Faeries Emojis & Symbols

⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠨⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⣀⣹⣿⣿⠃⠀⠀⠀⢄⠀⢠⣴⣿⡿⠃⠀⠀⠀⠰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣀⡀
⠀⠀⠸⠿⣿⣿⠟⠁⠀⠀⠀⠀⠉⢺⣿⣿⣤⣄⠀⠀⠀⢠⣿⣿⣿⣄⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠈⠑⣿⡿⣿⡧
⠀⠠⠤⣌⡿⠃⠀⠀⠀⠀⠀⠀⠀⠸⢿⣾⣍⡃⠀⠀⠀⠈⢿⣿⣟⡋⠀⠀⠀⠀⠀⢠⣿⣧⠔⠀⠀⠀⠀⠀⠀⠻⣷⣦⣉⠂
⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠤⢤⣄⡀⠀⠰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⠄⠀⠀⠀⠀⠀⠀⠈⠻⠿⣤⣄⣀⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣖⠊⠉
⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⡀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢀⣾⣶⡇
⠀⠀⠀⠀⢨⣿⡀⡀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⡿⠃
⠀⠀⠀⠀⢸⣿⣿⠉⠀⠀⠀⠀⠀⠀⠈⢿⡿⣿⠿⠆⠀⠀⠀⠀⠀⠀⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠠⢼⣿⣿⣧⣄
⠀⠀⠀⠀⠀⢹⣿⠗⠀⠀⠀⠀⠀⠀⠒⠘⢻⣷⠄⠀⠀⠀⠀⠀⣠⡖⠚⣿⣶⣀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⠟⠛
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⣀⣀
⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⠡⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣉⡲⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢻⣮⡳⣄⢠⣌⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⡀⠙⢿⣜⢷⣽⣧⢧⣤⣤⣴⣶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⣱⣶⣄⡆⢛⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢈⡹⠟⡿⢿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠍⠘⠷⣿⣯⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣒⣠⣭⣻⣿⣿⣿⣿⣿⣿⣷⣿⡟⢿⡯⠉⠀⠀⠀⠀⠀⠀⠀⢶⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⡭⡿⣿⣟⣿⡟⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠳⠿⣿⣟⠯⢶⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠩⢵⣾⣿⣿⣿⣿⣿⣿⡿⠋⠈⢿⣷⡀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢫⣉⡭⠙⢛⣹⣿⣿⣿⠏⠀⠀⠀⠀⠙⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠁⣀⣴⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣴⣿⣷⣶⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⡟⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⣙⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣾⣿⡿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣍
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⠉
⠀⠀⠀⠀⢰⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣷⡀
⠀⠀⠀⠀⢸⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⡇
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇
⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⣿
⠀⠀⠀⠃⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⡏
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⢸⣤⠃
⠀⠀⠀⠘⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠘⠃⢹⣤⠏⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢤⡀⢠⠟⡏⠈
⠀⠀⠀⠀⠸⣿⡆⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⠋⠀⠁
⠀⠀⠀⠀⢄⠙⢷⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣾⡏
⠀⠀⠀⠀⠈⠻⣮⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠳⣌⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⢀⠎⠼⠃⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠠⡀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢺⣷⣤⡠⠃⠀⠀⠀⡌⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⣰⡄⢀⡄⠀⠀⠀⣠⣿⣾⠃⢠⠁⠀⡼⠁⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠒⠰⣄⣾⠁⠷⠀⠀⠉⠉⠙⠃⡸⠀⢸⠃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡄⠀⠀⣤⠤⠐⢆⡟⠻⠆⠰⣞⣆⢰⡄⢴⠏⠁⢀⠇⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢋⢷⡀⠀⠀⠀⠸⠁⣠⠠⣄⡀⠀⣹⢸⡟⢧⡴⢀⣬⠞⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣿⣶⣠⠄⣠⡀⠹⣷⠄⠴⢾⣤⢠⠛⠎⣡⡞⠡⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⠿⢀⡼⠋⠈⠳⢌⢀⠐⢿⠁⣠⡀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠀⠀⣤⣿⠀⣠⡠⠄⠸⢀⠀⡈⡀⣏⠀⠈⠀⠀⠀⠰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏⠀⠀⠀⢻⡄⠡⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⢱⡏⠀⠀⠀⠀⠀⣷⣄⠺⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠁
⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⣿⠀⢸⠁⠆⠀⠀⠀⠀⣉⢿⠛⢦⡀⡀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣴⠟⠀⢸⠀⠀⡄⠀⠀⠀⣿⣳⣤⠀⣉⢳⡄⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠠⠈⡠⠀⢀⠔⠛⠙⡄⠐⣾⠀⠀⠸⣆⠀⠀⣿⡿⠉⠀⠀⠀⠙⠦⡀⣦⡸⠆⠀⠀⢋⣉⠙⠳⣄
⠀⠀⠀⠈⠀⠘⠷⣮⠑⡶⠀⠀⢠⠜⠻⠀⠀⠀⢸⡆⠀⠸⣧⠀⠀⢀⠀⠀⠀⠻⣏⠉⠁⠢⠄⣀⠈⠀⣀⡽⣆⠀⣀
⠀⠀⠀⣁⣠⠀⠰⢤⡟⡁⠀⢠⠊⠀⠀⠀⠀⠀⢀⡧⠀⠀⠻⣶⡞⠁⠀⠀⠀⠀⢸⠦⣕⢶⣤⣀⣬⣍⡉⠼⠟⠛⠻⢶⡤
⠐⢀⣀⣈⣀⣀⡴⠊⣼⡷⠖⠁⠀⠀⠀⠀⠠⠴⠿⢧⡀⠀⠀⠈⠳⠄⠀⠀⠤⠀⠘⠒⠬⠟⠛⠢⠄⠠⠦⠀⠀⠀⠀⠀⠼⠄
⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠔⢺⣭⠇⠀⠀⠀⢀⣀⣠⣶⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⢀⡄⠦⣿⣶⣉⣀⣠⣀⣪⡍⣽⣿⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣧⡄⠀⠀⣤⠻⣿⣶⣼⣿⣿⠍⢻⣾⣿⣶⠏⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⢃⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣟⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣿⣷⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⠯⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠇⠁⠀⣉⠉⠛⢻⣟⣷⠶⠤⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣿⣦⣀⣠⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣆⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠟⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⠛⠿⣿⣿⡿⠿⠿⠃⠄⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⡿⠷⣄⠙⠎⠹⠷⢤⡐⠲⣤⡀⠐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠙⢿⣿⣷⡀⠀⠁⠀⠄⠈⠀⠙⣦⠀⢡⡈⠣⢌⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣄⠀⠀⠀⠀⠔⢈⡄⠈⠁⠀⠀⠰⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⡀⠀⠀⠀⣸⡇⠀⠀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄⠀⢠⡟⣄⠀⠈⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⠋⡄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⠛⢿⡆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⡍⢀⠀⠘⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈

⢶⠿⠀⠈
⡆
⣄
⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠆
⠄
⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⣤⡴⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣾⣭⣾⠇⠀⠀⠀⠀⠀⢀
⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣄⡉⠙⣦⠀⠀⠀⠀⠀⣥⠔⠈⢸⣿⡇⠀⠀⠀⠀⠀⠀⠈
⠾⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠉⣁⠉⣿⡆⢠⣾⣧⡀⢠⡿⠤⢊⠸⠁
⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣿⣿⣿⡶⠀⠸⣯⣷⣾⠟⢉⣀⡀⡿⣧⡾
⠦⡤⠰⣤⡀⠀⠀⢠⣄⠀⠀⠀⠀⠋⠁⠀⠀⠀⢁⡌⢉⡿⠃⠀⠐⠁⠀⢸⣷⣿⣿⣧⡉
⠀⠀⠀⠛⠁⠀⠀⠀⡅⠘⠀⠀⠀⠁⠀⣠⠀⣰⣿⣷⣾⣁⣤⡤⣄⣠⣄⠀⠿⠿⢷⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠈⠹⢿⣿⣿⣿⣿⡆⣿⣿⣿⣶⣷⡤⢠⣄⠀⠀⠀⠀⢠⡀
⠛⠃⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡇⣿⣿⡅⠀⠀⠀⠀⠨⣧
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⡇⢸⣿⣷⠀⠀⠀⠀⢀⠛⠁
⠀⠀⠐⠉⠀⠀⢠⣧⠔⠊⠀⠂⠀⠀⠀⠶⠃⠀⠀⠀⠉⢹⣿⣷⣤⣿⣯⣷⣲⡦⠄⢀⠠⢜
⠀⠀⠀⠀⡄⠠⠞⢁⠀⠀⠀⠀⠀⠀⠂⢀⢀⡠⣤⠀⢿⣿⡉⠉⠉⣿⣿⣿⣿⣷⢀⡈⠆⣸⣦⡔⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶
⠀⢀⡁⠀⡇⣠⠀⠈⠀⠈⠀⠂⠈⠀⠀⠸⣯⣧⣬⣼⣠⣝⣻⣇⢀⣿⣿⣷⣷⣿⢻⣿⣦⡿⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠙⠛⠛⠛⠿⠷⣶⡶⠀⠀⡀⣀⣀⣤⣶⣿⢿⣿⡏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⠘⣿⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣼⣿⣿⣿⣿⣿⠀⢿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣷⠀⣿⣿⣷⣿⣿⣿⡏⣿⣿⡟⠀⢀⠠⠆⠀⠂
⠀⠀⠀⠀⠀⣀⡄⣀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠸⠁
⡤⠀⠀⢠⣴⣿⡿⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡀⣿⣿⣿⡿⠋⢩⡿⠛⢀⡀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠃⣤⡀
⠀⣀⣤⣼⣇⠀⠀⠀⣿⣿⣿⡿⣿⣅⣿⣿⣿⣿⣷⣿⣻⣿⣷⣸⣿⡟⠁⠀⠻⠀⠀⠀⠛⢶⠦⣸⣅⠄⠉⠁⢠⡤⠀⠉⢻⣄
⠉⠉⠉⠉⠻⠀⠀⣤⣿⠟⠁⢀⣼⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⠈⠀⠀⠓⠀⣽⣧⡀
⠶⠀⠀⠀⠀⢤⠸⣿⡇⢀⣰⡟⠉⠀⣿⠿⣿⣿⣿⣿⡎⢸⣿⣿⣿⣀⡄⢀⢶⣦⠀⠀⠀⠀⠘⡇⣠⣴⣾⢿⣀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣤⡄⢀⡀
⠀⠀⠀⠠⠤⠞⢶⡟⠀⠀⠈⢁⠀⠀⠁⠀⠻⣿⣿⣿⣴⡿⠿⠿⠋⠻⠃⠀⢾⣿⣤⣀⣿⣇⣰⠟⠛⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⢿⣿
⠘⠋⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⠖⠢⠄⠀⢀⣿⣿⣿⣿⣆⠀⠀⠀⢀⣴⣄⣼⢋⣤⠈⠉⠁⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣶⣿⢷⣄⡼⠀⠀⠀⠤⠀⠀⠁⠀⠉⣿⣿⡿⠋⠉⠉⠁⠛⠻⠟⠁⢰⠈⠉⠀⠀⠀⠀⠘⣿⣶⣤⠀⠠⠀⠀⠀⠀⠀⠀⢹⡶⠆⠀⠀⠀⣀
⢻⣿⡟⠛⣻⡶⠷⠓⠊⠘⠃⠀⠀⢀⢠⣶⣀⣿⣿⣧⠀⠀⠀⠀⠀⠀⢇⠁⣸⡂⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⡷⠀⠀⠁
⣤⣬⣡⣴⠏⠀⠀⠀⠀⠀⠰⠆⠚⠛⠛⠛⠛⣻⣿⣯⡤⠀⠀⠀⠀⠀⢨⣼⣻⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠙⡷⠷⢴⣶
⠋⣹⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠋⠀⠀⠀⣠⠿⠂⠉⠉⠉⠉⠉⠈⠘⠛⠷⠾⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⢰⣤⣤⣆⠰
⣿⣿⣏⣀⣠⣤⡤⠤⣴⣶⡇⠀⢶⣶⡄⠀⠀⠀⠀⠀⠀⣀⠐⠁⠐⠁⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠛⢻⣿⠧⠤
⠀⠈⢉⡉⠁⠀⠀⠀⠀⠀⣿⣄⣠⣯⣤⣀⣀⣀⣠⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⡾⡿⣿⢿⣿⣠⡀
⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠉⠉⢠⡝⠁⠀⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣧⣀⠀⠀⢀⣠⣴⣿⣿⣿⢀⣰⣿⣿⠋
⠀⠀⠀⡀⠀⠐⠀⠁⠀⠀⢰⡇⠀⠀⠘⠃⢾⣷⡦⠘⢈⣙⠛⢿⣧⣤⣤⠀⠈⠀⠀⠀⠟⠀⠐⠟⠻⣿⣿⣿⣿⣿⠛⠛⠛⠻⡟⠀⠈⢈⣿⡟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠤⠀⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡖⣭⣭⠛⢦
⠀⠀⠀⠀⠈⠀⣀⡈⠁⠼⡿⢏⣳⣤⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣖⣽⣽⡿⠛⢦⣶⣦⣵⠶
⠀⠀⠀⠀⠉⠚⠿⢿⡦⠒⠀⣾⣟⣛⣿⣦⡀⠀⢠⣶⣿⣿⡆⠀⢀⣴⣿⣩⣭⣿⠟⡓⠋⣻⡟⣋⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⣨⣿⡿⠿⢻⣿⣦⣼⣿⡿⢿⣷⣴⣾⣿⠋⠋⣻⣿⣦⣀⠠⣼⣷⠾⠂
⠀⠀⠀⠀⠀⠀⠈⠛⢻⣎⠛⢛⣙⢧⡴⠿⣿⣿⣻⠛⠀⣼⢏⣁⡈⠷⣤⠘⠃⣠⠀⠀⢘⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣏⡀⠀⠛⣚⢚⠛⠛⠃⠈⠀⠀⠈⠀⠙⢛⢗⡠⢀⣿⡃⠀⠠⢯⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠶⣿⡏⠀⠀⣰⡏⠋⠠⠂⣾⣧⠀⠀⠠⣿⣶⣅⠀⠀⠀⢻⡥⠜⢁⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠆⠈⠛⠀⠀⢠⡾⢿⣿⠀⢀⠐⠊⢿⣿⡷⠤⠔⠉⠡⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⢠⠟⠰⡾⢿⠀⠀⠀⠀⡈⠁⠀⠀⢶⠀⠂⢠⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠀⣠⠋⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡀⠈⢃⢲⣿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣂⠋⡹⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡄⢶⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⣿⣿⣷⣂⣠⡄⠀⠀⠀⠀⠀⠀⠶⡶⢢⣯⣾⠿⠟⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠟⠙⢿⠇⠀⠀⠀⡀⡠⠴⠂⠀⠳⠟⠉⠙⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⢀⣀⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠹⠛⠀⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠰
⠀⠀⠀⠀⠀⠐⠶⢷⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⣙⣿⣿⠋⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢟⣿⣾⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥⣾⣿⠗⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠛⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠘⢆⠀⠀⠙⠿⠟⠁⠲⠶⣦⣄⠀⠀⠀⠀⠙⣻⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡟⠋⠈⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣤⣶⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⢿⣿⣿⣷⣾⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⢻⣿⣿⣿⠉⠉⠙⠛⠛⠿⣶⣤⡶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠉⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡆⠀⠀⠀⠀⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡁⠀⠀⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠏
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⣀⣿⣿⣿⡟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⡌⢻⣷⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣿⠏⠻⣿⣿⣿⣿⣿⣿⣦⡝⢷⣄⡀⠀⠐⠀⠀⠀⠀⢠⣿⠿⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠉⠀⢸⣿⡟⠀⠀⢾⣿⣿⣿⣿⣟⠛⠁⠈⠉⠙⠁⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⠛⠃⠀⠀⠀⢸⣿⡀⠀⠀⠀⠉⠈⠛⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡅⢈⣿⣿⡀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠶⠛⠛⢿⣵⠦⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⠲⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣭⠆⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⢠⣤⣀⡀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢷⣌⠉⠀⡀⠀⡈⠒⡀
⠀⠀⠀⠀⠙⠛⣶⣤⡳⡄⠀⠀⠀⠀⢈⡀⠈⢻⣆⡉⢹⡉⡓⢢⣥⣽⢦⣵⡀⢄
⠀⠀⠀⠀⣄⠀⡀⠈⢷⠐⠦⠤⠄⠐⡈⢡⢶⡙⣽⣏⡑⣈⣞⢙⣦⣿⣷⣿⣿⣆⣡⡂
⠀⠀⠀⠀⠈⢿⣧⡹⣿⣆⣆⡀⡀⡀⠄⠾⡷⣿⣿⣿⣿⣿⣾⣧⣿⣿⣿⣿⣿⣿⣯⣿⡳
⠀⠀⠀⠀⠀⠰⣽⣿⣾⣿⣧⡮⢦⠰⣆⠔⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⡙⣿⣿⣿⣧⡾⢐⣵⣿⠶⣼⣍⣉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠑⣼⣿⣿⣿⣿⣯⡁⠉⠛⢦⣌⠛⠻⣷⡀⢛⢶⡻⣛⠻⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣥⣟⠿⣿⣦⠄⢀⡙⠇⢐⣯⠍⠘⣶⣄⡀⠰⠶⠶⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⣿⣷⣾⣿⣷⣶⣿⣶⠈⠃⠀⢀⡀⠈⣻⡄⣤⣀⣄⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣷⣤⣼⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠈⢹⣿⣿⠐⢶⡶⣾⠟⠁⠀⠀⠀⠀⠘⠁⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⣿⣿⣾⣿⣿⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠹⣄⠀⠀⠀⣿⡀⠀⣀⣤⣾⠫⡉⣹⣇⢸⣿⠟⢀⣀⣀⣤⣾⡇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠊⢿⣆⠀⢸⢣⣾⣯⡄⣀⠂⠁⣠⡽⢦⣿⣿⡿⣿⠟⣿⣿⡇⠀⠀⣴⡟⠁⠀⠀⢀⣠⣴⣿⡿
⠀⠀⠀⡌⣄⠀⢹⣧⣾⣷⣿⣿⣿⣿⠀⢸⣿⣿⠋⡹⠗⠋⠁⠀⢸⡿⢁⣤⣾⠿⠤⠶⠶⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣧⠸⡀⠻⣿⣿⣧⣿⣿⣛⢻⣷⣾⣿⣿⡶⠁⠀⠉⠀⠀⢼⡿⠛⠉⠀⠀⠀⢀⣠⡶⠚⠉⣡⡶⣀⠃
⠀⠀⠀⢨⡂⢠⡁⢬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡀⢀⣿⠃⢀⠀⠀⣠⣾⠟⠁⠀⠀⢸⣿⠟⡙
⠀⠀⠀⠘⣯⣉⠚⠛⠿⣿⣜⣿⣿⣇⣿⢿⣟⡷⠋⠁⠀⠀⣤⣾⣿⣿⣿⣿⣟⣟⡿⠀⠀⠀⠀⣸⣿⠎⠁
⠀⠀⠀⠀⣶⣭⠙⣓⡚⣻⣿⣿⣏⣿⣿⠿⠋⠀⠀⠀⠀⠀⠘⠋⠛⠿⠿⣿⣿⠟⠁⡀⠀⣀⣴⡿⡟⣸⡀
⠀⠀⠀⠀⠘⣿⢤⣀⠀⠉⠙⣛⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⣿⣿⠿⠛⠛⠛⠉⣸⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠘⢶⣶⣤⠀⣰⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠏⠀⠀⠀⢀⣴⣿⡿⢫⢿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⢶⣶⣲⣿⣿⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣿⣿⣿⡇⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠿⠘⢿⣿⡻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠛⠋⠉⣉⣵⡟⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⣬⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣁⠀⠀⢶⣿⣯⣽⡇⢸⠷⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣦⡀⠀⢿⣿⣿⠁⣶⣚⡠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠉⠛⠣⣄⠀⠁⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣦⠀⠉⢿⣤⣿⣼⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⣶⠀⠀⠀⠀⣭⡶⠒⠲⢶⣾⣿⠄⠻⣷⢾⡿⢿⡉⠑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠙⠲⢶⡿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⢼⣶⠟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣡⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣷⣶⣶⣤⣀⣀⣀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢻⣷⣶⣟⢷⣄⠙⠿⢿⣟⣿⣌⢻⣿⣿⢻⡇
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢿⣿⣿⠋⡻⣿⡋⠹⣦⣹⣷⣤⣄⣙⠛⠻⠿⠿⠷⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠉⠀⠀⠀⠘⠀⠀⠀⠛⠁⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⣀⠀⠁⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⣠⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠆⠀⢤⣾⡣⠆⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣮⡏⢉⠉⠫⠦⠤⡀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠶⡀⠉⠀⠀⠀⠀⠈⠀⢀⠀⠉⠝⠉⠁
⠀⠀⠀⢀⡠⠔⠒⠀⠀⠀⠀⠀⠬⠰⠦⠜⠒⠃⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢲⣶⠠⠀⣀⠠⠀⠀⠀⠀⠀⢀⡀
⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠤⠄⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠂⢄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⡎⢈⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⡀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⢻⣷⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⡟⢈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⣀⠀⠀⠀⠀⠁⠄⠀⠀⣶⡀
⠀⠐⠂⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠙⠋⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠣⠀⠁⠀⠀⠀⠀⠀⠀⠁
⣦⣀⢀⡴⠟⢀⣴⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⡄⠀⠀⣠⣶⣦⡶
⣔⣡⣌⣰⣿⢻⣿⣨⡇⢠⣾⠀⠠⠀⣀⣀⡐⠒⠐⠂⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣀⠀⠀⣼⠀⠻⣿⣟⡀⠲⢀
⠾⣿⢿⣿⣿⢿⣿⣟⣷⣿⣿⡀⠀⠤⠄⠀⠀⠀⢀⠀⡰⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠁⠀⠛⠷⢿⣿⡇⠠⠀⠀⠀⠻⣿⡄⠀⠈
⠿⣿⣿⣿⣹⣿⣿⣿⣿⡟⠛⠁⠀⠀⠀⠀⠆⠢⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⢧⣼⣿⡿⠃⠀⠀⠀⠀⠛⡀
⠀⠀⠈⠉⠛⠛⠛⠛⣉⡀⠀⠀⠀⠀⢠⠀⢀⣼⣿⣿⣴⡀⣤⡀⣠⣾⣶⣧⣴⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⡆⠘⠀⢰⠀⠈⣽⣷⣄⣀⠀⠘⠀⠀⠀⠰
⡦⠀⠀⠀⠀⠈⠀⠀⠠⠀⠐⠊⠉⠀⠠⢀⣾⣿⣫⣿⠛⣿⣿⣴⣿⡇⠉⠙⠛⢿⣷⡆⡀⠀⠀⠄⠀⠀⣤⣼⣁⣴⢶⣧⣤⡐⠁⠀⠳⢿⣿⡿⠓
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠙⠿⠿⠟⠀⠈⠻⢿⠟⠀⠀⠀⠀⠀⠉⠻⣷⣆⡤⠀⠀⠀⢿⣿⣿⢋⠙⠟⢻⣿⣦⠀⠀⢀⢠⠖⠀⠊⡀
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠈⢻⣿⡀⠀⢀⣼⣿⡿⠀⠀⠁⠀⠀⠀⠋⡰⢀⣄
⣿⣅⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⣠⡄⠋⠀⠀⠙⣄⠛⠋⡀⠀⠀⠀⡀⠀⠀⣀⣴⣼⢾⣿
⣿⣿⣷⣶⣾⣆⠀⠀⠀⠀⠀⠀⠀⠠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠛⠒⡒⠀⠚⠁⢀⡐⠀⠦⡀⠈⢇⠀⡀⠂⢄⠈⠁⣠⣾⣻⣟⣿⡟⣻
⣿⣿⡿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠋⠂⠀⠀⠀⠀⠦⣄⠀⠀⠀⣀⣤⡶⠜⠦⡀⠀⠀⠲⠄⠀⠘⠉⢿⠟⠛⠛⢻⣿
