Oyster Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠍⠅⠐⡀⠀⢀⠀⠐⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠐⠒⠂⠔⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠁⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠡⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠠⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠓⢄⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠁⠀⠀⠀⢀⠠⠣⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠑⠀⠀⠀⠀⠈⠄⠀⠀⢀⠀⠀⢠⠀⠀⠐⠀⠀⢀⠌⠀⠀⠀⡠⠂⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⢠⠁⠀⢠⠂⠀⠀⠠⠊⠀⠀⢀⠠⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠈⠐⠄⡀⠀⠑⢄⠀⠀⠀⠀⠆⠀⢸⠀⡄⠀⣰⠃⢀⡴⠂⠀⡀⠔⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡘⡄⢀⡀⠀⠀⠀⠈⠰⢄⡈⠳⡄⠘⡆⠘⠀⠀⢠⠁⣰⠃⣰⠋⣀⡴⠊⠀⠀⠀⠀⡀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⣄⡀⠙⢦⡙⠄⠡⠄⠀⠂⠸⢠⠃⠜⣥⠞⢁⡠⠄⠐⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⠉⠓⢄⣑⠊⠀⠀⠀⠀⠀⠀⠈⢎⡡⠚⠁⠀⣀⣀⢀⣤⣤⡤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠔⡐⠠⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⡩⠚⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣑⣄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡌⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣄⠉⣯⡉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⢄⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠈⠀⠉⠀⠈⢀⣉⡁⠀⡤⠭⠠⠂
⠀⠀⠀⠀⠀⠀⠈⠂⡀⣌⠷⠒⡶⠀⣀⠠⡀⠀⠀⣀⠀⠐⠀⠀⠀⠀⠊⠀⠀⢀⡀⡀⠀⢀⡔⠦⣤⣰⠀⡉⢀⢃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢔⡀⢂⢹⡉⠁⠈⡎⠰⠂⠀⠹⠦⢤⠒⠋⠙⠲⡄⠀⠕⠀⠈⠋⢰⠁⠀⠀⡜⡐⢀⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠂⠡⡀⢂⠰⡀⢡⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⡌⠠⠁⡐⡐⠠⠊
⠀⠀⠀⠈⠁⠁⠒⠒⠀⠀⠀⠀⠓⢮⣵⡄⠠⠱⡀⠡⠀⠰⠀⢁⠀⠀⠀⡆⠀⡀⠀⠁⡜⡀⢁⣔⡪⠦⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠰⠼⢄⠑⣄⢆⠀⡀⠀⠠⠀⠠⡡⢁⠼⠋⠄⠈⠁⠀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠉⠈⠑⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠠⠤⠤⠄⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⠶⣿⣿⣿⣿⡶⠶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⠩⠑⢁⠀⣾⠟⠋⠁⢀⡴⣾⣿⣿⡛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣿⣿⣿⡵⢃⣤⣴⣿⣷⣤⣤⣄⣀⣨⣠⣀⠙⠒⠛⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣵⡿⣿⣾⣿⣿⣿⠿⠻⠿⠿⠛⠛⠉⠛⠻⠻⠎⠁⠈⣌⣙⢶⣄
⠀⠀⠀⠀⠀⢀⣾⡿⣿⣿⠟⣻⣿⠃⣸⣿⡟⣙⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⢽⡄
⠀⠀⠀⠀⠀⣾⣿⠟⠉⠀⡼⡿⠁⠀⠉⠋⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆⢻
⠀⠀⠀⠀⣼⡿⠁⠀⠀⡔⠊⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡇⡾
⠀⠀⠀⢰⣿⣶⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣽⠃
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⢸⠋⣽⠁⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠎⠀⠉⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⢀⣠⡾⠃
⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⣿⣿⣛⣿⣷⣄
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠘⠉⣥⣴⠶⠿⠃⠘⢛⡉⢻⣆
⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠇⠀⠀⠀⣠⣾⠟⠋⠈⠀⠀⠀⣠⣭⡁⣸⣿⣇
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡇⣠⣦⣤⣤⠉⠀⠀⠀⠀⠀⠀⠴⠦⣤⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠙⢶⣴⣷⣶⣤⣴⡿⠯⢿⣿⣿⣾⣿⣿⣧⠈⠀⠀⠀⠀⢰⣶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⣿⡏⠂⢀⣾⡿⣿⣷⡞⠿⡏⠀⠀⢀⠀⠺⠻⣾⣿⡛⣋⣹⣿⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⣿⣿⣸⢁⣾⡿⠁⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⣿⣇⣾⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⢇⣿⣿⡿
⠀⠀⠀⠀⠀⡾⢜⣽⢿⣿⣿⣿⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢙⣿⡋⠈⠙⠁⢨⣿⡟⠁
⠀⠀⠀⠀⣸⠃⠿⣿⠟⣽⣿⡏⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢋⡤⠿⣽⣏⡿⢋⣴⣿⠟
⠀⠀⠀⠀⠏⠀⢠⡏⠀⣿⡛⢇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⣠⣿⡿⠉⣴⣿⡿⠋
⠀⠀⠀⢸⠀⠀⠸⡇⠀⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡟⢡⣞⣿⡟⠁
⠀⠀⠀⠘⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠏⢠⣿⣿⡟
⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⡿⠃⡰⣻⣿⠟
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⣜⣾⠟⠁
⠀⠀⠀⠀⠀⠙⢷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⢤⣄⣀⣀⣠⣤⠤⠶⠖⠚⠋⠉
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠤⠲⡆⠀⠁⣌⣫⣴⣿⡇⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣠⣴⢖⠀⠀⠀⠁⠀⠀⡇⠈⣿⣿⡆⠀⠂⠈⠍⣐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢡⣾⠟⠿⠆⠀⠀⠀⠀⠆⠀⠃⠀⠻⠻⠃⠀⠀⠐⣶⣭⣍⢙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣾⠋⠀⢀⣀⡤⠶⠋⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠛⠿⠁⠀⠁⢈⢛⡄
⠀⠀⠀⠀⠀⠀⠀⢀⢡⣿⣿⠟⠀⠀⢉⣀⣤⡤⠤⠴⠶⠤⠛⠛⠛⠛⠋⠻⣦⡀⠀⠀⠀⠂⣠⢀⣼⠁
⠀⠀⠀⠀⠀⠀⢀⢀⣾⡿⠋⠀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⡞⢱⣾⡇
⠀⠀⠀⠀⠀⢀⢀⡾⠋⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡿⠁⢀⡠⣻⡿⢸⣿⡇
⠀⠀⠀⠀⡐⠵⠋⢀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⣀⣶⣾⠋⣠⣶⣿⢟⣉⣤⣿⣿⡇
⠀⠀⠀⢠⠃⠀⡔⠟⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣴⣶⣿⣻⣿⢷⣿⣿⣿⣵⣿⣿⣿⣿⣿⠃
⠀⠀⠰⣧⢄⠀⣷⣦⣷⠒⠶⢤⣀⣀⣀⠴⣒⠭⣽⣾⣿⣿⣟⠁⠻⠄⢐⣍⣷⠿⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠘⢷⣥⣍⣛⡛⢭⣭⣭⣅⣤⣶⣿⣥⣮⣭⣛⣋⣭⣶⣿⣿⣿⣿⣿⡗⢂⣸⣿⡝⢁⡟
⠀⠀⠀⠀⠀⠙⠻⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⡿⠇⠀⣶⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠿⢿⣾⣿⣿⠿⠟⠋⠉⠋⡛⠉⣠⣤⣶⣶⠮⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠄⣤⠤⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠤⡤⠤⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡤⠖⠋⠀⠀⠈⠒⢦⠰⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣁⡤⠞⠋⠀⠀⠙⠲⢄⢱⠄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢞⠕⠉⠀⠀⠀⠀⠀⠀⠀⠀⠡⢳⣌⣂⠀⠀⠀⠀⠀⠀⠀⠀⡠⣪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢋⢷⣌⡀
⠀⠀⠀⠀⠀⢀⠞⡵⠁⠀⡀⠄⡴⠒⢒⠒⠠⠀⠀⠀⠣⡁⠉⠢⠀⠀⠀⠀⠀⣠⢊⡜⠁⠀⡠⢤⠶⠒⠒⠒⠤⠀⠀⠀⢦⠁⠙⠂
⠀⠀⠀⠀⠀⡇⡜⠀⠀⡜⢀⡾⠚⠉⠉⠉⠁⠈⠳⢄⡀⠙⢦⠐⠀⠀⠀⠀⢠⠃⡞⠀⢀⠎⢠⠖⠊⠉⠉⠉⠉⠙⠢⣄⠀⠑⢄⠱
⠀⠀⠀⠀⠀⡇⡇⠀⢀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀⠸⢁⠀⠀⠀⠀⢸⢸⠀⠀⢘⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠸
⠀⠀⠀⠀⠀⢳⡇⠀⢸⠃⠀⠀⠀⠀⠴⠒⠀⠠⢄⠀⠈⠀⠀⡎⡄⠀⠀⠀⠀⢿⡀⠀⢸⠁⠀⢀⠄⠠⠴⠒⠀⠠⣀⠀⢸⠀⢠⠇⡆
⠀⠀⠀⠀⠀⠘⡗⠀⡌⠀⠀⢰⠊⠀⠀⠀⠀⠀⠈⡅⢢⠀⡼⠀⠠⠀⠀⠀⠀⢸⠁⢀⠋⠀⠀⡸⠊⠀⠀⠀⠀⠀⠸⡁⠆⢠⠻⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⡇⠐⠀⡇⠀⡴⠋⠑⠲⠤⡀⠙⢤⡼⠁⠀⠁⠀⠀⠀⠀⢸⡇⠸⡀⡀⠀⡇⢀⠠⠐⠒⠲⢤⡀⠑⢆⡞⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣧⠀⠘⠃⠀⢃⡘⡠⠄⡤⣄⠀⠈⠀⠆⣳⡠⠊⠀⠀⠀⠀⠀⠀⣇⠀⠙⠀⠀⠇⢰⡤⢠⠤⣄⠀⠰⢀⠎⣷⡠⠂
⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⢀⠨⢚⠠⠞⠀⠀⠑⠠⣶⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⠀⠠⠂⢋⠀⠏⠀⠙⡗⠠⡆⠰⠟
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠂⢸⠀⡎⡴⠚⠒⡄⠀⠣⠏⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠠⠀⢸⠀⠐⠒⠀⡠⠣⠃⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠘⠈⢿⠁⣀⢀⠘⣴⠀⠐⠱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠆⠀⠀⠈⠱⠤⠔⡞⠀⢀⢔⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣣⢀⠈⢆⡇⠀⢡⡟⢰⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢆⠐⡀⠀⢠⢃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠑⠺⡟⠖⠚⣆⡈⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠑⢿⠒⠊⠘⢄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣸⡦⠤⠌⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣰⡄⢀⣨⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠩⠀⠀⠐⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠒⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠒⠒⠶⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⢀⣠⠯⠝⠮⢭⡓⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠘⠦⢽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠐⠀⠀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢻⠀⠀⢠⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡟⠀⢠⣏⣠⠞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠦⠀⠀⠀⠀⣠⠏⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠄⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠁⡴⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣤⠶⢲⡀
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡞⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡈⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠧
⠀⠀⠀⠀⠀⠀⠘⣇⡜⠉⠀⠀⠀⠀⠀⣀⣀⠠⢤⡴⠉⠀⠀⠀⠈⢿⠁
⠀⠀⠀⠀⠀⠀⣴⠻⠤⠤⠐⠛⠓⠚⠉⠉⠁⠀⣸⠀⠀⠀⠀⠀⠀⢸⣛⠛⠓⠒⠒⠶⠤⠤⣤⣀
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠉⣉⡩⠖⠒⡔⠊⠉⠀⠈⠣⣀⠀⠀⠀⡠⠃⠈⠉⠈⠳⡔⠒⠲⠴⠶⢬⠷⡄
⠀⠀⠀⠀⢀⡟⡗⠒⠢⠎⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠈⠛⡏⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⢦⠙⢦
⠀⠀⠀⠀⠸⡇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠛⠂⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣸
⠀⠀⠀⠀⠀⠳⡈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆⠀⠀⠀⢀⠀⠀⠀⠠⠞⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣍⠳⢤⣀⠀⠀⠀⠀⠐⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠴⠚⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠫⡙⠒⠮⡙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣀⣀⣀⠤⠞⠀⣠⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠈⠑⠒⠀⠉⠙⠓⠒⠒⠒⠊⠉⢠⣤⣤⡤⠤⠴⠒⠊⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠦⠤⠤⠤⠶⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⡠⠔⠂⠀⠐⠒⠒⠄⠂⠐⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠒⠢⠀⠀⠂
⠀⠀⢠⠊⠀⠀⣶⣿⣿⡿⠿⠟⠋⠙⠛⣷⣄⡑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡂
⠀⠀⠆⠀⠀⢸⣿⣷⣬⠀⠀⠀⠀⠀⠀⠈⠙⠛⡿⠆⠉⠀⠀⠀⠘⡀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⡜⢠⣿⣧
⠀⠀⠘⢄⠀⠈⠻⢿⣷⡀⣤⣀⣿⠷⠶⠖⢛⣉⡤⠶⠋⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠈⢀⣀⠀⠊⣠⣿⡟⠈
⠀⠀⠀⠀⢳⣤⣀⣀⣩⠭⠙⠒⠀⠀⢀⣤⣽⣷⠾⢻⠂⡙⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣛⣧⣴⣿⣿⣿⠇⢠⠆⠀⠀⠂
⠀⠀⠀⠀⠀⠉⢹⠋⠀⠀⠀⣀⣤⣶⠟⠁⣠⣤⡶⢃⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⠿⢿⣿⣟⠉⠀⠀⣿⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⡠⠃⢀⣴⣿⡟⢋⠄⠁⠀⣠⣿⠏⣰⠟⠀⢀⡠⠔⠂⠀⠀⠀⠀⡜⠀⢀⣠⣄⠀⠘⡇⠀⣇⡏⠀⠀⠀⡀⠀⢳
⠀⠀⠀⢠⠊⠀⠀⢸⣿⣿⣿⣃⣔⢠⣿⡿⠋⣰⢟⠀⠊⣡⠀⠀⠀⠀⠀⠀⡘⢀⣴⣿⣿⣿⣦⡀⠀⠀⢿⡇⠀⡀⠀⠙⢂⠘⣄
⠀⠀⠀⢸⡀⠀⠀⠀⠉⢻⣿⣿⣿⠟⣫⣴⣾⢯⠔⣡⢾⠃⡰⠁⠀⣀⡀⠀⡇⣸⣿⠟⠁⠉⢻⣷⡀⣾⡀⣿⠀⠙⠂⠀⢸⣷⢸
⠀⠀⠀⠀⠱⣀⠀⠀⠀⠀⠈⣉⣠⣾⣿⠿⣧⣴⣿⠏⣡⠞⠁⢀⡞⢁⠿⠀⠧⣻⡏⠀⠀⠀⠀⣿⡇⣿⡇⣿⣄⡀⠀⢰⣾⣿⣏⡇
⠀⠀⠀⠀⠀⠈⠉⠓⠒⠛⠻⠛⢋⠕⠉⢐⣾⣿⠱⠟⠁⠀⢠⣾⠁⠈⢲⡀⠲⣹⡇⠀⠀⠀⠀⢸⡇⣿⣆⠛⢿⣿⣿⣿⣿⠟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣴⣿⣿⠿⠗⠠⡄⣰⣿⡏⠀⢰⣿⣷⢀⠹⡹⣦⠀⠀⠀⠸⡇⣿⣿⣷⣆⠙⠛⠛⣡⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⣼⠟⠁⠀⢀⢌⣼⠿⣋⠀⢀⣿⣿⣿⢠⠀⠱⣌⡳⠀⠀⣠⡄⢻⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⢣⣾⠋⠀⠀⢠⠎⣼⠋⠀⣿⣿⣿⡿⣽⡏⠐⠀⠀⠈⠛⣶⡄⠙⠃⣸⣿⠛⠿⠿⠿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠜⠀⢠⣿⣰⡟⠁⠀⠀⠀⡔⠀⢹⣷⠀⠀⠉⢁⣼⣿⠇⢸⡆⠀⠀⠀⠘⣿⣦⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠊⢀⣴⣿⣻⡿⠁⠀⠀⠀⢠⠁⠀⢸⣿⣶⣴⣾⣿⣿⡟⠀⣾⡇⠀⠀⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⣘⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⠀⠀⠘⠛⠛⠛⠿⠟⠋⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⣀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣷⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣻⣛⣟⣙⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠶⠶⣿⣟⡭⠉⠛⠻⢉⠈⠛⠻⢟⠛⠲⣷⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡾⡿⢍⠁⠀⠀⠀⠱⠄⠀⠀⠈⡇⠀⠀⠀⠀⠀⡏⠈⢳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⢿⡀⠀⠀⠈⡇⠀⠀⠠⡰⠀⠀⢀⠐⠀⠀⢠⠆⢀⣠⣧⡼⠊
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠣⠀⢸⠁⠀⠀⢠⠀⢀⠀⠀⠀⢀⣠⢋⠀⢀⣠⣯⣤⠾⠻⣽⣦⠠
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢀⡴⠀⡸⣠⣴⣵⡷⣢⣵⡂⢀⣠⣾⣿⠷⠾⠛⠋⡁⡀⠠⠑⢾⣹⡀
⠀⠀⠀⠀⠀⢸⣧⣀⣴⣾⢟⣛⡛⣛⢛⠟⠉⠉⠉⡉⠉⢡⡀⠀⠐⠂⠂⣶⠁⠈⠀⠀⣠⢞⡜
⠀⠀⠀⠀⠀⠀⠙⠿⢧⣿⣻⡯⠿⣧⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡵⠋⠁⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠾⢍⡙⠻⣶⣦⣶⣶⣶⣾⣩⣠⣤⣶⣾⡯⠑⠋⡠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠚⠷⠾⠾⠭⠭⠵⠤⠀⢀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠌⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠴⠒⡋⢅⠀⠠⣶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣴⡮⠶⠚⠑⠖⠛⢿⣷⣙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢀⡟⠀⠃⠀⠀⠀⠀⠀⠘⠿⡦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⣗⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⢆⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⠀⠄⡆⠈⡆⢉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢦⠠⢻⠹⢦⠄⠀⠀⣀⠀⠈⠀⠂⠀⠘⢦⠀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢡⢢⠇⢀⠈⠙⠒⠊⠁⠀⠀⠀⢰⠀⢂⠀⡆⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⢢⡟⠀⠈⠓⠤⣤⠠⠆⠀⠀⢀⠃⠀⠀⡀⢣⡜⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡔⢁⡾⠀⠓⣄⠀⠀⠀⠀⠀⢀⠤⠋⠀⠀⣨⠇⡈⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢀⡾⢰⠇⠐⡀⠀⠀⠐⠀⠀⠚⠁⠀⠀⢀⡠⢟⡴⣿⠃⠀⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣼⡇⠘⠀⡀⠑⠒⠦⣀⣀⠀⢀⡴⠤⠖⠈⣠⠞⠁⣿⠀⠠⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠏⠁⠀⢧⠀⠦⣀⠀⠀⠈⠉⠁⠀⠀⣦⡴⠋⠀⣼⡿⠀⠀⠇⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡄⠀⠀⠈⠢⡀⠈⠉⠁⠒⠤⠄⠀⠈⠋⢀⠀⡼⡿⠁⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠱⡀⠀⠀⠀⠉⠳⠄⢀⡀⠀⠀⠀⠀⡤⠴⠖⠋⠀⠀⢠⡎⢀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠠⠶⠊⢱⣣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢁⠀⢷⠀⠀⠀⢄⣀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠖⠁⠀⣠⢞⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠳⡄⠀⠀⠀⠉⠒⠐⠀⠀⠀⠀⠀⠀⢀⡤⠞⠡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠑⠰⢧⣀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠓⠂⡀⣄⠀⠙⠚⠋⠉⠉⡉⠉⠀⣀⠤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢦⣤⡙⠈⠝⠶⠚⠟⢃⡀⠔⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠑⠔⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣠⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⣿⠏⣽⠛⠫⣯⠵⠛⣿⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⢀⡿⢛⣥⢤⣍⠢⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡯⣿⣷⡬⠭⣤⡶⠉⠻⢧⣜⣉⠿⠿⢿⣒⡦⡄⠀⠀⢸⢡⡿⠃⠈⠈⢳⡄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣶⣴⣟⣲⣆⣽⡻⢦⣾⢉⠙⣳⣦⡤⢋⣿⡇⠀⠀⣼⠀⡇⢀⠀⠀⠀⠙⢢⡀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⣿⣻⣟⣩⠴⣫⣭⣛⠒⣡⣾⣷⣾⡟⠁⠀⠀⡟⠀⣇⠀⠐⠉⠢⡀⠙⣿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠷⠚⠛⠿⠖⠋⠀⠀⠀⠀⠀⠀⡇⠀⠹⣇⠀⡀⠀⠀⠀⢸⡧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⠂⠀⠠⠤⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢹⡀⠀⠁⠀⢀⡾⣿⠀⠸
⠀⠀⠀⠀⠀⠠⠄⠐⠉⣀⣤⡤⠴⠒⠲⠶⡶⠖⠒⠒⠲⢤⡹⡟⢛⡆⠀⠀⠀⠀⠀⠀⠓⠄⠀⠉⠛⢧⣬⠭⠾⠃⠀⡌
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡄⢧⣀⣐⠀⠀⠀⠀⠀⢀⠔⣠⠇⣨⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠰⠦⣀⣠⠔⠀⠂
⠀⠀⠀⠀⠀⠒⣤⠈⠳⢦⡓⠶⢦⣤⡤⠀⠒⣚⠭⠕⣩⡵⠟⠁⠀⠀⠀⣾⠛⡗⠒⠲⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠸⣄⣀⣈⡙⠋⠛⠓⠚⣉⣁⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⢻⡁⣿⠉⠉⠓⠲⠤⣄⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠐⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⣄⠈⠁⠀⠀⢌⠙⠢⣀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠂⠁⠀⢤⡀⠀⠀⠻⣮⠳⡀⠀⢶⠦⠈⠀⠘⠳⣄
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢒⡫⢝⠀⠀⠀⠙⢧⡙⢦⡰⠀⠀⠀⠀⢀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠙⠶⢤⡀⠁⡀⠀⠀⠀⢁⢶⠆⠀⠁⠀⠬⠈⣶⣷⠀⠀⠀⠳⡈⢷⠀⢀⡀⠀⠀⣸⠇⠀⡰
⠀⠀⠀⠀⠀⠆⠀⢸⠃⠘⡀⠀⢄⠘⢆⡙⢄⠀⠸⣦⣢⠀⠀⠀⠀⠀⠄⣿⣿⡄⠀⠀⠀⠱⡀⠙⠛⠛⠓⠛⠋⠀⢀⡁
⠀⠀⠀⠀⠀⢡⡀⠸⣆⢇⠀⠄⠀⠀⠈⢷⡀⠀⠀⢻⢯⡀⢀⡀⠀⠀⠀⠨⠻⣿⠀⠀⠀⠀⠈⠓⠒⠒⠠⠄⠐⠒⠉
⠀⠀⠀⠀⠀⠀⢇⠀⢹⡀⢧⠀⠀⠀⠀⠉⢷⠐⡀⠈⢿⣿⣶⡴⣖⣠⡀⠴⢦⣿
⠀⠀⠀⠀⠀⠀⠈⠦⣄⠉⠲⢄⣀⠀⠀⠀⠀⢡⠱⠀⠀⠻⣷⣼⣿⣿⡶⠁⠈⣯⡇⠀⠀⡔⡁⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠙⠫⣦⣠⣤⡎⡡⣇⠀⠀⠀⠙⢿⣿⣿⣦⣤⢸⡇⠀⠀⠀⠈⠈⢲⠘⠋⠐⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⠭⠟⠚⠛⠁⠀⠀⠀⠀⠀⠈⠙⠛⠙⠋⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⡟⠋⣁⣀⣀⣀⣀⠠⠰⠞⠛⠋⠳⢦⣄⣀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⠀⠀⠉⠤⠄⠉⢉⢁⡠⠄⠤⠄⣄⣄⡀⠀⠙⠛⣶⡀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣀⣠⣰⣷⣄⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠉⣦⠀⠈⠹⡿⠀⠀⠀⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣷⣤⣀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠟⠀⣠⠟⠁⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠄⠈⠉⠉⠉⠛⠻⠿⠷⢶⣬⣕⡋⠁⠀⠀⠀⠀⠀⢀⣀⡴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠺⠟⢒⠻⠿⠏⠰⠶⠶⠶⠒⠒⠚⠳⢦⣀⠀⠀⠈⠉⠛⠶⣦⡤⠤⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣧⣾⣾⡀⠀⠀⠀⠀⠀⠐⠒⠒⠢⢀⡀⠉⠓⣄⣀⠀⠀⠀⠀⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠉⠙⠧⣄⠈⠙⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⣾⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⠈⣓⠄⠀⠀⠈⢄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢩⠄⠀⠀⢸⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠒⠐⠒⠒⠒⠒⠀⠀⠀⠀⠀⠉⠻⣿⣧⡀⠰⠒⠂⠐⠛⠁⠀⠀⠲⠏⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠉⠿⣿⣦⣄⣀⣀⣀⣈⡀⠉⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠀⠀⠀⠀⣠⣤⠴⠦⢄⣠⣤⣤⣤⣀⠀⠈⠢⣄⠀⠀⠀⠈⠳⠾⢿⣿⣿⣿⣿⣷⣦⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡛⠉⠀⣠⡴⠛⠃⠀⠀⠀⠈⠑⠦⣄⠀⠉⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⣾⡃⠀⢼⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠈⠙⢦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠠⠀⠈⣧⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠙⠉⠉⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⢻⡆⠀⠀⠙⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣴⣴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠀⠙⠦⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⠀⠈⠉⠛⠓⠖⢀⣤⣾⠿⠛⠛⠛⠛⠻⠛⠙⠛⠩⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠲⠾⠟⠛⠉

