Tipple Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢄⠐⠒⠠⠀⠀⢒⢁⣀⢀⠐⠈⠀⠀⠀⢀⡀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡀⠈⠀⠨⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠠⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠒⠁⠀⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⢠⡀⢀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⣀⢀⠁⠤⠀⠀⠒⢀⣤⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣄⣄⣀⣀⣀⣠⣄⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⠛⠋⠙⠉⣉⠙⠛⡉⠈⠓⠸⡓⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠍⡌⠋⢽⣿⢁⡹⠉⢆⠹⣍⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠐⠀⠀⢀⢀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣏⠠⡄⠄⠠⠀⠖⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⢘⠀⠀⠂⠀⠐⠂⠠⢦⡜⠠⠂⠀⠀⠐⠀⠄⠀⢡⣿⡅⠀⠀⣠⢦⣍⣢⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡⠄⠡⠁⠀⠠⠄⠀⠠⠀⠀⠰⠀⠀⠀⠀⠈⠠⢐⣮⡆⠁⠠⠀⠀⠂⠄⠀⠈⣽⡻⠧⠀⣨⣷⢿⡟⠻⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⢈⠀⢀⠀⠀⠀⠐⢸⠃⠓⠐⠀⠀⠀⡀⢀⣑⣾⣫⡄⠀⣊⣿⣧⣈⡠⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢐⠀⡂⣐⣀⡀⢁⠂⠐⣊⢐⢀⠉⠀⠀⠀⠈⠀⠙⠈⠀⠀⠀⠀⠀⠒⠀⣠⣿⡿⢝⠀⠢⣼⠟⢻⣦⣽⣄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠶⢴⡦⢴⡤⠤⠴⣤⠤⣄⣡⣀⣀⣁⣀⣀⣤⣀⣀⣄⣰⣄⣠⣔⣤⣤⡿⠳⡟⠛⢟⠛⣿⠚⠛⡿⠛⢋⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠛⢀⡰⠷⢀⣠⡌⠀⠀⡋⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢀⠖⡀⢮⣥⣽⣢⣤⣹⣟⣷⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠀⣠⠀⠀⢨⡄⢀⠖⠈⠉⠺⣿⣿⣿⣿⡟⣿⣿⡉⣿⣿⣿⣿⣿⣿⡗⢦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠁⠈⠻⠋⠲⠀⠀⢛⡀⠀⠀⠀⣿⣿⣿⣿⣿⡎⠙⠃⠛⢡⡟⣿⣿⣿⡇⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⣠⠜⠃⠙⢠⣤⡞⠁⠴⠐⠺⣿⣿⣿⣤⣴⡗⡄⠲⣀⡿⢿⣿⣿⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⣤⠀⠀⠀⠀⡿⠃⠐⢧⣀⠀⣿⣿⣿⣿⣿⣟⡃⣠⢃⡔⠚⣿⣿⣿⡇⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠠⠅⣸⠷⢠⡄⠾⠀⠀⠀⠸⠿⢀⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⡇⠈⡶⡿⢿⣿⣿⢿⣿⣿⠿⣏⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠀⠰⠄⠀⠛⠒⠶⠇⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠀⠓⠈⠫⠗⠘⠶⠽⠉⠡⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠛⠀⠀⠀⠈⠛⠳⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣡⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠙⢷⡄⠀⠈⠀⣀⣈⣠⡶⣆⠰⠆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠱⠾⠃⠼⠟⢉⡠⡄⢀⡸⠆⠀⠀⠀⠙⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢨⠀⠘⠃⠀⣈⢠⡄⠀⢀⡀⠀⠐⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠀⠀⠀⠀⠉⠓⠀⠀⠞⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⣶⣿⣿⣿⣿⠟⠛⠛⠻⣿⡏⠉⠙⡿⠛⠉⠉⠉⠻⣿⣿⣿⡿⠛⠋⠛⠻⣿⡿⠟⠻⣿⡿⠋⠉⠙⢿⣿⡶⣤⡀
⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⠘⣧⣤⣴⡇⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⢸⡄⠀⠀⢸⠁⠀⠀⠀⠀⣿⣀⣈⣿⣄
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⣠⡿⠿⠿⠷⣄⡀⠀⣀⠴⠟⠿⠿⣄⡀⠀⠀⢀⡾⠟⠓⠚⠿⢧⣀⣀⣀⡴⠛⠉⠙⠛⣿⣦
⠀⠀⠀⢠⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠶⢶⣶⣿⠃⠀⠀⠀⠀⠈⣿⡟⠁⠀⠀⠀⠀⠀⠙⢿⡿⠃⠀⠀⠀⠀⠀⠀⠉⢿⡏⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⣠⣾⠁⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⡾⠋⠉⠻⣶⣶⡶⠶⠿⢿⣿⣄⠀⠀⢀⣴⣿⡿⠶⠦⣤⣶⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣤⣤⣤⠞⠛⠛⢻⡇
⠀⠀⠀⣷⡀⠀⢀⣾⠁⠀⠀⠀⠀⠘⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⠀⣠⣼⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⠛⠉⠉⠙⢻⡏⠀⠀⠀⠀⣷
⠀⠀⢰⣿⣿⣿⣿⣯⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡶⠖⠛⠿⣿⣿⣿⠿⠛⠛⠛⠒⢾⣿⡇⠀⠀⠀⠀⠀⣿⣶⣤⣴⣾⣿
⠀⠀⢸⠏⠀⠀⠀⠙⣦⣀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢀⣠⡟⠁⠀⠀⠀⢻⡄
⠀⠀⣿⠀⠀⠀⠀⠀⣹⠟⠛⠛⠻⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢸⡇
⠀⠀⣿⣷⣤⣀⣠⣴⠃⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠉⠉⠛⠛⠿⣧⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠉⠉⠉⠙⢳⣶⡴⠶⣿⡇
⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣸⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⡀⠀⣀⣤⣾⠃⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣼⡇
⠀⢸⣿⠟⠁⠀⠀⠈⠻⣦⣄⣀⣤⠶⠛⠛⠙⠛⠻⣶⣶⣦⣶⣶⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠛⠉⠉⠙⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⢿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⢿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⣹⣧⡀⠀⠀⠀⠀⢀⣴⣿⣄⣀⣼⡇
⠀⠈⣧⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠉⠛⢿⣷⣤⣀⣀⣀⣠⣴⣿⣿⡄⠀⠀⠀⠀⡰⠛⠛⠛⠛⢶⣶⣾⠟⠁⠀⠀⠙⣿⠃
⠀⠀⣿⣷⡦⠤⣤⣴⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠹⣿⡿⠛⠉⠀⠀⠀⠀⠉⠳⢶⣶⡟⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⣼
⠀⠀⢹⡏⠀⠀⠀⣻⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣦⣤⣤⣴⡏
⠀⠀⠈⣷⣄⣀⣴⠿⠛⠛⠛⠛⢦⣄⡀⠀⠀⠀⣀⣴⡾⠟⠛⠉⠓⠲⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠈⠙⣿⠁
⠀⠀⠀⠹⣿⣿⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡶⠤⢤⣴⣾⣿⡄⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠻⣇⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠛⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠈⠻⣿⠿⠶⣶⣾⡟
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⣿⡿⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⠀⠀⠀⠀⠀⣧⡀⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠶⢶⣿⣿⣦⣀⣀⣀⣼⣿⣦⣀⣀⣀⣀⣤⣾⣯⡀⢀⣽⣶⣤⣄⣀⣀⣤⣴⣾⣿⣿⣷⣀⠀⠀⠀⣀⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠙⠛⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠁⠀⠀⠀⠍⠀⢤⡀⠸⢋⠋⠁⠀⠀⠀⠀⠀⠀⠀⠁⠉⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⢵⣖⣇⠈⠀⠁⠠⢾⣤⣬⢡⣞⣽⣿⣦⠀⣠⠀⠂⠀⠀⠀⠁⣀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣠⢤⣤⡀⠀⠀⠀⣖⣶⢀⡘⢿⣿⣿⠏⠰⠅⠾⠂⠀⣾⠆⠺⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⠇⣽⡇⡀⠉⣁⣮⡇⢀⡀⢠⣬⠻⠔⠀⠀⠀⠀⠄⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⠉⠁⡰⠿⠀⠒⠚⠛⢥⠀⣿⡿⠀⠀⠀⡔⢀⠀⠀⠀⠀⠐⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢰⣿⣿⡆⠀⢀⠀⠀⠤⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠙⣛⣥⡌⢀⠂⠀⠀⠀⠀⢐⠲⣤⣖⡀⢀⡐⡠⠜⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠈⢑⠂⠀⠻⣧⠅⣘⠅⢀⢀⡀⠠⠤⣤⠟⣈⣁⠛⣧⣼⠆⣠⡄⣀⠂⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠁⠀⠀⠣⠄⣼⣶⠠⠂⠠⠘⠀⠀⣂⢀⣿⣭⣿⣿⡈⢽⣿⠎⠿⠛⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣀⣠⣴⡢⡀⠀⢀⡀⠀⠀⠀⠀⢾⢠⣶⣦⡀⣾⣤⠀⢀⣿⣆⠻⢿⡿⠟⣠⡴⡖⠈⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣃⠀⢸⣾⠻⠟⣛⠏⠙⣧⣧⣔⡉⠮⠀⢀⠐⢨⣤⡘⠿⠿⢣⣾⣿⠆⠙⡓⠻⠳⣶⣠⡿⣿⡷⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣶⣾⡇⣴⣿⣿⣿⡆⢸⣧⠉⣍⠀⠀⠀⣣⣾⠟⠛⠷⠞⢛⣤⡻⡶⠀⠀⠀⠄⠰⠙⠱⢹⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠤⠼⣿⣇⠻⣿⣿⣿⠇⠀⣩⣤⡡⢔⣆⣰⡿⣱⣟⣿⣷⡄⢘⠛⠱⢳⣊⣶⠪⠴⠅⡀⢀⠄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢣⠔⢸⣷⣇⠈⠉⣤⣿⣧⡙⠟⢃⡀⠋⠻⣧⠹⣿⣿⣿⠇⣼⣿⡃⡰⠞⠀⠀⠀⠀⠈⢆⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠈⠐⢏⢧⢰⣿⡦⢻⣿⡟⡡⢴⠋⠉⠙⡔⢻⣷⣬⣭⣥⣾⣿⡳⠂⠘⠣⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢸⡊⠃⢉⣠⢴⡛⡥⣜⣸⢄⢀⣀⠮⠅⠘⠻⣿⣿⡟⠤⢇⠀⠀⢰⣦⢆⠤⠰⡊⠎⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡥⠔⣯⠽⡈⣣⠴⣅⡈⣏⡗⠎⢺⠬⢭⡒⣄⠀⣠⣷⣙⡾⢿⣣⠼⠃⠁⠈⢁⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠔⣿⣆⠻⠿⣼⣿⣬⢧⡄⠀⠀⠀⢹⢭⣩⡢⢻⣷⣜⠿⠣⣵⣿⣿⣟⠂⢤⡳⣼⡌⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⢺⠟⣛⣁⡚⠟⢿⡿⢎⣹⣄⠀⣀⣼⣵⡚⣡⣬⠋⠛⢁⣀⣉⠺⠿⢣⣶⣷⡴⠾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡄⢼⣿⣿⣿⡆⡦⣿⡿⠑⢺⠬⢿⣽⣿⣆⠻⠟⠘⠠⣿⣿⣿⡇⢠⣜⠿⠿⠣⠔⠁⠀⠀⠀⠀⠀⠀⢤⠄⢀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⡿⠿⠿⠿⠿⠿⡿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⢿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⡿⠿⠿⠿⠿⠿⠿⡿⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿
⣿⡇⠀⠀⠀⠀⠀⢰⣇⣖⣸⣇⠀⡵⠒⠠⠘⡀⢀⠈⢠⠀⢀⡞⠀⠘⠀⠀⠃⡀⠀⠇⠡⣀⡐⠆⣀⡏⢁⣤⠼⣷⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⢰⡏⠉⡷⠉⣩⠄⡀⠔⠀⠈⠇⠀⢀⡂⠀⢑⠐⢶⠀⢉⡆⠈⡠⠲⡔⠁⢢⡠⢇⣼⠉⣧⣀⣻⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⢸⠷⢾⡟⢒⡡⣀⠖⠈⡖⢰⠀⠁⠂⠈⡇⠳⡤⠀⠀⠀⠀⠁⠉⠒⢏⣻⣍⣷⢚⣯⣹⣅⣼⣧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠃⠀⠀⠀⠀⠀⢸⣤⣼⠗⣾⣅⡴⠈⠂⠉⡛⠑⣎⢙⡿⠳⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣿⣷⣯⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⢸⣇⣽⡖⣿⠤⡟⢲⠭⣇⣙⣋⣯⣹⡟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡄⠀⠀⠀⠀⠀⠀⣯⣹⡟⣳⠴⣏⡹⠒⣷⣮⣽⢶⣿⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠈⣯⣿⣹⣞⣧⣾⠶⣿⣷⣟⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣽⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣍⡷⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠙⣯⡀⣸⠶⡿⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⢾⡭⠿⢬⠿⢽⠋⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡁⠀⣳⠚⢣⢤⡜⢲⡾⠳⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣛⡏⣹⣏⠉⣷⠛⢺⣷⢺⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⣸⣇⣀⣿⣿⠶⠞⠛⣫⣿⣿⣶⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⣯⣿⣏⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣛⣧⡟⠶⠓⣾⡽⣿⣷⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣇⣨⣿⣏⢙⡟⣺⣿⢷⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⣾⣧⡼⣿⣟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⡽⢜⣹⣿⣿⣿⣻⡏⢱⢏⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣻⡟⢞⣢⠘⡶⢻⣿⣿⣿⢻⣇⠀⡕⠣⡜⢺⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡿⡿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣾⡓⢃⡈⠏⢙⣉⢿⣿⣿⣿⣿⣟⡞⣲⢸⣭⣾⣿⣿⣿⢻⣛⣏⠥⡓⢢⣼⢽⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣉⢆⣀⡄⢥⠘⢎⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⣢⠓⠃⠰⠂⡩⣩⣏⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⠤⡀⢀⠦⡔⠙⣄⡕⢣⢜⣂⢃⡷⠟⢺⢛⠉⠱⠚⠄⠹⣗⣁⠔⢢⢺⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡶⣏⠀⢡⠑⠲⡁⠨⠚⢤⠅⡠⠊⢊⢡⠅⢐⠄⠃⢠⣇⡼⢥⢷⣽⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣶⣶⣾⣾⣶⣿⣿⣶⣿⣶⣷⣿⣶⣷⣶⣾⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿
⣿⡟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠐⠽⣗⡶⣦⡖⠀⠠⡄⢰⠆⠐⠲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣲⠂⠘⡟⠉⢫⣹⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⢰⠤⠮⣀⣼⣷⣤⡔⠁⢺⡄⠀⢀⣿⣿⣿⣿⣟⣿⠟⠻⣟⡿⠵⡞⢻⣞⡻⣿⢿⣽⣿⣿⣿⣿⡟⠁⠉⢶⣴⠋⢲⣎⣻⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⡸⠤⡏⠉⠱⢤⠚⠛⠻⠃⠀⢘⢿⣿⣿⣿⣷⣒⡽⢄⣀⣽⣉⣾⣿⣉⣿⣿⣿⣿⣿⣿⠛⢲⠐⡟⠉⡛⠁⠀⢙⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⢠⢷⡌⢄⣀⡷⣴⠁⠺⡀⠀⣠⠃⡤⢧⣤⣿⣿⣿⣿⣠⣷⣾⣿⣷⣿⣿⣿⡿⠿⠛⠛⠋⠉⣹⣿⣿⣿⣿⣿⣿⣶⣶⣿⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⢸⣝⠃⠆⣀⡏⢹⣤⣤⠉⡉⠀⠈⣀⣀⠋⠉⢻⠿⢿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⢀⣼⣿⠿⡽⢿⠿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⢨⣼⠳⣦⡁⣘⠉⠁⠈⣋⢉⠖⠚⡿⠛⣳⠔⠻⡤⣼⡽⢿⣅⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣉⡏⢳⠛⠲⡟⢻⣿⣿⡇⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⢸⣩⡾⢤⠟⢿⡷⠤⢴⠁⠉⠄⣴⠟⠶⡟⢠⡴⠻⣯⠀⢀⣥⠼⠖⠀⠀⠀⠀⣠⣾⣿⡿⣷⢏⡨⠋⢲⠉⢽⣹⣻⣿⡇⠀⠀⠀⠀⠈⣿
⣿⡟⠀⠀⠀⠀⣿⡀⢀⣿⣇⡽⣦⡴⠃⠘⠤⠾⡀⣠⣟⠉⣧⠤⠛⠉⠁⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠓⡔⠈⢲⠋⠉⣆⣹⣥⣿⣇⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⣧⠿⣅⣭⣀⢟⣉⣧⡀⢀⠦⣠⣞⡳⠞⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⡾⣗⢙⡐⠊⢐⠈⢹⠤⣖⣿⣿⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⢠⢤⣾⣿⡅⢀⣿⡜⢷⠉⣷⡳⠊⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣽⣇⡴⢠⠊⠑⣝⡤⣯⢿⣿⡀⠀⠀⠀⠀⣿
⣿⣷⠀⠀⠀⠀⢰⣤⡇⠀⠈⣯⣀⣷⣛⣿⡷⠁⠀⠀⠀⠀⠀⢀⣤⣾⣿⢛⣭⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⡜⠂⣖⠉⡷⢾⣺⣿⡇⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢸⣀⡽⠒⠺⣤⡟⠛⣥⣻⠁⠀⠀⠀⢀⣴⣾⣿⣿⣙⣖⣧⣬⣷⣿⣿⣿⣟⣻⣻⣿⣿⣿⣿⣿⣽⣾⣀⡏⢷⣺⣽⣿⡇⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢸⣿⡀⠀⠀⡸⠗⣴⠧⡏⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣻⠛⣷⢻⣿⣿⣿⣿⣿⣼⣳⣿⣩⣷⣿⣿⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢘⠁⠙⣶⢺⣧⣤⠶⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠽⣛⠿⣿⣿⣿⣿⣿⣿⣟⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⢸⡷⣎⠀⣻⣀⣼⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡀⠀⣏⠀⠈⡟⣿⣿⣿⣿⣯⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣯⣀⣸⡋⢙⣿⣇⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠳⡆⠀⠙⡤⠃⠪⡀⡡⢜⢻⣿⣿⣿⣿⣶⣿⣿⣻⣽⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⣿⣿⣤⣿⣉⣹⡉⣛⣹⣇⡀⢀⡄⠐⠻⡀⠀⠀⡙⠊⠁⢲⠊⠃⣀⢠⡟⣧⣠⣅⠀⠻⣿⣿⣿⣧⣿⠯⣷⢛⠾⣿⣿⣿⡇⠀⠀⠀⣿
⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣀⡀⠦⢤⠔⠣⠇⠈⣡⣀⣀⠜⠁⠺⡀⠀⣻⠉⠙⡟⠙⠳⡜⠻⣿⣶⣿⣿⡎⠉⣶⢻⣿⣿⡇⠀⠀⠀⢿
⣿⣿⠀⠀⠀⢸⣯⡷⣾⠼⡡⠻⡴⣯⢷⣿⣏⠀⠀⢈⠷⠾⡷⠚⠛⢯⠤⡀⠀⠀⣈⠈⡹⠢⢔⠓⡤⢮⣑⠚⠛⢿⣿⣿⣯⣾⣉⣋⣿⣿⡇⠀⠀⠀⢸
⣿⣿⠀⠀⠀⢸⣟⣏⡵⠤⡗⠐⡤⠚⡞⣳⣿⣆⠐⢇⠀⣀⢆⠀⠀⡠⣀⣸⣑⡋⠀⠈⡀⠀⠀⡯⠦⢠⠿⡄⢀⡜⣿⣿⣷⣿⡯⣿⣷⣿⡇⠀⠀⠀⢸
⣿⣿⠀⠀⠀⢸⣿⣯⣷⢤⠃⡴⢣⠤⡿⣧⢿⣿⣷⡋⠁⢹⠊⠉⠹⡀⣸⣿⠋⢱⠤⢴⠋⠓⣾⡄⠀⠀⡖⠣⠠⡏⠈⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸
⣿⣿⡄⠀⠀⢸⣿⣿⣿⣾⣳⣟⣞⣿⣻⣿⣿⣿⣿⣿⣶⠃⠄⠀⠰⠟⠟⠘⠓⠛⠀⠘⠂⠀⠻⠿⠿⠿⠈⠇⠀⠸⠉⣹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸
⣿⣿⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢸
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣾
⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⡶⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠠⢼⢱⣯⠔⠁
⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠀⠀⠀⠀⢸⡿⠀⠀⠄⠀⠀⠛⠀⠀⠀⠀⠀⢀⡠⠤⠀⠀⠐
⠀⠀⠀⠸⢿⡧⠀⠐⠁⠑⠄⠀⠀⠀⡁⠀⠀⠀⡄⢨⢡⠀⠊⠀⠀⡀⠀⠀⠀⠠⠐⠆
⠀⠀⠀⠀⠀⠈⠓⡄⠠⠀⠀⠀⠀⠐⠉⠀⢰⣦⠀⢙⠀⢠⣴⣶⠈⢀⣠⠄⠀⡁⣶⠀⠀⠀⣴⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡐⣾⢠⣾⣾⣇⠀⠀⠀⠃⠀⠈⠀⣨⠃⠀⡰⢉⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠵⢦⠈⠻⣄⠙⢦⠀⢠⢀⠀⣴⣤⠀⠀⠐⣡⡾⢋⡈⠀⣀⠀⠔⠁
⠔⠀⠂⠦⠠⢒⡀⠀⠀⠀⣀⠂⠈⠓⠤⣵⠈⠇⢀⠀⠁⢀⣀⠒⡭⠐⠛⠛⠀⠀⠀⡀⠤⠤⠴⠂
⠀⠀⠀⠀⠀⠛⠿⠗⠢⠤⠿⡆⠠⣀⠈⠀⠀⠛⠞⠓⠘⢛⣛⣁⠠⢸⠧⠀⠀⠴⠿⣦⠀⣀⡀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⢤⣄⡀⣴⣤⡀⣸⣦⣚⣿⡞⠛⢠⡤⢤⣌⠁⠀⠀⠀⠙⠭⣍⣻⣷
⠀⠀⠀⢀⠀⠀⠀⠘⠏⠀⠀⠀⠘⠛⠛⠈⠁⠠⠂⡉⠻⢟⡁⠀⢀⡙⠶⠏⠀⠀⣤⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠶⠒⢤⡤⢆⣀⢀⠔⠁⠀⠸⣿⡄⠈⠟⠈⠒
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⠀⠀⣐⠍⠁⠀⢲⡖⠀⣤⡙⠁⠀⠐⡀⠀⠡⠀⠀⠿⠖
⠀⠀⠀⠀⣴⠖⣚⡷⠺⠟⠁⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠛⠹⠁⠀⠈⣷⡄⠄⠈⠐⠠⠄
⠀⠀⠐⠀⠙⠛⠋⠀⠀⠀⠀⠀⣠⢠⡴⢚⡗⣀⡀⠀⠈⠀⠛⢢⡤⠀⠿⠟⠀⠀⠰⡦⠀⠀⠀⠀⠸⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⣦⡌⠻⠟⣫⠜⠋⠐⠆⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠱⢾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠁⠀⠀⠀⠀⢈⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠃⢠⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣫⣭⣤⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⣿⣿⣿⣻⣟⣛⣛⣛⣛⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣼⣯⣤⣤⣭⣬⣭⣭⣭⣭⣥⣤⣥⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣛⣛⣛⣛⣛⠻⢿⡿⠟⣛⣛⣛⣛⣛⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢟⣭⣭⣭⣘⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣝⠻⠿⠿⢛⣥⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⣛⣛⣛⣋⣘⣛⣛⣛⣛⣛⣛⣛⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⡗⣿⠖⣿⠆⡆⢰⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠁⠀⠁⠀⠙⠈⠉

