Birds Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢆⣠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣴⣿⣿⣿⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣿⠟⠽⠋⠿⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠿⠚⠿⣤⠀⢐⣺⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠰⠃⡤⠵⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⢀⣮⡁⢠⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠀⢠⠦⣥⢦⣟⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⣾⣿⣏⠱⣬⡳⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡐⠁⠀⣿⡃⡌⢷⣌⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⠎⣀⣠⡞⠁⢀⣼⣿⡿⣮⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠿⠥⠤⣤⣀⣄⣹⣤⣨⣹⣟⣰⣿⣿⡿⣝⡾⠉⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢜⣿⡥⠂⠉⠉⠙⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠿⠻⣷⣾⡿⢾⡦⠆⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈⠁⠀⠀⠄⠀⠀⣀⣥⣬⣙⠚⠆⠀⠤⠄⠈⠳⡤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠡⠀⠂⢁⣀⣀⢤⢶⣿⣾⣿⡿⠟⠋⠈⢳⣾⣴⡎⢘⣷⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠰⣿⡀⠀⣀⣄⣤⢾⣿⣿⣦⣬⣿⠿⠋⠀⠀⠀⠀⠀⢻⡝⠻⠿⣥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣺⣿⣧⣂⢰⢿⢉⢻⢿⣿⣾⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⠒⠋⠁
⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣉⣧⣷⣾⣶⡿⠾⠛⠋
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣴⡿⠟⠻⣈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢫⣿⣿⡾⡿⠁⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠛⠉⢻⡷⢺⣷⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠃⠀⠀⣠⡆⣠⣇⠈⠉⠀⠀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⡴⢶⡿⠁⣿⢿⡿⢳⣿⢾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣉⡁⠘⠁⠘⠁⠀⢸⡇⢠⣿⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣌⠻⣿⠿⠗⠶⠄⢀⣸⡇⠈⣿⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡿⠿⣿⣿⡆⠛⠀⠀⠀⠀⠀⠀⠀⠀⠣⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⢼⠟⣁⢾⣟⣿⣿⠀⢰⣧⠀⠀⠀⣴⠀⠀⠰⠋⠈⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⢻⣄⠀⣶⡄⣿⡿⢻⣾⣿⡇⠈⠁⠀⠀⢸⣿⠆⣴⠇⠀⢿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⠉⠀⠛⣤⣬⡆⢾⠿⣉⠀⠀⠀⠀⠀⠸⠋⠠⢀⣶⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⢉⡤⢶⡿⠋⠀⢀⡄⠀⠀⠀⠀⠰⡟⢿⡀⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣠⡶⠀⣠⣶⢟⣧⣞⠃⢀⣠⣤⠅⠀⠀⠀⠠⡴⢱⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣠⡦⠀⠿⠁⢰⠟⣱⡿⣿⣿⠀⢸⠉⡈⠀⢴⠄⠀⠀⢠⣾⠇⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠛⣰⡟⢸⠇⢀⣴⣿⣿⣿⡿⠀⠿⠛⠁⠀⢠⣤⠀⠀⠈⠁⠀⠻⣿⣿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡿⢠⣾⣤⣿⣿⣿⣿⣿⠇⢰⡦⠤⠀⠀⠀⠂⠀⢰⡆⣀⠀⠀⣤⡿⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⢈⣹⣿⣿⣿⣿⣿⡟⢰⣷⣶⠆⠀⣀⠀⠖⢲⡀⠀⠋⢠⡀⠚⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠐⡇⢸⠃⢸⣿⣟⣼⣿⣿⣿⣡⣿⣿⠷⠀⠚⠇⠀⠀⠘⠃⠈⠀⠈⣽⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢰⠌⣻⣿⣿⣿⣿⣿⣧⣧⡀⠀⠀⠀⠀⠀⢠⠿⠇⠀⣶⠟⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣼⡇⣀⠀⣸⣟⣹⠟⣿⣿⣿⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⣴⣧⠐⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡀⢻⡇⣿⢰⣿⣿⣧⣾⣿⣿⡿⠋⠀⠙⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠀⢠⠉
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣬⣷⠚⠀⠟⣿⣿⣿⣿⣿⣿⣿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⣾⡟⢋⣭⡀⢀⢈⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠆⠀⢀⠊
⠀⠀⠀⠀⠀⠀⢠⡿⣇⠈⣿⣆⣿⢸⣿⣿⣿⣿⣿⡿⠹⠆⠀⠁⢸⠀⡀⠀⢠⠏⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠸⣿⠉⣡⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⠀⠈⠁⠀⠃⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⢠⣄⠶⠀⢸⣶⠁⣰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⢠⣾⣿⡶⠀⢸⠟⣰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣽⣿⡅⠀⢀⡀⣠⠈
⠀⠀⠀⠀⣿⣿⡟⠁⠀⢾⣿⣿⣿⣿⣿⣿⣿⣞⢁⣤⣀⣰⣿⣿⣏⠀⣠⣾⡇⠁
⠀⠀⠀⠀⠟⠋⠀⠁⣤⢹⣿⣿⣿⣿⣿⣿⣧⣽⠛⢋⣿⡏⠀⣿⣷⣾⠿⠛⡇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠰⢉⣾⣿⣿⣿⣿⣿⢳⠟⠀⠀⣼⡿⠇⠀⡀⠛⠛⢲⣾⣷⣄⡀⠀⠁⠐⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠋⡛⢄⡻⠂⣯⣀⣴⡟⠛⠛⠛⠛⠙⢻⠉⠉⣿⠉⠛⢛⡛⢻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠀⡸⠀⠃⠘⠋⣸⣿⡇⠀⠀⡇⠀⣷⢸⠀⠃⠃⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣧⣶⢣⣾⡀⠀⠀⠹⣿⡇⢠⠀⡇⠀⣿⡄⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠡⣿⣿⣿⣿⠁⠀⡀⠀⣿⡇⠸⣦⣧⠀⢿⡇⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⣴⣿⣿⣿⣿⠀⠀⣀⠀⢹⡇⠀⢸⡇⠀⢸⣇⡘⠀⠀⠀⡇⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⣸⣿⡟⢿⣿⡇⠀⠀⡀⠀⠘⣧⠀⢸⠃⠀⣼⣿⡇⠀⠀⠀⠃⠀⡏⠈
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⣰⣿⠋⢰⣾⡟⠀⠀⢠⡿⠀⢰⣿⠀⠘⠀⢀⡟⢿⠁⠀⠀⠀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⣿⡿⠀⣼⠃⠁⠀⢰⣿⠁⢀⡿⣿⠀⠀⠀⠘⠇⠸⠀⠀⠀⠀⠀⠀⠇⢸
⠀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣿⠁⣶⠛⠀⠀⠀⠺⣿⣧⣼⣇⣿⠀⠰⠀⠀⠆⠸⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⣼⠁⢀⣿⠇⠸⠉⠃⠀⠀⠀⠀⢻⡿⢹⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡟⠀⣸⡟⢀⠃⠀⠀⠀⠀⠀⠀⣸⡇⠀⠈⢿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢱⠛⠄⠀⠀⠀⠀⠀⠀⢼⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡻⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣟⡵⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣇⢀⠀⢤⣦⡤⠞⠉
⠀⠀⠀⠀⠀⢀⡿⡿⣳⣳⢫⢖⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣞⣽⡿⣣⡆⠚⠁
⠀⠀⠀⠀⠀⢸⢛⣿⣯⣿⣯⣣⠞⢻⢗⡶⣄⢀⣴⣿⡿⠿⠃⠰⣜⣴⠟⢡
⠀⠀⠀⠀⠀⣬⢮⣿⣿⡿⣵⡟⣵⣱⣋⡼⣾⣿⣿⡛⢏⢀⣀⠾⠛⢡⠖⠃
⠀⠀⠀⠀⠀⣯⣷⣽⣷⣿⣟⣿⠿⡿⣭⣽⣭⠋⠉⠁⠑⢸⡮⣡⠾⣇⢄⡀
⠀⠀⠀⠀⠀⠻⢿⠿⣿⡿⣫⡜⠋⠋⠁⠀⠀⠀⠀⢀⣼⣻⣿⣷⣾⣧⣾⣽⠐⠀⣠⠆
⠀⠀⠀⠀⠀⣀⣠⣾⣯⡜⠋⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡽⣿⣿⢻⣿⣿⣿⣷⠺⠏⠌
⠀⠀⠀⣠⣾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣾⣻⣿⣾⣿⡿⢎⠿⣫⠘⠁⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⢠⠿⠿⣴⠄⠒⠂⠀⠀⠀⠀⢠⣴⡿⢿⣿⢯⣿⣿⣴⣯⢾⡯⣷⢄⣤⣄⡀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣠⣤⣴⠆
⠀⠴⠿⠷⠂⠀⣄⣤⣤⣶⣶⣤⣴⣾⣟⣷⣾⣟⣿⣿⣿⣿⣿⣿⣏⣵⣯⣪⣽⣫⣟⠼⣋⣤⣟⣭⣟⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣮⣷⣮⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠐⠻⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡛⣛⡛⡛⠛⠛⠛⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣻⡻⠼⠴⠆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⠿⠦⠋

⠀⠀⠀⠀⣠⣶⣤⣀⡀
⠀⠀⠀⠘⠿⣿⣿⣿⣿⣷⣦⣄⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣷⣾⣿⣿⣶⣦⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣙⣿⣿⣷⣶⣾⣿⡿⠿⣿⣿⣿⣿⣶⣤⣀⠀⣀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣶⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀
⠀⠀⠀⠀⠠⣄⠀⢀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣌⡋⡻⢿⡿⠋⠁⠀⠘⠻⠿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠘⠿⣦⡀⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣾⣷⣶⠿⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢹⡃⠀⠀⠀⠀⠈⠻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠁⠀⣀⣠⠶⠚⠁
⠀⠀⠀⠀⠈⠳⡦⠀⢀⡎⢻⣶⣦⣄⠀⠀⠀⠈⠳⣄⣠⣽⣿⣿⣿⣿⣯⣉⣤⣴⣶⣦⡀⢻⡁
⠀⠀⠀⠀⠀⠀⠙⣮⠛⠀⠀⠙⠻⣿⣷⣤⣴⡿⣿⣿⣿⣯⣿⣿⣿⣿⡟⠉⠁⠀⠀⠻⠷⢷⣯⣖⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡤⡾⠿⠉⠉⠁⣁⣠⣼⣿⣿⣿⠟⠛⠋⠙⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠆
⠀⠀⠀⠀⠀⠀⢀⣼⣿⣟⢀⣀⣤⣤⣶⡼⣿⣟⣉⣉⡉⠁⠀⣀⣀⣀⣀⣠⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠙⠋⣽⣿⠟⠉⠁⠀⠀⠀⠉⠉⣩⢟⡿⠙⢧⡉⣦⠀⠙⠍⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠸⣿⡷⢷⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻⢿⣯⣟⣽⣿⠛⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⣧⠐⠛⠕⢹⣾⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠀⠀⢸⠋⡔⠀⢭⡴⠟⢻⣿⣷⣏⠁⠀⠈⢀⡀⠀⠈
⠀⠀⠀⠀⠀⢀⣿⠤⠓⠃⠉⠀⠀⣽⣿⣿⡌⠀⠀⠀⢀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠀⠀⣸⣿⣷⠃⢀⢠⣶⡻⣿⣿⣿⣿⢶⣵⣾⣧⣤⡀
⠀⠀⠀⠀⠀⠸⣿⣿⡄⣧⡏⣸⣰⣿⣿⣿⣿⡄⠙⢿⣿⣿⢷⡀⡀
⠀⠀⠀⠀⠀⠀⠹⣿⣯⣿⣿⣷⣿⣿⣿⣿⣿⣿⣤⡌⢿⠸⠁⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡾⠀⠀⢻⢻⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠁⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠗⣠⣾⠉⠁⠹⢿⣿⣿⣿⣿⣿⣧⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡞⣿⠀⣠⠺⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣺⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢾⠋⢣⣿⡟⠻⠿⠃⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡏⢿⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣷⣬⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣷⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣷⣽⣮⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⢿⣿⣿⣷
⠀⠀⠀⠀⠀⣠⣤⣶⣹⣿⣽⣿⣿⣿⣿⣽⣏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣛⢿⡙⠢⠙⣆
⠀⠀⠀⢀⣴⣿⣽⣿⣟⡿⣟⣯⣽⣿⣿⣿⣹⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣟⡲⣝⢧⠑⢈⡀⣟⡄
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣟⣾⣦⣵⣶⣿⡿⣷
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣟⣿⣿⣿⣾⠿⣫⣽⢿⣦⣄⣀⡀⣠⣾⣾⣿⣿⣿⣿⣿⣿⣿⣾⠓⠗⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣯⣿⣿⢾⣗⢻⣿⣿⣿⣿⣿⣿⣟⡛⠃⠹⣿⢿⣿⡷⠆⠰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣋⣮⣤⣆⣈⠁
⠀⠀⠀⠀⠀⠒⠲⠶⢶⣤⣤⣄⣀⣤⣄⣄⠟⠿⣿⡿⢟⡉⣙⣛⡛⡻⢿⣿⣿⣿⣟⣿⣿⣿⣿⡏⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⣿⣿⣿⢧⣞⢿⡾⢿⣾⡻⣷⠌⠐⡝⢿⣿⣿⣿⡿⣿⡿⠉⠑⠻⠷⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣿⣿⣆⣨⣿⣿⠿⣶⡼⠀⠘⢿⠛⢟⠉⣀⡤⠤⠄⠐⠂⠛⠓⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠨⢟⣊⢯⣰⠄⢀⣼⣄⣄⣹⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⠟⠛⠛⠛⠉⠉⠉⠁⡀⢀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⢀⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣫⠵⠦⠄⠄⡀⠁⠀⠁⠠⣤⣀⣀⣰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢀⠠⠂⢀⡼⠋⡤⠀⠀⠀⠚⠿⠟⠻⠿⠿⠛⠛⠋⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠴⠀⢉⠠⠊⣠⠀⠀⠀⠴⠷⠇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⣲⠶⣂⠀⠀⠀⠚⡠⠔⠁⡰⠊⢠⣾⣡⠄⣠⠟⠀⠀⠂⠀⠈⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⢊⣥⣦⡤⡖⠊⠛⠙⣠⢖⣀⠴⢚⡡⠈⡽⣷⡟⠁⠀⠀⠀⠐⠀⢠⠀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣞⣛⠭⣖⠯⠀⠀⢀⣠⣼⣏⣑⣚⠽⢚⡭⠖⠋⠀⠀⠄⠀⠀⢀⠀⢀⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠯⢍⠀⢈⣠⣤⣴⣾⣛⣻⣿⣿⣿⣖⣪⠟⠁⠀⠀⠀⠀⠀⠀⢀⠀⢳⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣋⠤⠶⠤⠭⣭⣍⣿⣷⣿⣶⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⣪⣭⠟⠁⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣇⣡⣤⣶⣶⣀⣶⣤⣤⢤⠴⠐⠒⠊⠉⠁⠀⠈⠀⠀⠀⠀⠀⢠⣤⡵⠗⠀⠀⠀⡀⠀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠟⠛⠛⣫⠽⠿⡟⢛⣭⢁⡇⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣾⡥⢀⣀⣀⠜⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠀⢀⡠⠔⠛⠁⠊⠉⠉⠑⠶⠙⠳⢆⡄⠐⠀⠛⠀⠀⠀⡀⠀⠀⠀⠀⠔⠛⠉⠀⠈⠉⣹⣿⡖⣡⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣪⠝⠁⠀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⡈⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣥⡶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣺⠕⠋⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⣠⢖⣭⠿⠋⠛⠛
⠀⠀⠀⠀⠀⠀⣀⠴⣊⠥⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣦⡀⢐⣡⡶⠛⠁
⠀⠀⠀⣀⠴⣊⡵⣚⢅⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣽⣵⣾⣿⡅
⢀⣴⣿⡿⢞⡽⢊⣤⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣲⣿⠿⣧⡭⠿⠛⠉
⠈⠱⣿⣾⣿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣭⠶⠛⠉
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡸⣧⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⣦⣬⡱⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠷⠒⠀⠀⠀⠤⣀⠀⠀⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⡤⠈⠉⠉⠛⠳⠶⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢠⠎⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣳⡄
⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⠄⠫⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢙⡇⠀⠀⠈⢎⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢛⠀⣠⣐⠝
⠀⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢨⡇⠀⠀⠀⠈⢂⡑⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⡍⠁
⣄⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢃⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠑⠢⡈⠁⠢⣄⠀⠀⠀⠀⠀⠀⡠⢚⡴⠛