⣿⣿⢻⢻⢿⣿⢻⢿⢿⡿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠈⠁⠀⣀⣤⢤⣥⣬⣄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢃⡴⠶⠖⠃⠈⠉⠀⠘⠋⠐⠹⣷⡀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠟⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣄⢸⡌⢷⣦⣀⠀⣀⣀⣀⡤⠀⠀⠐⣋⣤⣤⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣦⣌⣉⣉⣡⣤⣤⣾⣿⣿⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⠿⠿⠿⢿⣿⠿⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠴⠦⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣾⣛⣱⡶⠆⠠⢤⣠⡴⠶⠖⠲⢦⡀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡛⠟⠉⠀⠀⠀⠀⠙⠀⠭⠂⠰⣬⢹⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⠡⣷⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢘⣻⡆⠀⠀⠀⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣭⠂⠀⠀⣀⣴⠚⠀⠀⠀⢀⣤⣈⠝⠁⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣷⣶⡝⠕⠂⢐⣛⣋⣉⣠⣾⣶⡷⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⣿⣿⣶⣶⣿⠿⠛⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠶⠿⣿⣿⣿⣿⡷⢶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⢋⣴⣾⡿⠛⠉⠀⠀⡠⠔⡚⠉⢠⠟⠛⢛⡲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠝⠁⣴⠏⡟⢹⠇⠀⠀⠀⠈⡇⠀⡇⠀⠀⠀⠀⠁⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⢰⣿⠗⠁⣰⢿⡟⠰⡇⢸⠀⠀⠀⠀⠀⡇⠀⢧⡀⠐⠀⠀⣠⣶⣶⡿⠁
⠀⠀⠀⠀⠀⠀⣼⣿⠀⢰⡏⢘⡇⠀⢧⠘⣇⠀⠀⠀⠀⢙⢄⠀⠉⠒⣒⠿⣿⣿⠟
⠀⠀⠀⠀⠀⢀⣿⡏⠀⢼⡿⠘⣷⠀⠈⢧⡘⢦⡀⠀⠀⠀⠈⠉⠉⠉⣡⣼⣿⠏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⣿⠏⠀⠀⣿⠀⠀⠹⣧⡀⠀⠙⢦⣉⠛⠳⠾⠛⠛⣛⣾⣿⣼⠏⠀⠀⢀⣤⠾⠛⠉⠉⠉⠉⠉⠉⠙⠳⢦⡀
⠀⠀⠀⠀⢸⣿⡀⠀⠀⢿⡀⠀⠀⠉⠛⢦⡀⠀⠈⠳⠶⠶⠖⣛⣭⣿⣡⡟⠀⣠⠞⠋⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠙⠶⣄
⠀⠀⠀⠀⠀⢿⣇⠀⠀⠈⠳⡄⠀⠀⠀⠀⠈⠳⢤⣤⠴⢚⡽⠋⣠⡟⠉⠀⢰⡏⠀⣰⠊⠁⠀⠀⣀⣀⡬⠶⣶⢤⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⡴⡋⢠⡼⠋⠀⠀⠀⣿⠁⢰⠏⠀⣠⠶⠾⢿⡅⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠛⢤⣤⠔⢒⠋⠉⠉⢀⡴⠋⠀⠀⠀⢀⣾⠏⡰⠋⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠛⢷⣆⠀⠀⠀⠀⠀⠈⠀⠀⣀⡶⠋⠀⠀⠀⢀⣴⡟⠡⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠷⢤⣤⡤⠶⠚⠉⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢦⣸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠠⠐⠀⣸⠃⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢿⡄⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⠷⠾⠿⠛⠉⠀⣀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣯⣵⣾⣿⣄⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣠⣤⡾⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠻⠶⠶⠶⠦⠤⢤⣤⣤⠶⠶⠛⠛⠛⠋⠉



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣬⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣿⢀⠀⡀⢶⠀⢺⡗⠂⣀⣯⣇⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⡿⣿⢠⡇⡜⣧⢸⡇⣼⠵⢣⢸⡇⡜⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠁⢈⠽⡏⠙⠋⠉⠉⠉⠑⠁⠈⠋⠑⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠶⠲⠚⠉⡏⠁⠀⠊⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠠⣀⡀⠠⡄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⡠⢴⣋⠉⢓⠆⠒⠂⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢮⣷⣀
⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⣃⠼⠃⣎
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⢰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁⠀⠀⠯
⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢐⠣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⡟⠠⡀⠀⠀⡒⠦⠒⣀⠈⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⢸⠀⠀⡉⣡⠿⠤⠵⠊⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠒⠋⠉⠁⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠚⠓⠢⠤⠂⠚⠉⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⣀⣀⣀⣤⣤⣤⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⡠⠞⠉⠁⠀⠀⠈⠉⠻⣧⠀⠑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠤⠊⠀⠄⠊⠀⠀⠀⠀⠀⠀⣿⡇⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⢰⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠊⠀⠀⠀⠀⠀⠀⠀⡴⠔⠂⠀⠀⠀⣠⣾⢃⢄⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⢻⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣴⣶⣟⡛⢓⡵⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣞⣿⢶⣶⣤⣶⣶⣤⣶⣶⠾⠿⠷⣊⣭⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⠤⠴⣤⡤⠶⠚⠓⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠢⠤⠶⠚⠉⠁⠀⠀⣈⠑⠢⢤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣫⣄⡄⠐⠋⠉⠁⠈⠉⠋⠉⠉⠙⠳⣦⣀⡀⠀⠈⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣾⣦⡀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠉⠙⠛⠶⣄⡀⠄⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢤⡈⠻⣦⡨⠄⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠙⢤⣀⣀⠀⠀⠀⠀⠀⠀⢣⡀⢻⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⡳⢶⣤⡈⠉⠻⣦⠀⠀⠀⠀⠀⡷⢀⡷⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠤⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢏⠀⠈⡛⢶⣄⣈⠷⢶⠤⢖⣚⣡⡾⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⢶⢻⡄⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⡉⠛⠻⠶⠾⠟⠉⠁⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⢠⣿⣈⣿⣿⣿⣿⣷⣷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢄⣀⣠⠴⠖⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠹⠟⠿⠿⣿⣿⣭⣿⣶⣽⡣⣬⣶⣶⣿⣷⣤⣠⢄⡀⡀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣿⣟⠿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠿⠿⣿⣷⣯⣤⣶⣦⣬⣽⣶⡞⠢⡀
⠀⠀⠀⠀⠀⠀⠸⣍⡀⣆⣠⣭⣔⣢⠶⢆⣐⣲⣒⡦⡄⠀⠩⠙⢛⣛⣿⣿⣿⣿⡶⠼
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣾⡿⢿⣿⠿⠿⢋⡈⠩⠄⡸⣿⣶⣶⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⣶⣤⣤⣤⣤⣶⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠶⠛⠛⠛⠻⠟⣉⣴⠶⠶⠦⣤⣉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢛⣫⣭⡤⡶⠾⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠈⠙⠲⢤⡙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣫⣴⣾⣿⣿⣷⣦⠶⢖⣚⡛⠋⠉⠉⠉⠛⠛⠒⠲⠀⠀⠀⠀⠀⠉⠲⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣾⣿⣿⣿⣿⡿⠿⠚⠋⣉⣁⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⢛⣵⣿⣿⠿⠛⣉⣠⣴⡶⠿⠛⠛⠋⠉⠉⠉⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⣩⣴⡿⠛⠉⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⣴⡾⠛⠁⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠶⠒⠒⠒⠲⠶⠤⣄⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠃⣼⠏⠀⢀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠖⣊⣭⡥⢶⣶⣾⣷⣦⣤⡤⣤⣄⣀⠀⠉⠙⠶⣄⣠⡾
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⡏⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢀⣀⣤⠾⠛⣛⣛⠛⠙⠂⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⣄⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠃⢠⣿⠃⠀⠀⠀⠀⠀⢀⡤⠶⠞⣡⡴⠞⣉⡤⠗⠚⠋⠉⠉⠉⠉⠉⠙⠶⣤⣄⣀⠀⠀⠐⢄⠀⠈⠁⢤⡀⢻⡄
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⡴⠛⢡⠶⠛⡡⠖⣋⣥⠤⠶⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠶⣄⠀⠈⠲⣄⠀⠙⣦⣻
⠀⠀⠀⠀⠀⢠⣿⣯⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠏⠀⠴⢃⣠⡶⠞⣋⣥⠶⠞⠛⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠑⣄⠀⠈⢳⡀⠘⣎⢳⡄
⠀⠀⠀⠀⠀⢿⣿⡇⠀⢠⢀⡾⠁⠀⠀⢀⡼⢉⣠⡤⢾⡿⠗⣚⣋⣡⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢌⠲⡄⠈⠳⡄⠀⢳⡀⠈⢦⢹⡄
⠀⠀⠀⠀⠀⠈⣿⠃⢀⡞⢸⡇⠀⣠⠖⠋⡴⢋⣡⡴⣿⣾⣭⣭⣤⣤⣤⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⡆⠀⢻⠀⠀⢣⠀⠸⣾⡇
⠀⠀⠀⠀⠀⣼⣿⠀⡸⠁⢸⠇⡼⠁⠀⢊⣠⣿⠷⣛⣩⡤⠀⠚⠛⠻⢿⣟⡛⠦⠌⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⢸⡄⠀⠈⡇⠀⣿⣧
⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⣸⡾⠁⠀⠐⣫⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⠇⠀⠀⡇⠀⣿⢹
⠀⠀⠀⢀⣼⣿⡄⢀⠀⣠⠟⠀⠀⣰⣞⣉⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡟⠀⢀⡞⠀⣰⢃⡾
⠀⠀⠀⣼⣿⣿⡇⣞⡴⠃⣠⣶⣿⣿⡿⠿⠿⠶⠦⣄⡉⠙⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡾⠁⢠⠞⠀⣴⣡⠞
⠀⠀⠀⢻⣿⣿⡀⣿⢇⣼⣿⢟⡡⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡾⠁⣴⠋⢀⣾⠟⠁
⠀⠀⢠⣿⣿⣿⣿⣿⠞⠛⠉⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠀⠀⠀⠀⠀⠀⠀⣰⢣⠟⠁⣸⠃⢠⣾⠏
⠀⠀⣾⣿⣟⣹⡿⠃⢀⣤⠆⣠⡤⠖⠀⠀⠀⠀⠈⠉⠻⣄⣄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⡇⣿⠆⠀⠀⠀⠀⠀⢠⣿⠏⡴⠛⠁⢔⡵⠋
⠀⠀⢿⣿⣿⠋⢀⣠⣾⣵⠟⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⡌⡆⢘⡄⢸⡆⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⢀⣾⢋⡞⢀⡴⠛⠋
⠀⠀⢈⡿⡿⠀⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡄⠀⢸⠧⡇⢠⡇⣼⠃⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣠⡟⣡⣫⠶⠋
⠀⢀⣼⣾⠁⠀⣄⠀⠀⠒⠂⠀⠀⢀⠀⣰⣇⠀⣰⠁⠀⡼⣰⠃⡼⣱⠃⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⢀⡴⢋⠾⡟⠁
⠀⠸⣟⢸⡇⠀⠻⠀⠀⠀⠘⠛⣛⣫⠞⣛⣥⠞⠁⢀⠜⣱⢃⣾⠟⠁⠀⠀⠀⠀⣰⠟⠁⠀⠀⢀⡠⠖⣉⣔⣫⠼⠃
⠀⠀⠙⢿⣧⡿⠛⢦⡘⣛⣛⣋⣉⣵⠋⠁⠀⠀⠔⣩⠞⠳⠋⠁⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣀⣤⠶⠛⠉⠉
⠀⠀⠀⠈⠻⢿⣤⠴⣄⡉⢉⡉⠉⠀⢀⡠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠠⠴⠋⣀⣠⡴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠙⠷⣤⣤⠤⠤⠤⠶⠶⠶⠒⠚⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠖⠁⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⠤⠤⠶⠀⠀⠀⣀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⣰⢿⠛⠉⣼⠀⠀⠀⠀⡠⢺⣿⣧⡀⠀⠀⠀⠀⢀⣾⣇⠀⠀⠀⡠⠀⠀⠀⣀⠀⢀⣴⡄⣠⡌⠀⢀⣒⠶⠦⢀
⠀⠀⠀⢀⠀⠀⠁⠀⣀⠀⠙⠷⠤⣄⣀⡠⠞⢻⣿⣿⣦⡀⠀⠀⢸⣿⣿⣛⡷⠟⠒⠒⣛⣺⣷⣶⡦⠉⠄⠩⠿⠄⢿⣏⢤⠂⠸⠗⡀
⠀⠀⠀⣿⣀⣴⣾⡷⢿⡁⠀⢠⠄⠀⠈⠴⢶⢡⡻⣿⣿⣿⣄⠀⠀⠻⣿⣿⢧⡷⣼⣿⡷⢤⣼⣿⣵⣿⠛⠻⠶⢤⠈⠿⢾⡶⣄⣾⡇
⠀⠀⣴⡏⠉⠉⡈⡆⢀⣑⠚⠛⡷⢀⣠⡠⢀⡀⣿⣽⣿⣿⣿⠀⠀⠀⠀⠉⠻⣷⣯⣿⣗⣺⠤⣈⠻⣷⣿⠉⠉⢕⣀⠀⠈⠻⣿⣻⠁
⠀⠀⢹⡇⡀⠀⢠⣽⠟⠁⣀⡔⠓⣤⣿⡿⡁⠀⣸⣧⣽⣿⣿⡆⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣍⣉⣡⣿⣦⣀⣬⣾⣿⡿⠇
⠀⠀⣘⣴⣷⠾⠋⠀⠄⣠⣘⠿⢦⣼⡟⣁⢀⠈⢛⣿⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠷⠟⠛⠉
⠀⠀⢛⣉⣠⣀⠠⣠⣤⡾⠇⠠⣴⣿⣿⢛⢋⡜⢋⣻⣿⠟⠁
⠀⠀⠈⠙⠻⣿⣼⣿⣿⡗⠀⠀⣸⣿⣟⣡⡟⢙⢿⠟⠟
⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⣿⣿⡿⠿⠟⠁⢨⠆⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⢀⡠⠄⠀⠀⠒⠒⠒⠀⠒⠒⠢⠤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠯⠻⠛⠁⠁⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠒⠀⠀⣤⣤⡀⠀⠀⠈⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⣀⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⢿⠶⠟⠀⠀⣀⢀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢉⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡿⠃⠀⣀⣴⢊⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠔⠒⠒⢂⠉⠱⠈⢀⣴⣾⣿⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠠⠄⠴⠒⠎⡙⡄⡀⠄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴⠶⠶⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢦⣀⣀⡠⠤⠤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⣦⡶⢿⣿⡖⣉⣶⡀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⠘⣏⢀⣠⡿⢲⡗⣿⡇⠄⠘⣷⡈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣾⢣⣀⣿⢸⡟⠂⢿⣳⣿⠀⠀⠒⢹⣧⠀⠐⢲⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢴⣭⡿⣃⣜⣧⢻⣷⠀⢘⡿⢫⣿⠀⠀⠒⢩⣿⡀⠘⢛⣛⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⠘⠋⠉⠉⢫⣜⡛⠀⣽⣾⣿⣾⠁⣀⣾⠟⣤⡘⠤⠖⠋⣧⣿⣿⣄⣼⡟⣬⡿⣇⢹⣽⣲⣤⣴⡄
⠀⠀⠀⠀⠀⠀⢂⠀⠀⣀⠠⣵⡄⢘⡿⠛⢁⢸⡆⢿⢿⣝⡃⣈⣀⡛⢿⣍⡗⠀⣀⣤⣤⣌⣻⡉⠈⣼⣿⣾⣿⣠⡯⣹⡿⣿⣿⣆⣤⣀⣀⡀
⠀⢀⡤⠀⠀⠀⣈⠀⢀⣿⠐⢾⣿⠟⠉⢷⡈⢻⣿⣾⣿⠋⢀⣁⣉⡻⣎⠫⢦⠀⢈⣭⡽⠛⢿⡇⠁⡽⠋⠾⢩⣿⡇⠁⠀⠀⣨⣿⣿⣿⣿⡋⢶⣶⡀
⠀⠁⣿⠀⢠⠰⣽⣷⣌⣣⣕⢆⡉⠀⠀⡐⣷⣘⣋⣼⡏⠐⠉⠩⢙⣿⠼⣷⣤⢼⠟⢛⣿⣿⣮⡇⠀⣀⣠⣴⣿⣿⣿⡄⠀⠀⣿⢛⠌⢻⣿⣿⣾⣿⣷
⢈⢠⣘⣀⠨⣭⢂⠀⡈⠉⠀⠈⠛⣷⡘⣿⡿⠋⣿⣏⡀⠀⠛⠛⠿⠇⣯⣽⠧⠄⢈⣉⣭⡻⣿⣇⣬⠀⠈⠙⠫⡿⣿⣆⠀⡸⠋⠈⠀⢹⠿⢿⣟⢿⣇
⠈⠢⢌⠏⢱⢤⠀⠙⠃⠤⣀⠀⠸⣿⣧⡻⢇⣿⣿⠃⠱⠶⠶⢶⣤⠈⣽⣏⠀⢰⠿⢋⣛⣻⣿⣿⡿⠠⠞⠁⠀⢡⣾⡿⣷⣧⢀⠀⠀⠀⠀⠦⢿⣿⣿
⠀⠀⠀⣿⠁⠀⡄⠀⠀⢀⣼⣦⠰⣿⣯⢯⣷⣿⡥⠀⠀⠀⠀⠚⠻⣇⢸⣿⣿⡧⠄⠈⠉⣛⣻⣷⠀⠐⠀⢀⣴⣿⣿⢿⣿⣷⡟⠀⠀⠀⠀⠀⠠⠛⣿
⠀⢰⡐⠋⠠⡀⢸⡆⠀⣼⣿⣟⡧⢴⣦⣿⣿⡟⢁⡀⠀⠀⠀⠀⣄⢹⣟⣿⣟⣠⣴⢶⣷⣶⣯⠻⣇⢱⠟⠉⠀⠉⠙⠈⠛⣿⣿⣿⠁⡀⠀⠀⠁⣠⠏
⠀⠀⢣⠀⣶⣜⣼⣿⣦⣝⠋⣤⣤⣿⣿⣿⠏⠀⠀⠙⠳⢦⣤⣤⣌⢺⣿⣿⠿⣯⠗⠈⠛⢿⣿⣿⣷⡄⠂⠀⢀⡤⢾⣿⣿⡿⣿⣷⣿⣡⣆⣠⣼⠏
⠀⠀⠘⣿⣿⣿⣟⠿⡟⣿⣿⣾⣿⣿⣿⠟⣢⣀⡀⠀⠀⠐⣽⣿⣻⣾⣿⣿⠟⠁⠀⣠⣴⣾⣷⣼⣿⣃⠀⡀⢀⡐⠉⠀⠿⣳⣿⣿⣯⡶⠟⠋⠁
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠿⡯⠉⠤⠀⠙⠛⢛⢿⣦⣌⡙⠋⣿⣿⣏⣤⡴⡞⠉⠻⢿⣿⣿⣿⡏⠹⠁⠊⠀⣠⣴⣿⣿⣯⢙⡁
⠀⠀⠀⠀⠀⠉⠉⠀⠈⠛⢷⡦⠀⠀⣠⠀⡀⠈⠢⢀⠈⢹⣿⣿⣿⣿⣿⣿⣕⢠⠃⠐⣾⣿⣿⣿⣿⡟⣓⠺⡿⠟⠉⠀⢐⣿⣿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠉⠹⣦⡀⠂⠀⠀⢈⠸⣿⣿⣿⣿⣿⣯⠁⠛⣩⡾⠋⣿⣟⣿⣯⣧⣡⣌⠁⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠙⠳⢶⣦⡘⠄⠈⡊⣿⣿⣟⣯⠀⠻⠋⠀⣰⡿⣿⣿⣿⠿⠛⠛⠑⠄⣀⡀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⣀⠤⠒⠉⠿⣿⣶⣿⣿⣿⡿⠇⠘⠲⠒⠋⠍⣨⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢮⣿⣿⣿⠀⠀⠀⠀⠑⠯⢅⣀⣐⣋⠽⠉⠙⠛⠛⠁




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠀⠀⠀⠠⠄⠠⠠⠀⠠⠐⠀⠀⠠⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠲⠤⠶⠚⠉⢙⠒⠚⣛⣩⡭⢽⡿⠿⠟⠿⠭⢭⠛⣖⣢⢄⣀⣀⣀
⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⣀⣤⣤⡴⡿⠿⠏⠀⠉⠁⠀⠀⠀⠉⠀⠀⢈⡀⠀⠉⢢⡘⠛⢩⢽⣦⣤⡀
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠸⣿⡏⣍⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠆⠁⠀⠉⠃⠑⢀⠀⠀⠈⠉⠁⠙⠪⣷⣶⢤⡀
⠀⠀⠀⠀⠀⠀⠙⢷⣦⡀⠀⠈⢙⣿⣷⠀⠀⠀⠀⢀⣶⣿⣿⣿⡇⠀⢰⡄⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢉⠣⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠉⠛⡧⠀⠀⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⣿⣄⠀⠀⠀⠙⠻⢿⣿⣿⣶⣄⣀⣀⣀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠘⢇⣤⣄⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣷⣀⡈⠙⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣈⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣪⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣬⣽⣓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣒⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣯⣷⣦⣤⣄⣤⣤⣄⣠⣤⣼⣿⠿⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠋⠉⠉⠉⠉



⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⣀⠔⠂⠀⠀⠒⠠⣀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠘⠀⣠⣤⣤⠀⠀⣤⣤⣄⠀⠁⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⠉⠉⣡⣌⠉⠉⠉⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⡀⢢⠀⠀⠀⠛⠛⠀⠀⠀⡔⢀⡀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⡌⠀⠈⡆⢅⢀⡇⢀⡀⢸⡀⡨⢰⠁⠀⢡⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠰⠀⠀⡠⢈⡑⠂⠤⢀⣀⠤⠘⢊⡁⢄⠀⢀⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢀⠜⠚⢅⡀⠄⠛⢒⠤⠌⣁⠛⠠⢀⡨⠓⠫⡀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠘⡄⠀⠀⡔⠂⠉⠀⠀⠀⠀⠉⠐⢢⠀⠀⢠⠃⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠑⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⠊⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣄⡀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣀⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⢢⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⢙⣶⠤⠖⠒⠛⠉⠉⠉⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠉⠉⠉⠙⠓⠒⠦⢤⣴⠋
⠀⠀⠀⠀⠀⠀⢰⡋⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢣⡀⠀⠀⠀⠀⠀⠀⡰⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠸⡟⠦⣤⣀⣀⠀⠀⠀⡏⠀⠀⠀⠉⠓⠶⠤⠴⠖⠋⠀⠀⠀⠈⡇⠀⠀⢀⣀⣠⠤⢞⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠉⠉⠙⠛⠓⠒⠶⠶⠶⠦⠤⠤⠤⠤⠶⠶⠶⠖⠒⠛⠛⠉⠉⠁⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠶⠖⠒⠚⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⠤⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠟⠫⠙⠉⠉⠀⠀⠀⠈⠀⠀⠀⠈⠙⠿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢟⠛⠳⠁⠀⠀⠀⠀⡀⠤⠤⢀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣧⣶⣃⣀⠀⠀⣀⣠⣎⠀⠀⠀⠀⡱⠄⠀⠤⡀⠀⠠⡀⠀⠉⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⣿⠟⠈⢛⠟⠣⡴⠊⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠈⠒⠂⡐⢀⣘⠈⠙⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⣻⠀⢀⡎⠀⠀⢸⡇⠀⠀⠀⠀⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠢⡮⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⣉⣄⠘⣇⡀⠰⣾⣿⣦⣀⣳⣄⢈⣿⣆⡀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⠀⠅⠀⢈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣶⣶⣅⣄⣀⣀⡀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣮⡻⣿⡿⠻⣯⣽⣟⣻⣿⣿⣿⣿⠁⠀⠀⠀⠀⣛⣿⣿⣿⣿⣯⣭⣭⣿⡿⣛⣴⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠹⣿⣿⣿⣮⣹⣿⠿⢿⣿⣿⣿⡖⢤⠴⠒⠢⣼⡟⣻⣿⠿⢿⣯⣭⣿⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⡿⠟⠉⠀⠀⠀⠀⠋⠙⡿⢿⣿⣿⣆⠀⠹⠉⣿⡇⠀⠀⠀⢠⠉⢠⡿⠁⣠⣾⣿⡿⣋⣵⠟⠛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⣿⡿⢷⣝⠳⡈⠻⣦⠀⢀⣿⡇⠀⠀⠀⢸⡀⠘⠁⣴⣿⠟⣡⠞⡛⠛⠀⠀⠈⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⠿⣷⣬⡑⠮⡂⠈⠷⢸⣿⡇⠀⠀⠀⠈⡇⢀⡼⠋⠠⣪⠥⠬⣡⣤⣤⣤⣤⣤⣭⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣈⠁⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢹⡄⠀⠀⠈⠁⢀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⣾⣭⣭⣁⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢳⠀⠀⠀⠈⣉⣩⣭⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠉⠉⠀⠀⡀⢰⣿⣿⣿⣿⣄⠋⠁⡤⠄⠊⠃⠄⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠐⠀⠠⣀⣀⡀⠀⠻⠟⠉⠉⠙⢿⣷⣿⣤⣄⢀⣀⣄⣡⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢾⣥⣤⣤⣤⣤⣤⡽⠿⠿⠧⣤⣤⣤⡄⠈⠉⠀⠉⡙⢻⣿⣤⣬⣭⣽⣿⣶⣶⣶⣾⠷⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣟⣹⢛⣟⢋⣿⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣂⡐⢋⣥⣄⡀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⠹⣿⡇⠸⣿⣿⠃⢿⣿⡦⣀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠹⢿⡄⢻⠃⠈⠏⠉⠀⠻⢽⢡⣝⡽⢆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⣴⡏⠻⠛⡇⠀⠃⢀⣤⠀⣴⡆⢀⡀⠀⠟⠟⢠⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣿⣆⠀⠀⣦⠻⠀⠹⠀⠟⠁⠙⠃⣿⠆⣀⡀⠋⢀⣶⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣖⡀⠉⢦⡀⠊⠃⠀⠀⡄⡀⢠⠂⢤⠉⠢⢛⣠⠀⡶⠛⢃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢂⣞⡈⠙⠉⢦⡄⠉⢤⠸⡆⢧⣇⣅⠄⢰⢡⠂⡀⠈⠋⠁⠀⠤⢺⢆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡛⡷⢤⣄⠁⠐⣤⣳⣻⣾⣾⣿⣆⡒⡥⠂⡀⠄⠐⢋⢀⡔⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢨⣒⢮⣿⠟⠉⠀⠀⠉⠻⣷⣯⣤⠤⠼⠝⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠐⡀⠀⠀⣰⡆⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣠⣴⣾⣿⣿⣆⠛⠆⠘⠉⣡⣿⣿⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣴⣶⣾⣿⣿⣿⣭⣭⠿⣿⢻⣶⣶⣾⣽⡶⢾⡿⣳⠻⢿⣿⠿⢣⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣯⢻⣿⣿⣿⣯⣴⡞⠉⣾⠉⢹⣍⠹⣷⣶⣞⣿⣿⣿⠟⣾⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⠭⣭⣽⡻⢿⠿⣿⣿⣷⣿⣿⣿⠽⠿⠿⣃⣶⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⣿⣿⡐⣿⣏⣰⣞⣿⣭⢵⣷⣿⣿⣽⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠾⢿⣷⣿⣿⣿⣿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⢀⠀⠄⠒⠘⠐⠠⡄⠀⠀⠀⠀⠀⠀⢀⠤⠲⠘⠈⠁⠠⣄⠀⠀⠀⠀⠀⠀⣠⣀⣄⢀⡀⡀⢀⠀⣀⢀⣀
⠀⠀⠀⡘⠀⢠⠴⠚⣈⣉⠑⢤⠁⠀⠀⠀⠀⡐⠉⠀⣀⠀⠤⢄⡀⠁⡶⠀⠀⠀⠀⠀⠳⣸⢈⣸⡇⡇⢸⢨⣿⣿⠞
⠀⠀⢀⡁⡎⡡⠒⠋⠀⠀⠉⠊⢣⠆⠀⠀⠀⠍⢠⣷⠀⠀⠀⠀⠙⡀⠚⠀⠀⠀⠀⠀⠙⠉⠋⠉⠉⠛⠘⠋⠛⠘
⠀⠀⠀⠙⠴⠁⠀⠀⠀⡀⢀⠀⠅⠀⠀⠀⠀⠰⡈⠻⢔⠲⠦⠀⠈⢠⣮
⠀⠀⠀⠀⣙⠒⢶⡄⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⠷⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⠉⠀⠁⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠙⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠘⠋⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⡄⣀⠉⠓⢦⣀⡘⠿⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀
⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠘⠙⠛⠁⠀⠀⠀⠀⠀⢁⠴⢈⣤⢈⠉⠙⡐⡈⣇⠀⠀⠀⠀⢿⠁⠀⢀⣼⣿⣿⡷⡦⠄
⠀⠀⣀⠀⠀⠈⠓⠤⠦⠶⠄⠘⠂⠀⠀⠀⠀⡀⢠⣟⠂⠀⠀⣀⣩⡿⢛⣟⠉⠀⠀⠣⡀⠀⠁⠆⠠⢊⣿⠵⠸
⠀⠀⠩⡢⣀⡀⢀⠀⠀⠀⡀⠄⠃⠀⠀⠀⠀⠃⡈⠉⠑⢒⢈⡭⠄⠊⡈⠃⠀⠀⠀⠐⠽⢀⡆⢠⠀⡄⢰⣨⠆
⠀⠀⠀⠈⠈⠈⠘⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠁⠙⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠛⠛⠛⠛⠳⢶⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⣋⣉⣉⡛⠷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠀⠀⠈⠉⠳⣄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣼⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢤⠈⠂⠀⢀⡤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡾⠙⢷
⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⢠⡄⠀⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⡾⢁⣠⡴
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠘⠛⠛⠉
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⡾⠀⢀⡴⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⠒⠤⢄⡀⠀⠀⠀⠀⠀⢀⣠⠟⠁
⠀⠀⠀⣼⠁⠀⠈⠀⣰⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠙⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣦⡀
⠀⠀⢠⡿⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣀⡘⣧
⠀⠀⢸⡇⠀⢀⡞⣁⣠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⣤⡀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻
⠀⠀⠈⣷⢀⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠞⠉⠀⠀⠀⠀⠀⠙⢿⡉
⠀⠀⢀⣽⣾⠀⠀⢀⣠⣤⣤⣤⣤⡴⠖⠛⠛⠋⠁⢨⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣀⣀⣀⡀
⠀⠀⢿⡀⠙⠛⠋⠉⢀⣀⠀⠀⠀⠀⠀⢈⣀⡡⠤⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠧⣄⣠⣤⣀⠀⠉⠉⠉⠉⠛⠛⠷⢦⣄
⠀⢀⣼⠏⠀⠀⠉⠉⣉⣠⠔⠛⠛⣦⠚⠉⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡴⠃⠀⠈⠀⠀⠈⢳⡶⠛⠛⠳⠶⠶⠶⢦⣽⢻⡄
⢰⢿⡧⠤⢤⣀⣰⠋⠉⠁⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣶⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠘⢧⡈⠻⣦
⣿⢰⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⣇
⣿⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠃⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⡏
⠘⣧⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢲⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡴⠋⢀⡴⠟⠁
⠀⠘⢷⣄⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣠⠇⠀⣸⠃
⠀⠀⠀⠙⣷⢄⡈⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⠋⠉⠀⠀⢠⡟
⠀⠀⠀⠀⠘⢧⡉⠛⠓⠦⡈⠙⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⣀⣀⡤⠞⠁⠀⢀⡄⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠓⠦⣤⣈⠉⠛⠲⠶⠤⣤⣤⣤⣤⡤⠴⠞⠁⠀⠀⠈⠉⠉⠁⣀⣀⣠⡤⠖⠋⠠⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠐⢄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠛⠓⠒⠛⠛⠋⠉⠉⠀⠀⣀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠋⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⣤⣤⣀⣀⣀⣀⣀⣤⣤⡶⠞⠛⠉
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⠀⢠⣤⣀⠀⢠⣤⣤⣠⣤⣤⡄⠀⠀⢀⠀⠀⠀⢠⣤⣤⣤⣤⣤⡤⣤⣦⣴⡆⠀⢠⣶⣶⣶⣦⡶⣶⣦⣤⣤⣤⣤⣤⣴⢶⣴⣤⡤⠀⢀⡀
⠀⠀⠀⢠⣶⣤⣍⠀⣿⣿⡿⢛⣵⡜⠿⠿⠟⠋⠀⢀⣠⣌⠐⠠⢠⣿⣿⣿⣿⣯⣿⣬⣿⣛⣿⡀⠀⠈⠻⠿⣛⣽⣿⣿⣿⠻⣿⣯⣿⣟⢿⣷⡽⠟⢀⣿⣿⣧
⠀⠀⠠⣿⣿⡿⢁⣾⣿⣿⣾⣿⣿⣿⡀⠠⠀⡄⠀⠸⢿⣿⡠⠀⣿⠟⣩⣶⡘⢿⣿⣿⣿⣿⣾⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⡁⠐⢿⣿⣿⡛
⠀⠀⢠⠹⠏⠀⣿⣿⡿⣻⣭⣽⠻⠟⠀⠘⠿⢀⣶⣿⣦⣶⠙⠀⠀⣾⣿⣿⣷⣶⣿⣿⣿⢛⡛⠟⣁⠰⠐⠀⠻⣿⣿⣿⣏⣠⣿⣿⣾⣿⣿⣿⡷⠀⠘⣿⣿⡗
⠀⠀⢸⡇⠀⠘⣿⣟⣄⢿⣿⣿⣷⡄⠀⠀⣤⣾⣿⣿⣿⣷⡟⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⣿⡆⢲⣷⡀⠈⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠠⠘⢿⡇
⠀⠀⠘⣷⡄⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⢠⣿⢻⣿⣿⣿⣿⡿⠇⠀⣿⣿⣿⣿⣿⣿⣿⣎⣾⣿⣿⣟⡿⡎⠟⠃⠰⣷⡜⢿⣿⣿⣿⣏⣥⣮⣝⠻⠀⠀⣴⣮⠃
⠀⠀⠀⣿⣧⠀⠘⢛⣿⣿⣿⣿⣿⠂⠀⢘⣿⣿⣿⣿⣿⣿⡇⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢩⣻⣿⣿⣿⣿⣷⣷⣿⠛⠁⠀⣾⣿⡿⢂
⠀⠀⠸⢿⡇⠐⠀⣿⣿⣷⣿⣇⠀⠀⡆⠈⣿⣿⣿⣿⣿⣍⠀⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠀⡿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢻⣿⣿⣡⠏
⠀⠀⢰⣶⡀⠀⠀⢻⣿⢿⣿⡿⠀⠀⠁⠀⣿⣿⣿⣿⣽⡟⠀⡆⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠳⢿⣿⣿⣿⣿⣿⡷⠀⠀⠙⣿⣿⣿⠇
⠀⠀⢸⣿⣿⡄⠀⠐⣿⣿⣿⠇⠀⡀⠀⢈⣿⣿⣿⣽⣿⡧⠀⡇⠘⣿⣿⣿⡟⣭⣼⣭⢹⢛⣛⡿⢿⣻⣿⠁⠘⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠀⠀⠀⠘⣿⣿⡆
⠀⠀⣾⣿⣿⡇⠀⢀⠺⠿⢏⠀⢸⡇⠀⠈⣿⣿⣿⣿⣿⠃⢸⠀⠲⣿⣿⣿⣿⣿⣿⣿⡆⣄⠈⠁⠉⠱⢿⠀⠴⡄⢀⣼⢠⣦⣈⢹⣿⣿⣿⢀⣠⠀⠄⠸⣿⣦
⠀⠀⢹⣿⢋⣼⡇⠛⣴⢯⡿⢃⣼⣿⣄⠀⡙⢿⣯⣿⣏⠀⠹⠀⢀⢿⣿⣿⣿⣿⠿⢋⠀⠀⠀⠉⠀⣷⣶⣾⣆⢡⣿⣿⣾⣿⣼⣷⣽⣿⡟⢨⠁⠆⣈⣳⣝⣯
⠀⠀⠈⠉⠛⠛⠓⠺⠛⠳⠘⠟⠛⠛⠿⠄⠙⠒⡈⢉⣓⣀⠠⡐⠃⠄⠙⣻⠿⡿⢡⠅⠸⠐⢰⡄⣷⠌⠝⠿⠿⠧⠿⠛⠛⠛⠫⠿⠿⠫⠴⠺⠤⠼⠿⠛⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣟⡿⣽⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⣿⣿⢻⣿⣿⣿⣸⣿⢿⣿⢣⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣿⣿⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣥⣶⣾⣆
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠈⢿⡿⠿⠇⡿⠋⠉⠙⠻⣿⣿⡿⠟⣋⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣍⣛⡿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠈⣿⣶⣿⠿⠿⢛⣛⣭⣭⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠶⢦⡿⢿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠁⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣧⣄⣀⠀⠀⠀⠀⢹⣆⣀⣀⣤⣄⣀⣴⣿⡏⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣦⣴⣦⣤⣤⣽⣿⣿⣶⣤⣤⣴⣿⣿⡿⠿⠛⣿⡟⠛⢉⣠⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⣽⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣦⣤⣾⣿⢹⠉⢹⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣙⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⡁⠀⠀⠀⠀⠀⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣛⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⣛⣛⣻⣿⣿⣿⠃⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⣿⣿⣟⣉⡉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⣭⣭⣭⣭⣭⡿⠀⠀⠀⠀⢸⡇⣿⣿⣯⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡆⢸⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣦⣾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣆⣀⣤⣄⣀⡀
⢀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⡖⠚⠁⠸⣷⡆⠂⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣴⣶⣖⠀⠈⣿⣿⠀⠀⠘⣟⡃⠀⢹⣿⣶⣠⣤
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡋⠐⠈⠹⣿⢀⠀⢻⣿⠀⠀⢠⣿⠃⠀⢘⣿⠇⠈⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣐⠀⠙⠛⡷⠀⠀⠓⢷⡄⠀⢸⣿⡆⠀⠀⣿⠀⠀⢸⡿⠀⢀⣿⡿⠁⣷⣄
⠀⠀⠀⠀⠀⣴⣿⢄⠀⠀⠘⣧⡀⠀⠜⣧⠀⠘⣿⣷⠀⢀⣯⠀⠀⣼⠇⠀⢸⣿⠃⣠⣿⠿⢏⡀
⠀⠀⠀⠀⢰⠉⠉⠻⣦⡀⠀⠘⣷⡀⠀⢻⡆⠀⢻⣿⠀⢀⣿⠀⠀⡿⠀⠀⣸⠇⠀⣿⡟⠀⠘⢿
⠀⠀⠀⠀⣴⣷⡀⠀⠙⢷⡄⠀⠙⣧⠀⠈⣷⠀⢸⣿⠀⢈⣿⠀⢸⡇⠀⢸⡏⠀⣼⡏⠀⢠⣞⣻⡆
⠀⠀⠀⠀⣉⠉⠻⣧⡀⠈⠻⡄⠀⢙⣧⡄⢿⡇⠻⢿⠃⢻⡟⠀⣿⠀⣀⡿⢀⣼⠋⠀⢸⡿⢋⣽⡇
⠀⠀⠀⠀⠛⣳⣄⠈⠹⣦⡀⠹⣦⠙⠛⣇⠀⣷⠀⢿⡄⢸⡇⠀⡏⠀⣿⠁⣿⠏⣤⣴⠟⢀⣼⠋⡇
⠀⠀⠀⠀⠐⠀⠉⠣⣄⡘⠿⣄⠈⠱⡄⠹⡆⠹⡄⢸⡇⢸⡅⢸⠁⣸⠃⣰⠋⢠⡟⢻⣶⡏⠠⠛⠁
⠀⠀⠀⠀⠀⠈⠀⣘⠿⠟⢄⠈⠳⡄⠙⢆⠹⡄⢿⠸⡇⢸⠁⡟⢰⠏⣰⠃⣰⠟⣠⠟⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠀⠑⢦⡀⠢⣌⢣⡹⣼⡇⣧⢸⣰⣧⡟⣴⢃⡾⢁⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⠈⠢⣈⠢⣝⢿⣿⣿⣿⣽⣿⡿⣣⢞⣿⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⡈⠻⢾⣷⣽⣿⣿⣿⣿⣿⣷⠟⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠶⠭⠛⢦⣝⢿⣿⣿⣿⣿⡟⠁⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠒⠛⠛⠿⠿⠛⠛⠒⠒⠐⠒⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠋⠉⠛⠓⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠉⠈⠙⠻⠶⠶⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⣴⠂⠒⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠐⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠉⠁⠙⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠔
⠀⠀⠀⠀⠀⠀⠀⡀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡎⠀⠀⣠⠤⡀
⠀⠀⠀⠀⠀⠀⠚⢱⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠃⢠⠺
⠀⠀⠀⠀⢀⡏⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠈⠘⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⢰
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠛⠓⢿⠻⣶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣡⠎
⠀⠀⠀⠐⡀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢡⠉⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠔⠁
⢎⠴⡴⢢⠠⠬⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢰⢧⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⠇
⠠⡛⠲⢠⠑⠄⢏⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⣰⣞⣫⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠅⠄⠰⠸⢄⠂⠉⣙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣒⣂⣀⣭⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠄⠀⠊⠑⢄⡇
⠈⡄⡀⣸⠀⠀⠀⠉⠂⠒⡈⠉⠙⢗⠲⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠁
⠀⠁⢃⠻⠀⠀⠀⣀⣢⣤⡌⢉⠈⡂⡤⢑⠀⣀⠉⢆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠹⠃
⠀⠉⠂⠀⠑⠑⠂⠉⠀⠀⠁⠠⠑⠈⠲⠆⠄⢀⠠⠰⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⡴⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⢠⠄⢀⠀⠀⠂⢠⠙⠂⢑⣂⣠⣄⣒⣊⡥⠖⠁⠀⠙⢶⠤⠤⢀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠀⡀⠀⠀⠀⢀⡀⠀⠰⠦⠤⠛⣉⠨⠆⠐⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠴⠒⠒⠦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢉⠀⣋⡤⠤⠤⢤⣙⠀⡉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢀⣁⣠⠎⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠱⣄⣈⡀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣰⠋⠉⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠉⠙⣆⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠳⠄⠀⠈⠁⠀⠠⠞⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠠⣄⡀⠀⢀⡤⠶⠶⢤⡀⠀⢀⣠⠄⠀⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠄⠀⡏⢠⠖⠀⠀⢹⠀⠠⠤⠤⠤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠖⠒⠒⠒⠀⢧⡀⠀⠀⢀⡼⠀⠒⠒⠒⠒⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠤⣀⠀⣀⣀⠀⠀⢀⣀⠙⠓⠒⢋⣀⡀⠀⠀⣀⣀⠀⣀⠤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠉⠉⣉⠙⠛⠉⣉⠉⠓⠒⠉⣉⠉⠛⠋⣉⠉⠉⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠬⠷⢤⣀⣘⣦⣀⣀⣰⣃⣀⡤⠾⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠈⠁⠉⠁⠂

⠀⠀⡄⠀⠰⠂⠀⠀⠂⠀⢀⣀⠀⠀⠀⠄⠲⠀⠀⠀⠀⡀⠀⣠⠆⠀⠀⠆⠀⢀⠀⠀⢀⠀⠀⠀⠰⠀⠀⡠⠀⠀⡀
⠀⠀⠀⠀⠀⣀⠀⡀⠁⠀⠀⠁⠓⠀⠀⠁⡄⢀⠆⡰⢡⣷⣊⡄⠀⠔⠀⢀⠀⠀⠀⠄⠈⠀⠐⢤⣄⠀⠀⠀⠐⠈⠁⠀⢀
⠀⠀⠈⣄⡠⠁⠀⠀⠉⠀⠃⠀⡄⠀⡇⢰⠃⠈⢠⣶⡪⠥⠂⠀⠃⠀⠠⠀⠐⠀⠀⠔⠠⠠⠄⠀⠀⠉⠓⠤⡄⠀⠀⠀⠉
⠀⠈⠁⠀⠁⠀⠠⠤⠀⡀⢰⠀⠙⠸⠀⡀⢠⣶⠋⠄⠄⠒⠂⠐⢁⠀⠀⠀⠠⠆⠀⠀⠀⠠⣠⠎⢀⠀⠀⠀⠀⠺⢆⠀⠈
⠈⠀⣄⠀⠀⠀⠀⢰⠀⡇⠈⠀⡆⢸⡰⠷⠇⠀⠀⠀⠒⠒⠀⡀⠀⠀⢀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠘⠀⡤⠀⢠⠂⠘⠃
⠀⠀⠀⢄⠀⡆⠀⢀⡀⠀⢰⠀⣧⣈⣉⠓⠂⠀⠈⠒⠒⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠀⠠⠀⢀
⠀⡀⠰⠸⠀⢇⠀⠀⡇⢻⣘⠁⠀⠀⠀⠁⠀⠉⢉⡁⠀⣀⠀⠀⠄⠀⠀⠀⠀⠟⠂⠀⠀⠐⠀⠀⠀⠈⠂⠀⠀⠀⠂⠐⣄
⠀⠓⠀⠀⠀⢸⠀⠀⠣⠤⣬⡉⢀⠁⠒⠦⠄⠀⠞⠙⢦⡈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁
⠀⠀⠀⠀⣇⠘⣤⣘⣓⠒⠂⠠⠤⣈⠛⢂⡤⢨⢥⠛⡀⣁⠐⠆⠀⠀⠀⢀⠆⢀⠀⠀⠊⠀⠀⠐⠒⠀⠀⠿⠄⠀⠀⠀⢁
⠀⠀⠀⠀⠐⠓⠒⠀⠠⢄⣈⡑⠂⠀⠀⣉⠝⣇⠈⠆⠁⣨⠀⠘⠃⠈⡡⠄⢀⠀⠀⠈⠀⣀⠄⠀⠀⠀⠛⠀⠑⡄⠀⠀⠈
⠀⡠⠂⠀⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠈⠁⠀⢠⠀⠺⠀⠀⠀⠀⠀⠼⠃⠈⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠐
⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠘⠀⠘⠀⠀⠀⡠⠚⠀⠀⠀⠀⢀⠠⠋⢀
⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⣢⡄⠉⠀⠀⠀⠀⠄⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⡀⠔⠉⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⢆⠀⠀⠀⠀⣰⠦⠤⠄⠀⠀⠀⠤⠀⡀⠄⡀⠀⠀⠊⠀⠀⠉⠀⠀⠀⠈⠁⣀⠜⢏
⠀⠀⠀⢀⣀⠄⠈⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠⡜⠓⠀⠂⠀⢀⣀⠐⠀⠐⠃⢀⡠⠀⢰⠀⠁⠀⢀⡠⠀⠀⠰⠀⣠⠒⠀⠘
⠠⠤⠒⠊⠉⠀⠀⠈⢀⠀⠈⠀⠐⠄⠀⣴⣋⡁⠀⠀⠀⠀⢠⠅⠀⠀⠀⠀⠀⠀⠀⠈⠃⣠⠔⠋⠀⠂⠀⠔⠀⠀⠤⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠂⠀⠀⠰⠯⠀⠀⠀⡀⠀⡋⠀⠀⠂⠀⠀⠱⠀⠀⢠⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⡲⠄⠀⠰
⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢣⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠦⡀⠀⠊⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⡀⠜⠊⠀⠀⡀⠀⠀⢀⠔⣴⣾⣿⣿⡿⣿⣿⣿⣿⣷⣦
⠀⠀⠁⡀⠀⣐⠆⠂⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⠀⠡⠐⠀⠊⠀⠀⢆⠀⠀⠈⡠⠀⠁⢸⣿⣿⡿⢿⡷⣿⣿⡿⣿⢿⣿
⠀⠀⠃⠀⠀⠀⠿⠰⠙⠀⠒⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠱⠀⠀⠀⠀⠀⢸⣿⢸⡇⣻⡇⢹⠃⡆⣿⡸⣿
⠀⠀⠀⢀⠠⠊⠀⠄⠀⠀⠀⠀⠀⢠⠀⣤⢀⣀⣐⠀⠀⠀⠀⠀⠀⠈⠁⡠⢐⠊⠔⠃⠀⠘⣿⣼⣧⣻⣴⣾⣴⣇⣿⣣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠠⠀⠌⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⢯⠻⣿⢻⣯⢽⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡿⣯⣣⠙⠀⠀⠘⢿⡿⡿⠋⢘⣿⣿⣿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢯⣤⣤⣀⠀⣀⡤⠗⠚⠿⢟⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠻⣿⣿⣿⣒⡒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⣾⣿⣷⣤⣶⣿⣿⣿⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣟⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⣄⣀⣀⣀⠀⣠⣤⡀⢾⡿⣻⣿⣿⣽⣻⢟⣠⣶⣦⣬⣵⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣾⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠞⠉⠀⠀⠈⡍⠡⠀⡈⡝⠲⠶⠶⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠡⢀⣆⣦⢇⠀⠀⠀⠀⡇⠀⠀⠀⠁⡇⡀⠀⠀⢀⠙⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢉⢠⠀⠀⠈⢿⣿⣼⠀⠀⠀⠸⡇⠀⠀⠀⢰⣇⠁⠀⠀⠈⡴⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠐⢎⣯⠀⠀⠀⠠⣿⣧⡄⠀⠀⢸⡇⠀⠀⠀⢸⢿⠀⠀⠀⢰⡇⠀⠀⠀⠂⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⠀⠀⢈⣾⣧⠀⠀⠀⣿⣿⠈⠀⠀⢸⣧⠀⠀⠀⣸⡀⠀⠀⠀⣿⠓⠀⠀⠀⢠⠆⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⣿⣧⠀⠀⠙⠹⡆⠀⠀⣿⣿⣇⠀⠀⠘⣿⠁⠀⠀⣿⠁⠀⢀⣠⡿⠀⠀⠀⢴⡏⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢻⣧⠐⠀⠀⢻⡄⠀⠸⣷⣿⠐⠀⠀⣿⠀⠀⢠⡏⠀⠀⢰⡿⠁⠀⢀⢠⡟⠀⠀⢠⣶⠻⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠙⢶⡄⠈⠙⢷⡀⠀⠀⢻⡀⠈⢹⣿⡀⠀⠀⣷⡄⠀⢸⠁⠀⠀⣸⠇⠀⠀⢀⡟⠀⠀⣴⡿⠋⠄⣻
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠙⢙⢶⡀⠀⢻⣄⠀⠘⣧⠀⠀⣿⣇⠀⠀⣿⠃⢰⣾⡆⠀⢸⣿⠃⠀⢢⡿⠀⢀⡾⠋⠀⣠⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠉⠹⢦⡀⠹⣦⠀⠘⣇⠀⣿⣿⣤⠇⣿⠀⡘⡿⠀⠀⢸⡇⠀⣠⡿⠃⣠⡞⢁⣤⡾⢉⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠻⣄⠘⣧⡀⠹⡆⢸⢿⣇⣿⣿⢀⢻⡧⠀⢰⡟⠑⢈⡿⠁⡴⢋⣴⠟⢋⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣌⡳⣌⢷⣰⣿⡀⢿⣿⣿⣿⣼⣼⠃⣸⣿⡅⣸⡞⣡⡼⡵⣋⣽⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⣷⣿⡜⣧⠘⣿⣿⣿⡿⣿⢀⣽⣿⣶⣿⣵⣿⢏⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠻⣾⣷⢿⡏⣿⡇⡟⢺⣿⣿⣿⣿⢟⠁⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣤⣤⢼⣿⣷⣧⣿⢰⡇⣼⣿⡵⠿⠦⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⣀⣴⠶⠟⠛⠛⠛⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠉⠀⠀⠀⠀⠉⠛⠷⣦⣤⣴⠾⠋⠁⣀⣠⣤⣄⣀⡀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢀⣤⠶⠟⠛⠛⠶⣦⣄⡀⠀⢀⣠⣶⠟⠋⠉⠉⠉⠉⠻⣦⡀⠈⠿⠶⡶⠶⠶⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⠟⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠉⠛⢻⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣄⣀⣀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⢠⣴⠟⠋⠉⠉⠉⠉⠁⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠉⠉⠉⠉⠙⠻⣦⡀⠀⠸⣷
⠀⠀⠀⠀⣴⠟⠀⢀⣴⠾⠛⠛⠛⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⣿
⠀⠀⠀⢰⡟⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠸⣧⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠹⣷⣀
⠀⠀⠀⠀⢿⡄⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢷⣤⣀⠈⠛⢷⣄
⠀⠀⠀⠀⣼⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠈⠙⣷⡀⠀⠹⣧
⠀⠀⣴⡾⢋⣴⠾⠛⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢻
⢠⡾⠃⢠⡿⠁⠀⠀⠀⠀⠀⠙⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⣿⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸
⣾⠃⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣸⠀⠀⢠⡏⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⣾
⢿⡀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⢧⠀⠀⣿⠀⠀⣾⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⢀⣼⠏
⠘⣷⡀⠀⠹⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⣀⣼⡷⠶⠿⠶⢶⣧⡀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠉⣀⣤⡶⠛⠁
⠀⠈⠻⢶⣤⣀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡾⣋⡵⠖⢳⡄⠀⠀⠈⠻⣦⡴⠋⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⢠⡟⢰⠋⠀⣀⡼⠁⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢀⣴⠾⠋⣁⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣀⠈⣙⣷⣦⣤⣶⣦⣤⣼⡇⣨⡛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣷⠶⠶⠶⠿⢷⣤⣾⣋⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠟⠛⠉⠀⠀⠠⣤⣄⣹⣇⢧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣆⣀⣤⣤⣤⣤⣤⣀⡀⢀⣼⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤
⠀⠀⢀⣴⠶⠟⣻⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠷⠶⣦⣄
⠀⢠⣿⠁⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⡀⠹⣧
⠀⠸⣷⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇⠀⣿
⠀⠀⠻⣦⣀⠈⠛⠛⠷⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣴⡶⠶⠾⠛⠉⣀⣴⠏
⠀⠀⠀⠹⣟⠛⠷⣦⡀⠀⠙⢷⣤⣀⣀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⠀⠀⠀⢀⣠⣴⠟⠋⠉⣀⣀⣤⣤⣤⣤⡶⠶⢿⡿⠁
⠀⠀⠀⠀⠹⣧⣀⠈⣿⣆⡀⠀⠈⠉⠉⣉⣀⣀⠀⠈⠙⠻⢶⣤⣀⣀⣠⣤⡶⠟⠋⠉⠉⠉⠉⠙⠛⠛⠛⠋⠉⠀⣀⣴⢿⡿⠉⠁⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣟⠻⠟⠟⠛⠋⢿⡟⠛⢷⣤⡀⠀⠈⠉⠉⠉⠀⣀⣤⣶⣿⠟⠛⠷⠶⣦⣤⣤⣤⡶⠾⠋⢡⣾⣧⣤⣴⠶⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣤⣀⣀⣈⣻⣆⡀⠈⠛⠷⣦⣤⣴⠶⠟⠋⢡⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠻⣦⣄⣀⠀⠀⠀⣀⣠⡾⠟⠉⠛⠶⣦⣤⣤⣤⣤⣴⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡌⠁⠘⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠙⠈⢀⠀⢀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣼⡄⠀⠀⠀⠀⢠⠀⠂⢀⠀⠑⢦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢻⠿⣿⡇⠀⠀⠰⠀⠀⠂⢀⣀⠈⣷⡏⢳⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡞⠁⠀⠀⢘⡿⡄⠀⠀⢳⠀⠘⣮⠻⣿⣿⣿⡍⢻⡌⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⣾⡧⠀⠀⠀⢺⢿⠃⠀⠀⠀⠳⣄⠈⠲⣤⡤⠿⣶⠼⠃
⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⣰⣿⡃⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠈⠳⠤⠤⣤⣤⣀⡴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠉⠀⠀⡠⢊⡼⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⣊⡴⠛⠁⠀⠀⠀⢀⠀⠀⠠⠀⡀⣠⠤
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠶⠒⠚⠁⠀⠀⠀⡀⢠⢀⣠⠌⣂⠐⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣈⣐⠤⠌⢈⠁⠴⠖⠚⠋⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣤⣦⣉⣉⣤⣤⣄⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⣿⣿⣿⣿⣿⣟⡛⣻⣏⣉⣹⠚⠛⢓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠉⠙⠛⠻⠿⠷⣾⣿⣞⣋⡤⠀⣒⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⢦⡀⠀⠈⠙⠶⠾⠟⠚⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣀⣄⣠⡴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⣦⣀⠀⠀⠀⠈⠉⠒⠀⠀⠀⠁⠀⠀⠸⠇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⢦⣄⠀⠀⣼⠀⠀⠀⠈⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢤⣠⣴⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⣿⣵⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢂⠀⠀⠀⠀⠀⠀⠙⢷⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣀⠀⠀⠀⢀⠐⠢⣄⠀⠠⠈⠠⣄⠀⠀⠀⠊⠁⣾⡏⠀⠀⠀⠀⢠⡄⢀⣀⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣈⡀⠀⠈⠙⠀⠀⠀⠙⠃⡀⠀⠀⣸⡿⠂⠀⠴⠃⠀⠹⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠤⠂⠂⠉⠉⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠠⠶⣎⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠉⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⣼⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠀⠀⠀⢀⣠⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠉⠁⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡉⠉⠁⠀⠀⠀⠀⠀⠙⢿⣅
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠶⢶⣯⡴⠇⠀⠀⠤⠤⣿⣟⣠⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠒⢄⡈⠳⣌⢀⠠⣼⠓⠀⢀⣿⠃⠀⠀⠀⠘⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠈⢳⡄⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠈⠓⠀⠀⢹⣿⡟⠛⠛⠛⠓⠒⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⣿⡀⢀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣷⣿⠿⠆⠀⠀⠀⣴⣿⡷⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⣿⡟⠁⠀⠈⡄⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⣾⣿⠇⠀⠀⠀⠹⠃⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣶⣶⣾⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉


⣿⠿⠿⠿⡿⠟⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣟⣻⣟⣿⣿⣿⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢤⣤⢔⡪⠋⠉⢲⣄
⠀⠀⠀⠀⢀⣀⣠⡷⢰⢸⠈⠉⠉⠄⠘⠈⠌⠟⠟⠟⠏⠢
⠀⠀⢀⠊⠗⠛⠉⠈⠀⠘⠀⠀⠁⠀⠁⠰⠈⡈⠈⠎⢊⡀⢆⡀
⠀⠀⢸⡴⢠⢰⠠⢀⠀⠀⡄⠀⠀⢠⠀⠀⢡⢰⠐⠈⠀⠀⠰⠼⣕⡄
⡠⡺⠟⠑⠀⠀⠀⠈⢠⠀⠀⠀⠀⠈⡀⠀⠈⠀⠀⢀⠄⠁⠀⠀⣈⠇
⢡⣵⡐⠀⠁⢁⠁⠐⡈⠠⠠⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠨⡢⡀
⣾⠏⠳⠀⢂⠌⡀⠈⠁⠀⠀⠘⡤⠈⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡈⡀
⠫⢈⠐⣥⠀⠆⠀⠀⠊⢀⠁⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠚⠁
⠀⠀⠈⠒⠍⡎⢄⣀⡒⠁⠀⠀⡡⢔⠢⠤⣁⠂⡀⠀⢀⠤⣀⠔⠁
⠀⠀⠀⠀⠀⠈⠑⠢⠥⣀⣀⡌⡔⠀⠀⠀⢈⣧⣐⣀⣀⠔⠁
⠀⠀⠀⠒⣶⣶⣾⡿⡿⣛⣿⡄⠀⣀⣠⠤⢂⣮⣿⠷⡶⠖⠒⠒⢶⣦⣶⣶⠶⠖⢲⣶⣐⣀⣀⣀⣀⣠⡀⠀⢀⡀
⢀⣔⣈⣭⡠⡑⢕⣬⣺⢷⡾⠾⣲⣬⣭⡴⠟⢛⣁⠀⠈⠀⠒⠠⠀⢈⡹⣯⣽⣲⡒⠂⠐⣒⣒⠒⢀⡀⠐⠀⠒
⠀⢏⠉⠙⣿⣿⣋⠂⠉⠉⠀⠀⡰⠟⠉⠉⢀⢤⡶⠶⢦⡀⠀⢀⣠⣿⠋⢨⢓⣾⠅⣩⠍⠍⠉⠁
⠀⠘⢛⢶⣍⡻⢶⣶⠊⠙⠿⣦⣔⣉⣠⡺⠗⠍⠀⠀⠀⠈⡟⠛⠻⠀⢀⣠⡞⠒⢒⡒
⠀⠀⠀⠀⠉⠻⣄⠐⡁⠀⡄⠀⠀⠀⠈⠀⢈⠠⠂⠀⠀⡀⣡⣞⣛⣉⣩⣥⠠⠄
⠀⠀⠀⠀⠀⠀⠘⢳⣤⣄⣠⣤⡀⠀⠀⠐⢀⣴⣶⡲⠞⠋⠩⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠛⣳⠶⠶⠯⠅⠉⠁⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣫⣟⢦⣠⡞⠉⠉⢳⡴⠋⠀⠙⣆⡴⠛⠛⣦⢀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⣀⠀⣴⠋⠙⢾⠘⠃⠘⠋⣿⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⣼⠋⠀⠈⣷⢖⣒⢦
⠀⠀⠀⠀⢀⡞⠉⠹⡇⠀⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢀⡏⠀⠀⢠⠇⠋⠙⢺⡇
⠀⠀⠀⠀⠘⡇⠀⠀⢧⠀⠀⠀⣇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⠃⠀⠀⢸⠁⠀⢀⡟⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢷⠀⠀⠸⡄⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⡟⠀⠀⡼⠁⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢳⡀⠀⠈⣇⠀⠀⢸⠀⠀⠀⠀⡇⠀⢀⡇⠀⠀⣸⠃⠀⣰⠃⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⢳⡀⠀⠸⡄⠀⢸⡆⠀⠀⢸⡇⠀⢸⠁⠀⢠⡏⠀⢰⢃⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢧⢳⡀⠀⢳⡀⠈⡇⠀⠀⢸⠇⠀⡿⠀⠀⡾⠀⢠⢏⡏⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⢣⡹⣄⠀⢳⠀⢿⠀⠀⢸⠀⢠⡇⠀⣼⠁⢠⢯⠏⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠜⢦⡈⢷⡸⡆⠀⣼⠀⣼⠀⣰⠃⡴⠃⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣟⣛⣓⣶⣬⣦⣄⠱⣄⣳⣷⠀⣿⢠⠇⡼⣣⣾⣷⣋⡥⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡥⠤⠤⠭⠭⢿⣿⣿⣶⣽⣿⣷⣿⣿⣾⣿⣿⣯⣵⣶⣖⡒⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠾⠯⠭⠽⠟⠛⠋⠉⠉⠉⠀⠀⠉⠉⠙⠛⠲⠦⣬⣉⠉⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣷⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡿⠟⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⠻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⢈⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣶⣦⣴⣶⣿⣿⣿⣿⣶⣦⣤⣶⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠋⠙⣿⣿⣿⡇⠀⠀⢹⣿⣿⣿⠋⠙⠻⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡄⠀⠀⠀⠀⢠⣿⣿⣿⣧⣴⣦⣼⣿⣿⣿⡀⠀⠀⠀⠀⣠⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣯⡀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⣿⣿⣿⣿⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣶⣤⡀⢀⣴⣾⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⡿⠟⠋⠛⢿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠟⠉⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠃⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡃⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠀⠀⠀⠀⠀⢀
⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⢤⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆⠀⠀⠐⠃⡀
⠀⢀⣀⠀⠐⣬⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⣠⠏⠀⠀⠀⢀⠴⠀⠀⣁⠈⠀⠀⠀⢀⠀⠀⡴⠋
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⣠⠜⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⢀⣀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⢀⡈⢉⠀⠀⠉⡠⠆⠀⠃
⠀⠀⠀⣀⠀⠀⠀⠋⠀⠈⠛⠂⠴⠓⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠰⠃⠀⠀⡄
⠀⠀⠂⠌⠀⠀⣀⢀⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠀⠐⠐⠁
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠛⡒⠤⠀⠀⢸⠀⠀⠀⠀⠂⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠀⠀⡀⢀⡠⠚⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠄⠄⠀⠀⠀⠀⡀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢡⠀⠀⠀⡇⠀⠀⠀⢀⠌⠀⠠⠒⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠱⠄⠀⠣⡀⠐⠁⠿⠀⠀⠀⠀⠀⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠐⠀⣀⠀⡀⠀⠀⠀⠀⠋⠒⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⡴⠒⠀⠀⠀⠀⠀⠐⡌⠓⠀⠃⠀⠀⠀⠀⠀⠁⠂⠀⠀⠐⢢⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠐⣀⣀⡄⠐⠀⣠⠞
⠀⠀⠤⠀⠀⠓⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⣴⢟⣉⡁⢀⠀⡀⠀⠀⠀⠀⣉⡡⠤⠔⠋
⠀⣀⠀⠀⠀⠀⠀⠀⢠⠖⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠷⠀⠀⠀⢀⡤⠀⡆⠀⠔⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠒⠋⠁⠐⢀⣡⣌⠢⣀⡤⠟
⠀⠉⠀⠀⠀⠀⠀⢠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠚⠋⠉⠉⠈⠉⠈⡧⣈⠂
⠀⠀⠀⠀⠀⠘⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⣠⡄⠀⠹⡀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⡰⠃⠀⠀⠀⠄⠚⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⡀⠀⠈⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠋⠀⠀⠀⠀⠀⠀⠀⡀⢠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠢⢌⠙⢤⠀⠀⠀⠀⠀⠁⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⢇⠀⠀⠀⠀⢤⣀⠀⠀⠀⣠⠂⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠒⠒⠀⠀⠀⠀⠀⠐⢄⠀⠈⠀⠀⠀⠀⠀⢀⠀⡸⠁⠀⠀⠀⠀⠈⠗⢠⡎⠀⠀⠀⡀⠀⠀⠀⠀⠈⡦⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⡤⠒⢀⠀⠄⢀⠀⣀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠈⠤⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⢀⠤⠀⠀⠀⠀⠃⠀⠀⠠⠀⠸
⠀⠀⠀⠀⠁⠊⠀⠀⠀⠈⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣀⠀⢀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠐⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠶⣤⣀⠀⠀⢀⣤⠖⠛⠛⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⣤⣤⣤⣀⠉⠛⠛⢉⣠⡴⠶⠒⠶⢦⡀⠙⣦⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣰⠏⣠⠞⠉⠀⠀⠀⠈⠙⠳⡾⠋⠁⠀⠀⠀⠀⠀⠻⣦⣀⣉⣉⣀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢉⣉⣉⣡⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠀⠈⠉⠳⣄⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⡾⠋⠁⠈⠁⠈⠑⣄⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢸⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣠⠴⠻⢦⣄⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⢃⡼⠧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢸⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠙⣦⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠋⣴⠋⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢸⠀⠀⢠⠇⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠐⡇⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠘⡄⠀⢸⠀⠀⡞⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⣹⣤⣼⣤⣴⡃⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⢁⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣈⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⠈⠳⣤⢞⡭⠖⣦⠀⠈⠙⢦⡴⠁⠀⠀⠀⠀⣀⡴⠞⢋⣡⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣈⠙⠶⣄⡀⠀⠀⠀⢰⠇⢎⣀⠴⠃⠀⠀⠀⠈⣷⠀⣀⣀⡴⠛⣡⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⣦⠾⠛⠛⢋⡙⣻⣴⣷⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⠀⠈⠉⠉⢷⣁⣤⣤⣤⣤⣀⢀⣴⠟⠒⠂⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⣫⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡈⠛⠲⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⠴⠶⠞⠋⣠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡍⠛⣦⡀⠙⠷⠦⠤⠶⠖⠶⢦⣄⡀⠀⠀⠀⢀⣠⡤⠶⠦⣤⣤⣤⣤⡤⠞⠋⣁⣤⠴⠶⠶⠶⠚⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣼⣿⡶⣤⡤⠶⢶⡶⢦⣄⠈⠙⠓⠒⠛⠋⣁⣤⣴⢤⣤⣄⣀⣀⣀⣤⠞⣹⣏⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣤⣀⣀⣳⣄⠈⠛⠶⠤⠤⠶⠛⣩⣾⡁⠀⠀⠀⠉⠉⠁⣠⡴⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠛⠦⣤⣤⣤⡤⠞⠋⠈⠙⠳⠶⠶⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⡤⡄⢤⠦⠄⣤⠀⠀⠀⠀⠀⠀⠘⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠁⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⢻⣮⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣀⣠⡤⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠐⣿⣧⣤⣾⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⠟⠋⠉⢀⣤⣰⣿⣿⣿⣿⣿⣟⣻⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣔⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁⢀⣠⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣊⣽⣿⡿⢛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢫⡆⠀⠀⠀⢀⣾⣿⣿⣿⣿⢛⣯⣿⣿⣿⢿⣿⣿⡿⣿⡟⣫⣴⣾⣿⣿⠿⠟⠸⠿⠇
⠀⠀⠀⠀⠀⠀⡴⠄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠗⣨⣾⡵⢟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⠓⠂⠀⠀⠀⠀⠀⣿⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⢫⣞⣽⠟⠾⣿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠳⣤⡄⣰⠕⠲⣴⡎⢿⣿⣿⣿⣿⡿⢋⣴⣿⡷⠁
⠀⠀⠀⠀⠀⠀⠀⢘⠂⠀⠀⠈⠀⣯⠀⠀⡛⣁⣾⣿⣿⠿⣋⣴⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠀⠀⠀⠙⠿⣿⠿⠟⠋⣁⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡐⠶⣤⣶⢶⣶⣶⣾⠿⠿⠿⠯⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠛⠋

⣄⢌⠀⠰⡘⢃⠀⠀⠋⣰⠟⣱⡿⣻⣿⡟⠀⠀⣰⣿⣿⣿⣄⠀⠘⢀⢀⡄⠐⣶⣌⣿⣿⣿⠿⠟⠋⠁⣿
⡮⣑⣇⠀⠀⠈⠀⠀⠘⠃⠀⠋⣈⣼⣿⣀⣀⣀⣿⣿⣿⡻⡿⢷⣦⠀⠳⣿⣶⣾⣝⣿⣿⡇⠀⠀⠀⠀⠻
⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⠋⠿⣿⣿⣿⣿⡿⣿⡝⢿⡄⠈⡯⠳⠀⠀⠉⠛⠿⢿⣿⡏
⠀⠉⠢⠄⣈⢦⡀⠀⠀⠀⠠⠒⠁⠈⣦⣍⠙⠻⣷⠈⢿⡄⠀⠀⠀⠀⠤⠀⠀⣠⣄⠀⠈⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠙⠀⠀⠒⢦⣄⡀⠀⠘⢯⣝⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠉⢙⣧⣤⣄⣠⣼⣿⡿⢷
⠀⠀⠐⠂⠀⠀⠀⠀⠀⢠⣐⡽⠃⠀⠀⠀⠙⠻⠇⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠲⢿⣿⣿⡷⢞⡫⠅
⠂⠀⠀⠀⢐⣤⠀⢀⢨⣦⣿⡆⠀⣠⣶⣬⡄⣀⣀⠀⣀⠠⠔⠒⡫⠉⠁⠀⠀⠀⠀⠈⣉⠭⣭⡶⠋⢴⣿
⣴⠋⠀⡄⢹⣿⣦⣇⢿⣿⠟⠀⠀⠒⠂⢀⡬⢔⣈⠝⠀⢀⠀⠎⠀⠀⠀⡀⠀⠀⠸⣿⣿⡿⠋⠀⠀⣰⠟
⠇⣀⣴⣃⢀⡉⠙⠛⠈⠁⠀⠀⠀⠀⠐⠻⠶⠿⠟⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠦⡄⠀⠀⣼⡇⣿
⣾⣿⣿⡿⡼⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⢠⢀⢠⣀⡄⠀⢧⡀⠀⣿⠀⠀⠘⠛⠛
⠟⠸⠁⠃⠀⣿⣿⠀⠞⠙⠙⢣⣶⣶⣤⣀⠀⠀⠀⠀⠀⢿⢏⡿⣿⢸⡘⣿⠷⣦⢲⣽⣷⣾⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠍⠁⠀⠤⠤⠞⠫⠪⠫⣾⠇⠀⠀⠀⠀⡀⣤⣾⣷⢿⠜⠗⠀⠀⠘⢷⣗⣽⠇⠀⠀⠀⠈⠻
⠀⢀⠀⠀⠀⢀⡀⠀⠀⠤⣤⣾⠛⠉⣟⢿⡗⣄⡀⣠⡹⡌⣇⣿⣾⣦⠀⠀⠀⠀⠀⠀⣈⣠⠤⠀⠐⠂
⠀⢸⡇⡀⠀⠀⠀⠀⠀⠐⣛⣛⣡⣴⣷⣿⣛⠌⠻⢿⣷⠸⠸⢹⡿⠏⠀⠀⠀⢀⣤⣾⣿⠋⠙
⠀⠀⡇⠈⢢⠀⠐⠶⣬⣴⣒⡶⣶⣖⢰⡀⠉⠳⡄⠈⠁⠀⠀⠈⠃⠀⠀⠀⠸⡿⢏⡝⠁⣀⡍⣀⣠
⠀⠀⢡⠀⠈⠂⠀⠈⢺⣿⣿⠝⠛⠉⠈⠳⠦⠀⠀⠀⠀⠠⠀⠀⠀⡔⠁⠀⣘⢤⠩⠂⡴⢶⣞⣉⡀
⠄⠀⠞⡀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣦⠀⠀⠀⠀⠀⠘⣀⠴⠊⠀⠀⠠⢀⣀⣠⡿⠋⣠⢀⣼
⠄⣶⠀⢷⢹⡿⠀⣀⣴⣿⣦⡀⠀⠀⢠⡿⠭⢉⠀⠀⠀⠀⠀⠴⠋⠁⠀⠀⠀⠀⠀⠛⠻⢿⣵⣾⠷⠟⠁
⢸⡟⡄⠀⠻⢁⣾⡟⢸⣿⣿⣧⣀⣠⣤⣿⣿⣿⡦⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠛⠿⣦⡀
⠎⣿⠡⠄⠠⡿⠘⠁⠼⠿⢻⡿⢿⢿⢿⣿⠛⠻⠿⢿⡕⣠⠞⠀⠀⠀⠀⠀⠀⠻⣦⡈⠿⡿⣷⣶⣶
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡔⣍⣁⣀⣀⣈⣹⡵⣄⣤⣤⣤⣤⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠓⢸⣿⣿⣿⡟⣳⣾⣤⣠⡈⢀⠀⠀⣀⣱⣧⣼⢻⣿⣿⣿⡆⠈⠘⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⣿⢠⡟⠀⣹⢟⣣⣹⣏⣘⣿⣯⡉⢹⡄⣿⣿⣿⣧⠀⠀⠀⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⢧⡀⢤⣿⣿⣿⣿⡶⣟⣽⣿⡿⠟⢛⡛⠿⢿⣿⣿⣿⣇⣿⣿⣿⣿⡄⠠⢀⣯⣉⠹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢸⣿⣿⣿⡇⢠⣿⡟⠁⠐⠀⢸⠃⠀⠂⠉⢿⣷⡌⢸⣿⣿⣿⡇⠀⠀⠀⠸⡗⢻⣿⣷⣾⣤⡀
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠀⠛⠀⠀⢸⡀⠀⠀⠚⠈⢿⣷⢸⣿⣿⣿⡇⠀⠀⠀⢀⠟⣫⡿⠿⢿⡃⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠱⢤⠤⢼⣿⣿⣿⡿⣿⡇⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⣼⣿⣾⣿⣿⣿⡷⢤⡤⣤⣼⡼⡿⠁⠰⣟⣙⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⡇⢻⣿⡄⠿⠀⡀⠀⠀⢀⡀⠚⣰⣿⠇⣿⣿⣿⣿⠀⠀⠀⢸⣩⠼⠃⠠⣖⠻⣏⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢹⣿⣿⣿⡄⠻⣿⣶⣄⣁⠐⠂⣀⣥⣾⣿⠋⣸⣿⣿⣿⠇⠀⠀⠀⡘⠁⠀⣴⠲⣎⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠋⠹⠅⠛⠻⢿⡿⠿⠿⣛⡫⣹⠙⠛⠛⠛⠛⠉⠉⢉⡉⠁⣴⠶⡜⠓⢾⡗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣾⣽⢷⡞⣷⢠⡶⣦⢠⢶⡄⣿⢹⡆⠹⢦⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⠼⣿⡁⢸⣧⠟⠸⣠⡇⢙⣿⣇⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠚⠋⠉⠉⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃

⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠉⠙⠳⡤⠶⠠⠀⡀⠀⠀⢀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠠⣼⣿⣓⠦⣄⡀⠀⢀⣀⡀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⣀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⠳⣄⠉⠉⠀⠀⠉⠻⠋⠉⠩⠋⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠴⣚⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣾⠷⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢋⣥⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⣦⡀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠳⢦⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠛⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠁⠀⠀⠀⠀⠁⢐⡟⠀⠀⢀⣀⣠⡴⠖⠋⠁⠀⠀⠀⣨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠈⠛⠻⢯⣍⠀⠀⠀⠀⠀⠀⠀⠘⠓⠚⣻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⣀⣠⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠷⠶⠤⠄⠀⠀⠀⠀⠀⠀⢾⡛⠉⠋⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠃⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣤⢰⣴⣮⡀⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⡼⠃⢿⠹⣿⡏⠇⠙⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣠⢤⡖⠒⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⠉⠁⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣆⠀⡤⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⢀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠲⣝⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⡏⢸⡄⠀⠀⠀⠀⢀⣄⠀⠀⠀⢘⡆⠘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼⠀⠀⠙⠶⣤⣤⠶⠋⠹⣄⣀⣠⡾⠁⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠚⠓⡄⠀⢀⣠⠒⠉⠉⢓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢲⡦⠤⠒⡉⡆⠀⠀⢻⡝⠉⢀⢸⠀⠀⢸⡇⠏⠶⠒⠛⠁⠉⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⠀⠀⢹⡆⠀⠀⢱⠀⠀⣷⢱⠀⠈⢸⠀⠀⣾⢸⠀⠀⡆⠃⠀⢰⡿⡘⠳⠋⠉⢒⠄
⠀⠀⠀⠀⠀⠀⢀⣤⡄⠴⠃⢳⡀⠀⠘⡹⡄⠀⠸⠀⠀⢸⢸⠀⠀⢀⠀⠀⢻⣸⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⢊⢆
⠀⠀⠀⠀⠀⢰⠃⠀⠳⡄⡀⠀⢧⠀⠀⣤⢷⠆⠂⠇⠀⢸⡆⠀⠀⢾⣀⣁⡿⠃⠀⠀⠀⠀⢸⠋⠀⠀⠀⢀⡾⠃⢀⠄⠀⠓⠄
⠀⠀⠀⠀⢀⡼⠳⡀⠀⠹⡜⣀⠈⡆⠀⠘⢼⣄⠠⢰⠈⠹⢷⠓⠐⡇⠀⣸⣿⠦⡜⡾⢀⢄⡏⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⢼
⠀⢠⠞⢛⣏⠀⢈⠈⢦⠀⠙⣿⣆⠸⡄⠐⠙⡇⠀⢸⡄⠀⢸⠀⠀⠃⠀⡇⣁⠀⠀⠀⠀⠠⠳⡄⠀⠀⢠⢣⠀⠀⠀⢀⡤⡞⠚⣄
⠀⢸⠀⠈⠠⡳⣦⠉⠂⠁⠀⠘⣿⠆⠁⠀⠀⣿⠀⠸⡇⠀⠘⣀⠀⠁⢰⢰⡏⢰⠀⠀⣸⡏⠀⠁⠘⢡⠃⠀⠀⠀⢠⣾⠕⠁⢀⠔⠙⠆
⢠⢬⣷⣤⡀⠘⢎⢣⡀⠀⡐⡆⠘⢿⡐⠀⠀⢳⣦⠐⡇⠀⠀⡃⠀⠀⠘⣸⠁⠀⠀⠀⡏⢀⠀⠀⢠⠣⠚⢦⡀⢰⡏⠉⠀⠀⠀⠀⡠⢂⡄
⣿⢀⠀⠀⠛⣦⠀⠁⠻⣇⣠⠓⠀⠈⣷⠀⡀⠸⢹⡀⠏⠀⠘⡇⠀⠀⣀⠛⠀⠀⠀⢰⠀⡜⠀⢠⠃⠀⠀⢠⢽⡟⠁⠀⠀⠀⠀⡨⢚⠵⡆
⢨⠏⠁⢤⡀⠈⠙⣦⠀⠛⣧⠀⠐⡀⠈⠀⣧⠀⠀⡇⢸⠀⠀⢃⠀⣀⣉⠀⡇⠀⠀⢻⠀⠀⠀⠂⠀⠀⠀⣠⠎⠀⡀⠀⠀⣠⠮⠞⢁⠀⢇
⠰⣌⠀⠀⠙⢦⡀⠘⢿⠀⠈⠷⡀⡀⠀⠸⣾⣆⠀⣼⠘⠀⠀⡀⠀⢠⡟⠀⢰⠐⠻⡀⠀⠀⣠⠞⠀⢀⡜⠁⠈⠀⢠⠴⠿⡡⠀⠀⠀⢀⣸
⠀⠈⢦⠀⠙⠆⠉⠣⠀⠀⣦⡀⠙⠄⠀⡀⠘⣿⠀⠈⡇⡇⢸⣿⡇⠸⠁⢰⡏⠀⣰⡟⠀⡯⠶⠀⠀⠊⡎⠀⠀⠀⢉⡤⠊⠁⢀⠠⢒⢯⡔
⠀⠀⢸⠀⠀⠀⠙⠀⠀⠄⠈⠳⡄⠀⢴⣶⡀⠹⡇⠀⢻⠀⠀⣿⠃⠀⢰⣼⠃⢰⡿⠁⢠⠀⠀⢀⣀⠄⠀⣠⢊⠔⢋⠀⠠⣆⠴⠫⠔⢸⠃
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠠⡀⠙⢶⡀⠹⣥⠀⢸⠀⠘⠀⠀⢿⠀⠀⡞⠃⠀⣼⡟⢰⠏⠀⢠⡾⠋⡀⠈⠀⠁⣠⣴⠂⠈⠁⠀⠀⡰⠃
⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠈⠲⣄⠻⣄⠙⢧⠀⣧⠀⠀⠀⠀⠀⠾⠀⡄⢠⠁⢀⠋⠀⢠⠌⢀⠜⠑⠋⠀⠊⠁⢠⠖⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠈⠀⡈⠦⠈⠁⠸⡄⡀⠀⡆⠀⠀⢈⠀⡀⠀⡞⠀⢠⠏⠀⠁⠀⣀⠔⠀⠀⠀⣠⠀⣤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠤⣀⠀⠀⠐⠀⢥⠀⡆⠃⢠⠀⡇⠀⠀⡀⠀⠀⡠⠀⠀⢀⠔⠀⡠⠞⠁⠀⠀⠐⢏⣠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢦⢄⡀⠢⡀⡀⢸⡀⡀⡄⢠⢀⠀⠀⠀⠁⠀⢀⠊⡠⠊⠀⣀⣤⣦⣴⡞⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠛⢦⡈⠲⡘⠇⠁⠀⠂⠀⡀⠘⠀⠀⠀⠁⢀⣤⠶⠋⠉⠃⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⢢⡈⢀⠠⠀⠀⠀⠁⠀⠀⠀⣠⡖⠛⠁⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⣀⡴⠾⠋⠀⡀⠀⠀⡁⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠀⠀⠀⠀⠀⠀⠐⠒⠄⠈⠁⠀⠄⠀⠀⠂⠀⠀⢀⡴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠈⠄⠀⠀⠀⢠⣶⣿⡿⣿⣿⣿⣿⣿⡦⠀⠀⢠⣾⣿⣿⡿⠿⠿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣹⣳⢿⣿⣿⡄⠀⢾⣿⣿⠃⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⣿⣿⣿⣷⡿⠛⠉⠀⠀⠈⠛⣿⣦⡀⠀⣠⣲⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠠⡀⠀⠀⢰⣿⣿⣿⠟⠛⠻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁⠀⢈⣁⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠦⠀⠀⠀⢠⣭⣯⣿⣄⣀⣠⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣉⢉⠀⠈⢉⡀⠄⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠛⠛⠻⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣷⣶⡄⠀⠀⢀⢀⣿⣿⠿⠿⣿⣿⡀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠟⠛⠻⣿⣿⣿⣷⠀⠀⠘⢿⣿⡇⠀⠀⢈⣿⣿⠃⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⣠⣭⣭⣭⣄⣀⣠⣭⣭⣭⣄⠀⠀⠀⠾⢿⣿⣶⣶⣿⡿⠿⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠐⠙⠉⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢶
⠀⠀⠀⠀⠀⡀⠛⣍⠈⠁⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠄⠀⠀⠀⠀⠀⠀⢢⡀⠘⣿⠁⠀⠔⠁
⠀⠀⠀⠀⡀⠀⠀⠸⣄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠻⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⣿⠁⠁⠀⠊
⠀⠀⠀⠀⠙⠀⠂⠠⢸⡂⠠⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂⣿⡤
⠀⠀⠀⠀⠀⠂⣀⠀⠌⣇⢀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡄
⠀⠀⠀⠀⠀⠀⠁⢀⠀⢿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠚⠲⠒⠓⠚⠀⠛⠒⠓⠒⠒⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠰⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠐⠠⡀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⡄⠀⠀⣸⠅⠀⠀⠀⢤⠠⠀⠙⠀⠀⠁⠉⠃⠀⡀
⠀⠀⠂⠀⠀⠀⢲⡆⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⡼⠃⠀⠀⠉⠀⠀⠀⠳⢤⡄⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠠⠤⠤⢴⣾⡿⠃⠈⠉⠂⠀⠁⠀⠀⠀⢠⡼⠄⠀⠀⠀⠀⠀⠈⡓⠀⠈⠈⣙⡦⣄⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣟⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⡀⠐⠉⠹⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠟⠛⠛⠻⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⣀⣤⣴⡿⠉⠉⢿⣦⣤⣀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⣿⣿⣇⢸⡇⣸⣿⣿⠿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢠⣿⣄⢸⣿⣿⣿⣿⣿⣿⡇⣠⣿⡆⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡀⠀⠀⠀⢻⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣦⡀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⣴⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣿⣷⣤⣀⠙⠻⢿⣿⣿⡿⠟⠋⣀⣤⣾⣿⣿⡿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣄⠀⠀⣠⣴⣾⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣾⣟⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣶⠟⠻⢿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⣿⡿⠟⠋⠀⠀⠀⠀⠉⠻⢿⣿⣿⣦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢨⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠄⠀⠊⠀⠀⠀⠀⠀⠀⠙⠙⠂⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠒⠋⠃⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠐⣂⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⡖⠀⠀⠀⠀⠀⣼⡆⠀⠀⠰⡀⠁⠋⠀⢹⡵⠋⠀⠰⢀⠀⢀⡁⢀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡭⠁⠀⠃⠀⠈⠓⠀⠀⠉⠠⠄⠀⠀⠀⠀⠀⠊⠰⠄⠀⠀⠀⠀⠊⣭⢤⡼⢃⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⡇⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠠⠄⠀⠀⠀⠸⣦⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠛⠒⠒⢿⡣⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢴⡾⠛⠓⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣤⡄⢸⣷⠀⣿⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⡏⠉⠁⠐⠘⠀⠀⠀⣤⠠⠞⠃⠰⠆⢠⣤⣌⡶⡚⠀⡆⠀⢩⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⣼⠃⠁⠀⢰⠆⡴⠃⠰⠦⠐⠤⠄⠀⠃⠀⣀⣀⣤⢅⡆⠇⢴⡀⢈⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⣾⠗⠀⢠⡖⠐⠺⢦⡀⡦⠤⢿⣷⣶⣶⣦⣭⣥⣴⣾⣿⣷⣆⣀⠘⣿⣯⡴⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠻⣦⣒⠲⣽⣮⣻⣯⣙⣭⢙⠉⢻⣭⣿⣤⣯⣛⣛⡛⠨⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠚⠛⠉⠀⣯⣽⠙⣈⣁⢉⡞⠂⠨⣻⣿⡃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⢊⠉⠹⠻⡇⠳⢷⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠉⡛⣻⡆⢀⠤⡆⢄⢟⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣿⢸⠀⣘⡇⠸⡀⣴⣿⣨⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⣼⣿⡏⠗⢀⣀⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠃⠿⠿⠁⠶⠌⠁⠀⠙⠋⢹⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠰⠀⣀⠀⠀⠉⠙⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣐⣲⣿⣠⣄⢲⣴⣤⣤⣦⣬⣿⣟⡾⣠⣤⣀⣰⣽⣿⣵⣆⣀⠀⠠⠤⠶⡤⣦⣤⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠤⠤⠬⠿⠿⠶⠌⠐⠂⠒⠚⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠒
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⡐⠉⠗⠶⡶⢶⠶⢶⡖⠛⡿⢶⡀
⠀⠀⠰⠀⠀⠀⠈⠛⠛⠉⠉⠓⢿⡮⣛⣋⡉⠛⠒⣄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠲⢵⣘⣷⣤⢄⡀⠀⣀⣀⣀⠤⠶⠦⢤⡤⠤⠴⢒⡒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠥⣈⠉⠉⠖⠀⠀⢤⣤⠂⠉⠺⠦⠴⣖⡒⠀⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣼⡗⠒⠚⠶⣄⡤⠬⢽⣦⢁⠐⠂⠀⠢⡀
⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⡆⠀⠀⠈⠩⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣌⣑⣀⡢⠚⠃⠬⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⢭⡀⡈⢢⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠍⣢⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠠⢤⣤⣠⣄⠀⠀⢸⣆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⢀⠈⠀⠀⠀⠀⠉⠁⠁⠀⠀⠉⠛⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠷⣶⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠙
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⣿⡌
⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠴⣫⢛⠇
⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠛⣿⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⡰⠋
⠁⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⢴⡀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢈⣲
⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠘⢦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⣤⣄⠞⡰⠀⠦⠀⠁
⠀⠀⢆⢈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢀⡴⡊⠥⠒⠚⠓⠊⠁⠠⠈
⠀⠀⢀⠀⠑⢝⠦⣄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠮⡔⢰⡀⢠⡆⢀⠀⣀⢀⣀⣀⠠⠐⢻⠁⡀⠄⢀⠀⠒⠀⡁⠐
⠀⠀⠀⠁⢀⠀⡑⠈⠉⠓⠌⠀⠀⠀⠄⠈⠦⡑⠈⢻⣛⢤⡖⠴⣶⢶⢖⠾⠓⣴⣶⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠁⠈⡒⠢⠀⠰⠌⣙⣓⠒⢛⠟⠾⠂⠄⠓⠀⠁⠈⠁⠈⡀⠀⡀⠒⠀⡤⠀⠈⠁⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⢠⡀⠀⠀⠀⣀⠄⢆⡀⠀⠀⠀⠀⢀⠞⠂⠁⠀⠀⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠈⠉⠀⠀⠨⠀⠀⠂⠀⠙⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⢀⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢗⠴⠀⢛⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠻⡏⣽⣛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠄⠠⠔⠂⠈⠉⢡⠼⠉⠈⠙⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⡄⣀⢀⡀⢈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⣶⣿⣿⣿⣷⣮⣿⣦⣶⣭⣶⣶⣶⣖
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠀⠀⣤⡴⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⢲⡟⠁⠀⠀⠀⠀⡟⢀⠀⠀⠀⠀⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⡤⠀⠀⠘⡇⢰⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⡟⠁⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⢿⠀⡔⡀⠀⠀⡇⢸⠀⠀⠀⢸⠃⡎⠀⠀⠀⣸⠃⡼⠀⠀⠀⠀⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠀⠀⠀⠘⡆⢰⠁⠀⠀⡇⢸⠀⠀⠀⣼⠀⠇⠀⠀⢀⡏⣰⠁⠀⠀⢀⡼⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣿⠀⢃⠆⠀⠀⠀⢳⠀⠖⠀⠀⡇⢸⠀⠀⠀⡟⠀⠀⠀⠀⣼⢀⠃⠀⠀⢠⠞⠀⠀⡔⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠘⣆⠀⠞⠀⠀⠀⠘⡇⢀⠀⠀⣇⠈⠀⠀⢀⡇⠀⠀⠀⢰⠇⠎⠀⠀⣰⠏⠀⢀⠞⡀⢤⢱⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⠄⠀⠀⠘⡆⠀⠁⡀⠀⠀⢻⡘⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⠀⣴⠃⠀⡠⠋⠀⡸⣌⡾⠷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⢻⡄⢸⠀⠀⠀⠹⡄⠸⢀⠀⠀⠈⣧⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⢰⡇⠀⠀⣼⠃⠀⡼⠁⠀⢦⡵⠋⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⢠⡇⢄⠀⢿⡄⠸⠀⠀⠀⢹⡄⠸⠀⠀⠀⢻⡀⠀⢸⠀⠀⠀⣿⠀⠀⠀⣾⠀⠀⣸⠃⢀⡜⠀⢰⣼⠏⠀⢀⠄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠈⢦⠈⢻⣄⠸⠀⠀⠀⢻⡀⠉⡀⠀⠸⡇⠀⢸⠀⠀⢀⡇⠀⠀⢠⠇⠀⢰⠃⢀⠎⣰⣼⠟⠁⣠⠜⠁⢀⠀⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠣⡀⠘⣆⠀⠀⠀⠀⢳⠀⠁⠀⠀⣧⠀⢸⠀⠀⢸⡇⠀⠀⡼⠀⢠⠇⠀⠂⣠⠟⠁⡠⠚⠁⡀⢰⢸⣤⠿⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠐⠀⠘⣆⠀⠀⠀⠀⢧⢸⠀⠀⢻⠀⠘⠀⠀⢸⠃⠀⢠⠇⢀⠏⠀⢀⡴⠋⢀⠐⡁⠄⣦⣱⠞⠋⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⠀⠘⢦⠀⠀⠀⠘⡆⢰⠀⠸⡆⠀⠀⠀⣾⠀⠀⡼⢀⠊⠀⡰⠋⠀⠀⠁⣤⣧⠟⢋⡠⠐⠁⠀⡀⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢧⠀⠀⠀⠸⡀⠀⠀⣧⠀⠀⠀⡟⠀⠀⠃⠀⣠⠎⠁⠀⠀⣤⠷⠋⠀⠂⠁⢀⢀⢄⣦⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⢣⣀⢀⠀⠡⠀⠀⠸⠀⠀⠀⡇⠀⠀⢀⠜⠁⠀⢀⡴⠋⠁⠀⠀⢠⣀⣠⡾⠾⠛⠉⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⣄⠀⠻⡆⢣⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠠⠔⠁⠀⢀⣀⡤⠖⠋⠉⠄⠂⣀⣠⣴⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠙⢦⡀⠹⡈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠚⠉⠀⠀⠀⣠⣰⣤⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⢠⡀⠙⠦⣱⡈⢧⣤⡀⡀⢄⣀⢀⣠⠄⠀⠀⠀⠀⠀⠀⣀⣴⡶⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠉⠙⣻⢶⣾⣿⣶⣈⠉⠛⠻⠿⠿⠒⠊⡀⠠⠤⠖⣺⣿⠿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠤⣼⣯⣥⣤⣀⣤⣤⣀⡀⠀⠀⠀⠀⢀⠀⣰⣧⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠒⠒⠶⠶⠶⠶⠾⢿⣿⣿⣷⣶⣶⣿⡴⠿⣶⣶⣶⣾⣿⣿⣿⣿⣿⡶⠤⠤⠤⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢀⠀⠀⠈
⡈⠇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⢠⡡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣧⢸⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣆⡀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢮⣛⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣌⠳⣻⡵⡃⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠐
⠀⠦⡆⠘⢄⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠈⠯⢳⡵⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢰⡄⠙⠠⠂⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⢰⠀⠀⠁⠘⢃⠉⣀⡀⠛⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⣨⠃⠀⢲⣁⡀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠁⡴⠂⠈⠈⣌⠁⢋⣠⠆⠀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀
⢐⠀⠱⠀⠈⢑⠀⠈⠉⠀⠀⠠⠖⡄⠀⠞⠀⠀⠀⠀⠈⣄⠩⠖⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠾
⠠⠀⡀⠀⠀⣨⠆⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠂⢠⣶⣿⣿⠋⠀⠀⢸⣿⣿⣧⠀⠀⠐⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⣤⣶⣿⠟⠉⠀⢰⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣿⡆⠀⢀⠹⣿⣿⣿⣷⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣼⣿⢟⠅⢀⢀⣬⣿⣿⣿⣿⡀⠀⢀⣀⡘⠿⣿⣿⣷⣇⠈⠀⢹⣿⣿⣿⣿⣟⠹⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⢀⣼⣿⣿⠿⢃⡢⠄⠀⠠⢍⣛⠛⣛⣉⠔⠈⠉⠃⠁⠁⠢⣙⠛⣋⡡⠄⠤⢝⠻⣿⣿⣿⡳⠸⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠐⠚⠿⠟⣋⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠇⠀⠀⠠⠀⠀⠑⢌⠻⠿⠗⠓⠙⢿⣿⣿⣄
⠀⠀⠀⠀⠀⢀⣤⣾⠟⣡⠊⠁⠉⠉⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠉⢂⢻⣿⣿⣧
⠀⠀⠀⠀⠠⢠⠄⠂⠈⠀⠁⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠈⠀⢰⠇⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠻⠟⠻⣆
⠀⠀⠀⠀⠃⣘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠉⠉⠈