⠀⠀⡀⠠⣀⡠⠀⢠⣄⠀⣀⠀⡀⠄⠐⠒⠐⠒⠤⣀⡄⠒⠐⠒⠤⣀⠠⠀⠠⢄⠀⣀⣀⣀⠀⠀⠠⢀⠀⢀⠠⠤⣀⣀⠠⠀⠀⢄⡀⡀
⠀⢀⠔⠈⠉⠈⠙⢧⠀⠀⠈⢛⠤⠴⠤⣄⠀⠀⠀⠀⠙⣄⠀⠀⠀⣈⣆⡀⢀⠈⠁⠀⠀⠀⠉⠣⡀⢀⠛⠁⠀⠀⠀⠈⢣⡀⣠⠊⠉
⠀⠀⠀⠀⣀⣀⣀⣀⣇⣠⠊⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⢸⡤⠒⠉⠀⠀⠈⠙⠢⡀⠀⠀⢀⡤⠊⠉⠉⠉⠓⢦⡴⠊⠁⠀⠀⠉⠓⢆⢀⡀
⠀⠀⠒⠛⠳⢏⠁⠀⠀⠀⠙⢦⠀⠀⡠⠐⠊⠉⠛⢦⣐⣾⣁⣀⠀⠀⠀⠀⠀⠀⢹⡤⠢⠊⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⡨⠞⠓
⠀⠀⠀⠀⠀⠀⢱⢀⡀⠀⠀⠀⣧⣴⣀⣀⠀⢀⠔⠉⠁⠀⠀⠀⠉⠲⢤⣦⠔⠉⠉⠉⠑⢄⠀⢀⣀⡀⣠⠖⠉⠉⠀⠀⠉⠑⢦⡔
⠀⢀⠀⠀⢀⣀⣸⣿⣿⣯⠔⠉⠀⠀⠀⠈⠙⢧⡀⠀⠀⠀⠀⠀⠀⢀⠠⠟⠒⠒⠤⣀⡠⠚⠉⠉⠛⢿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠒⠲⢎⠀⠀⠀⠈⠑⢇⠀⠀⠀⠀⠀⠀⠀⣀⣧⣀⣀⠠⠀⢀⡔⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⣀⠷⠒⠒⠢⠤⣀⡀⠀⣀⡀⣷⣾⡂
⠀⠀⠀⠈⢣⠀⠀⠀⠀⠘⣆⣀⣤⣤⢲⠖⠉⠀⠀⠀⠈⣻⠿⠞⠒⠒⢄⠀⠀⠀⠀⠀⠀⣹⠀⠀⠌⠀⠀⠀⠀⠀⠀⢈⠗⠉⠀⠀⠀⠀⠙
⠀⠀⠀⠀⢸⠇⡠⣀⠔⠋⠁⠀⠀⠈⠙⢄⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⣆⡤⠒⠉⠉⠉⠑⠺⢄⠀⠀⠀⠀⠀⡰⠁
⠀⠀⡠⠔⠊⠉⠉⠓⠦⣀⠀⠀⠀⢀⣀⣈⣧⣠⣴⣸⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡠⠔⠓⠒⠒⠶⢤⡇
⠀⠎⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⠞⠉⠀⠀⠈⠉⠙⢿⣆⠀⠀⠀⠀⠀⢀⣸⣀⣀⣠⣀⣰⡶⠃⠀⠀⠀⠀⠀⠀⠀⠈⠢⡖⠀⠀⠒⢤⣤⣄
⢸⠀⠀⠀⠀⢀⠔⠉⠁⠀⠉⠃⢄⠀⠀⠀⠀⠀⠀⠈⢻⣿⡶⠶⠖⡶⠋⠉⠀⠀⢀⣤⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢳⠋
⠈⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣀⣀⣀⠀⢸⡏⠀⠀⡘⠀⠀⠀⠀⠀⠄⣉⠽⠖⠒⠂⠤⠄⢀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢈⡀
⠀⡱⠂⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠜⠋⠁⠀⠀⠈⠉⠓⢄⡀⠁⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⡰⠉⠑⡄⠀⢀⡼⠷
⢠⠁⠀⠀⠈⢳⣄⡀⠀⠀⠀⢀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⡟⠀⠀⠀⠘⡖⠉⠀⠾
⠀⢣⣀⠀⢀⠏⠀⠉⠓⠒⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢦⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⢠⡥⠒⠂
⠀⢂⠈⠉⠹⡀⠀⠀⠀⠀⠀⢀⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⠛⠚⠻⢾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⡴⠃
⠀⠈⠢⠄⣀⠵⠦⣀⣀⣀⡠⣞⣘⣉⠦⣀⠀⠀⠀⠀⣀⡤⠊⠀⠀⠀⠀⠀⠀⠉⠷⠦⣀⡀⠀⠀⣀⣀⠤⠚⠳⠦⠴⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⣀⢀⣀⡀⢀⢀⣀⣀⢀⣤⡀⣀⠀⣀
⠀⠀⠀⠑⠎⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⡤⠀⠀⠀⠀⠀⠀⠐⠄⡔⠀⠀⠀⠀⠀⠐⠀⠀⣰⣾⡿⠿⣛⠷⣿⢿⣿⡏⢛⣧⡭⣄⡿⡇
⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⢀⣼⡟⣱⣿⣿⣿⣮⢛⣛⢱⣿⣿⣷⣿⣷⡁
⠀⠀⠀⠋⠒⠕⠀⠠⠶⠂⠠⢦⣄⠀⠀⠠⡈⢉⠀⠀⠀⠀⠀⠀⠀⠾⡀⠀⠀⠀⠀⠀⠀⢀⢊⣹⣇⣿⡛⢿⣿⡿⢙⣟⡹⣟⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠊⠉⠺⠱⠐⠀⠐⠊⠀⠠⣌⠤⣀⡀⢀⡀⡄⠓⣿⢿⣿⡷⢩⣿⣏⣼⣿⣿⣿⡎⢛⡛⢻⣥⡅
⠀⠀⠀⠀⠀⠁⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠐⠠⠀⠀⠀⠀⢉⠈⠁⠁⣰⠊⠳⢟⠱⠛⢽⠟⠹⢭⣽⡿⠃⣟⣿⢘⠺⡃
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡊⠀⠀⠢⠓⠀⠀⠀⡀⠀⠀⠈⠁⢀⣝⣖⣆⣖⣰⣾⡶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠀⢀⠀⠀⠀⠃⠀⠀⠀⠀⠸⠀⠀⡀⠀⠀⠀⠨⡀⠐⠀⠊⢹⠆⠀⠀⠀⠀⠂⠀⡘⢎⣱⣽⣿⣿⡟⠛⣇⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠀⠀⠀⠀⡸⠀⠈⠀⠀⣠⣴⣴⣾⣿⣿⠋⠀⠙⢗⠔⠋⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠊⠀⠀⠂⠀⠀⡤⠠⢀⣿⢧⣮⣶⣿⡛⡿⠉⠀⠈⣷⠀⠀⠀⣸⣄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢠⣿⠟⠛⡴⠛⠉⠙⣦⢤⠞⠓⠳⢖⡾⢧⠼⠓⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠿⣻⠞⠓⢾⣁⠀⠀⢀⣼⠁⠀⠀⠀⠈⢳⡏⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠧⠀⢀⠀⠨⠰⡇⠀⠀⢀⡟⠓⣶⠋⠹⣄⠀⠀⠀⠀⣼⣱⣄⣠⠔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡽⠒⠺⢧⠟⠛⠱⡤⡴⠿⠧⡤⢲⡞⠉⠀⠀⠈⢳⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠠⠲⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢝⡇⠀⠀⠈⣦⠀⠀⣿⠀⠀⠀⠘⣯⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢀⣠⣴⣶⡟⣉⢪⡁⠂⠀⠈⢧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡽⠶⣴⢾⠝⠋⠉⠺⣄⡀⠀⣠⣷⡀⠀⠀⠀⠀⢠⡇
⠀⢀⣴⣾⠿⡛⡉⣂⡇⠥⢄⠁⠀⠈⠄⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣸⠏⠀⠀⠀⠀⠘⡯⠋⠁⠈⢻⡶⠤⣤⢶⡋⡂
⠀⣸⢟⡻⢿⠿⣁⡰⣤⢀⣀⠀⠀⠠⠀⠀⠂⠡⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡼⠛⠻⡄⠀⠀⠀⠀⢰⢧⠀⠀⠀⣸⣇⣀⡽⠋⠻⡅
⠀⢘⣿⣿⣷⣿⣿⣿⢝⠟⣭⣷⣼⢟⣴⢖⣠⣬⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠀⠀⠀⢹⣳⠦⠴⣾⠋⠉⠙⣿⠋⠀⠀⠈⢳⣄⣴⡃
⠀⢘⣌⢋⣽⣷⣇⣖⣯⣥⡻⠿⢋⣴⣶⣌⢡⣾⣿⡟⢟⡿⣆⢰⠂⠐⠲⠂⢤⡄⣿⣦⣀⣠⡾⡁⠀⠀⣼⢤⡤⠴⣇⠀⠀⠀⠀⠀⡧⣀⡇
⠀⢨⣷⠼⣿⠿⠻⢾⣿⣿⡷⣹⡘⣿⣿⡟⠮⣻⠿⡿⣎⣷⣝⠕⠀⠀⠁⠀⠀⢚⡂⠀⢩⡎⠙⠟⠛⢿⡧⠴⢧⣀⣽⣦⡀⠀⢀⣰⡋⠀⢱
⠀⢈⣰⣿⣿⣷⡼⢟⣭⣿⣶⣽⢿⣶⠍⣴⣿⣿⣮⠼⣿⣿⠹⣾⡮⠶⢀⠀⠀⡘⡋⣘⠋⠀⠙⡀⠀⢀⣥⠠⠧⣼⠚⠛⡉⠛⠛⠲⡷⠦⢾
⠀⢰⢿⣿⣿⣿⣿⣸⣿⣿⣿⣿⡷⣿⡟⣿⣿⣿⣿⣢⣦⣀⣿⡿⠻⣿⣖⣆⡄⢘⠉⠉⢰⠤⣔⠉⢙⣭⠁⠀⠀⠘⡦⢖⡁⠀⠀⠀⣉⠀⡸
⠀⢸⣠⣝⢟⣿⣦⣜⢿⣿⣿⡿⣳⣻⣿⣯⣹⣿⣽⣿⡿⣿⣶⢳⣿⣿⠿⣷⣾⣭⡋⢸⡁⠀⠀⡖⠈⠉⠁⣄⠠⡆⠀⠀⠐⣦⢠⠆⠉⠉⠳
⠀⢰⡻⠿⡸⣿⣿⣿⢱⡎⣽⣦⠛⡃⢿⣿⡿⠻⣿⣿⣷⡟⡛⣵⡭⠡⣾⣿⣿⣿⠋⠈⠑⠒⠾⢇⠀⠀⢀⡅⠀⠀⡦⠄⢴⠉⠘⣆⣀⣀⢼
⠀⠀⠉⠀⠚⠈⠛⠓⠒⠒⠊⠑⠁⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠋⠁⠛⠉⠙⠛⠛⠀⠀⠘⠛⠓⠚⠛⠛⠛⠛⠂⠘⠋⠉⠉⠁⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡇⠀⠀⢸⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⠶⣦⣄⡀⠀⠀⢀⣠⡴⠶⠛⠛⠛⠶⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠁⠀⢀⣤⣀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡶⢶⡄⠀⠀⠀⠀⠀⢿⣅⣽⠂⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣄⣀⣈⡛⠛⠁⠀⠀⠀⠀⠀⠀⠉⢀⣀⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠘⠛⠛⠋⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡾⢷⡄⠀⠀⠀⠀⠀⠀⠀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠃⠀⠀⣠⣄⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⡾⠻⣦⠀⠀⠀⠀⠀⣾⡉⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠈⠷⠶⠟⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠻⠟⠛⠛⢿⠻⣛⡿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠶⠶⠶⠶⣶⡄⠀⢰⡿⠿⠿⠿⠿⢿⣿⠿⠿⣷⠀⣶⡶⠶⠶⠶⠶⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣼⡇⠀⢸⣗⠀⠀⠀⠀⣼⡏⠀⢀⣿⠀⣿⣧⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢹⣿⠀⢸⡿⠛⠛⠛⠛⠛⠛⠛⠻⣿⠀⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⡇⢠⡄⠀⠛⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⠀⠛⠃⠀⠛⣿⡇⢸⡟⠀⠙⠃⠀⠀⣀⠀⠀⣿⠀⢸⣇⠀⠀⣀⡀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠗⠀⠾⠇⠀⠿⣿⡇⢸⡿⠀⠀⠀⠀⠀⠉⠀⠀⣿⠀⠈⣿⡀⠀⠉⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⣀⣀⣠⣾⠟⠀⢸⡗⠀⠘⠃⠀⠀⠀⠀⠀⣿⠀⠀⠸⣷⡀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡟⠉⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠶⠀⠀⣿⠀⠀⠀⠘⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠟⢿⡿⠛⢿⡿⠻⠟⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⠻⠛⠛⠛⠛⠻⠻⣿⠟⠻⣿⠟⠛⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣧⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⢿⠛⡿⢻⡿⢻⡿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣤⣤⣤⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣏⣉⡀⠈⣿⣧⣾⣏⠀⣽⣟⣙⣿⡆⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠿⣿⡀⠀⠀⢹⣿⣭⡤⠠⠿⠷⠾⠟⠀⠿⠏⢿⡍⠀⢀⣴⠟⣿⡆⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣩⡤⠀⠹⣷⠶⠓⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠛⠉⢻⣿⠁⠀⠙⣿⣦⡀
⠀⠀⠀⠀⢀⣤⣬⠉⠻⣿⣴⠟⠀⠀⠀⠀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠀⠀⠙⠀⢠⣾⠟⠉⠻⢂⡀
⠀⠀⠀⣰⠟⠉⠁⣴⠆⠈⠁⠀⠀⠀⠀⠀⠘⡇⠀⠉⠑⢦⡀⠁⠈⠀⡠⠚⠉⠀⢰⠃⠀⠀⠀⠀⠀⠉⠀⠀⣠⣼⡿⣆
⠀⠀⠀⢿⣦⣤⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠑⢰⡆⠊⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⠛⣧⠘⠇
⠀⠀⢠⡌⠉⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠘⠦⣀⣤⡤⠖⣾⡷⠤⣤⣄⣀⡴⠃⠀⠀⠀⠀⢀⡄⠀⠀⠀⠘⠓⠀⠀⢠⡄
⠀⠀⢸⡇⡇⠀⠀⠀⠀⠰⣶⣿⣷⡶⠀⠀⠀⠀⢰⠋⠀⠀⢀⠸⠡⡀⠀⠀⠙⡆⠀⠀⠀⠀⢶⣾⣿⣶⠆⠀⠀⠀⠀⠸⢸⡇
⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠼⠛⠻⠄⠀⠀⠀⠀⠸⣄⠀⢀⡼⠀⠀⣇⠀⠀⣠⠇⠀⠀⠀⠀⠠⠟⠛⠧⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣾⡛⢻⠆⠛⣿⠛⢿⣧⠛⣿⠛⢿⣧⠀⣻⡇⠘⣿⣦⡀⣿⡇⣰⡿⠛⠻⡿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠻⢷⣦⠀⣿⠷⠿⠋⠀⣿⢷⣿⠋⠀⣹⡇⠀⣿⠙⣷⣿⠇⢿⣇⠸⠛⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠋⠐⠛⠀⠀⠀⠐⠛⠆⠻⠷⠄⠛⠇⠰⠻⠀⠈⠻⠇⠈⠛⠷⠿⠻⠆
⠀⠀⠀⡀⢰⣄⠀⠀⣤⣄⣀⣀⣀⣤⠤⠖⠒⠛⡛⠓⠒⠠⠀⠀⠀⠀⠤⠒⠒⠒⠒⠲⠦⢤⣀⣀⠀⠀⣀⣠⠀⠀⢀⡄
⠀⠀⠈⢿⣤⡛⠷⠶⠤⠭⠛⠋⣉⡤⠖⠋⠉⠁⠀⠉⠉⠒⢤⡀⢀⡤⠒⠋⠉⠉⠉⠉⠒⠦⣌⡙⠛⠯⢭⣥⣤⡶⠟⣁⣾⠃
⠀⠀⠀⠀⠉⠛⠻⠶⠶⠖⠚⠋⠁⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⡱⢊⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠉⠓⠲⠶⠤⠶⠶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠊⠀⠀⠁⠒⠒⠂⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠉⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠂⠀⠤⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠛⠓⠒⣾⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠈⢉⣉⣛⣻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠰⡛⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣋⣙⣛⡛⠇
⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠀⠀⠀⠀⠀⠠⠀⡠⠉⠿⠃
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠿⠄⠀⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢺⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣤⣄⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢠⠀⢰⢀⠀⢀⠄⠀⡀⠀⡄⢨⠰⣭⣰⢾⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠙⢛⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠄⡠⢿⠈⠤⠼⠰⣦⠀⠀⠀⢠⣀⣤⠀⠀⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠤⠶⠒⠖⠲⠶⠒⡶⠾⠶⢶⡶⠘⠛⠛⠛⠛⠃⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠁⠀⠀⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠐⠂⠆⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⡀⣠⣴⣿⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⣠⣤⠀⠾⠁⠶⠴⣿⣿⣿⡇⠙⣿⣯⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⠿⠛⠛⠛⠉⠋⠂⠀⠀⠰⠒⠒⠒⠒⠲⢂⠈⢙⣙⡒⠒⠂⠀⠐⠚⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠙⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠭⠉⠁⠅⠀⣿⣯⣽⢿⢿⡿⣿⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠓⠂⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⣶⣶⣶⣶⣤⣌⡛⢿⣷⡀
⠀⠀⠀⢰⣿⣿⡿⠛⠛⠻⢿⣿⣿⡿⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⢀⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣣
⠀⠀⢘⠋⡎⠉⠀⠀⠀⣦⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠚⡉⠀⠀⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠆⠀⢀⠀⠀⠀⠘⣲⠺⠂⠀⠀⠀⠀⠀⣸⡇⠘⠂⠻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠈⠡⠀⠋⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠑⢄⠙⠻⢿⣷⠀⢰⣿⠿⠛⢉⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠒⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢄⢀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠸⠿⠗⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠮⣹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠐⠀⠀⠁⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⠀⠀⠐⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠴⠈⠄
⠀⠀⠀⠀⣠⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⢁⣠⠼⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⣈⠀⠒⠶⢦⣬⣤⣴⣬⣟⣃⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⣧
⠀⠀⠀⢀⣘⠛⣛⢿⣷⣶⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠐⢾⣿⣿⣿⣿⣿⣿⣻⣻⣿⢿⣟⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣇
⠀⠀⢀⣨⣿⡏⢹⣿⣿⣿⣽⡇⡟⣿⡏⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠉⠉⠙⡆
⠀⠀⠈⣿⣿⣷⣾⣷⣾⣿⣥⣧⣿⣽⣿⣿⣟⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠈⢓⡾
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡺⠤⠤⣶⣾⣧⡀
⠀⠀⣘⣿⣿⣿⣿⣿⡟⣛⣫⢭⢾⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠇⢀⣀⣉⣉⣛⣿⣷⣦
⠀⠀⢺⣿⣿⡇⠙⢸⡇⣿⣿⢸⢸⣷⡇⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠩⢽⣍
⠀⠀⢼⣿⣿⡇⣦⣼⡇⡾⣿⢈⢸⡟⡇⣿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⢠⣿
⠀⠀⠀⣿⣿⣷⣿⣶⣷⣶⣿⣾⣾⣷⣶⣿⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠰⠰⡍⢴⢸⣿
⠀⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠁⠀⠀⠀⠲⠀⠀⠲⠶⢶⠶⠾⠆⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⣰⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⣠⣤⣤⢴⣾⣯⣄⡀⠀⠀⠀⠀⠀⣨⣿⣿⡄
⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⢀⠍⠭⠩⡯⠀⠿⠛⠛⠫⡤⠠⢤⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠌⣽⠧⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠘⢛⡁⡃⠀⠐⠈⠉⠁⠒⠤⠀⡀
⠀⠀⢀⣠⣴⠇⠀⠐⢄⢘⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⢶⠷⠟⢁⣈⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⡄⠀⠀⠀⠀⠴⢿⣆⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡿⠤⠠⠀⠄⠀⠀⠈⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⢁⡤⢏⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠀⠀⠈⠳⢄⠠⠤⠄⠆⠀⢂⡈⠁⢢⡀⠈⢧⡀⠀⡀⠠⣀⠁⡀⠀⠡⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠁⢈⠀⠀⠀⣂⢀⠂⠀⠀⠀⠒⠦⡀⠀⠀⠀⠀⠑⡌⢀⠊⠀⠀⠈⠀⣀⣀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⢀⠂⠐⡰⠇⠄⠀⠀⠈⠱⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠀⠠⣄⠑⢆⠀⠰
⠰⣄⡆⠀⠀⡅⠀⠀⠀⢀⡤⡄⠀⢤⡄⠀⠀⠀⠀⠀⡆⠀⠀⠚⡅⠀⠀⡄⠀⠀⠀⡄⠀⠀⠰⠘⠲⠠⠄⠉⠀⢸⠀⡘
⠚⠻⣿⠋⠀⠁⠀⠀⠀⠀⣤⡄⠀⣄⡌⠀⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⢊⡀⠀⢰⢤⡀⠈⢠⣀⠀⠉⡡⠀⠀⠃⡐
⠀⠀⠈⠳⠦⠴⡀⠀⠀⠈⠘⠀⠐⠒⠀⠄⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠿⠀⢰⣦⣤⡼⠁⡀⢡⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠤⢒⣿⠀⠀⠀⠀⢠⢷⠀⠀⠡⢄⠀⡠⠄⢀⣼⣿⣾⢇⠀⠘⠒⠊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⣵⣤⡄⠀⠈⠋⠑⢀⡤⢄⡠⢴⣋⣿⠀⠀⠀⠀⣀⠀⠀⠀⣠⠋⢀⣨⠦⢀⣀⣀⡠⢒⡏⣀
⠀⠀⠐⢦⣤⣀⣻⠀⠉⠉⠒⠦⢖⠒⠉⡇⠀⠀⠀⠞⠟⠓⠂⠰⡎⡼⠁⠒⠈⠀⠰⠯⠥⠤⣤⡄⠀⢳⠈⠳⠟⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡟⢧⣰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠿⠴⠶⢲⡆⠀⠀⠀⣀⣴⠟⠁⠀⠈⠷⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣥⣤⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠉⠀⠀⡀⠖⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠡⠈⢁⠀⠀⡈⠀⡀⠀⠠⠀⠀⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠈⠁⠐⠀⠠⠦⠀⠀⠒⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⢀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⠉⠀⠀⠐⠀⠀⠀⡠⠤⠀⠀⠀⠀⢣
⣶⡶⠶⠶⠶⢶⢶⣾⢷⣾⣶⣶⣶⣶⣿⣶⣾⣶⣿⣾⣿⣶⣶⣶⣿⣶⣾⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⢐⡀⠀⣂⠀⠀⠀⠄⠀⠀⠀⢜⠀⠀⠆⠀⠀⠀⢤⢀⠠⡀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣤⠀⠀⠀⠠⠄⠄⣠⢀⠄⡀
⠠⠈⠂⠐⣈⠂⠀⡄⠂⠄⠀⠁⠉⠀⠂⡥⠆⠑⣈⢰⣇⡠⠋⠢⡀⣿⣰⣧⣝⣀⣇⣷⣅⣿⢀⠔⠀⠀⡤⠀⠠⡄⠀⢠⡔
⠀⡀⢀⠀⠄⠄⡀⢔⢊⠌⠀⡨⣀⠀⠀⠰⠀⠰⠢⢓⠀⢳⡄⠐⠀⠙⠛⠛⢻⡿⢛⡛⠛⠁⢨⠀⡳⠆⠀⣼⡄⢡⣶⡄⠁⢳⡇
⠀⠁⠁⠂⠀⠸⠅⠃⠀⡂⢀⠦⢀⠀⠁⢉⠀⠀⠾⠂⢠⠂⠀⡠⠊⠖⠀⡀⠈⠀⢠⡄⠀⠈⠀⠁⠠⠘⠠⣀⠔⠢⣀⠄⠁⠤
⠀⡠⢂⠅⠰⢈⢐⠀⡨⠇⣀⠪⠂⠁⢀⠰⢁⠀⠀⠠⠂⢰⡁⠄⠀⠀⠀⢧⢀⠀⡇⢩⠀⡞⢰
⠠⠠⠠⢐⠀⠂⠂⠂⠰⠃⠀⠄⠡⠂⠂⠀⠠⠃⠠⠄⠠⠊⠧⠤⠤⠀⠂⠈⠄⠀⠀⢸⠀⠇⢸⠀⡀⡤⠀⠠⡢⠀⠀⢰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⡀⠐⢖⠂⠡⠀⠁⠀⠀⠔⠀⠈⠀⠀⠈⣹⡄⠀⠀⠐⠤⠘⠀⡄⢼⠀⠀⠈⢀⠀⠀⠀⢀⠅⠘⢂
⢀⠄⠀⢠⢀⠀⠀⣉⣂⠐⠻⠲⡶⠲⡶⠆⠀⠰⢆⠂⢒⠶⡀⠁⡁⠈⠁⡀⠀⡐⠈⠀⠁⢀⠀⠀⠀⣄⠀⢠⠀⠀⠌⠈⡀
⢀⣤⣤⣄⡏⢄⢾⣿⣿⠀⠀⠀⠹⠜⡀⠈⠤⠂⠈⠳⠎⠀⠈⠒⠀⠀⠂⠂⠀⠀⠀⠀⠀⢀⠀⠀⣄⠈⠘⠈⠑⠠⠆⠀⠀⠠⠘
⠀⢿⣿⡟⠀⠀⠈⠋⠙⢆⠀⠀⠀⢀⢆⠀⠀⠠⠀⢀⠀⠲⠲⠒⠒⠒⠖⡤⢄⠀⠀⠀⠀⠞⠉⠀⠐⠄⠀⠀⠆⠀⠀⠀⣀⡀
⠀⠀⢖⠀⠀⠀⡠⠒⡀⠘⠣⢀⠠⠼⠸⠄⢀⠀⠀⡞⠀⢀⠀⢀⠐⠀⡆⠀⢘⠆⠘⠋⠁⠀⠀⠀⡀⠈⢆⠈⠀⠀⠃⡀⠉
⠀⠂⠛⠑⠀⠀⠅⠀⠀⢂⠀⠔⡀⠀⠀⠀⢰⡀⠀⡁⢈⠁⠄⠁⠈⠀⠁⠀⠀⠀⠀⠀⢢⡀⠀⠀⡀⡀⠀⠁⠀⠈⠂⠁⠀⠐
⠀⠀⠀⢰⡄⠸⡀⠀⠀⡘⢸⠠⠄⣁⠀⡀⠡⢸⠀⢃⢄⠀⠐⠐⠢⠀⠠⠄⢀⠇⠀⠀⠀⠀⠀⠤⠀⠸⠄⠀⠀⡀⠈⠁⠈⠂
⠈⡐⢠⠃⠱⡀⢡⠀⢀⠀⡸⣀⠀⠀⠀⠀⢐⠸⠀⠀⠘⠀⠤⠤⠤⠤⠄⠐⠁⠀⠰⠆⢰⣶⠞⠉⠁⠀⠙⢶⣶⠀⠀⠐⠈⠄⠁
⠀⠀⠎⠀⠀⠰⠀⠉⠈⠀⣈⠂⢩⠄⠀⠬⠀⠊⠀⣄⠀⠀⠀⢀⠀⠢⠉⠁⢐⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⢸⠁⢀⠠⠀⠒⢐⠢
⠀⢸⢀⡀⣀⠀⢇⠀⣸⠀⠉⠀⠙⠁⠀⠀⣠⠊⡀⠊⡀⣀⡂⠘⡬⢄⣲⠅⠈⡀⠓⠁⠀⠈⢀⡀⠀⠀⣠⠊⠀⢀⠘⠂⢒⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⠀⢀⢐⠀⠀⡰⠁⠀⠀⢠⡅⠃⠀⠀⠀⠀⠀⡀⢀⠅⠀⠴⠢⠀⣤⣬⣷⣯⣤⣇⠀⠀⠑⠀⠐⢂⠁
⠀⢀⡀⠙⢋⢄⠀⠈⠢⠈⠈⠈⠀⠀⠐⠧⠐⠂⠉⢀⡀⡀⠀⠀⠀⠉⠐⠒⠁⠐⡿⠠⡤⠤⡄⡉⠉⣉⣉⣀⣀⢀⠀⠠⠄⠺⠃
⢀⢮⣈⠻⠧⣔⠢⡄⠐⠠⠀⠀⠀⢀⠆⢀⠀⢠⠞⠁⠀⠰⠞⠉⠁⠀⠀⠀⠂⡀⠀⢠⣀⣀⡃⠀⠀⠀⠀⣰⠀⠀⠀⡀
⠘⠉⠀⠤⠀⠐⠀⠅⠐⠀⠊⠀⠈⢂⣀⡈⠀⠐⠀⠁⠂⢐⠆⠼⠀⠷⠀⠀⠀⠁⠀⠀⠀⠈⠁⠁⠈⠁⠈⠁⠉⠀⠁
⠈⠢⡀⠀⠀⣀⠤⠀⠚⠒⠀⠀⠂⠀⠙⠀⠀⠃⠒⠀⠀⠌⢸⠀⣇⠀⠍⠀⠀⠀⠘⠄⢱⠀⢉⠀⠀⠎⠂⠰⡰⢢⠚⠁
⠀⡠⠬⠉⠀⠀⢄⠀⢀⢆⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠌⠀⠀⠲⠀⢀⠃⠀⠀⠀⠀⠀⠈⠀⠀⠂⠐⠃⠀⠊⠀⠐⠂⠀⠈
⠀⠀⠀⡖⠲⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣰⣰⡌⢄⡸⠀⠀⡂⢀⠀⢀⠂⡂⠀⠚⠈⠇⢹⠈⠐⠄⠀⠡⠀⢡⠋
⠀⠒⠀⠀⠀⠀⠚⠀⠁⠂⠤⠁⠁⠀⠈⠉⠈⠒⠒⠒⠒⠒⠀⠂⠀⠁⠀⠑⠀⠈⠂⠘⠑⠃⠐⠂⠐⠁⠊⠐⠒⠀⠀⠀⠂⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠄⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣅⣀⣀⣰⣿⣿⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠛⠛⠛⠛⠙⠛⠋⠉⠉⠁⠹⣿⣾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢁⠤⢤⣄⠀⠀⣴⠶⠒⠢⠀⠀⠀⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡇⠀⠶⠿⢿⡇⠀⠀⠞⠛⠂⠀⠀⠀⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠺⢇⣀⠤⠂⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠰⢃⣀⣀⣀⣀⡤⠄⠁⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠠⠴⠀⠀⠀⠀⠀⢠⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠃⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠸⣿⡷⠦⣤⣤⠴⠒⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀
⠀⠀⠀⢠⣆⣀⣀⠀⠀⠀⠀⠀⠚⠉⠀⡇⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠐⠀⠀⠀⠀⠀⠃⠀⡄⠀⠙⢦⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠠⠏⠀⠀⠶⠀⠀⠠⠂⠀⡀⠀⠀⠀⠀⣰⠁⠀⠀⠀⢣⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⡔⠓⠒⠠⠤⠤⣀⣹⣿⣏⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢱⠠⠦⠤⠤⠤⠤⣤⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⣦⣤⣤⣤⣤⣄⣉⣠⠄
⠀⠀⠏⠻⠼⠁⠈⣇
⠀⢰⠀⠀⠀⠀⠀⢰
⠀⠘
⠀⠀⠁

⣿⣿⠟⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣶⠖
⠀⠀⠀⠀⠈⠁


⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣶
⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠃⠀⠀⠀⣀⣀⣀⣴⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠶⠿⠟⠋⠀⠀⠀⠀⢠⣖
⠀⠀⠀⣈⣿⣁⠀⠀⠰⣤⠉⠀⠰⣿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⠿⠿⠿⣿⣇⠀⠀⠘⠀⣆⣀⠠⠀⣤⡄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠰⠟⠉⢹⡇⠀⠀⠀⠀⠀⣼⠁⢸⡏⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠉⠃⡈⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⢀
⠀⠀⠀⠀⢀⠀⠀⣴⣤⣤⡀⠀⣿⣠⣤⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠛⠂⠀⠀⠀⠀⢙⠇⠀⠀⠀⠀⠀⠀⢳⡆⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⣠⣷⣦⣄⡘⢿⠿⢷⠀⣿⣿⣿⠏⡀⠀⠀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⣿⡇⠀⠀⠀⠀⠀⠀⣬⣁⣴⡿⠋⠁⠀⠀⠀⠀⠀⠸⣿⠄⣿
⠀⢠⣴⢛⡛⠛⠛⠁⠀⠀⢀⡀⠉⣶⣤⣶⣷⣤⡀⠙⢿⣧⡀⠀⠀⠿⠀⢸⣦⠀⣴⣦⣼⣃⠀⠀⠀⠀⠀⠀⠛⣙⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠛
⠘⣿⣿⣾⡏⢀⠀⠀⠀⠀⠀⠃⠀⠉⠘⠛⠉⠁⠀⠀⠀⠈⠻⠞⢿⡗⠀⠀⠋⣀⣿⣿⠟⠛⠻⣦⠀⠀⠀⠀⠘⣿⡦⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢰
⠠⣿⣿⣿⡿⠛⠛⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⡄⢷⠀⠀⠀⠀⠘⠀⠀⠀⠀⣿⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠛⢿⡗⠀⠀⠀⠿⣿⣆⣰⣜
⠀⠀⠉⠻⠀⠀⠀⠀⠀⠰⣾⣿⡇⠀⠀⠀⠀⠀⠀⡀⣧⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠠⠬⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠙⠃⠀⠀⠶⠂⠀⣠⣷⣼⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢸⣶⢦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣠⣾⣿⠟⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢶⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘
⢸⡁⠀⠀⠁⠀⣀⣠⣤⣤⣀⠀⠀⠀⢀⣿⣿⣿⡟⠀⠀⠀⠀⠀⣼⠃⠀⠀⣤⠀⠐⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠛⠃⠀⢶⡄⠀⠀⠀⠀⠀⣰⣆
⢸⣷⡀⠀⠀⠀⣽⣿⣿⣗⠀⠀⠀⠀⠸⠛⣿⡟⠀⠀⠀⠀⠀⠀⠈⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠆⠈⠁⠀⠀⠀⠀⠸⣿⣿⡆
⠈⢹⣷⡆⠀⠀⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡇
⢀⣾⡁⠀⠀⠀⠀⠘⢻⣿⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒
⢸⣟⢁⣌⣅⠀⠀⠀⠈⠻⢷⣦⢴⣿⣸⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⠀⠀⠀⠶⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢟⠁⠇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠋⠁
⢈⣉⠉⠻⢿⡇⠀⠀⠀⠀⠀⠀⠘⠋⣩⡄⠀⣀⡠⠘⠉⠀⣹⣿⠿⠿⠗⣀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠐⣷⠀⠘⢠⠄⠀⢈⣿⣂⠀⠀⠀⠀⠀⠀⠐
⠸⠿⣿⣦⣀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⠀⠀⠀⢠⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠀⠀⠉⠆⢩⡄⠀⠀⠀⠀⠙⠉⠛⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⣿⢻⣧⠀⠀⠀⠀⠠⣤⣄⠀⢾⠁⠀⠀⠀⠀⠈⣥⡀⢀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡇⢰⣶⣶⣷⣾⣶⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡿⢸⡇⠀⠀⠀⠀⠀⠀⢹⣷⣾⠇⠀⠀⠀⠀⠀⣿⣷⡈⠀⠀⠀⠀⠀⠀⣰⣶⣿⣿⣿⣍⣛⡛⣧⠟⠉⠛⠿⢿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠹
⠠⠀⠀⢀⡄⠀⢀⡀⢀⣀⣀⣀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣴⠏⠉⠈⠉⠻⣿⡿⠛⠿⠃⠀⠀⠀⠀⢹⣏⣽⡆
⠀⠀⠀⣼⣣⣶⣾⣿⣿⣿⣿⣿⣷⡶⠈⡅⠀⠀⠀⠀⠀⠀⠀⠈⢁⣄⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣿⣠⣤⠀⠀⠀⠀⠀⠀⣁⣀⡴⣶
⠀⠠⢾⡿⠿⠿⠿⠿⡿⠿⠟⠋⠁⠙⠰⠀⠀⠀⠀⠀⠀⠀⢾⢿⠿⠿⠿⢶⡦⡤⠏⠀⠀⠀⠀⠀⠸⠹⡿⣿⠀⠀⠀⠀⠀⠈⠉⠻⣧⢷⠀⠀⠀⠀⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣽⣿⡿⣏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⢟⣯⣿⣿⣿⣆⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⣒⠶⡀⠀⢈⣙⣷⠀⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠜⠛⠛⢧⠀⠀⠀⠀⠀⠀⠀⡠⠎⢁⣀⣀⣀⡟⡷⢷⣿⠿⣛⡟⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠒⠛⠛⠀⠀⠀⡀⠀⠀⠀⡀⣤⣾⣿⣿⣿⣿⡟⠀⠈⠋⠛⠝⠚⠛⠿⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢫⣟⡋⠿⠂⠀⠀⠀⠀⠀⠀⢦⠐⢛⢁⣀⣠⠿⠷⠿⢛⣟⣘⣿⠿⠗⠛⠑⠄⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠘⢬⣅⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢉⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⣶⠾⠓⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣦⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠉⢻⣦⣀⣀⠀⠀⢀⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠋⠀⠀⠀⠁⠋⠉⠉⠉⢹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠈⢣⡀⠀⠀⠀⠀⠀⠀⡰⡇⠀⠀⠒⠒⠒⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣁⠀⠀⠙⠢⢄⣀⡀⠀⠀⠀⠻⣴⣷⡶⠂⠀⠀⠀⠙⡖⠲⢄⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⡄⢰⠇⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠈⠣⣀⠀⠀⠀⠈⠉⠉⢛⣶⣄⢳⣿⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠈⠉⠀⠀⠀⠤⠾⡀⠸⣆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⢸⣧⡙⢻⣿⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠈⠳⣄⠀⠀⠀⠀⢷
⠀⠀⠀⠀⢀⠤⠒⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠚⠈⠻⢦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⡈⠓⠶⠶⠶⣻
⠀⠀⠀⡔⠁⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠈⢦⠑⢬⡊⠕⠢⠤⠖⠃
⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣁⠀⠀⠀⠀⠀⢀⣴⣊⣥⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢄⣉⣉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⢠⡾⠿⠷⠶⠤⠤⠤⠤⠴⠒⠛⠓⠒⠒⠒⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⠀⠀⡏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣤⣴⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣉⣈⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠩⢀⡤⠶⣾⣷⠶⣄⠀⠀⠀⠀⠀⢀⠀⠌⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠈⠻⣷⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠉⣼⠃⠀⠀⠀⠀⠀⠘⣿⡌⢧⠀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⡄⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⣿⢸⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠈⣧⠀⠀⠀⢸⣿⡝⢦⡀⠏⢸⠇⡁⠀⡁⠀⠀⠐⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⠀⣸⠘⣷⡀⢹⡇⢹⠀⠈⡊⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠹⣎⣡⠙⢦⡀⡟⣶⣿⣧⣼⠀⠸⡆⠀⠀⢆⡀⠀⠀⠀⠀⠈⠐⠢⢀
⠀⠀⠀⠐⡆⠀⡀⠀⠀⠀⠀⠀⠈⣻⣷⣤⣉⣓⣟⣉⣵⡇⠀⠀⢧⠀⡓⠒⠒⡄⠀⣀⡀⠀⠀⠀⠀⠙⢀
⠀⠀⣰⣇⠈⡏⠀⠀⠀⡐⠈⠀⣤⣇⠼⣟⣿⣯⣿⡋⡜⠀⠀⠀⠘⣆⠀⠀⠀⡇⠉⣄⠙⠋⠐⠒⠲⠄⠤
⠐⢠⠇⠸⡄⣇⠥⡀⠇⣘⠀⠐⠿⠿⣽⣧⣟⡗⣏⠟⠁⠀⠀⠀⠀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⣌⡉
⣠⡏⠀⠀⢹⡇⠃⠐⠀⠄⠾⠂⢼⡆⣼⣙⡮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡁⠀⠀⠉⠈⠒⠂
⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠃⠈⠋⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢂⡡
⠈⠓⠤⠔⠊⠀⠀⠀⠀⠀⠐⠆⠠⡶⠖⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢶⣤⡼⢃⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⣡⣤⣔⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣼⣿⣿⣿⣿⣧⣼⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣗⠂⠀⠀⠀⠀⠀⠀⠀⢨⡷⢘
⠀⠀⠀⠀⢻⣍⣏⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠇⠀⠀⠀⠀⠀⠀⡔⠊⠛⠋⠳⡄
⠀⠀⢠⣶⠞⠀⠀⠙⠂⠀⠀⠀⢰⣿⡿⠒⠁⠑⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⢤⡄⠟⠁
⠀⠀⢠⠀⠠⠖⠀⠀⡄⣤⠀⠀⢸⣿⠁⠀⠀⡂⣀⠀⠀⢀⠀⠀⡆⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠒
⠀⠀⠘⡀⠀⠀⠀⢠⠇⣟⠈⡁⠀⣿⣀⣀⠀⡇⠈⠛⠝⠋⠀⠠⣇⠀⠀⠀⠀⠀⢸⠀⠀⠀⣤⡤⠀⢠⡄⡀⣿⡒⡦⠇
⠀⠀⠀⠛⠛⠒⠂⠛⠀⠛⠉⠁⠀⠛⠳⠷⠒⠃⠀⠰⠓⠀⠀⠀⠚⠂⠐⠀⠀⠁⠘⠀⠀⠀⠓⠛⠛⠛⠓⠂⠚⠓⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⢀⣈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠉⠹⠿⠿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠒⠀⠒⠶⠾⠿⠿⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⣠⢽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡄⠄⢀⣤⣠⣠⣤⣄⡉⠀⠀⠈⠈⠈⠉⠉⣺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠉⠀⠒⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠿⠿⠟⠃⠀⡄⣀⢀⣤⣤⣶⢰⣾⡗⠀⠀⠀⠀⠀⠀⢲⣿⣇
⠀⠀⠀⠀⢠⣴⣶⡶⠶⠛⠓⠀⠀⠀⠈⠁⣈⣉⣛⣋⣏⠀⠿⢿⣗⣀⣀⢀⣀⣴⣿⣿⣿⣇⣀⡀
⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠤⠶⠶⠶⠶⠚⠛⠛⠩⠿⣿⣿⣽⣿⣿⠾⠎⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⣀⠀⠀⠆⠀⠀⣼⣤⣶⣶⣶⣶⣶⣾⣿⡖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠁⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⡶⢀⣠⡀
⠀⠀⠀⠈⢷⣘⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆
⠀⠀⠀⠀⠀⣿⣿⣟⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣤⠀⢀⡄
⠀⠀⢠⣤⣄⣿⣿⣯⣤⣀⢠⣠⣤⣀⡀⠀⢠⡏⣴⣦⠀⠀⠀⢀⣀⠀⡼⠀⠀⠀⣴⣦⠀⡿⠀⠈⣧⢰⣶⡆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⠡⣦⢹⡇⠀⠀⣷⣽⣭⣀⠀⠀⣿⣿⣾⡁⠀⣄⣀⣉⣭⣾⡁⠀⠀⠈⠳⣽⣶⣶⣆
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⡷⠶⣾⡇⠀⠀⢿⣿⣇⡽⠂⠐⠟⢻⣿⣿⣄⠀⠀⠀⣿⣿⣷⡀⠀⠀⠀⢸⣿⡿⠃
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢸⣇⣋⣼⡇⠀⢀⣼⠟⣿⡀⠀⠀⠀⠸⣿⣿⡿⠂⠀⠀⢿⡿⣿⠃⠀⠀⢠⡿⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣇⠀⠈⢿⣄⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⡇⠸⣧⠀⢠⡿⠁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣛⣀⣀⣀⣹⣃⣀⣀⣘⣛⣃⣀⣀⣀⣘⣇⣀⣹⣀⣙⣁⣀⣀⣛⣀⡀


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄
⠀⠀⠀⠀⠀⣴⣶⣆⣀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⡛⢛⣁
⠀⠀⠀⠀⣀⣙⣛⣩⡽⠇⠀⠀⣀⣛⣟⣁⠀⠀⡀⣸⢀⣼⣿⣧⣸⣿⡇
⠀⠀⠀⢾⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⡁⣿⠀⠈⢹⣿⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠘⠿⠿⠿⢻⣿⣿⣿⡻⠿⠿⠿⣿⣿⣿⣿⠻⠿⠿⠿⠿⠿⠿⠿⠟⠛⠿⠿⠃⠀⠀⠀⠀⠀⣨⡿⠟⠦⠀⣀⣀
⠀⠀⠀⢠⣿⠿⡿⢿⣷⠀⠀⢰⣿⠿⡿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣀⠀⣾⣿⣿⡇
⠀⠀⠀⢸⣿⠀⡇⢸⣿⠀⠀⢼⡏⠀⡇⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣄⠀⠀⣩⣿⣿⣮⡛⠛⠁
⠀⠀⠀⠀⠁⣀⣇⣀⠁⠀⠀⠈⢁⣀⣇⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⣿⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣾⣿⣿⣶⣾⣿⣿⣿⣿⣯⣤⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠈⣿⡇
⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⣈⠀⡅⠀⠀⠀⠀⡈⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣿⡇⠀⠀⠀⠀⠲⣲⣾⡆⠀⠀⠀⠶⣿⣿⡟⠃⠀⠀⠀⠀⠀⡁⣹⠃
⠀⠀⡄⣀⣤⣤⡼⠀⠀⠀⢂⠲⠇⠀⠀⠀⠀⠃⠻⣿⣇⠀⠀⠀⠀⠈⢛⠇⣀⠀⠀⠀⠀⣠⠀⠀⣦⡀
⠀⠀⡇⠙⠛⣻⡷⠀⣀⣀⣿⣿⣯⣉⡅⠀⠀⣄⣀⣻⣿⡇⠀⠈⠠⠖⠀⠀⠈⠇⠀⢀⣀⣤⣤⡠⢀⢤⣄
⠀⠀⡇⣿⣿⣿⡇⠀⣿⡏⣿⠟⠹⠿⡇⠀⠀⠈⢉⠉⢹⡇⠀⠀⠀⠀⠀⢰⠀⠀⠀⡎⣿⣿⡟⠁⠀⠀⢿⡆
⠀⠀⠗⠶⠿⣿⡇⠀⣿⣇⡟⡞⢟⣻⡇⠀⠀⣶⣶⣶⣾⡇⠀⠀⢀⠀⠘⣿⠇⡀⠀⣧⣿⣟⡃⠀⠀⣄⣼⡇
⠀⠀⠁⠀⢉⠉⢹⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⠻⣿⣿⠀⠀⠀⠀⠉⠉⠉⠁⠄⡆⠹⣝⢿⣉⣠⣤⣽⣿⠁
⠀⠀⡤⢇⡀⠳⡼⠀⢠⣤⣿⣯⣭⣭⡅⠀⣤⣾⣾⣿⣿⡆⠀⠀⡀⠀⠀⣀⢐⣰⠇⠀⠹⣷⢎⢻⣿⡿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡬⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⣁⡃⠀⠀⠀⠀⠀⢱⠚⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠺⡀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠀⠀⠁⠀⡀⠀⠀⣀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⢀⠠⠀⠀⠐⠀⠑⢄⠈⠂
⠀⠀⠀⠀⠀⢀⠀⢤⠤⠤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠔⢁⠄⠀⠈⠀⠀⠄⠀⠀⠠⠀⠔⠀⠀⢸⢀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢫⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡠⢂⡄⠀⢈⣨⡄⠉⠛⢄⠀⠀⠀⠀⢀⡀⠀⠀⠰⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣐⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡀⠀⠀⠈⠀⣀⡣⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠈⠀⠀⠀⠠⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠐⠞⠉
⠀⠀⢀⠂⠁⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠁⠈⡆⠀⡠
⠀⠀⡀⠁⠈⠀⠐⠀⠀⢳⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠄⠀⠀⠀⡯⡊⢀⠴⠁⠀⡀
⢸⡂⠧⠀⠀⠀⠀⠀⠀⢄⠁⠠⡄⠀⠀⠀⠗⢄⠀⢀⡀⠀⠀⠊⢀⡌⡀⠀⠀⠀⠀⢀⠇⠀⠁⠀⠈⠀⠀⡀
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢀⠀⠀⡀⠔⠑⠐⣦⣤⠀⠘⡀⠐⠁⠀⠀⠀⡠⠊⠉⠀⠐⠀⠀⠒⡤⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠀⠀⠀⢑⠶⠀⡀⠃⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢻⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢢⠀⢀⣔⡄
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢸⠠⠽⠚⠁⠀⠀⠀⠀⠀⠀⢰⢢
⠀⠀⢀⡀⠣⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⢤⣀⣀⣀⣀⣀⡀⠀⠀⢀⡴⢂⠭⠯⠭⢭⡔⢄⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠓⠒⠃⡬⠤⡀⠀⠀⠀⠀⠀⠀⠰⣉⣀⣀⣀⣀⣀⣀⣀⣈⡆⢀⣎⢐⣣⠒⠒⠔⠋⠙⠢⣧⠀⠀⣀⣀⣀⣀⣀⣘⣀⣀⡀
⠀⠀⠀⢰⠒⠁⠀⠀⠀⠀⠀⠈⠹⢤⠀⢅⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡤⠭⠭⠭⠭⠭⠭⠭⠭⡼⠀⠸⡄⠤⠤⠤⠼⠼⠤⠤⠤⡜
⠀⠀⠀⠀⣏⣉⣉⣉⣉⣁⣀⣉⣹⠇⠀⢸⠘⠐⠒⣲⣒⡒⠐⢻⠀⠀⡧⢤⣤⣀⣀⢀⣠⡤⢤⡇⠀⠀⡗⠒⠒⠒⡖⠖⠒⠒⢚⠃
⠀⠀⠀⠀⡆⠀⠀⣠⣀⣀⠀⠀⢸⠀⠀⢸⠀⠔⡞⠛⢛⠠⠀⡈⠀⠀⢰⠀⠀⠠⠄⣤⠀⠀⢰⠀⠀⠀⢃⢖⠉⡆⡇⡀⡁⣀⢸
⠀⠀⠀⠀⢠⠀⠀⠣⣉⠜⠀⠀⠸⠀⠀⠸⠀⡀⡄⠇⢠⢀⠀⡇⠀⠀⢘⠀⡄⣬⣥⣄⡄⡆⠘⠀⠀⠀⢸⠈⡝⢀⡇⡇⠤⠘⡀
⠀⠀⠀⠀⠸⠤⠀⢀⣈⡀⠀⠤⡄⠀⠀⠀⡆⠀⠉⠉⠉⠁⠰⠂⠀⠀⠀⡤⢉⣁⣀⣀⣁⠠⡆⠀⠀⠀⠀⡆⡈⠀⡇⡇⠀⠀⡇
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇⠀⠀⠉⠁⠅⢁⠃
⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠀⠉⠀⠀⠀⠀⠈⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠈⠁⠀⠒⠂⠀⠉⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣖⡲⢿⣶⣶⣶⣶⣷⣆
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠙⠃
⠀⠀⠀⠀⣰⣿⣿⣯⣿⣭⣯⣛⣿⣟⠋⢹⠍⡇
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⣤⣄⡀⠠
⠀⠀⢀⣷⣮⣝⣉⣛⣋⣛⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠌⠟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⢀⣀⡒⠒⠒⠒⠀⠀⢀⡀⢀⠐⣄⠒⣄
⠀⠀⠀⠘⣿⣿⣿⣿⠁⠁⠦⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠀⠐⠀⠀⠀⢈⠉⠀⠀⠀⠀⠀⠀⠀⢂⢨⣭⣄⡁⡝⡤⠆
⠀⠀⠀⠀⠈⢿⣿⣿⣦⣀⠴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠶⠋⠐⠒⠂⢄⡸⡿⠁⣠⣤⣤⣤⡄⠬⠉⢤⣀⣸⡣⣿⣗
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠲⠶⠼⠭⠿⢿⣿⣄⢠⣛⣉⣙⢃⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠘⢧⣄⢀⣀⣀⡀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢌⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⡀⠲⡶⠶⠒⠶⠋⠉⠳⠆
⠀⠀⠀⠀⠀⠀⠀⡴⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠛⢸⠃⠀⠁
⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡮⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀
⠀⠀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀
⢠⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⣀⢻⣄⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠑⠦⡀⠀⠀⠀⠀⢹⠀⢻⢢⡀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠈⠓⠦⣀⣀⡈⢇⠘⣷⣭⡳⢤⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠉⠙⢮⣣⣈⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⣶⣶⢶⡶⢶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣴⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠽⠽⠿⠍⠉⠀⠀⠀⠀⣀⣀⠀⢀⡀⠈⣠⡌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⣿⣿⣿⣌⣷⡦⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣸⣿⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⢿⣿⣿⣿⣿⣷⣌⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣿⣧⣊⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢦⣿⣿⠻⣿⣿⣿⣿⣿⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣴⠋⠉⠉⠁⠉⠉⠉⠁⠈⠙⠛⠛⠿⣿⡼⣸⣯⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⢛⣛⡿⢿⣿⣷⣶⣾⣷⣰⣶⣦⣤⣤⣯⣠⣀⣀⠀⠘⠛⢶⣭⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠚⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡍⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⢀⣤⣰⡄⠬⠭⠍⠉⣙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠟⠉⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣋⣉⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣷⣾⣿⡿⠿⢿⠿⠿⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡛⢻⠋⢀⣤⣤⢄⣀⣦⣤⣤⡀⠍⠛⢿⣿⣿⣿⣿⣿⣅⣙⠃⠘⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣮⣙⠿⣿⣿⣿⣿⣶⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠟⠛⠉⠙⠛⠛⠛⢯⣭⣿⣿⣿⣿⣿⣿⣿⣦⣀⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⡀⠀⣠⣴⣲⢤⣼⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⣠⣾⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢣⣶⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡾⢽⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⢹⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢃⡀⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣀⣠⠴⠶⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡀⢀⡤⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⠟⠹⣶⠋⠀⠀⠀⠻⣄⠀⢀⠀⠀⢠⡴⢰⣾⣇⡼⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢰⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡧⢠⣌⠳⠖⠋⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠏⢀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⠟⠀⠀⠀⢹⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣧⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠸⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠘⠛⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠿⠁⠀⠀⢀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢻⣆⠀⠀⠀⠀⠰⣤⡀⢀⣠⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⠀⢳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢷⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⢠⣀⠀⢉⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣾⣇⠀⠀⢿⣷⣤⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠉⠛⢳⡶⠤⠶⠚⠛⠉⠀⠀⠘⣏⠀⠴⠤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⠤⠚⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠃⠀⠀⣿⡉⣹⠏⠉⠉⠉⠉⢁⣀⠀⣼⠿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠏⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢰⡏⢉⣿⣿⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⢋⡴⠲⡄⠀⠀⣠⠖⠲⡄⠀⠀⠀⠉⠊⢹⡟⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠃⠈⠳⠞⠃⠀⠀⠛⣤⠾⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣄⠀⠘⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣼⠋⢙⡆⠀⠀⠀⠀⠀⠀⠘⢧⡴⠇⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⡀⠀⠀⠙⠓⠋⠀⠀⠀⢠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⡸⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢀⡷⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠉⠉⠀⠀⠀⠺⣅⣿
⠀⠀⠀⠀⠀⠀⢸⡿⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠿⠀⠀⠀⠀⠀⠀⢀⣀⢼⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠙⣿⡽⣟⠓⣂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⠾⣿⠛⠊⢳
⠀⠀⠀⠀⢠⡎⠀⠀⢰⡟⠀⡏⠀⠀⠀⠀⠈⠘⡿⢻⡄⠈⠉⠉⠉⠛⠛⢻⣿⠹⠉⢻⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠘⢧⣄⡀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢃⢸⡇⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠹⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠀⣀⣠⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠛⠿⠶⠶⠶⠶⠾⠿⣶⣶⣶⣶⣶⣶⣶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡀⣠⡀⢻⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠙⠁⠀⠈⣻⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣺⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣀⡜⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢀⣨⠟⠛⠷⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣅⣀⣀⢀⣀⣈⣉⣉⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⣿⠿⠿⡿⠿⠿⢏⠹⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⡟⣻⣛⠛⠛⠛⣛⡛⣛⢻⣿⣿⠛⠛⠛⠿⠿⠟⢻⠛⠛⠻⣿
⣿⣷⣿⣶⣶⣷⣾⣶⣾⣶⣾⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡴⠀⠠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠀⠀⢤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡚⠁⠀⠀⢀⣀⣠⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣴⢶⣠⠄⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠉⠁⠉⠉⠘⠡⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠀⠀⠒⠀⠤⠄⠀⢄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠰⡆⠉⠉⠁⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⢘⡶⠿⣶⣶⡶⠃⠀⣧⣤⣄⣀⣀⣀⣀⣀⣸⣿⡇⠀⢇⣀⣠⣀⣄⡠⠀⠠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⣆⠀⢈⣽⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠸⣿⡿⠋⠙⢷⣀⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⠿⠿⠿⣿⣿⣿⠃⠀⠀⠑⠪⣭⣛⣛⣛⣛⣛⣛⠽⠀⠀⠀⣿⣷⣤⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⣦⣤⣴⡿⠿⠟⠀⠀⠀⠀⣾⣿⡿⢻⣿⣿⠇⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⣹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠉⢩⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢻⣿⡆⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⢠⣿⡟⢻⣿⣆
⠀⠀⠀⠀⢀⣀⣀⠿⠟⠁⠀⢻⣿⠄⠀⠀⠀⠀⠀⣻⡿⠁⠀⠀⠀⠹⢿⣟⠀⠀⠀⠀⠀⣾⣿⠁⠀⠙⠿⣣⡀
⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⢀⣰⣷⠄⠀⠘⠳⠿⢿⡷⠀⠀⠀⠀⣀⣼⣿⡆⠀⠀⠶⣴⡶⠁⠀⠀⠀⠰⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

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

⠀⠀⠀⡀
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⡠⠀⠀⠀⠀⢀⡦⢄
⠀⠀⠀⢸⠀⡸⠁⠙⢉⠇⠠⠀⣠⢐⠚⣓⠢⢀⠴⠂⠀⣴⢿⡭⠭⡻⣧⠀⡀⠀⠀⠇⢰⠙⠒⡉⡂
⠀⠀⠀⢸⣰⠁⠀⠀⣌⠀⠐⠄⡁⠬⣶⡟⠁⣇⡄⠀⠸⢟⡸⡀⠀⡀⣸⠀⠀⠒⠊⢊⠓⠘⠉⢏⡿
⠀⠀⠀⢀⠏⠀⠀⠀⠘⡄⠀⠀⠉⢄⡀⢀⡜⠁⡂⠁⡄⠑⢯⢮⣯⠾⠋⡀⡀⠀⠀⠈⠦⠀⠠⠼⠃
⠀⠀⠀⠈⠉⠉⠉⠉⡉⢁⠀⡀⢠⣤⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡤⡌⣳⡁⢸⠂⡒⠄⠀⠀⠀⠒⡱
⠀⠀⠀⠀⠀⡲⢠⡀⠂⠠⠃⢱⡟⣷⡎⡆⠐⠂⢰⠀⢰⠅⠀⠀⠀⠈⠁⠉⠁⢸⠀⡃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠄⢵⠐⠃⠀⣀⢀⢀⢉⡒⠊⠀⠀⠀⠈⠀⠀⡀⠀⡀⢀⠀⡀⠀⠀⠀⠰⠥⠤⠤⠤⠤⠤⠱
⠀⠀⠀⠀⠀⠀⠤⣸⣆⠹⢯⣟⡿⠃⠀⠀⠀⠀⠀⠀⠈⠀⠇⠑⠈⠠⠥⠀⠀⠀⠀⠀⢀⠰⠄⡀
⠀⠀⠀⠀⠀⢀⣀⡈⣁⡀⠀⠀⠀⡀⠀⠀⠙⠰⠁⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠈⡍⣔⠁⠀⠀⠈⡴⡆
⠀⠀⠀⠀⡊⠁⠀⠀⠀⣈⠄⢀⠙⢋⡇⠀⠳⠀⣉⣀⣀⠈⢸⢓⣲⢀⢨⣿⣔⡁⠀⢿⠀⣴⣶⡀⠃⣤
⠀⠀⠀⠀⠀⠁⠉⠉⠉⠁⠀⠸⠤⠼⠆⠀⠀⠂⠡⣀⡠⠒⠉⠉⡀⠘⠿⠟⠉⠀⢁⠀⠈⠛⠞⠁⡀⠽
⠀⠀⠀⢸⡑⢒⠒⠠⢂⡄⠀⠰⢪⢍⢢⠀⠀⢶⠀⡀⠀⠀⠀⡼⣁⣥⠀⠀⠐⠉⠂⠉⢢⣀⣀⡜⠁
⠀⠀⠀⠘⠁⠀⠒⠒⠂⠝⠀⠰⢌⡈⠜⠐⠟⠀⠐⠘⠀⠀⠀⠀⠃⠔⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⣾⠏⣀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢿⣄⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣀⣿⣤⡟⠿⠁⢼⡟⠛⠛⣧⣀⣿⣆⠀⠀⠀⠀⢵
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣃⣼⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢠⠟⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠄⠀⢸⠃
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⡄⢀⡎
⠀⠀⠀⠀⠀⢹⣸⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠆⣸⠃
⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠼⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣦⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⠞⠛⣿⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠙⢿⣄⠈⠉⠙⠛⠚⠉⠉⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠛⠿⠶⢶⣤⣀⣀⣀⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣠⡛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⠶⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠶⣦⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢾⡷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣧⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⡀⢠⣴⠀⢶⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣾⡿⠿⠛⠉⠁⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢻⡃⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠄⢦⣤⠤⠴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠶⠶⠶⢶⣤⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣠⣤⠚⠛⠛⠛⠛⢳⣄⣼⠋⠁⠀⠀⠀⠀⠈⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⣧⣄⠀⠀⠀⠀⠀⠀⣨⣿⣤⣤⣀⣤⣤⣤⡤⠶⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⣶⣶⣾⡛⠾⠉⠈⠙⠛⠋⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⡀⠀⠀⠋⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⡠⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠷⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠈⠉⠛⢻⣷⠀⠀⠀⢀⣀⣠⣤⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠿⠛⠛⠉⠉⠁⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⣼⠇⣰⣷⣶⣦⣤⣀⠀⣿⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⢰⡟⢠⣿⣿⣿⣿⣿⡏⢰⡿⠘⣿⠀⠀⠀⣀⣤⣤⣶⣶⡆⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢀⣿⠁⣼⣿⣿⣿⣿⣿⠃⣾⠇⠀⣿⡆⢻⡄⢹⣿⣿⣿⣿⣿⡀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⣼⠇⢰⣿⣿⣿⣿⣿⡟⢰⣿⠀⠀⢸⣇⢸⣇⠘⣿⣿⣿⣿⣿⣧⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⡿⠀⣿⣿⣿⣿⣿⣿⠃⣾⡇⠀⠀⠘⣿⠀⣿⠀⢿⣿⣿⣿⣿⣿⡄⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⣿⣿⣿⣿⣿⡟⢰⡿⠀⠀⠀⠀⢿⡇⢹⡆⢸⣿⣿⣿⣿⣿⣧⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠘⣷⣸⣿⣿⣿⣿⡟⢠⣿⠃⠀⠀⠀⠀⢸⣷⠈⣷⠀⢿⣿⣿⣿⣿⣿⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣈⠙⠻⠿⠟⢋⣤⡿⠃⠀⠀⠀⠀⠀⠀⢿⡆⠹⣇⠘⣿⣿⣿⣿⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠶⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠙⢷⣼⣿⣿⠿⢃⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⣴⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⡾⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠷⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣡⣽⣿⡿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠛⠋⠉⠁

⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⡟⡉⣻⠿⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠲⠙⠓⠉⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⡎⠀⢀⣀⣀⣀⠤⣤⣤⠤⢂⣁⣀⡀⠀⠹⡄⢠⠦⠀⠀⠠⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢉⣉⠀⣤⣤⡄⠀⠀⠀⣤⡀⠀⣉⡑⠂⡗⠀⠋⠀⡀⠀⠑⠨⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡚⢡⣿⣿⣿⣷⣶⣤⡄⢠⣤⣶⣾⣿⣿⣿⡆⠙⡄⠀⠞⣡⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠀⠻⣿⣿⣽⣿⡿⠀⠀⢿⣿⣯⣿⣿⠟⠠⢚⣷⠀⠀⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⣚⠁⠈⠛⠛⠛⠑⠛⠚⠊⠛⠛⠋⠁⠀⣐⠃⡼⢂⡠⠀⠀⠀⠀⠀⠮⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢧⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣠⠏⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⠞⢉⠉⠒⠤⡙⢦⣄⣀⣀⠀⠀⠀⣀⣀⣤⠞⠁⠀⠀⠀⠊⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⡍⠀⡀⠀⠀⠀⠈⣦⠤⠽⢿⣉⣉⣹⠿⠷⠦⢤⠀⡀⠤⢄⠄⡀⠀⠀⠀⢀⠀⠤⠤⠒⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠄⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⠉⠉⠁⠀⠀⠀⡞⠘⠀⠠⠀⢀⠈⠆⠀⠀⠀⠳⣤⡀⠀⠀⠐⠴⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠐⠁⠠⠒⢄⠀⠀⣸⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠋⠀⠀⠠⢹⣦⣌⠛⠿⠶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠰⠀⠀⠀⠁⠐⠙⡇⠀⠀⢀⠘⠄⠀⠀⢠⡇⢀⠀⠀⠀⠂⠀⠀⠀⠁⠌⢉⡙⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣄⠤⢤⣿⣄⣀⡴⠛⣄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⠀⠀⢀⠢⣀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠘⠁⠀⠀⠀⠀⠉⠑⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⣿⠃⠉⠁⠀⢬⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠊⣾⠁⠈⠉⠉⠉⠉⠀⠀⠘⣿⣿⣉⠀⠀⠀⢀⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡷⠀⠀⠀⢠⣴⠀⠀⠀⠀⢿⠈⠻⢃⠂⠤⠞⠁⠀⠀⠤⠀⠴
⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⢹⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡂⠀⠀⠄⡀⠀⠈
⠀⠀⠀⠀⢸⠀⢀⠀⠈⠐⠄⣸⡆⠀⠀⠀⠀⠀⢀⣠⠊⣿⠀⠀⠀⠀⣾⠘⡆⠀⠀⠀⠀⢷⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⣸⣀⣀⣨⠉⠑⠉⠉⠀⠀⠉⡇⠀⠀⠖⠋⠀⢧⣤⡀⠤⠄⡞⠅⠀⠀⠀⠁⠘⠁⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠈⠻⠤⣄⣠⠶⣹⠋⠉⣹⠉⡁⠀⠰⠀⠀⣠⣖⣠⠀⡄⠀⠀⠀⠀⠀⠀⠄⣴⡒⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠈⠠⠴⠃⠠⠁⠀⠀⢀⣴⢋⡽⠁⠀⠀⠀⠀⠀⠀⡲⠘⠻⠶⠻⢦⣄⠀⠀⠃⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣷⡏⠀⠀⠀⠀⡀⠀⡜⠉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠙⡆⠀⠀⠀⠐⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠣⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠂⡠⠇

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠘⣄⡠⠊⢲⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠸⡀⠀⣀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⡯⠤⠒⠉⣀⡵⢀⠀⠀⠀⠀⠀⠀⠀⢧⠔⠉⠁⢀⣀⡨⠦⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠥⠤⠆⠉⠭⠀⠀⡠⠛⡄⠀⠀⠀⠀⣠⠊⠉⠀⠀⠀⠀⠀⣀⠴⢳⢄
⠀⠀⠀⠀⠀⠀⡴⢏⣀⠀⠀⠀⢀⡠⠔⠊⠀⣀⡸⢆⠀⠰⡏⣁⣀⡀⠉⠉⠈⠉⣁⣠⣒⡁⢬⠟
⠀⠀⠀⠀⠀⠀⠋⠒⠂⠬⠭⠭⠭⠬⠭⠭⠥⠐⠂⡇⠀⠀⢱⡀⢄⣀⣈⡉⣉⣀⣀⡠⢄⣀⡎
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢸⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⣀⠔⠒⠒⢂⡲⡖⠒⠒⠒⠈⢩⠀⠀⠀⢸⠀⠀⢠⠋⣰⡢⠢⢡⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡗⢂⠆⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠀⠈⢆⡻⠐⢁⠜⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠤⠤⠄⣀⡀⠀⢀⡆⠀⠀⠀⠀⠣⡀⠀⠀⠈⠉⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢱⠊⠁⠀⠀⠀⠀⠀⠈⠉⠁⡇⠀⠀⠀⠀⠀⠈⠒⠂⢤⢤⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⣖⠀⠃⠀⠃⠐⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⢤⠤⠤⠤⠤⡤⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣛⣛⣀⡀
⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣶⣶⣶⣾⣶⣶⣶⣶⣷⣶⣶⣶⣶⣿⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡀
⢸⣿⣿⣿⣿⣿⣿⣠⠴⣶⣶⣄⠀⠀⠀⠺⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠟⠃⠀⠀⠀⠐⣖⠘⠛⣿⣿⣿⣟⠛⠃
⢸⣿⣿⣿⣿⣿⣿⡁⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣤⣴⣶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡏⠑⠲⣿⣿⣿⡷⠖⠉
⢸⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡀⠀⣿⣿⣿⡇⠀⣠
⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⢀⣀⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⣾⣿⣿⣿⣿⣿⣿⡧⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⠲⢿⣷⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣤⣴⣶⣾⣿⣇⣶⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⠈⠙⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⢶⡶⠻⢻⣦⣤⣾⣿⡆⠀⣠⣴⣶⣶⡄⠀⠀⠀⢻⣿⣛⣽⣿⣮⣽⢸⣿⡛⢿⣿
⠿⠿⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣴⡾⠛⠓⢿⣯⣄⣠⡿⣣⣾⡿⠿⠛⠿⠀⠀⠀⠀⠈⢿⣿⣿⠿⠿⣿⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⣭⣟⣒⣒⣛⣻⣯⣵⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣾⣿⡿⠸⣿⣝⠛⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⡄⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣸
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢸⣿⣾⡇⣾⣿⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⣶⣿⡇
⣿⣿⣿⠀⠀⠈⠻⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⢹⡏⠁⠹⣿⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⣷⣶⣶⣶
⣿⣿⠏⠀⠀⠀⠀⠙⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⡟⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣼⠎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠷⠄⠀⠀⠀⠀⠀⢠⢧⠄⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠿⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⠿⠟⠃⠀⠀⢠⣏⡿⣆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⣑⠶⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣦⡀⠀⠀⣠⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠛⠊⠀⠀⣼⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣷⣶⣿⣿⣿⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠈⠙⠗⠀⠀⠀⡰⠳⡝⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣀⣤⣶⣶⣶⡄⠀⣀⠀⠀⠀⢠⡐⠟⢩⣄⡌⠉⣄⣤⠀⠈⣩⠄⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠌⠀⠐⠾⠿⠁⠐⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⠀⣀⣹⣿⣿⣿⣿⢿⡿⠋⠀⠀⠀⠈⢳⣾⣧⡿⣶⣿⡼⣧⣿⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣤
⠀⣿⣯⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢹⣿⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡗

⠀⠀⠀⠀⣠⡄⠐⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⣤⠀⠀⠀⣶⠀⠀⠀⡀
⠀⠀⠀⠀⠉⠀⠀⠀⠈⠂⢀⡄⠀⠀⠀⢠⠀⠀⠀⠿⠀⠀⠀⠀⠈⠓⠀⠀⠘⣿⣦⣾⡏⠀⠀⣠⣡⣄
⠀⠀⠀⠀⣠⣄⠀⠀⢡⠀⣾⣇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠙⠿⠁⠀⠀⡒⠄
⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⠈⠉⠀⣴⣶⣶⠀⠀⢸⣿⡇⠀⠀⠰⢿⣿⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡆
⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣿⡏⣿⡇⠀⠈⣻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⣿⡇
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⡇⠀⣴⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢹⡇
⠀⠀⠀⠀⠀⢄⠀⠀⠤⠀⠀⢀⠀⢻⠀⢹⡇⠀⠻⠛⣇⠀⠀⠀⡀⡀⠀⠀⢀⠀⠀⠀⢀⡀⠀⣀⠀⡀⠀⠟⠀⠸⠇

⣶⣶⡶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⡀⢀⠀⠀⠤⠤⡄⠤⠀⠀⡄⠀⠄⠀⠀⠒⠀⠀⠀⠀⠀⠠⠀⡄⠠⠄⣀
⠀⢀⠔⠁⠀⢀⣇⠆⠀⠀⣷⡀⠀⠀⡀⠀⠀⠀⢠⢠⠀⠀⠀⢀⠤⠴⢰⠃⠀⢀⠀⠈⢢⠀⡤⠐⠂⠠⢀
⠀⠃⠀⢰⠩⠀⠘⢀⠀⠀⠘⠂⠀⠀⣇⢰⠀⠀⠙⠒⡀⠀⠀⠈⠉⡁⣾⠀⠀⠛⠃⠀⢠⢾⠀⠀⣀⣀⡠
⠀⠀⠀⢸⡀⠀⠀⠸⠀⠀⢰⣲⡀⠀⢸⢸⠀⠀⠰⢶⠇⡄⠀⠸⠋⣠⡏⠀⠀⠀⣠⠔⠁⡙⣄⠀⠙⠢⡀
⠀⢃⠀⠀⠓⠀⢢⠀⡄⠀⠀⣏⡅⠀⣸⢸⡀⠀⢀⣀⠀⣇⣀⣀⣀⣿⡁⠀⣤⠀⠈⠀⡠⣁⣢⡱⠀⠀⡁
⠀⠀⠣⣄⣀⣀⡼⢇⠓⣒⣊⠧⠉⠈⠘⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠃⠄⠹⡝⠂⠴⢮⠁⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠉⠉⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠈⠁⠠⠗⠘⠒⠛⠛⠉
⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⣂⠭⠒⠁⠀⢠⠀⡌⠀⠀⠀⠀⠴
⠀⢀⡈⠁⠀⢀⠄⠀⠀⠀⠠⣄⠄⠀⡀⠲⠆⠆⠀⠴⠗⠂⠉⠀⠀⠀⠀⠀⠀⢣⠀⡠⠃⠀⠀⠀⠚⠃
⢠⣤⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠐⠂⠭⢐⢂⣄⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠠⡀⠀⠀⠐⠒⠒
⢠⡤⠄⠀⠀⠠⠂⠀⢠⠃⠀⠀⠀⠀⠀⢀⣺⣷⡀⡈⢆⢏⠀⠀⠀⠠⠄⠠⠋⠆⡇⠔⠁⠀⠀⠀⠐⠒⠒
⠀⢀⣤⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⣿⣿⢁⠀⠈⠢⡑⠠⠀⠀⠀⠄⢀⡼⠁⠈⠢⠀⠀⠀⡈⠛
⠀⠀⠀⠰⠀⠀⠀⠀⢸⡀⣴⣶⣶⣶⣶⣿⣿⢏⡎⠀⠀⠀⠈⠓⠦⠤⠤⡄⡏⠀⠈⢀⠀⠀⠀⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠳⣝⠻⠿⠿⠿⢛⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⡔⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠂⠂⠐⠀⠸⡵⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⠃⠀⠀⠰⠄⠈⠀⡀⠈⠀⢀⠀⠀⠂⠀⠀⠀⠀⣀⠵⠙⢉⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣊⣥⠶⠛⠉⠁
⠀⠀⠀⠀⠐⢯⣍⣓⠚⠹⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠤⠽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄⠀⠀⠀⣿⣿⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⣰⣿⣦
⠀⠀⠀⠀⠀⠀⣰⠆⣻⣾⠞⠀⠀⠀⠘⡛⠃⢰⡄⠀⠀⠀⠀⠀⣾⣿⣻⡇⠀⢸⡇⠀⠀⢸⣷⣿⡷
⠀⠀⠀⠀⢰⠟⢁⡠⠜⡃⠀⠀⠀⠀⣀⣤⠠⠌⣷⣄⣀⣴⠀⠀⠙⠷⣟⣷⣶⣿⠁⠀⠀⠀⠀⢐⣷⣾⣗⣦⣄
⠀⠀⠀⠀⠀⠀⣡⣶⡾⠁⠀⠀⠀⠀⠙⢿⣷⣃⣀⣀⣉⣭⠀⠀⠀⢠⣿⣿⣿⠇⢀⢀⠀⠀⠀⢰⣻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣼⣿⣿⡷⠦⡤⠄⠀⠀⠀⠀⠉⠙⠋⠉⠁⠉⠀⠀⠀⠾⠿⣿⠏⠀⠙⠛⠲⠶⠶⠿⠃⠘⣽⣯⣿
⠀⠀⠀⠀⢧⠙⢿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣿⡄
⠀⠀⠀⠀⠨⣷⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣿⠀⢸⣇⠀⠀⠀⠀⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠉⣽⣿⡟
⠀⠀⠀⠀⠀⢹⡇⠘⣿⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠈⣿⣷⡀⠀⢿⣿⣷⣤⣤⣤⣦⡀
⠀⠀⠀⠀⠀⠀⣧⠀⡿⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠳⣄⠀⠀⠀⠀⠘⣿⣿⡄⠀⠉⠛⠛⠻⠿⠟⢻
⠀⠀⢀⣤⠤⢤⣤⣠⣇⣤⠤⠤⠤⠀⠬⣽⡷⠦⢀⣤⠤⢤⣤⡀⢀⣼⡥⢤⣝⡇⠀⠀⢠⣖⠛⠛⣤⠀⣤⡀⢤⣤⠤⣤⣉⠀⣤⡤⠤⠤
⢀⣰⠞⠳⣶⡞⠉⣰⡟⠓⠒⠒⠀⠀⣰⡟⠀⠀⣼⡟⠲⢶⣟⠁⢿⣇⡀⠀⣸⡿⠀⠀⠀⠙⢷⣤⣿⠀⠘⣷⡌⢻⣖⠒⠛⣦⣌⠻⣶⡒⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠒⠚⢿⣿⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠒⣿⣿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣿⣿⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⣸⣿⣿⣷⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢀⣴⣿⣿⣿⣿⣿⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢴⡄⢠⡤⠀⣤⡀⢀⣤⠀⣤⡀⣠⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢈⣿⣟⠁⠀⠘⣿⣿⠁⠀⠘⣿⣏⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⠋⢻⡆⠀⣾⠟⢻⣦⠀⣼⠏⠹⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢻⣿⣿⣷⣶⣶⣾⣿⣿⣿⠏⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣈⣛⣿⣿⣿⣿⣿⣿⣿⣭⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⢻⡅⠈⢇⡴⠛⠻⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠈⣛⣓⣚⣓⡰⢦⣴⣚⣵⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠻⢿⠿⠛⠿⠷
⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢹⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡀⣀⢀⣀⡄
⠀⠀⠀⠀⠘⣿⣿⣿⡇⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⢠⣶⣶⣄
⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠀⣠⡝⠒⠶⠶⠶⠶⠶⠖⠒⣄⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣌⠙⠟⠁
⠀⠀⠀⠘⠿⢿⣷⣶⣦⣤⣤⣴⣶⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢻⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠈⠉⠙⠗⠲⠶⠶⠶⠶⠲⠖⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉
⠀⠀⠀⠀⠀⠀⠐


⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠘⠀⠇⡇
⠀⠀⠀⠀⠀⠈⠉⠂⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡓⠂⠀⢠⡿⡀
⠀⠀⠀⠰⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⠠⢦⣉⣉⣠⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣴⣤⠀⢰⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠻⢦⣬⣭⣭⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⣤⣤⠾⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⣰⡦⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠁⠀⠏⢀⣴⠞⠋⠉⠁⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠲⠶⠦⣄⡉⠀⠀⠀⢛⠃⠠⠀⠀⠈⠀⠀⠀⠀⠀⣤⣶⣾⢿⣦⡀
⠀⠀⢀⣤⣤⣤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠈⢿⡄⠀⢵⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢿⣇
⠀⠀⣾⡏⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡆⠘⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⠀⠀⠀⢸⣿
⠀⢸⣿⠀⠀⠀⢀⣄⣠⣀⣤⣶⣶⣿⣿⣿⣿⣿⡿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⣼⡟
⠀⠸⣿⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣠⣾⠟⠁
⠀⠀⢻⣷⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠙⠿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⠏

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⡞⡷⠀⠀⠀⠀⠀⣀⡀⢰⣴⡟⢀⡼⣿⠀⠀⠋
⠀⠀⠀⠀⠀⣴⠟⠉⢹⡇⣾⣸⢃⡄⢠⠶⡆⣼⡡⢃⣾⠇⠙⠻⠴⠟⠀⠈⠁
⠀⠀⠀⠀⣼⠇⢠⡴⠟⠀⣿⣿⣿⢁⣿⣞⡵⠿⠷⠋⠉
⠀⠀⠀⠀⣿⠀⠀⠀⣀⡴⡿⠙⠿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠙⠷⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⢠⡀⠀⠉⣼⣯⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠹⡄⠀⠀⣠⣷⡄⠙⣦⡀⣸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠌⠂⠐⠀⠈⠙⢧⣾⣦⣨⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣝⠃⠀⣰⠀⣴⣤⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⡀⢰⡇⢰⣭⣟⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣟⣽⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣶⣿⣿⣿⣿⡇⠹⠀⠀⣀⣴⡄⠸⡄⠀⢩⣴⣿⡄⠈⠢⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⣇⢠⣶⣿⢟⣿⣭⡀⢻⡀⣠⣿⣿⡇⠀⠀⠐⠔⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣿⠿⠶⠶⠚⠻⡀⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⠻⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠙⠌⠢⢀⣀⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠿⢿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠗⢁⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠂⠀⠀⠀⠀⠀⠀⠀⡀⣠⡿⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⢀⠤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦
⠀⠀⠀⠀⠀⠈⠲⠶⡉⠀⠛⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠛⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠓⣄⣀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢋⡀⠄⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠁

⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⣧⠀⢀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣤⡄⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠘⢷⣼⣦⣾⣷⣠⣿⠀⢐⣷⣶⠀⠀⣿⠀⠀⢀⡀⢰⣾⣆⣴⣦⠀⡀⢀⠀⠀⠀⢰⣿⣧⢀⣀⣤⠄
⠀⠀⢀⣤⢴⣿⣿⣴⢿⣿⣿⣿⣿⣯⣶⣾⣿⣯⣶⣾⣻⣷⣶⣏⣿⢟⣾⣿⣿⣏⠀⣿⣿⣷⣆⢀⣠⣿⣿⠋⠁
⠀⠀⠀⠙⣿⣿⣿⣿⡾⢿⣿⣿⣿⢉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣿⣿⣿⣿⣦⣿⣿⣿⣿⣾⣿⣿⣿⡇
⠀⠀⠀⠘⣿⣿⡿⣳⡇⣿⣿⣿⣿⠈⣾⣿⣿⣿⡟⣿⣿⣿⣿⣿⡝⢸⣿⣿⡟⢿⣿⡿⣹⣿⣿⡗⣿⣿⣿⣇
⠀⢀⣠⣤⣿⣿⣿⣿⣷⣿⣿⣿⣿⡎⣻⣿⣿⣿⣷⣿⢽⣿⣿⣿⢀⣼⣿⣿⣿⣹⣿⣱⣿⣿⣯⣷⣿⣿⣿⣧
⠀⢿⡟⠛⠿⣿⣿⡇⢸⣿⡿⢿⣿⡇⣿⣿⣿⣿⣿⢻⢻⣿⣿⣿⡈⢻⣿⣿⣾⡿⣯⡽⣿⣿⣿⣿⣿⡏⠻⣿⣧
⠀⠈⣧⠀⠀⣸⣿⠀⣾⣿⠃⢸⣿⣿⡿⣧⡿⢻⣿⠚⣿⣿⣿⡟⣷⠺⣿⣿⣿⣿⣿⢿⣿⠋⢉⣿⣿⡆⠀⢙⣿⡇
⠀⠀⣼⠃⠀⡏⢿⢂⢹⣿⠇⢸⣿⢻⣿⣿⠁⠈⣿⡇⢙⣻⣿⠁⡧⡐⢻⣿⡿⣿⣿⢿⡇⠀⣸⣿⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⠃⢸⡏⢠⣿⢳⠘⢈⠋⣿⡺⡆⠀⢿⡟⠈⣿⡿⠀⠰⠀⠈⣿⡿⢻⣿⣿⡃⠀⣿⠇⠸⣿⠄⢸⣿⡁
⠀⠀⠀⠀⠁⠀⢸⠁⠻⡇⠈⠀⠈⠀⣽⠅⠈⠀⢨⡟⠈⢹⡇⠀⠀⠀⠘⢻⡁⠀⠀⣁⠏⠀⣭⠀⠀⠀⠁⢸⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠸⠃⠀⠘⠇⠀⠀⠀⠀⠻⠁⠀⠀⠈⠀⠀⠙⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣧⢤⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⡇⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢹⠀⣿⣿⣿⣿⣿⣿⡆⢉⢻⣿⣿⣿⣿⠀⣿⣩⠟⣿⣿⣿⣿⣿⣿
⡇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⠀⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⠀⣿⣿⣾⣿⣿⣟⣟⡏⣹
⣿⣾⣿⣷⣦⣿⣿⣤⣤⣤⣤⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡍⣿⠀⣿⣿⣿⣿⣻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⠀⣿⣻⣿⣿⣾⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡛⢻⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⡟⢛⡟⠛⢛⡻⠟⠛⠛⠛⠋⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⠀⣿⠘⢿⣿⢿⣿⠿⣿⠏⠁⣸⣿⣿⠀⣷⣮⣿⡾⠋⣠⣴⢶⣶⣦⣄⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⠀⣿⠀⠈⠋⣡⠵⠯⠉⠁⠈⣱⣼⣿⠀⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⣿⣿⣿⣹⣾⣽⣿⣿⣿⠀⣿⢀⣾⣤⣁⣗⣀⡀⣠⣧⣉⣾⣿⠀⣿⣿⠃⣄⣴⢟⡛⠛⠲⡄⠀⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⡘⣿⣿⡿⠈⢭⠰⣿⣿⣿⣿⣿⠀⣿⣻⣇⣿⣷⣿⡏⠑⣄⣿⣾⣿
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⣿⣥⣿⣿⣷⢦⣌⣤⣹⣿⣿⣿⣿⠀⣿⣷⣿⢾⣿⣿⣿⣾⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣿⡟⡟⠈⣿⡟⣿⣿⣿⣿⠀⣿⠏⠉⣿⣿⠿⢿⣿⠿⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢱⣿⡇⣿⣾⣿⣇⣿⣿⣿⣿⠀⠛⠒⠛⠛⠛⠛⠛⠓⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠾⣠⡦⡀
⠀⠀⠀⠀⢠⠤⠤⠤⠠⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⡠⠤⣯⡇
⠀⠀⠀⠀⢸⣧⠶⠐⠖⠶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠟⠀⠠⠀⠤⠐⢚⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠀⢸⠃
⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠷⠄⠀⢤⡤⠾⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣤⣴⡿
⡀⠀⠀⠀⠀⠈⠒⠀⠐⠚⠁⠀⢇⡤⠀⠀⠀⠀⠀⠈⠛⠒⠶⠞⠋⠀⠀⠀⠀⠀⠀⠴⠖⠀⠀⠀⠀⠂⠉⠁⠈⠀⠀⠤⠤
⣙⣃⣀⣀⣠⣿⣬⣿⣄⣀⣰⣧⣀⣐⣶⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣐⣒⣀⣒⣂⣀⣀⣀⣀⣀⣉⣉⣓⣂⣀⣀⣀⣀⣀
⣿⣿⠋⢛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡂⠀⢀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⡿⠂⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠰⣶
⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⠆⠀⠀⠠
⠀⠀⠅⠀⠀⠀⠀⣶⡶⠸⡆⠀⣸⡀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢀⠀⠀⢦⡀⠀⢻⠟⠀⠙⢦⣽⣥⠾⠁⠀⣦⠀⢀⣇⠀⢰⠆⠀⣦⠀⢀⣆⠀⠀⠀⢠⡄⠀⣠⣇⠀⣾⠀⠀⢰⣀⠀⠀⣀⠀⡀⠀⠀⢀⣄⠀⢀⡆
⠀⠈⠀⣀⣻⣋⣠⠏⠀⠀⠀⢸⣿⣿⠀⠀⠀⠘⢦⣙⣟⣤⠟⠀⠀⠘⢦⣙⣟⣠⣄⡀⠀⠹⣦⣙⣯⡴⠃⠀⠀⣙⣿⣤⠴⠋⠀⠙⠶⣤⣘⣟⣠⠞⠁
⠀⠀⢸⡍⣿⣿⠀⠀⠀⠀⢀⣼⠋⠻⣄⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢀⣿⣿⡇⠀⠙⠂⠀⠀⢹⣿⡇⠀⠀⣞⠹⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⡇⢀⡀
⠀⠀⠀⠙⣿⣿⠀⠀⠀⢠⡾⠁⠀⠀⠙⣧⠀⠀⠀⣿⢻⡅⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢸⡟⡇⠀⠀⠙⠀⣿⠻⣆⠀⠀⠀⠀⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⢸⠇⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⢸⡿⡏
⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠘⠁⠀⠀⠀⠀⠀⠛⠘⠃⠀⠀⠀⠀⠀⠘⠃⠃⠀⠀⠀⠘⠃⠘⠋⠀⠀⠀⠀⠘⠋⠃

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠻⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠐⠀⠀⠐⠠⠌
⠀⠀⠀⠀⢠⡾⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣽⣵⣄⣀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠙⣿⢋⡿⠋⠙⠁⠀⠀⠀⠀⠀⢐⣯⣽⡛⣿⡏⠛⠁⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠀⠳⠀⠘⠀⢰⡃
⠀⠀⠀⠀⠀⠀⣿⡌⢿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⠛⢷⣄⠀⠠⠄⠃⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿⡄⠄⠀⣠⣤⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⣿⠁⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢠⡟⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣼⡀⠀⢈⣻⡆⠘⢿⣿⣿⣿⣷⡿⠁⠀⠂⠉
⠀⠀⠀⠀⠀⠘⠋⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠟⠁⠀⠀⠀⠀⣾⣿⡄⠀⠀⣾⣿⣇⠀⠀⡌⠻⣷⣶⣸⣿⣿⣏⣀⡀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠿⣦⣤⡻⢿⣋⣠⡄⢟⢆⢸⡿⣿⣻⡟⣿⣿⣿⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠐⢿⣿⣿⣿⣿⡏⢸⣿⣿⣿⣿⣯⡿⣧⣬⣭⣥⣴⡶⠟⠋
⠀⠀⠀⣠⣧⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣶⣿⣿⣟⣯⣴⣮⣥⣼⣿⢿⣿⠀⣿⣿⣿⣿⡀
⠀⠀⠀⢙⣛⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢛⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⢸⣿⣻⣷⣸⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣧
⠀⠀⠀⢸⣿⡷⠶⠶⠾⠀⠀⠸⠶⠶⢶⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⢸⣿⠙⣿⣹⣿⣿⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⡿⠿⣿⣿⣿⢸⣿⢸⣿⢹⣿⢻⣿⠟⠘⠿⠘⢛⣿⠟⣿⡏⠁
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⡿⠿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢸⣿⠁⠉⢸⣿⢸⣿⢸⣿⢸⣿⠀⠀⠀⠀⠀⣿⡇⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠐⠿⠋⠀⠀⠀⢸⣿⠀⠀⠈⠉⠀⠉⢸⣿⢸⣿⠀⠀⠀⠀⠐⠿⠃⠿⠃
⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠸⠿⠘⠿

⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡡⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠀⠀⢸⣯⣿⡇
⠀⠀⠀⠿⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠐⠶⠷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠂⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⣿⠛⠛⠛⣿⠀⠄⠁⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⣿⡀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠠⠀⠀⠀⠛⠋⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣨⣠⣤⣍⢥⠀⠐⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡄⠀⠀⡄⠀⠈
⠀⠀⠀⠁⣠⣤⣤⣀⡀⠀⠀⠓⠊⠀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠇
⠀⠀⠀⠀⠋⠉⢙⣿⡗⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⢀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣛⡛⣻⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⣄⣀⣀⣀⠀⠀⡇⠀⡇⠀⢀⣀⣀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⢈⠭⠉⠒⠒⠒⠀⠀⠁⠀⡀⡇⠀⡇⡀⠀⠀⠀⠀⠀⠒⠊⠭⢅⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⣄⡀⠙⠲⠦⢄⣀⣀⣀⣀⠀⠀⠙⠛⠛⠛⠁⠀⠀⣀⡀⠀⠀⠤⠦⠘⢁⡀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠈⠉⠁⠲⠖⠠⠤⠠⣤⣤⣤⣤⣭⣤⣤⣤⣤⡤⠠⠤⠐⠲⠖⠂⢉⣁⡄⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢢⠤⢉⣁⠀⠐⠒⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⠉⠥⢴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⡆⠀⡆⢠⣀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣔⠂⠉⠉⠀⠀⠀⠀⠀⠀⠇⠀⠇⠀⠀⠀⠀⠀⠈⠉⠉⢑⡦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠉⠃⠀⠸⠀⠶⠀⠄⠀⠄⠤⠤⠤⠶⠶⠶⠖⠀⠀⠉⠁⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢾⣷⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣾⣿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⣿⣿⣿⣿⣿⣿⣟⢿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⣷⣭⣿⣿⣏⣶⣶⣶⣶⢿⣿⣿⣿⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢿⣿⣿⣿⣿⣯⣷⣮⣝⣛⠿⣿⣝⣻⣯⣷⣿⣿⣿⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⡿⣾⣿⣿⣿⣿⣸⣿⣿⢿⡿⣿⣿⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⣿⣿⣿⣿⣧⣛⣛⣟⣛⣵⣿⣱⣿⣿⣿⣟⣿⣿⡇⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⣿⣿⣿⣿⣿⠿⣟⡻⣿⣷⣝⠿⢿⣛⣼⣿⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢹⣿⣿⡿⢟⣭⣶⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⣿⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠘⣿⡿⣾⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⣿⣿⣮⡻⣿⣻⣿⣿⠿⣣⣿⣿⣿⣿⣿⣿⡇⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⣿⣿⣿⣮⣽⣽⣶⣿⣿⣿⣿⣿⣿⣿⡿⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠙⠀⠤⠤⠀⠀⣀⣀⡀⠀⠤⠤⠖⠘⠃⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠒⠒⠒⠒⠂⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⣤⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣁⣤⣤⣤⣀⣉⡙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⡿⠿⠛⠉⠈⠛⠻⠿⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⢿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠉⠃⠈⠃⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠅⡀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠘⠈⠁⠀⢀
⠀⠀⠀⠀⡕⢢⠙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⢱⠿⠠⠂⠀⢀⠀⠀⠀⠀⢒⡶⢶⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⡗⠚⢾⡔
⠀⠀⠀⠀⠠⢈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠀⠆⠀⠁⢄⢀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⡙⠿⣷⣦⣤⣄⠀⢀⣴⣶⣾⡿⠟⢉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠂⣀⠉⠙⠋⠀⠜⠙⠋⠀⢠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⡐⢂⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠘⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠥
⠀⠀⠀⠄⡀⠑⠂⠀⠌⠀⠀⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠂⠂
⠀⠀⠈⠁⠜⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⡤⡀⠀⣀⣴⣞⣡⡄
⠀⠀⠀⠀⠀⠀⡔⣿⣿⣮⢞⡵⣫⠝⠁
⠀⠀⠀⠀⢠⢖⡣⣞⡿⠁⢀⣜⣡
⠀⠀⠀⠀⣎⣿⣿⣿⣷⣠⠜⠻⡀⠀⠀⢀⡀⠀⠀⠀⠈⠁⢀⣀⡀
⠀⠀⠀⠀⠙⠿⣿⣿⠿⣇⠀⠀⠱⣄⠀⢉⡤⢤⣀⠀⢀⠞⠁⠀⢸⠂
⠀⠀⠀⠀⠙⠓⣼⣅⠀⠀⠑⢢⡄⠀⠙⠺⣆⠀⠈⠳⣼⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⢠⡎⢻⣷⣄⠀⢀⣭⡓⢤⡀⠈⢣⡀⠀⠈⢳⡭⠥⠤⠤⠤⠤⠔⠒⠚⠉⠁⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠳⢿⡈⢮⣑⢮⡀⠈⠙⢿⣦⡀⠙⠲⢤⠤⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠁⢀⡤⢔⡛⠿⠆⠀⠀⠀⠃⠀⠀⢹⡉⠁⠐⠒⠒⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⠋⣉⠴⠚⢇⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈⢆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠻⠻⡖⠚⢻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣚⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⢣⠀⠀⠙⠛⠓⠿⣶⡄⢀⣤⣶⠿⠛⣁⣸⠾⠤⠤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠹⡭⠞⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠧⣄⠀⠀⣾⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠷⢖⠤⣈⣋⢿⠢⣄⠀⠀⠀⠀⠀⠀⠀⠸⡁⢠⡼⠂⠀⠙⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠔⣊⡥⠄⠒⠂⠀⠀⢲⡀⢱⠀⠁⠀⠱⢬⡑⢤⡀⠀⠀⠀⠀⠀⣈⡦⡇⠀⠀⠈⠳⡞⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣈⠳⢄⡠⢊⡵⠊⠁⠀⠀⠀⠀⠀⢰⡣⠷⠞⠀⠀⠀⠀⠈⢯⠓⢬⡒⠒⣒⡩⠥⠒⠊⠹⣤⣀⡀⠠⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠑⣒⣻⡀⠀⣀⠉⣉⣉⡹⠂⠀⠀⠀⢠⡗⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡩⠕⠊⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⣤⣀⣀⣀⣀⣀⣀⢤⠤⣴⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣻⠧⡔⠈⢋⠫⠏⠉⢖⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣍⠋⢀⢿⣾⣿⡛⠺⣧⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣷⣶⠷⠞⠿⢳⣾⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⣀⣀⡴⢩⠃⠀⠀⣠⠎⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⡀⠈⣛⣦⣀⡀⣀⣨⠇⣴⡉⠀⠀⢉⣑⣾⣖⠒⣒⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣏⡫⢟⣽⡩⠊⠀⠀⠉⠒⠭⣓⠿⣏⡩⣟⣿⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠽⠷⠟⠉

⠀⠀⠀⢀
⠀⠀⠀⡄⢀⡘⣄⠀⢹⡀⢀⣀⠀⣠⠃⠀⠀⠀⠸⡟⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢣
⠀⠀⠀⢷⣿⣷⡾⠀⠀⠳⣾⣯⡾⠁⠀⢀⠀⢰⣿⣿⣷⠀⠀⠀⠹⣦⣶⣦⠀⠀⠀⠀⠘⢷⣿⡶⠃
⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣿⡆⠀⠀⠀⠙⠛⣿⣿⡟⢧⠀⠀⠀⣿⣿⠟⠲⠶⠄⠄⠀⠸⣿⡇
⠀⠀⠀⢈⣿⣯⠁⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿⡇⠀⠃⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⢰⣿⣇
⠀⠀⠀⢸⣿⣿⣧⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢠⣿⣿⠇⠀⠀⠀⠀⠀⠹⣿⣿⡀
⠀⠀⠀⠈⠉⣿⣿⠀⠀⠘⣿⠁⠀⠀⠀⠀⠀⠀⣿⢹⡇⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠰⣿⡟⠁
⠀⠀⠀⠀⢰⢣⡃⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠘⠁⡇
⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣾⢿⣶⣶⡆⠀⠈⠀⠀⠀⠀⠀⢀⡷⣿⣿⣿
⣤⣤⣤⣤⣤⣤⣽⣭⣤⣤⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡿⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿
⠈⠻⣌⠻⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⢹⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠋⠀⠀⡀
⠀⠀⠈⢿⣽⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣜⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅⠀⠀⣞⠋
⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⠿⠋⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢻⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⢀⣤⣤⣤⣤
⠀⠀⠀⠒⠚⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠓⠂⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠂⠈⠛⠛⠛⠛


⠀⠀⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠸⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣙⠃
⠀⠀⠀⣺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡛⣛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⣴⣿⣿⣷⡄⢠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⣿⡇⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⣩⣿⣿⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⡇⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢻⣿⡏⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢀⣀⣀
⠀⠀⣿⣿⣿⣿⡇⠀⠘⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⣿⣿⣿⣿⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⣒⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠁⠈⠉⠉


⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⣀⢸⣷⢀⡀⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣾⣧⣿⣧⣿⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣷⣾⣿⣦⣤
⠀⠀⣰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢟⣭⣽⣾⣵⣶⣤
⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣽⣿⣿⣽⡿⢿⣭⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣡⣿⣿⣿⣿⣇⠀⢹
⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⠛⠛⠛⣵⣿⣿⣎⡟⠛⠛⠆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⡾
⠀⠀⣿⣿⣿⣿⢀⣠⣤⣶⡷⠀⠀⠀⠀⠻⣷⣶⣶⣦⡸⢿⡿⢏⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠾⠷⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠛⠛⠛⠋⠈⠙⠛⠋⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠸⠇⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀⠀⢀⣀⠀⠀⣀⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠯⣿⠽⣷⠀⠀⠀⠀⠀⢸⡏⢹⡇⣸⢿⡀⢸⡏⣿⠈⢹⡏⠉⣷⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⣾⣿⠀⠀⠀⠀⠀⢸⡷⠾⠃⣿⢸⡇⢸⡷⣿⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣾⡄⣿⠀⠀⠀⠀⠀⠸⡇⠀⠰⡟⠙⡧⢸⠇⢿⠀⢸⡇⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠑⠛⠀⠀⠀⠀⠀⢠⣤⣤⣤⡄⣤⡄⣤⣤⠀⢠⣤⡄⢠⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠁⠀⣿⡇⣿⣿⡆⢸⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡹⡀⠀⢀⣀⡆⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⣿⡽⣇⣿⢻⡇⢸⡿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⠂⠀⠀⡈⣆⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⣿⡇⣿⡏⢸⡇⢸⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠘⠉⠐⠒⠃⠀⠈⠑⣄⠀⠀⠀⠀⠛⠀⠀⠙⠁⠙⠁⠙⠁⠘⠃⠘⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡀⠌
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⢀⠜⠃⠀⠠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠈⠗⠀⢀⠤⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⠀⠀⠀⠀⠀⣀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣼⠛⠿⠏
⠀⠀⠀⠀⠀⣴⠾⠿⠛⡙⣩⣥⣾⣿⡟⢿⡄
⠀⠀⠀⠀⢀⣥⣀⣶⣿⣿⣿⣿⣿⣿⣿⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡄
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⠀⠀⢤⣤⡄⠄⠀⠀⠀⣄⣴⣷⠶⣏⣠⠼⠗
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠋⠀⠀⢵⠴⢿⡇⠀⠂⠐⠀⠈⠹⣟⣾⣿⣷⡆⣠⡆
⠀⠀⠈⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡔⢂⣀⠀⢈⣭⣿⣿⣿⣿⠇⣿⠔
⠀⠀⠀⢹⣿⣿⣿⡇⠞⠳⠈⠛⠻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⡾⠿⠻⠀⠸⠭⠒⠀⠉⠑⠆⠀⢀
⠀⠀⠀⠈⣿⣿⣿⣷⡀⠴⠶⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠊⣩⠥⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡠
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣦⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣶⣄⠀⠀⡸⠂
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⠃⠛⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣻⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠀⠉⠛⠶⣶⠶⡄⢂⡀⠐⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠐⠒⠉⠁⠀⠀⠳⠶⠤⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠒⠒⠶⠤⠤⠤⠤⠶⠤⠤⠱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⢀⠴⠒⠋⠉⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⠶⠶⠶⠶⠶⠤⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⠉⠉⠉⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠈⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⣴⠟⠉⠀⠀⢀⣀⣛⣛⡛⠻⠿⢯⣛⠿⢦⠀⢦⣼
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⢁⣀⠀⠀⠚⠋⣉⣩⣭⣟⠛⠶⠀⠈⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⣿⣡⠴⠀⠀⠸⠟⣦⣀⣀⣠⣴⣶⡄⠀⠀⠀⠀⣠⡶⠶⣤⠀⣿
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢠⡟⣡⠾⠋⠀⠀⠀⠀⢉⣩⢴⣿⡽⠂⠀⠀⠠⡾⠋⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⢸⠃⣠⠆⠀⠀⣀⣾⣯⣶⣯⡵⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢻⣇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⣴⠶⣠⣾⠃⠀⠀⠀⣉⡵⢞⣽⠏⠀⠀⠀⠀⠀⠀⠀⠿⠶⡶⠿⠦⠀⠙⢷⣄⠀⠀⣠⣄
⠀⠀⠀⣀⡴⠿⠀⠀⣠⢴⡟⠀⢸⠇⡿⠀⠀⠀⠀⠉⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⡟⠉⠻⢶⣤⡶⢶⣄⣀
⠀⠀⣾⠋⣰⠟⢡⣾⡟⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡟⠀⠀⠀⠀⠙⠷⢾⣧⣽⡿⣦⠀⠀⣠⠶⡆
⠀⠸⣏⠀⠿⠀⡿⠘⠇⠀⠀⠰⣄⡀⠀⠀⠀⠲⣦⣀⣀⣀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢿⣶⣞⣡⠞
⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣦⣤⣴⢿⣯⣩⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢦⣄
⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⣙⠶⢤⣤⣤⠶⠋⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⢧⣤⣤⣤⡤⠤⠤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡖⢀⡟
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠙⠻⠶⠖⠂⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠿⣤⡌⠻⠓⢶⣦⣤⣀⡀⠀⠀⠀⢠⡏⢀⡾
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⣴⠏⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡅⠀⠀⠈⢻⡄⠀⠀⠓⠀⠈⣹⠟⠲⠦⣼⣁⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⣶⣶⠶⠛⠁⠀⣀⣼⣄⣠⣄⡀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⣿⠆⠀⠀⢀⡾⠁⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠉⠛⠉⠈⠙⢷⣶⣶⣤⣴⣟⠉⠙⠛⠓⠶⠶⠶⠶⠿⠋⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠄⠶⢦⣤⣄⣀⠀⠀⠙⠿⠟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠙⠻⠶⣤⡀⠀⠸⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢃⣴⠶⠛⠂⠀⠀⣀⠀⠀⠈⠙⢦⠀⣤⣿⣧⡀⠀⣀⡠⣄⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠴⠋⠀⠀⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⢶⣶⡉⣿⠟⢋⣡⡏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⠗⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠀⠀⠀⠈⢻⣽⡏⠀⡈⢻⡿⠳⠶⢦⣤⣤⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣿⠛⠁⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⠛⣅⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⣼⠂⠀⠀⠀⠀⢿⣿⣋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣗⠀⠈⠙⠷⣦⡀⠀⠀⢀⣴⣿⠋⠀⠚⠁⠀⠀⠀⠀⠀⠈⢿⡉⠉⠛⠛⠻⠶⠶⠶⠶⠶⠶⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠛⠛⣷⠀⠈⠙⠆⣴⡟⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠙⠳⢶⡶⣦⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⣰⠿⡅⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⠀⠀⠀⣼⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⢿⣀⡞⠁⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⣀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⣀
⠀⠘⠛⠁⣾⡆⠀⠀⠀⠀⠄⠀⠀⠀⠘⠛⠋⢀⡙⠛⠀⠀⠀⠘⠛⠁⣿⡆⠀⠀⠀⠀⠄⠀⠀⠀⠘⠛⠋⢀⡙⠛
⠀⠐⠾⠖⠀⠀⠀⠀⢀⠠⠀⠀⠀⡀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠐⠾⠶⠀⠀⠀⠀⢀⠠⠠⠀⠀⡀⠀⠀⠀⠿⠏
⠀⠀⠀⠀⠀⠀⠀⡐⡁⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡁⠀⠂⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠐⠃⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠂⠀⠀⠀⠀⠈⠁⡢
⠀⠀⠀⠀⠀⠀⠀⢸⣶⣶⣶⣷⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣾⣶⣿⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢤⣄⠀⠀⣻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠘⣻⣿⣿⣿⣿⣿
⠀⣀⣠⡄⠈⠉⠀⠀⢻⣿⣿⣿⣿⠁⠀⠰⣶⡦⢀⣤⡄⠀⠀⢀⣤⡄⠈⠉⠀⠀⢻⣿⣿⣿⣿⠁⠀⠰⣶⡦⢀⣤⡄
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠉⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣷⠀⠀⠀⠀⣀
⠀⠀⠀⠉⠀⠀⠀⣾⡿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣡⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠐⠉⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⢰⡟⢀⣠⣿⣷⠀⠀⠀⠀⠀⠀⠀⢠⣥⡶⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠀⠀⠀⠀⠀⣀⠀⠀⢁⣀⣀⣀
⠀⠀⠀⠀⣠⣴⢿⣧⣿⣿⣿⣿⠀⢀⣴⡀⠀⠀⠀⠈⢻⣿⠛⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠄⠋⠀⠀⠀⠹⣿⣿
⠀⠀⠀⠀⢈⠁⠂⣿⣿⣿⣿⣅⠀⠀⠉⠀⠀⢀⡴⠱⣿⣿⣧⡀⣰⡏⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⣼⣧⡀⠹⠟
⠀⠀⠀⠀⠙⠛⠛⢻⣿⣿⣿⣿⡷⠀⠀⠀⠀⢾⠙⠗⣉⠻⣿⣿⣿⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⡿
⠘⠃⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠓⣀⠉⠀⠛⠉⠁⠀⠀⠀⠀⣀⣀⣀⠢⡤⠄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠛
⠘⠛⠂⠀⠀⣴⣿⡿⠃⣿⣿⣿⠁⠀⠀⠈⡘⠆⡈⠉⣡⡄⠀⠀⣄⠀⠀⠀⠀⠣⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣷⡀⠛⣓
⠀⠈⠁⠀⠰⠿⢿⣿⣷⣿⣿⣧⣦⠀⠀⠘⠉⡀⠓⣠⡅⢡⡀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⢿⣿⣿⡿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠸⠀⠀⠀⠀⣛⣠⡤⢀⣬⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠒⠀⠀⠀⢸⣿⡆⠀⠶⠀⠀⢸⣿⠏⠀⣾⣿⠃
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠈⣿⡇⠀⠶⠦⠀⠘⣿⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠀⠓⠀⠀⠻⠄⠀⣿⠀⠀⠀⠀⠀⠀⠰⣹⣅⠀⠈⠉⠉⠁⠀⠀⠀⠿⠀⠰
⢀⡤⠤⠀⠀⠀⠀⠀⠒⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤⡲⠄⠀⠈⠁⠀⢀
⢸⡧⠄⣰⠊⢳⡀⡖⢠⣎⣱⠔⡆⢰⢢⣎⣱⠄⡖⠁⠀⠙⡞⢡⡞⠑⡆⣶⠀⣶⢰⡆⢳⡄⡖⠑⣶⠀⠀⡧⠴⠃⣈⢳⠀⡖⠁⣿⠁⢦⢠⠆
⠘⠃⠀⠘⠲⠚⠀⠃⠈⠳⠶⠀⠘⠃⠈⠳⠶⠀⠃⠀⠀⠀⠃⠈⠳⠖⠃⠘⠖⠛⠘⠃⠘⠃⠓⢖⡇⠀⠀⠃⠀⠀⠳⠚⠀⠃⠀⠙⠆⢈⡟

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⡚⣛⠛⡳⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣯⠐⠉⠁⠰⠙⠀⠀⠈⠰⠬⡛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⡞⠠⠀⠴⠠⠔⢀⠁⡹⣶⠶⠶⠶⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⢟⠲⣿⠶⠶⢶⠶⣶⣴⢶⢷⣼⣧⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠒⠒⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠈⠣⣸⡮⠤⠒⠉⠐⠠⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⣟⠗⠒⢤⠭⣄⣈⡐⢰⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣟⡉⠉⠉⢣⡠⠐⠀⠈⢹⢇⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠓⢲⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠛⢛⡷⠾⠻⣾⣧⣴⣮⣾⣧⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠴⠶⠖⠒⠋⠉⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⡞⠋⠀⠀⠀⠀⢻⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣿⡠⠀⠀⠀⠀⡀⣠⠃⠆⠐⠂⠀⠀⠉⠡⣵⢀⡾⠫⠙⠢⣄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣷⣤⠠⠒⣉⣤⡏⠻⡄⢠⣤⣀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣯⠄⠒⠉⠀⠀⠹⣄⡽⠃⣽⠦⢍⡒⠶⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡅⠀⠀⠀⠀⠀⢀⡟⢁⠞⠁⠀⠀⠉⠓⠮⣕⣆⠀⠀⠀⠀⠀⠀⣠⣿⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⣷⣀⠀⠀⠀⢀⠏⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠀⠀⠀⢀⣼⣿⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣷⣦⢀⣠⣈⣶⢻⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⡲⣭⣀⣱⣟⣿⣿⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣿⣾⣿⣿⣿⣮⣳⢄⡀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣮⣽⣾⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⢷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠷⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠶⠶⠖⠛⠋⠁

⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⣤⣄⠀⠀⠀⠀⠀⠀⣀⠀⢀⣴⡆⠀⢀
⠀⠀⠘⠿⣶⣿⣇⣴⡂⣰⠛⠀⠁⠈⠑⢶⣤⠀⠀⠀⣨⣽⠆⡛⡵⠚⠛⠃
⠠⡶⠶⣤⣻⣿⢿⡿⢺⣇⣀⣠⠁⣤⣤⡼⣞⣷⡀⠖⠋⢡⢎⡇⠙⢿⡆
⠀⠀⢠⣤⡍⣾⢀⠻⣿⣿⠛⠁⠀⠈⠉⠀⠹⣿⡇⠀⠀⠉⠘⡇⠀⠀⠁
⠀⠀⡿⠃⠀⣼⡌⠧⠸⠛⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⣰⠂⠀⠰⡦⠐⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⡇⠀⠀⠀⣤
⠀⠀⠀⠀⠀⣿⠀⠁⠀⠈⣸⠿⠀⠀⠀⠀⠈⠉⠓⢤⠀⢀⠀⣷⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⢀⣀⢀⠈
⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⠋⠀⠀⠀⠆⠀⠀⠀⠀⢸⠀⠀⢡⣿⣯⣾⣷⠀⠀⣠⡄⠀⠀⡠⠾⡦⠀⠈⠁
⠀⠀⠀⠀⠀⣿⠁⠀⠔⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠖⠋⠀⠀⠀⠊⢠⣨⡇
⠀⠀⠀⠀⠀⠇⠀⠀⡷⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⢤⡄⠀⠸⡏⠀⠀⠀⠀⠀⠀⠀⡀⣸⢷⣦⣤⣤⡆
⢀⠀⠀⢰⠊⠀⠀⠠⠁⠈⠢⣀⣀⣴⣀⡠⠖⠛⠁⠀⠀⡀⠀⠀⠙⣄⣠⠗⠁⠀⢀⠀⣤⠈⣶⢶⢦⣬⠅⠀⢀
⣿⣟⣤⠆⠀⠀⠀⣼⣧⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠁⠀⠀⠀⣠⣫⠄⢹⣷⣯⢾⣸⣿⣷⣦⣼
⣿⣿⣿⡄⠀⠀⢀⣿⣿⣦⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⢀⣿⣷⣄⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣴⣿⣿⣿⣿⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣠⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢲⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⣰⠏⢷⡀⠀⠀⠀⠀⠀⠀⡘⠋⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⣄⡀⢠⡶⠛⠛⠻⠶⣦⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⠙⡿⠁⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⢀⣿⡁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣴⠟⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⣙⣉⣁⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢰⠞⠉⠈⠙⢷⡀⢀⣤⣤⣤⣄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣟⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣧⠀⣰⣾⢲⣀⡿⠀⠀⣀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⢦⣿⣿⣿⣿⡇⠀⣾⣿⣋⣥⢸⡇⠀⠀⠀⠀⠀⣦⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠉⠉⠉⠀⠹⣆⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣰⣿⣆⠀⠀⠘⠓⣲⡶
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⢰⡏⠈⠻⠿⣶⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣗⡀⠀⠀⠀⣰⣏⣀⣀⣀⣀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣦⠀⢰⡟⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣿⡀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⣇⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⡿⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠘
⠀⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠀⠀⠀⣰⠋⠛⠀⠀⢠⣏⠉⠉⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⢀⣤⠾⢷⣄⠀⠀⠀⠀⢻⡄⠀⢀⣴⠟⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⡾
⠀⠀⠀⢀⡴⠋⠁⠀⠈⠛⢷⣦⣤⣤⣤⣿⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⢀⣿⠁
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢇⠀⠀⠀⣾⠁
⢠⡾⢀⣤⣾⠇⠀⠀⠀⠀⠀⠀⢀⡿⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢻⡏⢸⡆⠀⠀⣿
⣮⣴⠟⢡⡟⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣿⠀⢸⡇⢸⡇⠀⠀⢹
⠛⠁⠀⢸⡇⢀⡄⠀⠀⣤⠀⠀⣿⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⡿⢀⡿⠀⠀⣿⠀⠀⣿
⠀⠀⠀⢸⡇⣼⢷⡀⠀⣿⡄⢠⡟⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⣀⡈⣃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠋⠀⠀⠀⢠⠇⣼⠃⠀⠀⣿⠀⢀⡟
⠀⠀⠀⠘⣷⡿⠘⣧⠀⡏⢳⣀⡇⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⠉⠛⢻⡄⠀⣿⡿⠶⠶⠶⠾⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣀⣠⡏⣼⠏⠀⠀⢰⡏⠀⣸⠇
⠀⠀⠀⠀⠙⠗⠀⠘⢷⣧⠀⠛⠓⠀⠀⠀⠀⠀⠀⣸⠇⢸⡏⠀⠀⠸⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠛⣧⠀⠀⠀⣸⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⣰⠏⢠⡿⠀⠀⠀⠀⣿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠴⣶⣠⠏⢠⡶⠾⠷⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠛⠶⣿⡁⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠀⢸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣀⠀⢀⣾⠃⠀⠀⠀⢸⡏⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⣷⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠻⣧⡞⠁⠀⢀⣀⣀⣸⡃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡏⠉⠛⠻⠿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣄⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⢻⣄⡾⠁
⣀⣀⣀⣄⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣈⣡⣀⣀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⡿⠿⣿⣿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣼⣧⣠⣧⣠⣶⣦⣥⣼⣿⣱⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⠿⠿⢟⣛⣛⣭⣭⣭⣭⣝⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢗⡲⣶⣿⣿⣿⣿⠫⣵⡤⣶⠝⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣮⣵⣿⣿⡿⠿⢿⣿⣮⡃⠫⠾⠇⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣛⣫⢍⢠⣚⠿⢸⣿⣿⣿⣮⡻⢏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣝⡋⠾⠇⢟⣣⣿⣿⣿⠿⠿⣿⣿⣿⣀⠀⣤⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡸⢟⣼⣿⣿⣿⡤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⢰⣭⡝⠻⠿⣿⣿⣿⣿⠛⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⣣⣿⣿⣌⢵⡞⣴⡹⣶⢎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣤⣭⣥⣶⣿⣿⡿