Puffball Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⣀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡆⠀⠀⢠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⠇⠀⢸⠇⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠳⡞⠁⠀⠀⠞⠓⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡔⡛⢒⣤⠀⣀⠟⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠙⡦⢀
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠉⠀⢲⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡆⠀⠀⡿⠬⢷⠐⠢⢤⠂
⠀⠀⠀⠀⠀⢀⠋⠙⠃⠀⢐⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⡤⠏⠃⠀⢸⣧⣄⣠⣀⣤⡤⠀⠀⠀⢀⠘⠋
⠀⠀⠀⠀⠀⠋⠈⠁⠑⠖⣛⡿⡀⠠⢠⣤⣀⡀⣠⣶⡿⢡⠀⡆⠘⣾⣷⣿⣿⣻⣅⣤⣾⣷⣢⣼⠣⣠⣄⢀⣄⡀
⠀⠀⠀⠀⣀⡀⢀⠶⠶⠚⡻⣿⣇⠀⠀⠀⠀⠁⠙⠀⠁⠀⠈⢠⣰⣿⣿⠟⠋⠋⠉⠀⢈⠹⠟⠛⣴⣽⡥⠚⠛⠁
⠀⠀⠀⠈⠉⢓⣤⠀⣠⣾⡾⠟⠛⢷⠄⠀⠀⠀⠀⠀⡤⠀⢀⣽⣿⣿⣿⠄⠀⠄⠀⠙⣿⣱⣿⠶
⠀⠀⠀⣶⣐⠙⠓⠺⠛⠁⠀⠀⠀⠊⣀⠀⠐⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⡆⢀⣠⣾⡷⣈⡉
⠀⠀⠀⠉⠁⠀⠀⠀⠀⣓⣲⠂⠠⡀⠁⠀⡀⢀⣴⣺⣿⢿⣿⣿⣿⠿⠟⠃⠤⠝⠀⠈⠁
⠀⠀⠀⠀⠀⠤⠖⠚⠛⠁⠀⠀⣀⣰⣦⣤⣤⠀⢹⡏⠀⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠛⠛⠋⢶⣿⣤⣿⣯⣷⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠐⠀⠙⠙⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢠⡀⠀⣴⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣤⣀⣠⣿⣧⠐⢿⡤⠴⠛⢿⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣧⠀⠀⠈⣻⣿⠛⠛⠿⣿⣿⣷⡄⣀⣤⠀⠈⠛⠢⣤⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣈⣹⣯⡀⠈⠙⣤⣀⣀⣀⠈⢿⣿⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠛⠉⣽⣿⣿⣏⡉⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠀⠻⡿⠗⠀⠙⡿⠻⢶⣶⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣶⣾⣿⡟⠙⣿⣿⣦⣴⣶⣾⣿⡿⠋⠀⠹⣿⣿⠛⠻⣿⣿⣿⣄⣤⣀⠀⠳⡄⢀⣿⣿⠟⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⣿⠟⠉⢻⣿⡀⣰⣿⣿⠟⠛⢻⣿⣿⣿⣶⣤⣀⣿⣿⣀⣀⣬⣿⣿⠋⣿⣿⡿⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣀⣸⣿⣿⣿⣿⣿⡄⠀⣀⣿⣿⡋⠙⠛⠛⢿⣿⣿⡿⠛⠻⣿⣤⠛⠉⠀⣤⣶⣶⡙⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠿⠿⣿⣿⣿⠋⠙⢿⣿⣿⠿⠿⢛⣿⡛⠒⠠⣤⡼⠻⠿⠷⢀⣤⣼⠇⠀⠀⠈⠛⠛⠋⠛⠛⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣧⡀⣠⡴⠇⠀⠀⢴⣿⣿⣷⡀⠀⠀⣤⣴⣶⡄⠀⠀⠀⢠⣤⠶⠦⢤⣴⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⡿⠋⠀⠉⣀⡀⠀⢠⣤⣿⣿⣿⣷⣶⣾⣏⡀⠀⢸⣿⠟⠉⢿⣦⣄⣀⡀⢙⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢾⣿⣋⠀⠀⣀⣹⣿⣿⡏⠉⠉⠙⢿⣿⣿⣿⠋⠀⠀⢈⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⢿⣿⣷⣶⣦⣤⣼⣿⡿⠿⠷⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠶⠦⠤⠤⣤⣤⣀⣀⣠⣤⣤⣴⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣿⣯⣝⡛⠿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣦⠀⠈⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡿⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠃⠀⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⢰⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠤⠤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠙⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⣼⣦⣴⡐⠀⢀⠀⣀⡲⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢐⡀⣲⠟⢉⣍⣤⣶⣶⣶⣶⣶⣤⣤⣌⣙⠓⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡃⠂⠀⠄⠤⠄⠁⠠⠀⠙⠀⢤⠄⠐⠪⠄⢘⣛⣾⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠐⠀⣤⡄⠀⠀⠒⠈⠠⠠⠥⢾⣧⣽⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣤⡐⠂⠀⠠⠀⠈⠃⡚⠁⢀⠛⣞⣻⣿⣿⣿⣧⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣄⣁⢀⡚⡀⢠⣈⣤⣟⣿⣿⣿⣿⡿⠟⠙⠲⢭⣉⣉⣉⢀⣈⣭⣈⣉⡉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠺⣼⣷⣿⣿⣿⣿⣿⠛⠉⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣿⣇⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠛⠿⣿⣿⣿⡿⠿⢷⡄⠀⠀⠠⠋⠻⣿⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡶⠾⠛⠉⠁⠁⠀⠒⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠉⠀⠑⣄⠀⠀⠀⠀⠀⢀⣠⡴⠒⠂⠁⠁⠀⠀⠉⠉⠵⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢆⠀⠀⠉⠳⡀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠽⡳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⣡⡀⠘⢏⠃⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⢠⡛⠆⠀⠀⡃⠀⠀⠀⠀⠇⠀⢀⣠⠄⠠⠀⠀⡅⡠⠋⠀⠀⣺⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌
⠀⠀⠀⠀⠀⠀⢸⡀⢀⣀⣈⡤⢀⣀⡀⠐⠂⠉⣠⠀⠀⠺⠁⢸⠔⠁⠀⠀⠀⣯⠍⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠑⢀⠈⠛⢀⠀⠉⠙⠀⠀⠀⠸⠀⠀⠀⢠⡎⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠌⠉⠓⠀⠀⠀⣘⠀⠤⠀⠀⠼⠇⠀⠀⠀⠀⢀⠃⢻⣆⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠁⠀⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠂⠀⢣⡸⣯⣄⡀⠀⡀⠀⠀⠀⠄⠀⠀⣀⠀⠀⠰⢤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⡆⠉⠉⠙⠢⣵⠘⠂⡤⠀⠁⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⣯⣻⠏⣍⣄⡤⠀⣦⣐⣠⢀⣀⣨⠞⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠀⠀⠀⠄⠚⠘⡆⠸⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢼⣠⠻⣿⣿⣿⣦⣽⠶⠿⠎⠛⠉
⠀⠀⠀⠀⠀⡌⠀⠀⠀⣈⣤⠀⠁⠀⠐⠀⠰⠀⠁⠀⠀⠆⠀⠀⠀⠀⠠⠀⢀⠁⠀⠀⠀⠁⣼⠟⠁
⠀⠀⠀⠀⣠⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⣆⠤⠖⢚⡟⠋⠛⣦⣀⠀⢠⠆⠀⠀⠂⠀⡶⠂
⠀⣠⡴⣾⣿⠈⠀⠀⠈⠛⠋⠀⠀⠉⣩⠖⠉⠀⠀⠀⡞⠀⣠⠉⢱⠈⣂⠀⠀⠀⠀⠀⢠⠃
⢀⠀⠁⠈⢻⣄⠀⠀⠀⣴⠀⠀⠀⡐⠀⠀⠀⢃⠀⠀⠃⢠⢅⠀⠀⢠⠄
⠁⢀⠈⠀⠠⣿⣶⣄⠁⠘⠀⠀⡐⠀⠐⠀⠀⠄⠀⠀⠀⠀⡀⠚⠀⠀⠘⡘⠀⠀⠀⣤⣴⣴⡄⢀⢀⠀⢀
⠀⠀⠰⠆⠦⠳⢬⡫⠹⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⡄⠀⠈⠀⡤⠈⠑⢁⢀⣴⠟⠉⢸⢁⡈⣭⠞⠘⠃
⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⣽⣿⡇⠀⠀⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⣾⣿⠇⠯⠙⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⢿⢷⡄⠀⠀⠠⣀⠀⠠⠀⠀⠀⠁⠀⠂⢐⣴⠋⠿⠁⠶⠢⠄⢀
⠀⠀⠀⠀⠀⠀⠂⠈⠀⠃⠀⠘⡚⢿⣦⣀⡈⠁⣀⣀⣀⣤⣤⣤⣤⣶⣿⢟⡅⠠⣦⠀⠒⠈⠁
⠀⠀⠀⢀⠀⠀⠊⠀⠀⠀⠐⠀⠄⠐⢍⠛⣻⠛⢿⢛⢛⡉⣭⡽⠝⠋⠁
⣤⣤⣬⣬⣿⣿⣧⣾⣬⣿⣿⣬⣬⣤⣤⣤⣬⣬⣽⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣅⣍⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢤⡜⣫⠀⡶⠁⢤⠠⠄⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢄⣋⣘⠀⢲⣀⡶⠁⣋⠲⠆⣤⠓⠀⠀⠀⠀⢀⠈⠉⠁
⠀⢠⣽⣾⣽⣶⣾⣿⣷⣾⣶⣿⣶⣿⣶⣿⠤⠀⠀⠀⠘⠷⠶⠞⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⢶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠦⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠠⢤⣤⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⣤⡄⢠⡀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢀⢀⠤⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡴⣸⣧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢠⠠⠀⠐⠀⠀⠀⠀⠀⢤⠤⢤⡀⠀⠉⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⠀⠉⠀⠀⢀⠀⠀⠘⠀⡀⠁⠀⠀⠀⠀⠀⠀⠐⡓⠄
⠀⣠⡦⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⢀⢎⠴⣓
⣴⣿⣇⠧⢸⠄⠀⠀⠀⠀⠀⠀⠀⢠⠴⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠂⠀⠠⠁⣨⠺
⢻⣟⢺⢰⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢂⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⡇
⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠀⠀⠀⠀⠀⢰⠤⠄⠀⠂⠀⠴⠾⣾⣿⣿⣿⡄⠀⠠⡇
⠀⠀⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠃⠀⠀⠀⠉⠚⠋⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣶⣦⣀⠀⠄⠠
⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⣀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠆⠀⢀⣤⣶⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠙⠛⠙⠉⠉⠉⠩⠛
⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠶⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡀

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠠⠁⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⠀⠄⠂⠁⠀⠀⡔⠁⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠖⠂⠀⠀⠀⠀⠀⠂⠔⠀⠀⠀⠈⠑⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠠⠄⠠⠄⠀⠒
⠀⠀⢀⠠⠀⠀⠈⠀⢀⡤⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠄⠀⠀⠶⡌⠁⠀⠀⠸⠁⠐⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘
⠀⠀⠐⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠈⠰⠀⠀⡠⠄⠀⠀⠀⠀⣄⠀⢐⠥⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠂⠀⠀⠂⠈⠀⢀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠈⠉⠉⠁⠀⠀⠀⠉
⠀⠰⡀⠀⠰⢀⠀⢔
⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠰
⠀⠀⠀⠀⢗⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠥⠀⢀⣈⠀⡀⡀⠀⠀⠀⠀⠀⠂⠁⠀⢣⠖
⠀⠀⠀⠀⠚⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠙⠀⠋⠉⠃⣼⣈
⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢀⠀⠀⠀⠀⣀⠄⢠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⢀⣿⠀⢀⠔⠁⠀⣄⠀⠰⡀⠫⣧⡀
⠀⠀⠀⠀⠀⠀⡀⢸⡄⡏⢸⡇⣸⠃⠰⠀⠀⠀⠀⠢⠈⢩⡷⣿⡆⢀⣀
⠀⠀⠀⠀⠀⠀⠑⣄⠻⣇⡟⢀⡿⣰⠀⠐⠀⠀⢀⠀⢠⢸⠿⣳⠋⢯⠋⣿⡝⣟⡶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣄⣸⣧⣿⢰⡄⠀⠀⠀⠀⠀⡨⡾⠛⠃⠀⡀⢀⡑⡥⣷⢷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣭⣧⣸⡏⣸⠇⠀⠀⢀⢀⣽⣴⠃⠀⠛⢁⡈⡅⢀⡨⢿⣞⡽⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣮⣽⣷⣿⠀⠐⠀⣬⢢⢥⣻⠀⢀⠀⠀⠁⠢⠀⢀⠡⠳⢂⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⣍⣣⣹⣶⢦⠮⢅⣿⠟⡇⠈⠀⠀⠀⠠⠋⠀⠀⢦⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⡹⣎⣷⣯⣧⠄⠑⣄⠂⠀⠀⠀⠀⠀⢈⣭⢏⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣟⣶⣾⠗⠀⠀⢩⡟⢰⢲⣲⣾⣿⣏⣵⣛⣥⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠘⠓⠘⠛⣿⣆⠀⠀⣼⡇⣿⣾⣿⠿⣭⡙⢦⡙⢿⣝⡲⢦⣍⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢼⣿⡟⠀⢰⣿⣛⢧⣻⣟⠳⣌⠻⣆⡙⢦⡈⠛⠦⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠐⣤⣀⣀⡹⠏⠀⠀⢸⠋⠈⢁⠏⠈⠉⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡐⢠⣿⣤⣶⣿⣿⣿⣦⣄⣼⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢈⢿⣿⡏⠛⠿⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⢶⡞⠋⠋⠍⠉⣿⣲⣲⣶⡦⠤⠤⠤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⢀⣠⠶⠋⠉⠀⢀⠀⠀⠀⠀⠀⢁⠀⠀⠉⠀⠀⠀⠀⡀⠀⠈⠀⠈⠟⠳⢦⣄⡀
⠀⠀⠀⢀⣴⡋⠀⠀⠀⠈⠃⠀⠄⡀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠃⠈⠛⣦⡀⠀⠀⣀⣀⣠⣤⢤⣤⣤⣀⣀
⠀⠀⣰⠋⠤⠀⠀⠈⠁⢀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠄⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⢠⠲⣾⠿⣶⣻⡌⠓⠀⠠⣄⠀⣠⢀⣨⢍⣲⡶⢤⣀
⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⢠⡖⢠⣔⣬⣶⠾⠛⠋⠉⢒⣠⣄⣤⣀⣠⣀⣀⣀⣐⠀⠈⢁⠙⠛⣻⠶⣄
⣼⠀⠀⠀⠇⠶⠀⠀⠀⠰⣦⠀⠈⠁⢀⣀⠀⠀⠈⠉⠁⠀⠐⠀⣤⠀⠆⡚⣠⣦⢡⣬⡟⠀⣁⣖⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠉⠊⢦
⣏⠀⠓⠀⠠⠠⠀⡀⠀⠀⠁⠀⢴⠀⠀⠸⠋⠀⣀⣤⡄⡀⠒⢃⣭⣄⠐⠛⢛⡳⢶⠏⢡⣴⣿⣯⣿⣿⣷⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣧⠀⠈
⢻⡆⠄⠀⠀⡀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⣤⡀⠇⠀⠁⠀⠴⢀⠶⣾⡙⢟⣿⣿⠃⣾⣿⣿⣿⣿⣿⣿⡿⣿⣭⣿⣿⣿⣿⡿⣿⣿⣿⣾⣿⣿⣿⣿
⠈⢳⣈⡀⠈⠑⠂⠄⠀⠰⠶⠷⠄⠀⠀⠀⠀⠉⠁⠀⠘⠀⠒⠛⠛⣻⣷⣷⣾⣽⣿⠠⡹⣿⡿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢠
⠀⠀⢳⡀⣠⠐⠢⠦⠀⠀⢀⡀⠀⠀⠙⢣⠬⠔⠛⠁⠀⠚⠟⢰⠾⢿⡇⢂⢋⣻⣛⣧⣁⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⢀⠎
⠀⠀⠀⠻⣟⠓⠀⠈⠢⠀⣀⡐⠀⠀⣀⡀⠰⠆⢀⡬⠙⢡⣈⡯⣾⣿⣿⣿⣿⣿⣿⣿⣷⣕⢌⠻⠿⣿⣿⣿⣿⡿⠿⠟⠿⠿⠿⠿⣿⣿⣿⡿⠟⠁⡠⠋
⠀⠀⠀⠀⠙⠳⠶⣏⣶⢀⡬⢀⠀⣈⠉⠀⠐⠶⣀⠴⣦⡖⣬⣷⣿⣿⣿⣿⣿⣿⡿⠟⠉⠈⠓⠺⢶⣀⣀⡀⠁⣠⡤⠄⠠⣔⣐⣂⣠⡄⠈⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣼⠡⢛⣋⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣯⣶⣿⣿⣿⡿⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣿⢿⣾⣿⣿⣿⢟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠻⠛⢻⣿⣿⣿⣿⣿⣿⣿⣭⣷⣿⣿⣇⣄⠀⠀⠀⠀⠀⠀⠀⣿⣻⣽⣏⣭⣭⣷⡶⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠉⠛⢻⣿⣿⣿⣿⣿⡟⠛⠛⢿⣷⠀⠀⠀⠀⠀⠞⠃⠀⠸⡿⣿⢿⣇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠾⠀⠛⢺⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠗
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠄⠠⠆⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣡⠀⡈⠡⠀⠀⣀⣠⠜⢧⡀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣤⣠⣰⠿⢀⠛⠉⠳⢤⡎⠀⠈⠷⠀⠉⠉⡇⣠⣄⣀⠄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⣿⣄⡘⠳⡄⠀⠀⠀⣾⣠⠺⡄⢠⡄⣀⠀⣿⣀⣤⠗⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣶⣿⢿⡯⢃⣿⣿⣿⣿⣾⣷⣿⣶⣿⣿⣿⣿⣿⣷⣿⣾⣿⠟⠃⠸⣶⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠴⡻⠟⠛⠉⠁⠈⢀⠈⠁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⢹⡏⠀⢠⣿⡞⡝⣿⢷⣆⡀
⠀⠀⠀⠀⠀⠀⣎⡑⢪⡅⠄⠾⠁⡳⠄⡀⠀⠀⠀⠹⠿⠟⠛⠃⡈⠙⠟⠉⠀⠰⠀⠀⠋⠀⠀⠘⠿⢧⣸⡟⣯⣿⣿⣄
⠀⠀⠀⢀⡄⣵⣻⠔⢣⡦⠀⣠⠖⠀⠀⠠⣼⣶⡀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠠⠀⡀⠀⠀⢀⠀⣀⠂⠀⠟⢹⣿⢻⣿⣧⡀
⠀⠀⣴⢋⣷⠘⠇⠰⠋⠀⠠⢉⠀⠀⠈⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠕⠀⠂⠸⡆⠀⠐⠈⣉⣧⢻⣟⣧
⠀⢸⠃⢿⠻⡗⠀⠀⠢⢄⠠⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⢀⡈⢶⠀⠁⣹⣿⣇⣿⣿⡀
⠀⣿⣸⣇⢳⠀⠦⢕⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢟⣩⣞⠠⠈⣷⣙⣿⣿⡇
⠀⢹⣟⣿⡈⠑⢦⣈⡐⠇⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣩⡆⡀⣿⣿⡿⣿⠇
⠀⠘⣿⡃⢧⣠⡾⣏⠀⠀⠀⠘⠂⠄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢀⠀⠀⠀⠀⠀⢰⠀⣻⠁⣿⣿⣿⡿
⠀⠀⠘⢿⡓⣭⣙⣯⣤⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⢸⡎⠁⣼⣿⣿⠟
⠀⠀⠀⠈⠻⢿⣮⣹⡟⣿⣶⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡈⠇⠀⠃⢘⣧⣲⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⢦⡆⣀⠀⠐⠀⡀⢃⢰⠁⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢇⡀⢠⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠍⠻⠻⢿⣿⣽⣆⠹⣣⠀⠈⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣷⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢣⡈⡠⠀⠁⠀⠀⠀⠀⠀⠀⠘⣄⣰⢤⣀⢄⣤⣴⣶⣿⡿⡿⡟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡆⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⡿⠾⠁⡍⠁⡞⢰⠛⢰⣿⣿⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠑⠆⡀⠀⢺⠁⣸⠀⣼⣿⢻⢸⡼⠇⣼⠇⡀⢸⡇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣧⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠇⢸⠑⠀⠻⠐⢿⣿⣸⢸⣇⣿⡿⢰⠇⢸⠏⣼⢱⡇
⠀⠀⠀⣀⡀⠠⠀⠳⡀⢦⠱⡜⢿⣟⠠⢀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⢺⡏⣿⢸⣿⣿⣷⣿⣤⣿⣶⡏⡿
⠀⠀⠀⠈⠙⠷⢷⣄⣿⡷⡕⣿⣾⣿⡀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⡠⠼⠀⢀⡆⡆⣸⣸⢣⣿⣼⣿⣿⣿⣿⣿⣾⣿⣷⣇
⠀⠀⠀⠐⣄⠙⠢⣍⢿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠘⠛⠀⠀⠀⠀⠀⠀⢰⡄⢸⢡⡇⣏⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡏
⠀⠀⠀⠀⢋⠻⢷⣽⣿⣿⣿⣿⢿⣿⣿⣷⡀⠀⢃⡀⠈⠀⠀⠀⠀⠀⠀⠀⢦⡀⣿⡈⣾⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⠇
⠀⠀⠀⣀⠻⣿⣶⣿⣷⣽⣿⣿⣿⣿⣿⣿⣟⣶⣼⣷⣦⢰⣤⠁⠀⠀⠀⡀⢀⡿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠙⢷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠆
⠀⠀⠀⠙⢦⡙⣮⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣤
⠀⠀⠀⠀⠀⣨⣼⣿⠟⠛⢛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣽⣿⣿⠿⠿⠿⠷⠦⠄⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠛⠿⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠛⠋⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡟⣋⠩⠏⠁⠆⠙⡹⠿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠂⠁⠀⠂⠀⠀⠦⢈⠁⠴⢹⣹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠿⠞⠁⠄⢀⠁⠀⠀⠐⠀⡀⢼⢃⣥⣧⠀⢀⣠⣤⡤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢹⣷⠆⢡⡀⠀⠈⠀⠀⠀⠀⣡⠟⡼⣯⣷⣾⠫⠍⠁⠀⡐⠐⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢳⡶⡶⣄⠃⡀⠅⣀⡍⢀⢡⣴⠔⢻⡏⡉⡀⠀⠠⠀⠀⠈⢀⠑⢦⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⡶⢾⢧⢼⡧⠴⢾⡿⣿⡂⠸⡻⢧⣁⠐⠠⠀⠀⠠⠀⢛⡀⡔⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⡀⠈⠀⠀⠸⠛⠟⡇⠀⢻⣶⠶⠀⠆⠀⠀⠀⠀⠄⠢⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⡀⢼⣧⠀⢸⡿⢻⣿⣤⢓⣰⣶⣒⣭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⣿⣄⣿⠀⠀⠀⠉⠉⠹⡿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠈⣽⡯⠀⠀⠀⠀⠀⠀⠁⠜⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢨⡿⠃⠀⠀⠀⠀⡀⠀⠰⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣽⠏⠀⠀⠀⠀⠀⠀⠀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⣤⠤⠤⠴⢿⡀⠀⠀⠀⠀⠀⠐⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⠤⠤⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣈⣉⣿⣶⣾⣾⣿⣿⣾⣿⣶⣧⣯⣧⣶⣿⣿⡿⠿⠛⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠙⠛⠉⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⢀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⢰⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣀⣤⣥⢜⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠻⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⢟⣳⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡙⣫⣿⣶⢞⡾⣯⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⢶⣋⣹⣾⣻⣟⣯⣿⣾⣿⣺⡦⡷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⣠⢺⡟⣿⢿⣯⢿⡏⣟⣿⢴⡿⣻⣩⣜⠷⢭⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⣴⣛⣿⣿⣿⡟⢛⡀⢁⠈⠏⠈⢗⠙⠈⠉⠓⢰⣿⣷⣰⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⡼⣭⣿⡽⣿⠛⢧⠀⠦⠀⠀⠆⠀⠈⠀⠀⠀⠀⠀⠈⠻⠛⣶⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣸⣿⣯⣼⠇⠈⢆⣤⠃⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡿⣿⣿⡁⠙⡢⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢰⡟⣿⣥⠉⢰⡀⠡⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣯⡷⣄⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⣿⣿⢇⠈⣧⡷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⣿⣿⣌⠓⣿⣿⣿⣄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣧⡘⣿⣿⣿⣿⣥⡰⣦⣄⣀⢀⣀⠀⠀⠀⢀⡀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⡃⣿⢿⣟⣙⣿⣿⣿⣿⣷⣾⣷⡦⡴⢦⢾⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠻⠗⡸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡮⢛⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡥⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⣽⢯⡞⠉⣿⣿⣿⣿⣿⣿⣂⣗⣿⣿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡟⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣴⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠋⣂⣀⣀⠀⠀⣤⣙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢶⣶⠫⠭⢤⣌⣉⠉⠁⠀⠀⣠⣤⢐⣊⣩⠙⠛⣷⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣤⣞⣝⣒⠤⠉⠉⣑⡒⠲⡤⠭⠙⠒⠀⠀⠐⣊⣉⣁⠤⠋⠹⣆⣒⣊⣵⣫⣏⣹⠖⣦⣄
