Athletic Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⢠⣤⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣷⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠂
⠀⠀⠀⠀⠀⡀⠸⢖⠀⢜⡷⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢤⡀⠀⢿⣇⢨⠻⣿⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢈⡦⣤⣜⣭⡺⠗⠚⢶⣶⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⣷⣾⣦⠟⠋⠀⠀⠀⠀⠀⠀⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠛⠉⣤⡖⠀⠀⠀⠀⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⣰⠀⠀⠀⠀⠀⠠⠐⠂⠀⠒
⠀⠀⠀⠀⠀⢠⣤⣶⡀⡿⣘⡟⠃⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠸⡛⢹⡿⣿⠻⠀⠀⠀⠀⠀⢫⡤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⢆⣻⢿⡀
⠀⠀⠀⠀⠀⠐⠐⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠛⠛⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⢀⡀⠘⠇
⠀⠀⠀⠀⠀⠠⠋⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡀⠀⢨⣹⣄⡀⠀⡀
⠀⠀⠀⠀⣄⣀⠀⢘⡏⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢴⡆⠐⠊⡤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠟⣙⢉⡟⠃⠀⠁
⠀⠀⠀⠀⠛⠛⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣾⣾⣷⣿⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣄⠴⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠈⣀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢿⡿⠿⠿⣿⣿⡛⣹⣏⠁⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢋⣩⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠐⣿⣿⣿⡿⠻⡷⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠊⣿⣿⡅
⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⣿⣿⠃⠀⠀⠀⠁⢀⡰⠎⢠⠄⠀⠀⠀⠀⠀⠀⠐⠏⠀⠉⠹⠃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣣⠎
⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡤⣶⣦⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⣀⣴⣥⠞⠀⠀⢀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠒⠲⣆⠀⠀⠀⠀⠀⠀⣿⣁⡈⢥⣈⡀
⠀⠀⠀⠀⠀⢸⣾⠟⠁⠀⠀⠀⠸⣾⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⠉⠁⢀⣀
⠀⠀⠀⠘⠳⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡙⠀⢈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠓⠀⠐⢢⣦⣷
⠀⠀⠀⠀⠀⠰⣿⣧⣀⡀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣿⡁⠀⢿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡔⠂⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⡿⢿⣿⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠤⠴⠆⢠
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⣿⠃⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠠⢜⢿⡿⠋⠙⠻⢤⡀⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣽⣾⣷⡀⠀⠂⢴⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢽⣶⠶⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠪⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠘⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠈⠣⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⢀⣨⠽⠓⣶⣦⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⡖⠛⠉⠁⠀⠀⠸⣿⣿⠃⠀⠀⠀⢸⣇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⢀⣿⡈⠀⠀⠀⠀⠈⠙⢦⣄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⣸⠿⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠀⠀⠀⠀⠀⠀⢀⠟⠂⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⣿⣿⣷⣴⣶⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠉⠉⠉⠉⠉⠛⠻⠿⠿⠶⠶⠶⣦⣦⡀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠚⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢰
⠀⠀⠀⠀⠀⢰⣶⡶⠀⣀⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠙⣿⢳⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠨⣄
⠀⠀⠀⠀⠀⣼⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢻⣶⣿⢿⣶⡆⠄
⠀⠀⠀⠀⠈⠉

⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠿⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣶⣿⣾⡿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣾⣿⣿⣿⠿⠿⠿⠟⠛⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⠰⠿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣷⣶⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⡄⠀⡘⠀⠀⠀⠀⠀⣀⣀⡀⠄⠒⠀⣙⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣰⣿⠟⣻⣿⣍⠙⣿⣿⣿⣿⣿⣃⡄⢀⣀⠴⠒⠉⠁⣀⢨⣷⣾⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠾⣿⡏⠸⣿⣿⣿⡇⢸⣿⣿⣿⣶⠚⢡⣿⠀⢰⣶⣧⣾⣿⣿⣿⣎⡻⠿⠋
⠀⠀⠀⠀⠀⠀⡈⠀⠀⣉⣛⣛⣿⣮⡛⣿⣿⣁⠀⢨⡟⠀⠀⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢀⠄⠀⠀⢀⣼⣿⣿⣿⣿⣿⠿⠓⠲⠞⠛⠋⠀⠀⠀⢹⣿⣿⢿⠁
⠀⣠⡞⠁⠀⠀⢀⠸⣿⣿⣿⡿⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⣿⢿⡌
⢰⠏⠀⠀⠀⣠⠟⢹⣾⣿⣿⠁⠀⣠⣾⣶⣿⡇⠀⠀⠀⠀⣾⣿⣿⢿⡇
⣏⠀⠀⠀⠞⠋⠀⢸⡿⠛⣿⣤⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⣿⣿⢧⢺⡅
⠙⠷⠶⠤⣤⠔⠋⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⡀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣦⡀⠀⠛⢿⣿⡇⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣿⣿⡀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣤⣼⣿⣿⣧⣀⡴⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣿⣿⣿⣿⣻⣻⣛⣛⣛⣫⣭⣥⡀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣿⠋⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠷⠈⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠁⠀⠀⠀⠀⢀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠻⣿⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⠀⣀⣠⣬⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⡟⠁⠿⢿⣿⣻⢿⣿⣿⣿⡿⣫⡶⠟⠉⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⢹⣿⣧⢻⡿⠿⢟⣩⣦⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠻⣿⣷⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⣸⡿⠁⠀⠀⠀⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠃⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡋⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣄⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⡀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠿⠟⣫⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⣛⣥⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣭⡉⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣭⡶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢛⣻⣿⣦⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣀⣤⣀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠓⠿⠛⠻⠟⣉⣬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣛⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⠿⠷⠶⠞⠋⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⡿⢿⠟⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠷⣤⣤⣤⣤⣶⠯⠓
⣤⣤⣤⣄⣠⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣜⣷⣶⣿⣛⣩⣄⣠⣄⣠⣄⣤⣤⣤⣤
⣿⠿⡿⠿⡿⢻⠿⢏⠻⠿⣿⠿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣟⣛⣛⣛⣛⣛⣿⣿⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⣀⡀
⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢲⠰⢂⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⡀⣀⡀⠀⢀⣾⠫⠇⡀⠀⠀⠂⢺⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠘⡏⢼⡏⢠⡿⢖⣻⠠⢄⠃⠄⠐⡘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⢀⡴⣤⠀⢡⢼⡇⣼⠃⣄⠃⠠⠌⠀⠀⠨⠀⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠸⢦⠈⠆⢸⢸⡇⣿⠍⣔⠃⠥⠀⡀⠀⠅⢠⣿
⠀⠀⠘⢧⠘⣼⡎⡇⢹⣆⠣⠀⢀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠈⢣⠘⢧⡇⠈⢷⣞⡈⠦⠂⡄⠆⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⢣⡸⣿⡀⠈⢟⣦⣬⣐⡘⣾⠋
⠀⠀⠀⠀⠀⢸⢳⡄⠱⡀⠀⡆⠉⡵⠋
⠀⠀⠀⠀⠀⢸⠀⢳⡀⠙⣆⡇⢰
⠀⠀⠀⠀⠀⢸⠀⠈⢳⠀⠈⠳⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠘⣆⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⢀⠴⠎⠉⠀⠀⠀⠁⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⠸⠀⠀⠀⡈⣄⠀⠀⠈⢢⠀⠀⠀⠀⡰⠋⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⣿⡆⠀⠀⠀⢣⠀⠀⢰⠁⠀⢀⠠⠊⠀⠀⠀⢀⠄⠂⠀⠀⢀⣠⠧⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⡆⠀⠀⠀⠳⡤⠒⠲⡈⠀⠀⠀⠀⠀⠐⢇⠀⠀⢠⠖⠙⡀⠀⠀⠀⠐⠓⢌⠐⣬
⠀⠀⠀⠀⠀⠀⡆⠀⠀⣿⡏⠉⠻⣦⠀⠀⠀⠱⡀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠑⡴⠁⠀⣰⣿⣿⣶⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⡇⠀⡔⠛⠑⢦⡀⠈⠣⡀⠀⠀⠘⡄⠀⢫⠀⠀⣐⠀⠀⠀⢀⣦⡠⠐⠙⢿⡿⠟⠀⡀⠀⡀⠐⠻⢿
⠀⠀⡜⠋⠉⠈⣧⠠⠷⠀⡀⠀⠉⠲⠄⠑⡄⠀⠀⠘⢄⠘⠹⡁⠀⠐⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡁⠀⠀⠀⢹⣄⡀⠀⠈⠑⠒⠀⠀⠀⢻⣄⠀⠀⠈⢢⣄⡑⢄⣀⣀⡀⠀⠓⠤⣀⣠⣴⣄⠀⠀⠀⡄⠀⠀⠀⢰
⠀⠀⠣⡄⠀⠀⢘⡈⢯⡢⣀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠱⠱⡄⠀⡠⠒⠂⠑⡄⠘⣿⣿⣿⠗⠀⠀⠀⠀⢠⣤⠂
⠀⠀⠀⠀⠀⠉⠀⠑⠆⠷⠏⡀⠀⠀⠀⠄⠀⠀⠐⡳⢄⣤⣨⠗⠃⠀⠁⠀⠀⠈⠉⠲⢉⠉⠙⢄⠀⠀⠀⡀⠞⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠋⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠻⣿⣿⣿⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂
⣿⣿⡏⠸⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣆⢻⣿⣿⠀⠀⡄⠰⠓⠐⠚⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠘⢷⠰⡀
⣿⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠮⠭⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢻⣿⠀⡜⠀⠀⡆⠀⠀⠀⢀⣀⢀⣀⣠⠳⠤⠴⢇⣀⣀⠀⠀⠀⠀⠀⠀⡎⠀⠑
⣿⠀⠀⠀⠙⠉⠉⠙⠁⠀⠀⠈⠉⠁⠀⠀⠀⠉⠁⠀⠀⢈⠉⠉⠉⠻⠀⠀⠀⣿⢠⠀⠀⠀⠋⠉⠉⠙⠁⠀⠀⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⡉⠉⠉⠙⠄⠀⠀⢁
⣿⣄⣀⣀⡀⠒⠒⠚⢛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠒⠒⠀⠀⣀⣠⣿⠘⢤⣀⣀⡀⠒⠒⠚⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠓⠒⠂⢀⣀⣠⠄
⣿⣿⣿⣿⣷⣶⣶⣿⣿⠉⡱⢤⣤⣴⡂⠘⣦⣤⣤⠶⠋⢹⣷⣶⣶⣶⣿⣿⣿⣿⠀⠀⠈⠉⠓⠒⠚⠙⣏⢈⠳⠤⣤⣶⡁⢑⣦⣤⠴⢚⣉⡿⠓⠒⠒⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡦⠋⠀⠀⡀⠀⠙⡇⣰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣴⡜⡉⠀⠀⠀⠀⢙⢆⣦⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠛⠀⠀⠄⠀⠘⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⠋⠁⠀⠀⠈⠛⢸⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠸⠂⠀⠀⠐⠺⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⢛⠁⠀⠀⠘⠏⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣷⣦⣤⣦⣴⣷⡞⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠻⣿⣷⣴⣶⣾⣿⠛⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡀⠈⣿⣿⣿⣿⡏⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⡿⣿⡿⣿⠀⠀⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡇⠘⡏⠸⡇⠀⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡏⠀⢸⠁⢸⠀⢹⠀⠘⡟⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠀⠀⠁⢀⡇⠀⠁⠀⠸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢱⠀⠀⠀⡸⡀⠀⠀⠀⠁⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣦⠀⢠⣾⣿⣄⠀⢠⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⡆⠀⣼⠁⢹⡄⠰⡞⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠀⢸⣿⣿⣏⠀⠈⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡏⠀⢸⡇⠀⠀⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⣸⣿⣿⣿⡀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⢠⡇⠀⠀⣧⠰⢾⡅
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠠⣿⣿⣿⣿⠇⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠘⣷⠀⢰⡟⠀⠈⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠘⣿⣿⡏⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠘⠀⠸⡀⠀⢠⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣸⣿⣿⣷⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣤⠁⠀⠀⢣⡄⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢠⣤⣼⣧⡀⠀⠀⠀⠀⠀⠒⢦⣤⡶⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢀⣀⣼⣆⣠⠀⠀⠀⢈⣧⣤⡀⠀⠀⠀⠀⢀⠐⠶⣤⣴⡆⠀⠀⠀⣴⣧
⠀⠀⠀⢠⣿⡏⠙⠢⠀⠀⠀⠀⠀⣼⣿⠛⠒⠓⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠉⠀⣿⣧⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠈⠀⠀⣸⣿⡇⠀⠀⢀⣿⡏
⠀⠀⠀⣼⠛⠻⡄⠀⠀⠀⠀⢠⠾⠋⠻⣄⠀⠀⠀⠀⠀⠀⡾⢿⠀⠀⠀⠀⠀⢰⡏⠹⡆⠀⠀⠀⠀⢙⡿⣷⠀⠀⠀⠀⠀⠀⣾⣿⠟⠁⠀⠀⠘⣿⣇
⠀⠀⠋⠁⠀⠀⠹⡄⠀⠀⠴⠋⠀⠀⠀⠈⠳⠄⠀⠀⠀⢠⠃⠘⡅⠀⠀⠀⠀⢸⠀⠀⠃⠀⠀⠀⠀⢼⠀⠸⠆⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠴⠟⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⢀⡀⢀⣠⠆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣄⣀⣀⡀⠀⡄⢀⡀
⠀⠴⠒⠒⣶⣿⣅⡀⠀⠀⢀⣼⣷⠒⠋⠁⢀⡴⢿⣿⡟⠋⠛⠀⠒⠒⢾⣷⠚⠉⠀⠀⣼⣿⣏⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⣿⣿⡏⠀⠀⠘⢳⣿⡶⠤⠄
⠀⠀⠀⢠⣿⡁⠀⠉⠀⠀⢉⣿⣧⡀⠀⠀⠈⠀⢸⣿⠇⠀⠀⠀⠀⠀⢿⣯⠀⠀⠀⠀⢩⣿⣿⠀⠀⠀⠀⢰⣿⡟⠉⠁⠀⠀⢠⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⣠⠏⢹⠀⠀⠀⠀⣼⠁⠈⢻⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⡾⢹⠃⠀⠀⠀⡟⠁⠙⣧⠀⠀⠀⡾⣹⠇⠀⠀⠀⠀⢸⠙⡆⠀⠀⠀⠠⡏⡇
⠀⠀⠛⠀⠀⠘⠃⠀⠀⠀⠃⠀⠀⠘⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠓⠘⠂⠀⠀⠐⠃⠀⠀⠘⠂⠀⠀⠃⠉⠀⠀⠀⠀⠀⠉⠀⠃⠀⠀⠀⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣤⣴⡆⠀⠀⢠⣶⣶⡶⣶⠀⠀⠀⣰⠆⠀⡀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢀⠀⠀⠀⠀⢀
⠀⠀⢣⡀⠀⡇⠀⠀⠀⠀⠀⠂⠀⠀⢹⡿⠯⠿⠀⠀⠙⠃⠈⡾⠁⠀⣠⣴⣿⠄⠀⠈⣾⣿⣹⣥⡏⠀⠀⠀⠀⢺⣴⠔⠁⠀⠀⢶⣵⠀⠀⣠⣤⣽⡿⠿⡇
⠀⠀⠀⢿⣾⣁⣤⣠⣤⠤⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠻⣯⣼⡄⠀⠀⠉⠀⠀⠋⠂⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠈⣿⣦⠀⢻⠈⢿
⠀⠀⠀⠀⠙⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⢺⠛⢿⠀⠀⠀⠀⠀⠀⠭⢹⡄⠛⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡶⠀⠀⠀⠀⠀⣰⣆⠀⠸⠂⠀⠙⢦⠀⢠⡄⠀⠀⠘⡇
⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⣿⣄⡀⠀⠀⠀⠀⠀⡠⣾⡷⢆⠀⠀⠀⠀⠂⣾⣏⣉⠢⠀⠀⠀⢸⣿⡿⣄⣀⠀⠀⠀⠀⣰⡿⠇⠀⠀⠀⠁⠀⠠⢶⣷⡦⡄
⢀⣀⠀⣿⡿⠷⣯⡀⠀⢀⠴⢻⡇⠉⠄⠀⠀⠀⠸⠴⠛⣷⠀⠀⠀⠀⢠⣤⣈⣹⣏⡀⠀⢀⡴⠟⠉⠀⠈⢡⠀⠀⠐⠓⠿⢛⠒⡦⢤⠆⠀⣀⣠⣾⣛⣁
⠈⠉⠙⣿⣤⣄⡀⠙⠁⠀⣠⢾⡿⠴⠀⠀⠀⠀⠛⠉⠉⢹⠉⠁⠀⠀⢰⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠨⠉⠁⠀⠀⠀⠀⠨⠆⠀⠈
⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡿⠈⠀⠤⠄⠀⠀⠀⣆⠀⢀⣘⡥⠀⠀⡀⢸⠀⢀⣠⡤⠀⠀⠀⣀⣀⣀⡀⠤⠤⠀⠀⠀⠀⠀⣘⣀⡀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⣠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠉⠈⠉⠈⠉
⠀⣷⣶⠀⠀⠀⠀⣏⣰⡄⠀⠀⢼⢀⡄⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠤⠄
⠀⣿⡟⢆⠀⠀⠀⠘⣿⡟⠒⠀⠀⢿⣷⠃⠀⠀⢠⢿⣿⡟⠀⠀⣠⣿⣇⠀⠀⠀⢿⣿⠛⠦⠀⠀⠀⠀⢀⣀⣀⡀⠈⣻⠿⣷⣄⠀⠀⠀⠀⢀⣤⣄⡀
⢸⡟⣻⡆⠉⠀⢀⡼⣿⡇⠀⠀⠀⢸⡿⢧⡀⠀⠁⢸⡿⣧⠀⠈⢠⣿⣷⠄⠀⠠⡴⢿⡀⠃⠀⠀⢀⣾⣟⢻⠉⠁⠀⠁⠀⣾⣧⡀⠀⠀⠀⢠⣫⣿⣿⣀
⠀⡇⠀⠀⠀⠀⠋⠀⠹⠖⠂⠀⠐⠾⠃⢘⠁⠀⠀⠈⡇⠉⠀⠀⢸⠀⠋⠀⠀⠀⠙⡎⠳⣄⠐⠛⠛⠋⠉⠰⠀⠀⠀⠀⠀⠀⠈⠛⠷⠠⠆⠘⠈⠈⠟⠙
⠀⠘⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢠⠶⣾⣧⠀⠀⠀⢀⣼⣇⠆⠀⠀⠀⢀⣼⡄⠀⠀⠀⣿⡀⠀⠀⠀⣤⣼⡇⠀⠀⠀⣺⡇⠀⠀⠀⣼⣧⡀⠀⢀⣼⣧⡀⠀⠀⢀⣀⣾⠀⠀⡄⣷⣀
⠀⠃⣼⣿⠃⠀⠀⠀⣿⣿⡟⠁⠀⠀⠀⢻⣿⡗⠀⠀⠠⣿⡟⠀⠀⠐⣼⡿⠃⠀⠀⠀⣿⠏⠀⠀⠘⣿⣏⡃⠀⠙⢾⣿⠁⠀⠀⠙⣿⡿⠀⠀⠘⣿⡏⡇
⠀⠠⣿⠏⠀⠀⠀⣀⣼⠟⠙⢷⠀⠀⠀⣸⡿⡇⠀⠀⢰⡟⡇⠀⠀⠠⣿⠃⠀⠀⠀⢸⡟⣷⡀⠀⠀⣿⡿⠀⠀⠀⢸⣟⠀⠀⠀⢠⡿⢧⠀⠰⡞⠛⢿⡉
⠀⠰⠁⠀⠀⠸⠋⠉⠀⠀⠀⠘⠆⠀⠈⠀⠀⣇⠀⠀⠀⠧⢹⠀⠀⠘⠃⠀⠀⠀⣰⠛⠎⠁⠀⠀⠀⠀⠇⠀⠀⠠⠏⢻⡀⠐⠊⠉⠀⢉⠀⠀⠷⠀⠀⠉⠲

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⣀⣴⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⣶⣿⣿⠿⣷⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠈⠉⠁⠀⠸⠀⠀⠉⠉⠛⠻⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⢿⡟⠀⡴⣤⣀⣀⣀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠏⠁⠈⠉⢹⠟⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠂⠰⠾⠅⠀⠴⡤⠂⢺⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⢰⣦⣄⠀⣐⣤⣤⠄⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢀⡀⠀⠒⠀⣿⠈⠈⣳⣦⣴⣷⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠈⠀⢀⠀⢰⣿⣇⣷⡿⣻⣿⢿⣿⠧⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢰⡄⣠⢏⡞⢸⣿⠛⣿⠇⠃⡃⣹⢣⠀⠀⠙⢗⠀⠀⠀⠂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⢷⣿⣼⠀⣿⣿⡆⠁⡀⠀⣼⠃⡿⠀⠀⢐⡤⠛⠀⠀⠀⠈⣙⠂⠀⠀⠀⠀⠀⠀⠀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⣿⣿⠀⢿⣿⣿⠠⠁⣸⡏⡄⣿⠀⣴⠋⢀⡀⠀⠀⠀⠀⣌⠳⡄⠀⠀⠀⢀⣤⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠈⢛⣿⠀⢈⣿⡇⠀⡸⢸⡟⡇⢻⢰⡇⣠⡞⠁⢀⡴⢠⠆⠈⠱⣿⠀⢠⣾⣿⡿⠋⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⢠⣸⣿⡾⣯⣿⠀⠰⠁⠀⠀⡙⣾⣿⣇⣿⠀⠀⣾⢁⣿⡄⠀⠀⢦⡾⠏⠁⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⣿⣿⣿⣷⡈⠻⢰⣶⣆⠙⢆⠻⣿⣿⣿⣿⡇⣷⡛⣈⠉⠁⠀⠀⠀⢁⠺⠀⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⣸⣿⣿⠟⠉⠀⠸⠉⢻⣿⣦⠲⣄⠸⣿⣿⣿⣷⣿⠇⣿⡄⠀⢀⣀⣀⣀⡀⣦⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢿⡁⠀⠀⠀⠀⢳⢾⡿⣿⣷⣌⡻⢿⣿⣿⣿⣯⡀⠻⠷⠛⠉⣉⣬⡭⢍⠳⣌⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⢹⣶⣄⠀⠈⣯⣙⣷⠀⠿⣿⣿⠓⠻⣿⣿⣻⣧⠰⠟⣡⡿⣉⠴⠂⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢞⣾⡞⠁⠈⠙⠿⠶⡌⢿⣿⣿⣿⠾⠿⢿⣿⣿⣿⣿⣿⡆⢰⣿⠾⠋⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⡿⠿⠿⢟⣛⣛⣩⣽⣷⠘⠃⣤⣾⣶⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⠶⠶⣛⣻⣿⡿⠿⣟⣛⣿⡇⠈⡋⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠛⠋⡉⠉⠹⠟⠀⠠⡾⠁

⠀⠀⣤⢂⢛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢠⣄⣠⡉⠁⢹⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣄⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⡀
⠀⠛⠿⠀⠀⠈⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠿⠻⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣴⢖⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠛⢟⣉⢀⣼⡟⣉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⣿⣿⡇⢸⠃⠀⠀⠀⠀⠀⢀⣾⣡⣼⢿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠘⣿⣿⢧⠀⠀⠀⠀⠀⣠⡴⢶⣎⠱⡈⢿⢀⣃⠴⡤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠈⠀⠀⠀⠀⠀⣠⡿⠛⠉⣰⠟⠉⠁⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⣿⢿⡈⠁⠀⠀⠀⠀⣯⠰⡟⠘⡃⢈⡾⠃⣿⠠⣿⠄⡇⠀⠀⠀⢀⣠⣄⠀⢸⡏⢷⡄⠀⠀⠀⠀⠀⠰⠏⠀⢠⣾⢁⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠘⢯⡛⢦⣀⠀⠀⠀⠘⠲⠴⠞⠀⠀⠀⠀⠙⠳⠶⠞⠁⠀⠀⠀⠸⣿⠟⣠⠟⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠶⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠟
⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⣄⣰⠶⠶⠶⢶⣾⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢇⣴⣶⣄
⠀⠀⠀⠀⣄⣲⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⣄⡀⠀⠀⠀⠀⠠⣿⣿⣿⣿⡄⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣯⣉⣿⣿⡿⠛⠃
⠀⠀⠀⠀⠀⠀⢙⣿⡟⠀⠀⠀⢠⣤⣤⣀⡀⠀⠀⢐⣀⣤⣾⣿⠿⠃⠈⠁⠀⠀⠀⠀⠀⠀⣼⡟⢿⣷⠀⠀⠀⠀⠀⢠⡾⢻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣾⠟⣿⡀⠀⠀⠀⠀⠀⠙⠻⢶⠿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⣹⠆⠀⠀⠀⠰⠿⠇⠈⠛⣿⡿⠿⠾⠛⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣿⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣀⡀⠀⢀⡿⣸⠃
⠀⠀⠀⠀⠀⠴⠿⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⠈⠋⠈⠁
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣄⠀⢀⣻⣇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿⡀⠘⣏⠀⠀⠀⠀⡶⠀⢰⣿
⠀⠀⣠⣾⣿⣿⠛⠛⠛⠉⠉⠁⠈⠁⠀⠀⠀⠀⠈⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢶⣶⣾⣷⢶⣤⣤⣤⠤⠆⠀⠀⠀⠀⠉⠀⠀⢹⣦⣶⢄⣼⠁⠀⠙⠋
⠀⢰⠏⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠈⢀⣿⠿⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⢿⡆⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⡀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⢀⡼⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣸⡏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⢿⡆
⣰⡋⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠈⣇
⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠳⠄⠀⠘⠂⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⡐⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣶⣶⡄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⣧⣀⣠⣞⣠⠁⠀⠀⠀⠀⠀⢀⡤⠶⠖⢛⣿⣿⣏⠀⣠
⠀⠀⠀⠀⠀⣶⡆⢀⡞⢁⠀⠀⠀⠀⠀⠀⠀⣾⠟⣿⣿⣿⡟⠻⣆⠀⣀⡰⠏⠀⠀⠀⠠⠆⣿⣿⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠻⠷⠃
⠀⠀⠀⢀⣾⣿⣿⣯⣤⠤⣤⠀⠀⠀⠀⠀⠀⢹⡄⢽⣿⣿⣷⡈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣶⣶⡀
⠀⠀⠀⠸⣿⣿⠂⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠈⠻⡏⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠻⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠀⠀⣴⠏
⠀⠶⢶⢶⠿⣿⣿⢿⢶⠆⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⠘⠷
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣛⣀⣀⣀⣀⣙⣀⣀⣀⣀⣀⣀⣀⣀⣴⣁⣀⣀⣀⣀⣳⣄⣀⣀⣀⣀⣀⣙⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠉⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⡶⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡖
⠀⠀⠀⠀⢠⢶⣿⣄⠀⠀⠀⠀⠀⠀⠼⢻⣿⣅⠀⠀⠀⠀⠀⠀⣠⣶⠿⠤⠄⠀⠀⠀⠀⠀⠀⣾⣟⡓⠚⠀⠀⠀⠠⣶⠿⠛⠻⢾⣃
⠀⠀⠀⠀⠈⢸⣯⡈⠁⠀⠀⠀⠀⢀⣀⣿⣇⠈⠉⠀⠀⠀⠀⠐⠿⠟⠲⠶⠄⠀⠀⠀⠀⠙⠛⢡⠀⠙⠃⠀⠀⠀⠘⠛⠇⢸⠀⠀⠈
⠀⠀⠀⠀⠠⠟⠈⠧⠀⠀⠀⠀⠀⠀⠉⠁⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬

⠀⠀⠀⠀⠀⢠⡄⠀⢀⠠⠀⠀⠀⣠⡄⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⢠⡄⠀⢀⣄⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⣀⡗⣤⡀
⠀⠀⠀⢠⢶⣾⡗⠛⠉⠀⠀⠀⠀⠨⣷⡟⠉⠁⠀⠀⠀⠀⢀⣠⢴⣾⠗⠚⠃⠀⠀⠀⠀⠙⠤⣽⣴⠖⠒⠀⠀⠀⠈⠹⣿⣥⢾⣄
⠀⠀⠀⣨⠾⣏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠈⢁⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⢂
⠀⠀⠈⠁⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⢳⡀⠀⠀⠀⠀⠐⠛⠁⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢳⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢶⣠⣄⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠠⢤⣤⣼⡃⠀⠀⠀⠀⠀⢀⡀⠉⢳⣶⡛⠀⠀⠀⠀⠀⣀⡀⢻⡟⢀⣀⠀⠀⠀⠀⢀⡴⠾⢻⡉⠀⠀⠀⠀⠀⠀⣾⣆⣀⣀
⠀⠀⠀⠀⠀⢹⣿⣤⡀⠀⠀⠀⠀⠉⠛⠿⠿⡷⠶⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠒⠋⠀⠀⣤⣷⣤⣄⠀⠀⠀⠀⠃⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠏⠉⠀⠁⠀⠀⠀⠀⠶⠷⠶⠶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠆⠰⣤⡴⠂⠰⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠤⢴⣧⠤⠀⠀⠀⠀⠀⠈⠓⡟⠁⠀⠀⠀⠀⠀⠀⠀⠒⢲⣾⠷⠒⠂⠀⠀⠀⠒⠢⣼⣧⠔⢲⡀⠀⠀⠴⠞⠋⠛⠻⡟⠃
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣀⣇⣀⣀⣀⡀⠀⠀⠐⠖⠒⠚⠛⠒⠲⠂⠀⠀⠉⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣤⣽⣤⡀
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠄

⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⠀⠀⠀⠀⠀⣼⣀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣰⢰⠆⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢀⣿⣴⠶⠈⠀⠀⠀⠀⠀⢹⡿⠖⠋⠀⠀⠀⠀⠀⠀⣿⡷⠋⠀⠀⠀⠀⠀⡀⠈⢳⣾⣃⡄⠀⠀⠀⠀⠀⠸⠿⢿⡾⠁
⠀⠀⠀⠀⣠⡾⠧⣄⠀⠀⠀⠀⠀⠀⣴⢿⡇⠀⠀⠀⠀⠀⠐⠦⣤⠿⠧⣤⠀⠀⠀⠀⠀⠙⠶⢿⠇⠀⠀⠀⠀⠀⠀⡒⠒⠒⠛⢛⡖
⠀⠀⠀⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡄⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⢰⢀⡞⠀⠀⢠⠀⠀⠀⠐⠀⠀⢀⢀⣠⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠋⠹⠟⠳⢆⣠⡆⠠⡀⠀⠀⠀⠀⠀⠲⢶⡄⠀⠉⢰⣾⣿⡶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣅⢴⡿⣿⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⣿⡟⠁⠀⢀
⠀⠀⠀⠀⣦⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠾⠿⣿⠀⠀⢀⣦⣿⡀
⠀⠀⠀⠀⢹⣿⡄⠀⣀⣤⠤⠛⠓⠾⠿⢮⣕⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠉⠁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠁⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⠀⠀⠻⡄⠀⠀⠀⠀⢀⡰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣧⣾⠃⠀⠀⠰⣿⡄⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡶⠉⠲⠖⠉⣷⠀⢀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣷⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⡈⠙⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿⠇⠀⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⢠⣤⠤⠶⠶⠒⠂⠀⠤⠀⠸⡟⠒⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣿⣼⣿⣦⣴⣼⡟⠀⠀⠀⠀⠀⢰⠟⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠐⠀⠀⣠⣴⣚⡿⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣾⣉⣢
⠀⠀⢀⣀⣀⢀⣼⠟⠉⠉⠉⠛⠻⣿⣿⣿⠃⠀⠀⢀⣀⣀⠉⣦⣤⣤⡄⠀⠀⠀⢀⠀⠀⢙⠂⠀⠀⠘⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⢠⠏⠉⠉
⠀⠀⠘⠛⠻⠾⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠈⠉⠉⠈⠛⠿⣶⣀⣀⠐⠋⢺⠆⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⢸⡶⠿⢯⣭⣯⣽⣾⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⡆⠀⠀⣼⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠇⠀⢙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⢩⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢴⠝⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⣽⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠐⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣾⣿⣶⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆⠈⠿⠿⢷⡺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣶⠄⠀⠀⠀⣠⣾⣿⣿⠟⠙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣦⣤⠀⢾⡟
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢣⣟⠀⠀⠀⢠⣿⣿⠋⠁⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠛⠁
⠀⢰⡠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⢹⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣔⣶⡆
⠀⠘⠃⠀⢶⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠄⠀⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠘⠻⠁
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠋⠉⢉⣥⠞⠋⠁⠉⠉⠉⠛⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⣷⣄⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⣀⣀⣀⣤⣤⡤⠶⠶⠶⠶⠶⠖⠒⠒⢀⣄⣀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⠙⠛⠷⠶⠶⣶⣿⠷⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⣀⣠⣤⣤⣶⣶⣾⣷⣶⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣡⣴⠾⠟⠛⠉⠉⠁⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠿⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠾⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣦⣄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⣶⠟⠋⠉⠀⠀⣿⡇⠀⠀⠀⠉⠙⠻⢷⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠸⡇⢀⣴⠟⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣾⠏⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⡀⠀⠀⠀⠀⢀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣆⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⣠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣦⣄⣀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣿⣿⣯⣷⣦⣄⠀⠈⢿⣆⠀⢀⠀⠀⣀⣀⠤⣀⡠⣀⣠⣶⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣧⣽⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⠛⠟⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠋⠹⠏⠙⠛⠛⠛⠟⠛⠛⠛⠛⠉⠙⠏⠙⠫

⠀⠀⠀⠀⠀⠸⠗⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠠⡶⠉⠀⠀⠀⠀⠀⠀⠀⢀⡴⠆⢲⠶⢝⣠⠀⡈⠀⠀⠀⠀⠂⠈⠑⢩⣶⣤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔⠀⠀⠀⠰⠔⠿⠟⠀⠁⠀⠀⠀⠀⠠⣄⣠⣿⠋⠀⢰⡀
⠀⠐⣿⡀⠀⠀⢀⠤⣤⣼⣫⣽⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠷⠂⠀⠀⠀⠀⠀⠀⠙⠙⠛⠀⠀⠿⠇
⠀⠀⠘⠗⠀⠉⠀⠀⠀⠀⠉⠺⣿⡄⠀⠀⠀⠀⠀⠀⠘⡿⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡆⠀⠀⣧⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠸⠞⠐⠚⢹⡄⣠⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠖⠀⠀⢀⢀⣀⣀⡀⢀⠀⢠⣌⠛⠃
⠀⠀⠀⠀⣀⣀⣶⣼⣧⣤⣿⠃⠀⠈⠁⣴⠖⠀⠀⠀⠀⠀⢀⡀⠠⠀⢀⣠⣴⠏⠁⠀⠀⠀⠀⠀⠈⢰⣒⣀⠈
⢀⣤⣄⠜⠋⠈⠉⠛⢻⣿⡏⠀⠀⢤⡤⢹⣶⡶⠀⢀⣀⡀⠀⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⢠⣆⠀⠀⠌⠉⠁
⠀⠀⠉⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣭⣿⣶⣶⣠⣭⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠋
⠀⠀⠀⠀⡴⣶⡆⠀⠨⠟⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠛⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠨⡿⠤⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠈⣿⣤⡀
⠀⠀⠀⠀⠀⠘⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢲⣿⣿⣧
⠀⠀⠀⠀⠀⠈⢫⣦⡄⠀⠀⢠⣾⣿⠟⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣤⡄⠼⠏
⡀⠀⠀⠀⠀⠀⠈⢻⡁⠀⢀⣿⡏⠁⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⢀⣛⡁⠀⠉
⠏⠀⣤⡀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠸⢿⠁
⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣛⠋⠉⠉⠉⠉⠙⠛⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠈⠙⠻⢶⣤⡀⠀⠀⠀⢸⡟⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⣠⡿⠁⠀⠀⠀⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣴⠿⢿⣄⠀⢀⣠⡶⠟⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣤⡶⠶⠟⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⢀⣻⣶⠟⠉⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⠙⣷⡀⣀⣀⣀⣀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⢀⣠⡾⣿⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⠀⠀⢠⡾⠋⠀⢹⣇⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⣠⡴⠟⠋⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠈⣿⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⣿⡆⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⣿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣦⣴⣾⣏⣠⣤⣤⣴⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣾⡶⠶⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠌⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⠀⣜⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣤⣘⠦⢏⡀⠀⠰⠂⠀⠀⠉⠀⢀⠀⠤⠤⣴⠆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⡉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⠁⠀⣏⠙⢳⣴⡯⢐⡱⠀⠀⠀⠀⠀⠑⠳⠀⠀⢀⣿⣿⠿⠋⠢
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠶⠚⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⣼⣿⠁⠀⠀⢧⡄
⠀⠀⢺⣿⡀⠀⠀⠀⠀⣠⣄⣢⣜⢿⣛⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣶⡶⠂⡐⠀⠀⠀⠀⠀⠀⠀⠙⠝⠻⠟⠀⠀⣼⣿⠇
⠀⠀⠀⣿⣧⠀⠰⠒⠚⠁⠈⠙⠛⠛⠪⣻⣗⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠌
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣽⠆⠀⠀⢹⣆⠀⠀⠀⢀⡒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣂⡦⣄⡠⠿⣏⠀⠀⠀⠈⢿⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠁⢀⣿⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠺⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠏⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠠⠜⠃⠀⠀⢀⣀⣠⣀⣀⣠⠠⢐⡈⠀⣿⣤⡼⠋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣰⣿⣄⣠⣾⡿⠀⠀⠈⠁⠀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠋⠀⠀⠀⠀⠀⠀⡆⠄⠈⢠⡖⢦⠈⠈⠁
⠀⠀⠀⠀⢠⣾⠓⠛⠻⠿⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠃⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠈⠁⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣸⠛⠚⠙⠁
⠐⠿⠿⣤⠋⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠘⠀⠀⠀⢹⣏⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣄⣠⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣦⣾⢾⣟⣳⣢⣴⣦⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠟⠁
⠀⠀⠀⠀⠀⢠⣤⣶⣆⠀⠀⣨⡿⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠉⠛⠛⠻⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⠋
⠀⠀⠀⠀⠀⠯⠀⠹⣿⡀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡜⠃
⠀⠀⠀⠀⠀⠀⣥⣀⠷⡇⠀⠀⠀⠀⠀⠀⠀⠀⣘⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⢡⣦⡌
⠀⠀⠀⠀⠀⠀⢼⡿⠆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⠜⠢⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣿⣷⡶⣿
⠀⠀⠀⠀⠀⠁⠀⣀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠈⠿⠿⢗⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣶⠆⠀⠀⠀⣴⣿⣿⡿⠋⠹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣶⡦⠀⠿⠇
⠀⠀⣠⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⢰⣿⡟⠁⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⠀⠀⠉⠁
⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷
⠀⠀⣬⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠘⢿⣿⠆⠀⠀⠀⠀⠀⠀⠈⣙⡅
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣀⡘⣁⡀⢀⣾⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠾⠟⠀⠀⣠⣾⣿⣿⣿⠟⣿⠿⢿⣿⣿⣧⣌⣉⣩⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡎⠉⣩⣤⣔⠀⠈⠀⢸⣿⣿⡟⠉⠀⠀⠀⠀⠈⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡇⣿⣿⣩⣽⣦⠑⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠦⠀⠀⣨⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣮⣿⣿⠿⠿⣲⣄⠸⣿⣿⣿⡖⣀⡀⠀⢠⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣩⡽⠛⣋⣵⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⢻⣿⣿⠿⣿⣿⣿⣿⠋⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⡿⠋⣿⣿⠋⠵⠚⠉⠁⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠻⠿⠧⠈⠩⠀⠀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣅⠀⡰⠀⠀⢠⣠⣾⣿⣄⣴⠾⢋⠁⣁⣀⠀⠤⠛⣁⢀⢰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣸⣤⣾⣿⣿⣿⢏⣁⠴⠊⠀⢹⡄⠡⠠⢾⣟⢢⣸⣿⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⣭⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠉⠂⢀⣐⣢⡶⣾⡇⢸⣿⣨⣿⣿⠿⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⣿⠏⠁⢸⡏⠙⠛⠟⠛⠋⢀⣉⣠⣽⣿⣷⣦⣍⡤⠿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⢤⣷⣤⡀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣿⣿⡇⢠⣿⣬⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣭⣶⣾⣾⡇⣤⠀⠈⠋⠁⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⠀⢾⣿⠇⣀⣴⢀⡀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣦⣉⣀⣿⣿⣿⣷⣤⣶⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠠⠘⠟⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠹⠿⠿⠿⠿⠒⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡛⠻⠿⢿⡿⠿⠷⠄⠜⠛⠛⣿⣿⣶⡞⠃⠀⠘⣿⠿⣿⣋⣉⠚⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣿⣿⣦⡜⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⡀⢭⠔⠀⠀⠉⠁⠀⢹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⡀⠀⢀⠴⢤⠄⢀⣴⣋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⢿⣿⣶⣄⣤⠀⢀⣀⣼⣿⣿⡿⠛⠁⣠⢄⠱⣦⣄⣴⡿⢟⣛⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣶⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣾⡿⢟⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠿⣿⣧⠉⠙⠛⠋⠺⠷⠿⠛⠻⣽⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡀⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠉⠻⠿⠿⢿⣿⣿
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠻⣿⣿⣶⣤⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠘⠛⠻⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢠⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣤⣀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⠀⠀⠀⢀⣠⣤⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⡻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⠿⠿⠿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣻⣯⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣴⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⠉⠉⠉⠙⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣇⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠟⠿⣿⠟⢻⠿⢿⢿⠻⣿⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣷⣾⣷⣿⣶⣿⣷⣾⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠉⠉⠛⠻⢿⣿⣿⡿⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣉⣻⣿⣷⣬⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⠛⠛⢿⣿⡁⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣠⡾⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⡀⠈⠻⣿⣄⠀⣘⣿⣿⣦
⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⣤⣿⣿⣿⣿⣿⡿⣿⣷⡀
⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⡿⠿⣿⣏⠁⠀⠹⣿⣷⡀
⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⠿⠿⣿⡇⠀⠀⢻⣿⣆⣀⣀⠹⣿⣷
⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⡿⠟⠛⠉⠀⢀⢈⣿⡇⣠⣴⣿⢿⣿⡟⠿⣿⣿⣿
⠀⢸⡏⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⡜⢸⣿⣿⡿⠋⠀⠀⣿⣿⡄⠀⠹⣿
⠀⣾⡇⣀⣤⣶⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣾⣿⡟⠁⠀⠀⠀⢿⣿⣷⠀⠀⢻
⠀⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⢀⣾⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⣸
⠀⢸⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⣿⣿⠟⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣿
⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿
⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢁⣴⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣿⡿
⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣠⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⢀⣴⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷⣦⣤⣤⣀⣀⣀⣀⣀⣤⣤⣴⡾⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡤⠤⠤⠤⠀⠒⠒⠒⠒⠒⠒⠀⠤⠤⠤⢤⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠖⡩⢄⢱⠀⠀⠀⠈⢢⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡔⠁⠀⠀⠀⡎⡠⢍⠲⡀
⠀⠀⠀⠀⠀⢠⠃⠼⣀⠸⣼⡀⠤⣰⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⢆⠤⢀⣧⢇⣀⠧⠘⡄
⠀⠀⠀⠀⠀⡆⠰⠐⠂⠐⠂⠒⢨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠒⠐⠒⠐⠊⠆⢰
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢁⠸⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⠇⡈
⠀⠀⠀⠀⠀⠈⡄⢣⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⡜⢠⠁
⠀⠀⠀⠀⠀⠀⠘⡄⠣⡀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⢀⡜⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠑⢄⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡠⠊⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠳⣄⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠑⢄⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡠⠊⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡉⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⣉⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠑⠄⣣⠀⠀⠀⠀⠀⠀⣔⣡⠈⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠒⣁⡷⠀⠀⠀⠀⠀⠀⢾⣍⠒⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼

⠀⠀⢀⠤⡂⠀⠉⠙⡌⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠉⠙⡆
⢀⠔⠁⠀⠈⠓⣄⠀⠸⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠙⢗⠢⢌⣒⠒⠁⠀⣀⣀
⡄⠀⠀⠀⠀⠀⠈⣇⠀⡇⡼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⣚⡉⣉⣥⣴⢾
⡧⢄⣀⠀⠀⠀⠀⢸⠀⣧⣃⡠⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⣩⠖⠊⠉
⢱⡀⠀⠈⠉⠐⠒⡞⢺⠛⡅⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⡴⠁
⠀⠑⢤⣀⠀⣠⠞⠀⡞⢀⡹⠖⠁⠀⠀⠀⠀⣠⠔⠐⠒⢔⠒⠦⢤⡀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟
⠀⠀⠀⠀⠉⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⢠⠎⢀⡉⢦⡀⠈⢂⠀⢡⠈⢣⡀⠘⠢⢝⡒⠶⠤⠤⠴⣒⡪⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡔⠁⢀⡠⢷⠀⢸⠀⠘⠀⢸⢳⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⡠⠃⠀⣀⣄⢸⠀⢠⣀⣸⠼⡇⠀⠀⠀⠀⠀⠀⠀⡠⢚⣅⡢⠾⣿⠷⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠸⡇⠀⠁⢀⠾⢄⣀⠈⠉⠉⢀⣀⢴⠇⠀⠀⠀⠀⠀⢠⢊⣴⢷⡏⢨⠞⢄⡜⠃⢼⠸⡆
⠀⠀⠀⣠⠔⠛⢧⡔⠋⠀⠈⢟⠦⡀⠀⠱⣄⠀⢸⠤⣀⠈⠉⠉⢉⣁⣠⠞⠀⠀⠀⠀⠀⠠⢡⣳⠗⢅⡜⠑⠦⣎⠈⡰⢺⠀⡇
⠀⠀⡌⠀⠀⠀⢠⡀⠀⠀⠀⡘⠲⣝⣆⠀⠈⠑⠮⢷⣤⣬⣭⡭⠽⠛⠁⠀⠀⠀⠀⠀⢠⢣⡿⠋⡦⡂⢈⠖⢄⡠⠋⡲⡞⢰⡇
⠀⡜⣸⠦⠤⡖⠁⠙⠦⡠⠴⣄⠀⣸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⢉⠶⣀⡼⠓⣤⡊⢩⠾⡕⡡⢣⠁
⠀⣏⠀⠀⠀⠱⣄⣀⡜⠀⠀⢠⠋⠈⢦⡇⠀⠀⠀⠀⠀⢀⠤⠀⠲⠤⡀⠀⠀⠀⠀⠀⡇⡧⠉⣦⡎⠑⠴⣁⡜⡳⢊⠔⡱⠃
⠀⠘⣄⢀⣀⠊⠀⠀⠑⢢⠖⠣⣀⠀⡼⠁⠀⠀⠀⠀⢠⠊⠐⠢⡀⠀⢈⣆⠀⠀⠀⢐⢠⡘⠶⣁⣨⡳⠤⢛⡩⢔⡡⠊
⠀⠀⠈⠳⢄⣇⠀⢀⠠⠯⣄⣰⠧⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠆⠊⢸⠀⠀⠀⠌⣬⠶⢋⡠⠴⡒⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠔⠋⢉⡷⠃⠀⣀⡌⢈⡵⢊⠕⠊⠁⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠲⢤⠤⣀⡀⠀⠀⠈⠉⠉⠁⠀⢀⠾⠉⢍⢫⠈⠀⠀⠀⠀⢸⡀⣀⣸
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣀⣠⢤⠴⠤⢦⠈⡍⡗⠀⠀⠀⠀⠀⡰⡉⢦⢱⢸⠎⠀⠀⠀⠀⢀⣼⣖⣗⣮⣄
⠀⠀⠀⠀⢀⠜⢄⠀⡠⠴⠎⠙⠉⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠀⡾⠤⡙⡼⡤⠃⠀⠀⠀⠀⠀⣾⢏⡏⢿⢙⢿⣄
⠀⠀⠀⠀⠊⠡⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠑⢄⣈⡷⠁⠀⠀⠀⠀⢀⣎⡟⣹⠓⢼⠉⣍⣷⢆
⠀⠀⠀⠀⠀⠀⢱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠾⢁⣼⠑⣾⠁⣤⢹⠛⠧⡀
⠀⠀⠀⠈⠢⢄⣼⣠⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣅⠘⡹⠃⠫⣀⣨

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣠⣌⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠁⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⣴⣦⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠈⠁⠀⣙⠋⠀⠂
⠀⠀⠀⠀⢠⣾⠛⠉⠀⠈⠉⠺⣷⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠉⠉⠉⠛⢿⣆
⠀⠀⠀⠀⣿⠃⠀⠀⠀⢀⠀⠀⠸⣷⠀⠘⠓⠶⠀⣀⠀⠀⢸⡟⠀⠀⠀⠀⡀⠀⠀⢻⡆
⠀⠀⠀⠀⣿⡀⠀⠀⠐⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠘⢿⣄⡀⠀⠀⢀⣤⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⣀⣴⠟⠀⠀⠀⠀⠀⠀⠑⠲⢤⠤⠤⠤⠤⠤⠀⠒⠶⠶⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⠛⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⡐⣤⡄⢸⡆⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⣀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠛⢦⢧⠈⠁⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠈⠘⠀⠄⡀⠀⠀⠂⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡌⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠛⠢⠀⠠⠔⠊⠀⠀⢀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠊
⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⡁
⠀⠀⠀⠀⠀⠆⠈⠋⠀⠴⠃⠶⠉⠀⠴⠇⠠⢺⡄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⡾⠛⢁⣠⣴⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠰⣿⣿⠟⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⠀⢀⡀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠁⠰⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣈⠙⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠀⣴⣶⣤⣄⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠛⠋⡉⠁⠸⢿⣿⣿⣿⣿⣷⣦⣄⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⣠⣾⣿⡏⢠⣶⣄⠙⢿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠀⣼⣿⣿⣧⡈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡏⢠⣿⣿⣿⣿⣷⡄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣆⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠈⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⡀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠻⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠈⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠼⢧⣯⣤⣤⡿⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠉⢀⣀⣠⣄⠒⠪⠙⠢⢤⣄⠀⢀⠈⠁⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠚⠓⠲⠟⠓⠉⠉⠉⠀⠀⠀⠀⠁⠀⠉⠉⠘⠛⠓⠲⣄⡈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡴⠠⠄⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠲⡌⠹⣦⠂
⠀⠀⠀⢀⡀⠠⠄⢀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠐⠀⠈⢳⡀⠀⠀⠀⢀⣀⣀
⠀⡠⠊⠁⠀⠀⠀⠀⠈⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠈⣱⡄⠀⠁⠀⠀⠀⠀⠁⠢⡀
⡐⠀⠀⠀⢀⠄⠠⢀⠀⠀⡏⠃⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⢀⡤⢘⠹⠟⡀⢀⠤⠐⠠⢄⠀⠀⠈
⡇⠀⠀⠀⠂⠀⠀⠀⠑⣤⡇⢈⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣜⡤⠞⠁⠀⠀⠀⠈
⠇⠀⠀⠀⡀⠀⠀⠀⠃⢨⠷⠸⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⡇
⠘⡀⠀⠀⠠⠀⠀⠀⠑⢼⣄⠠⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠣⡀⣃⠂⠀⠀⠀⠀⢠
⠀⠐⡄⠀⠀⠣⡀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢢⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠃⠀⢀⠜
⠀⠀⠈⠢⡀⠀⠀⠠⡀⠈⠰⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢈⠀⠀⠀⠀⠀⢰⠀⠀⠀⠄⠊⠀⠀⡠⠂
⠀⠀⠀⠀⠈⠢⡀⠀⠙⢢⡄⠂⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⡘⠀⠄⠀⠀⠀⣤⠔⠉⠀⠀⢀⠔⠈
⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠹⢌⠄⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠐⠀⡘⠀⠀⠀⢼⠃⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢷⠈⢄⠀⢄⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⡠⠊⢀⠜⠀⠀⠀⠀⡎⠀⠀⠸⡀⠀⠀⣀⢀
⠀⠀⠀⢍⠁⠀⠀⠉⠀⠀⠀⡘⢠⠈⢢⣀⠁⠦⢤⠐⠒⠈⠀⠀⢀⠤⠊⢀⡠⠂⠀⠀⠀⠀⡐⢃⠀⠀⠀⠈⠉⠉⠀⡸
⠀⠀⠀⠀⠑⠠⢀⡀⢀⠠⠊⠀⠀⠳⢅⡀⠖⠰⠦⠤⡭⠬⠭⣥⠐⠖⠒⠉⠁⠉⠀⠀⡨⠛⠀⠀⠑⠠⠀⠀⠠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⠦⢄⣠⡐⠒⠒⢀⣤⡀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡗⠙⠦⠆⠉⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢾⠠⠄⠠⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠸⠁
⣿⣿⡿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠀⣰⣾⠤⠀⠀⠀⠀⡠⠄
⠀⠈⢄⠀⢀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣀⠔⠉
⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠡⣀⠀⠀⠀⠔⠀⠀⠠⢂⠴⠊⠀⠀⠀⣠⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡠⠈⠀⢀⠀⠀⠉⠈⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠰⠛⠂⠀⢴⣢⣄⠀⠀⠉⠁⠀⠈⠀⠀⠀⠙⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠀⠀⠈⢪⠘⣆⠀⠀⠀⠀⣦⠀⠀⠀⠠⠀⠀⢀⠔⠉⠀⠀⠀⠞⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣺⠟⠓⠲⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡀
⠀⠀⠀⠐⠤⣤⠠⠔⠀⠀⠑⠤⠀⠀⢠⠏⠓⣾⠇⠀⢇⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠁⠀⣠⠞⠀⠀⠀⠀⠀⡰⠀⠐⠂⠀⠀⠀⠀⠀⠃⠀⢠⡞⠀⠀⠀⠀⢐
⠀⠀⠀⡯⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⢠⣼⡦⠀⠀⢃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠈⠲⢄⡀⠈⠀⠀⠀⣀
⠀⢸⡀⠀⠀⠀⠀⣠⣾⡀⣀⣀⠀⠀⠀⠀⠈⠁⠘⠉⠉⠀⠀⠀⠀⠴⠠⢰⠄⠀⠀⠀⡠⠀⢀⡴⠶⠀⠀⠒⠁⠀⢠⠀⡄
⠀⠀⠳⢤⠀⠐⠛⠛⠻⠻⣛⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢀⡀⠀⠀⡐⠂⠀⢀⣠⠤⢤⠀⢀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠁⠀⢀⣿⠇⠰⣿⣿⡆⠃⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣶⣤⠄⡘⠁⠀⠀⠈⡉⣄⣀⠉⠩⣠⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⢿⣿⠂⡴⠿⠁⠀⠀⠀⡇
⠀⠀⠁⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⡀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⡄⢶⣦⣄⡀
⠀⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣷⣦⡀
⠀⠀⢀⣴⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣦
⠀⢠⣾⣿⣿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷
⢀⣿⣿⣿⣿⣿⣷⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⢰⡀
⣸⣿⣿⣿⣿⣿⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⣸⣇
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⢡⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀
⣏⢿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⢻⣦⡹⢿⣿⣿⣿⣿⣿⣿⣿⡌⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⡟⣡⣿⣿⣿⡟⠈⢳⡀⠀⠀⠀⠀⠀⡾⠁⢹⡆⠀⠈⢙⣿⣿⣿⣿⣷⣤⡀
⠘⣿⣿⣦⡙⠿⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣿⢸⣿⣿⣿⠏⣴⣿⣿⣿⡿⠁⠀⠈⢷⡀⠀⠀⠀⣼⠁⠀⠀⣷⠀⠀⣾⠁⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠘⣿⣿⣿⣷⣦⣍⡛⠿⢿⣿⡇⣿⣿⣿⡇⣼⣿⣿⠏⣼⣿⡿⠟⣫⡄⠀⠀⠀⠈⣷⠀⠀⢰⠇⠀⠀⠀⢹⡄⢰⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣶⢠⣬⣭⣍⢁⣍⣭⡉⣨⣭⣴⣶⡟⢹⠃⠀⠀⠀⠀⠸⡇⢀⡟⠀⠀⠀⠀⢸⣇⣼⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠁
⠀⠀⣸⣿⡻⣿⡿⠿⠿⠿⢟⣡⣿⣿⣿⠇⣾⣿⣿⡇⣿⣿⠟⢯⡀⢸⠀⠀⠀⠀⠀⠀⢻⣼⠃⠀⣀⣤⣶⣿⣿⡿⣿⠋
⠀⠀⠸⣿⣿⣌⠻⢿⡛⠿⣿⣿⣿⣿⠏⣼⣿⣿⠿⠟⠈⠁⠀⠈⢳⣼⠀⠀⠀⢀⣀⣤⣼⣿⣾⣿⢿⠟⣿⡇⣸⣷⠏
⠀⠀⠀⠈⠛⢿⣷⣶⣿⣦⣽⣆⣀⣀⣀⣀⣙⣿⣆⣀⣠⣤⣤⣤⣴⣿⣶⣾⠿⢿⡿⣿⡉⣽⠋⣸⢿⣤⡏⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠙⢿⡟⣿⠿⣿⣻⠿⣟⢻⡿⣟⠛⠛⣻⠿⣍⠉⣩⠏⠙⢦⣠⠟⠀⢀⡿⢧⣴⠏⢠⡟⢧⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡷⣈⡿⢦⣈⡿⣦⡈⠻⣶⡃⠀⠈⣿⢯⡀⠀⣰⠛⠷⣤⡞⠁⣰⠛⢳⣾⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣧⠈⣿⢧⡀⠙⣾⣇⠙⢳⣾⠃⠀⢙⣾⣇⠀⣠⠟⠳⣶⡃⠀⡾⠉⠳⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣻⣇⠀⠙⣾⡃⠙⢳⣞⠉⠛⢦⣞⠁⠈⣻⢯⣀⣼⠃⠙⣾⣅⠀⣸⠏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⢧⡼⠃⠙⣶⡏⠈⠳⣦⡟⠙⢳⣼⡃⠀⡽⠷⣄⣸⠃⠈⢳⣿⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⠛⣆⣴⠋⠙⢦⣴⠋⠙⢦⣾⠉⢛⣾⡁⠀⣸⠿⢦⡀⣼⠋⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⡼⠳⣄⢀⡼⠛⢦⣠⡟⠉⢳⣾⠁⠙⢳⣏⡀⠀⢹⡿⣄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁⠀⢙⣿⡁⠀⣰⠟⢦⣀⡞⠉⠳⢦⡟⠈⠙⢦⣾⠀⠈⠳⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢠⠏⠀⠻⣾⠃⠀⢀⡟⢧⡀⢀⡞⠙⢦⡀⢰⠏⠳⣄⢠⠏⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⢇⠀⢀⡜⠉⠣⣀⠞⠀⠀⠙⣾⡀⠀⠀⠈⡿⣄⠀⠀⢹⠦⡀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠈⢳⡞⠀⠀⢠⠟⢦⡀⠀⡼⠁⠙⢦⡀⣸⠁⠈⠑⢤⡟⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠏⠹⣄⢠⠏⠀⠀⠳⣼⠃⠀⠀⠀⠙⣯⠀⠀⠀⢈⣷⡀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠙⣏⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⡟⢦⡀⠀⣸⠁⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣠⡾⠻⣦⠀⠀⣰⠇⠻⣆⠀⠀⣸⠇⠀⠙⠛⠁

⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⡔⠋⠀⢂⢀⠉⠓⡄⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⡴⢏⠀⠀⠀⢈⠷⡄⠀⠀⠀⢠⣜⢿⣿⣿⣿⢟⣤
⠀⢸⠀⠀⠀⠀⠠⠀⡀⢹⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣘⠀⠈⡆⠀⠀⡎⠀⠘⠀⠀⠀⣿⣿⡞⣿⣿⡟⣾⣿⡇
⠀⠰⡀⠀⠀⠠⠀⠀⠀⡼⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠰⡀⢠⠃⠀⠀⢣⠀⡰⠀⠀⠀⢿⣿⢣⣿⣿⣧⢿⣿⠇
⠀⠀⠙⠮⣄⢀⣀⡤⠞⠁⠀⠀⠀⠀⠻⠿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠑⠧⣀⠀⢀⡠⠗⠁⠀⠀⠀⠈⠱⢿⣿⣿⣿⠧⠉

⠀⠀⠀⣀⠤⠀⠠⢄⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠤⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀
⠀⢀⠎⠀⠀⠀⠀⠀⢙⣄⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢷⠀⠀⠀⢠⣪⠖⠉⠉⠲⡄⠈⢢⠀⠀⠀⣴⢫⣵⣶⣮⡝⣿⣷⡄
⠀⠘⠀⠈⠉⠀⠀⠀⠠⢬⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣻⡇⠀⠀⠐⡇⠀⠀⠀⠀⠸⡄⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⡼⣿⣧
⠀⠘⣄⠀⠠⠀⠀⠈⠙⡜⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠰⡱⡀⠀⠀⠀⠀⠙⡶⠀⠀⠀⢷⡹⣿⣿⣿⣿⣿⣬⠃
⠀⠀⠈⠒⠤⢠⠠⠖⠊⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁⠀⠀⠀⠀⠈⠺⠦⣄⠀⠤⠊⠀⠀⠀⠀⠀⠙⠮⠛⠿⠿⠟⠁

⠀⠀⢀⢤⣖⡀⠐⠦⣄⠀⠀⠀⠀⠀⣠⠴⠶⣶⣦⡀⠀⠀⠀⠀⠀⢀⠔⠂⠀⣀⠒⢄⠀⠀⠀⠀⠀⣠⣶⣾⡿⣶⣦⡀
⠀⢠⠃⢿⡿⠇⠀⠀⠈⢣⠀⠀⠀⣼⣇⢀⣀⣿⣿⣿⣶⡄⠀⠀⠰⠁⠀⠀⠀⠉⠲⠂⠱⠀⠀⠀⣾⣿⣿⣿⣷⣾⣹⣿⡆
⠀⢀⠀⠀⠀⠀⠀⣿⣿⢄⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⢠⡆⠀⢸⣿⣿⣿⣏⣹⣿⣿⣿⣷
⠀⠈⢾⣿⣶⠀⠀⠉⣋⠎⠀⠀⠀⠹⣅⠈⢻⣿⣿⣶⡿⠁⠀⠀⠘⢄⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠉⠒⠒⠒⠊⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠑⠂⠠⠀⠚⠉⠀⠀⠀⠀⠀⠈⠛⠛⠟⠛⠋