⢛⣷⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠉⠀⠄⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡑⢄⣀⣀⢠⢟⠐⠋
⠀⠀⠉⠛⢣⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⣿⣿⣶⣶⣤⣀⡀⢀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⠿⡈⢛⠃
⠀⠀⠀⠀⠀⠈⠙⠋⢻⠯⣁⣶⠶⠶⠶⠶⠶⠻⠳⢶⠊⠀⢀⣉⢻⣿⣿⡯⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣐⡞
⠀⠀⠀⠀⠀⠀⡠⢾⠟⠃⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠈⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⣰⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠐⢚⠿⠛⠃⠑⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⡘⢯⠉⠛⠳⠶⠼⠶⣖⢦⡇⢠⡇
⢴⠋⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠂⣠⡀⣤⠄⠄⠀⠀⠉⠀⠈⢳⡀⠀⠀⠀⠀⠈⡁⢀⡟⠁
⠃⠀⠀⠀⠀⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠃⠀⠀⠀⠀⠠⠀⠐⡀⠀⠈⠀⠋⠀⠀⠀⠀⠰⣡⠞
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⠸⠂⠀⠀⢤⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢻⣄⢠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢻⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠘⠆⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄⠠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⢰⡟⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠠⠈⢃⣻⠇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢠⣀⠀⠀⠐⠖⠃⢠⣾⡅⠀⢸⠋⠤⠀⠒⠒⢲⢿⢙⠁⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠠⣄⣴⣦⣟⢻⣧⣿⣷⠸⠀⠀⠀⠀⠀⠀⠀⠸⣆⠠⢀⠸⣦⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⣿⣿⣿⠻⣄⣘⢿⠠⡟⢹⣽⠉⠀⢀⣰⠂⠀⠀⣘⣃⣠⡿⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⡶⢺⠃⠈⠛⣧⠸⡆⠈⠉⣈⡃⣠⣼⣿⡀⠀⢸⣗⣇⣸⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠉⠨⣿⠀⠛⠉⠁⡀⠀⡀⠀⣿⡆⠛⣤⢁⡜⠉⢏⠷⠲⠞⠉⠈⠉
⠀⣀⣀⢄⣀⠀⢀⣀⡠⠔⠒⡀⠒⠒⠂⢈⣍⠀⢰⣦⡶⡀⠀⠈⢀⣰⣷⠓⠲⣟⠶⣉⠃⣨⢋⡿⠁⠀⠈⡄
⠘⠿⠿⠿⠿⠀⣀⣈⣀⡾⠾⠇⠀⠤⠾⠛⠷⣀⡾⠛⠛⠛⠛⢀⠌⠁⠀⠀⠀⠈⠛⠛⣾⣯⠛⣿⡄⠀⠀⢿⣄⣀
⠐⠒⠉⠀⠂⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢧⠀⠀⡘⢀⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢸⣇⠀⠀⢸⡏⠙⠛⠷⠶⢦⣤⣀⡤⢄⣀⣀⡀⠠⠤⢤⣤
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠸⠦⣀⣣⠈⠀⠀⠀⠀⠀⠀⢀⣴⡏⠁⢇⠀⠘⣿⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠂⢍⡙⢧⡄⠒⠂⠉
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠄⣼⡏⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠘⡆⠀⠙⡇⠀⠐⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣦
⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠉⠀⠀⠀⢶⢹⠀⠀⠀⠀⠀⠀⡔⠿⠁⠀⠀⠀⠀⠘⢆⡀⢹⠇⠀⠚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡣⡀
⠀⠀⠀⡄⣰⠀⠀⠀⠀⣀⣀⣀⣼⣇⣰⠀⠀⠀⠀⣶⠇⠀⠀⠀⠀⣠⣎⡜⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⣷⠀⠐⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣮
⠀⠀⠀⡿⣿⠀⠈⠉⠉⢀⠠⢊⡙⠀⢠⠀⠀⠀⢠⡟⠀⢀⠠⠔⣩⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⢠⠀⡟⠀⠀⠀⢀⠎⠀⡌⠀⠀⠐⠀⠀⢀⣾⡡⠚⠁⢀⣜⠏⠠⠏⡳⢤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⢃⠀⢳⡄
⠀⠀⠀⣨⠷⢤⣌⣙⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⣰⠋⠳⣤⠿⠀⠀⠂⠀⠻⣄⣀⣠⣤⠤⠶⠖⠚⠛⠛⠛⠓⠲⢤⡀
⠀⠀⠀⢷⣀⠀⠀⠈⠉⠓⠲⠭⣝⣒⠦⢤⣤⣼⠁⠁⠄⠈⠳⢦⡤⠶⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢻⡄
⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⣀⣀⣈⣉⣿⠀⡜⠀⠀⠀⠐⢄⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠾⠄⠘⣯⡽⠛⠁
⠀⠀⠀⠀⠀⠀⠸⣏⠁⠀⠀⠀⠀⣀⣀⡇⠀⢧⠀⠀⠀⠀⠀⠀⠀⡄⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠉⠹⡇⠀⠈⢣⡀⠀⠈⠀⠀⢠⠁⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠉⠲⢄⣀⠀⠐⠀⠀⠀⠀⣠⠇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠈⠙⠒⠳⠶⠖⠚⠁⠀⠀⢀⠈⠀⢀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⣄⡀⠰⡟⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣳⣦⣬⣉⡛⣻⡦⢤⣤⣤⣤⣤⡤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢻⣿⡿⢿⣴⣿⡆⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠷⢷⣄⡀⠀⣀⡤⠞⠉⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡯⣤⠶⠲⢄⠀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠳⡌⢿⡀⢀⣸⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⢳⠖⠛⡏⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠘⣦⡤⢿⡏⠃⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢰⠇⠀⠀⢷⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠖⠋⠀⠀⠀⠀⠙⠲⠚⠃

⠀⠀⠀⠀⠀⠀⣤⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢻⡈⢳⣼⣧⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢸⡷⢌⣇⠙⢧⡀⠀⠀⣿⣇⣴
⠀⠀⠀⠀⠀⠀⠀⣿⣄⠙⢦⡀⠙⢷⣄⣿⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣌⠓⢤⡉⠢⣄⠈⠻⣦⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠰⣿⡷⢄⣈⠑⢷⠀⠀⠈⠻⣄⢛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣃⡀⠉⠚⢷⠀⠀⠀⢹⣮⡗⠙⢧⡀⠀⠀⠀⠀⣠⠶⠒⠛⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣉⣉⣉⣉⣹⣄⠀⠈⢷⣿⠄⠀⢳⣀⣀⣤⠞⠁⠀⣶⡆⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⣤⡤⠤⠤⠚⣆⠀⠀⠙⢦⣀⠀⠙⠀⠀⠀⠀⠀⠉⠁⣀⣾⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡖⠂⣉⠬⠋⣠⣀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣠⠔⢋⣔⡡⠊⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⡼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⣀⢀⡀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢋⡵⠁⣠⣋⣟⣰⣶⠖⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣉⣤⡾⠋⣠⠊⡨⢋⠟⣹⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⢞⡿⠋⢀⠔⠁⡔⢁⠎⣼⠃
⠀⠀⠀⠀⠀⠀⣠⠼⠋⣡⠋⢀⣴⠋⢠⠎⢠⣏⡼⠁
⠀⠀⠀⠀⠀⠀⠛⠒⠛⠙⠛⠉⠈⣻⠃⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣄⠀⠀⠀⢀⣤⣶⠟⢛⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣿⣿⣿⣿⡇⢀⣴⣿⣿⣿⣷⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣥⣿⠛⠉⠛⣿⡇⡘⢿⣏⢶⣽⣿⣿⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⣿⣿⣿⣿⠀⠀⢀⣿⡇⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠈⠻⠿⠿⡧⢠⢿⡿⠟⠙⠏⠻⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣦⡀⠀⠀⠀⠀⠀⠀⣀⣾⣼⠇⠈⣦⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣷⣦⣤⣤⣶⣾⣿⣿⡿⠀⠀⠈⢿⣿⣷⣤⣠⣆⣀⣤⣿⣦⣾⡿
⠀⢀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣦⣄⣩⣥⣥⣬⣍⣡⣍⣉⣁⣄⣀⣀⣈⣋⣙⣏⣉⣉⣉⣹⣥⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⠏
⣷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡏
⣷⡄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⠁
⣿⣿⣶⣀⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⢀⠞⠻⣿⣿⣶⣄⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⣪⢽⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⣚⠻⢸⢿⣿⣿⡟⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣯⣙⣊⣶⣽⡷⠿⠛⠛⠉⠀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⢿⣶⣬⠿⠿⠏⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢛⣩⣿⢟⠿⠋⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⢀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡽⣣⣤⠄⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⢠⣾⢻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⡿⢡⠀⡀⣀⡀⠀⡀⠀⣀⣀⣶⠷⢚⣥⣶⢻⣟⡿⠁
⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⢿⣿⣿⣍⠁⠀⡀⠈⢉⣻⢿⣛⣛⡻⣿⣿⣿⠭⢽⣯⣤⣿⣿⣯⣷⣿⣿⣿⣿⣴⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡷⠒⠉⢠⣾⣿⡿⠟⠛⠉⠀⠀⣼⣿⣞⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠁⣁⣴⠿⠛⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣽⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡶⡐⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠆⠭⣤⡝⢻⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⢋⣠⣤⠀⣀⡍⠀⠀⠀⠠⠄⠀⠀⠀⡸⢀⠀⢀
⠀⠉⠻⠒⠲⣶⠲⣤⣀⢀⣠⣤⣶⣿⡿⠁⡀⠈⠉⢿⡦⡀⠀⠀⠀⠀⢻⡅⠀⣠⣿⣷⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣴⠚⠛⢿⣿⠈⠉⠀⣸⣷⡄⠀⢸⣇⢀⣦⠒⣦⣒⣾⣷⡾⣿⣿⣿⣿⣏⠍⠉⠃
⣀⣀⣠⣤⣶⠟⠉⠉⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⡤⣿⣿⣿⣿⣾⣷⣸⣿⣿⣾⣿⣿⣿⣿⣿⡀
⠉⠻⢯⠀⠀⠀⠀⣠⠆⠠⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠁⠀⠀⢠⣿⣠⣤⣤⣶⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡤
⠽⠿⢿⡿⠿⣿⣿⣿⣶⣾⣯⣾⣿⣿⣿⣻⣿⢛⣟⠿⣿⣿⣿⣿⣿⡿⣿⡛⣂⣈⣷⣿⣿⣭⣃⡀
⠀⠀⠀⠀⠀⠠⣾⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⢿⣿⣿⣻⣿⣿⣿⣿⣟⣛⣿⣆⣐
⠀⠀⠀⠀⣀⡐⢿⡇⣙⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣈
⡐⣒⣶⣶⣿⣷⣿⣧⣬⣼⣿⣿⣍⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠙⣗⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⡄⠀⠛⠿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠈⠀⢠⠀⠈⠙⢿⣿⣿⣿⣶⣶⣶⣶⣶⣤⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠉
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠹⢽⣍⣍⣍⣽⣿⣿⣿⣿⣋⣍⠳⠴⡆⠄⢀⠀⢀⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠮⡻⣿⣿⣿⣿⣿⣿⣿⣷⣯⣿⡺⣝⠲⠾⣥⡿⣿⠙⠻⠷⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⠿⣙⣻⠧⠈⠷⠲⠿⣿⣛⠶⠶⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠢⢀⣀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢻⡿⣋⣠⠨⠇⣠⠿⠁⠀⠠⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣏⣫⡿⡀⡨⠃⠀⡰⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⠋⠠⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣿⣉⠉⢹⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⢷⣼⠠⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠹⣌⢦⡏⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠘⢎⣿⡄⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠢⢈⢻⣧⠀⡇⠀⡀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⠀⠈⠻⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠓⢄⠀⣤⢽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠻⡿⢿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢛⣃⣩⣭⡗⠊
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡼⢀⠐⠶⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢠⠂⣄⠀⠀⢤⣤⠊⠖⠙⠋⠀⠀⠀⠁⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣤⠀⣤⢀⠐⢰⣾⡧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿⣏⣿⡄⠀⠀⠀⠀⢀
⠀⠀⠀⠻⠥⠟⢻⣷⣧⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣻⠀⠀⠀⢀⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣀⡿⡿⠀⠀⠀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣚⣿⠇⠀⠀⣠⠆
⠀⠀⠀⣠⣀⠀⠴⣐⢻⡆⢡⡀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢠⣽⠟⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠁⠀⠙⠛⠪⠛⡛⢧⣦⡞⢻⡦⠀⠀⣨⣤⠞⠁⠀⠀⣠⡶⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⠛⠀⠀⠈⣿⣵⣷⡦⡍⠳⡀⠀⠀⢀⡽⠁⣾⣑⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⣿⡁⢫⣜⣧⣼⣿⣁⣀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠿⣿⣿⢻⡷⢎⣼⣧⣽⣿⣬⣿⣿⣻⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⣧⣼⣿⣧⣾⣿⣋⣿⡝⣿⡋⣿⣿⣿⣷⠀⠀⠀⢀⣠⣶⠾⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣵⣿⣿⣿⣿⣿⣏⣠⣴⠾⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⣿⣿⣿⣧⣽⣿⠿⣭⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡿⡟⣿⣿⣿⣿⣧⣼⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡞⣿⣿⣿⣿⣿⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢹⠿⠟⢻⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⣁⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⡞⠙⠗⡀⠰⣿⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢼⣷⣾⣭⣯⣧⢨⠝⣶⡌⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣶⡿⠀⠙⠛⠁⠙⡸⠿⠃⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄

⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠄⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣶⣶⣶⣤⣦⣄⠀⠠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⣶⣿⣤⣾⣶⣼⣶⣾⣏⡵⠾⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢿⣿⣿⣿⣿⣿⣷⣤⣈⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⣴⡟⣻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⢠⣾⣾⣿⣾⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣧⣻⣶⡄⡀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧⣽⣷⠘⠶⠂⠀⣸⡇
⠀⠀⡇⠀⠀⡀⢠⣿⣽⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣟⣿⡆⠀⣇⠘⢿⡇
⠀⠀⡇⠀⢠⣿⣯⣾⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⣶⢻⠀⢸⡇
⠀⠀⡇⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣧⡟⡟⠀⢸⡇
⠀⠀⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⢸⡇
⠀⠀⡇⠀⢴⣿⣿⣿⣿⣿⣿⣿⢿⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠻⠋⠀⢸⡇
⠀⠀⡇⠀⡘⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⢿⡟⢳⡞⠀⢸⡇
⠀⠀⡇⠀⢳⡈⣿⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⣿⣿⡞⠁⠀⢸⡇
⠀⠀⡇⠀⣦⣴⢽⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣱⣿⣿⣿⣾⣷⡿⣦⠀⠀⢸⡇
⠀⠀⡇⠀⠈⢿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠙⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠂⢠⣤⣤⡀⠀⣀⢀⣀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⡙⣿⣽⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣻⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣀⠁⠞⠶⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠈⢤⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⡽⠾⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡌⠻⢿⣿⣿⣿⡏⠿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢏⣰⣿⣿⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠙⠿⢿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⡿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠊⠽⢿⣿⣿⡿⣿⣿⣿⣛⣛⣛⠛⠁⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣌⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠁⠁⠁⠉⠈⠁⠁⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣰⠃⣰⠏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣿⠀⢠⢏⣼⠋⣠⠖⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⣰⣫⣞⣵⣮⣷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣿⣿⡿⠟⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⠾⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠂⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢒⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠒⢶⣦⣀⡀⠀⠀⠀⣀⣤⣴⣶⣾⣶⣶⣾⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⢦⣤⣤⡶⠟⣛⣉⣍⣁⣤⣶⣿⣿⣿⣿⣿⣿⢿⡿⠫⠙⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠉⠁⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠿⣋⣀⠀⠀⠰⣿⣿⡿⠻⢿⠛⣿⣿⣿⣿⣿⣿⣦⣄⣀⡀⠀⠴⠀⠴⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠲⠿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡷⠌⠙⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠛⠻⡟⠚⠟⣿⡛⢍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠂⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠐⠛⠛⢶⣿⣿⡺⠿⠆⣹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠛⠿⠫⠉⠛⣦⡀⠀⠀⠀⠀⠐⢀⠀⠢⠘⠆
⠀⠀⠀⠀⠀⠀⠠⠀⠀⢄⠀⠀⣀⡀⠀⠀⠁⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠠⢄⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠛⢆⠠⡀⠀⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣤⣤⣬⣥⣄⣀⠄⠀⣙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⣶⣮⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠋⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣉⠓⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⣿⣿⣿⠿⠋⠛⢿⣿⣿⣿⣿⣿⡄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠋⠙⠁⠀⢀⣠⡴⠟⠉⠙⠻⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡌⠀⠈⠁⠀⠀⠀⢄⣴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢅⠠⠊⠄⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠂⠁⠠⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠛⠿⠟⠋⠁⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠆⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⢀⡀⠤⠤⡤⠠⠤⠤⠤⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⠀⠺⡍⢀⡀⠄⠀⠀⠀⡠⠀⣀⠐⡸⠀⠀⠀⠀⢀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠋⠀⠀⠀⠀⠀⠈⠢⡔⠂⠀⠀⠀⠒⠈⡤⠜⠁⠀⠀⠀⠀⠈⢻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠻⠿⠿⠿⠿⠿⠟⠛⠋⠉⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏⠉⠛⠛⠿⠿⠿⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡚⠝⡻⠿⣿⣿⣿⣿⣿⣿⢿⠛⠉⠁⠈⠙⠿⣿⣿⣿⣿⣿⣿⡿⠿⡛⠅⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠌⠁⠀⢠⡁⠂⠘⣉⣭⣂⣒⠈⡄⠀⠀⠀⢠⡉⠀⠂⠀⣬⠍⢀⡐⠊⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠐⠦⠐⠠⠔⠉⠐⠤⠦⠃⠀⠀⠀⠈⠳⠶⠊⠉⠲⢤⠐⠤⠂⠁⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⣰⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠛⠛⠋⠣⡀⠑⠩⡙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠊⠀⠀⠀⠀⠐⠄⠀⠈⠢⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠈⣀⣀⣤⠴⠒⠋⠉⠳⡄⠀⠀⠀⠀⣠⠴⠖⠋⠉⠉⠉⠛⢲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣸⣀⣠⣴⠤⠴⠖⠛⠉⠁⠀⠀⠀⠀⣀⡤⢿⡀⠀⣠⠞⠁⠀⠇⠀⠀⠀⡠⠊⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠖⠛⠉⠁⠀⠀⠙⣶⢇⠀⠀⠀⠰⡀⠀⠐⠀⠀⢀⣀⠀⢀⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠦⠴⠶⠶⠶⠒⠛⠋⠉⠃⢄⠀⠀⠀⠀⢀⠔⠒⠒⢠⡏⠀⠑⠠⣀⠀⠑⢄⣀⠔⠊⠁⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣉⠝⡛⢍⠁⠐⠠⣀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠘⠀⠀⠀⠸⠗⠀⠀⠀⢸⢀⠀⢀⡃⠀⠸⡇⠀⢻⡂⠀⠀⠒⠈⠀⢀⠃⠀⠡⠀⠀⠀⠣⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣧⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⢎⠁⠀⠀⢩⠂⠀⠀⠀⠀⠷⣄⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠙⠳⠾⣄⣀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⢀⡟⠳⢤⣊⠀⠀⠀⠀⢀⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠛⠀⠐⠀⠀⠀⠉⠙⠓⠲⠦⣤⣄⣀⣀⣀⣀⠀⣀⣀⣀⣀⣌⣼⣥⠀⠀⣼⠉⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⣀⠉⠀⠘⠃⡀⠁⢀⣀⠀⢤⠀⠀⠀⢈⠉⠉⠉⣉⠉⠉⠀⠀⠔⢸⠇⢀⡏⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⡾⠉⠙⠲⠦⣤⣀⡀⠈⠁⠠⠄⠀⠛⠀⡈⠀⠾⠀⣀⠀⠘⠋⠀⠤⢸⡀⢸⠁⡿
⠀⠀⠀⠀⠀⠀⢠⠖⢋⡠⠄⠐⢺⠃⠀⠀⠀⡘⠀⠈⠉⠛⠓⠲⠦⠤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⢶⡻⣄⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠻⣼⣁⣀⠀⢀⡟⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡗⢸⡧⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡇⡾⣆⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣸⢻⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡟⠘⠲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠙⢦⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠈⠁⠒⠠⠄⠄⠊⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠧⠴⣄⣼⠃⠀⠀⠀⠈⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡷⠒⠒⠚⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠓⠚⠙⠋

⠀⠀⠀⠀⠀⠀⢠⡄⠈⡀⠀⠀⠀⢀⠄⠒⠔⠀⠐⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⢀⢶⣶⣶⣶⡖⢁⣾⣁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠉⠁⠀⠀⠀⠑⠒⢦⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠱⡤⠒⠄⠴⣱⠞⠁⠀⠀⠀⠀⠀⠀⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⡞⠁⠀⠀⠀⢠⡀⠀⠂⢠⠀⡀⠈⠑⠒⠢⢄⠠⠤⢤⠔⠚⠃
⠀⠀⠀⢀⣤⣴⣎⠉⢳⡞⣦⡀⠀⠀⠀⠈⠀⣜⠀⠀⠀⠜⠁⠀⠀⠐⠀⠀⠠⠄
⠀⠀⠀⠈⢶⣾⣟⡛⢚⣿⣩⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠙⠚⠛⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢖⡐⢤⢀⠀⠀⠀⠀⠀⠀⠀⡎⠁⠁⠀⠀⢤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣤⠊⠐⣤⢏⣀⣞⡂⠀⠐⢿⣿⠿⠷⠂⠀⠀⠾⠟⠁⠀⠀⠐⠀⠀⡀⠀⠀⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠈⢜⠷⠞⠚⠒⠓⠺⢭⣨⡅⠀⠀⠸⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠕⠐⠀⠆⡌⠀⠀⠀⠀⠀⠿⢿⣶⣿⣶⣶
⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⢦⠇⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡂⠠⢀⡀⠠⠄⠀⠠⣔⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⡂⠠⠶⠂⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠂⠚⠉⠉⠀⠦⠀⠀⠠⠌⠃
⠀⠀⠀⠈⠂⢀⠈⠁⠀⠀⠠⣁⠖⡀⡡⢀⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⠄⠙⢁⡦⠀⠀⠀⠽⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠈⡆
⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⣤⣤⠷⢶⡶⣄⣨⡵
⠀⠀⠀⠀⢠⠄⢄⠀⠀⠀⠀⠀⠀⣴⣦⡴⣶⡶⡆⠀⠀⠀⠀⣠⠄⠒⠂⠉⠉⠻⣶⡀⠀⣼⡟⠀⠀⠀⠀⢻⡏
⠀⠠⠢⢀⡈⣆⠈⠖⠐⠀⠀⠀⣴⣾⣿⣿⣾⣿⣿⣤⠀⠀⠸⣯⡀⠀⠀⠀⢀⢸⡟⠀⠀⠙⣧⠀⠀⠀⣠⣾⠇
⠀⠀⠉⠄⠘⠀⢆⠓⠀⠀⠀⠀⠈⠁⠈⠓⠋⠋⠙⠛⠧⠀⠀⠈⠻⠶⠚⠑⠺⠛⠁⠀⠀⠀⠈⠛⠛⢷⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠛⠛⠋⠉⠉⢉⣿⠿⠛⣻⣿⣿⣿⡿⢾⣿⠛⠛⠙⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠿⠟⠋⠁⠀⠀⡇⠀⠀⠀⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠁⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⡟⠁⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⢄⣀⡾⠋⠀⣰⠟⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢁⡆⠀⢿⠀⢀⣴⠗⠒
⠀⠀⠀⠀⠀⠀⠈⠿⠑⠶⡚⠢⣤⡶⠆⠤⠤⠠⠤⠀⠀⠀⠀⠈⣧⠙⠁⠀⠘⣱⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣡⣤⣀⣀⣿⡆⠀⡀⠀⠤⠄⠀⠀⢀⣿⡦⠀⢀⣼⠿⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⣿⣿⣿⢟⣿⣇⠀⡄⠀⠀⠠⠶⠶⣿⠿⠦⠶⠛⠁⠀⠀⠋⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⣠⡟⡽⢻⡿⠃⣼⣿⣿⠀⢹⣄⣀⠂⠆⠂⠀⠰⠀⢀⣀⠀⠀⠀⣤⡀⠈⠉⢀
⠀⠀⠀⠀⠀⠀⠘⠋⠰⠇⠘⠁⢸⣿⢹⣿⣿⣧⠉⣙⣿⡷⢷⣶⣶⣾⣿⣿⠇⠖⠀⢸⣏⣠⣴⢛⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⣿⣿⣿⢷⣿⣿⠇⠀⠙⠉⣠⣾⣧⠀⠀⠀⠘⠿⠙⢃⣼⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠋⡀⠀⢄⣤⢹⣿⠃⠀⠀⠀⠁⣀⡴⠛⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠟⠃⠀⠀⠀⠰⣠⣿⡿⠁⠀⣿⣷⠖⣿⣿⠡⢆⢠⣾⠔⠚⠂⠀⠀⠀⠀⠀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⣼⣿⡿⠋⠀⢀⠀⠀⠂⠀⠀⠀⡜⠁⠀⠠⠙⠁⠀⢿⣿⡀⠘⠿⠁⠂⠀⡀⠀⢀⣤⣴⡾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠻⠋⡀⠀⠐⣍⣼⠇⠀⠀⠀⢊⣤⡀⠀⢀⣤⣤⡀⠘⠟⠋⠀⠀⠀⠀⣀⣶⣾⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣠⣄⡀⢸⣿⡟⠀⠈⠀⠀⠂⠀⠀⣳⣯⠀⡖⣿⣟⡿⠃⠀⠀⠀⠀⢀⣴⡿⠻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⡽⠁⠈⠋⢀⡤⣀⠀⠀⠀⠀⠀⠨⡿⣤⣿⣿⡭⠁⠀⠀⠀⢀⣴⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⡀⠀⠀⠀⠈⣧⡌⠀⠀⠀⠀⠀⠀⠙⢼⣿⠿⠇⠀⠀⠶⠞⢳⣾⡿⠇⢀⣴⣦⡀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢥⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣀⠀⠀⠀⣀⠀⠈⠁⣾⣿⣿⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠛⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⠀⠀⠀⠐⠁⢠⠀⢹⣿⡿⠇⣀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠋⣾⠇⠀⢀⣀⠰⢀⡷⢴⣦⣨⣾⣿⣾⣿⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠄⠠⠄⠂⠀⠲⣦⠀⢀⣾⠛⠁⢀⠞⠁⠠⠐⠀⣈⣈⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⣦⡈⠁⢀⡔⠋⠀⠀⠀⠀⠀⠠⢩⣭⣽⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⡿⢆⠘
⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⡏⠰⡀⠘⣧⠀⠀⠀⣀⣀⣤⣤⣼⣿⣿⣿⠿⢿⠿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠾⠟⠛⠓⠀⠀⢄⠻⣟⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠴⠆⠴⢟⢻⠟⠛⠛⠋⠋⠋⠍⠙⠉⠹⠛⠟⠛⠛⠛⠉⠙⠃⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⢀⣀⢤⣶⣾⣿⣿⣷⣶⣶⣶⣤⣤⣀⣀⡀
⠀⠈⠉⠛⣛⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⣶⠆
⠀⠀⠀⠀⠀⢀⣠⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠉⠉⠉⣩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣀⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠛⠛⣿⣿⣿⠿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠋⠀⠀⠀⠈⠁⠀⠀⠀⠘⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢼⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡶⠶⠶⠶⢶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⢉⣠⣤⣤⣶⣶⣤⣤⣬⣙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢁⣴⣾⡿⠟⣛⣉⣩⣭⣍⣉⣛⠻⢿⣦⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⣰⣿⠟⣩⣶⡿⠿⠛⣛⣛⣛⠛⠻⢿⣶⣌⠻⣦⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣸⣿⢣⣾⠟⣥⣶⠾⠛⠛⠛⠛⠿⣿⣦⡙⣿⣆⠸⣇⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠇⣾⢏⡾⢫⠴⠞⠛⠿⢿⣶⣄⠈⢿⣧⢸⣿⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⢠⡟⠸⢁⣴⣾⣿⣿⣷⣦⡙⣿⡄⢸⣿⢈⣿⠀⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠘⡇⢰⣿⡿⠉⠀⠀⠈⢹⣧⢸⡇⢸⡏⢸⡿⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣛⠛⠲⠄⡠⠋⢉⣉⠙⢆⡰⠚⠛⠻⣆⠀⠀⠀⠀⠸⡄⠸⡇⠇⣿⣿⠀⠀⠀⠀⠀⢸⡟⣸⢃⡿⠀⣼⠃⢰⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣾⠉⠀⡀⠈⠁⢰⠏⠉⡧⢀⠠⢾⡉⠢⠙⠀⠀⠀⠀⠀⠱⡄⢻⡄⢻⣿⠀⠀⠀⠀⠀⡼⢡⠏⡼⠁⣸⠋⠀⠾⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⣷⢯⣭⡀⠉⠙⣷⣬⡴⠋⣀⣘⣧⣴⣶⣦⣤⣤⣤⣀⣀⠙⢦⠻⣌⢻⡄⠀⠀⠀⠈⠀⠋⠈⠀⠘⠁⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣧⣀⣨⣿⠀⠀⠀⣠⠶⠟⢋⠉⡉⠁⠀⠈⠉⠉⠉⠛⠻⠿⢾⣷⣽⣦⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣬⡽⠿⠷⠶⠶⡾⠅⣀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣽⠛⠉⣀⡤⣤⣤⡀⢀⡗⠀⠀⠀⠠⠄⠀⣠⣴⡿⠟⠛⠲⠶⣤⣀⡀⠀⠀⠀⠀⠙⠿⣷⡀
⠀⠀⠀⠀⠀⢀⠔⠁⢨⡇⠰⠙⠿⠿⠆⢁⣾⠁⢈⡉⠐⠲⠄⢰⣿⠏⠀⠔⠚⢶⣄⢠⡉⣉⠛⣛⠲⠶⢤⣀⡉⢿⡄
⠀⠀⠀⠀⠐⠿⠈⠀⠀⣵⣀⣀⣀⣠⣴⡿⠋⠲⢤⣈⠛⠦⠀⣿⣿⠀⠀⠀⢀⠀⣿⡄⢁⠑⣾⡥⢃⡶⣀⣌⡙⠾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⠉⠉⠙⠋⠡⠀⠄⠱⣦⠉⠓⠂⠀⢹⣿⡀⠀⠀⠣⡴⠏⢰⠀⠀⡼⢷⢾⣏⠡⣜⠐⡂⠘⠄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢠⠀⢷⠈⢦⠀⠀⢀⠀⠀⠈⠿⣷⣄⠀⠀⣀⣴⠇⡐⠁⠀⢼⡟⠻⣤⣀⠔⣡⠎⡌
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡻⡻⡟⠋⢁⠄⡰⠒⣂⠸⣤⠂⠀⠙⢧⢐⠚⠷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡞⢩⢝⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡑⢮⡑⠾⣭⣹⣐⢁⠀⠛⠀⠀⠀⠈⣸⠈⡆⢐
⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠐⢣⣌⠘⣧⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢷⡽⣦⡉⠳⢦⣭⣙⠛⠿⠶⣶⣦⣬⣭⣉⣚⣁
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⠤⠖⢃⠈⢣⡘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡛⢷⣤⣈⡙⠛⠿⢶⣶⣤⣬⣭⣉⣛⣛⣋⡁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⢸⠇⠈⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣬⣙⠻⠿⣶⣶⣤⣬⣍⣉⣛⣛⣛⣋⡁⣇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠋⠀⠀⠀⠀⠙⠻⠷⣶⣦⣤⣤⣤⣤⣤⣴⣾⡿⠿⠿⣶⣶⣤⣭⣍⣛⣛⣛⣛⣛⣉⣁⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠉⠩⣉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠈⠉⠒⢤⠄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠶⠚⠛⠉⣠⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⢀⡈⢩⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⠞⡆⠁⠀⠁⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠻⠐⠇⠀⣀⠇⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⡛⠁⠀⡤⠶⣌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢡⠞⣱⠃⠎⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠛⢿⣿⣿⣿⣿⣿⠟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⢤⣀⠀⠀⣿⣿⡇⠀⢀⣤⢀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠸⣾⡿⠀⢀⣿⣿⣇⠀⠸⣿⡾⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠘⣿⣿⡟⠀⠀⠀⠀⠀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠏⠉⠻⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠶⢿⣿⣿⡿⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠰⠁⠀⠀⢻⡟⠀⠀⠈⡆⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠚⠻⣿⣿⠛⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠰⠿⠀⣼⣧⠀⠿⠆⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣦⡄⣠⡄⢸⡇⢠⡄⢡⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠙⠋⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢻⣿⣿⡷⠈⠀⠻⠇⠈⠁⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠁
⠀⠀⠀⠀⠈⠻⣿⠃⠀⠀⠀⠀⠀⠀⠘⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣄⡀⠀⠀⠀⣀⣤⣤⣤⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⢆⠀⣀⣀⠀⠀⣀⡀⠀⣼⣿⣿⣿⣿⠷⣶⣶⡶⢿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠥⠼⠿⠟⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠸⢿⣿⡿⠦⠴⢿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⡄⠀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖
⠀⠀⠀⠀⠀⠀⣖⣚⠑⣒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⣠⠤⠐⠒⠒⠤⣄⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠓⢄⡀⣀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠑⢄⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⡎⠀⠀⠀⠀⠀⠀⠈⠀⠀⠱⠒⠪⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡞⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠑⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⡄⢀⡀
⠀⠀⠀⠀⠀⣰⠖⢆⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠊⠠⣐⡄⠠⠄⠀⠀⠀⠀⠀⠈⡀⠬⠖⠦⢄⡀⠀⠀⠀⠀⠰⠥⢤
⠀⡤⠒⠲⠎⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠑⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⠒⠢⡄
⠘⢦⢀⡤⣄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢱
⠀⠀⠈⠀⠀⠉⠉⠋⠹⣁⣀⠉⠈⠉⠁⢀⣀⠉⠛⠀⠛⠉⢁⠘⠒⠐⠂⠒⠁⠉⠘⠛⠋⡛⠒⠻⠭⠭⠤⠔⠒⠙⠒⠂⠐⠒⠛⢯⣲⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠉⠀⢽⣉⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠃⠀⠀⢄⢀⡀⣀⠀⠀⠀⢀⡠⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠑⠒⠒⠚⡩⠽⠖⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠚⠻⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢼⡉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗
⠀⠒⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠤⠤⠊⠁⠀⠀⠈⠢⠤⠀⠀⡀⠉⠉⠋⠉⠉⠁⠈⠉⠁⠈⠉⠒⠒⠒⠉⠙⠛⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢂⣀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡤⣀⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡠⠠⡔⠒⢄⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣀⡤⢄⣀⣀⢄⣀⣀⣀⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣒⣒⣢⣤⣔⣒⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣌⡘⣿⣿⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⢂⢠⣑⣾⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢾⣿⣿⣿⠻⠟⠿⣿⡿⢿⣷⣦⣴⡖⠲⢦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠿⠘⠻⠦⠀⠀⠀⠀⠙⠧⠈⠈⠻⠛⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⣄⣒⡂⢒⣢⣶⣷⣆⠀⠀⠀⠀⠀⣿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠤⠈⣉⣉⣉⣙⣿⣲⣿⢇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⠃⠀⣠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡾⢿⣻⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⠃⠀⠔⠀⠀⠀⠀⡶⠖⠒⠘⠛⠻⢿⣿⠁
⠀⠀⠀⠀⢀⣠⢀⡋⢼⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠉⠀⠀⠀⡀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠾⠃
⠀⠀⠀⠀⣼⡟⣿⣃⣰⣿⣿⣿⣿⠟⣿⠀⠃⡴⠀⠀⠀⠲⠐⡶⣿⣶⣿⣿⣧⢄
⠀⠀⠀⠀⠉⢠⣿⣿⣿⣿⣿⣟⣡⠖⠋⠐⡠⠂⢠⠀⣤⣀⣶⣧⣹⣬⣟⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠈⣹⣿⡿⠻⢋⣸⠈⠀⠐⠉⢰⣶⡈⣼⣿⣿⣿⣿⡿⣿⣿⣞⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⢹⡏⢀⣴⠏⡉⢤⣄⢠⠂⡋⠙⠿⣿⣷⣿⣿⣿⣿⣮⣻⣿⣦⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⢁⡔⢷⣌⣿⢢⠰⡿⢢⣀⠘⣿⠏⠙⣿⢿⡻⡟⢿⣻⣧⢷⡙⡄
⠀⠀⠀⠀⠀⡰⢃⡞⠃⠀⠀⠀⠀⠉⠛⠷⣧⡀⠹⣧⣄⠀⢤⡈⠀⠙⠶⣌⡻⢿⣄⢻⣿⠶
⠀⠀⠀⠀⢸⠃⠈⠁⠀⠀⠀⠘⠒⠀⠀⠀⠈⠙⢦⣙⣿⣿⣿⣿⣤⡈⠻⢦⣅⡀⢸⣧⣿⣄⢰⣄
⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⢿⡟⠻⣷⡄⢸⣿⣿⣾⢯⡻⢿⣆⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠿⣽⡿⠗⠈⢻⣆⠁⣶⣿⣾⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠈⠛⠛⠉⠛⠛⠛⠃





⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠃⠀⠀⠘⠒⢲⣀⣀⠀⠀⠀⠀⢰⠉⠉⠑⠂⠐⡄⠀⠀⠀⠀⣀⣀⣰⠋⠀⠀⠈⢀⣀⣀
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢷⡀⠀⠀⢫⠀⠀⠀⠀⠀⠀⣰⠀⠀⡸⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠒⠐⢄⡀
⠀⠀⠀⠀⠀⠑⠤⠤⠤⠶⣤⣤⣤⣴⠴⠦⠤⠴⠞⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠉⠀⠀⠉⠛⠒⠒⠲⠶⠶⠶⠒⠒⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠚⠉⠁⠉⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠒⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠄⡀⠀⠀⠀⠀⠀⢀⡄⠒⠁⠀⠀⠀⠀⠀⠀⠉⣠⠵
⠀⠀⠀⠀⠀⠀⡀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢄⠀⠀⠀⠀⠙⠒⠒⠠⠔⠒⠢⠤⠚⠉⠉
⠀⠀⠀⠀⠀⠈⠒⠀⠠⠤⠤⢤⣀⣀⣀⣀⣀⡤⠤⠤⠤⠟⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠤⢤⠋⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⣦⣀
⠀⠀⠀⠀⠀⠀⢠⠴⠚⠀⠀⠀⠀⠹⠒⢦⠖⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠀⠀⠀⢀⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠘⠦⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠤⠐⠁⠀⠀⠀⠈⠒⠂⠀⠤⠤⠤⢄⠀⠀⠀⢀⣤⡤⣄⠀⠀⠀⣠⠤⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠤⠔⠒⠚⠐⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋⠀⠀⠈⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠾⠿⠿⠶⣷⣴⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣤⣶⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠟⠟⠋⠉⠁⠠⠌⠙⠻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠐⠂⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠚⢤⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠐⢦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣼⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧⣼⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣵⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠈⢿⣧⡼⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⡀⠈⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⣽⣇⠀⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠈⢿⣧⣠⣄⠀⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⣸⣿⣿⢿⡿⠦⠶⠀⣮⠁⠈⠀⢿⠂⠀⠀⠀⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢁⣀⣄⣼⣿⣿⣧⣄⡀⠀⠸⡟⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠹⣯⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣼⣿⣿⣿⣿⣿⠿⠟⠃⠀⠘⠶⠆⠰⢆⣤⣶⣿⣷⣤⡄⠴⣶⠿⢿⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠘⠛⠉⠉⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⡀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣀⣤⡸⣿⣿⡏⠉⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⣹⣿⠉⠛⠻⠿⢈⡿⣿⣇⠶⡞⢻⠿⣶⣦⣤⣤⣼⣿⠖⠓⠛⠃
⠀⠀⣴⣤⣄⣰⣿⡦⣴⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⣰⣼⣿⣿⣋⡁⣘⣷⣿⣇⢼⣼⡿⢿⣿⣿⠏
⠀⠀⠉⠻⣿⣿⣿⣿⡟⠁⢼⢿⣇⣀⠀⠀⠀⠀⠀⠀⢠⣶⣤⡿⢿⣿⣿⡟⠹⠿⠛⣻⣏⣽⣏⠀⠹⣶⣿⣿⣿⣶⣶⣄⡀
⠀⢀⣀⢰⣿⣿⣿⣿⣯⣶⣟⣀⣿⣿⠃⠀⠀⠀⠀⢰⠄⢢⠖⠀⣶⢍⣉⡉⠻⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⠿⠿⠷⠖⡂
⢠⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣾⡃⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠙⢻⣿⣿⣏⣹⣿⣿⡿⣿⣤⡉⠛⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⣿⡯
⠀⠠⣾⠛⣿⡟⠛⢼⣿⠇⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⣿⣫⡝⣿⣿⣿⣟⠀⠀⠠⠛⠋
⠀⠰⢿⡿⠏⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⣩⣿⣿⣿⣿⣯⣳⣿⡀⣀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣆⠀⣉⣿⣿⠿⠿⠙⣙⠛⣃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⣿⣿⣾⣿⣧⣿⣿⣿⣿⣿⣿⣦⣄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣤⣿⣿⣿⣯⣙⡿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣟⠟⠓
⠀⠀⠀⠀⠀⢀⣀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⣹⣿⣿⣿⣿⣿⢋⣿⣿⣛⣿⣿⢛⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⣤⣾⣦
⠀⠀⠀⠀⠀⢘⣿⣿⣿⡤⠀⠀⠀⢠⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡞⣶⣿⣿⣿⣿⣧⣴⣶⣼⣷⣿⣿⣿⣿⣿⣿⣦⣀⡀⠀⠀⢠⣼⣿⣿
⠀⠀⠀⠀⢴⣿⣿⣿⣿⡀⠀⣘⡇⢰⣾⣷⣄⣀⣠⡾⣿⣶⣶⣴⣷⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡋⠀⢰⣿⣿⣿⠄
⠀⠀⠀⠀⠈⠋⠠⣿⣿⣿⣶⣩⣥⣾⡟⡶⣿⣿⣿⣿⣿⣿⡿⠿⣿⡿⠻⢿⣿⣧⡏⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠟⠻⣦⣭⡅⡛⢻⣿⣻⣏⡘⠏⠰⠾⠻⣾⣿⠐⠈⢿⣁⣬⣯⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠂⠀⢀⣴⣿⣿⣿⣿⣶⣠⠒⠀⠒⠀⠀⠂⡘⢿⣿⣿⣿⣿⠛⢻⣯⣴⣿⣿⣇⣠⣥⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠸⣽⣿⣿⣿⣿⣿⣿⠋⣻⣷⠟⠃⠀⠰⣾⣿⣿⣿⣷⡤⣼⣿⣿⣿⣿⣿⠛⡿⣿⡛⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣍⠁⠀⠈⠀⢀⣤⣿⣟⠋⠎⠀⠈⣿⣿⣿⣿⡟⢀⢀⢻⣿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⣠⣯⡉⠙⣿⢿⢿⡿⣷⡶⠶⠿⠛⠻⢿⣿⡷⠤⣤⣶⣿⣿⣿⡇⣀⡀⠀⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠾⢟⣿⣿⣶⣤⣦⡤⠴⠍⠶⠀⠀⠀⠱⠾⠓⣦⠀⠐⢸⣿⣿⣿⡿⠿⠛⣡⣥⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠠⢸⠡⣿⠿⡿⠓⠀⠂⠂⠀⢀⣴⣷⣄⠀⠠⠁⠀⢀⣼⣿⣿⣝⣀⠀⠀⡿⠿⠋⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠈⠀⠡⣵⣮⣠⣌⣀⣠⢰⡾⣿⣿⣿⣿⣿⣿⡷⠶⣾⠟⣻⢛⣿⣟⣿⠛⠑⠄⣄⣀⣠⣾⡆⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣾⢯⣿⣻⣿⣿⠀⠈⠃⠘⠿⠙⣿⢿⡫⡵⡼⣿⣇⢋⠺⣛⠿⠂⠀⠀⢰⣿⡿⠟⠋⠻⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠐⠀⠉⢎⠠⣙⢏⠱⠈⠂⣄⣀⢀⡀⠀⠋⠀⠁⠀⠉⠀⠀⠀⣠⣾⣿⣶⣦⣈⠁⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⣀⣈⡄⠀⢀⣀⠀⠀⢀⣾⡿⣿⣿⣿⣶⣦⢴⣶⠆⠀⠀⢠⠟⣿⠙⠟⣽⣿⣿⡏⣁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣰⣿⣿⣿⡿⠻⠋⡄⠀⣘⢻⠱⣟⠿⣛⠻⠛⡆⠀⠀⠀⠈⠁⠀⡀⡜⢼⠿⡿⣿⠛⠛⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⠸⢏⠲⠽⡢⠄⠀⠀⠀⠀⠀⠙⠒⡓⢒⣾⠤⠃⠀⠀⠀⠀⠀⠀⠀⠈⠂⠭⠉⠑⠂⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣥⣶⡄⠂⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣜⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⢿⣿⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⢫⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⣤⣷⣴⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠟⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠀⠀⣾⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠉⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⡿⢠⣶⣤⣤⠀⢀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⠀⠈⠀⠒⢻⣿⣶⠶⢁⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡖⠠⠀⠀⠠⠐⢠⣿⣄⣠⣤⠤⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡴⠀⢈⠥⠄⠀⢰⣿⣍⣁⣚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡏⡀⠀⠀⠀⠀⣐⣿⣏⣩⣭⠏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⢀⡤⢒⣾⠟⠋⠁⠀⠀⠀⢨⣿⡿⢷⣾⡛⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⣈⠷⠆⠚⢓⠒⠿⣷⣤⣄⣀⠠⠖⠋⠚⠈⠁⠀⠀⠀⠀⠀⠀⣀⣶⣧⣈⣷
⠀⠀⠀⠀⠀⠀⠀⠈⣁⣀⢸⣖⠀⠜⣃⠀⠔⣿⡿⠛⠁⠀⠀⠀⣤⠴⠢⠖⡰⠂⢀⢀⡿⢿⣯⣤⡶⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣾⣿⣶⣭⣠⣞⣋⠀⠀⢠⣄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠺⢿⣥⣤⡠⢤⣄⠐⣤⣴⠖⢲⣶⠒⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣷⡧⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠛⣏⢿⣯⣬⣿⣇⡙⣿⡦⠀⢿⣆⠀⠻⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⠀⠀⠀⠄⠀⠦⢀⡃⢈⠀⡁⠔⠀⠀⠀⠀⠀⠀⠠⠈⣿⠿⣿⣿⣿⣂⣿⣿⠀⢘⣳⠀⠀⣿⣷⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠀⠀⠀⠉⠉⠀⠃⠼⠐⢆⣀⣄⠰⠀⠀⠀⢐⢴⣄⠓⠾⡩⠟⠿⣭⣴⡟⠉⢩⣷⡶⠿⣥⣠⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⠒⢿⡄⢛⠳⠶⠈⠙⠻⢿⣷⣾⡻⠉⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡐⠟⠉⠛⠂⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⠟⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠁⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⢀⣄⠀⠀⣠⡀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠈⠛⠀⠀⠙⠃⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠇⠀⠀⠀⠀⠠⣶⣶⣶⣦⠀⠀⠀⠀⠸⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⣿⣿⣿⣿⣿⠏⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠏⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⠏⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⢻⣿⣿⣿
⠀⠀⠀⢿⣿⠋⠀⠀⠸⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠻⣿⡟
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⠶⠟⠛⠋⠁⠀⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣾⣿⣿⣿⠃⠀⠀⠈⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⡿⠁⠀⠀⠀⠀⠘⠿⠛⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣟⣴⡾⣃⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣴⣿⣿⣿⣿⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡿⠋⠽⠋⠼⢟⣻⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠷⠻⢿⣦⡀⠄⣚⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⣴⡟⣀⣙⣽⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⢀⡴⠂⠠⠬⣦⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⣀⠾⠛⢢⣝⣟⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠛⠇⢀⣀⣠⣠⣟⠳⣎⣹⣶⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⢠⡆⠈⣿⣿⣯⡈⢳⣮⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣰⠯⠀⢈⣿⡇⣈⠻⣷⣜⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣋⢀⣠⡾⠃⠀⢀⣼⣿⣿⣦⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣾⣗⣀⠀⡀⠀⠀⢺⣇⡈⠛⢿⣷⠬⣾⣿⡿⣿⠿⣬⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⡈⠉⠉⠛⢿⣿⡞⢛⠳⠿⠿⢷⣾⣿⣿⣿⣯⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣶⣄⡀⣷⣿⡇⡁⠀⠀⠀⠀⠙⢿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠞⠻⠏⠻⢿⣿⡿⠿⣶⡇⣃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠊⠉⠀⢀⣠⠤⠀⠀⢀⣀⢨⣬⣍⡙⠻⢇⡐⢰⠆⠐⠈⠹⡖⡢⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠁⠠⠴⠒⢃⡉⠀⣀⡰⣮⣿⣿⣿⣿⣿⠟⠃⠈⢳⣶⣏⣴⡍⠹⣿⠊⠀⣠⠟
⠀⠀⠀⠀⠀⠀⣀⡔⣵⣿⠁⠀⠀⢀⢀⣠⣼⣿⣿⣧⣌⢻⣵⣿⡟⠃⠀⠀⠀⠀⠀⢻⣿⢿⣷⣿⡟⣥⠞⠁
⠀⠀⠀⠀⣤⣾⣿⢋⠝⣿⣦⡶⣾⣿⣿⣽⢹⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢎⣁⣣⡵⠟⠁
⠀⠀⠀⣠⣶⣿⣿⣧⣦⠟⡛⢰⣔⢻⣿⣿⣾⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⣠⣾⣿⣿⣷⣌⣾⣾⠿⠾⠿⠛⠋⠉⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣶⣶⣤⣀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠙⠛⠛⠛⠋⠉⠈⠻⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠟⠛⠛⠿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣭⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠛⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⢷⣬⣻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⣽⣿⣽⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣻⢿⣯⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣟⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣟⢿⣿⣿⣷⡿⣷⣸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⡿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡙⣿⣿⣿⣿⣷⣿⣿⣿⣇⣿⣧⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣝⠻⢿⣌⠻⠷⡜⢿⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣻⣷⣿⣿⣿⣿⢿⣿⠿⣿⣯⣿⡿⣯⢹⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠼⣮⡙⢿⣎⠹⣆⠐⡄⢸⣿⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣾⣟⢷⣿⡿⢯⣿⣟⣿⣿⣾⣿⣹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣻⢿⣦⡙⢧⡙⢧⡀⠰⢦⣨⣷⣧
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣟⢿⣾⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣮⣛⣲⣷⣶⣿⣿⣿⣿⡿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⡿⣿⣿⣿⣿⣯⣿⣿⡿⣯⣻⣿⣦⣀⡀⠀⠀⠀⠀⣴⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢼⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣿⣿⣿⣿⡿⣵⣿⣛⣻⣭⣽⣿⣿⢿⣶⣦⣾⣿⣿⣿⣿⣿⢿⡿⠿⢿⣿⣿⣿⣿⣿⡀⠈⣁⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣯⣝⣿⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣄⣀⣀⣛⣿⣟⡿⢿⠟⠟⠀⠚⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠿⠿⣟⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡛⡛⣛⣃⣠⣞⣉⣸⠆
⠀⠀⠀⠀⠀⠀⠤⠤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣽⣿⣿⣿⣿⣷⣛⡻⢿⣿⢻⡿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⢟⡛⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣷⣾⣿⣻⣿⣷⣧⣬⡉⢿⣿⢍⡱⣦⡻⣶⣮⣦⡐⢯⡻⣿⣿⣿⣿⣿⡶⣿⣿⣿⣿⣿⣿⣦⣁⣠⣄⡉⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣿⣷⣿⡟⢷⣭⣻⣿⠟⢻⣿⣏⡻⣿⠿⢄⠙⢦⡙⣿⣿⣿⣿⣿⡟⣾⣿⡿⠗⠠⠌⢙⠿⠟⢛⣽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣷⣿⣿⣿⡄⣀⣿⣿⣿⣾⢷⣦⣸⡫⠀⠹⢿⡿⠿⠿⠋⡇⠀⣡⣀⣤⠠⠤⠴⠂⠿⠿⠷⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢟⠈⠛⢿⢿⡄⣴⡟⢁⡐⠄⠀⢸⡇⠀⠙⢷⣾⣿⣃⡀⠠⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢸⣎⣟⣾⡍⠀⣰⢾⣷⣽⣷⣾⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣝⠦⣑⢄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣷⣮⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⣿⣧⢻⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣷⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣳⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢻⠿⣿⠿⠛⠛⠉⠉⠀⠀⠸⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣠⠖⢦⡄⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣰⡞⠛⢲⡄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣷⣤⣼⣿⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣶⣿⡇⠀⠀⠘⣿⡿⠛⠋⠉⠙⠛⣿⠀⠀⠙⠿⠿⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠈⠙⠓⠲⠶⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠶⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠷⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⡷⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣷⣄⣈⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣭⣉⣠⣴⣿⣿⣿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢰⣄⠀⢰⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡎⣿⡄⢸⣿⡄⣿⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢹⣿⣿⣿⣸⣿⡿⣿⡟⢻⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣞⣿⠟⠋⠁⠈⠁⠋⢀⠻⣭⡽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⣀⣀⣄⣀⡤⠀⠘⠀⠑⠚⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡴⠞⠟⠛⠻⢋⠁⠀⠈⠀⠀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣤⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠤⣤⣀⡀⠀⡸⠇⠀⠀⣀⣀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠶⣶⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣶⣿⣿⣿⣁⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⣿⣧⣬⣉⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠈⠀⠰⠲⡀⢶⣶⣤⣤⣄⠀⣄⣴⣶⣿⠳⠄⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠁⠀⠀⠀⠀⠈⡴⢡⠠⠠⠀⣿⡿⢧⠘⣏⣻⣧⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠏⠀⣀⠀⡀⠀⠈⢥⡃⠆⡀⢸⡟⠀⠀⠢⣽⡿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⢉⡲⠁⠀⠀⠀⠙⠶⣅⣿⡇⠀⠀⠀⠈⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠒⠁⡄⠀⠀⠑⠀⠐⡀⠂⠀⠠⢀⣲⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣖⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⠂⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠈⢡⣼⡿⠋
⠀⠀⠀⠀⠀⠀⣄⠲⠤⠶⢋⣅⡆⡄⠀⠀⠀⠤⠀⠄⠀⡀⠀⣀⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠻⠆⢀⡴⠟⣽⣿⢯⣭⡒⣠⣀⠀⠀⠀⣂⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣠⡾⠟⢡⣿⢿⣿⣿⣷⣶⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠋⣸⣿⣿⣿⠋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠟⠃⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⠉⠉⠉⠙⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⣀⣀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠘⠦⣄⣀⣠⠴⠒⠶⠶⠒⠋⠳⠤⣀⣀⣀⣠⠴⠊

⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⡞⠀⣠⡄⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠢⠀⠈⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⡠⠄⠻⢄⡀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡗⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⢦⣀
⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢹⡇⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠀⠀⠀⠀⠈⢳⣴⣿⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⠀⠀⠀⢹⣇⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣵⡀⠀⠀⡐⠂⠀⠤⠤⠴⠞⠻⣍⠛⠳⠶⢾⣿⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠙⠓⠶⠤⣤⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠜⠀⣒⠉⢀⠄⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⢄⠀⠀⠰⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠡⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⢀⡐⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠡⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠧⠀⠀⣈⡖⢂⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢀⠀⠈⡆⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⢀⣤⠀⠀⠀⠀⠑⢤⡀⠀⠀⠈⠁⠀⠀⡁⠐⠐⢦⡀⠀⠀⠁⢢
⠀⠀⠀⠀⠀⢀⢠⠖⠃⠀⠤⠀⢅⣄⠀⠀⠀⠐⠂⠀⠚⠲⣄⠀⢀⣤⠀⣀⣀⣄⣀⣀⣀⣰⠤⡀⠓⢀⡀
⠀⠀⠐⠒⠓⠒⠒⠲⠢⠴⠚⠒⠒⠒⠤⢤⣀⣤⣉⡤⠤⠤⠤⠭⠡⠑⣟⡠⠠⠀⠀⠀⠀⠀⠁⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠀⠀⢰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣤⣤⣠⣤⣄⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠤⠦⠄⠀⠀⠀⠀⠀⠀⣠⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⣀⣼⣿⣿⣿⣿⣤⣀⣀⣀⢰⣄⡠⣀⣀⣀⣠⣤⣤⣾⣿⣿⣿⣷⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠛⠿⠿⣿⡿⠋⠉⠁⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⡄⢀⣴⣿⣿⣷⣶⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠉⠉⠁⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠿⢿⣿⣿⡿⠟⠋⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⠉⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠛⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⣸⣿⣿⣿⣿⠁⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰⡗⢠⣿⡿⠻⣿⣟⠀⡄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⣿⣦⣠⣿⣿⠈⠋⠀⠸⡏
⠀⠀⠀⠀⢰⣃⠈⠀⠀⠐⠢⣄⡀⠀⠀⣠⣾⣄⠀⠀⠀⠙⠿⢿⡿⠿⠋⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠉⠳⡤⠄⠀⠀⠀⠈⣷⡶⠏⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠶⠂⠂⠀⠀⠘⠲⠤
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣠⡠⣄⣴⣹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢻⣿⣧⣀⣀⡠⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⣯⣁⢈⡀⠀⠀⠀⠀⠀⠀⢄⠆⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠚⠸⣿⣿⣗⣺⠄⡔⠀⠀⢀⣠⡾⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢼⣿⣿⣿⣿⣴⣶⣷⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢻⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢹⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢺⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣃
⠀⠀⠀⠀⠀⠀⢠⡏⠹⣿⡉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠿⣦
⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠿⡀
⠀⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿
⠀⠀⠀⠀⠐⣣⠐⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠿⣟⡂
⠀⠀⠀⠀⢸⠦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡗
⠀⠀⠀⠀⢠⠆⠘⠁⠀⠀⡴⠀⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢴⣿⣿
⠀⠀⠀⠀⠀⢄⠈⠀⠀⠀⣶⣄⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⡏
⠀⠀⠀⠀⠀⠐⣠⠂⠀⠀⠈⢣⡞⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⡶
⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠈⢻⣿⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⠀⠀⠀⣄⢦⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡷⠄⠀⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣣⣵⠄⠀⠀⠀⣈⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⣰⣿⣿⣿⣿⣿⣧⣆⣠⠀⠠⡀⠀⠀⢀⠀⢀⡄⢀⣤⣟⡅⡀⢀⣤⣹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣯⣦⣴⣮⣤⣼⣶⣿⣿⣹⣷⣿⣮⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⣻⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⢿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠻⠛⠉⠿⠛⠁⠀⠀⠈⠀⠀⠀⠀⠘⠻⠶⠛⠷⠿⠛⠋⠁

⣿⠿⡿⢿⠿⠿⠟⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⠿⢿⣿⡿⠿⠿⣿⣿⡿⠿
⣿⣿⣿⣶⣿⣿⣿⣿⣾⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣠⣿⡿⣡⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣾⣿⣏⡀
⠀⠀⠀⠀⠀⠀⢰⣆⠀⠀⣰⠏⢏⣉⠉⣭⢭⣾⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠀⢀⠏⠐⣨⢉⣐⢒⣩⣽⣿⣿
⠀⠀⠀⢀⠀⢀⣿⣿⣶⡐⣐⠃⠀⠭⢏⣒⡲⠚⣛⣿⡆
⠀⠀⠀⢸⣷⢸⣿⣿⡿⠠⡎⠈⣛⣥⠼⢩⣭⡭⢻⣿⣇
⠀⠀⠀⠸⣿⣨⣿⣿⠁⠀⡡⠀⠉⣉⡓⠚⠤⡬⠿⠿⡏
⠀⠀⠀⠀⣿⣿⣿⣿⢘⠹⣡⡶⠀⠠⣬⡛⢯⣼⣻⣿⡗
⠀⠀⠀⠀⢻⣿⣿⡏⠐⠆⠶⣴⢮⡆⠢⣍⡳⣴⣾⣿⡇
⠀⠀⠀⠀⠀⢿⣿⡇⢈⢈⠃⠌⣿⠃⢹⣸⣭⡷⣭⣿⡅
⠀⠀⠀⠀⠀⠐⣿⡇⠀⣤⡉⠰⠉⠃⠠⣺⢮⡳⣝⣿⠁
⠀⠀⠀⠀⠀⠐⣿⣿⣦⡈⠳⣴⣼⠫⡖⠳⣿⣞⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄⠉⠞⠴⣀⠲⢍⡀⢹⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠠⠼⢀⠠⡹⣞⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢴⢨⡗⠎⡸⣿⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⡆⠀⠀⠀⢠⠒⢺⡄⣈⣿⣬⡻⣷
⠀⠀⠀⠀⠀⠀⠀⣤⠟⢼⣿⡋⠽⣿⣆⠐⠂⠍⣁⡈⠉⢓⣬⣻⣷
⠀⠀⠀⠀⠀⠀⢰⣯⣾⣷⢿⡯⠈⣿⣿⠄⠈⠂⠊⠡⠀⢾⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⢛⣿⡇⠈⠙⠻⢿⣿⣿⣥⣤⡀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠉⠙⠻⣿⣿⣆⠀⠀⠀⣾⣿⣄⠀⠀⠺⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠆⠀⢰⣿⣿⣿⣶⣾⣆⣹⣿⣿⣿⣶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⣈⣷⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡳⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣶⣄⡀⣠⡾⠛⣩⣷⡿⠇⠁⠀⢸⣿⣿⣿⣿⢻⣿⣿⣿⣿⢦⣁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⠛⣿⣿⣻⢤⡼⠋⠀⠀⠀⠀⠀⠸⣿⣿⡏⠿⡇⠙⢿⠙⣷⣖⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣁⡼⢻⡟⢷⣄⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢀⠀⠀⠬⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠾⠁⠀⠀⢳⡤⣤⠀⠀⢠⡞⠁⠀⠀⢛⡆⠀⠀⢀⣠⣤⣤⡾⠋⠀⠀⠀⠉⢳⣀⣀⡀
⠀⠀⠀⠀⠀⣰⠞⠛⠂⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠉⠉⠛⠉⠉⠁⠀⢰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⢧
⠀⠀⠀⠀⠀⠹⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣸⠃⠀⠀⠀⠀⠀⣠⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠈⠉⠻⠶⠞⢦⣤⡤⠶⠶⠞⠉⠁⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠛⢦⡄⠀⢹⣆⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣀⡀⢀⠀⠀⢛⡆⠀⠙⠳⠶⠴⠞⠦⣄⣀⣀⣹⣄⠀⠀⠀⠀⢀⡰⠦⠤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⣤⣀⡤⠤⣄⡈⠉⠓⠚⠛⠋⠉⠁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠛⠶⠶⠖⠋
⠀⠀⠀⠀⠀⠀⣀⣰⠁⠀⠀⠀⠛⠀⠀⠀⢷⣀⠀⠀⠀⠀⣴⠒⠏⠉⠛⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠓⣄
⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢳⡄⠀⠰⣏⣠⠀⠀⠀⣁⣹⠇⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠁⠀⠀⠀⠸⠦⣄
⠀⠀⠀⠀⠀⠘⢦⣄⣤⠀⠀⠀⢠⡄⠀⢀⣄⣠⡼⠁⠀⠀⠈⠙⠛⠛⠛⠉⠁⠀⣠⣤⣀⠀⠀⠀⠀⠹⢦⣄⣀⣀⣀⣀⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⠤⠞⠛⠒⠋⠀⢀⣀⣠⣤⣄⠀⠀⠀⠀⠀⢀⡴⠚⠀⠀⠉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡴⠿⠀⠀⠀⠈⠛⣧⠀⠀⠀⠘⢧⣄⠀⣄⠀⣲⡼⠇⠀⠀⢀⣤⠶⢦⣤⠤⣤
⠀⠀⠀⠀⠀⣠⠟⠉⠁⠉⢳⠀⠀⠀⠀⠈⠷⣤⣤⣤⣤⣤⡤⠼⠃⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⡞⠉⠁⠀⠀⠁⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠙⠚⠦⠶⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠛⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠳⢴⡀⠀⠀⣀⠀⣠⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⠀⠀⠇⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢴⡞⠉⠀⠈⠻⣤⠤⣤⡀⠀⢷⣀⡆⠀⠀⡀⠀⠀⠀⢀⣸⠇⠀⢀⡴⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⣠⣴⠇⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢻⡆⠀⠈⠳⣤⣤⠷⣤⣤⡴⠋⢀⣀⣴⠋⠀⠀⠀⠀⠙⣧⠴⠶⣤⡀
⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠻⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠹⣧⡀⠀⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⢀⣴⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣄⠀⠀⣀⣀⠀⣀⡼⠛⠛⠉⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


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

⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⢿⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣯⣤⡶⠀⠂⠀⣄⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣴⣶⣿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣨⡷⠄⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠟⠛⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣤⡀⠀⠀⢹⢡⠙⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣠⣄⠄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⡙⢂⡆⠀⠄⠀⠀⠂⠈⠠⠀⠀⠀⠀⠀⠀⡤⠞⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣷⣾⠃⠄⠀⠀⠀⠂⠁⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣳⣀⣹⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣼⣵⢧⠋⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣛⠛⠛⡂⠀⣀⣀⣤⣾⣿⣿⣶⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠀⢀⣀⠛⠛⠉⠀⠀⠿⡿⣿⡏⢍⡊⠼⢽⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢝⠧⠻⣷⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣄⠰⡱⡑⣕⢮⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⣶⠔⠔⣱⢲⢿⢸⢹⠈⢢⣤⢀⡆⠈⠁⠓⠚⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠴⢪⢦⢃⣇⣎⣄⢤⠖⠉⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡈⠀⢄⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⠑⠋⠊⠉

⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣷⠸⣧⠀⠀⣾⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡆
⠀⠀⠀⠀⠀⠀⢿⠀⠙⣦⠀⣿⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⡇
⠀⠀⠀⠀⢰⣦⡘⣧⠀⠈⢷⣼⡆⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣰⠏⢀⡿⣠⡄
⠀⠀⠀⠀⠸⣇⠙⢿⣧⡀⠀⠙⢿⣄⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢿⡼⠋⠀⣼⠟⢹⡇
⠀⠀⠀⠀⠀⠻⣆⠀⠉⠻⣦⡀⠀⠙⢦⣀⣾⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢋⣴⠋⠀⣠⡾⠋⢀⡾⠁
⠀⠀⠀⠀⠀⣤⣝⣷⣄⠀⠀⠙⠲⢤⡀⠈⣿⠣⡀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢉⠔⢋⡿⡠⠞⠁⠀⣠⣞⡁
⠀⠀⠀⠀⠀⢻⣍⠉⠛⠷⢦⣄⣀⠀⠈⠑⣿⣦⡈⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠀⠔⢻⡏⠀⣀⡤⠞⠛⣹⠇
⠀⠀⠀⠀⠀⠀⠙⠷⣄⡀⠀⠀⠉⠉⠓⠒⢿⣄⠙⠦⣄⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣁⣤⣄⣈⣀⡴⠟⠂⠉⠀⢀⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⠿⠶⠦⠤⠤⠤⠤⠽⠷⣤⣀⡀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣠⡶⠋⠉⠀⣀⡀⠉⠙⢷⣤⠴⠶⠾⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣉⣿⣅⡀⠀⠀⣿⠀⠀⠀⠀⣰⠏⠀⠀⠀⠈⠁⠈⠉⠲⡄⠹⣦⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣿⣭⣁⣀⠠⠤⠤⢿⣄⣀⣀⡀⠀⠘⣧⣀⣀⣰⡏⠀⠀⠀⠐⠶⣾⣿⠄⠀⢹⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⣀⣀⣀⡠⠤⢼⡏⠁⠀⠀⠀⠈⠛⢿⣍⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣺⠟⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠋⠉⠀⠀⠀⣀⠿⢦⣀⡀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⣶⠛⠉⠀⣀⠄⠉⣉⠋⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢶⣋⣁⡠⠞⠁⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠷⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠛⠉⢿⣄⠀⠀⠀⠀⣠⠀⠀⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡶⠞⡋⠄⠀⡀⠀⠀⠀⢠⣯⣤⣶⢶⡟⠓⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣻⡿⠗⡉⠀⣠⠞⢀⠄⢀⠀⣼⠹⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠶⣶⣾⣿⠟⠋⠐⠈⣠⠞⠁⣠⠊⢀⠎⣰⢿⡄⣷⢹⠀⣇
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠈⢀⣠⠞⠁⢀⡜⠁⣠⠏⢰⠏⠘⣧⣿⠈⠳⠟
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⣹⠇⢀⣴⠋⠀⣴⠃⢠⡟⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠗⠛⢿⡶⠞⠻⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡐⢤⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢁⣀⣱⣮⡀⢸⡀⠀⠀⠀⠠⠝⠿⣄⠁⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡁⠀⠀⠈⢁⡨⠿⠯⠀⠀⠀⠀⠀⠀⠀⠛⣇⣀⣠⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠟⠃⠀⠀⠀⠻⠀⠀⠀⠈⢈⡧⠀⠀⠀⠀⠀⠘⠿⠛⠛⠛⠛⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠐⣀⣠⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⣤⣦⡀⠀⢀⣄⠀⠀⣸⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣶⣶⣤⣶⣾⣿⣶⣶⣦⣀⣠⣴⣿⣿⣷⣶⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⢿⣿⣿⣿⡿⠿⠿⠿⢿⣿⠿⠛⠋⠁⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⢻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⡀⠄⠀⠀⠘⠓⠲⣾⣿⣏⣀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠶⠒⠇⠀⠀⠀⠐⠚⠃⠀⠀⠀⠀⠀⠀⢼⣿⣿⠟⠉⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢈⣿⣇⣀⣀⣠⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠙⠛⠓⠒⢶⣿⣶⣾⣿⣿⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢰⡷⣦⣄⣀⠀⠀⠀⢀⣠⣤⣤⣤⣀
⠀⠀⠀⠈⣿⣮⣻⣿⣷⣄⣼⣿⢟⣵⣿⡟⠁
⠀⠀⠀⠀⠘⣿⣿⣾⢻⣿⢽⣯⣿⣿⡟
⠀⠀⠀⠀⠀⠈⠙⠿⠏⢫⣼⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢀⣠⣴⣦⢶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢀⣴⣾⣿⣿⠀⠀⠀⣠⣶⣿⣿⣿⣿⣮⡻⣿⠿⢿⣷⡀
⠀⠀⠀⠀⢀⣀⣀⣿⠀⠀⠀⠀⣿⣿⣿⡿⠃⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣩⣟⡛⠛⣠⡀
⠀⠀⠀⠀⠀⠉⠛⢿⡆⢀⡤⠀⠈⠉⠁⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣟⢩⣼⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣼⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢠⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣺⣿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣧⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⢿⣦⣤⡀⢀⣤
⠀⠀⠀⠀⠀⠀⣠⣴⡿⢿⣿⡿⠛⠉⠉⠙⣿⣿⣿⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⠁⢻⡆⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢟⣟⢿⣽⣟⣷⣿⣿
⠀⠀⠀⠀⢀⣾⡟⠉⠀⣾⠋⠀⠀⠀⠀⠀⢻⣿⡿⠀⣿⡄⠙⣿⣿⣿⣿⣿⣿⣿⠀⢸⣷⡈⠛⠛⢀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣟⣯⡿⣿⡿⣾⣿⣿⣿
⠀⠀⠀⢠⣿⠏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣠⣼⣿⣧⠠⣌⠻⣿⣿⣿⣿⣿⢸⠀⣿⣿⣷⣶⣿⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣷⣿⣾⣧⠙⠿⠟⠃
⠀⠀⠀⣼⣿⣷⠀⠀⠀⢻⠀⠀⢀⣀⣀⡀⠀⠈⢻⣿⣿⣿⠀⣿⣦⠈⠻⣿⣿⣿⠘⣇⠘⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠟⠀⣦
⠀⠀⠀⣿⡏⢿⣇⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⡄⠀⠹⣿⠿⠀⣼⣿⣧⠀⡘⢿⣿⠀⢻⣦⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⠄
⠀⠀⠀⢹⣷⠀⠙⢧⡀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠀⠀⢀⣴⣿⣿⣿⡇⢹⣆⠹⠀⠈⠻⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⢠⢿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠀⠀⠀⠀⠀⠙⠿⣿⡿⠁⣼⣿⡆⢠⠸⣦⣈⢉⣀⣶⣄⠀⠀⠀⣀⣀⣤⠖⠀⠀⠀⠀⠀⣠⡾⠃
⠀⢀⡴⠋⠀⠻⣦⠀⠀⠀⠀⢰⣿⣿⣿⣷⣶⣄⣀⡀⠀⠀⠀⠀⠀⠠⢾⣿⣿⡗⢸⣧⡙⠿⠿⠉⠻⢿⣿⣾⣽⣟⣋⣁⣀⣀⣀⣤⣴⡾⠟⠁
⠀⠉⠀⠀⠀⠀⠘⠳⣄⡀⠀⣿⣿⣿⡿⣿⣯⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠈⢫⣥⣾⣿⣿⣶⣶⡄⠀⠀⠈⠉⠛⠛⠻⠿⠿⢿⣛⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢘⡛⢿⣶⣿⣿⣮⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡽⣿⣻⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢿⡏⣿⣯⡻⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠾⠿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢹⣿⣷⡘⢯⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⠈⣿⣿⣿⡌⢳⡝⣷⣄⠀⠀⠀⠀⠀⠀⣼⣿⣿⡌⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢹⡆⠸⣿⣿⣷⡀⢻⣎⢿⣷⣄⠀⠀⠀⠀⢿⣯⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡘⣷⡀⠹⣿⣿⣧⠀⢿⣆⠻⣿⢷⣄⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡇⢻⣧⡀⠙⢿⣿⠀⠘⣿⡆⠹⣧⡙⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻⡄⠹⣷⣄⠀⠁⠀⠀⣿⣿⠀⢹⣷⠈⢿⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠻⠀⠙⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠘⠃⠀⠃

⠀⠀⠀⠀⢰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠂
⠀⠀⠀⠀⣶⣽⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣈⣻⣿⡛⠿⠿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢾⣿⡿⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠈⣛⣿⣟⠓⠲⠤⢌⠛⠿⠿⠶⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠛⠛⠩⠒⢉⣰⣾⣿⢿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣧⣤⣤⣄⡀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠁⠀⠀⠀⠀⠀⢒⣛⣋⣉⣵⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢷⣶⣶⣶⡮⠤⠀⣀⠀⢀⣀⠀⠈⠻⢦⡀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⢀⠀⠀⠤⣤⣽⣶⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣷⣊⣩⡤⢖⡫⠀⡠⠀⠀⠀⡠⠀⢄⠀⢸⠁⠀⠀⠀⠀⠀⠐⠭⣕⣲⣶⣶⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣛⣷⡞⢉⡴⢉⠀⢀⠀⠸⣷⣤⠂⣁⣈⠀⠀⠀⠀⢀⠈⠒⢤⣦⣭⣿⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣵⣯⣶⡃⢀⡞⡄⣰⡅⢊⣾⣿⣿⣷⡀⠀⢄⠈⢷⣝⢦⣙⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣼⣡⣿⣿⣿⣿⣿⣿⣿⣧⠠⡀⢳⡘⣿⡳⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⠟⠈⠀⠹⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠎⠸⠛⣉⠀⡀⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣟⡩⠀⢀⡄⠀⠁⠀⠀⠀⠐⢄⠈⡑⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣥⢖⡡⣂⢀⡆⡄⢀⢣⠐⡌⢳⣼⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠾⣧⣾⣵⣇⣴⣬⣷⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣢⡴⣬⣴⡋⠉⠐⠶⠮⢩⠵⡖⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⡄⣾⣿⣉⣿⣷⠿⠂⠀⠤⠀⠀⠀⠀⠒⠛⠁⠐⠾⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣯⣶⣿⣿⣿⣷⡛⠉⠉⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠁⠦⠀⠠⣍⠙⢶⣄
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣥⣾⣿⣻⣿⢿⣿⣿⠗⠞⠋⠀⠀⠀⠀⠀⠠⠤⠤⢼⣿⣦
⠀⠀⠀⠀⠀⡮⠉⠾⠁⢹⢾⠛⠻⣿⣿⣿⠟⣹⣿⡼⢃⠍⠀⠀⠀⠀⠀⠀⠐⠂⠀⠰⠦⣶⣶⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠀⠀⠀⢠⣤⣍⣉⣍⣨⣶⣯⣤⣄⣤⢄⣀⠀⠀⠀⠀⠀⠺⠿⠿⢿⣛⣽⣇
⠀⠀⠀⡔⠀⠐⣜⠛⠐⠙⠗⠀⣀⣼⣿⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⠀⠀⠀⠁⠂⠴⣶⣌⠛⣟⢿⡆
⠀⠀⢰⣿⣶⣿⣧⠄⣤⣶⣶⠟⠙⠒⠚⠻⠿⢟⣿⣿⡿⠿⣿⣿⣿⣶⣿⡛⠲⣶⣄⣈⠓⠛⡿⠃⣿⣧⣷
⠀⠀⠸⣿⣿⡿⠿⠿⠟⠛⢳⣾⣿⣿⣷⣷⣤⣡⡀⠀⠀⠀⠛⠙⠛⢿⣿⡶⡲⣬⢕⡙⠻⢿⣾⣿⣿⡿⣿
⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⢀⣿⣷⣏⢿⠛⡻⢿⡷⣄⠀⠀⠀⠀⠀⠢⡙⠷⣦⣹⡶⣽⡷⡆⢮⣿⣟⣟⣾
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣴⣮⠀⢃⠈⠣⡀⠀⠀⠀⠀⠠⡀⠘⢿⢿⡌⢣⣸⠿⣿⣷⣿⡏
⠀⠀⠀⠀⠀⠘⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡌⠈⠹⡙⢻⢣⣙⣧⣽⠇
⠀⠀⠀⠀⠀⠀⠈⠱⠿⠿⠿⢿⣿⢿⡽⢿⡟⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⢹⠀⠀⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡸⢿⠟⣥⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⣧⢐⣀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠟⢁⠔⠤⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡄⠀⢿⠀⠀⢸⡆⠈⠨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠸⠁⠀⡨⠑


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⠀⠀⢀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⣈⠉⡁⠢
⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⠈⠹⠂⢡⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠅⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠄⠀⠐⣁⠈
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⡄⠀⠀⠀⠀⠠⠐⡒⠠
⠀⠀⠀⠀⠀⢄⠢⣀⠄⠀⠠⢀⠌⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠙⢀⠤⡊⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢔⠀⠈⠘⠛⠂
⠀⠀⠀⠀⠀⠀⠑⠄⠁⠀⢀⡈⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⢰⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠄⠢⠐⢂⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⢠⡃⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⠀⠀⠀⠀⢤⣼⣶⣤⣄⡐⠦⣤⣶⣴
⠀⠀⠀⠀⢠⣀⡨⠖⠛⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⢄⠰⢁⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⢀⣸⣻⣿⣿⣿⠉⠉⠛⠓⠏
⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠚⠂⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⣀⣉⣉⣉⣉⣹⠇
⠀⠀⠀⠀⢀⡀⠀⣁⠀⡀⡠⠌⢂⣤⣴⣈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⣀⠀⠀⠀⠀⣀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⡅⠈⠉⠋⡩⠜⢀
⠀⠀⠀⠀⠨⠁⠀⠀⠀⠘⢾⣃⣈⡛⡛⣲⠄⡀⠀⠀⠀⠀⠀⠀⢹⣛⣛⢓⣲⠿⠋⠉⠉⠩⢁⠀⠀⠀⠀⠀⠀⠀⣔⠫⠛⠛⠀⡤⠵⠖⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠚⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡉⠁⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠈⠀⠀⠀⠀⠈⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠑⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠒⠂⠴⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣾⠓⠒⠒⠒⠲⠒⠖⠖⠖⠓⠲⠓⠲⠶⠖⠳⠒⠒⠒⠺⠖⠳⡖⠚⣶⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠊⠀⢀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⡀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⢀⣠⣎⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡔⠉⠀⠉⠁⠀⠀⠑⣄⠀⠀⠀⠀⢀⠔⠒⠢⠤⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠈⠒⠠⡄
⠀⠀⢀⡴⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠼⠀⠀⠀⠀⠀⠀⠑⠒⠦⠒⠢⣀⡀⠀⠀⠀⠀⠀⠀⣀⡠⠎⠁⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠆⠀⠀⠘⠛⠛⠦⠰⢄⠀⠀⠀⠀⠀⣀⢀⡠⠬⠭⠭⠕
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⣁⠤⠤⡀
⠀⠀⢠⠊⠉⠉⢢⣀⣀⠤⠒⠤⡀⠀⠀⢀⠤⠤⠊⠀⠈⢳⠀⠀⠀⠀⣀⠤⠤⠤⠔⠉⠉⢢⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠚⠁⠀⠀⠈⠦⢄⡀
⠀⢀⠎⠀⠀⠀⠀⠀⠈⠀⠀⠀⠳⡄⠀⡇⠀⠀⠀⠀⠀⠸⡁⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⡀⠀⠀⠀⢀⡴⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢
⠀⠀⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠁⠀⠈⢇⠀⠀⡀⠀⠀⡇⠀⠘⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡧⠀⠀⠱⢄⠀⢀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠜
⠀⠀⠀⠀⠀⠀⢀⡤⠒⠢⡀⠀⠀⠀⠀⠀⠈⠒⠊⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠓⠠⠔⠊
⠀⠀⠀⡰⠉⠉⠋⠀⠀⠀⢸⣄⣀⠀⠀⠀⠀⢀⠔⠊⠉⠒⢄⠠⢄⡀⠀⠀⠀⢀⣀⡔⠁⠀⠈⢢⢀⡀⠀⠀⠀⠀⠀⣀⢀⠔⠉⠉⠑⢄
⠀⠠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⠀⡰⠉⠀⠉⠀⠀⠀⠀⢸⠒⠒⢯⣉⡉⢡
⠀⠈⠢⢄⣀⣀⣀⣀⣀⣀⣀⣀⠴⠃⠀⣏⣁⠀⡀⠀⠀⠀⠀⠀⠀⢀⠇⠸⣄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠱⣀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠁⠈⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠒⠀⠒⠉⠒⠊⠁⠀⠀⠀⠀⠑⠒⢆⡀⢀⡠⢀⣀⣀⠜⠀⠀⠈⡀⠀⠀⣀⠀⠀⢀⠆⠀⠀⠀⡤⠃
⠀⠀⠀⡠⠒⠠⣀⣀⡰⠊⣉⠵⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠉⠘⠣⠤⠒⠤⠴⠃
⠀⢀⠜⠀⠀⠀⢈⠔⠈⠒⠁⠀⠀⠈⣆⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⢁⠔⠋⠑⠒⠉⠉⠐⢄⠀⠀⠀⠀⢠⠂⠉⠑⠴⠊⠉⠲⡄
⠀⠈⠢⠤⠤⡔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⢴⠊⠀⠀⠀⢀⡔⠊⠋⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⢀⡤⠶⠤⠴⠉⠉⠑⣄⣀⠱⠒⠢⠔⠒⢢
⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠉⠉⠉⢳⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⣀⡀⣀⡀⡉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⠒⠒⠂⠀⠀⠈⠉⠀⠀⢧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣹
⠀⠀⠀⠀⠀⡠⠴⠁⠀⠀⠙⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡰⠂⠁⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠒⠒⠢⡀
⠀⠀⢠⠔⠊⠀⠀⠀⠀⡠⠤⠎⠁⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⡤⠒⠓⢤⡀⢀⡠⢄⡀⠀⠀⠀⠀⠀⣀⠤⠀⠤⡎⠀⠈⠀⠀⠀⠀⠥⠤⡀
⠀⠀⡇⠀⠀⠀⢀⠔⠚⠀⠀⠀⠀⠀⠀⠀⠀⠱⠤⡀⠀⠀⠀⠈⢲⠀⡠⠜⠀⠀⠀⠀⠁⠉⡆⠀⠉⠙⡆⠀⠀⢀⠃⠀⠀⠀⠘⠒⠢⡔⠋⠉⠉⢢⠀⢈
⠀⠀⠘⠦⢀⠰⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢠⡤⣤⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠾⣁⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡎
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠓⠔⠢⢄⣀⠤⠊⠈⠁⠀⠀⠀⠀⠉⠉⠑⠈⠢⠄⠀⠤⣀⡠⠆⠔⠊⠉⠉⠉⠉⠀⠀⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠤⠤⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠁⠘⠻⣿⣿⣿⣿⣿⣿⣿⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠇⠀⠀⠀⠀⠀⠙⠿⣿⣿⠁⢸⡇⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢰⠀⠀⠀⠀⠀⠀⢀⡴⢫⣿⣶⣼⣷⡀⢧⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠁⠀⡆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠟⣿⣀⢺⣹⣇⣽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⡰⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⢸⣿⡟⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡉⠉⠓⢄⠀⠀⢀⣾⣿⣿⣿⡿⣻⡿⣿⣿⣿⣿⣿⣼⣟⣿⡟⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⡼⠓⢤⣿⡟⠻⣿⣿⣿⡀⠛⠺⣝⠻⣿⣿⢿⣿⣯⢡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⠞⠀⠀⠀⠙⢷⣌⣿⣟⣿⣿⣼⣃⡀⣴⡟⢿⡝⣿⣾⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡰⠃⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧⡛⣼⢯⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣯⣹⣿⠏⣽⣷⣿⣿⣿⢛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣷⣄⣟⣾⣿⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣟⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣾⣿⣿⣿⢹⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⣀⣤⣴⣿⣿⣿⣿⡍⣣⣿⣿⣦⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣿⣛⣿⣿⣿⣷⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢧⣉⣽⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡔⣿⣿⢦⣉⣹⣿⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢹⡀⠀⢹⡟⣿⣿⣿⣿⣿⣧⣿⣿⣿⠟⣫⡿⣿⢿⡿⢛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣇⠀⠈⣧⠀⢸⣷⣾⣿⣿⣿⡏⣿⡿⠁⣼⣿⣿⡔⠻⢸⣿⠀⠀⡜⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⣰⠟⠁⠘⠋⢆⠀⢸⡀⣸⣯⣿⡿⣽⡏⣿⠟⣠⡾⣯⣚⣿⢿⣦⡸⣿⠀⡴⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢀⠞⠁⠀⠀⢸⠀⠀⢂⠀⣇⣿⣿⣟⡞⡡⢘⣿⣾⢿⣿⣿⣧⢿⣞⣿⡁⣿⣾⣱⣿⣜⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡜⠁⠀⠀⠀⠀⡀⠀⠀⠀⢣⣸⣿⣯⡟⢁⣠⣿⣿⣿⣀⠈⡹⣿⣼⣿⡝⣧⣿⣿⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢀⡅⠀⠀⠀⠀⢹⣿⢏⣴⡟⢿⠟⠹⣷⣻⢦⡔⣮⢻⡿⣷⣽⣿⣿⣿⣽⣷⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠑⡀⠀⠀⢨⡇⠀⠀⠀⠀⢸⣿⣿⣏⠣⠈⢱⣄⠈⠙⢣⣾⣿⣶⣻⣌⠻⣿⣯⣹⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⣼⠙⣄⠀⠀⠀⣿⡁⣹⡞⡇⠀⢈⢿⠀⠈⢾⣯⣿⣿⣿⡿⠿⠟⡋⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠒⠤⠤⣀⣀⠈⢢⡿⠀⠈⢆⠀⢠⡿⠿⣿⣿⣗⠄⣀⣌⣷⠤⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠂⠬⢧⣸⣯⣹⣷⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⢀⣀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⢤⣤⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⠟⢧⣤⡄⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠙⠛⠻⠿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠂⠀⠀⠀⠀⠀⠀⠠⣤⣶⠟⠙⠿⠏
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⣧⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀
⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠙⢦⣽⣷⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⣃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠹⠟⠃⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⢤⣤⣽⣿⣾⣿⡿⠃⠀⠀⠀⠀⠀⢠⣤⣤⣾⡿⠻⠏⠀⠀⠀⠀⢾⣇⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⣤⡶⠖⠀⠀⠀⠀⠀⠀⠈⠉⠻⠷⠒⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣭⣙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣀⣀⣤⡴⠀⠀⠀⠀⢤⣾⣿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡋⠉⠀⠀⠀⠀⠀⢨⣿⡀⠀⠀⠀⠀⠀⠠⣄⣤⣶⣶⠦⠀⠀⢼⠋⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠻⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⣴⡿⠛⠻⡦
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣋⣀⣀⣀⣀
⣿⣿⡏⠉⠙⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⡀⠘⢮⡳⣌⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣋⡴⣫⠞⠀⡀
⠀⠀⠀⠈⠛⢒⣚⣃⣄⡀⠒⠠⠀⠑⠲⠤⣤⡀⠀⢠⠀⠀⣠⡤⠴⠒⠁⠀⠐⠒⣀⣄⣛⣒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠘⠻⢻⢿⣶⣆⣄⣀⣀⣀⡝⠀⠀⠀⠘⣅⣀⣀⣠⣄⣴⣾⢿⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⠙⠉⠉⠁⢀⠀⡀⡀⢀⠀⠉⠉⠙⠙⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣦⠁⢡⣶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠟⠗⠟⠏⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⢿⡿⠷⢦⡄⠀⣨⣿⡿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⡿⠀⠀⠘⠀⠸⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣶⣿⡟⠂⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢦⣄⢠⣤⣤⣄⣿⢶⣴⠟⣽⣿⣿⣿⣿⣶⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣽⣦⠾⠿⠿⠿⡿⣿⠷⣿⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠤⠄⠀⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠃⠁⠀⣀⣤⡤⠒⠚⠐⠂⠀⣀⣤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣭⣽⣶⣶⠶⠶⠶⠤⠶⠶⠶⠶⠾⠖⠛⠋⠁⠀⠰⢀⣀⡀⠀⠑⠘⠂
⠀⠀⠀⠀⠀⠀⠤⠤⠴⠖⠒⠚⠛⠉⠭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⡽⠟⠉⡧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠒⠒⠚⠒⠒⠛⠛⠛⠉⠤⠰⠖⠉⠉⠉⠉⠁⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣠⠤⠶⠶⠛⣛⣩⣤⣤⣶⠶⠶⡶⢶⠿⠶⠖⢾⡶⣦⣤⣤⣤⣠⣤⣠⡤⠴⢶⡾⣏⣁⣲⡄⠀⣀⠠⠄⠂⠁⠠
⠀⠠⠀⡤⠠⠤⠤⠤⠴⠖⠒⠒⠒⠂⠉⠉⠁⠁⠀⠀⠀⣀⣀⡬⠤⠄⣒⡤⠖⠒⠉⠀⠀⣀⠠⠒⠋⠁⠀⠀⠀⠈⠉⠉⠀⣶⡀⣀⡀⣤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠒⠚⢋⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡠⢀⢠⣠⣀⠸⣿⠏⠿⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⢀⣠⡤⠖⠐⠉⠁⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠤⠊⡴⠾⢷⡛⠒⠾⡧⠀⣲⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠙⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠈⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣀⣴⣷⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠞⠉⠻⣿⣿⠛⢄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠂⢠⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠃
⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⢹⣿⣯⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶
⠀⠀⠀⠀⠀⠀⢿⣿⣧⣤⣴⣦⠀⣤⠄⠐⠊⠀⠀⢠⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣏⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣀⣠⣀⣀⣯⣿⣿⣿⣿⡟⠀⠈⣷⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢉⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣼⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣿⡛⠻⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣶⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⡛⢁⣿⡇⠀⠀⣽⣿⣷⣄⣀⣀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣴⣤⣶⣶⣾⣾⣿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⡇⠀⠖⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣶⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⣠⣴⣾⡏⣾⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⡟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⣻⣿⣿⣿⣤⣤⣠⣤⣤⣄⣠⣤⣤⡀
⠀⠀⠀⢀⣀⣿⣿⣿⡃⠉⢻⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢠⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⣴⣿⠏⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⣠⠟⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⣀⣶⠁⠈⠑⠀⢹⣷⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⡇⠀⣼⠉⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠋⠁⠀⠀⠀⠀⠀⣿⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢠⡼⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⡀⠀⠀⠀⠀⠀⠀⢿⣷⣆⢸⣿⣿⣿⣿⣿⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣾⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⣿⣦⠀⠀⠀⠀⠠⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣰⣿⣿⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⡑⣤⢀⣷⡄⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠘⣿⣧⠀⠙⢿⣿⣷⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠁
⠀⠘⣿⣧⡀⠀⠻⣿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠘⢿⣿⡄⠀⠈⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡇⠑⠀⠀⢹⣿⣿⠻⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢻⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠙⣿⣇⠈⢿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⠠⠀⠀⠀⠀⠈⣿⣿⡄⠀⠙⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣷⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠘⣿⣿⣄⠀⢻⡇⡈⣋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⢀⣼⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣷⡀⠐⢦⣄⢿⣿⣿⣿⣿⣷⣧⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣦⡀
⣿⣤⡀⠹⣿⣿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣄⠹⣿⣿⣿⣿⣿⡄⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠰⢀
⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⠘⢻⣿⣿⣿⣟⡀⠹⣿⣿⣿⣿⣿⣆⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢾⣷⣄
⣿⢿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣷⣤⣸⣿⣿⣿⠏⡛⢻⣿⣿⣿⣿⣿⣦⠀⠉⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠀⠀⠀⠀⠀⢙⠻⣻⣦
⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⠈⢻⣿⣮⡻⣿⣿⣿⣥⢿⣿⣿⣿⣿⣿⣿⣧⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡀⠀⠀⠀⠀⠻⢿⣦⡍⣠⡀
⣿⣿⣿⣿⡛⢿⣷⡻⣿⢻⣧⡄⠹⠿⣿⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢀⠀⠈⠉⠛⣅
⣿⡿⢿⣿⣄⡀⠙⢷⡙⢿⣋⡿⣦⡴⢆⣹⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⣠⣿⡷⠶⠆⢸⣿⡆
⣿⣧⣮⣿⣿⣿⣷⣄⠘⣄⠻⣄⠦⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⠁⠀⠐⣿⠟

⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⣀
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣌⡛⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠐⠊⠁
⠀⠀⠀⠀⠠⢄⣦⣀⣀⠀⠀⠀⠀⣰⡉⢉⣉⡻⠋⠀⠀⠀⠀⠀⢀⣤⣶⣶⡏⠀⠀⠀⠀⠠⢤⣤⡀
⠀⠀⠀⠈⠐⠋⠁⠁⠀⠈⢉⣤⣾⠿⠻⡯⡭⠀⠀⠀⠀⢠⣶⣶⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠄
⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡠⠒⠉⠀⠀⠈⠡⠄⠄⠀⠀⠀⠀⠀⠁⠀⠠⢶⣄
⠀⠀⠶⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁⠀⠠⡀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢈⠋⠀⠁⠀⠀⠀⣀⣤⣦⣴⡶⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣤⡀⠀⠴⠒⠂⠟⠛⠛⠛⠛⠛⠀⠀⠀⠤⠀⠀⠉⠉⠉⠡⢤⠐⠠⢦⠄
⠀⠀⠀⠀⠀⢢⣀⠉⠋⠀⠈⠉⠐⠀⠀⠀⠀⠀⢰⠆⢀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠐⢲⠎
⠀⢀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠀⠀⡀⡀⠀⠀⢀
⠀⠘⠀⠁⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠳⠖⢤⣶⣄
⠀⠀⠀⠀⠀⠈⠛⠻⡿⠷⠤⡀⠀⢀⡀⠀⠤⠩⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⣿⣿⣦
⠀⠀⠀⠀⠒⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠈⠙⢷⡄
⠀⠀⠀⠐⠓⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⣼⠿⠿⠿⠛⠫⠿⠿⢾⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⣉⣥⣴⣴⣶⡶⠶⠿⢷⣾⡿⢿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠈⢷⣿⣿⣿⣿⡿⠃⠀⣤⣤⠤⠀⠻⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⢈⠀⠀⢁⣀⣛⣹⡿⣷⣿⣿⠿⠛⠀⠐⠯⣟⣿⣿⣎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠿⠟⠛⠿⠯⣹⣿⣿⣿⣿⣿⣯⡙⠒⠄⢀⠨⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠈⠙⡿⢻⡿⡿⡿⠻⠖⠄⠐⣌⣙⡛⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡁⢀⣄⡴⣀⣠⣤⢤⠀⠈⠙⢿⣝⢮⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⣲⢼⡼⣿⣰⣏⣿⡿⣽⢟⣭⡷⣢⣿⣼⣿⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢏⣯⣷⣿⣱⣯⣾⡋⢼⣿⢻⣾⢫⣿⣽⣿⣭⣿⠃⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠋⣽⣿⡞⣽⣧⡾⣩⡴⢟⣾⣿⣿⣿⠏⠈⢡⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⡼⣯⣹⡆⣴⡏⢰⣿⡾⣼⣿⣿⠙⠟⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣼⣿⣿⣴⠟⠑⣿⡿⢱⣿⣿⠃⠀⠀⢠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣠⠟⣡⣿⡿⠃⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⣴⡿⠋⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⢻⡿⠉⠉⠉⠉⣹⠟⠛⢻⣿⣿⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡿⣿⡅⠈⣧⢴⣿⠟⣤⣿⣿⣦⣾⣤⣨⡿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣍⣹⡷⣾⣀⣽⣿⣾⣿⣤⣿⣿⣿⣿⣿⣽⣿⡄⠀⢹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣏⣙⣟⣿⣟⣿⣿⣹⡿⣿⣏⡿⡿⠋⠙⠉⠀⢹⣶⡟⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣻⣏⣩⣿⣿⣿⣷⣿⠋⠠⠈⠙⠃⠀⠀⠀⢀⣾⣷⣜⣷⢀⣨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⣾⣏⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠐⢠⠠⠀⠀⠀⢀⣜⣻⡟⠀⣙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢅⠀⢴⠆⢺⡟⠛⣿⣿⣿⣽⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⢰⡜⢀⣤⣶⣾⣇⣾⣋⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢠⣶⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣼⠇⠀⠀⠀⠰⢀⣿⣿⡿⢿⡿⢿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠾⠟⠁⠀⠰⠀⠀⠜⠉⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣠⠀⠀⠀⠉⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢾⠃⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠉⢹⡿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⣿⢦⠉⠹⠋⢩⣿⠻⣿⣿⣿⣡⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠇⢻⠟⠿⠁⠌⠀⠂⠀⠘⠁⢡⠟⢻⣿⣧⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣄⠐⠄⠀⠀⠀⠀⠀⠀⢀⡃⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣦⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⢹⣿⠟⠉⠀⠀⠀⢀⣤⣼⣿⠃⢋⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⡀⠀⠀⡀⠀⠀⠁⠀⠀⠈⣷⠀⠀⠀⠀⢀⣠⠊⠙⠿⢿⡿⡿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣯⣀⡀⢂⡀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⡀⢀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠁⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⢿⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠻⣿⣾⣿⣿⣿⣿⣿⣿⡋⠁⠀⠀⠀⠀⠀⠀⠔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠤⠀⠁⠉⣿⡿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠙⠁⠀⠀⣀⣴⣿⠷⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⢀⡿⠟⠀⠀⠀⢺⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣿⠂
⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⣀⣀⣀⢀⣠⣴⣶⡶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠈⠙⠿⣿⣿⣿⣿⢷⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠘⠙⠋⠁⣸⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠒⠉⠈⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⣀⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⣀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠠⣤⣶⣶⣦⣀⠀⠉⠉⠛⠛⠻⠿⠿⢿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠚⠄⠀⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣙⠛⠿⣷⡅
⠀⠀⠀⠀⠀⠀⠑⣠⣄⠀⠙⠛⠲⡤⠄⠒⠠⢀⣠⡴⠋⠙⠛⠛⢿⣿⣯⠉⠉⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠐⣀⣠⡶⠛⠁⡄⠀⠀⠀⠀⠀⣽⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣾⣿⡿⠿⣦⣄⡀⠀⠀⠀⠀⠀⣰⡏⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢿⡿⠀⠀⠘⣌⠁⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⠀⠙⠓⠄⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⣴⣶⣿⣿⠿⣾⡉⠁⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⠋⠀⠤⠺⣿⣦⡀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⡀⠀⠐⠂⢀⡼⠀⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠹⣿⠇⠄⣄⠠⣬⠻⣿⣦⡀⠰⣾⣷⣶⢆⡀⠀⣟⠃⠀⠀⠀⠉⠀⡄⢄⣀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⣦⣄⠀⢀⣀⠙⠀⠀⠉⠀⢆⡈⢿⣿⣷⡀⠡⣽⠟⠟⠫⢚⡿⠦⠤⠀⠀⠂⠐⠛⠋⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠻⣿⣿⣀⣼⣟⣯⣿⣿⣷⣄⠻⡁⢎⠻⣿⣧⠀⠀⠀⠁⠈⠁⠀⠠⢄⠸⠇⠐⠶⠦⢤⡤⠄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢉⣾⣿⣿⣿⣿⣿⣿⡿⠆⠇⠀⡀⠙⣿⣷⡀⠀⠀⢠⣤⣤⠤⣂⣴⣾⡟⠛⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠈⠄⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⢿⣧⠀⠘⠋⠁⠐⠾⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀⠘⠅⢀⡠⢖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⢄⡀⠀⠀⠀⠀⢺⡀⠄⢣⠘⡁⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢤⡌⢢⡀⠀⠀⢰⠯⠔⠧⠋⠁⠫⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠀⡰⠽⠟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣠⠔⢛⢶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⢸⡘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⢀⠔⠁⠀⠀⠀⠣
⠀⠀⠀⠀⠀⢀⡠⠔⠋⠁⠀⡨⢳⠦⢼⡬⢼⡅⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⠖⢁⠂⠀⠀⠀⠀⠀⡐⡀
⠀⢀⣤⡒⠊⠁⡀⢀⢔⣵⠎⡴⠃⠀⠈⣗⣊⠹⣆⡀⢠⡀⠀⠀⠀⣀⡰⡀⠑⠘⠦⠀⢀⠀⠀⠀⠰⡘⢦⡀
⣠⣞⣁⠤⢒⡡⢚⣵⠟⡡⠊⠀⠀⠀⠀⡯⠔⢂⡹⣦⠄⠉⠒⠖⢒⡫⢖⢷⠀⠀⠀⠀⠀⠀⠀⠑⠠⠱⢄⣫⡢⢄
⠉⠙⠖⠾⢽⣞⣛⡥⠊⠀⠀⠀⠀⠀⠀⡷⠊⡡⠒⢉⢳⣞⡅⠀⠀⠀⠪⠔⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠓⠊⠉
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢀⠜⠁⠀⠉⠷⣇⡼⡀⠀⠀⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⠁⡠⠊⡐⠁⠀⠨⠓⢅⣰⢠⢰⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡞⢁⠜⢀⠎⢀⠁⢠⠀⠉⠉⠉⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠎⢀⠎⠀⠀⠀⡎⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢎⠀⠰⠀⢠⠁⢀⠀⠈⠀⢠⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠇⠀⠸⠀⢈⠀⠀⡇⠀⡧⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣧⠀⢸⠀⠀⣇⠀⠸⠀⢄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⣄⢸⢁⠀⣸⢦⡀⢧⠀⠑⠚