⣿⣿⣷⣿⣿⣿⡏⢰⡦⣀⣀⣀⣀⢺⣿⣷⣤⣤⣤⣤⣀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠤⠉⠊⠉⠁⠀⠀⠀⠈⠀⠂⢐⣶⣶⣿⣷⣶⣖⣶⣶⣾⣿
⣻⣷⣿⣿⡟⣿⣅⣿⣷⣽⣷⣾⡏⢸⣿⣽⣿⣿⢿⣿⡿⠃⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠂⠤⠤⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧⣼⣿⣿⣿⣟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠂⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣤⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⢷⡿⢋⣽⣿⣿⣿⣿⣿⣦⣰⣶⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣻⣿⣿⣿⢿⠿⣞⢻⣏⣿⣿⣿⣿⣿⣿⣽⡞⣿⡿⣀⣒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣿⣿⣿⡿⢹⣿⣿⣿⣿⡿⣧⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠻⣷⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣽⣯⣘⣺⣶⠆⢀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿
⣿⠿⠿⡿⠉⠀⠀⠀⠀⠠⠖⠸⢻⣿⣿⣿⣧⣜⣫⣿⣿⣿⣿⣿⣯⣿⣿⣿⡟⢠⣿⣿⣿⣶⣤⣤⣶⣦⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⢿⣮
⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢌⠛⠛⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣯⣻⣿⠋⠉⠉⠉⠉⠱
⡟⣤⣼⣿⣆⡀⠀⡀⠀⣠⡐⠁⠀⠀⠀⣾⣟⣿⣿⣿⣧⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡧
⣷⡞⢻⣛⠟⠃⢠⣿⣴⢇⢾⣷⣠⣀⢀⡿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡾⣿⣿⣽⣿⣿⣷⣿⣿⣿⣿⣿⣽⣿⣿⣿⡿⢷⣤⡤⡶⣷⣶⣆
⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⣿⣿⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⢹⣏⣭⢯⠻⠉⢹⡩⡙⣿⣿⠟⢻⡟⣿⠟⣻⠏⡩⠟⠛⠋⠉⠙⠙⠿⡉⣵⣾⣬⡁⣬⠃⢤⡌⡭⣿⠉⠉⣨⣽⣿⣧⣞
⣿⡏⢟⡏⢈⣐⠀⡌⢷⣄⣱⣟⣿⣛⣶⡈⠁⣰⡞⠗⣤⡄⢀⣀⣀⡀⣀⡘⠋⢁⣂⣠⣬⣾⡇⣈⢃⡁⠈⠋⠉⠀⠀⠈⠉
⣿⣷⡿⠁⠀⠀⠂⠙⣿⣟⠉⢫⣛⡿⢿⢷⣿⠏⢛⡛⢶⣀⡀⠀⠌⢻⡿⠿⠒⠋⠁⠘⢿⣿⡿⢿⡶⢁⣀⣤⢒⣀⠈⣘⡓
⡿⠏⠓⠲⢠⠴⠧⠐⠉⣀⠐⠰⣏⣙⢯⣺⣿⣴⣦⢠⣙⣟⠩⣷⣄⣉⠋⣿⣢⡾⠷⣎⠙⢻⡛⠋⢉⡺⠶⣿⣿⠟⠉⠙⢿
⡇⠀⠉⠐⠛⠁⠀⢀⢠⠀⡀⠀⠺⣟⡆⠄⠙⠻⣯⣿⣿⣿⠀⢗⣂⡏⢀⣿⠿⣷⣶⠏⢻⠋⠀⡀⠘⠷⣶⠻⣏⠀⢠⢀⠸
⡇⠀⠀⣶⣶⡈⠂⢿⣬⠤⡝⠀⠀⣀⠀⡖⠀⠚⠋⢘⠟⠟⢧⢸⡿⢁⡺⣿⡀⠈⠛⠃⠍⠀⠀⠇⣴⣾⡧⢀⣶⠋⠀⠘⣃
⡅⠀⡀⠀⣸⡗⠂⠀⠻⠁⠑⢄⠰⠿⠆⡑⠀⢃⢠⡘⠀⠂⢨⠉⠁⢚⠃⠙⠓⠈⢯⠜⠁⣐⠊⢦⡘⠇⠀⠘⠡⡤⠈⠉
⠉⠀⠀⢐⣛⣃⢴⢤⡶⠄⠘⠦⠌⣉⠷⠀⠀⠸⢷⠀⡷⠄⠈⠂⠀⠀⠆⠂⠜⢨⡘⠄⢠⠟⠐⠀⠨⢴⠀⢠⠄⠀⠀⡀
⡄⠓⠄⠊⠈⠁⠈⠀⡀⠀⡀⢀⢼⠑⣧⡠⡀⢠⠂⠀⢠⢠⡀⡀⠀⠀⢀⠄⢠⠀⢀⢀⠀⠄⢀⠀⠀⠀⠀⠁⠈⠙⢲⡠⣄
⠇⠀⣄⠀⠀⠀⠀⠀⠤⡏⢡⠠⡄⠀⠀⣄⠴⠊⠁⠀⠚⢤⡑⡄⠀⠈⣁⣰⣯⠶⠛⢿⠐⠀⠈⠀⢹⡷⢂⡐⠀⢀⣀⠣⣽
⠐⠷⠤⣴⡀⣤⣄⠀⠀⣤⠞⡷⡇⠠⢎⣪⣽⣿⠆⠀⠀⠀⠁⠀⠈⠲⠡⠛⠏⠀⠀⠈⠘⠢⢄⣅⣠⠁⠐⠀⢊⠉⠉⠀⠈
⠀⣈⠻⣿⡿⠟⢛⠀⠒⣥⠐⠺⠷⡄⠀⠈⠉⠉⠀⣀⠀⡠⠴⠄⠠⣀⣂⣄⠀⠈⡄⠠⡊⠤⠀⠀⠄⠃⠀⠀⠰
⠒⠙⠛⠀⡔⠁⢸⠀⠀⠐⣮⣶⢶⠒⠚⢶⠷⠄⠀⠉⠉⠀⠱⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠘⠈
⠀⠀⠀⠊⠀⠠⣞⡗⠀⠀⠀⡙⢃⡀⣰⠁⠀⠀⠀⠀⠀⢀⠐⠂⠀⢄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠓⠂
⡆⠀⠀⠀⠼⠓⠎⢶⡄⠸⠲⠃⠬⠡⠌⡙⠶⠴⢰⠃⣒⠂⣁⠀⠀⠀⢳⡴⠀⣀⡄
⡇⠀⡁⠀⠀⠀⠈⠑⠁⡧⢀⡰⢀⠠⠈⠉⠀⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠙⢸⡇⠀⠀⠀⠈⠀⠀⠀⠀⠠⢀⠶⠂
⡇⠀⠈⠀⠀⠀⠀⢀⠈⠀⠉⠋⠀⠠⠀⠒⣄⢀⡖⡀⢠⣤⠿⠶⢄⣤⡀⠀⡚⠀⠀⠀⠀⠚⠛⠳⠟⠛⠻⠋⠟⠀⠀⡀
⡇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠒⡧⣼⣷⢾⣩⣌⠉⠋⠀⠀⠀⠀⠀⠀⠌⠈⠓⣀⡀⢠⠠⣴⣾⣆⡀⠀⠀⠀⠀⡀⠈⠁
⣤⣦⣬⣤⣴⣦⣴⣤⣴⣤⣤⣿⣭⣶⣤⣠⣾⣿⣦⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣽⣤⣼⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣼
⣿⣿⣉⣩⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⣀⣄⣄⢀⡀⠀⠀⣲⡀⠀⠀⠀⠲⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠈⠉⠀⠀⢠⣄⠀⠙⠂⢀⠈⠀⠁⣉⠋⣷⡆⠀⠀⣬⠖⠠⣇⠀⠈⡲⠄⡉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⣄⠀⠀⣀⠀⠀⠄⣆⢬⡼⠓⠀⢠⣰⡧⣀⣀⣻⣇⡀⠀⠀⠀⢀⠉⢀⠙⣄⠈⠉⡒⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡻⠀⣧⡄⡈⣥⢾⣜⡓⢉⣁⡀⠀⡀⠈⠁⠀⠀⠀⠈⢛⣁⠐⠤⠤⢀⡉⠁⠳⢽⡄⠀⡇⠀⠀⣷⣼
⠀⠀⠀⠀⠀⠀⠀⢰⡈⢳⣶⣿⣦⣿⠿⢋⠁⡠⢈⣨⠍⣁⠐⠦⠀⠀⠀⠀⠀⠈⢥⡆⢀⠀⠨⡇⠢⣀⠀⣿⠢⣅⠀⠀⠹⣿⠀⢀⡄
⠀⠀⠀⢀⠀⠀⠀⠎⠀⣠⣿⡏⢱⢋⣴⠥⠐⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⢀⡆⣷⢀⠈⠀⢹⠆⢺⣷⡀⢣⢹⣦⠞⢹⡆
⠀⠀⠀⢸⡄⠀⡄⠀⣸⠿⣸⠷⣿⠁⣀⢂⡂⣔⣬⣰⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⢿⠓⠾⢬⣤⡈⢧⡑⣼⢠⣿⣦⣿⠁
⠀⠀⠈⡀⢧⣠⠓⣤⣿⢟⡟⣸⡿⠬⠋⠣⠑⠈⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣳⠌⢳⣄⠸⡋⠙⢫⣷⣼⠁⢸⣿⠁⢰
⠀⢢⠀⢧⠸⣧⠑⣾⢧⡾⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀⠀⠙⣿⡗⢾⣿⣊⣋⠄
⠀⠈⣧⢨⠃⠸⣿⣣⡾⢛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠶⣄⣿⣧⠸⣿⠈⣯⠆
⠀⠀⠘⣷⣀⠀⢻⣿⡡⣺⣻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣌⢻⣿⠀⣿⣇⡳⠂
⢦⡀⢰⡛⣿⣦⣾⣯⣠⣿⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣍⠹⣿⠳⣿⣧⣟⠁
⠈⢷⡌⡾⠰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣱⣿⣧⣿⣿⠃
⠀⠈⢷⡸⡇⢸⣿⣿⠁⣀⡀⠀⣤⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢹⣿⣿⣿⠇
⠀⠀⠘⣿⠷⣸⣿⣿⠘⡿⠛⡸⡯⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣤⠀⠈⠣⣼⣿⣿⣿⣆
⠠⣼⡇⣹⣯⢸⣿⣿⡏⠀⠐⣮⣿⢅⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢷⣫⣴⣶⣼⣿⣿⡇⢸
⠀⢨⣟⡵⢻⣿⣿⣿⣇⠀⢐⡫⢷⢝⡂⠀⢄⣧⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢭⣷⡋⠡⣼⣿⣿⡇⢸
⠀⠀⢻⣫⠌⢻⣿⣿⣿⡆⠀⢩⠽⣝⣃⣨⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣗⡂⢀⣿⣿⣿⡃⣾⠀⢸⠁
⠀⠀⠈⠧⣞⢄⢹⣿⣿⣿⣄⠈⠬⣯⢲⣵⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠬⢽⣿⠒⣾⣿⣿⣿⣶⠃⠀⠅
⠀⠀⠀⠀⠻⢎⢔⣿⣿⣿⣿⣦⠨⣧⣾⣿⢓⣂⣤⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣯⣄⠀⡄⠀⢀⣦⣄⢺⣿⣃⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠈⢏⣿⣿⣿⣿⣿⣷⣾⣿⠻⠛⢻⣟⣿⣿⣤⣤⠀⠀⠀⢀⠀⠀⠐⡩⣻⣽⠇⡇⣇⡼⣻⠏⡽⢷⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠿⣿⣿⣷⣬⣧⣼⣽⣽⣿⣿⣀⠀⣈⣄⣄⣀⣈⡴⠖⠓⣚⡛⠛⣿⣾⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⠷⣤⣀⡀⢹⣿⣟⠁⠀⠀⠀⡀⢀⠌⠋⠙⠻⣧⡊⠁⢀⣅⠚⢻⣛⣠⣿⣿⣿⣿⣿⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣽⠿⢿⣯⣭⣭⠾⢿⣤⣀⢀⠠⠊⢀⣱⡤⢶⣿⠖⡛⠛⠋⠛⠛⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⣿⣿⣧⣼⣾⣾⠾⠷⣦⠾⡉⠐⢣⡏⠀⠀⠀⣈⡥⠖⠚⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣙⣇⣀⣜⣁⣀⣈⣠⣎⣈⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⡛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠀⠉⠉⠁⠒⠢⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠦⠀⠙⢆⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠣⢄⡀⡰⠁⡴⠁⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠾⡀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡏⢀⡵⣕⠀⠀⠀⠀⠈⢢⣀⡠⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⡏
⠀⠁⠢⠀⠀⠒⠄⡀⢀⠔⠀⠈⠊⠐⣈⡻⠃⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢰⠁
⠀⠈⠉⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠌⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡇⠀⠀⠀⠀⢸
⠀⢰⠁⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⡠⢂⡔⠛⠃⠄⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⡇⠀⠀⠀⠀⢸
⠀⠘⠦⣀⡀⢀⡠⠃⠀⠀⠀⢀⠤⣚⡐⢛⣤⠤⣀⡀⠀⠈⠁⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⠀⠀⢧⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠞⠁⠄⠀⠣⠄⢠⠤⠊⠉⠓⠢⢤⠴⠃⠀⠀⠀⠀⢀⣀⡤⡞⠁⠀⣸⠀⠀⠈⠳⢤⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠀⠀⠀⠀⢰⠀⠀⡀⠀⠀⠀⠀⠀⠀⡤⠒⢉⣉⡀⠔⠀⡠⠔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠐⠁⠀⠀⠀⠀⡸⠀⠀⡱⠦⠴⡄⠀⠀⢰⡆⠠⣥⠀⢀⡀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡇
⠀⠀⠀⠀⡄⠐⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣀⣇⣀⣀⣧⠀⠀⠀⡇⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⢸
⠀⠀⠀⠐⠲⢄⣄⣠⢶⠊⠉⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣄⠀⣰⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠐⠄⠀⢀⠀⠀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⢢⣲⢟⠻⢁⡝⣱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⠛⡍⢁⠸⢺⣿⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⡙⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡌⡅⠀⡴⠛⡿⠢⣄⠀⠀⠀⠀⠀⠀⣀⡀⢁⣀⣣⣜⠄⠀⠀⠀⠀⠀⣇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣇⠀⢻⠀⠀⠀⠀⠀⠀⣁⠄⠁⠀⠀⠈⠀⠀⣀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⣿⠀⠸⠀⠀⠀⠀⠀⣀⡾⠀⠀⠀⠀⠀⣀⡤⠙⣲⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⠮⠀⠘⠈⠀⣧⠀⡇⣎⣷⣻⣾⠉⠀⠀⠀⣀⠀⣤⣄⡤⡊⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠿⠶⣄⣆⢰⢠⣿⠀⠁⠠⣾⢥⠅⠀⠀⢠⣾⢸⡗⠲⣯⣾⣱⠚⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠈⠀⠙⢛⡶⣾⣿⡆⢸⠶⠠⢀⡘⠀⣴⣿⡉⣋⠉⠆⡹⠟⠚
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⢤⡐⠆⠀⠀⢀⣽⡿⠇⢸⣀⣄⣂⡤⠍⢉⠄⠳⠟⠂
⠀⠀⠀⠀⠀⠐⠈⠀⠢⡀⠀⠀⡀⠀⠀⣣⡀⠰⣤⡞⢩⣴⣰⣶⡿⣿⢉⣴⢲⣏⠒⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢩⠁⠀⢸⠛⢠⣠⣽⡶⢿⣿⡉⠋⡁⣬⠑⢶⣾⡻⡷⠾⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⢀⡾⠀⢀⣯⢠⢸⠇⠀⡴⢦⠀⠁⠀⠀⢎⠽⠘⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢠⣿⡿⡿⠈⠀⡸⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⢰⣧⡏⠃⠸⠃⠐⠛⠁
⠀⣄⠀⠀⠀⠀⠀⢠⣾⠀⣾⠟⠀⠀⠀⠛⠃
⢠⠜⣥⣂⠄⠰⢣⣭⡇⠀⡷⠄
⠘⠁⠀⠀⠓⢢⢿⢻⠇⡔⠁
⠀⠀⠀⠀⠈⡌⠀⢿⡴⠃
⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⡀⡀⠀⠐⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠠
⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠆⠂⠀⠐⠄⣀⠀⣀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠠⠀⠀⢀⠀⠘⠖⠀⠀⠀⠰⠐⡂⠐
⠀⠀⠀⠀⠀⠀⠤⡌⠀⠀⠀⠀⠀⠀⠷⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢈⡄⠀⠐⠈⠒⠉
⠀⠀⠀⠢⠂⠀⠀⡄⣀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⡠⠀⠀⠤⠩⠀⠀⠀⢄⠀⠀⠀⠀⡠⠀⡀⠀⠁⠀⠀⠀⠀⠠⠈⢁
⠀⠠⡀⠀⠀⠀⢀⠀⢈⠁⠀⠀⠀⠈⠀⣤⠄⠀⠘⠤⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠀⠀⠂⠈⠀⣀⣁⠈⠀⠀⠘⠁
⠹⠆⠀⠀⠐⠀⣁⣠⢀⠄⠈⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢨⣤⠠⣄⡖⠢⠀⣀⠀⡀⠀⠈⠹⢆⡀⠀⠀⡀⡐⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡦⢀⠄⠀⠀⠀⠀⠙⠁⠐⠀⠀⠆⠈⠀⠁⠀⠀⠀⠠⠴⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⢀⡂
⠀⠀⠀⠀⠁⠀⢀⣄⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠳⡂⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠚⠛⢄⡀⣀⠀⠛
⠀⢘⢀⠀⠀⡼⡏⠁⠀⠀⡀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠄⡀⠐⠂⠀⡓⢠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠀⠦⠄
⠀⠦⠀⠡⠔⠀⠸⠀⠀⠀⠀⠀⠀⢦⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠀⠈⠁⠗⠀⢦⡀
⠀⠠⣄⠓⠀⣆⠸⢧⢀⣀⣀⣀⠸⠏⠁⠀⠀⠀⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠈⠢⠄⠐⠿⠀⠀⢠⠀⠀⠀⠀⠰⡦⠃
⠀⠀⠈⠁⠈⠁⠀⠤⠴⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠴⠆⠀⠰⠦⠆⠈⠃⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣰⢁⡁⠀⣰⣆⠀⢰⡖⢀⡀⣀⠀⣸

⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⣸⣇⢃⣀⣄⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠛⠀⠀⠀⠙⣿⣴⠋⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⡠⠄⠈⠁⠀⠀⢰⡆⠀⠀⣼⣿⢰⡏⠀⠀⠀⠀⠀⡄
⡄⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⣿⣁⣤⠖⠀⠀⠀⠀⠀⠀⠀⢀⠚⣷⣤⢀⣱⣿⣸⡇⢛⣀⠀⠀⠘⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠿⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠟⢥⣀⠀⣿⢸⢋⡀⢸⣇⡄⠀⡀⠲⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⣠⡿⠆⢹⣾⢿⡏⣿⠓⢀⣿⣀⣄⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣤⡤⠀⠀⠀⠀⡄
⠀⠀⠘⠀⠀⠀⠀⢀⡜⣄⠀⠀⠀⠀⠀⠀⠀⠈⣤⡘⢠⣼⡟⣳⡷⢸⣧⠈⣿⣤⣷⡾⠋⢀⣼⢤⣂⣀⣠⠀⡤⠄⢠⡀⠀⠺⢂⣾⡟⣋⡁⠀⠐⢃
⠀⠀⢂⠀⡄⠒⢀⡤⠟⠸⠀⡉⠋⠛⢢⡀⠀⢰⠌⡹⣦⣽⣿⣿⣿⠛⠛⣶⣿⣿⣿⣶⠶⡟⣧⠀⠉⠙⠧⠀⠁⠀⡸⢃⣼⣿⠛⠛⠉⠇⠀⢳⡾⠏⠁
⠀⠀⠈⠒⠑⣀⣼⠀⠀⠠⠦⠏⠁⠈⠀⠙⠆⠈⣠⣀⣡⠘⢿⡿⠽⢶⣦⣽⣿⣿⢻⣿⣦⣿⣿⡷⡦⠁⠀⠰⢀⣲⣆⣀⣼⣿⣧⣀⣠⠀⠀⠈
⠀⠀⠀⠀⡄⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠶⠆⠰⢿⣿⠓⣬⣳⣤⣶⣄⣿⣛⣿⣶⣗⡛⣉⣿⡷⠶⣤⣤⣄⠸⣿⣿⣷⣿⣬⠿⠿⢧⡤⠀⠀⠀⠀⠻⠶⠆
⠀⠰⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⣘⠃⠰⠀⠀⣴⡿⣿⣿⣷⣽⣿⣟⠉⠀⣈⣼⣟⢧⣁⡀⣬⣿⣿⡾⣿⣿⣿⣿⣷⡞⠀⠀⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⠿⢷⣀⣿⣿⣿⠷⠿⢿⣿⣿⣿⡏⢺⡿⡯⠒⠉⣠⣿⣿⣷⣿⡿⣿⣿⣿⣿⣿⣿⠁⢤⣶⡇⢀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠁⠀⢀⣠⢾⣿⠅⠀⠀⢐⣶⣿⣿⣉⠻⣿⣾⡿⣶⣴⡾⠿⠟⠻⠿⠿⠏⢙⣿⣿⣷⣿⣶⣷⣼⡋⠀⠚⠁
⠀⠀⠀⠀⠀⠛⢶⠆⠰⡄⠚⠣⣶⣖⣦⣤⣱⠞⠁⠀⠀⢸⣿⣿⣿⢿⣿⣷⣜⣿⣷⣿⣿⣆⣀⣤⣤⣤⡤⡐⠈⠙⡛⢿⣿⣷⣾⡿⢏⡀⠀⢀⣤⠔
⠾⢿⠿⣿⠛⢣⡴⣶⢳⡶⠖⣻⡿⠿⢿⡟⣿⡀⠀⢀⠔⠿⠛⠻⢤⣀⣽⣿⠏⠹⣿⣿⣿⡟⢡⣿⡿⡀⢐⡀⡈⠀⢻⠀⠹⣯⢹⣶⣎⣿⡤⠾⠁
⠀⠀⠀⠠⣛⡿⠓⠉⠉⠐⠾⣋⠠⠦⢾⡇⠉⠀⠀⢀⡆⠀⠀⠀⠀⠹⣿⠁⢰⣀⣿⣿⡟⣑⠿⢿⣿⣿⡿⣿⣿⠀⢾⠀⢻⣿⢼⠋⠉⠈
⠀⠀⠀⠀⢿⣦⠀⠀⠀⠀⢀⡝⣴⠷⡾⠃⢀⣴⢊⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⡿⣶⡈⢿⢛⠥⠋⣉⠰⠞⢀⣳⣿⠾⢶⣦⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⢿⣯⠱⠀⠀⠀⠘⢿⣈⣄⠀⠀⢻⡋⠛⣷⡀⠀⠀⠀⠀⣸⡅⡀⠴⣾⣿⣿⣿⡗⢿⣷⡀⠻⣤⣎⣯⡥⠖⢘⣿⣿⣴⣦⢽⠃⠉
⣤⣀⡉⠳⠘⢻⣷⡀⠀⠀⠀⠀⠙⢧⡌⠢⠈⠁⠀⠠⠁⢀⣀⣀⣾⣿⣿⡿⣤⣿⣿⣿⣿⣿⣮⣙⣷⠀⠈⣿⠟⠀⠀⠛⣟⢿⣿⠏⠛
⡏⠹⢿⣦⡐⣾⢻⣿⡰⡾⠃⠀⠀⡿⢁⣤⣃⣐⠚⠀⠀⠘⢓⡙⣿⣿⡿⠃⠙⢿⣿⣯⣾⠿⣻⣿⣿⡿⣾⣶⣤⣶⣶⣾⣿⠟⢁⣴⢇⣀⣶⣇
⣦⡶⢿⣀⡐⡷⢮⣟⢻⣇⡀⠀⠈⠃⢸⣿⣿⠟⠛⢰⣄⠰⣤⡄⠉⠩⣤⡶⠀⢾⠿⣿⢹⣿⣿⠛⣀⣁⣉⡙⣿⣿⣿⣟⣽⡇⢾⣿⡟⡯⢠⣾⣷⣀⣀⣀⣀
⢦⡄⠀⠙⢿⣽⣰⣻⣶⣏⢳⣄⠀⠀⠸⢿⣷⠀⠀⠀⠹⠆⠀⠉⣿⣦⣈⠛⠀⠸⣶⣿⣴⣌⡛⢯⣿⣟⠛⣿⣿⣾⣻⣿⣿⣁⣸⣿⣥⣿⣿⡿⢿⠿⠟⠛⠋
⠘⠃⠀⠠⡾⣻⠛⠁⠸⣿⣿⣿⣦⡀⠀⠈⢻⣶⣦⣄⣀⣀⣠⣾⠿⠛⠙⠓⣶⣴⣿⣿⣿⣿⣷⣶⣿⣿⣿⡼⠛⠽⠿⠛⣿⡛⠛⠋⠉⠉⠈
⠀⢤⣦⣤⢤⣿⣄⠀⣀⣹⣿⣿⣿⣿⣦⣤⣼⣯⣿⢿⣛⣻⡛⠁⢀⣴⣶⣾⡿⢿⣿⣿⣽⣶⣬⠛⠙⠻⣿⣷⣤⡴⣶⣀⣻⣿⣆⠀⠀⢀
⠀⠀⠉⠀⠀⣿⢿⣤⣈⡹⢿⣿⣿⣿⣶⣿⣿⣿⣾⣾⣿⣿⣷⣿⣿⣿⣿⣿⣷⡷⠛⣿⣿⣿⣿⣷⣿⠀⢸⣿⣿⣷⣿⣿⣿⣻⣿⣿⠃⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⡀⠀⠀⢾⣿⣿⣷⣿⣿⣿⡿⡿⠃⢹⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠘⣿⣿⣿⣿⣿⣿⠇⠈⣿⣿⠿⠹⣿⣿⣿⣿⣤⡄⠀⠀⠀⠴⢷⣴⣿⡿
⠀⠀⠀⠀⢰⠟⢿⣿⣿⣿⣿⣿⡟⢰⣇⣼⣿⣿⣿⣿⣿⣦⡙⢿⢟⣿⠿⠿⠻⠀⢀⣾⣿⠿⠿⠛⠁⢀⣴⣿⣟⣿⣿⢉⣿⣿⣿⣏⣡⠈⡀⠉⣀⣾⣿⣿⡭
⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⢿⣿⣿⣿⡿⣶⣬⣥⣶⣶⣶⣶⣾⣿⣿⣷⣾⣷⣾⣿⣟⢻⣿⣿⣿⣿⡿⢛⠻⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣷⣼⣿⣿⢿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⣿⣟⠛⠿⠃⠈⠋⢉⣾⣿⣿⠿⠿⠂⣿⣿⣯⣿⠁⠂⣾⣿⡟⣛⣭⣵⣿⠉⠈⢻⣿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⢙⣿⣿⣯⣿⠿⣿⣿⣿⣿⣷⣹⡿⢛⣿⣷⣶⣶⣿⣿⡛⠋⠉⠀⣠⣾⣿⣿⣿⢿⣳⣿⣿⡁⠘⠛⠛⠛⢉⣀⣴⣯⣥
⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⠿⣿⣽⣿⣿⣿⣷⣾⣷⣿⣿⣿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⣿⢿⣿⣿⡿⠶⢶⣦⡔⣾⣽⡿⠈⠉
⠀⠀⠀⠀⠀⠟⠻⢿⢃⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢿⣻⣿⣧⣙⣻⠿⠿⠿⢸⡿⠿⢈⣾⣿⣿⣿⠄⠀⠘⢷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣯⠴⣯⢛⣉⣷⣶⣿⠆⣼⣻⣿⣿⣿⣿⠀⠂⠰⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠨⠏⠀⠙⠛⠻⠟⠙⣿⣿⣿⠀⠸⠃⠘⣿⢿⣿⣿⣿⣿⣯⣿⡛⠏⠉⢹⡖⣿⢿⡝⠻⣫⡿⣶⡿⠿⣿⡿⣿⣿⡀⠀⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠈⢿⣤⣴⢿⣿⣇⠀⢻⣿⣿⣿⣿⠌⠀⠀⠀⠀⠀⠀⣸⡽⢶⣶⣴⢾⣿⣶⣿⣧⣍⣿⡿⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠸⠆⠀⠁⠀⠸⣟⣟⣹⡿⠀⠀⠀⠀⠀⢉⣠⡬⣶⠉⡅⠀⢈⠰⠾⠇⠙⠞⠉⠻⢾⠷⡇

⠀⠀⢠⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢰⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠀⠆⠘⠃⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⣇⠀⠀⣿⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠈⠁⠈⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⣀⣠⣦⡀⠀⢀⠀⠀⠀⠈⠉⠀⡀⠀⠀⠀⠄⡀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠈⠙⢉⡟⠁⠀⠀⠀⠀⢤⠀⠀⠀⣷⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⢰⣤⣠⠀⠀⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⢀⣿⡄⠀⢰⣿⡇⠀⠀⢠⠀⠀⠀⣀⣂
⠀⠀⠁⠀⠀⠀⠀⠀⠐⠿⠂⠀⠀⠀⢰⠛⡚⣅⠀⠀⠀⠀⢸⣿⡯⠀⠀⡞⠃⠀⠀⣿⣧⠀⠀⠈⠉
⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣤⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠻⠉
⠀⠀⠀⠀⢀⠀⠀⠀⠁⢶⣀⠀⠀⠀⠀⠹⡿⡛⣻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠄⠈⢆⡀⠀⠀⠐⢍⣣⡀⠀⠻⠿⠿⢷⣿⣽⠄⠀⠘⠀⠶⠀⠀⠀⠄⡠⠀⠀⠢⣄⣠⡆⠉
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠹⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠉⠁⠀⠀⣀
⠀⠀⠀⠐⠀⠀⠀⠀⠖⠛⠇⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠁⠀⢈⣴⠛⣹⣤⠀⠀⠀⣠⠖⢀⠠⠀⠀⠀⠀⠀⠀⠀⢠⡩⠘⠁
⠀⠀⢠⠀⠀⠀⠀⠀⢼⣦⡄⠀⠀⠀⠀⠈⢹⡿⣿⡇⠀⠀⢀⣒⣀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠛⠀⠀⢰⠶⠶⠆⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠶⠦⠄⠀⠀⠀⣅
⠀⠀⠀⠀⠰⣛⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠛⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠉⠈⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣳⣶⣾⣾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⡄⠀⠀⠀⢠⡾⣻⠏⣿⢻⣇
⠀⠀⠀⠀⢀⣸⢃⣿⠀⣿⠦⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⢏⠹⡇⡹⡆⠋⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠈⠋⠸⠟⠀⣴⣿⣿⠛⢿⣿⣦⡀⠀⠀⠀⠀⡄⡆⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⡟⠀⠈⢻⣿⣿⣦⡀⠀⢀⡡⠐⠊⠁⠀⠀⢀⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣷⠀⠀⣀⣹⣿⣿⣤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣯⠀⠀⢹⠀⠀⠀⢸⣿⣿⣿⠀⣠⣄⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⡄⠀⠀⣀⠀⠀⢸⣿⣿⠿⢐⣇⣈⡄⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠾⢿⣿⣿⠦⣀⠀⢀⣠⠞⣿⡿⠀⠘⠉⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠛⠉⠁⠀⠀⠀⠃⠀⡀⠠⣥
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠛⣿⠀⠠⠄⠀⠄⠄⠤⠂⠈⠀⠀⠀⠑⣄⢚⡆
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣛
⠀⠀⠀⠀⠀⠀⠀⡋
⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡀⠀⠔
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣇⡀
⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠋⠘
⣿⣶⠀⡀⢀⠀⣾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠀⣠⠞⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢼⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠠⠁⣀⣠⣂⠀⡤⠃⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⢀⠈⢸⡆⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡰⠃⠄⠀⠁⠀⠁⠏⠀⠀⡤⠂⣠⠾⠅⠀⠀⠊⠭⠋⠀⣈⠉⠛⢤⡀⣧⡀⠀⠘⣿⢃
⠀⠀⠀⠀⠀⠀⠀⢠⢦⡟⠀⢁⠞⠁⠀⣢⠎⠀⣂⣴⡤⠊⠁⠀⠈⠈⠁⠀⠀⠀⣸⣿⣿⡧⠀⠙⢿⡄⠑⢀⣿⡈⣞
⠀⠀⠀⠀⠀⠀⡰⠃⡿⠀⢸⠃⣠⡶⢋⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣷⡄⠘⠀⠉⣷⣾⣿⣧⡿⡄
⠀⠀⠀⠀⠀⣰⠁⢰⡇⠀⣿⣟⠁⣰⣿⠡⠄⠀⠀⠀⠀⠀⠀⠒⠠⠤⣭⡽⠦⠤⡿⣿⣿⣿⣧⠀⠀⠀⣌⣿⣿⣿⡇⠘⡄⠀⡀
⠀⠀⠀⠀⢠⣿⠀⢸⣧⣰⡿⠈⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⠀⠀⠀⠘⢿⠿⢦⡀⠀⠀⣿⣿⣿⡇⠀⢸⣠⠃
⠀⠀⠀⠀⠘⣿⠀⠀⣿⣿⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣾⣿⣿⣿⡇⢀⡾⠁