⠀⠀⠀⠀⡠⠔⠲⢦⡒⡄⠀⠀⠀⠀⠀⢀⣤⣦⣔⠲⣶⠀⠀⠀⠀⠀⠀⢀⡠⠒⢒⢄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⢤⡀
⠀⠀⠠⠊⠀⢀⣴⠄⠹⡇⠀⠀⠀⠀⣴⣿⣿⠟⢙⣷⡜⠀⠀⠀⠀⠀⡔⠁⠀⢀⠎⠀⡇⠀⠀⠀⠀⢀⣴⣿⣿⣫⣿⡿
⠀⠀⣧⠀⢴⠟⠁⠀⡰⠁⠀⠀⠀⢈⢿⣟⢁⣴⣿⣿⠏⠀⠀⠀⠀⡎⠀⣀⠔⠁⢀⠎⠀⠀⠀⠀⢠⣿⡿⣻⣿⣿⠟⠁
⠀⠀⣎⡳⣤⣀⠤⠊⠀⠀⠀⠀⠀⢸⣦⣙⠿⡿⠟⠁⠀⠀⠀⠀⠀⠑⢍⣀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠳⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠠
⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠈⢷⠀⠀⠀⡏
⠀⠀⠀⣴⡡⠤⠒⠒⠒⠒⠲⡾⠤⣄⣸⠃⠀⢀⡴⠦⡀
⠀⢀⠈⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡟⠙⠲⣾⡅⠀⠸⡀
⠀⠘⠀⠀⠀⠀⠀⣠⠶⠉⠀⠀⠀⣸⠃⠀⠀⡧⠈⠑⢄⢠
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⢠⡏⠀⠀⠀⢷⠀⠀⠀⢻⡄
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢸⡆⠀⠀⢘
⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠈⡇⠀⠀⡘
⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡧⠀⡰⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡧⠞⠁⠀⢀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠒⠄⣰⣋⡀⠀⡀⢀⡀⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠖⠚⠋⠉⠉⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⠓⠲⠤⣤⣀⣀⣀⣀⣀⣀⣠⡤⠖⠊⢁⠞
⠀⠀⠀⠀⠀⠐⠒⠒⠲⠤⠤⠤⣴⣦⠤⠒⠒⠋⠀⠠⡄⠀⠈⠢⡀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣀⣀⣀⣀⣀⠀⠐⠄⠀⠀⠀⠙⠒⠠⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠂⠀⣀⡤⠖⠒⠊⠉⠉⠈⢹⡏⠁⠀⠀⠈⠉⠓⠲⡄
⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⣀⡤⢴⣷⠦⢤⣄⡀⠀⠀⠀⠈
⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡧⠀⠀⠈⠙⢆
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⠄⡀
⠀⠀⠀⢆⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡃⠁
⠀⠀⠀⠀⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⠄⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠃
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣀
⣿⣀⣀
⣿⣿⣿⣷⣀
⢻⣿⣿⣿⣿⣀⠀⠀⠀⣠⣄
⣶⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣷⣦⣄⣄⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀⣄⡀⣀
⡿⣿⡄⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⢴⣤⣄⣀⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣄
⠋⠀⠀⠈⢸⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣾⢻⣿⡿⣿⡟⠏⡟⣿⣿⣿⠛⣿⣿⣯⢽⣿⣯⣿⣟⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⢾⡟
⠀⠐⠀⠀⠀⢸⠟⠹⣿⡇⠱⠆⠸⠻⠏⢩⡝⠛⠏⢁⠃⠀⠈⠓⠠⠿⠶⠤⢬⣽⢓⠞⡌⣻⢻⠥⣽⣿⣿⣟⡋⠀⠩⢷⡻⠽⠟
⠀⠀⠀⠀⠀⠀⠃⠐⢪⡇⠠⣤⡃⣷⠅⠀⠁⢢⠀⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠁⠐⠃⡁⠂⣀⠋⣀⢒⡐⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠀⠈⠀⢠⠂⢀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⠓
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⡀⠀⢀⣀⣠⣤⣶⣶⣿⣟⣉⣉⣀⣤⣀⣀
⣿⣿⢛⠿⡛⠛⢿⠟⣻⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠖⠋⠉⠈⠉⠒⢄⠀⠀⠀⣠⠔⠋⠉⠉⠉⠒⢤⠀⠀⠀⢀⠔⠚⠉⠁⠉⠓⢤⡀⠀⠀⢠⠖⠉⠒⠢⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⡰⠁⠘⢡⡄⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠑⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡇⠴⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢣⡀⢿⠧⠀⠀⠀⠀⠀⡼⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠑⢄⠀⠀⠂⠀⠀⣀⠇
⠀⠀⠀⠑⠤⣀⣀⣀⣀⡤⠊⠀⠀⠀⠙⠦⣀⣀⣀⣀⡠⠚⠁⠀⠀⠈⠢⣀⣀⣀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠙⠢⠤⠠⠔⠁
⠀⠀⢀⡀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡤⠄⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⡇⠀⠀⣠⠄⠉⠲⢄⠀⠀⠀⢀⣴⠁⢀⣴⣶⣄⠀⢱⣄⠀⢀⡔⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠋⠀⠀⠀⠀⡈⠙⢆
⠀⠀⢧⠀⠀⠀⠑⢕⠄⠀⠱⡀⠀⣾⠃⠀⠈⠿⠿⠏⠀⠀⠹⡄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠁⠀⠄⠀⢠⠀⡀⠂⠈⡆
⠀⠀⠈⢆⠀⠀⠀⠀⠈⡦⠀⢱⠀⠇⢠⣤⣄⠀⠀⠀⣠⣤⢀⠇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠐⠑⠁⠀⠇⠈⠌⠀⢂⡇
⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⢘⠀⠈⢾⣿⣿⠆⠀⠐⢿⣿⠏⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠳⡀⠀⠀⠀⡐⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠀⠀⠀⠉⠒⠢⠤⠔⠊⠁⠀⠀⠀⠀⠉⠒⠶⠤⠔⠚⠁⠀⠀⠀⠀⠉⠓⠲⠦⠖⠚⠁
⠀⠀⠀⢀⠴⠂⠉⠉⠑⠢⣄⠀⠀⠀⢀⡤⢆⣉⠉⠓⠢⣄⠀⠀⠀⢀⡴⠒⠻⠉⠓⠲⢄⠀⠀⠀⠀⠀⠀⡐⠳⢤
⠀⠀⡰⣡⡶⡻⣶⡄⠀⠀⠈⢣⠀⠰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢰⠄⠀⠀⠙⢄⡀
⠀⠀⡃⣿⠰⢡⣾⡇⠀⠀⠀⠈⠃⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣿⣦⡀⠀⠀⠀⠑⣄
⠀⠀⢣⠈⠛⠟⠋⠀⠀⠀⠀⣰⠁⢣⠀⠀⢀⠀⠁⠀⠀⠀⣰⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⢴⣿⣿⣿⣿⣦⣄⠤⠔⠊
⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠑⢄⡁⠀⠀⣀⣀⠔⠁⠀⠀⠑⢤⣀⠀⠀⠀⣠⠔⠁⠀⢀⠀⠙⡿⠛⠉
⠀⠀⠀⠀⠀⣉⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠁⠈⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠢⡀⠀⠀⡴⠋⠀⠘⣄⠀⠈⠲⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠓⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠑⣄
⠀⠀⡞⢰⡷⠀⠀⠀⠐⣷⠆⢹⠀⡜⠀⠀⠀⠀⠈⠱⢄⡀⢘⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠒⠠⣀⠀⠀⠀⠀⠈⠉⠇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⣄⠀⠀⣠⣀⠀⠀⢀⠎⠀⠘⣄⠀⠈⠳⡄⠀⠀⢀⠜⠀⠘⢄⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠈⠑⠶⠬⠥⠤⠚⠁⠀⠀⠀⠈⠑⠢⠄⠼⠤⠒⠁⠀⠀⠀⠀⠑⠢⠤⠤⠤⠒⠉⠀⠀⠀⠀⠑⠢⠤⠀⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⣻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⣷⣴⣿⣿⣿⣿⡿⢿⣷⣄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣾⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⢿⣿⣿⣿⣿⣿⡿⠁⠀⠈⢿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠀⣨⣿⣿⣿⣿⣿⣧⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡤⢾⣿⣿⣿⡟⠉⠉⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠝⢀⣬⣷⣿⣿⠃⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⡿⣿⠗⣄⣠⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣴⣶⠋⠁⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣧⣄⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠟⠀⠀⠀⠀⠀⢻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠰⡄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠉⠉⠓⠢⣄⡵⠶⡉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡤⠴⠒⠒⠲⣤⣔⠋⠙⢦⡈⠑⠠⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣐⡠⠐⠒⠐⠒⠤⣧⡖⠁⠀⠀⠀⢀⡴⠋⠈⢧⡀⠀⠙⢆⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠲⢄⠀⣠⠏⠀⠀⠀⠈⣇⠀⠀⠈⢣⠀⠀⡄
⠀⠀⠀⠀⠀⡰⠱⣿⣿⡿⠟⠉⠁⠀⠄⡀⠀⣀⣀⣀⠀⠀⠰⡃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠇
⠀⠀⠀⠀⠘⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠡⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⡜
⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡷⠄⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⡸⠁
⠀⠀⠀⢸⠀⠀⠀⣠⣦⣄⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⠟⠁⠀⠀⡇⠀⠀⠀⣸⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠧⠄⠀⠠⠧⠄⠒⠉
⠀⠀⠀⠀⠠⡀⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠑⠄⠈⠿⠿⠟⠛⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⠌
⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠤⠤⠚⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⠿⠿⠛⠛⠋⠉⠉⠻⣿⣟⠛⠿⢷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣫⣥⣤⣤⣤⣤⣄⣀⣀⠀⢀⣾⡿⠀⠀⠀⠈⠙⣻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠿⣿⣿⣿⣥⡀⠀⠀⣠⣾⡿⠋⠙⢿⣷⣄
⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⠟⠋⠉⠛⢿⣷⣾⡟⠉⠀⠀⠀⠀⠙⢿⣧⡀
⠀⠀⠀⣠⣿⠟⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⠿⠟⠛⠉⠀⠀⠀⠀⢀⣾⡿⠛⢿⣦⣄⠀⢀⣴⣾⣿⣿⣷⡄
⠀⠀⢠⣿⠏⠀⠀⢀⣠⣶⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠙⢿⣷⣾⡿⠁⠀⠀⠹⣿⡄
⠀⠀⣾⡏⠀⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣦⠀⠀⠀⠀⢹⣿
⠀⢰⣿⢁⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢿⣷⡀⠀⠀⠀⣿⡇
⠀⣾⣿⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢻⣷⡀⠀⠀⢸⣿
⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢻⣷⡀⠀⢸⣿
⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⢿⣧⠀⢸⡿
⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⡄⣿⡇
⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠈⣿⣷⡿
⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠘⣿⣦⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⣸⣿⠋
⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⣠⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣁⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢷⣶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣡⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⢽⣒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣾⣣⡼⠖⠋⠉⢑⡜⠂⠀⠀⠀⠉⠉⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢙⣲⣤⠤⠤⠤⣿⠄⠀⣀⣀⠞⠁⠀⠀⠀⢀⡀⠀⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⠄⠀⠁⠀⠀⠘⠉⠉⠁⠉⠁⠀⠀⠀⣠⣏⠏⠁⠀⣁⡤⣤⣄⠹⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⢛⡶⠋⠀⠀⣀⣀⣉⣁⣠⠼⠁⠀⣠⠀⠀⣦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠉⠒⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠘⢉⡴⠚⠑⠺⣄⠀⠀⠈⠑⠢⠤⠴⢻⣀⣠⠿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠚⠙⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡖⠒⢜⣯⣍⣠⣟⠔⠊⣉⠆⠀⠀⠀⠀⠀⠀⠀⣠⣞
⠀⠀⠀⠀⢀⣴⣾⠏⠁⠉⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠸⣉⣱⢇⡠⠤⣽⠏⠇⠠⢐⠄⠁⠀⠀⠀⠈⣦⡠⢾⡿⢿
⢠⠞⡑⠠⡟⠋⠑⣄⣀⣀⡠⠋⠙⠻⣆⠀⠀⣀⠤⡄⠀⢉⣒⣊⣀⣴⠃⠀⠢⢈⡁⠔⠁⠀⠀⠀⠀⣧⢠⠼⠀⠘⠓⡄
⡄⠀⠇⣶⡅⠀⢠⣿⣿⣿⣇⠀⠀⠀⣻⡀⢈⡟⢠⢞⣆⣸⡄⠀⠀⠈⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢻⠈⢆⣠⠤⠚⠁
⠃⠀⠀⢻⡟⠀⠙⢿⣿⠿⠛⠒⢤⣴⣿⠁⠈⠻⡈⢧⠚⢇⣻⣄⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠚⠛⠦⡄⢡⣊⠙⢶⡄⠀⠀⢄
⠀⠀⠀⠈⢅⠀⠀⠀⡇⠀⠀⠀⢰⣿⠃⠀⠀⠸⢵⡌⠁⠛⠁⠈⠳⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⢺⠉⣇⠀⢣⠀⠈⡆⠱⠀⠀⠑
⠀⠀⠀⠀⠀⠑⠲⠾⢿⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠒⠊⠁⠀⠱⣄⠀⠀⠀⠀⢀⣴⣿⣁⡼⢾⡽⣇⢈⡿⠀⠁⠀⠄⠘⠃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠤⠖⠉⠀⠀⠀⠀⠈⠫⣈⡼⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢠⠀⡄⢀⠀⡄⢀⠀⡄⠀⡀⠀⡀⠀⡀⢀⠀⡀

⠀⠀⠀⢀⡴⠒⡿⠛⠛⠒⢤⡀⠀⠀⠀⠀⠀⠀⣠⠶⠒⡶⠒⠒⠦⡀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠑⠲⣄
⠀⠀⢰⠋⢀⠰⢆⠀⠀⠀⠀⢷⡀⠀⠀⠀⢀⠞⢩⠀⠎⠀⠀⠀⠀⢘⣆⠀⠀⠀⠀⢠⠋⠀⠀⠀⢠⡄⠰⡆⠹⡄⠀⠀⠀⢠⠞⠀⢀⡠⠤⠤⢄⠀⠈⢧
⠀⠀⣼⠰⠁⠀⠀⢹⠉⠉⠙⢮⡇⠀⠀⠀⢸⡄⠀⠻⠤⠀⠠⠤⠊⠁⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣒⠄⠀⡇⠀⠀⠀⢸⠀⢠⠏⢰⡿⣷⠀⢳⠀⢸⡀
⠀⠀⢹⠀⡱⠀⠀⢎⠀⠀⠀⢸⠃⠀⠀⠀⠘⣎⠒⢦⢀⣀⠀⢀⣀⠤⡾⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠸⡄⠈⢧⡘⠿⠟⣠⠞⠀⡼
⠀⠀⠈⠳⢧⣀⠀⢀⣳⣲⠾⠃⠀⠀⠀⠀⠀⠈⠳⢾⣦⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠈⠳⣄⣀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠘⠦⣄⡉⠉⢉⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⣠⠤⢲⡖⠒⠤⣄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⢒⡖⠤⣄
⠀⠀⢠⠞⣡⠖⠋⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⡴⢻⡂⠀⠀⠀⠀⢸⡷⡄⠀⠀⠀⠀⢠⠞⢁⡤⠚⢅⠉⠲⡉⢧⠀⠀⠀⢀⣠⡔⠒⠛⠉⠉⠉⠉⠓⢦⣄
⠀⠀⡞⢸⠇⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⡯⠰⠁⠑⢄⠀⠱⡀⠘⡋⡇⠀⣰⢋⡹⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢳
⠀⠀⣇⠘⣆⠀⠀⠀⠀⢠⡞⢡⠇⠀⠀⠀⠸⡄⠈⠷⣄⣀⣀⣠⠞⠀⡞⠀⠀⠀⠀⢧⠄⠧⠀⠀⠢⠀⠘⣀⢷⠇⠀⠹⣭⣻⠚⠒⠒⠒⠒⠒⠒⠛⣻⡿⠏
⠀⠀⠘⢦⡉⠻⠲⠶⠖⢋⡤⠋⠀⠀⠀⠀⠀⠙⢤⡀⠀⠈⠁⠀⣠⠞⠁⠀⠀⠀⠀⠈⢧⣤⠋⠢⡤⠷⢒⣡⠏⠀⠀⠀⠀⠉⠓⠶⠤⠤⠤⠤⠖⠚⠉
⠀⠀⠀⠀⠉⠙⠒⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠚⠓⠚⠉

⠀⠀⠀⠀⠀⣀⡠⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢤⡔⡶⢶⣶⠶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⡀
⠀⠀⢀⡴⠋⡹⠛⠚⡇⠙⠗⢆⠀⠀⠀⠀⠀⣀⡠⠤⠤⠶⠶⠤⢤⣀⠀⠀⠀⠀⠀⠀⢽⡖⡆⢽⠀⣇⣹⢨⡇⠀⠀⠀⠀⠀⣠⠾⡍⠽⢀⣁⠺⢋⢷⡄
⠀⠀⡼⠀⢠⠁⠀⠀⠀⠀⠈⡈⡆⠀⠀⠀⡾⠅⢀⡀⠁⠀⠐⠀⣀⠨⠽⡄⠀⠀⠀⠀⠸⣿⡇⢸⠐⡇⡇⣼⠁⠀⠀⠀⠀⢰⡏⠳⡙⠍⢢⣢⢙⡁⠧⢻
⠀⠀⢧⠀⢸⠀⠀⠀⢸⠀⠀⣷⠇⠀⠀⠀⣇⠀⠀⠀⠀⠈⠈⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠻⣿⣼⣾⣶⡷⠃⠀⠀⠀⠀⠀⢸⡲⢱⣅⢫⡠⣒⢕⣘⠵⣼
⠀⠀⠘⢦⡘⡄⠀⠀⠈⠀⣠⠟⠀⠀⠀⠀⠈⠒⠤⢤⣤⣤⣤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⡤⡔⢃⣴⣈⠚⣲⠏
⠀⠀⠀⠀⠉⠛⠲⠤⠗⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣋⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠥⠶⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠙⣾⠁⣠⠞
⠀⠂⠀⡴⠚⠉⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⣠⠤⢴⡋⠁⠀⡠⠶⢦⡉⢢⡀⠀⠀⠀⠀⠀⠀⠀⣷⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⣾⣿⠁
⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠙⢶⡤⣬⣆⠀⢧⣱⡶⠃⠀⣷⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡡⠤⠤⢬⡙⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢠⣖⣶⣷⡶⠼⣦⡴⠚⢉⡷⠀⣿⠀⠀⠀⠀⠀⠀⣰⡃⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⡏⡐⠁⡔⠉⠑⠀⠘⡈⡇
⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢸⣯⣏⣿⣏⣹⣛⠏⠉⠉⠀⢠⠃⠀⠀⠀⠀⠀⠀⣯⣉⣉⣩⡇⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠈⡅⢐⠀⢀⢃⡇
⠀⠀⠀⠈⠓⠢⠤⠤⠶⠒⠋⠀⠀⠀⠀⠈⠓⠳⠼⠭⠿⠿⠒⠚⠋⠓⠋⠀⠀⠀⠀⠀⠀⠀⢯⠁⠀⢨⠇⠀⠀⠀⠀⠀⠀⠀⠙⢦⡑⠢⠬⠅⠒⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⣁⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⢠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⣁⣤⡶⢾⣛⣿⡿⡿⢛⢿⣟⣳⣦⡀⠀⠀⠀⠀⠀⢀⡶⠴⠒⠒⢏⢡⢠⡙⢦⡀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⢸⣿⣫⣼⠟⠛⠉⠀⠀⠀⠀⠘⢿⣿⣧⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⢘⡏⠑⢦⠀⢻⡄⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠙⡿⠀⠀⠀⠀⠀⣠⣦⠀⠀⢸⣿⣿⠀⠀⣀⣤⠾⠛⠻⡤⠚⠃⢉⠛⠒⠶⠞⠛⣧⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣠⡤⠀⠀⢈⠁⢠⣦⠀⠀⣿⠋⡍⠿⠋⠀⠀⢀⠞⠁⠀⢠⡎⠀⠀⢸⠀⠀⠸⣆⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⣠⡄⠀⢸⡀⠀⠉⠀⠀⠀⠀⠙⣧⠀⢀⡔⠃⠀⠀⢠⡟⠀⠀⠀⢸⡆⠀⠀⢻⡄⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠁⠀⠸⠇⠀⠠⣤⡀⠀⠐⡟⣁⠔⠋⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⣷⠀⠀⠘⣷⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⡸⠛⠁⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠾⠛⢷⣀⠀⠀⠤⠔⠋⣀⡤⠖⠉⠀⠀⠀⠀⢀⣤⠿⠃⠒⠒⠒⠒⠒⠛⠛⠉⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠀⠀⠀⠙⠶⣤⣤⠴⠛⠉⠀⠀⠀⠀⠀⢀⣴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⢀⣠⣶⣟⠁⠀⢠⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⡶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⣼⡏⣷⣄⠈⣇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⢀⣠⡴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣠⡴⢟⡿⠋⠀⠀⠸⣇⣿⠛⣦⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⢀⣤⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠉⠁⠔⢋⡴⠀⠀⠀⠀⢻⣿⠀⣿⣷⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡾
⣼⡿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⠀⡴⠋⠀⠀⠀⠀⠀⠈⢿⣠⣿⡏⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⡼⠁
⢸⡆⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣅⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠃⠀⢻⡀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⢀⡾⠁
⠀⠻⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⡀⠀⠀⣠⠔⢲⡾⢿⠟⣹⣆⠀⣠⠞⠁⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣹⣟⣿⣟⣼⡫⢿⣎⣿⣿⡁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠷⣿⢎⡵⣯⣚⣽⡦⣿⠏⣻⣷⡀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢯⣿⣧⣛⣼⡷⢋⣾⡿⣿⣛⢌⣻⣧⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠽⣳⢞⡶⣩⣿⣵⡿⣻⢎⡳⢉⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢏⠶⣭⢻⣵⡿⣻⢏⡱⢯⠛⣽⢻⣝⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢪⡟⣴⡳⢾⡷⣭⢳⡜⡳⣛⠾⣱⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡳⢎⡲⣉⣾⣟⠶⣙⢮⡵⢯⡚⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡷⢫⣝⠤⣿⢎⡳⣭⠛⣜⣣⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠚⠉⢹⡟⣬⡳⢎⠷⣿⢻⣝⡦⣛⢾⣽⢯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠞⠛⠛⠉⠉⠀⠀⠀⠀⠀⠸⣟⡶⡝⢫⣾⡾⣳⢎⡱⢯⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⢼⡳⣟⠶⣉⢲⣝⡿⣋⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣮⣽⢯⣝⡦⣋⣾⡱⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⢹⡳⢌⣱⣿⠳⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣴⣿⡓⢮⣛⢤⣹⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣝⡳⣏⣳⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢺⣿⣿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣟⣉⠛⠛⢋⣩⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠫⠉⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠍⠊⠀⠀⠀⠀⠈⠒⢍⡻⢿⣿⣿⣿⡿⠟⢋⠡⠐⠀⠀⠀⠀⠉⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠟⡁⠤⠈⢐⡀⠀⢀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀⠄⠀⠒⡈⠈⠀⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠟⡡⠊⠀⠀⠀⠀⠀⠁⠀⠒⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⠄⠒⠀⠈⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⢸⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠁
⠀⠀⠀⠸⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠉⠉⠉⠉⠉⠉⠩⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠇
⠀⠀⠀⠀⢳⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⣤⣤⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠬⠥⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⡶⠞⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠳⢶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣮⣟⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⠮⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠵⣣⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⠀⠀⠀⠀⠀⠀⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⠀⠀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠷⠀⠀⠾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣋⣀⣀⣀⣀⣙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠻⠛⠓⠚⠛⠟⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⠤⠂⠀⠙⢏⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⢄⡀
⠀⠴⠋⠉⠈⠁⠀⠀⠀⠀⢀⡷⡀⠀⠀⠀⢀⣾⣽⣿⣿⣽⣇⣻⠿⣿⣆⠀⠀⠀⠀⣠⡞⠥⠀⠤⠤⢵⡀⡉⠑⣄
⡌⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠁⠀⠀⠀⣰⣿⣿⣿⣿⣿⢟⣽⣿⣹⣦⣝⣧⠀⠀⡸⠁⠀⠀⠀⠀⣠⠎⢈⠓⠮⡈⢆
⡇⠀⣀⡴⠊⠁⠀⠀⢰⠀⢰⠀⠀⠀⠀⣿⣿⣿⢟⣯⣾⣿⣿⡏⣿⣟⣿⣯⠀⠀⡇⠀⠀⣀⠴⠊⠁⠀⡸⠀⠀⠈⢾
⢣⡴⠁⠀⠀⠀⠀⢠⠏⠀⠀⡆⡰⠀⠀⢹⣟⣵⣿⣿⣿⣿⡿⣹⣿⣿⣿⡟⠀⠀⢧⢀⠞⠁⠀⠀⠀⢠⠇⠀⢠⠀⡼
⠀⠣⡀⠀⠀⠀⣠⠏⠀⠀⣼⠔⠁⠀⠀⠀⠹⣿⣿⣿⣿⡿⣳⣿⣿⣿⠟⠁⠀⠀⠈⢧⡀⠀⠀⠀⢀⠎⠀⠀⣈⡴⠁
⠀⠀⠈⠑⠢⠶⠥⠤⠄⠊⠁⠀⠀⠀⠀⠀⠀⠈⠛⠿⠻⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠙⠢⢤⣔⣁⣀⡠⠞⠋

⠀⠀⠀⢀⡠⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⢄⡀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⡔⠁⢧⠒⠊⠉⠉⠣⡀⠀⠀⣴⣿⣩⣽⣶⣾⣿⣦⠀⠀⢀⠾⡀⠀⣠⠤⠠⢝⢆⠀⢀⣜⢿⡿⢟⣯⣭⣛⢆
⠀⢸⢆⠀⠈⠢⡤⠄⣤⠤⡇⠀⢈⢿⣿⣷⣝⣻⣿⡟⣟⡃⠀⡌⠀⠈⠁⠀⠀⠀⠈⡞⠀⣼⣿⣿⣿⣿⣿⣿⣿⣞⡆
⠀⠘⡄⠑⢄⡼⠀⠀⡏⢠⠇⠀⠘⣷⣝⡿⢧⣿⣿⢷⣿⠃⠀⠰⣀⠤⠤⢤⣀⢀⡴⡹⠀⠸⡿⣛⣛⡿⢿⣿⡿⡽
⠀⠀⠈⠲⢎⣀⣀⡼⠕⠁⠀⠀⠀⠈⠛⢵⣿⣿⠟⠞⠁⠀⠀⠀⠙⠢⢄⣀⣀⠤⠚⠁⠀⠀⠘⠻⢿⣿⣷⠶⠛⠁
⠀⠀⠀⢀⣀⡀⢀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⣤⡀
⠀⠀⠰⠛⢁⣀⡀⠛⠻⠂⠀⠀⠀⣠⣤⡿⣿⢿⣤⣄⣄⠀⠀⠀⠀⠀⠀⠀⢈⡛⢦⡀⠀⠀⠀⢠⣾⣿⡿⢦⣍⠓
⠀⢠⡀⠀⣼⣿⣷⠀⠀⣠⡀⠀⠐⢿⣿⠁⠀⠘⣿⣿⠟⠆⠀⠀⠀⠀⠀⣰⡟⠁⠀⠉⠀⠀⣰⣿⣿⠏⢠⣾⣿⡇
⠀⠘⣿⠀⠀⠉⠁⠀⢸⣿⠇⠀⠀⢰⣿⣿⣶⣿⣿⡇⠀⠀⠀⠰⣄⡀⠾⠋⠀⠀⠀⠀⠀⠀⠿⣿⣃⣴⣿⣿⠏
⠀⠀⠀⠀⢰⣾⣶⡄⠀⠀⠀⠀⠀⠈⠻⠟⠉⠙⡿⠟⠁⠀⠀⠀⠈⠙⢦⡄⠀⠀⠀⠀⠀⠈⢷⣌⡛⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⠀⠀⠀⠀⢀⡤⠂⠀⠀⠂⣀⠀⠀⠀⢀⢠⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⣿⣿⣿⣿⣷⡀⠀⠠⢁⠇⠀⢀⠔⠋⠉⢁⠀⢠⡟⣸⣿⡿⣣⣶⣿⣧
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠅⠀⢠⠎⠀⠀⣿⠀⠀⠀⢀⠀⠘⣰⣿⣿⡃⣿⣿⣿⣿
⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠀⠀⠈⢄⠀⠀⠹⣄⠀⢀⠀⠀⠈⢿⣿⣿⣇⠻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⢁⡒⠀⠘⠋⠀⠀⠀⠀⠀⢉⡛⠛⠓⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣈⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟
⠀⢻⣦⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢾⠀⠀⠀⠀⠀⠀⠀⠀⣨⡅
⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⢂⡉⠉⠑⢤⠀⠀⢠⣿⣿⣴⢾⣿⣶⣄
⠀⠀⠀⠩⠑⠀⠄⠐⠈⠀⠲⠀⡀⢻⣦⣄⣤⣶⣞⣿⣧⣤⣄⠀⠀⢘⠀⠠⠐⠂⠀⠀⠥⠀⢻⣷⣿⣟⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠐⠀⠀⠀⠉⢠⡀⠀⠁⠀⠹⣿⣿⣿⣷⡿⠿⠿⠛⠀⠀⣘⡀⠀⠀⠀⠀⢀⠎⠀⠘⣯⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠄⠠⠐⠁⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣿⣛⣻⣟⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡈⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⡂⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⡃⠀⠀⢀⠇
⠀⠀⠠⠀⠂⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⢸⠅⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠄⠀⠀⠀⠀⣸⡆⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠁⠀⠂⠄⠀⠀⠀⠀⠀⠀⠨⡇⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢈⣼⠔⠀⠀⠀⠀⠀⢀⠀⠀⡼⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠠⠀⡀⠐⠂⡥⢘⡩⠀⠀⠀⠀⠀⠀⠊⠀⢰⢧⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⢀⠄⣀⠐⢢⠇⡾⠘⠔⢌⣀⠀⠀⠀⠀⠶⠲⠂⠀⠶⠖⠐⠱⠁⠂⢀⣀⣀⡠⠤⣀⣒⠠⠬
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠻⠓⠂⠚⠁⠀⠀⠀⠀⠁⠀⠀⠀⠂⠐⠒⠒⠒⠐⠒⠀⠐⠒⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠡⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⢂⠂⠀⠀⠀⠐⠊⠀⠀⡀⠀⢠⠇⠀⠒⠒⠀⠐⠀⠐⠂⠀⠂⠀⠘⠀⠂⠂⠀⠒⠀⠀⠀⠀⠂⠀⠐⠒⠒
⠠⣀⢄⡁⠀⢄⣄⠤⠀⢀⠀⣀⠀⢀⠒⢈⠂⣀⣠⢸⣔⢀⠀⠀⠠⡀⢡⢠⠄⠄⠀⡄⢆⠄⢀⢐⠀⠀⠀⠀⢐⠆⢀⠀⠀⠀⠄⠀⡌⢀
⠀⠈⠛⢃⠔⠂⠀⠀⠡⠀⠆⠦⠂⠠⠐⠶⠁⠁⠉⠉⠈⠳⣪⠄⢢⡜⠊⠈⠕⡤⡰⠀⠀⠈⠀⠱⠐⡀⠀⠐⠰⠟⠐⠈⠜⠦⠁⠆⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠖⠛⠉⠀⠀⠘⢧⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠲⣄⣤⡴⢞⡛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠚⢹⣦⢀⣀⣤⡍⠀⠠⢏⣴⣥⣤⣴⣞⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠇⠀⡀⢀⡍⠉⠹⠖⠃⣠⣾⡟⠁⠰⢚⣭⠽⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢈⣿⠏⡱⠔⢾⡏⠋⠀⠀⠀⡴⣤⣿⡻⠧⠀⠘⠩⠖⢻⣽⠿⣿⣧
⠀⠀⠀⠀⠀⢠⣾⠃⢸⣅⠤⢼⡇⠀⠀⠀⢰⢰⡏⠀⢀⡀⠀⠀⠀⠀⠀⠐⠛⠉⣿⡠
⠀⠀⠀⠀⠀⢸⡏⠀⢸⡁⠀⠈⣧⣀⠀⠀⠸⣾⠃⠀⠉⠹⡆⠀⠀⣠⣴⠄⠰⠟⣿⣤
⠀⠀⠀⠀⠀⠘⣧⡀⢸⡇⠀⠀⠻⡌⠙⠲⢤⣸⠆⠀⢰⡦⠁⠀⠀⣡⡀⠀⠀⢠⢟⠟
⠀⠀⠀⠀⠀⠀⠈⠛⠦⢿⡀⠀⠀⠙⠢⣄⡀⠘⡇⠀⠀⠀⡀⡀⠀⠙⠁⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠉⠑⠻⠦⠤⣴⢁⣇⣀⡀⠀⠔⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄⣀⣀⡀⠀⢀⣀⣀⣀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⠙⠛⠛⠉⠉⠩⣤⣀⣀⡀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣀⡀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⢸⡏⠙⣿⠼⢿⡟⢷⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⡿⠛⢿⠓⢺⡟⠚⣿⠓⢿⡟⠻⡿⠛⠶⠦⢤⣀
⠀⠀⠀⠀⠀⣀⡤⠾⣿⠷⢾⡿⠨⣿⠇⢿⡿⠬⣿⠥⢿⠧⢼⡿⠴⡿⡧⢾⡿⠿⣷⠦⣤⣀
⠀⠀⢀⣤⡾⢿⣷⠲⣿⡖⢾⡷⠒⣿⢖⣾⣷⣶⡿⠶⠿⣶⣾⣶⣐⣷⡆⣼⣷⣤⣿⣤⣼⣭⢽⣦⣄⡀
⠀⣴⡿⣿⡦⣼⣧⣤⣿⡦⣼⣧⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣦⣼⣧⢠⣿⣀⣼⣤⣀⣭⣈⣹⣦
⢸⣏⣀⣼⣇⣸⣇⣀⣿⣅⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣏⠈⣯⡁⣹⣉⠈⣻⡉⢹⡏⣷
⠈⠿⣄⣹⡇⢹⣟⠈⣻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣿⠋⢻⡛⠙⢻⠋⠻⣿⣿
⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⣿⣟⠚⣿⣗⣾⠟⠁
⠀⠀⠀⠀⠀⠳⡶⠶⠶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣝⡛⠿⣿⠟⠉⠉
⠀⠀⠀⠀⠀⠈⣷⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⢈⡽⠋⠁
⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢸⡆⢸⠃⠀⠀⠀⠀⠀⠰⣇⣨⠇⠀⠀⠀⠀⠀⣀⣴⠃⣠⠞⠁
⠀⢀⣤⣤⣴⡚⠉⠓⣾⣶⡀⠀⠀⠀⠀⠀⣼⢇⠀⠀⠀⠀⠀⢸⣿⣏⡉⠉⢲⣤⣤⣄
⠠⠾⠿⠿⠿⠿⠿⠿⠿⠿⠧⠴⠤⠤⠤⠴⠧⠼⠦⠤⠤⠤⠤⠬⠥⠥⠬⠿⢿⢿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠻⣿⡟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢿⠓⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢠⣟⠆⠀⠀⠈⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣽⡶⢀⣀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠁⠀⣤⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⠀⠀⠀⢰⣿⡵⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣆⠀⠀⠀⢼⡿⠗⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠏⠙⠛⣶⠟⣷⠆⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⢻⣧⠀⣤⢀⠄⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⣧⣴⢛⣿⣶⣾⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠸⣿⣿⡟⣻⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⠁⠀⠀⠀⠀⠀⢰⣿⡿⢁⣼⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⣿⣿⢃⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⢂⣾⣿⡏⣼⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣾⣿⡿⠠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢂⣿⣿⣿⠃⢾⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢰⠰⣿⣿⡏⣀⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠘⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣀⢹⣿⣿⣿⣧⠀⠀⠀⠀⢀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠒⠀⠒⠒⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠀⢸⢻⣿⣿⡇⠀⣿⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾⣿⣿⣇⠀⢻⣰⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⠙⠉⠉⠃⡀⢿⣿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠘⠂⠐⠒⠒⠐⠀⠀⠀⠀⠀⠀⠐⠆⠴⠄⢀⡀⣠⣤⣦⣾⣿⣷⠞⠂⠚⠁⣴⣶⣶⣦⣤⣤⣤⣤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠏⠀⠀⠀⠰⠶⠀⠉⠉⠉⠁


⣶⣶⡶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣇⣴⣦⡞⢹⣷⣶⠛⢷⣴⣷⣶⠞⣿⠛⠻⢿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣥⠾⠛⣿⣧⠾⠙⢿⣤⡼⠏⢻⡿⠀⢿⡶⠻⣿⡤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠄⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢴⡿⠟⠀⠀⠀⠀⠒⢄⢸⣇⣾⡯⠴⠒⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⣒⡲⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠙⢟⠙⢦⡂⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⠩⠄⢢⠠⠈⠉⢷⣆⣄⠀⠀⠁⣾⣿⣿⣶⠀⠀⠀⠸⣷⡄⠈⠑⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠀⠀⠀⡈⣀⣤⣴⣾⣾⠉⣿⡄⠀⠀⠛⠻⠿⠃⠀⠀⣸⣷⡘⢿⣶⠞⠁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠁⠀⠀⢀⣼⡿⠋⣀⠄⠠⠈⠐⣹⣷⣀⡢⠀⠀⡀⠀⢀⣤⣯⣿⣷⠀⢉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⣿⣿⣠⠎⠀⠀⠀⣰⣷⡿⠋⠉⣵⣿⡏⡀⢰⠟⠁⠀⠈⣿⣿⣧⣀⣄⠀⠀⣤⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⡁⠀⠀⢠⣾⢻⣿⣟⠂⣸⣿⣿⣿⣧⠂⠀⠀⠀⣠⣿⠛⢿⣿⣧⣤⣴⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠳⣶⣾⣾⣯⣿⡿⠾⠿⣿⣿⣯⡹⣿⣽⣀⣠⣴⣿⠃⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣼⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⠋⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠒⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣤⣴⣤⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠂⠀⣠⣤⣶⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠊⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠠⣦⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠⡦⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡗⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠉⠑⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⣿⣿⣿⣿⣿⣯⣿⣿⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠻⣿⣿⣿⣿⣿⡿⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠋⠛⠛⠛⠛⠃

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠀⠀⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢂⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠠⠤⠐⠒⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⢀⣤⣶⣤⡤⠀⠀⠚⠛⠻⠆⠹⣿⣿⣿⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⣶⣿⣿⣷⣶⠄⠀⠈⠀⢤⣤⠃⠀⠀⠀⠰⠶⠀⠀⠙⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠸⡏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⢡⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠁⠀⢀⠀⠀⠰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠛⢃⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢤⡀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄⡀⠀⣀⣀⣀⣀⣀⠼⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⢄⣡⠊⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣿⡏⠛⠛⠉⠉⠉⢉⠉⠀⠀⠀⠛⠀⠤⠄⠀⠀⠀⠀⠀⠀⠆⠀⢀⡠⣌⠁⣈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠁⠠⢤⡈⠠⠖⠀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⢸⠄⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⠁⠀⠀⠈⠉⠠⢄⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠰⠆⢀⡴⠀⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢆⠀⢀⠆⠀⠻⠖⠐⠶⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠌⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠽⠀⠀⠀⠀⠀⠀⢀⣤⠞⠳⠋⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠐⢢⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⢃⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⣽⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠑⣄⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⠻⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡈⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠃
⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠠⠄⢀⣀⡀⡀⠀⠀⠀⠀⣀⣀⣀⡠⠆⠈⡹
⠀⠀⠀⠔⢉⠀⠀⠤⠄⠈⠀⠀⢈⠛⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀
⠀⠀⠀⠸⠁⡀⠊⠀⠰⡖⠒⣦⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠣⠀⡜⠁⡋⠈⡟⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠠⢀⠀⠀⠠⠎⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠈⢦⠀⠧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠆⢀⠀⠃⠀⠀⠐
⠀⠘⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡇⠀⠀⠀⠈⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢀⣤⣶⠟⠁⠀⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠉⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⡀⠀⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠒⠤⠤⡤⠤⣤⡤⠟⠀⠀⠀⠀⠘⠦⠤⠘
⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⢠⡇⠀⠀⠀⠀⠘⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤
⠀⢼⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂
⠀⠈⠳⠄⣀⣀⣀⣀⣠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠛⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠛⠛⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠟⠁⠀⠀⠠⠄⠀⠀⠀⢠⣤⣄⠈⠛⠷⣦⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠁⣴⣤⠶⠒⠛⠙⠋⠉⠉⠉⢻⣿⡿⠶⣤⡄⠈⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢻⣷⣤⣤⣤⣤⣴⣶⣶⣶⣬⣿⣧⣤⡴⠛⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣿⡎⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢘⣆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢸⡇⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⡏⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣷⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢸⡇⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⠃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⣾⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⣈⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⣀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣴⣿⣶⡄⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠝⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⠆⠨⠥⠄⠀⠀⣶⣶⣄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠋⠐⠚⠓⠒⠀⠀⠀⠀⠛⠛⠛

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣻⣛⣟⣿⣋⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⡄⠀⢀⡀⢠⠀⠀⠀⠀⠀⠠⠖⡲⢦⡀⠀⠀⠀⠠⡊⢳⣿⣿⣿⡏⢑
⠀⠀⠀⠀⣤⣷⣿⠇⠀⠀⠀⠀⠀⠈⢃⠀⣾⡿⠟⠀⠀⠀⠀⣈⠂⠜⢠⠃⢽⠀⠀⠀⠀⠙⠺⣿⣿⣿⠗⠁
⠀⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⠀⠠⣶⣾⡟⢀⣤⡀⠀⠀⠀⠀⠘⢕⣒⣣⡈⠎⠀⠀⠀⠀⠀⠀⢉⣿⠁
⠀⠀⠀⠀⠘⠿⠛⠁⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⢀⠀⢀⣤⡄⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⣰⣧⡂⠀⠀⠀⠀⠀⢠⣤⡄⣀⡀
⠀⠀⠀⠀⢨⣿⣿⡿⠋⠀⠀⠘⢟⢿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⣰⣿⡣⣹⣦⡄⠀⠀⠀⢻⣿⣧⣿⡿⣦
⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠻⡟⠉⠉⠉⠀⠀⠀⠀⠀⠰⢟⣱⠿⢛⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣾⡥
⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠞⠋⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⢀
⠀⠀⢀⡈⣶⣷⠆⢳⡀⠀⠀⠀⢀⡤⡴⠗⠳⠤⢤⠄⠀⠀⠀⢠⣿⢃⡼⣭⣤⠀⠀⠀⠀⠀⣼⡙⣎⣣⠶⣀
⠀⠀⠘⢇⣈⠉⢰⣶⠀⠀⠀⠀⠀⠙⣷⠀⠀⣴⠃⠀⠀⠀⠀⠰⣟⣶⣽⣷⡝⠀⠀⠀⠀⡼⠡⠗⢉⣤⠶⣉⡄
⠀⠀⠀⠈⠛⠃⠈⠁⠀⠀⠀⠀⠀⠰⠿⠛⠛⠾⠀⠀⠀⠀⠀⠀⠈⠛⠓⠉⠀⠀⠀⠀⠘⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⡿⣶⠄⠀⠀⠀⠀⠀⠘⣿⠇⣴⡿⠃⠀⠀⠀⢐⣴⣿⠟⢻⢹⣿
⠀⠠⠞⠋⡳⠤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡋⠀⠀⠀⠀⣶⣿⣿⡾⠋⠀⠀⠀⠀⠀⣾⢏⠘⡐⣼⡼⡋
⠀⠀⠀⠀⠁⠢⢈⠗⢦⡄⠀⠀⠀⠀⠛⠛⠟⠃⠀⠀⠀⠀⠀⢻⣿⣿⠟⠀⠀⠀⠀⠀⠀⠿⠦⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠶⢸⣏⠀⣿⣿⣿⣿⣿⣿⡗⠶⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⣉⣿⢸⣿⠁⠀⠀⠀⠀⠀⣿⣉⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣧⡀⣿⠀⣿⣿⣿⣿⣿⣿⢀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠘⣿⠀⣿⡀⠀⠀⠀⠀⠀⣿⠃⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠹⣇⢹⣿⣿⣿⣿⠏⠞⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣌⣇⠹⡇⠀⠀⠀⠀⣸⣡⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣟⣛⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠤⠤⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠛⠛⠛⠛⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣤⣤⣤⣤⡄⢠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⣀⣄⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢂⠀⠀⠐⠀⠁⠀⠀⣈⣀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠈⠀⠀⠀⠚⠃⠀⠀⢈⡀⢸
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠅⢸
⠈⠛⠋⠁⢀⣀⠀⠀⠀⠀⠀⠀⢠⣐⣆⣀⠀⠄⢶⡆⢀⠀⢀⡀⠀⠈⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠚⠂⠙⠀⢸
⠀⠀⠀⠈⠾⠯⠁⠀⠀⠀⠀⠀⠀⠰⠾⠏⠀⠀⠛⠂⠐⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠮⠳⠦⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠀⢠⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢑⠔⢰⣃⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⠘⠀⢼⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠠⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⠀⠀⠀⠲⠂⠀⢰⣷⡀
⠀⠀⠀⠀⢀⣀⣄⠀⣡⣘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣄⢤⡄⠀⠀⡀⠀⣦⢁⡂⠂⢠⣤⣾⣿⡘⡷
⠀⠀⠀⢠⣿⣿⣿⣿⣦⣿⣿⡃⠀⡀⢀⠀⣀⡀⣀⢀⠼⡿⠿⡻⣦⡏⠀⠀⢧⣤⣽⣴⡟⠻⠿⠛⠉⠑⢀⣤
⠀⠀⠀⠀⠻⣿⣿⣻⡙⢻⣉⠉⢀⡀⠀⠄⠀⠀⠀⠀⠀⠁⠈⠀⢈⣀⡲⠤⠤⠩⠽⠭⠠⡆⠤⢴⠶⢠⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⣿⠓⢀⣀⠀⠀⠀⠒⠉⣉⣡⣤⠤⠤⠤⠤⠤⢄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠋⠉⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠒⠒⠒⠦
⠀⠀⠀⠀⠀⡀⠤⠐⠀⠀⣀⣠⡤⠤⠤⠶⠶⠦⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢄⣶⡿⡫⡠⡦⣮⣲⣢⣶⣤⣬⣯⣖⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⠙⢡⣾⠃⠀⠀⠀⠀⠈⠠⠪⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠟⠋⠁⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⢟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⢵⢟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⡿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⢑⢟⢗⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢾⣯⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⠀⢑⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⢠⣟⣽⣷⣄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡟⠀⠀⠀⣀⣀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢐⣽⣿⡿⠈⠿⣦⠀⠀⠀⠀⣿⡄⣀⣠⣴⣿⡷⠶⠟⠛⠛⠉⠉⠉⢽⣕⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢱⣿⣿⢑⢀⣬⢻⣧⣀⣠⣴⣿⡟⠛⠉⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢁⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣵⣿⣟⠉⠀⢸⣧⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣟⣵⣻⣿⠀⠀⠀⣿⡄⠀⠀⠀⠙⢿⣦⡀⠀⠀⣀⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢴⣐⠀⢻⣷⠀⠀⠀⠀⠈⡹⢿⣶⣴⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣧⢴⣴⣷⢝⣵⢟⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣜⣿⣇⣽⣷⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢜⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢜⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠳⢤⡀⠀⠀⢸⣃⢀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣵⣦⣶⡄⠀⠘⢿⠢⣧⡄⠀⠸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠃⠀⠀⠀⢱⡿⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠈⠀⠀⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠀⠀⠀⣶⡦⣬⠐⠄
⠀⠀⠀⠀⠀⠀⠀⣤⣰⡀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿⣟⠉⠁⠀⠀⣈⣉⣁⣬⣤⣴⣤⣤⡈⠙⢛⣛⣿⣿⢇⡤
⠀⠀⠀⠀⠀⠀⠘⢿⣆⠹⣆⠀⠀⠀⠀⠑⠉⡙⠛⠛⠫⠾⠿⢄⣤⣈⣉⢙⣿⣽⣄⡙⠿⠿⠖⠛⠛⣋⣤⣾⣷⡾⠧⠯⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⢧⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠶⢦⣶⣶⣿⣿⣿⢿⡿⠓⠒⢄⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣦⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣤⣿⣿⣿⣿⢁⡾⠀⠀⠀⠨⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠋⡄⢿⡃⠀⠀⣠⠇⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠙⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢠⣶⠋⣼⣿⠿⠏⢀⣀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⢳⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠨⡍⢰⣷⣉⣠⣬⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⣦⡈⢤⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣭⣉⣑⠈⠉⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⡴⠞⠉⠙⠻⢦⡑⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⡒⢀⣴⣿⣿⣿⣿⣿⣩⠙⠲⢄⠀⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡾⠁⠀⠀⠀⠀⡴⡀⠈⠺⣿⣿⡆⠤⢄⣀⣀⣀⠀⠀⣐⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⠰⠋⠀⠀⢡⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠘⠐⠀⠘⠅⠀⠀⠀⠰⣎⣰⣋⡠⣀⠈⠻⣏⠢⠤⠍⠛⡉⠻⠿⠟⠛⠛⠿⠟⠛⠛⠋⠈⠁⠀⠀⠀⢀⠠⠃⣴⣟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠓⠆⠀⠀⠙⠲⠖⠃⠀⠀⠸⢆⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⣔⡻⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣀⠀⡀⠀⠀⠀⠀⠀⢄⣶⣦⣄⣀⣀⢹⠟⠒⠂⠀⠀⠀⠀⠁⠀⠀⡠⠤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠛⠛⠛⠟⡻⢿⠿⣯⠾⠿⠟⠃⠀⠘⠛⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠿⠟⠛⠋⠉⠉⠉⠛⣿⡟⠛⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣯⣥⣤⣤⣤⣤⣤⣤⣀⣀⢠⣿⡏⠀⠀⠀⠈⢙⣿⣷⣤⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠉⠉⠉⠁⠀⠈⠉⠉⠙⢛⣿⣿⣿⣦⣀⠀⣠⣶⡿⠋⠀⠙⢿⣦⡀
⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⠿⠋⠁⠈⠙⢻⣿⣿⣋⠀⠀⠀⠀⠀⣹⣿⣆
⠀⠀⠀⣼⡿⠁⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠋⠉⠀⠀⠀⠀⢀⣴⣿⠋⠙⢿⣷⣄⠀⣴⣿⠟⠿⣿⣧
⠀⠀⣸⡿⠁⠀⢀⣴⣾⡿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠉⢿⣿⡿⠁⠀⠀⠈⢿⣧
⠀⢠⣿⠃⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⠀⠀⠀⠘⣿⡆
⠀⣸⣟⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢿⣧⠀⠀⠀⢹⣧
⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⢿⣧⠀⠀⢸⣿
⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠈⣿⣧⠀⢸⣿
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠸⣿⡄⣸⡟
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⣿⣇⣿⠃
⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠈⠻⣷⣄⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣡⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢷⣦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣼⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠁⠒⠤⠀⠀⣠⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠰⣿⡆⢠⣾⠁⠀⠈⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠑⠄⠠⠿⠛⠃⠀⠀⠀⠀⠀⠘⠀⠀⠠⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠉⢀⡀⠤⣿⣤⡌⠐⠠⠀⢀⣤⣤⣀⡀⠠⢤⠀⠙⠢
⣠⡤⠶⠶⠀⢀⡁⠀⠘⡉⢀⡠⢶⣦⣤⣐⣂⠈⠛⠇⠠⠤⠼⢿⣿⣿⣿⠿⠿⠶⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢂⡀
⠻⢶⣄⡀⠀⠀⠹⢷⡒⠂⡏⠉⠃⠀⠀⠀⠀⣀⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡥⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠳⢯⣀⢀⣾⣿⣿⠀⠀⠀⠀⢀
⠀⠀⠈⠙⠳⢦⣤⣸⡉⠀⠰⡀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣵⣾⣿⣶⣦⡄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡻⠻⠟⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠙⢻⠷⢦⣵⣄⣀⠀⠈⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⣵⣿⣿⣿⣿⣻⣿⣦⡬⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡬⠵⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠙⡍⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣿⣿⣿⣿⣟⣿⣿⣧⣠⣦⣤⣤⡤⠤⠶⠖⠒⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⡄⠿⠆⠈⣝⢿⣿⣿⣿⢟⣽⣿⣿⣿⣿⣿⣿⢏⣽⣿⣿⣿⣧⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠈⠀⠒⠠⠄⢬⣖⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣮⣙⣤⣬⣩⣯⣵⣿⣹⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⣦⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠡⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣦
⠀⠀⠀⠀⠀⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷
⠀⠀⠀⠀⠀⠬⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡀⠀⠀⠀⢀⣈⡁⠈⠀⠀⠂⠀⠀⠀⠀⠘⠀⠀⠁⠸⠁⠀⠀⠀⠀⢀⠀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⣶⡿⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠟⠛⠣⡀⠀⠀⠀⢀⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⢨⣶⣤⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⣀⡠⣿⣿⣿⣿⡆⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡆⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠈⢻⠛⠁⠀⠑⢴⡿⠀⠁⢄⠢⠒⠀⠀⠀⠂⠈⠀⢀⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⣀⣀⣘⠀⠀⠀⡠⠎⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⠦⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠟⠛⢋⠉⠁⠙⠛⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⣀⣴⣶⣟⢻⡿⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠻⢿⢶⡶⠟⠿⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠈⠢⡀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈



⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢴⣦⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⠆⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⢀⣼⣷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⣶⡦⠀⠀⠀⠀⠀⠰⢫⣿⠓⠞⣁⣰⡿⢆⠀⣠⣴⣯⣁⡀
⠀⠀⠘⣻⣿⠀⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⡋⠳⠚⠀⠀⠀⠀⠀⢀⡾⠙⢻⠞⣹⣿⠛⠋⠀⣡⣴⡀
⠀⠀⠀⢹⢿⠀⠀⠰⢻⡿⠟⠆⠀⠀⠀⠀⢠⣄⠴⠋⠀⠞⠃⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣠⠏⠉⠻⠀⠚⠋⠘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⣀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣠⣼⣏⡀⠀⣀⣤⣼⡏⠀⠀⠀⠀⠀⠀⢫⣿⣟⡷⠃⠀⠀⠀⠀⠀⠈⢯⣿⣇⣀⡀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠈⣼⠿⣬⠉⠀⠉⣼⢿⣭⠉⠀⠀⠀⠀⣠⠟⠉⠉⡍⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠉⠀⠀⠀⠀⠀⠚⠋⣹⣿⠛
⠀⠀⠰⠞⠁⠀⠉⠀⠀⠞⠃⠀⠃⠀⠀⠀⠀⠀⠁⣀⣀⣠⠇⠀⠀⠀⠀⠀⠀⠠⠞⠁⠈⢿⠀⠀⠀⠀⠀⠀⣤⠟⠉⠳⠄

⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠦⣤⣴⣶⣦⡄⠀⢀
⠀⠀⠀⠀⠀⠀⣹⣏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣬⣿⣧⡤⠀⠀⠀⠀⠀⠀⣸⣿⣇⣀⡀⠀⠀⡇⠀⠰⣿⠛⣶⠋⠋⠀⢘
⠀⠀⠀⠀⠀⠞⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⣃⣀⣀⣀⣀⣀⣀⣀⣀⣐
⠀⠈⠀⢀⠀⢸⠇⠘⡇⢀⠀⠀⠀⠀⡀⠘⠛⠀⢻⡄⡀⠀⠀⠀⡀⠀⢀⡾⢹⡇⢀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠈⠀⠈⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠁⠀⠀⠀⠁⠀⠀⠀⠈⠁⠈⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠
⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠟
⠀⠀⠀⠀⠀⠀⢸⣿⣀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠷⣶⠆
⠀⠀⠀⠀⠀⠀⡾⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⢀⠴⠖⠒⠶⠶⠒⠒⠶⠶⠒⠒⠒⠶⠶⠆
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠖⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠖⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⢀⠜⠁⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢠⠋⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⢰⠃⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢧
⠀⠀⢠⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢇
⠀⠀⡜⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠘⡄
⠀⢀⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠃
⠀⢸⡀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⢷⡀⠀⠀⠀⠘⣧⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡌⢳⣄⠀⠀⠀⢻⡆⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⢡⠀⠙⠷⣶⣤⣼⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠈⡆⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⣀⣤⡶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⢈⣷⣤⣴⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞
⠀⠀⠀⠀⠘⢤⣤⡶⠶⠟⠛⠋⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠞⢋⠞
⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⡴⠶⠶⠾⠛⠛⠋⠉⠉⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⣴⠟⠛⠿⣍⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣾⠃⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⢄⣀⠀⠀⠈⠙⠲⠤⣄⡀⣀⣀⠤⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣛⣟⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠸⠁⠀⢀⠁⠀⠀⠤⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⢀⠀⠁⠀⠀⠀⠀⢸⠂⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠟⣄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡁⣛⡣⠀⠀⠀⠀⠀⠘⠀⢰⠙⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠠⠀⡐⢂⠇⡴⠒⠴⢍⡀⠀⠀⠀⠀⠀⠀⠀⣔⠆⠀⡌
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠛⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠁⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠈⠄⠀⠂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠀⠀⡖⠀⠀⠀⠀⠈⡀⠀⠀⠀⠐⠚⠀⠈⠁⠀⠁⠀⠀⠀⠈⠀⠈
⠐⢤⣠⡀⠐⠤⠔⠀⠀⠀⠤⠀⠠⠁⢬⠠⢴⣄⣷⡥⢀⠀⠐⢄⣢⣐⠐⠀⡢⠨⢀⢠⡠⠀⠀⠀⡀⣑⡀⢄⢀⠀⡀⠀⠄⡄
⠀⠀⠈⠘⠈⠀⠀⠀⠀⠁⠁⠁⠁⠉⠁⠀⠀⠀⠀⠈⠲⠂⠙⠃⠀⠐⠘⠐⠃⠀⠀⠀⠈⠀⠀⠀⠁⠉⠀⠁⠀⠑⠀⠃⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣆⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠑⠂⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣦⣤⣖⡄⠀⠀⠀⠀⠀⠉⠳⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠤⠠⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠈⠉⠙⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠙⠛⠻⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⣉⡑⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠠⠴⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠒⠀⠀⠀⠛⠇⠀⠀⠀⠀⠠⣏⡿⠂⡗⠾⢿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠉⠉⠁⠁⠒⠒⠦⠤⢄⣀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡋⣀⣅⣀⣸⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠀⠀⠀⠀⠀⡠⠀⣠⣰⣮⣽⡿⠿⠿⡟⠛⣛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⡿⠛⣿⡉⠭⠑⠒⠈⠃⠁⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⡵⠶⠓⠉⠀⠀⢀⣀⣤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞⠋⠁⠀⠀⠀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⣤⣤⣴⣧⠄⠊⡀⢰⣦⡄⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢸⡿⠛⠻⠋⠳⠟⢀⣈⣽⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴
⠀⠀⠀⢀⣀⣀⣀⣀⣠⣬⠤⠶⠶⠒⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁



⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⢀⣤⣤⡀⠀⣤⡤⠀⠀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠠⣼⡅⣿⡃⠀⣿⠄⢾⡇⢀⠀⢸⣧⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠃⠈⠓⠚⠋⠀⠛⠓⠚⠀⠚⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠀⡀⠄⠒⠄⠠⠀⠂⡴⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣤⠐⡶⣦⣤⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣾⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣾⣿⣤⣦⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣴
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⡿⠛⠻⡿⢿⠟⡛⠿⠟⠿⡿⢿⠿⠃⠻⠛⠛⢻⣿⣿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠂⠤⢁⠀⠸⢃⡀⠻⠇⣠⠀⢼⡇⠟⡁⠠⠆⣩⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣷⣿⣿⡆⣺⣿⣿⣷⣾⣿⣿⣿⠷⠟⠛⠒⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡿⢻⠁⡔⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠛⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⠀⠀⠀⠈⠋⠀⠈⠁⠀⠀⠺⡞⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠟⠉⠉⠛⠉⠻⢿⡟⠛⠁⡠⠐⠀⠀⠀⠙⢢⣤⡀⠀⢰⣾⣷⡆⠀⠀⠱⣄⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣠⠴⠶⠿⠸⣧⠀⠀⠙⠛⠁⠀⣼⣇⠙⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡔⠃⠀⠀⣠⡟⡠⠀⠀⠀⣐⡿⠒⠀⣤⠀⢀⠘⠉⠻⣧⣄⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣏⠀⠀⢠⢺⣿⠀⢰⣾⣿⡄⠀⠀⠀⢰⡟⠻⠷⣤⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠷⠷⠟⠃⠙⠛⠛⠊⠿⣧⣤⡶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠙⠒⠲⠦⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⡤⠤⠤⠤⠤⠤⠖⠒⠊⠙⡆
⠀⠀⠀⠀⠀⠀⣠⠤⠶⠶⠖⢾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀
⠀⠀⠀⠀⠀⣼⠀⠀⣀⠤⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⠉⠉⠙⢦
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠓⡆⠀⠘⡇
⠀⠀⠀⠀⠀⢿⠀⠀⣇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⡇⠀⢠⡇
⠀⠀⠀⠀⠀⠘⡆⠀⠸⡄⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠃⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠱⡄⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠙⣄⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⢠⠏⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠳⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡴⠃⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⡀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠉⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣁⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠦⠤⣤⠤⠤⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⠒⢿⣀⣀⣀⣀⣀⣀⣀⣸⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠤⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠴⠒⠋

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

⠀⠀⠀⠀⠀⣠⡀
⠀⠀⢀⠀⠣⠆⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠈⣿⡇⠀⠘⣷⠗⠀⠀⠀⠀⠀⠀⠀⠈⢛⡿⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣀⠀⣰⣿⣿⣦⡀⠀⠀⠀⠀⠀⢀⡴⢋⡤⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠚⣟⠁⣰⠟⣩⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⣿⣿⢸⠇⠀⠀⠀⠀⢠⣞⣤⢿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⡄⠀⠀⠀⢠⠖⣲⢏⠢⠹⡄⡥⠲⡲⢄⠀⠀⠀⠀⠀⠀⢠⣿⠋⠈⠀⠀⠀⠀⣰⠟⠉⣠⠞⠉⢀⣠⡤⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢿⢧⡀⠀⠀⠀⠸⣖⢋⡹⠀⠚⠁⢯⣝⣋⡽⠀⠀⠀⣴⣶⠀⡾⠈⢷⡀⠀⠀⠀⠀⠃⣀⣼⡧⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡍⡶⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⠉⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⢄⠴⠶⠶⢶⣾⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢧⣴⣶⡄
⠀⠀⠀⠀⠀⠢⠴⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠢⣄⠀⠀⠀⠀⠿⢻⣿⣧⠀⠀⠀⠀⠀⢀⣀⣤⣼⣿⣤⣿⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠐⠛⠶⣦⣀⣤⣴⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣿⠋⠻⣇⠀⠀⠀⣠⣿⠸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⣄⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢰⠃⠀⠀⠀⠈⠁⠀⢰⣿⠁⠉⠀⡿⠹
⠀⠀⠀⠀⠀⠀⠤⠿⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⠐⠚⠑⠁
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣄⠀⢀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣼⣯⣄⣀⣀⣤⡀⠀⠀⠀⠈⠘⢦⣀⣼⣷⡄⠀⠀⠀⠀⠀⠀⣀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠘⢿⠀⠸⡇⢀⠀⢠⠇⢰⣿⠇
⠀⠀⠀⢠⢟⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⠻⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⢻⣿⣷⠟
⠀⠀⠀⠋⢰⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟
⠀⠀⠀⣠⡿⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢸⡇⠀⠀⠀⠀⠀⢀⡿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⢻⡄
⠀⠀⢼⡋⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣇⠀⠀⠀⠀⠀⣸⠃⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠁⠀⠀⣴⡆⠀⠀⠀⠀⠉⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠆⠋⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⣤⡄⢠⠆⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢷⡀⠀⢀⡾⠀⠀⠀⡄⣾⣿⠟⠛⠛⠀⠀⠀⠀⠀⠘⠉⠀⣰⣿⡿⢷⡼⠁
⠀⠀⠀⠀⠀⣴⣿⣶⣯⣤⣤⠀⠀⠀⠀⠀⢻⡄⣿⣿⣷⡒⠙⠀⠀⠀⠀⠀⠀⠀⣽⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣶⣤
⠀⠀⠀⠀⠀⢿⣿⣀⡀⠀⠀⢲⡄⠀⠀⠀⠀⠀⠘⢿⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠻⣆⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠃
⠀⠀⠐⠛⠛⠛⠛⠛⠛⠛⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢿⡀⠀⠀⠀⠀⢾⠉⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠶⠶⠶⠶⠶⠶⠤⠤⢤⣤⣀⣀
⠀⣠⠤⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠤

⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡰
⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠈⠀⠀⠀⠀⠀⠀⢁⠒⠠⡀⢀⡇
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⡁
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠈⢆
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⢿⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠑⠠⢀⠀⠀⢀⠤⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠐⠂⠉⠀⠀⠀⠉⠁⠀⠈⡉
⠀⠀⠀⠀⢻⣀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⣀⣈⣉⣳⣶⣦⣯⣤⡀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢀⡔
⡔⠉⢔⠋⠙⢇⣠⠋⠘⢄⣀⠜⠉⠂⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⢨⠉⠒⢄
⠸⡐⠒⢭⢶⠞⢙⣢⡤⢼⣽⣦⡤⠴⠡⡬⣰⠶⠤⠤⡤⢤⡶⣖⣃⣚⢳⠮⠅⠒⡺⠁
⠀⠙⢦⡠⠻⣄⢀⡽⣄⠀⠈⣿⠁⠀⢧⡸⠁⠈⣯⠁⠀⠈⣧⠃⠀⣸⣎⠀⢀⠜⠁
⠀⠀⠀⢱⠀⡨⢯⠀⠈⣳⠊⠀⠑⣤⠊⠱⣄⠞⠉⢢⢠⠎⠈⢣⡔⠁⠈⣦⠃
⠀⠀⠀⠀⢷⠁⠘⣆⡜⠀⢆⠀⢰⠛⡄⢠⠿⡀⠀⢈⡇⠀⠀⡸⡆⠀⢀⠎
⠀⠀⠀⠀⠀⢣⠀⡸⣇⠀⠈⢦⠇⠀⢱⡜⠀⠳⣀⠎⠸⡀⡰⠁⠹⣀⠎
⠀⠀⠀⠀⠀⠀⠻⣅⠈⢦⣠⠛⢆⠀⡎⢱⡀⡠⠿⡀⠀⣱⡃⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢆⡴⠹⡄⡠⠻⡀⢀⡟⢄⠀⠘⡞⠀⠱⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢻⠁⠀⢣⡸⠀⠈⡆⣸⠳⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠋⠱⣄⠞⢣⡀⠀⡼⡁⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⢠⠟⡄⠀⠳⡜⠁⠘⡞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠛⠛⠛⠂⠀⠀⣰⣾⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠀⠀⠀⠀⣰⡿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣶⣦⣤⣶⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⣿⠿⠛⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠛⠻⣿⣿⣿⣶⣄⣠⣶⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⡾⠋⠀⠀⠀⠀⠀⠈⣿⠛⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⣻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⢀⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣼⣷⣄
⠀⠀⠀⠀⠀⣀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣄
⠀⠀⣠⣴⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠻⣿⣿⣿⣿⣷⡄
⣤⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⣰⣿⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⣿⣿⢿⣿⣿⣿⡄
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡇⢀⣾⡇⠀⠀⠀⠀⠁⠀⢰⣿⡇⢸⣿⣿⣿⡇
⠘⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣿⡟⠀⠀⠀⠀⠀⢀⠀⢸⣿⠁⠈⣿⣿⣿⣿
⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇⠀⠀⠀⠀⢀⡿⠀⣼⡟⠀⠀⣿⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡇⠀⠀⠀⢠⣾⠇⢰⣿⡇⠀⢀⣿⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣴⣿⡏⣠⣼⣿⠀⠀⣸⣿⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣾⣿⡏⣰⣿⠏⠋⣰⡆⣿⡟⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣴⣿⡿⠀⣼⣿⣿⣿⠁⠀⢠⡀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣼⣿⡏⠻⠃⣶⠀⢸⡇⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣾⣿⣀⣾⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡟⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⣦⣄⡀⠉⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⡉⠉⠉⠛⠋⠀⠀⠀⠀⣼⣿⢋⣿⡿⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣸⣿⠁⢠⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣦⠀⠀⠰⠛⠁⠀⢰⣿⡇⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⢀⣾⣿⣀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⢛⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣶⣾⡿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠿⠿⠀⠀⠀⠚⠛⣿⣿⣿⣿⣿⣯⡄⠀⠀⠈⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠛⣻⣿⣿⣿⣿⣿⣷⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⠋⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡏⠻⠇⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⣿⣄⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠛⢿⣿⣷⠀⠀⠀⠀⣿⣿⠇⢨⣿⣿⣿⡇⠘⢿⣆⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠉⠁⠀⠀⠀⢀⣿⡏⠀⢸⣿⣿⣿⣷⠀⣈⠉⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠀⠀⣾⣿⣿⣿⣿⡄⢻⣷⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃⠀⠀⣿⣿⣿⠟⠻⣷⠈⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⢠⣾⣿⣿⠏⠀⠀⠀⣿⣿⡟⠀⠀⢹⣧⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⣠⣿⣿⣿⠏⠀⠀⠀⠀⣿⣿⠇⠀⠀⠈⠻⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⠀⠀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠻⣿⣿⣿⠀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⡟⠀⠀⠈⠻⢿⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⣀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⢿⣿⡟⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠀⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠉⢉⣡⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⠀⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠁⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣆⠉⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠟⠋