Hein Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⡀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠛⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠀⣈⣿⣿⣿⣿⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠶⢶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠘⣿⣿⣿⣿⣿⣿⣥⣬⣽⣷⣬⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢻⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣻⡎⣿⡄⠀⠀⢿⣿⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠟⠋⠉⠀⣿⣿⣿⣿⣇⠘⡇⢿⣿⡄⠀⠈⢿⡿⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣤⣀⠀⠀⣿⣿⣿⣿⣿⡄⢻⢸⣿⡜⠂⠀⠀⡱⣜⠆⢻⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣍⠙⢶⠿⠿⠿⠿⣿⣿⠊⣿⣿⣿⣦⡀⢀⣷⣿⡞⡈⣿⣇⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿⠂⠀⠀⠀⣴⣿⡿⠀⠸⢿⣿⣿⡿⠞⠛⠋⠙⠉⠈⢻⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢀⣤⣶⣷⣾⣿⣿⣧⡤⣀⣸⣿⣿⣿⣷⣾⣿⣄⣦⣀⠈⢻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡟⠟⣿⣿⣿⣇⢘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⡿⠛⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡹⣻⣿⣯⠄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⣽⣿⣿⣿⣿⠎⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⡆⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⢻⣿⣿⡄⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣤⡼⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠈⠉⠿⣿⣶⣌⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠃⠀⠀⠀⢸⣿⡏⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡡⠋⠀⠀⠙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿

⠀⠀⠀⣾⣷⡓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣿⣶⡄
⠀⠀⢀⠳⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇
⠀⠀⠘⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠺⠂⠀⠀⠁⠀⠀⢀⣤⣰⣶⠀⣶⣦⡄⢀⣀⠀⢠⣶⠀⠀⢀⠀⢠⣤⣤⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣄⣠⣀⡴⠁⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢹⣿⡿⣧⣾⣿⢟⣰⣿⣿⣿⡟⠉⢿⣿⣿⣧⣜⢿⣿⣦⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠒⠶⠒⠂⠀⠀⠀⠀⢸⠃⠙⠋⠀⠀⠀⠐⠚⠢⣄⠀⠀⠀⠀⠀⠀⠙⣻⣿⡿⣡⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⡄⣿⣿⡿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣧⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⢿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡿⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠛⢿⣏⠻⣽⡿⠋⠁⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣼⠛⣠⣿⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣾⣭⠻⢗⣰⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠛⠉⠠⣿⠉⢳⣦⡀⠀⢀⠀⠀⣀⣴⣎⠉⠙⣷⠘⠛⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣻⣿⡄⠀⠀⠀⠷⢿⣿⣿⡦⢹⣷⣾⢛⣿⡟⠓⠊⠁⠀⠀⣸⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡽⣿⡄⠀⠀⠀⠻⠟⠙⡇⢿⠏⣟⠈⠛⠿⠀⠀⠀⠀⣾⠿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠖⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⣠⣤⠀⠀⠀⠀⢄⣿⡆⠉⠄⠀⠀⠀⢴⣄⠐⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣩⡆⠰⠀⠘⡆⣿⡇⠀⣄⠀⠔⠀⣿⣼⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣫⢶⡀⠀⠀⠁⠛⠁⢀⠃⠀⣠⣾⠃⢿⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⣎⡙⢳⠀⠀⠀⠀⠈⠈⡏⣉⣈⣛⠋
⠀⠀⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⡹⠛⠟⢿⠀⠀⠀⠀⠀⠰⡿⠛⠙⣇
⠀⠰⣿⡿⣵⣾⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢾⣶⣿⣿⠗⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⣿⣿⣼⣷⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⣴⣿⣹⣯⠃
⠀⠀⢸⣿⠛⠛⠉⢹⣁⣠⡶⠆⠀⢀⠀⠀⢠⣤⡈⣉⡁⠀⠛⠛⣷
⠀⠀⠸⣿⠀⠰⣟⠉⠉⠁⣢⠀⠰⠋⠶⠀⠀⢠⡟⠛⢻⣷⠄⣰⡿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠈⣁⣤⡶⣶⡦⠀⠙⠃⠀⠀⠀⠀⠀⠀⣡⢠⣼⣷⡄⠉⠀⠀⠀⠀⠀⠀⠀⣇⠰⠿⠀⠀⠒⠀⠠⣶⣾⣚⠅⠀⠒⠠⠀⠻⠀⡼
⠀⠀⠀⠀⣿⢣⡀⠉⠀⠛⠀⠀⠘⣤⣤⠀⠀⠀⡄⠈⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⢼⠿⠉⠓⠤⠀⠀⠀⢠⠞
⠀⠀⠀⠈⣧⠈⡅⠐⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⣤⠄⠀⠃⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⡜⠻⠀⠀⠀⠀⢰⡎⠁
⠀⠀⠀⠀⠸⣦⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⢸⠀⠀⢇⠀⠀⣠⠜⠃
⠀⠀⠀⠀⠀⠈⠹⠦⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠔⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤
⠀⠀⠠⠚⠀⠹⠆⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠛⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⡿⣟⣽⣶⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠚⠿⠆⣿⡄
⠀⠀⠐⠋⠀⠀⠰⠟⠋⣱⡦⠀⠀⠀⠀⠀⣤⠙⠻⠗⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠉⣁⠰⠿⢓⣤⠀⠀⠀⠀⢀⣄⠛⠿⠆⠀⠀⣿
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠡⡦⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠰⢂⡀⠛⠁⠬⢀⡆⣄⠀⠈⠓⣠⡀⠓⢀⡿
⠀⠀⠀⠀⠀⠠⠄⠤⣄⠀⠀⠸⠀⠀⡃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢼⣷⣶⣾⣶⡏⢼⢷⣴⣦⣴⣿⡷⢤⠈
⠀⠀⠀⠀⠀⢠⡐⣷⠈⠆⠀⠸⢷⡶⠻⡶⠛⢶⠀⡖⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠇⠘⢿⣿⣿⡶⢷⠠⡞⢻⣿⣿⠿⠃⢸⣷
⠀⠀⠀⠀⠐⡄⠙⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠈⠀⠉⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⣶⠀⠈⢿⡻⢡⠜⢦⠳⠌⢛⠟⠁⢀⣟⡾
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣷⡦⣠⡁⠉⠀⠀⠀⠀⠀⣀⣤⣿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⡉⢿⣿⣤⣤⣤⡾⠋⣺⣇⣠⠊⣹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢀⠴⠿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠃⠘⢻⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠻⢿⣷⣶⠛⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣿⡿⠛⠉⠀⠀⠀⠀⢀⢀⣠⣾⣷⣶⣿⣷⣿⣶⣶⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡗⠀⢰⣿⠃⠀⠀⠀⠀⡤⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣀⡴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣤⣿⣏⠀⠀⠀⢀⡤⠐⣸⣿⣿⠋⢨⣿⣿⣿⡿⠿⠟⠒⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉⠀⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⠴⠛⠉⠀⠉⣷⠀⢀⡾⢴⠞⣻⣿⣣⣼⡏⠉⠈⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⠀⢀⣀⢀⣀⣀⣠⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠈⣿⠿⠶⢦⠚⢸⣿⠟⠁⠀⠀⠀⢀⡄⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⢷⣾⠿⣲⡄⡤⣀⣤⣤⣿⣾⣿⣿⠿⣿⣟⣋⣉⡟⠉⠉⠉⠁⠀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⢠⣾⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣁⣤⣹⣿⣿⡟⣻⣇⡀⠀⠀⠈⠚⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⣿⡿⠃⠀⢀⠟⠻⣿⠿⠿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣩⣏⢷⡀
⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⣶⣤⣴⣿⠃⠀⠀⠆⠀⠀⣿⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣾⣿⣿⣿⠀⠀⠀⠀⣿⣿⣾⣧
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠁⠀⢀⠀⠀⠘⣷⣆⢸⣿⣿⠁⠀⠀⠀⠠⡀⠀⠙⡆⠀⠀⠀⠀⢰⣾⣿⣿⣿⡟⣼⣿⣹⣇⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⢀⣼⣿⣿⠯⠉⠣⠔⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⢿⡋⣼⢣⣿⣿⣿⣿⣶⣶⣿⣹⣿⣿⣿
⠀⠀⠀⣾⣿⠀⠀⠀⢰⠀⠀⢠⣌⢻⣶⣶⣿⡿⠇⠀⠀⠀⠀⢺⡇⠳⢠⣤⠖⠀⠂⠈⢁⣾⣿⡽⠛⣿⢼⡌⣿⡜⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠸⣯⢿⣦⣤⣈⣷⣤⣤⣿⡿⠛⣏⠉⢻⡄⠀⠀⠀⠀⠞⠧⠀⠀⠻⠤⠀⠀⠀⣾⢋⡿⠀⠀⡟⣨⣿⣼⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢿⣿⣿⡻⢿⣿⣿⣏⡠⠄⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⣾⢣⠟⠇⠀⢸⢸⣿⡏⢻⡅⠙⠟⢿⣿⣿⣿⣿⣷⣿⣿⡄
⠀⠀⠀⠀⠈⢿⣿⣿⣿⣦⡈⣉⡃⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⡉⡔⠀⠀⠀⢸⢺⣧⠇⠀⢹⡄⡀⠀⢩⣯⡙⢛⢻⣿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠉⣧⡏⠙⣦⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⠀⠛⠀⠀⠀⡜⢁⢻⣶⡀⠈⠀⠛⠐⠢⡙⠉⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⡟⣿⠏⢠⣿⠃⡀⢸⠁⠑⠌⡲⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢛⠀⠀⠀⠀⠀⢳⡏⠀⢿⣿⡀⠀⠀⠂⠀⠀⠀⠳⣮⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣧⡿⣦⠈⠁⡇⠀⠀⢃⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⢀⣸⠁⠀⣾⣾⡇⠀⠀⠀⠀⠀⠀⢷⡾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣿⡿⣿⢇⠀⣷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⢸⠀⢀⣿⠩⡳⡀⠀⣀⡐⠂⠙⢻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣷⣿⡼⢠⡈⢣⡀⠀⠀⠀⠀⠀⠈⠏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⢸⠿⡀⠀⠙⠻⣧⢉⣦⣹⣧⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣽⡿⣾⣧⠈⣿⠦⡀⠀⠀⠀⠀⠘⠀⠀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡜⠀⠀⢾⡆⠑⠄⠀⢠⡌⢛⢶⣹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⣿⣚⠀⣷⡀⠀⠉⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⠀⠀⢷⡀⠀⠀⡀⢰⡛⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢿⡀⠻⠙⡆⠀⡄⠀⢒⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⡀⠀⠈⣧⡄⠀⠀⠸⢻⣉⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢻⡀⠑⡄⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⡇⠀⠁⠀⠀⣿⠹⡎⢠⡍⠘⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢘⣷⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⡟⠀⠀⠀⢠⣿⠀⡇⢸⠶⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡞⢻⡀⠸⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠱⠀⠀⢰⡇⢻⠀⡾⣹⢷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢋⠀⢻⡈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠎⠀⠀⠀⡈⠁⢸⣀⡶⣿⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣆⠀⣠⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⣠⠀⠀⠀⢉⣤⠘⡛⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢾⣓⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠁⠂⠀⠀⡀⠀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣜⣦⡄⠴⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠃⠀⢠⣤⣁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣕⠲⢀⠀⠀⠀⠀⠛⠆⠀⣰⠆⠀⠸⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣉⣀⣤⡅⢃⣈⣀⣅⣨⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⢿⡿⠀⢀⠤⢤⣍⡀⢶⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⢀⡞⠀⠀⠀⠀⢰⣿⣿⣦⡟⢧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣋⡛⠽⢦⣞⠃⠀⠀⠀⠀⠀⣿⣿⣿⡿⣾⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⣿⠃⠺⢾⠛⠀⠀⠀⠀⠀⠀⢡⣿⠿⠂⠹⣧⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⢿⡄⠱
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢠⠀⢀⣡⡀⠀⠀⠀⢠⣄⢀⣼⣟⠃⠀⠀⢸⣇⢐⡂
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆⢿⣶⣾⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣤⢠⣄⣿⣿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠓⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⡀⢸⠿⠟⠿⠟⠄⠀⠀⠀⠴⠀⠺⢿⣿⠟⠉⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⣿⠿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣴⣀⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠲⣶⣤⣴⣾⣿⣿⡿⠛⠿⠿⣃⣹⣾⣷⣾⡶⣢⣄⣀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠋⠉⠁⠀⠀⠀⠉⠙⠎⣿⡠⣏⠯⠟⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠸⠂⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢉⢧⣈⣉⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣏⣿⣿⠀⠀⠀⢀⠀⠀⠀⠘⠷⣤⣤⣤⣤⣄⣤⣾⠁⢰⡟⠁
⠀⠀⠀⠀⠠⠀⠀⠀⣀⡴⣿⡿⠋⠸⢿⡄⠀⢀⠊⠀⠀⠀⠀⠀⠻⣿⡟⠛⠛⠋⠉⠀⠀⠁
⠀⠀⠀⢀⠆⠀⠀⢰⠟⢨⡟⠁⠀⠀⣼⡃⢀⠆⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡄⠄⠀⠐⠀⠝⠀⢀⠀⠀⣿⣇⠈
⠀⠀⡑⡸⠁⠀⠀⠀⠂⠀⠀⠘⢀⠨⡿
⠀⠀⢣⠇⠀⠀⠀⠀⠀⠀⣤⡄⠈⢸⠇
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡠⠐⠁⠀⠀⠀⣠⡞⣡⣿⣿⠿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡞⡭⠊⢠⠀⠀⡇⣠⣿⣿⣿⣿⡉⠛⠀⠀⣄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠀⢰⡋⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⣴⡌⢻⣤⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢻⡏⠀⠀⣿⠁⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣄⢸⣷⠘⣿⣿⣿⣧⡘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣴⡀⡀⡿⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣰⡀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⣿⢿⣿⣿⣿⠟⠻⠆⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠘⡿⠀⠀⠴⠶⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡋⠀⠀⠀⢠⣤⣤⡾⣿⠿⡿⢶⠀⠀⠀⠀⣠⣶⣾⡷⣀⣀⣈⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠻⠃⠉⠉⠀⡉⠸⠇⠀⠀⠀⠋⠃⠳⠛⠠⠿⠅⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣀⣀⡄⢠⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⡀⠀⠀⠀⠈⠁⠀⠈⠻⠛⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠛⢄⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⡿⠁⣸⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⣠⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡖⠋⣉⣉⣭⣿⣦⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⡶⠒⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠁⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⠀⠐⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡁⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢚⣴⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣁⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠟⠛⠋⠸⣷⣿⣿⣿⣿⣿⣿⣿⣟⣽⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⣿⣿⣿⢛⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⠀⠸⡶⠟⢻⣾⣿⣿⣿⣶⣿⣶⣶⣀⣤⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠆⠀⠾⡮⠡⢄⣛⠘⠿⢯⠍⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠦⣀⣐⣶⣶⣿⣤⣌⣙⠳⢷⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⣣⣶⡻⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠛⠟⠙⠻⢿⣿⠙⠿⣿⣏⠻⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡽⠁⠀⠀⠉⠛⠻⢿⣛⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣌⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⢰⣦⣄⠙⡿⣆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠇⠈⠙⠻⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢿⣿⠂⠉⠙⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡀⠀⠠⣶⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡞⠒⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣷⣢⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠔⠀⠄⡐⢶⣶⠠⣼⣦⣝⡷⣶⡄⠀⢸⣿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⢀⠀⠀⠀⠀⠃⢀⡀⠉⠛⢻⣿⡗⠀⢀⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢃⣴⠀⠀⠹⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢠⣾⠋⢀⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠠⠴⣿⣧⣡⠌⢩⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠓⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠛⢿⣅⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⣀⠤⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣦⠀⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣤⡈⣹⠴⡟⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⡄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣌⢳⠌⢋⣥⣠⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣠⣴⠿⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠩⣊⡼⢫⣿⡿⠀⠇⠀⠀⢀⣴⠞⠋⣠⣾⠟⢉⣤⡾⠟⠁⠀⠠⢀⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠐⡛⠉⣰⡟⠧⣴⡶⠒⣠⣴⠟⢁⣤⡾⠋⣀⣴⡿⠋⠀⢀⠤⠀⣠⡿⠟⢉⣠⣾⠗⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠂⠉⣠⡾⠛⢁⣴⡿⠋⣠⡾⠛⠉⠀⣀⠘⢀⣴⠾⠋⣠⡶⢟⣋⢥⠄⠠⠞⠃

⣿⣿⣿⣦⠀⠈
⣿⣿⢿⣍⣷⡄
⣿⣿⡷⣶⣿⠦⣍⣀⠲⠂
⣿⣷⢤⣺⣿⣂⠈⠙
⣿⣿⡀⢁
⣿⣯⠻⣆
⣿⣿⣷⣿⣤⢤⣶⣶⣶⠀⠀⣤
⣿⣿⣿⣿⣿⣿⣿⣶⣴⣆
⣿⣿⣿⠿⠛⠛⠻⠋⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠘⠙⠧⠾⠶⠆
⣿⣿⣿⣿⡿⠿⣿⠿⠿⢿⣿⣿⣿⠿⠛⠿⠿⠹⠏
⣿⣿⣿⣿⡿⢷⣷⣤⠀⠀⠉⠙⠛⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣿⣿⣿⣿⣿⣆⡉⠀⣮⣘⡶⣴⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣶⣶⣏⣻⣻⡏⠙⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⣿⣿⡿⠿⣿⣿⠿⣿⣿⠿⣿⣿⡆⣀⠤⠄⠀⠀⠐⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣶⣾⣽⡀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣿⣿⣿⣿⣿⣿⣷⣾⣷⣶⣶⣤⣤⣄⡠⢀⠀⠀⠀⠀⠀⠀⠀⣈⡓⣦⣄⠈⠁⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣧⣶⣸⣇⠈⠉⠉⠳⣍⣿⣁⠀⠀⠙⠛⠀⢀⣤⣾⠏⠄⣬⠛⣿⡆⢰⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣿⣿⣿⣿⣿⣍⢹⣟⣓⠀⠀⠑⠀⠉⠁⠀⠀⠀⣠⣤⡟⠻⢷⠖⠶⠈⠀⠋⠛⠙⡓⢀⠀⠀⠀⢉⡁
⣿⣿⣿⣿⣿⡿⣜⣯⣍⣷⣲⣶⣾⢄⣾⣿⣷⣈⠁⠀⠀⠀⠚⠓⠛⠀⣀⣀⡒⠀⣩⣜⣿⣤⣤⣿⢇⠀⠀⢀⡀⡀⢀⣀⣀⠀⢀⣀
⣿⣿⣿⣿⣿⣟⣟⣿⣿⣹⣿⣿⣿⡿⠄⠉⠛⠉⢁⡌⠁⠨⣉⢌⠉⠥⠁⣤⣨⠉⢹⣿⣿⣿⠻⣿⣿⣶⣷⡎⠉⠀⠀⡀⠀⠀⠈⠉
⣿⣿⣿⣿⣿⣿⣿⡟⣟⣿⣿⣿⢮⣷⣌⠀⣀⠀⠺⠧⠀⠀⠭⠩⠭⠅⠀⢿⡇⠀⢸⣿⣿⣿⠛⣿⣛⢻⣿⣷⠀⡀⢠⣧
⣿⣿⣿⣶⣿⣬⣉⠻⣾⣿⠹⣿⣦⡘⢿⡷⢿⣿⣶⣶⣴⣶⣿⣾⣿⣿⢶⠀⠀⣤⣽⣿⣿⣿⠳⣯⣿⡈⣻⣿⣟⢿⣿⢹⣀⣾⣶⡾
⣿⣿⣿⣿⣿⣸⣿⣿⣾⣯⣤⡘⠛⢣⡌⠛⣰⣾⡧⠾⠿⠿⣿⣿⣿⣿⡿⠀⠒⠒⢺⣿⣿⣿⣶⣷⣷⣎⣩⣿⣗⢺⣿⣼⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣟⣿⡙⠻⠿⣿⣷⣼⣿⡇⠀⠀⠘⣯⠌⡷⠄⠀⠀⢀⣼⣿⣿⣿⣿⣸⣿⣿⡿⢟⠿⠿⠋⠁⠁⠀⠉⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣃⣻⡏⠁⠛⢒⠀⠘⠹⢿⠿⠇⠀⠀⠀⣿⠚⣋⡶⠐⠐⠒⡇⢸⣿⣿⣿⠉⠩⡏⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⣀⠘⠀⠀⠈⠉⠛⠙⠃⠸⠿⠛⠋⠒⠀⠲⠚⠛⠻⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣿⣏⠛⣿⢶⠶⣶⠶⣤⢤⣦⣦⡶⠂⠀⠀⠀⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡶⢃
⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣾⣶⣭⡁⠈⠻⠴⠌⠛⠝⢦⡌⠉⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣁⠊⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢁⡀⣀⣠⣤⣄⢤⠄⠤⠤⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⡿⢠⣄⡈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣈⣿⣯⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣿⣼⣿⣿⣧⣀
⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⠿⣿⡿⠛⠛⠿⣿⣿⣿⠿⠁⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣤⣶⡶⠾⠛⢋⣠⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣯⣀⣤⣔⣼⣿⣇⣤⣤⣤⣀⣀⠠⢤⣤⣶⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠙⠛⠛⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣏⣉⣙⣿⣿⣻⣿⣿⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣴⡒⠠⠴⣦⣤⣀⣴⠶⣾⣶⣾⣽⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠉⠀⣴⡾⣿⣿⣿⣾⣿⣿⣿⡟⣶⣶⣽⣿⣾⣿⣿⣿⣦⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠿⠂⠀⣠⣾⠃⠈⠉⠛⣿⣿⣿⣿⡇⠘⢻⣿⣿⣯⡻⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠿⠉⠀⠀⠙⢿⣿⠿⣧⣤⠀⣼⣬⣿⣿⣼⡆⢰⣿⣿⡟⠻⣧⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠹⠉⢻⣿⣟⠻⠿⡏⠁⠀⠓⢿⣿⣿⣶⡀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⠀⠀⠀⠀⠀⢾⣯⣽⣿⡇⠀⣿⠠⣿⣿⡇⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣂⣰⣧⡀⠀⢀⣻⣿⣿⣿⣿⣤⣿⣴⡟⢹⣿⡂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⠘⣿⣿⣷⣿⣿⠿⡿⣿⣿⡏⣿⣿⡇⠀⣿⣷⣦⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣆⠀⠀⠀⠐⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣿⣄⡀⣸⣿⣱⣿⣿⡇⢀⠏⢸⣿⣿⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡆⠀⠀⠀⢰⣿⡿⠛⠳⠄⠀⠀⠀⠀⠀⢸⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢡⣹⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢴⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⡟⠋⠙⢿⣿⡏⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣤⠀⠘⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠛⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢩⠀⠀⢻⣿⣷⢿⣿⣿⣿⣿⣿⣿⠁⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⢰⠀⠀⠈⣿⣿⣿⡿⣿⣿⣿⣿⣷⡄⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠘⠁⠀⠀⠀⡀⠀⡀⢸⣿⣿⣷⡿⣿⣿⣿⣿⣟⣼⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣴⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⣯⠾⣿⡏⠹⣿⣿⣿⣿⣿⣿⣿⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡀⠐⠃⢠⣤⣼⡿⣃⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠹⣇⠂⠹⣿⣿⣿⣿⣿⠿⣰⡇⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⢼⣿⣄⠐⠶⣌⠛⠋⠉⠻⣿⣦⣘⣆⡀⠀⠀⠀⠀⠀⣄⢸⡀⠀⢹⣿⣿⣿⣿⡆⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣾⣿⣿⡿⢾⣧⢻⣿⣷⠆⢈⠙⠲⠶⠦⠘⠏⢸⣿⠃⠀⠀⠀⠀⠀⣿⣿⡇⢠⣼⣿⣿⣿⣿⣿⣿⡧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠡⠈⡿⢹⣿⠋⠳⠃⠀⠀⠀⠀⠰⣦⣤⣁⣀⢀⡀⠀⠀⠐⣧⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣧⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⣿⣿⠀⠀⠁⠘⣧⡂⢀⠀⠀⠀⠀⠀⠀⢠⠉⢋⡉⣯⡇⠀⠀⠀⠈⢸⣷⣸⣿⣿⣿⣿⣿⠻⣿⣯⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡇⣿⠀⠙⠁⠀⠀⡧⢰⡿⠏⣿⠀⠂⡀⠀⠀⠀⢀⡛⣿⡗⠿⣇⣤⡀⠀⢰⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢐⣷⡾⠄⢢⠈⠁⠀⠀⠀⠀⠀⣸⡿⡟⠋⣠⣾⢹⣇⢰⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡟⣿⣿⣿⣷⣠⣿⣶⣦⣾⣿⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⣹⡇⠁⣠⠏⠃⢘⢋⠸⣿⣻⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠘⠃⠀⢠⡈⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢉⡟⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⢘⣿⣿⣿⣿⣿⣿⢋⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠁⣸⣿⣻⠟⢿⣿⣿⣿⡏⣹⣿⣿⡟⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⣴⣿⣿⣿⣿⣿⠉⠀⠉⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢃⠄⠈⠙⠃⢀⠀⠻⣿⣿⣤⣻⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁⢿⣿⣿⠿⠟⠁⠒⠀⠴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⣇⠀⠀⠀⠀⠀⠐⢁⣿⣿⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠋⠀⠀⠀⠀⠀⠀⠘⣿⠟⠁⡿⠛⠋⠛⠣