⠀⠀⠀⠀⠀⢻⡄⠀⣿⣿⣿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣷⣼⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⣿⡇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣦⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⠃⠀⡊⣉⣷⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⣧⠀⢀⠀⠀⠀⠀⠀⢈⣽⡇⠀⣿⠀⠀⣄⡟⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠟⠛⠻⠿⡿⠶⡷⠶⠶⣤⣬⣾⣿⣸⣿⣶⡶⠿⡿⣿⠟⠋⠉⢙⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠿⣧⣤⣤⣴⣷⣤⣧⣦⣀⠠⠬⣿⣿⠛⡲⣗⡂⢠⢠⣿⠀⠀⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣨⣿⣾⣿⠁⠀⠀⠴⠿⠿⢭⣭⡭⠀⠼⠿⠷⠦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠼⠶⠒⠾⠖⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⣄⣸⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣯⣄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡍⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⢿⣿⣿⣿⣿⣿⣿⡇⠀⠈⡇⠀⠀⠀⢀⣠⣶⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⢿⣿⣿⣄⠈⡀⠀⣠⣴⣻⣿⣧⣀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣹⣿⣷⣬⣿⠿⠿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⢡⣿⣿⣿⣏⣻⡿⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣛⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⢻⣿⡻⣿⣿⣿⣿⣿⣟⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⣿⣿⣇⠀⠀⢻⣷⣌⠻⣿⣿⣿⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠟⢀⣿⣿⣿⣿⣿⣦⣀⠀⠹⣿⣆⠀⠈⠁⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⣦⡀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢦⠘⣿⡦⠀⠀⠀⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⡄⠀⢸⠁⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⣿⢻⣿⣿⣿⣿⣿⡟⢿⣿⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠉⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠠⠖⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠈⢻⣿⣌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠙⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⣤⣤⠀⠀⣠⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⢀⣀⠀⠀⠀⣀⣰⣾⢿⣷⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⢸⡟⢻⣷⣴⣄⣻⣿⣆⠀⢠⣿⠛⣿⡄⣿⣿⠀⠀⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⣿⣿⢿⣿⣿⣿⣿⡿⣿⣦⣾⡏⠀⣿⣇⣿⡿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣷⣿⡇⠈⣿⡇⢻⣷⠀⠘⣿⠇⠀⠀⢹⣿⡿⠿⠇⠀⠰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⠿⢷⣤⣴⣶⣬⡟⢃⣶⡿⠃⠀⢸⣿⠀⠀⢸⣿⣤⣾⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣏⣀⣾⡟⠉⠉⢻⣿⣾⡟⠀⠀⠀⠀⣤⣄⣀⣾⡿⢿⣿⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠋⠋⢿⣧⣀⣀⣼⣿⠻⣿⣄⣀⣀⣤⣨⣿⣿⣿⣥⡄⣈⣉⣹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡉⠀⢠⣤⣽⣿⣟⣿⣥⣶⣾⠛⠛⣿⣯⣿⠟⢿⣿⠛⣋⣿⡛⠛⡏
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⢸⣿⣇⣹⣿⢿⣿⣀⣀⠀⢀⣿⡋⣿⡇⢸⣿⣾⣟⠋⡀⠀⠁
⠀⠀⠀⠀⠀⠐⣿⣇⠀⠀⢀⢹⣿⣿⣿⣁⠸⣿⡟⠉⠀⢸⣿⠿⣿⣧⢸⣿⠘⣿⣿⠛⢀⠀⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⡇⠙⢿⡷⣿⣷⣿⠿⣿⡯⠀⢚⣿⣼⣿⠀⠘⣿⡿⠟⠁

⠀⢀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠸⠀⢀⡄⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢮⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠉⢣⡄⠀⠀⠀⣤⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⢀⠀⠈⠁⠠⠂⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢀⣀⡴⡼⠁⠀⡠⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⠀⠀⢀⠊⢹⣷⠻⡆⠸⣴⠏⠠⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⡄⠰⠏⠀⠀⡀⠀⠀⠀⢀⠀⠀⣿⢷⣿⣿⣱⢿⣾⢷⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠆⠀⠀⠀⠂⠀⠀⠁⠀⠀⢀⠈⠀⠀⡹⣾⣿⡿⠁⣾⢃⠌⣰⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⢀⡀
⠀⢸⣀⡆⠀⡀⠀⠀⠀⢀⠤⡀⠀⠑⠀⠀⠁⢿⣿⡇⢠⠛⡏⣴⠿⠒⠀⣴⠁⠀⠄⢠⠈⠀⠀⠈⠙
⠀⠐⠀⠀⠈⠇⠀⢀⠔⣋⡀⠄⠁⠷⠛⢷⡉⠷⣼⣷⡶⢻⣠⡿⣢⠴⠆⠀⠀⠀⢶⠂⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠖⠈⣩⠖⠈⠀⠀⠀⠀⠀⠀⠚⠳⣦⣴⡬⠛⠛⠿⣵⠄⠀⠀⠛⠓⠚⣁⡴⠆⠰⢤⠤⠄
⠀⠰⡄⢰⠀⡔⠁⠀⠐⣦⡠⡄⠀⠀⠀⠀⠐⠉⣙⡕⠊⠉⠑⣖⠓⣞⡤⡶⠐⣊⠁
⠀⠀⠄⡌⡀⠀⠀⠀⠀⠈⠻⣶⣃⡀⠀⠀⠀⠀⡿⠀⠴⠀⠛⠹⡆⣉⢷⡄⢀⢈⠙⠢⣀⠀⠀⠀⣠⠄⠶⠀⡄⠀⠀⠀⠄
⠀⠰⡎⢀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠇⠀⠀⢀⡝⠳⡄⠰⠟⠀⢿⣸⠎⠷⠆⢘⣋⡄⢹⠗⠂⠉⠐⠖⠒⠈⠀⢠⠾⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠰⠀⠉⠒⣰⣷⡶⠟⠊⣠⣔⣻⣧⢈⣁⣙⣧⠳⣄⢦⣄⠠⠀⠉⣸⢇⣀⡀
⠀⢰⠆⡄⠷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⣼⠆⣠⣤⡀⢠⠞⢢⣴⢇⣼⠿⠿⡄⣉⡻⣹⡆⠈⠴⣉⣦⡰⠞⠁⠀⠈⢉⠆
⠀⢨⡂⡇⠘⢇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⡧⠴⠇⠀⠳⢶⣢⣶⣶⣿⣷⠄⣀⠿⢿⠙⢦⣛⡶⡘⠗⣤⠸⣄⠀⠰⠆⠈
⠀⢠⠍⠀⠀⠈⠷⢀⠀⠀⠀⠀⠀⠈⢉⠅⡲⣓⢛⠶⠹⠃⠌⢹⡄⣀⠙⠶⡫⠄⠰⢇⠠⡏⢳⠚⣤⠀⠙⢿⡓⠳⣄
⠀⢠⠀⣆⠐⠀⣄⠀⠙⠙⠢⠄⣠⣖⣁⡴⢉⠋⡽⢛⢄⣼⢳⣷⡾⠙⠥⠀⡿⣤⡚⠁⠀⠿⠀⠃⢻⡉⠩⣈⠣⢤⣁
⠀⠈⠀⠉⠠⡀⠈⠀⠀⢀⠀⠉⠉⣠⠞⣴⠋⣼⠳⠟⣘⡣⠙⠁⢀⡐⠂⠤⠭⢙⣡⡴⠂⠀⠀⠙⢮⣹⠶⠺⢿⠂⣉⠓⡀
⠀⠀⢀⠀⠰⠁⠀⠀⠴⣿⠀⠀⠀⠀⠘⠀⠢⠙⠒⠎⠁⠴⠣⠾⣦⡀⠀⠀⠀⠈⠳⠶⢞⡉⣤⡄⡀⠈⢷⡒⠦⠀⢉⡜
⠀⠀⠈⠂⠀⠀⠀⠀⢠⣄⠙⠆⣰⡆⠀⠂⠀⠀⠈⠀⠀⠀⢰⡞⢻⣿⡧⠀⠀⠀⠀⠀⠈⡡⠟⠐⣻⣿⡀⠘⢦⠁⠈⠓
⠀⠈⠀⢀⠄⢰⡆⠀⠰⣿⠀⠘⠉⠇⠀⠀⠀⠀⡄⠀⠀⠀⢠⡿⢇⣈⣿⣀⠀⠀⠀⠀⠀⢀⠀⠀⣿⠏⠿⣄⠈⠼
⠀⠀⠀⠈⢀⣈⠹⠀⡀⠁⣾⠀⠀⠃⠐⠀⠀⠀⠃⠀⠀⠀⢷⣄⣤⠙⣟⠻⡶⣦⣤⣄⣰⡁⠀⢸⣿⣿⠛⡇⢠⠀⠀⠀⡀
⠀⠀⡄⠀⠈⠟⠀⢠⡁⠢⠘⠄⠀⠀⠐⣦⠠⡀⠀⢀⠀⠀⠀⠀⢠⡁⣘⣇⠠⡏⡹⢏⡿⢻⢦⡏⠀⠀⠀⡀
⠀⠀⠁⢀⡈⠀⣀⣾⠑⠊⠀⠀⠀⠀⠈⠈⡠⡏⠀⠘⠀⠀⠀⠀⠀⣉⣹⠟⢻⡷⠇⠈⠱⡟⢬⣿⣶⠟⣿⣀
⠀⠀⠀⠸⠓⠁⠙⢉⠀⠀⠠⢦⠀⢠⡀⠄⠡⣁⠀⠀⠀⠀⠀⠀⠈⠉⠀⣤⣼⢁⢦⡀⠆⢻⡆⣿⠻⣿⣽⣯⣶⡖⠀⠠
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠈⣇⠄⠻⢠⠀⢈⡇⠀⠐⠀⠀⠀⠀⠀⠀⠤⠜⣡⡜⠀⠈⠀⡌⢛⣦⡁⠈⠉⢻⡎⣠⠀⠇
⠀⠀⠀⠀⢳⡤⠤⣄⡈⠀⠀⠈⠀⡀⣦⠈⢀⣊⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠁⠻⡄⠀⠀⠸⡇⠙⠻⣦⣡⣤⢹⡗⠀⡀
⠀⠠⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠞⠉⢈⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠙⠄⠀⠀⣷⢀⡀⠟⣀⠀⠀⠀⠀⠁
⠀⢰⣤⠐⠒⣋⠟⣚⣡⡾⠲⣟⣵⠞⣍⣓⣶⣤⣼⡵⠦⠠⠀⢠⠀⠠⠄⠠⠤⠆⠤⠤⠔⠉⣉⣠⣾⠷⢦⠟⠷⣦⡄
⠀⢸⡟⠛⠛⠟⠛⠉⠏⠑⠛⠋⠉⠀⡁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠙⠋⠴⠁
⠀⠸⡇⣴⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⣀⡀⢘⡀⢀⠐⠀⡀
⠀⢠⡀⡀⢀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠋⠁⠁⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠋⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⡀⢀⠀⡔⠀⠢⠀⠀⡀⣀⠀⠀⣀⠀⠀⠀⢰⠄⢰⠂⢀⠆⠀⢲⠀⣀⠀⠀⠔⠒⠒⡄⠀⣀⠀⠀⡠⠄⠤⡀
⠘⠁⠃⠀⠀⠁⠀⠀⠀⠄⠘⠀⠀⠑⠂⣤⡈⢿⣵⣮⠣⡈⣄⠾⠲⣤⣘⡠⢃⣀⠀⠀⠀⠀⠘⠨⠁⠀⠈⠀⠀⠀⠸⠀⠀⠀⠉
⠀⠤⢄⠀⠆⠉⠁⠀⠀⠀⢀⠀⠰⠃⠀⢻⣝⣶⣇⣉⢦⢋⣤⣶⣶⣄⣈⡶⣋⠏⣄⠀⠴⠤⠀⠔⠀⠀⠈⢢⠠⠄⠀⠂⠈⠁⠂
⠀⠤⠄⠀⠀⢂⠀⠘⠁⠀⠀⠀⠠⡔⢂⢦⣽⣿⣿⣄⣶⠸⣿⣟⠛⠛⣇⡶⠧⠾⣥⣄⣀⠀⠀⡀⢐⠈⠉⠀⠀⠀⠀⠉⡀⠀⠁
⠀⠀⣠⠴⣶⣿⣿⣷⣶⣤⡀⠀⠀⠙⠣⠿⢿⠛⠛⣏⣉⠭⣿⣶⣾⠟⠃⣸⠒⢾⣿⣿⣿⣷⠄⠀⠀⠱⣠⣶⣶⣾⣿⣳⣶⣤⡀
⢠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣛⣀⡤⠞⣹⠉⡸⣧⢴⣇⣴⡦⠻⣎⢙⠉⠙⠈⠛⢯⣀⣀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⡽
⢻⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⠁⠐⠃⠚⠅⡾⢳⡈⠁⠈⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⣶⣄⠀⠀⠀⠀⢠⠞⠀⠀⠙⣄⠀⠀⠀⠀⢠⣴⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠻⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣰⠋⠀⠀⠀⠀⠈⢦⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠁⠸⡻⠟⠋⠹⡄⡴⠁⠀⠀⠀⠀⠀⠀⠈⢧⢀⠟⠉⠛⠛⡏⠉⠛⠛⠛⠛⠛⠋⠁⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⣀⣳⡀⠀⡇⠀⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠄⡀⠀⠲⣭⣯
⠠⣶⣦⢠⣤⣄⢀⣄⡀⠙⠿⠋⢀⡀⣧⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠘⢾⠁⠀⠀⠀⡏
⠀⢹⠇⠂⠻⠃⠘⠿⠃⠒⠚⠒⢻⠟⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸⣇⣀⣀⣠⣤⣤⠴⢻⠲⠤⠒⠊⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⢾⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠉⠉⠉⠉⠉⠉⠉⠀⠐⠀⠐⠒
⢠⣀⣀⣄⡄⣀⢠⣀⣀⣀⣤⣤⠤⠖⣺⡆⠀⢨⡴⣖⣋⠉⠉⠉⠉⠉⠉⣓⣲⢤⢰⠀⠀⣼⠆⣄⡀⡀⣀⣀⣀⠸⠴⡇⠀⠤⠤
⣼⣿⣿⠿⠿⠟⠛⣋⡉⠀⠢⠃⠫⣯⢽⢳⠀⡿⢸⣏⡛⠻⠿⠿⠿⠿⠛⣋⣿⡜⣿⠀⢠⣿⠈⣐⡪⣳⠐⢚⡀⢀⠐⠀⢀⠊⢰
⢀⢢⢤⠤⣷⡉⠉⢤⠐⣾⢫⡶⡂⢠⢸⡘⣷⠇⢸⣿⣿⡆⠀⠀⠀⢠⣾⣿⣿⡇⢻⡄⡼⢸⢰⢢⠢⠄⠡⠠⣤⠳⣞⣀⡸⠱⣅
⠐⢽⡉⢡⣈⡟⣉⣉⣇⣼⠐⠀⠅⣀⠘⣷⡇⠀⠘⠿⢿⣧⣶⣤⣦⣼⣿⣿⡿⠁⠀⢻⡇⣾⢀⠝⢁⡂⣸⡀⠨⠤⠏⠋⣠⣀⣤
⠰⣄⡈⠣⡜⡷⣨⠃⡦⢤⠼⠲⠠⠁⠄⣟⠧⣴⣶⣾⡿⠿⠿⠿⠿⠿⠿⣿⣷⣶⡦⠞⢡⡇⡜⠋⢯⢔⡹⠈⠞⠆⢒⠫⣥⡽⠥
⠀⡁⢈⠀⢿⢻⡁⡈⢤⡾⣤⢺⡜⡼⠀⡜⢦⡀⠉⢪⡻⠭⠭⠭⠭⠭⠭⢋⡞⠁⠀⣠⠏⠄⢧⠆⣸⣁⡄⠁⡘⠧⣾⠤⢃⠰⢰
⠀⡧⢸⠎⠉⢸⡈⢣⡘⠿⠛⡯⡠⠃⡂⠹⣄⠳⢄⡀⠳⣄⣀⣀⣀⣀⣠⠞⠀⡠⢚⡁⠐⣞⡎⢉⠓⢴⠘⣒⢡⣥⡌⣢⢌⠄⢛
⠀⠎⡆⢼⠫⣳⠙⡼⣗⠓⢊⡅⡀⠼⣡⠀⡏⠳⣄⠉⠓⠦⣍⠉⣉⠭⠖⠒⠚⠱⡱⠀⠆⢻⠘⢤⡾⠀⡣⣠⠸⣍⡆⠣⡜⢢⠐
⠈⡄⡱⠼⣥⡥⡜⢫⣘⢾⠄⣹⠁⡄⠀⡴⡁⠸⡈⣉⠙⢲⢬⡛⢁⡴⡞⠛⡩⢠⠁⣶⠀⣟⢉⡌⠩⢹⢴⠈⢏⢸⠰⣃⡨⢶⢸
⠀⠻⠆⠲⠑⠃⠣⠇⠋⠖⠈⠻⠄⠫⠁⠠⠅⠸⠰⠇⠎⠀⠰⠟⠻⠴⠁⠘⠓⠚⠷⠠⠆⠨⠉⠃⠰⠚⠖⠖⠉⠾⠰⠡⠰⠟⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⢀⡀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⣠⣶⣾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿
⠈⠡⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⠀⢺⣿⣿⣿⣿⣿⣿⣟⣠⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢸⣷⣄⠹⣿⣿⣿⣧⠀⠸⣿⣿⡿⠿⠃⠈⠉⠛⠂⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⣷⣦⡀⢸⣿⣿⣷⣾⣿⣿⣿⣷⣾⣿⣿⣷⡀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠈⠙⣷⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠸⣷⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣁
⠀⠀⠀⠀⠀⠀⠻⣿⣾⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⢸⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣄⠀⠈⣿⣿⡍⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠻⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠈⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠘⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⣶⠖⠒⠛⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⡞⠋⠉⠉⠓⠦⣄⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣁⣠⠞⠀⠀⠀⠀⠀⠀⠈⠙⠲⣾⡉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⣰⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢰⡇⠀⠀⠀⢀⡴⠶⠶⢤⣀⠀⢀⣀⣀⣀⣀⡟⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⡾⠀⠀⠀⢰⡏⠀⠀⣴⣚⠉⠉⠻⢧⡀⠀⠈⠉⠳⣄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⠀⠀⣼⠁⠀⠀⠀⠈⢷⡀⢸⡇⠈⠙⢦⠀⠈⢷⠀⠀⠀⠀⠘⡆⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠹⡄⠀⠀⠀⠀⠈⢷⠀⠓⠦⣤⠞⠀⠀⣸⠀⠀⠀⠀⢀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡟⠲⢤⣀⡀⠀⠈⣧⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⡞⢀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣘⡇⠀⠀⠈⠉⠛⢲⡇⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⣸⠋⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠀⠀⢀⣠⠶⠚⠁⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢈⣩⠷⠒⠒⠋⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠈⠙⣳⠖⠛⠻⣍⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⢻⠀⠀⢠⡟⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⡟⢸⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸⠀⣾⠃⣀⡤⠶⠒⠛⢛⣋⣩⡽⢿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣷⣸⣾⢗⣋⡤⠴⠖⠚⠋⠉⢁⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡿⢟⣉⣠⣤⡤⠤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠾⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣤⢀⣠⣤
⠀⠀⠀⠀⢰⣼⣿⣷⣦⡀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠰⣶⣶⣤⡀⠀⣿⣿⣿⡅⢀⣤⣶⣶⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣩⣿⣿⣿⣷⡶⠂
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣮⣼⣿⠛⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣦⣹⣿⣤⣴⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠩⣿⣿⣿⣿⣿⣇⠀⠀⣀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠐⠒⠲⣄⣻⣿⣿⣿⣿⠏⠉
⠀⠀⠘⠀⠀⠀⠀⢠⣾⣿⣿⣿⠟⢿⣶⠞⠉⠁⠀⠀⠀⠀⠀⠀⢩⣟⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠚⠛⠹⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣀⢀⣀⣤⣭⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣡⣽⣿⣿⣿⣿⣿⠿⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠠⠞⠛⠉⠉⠉⠉⢩⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⠶⠶⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦
⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⢋⣬⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⣋⣴⣿⣿⣿⣿⣿⢟⣿⣯⣿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⢇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣗⣿⣽⣿⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣌⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣿⠿⣿⣿⣿⣿⣿⣿⣿⣇⠐⢿⣿⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣒⣸⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⣿⣿⣿⣟⠋⡽⠋⠉⢻⣿⡿⠿⣿⡿⠛⣿⣿⣿⣿⣿⡟⣭⢹⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⣓⣽⣿⣿⣿⣿⣇⠀⠀⣼⡿⠁⠀⢈⡇⢀⣼⡷⠿⢿⣿⣿⣶⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⠋⠈⠙⣿⣷⣾⣿⣷⣿⣿⠿⠿⠿⠋⢀⡴⢊⣽⣿⣿⡿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⣠⠴⠚⠙⠻⣿⣿⣿⣿⣿⣷⣶⡶⠋⠉⠉⠁⢿⣿⣿⣆⣣⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣧⣿⣿⣷⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣾⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣷⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⢈⡤⠴⠇⠑⠦⣈⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠐⣀⠂⠀⠀⠰⠚⠀⡡⠌⣱⠀⠀⢀⡀⠀⠀⠠
⡔⠁⠀⠀⠀⠁⠀⠈⠉⠛⠓⠀⢀⠀⠀⠈⠀⠐⢂⠀⠀⠠⠁⠀⠚⠋⠁⠀⠀⠀⠀⠈⠉⠛⣁⠈⠊⡁
⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢄⠀⠠⢀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠈⠂⠈⠁
⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⡀
⠀⢀⡁⠒⡤⢀⣀⣀⣠⠤⠚⠙⠄⠀⠀⠀⠠⠀⠀⡀⠀⠀⠁⠀⠀⠀⠉⢀⠀⠀⠀⢀⠀⠀⠀⠒⠢
⡔⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠎⠁⠁⠀⢠⠠⢤⠀⠛⠀⠸⠄⠀⠀⠉⠀⠐⠀⠠⠀⠀⠀⡆⠀⠀⠀⠑⠂⠤⢄
⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠈⠠⠀⠌⢰⠈⠑⢂⡀⡐⠀⠉⠡⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠎⠀⠐⠁⠀⢀
⢄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠨⠀⠀⠠⠀⠀⠀⠘⠀⠀⠀⠀⡄⠀⢠⡝⠙⠀⠀⠀⠀⠄⠀⠈⠐⠀⢀⡀
⠀⠁⡶⠚⠙⠣⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠈⠀⠁⠀⠐⠀⠀⠀⣆⠀⠤⠀⠆⠉⠇⠀⠈⠀⢀⢀⠀⠀⠈⠁⠐⢢
⠀⣀⡇⠀⠘⢡⡚⢂⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢁⠠⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠈
⠈⠁⢸⡀⠀⢜⠁⢸⡁⠂⠈⠀⠀⠀⢸⡀⠂⣀⡀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠁⠸⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢳⣄⡈⠀⠀⣙⣀⢀⠀⡤⠀⢠⡎⠃⠘⠛⠂⠀⠀⠀⠐⠚⠓⠘⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠄
⠄⡀⢀⡝⠈⠙⠲⠞⠉⠈⠙⢷⠧⠔⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸⣤⡀⢑⠀⠀⠀⠀⠀⠁⡄⢀
⢀⠜⡅⠀⠀⠀⠀⠱⣄⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⣈⡿⠁⠨⡄⠀⠀⠀⠐⠂⠀⠀⠑
⡇⠀⠈⠢⢤⣀⠀⠔⠻⣁⠖⠒⠀⠐⣄⠀⡀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣼⡇⢏⠁⢀⡤⠋⠉⠄⣀⠀⣰⠤⠂⠚
⠀⠀⠈⠀⣶⠄⢀⠀⠀⢹⠀⠈⠀⠀⠈⠂⢹⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡝⠀⣾⠈⠅⠀⠀⠀⠀⠉⠉⠑⡄
⡄⠀⠀⠠⠂⡠⠂⠀⠀⢸⠀⠀⠈⠀⠀⠀⠿⣧⢉⠂⠀⠀⠀⠀⠀⠁⠀⠘⠁⠀⠁⠀⠘
⠙⢦⣀⠀⠈⡅⠀⢀⡴⠃⠀⠀⠐⠀⠀⠄⠐⡿⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠎
⡠⠤⠽⠍⣒⠓⠉⠉⠀⠈⠢⠄⠀⣀⠀⡠⠄⠣⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠁⠀⢀⣀⡤⠊⠁⠒⠒
⠀⠀⠀⠀⠀⠙⢂⡀⠀⠀⠀⠀⠀⠼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠐⠀⠌⠣⠀⡄⠀⠀⡤⠊
⠀⠀⠀⠀⠀⠀⢀⠗⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠒⢲⠁
⠀⠀⠀⠀⠀⢰⠃⢀⠂⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⡠⠴⠦⠄⣤⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⢰⣄
⠀⠀⠀⠀⠘⢡⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘
⠀⠀⠀⠀⡄⠀⠙⠀⠀⠀⠀⠀⠘⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡔
⠐⣶⣶⡄⡀⠠⣶⠀⠀⠀⠀⠀⡄⣀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⣤
⠀⠙⠋⠀⠀⡀⠀⠀⠀⠀⠠⠄⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈
⢤⡤⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⢰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⢲⠁⠐⠋⠀⣀⡀⠉⢷⠉⡴⢟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⢀⣎⣀⢠⠔⠀⠀⠀⠀⡀⠀⠀⢁⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠓⠉⠎⡉⠣⠀⡀⣤⠔⠒⠉⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠘⠛⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⠚⠒⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠰⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⣤⣤⣤⣤⣴⡦⠀⠀⠒⠀⢰⣦⣤⣤⣤⣤⣤⣤⣤⣤⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀
⠀⠀⠀⠀⠀⠀⣀⣤⠐⠶⠒⠈⠛⠓⠒⠶⠶⠖⠒⠒⠲⠶⠿⠷⠀⠒⠒⠂⠸⠿⠒⠶⠶⠶⠶⠖⠲⠶⠶⠒⠟⠋⠁⠲⠶⢤⣤⡀
⠀⠀⣠⡤⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢢⣄
⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⡀⠀⠀⠀⠤⠤⠤⠤⢤⣤⡤⣤⣀⡀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣄
⠀⠓⠶⠖⠒⠒⠒⠘⣿⣯⡽⠋⠉⠉⢉⣉⡹⠛⠉⠁⠀⠀⠀⠐⠂⠀⠀⠘⠛⠓⢏⣉⡿⠟⢉⣉⣉⣉⣁⣤⣤⣶⣶⡶⠲⣤⠄⢀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠙⠿⠟⠀⠈⢉⡅⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠛⠀⠀⠀⠀⠀⠛⠛⠉⠉⢩⣍⠉⠛⠻⣍⠙⢧⡽
⠀⠀⠀⠀⠀⠀⣀⣿⠇⠀⢸⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⢻⡄⠈⢱⡟
⠀⠀⠀⠀⠀⠀⠻⡇⠀⢠⡇⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠸⡇⠀⢹⠀⣿
⠀⠀⠀⠀⠀⣼⠀⣿⠀⢸⡇⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢰⠀⢸⡆⠈⣄
⠀⠀⠀⠀⠀⠙⠀⠟⠀⢸⡇⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠸⣿⠀⡟
⠀⠀⠀⠀⠀⣠⣶⠀⠀⣼⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⠀⠀⣿⠀⠹
⠀⠀⠀⠀⠀⣿⣅⣴⠟⠉⢀⠀⠀⢠⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⠀⠀⢠⡿⢧⡀⠹⣷⣦
⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⢸⡇⠀⣾⠙⠳⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠁⣽⠀⠀⣸⡇⠀⠙⣶⣿⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡈⠻⣀⣿⠀⠀⠀⠈⠉⢷⡶⠦⠄⣀⣀⣠⣤⣤⣄⡀⠠⠶⠶⣾⠋⠀⠀⠀⠀⣿⠀⡀⢿⡇⠀⠀⢸⣯
⠀⠀⠀⠀⠀⣠⠈⣿⠀⢸⠇⢰⡏⠁⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣶⡿⠀⠇⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠹⣶⠏⢠⡆⢀⡜⠀⠀⠀⠀⠀⠀⠀⣼⠟⠢⣤⣤⠀⠀⠀⠀⠀⣠⣤⡴⠛⠛⣦⠠⣤⣄⠀⠀⠀⠁⡀⢰⡄⠀⣀⣟
⠀⠀⠀⠀⠀⠀⠸⢶⣯⣠⡞⠁⠀⠀⠀⢀⡤⠜⢛⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠛⡀⠀⠈⠳⣄⠀⠈⢇⡈⢣⡀⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣀⣀⣀⣠⣴⠋⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠉⢷⡀⠈⢳⡾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠟⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢻⣤⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⢀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠁⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡄⠀⠀⠀⠀⠸⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠠⣾⣅⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⣠⣤⡼⠛
⠀⠀⣀⠴⠋⠉⠛⠶⢄⡀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⠶⠎⠉⠀⠈⠹⣆⡀
⣤⠞⠁⠀⠀⠀⠀⠀⠈⠉⠲⣦⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦
⠛⢠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⣄⣠
⠀⠨⡧⣼⡏⠀⣤⠀⣀⣀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⣀⠀⢶⣄⣹⠇