⠀⠀⠀⢀⣠⡞⠓⣒⣢⣀⢀⠉⠙⠛⠒⠀⠁⠁⠀⢠⠀⠀⠀⠒⠚⠉⠁⠀⠘⠋⠉⠁⢉⣈⣉⢛⣳⣿⡛⣩⣽⣦⡀
⠀⠀⣠⡞⠡⠤⠉⢀⣀⣈⣉⡴⣤⣔⡀⠀⠀⠀⠙⢦⡀⢠⢠⠀⠀⠀⠀⠀⠀⠀⡼⠋⠉⠀⠀⢤⣌⣤⡽⢛⣬⠥⣽⣦
⠀⣰⠟⡀⢍⣱⠂⠀⠀⢀⡍⠉⠀⠀⠁⠀⠀⠀⠀⠈⢇⢸⢼⢠⠇⠀⠀⢀⡄⠀⠀⠀⠘⣄⣐⣁⣀⠵⠛⠒⠒⢯⠥⠼⣇
⢰⠗⣀⡐⠂⠤⢥⠤⠤⠚⠒⠒⠒⠂⠀⢠⠈⡇⢀⢸⠈⡈⠀⢸⠀⠀⠀⣼⠔⠀⠀⠀⠰⠋⠁⠀⠠⠊⠙⠓⢒⠒⠖⣋⣹
⣿⣉⠵⠐⠒⠒⠒⠠⠴⠲⠶⣖⠍⠀⢀⢠⠳⡇⠘⡇⢠⢱⢡⠃⡄⡼⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢤⡲⠤⣌⠉⠙⠒⠤⣼
⣯⢋⠥⠶⠒⣁⠀⠀⢀⣀⣀⣀⡠⠀⠈⠦⣳⠸⡀⠹⢸⢸⢾⠀⡞⠀⠄⢨⠃⠀⠀⠀⠀⢮⣳⣄⡠⠄⠀⠀⡉⠒⢗⠒⣾
⣿⢀⠔⢒⣺⣧⣮⢎⡀⠀⠉⢁⠀⡀⣀⠲⡟⢆⠱⣤⠸⣬⠈⣼⠀⣼⢀⠇⣠⠇⠀⣆⣄⠀⡠⣄⣀⣄⣀⠳⣌⠳⡦⢍⡿
⢸⣇⡞⣁⠔⢉⡶⠋⡆⡄⢰⡡⢀⠈⢿⡀⢻⠘⡇⣹⡄⠟⠂⠿⠀⡏⠜⣰⢁⠀⢀⠸⠋⠀⠸⢍⠀⢠⣀⢵⢸⠀⡘⣾⠇
⠀⢻⣷⢃⡤⢊⡄⢀⡷⠁⡸⢁⡿⣠⡤⠳⣀⡷⣇⣇⢱⣰⡏⣠⡆⡜⡔⢱⠏⣼⡼⣃⠄⡄⠠⢉⢳⡜⡏⢸⠸⡀⢳⡏
⠀⠀⢿⣿⣀⠊⡠⢊⠜⡎⠀⢸⢠⡋⣾⠠⠀⢻⣿⣾⣼⢿⣼⢴⣴⣝⣿⢡⡎⡐⣰⠋⣰⠃⣦⣯⠾⡌⡆⡆⣠⣷⠏
⠀⠀⠀⠙⠻⣿⡵⢾⣺⣷⡦⣤⣸⣿⣼⡾⣷⡏⡇⡓⠙⣑⡗⣋⢷⣯⣿⡿⢴⣋⣽⢩⠅⡿⡉⣸⢃⣻⣃⡥⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣹⠂⣿⢏⠱⣦⣥⢼⣿⠻⠗⠚⡥⠣⠌⡂⣛⠋⡞⢉⠉⡯⡿⣷⡳⣾⣿⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠣⠳⠌⡴⢛⢗⢸⣧⢃⣧⢉⡴⢟⣦⠐⡇⠟⣉⣙⠋⣕⢁⠝⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣳⠚⣥⡭⣸⠉⢶⡍⡤⠎⢣⣴⠢⢈⣿⠁⠉⢆⣘⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡩⢈⡧⢆⣥⠳⠄⠒⠲⡔⠁⠞⢠⠈⠛⢐⣏⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⣮⣇⣟⠀⠙⡞⡟⠀⣸⠦⣩⣌⣹⣧⣎⢳⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⢩⣼⢲⡏⠄⠃⣰⠛⣭⣛⠸⡇⠤⣬⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣋⣇⣸⠷⠑⠌⣻⢀⣿⣥⡆⣯⡼⣭⣿⠀⡄⢠⠀⠤⠠⠄⠤⠤⠨⡽⠺⡅⢯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣧⡞⡿⣳⣿⣮⣰⢨⣏⢏⣍⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢯⣏⣉⣗⣧⡶⠷⣯⠾⠛⠁
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⢀⠀⠀⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡖⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣽⣿⣿⣿⣤⣤⣤⣤⣤⣄⣀⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⣤⣀⣦⣄⣘⣖⣢⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣴⣶⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠋⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢛⠻⠿⠟⠛⣿⣿⡿⠿⠋⠉⠉⠉⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠄⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠂
⠀⠀⠈⠉⠿⠿⠿⠿⠿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⢻⡿⠟⡛⠿⡖⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠍⠐⠀⠈⠀⠈⠙⠀⣴⠉⠻⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢛⣁⡄⠀⢈⠁⢀⠀⠑⠀⡄⠀⣋⠶⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠺⠈⠁⠄⢀⡀⠐⠀⠀⡀⠃⣀⠀⢾⢑⣦⣿⡄⠀⣀⣤⣤⣤⠴⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡆⣩⣀⠈⣀⠀⠀⠀⠀⠀⢉⣠⠓⣬⢯⡿⣴⡞⠫⡉⠅⠄⠀⠀⠐⠊⢟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣡⣦⣬⡅⠦⠀⠐⠀⠀⠲⢠⠾⢀⣬⣡⣿⢿⠹⠃⠀⠄⢀⡀⠉⠀⠂⠈⢫⡍⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣧⣻⣣⣶⠈⣤⡞⣋⣀⣾⣿⣿⠀⠸⣯⣖⠐⠂⣀⠀⠀⠀⠀⠀⠙⣤⠈⡛⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠝⡟⠹⠛⠛⠍⣿⣽⣿⡄⠈⣿⣝⠓⠆⠀⡈⢀⠈⠀⠉⠄⠰⡆⠽⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡐⠠⠀⠀⠁⠈⠀⠋⠝⣹⡇⠀⠈⣿⡟⣇⡜⠃⠀⠐⠂⠐⠠⣆⣝⣲⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡀⠄⠁⠀⠀⠀⢃⠐⣿⣷⠀⢰⣿⠛⡷⣽⣶⢛⣘⣺⣞⣫⣬⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠰⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡄⣿⠅⡀⠂⠈⠉⠉⠏⣿⡿⢿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠳⠀⠁⠀⠀⠀⠀⠀⠈⢐⣻⣿⠋⠀⠀⠀⠁⠀⠀⡐⢈⣧⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠠⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠠⠀⠄⣁⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢠⣾⡏⠅⠀⡀⠀⢠⠀⠀⢀⠀⢁⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠤⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠤⠤⠴⠶⠶⠚⣧⡁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⣤⣄⣠⣤⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣄⣤⣂⣲⣦⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠴⠿⣿⣿⡯⡛⣿⡻⣟⣿⣿⢻⣿⠿⣿⢿⢿⡻⢟⢿⣿⣿⣿⣿⣯⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣟⣛⠂⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠙⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠻⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣾⣿⣿⣿⣿⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⣶⣄⠀⠀⠀⠰⣿⣿⣥⣿⣿⣿⣿⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⠉⠉⠉⡏⠉⢹⡇⠀⣠⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣿⣿⣿⣯⣿⣿⠀⢀⣴⡾⣷⣾⡇⠀⢸⣥⣾⣿⣯⣨⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⢹⠋⠉⠁⣴⣿⣿⣦⣾⣿⣿⡄⣾⣿⣿⣿⣿⡿⢿⣿⡿⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⢲⣄⠀⢁⠀⡆⠀⣾⣿⣿⣿⢿⣿⣿⣿⣿⣸⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣿⢿⣿⣧⠈⡄⠇⠸⣿⣿⣿⣿⣮⣿⣿⣿⣿⢳⠅⠀⡙⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢛⠛⠋⢐⣀⡀⡆⠀⠀⠀⢸⠀⢈⡇⠀⠀⢸⠀⢀⠇⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⢣⠈⢣⠀⠀⠀⡀⠀⢰⠀⢀⣀⠞⠀⡜⠀⢀⣾⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣬⡆⠀⡆⠀⠀⠇⠀⣨⣾⣻⣿⣆⢰⠁⢰⣿⣽⣷⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⡿⢷⠀⠠⣀⣸⠀⢸⡿⣿⢯⣿⣿⠀⠀⢀⠏⠉⠉⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠘⡄⢱⠈⡇⣼⣿⣿⡆⢸⢃⠃⢰⡠⠋⠀⢀⡎⡐⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡶⣧⡱⠈⢣⢣⠀⡃⢸⠀⣼⠎⢠⡏⠀⢀⢀⠘⢆⡀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢷⠛⡅⢣⠈⢻⡀⡇⢸⢀⡟⠀⡞⠀⠔⠱⢋⠄⢀⠕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢢⡘⢄⠳⡀⠙⠇⠸⠊⠀⡼⠅⠐⠀⠈⢀⠔⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣿⠄⠀⠀⣉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣁⠀⠀⠰⣿⣄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠂⢀⣿⣷⡄⠀⠀⢀⣤⣶⣤⡀⠀⠀⢠⣾⣿⡀⠐⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠁⠐⠀⣿⣿⣿⣿⣷⠀⠂⠘⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡘⢡
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣛⡹⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢀⣬⠑⠀⠀⣠⣿⣼⣇
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⠀⠀⠀⠀⢠⠿⠿⠁⠀⠀⡀⠀⠀⠀⢀⡡⠤⠀⠀⠀⠀⠐⣤⠇⠸⢿⠀⠂⠔⣿⣟⡠⠞⣷
⠀⠀⠀⠀⠀⠀⢸⡰⡆⠀⠀⠀⠀⢀⣶⡶⠀⠀⠀⠀⣳⡜⠀⠀⢘⡁⢠⡦⠀⡀⠀⠸⡆⡾⠀⠣⠈⠠⣼⣿⠟⠁⠀⣠⡇
⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⡀⠀⠀⠚⡉⠀⠀⠤⡄⠀⡶⢰⡖⣠⡜⠃⠀⠁⠘⢇⣠⡴⠇⠽⢤⣀⠀⡇⠿⢃⣤⠶⠋⠁⣿⠀⢸⡀
⠀⠀⠠⠁⣿⣿⠟⢉⡁⢈⣯⡠⠰⡿⠇⠀⢀⣶⡇⢠⣧⣼⡅⠘⠀⣤⣼⡟⣴⣿⣁⣀⣀⣀⣤⡬⠷⠁⢴⠛⣵⠀⣀⠴⠀⢀⣎⢸⡄
⠀⠀⣇⢰⣏⡩⣶⢁⠀⡂⠀⡱⣤⠁⠸⠿⠤⠙⠁⡞⠁⠈⣧⠀⠀⠏⠙⠃⣾⣅⡀⣀⡀⣀⡄⡠⣾⣷⠈⢆⣿⠊⢉⠄⠀⢘⡁⢀⣙
⠀⠀⠸⡢⣿⣧⡜⢂⣉⠔⣤⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⡀⠀⠀⠀⢻⣯⣽⣯⣼⣧⠯⠻⢿⠇⠀⠀⠡⣔⠉⠀⠀⣫⡀⢀⣈⣇
⠀⠀⠀⠘⠻⢿⡿⠧⠁⠈⠔⠁⠀⠀⢀⣼⣤⣀⣀⠀⠀⠀⠀⡟⢃⠀⠀⢀⠀⠙⠶⣶⠶⠖⢒⡾⠋⠀⠀⠀⠀⠀⠀⣠⣾⣯⣥⣤⣤⣭⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣏⠉⢄⠀⠀⢰⠃⠀⠉⠉⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠉⠂⠘⣧⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠌⡌⠐⠀⠆⠱⠈⠆⡆⠀⢘⠀⠠⠋⠰⠈⠀⠘⢠⠁⠚⠛⢠⢰⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣤⠴⠒⠂⠀⠉⠓⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠉⣠⠔⠊⣀⣀⣈⢉⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⠞⣁⡴⢧⣄⠠⠶⠶⠤⠰⠶⣄⡀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡾⢁⠀⢀⣿⠷⠆⠀⠤⣭⠏⢴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⣿⠸⠋⠀⣀⣉⣫⡥⠩⣟⣓⡉⠙⠓⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢳⠈⢸⠘⠀⡄⠀⢀⠐⢒⡂⣄⣓⠂⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠀⠀⠀⠸⠰⣇⠋⠰⠈⡀⠡⠍⠰⠸⢀⡆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣇⠁⡀⡀⠃⠘⠧⠙⢳⣤⣤⣤⣤⡾⢋⡹⠂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠸⡀⢃⡅⢰⡘⠦⠠⠀⡿⠁⠈⢻⠧⠠⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠁⣸⣿⣄⠉⠂⡴⠀⡇⠀⠀⠨⠗⠄⠞⢃⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡍⠻⣷⣾⠁⠀⠐⢦⡴⠂⠀⣸⣾⠛⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣾⣿⡉⢦⡀⠈⠳⢦⠀⠚⠃⢠⡴⣋⣠⢊⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢸⡟⢳⡄⠁⠀⠀⠈⠛⠾⠟⢩⡾⠋⢉⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣟⠘⠻⣦⡀⠠⣀⣀⠀⠠⠏⠀⠐⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣄⡀⠀⠙⢦⡈⠙⠛⠃⠀⠀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠢⡄⠁⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣀⠀⠀⠀⠁⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⢰⠀⠀⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣈⣠⡿⠐⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠘⠛⠛⠛⣧⠀⢰⡆
⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠆⢠⠄⣀⠀⡀⠀⡄⢠⠀⠲⠈⠁⢀⣿⡇⠀⢠⣤⣄⠈
⠀⠀⠈⣡⡤⢠⠄⠀⣴⡇⠀⠀⠀⠀⠀⢠⡄⠀⡤⡀⢄⠀⠁⠀⠁⠀⠀⠀⠀⠀⣸⣿⣿⠀⠹⠿⠟⠀⠀⢀⣄
⠀⠀⠀⠀⠰⠈⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⡿⣄⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠉⠁⠈⠉⠀⠁⠡⠶⢿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠐⡅⠀⢠⠃⠀⠀⠃⠈⢀⣴⣷⡀⠛⠛⢁⣀⣀⡀⠀⢀⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠑⢸⠘⠀⠀⠐⠰⠾⣿⡿⠆⠀⠀⠀⠀⠈⠀⠈⠁⠀⠐⢛⡉⣉⢳⡀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣀⠀⠀⢠⣈⣠⠀⠀⠀⠉⠄⠀⠀⠀⠀⠀⠀⠖⢳⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠈⠀⠀⠀⠀⠀⠙⠀⠀⠘⠖⠃⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⢀⣠⣶⡀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠠⢄⡒⠀⠀⠸⠀⠀⠀⠚⠁⠀⠛
⠀⠀⠀⠛⠛⠿⠇⠀⠛⠋⣤⡶⠀⡀⠀⠀⠈⠥⠀⠘⠂⠀⠀⢄⡐⠀⠀⠀⠸⡀⠀⠀⠀⡀⠀⠀⠛⠀⢀⢀⣀
⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⡀⠀⠀⠁⢀⠀⠀⠀⠛⠛⠛⠃
⠀⠀⠻⣿⣿⣿⣄⠀⠀⣶⡄⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠸⠇⡌
⠀⠀⠀⠻⠿⢿⣿⡷⠾⠛⠀⠀⢨⣯⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⡎⠀⠀⠀⣌⠀⠤⠀⠀⢠⡄⣬⠀⠀⢠⡆⠅
⠀⠀⠀⠈⠁⠈⠀⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠈⠋⠀⠒⠉⠉⠉⠘⠛⠛⠿⠿⠿⠿⠿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠁⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠢⡀
⠀⠀⠀⠀⣠⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⣴⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡖⠉⠉⣿⣿⣿⣶⣖
⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣿⣿⣿⣧⠀⢀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⡀
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⣻⣿⣿⣿⡄⠀⠀⣿⣿⠃⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠛⠿⠇⠀⢀⢀⣁⣄⣿⣿⡟⠀⠀⠀⢀⡀⠞⠻⣿⣿⣿⠏⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣤⠺⠋⣺⣿⣿⠉⠁⠀⠀⢀⣤⡻⡿⠶⠌⠉⠉⠿⠿⣽⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⣻⣿⡄⠀⠀⠀⢈⣙⠛⠳⠀⠀⣀⠀⠀⢀⢙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢴⣿⣷⠀⠀⠀⠈⠛⠠⠶⠀⠀⠉⢳⠀⣿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣸⣹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⣦⣄⣰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣻⣿⣧⠀⠀⠀⠀⠀⢀⠀⡀⠀⠈⠉⢘⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡀⠀⠀⠀⠀⠀⢈⣿⠆⠀⠀⠘⠿⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⢿⣿⡇⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢝⣩⣿⣿⡀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣿⣿⣇⠀⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠿⣿⣿⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠯⠄⣽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢛⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢷⢯⡘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢷⡟⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢟⣰⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣈⣶⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⣿⡌⠂⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⢣⣸⠀⠈⠙⠃⠀⠐⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠴⣍⡆⠀⠀⠀⠠⠀⠀⠈⠀⡏⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⠀⠉⢸⠆⠀⠀⠀⢁⡦⠀⠀⢇⡏⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡤⢿⡷⠐⣏⣠⣖⡒⠉⡁⠀⣠⠊⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡖⢠⡾⢋⠅⡼⢟⣡⣄⠄⣰⠃⡀⠀⡀⠉⠳⣄⠐⠁⠀⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣮⡰⣇⠈⢧⠸⢊⣴⢋⡜⠁⣼⠃⢸⠀⠀⠈⠛⢆⠈⠢⠄⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⡲⣆⠹⣿⠇⢸⢠⢺⡎⢸⢀⢸⡎⢀⡟⠀⠈⠉⠒⠄⠀⠀⡀⣀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠤⡉⠟⣸⠇⠠⠟⠈⢯⠃⣟⢸⠸⡆⠸⠁⠀⠀⢀⠀⠀⠀⠴⣿⠏⠀⠰
⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⠀⠓⣀⠀⠈⣦⠀⠀⠀⠈⢧⠙⠮⠧⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣠⣤⣦⠀⠶⠆⠛⠛⠋⠀⣤⣤⡤⠀⠙⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⢀⣄⠤⠠⠖⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⣴⡖⠠⡴⠦⠄⡀⣲⣶⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠷⠖⠿⠀⠉⠀⣤⡂⠈⠟⠛⠻⢻⡏⠋⠙⠛⠳⣦⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡯⢌⣥⣤⠀⠤⣤⠄⠀⣹⠛⠤⠀⠁⠀⡀⠀⠁⠶⠲⠤⠈⠛⣷⣄
⠀⠀⠀⠀⣠⠖⣽⣿⠿⢺⣧⣤⡚⠛⠉⠀⠀⠀⠠⠿⢧⡀⠀⢤⣄⠀⠁⠀⠀⠀⠒⠢⠀⠊⢸⡙⣷⡄
⠀⠀⢠⣮⢁⡄⢁⣤⠤⢞⡙⠀⡀⠀⠰⢧⣤⠤⠀⠀⠈⠁⢀⢀⡀⢠⢐⢿⡆⠀⠀⠀⠀⡀⠀⢟⣹⢷⣄⣀⣀⡀
⠀⠀⣿⢻⡆⠈⠠⣅⡠⣺⢍⠃⠁⠀⠄⢀⠀⠀⠀⡄⠠⠊⠀⠠⣠⠀⠀⠀⠃⠂⠀⠢⡭⢌⠜⠚⣷⣿⠻⠆⠈⠙⢓⠤⣀
⠀⠀⣿⡞⠟⢇⣓⡓⠨⣟⠏⢶⠦⠀⢀⣌⣄⣤⣄⣀⣦⣤⣤⣀⣉⣀⣀⣈⣦⣥⣤⣤⣄⣤⣴⣿⠿⠃⠀⠐⠉⠐⠦⣠⠈⠻⢄
⠀⠀⢻⣿⠀⣠⣤⡿⢿⣿⢶⣶⣿⣿⣷⣷⣾⢿⣿⣟⠿⠟⡟⠻⠛⠻⠋⠉⠉⠉⣉⣩⣿⣿⣿⣿⣿⣺⣇⣀⡀⠀⠀⠨⠀⠢⢘⣧⡀
⠀⠀⠀⠻⢿⣿⣯⣎⡁⡤⡀⠀⠀⠀⠉⠀⠆⠜⣿⠍⠀⠀⠀⠱⣴⣶⣶⣶⡾⠿⠿⠛⠉⠉⠉⢹⣿⡿⠙⠛⠻⢿⣷⣶⣦⡀⢈⡹⣧
⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣦⣤⣤⣤⣤⣤⣤⣤⣌⡛⠘⠀⡀⠀⢷⣀⣀⣀⣀⣀⣄⣀⣀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣨⣥⣽⣿⣿⣿⡿⣿⣿⣿⣿⣻⣔⠃⢆⠰⡁⡿⠟⠛⠛⠻⠟⠛⠻⢿⡟⠀⠀⠀⢀⠀⠀⣠⣀⣲⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢀⣴⡾⣿⠿⠿⣿⡟⠿⣿⣿⣿⣹⣿⣿⣽⡾⣤⠆⠁⠀⠑⡶⢶⠂⠀⠀⣀⣴⣾⣷⣿⢷⢶⣶⣶⣾⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⣴⣻⣋⣌⣉⣉⣴⣭⣼⣷⣿⣾⣿⣿⣿⣿⡿⣿⠃⠀⠁⠄⠀⢽⣶⢀⣴⣿⣿⣿⣿⣿⣶⣤⣈⠀⠉⠄⢻⣦⣄
⠀⠀⢸⣿⣿⣿⣿⡿⠟⠛⠙⠟⠀⠀⠀⠀⠀⠀⣿⣷⡽⠂⠀⠀⠀⠀⢠⢿⣿⣿⠟⠉⠀⠀⢀⠔⠉⠙⠛⠻⢷⣼⣷⣿⣷⣄
⠀⠀⠈⢻⣿⣋⠁⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⡀⢻⣿⣦⠇⠀⠀⠀⠀⠀⢠⡻⣿⣤⣀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠉⠛⠛⠲⢦⡤⢶⣧⡆⠀⠀⠀⠘⠃⢸⡿⣿⣌⠃⠀⠀⠀⠀⠘⣇⠀⠙⡿⣿⠇⠀⠰⠃⠀⡄⠀⠀⠀⠀⠀⣉⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠘⣿⣹⡅⡄⠀⠀⠀⠀⠀⠹⠀⠀⢰⡟⠀⠀⠃⠀⠸⠿⢷⣷⣶⣶⣶⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⢈⠈⣿⣿⠀⠀⠀⠀⠀⠀⠒⣃⢰⣿⠁⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠂⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠈⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠂⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣤⠆⠀⠀⠠⠀⠀⢹⡟⠀⠀⠀⠀⠀⢹⣄⣀⡀
⠀⠀⠀⠀⠰⣤⣶⣴⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡴⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣄⢄⠀⠘⣿⣶⣽⢲⣀
⠀⠀⠀⠀⠀⠙⣻⣿⣵⣈⣀⡈⣧⣀⣀⣠⣀⣄⣰⣿⣿⣷⣻⣰⣀⣀⡀⠀⠀⣠⣽⣷⣶⣿⣾⣧⣀⣹⡿⡿⢿⣿⢛⠃
⠀⠀⠀⠀⠀⠤⡘⢨⠙⠻⠿⠛⢿⣿⢿⣿⣿⡿⢿⡻⢿⡟⢻⢆⣫⣐⢈⡿⠓⠹⢲⣭⣝⠟⣩⠙⢗⢥⡐⢣⣤⠈⠙⠊⢠⡀
⠀⠀⠀⠀⠁⠁⠁⠀⠀⠀⠀⠀⠈⢁⠛⠉⠉⠿⢼⠟⠾⠛⠛⠂⠰⢞⡚⠧⢿⡿⠿⠿⠿⢿⢿⡿⠿⠞⠉⠈⠤⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂

⣀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠿⠿⣿⠿⠗⠀⠀⠀⠀⠉⠀⠀⢀⣴⣄⠀⠹⠇⢀⡀⠀⢰⣿⣿⣷
⣿⡀⠀⠀⣿⣿⣿⠄⠀⠆⠀⣴⡁⠀⠀⠀⣾⣿⣷⠄⠦⠀⣾⣿⣿⠆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠉
⣿⣷⡀⢴⣿⣿⣿⣧⠀⠀⣼⣿⣷⡀⠀⣠⣿⣿⣿⡄⠀⠰⣿⣿⣿⡷⠀⢀⣄⣀⠺⠿⢿⣿⠿⠟⠂
⠻⠋⠁⣼⡿⣿⣿⣷⠀⢀⣼⣿⣷⡀⠀⣹⣿⣿⣿⡏⠀⠼⠿⣿⠿⠿⠄⣹⣿⣿⠀⠀⠈⠁⠀⠀⠀⠀⢈
⠀⠠⠄⠀⠀⠙⢠⡄⠀⢉⣿⣿⣿⡉⠀⠛⠛⣿⠛⠛⠂⣄⢠⠘⠀⠀⠰⣿⣿⣿⠧⠀⠉⣀⠀⠐⢻⣤⡞
⣤⣤⣤⠄⠀⠀⣾⣿⡀⠚⠛⣿⠛⠛⠂⠀⠀⠉⠀⣸⡄⣏⢹⠀⣼⡆⠼⠿⣿⠟⠷⢀⣾⣿⣷⡀⠿⠛⣿
⠿⢿⢿⠀⠀⣼⣿⣿⣧⠀⠀⠉⠀⢦⡼⣆⠀⠃⢠⣿⣧⠈⠉⣸⣿⣿⡀⠀⡉⠀⠀⣸⣿⣿⣿⡇⠀⠀⣿
⠀⠸⢸⠀⣰⣿⣿⣿⣿⣇⠀⠋⠀⣾⡿⠉⣀⠀⠛⢿⠿⠓⢰⣿⣿⣿⣷⠀⠉⠀⠰⢿⡿⣷⣿⡿⠆⠀⠏
⠀⠀⣰⡀⠛⠛⢻⡿⠛⠋⠀⠀⠀⠙⠶⠿⢫⣾⣿⣄⠀⢀⣿⣿⣿⣿⣿⣇⢰⡆⢰⣿⣿⣿⣿⣿⡆
⠀⣰⣿⣷⡀⠀⡀⡀⠀⠀⣴⣶⣶⡀⠀⠀⢿⣿⣿⣿⠆⠀⠀⠉⠻⠉⠁⠀⣾⣷⡀⠀⠀⠟⠀⠀⠀⠀⠈
⠀⣹⣿⣿⡉⢰⡍⣵⠀⢈⣿⣿⣿⡅⠀⣰⣿⣿⣿⣿⢧⠠⢶⣀⡴⠃⠀⠾⢿⡿⠷⠀⢀⡄⠀⣼⣶⣶⡀
⠴⣿⣿⣿⡿⠆⠉⠁⠀⢾⣿⣿⣿⡿⠄⣹⣿⣽⣿⣽⣯⠀⣾⢿⣷⣀⣁⠀⢀⡀⠀⢠⠀⠀⢠⣿⣿⣿⡄
⣰⣿⣿⣿⣷⣄⠀⠂⢀⣾⣿⣿⣿⣷⡄⠛⠛⢻⡟⠛⠛⠃⡄⠀⣿⣿⣿⣿⣿⠀⢠⣿⣧⡀⢻⣿⣿⣿⡟
⠀⠀⢻⠁⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⣄⠀⢄⠈⠁⢀⠀⢀⣿⡀⡟⡇⠀⢸⢸⠀⣠⣿⣿⡀⠟⠛⣿⠛⠛
⠀⣧⠀⠀⠂⣰⣶⣶⡀⠀⠃⠀⠀⠀⣸⣶⣏⠁⠀⣀⢀⣾⣿⣧⠀⠁⠀⠈⠈⠘⣻⣿⣿⡛⠀⠀⣉⠀⠤
⣼⣿⣆⠀⠀⣽⣿⣿⡏⠀⣶⣶⣶⣶⣿⠋⠙⠁⢠⣿⣎⠙⠟⠉⠁⢠⣾⣦⡀⠘⠛⢻⠛⠛⠂⣴⣿⣷⡀
⠻⡿⠟⠂⢾⣿⣿⣿⡿⠄⡟⡟⠛⢻⣿⠀⠀⠐⢻⣿⣟⠂⠀⠀⢰⣯⣿⣿⣷⡀⠀⡈⠀⠀⢠⣿⣿⣿⡅
⢠⡄⣀⠀⣼⣿⣿⣿⣿⣄⠧⠃⣤⠘⠟⠀⢀⣠⣿⣿⣿⣦⡀⠀⣴⢿⣷⢿⣿⣄⢰⡟⣳⠀⣹⣿⣿⣿⣏
⢾⣽⡅⠈⠁⠀⢹⠁⠀⠀⠀⣸⣿⡆⠀⠤⠀⣰⣿⣿⣿⣆⠀⠸⣿⡿⣿⣿⢿⡿⠀⠓⠃⡄⠉⠉⢿⠉⠉
⣤⣙⡟⠀⠐⠂⠀⠀⠀⠀⢰⣿⣿⣿⡄⠀⠘⠋⠉⣿⠉⠉⠃⣰⣿⣿⣿⣿⣿⣿⡆⠀⣸⣿⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⢁⠠⠐⣶⡄⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣟⡉⠫⣽⡚⣿⣴⡂⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠈⠝⠛⣿⡿⢛⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⣤⣤⣠⣴⠾⡷⠁⠙⢿⣾⣏⠖⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠈⠁⠈⠀⠙⠃⠀⠀⠀⠈⠉⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣶⠙⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠣⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡅⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣿⣷⣤⣄⣤⡄⣀⣀⣢⡶⠂⠀⠀⣀⢻⣿⣿⣿⣷⡀⠀⠀⠀⠀⢀⣀⣴⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠉⠙⠛⢻⣶⣷⣿⣿⣿⣿⣿⡿⠿⠟⠁⠀⠹⣿⣿⣶⣶⠦⣻⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡶⠒⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣄⣀⠜⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠏⠃⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣶⣧⣬⣵⣄⡀⠀⠀⣀⣀⣀⢈⣿⣿⣿⣞⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿⢿⣿⣿⣢⡞⠉⠀⠁⠀⠙⢿⣿⡟⢠⠀⠀⠀⠡
⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⣿⠿⠋⠀⢸⠿⠛⣿⣧⣀⡀⠀⡀⠀⠀⢻⣇⠈⠀⠀⠀⠈⡄
⠀⣠⣿⡟⠀⠀⠀⠀⣴⣿⡟⠁⠀⠀⢸⠀⠀⠈⣹⣿⡿⢿⣧⣀⣠⣾⣏⣀⣠⡀⠀⠀⠙⢆
⢰⣿⡟⠀⠀⢠⣤⣴⣿⣿⡇⠀⠀⠀⠈⣆⠀⣰⣿⡏⠁⠀⠉⣿⣿⣿⡇⠈⠙⠻⠀⠀⠀⠀⠣⠀⠀⠀⠀⠆⢴
⠀⢿⣯⣀⣿⣿⠇⠈⠛⢿⣷⣤⣤⣄⣀⣿⣰⣿⡁⠀⠀⠀⣀⣹⣿⣿⡀⠖⠛⠳⡄⠀⢀⠀⠀⢁⠀⠀⡘⢠⡇⠀⡠⠄⠀⠁
⠀⠀⠉⠁⠈⣿⣇⢀⣠⣴⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣦⠀⠀⠁⣤⣤⣶⣾⣾⣴⣄⣁⣴⡇⣠⣶⣦⣦⡄⢀⡠
⠀⠀⠀⠀⠀⠸⠿⠿⢟⢏⡉⠁⠀⢿⠛⠟⠟⠋⠉⠿⠛⠛⠗⠞⠋⠉⠛⢿⡿⣷⠿⣿⢾⠿⢿⡿⣿⢾⡿⠏⠻⠋⠹⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡟⠉⣽⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣾⣿⣿⣿⣿⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣇⣀⣈⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⢋⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠛⠻⠯⢍⡛⢍⠙⢯⢹⠉⠛⡙⣋⡽⢛⠿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⡋⠩⠬⠍⠑⠒⠲⢦⣬⣶⣷⣤⣥⣮⣬⡾⡡⡂⢕⡊⠁⣐⡉⠉⠙⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣤⣄⣒⣈⣉⠅⠀⡛⣿⡏⡢⣄⣀⠁⢹⣿⣾⣶⣿⣿⣶⣶⣒⣲⡶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⢻⡗⠧⠀⠐⠠⢹⡟⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡃⢀⣀⢀⣁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠷⠛⠛⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠠⠀⠀⡇⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡘⠀⠀⠘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡞⠀⠐⠀⠀⠀⠰⠈⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠁⡾⠀⠠⠀⢠⡟⠀⠀⠀⣧⠀⡴⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⢸⣾⠃⣰⡆⢰⢯⡇⠀⢠⣶⣿⡼⠁⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡇⠀⣿⡇⡟⢸⢠⣆⣞⣤⣿⡴⢚⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣸⡟⢿⢶⡇⣿⠃⣿⡿⣿⠟⣿⠏⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣏⠓⠈⠇⠁⠈⠀⠛⠀⠁⠀⠀⠀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣠⣿⡇⠀⠀⠀⠀⠀⡇⠰⠖⠘⡸⢙⡤⢤⢐
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠙⠋⠀⠀⠀⠀⠘⠐⠻⠈⠒⠀⠱⢎⣀⢒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⠀⠀⠀⠀⣰⣀⡀⢠⡴⡴⢤⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠖
⠀⠀⠀⠀⠀⠀⠀⣿⣾⢿⣽⠀⠀⠀⠀⠩⠸⠿⣿⡖⣲⡀⠿⡗⠳⠗
⠀⠀⠀⠀⠀⠀⣴⣯⣿⠿⣷⠆⠀⠀⠀⢴⣶⢶⡂⣟⣸⣶⣶⡶⠲⠶⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠀⡀
⠀⠀⠀⠀⠀⠐⣿⣯⣿⢿⣿⣾⠄⠀⠀⢸⡿⣢⣴⣷⣶⣶⣶⡔⠤⠖⠀⠀⠀⠀⡼⠀⠀⠀⠔⠀⡰
⠀⠀⠀⠀⠀⣼⣹⣹⣿⣿⢻⣿⣦⠀⠀⠐⣪⣼⣿⣿⠟⢻⣿⡇⠄⡀⠀⠤⠀⠈⠑⠀⢀⣴⠃⢀
⠀⠀⠀⠀⢠⣟⣻⣿⣿⢿⣿⢿⡾⢀⠀⠈⠁⠐⣿⠁⢀⡈⢻⣇⠈⡇⢀⡰⢄⣠⣦⣾⢟⠋⢀⠄
⠀⠀⠀⠀⡷⣿⣿⣿⡏⠈⣹⣗⣿⠟⠢⣤⡀⠈⡿⡬⠿⠧⣿⡇⠀⠃⠀⠀⠈⠀⠁⢸⠁⣆⠊⣀⡀
⠀⠀⠀⠀⢛⠻⠁⢘⠟⢱⡏⠛⠀⡟⢰⡄⢀⠠⡴⣦⣾⡧⠾⠭⣦⠀⠀⠀⠀⠀⠀⠈⡇⡃⠀⠈⣤⠰⡃
⠀⠀⠀⠀⢷⡤⠀⣰⠀⠈⠀⢀⣿⡟⣾⠁⣸⡇⠠⠾⠿⠷⠶⠶⠶⠶⠺⠁⠀⠂⡤⠈⠁⠀⣿⠀⢻⠞⣧
⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠉⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⠀⣤⣤⠄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠮⡥⠉⢡⡽⠷⡜⠁⣤⢾⣷⡈⠟⡃⣸⡷⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⢈⣧⣶⢶⡰⣟⣽⢳⣧⢻⣿⢯⣶⣷⣾⡽⣿⡿⠈⠽⠛⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠘⣷⣏⣏⡬⢿⣿⠛⠓⠈⠀⠀⢀⣀⣠⣤⣀⣉⡉⠁⠻⣿⣃⣴⣿⣼⠏⢤⡀
⠀⠀⠀⠀⠀⠀⣼⣃⡏⢰⣿⣶⠊⠁⠀⢀⣤⢞⡟⢋⣹⠯⣌⣿⣷⣿⣂⣦⠀⠐⡍⢻⣟⠁⢃⠷⡀
⠀⠀⠀⠀⠀⢸⡩⢮⡕⠆⣻⠁⠀⣠⡾⡹⢿⣭⡽⣶⣿⣿⡶⠟⡛⠈⠀⠁⠀⠀⠐⠀⠘⣿⣯⠚⡧
⠀⠀⠀⠀⠀⢺⣶⢄⡟⢈⣿⠀⠀⣿⣟⣷⢭⣍⡁⠻⣿⣿⣷⣋⢽⣀⡀⠀⢀⡔⠀⠈⠀⢸⡷⢾⡶
⠀⠀⠀⠀⠀⢨⡹⠾⣷⡈⠿⣦⡴⣿⡋⠀⠠⣄⠴⠷⣌⠳⣤⡘⢦⣈⠙⠓⠋⠀⠀⠃⠀⠐⡷⢶⡁
⠀⠀⠀⠀⠀⠀⢻⡄⠾⣛⠛⣫⣽⡏⠂⢀⣲⡃⠀⠀⠀⠙⠚⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⣼⡯⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠨⢙⠘⠛⠛⠊⠁⠀⣤⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⣸⡷⣷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠆⠐⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡞⣩⣹⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣻⢯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣠⠆⠳⣿⣵⡸⠏⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣤⣴⣴⢶⣾⣾⠉⢿⣿⣣⣾⣾⣿⣿⣿⣿⣿⡶⢦⢤⣀⡀
⠀⠀⠀⠀⠈⠀⠀⢠⡤⣦⢦⣼⠿⢛⡻⠟⣷⣀⣪⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣸⣿⣟⡷⢼⠯⠇⢀⣦⣄
⠀⠀⠈⢀⣤⣶⠿⠿⠳⣷⣿⠝⠓⠉⠉⢁⡴⢻⡟⣿⣿⡿⠟⠋⠉⡁⠀⠠⠠⠠⠠⠡⠉⠛⠷⣤⣶⢿⣮⣛⣉⣡⡄
⠀⢀⠶⣿⠯⣍⢿⣿⠟⠉⠀⠀⠀⠀⢠⣯⣷⣮⣛⡿⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢿⡃⠛⠋⢩⣿⡆
⠀⣼⠲⠞⠧⠙⣼⡏⠀⠀⠀⠀⠀⠀⣾⡌⢻⣿⣿⣇⡀⠀⠀⢀⣀⣠⣤⡔⢲⣖⣶⣶⣶⡄⣀⠀⠀⠈⡿⣿⣿⡷⣿⢛
⠀⣿⠛⡿⢷⡞⢿⠃⠀⠀⠀⠀⠀⠀⢺⣵⡆⢥⡄⣿⣷⠶⢋⡽⠛⢛⣯⠒⣩⡭⣬⠉⣷⠀⣾⣗⣄⠀⣷⣮⣁⠸⢿⡞
⠀⢹⣺⢳⣿⢏⡐⣜⡦⠀⠀⠀⠀⠀⠈⢷⣶⣗⣲⢿⡶⢃⣴⣲⣮⣅⣀⡼⠿⠿⠯⢿⣯⠈⢀⣡⣥⣿⣭⣻⣭⣧⡍⣠
⠀⠈⢿⣿⣥⢾⣿⢩⢮⢀⣷⣄⡀⠀⠀⠀⠉⠻⠿⢮⢡⣼⠽⠿⠛⠉⠁⠀⠀⠀⠀⢀⣨⣿⢮⡽⢿⣿⣆⣫⣭⠟⢠⠇
⠀⠀⠀⠻⣿⢛⣏⡘⢫⡠⣽⡶⢶⣝⠒⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣾⣯⠿⣉⡧⠔⠘⣇⢻⡟⠋⣰⠃
⠀⠀⠀⠀⠈⠙⡯⣿⣌⣱⣮⠿⣑⡛⢈⡳⢠⣾⢻⣿⠱⡴⢇⣾⣉⣻⣻⣿⣿⣿⣿⣿⣷⣯⣼⢀⡜⣿⣶⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠍⡙⠓⠾⢿⣿⣌⣡⣿⡴⢟⣙⡃⢀⣾⣟⣻⣿⣓⡿⣿⣿⣿⣿⠿⠞⢹⠌⢻⡾⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⢀⡉⠉⠁⠙⠛⠻⠿⠛⠿⠟⠟⠇⠃⠛⠉⢉⡀⠄⠒⡧⢀⣼⡃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠂⠀⠈⠈⠀⠀⠀⢜⡑⢿⢐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠃⡀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣠⢦⣤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢀⠶⣆⣰⣯⡿⠟⢻⣽⣿⣦⣅⠀⢀⡠⢲⣭⣵⣟⣻⡿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢽⣿⢿⣁⣀⣀⣀⣀⣩⣥⡿⣸⡆⢣⣵⣟⣻⣾⡡⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⣻⣖⣎⣝⢣⠮⣝⣿⣿⡆⡾⣷⡸⣃⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⣉⡛⣩⣟⡿⠛⠉⣿⣿⢚⡷⠟⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⣀⣼⣫⠇⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠒⠂

⠀⠀⠀⡀
⠀⠀⠀⠋⠀⠀⠀⠂⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠐⠄⠃⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠭⢹⠀⠀⠀⢘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⢁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠀⠄⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠹⠒⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠀⠀⠀⢰⠄⠀⢀⠀⠀⠠⢠⠀⠀⠀⢀⠀⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠐⠀⠀⠀⠀⠠⠉⠀⡀⠀⡄⠀⡀⠘⠀⠀⠘⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡈⡴⢶⡎⣀⠀⠠⠄⠀⠔⠁⠑⠄⠀⠁⠀⠀⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡙⡚⠣⡀⠀⠀⠀⠀⠢⠀⠠⠃⠀⠀⠀⠀⠁⠀⠀⠲⠀⣨⠡⡠⠆⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢴⣟⣳⠡⠠⠤⠀⠀⠀⠀⠀⠁
⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡄⠀⠀⠀⢀⠀⠀⣠⠐⠫⠁⠃⡄⠀⢠⠀⢀
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃
⠀⠀⠀⠠⢔⡀⠄⠀⠀⠤⠀⠌⢢⣀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠐⢨⢇⠜⡕⠂⠀⠀⣀⠀⠀⠈⠑⠀⠀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⠄⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⡱⠒⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⢀⡀⠀⢐⠀⠀⢀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠖⠒⢂⢤⣤⡀⠀⠀⠀⠀⢀⣾⠋⡇⠆
⠀⠀⠆⠘⠅⢢⠐⠀⡆⠏⢐⠀⠀⠀⠠⢃⡤⠤⠄⣀⠀⡄⠀⠀⠀⢀⣾⠟⠉⠀⠀⠀⠈⠡⢷⡄⠀⠀⠀⡎⣬⠊⡄⣷
⠀⠀⠙⠀⠀⢸⠠⠀⠃⠀⡄⠀⠀⠀⠀⠋⡆⠀⣴⡭⠿⠃⠀⠀⠀⠸⣿⣆⣲⡂⣀⣀⣠⣾⣿⠇⠀⠀⢰⢧⠉⠀⠇⡿
⠀⠀⠀⢂⣬⡥⠘⡄⡐⢀⠀⠀⠀⠀⠀⠀⠉⠀⣈⣀⢠⠄⠀⢀⠲⠀⠀⠉⠛⠛⠛⡛⠛⠉⠀⠀⠀⠀⠈⠛⠦⠴⡴⠃
⠀⠀⠈⠒⣎⣇⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⢠⠎⠉⠙⠈⠃⠈⡳⠟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠸⢻⣴⢠⡇⡞⠈⠀⠀⠀⠀⢀⣽⣤⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈
⠀⠀⠀⠀⠀⢸⣿⢸⣧⢧⠁⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠈⠿⠸⠿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣶⣒⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣶⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣀⣛⡋⠉⢉⣭⣙⣋⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⣷⣿⡟
⠀⠀⠀⠀⠀⡐⠚⣛⠛⠁⠙⠢⠀⠀⠀⢠⣶⠾⠟⠛⣛⣭⡦⣯⡈⠉⠉⠉⠉⠙⠛⠆⠀⠀⢀⣀⣀⡀⠘⠻⣿⠁
⠀⠀⠀⠋⠉⠀⠀⠈⠰⣳⠆⢀⣄⠀⠀⠈⠁⠀⠀⠹⠟⠚⠛⠛⣛⣠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠀⠀⠉
⠀⠀⠀⠠⠶⠻⠂⢀⡀⠙⠶⢯⠆⠀⠀⠀⠀⠀⠁⠀⠀⡀⢰⣟⢻⣿⠀⠀⣿⣦⣼⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⢷⠄⠀⠀⠀⠀⠀⠀⢆⣰⡀⠀⣦⡬⢿⣿⡀⠀⢿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠸⡇⠀⠈⠃⢸⣧⡆⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠤⠤⠖⠲⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠧⠀⠰⣄⣾⠛⠁⠠⠆⠀⠀⠀⠀⣀⡀⠀⢠⣿⣤⣄⣠⣤⣤⣾⡾
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⣡⣬⡿⠂⠀⠁⠀⡏⠉⠉⠋⠁
⠀⠀⠀⠀⠉⠓⠷⣤⢀⠀⠀⠡⠖⠚⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠢⠐⠒⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢈⠀⢐⣿⡿⠆⢰⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⡶⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⡅⠀⠀⠐⠿⠀⠰⠄⠀⠀⠀⠀⠀⣠⠀⠀⢀⠂⠐⡓⠻⠁⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢐⠇⠀⠁⠀⠆⡉⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠀⠀⢀⡐⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⠀⠀⢶⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⣀⣀⣼⠃⠀⣄⠠⠤⢶⢀⡉⠀⠀⢀⣦⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⣿⠀⢰⣿⠀⠀⠀⠋⠈⠁⠈⡤⠟⢻⣿⡇⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢻⡀⡙⣄⢼⣏⡀⠀⠀⠠⠑⢲⡁⠠⠾⠾⢿⡇⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠸⡇⠀⠘⣿⣦⣤⡀⠀⠨⠦⠀⠄⠀⠀⠉⠙⠁⠉⡏⣶⠂⠀⠀⠁⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢳⣭⣇⠀⢷⡀⠀⠘⣿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠱⣦⣤⣄⠀⠀⠀⢀⣾⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⢸⣿⣤⣀⠹⣿⣿⢷⣄⠀⠀⠀⠀⠀⠀⠰⠟⠛⠁⠀⢀⣠⡾⠟⠛⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠄⠀⠈⠛⠷⣝⣟⠻⣷⣟⣷⣤⡀⠀⠂⠁⠀⠀⢀⣠⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⣀⠀⠈⠛⠿⣿⣿⣏⣉⠀⠀⠀⢀⣠⣴⣿⣿⠋⠀⢀⣤⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠑⢦⣀⡀⠈⠛⢿⣿⣷⣤⣀⣀⣀⣴⡿⢁⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀⠀⠀⠁⣽⣦⣤⣤⠛⣩⣿⡿⠿⠿⠛⠂⠉⡉⠁⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠙⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠚⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢲⠞⠛⠉⠉⠉⠙⠛⡖⠲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠉⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⡀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠤⡀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡡⠔⠋⠀⠀⢀⡤⠐⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠙⡀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⣀⠤⠐⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠶⠢⢄⠀⢳⡀
⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢣⠀⢳⡀
⠀⠀⡾⢀⠔⠋⠉⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⢀⡎⠀⠈⢷
⠀⣸⠃⡎⠀⠀⠀⠀⠀⠀⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠒⠉⠀⠀⠀⠘⣇
⢀⡟⠀⢇⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⢸⠃⠀⠈⠲⢄⣀⣀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣾
⡟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⠄
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡞
⠘⣧⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⣣⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣀⣀⣀⣀⣴⠋⠀⠀⠀⠀⠀⡠⠀⠀⠀⣀⠀⠀⠀⠀⣴
⠀⠈⠓⢦⣄⣀⣀⣠⠬⠿⠶⠖⠒⠚⠛⠉⠉⠉⡏⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⠉⠉⠙⠛⠓⠒⠛⠶⠤⠴⢾⣁⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣹⣦⣀⣀⡀⠀⠀⢀⣀⣠⡼⢧⡤⠴⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⣀⣤⠶⠂⠐⠁
⠀⠀⠀⠀⠀⠀⠤⠤⢀⢀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⡔
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠄⡀⠂⠀⠄⠀⢀⠀⢘⡀⠀⠀⠀⡠⠀⠀⠨⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠊⢁⡀⡋⠁⠀⠈⠂⠀⠀⢠⠃⠀⠀⠉⠀⢤⠈⠀⠀⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢄⠀⠀⠀⢐⢯⠀⠀⠀⠀⠖⠀⠀⣀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣶⣾⣧⣜⢦⠀⠀⣞⠀⣬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠁⠀⠀⠀⠈⠙⢦⣠⠻⣧⣾⠇⠀⠀⠀⠀⡀⠀⠠⠂⢀⠀⠀⠀⠺
⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠇⠀⠀⠀⢰⣧⣼⡦⢿⣦⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠠⠀⠀⠀⠀⠁
⠀⠹⢄⠀⠀⠀⡄⠀⢳⠀⠀⠀⠀⣠⡆⠀⣨⣍⣭⠖⣩⡿⠉⠀⠀⠀⠀⢀⠀⠀⠀⣀⠶⠁⠀⠀⠐⠁
⠀⠀⠀⠛⠢⡈⢧⠀⠘⠀⠀⠀⠀⠉⢃⠀⠙⠛⢿⣶⣿⡇⠀⠀⠀⢀⠔⣙⣀⡚⣀⢠⠀⠀⢀⣠⡦⣉⣠⠊
⠠⡄⠀⢀⠻⣤⠿⠗⠀⠘⠄⠀⠀⠀⠀⣠⠋⠀⠀⠀⢸⡷⠀⣠⡴⠟⠛⠛⠙⡁⠉⣠⣦⣴⣿⢿⠔⣠⠃
⠀⠀⠀⠀⠄⠻⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢠⡄⣦⡀⠸⢿⠟⠢⢀⠀⠐⠗⠀⠻⠛⠋⣹⣿⡿⠀⢀⡿⠤⠄
⠀⠀⠀⠄⠈⠂⢒⣇⠀⠀⠀⠠⠀⣼⣷⠄⠘⠿⠟⢹⣶⣿⠂⠀⠀⠀⠀⠀⠠⡀⠀⠀⠸⡟⢐⠀⠈⠁
⠀⠀⠀⠀⠐⠌⣻⣿⣦⡀⣠⢤⡾⠋⠀⠀⠀⣶⣤⣷⡟⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠹⣍⡻⠦⠈⠁⠐⠒⠀⠀⠈⠉⠉⣶⣤⣿⠀⠀⠀⣀⠀⠀⠠⠐⢀⠀⠀⠀⠝
⠀⠀⠀⠀⠀⠀⣶⡿⠃⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢠⡞⠉⠁⠀⠀⠀⠛⢠⣤⣀⠀⠀⠙⠀⠀⠀⢀⡠⠔
⠀⠀⠀⠁⠀⢸⡟⡤⠄⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢠⣿⢿⠣⣀⣀⡈⢒⠀⠐⠀⠀⠀⡀⠀⠊
⠠⠒⠂⠀⣰⠏⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠂⡾⠀⢀⣄⣶⣤⣀⣦⢀⡾⠁⡌⠸⠛⠉⠠⠘⠋⠀⠀⠄⠘
⠀⠀⠀⠐⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢈⣻⡿⠍⢻⣿⡊⠀⠀⠀⠊⠁⠠⢄⠀⡠⠒⠀⠉⠛
⠀⠀⠀⢇⠄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⡁⣨⡴⠚⠍⠉⢐⠦⣄⠈⠑⢦⡀⠀⠤⠒⠉⠉⠑⠒
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠍⠀⢾⣦⡷⠻⣾⡿⠀⠀⠀⠁⠀⠢⣤⣞⡁
⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠷⠲⢤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⢠⣶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢬⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡦⠄

⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢠⡞⠁⠐⠂⠬⢍⣒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢖⣋⠭⠄⠂⠈⠙⣦
⠀⠀⠀⢸⡖⠚⢋⣉⣓⡒⠦⢝⡮⠷⣄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠴⠭⣞⠥⠖⣚⣋⣙⠛⠒⣿
⠀⠀⠀⠈⢷⡉⣉⠀⠨⢭⣍⣻⡄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣼⣋⣭⠍⠀⣈⡉⣸⠇
⠀⠀⠀⠀⠘⢧⡠⠤⠖⠒⠒⠒⠛⠦⣄⠀⠀⠙⢦⠀⠀⠀⠀⣠⠞⠁⠀⣀⡴⠞⠒⠒⠒⠒⠦⠄⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣐⠊⠉⢉⣉⣉⡙⠛⠛⠒⠦⣄⡳⣐⣢⡴⣁⡤⠖⠛⠛⠛⣉⣉⣉⠉⠉⢒⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠧⣤⣤⠬⣭⣭⣭⠯⠭⢽⣏⢘⣿⠭⠭⣯⣭⣭⡭⢤⣤⡬⠿⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣁⠤⠄⠐⣶⠤⠒⠊⡝⣿⢾⡿⡑⠒⠤⢴⡒⠀⠤⢄⣉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⣉⠤⠒⢈⡏⠀⠀⣠⢾⢀⡏⡘⡇⠹⢆⡀⠀⠙⣏⠐⠢⢌⠓⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢀⠤⠂⡩⠳⡴⢺⠁⠀⣸⢀⢃⢹⠸⠀⢻⠲⡴⠫⡁⠢⣄⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⢠⠊⠀⡜⠀⠇⠀⢰⠇⠈⠃⠈⣿⠀⠈⡄⠑⡀⠈⢦⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣇⠀⡘⠀⠀⡆⢀⡞⠀⠀⠀⠀⠸⣆⠀⠇⠀⢸⠀⢈⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠦⠶⠚⠋⠀⠀⠀⠀⠀⠀⠈⠛⠶⠤⠶⠚⠉

⠀⠀⠀⠀⠀⠄⠀⠐⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠻⢤⣤⣤⣴⠟⠁⠀⠀⣠⡄⠀⣄⠀⠀⠀⠀⠀⠀⡔⠃⠀⡠⠀⢀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢪⡀⠀⠀⠈⣵⠀⠀⠀⠀⢰⡷⠟⠟⣛⣺⡿⢷⣃⡇⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠰⠀⠀⠀⡇⠀⠀⠀⠉⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠐⠚⠁⠀⠳⠀⠉⠁⠀⠘⣧⣈⡓⢄⠀⠂
⠀⠀⠀⠀⢀⠃⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠁⠀⢹⡏⡇⢀⡀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠺⣧⡀⠀⠐⠈⠻⣄⣾⡿
⠀⠀⠀⠀⠈⠳⠦⣤⡤⠔⠁⠑⠤⠀⢠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⣠⢍⠙⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣙⣌⣷⠂⠘⠰⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⡀⡽⠒⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡉⠉⠁⠀⠀⠁⠠⡔⠷⣄⣀⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⣼⢥⣤⠽⠃⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⢠⠊⢉⡷⠐⠀⠐⡀
⠀⠀⠀⠀⠀⠠⠖⠀⠸⠃⣯⢱⢠⡇⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢸⡯⣘⠇⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠓⢤⣄⣀⣀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⣱⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⠁⠀⠀⠈⠁⠀⠀⢀⢖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢹⠘⡇⡆⠀⠀⠀⠀⠀⡀⠀⠀⢀⣔⡊⠀⢲⠀⠀⠀⠰⢤⣨⡇⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠂⢸⠀⣿⠁⠀⠀⠀⠀⠀⠉⠙⠉⢁⠉⡉⠙⠋⠀⢐⢂⠀⡔⢠⠇⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⡾⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢴⡀⠑⢄⣠⠋⢀⠟⠁⠀⡈⢄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⢀⣃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠃⠀⡘⢀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⠀⠈⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠒⠒⠀⢀⠖⠒⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⡁⢀⠀⠀⠔⢢⠀⠀⠁⠀⠀⠀⢀⡤⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡀⠄⠀⢁⠉⠀⢀⠀⠀⠀⡠⠄⠀⢀⠀⠀⠣⠤⠇⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⠔⢁⠄⠂⠀⠀⠀⠑⡝⠂⠀⠀⢇⠀⠀⣸⠆⠀⡀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠁⡌⠀⠈⠀⠈⢠⡰⠀⡇⢰⢀⡤⣄⠀⠈⠉⠁⠀⠀⠈⢁⣠⣾⠏
⠀⠀⠀⠀⠀⠀⢠⣄⠳⣄⠀⢰⡀⠐⠤⠀⠔⢁⡞⠘⠤⠎⣀⣤⣶⣶⠾⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⢸⠈⠩⢐⡉⠈⠱⠦⠤⠤⠴⠊⢀⣠⡶⠟⠛⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⢀⣀⣤⠶⢺⡏⠁⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢧⢸⠀⢠⡄⢀⠴⠂⠀⢸⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢸⠈⡷⢱⡶⢁⡧⢲⣣⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡈⠁⠘⠁⢈⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠾⠿⠟⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⡴⠶⠶⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⡰⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⡀
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⣤⠒⠋⠉⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡀⠬⠳⡀
⠀⠀⠀⠀⡼⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠀⢤⡀⠀⠀⣠⡇⠀⠲⣄⠀⠀⣠⡟⣡⡞⢀⡀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⣇⢠⣀⠀⣀⣹⢦⡀⠻⣄⣴⠟⡇⠀⠀⠹⡳⡜⡝⡴⢹⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⣿⢀⣙⠻⡦⣹⣾⣌⡦⠾⠴⠴⠣⠶⣦⣄⣸⢠⡷⢁⠟⣿⡤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⢹⡀⠉⢳⣝⣜⢿⠏⠀⠀⠀⠠⠊⠀⠈⢷⡇⢸⡇⡜⢸⣿⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⢱⠛⠍⠉⢻⣾⡀⠀⠀⠀⠀⠀⠀⣀⣼⣇⣸⣷⠋⠉⠉⣹⠉⢹⡄⠀⡇⠀⢀⣀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠓⠚⠁⠀⠉⠑⠚⠉⠉⠉⠉⠀⠀⠈⠁⠈⠓⠒⠚⠉⠀⠉⠉⠉⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⡀⡀⡀⠀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠛⠓⠲⠶⠶⠠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⢀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠐⠉⠁⠀⠀⠀⠛⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣠⡤⠖⠀⠂⠀⠀⢸⠀⠀⠀⠀⣠⣤⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠀⠀⠀⠀⠀⢀⣀⠼⡄⠀⣰⣾⡿⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠒⠚⢛⣻⣿⣟⠀⢸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠃⠡⠭⠩⠀⠸⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡏⠀⠀⠀⠘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠿⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣡⣤⣤⢌⠀⢀⠀⡀⠀⠐⢠⡾⠁⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠺⠀⠿⠟⠉⠛⠀⠀⡿⢗⠁⣠⡤⣈⢈⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠻⠏⠀⠹⠗⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⢰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣧⡀⣀⡼

⣿⣟⣛⣟⣛⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣀⠀⢀⣀⡀
⠀⠀⠿⠯⢠⣾⣧⡀⠐⠀⠀⣾⠁⢨⡷⠀⣴⢒⣦⠀⢉⡀⢰⢿⠂⣰⡿⣛⠻⣇⠀⡞⠀⠈⢳⠋⠀⠙⡆
⠀⠀⠀⣰⣿⡟⣿⣷⡀⠀⠀⠹⢤⣀⡀⠀⢻⣌⣉⢰⠫⣽⠂⠀⠀⠹⣷⣭⣾⡏⠀⠳⡀⠀⢺⠀⠀⡸⠃
⠀⠀⣴⣿⣷⡅⣥⣿⣿⣄⠀⣶⠆⢨⠟⣉⠉⠉⡉⠙⢳⡀⠀⠃⠀⠐⢶⠭⠶⠆⠀⠀⠙⢦⣸⣠⠞⠁
⠀⠈⢉⣽⡿⠇⡿⣿⣏⠉⠀⡀⠀⠘⠦⠿⠀⠀⣉⣀⣰⣧⣄⣀⣀⡘⠻⠖⠿⠃⢀⣤⡀⠀⠙⣡⡶⣤⡀
⠀⠠⠾⢿⣿⡆⣾⣿⡿⠧⠀⡁⢀⠀⠀⣀⣶⣀⠙⣿⣿⣿⣿⣿⡿⠟⠀⠄⠀⠀⠈⠛⠃⠀⠸⣕⡿⣫⡇
⠀⠀⣠⣿⣯⡃⢋⣿⣷⡀⢸⣎⣹⡁⠐⣺⣿⣗⠂⢸⣿⣿⡟⠁⠀⢠⡄⢶⢦⣤⠀⠸⣿⠟⠀⢉⣉⠉
⠀⠐⢛⠛⠛⣷⡟⠛⠛⠛⠈⠑⣋⡁⠀⠀⠛⠀⠀⢸⣿⣿⣧⠀⢸⡗⢠⣶⣦⠐⣷⠀⠀⠀⣰⢿⣿⡷
⠀⠀⣉⠠⡄⢉⠁⠀⣠⡀⠠⣾⣿⣿⣆⠀⠀⠙⠀⢸⡿⡟⡿⣧⠘⢫⡄⣩⣡⣬⠋⠀⢼⣿⣇⣠⡿⠃
⠀⢠⣭⡷⢳⣯⣄⠘⠿⠏⠀⠘⢿⣄⣸⣿⣯⠄⢀⣿⡁⠀⠀⣽⣧⠀⠀⠉⠁⠀⠀⠄⠀⠀⠉⣉⠀⢠
⠀⠀⠴⢋⡏⠳⠀⠀⢀⠀⠀⢀⣀⣉⣉⣉⣤⣶⣿⡿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣦⠀⠀⢀⣼⣿⣦
⠀⠀⠀⢠⣤⡄⠘⠁⠘⢷⣾⣿⣿⣿⠿⠻⣻⣿⡃⠀⢐⣻⣿⣿⡿⠛⠉⢀⣿⠏⣿⡆⢠⣾⣯⣩⣿⣷⡀
⠀⢰⣿⡟⣙⢻⣷⠄⠀⢸⣿⣿⣿⣧⣀⢀⢸⣿⣷⣦⣷⣿⡿⠋⠀⠘⠃⢸⡿⠀⣿⠇⢀⣼⣿⢙⣿⣧⡀
⠀⠀⢽⣧⣭⣾⡏⠀⠶⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣁⡤⢤⣀⠀⣰⡞⠁⣴⠟⠀⠀⣼⣟⠛⣿⣇
⠀⢰⣶⣭⢀⣭⣶⡆⠀⢰⣿⣿⣿⣿⠋⠉⠀⣠⣀⠀⡼⢃⢮⣷⡂⢧⠈⡀⠀⢡⡀⠀⠈⠉⠉⡿⠉⠉⠁
⠀⠈⠻⠿⠿⠿⠛⠀⢠⣿⢣⣿⡟⠁⠖⠀⠀⠛⠋⠀⠸⣆⡛⢟⣂⠏⠈⠁⣰⣿⣿⡄⢀⡄⠀⠠⡞⢳⠄
⠀⠀⠠⠀⠀⠒⣠⣴⡿⠃⣼⡟⠀⣰⠚⣶⣴⣾⣶⡀⠀⠀⣍⢉⠀⠀⠀⠐⣿⣧⣾⡟⠂⠀⢀⡀⢉⠉
⠀⢠⡶⣲⢶⣰⡿⠋⣠⣾⠟⢁⠀⠹⣾⣿⣿⣿⡿⠀⠐⢶⡿⢻⡷⠂⠀⠰⣿⣧⣾⡿⠂⠐⠶⢟⡻⠶⠂
⠀⠸⣞⣿⣳⠇⠀⠀⡿⠋⠐⣏⣱⠀⠘⢿⡿⠋⠀⠆⠐⠛⡿⢿⠛⠃⠀⠰⠛⢧⡾⠿⠂⠐⢋⡶⢾⠙
⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⢒⡲⠶⠶⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣶⢿⣯⣷⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⡀⠄⡾⢣⡤⠚⠁⠀⠀⣀⣀⠀⠀⠱⡀
⠀⠀⠀⢀⣤⣶⣞⣻⡟⡽⣿⣿⣿⣵⣿⣿⣿⣿⣿⣦⣶⣭⢻⣿⠀⠀⠀⠀⢀⡴⠚⠉⠰⠤⢰⠿⠒⢀⠔⠉⡀⠀⠀⠀⡀⠈⠉⠙⣦⠀⠁
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⢀⠔⠁⠄⠀⠁⣀⠀⠀⠀⠀⠐⠀⢁⡄
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⣬⣏⠀⠚⠂⠀⢀⠀⠈
⠀⠀⠈⢿⣿⣟⣿⣿⣿⣿⣿⣯⣿⣿⣷⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⡲⠄⠀⠀⠀⠈⠉⢆⠀⠈⠁⠀⠂⠀⠄⠈
⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⢿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠤⠄⠠⠈⣄⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣷⣄⣀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠁⠙⢻⢦⣴⡆⢤⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣭⣯⣿⣻⡿⣿⣷⣿⣿⣿⣿⣼⠶⠚⣭⣦⣤⠀⠀⠀⠀⠀⠀⠀⠰⢷⢛⠘⠁⡆⢤⡀⠈⠁⠈⠀⠠⢩⣽⣿⠛⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣾⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⠀⠡⠀⠉⢑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣯⣯⣹⣟⣿⣯⠿⠃⠀⠀⠀⠀⠀⠀⢤⠀⠀⠁⠠⢀⠀⠀⠀⠀⠀⠀⠘⠙⠒⠲⠒⡒⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⢻⣿⠛⠟⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠉⠀⠀⡀⠂⠀⠒⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⢀⠀⠀⠋⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢦⠀⠀⠀⠀⠀⠀⠈⠀⠀⢁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⡄⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⡁⡀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡄⢠⠀⠀⠀⢠⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣿⡿⣿⣿⣿⣿⣿⣷⡷⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⢧⣹⠀⢴⡀⢸⠀⠀⠈⠢⣑⣤⡤⢄⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠙⠠⠟⢿⣿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠐⢌⣛⣮⣷⣿⣾⡄⢀⠟⠋⠁⢐⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠑⢺⣇⣀⣀⣀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣘⣿⣿⣿⣿⣿⣷⣦⣍⢁⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⡖⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣦⣤⣤⣄⣀⣀⡀⠀⠀⠀⢀⠀⢀⣱⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣤⣤⣶⣾⣿⣿⣿⣽⣭⣯⣧⢽⣿⣴⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣞⣿⣿⣿⣿⣿⡿⠉
⠀⠀⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠁⠀⠻⠻⢿⣿⣽⣿⣿⣛⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣽⣾⣿⠿⠋
⠀⢺⣿⣿⣿⣿⣿⢛⣻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣯⣭⣿⣟⣿⣿⣿⣿⣿⠙⠛⠟⠛⠉⢀⠀⠀⠁
⠀⠀⠁⠻⣿⣿⣿⣶⣾⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣼⣿⣹⣿⣿⣿⣿⣾⣿⣿⣿⠿⠟⠙⠉
⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣛⣉⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⡀⠀⠀⠘⠻⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠘⠛⠶⣦⣤⣀⣠⣝⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⡂⠀⠀⠀⢀⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠉⠀⠉⠁⠈⠉⠉⠉⠉⠉⠋
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠓⠚⠋⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠼⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⣤⣄⣤⣴⣤⣶⣤⣶⣿⣶⣷⣶⣾⣶⣶⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⠿⠿⢿⢿⠿⢿⠟⡿⡛⡻⠟⠛⠙⠛⡙⢋⠛⠛⢟⠟⠻⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠉⠙⠋⠋⠘⢓⡁⠈⠈⠃⠀⠀⠁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠠⢜⣅⣹⣿⣿⣿⣿⡟⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠀⠀⠁⠁⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠐⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠠⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠀⠠⢳⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠉⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⡄⠀⠀⠀⠀⠀⠀⠀⣂⠸⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⣼⣇⠀⡀⣀⣠⡴⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠛⠻⠟⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⣀⣽⣿⣿⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⠟⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⣙⣿⣿⡟⣿⡟⠗⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠹⣿⣿⣿⣿⠟⠛⠙⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⠹⡟⡇⠔⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡰⣶⣿⣿⠿⢁⣀⠀⢶⣿⣿⡿⠟⠛⣿⣿⡿⠿⠛⠉⠈⠀⠀⠀⠈⠀⠀⠀⠢⠀⡀⠀⡀⠐⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡆⢸⡿⠿⠖⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠄⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣠⡄⠀⠐⢿⣿⣿⣿⣿⣷⣾⠿⡛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣏⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀⠀⠀⠀⢀⠀⠖⢠⡄⢰⣾⣿⣿⣾⣇⢩⣾⠿⠟⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠙⠲⢤⠄⠀⠀⠀⠀⠀⠀⣾⣿⣼⠋⢀⠄⣤⣴⣾⣿⣞⢿⣿⡿⠟⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⡆⠨⣿⣿⣿⠿⠛⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠉⡙⠟⠛⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣼⠁⠀⢠⠀⠀⠀⠀⠀⠀⠋⠉⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠰⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠠⠤⠖⠁⠀⣀⣀⢀⠤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⢶⡶⣿⣿⣿⢿⣿⣿⡆⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⢿⣻⣿⣿⣿⣷⣿⣿⣿⠁⠀⠄⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣰⡟⠉⢋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⢢⠁⠀⠀⠈⠀⠀⠀⠀⠀⣀⠀⣶⡄⣼⣿⡟⠠⢃⡀⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣼⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠂⢀⠄⠀⠀⠀⡀⠀⣠⣿⣿⣿⣿⣧⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠿⡿⣿⠀⠀⠱⠀⠀⠀⠀⠘⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠿⠶⠐⠃⢸⡆⠀⠃⠀⢰⣠⠀⠀⡐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣄⣀⣰⣠⣴⣿⡇⠀⢠⠀⠀⣿⡀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠘⢧⠀⠙⣿⣄⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢸⣧⣀⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠉⠙⠛⠶⢶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⣦⡀
⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠰⠆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠁⠀⠀⢀⣠⡶⠟⠋⠉⠁⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠞⠋⠁⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⣦⣤⣴⣶⠶⠒⠚⠛⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⢀⣠⡴⠖⠒⠶⣄⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⢀⣀⠘⣧⠀⠀⠀⠀⠀⣰⠏⠀⢀⡀⠄⠀⢀⡴⠛⠁⠀⠀⠀⠀⠹⠄⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⠶⠶⠖⠋⠉⠳⠿⣷⣤⣀⡀⣸⠃⠀⠀⠈⠁⠀⣰⠏⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠀⢀⡼⠁⠀⠰⡄⠀⡏⢀⡴⠂⠈⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠹⣆⠀⠀⠀⡾⠁⠀⠀⠀⢳⣸⣧⣾⣡⣚⣥⠀⠀⣰⠏
⠀⠀⠀⠀⠀⣰⡟⠁⠀⣀⡴⠞⢋⠉⠀⠀⣠⠀⠀⠀⠀⠉⠛⠦⣄⠀⠀⠘⢧⠀⢸⠇⠀⠈⠓⠶⣼⠋⠁⠉⠙⠻⢦⣤⣴⣏⣀⣠⣤⣤⡀
⠀⠀⠀⠀⣸⡏⠀⠀⡴⠋⠀⠀⠀⢳⡀⠀⣿⠀⠀⢀⡴⠁⠀⠀⠈⢳⡀⠀⠘⡇⢸⠀⠀⠀⢠⠤⠿⣤⡀⠀⠀⠀⠀⠈⠉⠉⣩⠟⠁⠙⣷
⠀⠀⠀⠀⣿⠀⠀⣼⠁⠀⠀⠀⠀⠀⣹⡦⢧⣄⣠⠏⠀⣀⡤⠖⠀⠀⣧⠀⠀⣿⠘⣇⠀⠀⠀⠀⣰⠋⢹⢦⡀⠀⠀⠀⢀⡞⠁⠀⠀⢀⡿
⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠈⠉⠉⢹⡇⠀⠀⠈⢻⡤⠞⠉⠀⠀⠀⠀⣿⠀⢀⡟⠀⠘⢦⠀⠀⠀⠀⠀⠉⣠⡿⣦⡀⠀⡾⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠹⣆⠀⠸⣆⠀⠒⠋⠉⣽⠁⠀⠀⠀⡼⠤⠤⢤⠀⠀⠀⣰⠃⠀⣼⣅⠀⠀⠀⠀⠀⠉⠀⣠⡾⠋⠀⠈⢻⣤⣇⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠙⠷⣦⣬⣙⣛⣷⣾⣇⡀⠀⠀⠀⣇⡀⠀⠀⣀⣠⠔⠃⢀⡾⠋⠙⠷⢦⣤⣤⡴⠶⠟⠉⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠉⠀⠀⠈⠉⠳⢦⣀⣯⠉⠛⠋⢉⣁⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠂⣀⠀⠀⠈⢿⡟⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⡀⠀⠀⠀⠀⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣦⡭⢉⣵⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠻⠘⠟⠛⠋⢀⣀⣤⣴⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⣉⣉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣉⠀⣾⣿⣿⣷⠈⠉⠉⠉⠉⠉⠉⠉⢉⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣧⡘⠻⠟⢋⣴⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⣶⣶⣤⡙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠃⢸⣿⣿⣿⣿⡧⠈⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⡈⠿⣿⣿⡿⢃⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠻⢀⠘⡀⠃⠀⠶⠧⠊⠃⢡⡴⠀⠶
⠀⠡⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣼⠁⠀⠠⣤⢠⣆⣃⠦⢯⢡⠑⣄⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠉⠓⠁⠈⠀⠀⠘⠑⠀⡄⠉⠉⢠⠀⠊⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠩⠁⠀⠒⠐⠊
⠀⠀⢠⡀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢰⡀⠡⠀⠀⠀⠂⠀⠀⠀⠠⣤⡄
⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣦⢤⡶⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠤⡨⠂⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢰⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠑⠀⠀⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡤⠔⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⠶⢶⣄⡀⠀⠠⠀⠄⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣽⠇⠀⠀⣐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⢀⡀⠀⠀⠀⠦⠹⠴⠋
⠀⠀⠀⠀⢷⡶⠃⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠻⡆⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠖⠀⠀⠀⠀⣴⡀⠀⠀⠓⠚⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠉⠉⠤⠄⠈⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠟⡆⠀⠀⠀⠀⠀⠀⠉⠀⡄⠀⠀⠀⡰⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⠀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠑⠐⠂⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠆⠀⠀⠀⠀⢄⣶⣿⣿⣤⣰⣿⡇
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣟⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣿⣿⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⠀⢀⠀⠐⢠⡄⠀⠀⠀⠠⣸⣀⠤⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢄⣸⣠⠄⠀⠀⠀⢀⡄⢂⠀⡀⠀⢀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠱⠘⠀⠀⠀⠠⠄⣻⠄⠑⠆⣲⢒⡐⠂⠀⠐⠛⡝⠒⠀⠈⠒⣀⢀⠰⢀⠄⡠⠎⠀⠐⠊⡍⠓⠀⠀⠀⢂⡚⢖⠀⠎⠡⣞⠠⠄⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠉⠀⠀⠂⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠲⢈⡁⠆⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠐⠀⠀⠈⠁⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠠⠀⠀⠀⠘⢘⡠⢣⢜⡳⠜⠀⠁⠁⠈⠀⠀⠄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠛⠀⠀⠄⡹⠃⡭⠀⠀⠀⢀⠐⠑⠠⠍⠂⠑⢦⡀⠀⠀⠨⢐⠟⡠⠀⠀⠛⠀⢀⠀⠀⢠⠀⠀⠀⠀⡀⠠⠄
⠀⢀⠀⢿⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠐⠂⠀⠀⠐⠄⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⡠⠀⠀⠀⠀⠀⠂⣼⠆
⠀⠀⠂⠠⡄⠀⠀⡀⠀⠀⠠⡇⠀⠆⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠰⠆⢨⠅⠀⠀⠀⠀⠀⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⠤⠢⠀⢠⠀⠀⠐⠀⢰⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⣆⠀⠀⠀⠀⢠⡆⠀⠀⠀⠈⡠⠀⠔⠤⠤⠄⠀⡄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠐⠖⠂⠒⠀⣷⠀⠀⠀⠑⣄⠀⣀⠞⠁⠀⠀⠀⠀⠈⠳⣄⠀⣀⠌⠀⠀⠀⣴⠀⠔⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠒⣀⠋⠰⠀⠀⠙⠛⠀⢣⠆⠀⠀⠀⡄⠀⠀⠐⡞⠂⠛⠊⠀⠀⠒⠘⣀⠐⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⡄
⠀⠀⠀⠀⢤⠘⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠌⠀⠀⠀⠀⠀⠘⠁⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⣠⠁
⠀⠀⠀⠀⠀⡀⠊⢠⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠤⠑⢂⠄
⠀⠀⠀⠀⠀⢀⠐⠆⢀⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠘⠢⢜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⡰⠀⠀⠀⠀⡀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢃⠔⠛
⠀⠀⢠⡄⠀⠀⠀⠉⡈⠂⡑⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⠐⢃⡈⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⣶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢰⠀⠈⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠤⠄
⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠
⠀⠀⠀⠄⠄⠡⠄⠤⠀⠀⠠⠀⠀⠂⠀⠀⠀⡈⢀⣀⠀⠄⢂⠀⠀⠀⠀⣀⡰⠤⡀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠃⠀⠀⠀⠀⠂⠀⠄⠀⠀⠤⠠⠤⠀⠠
⠀⠀⠀⠐⠈⢀⠂⠀⠀⠀⠀⠘⢣⣽⡀⣁⢒⢡⡤⠭⠒⢠⠊⠉⠀⠀⠀⢉⣇⣀⠟⠀⠀⠀⠘⡙⡄⠐⠭⠤⡌⠐⣀⢀⣼⣔⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠐⠋⣕⠉⠉⠀⡃⠠⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⠘⡉⢢⠙⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⠔⢲⢺⠀⡉⠀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡗⢒⠢⠔⠂⠀⠀⠀⠠⡄
⠀⠀⠀⢈⣰⠀⠀⠀⠀⠂⠒⢋⠁⡈⠐⠀⠀⠀⠀⠀⠀⠀⠗⠔⠃⠠⠄⠀⡀⠀⠀⠃⠘⠀⠃⠌⠀⠀⠀⠀⠀⠀⠈⠐⢉⠀⡉⠒⠀⠂⠀⠀⠁⣦⠀⠆
⠀⠀⠀⠈⡃⡄⠀⠀⠀⠈⠄⢨⠱⡀⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⡁⠠⠁⠀⠀⠀⢁⢌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⠀⠄⠀⠄⠀⠠⠀⠀⠐⠀⠀⠀⠄⢰⡈⠣⢀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠓⡄⠀⠀⠀⠀⠐⠂⢲⠀⣤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⠀⠁⠀⠀⢀⠞⠁⠃⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣦⣶⡇⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠋⢟⠁⠃⠀⠛⠷⠈⡘⠛⠻⢀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠁⠀⢀⠈⠁⠘⣏⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠈⠁⠀⠀⠿⠄⣾⣧⣢⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢧⠘⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡋⠉⠻⠍
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡟⠀⠁⠀⠀⠀⠀⠀⢀⡤⠈⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠄⣰⠀⡀
⠀⠀⠀⠀⠀⣠⢠⠐⠏⢀⠀⠸⠆⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠈⢷⡗
⠀⠀⠀⠀⠀⢸⣿⡀⠀⢿⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠈⡣⠀⠀⠀⠀⠀⠄⠀⢸⣷⡄
⠀⠀⠀⠀⢰⣿⡇⠁⠄⠠⠅⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠐⠃⠀⠀⠀⢘⣀⣀⣤⡶⢿⣿⠿⠋⠉⠀⠰⠀⢣⠀⣬⣿⣷
⠀⠀⠀⠀⣸⣟⠃⠰⠀⠀⠀⠀⠀⠀⢰⠿⡞⠉⢀⣤⣤⠤⡴⢲⢺⣿⣿⣿⡿⣺⠮⠑⠈⠀⠀⠀⠐⠁⢀⠚⠀⢸⣿⣿
⠀⠀⠀⠀⢸⣿⡇⠐⠀⠀⠀⠀⢀⣰⣟⣀⡤⠾⢉⣻⣿⣷⣼⣾⣿⣿⣿⢟⣁⣴⣮⣤⣤⣤⡠⠀⠀⠀⠀⠈⡇⣼⣿⣿
⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⣰⣿⣿⣿⡿⠷⠚⠛⠭⠝⣫⠿⣋⣉⣁⢀⣴⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿
⠀⠀⠀⠀⠘⣿⣿⠘⠀⠀⠀⠻⢿⣿⣿⣶⣄⠀⠀⣠⣶⣷⣿⣿⣿⠋⠋⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⡀⠀⠈⠉⠉⠉⠉⠉⠋⠁⠀⠈⠀⢹⡄⠀⠀⠀⠐⢤⠀⠂⠀⠀⠀⠀⠀⠀⣸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢿⣿⣖⠤⣀⠁⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⠘⡇⠀⠀⢀⡤⠖⠁⠁⠀⠀⠀⠈⢀⣤⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣬⠁⢄⡈⠂⠈⡆⡄⠀⠀⠀⠀⠀⡆⢀⣧⠀⢠⣿⣿⣀⠀⠀⠀⠀⠀⠚⠁⣿⠁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡀⣘⡓⢠⡔⢸⣿⣇⣀⣤⡄⠀⡃⠀⢹⡷⠀⠹⡿⢷⣦⣤⣀⣀⣀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡩⠷⣶⣿⢛⣿⣿⢛⠻⠿⣵⡃⠀⢸⠇⠀⠄⠀⣼⠋⣿⢈⢁⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣾⣿⣎⠙⠻⢠⣠⣾⣷⣦⣴⡂⠀⢀⣰⣇⣈⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠂⠸⠿⣿⡿⠿⠿⠿⠴⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⣠⠐⠁⠀⢀⠁⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠉⠁⠒⠒⠂⣄⣀⡀
⠀⠀⠀⡀⠎⠁⠀⢀⠀⢀⠀⠀⠀⢙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠀⠂⢄⠀⠀⣀⣀⣀⣀⡀⠀⣀⣉⣀⡀⠀⠱⡄
⠀⠀⡔⠁⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⠔⠀⠁⠀⡴⠊⢁⡠⡔⠢⣌⠉⠀⢀⡤⢤⡀⢀
⠀⠀⡆⢎⡀⠙⢭⣍⠉⠩⠉⣘⣉⡩⢓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠊⠀⢀⡨⡇⡇⢸⠀⡵⠴⠋⠀⠀⠉⠁
⠀⠀⠀⠐⠫⠝⠛⠃⠀⠀⢠⠴⠔⠢⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠑⠤⠤⠐⠚⡁⢇⢹⠀⢸⠸⣠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠌⠞⠘⠺⡜⠁
⠀⠀⠀⠀⠀⠀⠰⣤⠠⡀⠤⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⡜⠈⠡⠀⢈⠀⠢⡀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠠⠎⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⢷⠀⠀⠀⠀⣗⣼⣀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠐⠀⠀⠂⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀⣉⠉⣉⢛⡛⣩⣽⡶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠬⠄⠰⣯⢠⣶⡌⠁⡾⠆⣴⡆⢉⣲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠰⠆⠀⠀⠀⡄⠂⠠⠀⠒⠒⡃⠦⢤⣬⣍
⠀⠀⠀⠀⡠⠐⡨⠀⠈⠈⠀⠈⠉⠒⠠⣀⠀⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⡐⠁⠀⠊⠀⢀⣀⣤⣴⣤⣤⣄⣀⣈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠁⠐
⠀⠐⣶⣶⣶⣾⣿⣿⡿⠿⠻⣿⡿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⣀⡀
⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠘⠁⠀⠉⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⡀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⠉⢀⣀⡀⠠⠄⠒⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠈⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡯⠄⠀⠀⠀⠀⠀⠘⢻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⠛⠂⠉⠀⣀⣀⣀⣀⣀⡀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢀⣤⠶⡾⡛⣻⡏⠉⢿⡝⡉⣿⣿⣷⣄⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢾⡵⣾⢿⡽⢸⡏⠀⠹⡏⡞⣮⣿⣿⠟⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠳⢦⣉⣛⠻⠷⢶⡇⠀⠀⣟⣛⣫⡭⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⠙⣦⠀⠀⠀⠀⠀⠈⠉⠉⠉⢻⠃⠀⢰⡏⠁⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣛⠂⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢸⠁⠀⠀⢀⣠⣴⠟⠉⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⢀⣀⣀⣀⣀⣀⣀⠈⠘⠻⢦⠀⠀⠀⠀⠀⣸⠃⠀⠀⡟⠀⣰⢾⡋⠍⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡐⠿⣿⣿⣻⠉⣿⣻⣏⣷⡆⢨⡇⠀⠀⠀⢠⡏⠀⠀⣸⠃⠀⣟⠀⠈⠀⠀⠀⠀⠀⠀⠈⠽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠦⡏⠀⣩⣭⡭⠥⠖⠋⠀⠀⠀⢀⡟⠀⠀⢠⡏⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⢑⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⣽⠟⣲⠶⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠸⡆⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣸⠃⠀⠀⠀⠀⠀⡼⠃⢀⡟⠀⠀⠀⢀⣠⡴⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢻⡀⠀⠀⠀⠀⢀⡟⠀⠀⢀⡏⠀⠀⠀⠀⢀⡾⠁⠀⡼⠁⣠⠖⠛⠋⠁⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢷⡀⠀⢳⡄⠀⠀⠀⢸⠃⠀⠀⢸⠇⠀⠀⠀⢀⡞⠀⠀⡼⠁⠀⣧⠰⣖⢲⣦⢤⣀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠈⠙⢶⣄⡈⢷⡀⠀⠹⣆⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⢀⡞⠀⠀⡼⠁⠀⠀⠈⠓⢮⡿⠃⣼⣯⣳⡄⢸⠇
⠀⠀⠀⠀⠀⠀⠀⣴⡟⠅⠀⠀⠀⠀⠈⡻⡆⢳⡀⠀⠈⢧⡀⣿⠃⠀⡀⣿⠀⠀⠀⣼⠁⠀⢸⠇⠀⠀⠀⠀⣠⠞⠁⣰⠛⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠘⢯⣀⣀⠀⠀⣀⣀⡤⠞⠁⠀⢳⡀⠀⠀⣻⣯⠃⠀⣧⣿⡀⠀⢰⣿⢠⠠⣿⠀⠀⢀⣤⠞⠁⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⡏⠙⣇⠀⠀⠀⠀⠈⢷⢠⠀⠜⣿⣇⢸⢼⣾⣇⠀⣸⡏⣏⠈⣿⣠⡴⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⢧⣄⡀⣀⠀⠘⣇⡎⡜⢻⣧⡎⡞⡟⣿⡄⣿⡏⡏⢀⢿⣏⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢘⠽⠿⠟⣷⢹⣴⢰⠂⣿⡔⡜⡜⢜⡇⢹⡇⡇⠾⣽⡗⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣿⣿⠿⠟⠘⣧⡔⠸⢸⡿⠾⠶⠞⠁⠘⣿⡐⢃⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡴⠞⠁⠀⠀⠀⠀⠀⠈⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠘⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⠀⠀⠈⠈⠲⠄⠉⠐⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣢
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⣌⣀⣁⠀⠐⢀⠘⡁⠀⠀⡐⠚⠓⠙⠛⠃
⠀⠀⠀⡐⠀⠀⠀⠀⢀⠶⠲⠀⠐⠀⠂⠐⢀⠀⠀⠀⢀⠀⠀⣅⡉⠉⠻⢿⡇
⠀⠀⡰⠁⠀⠀⠀⠀⣚⡷⠋⠉⠉⢻⡷⠆⠈⢁⠀⠀⠈⡄⠀⠃⠁⠠⠤⢶⡟
⠀⢀⠃⠀⠀⠀⠀⣸⣰⡇⠀⠀⠀⠈⢻⡆⠠⠬⡇⠀⠀⠁⠀⠀⠀⠀⠀⢸⠇
⠀⠘⡀⠀⠀⠀⠀⠿⡿⣷⠀⠠⠀⠀⠀⣿⡅⢐⠒⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠑⠀⠀⠀⠀⠀⣅⣹⡆⠀⠀⠀⠛⡧⠈⠁⠀⢀⠐⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠄⠀⠀⠁⠀⠀⢰⡆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠀⠀⢰⠀⠀⠀⠠⡀⠀⣼⠁⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⢄⠀⡠⠃⠀⠀⠀⠠⠊⠀⠀⠀⠀⠸⢶⣾⠇⡆⠀⡀⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠀⢀⢠⠞⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠙⣧⡰⢰⣴⢿⠜⠑
⠀⠀⠀⠀⠀⠀⢨⡎⡀⠀⠀⠀⠀⠀⠴⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡌⣷⡆⡇
⠀⠀⠀⠀⢀⠀⢺⠃⠈⡆⢸⣀⡄⣴⢃⣇⢉⠀⠂⠀⡀⢀⣠⢞⡳⡙⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠙⠷⠏⠃⠵⠏⠸⢁⢋⢀⢐⣴⡟⢺⠁⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠋⠉⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⣤⣾⣿⣿⣿⡿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⢰⣿⣿⡟⣿⣿⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⣿⣦⡀⠀⠀⠈⠛⠿⠿⠿⣿⠿⠿⣿⠛⠛⢉⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⢿⣿⣿⣿⢿⣿⣆⠀⠀⢀⣀⣀⡀⡇⠀⠀⡿⢀⣴⣿⠟⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠟⢀⣴⣿⠟⢿⣿⣧⡀⠀⣷⣿⣿⣿⣷⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡇⠀⠀⣰⣿⣿⣿⣷⣿⣿⣿⣷⢸⣿⡿⢿⣿⣿⡟⣻⣿⣿⣹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣹⣦⡀⢸⠀⢸⠀⣰⣿⣿⣿⣿⢻⣿⣿⣿⣿⣇⣿⠿⣿⣿⠿⠿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣿⡿⢟⣿⣷⡌⣦⠸⡀⠻⢿⠿⠿⣿⣶⣿⣿⣿⣿⠏⡏⠀⢀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠉⢑⣀⣀⢴⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢀⡇⠀⣸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢹⡀⠙⡄⠀⠀⠀⡇⠀⢸⠃⠀⣀⢸⠀⢠⠇⠀⣠⣿⣾⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣱⣾⣷⣄⣷⠀⢱⠀⠀⢠⠇⠀⢈⣴⣻⣿⣧⠀⡞⠀⣼⣿⣿⣧⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣹⣿⣿⣿⡄⠀⠆⠀⣸⠀⢰⣿⣿⣿⣹⣿⡇⠀⠀⢨⠏⠙⠉⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠙⡏⢹⡀⢣⠀⣼⢿⣷⡀⢸⠇⠏⢈⡏⡹⠃⠀⢀⡎⡸⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣇⢱⠈⢧⠸⡈⠛⠏⡟⠁⣸⡸⠀⣰⠋⠀⢀⢀⢮⠐⠁⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢣⠀⢧⡇⢰⡀⡇⠀⣿⠃⢼⠇⠀⡠⣢⠆⠈⠳⣤⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡄⠱⡈⢧⠀⠳⣼⡇⡇⡰⠃⢠⡏⢀⣉⢊⠥⠊⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⠀⠀⠘⠦⠙⠈⠳⠀⠈⠁⠋⠀⠀⠛⠉⠁⠀⢀⡠⠞⠁⠀⢀⣴⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡿⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠸⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠁⢀⣿⣿⣧⠀⠀⠀⣤⣶⣷⣶⡄⠀⠀⠀⣿⣿⣧⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣿⡿⠃⠀⠂⠸⣿⣿⣿⣿⣿⠀⠂⠀⠻⢿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿⣶⣦⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣿⣿⢿⠿⠟⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⣀⣀⡤⠖⠒⠛⡛⠻⣟⣛⣒⠂⣤⠄⢠⠀⠀⠀⠐⠞⠋⠙⢟⣓⠚⠿⠿⠃
⠀⢀⣠⣤⣤⣴⣶⣿⣿⣿⣿⣽⣿⣿⣿⣯⣷⣶⡶⠛⠛⠙⠉⠛⠛⣿⣻⣿⠋⠩⣒⢠⣴⣭⣤⢾⣾⣽⣾⠟⠉
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣿⣿⣿⡷⠷⠚⠂⠁
⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⠟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿⡟⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠓⠛⠢⣤⣀⣈⣙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢀⣳⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠉⠐⠉⠈⠉⠉⠋⠙⠛⠉

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⡀⠛⠻⣛⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠿⠿⢷⣄
⠀⠀⠀⠀⢰⣧⣤⣴⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⡾⠛⣿⣷⠞⡠⠐⠲⢦⣄⡀
⠀⠀⠀⠀⠀⣈⣭⣥⣀⢾⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⢻⣿⣶⣿⣿⣾⣿⣦⣤⣼⣿⣿⡀
⠀⠀⠀⠀⠊⢀⣤⣄⠈⡇⢈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠻⢿⣿⣿⡉⠉⠉⠙⢻⣿⣿⣿⡿⠿⠿⠟
⠀⠀⠀⠀⠀⠸⡟⠻⣧⠀⢀⣿⣷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣴⣾⣿⣿⣿⣶⣴⣴⣾⣿⣿⣿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠐⣬⣻⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡽⣯⣿⠶⢿⣿⣿⢻⣭⣿⣿⣧⣾⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣷⣤⣼⣀⣰⠛⣻⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣻⣦⡾⣿⣶⡾⣿⣿⣾⡶⣿⡿⠿⣿⣟⣻⣿⠇
⠀⠀⠠⣶⣿⣿⣻⣿⣿⣿⣿⣶⣿⣧⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⠵⢻⣿⡶⣻⣷⠟⣷⣿⢟⣿⠿⠏⠛⠁
⠀⠀⠀⠙⠙⠛⠛⠻⠿⠿⠿⠟⠛⠛⠛⠉⠁⠀⠀⠀⠀⣀⣀⡀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠴⣋⠉⠈⠛⠋⢉⣻⡏⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⢷⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⢴⣴⣿⣾⣿⠏⢹⣿⣿⣿⣿⢣⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⣿⢷⡀⠀⠀⠀⠀⠙⢿⣿⡟⠀⠀⠛⢻⣿⣇⠸⣿⣽⣟⡳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡸⡡⣟⠟⣮⣧⠀⠀⠀⠀⠀⢤⣿⢷⠀⢈⣧⠘⣿⠱⢂⣿⠟⣉⠀⠁⠀⠀⠀⠀⢰⠖⡠⠞⠐⢎⣉⣖⢐⡑⢆⡠
⠀⠀⠀⠀⠀⠀⠀⠰⣷⣫⡜⣸⢯⡏⡄⠀⠀⠀⣩⣩⣤⣯⣠⣾⣿⣄⣿⣶⣿⣿⣿⣤⣀⠀⠀⠀⠀⣁⣤⣤⡓⣫⣍⣰⣠⡽⢲⡷⣶⠐
⠀⠀⠀⠀⠀⠀⠀⣶⠙⣿⢣⣻⣾⢻⡇⠀⠀⠘⠻⠿⠿⢻⣿⡿⣿⡿⣿⣿⣤⣉⣻⡿⠉⠁⠀⠀⠀⠸⣷⣾⠟⣿⣴⣿⣦⣾⣿⡷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⢻⣘⣿⣽⡽⡿⠟⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠊⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢟⣿⣿⣻⣤⡿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⣠⣄⠀⡳⢾⣜⢮⣯⣿⣿⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣯⡿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣘⡿⣷⣛⣼⡟⣿⣾⡏⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣛⠻⣇⣿⣿⣿⣿⡟
⠀⠀⠀⠀⡷⣼⢽⡇⠙⡏⢱⣿⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠿⣿⠀⢠⡄⠘⢿⣿⣴⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢠⣶⢿⠷⣅⣀⠀⢀⡽⣿⢿⣃⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠉⠀⠾⠊⠻⠭⠜⢳⡬⠭⠟⠉⠸⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀⣀⣠⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠈⠉⠈⠁⠉⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡯⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣦⣙⣳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢀⡙⠿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣤⣴⣿⣷⣆⠈⠁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠻⠟⢻⣿⠿⣿⣿⣿⣿⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⣠⠴⠛⠃⣤⣾⣿⣿⣿⣬⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⣧⠄⢀⣴⣿⣅⢀⣿⣿⣍⡻⡟⣳⣤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⢆⢀⡷⣼⣿⣿⡿⠟⢁⡰⠎⣩⣿⣿⠏⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠛⢿⣿⢿⣿⣿⡟⠉⣃⢲⡛⠛⠛⠻⢆⣈⣤⣤⠴⠋⣹⠇⠀⠀⢀⣶⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⢀⢉⡼⠋⠁⢠⣾⣿⣯⠝⢶⡤⡴⠾⠿⠟⢽⢳⣾⣥⣤⣤⣶⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⢶⠞⠛⣿⡀⢀⡀⣀⣾⣿⣿⣿⣶⣤⣤⣤⣷⣶⣶⡘⣟⣿⢿⣿⠿⢿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣁⢀⣤⠂⠀⢀⣨⣷⣨⣿⣿⣿⣽⣿⣿⣿⡇⠀⢹⣿⡟⠿⡿⢧⣿⣶⣤⣴⣿⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢁⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣯⣒⠉⠙⠓⠲⠧⢬⣁⠛⠉⣫⣼⣄⡠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡿⠛⢹⢿⣻⣽⠟⠋⢱⣾⠿⢿⣦⡽⠟⠛⠿⣿⡷⣶⡄⢠⡼⢋⢀⡀⠀⣠⣶⢶⡌⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡂⣀⠠⣦⠰⠟⠉⠀⣀⣠⣾⣿⣄⣼⣿⣖⠶⢤⣤⡬⢿⣶⠊⠁⠙⠿⣿⣿⣶⣟⣁⣾⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣴⡆⣉⢩⣅⠀⠀⠘⣧⡀⣠⣾⠟⠛⠋⣠⣦⠀⠛⢿⣦⣆⡀⠀⠀⠙⠿⣷⣠⣤⣀⠈⠉⣉⠉⣭⣁⣴⣶⣾⠇
⠀⠀⠀⠀⠀⡸⢸⣿⣿⣦⡆⠀⠙⠣⣾⠟⣿⡷⢀⣷⣷⡟⠾⠁⣀⣴⣿⠧⣿⣷⡀⠀⠙⣿⣿⠂⠀⠀⡖⠉⢻⡟⣿⣿⣶⣿⣶⡏⠉⠀⠀⠈⠁
⠀⠀⠀⠀⠐⣷⣾⡿⢣⡟⠀⠀⠀⠀⠀⠈⠻⣿⣿⡟⠁⠀⠀⢻⣿⣥⣿⠒⢿⣿⡇⠀⠀⠈⢿⡿⣿⣇⠰⢲⠞⠀⠀⠙⢿⣇⡁⠀⠀⠀⠀⠀⣠⡄⢰⣞
⠀⠀⠀⠀⢀⡿⠙⢱⡟⠀⣴⣶⣶⣶⠀⣠⣴⠿⠋⢀⡰⣷⣾⣿⣿⠿⢿⠦⢾⣿⣷⣶⠀⢀⠈⠙⠻⢿⣿⣛⢿⡷⢦⣄⣠⣈⢉⠓⢦⣤⣤⣾⢣⢿⣼⣿⡇
⠀⠀⠀⠀⢸⠃⠀⡾⢸⣿⡘⣿⣿⣿⡟⢉⠀⠀⣰⣿⣷⡸⣿⣷⣿⣤⣴⣄⣴⣿⣟⣻⣾⣿⡆⢰⣶⣾⣯⠉⠳⣄⠸⠟⠛⠛⢾⣿⣤⣦⣤⣿⣿⡜⠛⡿⠤⠴⠂
⠀⠀⠀⠀⢸⡆⢸⡇⠀⣿⣧⠹⣿⣿⣿⡘⢷⣿⠟⢛⣉⢷⣿⣿⣿⣿⣿⠿⠛⠉⠉⠉⠙⣿⠇⣾⣿⣿⡟⢀⡼⠋⣀⣀⠀⠀⠀⣽⢻⡽⠿⣿⡍⠀⢸⠷⠿⠂
⠀⠀⠀⠀⢸⣇⠈⢷⡖⠛⡿⠃⠉⣀⣀⣰⣞⢳⣼⣿⣿⠿⠿⢿⣿⣿⡇⠀⢀⣤⣤⣄⢸⠟⠀⠿⢿⣿⣧⣾⣵⣿⠟⠻⣿⡆⠀⣷⢰⣇⣴⡟⣠⢆⡇
⠀⠀⢀⡴⠾⣷⡀⠈⢿⡾⣧⣴⣿⣿⠋⠉⣿⣿⣿⣿⡇⠀⠀⠸⡏⢿⡇⣰⠿⠋⠉⢻⡟⠶⠦⠶⣦⣿⣿⣿⣿⠀⠀⠀⢹⣿⢠⡷⠖⢛⣫⣾⣡⣾⣀⡀
⠀⠀⠈⢰⣿⣿⠧⡀⢨⢷⣿⣿⠷⠛⠀⠀⣸⣿⣿⡏⠤⠀⠀⠀⣿⡘⣹⣿⠀⠀⠀⣼⠀⠁⠀⠀⣜⢹⣿⠟⣿⠄⠰⠆⣼⣾⣯⠶⢒⣻⣿⣟⣥⣤⢀⡉⠁⣀
⠀⠀⠀⠀⠉⠃⠀⢻⣿⣼⣟⢣⣀⠐⣠⢤⣿⣿⣿⣿⠔⣄⠶⢰⣿⠇⢻⣿⠣⡜⠃⠻⡇⠶⠀⠀⠉⣻⣟⢦⡀⠶⣤⠴⠂⣠⢿⣾⡟⠋⢠⡅⠈⡿⠏⠛⠃
⠀⠀⠀⠀⠀⣼⢻⡏⠙⢯⠙⠿⢿⣯⣤⡞⠁⣉⣿⡀⠀⠀⠑⠉⠉⠀⣸⡈⣳⣶⣶⡾⠿⢆⠰⡶⠚⢿⣿⣿⣿⣶⣦⣀⠶⠋⣀⠬⠛⢁⡾⠶
⠀⠀⠀⠀⢰⡇⡀⠃⣀⢈⣷⣀⣀⠈⢩⡙⠚⠻⠿⣷⣤⣀⣀⣀⣄⣰⣿⣿⣯⢿⣿⣇⢀⡜⠉⢱⣄⣼⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⣸⣧⣄⠀⠈⡄
⠀⠀⠀⠀⣸⢟⣡⠋⣹⣿⣿⣿⡿⢣⢸⡏⡷⣶⠀⠀⠉⠉⠻⠿⠿⢿⣿⣿⡿⠿⠿⠿⠛⠛⠀⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⠀⠀⡧
⠀⠀⠀⢀⡾⢹⡇⠀⢿⢻⠝⡏⠀⠀⢸⡇⡇⣿⠀⣤⢀⠀⠀⠀⠀⠀⠀⠀⣤⢸⠆⢶⠀⠀⠀⠀⠀⠀⠀⣤⢀⣀⣠⡄⣲⠴⢷⠀⣸⡿⢥⣠⠔⠋
⠀⠀⠀⢘⣷⠟⢳⡀⢀⡟⣰⠀⠀⠀⠘⠃⠗⠿⢸⡏⢸⡇⢸⡇⢰⡆⡆⠀⣿⠀⠀⢼⠐⡆⣷⠁⣷⠛⣤⣿⢸⣿⣿⠷⠟⠃⠈⢠⡏⠀⢠⡗
⠀⠀⠀⢸⡏⠀⢸⣿⣼⠁⠈⠉⠛⠲⠦⠤⣤⣀⣀⡀⠈⠁⠈⠃⠈⠃⡇⠀⠙⠀⠀⠚⠚⠃⠛⠀⠉⠀⠉⠀⠀⠀⣀⣀⠤⠤⠒⠉⢷⣠⠟
⠀⠀⠀⠈⢿⣀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣄⣀⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⢴⡦⠶⠶⠖⠙⠃⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⠁⠀⢹⣿⣿⣿⣿⡿⠷⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣾⣿⣿⣿⣿⣇⡀⣠⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠋⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⣤⣴⣿⣿⣿⣿⡿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⠋⠉⠹⣿⣿⣿⣿⠟⠉⠙⢿⣦
⠀⠀⠀⠀⠀⣰⣿⣿⣿⡀⠀⢀⣿⣿⣿⠟⠉⠻⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣦⣤⣼⣿⣿⣿⣿⣄⠀⢀⣼⣿⣧
⠀⠀⠀⠀⠘⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣦⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉




⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡶⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣿⣧⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⢠⣿⣄⣼⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⡉⣹⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠹⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⢀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⠀⠀⠀⢀⣶⠀⣤⠀⢀⣼⡟⠀⠀⣰⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣿⣧⡀⠀⣾⠁⠀⣠⡾⠃⣠⣿⣧⣾⣿⣇⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⢿⣿⣿⣾⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣀⣤⣶⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⢀
⠀⠀⢀⣾⣿⡛⢿⣗⡀⠠⠉⠢⡀⠀⠀⠀⠀⠀⢠⡉⢬⣯⣒⢀⡀⠀⠀⠔⠀⡤⠈⠒⠀⠀⠀⠀⣼⣽⣋⣿⡄⢈⣨⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣦
⠀⠀⠸⠛⠛⠛⡶⠾⣥⣀⡀⠀⠹⣄⠀⠀⠀⠀⡈⠿⠩⠽⠦⠬⢥⡀⠀⠀⠁⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⢻⣷⣾⣿⡇⠀⢀⣜⣐⡆⠀⡾⠃⠀⢀⣈⡟⢏⣄⡀
⠀⠀⠀⠀⠀⠄⠁⠁⠈⡍⠛⠶⣦⣽⡆⠀⠀⣒⠃⢲⣒⣲⣒⣉⣊⡀⠀⠀⠘⠀⢠⡔⠲⢦⡄⢸⣷⣿⢵⣿⣿⣿⣿⡇⠀⠈⠉⠙⠁⠀⠀⠀⠀⠈⠋⠁⠀⠛⠃
⠀⠀⠀⢀⠊⠀⠀⠀⠸⠁⠀⠀⠀⠙⠁⠲⢄⣉⣩⣥⢥⣞⣃⣐⣎⣥⡀⠀⠀⠀⠀⠁⠀⠶⠋⠈⠙⠻⠛⠟⠀⠋⠉⠀⠀⠀⠀⠂⠒⠓⡀
⠀⠀⠠⠁⠀⠀⠀⠀⢠⠀⣴⠞⠉⢿⣦⠀⠀⠙⢮⡭⠴⢏⣑⠫⠘⠉⠁⠈⠛⠛⠋⠀⠀⣷⣤⣤⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢰⣤⣶⣿⠂⠈⠀⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠿⠃⠀⠈⢉⡙⣲⣒⡛⠓⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠋⠀⠀⠀⣀⣠⣤⣭⣁⠉⠀⠀⠈⠉⠉⣡⡆⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣻⣆⠀⣴⣿⠁⢹⣭⣪⡿⠁⣢⠘⠃
⠀⠀⠀⠑⢶⣤⣤⡤⠄⠀⠈⠐⠶⠂⠁⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⠇⣿⠀⠫⣁⣠⠞
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠘⢿⣙⡝⠿⢛⣣⡴⠃⠀⠀⠀⠀⠀⠀⣀⣁⠉⠁⠉
⠀⠀⠀⠀⢀⢠⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣈⠛⠟⠀⠀⠀⠐⣮⣣⣒⠄⠀⡔⠁⠀⠀⠈⢂⣀⠀⠉⠉⠉⢉⣥⣤⣦⣀⠀⠀⠾⣿⣿⣿⣟⠁⠰⣿⣦⣤
⠀⠀⠀⠀⠿⠆⠙⠁⠀⠀⠁⠀⠀⡀⠀⢸⣿⣿⣿⣿⣷⣴⣤⡀⠀⢸⣿⣿⣿⠋⠀⢧⠤⠤⠦⢶⣾⡀⢑⠀⠀⠀⣿⡁⠧⣡⠽⠧⠀⠀⠙⣛⣏⣀⡀⢀⡈⠉⠉
⠀⠀⠐⠀⠀⠀⠀⠀⠀⣀⠐⠆⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡧⠀⠈⣿⣿⡿⠀⠀⠀⢸⡇⠈⠈⠁⠰⠏⠰⠒⡶⠈⠻⠶⠿⣆⡓⠀⡃⠰⣿⡽⢿⣿⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠀⠀⠀⢿⣿⠃⠀⠀⠀⠀⠀⣠⠄⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠃⠀⠈⠡⠌⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠁⠀⠀⠀⠀⠁⠀⠉⠋⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡟⠉⠀⠀⠈⠙⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⣠⣴⣿⣿⣿⣿⣿⡿⠿⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠉⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣤⣴⣾⣿⡿⠁⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠿⢿⣿⣶⣦⣤⣾⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠿⢿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⠀⠀⣠⣿⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⡄⠀⠀⢹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠻⣟⣻⣛⣛⣛⣛⣛⣛⣛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠷⠶⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠳⠾⠿⠿⠿⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢋⣩⣤⣤⡉⠻⣿⣿⣿⣿⠛⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣽⣿⢯⣶⣶⣤⠀⠀⢀⣾⣿⣿⡿⢀⣿⡿⠛⢿⣿⡆⢿⣿⣿⣿⣤⣿⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⠿⣿⠿⣟⡋⠀⠀⠀⣾⣿⣿⣿⡇⢸⣿⠅⠀⠀⣿⣇⢸⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣦⠀⣼⡏⠀⢹⣿⣇⢸⣿⠄⠀⠀⣿⣿⢸⣿⡟⠉⢻⡄⠀⢀⠘⢿⣷⣿⣿⣻⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣿⣝⣿⣄⡄⣠⣤⣶⣿⣿⣾⣿⣿⣿⢸⣿⣥⣤⣤⣿⡯⣸⣿⣿⣶⣿⡇⣷⣿⢠⡄⠙⢻⣿⠛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣯⡿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⡟⢻⡟⠛⠛⡇⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣾⣷⣿⣾⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣸⣇⣰⣹⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⢀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠛⠁⠀⠛⠁⠀⠀⠀⢠⡬⠢
⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠛⠉⠁⠀⢀⡀⠀⠀⠀⢠⣆⠀⠀⠀⣀⠀⠉⠒⠐⠀⠠⠄⠀⡀
⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⣀⠀⠀⢀⡀⠀⠘⠁⠀⠀⠀⠈⠁⠀⢀⠀⠉⠀⠀⠰⠆⠀⠀⠀⠀⠀⠈⠑⣆
⠀⠀⠀⠘⢿⣦⣄⡀⠀⠀⠁⠀⠀⠘⠁⠀⠀⠀⠀⠃⠀⢀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⢓⡲⠋
⠀⠀⠀⠀⠀⠉⠛⢿⣟⣶⠶⢤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠤⢤⠤⠔⠒⠐⠚⡉⠵⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠿⢶⣤⣍⣷⣌⣁⣀⣨⡏⢀⣤⡶⠶⠗⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡏⠉⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣽⣥⣀⣀⣠⠴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠐⣷⣤⣆⡤⠼⠲⡅⠀⠀⢠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢆⢰⠀⣿⡟⠀⠀⠀⠀⢣⠀⢀⡇⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢑⢌⣿⣀⣿⠁⠀⠀⠀⠀⠈⠇⢸⢁⡎⡹⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⢸⣧⢹⣿⠀⠀⠀⠀⠀⠀⠀⣼⡸⢀⡧⢀⡔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡟⣼⣿⡟⡇⢀⠀⠘⠀⠀⠀⢻⡗⣹⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢁⡇⣿⣋⡗⠸⠀⠄⠀⠀⠀⣼⣡⠇⣼⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⣧⡇⢸⠀⡇⠀⡆⣿⣿⣾⣋⠥⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⣿⣷⣼⣆⣧⣷⡿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠄⠠⠠⠤⠀⢀
⠀⢀⣰⣶⣟⠲⢶⢖⠳⣘⣫⣶⠿⠿⠞⠛⢓⢹⠄⠂⡈⣡⢰⣺⣦⣷⣠⡄⠒⠚⠚⠇⡴⡂⢄⣃⣀⡀⢀
⠀⣠⡼⢖⣀⣤⣯⣥⠴⠂⣿⠃⠀⣀⠀⣄⣾⢻⠀⣀⠝⠨⠧⢸⡽⣏⠉⠳⢄⣄⣤⢤⡀⣤⣴⣛⣃⢼⣌⡙⣤
⢀⠏⡀⠈⠁⢈⢇⣘⣆⢀⡼⡆⠒⠉⢣⣀⢯⡘⡆⠀⠠⠤⠀⠈⢃⡷⠲⠆⠀⠀⠐⠊⠏⠁⠀⠉⠹⡀⠀⢈⣛⠁
⠊⢿⡾⠠⣿⣾⣹⡽⣝⢧⣀⣈⣀⠀⣉⡠⠀⠘⣇⠀⠐⢠⣯⠴⣟⢲⣄⢀⣐⣆⡀⢀⠶⠂⠈⠛⠂⠀⠃⣀⠉
⠀⢀⠇⠀⠀⡞⡩⠙⣶⡾⠟⠛⠻⡛⠻⣟⡣⢾⣿⣭⣶⣿⠋⠛⠉⣠⣟⠦⢌⠿⣻⣿⣾⠛⡯⡀⠠⠂⠀⠀⢀
⠀⢨⠔⠊⡰⣠⠇⢡⣿⠷⣶⡶⠀⠹⢷⡟⠙⠷⢼⣮⢻⣻⣧⠄⠞⢻⣯⣿⣃⣼⠇⠀⠈⠛⣧⣼⡁⠀⠀⠀⢨⢰⡀
⠰⠖⠶⠂⠤⠀⠀⠈⣩⡗⠈⠀⢴⡄⠀⡓⢊⠒⠶⠶⣾⡽⡈⠋⣄⣀⡏⣩⣷⡋⠀⢀⣀⣀⢸⣿⣖⡆⢸⣄⠱⣣⠂
⠀⠰⠀⠀⢘⠄⠀⠀⠀⠱⢤⣀⠤⡄⣰⡀⠙⢂⠀⠀⢪⡄⠇⡠⡤⠀⣤⢸⡂⠛⠃⢀⠁⢉⠾⣟⣿⠇⠀⠈⣄⠉
⠀⠃⣈⠉⠩⠓⠒⢲⡞⢁⠘⢿⣟⣻⠷⠖⢶⣿⣦⣐⣿⣧⣤⣀⢀⡼⠗⠛⠗⠢⡄⠛⠋⠜⠃⣁⣀⡀⠀⣀⡄⢀
⠀⢶⠉⡚⡶⠞⠶⡖⡽⠎⠃⠼⢸⣵⠴⠚⠛⠛⠿⢏⡙⠛⡛⠛⣻⠆⠀⠀⠀⠀⠙⢶⣵⣧⣼⡿⣭⣙⢷⣄⣀⠆⣃
⠀⠈⢻⠉⠀⠀⠃⠞⠀⠀⠀⠀⢈⢻⣾⣷⠄⠺⣿⡆⣿⡇⠉⣉⣿⠆⠂⠰⠶⠈⠐⢀⣏⣽⠻⠋⣹⡛⠿⠻⡆⠀⣹
⠀⠀⢩⠁⠀⠀⠀⠀⣀⡀⡵⠖⣶⠞⣇⠀⢀⣀⠀⢀⡋⠀⠘⠛⠣⡄⠷⠀⠀⠀⠀⠊⠉⠈⡆⣀⣭⣩⠀⠜⠀⠙⠃
⠀⠀⢨⡇⠤⢖⠨⠱⢠⠀⠀⠀⠀⠀⣨⣁⠠⣠⠀⡈⠀⠀⠀⠀⣰⣬⠀⠂⢀⠀⠀⠀⠀⠀⣙⢏⣁⠤⠀⠰
⠀⠀⣚⠦⣿⡴⣴⠂⠟⠀⠸⠁⠀⠀⣸⣭⠓⠻⢺⣾⡃⢤⢠⢤⣼⣿⣲⢶⠃⠘⠿⠗⢀⣶⠳⠾⠲⣄⠀⠒⠿⢻
⠀⠀⠀⡞⢿⣠⣤⠠⠒⠓⡆⠈⠀⣜⢟⣁⡈⠚⠓⢋⣷⠻⠇⡆⠨⡿⢿⠀⠀⠀⠀⠀⠀⣹⠈⠁⠰⠟⠛⡖⣺⣿⢧
⠀⠀⠀⢀⠈⠎⠘⢲⣥⠀⢈⢶⣺⢶⠊⠛⡛⡁⠸⡈⢻⠀⠀⠓⠼⠀⢸⡏⣡⠶⠄⠤⣸⠛⠟⢀⡄⠀⠀⢰⠿⠛⠈
⠀⠀⠀⠉⢠⡔⠓⠂⣶⠒⡡⡶⠀⠛⠀⠀⢀⡁⠀⢉⢠⠀⠀⠚⠻⢂⡞⠼⢿⡷⢶⣟⡹⢠⡀⠀⠀⠊⡠⠀⠆⠄
⠀⠀⠀⠀⠠⠱⡤⠀⠄⠀⡄⠀⠉⣥⠀⡠⢬⣁⣀⡠⠈⠀⠀⠤⠀⠈⠊⠉⠸⠉⠓⠛⠀⠀⠘⠃⠄⠀⠐⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣴⣶⣿⡿⠛⣿⠟⣶⣦⣄⡀⠀⠀⠀⠀⢴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⣾⣯⣿⣭⣿⡟⣿⣿⣟⣻⣯⡼⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣋⣿⣿⠿⠿⠿⠿⣿⠿⠿⣿⣶⣿⣏⣽⣦⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⣯⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠯⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⢀⠀⢄⣀⣤⣦⣾⡴⡂⠀⠀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣝⣳⣿⠋⠁⠀⠘⢿⣦⣄⣀⠀⠀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡏⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⣀⠀⢀⡀⠠⠧⡀⠀⣠⣾⢿⣾⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠄⠛⢿⣿⣯⣽⡿⣾⣿⣇
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡽⠆⠀⠀⠂⠀⠀⠀⠀⠊⠓⢺⠉⠊⠀⢤⣉⡉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠈⠛⠛⠛⢛⡉⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⣀⢀⣇⠠⣅⡀⠀⢀⣠⢀⣰⡇⣠⡀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣷⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⡴⠶⠶⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢊⣽⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⣴⣿⣿⠋⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣴⣤⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣡⢀⣀⣤⣿⣿⣿⣶⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠟⠿⠑⠆⠀⠀⠈⡛⠛⠙⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⢀⠤⠒⠚⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠟⠳⠶⢤⡀
⠀⠀⠀⠀⢀⣜⣥⣀⠀⠀⠀⢸⣿⡄⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⢙⢦
⠀⠀⠀⢠⣾⡿⠻⣿⣧⣴⣾⣿⣿⣿⠀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠈⠀⠀⣠⣶⣶⣶⣾⣿⣷
⠀⠀⠀⣾⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣟⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢀⣿⣿⣄⣀⡀⣰⣿⠋⠀⠈⢻⣿⣿⡇
⠀⠀⠀⣻⣷⡀⣠⣿⡿⣿⣻⡿⣿⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⡿⡗
⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣶⣤⣤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣾⣿⡿⣿⠿⣟⣛⣿⣧⣄⣠⣼⣿⣿⠇
⠀⠀⠀⠈⢿⡻⢾⣿⣿⣿⣿⣿⠟⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣛⢿⡟
⠀⠀⠀⠀⠀⠙⠶⣾⣾⡿⠟⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠈⠛⠿⠿⠿⠶⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⡀⣠⣴⣶⣶⢶⣦⡀⣑⣒⣀⣀⣠⣶⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣴⣶⡎⣿⡏⣹⠛⠀⡌⠓⠺⠛⠛⢛⠛⣛⢻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢋⣿⣏⣱⣷⣪⠭⠂⠀⠒⠀⢲⡤⠤⠤⡽⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠋⠉⢀⡄⠀⠀⠀⠀⠉⠀⠀⠀⠈⠲⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣡⣶⣴⡳⠞⢶⠆⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠷⠦⡄⠁⡠⠀⠠⠋⢀⠀⠀⠀⠀⡄⠂⠀⢀⠀⠀⠀⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠖⠚⡗⠒⠛⢶⠤⣄⣀⣁⣠⣄⣤⡵⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠘⠂⠤⡄⢀⡇⣀⠀⣉⠈⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠀⠀⠀⠀⠈⠙⠻⠄⡀⡀⢨⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠦⢤⣀⡀⠀⡇⠀⠀⣀⡤⠚⠁⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠓⢻⠚⡏⠉⡇⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠇⠀⠇⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠇⠀⢠⠀⠀⢠⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣾⣿⣆⠘⢰⡀⠂⠀⢀⣾⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⣿⣿⣿⣿⣶⣾⣿⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⣿⣿⣿⣿⡿⠻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⢶⠶⢶⡶⠒⠄⠄⣠⠀⢀⠀⠀⠀⠀⡶⠟⠛⠿⠶⠤⢾⣞⠿
⠀⠀⢀⣀⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣭⣦⡶⠶⠛⠒⠺⠿⠿⣿⢾⣿⠖⢟⠥⢀⣀⣜⣀⣤⣆⣜⣹⡷⠖⠂
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣦⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⣴⣿⣿⣿⣿⣿⠭⠯⠛⠉
⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣆⣀⠀⠙⠻⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⠿⣶⣾⠇⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⢐⣆⠀⡀⢠⣿⣿⣿⣿⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⣀⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣀⡴⠒⠊⢻⠉⠙⡏⠉⠙⠓⢦⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠦⡸⡀⢯⣄⣄⡀⢸⠀⠀⡇⣄⣄⡬⢎⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠙⢌⡓⠦⠭⠭⢽⠀⠀⡯⠗⢒⣋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⣠⠤⢤⠄⢤⠤⣤⣀⠙⢦⠀⠀⡞⠀⢀⣇⡴⠉⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡻⣄⣀⣸⠀⢸⣇⡀⣨⠗⡸⠀⢰⠃⠀⢸⠁⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡉⠒⠺⢽⡄⠈⡯⠭⠖⠊⠀⢀⠏⠀⢀⢧⣒⠚⢻⠀⢲⠒⠤⢄⠈⡇
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠈⠳⢄⠀⡇⠀⢣⠀⠀⠀⢀⡞⠀⠀⡜⠒⠬⠭⡝⠀⢘⣛⣫⠭⠖⠁
⠀⠀⠀⠀⢠⠋⣀⣀⣤⠠⡤⢤⡀⢱⢸⡀⠘⣆⠀⠀⡼⠀⠀⢰⠁⠀⠀⢠⠇⠀⡏
⠀⠀⠀⠀⠘⠦⣓⣒⣧⠀⣗⣋⠥⠊⠀⢧⠀⠘⡆⢠⠇⠀⢀⡏⠀⠀⢀⠎⡠⠔⠋⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢣⠀⠀⠀⠀⠈⣇⠀⠘⡾⠀⠀⢸⠀⠀⣠⠊⡎⣠⠤⡤⢀⣀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢣⡀⠀⠀⠀⠘⡄⠀⠸⡄⠀⢸⢀⡔⠁⠀⣸⠞⢽⠇⢰⣞⣛⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠙⢦⠀⠀⠀⢹⠀⠀⢱⡀⢸⠋⠀⢀⡜⠁⣠⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠱⣄⠀⠈⡇⠀⠀⢇⢸⡀⢠⠎⢀⠞⠁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠘⢦⠀⢻⠀⠀⢸⠀⢷⠃⡴⠃⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⡇⣾⠀⠀⣸⠀⢸⣾⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠃⠙⠤⠴⠓⠒⠋⠘⠦⠞

⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢰⠖⠟⠳⢲⡆⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⣰⣿⣿⣿⣿⣿⣦⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⢰⣶⣶⡆⢱⣶⣶⡆⠀⠀⠀⠀⠀⠇⠀⠀⡀⠀⠾⢽⢯⠷⢀⢀⠆
⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⡷⠀⠀⢸⣿⣿⡇⢸⣿⣿⠁⠀⠀⢾⣦⡀⠀⠀⠀⠈⠀⢰⣿⢾⣿⣄
⠀⠹⣿⣿⣿⣿⣿⠟⢀⣴⣿⣿⣿⣿⣿⣇⠀⠀⢸⣿⣿⡇⢸⣿⣿⠀⠀⠀⣸⣿⣧⣤⣄⠀⠀⠐⢿⠿⠀⠿⠿⠁
⠀⠀⠀⠉⠉⠉⠀⠀⠸⣿⣿⣿⠿⠋⠁⠀⠀⠀⠸⠿⠿⠇⠸⠿⠿⠀⠀⠿⠟⠛⠉⠛⠛⠃⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⣉⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⡁
⠀⠀⠀⠀⣸⣇⠀⠀⠀⠈⣚⣷⡴⠀⠀⢦⣴⣗⠚⠀⠋⣤⠀⠀⠀⠀⠀⢈⡉⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⢀⡻⢟⡀⠀⠀⠈⠁⢖⣿⣶⣶⣾⣳⠋⠁⠀⠈⢃⡄⠀⠀⠀⠀⣈⡠⠄⠀⠸⠿⠆⠀⣀⠂⠀⡀⠀⠺⠿⠂
⠀⢻⣿⣸⣿⣿⣇⣿⡿⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⣤⠄⠀⢀⣈⣁⡤⠖⠀⠀⠀⠀⠀⠈⠉⠹⣷
⠀⠈⠿⠋⠀⠀⠙⠻⠋⠀⠀⠈⣿⠛⠛⢻⠏⠀⢠⣦⠀⠀⢡⡆⠀⠈⣡⣀⡠⠵⠚⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⣰⣾⣶⣶⣧⣄⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠀⠎⠸⠂⠀⠈⠈⠁⠀⠂⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣠⣶⣿⠓⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⣠⣾⣿⣿⠟⠃⠀⠀⠀⣼⣿⣿⣿⡄⠀⠀⠀⠐⠻⠿⠿⠀⠀⣤⣄⣻⣿⣿⣟⣠⣠⠀⠀⠀⢀⣴⣄⡘⠇
⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠂⠀⠀⠙⢿⣿⣷⣆⣹⡿⠋⠀⠀⣴⣿⣤⣬⣿⡆
⠀⠰⠿⠿⠿⠿⢿⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣄⠀⠀⠀⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢼⣿⡿⠋⠈⠿⢿⠷⠀⠀⠿⠿⠀⠸⠿⠇
⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⢀⣬⠁⠱⠆⠀⠀⠀⠀⠀⠀⠀⣔⣰⣶⣠⣾⣧⢀⣤⠀⠀⣴⡟⠛⠉⠉⠙⢳⣆⠀⠀⢀⣴⣾⣦⡀
⠀⠀⠘⢆⡛⣡⡿⡿⠳⠇⢀⠀⠠⠀⠀⢀⣹⣷⣽⣿⣿⣿⣿⠃⠀⢘⣿⠀⠀⠀⠀⠀⠀⣯⠀⢠⣿⡿⠟⢻⣿⡆
⠀⠘⠛⣤⠢⣤⠷⢆⠈⣘⠋⠀⢇⠀⠀⠸⢿⣿⣿⠿⠿⠟⠃⠀⠀⠀⢿⣤⣀⡀⢀⡀⣰⡟⠀⢺⣿⣭⢀⠰⣿⣿
⠀⠀⠀⠉⠀⠀⠀⠘⠁⠙⠁⠀⠀⠑⠂⠤⠤⠤⠤⠤⠄⠂⠀⠀⠀⠀⠀⠉⠙⡿⡿⠛⠉⠀⠀⠈⠻⢿⣿⣿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠴⠒⠚⠉⠉⢉⡭⠭⠭⢭⡛⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⡡⠤⢤⡀⠀⠀⠀⠀⠈⠆⡀⣀⡠⠟⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣫⠤⠤⣀⠀⠑⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⡔⠉⠀⠀⠀⠈⡧⠀⠀⢠⠖⠉⠉⠉⠉⠉⠙⡆⠀⠀⢠⠞⠉⠉⠉⠉⠲⡀⠈⢦
⠀⠀⠀⠀⠀⣠⠋⠀⠀⠣⠤⠤⠤⠖⠊⠁⠀⠀⡇⠀⠀⠀⠀⢀⣀⠴⠁⠀⠀⢸⠀⠀⠀⠀⠀⢀⠇⠀⠀⠱⣄
⠀⠀⠀⠀⣰⠃⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠑⠒⠒⠉⠉⠉⠀⠀⢀⣠⣀⡀⠓⠤⠤⠖⠚⠁⠀⠀⢀⠤⣌⢧
⠀⠀⠀⢰⠃⠀⠀⡴⠋⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣀⡽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢘⡞⡆
⠀⠀⠀⢸⠀⠀⠀⢇⠀⠀⠀⠀⣀⡤⠔⠋⣀⣀⣠⡤⢤⡤⢤⢤⠤⢤⠤⣤⠤⢤⣤⣀⡀⠀⢰⠂⠒⢦⠀⠈⠉⠀⡇
⠀⠀⠀⢸⡄⠀⠀⠀⠁⠀⠈⠉⣠⡴⠚⠛⢅⡀⠈⢦⠀⡇⢸⢸⢀⡆⡴⠁⡠⢊⠴⢊⡩⠓⢦⡑⠦⠞⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠳⡀⠞⠉⢱⠀⣠⠚⠡⢤⣉⡒⠦⢌⡓⢄⠣⣡⢈⣘⣸⡜⣡⡮⢖⡥⣚⡥⠔⠊⣁⡙⣦⠀⢀⠖⣳⡼⠁
⠀⠀⠀⠀⠀⠘⠢⡀⠁⠀⣇⣉⣉⣐⣒⣺⣷⣶⣦⡭⠟⠋⠉⠉⠉⠉⠙⠛⢯⣭⡶⠶⠟⠛⡒⠒⢸⡇⠐⣲⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣙⣦⡐⠒⢒⣒⡦⠽⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣭⣉⣉⣩⣭⡴⠾⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠒⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠸⠀⠁⠈⠀⠀⠀⠀⠡⠀⠀⠀⠀⢠⠃⠀⠄
⠀⠀⠀⢠⣄⣀⠀⢪⣩⣆⢠⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⡇⠰⠀⠀⠀⠐⠀⡔⠀⠁⠀⡁⠀⠆
⠀⣀⠐⢦⡀⠙⣷⣘⢳⠏⢰⢀⠏⠰⠀⠀⠀⠀⠀⡔⠡⡐⢤⠤⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢣⠀⠀⠈
⠀⠸⣆⠀⠙⠄⠈⣿⢼⠇⠰⣎⡴⠓⠄⢀⢔⣤⡞⠀⢀⠇⠈⡆⢠⠔⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸
⠀⠀⠈⠑⢄⢸⣆⢻⣶⡆⠐⣽⠁⠀⠀⢈⠘⣿⡇⠀⢨⡆⠀⢿⢀⡟⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⢦⠻⡈⣻⢃⢣⡏⠀⠀⠀⠀⠄⠻⣷⡄⠸⣆⣀⡞⣼⢁⠇⠀⠀⠀⠀⠀⢀⡀⠀⠂⠀⠀⡄⠀⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠣⣷⣿⣾⠋⠀⠀⠀⠀⠀⠀⠣⠼⡿⠿⠋⠜⡿⠑⠁⠀⠀⠀⠀⠀⠀⠈⠯⣤⠀⢳⠆⠀⢰⡟⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣸⠘⠀⠀⠀⠀⠀⠀⡴⠒⠂⣴⠶⢄⡀⢳⣼⣇⣼⣿⠷⠚⠃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠑⣄⡀⠈⠷⣇⡙⢶⣍⢯⠉⣤⣤⡶⠿⢛⡋⡳
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⢀⠀⠹⢦⣦⡭⣷⣶⠽⠷⠦⣤⠤⡤⠀⠉
⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣉⣀⣀⣑⣀⣐⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⠻⣟⡛⢻⡟⣻⢿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠛⠛⠋⠓⠒⣠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⢁⣸⣤⣶⣶⣶⣤⣦⣕⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡙⣮⣿⡟⠛⢿⣿⣿⣷⣾⡧⠞⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠜⣾⢿⡇⠀⣿⣿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣦⣄⡀⠀⠀⠀⢂⡸⢧⣼⣿⣍⡏⠃
⠀⠀⠀⠀⠀⠀⣠⣴⣗⣤⣴⣾⣿⣝⣭⣴⡿⠽⠀⠆⠐⠋⣺⣿⢿⠿⢿⣯⡀⠠⠄
⠀⠀⠀⠀⠀⠜⠛⢽⣿⣷⣿⣿⣿⠿⠿⠏⠁⡐⠀⠈⣡⣿⡿⢾⣿⣛⠻⠿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⢻⣯⡇⠀⠀⠀⢀⠀⠐⠋⣿⣿⣿⣿⣿⣿⠄⠈⢹⣿⣿⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⡈⠁⠀⢀⣀⣤⣾⣿⣄⠀⠚⠛⣿⢘⠟⠉⠀⣠⣾⣿⠟⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠀⠀⡚⢿⣿⣿⣿⣿⠗⢠⡤⠠⠈⠠⣴⣾⣿⡿⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠿⠃⠀⠀⠟⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠐⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢁⠀⡠⠀⠀⠀⠘⠻⢿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠀⣠⢐⡄⣾⣿⢃⣤⠇⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡀⣿⣎⣿⢿⠃⣿⡿⣼⣿⡀⠀⠀⠀⠀⢳⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣼⣾⣿⢳⣿⣿⡇⢀⣤⣴⣶⣿⣿⣿⣶⡄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠴⢞⣿⣿⠿⡟⠻⣿⠿⣿⣿⣿⣿⡿⢛⡟⠁⠀⠛⡇⢸⡇⠙⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠏⠁⢸⢲⠁⠀⠓⡀⠹⡀⢻⡈⠁⠸⠁⠀⠾⠚⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠐⠊⠂⠈⠀⠚⠀⢰⠁⠆⢨⡀⢆⡀⠞⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢀⠀⢤⢿⣿⣶⣶⣶⣶⣶⣶⣦⣤⣤⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣿⠿⠿⠿⠟⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⢀⡀⠀⠠⣤⣮⣤⣔⡒⣶⠖⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⡀⢀⡀⠈⢿⣶⣤⣄⣀⣀⣈⣉⠐⠀⠀⠀⢀⣩⣭⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢿⣾⣿⣿⣿⣶⡦⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣬⠇⠀⣸⣿⣿⣿⣿⡿⢿⣷⡆⠀⠀⠀⠀⢿⣿⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣡⣤⣶⣾⣿⣿⣿⣿⣯⣽⢶⡇⠀⢠⡿⠟⢉⣻⣯⣤⣾⡿⠓⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠘⣿⣿⠿⠁⠘⠇⠀⠘⢁⣴⣏⣉⣹⣿⡍⠒⡄⠀⠀⣔⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠉⠛⠛⠩⣤⡈⠀⠀⠀⠀⠠⠀⢼⣿⡿⣿⢿⡟⠓⠀⠀⠀⢠⣾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⢏⠑⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⢦⠀⢸⡀⠀⠀⠀⣞⢀⠀⢠⠀⠀⠀⠀⠀⠀⠴⠋⣄⡀
⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠒⢶⣿⣿⣶⣄⣁⠀⠀⢠⣟⣿⡀⠈⣶⡄⠀⣄⣰⠋⠀⠈⠁
⠀⠀⠀⠀⠀⠉⠀⠉⣤⠀⣾⡀⠙⠉⣻⣿⣧⢤⢶⣾⣿⣿⣿⣶⣿⣶⡏⣾⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠸⡿⠙⠙⣻⣷⠿⣿⣿⣿⣿⣆⢹⣿⣿⠛⠙⠋⢸⠛⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⠂⠹⢿⡿⢯⣾⣿⡿⠿⣿⢿⠗⠈⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠀⠀⢽⣶⡄⠀⠀⠃⠾⠏⠻⠄⠱⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠰⠇⡛⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣰⣿⠀⠀⣿⣄⡀⢸⡀⠀⠀⠱⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⣿⡏⠉⠉⠀⠀⠀⢸⣏⠉⠉⠙⠛⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⢀⣴⣿⡟⠉⠀⠀⠈⣿⡄⠀⠀⠀⠀⠈⠻⢷⣦⣶⠾⠋⠀⣨⣿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠙⣷⣄⣀⣀⣴⡿⠀⠀⢠⣴⠿⠛⢷⣦⡀⠀⠀⠀⢰⡿⠁⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⡀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⣿⠁⠀⠀⠀⢹⣷⠀⠀⠀⠘⣿⣄⡀⠀⣠⣿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠉⢻⡆⠀⠀⣀⣀⣀⣤⣤⣤⣤⣿⣦⣤⣤⣤⣾⣯⣤⣀⣀⣀⠈⠙⠛⠛⠋⠁⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⣀⣤⣾⣿⠿⢿⡛⠛⣿⡋⠉⢹⡇⠀⠀⢿⠁⠀⠀⢹⡏⠉⠉⣿⠛⠻⢷⣶⣤⣄⡀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠙⢧⡀⠈⠳⠀⠘⠃⠀⠀⠃⠀⠀⠘⠀⠀⠀⠛⠀⠀⠠⠋⠀⢀⡾⠁⠈⣹⡿⢷⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣤⣤⡄⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠀⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⣀⣀⡴⠀⠀⠀⣠⠀⠀⢸⣿⠀⠀⠀⠀⠀⢾⣧⠀⠀⢀⠀⠀⢤⡀⢀⣹⣶⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣶⣶⣴⣯⣤⣄⣼⣿⠀⠀⠀⠀⠀⢸⣿⣀⣤⣼⣦⣶⣾⣿⡿⠿⠛⠉
⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⣿⡏⠀⠀⠀⠀⢰⠘⣿⡟⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢸⡆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⢸⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⢠⢣⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⢠⠏⣸⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⡞⠀⠃⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⣸⠃⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠘⠁⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⣀⣀⣀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠻⢷⡶⢤⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⠁⠀⠀⠈⠀⠉⠒⠌⠀⠀⠉⡓⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠘⠀⠀⠀⠀⠀⢀⣠⣶⠿⢛⡷⢾⣆⡘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠃⠀⠀⠀⢀⣀⠤⠒⠻⠁⠈⠀⠀⠀⠜⠛⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢀⣠⣴⠦⠘⠫⠁⠀⠀⡖⠀⠀⠀⠀⠀⣀⣤⡶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢄⣿⠀⠀⠀⣴⣾⣿⡿⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⣀⡀⢀⣀⣤⣤⣤⣴⣾⡎⠙⢳⣤⣾⣿⣟⡃⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⣉⠈⣀⣶⣿⣿⣿⡇⠀⢸⣿⣟⣩⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠒⠟⠺⡇⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠈⠘⢿⡴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡇⠀⠀⠀⠁⠀⠓⠁⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣾⠀⣀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⣠⡿⠉⠁⠻⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣤⣄⣦⣌⡗⠤⡀⠈⠄⠀⠀⠸⠀⣼⠏⠀⠀⠀⠀⠈⢅⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠽⣿⠁⠹⣶⣿⠷⢾⠞⠀⠀⣿⣿⡔⠊⠈⡁⠐⠐⠻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠈⠀⠀⠀⠀⠹⠜⡶⣶⡿⠉⢹⣶⣞⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠃⢠⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠹⡀⠀⠰⠀⠀⠀⠀⠀⠀⢰⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠙⢦⡀⠆⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⡇⠀⠀⠀⠀⣦⠞⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⢹⡀⠀⠀⢸⣧⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠳⠦⠴⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⢀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⢀⠀⢀

⠀⠐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢡⢠
⠀⠀⠘⠇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠆⠰⠈
⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠛⡘⠛⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠰⡇⠀⠀⠠⠤⠖⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠠⠄⠀⠀⠠⠠⠄⡄⠀⠀⠀⠀⡂⠆⠅⠀⠀⠀⠰⠰⠀⠀⠀⠀⡀⠀⠀⣠⢭⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠰⠠⠀⣃⣀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⡀⠀⠀⡀⢀⢀⠀⠀⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⢸⣇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⡀⢇⣃⠸⡤⡀⣧⣅⠈⣆⠀⠀⠀⠁⠀⠀⠸⡿⠇⠀⠀⠀⠀⠀⡀
⠀⠀⣩⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⢀⡄⡄⠀⣄⠀⢠⠀⡀⡀⢠⡀⠀⡄⠀⠀⢠⠀⠀⠀⢠⠀⠀⠀⢀⣿⡇⠀⠈⠫⣶⡽⠃
⠀⠀⡡⢖⠀⠀⠀⠀⠀⠀⢀⠀⠀⠣⡀⠐⠇⠧⠀⠷⠄⠸⠀⠬⠄⠀⠧⠀⠧⠉⠆⠮⠐⠦⠰⠥⠀⠀⠠⠻⠛⠃⠀⠀⠀⣿
⠀⠀⠖⠉⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠁⠈⠁⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠁⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⢴⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠄⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠂
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢄⠐⠀⣠⠆⠀⠀⠀⡆⠀⡆⠀⢀⢀⠀⠀⠀⢀⣀⡀⠀⠀⣄⠀⠀⠀⢰⡿⠁⠀⢈⠀⠄⠀⠀⠁⢈⠄
⠀⠀⠀⠛⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠐⠉⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢀⠀⠀⠔⠋⠈
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⡀⠀⠀⣧⡀⠀⠀⠀⠀⡀⣀⠀⠰⠆⠀⠀⠀⠀⠙⠙⠓⡚⠋⠤⠴⠶⠶⠦⢄⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠈⠡⠤⣤⣀⣀⣀⡢⢄
⠀⠀⠠⠄⢇⠀⢐⣻⣶⠆⠀⢦⡳⠭⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⢐⡀⠀⡿⠉⠋⠉
⠀⠀⠀⠀⠐⣀⠀⠹⠿⡾⠁⠳⣭⠍⠀⣠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⢀⡐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀⠀⠴⣠⠀⠀⢤⠀⠰⠖
⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⢠⡁⢁⠀⡈⠁⠀⠁⠀⡠⠁⠀⠠⠤⠔⠰⡖⠀⢠⠆⠔⢃⠠⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⣔⠀⠠⡵⠉⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠉⢀⠀⠂⢰⡀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠈⠀⠉⠑⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⢀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢐⣟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠀⠀⠐⠀⠀⠀⠀⠐⠁⢢⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠅⠺⠇
⠀⠀⠀⠐⠐⡆⠀⠀⠁⠠⠻⠀⠀⠀⠀⡴⠢⠄⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠘⠗⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠈⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠘⠀⡀⠀⠀⠀⠀⡁⠀⠉⠉⠉⠲⣄⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣤⣀⢀⣄⠀⠀⠀⠈⠩⠁⠀⠛⠀⠢
⠀⠀⠀⠀⡀⢀⡀⣀⠂⠀⠀⠀⠀⡀⢀⠐⢀⠁⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠙⠲⠤⠤⣄⠀⠈⢀⠀⠀⠀⠀⢩⡼⣻⣘⠛⢺⡏⣻⣷⢦⣵⣴
⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠘⠁⡀⠅⠀⠀⠀⠀⠈⠀⠙⠻⠀⠈⠙⣧⡀⠀⢉⣧⣠
⠀⠀⠁⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣉⣼⣿⢿⣿⡏
⠀⠀⠀⠂⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠐⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠿⣽⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠈⠻⡿⠃⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⣐⣈⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠨⢄⠀⠀⠀⠘⠉⠁⠀⠀⠀⠀⣠⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠆⠈⠀⠀⠀⠀⠀⠀⠠⠄⠈⠀⠆
⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠉⠀⠀⠀⣄⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⡎⠩⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⠀⡀⣰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠐⠐⠀⠖⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠊⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠈⠂⢀⠄

⠴⡈⡧⢩⠦⡅⠀⠀⠀⠀⠀⣼⠁⠀⠀⠰⠆⠀⡀⠀⠉⠀⠀⠀⠀⠀⢁⣸⣦⠤⠛⠉⠀⣾⡿⣿⣿⣿⣟⠛⠛
⠁⠀⠀⠀⠄⠴⠀⠈⠀⢀⡼⠁⠴⢦⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠘⣾⡷⡀⠸⠀⠀⢀⡟⠀⠠⡏⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠁⡀⠀⠀⠀⠀⡁⠀⡠
⠀⣰⣿⠀⠁⠀⠀⢀⣾⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢐⡄⠀⠀⢰⡇⠀⠷⣄
⢠⣿⡏⠀⢀⡄⠀⠈⢿⣦⣀⣰⡏⠀⠁⠀⠀⠀⠀⠀⢀⣰⣦⣀⠀⠀⠀⠀⠀⡀⠀⠰⠟⠀⠀⣾⡅
⢸⣿⣷⠀⠈⠷⠄⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⠈⣷⡀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠛⠁
⠘⣿⣿⡇⠀⢀⣠⣴⣿⣿⣾⣿⣿⣶⣦⣄⠀⠀⠉⠀⠀⠇⢠⡜⠃⠀⠀⠀⠉⠀⢀⣴⣶⣶⣿⣿⣿⣷⣦⣄
⠀⠈⠉⡀⠀⠻⠷⠛⠛⠛⠛⠋⠛⢛⡛⣿⠿⠆⠀⠀⢰⡇⢼⠆⠚⡇⠀⠀⠀⠰⠿⠟⠋⢉⣩⣀⠭⠀⠐⢡
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠛⠓⠊⠉⠀⠁⢀⡆⠀⢰⠇⣶⠄⠘⡃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⡀⢡⡭⠃⠀⠀⢀⡀⠀⠀⠀⠀⢀⣰⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣷⠰⠂⠀⠀⢠⠄⣠⠀⢀⡈⡉⠀⢀⡀⠀⠀⠈⠙⠂⠀⠀⠀⠀⢻⡇⠀⠀⠀⠰
⣠⣦⣾⣄⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡶⣶⣾⣾⣷⣾⣻⣿⠧⡄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇
⠉⠀⠈⠁⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠀⠡⣼⣿⣿⣿⣿⣿⣧⣀⠀⠀⠉⠁⠀⡐⡀⣄⠀⠀⠷⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⡿⠀⠀⠀⠀⠀⠀⢫⣽⠟⠃⠙⠿⠏⠀⣠⠀⠀⠀⠀⣄⡈⠀⠀⠀⠆⠀⠐⠀⠈
⠀⠁⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⣧⠀⠀⠀⠚⡄⠀⡄⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⣀⠀⠀⠀⢠⢷⢢⠂⡀⠀⠀⣞⠂⠸⡄⠀⠀⠉⠉⠘⠉⠉
⠈⠻⠋⠀⠀⠀⢀⡴⠾⡄⠀⠀⠀⠀⠛⠒⠀⢠⣿⡄⠀⠀⡒⠀⣀⡀⠀⠀⠀⣠⠇⠀⢹⠀⠀⠀⠠⠤⠄
⠀⢀⠀⠀⠀⡴⢩⠄⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⢯⠀⠀⢸⡇⠀⠀⢠
⠀⠀⠀⠀⢠⠐⢈⣘⠀⠀⠀⢀⢀⡀⢀⣄⠀⣾⣿⠁⠀⢠⠀⠀⣽⣇⠀⠀⠀⠘⠶⠶⠟⢁⠀⠀⢿⡤
⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⢴⠃⡊⡥⠀⠀⠀⠛⠋⠀⠀⠀⠀⢠⣿⡏⠀⡀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠒
⠀⠀⡀⢰⢄⡀⠸⣄⠁⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠉⠀⡀⠀⠀⠀⡓⠂⠀⠀⣤⣶⣷⣴⢆⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⡠⠔⠀⠁⠀⠀⠀⠀⠂⠈⠁⠐⠀⡀
⠀⠀⠀⠀⠐⠄⠀⠀⠂⠀⠠⠀⠠⡀⠀⠀⡀⠀⡀
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⢀⠄⠀⠀⠤⢀⠘⠍⠁⡁⠈⠈⠀⠁⠈⠈⡉⠐⢂⠤⠀⠈⠂
⠀⠘⠀⡂⠀⠀⠀⠨⢀⣠⠖⠊⠁⠉⠑⠳⣤⣠⠁⠀⠀⢀⡃⡈
⠀⠀⠀⠈⠐⢤⣤⣣⣋⣧⠀⠀⠀⠀⠀⠀⣼⣛⣒⣤⢴⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠀⠘⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢶⣶⢶⡀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⣧⣾⣿⣛⣿⣿⣩⣯⢞⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠄⠀⠀⠀⠀⠀⣠⣾⣷⣤⣴⣋⣻⣏⠉⠉⣿⣻⣬⣯⣬⣿⣿⣿⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⠿⠿⢿⢻⠛⠶⠾⢿⠿⢿⣿⣿⣦⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⠉⠒⠠⢁⣪⣢⣮⣾⣼⣴⣕⣥⡂⠡⠐⠋⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⣀⠀⢨⣽⣿⡿⠁⠀⠀⠀⠀⠈⢻⣿⣧⠄⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⠷⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡋⢻⡙⠳⣤⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣼⡟⠛⢛⡴⠞⠱⣴⡿⢵⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡗⠉⢠⣤⡿⢻⣿⣧⣀⡀⢝⢋⣶⣿⡿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿⠇⣤⡞⣃⠀⠹⣭⣏⣹⣿⠜⣠⢻⡻⡤⠿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣏⣀⡔⣋⣥⡤⣭⣭⣷⣮⣭⡽⣷⣾⣫⣭⣤⡞⡷⣿⣦⢀
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣾⢋⣹⣟⣩⣙⣧⣶⣴⡯⣬⣿⣽⣯⣭⣤⣥⣽⣿⣿⢧⡤⣤⣝⣆
⠀⠀⠀⠀⢰⠊⣫⢏⣠⣴⣻⠿⠋⠉⢝⡋⠉⠀⠀⠀⠁⠀⠐⣠⠖⠉⠉⠛⠢⣮⣿⣿⣬⢽⣧
⠀⠀⠀⠀⠘⣌⢹⢷⣮⣀⠀⠠⡶⢿⢿⢷⡀⠀⢀⠀⠀⡅⣸⣿⣥⡀⠀⠐⠛⠃⠠⢞⣥⣶⠟
⠀⠀⠀⠀⠀⠈⠛⠢⢐⠩⠛⠚⣒⠷⣤⣠⡇⠀⢸⠂⠆⠇⣿⠀⣙⣁⠭⢐⠂⣭⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈⠙⠒⠹⠁⠀⠼⠀⠀⢠⣿⡧⠤⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠸⡆⠀⠑⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠇⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠘⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡀⢹⣄⠀⠀⠀⠀⡄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣽⣳⣤⣤⡾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣔⣂⣀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣶⣣⣾⣀⣴⡅
⠀⠀⠀⠀⠀⠀⠲⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣠⢀⠆
⠀⠀⠀⠀⠀⠀⠘⢾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣯⡻⣿⣿⣿⣿⣿⡶⠟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢟⣽⣿⣿⣿⣿⣿⣿⡜⣿⣿⣿⡎⡻⣿⣿⣿⣶⣶⠶⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⢿⣿⣿⣿⠿⣿⣿⣷⢸⣿⣿⣿⣷⡝⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⣿⠏⠀⣸⣿⡿⣼⣿⣿⣿⣛⣋⣈⢻⡿⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡷⢠⣿⣿⠿⣛⣿⣿⣿⣿⣿⣿⣧⣽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣈⠉⠁⣰⣾⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⣄⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣄
⠀⠀⢀⣠⣴⣾⣿⡟⢫⣍⣙⣿⢿⣿⣿⣷⡼⠛⣩⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣟⣿⣍⠛⢻⣿⢿⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿
⢠⣾⣿⡟⠉⠀⣼⣿⣿⣿⠿⢿⣷⣾⣭⣭⣥⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣮⢻⡆⠊⠙⠀⠀⡿⠸⢹⣿⣿⡿⣿⣿⢻⣿⣿⣿
⢻⣿⣯⠁⣠⣾⣿⠋⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⠀⠀⠀⠜⠁⠤⠟⣿⣿⡇⣿⡏⣼⣿⣿⣿
⡞⢡⡟⢠⡛⢻⣿⠆⠀⠀⠀⠀⠀⠀⠈⠙⠻⠟⠋⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⢹⡿⣿⢠⡿⢹⡏⠁
⠻⠞⠀⠀⢹⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠩⣿⣿⣿⣎⠻⡼⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠈⠃⠑⠝⠦⠓⠈⠷⠄
⠀⠀⠀⠀⢸⡋⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣿⣶⡶⢦
⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣷⣯⣿⣿⠿⢿⣿⣿⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣧⠀⠈⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣀⠈⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡷⡏⠟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⡗⣴⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⣿⡏⢸⠹⠿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠈⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡛⠀⠠⠦⠀⠀⠀⠈⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⣀⣤⡶⠶⠛⠋⠉⠁⠀⠀⠀⠀⢠⡶⠀⠀⠀⠀⠮⠇⠀⠀⠀⠰⠂⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⣄
⠀⠀⠀⢠⣾⡋⠀⠀⠀⠀⠀⠤⠀⠀⢠⠄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⡀⣀⣷
⠀⠀⠀⠀⠙⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⡶⢚⡩⠝⠉
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣽⣟⠲⠶⣦⢤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠲⠖⠒⠂⡈⠉⠁⠤⠚⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠾⠷⢶⣤⣽⣦⣄⣀⣀⣴⡀⢠⡴⠷⠞⠓⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡇⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⡛⢓⠒⠀⢂⣩⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀⢠⠀⢹⣿⡟⠚⠃⠉⠙⡆⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠱⣀⠂⢸⡟⠀⠀⠀⠀⠀⢹⠀⠀⡏⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⠠⡽⢦⣸⠇⠀⠀⠀⠀⠀⠀⠃⢸⢃⡎⢰⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⣃⣷⡄⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⢸⠀⡮⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢢⣿⣿⢻⠀⣀⠀⡈⠀⠀⠀⢸⡿⢸⠁⣸⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣾⢸⡇⣸⠃⡏⢠⠀⠀⠀⠀⢸⡇⡞⢠⠇⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣻⣿⣧⢻⣤⠁⡄⠀⡄⠀⡄⣾⣟⣴⢟⣋⠡⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⢿⣿⣾⣿⣆⢱⠀⣇⣤⣷⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⣴⣆⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⢠⣶⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣴⢾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⣫⡖⣦⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢟⠵⣯⣼⣿⣿⢸⣧⡴⢞⠛⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡔⢻⣶⣿⣿⡿⣟⠈⢷⠄⠙⠃⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣼⢋⣴⣉⣿⣿⠿⠋⣵⠄⠉⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣴⠟⣵⣧⣼⡿⢋⡅⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⠀⠀⠀⠀⢀⡾⢫⣾⣿⠟⠉⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⡇⠀⠀⠀⣠⡟⣱⣿⡿⣁⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡇⠀⠀⣰⡟⣼⡿⢫⡄⠉⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⠀⠀⠀⠀⠀⢸⣟⢱⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣧⠀⢰⡟⣼⠟⣥⡌⠀⠀⠀⠀⠀⣠⣾⡿⠿⣿⣿⣿⠿⣿⣿⣶⣄⠀⠀⣿⣿⣋⣣⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡻⣤⣿⢱⠏⣦⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⠀⢸⣿⣿⠀⠈⢻⣿⣿⣦⣼⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⡴⠛⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣿⡿⠋⠀⠀⢸⣿⣿⣿⣿⣿⣿⣏⣹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⣦⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠛⡟⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢈⣿⣶⡄⠈⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣟⢛⡿⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠉⣷⣦⣤⣟⢷⣭⡻⢶⣦⣤⣤⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⡋⣻⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠚⠉⢻⣿⣿⣷⣬⡛⢶⣭⣛⠿⣍⣽⣿⠛⣿⠟⣿⠟⣿⣏⣹⣧⠼⣛⣵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠈⠙⠻⠶⣮⣭⣽⣛⣓⣛⣛⣫⣭⣭⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠐⠒⡒⠒⠒⠂⠄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠠⠒⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠀⠀⠀⠀⢑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⡄⠀⠀⠀⠠⣀⡠⠂⠀⠀⢠⠂⠈⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢁⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠘⠀⠀⠀⠀⠀⠔⠒⢢⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠀⠒⠲⠌⠉⣁⡈⠁⠀⠀⠀⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠐⠁⠉⣄⣀⠤⡄⠂⠂⠉⠀⠀⠀⠀⠀⠀⠀⠗⢄⠀⠀⠀⠉⢉⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠑⣀⠠⠐⠋⠁⢀⠔⠁⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢈⡐⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⢁⠀⣀⠄⠊⠁⢀⠜⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠀⠰⠀⠀⠀⠄⠀⠰⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠙⠒⠀⠀⠀⠐⡆⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣷⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣶⣶⣾⣷⣷⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣴⣶⣤⣤⣦⣤⣀⢀⡀⢀⣄⠿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣭⡟⠻⠟⠻⠿⡿⠻⠿⠿⡿⣿⣿⣿⠿⣿⣿⣿⡏⠉⠙⠙⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠾⠟⢶⣴⡶⣮⣥⣽⠏⠹⠓⠒⢻⣿⣿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠭⠉⠉⠿⠿⠀⠀⠀⠀⢸⣿⣯⣻⣿⣿⣿⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠘⠒⢺⣿⣿⣿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⣥⣀⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠘⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣥⣄⣀⡀⠀⣶⢖⣦⣶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠿⠿⠿⠿⠿⣿⣿⣿⣻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⢪⡙⣿⣿⣿⡆⠀⣼⡟⠀⠀⠔⢾⣿⣷⡆⣤⠀⣠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠈⠱⣾⣿⠄⠀⠀⠀⠀⠀⢀⣠⣿⣼⣿⡀⣄⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠄⠐⢻⣿⠃⠸⠟⠂⠀⠀⠙⠿⠿⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣤⣤⣄⣴⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣴⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣔⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠻⠿⠿⣷⢛⣿⣿⣿⣿⡟⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣟⠟⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠿⠉⠈⠉⠁⠼⠛⠛⣿⣿⣿⣿⣿⠿⣿⡿⢿⣟⡿⠿⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⠳⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠚⠉⠉⠉⠉⠀⠈⠉⠉⠙⠛⠛⠛⠃⠈⠉⠓⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣼⢿⡇⠀⢠⣀⠀⢀⠀⣠⣄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠟⠛⠁⠺⡍⠀⠃⠂⠘⠻⢿⠀⡌⠉⠉⠟⢀⠀⠀⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⢸⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⡀⠁⠀⠀⠀⠌⠃⡀⢿⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⢰⠿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠃⠀⠀⠀⠓⠇⠀⠀⢸⣷⠀⣰⣾⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠈⠁⠁⡿⣽⣷⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠘⣇⠈⡄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣄⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡦⠀⠠⡅⠀⠈⡟⡂⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⢸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠊⠀⠀⠈⠀⠐⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠚⡃⠀⠀⠀⠀⠀⠀⡇⢸⡀⢀
⠀⠀⠀⠀⠀⠀⢀⡄⢀⠰⡏⠂⡀⠀⢰⡆⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⠀⢀⠀⠐⠅⠀⠀⠀⠀⠀⠀⠠⠈⢳⣼⠃
⠀⠀⠀⠀⠀⠀⠘⣷⡞⢀⠁⢠⡇⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠑⡦⢀⠀⠀⠀⠀⠀⢠⢠⠀⠸⣿⣀
⠀⠀⠀⠀⠀⠀⣤⣿⡟⠆⠀⠘⢰⠀⠀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⣠⣦⣤⣼⣥⢤⠤⠄⢀⡀⠈⡌⠀⠠⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠂⠀⠾⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⣀⣤⣤⣴⣶⡿⣛⡿⢿⡛⠩⠀⠀⠒⣰⠃⠀⢣⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣏⣛⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣸⢛⣯⠉⠀⣠⣴⣶⠶⢶⠒⢻⢻⣏⣿⣿⣿⡿⣊⡽⠷⠊⠁⠀⠀⠀⠀⠐⠁⠀⡀⠈⠀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠐⠀⠀⠀⠀⠀⠀⢀⣼⡟⢀⣠⣴⡏⠩⢿⣿⣷⣦⣷⣾⣿⣿⣿⣿⠟⣋⢥⣴⣏⣤⣄⣀⣂⡠⠈⠀⠀⠀⠀⠈⣧⡀⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡄⠰⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠧⣥⡶⠾⠿⣿⡿⢿⣿⠿⠿⠛⠋⠁⣈⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠃⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣿⡇⢰⡀⠀⠀⠀⣿⣿⣿⣿⣟⣃⠉⠀⠀⡀⠉⢡⣾⣋⣴⣶⣶⣶⢦⡼⣿⣿⣿⠿⠋⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣇⠈⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣷⣔⣈⢢⣔⣿⡿⣿⣿⣿⣿⡍⠈⠁⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠘⠇⠀⠀⠀⠀⠀⢹⣼⠀⠀⠀⠀⠀⠰⡆⠀⣂⠀⠀⠀⡄⠀⠀⠀⢀⣼⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⢤⣄⠘⠐⡀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠀⠀⠀⠆⢀⠀⣏⠀⠀⠄⠀⣀⡰⠞⠀⠛⠀⠀⠀⠀⢨⠀⣠⢸⡿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⠈⠳⠄⡀⠈⠧⡀⠠⣀⢀⠀⠀⠈⠉⠁⠂⢄⡎⠀⣿⠀⢀⢂⣼⣿⡆⠀⠀⠀⠀⠀⠀⠘⣤⠞⢹⣿⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⢉⣷⡀⠀⢀⠃⣿⣄⠀⠀⠀⠀⠀⢈⡇⠆⢻⣇⡈⠘⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⣸⠗⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣄⣮⣭⠀⢳⡎⢸⣿⣿⣪⣤⣶⣶⡀⠉⡄⠀⠈⣿⠗⠀⠈⠿⠛⣻⣿⣷⣤⣤⣤⡤⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡠⠷⢿⣶⣿⠏⣿⣿⡿⢋⠹⡿⢮⣶⠀⠀⢀⡿⠃⡀⠄⠀⢠⡿⠉⣿⡎⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⣷⣿⣿⡇⢉⠻⠯⢅⣐⣲⣿⣧⣦⣈⣴⠊⡁⢀⢀⣾⠅⢘⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠆⢇⡀⢸⣿⣿⣿⣿⣿⣇⣿⣶⣰⣾⡿⠿⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠉⠉⠉⠉⠁

⠀⠀⣤⣤⣀⠀⣀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⢀⣀⣀⡠⠀⢤⣤⣤
⠀⠀⢹⣿⠋⠀⠹⣿⡏⠀⠹⣿⡯⠉⢽⣿⡷⠀⠲⣿⡶⠂⠰⣾⡶⠂⠰⣟⡶⠃⠈⢻⡿⠏⠈⢻⡿⠇⠀⠛⣿⠇⠀⠐⢿⠏
⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠛⠀⠀⠈⠛⠀⠀⠀⠻⠃⠀⠀⠸⠃⠀⠀⠸⠇⠀⠀⠈⠏⠀⠀⠀⠛⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣀⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⡀⠀⠀⠈⠛⠀⠀⠀⠀⠀⣀⡴⠄
⠀⠀⠀⠀⢛⣼⡺⠀⠀⠀⠀⠀⣻⣿⣿⠋⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣿⣿⡟⠁⠀⠀⠀⠀⢠⣆⣀⠀⢿⣿⡘
⠀⠀⠀⠀⠀⠉⠁⠐⣶⣾⣇⠀⠈⠀⠉⠀⣾⣿⣿⠀⠀⠀⠀⠀⢴⣿⣿⡃⠈⠁⠈⠁⠀⠀⣀⠀⠘⢻⣿⣿
⠀⠀⠰⠖⠀⠀⠀⠰⠿⢿⡏⠁⠀⠀⠀⠀⠈⠉⠉⠀⢠⣾⣷⠀⠀⠛⠉⠁⠀⠀⠀⠀⠀⡞⡹⠇⠀⠈⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠀⠀⠀⠀⠘⠿⠾⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠒⠀⠀⠀⠀⠀⢀⠀⠁⠁
⠀⠀⠀⢀⡾⠿⠞⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣤⣦⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠖⠄⠉⠃
⠀⠀⠀⣿⣧⠀⠀⢈⣿⡇⠀⠀⣀⣀⡀⠀⠀⡀⠀⡀⠀⣀⡀⢘⢋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣡⡀⢀⡀⣀
⠀⠀⠀⠻⣿⠀⠔⢺⡿⠃⠀⠨⢛⣧⣧⠀⠀⡟⠶⢻⠀⡧⠄⢸⢴⠃⣯⢼⠁⢱⠎⠀⠀⠈⠁⠁⠀⠀⢛⠋⣉⣡⣟⡃
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠈⢀⠃⠃⠈⠀⡁⠁⠈⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠊⠀⠀⠀⠀⠙⠁⠘⠃⠑⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠈⣠⣤⡀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣛⣛⣂⠀⠈
⠀⠀⠀⠰⣾⣷⣾⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡄⠀⠀⠐⠽⠿⠯⠛⠄
⠀⠀⠀⢀⣝⣛⡿⠿⢦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⡖⠀⠀⠠⣄⣀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠉⠹⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠠⡾⠃⠀⠀⠀⡞⠛⠆⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⠤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⣀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠤⠄⠀⠀⠠⠤⠀⠄⠀⠀⠀⠀⠿⠿⠶⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⣀⡀⢮⣷⡄⠀⠀⠀⠀⢀⡴⣆⣠⣄⣤⣀⡄⠀⠀⠀⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢮⠅⠘⠛⠀⠀⠀⠈⠀⠠⢤⣀⠀⢠⣀⣈⡀⠀⠁⠠⠄⢀⣠⡼⠛⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣸⣿⣇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠈⠀⠀⠀⠁
⠀⠐⠀⠀⠄⠤⠀⠀⠀⢀⣠⡝⠁⡈⣿⠇⠀⠀⠀⠀⠀⠀⢠⣦⣴⣦⣴⣦⣤⠀⠀⠀⠀⢿⡏⠋⠁⠀⠀⠀⠀⠒⠒⠲⠆⠐
⠀⠀⠀⠀⡀⠀⠙⠿⠛⠛⣿⣇⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠒⠒⠒⠘⠋⠙⠀⠀⠀⠀⠀⠐⠋⠋⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠙⠋⠉⠃⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡤⠴⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠾⢭⡀⠀⠈⠢⠱⡀⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠡⠤⠀⠀⠈⠒⢄⠀⠑⡌⢈⠄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠃⠀⠒⠂⠂⠐⠢⠦⢤⣤⣈⣼⡇⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠛⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠤⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⣠⡴⠂⠈⢉⢛⡶⢠⣄⣀⠀⢀⠀⣀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⢶⠶⠊⠉⠁⠀⠀⠀⠁⠞⢠⠋⠀⢀⡀⣉⢁⡀⠀⢀⠤⠀⠀⠀⣀⣠⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⡠⠞⠁⠈⠀⠀⠄⠁⠀⢀⣀⠤⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⠀⠀⠀⠘⢀⡎⢠⠀⠀⠀⠐⠀⠤⠄⣤⢞⡉⢉⡄⠀⠐⠒⠀⠙⠲⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⣸⠁⢸⠀⠂⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠘⣧⠀⠀⠀⢠⣿⠀⠈⠧⣤⡀⠀⠀⠀⢀⣀⡀⡄⠀⠀⣀⠀⠀⢀⣀⡤⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⢻⣇⠀⠸⣸⡿⠀⢰⠀⠀⠉⣡⠉⠉⠱⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠘⣿⣇⠀⢿⣷⠀⢸⠀⠀⢠⡇⠀⠀⢠⠂⠀⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⡞⣄⢸⣿⡄⣸⠀⣠⡟⠀⢀⢄⡌⡘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠈⢿⣦⡹⣆⡻⠱⣯⠼⠟⠀⢀⣡⡾⢊⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⠛⡛⠁⠘⢵⠎⠠⢴⠿⠿⠟⠋⠑⠊⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣲⣞⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡔⠐⠀⡀⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⠀⢠⡀⠤⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⣄⡔⠁⢁⠠⠔⠻⢧⣰⡾⠀⠀⠀⠀⠀⠀⠀⠆⡀⠈⠀⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢨⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠢⡔⠠⢄
⠀⠀⠀⠠⠹⢷⡀⢁⢀⠀⣀⠄⠩⢥⡀⢠⠄⠀⠀⢀⢰⢮⠀⠀⠀⠂⠐⠢⠀⠀⠀⠀⠀⠀⠀⠄⢀⠘⢥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠆⠃
⠀⠀⠀⠁⠀⢀⣯⣄⢰⡟⠉⠗⠐⡺⠁⠟⠂⠀⠀⠈⣿⠀⠀⠀⠈⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠈⠉⣳⡀
⠀⠀⠀⠀⠀⠙⢿⣧⡦⣙⠉⣋⠶⠆⢶⣺⠂⡄⠀⠀⣙⠂⠀⠀⢰⠢⡔⣼⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡧⠠⠄⠀⠀⠀⢀⠀⠀⠈⢀⣀⢁⠄⢀⣯⠄
⠀⠀⠀⠀⢠⠀⠀⠙⣷⠹⠀⡏⡲⠊⣿⣉⠾⢋⡄⠀⢨⡄⠀⠀⠈⠲⡇⢸⠦⡄⠄⢀⠏⡁⠀⣠⠀⢠⣀⠳⠀⠀⠀⢐⣾⠀⠈⢉⠘⠈⢐⣀⢈⠉⠓⠄
⠀⠀⠀⠀⢂⠀⠀⠀⠈⠑⠠⠆⠰⠢⠿⠗⠊⠌⠀⠀⠀⠙⢄⠀⢲⠃⢸⡟⢺⠁⣴⢯⡀⠂⡠⠀⠀⠘⢫⢧⡤⠐⡄⠨⣯⠀⠀⠀⢀⠀⢸⣷⠗⠋
⠀⠀⠀⠀⠀⠑⠠⡀⢀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠱⠄⣧⣸⢇⡆⡜⢀⢿⠶⠂⠀⠀⠀⠀⠀⠛⠰⠦⣤⣛⣻⠀⡐⡂⢊⠤⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⢀⠀⡖⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀
⠀⠀⠀⠀⢰⣷⢙⡇⠿⠻⢑⡩⠄⢛⣂⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⡁⢀⠀⡀⠀⠀⠑⠦⠀⠀⠀⠀⠀⠀⠀⢀⠄⠲⠂⠂⠀⠀⠀⠁⠂⢄
⠀⠀⠀⢀⣸⡔⠁⠁⢀⠔⠊⠀⣴⢿⡿⡆⠀⠀⠀⠀⠀⠴⠄⠀⡃⣺⣾⣁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⣧⢠
⠀⠀⣠⣌⡝⠀⠀⢰⣾⣤⡸⢒⡆⢩⣥⣦⠀⠀⠀⠀⠈⠀⠀⠀⣸⡿⢿⣿⣷⡠⣴⠤⠀⠀⠀⠀⠀⣠⡈⣀⠀⠠⠄⠀⠳⡄⠀⠀⠀⠀⡆⠀⠈⢸⡄
⠀⠀⠻⢻⡇⠀⠀⠻⠏⢛⢠⣾⡿⣏⡿⢻⠀⣤⠀⠀⠀⠀⢁⢠⣇⣻⡦⡨⡈⠃⣸⣿⣦⡀⠀⠀⠀⣿⡄⠀⠬⡥⠈⠃⠀⢿⣄⠔⡀⠘⢥⠀⠀⣿⡇
⠀⠀⠀⣼⡟⠀⠀⠀⢄⠳⡉⠉⣧⠞⣶⣆⠺⠁⠄⠀⠀⠉⠜⣯⢧⠀⠐⣿⣿⢄⣙⣋⠍⣄⣂⠀⠀⢹⡆⢾⠃⠁⠀⠄⠀⠁⣯⠄⠾⡂⠀⠰⣄⡞
⠀⠀⠉⠟⠙⡄⠀⠀⠈⠳⢧⣰⡇⣰⢩⣇⣤⢠⠆⠀⠐⡄⠀⠒⠛⠢⡀⠀⣡⣜⣿⠋⢼⣿⠏⠀⠀⠀⠑⡈⠂⠀⣠⣈⠙⡧⣼⣤⡸⠷⡏⠃⣋⣈⠆
⠀⠀⠀⠀⠈⠈⠢⢄⠀⠠⣀⠈⠉⠉⠉⢋⠔⠁⠀⠀⠀⠈⠢⡀⢀⠀⠈⠙⠒⠧⠾⠤⠆⠟⠀⠀⠀⠀⠂⢤⢫⡐⢸⠋⠀⠅⠉⠁⢀⢀⡧⠿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠂⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠠⠤⠤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠳⠶⠦⠴⠑⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠛⠙⠉⠉⠉⠉⠀⡩⡶⠒⠒⠒⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣭⣍⣩⣤⣤⣤⣄⠀⢠⡼⠀⠁⠀⠀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣛⣛⣹⣿⣯⣭⣭⣾⣏⣉⣁⣤⣈⣐⠒⠂⢤⣀⣀⠀⠈⠑⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⠟⢿⣿⣿⣿⣿⣦⣴⡛⠶⣀⡀⡀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣶⣮⡁⣀⡀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣧⠁⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡴⠃
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣮⡏⠉⠓⠉⣿⣆⢤⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉⠉⠉
⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠫⠏⣉⡟⠂⠀⠑⠀⠂⠀⠚⠉⢳⣆⠈⢳⠀⠀⠀⠀⠐⠀⠀⠀⣀⣤⣤⣴⣶⣦⡤⣤⣀
⠀⠀⢀⣴⣿⣿⣿⡿⠤⠀⠀⠀⠋⠀⠀⠀⠠⠠⢶⣄⡀⡈⢰⣿⢇⢹⡇⠀⠀⠀⠀⠀⣴⣿⣻⣟⠽⠅⠉⠉⠙⠑⠈⠙⠲⣄
⠀⢠⣿⣿⣿⣿⡏⠁⢀⡀⠀⠀⠀⠀⣀⣂⣴⣿⣿⣿⣷⣍⠻⢟⣾⣽⢿⠀⠀⠀⠠⣹⣯⣽⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣
⠀⣿⣿⣿⣿⣿⣶⣴⣿⣇⠀⣥⣠⢶⠋⠉⠻⣿⣿⣿⣿⣿⣿⠛⠖⠀⠈⠀⠀⢠⣺⣿⣟⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣵⣾⠅⠂⠀⠀⡀⠀⠹⣿⣿⣿⣟⠑⠃⠀⠀⠀⠀⠀⣷⣯⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣹⣶⠿⠿⠿⠿⣷⢶⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠙⠙⠳⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚
⠀⠀⠻⣿⣿⡷⠿⢿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡿⠶⣶⣶⣼⣧⣤⣤⡀⠀⢠⠀⠀⠀⠀⢤⡙
⠀⠀⠀⠀⠁⠀⠀⠘⣷⣿⣿⡟⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠐⣰⠀⠀⠀⠀⠀⠀⢀⣷⣿⠟⠻⣇⠀⢈⠀⠉⠘⠢⢆⢠⣤⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣶⣆⢰⣆⡄⡇⠀⢠⣜⣘⣿⣿⣇⠀⠀⠀⠀⡀⢸⣿⣿⠊⠀⠀⠉⠈⠃⢀⠲⢄⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧⠂⠀⠀⠀⠈⡉⢹⠛⠀⣠⡶⣸⣧⣾⣿⣏⡓⠀⠀⠀⠀⠀⠠⢨⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠸⣿⣟⠀⠀⠀⢰⣿⣹⣆⣴⣟⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠲⠶⠶⠼⣿⣟⣡⣿⣿⣧⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠃⠒⠋⠒⠲⠶⠦⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

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