀⠀⠀⠀⠀⡀⠷⢾⠛⢄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⢦⣀⡠⠔⠊⠀⠇⢠⠡⡀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡸⠀⠙⠒⠠⠼⠆⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠐⠒⠒⠒⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠄⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⠄⠀⠀⠀⠉⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡠⢄⠉⠐⠂⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠷⣭⣢⠀⠀⠀⠀⠀⠀⠀⢷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠲⠶⠣⣮⡶⠋⠀⠀⢀⣀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠷⣬⡷⠀⠀⠀⢹⢹⠄⡀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣨⠝⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠈⢆
⠀⠀⠀⠀⢀⡔⡀⠀⡠⢀⣽⠇⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣨⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢀⠀⡀⠀⠀⠀⠀⢠⡄⠱⡀
⠀⠀⠀⢀⠎⠰⠃⠐⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⣿⠃⠀⠀⠀⢠⣲⡇⠀⠱
⠀⠀⠀⠈⣀⠀⠀⠀⠀⢰⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⢇
⠀⠀⡜⣡⠞⠃⠀⠀⠀⠸⠾⠀⠀⠀⠀⠀⠀⠀⣠⣶⣽⣷⣿⣷⣶⣶⣴⣆⣴⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⡂⡄⠘⡄
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⢿⣅⡅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠈⠙⠃⠀⢠
⠀⢀⠇⠀⠀⠀⠀⢀⢼⣇⡀⢤⣤⡄⠀⠀⠀⠀⠀⠀⠙⠿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡹⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣿⠀⠀⠀⠀⠀⢀⣴⠀⡀
⠀⠘⠒⠀⢀⡀⠀⠸⠟⠘⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠛⠓⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠞⠛⠛⠀⠀⠀⠀⠠⣾⠇⡰⠁
⠀⢠⢸⡀⠛⣻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⠞⣹⡇
⠀⠀⠣⣵⣄⡈⢻⣿⣦⣄⠀⢠⠄⠠⢤⡀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣴⣒⣂⣰⠀⠀⠀⠀⠀⢸⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠏⢉⣵⣾⣿⠇
⠀⠀⠀⠟⠻⣿⣦⡩⣻⣿⠛⠦⣭⡓⠊⠀⠀⠀⣔⣸⠃⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡠⣺⣿⣿⣽⡛⠀⢀⣿⣿⣿⡿
⠀⠀⠀⢰⠀⠈⠙⢳⣬⣍⡀⠀⠀⣥⣤⣤⠄⢠⠈⣥⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣶⣦⣤⢛⣿⠟⠍⠁⠀⣠⣾⣿⣿⡟⠁
⠀⠀⠀⠀⠱⡄⠀⠀⠙⠿⣿⣦⡀⠀⠈⠋⠉⠀⠈⠙⠻⣿⣿⣿⣿⣭⣯⣿⣿⠿⣿⣿⡿⠿⠯⢝⣛⠙⠋⠉⠛⠉⠉⢁⣠⣴⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠈⠉⠛⠿⢿⣿⣿⣶⣶⣦⡀⠀⠘⠛⠻⠛⠛⠉⣀⣤⣤⣤⣶⣶⣾⣿⣿⣷⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣄⠀⠀⢀⣴⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⣀⣀⠀⠀⠀⠀⠉⠟⠱⠛⠛⠉⡀⣠⣬⣤⣤⣤⣿⣿⣯⣽⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠶⠀⠤⠤⠀⠤⣀⣽⣿⣵⣾⡷⠶⠭⠿⠿⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢲⣶⠀⢀⣀⣀⠀⢰⣶⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⠖⠉⢹⠴⠋⠀⠈⠳⢼⠉⠑⢦⡈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠎⠀⠀⠀⠸⡀⠀⠀⠀⠀⡸⠀⠀⠀⠘⢄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡠⠃⠀⠀⠀⠀⠀⠳⣦⣤⣤⡶⠁⠀⠀⠀⠀⠀⠣⡘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⠜⠁⡀⡀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⡀⠑⢌⡢⡀
⠀⠀⠀⠀⠀⢀⡠⢦⣾⠝⠁⠀⠀⠻⡀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠙⢾⣧⠤⣀
⠀⠀⠀⣠⢚⠥⠚⠁⠘⠂⠀⠀⠀⠀⠈⠀⣀⠤⠒⠉⠉⠉⠉⠉⠐⠢⢤⣄⡈⠀⠀⠀⠀⠀⠛⠁⠙⠢⢝⢢⡀
⠀⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠤⠀⢀⡴⡺⠃⢀⣀⣠⣭⣭⣭⣥⣀⣈⣹⠇⡍⢢⠀⠠⠄⠀⠀⠀⠀⠀⢀⡙⡇
⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⣠⡏⣠⣾⣛⣉⡥⣄⣀⣀⣤⣶⠿⣟⢿⣦⣇⣼⣧⡀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠙⢷⣶⣤⣠⡀⠸⣿⣿⢽⣿⣿⣿⣿⣿⣿⠋⠻⢥⣶⣿⣧⡉⢿⢿⣿⡿⠀⣠⣠⣴⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⠗⣻⣻⡽⣿⣿⣿⣿⠏⠄⠀⠈⠡⡍⢠⠀⣾⠈⣿⣿⣿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡞⢠⡾⢧⠿⣮⠙⠋⢁⠔⠀⠐⢄⠀⠀⢠⡾⢻⠀⢸⣿⡚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢾⡸⣷⡟⠀⢆⠀⡠⠶⠻⢦⠾⠻⠦⡀⢀⠆⡀⣷⣏⡇⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠙⢧⣤⡄⠺⠯⣴⣺⣧⢤⠤⣿⣲⣬⠿⠂⣼⣿⡉⢹⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢷⡀⠻⢋⢀⡝⠛⠛⢄⢉⡻⠃⣰⠏⣸⠃⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠳⣄⠋⢉⠀⡆⠀⢈⠉⢃⡴⢃⡾⠛⡤⠴⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢍⠀⠿⡄⠀⣄⡼⠇⢈⠵⠊⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⢿⣾⣿⠷⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡙⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠈⡉⠉⠉⠉⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠐⠂⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⡀⣀⣀⣀⣀⣈⣁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⡂⠉⠀⠀⠀⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⠤⣄⣀⣈⣠⣤⠤⠀⠌⠥⠴⣺⠁⠀⠀⠀⠀⢀⠀⠔⠁⠀⠀⠀⢀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠒⠀⠀⠀⠀⠐⠂⠀⠣⠁⠎⠀⠀⠀⡸⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠰⠤⣀⠀⠀⢀⡴⠁⠀⠀⠀⣀⠤⠊⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⡠⠖⠉⣀⡤⠒⠈⢁⡠⢀⠔⠀⠀⠉⠉⡁⠀⠀⠁⢉⠁⠀⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡴⠃⠀⡠⠊⠁⠀⢀⠔⠉⣠⠃⠀⠀⠀⠄⢠⠀⠸⠀⠀⠸⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⠞⡑⠀⣠⠊⢀⠂⠀⡰⠃⠀⡴⠁⠀⡜⠀⠀⠀⡎⢠⠃⡇⠀⢠⡙⡄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡔⠁⠀⠀⢀⠜⠁⡠⠊⠀⢰⢰⠁⠀⠀⡜⠀⡜⠀⣻⠀⢸⢡⠸⡄⠘⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠎⠀⢀⡠⠚⠁⠀⠀⡠⠨⠀⠸⢸⠀⠀⡜⠀⡜⠀⢈⠉⡆⢸⢸⠀⢣⠀⠸⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⠁⡰⠁⢀⡰⠋⡰⠁⠀⡇⡆⠀⢈⠀⡜⠀⠀⢸⠀⡇⠸⢸⠀⠸⡀⢠⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠜⣁⠔⠉⢀⠜⠁⢀⠜⢰⠁⠄⣡⠞⠀⡀⢀⣾⠀⡇⡇⢸⠀⠀⡇⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⢀⠔⢃⠠⠔⠁⣠⠃⡠⠊⠁⠀⡜⠀⡜⢸⢠⢡⠁⡌⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠋⡇⢰⠁⠀⠀⠌⠀⡼⠀⢸⠀⠆⢠⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠃⠀⠐⠀⠘⠀⠀⡆⠈⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⡀⠀⠀⠀⠀⠰⠁⢀⠊



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢦⠀⠀⢀⣀⣀⡀⠤⠤⠤⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠓⠈⠉⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠤⠄⠐⠒⠒⠒⠀⠠⠴⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠤⠤⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠓⠢⢄⡀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⣀⣠⡤⡇⠀⠐⢆⠀⠀⠐⢢⠎⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⡀⠀⠀⠀⠐⠀⢀⠀⠀⠀⢺⣤⣄⣀
⠀⢀⣤⠖⠛⠉⠀⠀⠐⠄⠀⠀⠁⠀⠀⢸⡄⠀⠁⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠄⠀⠉⠀⠀⠀⠁⠀⠀⠀⠟⢆⢠⠀⠀⠔⠀⠈⠀⠁⢠⠀⠀⠉⠛⠲⣤⡀
⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⢾⠀⢄⡠⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠋⠠⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢻
⠻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⠢⠀⠠⠠⠎⠀⠀⢀⠀⢀⠈⠀⠈⠀⠁⠀⠀⡰⠣⠤⠤⠤⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠈⠛⠷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠾⠛⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠶⢶⣤⣤⣤⣤⣀⣀⣀⣉⡒⠢⠤⠤⠤⠤⠤⠤⠤⠀⠒⠒⠊⠉⠁⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣴⡶⠶⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁




⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡚⠛⠉⠉⠉⠉⠒⠤⣔⣲⠶⣤⡀⠀⠀⠀⠀⠀⢀⣤⠞⣉⣉⣽⡶⠆
⠀⠀⠀⠀⠙⠛⠿⣯⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣽⣗⡢⢄⣀⠀⠠⢀⡀⠈⠉⠓⡓⠤⢤⣀⣴⣾⣯⣷⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⣿⠒⠦⣌⡙⠓⠦⣄⣀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣈⣉⣲⣤⣤⠝⠒⠋⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆
⠀⠀⠀⠀⠀⢸⡷⠦⣄⡉⠙⠲⢤⣉⡛⢒⣛⣛⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢰⡀⣻⡿⠿⠿⠿⠿⠯⠅⠀⠐⠒
⠀⠀⠀⠀⠀⠈⣗⠀⠀⠀⠀⠀⠀⠀⠉⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣻⣷⣤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠠⣟⡒⠤⣀⠀⠀⠀⠀⣹⣟⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠸⢍⡓⠦⢍⣀⠒⠠⠄⠙⣩⠵⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⠀⠀⠀⠈⠁⠀⠉⠓⠢⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⣍⣙⠒⠒⠖⠋⣱⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⠤⠐⠒⠛⢣⣄⡀⠀⠈⠐⠛⠷⠆⠀⠀⠀⣀⣠⣤⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠚⠃⠀⠀⠀⠀⠀⠀⠀⢟⠂⠤⣀⠀⠀⠀⠀⠀⠈⠙⢋⣉⣐⢤⡶⠒⠋⠀⣻⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢤⣤⣤⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⠀⠉⠒⠲⠶⣾⣿⣿⣿⣿⣿⣽⡇⢀⣀⣤⣿⣿⣯⣴⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠈⣉⡒⠾⣿⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠤⣀⠀⠈⢿⣿⣿⣿⠿⠟⢛⣳⣯⠛⠉⢁⣼⣿⣯⡽
⠀⠀⠀⠀⠀⢈⡀⠠⢤⣴⣿⠢⠬⣭⣿⣿⣿⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⡠⠿⢤⣀⡈⠀⠒⠛⠉⢀⣴⣾⣿⣿⡿⠒⠚⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠬⢭⣛⣿⣊⣁⡒⠺⣿⡿⣿⡀⠤⢤⣴⣿⡆⠀⠀⠀⠀⢸⡏⠑⠒⠤⢍⣀⡐⠀⠀⢼⣻⣿⣿⣿⡷
⠀⠀⠀⠀⠀⢀⣑⠒⠶⣿⡟⠢⢬⣽⣿⣿⣿⠌⣙⣻⣿⣛⣧⠄⠀⠀⠀⠈⠣⠄⣀⠀⠀⠀⠉⠉⣒⣭⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠬⢉⣛⣿⣛⡁⠒⠲⢿⣯⣿⡓⠲⢾⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠒⠶⠚⠁
⠀⠀⠀⠀⠀⠐⠓⠶⢾⣿⣧⠬⢐⣒⣶⣿⣿⡅⠙⠛⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠲⢾⣿⣿⠿






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠿⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠈⣿⠈⠲⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⢒⣛⡓⣒⠒⠒⠛⠒⠒⠶⠶⠾⠶⠶⠶⠶⠶⡀⠔⠊⠀⣷⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣙⣿⣿⣿⣿⣿⣿⣿⡟⠿⡷⠶⣶⣾⣷⣶⣶⣶⣶⠿⠶⠶⠶⠼⠶⠶⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⠿⠿⠿⠿⡿⠿⠿⣿⡿⢋⣴⣿⣿⣿⡿⠿⠿⠛⢻⡿⠿⠿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠒⣒⣚⣛⣛⣒⣒⣒⣒⣒⣻⣟⠒⠒⢲⣶⠿⠛⠛⠛⠒⠒⠒⠒⠀⣼⡇⠀⠀⠀⠈⠙⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⣿⣿⣿⣿⣿⣛⣛⣛⣛⣛⢛⣛⣓⣒⣒⣒⣒⣒⣒⣲⠈⠓⢲⡶⡶⠶⠶⠶⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠈⠙⠿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠙⣿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠉⠛⢛⡿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀⣤⠶⣛⢅⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡏⠀⠀⠀⣠⠾⣫⣵⡿⢣⣾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣻⡇⠀⠀⠈⠛⢛⣻⣿⣿⣿⣿⣿⡇⣿⣿⣿⡟⠀⢀⣴⢟⣵⣾⣿⡟⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠯⢿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⠟⢀⡴⢋⣵⣿⣿⣿⠏⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⢋⡴⣋⣴⠿⠿⠿⠿⠋⢰⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⢳⣶⠀⠀⢀⣛⣻⣿⣿⣿⣿⣿⣿⡿⢡⣟⡵⢋⣡⠥⢶⣒⡚⣛⠃⠀⠀⣹⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⣽⡇⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⣠⠿⢛⣫⣵⣶⡿⠟⣋⣱⠃⢠⣆⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣛⣿⣥⠾⠟⠋⠉⠁⢸⡟⢀⡤⢖⣫⣵⣾⡿⣟⣫⣵⣾⣿⢋⡟⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣾⡟⠀⠀⠀⠀⠀⢀⡿⣛⣥⣾⣿⣿⣿⣷⣿⣿⣿⣿⣶⡟⣾⠁⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣭⠿⠧⠤⠤⠤⠤⠤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣹⠇⠀⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣋⣭⣴⣶⣿⠛⣛⡛⢛⣟⣳⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳⠏⠀⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣦⣿⣧⣼⢟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢣⠏⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⠏⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡽⠁⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠓⠋⠠⠴⠿⠿⠟⠛⠛⠛⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠐⠒⠺⠿⠿⠿⠿⠷⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠩⠽⠿⠿⠿⠿⠿⣿⣿⣶⣶⣶⣶⣶⣶⣾⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⡴⠶⠶⠶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣛⣉⣭⣭⣤⠤⠤⢤⣤⣤⠀⠀⠀⢤⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠕⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⡩⠥⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⠶⠒⠉⠀⠀⢠⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠉⠙⢏⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⠶⠶⠶⠶⠶⢶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡥⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⡤⠸⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⠶⠞⠛⠋⠉
⠀⠀⠀⠙⠛⠲⠶⠦⣤⣤⣤⣤⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠒⠛⠛⠛⠛⠉⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣧⣴⡖⢤⣀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⣦⡀⠀⠀⠀⢸⡆⠀⠀⠸⣦⣄⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡷⠘⣏⠑⢄⡀⢸⡇⠀⠀⠀⣿⡿⠁⠢⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠘⠀⠘⡀⠀⠙⢾⣧⠀⠀⠀⡏⠁⠀⠀⠈⢂⣄⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⡏⠀⠀⢀⠄⠀⠀⠀⠀⠀⢉⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡆⠀⢸⠀⠀⠀⠀⠀⠀⠋⢹⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢼⡇⢀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢹⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⣾⢷⢸⠀⠆⠀⠀⠀⠀⠀⠀⠀⠐⡿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠏⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠃⠀⠀⠀⠀⠀⠁⢈⣷⣶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢻⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⡛⠠⠁⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠈⠢⣿⢿⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣼⠀⢣⠀⠀⠀⠀⠀⠀⠈⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⠀⠀⠀⠒⠻⣿⠟⣋⠴⣶⠶⠦⠄⠀⠀⠀⠀⠀⠾⢶⡁⠀⠀⠀⠐⠂⢀⣠⣶⡟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⣟⣀⠠⠘⣁⡀⣀⣀⣀⠤⣈⣿⣿⡿⢿⣿⠀⠒⠒⠛⠀⠀⠀⠀⠿⠿⠿⠶⢀⣠⣴⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠐⠛⠐⠊⠤⠤⠤⠄⠀⠀⠀⠀⠉⣻⣻⡿⣾⣿⡆⠀⠀⠀⠀⠀⢠⡤⣟⣭⣴⣾⣿⣿⣿⣿⣿⣿⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡃⠀⣸⣿⠏⢀⣀⣤⣶⣾⣿⡿⢿⣿⠿⠛⠃⠀⠈⠙⠃⡠⠎
⣀⣀⣀⣀⠀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣤⣼⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣬⣭⣶⡿⠛⠀⣴⡆⠀⣺⣿⠂⢀⣄
⣿⣿⣿⣿⣿⣖⠀⠑⠪⠄⡀⢀⣤⣀⠀⠛⢿⣿⣿⣶⣶⣶⣿⣿⣿⣿⠿⣟⣻⣿⣭⠉⠁⠉⠉⣴⣶⣾⣿⣿⡇⢸⣿⡿⠗⢸⣿⡷⣄⣀⡀⠀⢀⡠⠀⡀
⠛⠛⠛⠳⠆⡀⠀⠀⠀⢰⣬⣀⠉⠙⠛⢶⣶⣭⣭⣭⣭⣭⣿⣿⣭⣤⣼⣿⣿⣿⣿⣷⣦⣿⢿⣿⣿⣿⣿⣿⡟⠘⢿⠀⠀⢾⣿⣿⣏⠻⢿⣿⣷⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣤⣄⠈⠉⠻⠿⠟⠻⣯⣝⣛⡛⠛⠋⠉⠀⠀⠈⠉⠙⠷⠀⠉⠉⠉⠀⠀⠀⠀⠀⠆⠸⣿⣿⣛⠂⠀⢨⣝⣛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣯⣄⡀⠀⠙⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⡶⠶⠿⠟⠛⠛⠻⠿⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣠⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⢿⣿⣿⣶⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣄⡀
⠀⣠⣾⠟⠁⠀⠀⠀⢠⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡇⠈⠙⠻⢷⣤⡀
⣰⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠻⢷⣄
⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠶⠶⣶⣶⣶⣤⣤⣤⣤⣤⣶⣶⣶⡶⠶⠶⠿⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡶⠶⠿⠿⠟⠛⠛⠿⠿⢿⣷⣶⣤⣤⡴⠶⠶⠿⠿⢿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⣛⣩⣥⣶⣶⣾⣿⠿⢿⣿⣿⣿⣿⣶⣌⠉⣥⣴⣶⣶⣶⣶⣶⡆⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢟⣡⣶⣿⣿⣿⣿⣿⣿⠋⠀⠀⢀⠙⣿⠟⢻⣿⣿⣆⢻⣿⣿⣿⣿⣿⡿⠃⣼⣿
⢀⣤⣴⣶⣤⣤⣀⡀⠀⠀⣠⣶⡿⢋⣴⣿⣿⣿⣿⣿⣿⡟⢻⣿⣄⠿⢠⠛⢱⣟⣥⣶⣿⣿⣿⣆⢻⣿⣿⣿⠟⣡⣾⣿⠃
⣿⡋⢡⣤⣬⣭⣝⡻⢷⣾⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣬⣽⣓⡲⠾⠫⢾⣿⣿⣿⣿⣿⣿⣦⣙⣉⣴⣾⣿⡿⠁
⢻⣧⠘⣿⣿⣿⣿⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⣵⣾⣿⣷⣦⣠⣾⣿⣿⣿⣿⣿⠿⣿⣿⡟⠁
⠀⢿⣧⡈⠻⣿⣿⠟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠈⢿⣿⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡛⠉⠩⠴⠚⣋⣩⣵⡶⢿⣟⡉⠉⠛⠻⢿⣭⣙⡿⠁
⠀⠀⠈⢿⣿⣿⣿⠿⢟⡿⣿⣿⣿⣿⣿⣿⣿⠷⠟⠋⠁⣀⣤⣶⠿⢻⡿⠟⠻⢷⣾⣿⠷⣦⣄⣴⣿⣿
⠀⠀⠀⠈⢻⣿⣿⣯⣥⣚⡿⣿⣿⣿⣿⠟⠻⣦⣶⣾⣿⣏⣡⠀⢠⣬⡀⣠⣭⣽⣿⣍⠀⠈⢽⣿⣶⣷
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣾⣿⣿⣿⡇⣾⡻⣮⢻⣿⣿⡿⠋⠀⠉⠚⠉⠙⠛⠛⣻⠿⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⢟⣿⣿⠸⣴⡇⠀⢿⣿⡇⢦⠀⠀⠀⠀⠀⠒⠋⠁⣠⡖⠓⠀⠈⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠵⣻⣹⣿⣷⣽⠷⣄⠈⠉⠀⠈⢷⡀⠀⠀⠀⢲⣄⣀⣉⣷⣶⡒⣦⣾⣁
⠀⠀⠀⠀⠀⠀⠀⠺⣯⣀⡀⠀⣱⣿⢿⡿⡿⢰⣿⣇⠀⠀⠀⠈⣧⠀⡞⢰⠆⣽⡿⠿⠿⣿⣥⠟⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣅⣠⣿⣇⠿⠞⣡⡿⢻⣿⣄⠀⠀⠀⠀⠀⣇⢸⠀⠠⣝⠲⠶⠶⠿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠁⠙⠿⠿⠟⠁⠀⠻⢿⣦⣀⠀⢠⣀⠘⠈⠀⠀⠀⠙⠲⠶⠞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣿⣦⣄⠀⠀⠀⠀⠀⢀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⣴⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠀⠀⠀⠀⢀⣀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⡿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡇⠀⠀⠀⠐⠃⠀⢠⠀⠀⠒⠀⠀⠃⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠀⠀⠀⠀⢀⡠⠖⠋⢥⣄⢰⣴⡀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠐⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠊⠀⢀⡴⣫⣤⣄⣀⣸⠏⣼⣿⡇⠀⢀⢸⡇⠀⠀⠀⠀⠐⠄⠀⠀⢿⣄⠀⠀⠶⠀⠀⠀⡀
⠀⠀⠀⠀⣰⡆⣠⠆⠀⠀⠀⠀⠀⢀⣤⣤⣿⣸⣿⣿⣿⣿⣷⣿⣿⣿⣷⣀⠞⠛⠁⠀⠀⠀⠀⠀⠀⠟⢆⠘⢿⣦⣄⣠⡀
⠀⠀⠀⣸⣿⣼⠏⠀⡰⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣄⣠⣤⡀⠂⠀⠀⠀⠀⠀⢤⡕⡌⠻⣿⣿⣿⠀⠀⠒
⠀⠀⠀⣿⣷⡀⠀⠀⢣⡀⣀⣀⡠⠔⠉⠀⠀⠀⠉⠁⠉⠉⢀⡰⠾⠊⠉⠁⣩⢻⠿⣿⣦⡀⢶⡄⠀⠐⣦⢲⣧⠀⠈⠙⡋⠀⠀⡀
⠀⠀⠀⢻⣿⣿⣖⠆⠀⠈⠉⠁⠀⠀⠀⠀⠀⡇⠀⠀⠠⠔⠁⠀⠀⠀⢀⠈⠁⠀⠀⣹⣿⣿⣄⠈⠀⡈⢩⣷⣿⠀⠀⠀⠀⠀⠀⠁⠀⠈
⠀⠀⠀⡈⢙⠿⣿⣿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦⣌⣯⣶⣏⠁⠀⠀⠚⠀⠀⠀⢀
⠀⠀⠀⢠⡀⠙⠪⠛⣿⣿⣿⣾⣄⣠⣠⣀⠀⠀⠀⠀⢀⣀⣀⣄⣠⣄⣤⡴⠾⠞⠛⠋⠡⣿⡿⣧⣨⣿⣿⠙⣧⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠙⢦⡀⠀⠉⠂⠉⠛⠛⢛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⣿⣿⣿⠃⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⣄⣀⡀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠚⠋⠁⠀⠈⠛⢉⠀⠀⠀⠀⡀⠙⠒⠒⠒⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠲⠶⠶⠶⠦⠴⠶⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⣀⡉⠀⠐⠊

⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣤⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠛⠛⠇⠈⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⣀⣤⣤⣴⢤⡄⣰⣠⣶⣠⣞⣹⣿⣡⣤⣀⠐⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣇⣠⣶⣿⣿⠟⣋⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣟⣰⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠟⣾⣿⣿⣿⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡚⠻⣿⡯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠏⣼⣿⣿⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⠿⣿⣿⣿⣿⣿⣯⡿⡍⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠏⣼⣿⣿⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣥⣦⣽⣷⣾⣿⣿⣿⣿⣿⣿⣛⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢸⡿⢟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡖⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣿⣶⣷⣿⣿⣿⠿⠟⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⣀⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⡤⠀⠀⠊⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⡷⣿⣿⣿⣿⣿⡿⠻⢶⣶⣿⣿⣿⣿⣦⡘⣧⠀⡀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⣿⣿⣿⣿⣶⡄⠘⠛⠛⢿⣿⣿⣿⠋⣿⣧⣼⣄⠀⠀⠌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠒⠒⢛⢡⣿⣿⡇⠀⠀⠒⠈⠹⠟⠃⠘⠿⠟⠋⢙⣫⢄⡰⠂⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣀⠛⠻⠿⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⡀⠀⠀⠀⠰⠿⣿⠿⢶⣤⠲⠞⣰⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⡉⠉⠛⠛⠓⠐⠙⠋⢡⠀⢠⡟⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠈⠀⠒⠐⠂⠀⣠⠔⠛⠁⠀⢠⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣤⣤⡀⡄⢤⡄⠉⠀⠀⠀⠀⢀⣾⠟⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠉⠀⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣀⣀⣠⣤⠴⠒⠉⠉⠒⡠⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣴⠎⣟⣣⡽⠛⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣯⣿⡟⠀⠀⠀⠀⠈⣵⢀⣄⠀⢀⣤⣤⣴⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⣿⣿⣿⣿⣿⣿⠟⣵⠏⠀⠀⠀⠀⠀⣠⣿⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢡⣿⣿⢿⣿⣿⣿⠯⢁⡾⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⣿⢿⣿⣿⣿⣿⣿⣡⣼⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠛⣻⡿⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⣼⢫⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢻⢟⣿⡟⣸⢣⠟⢰⣿⢻⡟⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⠃⠃⠈⠉⠠⠃⠏⠀⠉⠁⠜⠰⠁⡟⣿⣿⡟⢹⡿⡇⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠃⠸⠁⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢌⣆⠤⢆⡄⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠰⠀⠀⠀⠌⠢⠈⠢⡈⠂⠂⠚⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡵⡇⠀⠄⣀⠂⢄⠢⠀⢈⠀⣀⠠⠄⠤⢱⣄⣣⡀
⠚⠁⠀⠀⠀⠉⠒⠤⣀⠀⠀⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠳⢄⡀
⠀⠀⠀⠀⠓⠤⢄⡀⠈⠙⠢⢤⣀⠀⠀⠀⠀⠘⣠⠴⠊⠁⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠸⢄⠀⠀⠀⠀⠀⠁⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⣁⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠿⠘⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⡠⠖⠀⡠⠒⠀⠀⣏⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡠⢹⠀⠀⠀⠀⠀⠨⠭⠄⡠⠾⢑⠟
⠀⠀⠀⠀⠀⠀⢸⠤⠒⡉⡀⢀⠌⠀⠀⠀⠀⢣⠄⠀⡀⠀⠄⠈⠙⢁⡘⠤⠐⠈⠁⠀⠀⠑⢌⠠⠀⠐⠄⡴⠊⣠⠔⠁
⠀⠀⠀⠀⠀⠀⣁⡠⢊⠊⢠⠊⠀⠀⠀⠀⠀⢹⡒⠁⢀⣀⣀⠴⠈⣡⢀⠄⠠⠠⠴⠄⢀⣀⢀⡉⠀⡄⠐⠒⠉
⠀⠀⠀⠀⠀⠀⡝⡰⢁⡴⠁⠀⠀⠀⠀⠀⠐⠓⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⣠⣟⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠹⡈⠀⢀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡴⠂⠀⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠊⠀⠀⠀⠀⡠⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣶⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠛⢾⠗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠿⣿⢤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡖⢻⡆⠠⠈
⠀⠀⠀⠀⠀⠀⠙⢗⠒⠒⠶⠼⣧⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠁⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠉⠁⣿⡿
⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣴⣿⣛⣻⣿⡿⣿⡾⣦⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⡵⢌⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠒⠲⠤⠶⣼⠇
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡀
⠀⠀⠀⠀⠀⠰⠧⠐⠒⠲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡠⢐⣃⣹⣿⣠⡤⠬⣽⡿⠶⠖⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠉⢀⣿⣄⣠⣴⣿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣻⡷⣧⣙⠤⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⠾⢿⣛⣿⣿⣿⣿⣍⡉⣴⣜⣯⡾⠞⠓⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⠀⢻⣿⣟⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣿⠛⠻⣿⣍⢙⡿⢿⣿⡇⢠⣿⡿⣼⢀
⠀⠀⠀⠀⠀⠀⣨⠿⣿⣿⣿⣯⣮⡀⠛⢛⣤⢀⠈⠛⠲⢿⡟⠓⢿⣇⠳⣄⡀
⠀⠀⠀⠀⢠⣶⣿⠆⠠⠀⠘⠉⠉⠉⠒⠀⠀⠈⠠⠂⠀⠠⠄⠀⢘⠻⠿⢮⣉⠁⢤⣴⣶⣤⡤
⠀⠀⠀⠚⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣤⣽⣄⣈⠙⠳⠦⠀⠀⢀⣠⣤⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠛⠋⠉⠐⠀⠈⠙⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣧⣸⣯⣝⣿⣭⣉⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⣀⣴⠶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⢀⣴⠟⠋⠀⠀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡟⠷⣦⣄
⠀⠀⣰⠟⠁⠀⠀⠀⠀⡇⠀⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠊⠁⠀⡇⠀⠀⠉⠻⣦⣄
⠀⢠⡿⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠒⠒⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠒⠚⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿
⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⢴⡶⢫⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣭⡍⢩⡍⡉⠉⡿⠉⡍⢩⣷⣿⣿⣿⠿⡟⢸⣿⠟⣥⣯⣭⣭⣿⣿⣿⣿⣿⣿⣶⠸⣧⠻⠴⠛⠚⢻⢀⣿⡿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣣⠾⠛⠓⠈⠁⢀⠈⠉⠁⠈⠉⣉⠝⣡⡖⢁⡇⡟⣵⠋⠀⢋⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠛⠓⠀⢸⢸⢿⡵⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠜⠁⠀⠀⢀⣤⠠⢄⠀⠀⠀⠀⡜⣠⣾⣿⢃⡼⢱⡿⣿⠀⠀⢸⠀⠀⠀⠀⣄⣠⣤⣠⢤⠀⠈⠀⠀⠀⠀⡜⡜⣼⣏⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢎⡴⠁⠠⠄⠀⠀⠸⣌⣉⠴⠀⠀⠀⢸⣵⣿⣟⣡⢃⡆⣼⣼⣿⡄⠀⡼⠀⠀⠀⣸⢻⣿⠿⢯⠏⠀⠀⠀⠀⠀⡰⢱⠡⣽⣾⣏⡝
⠀⠀⠀⠀⠀⠀⡠⢣⠎⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠘⡻⣿⡿⢁⢾⡇⣿⣾⡿⠁⡴⠁⠀⠀⣆⠀⢀⠄⣀⡀⣤⣤⣤⣴⡖⣰⢣⢣⡜⣽⣿⡿⠁
⠀⠀⠀⠀⠀⡔⣱⣿⣶⣶⣶⡖⠰⡆⣴⣠⣞⣪⠉⡥⣤⣤⣤⣿⡟⢠⢋⣾⡇⣿⣧⣶⡧⢤⣤⢤⣤⢀⣤⣤⠄⠤⢤⡥⣭⠵⠒⠺⢃⣯⢻⣰⣿⡿⠁
⠀⠀⠀⠀⢾⢤⠴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⣤⣤⣭⣉⣉⣉⣹⣡⣏⣤⣼⣄⣯⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣭⣭⣭⡴⠶⠒⣒⣛⡿⣱⣷⣿⡿⠁
⠀⠀⠀⠀⢘⡿⣿⡿⢗⣒⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡷⠤⢤⣬⣿⣿⣋⠈⠀⡐⠒⠲⠶⠶⠶⢴⣶⣿⣽⣿⠃
⠀⠀⠀⠀⠸⣯⣭⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣽⢿⣯⠿⠿⠿⠿⠿⠿⠦⠶⠶⠿⢿⣾⣿⣿⣿⣿⣿⢿⣻⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠎
⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣻⠋
⣀⣤⢤⣶⣾⡿⠿⢻⡿⢋⣭⠉⠁⣀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠛⠛⠻⠿⣶⣦⣤⣤⣤⣴⣿⣿⡽⠃
⠻⣿⠀⠫⢥⣀⠒⠛⠳⣿⣿⡀⠀⠀⠀⠌⠉⠁⠃⠀⠁⠀⠁⠁⠉⠉⠁⠀⠄⠀⠠⠀⠤⠰⠨⠭⠭⠭⣼⣿⣏
⠀⠈⠓⠦⣠⠒⠒⠾⡶⠠⠒⠂⠀⠠⡄⠀⡀⠀⢂⠉⢀⣀⣀⡄⠀⣀⣀⠀⠀⠈⠀⢀⣀⡤⣶⠟⠛⠓⢾⣿⣿⣦
⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣴⣶⣶⣤⣤⣤⣤⣤⣄⣤⣤⣤⠤⠤⠤⠤⠤⠤⣶⣖⣚⣩⣵⠞⠁⠀⠀⠀⠀⠙⢮⣋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡏⠀⠀⠀⠀⠀⠈⠻⢿⠀⠀⠀⠀⠀⠀⠀⠙⠾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⡶⠖⠋⠉⠉⠙⠒⠶⣶⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠁⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠾⠗⠒⠒⠚⠉⢁⣀⣠⠤⣄⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠉⠛⠀⠂⠒⠚⠛⠦⣄⡀
⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⣴⠋⠉⣁⣠⡤⠞⠀⠀⠀⡇⢈⡇⠀⠀⠀⠀⢀⣠⠖⠓⡆⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⢰⡇⢰⡇⠀⠀⢀⠔⠋⢀⡴⠊⠁⠀⢀⠀⠀⠀⢿⢦⡀
⠀⠀⢀⡞⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣸⠁⠀⠀⠘⠲⠚⠁⠀⠀⠀⢸⡉⠳⡀⠀⢸⠀⢳⡀
⠀⢀⡞⠀⣸⠀⠀⠀⠀⣴⠒⠚⠛⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠹⡄⠘⣇⠀⢳⡀
⠀⡼⠀⣰⠃⠀⠀⠀⠀⠈⠓⠛⠋⠉⠉⠁⠀⠀⢠⡾⠿⠒⠦⠤⠴⠚⠯⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣀⡽⠀⠈⠳⣌⣧
⢠⢇⡜⠁⢠⠖⢦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠻
⢸⠋⠀⠀⠈⢦⠀⢣⡀⠀⠀⠀⠀⣰⢋⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⠀⠀⠀⠀⠀⠀⣠⠄⢦
⡏⠀⠀⠀⠀⠈⢣⡀⢳⠀⠀⠀⠀⢻⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢀⡴⠋⢀⡴⠋
⢷⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠯⣠⠖⠁⠀⠀⠀⠀⡰
⢸⢧⠀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⢸⡀⢳⡀⠀⠀⠀⣰⠃⢠⠇⠀⠀⢸⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣡⠟⠀⠀⠀⠀⠀⣰⠚⣦⢸⠁⢀
⠀⡇⠀⢳⠀⠀⡴⠁⣰⠃⠀⠀⠀⠘⠦⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⢰⠃⢠⠇⢸⡄⢸
⠀⢹⠀⢸⠀⠀⠳⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⣠⠲⢄⡀⠀⠀⢠⡏⢀⡏⠀⠀⢣⡏
⠀⠀⢇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡷⠶⠚⠉⠓⠶⣏⣡⠎⠀⠀⠘⠢⡀⠙⠢⡀⠀⠑⠋⠀⠀⠀⢸
⠀⠀⠈⢿⠀⠀⢀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣠⢤⡀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡀⠀⠀⠈⠳⣄⡼⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠈⣇⠀⢏⣀⣀⣀⣀⣀⠟⠀⠀⠀⣠⠞⢁⡰⠃⠀⠀⠀⠀⠀⠀⠀⡇⢰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⡼⠁
⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠇⠀⠀⠀⠀⠀⠀⣸⠁⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠓⢶⡶⠤⠤⣄⣀⡀⠉⠉⠀⠀⠀⣴⠓⠢⢤⡀⠀⠀⠘⢧⠼⠀⠀⠀⠀⠀⠀⢀⣏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⣀⠈⠉⠓⢤⡀⠀⠀⠈⠓⠦⣄⡈⠑⡆⠀⠀⣀⣤⣄⣀⣀⢀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣈⣓⣤⣀⠀⠀⠀⠉⠛⢁⡴⠊⢁⣀⡠⠴⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣽⣶⣶⣶⣶⣿⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⠒⢶⣦⣤⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢰⣤⣤⣽⣿⣿⣿⢖⣦⣾⣿⡿⣿⣿⡟⢸⢞⢲⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠠⣤⣭⣽⣿⡿⢋⣵⣾⣿⣿⣿⢳⣿⡿⡇⠹⣙⡢⠋⠁⡬⠛⠁
⠀⠀⠀⠀⠀⠤⠶⣾⡿⢋⣴⣿⣿⣿⣿⣿⠃⠙⡛⠀⠀⠠⡀⠀⠀⠈⠉⢙⠇
⠀⠀⠀⠀⠄⠲⣿⠏⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠁⡀⠀⠀⠨⡀⠀⢁⠤⠃
⠀⠀⠀⠠⠶⢾⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢡⡀⠀⠂⠈⠳⠤⠝⠁
⠀⠀⠀⢀⣴⡟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠋
⠀⠀⠀⢠⣼⠣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢀⢀⡠⠄
⠀⠀⠀⢀⣼⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⠀⣴⣤⣶⡿⠋
⠀⠀⠀⠁⣽⠨⣿⣿⣿⣿⣿⣿⣝⡛⣿⢕⣣⢴⠟⠛⠉
⠀⠀⠀⠀⢰⡆⣿⣿⣿⣿⣿⣿⡀⠅⠈⠉⠉⠁⠀⠀⠀⢀⣀⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠐⠀⠱⠹⣿⣿⣿⣿⣯⢢⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠻⠿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠸⣄⢣⢽⣿⣿⣿⣿⣫⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢯⠙⢿⣧⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠹⡆⣬⡻⣿⣿⣿⣷⣷⣄⢀⣠⠾⠟⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠉⣿⠑⣮⠿⣿⣿⣿⣿⣀⠉⠋⠉⠙⠶⠆⠀⠀⠀⠀⠀⠀⠐⣘⣧⣤⣶⣾⠟
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠜⡓⢮⡛⢿⣿⣿⣧⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠙⠀⢤⣄⣉⡛⠿⠿⣿⣶⣶⣶⣶⣾⣷⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡟⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠉⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠆⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠚⠈⠀⠈⠀⠀⠀⠀⠀⠀⠉⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣀⣀⣄⠀⠀⢄⣂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡴⣿⢻⣏⢙⣁⠀⠈⠈⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠵⠓⠛⠋⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⡠⠀⣸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⣀⣠⣤⡄⠀⠈⣤⡀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⣀⣴⣺⡥⢿⣤⣉⣰⣄⣠⣨⣦⣼⣅⣀⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣼⡴⠞⣛⣩⡤⠷⠞⢻⠚⠛⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⢿⠚⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣃⣢⠘⠺⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡄⣘⡿⣿⣿⣿⣿⣯⣝⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣴⠞⠁⣹⣁⡟⣻⣔⣻⣿⣿⣿⣷⣮⣟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣀⣦⣤⢿⣻⣮⣷⣿⣿⣿⣿⣿⣿⣿⡏⠈
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⠀⠛⠉⠉⠙⢿⡿⠋⠄
⠀⡂⠀⠀⠀⠀⢤⠤⣤⣤⠤⠶⠦⢤⣤⣬⣿⢿⠿⢁⣤⡀⠀⢀⢢⣼⡇⠀⡄⠖⠠⣤⠄⠀⠀⠀⠠⠶⠦⠠
⠀⠂⠀⠀⠀⠀⠀⠀⠡⢶⣾⢂⣶⢶⣦⣤⡤⠀⠐⢺⣽⡿⠈⣽⣾⣿⣇⣀⠻⢿⣿⣃⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⢶⣶⡬⢙⣠⣴⠚⠿⠃⠀⠈⣛⣯⡌⠻⣿⣦⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠛⠛⠒⠶⠶⠤⢴⣶⡶⠾⠛⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠑⠦⠤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣒⠲⣤
⠀⠀⠀⠀⠀⠀⢀⡏⠈⠑⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⣧⠀⠀⠈⣷⠘
⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠚⠛⢏⠉⠁⠀⠈⡇⠀⠀⠀⢸⡄⢀⡴⠃⢠
⠀⠀⣠⠶⠛⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⣀⡷⠋⠀⠀⣼
⢠⠞⠁⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣀⣆⣴⠞⠉⠀⠀⠀⣠⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠓⠒⢋⣭⠟⠋⠀⠀⠀⢀⣠⠞⠃
⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠋⠀⠀⣀⣠⠴⠚⠋
⠀⠀⠉⠛⠲⠦⣤⣄⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⠶⢟⣛⣋⣥⡤⠴⠖⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⣲⠶⠤⢤⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠛⠉⠉⠀⠀⠰⡅⢃⠴⣣⡼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠲⠭⠙⠛⠫⠷⢦⣄⠀⠀⣠⣾⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⡶⠦⣤⣀⣀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⡿⠥⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡐⠌⠊⠁⠀⠣⡠⠀⠐⢉⠝⣓⡶⢤⢠⠐⣸⣿⣿⠿⠏⣆⣴⣾⡆
⠀⠀⠀⠀⠀⣠⠐⢊⣥⠞⠁⠛⠀⠀⠀⠀⠀⠈⠉⠉⢉⢥⡪⣾⣿⡀⣳⣯⣿⢍⣡⣾⣿⣿⢏⡇
⠀⠀⠀⠀⠀⣾⢛⠒⠤⣼⣒⠤⣁⠀⡀⠤⣤⠄⢤⣲⣵⣿⣿⣿⡿⢐⣿⡛⠈⠀⡉⣿⡾⣻⢏⡀
⠀⠀⠀⠀⠀⡷⠈⠹⠷⣒⠭⠹⣶⣯⣠⣚⣭⣶⣽⣾⣿⣿⣿⡿⢟⣋⣀⡀⢠⢴⡃⣿⣿⠿⣤⠉⠹⡖⠤⢄⡀
⠀⠀⠀⠀⠀⠑⠢⠤⣐⡢⠭⣝⡂⠝⡓⠮⣿⣯⣷⣿⠿⠋⢉⣁⡠⢴⢻⣻⢮⠷⢻⡿⠖⠚⠹⣧⢤⡐⢋⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣗⡲⢬⣙⡒⢧⣿⣿⡋⠭⠃⠀⠀⠀⡔⣵⣿⣿⠲⣱⡋⠀⢀⠔⠀⠀⠁⢚⠻⠿⡿⢿⡀
⠀⠀⠀⠀⠀⠀⣤⠒⠩⠍⠁⣂⡐⠀⢍⡛⠑⠁⠀⣀⢀⣀⡠⣪⣾⣿⣿⣿⣼⠁⠀⠒⠁⠀⠀⠀⠤⠀⠀⠀⠃⠰⣨⠉⣲⡄
⠀⠀⠀⠀⠀⠠⣗⠂⠈⠩⠥⠐⣂⡈⠓⠲⠤⢔⣉⣐⣊⢔⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⣀⢕⡤⢖⡫⢔⣷
⠀⠀⠀⠀⠀⠀⢿⣊⡠⠬⠍⠁⠐⠒⢒⣀⣤⣬⢐⣶⣂⣭⣾⣿⣿⣧⣤⡠⠤⣤⠄⣀⠀⢀⡀⠠⣐⡮⢝⣪⣵⣾⣿⣾⢿⡃
⠀⠀⠀⠀⠀⠀⠉⣶⢖⠶⡮⢭⠥⡤⡥⣤⢶⢖⡶⡶⣲⢻⣿⡫⡃⠀⠙⠻⣖⠬⣙⣠⣭⣶⣦⣟⣾⣿⣿⡿⣟⣊⣥⣒⣿
⠀⠀⠀⠀⠀⠀⠈⡯⠁⠑⠊⠂⢄⣈⡘⠐⠑⠠⠊⠄⠔⠔⡠⣻⣮⡀⠀⠀⠉⠯⢃⣬⣭⣍⣬⣿⣻⢿⣷⣾⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠈⢻⢦⡉⠭⠽⢓⣾⡿⣿⣿⣿⢿⠶⠄⡀⠀⠰⣷⡶⣶⡷⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⠀⠀⠀⢯⢍⠑⢾⣻⣷⣿⣿⠟⠋⠁⠀⠀⠀⠑⠒⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢀⠃⠐⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠢⠰⢀⠄⡠⡀⢀⠀⠀⢐⠅⡨⢀⠀⡠⠀⠔⠁⠀⢀⠔⠡⠊⠀⢀⢀⠑⡡⠂⢀⠔⡁⡠⢀⢔⢕⡔⡰⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠐⠡⠪⠀⠅⠨⠀⠕⡨⢂⠔⠀⡨⠀⠀⡠⠪⢀⢀⠔⡡⠂⠂⠐⠁⠊⠌⠐⠁⠃⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣇⡴⠪⡀⣨⡶⢋⡵⠊
⠀⠀⠀⠀⠀⣿⠗⣠⢟⡡⢞⠍⢀⠂⠀⠀⢈⠤⠀⢀⣤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢺⣷⢷⣷
⠀⠀⠀⠀⠀⣿⣾⡫⠋⠈⣀⣤⡶⠞⢁⠆⢀⣤⠔⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢾⣿
⠀⠀⠀⠀⠀⣿⣿⣴⠞⢛⣉⣵⣶⣤⠾⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣿
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⢻⣿⣼⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⢹⣿⢺⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡏⢣⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣦
⠀⠀⢻⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣬⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀
⠀⠀⠘⣿⣿⣿⣿⣆⣀⣀⣀⣰⣶⣶⣶⣶⣾⡿⠿⢿⡟⠛⣿⠛⢛⠉⠻⠏⠁⠉⠙⠟⠛⠛⠿⠻⢿⢿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⠿⠭⠿⠀⠠⠾⠀⠀⠒⠒⠒⠂⠀⠙⠀⠀⠀⠉⠀⠀⠴⠀⠐⠇⠚⠛⠛⠿⠻⠿⠿⠩⣿⣿⣿⣶⣤⡀
⠀⠀⠀⣿⣿⣿⣷⡤⠄⠀⠀⠀⢀⣀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣦⠠⠄⠀⠤⣤⠀⣈⠉⠹⠯⠄
⠀⠀⣸⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠿⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠛⠋⠉
⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⣀⣉⣉⣉⣉⣉⡁⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣠⣶⣿⣿⣿⣿⡿⢿⣿⣿⣿⣦⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢰⣿⣿⣿⠏⢁⣀⠀⢀⣀⠙⣿⣿⣧⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⢉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⠀⢿⣿⢇⠻⠿⢃⣿⣿⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢂⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠸⣿⣿⣿⣷⡦⠀⠿⠓⢐⣿⣿⣿⡇⢸⠁⠀⠀⢀⠠⢸⡆⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠈⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠹⣆⢻⣿⣿⣿⣷⣷⣾⣾⣾⣿⣿⡟⠀⠃⠀⣒⣈⣤⠆⡼⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠙⠒⠄⠤⠀⠀⣋⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⣩⣭⡍⠙⡟⣛⡛⠻⠏⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣙⡿⠟⠋⠙⣿⣿⣿⣿⣿⠇⠀⠀⠁⠀⠀⠰⠃⠈⠉⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⡠⠐⠠⡀⠀⢀⡀⣀⠀⠀⠀⠉⠀⠀⡄⠀⠀⠈⠙⣿⠏⠀⠀⠀⠀⢠⣄⠀⣤⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⢰⠁
⠀⢀⠀⢀⡇⠀⠀⠘⣴⠋⠀⠘⠇⠀⠀⠀⠀⠀⣧⠀⢀⠘⡂⠀⢠⣤⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡌⢀⠂
⠀⡁⠀⠀⠈⠀⠀⠀⠰⡀⠀⣏⠀⠀⠀⠀⠀⡌⠀⠀⠉⡏⡇⠀⠀⠁⠐⢶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢈⡁⠐⠺⢤⡂
⠀⠱⠄⣀⣀⠀⠀⠀⠐⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⣄⣀⡐⠀⠙⠻⢿⣷⡶⠤⣤⣤⠶⠞⠷⠄⠀⠀⠀⠀⠀⣠⠊⠁⠁⠚⡁⠁⢢⡀⠉⢢⡀
⡄⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⡴⠒⠄⡀⠀⠈⠀⠁⠐⣼⠀⠀⣿⣿⣷⣦⣤⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⡐⠂⢇⠘⡆⠀⠱
⠈⠒⠒⠊⠉⠠⣀⣀⣠⡔⠀⠀⠀⠀⠀⠀⠑⠀⠀⠔⠣⠘⢆⠀⠘⡿⠋⠉⠙⠉⠉⠑⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠔⢶⠀⠀⠁⣶⣎⣀⣀⡠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠉⠂⠒⠀⠀⠑⠄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⠚⠁⠀⠀⡆⠀⢠⡥⠄⣀⠠⡅⠀⡌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡉⠢⢄⡀⠀⠀⠀⠀⢀⡴⠊⢦⠀⠀⣸⠟⢦⠀⠀⠀⠀⣀⣆⠑⠠⣤⠐⡡⠊⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣗⣠⡀⠀⠤⣊⠔⡰⠀⠻⠁⠀⠀⠘⠂⠀⠀⢸⡋⠐⠂⠤⠤⠤⠀⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⡇⠀⠀⠱⠈⠀⣀⠀⢣⠀⠀⠀⠂⠀⠀⢸⠃⠂⠤⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢢⢀⠀⡇⠀⠀⠀⠃⠻⠠⠁⠀⢃⠀⢀⡄⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠈⠀⠁⠀⠀⠀⢰⠰⠆⠰⠀⠸⠒⣿⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⡀⢸⠀⠀⠀⠀⠈⠀⣄⠤⡀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⡷⠤⣀⡀⠀⠀⠀⠀⠐⠉⢆⠁⣬⣤⣤⡀⢠⣤⡇⠉⠒⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠈⠙⠛⠶⢶⣤⣤⡄⢬⣴⣿⣿⣿⣿⠆⣿⣷⠷⠟⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⠉⠛⠛⠋⠡⠊⠉⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡆⠀⠀⠀⠀⠀⠀⠀⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠇⣾⣿⣷⣶⣶⣶⣾⣿⣿⣿⡀⠳⣤⣀⣀⣀⣀⣤⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠘⢮⣥⢬⢤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠁⠀⢸⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⡀⠈⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣧⠀⢘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠘⠢⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡾⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠿⠛⠋⠉⠀⠀⠈⣿⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⣠⣴⠶⠿⠿⠿⢿⣤⡀⠀⠀⡀⠀⠀⠀⠀⣉⣉⣠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⢿⡿⢋⠀⠰⠄⠀⠀⠒⣈⣉⣤⣤⣴⣶⠶⠿⠟⠛⠋⢿⣇
⠀⠀⠀⠀⠀⠀⣈⣤⣴⣶⠿⠛⠋⠁⠀⢀⣉⣉⣥⣤⣶⡶⠾⠛⠛⠿⢿⣭⡁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⠉⣡⣤⣤⣶⡶⠿⠿⠛⠛⠉⣹⡟⡁⠀⢐⣦⣀⠀⠀⠈⢻⣆⠀⠀⡀⢀⣀⡀⡀⣿⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠉⣩⣥⣤⣤⣤⣤⣤⣬⣭⣬⣤⣤⣭⣭⣬⣥⣤⣤⣥⣭⣤⣤⣥⣭⣭⣥⣥⣭⣭⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⣿⣟⠉⠉⠉⠉⠉⠉⠉⠉⣩⡿⠛⠉⠉⡉⠉⠙⠻⣯⡉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡿⠀⠀⢀⡀⠀⠀⠀⣼⠋⠀⠀⣠⠾⡗⣦⠀⠀⠈⢿⡄⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⣿⡟⠀⢠⣶⣦⠀⠀⢸⡏⠀⠀⠀⠻⣶⣇⡉⠀⠀⠀⠘⣷⠀⠀⢠⣶⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⣿⣷⠀⠈⠛⠋⠀⠀⢸⣧⠀⠀⠀⣤⡌⡟⢻⡆⠀⠀⢠⡿⠀⠀⠈⠛⠋⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠙⠲⡷⠛⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣠⣀⣀⣀⣀⣀⣀⣀⣹⣷⣦⣄⣀⣁⣀⣤⣾⣟⣡⣄⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠦⠶⠶⠶⠶⠶⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠚⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠤⠤⣄⡀
⠀⠀⠀⠀⢀⠴⠊⠉⠀⠀⠀⢾⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢖⡡⠃⠀⠀⠀⠉⠢⡀
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠓⠤⣈⡉⠓⠒⠒⠒⠚⢛⣉⣉⡥⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠦⠤⠤⠤⢤⣤⣤⣀⣀⣠⡤⠤⠤⠤⠶⠒⠚⠉⠁

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠈⢣⡄⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⡰⠁⠠
⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠈⠀⠀⡂⢀⠢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠂⠀⠀⠁⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑⠠⢐⠠⠂⢁⠠⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⡀⠀⠈⠀⠀⠀⡐⠀⠀⢀⠔⠁
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠂⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠠⠀⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠌⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠐⠠⠀⠑⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⠀⠈⠀⡠⠊⠀⠀⠀⠀⠀⠁⠐⠀⢀⠠⠐⠈
⠀⠀⠁⠂⢀⠀⠀⠀⠠⠀⠀⠁⢄⠀⠀⠀⠀⠡⡂⠀⠁⠀⠀⠁⠀⠘⠀⡄⠀⡈⠀⠀⢀⠇⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠘⢆⠠⠄⠀⠄⠠⠠⠄⠁⠠⠀⠀⣠⠃⠀⠀⠀⡠⠈⠀⠀⠀⠀⠂⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢠⠄⣑⢄⠠⠀⠈⠁⠀⢀⠀⠁⠀⠀⠉⠀⠒⠤⢄⠀⠜⡢⠈⠀⠀⠠⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠐⠀⠀⠤⢀⠀⠀⠂⠀⠀⠀⠀⠂⢄⠈⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⡀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠤
⠀⠄⠀⠀⠀⠀⠈⠐⠠⠄⣈⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⢀⣀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠂⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠠⠠⠤⠄⢄⢤⢠⢀⢠⢠⠠⣠⠀⡄⣤⣦⠧⠤⠤⠤⠤⠤⠤⠸⠠⠠⠤⠤⠤⠦⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⠒⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠈⠁
⢀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⢉⡁⢰⣶⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠒⠒⠋⣉⡠⠤⠖⠒⠋⠉⠈⢇⠀⠿⠇⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠔⠒⠚⣉⣁⠤⠤⠶⢊⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠘⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⠂⣉⣉⡤⠤⠔⢂⠟⠋⠁⢀⣤⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧
⠀⠀⠀⠀⠀⡔⠒⢈⠉⢠⡤⠤⠒⠊⠉⠁⠀⠀⠀⡔⠁⠀⠀⣴⣾⣿⡿⣷⡆⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⡀
⠀⠀⠀⠀⠀⢱⠀⠘⣢⠀⡇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⣿⡁⢹⣷⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡇
⠀⠀⠀⠀⠀⠘⡆⢀⠥⠔⠁⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠈⠻⣿⣾⣿⣦⣤⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢹
⠀⠀⠀⠀⠀⠀⢡⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⢻⣷⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠀⠏⡇
⠀⠀⠀⠀⠀⠀⠈⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢠⣿⡇⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠰⡇⠀⢧⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠐⠿⣷⣶⣿⣿⡿⠋⠀⠀⣠⠋⠀⠀⠀⠀⣀⣠⠤⠄⠓⠀⣉⣠⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠈⠛⠁⢀⣠⣞⠥⠤⠒⠒⢉⣁⠤⠤⠖⠒⠈⠉⠀⣳⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠈⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⡲⠶⠶⠒⠊⣉⣁⠤⠔⠒⢺⠋⠁⠀⠀⠀⠀⠀⢀⠞⢁⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢴⠀⠙⡆⠀⣀⣀⠤⠄⠒⠚⢉⣁⠠⠤⠶⣶⣿⣿⠟⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⢀⢸⡄⠈⡳⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⠇⠀⢉⣉⡁⠤⠔⠒⠋⠉⠓⢤⣀⠀⠀⠀⠈⠉⢀⣀⣴⣋⡠⠤⠤⠒⠒⠂⢋⣉⣉⡡⠤⠤⠰⡖⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠧⢲⠊⠙⡍⠀⠀⠀⠀⠀⣀⣀⣀⠠⠬⠽⠛⠒⠊⢉⣉⣁⡠⠤⠤⢲⠖⠂⠉⠉⠉⠀⠀⠀⠀⣀⣀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⡷⠀⠻⠒⠒⠊⣉⣁⣤⠤⠤⢤⠒⠒⠊⠉⠻⠟⠉⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⠀⢠⠊⢠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠦⠤⠖⠲⠶⠬⠭⠥⠤⠤⠤⠤⠤⠤⠭⠶⠦⠤⠤⠤⠤⠶⠮⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⠨⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢽⡀⠀⠀⠀⠀⡀⠦⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠲⠶⠚⠃⠀⠀⠀⠀⠀⠀⠉⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠯⠸⠟⠋
⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠈⠻⣿⡭⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣀⡐⠦⠄⠀⠰⠟
⠀⠀⣶⣿⣧⣈⣉⣉⣀⣤⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠄⠀⠐
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡿⠿⠟⠛⠛⠛⠛⠛⠿⠶⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⣳⠄⠈⠉⠑⠲⠤⠤⢤⣤⣤⡤⠤⠤⠶⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣓⠲⠿⡧⢤⣄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠦⣌⣹⣖⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣝⡲⠭⠍⠙⠒⠒⠀⠠⣤⣤⣤⣀⣀⣀⣠⣄⣀⠤⠴⠚⠀⠀⢻⠀⠀⠀⠀⢀⠴⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⠲⢭⡙⠀⠰⠤⠀⠀⢀⣀⣀⣀⡀⠀⢀⠀⠀⠀⣴⠶⠦⣄⠀⠸⡇⠀⢀⡾⠁⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠮⣍⡓⠠⢤⣀⣀⢀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠀⣼⠃⠀⠀⠈⠳⣄⣷⣰⠏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣗⡶⢤⣉⡁⠀⠀⠀⠉⠉⠋⠉⠙⠒⠒⠒⠒⠂⠀⢿⣄⣀⠀⠀⠀⢈⣻⣯⣀⣤⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⣟⣡⡈⠙⠳⠮⣭⣛⣒⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣛⣩⣿⣿⣿⡿⢦⣄⣀
⠀⠀⢀⣀⣀⣤⣤⡶⠶⡟⠻⣝⠯⣕⠮⠉⠙⠓⠦⣤⣀⡀⠉⠉⠛⠓⠾⠿⠿⣿⣿⣿⣶⣶⣿⣟⡋⠩⠽⠛⠛⠉⢸⡏⠙⢦⡀⠉⠛⠳⠶⢤⣄⣀⡀
⣶⢿⡉⠯⣉⠙⠦⣙⠢⣌⠓⠎⠉⠲⢤⣀⣀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠲⢦⣤⣤⡤⠤⣤⣠⡤⢤⣤⡤⠴⠒⠃⠀⢸⡇⠀⠀⢳⣠⠞⠀⠀⠀⠀⢩⡍⠙⢻
⠙⣦⡙⠢⠈⠑⠀⠀⠀⠈⠓⠦⢤⣀⣀⣀⠀⠀⠀⠀⠙⠲⠤⠶⠶⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠀⢈⡗⠚⠁⠸⠏⠀⠀⠀⣠⠖⠋⢀⣴⡟
⠀⠈⠛⠶⣦⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠈⠙⠛⠒⠒⠦⠤⢀⡀⠀⠀⠈⠉⠀⠀⠈⠙⠋⠁⠈⠉⠉⠁⠀⠁⠀⠀⠾⠃⠀⠀⢀⣠⡤⠖⠋⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠶⠶⠶⠶⢶⣮⣝⡛⠛⠛⠒⠲⠤⠤⠄⠠⢤⣤⠤⠴⠤⠤⠂⠀⠀⠒⠚⢉⣀⣀⣤⣴⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⠶⠶⠾⠿⠛⠛⠛⠉⠉⠁
⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠒⠒⠋⠉⠉⠉⠉⠉⠙⠒⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠉⣀⠠⠄⠒⠒⠒⣀⣀⣀⣀⠀⠀⠀⢀⣀⣠⣬⡟⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠁⢀⣠⡬⠴⠒⠚⠉⠉⠁⠀⠀⠀⠀⠉⠉⠉⠛⠛⠋⠉⠀⠀⠐⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠷⠚⠉⠀⠀⠀⢀⣀⣀⣀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⣹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠠⠖⠚⠛⠛⡿⠛⠉⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣒⣛⣫⣭⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡀⠀⠀⠀⠤⠠⠁⠤⠤⢤⡤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⢿⣿⣿⣿⣦⡀⠀⠲⣾⣿⣿⣿⠟⠛⠛⠋⠉⠉⠀⠀⠀⢀⣀⣀⣠⣴⡿⠿⠛⢡⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢙⣿⣎⡻⣿⣿⣿⣦⡐⠛⠛⣛⣁⣠⣤⣤⣤⣤⣉⣁⣈⣉⡉⠑⠒⡠⢄⠀⠀⢐⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣴⠿⢛⣋⣩⣤⣙⣿⣿⣿⣷⣾⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⢭⣗⣲⣤⡉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⢸⣖⣫⠿⠛⠛⠋⠉⠛⠛⠿⢿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⠴⣾⡏⠏⢸⡟⠁⢠⠀⣴⣶⠒⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠒⠻⢿⡄⢢⣸⠀⢹
⠀⠀⠀⠀⠀⣠⠶⢋⡡⠞⢛⣩⠷⢛⣯⣿⡄⣾⠁⣆⡾⢐⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣤⣤⣤⣈⡁⠐⠀⠈⠀⠹⣦⠈
⠀⠀⠀⢠⠞⢁⠴⢋⣀⠠⠤⠴⣞⣯⣿⠟⠚⣇⢠⡟⢚⣽⣶⣶⣬⣍⣀⡀⠀⢀⠀⣠⠶⣿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡐⠀⠀⠘⣿
⠠⠤⠾⡥⠴⠞⠊⠉⡠⠲⠟⠋⠁⣰⠃⠀⠀⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⠇⢠⡋⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⢆⠿
⠀⠀⠀⠀⠀⠀⠀⡞⢀⡞⠀⣠⣾⣟⣠⡶⠛⠙⢿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣠⣷⣄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣦
⠀⠀⠀⠀⠀⠀⢸⢀⡾⢃⡾⠟⠗⣿⡿⠁⠀⠀⡼⣆⠹⣿⠻⠻⠿⠿⠟⢛⣉⠽⠁⣸⣿⣿⣿⡄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⡎⡞⡱⠋⠀⡘⢰⣿⡇⠀⠀⣸⣹⣿⣄⡈⠓⠒⠶⠤⢒⠥⢶⡟⢀⣿⣿⣿⣿⣿⣄⢠⣙⠿⣿⣿⣿⣿⣿⣿⡿⠋⠠⢞⡩
⠀⠀⠀⠀⡠⠊⡼⠀⠁⠀⠀⣿⠈⣿⣧⠀⠀⠸⣜⠷⣦⣀⠀⠀⠀⢀⠀⠀⠘⠁⢾⣿⣿⡟⠿⠓⣿⡄⠉⠀⠀⠈⠉⣉⣩⣥⣴⠖⠚⠉
⠀⠀⣠⠊⢀⠞⠁⠀⠀⢀⣴⠟⠀⠘⢿⣦⣀⣀⡀⠉⠲⡶⣶⣦⣀⠈⢷⠀⠀⠰⠈⠳⠟⠋⠙⠚⢳⣵⠆⠀⡼⣡⣾⣿⡯⢋⡇
⢠⣾⣋⣠⠤⠤⠒⠒⠚⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠐⡜⠆⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⢸⠁
⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢹⣿⡿⣃⢼⡆⢀⡀⢠⣠⢆⣀⡄⣤⣴⢀⢻⡄⣿⣿⠇⣾
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢿⣷⣿⣸⡆⣨⡇⠸⣥⠡⣬⡇⢡⡄⣼⣤⣿⡿⡋⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⢹⡛⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿⣿⣿⠿⣿⠁⠈⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⡴⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⢿⣿⡍⣻⠉⣿⠋⣽⡇⢠⣿⣼⠏⠀⠈⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠈⡱⡟⠰⠿⣦⠟⠉⠁⠄⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢧⡂⠀⠀⠀⣠⣴⡶⠂⠀⣀⠔⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣿⣶⣾⣭⣭⣭⣭⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⠟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠠⡀⢷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⢉⡵⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣼⣿⣿⣾⣷⣿⣿⣿⣿⣿⣷⣦⣄⣠⣴⣛⣉⢀⡴⠋⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡴
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⡼⠃
⠀⠀⠀⣀⣤⣶⣿⣟⣛⡟⠙⢻⣾⡿⠽⠟⠒⠛⠺⠿⣿⠹⠟⠹⡟⢰⡟⢠⡟⢣⣾⠋⡾⢋⡿⢻⣿⠛⣿⡿⢿⣿⣾⣿⣶⣶⣠⣄⣀⠀⠀⠀⣰⣏⣀⣰⠇
⠠⢶⡿⠋⠉⢰⣴⣿⡧⠇⠀⠀⠈⣻⣶⣶⣶⣞⣿⣿⢻⠿⢿⡟⠛⠳⢶⡾⣶⣦⠀⠀⠀⠈⠀⠚⠃⠀⠉⠁⠘⠋⠘⠛⠟⠛⠛⠻⠟⢿⣿⣾⣿⣿⣻⡏
⣤⣠⠹⢦⠠⣌⠛⠛⠁⠁⠀⠰⠀⣸⡗⢹⠉⠀⠀⠀⣀⣀⣠⡤⠤⠴⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡶⢾⠟⠛⠛⠛⢿⣿⣿⣼⣧
⠈⠙⠻⠶⣾⣭⣀⣀⠀⠀⠀⣾⣆⣾⣧⠭⣷⣶⣞⣁⣀⣀⣒⣀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⠿⠛⠀⠁⠀⠀⠀⠀⠀⠀⠹⣮⠉⠹⡆
⠀⠀⠀⠀⠀⠉⠙⠛⠾⢯⣭⣿⣿⣟⣛⣛⣭⡽⠿⢿⣷⣶⣶⣶⣶⣶⣶⣿⡿⣯⣭⣭⣵⣶⣿⣿⣿⡿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⢳⣾⣿⣯⣭⡭⠿⠿⠿⠿⠿⠿⠟⠛⠛⠉⠙⠻⣿⡛⠛⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣽⣦⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⣴⠆⠰⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠀⠀⣾⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢲⡯⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠋⠉⠀⠂⡀⠉⠀⠒⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⠠⠄⢀⣔⡏⠀⢰⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⢿⣇⣀⢸⡇⢰⡶⣄⣀⠐⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠈⠙⡛⣸⠃⠀⠉⠙⠻⡾⠧⣤⣛⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡆⠀⠀⠀⣤⠄⠀⠀⢨⣵⡏⠀⠀⠀⠀⠀⢹⡄⠀⠈⢿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠚⠂⠲⠀⡜⣨⡿⠀⣀⣠⣿⡿⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⠛⠛⠿⠿⣿⣿⣾⣶⣿⣥⣶⣿⣿⡛⠛⠻⠿⣿⣆⡉⢰⡦⢿⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢩⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠯⠿⣿⠃⠀⠀⠀⠀⠀⠉⠙⠳⢾⡧⣤⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠚⣇⠀⠀⠀⠀⠀⠀⡠⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⣠⣾⠟⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠈⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣰⣃⢋⢌⣼⣦⣗⣴⣷⣧⣾⣿⠋⠀⠐⠇⠴⠃⠀⠀⠀⠀⠀⢰⣽⠃⠀⣀⣀⠐⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣤⣬⣅⣟⣻⣟⣿⡿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣶⣤⡀⠀⡀⣀⢀⣼⡿⢀⣴⠟⠉⠓⢢⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣉⡉⠛⢻⡿⣿⣿⣷⣮⣿⠋⠰⢟⡿⠿⡿⣿⡿⣿⣷⣦⣤⣤⣽⣿⣿⣷⣿⣧⣦⣤⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣻⠷⢤⢄⣉⠛⠃⠀⠀⢈⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣣⠆
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣀⣠⣞⣛⢻⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠏⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⢀⣀⣸⠃⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⡅⡟⠋⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣭⡤⣄⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠈⢂⠀⠀⠀⢴⢞⣛⣩⣥⣶⣤⣿⣿⣿⣿⣿⡿⣿⢟⣩⣾⣏⠀⠀⠉⠿⢿⢿⡿⠿⠿⢿⡅⠾⢿⠀⠀⠂⠀⢀⣤⡞
⠀⠀⠀⠀⠀⠈⠳⣤⣀⣐⣿⣿⡿⣿⡿⠿⣿⣿⣿⣿⣿⣟⣛⠛⠿⣿⡅⠀⠀⠀⠀⢠⠷⣍⠢⣶⣿⣧⣀⢤⣀⣀⣰⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠓⠐⠓⠶⢚⣋⣩⣭⣭⣭⣏⣉⡛⠿⣦⣝⠂⠀⢐⣕⣾⣶⣮⣷⣮⣽⣿⣿⣿⣿⣿⣟⡯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣄⣀⣁⣐⣨⣽⣿⣿⣿⣿⣿⣿⣿⣄⠈⠛⢷⣄⠀⠉⡛⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣭⣭⣤⣶⣾⣿⣿⣿⣿⢿⡏⠁⢾⣆⠙⠶⣤⣈⣛⠿⠿⠿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢈⠙⢿⣿⣧⡍⠙⠀⠶⢿⣷⣴⣤⣉⣉⣛⡛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠻⠶⠦⠉⠀⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣒⢀⣩⣭⣁⣔⡖⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣤⣍⠘⢶⣏⣀⠀⠀⠈⠉⣽⣿⣿⣿⡕⢀⠠⠖⠒⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠘⠛⠛⠉⠳⣄⠙⠿⣿⣄⠀⣀⣴⣿⣿⣿⡟⢈⣠⣶⠿⢶⣦⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡘⢿⣿⣾⣾⣿⠿⣉⡴⠟⠉⠀⠀⠀⠀⠀⢳⣄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠩⣴⡚⡿⢀⠀⠀⠀⠀⢠⣠⣤⣄⠉⠳⢦⣕⡢⢄⡀
⠀⠀⠀⠀⢀⣶⡟⠁⠀⢀⠀⠀⢠⡀⠐⢀⣐⣄⣴⣤⣤⣴⣤⣤⣴⣿⣧⣿⣽⣿⣿⣾⣾⣾⣿⣿⣿⣿⣿⣿⣿⣾⣭⣝⣳⣝⠢
⠀⠀⡀⠀⠊⡀⠁⢄⠀⣀⣀⣣⣨⣿⣶⣾⡿⠿⠟⢛⣿⠿⠍⢉⡉⢉⣉⣍⣛⠻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣦
⠀⠈⠀⠀⢀⡙⣾⢠⣰⣿⣿⣿⣿⣿⣿⣥⡴⠞⠛⠓⠒⠒⠛⠋⠙⢿⡿⠛⠛⠛⠶⠶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠉
⠀⢰⠀⠂⣈⠸⣶⣿⣿⣿⣿⡟⣽⣿⡇⠰⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⣄⠀⠀⢀⠀⠈⠛⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡆⠀⢰⡆
⠀⢸⡀⠀⠈⢰⡚⣻⣿⣿⣿⣷⣿⡿⠃⣠⣶⣾⣶⣶⣶⣖⠒⠀⠀⠀⠃⠙⣿⣿⣷⣿⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢸⡇
⠀⠘⣷⠆⣀⣉⣻⣾⢿⣿⣿⣿⣿⡿⣾⣿⣿⣿⢿⣿⣿⣿⡄⠀⠀⠀⠀⢀⢆⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠀⠀⣾⠁
⠀⠀⠀⠀⠙⠙⢟⣛⣿⣿⢿⣿⣿⢷⣾⣿⣿⡏⢀⠼⣾⣿⠀⠀⣀⣀⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⡼⠃
⠀⠀⠘⠢⢀⠀⠀⠑⠲⠙⠋⢉⣿⡯⠻⣿⣿⣿⣿⡿⠷⠋⠀⣼⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣟⠛⠟⢛⠉⠉⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢀⣀⠀⠛⢸⣿⣿⣿⣿⣿⡑⠏⠙⢿⡿⠿⠿⡛⢻⡏⠀⣷⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣦⡀⠀⣀⡀⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠁⢀⣀⣈⣈⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣌⡻⣷⣴⡇⠈⠙⢻⡏⠙⠋⠉⠀⢠⡾⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠀⠀⠀⠈⡏⢿⣿⣷⣾⣿⢧⠀⡄⠀⠀⠀⠀⠀⠀⠘⢡⣿⣿⣿⣿⠏
⠀⠀⠀⠀⢀⠂⣞⡷⢸⡆⠀⠀⠀⠀⠀⢘⠸⣿⣏⢻⠿⡟⢲⠓⡗⢲⡗⢾⢲⡗⡶⣾⠋⣽⣋⠃⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⣶⣤
⠀⠀⠀⠀⠾⠳⢾⣷⡏⢳⡀⠀⠀⠀⠀⠸⡇⡏⠙⠶⣦⣧⣀⣰⣇⣸⣀⣤⣀⣇⣷⡿⠼⠟⣻⡄⠀⠀⠀⠀⠀⣠⡖⢃⣠⣴⣾⡿⠋
⠀⠀⠀⢸⠇⠀⠀⠛⣿⡏⠙⠂⠀⠀⠀⠀⡋⡇⠀⠀⠈⢅⣿⢹⣽⣏⣯⡏⣟⣿⡳⠃⠀⢰⠋⠇⠀⠀⠀⢀⣼⠏⣴⣿⠟⠉
⠀⠀⠀⢀⡄⠀⠀⠀⠘⢿⣿⡀⢣⡀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠉⢉⠁⠀⠀⣠⠞⠁⣀⣀⣠⠎⣡⣾⣿⠏
⠀⠀⠀⢰⡁⠀⠀⠀⠀⠈⢿⣿⣀⡹⢆⣴⠉⡆⠑⢦⣄⠀⠘⠆⠀⠀⠀⠀⠀⠹⣀⣴⠟⠁⠰⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠻⡆⣀⠀⠀⠀⠈⣹⣿⡷⠋⠙⣏⠁⠀⠀⠈⠙⠛⠛⠻⠶⠶⠶⠂⠊⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠛⠳⢷⡤⠃⠈⠠⡀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣠⠆⠀⠀⠽⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⢁⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣤⠞⠋⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣄⠀⠀⠀⠀⠀⠈⠑⠲⢇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡼⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠙⣈⡤⠟⠛⠉⠓⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄
⣀⣀⠀⠀⠀⠒⢀⣀⣤⠤⠶⠶⠛⠋⣁⡠⠤⠄⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠉⠁⠀⠒⠀⠀⠀⠀⠒⠂⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⠄⠀⣀⣀⣀⣄⣀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣠⣤⣤⠤⠤⠤⣤⠤⣤⣀⡀⣀⣤⣰⣦⣴⠆
⠀⠀⠀⠀⠀⠀⠀⣰⠿⡿⣍⠻⣗⡒⠰⣟⡳⢉⣽⣟⢿⣿⡿⣶⣭⣟⠯⣫⡿⢿⣶⣆
⠀⠀⠀⠀⠀⠀⠀⢿⢸⠐⣽⡧⠹⡿⠑⠺⡻⡿⠻⠍⡣⢛⢿⣿⡿⣿⣿⣮⣔⢩⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢘⣏⣶⣘⣿⡀⠇⢠⢀⡇⠀⠀⠐⠀⢸⣆⡠⢃⣿⣿⣿⣿⣷⣻⣽⣿⡀
⠀⠀⠀⠀⠀⠀⣴⢛⣭⣍⣉⡉⠘⠒⠋⢙⣷⠦⠤⠤⢔⣘⣠⠔⣨⠀⢛⣹⣿⣿⣿⣞⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠦⣦⡐⠀⠀⠉⠉⣛⣯⣤⣤⣄⣤⣾⣉⣁⠚⣠⣿⣿⣾⣽⣿⡿⣿⡾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣴⡖⠮⢿⣿⡿⢿⡛⠷⠊⠁⠀⠙⢿⣿⣿⡿⢛⣭⣬⣿⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣴⣤⣄⣀⡀⠀⠈⠁⠀⠙⢛⣫⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣤⡀⠀⠙⣿⣯⡹⣿⠀⠀⠀⠀⠐⣿⢿⢿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡿⠓⠦⢼⣾⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠘⢿⣿⡄⣤⣄⡀⢰⣾⢷⡏⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢀⣀⣀⣀⣀⣀⡼⠃⣽⡿⣷⣝⣯⣿⡾⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⠿⣿⣿⠀⠀⢹⣤⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡯⢿⣿⣤⡐⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣽⢿⣿⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⣡⢿⡿⠁⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡶⢲⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⠉⠁⠀⠐⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠴⠂⢠⡤⢐⡺⡭⣭⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠸⠓⠲⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠⣤⣤⢶⣶⢄⠀⠀⠀⣤⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⡾⠛⣿⣏⠳⣾⡥⠤⠾⠛⠓⣿⡶⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣾⠃⢀⣠⠿⠒⠚⠉⠉⢻⠏⠀⠀⠀⠀⠀⠀⠀⣰⣦⡤⢶⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⡉⠀⡀⠛⡾⠋⠀⠀⠀⠀⠀⠀⣿⣀⣀⣤⣀⠀⠀⠀⣿⣿⠁⠀⢸⡇
⠀⠀⠀⡀⠀⠀⠀⠀⣾⣏⡉⣉⣤⡟⠀⢠⣦⠴⠶⢳⣶⣶⣜⣩⣭⣭⣿⣿⣶⣤⣹⣿⡄⠀⣼⡀
⠀⠀⠀⠈⠲⢤⡀⠀⢿⣿⣾⣿⠿⢇⣴⠯⠤⠴⠾⠛⠛⢿⣾⣿⠿⠟⠛⠛⠻⠿⣿⣿⡇⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠙⠢⣌⠻⡏⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⢧⣠⣦⣶⣦⣤⣀⡀⠀⣿⣷⣶⣦⣄⠀⠀⠁⣿⣿⣿⣶⣿⣿⣷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣽⣁⣀⣄⡀⣿⡿⣦⡉⠀⠈⢙⣿⠷⣦⡀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢿⣿⣿⢿⣿⣿⣿⣿⣧⣬⡙⠦⡄⠀⣿⠀⠹⣿⣷⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⣯⢷⡿⢟⡛⠙⣿⣷⣯⣟⣿⢻⠶⠿⡷⣾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⣍⣻⣶⣿⣿⣿⣌⡿⢉⣿⡟⣷⣶⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⢬⡘⢪⣿⡏⢉⠛⠻⡟⢻⣿⡿⢿⣿⣿⡿⢻⣿⠿⠟⢻⢿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠈⠀⠈⠉⠁⠀⠀⠐⠃⠀⠉⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠻⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢴⠊⢿⣿⣿⣿⣧⡈⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⢁⠍⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣼⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢾⣅⡐⣿⣿⣿⣿⣯⡐⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡨⢉⣤⣿⠏⠙⣿⣿⣿⣿⣿⣇⠀⠀⠰⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⢀⡾⢃⣾⣿⠤⠤⣾⣇⠉⠻⢿⣿⣿⣷⡀⠀⠐⠆⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠸⠇⠸⣿⣠⠀⠠⠋⠙⠃⠀⠀⠉⠻⣿⣷⣶⣾⣶⠛⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣠⣬⣿⣄⣀⣤⡾⡿⢿⣾⠃⣀⣉⠐⢆⠀⠀⠀⠈⣿⣿⣿⣿⡀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡍⢻⣦⢉⣿⣿⠀⠀⠰⢾⡏⠀⠛⠉⠀⠀⠀⠀⣀⣄⠘⣿⣿⣿⣿⣷⣾⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣿⡷⣾⠋⠉⠶⠄⠀⡸⠰⡟⠀⠀⠀⢀⣦⣄⣤⠀⣄⠀⠟⠻⣿⣿⣿⣆⠀⣉⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠘⡟⢀⣴⣤⣷⡴⣿⠈⢻⣶⣶⣄⣠⣿⣿⡶⠀⢸⡇⠀⠀⠉⢹⣿⣿⣿⡟⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠘⢷⣏⣙⣿⠋⠛⠷⣤⣤⣄⢿⣿⠛⠛⡿⢿⢷⡎⠀⠈⠰⣤⣷⣶⢿⣿⣿⣄⢀⠰⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠰⠛⠻⠿⣤⣶⣤⠀⠈⠟⣠⣿⠇⣠⣿⠒⡿⠃⣀⣛⣿⠿⠿⠏⠀⢹⣿⡿⠿⢷⣴⠖⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠆⠀⠀⢀⡀⢀⣄⠀⣀⣤⠀⢺⡿⢿⣿⣿⣿⣧⡀⢰⣿⡟⡟⠃⠀⠐⠛⠿⣿⡇⢀⣾⡅⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣤⠀⣴⠞⣿⢧⣤⣶⣿⣏⡉⠻⣧⠽⣿⣿⣿⣿⣿⣿⣿⣽⢻⣤⢛⣼⣿⣷⠋⣠⣾⣿⣷⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣷⣤⣿⡄⠉⠐⠉⢿⣷⡿⠋⣀⣹⣶⣾⣿⣿⣽⠟⣭⣿⠏⠸⣿⣿⡝⠋⣹⣤⣽⡛⠛⣯⡀⠀⢈⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⣿⣿⣷⢆⣤⣥⣿⣿⣿⡀⢉⡀⢺⣿⣟⣩⣿⣿⣿⣿⣶⣿⣿⣶⠷⢎⡁⠀⠚⠉⠙⠛⢿⡿⠄⢁⣤⠺⣿⠆
⠀⠀⠀⠀⠀⠀⢠⡶⠀⠀⠈⣿⣿⣿⣤⣤⣈⣀⣨⡉⢿⡿⠉⢻⡿⢿⣿⠟⠛⠿⢟⢻⡇⠻⠛⢠⣀⣿⠀⠀⠀⢺⡆⢼⣧⣄⡈⢉⣥⡀
⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣯⣥⠘⣷⣀⣼⠓⢂⣠⣶⠶⡶⠋⠈⠿⠀⠀⣟⠻⢻⣤⠄⠀⠀⠛⣮⣉⣾⣧⣤⣽⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⢀⠘⠙⢿⣿⣿⣿⣿⣿⢿⣿⣿⣾⣿⣿⣿⡷⢏⣹⣿⣿⣧⣼⣇⠀⠀⠾⣧⣴⣿⡁⠀⠀⠀⠀⠉⠁⢩⣙⣿⠟⠉⠀⣡
⠀⠀⠀⠀⡀⢀⡇⣀⣼⣧⣷⣤⣿⣿⡿⠛⣿⣿⠉⣿⣿⡿⠛⢿⣤⣾⣿⣿⣿⣿⣿⡻⣿⣶⣾⡏⢁⣄⠙⢓⣤⣤⠄⣤⣴⠛⠛⠻⣶⣾⣷⡿
⠀⠀⠀⠀⣷⣾⣿⣿⣿⣿⣿⡙⣻⣿⢁⣀⠁⠙⡟⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡴⠟⢻⣧⣤⣶⣤⣉⣋⣛⣯⣤⣾⡇⢠⣿⣿⣿⣇
⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣉⣿⣿⠖⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢙⣷⠂⣹⣿⢉⠽⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣷⣶⣷⣄
⠈⠿⠿⢻⣿⣿⣿⠛⡜⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⡿⠋⠙⣿⣶⣿⡁⢙⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⠟
⠀⠀⠀⠀⠉⠙⠃⠰⠿⠿⢿⣿⣯⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣿⣿⣿⣿⡿⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⢿⣿⡿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⣾⠿⠿⣿⡟⣛⣀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠈⠛⠿⠟⠿⠿⠛⠛⠛⠻⣿⣿⠷⠿⣿⣟⣿⠖⠋⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⠏⠙⢻⡏⠉⣙⠛⢛⣏⠛⢹⣽⠟⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣀⡀
⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠁⢀⡀⣀⡄⠉⠓⢶⣦⣄
⠀⢸⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠼⠁⠉⢹⡂⢐⣇⣀⠀⠀⠀⠙⢿⣧⡀
⠀⢸⣦⢻⡀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠚⠉⠉⣉⣀⣉⣀⣀⣀⡀⠀⠉⠙⠛⠒⠶⡄
⠀⢸⣿⡆⢧⠀⠀⠀⠀⠀⠀⠈⠧⢤⡴⠶⣒⣾⣿⣿⠉⠀⠏⠚⠢⢤⢌⣦⠀⠀⠀⠀⠀⡟
⠀⠘⡇⣷⠸⡆⠀⠀⠀⠀⠀⠀⠀⢹⡵⣿⣿⣿⡿⠁⠀⠲⣭⠭⢏⡂⠀⠉⢳⣤⣤⠴⠞⠁
⠀⠀⢷⢹⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⠿⢿⠁⠀⠒⢲⡌⠓⠋⠀⠀⠐⡿⢧⠘⡆
⠀⠀⢸⡌⣿⡘⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⢏⢹⠀⢠⣤⠞⠣⣄⠀⠀⠀⢀⣿⢴⣵⡃
⠀⠀⠀⣧⢹⢇⢹⡀⠀⠀⠀⠀⠀⠀⢴⣥⣢⣾⣳⠿⠷⣽⣂⠸⡆⠀⠀⢎⢿⡻⡃⠘⢆
⠀⠀⠀⢸⡌⡾⡈⣇⠀⠀⠀⠀⠀⠀⢿⡸⡿⠁⠀⠀⠀⠀⠑⢣⣵⣤⠴⣿⠑⢮⡁⠀⠀⠑⢄⡀
⠀⠀⠀⠀⣧⢱⢃⢹⡀⠀⠀⠀⠀⠀⠘⠀⣇⠀⠀⠀⠀⠀⢀⡼⠛⠀⠀⣷⠀⠀⠙⠢⡀⠀⠀⠈⢑⡦
⠀⠀⠀⠀⠸⡌⡞⡄⣧⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⣖⡾⠋⠀⠀⠀⠀⠻⣦⣄⣀⠀⠈⠓⢖⡉⠁
⠀⠀⠀⠀⠀⢷⢱⢣⢸⡄⢀⣤⡀⠀⠀⠀⠀⣠⠞⢉⡽⠁⠀⠀⠀⠀⠀⠀⣘⣦⠉⠙⠲⠶⣖⡉⠁
⠀⠀⠀⠀⠀⠸⡆⡎⡆⣧⣿⣠⠇⠀⠀⢀⡞⠁⢀⣾⡀⠀⠀⠀⠀⠀⠀⢤⣻⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢳⣰⠟⠁⣸⣁⣤⡀⢠⡟⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⣿⡍⠁⠀⣀⠀⣤⠀⡷⣼⡆
⠀⠀⠀⠀⠀⢀⡼⡥⠖⣋⣥⢼⡇⢹⡟⠀⠀⠀⠀⢹⣇⣤⢀⡀⣄⢸⠛⠇⠁⠀⡴⠻⣤⡟⢷⡇⢿⠛
⠀⠀⠀⠀⢀⣾⣹⠁⠚⣉⡤⠴⢿⡼⠃⠀⠀⠀⠀⠸⠏⣿⡼⠹⠿⠻⠇⠀⢀⡞⠁⠀⠙⠀⠈⠃⢸
⠀⠀⠀⠀⢾⣀⡿⡆⠈⣡⡤⠆⢸⡄⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠋⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⠠⠀⠀⠀⠀⠀⠀⠀⠀⣻⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠠⠒⠈⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣀⢴⡤⠴⠓⠀⠀⠀⠀⢸⠀⠀⣀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠤⠐⠛⠉⠀⠀⠀⢀⡀⡀⠀⣈⣺⠼⠓⠚⠉⠙⣦⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠒⠀⠀⠀⠀⠀⠀⢀⡰⠍⠋⠁⠀⢈⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠊⠉⠀⠀⠀⠀⠀⠀⢺⡇⠀⢈⣤⣤⣶⠶⠓⠒⠚⡂⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠿⠑⠛⠉⠁⠀⠀⠀⠀⠠⡄⣀⣠⣿⠇⠀⠋⠁⠀⠀⠀⠀⠀⠀⢰⣄⠀⠛⢧⣀⣀⣀⡀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣇⡯⠴⠾⣟⡋⠉⠩⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠈⠉⠋⠉⢁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⡬⠉⢒⠈⠁⣥⠟⠷⠛⠁⠀⠀⠀⠀⢀⣀⣀⣤⣤⠾⡾⠿⢿⣀⣀⣀⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠲⠶⠖⢒⡫⠈⠀⠀⠀⢸⡆⠀⡀⡴⢀⡄⠴⠒⠋⢹⡅⠹⣿⣤⣶⣾⣿⣿⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠻⠈⠀⠀⢤⠀⠘⠀⠀⠀⠀⠀⠀⣷⠀⠀⡴⠛⠀⠀⠀⠀⣠⣷⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣫⣠⢴⣶⣶⣿⣿⣿⡿⢻⠿⠿⣿⣿⣿⣵⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣯⣼⣯⣤⣥⣾⣷⣖⡚⠻⢉⣑⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣭⣤⣤⣬⡁⠈⣿⣿⣿⣿⣦⡄
⠀⠀⠀⣶⣶⣤⣀⡀⠀⠀⠙⣶⣶⠒⢀⡤⣤⣶⣶⣶⣿⣿⣟⣛⣛⣛⠻⠿⠿⠟⠻⣂⣽⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣏
⠀⠀⠀⠈⠽⠟⠛⠿⢷⣶⣤⣬⣭⣶⠿⠿⠖⢸⡋⠋⠀⡉⣈⡉⠉⢉⣩⣄⣀⣴⣿⣿⣿⡿⣿⣿⣿⣿⠃⠟⠉⠉⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠁⠀⠀⠀⠓⠂⠈⠉⠁⠀⠈⠉⠉⠉⠉⠉⠈⠻⠛⠿⠛⢫⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⢿⡿⣿⣿⣶⣶⣦⣤⣤⡄⢠⣄⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣘⣿⣯⣲⣦⠀⠀⠀⠀⢀⠄⠁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠈⣧⠀⠀⢠⣿⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡈⢿⣿⣿⣆⣿⡇⢠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⣉⣽⣿⣿⣿⣿⣿⠷⠚⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠋⣩⣿⣿⣿⣾⣶⡶⢲⣾⡑⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣧⣤⣾⣿⣿⣿⣿⠟⣸⠃⠈⠻⡿⣄⠁⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⡀⠋⢠⡄⣤⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠋⠀⢀⣴⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣽⣯⠀⠀⣿⣿⡟⡇⣿⠿⠛⠃⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡗⡀⠀⢿⣿⣿⣧⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⡇⠀⠀⠛⢹⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠀⢠⣤⣿⢸⡟⣿⣿⡄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⢈⣹⣻⣿⣿⣷⡶⠶⢦⣤⣄⣀⣀⡀⠀⠀⠀⢀⠠⠀⠀⠀⢀⣿⣿⣼⣧⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣤⣮⣿⣿⣿⣿⣿⣿⣷⣂⣀⡤⠖⣾⣿⣏⢻⣿⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠁⢷
⠀⠀⠀⠀⢶⣾⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣽⣿⠿⢛⣁⣀⣀⣿⣿⣷⡬⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠛⠛⣿⣯⣍⣉⣉⣉⡙⠛⠛⠻⠿⡯⠔⠚⠋⠉⠉⣙⡻⣿⣿⣧⢀⡀⢰⣁⠀⠀⠀⡆⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠐⠿⠿⠿⣿⣿⣿⣻⣻⣶⣶⣦⣤⣄⣀⡀⠐⠉⠁⣀⣿⣿⣿⣧⡬⠙⠿⠿⡿⠀⣂⣠⡤⠞⠋⠁
⠀⠀⠀⠀⢸⣶⣦⣤⣀⣀⡀⠊⠉⠉⠛⠛⠻⠿⣿⣯⡤⠴⣚⣉⣽⡿⠿⣿⣿⣿⣿⣿⣻⣟⠁⡿⠟
⠀⠀⠀⠀⠙⠿⠷⠾⣯⣿⣿⣿⣶⣶⣦⣤⣀⣤⠼⠗⠊⢉⣉⡠⠼⠁⠀⠀⣌⠁⠀⠀⢠⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⣿⣧⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⡼⢽⡧⠀⠼⣏⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣻⡗⢀⡚⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠷⢦⣯⠤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⡤⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣒⣒⣢⠉⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠄⠀⠀⢠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣄⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠀⣻⠀⠀⠀⠀⣸⠇⠀⠀⠀⢠⡴⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠠⠔⠃⠀⠀⢀⠴⠋⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠉⠉⠛⠛⠛⠶⠀⠀⠈⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠚⠛⠛⠛⠳⠀⢀⡤⠶⠛⠋⠉⠉⠉⠉⠙⠛⠷⣦⣀⠀⠘⠉⠀⢀⡴⠋⠀⠙⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠀⠀⢠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠰⠋⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⠀⠀⣠⠞⠁⠀⠰⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡴⠋⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠓⠒⠛⠒⠒⠒⠲⠶⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⣀⣤⣤⠤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⢛⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠐⢂⡄⠉⠉⠉⠁⠐⠒⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣊⣥⣄⣀⠀⠀⠺⣿⢶⣿⣿⠄⣠⣶⣶⠀⠀⠉⠑⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣼⣿⣿⣿⣿⣧⠀⢀⣠⣀⡠⢋⣾⣿⣿⡿⣣⣼⣿⠟⢠⣄⣤⣬⡉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⡴⠀⣼⠟⡿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⠉⠉⠋⣾⣿⣿⣿⠙⠿⠿⠜⠛⠛⢂⠤⣜⡂⠤
⠀⠀⠀⠀⠀⠀⠀⣰⣫⣾⣿⢃⣼⡟⡌⠁⠋⢻⣿⣿⣿⠏⢰⠰⣴⡀⠀⠰⠀⠁⢀⠴⠒⠒⠒⠒⠀⠀⠀⢴⣿⣿⣶⡶⠖⠒⢷
⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⣷⣾⠛⢡⠀⠀⠀⠸⣻⣿⠃⢀⡟⣧⣿⣧⡀⠀⠀⠠⠃⠀⠀⡀⠊⠐⠀⣠⣶⣿⣿⣿⣿⠿⣆
⠀⠀⠀⠀⣠⣴⣷⣿⣿⣿⣿⡏⠀⢠⠀⠀⡀⡀⠈⠃⢰⣼⡇⠸⣿⣿⣷⢀⠀⠀⠀⡠⠊⠉⠀⢢⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⢰
⠀⢀⡴⢋⠝⣫⣿⣿⣿⣿⣿⠇⠀⢸⡀⠀⠱⡐⣄⠀⢸⣿⣿⣄⠈⠛⠿⣿⣦⠀⢸⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⡏
⣰⠋⠀⠀⠊⣡⣿⣿⣿⣿⡵⠂⢀⣾⡇⠐⢄⠘⠮⡳⢎⢿⣿⣿⣷⣦⣄⣀⠈⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⡃⡜⠀⡞
⢻⣀⠀⠀⠀⠺⣛⣭⠞⢁⣠⣶⡿⠃⠘⡄⠈⠓⢤⣘⣆⠳⢍⡻⠿⠿⠿⢿⣿⡦⠀⠀⠀⡈⠉⠒⠄⡈⠙⠻⠿⣿⠞⠀⡸
⠀⠈⠙⣶⡾⠟⠋⠀⠀⣿⡟⠉⠀⠀⠀⠈⠦⡐⢤⣈⠉⢅⣀⠈⠉⠉⠉⠐⠢⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠤⣉⡒⠇
⠀⠀⢸⡇⠀⢀⣴⣾⣷⠋⣽⣗⡀⡀⢀⣠⣤⣌⡒⠤⣉⠉⠩⠕⠲⣄⡀⠀⠀⠀⠉⠒⠦⠤⠤⣤⣀⣀⣀⣀⠀⠀⠀⠉⣢⠁
⠀⠀⠘⣇⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣤⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠭⠀⠀⠐⠊⠁
⠀⡴⠋⠁⠩⠾⠟⣻⣿⣎⣝⣛⣷⣆⣀⣈⣉⠉⠛⣿⣿⣿⣿⣿⠀⣶⣤⣄⣀⣤⡤⠤⠄⠒⠈⠁
⠾⠒⠒⢶⡠⡄⣍⠹⣿⠟⢿⣿⣿⣿⣫⡞⠁⠀⢸⣿⣿⣿⣿⡏⢸⣿⡟⢿⡿⠙⠀⢀⠁
⠀⠀⠀⠀⠳⡱⣌⢷⡅⠀⠸⢟⢭⣻⡿⣷⣄⠀⠀⠈⠙⠿⠟⢀⣿⣿⣿⢸⣷⡦⢔⡀⢀⣠⣤⣤⣤⣤⣤⣤⣀⣀⣀⡀⣠⣤⠄
⠀⠀⠀⠀⠀⠙⢾⡶⠇⡠⠶⣶⣦⣬⣉⡛⠻⢷⡀⣤⣀⠀⠠⣿⣿⣿⡟⣼⣿⣷⢰⣍⣾⣿⣿⣿⣿⣿⣿⠧⣀⡴⣨⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠂⢬⣛⣶⣮⠉⠛⠛⠛⠚⢻⣦⣝⢷⣤⡀⠈⠉⠰⣿⣿⠏⣰⣿⣿⣿⣿⣿⣿⣿⡇⢰⢋⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣶⣦⣽⣿⣿⣿⣿⣶⣄⣀⠈⠁⠀⠹⠟⠙⠿⢿⣿⣿⣿⢧⣓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⠙⠋⠁⣠⠞⠉⠻⢿⣿⣿⣿⣷⣦⣄⡀⠀⣠⣴⣿⣿⣿⠛⡗⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⡼⠁⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠹⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣯⣻⣿⣷⣿⡏⠒⠄⢹⡈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣝⢷⡝⢟⢮⡋⠓⠤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠠⣀⠙⠦⡀⢨⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢔⡀⠑⢄⣸⠿⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠪⠅⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠐⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠁⠀⠁⣀⠛⠀⠐⠃⠀⢸⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⢠⠟⠀⠀⠘⠂⠀⠀⠒⠲⠂⠶⠶⠟⠀⠀⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠚⠀⠀⠀⠀⠀⠀⠘⠃⠀⠶⠀⠀⠃⡖⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠂⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢴⠀⠀⣤⡀⠀⠶⠂⠀⠀⠀⠂⢀⡼⠝⡄⣀⢀⡂⡇⠀⠠⠒⢁⠀⠊⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠛⠀⠀⠀⢀⡀⠄⠈⢀⡴⠛⠒⠂⠀⠀⠄⠀⠀⠒⠂⠁⠀⠀⠀⠤⠀⠁⠀⠐
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠀⠀⠈⢁⠀⠀⠂⠙⠤⣀⠀⠐⠄⠀⠔⠀⠤⢄⡀⠀⠈⠁⡀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠒⠂⢀⣀⣀⡠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠒⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠐⠒⠒⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡵⠋⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢶⡺⠛⠛⠓⠶⣤⠀⠀⢸⡟⠲⠦⠤⠤⠴⢶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣋⠀⠀⠀⠀⣀⣼⡇⠀⣿⣝⠷⠶⠤⠤⠶⣾⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⣧⣉⡙⠛⠛⢋⣹⡿⠃⠀⢻⣍⡙⠛⠛⠛⠋⣿⡟⠀⠀⠀⣀⣤⠴⠚⣋⣥⣙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⣥⣍⣉⣛⣫⣽⣿⠀⠀⠀⠀⠈⠙⠛⢛⣛⣫⣥⣤⠶⠚⠋⠉⠀⠀⠉⠁⠀⠉⠓⠌⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣤⣀⣀⣤⠾⠛⢀⣀⣀⣤⡤⠶⠒⠛⠻⣿⣿⣿⣷⣄⠐⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣟⠋⠉⠉⠀⠀⠀⠀⠀⠀⣷⠈⠻⣿⣿⣿⣷⣄⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣀⡴⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡶⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⣿⣿⣿⣿⣦⡛⠀⠀⠀⢀⣠⡴⢚⣩⡴⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⡛⢦⡀⠈⠳⣄⡀⢄⡀⠀⠀⠀⠉⠻⠷⣦⣬⠙⢿⣿⣿⣿⣦⡴⢞⣫⡵⣞⣫⠵⢶⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⡛⢦⡙⢦⣄⠈⠛⢦⡙⠶⣤⣤⠤⠤⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⠿⢛⣋⡭⠶⣚⣻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡙⢦⡙⢦⡙⠷⣄⠀⠙⠶⠶⠶⣒⣛⣯⠽⣷⣿⡿⢿⣿⣿⣿⠶⣛⣭⠵⠖⠋⢩⡴⣞⠛⠛⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣝⢦⣙⢦⣈⠳⠶⠖⢛⣋⣭⠵⠖⣛⣭⠴⢖⣻⣿⣿⠟⠋⠁⠀⠀⠀⠀⣿⣜⣃⠀⠀⠀⣀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⢦⣝⠷⠶⢚⣋⣥⠴⢞⣫⡭⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⢻⣤⣉⠙⠛⢋⣉⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣝⠷⣚⣫⡥⠖⠛⠉⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⡀⠀⠀⣿⠦⣤⣍⣉⣩⣭⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢹⠒⠁⠀⠀⠈⠙⣆⠀⠙⠶⣤⣤⣤⣤⡤⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⢤⣤⠤⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣤⡤⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢛⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣖⣯⠭⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠒⢲⡄⣿⠁⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣖⣾⣿⣧⡿⠀⠀⠀⢀⣀⣸⣧⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⢀⣀⣿⡷⠶⢚⣛⣉⣉⣀⣠⣤⣤⣤⣤⣤⣤⣤⣽⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣀⣰⡏⠉⣟⣋⣉⣭⣭⣥⢤⢤⢶⢶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣬⣭⣉⠙⡇
⠀⠀⠀⠀⠀⠀⢀⡤⠤⠴⠶⠖⠚⡛⠉⠩⢹⡇⠀⡿⠷⠷⠓⣛⣛⣛⣉⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣍⣉⣛⣛⣤⠇
⠀⠀⠀⠀⠀⠀⢸⣐⣀⣁⣩⣬⠤⠴⢖⣒⣿⢀⠉⡟⢛⣛⣉⣉⣩⣭⣤⢤⡤⣤⢤⡤⡤⡤⣤⢤⢤⣤⣤⣬⣽⡇
⠀⠀⠀⠀⠀⣯⣭⣵⣶⠶⠶⠶⠿⢟⣛⣛⡿⠀⠤⣿⢹⡇⡇⢸⣀⣇⣸⣸⣇⣿⣼⣥⣧⣧⣿⣸⣼⣿⣷⠿⠿⠗⠚⠲⣦⡀
⠀⢀⣤⣴⠚⡓⢒⠒⡉⣉⢉⡉⣉⠁⠄⠠⠀⠆⢀⡟⠛⠛⠋⠉⢉⣁⣀⣤⣤⠤⠶⠶⠚⠛⠋⠉⠉⠀⠀⠀⢀⣀⣀⣀⣟⣿⠆
⠀⠘⠦⣤⠤⠥⠬⠤⠤⠤⠶⠶⠶⠖⠒⠒⠛⠉⠉⠙⠒⠚⠛⠋⠉⠉⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠒⢚⠋⠉⠉⣁⣸⡿⠃
⠀⠀⠀⣇⣐⣀⠀⠀⠀⠀⠰⠐⠐⠂⠂⠂⢀⣀⣁⣈⣀⣠⡤⠤⠤⠶⠖⠒⢛⣉⣉⣡⣤⠤⠴⠶⠒⢾⣿⡟⠉⠉⠉⡿⠁
⠀⣀⣀⢸⡟⠛⠛⠿⠿⠿⠿⠭⠭⠭⠭⠭⠭⠭⠴⠶⠖⠒⠒⠒⠚⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠳⣌⣯⣤⡄⠀⢸⠇
⠈⠉⠉⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡿
⠚⣉⣱⣢⣿⣽⠷⠲⢶⣦⠤⢤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡷⢤⣤⠖⠲
⠚⠁⠀⠈⠑⠦⣀⣀⡞⠙⠻⢷⡲⠶⠋⢈⡽⠿⠭⣭⣽⠛⠻⢷⣒⠒⢒⡖⠶⣖⠒⠲⢲⡶⢶⡒⠒⢒⣿⡿⠭⣿⠛⠳⣤⣀⣀
⠀⠀⣠⢤⣀⡀⠀⠀⠀⠀⠀⠀⣈⡛⠒⠋⠀⠀⠀⠀⢈⣉⡓⠒⠚⠛⠋⢀⣀⠈⠙⠲⠶⠃⠀⠉⠳⢮⣍⡽⠂⠈⡷⠦⢤⡉⠁
⠉⠉⠁⠀⠀⢈⣉⣉⣉⡒⠒⠚⠉⠉⠙⠒⠦⠤⣄⡤⠞⠉⠉⠛⠲⠦⠤⠜⠉⠙⠒⠦⣄⣀⣠⡟⠙⠲⠤⣄⣠⠴⠃⠀⠀⠙⢦
⠀⣀⣠⡴⠞⠋⠁⠀⠀⠉⠙⠒⠤⣄⣀⠀⠀⠀⢀⣀⡤⠖⠋⠉⠉⠒⠒⠤⢤⣀⡀⠀⠀⠀⠀⠀⣀⡤⠚⠉⠉⠓⠒⠤⢄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠐⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠑⢤⡀⠙⢄⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡙⢮⠢⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣀⣈⣈⠀⠐⢅⠈⠀⢁⡤⠞⢁⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠍⢢⠔⢁⠔⠉⠀⠀⠀⠈⠒⢄⠡⡒⠁⠐⢊⠡⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠥⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠠⠀⢈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠚⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⡀⠀⠀⠈⠀⢣⣀⣤⣤⣄⣀⡤⠴⠚⠉⠉⠉⠉⠉⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠚⠫⣉⢀⣀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠤⠴⠚⢉⠰⠲⠁⢀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⡻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢮⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢰⣐⣋⡂⠆⠀⠀⠐⢉⠀⠜⠅⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠦⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠍⠭⠽⡟⠉⠉⢰⡄⠒⠐⠒⠒⠋⠉⠒⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠻⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢲⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠛⠛⠿⠿⢷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣶⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣾⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢀⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⣿⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣸⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⡶⢒⠀⠈⢰⢠⣷⣄⠴⠖⣲⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣞⠫⣅⠢⣷⡓⠀⠀⠃⠠⣼⠟⢿⣿⣿⣿⠿⣍⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢝⣥⣏⡙⠛⣿⣧⣽⡧⢤⡤⢀⣽⣧⠀⣉⠁⡈⢰⣿⡿⡟⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣞⣥⡾⣿⣦⣩⡿⢶⠏⣻⣷⣽⣿⣓⣚⠋⠻⣷⡟⢻⡇⠀⣗⠀⣼⣉⣳⣿⣦⡀
⠀⠀⠀⠀⠀⣀⣀⡀⣀⣀⠜⡋⣹⣾⡟⣃⢌⣹⣽⠟⣿⠂⠿⣿⡟⢿⣿⢿⣿⣹⣬⡻⣄⣷⣤⣉⡄⠙⣿⣏⢅⠙⢿⣶⡄
⠀⠀⠀⠰⠟⠩⣍⣩⡿⢧⡡⣿⡿⢿⣼⠃⠀⣀⣨⠀⣿⡜⣿⣷⡷⣾⣿⣿⠿⣿⣿⣿⣮⣿⡟⠻⠿⣶⣎⢻⣎⡙⢾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠛⠀⠀⠀⠀⠙⠛⠋⢗⢈⣇⠸⣿⣷⡹⢻⣿⣶⣮⣿⡿⣿⣿⣷⣿⣷⣾⣿⣿⣿⣩⣄⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡉⢻⡓⠉⠹⢿⠿⣿⣿⣞⣷⣶⣌⠻⠿⠿⣿⡿⠯⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠨⠀⠀⢀⠘⠀⠛⠩⠻⢿⣏⡿⢷⣿⠓⠪⠱⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠊⠀⠀⠉⠉


⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣦⣦⣷⣴⣤⣶⣤⣾⣵⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉
⣏⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦
⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠻⠟⠿⢻⣿⡿⠛⠛⠛⠛⠛⠛⢛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⢿⣿⣿⣤⣶⣤⣦⣤⣤⣤⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣿⣿⡿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠁⠀⠰⣷⠀⠀⠀⠀⠀⢐⡈⠙⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⢰⣶⣿⣦⣀⠀⠀⠉⣉⣋⣀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠛⠛⢚⣳⣿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⢸⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⢀⠗⠀⣀⠴⠒⠉⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢾⠘⣿⠳⣄⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠠⠀⠃⠹⣧⠈⢃⠄⠀⠀⠁⠀⠀⠀⠀⠀⢠⣿⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣼⡏⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠁⠐⠀⠀⢻⣆⠀⠀⣱⣇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣶⣶⣦⣰⣤⡠⢤⣤⣌⣿⣿⣷⡄
⠀⠀⠀⠀⡘⠀⠘⡄⢀⣀⣻⣄⣴⠟⣨⠓⢄⠀⠀⠀⠿⠿⢻⣟⣿⣿⢹⣿⠿⠿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣠⣇⠀⠰⢃⣼⣿⣿⣿⣿⣧⢹⣧⣌⠳⣄⠈⠼⡀⣾⣿⣿⣿⣸⣿⡇⠀⠈⠙⠻⠝⡻⢿⣿⣿⣿⣿⣧⠀⠀⠄
⠀⢀⣴⡿⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣳⣆⣥⣿⣿⣿⣿⣿⣿⣷⣶⣴⣦⢤⣀⣉⡳⠿⠿⣿⣿⣿⣧⠀⠰⡄
⠀⣸⡟⠀⠀⠀⣾⠟⢩⣿⠟⠉⠹⣿⣿⣿⣿⡟⢹⣿⣿⣿⣿⣿⣿⡟⠛⠁⠀⠀⢸⣿⠀⠈⠉⠙⠛⠿⢾⣿⣿⣿⣧⢰⣿⣦
⠀⣿⠁⢴⠀⣠⣸⣇⣛⣀⣀⣀⣀⣹⣆⣏⣛⣤⣿⣿⣿⣿⣯⣥⣼⣧⣤⣤⣄⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣈⣙⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠙⢉⣿⣿⣿⡿⠿⠻⠏⢿⣿⡏⠉⠉⠉⠉⡻⠏⠉⡟⠻⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⢷⡀
⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⡴⠁⠀⠺⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢨⡏⠋⠉⢉⣉
⣧⣴⣦⣤⣤⣤⣼⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⢦⡤⠤⠴⡦⠶⠤⡶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣌⣾⣇⠀⠀⠀⠁
⢥⠀⠀⠀⠀⠀⢰⡇⢶⡀⠀⠀⠀⠀⠀⠀⢿⣿⡄⢀⠶⣤⡄⠀⠀⠀⠀⠀⠀⠀⢰⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠈⣿⣿⡏
⠀⠃⠀⠀⠀⠀⠈⣿⣶⣾⢶⣤⠀⣠⣦⣤⣾⣻⣿⣟⣴⣯⣤⣤⣴⣤⣶⣶⣶⣶⣷⣶⣦⣤⣤⣀⣴⣷⣶⣤⡜⣧⣿⣿⣿⠁
⠀⠀⠀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠹⣿⣿⠿⠟⠉⣿⡟⠻⠿⣿⠏⢙⣿⣿⣿⠿⠻⢿⣿⣿⠿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⢹⠗⡄⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣀⣿⣧⡄⠀⢀⡀⢸⣿⠘⡀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⢀⣶⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣧⣾⣿⣤⣦⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡖⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣄⣀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⠶⠀⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡉⠉⠛⠛⠟⠛⠉⠛⠢⠦⣤⣀⡀
⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣤⣤⣄⣤⣄
⠀⠀⠀⠀⠈⠉⣛⠿⠛⠛⣉⣉⣀⣀⡈⠋⠙⠛⠛⠛⠛⠛⣻⣿⣿⠛⠻⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠻⣿⣋⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣴⣾⡿⠟⠋⠉⠛⠻⠷⢶⣦⣴⡶⠾⠿⠛⠿⠟⠛⠛⠿⠿⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣴⡟⠀⡀⣿⣷⣄⢀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠥⠂
⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⣴⣿⣿⣷⣥⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠑⠄
⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⣽⣿⣿⣿⣿⣿⡆⠀⠀⠀⠁⠀⠀⠀⠀⡜⢩⠂⠁⠀⠀⠑
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠀⠀⢿⣿⣿⡷⠾⠇⠘⠁⢎⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀⠀⡸⢠⠃
⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠉⣿⣿⣟⣋⣙⣸⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣰⠃⡰⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⡆⠀⠸⡀⠀⠀⠀⠀⠀⠀⢀⡐⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⠲⠤⡀⠀⠀⠈⠀⠀⢠⠃⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀⢻⠀⠀⢇⠀⠀⠀⠀⠀⡵⠉⠉⠀⠀⠈⠛⠿⠛⠻⠛⠋⠁⠀⠐⢧⣾⣿⣷⠀⢐⠊⠀⠘⢇⠀⠀⠀⠀⠀⢰⣷⡦
⠀⠊⠈⢇⠀⢸⠄⠀⢀⣀⣼⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠋⠨⣿⣿⠟⠉⠀⠀⢀⣀⠴⠘⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠈⡆⠈⣦⡔⠉⠀⠀⢀⠀⣀⣴⡆⠀⠈⢷⠀⡤⠞⠀⠀⠀⠀⣠⠘⠉⠀⠀⠀⣤⣴⣶⣶⣶⣦⣄⣠⣶⣿⠟⠋⠙⠁
⠀⠀⠀⠀⢹⣄⢀⣠⠂⠀⢀⡈⣿⠇⣿⡇⠀⠀⣠⠞⠂⡶⠂⠀⠀⠀⠂⢀⣦⡀⢘⣷⣾⣿⣿⣿⣿⣿⣿⡏⠟⠉
⠀⠀⠀⠀⣿⡏⠙⢰⢀⣷⣾⢿⠋⠀⡿⠟⣠⠜⠁⢀⡌⢙⣦⣴⣤⠷⣌⣩⠟⠉⢹⣿⡿⠿⣿⣿⡟⢿⣿⣷
⠀⠀⠀⢀⠘⣿⡶⠆⠴⠟⠉⠁⠀⢸⠕⠴⠐⠃⠠⢊⠃⠈⢰⣎⠻⣷⠀⠁⠀⠀⢸⣯⣀⣴⡟⣿⣤⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⢸⠷⠶⠶⢶⣮⣨⣀⣭⣴⣿⣄⠈⠧⣀⡀⠀⡘⡛⣿⣿⣷⣿⣿⡟⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡚⠁⠀⠀⢀⠀⠙⠋⢡⡁⠀⢉⠻⣶⡀⠉⠚⠺⢷⡄⠿⢿⡿⠿⡿⠇⢰⣿⣷⣄⣀⣀⣀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠄⣠⠝⠋⠀⢀⠤⠂⠁⡀⠀⠀⠈⢷⠀⠀⠀⡈⠻⣦⡀⠀⣠⠛⢆⠂⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠁
⠀⠀⠰⠀⠀⠀⠀⠔⠋⢀⣠⣴⡾⠁⠀⠀⠀⠙⣷⣤⣄⢸⣧⠂⠀⠀⠀⠹⡿⡇⠀⠰⢉⣧⠀⠀⠀⠀⠀⠀⢺⣿⡿⠁
⠀⠀⠀⠀⡀⠀⠰⢀⣶⣿⣿⡿⣧⣤⣄⠀⠀⣄⣿⣿⣿⣿⡃⠀⠀⠀⠀⠂⠱⣜⡋⢀⠘⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⢀⠆⣼⠋⠉⢾⣼⣿⣿⣿⡄⡀⣎⠉⠉⢙⠛⠿⣄⠀⠀⣀⡀⢳⡘⣗⣡⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠃⡃⠀⠈⣾⣿⣿⣿⣿⡇⡽⠃⠀⠁⠀⠀⢀⢺⠂⠀⠹⣧⡿⣷⡸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⣸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⣸⠛⡂⠠⠷⢼⠈⡳⠿⡩⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠉⣿⡘⠀⠘⢆⣽⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠠⠀⠀⠀⠀⠀⠀⠀⠧⣄⣀⣀⣿⣧⠀⠀⠀⠠⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠁⠀⠀⠀⠀⠀⠉⠀⠀⠐⠀⠐⠎⠉⠉⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⣇
⠀⠀⠀⣀⣤⣤⣶⣶⣾⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠀⠰⠄⠀⠁⠀⣤⣤⣀⣀
⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠶⠿⠿⠟⠛⠁

⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠀⠀⠀⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠳⣒⠤⠤⠤⠔⠒⢊⣡⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡀⠀⠀⠊⠀⠀⠀⠀⠏⠈⠀⠀⠁⢀⠂⠀⣀⡰⠄⠀⠈⠀⠀⠀⠀⠀⠀⠉⠙⠓⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢄
⢀⣁⠤⠊⠀⠀⠀⠀⠀⠀⠐⠀⠤⣀⠂⠀⠀⠠⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠒⠒⢀⣀⠀⠀⠀⠀⠑⠤⣄⣠
⠁⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⣀⡴⠀⠀⠀⠀⠀⠀⡀⢊⡀⠄⠐⢊⡩⠕⠛⠃⢀⠀⠀⠀⠀⠀⠈
⠤⠊⠅⢀⡴⠛⢙⠶⡦⠀⠀⠀⠀⠈⠐⠀⠤⢀⣀⣁⡠⠞⠋⠀⠀⠀⠀⠀⢸⡾⠊⡠⠤⠴⠶⠋⣤⠂⡁⠬⠭⠗
⣘⡥⠖⣁⡤⠚⡡⠞⠁⡠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣉⠀⠤⢤⠀⠉⡳⠋⡀⠤⠴⢶⡄⠀⠀⠀⣰
⠁⠒⠋⠁⢔⣊⣀⠀⠊⡴⠃⠀⠀⠀⠀⠀⡀⠤⠒⢂⠀⠀⠀⠀⠀⠀⠀⠀⢜⢋⠀⡠⠊⠉⠀⠘⢁⡡⠔⠊⢉⣽⠀⠀⠀⢰⠁
⠀⠀⠈⠈⠀⡽⢷⠖⠋⡠⢞⠗⠀⠀⢀⠌⠐⠁⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⢢⡹⣤⡔⡀⢀⠖⣩⠄⠂⡉⢩⠂⠀⠀⠀⡇⣐
⠤⠐⢊⡵⠉⣠⠖⠒⢉⠔⠃⡀⠀⠀⡆⠘⠀⠀⠀⠀⠠⢀⠀⢈⠀⠀⠂⠀⠀⠀⠉⠲⠤⣀⣈⣉⣁⡠⠄⠊⠀⠀⠀⠀⠀⣇⢻
⡠⠔⠈⣀⠔⠁⡠⠖⢁⡠⠞⠁⠀⠀⡇⢠⠀⠀⡠⠀⠃⠀⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠃⠉⣡⡤⠔⠈⠤⠐⡉⠀⠀⠀⠀⠀⢣⠈⡄⠀⠁⠀⠐⢤⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⢄⠂⠈⠁⠀⠆⠈⠁⠀⠀⠀⠀⢀⠄⠀⠠⠎⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣀⠀⠀⢠⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠭⠷⠒⠂⠀⠀⠀⢀⠇⠠⠁⠀⠠⠀⠀⠂⠠⠀⠘⠀⠀⠀⢀⠀⠁⢠
⠠⠄⡀⠚⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠀⠀⡄⢸⡀⠂⠀⠈⠄⠀⠀⠀⠡⢠⠀⠀⢀
⠀⠠⠋⠈⠁⠀⠉⣸⠀⠀⠀⢀⠐⡀⠔⠊⢉⡀⠤⠄⣠⡄⠀⠀⠀⠀⠀⠈⢷⠀⠈⠀⠀⠀⠁⠀⠙⠀⠀⠀⠀⡄⠀⠀⠀⠀⠢
⣂⠀⠈⠁⠀⢀⡔⡵⠀⠀⢀⡠⡀⠄⠒⠉⢀⡤⠚⣉⠀⠤⠤⣄⠀⠀⠀⠀⠈⠣⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢳⢄⠀⠀⠁
⡀⢀⣀⠤⠚⣩⠞⠁⠀⠀⠘⢋⠠⠒⠉⠋⠉⢬⣟⡥⠐⠊⠉⢉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡁⠤⠀⢀
⠠⠥⠤⠒⠋⠀⠀⠀⠀⠀⣠⡤⠔⠀⣉⠽⠀⠀⠊⠄⠐⣈⠭⢽⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡑⢤⠊⢀⡀⠀⢀⡤⠒⣉⢠⠴⡚⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⡄⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⡛⠋⠤⠤⠥⢖⠉⠀⣂⠔⠁⠀⠀⠀⠀⠒⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⢧
⠂⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠂⠁⠀⠍⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠘⠒⠂⠀⢒⠀⠀⢀⠄⢰⠀⠀⠀⠀⠀⡈
⠀⠁⠆⠐⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠀⠈⢀⠀⠀⢀⠔⠁⡠⠃⠀⠀⠀⠀⡀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢿⣿⣿⠿⢿⣿⠟⣣⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⣵⣶⣶⣶⠞⣢⣴⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⣡⣾⣿⠿⣋⣵⣾⣿⠟⣋⡛⠿⠿⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣋⣼⡿⢋⣵⣾⣿⠟⣋⣴⣿⣿⣿⣿⣿⣷⡌⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⢛⣥⣾⣿⣿⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢟⣡⣾⣿⣿⣿⠻⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣱⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢟⣩⣶⣿⡿⢛⣥⣿⣿⠃⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢠⣾⠟⣫⣴⣿⡿⢟⣡⣶⣿⣿⣿⣿⣄⠙⠛⢉⣉⠙⢿⣿⣿⣿⣿⣿⠿⣋⣴⣿⡿⠟⠉
⠀⠀⠀⢀⣴⡿⢱⣿⣿⣿⣿⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⠿⣋⣵⣾⣿⠟⠉
⠀⠀⠀⣾⣿⣿⡘⣿⣿⣿⡿⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣉⡀⢺⣿⣿⣾⣿⠟⠋
⠀⠀⠀⠻⣿⣿⣿⣌⢿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣾⣿⠿⠋
⠀⠀⠀⠀⠈⠙⢿⣿⣦⡹⣿⣷⡜⢿⣿⣿⣿⣿⣿⡿⢟⣩⣶⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡜⢿⣿⣮⣥⣤⣭⡛⣩⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣎⠻⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠿⠿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠈⠛⢿⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⢸⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡾⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⠟⠻⢷⣦⣄⡀
⠀⠀⠀⠀⢻⣧⠈⠉⠻⢷⣦⡀⢀⣠⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠈⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢀⣴⣦⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠛⠀⠀⠀⠀⣹⡇
⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢀⣴⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⣴⡟⠁⣀⣤⣾⠟⠋⠈⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⢸⣿⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣦⣴⡾⠟⠋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⢀⣠⣴⡿⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠰⣏⣝⣢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣍⣹
⠀⠈⠻⠻⣿⣵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⡿⠟⠟
⠀⠀⠀⠀⠀⠻⢯⡷⣄⡀⢀⢠⠠⢀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡠⠤⡄⡀⢀⣴⢮⡽⠋
⠀⠀⠀⠀⠀⠀⠈⠹⢿⣏⠟⣴⠟⠛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣦⠻⣙⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣣⣼⢟⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣻⣧⡜⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠋⠈⢻⣯⡻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡟⠁⠙⣌⠂
⠀⠀⠀⠀⠀⠀⠀⡺⠀⠀⠀⠀⠙⢿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠋⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⠑⢄⠀⠀⡠⠊⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢎⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢿⣧⣀⠙⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⢻⣧⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠙⢿⣦⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣶⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⠀⠀⡠⠊⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠑⢄
⠀⠀⠀⠀⢠⣾⢞⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⡳⣷⡄
⠀⠀⠀⣰⠟⣱⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣎⠻⢆
⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣠
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠐⠈⠉⠀⠰⠀⢀⠂⠀⢶⡔⠡⠐⡄⠀⠀⠒⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠁⣀⠐⣄⠈⠲⣦⠀⠀⠆⠀⠀⠀⢀⠈⠀⡐⠀⠈⠂⠀⠀⠄⢮⣱
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠊⡀⠀⠀⠀⠘⠂⠀⠈⠁⢀⣀⠀⠀⠄⠡⠀⠀⠀⠁⠀⠐⠀⠀⢀⠈⠱⡃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⣀⠑⢐⣀⣀⠠⠀⢠⠁⠁⠀⡀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠱⡄⠀⠂⠀⠀⠀⠀⠀⠈⠁⠀⣀⣁⣤⣼⡶⠂⠀⠀⠀⠈⢐⡄⠀⠀⠐⠀⢀⠄
⠀⠀⠀⠀⠀⡀⡀⠀⠈⠀⠀⠀⠀⠄⠀⠈⠉⠘⠿⠋⠊⣀⠀⠘⠀⣄⠀⠁⠀⠀⡀⠀⠀⠀⢠⠐⠀⠤⠀⠀⠀⠀⢐⠀⠀⢀
⠀⠀⠀⠀⢐⠀⠀⠀⡄⠀⠀⠀⠀⠒⠃⠉⠀⠀⠀⠁⠐⠡⠀⠀⠀⠀⠂⠀⠈⠘⡤⢔⡊⠁⠘⠠⠀⠀⠀⠂⠀⠀⠈⠠⡀⠘
⠠⠀⠌⠴⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠁⢈⣦⠞⠡⠀⠉⢦⡀⠀⠈⠁⠀⠀⠠⠑⠂
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⢈⣠⠄⠀⠀⠀⢠⠈⣂⠀⠀⠀⠀⠀⠀⠀⣴⡟⠣⠀⠀⠀⠀⢀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠒⠂⣤⣀⣀⣀⠀⠀⠀⠠⠀⠀⠐⠂⠀⠂⠀⠀⠁⠀⢀⣴⣾⡏⠁⠀⠀⠀⠀⠀⠉⠉⠙⢦⠀⠀⢀⠀⡄⠀⠤⢀⠠
⠀⠀⠀⠀⠈⠙⠛⣿⣿⡆⠀⠀⠀⠀⠐⠠⠄⠂⠀⠐⠒⠠⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣰⣶⠶⣶⣶⣦⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠉⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠖⠚⠛⠛⠋⠉⠉⠙⠛⠛⠛⠓⠲⠶⠶⠶⣤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠳⣤⣉⣉⣙⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⢀⣠⡟⠀⠀⠀⠀⠀⠀⠳⣄⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⢀⡴⠋⡼⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⣀⡴⠞⠋⠉⠛⠲⣤⣀⡀⣀⣤⠊⠉⠉⠉⠛⠋⠁⠀⠈⢻⡆⠀⠀⣟⠳⢦⣄⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⣰⠟⠀⣾⠁⠀⢀⡤⠖⣢⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⠢⣀⠙⢷⠀⠀⠀⠀⠀⠈⢷⡀
⠀⣸⡏⠀⠀⢹⡀⡴⠋⣠⠞⠁⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠤⢤⣤⣤⣄⡀⠀⠈⢉⡓⢦⣀⠈⢧⡀⠀⠀⠈⠓⠞⠀⠀⠀⠀⠀⠀⠸⣧
⢰⡿⠀⠀⠀⡼⠁⠙⠋⠵⠋⠀⠀⠀⠀⠀⣸⠀⠀⣀⡴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⡙⢲⡜⢧⠀⠉⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆
⣾⠃⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠌⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⣿
⣿⠀⠀⠀⠘⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡴⠃⠀⠀⠀⠀⠀⠀⢠⡞⢹⡄⡴⠃⠀⢸
⢿⡄⢸⡆⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠷⠋⠀⣠⠖⡆⠀⠀⠀⠀⡞⢠⠞⠈⣇⠀⡄⢸
⠸⣧⠀⣷⠀⠀⠀⢀⡾⠁⠀⠀⠀⣀⡤⠤⡀⠀⠀⠉⠓⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠛⠉⠁⠀⢀⡾⠁⡜⠀⠀⠀⠀⠀⠓⠉⢀⣠⠟⡼⠀⣿
⠀⢻⣆⠘⣧⠀⣰⠋⠀⠀⣠⠖⠊⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠧⠊⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⡼⠁⣸⡏
⠀⠀⠹⣆⠘⣧⣹⡀⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⡤⠖⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡼⠃⣰⡟
⠀⠀⠀⠙⢧⡈⠳⣝⠲⠞⠁⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⢠⣞⣡⠴⠊⠁⠀⠀⠀⠀⠰⣏⣉⣉⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⡾⠁⣴⠏
⠀⠀⠀⠀⠀⠙⢦⣈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠒⠒⠒⠲⢄⣀⣠⡾⢃⣴⠟⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣤⣤⣤⣄⡀⠀⢀⡤⠖⢦⠀⠀⠀⣰⠞⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⠴⣋⡵⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠈⢧⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠦⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⣀⣀⣠⣤⠶⠖⠛⠉

⠀⠀⢠⣶⣦⣄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣴⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⣾⣿⣿⣿⣷⣶⣿⣿⣿⣶⣄⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣼⣿⣇⠙⠛⠿⠛⠛⠁⣸⣿⣦⢠⣴⣶⣶⣿⣛⣛⣛⣛⣫⡅⣶⣦⣄⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠙⠻⠿⠿⠿⠿⣿⣿⣿⣧⡀⠀⠙⠻⣿⣷⣦⣤⣴⣶⣿⣿⠟⠃⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⠃⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣦⡄⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣴⣶⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣴⣿⠇
⠀⠀⠀⣀⠿⣿⣿⣿⣿⣿⣿⠿⣀⠀⠀⠀⢀⣠⣌⠙⠛⠛⠛⠛⣠⣀⡀⠀⢀⣴⣶⣶⣦⣤⣤⣀⣉⣁⣀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢾⣿⣦⣄⣉⣉⣉⣉⣠⣴⣿⡷⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⠋⠁⠀⠀⠙⠻⣿⣿⣿⣿⣷⠄⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠠⣄⡀⠀⣠⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣙⠿⠿⠿⠿⢟⣃⡀⠀⠀⠈⠻⣿⣯⣛⢿⣿⣿⣿⠂
⠀⠀⣠⣄⠛⠻⠿⠿⠿⠿⠛⢉⣿⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⠻⣿⣷⣮⣟⠋
⠀⠈⠻⢿⣿⣶⣶⣶⣶⣶⠾⠛⠁⠀⠀⠀⠀⠉⠙⠛⠉⠉⠛⠻⠿⠿⠿⠋⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣶⣶⡀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠷
⠀⠀⣠⣾⠛⠛⠿⠿⠿⢿⣻⣤⡄⠀⠀⢀⣸⣿⡿⠿⠿⠿⠛⠛⠛⢉⣀⡀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⣾⣶⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣄
⠀⠐⣿⣿⣶⣤⣤⣴⡾⠿⠛⠋⠀⠀⠀⠘⠻⠿⠿⠟⠚⠛⠛⠛⠛⠉⠉⠉⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⠿⠏⠀⠙⠿⠿⠿⠿⠛⠛⠛⠻⠿⠿⠛⠁
⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣦
⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠐⢤⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠉⠛⠶⣍⡻⢿⣿⣿⡿⠁
⠀⠠⣿⣿⣧⣄⣈⣉⣉⣉⣠⣿⣿⡇⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣟⣄⣀⠀⠀⠻⣷⣬⣭⣭⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠛⢶⣬⣉⡁
⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠂
⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠠⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡤⠤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⡠⠶⠋⢁⠀⠀⠉⠢⡀⠀⠈⠐⢤⣀⣀⡀⠄⠊⠁⡘⠀⣶⠀⠀⠈⠆⠉⠢⡀
⠀⠀⠀⠀⣠⠞⠀⠂⠀⠸⣧⠀⠀⢀⣈⠑⠒⠊⠀⣠⡑⢄⠀⠀⣠⠁⠀⠉⠀⠀⠀⢠⠀⠀⠈⢆
⠀⠀⠀⢰⡏⠀⠀⠈⢆⠀⠀⠀⠀⠈⠙⠃⢀⣀⠘⠛⠀⠀⠉⣉⠁⠀⠀⢤⣶⠆⠀⠀⠉⠉⡉⠘⡆
⠀⠀⠀⣎⡇⠀⠀⠀⠀⢡⠀⠀⠀⢀⡔⡿⠟⠈⠉⠐⠐⠊⠁⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠸⡧⠀⣽
⠀⠀⠀⣧⡇⠀⣀⠠⠄⠂⠠⡿⠁⠸⢌⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢤⡶⠆⠀⠀⠀⢀⠔⡿
⠀⠀⠀⢰⠚⣴⠁⠀⢀⡀⠀⠀⠀⠘⣖⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢠⠂⠀⠀⠈⠉⠀⣼⠇
⠀⠀⠀⠀⠳⠙⣦⡀⠛⣋⡀⣀⠀⣀⡈⠳⢗⣄⠀⠀⠀⠀⠀⢀⡠⠊⢠⡆⠸⠀⠀⠀⠀⢀⣜⠎
⠀⠀⠀⠀⠀⠙⢄⠽⢄⠀⠀⠀⠇⠙⠷⠀⠀⠉⠳⢄⣀⠠⠒⠁⠀⠀⠘⠃⠀⡆⠀⢀⠔⡧⠋
⠀⠀⠀⠀⠀⠀⠈⠳⢎⠑⡦⡀⢃⠀⠀⠀⢠⡿⠀⡀⠤⠄⣀⠚⠷⠆⠀⠀⢀⡧⢔⣩⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⡸⠙⢳⢤⠤⠤⠐⠉⠀⠀⠀⠀⠉⠒⠀⡤⢶⢉⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠧⣀⠃⠢⣄⠀⠀⠀⠀⢀⣠⠰⢉⡧⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠴⡀⡙⠐⡖⠘⡃⡸⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠃⠚⠁