⠀⠀⠀⠘⠇⠾⠿⣀⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣿⣄⣸⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⢴⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡧
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠶⠶⠶⠶⠶⠶⠤⣤⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⣀⡀⠀⠠⠤⠶⠶⠶⠶⠶⠶⠂⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠚⠚⠛⠻⠛⢿⡏⠈⠉⠈⢹⡟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠖⣿⡇⠀⠀⠀⢸⣷⠴⣶⠶⠶⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠶⠶⠶⠶⠖⢿⡇⠀⠀⠀⣸⣿⠙⠛⠟⠛⠷⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠂⣤⡤⠔⠚⠛⢻⠀⠀⠀⠋⠉⠛⠓⠀⠤⠔⠚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠃⠀⠀⠀⠀⢀⣀⣠⣴⠿⣀⢀⣸⠃⠀⠀⠀⢰⣄⠀⣠⠿⣦⣀⣀⠀⠀⠀⠀⠘⠳⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⣀⣤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⣇⢸⣻⣿⡇⠙⢶⡀⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠉⠈⣏⠟⠓⠲⣾⣿⣷⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⡀⢐⡶⢿⡗⢠⡄⠀⠙⢻⡟⣉⣉⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣨⣃⡈⠙⠊⢱⠀⢠⡄⢀⡬⡟⢿⡟⠷⠀⠀⢠⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠬⢭⣀⠀⠀⢢⠀⠀⣈⣿⡟⡮⢁⣄⠐⣜⣳⣿⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⣿⢽⣛⡟⠡⠴⠷⣾⣿⣿⣎⠝⠋⠀⢀⡜⠿⢽⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⣾⣧⣞⠉⠀⢤⢞⠀⠀⠈⢉⣀⡤⡄⠐⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠈⠋⠤⠴⠿⠿⢟⢵⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⣽⣿⣷⣦⠀⢀⡀⠀⣀⠜⠫⠄⠂⢈⣷⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣻⡯⢾⡟⠉⠀⠀⡐⢈⢂⣾⣿⣮⡽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠙⢳⣸⠃⣍⣷⣤⣤⠾⠛⠁⠈⣻⠟⠉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⠟⠛⠋⠉⠀⠀⠀⠀⠼⠣⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⠁⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⠀⠰⠄⠀⠀⠀⠀⠈⠻⣿⠿⠀⠐⠾⣻⢓⣧⠀⠀⠀⢠⠤⢀⠀⠀⠀⠀⣀⠒⢀⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢤⡀⠰⠦⠐⠓⠸⠃⠶⢰⠦⠒⠘⠛⢶⣶⣿⡄⡀⡍⡉⡁⠰⠃⠀⠀⠀⣴⣦⡤⠀⠀⠙⣉⣿⣿⠏⠁
⠠⢊⣠⠀⠀⠀⠤⠀⠆⠐⠂⠛⡀⠛⢠⣮⣤⠆⠀⠀⠀⠈⢹⠇⠆⠻⠃⠓⠀⠀⠀⠀⠀⣿⡟⠁⠀⠠⠾⠟⠛⠁
⠀⠈⠀⠀⠰⢠⡄⠰⢂⠐⠃⣈⠁⠶⠂⣈⠙⠀⠀⠀⠀⠀⣾⠀⠀⣀⣀⠀⣵⣦⡤⣴⢶⣿⠈⠠⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⡄⠗⠈⢁⣄⠋⡀⠒⠀⠉⣀⠉⠁⢠⡤⠚⠽⣏⣻⠟⣡⣼⣿⣿⣿⡇⣓⠿⡻⠆⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈⠋⡀⠛⣀⠁⠘⠋⣠⣏⣭⣽⢛⣛⠭⠥⢲⣿⠿⣿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠛⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡝⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠘⣿⣷⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣴⣿⣮⡻⣧⡈⢷⡀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⠀⢈⣷⣝⢿⣎⢷⡀⣰⣿⣿⣿⣿⣶⢠⣿⠇⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣻⠿⢧⡹⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⣤⣦⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⣿⣿⣿⣿⣶⣌⣿⣇⠹⣿⣿⣿⣿⡿⣿⣿⣿⣥⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⢴⣿⣿⣿⣿⣿⣿⣿⠿⠄⣼⣿⣿⣿⡷⣮⣉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣋⣤⣶⣿⠟⣫⡿⣻⣿⠟⠀⢀⣼⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢃⣼⣿⣿⣿⣯⣾⣟⣼⣿⡟⠀⢠⣾⠟⠁⣾⣿⡿⠋⠀⠀⠀⢀⠀⠐⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⣾⡿⠿⣿⣿⣿⣿⣿⣿⡿⠁⣴⠟⣡⣴⣾⣿⣿⠀⠀⠀⠐⠲⣾⣤⡐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠉⠀⠀⠿⠋⠉⠀⠈⠉⢠⡾⠁⢼⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠙⢻⣿⠟⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠀⣸⣿⣿⣿⣿⣿⣿⡿⠗⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠆⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⠀⠃⠀⠀⠀⠊⢀⢀⠄⠰⡆⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡀⢠⡀⠀⠀⠀⠀⠀⠀⠐⠛⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠢⠂⠀⡀⠂⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⣠⠔⠀⠀⠀⠀⠈⠑⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⣀⡠⠀⠀⠀⠀⠂⠀⠀⠀⠀⡌⠀⠀⠁⠀⠀⠀⠀⢀⡴⠇
⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠃⠐⠧⠒⠒⠈⠀⠈⠒⠂⠚⠀⠄⠄⠐⠈⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠁⠀⡈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣀⠤⠀⠤⠄⠠⠤⠤⠄⠀⣀⡀⠀⠀⠀⠀⣀⡠⠔⠀⠀⠀⠀⠀⠉⠓⢄⡀⡀⠀⠀⠀⠀⠴⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠠⡀⣤⠷⢲⠀⠻⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠾⢯⠢⠆⢩⢱⢾⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡒⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡈⠸⠀⠂⣀⣾⣼⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠩⠤⡀⡤⠀⠀⠀⠀⠀⠀⠰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠋⠀⡾⠀⢹⠀⠀⠀⠀⠀⠀⠀⠖⠎⠰⠒⠒⠛⠾⠀⠀⠀⠀⠀⢠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢻⠀⠘⡇⠀⠀⠀⠀⠀⠀⢢⠃⠀⠀⠀⠀⠀⠀⠀⢶⠀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢑⡀⡠⠀⢠⢀⠿⠄⠀⡇⠀⣀⣀⡀⢠⡴⠟⠀⠀⠀⠀⠀⠐⠴⠃⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡍⠒⠀⠈⠀⠀⣻⠄⢰⠸⠼⣏⠼⡿⠀⠀⠀⠀⡀⣄⠐⢴⡤⣊⠅⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠴⡿⠵⣄⣇⢘⢀⣿⠀⠈⠀⠔⡶⠔⡁⠀⠀⣠⣾⢐⢿⡋⢛⣴⡯⢽⠊⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠁⠀⠙⢛⠶⣮⣿⣧⠀⡗⠒⠢⢀⡌⠀⣼⡿⡇⢈⡉⠹⠂⡔⠛
⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⡀⠀⠀⠠⣄⠲⠆⠀⠀⢀⣽⣿⣿⠀⣇⣨⠔⣀⣤⠍⢉⠀⠻⠞⠃⠈
⠀⠀⠀⠀⠀⠀⠄⠐⠘⢄⠀⠀⠀⠀⠀⠀⠘⡄⠀⠠⣤⡾⢹⣠⡀⡤⣾⠿⡯⢉⣤⢲⡞⠠⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⡃⠀⠀⡟⠋⣐⢀⣯⣤⣾⡿⡟⠻⠛⢉⡙⢨⡴⣼⠽⢣⣠⡳⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡾⠀⣰⠃⠀⢠⠇⡀⣻⠋⠉⣁⣬⠉⠑⠀⠈⢨⠛⡆⠻⣭⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠃⠰⢣⠀⣰⣿⣧⡇⠹⠘⢠⠇⠈⠂⠈⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡰⢿⠚⣽⠁⣀⣴⠃⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢸⣳⠟⠀⠀⢡⣤
⠀⠢⢀⠀⠀⠀⣀⠀⡜⡏⠀⡿⠋
⢰⡎⠲⠥⢀⠀⢛⣴⣾⡇⠀⡟⠃
⠘⠀⠀⠀⠈⠋⡼⠿⡙⠀⡞⠁
⠀⠀⠀⠀⢀⢡⠁⠀⣧⡴⠁
⠀⠀⠀⠀⠸⣾
⠀⠀⠀⠀⢰⣽
⠀⠀⠀⠀⠈⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠠⠀⣴⣶⣶⣶⡄⠉⠀⠀⠀⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⣰⡀⠻⠿⠿⠟⠃⠸⠂⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠘⢿⠟⠀⠀⠀⠰⠆⠀⠀⠘⠁⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣤⣾⣿⠟⠀⠃⠀⠈⠀⣠⣶⣶⣦⣄⠀⠀⠀⠀⠀⠻⣿⣷⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠉⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣄⠀⢰⠄⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠆⠀⠈⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠺⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠛⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⠟⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⡀⠀⠈⠙⠛⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⡟⠀⠀⠀⠀⣠⡀⠘⠋⠀⣤⠀⠀⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠀⠀⠀⠺⣿⠗⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣽⣿⣧⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⠾⠿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣭⣥⣤⣶⠾⠃⠀⠀⠀⠈⠻⠿⠋⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣛⣛⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢠⠀⢠⠞⢉⢤⡈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡄⠀⢠⡏⠹⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⡑⠀⢀⠉⠀⠀⠀⡇⠀⠀⢰
⠀⠀⠀⠀⠀⠀⣠⡄⢰⣅⢀⡀⠀⢀⠙⠀⠀⠀⣠⠀⠀⠀⠀⠂⢹⠃⢀⡾⠁⠀⠛⠍⠀⠀⣀⣷
⠀⠀⠀⠠⢀⡸⠀⡠⠆⡠⠼⠿⠨⠦⣤⣄⡙⠲⣼⡄⠀⠂⣤⣤⣿⠶⠯⠴⠿⣿⠶⢤⣤⣤⣭⠟
⠀⠀⠀⠀⠙⠀⠀⠇⣀⠃⢀⣨⣤⡶⠶⣿⣿⣷⡘⣿⣆⣼⡿⠛⠉⠐⠀⠈⡀⢈⣳⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⠛⡸⠋⠀⠁⡀⣀⡀⠈⢿⣿⣾⣿⢏⣴⠶⠛⠛⠓⢒⡀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠻⠆⡮⠚⢿⡰⣶⠀⠙⣿⣿⡟⢀⣠⠘⡄⠁⢈⣉⠀⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⠋⡀⠉⠁⠂⠀⢘⣿⠇⠐⠂⠁⠀⠀⠉⠘⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⡿⣹⡆⠀⠉⠀⠈⠀⢨⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⠄⣸⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠟⡁⠀⣼⠞⣧⡤⢸⣦⠀⣳⡶⠶⠀⣬⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠁⡸⡝⠀⣈⣵⡄⠈⢬⠃⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠘⣤⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠱⡄⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣶⣷⣦⡴⠛⢲⠔⠋⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⢠⣾⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠇⣴⣁⡠⠔⠒⠈⣙⡆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠿⢏⡜⠁⢸⡤⠖⠊⠉⢉⣹⠆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣧⣀⡞⠀⢀⡾⢒⠊⠉⠁⢉⡷
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⠔⠋⠉⠉⠉⣉⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣴⠶⠶⢤⣄⣀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣒⠶⢿⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⡿⣿⣿⣿⣿⠀⣿⣿⡦⡙⠦⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡀⣇⢹⣷⣹⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣼⣿⣿⣿⣷⡌⠉⢻⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣻⣆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢯⣽⣿⣭⡿⢿⣿⡋⣻⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⢿⣿⡟⢿⣿⣿⣻⣿⣶⣶⣿⣽⣿⣄⣠⣿⠿⣿⣿⣿⣧⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⢸⣿⡇⠀⠉⠛⠛⠻⠿⢿⣿⣿⣟⡿⠿⠿⣶⣾⣿⣷⣤⣿⣿⣿⣿⠿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣿⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⢱⣿⣿⡇⠀⣠⣴⢶⣶⣶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡧⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣿⣿⠀⢸⣿⣿⣿⠛⣿⣷⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⠀⠀⠹⢶⣿⣿⣿⡏⢻⣷⣾⣿⣶⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⣿⣿⣿⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣤⣿⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡀⠀⠀⠀⢰⣿⣿⣽⠋⠈⠉⠉⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⣇⣿⣇⠀⠀⠀⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠸⣿⡘⣿⡄⠀⣸⣿⡟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⢻⣧⡸⣿⣄⣿⣿⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣿⣿⡿⠿⠟⠿⠿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡙⣿⣿⣿⡀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⢿⣿⣿⣿⣧⠹⣳⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⣳⣿⣿⡿⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣷⣯⡻⣿⣿⡧⣀⠛⣧⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠟⢁⠄⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠱⡄⠹⣿⣿⣿⣞⢿⣇⣈⣿⣿⠿⣤⣶⣿⣶
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣾⣿⡿⠃⣠⠂⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠙⢆⠘⢿⣿⣿⣷⡽⠟⢉⠄⠊⠍⢿⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣻⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⡿⣶⣶⣿⣿⣿⣷⣾⣿⣟⣉⣤⠘⠛⠀⠛⠂⣾
⠀⠀⠀⠀⢸⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡰⠈⠻⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡆⢠⣶⡄⡸⠿⢃
⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠌⠀⠀⠀⠘⡿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⡼⢿⠆⠁⠰⠯⠄⢨⡄
⠀⠀⠀⠀⢻⣿⣿⣿⡇⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣾⣿⣿⣇⠼⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⠠⡜⡃⠀⠀⢳⣴⠟
⠀⠀⠀⠀⢸⣿⣿⣿⣷⢿⣿⡇⢨⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠈⠻⣿⡿⠟⢹⣿⡋⠀⠀⠀⠘⠃⠀⠈⠀⠛⣄
⠀⠀⠀⠀⠈⢿⣿⣿⣿⡼⣿⣿⡸⠀⠈⠛⠿⠝⠙⢻⣿⣿⢿⡿⣿⣿⣿⣿⡿⠛⢦⠀⠈⠳⣀⣿⣿⣅⣇⡀⢀⣠⣤⡤⠘⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡻⣿⣿⣦⣄⡀⢠⣤⣤⣈⠙⠛⢃⡀⣀⠿⠿⢿⣀⣀⡈⠳⣄⠀⣼⣿⣿⠿⠿⣿⣿⠏⣿⣷⡆⡾⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣀⣸⣿⣿⣇⣀⡘⣿⣿⣿⣤⣿⣷⣌⣹⡅⠀⠀⠸⣿⣸⡟⠁⣡⡔⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠃⠀⠛⣻⠤⠚⠁⡆⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢷⣦⣤⣄⣠⠞⠁⠀⠀⢠⠃⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠟⠋⣽⣿⣿⣷⣶⣶⠄⣷⠀⠀⠀⠀⣜⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣧⡂⢻⡆⠀⠀⣴⠋⢁⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⢻⣿⣿⣿⣿⣿⣧⠈⣣⠀⡘⠃⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠟⠀⠉⠫⣸⣿⣿⣿⣿⣿⣿⣿⠙⣆⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠹⣿⠿⠟⠛⠛⠛⢿⣷⣥⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠈⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⢀⠔⠁⣾⠀⠀⠀⠀⠀⠇⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢀⠔⠁⠀⠀⣿⠀⠀⠀⠀⠘⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡐⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢠⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⢳⡀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⣇⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⢹⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠸⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⢀⡀⠀⠐⠀⡇⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠆⠀⢻⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠘⠀⠀⠒⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡄⣠⡀⠀⠀⠀⠀⡀⠑⠲⠴⠒⠁⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠁⠀⠁⠈⠐⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⢀⠄⠀⢄⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠉⠢⢄⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠘⡀⠀⠀⠄⠁⢠⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⡐⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⡄⠀⠀⠀⠀⠡⣬⠀⠸⡉⠀⡜⠀⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⢀⡜
⠳⠄⠀⠜⠁⠐⠀⡅⠉⠉⠈⠀⠀⠀⠀⠀⡠⠊⡜⠓⠂⠤⢤⠊⠉⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢇⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⢠⡄⠠⠤⠔⠈⣠⣾⣤⣀⡀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠈⢦⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠤⠤⠐⠋⠁⠙⠁⡐⠀⢹⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡄⠀⠀⠀⠑⢤⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠘⢦⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣀⠵⠀⠀⢀⡤⠊⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠂⠤⠤⠤⠄⣀⡨⠀⢨⠁⠒⠚⡄⠱⡤⠤⠄⠒⠊⡡⠐⠂⠉⠀⠀⢀⠔⠋⡠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠄⢀⠀⢠⠤⠤⠀⣠⠖⠒⠚⠒⠒⠢⢼⡄⠱⡤⣄⣄⣀⡟⠉⠉⠒⠒⠞⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢾⣀⣼⣀⡸⠀⠀⠀⡇⠀⡄⠀⠀⡀⠀⣄⠈⠂⠘⢄⠀⠸⣀⡆⠀⡄⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣆⣀⣀⡧⠤⠇⠈⠒⠢⠚⠀⠀⠀⠈⠉⠙⠒⠚⠁
⣿⣿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣰⢣⠃⠀⠀⢀⣸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⡇⡇⠀⡌⠀⠀⣛⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡇⠃⢰⠀⠀⠀⠸⣿⣿⡗⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⣽⡄
⠄⠀⠀⠀⠀⢻⡀⢰⠀⠀⠀⠀⣻⣿⣿⣄⠀⠀⠀⣿⣦⡀⡀⡀⣾⡟⠀⠀⣠⣼⣿⣿⣧
⣧⠀⠀⠀⠀⠀⢳⡀⢆⠀⢀⣶⣿⣿⣿⣿⣤⣤⣤⣻⣿⡿⣷⣿⣿⣅⣶⣿⣿⡟⣿⣿⢯⢃⡀
⣿⣧⡀⠀⠀⠀⠀⠙⢌⢲⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⡿⡌⠛⢸⣿⡿⠉⠙⠜⢋⣵⣿⣿⣷
⣿⣿⣷⣶⣦⣄⣀⢀⣼⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⠻⢿⣿⣾⣤⣼⠿⢤⣤⡴⣚⠛⢛⣛⡿⠟
⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠅⠛⠀⠀⠈⣿⣿⣯⣿⠿⣿⣿⠁⠀⠀⠉⢅
⠀⠀⠀⠀⠅⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⡀⠀⣠⣿⣿⣟⢃⣶⣤⣭⣔⢦⣄⠀⠀⠱⡀
⠀⠀⢠⡄⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠞⢫⣿⣿⣿⢿⣿⣷⡄⠀⠐⡀
⠀⠀⢼⢷⣿⣿⣿⣿⣿⣿⡿⠙⠛⠉⠙⣿⣿⣿⣿⣿⣟⣿⣿⣿⣦⣾⣿⣿⢹⣿⣿⣿⠛⠀⠀⠱⠀⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⢈⣿⣿⣿⠿⠛⠉⠻⢧⣅⣈⣿⣿⣤⣿⣿⠃⠀⢀⣴⣶⣿⠇
⠀⢠⠷⣿⣿⣻⣿⣿⠟⡃⠀⠀⠀⠀⠀⡿⠛⠃⣥⣿⣧⡄⠀⠀⡀⢰⣼⣿⣿⣿⣹⣾⣟⠞⣻⡿⡏
⠀⠀⢠⠟⢿⣿⣿⣧⣦⢉⠁⢀⠆⢸⣾⣃⠀⠀⠉⠿⠏⠁⠀⣼⠇⣸⣿⣿⣿⣿⣶⠁⠈⠉⠁⠀⡁
⠀⢰⣧⣦⣶⣿⣿⣻⠇⠘⠐⣿⠀⠀⢻⣿⢷⢦⡀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠋⣠⡖⠀⠀⠀⠀⠀⠃
⠀⢰⣿⣿⣿⣿⣿⡁⠀⣆⢤⠉⠀⠠⣾⣿⣅⣁⢡⠀⠀⢀⣤⣆⢀⣙⠋⠉⠀⠁⠀⠀⠀⠀⠀⠰
⠀⠈⠇⣿⣹⣿⣿⣿⣦⡘⣤⡀⠀⣼⣊⣿⣝⠈⠳⠀⠀⣿⡻⣿⠇⠀⠀⠀⠠⣀⢠⣀⠀⠀⢠⠁
⠀⠀⠀⢻⣷⣿⣿⣿⣿⣿⢛⣿⠛⠿⣿⣿⣿⣆⠀⠀⠐⣿⡿⠯⠀⠢⠀⠀⠀⢽⣿⣿⣷⣶⣧⣤⣀⡀
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠆⢤⡤⢘⠻⣷⣄⡀⠀⠀⠙⣷⢂⠀⠈⠢⡀⠀⠀⢋⠟⠿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⢰⢿⢿⠿⣿⣫⣿⣿⣿⣿⣆⣿⣿⣾⣿⣿⣦⣈⢿⣿⣿⣆⡀⠀⠀⡡⠖⡁⠀⠀⠈⠈⠙⢿⣿
⠀⠀⠀⠀⢸⣆⡴⣷⡿⣿⣿⣿⣟⢿⣿⣿⡿⣿⡝⡿⣿⣿⣿⣿⣿⣿⣶⡶⠉⠀⠈⢊⠲⡀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠟⢰⢻⢹⢻⢉⣎⡾⢿⣿⣿⠻⢿⡿⡸⢹⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢃⠙⣆
⠀⢤⠄⠀⠀⠀⠘⡆⡸⣼⠞⠋⠀⢠⢏⡿⠀⠀⠙⣧⡼⠏⠉⠀⠈⠹⣿⣿⡇⠀⠀⠀⠘⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢼⠃⠀⠀⠀⢸⠞⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⢰⠀⢠⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⣿⣿⣯⡅⠀⠀⠂⠀⡌⡞

⠀⠀⠀⠀⠀⠀⠀⡌⠀⡀⠀⡥⠒⠉⠀⢀⡤⢲⠎⢀⠴⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⣟⠀⠀⠙⣆⠠⡇⡇⢀⡔⠁
⠀⠀⠀⠀⡀⠀⣸⠃⢀⠀⢰⠁⢀⣀⣰⣋⣠⣤⡴⠧⠒⠀⠀⢀⡠⠤⠒⣀⠀⠀⠐⠂⢀⡂⠄⠀⡀⠀⠀⣿⠀⠀⠀⠘⣿⣷⢷⣞⡀⠀⢀
⠀⢘⡐⢰⠃⡰⠃⢀⠈⡠⠃⠀⠘⠁⡝⠁⠀⢀⣄⠤⠀⢀⡴⣋⠀⠀⠀⠰⠋⣩⠟⠉⠉⠉⠉⠛⠧⣄⠀⢸⡇⠀⠀⠀⠸⣿⡾⡁⠀⣠⠋
⠀⢸⡇⣿⠞⠁⠶⢉⡼⠀⠀⣈⠀⡼⠁⣄⠐⣠⠃⣀⡶⠫⠤⠅⡦⠤⠖⢊⡉⠀⠀⠀⣰⣷⣶⣶⡊⠈⠳⣌⣇⠓⠠⣄⠈⣿⡇⢧⡴⠁
⢠⡞⣸⡏⠀⣀⡰⠋⠀⠀⣀⡵⠊⠁⣀⣭⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣿⣿⠃⠀⠀⡈⠻⣦⣀⠉⢠⣿⣧⠘⡇
⠏⢰⡿⠀⠀⣽⠁⢀⣴⠟⢉⣴⡶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⡄⠈⠉⠐⠀⣹⣷⣿⣿⣿⣆⡿⣄
⠀⢸⡇⠀⢰⣿⡾⠋⠀⣰⣿⢋⣤⡁⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⣉⣉⣉⣙⣳⣤⣀⣘⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⡈⢹⣿⣿⣿⣿⠇⠙⣆
⠀⢸⡇⠀⣼⣿⠱⠆⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⣤⠤⠤⠤⠞⠋⠉⠁⠈⠉⠉⠛⠻⣿⣿⣿⡀⠀⠀⠀⠉⠹⣿⣿⣿⣿⠀⠀⠸
⠀⢸⣿⣰⣿⡏⢀⣶⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠙⢶⣄⠀⠀⢠⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣶⣿⣿⣿⣿⣿⡆⠀⣸