⠀⠀⠀⠀⠈⠓⠀⠀⠂⠄⠀⣀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⡀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢀⠠⠶⣾⣿⣿⣷⣶⣀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠒⡀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣯⡅⠀⠈⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡀⠀⠈⢳
⠀⠀⠀⠀⠐⠊⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⠁⠀⠀⢈⡇⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠸⠙⣿⡟⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣤⣿⣿⣿⣿⡗⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠠⣠⣤⣤⣤⣴⣿⣿⡇
⠀⠀⠀⠀⠀⢀⠹⣿⡄⢀⣠⣤⡀⠡⠀⠀⠀⢀⣀⡀⠀⠆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠠⡸⠀⠀⠀⢀⠄⣀⡀⢀⣴⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄⢙⢿⣿⣄⠂⠀⢀⣴⣦⣌⠁⠀⢜⣿⣿⠿⠛⠛⠻⢿⣿⣿⣿⠋⠄⣀⣚⡛⠃⠀⠀⢀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⡈⠻⣿⣦⠀⠈⠺⣿⣿⣷⠦⠈⠁⢴⣾⣿⣿⠀⠀⠉⠉⣤⣶⣿⣿⣿⣿⠇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣜⢿⣷⣄⠀⠈⠈⠿⣷⡄⠀⠈⣿⣿⣝⠀⠀⠀⠰⣹⣿⣿⡿⣿⠋⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡌⠻⢕⢀⠀⠀⠈⠻⣦⠀⠙⢿⣯⠀⠀⠀⢰⣿⣿⣿⠟⠂⢠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣯⣦⡄⠀⠈⠳⡀⠀⠹⡄⠀⡠⣾⣿⣿⠋⢀⣴⣿⣿⣿⡿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠀⠠⠄⠀⣵⣿⠿⠇⠠⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡠⠤⣀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠴⢦⠸⡁⠘⡉⡇⠀⡏⡇⠀⡎⡟⠀⡦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣼⠉⠏⡆⠘⡄⠇⠀⡇⡇⠀⠇⡇⢰⠇⠃⢀⢃⠇⢠⣳⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⡄⠱⢡⠘⢸⠀⢱⠸⡀⠇⢃⢸⠀⡇⢸⢸⠀⡘⠘⢀⢣⠇⢰⢳⢆
⠀⠀⠀⠀⠀⠀⠀⡸⡌⢧⡘⣄⢣⢧⢃⢃⠀⡇⢃⢸⠘⢸⢰⠀⡆⡆⢀⢇⠃⡎⡌⢠⢃⠂⢜⡦
⠀⠀⠀⠀⠀⠀⡘⠙⢿⣆⠱⡌⢆⢫⡄⠊⡆⢡⠈⠘⢸⢸⢸⠀⠃⠁⡘⠘⡸⡐⢀⢆⢆⠎⠜⢠⢧⡀
⠀⠀⠀⠀⠀⠀⡚⢦⡈⢫⡣⡘⢌⢆⢞⠌⡜⡀⡆⡀⠈⢸⡀⠐⠰⢠⢁⠣⢁⠃⠎⢌⢌⠎⡰⡡⢁⠶
⠀⠀⠀⠀⠀⢠⡗⢄⠙⡄⠱⡘⣌⢊⢈⠐⠘⣣⠀⠀⠀⠘⠁⠀⠃⡘⡌⠄⠄⡜⠜⢌⠎⡜⠔⣠⢋⠔⡆
⠀⠀⠀⠀⠀⢸⡈⠢⡑⢌⠢⡘⢌⢂⠠⡌⢧⠠⠀⠀⠀⡀⠀⠀⢀⠁⡀⠌⠐⠀⢘⢊⢎⠎⡔⡵⣡⢞⡇
⠀⠀⠀⠀⠀⠈⡌⠢⡈⠂⠑⢜⢎⠣⠣⢹⡆⠂⢸⡄⣷⡇⡀⢘⠈⠆⡈⠠⠀⠀⡠⣡⠫⣮⢊⠞⡡⠫⠁
⠀⠀⠀⠀⠀⠀⠈⠢⢈⠣⢄⠙⣧⡱⡑⡡⠹⣦⡘⣇⢻⣷⠁⡦⠁⡄⢠⠁⠀⠔⠐⢑⣴⠟⡡⢊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡑⢌⠫⣸⣜⢧⢻⣧⣿⢸⣿⢰⡇⢰⠃⢡⡔⣑⣬⣾⢕⢕⡩⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢕⡈⠻⣯⢿⣿⣿⣿⣿⣾⢇⡿⢠⣟⣾⣿⣿⠗⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠛⢿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡈⢻⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⡯⠟⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣰⢋⡾⢘⡊⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⣀⡴⠋⠩⢈⡿⣠⣎⣤⡤⣖⠶⣴⣐⠢⠒⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠛⠤⠶⢠⢞⡴⠛⠉⠁⠀⠀⠀⠀⠈⠀⠈⠉⠋⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢃⠄⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣚⡼⣡⣎⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣢
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⣐⠺⢗⣪⣭⣕⡡⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⣰⢏⡶⠗⠛⠛⠛⡟⠉⠁⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡆⠀⡠⡾⠑⠁⠀⠀⠀⠀⢨⢠⠀⠂⠀⠀⣺⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡴⠊
⠀⠀⢾⡼⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡙⣆⣺⠟⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⢟⠇
⠀⢰⡿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⡴⠊⠁
⢐⣫⡿⢮⡻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⠔⠛⠐⠉⠁
⠹⡻⣽⡾⣿⣷⣄⠀⠀⢀⣠⠆⣒⡒⠆⢀⠀⠀⢀⡀⠀⡀⢀⠞⠋⠑⠶⠶⠂⠚
⠀⠌⠠⠴⠚⠻⣖⣤⡴⠍⠙⠉⠉⠉⠲⢄⣄⠉⢄⣠⡴⠎⠁

⠀⠀⠄⠙⠒⠶⠲⠶⠶⣖⠚⣻⢷⡀
⠀⠀⠒⠀⠀⠀⠉⠛⠁⠉⠙⠺⠇⢙⣭⡌⠉⡰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⢽⡷⣦⣤⠀⣀⣠⡤⠔⠒⠒⠢⠴⡒⠚⠋⠑⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠉⢂⠀⢰⡖⢀⣉⠙⠒⠺⢥⡀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠋⠉⠉⠓⠐⠒⠓⢻⡖⠌⢁⠐⠘⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠂⠀⠀⠀⣂⡜⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠬⣄⠤⠐⢮⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⣔⠦⡱⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠉⢷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠒⠂⠂⠀⠐⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠙⢻⣿⣤⣤⡾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⣿⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯
⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⡻⠻
⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢹⠗⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠴⠃
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠰⣆⠀⡙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⢈⡶
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⢀⢱⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠠⢤⡤⢋⠔⠀⠂
⠀⠀⠂⢈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠂⠉⠉⠉⠁⠀⠈
⠀⠀⠀⠀⠑⢉⡲⢤⣀⠠⡀⠀⠀⠀⢀⠀⠀⢀⣈⡲⠀⠷⠀⠖⢀⢀⣤⡠⠤⠀⠐⠈⠹⠠⠀⠀⠄⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢄⡀⠀⠁⠠⣄⣀⠀⣤⣓⢌⡀⠐⡐⠿⠉⠹⠻⠭⠋⠁⠚⠛⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠂⠐⠤⠐⠉⠉⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠲⠄⠐⠀⠄⠀⢀⠰⠩
⣤⡤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣷⣽⣷⣬⣿⣶⣥⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠁⠀⠀⠀⠈⠁⠈⠀⠐⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⢀⣀⣄⣀⡴⠦⢄⣤⣤⡀⢀⣄⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⡀⢀⡞⠋⠒⣛⠭⢀⠐⠒⠂⠀⠀⠀⠉⠉⠙⠓⠊⠓⣄⣈⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡏⠉⠀⣀⡤⠊⣀⣠⡤⠶⡔⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢠⠀⠄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣤⡀⠈⢧⣞⠙⢴⣾⣿⣾⠶⠶⢶⣄⣀⣈⣃⡀⠀⢓⠉⠥⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠒⠈⠁⠲⣿⣻⣛⣽⡏⠀⠀⠀⠀⣹⠻⣿⣟⡛⠉⠉⠐⠓⢆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢤⡀⠀⠘⣤⣎⣷⣿⣿⣿⣞⡀⠀⠀⢀⣿⣽⣿⣿⣷⡶⠴⢢⣄⣶⣿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢈⠱⣄⣤⡘⠋⠉⠋⢛⢻⣿⣿⣶⡶⣿⣫⡽⠿⠒⠒⠐⠂⢀⡠⠼⣫⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⣶⠀⡉⠉⢢⠄⠄⢀⣬⣍⣙⣿⡋⢁⣤⣠⡶⠂⢐⠋⣩⢰⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⣧⠀⡏⠀⠀⠀⠀⠀⢀⠀⢀⠀⠰⠂⢀⠇⣼⢀⣯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⣸⡄⢸⡄⢸⣆⣸⠀⡀⠀⣜⠀⡼⣠⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠺⠿⢦⣿⣿⣶⣧⣴⠷⠾⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣶⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⡿⠿⠿⠟⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⢸⣿⣷⣄⣀⣀⣠⣾⠟⢻⣷⣄⣀⣀⣠⣿⣿⡇⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠙⣿⣿⣿⣿⣿⣿⡟⠀⠀⢻⣿⣿⣿⣿⣿⣿⠋⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣿⣷⣌⠙⠛⠻⣿⣿⣿⣾⣷⣿⣿⣿⠟⠛⠋⣡⣾⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣟⠛⠛⠓⠀⠀⣿⢻⡟⣿⣿⢻⡟⣿⠀⠀⠚⠛⠛⣻⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠛⠷⣦⣀⠀⠀⠀⠈⠁⠉⠉⠈⠁⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⣉⠳⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢉⣀⡄⢰⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⣿⡆⠀⣿⣷⠀⢰⣶⡀⣶⡆⢠⣶⣾⣦⢸⣿⠛⠃⢸⣟⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣼⡇⣸⣟⣿⡄⢸⡿⣷⣿⡇⣿⡏⣶⣿⢸⣿⠿⠇⢸⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠛⠁⠿⠋⠻⠷⠸⠇⠘⠿⠃⠙⠿⠿⠿⠸⠿⠿⠆⠸⠟⠘⠿

⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠁⠀⠀⠤⠀⢀⣄⠸⠀⠀⠀⣤⣶⡄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠻⣿⠗⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢰⡖⢰⣶⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⡽⠟⠛⠘⡅⢆⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠤
⢀⡌⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠐⠆⠀⡀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠦⣄⠀⠄
⣿⠇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⢤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠐⠤⠄
⠋⠀⠘⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⢀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢄⠀⠠⠀⡀⠀⠑⠩⣢⡀⠀⢀⠀⠀⠀⠀⠀⠀⢠⡔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀
⠈⠀⠀⣦⡈⡢⠄⡀⠀⠀⠀⠀⠀⢉⣉⣙⡛⠙⠋⠀⠀⠀⠀⠐⠠⢀⠀⡠⡀⠰⡃⠀⠠⠎⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠂⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢶⢶⣦⣃⡀⠀⣤⣀⣉⣩⣭⣍⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠲⣆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹
⠀⠀⠀⡃⠸⣿⢿⣿⣿⣿⣿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠉⠁⠘⣄⠀⠄⠀⢀⡀⠀⠀⠀⢀⠀⠂⠁⠠⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⢀⠀⠀⠘⠛⢿⣏⣛⠙⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢈⡁⠀⢀⡀⠈⠧⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⢤⢀⠀⠀⠀⠀⠀⠈⠙⠳⠀⠀⠀⠀⠐⠀⠀⠀⠀⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢐⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁
⠺⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠰⠂
⣄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠛⣦⡀⣿⣄⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠘⣣⠈⠓⠘⡦⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢰⠀⠀⠀⠀⠢⠀⠐⢄⠐⠀⠀⠀⠘⠋⠁⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠄⠘⡀⣄⢀⢃⠀⠀⠀⠀⠁⡞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠂⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⢠⠼⠏⡐⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠈⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⡈⠀⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠰⣟⣿⣿⣿⡿⠟⡻⠶⠖⠲⢆⣀⣀⠀⠀⠀⢃⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠠⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⣠⣾⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠃
⠀⠀⠀⠀⠀⠇⠀⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣄⠀⠀⠀⠀⠀⠃⢠
⠀⠀⠂⢄⣀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣄⠘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉
⠀⠀⠀⠀⠀⠙⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⢀⠔⣩⠖⠈⠉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢠⣶⠠⠀⠀⠀⠀⠀⠁⠠⢁⠎⠀⢀⠔⡀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⢋⠞⡴⠁⢀
⠉⠀⠀⠀⠀⣘⠿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡌⠐⠀⢠⠃⡠⢀
⠀⠁⠀⠀⠀⠉⠲⠠⠄⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠂⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠰⢁⠂
⠀⠀⠀⠀⠀⢀⡄⣧⠀⠀⠁⠄⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⢸⠇⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⠶⠲⠤⣤⡴⠠⠴⠦⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠈⠈⠁⠉⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠀⠑⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠚⡩⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⠁⢺⠀⠉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠖⢁⡄⠀⠘⠑⠀⠉⠀⠀⠀⠀⠀⢀⠀⠀⠂⢀⣓⡠⠄⢀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠁⠐⡠⠄⠀⡀⠀⠀⢀⡀⢀⢀⠠⠴⣷⡒⠿⣷⡦⠀⠀⠀⡆⢠⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠀⠀⠈⠈⠀⠴⣺⠀⣾⣯⣄⣴⣶⣾⣿⣿⣿⠀⠀⢀⠌⠉⠀⠀⠀⣀⡀⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⠄⢀⡘⠾⠱⢎⣿⣿⡿⠟⠛⠋⠉⠋⡲⢦⡀⠀⠄⠒⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢮⢟⡍⣀⠀⠀⠀⠀⢺⣷⣄⡻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⡏⢻⣯⠀⠀⠀⢀⡠⣻⣿⠟⠻⠀⠀⠐⠒⠀⠀⢀⣀⡠⠀⠉⠑⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠐⠒⠀⠀⠀⠀⣰⣿⣧⠘⢿⢶⣟⣾⢏⡙⠛⠁⠀⣾⣿⣿⣿⠟⠋⠉⠃⠀⠀⠂⠀⠀⠀⠀⠀⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⢁⠐⠂⠀⠀⠀⠀⠁⠀⢿⣿⣷⡀⢿⣿⢷⣿⡿⠋⢀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡤⠠⣄⠀⠀⠀⠀⠀⠙⠛⠻⢶⣤⣤⣤⣶⣶⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠿⣙⠉⢳⣇⠐⠠⠀⠀⣀⣄⠀⠀⠈⠉⣠⣀⣀⡀⠀⣀⣠⡄⠀⠄⠤⢠⣶⡿⠓⢀⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠬⠿⣤⣀⣀⣴⣿⣿⡬⣍⣽⣿⣿⣿⢃⢀⠀⣽⣿⠃⢠⢰⣼⣿⢿⣥⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠿⢷⠿⠿⢿⣷⣿⡷⠾⡿⠿⠶⠟⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠮⡭⠽⣖⠤⠠⣴⠿⠒⢻⠒⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢴⡿⠁⠀⢣⠀⠀⠉⢻⠁⠀⠀⢸⠀⠈⠓⣭⠷⠾⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠛⠛⠻⡀⠀⠀⠸⡄⠀⠀⢸⠀⠀⠀⣼⠀⠀⣰⠁⠀⢀⡞⢸⡇
⠀⠀⠀⠀⠀⠀⣰⡟⠀⢳⡀⠀⠀⢣⠀⠀⠀⢧⠀⠀⢸⠀⠀⠀⡇⠀⢠⠇⠀⢀⠎⠀⠘⣷⣤⣀
⠀⠀⠀⠀⢀⣴⣟⠀⠀⠀⢳⡀⠀⠈⢧⠀⠀⠸⡀⠀⢸⠀⠀⠀⡇⠀⡼⠀⢠⠏⠀⢠⠎⠀⠀⣙⣷⡀
⠀⠀⠀⢠⡟⠁⠈⢦⠀⠀⠀⠱⡄⠀⠈⣆⠀⠀⣇⠀⢸⠀⠀⢰⠇⢰⠃⣰⠃⢀⡴⠁⠀⡠⠚⢁⣿⠃
⠀⠀⠀⠀⢷⡄⠀⠀⠳⣄⠀⠀⠘⢆⠀⠘⣆⠀⢸⡀⢸⠀⠀⢸⢀⡏⣰⠃⡠⠊⢀⡴⠋⢀⡠⠞⣿⡀
⠀⠀⠀⠀⠈⠛⢷⣤⣀⠈⠳⢄⡀⠈⠳⡀⠘⡆⠀⡇⢸⠀⠀⡏⡼⡰⢡⠞⡠⠖⢁⡠⠖⠉⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣉⠲⢄⡙⢄⠘⣆⡵⠼⠦⢼⣱⣷⣵⡷⢊⡠⠖⠉⢀⣀⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣭⣽⣿⣿⢛⣿⠃⠀⠀⠀⠀⠘⡿⡟⠲⡿⢶⣾⣛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⠋⠉⠁⠀⠉⠉⠉⢻⢨⡄⠀⠀⠀⢀⡟⠉⠉⠉⠉⠉⠉⠛⢶⣄⡀
⠀⠀⠀⠀⣠⠞⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢭⣻⡤⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢳⡙⠦⣄
⠀⠀⠀⢸⢧⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠈⢳⡀
⠀⠀⠀⠘⣮⢽⣷⠭⡭⣷⣖⠦⠤⢤⣤⣄⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⢾⣴⣿⣦⣾⠇
⠀⠀⠀⠀⠘⢦⡀⠀⢇⠈⠻⣷⣖⣒⡲⠿⢲⠒⠛⠿⣭⣭⣭⣿⠿⠿⡿⠷⣦⣬⣵⠿⠛⡏⠀⣠⠿⠃
⠀⠀⠀⠀⠀⠀⠉⠑⠺⣄⠀⣇⠀⠀⠀⠀⣼⠀⠀⠀⠈⡏⠉⠀⠀⢸⠁⠀⠀⢸⠀⢀⡴⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠤⠴⠒⠒⠻⣄⡀⠀⡼⢀⣠⠤⠤⠾⢤⣀⣠⠯⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⠴⠚⠻⠉⠉⠙⠖⠛⠲⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⡏⠀⢠⠀⠀⠆⠀⠀⡈⠀⠀⡜⠀⠀⠸⠋⠒⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢻⠀⠀⡇⠀⠸⠀⠰⠀⠀⡜⠀⠀⡜⠀⠀⡠⠃⠀⡰⠁⢀⡹⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⠀⠀⡆⠀⡇⠀⢰⠀⠄⠀⠌⠀⠀⡜⠀⠀⡐⠁⢀⠎⠀⡠⠊⠀⢀⠼⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣇⢸⠀⠀⠇⠀⡇⠀⡀⠘⠀⡌⠀⠀⡜⠀⢀⠜⠀⡔⠁⡠⠊⠀⢀⠔⠁⠀⠀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⢸⠀⡇⠀⠀⠀⡇⠀⡇⠀⡈⠀⠀⡜⠀⠠⠊⢠⠂⢀⠈⠀⡠⠒⠁⠀⡠⠔⠉⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⡆⠇⢸⠀⠀⠀⠃⢰⠇⡐⠀⢀⠌⠀⠔⢁⠔⠁⠄⠁⡠⠊⠀⠠⠐⠁⠀⠀⠀⢀⠠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⢇⢃⢸⠈⡆⠘⢀⠀⡟⠠⠁⢀⠎⢀⠊⡠⠂⡰⠀⡠⠊⡠⠔⠁⠀⠀⠀⡀⠔⠊⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡯⠸⡸⠘⠀⠁⠀⠘⠰⠁⠁⢀⠂⡰⢁⠔⠡⠊⠠⢊⠔⠉⠀⠀⢀⠀⠀⠁⠀⢀⠠⠔⠂⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡇⡇⡇⠁⡆⢀⠃⢁⠂⢠⠂⡜⠠⢊⠄⡐⡩⠊⠁⠀⡠⠐⠈⠀⠀⠠⠀⠈⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢩⠁⣆⡄⣜⠆⡌⢠⢃⠌⡔⡡⡠⡪⠊⢀⡠⠂⠁⢀⠠⠀⠈⠀⠀⢀⡀⠤⠐⠂⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣄⠀⢏⢀⡿⡐⢀⢣⢊⢊⡴⢊⠌⡠⠔⠁⡀⠄⠈⢀⣀⠤⠐⠊⠉⠀⠀⢀⠀⠠⠔⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢹⠘⣹⢠⠳⢠⠣⠣⡡⢂⢔⠁⠊⡠⠂⢁⡠⠄⠊⠁⠀⣀⠠⠀⠀⠀⠁⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢠⢤⡆⡇⣮⢣⡟⣱⠕⢕⠁⡡⠐⣡⠔⠊⠁⢀⠠⠐⠀⢀⣀⡤⠐⠂⠀⠠⠄⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣿⠈⣸⡿⢠⣣⡗⣴⠏⢔⠕⣊⠔⠈⡀⠄⣂⣡⠴⠂⠉⠋⠉⠀⠀⠀⠒⠀⠀⠁⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⣿⣧⣿⡡⣾⣯⡞⣡⢞⡡⢊⣠⡴⠷⠲⠛⠉⠀⠀⣀⠠⠤⠤⠒⠒⠒⠒⠀⣈⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣷⢱⣿⠋⠊⠔⣡⠖⠙⢀⠀⠤⠶⠒⠓⠋⠉⢉⣀⠀⠉⠉⢉⣡⠞⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣇⣋⠕⣴⠕⠊⠅⢒⣈⣤⡴⢶⡶⠟⠋⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣶⡒⠋⠉⢁⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⣳⣤⠄⢀⣠⣴⡾⠛⠚⠛⢶⣶⣤⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⢕⢶⣿⡿⠟⢻⠿⠒⠣⠀⠀⠀⢉⠙⢿⣤⡄⠲⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢾⣷⠋⠀⠃⠁⠀⠀⣀⣄⡀⠀⠀⠽⣿⠿⣾⣷⠀⠀⠀⣀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡿⠖⠀⠀⠀⠔⠉⠀⠀⠈⠁⠀⠀⠈⠀⠈⢻⣶⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⡿⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⢿⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠼⡇⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣇⠀⠀⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣼⡢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⣫⣷⡁⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡿⠉⠛⠦⣌⠐
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢫⡻⢿⣿⣷⡷⣄⠙⠚⠳⢤⣤⠒⠅⠀⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠉⠛⠚⠛⠛⠋⠙⠿⠿⠿⠿⠛⠓⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠓⠒⠒⠊⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠃⠻⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠟⠀⢀⣶⣿⣷⣆⠀⠘⢿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠘⣿⣿⣿⡟⠀⠀⠈⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠋⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠈⢻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠀⠀⠻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆
⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣆
⠀⢠⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧
⣠⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣧
⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣿⣿⣿⡏⡛⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣏⠻⣶⣮⡝⢿⣿⣿⡌⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠀⠈⢻⠀⠀⠈⠪⣁⡒⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠀⠀⠈⡆⠀⠀⠀⠘⣿⣷⣶⣶⣶⣦⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢤⣤⣦⣴⣿⠇⠀⠀⡠⠷⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⡿⣧⡂⢀⠠⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣞⠽⠓⠋⠉⠉⢉⣩⣭⠤⠆⠈⢳⡀⠀⢠⣤⠀⡀⢹⣿⡟⠛⠋⣹⣿⣇⠀⠙⢶⣕
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢁⠀⠀⢀⣠⠴⠛⠉⠉⠀⠀⠀⠰⣿⣀⣹⣿⣦⠀⢻⡇⢠⣿⣿⣧⢣⡌⠈⠍⠃⠀⠀⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠁⠀⠀⠀⠀⠀⣄⠶⠴⠒⠙⣏⣽⣿⣨⠖⠻⣷⠄⢸⣿⣿⣦⣡⢠⡏⠀⠀⢠⣾⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢨⡁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠙⠛⣿⣷⣴⣿⣿⢻⣿⣿⣟⣷⣿⠆⠴⠶⠿⠿⠃⠓⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⡿⠿⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⡋⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠄⠀⠠⡅
⠀⠀⠀⠀⠀⠀⢀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣦⣜⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⢲⡄⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡧⢻⡏⡬⠀⠘⠁⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⠙⡢⡀⠀⠀⠀⠀⠀⠀⢠⠶⢋⣳⣓⠇⡐⣠⡀⠀⠀⡄⠀⠀⠀⣁⠀⠀⠁⡄⠇⠸⠀⠀⠨⠀⠀⠈⠐⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠹⠶⣤⣄⣀⣄⣨⣷⠂⢁⢰⡄⠹⣀⣊⡃⢀⣉⡀⠀⢴⡄⢹⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣬⡄⢐⢯⢷⣇⠈⢀⡿⠀⠨⣴⠤⠄⣐⣶⠄⠻⢷⣄⠀⠀⠀⠀⠀⠀⠈⣿⡟⡴⢶⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠿⣿⣿⡅⣼⡟⠉⢞⡓⢘⡧⠀⠀⢸⣷⠀⠈⠉⠇⠀⠐⠙⡵⡢⠂⡀⠀⠀⠀⠈⠁⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡗⠀⢀⢙⠿⠙⠸⡇⠀⠀⠒⣿⡄⠈⠀⡠⢵⠂⠀⠀⠚⠀⠚⢋⣴⣿⣶⣶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣮⣟⢦⡒⣿⡧⠉⠀⠠⢽⣯⡁⠀⠐⠩⠯⡤⠀⠠⠀⠀⢈⡟⢻⣯⣿⣽⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⡄⢁⡬⣷⠍⢀⣐⡒⢺⣄⠀⡠⠀⠉⡯⣀⡀⠀⠀⠈⡁⠜⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⣿⣾⣿⣄⠁⠲⢿⣿⣿⡶⠁⠀⠀⣢⣿⣿⣷⣤⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢽⠿⢰⣶⣾⣿⣽⣿⠆⢠⣿⠿⠟⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠾⠟⠉⠙⠉⠉⠳⠞⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣻⣛⣻⣉⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣯⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣶⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⢈⡏⠁⣠⣆⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠢⠒⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠁⠃⣨⠙⢉⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠶⠀⠸⠇⢀⡾⢡⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⣱⠄⠀⠀⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⡱⣦⠀⢀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠧⠀⠀⠋⠠⠋⠔⠁⠀⠀⠀⠀⠀⠀⢀⡀⡠⠀⠃⠀⠀⠀⢠⣄⣂⣀⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠰⡄⠇⠀⠀⠁⠀⠀⠀⠉⠙⠛⠻⠷⠿⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣴⣀⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠒⠶⠲⢀⣤⠄⣠⠴⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⠟⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠰⠀⠐⠂⠘⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠂⠙⢻⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⣀⡠⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠆⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠞⠢⠄⠀⠤⠊⠉⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢄⡀⣀⡀⠠⢀⠐⠃
⠀⠀⠀⠀⠀⠀⡬⢀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣷⡾⠷⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣈⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠰⣤⡀⠀⠀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠀⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⠀⠀⡄⠐⠃⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⢤⡀⢤⠈⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠟
⠀⣠⠀⠀⠀⠀⠈⠀⣀⠒⣀⠀⠂⡐⢂⣤⡄⠀⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⡀⣀⠀⡀⣀⡠⠆
⠘⢋⡙⠉⠉⠁⡀⠀⠻⠿⣃⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠁⠂⠀⢀⠀⠀⠀⠀⠀⠀⠚⠃⠰⠂⠉⠀⠀⠀⠀⠀⠀⠁⠒
⠀⠼⢇⡨⠍⠀⠀⢀⠄⠀⠛⠀⠀⠠⠀⠐⢢⡤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡃⠐⢀⠀⠀⡁⢠⡤⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠛⠂⠀⠀⠘⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡼⠃⠀⠚⠃⠀⠀⠀⠁⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢀⡀⢠⣤
⡄⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠋⠁
⠁⠀⠀⢠⡀⢃⡰⠐⠶⠀⠑⠲⢄⣠⡄⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡰⠚⠎⠀⣠⣀⠀⠀⠀⠀⣀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠘⡀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡀
⠀⠀⠐⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠉⠀⠀⡰⠀⠀⠀⠀⠠⠤⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣀
⣀⣀⣀⣀⣈⣉⣉⣁⣀⣘⣢⣄⣀⣀⣐⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⡶⣻⣽⣏⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠴⠒⠛⠲⠖⠒⠐⠲⡤⠴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠚⠉⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠈⠳⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠐⠀⠀⠀⢀⠀⠀⢷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠃⠀⠀⠀⠘⢄⠀⠀⠀⢁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⠀⢳
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠑⠄⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⢠⠈⡇
⠀⠀⠀⠀⠀⠀⡰⠷⠀⠀⠀⠀⠈⠢⡀⠀⠀⠁⠀⠀⠈⠆⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⢠⠃⡘⣌⢿
⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⡀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢠⣷⣹⣿
⠀⠀⠀⠀⠀⣰⢄⠀⠀⠀⠈⠢⣄⠀⠀⠀⠣⡀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠀⡀⢸⣾⣿⠏
⠀⠀⠀⠀⠀⣇⠀⠈⠢⡀⠀⠀⠈⠳⣄⠀⠀⠈⢂⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⡟⣿⠏
⠀⠀⠀⠀⠀⢨⢄⠀⠀⠀⠑⢄⠀⠀⠈⠑⢄⠀⠀⠑⡄⠀⠡⠀⠀⠀⠀⡆⠐⠀⠠⢸⢸⣿⡇
⠀⠀⠀⠀⠀⠸⡀⠀⠑⠂⢄⡀⠈⠐⠄⡀⠀⠑⢄⠀⠈⢢⡀⠑⠀⠀⠀⠰⠀⠀⠀⢤⣿⠕⠹⡄
⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠈⠐⠤⣀⠈⠐⠠⣀⠀⠠⡀⠀⠄⠑⠀⠀⠀⠆⠀⢀⣼⢁⠔⠰⢯
⠀⠀⠀⠀⠀⠀⠀⠑⢆⠠⠈⠑⠢⠤⠈⠉⠒⠄⣀⠁⠠⢈⠳⠄⠠⠀⠀⠃⠀⠀⣸⡏⠀⠀⢀⣼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡈⠀⡐⢠⡤⣀⡤⣨⣖⣦⣀⠀⠁⠈⠢⣠⡰⣑⣿⡧⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠺⠟⠻⠉⠉⠉⠳⠶⣾⡿⠷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⠄⠄⠂⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠖⠚⠋⠁

⡀⠀⢹⡄⠀⠀⠀⢀⡀⡂⠀⠀⠀⣠⡄⠊⠀⠀⠀⡀⠈⢿⣇⠀⠃⢀⣀⠀⠛⠻⣐⠿⢤⠔⠒⠤⠄⣿⡾⠟⠀⢊⣿⣿⣿⠏⠈⣯⡐⢀⠀⠀⠀⠈⠒⠤⣤
⠁⠀⠀⠙⡄⠀⠀⣤⡀⠡⠶⢶⣼⣯⠕⠀⠀⢂⠀⠀⡀⣸⡿⠀⠘⠀⢩⣄⠀⠀⠎⣤⢤⡴⠶⠟⠘⠈⠀⠀⢀⣌⢶⡟⡻⣄⠀⠋⠐⢦⣣⣄⡀⠀⠀⢀⣀
⡀⠀⢀⠀⠙⠂⠒⢿⡈⠳⢤⡈⠛⠻⠀⡀⠠⡤⠀⠀⠀⡿⠃⠐⠀⠀⢾⠋⡥⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠊⠁⠈⠋⠑⠫⡀⠀⠀⠈⠉⠻⠻⠆⠉⠉
⡇⠀⠸⣆⠀⠀⡀⢈⣷⡀⠀⠙⠣⢀⠈⠈⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⣇⢀⠀⠘⡆⠀⣇⠀⢿⠛⠢⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠟⡌⠀⠀⠋⠂⠄⠀⢠⡀⠀⠈⠓⢆⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⠠
⠀⡀⠀⠀⠠⠀⠀⠀⠀⠈⠒⠄⡀⠀⠉⠓⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⣘⠀⠀⠀⢀⠀⢂⡐
⠟⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠠⡀
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢻⡄⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣿⣦⣤⣀⠉⢷
⠀⠀⠀⠀⠃⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠈⣉⣙⠛⠶⢬
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢠⡤
⢆⠀⠀⠀⣶⣠⣥⡀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣬⡀
⡛⣷⠀⠀⣮⣽⣷⠞⠁⡀⠀⣠⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠤⠄⠀⠀⠀⢀
⡻⣾⡇⢸⣿⣯⣿⠀⣸⠗⠊⠖⠉⠉⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠘⠓⠲
⣿⠿⣿⣼⡇⣸⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢹⢶⣽⠙⣷⣿⣥⣄⣤⡴⣆⠀⢰⠴⣦⠠⢄⡀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠠⡀⢀⣦⠀⠀⠀⠠⠀⣴⣟⠉
⠀⣿⣽⣷⡌⠻⠟⠛⠉⠁⠀⠀⡀⢅⣨⣤⡀⠀⠀⠀⠀⢀⠃⠀⠀⠂⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⣿⣃⠀⠀⠀⠀⢀⠀⠈⠻
⠂⢿⡏⠙⠟⠧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⡄⠀⠀⡜⠀⡠⠀⣼⠏⠀⠀⢀⡤⠤⠎⠀⠀⠀⠀⠀⠁⠀⢀⡅⠀⠀⠀⠀⠀⠀⢰⡖⠒⠁⠊⢄⠈⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⢀⣉⣉⡀⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡡⠒⠉⠀⠀⠀⠀⠉⠒⢌⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⠩⠭⠈⠩⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠍⠁⠬⠅⡂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠱⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠎⡄
⠀⠀⠀⠀⠀⢀⠔⡂⠕⠃⢀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠘⠪⢐⠢⡀
⠀⠀⠀⠀⠀⢎⠊⠀⠀⠀⠈⠄⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⠁⠀⠀⠀⠑⡱
⠀⠀⠀⡀⢔⠧⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠴⡢⢄
⠀⠀⣌⠊⠀⠀⠀⠣⡀⠀⠀⠀⠘⡄⠀⠀⠀⠘⡀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢀⠎⠀⠀⠀⠀⡐⠀⠀⠀⠑⣡
⠀⠀⢻⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠘⢄⠀⠀⠀⢃⠀⠀⠀⠀⡘⠀⠀⠀⢀⠊⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⡟
⠀⠀⠈⢇⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⢂⠀⠀⢈⡄⠠⠄⢠⡁⠀⠀⡠⠃⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⡸⠁
⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⢣⠒⠁⠀⠀⠀⠀⣈⡒⡔⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠑⢆⣠⠁⠀⠀⠀⠀⠀⠐⠧⠄⠈⡄⡠⠊⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣀⣀⣘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣃⣀⣀⣠⠔⠊
⠀⠀⠀⢀⡠⠤⠐⠒⠈⠈⣉⡀⠠⠄⠀⣀⣈⠦⣀⠀⠀⠀⠀⣀⠴⣁⢀⡐⠀⠤⢀⣉⡁⠁⠒⠂⠤⢄⡀
⠀⠀⠸⡥⠖⣀⢄⡈⠉⠀⡀⠤⠐⠂⠁⠈⡱⠂⠀⠨⠥⠬⠥⠀⠰⡎⠁⠀⠁⠒⠤⠄⠀⠉⠉⣀⢀⠒⢬⠇
⠀⠀⠀⠑⢄⠀⠉⠀⠚⠚⠴⠂⠶⢆⡀⢐⡡⠀⣀⠀⠀⡀⠀⣀⠀⢬⡂⠀⠀⠰⠆⠰⠜⠚⠂⠈⠁⡰⠊
⠀⠀⠀⠀⠀⠑⠂⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠐⠂⠒⠤⠀⣀⣀⠀⠠⠄⢀⣀⣀⠀⠤⠒⠐⠂⠐⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣤⣄⣀⣴⡿⠛⠉⠉⠙⠻⣦⣀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁⠉⠉⠛⠋⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠙⠻⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠈⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠿⠛⠃⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠘⠛⠻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⡿⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⣧⠀⠀⢸⠃⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢸⠀⠀⡎⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠋⠀⠻⣦⡀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠁⠀⠀⢀⠎⠀⠀⠀⠀⠀⢀⣴⠿⠀⠈⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢀⣴⠋⢉⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠁⠀⠀⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡹⣿⣿⣿⣿⣿⣿⣿⣿⢏⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠚⠛⠛⠋⠉⠁⠈⠛⠿⠿⠿⠿⠛⠁⠈⠉⠉⠛⠛⠓⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢿⣿⣿⣿⡿⢷⣶⣶⣶⣶⣶⣶⡾⠿⣿⣿⣿⡿⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋




⡿⢿⠿⠿⠿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡦⣤⡴⠲⢦⡶⢦⣤⢤⣄⢀⣀
⠀⠀⠀⠀⠀⠀⢀⡶⠶⡏⠉⠀⡆⠀⢀⢠⣾⠀⠀⡀⡀⠈⠉⠉⠳⠤⢤⡀
⠀⠀⠀⠀⢰⡆⡿⠀⢰⢡⠧⣴⣧⣄⣀⣼⡿⠟⢷⣷⢥⣧⣀⡼⠁⠀⠀⠙⠦⢤⡀
⠀⠀⠀⢸⠏⡼⣿⣤⢾⡿⠃⢸⠁⠈⢩⣿⠀⠀⣿⠇⡾⢬⡿⠓⠀⢀⣠⠞⠀⠈⠣⢤
⠀⠀⠀⣤⣧⢿⣿⠀⠸⡟⠀⣿⠀⠀⣾⡇⠀⢴⡟⠀⠀⣼⠃⠀⠀⣼⡿⢷⡴⡢⢂⠈⠧⣄
⠀⠀⢀⣿⣿⡟⣿⡆⠀⣷⠀⣿⠀⢸⣿⣇⠀⣸⠃⠀⡼⠁⠀⠀⡾⠃⠹⢮⣾⡿⣇⠀⣠⡿⣦⡀
⠀⠀⠸⣿⣿⣷⠈⢷⠀⢾⡏⠩⢀⣼⣏⠀⢠⠏⠈⠹⠃⠀⣀⠞⠀⢀⣴⠻⠋⠀⣿⣿⠟⠀⢸⣇
⠀⠀⠀⢻⣿⡟⣷⡜⣦⢚⡴⠘⠉⠉⠉⠉⡝⠑⢶⠃⠀⡰⠉⠓⣴⠟⠉⠀⣠⣴⠿⡅⠀⠀⣀⠘⡇
⠀⠀⠀⠀⠹⣿⣿⣿⡽⡏⢲⠀⠀⠀⠀⢰⠀⢠⠀⠀⠜⠓⢤⠞⠁⣤⣤⣾⠟⠁⠀⢙⣿⡞⠁⡀⢳
⠀⠀⠀⠀⠀⠈⢻⣿⣷⡱⠸⡆⠀⣴⠀⠆⢀⠎⡰⠃⢀⠔⠁⢳⣤⡾⠟⠁⢀⣤⠔⠋⢠⡴⣫⠖⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡀⢣⠀⠁⠀⢀⣜⠞⠀⡄⠉⠀⠀⠀⠻⣁⠠⡖⠋⣀⣤⡶⠿⡏⠀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠈⣇⣇⢀⢾⠎⠀⢀⡼⠆⠀⠀⠀⢀⡅⡴⠿⠈⠉⠁⣀⣤⡿⢾⢧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣇⢻⠋⠀⣠⣤⢂⣀⣠⡤⠶⢒⣋⣸⣥⠤⠤⠶⠺⠟⢿⡟⢋⢡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣽⡆⠀⢠⣛⣋⣡⣤⣤⣤⣬⣤⣶⣶⣾⣾⣿⣶⣾⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣯⣽⣟⣷⡿⠋⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠁⠄⠁⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣿⣷⡆⠀⠀⢀⢀⠁⠁⡶⠳⠢⠶⡆⣠⠤⢶⠴⠤⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡟⠘⡇⠀⠀⠃⣚⢃⠀⡁⣴⣟⣈⣹⡇⢦⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⣰⢤⣾⣋⡴⢤⡗⣸⡁⣩⡟⠉⣻⡇⢸⣆⢠⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡶⢿⠒⢤⡙⣷⣿⠿⠁⠉⠀⢻⣿⣿⠁⠀⠹⣿⣷⢸⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⢟⢸⠉⠢⢬⣿⠉⠀⠀⠀⠀⠀⠉⠃⠀⠀⠀⣸⡗⢀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⡚⠘⠆⣽⣿⣿⡇⠀⢺⡆⠀⠀⠰⠀⠀⠀⢠⣯⣵⠇
⠀⠀⠀⠀⠀⠀⠠⠒⢻⣷⡟⢦⠄⡾⠙⠛⡿⠶⣞⠛⠛⠟⢛⣷⣶⣶⣿⡇⠘
⠀⠀⠀⠀⠀⠀⢠⢳⠚⢿⠀⠈⠀⢀⡴⢆⠟⠤⠂⠠⠌⠡⠽⣿⢿⣿⠙⢌⡒
⠀⠀⠀⠀⠀⠀⢘⣋⠅⠈⠂⠀⠀⢸⡏⠓⠲⠁⢠⠀⠀⠀⡆⠰⠖⢿⣓⠄⠙⢺⡄⣤
⠀⠀⠀⠀⠀⠀⣼⠁⠀⢠⠀⠀⠀⢰⠇⠀⠀⠄⢸⡄⠀⢠⣿⡙⠀⠈⣿⠀⠀⣸⢻⠟⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⢀⣤⣬⠂⠀⠀⢸⣿⠿⢠⣿⣿⡁⠀⠀⢻⣧⣴⣿⢻⢺⠁
⠀⠀⠀⠀⠀⠀⡛⢀⡴⢣⣶⣿⣿⣿⠀⠄⠀⢸⡟⠀⠀⠹⣿⣇⠀⠀⠀⢻⣿⣯⡰⠟
⠀⠀⠀⠀⠀⠀⠙⢾⣵⠟⢻⡟⠁⢿⠀⠀⠀⢾⡇⠀⠀⠀⣸⣹⡆⠁⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⣸⠀⠀⢸⠓⠶⣶⢿⣿⠀⠀⡼⡿⢛⢻⡟⠛⠫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠀⠀⠸⡏⣱⣿⣾⡏⠰⣶⣗⣡⠊⢠⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣇⠙⢹⣾⣿⣶⣧⠭⢥⣠⣾⣞⡛⠲⣆
⠀⠀⠀⠀⠀⠀⢀⣸⣿⣴⣶⣿⣿⣿⡿⣤⠯⠞⣿⣿⣿⠀⠀⠈⠈⢿⣿⣿⣷⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠶⣯⣿⣿⣿⣿⣿⣿⡿⠀⠁⠀⠀⠸⣿⣿⣦⣀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠿⠂
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷⣦⣤⣼⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣤⣾⣿⣿⠿⠟⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⠏⠁
⠀⡀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠉⠙⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠉⠙⠛⠛⠟⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⣤⣀⠀⠀⠀⢀⣠⠶⠞⠛⠛⠳⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢀⣀⣀⣀⡉⠛⠲⠶⠛⢁⣤⠴⠶⠶⣤⡀⠙⣧⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⠞⠋⠁⠀⠈⠙⠳⠶⡶⠞⠋⠀⠀⠀⠀⠀⠻⣆⡈⠉⠉⠉⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⠛⢁⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠉⠉⠙⠳⣦⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣠⡶⠛⠉⠛⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⣧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⣦⡀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢸⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠙⢧⡀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠉⠉⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢸⠀⠀⠀⡞⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢀⡟⠀⣾⠁⠀⠀⠀⠀⠀⠈⠓⢦⡀⠀⠀⠀⠀⠀⢷⠀⠀⢸⠀⠀⣸⠁⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠘⣇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠈⢧⠀⢸⠀⢀⡇⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⣠⠾⣛⡛⠛⠻⢦⡀⢀⡼⠁⠀⠀⠀⠀⠀⣀⣤⠶⠛⣉⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣌⡉⠛⠶⣄⡀⠀⠀⠀⠀⠀⢸⡟⡵⠋⢁⡟⠀⠀⠀⠹⣏⠀⠀⠀⠀⣠⡴⠞⣉⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡈⠙⣶⣤⣤⣤⣤⣾⢀⡳⠒⠋⠀⠀⠀⠀⠀⢹⣤⡤⠴⢾⣥⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠛⠉⠀⠀⠤⢤⣿⡾⠟⠀⠀⠀⠀⠀⠀⠀⣾⣀⣀⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠖⠒⠒⠲⠦⢤⡾⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣈⡉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠶⠖⠛⠛⣉⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡉⠹⣦⠀⠙⠛⠒⠚⠛⠛⠛⠲⣦⣀⠀⠀⣀⣠⡤⠶⠛⠓⠶⠶⠦⠴⠶⠛⠉⣠⣴⡖⠛⠛⠛⠋⣩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⠿⣿⡶⠶⠶⠛⢻⡛⠶⣤⡀⠉⠉⠉⠉⢁⣠⣴⡶⠶⠶⢤⣤⣤⣤⡴⠞⣡⣿⣤⣤⡴⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣬⣷⣄⡈⠙⠲⠶⠶⠚⠋⣡⣾⣁⠀⠀⠀⠀⠀⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣤⠶⠞⠁⠀⠉⠛⠓⠒⠚⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⡿⠿⠿⠟⠛⠛⠛⠿⠿⢿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠋⠉⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⡄⠀⠈⠙⠻⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠉⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡟⠁⢀⣄⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⣾⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⡄⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁⠀⠀⢀⣀⣄⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡄⢹⣟⣷⣠⣤⣤⣤⣀⠀⠀⠀⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡿⢟⣫⠽⠚⠉⠀⠀⠀⠀⠈⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣛⣛⡿⠿⠿⢟⣛⣻⣭⡷⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠹⣿⣝⣻⣿⣿⣿⣿⣛⣽⡿⠀⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡈⠉⠛⣻⠃⣿⠛⠛⣋⣀⣰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⠀⠞⠢⣀⣡⠟⠎⢿⠿⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣦⢠⠆⠤⢽⡯⣤⣴⢆⡔⢹⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⡿⠨⣪⡀⠶⠛⠛⠿⢋⡾⡰⠁⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⢀⣠⣾⣿⣿⠃⣇⠀⠙⢿⣦⣤⣶⣶⣿⠟⠁⢠⣿⢻⣿⣿⣿⣿⢿⣶⡒⢢⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣤⣶⡾⢫⣿⣿⣿⣿⡄⣿⠄⠀⢀⣽⣿⣿⣋⠁⠀⣠⣿⡿⣼⣿⣿⣿⣿⣥⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣾⣿⣿⣿⣿⣇⣿⡈⠶⣿⣿⣿⣿⣿⠷⡼⢡⣿⢷⣿⣿⣿⣿⠟⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣿⣿⣿⣿⣼⣇⠀⠹⣟⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣆⣴⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠿⠛⠛⠓⢦⣙⣿⣿⣿⣿⣋⡴⠚⠛⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⡿⢿⣿⣿⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡟⢩⣝⣦⡀⠀⠀⠀⠀⢀⣿⣷⣄⣀⣀⣀⡴⢋⣼⣿⠁⠈⣿⣧⡙⢦⣀⣀⣀⣠⣾⣿⡀⠀⠀⠀⠀⢀⣴⣫⡍⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣿⣿⣷⠀⠀⠀⠀⢸⣿⣿⣿⣬⣭⣴⣾⣿⣿⡇⠀⠀⢸⣿⣿⣷⣦⣭⣥⣾⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⣷⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⣿⣿⣿⣧⡀⠀⠀⠘⢿⣿⣿⣿⢿⣿⣿⣿⣿⣤⣴⣦⣤⣿⣿⣿⣿⡿⢿⣿⣿⣿⠃⠀⠀⢀⣼⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠉⠉⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠉⠉⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠸⣿⣿⡿⣿⣿⣿⣿⣿⣿⢿⣿⣿⠇⠀⢀⣠⣶⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣶⣄⡈⠉⠀⠈⠉⠁⠈⠉⠁⠀⠉⢁⣠⣶⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣷⣦⣀⠾⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢿⣿⣿⣿⣿⣿⣶⣍⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣿⣿⣷⣬⣙⠿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣮⣴⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣵⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⡤⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⣀⡀⣠⢤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠃⠀⢀⡀⠄⠀⠀⠀⠁⠀⠀⠐⢦⡤⠾⠋⠀⠘⣁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⡄⠀⠂⠀⠀⣼⠫⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⣠⠌⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠾⠶⠤⠀⢄⣴⠋⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠂⠈⢡⡄⠀⠀⠀⠀⠛⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠀⠄⣼⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠿⠀⠂⠀⢀⠁⠀⠀⠀⠀⠀⠐⠀⠈⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠲⣌⠁⡰⠛⠋⠉⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣦⣀⠀⠈⠀⠀⣼⠀⢄⡀⠀⠀⣀⠀⠐⢦⣶⡾⠋⠉⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣾⣧⡧⢾⣦⣼⣶⠿⠷⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡆⢾⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡦⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣀⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠒⠀⠀⠀⠀⠀⠀⢺⣿⠓⢶⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⣟⠂⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⣿⠏⠁⣤⠆⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⣛⣿⡿⠀⠀⣾⠠⣤⠀⠀⠀⠀⠀⡄
⠀⠀⢀⣤⣀⠀⠒⠀⠀⠔⠀⠀⠀⠳⠒⠶⠟⠷⢄⡀⣹⠀⠉⠀⠰⠄⠀⢀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⣽⣽⣿⣷⣿⣤⣽⣤⣴⣀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠀⠇⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁
⠀⠘⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠁⠙⠛⠖⠺⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠒⡀⠐⠒⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣌⡙⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠉⠉⠉⠉⠁
⣿⣿⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣦⣶⣶⣶⣦⣾⣦⣾⣧⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠘⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠹⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⠁⢻⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⡿⢡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⠀⢸⣿⣿⣿⠇⣸⣿⣿⣿⣿⡟⣰⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⣇⠸⣿⣿⣿⠀⢸⣿⣿⡟⢀⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⠻⣿⣿⣿⣿⣿⣄⠻⣿⣿⣿⡀⢻⣿⣿⠀⢸⣿⣿⠃⣸⣿⣿⡿⢃⣼⣿⣿⣿⣿⡿⢛⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣌⠙⢿⣿⣿⣿⣆⠙⣿⣿⣧⠈⣿⣿⠀⢸⣿⡏⢠⣿⣿⡟⢁⣾⣿⣿⣿⠟⢉⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣄⠙⠻⣿⣿⣧⠘⢿⣿⡄⢹⣿⠀⢸⣿⠁⣼⣿⠟⢠⣾⣿⡿⠛⢁⣴⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠻⣿⣧⡈⠻⣧⠈⣿⠀⢸⡏⢠⡿⠋⣠⣿⡿⠋⣠⣶⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣬⣉⠛⠻⢿⣿⣿⣦⡀⠙⢷⡀⠙⠀⠈⠀⠈⠀⠘⠁⣰⠟⠁⣠⣾⣿⣿⠿⠛⢋⣩⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣦⣄⡉⠙⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠉⣁⣤⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠛⠛⠓⠂⠀⠀⠀⢀⣤⡴⠶⠶⣤⣄⠀⠀⠀⠀⠒⠛⠛⠛⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣷⣦⣄⠀⣰⠟⠁⠀⠀⠀⠀⠙⢷⡀⢀⣤⣶⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣧⣿⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⣿⠋⠁
⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠶⠶⠶⠒⠒⠒⠒⠒⠒⠛⠋⢿⡀⠀⠀⠀⠀⠀⠀⣸⠏⠛⠓⠒⠒⠒⠒⠒⠲⠶⠶⠦⠤⠤⣤⣄⣀
⠀⠀⠀⠀⢿⣷⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⣕⣒⣒⣭⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣴⣿⠇
⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣿⣿⣿⣷⣦⣤⣤⣶⣶⣿⣿⣿⣶⣬⣭⣭⣴⣾⣿⣿⣷⣶⣦⣤⣤⣶⣿⣿⣿⣿⣿⡿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠿⠿⠿⠟⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⢰⠤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠉⠙⢶⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣴⣀⠀⠀⢠⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢓⠳⣤⠀⡀⠀⠈⠛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⣆⣿⣊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡘⠷⣧⣴⡂⣀⠻⡇⠀⠀⠀⠀⠘⠠⡀⠀⠘⠶⢦⣴⣿⡧⠔⠁⠀⠀⢳⣸⣿⣿⣥⣄⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠛⠻⣥⣈⣻⣿⣿⡾⠀⢠⠤⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣿⣗⠀⠀⠀⠀⢹⣿⣃⡠⠉⠙⠋
⠀⠀⠀⠀⠀⢠⢁⡤⠴⠆⠶⠶⠿⣿⠿⣿⣧⡠⣷⠀⢀⠀⠀⠁⠀⣠⣤⠶⠋⠀⠀⠛⢆⡀⠀⣤⣾⣿⢻⣍⢛⠠⢄
⠀⠀⠀⠀⠀⠈⠈⠒⠤⠀⢀⣤⠤⠾⣄⠀⠉⠧⢿⣄⡽⠒⠀⠂⠀⠑⢆⠀⠀⠀⠀⠀⠈⠁⠀⠘⠿⠿⠏⠉⠙⠷⠬⢵⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⣟⢠⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⡀⢀⠀⡀⢀⠄⡄⡄
⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡝⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢣⠈⣄⠀⠀⠀⡞⠀⡄
⠀⠀⠀⠀⠁⢀⠀⢀⠀⢀⠄⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⣼⡀
⠀⠀⣀⠀⠀⠈⠁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢄⠀⠀⠀⢨⣀⣴⣿⠷⣠⠀⣠⠔
⠀⠈⠀⠀⣀⠀⠀⢧⠀⠀⠀⠀⠐⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠄⢀⣄⠤⣙⣿⣿⣿⣷⣶⡈⠀⠀⠂
⠀⢀⡠⠜⠁⠀⠀⠼⡖⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡇⣆⡏⠀⡀⠀⠀⠀⠐⠂⠀⠉⠿⡿⠟⠿⠿⣿⡂
⠀⠸⠀⠀⣄⠍⠀⠀⠉⠀⡀⠀⠀⠲⠄⠀⠀⠀⢀⡃⢀⠹⣿⣠⣨⣾⡶⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢼
⠀⠀⠀⠀⠁⠀⠀⢀⠀⢁⠉⠚⠉⠀⠀⠀⠀⠀⠰⢄⡪⣤⣰⢳⣿⣿⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⠀⠠⣄⠈⣑⢿⡻⡘⣿⣗⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠰⡇⠀⢸⠀⠀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⡰⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⡀⠘⣿⠇⢇⠘⢻⣃⠌⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⠃⡀
⠀⠀⠀⠀⣤⡀⠆⠸⠀⢐⡤⠂⢐⠀⠀⠀⠀⠀⠠⠄⠉⠙⢳⠄⢧⡘⡵⢊⠤⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠠⣸⣿⣰⠁⡠
⠀⠀⠂⠀⠨⣍⠳⡟⢤⣠⣦⠄⠚⠚⠀⠘⠁⠀⠀⠀⠄⠚⣉⡥⠄⢣⣘⣕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡟⣿⣇⠎
⠀⠀⠀⠀⢢⣝⣦⣶⠈⣿⣡⣴⣷⡶⠂⢀⠀⡀⠀⠀⠀⠈⠐⠚⠿⠶⣛⠅⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⢈⣻⣷⣿⠃
⠠⠀⠀⠀⠀⢉⡽⠿⠿⣿⡿⠑⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠁⢀⡀⠀⠀⠀⠬⢝⣿⡏
⠀⠀⠀⠀⠀⠨⣀⠀⠠⡿⠛⣦⣤⣴⣶⣍⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⠶⠦⠤⠴⠾⢿⡦⠂
⠀⠀⠀⠀⠀⠀⣦⠀⠀⣏⠉⣡⠋⠯⠙⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⣤⣤⣤⣤⣤⣤⣬⣤⣤⣥⣤⣤⣤⣤⣤⣦⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤
⣿⣿⣁⣍⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⢿⡄⠈⢿⡙⡆⠐⣦
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⢠⡀⢀⠙⣦⠈⢣⡀⠀⠸⡄⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠒⠐⠛⠒⠶⡦⣠⡀
⠀⠀⠀⠀⠀⠀⠴⢃⠀⡀⠈⠻⢦⡀⠈⠣⠀⢻⣄⠀⢡⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠉⠀⠀⠀⠀⠀⠄⠀⡄⠈⡉⠰⣄
⠀⠀⠀⠀⠀⣠⠀⠀⠙⠪⣷⣄⠀⠻⣦⠀⢆⠀⢻⡄⠘⡇⢀⠨⡇⠀⠀⠀⠀⠀⠀⠀⡔⢁⠀⠐⡀⠈⠀⠀⠀⠀⠀⠀⠈⠀⣀⠉⢲⡀
⠀⠀⠀⠀⢀⠁⠈⠐⠦⣄⠈⠻⢷⣄⠈⢧⡄⢷⡚⢿⠀⣷⢙⢱⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠈⠁⠠⠈⢱⡀
⠀⠀⠀⠀⠀⠓⠴⠶⣤⣈⠙⠲⢤⡉⢷⣌⠻⣄⢳⡘⣇⢿⣾⡾⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⣀⣠⡤⣴⡋⠀⠉⠑⣄⠀⠀⠀⠀⠀⡀⠀⡃
⠀⠀⠀⠀⠀⠱⠄⠠⣤⣈⡙⠳⣦⣍⠒⢽⣷⡙⣦⢷⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰⠋⠁⠀⠤⠁⡄⠀⠀⠈⢢⡀⠀⠀⠀⢀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⣛⣲⣯⣶⣽⣿⣿⣿⣿⣿⢇⣱⠀⠀⠀⠀⠀⠀⠀⠈⠄⢻⣴⣠⠔⠊⠁⢰⠀⠀⠀⠀⠁⠀⠀⠀⡀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠒⠒⡂⢬⣭⣽⣿⣿⣿⡿⣿⠵⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠙⠖⣄⠀⠀⠈⠷⢤⠀⠀⠀⠀⠀⡀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠐⢄⡰⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠘⠖⡤⠒⠂⠁⠀⠀⠀⠀⢀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⠆⠠⠤⠤⣐⠤⡤⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⠁⠀⡜⠀⢸⠏⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠀⢀⠇⠀⡸⠀⠀⣰⠯⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡇⠀⢿⠀⢀⠃⠀⠀⡯⠀⠀⠀⣦⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡅⠀⡇⠀⡆⠀⢬⠀⠀⣼⠁⠀⢀⣾⠃⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⡃⠀⠠⡇⢸⡇⠀⠇⠀⣾⠇⠀⣠⠞⠁⣠⡝⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠸⠀⣷⢸⡇⣼⠀⣼⠏⠀⡰⠃⣠⡞⢋⠔⢁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⢃⣿⣸⣧⡏⣸⡟⣠⡾⢀⡴⠋⠔⠁⢀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣾⣿⣿⣷⣿⡴⢋⣴⡯⠂⢁⠀⡠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⡙⢿⣿⣿⣿⣿⣛⣁⣄⠨⠡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⢭⣉⢉⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⣶⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣮⠻⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⠟⣵⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⢿⣿⣿⣿⣇⢸⣿⣿⣿⣿⡇⣸⣿⣿⣿⡿⢃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢿⣿⣿⣿⠘⣿⣿⣿⣿⠁⣿⣿⣿⡿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⡛⠿⣿⣿⣿⣿⣿⣿⣧⠘⣿⣿⣿⡀⣿⣿⣿⣿⢀⣿⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⠿⢛⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⣤⡙⠻⣿⣿⣿⣿⣧⢹⣿⣿⡇⣿⣿⣿⣿⢸⣿⣿⡇⣼⣿⣿⣿⣿⠟⢉⣤⣾⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⣿⣿⡆⣿⣿⡇⣿⣿⣿⡟⢸⣿⣿⢰⣿⣿⣿⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⣿⣿⣿⣸⣿⡇⢸⣿⣿⡇⣸⣿⣇⣿⣿⣿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣛⣛⠻⠿⣿⣿⣿⣿⣿⣜⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⣣⣾⣿⣿⣿⣿⠿⠟⣛⣛⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣶⣭⣛⢿⣿⣿⣮⣿⣿⣿⡿⢿⣟⣻⡟⢿⣿⣿⣿⣿⣿⣿⡿⣛⣭⣶⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⡿⣽⣾⣿⣿⣿⣿⣷⣯⢿⣿⣿⣿⣷⣿⣿⠿⠿⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⡿⣾⣿⣿⣿⣿⣿⡁⠀⢹⣷⢿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⢺⣇
⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣴⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠯⣿⣿⣷⣦⣤⣤⣤⣀⣀⣀⣀⣀⣠⣤⣄⣀⣀⡀
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠈⠛⠿⠿⠿⠿⠛⠁⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠰⡼⣿⣿⣿⣿⣿⡟⠛⠛⣛⣉⣭⣴⣿⣿⣿⣿⢟⣽⣿⣿⣷⣶⣶⣶⣶⣶⣿⣿⣯⠻⣿⣿⣿⣿⣶⣭⣉⣛⠛⠛⢻⣿⣿⣿⣿⣿⢷⠆
⠀⠀⠀⠙⣮⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣋⣵⣿⣿⣿⣣⣿⣿⣿⣿⣿⣿⣜⢿⣿⣿⣮⣙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣯⣵⠋
⠀⠀⠀⠀⠈⠻⢿⣮⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣟⣵⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣮⣟⣛⣿⣽⣮⣻⢿⣿⡿⢟⣵⣾⡿⢿⣿⣿⡿⢟⣷⣮⡻⢿⣿⡿⢟⣵⣯⣿⣛⣻⣵⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠉⠙⠻⠿⠿⠿⠟⠛⠿⣿⣿⣿⣿⠿⠛⠻⠿⠿⠿⠟⠋⠉⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⠀⠠⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣀⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠈⠑⢤⡀⠀⠀⠀⣀⠔⠈⣀⠔⠊
⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⠀⠔⠀⠀⠀⠀⠀⣇⡠⠔⠉⣀⠔⠊
⠀⠀⢀⠴⠊⠁⠀⠀⠀⠀⠀⠀⡠⠔⠁⡠⠔⠉⠑⠖⠊
⡠⠂⠀⠀⠠⠶⠀⠀⠀⡠⠔⠉⢀⠔⠊
⡇⠑⠤⡀⠀⠀⠀⠐⠉⢀⠤⠊⠁
⠑⣄⠀⠈⡆⠀⢀⠤⠊⠁
⣤⣬⣵⣶⣶⣪⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣄⠐⣶⣤⡄⠀⠀⣀⢰⡶⣾⣶⠀⠀⠀⢠⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⢀⣆⡀
⠀⠀⠀⠀⠀⠀⣼⡞⠀⠀⠀⠀⡀⢿⣼⢟⣿⣛⡆⠀⣤⢹⣼⣿⣿⡩⠄⣠⣴⣿⣿⣿⣿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣿⣿⡿⡃⠀⠀⠐⢿⣷⡄
⠀⠀⠀⠀⠀⠲⣿⠂⠀⠀⠀⠀⠻⣾⣫⣾⠿⢦⠀⢰⣸⣿⣿⣽⣿⡗⠀⣿⣿⣿⣿⣿⣷⣿⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣯⣴⡶⠂⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡟⢯⠷⠀⠀⠀⠉⠙⠿⠟⠃⠀⠀⠻⢿⣿⣧⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠙⣿⡓⠀⠀⠀⠠⣾⣿⡇
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⡡⠄⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠹⢿
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⢠⣿⠂⠀⠀⠀⠀⢀⡴⡛⠈⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⡀⣄⣀⠀⠀⢠⣄⠀⠀⠀⠀⣀
⠀⠀⠸⢿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢷⣦⣿⡟⢁⣴⡿⠀⠀⢀⠎⡀⠁⠀⡀⠄⠀⢸⣿⡆⠀⠉⠀⠠⣉⣻⡥⢆⡀⠘⣿⣷⠀⣠⣾⣿⠀⠀⠀⢀
⠀⠀⢻⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⢬⣷⡿⣋⣤⡤⠀⠀⠘⠁⠀⠀⠀⠀⠀⠒⠓⠒⠀⠀⢐⠲⠶⣽⠋⠩⠀⠀⠈⠛⠃⣿⣿⠟
⠀⠀⠻⣿⣿⣿⡗⠀⣤⣤⠀⣼⣷⠀⣠⣤⠸⠿⡶⠦⣿⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠻⠀⠀⠠⠭⢭⣯⠴⠞⣋⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠁
⠀⠀⠈⢿⣿⠆⠀⠀⡻⣿⣷⣿⣿⣾⠿⠋⠀⠀⠀⠀⠀⡆⠀⠀⠀⠁⠘⠀⠀⡀⠀⠀⢠⠀⠐⣻⡭⠀⠠⣤⣀⡒⣶⣚⣯⠗
⠀⠀⢠⣀⠀⠀⠀⠘⠿⠿⣿⣿⣿⡷⠿⠷⠀⠀⠀⠀⠀⠃⢀⠀⠀⠀⠐⢢⡤⠙⠀⠄⠀⠀⠀⢻⡯⠀⠲⠦⣤⣄⣋⣡⠴⠚
⠀⠀⠸⣿⣿⣦⠀⠀⠀⠰⠿⠛⠻⠿⠀⣴⣄⢠⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠂⠐⠠⠀⠀⠀⠁⠀⠑⠶⢦⣄⣉⣴⠾⠋⠀⡀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⢰⠀⠀⠀⣬⣽⠿⡇⠀⠀⠀⠀⠀⠀⣾⡆⠾⠡⡤⠆⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠁⠀⠀⠀⠛⠏
⠀⠀⠀⠀⠈⠉⠁⠀⢠⡀⣴⠀⢁⣀⣠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠹⣦⡘⠳⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⡀⠀⢰⣴⡀⠀⣄⠀⠀⣶⡧
⠀⠀⠀⠀⠀⣀⠸⣆⢸⠇⣯⣴⣿⡉⠁⠀⠀⢀⡄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠿⠛⠀⢼⣯⠄⠈⠉⠀⠀⠿⢀⡀⠈⠁⠀⢠⡐⣴⢃
⠀⠀⠀⠀⢦⡙⢦⠻⣼⣿⠯⠄⠀⢀⢠⣶⠶⠘⣿⣖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⢀⣀⡙⠶⣋⣤⠄⠀⠀⠲⣝⣗⣫⣔
⠀⠀⠀⠳⣮⡻⣦⣍⣛⡛⠁⠀⠠⢜⠛⠁⣀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⣿⡇⠀⢠⣛⠯⢈⠥⠶⡂⠀⣈⡻⢮⡭⢟⡋
⠀⠀⠐⠲⣞⣿⡓⢦⡄⠀⠀⠀⠀⠀⣶⡿⣿⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢩⡿⠇⠀⠀⢭⣍⡀⢚⡋⠁⠀⠠⣭⣹⣛⡭
⠀⠀⠀⠉⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠱⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠈⠑⠋⠁⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣤⣭⣥⣦⣭⣷⣥⣿⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠀⢰⣿⡿⠿⠀⢀⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣾⣄⠀⣿⡇⠶⣿⡇⢸⣿⣛⣛⠀⣼⣿⣦⣿⠇⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣾⣿⣆⠙⠻⠿⠛⠁⠈⠉⠉⠛⠀⠛⠃⠿⡇⠀⣿⡏⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣄⡀⠹⣷⠄⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣶⠿⠋⣠⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁⢀⣴⡦
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡈⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠋⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣾⣏⡀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⣿⡟⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⠃⠀⣀⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⡀⠀⠘⢷⣿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆⠀⠀⢸⣿⣿⣿⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡆⠀⠀⣰⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣄⠈⣿⣿⠁⠀⠀⠈⣿⣿⠁⠀⠀⠈⣿⣿⠁⣠⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠿⢿⣿⡗⣸⣿⣦⣤⣤⣼⠟⠻⣧⣤⣠⣴⣿⣇⢸⣿⡿⠿⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⢿⣿⣿⣿⣿⣀⣀⣿⣿⣿⣿⡿⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⣿⣿⣿⣿⣿⣿⣩⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣽⢽⣉⣉⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⠆⠻⣿⣾⣶⣶⣷⣿⠟⠰⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣴⣶⣿⣿⠿⠋⠁⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠈⠙⠿⣿⣿⣶⣦⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣠⣾⣷⡆⠀⣀⡀⠀⠀⠀⠀⢀⣀⠀⢰⣾⣷⣄⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⢿⣧⣿⠃⢀⣿⡟⣿⣧⢸⣿⣟⣻⡀⠘⣿⣼⡿⡷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⢸⣿⢻⣿⠋⠀⣿⣯⣭⡅⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠀⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