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠈⢆⠀⠀⢻⡀⠀⠈⡇⠀⢠⡏⠀⠀⠀⡇⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⣄⠀⠀⠈⢣⡀⠈⣿⡆⠀⣧⠀⢘⣷⠀⠀⢸⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠉⠿⣶⡄⠈⣿⠦⠼⠧⠴⠿⠧⣼⡟⠀⢀⡾⠀⢰⡿⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⢦⣀⠀⠀⠈⠻⣶⣶⠼⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⢷⣴⠿⠃⠀⢰⠋⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢷⣄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂⣀⡴⠆⣠⡶⠋⠉
⠀⠀⠀⠀⠀⠉⠙⠒⠲⠶⣤⣴⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⠛⠛⠁⠀⡠⠖⠋
⠀⠀⠀⠀⣀⣀⠀⠀⣀⣠⡈⠟⠀⠀⠠⠴⠒⠒⠒⠲⢦⠀⠀⠀⠀⣠⡴⠿⣾⠟⢷⣶⡄⠀⢸⣤⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢙⠃⠀⠀⢀⣀⣤⣶⣶⣦⡄⠁⠀⠀⠀⣿⠃⠰⣤⣶⣼⢿⡇⠀⠀⣯⣤⣶⡾⠛⠉⠋⠁
⠀⠀⠀⢀⣀⠤⠤⠴⠶⠺⢿⠀⠀⠀⠀⠉⠀⠈⠁⠀⠓⠀⠀⠀⠈⣟⣸⡦⠈⠉⣁⠘⠟⠀⠀⠘⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠖⠂⠉⠉⠁
⠀⠀⠀⠀⠴⠒⠒⠛⠉⢀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡆⢶⣜⠉⠃⠀⠀⠀⠀⠀⠀⠀⠛⢶⣤⣤⡤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⠶⠉⠁⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠬⠰⠾⣦⣤⠀⡄⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⣰⡿⠟⣆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠾⠿⢅⠀⠀⠀⠀⠀⠠⣾⡉⠛⠒⠒⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⢀⡾⠗⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⣴⡅⠈⠻⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢠⡾⠁⢀⣴⠛⣶⣤⡤⢀⣀⣀⢤⣶⣶⣶⣶⠂⠀⣿⡈⠛⠲⣄⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢸⠃⠀⣹⠁⢠⡞⠙⣏⠀⠀⣿⠈⠻⣆⠀⠙⠳⣆⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠏⠀⠀⡟⠀⢸⣇⠀⢹⡂⠀⠙⢷⠀⠘⢦⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠃⠀⠀⡏⠀⠘⡇⠀⠀⠨⡀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠇⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣾⣿⣆⠀⢰⡀⠀⠀⠀⠀⠀⡆⣿⣦
⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠘⣇⠀⠀⠠⠀⠀⠁⣿⣿⣷⣤
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⠀⠙⢶⣅⠀⠚⢀⡇⣿⣿⣿⣿⣷⡀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠚⣿⣿⣿⠿⣿⡇⢰⠈⡀⠀⢰⠀⠀⢙⠛⢿⣿⣿⣿⣆⡈⠀⠁⢄
⠀⠀⠀⠀⠀⢻⣿⡿⠀⣼⣇⡖⠀⠀⠀⠀⠸⠀⠑⢊⠂⠴⠀⠨⠍⣻⢞⣓⣦⠗⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠸⣿⠃⣸⠿⣿⡄⠀⠀⠀⠀⠐⣃⢄⠐⣥⢁⡄⢀⣤⠿⢿⣿⠿⣿⣦⠀⠐
⠀⠀⠄⠀⠀⠀⡿⠀⠀⣠⣾⣷⣤⠀⠠⠀⠀⢬⡀⠆⢈⡀⣁⣿⣴⡗⠦⠙⣻⢿⣿⣤⣦⡀
⠀⠀⠀⠀⠀⠀⡇⠀⣸⣿⣿⣿⠁⠀⠀⠀⠀⢀⣨⣵⣾⣶⣤⣬⣿⡟⢚⡅⠒⢜⠁⠐⡌⣿⡅
⠀⠀⠀⠀⠀⠀⣇⠀⣿⣿⡿⠻⣦⣄⣠⣤⣴⣿⣿⣿⣿⣿⣿⡿⠿⢧⣄⣿⣷⣀⠐⣚⣡⣿⣁⠀⢀⣀⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠹⣆⠘⠻⡄⠀⠈⢿⣿⣿⣿⣿⠟⠙⠀⢀⣀⠤⠄⠒⠉⣩⣥⣿⣿⣿⣯⠩⠛⢷⣄⠘⢿⣿⣍⣴⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠸⡟⠢⠀⢸⡆⠂⣼⣿⣿⡿⠋⠀⢘⣫⣵⣶⢚⠛⢹⣾⣿⡄⠈⠛⣹⣽⣷⠎⡬⢿⣷⣄⠻⡛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠈⢷⣾⣿⠿⠃⠀⠀⢐⡿⠿⠁⠀⣸⡿⣤⢿⠟⠁⠀⠀⣁⣈⣿⠀⠈⠳⣝⢿⡦⣈⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣟⠁⢹⡏⠀⠀⠀⡀⠉⠉⣉⠀⠀⣀⠘⣮⣷⣄⠙⢦⡈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢹⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣖⡿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣤⣤⣠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤
⠀⠀⠀⠀⠀⠡⠀⠰⠒⠀⠿⣿⠀⡏⡀⢠⠀⠀⠀⠀⠠⠶⢠⣶⣦⠀⠀⠀⠀⠀⠀⢦⡀⠙⡖⡁⠀⠘⠆⠀⡔
⠀⠀⠀⠀⠀⠀⠐⠀⠀⢢⠀⠀⠀⠀⢀⣤⠀⠀⡟⠂⠀⢠⣿⠉⢙⣆⣀⠀⢙⡗⠀⢠⡁⠀⠀⠀⠀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣄⣀⠈⣿⡻⢠⠁⠠⠊⣿⣿⡃⢻⣿⠏⡄⠈⠀⣰⣑⠁⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢷⣄⠀⠀⠀⣤⣩⣷⣿⣥⡄⠀⠀⣠⣼⠟⢠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠙⢦⡀⠀⢭⡭⣿⣿⠤⠆⠀⣴⢧⣤⠿⠟⠛⠉⠀⠀⠀⠀⠀⢆⠀⠀⣄⠀⠀⠀⠀⠀⢠⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠒⣿⣟⠛⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠈⢄⠀⠀⠀⣠⠋⣠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⣿⣿⣿⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⡷⠀⠼⠃⠀⠁
⠀⠀⠀⠀⠀⠀⢠⡿⠒⠀⠀⠀⠀⠀⢀⣹⣿⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠶⢐⡄⠄⠴⡆⠶
⠀⠀⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡾⠁⡒⠉⠋⠐⢶⡀⠁
⠀⠀⠀⠘⡀⠂⠘⣧⠀⠀⢀⡀⣈⣀⣠⡾⠃⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⢚⣛⣛⣂⠀⣇
⠀⠀⠀⠀⢱⡀⠀⠘⣿⣿⣿⠿⠿⢿⢻⡇⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠉⠥⠷⠀⠀⠈
⠀⠀⠀⠀⠀⠳⡀⠀⢨⡾⣿⠧⢴⣾⣼⠃⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠙⢤⣄⣼⣧⣀⠀⢀⣿⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄⢲⣦⡌⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣼⠋⡌⢻⣁⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠻⢛⠻⠧⠔
⠀⠀⠀⠀⠀⠀⠀⡰⠚⣿⣿⣿⣿⣟⣿⡟⢣⠀⠀⠀⠀⠀⠐⠛⠛⠉⠁⠒⢢⡤⠄⣀⣤⣔⠐⠲⠶⣆⣶⣶⠆⠀⠀⢠⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡟⠙⡿⠃⢿⡇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢩⣍⣥⣿⣷⣿⡿⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⠀⡃⠀⢸⢷⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢠⢄⣚⣉⠛⠁⠀⡏⣭⠂⢉⡶⣬⣔⣤⣦⣤⠤⠤⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠻⢦⣴⣇⣤⢋⡄⠘⠀⠀⠀⠀⢀⣤⣠⡴⠾⠛⠛⠉⠀⠆⠈⠟⠵⠀⠀⠀⢰⡿⠀⠐⠀⠄⠫⢉⠉⠒⠾⠛⢿⡻⠶⢻⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢈⠀⠀⠀⠀⠀⠀⠁⠄⠉⠀⡄⢀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠘⣇⠀⠉⠀⠀⠀⠂⠄⠀⠂⠀⠀⠐⠐⠈⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⣀⣀⡀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⣀⣴⣿⣴⣿⣦⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢛⢾⣿⡯⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⢚⣃⣁⣈⡙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡏⢨⣭⣿⠽⡇⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢳⣾⣿⠁⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠤⢤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠻⠻⢿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠉⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣤⣶⣾⣤⣤⣀⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⠇⢀⣴⣶⣿⣿⣿⣿⣿⣿⣧⢠⣤⣠⣤⡈⣹⣿⣿⣿⣿⣿⣿⣿⣶⣄⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢰⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⢹⡀⢹⠁⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡄⣾⣿⣾⡄⠀⢺⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⡏
⠀⠀⠀⠀⢦⣴⣿⣇⠀⢸⣧⣽⣿⣿⣿⠟⣡⣤⣿⣾⣿⣿⣿⣿⣾⢿⣿⣝⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠙⠛⠋⠀⠈⠩⠙⠍⣉⣀⠬⢥⣶⣿⣿⣿⣿⣿⣿⣿⣿⡧⠍⠙⠛⠛⠛⠁⠀⠊⣹⣿⣿⠃
⠀⠀⠀⠀⠀⣀⠀⠐⠀⠀⠀⡠⠚⠁⠀⠀⢠⡏⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣧⡦⠀⠷⣦⠀⢴⡿⣿⠟
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠘⠃⠀⠀⠀⠀⠀⢿⡇⠻⣻⡿⢻⣿⣿⢃⣿⡿⠏⠉⠀⠀⠀⠈⣧⣻⣾⣿
⠀⠀⠀⠀⠀⠘⢿⡈⠿⡿⣿⣿⣷⣦⡀⠀⢼⠁⠂⠸⣦⢤⢭⡿⡟⣿⣧⠀⢀⣤⣶⣿⣿⣾⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⡇⠀⠀⠞⠀⠀⠀⠁⠘⠈⡄⠠⠸⣿⠀⣸⣿⣿⣿⢡⣟⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⣿⣷⣦⢰⠀⠀⠀⠀⠀⠠⠀⡄⢀⢰⣳⣦⣿⣿⣿⣿⠉⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡙⠿⢶⠗⣶⠷⢶⡷⢤⡧⣼⣾⢻⣿⣽⡟⣹⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣝⡷⣾⣤⣾⣀⣼⣀⣼⣄⣿⣾⣾⡿⠋⢠⡟⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠾⣽⣤⣍⣭⣽⣩⣇⣿⡿⠛⠁⢤⠏⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⡤⣤⠵⣬⣶⢃⣉⡀⠀⠉⢀⣴⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣿⣤⣠⣴⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⢺⡆⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢸⣿⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣽⡇⠀⠀⠀⠀⢸⣷⠀⠀⢸⡇⢸⣿⠀⠀⠀⢸⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⣿⡇⣸⡿⠀⠀⠀⢸⣿⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇⠀⠀⠀⠀⢸⣧⠀⠀⣿⣧⣿⠇⠀⠀⠀⠀⣿⡆⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡿⢿⣦⡀⠀⠀⣼⡟⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⠘⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠙⣿⡆⠀⢿⣿⠀⠸⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⠀⠈⣿⡆
⠀⠀⠀⣠⠤⣄⡀⠈⣿⢤⠀⠀⢀⣿⡟⠀⣸⡇⠁⢠⣿⢿⡇⢤⠐⠛⠻⣼⣿⠘⣧⠀⣨⡏⠀⠀⠀⣢⣄
⠀⠀⠀⣇⠀⠀⢙⣿⣿⠀⠁⣠⡾⠛⠁⠀⣿⣧⣠⢿⡇⠘⠻⣌⠷⣤⣀⡿⡿⣧⠘⠳⢬⣇⣘⣄⡀⠀⠉
⠀⠀⠀⠀⠉⢀⣼⣠⣿⠀⠀⡾⠀⠀⠀⣼⢻⡟⠃⠘⣷⢠⠀⠘⢦⢄⡀⣰⢣⡿⡆⠀⠀⡀⠀⠈⡙⣦
⠀⠀⠀⢀⣠⠞⠀⣸⡙⠀⢰⡇⠐⢀⠞⡱⢻⠇⠀⠀⣸⣿⢷⡠⢼⠀⠈⢉⠞⠻⣟⠘⠂⠃⠀⠀⢡⠈⡇
⢀⣤⡶⠛⢁⣀⢠⠇⣠⠶⡿⠁⡄⡌⢰⠀⠈⠀⣰⠎⠀⠈⢻⣧⡟⠀⠀⠊⡄⠀⡿⡱⢢⣞⣉⠛⣦⢀⡇
⠈⠁⠀⠰⠋⠉⣿⠰⢏⡴⢁⠀⠧⠆⡞⢀⠀⠄⢹⣤⣴⠤⣾⠯⡀⠁⡃⠀⠀⢠⡿⠁⠸⢇⣉⣻⣿⠿⠿
⠀⠀⠀⠸⠀⠊⠈⠀⣿⢁⠎⠀⣴⢀⡏⠈⠀⠀⠘⣆⠤⠞⠀⠀⠘⠦⣇⣀⡼⠟⢦⣤⡤⠤⠚⠋⣽⡄⢀
⠀⠀⠀⠀⠀⠀⠰⠀⡿⠊⠀⠀⠰⠈⢁⢷⠀⠀⡰⠉⠳⣄⡀⠀⠀⣠⠟⠿⢁⠀⠚⠀⠇⠖⠀⠀⣾
⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⠀⠀⠀⠀⠓⡶⠯⣤⡀⠒⡄⠀⠀⠀⢸⠀⠀⠀⡿
⠀⠀⠀⣨⠛⠀⡔⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⡆⠀⠒⠒⢶⣮⠁⠀⠀⠙⣤⣀⠀⠀⠀⠀⠁⠐⡾⠣
⠀⠀⠀⠙⠓⠁⠀⠀⣀⠄⠀⠀⠴⠛⠁⠀⢠⡇⠀⢀⡴⠹⣧⣀⣀⣤⠞⢉⡆⠁⢶⣤⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠃⠀⠈⢁⢀⠀⠉⠉⠀⢠⡞⡅⠀⠈⠁⠒⣩
⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⣀⣀⣀⣈⣩⡭⠤⠶⠒⠂⠹⣄⠀⠀⠀⣠⠞⠀⠀⠢⢄⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣀⡠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣿⣿⣿⣷⣶⣤⣤⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠉⠻⢻⠟⠿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡞⠀⢻⣾⣿⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣶⢍⠻⣿⣿⣿⣧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⣴⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠀⠀⢹⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⠁⠀⣤⣤⣴⣦⣤⡦⠀⠀⢀⣤⣄⣤⣤⡀⣾⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⠃⠀⠘⢫⣤⣶⠿⣿⠆⠀⠀⣼⣿⣟⣯⣻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢉⣿⣿⣿⣩⣿⡟⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠈⡀⠉⠿⠏⠛⢹⣿⣿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⣰⣿⣿⣿⡛⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⡷⢿⣧⠀⠀⠀⠀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢉⣽⣿⣿⣅⣀⡁⠀⠀⠀⠀⠀⢀⡤⠶⠿⠟⢿⣿⣿⣤⡀⢀⠀⣿⡿⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠻⠳⠶⠚⠻⠿⠿⢿⣿⣿⠈⢣⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⡀⠀⠀⠀⠀⠻⢶⣶⣶⠆⠙⠃⠀⣸⣻⣿⣿⢿⡂
⠀⠀⠀⠀⠀⢰⡇⠀⠈⣷⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢠⠂⣀⣀⣀⣤⣾⣿⣏⣀⡀⣀⣾⣿⣿⣿⣿⠎⢷
⠀⠀⠀⠀⠀⠀⠁⢬⣀⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠜
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠪⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣥
⠀⠀⠀⠀⠀⠀⢦⢀⣀⠒⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⣀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠟⠁⢻⣿⡿⠋⢿⣿⣿⣿⣿⣷⣬⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⣿⣄⡀⠈⢿⣿⠀⢸⣟⣻⣿⣿⡿⣿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠊⠙⠄⠛⠊⢿⠀⢸⣿⣿⣹⠋⠰⠃⠈⢿⣿⣿⢿⣿⣿⡿⢿⡿⣿⡧⠈⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣼⡟⠁⠀⠀⢠⡆⠀⠀⠻⣿⡧⣿⣿⠀⣸⠃⠙⠃⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠓⠀⠀⠀⠘⠇⢹⣿⣄⠁⣤⡽⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡄⢠⣀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣷⣿⣿⣿⣿⡿⣋⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣲⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣾⣷⣖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡿⣿⡿⢿⢿⡙⣿⡯⠻⣿⠻⣿⣿⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢾⠟⣜⠁⠀⠋⠳⠈⠳⡰⣌⠣⣿⣹⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡀⠀⠀⠠⣤⣤⣤⠀⠀⠀⢠⣤⣤⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠠⣤⣤⠄⠀⠀⠀⠂⢠⣤⣤⣬⠍⠛⣿⣿⣟⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠜⠧⠜⠛⠀⠀⠀⠀⠀⢼⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣙⣙⣉⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡀⠀⠀⠈⠈⠒⠄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⣸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣄⣠⣷⣀⣴⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠋⠹⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣮⡳⣞⡵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠖⢿⣤⣉⣉⣤⣿⠲⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⣀⣤⣀⠀⠀⣸⢧⡈⠓⠤⣏⠀⠀⣀⣤⣀⠉⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠐⡇⠀⠈⠉⠉⠹⣆⠉⠲⢤⡏⠉⠉⠁⠀⢸⡂⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡖⠒⠋⠀⠀⠀⠀⠀⠀⢯⣑⣂⡼⠀⠀⠀⠀⠀⠀⠙⠒⢲⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠉⠉⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠉⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡏⠙⢦⣄⠀⠀⠈⠧⠤⠴⠒⠒⢦⠀⠀⠀⠀⡴⠒⠒⠦⠤⠼⠁⠀⠀⣠⡴⠋⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⣇⠀⠀⠈⠙⠲⠦⣤⣀⣀⠀⠀⠈⠑⠶⠖⠋⠁⠀⠀⣀⣀⣤⠴⠖⠋⠁⠀⠀⣸⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⡟⠓⠚⠙⣆⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⣰⠛⠓⠚⢻⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢿⡀⠀⠀⠘⣆⢀⡴⠛⢦⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⡴⠛⢦⡀⣰⠃⠀⠀⢀⡿⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⢷⣄⠀⠀⠈⠉⠁⠀⠘⢧⣀⣠⡞⠁⠀⢳⣄⣀⡼⠃⠀⠈⠉⠁⠀⠀⣠⡾⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣆⠀⠀⠀⠀⠀⠉⠳⠦⣄⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣀⣠⠴⠚⠉⠀⠀⠀⠀⠀⣰⠟⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠤⠤⢤⣤⠤⠤⠶⠶⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠙⠳⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⢦⣀⣀⣤⣄⠀⠀⠀⠉⠉⠛⠒⠶⠦⢤⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⡤⠴⠶⠒⠛⠉⠉⠀⠀⠀⣠⣤⣀⣀⡴⣇
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠉⠉⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠉⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠘⣧⠀⠀⣰⠞⠛⣦⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⣴⠛⠳⣆⠀⠀⡼⠃⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠈⠓⠞⠃⠀⠀⠸⣆⠀⠀⠀⢠⡟⠀⠀⢻⡄⠀⠀⠀⣸⠇⠀⠀⠘⠳⠚⠁⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠋⠀⠀⠀⠀⠙⠶⠴⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠦⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡄⠀⠀⢀⣾⠻⣧⡀⠀⠀⣠⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⢈⣿⠀⠀⢸⣇⣀⣽⠇⠀⠠⣿⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⢀⣉⣛⣉⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢻⣧⠀⠀⢸⣿⠉⣿⠀⠀⠀⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣯⣀⣠⣼⣿⠀⣿⣤⣄⣀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠾⢿⣇⣸⣿⠉⠉⠹⣿⣤⣿⠉⠉⠉⣿⣄⣸⡟⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣌⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣉⣡⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣯⠉⠛⠛⢻⣿⠛⠻⠿⠟⢻⣿⡟⠛⠛⢉⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠙⠷⣦⣴⠟⠻⣷⣤⣀⣤⡾⠛⠷⣦⣴⠾⠃⠈⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠟⢻⣧⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣼⠟⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣀⡈⠉⠛⠛⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⠛⠉⣁⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡟⠛⠿⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠟⠛⢻⣯⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠾⠛⣷⡀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⣸⣿⣄⠀⠀⠀⢠⣾⠻⢷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠻⠷⠶⠾⠋⠈⠻⣶⣤⣤⣤⡾⠟⠈⠛⠷⠶⠾⠛⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠟⠛⠛⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀⠀⡀⠀⡀⠀⠀⣀⡀⠀⠀⠀⣀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⢸⣧⣼⠀⣼⣷⠀⣿⢹⡆⣿⢹⡎⢷⣴⠏⠀⠀⣿⣽⡇⣿⢸⡏⣷⠉⣿⠋⢸⣧⣼⡇⡟⢻⡆⣸⣷⡙⣧⡾⠁
⠀⠀⠸⠇⠹⠰⠟⠛⠇⠿⠋⠁⠿⠋⠁⠀⠿⠀⠀⠀⠿⠾⠇⠿⠸⠟⠿⠀⠿⠀⠸⠇⠹⠃⠷⠾⠳⠟⠛⠧⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠦⡄⠀⠀⠀⠀⠀⠀⠙⠲⡀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⢀⡟⠀⠀⠀⠀⠘⠄⣀⣴⠇⠀⠀⠀⠀⠠⡊⠉⢉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠄⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⢠⢥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⣧⠀⠀⢃⢾⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠄⢺⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⡟⠀⠀⢣⣾⠟⠿⠶⣶⡀
⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠠⣒⣦⣆⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠢⠀⠐⠂⠉⠀⠀⠀⠀⠀⢁⠤⠖⠛⠁⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⠍⠻⠿⡁⡀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠄⠀⠐⠈⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⢸⡧⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠊⠀⡅
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⡤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠐⠒⠂⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣼
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠤⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠰⡄⡐⠘⡄⠀⠸⣿
⠀⠀⠀⣇⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⢸
⠀⠀⠀⢹⣶⡇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⠊⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡤⠋
⠀⠀⠀⠈⡿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⣼
⠀⠀⠀⠀⡟⠔⠀⠀⠂⢦⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⣰⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠀⠐⠀⢀⣀⡀⠄⠂⠐⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢩⡟
⠀⠀⠀⠀⠸⢠⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠤⠔⠂⠁⠀⠀⠀⠀⠀⠁⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣡⡏⠛⠐
⠁⠀⠀⠀⠀⠀⠺⠗⣣⠘⢸⣶⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡴⠞⠉
⠀⠀⠀⠀⠰⠀⠀⠀⠈⠛⠢⢄⡁⠙⠛⠳⠶⠦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠈⠉⠒⠲⠤⣄⠀⠈⠈⠉⠉⠑⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠤⠶⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣀⣈⣄⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⠤⠤⠤⡀⢠⠔⠒⠒⢤⡀⡠⠤⠠⢤⡀⠀⣀⣀⣀
⠀⠀⠀⠀⣀⡤⢄⡀⣰⣋⣀⡤⠽⠖⠓⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠓⠾⠥⢄⣈⣱⡄⣀⠤⢄⡀
⠀⠀⠀⡞⠁⢀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢄⠀⠘⡆
⠀⠀⠀⣧⠒⠋⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠋⠓⢦⡇
⠀⠀⠀⡇⠀⠀⠀⠀⣷⠊⠉⠉⠑⣄⣠⠔⠒⠦⡀⠀⣠⠤⢤⡀⢀⡤⠖⠒⢤⣀⡔⠉⠉⠉⢲⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠑⢄⣀⣀⠴⣇⠀⠀⠀⠀⣸⠁⠀⠀⠀⢸⡏⠀⠀⠀⠈⡟⠀⠀⠀⠀⢹⡀⠀⠀⠀⢀⡷⢄⣀⣀⠔⠁
⠀⠀⠀⠀⠐⡅⠀⠀⠈⠢⠤⠤⠖⠙⢆⡀⠀⣀⡼⢧⠀⠀⠀⣠⠿⣄⠀⠀⣠⠞⠑⠤⠤⠴⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⢀⡴⠳⡄⠀⠀⠀⠉⠉⠁⠀⠀⠉⢒⠊⠁⠀⠀⠉⠉⠀⠀⠀⢀⡴⠳⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡗⠒⠊⠁⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⢀⡴⠋⠳⣄⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠙⠒⠒⡇
⠀⠀⠀⠀⠀⡟⠢⢄⣀⠀⠀⠀⠀⠉⠲⠤⠤⠤⠖⠉⠀⠀⠀⠈⠑⠦⠤⠤⠴⠊⠀⠀⠀⠀⢀⣀⠤⠚⡇
⠀⠀⠀⠀⣠⡏⠒⠤⣈⡉⠓⠒⠦⠤⠤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠖⠒⠋⣉⡠⠔⠊⣧⡀
⠀⠀⠀⢰⠁⡇⠀⠀⠀⠉⠉⠒⠒⠦⠤⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠖⠒⠊⠉⠁⠀⠀⠀⡇⢹
⠀⠀⠀⠈⠢⣈⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⢊⡡⠊
⠀⠀⠀⠀⠀⠈⠓⠦⣤⣈⣉⠙⠒⠒⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠚⢉⣉⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠷⣖⣒⣶⠤⠤⠤⠤⠤⠤⠤⠤⠤⣴⣖⣒⡶⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡭⠭⢿⠉⠉⠉⢹⠯⠭⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠈⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠈⠙⠒⠚⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠶⠶⠶⠿⠿⠟⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡵
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡄⢸⢚⡲⠄⡄⣄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠈⠀⠀⠙⡟⠂⢠⣀⢂⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣉⣤⣀⣀⣀⣀⣈⣀⣀⣭⡶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠉⠉⢉⡉⠉⠉⠉⠁⠀⢀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠁⠠⠤⠀⠀⠀⠤⠄⠐⣀⠀⠀⠂⠐⠀⠀⢄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠠⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡀⢀⢀⠀⠀⠐⠀⠀⠀⠀⠆⠀⠈⠁⠀⣀⠀⠀⠄⠀⠐⢄⢖⡠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣧⣴⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⣁⣀⣀⣤⣤⠶⠞⠛⡃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣤⣤⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⡀⠄⢂⠀⠡⠂⠄⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠢⠍⠀⠀⠈⢵⡄⠈⠐⠢⢦⣶⣄⣀⣀⢀⡀⠀⠀⣀⢀⣀⠀⣤⠴⣖⡞⢀⠀⠀⠀⠀⠘⣡⠀⠀⠑
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠑⠂⠀⠱⠀⣀⣋⠀⢠⢙⠃⢠⠀⢼⠉⠷⢶⠀⡀⠠⢐⠃⠀⠊⠙⠂⢠⠀⠐⠤⠌⠀⢀⠀⣀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⡷⠧⠉⠒⠄⠁⣄⡀⠂⠈⠀⠠⠀⠀⠂⠀⠙⠀⠁⠀⠈⠀⠀⠀⠁⠀⡀⠀⠀⠂⡙⠃⠢⠐⢉⡂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣇⡀⠀⠀⢀⠐⠒⠄⠈⠁⢀⠐⡀⢠⠀⠀⠀⠐⠀⠐⠀⠀⠀⢤⠦⠀⡄⠀⠀⠀⢐⣤⣶⣶⣿⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠉⠛⠻⣶⣦⣤⣤⣄⡂⠀⠀⠀⠀⡉⠀⠑⠂⠀⠀⠀⢈⠀⠀⠀⣀⣠⣤⣦⣴⣶⣿⡿⠛⠋⠁⡐
⡇⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠑⠀⢬⡀⠀⠍⠉⠛⠻⠿⢿⠿⠿⢶⣶⣶⣶⣶⣶⣿⣿⡿⠿⡿⠟⠛⠛⠉⠉⢁⠀⢈⠀⠄⠊⠀⠐⢄
⡇⠀⠀⠀⠀⠀⠀⠈⢻⡀⣀⠀⠀⠀⠀⠉⠒⠀⠀⢤⣀⡀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣈⢀⡀⠀⠀⠀⠉⠀⠀⠀⠀⢠⠤⠾
⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⣤⡤⣆⠀⢀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⠒⠃⠀⠀⠒⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⢠⡖⠟
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢧⡤⢤⣀⣠⡀⠀⣀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀⠀⢀⣀⢀⡦⣄⡞⠲⠾⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠒⠈⠉⠉⢛⠋⠉⠷⢞⢹⡶⠎⠳⢶⠉⢲⠋⡙⠛⢋⠛⠁⠉⠠⠤⠤⠤⠤⠀⣀
⡇⠀⣀⠠⠀⠀⠒⠈⠉⠉⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁⠀⣀⠄⠂⠘⢀⠀⢀⡘⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⢠⢀
⡇⠁⠀⠀⡤⠀⠀⠀⠀⠤⠀⠀⠀⠤⡦⠀⠀⠀⠀⢠⠄⠰⡀⠀⠀⠩⠄⡆⠠⠣⠈⡃⠀⠐⠀⡠⠀⠀⣀⠐⡄⠀⠶⣀⠀⣀⢀⠀⢀⡀⢀⠀⠀⢀⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡂⠀⠀⠀⠀⠐⠂⠀⠂⠀⠀⠀⠀⠂⠀⠤⠬⢀⠤⠄⠀⠀⠐⠁⠀⠀⢀⠀⠀⠐
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠀⠐⣨⠂⠀⠘⣈⠂⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠶⠀⠘⠠⠒⠐⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑
⠏⠀⠀⠀⠀⠌⠁⠀⠀⠀⠀⠌⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠠⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⡔⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⣧⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢼⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣅⣀⣀⣀⣸⣄⣀⣀⣀⣀⣀⣀⣀⣇⣀⣀⣸⡀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣠⡾⠛⠉⠭⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠩⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠷⠶⢤⣤⣤⣀
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣀⣀⡀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⣧⡀
⠀⠀⠀⠀⠘⣧⠀⠀⠀⣠⡞⠉⠉⠻⣦⡀⢀⣴⠞⠛⠛⢶⡀⠀⠀⠐⠀⣠⡶⠛⠋⠙⠳⣦⡀⠀⠀⠠⠀⣠⣤⢤⣤⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢷⠶⠟⠉⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠙⠶⣤⠴⠞⠉⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠛⠉⠀⠀⠈⠻⢦⣤⡾⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣸⡀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⣤⠶⠛⠛⠛⠉⠉⠉⠉⠉⡉⡉⠛⠛⠛⠓⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣦⣤⣀
⠀⢸⡇⠠⠄⠀⠀⠀⠀⣀⣀⣀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠉⢷⡀
⠀⠘⢷⣀⠀⢀⣀⣴⠟⠉⠉⠉⠻⢦⣄⣀⣤⡴⠖⠛⠲⢶⣄⡀⠀⠀⣀⣠⣤⠤⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠖⢀⣀⣀⣀⡀⠀⠀⠀⢸⡇
⠀⠀⠀⢹⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠁⠀⠀⠀⠀⠈⠙⠷⣤⣤⣤⣤⠶⠞⠋⠉⠀⠈⠙⠳⠶⣶⠞⠃
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣴⠞⠉⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠓⢦
⣼⣿⠀⠀⠀⠙⠷⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠶⠛⠁⠀⢸
⣿⠙⠷⣤⣀⡀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠳⠶⠶⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠞⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⣠⣾
⠘⠷⣤⣀⠈⠙⠛⠶⠶⢦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠶⢛⣩⠞
⠀⠀⠀⠙⠳⠶⣤⣄⣀⡀⠀⠈⠉⠉⠛⠛⠛⠳⠶⠶⠶⠶⠶⠶⢤⣤⠤⠦⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠋⢉⣉⣡⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠶⠚⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠙⢄⣠⠴⠚⢉⡀⠙⠒⢦⣀⡠⠞⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⣄⡞⢁⣤⣶⣄⠀⠙⣦⣾⣿⣿⣷⣆⡜⠁⣠⣤⡀⠈⠳⣀⣠⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢁⡰⠃⠈⣵⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⠞⠉⢀⡴⠈⠳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⢋⡱⣄⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡘⠉⢀⠔⠋⢠⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠒⢋⡔⣈⢦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠏⡀⢠⠎⢸⠄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣠⠤⠖⡉⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠖⠙⠲⢤⣁⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠈⡞⠁⠤⠚⢡⠄⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡟⠁⠀⡇⢲⠀⠈⣷⠛⡇
⠀⠀⠀⠀⠀⠀⠀⡿⡀⣇⠀⣀⡤⠊⠀⢀⠜⠋⡰⠉⠻⢿⡿⠿⣿⡿⠛⢿⣿⢿⡿⣿⠟⠋⠰⢈⠙⢧⠀⠊⢀⡜⠀⠀⡼⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠘⢾⡦⣄⣀⢀⣰⠃⢀⡴⠃⡟⠀⠀⢻⠞⠁⢠⠏⠀⠙⢦⡾⠁⢀⡰⠃⢸⠀⠀⣇⣀⠀⣀⡠⠞⣡⠋⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠓⢬⣁⠹⣄⠀⢀⠔⠁⠀⢀⣾⠐⠚⠁⠠⠃⠀⠀⣧⡀⠈⠀⠰⠃⠀⣠⠏⠉⠉⢁⡠⠞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠠⣄⡀⠀⠀⠈⠑⠺⠷⣶⣶⠖⠒⠋⠘⠦⣀⣀⣀⣀⡤⠞⠀⠉⠒⠲⠶⠖⠚⢁⣠⠤⠒⠉⠀⢀⡠⠄⡇
⠀⠀⠀⠀⠀⠀⠀⡇⣄⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠒⠒⠒⠶⠶⠶⠶⠶⠖⠒⠒⠚⠋⠉⠁⢀⡴⠒⠒⢀⣞⢀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠈⠙⠒⠒⠒⣞⠉⠉⠓⠦⡀⢀⣀⣠⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠖⠒⠂⠀⣮⣀⣀⡠⠞⠀⠉⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠈⠙⠲⠤⠤⠖⠻⣄⡀⠀⠉⢓⣠⠖⠉⠉⠉⠃⣀⠜⠡⠤⠤⠴⠊⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠓⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠴⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡀⡀⠀⢀⢀⡀⣀⣵⣺⣣⢒⡒⡢⢀⣀⣯⣿⣭⣽⣽⣟⣫⣽⣿⣶⣶⣀⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡵⢚⣉⣦⣖⣍⣯⣸⣳⣦⣦⣶⣎⣴⣽⠉⢉⣯⡭⠀⠉⠀⠉⠉⠉⠁⢩⠀⢻⡿⡟⢁
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣧⣿⣿⣿⣥⣷⣿⣿⣿⣿⣿⣿⣿⣿⢳⠇⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⣷⡂⡡⠄⢀⡀
⠀⠀⠀⠀⢀⣄⣶⣶⣿⣿⣿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠀⠀⢯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠹⣷⡃⠘⠊
⠀⠀⠀⠀⠰⣻⢿⣿⣿⣿⠁⠀⠀⠀⠛⠛⣿⣿⣿⣿⣿⣿⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⢲⣞⣥
⠀⠀⠀⠀⠈⣻⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣸⡇⠀⢰⣾⣶⣶⣶⣶⣶⣶⣶⣿⣷⠀⣿⡆⠀⣺⣿⣏⣻⣒⢀
⠀⠀⠀⠀⠠⣿⣿⣿⣿⠁⠀⢀⣤⡀⠀⠀⠀⢹⣿⣿⣿⣿⢹⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠁⠀⣻⣿⣿⣿⡇
⠀⠀⠀⠀⢐⣺⣿⣿⡇⠀⠀⠀⠘⠀⢀⠀⢰⣴⡅⠈⣿⣿⢸⣿⣟⡄⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⠀⢀⣿⣿⣿⣿⠏
⠀⠀⠀⠀⡨⡿⣿⣿⠀⣤⠀⠀⠀⠀⠁⠀⠀⠀⣥⠀⢿⣿⣇⢻⣿⣷⡹⣻⣿⣿⣿⣿⣿⣿⣿⣟⣼⣏⡼⣰⣿⣿⣿⣿⣾⠂
⠀⠀⠀⢼⣭⣿⣿⡟⢀⠁⠀⠀⠀⠀⠀⠀⠠⠚⠯⣄⣨⣿⣿⣯⡿⣿⣿⣷⣾⣿⠛⢻⣿⣿⣿⣾⣿⣿⣽⣿⣿⣯⢛⢟⣴⠕
⠀⠀⠠⣴⣿⣿⣿⠁⠘⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢺⣿⣿⣿⣿⣿⣿⣟⠻⣷⣿⣓⣐⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣟⡩⠶⠄
⠀⠀⢴⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⡿⢿⣿⣿⣿⣭⣛⠻⣿⡿⠿⢛⣟⣽⣿⣿⣿⣿⣿⣿⣿⣻⢏⡑⠉
⠀⢠⣴⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢠⡀⠐⠘⠁⠈⢻⣿⣿⣿⣿⣿⣽⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡎⠐
⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢹⣦⠀⠠⠡⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⡛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⢯⣭
⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡘⠟⠃⠀⠌⠀⠀⠀⠀⠀⠀⢿⣻⡉⠻⣿⣿⣿⣏⠀⠳⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣅
⠀⢼⣿⣿⣿⡇⠆⠒⡄⠀⠀⠀⠀⠀⠁⠸⡗⣠⠤⠀⡀⠀⠀⠀⠀⡘⣠⡾⢻⠉⣿⣇⣽⣶⣶⣿⠭⢉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀
⠀⣾⣿⣿⣿⠏⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢋⡇⠀⠀⡁⠀⠀⠀⣞⠀⣻⣧⣈⣀⣿⣿⠘⠻⡇⠀⠀⠀⠡⣴⡿⢻⡏⠙⠻⣿⣿⣿⣿⣿⣧⡀
⠀⣿⣿⣿⡟⠀⠀⠜⠁⠀⠀⠀⠀⠀⠲⠀⠈⠀⠀⠠⢶⠀⠀⢸⣧⣿⣷⣿⠿⣧⣾⣿⣶⣶⣿⠦⠤⣤⣶⣾⡇⠈⠃⣀⡀⣿⣿⣿⣿⣿⣿⣷⣀
⠀⣿⣿⣿⠏⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠈⣿⠖⠉⡀⠈⠀⠀⠀⠙⣿⢿⣿⣧⣻⣿⣿⡇⣿⢁⣁⢀⡽⠟⢻⣹⣷⡜⠻⡇⠈⢻⠿⠉⠙⢿⣿⣿⣶⣆
⠀⣽⣿⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣠⡟⠀⠀⠻⡆⠀⠀⠀⢀⣻⣿⣿⣿⣿⣟⣿⣜⣿⣶⣿⣿⠀⢡⣤⠛⠛⢇⠀⠐⢠⣿⠋⠉⠀⣼⠛⠛⠿⡿⣷
⠀⠹⠿⠿⠷⢦⢤⣤⣀⣀⡀⢀⠀⢀⣤⣽⡇⠀⠀⠀⠁⢀⣀⣀⣨⣿⣿⣿⡿⣿⠇⣿⡿⣿⣿⣿⣿⡄⡇⠘⠄⠀⢸⣖⠶⢶⠉⠇⠀⠀⣟⠒⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⡛⠟⠛⠋⠙⡛⠛⠉⣉⣁⣯⠞⠻⣾⣿⣍⣝⢯⣿⣿⣿⣿⣦⣤⣤⣾⣿⡿⡿⣷⢦⡴⢾⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⣒⣶⣒⣒⣒⣒⡒⢰⣶⣿⣿⣍⡀⠓⢋⣶⣷⣐⠞⠳⠞⠛⣛⣿⠾⠁⠙⠛⠻⠒⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠁⠀⠈⠉⠉⠛⠛⠋⠙⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⣀⢛⡦⢄⠀⣀⣀⣀⡳⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠑⠚⠛⢇⣀⣭⠙⠤⡶⠆⠸⢉⣵⡐⠲⠗⠉⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢣⠖⣂⠄⠀⢁⡀⠀⡀⢀⠀⡀⣤⠄⡤⠤⣢⡎⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⠤⢬⡕⣴⣆⡿⠁⢈⢰⠴⡇⣉⣣⣇⣰⠀⡷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠶⡒⣄⡀⣫⠈⢉⢁⢸⣆⣀⣏⣉⣉⣀⣤⣭⣷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⣁⠒⠤⠤⣌⣉⣈⣂⠉⡙⠉⠉⠈⠉⣉⣉⣠⣥⠴⠞⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣄⣉⡸⣂⠤⣠⡄⢤⣈⡩⣉⣉⣉⢉⣉⣠⢁⠄⢰⡆⠚⢠⣅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢶⡌⢒⠛⠛⠲⠨⢤⣼⣤⣙⣁⣼⣀⣹⣌⣧⣭⣭⣲⠟⢟⢛⣹
⠀⠀⠀⠀⠀⠀⣀⣠⣸⠧⣄⠉⠃⠙⠓⠖⠶⢲⡤⡬⣀⣩⣭⣥⣤⣤⣽⡴⠶⠶⢛⢋⣩⣀
⠀⠠⡀⠒⠐⢅⠀⠀⠙⠲⢄⣁⠘⠙⠼⠶⠤⢤⣤⣨⣄⣄⡤⣤⣤⣤⣔⣦⢴⠶⠟⣋⣹⠏⠉⢓⠅⣀⠂⠄
⠀⠀⠀⠁⠐⠀⠠⠄⠀⠀⠀⠀⠉⠀⠂⠠⠄⠤⠤⠤⠤⠤⠀⠠⠤⠤⠤⠶⠒⠒⠉⠁⠀⠀⠀⠁⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⡀⡀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡩⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠓⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠛⠉⢹⠁⠀⠀⠀⡏⠉⠛⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⠜⠁⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡠⢄⠀⠀⡠⠤⠀⢀⡀⢀⡀⠀⠀⢀⠀⣠⠲⡄⡠⡀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣠⠇⢸⠀⢰⠃⠀⠒⠋⡇⡏⠘⠀⠀⡏⠉⠁⠀⠧⠇⢸⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠸⢤⠼⠀⠀⠀⢠⠃⢹⠀⡄⠀⡇⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⢹⠀⠀⠀⠀⠀⠀⠀⠘⣤⠎⠀⠳⠴⠃⠀⠀⠀⠀⠀⠀⡏⠉⠓⠦⣀
⠀⠀⠀⠀⠀⢀⣞⠁⢰⠉⡳⠈⠳⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⢞⠉⡦⠈⡳⡀
⠀⠀⠀⠀⠀⠸⡌⠂⠀⠀⠀⠀⠀⠀⠀⠉⠩⡙⠓⠒⠒⠒⠒⠒⠒⠒⢋⠍⠉⠀⠀⠀⠀⠀⠈⡠⠚⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⡀⢢⠀⠀⠀⡀⠀⠘⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢠⠇⡜⠁⢸⠀⠀⠀⠈⠋⠀⢱⠀⠀⠀⠀⢰⠙⢦⡀⠀⠀⢀⠀⠀⠀⣼⠵⠤⣄⠀⣴⡸⡀
⠀⠀⠀⠀⠀⠈⢲⠃⠀⠸⡄⢀⠆⠀⠀⠀⣼⠀⠀⢀⡀⢸⠀⠀⡇⠀⠰⠛⣆⠀⢀⡇⠀⠀⠘⠚⢹⠃⢣
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠑⠊⠀⠀⠀⠀⢹⠀⠀⣼⠐⠁⠀⠀⢧⣠⠇⠀⣿⠀⠈⡇⠀⠀⠀⠀⠸⣆⡼
⠀⠀⠀⡠⠤⡴⢺⡁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠞⠁⠀⠀⠀⠀⢈⡗⢦⠤⣄
⢀⣠⠞⠧⠴⠃⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⡤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃⠘⠦⠼⠳⣄
⡏⢯⠀⠀⠀⠀⠀⠀⣙⣒⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⣒⣋⠀⠀⠀⠀⠀⠀⡽
⠘⢆⡓⢤⣀⠀⠀⠐⢅⡨⠂⠀⠈⠉⠉⠓⠒⢒⣶⠶⠶⠶⠶⠶⠶⣶⡒⠒⠚⠋⠉⠁⠀⠐⢅⡨⠂⠀⠀⣀⡤⢚⡱
⠀⠀⠙⠲⢬⣙⠓⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⢧⣠⠇⠀⠀⠀⠀⠸⣤⡼⠀⠀⠀⠀⢀⣀⣀⣤⠤⠶⠒⣋⡥⠖⠋
⠀⠀⠀⠀⠀⠈⠉⠒⠶⠤⢤⣉⣉⡉⠙⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⢉⣉⣉⡤⠤⠖⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠂⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⢠
⠀⠀⠀⣆⣀⠈⠀⠀⢈⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠀⠀⠀⠀⢀⣧⣀⣀⣀⣛⣀⡼⡀
⠀⠀⠀⢿⣿⡟⠀⠘⣿⣿⠇⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⢿⣶⣶⣶⣶⣶⣶⠀⢠⣶⣶⣿⠂⠀⠸⡇⠀⠉⠉⠀⠀⠀⠇
⠀⠀⠀⠈⢿⣿⠄⢸⣿⠟⠀⠀⠀⢿⡞⠒⠲⠶⠶⠶⠶⠶⠖⠒⠒⠉⢸⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⡄⢰⣿⡿⠃⠀⠀⠀⠳⡀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠙⠣⠽⠁⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠃⠈⠉⠀⠀⠀⠀⠀⠀⠈⠐⠶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠃⠀⠦⣈⣀⣈⡠⠞⣁⠀⢸⡇⠀⠀⠀⠨⣅⣀⣀⡠⠽⢡⠀⠀⠀⣿⠀⠘⠀⠀⠀⡷⡍⣁⣀⣩⡵⠀⢹⠀⠸⠀⠀⠸⢯⣀⣀⣈⡭⣇
⠀⠀⠀⡆⡀⣾⠀⠰⠀⠀⠈⢀⢸⠇⠀⠠⣦⠀⠀⢸⡇⠀⠀⠈⠉⠀⠀⡇⠀⠘⠔⠀⠀⣇⠂⠀⠀⠀⠁⠀⢸⠀⠀⠠⣤⠀⠘⢸⠃⠀⠀⠉
⠤⠤⠀⠤⠧⡿⡤⠤⠼⠀⠤⠼⠼⣤⣤⣤⢤⣥⠤⡼⠧⠄⠠⠤⠴⠤⢠⡧⠄⠄⠠⠼⣶⠿⠤⠴⠧⠠⠤⠦⠼⣤⣤⡤⠼⢤⠤⡼⠤⠄⠤⠤⠧⠄⠄

⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡼⣻⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⣈⠀⠈⠘⣿⣿⡇⠀⠀⠀⠀⢀⠚⠁⠀⠀⠀⠀⢉⢑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣢⡌⢈⡉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇⠀⢻⣿⣿⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠓⠚⠋⢸⣿
⠀⠀⡠⠖⢒⣛⣛⣛⠒⠲⡄⠀⠀⠀⣾⣿⡿⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢷⡦⢐⣷⡆
⠀⠀⡓⠶⠤⠭⠭⢭⢤⠔⣻⠀⠀⠈⠻⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡴⡂⠀⠐⠹⠆⠀⠀⠀⠈⠢⡑⢄⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠇
⠀⠀⢏⠒⠦⠤⠤⠤⠀⢚⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠾⠗⠷⠚⠛⠒⣄⠀⠀⠀⠀⠀⠀⠈⠲⡕⠄⠀⠀⠀⢸⣿⣶⣿⡿
⠀⠀⠀⠁⠒⠀⠂⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠸⡇⢠⠤⠔⠒⢲⡄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠀⠸⡜⢋⢛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⢹⠀⠀⠀⠀⠀⠀⢻⡈⠉⠁⠀⠈⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠄⠄
⠀⠀⠀⠀⠀⠰⡶⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⣀⣰⢶⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠖⠚⠁⠀⠀⠀⠀⠀⠈⠻⣏⣀⡇
⢀⠤⠚⠆⠀⠀⠀⠀⠀⣴⣾⣿⣦⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢹⠀⠀⠀⠀⠀⢿⣿⣿⣿⠀⢠⠑⠁⠀⠈⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠰⡒
⠀⠀⠄⠀⢃⠀⠀⠀⠀⠀⠈⠉⠛⢦⣘⣷⠀⡀⣠⣛⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠰⠋⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣰⢾⣿⣿⣿⣦⠀⠀⠀⠘
⢂⠉⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡈⠙⠛⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠂⠁⠀⠀⠀⠀⠀⠀⠀⢷⣺⣿⣿⣿⡿
⠀⢣⠄⢀⠈⢸⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠀⠀⠀⠀⠀⠀⠀⠀⢙⠿⠛⠛
⠀⠀⠳⡤⢤⠊⡧⠀⠀⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠙⠒⠋⠁⠀⠀⠀⠀⢀⠀⠰⠷⣶⣶⡶⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠁⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠘⡝⢶⡆⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠀⠊⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠤⠒⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡉⠢⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠈⠑⠒⠀⠉
⠀⢰⠀⠀⠀⠀⠀⠤⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⣠⠊⡀⢀⠀⠼⠂⣈⠢⡀⠀⠀⠀⠀⠀⠀⠀⡾⠑⠒⠓⠒⠬⣉⠹
⠀⠀⠓⠀⠀⠀⠀⡀⠙⢄⡠⠀⠀⠀⠀⠀⠀⡔⠁⢠⠀⠈⡀⢀⢠⠀⢑⡬⠂⠀⠀⠀⠀⠀⠀⠁⣀⣠⣶⣒⣓⣓⣶⣀
⠀⠀⠀⠀⢀⡴⠛⠷⡤⠊⠀⠀⠀⠀⠀⢠⠎⠠⠀⠀⠠⠄⠀⢠⡤⠒⠁⠀⠀⠀⣀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠢⡀⠀⠒⢀⡤⠚⠁⠀⠀⠀⠀⢀⣸⡍⢆⠀⠀⠀⠘⣿⣿⣉⣉⡉⠛⠛⣿⣿⡇⠀⠀⠀⢠⣤⡴⢦
⠀⠀⠘⢄⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡒⢈⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠊⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠈⢧⣠⠏
⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣁⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⠠⠅⠀⣠⠄⡂⠀⠀⠀⠅⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡙⡃⠘⠀⠘⣛⣛⡀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣈⣉⣉⣉⣉⣉⡅⠠⠂⢸⣿⣿⠇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠁⠀⠈⠁⢠⡆⣼⣿⣿⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠆⠨⣤⠄⣀⣴⣪⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣌⢛⠋⣡⠎⠙⡽⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢴⠃⠀⠀⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢿⠀⠀⣀⣼⣿⣴⣶⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢸⠋⢹⡏⠐⠛⠉⡿⠿⠧⣤⣄⣰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⣞⣹⡂⣀⢀⡀⠀⢸⡇⠀⣇⠀⢸⡇⢀⠀⠀⠀⠠⠀⠈⠀⠉⠛⠃
⠀⠀⠀⠀⠀⠀⣠⣾⣿⢛⣿⡿⠿⣾⣿⣽⣦⣯⣼⣇⣿⣿⠀⣿⣿⡈⠀⠀⠀⠀⠀⠀⡀⠈⠹⣷⠜⠛⢦⣾⣄⡤⡄
⠀⠀⠀⢀⣶⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣏⣿⡿⡧⢱⣟⠇⠀⠀⢐⣦⡀⠀⣀⠀⢸⣿⡟⠲⣾⠁⢻⡌⢶⠚⠀⠀⡀
⠀⢠⣾⣿⣿⣿⣟⣿⣿⣿⣶⡿⢻⣾⣿⠟⣿⣋⠏⠃⢆⠀⠀⠀⠟⠒⠶⠆⢀⣡⣂⣠⣤⡼⠛⢏⡉⠿⠉⣹⡟⢢⣼⡷⠒⢡⣾⡅
⠀⠈⠛⠻⣿⣿⣿⣿⣹⣿⣼⣿⣶⣿⣿⣿⡏⠋⣐⣶⣨⣤⡀⠈⠛⠛⣙⣿⣿⣿⡟⠁⠉⣿⣿⠈⠁⣄⣸⡉⣳⡋⢻⣷⢦⣾⡿⠁
⠀⠀⠀⠀⠛⢛⣿⣿⣿⣿⣿⣿⣦⣿⣬⣄⣁⣀⣀⡐⣀⣉⣉⣰⡟⠹⣿⣿⣿⣿⡷⠠⣾⣿⣿⣦⣤⣷⣾⣭⣿⣯⣽⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠚⠉⠛⠃⠉⠉⠉⢩⢀⠁⠉⠩⠤⠖⠌⠻⠄⠰⠶⠂⠲⠛⠛⠛⠏⠡⠤⢿⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠉⠉⠉⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣝⠦⣄⣀⡀⠀⠀⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡷⠤⣍⣉⣉⣩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⠀⠀⡾⠀⠀⠀⠀⢀⡤⠖⠋⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢸⠃⠀⠀⢀⠔⠁⠀⠀⣠⠞⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡟⠀⠀⢰⣏⣀⣤⠴⠚⣡⠾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣼⠁⠀⠀⠀⠙⠦⠴⠶⣏⠁⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⡞⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⢰⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⡼⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠹⡄
⠀⠀⠀⠀⢰⡧⢤⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠙⢦
⠀⠀⠀⠀⠈⣇⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⠔⢊⢧
⠀⠀⠀⠀⠀⢿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⢻⡆
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠐⠀⠐⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡻⡀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡘⠀⠀⠀⠀⠀⠈⡀⡀⠄⠀⠀⠃⠠⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡄⠀⢠⢇⡓⢤⡄⠐⠒⠐⡞⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣠⠁⠀⠀⡘⠈⠃⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠐⠁⠀⠊⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢄⠀⠀⠀⠀⠀⠆⠀⠘⠀⠀⠀⠀⡐⠹⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠐⠦⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠙⠉⠉⢣⠰⢎⣠⠀⠀⠀⠀⠄⠀⠘⡀⠀⠀⠀⠈⠂
⠀⠀⢰⠁⢅⠀⠀⠀⠀⠉⠉⠉⠛⠓⠂⠠⠶⡈⠀⠀⠀⡀⠀⣷⠀⠈⠢⣀⣠⡤⣤⠤⠤⢷⡶⠒⠃⠇⠴⢠⡠⢀
⠀⠀⠠⠀⠀⠑⠢⢤⡤⠤⠤⠤⠔⠀⠄⠠⠤⡁⠀⡀⠚⡐⠚⠸⡍⠢⡀⢁⠘⠀⢤⢠⡄⠼⠀⢐⡀⠯⠷⠠⣄⠱⠍⡄
⠀⠀⠀⠗⠢⢄⡀⠀⠀⠉⠑⠂⠲⠤⠤⣀⡀⠱⡄⣈⣠⠔⠒⠀⠁⠀⣄⣈⣦⠂⣠⣊⠠⠌⠄⠂⠚⠉⠈⠁⠀⢀⣠⠃
⠀⠀⠀⠘⡀⠀⠀⠉⠐⠀⠤⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⣀⣀⡀⠤⠄⠐⠂⠉⠀⠆
⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠒⠂⠀⠀⠀⠐⠒⠀⠒⠒⠂⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢰⠎
⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⡠⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠉⢳⠣⢀⠀⠑⣄⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠤⠠⠄⠒⠨⠋⠉⡱⠈⠠⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⢁⠤⠊⠐⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⢠⠔⠈⠹⡄
⠀⠀⠀⠀⢸⣏⠢⣄⠀⠀⠀⢀⢿⣆⡀⠀⢣
⠀⠀⠀⠀⢸⣿⠂⠘⠲⠖⠒⠺⠈⢿⣾⡅⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠤⠤⠀⠀⠀⠀⠂⠀⠀⠠⠤⠀⣀⡀
⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠟⠉⠘⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⡀
⠀⠀⠀⠀⡻⠁⠀⠀⢲⣾⣷⡀⠀⠐⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦
⣀⣄⣤⡊⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⣿⠗⣿⠏⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠿⢶⣟⣛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠈⠙⠓⠒⢶⣀⠀⠀⢀⠄⠀⢀⢠⠴⡄⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⡀⠁⠀⠌⣠⣁⣌⡀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠋⣡⡀⠁⠀⠀⠀⠀⢀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣾⣧⣽⣿⣿⡘⠧⠀⠀⠀⠀⢰⡟⠈⣖⣡⣴⣾⣥⣤⡄⠀⠀⣠⣤⣿⣿⡄⠀⠀⠀⢀⠊⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣿⠻⠂⠀⠀⠀⡰⠛⠛⠿⠿⠿⠟⠿⢿⣿⣴⡷⠟⠛⠉⢻⣿⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠿⢀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⡇⢇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⣨⠀⢸⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⡼⡟⢀⠃⠀⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⡾⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠟⠁⢠⣾⣦⡾⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⣿⣧⣠⠾⠇
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⢀⡠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡎⠀⠇⠀⢀⠔⠉⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⡌⠙⠒⢼⠠⠐⠛⠠⡟⢀⣔⡿⠒⠒⡤⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠡⡀⠀⣠⠏⠸⠁⢈⡒⢢⡀⠘⣄⠀⠁⠀⠀⠀⠀⠘⢏⠉⠁⢈⡸⠁⠑⣏⠈⣧⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣣⣱⡯⠣⢴⡆⠙⡤⠸⠊⢙⣶⣿⣷⣶⣄⠀⠈⠀⠀⠸⠶⣈⣁⠴⡄⢴⠁⠈⢦⢠⡯⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⢤⣾⠑⢦⢸⠀⠋⢁⡔⠛⣾⡿⣺⣘⠽⢼⡽⣦⣄⢤⡧⢊⡥⢖⠛⠃⠀⠱⠀⠣⡀⣁⠸⣆⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠒⡇⠀⠀⠀⠀⠀⡰⢻⡿⠒⣤⢞⣿⢿⢷⣷⣲⣏⢏⢱⣳⡿⠼⠒⠛⠲⠧⢄⣠⠁⠉⡤⠐⢇⠈⡦⢜⠉⠓⢤
⠀⠀⠀⠀⣠⠊⠁⠐⠀⠱⡀⠀⠀⠀⢠⡀⠀⠀⣼⣯⠮⣗⣏⠿⣻⣟⣿⣻⡾⠋⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠓⡄⠸⠣⣀⡸⠈⠃⠀⠱⡀
⠀⠀⠀⡸⠁⠀⠀⠀⣀⡀⠈⠑⠒⠒⠟⠳⢄⣀⣿⣟⣿⣿⢎⣸⣛⣿⢻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡴⠛⠶⠚⠁⠀⠀⠀⣀⠀⠇
⠀⠀⠀⡇⠀⠀⠃⠀⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠙⢿⣻⣮⣾⣿⠿⠃⠸⣏⡒⠄⢒⣒⣒⠦⣄⠀⠀⠀⠀⠀⢹⡄⠀⠀⠪⠤⠷⠀⠀⠀⡄
⠀⠀⠀⡼⣄⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣀⡈⠉⠉⠀⠈⠀⠉⠢⣕⠢⣄⠀⠀⢠⡇⠀⠀⠤⠄⠀⠀⢀⠔⡇
⠀⠀⠀⡇⠈⠳⡌⠀⠄⠀⠠⠤⠀⠀⢀⣀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠙⠲⠭⠿⠟⠁⠀⠀⠀⠀⠀⢠⠃⢀⠇
⠀⠀⠀⡇⠀⠀⠈⢦⡀⣀⢀⡀⠀⠠⠌⢉⡀⡀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠶⠶⠄⠀⠀⠀⡰⠆⠠⠀⠀⠈⠁⣀⠤⠒⠒⠊⠁⣰⢻
⠀⠀⠀⠘⣄⣀⡀⠀⠀⠀⠀⠈⠒⠒⠒⠦⡀⣁⣉⣀⣁⢀⣀⠈⠀⠉⠀⠁⠒⠂⠀⢀⠀⠀⠀⠀⣀⣀⣀⣀⣴⡿⢦⡀⣰⠿⠛⠛⠉⠨⡇
⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠤⠀⠀⠄⠊⠁⠈⠁⠀⠉⠀⠀⢀⡾⠁⠠⠀⡇⢻⡀⠄⢀⠔⠁⣧⣀
⠀⠀⣠⢴⠈⠣⣉⡇⠀⣀⣤⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠒⢒⢶⣄⣀⣀⣀⠀⠀⣾⠀⠀⠀⠀⡇⢸⣢⠐⠀⠀⠀⡧⠈⠓⣄
⢀⡔⠁⣸⠀⠀⠈⡇⠠⡟⢘⠥⡂⢙⢿⣷⣄⠀⠀⠀⠀⠀⣠⠖⠏⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⡧⠀⢿⠀⠀⣀⠠⡷⡏⠀⠀⠀⢀⣠⠃⠀⠀⠈⢦
⡜⠀⠀⠈⣦⣄⠀⠑⠚⠀⠉⠀⠛⠐⠀⠨⢿⡆⠀⠀⠀⢸⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⠀⠃⣿⢠⠞⠛⠉⣻⠀⠀⠀⠀⠈
⡇⠀⠀⠀⡯⠘⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⡎⠀⠀⠀⢀⣇⣸⠎⠀⠀⠀⣹
⢳⠀⠀⠀⠓⢦⡀⠙⢿⣦⣀⣀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⡇⢀⣠⠞⠉⠉⠁⠀⠀⢀⡴⠉⠀⠀⠀⠀⡸
⠀⠳⡀⠀⠀⠑⠉⠢⣀⠉⠛⠿⠷⣦⡤⠒⠶⣾⠀⠀⢷⣤⣤⣤⣤⡀⠀⠀⣀⣤⣤⡤⠞⠋⠉⠓⣏⠀⡸⠁⠁⠀⠀⠀⠀⢀⠴⡪⠊⠀⠀⠀⢀⡔⠁
⠀⠀⠈⠢⣄⠀⠀⠀⠀⠉⡒⠤⣀⠀⠀⠀⠀⢸⡀⢀⡎⠀⠉⠉⠙⠻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⡀⠄⡪⠁⠀⠀⠀⠀⣀⠔⠋
⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠁⠀⠉⠔⠒⠠⢬⡅⠸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠄⠒⠆⠁⠀⠁⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢄⣀⡀⠀⠀⢸⠁⠀⡧⠋⠁⠈⠩⠟⠭⠬⠬⠛⠋⠀⠈⠙⠀⠀⠁⠀⠀⠀⢀⣀⣠⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⡏⠀⠀⢸⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⡶⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⠜⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠤⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠆⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⠄⠀⠈⢄⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠢⠀⠀⢂⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠐⠄⠀⠀⠑⢄⠀⠁⢀⠀⠢⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⡌⠀⡀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡀⠀⠀⠢⢄⠀⠑⢄⠀⠢⡀⠡⠀⢂⠀⠄⠀⠀⠀⠈⠠⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⠀⠀⠀⠑⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠑⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂
⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠠⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠢⠤⠤⠄⣾⣄⣤⣤⢤⠦⡤⠴⡼⠇⠀⢀⠀⠀⠀⠀⠀⢸⠀⠐⠖⠰⠤⠴⠤⠦⠂⠒⠒⠚⡅⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢣⠀⠀⠀⠀⢻⡳⠤⣤⣠⣄⣤⣀⣄⣤⣠⡤⣠⠤⡔⡆⠀⠈⠀⠐⠂⠀⠀⠘⠓⠆⢤⠀⠀⡘⠇⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠈⢦⡜⣧⠀⠀⠁⠈⡒⠉⠠⠀⠀⠐⢌⠲⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠰⠀⡄
⠀⢀⠀⠄⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⠠⠤⣼⣧⡀⠈⠀⠁⠀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⢀⡇⠀⠀⠀⠀⠀⠀⠂⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠈⠂⠄⠀⣀⣀⣠⠬⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠋⡇⠀⠈⠐⠂
⠀⠀⠀⠀⠐⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠰⢦⡇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⣀⠀⠀⢀⡀⠀⣨⠤⠀⠈⡇⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆⠁⠉⠉⡁⠀⠐⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠤⠀⠀⠀⠀⠀⠀⠠⠤⠠⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⠶⠿⠛⠛⠛⠛⠛⠛⠿⠶⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣀⣠⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉
⣶⣶⣶⢶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠰⠍⠉⠉⠋⠙⠭⡆⠀⠀⡠⠤⠤⣤⠀⠀⠀⢀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡆⠀⠀⣰⠂⠀⠰⠂⠀⠀⠀⠀⠋⠁
⠀⠀⠀⣠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢲⡐⣀⣿⠀⠀⠈⢹⠁⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠄⠁⠀⣠⢀⡀⠀⠀⠈⠀⠁⠁⠀⠀⠀⢸
⠀⢀⡴⠀⢀⠠⡀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠂⠈⠉⠀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠠⠆⠀⠀⠀⢸⠀⠀⠀⠁⠀⠁⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠤⠰⠃⠀⠀⠐⠀⠀⠀⠠⣂⠄⠀⠀⢰⠀⠀⠃⠀⠀⠰⠘⡄⠀⠀⣗⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⡠⠄⠈⠁⠀⠀⠀⠘⠀⠀⡀⠀⠀⠀⠹⠁⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠁⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⡀⠀⠀⠀⠀⢀⡠⠒⠉⣠⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄
⠀⠀⠀⠀⠀⡀⠀⠁⠀⠌⠂⠀⠀⠀⣀⠔⠉⢀⡴⠊⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇⠄⠂
⠀⠀⠀⠄⠈⠀⠠⠀⠀⠀⠀⢀⠤⠊⢀⡠⠞⠁⠀⡠⠔⠋⠀⠀⠀⠀⠀⠀⣀⡠⡠⠁
⠤⠈⠀⡀⠄⠀⠀⠀⠀⡠⠖⠁⣀⠔⠋⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠕⠂⠀⠀⠀⠒⠀⠀⠀⣴⣾⣧
⠘⠁⠀⠀⠀⠀⣀⠔⠊⢀⡤⠚⠁⠀⠀⠂⠀⠀⢀⠔⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠃⠀⠸⣿⣿⡡
⠀⠀⠀⠀⡠⠞⣡⢀⠖⠉⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠰⠋⠁⠀⣠⠺⠁⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁⠀⠀⠀⠀⠀⠀⢈⠀⣀⠔⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⣀⡀⡄⠀⣀⠈
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣤⣤⠤⠔⠒⠈⠉⠁⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠵⠒⠚⠛⠒⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣉⣩⣭⣙⡲⡄⢀⢼⡋⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁⠀⢀⣀⡀⠀⠀⠙⣆⣷⠀⠀⠀⠀⠀⠀⠀⡼⢡⡴⠛⠉⠀⠀⠈⠙⢿⣸⠀⠙⢦⣀⠡⠠⠌⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⡴⠋⠉⠙⢷⠀⠀⢹⡟⠀⠀⠀⠀⠀⠀⠀⣷⠋⠀⣠⠖⠛⠛⢳⡄⢸⠏⢧⣄⠀⠈⠙⠲⠦⣤⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣸⠁⠀⠀⠀⢸⠇⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⡏⠀⠀⠀⠀⢳⠞⢠⠏⠈⠛⠶⢤⣄⣀⣀⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡼⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣽⡄⠀⠀⠀⠈⣷⠏⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⢠⠏⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⠏⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣠⠏⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣰⠏⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⣠⠏⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣰⠏⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢠⠿⡄⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⠏⠀⢻⡀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢠⠏⠀⠀⠈⣧⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢠⡏⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢷
⠀⠀⠀⠀⢸⡇⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠈⣧⠀⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⢸⡇
⠀⠀⠀⠀⠀⢿⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡇
⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠁


⣿⣿⠿⠿⡿⢻⠿⢿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠒⠄⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⢄⠠⠊⠀⠠⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠁⠠⠐⣂⠠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣉⡀⢀⢈⠉⣀⡠⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠈⠁⠃⠉⠀⠀⢀⢠⢄⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠠⡐⢀⠁⢀⠥⠼⡁⢄⠑⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⡗⡌⢄⡤⢾⢉⠉⠉⠑⡓⠀⠀⠘⡦⢀⣲⠃⠀⠀⠀⠈⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠄⠀⠀⠈⣉⣁⣈⣂⣀⣀⣒⡀⠀⠀⠀⠉⠠⠀⠀⠐⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠠⠤⡀⠀⠀⠀⠠⡀⢀⠀⠐⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠈⠒⠤⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠤⠐⠈⠁⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠐⠒⠂⠀⠤⠤⠀⠀⠀⠤⠤⠀⠒⠒⠒⠂⠉⠉⠁


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

⠀⠀⠀⠀⠀⢰⠂⢺⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠒⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢲⡆
⠀⠀⠀⠀⠀⢸⠁⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠘⡄
⠀⠀⠀⠀⠀⢰⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡠⣴⠃
⠀⠀⠀⠀⠀⢼⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿
⠀⠀⠀⢀⣠⣧⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⣀⣼⡅⢛⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠸⡆
⠀⠀⢀⣿⢱⣿⣿⢻⣿⣿⣿⡄⠀⠀⠀⣼⠃⣿⡇⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠟⢁⠐⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠊⢹⡆⡀
⠀⠀⢸⡏⢸⣿⣿⣸⣿⣿⡟⡇⠀⠀⠀⣿⣄⣿⠇⢤⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⢀⣾⣾⣦⣼⣷⣧
⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⠀⠀⠄⠀⠀⠀⠀⠛⣿⣿⡄⠀⠀⠀⠀⠀⣼⢳⣿⣷⢸⣿⣿⣇
⠀⠀⠸⠇⠘⠛⠛⠻⠟⠿⠷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣼⣸⠀⠀⠀⠀⠘⢿⣿⣿⠀⠀⠀⠀⣸⠃⢸⣿⣿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣤⣀⣿⣀⣤⣿⣿⣿⡤⠀⠀⠀⡿⠀⢸⣿⣿⢈⣹⣿⣿⣧
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢸⣿⡟⢸⣿⣿⣿⣿
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⣶⣶⣤⣤⣤⣤⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠘⠛⠀⠛⠛⠃⠒⠛⠛⠛⠛
⠀⠀⢸⡇⢸⣿⣷⣾⣿⣿⣿⡇⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁
⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⢸⣿⣿⣾⣿⣿⣿⣿⠀⠀⢶⣤⣤⢀⣀⣀⣀⣀⣀⣤⣤⣵⠂
⠀⠀⢸⡇⣺⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣾⣿⣿⣠⣿⣿⣿⣿⠀⠀⢸⣿⡃⠀⢙⣟⡓⠀⢹⣿⣿⣿
⠀⠀⢸⣷⣻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⡇⠀⣀⣿⣥⣀⣸⣿⣿⡇
⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠹⣿⣿⣿⣿⣷⣿⣿⣿⡿⠁⠀⠀⠘⠷⣶⣤⣤⣤⣤⣤⣤⣶⠾⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⡠⠒⠒⠋⠉⠉⠉⢒⠶⣤⡀
⠀⠘⠁⠀⠀⡤⠒⠒⣒⣚⣷⢦⣾⣿⣷⠂⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢰⣦⣄⡀
⠀⠀⠀⢀⡜⢀⠔⠁⠀⡰⠃⠿⠿⠿⠁⠀⣀⠰⣶⣤⠝⣟⡛⠉⠐⠠⠄⣑⠦⡈⠑⠤⡀
⠀⠀⠀⠀⠈⠳⣄⢔⡝⠁⠀⣠⠼⠆⠀⠀⠉⠹⣷⠆⠀⡀⠉⠉⠐⠒⠤⡀⠉⠺⢶⣀⠈⠢⡀
⠀⠀⠀⠀⣀⣀⡴⠋⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠉⠑⠒⢤⡀⠀⠁⠀⠀⠈⠀⠀⠈⠒⢄⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣉⠉⠒⠦⠤⠤⠽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠤⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣳⡀⠀⠀⢠⣓⡄⠀⠀⢀⣟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣼⠇⠀⠀⣿⣩⡇⠀⠀⢿⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣭⡇⠀⠀⣟⢩⡇⠀⠀⣟⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣵⡇⠀⠀⣟⢥⡇⠀⠀⣟⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣴⡇⠀⠀⣯⣵⡇⠀⠀⣿⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⣴⡇⠀⠀⣯⣴⡇⠀⠀⣯⣵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠆⠒⠛⠛⠓⠀⠉⠉⠉⠉⠉⠁⠉⠛⠓⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡖⠉⢢⢀⡴⡄⡎⢸⠀⢸⠁⢸⠀⡇⢡⠀⡎⠀⢱⢠⠒⠀⢴⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢴⠁⠀⢸⠈⡇⠓⠂⢸⠀⠸⠀⢼⠀⢃⠼⠀⡇⠀⣆⢈⠀⠓⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠒⠁⠀⠀⢨⠀⣸⠃⠈⠒⠂⡇⠀⢸⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣦⠤⠤⠆⠒⠒⠒⠚⠛⠉⠀⠉⠉⠈⠁⠒⠚⠒⠂⠤⠤⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠒⢤⠀⢀⣀⢀⡊⢦⠀⡰⠋⢱⡀⡎⠱⠀⢠⠊⠑⡄⢀⢤⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⠇⠀⢰⠀⢂⠸⣠⠆⠘⠀⢃⠀⢘⠃⢣⢀⠀⢣⠀⠀⡇⢸⠐⠤⣼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠈⣄⡸⠀⠀⠀⡇⠀⠸⠀⠹⣀⡸⢸⠀⠈⡄⠀⠣⠞⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠼⡄⠀⠀⠀⠀⠀⠀⠀⠳⠤⠚⠀⠀⠀⠀⣈⠀⠀⡇⠀⠀⠀⠀⢠⠧⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⢠⠒⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠁⠀⠀⠀⠀⡜⠒⡄⠀⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠈⠲⠤⠍⣐⣒⠒⠐⠲⠆⠀⠀⠤⠤⠤⠄⠀⠀⠐⠶⠒⠒⣒⣒⡩⠤⠖⠁⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠬⠭⠍⣉⣉⣁⣀⣀⣀⣀⣈⣉⣉⠉⠭⠥⠤⠤⠐⠒⠂⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠑⡄⠀⠀⠀⠀⢠⠃⢆⢣⠀⠀⠀⢀⠇⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡆⢱⠀⠀⠀⠀⢯⢸⢸⢸⠀⠀⠀⢸⢠⠱⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣄⣄⡜⠀⠀⠀⠀⠈⣻⣿⢿⡄⠀⠀⠘⣼⣾⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⣯⢲⡀⠀⠀⠀⡞⠛⢩⣼⠃⠀⠀⡴⠻⠛⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣴⢸⡇⠀⠀⢸⣴⣿⡗⣿⠀⠀⠀⣷⣰⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢚⣿⡇⣀⡀⢈⣿⠦⣌⣏⠀⣀⠀⢸⣿⠽⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣤⠶⣤⣶⣾⣿⣻⡏⠭⠛⣿⣿⣭⣿⡯⣑⠢⣿⣿⣭⣿⣿⠿⣋⣿⣷⣶⣶⡴⢶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⢀⣤⢾⣿⣷⣿⣛⣛⠋⠉⠉⠉⣸⡧⢐⣫⣧⠀⠀⢰⡗⠠⢍⣿⠀⠀⠀⣿⣈⠤⣿⠉⠙⠛⠛⠛⣷⣶⣽⣟⣛⣳⣆
⠀⠀⠀⠀⠀⢾⣷⢻⣽⣶⣿⣭⣭⣿⣶⡞⠲⣮⣩⣭⣭⣥⣤⡶⢮⣭⣛⣋⣭⣴⡶⢦⣭⡿⡿⢭⣴⡞⣛⣛⠺⣿⣷⣴⡟⣿⢻⡿
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣯⣚⣿⡸⣿⣿⣿⣿⣾⣽⣶⢶⣾⣿⣿⣶⣮⣻⢯⣯⣭⣿⣶⣿⣯⣿⣷⣭⣿⡿⣿⠿⢿⣿⣿⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡯⣽⣽⠿⣿⣿⣿⣮⣿⣯⣿⣾⣿⣯⣽⣿⣽⣿⣿⣟⣟⣫⣾⣯⣿⣷⣿⣷⣷⣿⡿⣿⣻⣿⣟⡾⢻
⠀⠀⠀⠀⠀⠀⠀⢸⣏⡷⡏⣵⠟⣿⣿⣿⣞⣭⡭⢿⣟⣯⣿⣿⣟⣯⣭⢯⣟⣿⣯⣷⣿⣿⣷⡿⠞⢾⣿⢿⣿⢫⡷⣿⢧⡞
⠀⠀⠀⠀⠀⠀⠀⠘⣯⠉⣸⠇⠀⢹⡟⣿⠉⣿⡝⣹⠟⠉⣿⣯⡿⠁⠹⣟⣿⣿⠿⣿⡿⢡⡟⠀⠀⠀⢻⣏⢃⡾⠀⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡋⠀⠀⠘⣷⣻⡿⠃⣴⠏⠀⠀⠉⠉⠁⠀⠀⢿⣭⣿⣠⣿⠇⣼⠃⠀⠀⠀⠀⠙⠛⠁⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣦⣤⣘⣻⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣹⠟⢁⣴⡟⠀⣀⣀⣀⣠⣤⣴⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣭⣿⣿⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⠿⣫⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠩⠻⣯⡛⠿⠗⢒⣺⣾⠶⢾⣖⠾⣿⣿⠿⢟⣫⡶⠞⠛⠻⢮⣟⣟⣩⡴⠞⠉⠉⣇⡀
⠀⠀⠀⠀⠀⣀⣀⣠⠶⣻⣄⣀⡀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠉⠓⠶⠶⠚⠋⠁⠀⠀⠀⠀⠀⠈⠉⠁⣀⣀⣠⡶⠿⣟⣷⣄⣀⡀
⢀⣠⠴⠞⠋⠉⠉⢿⡟⣽⢿⣯⣟⢛⣶⡶⢦⣤⣤⣄⣤⣤⣀⣤⣤⣤⣀⣀⡀⢀⣤⣤⣤⣤⣤⣤⣶⣶⣿⣻⣭⣽⣿⣿⣿⡿⠋⠉⠉⠛⠶⢤⣀
⣾⣧⣄⡀⠀⠀⠀⠈⠻⠾⢿⣽⣿⡿⡷⣿⣿⣷⣷⣿⢿⣿⣫⣴⣶⣮⣿⣿⣛⣿⣭⣴⣦⣽⣷⡾⠿⣶⣾⢿⣻⣿⣵⠟⠋⠀⠀⠀⠀⠀⠀⣀⣽
⠈⠛⠿⣿⣷⣶⣦⣤⣄⣀⣀⣀⠈⠛⠿⠿⠻⠷⡿⠿⣿⣯⣿⢾⣿⣿⠿⣯⣭⣽⡽⠿⣿⡿⠟⠛⠛⠛⠛⠛⠉⠀⢀⣀⣀⣀⣤⣤⣶⣶⡿⠟⠋
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢡⠇⠀⠀⠀⠀⠀⠀⠰⣶
⠀⠀⠀⣄⠀⠀⠀⢷⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⢀⡴
⠀⠀⠀⠸⡄⠀⠀⠸⡆⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⢷⠀⠀⠀⢻⡀⠀⠀⠀⠈⠳⣄⣀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⡼⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠈⣧⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢩⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⠃⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⢹⡄⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣰⠇⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢷⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⡰⠋⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠛⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠉⠀⠀⠀⠀⣎
⠀⠀⠀⠀⠀⣠⠖⠁⠹⡄⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⡇⠐⣆
⠀⠀⠀⠀⡼⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣰⢃⠄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⢧
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢀⡘⣆⠀⠀⠰⡷⢋⠆⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠈⣧
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⢷⣜⣦⠀⠀⠀⡼⢀⡶⠀⡄⢀⡀⣸⠁⠀⠀⠀⠀⢸⡆
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⣄⠸⡄⠉⠀⠀⢠⡇⣸⠇⢼⠃⣾⢁⠇⠀⠀⠀⠀⠀⠀⣷
⠀⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⢻⠰⡆⢹⡆⢻⡄⢡⠀⠀⠀⠸⠠⢹⠠⢿⢰⢹⠸⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⣼⠀⠀⠀⠀⠀⡴⠹⡆⠀⠈⣇⣿⡈⡿⡌⣿⣼⠀⠀⠀⠀⠃⠨⠜⠀⠌⠈⠀⠀⢰⠣⡄⠀⠀⠀⠀⣧
⠀⣿⠀⠀⠀⠀⠀⡇⠀⢿⠀⠀⢩⡇⢣⢳⢳⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢻⡀⠀⠀⠀⢹⡄
⠀⡏⠀⠀⠀⠀⢸⠇⠀⠸⡆⠀⠀⠀⠘⠞⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠘⡇⠀⠀⠀⠈⣇
⢰⡇⠀⠀⠀⠀⣼⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⢻⠀⠀⠀⠀⢻⡀
⢸⠇⠀⠀⠀⠀⡟⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠸⡆⠀⠀⠀⠸⡇
⢸⠀⠀⠀⠀⢰⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⢿
⡏⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⠀⠀⠸
⠃⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸
⠀⠀⠀⠤⣤⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠾⠟⠛⠛⠛⠉⠉⠉⠉⢉⣛⠛⠛⠻⠷⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠉⠉⢛⡛⠛⠂
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠓⠀⠉⠉⠉⠁⠀⠀⠀⠉⠉⠀⠐⠂⠀⠤⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡠⠤⠚

⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⣤⣄⣀⣀⡀
⠀⠀⠀⢰⡸⠭⠇⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠉⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⣀⠠⠤⢺
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠇⠠⡤⠔⠃
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⡄⠠⠦⠠⢦⠁⣤⢟⢧⣷⢠⠴⠀⣤⢤⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡀⠁⠞⠁⠐⡒⠄⠙⠸⠜⡧⠓⠯⠥⡈⠠⠭⠄⠀⠀⠠⠬⠐⠴⠤⠉⠂⠀⠾⠂⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⡸⣆⠀⠠⢸⠋⠒⠢⠤⠄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠢⢠⣀⡀⠀⠊⠀⠀⠃⡀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠂⡴⠁⠸⡄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⠙⠀⠀⠀⢃⠀⠀⠉⠐⠒⠂⠶⠬⣀⢒⡞
⠀⠀⠀⠀⠀⠀⠀⡆⠈⣼⠀⠀⠀⠱⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡎⠳⡄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⢧⠀⠀⠀⠀⠱⡈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⣆⠀⠘⣄⠚⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠓⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠐⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠘⠂⠇⠀⠀⠈⠈⡃⠀⠀⠀⠃⠀⠀⠀⢀⠄⠼⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠤⠀⠄⠂⠀⠀⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣤
⠀⠀⠴⠒⠒⠒⠒⠲⠒⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠉⢱⡆⠀⠀⠀⠀⠀⠄
⠀⠀⡇⢀⠖⠉⠁⠉⠢⡀⢰⠉⠉⡆⡊⠀⠙⠄⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠞⠃⠀⢀⠀⠀⠀⢀
⠀⠀⡇⡌⠀⠀⠀⠀⠀⠱⠈⡲⣒⠁⢙⡶⡊⠀⠀⠀⠀⠀⠀⠀⣀⠀⣃⠀⠀⠠⢀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠛⠄
⠀⠀⡇⠡⡀⠀⠀⠀⠀⡔⢸⠀⠀⡇⣇⠀⢈⠀⠀⠀⠀⠀⠀⠀⡅⠀⠬⠀⠀⢀⣀⣀⡀⠀⠠⠀⠰⠀⠀⠀⠀⠁⠤⠀⠄
⠀⠀⣇⠀⠑⠢⡤⠄⠊⠀⠀⠉⠉⠀⠈⠉⠁⢸⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⠀⠤⠤⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⠌⡆⠀⠀⠀⡇⠘⠀⠀⠀⠀⠅⡀⠀⠀⠀⠀⡈⠂⠀⠰⠰⠂⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠁⠀⠀⢮⡰⠃⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⡾⢩⠏⢿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢤⢀⠠⢄⡀
⠀⠀⠀⠆⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠄⠀⠀⠀⡅⡆⠀⠹⢍⡘⢹⠀⢀⠔⠛⠋⠲⡄⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡁⠀⠀⢻⡀⢀⢈⡬⢷⡾⠀⢸⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⠀⢠⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠔⠢⡉⠓⠛⠛⠉⠀⠀⠀⠓⠤⠤⠖⠁⠀⠀⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⢰⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠈⢢⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠠⢄⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠒⠀⠘⠀⠀⠀⠀⠸⠤⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠔⠀⠀⡇⠀⢸⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄⠀⠄⠀⠀⠀⠀⡀⠀⠀⠡⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠘⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡯⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⠤⠤⠴⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠙⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠆⠀⠒⠛⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⣶⣶⡟⠒⠀⠀⡀⠀⠀⠀⣴⢯⠘⡍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⣿⠟⠣⠄⢂⠀⠄⠀⠀⠀⠾⣞⣾⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⢈⣙⠧⠻⣀⠀⠀⣿⣷⡆⠀⠀⠀⠀⣌⠉
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⣄⣸⡿⠿⢤⡄⣡⢐⣿⣿⣿⡀⠀⠀⠀⠀⠈⢈⣁⠁
⠀⠀⠀⠀⠀⠀⢀⣷⠾⠿⡁⠉⠀⢹⣶⠂⢹⣿⣽⣏⠀⠁⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢤⣾⠻⠆⠀⢷⡎⠐⠛⠫⣴⣿⣟⠁⢭⣤⡃⠀⢸⠋⠀⠀⠀⠀⢀⣠⣶⡿⠈⢀
⠀⠰⠐⠀⣦⠛⠁⢹⣄⢸⣿⠧⠀⠠⣤⡿⡧⣭⡤⠓⢩⢧⠀⠀⠀⠀⠀⠀⠐⠛⠉⡅⠂⠀⠀⠀⠀⢲⠒⠀⠂
⠀⠀⠀⠀⡇⠁⠀⠉⠙⢻⠉⠓⢶⠈⢈⡏⠉⠉⠁⠀⢸⠀⠘⠖⠀⡄⠤⠀⠀⠢⢠⡄⣰⣷⣶⣤⣄⡘⠈⠁
⣀⣂⣒⣛⢓⣒⣒⣐⠒⠚⠒⣒⣒⣒⣚⣓⣀⣒⣒⡂⣖⣀⣀⡐⣚⣓⣒⣓⡐⠚⠚⠒⣓⣛⣛⣛⣓⣀⣒⣒⣂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠉⠀⠀⠠⣄⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⣀⠀⠀⠠⠙⠶⡄⠀⠙⠢⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⢄⠀⠈⠁⠈⢧⠀⠀⠀⠀⠹⡀⠀⠀⠘⢏⠉⠉⡇⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠘⣆⠳⣄⠀⠀⠀⠱⡀⠀⠀⠀⠱⣄⠀⠀⠘⢶⠒⠐⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⣄⠀⢈⣑⢌⠳⡄⠀⠀⠘⢆⠀⠀⠀⠹⣧⡀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⢜⠢⣜⢿⡷⠁⠘⢆⠀⣠⣤⣠⡀⠀⠀⢳⣳⡀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⠁⠀⠀⣠⣤⠶⠞⠳⣝⣿⣿⠗⢸⣿⣦⠇⠳⡜⣿⡧
⠀⠀⠀⠀⠀⠀⠀⣀⢀⠔⠋⠀⠀⠀⣠⢶⠿⠋⠁⠆⠀⠀⢈⣈⡥⠄⡈⢿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣾⠟⠁⠀⠀⠀⡠⡪⠒⠁⠀⠀⢀⣠⠴⠚⠉⣡⠴⠋⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠋
⠀⠀⠀⠀⠀⢠⣴⣶⠀⢀⡴⠋⠀⠀⠀⢀⡠⠚⠉⠀⢀⡴⠟⠁⠐⠀⢀⣠⡤⠶⠒⠛⠉⠉
⠀⠀⠀⠀⠰⡿⠟⠡⢞⣽⣾⡆⠀⣠⠴⠋⠀⢀⡠⠞⠉⠀⠀⢀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣫⠴⠋⠁⠀⢀⣤⣶⣦⠀⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠯⠵⠚⠉

⠀⠀⢀⠄⡀⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠻⣄⠈⠁⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠿⠛⠂⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠸⢰
⠀⠀⠴⢲⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠤⠴⠶⠚⠉⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠦⠤⠤⠄⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⢀⠀⠐⢦⡀⣀⢸⠀⠀⠀⠀⠀⠃⣠⠖⠉⠀⣀⢀⣀⠀⠀⠢⡄⣠
⠀⢠⡇⠀⣀⠀⠤⠤⠄⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢰⠋⢧⡀⠀⠀⠀⠀⠀⠀⠉⣀⡼⢺⠀⠀⠀⠀⠀⠀⠳⠄⡀⠀⠀⠀⠀⠗⠠⢀⣷⡏
⠀⢸⣷⠋⡀⠛⠀⠀⢂⡴⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠛⠓⠒⠒⠒⠚⠋⠉⠀⡾⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠤⣐⣽⠟
⠀⠈⢿⡋⠃⠜⠀⠒⠒⠈⣱⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠦⣌⣙⠒⠚⠛⣋⣡⢄⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢉⠷⠶⠲⠶⡛⠉
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢭⣭⣭⣽⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⠐⠂⠁
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⡸⡄
⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠇
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⣀⡀⠀⠠⠚⠁⠁⠀⠮⣰⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡐⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⠦
⠘⠁⣬⠁⠀⠀⠀⠀⠀⠀⠘⠮⣏⠄⠀⠀⠀⠀⠀⠀⠀⠉⠂⠚⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠔⡁⠈⠃
⠀⠀⠀⠒⠲⠶⠶⠶⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⠤⠴⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠦⠤⠤⠴⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣟⠉⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⡿⠿⠛⠋⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠐⠰⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡁⢘⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠰⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡈⠀⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠄⠀⡃⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡍⠀⠀⠆⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠀⠀⠀⠀⠂⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂
⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣶⣶⣶⣶⣶⣶⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠉⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⠉⢿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⢿⠟⡻⠿⣟⠟⣿⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⣻⠿⣭⢭⣍⣉⡿⣿⡄
⠀⠀⠀⠀⠀⠀⣼⣿⠉⠶⢻⡽⢯⣽⣿⠶⠿⣡⣿⠇
⠀⠀⠀⠀⠀⠀⢹⣿⣛⣛⠛⠛⣋⣉⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠻⣿⣿⡹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡅⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⣻⡶⠶⣍⠻⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢠⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣍⣻⣷⡤⠜⢧⣹⣿⣛⣿⡿⠿⠿⡿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣯⠒⣿⢿⠘⡇⢿⡿⠿⠿⠿⠶⢶⣶⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣜⠛⢁⣟⣿⡇⠀⠀⠀⢀⣾⠏⢁⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠋⠀⠀⠀⠀⣾⣯⣦⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣼⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠌⠉⠭⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠏⠉⠔⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠿⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣟⣧⡀⢾⣛⢧⡌⣙⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⡛⢦⣅⣐⡋⠹⠆⠈⣻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠫⠥⢶⣞⣖⣢⡍⢛⠿⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣶⣤⣍⣙⣛⣓⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠷⣬⣉⡫⠽⠖⠗⠩⠄⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⣽⡟⠉⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣭⣙⣛⣛⠛⠛⣛⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡾⠟⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠛⠛⣿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⢋⣡⣤⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⣷⣀⣸⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣏⠻⠱⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠭⣭⣭⠉⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⡘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣷⣾⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠐⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠹⡿⠋⠀⠀⢀⠀⠚⠉⡛⠻⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣀⠀⠀⠠⠱⡱⣠⣠⠃⠀⠐⠄⠃⠀⠀⠈⢛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠦⣄⣣⣿⣿⣿⠃⣠⡀⠀⠀⠀⠲⢄⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠉⠉⢻⣿⡀⢹⣷⡀⠀⠀⠃⠈⠴⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡤⠶⠟⠛⠾⠟⠋⠉⣩⡽⠟⠋
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠙⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠈⠻⠿⠿⠿⠷⠦⠄⣀⣤⠔⠉⠒⠦⡤⠠⠄⠀⠀⠛⠿⠷⠟⠛⠁⠀⠀⠀⠀⠀⣴⣾⣿⣷⣆⡀
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠐⠶⣖⠀⢡⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡠
⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣈⠲⣄⠀⠀⠀⠀⠀⠀⣸⣷⣦⣄⣀⣀⠤⠀⠀⠀⠀⠀⠀⠀⡀⠿⠛⣇⠈⢻⡇⠰⡀
⠀⠀⠀⠀⠀⢹⣶⣄⡀⠠⡀⠀⠁⠀⣱⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⢰⣿⠀⠘⠀⠀⡇
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣶⣬⣲⣤⣰⡾⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢿⡄⠀⠀⣸⠇
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣃⣡⣼⡿⠿⠿⠿⠿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⢠⠃⢡⠀⠀⢻
⠐⠿⠿⠟⠫⣿⣿⣿⣿⣿⣿⣿⡛⢻⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠈⠳⠄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⣸⠁⠀⠘
⣀⣀⣀⣀⣀⣉⣛⣉⣀⣙⣛⣛⣃⣻⣿⣿⣿⣿⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣋⣀⣀⣀⣃⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣯⣿⣯⣽⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⣤⣄⠀⠀⠀⢀⡀⠐⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠿⣇⢀⣠⣜⣷⣤⣾⣯⣤⣤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠜⣿⣟⡿⢻⣿⢟⣽⣏⠛⠿⣿⣿⢉⠛⢻⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡜⠻⠟⠿⢾⣧⣾⡏⠃⢀⣴⡄⠻⢾⣶⣶⡴⠀⢀⡤⢶⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠘⢀⣀⣤⣨⣿⠓⠶⠛⢙⣿⣷⡬⠭⢄⡛⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣤⣛⠁⢀⣹⣿⣿⣿⣿⣳⡐⠞⠛⢻⣧⣄⠀⠰⠼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣾⣷⣬⣿⣿⡿⢏⣿⣿⣿⣿⣦⣞⣽⣿⣿⣿⣾⣿⣧⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢫⣭⣽⣷⣦⣩⡁⠘⠿⣿⣿⣿⣿⣿⣿⣿⢏⠀⠈⠉⠛⠛⠛⠹⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣃⣴⣿⡿⠿⠿⢿⡿⠿⠉⠛⢋⣻⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠤⠒⠀⠠⠂⠀⡚⣲⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣯⠉
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠼⢯⣿⣿⢻⣿⣿⡟⣿⣿⡿⠻⠛⠿⠋⠻⣟⣷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢟⢸⣽⡽⠳⠿⠟⠀⠀⠀⠀⠐⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⢉⣵⡆
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠈⠀⢀⠀⠘⢂⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠈⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⣿⣯⣽⣶⣌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣽⣿⣽⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⢢⣶⣤⢀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⣾⢾⣷⢧⣴⣤⣤⣌⣿⢮⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠸⢿⣊⣾⣿⣿⣿⢿⣾⣿⣷⣿⣿⣿⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⣧⣭⣿⣿⣿⣿⣿⣿⣽⣿⣻⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣧⢀⣀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢹⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛⢶⣾⠁⠀⠙⠷⣤⡀
⠀⠀⠀⠀⢰⣿⡿⠛⡛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠻⣄⠈⠛⠷⣦
⠀⠀⠀⠀⢸⣿⣧⡀⠈⣨⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡳⣴⠿⣦⠀⠀⣿
⠀⠀⠀⠀⢸⣟⢿⣟⣛⢉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠳⢤⣄⣀⡀⠀⠀⢸⢇⡴⠆⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠉⠉⡹⣦⠘⠷⣴⠏
⠀⠀⠀⠀⠀⠹⣦⡌⠙⠋⠉⠙⠛⠛⠛⠛⠛⠛⠟⢛⡟⠀⠀⠀⠀⠈⠉⠉⠉⠉⠠⢦⡄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠳⣦⡿⠘⢷⣄
⠀⠀⠀⠀⢀⣾⡋⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⡟⢠⠞⠁⠀⡴⠋⢷⠀⣾⠳⢄⠈⠳⡄⠀⠙⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⠁⠀⣿⠀⠹⡄⠈⡇⠀⠹⡄⠀⠈⣿
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠒⠈⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⣤⡴⠃⠀⠀⠉⠛⠁⠀⢠⠇⠀⠀⣾⠁
⠀⠀⠀⣿⠂⠁⠀⠀⠠⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣄⠀⠀⠀⠈⠉⠓⠲⠶⠶⠶⠖⠊⠁⠀⠀⢀⡾⠋
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣦⣤⣤⣤⣤⣤⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡤⠀⣀⣤⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⡏⠉⠓⣦⣤⣤⣀⣀⣠⡇⠀⠀⠀⣼⡗⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣦⣉⣽⡄⠀⠉⠉⠉⣧⠀⠀⢰⡿⣷⠄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢙⣧⠉⠉⠀⠀⠀⠀⠀⢻⣅⠀⢸⡇⠿⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⢰⡒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠙⢦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠞⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠴⠶⠶⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣠⠖⡄⢸
⠀⠀⠀⠀⠀⢀⠤⢤⡰⠊⠁⠀⠈⠙⠁⠀⠀⠀⠀⢡⡔⠊⠉⠉⠐⠖⠒⢢⣀⠤⠤⣄⢀⠠⢤⡀⠀⡠⣇⠸⠧⢤⣄
⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠙⢫⢣⠜⢰⢀⣦⢼
⣠⠊⢉⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢤⣈⣀⢏⠈
⡇⠀⠣⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠤⠤⠤⠒⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡤⠖⠁
⠀⢸⡉⠑⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⢸⡑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠊⣆⡀
⠀⠀⢀⡽⠀⠈⢉⠗⠒⠲⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠶⠒⠊⠉⣳⠀⠀⢀⣹
⠀⠀⠘⠤⣀⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠠⠴⠤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⢻⠿⠋⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⡠⠶⠖⠀⠖⠀⠆⠀⠆⠀⠛⠒⠠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⣀⣈⠁⠀⢠⠂⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⣀⠀⢐⠀⠀⠁⠀⠀⠄⢀⣠⡆⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠐⠄⣀⠀⠀⠀⠀⠀⠀⠀⡄⢀⢀⡯⠖⠀⠀⢹⠁
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⢸⠉⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⢒⠒⠐⠂⠀⠀⠀⠈⠉⠓⠒⠒⠒⠒⠓⠋⠉
⠀⠀⠀⠀⠠⡀⢰⠀⠀⡆⡆⠘⠀⠀⡇⠀⠀⡏⠀⠀⡔⢸⣽⢲⡄⢁⣈⠀⠀⢀⡀⠐⠃⠀⢀⠉⠀⠀⠘⡁⠀⠒⠒⢀⡀
⠀⠀⠀⠀⠀⠁⠈⢀⠟⠛⢻⡁⠀⢠⡇⠄⢠⠃⠀⠸⣚⣚⣗⣰⡇⢀⠀⠀⠀⠈⠀⠀⠀⣰⠛⢿⡄⣴⠀⠀⠀⠀⣄⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⢸⡇⠀⠈⠁⠀⠀⠙⢋⠏⠃⠄⢀⠁⠀⡀⢠⠀⠀⠀⠘⠳⠾⠃⢸⠀⠀⠀⢰⡟⠊⠃⠀⢠
⠀⠀⠀⠀⠄⠀⠀⢸⠀⠀⠀⣿⠀⠘⠓⠀⠀⢸⢠⡖⠀⠸⠀⢰⠀⢸⠀⢀⣇⢀⠀⠀⢰⠛⠉⠀⠀⠁⠀⠀⠀⢸⡇⠀⠀⠈
⠀⠀⠊⠀⠀⠀⠀⠈⠆⠀⢰⠏⣀⠯⠀⠀⠀⠈⢸⡇⠀⠀⠀⢸⠀⠈⠀⠐⠶⢺⡀⢀⡀⠧⠔⠰⠶⠆⠀⠀⠐⠀⡇⠀⣷
⠀⠀⠀⠀⠀⠐⠀⠀⠀⢒⡋⣀⢹⠀⠀⡄⠀⠀⠠⠁⠆⡏⠀⢸⠀⠀⢠⡆⠀⠀⠀⠈⢱⢸⠀⠀⢆⠉⠀⠀⠀⢰⡇⠀⠁
⠀⠀⠛⠒⠋⠁⠀⠀⠀⠸⡝⠹⢸⡀⠀⠳⠀⠀⡄⠀⠀⠁⠴⠞⠀⠤⢹⠀⠘⠌⠀⠀⠈⢘⠀⠀⢸
⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠑⠃⠀⢡⣤⣤⡄⢰⠁⠀⠀⠀⢠⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠇⠀⠀⠀⣀
⠐⠒⠒⠒⠒⠒⠒⠀⠒⠒⠓⠒⢢⣤⣤⣤⡄⢸⠀⠀⠀⠀⠘⡇⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⡄⠀⢀⡃⠀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠘⠀⠀⠂⠀⠀⠏⠀⠀⠀⠀⠀⠋⠀⠀⠄⠀⠃⠀⠀⠳⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣵⣷⣬⣍⣉⣥⣤⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣿⣿⣿⣿⣛⡒⠒⠒⠒⢚⣟⣫⣽⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣛⡛⠛⠛⠛⡛⠿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠤⢀⣀⢀⣀⣀⣀⣀⣀⣀⠠⠤⢴⣶⣿⣽⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡻⠷⠶⠶⢶⣤⣤⡴⠶⠶⠶⠛⢛⡫⠭⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣿⣿⡏⠁⠀⠐⠒⠒⠒⠒⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⢀⣀⣀⠀⠀⠀⠀⣀⣀⠠⢤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣟⣛⣓⡀⡀⢀⡀⠤⣶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡄⢠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⢠⢶⣾⣲⢤⣄⣄⣤⣒⣐⣒⣤⣤⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠘⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠒⠚⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠠⠄⠀⠀⠀⠀⠀⠀⠈⢳⠤⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠉⠙⢶⡄⠀⢠⡀⣀⠀⠀⠀⡂⠀⠀⠀⡠⠐⠁⢀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠌⠋⡠⠀⠼⠠⠄⠁⢀⣰⣄⡀⠀⠀⢂⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠈⣷⡴⡁⢀⣀⢼⡿⠷⡈⢿⣋⢈⡄⠀⡠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣥⠀⠙⡿⠁⠀⠀⠀⣤⣸⣿⣧⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠷⣽⣶⣲⣦⣤⣶⣽⡻⠆⣸⡟⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠁⢊⠳⠒⡱⠿⣯⠻⣿⣿⣷⣔⠋⠈⡧⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠉⡠⠘⠣⠀⠉⣹⣿⠛⢳⡨⠶⠎⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡤⠂⠄⠀⠉⠙⠀⠑⠄⠀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠠⡾⠋⠡⠁⢈⢹⠆⠀⠀⠀⠀⣿⣿⢻⠃⠀⠀⠀⠀⠀⠀⡇⢻⣿
⠀⠀⢰⡦⠤⠴⢶⡆⠀⠀⡘⠀⠈⠈⠉⠙⢇⠀⠀⠀⠀⠀⠀⣩⣗⡢⢀⡝⣈⠔⠲⣄⠀⠀⣿⣿⣨⠇⠀⠀⠀⠀⠀⠀⡇⢸⣿
⠀⠀⢸⡇⠂⠺⢻⡇⠀⠀⠁⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢠⠾⠀⣠⡶⢻⡃⠸⣿⠟⠺⠄⠀⡏⠛⠸⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿
⠀⠀⠘⣇⠀⠀⣸⠁⠀⠸⣄⡀⠀⠀⣀⣀⣸⠃⠀⠀⢠⡏⡀⠚⠿⠁⠘⣿⣶⢿⣁⠌⠀⠀⡄⢸⢸⠀⠀⠀⠀⠀⠀⣠⣧⣼⣿⣄
⠀⠀⠀⠙⢷⣾⠃⠀⠀⠀⠙⠷⠶⣶⣾⡿⠋⠀⠀⠀⠸⠏⠀⠀⠀⠀⠒⠈⢳⠴⠋⠀⠀⠀⡅⢸⠠⠀⠀⠀⠀⢀⡾⠋⢉⣿⣿⡟⣳⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠲⠂⠀⢀⠒⢋⣦⡀⠀⠀⠀⠀⣠⡴⢴⡦⢔⠀⠀⠀⠀⡇⣿⠘⡀⠀⠀⠀⢸⠁⠀⣼⣿⣿⡇⣹⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠢⢾⠉⢙⠇⠀⠀⢤⣐⢃⠎⢹⡖⠁⠀⠀⣸⠷⢿⠔⢧⠀⠀⠀⢸⠀⠀⠻⠿⣿⡗⠛⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢹⡆⢱⡞⢀⠀⠀⢰⣇⠀⢠⠀⠈⣆⠀⠀⢸⠀⠀⠀⠀⣾⠀⠀⣿
⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠰⢿⣭⣷⠂⠀⠀⠀⠀⠀⠠⣤⡄⠀⠁⠀⠈⠀⠀⠀⢠⣯⣿⡆⢸⡇⠀⢹⣆⠀⢸⠀⠀⠀⠀⣙⠀⠀⣿
⠀⠀⠀⠘⠯⠿⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣠⠀⣚⣫⡽⣧⡆⡀⠀⠀⠀⠀⣸⣿⣇⣙⣿⡏⠀⠈⣿⠀⢸⠀⠀⠀⠀⢩⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⠴⣚⡿⡬⢇⣰⡼⡵⢫⠇⠀⠀⠀⣿⡿⠉⠀⢸⡏⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡄⢐⡆⠀⠠⠼⠃⠀⠀⠀⠀⢰⣟⣺⣽⡫⢞⣸⢛⣿⣉⣻⢤⣟⢷⣀⠀⠀⠀⣿⣇⠀⠀⢸⡇⠀⠀⣿⠀⢸⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢸⣩⠗⢣⣎⣸⣶⣾⣉⣻⡷⣦⣉⣗⣆⠮⣭⡶⠀⠀⣿⣿⠦⣤⣼⡇⠀⠀⣿⠀⢸⡇⠀⢺⣿⣿⡟⣿⢷
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣛⡶⠷⠳⢲⡳⢾⢭⠷⣿⣧⡶⠏⢿⣭⡮⢽⣒⡿⠿⠧⠀⣿⣿⣠⣴⣾⡷⣄⣰⣿⠀⢸⡇⠀⢸⣿⣿⣇⢿⣿
⠀⠀⠀⠀⠀⠀⠘⠓⠂⠀⠀⠀⠀⠀⠁⠀⠮⠗⠑⠚⠣⣇⣶⣷⠞⠛⠙⠻⠦⠀⠀⢿⣿⣿⣿⣼⡇⠈⢻⣿⠀⢸⣇⣀⣸⣿⣿⣿⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠓⠒⠛⠉⠀⠀⠉⠛⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣦⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣧⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⡇⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⣷⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⣠⣾⣷⣄⠀⠀⠀⠀⠀⠈⢿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡿⠋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡔⡻⢶⢀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⠓⠒⠦⢄⡀
⠀⠀⠀⠀⠳⣌⣊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣵⢤⠤⠐⠒⣻
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠫⠼⠀⢤⡤⠞⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠆⠀⢀⠸⠸⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⠀⢀⠐⠂⠀⠀⠀⠈⠀⠀⠎⠃⠂⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⣰⡆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡊⠀⣴⠀⠀⢐⢑⡶⠤⣄⣈⠀⠀⠀⠁⠀⡀⠀⢀⠀⢠⠀⠀⠀⢀⠀⡀⠅⠀⠀⠄⢀⠀⢠⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⡰⠉⣧⠀⠀⣼⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠒⠒⣶⢆⠀⠀⠈⢰⠋⠙⠒⠤⠀⣀⡀⢀⡑⠄⢀⡌⣻
⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢀⡜⠁⠀⠸⣄⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⣄⠀⠈⣆⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠁⠀⠀⠀⠘⡆⢀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠇⠈⢣⡠⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢺⡀⠀⠀⠀⠀⠙⣌⣚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠘⣦⠀⠀⠳⡌⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠚⠋

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢨⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⢀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠤⠤⠤⠤⠤⣤⠤⠤⠤⡤⠤⠤⠤⠤⡤⠤⠤⠤⠤⠤⡤⠬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠄⠀⠠⠀⣠⠕⠒⠘⠒⠤⡁⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⠀⢠⠒⠉⠉⠑⠚⠀⠀⠀⠀⠀⠀⠈⠦⠔⠒⠬⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠠⢡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤⣬⠉⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠸⡀⢸⠛⠈⠺⡇⠀⠀⠀⠀⢠⠦⠄⠤⠔⠉⠓⠒⢲⢸⠗⠁⠋⣧⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⢀⠞⢁⠆⠀⡠⠊⠀⢰⠢⠤⠐⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠳⢄⠀⠹⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡸⡀⡜⠀⡰⣳⠤⠴⠃⠀⠀⠀⠀⡇⠀⠀⠀⠘⡇⠀⠀⢸⠀⠸⡆⠀⢡⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠠⠀⠁⢸⠀⠀⡿⣹⠀⠀⠀⠈⢦⣤⠴⠃⠀⠘⣲⠀⠙⠲⠒⠃⠀⢰⢱⠀⠈⡆⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⢸⠀⠀⢇⢈⢆⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⠀⠀⡀⠀⢀⢀⠎⡟⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠄⢀⠈⠘⡆⠀⠈⢢⡈⢆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰⢋⡔⠁⠀⢰⢁⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⡄⠀⠀⠈⠒⠱⠦⢄⣀⣀⡀⢀⣀⣀⣀⠤⠶⠏⠒⠁⠀⠀⣠⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠰⠀⠂⠘⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠐⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠘⠀⠈⠀⠐⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⡇
⠀⢸⡇⠀⠀⠀⠀⠁⠠⠀⠁⠀⠀⠃⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⢸⡇⠀⠀⢀⠘⠀⠀⠀⢈⠀⡀⢀⠁⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠂⢀⠀⠀⠀⢀⡀⠈⠀⠈⠀⡀⠀⢀⢘⡇
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡒⠡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡆⢤⠰⠄⠀⠀⢠⠀⢨⠊⠀⢀
⠀⠐⣤⠀⠠⠀⠄⠀⠀⠀⠢⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠙⠒⠚⠛⢃⠐⢀
⠀⡀⠂⠀⡤⠀⠈⠀⠈⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⢰⡌⠄⠀⠠⠀⠀⠀⢄⠀⠀⠀⣁⠈⠀⠀⠀⠊⠁⠐
⠀⠀⠀⠀⠀⠄⠃⠨⠀⢂⠀⠐⠐⠀⠠⠀⠀⠀⠀⠈⠀⠁⣀⠀⣀⡤⠦⠀⠄⠂⠀⠀⠀⠂⠀⠀⠂⠤⠄⡁⠀⠀⠀⣀⠀⢀⠀⠄⠀⠀⡀
⠀⠂⠀⣠⠴⠲⣦⠀⠀⠈⠆⢰⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣽⣤⠄⢐⠀⠃⡡⠄⠀⠀⠇⡀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠁⠀⢤⡤⠀⠀⢀⣴⣄⣀⠀⠀⢀⠁⢰⣶⢶⣶⡗⠙⠛⠉⠉⠉⠈⠑⠒⣀⣸⣡⣿⣶⢷⣤⣄⠀⠎⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢄⠀⠀⢈⣿⡿⣯⡀⣤⡭⠃⠀⢀⠰⣾⣿⣿⣿⣿⣿⣀⡆⠘⠃⠿⣶⢾⠛⢹⣿⠓⢩⢌⣉⠍⣻⣦⣄⡀⡀
⠀⠀⠀⠸⡆⠄⠁⠈⠉⠁⠀⣠⠂⠰⠌⠀⢸⠿⠛⠿⠯⠿⠿⠯⠄⠀⠀⠀⠀⠠⠘⣽⠣⢹⣮⣸⡷⠆⢺⡏⠋⠁⢃⡀⠄
⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣄⠤⢤⡲⣻⣆⠀⠹⢿⡏⠹⠘⠃⠀⢀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⢀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⣢⠴⠗⠹⠷⠴⠷⠰⠄⠀⠀⠀⢠⡆⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠨⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣾⡥⣄⠀⠀⢄⠀⢀⣠⠀⠀⠉⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠁⠀⠀⢦⠀⠀⠱⠂⠀⢀⣀⣀⣨⠀⠀⠀⠀⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠱⠶⠖⠁⠀⠙⠛⠁⠀⢀⣀⠁⠊⡠⠂⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢿⣸⡇⠈⠀⠀⠋⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⠉⠛⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢀⣼⡇⠀⠀⠐⠒⠉⡄⠀⠸⣠⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠀⠸⠿⠇⠀⠀⠀⠀⠀⠸⠆⠀⠛⠀⠀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⠯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠚⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠠⠀⠁⠀⠀⠀⠡⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⡀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠚⠛⠃⠀⠀⠀⠈⠀⠀⣀⡜⠁⣀⠆⠀⠀⠀⠀⠀⠘⣄⡀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣠⠆⠘⠀⠀⠀⠀⠀⢄⡀⠀⠌⠀⣶⡄⠀⠀⠀⠀⠀⢀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⢀⣤⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠀⠘⢳⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡱⠀⠀⢀⠴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⠈⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀⠀⠀⣀⠤⠤⠤⣀⡀⠀⠀⣠⡔⣃⠙⠃⢀⢤⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⡇⢀⣤⣀⡀⠀⢱
⢀⠔⠋⠙⠲⢤⡀⢀⡐⠉⠙⠞⠀⠀⠈⠖⠉⠀⠀⠀⠀⠀⠙⢆⠀⠀⠿⠛⠁⠀⠈⢛⡀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠑⠊⠀⠀⢹⡀⢸
⡌⠀⠀⠀⠀⠀⠈⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠈⠱⡀⠀⠀⠀⢿⠀⠀⠀⢸⡇⢸
⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠰⠆⠉⠉⠳⡁⠀⠀⠀⠀⠀⠀⠀⡷⠒⠒⢄⠀⠀⠀⡀⣔⢀⢸
⠈⠢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠤⠄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⢱⠯⣷⣊⣨⣧⣤⣀⡀
⠀⠀⠈⢇⠉⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠒⠒⠒⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠙
⠀⠀⠀⢸⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠤⠄⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠘⠐⠈⠂⠒⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠽⠂⠩⢥⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⡀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠠⠀⢠⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠟⠁⠀⢀⡸⠄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀⡜⠀⠀⢀⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡜⠀⣄⣸⠆⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢡⠀⢹⣤⠃
⣶⣶⣶⣶⣶⣶⣴⣶⣶⣶⣴⣼⣿⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣿⣽⣾⣧⣶⣶⣶⣶⣴⣶⣶⣶⣶
⣿⣿⣥⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠙⣷⣶⣦⡀⣦⣀⢀⡀⠀⠀⠀⠀⠀⢼⣿⢘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣴⣦⡀⠀⠀⠀⠀⣠⠸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⡗⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢟⣵⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡌⣿⢾⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⢎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣭⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⢸⣿⣿⡟⣿⣿⣷⠀⠀⢸⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣼⣿⣿⢠⣿⣿⣿⠀⠀⢸⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠿⠟⠿⠛⠛⠛⠛⠿⠅⠛⠿⠀⠒⠈⠛⠈⠓

⠀⠀⠀⠀⢀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣶⣶⣶⣶⣶⣶⣶⡾⣻⡿⠃
⠀⢀⣴⣾⣿⢿⣿⠿⣻⠝⣻⢟⣿⠟⢋⣽⣿⡽⣿⢟⣽⣾⢿⣫⢟⣯⣿⡽⣛⠝⡫⢞⠵⣿⠟⠙⣫⣾⠏
⣠⣾⡷⠿⠖⠛⠷⠚⠕⠋⠕⠋⠀⠚⠋⠘⠁⠈⠀⠉⠁⠀⠈⠁⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⣇⠀⠀⠀⠀⠀⠆⠀⠀⠻⠀⠐⡀⠀⠉⠀⠸⠀⠰⠄⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⣿⠀⠀⠆⠀⠀⠀⡄⠀⠀⠀⠁⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠀⢸⡇
⠀⣿⠀⠀⠀⠀⠀⠀⠈⢠⠀⠃⠀⠀⠰⡆⢐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠘⣏
⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢃⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠈⠄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢹⡇⠀⠀⠀⠀⠂⠀⢰⠀⠄⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⢸⡇⢰⠀⠀⡄⢀⠀⠀⠃⠀⠀⠐⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠘⣿⠀⠀⠁⠁⠈⠀⠃⠀⠀⠀⡆⠀⠀⠐⡄⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⣿⠀⡄⠂⠁⠐⠆⢆⢰⠸⡄⠃⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿
⠀⠀⣿⡇⠀⢆⢲⠀⠀⡄⠈⠀⠁⠀⡄⠀⡀⡄⠃⢦⠀⠠⠀⠀⢻⠀⠀⠸⣤⢀⠀⢰⡄⠀⡃⠘⠈⢠⡆⠀⣿⡄
⠀⠀⢹⣧⠀⠈⠬⠁⠀⠐⠐⡶⠀⠃⠙⣆⣁⠀⠀⠀⠀⠠⠄⠨⠄⠀⠤⠀⢀⣈⣀⣀⣀⣀⣀⣀⣀⣀⢐⣂⡿⠛⢷⣄
⠀⠀⠘⣿⣦⡷⣾⣷⣦⣤⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⡿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣦⠀⠙⣧⡀
⠀⠀⠀⣿⣿⣷⣜⢿⣿⣿⣾⣟⢛⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣤⣄⠊⠻⣦⡀
⠀⠀⠀⣿⢻⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣿⣿⣿⣿⣿⣷⣿⣿⣿⣃⣀⠈⢻⣄
⠀⠀⠀⣿⠀⡿⣿⢿⡿⣦⡻⣿⣿⣍⣻⣷⣶⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣛⣛⣛⣛⣛⣿⣛⣟⣻⣛⣻⣿⣿⣿⣧⠈⠙⢷⡄
⠀⠀⢸⣿⠀⢘⣻⣯⢻⣿⣷⣜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻⣿⡿⢿⣷⡄⣹⣦⡀
⠀⠀⢸⣿⠀⣌⡛⠙⢇⢻⣮⣿⣷⡝⢿⣿⣿⣿⣭⣽⣿⣿⣿⣷⣾⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠟⠻⠛⠛⢻
⠀⠀⠈⢻⣤⣿⡉⠻⣜⣧⢻⣿⣿⡟⠀⠍⠉⣁⣀⠀⢀⣀⠀⠀⣄⡄⠀⠀⣀⣤⣤⣶⣶⣶⣶⠞⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣿⣆⡵⣜⣷⣿⣿⡇⠀⠚⠉⠁⠀⠈⠉⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⡿⣿⣿⣶⣶⠌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢻⣿⡖⡝⣾⣿⣿⣇⣴⣶⠉⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⡿⢿⣿⡙⣿⣷⡸⠿⢻⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⠇⠀⠔⢃⣠⣴⣾⣿⣿⣿⣿⣿⡿⣿⣷⢶⣙⢷⣌⣿⣷⣾⣿⣿⣿⡿⣿⣍⣾⣿⣿⣿⡿⠟⠋⠁⠚⠉⢀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⡄⢾⣍⣈⣿⣾⣿⣶⣿⣿⣷⣿⣿⣯⣼⣷⣿⣿⣿⡿⠟⢃⣤⠀⠐⠀⣞⠛⠯⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣟⣭⣶⣶⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣉⣻⣉⣝⢛⣟⢋⣿⢻⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠀⠒⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠞⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠤⠴⠖⠒⠲⡒⠉⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⢼⠉⠑⢌⠻⢆⡀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣣⣀⣨⠶⣀⡸⠀⠀⢉⣲⠖⠂⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⣟⢰⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⠤⠤⠴⠆⠀⠀⣠⣿⠘⡀⠀⢠⣗⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠥⠤⠒⠊⠈⣿⠀⣇⢤⠖⠥⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡖⢾⣻⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⢭⢈⣀⣉⠶⢢⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⡀⠀⢨⠀⢈⣱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⣰⠚⠧⡀⠸⣤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⡏⠬
⠀⠀⠀⠀⠀⠀⠀⠳⣚⣉⣽⡙⠒⣳⠸⡿⢣⡌⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠿⠓⢤⠇
⡰⠊⠉⠉⠉⠉⠉⠉⠉⠀⠈⠛⠉⠑⡖⠀⡤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⢂⠔⠀⠰⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢠⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠇⣯⣄⣠⠀⢧
⠈⠢⣄⣀⣀⣠⡤⠔⢪⠋⠀⠀⠀⡸⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⢽⠗⡸⢁⣠⠔⠊⠉⠉⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⡠⠞⠁⠀⠀⢠⠀⠀⠀⠀⠀⢠⡄⠤⢴⣒⠒⢡⡪⢊⠜⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⣀⣀⣀⡀⢸⢳⠊⠢⣀⡀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠱⣤⣄⡀⠀⠀⠑⠒⠐⠒⠒⠊⠉⠉⠙⠦⠬⠥⡀⢹⠳⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢫⣉⣉⣉⣙⣉⣍⠁⠀⠀⠀⠈⠋⠰⠢
⠀⠀⠀⠀⠀⡅⠀⠑⠂⠄⠀⠀⠀⠀⠀⢀⣉⠁⠂⠆⢠⣀⣀⣀⣀⠀⠀⣀⣠⠇⠆⣸⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠒⠢⠤⢤⠀⣀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⠘⡀⠀⠀⠀⠀⠀⠀⠈⠓⠓⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢍⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣄⡀⡰⠲⡄⠀⠀⠀⠀⠀⡆⢀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠎⠀⠀⠈⢡⠎⣀⠀⠀⠀⠀⠀⡇⠒⠒⠒⠂⠠⠬⢁⣀⣀⣀⡀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠑⡄⠰⢦⠀⠜⠀⠀⠀⠀⠀⠓⠊⠉⠉⠉⠁⠒⠒⠂⠤⠤⠀⠀⢠⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚

⠀⠀⠀⠐⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⠀⢰⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠄⠤⢤⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢰⠃⠀⠀⠈⡇⠑⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣤⣄⠐⠈⢀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠸⡄⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⢿⣿⡆⠤⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠘⢄⡐⣄⢀⢀⡠⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠠⠄⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠈⠉⠹⡏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠠⠀⠠⠄⠂⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶⠀⢰⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⠄⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⢠⡇⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠐⠠⠀⠀⠀⢀⠀⣃⣤⠀⠀⠈⠀⠀⣿⠀⢸⡇⠀⠀⠘⣇⠰⡀⡀⠀⠀⣸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠚⢿⡟⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠨⡷⣬⡿⢶⠛⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⣠⠀⠸⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢠⣄⡴⠀⢱⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣿
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠀⠘⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠟

⠀⠀⡠⠐⠈⠉⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⢀⠎⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠀⠘⠛⠛⠛⠛⠁⠀⠀⠶⢶⣤⣀
⡾⠀⠀⠀⠀⣀⠤⠀⠉⠀⠀⠀⠀⠀⡀⠀⠤⠄⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⢿⣶⣦⡄
⣇⠀⣤⡾⠋⠁⠀⢰⠁⠀⠀⠀⠀⠀⠈
⠹⣄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡠⠒⠊⠉
⠀⠙⠿⣦⡀⠰⠊⡀⠀⠀⠀⢀⣤⡆⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠘⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⣠⣤⠤⣤⠀⠀⠀⠈⠑⠲⠶⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⠛⠛⠉⠁
⣼⣿⢱⡟⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠈⠳⢤⣀⣀⡀⠀⢀⣀⣀⣀⣠⣤⡤⠤⠤⠴⠶⠒⠒⠒⠛⠛⠛⠛
⠹⣿⣌⠛⢁⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠉⠉⠁⠀⠱⣄⠀⠀⠀⠀⣀⠤⠖⠋

⠀⠀⠀⣠⣤⣠⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⢻⡿⠿⠿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⣿⣿⡇⠀⢰⡇
⠀⠀⠀⢸⣷⣶⣶⣶⣾⠃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠅⠀⠀⠀⣿⣿⣷⠀⠸⠇⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⣤⣾⠀⠀⣶⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡏
⠀⠀⠀⠀⣿⣿⣿⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⡞⠂⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡷⣿⠀⠀⡇⠀⠀⠀⠀⢸⣿⡇⡇
⠀⠀⠀⠀⣿⣿⣿⡇⡇⠀⠀⠀⠀⠀⣴⣧⠀⣿⡇⠀⠀⠀⠀⣿⣇⡀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣷⣿⢰⣾⣷⡆⠀⠀⢀⣘⣿⣧⣇⡀
⠀⠀⠀⠀⢿⣿⣿⠃⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⣿⣿⢸⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⢠⡇⠀⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⠀⢸⣿⡇⡇⢸⣿⢈⡇⠀⠀⠘⣿⠂⠀⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⠘⣿⣿⣿⣟⡀
⠀⠀⠀⠀⠈⣿⣿⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠
⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⣿⣿⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⣀⣤⣄⣀
⠀⠀⠀⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢹⣶⣾⣿⣿⣿⣿⣷⣶⣶
⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢀⡾⠋⠀⠀⠙⢷⡄⠀⠀⠀⠀⣀⣙⣛⣛⣛⣛⣛⣋⣠⠀⠀⠀⠈⠻⠟⠁⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⣾⠁⠀⠀⠀⠀⠘⣷
⠀⠀⠀⣿⣿⢶⣶⣶⠀⠀⣿⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠉⠉⠉⠉⠋
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⢹⡄⠀⠀⠀⢀⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠻⠲⠒⠓⠓⠚
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠙⢶⣤⣾⡷⠏⠀⠀⢀⣤⣤⣤⡄
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣼⣧⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠾⠿⢿⣿⣿⣿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⣿⣿⢚⣛⣾⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⠀⣾⠉⠙⠻⣿⣿⠟⠋⠉⣷
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠘⡆⠀⠀⠀⢠⣾⣿⣿⣧⡀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠦⣄⣰⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠘⢷⣤⣀⣀⣀⣀⣴⡾⠃
⠀⠀⠀⠿⠿⠛⠛⠛⠁⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠙⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⣀⣠⣄⣀
⠀⠀⠀⣰⠋⠀⠀⠀⠈⠑⡄
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⡴⠒⠚⠒⠲⡄
⠀⠀⢸⡇⠀⠀⠀⠼⠥⠤⠤⠤⠤⠼⠄
⠀⠀⢸⡇
⠀⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠋⠀⠀⡀⠉⠉⠉⠒⠒⢦⠞⠉⠉⠙⢦⣀⣀⠀⣀⣀
⠀⠀⢸⡇⠀⠀⠀⣠⠖⠒⠲⠴⣒⠃⠀⢠⡦⢀⡈⠋⠀⠘⠃⠀⠀⠶⠆⠀⠰⠆⠀⡀⠈⠋⠁⠈⠙⢦
⠀⠀⢸⡇⢀⡤⠚⠁⠰⠇⠀⢀⣍⠀⠀⢀⡀⠈⢁⡀⠠⡦⠀⠀⢠⡄⠀⢀⣄⠀⠈⠛⣀⠀⢤⡄⠀⢉⡉⠓⣄
⠀⠀⢸⡇⡏⠀⠀⠀⠀⠠⡦⠀⠁⢤⠄⠘⠁⣤⠈⠉⠀⠀⠴⠆⠈⠀⠶⠀⠁⠠⡦⠀⠛⠀⠀⠰⠆⠈⠁⠀⠚⠒⠲⣄
⠀⣠⠚⠓⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⢢
⠀⠙⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠜⠁
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⠤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⡤⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋
⡖⠒⢲⠶⡶⠶⠶⢶⠶⡶⠶⢶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⠶⢶⣶⣶⢶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣞⣷⠦⣀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢦⡈⠉⠳⢬⡙⢾⣅⢦⣀⠀⠀⠁⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⢄⠈⠛⠯⡙⠣⣝⢳⣄⠀⠀⠀⠔⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠹⢦⣍⠻⣌⢷⡀⠀⢀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠛⠉⠙⠛⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⣦⣄⠈⠙⢦⣀⠙⠳⠈⠀⢷⡈⢩⢟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⢠⡆⠀⠀⠀⠹⣷⡀⠀⠀⠈⠀⠀⠀⠀⠀⠘⡄⠘⠈⠷⣄⠀⠁⠀⠀⠀⠀⠘⢷⡉⢀⣿
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣸⣿⠃⠀⢠⠿⣦⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⠀⠀⠀⠀⠈⢂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠈⣧⠉⠯
⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⣿⡇⠀⢰⣃⠜⠁⠀⠀⠀⠀⢸⣇⠀⠀⠀⠉⠛⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣶⡧
⠀⠀⠀⠀⠀⠀⠌⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⡇⢀⡀⠈⢁⡟⠹⠀⠀⠀⠀⠀⠀⠀⠈⣳⣄⡀⠀⠀⠀⠀⠀⠀⣿⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⢀⠀⠀⠀⠀⠀⠀⢸⣿⣴⠋⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⡷⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠰⠄⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣵⣿⣿⠀⠀⠀⠀⢀⠏⢠⠓⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠙⠏⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢿⡇⠀⠀⢠⡾⠋⣽⠎⠀⠴⢣⣾⣿⣿⠇⡀⠀⠐⠁⠀⠠⣂⣾⣿⢿⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠐⣄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⢿⣥⠾⣇⠀⢐⣵⣮⣿⣿⣁⠀⠀⠀⠀⢀⠔⠀⠹⡎⠁⢹⣷⡀
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠹⡪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⣠⢒⣀⣭⣶⠿⠚⣧⠙⠂⠀⠀⠂⠀⠀⠀⠀⠀⢀⣼⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⣻⠖⠥⠶⣣⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣐⠈⠒⠉⠙⣇⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃


⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⠤⠤⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⣀⣀⣀⣀⡐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⡊⠀⠠⠄⠤⢾⣷⣶⣒⡂⠄⠙⣆⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠀⠀⠀⣀⣀⡼⠀⠀⠀⠀⠀⠀⡀⠀⠰⢮⣝⣀⣐⢧⡽⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠙⢄⠂⠠⠤⠤⠭⠥⠤⠤⢤⡶⠃⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣶⠚⠋⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⣤⣬⣭⣭⣥⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡛⠛⢺⣧⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡂⠠⠬⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢯⡑⣲⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠢⠍⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡶⠬⢭⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⣁⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠳⠉⠹⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡂⠄⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⣍⣓⢲⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠬⢁⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡑⠈⠽⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⠂⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡁⢲⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢁⣐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠙⠤⣬⡷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣑⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡲⠤⡴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃



⣿⣿⣿⠿⣿⡿⢿⢿⡿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣦⣶⣷⣶⣿⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠀⠀⠀⠠⠤⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣾⡿⡫⣄
⠀⠀⠰⣌⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⡉⠀⠀⠘⠿⠞⠁
⠀⠀⠀⠈⠉⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⢩⡿⠁⠒⠲⠀⠈
⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣦
⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡤⠀⠀⠀⠀⠀⡐⠺⠦⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠉⠀⠀⠀⠀⠀⢠⣴⡾⢏⣀⣀⡀⠠⠤⠀⠐⠠⢀⠀⠀⢀⢁⣨⡋⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠿⣷⣶⣾⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣻⣿⣿⣿⣩⣿⣛⠀⠂⠀⠀⠄⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣾⣿⡿⠉⠉⠙⠛⠛⠛⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⠿⢇⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡂⠈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠁⠀⠀⢾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣸⣁⠀⠀⠸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣠⡀
⠀⠀⠀⠰⠒⠒⠲⠆⠐⠒⠎⠛⠉⠛⠚⠃⠈⠉⠉⠉⠙⠉⠙⠓⠂⠀⠀⠀⠉⠛⠉⠁⠉⠉⠉⠉⠉⠀⠈⠛⠒⠒⠒⠢⠤⠤


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⣿⡿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣿⣷⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠹⣾⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣵⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢮⣛⠿⢿⣿⣿⣿⣯⣭⡵⠶⢛⣥⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣤⣄⠀⢤⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣉⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠶⠒⠋⠐⠒⠂⣨⡿⠷⠖⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠒⠶⠶⠶⠶⠶⠖⠛⠛⠛⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⣲⣿⣿⠀⠁⠈⠉⠀⠀⠀⠀⠀⠂
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⡟⠀⢄
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢞⠟⠁⠀⠐⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠
⢠⠀⠁⠀⠀⡀⠀⠀⠀⣤⡒⠂⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⢁⡄⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠊⠀⠀⠈⢂⠀⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⣠⣤⣴⣤⣄⣴⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢳⠀⠁⠁⠀⠀⠀⢀⠀⠀⠀⠈⠉⠙⠛⢿⣿⣿⣷⣶
⠓⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠁⢀⠀⠄⠂⠀⠀⢹⡿⠃⠙⠉
⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠀⢄⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠘⡁⠀⠂⠀⡐⢤⣄⠀⠀⠀⠀⠀⠀⢻⣿⣇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⠀⠈⠉⣺⣿⣿⣠⣿⣶⣾⠆⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣻⣿⠛⠛⠃⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠀⣰⠁⠀⠶
⠀⠁⢰⡀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠤⠀⣲⠞⠛⠻⣇⠐⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⢠⠋⠁⠊⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠤⠶⢦⣤⡄
⠀⠁⠀⠀⡀⠀⠀⠀⠠⠤⣀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠉⢘⡄⠀⢀⠀⠀⠀⣠
⠀⠀⠀⠆⠠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹⣄⣸⣦⣶⣾⠋
⠒⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣿⣦⠀⠀⠋⠀⠀⣠⣖⠀⠀⠀⠀⠀⠀⢈⣿⣿⡛⠉
⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⢄⡀⠀⠸⣿⣷⣤⡆⢠⣾⡿⠋⠄⠀⠀⠀⠐⠀⠈⠁⠀⠙⠦⣀⡒
⠁⠀⠀⠀⠀⠀⠀⢠⠀⠠⠁⠀⠀⠀⠀⠀⢀⣀⣀⣤⣌⣙⠇⠀⡸⣿⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠔⠒⠊⠩⠛⠛⠛⡿⡿⢉⣤⢚⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⢰⠄⠀⠀⢠⠄⠤⠤⠤⠤⢼⠰⡰⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠁⠙⠁⠙⢸⣄⡀⣀⡀⠰⡆⡶⠀⢲⢸⢠⣢⢸⡄⠀⠀⠀⠀⣀⠀⠀⠆⢠⣀⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠤⠁⠐⠲⠀⠲⢸⠁⠄⠀⠀⠘⠃⠛⠀⠛⢸⢠⢤⢸⠀⠖⠰⠂⠖⢸⠀⠀⡁⢰⠀⡆⢀⠀⢤⠀⡀
⠀⠀⠀⠀⡲⠒⠒⠒⠒⠚⠒⢶⠀⠀⠒⠂⠒⢸⠀⢾⡶⣄⠈⠁⠙⠀⠉⢸⢀⣉⢸⠐⡰⠀⠒⢀⡙⢀⠘⡃⠰⠀⠆⢀⡀⠉⠀⢉⡍⡇
⠀⠀⠀⠀⠂⢰⠀⢰⠀⠐⠀⠸⠐⢃⣀⢀⣀⣘⠃⠏⠀⠂⠚⠐⠇⢸⠀⢸⠈⠊⢸⠄⢀⠀⡄⠀⠀⠀⠀⡍⢩⠉⠍⠩⠉⡁⢀⡘⠇⣇
⠀⠀⠀⠀⠀⠘⠁⠘⠁⠘⠁⢰⠀⢸⣙⢋⣏⣸⠀⠀⠺⠲⠷⠐⠿⠸⡶⠾⠶⠶⣾⠀⣬⠆⠈⠉⠁⠉⠁⠁⡶⢶⢶⠶⠶⠋⢈⣍⣈⣉⣉⡄⠉
⠀⠀⠀⠀⠀⠀⠅⠅⠀⡭⠀⢐⠀⢸⢤⢬⡥⢬⠀⠀⠲⠲⠶⠶⠶⢈⡿⠀⡇⢸⣿⠀⠦⠆⠐⠘⠂⠚⠂⡀⠗⠶⠶⠶⠶⠀⠀⡥⢬⠥⣤⡅⠀⡃
⠀⠀⢰⠿⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠷⠿⠿⠿⠿⠟⠯⠯⠷⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠽⠿⠽⣷
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠘⢤⡀⡠⢺⠢⡈⣀⠘⡓⢄⢀⡄⡟⠠⡈⡠⢺⠃⢄⣀⠔⡗⢄⡈⡠⢺⠢⣀⣠⠜⡗⢄⢁⠔⢻⠢⡀⡠⠚⠳⣄⢁⠔⡛⢄⠀⡠⢺⠢⡀⣡⠞
⠀⠀⠀⠀⠙⠀⠘⠀⠈⠁⠀⡇⠀⠉⠀⡅⠀⠈⠀⢸⠀⠀⠁⠀⡇⠀⠉⠀⢸⠀⠈⠀⠀⡇⠀⠁⠀⢸⠀⠈⠀⠀⠀⠈⠁⠀⡇⠀⠉⠀⠸⠀⠘⠁
⠀⠀⠀⠀⢰⠀⢰⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢠
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⣅⠀⠀⠀⢸⡀⠀⠀⠀⣇⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⠀⠀⣸⠀⠀⠀⢀⡀⠀⠀⠀⣇⠀⠀⠀⢸⡀⢸
⠀⠀⢀⣀⣸⠔⣡⡙⢦⡴⢋⣌⠢⡠⠊⣤⡑⢄⠔⣡⣝⠢⡤⢊⢌⠳⢤⠔⡠⡐⢄⠴⢂⢌⠢⡠⠊⣤⡑⢄⠔⣡⣌⠢⡤⢊⣌⠳⣤⠞⣡⡑⢼⣀⣀
⠀⠀⢸⠤⠤⠊⣀⡈⠢⠖⠁⠀⠑⠴⠋⣀⠉⠢⠚⠁⠈⠣⠔⢁⣀⠑⠤⠊⠀⠈⠢⠔⢁⡀⠑⠴⠋⠀⠙⠢⠚⢁⡈⠳⠔⠁⠀⠑⠤⠊⣀⡈⠢⠤⢼
⠀⠀⢸⠀⠀⠀⠧⠝⠀⠀⠀⠀⠀⠀⠈⠤⠇⠀⠀⠀⠀⠀⠀⠠⠤⠂⠀⠀⠀⠀⠀⠀⠣⠼⠀⠀⠀⠀⠀⠀⠀⠣⠌⠀⠀⠀⠀⠀⠀⠘⠦⠇⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠈⡉⠀⠀⠀⣀⠀⠀⠀⢉⡁⠀⠀⠀⡀⠀⠀⠀⢁⠁⠀⠀⠀⡀⠀⠀⠀⣉⠁⠀⠀⢀⡀⠀⠀⠈⡉⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⢸⠣⣄⠞⠁⠑⢄⡴⠋⠈⠢⣠⠞⠈⠑⢄⠔⠁⠙⠦⡠⠊⠈⠳⣤⠔⠁⠑⢄⡴⠋⠈⠢⣠⠞⠁⠑⢄⠔⠋⠙⠦⡠⠊⠈⠳⣄⠔⠁⠑⢤⡴⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠙⠊⠓⠁⠀⠀⠀⠀⠀⠑⠉⠊⠀⠀⠀⠀⠀⠈⠚⠙⠊⠀⠀⠀⠀⠀⠈⠚⠙⠂⠀⠀⠀⠀⠀⠘⠊⠑⠁⠀⠀⠀⠀⠀⠑⠋⠚⠁⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⢀⡠⣀⡄⠀⠀⠀⠀⠀⢄⡠⣀⠀⠀⠀⠀⠀⠀⣀⡄⣀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⢸⠳⣄⠔⣡⡑⢄⡠⢊⣍⠢⣤⠞⣩⡑⢄⠔⣩⡙⠦⡤⢊⣍⠳⣤⠔⣡⡑⢤⡴⢋⣍⠢⡠⠚⣩⡑⢄⠔⣩⡙⠦⡤⢊⣍⠲⣤⠔⣡⡑⢄⡴⢻
⠀⠀⢸⠑⠦⠊⠀⠈⠢⠔⠁⠀⠑⠤⠊⠀⠙⠦⠞⠁⠈⠳⠔⠁⠀⠑⠤⠊⠀⠈⠢⠔⠁⠈⠑⠴⠋⠀⠑⠢⠚⠁⠈⠢⠔⠁⠀⠑⠤⠊⠀⠈⠢⠞⢹
⠀⠀⢸⠀⠀⠀⣖⡢⠀⠀⠀⠀⠀⠀⢰⣒⡀⠀⠀⠀⠀⠀⠀⢰⣲⠀⠀⠀⠀⠀⠀⠀⢐⣲⠀⠀⠀⠀⠀⠀⠀⣒⣦⠀⠀⠀⠀⠀⠀⠀⣖⡆⠀⠀⢸
⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⢸⣹⣩⣹⣉⣹⣉⣹⣉⣩⣉⣉⣉⣏⣉⣏⣉⣏⣍⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣏⣍⣏⣉⣏⣉⣏⣉⣉⣩⣉⣹⣉⣹⣉⣹⣹⣹⣹

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⣄⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠖⠲⢶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⡏⢉⣶⣿⣿⣿⣿⣵⣶⣮⡝⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⣻⣿⣿⣾⣿⠁⠀⠁⠀⣛⡛⠻⣿⣆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠆⢀⣾⡿⣿⣿⣿⣿⣧⣿⣧⣾⡿⢿⣦⡹⣿⣼⡷
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⣻⣥⣶⣿⠏⠀⣹⡿⠋⠉⠉⠙⠛⢿⣤⣌⣹⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣶⡿⠛⣽⣿⣿⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⣿⡟⢿⣧
⠀⠀⠀⠀⢀⣤⠾⣿⣿⠀⣼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣻⣿⣧⠀⠙⡇
⠀⠀⠀⣴⣿⠀⣠⣶⣿⢰⣿⣿⠁⠀⠀⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⠈⣇⢹⣿⣿⣦⣷⣷
⠀⠀⢸⣿⡁⣠⠶⢿⣧⠞⣽⠃⠀⠻⡿⢻⣷⣍⠃⠀⠀⠀⠀⣀⣀⣀⣈⣿⣯⠻⣿⣿⣿
⠀⠀⢸⡿⠸⣿⢲⣿⢁⣾⣿⠀⠀⠀⠘⠒⠛⠁⠀⠀⠀⠀⠀⠩⢿⣷⣄⡈⢻⣷⡌⠙⣿
⠀⠀⢸⣷⣄⢿⣼⣿⣼⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠿⠿⠛⠃⣸⡟⠹⣆⢸
⠀⢠⡿⠛⣿⣿⠟⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⠚⢻⣿
⠀⠸⣷⠀⣹⢻⣤⣬⣿⣦⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣍⠉⢳⣿⣿⡇
⠀⠀⢿⣷⡉⠺⣭⣽⡟⢹⣷⠀⠀⠀⠀⠒⠀⢤⣄⠀⢄⠀⠀⠀⠀⠀⠀⢨⣟⡂⠐⣿⣿⡇
⠀⠀⠀⠈⠙⠛⠻⣿⠇⢸⣿⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡉⢙⣿⡆⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⡏⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⢿⣆⢈⣿⠇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠁⠀⠀⠀⢀⣠⣤⣶⣿⣟⣋⣵⣾⠏⣸⣯⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⡇⠈⠛⠻⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄
⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠃⠐⠀⠄⢀⣀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠉⠑⠂⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠈⠑⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢱⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢠⡞⠀⠀⠀⠀⠀⠀⠹⣆⠀⣠⠇⠀⠀⠀⠀⠀⠀⠸⡄⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⣫⣍⡀⠀⠀⠀⠀⠀⠀⣴⠞⣛⣥⡀⠀⠀⠀⠀⠀⠀⣀⣩⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡉⠑⠚⢿⠀⠀⠀⠀⠀⠀⣯⠉⠉⠙⡏⠀⠀⠀⠀⠀⠀⣻⠳⠾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠒⠶⢾⠀⠀⠀⠀⠀⠀⡏⠉⠓⠲⡇⠀⠀⠀⠀⠀⠀⣿⠲⠦⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢦⣄⣀⣸⠀⠀⠀⠀⠀⠀⡿⠢⣄⣀⣯⠀⠀⠀⠀⠀⠀⣿⣄⡀⠀⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⢻⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⡿⠀⠀⠀⠀⠀⠀⣿⠀⠉⠒⢿
⠀⠀⠀⠀⠀⢀⡴⠢⣤⣀⠀⠀⠀⣀⣀⣀⣿⣦⣄⣀⣸⣄⣤⣤⣤⣄⠀⡿⢢⣤⣤⠷⠤⣤⣀⠀⠀⣀⣿⣄⡀⠀⣸⠀⣀⣀⣤⣀⡀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠉⢳⣴⠛⠉⠉⠉⠉⠑⣦⡝⠋⠁⠀⠀⠀⠈⢻⣧⡞⠉⠀⠀⠀⠀⠉⢧⡞⠉⠀⠈⠙⢿⣿⡞⠉⠁⠀⠈⠙⣦⠞⠉⠉⠉⣷
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠙⣷⠤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⠷⣄⣀⣀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠉⠀⠉⠛⠲⠤⠴⠚⠋⠉⠓⠤⠤⢄⣤⠤⠞⠋⠙⢦⣀⣀⣀⣀⡤⠞⠻⢤⣀⣀⣠⠞⠉⠓⠦⠤⠴⠋⠀⠈⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢀⣤⣄⡀⠀⡇⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡼⠋⠀⠀⠉⠛⢧⣤⠞⠁⠀⠀⠀⠀⠈⠻⣄⣠⠶⠛⠋⠉⠉⠉⠛⢶⡄⢀⡴⠚⠋⠁⠀⠈⠉⠻⣆⣠⠶⠊⠙⠋⠑⠶⣄⣠⠶⠚⠛⠛⠛⠛⢶⣜⡶⠞⠋⠙⠶
⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠟
⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⢀⣼
⠀⠀⣟⠉⠉⠓⠚⠋⠙⠶⢤⣤⣤⣤⣤⡤⠖⠲⢦⣤⡀⠀⠀⠀⣀⣤⡴⠖⢶⣀⠀⠀⠀⠀⠀⣀⡤⠾⣄⡀⠀⠀⠀⣠⠞⠳⣤⣄⣀⣠⣴⠏⠀⠙⠒⠒⠚⠉⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠑⠒⠶⠖⠊⠉⠀⠀⠈⠉⠓⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢠⡞⠛⠛⠳⢦⡀⠀⢀⣀⠤⠤⠦⠤⣤⡀⠀⢀⣀⠴⠶⠶⠖⠶⠦⣄⠀⠀⣀⡴⠆⠛⠛⠓⠶⢦⡀⠀⣠⡤⠤⠤⢄⡀⠀⠀⢀⡤⠶⠶⠶⠦⣄⠀⢀⡤⠦⠤⣮
⣯⠀⠀⠀⠀⠀⠙⣶⠋⠁⠀⠀⠀⠀⠀⠙⠷⠛⠀⠀⠀⠀⠀⠀⠀⠈⢣⡼⠋⠀⠀⠀⠀⠀⠀⠀⢹⡾⠁⠀⠀⠀⠀⠙⣦⡼⠋⠀⠀⠀⠀⠀⠘⣷⠋⠀⠀⠀⠈
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠
⠀⠈⠛⠲⠶⠤⠤⠶⢦⣄⡀⠀⠀⠀⠀⢀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⢀⡜⠛⠶⣤⣤⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠉⠛⠓⠶⠶⠶⠞⠛⠁⠀⠈⠳⠆⣀⣠⣠⡴⠞⠋⠈⠛⠶⠦⠤⠶⠛⠁⠀⠙⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⣦⢠⠴⠴⢤⠈⠢⠤⠤⠤⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢸⠱⠶⠆⠁⠀⠀⢸⡂⣀⣛⢸⣀⣀⣺⠀⠀⠀⢀⠶⡀⠀⠤⢼⣍⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⠀⣚⡊⢉⠊⠙⣻⠤⠤⣾⢸⠤⠤⢼⠀⠀⠀⠘⣦⠃⠘⠒⠻⢈⢸⠁⣀⣞⠃⠜⠓⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡖⣶⢶⢶⡇⠠⡅⠀⠀⠜⠀⠋⠁⠈⢡⠌⢹⡂⠂⣿⢸⣒⣂⣺⠒⣶⣶⠒⣿⣀⣨⣋⣹⢈⢸⣤⣬⣤⣦⣤⣭⣤
⠀⠀⠀⠀⠀⠘⠀⠘⠀⠦⠶⠶⠇⢩⢉⠉⡍⠉⠘⠃⠀⠃⢠⠸⠇⣸⠄⠤⠿⠸⢾⠹⠍⠏⠹⠉⡣⠟⣘⣛⣋⢚⡈⢸⠠⠤⠀⠦⠠⠦⠈
⠀⠀⠀⠀⢰⠀⣀⣈⣈⣁⡁⠘⡧⠤⡤⠤⣤⠄⠘⠂⠀⠃⠀⢸⡃⢸⣖⠒⡛⢛⣿⠶⠶⠶⠶⠆⠀⠀⣍⣍⣩⣻⡁⢸⠀⠒⠀⠗⠐⠒⢠
⠀⠀⠀⠀⢸⠀⢡⣥⣉⣭⣇⠀⡷⠶⠶⠶⠶⠆⠐⠂⠀⠓⠀⠘⠁⢘⣿⠀⡇⢸⣿⠶⠶⠖⠶⠶⡁⠀⣬⣌⣡⣽⡀⢸⠀⢀⡁⠀⣏⠀⢘
⠀⠀⠀⣼⠯⠿⠽⠭⠯⠿⠽⠭⠯⠯⠽⠽⠯⠯⠽⠹⠭⠯⠿⠽⠭⠯⠯⠽⠭⠯⠿⠽⠽⠯⠯⠽⠽⠯⠯⠭⠽⠭⠯⠿⠽⠽⠯⠯⠽⠭⠏⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠒⢄⡠⠊⠑⢌⣡⠞⠑⢄⡠⠞⠑⠤⣉⠔⠙⠢⣀⠴⠋⠢⣉⡴⠈⠢⢄⡠⠊⠳⢌⡡⠊⠑⢄⡠⠊⠑⢎⣁⠔⠙⢦⣀⠔⠉⠢⣉⠴


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡤⠤⠤⠤⠤⢤⠀⠰⣉⣑⠄⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣈⣡⠀⣀⡏⣻⣿⣏⣁⣀⢀⢀⣄⠘⠛⠛⡓⢠⡀⣀⠔⢤⡀⠀⣎⣀⣀⣉⡄⠀⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠁⢀⡀⠁⠀⠧⡤⠄⠤⠄⠤⠿⠀⠤⡤⠠⠄⠤⠤⢤⠾⣥⢤⣤⠼⢶⠉⠠⡄⠈⣁⡴⠛⠦⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣄⡀⢠⣁⣀⣀⣘⡀⠀⣀⣏⣛⣛⣋⣀⣀⢀⣀⣅⣈⣃⣀⣁⣈⣳⠛⢛⡛⠛⣞⣒⣒⣒⣒⣛⠖⠒⠒⢻⡀⠣⠀⠃
⠀⠀⠀⠀⡤⠮⠈⠤⠡⣾⢠⠄⢠⠄⣶⡐⢣⡄⣀⠀⡀⠠⠟⢘⡛⠘⠃⢫⠴⢄⠋⡸⠋⠀⠀⠙⢮⠀⠤⠴⠀⡅⢠⣷⣄⢸⣧⡔⢣⡤⢤
⠀⠀⠀⠀⠣⠄⡀⢀⡄⢿⢸⢆⡰⡂⣽⠿⠭⡀⢻⢈⡃⠰⠄⢨⠉⡚⢶⠇⠤⢤⡷⠆⠀⠀⠀⠀⠼⠀⡂⢴⠀⠷⠤⠀⠆⢦⡷⠈⠁⠰⡼
⠀⠀⠀⠀⠀⠀⡥⠈⡅⠚⢨⠄⢠⡅⢻⣿⣼⠃⠶⠰⠆⠰⠖⢊⠀⡇⣨⠈⡯⢠⡄⠘⠛⠛⠛⠫⠒⠀⠆⠲⠀⠀⡿⠿⢻⢠⠄⡯⢈
⠀⠀⠀⣰⠿⢿⢹⠼⠧⡿⠼⠥⡯⠯⠿⠭⠿⠷⢶⠰⠶⠿⢷⠹⠭⡿⠯⠽⠽⠯⠿⠷⠰⡦⠶⠶⢹⠷⠦⠶⠼⠯⡯⠿⢽⠼⠯⠿⠿⢹⠯⢾⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠢⢄⡠⠊⠳⢌⣡⠔⠑⢄⣀⠔⠑⢄⣉⠔⠑⢦⣀⠴⠊⠢⣉⡠⠈⠢⣀⡠⠊⠣⣈⡡⠖⠑⢄⣠⠔⠑⢌⣁⠔⠑⢤⣀⠔⠉⠢⣉⠰⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀⠀⠀⠀⠀⠀⠠⠄⠤⢤⠀⠀⢀⡴⠒⢢⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠠⠀⢄⠑⠒⠁⡀⠀⠀⠀⠀⡄⠀⡆⢸⠀⠀⠯⠀⠄⢠⠔⠀⠀⠀⠀⠀⠖⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠒⠚⠐⠒⠀⠖⢸⠀⠀⢈⠀⡄⢠⡄⢤⠀⠀⡄⢸⣀⣀⣀⣀⣀⡀⠀⠐⠂⣀⣀⠚⠂⢤⢈
⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠭⠀⠰⠴⠀⠖⢘⢠⡁⠈⠀⡃⢘⠃⠛⠀⠀⡅⢸⠰⠆⠴⠒⠀⠀⠀⠀⠠⠅⢡⠀⡁⢭⠐⡀
⠀⠀⠀⠀⢰⠤⡤⠤⠤⠝⡢⢶⠀⠰⠤⠀⠦⠸⠢⣴⢠⠀⠁⠈⢁⠉⠀⠀⡅⢸⠰⠆⠀⠊⡉⠉⢈⢃⡐⠂⠚⢀⣀⠉⠠⠁⣌⠁
⠀⠀⠀⠀⢸⠐⠖⠀⠆⠰⠄⢸⠐⣃⣐⣀⣀⡛⠒⠁⠚⠐⠃⠟⢸⠀⠀⠀⠃⢸⠄⠄⠰⠀⡁⠀⣈⠈⠉⠥⠉⠸⠉⡇⢐⠀⠚⡀
⠀⠀⠀⠀⢸⠈⠃⡈⠃⡐⠂⢸⠀⣭⣭⣭⣭⡅⠀⠐⠛⠛⠛⠛⢻⣋⠙⡛⢋⣿⠀⣼⠀⠈⡁⠈⠉⠀⠿⠿⠯⠷⠕⡏⢉⣭⣭⣭⣭⡉⢹
⠀⠀⠀⢀⣸⣀⢀⣃⣀⣃⢀⣘⡀⣢⣰⣄⣤⣂⣀⣘⠚⠛⢓⠓⣻⣾⣀⣇⣸⣟⣀⣐⣀⣈⣁⣈⢉⣀⡒⡓⢚⠛⣒⣇⣀⣆⣖⣰⣴⣀⣨⣀
⠀⠀⠀⠽⠯⠿⠽⠽⠯⠿⠹⠭⠯⠿⠿⠹⠯⠯⠿⠽⠭⠏⠿⠽⠭⠯⠯⠽⠭⠏⠿⠿⠽⠯⠿⠿⠽⠏⠯⠿⠽⠯⠯⠿⠽⠽⠯⠿⠿⠽⠯⠿⠅
⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⢀⡀⠀⠁⠀⡀⠀⠀⠀⣀⠀⠉⠀⣀⠀⠀⠀⢀⠀⠈⠁⢀⡀⢀⠀⢀⡀⠈⠁⢀⡀⠀⠀⠀⣀⠀⠁⠀⣀
⠀⠀⠀⠙⢦⠔⠁⠈⢢⡔⠉⠉⠢⡤⠋⠈⠳⡤⠚⠉⠓⠤⠊⠈⠑⢤⠜⠁⠙⢦⠔⠉⠁⢠⠔⠁⠙⠢⡴⠊⠈⠣⡤⠊⠈⠲⡤⠊⠈⠑⢤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⠉⠁⢀⣀⡀⠀⠠⠍⢁⠐⠄⠀⠀⠀⠀⠀⠉⠉⠁⠐⠘⠤⢌⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⡤⠒⠉⠉⠀⠀⣀⣀⡀⠀⠈⡀⠀⠀⠀⠀⠀⠠⢄⡀⠁⠐⠠⡀⠁⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⠎⠐⠁⠀⠀⠀⠀⠀⠂⢀⡒⠨⠑⢄⠎⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⢀⠨⡢⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠃⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠛⠷⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣕⢕⢌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠀⠀⡴⠀⢀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⢴⣄⠀⠀⠀⠀⠀⠀⠀⠈⢫⡣⢢⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡰⠀⠀⠀⢸⠇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⢄⠀⠀⠀⠀⠀⠀⠑⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡼⠁⠀⡄⠀⣾⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠂⠀⡇⠀⡏⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠊⠀⠀⠀⠸⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠦⢄⠀⠀⠙⢦⣀⡀⠀⠀⠹⡆⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⡇⢁⢀⠀⠀⢀⡠⠶⠦⡄⠀⠀⠀⠀⠀⠀⣠⠴⠾⠿⣟⠋⠀⠀⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⢁⠸⠌⡀⠺⣿⢧⡾⠗⠺⡆⠀⠀⠀⠀⠈⠁⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⣎⠀⠹⡼⡄⢿⣿⡄⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡿⠀⡀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢦⠀⢳⡟⡞⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠡⠇⢠⠇⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣸⡇⢹⡘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢠⠏⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢸⡇⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣰⠋⠀⢸⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡻⠁⠀⢸⠁⠀⠘⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣰⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡜⠀⠀⠀⡀⠀⠀⡀⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠁⠀⠀⢰⠇⠀⠀⡇⣿⣿⣿⣿⡄⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢸⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⡞⡆⠀⠀⣷⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇⠀⠀⢹⣇⢄⢀⠀⣇⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⠀⢃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣶⣿⣿⣿⡿⢧⠀⠀⠈⠫⠪⠓⢹⡏⠀⣧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣷⣈⣧⣀⣤⣠⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣀⣸⣦⣀⣠⣤⣤⣶⣋⣀⣼⣇
⣿⣻⣛⣟⣿⣋⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣭⣿⣿⣽⣯⣽⣽⣿⣿⣿⣿