⠀⢸⣿⣿⣿⣿⡏⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣇⣰⡟
⠀⠈⣿⣿⣿⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⡄⠀⣿⣿⣿⣿⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡀
⢷⣴⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⢹⣿⣿⣿⣿⣿⣷⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠄⠀⣿⣿⣿⣿⣿⡇
⠀⠘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⠍⢸⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣿⡿⢣⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣃⡤⠾⡟⣻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢀⠀⠀⠀⠀⢀⡞⠀⠀⡇⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⣿⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢛⣻⣧⠀⢰⣯⠄⠀⠀⡞⣼⠃⣠⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣧⣤⣦⣤⣀⣀⠀⢀⢒⣻⣷⡒⣶⣻⣧⣤⣤⣿⣿⣾⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣴⣿⣿⠟⠉⢹⣿⡏⣿⣿⡃⠀⠈⠉⠉⡿⡉⠉⡏⢹⠛⠛⠻⠷⠭⣽⣯⣷⣿⠿⣿⠟⠛⠛⡛⢻⡿⠁⠀⠀⠀⣠⣿⣿⠿⠛⠛⠛⠻⠿⠿
⠛⠉⠀⠀⠀⠀⢿⣷⣜⢿⣷⣦⣤⣤⣴⣷⣷⣶⣷⣾⣤⣀⠀⠶⠶⣿⣿⡛⢛⡛⣟⣋⡁⢰⡇⣼⣇⠀⠐⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠉⠀⠀⣿⣷⣾⣿⡏⠀⠈⠉⣉⣽⣿⣿⣝⣋⣛⣏⡁⢀⣼⣷⢿⣿⣤⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠄⢄⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠉⠀⠀⠀⠀⠈⠓⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⣀⠠⢤⣀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠴⠊⠁⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⢲⠒⠄⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠐⢄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠀⠀⠀⠉⠁⠀⠲⢄⢀⠞
⠀⠀⠀⠀⠀⠀⠀⡈⢀⠴⠀⠀⠡⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢣⠀⠀⠈⣀⣀⡀⠼⡀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠔⢪⡁⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⢄⠁⠀⠀⠀⠀⠀⠀⠘⠁⠂⢢⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠈⣤⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⣄⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢸⡀
⠀⡐⠂⠀⠂⢴⠂⠉⠀⠀⠀⠘⠿⠋⠀⢸⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣀⡇
⠀⡇⠰⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⡀⠀⠀⠀⠀⠀⠀⠤⠴⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⡐⠁
⠀⠙⠒⢳⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⣰⡿⠋⡅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠈⠓⠲⠤⠤⠶⠒⠻⣁⡠⠊⠉⠳⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠂⠀⡐⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣧⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠥⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠀⠀⠀⡜⣸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡴⠋⣠⠀⠀⠀⡜⢀⠇⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⢋⠔⠊⠀⠀⠀⡘⡠⠎⠀⠀⠀⡴⠁
⠀⠀⠀⢀⢀⣀⣀⣀⠠⠄⠒⠂⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⢄⡀⠀⠀⠀⠀⠘⡅⠀⠀⠙⠢⠴⣅⠴⡴⠋⠀⠀⠀⠀⢸⠁
⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠓⠚⠉⠉⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠸⢆⡠⠀⠀⡀⣠⠏
⠀⠀⠸⢅⣀⣤⠤⣤⠤⠤⠠⠤⠔⠒⠒⠚⠛⠉⢉⠭⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⣀⡀⠀⢀⡀⣀⡤⠒⠚⠉⠁

⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⢸⡁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹
⠀⠀⠀⢸⢁⠿⣎⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡟
⠀⠀⠀⢣⡏⠐⠿⠋⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠺⠃
⠀⠀⠀⡘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇
⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠘⡎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⡐⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣿⡟⠛⠁⣄⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣷⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠁⠀⠀⣙⡄⠀⠀⠀⠀⢠⣶⠛⠀⠐⢿⠀⣄⠀⠀⠀⠰
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⢀⣾⣿⣿⡄⠀⠠⣼⠃⠀⠀⠀⡄⢠⠙⢻⡆⠀⠀⠁
⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⡁⠀⢀⣾⣿⣿⣿⣿⠀⠀⠉⡀⠀⠀⣤⣴⣿⠀⠋⣿⠀⠀⠀⡇
⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⠀⢸⣿⡇⣚⣿⠿⢿⣻⡿⠟⠀⠀⠀⢙⠆⢰⣿⣿⣿⠆⠐⣿⠀⠀⠄⠁
⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠈⡿⠛⠉⠁⠀⠀⠑⡀⠀⠀⠀⠀⠘⣂⠀⠟⠛⠋⠘⠂⡿⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⢹⣷⣀⣤⣶⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⡀⠀⡟⠹⠃⠀⡌⠃
⠀⠀⠀⠀⠀⢀⣤⣾⠟⠛⠋⠉⠉⠛⠳⣤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠈⠃⢠⣠⡆⠀⠘⡙
⠀⠀⠀⠀⠀⠸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠁⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣸⠃⠀⠆
⠀⠀⠀⠀⠀⠀⠈⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠹⣗⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⡋⣠⢎
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢿⡆⠀⠀⠀⠀⠀⠀⠀⠙⣾⣄⡀⣄⠀⠀⠀⢸⠁⠀⣼⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢻⣤⣄⡀⠀⠀⡀⣀⠀⠘⠀⠣⠀⠀⠀⠀⢸⠀⠀⢻⣿⡵
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⢿⣿⠧⣴⣿⣿⣿⠆⠀⢀⠀⠀⠀⠀⠸⡄⠀⠈⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢺⣿⣆⠈⠛⢻⡿⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠆⢷⠀⠀⠀⠀⠿⣿⣶⣤⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣼⠀⠀⠀⠀⠀⠈⠙⠻⠏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣀⠤⢺⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠸⠟⠋⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠁⡠⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠠⡀
⠀⠀⢀⢀⠀⠁⠀⠀⠀⠀⡀⠄⠢⣴⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢹⠿⣷⡀
⠐⠀⣾⡷⣧⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠓⠂⠀⠀⠀⠐⠂⠁⠀⠀⠘⢿⣿⠇
⡇⢸⣿⡴⣄⣴⠞⠀⠀⠀⠀⠀⠀⠈⠢⠤⠀⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠣⡉⢋⣵⢿⣧⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠉⣿⠄⠙⢿⣦⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢣⡀
⠀⠀⠀⢿⣦⡴⣨⣿⣷⣶⣤⠃⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢲⡿⠟⠃
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⣀⣴⠞⠻⣦⣀⣀⣄⣤⣤⣤⣤⠟⠂⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⣦⣀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠈⠉⠉⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⠶⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠠⠆⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⡿⠃
⠀⠀⠀⠰⠛⠈⠁⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡄⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣄⡀⠈⠻⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠂⠀⠈⠻⣿⣿⣿⠁⡄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⡇⠀⠀⠀⢠⠃⠀⠠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣠⡆⠐⠁⠀⠀⠀⣀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⢀⣿⠥⠀⠀⠀⠈⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣼⣿⣄⠀⠀⠀⠀⠀⠀⠁⠀⠘⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠁⠀⠀⠄⣾⡇⠙⣆⢀⠀⠀⠐⠂⠠⢤⡀⠀⡇⠀⡆
⠀⠀⠀⠀⠆⢠⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⡇⢡⠀⠘⢄⠀⢀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⢀⠇⠸⠀⠀⠀⠳⣌⡓⠠⠀⠠⢂⡜⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⣅⠀⠀⠉⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠱⡄⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢤⠀⠀⠀⠀⠀⠲⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⣀⠠⠤⣄⠀⠀⣠⠤⠆⠱⡦⠆⢀⠀⠀⡄⠀⠴⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠁⠑⠂⢰⠀⠀⠀⠐⠂⡀⠄⢢⡀⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⢠⡄⠀⠀⠀⠀⠙⣤⠀⢠⡅⣄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠈⢧⠘⣇⠘⣆⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡈⠀⠘⡄⠀⠀⠁⡀⠀⠀⠀⠀⠀⢸⡅⠹⡆⢿⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠄⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣯⣿⠁
⠀⠀⠀⠀⠀⠀⣾⣿⡿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⣶⣻⣦⣀⠀⠀⣠⠖⣋⡉⠁⠀⣀⣤⠤⠶⠒⠒⠄
⠀⠀⢤⣤⣤⣼⣿⡿⠿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣽⣿⣿⣿⣿⣿⣿⣿⣿⣷⠞⣱⡾⣋⣵⡶⣋⣭⣶⡾⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡿⣏⣽⡾⠟⠋⠁⠀⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠿⢶⣄⠀⠀⠀⠀⠀⠈⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣵⣾⣯⡁⢺⣀⣀⡉⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠻⠿⠻⣿⣯⣻⣿⣿⣿⣿⡾⢟⣋⣽⣿⡿⣿⠿⠟⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣯⣧⣾⣥⡛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⡋⢉⣉⡉⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣶⣿⠟⠋⠹⣿⣿⣿⣿⣿⣿⣿⣟⠋⣿⡿⣻⣿⣶⢭⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣟⠳⣝⢿⣿⣿⡟⢢⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⢿⣿⣿⣧⣌⠻⣟⠻⠟⠋⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⡙⢿⡻⢳⣍⢀⡛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⠀⡝⠢⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡁⠈⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣄⠀⢾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠟⢻⣯⣵⣤⣄⢀⡀⠀⣀⣀⣤⠀⠀⠀⠀⠀⠀⢰⠏⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⠈⣿⣿⡇⠈⢿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠁⠀⠘⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡇⠀⣾⣿⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢷⣄⠀⠀⠀⢰⣿⠀⠀⠀⣀⣤⣿⢋⡋⢹⣿⡀⢸⣿⡿⠁⠀⣀⣀⣠⣶⣿⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠈⠉⣷⣄⠀⠀⣿⢇⣴⣿⡋⠀⠁⠄⠁⠈⣉⣷⣜⣿⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⣀⣴⣿⡟
⠀⠀⠀⠀⢦⠀⠈⢻⣧⢀⣿⣼⣿⣿⣿⣶⡿⠀⢰⣾⣿⣧⡝⢿⣛⣭⠾⠋⠘⢹⣿⡿⠀⢀⣴⣿⢋⣀⣀⣤⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⢸⠘⢷⠀⣤⣿⣧⢿⡿⣿⣿⣿⣽⣷⡀⢸⣿⣿⡇⢁⡞⠁⢠⣄⠀⠀⢸⣿⣤⣶⠟⠛⠋⠉⠉⠉⠛⠛⣿⣿⡿⢿⣛⠀⡈
⠀⠀⠀⠘⣇⠈⠢⡘⠿⣿⣿⣷⣽⣿⣧⣬⣹⣿⣿⣿⣿⣧⡛⠀⠀⠀⠀⠀⠐⣿⣿⠋⠀⠀⠀⠀⠀⣀⣴⡾⠟⠉⠀⣴⣾⣇⣴⠁
⠀⠀⠀⠀⣬⣃⠴⣌⠰⣦⣿⡿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣷⣄⠀⠀⠠⠀⢠⣿⣃⣀⣀⠀⢀⣠⣾⡿⠋⠀⠀⠀⢸⣿⡿⢃⡄
⠀⠀⠀⠀⠹⣧⣈⠒⠚⠿⢿⣿⣌⣷⣿⣿⡐⣿⠿⣿⣻⡷⠉⠁⠀⠀⢀⣴⣿⣻⣿⣿⣿⣿⡿⣿⣿⠃⠀⠀⠀⠀⡰⣯⡷⡜⠁
⠀⠀⠀⠀⠀⣬⣛⠛⠶⠦⢾⣿⣿⣿⣟⠻⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠻⠛⠿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⣠⣿⣿⠇⣧
⠀⠀⠀⠀⠀⢻⣿⡉⠁⠐⠘⠲⠶⣿⣿⣿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⡤⢸⡿⣯⣤⣤⣶⣶⡶⠞⢩⣿⣾⣴⣿⡆
⠀⠀⠀⠀⠀⠈⢪⡙⡓⠆⠀⠀⠀⣤⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⣵⡿⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠞⠀⠬⢛⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⢀⣤⣿⣿⡿⠋⠞⡿
⠀⠀⠀⠀⠀⠀⠀⢀⢹⡿⣿⣿⣻⣯⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⡏⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢹⣇⡹⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠛⠛⠛⠋⢉⣥⣿⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣎⡻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠡⠀⠀⢠⣶⣞⣟⣻⣿⠀⣿⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣞⠁⠈⣿⣿⣿⣿⣿⣾⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣿⡿⢿⣿⠿⠀⠀⠠⠀⠀⠀⢀⠀⠀⠐⠘⠀⠚⠻⡿⣛⣛⠗⠀⠘⢿⣿⣿⣽⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠉⠙⢧⠀⠀⠀⠠⠀⠀⠈⠈⠉⢁⣴⠿⠿⠿⠿⠿⢿⣿⠂⣀⣙⠿⢿⣿⣿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣿⡄⠀⠀⠀⠀⣴⠾⠋⠙⠛⠿⣿⣿⡢⠈⠻⣿⠷⣿⠿⢏⡉⠑⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠈⠷⠶⣶⣿⣧⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠀⠠⣽⣶⡿⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣽⢷⣦⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣟⠛⣿⣿⡛⠟⢿⣿⣿⣿⣶⣶⠲⣶⣾⣿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡯⣻⣿⣿⣿⡝⢿⣦⡀⠙⠛⢿⣿⣿⣷⣜⣯⣿⠟⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⢿⣿⣿⠏⠉⡟⣿⡏⠉⠻⢆⣹⣿⣶⣯⣄⣈⡙⠛⠻⠿⠿⠗⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠍⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠈⠛⠀⠈⠋⠈⠉⠉⠁⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣷⣤⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⠇⢵⠈⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡃⠀⡀⠀⠌⣿⣿⣿⣧⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢸⣿⣷⣆⣁⣀⡀⣿⣿⣿⣿⣿⣿⡛⣴⣷⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⣿⣿⣿⢆⣿⣿⠻⠿⣿⡟⠉⠘⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢁⣿⣷⢿⡿⠋⢸⣿⠋⠀⠀⠀⠛⣤⡀⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣾⡞⠁⠎⢡⡄⠈⣿⡇⠀⠈⢀⣼⣿⣿⣄⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⢀⡇⠘⣧⣀⣹⡇⡠⢾⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣸⠇⠀⠈⢻⣿⡟⢣⣿⣿⣿⣿⣿⣿⣽⣦⡓⠂⢀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠐⡉⡛⢻⣿⣿⡿⣿⣿⣿⡟⠘⠳⢛⣦⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣷⠀⣾⣿⣿⣿⡿⡇⢹⢻⠄⠀⠈⠀⢪⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣶⣿⣿⢉⣿⡸⢇⡿⠀⠀⠀⠈⣿⣦⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠞⣡⣿⣁⡼⣿⠋⠙⠃⠛⣠⠆⠀⠈⡄⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣀⢼⡏⡽⢻⡿⡀⠀⠀⠀⠃⠰⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢣⡁⣨⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣤⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⣿⡉⣹⡏⠸⣇⣡⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣍⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⢸⣿⠁⣇⠸⢰⠀⠀⠀⠀⠀⠀⢶⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢹⣎⡃⠘⠀⣸⠀⠀⠀⠀⠀⠀⠘⢡⠘⠀⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⢺⣿⣯⣽⠀⣿⠀⠀⠀⠀⠀⠀⣧⠀⢣⡀⢼⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢰⣿⣿⣿⠈⡿⢯⠉⢰⣿⠀⠀⠀⠀⠀⠀⠃⠀⠀⠱⣾⣿⡟⣿⣿⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣸⠇⠸⢿⣿⣿⠀⠀⠀⡀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣿⣿⠆⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⣿⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⠓⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⢦⣤⣄⢀⣿⣿⣺⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢘⣄⣭⢾⠿⢿⠿⠎⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣀⢤
⠀⠀⠀⠀⠀⢸⣿⡅⠘⢿⣶⣾⣦⡀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠁
⠀⠀⠀⠀⠀⣾⣿⣿⣷⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⢀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏
⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⠟⠋⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣏⣹⠟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠟⠛⠃
⠀⠀⠀⠀⠀⠀⣿⡙⣿⣻⣿⡖⢢⣤⢸⣷⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢹⡇⡟⢩⣿⡇⢸⡟⢸⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠘⣡⣦⡼⢿⣇⣼⣧⣾⡟⢉⡿⠋⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠻⠿⠿⠤⠸⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⢉⣠⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⣠⣴⣾⣿⣿⣿⡟
⠀⠀⠀⠀⠀⢀⣦⠀⢀⣤⠶⠿⠿⠷⣶⣭⣝⠲⣄⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⢁⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣤⣤⣾⣿⣶⠿⠿⠷⣶⣤⣤⣢⡍⠛⠛⢹⡿⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢉⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⣈⡿⣿⠟⢯⣙⠲⣄⡀⠀⠀⠉⠙⠀⢸⣿⣧⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⣠⠞⠁⠀⠉⠀⠀⠉⠳⠶⣭⣓⣆⠀⠀⠀⢸⣷⢻⢷⣶⣞⠁⠀⠀⠀⠀⠀⠀⣠⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠁
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠀⢦⢸⣿⡆⠈⡏⠙⢦⠀⠀⠀⠀⠀⣰⠟⢁⣼⣿⣿⣿⣿⣿⡿⠟⣩⣵⣾⣿⠋
⢿⣄⡄⢸⡇⣀⠀⠀⠳⢤⣐⠲⠶⠖⠀⢀⡲⢥⣿⡇⢸⡇⠀⣾⡇⠀⠀⠀⣼⠏⣠⣿⣿⣿⣿⣿⠟⣫⣴⣿⣿⣿⣿⠿⠧⠤⢤⣤⣤⣀⣀
⠀⠹⣇⠸⣧⣻⣶⣤⣤⣶⣾⣿⣤⣴⠄⣿⣿⠈⣿⣧⡿⠀⢠⡟⠀⠀⠀⣼⠃⣰⣿⣿⣿⣿⣟⣥⠾⠟⠛⠉⠉⢀⣠⣤⣶⣶⣿⣿⣿⡿⠟
⠀⠀⠈⠓⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠟⠁⣸⣿⡿⠶⠶⠞⣴⠀⠀⣼⠇⣼⣿⣿⣿⠿⠛⣉⣤⣶⣗⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣿⣏⣴⣶⣾⣿⣿⣿⣶⣶⣿⠞⠀⢰⠏⣼⣿⡿⠋⣡⣴⣿⠿⠟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠉⠉⠁⠀⠀⠀⢟⣼⠟⢁⡴⠟⣋⣥⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣠⠞⠁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠋⢻⣿⣿⣿⣿⣷⣄⣤⣤⣤⣀⠰⢾⣿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⣿⣿⣿⣿⡟⠉⠀⠀⠀⠙⢳⣄⣴⣶⣿⣿⣶⣶⣶⡦⠈⠉⠉⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⣟⣿⣿⠀⣴⣿⣿⣶⣄⠀⢻⡌⠉⣩⣭⣭⣥⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣿⣾⣿⡇⠀⣿⣿⣿⣿⣿⣷⠀⣷⠰⣶⣶⣶⣶⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⣿⣿⡇⢸⡟⣦⡙⢿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡀⠘⣿⣿⣿⣿⣿⡇⢸⣇⠈⠻⣷⣌⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⣼⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⣧⠀⣿⠀⠀⠙⢿⣷⣌⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⠀⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢹⣿⣿⣿⣿⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠶⣤⣄⣽⣿⣿⣿⣿⣿⣿⣿⠇⠀⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⡿⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⡀⠈⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣽⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢊⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⡢⡤⠤⠠⠄⠐⣁⡀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠢⠄⣀⠀⠀⠀⠀⠒⠢⢄⣠⠋⠀⡀⠀⠀⠊⣁⠂⡍⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠢⠀⠈⢒⠛⠙⡉⢨⠀⠈⠃⠄⠀⡿⠃⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢀⠔⠒⠊⠉⠩⠄⠈⢅⢀⠇⠙⠀⠀⣴⠃⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠀⡴⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢴⡀⠁⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⣄⣀⠀⢀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠀⠹⠀⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠠⡂⠀⠈⠀⢠⡴⠴⡄⠀⠀⠐⠀⠀⠀⢠⡁⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠈⠓⢴⠀⠀⠀⠀⢠⠎⠜⠃⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠅⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢀⠀⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠈⠁
⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠪⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠃⣤⠀⠱⡄⠄⢸⠄⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⡄⠐⠂⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠱⡄⠀⠀⠀⠇⠀⠀⠀⠰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠖⣀⠙⠉⠒⠂⠀⠈⠀⢈⠀⠁⢇⣢⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⠀⠲⡾⣸⣠⣠⠀⠀⢂⢸⠆⢗⣫⡽⠁⠐⡌⢒⠀⠀⢀⠔⠂⣖⣒⠀⠉⢢⡤⠀⢠⣀⢀
⠀⢀⠠⠤⠔⠂⠤⣔⣢⠅⢚⣒⣒⡌⠙⠍⠀⠬⡑⢉⠶⠀⣄⡘⠏⣦⠷⡂⠀⡻⠶⠊⠀⢀⡀⣠⠤⣶⡁⠂⣃⠑⢒⡠⠔
⠀⠀⠀⢄⠞⠂⠖⠈⠉⣶⣄⢰⣅⢀⠀⠀⠈⠐⠀⠈⢿⡆⢨⢤⠀⠍⠉⠊⠁⠀⠀⠀⠀⠻⠄⠛⠃⢀⡲⠭⠕⠊⠁
⠀⠀⠀⠀⠉⠂⢄⡐⠤⣭⠑⠛⠫⠞⠁⠀⠀⠀⠀⠀⠀⠙⢶⣦⠄⠠⠄⠀⠀⠀⠠⠤⠄⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠘⠣⠀⠠⠄⠀⠀⠤⠤⠄⠐⠀⠙⣥⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠻⣧⡀⠀⠀⠀⠀⠀⠀⡀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠘⢿⣄⠀⠀⠀⠀⠀⡉⢠⣿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠈⢻⣦⠀⠀⠀⡰⠁⣾⠃⠀⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠈⠛⢀⡠⠀⠈⢿⡄⠀⢻⣧⠀⢰⠇⣸⡏⠀⢸⣿⠀⠀⠀⢂⠄⠉
⠀⠀⠀⠀⠻⣿⣛⠛⠻⠷⢶⣤⣄⡒⠉⠲⣄⠘⣿⡀⡄⢻⣆⠈⢰⡿⢀⠆⣼⡇⠀⣠⠔⠁⠀⣀⣤⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠈⠉⠻⢷⣤⡈⠳⣼⣧⠰⠈⣿⣧⣿⢁⡞⣠⡿⢀⠜⣁⣤⡶⠟⠋⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠠⣀⠉⠻⣦⡙⣿⡆⠃⢸⣿⡇⠎⢠⣿⣷⡷⠟⢋⠁⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡙⢦⡈⠻⣿⣷⣿⣿⣿⣿⣶⣿⠟⢋⡠⠖⣁⣴⠿⠋⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⢲⣿⣿⠏⣁⠉⠻⣿⣿⡎⣁⣤⣾⠿⠥⠖⠚⠉⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠒⢂⣴⣿⣿⣿⡏⢀⣭⠖⢀⣿⣿⣿⠿⠿⠿⠶⠶⠶⠶⢶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠁⠀⢀⣤⣾⠟⠋⡩⣿⣿⣷⣤⣥⣤⣾⣿⡿⠉⠈⠛⠒⠀⠀⠂⠀⠀⠀⠀⣉⣉⣛⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⣶⠿⠋⠔⣪⣵⠞⣻⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⠶⠶⠶⠶⠶⠿⠿⠿⠟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢁⣤⡶⠛⠉⢀⣴⣿⢃⠏⣸⣿⣿⢰⠀⢿⣿⣌⠲⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠞⠋⠁⠀⠀⡠⠊⣼⡏⡎⢰⡟⢸⣿⡀⠆⠸⣿⢻⣆⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⢀⠜⠁⢰⡿⡘⢠⡿⠀⡟⢻⣧⠘⡀⢿⡄⢿⡄⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠑⠀⠀⣾⠃⢡⡿⠁⢠⠇⠈⣿⣆⠀⢸⡇⠸⣷⠀⠀⠳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⡿⢠⡿⠁⠀⡞⠀⠀⠘⣿⡄⢸⡇⠀⣿⡆⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢇⣾⠁⠀⠰⠃⠀⠀⠀⠹⣿⣸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠃⠀⠀⠀⠀⠀⢹⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣼⡇⠀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢷⢀⣿⣡⠞⠫⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢸⣿⡿⠋⠆⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⣸⣿⠁⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣿⣧⡾⠋
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠉
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⢀
⠀⢈
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⢠⠏⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠈⠙⢲⣤⠶⠋⠀⢸⡇⠀⢾
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⡀⠀⠀⡄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣀⣀⣤⠟
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣼⡟⢡⣄⡖⢶⠛⠾⠫⢅⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠙⢻⡟⢻⡆⠀⣠⣴⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⡇⠨⠛⣿⣿⣿⣦⣀⣾⢏⡀⠈⣵⣛⣣⠗⢠⡀⣯⡸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢻⡟⣵⣾⣿⣿⣿⡕⢹⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⡤⢐⡈⢿⣿⣿⣿⣋⢯⡇⣠⢧⣀⠘⣦⡴⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⣷⠀⠀⠈⣿⣿⣿⡯⣿⣧⡤⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⡇⣄⡀⢢⣿⣿⡇⡷⣠⣱⡇⠀⣹⠋⠁⢀⣠⡤⠶⠤⣤⣤⡤⠴⠖⠋⠁⠀⠀⠘⣧⠀⠀⢹⣿⣿⣿⣿⣉⣶⣿⣿⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣧⠀⠙⡽⣿⣿⣷⡿⢻⣿⡇⠠⡇⠀⠀⡀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠸⠿⠃⢸⣧⠀⣸⣟⣼⣿⠷⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⢩⠂⣿⣿⣿⣿⠛⠻⣷⡀⠇⠐⣦⡿⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣼⣇⣴⣿⣿⣹⡳⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣿⣿⣿⣿⡄⠀⠢⢳⠀⠀⠹⡇⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⢿⡯⣽⠻⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⣯⡟⠛⠿⣿⣿⣦⣀⣹⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⡀⠀⠛⠛⠘⠁⣿⣂⣼⠿⠿⡛⢿⡟⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣤⠘⢿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠀⠀⠀⠀⣠⡿⠋⢀⠂⢀⣬⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣤⣙⠿⢿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡾⠋⠀⠐⣁⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⠉⠉⠉⢙⣳⣶⣿⣿⣿⣿⡿⠿⠻⣿⡋⠉⠉⠉⠛⣿⢿⣿⡷⠖⠐⠚⠉⠉⠉⢁⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣷⣶⡶⠶⠿⣿⣿⣿⣿⣿⡀⠀⠠⣿⠷⣦⣀⣀⣴⣿⣆⢟⡷⣶⡶⠶⣶⣶⣾⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⣛⣗⠀⣶⣿⣿⣿⣿⠟⠁⠈⣳⣶⣿⣾⠀⡊⠙⠑⢬⣯⠞⠀⠈⠹⣆⠀⠹⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⠈⠢⠤⣼⣿⡿⠃⠀⣠⣾⣿⡿⢿⣙⠯⣁⡜⣠⢯⡿⣿⣦⣄⡀⠘⠳⠾⣿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢶⣶⡟⣋⣅⠀⢠⣾⣿⡿⡯⠃⠀⠈⠙⠲⣝⣱⠋⠀⠈⢻⣿⣿⣆⢰⢶⡼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣏⣟⣴⣿⣿⢫⠖⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢻⣿⣿⣷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡟⣱⡟⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣤⣴⣯⣤⣷⣀⣧⣀⣰⣇⣼⣦⠷⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣿⠏⠉⢹⣿⣫⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⢸⡀⠀⠀⣿⣟⡉⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣶⣾⣧⣐⣀⣸⣿⣿⣿⣿⣿⠦⣄⣈⣷⣶⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠁⢀⡤⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣰⠀⢰⠀⠀⠀⢀⠀⠶⢀⡀⠀⠀⠛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢈⠅⠀⠀⢰⡀⠀⠀⢠⡀⠀⢠
⠀⠀⡅⠉⡄⠈⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠃⢠⠞⠉⠂⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢉⠀⠠⡐⠋⠘⠃⠈⠀⠏⠀⠈⠒⢀⠀⢨
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠾⠒⠲⣄⠈⠳⡀⠀⣠⡄⠀⠀⠀⠀⠀⣄⠀⠀⠉⣤⣀⡂⠐⢤⣄⠀⠀⠘⠋⠁⠘
⠀⠀⡄⠀⠀⠀⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⡾⠃⠀⣤⢀⠁⠀⠀⢿⡆⠘⠀⠀⠀⠀⠠⠒⠉⠀⢠⠻⠋⣉⣹⣶⠀⠙⣷⠀⠀⠋⢱⣾
⠀⠀⡇⠀⠀⠀⠀⠐⣏⠀⠀⠀⢱⠀⠀⠀⠀⠈⢡⠃⠀⠀⠰⠃⣠⠜⠀⠀⠚⠀⠷⠀⢷⡀⠀⢠⠀⠰⠛⢐⠤⠀⠀⠀⠀⠉⠀⢸⣧⠀⢸⡇⠀⠖⢱⣾
⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠃⡤⠂⡲⣘⠛⣗⣤⠘⠣⠀⠙⠀⢀⣄⠙⠃⠀⠠⡀⠀⣠⡄⣠⡏⠀⣸⠃⠀⢤⡄⣩
⠀⠀⡇⢠⠆⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⣀⠀⠘⣠⠂⣠⠟⠁⠈⢁⣈⠁⠐⢀⡆⠀⠀⠛⢁⠑⡶⠀⠀⣿⡛⠿⠿⠋⢀⣰⠏⠀⠀⣀⡸⢿
⠀⠀⡇⠉⡆⠀⠀⠀⠰⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣥⠀⠐⠃⢰⣯⠀⠀⠀⠀⠀⢀⣶⡄⣥⠀⠀⠀⠈⠣⡀⣀⠀⠀⠑⠒⠲⠚⠛⠁⠀⠤⡀⠸⣀⣈
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠦⠀⡄⠀⠀⠀⢀⠀⣀⣾⠇⠀⠛⠇⠀⠐⠿⣿⡇⢸⣇⠀⣿⢆⣠⠀⠉⣀⠴⡂⢀⠂⡄⠀⣷⠀⡜⠋⠛⠉⡸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⡖⠁⠀⠀⠀⠀⣰⣧⡟⢰⠄⠀⠀⡀⠄⠠⣿⢠⡇⣿⠀⠑⠈⠉⠂⡀⠋⣤⠈⣸⡇⣧⡜⠀⠐⠶⣠⣶⣾⢷
⠀⠀⡇⠀⠀⣰⠃⠀⠀⠀⠀⢀⡀⠀⠀⢀⡶⠀⠘⠀⠀⢠⡟⠁⢃⣼⡀⠀⠀⠀⠀⣸⣷⡌⢃⠘⡆⢻⣏⣀⣀⡀⠚⠛⣦⠀⣤⡈⠁⣦⣽⣦⠟⢻⡇⢘
⠀⠀⡅⠀⠀⠈⠃⠀⠀⠀⠘⠄⠀⣠⣶⡏⠀⠀⠀⠰⡄⠁⣰⣶⠚⢅⠱⢦⣀⣤⣴⡉⠏⣛⠉⠴⣣⠀⢉⠉⠀⣄⢹⠀⠘⢦⠛⡧⠘⠃⠙⠃⢴⣬⠀⢸
⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⡶⠀⢻⣿⡌⠛⠂⠀⡀⢾⠀⣀⠀⠘⠋⢼⣿⠀⣉⣤⢷⡆⢸⡍⠛⣀⠙⢆⡀⠁⠀⢿⣥⡄⠀⡀⠀⠇⠘⠀⠀⢀⠾⢋⡀⢸
⠀⠀⣯⠀⠀⠀⠀⢈⡷⠀⡴⠀⠀⣀⣤⠐⠒⠋⢤⡈⠀⡀⠉⠀⠀⠀⢻⡇⠀⣨⡼⣷⣾⠕⠊⡙⠰⠚⠉⢂⡀⠀⠀⠰⣶⡅⡀⠀⠀⠀⠀⠈⢰⡾⠃⢸
⠀⠀⠉⠀⠀⠀⠀⠈⠁⠸⢧⠄⠒⠂⢠⠀⠳⢄⠀⠹⣆⠀⠀⠀⠈⠂⢸⣇⠀⢈⠀⢻⡃⠀⣹⡍⠂⢀⡤⠚⠡⣄⠀⠘⣶⣴⡍⠆⠀⠀⠀⠀⢩⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡼⠖⠀⠀⠀⢋⣶⡄⠓⡀⠉⠳⢦⡁⢀⠀⢸⡟⠀⠈⠀⣸⠁⡂⢹⣇⡀⠂⣠⡈⣰⠛⣀⠀⠿⣤⠀⢄⠀⠀⠀⣠⠞⠁⡄⣠
⠀⠀⡄⠀⠀⠀⠀⡾⢠⣾⠃⢲⣶⣟⠀⠈⠁⣿⠀⠀⣠⣤⣄⠙⢦⡀⢨⡗⠂⠔⠸⢷⣈⣱⡞⢋⠀⣤⠼⠈⢉⡴⢇⠀⠂⠀⠀⢸⠁⠀⣐⠁⠀⠈⠓⠹
⠀⠀⡧⠤⠔⠀⠤⢇⣼⠟⠀⣿⢿⣿⠚⣤⠀⠙⣧⡀⢀⠋⠀⠛⠲⠷⣌⡛⠛⠛⠛⠂⡉⢡⡀⣹⣶⠏⠠⢠⡾⡇⠀⢣⡾⠦⣀⠈⣦⠀⡉⠠⠤⠤⠄⣸
⠀⠀⠃⠀⠀⠀⣴⣾⠇⠀⠀⣹⢿⣿⠐⢸⡄⠀⠈⢷⡀⠻⣷⣄⡀⠀⠈⠙⠛⠧⠼⠶⠷⠾⠿⠿⠟⠓⣐⠿⡄⠈⠒⡋⠀⠞⠛⠛⡤⠟⠀⠀⠀⠀⠀⢿
⠀⠀⡆⠀⠀⠈⢻⠋⠀⣤⠀⠇⢸⣿⠀⠈⠀⠀⠀⠈⠳⣄⠈⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠚⠟⣇⠈⡖⠀⠀⣾⠀⠘⣾⠤⠀⠦⠀⠀⠀⠀⣸
⠀⠀⡇⢠⠀⠀⠇⢀⡴⠉⠀⠃⣼⠇⠀⠀⢸⡀⠀⡀⠀⠻⣆⣀⠈⠛⢿⣶⣄⡀⠀⠀⠤⠄⢀⣀⣀⡀⠀⡈⠓⠀⡀⢸⣿⣧⣀⡀⠀⠋⠀⠀⠀⢀⠀⢨
⠀⠀⡁⠘⠆⠀⣀⡌⠁⠀⠁⠀⣿⠀⠀⠄⠀⣷⠀⠹⡀⠀⠙⢾⡦⠀⠈⠻⣿⣿⣷⣤⣤⣤⣴⡾⠋⣡⡞⠃⣠⡾⠃⢸⣿⣿⣿⠀⠀⠀⠀⠆⠀⠀⠀⢼
⠀⠀⠟⠃⠘⠃⢸⠀⢀⣿⣠⡄⢩⣷⡄⠀⠀⠘⣧⠀⠀⡀⠀⠈⠷⡼⢆⡀⠈⠙⠷⠾⠟⠉⠁⢀⣼⡟⠀⣠⠁⠘⠃⠸⣿⣿⠏⠠⠖⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠎⠀⠈⠁⠀⠶⠾⢻⣿⢀⡄⢻⣷⠀⠀⠀⢹⣄⠀⠘⠀⠀⠀⠈⠺⢏⠂⡀⠀⠀⠀⡀⠀⢠⡿⠀⡄⢙⡂⠀⠀⠁⠈⢡⣤⡀⢀⠀⠀⠀⠞⠁⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣹⣿⣆⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠈⠀⠉⠀⠈⠀⣈⣡⠏⠱⠀⠀⠀⣠⣌⠟⠻⢮⣧⣆⣰⠇⠀⢀⣿
⠀⠀⠇⠀⠀⠀⠠⠀⠀⣰⠊⠙⢿⣿⣿⡟⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣿⡏⠀⢠⠀⢠⡎⠁⢈⣅⢠⣼⠋⠙⢿⡎⢉⠉⢻
⠀⠀⠄⠀⠀⢠⠆⣠⠎⡉⠁⠀⠀⠙⢿⡇⠀⢆⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠈⠒⠄⠀⠠⠒⠋⣠⣼⠟⠀⢠⡞⠂⠺⣁⠀⠸⢧⣸⣧⡀⢀⠸⢅⣸⣇⣤
⠀⠀⠆⠀⢸⡅⠀⠹⡄⠁⠤⠀⠘⠇⣠⠀⠀⠘⢄⠀⠀⠀⠀⠐⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠠⡾⢂⠀⠛⣷⣤⣤⣠⡄⠀⠀⠀⢤⡄⠀⠆⢸
⠀⠀⠂⠀⠃⠰⣦⡄⠈⢑⣆⣠⣤⡄⢸⠀⠀⠀⠀⢂⠀⠀⠀⠄⡀⠀⠀⠀⠀⠈⠉⠿⠿⠛⠉⠀⠀⣠⠀⣷⠈⡆⠀⠸⠁⢘⡁⢀⠀⠄⠀⠀⠉⡀⠀⢘
⠀⠀⠁⠐⠂⠄⠘⢷⠟⠛⠋⠉⠀⠀⢸⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠰⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⢱⡄⠀⣦⠎⠁⠐⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⢾⣧⣀⡀⣀⣄⠀⠀⠸⢂⡀⠀⠀⢮⠀⣤⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⢀⡜⠀⠀⢸⠖⢲⠆⠀⠐⠛⠁⠀⠀⠀⣦⡀⠀⡤⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠉⠁⠀⢠⡆⠈⠷⠀⠀⠆⠀⡏⠑⣄⣀⢀⠀⠀⠀⠢⠀⠀⠀⣠⡞⠀⠀⠀⡞⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠂⢀⡄⠀⠀⠸⠁⠀⠀⢆⠀⠐⠀⡇⠸⠟⠉⠉⠻⠷⣤⣀⠀⠀⠈⠀⠀⠀⢀⡰⢿⠄⠀⡅⠀⠁⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⡄⠀⠀⠀⠐⡀⠀⠀⣀⣴⡄⠀⡇⠀⣿⡟⠳⠦⣤⣤⡾⠛⠙⠀⢸⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⣁⣀⣀⠀⠠⠤⠤⠤⠀⠀⠀⢰⣿⣟⠀⢱⡄⣠⠀⠀⢹⣿⠀⢀⠀⠀⠿⣿⠀⠀⠘⠁⡇⠀⢸⡆⠘⣷⠀⠀⡀⠤⠛⠀⠓⠀⠀⠤⠤⠤⢀⡀⠀⢨
⠀⠀⣙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠈⠁⠈⠁⠀⠦⠀⠀⠛⠀⠸⠀⠀⠂⣿⠀⠀⠀⠀⢱⠀⢸⡇⠀⢻⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠄⠀⠀⠀⢳⣄⠀⠀⠄⡇⠀⠀⢹⡀⠀⠀⠀⢸⡀⠈⣷⠀⠀⢻⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠒⣰⣤⡀⠀⠀⢸⣿⠟⠉⠀⣷⠀⠰⣿⣧⠀⠀⠀⢸⠃⠀⣹⣦⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠞⠛⠁⠙⠁⠶⠚⠛⠦⠤⠴⠿⠀⠀⠿⠗⠀⠀⠐⠚⠓⠘⠛⠹⠶⢶⡴⠆⠀⠀⠀⠀⠀⠄⠀⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠻⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰
⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⡤
⠀⠀⠀⢠⠂⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢸⡇
⠀⠀⠀⠈⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠃
⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⣇⡜
⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣿⣷⠄
⠀⠀⠀⠶⡀⣿⠀⡾⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⢄⠀⠀⣼⣿⠏⠂
⠀⠀⠀⠀⠀⣾⣼⡇⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢹⣄⡼⠃⠸
⠀⠀⠀⠀⠀⠘⣿⣇⠀⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⡿⠁
⠀⠀⠀⠀⠐⢦⡈⢻⣆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⢣⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠙⢧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⡟⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡌⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣡⠟⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣠⣿⣀⠀⢠⠊⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⣿⡙⣷⡀⠀⠄⠀⢀⡶⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢆⠀⢊⠆⠀⣶⢠⠀⠀⠀⢴⣿⢿⣯⠀⠼⠀⠀⡾⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠃⠸⣦⣹⡄⠈⠃⡀⠀⠀⠈⠀⠉⣠⡇⠀⢸⠁⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢐⠄⠀⠀⢦⡴⠆⠘⡧⡟⠙⠗⠀⠾⢍⣄⠸⣦⠰⠞⠁⠀⢀⠇⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡙⢷⡀⠀⠀⠀⠀⠟⠀⢡⡀⡄⠀⠀⠈⣽⠀⡟⢳⣠⠆⠀⣈⡔⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣈⣿⣤⠂⣀⠀⢈⠀⠻⣷⣌⢁⣤⣶⣩⠀⡿⢆⠙⢀⡞⣉⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣿⡙⢁⡴⠟⠐⢦⡘⢀⠄⢡⣿⠋⢁⠀⠀⣶⠟⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢠⡞⠁⠀⠀⠀⠈⡀⠂⠀⠟⠀⣼⠇⠀⠟⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢢⣿⣇⣠⡶⠊⠂⠀⢡⡄⢸⢠⡆⣿⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣻⡿⠃⠀⠀⠀⠘⢧⠈⠄⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡿⠃⣿⠁⠀⠀⠀⠀⠀⢸⣶⣄⠛⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⣿⠃⠀⡟⠀⠄⠀⠀⠀⠀⢈⣡⡟⠛⢦⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣰⡿⠆⢸⡇⠀⠠⡀⠀⠀⠀⢸⡿⢤⣀⠀⠙⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀
⠀⠀⠀⠀⠀⢀⠐⠁⠠⠀⠀⣠⠾⢻⠀⠀⣤⡇⠀⠀⢳⡀⠀⠀⠸⣿⣿⠛⠀⠈⠉⠛⢦⡀⠀⣀⠀⣤⠀⠀⠀⡴⠚⠿⢦⣀
⠀⠀⠀⠀⠔⠁⢠⣎⣠⠔⣊⣀⠀⠀⠃⢀⣼⡇⠀⠀⠀⢷⡀⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠉⣤⡟⠳⠄⠀⠀⠀⠐⠲⠂⠀⠙⣦
⠀⠀⠀⢀⢀⡀⠀⠉⠷⢣⣼⠋⠀⠀⣰⠃⠘⠁⠀⠀⠀⠘⡇⠀⠀⢹⣏⠀⢀⡤⠀⠀⠀⠀⠉⢳⡀⢄⡀⠘⠀⠤⠤⣤⣤⠶⣹⣦⣀⣤⡀
⠀⠀⠀⠠⣦⠅⠀⠐⢳⡾⢡⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⣠⡟⠂⠀⠀⠙⢿⡋⠀⠀⠀⠀⠀⠀⢸⡛⢦⣝⣿⡦⠤⠶⠶⠤⠐⠛⠉⠉⠙⠛⢷⠒
⠈⠤⣤⣀⣀⣀⣠⡔⠋⣰⡿⠖⠉⠀⠀⠀⠀⠀⠁⠒⠚⠿⠷⢤⠀⠀⠀⠀⠹⠆⠀⠀⠐⠒⠀⠘⠛⠀⠼⠋⠙⠲⠂⠀⠶⠀⠀⠀⠀⠀⠀⠰⠗
⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⣴⣇⣿⡿⣶⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢠⣿⢟⠉⠀⣉⠻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⢠⠻⣷⡌⠀⠀⠉⣴⡟⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠟⠀⠀⠀⠀⣾⡆⠻⣿⠆⢠⣾⠟⢠⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣃⡀⠀⠰⣿⣿⣿⣿⠧⢠⣌⣵⣶⣾⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⣿⣿⡀⡴⠛⢻⣿⣻⣷⣶⣾⣿⣿⣿⣿⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣴⠏⣸⡗⠀⢶⠚⠿⢿⣿⣝⣿⣿⠟⢹⣀⡍⠈⠛⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣾⡟⣼⣧⢾⣦⡄⠈⣿⣿⣷⠁⠂⣾⣿⢻⣆⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠛⢉⣾⣿⣿⣾⣿⡇⢰⣼⣿⣯⣷⣾⣿⣧⠘⣿⣷⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠸⡿⠿⣋⣽⢿⣭⡶⢶⣿⣿⠇⠈⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣭⣽⠱⣿⢿⣿⣿⢣⡤⢞⣩⣾⡿⠟⠛⠀⠀⢀⠀⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣿⡿⢡⣿⠂⣤⣬⣟⠋⠀⠀⡀⣰⡇⢸⡇⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⣾⣿⣦⣬⣽⣿⣧⣶⣾⣷⣿⡟⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⣼⣿⣿⣿⣏⣿⣿⣿⣿⣿⢟⣽⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣷⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠋⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠏⠹⠋⠉⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣟⣠⠀⣼⡏⣿⣿⠛⣿⣧⣙⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠀⣰⣄⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢹⣿⢿⣼⡿⡏⣿⣿⡿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢠⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⣾⠟⠁⠇⣿⣿⠇⡏⠙⠏⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⡼⠁⠐⠀⢠⣿⡇⣰⣡⣷⣶⢼⣿⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢏⠀⢠⡤⣤⣶⣿⣿⣿⣿⠷⣿⡟⣾⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣣⠏⣴⡟⢸⡿⣿⣿⣿⣿⠸⣾⣿⣿⠃⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣸⡟⠷⠛⠀⣿⣿⣿⡟⢠⡿⠿⢿⣷⣤⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢾⣿⣷⣶⣦⣿⡿⠟⠿⠅⠸⠿⣿⣿⣿⣷⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠸⠘⠇
⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠀⠀⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣄⠀⢀⠸⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣈⣁⣀⣢⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢾⢿⡋⠛⠋⠉⢚⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⣴⣿⡀⠀⠈⠣⡀⠘⠁⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⠆⠀⠀⠀⣠⣾⣿⣿⣿⣷⣤⡀⠻⠐⠃⢀⣴⣮⣿⢹⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣛⣿⣷⠀⠀⣴⣟⣻⣿⣿⣿⣿⣿⣿⣦⠰⣿⣸⣿⣶⡿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣦⣾⣿⣿⣽⣿⣿⣿⣿⣽⣿⣿⣷⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣿⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠙⢿⣿⣿⣮⣽⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⠀⠀⠉⠻⢷⢿⣿⣿⣿⣿⡏⠈⠃⠈⢻⣿⣿⣿⣿⣏⠙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⣿⠙⡏⠉⠀⠀⠀⠀⠀⣹⣿⣿⣧⣼⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⣠⢿⠏⠀⠀⠀⠀⠀⢀⣄⠂⠈⠛⢿⡿⠻⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⡿⢸⠀⠀⠀⠀⠀⠀⠺⡉⢷⠘⣆⡀⢀⠀⢸⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠀⣼⠇⠸⠀⠀⠀⠀⠀⠀⠀⠐⠶⠛⢿⣧⣤⡄⠸⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠠⢯⠀⠘⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠀⠘⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣷⡂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣋⡤⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⣤⠅⢰⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⢘⣿⣿⠀⠈⢡⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠸⣷⣿⡟⠁⠀⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣾⣷⣶⣧⡟⠀⠀⠀⠈⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠸⣇⣀⡀⠀⠀⣀⣀⣀⣀⣀⣤⣀⣤⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠙⣿⣟⣿⣿⣿⣯⣿⣯⣿⣿⣿⣷⣷⠛⢀
⠀⠀⠀⠀⠀⡀⠤⢸⣿⢿⣿⣿⣿⣿⣿⠿⣿⣿⣷⡄⠀⠀⢹⣿⣿⡟⠿⢛⣿⣿⠛⠛⠋⠒⠚⠓⠒⠒⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⡀⠀⠻⠙⢿⣟⠀⣸⣿⣿⣗⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⣐⣿⣆⣿⣿⣿⣿⣿⣽⣿⣷⡟⢋⡀⠀⡀⢀⡼⡿⣷⣿⣿⣏⡈⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⣿⣿⣿⡍⠉⢽⣏⠙⠐⣾⣿⠏⠁⠀⠉⠉⠉⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠛⠛⠃⠈⠁

⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢲⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿
⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣿⣿⣿⠿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣄⣴⣿⣿⣷⣾⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⣼⠟⢒⡆⠀⠀⠀⠀⣀⣀⣄⣀⣤⣾⡷⢿⣿⣿⡿⠟⠉⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣄
⡟⠀⣿⣵⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣄⣨⣿⣿⣷⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⠿⢿⡿⣷⡀
⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣟⠛⠛⠛⢙⡃⠀⠀⢁⣿⣿⣿⣶⣶⠶⠆
⠀⢠⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣷⣾⣿⣿⣷⣿⣿⣿⣿⣿⣟⠁
⣴⡾⡗⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡀
⣽⠟⢿⣶⡆⠀⠀⠀⠀⠀⠀⢸⣟⣹⣿⣟⠁⠛⣿⡏⠉⠻⣿⣿⣿⣧⣀⡴⢾⣿⣷⣴⣾⢿⡿⣿⣿⣿⡿⠛⠹⢿⣟⢾⣿⣿⡇⠀⠀⠀⠀⠀⢦⡻⠄
⡟⡀⠀⢈⠀⠀⠀⠀⠀⠀⠀⢸⡿⠻⣾⣽⣶⣤⣭⣳⣠⣼⠺⣿⡟⣹⣿⡦⣴⣿⣴⠿⠇⣯⣿⣿⣫⣯⣤⠠⣤⣶⣛⡮⣿⡻⠀⠀⠀⠀⠀⠀⢾⣷⡀
⠤⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⢠⣽⣿⣿⡿⣏⣾⡟⢿⣧⣿⣻⣾⡀⢈⣛⢿⣿⣿⣿⣿⠿⢫⣙⢿⣿⡿⢛⣿⣷⣼⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄
⡿⠂⠰⢿⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣟⣺⡿⢧⣌⢁⡀⠀⢿⢿⣿⣿⣶⣾⣿⣿⣟⡍⣤⣴⣿⣿⢛⣅⣀⣿⣧⣆⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦
⠓⠀⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡽⠧⢥⠡⣟⢢⣙⣿⣿⣿⣿⡿⣿⣿⣿⠿⠯⢽⣿⣿⣿⣿⣿⣈⣮⣿⠵⠼⣿⠵⠶⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣌⣸⣶⣠⣿⡶⠿⣿⠿⠛⢛⣉⡉⢰⣿⣿⣿⣿⣿⡷⠀⠀⠙⠻⠿⠷⣷⣿⣾⣮⣷⣾⣁⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⣿⣿⡿⢀⣴⣇⠀⠀⠀⠀⢙⣿⣿⣋⣉⣁⣸⣟⡏⠀⠘⣛⡏⠀⢀⣉⣭⣿⣿⣿⣿⣿⣶⣄⢠⣀⣈⣿⣿⣻⣿⡿⣿⣾⣿⣿⠓
⢸⣿⣿⠇⣾⣿⡿⠀⠀⠀⠀⠀⠉⠛⢿⣟⠻⣿⣷⣶⡶⣶⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⡾⠿⠋⠀⣭⡉⠉⠉⢁⣘⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⣀
⣾⣿⣿⣸⣿⣿⠇⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⢿⣿⣷⣗⣷⣾⣿⣿⣿⣿⣿⡿⠏⠀⠀⢀⣤⣤⣤⣤⣤⡿⣿⣿⠋⠟⣡⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣇
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⣿⡷⠀⠀⠀⠀⣿⢿⣯⢹⣿⣿⣿⣿⣿⣿⣶⣴⣤⣴⣾⡿⠿⣿⡟⣹⣶⣿⠟⠁⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿
⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠀⠰⠆⠀⠀⣿⠀⡙⢾⣿⠟⣿⣿⠁⡘⣿⢿⠋⣼⣯⣴⣷⡟⠩⣿⣿⣿⠀⠐⠋⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡆
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⣠⠀⣿⠀⣀⣼⣿⣿⣿⣿⣾⡇⢹⠘⡀⣸⣽⣿⣿⣿⡿⠋⢹⣿⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠤⠀⠻⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣱⣿⣿⣿⣿⣿⠄⠀⢸⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⣶⣿⡄⠈⢹⣿⣿⣿⡿⣿⣿⣿⠀⢿⣿⣿⠟⠉⠃⠀⠀⣸⣿⠈⠛⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣿
⣿⣿⣿⣿⡇⢀⣴⣦⡀⠀⠀⠀⠀⠀⠃⠀⠀⠻⡿⠃⠀⠘⣿⣿⣿⡇⠘⣿⣿⣶⣾⡿⠁⠀⠀⢀⠀⠀⣻⡿⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣟
⣿⣿⣿⣿⣧⣾⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⣿⣿⣿⡇⠀⣿⢿⣿⠏⣹⣷⣷⡞⠁⠀⠀⠿⠁⠠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣿⣿⣿⡇⠀⣿⡇⠁⠀⠹⠟⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠄⠀⠉⠉⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢢⠈⢟⠄⠀⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣯⣴⣊⠠⠂⣐⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡿⣟⠳⣝⢿⡛⢄⣙⠵⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠈⣀⡏⠗⠛⢆⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⢀⡀⠀⠀⢛⡀⠠⠄⠀⠉⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⢀⢰⣿⣿⣿⣿⡇⠹⣿⠠⠾⠟⣋⠀⢠⣄⠀⣠⡀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠺⣦⡀⠀⣼⣿⣿⣿⣿⣧⡀⠀⠴⠿⠛⢿⣷⣾⣿⣦⣉⠁⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠑⢴⣿⣿⣿⣿⣿⣿⡗⢀⣶⠗⢀⣘⣚⣻⣿⣿⣿⡇⠀⠺⠂⢀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠑⠂⠀⠘⠻⣿⣿⣿⣿⣿⣿⣷⣴⣤⢠⠟⡃⠨⢹⣿⣿⣿⣄⡀⣀⠴⠀⠀⠀⠀⠀⠀⢠⠤⡀
⠀⠀⠀⠀⠀⠀⠈⠀⣀⠀⠈⠐⠾⠶⠤⠀⠍⢹⢿⣿⡿⠿⢿⣯⡁⠞⡁⢄⠈⣹⣿⣟⣨⣿⣇⠀⡤⠤⡀⢴⣴⣶⣂⣠⠃⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠏⢻⢡⠈⠀⠟⢖⡐⡲⠈⡟⣾⣏⢀⢹⣿⣿⣿⠋⠀⠙⢷⡀⠀⣹⣼⠋⠀⠈⠳⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠢⠠⠅⠐⣣⠀⠏⢹⢁⠄⠁⢸⣿⣿⣷⡾⣋⣽⣿⣷⣂⣀⣻⣴⠿⠿⢄⢠⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀⡂⠆⠁⠍⣠⠗⠘⠿⣿⣿⣿⣿⣿⡿⣟⡻⢿⡋⠁⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠉⡭⣤⠁⠸⠹⡶⠹⠟⠛⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡃⠑⠜⢠⠄⠹⢂⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠾⠷⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣧⡄⢀⣠⣶⡾⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣟⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⡿⡟⣿⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣍⣿⣿⣿⣷⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣟⠿⠟⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡀⣠⣤⠄⢠⣼⣿⣿⡟⡙⠀⠀⠀⠀⢠⣶⣼⡿⣷⣾⠓⠦⡀
⠀⠀⠀⠠⣀⡉⠰⢿⣷⣿⣿⣿⣽⣷⣿⣿⠀⠀⢀⡿⣿⢋⣿⣿⢷⣀⢀⣴
⠀⠀⠀⠀⠀⠉⠁⠀⢉⠛⠿⣭⣿⠟⠉⠁⠀⡠⣻⡟⣿⣻⣿⣷⡶⠋⠁⠹
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⣿⣏⣽⣴⣿⠏⡀⠈⠂⠁⠀⣇
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷⣄⠀⢐⣌⡿⠇⠁⠊⠀⠐⠀⠀⠀⢿
⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⢐⣦⣼⣿⣿⣆⣾⣿⠆⠐⣲⡄⠀⢀⠀⠀⠀⢾⣆
⠀⠀⠀⣾⣽⣷⣤⢀⣴⠾⣿⣿⠉⢉⣉⣿⣿⣿⡇⠀⠀⢉⡿⠆⠈⣳⠀⠀⠞⠉
⠀⠀⢠⣿⣿⣿⣿⣿⠿⠀⠉⠉⠟⠉⢴⣿⠛⣿⠃⠀⠀⢈⣽⠀⣼⠏⠃
⠀⠀⢸⣿⣿⡿⣣⠉⠀⠀⠀⠀⠀⠠⢺⣗⢲⣿⡍⠀⢠⣁⠀⠀⢀⣴⠂⠀⠀⡆
⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⠛⣦⣿⡇⠀⠞⠁⠀⣠⠎⠃⠀⠀⠀⡇
⠀⠀⠀⠸⣿⣿⣿⣷⡀⠀⠀⢀⠀⠀⡄⠀⠀⣿⣷⡀⣀⣴⣿⠃⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⢸⣿⣿⣛⣿⢏⢰⣾⣶⣾⣇⠀⠀⠸⣿⡟⣿⡿⣿⡕⢦⣤⣀⠀⢠⠟⢆
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣶⣄⣀⣿⡇⠀⠀⠀⠙⢲⡙⠿⣤⡘⠀⠈⢂
⠀⠀⠀⠀⠀⠈⢿⠿⣿⠿⠿⣿⣿⣿⡿⣿⣿⣿⣿⣳⡔⢿⠢⣆⣈⣙⣾⣿⡃⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⣠⣇⣀⣰⣮⣿⣿⣿⣿⣿⢯⣿⣿⡇⣜⣇⣹⣝⣿⣿⣿⣷⡄⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⠋⠹⣯⣿⣿⣏⣺⣿⣿⣷⣾⣿⣏⣹⣿⣿⣿⡟⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠟⠿⠿⠿⠿⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⢀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠚⠑⠂⠁⠁⠡⠋⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠳⣿
⠀⢐⡊⠀⢐⠉⠄⢀⣀⠃⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢳⣆⣤⣤⣄⠀⠀⠀⡰⠟⠻⢵⠴⣽
⠀⢙⡃⢂⢀⠈⢀⢘⠦⠬⠖⠒⠠⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢳⣿⣿⣿⡇⢀⠞⠀⠊⠀⢠⣶⡇⢀⠤⢛⠏
⠀⠀⠉⠁⠀⠁⠒⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠙⠿⠿⢯⡎⢴⡀⡀⠀⢀⡿⠆⠉⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠳⣈⠐⠒⠒⠒⠀⠀⠀⢱⠗⠀⠀⢸⠁⠀⠉⠁⠀⠉⢠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠣⣼⠊⠉⠁⣤⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⢤⣤⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣈⠈⠑⠐⠂⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢄⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠳⡦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢛⠥⡀⠀⠀⠀⠐⠂⠠⣀⠀⠀⣀⠶⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡌⠕⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⣀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠶⠷⠆⠀⠀⣀⠐⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢄⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣈⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠠⠞⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠾⠾⠶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⠁⠀⠀⠀⠔⢂⣈⡭⠥⠤⢤⣀⡐⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⡷⠋⠀⠀⣀⣤⠴⠄⠒⠋⠉⠁⠀⠀⡄⠀⠀⢉⠛⠆⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡛⡴⠖⠒⠉⣁⣀⣀⣤⣴⣦⣾⣴⣿⣿⣿⣷⣷⠟⠋⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠯⢱⣤⠐⢠⡉⢹⣿⠟⠉⠉⠀⠉⠉⠋⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣩⣷⠂⠀⠀⡜⠁⠀⣀⠤⠴⠤⢄⠀⠀⠒⢓⣛⣒⣒⣚⣛⡒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣒⣸⡆⠂⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠉⠛⠋⡩⡭⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣨⢿⠞⢫⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠦⠰⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⡿⠀⢀⡀⢀⣠⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⡄⠀⠀⠀⠀⢀⡾⠀⢰⣧⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣾⣿⡿⠟⠿⠷⠷⠼⢼⠟⠁⠈⢨⠿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠃⠂⠀⠀⠀⠀⠀⠘⠀⠈⠀⠁⠁

⠀⠀⠀⠀⢠⣄⠀⣼⣿⣦⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣶⣦⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢠⣶⣆⠀⢠⡄
⠀⠀⠀⠀⠘⠀⠀⠻⡿⢿⡇⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣾⣿⣿⣾⡟
⠀⠀⠀⠀⠀⠀⢰⠆⠀⣻⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡋⢹⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⠀⢸⡟⠋
⠀⠀⠀⠀⠀⣴⠏⠀⠁⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠈⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⣀⣀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇
⠀⢀⡟⠉⠉⢿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁
⠀⠘⢿⠆⠀⠀⢹⣿⣶⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣄⡄⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢴⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣬⣷⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣤⡾⠀⠀⠀⠀⠀⠀⠀⡎⠀⠻⣿⡿⠁⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⣿⣭⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠉⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡏⢹⠲⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⠰⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡

⢠⣤⣤⣤⣤⣤⣤⠤⢤⣤⣤⣤⣤⣤⣤⠤⠀⡤⠤⠀⠀⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣦⠒⠲⠖⣶⣤⣤⠤⡤⣖⢒⠶⠂⠒⠒⠒⡒⠒⠂⢸⣶
⢘⢿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡟⣿⣿⡆⠃⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣴⡇⠈⢛⣿⣿⠗⠀⠀⠀⠀⠀⠀⠅⠀⠀⢸⣿
⢰⢾⣿⣾⣿⣿⣿⡇⢸⡗⢿⣿⣧⣿⣿⣿⡄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠾⠃⠀⢠⣿⡽⣴⣿⡟⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢘⢸⣿⣿⣿⣿⣿⢀⠸⢟⣨⣭⡳⣿⣿⣿⣿⣄⠂⢀⡑⢄⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠠⠔⣀⣾⣿⣿⣇⣯⢊⣀⣠⠀⠀⠀⡼⠀⠀⢀⠊⢸⢸⣿
⢸⢸⣿⣿⣿⣯⣿⠐⢸⣿⢉⣿⣿⣾⣿⣿⣿⣿⣧⣀⢐⣢⣾⣿⣿⣦⣁⣀⣄⡀⡋⠀⠀⢠⣀⢴⣿⣿⠋⡟⣼⣿⣿⡏⠀⠀⣠⡞⠁⠀⠀⠸⠀⣠⢸⣿
⢸⣼⣿⣟⣩⣭⣭⡇⢸⣿⢸⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⡅⠰⡀⣠⡶⣿⣿⣿⣿⠜⠟⣹⢫⡿⣠⣶⣿⣿⠃⠀⠀⠀⢣⠆⣾⢸⣿
⢰⣿⣿⣿⡿⣿⣿⡀⣼⣿⣙⣿⠿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⢿⣿⣿⡟⢷⣤⣾⣿⣿⣯⣿⠿⠛⠳⣬⣷⣿⣿⣿⣿⣿⡇⠀⢀⠀⣼⡾⢠⣿⢸⣿
⢹⣼⣿⣿⣿⢹⣿⡇⢸⡟⢢⣿⡷⠋⣻⣿⣿⣿⣿⣿⣿⣿⡏⠀⡟⣸⣿⣿⣿⣾⣿⣿⣿⣿⣽⣿⣆⠀⠀⠈⣿⣿⣿⣿⣿⣿⠃⣠⣿⣸⣿⣇⣿⣿⢸⣿
⢸⣸⣿⣿⣇⣼⡿⢧⣛⣻⢿⣿⡶⢦⡿⣿⣿⣿⢿⣿⣿⣿⣅⢻⣿⣽⡿⢯⣿⣿⣿⣿⢿⣿⣿⣿⣿⣦⣤⣾⢟⣿⢿⣿⣿⣿⣴⣿⣿⣿⣿⣾⣿⢾⢸⣿
⢸⢯⣽⣿⣿⣿⣿⣿⣿⣿⣯⣽⠟⠿⢷⣾⠾⣿⣾⣿⡿⣿⣟⢿⣿⡿⠉⠛⣿⣿⣿⣿⡋⢊⣍⣼⣿⡿⣿⣿⣞⡙⢿⣥⣿⣿⣿⣿⣿⣿⣿⣟⣈⣾⢸⣿
⢸⢸⣿⣿⣟⣿⣿⣿⣿⣶⡿⢫⣧⣿⣤⡿⠿⣿⣿⣿⣿⣿⣿⣫⢿⣷⣴⣆⣿⣿⡟⠏⠻⢾⡟⣿⡟⢿⠏⡿⡿⣷⣆⢿⣿⣿⣿⣿⣿⣿⣿⣽⠾⢻⢸⣿
⢸⠘⣿⣷⢼⣿⠋⣹⣿⣿⣿⠛⣳⣦⣼⣷⠒⣛⣿⣿⣿⣿⡟⣷⣄⣾⣿⣿⣿⣿⠁⠀⠀⡸⠘⣿⣣⣾⣯⣿⣿⠿⠉⢾⣿⣿⣽⣿⣿⣿⠗⢱⡶⣽⢸⣿
⢸⣦⣿⢆⣤⣿⠻⠉⣰⡟⢛⡿⢛⣀⣿⠿⢻⣭⣿⣿⡟⣿⠀⣿⣿⠛⣿⣿⣸⣷⠄⢰⡗⢀⠜⠛⢿⣿⣯⠭⣘⣇⣒⣎⢶⡿⠟⢫⣿⣿⢿⣾⡿⢿⢸⣿
⢸⡿⢳⣤⣬⣿⣿⢶⣿⣯⠭⠽⡻⣿⣬⢦⡀⣿⣿⣿⣷⣿⠀⢹⡟⠀⠸⠟⠈⣗⢀⡾⣓⡈⣠⣶⣽⣿⣿⣿⣿⣿⣿⣯⠦⠄⠙⠀⠍⡷⢯⣿⣿⣾⢸⣿
⢸⣷⣻⣿⠿⠿⢿⣿⠏⣶⣷⣿⣷⣿⣿⣿⣿⢻⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⠾⢿⡿⢵⣿⡧⣾⣿⣿⣿⣿⣿⡿⣿⣿⠉⠂⡄⢠⠀⠀⠛⠦⢿⢻⣿⢸⣿
⢸⢿⡟⡴⣦⣾⣯⣰⣆⡈⢻⣿⡿⣿⣿⣿⣿⣦⢿⣿⣿⣇⣠⣼⣷⣶⡿⣷⠺⡾⣵⣿⣿⣿⠄⡘⢻⣿⡿⠟⠿⠿⣯⠀⠈⠂⠀⠁⠀⠀⠀⠀⠀⢸⢸⣿
⢸⣾⡇⣶⡿⠙⣿⣟⣿⣿⣾⣷⣾⣿⣿⠟⢹⢹⢠⣿⣯⣿⣿⢶⣶⣾⢟⣿⡿⣽⣿⣿⣿⡿⣦⠿⠋⣤⣀⠾⠀⢀⢾⣼⠲⢄⣤⠀⡀⠁⠀⢀⡀⢸⢸⣿
⢸⣿⢿⡷⢜⡛⡋⢹⢋⣹⣉⣹⣟⡿⢿⣧⣼⣭⣼⢿⣿⣛⣿⣿⢿⣿⠸⡿⠹⣿⣿⣿⣿⣷⣤⡀⢤⣼⣿⣀⡜⢴⢻⠶⡠⣿⣯⣘⡆⢀⠀⡀⡀⣸⢸⣿
⢸⣧⠄⢱⡼⢶⣶⡟⠉⠉⠈⡙⢿⢛⠳⠿⣿⣁⠘⣿⣀⠐⠿⣿⣀⣺⡷⣨⣤⠈⠛⢿⣟⢻⣿⣿⣿⠟⠛⠋⢼⡾⣤⣹⢼⢛⣏⣹⢴⣃⡰⣟⣶⣽⢸⣿
⢸⣛⣃⣈⣥⣈⣿⣿⣿⣶⣮⣀⣌⣏⣠⣴⣰⣿⣭⣿⣿⣷⣒⣾⣉⣻⣿⣒⣶⣗⣻⣿⣿⣭⣿⠛⠂⠂⠄⠀⢸⣗⢾⣿⣟⣛⣿⣿⡿⠛⡿⠿⠋⢻⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠘⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣃⠀⠀⠀⠀⠀⠙⠋⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠘⢀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠙⠀⢀⠈⠁⠀⠺⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢛⢫⡀
⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⢀⡀⠀⠀⠀⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠒⠉⠁⠛⠑⢯
⣶⣼⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⡀⣿⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢣⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢠⡶⠖⠒⠀⠀⠀⠸⠆⢀⡀⠤⠀⠒⠐⠒⠂⠠⢸⠁
⣎⣿⡇⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢷⠇⠀⠀⠀⠀⠀⠸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣍⣩⡆⠀⠀⠀⠀⠀⠴⡀⠣⡀⢠⡄⠀⠀⠀⢴⡄⠀⠈⠀⠈⠀⠁⠀⠀⠀⠀⠘⠀⠀⠀⠠⣀⣤⡀⠀⠀⠀⠸
⠉⠀⠀⠀⢀⠀⢀⡀⠈⠅⠀⠀⢀⠀⢠⠼⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⠋⠁⠀⠀⠀⢀⡀⢐
⠀⠀⠐⠇⢀⡠⠀⠀⠀⠀⢀⡀⠛⠛⠛⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣸⢒⣶⠶⠆⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⡀⡠⠀⠠⢰⡀⣴⠶⠶⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠛⠪⠟⠷⢆⣤⠦⣀⣤⠂⠀⢦
⠀⢀⣤⠀⠀⠀⠓⠀⢤⠀⣌⡁⠀⠀⢊⣡⡍⠀⠀⣠⠀⠀⠀⠀⠀⠀⢀⣴⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠓⣄⠀⠀⡀
⠀⠈⠙⣄⢀⠀⠈⠁⠀⠀⡏⠉⠀⠀⠨⠉⠹⠶⣤⣿⣤⣴⠆⠠⠲⢲⠚⠧⠤⠤⢀⣀⠀⣿⢀⢀⡄⣀⡀⠢⣠⡏
⠀⠀⠀⠍⣧⢀⠀⡈⠁⣁⣁⠀⠀⠀⠀⠀⠁⠀⠈⡈⠁⠀⠊⠀⠀⠈⠀⠀⠀⠀⠀⢈⡧⠒⠻⠙⣷⠉⠻⣏⠑⠂
⠀⠀⢠⠆⠁⠘⠉⠉⠻⢿⣿⠃⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⣀⣀⣀⡀⠀⡀⠀⠀⠀⠚⠁⠀⠀⠀⠉⠁⠀⠈⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠡⡀⠀⢈⠁⡈⠐⡾⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣰⡟⢿⡄⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⢾⠋⠩
⠀⠀⠀⠈⠉⠀⠀⢉⠖⠤⠀⢃⠀⠀⠀⠀⠠⡀⠀⠀⠀⢠⠋⠀⠀⠹⡄⠀⠀⢀⠶⠦⠐⡀
⠀⠀⠀⠀⠀⠙⠀⠀⢟⣿⠶⣾⡄⠀⠰⡖⠋⢹⠀⢀⣴⠁⣠⡄⢀⡀⢰⡀⢠⠿⠀⠂⡐⠉⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢠⠒⠒⠉⣿⣾⡟⢼⡧⠤⠧⠀⠺⣤⠟⠁⠚⠛⣣⠾⠿⠈⠛⠋⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣴⣤⠈
⡀⣀⣠⠀⠀⣤⡄⠀⠀⠙⠛⠻⠦⠄⠰⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠈⠛⠷⠚⠀⠀⠀⠀⠀⠸⠋⠠⠤⠀⠂
⠅⠈⠀⠂⠀⠼⠤⠊⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠖⠖⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡠⠤⠀⠤⠀⠀⠀⠀⠀⠀⠀⠐⠟⠁⠂
⠀⣶⠄⠁⠀⣀⣥⠰⠃⠀⠈⢡⠀⠌⠀⠀⠀⠀⢀⣀⢀⢀⣠⣴⠶⢶⡀⢀⡀⠀⠀⠀⠀⠀⠰⠀⠶⠂⠀⠀⠀⠀⡀⠀⠀⠓
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⢀⠀⠤⠈⠑⠒⠒⠦⢆⡀⠘⢂⡀⣤⠈⠫⡒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣰⣆⠀⢀⠀⢈⣡⣴⡴⠂⢀⣠⡅⣈⣳⣀⠉⠀⠈⠐⠐⢆⠀⠲⠿⢦
⠀⠀⠀⠀⠀⠈⢧⡃⣿⢂⣡⣶⠾⢁⡥⣄⠀⠈⠀⠀⠀⠈⢥⠀⠒⠠⡄⠈⠺⣆⠐⠀⠨⣷⡇
⠀⠀⠀⠀⠀⡸⠂⢻⡿⢻⢋⣀⡠⠁⠒⠐⠆⠉⠀⠀⠀⠀⠈⢣⡆⣀⣧⠐⠂⢼⠙⣦⠀⡼⣇⡰⣆
⠀⠀⢸⡀⢠⠀⡸⢟⡷⣟⢩⣀⣀⣠⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠴⢼⣄⢹⡁⡄⣿⣧⡟
⠀⠀⡄⢧⣊⢲⡟⡿⣼⠟⠉⠑⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠵⠊⠦⣘⡁⠻⢷⣿⣸⣿⢠⠄
⠀⢣⠰⠞⣧⣟⡽⢷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢤⡀⣿⠹⣿⢲⡂
⠀⠈⣷⡀⢸⣿⡡⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣙⣿⡇⣿⣔⠇
⢶⡀⣏⢿⣾⣿⣼⣯⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⢹⡟⣿⣿⠅
⠀⢻⣸⡎⣿⣿⠋⠁⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣑⢾⣿⣿⠏
⠀⠈⣿⣧⣿⣿⢸⠟⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠈⠫⣼⣿⣿⡄
⠠⣿⡼⣧⣿⣿⡅⠀⣻⣟⡆⠀⢀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡶⠷⣾⣿⡇⣷
⠀⢺⣿⡹⣿⣿⣧⠀⢊⣿⣷⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡿⢈⣿⣿⡇⡇⠀⠄
⠀⠀⠩⣾⠘⣿⣿⣧⠘⣿⡏⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣽⡗⣾⣿⣿⣷⠁⠈
⠀⠀⠀⠙⣧⣿⣿⣿⣷⣜⣾⣿⣭⣶⣼⡆⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⣰⣶⢽⡟⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣯⣄⣻⣺⣿⣿⣦⠀⠀⠀⠀⠈⣻⠻⠾⠷⢧⣧⣯⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠙⢻⠿⢄⣀⢹⣿⣋⡉⠉⢉⠉⠙⠛⠻⣥⠋⢁⡔⢺⣓⣰⣿⣿⣿⡟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⢿⣶⡞⣛⣶⣤⡔⣀⡤⣷⡺⠛⠉⠉⢉⣉⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠻⡟⠉⠝⠉⠂⠨⠃⠀⠐⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⢰⡇⡀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣾⣿⠇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡜⣡⣾⣿⣿⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⣴⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⣄⢠⢣⣿⣿⣿⣿⠟⠋
⠀⣸⣦⡄⠀⠀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⡤⢊⣡⣴⣶⣿⣿⣿⣿⣶⣝⣿⣿⣿⠟⠁
⠀⢿⣿⣿⣿⣿⣷⣦⣭⣑⣒⣲⡄⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⢎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⣤⣥⣵⣿⣿⣿⣿⣿⣿⣿⡇⠀⡰⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠉⠛⠙⠛⠉⠉⠀⣹⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢔⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣟⣿⣿⠏⠀⣿⣿⣿⣝⣿⣿⡟⣿⣿⣬⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⣇⠀⠐⣿⣿⡿⢿⣿⣿⣿⣝⠏⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡞⣿⣿⣿⣿⣿⣷⢠⡀
⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣯⠝⠋⡀⠀⠹⣎⣛⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⡟⢸⠁⣽⣿⣿⣿⠹⡟⢰⡇
