Giggle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣻⠟⠉⠁⠀⠈⠉⠛⣷⣄⠀⠀⢀⡴⠋⠁⠀⠠⣤⣔⣚⣩⣤⢞⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡼⠃⠀⢀⣠⠤⠴⠂⠀⢹⣿⣧⣀⡞⢀⡔⠊⠉⠉⠛⢌⠒⠚⠁⠈⠳⡄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣼⢡⣧⠞⠉⠀⠀⠀⠀⠀⢀⢽⣿⡟⣰⠋⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠢⡈⢢⡀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠇⣸⠃⢀⣀⠀⠀⠀⠀⠐⠁⠀⣸⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠐⡄⢻⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⠀⣿⡤⠋⠀⠀⠀⠀⠀⢠⠖⠛⣿⡇⠀⠐⠀⠐⠄⠀⠀⠀⠖⠒⠒⠲⠀⠀⠘⠈⣧⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣾⠀⣿⠁⢀⣠⠖⠛⠀⠀⠘⢄⠀⢸⡇⠀⠒⠂⠰⠄⠀⠀⠀⠤⠀⠀⠠⠄⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡟⢰⣿⠀⠉⠁⠀⠀⢀⣶⢤⣤⠷⣸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢰⢀⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡇⢸⣿⠀⠀⠀⠀⣀⢡⠏⢸⢹⣮⣟⠻⡀⢀⡀⠙⢤⠀⢀⣜⣀⠀⠀⠀⢀⣿⠊⣾⣻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠃⢸⡏⢛⢦⠀⢀⠂⠎⠀⠃⣾⡟⣿⠀⠣⠀⠿⢁⡆⠚⠉⡠⠞⢡⡾⢀⣼⠋⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⢸⡆⠑⠃⠀⠀⠐⢠⡿⢰⣿⠁⠘⠶⠊⠀⣱⠖⠉⢀⡴⠛⡽⢫⠋⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⡇⠘⣷⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠐⠈⠁⣀⡴⠋⡠⠚⣡⡀⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡏⣇⠀⠀⠀⠀⢀⡴⠃⡄⠀⠀⠀⠀⠴⠟⣉⢔⡋⢀⡠⠞⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⢸⣡⡏⠀⠀⠀⡠⠊⠀⠀⡇⠀⠀⠀⠀⠀⠈⢉⣡⠔⠋⠀⢀⡼⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⢀⣿⡏⠀⠀⠀⣼⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣠⠞⠁⠀⡠⠞⠉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⣰⠃⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⠁⠀⠀⠊⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠦⠠⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣦⣀⡀⢀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢃⠇⡞⣩⠋⡄⢱⠶⡄
⠀⠀⠀⠀⣀⡤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣾⣰⢴⣃⣜⣁⠀⠠⣿⠄⠐⡤⠤⠤⠤⢤⣀⣀⡀
⠀⢀⡴⠋⠁⢀⣀⣄⠀⠡⡀⠀⠀⠀⢀⡀⡜⠁⠈⠉⢋⠏⡩⠁⠀⡃⠺⣜⡜⠁⠀⠂⠀⢀⡀⠀⠉⠉⠓⠦⣄⡀
⢠⠋⠀⠀⡞⠁⠀⠀⢣⠀⢡⠀⠀⡎⠁⢉⠉⠢⡄⠀⢸⢰⣥⡀⢣⢻⣦⠗⢥⠢⡀⠀⠀⠀⠈⠁⠒⣤⡀⠀⠀⠙⢦
⡏⠀⠀⠀⡇⠀⠀⠀⠀⡇⢸⠀⠀⢇⠀⠀⡩⠲⢸⢃⣸⠈⠛⠥⠊⠬⠑⠋⠲⡓⠧⠄⠀⠐⠒⢈⣁⣀⡠⠤⠴⠒⠊
⢧⡀⢀⡴⠁⠀⠀⠀⠀⠁⢸⠀⠀⠈⠢⣀⠈⢀⡌⠘⢄⠘⠳⣄⠀⠀⠀⠀⠀⢳⠒⠒⠒⠉⠉⠉
⠀⠉⠉⠀⠀⢀⠤⠤⠤⠶⠶⠤⠤⠤⠄⠒⠛⠋⡇⠦⠰⠛⠒⠤⡉⠛⠚⠁⢀⡌
⠀⠀⠀⢠⣊⠀⠀⠀⠈⠢⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠑⠦⠒⠉
⠀⠀⠀⠀⠧⣀⣀⣀⣀⡀⠈⣳⠔⢡⡀⢹⠀⠀⠀⠀⡠⠲⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡝⠀⢠⠋⠈⠁⢰⠀⢀⡜⠀⠀⠈⢢⠈⢆⠀⠀⠀⠀⣀⣠⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⡎⠀⠀⠀⡇⢀⡞⠀⠀⠀⠀⠀⢣⠙⠢⡀⡠⠊⠁⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡀⠡⡀⠀⠀⡇⠸⠀⠀⠀⠗⢄⠀⠀⢣⠀⠈⢣⡀⡴⠂⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡷⠶⠷⠀⠀⢧⠀⠀⠀⢸⠀⠈⢢⠀⠈⡆⠀⠀⠹⡀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠑⢄⡀⠀⠀⠘⣆⠀⠀⠘⠀⠀⠈⡆⠀⢳⡀⠀⠀⡇⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠘⠓⠒⠋⠉⠢⣀⠀⠀⠀⡖⡧⢄⠀⡧⣀⢜⡵⠋⠳⠋⣳⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠦⡀⠀⠀⠈⠢⣤⡖⢸⢀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠳⡘⢖⢜⢇⣸⣼⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠎⠇⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⢸⡿⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣷⣋⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣠⣤⣤⣶⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣄⠀⣴⣶⡆⢀⣴⣾⣿⣶⠆⠀⣴⣾⣿⣿⠗⠀⢸⣿⣿⠀⠀⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⣋⣀⣀⢿⣿⣧⣾⣿⠁⣨⣤⣤⣼⣿⣏⠀⣴⣶⣶⢸⣿⣿⣀⣀⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠸⣿⣧⣬⣿⣿⣿⣸⣿⣿⠻⣿⣷⣾⣿⣿⡏⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠻⠿⠟⠻⠛⠛⠀⠈⠉⠋⠛⠛⠁⠀⠀⠉⠉⠋⠉⠀⠉⠉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣠⣄⢀⣤⣶⡄⠀⣶⣶⣦⠀⣶⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢿⣿⡿⠿⢿⣿⣿⠘⣿⣿⣿⣼⣿⣿⣿⠀⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⢸⣿⣿⡄⣿⣿⢿⣿⠟⣿⣿⡆⣿⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠀⠸⣿⠿⠷⠿⠛⠀⠀⠀⠛⠛⠋⠟⠛⠛⠛⠛⠃







⣶⣶⠶⣶⠶⡶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⣠⣶⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠁⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⠞⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠟⠛⠋⠹⣿⣿⣿⣿⣿⣿⣋⣟
⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣍⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⠄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠟
⠀⠀⠈⠙⠋⠁⠙⢿⣿⡿⢿⣿⡿⠉⠀⠀⠀⠀⠀⠀⠒⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠻⠿⠟⠻⠿⠿⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠙⠢⣄⣤⠰⠁⠋⠸⠖⡿⠗⢉⣧⡜⠁⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⢀⣀⣐⣻⣄⠀⣀⠔⠁⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠧⡤⠩⠥⢦⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠆⠀⠑⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠘⡀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡁⠀⠀⠀⠀⠀⢀⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⢠⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠘⠿⠿⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣽⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣶⣛⣹⣏⣹⣇⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠉⠃⠀⢀⣤⣶⣶⣾⣷⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢤⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠞⠀⠀⠀⣼⣿⣿⣿⡿⠻⣿⣿⣿⣿⣿⣿⡿⣛⢿⣿⡄⠀⣀⡽⣞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⡏⠓⠂⠀⢰⣿⣿⣿⠿⠦⡄⠈⠉⠡⠞⠻⢯⡾⢫⣾⣿⣷⠀⠡⠤⢼⡶⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠛⠒⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡄⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⣿⣿⡤⠚⠛⠆⠀⠀⠐⠋⠙⢦⢸⣿⢿⣿⡿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⡀⣠⣴⢺⣿⡟⠟⣦⠀⠀⠈⣁⣠⡿⠃⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣏⠁⠸⣍⢀⣀⣸⣇⣠⣾⣿⠋⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⡿⠿⣿⣿⣿⠿⠛⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⡇⢸⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣸⡇⢸⣇⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣾⣽⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣶⠟⢯⣭⣿⣼⣾⠛⠷⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣀⣀⣀⡽⠛⢹⣀⣀⣀⣸⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠩⠾⠿⢿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡜⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⣿⣿⣿⣋⣭⣙⣿⣿⣿⣿⣿⣟⣩⣍⣻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠻⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⠟⠛⠛⢿⣿⡿⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡅⢤⣭⣭⣭⣭⣭⣥⠄⣽⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⡲⠦⣤⡤⠶⣂⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣶⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣶⣤⣭⣥⣴⣶⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣶⣬⣉⣉⣉⣩⣴⣾⣿⣷⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⣀⢀⣤⠀⣠⡤⢠⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡆⡗⢿⣰⡞⢧⣶⢸⣄⢸⣗



⣿⠿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣿⣟⣻⣟⣿⣻⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠚⠛⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⣤⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣦⠈⢻⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠟⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣴⣾⣿⣿⣿⣿⣿⠛⢉⣀⣉⡉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢉⣉⣁⡉⠛⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠈⠻⣿⣿⣿⣿⡿⠿⠿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠻⠿⢿⣿⣿⣿⣿⠿⠁⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣆⢈⣿⣿⣿⣷⣶⣶⣀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣀⣀⣤⣼⣿⣿⣿⡇⣰⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡏⠉⡉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⢹⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⠀⢿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⡿⠀⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣧⡀⠀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⠀⢀⣼⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣈⠙⠛⠿⠿⠿⠿⠛⠋⣁⣴⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣶⣶⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⣰⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣶⣤⣈⡉⠉⠛⠛⠉⢉⣁⣠⣴⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠀⣿⠆⣠⡾⠛⠛⠀⣠⡶⠛⠛⠀⣺⡇⠀⠀⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠘⣷⠀⣿⡀⣿⡅⠘⣿⡆⣿⡇⠐⣷⡆⢸⡇⠀⠀⣿⡟⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣩⣿⣶⣿⣤⣿⣧⣬⣿⣷⣿⣧⣬⣿⣷⣿⣧⣼⣿⣿⣦⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠛⠿⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠉⠉⠉⠉⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⢀⣤⣤⣶⣟⠁⠀⠀⠀⠀⢙⣷⣦⣤⣄⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⠀⠀⠛⠁⠀⠈⠙⠃⠀⠀⠀⠀⠛⠉⠀⠀⠙⠃⠀⠀⠀⠈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠠⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⣿⠀⠀⠀⣴⣶⣤⣀⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣀⣠⣴⣶⡄⠀⠀⢸⡇⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣿⡀⠀⠀⠸⣧⠻⣿⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠛⢻⡿⢣⡿⠀⠀⠀⣸⡷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠹⣧⡙⠛⠷⠶⢶⣶⣦⣶⣶⠶⠶⠿⠛⣡⡿⠁⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠈⠻⣦⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠋⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠈⠙⠛⠷⠶⠶⠶⠶⠟⠛⠉⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠉⠛⠻⠶⢶⣦⣤⣤⣶⠶⠾⠛⠋⠁⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣦⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠗⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡄⠘⣷⠶⠶⠿⠛⠛⠛⠶⠶⠶⠶⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠀⣀⣀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠛⠉⢻⡄⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⡀⠀⠀⣠⠞⠉⢠⡴⠟⠉⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢶⡾⠁⠀⠀⠀⠀⠀⢿⡀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡾⠁⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⡟⠁
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⣀⡷⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣠⣿⠛⠷⣦⣄
⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⡀⠀⠀⠀⠀⢻⡆
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢸⡇⠀⠀⠀⠀⠀⢿⡄
⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠖⠁⠀⠀⢠⡾⠁⠀⠀⡇⠀⠀⣷⠀⠀⠘⣷⣦
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⣠⠋⠀⠀⠀⠀⣠⣾⠃⠀⠀⣠⡶⠋⠀⠀⠀⠀⢻⡀⠀⣿⠀⢀⡾⣡⠉
⠀⠀⣿⠀⠀⠸⣷⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⣀⣀⡴⠛⣄⠀⢀⣠⠞⠁⠙⠲⠴⠚⠉⠀⠀⠀⠀⠀⠀⠈⠷⠼⣯⣴⠋⣼⡇
⠀⠀⣿⠀⠀⠀⠹⣞⢷⡀⠀⠀⠀⠀⠸⠀⠀⢀⡿⠁⠉⠉⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⣼⡟
⠀⠀⢸⡇⠀⠀⠀⠘⣆⠙⢦⣀⠀⠀⠀⣷⣄⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⡟
⠀⠀⠘⣷⠀⠀⠀⠀⢹⡀⠀⠙⠳⣄⠀⢃⠀⠀⠀⠀⠀⠀⣀⣤⠴⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠳⢦⡀⠀⣿⠀⢿⡀
⠀⠀⠀⣿⡄⠀⠀⠀⠈⢷⡀⠘⢶⡈⠳⠾⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠈⠿⠀⢹⡄⠈⠻⢶⣤⣤⣤⣄
⠀⢀⣠⡿⠁⠀⠀⠀⠀⠀⠙⢦⣤⣤⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠖⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢠⣄⣿
⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⣹
⠙⠳⠶⠶⢶⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡴⠻⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣼⠀⠀⠀⠀⠀⢀⣀⣠⣄⠀⠀⣿⠀⠀⠀⣤⣤⡴⠋
⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⡶⠓⠀⠀⠀⠀⢀⡿⠁⠸⠃⠼⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⠀⠀⠀⣾⠻⣯⠙⠇⠸⣇⣼⢯⣀⣀⣤⠿
⠀⠀⠀⠀⠈⠻⠶⠛⠋⠙⠷⣤⣤⣤⢶⣾⠛⢷⣄⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⢿⠀⠁⠀⠀⠀⢸⣇⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⣄⡀⠈⠻⠶⢤⣼⣏⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣠⣤⡶⠞⠋⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠙⠶⢦⣤⣤⠟⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⢀⣘⣧⣠⣤⡴⠶⠚⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣀⠀⠀⠀⠀⠀⢸⡟⢲⡖⠒⠒⠒⠒⠒⠒⠒⠒⠓⠚⡛⢿⠋⠉⠻⠋⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠙⢦⠀⠀⠀⠀⠑⢸⡇⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⢸⡇⠀⠀⠀⠀⠀⠀⣠⡶⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢸⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠰⠇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣼⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠰⣆⣠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⡀⠀⠀⠉⠉⠙⠛⠋⠛⠉⠉⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⣀⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⠤⠴⠶⠖⠂⠀⠀⠀⢴⡾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⠀⠀⠀⠀⠀⢀⣤⣤⠤⠀⠒⠒⠂⠀⠀⠀⠀⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡤⠀⠀⠀⠀⠀⠀⠤⢤⡤⠤⠤⠀⠀⠀⠀⠀⠀⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⢻⡟⠛⣿⡂⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣰⠟⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠴⠶⠤⣴⣇⠀⠀⠀⠀⠹⣦⠤⠶⡤⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣇⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⢸⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠉⠉⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣠⣹⡇
⣦⣴⣶⣴⣶⣶⣶⣤⣦⣴⣶⣶⣶⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣧⣤⣶⣶⣶⣤⣶⣤⣿⣿⣿⣯⣭⣭⣭⣭⣴⣦⣤⣶⣶⣶⣶⣦⣴⣶⣶⣶
⣿⣿⣿⡟⠉⠛⠉⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⣤⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⠀⠀⢀⣠⣴⣶⣶⡄⠀⣶⣶⡄⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣾⣿⠟⠛⠉⠩⣶⣶⣿⣿⡇⠀⣴⣿⡿⠿⠿⠀⣠⣿⣿⠟⠛⠛⠁⠀⣿⣿⡇⠀⠀⠀⢿⣿⡟⠛⠛⠛⠛
⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠉⢿⣏⠀⠀⢸⣿⡏⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡆
⠀⠀⠀⣿⡇⠀⣀⣤⣤⡀⠀⢸⣿⠀⠀⢸⣿⢀⣤⣶⣶⡜⣿⣿⠀⣀⣤⣴⣶⡄⣿⣿⡇⠀⠀⠀⢸⣿⣟⠉
⠀⠀⠀⢻⣿⣄⣙⣻⣿⡇⠀⢸⣿⣀⣀⠈⢿⣷⣭⣿⣿⡇⢻⣿⣦⣽⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡞⣿⣿⣶⣶⣶⣶
⠀⠀⠀⠀⠙⠿⠿⢿⣿⣧⢸⣿⣿⣿⣿⠀⠀⠙⠛⠛⣿⠟⠀⠉⠛⠛⠋⠉⣿⡇⠙⠛⠻⠿⠟⠛⠁⠻⠿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⣤⣶⣶⣶⣶⣶⡆⣴⣶⣶⣾⣿⡆⠀⢠⣴⣦⠀⠀⣠⣴⡄⢠⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠿⠿⠃⠘⠻⣿⡏⠉⠁⠀⣸⣿⣿⡆⢠⣿⣿⣿⠈⣿⣿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣿⣿⠾⠿⣿⡿⡆⣿⣿⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣧⠀⠀⢠⣿⡟⢹⣿⣶⠆⠲⣶⠾⢿⢿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⢸⣿⠇⠀⠛⠛⠁⢹⣿⣿⢻⣿⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠉⠉⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈⠛⠛⠈⠛⠛⠉⠉⠁

⠀⠀⠀⢀⣠⣤⣤⣄⣀⣤⣤⣤⣄⣀⣤⣴⣶⣾⣿⡿⠿⠿⠿⢿⣿⣷⣶⣦⣤⣀
⠀⠀⣰⣿⠟⠋⠙⠻⠿⠛⠋⠛⢿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣶⣄
⠀⠀⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣦⡀
⠀⠀⠻⣿⣦⡀⠀⠀⠀⠀⠀⢀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠈⢻⣿⣦⡀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀
⠀⠀⠀⢀⣾⡿⠻⣿⣶⣿⠟⣁⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⣼⣿⠁⠀⠈⠛⢁⣾⡿⠛⠛⠛⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠛⠛⢿⣷⡄⠀⠀⠀⠈⣿⣷
⠀⠀⢰⣿⡇⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⣾⣿⠀⢀⣴⣿⣿⣿⣦⡀⠀⣠⣶⣿⠿⣿⣶⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⢸⣿⠀⣼⣿⠁⠀⠈⣿⣧⣾⡿⠋⠀⠀⢈⣿⣿⡿⠿⣿⣦⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠸⣿⡇⣿⣿⠀⠀⠀⣿⡿⠋⠀⠀⢀⣴⣿⠿⠋⠀⠀⢸⣿⡇⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⣠⣴⣿⡟⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠈⢿⣿⡿⠀⠀⠀⠀⠀⠀⠿⠟⠁⠀⠀⣠⣾⡿⠟⠿⣿⣦⣀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢀⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⢀⣀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⣾⣿⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋
⠀⢿⣿⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⡿⠟⠁
⠀⠘⣿⣧⠀⠻⣿⣶⣄⣀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣿⣿⠿⠟⠋⠁
⠀⠀⠘⢿⣷⣄⡈⠙⠻⠿⠿⣿⠿⠿⠛⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠙⠻⠿⣿⣶⣶⣾⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣴⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣦⣀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢰⣿⠉
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡾⠛⠉⠀⠀⠀⠀⠀⠀⢠⡶⠿⠿⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠿⠿⣶⡆⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣼⡏⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠙⢿⡆⠀⠀⠀⠘⠛⠋⢻⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢹⣿⡿⠿⠃⠀⠀⠀⢰⡿⠋⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣆⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠘⣿⡛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠛⠛⠛⠛⠛⢛⣿⠃⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠹⣷⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣾⠏⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠘⢿⣯⡉⠉⠉⠛⠛⠛⠛⠛⠉⢉⣽⡿⠋⠀⠀⠀⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠀⠙⠻⢷⣦⣤⣤⣤⣤⣴⠾⠟⠋⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡁⠈⠉⠛⠛⠛⠿⠟⠻⠟⠛⠛⠛⠉⠁⢈⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⡾⠋⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⢿⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣤⣤⣤⣤⣴⣶⠿⠛⠉⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣦⣀
⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⢀⣀⣀⣀⠀⣹⣇⠀⣀⣀⣀⣀⣀⣀⡀⢀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⡀⣀⡀⣀⣀⣀⣀⣸⣟⡀⣀⡀⣀⡀⠀⠀⠀⠀⢸
⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢀⢠⠤⠤⠠⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠮⣳⠂⢢⠀⠀⠀⠀⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢪⠉⠆⠀⠀⠀⠀⠀⢈⣯⣶⠖⢖⡒⢄⡀⠀⠀⠀⢸⣆⣀⠀⠀⠀⢀⠤⢄
⠀⢰⠒⢄⠀⠀⠀⠀⣀⠇⡄⠀⠀⠀⠀⠈⡙⣿⠀⢀⠈⠉⠊⢏⠲⣤⠖⠉⠀⠀⢣⠀⠀⠊⡠⠂
⠀⠈⠒⠑⠤⡀⡀⠉⠀⠠⡃⠀⠀⠀⠀⣾⣹⣉⠙⣏⠰⠦⡄⠈⠄⢣⠐⢅⠀⠀⡆⠀⠀⢀⠈⠢⢄⣀⠀⠠⠒
⠀⠀⠀⠁⠣⣀⣀⣂⣀⣠⠇⠀⠀⠀⠀⠀⡛⠁⠀⠈⠃⢀⡠⢞⠆⠀⢆⢁⡀⠞⢦⠀⠀⡎⠐⠄⠀⠢⢒⢈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡢⣄⣤⣶⣟⣔⡁⠀⢤⣼⠦⢤⣀⣀⡇⠀⠙⠉⠐⠢⠄⠈
⠀⣀⠤⠄⠀⠤⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⢧⣏⡀⢪⠊⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠃
⡜⠁⠀⢀⡔⠒⠒⢷⠀⢸⠀⠀⠀⢀⠔⠋⢀⣱⠢⠖⠁⠀⠀⠀⡙⠢⡀⠈⠢⡀⠀⠀⢘
⠳⠤⠴⠊⠀⠀⠀⣄⠀⠸⠀⢸⠂⠁⢀⠔⠁⠈⠀⠀⠀⠤⡀⠀⢇⠀⠈⠂⠀⠙⠞⠀⠠
⠀⠀⠀⠀⠀⠀⢰⠿⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⡴⠁⠀⠈⢆⠘⡄⠀⠀⠀⢄⠀⠀⢨⣄⡀
⠀⠀⠀⠀⠀⠀⡎⠇⠀⠃⠀⠊⠀⠀⠀⠀⢰⠀⡼⠁⠀⡀⠀⠈⢣⠙⢤⣀⠖⠉⠑⠀⠙⢸⠁
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠃⠀⠀⡝⢦⠀⠀⢇⠀⠙⢆⡠⠊⠀⠀⡞
⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠰⠀⠀⢣⠀⠘⡄⠀⠈⠇⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠱⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⡇⠀⢣⠀⠀⡰⢀⣰⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠪⢔⠤⢀⠀⠀⢀⣀⣤⠶⠛⠢⢄⣀⣀⣏⠓⠢⠒⠳⣞⠕⢁⠈⠁⡈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠑⢄⠠⡀⡈⡳⣿⢠⢠⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠑⣌⢎⢊⣾⠿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡏⣦⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡸⠧⠏
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠤⠶⣶⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣄⠸⠀⢠⡄⡃⠀⠀⠀⠀⣤
⠐⠀⠉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⡄⠒⠟⠀⢠⣤⡄⠀⠉⠙⠂⠀⠀⠀⠐⠈⠁⡃⠀⠀⢒⡢⠀⠀⠀⠀⠛
⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢀⡠⠀⡀⠀⠀⠀⢀⠈⣸⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⡶⠋⠁⠀⠀⠀⠀⠒⠀⠉⠠⠔⠒⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⠄⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄
⡄⠀⠀⠀⠀⠀⠀⢀⢀⣾⡌⣂⡐⠷⠂⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⡀⠀⢄⡀⠀⠀⠠⠶⠂⠀⠀⠀⠠⢤⡀⠀⠀⠀⢠⡌
⢁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠐⠓⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠁
⢰⣦⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠻⠟⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢠⡆
⠸⢿⠄⠀⠀⠀⡉⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⢠⡦⠁⠉⣤⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠎⠀⠀⠈⠙⠢⠄⠀⣤⠤⠄⠀⠀⠀⢸⠟⠉⠉⠻⠆
⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⠀⡀⣀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢦⣀⣀⣠⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠉⠀⠀⢀⣀⡀
⠀⢀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠂⠐⠶⠛⠛⠛⠁⠀⠀⠀⢠⡀⠀⠛⠷
⠀⠈⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠐⠛⠶⠀⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠃⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⠆⠀⢀⣤⠀⠀⠀⠀⠆⡀⠘⡃⠀⠀⠙⡂⠀⠀⠀⠶⠋⠀⠀⠈⡳⠆⠀⠀⠀⢈⠉⠀⠁⠂⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠁⠐⠀⠀⠀⠀⠈⣹⣤⣀⣌⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈
⠤⣄⠀⠠⠞⠓⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⡀⠋⠉⠉⠭⠃⠀⠀⠀⠀⠀⡹⠣⠀⠜⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡀
⡺⠀⠀⠀⠛⠀⠠⠆⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣲⠾⢁⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⣉⠀⠈⠛⠶⠤
⠀⢩⣄⣀⣴⡚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⣿⣷⣄⡇⠀⠀⠀⠀⠈⠄⠄⠀⣀⠄⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠘⠃
⠐⠋⠍⠉⠛⠃⡀⠀⠀⠀⠀⠀⠀⡴⠀⣉⣁⣀⣀⣉⡉⠰⣦⠀⠀⠀⠀⢾⠆⠀⠡⠖⠄⠀⠀⠀⠀⠐⠀⣠⣀⣀⣀⡀⠐
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⡶⠂⠀⠀⠀⠀⠀⠿⠛⠛⠻⠃⠀⠀⣠⠤
⠀⠀⠀⠈⠁⠀⠀⠀⠠⠴⠃⠀⠀⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢧⣤⣤⣴⣾⠁
⡆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⢀⣠⠀⣀⠤⠶⠄⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠐⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠤⠄⠀⠀⠀⠀⠀⠰⡆⡀⠀⠷⠀⠀⠀⠀⠐⠲⠖⠉⡅⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠛⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠚⠛⠛⠛⠛⠓⠲⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣴⠏⠀⠀⢀⣠⠶⠛⠛⣛⣛⣛⣲⣦⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣚⣛⣛⡛⠛⠶⣤⡀⠀⠀⠹⣦
⠀⠀⢀⡾⠁⠀⠀⣠⣿⠵⠖⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠳⠶⣿⣦⠀⠀⠈⢷⡀
⠀⢀⡾⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢷⡀
⠀⣼⠃⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⣤⣀⡀⠀⠀⠀⠀⠀⠘⣧
⢠⡏⠀⠀⠀⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⠀⠀⠀⢹
⣸⠁⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈
⣿⠀⠀⠀⢀⡏⠀⠀⠀⠀⣴⣿⣿⣿⣦⡀⠀⠀⠀⢿⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣾⣿⣿⣷⡄⠀⠀⠀⢹⡄
⣿⠀⠀⠀⠸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣼⠀⠀⠀⠀⣿⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⢸⡇
⢹⡀⠀⠀⠀⢿⡀⠀⠀⠀⠈⠛⠿⠛⠋⠀⠀⠀⢠⡟⠀⠀⠀⠀⢹⡄⠀⠀⠀⠈⠛⠿⠿⠋⠀⠀⠀⠀⡾⠀⠀⠀⢀
⠘⡇⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸
⠀⢻⡄⠀⠀⠀⠀⠈⠓⠦⣤⣄⣀⣠⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣀⣤⡴⠞⠋⠀⠀⠀⠀⢠⡟
⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠦⠤⣤⣤⣤⣤⡤⠴⠶⠚⠛⠉⠁

⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⡤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⣀⣠⣤⢀⣀⣤⠤⣄⣀⣠⣤⣶⣦⣼⣿⣿⣿⣤⣀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⡼⡶⣿⣿⣿⠋⠁⠀⠀⠀⠁⠀⣼⣿⣿⣿⡟⠁⠀⠀⠈⠳⣄⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢻⣿⣿⣶⠆⡄⡐⢀⠄⠀⢠⠀⠈⡌⠉⠙⣼⠀⠀⠀⠀⠀⠀⠘⣆⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠸⠿⢻⠃⠰⠀⠇⡾⠀⠀⠘⣇⠀⢰⢠⠀⠸⣧⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⡨⠇⢳⠀⠀⠀⢛⣅⣘⡮⢦⠀⡇⠱⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠠⠋⡼⣿⢾⢆⠀⠰⠛⡿⠹⣿⡤⠀⡷⢤⡟⢦⢀⠀⠀⠀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢿⡄⠸⠴⡇⠁⣬⠀⠀⠂⠀⠨⠄⠈⠹⣿⠉⢸⠇⠈⣧⡆⢀⣀⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢸⣸⠳⣾⡤⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⣽⠔⠋⠀⠀⣿⡇⢸⡿⢹
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢧⠸⣿⠀⢰⣏⢤⣄⡀⠀⠀⠀⠀⢀⣤⣚⠻⣿⠀⠀⠀⠼⢋⡧⠞⠁⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠓⢿⠓⢈⣧⣑⣴⣛⣛⢿⣦⡀⢨⣿⣿⣷⡄⠀⠀⠀⠀⠈⠁⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠣⢀⣀⣀⡀⣰⡻⢥⠊⠙⢑⡐⣿⣿⣷⣿⣿⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠢⢄⡀⠀⠀⠀⣀⠴⠋⠉⠀⠀⡛⠋⠠⠔⢲⡄⣴⠭⠤⠤⠉⠛⢿⣛⣽⠾⠛⠉⠉⠉⡄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡰⢷⣿⣅⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠈⠉⢹⣦⡀⠀⠀⣨⣷⣄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣦⠴⠚⠛⠉⠉⠉⠙⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠳⣄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⢠⠞⠉⠙⢦⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢠⡖⠒⠒⠒⠒⠒⠒⠒⠒⣦⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠙⠒⠒⠖⠚⠉⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⣤⠤⠤⠶⠚⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠒⠦⣴⠚⠳⠜⠉⢳⣠⠊⠙⡶⠋⠙⡦⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠃⠈⠀⢀⣀⣀⠀⠀⠀⠀⠀⠁⠀⠀⣁⠀⠈⠁⠀⠀⣼⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠞⠉⠀⢀⣰⠋⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⣧⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⠁⠀⠀⠰⠁⠘⠃⠀⠀⢠⠖⠢⠀⠀⠀⠀⠀⠀⠐⢯⠽⠳⣆⠀⠀⠁⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⡴⠋⠁⢀⡔⠚⠂⠀⠀⣴⠋⠙⠂⠀⠀⠀⡰⠀⠐⠲⢤⡀⠀⠀⠸⠓⢦⠀⠀⠀⣾⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡷⠀⡔⢹⣳⠀⠀⢠⠊⠻⠂⠀⠀⢀⣤⠞⣇⠀⠀⡀⣴⣷⢦⡀⠀⠀⠈⢷⡀⠀⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣿⠁⠘⠀⠀⠀⢀⠴⠲⠄⠀⠀⠀⠀⢀⡟⠀⠙⠶⢾⡇⠈⠁⠀⣽⠶⣄⠀⠸⠏⠳⡄⢀⣾⣤⡀
⠀⠀⠀⠀⠘⢿⣶⣤⣀⢸⣰⡇⠀⠀⠀⢀⢴⡷⡄⠀⠀⠀⠀⣰⠖⠋⠀⠀⠀⠀⠈⠷⡤⣶⠀⠀⡀⠈⢷⡀⠀⠀⣿⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⣙⣿⣿⡿⠧⠀⠀⠀⠁⠈⡛⠁⢀⡠⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⢺⡇⠀⠈⠋⠹⡄⠉⠈⢳⣴⣇
⠀⠀⠀⠀⠀⠀⠛⠛⠛⡟⠀⠀⣴⠒⠦⣄⣠⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠶⣶⠀⠿⠳⣆⠘⡇⠈⡇⢀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢠⠖⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢿⡀⣠⡄⠘⡄⠃⣸⣷⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡞⣱⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠉⠹⣧⣀⡄⠈⢻⡿⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⣸⣷⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⢸⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⣹
⠀⠀⠀⠀⠀⠀⣼⠁⢹⣾⡗⠘⣇⠀⠀⠀⢰⣤⣶⣶⣴⣶⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣷⡄⠀⠀⠀⣸⡟⠙⢦⠉⢷⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠸⣏⣠⠀⢿⡀⠀⠀⠘⠛⠛⠛⠛⠛⠋⠀⠀⠸⠧⠤⠀⠀⠀⠀⠀⠀⠻⠿⠛⠛⠛⠋⠀⠀⢀⣿⢃⣄⣸⠀⣸⠇
⠀⠀⠀⠀⠀⠀⡸⠧⠀⢸⠃⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠤⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠸⣾⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠑⡆⢀⡘⠷⣃⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠛⡧⢀⣽
⠀⠀⠀⠀⠀⠀⠀⠛⠛⣧⡴⠛⢦⡼⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢀⣠⠞⠃⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⢻⡤⢶⣠⠷⠴⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡄⣠⣤⣀⣴⠀⠀⠀⠀⠀⠀⠠⣤⣶⡖⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠙⠉⠉⠁⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣧⢰⡶⠄⠀⠀⠀⠾⠀⠀⠀⠀⠺⠛⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣀⣠⡴⠂⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠶⣤⣀⡀⠁⠀⠀⣀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⠀⠀⡀⠀⢀⡿⡆⣠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠸⣷⣄⠀⢳⣰⡇⢠⣸⠁⣷⠇⢸⠁⣤⣴⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⢦⡀⠢⡈⠲⣿⡙⢶⡀⢻⣇⠀⡏⠀⠙⠀⣿⠀⠁⣾⠞⣰⢁⣤⢔⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠢⣌⡃⠙⢦⡈⡦⠈⢅⡄⠀⢧⠉⢸⠀⠀⡤⠀⠀⢰⠆⠏⢀⠀⠚⠁⣽⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⣥⣀⣬⠿⠶⠱⣦⠙⠆⠀⠉⠀⠠⠄⠈⠀⠇⠀⠸⠀⠈⡠⠀⠋⢠⠀⢘⡿⠋⣀⠄
⠀⠀⠀⠀⢤⣀⠀⠀⠒⠮⣿⡌⢍⠁⠠⡀⠈⠀⢢⡀⠁⠀⠀⠄⠀⢀⠀⠂⠀⡀⢀⠁⢠⠀⠁⠀⠠⠴⠯⠗⣶⠃
⠀⠀⠀⠀⠀⠈⠙⠳⠦⠍⠒⠦⣄⠁⠀⠉⠀⠠⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣷⡀⠀⣾⡴⠀⠀⠀⠀⠀⠬⣭⣤⠄
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠲⠀⠠⡀⠀⠀⠙⠂⠀⡀⠀⠀⠀⠀⠐⠀⠀⠀⠠⣄⣿⠳⠀⠻⠁⠀⠀⠀⠀⠐⢻⠯
⠀⠀⠀⣤⣤⣽⠿⠧⣤⠀⠒⠚⠛⠂⠈⠁⣀⠀⠀⠀⠈⠀⠀⠀⠠⣤⣬⣆⠈⠛⠀⠀⢀⣄⠀⠀⢠⣀⢸⡇
⠀⠀⠀⠀⢀⣀⣠⡤⠀⢉⠿⠒⠀⠠⠄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢙⣦⠀⠀⠀⠀⠀⠋⠥⡀⠀⠀⠪⢸⡇
⠀⠀⠀⠀⠀⠰⠶⣤⣤⡄⠐⣋⡀⠀⡴⠀⠠⠀⠀⠀⠒⠀⠀⠀⠸⣏⠀⠀⠀⠀⠀⠀⠰⠇⡇⠀⢞⢣⢺⠇
⠀⠀⠀⠀⠀⠐⠋⠉⠐⠚⠉⡉⠁⠊⠀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⡞⠀⠀⢀⣀⣀⡀⠈⠒⠃⢀⠬⠛⠛⠛⠻⢶⡞⢻⣦
⠀⠀⠀⠀⠒⠚⣋⣴⡾⠀⠚⠇⠀⠚⠀⡄⠀⠆⠀⠀⠂⠀⠀⠀⠈⠃⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠤⠚⢩⣟⣥⡶⢀⣠⣀⣄⡀⡁⠰⠀⠉⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⠀⠀⢀⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠠⠟⠋⢰⡷⠛⢛⠁⠀⠉⠀⠀⣄⣀⣀⣀⡀⣶⠀⠀⠀⠀⠀⠉⠉⠽⣦⣠⣶⣭⡽⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣴⣤⣤⡼⠶⠏⠈⠉⠉⠁⠙⠳⠶⠶⢶⣦⣀⣠⣤⣾⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⠿⠿⠿⠿⠿⠿⠿⢷⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣦⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠉⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠷⢶⠾⠿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⢚⠉⢑⠰⠻⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠉⠁⠒⠒⠀⠤⠄⠠⣧⠬⠄⠚⡂⡤⢏⡗⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⠄⠐⠂⠒⣿⢸⠀⠀⠠⠒⢧⡟⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡹⣍⢠⠱⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢈⠉⠢⠀⠀⡊⠀⠀⢻⠀⠀⢀⠄⠀⠀⡌⠁⢱⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⣀⠙⣑⠀⣇⠄⢌⠁⡟⠀⠁⠉⢋⡠⠠⠀⢱⢐⡑⢁⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠴⠀⠀⠯⢔⠖⠉⠀⠀⠀⠀⠎⢐⣓⠤⠎⠈⠐⡂⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠔⠹⢀⢀⢠⡎⠀⠀⠀⠀⠀⠀⡄⠑⢄⢑⠀⠀⠘⠈⠢⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠣⡀⠀⠀⠀⢀⠔⠁⡘⠀⢫⠃⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣀⣉⣔⠉⢁⡀⢠⠃⡰⠃⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢀⠊⠀⠀⠘⠓⠁⠈⡡⠊⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡑⡈⠢⠔⠁⢰⠀⠀⡔⠁⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢼⠓⢄⣀⡘⢀⡊⣀⠴⠶⣒⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠈⠣⠘⠁⠁⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⡀⠀⠀⠀⡩⣀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠠⡤⢷
⠀⠀⠀⠀⠀⠀⠀⠪⡑⢄⣈⣦⡀⡸⠀⠀⠑⢤⣁⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠁⢘⠻⣅⠀⠀⠀⠈⠀⠉⠒⠦⡤⠌⠡⠤⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⡦⡀⠀⠦⢄⠀⢀⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⢕⢝⣶⡙⠀⠉⡟⠛⠂⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠛⠀⠀⠀⢱⣀⣀⣀⣀⣀⣐⣦⣤⣤⣤⠠⠄⢄
⠀⠀⠀⠀⠀⠀⠀⠐⠙⠊⠉⠋⠉⠉⠉⠋⠛⠋⠉⠀⠈⠋⠉⠉⠭⠟⠓⠒⠓⠒⠘⠓⠢⠝⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢹⠀⠀⠀⠀⢠⣞⣙⡷⠤⠤⠶⠾⠾⠶⠦⠤⣤⣤⣒⠙⢦⠀⠀⠀⢀⡔⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡼⣧⣠⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠛⠿⣄⡀⢀⡜⣀⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠀⠀⡴⠚⢶⠀⠀⢀⣠⣀⠀⠀⠀⠠⠊⠉⠇⠀⠀⠀⠈⠙⢿⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⠞⠁⠀⠀⠀⠀⠉⠀⠀⠀⠸⣇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡠⠒⢳⡄⠀⡹⠃⠙⣦
⠀⠀⣀⡀⠀⠀⠀⠐⣧⠚⠁⣠⠴⢤⡀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠞⠀⠚⠁⣠⠖⢷⡒⠋⢻⠆
⠀⣼⠁⢹⣀⠀⠀⡜⠁⠀⠸⣥⣀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠋
⠀⠘⠷⠶⢽⣷⡞⠀⠀⠀⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡌⢷⡀⠀⠀⣠⠶⠶⣄
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⡸⠀⠀⠀⠀⠀⠀⠒⢉⡀⢀⡿⠘⣧⢴⣊⣥⣤⣄⡏
⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣿⣷⡀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⢹⡏⠁⠀⠀⠈
⠀⠀⠀⠠⣯⡇⠀⠰⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⣿⣿⠇⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⠋⢹⠆
⢀⣀⣀⣀⣙⠁⠀⠀⢳⡈⠉⠛⢶⣶⣦⠤⣤⠄⠀⠈⠈⠙⠒⠦⠤⠤⠤⠤⠖⢉⡤⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⣦⠀⠀⠉⣿⢉⣀
⣎⡀⢈⠥⢼⡆⠀⠀⠀⠳⣄⡀⢈⣛⠯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠋⠀⠀⠐⢿⣀⢹⠆
⠈⠉⠀⢀⠔⡇⠀⠀⠘⠦⣌⣉⡡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠒⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠘⠦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡤⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠸⡆⢠⡏⠀⠀⠀⠀⠀⠀⠀⠰⣏⠈⢧⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠜⠀⠀⢠⡀⢹⡶⢤⡀⠀⠀⢳⡄
⠀⠀⠀⠀⠀⡔⠹⡌⠙⠛⠀⠀⢠⡔⠹⣆⢀⡹⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢃⡟⠀⠀⢧⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠳⠤⠿⣆⠀⠀⠀⠀⢹⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⣦⠀⠈⠳⢤⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣦⠰⠽⢦⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣠⠛⣧⠰⠋⠀⠀⠀⢀⡟⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⢹⠀⣠⠜⣻⣦⡀⠀⠀⠀⠀⡟⠀⣷⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠦⠴⠃⠀⠀⣠⡞⢥⡀⠸⡆⠀⠀⠀⠀⢸⠀⣤⡀⢰⡈⡇
⠀⠀⠀⠀⠀⣴⠊⠉⢀⡞⠁⠀⢙⡶⣄⡀⠀⠻⠖⠋⠀⠀⠀⠀⢸⣄⣸⠄⠀⠀⠀⠀⠀⠀⣀⣠⠴⠊⠁⢹⡄⠙⡆⢿⠀⠀⠀⠀⢸⡼⠁⠹⣄⡿⡗
⠀⠀⠀⠀⠀⠘⠳⠶⢞⠁⠀⠀⠸⡇⠀⠉⢳⠶⣦⣤⣤⣠⣀⣀⣀⣉⣡⣤⣤⣤⠴⠶⡖⡟⠧⠙⢦⡀⠀⠈⠛⠊⠁⢘⡀⠀⠀⠀⠈⠁⠀⠀⣿⠀⠁
⠀⠀⠀⠀⠀⠀⡞⢀⡎⠀⠀⠀⠀⣷⠀⡴⠋⣼⠁⠀⣰⠁⠜⠁⠹⢡⠀⠘⠒⠂⠀⠀⡇⠙⢶⡄⠀⠙⢦⡀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠇⠈⢶⠲⡄⠀⠀⢻⡘⠷⠤⠟⠀⣰⠃⠀⠀⠀⣇⣸⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⠀⠀⠙⠳⡶⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⠀⣆⢸⣇⡸⠀⠀⠘⣧⠀⠀⢀⡴⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢹⡄
⠀⠀⠀⠀⣄⠹⠦⣬⣷⡼⠟⠀⠀⠀⣽⣂⠀⠈⠳⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣇⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠳⢦⡀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⢉⡵⠞⠛⠶⠲⠦⠟⣠⠼⠦⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠇⠀⠀⠤⠍⡁⠶⠂⠴⣤⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⢼⡟⣥⠶⣄⣀⡴⠒⠦⣤⡤⠛⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠭⠔⠒⢦⣀⣠⠔⠒⢠⡀⠀⢀⡨⣷⣄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠿⠊⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢧⣽⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶
⣿⣿⣇⣀⣀⣀⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢤⡟⠁⠄⠀⠠⠀⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣽⠇⠀⠐⠂⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠿⠷⣝⠀⠀⠀⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠐⠐⠆⠀⢈⠀⠀⢀⠀⠀⠀⠀⢀⡄⡀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠂⠀⠓⠀⠀⠀⠐⠄⠀⠀⠀⠀⠈⠙⢧⢑⠈⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡄⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⣐⣧⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡟⢸⡇⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⣶⡐⠀⠓⢋⠨⡧⡄⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠛⠐⠀⠀⠈⢳⠆⠀⢹⡇⡀⠀⠀⠙⠄⠛⠃⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣷⡀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣠⡄⢸⣿⣿⢿⣿⣿⣷⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣟⡇⠀⠀⣿⠁⠈⣿⣿⣾⣿⣿⡏⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠄⠠⣿⠃⠀⢿⠿⢿⣿⢻⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠠⠀⢰⠟⠃⠀⢸⣷⢸⣿⠁⠀⠀⡴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢘⡄⣄⠀⠈⣿⢸⣿⠀⠀⠄⣀⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣁⣀⡀⠈⢷⣿⢂⣿⣿⡿⠿⣆⡀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣦⣤⣿⡿⢯⣀⡈⠷⠂⢈⣉⣸⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠭⠬⠻⠿⠑⠒⠁⠴⢾⢿⣿⣿⢿⣿⣿⡿⠏⠁⠒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠛⠓⠚⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠯⠉⠉⣙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠔⠒⠒⠒⠒⠲⢤⣀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠀⣀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣍⠂⠲⠤⠖⠒⢋⡍⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢿⣷⣶⣤⣶⣶⡟⠁⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠛⠿⠿⠟⠋⠀⠀⠀⢠⡏⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢄⡀⠈⠉⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠝⠓⠲⠤⠤⠤⢴⠞⠫⢄⠀⠀⠉⠲⡀⠀⠀⠀⠙⢦
⠀⠀⢀⠤⠐⠒⠒⠒⠤⢄⣀⣀⣀⣠⡤⡾⠁⠀⠀⣀⣠⠴⠚⠁⠀⠀⢠⠇⠀⠀⠀⠙⣆⠀⠀⠀⠀⢳⡄
⡖⢶⠡⣆⣀⡹⢤⣼⣃⡀⢠⠤⠤⠤⠴⠷⠒⠚⠙⠍⠀⠀⠀⠀⢀⡴⠋⢀⠠⠂⠀⠀⠘⡆⠀⠀⠤⠊⢹
⠓⠒⠶⠮⠭⠥⠤⢤⣀⠉⠙⢧⡔⠁⢀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠛⠂⠉⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⡗⠺⢿⡹⣄⣀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠸⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠈⡗⢬⣃⡇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⡇⢰⠀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⡇⠀⠀⠀⢧⠀⠈⠀⠀⠀⠐⠂⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⢠⡇⡸⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⣸⢁⡇⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡷⠀⠀⠀⠀⠈⣧⣄⠀⠀⠤⠤⠤⠤⠤⠶⠒⠉⠀⠀⢀⡴⢧⣼⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⢀⡞⠀⠈⠙⠓⠦⢤⣄⣀⣀⣀⣀⣠⣤⠖⠋⠀⠀⠈⠉⠛⠛⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢀⣠⠤⣤⣇⣇⣠⠤⢤⡏⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⣠⡤⠖⠁
⠀⠀⠀⠀⠀⡜⠁⣜⣆⠀⢀⡞⢁⣤⡀⢻⠒⠾⠦⢤⠼⠤⠼⠶⠤⣤⣄⣀⣀⠤⠤⢄⡈⠉⠉
⠀⠀⠀⠀⠀⢧⠀⠛⢉⡗⢺⠀⢸⡠⠃⢸⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡔⢢⠀⢷
⠀⠀⠀⠀⠀⠀⠙⠒⠋⠀⠘⣆⠀⠀⣠⠏⠀⠈⠉⠉⠉⠉⠉⠙⠒⢶⠒⢿⠀⠳⠁⢀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⠀⠀⠀⠀⣠⣴⡾⣿⣿⡿⠿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⢸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⢀⣼⣿⣵⣺⣽⢶⣟⣛⣛⣛⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢳⣾⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣏⣇⡤⢦⡀
⠀⠀⠀⠀⠀⠀⢠⡄⠀⢣⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣭⣿⡿⠿⠿⠟⠋⢻⣿⣿⣿⣿⠀⠀⠀⠀⣿⢋⡴⠋
⠀⠀⠀⠀⠈⠻⢾⣿⡀⠀⠙⠁⠀⠀⠀⢰⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⠀⠀⠀⠛⠁⠀⠤⠴⠂
⠀⠀⠀⢀⣀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⢀⢼⣤⠀⠀⠀⢠⣴⣧⠄⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠦
⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⢿⠿⠃⠀⠀⠐⠾⠯⠗⠀⢸⣿⡇⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣶⣣⣄⠀⠀⣸⣿⣿⠀⠐⠚⠛⠀⠰⡤⠀⠒⠖⠂⠀⢸⣿⡇⠀⠀⠀⣼⢶⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢊⠝⠇⠀⠀⣿⣿⣿⣧⠀⠀⢠⡀⠀⠀⢠⠀⠀⠀⢀⣾⣿⡇⠀⠀⠰⠋⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣄⡀⠙⠒⠒⠁⠀⣀⣤⣾⣿⣿⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣉⣉⣛⣛⣛⣛⣛⣛⣓⣚⣛⣛⣛⣛⣋⣉⡩⠭⠽⠧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⣠⠤⠬⠿⢦⡀⠉⠉⡩⠭⢍⡀⠀⠀⣴⣛⣛⠒⠲⢄⡈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⡔⢋⡠⠖⢊⣉⠭⢷⡀⣞⠰⣿⡆⢹⣀⠼⠓⠲⠤⡉⠲⣄⠙⣄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠋⠀⠊⣠⠞⠉⣀⠤⠦⢬⠟⢦⣙⡡⠞⠳⠒⠛⠲⣄⠈⠳⠈⠁⠈⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⠾⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⢧⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⠚⢳⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣶⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠛⠉⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠛⠋⢻⣿⣿⣿⣿⣿⡟⢻⡇
⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣯⣽⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠈⠉⠋⠉⠈⠻⣿⡿⠿⣿⡿⠏⢀⠀⠀⠀⠀⠀⠒⡊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠻⠿⠟⠻⠿⠟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣤⡴⠐⠋⠰⠖⡿⠖⢹⣤⡎⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠰⢄⣀⣐⣻⡄⠀⣀⠔⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠾⠧⡤⠭⠗⢶⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠐⠀⠀⠌⠀⠙⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠁⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡁⠀⠀⠀⠀⠀⢠⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⢰⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠘⠛⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠣⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡃⠘⢦⡀⠀⣀⠾⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠇⠐⠀⠈⠉⠀⡄⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠈⠄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣏⡀⠐⠂⠀⠀⠀⣀⠠⠀⠀⠀⠳⠤⠤⢶⡞⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣠⠋⢫⠀⠀⢀⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠈⢲⡀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⣰⠏⢃⠀⢀⠜⠁⠀⠀⠉⠉⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠡⠀⠀⠀⠀⢀⣃⣀⣀⠀⠀⠀⡰⠁⠀⠀⠑⠁⠀⢀⠁⠀⠈⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡋⠀⠀⠀⠀⢀⣠⠤⢄⠀⡞⠁⠁⠀⠙⣆⡜⣁⣄⠀⠀⢀⠀⠂⠀⠁⠐⠠⢈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠦⢤⣸⠀⠀⠀⡽⣇⠀⠀⠀⠀⡼⡏⠀⠀⢱⠊⠀⠀⠀⡠⠖⠊⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠙⠒⠊⠀⠈⠓⠤⠤⠚⠁⠑⠤⠤⠊⠑⢤⣀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢀⡤⠤⣄⠀⠀⠀⠀⠀⣀⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⡠⠖⠁⠀⠀⠈⠳⣄⢀⡠⠞⠁⠀⠀⠈⢦⠀⠀⡠⠤⢄⡀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠛⠉⠉⠁⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠋⠁⠀⠀⠉⠒⠶⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠸⠿⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⡄⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⢶⠆⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠾⠂⠀⠀⠀⠀⠶⠄⠀⠀⠰⠆⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⡀⣀⡇⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠸⡁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠙⠓⡲
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠚⠛⠋⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣽⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⣶⣾⣿⣿⣷⣾⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡄⢀⣀⣤⣤⣶⣶⣶⣯⣤⣤⣀⡀
⠀⠀⠀⠀⠸⢷⣦⡀⠀⠀⠀⠀⠀⠁⠀⣰⢋⣴⣿⣿⣥⣤⣈⣉⡙⠛⠛⢻⣿⣿⣵⡢⣄
⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⠀⠀⠀⣠⣤⣽⣿⣿⡿⠟⠟⠻⢻⣿⣯⣶⡶⠇⠈⠉⠽⣿⣷⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣠⣿⣥⣴⣶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣒⠀⠙⠻⣿⣷⣕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡴⠋⣉⣠⣤⣤⠤⠤⣴⣤⣀⣟⣻⠻⣿⣿⣿⣿⣿⡦⠍⠺⢿⣿⣿⣧
⠀⠀⠀⠾⠿⠛⠛⠋⠉⠁⢠⢾⠃⠈⣁⣀⣠⣤⣤⣤⣤⣤⣏⣉⡗⢀⢸⡏⠉⣻⣿⣿⣷⡉⠳⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡟⠀⠉⢉⣁⣤⣤⣤⣭⣴⡿⢷⡄⠀⠛⠛⠁⢠⣾⣿⠟⠿⣶⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢡⡶⠶⠛⠉⢩⡿⢯⠻⢿⣿⣦⣉⠀⠀⠀⢠⣿⣿⠟⠃⡀⠠⣽⣿⣿⡽
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣼⣧⡰⣤⣸⣆⠀⠈⠃⠀⠁⠈⠋⠁⠀⠀⠘⣿⣷⡀⠀⣙⣷⣿⣿⡿⠁
⠀⠀⠀⠀⠀⢀⡼⠀⠀⢀⣴⣿⡿⠥⡈⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣸⢯⣤⣽⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⢧⡀⠳⣿⣿⣿⣋⣉⣓⠢⡄⠀⣿⠀⠀⠀⠀⣀⣴⡿⠇⠘⠿⠿⢫⣵⣶⡌⡏
⠀⠀⠀⠀⠀⠀⠀⠙⠲⠿⣿⣭⣍⣉⡉⠙⢿⠃⢋⠀⠀⢀⡼⠋⠁⠀⠀⢀⣠⣶⢾⣿⢟⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠠⠴⠖⠋⠀⠀⠊⠀⠀⠀⢀⣼⡿⢧⣌⡥⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣦⣀⣀⣀⠀⠀⠀⡀⣀⣀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢼⠢⣄⠀⠰⡖⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠉⠒⠼⠆⠀⠀⠉⠒⠢⠤⠤⠤⠤⠶⠒⠒⠚⠛⠋⠉⠉⢳⡀⣀⣀
⠀⠀⠀⢠⣄⣀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢅⠠⠼
⠀⠀⠀⠈⢧⣈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠈⣛⠶⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⡴⠂⠀⠀⢀⣀⠤⡴⠂⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⡳⢛⡫⠿⠉⠁⠀⠈⠉⠉⠉⠉⠁⠀⠸⣇⢶⠀⠀⢀⡤⠞⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢡⠖⠛⠓⢦⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠈⠈⠙⠚⢁⣤⣒⣲⠆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠏⠀⠀⠀⠀⣳⣀⠤⠤⣌⠓⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠛⠋⠉⠙⠿⡽⠉⠒⣤⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠣⡀⠀⠀⣀⢧⡀⠀⠉⢀⠇⠀⠀⠀⠀⠀⠀⠀⡠⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣨⠍⡉⠁⠀⠙⠣⠉⠁⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠣⠤⠆⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⢠⠀⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣂⠔⠋⠀⠀⠀⠀⢸⣉⡹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢄⣀⣀⣀⣠⠤⠖⠋⡙⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⣀⣀⡀⠀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠈⡏⡟⠉⠈⠉⠉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⡟⠗⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣉⡿⠗⢾⣦⣀⣀⡀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀⠀⠀⢀⣴⣾⣿⣿⣿⣶⣄⠀⢀⡀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣶⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠁⠘⠻⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠛⠋⢿⣿⣿⣿⣿⣿⡟⣿⣃⡀
⠀⠀⢀⣶⣾⣿⣿⣿⣿⣟⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣯⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀
⠀⣾⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡇
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡠⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠉⠛⠛⠉⠹⣿⣿⡿⣿⣿⡿⠁⠄⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠙⣿⣿⠿⠿⡿⠟⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢄⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠈⠳⢄⣀⡤⢯⠹⠎⡷⢜⡛⠖⢻⣤⠌⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠴⢾⡄⠀⣈⠝⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⢠⡿⣥⣈⣋⣉⣙⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢪⠂⠀⠂⠀⠀⠆⠀⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢂⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠠⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⣀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⣠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠻⣿⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢶⣻⣿⣿⣶⡄
⠀⠀⠀⠀⠀⣤⢠⡄⣀⢀⣀⡀⠀⣀⡀⡀⣀⢀⡀⠀⡄⠀⣀⡀⡀⢀⢀⣀⢠⣄⢀⣄⢀⣀⢀⡀⠀⠀⠀⠀⣰⡾⣩⡞⢁⣿⣿⡉⠟⠁
⠀⠀⠀⠀⠰⠟⠿⠻⠏⠾⠋⠻⠛⠿⠇⠿⠿⠾⠁⠰⠷⠾⠿⠇⠿⠻⡿⡿⠻⠻⠸⠧⠿⠿⠟⠁⠀⠀⠀⣼⡏⠀⣿⠀⠈⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⣿⠀⠀⣤⡽⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠦⠤⠤⠴⠤⡤⠄⠠⠶⠤⠤⠶⠤⠆⠰⠤⠴⠀⠴⠄⠲⠤⠤⠤⠄⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⢀⠿⠙⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⠀⠀⠀⠀⠀⠠⠀⠀⣁⣴⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⣀⣠⣴⡾⠛⠁⢞⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠉⠁⠘⠓⠒⠉⠉⠉⠁⠀⣤⣶⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣛⠻⠶⣶⣦⣤⣤⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠙⠛⠛⠛⠋⠉⠉⢁⡀⠠⢝⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠶⠶⠆⠰⠶⠶⠶⠶⠰⠆⠀⠀⠀⠀⠀⠀⣿⣯⣿⣤⣀⠀⠀⠀⠀⠀⠉⡉⣠⣴⠾⣻⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⠉⠻⠿⠛⠟⠻⠟⡻⠿⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⡷⠶⣤⣤⣤⣤⣄⣀⣀⣠⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⢠⣄⢠⣉⡍⠙⣫⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣶⠶⠿⢿⢿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠁⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣷⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠎⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⣿⢋⣉⣉⡀⠀⠀⠀⠶⢛⡂⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⡴⠟⣹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠋⠙⠀⠀⠐⠾⠿⠟⠗⠀⠀⠀⠀⢹⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠃⠀⠀⢠⣾⡿⣻⣿⡆⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠐⠒⠈⣠⣾⣿⢿⣷⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠠⠄⠰⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⢁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠂⠀⠀⠀⠈⠉⢈⣵⢟⣿⡆⠀⠀⠀⠀⠀⠀⢿⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⢀⣤⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣀⣠⠐⠂⠀⠉⣀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠤⠄⠚⠁⠀⠀⠀⠀⢀⡞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⠆⠐⠒⠒⠋⠁⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⣤⠀⠀⡀⠀⣠⠋⠀⠀⠀⠀⣀⣤⡴⠞⢻⠟⠟⠓⢤⣤⢀⣀⣠⠔⠚⠁⠀⠀⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⠀⠈⣁⢼⣿⡀⣀⣤⠴⠚⠉⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⣿⣿⡍⠉⠒⠢⢤⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠞⠛⠁⠀⠀⠁⢸⡟⢻⡍⠀⠀⠀⠀⠀⠀⠄⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⢻⡆⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡤⠶⠞⢋⠀⢸⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⠀⠀⠀⠀⠀⠀⠀⡀⠤⠋⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡀⠴⠛⠏⠐⠛⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠀⠀⠈⠉⠀⢀⣰⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⠆⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠶⠶⠶⠦⠴⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡄⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠎⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠈⢆⠀⠀⠀⠀⠏⠑⡄⠀⠀⠀⠀⠀⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⡂⠀⠇⠀⠀⡘⠀⢠⠁⠀⠀⠀⠀⢀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠒⠂⠐⠂⠠⢎⢀⡈⠀⠀⣰⠷⠀⠊⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⠚⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡴⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠒⠀⠉⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠠⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸
⠀⠀⠀⠀⡄⠀⢌⠢⠀⠀⠀⠀⠀⠀⠔⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠡⠌⠦⢕⠚⠂⠀⠀⠀⡌⠀⠀⠀⠀⠀⢠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠀⠈⠢⠐⠀⠀⠀⣀⡼⠀⠀⠀⣀⠠⠄⠚⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⣨⢃⡹⠊⠁⠀⠀⣀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢘⡠⢶⠟⢹⠀⠀⠄⠒⠈⠁⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡖⠀⠀⠀⠀⠀⣸⠂⠁⠄⢜⠁⠀⠀⠀⣀⣀⣀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢡⡶⢀⠀⠤⠞⢙⣈⣤⣠⢭⠉⠁⠒⠤⢠⢡⠀⠀⠤⠠⡔⠠⠀⡠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡼
⠈⢆⠀⠀⠀⠀⠀⣐⡏⡀⠘⠀⠀⠀⠀⣈⡀⠀⠀⠀⢀⠟⠄⢊⣀⠐⡈⠑⣀⠌⠀⢠⠂⢄⠀⢀⠂⠀⠀⠀⠀⠀⡇
⠀⠘⣆⠀⠀⠀⡰⢹⡠⠉⠙⠂⠨⣉⢡⠀⠈⠦⠒⡖⠁⠀⠀⠀⢠⠀⠀⠉⠓⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢄⠀⠀⠄⠃⢇⠀⠀⠀⠀⠀⡄⢰⡄⠳⠈⠀⠀⠀⠀⠀⠘⡅⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠇
⠀⠀⠀⠀⠒⠰⠀⠐⣚⣧⣐⣀⣀⣀⣈⠁⠈⠊⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⠀⠀⠀⠀⡘⠣⣶⣦⡀
⠠⡼⠓⠂⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⣸⡁⠠⠀⠀⠀⠀⠀⠀⢀⡿⢡⠌⢉⣙⠄⠠⠄⢀⠁⠘⠻⣿⡇
⠀⠒⠀⠂⣠⡴⡻⢏⠱⢄⠀⠀⠀⠀⠠⠀⢠⣀⣴⣦⣤⠈⠜⠛⠤⠀⢹⠇⠂⠀⠀⠈⠀⠀⠀⠈⢩⠀⢀⡀⠈⡁⠀⢈⣟⠁
⠀⣷⡄⠐⣿⡆⠤⠀⢈⡛⠀⠀⠀⡆⠀⢹⡜⣿⡇⠩⠀⠇⠀⠈⣦⣀⣸⡀⠀⠀⠀⢀⣆⡈⠛⣠⣱⣶⡖⢀⣀⣤⢶⣡⣾⣷⣤⣀⢀⠀⠱⣦⡀
⠀⣿⣿⣿⣷⡀⠄⡀⠙⠀⢀⠄⠈⡏⠉⠁⣧⠘⢅⠀⠰⣴⠊⠀⠀⠉⠏⢛⡳⠤⠖⢫⠘⣛⣟⡏⢻⠉⢹⣽⢛⣋⡒⠻⠿⠛⢛⠟⡀⠀⠀⠈⠁
⢠⣾⣿⡏⠞⠑⢤⡦⠒⢌⣑⠄⠀⢹⠀⠀⠈⢆⣸⣄⢰⣿⡆⠈⠄⠀⠂⢃⠀⡚⠀⢸⢨⣿⢹⠀⢉⠃⡸⣵⡿⣿⣿⣿⣷⣈⣡⣮⣶⠁⡄
⠘⢟⡂⠀⠀⠀⠀⠀⠈⢸⠋⠀⠀⠈⢦⠀⠀⠀⢿⣿⣷⣿⡿⠀⠀⠀⠀⢀⠃⢡⠀⠘⢸⡏⠀⠀⢸⠈⠘⡿⠭⠀⢐⡂⣿⣿⡏⢿⠇⠀⠁⠀⠐
⠀⠀⢁⡀⠀⠀⠀⠈⠂⢠⠇⣠⣶⣲⣾⣷⣦⣄⠈⠿⠛⢿⢿⡕⠀⠀⠐⢸⠀⡀⠀⠀⣼⣷⠀⣠⠣⠞⢠⡇⣀⣄⠀⠀⢻⣿⡟
⠀⢰⠎⠳⣖⡀⠠⠔⠀⣟⣼⠟⠁⠀⠉⠛⠛⢿⣦⠀⠀⡌⠀⠹⡄⠁⠀⣮⣳⣽⠷⢤⣿⣿⡖⠁⠀⠊⡌⠁⣈⡁⠀⣼⣿⠏⢀⠔
⠀⡀⠀⠀⠈⠻⣤⠀⠀⡞⠃⠐⣶⠀⠀⡀⠀⢈⣿⠀⡠⠁⠀⠀⠀⣢⢜⣛⠿⠷⠦⢤⣭⡻⡀⠰⢁⣰⣁⢦⣀⡤⢎⡷⣤⡶⣷⣤⣤⡄
⠀⢀⠀⡀⠀⠠⠟⠒⠊⣿⠀⠀⠀⠀⠐⠋⠁⣸⡇⠀⠀⣤⣶⣀⡞⣵⠟⢁⡒⠢⠀⠀⡀⠙⣷⠊⠉⡔⣱⠁⠀⡠⠊⣘⡭⠾⠟⠉⠉⣳⡀
⠀⠎⠘⢋⠀⠉⠀⠀⠀⡇⠈⣄⠀⠀⠀⠀⠀⠛⢙⠀⢸⣿⣿⣿⢶⠅⠀⠈⠁⠀⠠⢀⡉⢂⣼⠀⠀⠂⡏⠡⠊⠀⠀⣿⠀⠠⠄⠀⠨⠄⢹⡀
⠈⡆⠀⣉⣄⠐⠂⠀⢀⣷⠀⠪⠍⢉⠀⠀⠁⠀⠸⡄⣾⣿⣛⣫⡏⠀⠀⣠⣀⢤⠀⠀⠉⣼⠋⠀⠀⣸⠗⠀⠀⠀⠀⢹⠇⠀⠛⠀⠐⢆⠀⠂
⠀⠘⡀⠀⢸⣤⣤⣴⠟⠘⡷⠤⠤⠀⢸⡄⠀⠀⠀⠇⠈⢿⣿⣿⣷⠀⠀⣿⣿⡙⠀⠀⡴⣻⣿⠀⡄⡏⠀⠀⠀⢀⢀⡼⠓⢤⡀⠀⠶⢾⠁⣄⡀
⠀⠀⣿⡀⢻⡞⢰⠃⣆⣈⡻⢶⡋⢠⣾⣿⣦⣄⠀⠈⠀⠀⠙⣻⡟⠠⡀⠈⠉⢀⣠⣾⣿⡿⠛⣆⣅⣀⣀⡈⣠⢱⠞⠀⠀⠀⣷⡀⠀⠠⠴⠋⢿
⠀⠀⠏⠀⠀⢡⠀⠀⠀⠀⠀⠈⠙⠸⠛⠛⠛⠛⠛⠒⠀⠚⠉⠁⠁⠀⠈⠛⠛⠛⠁⠘⠁⠀⠀⠛⠛⠉⠉⠉⠐⠋⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⡾⠿⢿⣿⣿⢿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠫⣝⣿⡭⠶⣶⡿⠟⠻⠿⠿⢶⣭⣍⠛⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢟⣡⣔⡲⠛⠉⠓⠂⠀⠀⠐⠦⡀⠀⠶⢶⣬⣉⠁⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣴⣏⣈⣥⣦⣤⣤⡤⣴⡶⣮⣿⣛⢶⣦⡀⣤⡀⠈⠁⠠⡄⡙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⣷⣾⠿⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⡶⢀⠀⠈⢳⡜⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠞⢹⣿⡅⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠦⠈⠛⠛⠛⣿⣿⣿⣄⠀⣿⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣷⠹⠀⢼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⢰⣿⣿⠀⠀⣀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠸⣿⣿⣷⠀⣎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⣾⣿⡟⠠⠚⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠳⡄⠀⣧⢹⣿⣿⢻⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⠃⠀⠒⡉⣀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⡀⠁⠀⣼⣷⣿⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠻⡿⠀⠀⠁⡀⣀⠐⣀⣽⠀⠀⠀⢰⡋⡤⠐⠠⠐⢄⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⡐⠀⠀⠉⠐⠺⠿⠟⠋⠁⠀⠀⠀⠀⠳⢶⣿⡆⣀⣠⡄⠀⠀⠀⢻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠰⣼⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡍⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠞⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢴⠟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠈⣝⠳⠖⣿⠇⠀⠀⠀⠀⢀⢺⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠻⣷⣄⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⣰⣭⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠐⢀⡙⢷⣤⣀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢋⠀⠀⠀⠀⠈⢻⣄⠈⠙⠳⢶⣶⣶⣾⢿⡿⠋⡐⣹⣿⣯⣻⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⣆⡀⠀⠀⠀⠀⠀⠙⠳⠄⠀⠀⠀⠀⠀⠉⠀⠀⡿⢁⣠⣿⣟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡿⠿⠀⠀⠉⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠴⠛⠛⡿⢛⠁⢀⣿⡿⠹⡈⢿⣿⢻⠻⢿⣿
⣴⠾⠛⠉⠉⠉⠉⠙⠛⠛⠻⠶⠆⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⡶⠚⠛⠉⠀⠀⠀⠼⠋⠀⢃⣌⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⣭⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⠕⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠖⠋⠉⢀⣬⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠤⠄⠀⠀⠀⠀⠉⠉⠀⠀⢀⣠⠴⠚⠋⠁⢀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⠚⠉⠀⠀⠀⣠⣶⣿⣿⣿
⣦⣴⣦⣦⣶⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣯⣴⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣆⡠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⠤⠤⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠥⠟⢶⣉⣓⠻⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠭⣑⠢⣄⠀⠔⠊⠀⠐⠊⢭⡈⠳⡄⠈⠳⣄
⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡑⣄⠀⠀⠀⠀⠀⠈⠑⢄⠀⠳⡈⢆
⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡌⢆⠀⠀⠀⠀⠀⠀⠀⠱⡄⡃⢸
⢰⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⣄⠀⠀⠀⠀⠀⠀⠀⣠⢃⠏
⢘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠳⣄⠀⠀⠐⠒⠋⡜⢣⠋
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢤⣈⡳⡄⣆⡆⣼⡵⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢉⠟⠁
⠈⡆⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⠒
⠀⢣⠀⠀⠀⠀⡓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠘⡆⠀⠀⢸⣇⠀⠈⠉⠒⠒⠶⠶⠶⠖⠒⠚⠓⣤⣀⣀⠳⡀⠀⠀⡄⠀⠀⠀⠀⠀⡸
⠀⠀⠹⡀⢠⢸⣽⠖⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⣹⣿⡉⢱⡇⠀⠀⡀⠀⢠⠇
⠀⠀⠀⠱⡈⡆⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠁⣾⠁⢀⡼⠁⢠⠎
⠀⠀⠀⠀⠙⣧⢹⣙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡞⡜⣠⣾⠁⡴⠃
⠀⠀⠀⠀⠀⠘⢆⢫⠙⠲⠤⣄⣀⣀⣈⣁⣀⣀⡠⠤⠴⠒⠋⡿⣱⠕⢱⡧⠊
⠀⠀⠀⠀⠀⠀⠈⢦⢣⠀⠀⠀⠀⠀⢀⡝⠈⠂⣠⠠⣄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣰⣡⠈⠉⠉⠀⢤⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡹⠀⠀⠀⠀⠈⢆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⡏⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣿⡛⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⡗⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣾⠏⡁
⠀⣠⣴⣶⣶⣆⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠓⣠⡿⣼⡗⠀⠀⠆⠀⣀⣤⣤⣶⣶⣶⣶⣶⣤⣄⡀
⣾⢟⣹⡿⣻⣿⣾⣕⢌⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣀⠸⣧⣿⠆⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⣷⣿⣿⣿⣿⠟⡘⣿⣷⣅⢈⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠛⠀⢻⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡳⣄⠈⠻⣆⡀⠀⠀⠀⣸⠃⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣮⡳⣄⠹⡝⠶⣄⢠⠃⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⡷⢙⢿⣾⣧⡈⢆⠹⣯⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣮⣂⠽⣿⢡⠈⡧⠈⢷⡀⠀⢠⠛⣿⣿⣿⣿⣿⣿⣿⣿⢛⣟⣛⠛⠛⠻⠿⠋⠛⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣾⡟⡘⠀⣤⠀⠀⢻⣄⡜⠀⠸⣿⣿⣿⣿⣿⢿⣿⠋⠉⠉⠉⠛⠳⣄⠀⠀⠀⠀⠀⠉⢛⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⣾⢿⡿⠛⠀⠀⠀⠙⣧⡀⠀⣿⣿⠙⡿⠀⠀⣇⠐⠒⠶⡿⢲⣄⣿⠉⠛⣶⡤⠶⢤⣤⡀⠉⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⢳⡄⠀⠀⠀⠀⠈⢻⣄⢿⣿⣿⠁⠀⠀⠙⢦⣄⣒⣀⡾⡿⠃⠀⢠⣿⢤⣤⣀⠀⠙⢳⣌⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢿⡀⠀⠀⠀⠀⣠⡿⢻⠿⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠈⠉⠁⠀⢀⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⣧⠀⠀⠀⢨⣿⢀⡇⠀⢠⣿⠀⠀⢀⡀⠀⠘⠃⠦⠀⣀⡀⠀⠀⠙⠶⢤⡤⠖⠋⢸⢟⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⡋⠀⠀⠀⠈⠿⡀⠀⠚⠀⠀⣀⣀⣀⡀⠈⠁⠀⠀⣦⠀⠀⠀⢀⡴⠾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢀⣸⡁⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠠⣉⣙⠛⠒⠂⠀⠛⠀⣀⡾⠓⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣀⣠⣤⣾⡾⢛⡿⢻⣇⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠉⠁⠀⠀⣠⠤⠚⠉⠀⠀⠀⠀⠀⢀⢾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⣤⣾⣿⣿⡿⢻⠏⢠⡟⢀⢾⣿⣆⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⢘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣀⣠⣴⣿⣿⣿⣿⣯⣤⠟⠰⠿⣠⡟⢀⣿⣿⣆⠀⠀⠀⠐⠂⠈⣿⣷⣤⣤⣤⣤⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣽⣿⣿⣿⠿⠶⢤⣄⡙⠢⡀⠙⣤⡟⢀⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣏⠽⠦⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠓⠒⠲⠦⣤⣙⠦⢘⢷⡈⢦⡞⢹⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣾⣾⣿⣟⡭⠞⠓⠒⠒⠛⢧
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠋⠙⠛⠓⠦⣉⠓⠮⣟⠻⣿⣶⣟⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢰⣿⠗⠙⣉⡤⠖⠛⠛⠋⠁⠺
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣹⣋⠉⠙⠛⠶⠦⣬⣙⡒⠈⡱⢌⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣸⣇⣴⠟⠓⠚⣉⣉⣉⣉⣙⠛
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣻⣛⣿⣷⣶⣦⣤⣌⣉⣓⣴⣷⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣯⣴⣶⣛⣋⣉⣉⣉⣉⡉⢹
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡜⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣉⣉⣉⣉⣀⣀⣀⠈⠉⢙⡿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⢹⡆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⣿⣿⣿⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠉⢙⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠘⣧⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⡀⠀⢀⣠⠟⠙⠷⣄⣀⣀⣀⣀⣀⡴⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣤⡄⠀⠀⠀⢀⣤⣄⢰⡆⠀⢀⣤⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣤⣅⠁⠀⠀⠀⢀⣀⠐
⠀⠀⣰⣮⣿⣃⡀⠀⢰⣧⣯⣯⠞⠀⢠⣞⣯⣥⣄⡀⠀⢠⣽⣿⣇⠀⠀⣬⣿⣿⡀⠀⢠⣦⣯⣏⡀⠀⠀⣠⣻⣏⣻⠶
⠀⠀⠋⢹⣿⣿⣷⠀⠉⢹⣿⣿⠀⠀⠈⠉⣿⣿⣯⡿⠀⠛⠋⢉⣿⡷⠘⠛⣿⣿⡟⠂⠙⢹⣿⣿⣿⠀⠘⠛⢻⣿⣷
⠀⠀⠀⢸⣿⣿⠿⠀⠀⢸⡿⣿⣄⣠⡀⠀⣿⡿⣿⠁⠀⠀⣶⣴⣿⠀⠀⢠⣿⣿⡇⠀⠀⢸⣿⣿⠿⠀⠀⠀⢸⡿⣿
⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⡇⠸⠟⠋⠀⢠⣿⠀⣿⠀⠀⠀⣿⢸⣿⠀⠀⢸⡏⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⢀⣿⠃⣿⠆
⠀⠀⠀⢸⡿⡿⠀⠀⠀⢿⠇⠀⠀⠀⠠⡿⠃⠰⡿⠀⠀⠰⡿⠸⡿⠀⠀⢼⠇⢿⠇⠀⠀⠸⡿⣿⠀⠀⠀⠾⠋⠠⡿

⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡘⢿⠗⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⢀⣿⣻⣦⠄⠀⣤⣿⣿⣤⣀⠀⠀⢠⣼⣿⣧⣄⡀⠀⠀⣤⣿⣿⣤⣀⣉⡤⠀⠀⠀⠀⠀⢀⠀⣿⣿⣾⣇
⠀⠀⠸⣿⣿⣿⡀⠘⠋⢹⣿⣿⡛⠁⠀⠛⠉⣿⣿⣯⡟⠀⠈⠋⠹⣿⣿⡍⠉⠀⠀⠀⠀⠰⠚⠀⢰⠏⢹⣿⣿⡄
⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⣿⢻⣇⠀⠀⠀⠀⣼⡟⣿⠁⠀⠀⠀⠀⣿⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⣾⠏⣿⣧⣤⡄
⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⣿⠘⣿⠀⠀⠀⠀⣿⡇⢻⡇⠀⠀⠀⢰⣿⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠈⠉⠉⠁
⠀⠀⠀⠀⠿⠿⠇⠀⠀⠸⠏⠀⠻⠇⠀⠀⠀⠿⠀⠸⠷⠀⠀⠀⠸⠟⠀⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢔⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠁⠁
⠀⠀⠀⠀⣠⣤⠀⠄⠀⠀⠈⡛⠉⠀⣠⣦⡄⠀⠀⠀⠀⠀⡠⠒⠀⠀⠀⢈⠁⠀⠀⢠⣶⡄⠀⠀⠘⢷⡀⣾⣷⣄
⠀⠀⠀⢠⣽⣿⣷⣄⠀⠀⠈⠉⠛⢳⣾⣿⣆⠀⠀⠀⠀⣀⣀⢿⡿⠈⠉⠀⠀⠀⠀⣤⣝⣷⠀⠀⠀⠀⠙⣾⣷⡿
⠀⠀⠀⠛⢹⣿⣿⡉⠀⠀⠀⠀⠀⢸⣿⣿⢿⠀⠀⠀⣴⣿⢿⣀⣤⠀⠀⠀⠀⠀⢸⣿⠏⠉⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⢸⣿⣿⠙⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢠⡿⠁⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⣰⡿⢹⣧⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠛⠁⠀⠛⠁⠀⠀⠀⠀⠘⠋⠛⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠈⠛⠀⠀⠀⠀⠀⠛⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢁⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠀⠷⠀⠓⠴⠃⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⠔⠋⢁⠤⡄⢠⠈⠒⢄⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠏⠀⠀⡎⠀⡇⠈⢆⣀⣀⡑⣄⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⠋⣄⠀⠀⠓⠢⡱⣄⠀⠁⠀⡹⠈⡆⠀⠀⠀⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⠀⠈⠓⠤⠤⠚⠁⠀⠉⠒⠊⠁⠀⠸⡀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈⢇⠀⠸⡐⠻⠿⠟⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⡇⠀⠀⣀⠀⣀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠃⠀⢠⠎⠑⡼⠀⢀⡇⠸⣿⢀⠠⠀⠀⠀⠸⣿⠀⠀⠀⢀⠇⠀⡸⣡⠄⠀⢣⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠀⠀⡜⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠴⣊⡠⠔⠁⠻⠂⡠⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠠⡀⠀⠀⠀⡏⢿⡧⠀⠀⡀⠀⠀⠀⠀⠠⣿⠏⠀⠀⠀⢀⠏⠙⠉⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠤⠤⠜⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⢠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣥⠀⠀⠀⠀⢸⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⣀⡙⠤⠤⠤⠤⢛⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠸⣅⣀⣠⣄⠀⡨⠁⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠉⣀⡀⠉⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⢀⡀⠀⠀⠀⠰⠆⠀⠀⢀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣆⡤⠔⠒⢈⣽⣋⡤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠡⠞⠁⠀⠀⠈⠉⠀⠀⠀⢀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡼⠀⠀⠀⠀⠀⠀⠀⢛⣉⡠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠧⣀⢀⣴⠒⠒⠲⡍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⠀⠀⢰⠃⠀⠀⠠⠂⡜⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣰⡁⠀⣇⠀⠀⡠⠔⠓⠒⢤⠄⠒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠻⣿⠀⡎⠀⠀⠀⢀⣾⣄⣀⡠⠓⠒⠢⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢉⣉⠛⠟⠀⠑⠦⠘⠻⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠈⣀⡄⡀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠈⢀⠈⡧⡾⠉⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠸⣥⠀⠀⠀⡏⠉⠁⠛⠓⠒⠒⠒⠀⠀⠒⠀⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠸⡆⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⡠⠤⡀⠰⠒⠄⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢠⠀⠀⠀⡘⠁
⠀⠀⠀⠀⠀⢇⠀⠑⠸⡀⢰⠀⡰⢁⣍⡓⠦⣀⣤⣀⠀⡵⠠⣙⠇⠀⢀⠜
⠀⠀⠀⢰⠋⠛⢶⣄⡐⠧⢸⡰⢁⠂⠀⠉⠓⡞⠀⠈⠉⠳⡀⠙⢦⣀⣈⣣⣄⣀⠀⠀⠀⢀
⠀⠀⠀⠈⠑⠢⠤⠬⠥⠤⢄⡀⠚⡤⠀⠀⠘⠳⣤⡀⠀⢀⠳⠔⢣⡙⢹⢻⠁⠀⣷⡨⠭⠄⡁⠀⠀⠀⡴⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⣅⠀⠀⣠⠛⠉⢧⣠⠾⠠⠄⠀⣙⠣⠤⡆⡰⠁⠀⠀⠀⠀⠀⠀⡰⠁⡰⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⢄⡨⢶⡊⠫⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⡠⠄⠲⠊⣠⠴⠓⠉⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣽⠖⠦⣬⣉⣐⣀⠀⠀⠤⠄⠀⠐⢂⣹⠀⣀⡼⢃⠴⢤⡐⠶⠶⠦⠖⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⠀⠶⠀⢈⡁⢽⡏⢉⡉⠻⠟⢉⣉⡟⠸⣀⡠⠖⠀⠀⠈⠓⠤⣀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣥⠀⠛⠁⣤⡄⠘⠃⣠⡄⠘⠋⢠⣤⠘⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠶⢆⣉⡁⣼⡟⠛⣟⣻⣿⣉⠩⠴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣒⠒⢒⣒⡀⣴⠛⡿⠭⠭⠭⢥⣏⣿⠠⠤⠔⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠉⠉⠉⠉⠉⢹⣂⠊⠉⠉⠩⢤⠖⠃
⠀⠀⣀⣀⣀⣀⣠⠄⠴⡒⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠞⠋⠀⠀⠀⡒⠶⠒⣲⣄⣀⣀⡀
⠴⡭⠥⠀⠀⠈⠭⠭⠙⠴⠤⣤⠤⠤⣤⣤⣀⣀⠀⡀⠀⠀⢀⣠⣤⠶⡛⣩⣩⣁⣀⣀⣀⣀⡽⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠀⠈⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⣀⡀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⠊⠁⠀⠀⠀⠀⠀⠀⠈⢻⣟⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⠁⠀⠀⠀⢸⡆⠀⠀⠀⢸⡆⠀⠛⢈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⡄⠀⠀⠀⠈⠁⢀⣄⠀⠈⠁⠀⠀⡞⢹⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠘⢦⡀⠀⠀⠀⠈⠈⠀⠀⣤⠀⣼⡀⠸⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣈⠓⠤⠤⡴⢊⣭⢾⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡒⠤⠶⠉⠀⠈⢻⣿⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⢀⣴⠒⠢⣀⣤⠇⠀⠈⠦⢄⠖⢢⡙⢿⣿⡛⠿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢀⣠⠏⠁⢉⠙⢛⡏⠀⠑⢄⠈⠳⣄⠀⠀⣰⠋⣠⠏⠀⠀⠙⠻⠀⣀⠈⠻⣿⣯⠻
⠀⠀⠀⠀⢾⢀⢠⠀⠸⣠⣰⢷⠀⠀⠀⠑⢄⠈⠳⣔⠃⡰⠃⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⢿⡇
⠀⠀⠀⢸⠈⣯⣆⢧⣀⣹⣧⠜⠀⠀⠀⠀⠀⠑⠄⠈⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⡃
⠀⠀⠀⠸⠀⠈⠙⠯⠭⣎⢹⠘⢧⠀⠀⠀⠀⠀⠈⢂⠈⢇⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⢣⠀⠀⠀⢠⢏⡎⠀⠈⣇⠀⠀⠀⠀⠀⠀⡀⠸⡄⠀⠀⢠⠇⠈⠳⣄⠀⠀⠀⠀⠀⢨⡆
⠀⠀⠀⠀⠀⠓⠤⠼⠷⠋⠀⠀⠀⢸⡆⠀⠀⡀⠀⠀⢁⣀⡇⠀⠀⢸⡀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠻⠿⠿⠿⠿⠿⣿⣿⣿⣳⣶⣶⣿⣷⠀⡾⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⣀⣀⣀⣀⣤⣶⠟⢉⡌⢸⡇⠀⠁⠀⠀⠙⠿⢳⠶⠤⣀⡀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠈⠛⢿⣿⣿⠁⠀⡞⢆⢸⡇⢠⠄⠤⠤⠤⠶⠋⠀⡜⠁⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢩⠟⠉⠒⠢⠤⣀⢀⡼⠃⠀⡼⠹⡄⢻⣇⢸⡆⠀⠀⠀⠀⢀⣸⠓⠂
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠁⠀⠉⣿⣿⢻⡽⡰⠯⠭⠉⠉⠤⠒⡎
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡙⢿⡆⠟⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⢤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣤⣀⣀⣀⣠⠴⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣛⣻⣛⣛⣙⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠉⠉⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣤⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⠄
⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠆⠀⠀⠀⠀⠠⠐⠋⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣀⠀⠀⢀⡄⡀⠀⠠⠀⢠
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⡤⠚⠁⠚⠀⠀⠀⠈⠁⠀⠀⣬⠃
⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠈⠃⠀⠗⠁⠀⠋⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠂⠀⠀⠀⡀⠀⠛⠢⠄⠀⠀⠀⠀⠴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣁⣡⠄⡀⠈⠢⠄⡀⠀⠀⠀⣀⠐⠁⠀⠀⠀⠒⠂⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠉⠁⠠⡄⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠖⠢⠤⠠⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⡐⠀⠀⠐⠀⠀⠀⠈⠀⠐⢂⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠤⠄⠒⠂⠈⠉⠀⠀⠈⠉⠉⠉⠉⠉⠑⠒⠀⠀⠄⠠⠀⠐⠂⠀⠀⠉⠉⠁⠀⠈⠉⠉⠀⠀⠀⠉⠒⠒⠒⠢⡀
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡏⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢼
⠀⠀⠀⠀⣷⡄⠐⠢⠤⠤⠤⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⠁⠀⠀⠀⢨
⠀⠀⠀⠀⢣⠑⠤⢀⣀⣀⣀⠀⣀⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠀⠀⣀⠤⠖⠂⠉⠉⢉⠏
⠀⠀⠀⠀⠘⢮⢢⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠐⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠒⠒⠂⠀⠒⠊⠁⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠈⠙⢮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢀⡼⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⣀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣀⣸⣦⣀⣰⣄
⠀⠀⢸⠁⢸⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣅
⠀⠀⡯⠤⡾⠀⡴⡆⠀⠀⡏⠀⢸⠀⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠠⠃⠘⠓⠇⠀⢀⡇⠀⣼⢠⣿⣿⣿⣿⣧⣿⣿⣯⣾⣿⣯⣯⣽⣿⣿⣿⣿⣿⣷⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠁⡟⣾⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⣰⠃⣠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡟⠘⣿⣿⣿⣿⣿⣏⠻⣿⣇⡿⢿⣶⡮⠙⢿⣿⣿⢸⣿⠧⢠⠃⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠃⠀⠸⣿⣿⠯⣭⣿⡆⠈⠛⠓⣚⡯⠖⠀⢸⣿⣇⣼⣿⠀⠋⢀⡄⠀⠀⡼⠀⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⢻⣿⡶⠲⢾⠃⠀⡠⠀⠉⠀⠀⠀⣼⣿⣿⣿⣿⠷⠴⠣⠇⠀⣰⠁⣸⠁
⠀⠀⢀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠖⢹⣿⡟⠀⠘⠶⡀⢈⣁⣀⡀⠀⢀⡟⣿⣿⣿⣿⠇⠀⠀⠀⠀⠁⠹⠃⣠⡀⠀⢀⡞⡀
⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠐⠿⢿⣷⣄⠠⣶⣶⣭⣾⡿⠃⢀⡞⡀⢸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⠙⠁⢀⡞⣰⠃
⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠸⢿⣿⣷⣬⡉⠟⢁⣀⡴⠋⠀⠁⠀⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢱⣃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠲⢶⣤⠴⠶⠶⢒⣚⡟⠀⠉⢟⠉⠁⠀⠀⠀⠀⠀⣠⠤⠵⠶⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠈⠻⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠉⠀⠀⠀⠀⠀⠁⠈⠳⡀⠀⢠⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⡀⠀⠀⠀⠀⠀⠈
⠱⣶⡦⠄⠤⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠈⠛⢷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢣⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣤⣤⡄⠀⠀⠀⣠⢞⠃⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠁⠀⠀⠙⢛⣁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⣤⣈⣉⣉⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢁⣀⢨⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡅⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠐⡄⠠⡀⠁
⠀⠐⠠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢄⠈⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠄
⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠂⢁⡠⠜⠀⠀⠀⠠⠁⡰⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠂⠤⠐⠈⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠇⠀⠠
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄⠀⢀⠀⡈⠀⠀⠀⠀⠀⠀⢰⠠⢀⠌⠀⠀⠀⠀⠀⠀⠀⡇⡠⠂
⠐⠀⠁⠀⠌⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⡇⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠉
⢠⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⢤⠊
⠀⠀⠐⠁⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠄⠒⣆⠁⠀⢀⠀⠀⠀⠒⠂⠀⠀⠀⠐⠒⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠔⠲⠏⠉⠉⠍⠐⢢⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠁⡀⠀⠀⠗⠀⠀⠂⠀⡇⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⣀⣠⣀⣰⣆⣀⣰⣆⣨⡀⡇⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡊⠀⢠⡀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠑⡆⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠈⠳⣄⠀⠉⠀⠀⠀⠀⠀⠈⠉⢀⣠⠞⠑⢆⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡆⠀⠀⠈⠳⣤⣀⠀⢀⣀⣤⠴⠚⠋⠀⠀⠀⠸⡄⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠉⢙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⣠⠞⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⢸⠀⠀⢸⠀⢀⠞⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠘⡄⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢸⠀⠀⡘⢠⣿⣿⣿⣿⣿⣿⣿⣆⠰⡄⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠀⠀⡇⢸⣿⡿⠛⠛⠋⠙⢿⣿⠀⡇⠀⠀⠀⠀⠀⡇⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢸⠀⠀⢻⠸⡿⠶⠶⠶⠶⠶⠾⡿⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠀⠀⠘⡄⠻⣄⠀⠀⠀⠀⡴⠃⡀⠀⠀⠘⡆⠀⠀⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠈⠛⠶⠶⠾⠗⠋⠀⠀⠀⢀⠇⠀⢠⡇⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣤⡴⠃⠀⠀⠚⠀⠀⡾⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠉⠉⠈⠳⣄⠀⠀⠀⢀⡼⠃⠀⠀⠀⡇⠀⠀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡼⢆⡀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠙⠶⠤⠶⢻⠀⠀⠀⠀⠀⡇⠀⠀⠈⢆⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠃⠈⠙⠲⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢹⠀⠀⠀⢸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⢸⡄⠀⢀⡞⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠸⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣧⠔⠉⠀⢀⡞
⠀⡠⠒⠂⠒⠒⠒⠀⠀⢼⠁⠀⠀⠀⠀⣇⢸⠀⠀⢰⡄⠀⠀⠠⠶⠂⠀⠀⠘⡄⠀⠀⣧⠀⠀⠀⠀⠀⠸⡦⠤⠖⠛⠉⣉⠱⡄
⡜⢀⠎⠀⡰⠒⠁⠀⢠⡏⠀⠀⠀⠀⠀⢸⣼⠀⠀⠀⣧⠀⠀⠀⠀⡇⠀⠀⠀⠹⡄⠀⡟⠀⠀⠀⠀⠀⠀⠘⢦⣀⠐⡄⠈⢧⡱
⠙⠺⢤⣤⣇⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢹⠀⠀⠀⠀⣧⠀⠀⠀⠀⠹⣆⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢾⡀
⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⢀⣠⠤⠤⠏⠀⠀⠀⠀⢸⣦⣤⣤⣤⣷⠤⠀⠀⠀⠀⠈⠛⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙
⠀⠀⠸⣄⠀⠀⠀⠀⠀⣀⡴⠋⡀⠀⠀⢀⠀⠀⠀⢀⠈⠳⡄⠀⡞⠁⠀⢰⠂⠀⠀⠀⡀⠀⠀⡄⠘⣆⠀⠀⠀⠀⠀⣀⣠⡴⠟
⠀⠀⠀⠈⠛⠛⠚⠛⠛⠙⠦⢼⣃⠀⠀⢸⡀⠀⠀⢸⣄⣀⣿⠀⠳⠤⠴⢿⣄⣀⣀⣀⣇⣀⣀⡽⠒⠋⠙⠛⠛⠛⠛⠉⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣷⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠬⠷⠘⠙⠚⠉⠛⠒⠒⠢⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡎⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠈⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛
⠀⠀⠀⠀⠀⢰⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⢀⣐⣂⣀⡀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⣇⣤⠀⠀⢀⡀⠀⢸⢇⢖⡴⠋⠀⠋⠁⠀⠀⠀⠈⢳⡀⠀⠀⠀⣀⣈⣸
⠀⠀⠀⠀⠀⠁⠀⠀⣠⠌⢯⡑⠒⠈⠋⠀⠀⢀⣠⠤⠤⠄⠀⠀⠀⣧⠀⠀⡾⠠⠢⠙⡄
⠀⠀⠀⠀⠀⠀⢰⢸⡇⠉⢉⣀⡀⠀⠀⠀⠞⢉⡠⠔⢺⡍⠃⠀⠀⠸⡆⢀⢧⠐⣲⠄⡇
⠀⠀⠀⠀⠀⠀⠀⢳⡅⢼⣻⡉⢭⡀⠀⠀⠀⠉⠻⣿⣿⠇⠆⠀⠀⠀⠙⠦⠞⠸⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡉⠛⠛⢁⣤⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡠⠀⢏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠠⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠙⠛⠃⠀⠀⠀⢀⡴⠁⠀⠀⠀⣠⠎⡇⠀⠐⠒⠒⠋⠉⠉⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠂⠑⣄⠀⠠⢄⣀⡠⠔⠉⠀⠀⠀⣠⠞⠡⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣
⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⠑⢦⣀⡀⠒⠂⠀⣀⠴⠋⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⡀
⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⡉⠲⠙⠀⠄⠠⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠐⢹⢦⡀
⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠢⠑⣄
⠀⠀⠀⠀⢠⠇⠀⡀⠀⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⡴⡡⠒⠀⠁⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣠⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⡜⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢩⡝⡂⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⡗⠾⠶⠖⠋⠀⠀⠀⠀⠀⠀⠈⢆
⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⡣⡀
⠀⡄⠀⠀⠀⠀⠀⠀⣀⣠⡭⠴⢂⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡽
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⡈⡎
⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡈⢃⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠠⠀⡰⠁
⠀⠀⠑⢄⠀⠀⡀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢱⡀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⢤⡀⠀⠀⠀⠒⠒⣒⡒⠒⠪⠍⠀⠀⠀⠀⢧⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣸⠁⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠀⠀⠀⢌⣂⢪⠇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡠⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠒⠒⠊⠉⠀⠀⠁⠀⠀⠀⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⢂⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡞⡩⢙⡄⠀⠀⢀⣀⠤⠚⠡⠀⢀⡡⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⠀⠈⠉⠉⠒⠂⠤⠤⠄⢀⠳⢬⡞⠡⠔⠚⠉⠄⠀⢈⠤⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠄⣀⠀⠂⠀⠀⠀⠀⠀⠀⡆⠀⢀⣀⠥⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠐⠒⠒⠊⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢌⡉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠀⠤⠄⠤⠤⠤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠉⠒⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢪⢋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉⠁⠀⡴⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⢀⠎⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠎⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⡆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣀⢀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠭⡯⠯⠛⣛⣒⣒⣒⣋⣉⣟⡛⠒⠒⣒⣓⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠁⠐⠒⠒⠀⠀⠤⠤⣷⠤⠭⠤⠤⠤⢼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠦⠬⠍⢁⣀⡐⠒⠒⠒⠘⠛⢲⡶⢾⣋⣉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡑⠒⠤⠤⠤⠤⠄⠀⣀⣀⣀⣈⡇⢠⣄⠤⢼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠒⢲⢲⠒⠒⠒⠒⠒⠒⡇⠀⠙⣏⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸⠸⡀⠀⠀⠀⠀⢰⠃⠀⠀⢥⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⡇⡇⠀⠀⠀⠀⠈⠓⠒⠋⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣉⠓⠠⠍⠘⠛⣟⠒⠟⠁⠀⠀⢀⣠⠴⠚⠉⣆⡀
⠀⠀⠀⠀⠀⠀⢀⡔⡫⠛⠉⣩⠭⠿⣖⡦⠤⢀⣀⠈⢹⠒⠒⠒⠉⠉⠁⠀⣀⣠⢤⡶⠯⠭⢬⡑⢄
⠀⠀⠀⠀⠀⠀⢸⡚⠦⠾⠥⠧⠤⠤⠬⠅⠂⠨⠶⠾⢭⡯⠽⠯⠽⠿⠭⠭⠤⣈⣉⣸⣓⣀⣀⣙⣼
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈⠑⠂⠉⠉⠉⠁⠐⠒⠒⠤⠤⠤⠭⠥⠴⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡴⠗⠀⠀⠀⠰⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠀⠀⠀⠘⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠂⠀⠀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠁
⠀⠀⠀⠐⠀⠀⠠⠆⠀⠆⠆⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠶⠀⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⡦
⠀⠠⢤⠀⠀⠀⠀⠠⠊⠑⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠀⠠⠊⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠈⢢⢀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠂⠉⠑⠂⠤⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⣀⠤⠊
⠀⠀⠀⠀⢄⠉⠀⠂⠀⠂⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠈⠹⢅⠈⠁⠈⡁⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠤⠀⢢⡀


⠀⠀⠀⠀⠀⢠⣤⠤⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⡀⠀⡀⠈⡑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠈⠁
⠀⠀⡠⠄⠀⠉⠉⠀⠀⠀⠀⠉⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡑⣠⣤⣤
⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢀⠄⠀⠀⠀⠀⠤⠄⠀⡈⠻⣿⠟⢡⡙⠻⠛⠀⠀⠐⠒
⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠁⠈⠁⠀⠀⠀⠀⠀⠈⡄⠀⠈⠀⠀⠂⠁⠈⠁⠈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠀⠤⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⢠⣄
⠀⠀⠀⢤⣤⣤⣀⡀⠀⠀⠀⣀⣠⣤⠶⡦⠀⠀⠀⠀⠀⣠⠴⠞⠋⢠⠜⠓⠦⠄⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠐⡀⠈⠛⠛⠛⠿⡛⠻⠁⠀⢀⠇⠀⠀⠀⠀⠸⡀⠐⠲⠂⠀⡀⠒⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠘⣠⡀
⠀⠀⠀⠀⠉⠂⠀⠐⠊⠀⠈⠂⠀⠐⠁⠀⠀⠀⠀⠀⠀⠑⠤⠀⠀⠊⠁⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡰⢀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⡀⠄⢰⠀⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠈
⠀⠀⠀⠀⡰⠯⠤⠤⠘⠀⠔⠒⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡀⠇⠄⠄⠒⠒⠒⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠉⠉⢉⣉⣥⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⣿⣦⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⠀⠛⠛⢻⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠅⠀⠀⠀⠀⢰⣣⡄⠸⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢸⣿⡉⣧⠀⠀⠀⠠⠀⠀⡀⢀⣀⠠⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠱⠴⠖⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢻⡁⠜⢦⡀⠀⠀⣄⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣎⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠈⠳⢶⡄⢙⢦⣀⠁⠙⡶⠾⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠐⡨⡹⡆
⠀⠀⠀⠀⠀⠀⠙⣮⣠⣌⣷⠞⠀⠀⠀⠠⣉⢻⣶⠒⠒⣀⣤⣶⢋⠁⠁⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠂⢠⢈⠄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠟⠓⠒⠦⠀⠀⠀⠀⠙⢿⣷⣛⣁⡘⣿⠀⣴⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣍⡀⠀⠰⠶⢷
⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣷⣤⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣦⡂⠨⠘⡀
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠏⠉⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡹⣿⣿⡟⡁⠁⠠⠘⡄
⠀⠀⠀⠀⠀⡠⠃⠀⠀⢄⢀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣆⠀⠀⠘⠘⡄
⠀⠀⠀⠀⠘⢁⣀⠀⠀⠘⢾⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠰⠱⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠙⣦⡠⠌⠐⠙⣄
⠀⠀⠀⡰⣒⣲⣤⡑⣄⠀⣼⣠⠿⠶⠶⠖⠓⠀⠀⠀⠀⠀⠀⡇⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣸⠿⠳⢶⢦⢸
⠀⠀⢀⠄⠀⠀⠈⠙⢾⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⡖⠑⡈⠠⠄⡡⠃
⠀⠀⠈⣶⠞⠛⠛⠓⢲⡏⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣰⠫⠔⠊⠨⡤⠊⠀⠀⠀⠀⢀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠻⣷⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠹⡇⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⣀⠀⡇⠀⢰⠀⠀⠀⠀⠀⠀⣾⣯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣶⠇⠀⢸⠀⠀⠀⠀⠀⠀⠙⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢉⠟⡀⠀⠸⡄⠀⠀⠀⠀⠀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠓⠲⢦⣄⣀⣤⠞⠋⠉⠉⠁⠘⠦⣤⡤⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢀⡀⠀⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⢆⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠞⠁⠋⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣄⣀⣀⣀⣀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⢠⣤⡤⠴⠶⠲⡆
⠀⠀⢸⣇⣀⣷⠀⠀⠲⣦⡴⠲⣦⡴⢦⡀⠀⠘⢻⡇⠀⠀⠀⣠⢤⣀⣀⣄⣀⣀⣀⣠⣀⣀⣀⠀⢠⡤⣄⠀⢀⣠⣤⣠⠄⣀⣤⣄⣤⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⢻⡏⠉⠿⢀⡄⠀⣿⠀⠀⣿⠀⠀⢻⡄⠀⢸⡷⠶⢦⡀⣁⣤⣿⠈⣿⠁⠈⠋⢸⡟⠁⠙⠁⢈⣠⣼⡇⣿⢤⣤⣿⠘⡧⣤⣼⡇⣴⠋⠛⣦⣴⠋⠙⣿
⠀⠠⠼⠷⠤⢤⣼⠇⣠⣿⣄⢀⣿⣄⠀⣈⣧⣀⢸⣁⠀⣸⢿⣁⢀⣿⣀⣿⣀⠀⠀⣈⣇⡀⠀⠀⣏⡀⣸⡃⡿⠦⠤⠟⢸⡷⠤⠴⠋⢿⡋⠙⡛⣧⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠈⠉⠉⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣴⣻⢿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣵⠟⠋⠀⠈⢧⠱⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠐⠉⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢎⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣻⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠋⠀⠀⠀⣀⣠⣶⠶⠶⠐⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠲⠟⢋⠜⠁⠀⠀⠀⠀⣰⠧⠹⣿⠀⠀⠀⠀⢀⡜⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣆⠀⠰⠋⠀⠀⠀⠀⢀⡴⠋⠀⠒⠀⢱⣄⠤⣴⠃⠀⠄⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠷⠀⠀⠀⠀⠀⠀⢸⠇⢀⡤⠶⠿⠋⠀⠀⠻⠷⣄⡀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠂⣿⠀⣠⠶⣶⢶⡀⠀⠀⢀⡤⠦⣿⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⣼⡇⣸⠁⢸⣿⠇⣇⠀⠀⣿⣷⡆⠀⢹⡄⠀⠐⣠⠄⠀⠀⠀⠀⠀⠀⠘⣦⣌⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⢠⣿⣿⢿⡗⢮⣍⣴⣿⡆⠀⡙⠿⠁⢀⡾⠁⠐⢃⢻⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⠄⠀⠀⠀⢸⠇⠙⢦⡈⢠⣾⠿⠟⢷⡀⠛⠒⠚⠋⠁⠀⢀⠞⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣆⠀⠀⢸⠀⢀⠈⠓⠾⢇⠀⢀⡀⢹⣦⡀⠀⠀⢀⡤⠁⠀⢰⡄⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢦⠁⠈⡇⠈⠻⣶⣦⣤⣍⣛⣋⠉⠀⠈⠛⠚⡉⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⣰⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠷⣄⠸⣆⠀⠘⢷⡀⠉⠉⠉⠁⣒⣲⠶⠊⠉⠀⠈⠁⠀⠀⠀⠀⠀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣿⣧⣿⣆⠀⠀⠛⠶⠦⠤⠞⠋⠁⠀⢶⡀⠓⠛⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣬⡉⠙⣿⣷⣄⡀⠀⠀⠀⠀⠀⣀⣤⣾⠿⣷⠶⣾⢿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠿⠋⠉⠉⠙⢿⣷⣈⠹⣟⣿⡷⡶⢶⣿⣿⡿⠋⠁⠀⣿⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠈⣿⣿⣧⡙⢿⡟⠛⠋⠉⠁⠀⠀⠀⢰⡟⢁⣠⣴⣶⡷⠷⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠁⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣤⣤⣶⡶⠷⣾⣿⣿⡿⣿⡄⢀⠀⠀⠀⡀⠀⢠⣿⣿⣿⣟⡟⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⠞⠋⠀⠀⠀⠀⠀⣤⡙⢻⣍⣹⣇⠈⠀⠀⠂⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⢀⠰⠦⡀⠙⢦⣿⡟⠙⠛⠿⣷⡀⢀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠠⣤⣤⣤⣄⡉⠳⣄⠈⢂⠈⣿⡛⠃⠀⠀⠈⢷⣾⣿⣿⣿⡿⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⣿⠉⣿⣷⣮⣷⣤⣶⣍⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⢰⣿⠁⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⠀⠀⣸⡿⣠⣿⣿⡏⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⣾⣿⣿⠇⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢸⡃⣿⣿⡇⠁⠀⠸⣿⣿⣷⣄⠀⠀⠀⠀⢹⣿⡏⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⢸⡇⣿⣿⡇⠀⠀⠘⢻⣿⣦⠻⣷⠀⠀⠀⠀⠹⣿⠂⠀⢿⡏⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢸⡷⠀⠀⠀⠀⠀⢸⡇⢿⣿⣧⠀⠀⠀⠀⣿⠁⠀⠹⣆⠀⠀⠀⠀⠙⣿⡁⣼⡇⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠘⠃⡀⢿⣿⡄⡀⠀⠀⠙⡄⠀⣠⣿⡀⠀⠀⠀⠀⠙⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣤⣄⠀⢣⣴⡟⠹⣿⡀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⢻⣿⣿⣿⣿⣶⣾⣭⣘⣦⢼⣧⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣸⣿⣿⠃⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⢹⣤⣀⢀⣠⣾⣿⠟⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⣷⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠉⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣄⠀⠀⠀⠀⠀⢀⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⣿⣿⠿⣿⠿⢶⣄⡀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠁⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠠⡄⣤⣀⣴⣶⣤⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⠧⡇⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠻⣿⣿⣿⣿⣛⡞⠁⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢤⡞⠉⠓⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠙⢄⠀⠀⠈⠻⣿⠟⠉⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣦⡀⠀⠀⠑⢄⣀⣀⣀⣀⣤⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣷⠠⡏⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⡛⠉⢉⣀⡀⢸⣿⣿⡆⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⢄⡀⣠⠀⢀⣀⡀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠠⠩⠀⠀⠀⠉⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣆⡀⠀⠢⠤⠞⢀⣠⢎⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣎⠛⢿⡋⠉⢻⡟⠁⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠹⡅⠀⣷⠀⡼⠀⢰⠿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⣼⠉⡇⠀⡟⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡟⠀⡟⠀⡇⠀⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢀⡇⢠⡇⠀⠃⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠸⠃⢸⠁⢸⡇⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢀⡏⠀⠘⡇⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠞⠀⠀⠀⠳⠤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣶⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢿⣿⣿⣿⡄⠀⠀⠉⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⢠⣶⡄⠀⠉⠀⢀⣴⣤⡀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣇⠀⣠⣴⣿⣿⡟⠀⠀⠀⠸⣿⣿⣷⣶⣴⣶⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠸⢿⣿⠿⢋⡀⠀⠀⠀⠀⢈⠛⠿⠟⠛⠃⠀⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⣿⡿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣧⣸⢻⡄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣎⠿⡇⠀⠀⠀⡀⠀⠻⣦⡤⠖⠀⠀⠀⣈⣷⣤⡀⣿⢣⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣷⢀⡤⠾⣧⣤⣤⣤⣤⡤⠶⠚⣋⣽⡏⠈⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢿⣶⣤⣤⣤⣴⣶⠿⣿⡟⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠘⢷⣄⣉⣉⣀⣀⣤⠿⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠉⠉⠉⠉⠀⠀⢀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠦⣤⣀⣀⣀⣀⣤⠴⠋⢹⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⢾⡆⠀⠀⠀⠁⠀⠀⠀⠀⢸⣅⣩⠍⠉⠛⠳⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠰⢾⣇⡀⠀⠀⠀⠀⣀⣤⣖⣋⣭⣤⠤⠶⠶⠚⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠦⢤⣤⣤⣤⣈⣙⣷⣤⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⢀⡴⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⢸⡇⢰⡇⠀⠀⠀⠀⠉⠤⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠙⢚⠇⠀⠀⠀⢸⣿⣦⡀⠀⠀⠀⢸⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣤⠆⠃⠀⠀⠀⢸⣿⠻⣿⣦⣀⠀⢸⠀⡈⠀⣠⡆⠀⠀⠀⠀⠀⣀⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠈⠻⢿⣿⣶⣶⣶⡾⢿⣿⣤⣤⣴⣶⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⢀⣠⣿⠇⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠉⠉⠉⢠⣿⡇⠀⢰⡖⠛⠉⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠻⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⠴⠒⠚⠛⠷⢤⣀⣧⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⢹⣷⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡞⠉⢷⡾⡁⢀⣤⣤⣄⡀⢠⣟⢹⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣉⣁⣀⣠⡾⠿⠶⠶⢤⣸⣿⣝⢦⡀⠀⢠⡇⠀⠈⢿⣢⡟⢩⡴⠛⠛⠉⠉⣿⣇⠀⢸⣄⣀
⠀⠀⣀⣀⣄⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⢀⣀⠀⠀⠈⢻⣿⠀⠹⣆⣸⠁⠀⠀⠘⣿⣧⡀⣷⣠⡤⠀⠀⡿⣿⠖⠚⣇⡀
⠀⠀⢸⣃⢈⠄⠀⠀⠀⠀⢀⣿⣿⢤⡾⠛⠛⠻⣦⠀⠀⢻⣔⡀⠈⠛⠀⢠⡆⠀⠸⣎⠉⠛⠉⠀⢀⡼⢃⣽⣀⣤⣽⠛⠛⠓⠂
⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣼⣿⣿⣿⡁⠀⠀⠀⠘⡇⠀⢸⡯⣿⣄⠀⠀⢸⢿⡄⠀⢹⣶⣶⣶⠾⠛⣽⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⢷⣄⣀⣠⠾⠁⠀⣸⣿⣿⠙⢷⣤⡿⠀⠿⠛⠛⠁⠀⢄⠀⠀⠹⣄⠀⠀⠀⠀⡐⠀⠆
⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⡴⠛⣿⣿⣷⠈⠉⠁⠀⢀⣴⣿⠿⠟⠃⠈⠉⠀⠀⠀⠀⠀⢀⠵⣎⠻⣍⠉⠉⠁⠀⠀⠀⠀⠶⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠋⣠⠞⠁⠙⢻⡿⠶⠶⠿⠛⠙⢿⣆⠀⠀⠀⠀⣠⡤⠤⠤⣄⢸⠀⠘⢦⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢠⠞⠁⣀⣠⣄⡀⠙⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠖⠀⠀⠀⠀⠀⠀⣿⢰⠏⠀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢀⠝⠀⠀⠀⠀⠀⢹⡏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠻⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣦⣤⣄⡀⠀⠀⠀⠀⣀⣤⢴⣶⡶⣤⡤⡀⠀⠀⠀⣀⣀⡠⢴⡊⠠⡁⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠾⠋⠉⠙⣛⠛⠒⢀⡾⠉⠙⠛⠻⢿⣶⠂⡴⠒⢋⣐⠛⡉⢘⣋⠀⢸⠰⠟⠛⠋⠉⠀⠀⠀⠀⠱⠀⠀⠀⠉⠐⠢⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠠⢄⠉⡞⠂⠀⠀⠀⠈⠉⠀⠈⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠦⢤⣄⡀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠂⢼⠃⠀⢀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⣠⣤⣤⠤⡀⠀⢀⡶⣦⣤⣀⣀⠈
⠀⠀⠀⠀⠀⡐⠤⢤⣤⣀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⠿⢟⡛⠉⠐⠂⠀⢄⡾⠁⣀⠀⠉⠉⠳⡄
⠀⠀⠀⠀⠠⠡⠄⠀⠐⠃⠈⠉⠁⢀⡴⠒⢦⡒⢶⣦⣄⠀⠈⠠⠀⠀⠶⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣠⠎⠀⠀⢂⡙⠆⠙⢿⡿⠀⠁⠀⠀⠀⠀⠀⠀⡆⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠷⠋⠴⢶⣶⣶⠲⠴⣦⣄⡀⠀⠀⠀⠀⢀⣀⣀⣠⡀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⡄⠀⢠⠤⠈⠀⠤⠄⠀⠐⠀⣴⣶⠟⠛⠙⠋⣁⡐⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⡭⠭⢤⡶⠛⠉⠄⠀⠀⠀⠂⠀⠀⠐⠂⠀⠀⠑⡿⠁⠀⢋⡉⠀⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠋⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⡠⠤⡾⠿⠛⠛⠋⠉⣿⠀⠀⠰⠶⠶⠶⠶⣶⣤⣤⠶⡄⠉⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠒⣀⡄
⠀⠀⠀⠀⠀⠀⡳⢛⠛⢋⡉⠉⠀⡇⣠⠦⠀⡋⠅⠀⠃⠘⢿⢠⠁⢀⣀⡀⠀⠉⣀⣀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠋⠃
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠁⠀⠀⢠⠏⠀⠉⠭⠁⠀⠀⠨⠍⠁⠈⢷⠀⠀⠀⣠⣤⣾⣶⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢨⠁⠀⠀⠀⠀⠀⠂⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⣠⠀⠀⠠⣄⠀⢒⣈⡴⠊⠀⠀⠀⡀⠀⠀⠂⠂⠀⠀⡀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠈⠘⢶⡖⠛⢚⡻⠁⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠙⣮⣉⣀⣉⣡⡜⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠐⠀⠀⠀⡿⠆⠘⣛⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢉⠁⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡆⠀⠀⠀⠀⠀⢰⣟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⠀⠀⠀⠀⠀⠸⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢀⡠⠤⠀⣰⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡴⠋⠀⠀⠐⠉⠁⠀⠀⣀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⣀⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠏⠀⠀⠀⠀⠀⠀⣉⡁⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠄⠴⠆⠉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⢠⠀⠩⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠤⢄⡆⠀⠀⠀⠀⠀⡸⠀⠀⣱⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠼⠇⠀⠀⠠⢦⠞⠙⠓⠋⠁⠀⠀⠈⠉⠙⠒⠲⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠈⠑⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢰⠙⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠉⠉⠙⠒⠐⠒⠒⠒⠀⠀⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⢀⠐⠢⡌⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢤⣀⠀⢠⠀⠀⠀⠀⠁⠀⢄⠋⠀⠒⢯⠀⠳⡀⠀⠺⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣌⢛⣀⣀⡁⠴⡄⠀⠀⠸⢤⡀⠀⠀⠧⠐⢏⠟⡹⡟⠉⢳⡒⡶⡉⠂⠀⠀⢀⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠰⡁⠀⠀⡐⠋⢡⣀⠼⠀⢀⣀⣙⡀⢤⣀⠏⠀⠀⠀⠀⠀⢀⠎⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠌⢶⠙⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠤⠀⠒⠩⠴⠇⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠒⠀⢠⣭⣄⡀⠒⠂⠀⣀⣈⢠⠀⡤⠜⢢⠂⠠⡘⠓⠒⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢀⣤⠘⠃⢠⣤⠘⠃⢠⡄⠙⠉⣤⠀⠘⠦⠐⠃⠀⠀⠈⠑⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠦⢀⣀⠀⠶⢀⣄⣴⣦⣄⣀⣴⠆⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠀⠈⣉⡲⢶⡾⠿⢤⣓⡿⣗⣒⣈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⠂⠉⠒⠒⠣⠾⠥⠤⡌⠝⠒⠓⠒⣉⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢠⡄⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠟⠀⠀⠀⣬⡥⠤⣄
⠀⠀⠀⠀⠀⠀⠠⡊⠉⠁⠈⠉⣭⡒⠤⣥⣄⡀⢀⣀⣀⡀⠀⢀⠀⠀⢈⣀⣭⠄⢒⣞⡋⠁⣉⢉⣁⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠒⠀⠐⠀⠀⠆⠂⠒⠀⠀⠂⠀⠀⠐⠐⠐⠂⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠂⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⢠⡿⠿⠀⠀⠀⠰⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⡄⠀⠀⣶⠶⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠰⠶⢤⡀
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠁⣀⠔⠀⠀⠀⡀
⢀⠁⠀⠀⠀⠈⠆⢀⠀⠀⠈⠄⠀⠀⠶⡿⣶⣤⣄⣀⣤⣀⣤⣴⣾⠟⠛⢫⠉⠀⡇⠀⠀⠀⣄⠸⢠⡄⠀⠀⢰⠀⠀⠈⠀⠀⠀⠀⠈⠃⠛
⠸⡀⠀⢿⠂⠀⡇⠸⠃⠀⢀⠆⠀⠀⠀⠀⠀⠀⠘⠋⠙⢍⠁⠈⠁⠀⢀⠎⠀⠀⢇⠀⠀⠘⠿⢠⠘⠏⠀⠀⡌⠀⠀⠠⠀⠀⠀⣠⠀⡀⠀⠀⠀⢠
⠀⠣⡀⠀⠀⡠⠃⠄⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠒⠐⠂⠁⠀⠀⠀⠈⠢⡀⠀⠠⠊⠑⠀⠀⠊⠀⠀⠀⠀⠠⠀⠀⠛⠌⠁⠀⠀⠐
⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠦⣄⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⢀⣤⡄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠢⡀⠀⠀⠠⠞⠉⠁⠀⠀⠀⠀⠁
⠀⢀⣠⡴⠛⠉⢠⡐⠉⠳⢤⣀⠀⠀⠀⠤⠴⢤⣤⣀⠀⠀⠌⠀⠀⠀⠁⡀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠊⠀⠀⠀⠀⠋⠐
⠐⣏⠁⢠⣤⠀⠀⡇⢤⡄⠀⠀⡍⠀⠀⠀⠀⠀⠈⠹⣷⢸⠀⢠⣄⠀⠀⠁⠀⠀⠆⠀⠀⢀⣶⡆⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⢸
⠀⢩⠄⠀⠀⠀⠀⣇⠀⠀⠀⢀⠇⠀⠀⠰⠦⠦⢌⣦⠈⠀⢆⠸⠟⠀⢀⠂⠀⠀⠡⡀⠀⠀⢈⡁⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠘⠃
⠀⠀⠢⣐⠀⢀⠖⠉⠠⠀⠠⠋⠀⠀⠀⠀⠀⠉⠀⠉⠁⠀⠈⠒⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠙⠂⠀⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠂⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⣰⡿⠆⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠘⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡠⠞⠃⣀⠉⠙⠷⢤⣀
⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠆⠀⠀⠀⠀⠰⢡⡴⣦⣤⠘⠁⣴⢷⠶⠐⡀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢰⡆⢶⠀⠀⠀⢸⠁
⠀⠠⠀⠀⠀⢀⡆⠀⠀⠀⠠⡶⠀⠀⠀⠀⡇⣠⣽⣷⡅⢀⠀⣙⣿⣿⠆⡇⠀⠀⡤⠀⠠⠄⠀⡄⠠⠄⠀⠐⠀⠀⠀⠀⠆⠀⠀⠀⣠⠀⠀⠀⠀⠌
⠀⠀⠃⠀⠀⢀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⡀⠉⠋⢁⠊⠡⠉⠈⠁⡐⠁⠀⠀⠐⠀⠀⠀⡸⠉⠀⠀⠀⠆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠑⠄⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠛⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⣴⣿⡆⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠰⠟⠃⠀⠀⠰⢤
⠀⢠⣿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⢠⠀⠀⠉⠭⢙⠆⠀⠀⠀⢈⡥⠴⠂⣄⠀⢀⠬⠍⣇⠀⠀⠀⠀⢀⠴⠀⠰⠄
⠀⠉⠀⠀⠀⠀⠀⠀⠺⠿⠻⠷⠆⠀⠀⠀⡌⢀⢒⡓⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢀⣤⣶⠀⠀⠀⣴⡆⠀⠀⠀⠆⠀⠀⠀⢀⡀⠀⠀⠀⠄
⡆⠀⠀⠲⠶⠀⢐⠀⠀⠶⠶⠒⠀⡆⠀⠀⡇⠉⠙⠉⠀⠄⡆⢰⠋⠀⣰⠆⠀⠈⡇⠀⠀⠘⠿⢸⠂⠀⠀⠉⠑⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠈⠀⠀⠀⠀⠀⠈⠁⠀⠠⠄⠴⠚⠀⠀⠀⠹⡄⠬⡭⠔⣴⣎⠛⠯⢚⡟⠀⠀⠀⢳⡄⠀⠀⢀⡼⢄⡀⢀⡴⠇⠀⠀⠀⢃⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠖⠚⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠂⠰⠋⠀⠈⠙⠉⠀⠀⠀⠀⠀⠈⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⣠⣤⣄⢀⣴⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⡿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀
⠀⠀⣠⣾⣿⡿⠋⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦
⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀
⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷
⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠈⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⣿⣿⣿⣷⣶⣤⣴⣶⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡿⠈⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠛⠋⠉⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠙⠛⠻⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠿⠿⠿⠿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠀⢀⣀⣀⣀⡀⠀⠈⠙⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠠⣾⣿⡿⠿⢿⣿⣷⠄⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠿⠿⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡟⠁⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠊⠁⣸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠹⠀⠹⣿⢿⣷⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⢀⣠⣤⣤⣀⠀⠁⠄⠀⢸⠄⠻⡄⠀⡏⠛⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣸⣿⣿⡇⠀⡀⠀⢼⣿⣿⢦⣄⠀⠁⠀⠀⠉⠀⠃⠀⠀⠙⢆⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⡇⠀⠈⠓⠦⠭⠴⠤⠙⠀⠀⠀⠀⠀⠀⢠⠶⠛⢻⣿⣿⡛⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢦⣼⣋⣥⣠⠔⣹⣿⣿⣿⢟⡉⢻⣿
⠀⠀⠀⠀⢠⣴⣶⣷⣦⣀⡀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠟⠝⠀⠀⣼⠙
⠀⠀⣶⣶⣼⣯⣿⣿⣿⣻⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠀⡰⡇
⠀⠀⠘⢻⢿⣿⣿⣽⢿⠿⡄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⢃⠀⠀⡀⠘⣇⠑
⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⣤⠺⣇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠃⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠂⠀⠻⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢀⠄⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣺⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠈⠁⠈
⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠁

⠀⠀⠀⢠⣦⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⢤⡄
⠀⠀⢠⣶⣷⣦⠀⣾⣿⡇⠀⠀⠀⠀⣤⣶⣶⣦⡄⠀⣰⣶⡄⠀⠀⠀⠀⣀⢶⣾⣦⠀⠀⠙⢻⣿⡶⡄
⠀⠀⠸⣿⣿⠿⠀⢹⣿⡇⠀⠀⠀⠀⠈⣽⢿⡈⠀⠀⣿⣿⡷⠀⠀⠀⠀⠀⠀⣿⣧⠉⠀⠀⣿⢿⡟⠁
⠀⠀⠀⢸⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⢸⡇⠀⢰⡏⠀⢻⡄⠀⠀⠀⠀⣸⠇⠸⡇⠀⠀⣿⠘⣧
⠀⠀⠀⡈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠈⠁⠀⠈⠁⡀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠈⠀⠈⠁

⠀⠀⢀⠀⢀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡐
⠀⠀⠀⠑⢮⣭⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣅⠀⠀⣸⣿⡀⠀⠀⠀⠀⣨⣿⡄⢀⣀⣹⣥⡴⠞⡄
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠓⠒⢿⣿⠗⠀⠀⠀⠀⢹⣿⡃⠈⠁⢸⣿⡇
⠀⠀⠀⠀⡿⠸⣇⠀⠀⣴⣶⣤⡀⣠⠄⠀⠀⠈⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⣼⢹⡇⠀⠀⣼⠋⡇
⠀⠀⠀⠸⠃⠀⠻⠀⠰⠾⠿⠞⠿⠧⠀⠀⠀⠀⠟⠇⠀⠀⠺⠹⠃⠀⠀⠀⠀⠿⠘⠇⠀⠰⠋⠸⠇
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐

⠀⠀⣀⣤⣄⡀⢠⢄⣤⠀⡴⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⢠⣤
⠀⠀⠑⣶⡖⠁⠘⢲⣷⡞⡁⠀⠀⠀⠰⢤⣶⣾⡄⠀⢰⣿⣧⠀⠀⠀⠀⠀⣰⣾⣶⡄⢠⢶⣷⢦⡀
⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡿⣿⠁⠀⢸⣿⡇⠀⠀⠀⠀⠘⠉⣿⣿⠇⠈⢿⢿⡟
⠀⠀⠀⣿⡇⠀⠀⢸⢹⡇⠀⠀⠀⠀⠀⠘⡇⢻⡄⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⢸⡇
⠀⠀⠀⠋⠃⠀⠀⠙⠈⠃⠀⠀⠀⠀⠀⠀⠋⠀⠉⠀⠈⠋⠃⠀⠀⠀⠀⠀⠀⠋⠋⠀⠀⠉⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣭⣽⣿⣶⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⢀⣀⡀⢠⡬⠿⣄
⠀⠀⠀⠀⢀⣾⣿⣿⡿⠂⠁⠈⠁⠬⠿⠆⠘⡄⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⢀⣾⣿⣿⣝⠀⠀⠀⠀⠀⢀⣀⡤⠀⡇⠀⠀⢀⣴⣿⣿⣿⣿⡿⣿⣦⡀
⠀⠀⠐⠻⣿⣿⣿⣿⣆⠀⠀⠘⠿⣿⠿⠃⣰⠃⠀⢠⡿⣙⢉⡀⠀⠠⠄⠈⢻⣿⣄⡀
⠀⠀⠀⠀⠀⣉⣿⣿⡿⣷⣤⣄⣀⣀⣤⡴⠁⠀⠀⢸⣿⡤⣶⣇⣶⡆⠀⢀⡄⣿⣿⣿⣿⣦⣄
⠀⠀⠀⣴⣿⣿⣿⣿⣷⣶⣿⠿⢿⠿⠿⣷⡄⠀⠀⠸⣿⠁⢠⣧⠸⠰⣎⠀⢀⣿⣿⢻⣿⣿⣿⣷⡀
⠀⠀⢸⣿⣿⣿⡟⠉⠙⢯⣀⣀⣀⣀⣠⣼⡇⠀⠀⠀⢸⣦⠙⣷⡀⢠⣷⣶⣿⣿⣿⣟⣿⣿⣿⣿⣿⡄
⠀⢀⢘⣯⡙⠛⠿⠶⣤⣄⣀⣀⣠⣿⠛⠉⠀⠀⠀⠀⠘⠿⣦⣈⣿⡄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⣿⠩⣿⡿⣷⣤⣄⡀⠀⠀⠀⠈⠉⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣶⣾⣿⣿⣿⡿⠿⠛⣻⣿⡿⠥⣤
⡿⢰⣿⣄⠀⢹⣿⣷⡄⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⣠⠟⠋⠿⣿⠏⠉⠽⠶⠛⠙⠋⢩⣿⣯⣿
⠻⣀⣉⣙⣷⣼⣿⣏⣭⣶⣶⣶⣷⣤⣤⣷⡟⠀⠀⠀⠀⠀⠀⠻⣤⣄⣘⣿⣦⣠⣤⣦⣶⣶⣶⠿⢻⣯⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠠⠂⠀⠀⠄⠀⠐⠠⡀⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠂⠐⠀⠀⠤⠀⠠⢄⠀⠀⠀⢀⡀⢀⠀⠈⢆⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠠⠀⡇⠀⠇⠀⠄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠈⠚⠃⠐⠄⠀⠔⠻⠎⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠤⠠⠄⣠⠄⠀⠠⢤⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢰⠀⠀⠈⠂⡀⠀⠀⣀⠜⠁⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠋⠉⠀⠀⠀⠈⠀⠀⠀⠀⠙⠉⠛⠁⠀⠚⠃⠀⠀⠒⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠐⣤⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⣠⠈⠀⠰⢀⣄⠒⠈⢧⡀⠀⠀⣤⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢱⡤⠂⠛⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠠⡤⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠓⠠⢀⠀⠀⠀⠁⠆⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⣎⡠⠂⠉⠁⠀⡄⠀⡃⠀⠠⡆⠀⠀⢀⡀⠀⠈⠁⣴⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡠⠜⠀⢃⠀⠀⠈⠀⢀⡀⠀⠠⡄⠋⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠒⠤⠔⠊⠀⠀⠀⠈⠄⠀⠀⠀⠸⡏⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠄⠁⣸⠇⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⡀⢈⡀⡾⠀⠸⠀⠀⢀⣀⠰⠈⠁⠀⠀⠀⠀⠇⠀⢠⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠠⣤⠒⠉⠀⠀⠃⠀⠀⢆⡰⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⡠⠤⠤⢀⠀⠀⠀⢠⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢃⠀⠀⡇⠀⠀⢀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠄⠀⠀⠀⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠘⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠀⠀⠀⠀⠸⠀⠈⠢⣀⠀⣀⡴⠀⠀⠀⠀⠀⡀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⡰⠦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⣟
⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⣀⣀⠀⠈⠐⠂⠐⠀⠀⠀⠀⠉⠐⠒⣶⡄⠒⠀⡛⠀⢀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠐⠂⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠀⠁⠀⠀⠘⣄⡀
⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡞⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀
⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⡱⠆⢾⣋⣉⣉⣉⣉⣉⣉⣉⣙⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡘⢄⠀⠀⠀⠀⠀⢠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣀⣤⣤⣀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢠⣿⠿⠋⠀⠙⠛⠛⠛⠷⣦⡀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠉⠒⢿⡃⠀⠀⠀⠀⢀⣤⠤⠀⠈⠳⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣠⠤⠤⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣀⢀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣊⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠈⠱⡀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣤⣀⣀⣀⣠⠀⠀⠀⠉⢦⡀⠀⠈⠉⢢⠀⠀⠀⠀⠀⠻⢶⡄⠀⠀⢀⡗⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠈⠉⠉⠀⠀⠀⠀⠀⢀⣸⠟⠛⡗⠀⠀⠀⠀⠀⠀⠀⠿⠅⠀⠀⠈⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣼⠉⠉⠃⢲⠞⠛⠉⠉⠄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠠⣄⣀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⡆⠀⠀⠀⠈⠀⠀⢀⣠⣾⣿⠀⠀⠀⠀⠀⡟⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⣤⣴⣄⣀⣤⣴⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⡿⠿⠛⠋⠉⡿⠁⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠉⠀⠀⠉⠀⠀⠀⢀⣾⠃⣀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠈⠓⠢⠤⠤⡤⠤⠴⠖⠛⢁⡾⠁⠀⠀⠀⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⡀⠀⠀⠀⢀⣀⣴⠟⠀⠀⠀⠀⢠⡏⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⠉⠉⠉⠉⠉⠉⠉⠀⢀⣠⠀⠀⢀⡼⠁⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁⠘⢦⣀⡀⠀⠀⠀⣀⣴⠟⢁⣠⡴⠋⠀⠀⠀⠀⡟⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣹⡟⠀⠀⠀⠘⢭⠉⠙⠛⠛⣩⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⣿⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠁⠀⡿⠀⠀⠀⠀⣠⡬⢷⣠⡴⢋⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡇⠀⠀⢠⣾⣿⣷⣾⣏⣈⣉⣤⣷⣄⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⢠⡖⠀⠙⢦⣠⡟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⢠⡿⠁⠀⠀⠀⠈⠀⠀⣸⣿⣿⣿⠛⠟⠛⠉⠀⠀⠳⢄⣀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀
⠀⠀⠀⢀⣠⣶⣶⣶⣶⣦⣌⠉⢻⣦⡀⠀⠀⢀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠉⠁⠀⠀⠀⠉⠙⢿⣷⠀⢻⣿⠖⠂⠘⠁⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣦⣀⡀
⠀⠀⡀⠀⠀⢀⠀⢀⠀⠀⠈⣿⡇⠘⣿⠄⡀⠀⠀⠠⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⡄
⠀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⣿⡿⠛⢿⣿⡄
⠀⠀⠀⠀⢠⡤⠶⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⣿⡇
⠀⠰⢿⣄⡀⠀⣀⡠⠖⠁⢸⣿⣿⣿⣆⠀⠀⠄⠀⠀⠀⠛⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⡗
⠀⠀⠀⠉⠛⠟⠿⠇⠀⠀⠟⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⣤⡀⠘⠂⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣦⣤⠀⠀⠀⢠⣿⣿⠿⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠋⠁⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠸⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