⠀⠀⠀⠀⠀⠀⠸⠛⣿⡿⠇⠀⠐⠀⠀⠀⠹⠿⠟⠻⣿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠃⡟⠀⢱⣿⠻⣿⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄⠀⠀⡌⢠⣿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⡸⠁⠀⠀⢳⡀⠙⠃⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠻⠿⠿⠃⢀⠜⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢡⡟⠁⠀⠀⠀⠀⠙⢦⣀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠲⡟⠲⡞⢇⢀⣼⣿⣿⣿⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡄⠀⡘⣦⡙⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⣿⡿⣉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣾⣿⣿⡼⣿⣿⣿⣿⣿⣧⢻⣿⡿⣫⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢤⣴⣾⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣎⠭⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠐⠀⠘⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠺⣻⣿⠿⢿⣛⣻⣯⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠾⣄⣀⡰⠤⠜
⠀⠀⢪⡀⠀⠀⠀⣀⠄⠐⠀⠁⠀⠀⠀⠀⠀⠀⠛⠉⢩⢤⡍⠘⠩⠞⠁
⠀⠀⠀⠈⠉⠓⠚⠁⠰⡀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠼⠤⠂
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⠶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⢤⡀⠀⠀⠘⠂⣵⣀⣄⡀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⠇⠀⠈⠓⠲⢍⡔⢳⣿⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠉⠛⠷⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣻⠿⣿⡿⣿⢿⠥⠁⢀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢴⠀⠈⠀⠈⠁⠡⠒⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠌⠀⡐⠁⠀⠀⠠⠀⠀⠀⡠⠆
⠀⠀⠀⠀⠄⠀⢀⠔⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠠⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⡏⠉⠁⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠠⠀⣀⣀
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚⠂⠀⠀⠊⠀⢄⣀⡵⠒⠁⠀⠀⠀⠀⠀⢀
⣜⣃⠂⠀⠀⢀⡲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠇⡊⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡁⠢⣄⡀
⠃⠀⠀⢀⠜⠉⠀⠘⠢⢄⠈⠀⠀⠀⠀⠠⣀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⡴⠏⠁⠀⠀⢀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⣁⡤⠞⢋⡦⠤⣠⠀⠀⠀⠐⠀⠈⠀⠀⠠⢡⡞⠥⠖⢛⠋⢈⣡⠔⠋⠁⠐
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠂⠊⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣁⠀⣴⠛⠀⠀⣀⡮⠖⠉⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠥⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠇⠐⡌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⣉⠐⠢⠄⣀⣀⣴⠏⠖⡙⡉⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠈⠩⠓⠬⣡⡝⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠱⣦⣄⡀⠁⠚⠉⠟⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠉⢩⡤⠎⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠑⠃⠙⣎⠯⡓⠦⠤⣀⠀⠀⠈⠲⣄⠀⠀⠀⠀⢀⣠⠴⠊⠑⠋⠀⠀⠀⠈⠛⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠀⠀⣻⠣⣀⠀⠀⠀⠉⠒⠠⠐⠺⢦⡴⠒⠚⢉⡀⡄⠀⠀⢠⡦⡆⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡏⣡⣦⣌⠷⠦⢤⣀⣀⠀⠀⠀⠀⠈⠛⠋⠻⣛⠓⠲⢤⡈⠓⠂⠀⠐⠂⣽⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢹⠉⣸⡟⠀⣀⣀⠀⠉⠁⠀⠀⠈⣁⠉⠉⠒⢌⢻⡦⡀⣄⡶⠠⠄⣀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⢀⣴⡿⣿⣁⠀⠳⠍⠀⢀⡘⣦⠀⠀⠙⠉⠀⠀⠠⢌⡇⠈⣾⡿⠀⠀⢾⡹⠶⠄⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣾⢶⣶⣿⣧⡀⠈⠒⠤⣀⠐⡤⠨⠀⠁⠀⠀⠀⠀⢠⡄⣳⠀⠈⠒⢬⡃⠀⠀⢀⡀⠀⠀⠘⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⣿⡇⠘⣦⢤⣀⠀⠉⠲⢅⡑⠰⣤⠀⠐⠄⠀⠀⠉⠀⠀⠀⠀⠤⠆⠀⠈⣥⠟⠀⠀⣦⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠠⣹⣿⣿⣿⡟⣀⠜⠁⠀⠈⠙⠲⣄⣀⠉⠓⢤⡘⠄⠀⠐⠀⠀⢰⡐⠖⠀⠀⢀⠀⡀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⢀⠤⢻⣿⡟⣿⡏⡡⠚⠉⠉⠉⠉⠉⠉⠉⠛⠁⠆⠙⢶⡷⢠⠀⠀⠀⠉⠀⠀⠀⠸⠮⢁⣀⢻
⠠⣐⢒⠒⣂⠀⢠⠐⢀⢀⣙⣀⣔⣸⣇⣿⣿⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠛⣶⢤⣀⣦⣠⠴⠚⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠁⠈⠐⠒⠚⠀⠡⠾⠥⠍⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠈⢣⡈⠁⠀⣤⣀⠀⢀⣠⣤⣤⣤⢤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠤⠤⠤⠄⠀⠀⠀⢳⣒⣲⣿⣿⠙⡛⠉⠉⠙⠛⠛⠛⠻⣟⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡠⠤⠀⠀⠀⠀⠀⢷⡀⠙⠓⠚⠿⣛⡿⠶⠤⠉⠽⢿⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠸⠉⠳⣄⠀⠀⠀⠈⠙⠒⠶⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⠀⠀⠀⠀⠀⠀⠙⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠕⠀⠀⠀⠤⠴⠄⠀⠀⠆⠀⠠⡀⢀⠳⡄⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣝⠶⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠷⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⣘⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⠔⠃⠀⠀⠀⠙⢦⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡄⠊⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠉⠂⠀⠀⠀⠘⢉⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠸⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣤⠞⠁

⠀⠀⠀⠀⠺⣟⡛⠛⢲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠙⢾⡓⠿⠉⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣤⢤⣤⢬⣟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⠦⠉⠃⠉⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢈⣛⣶⣤⣀⣀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⢸⡏⠉⠛⠿⣍⡉⠉⠛⠛⢩⣭⡭⠿⣿⣿⠳⢶⡶⠶⠶⠒⣶⣤⠴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⢀⡀⠀⠀⢸⡇⠀⠀⣤⣈⡙⢄⠀⠀⠀⠬⣽⣿⡗⠀⠀⠀⢀⣤⡶⠛⠉⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠜⠉⠀⠀⠀⠀⢷⠀⠀⠞⠀⠀⠀⠀⠀⠀⠈⣿⡋⠉⠀⠀⠀⠘⠁⢀⣠⡤⠀⢠⡟
⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡟
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢻⣻⣿⡹⠀⠀⠀⠀⠀⢸⣿⣿⣳⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⣿⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠉⠉⠁⠀⡀⠀⢀⠀⠠⠉⠉⠁⠀⠀⢀⣴⠏
⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀⠘⠀⣀⣉⣶⣏⣀⠐⠀⠂⢀⣠⠴⣿⠋
⠀⠀⠀⢀⣾⠇⠀⠀⠀⣸⣿⣦⡀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠉⠓⢦⣄⣀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠘⢧⡀
⠀⢀⡤⠞⠁⢀⡀⠀⢰⡟⠀⠈⠻⣦⡀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣴⢋⡤⠖⢠⡏⠀⢀⡿⠀⠀⠀⠀⢸⣿⡋⣀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠈⠙⠷⠤⠴⠛⠛⠛⠁⠀⠀⣀⡴⠋⠁⠘⢛⣷⡄⠀⠙⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠀⠀⠀⠀⢀⣰⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠉⠀⣢⡀⠀⣨⣿⠿⠒⠛⠋⠛⠓⠶⠶⠶⠶⠚⠛⠋⠉⠛⠻⣦⡀⠀⢀⡐⢿⣿⠑⠀⠈⠛⢦⣄⡀
⠀⢀⡤⠶⠖⠚⠛⠉⠁⠀⠀⠀⢀⣾⠿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⡟⠃⠀⠀⠀⣶⣄⣈⠙⠳⠶⣤⣀⡀
⠀⠿⠧⣤⠖⠀⠀⣠⡆⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣼⡋⠀⠁⢀⠀⢠⣄⡀⠉⠙⢶
⠀⠀⠀⠻⠤⠴⠚⠁⠹⡶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⡅⠈⣳⣄⢈⡿⠶⠖⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣯⣥⣬⣽⣧⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⣢⣭⡭⣭⣭⣫⢭⢭⣭⣅⡓⢄⡀
⠀⠀⠀⢀⣤⣶⣶⣿⣶⣦⣀⠀⠀⠀⠀⣀⡴⣿⣷⣿⣿⡙⠋⢟⣱⡮⡽⠟⡕⣿⣾⣾⣿⢦⣄⡀⠀⠀⠀⢀⣤⣶⣾⣶⣶⣤⣀
⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣙⣫⣷⣿⣿⠟⠛⣿⣷⢀⡄⠚⡛⠠⢀⣷⣿⠋⠛⣿⣷⣾⣭⣋⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⣿⣿⣿⣿⣿⣿⣿⠏⢘⠽⣦⣫⣽⣿⣿⣟⣩⣿⢶⠂⠀⡻⠿⣶⣦⣾⣾⠿⣟⠀⠐⣶⣿⣭⣿⣿⣿⣯⣭⣅⡟⠓⣈⣻⣿⣿⣿⣿⣿
⠙⣿⣿⣿⣿⣿⣷⣼⣷⣛⣽⣻⣴⠿⢯⡁⠀⠙⠏⢿⡀⠁⡀⠘⢻⡿⠡⢐⠉⢀⡿⠝⠉⠀⢈⡽⠿⢶⣟⣯⡟⣿⣯⣷⣿⣿⣿⣿⣿
⠀⠀⠛⢿⣿⣿⣿⣿⣯⣵⣮⣾⣷⣖⣢⡈⢂⠀⠈⡇⠹⣟⡛⢷⡃⢨⡞⡋⢟⡋⣬⠀⠀⠐⢁⣔⣲⣾⣷⣽⣧⣭⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⡿⣿⠻⣿⣿⣿⡄⡇⠀⣷⠀⢀⣟⣼⣁⣈⣧⣯⡜⠀⣾⠀⢸⢠⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠀⠀⠀⣿⡿⢿⡿⣧⡀⠟⡠⡾⠁⠀⠀⠀⠀⠉⢳⢄⠁⢀⣾⢻⡿⢿⣿⠀⠀⠀⠙⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢳⠸⣷⡘⣇⠠⢀⠤⣒⠋⠙⠛⠑⣒⢌⡀⠀⡾⢡⡾⣇⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣇⠀⢣⣸⢳⢹⡶⣭⠀⠉⠀⣆⣡⠄⠉⠑⣝⢦⡇⡾⣷⡼⠀⢸⢄
⠀⠀⠀⠀⠀⠀⠀⡀⢠⣾⠏⣩⠗⣻⡄⠈⣿⣼⣼⢱⡀⢰⡖⠈⠀⠀⠉⠰⡄⢀⡟⣷⡇⣿⠁⢀⣟⢹⣉⡝⣷⡦⢄
⠀⠀⠀⠀⠀⠀⣸⡴⠟⠛⠋⢿⢸⠁⠻⣄⣿⣿⢻⠁⣈⠃⠱⠤⠄⠤⠤⠌⢸⡁⣁⡇⣿⣿⣠⠞⠀⣇⡿⠙⠛⠳⢦⣡
⠀⠀⠀⢀⠋⠀⡟⢀⠀⠀⠀⢸⡾⡄⠀⠈⣿⢹⣼⣫⠥⠖⠓⠋⠉⠉⠙⠛⠺⠮⣟⣣⣿⣿⠁⠀⢠⣿⠇⠀⠀⠀⡈⢷⡀⠑⡆
⠀⠀⠀⠁⠀⠘⢷⡀⣸⡆⢀⡼⠃⢽⠂⠒⣾⣿⠋⠀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠙⣿⣿⠒⢐⡏⠼⠦⣀⠠⠾⣀⡾⠃⠈⠉⡄
⠀⠀⠀⣆⠀⠀⢨⠈⠀⠀⢈⣧⡟⠷⠃⣼⣹⠇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠸⣿⣧⠘⠾⢻⡼⣇⠀⠀⠀⢕⠀⠀⣀⠁
⠀⠀⠀⡏⠑⠓⡻⠂⠶⣀⠉⠀⢨⠎⠉⢹⡿⠀⣸⣿⣿⣿⣷⡙⣿⣿⢫⣿⣿⣿⣾⣧⠀⢻⠋⠉⢻⡁⠀⠉⢩⠖⠂⢞⠓⠪⢹⠂
⠀⠀⠀⢹⣤⢰⡀⠀⢀⠘⡶⠖⠻⢇⠠⣼⡇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡿⠀⢸⡇⠄⡸⠗⠲⢶⠁⡀⠀⢀⣅⣠⡄
⠀⠀⠀⡇⠀⢀⡵⢤⡏⠩⠰⠐⠀⣄⡷⣽⢻⡀⠀⠙⡻⢿⡿⠿⠿⠿⠿⠿⡿⢟⠋⠀⢀⡞⣿⢿⡤⠀⠀⠈⡉⠰⡤⣮⡁⠀⢸
⠀⠀⠀⢳⠀⢸⠀⢈⣷⠀⠀⡀⠀⠐⠛⢸⡷⠿⠿⣿⣿⣦⣄⡀⠀⠀⠀⣀⣤⣿⣿⠿⠿⢾⠃⠿⠀⣀⣈⠁⢀⣿⡀⠀⡇⠀⡚
⠀⠀⠀⡴⠓⠛⠻⢹⠀⠱⡶⠀⠀⡓⠐⣸⠃⠀⣴⠳⢼⣾⡍⠉⠉⠉⠉⢉⣷⠋⠲⡆⠀⠘⡆⠀⢎⠀⠠⢷⠀⢀⡟⡟⠓⠛⢳
⠀⠀⠀⣧⡀⣰⠞⢸⡗⠀⠀⠠⢎⠄⡰⠙⣄⢀⡿⠀⠚⢇⠙⠶⣤⣤⠔⠋⡸⠃⠁⢦⣀⣰⠹⣄⠨⠓⠠⠈⠠⣶⠗⠚⡄⠀⣸
⠀⠀⠞⠉⠉⡻⣄⣠⣝⣚⣓⡠⠄⠈⣇⣀⡟⠈⠀⠀⠀⠀⠉⢲⡇⢸⡗⠉⠀⠀⠀⠀⠉⢻⣄⣸⢁⠬⡀⣐⣛⢫⣄⡴⡟⠉⠁⢁
⠀⠀⡇⠀⠀⣐⠀⠀⠈⡸⠛⠙⠷⣋⡽⢅⣸⠀⠀⠀⠀⠀⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⣧⣠⢿⣈⠻⠛⠙⣇⠀⠀⠁⣃⠀⠀⢈
⠀⠀⠈⠀⠈⠿⡦⡀⢾⠁⢀⠀⣸⡝⠈⠀⣀⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢉⠀⡀⠩⣇⠀⠀⠘⡷⠀⣴⣿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠑⠀⣛⣒⠪⠊⠈⠁⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠁⠀⠀⠃⠙⠗⡒⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⡄⠀⠀⠀⠀⣰⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⠀⣸⡏⠈⢷⠀⠀⠀⣴⡿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡟⣰⠏⡀⠀⢸⠀⣠⡾⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣷⡟⢢⣀⣱⣼⠿⡋⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢿⣿⣧⡟⠘⣱⠾⢏⡑⡂⠘⡄⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⡾⠶⣄⠀⠀⠀⠀⢀⣠⣿⣿⣿⣾⣅⠀⠩⠧⢼⡀⢑⢠⡾⠁
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⢀⣴⣿⣿⣿⣿⣿⣿⢿⣷⣆⣛⠒⣅⣼⡋
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠋⠁⠿⣿⣿⣿⣿⣞⠛⢻⣿⡟⡅⠘⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣙⣿⣔⣀⡅⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣆⣟⣿⣷⣤⣤⣤⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣥⣤⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠸⣿⣯⡉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧

⠀⠀⠀⠀⣠⣤⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡄⠘⡄⢠⢃⡨⠛⠧⣽⡤⠃⠠⠿⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣧⡟⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡈⠛⠯⠤⠤⠔⠊⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠰⢿⠀⠀⠀⣄⠫⠑⠀⡀⠀⠀⠀⠠⠀⠀⠈⠇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟
⠀⠀⠀⠀⠀⢀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⠀⠀⠀⢿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠐⠐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠉⠉
⠀⠀⠄⠀⠀⠀⠀⠾⠆⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠀⠄⢀⣴⣿⠟⠁
⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠁⠤⢘⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⢻⣿⣿⡀
⠀⠀⠘⠀⡒⠈⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⡀⠀⠀⠀⣾⡇⠀⠀⢠⡶⣆⠀⠀⠀⠛⠛⠛
⠀⠐⠂⠀⠀⠀⠀⠳⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠋⠀⠀⠀⠈⠀⠀⠀⢸⣽⣿⡇⠀⢀⢀
⠀⠀⠀⠀⠀⣀⡀⠳⠆⠀⠀⠀⠈⠀⠁⠀⠀⠀⡀⠀⠀⣠⣶⡧⠀⠀⠀⠀⠀⠻⠓⠀⠙⠛
⠠⠀⠀⠀⠀⢙⡃⠀⠀⠀⠐⠂⠀⢠⣤⡄⠀⠻⡟⠀⠀⠘⠁⠀⠀⡀⠀⢀⡴⡶⢢⠀⢀⣦⡀⠀⠀⠀⠀⢴⡷
⠀⠐⠢⠀⢰⣏⣷⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣤⣤⡄⠀⣾⠿⠀⠘⢷⣏⡿⠁⠈⠰
⠀⢈⠡⠀⠹⣾⡿⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠑⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠾⠮⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠛⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠆⠀⢰⠀⠀⠆⠀⠀⠈⠀⢀⡤⠒⡻⣏⠑⠢⣄⠀⠀⠀⠀⢒⠀⠗⠀⣀
⣐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⣏⣤⣤⡧⠼⣦⠤⢬⣇⠀⠀⠠⠈⠀⠀⠀⠐⠀⠀⢀⠠
⠀⠅⠀⠂⠀⢲⡶⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠉⠲⣾⣀⠀⣹⡖⠉⢸⠀⠀⠙⠀⠀⠀⠁⣰⣰⡀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣄⠀⠙⠻⢿⡿⠿⠛⣻⣿⣿⡿⢿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⢔⣋⠀⠀⢸⣿⠿⠛⣴⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⠻⠷⣄⣶⡄⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣆⡀⠙⢿⣿⣿⣿⣿⠀⢹⠻⣿⣿⣦⣿⣿⣿⣿⣿⡋⡇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢐⣡⠀⣰⣿⣿⣿⣿⣿⣧⣶⡀⠟⠟⠛⣿⣿⣿⣿⡂⠢⣯⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⢡⣄⣈⠉⣿⣿⠏⢀⣿⣿⠄⢀⡄⠀⣿⣿⣿⣿⠿⠀⠿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣞⠛⢿⣿⣾⡏⢰⣿⣿⣿⣷⣿⡇⠀⣿⣿⣿⡷⢠⣶⡀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣷⡀⣾⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⢩⣆⣿⡇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠟⠛⣩⣤⣤⣄⣀⣀⣴⣿⠿⠃⠘⢿⣿⠇⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⡿⢟⣶⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡟⡃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠸⢿⢿⣾⣿⣶⣿⣾⣿⣿⣿⠿⢿⣿⣿⣟⡿⠻⡿⢛⡿⠄⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠸⠦⠬⠛⠻⠿⠯⠙⠻⠆⠘⠛⠿⠟⠛⠻⠷⠴⠚⠋⠁⠁⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⠂
⠀⣀⣀⣀⣠⡄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣼⣶⣿⣶⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠙⣆⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⣸⣀⠀⠀⠀⠀⠀⡰⣻⡳⣀⠳⣦⣄⠀⢀⣠⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡖⠊⠁⠈⠉⠓⠒⠒⠚⠣⢿⢫⣴⣵⠘⠀⠋⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⡗⠪⠒⠛⠋⠀⢀⠀⠀⢠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⢄⣀⣀⡀⢤⣾⠇⠀⠀⠀⠀⠀⠈⡗⠒⠁⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠬⠒⠒⣒⠮⢡⠎⠀⠀⢀⡰⠃⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠉⠀⠛⡄⠀⠀⠀⠀⠀⢠⠃⠀⠀⡐⠑⣂⡥⠚⠒⢚⡿⠀⠀⠀⠀⣿
⠀⠀⠀⡄⠰⢆⠀⢸⣀⡀⠀⢠⠃⡤⠤⢀⡀⠀⠈⠒⠒⢊⠠⠒⠉⠉⡧⠖⣋⠀⠀⠀⠀⠀⡿
⠀⠀⡜⠻⠁⠘⠋⠉⠈⠉⠑⡇⢘⠄⠀⠀⢳⠀⢀⠔⠊⠁⣠⠔⢊⠼⠚⠉⡀⢱⠀⠀⠀⠀⡇
⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠑⣭⢞⣀⡠⠁⣸⠐⣕⡀⢰⡟⡡⠚⠁⠀⣠⠜⣱⠃⠀⠀⠀⢰⠃
⣤⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⡤⠞⠁⠀⠀⠉⢹⣞⢁⠄⢠⠚⠁⠔⠁⠀⠀⠀⠀⡞
⡍⠇⠀⠀⠀⠀⠀⡴⢋⠀⢀⣴⠋⢳⣀⡀⠀⠀⢀⡤⠋⠈⠘⢲⠋⠀⠀⠀⠀⠀⠀⠀⡸⠁
⠇⠀⠀⠀⠀⠀⠀⠈⠛⡤⠄⠈⠆⢸⠀⠙⣶⠊⠁⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠸⡀⠀⠀⠀⢀⠀⡸⠳⠇⠀⠀⢸⠂⠀⠀⡽⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠱⣄⠀⠀⠘⡗⠁⠀⠀⠀⠀⠀⠀⢀⡴⠃⣀⡠⠔⠊⠀⠀⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠈⠑⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠙⠢⣄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢸⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠈⠀⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠑
⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⠀⠀⠰⢶⠱⣾⡄⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⢸⣆⠀⠠⡄⢰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢠⣦⡈⡵⡞⣧⠀⠀⠀⠌⠉⠉⠀⠀⠀⠀⠀⢸⠛⠀⠘⠃⠸⠛⣧⠀⠐⠀⠀⠀⠀⠀⢀⣆⠀⠐⣇⠀⠀⠀⠀⡆⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⢈⢶⠀⠀⢠⣏⠀⢡⠞⣷⣿⡆⠀⠐⣢⣤⣴⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣤⠛⠀⠀⠈⣄⢀⣀⣠⢼⣿⣆⠠⠀⠀⣷⣇⠀⢃⣷⣿⣿⣿
⠀⠀⠀⠀⢘⡷⠰⠁⡁⠀⢸⠏⣷⣌⣘⠻⠂⢙⠀⡀⣹⣷⣯⡼⣦⠀⠀⠀⢸⠀⠀⠀⠀⠘⡯⠄⠀⣷⢦⣿⡏⠏⡌⠉⣿⠉⠟⠚⠈⢉⣵⠀⠈⠙⢻⣿⣿
⠀⠀⠀⠀⠉⠱⠀⢀⣿⠀⣿⣀⠫⠹⡿⣏⣼⣿⠀⣶⣿⣟⢁⡀⣻⣅⠀⠀⢸⠀⠀⠒⠒⢀⣭⣦⡴⠁⣴⡶⡀⠀⠁⠈⠀⠀⠀⣀⡠⣾⣿⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠰⢍⡀⠈⠀⠡⠿⣽⢻⠀⠿⣿⠛⡏⠻⠿⣿⣷⠀⢸⠀⣴⡂⢰⡏⠉⣾⣋⡁⣿⣿⡇⠀⢸⣆⠀⠀⠀⠀⢠⣿⢿⠀⠀⠀⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠉⠉⠃⢀⠀⠉⠆⠀⠠⢠⠤⢥⢰⠀⠀⠀⠛⠙⠀⢸⡀⠛⢁⠈⠃⣴⠅⢸⣽⣿⣿⡿⠀⠀⠈⢷⡀⢀⣺⠖⣏⣿⡇⠀⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⠇⠀⡀⠋⠀⢀⣶⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⢤⣼⡤⠆⠀⠠⠙⣯⣾⣿⡄⠀⠀⠀⣸⣿⡆⠀⠀⢸⣿⠅⠀⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣀⠀⠀⢸⡇⠈⠀⣤⣄⣴⣶⣾⣿⣿⣿⣀⢀⣤⣾⠿⠋⠀⠀⠀⠈⠁⠀⠰⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢸⣷⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠔⠈⣿⣿⣷⣅⣤⣴⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡡⠐⠁⠀⢀⣻⣿⠛⢉⠀⣸⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣤⣀⣀⣠⣚⡅⠨⠠⠀⠈⣰⡿⢻⣿⠀⠛⠀⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣤⣶⣿⣿⣾⢿⣿⠆⡀⠐⠺⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠐⠄⠂⡄⠀⢀⣠⠴⠚⠁⠀⠀⠘⣯⣾⣏⣾⡿⠘⡋⠀⣀⣼⠧⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠳⣄⣀⣒⣪⡤⠞⠉⠀⠀⠀⣀⣀⣤⠤⠖⠊⢉⡠⠞⠁⠀⠀⠹⡏⠀⠀⠀⠈⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠚⠉⠉⠀⣀⣠⣴⣾⡿⠁⠀⠀⢄⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣑⣄⣤⣤⣤⣬⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⡇⠀⣀⣤⣼⣆⠀⠀⠓⠄⡀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣆⠀⠈⣄⠈⠂⢄⡀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢹⠀⠀⠀⠀⢈⠿⠛⢧⡀⠈⠹⠢⣄⡈⠐⢤⡀⠀⠀⠀⠱⡀
⠀⠀⣸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢠⣷⣶⣤⣤⡠⠊⠀⠀⠀⠑⢦⣀⡀⠈⠫⡐⠢⢈⡒⠄⡀⠀⣣⣠⣴⠼⠃
⠀⠀⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⠛⡟⠛⠋⠉⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⢄⡀⠙⠆⣸⡤⢿⠉
⠀⠀⠟⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣶⡆⠂⠀⣀⡉⠓⠒⠒⠠⣄⢤⡔⠛⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠉⠀⠀⠀⠀⠀⠁⠚⠳⢏⠀⠀⢈⠆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⣀⠤⠂⠀⠀⣡⠔⠋⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠉⠛⣻⡿⠶⠒⠒⠒⠛⣻⠟⠁⠀⠀⣀⠖⠁⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⢆⠀⣀⣤⠶⠖⠋⠁⠀⠀⠀⠀⠀⡼⠃⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⣾⣙⣔⠞⠻⡉⠀⠀⠀⠀⠀⠀⢀⣠⣀⡼⠁⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⢀⣀⣠⡤⢴⡴⠟⢋⡉⢡⣤⣶⣶⠟⠛⠓⠀⠀⠘⠦⡀⠀⠀⠀⠐⣫⢟⠍⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⢀⣤⠡⢜⢥⣴⠞⠋⢑⣿⣫⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠀⢾⡧⢊⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠈⣶⠴⢖⡫⡠⣴⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡆⠈⡠⠚⠢⠤⠤⠤⠤⠖⠒⠚⠉
⠀⠀⠀⢘⣿⣭⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⢋⡠⠊
⠀⠀⠀⠈⠀⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠩⠂