Call up Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⢭⣝⡳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣾⣷⣄⣿⣉⠻⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠋⠙⠛⠿⠟⠛⢻⣯⣥⠀⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠧⡄⠀⠀⠀⠀⠀⠀⠀⠈⣿⠂⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⡿⢳⡄⠀⠀⠐⠒⠀⠀⠀⠀⣯⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣿⠁⠀⠈⠀⠀⠀⠿⠆⠀⠀⢀⣿⡄⠸⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠀⢘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⣯⡍⣡⠆⠀⠀⠀⠠⣶⣶⣳⣶⣏⢉⡹⡝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣙⠛⠁⠈⣱⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣉⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⣮⣛⣛⣿⡿⢿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣭⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠜⠁⠀⠀⠈⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⢢⣂⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠘⠃⠐⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠰⣶⠺⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠖⠴⣿⣷⣦⣄⠠⡀⠀⠀⠡⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠴⠌⢣⡙⢷⡮⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⢀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣤⢀⠀⠓⠤⡀⢲⢦⠄⣀⣤⡒⠈⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣥⣤⣀⣀⣤⣬⣼⠃⢁⣬⣷⠀⠁⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠻⠟⣿⣿⠀⠀⣿⠛⣿⡟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⢄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣽⣾⣿⣿⣕⣚⣦⡀⠛⢆⡘⣛⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣎⡿⢛⠻⣿⣍⣽⡿⣷⠀⣩⡴⠾⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣝⢷⡏⠀⠀⠘⣇⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠿⠿⠿⠿⠿⠶⠾⠿⠿⠿⠿⢿⣿⡄⠀⠀⠀⢿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠳⢦⡶⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⡀⢀⣀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⡀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⢦
⠀⠀⠀⠀⠀⠠⠁⠀⠀⡜⠄⢢⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡍⡇
⠀⠀⠀⠀⠀⠸⠀⠀⠀⢈⡟⢸⠀⠀⠀⢰⡀⠀⠀⠀⠀⡀⠤⣀⠀⠀⢀⠤⢤⡀⠀⠀⠀⠘⢸⠀⠇
⠀⠀⠀⠀⠀⠠⣀⠀⠉⠉⡇⠀⠇⠀⠀⢸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢐⡃
⠀⠀⠀⠀⠀⠀⠀⠙⠲⣯⣖⡅⠇⠀⠀⢸⡇⡕⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡅⠰⠀⠀⣘⠅⠹⠀⡀⠈⠢⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⡏⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠱⢄⣼⠂⠀⢂⠸⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠄⡿⠅⠀⡀⠐⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠉⠀⠀⠈⠐⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢿⡅⡦⡈⠉⣹⠊⢂⡐⢒⠢⠀⡤⠀⠠⢄⠀⠀⢠⡝⡖⠛⠚⠛⠤⢄⡀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂⢷⣉⣉⢠⡥⡈⠉⢢⣱⡄⡇⣌⡈⠀⡃⠀⢸⠇⡇⠀⠀⠀⢀⠄⠁⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⣌⠛⠛⢲⣨⠞⠛⠉⠁⠀⡑⠒⠶⠄⠃⠀⢸⠆⡇⠀⢀⠔⠁⡀⣨⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠘⡏⠀⠤⢈⢟⣠⡌⣀⢸⠀⢳⣦⡈⢱⠀⠀⢸⢅⡇⣠⠂⢀⣄⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⢀⠘⠙⡅⠉⠈⠳⠾⢷⣾⡀⢮⣛⣓⡳⠀⠀⢸⠕⡖⣍⣢⣈⣅⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠨⠂⠣⠪⢀⠀⠀⠘⠌⢳⡗⠀⠀⠀⠀⠀⠐⢸⢿⣿⠇⠉⠙⣷⡾⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠅⠉⠀⠒⠢⠤⣄⣀⡂⠀⣓⢛⠂⠀⠀⠀⢐⢘⢛⡋⠛⠀⣺⣾⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠰⡁⠈⠉⠳⠩⠉⠋⠋⠉⠓⠊⢸⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠍⣻⠀⠒⠂⠹⠷⠶⢏⣐⣀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠘⢆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣻⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⠉⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠠⠀⣇
⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠈⠳⢦⠤⢤⠴⠚⠛⢁⠅⢂⠨⠀⡈⠐⡇⠀⠀⠀⠱⡀⢄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠚⢦⣀⣀⠔⠀⠀⠀⠀⠄⡀⠄⠠⠀⠄⠠⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⠤⠤⠤⠤⠶⠶⠤⠤⠤⠴⠶⠖⠒⠒⠒⠒⠊⢈⢉⡍⢉⢉⡉⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡱⡋⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠕⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣄⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣸⠀⠀⠀⠀⠀⠀⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡇⠀⠀⠀⣤⣤⣤⣤⣤⠤⣄⣀⠜⠁⠀⣟⣀⣀⣀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⣏⠀⠀⠀⠀⡿⢧⠤⢬⡇⠀⠀⡏⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡇⠀⠀⠀⢻⣿⣿⣿⡇⠀⣷⠀⠀⢀⡼⢡⡷⠤⢤⡏⠀⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⠀⠀⠀⢼⣿⣿⣿⠁⠀⣧⣄⡀⠈⠀⢨⠷⠶⣶⠃⠀⠀⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢸⠀⠀⠀⠘⠛⠛⠋⠉⠉⠁⠀⠉⠙⠓⠉⠒⠒⠉⠀⠀⢸⡇⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⡇⢨⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⢿⠀⣫⢎⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡂⠘⠦⠤⠤⠬⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⡚⠛⢲⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣤⣤⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠷⠧⠾⠴⠵⠮⠾⠴⠧⠮⠼⠴⠥⠯⠾⠾⠵⠧⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣷⣶⣶⣬⡙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⡄⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡄⠘⣿⡞⣿⡗⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⠁⣿⣧⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢠⣿⡏⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣹⣿⣿⣿⣿⣿⣿⣷⡀⠀⢀⣾⠟⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣟⣡⣴⣿⣃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠙⠻⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠙⠿⠙⠿⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣰⣄⣀⡨⠿⣿⣿⣿⣿⡝⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠃⠀⢰⣦⣤⣤⣼⣿⠋⠀⠀⠛⠛⣿⣿⡇⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⠀⠀⠸⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠉⠈⠀⠙⠛⡿⢦⣀⣀⣀⣀⣤⡖⢀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⣄⠀⢤⣀⠀⠀⠀⠈⣠⠀⠉⠙⢿⣦⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢺⣷⣶⣭⣟⡢⠤⣴⣯⣰⣿⠀⣸⣿⣧⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠙⢿⣦⡄⡄⠈⠙⠲⢿⣿⣿⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠈⠀⠀⠀⠀⢲⡠⣤⣤⣤⠄⢀⣉⣩⣿⡆⢀⣿⡆⢹⣧⣄⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢻⡇⠈⠀⠙⠃⠀⠉⣹⠛⠷⠘⢿⣀⣾⡿⠿⠛⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣿⣿⡙⠷⣤⢀⣼⡿⠏⠀⢀⠆⠈⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⢀⣤⣿⣿⣿⣿⡛⢿⡿⣿⣿⡿⠓⠶⠋⠁⠀⠀⠀⢺⣄⠀⢸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⠠⠸⣿⣿⣷⣄⠹⣿⣦⡀⠀⠀⠀⣴⣷⡀⢻⣇⠀⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠃⠄⠀⠐⠐⢿⣿⣿⣿⡆⢹⣿⡇⠠⠀⠔⢻⣿⣿⣦⠙⠀⠀⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢨⠀⠀⢀⢠⣿⣿⣿⣿⣧⢈⣿⠃⠐⠊⣴⣿⣿⡿⠁⠀⠀⠀⠀⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⢂⠐⢀⢿⣿⣿⣿⣿⠇⣸⡿⢀⡀⠈⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⣿⣷⣄⠀⠀⠈⢻⣿⠟⢁⣼⣿⢣⣿⠃⠀⠀⠈⢻⣿⣿⣄⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣯⣀⣈⣡⣤⣶⣿⠟⠃⣫⠿⣄⠀⠀⠀⠀⢻⣿⣿⣦⠀⣴⣦⡀⠀⠀⠻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢟⡻⠿⠿⠿⠛⠋⣁⡴⠞⠁⠀⠹⣆⠀⠀⠀⠀⠻⣿⣿⣷⣼⣿⣿⣦⡀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿⡿⠉⠁⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣷⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⣭⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣬⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣼⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣛⣿⣽⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣀⣀⣀⣀⣀⣈⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠂⠀⠉⠉⠉⠉⠉⠉⠉⡍⠉⠉⠉⠉⠉⠉⠉⠉⠑⢶⣄
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢄⠀⡂⠀⠀⠀⠀⠆⠐⠀⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⡆
⠀⠀⠀⠀⠀⠀⠅⠀⠀⠐⢥⣇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠃⠀⠠⠬⢤⡇⠸⡄⠀⠀⠀⡷⠀⠀⠀⠀⠠⠒⠒⠢⠀⠀⠀⠒⠈⠢⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠑⢤⡀⢀⡠⢃⠄⡅⠀⠀⠀⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣯⡆⠃⠀⠀⢰⣧⢰⡁⠀⠀⠤⠀⠀⠀⠀⠒⠒⠒⠀⡠⠀⠀⠀⠀⠀⣗⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠘⡀⠀⢸⡱⠀⠃⢢⡀⠈⠢⢀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⡇⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⠢⣼⠟⠐⠐⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣩⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡄⠀⢸⠯⠈⢀⠈⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⠈⠉⠒⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢿⣇⢳⢌⠉⠉⣳⠐⠂⡐⢒⠢⡄⡠⠤⠠⢄⡀⠀⢀⢨⠓⡗⠚⠚⠛⠦⢤⡀⠀⠑⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⣈⠁⠁⣸⢄⠈⠃⢢⣱⡇⠧⢐⢀⠂⢸⠀⠀⢸⠡⡇⠀⠀⠀⠀⡀⠍⠀⡀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⣀⠓⠛⠫⣮⢪⡽⠖⡚⠓⠁⠐⠬⠥⢤⠸⠀⠀⢸⠀⡇⠀⠀⠠⠊⠀⡀⣪⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢽⠃⢀⡁⠚⡿⢀⡐⠀⠘⡀⠪⣄⠀⠒⡄⠀⠀⢸⡢⡇⢀⠔⠀⠀⡨⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠆⢈⠖⠖⠐⠃⠻⢯⣡⣟⣠⠀⠇⠿⠿⡶⡇⠀⠀⢸⢈⣥⡋⢄⡐⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠀⢂⠠⢈⢆⠀⠀⠂⠠⠘⣶⣷⡌⠛⠛⠚⠁⠀⣆⢼⢶⣷⡷⠾⠿⣿⣿⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠦⢡⠄⢁⣁⠀⠀⠁⠀⠁⣘⣧⣠⡀⠀⠀⠀⠀⡄⣸⢠⡧⣤⠀⣀⣽⣟⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠂⠂⠀⠈⠈⠈⢹⠶⠶⢦⣼⣧⣬⣬⣄⣀⣀⣀⡅⣨⣬⡀⠈⠁⠀⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⣀⠀⡀⠘⣀⣤⢀⣈⠈⠀⠀⠀⠀⠀⠀⠀⢸⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⡀⡏⠀⠉⠀⠉⠉⠛⠃⠤⠤⣄⣀⣀⡀⣀⡤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠑⡄⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⡀⠀⢂⠐⠀⠀⢀⠀⠀⠀⠀⠁⠐⠺⣀
⠀⠀⠀⠀⠀⠈⠀⠨⠀⠀⠁⠂⠉⢀⠱⠾⠴⢤⠴⢖⠫⠋⠀⠄⢊⠡⠐⡡⠄⣫⠀⠀⠁⡈⢲⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠁⠐⠁⠚⠧⣀⣀⡠⡚⠀⠀⠄⠂⠄⠂⠄⠂⠄⠀⠀⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣛⣩⣥⡄⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣮⣭⣳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢞⣭⣶⣿⣿⣿⣿⠷⣽⡿⠿⠿⠿⠿⠿⠿⠟⠁⢿⣿⣿⣿⣷⣮⣗⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠉⣭⣿⣿⣿⢇⡀⠑⢷⣦⣤⣤⣤⣤⡴⠀⠀⣠⣌⣋⣁⢀⣀⣉⣃⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣿⠇⣠⣾⠿⣻⣽⣿⣷⠀⢸⣿⣿⣿⣿⣿⣿⡀⢰⣿⣮⣝⡻⢿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠠⣭⣤⣽⢾⣿⣿⣿⠟⠁⠀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠛⢿⣿⣿⣿⣶⣿⣯⣵⠇
⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⠀⠙⠿⠿⠷⠛⠉⠀⠀⠀⠀⠟⢫⠿⣦⢰⠛⡯⣩⡢⢄⠀⠈⠉⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠯⣿⣷⣶⣖⣀⡀⠀⠀⠀⠀⠀⠀⠀⡴⠻⢆⣴⠗⠀⠑⠶⣗⠈⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⡏⠆⠀⠀⠀⣤⡦⢠⣤⡀⣾⠁⠀⠀⠀⠀⠘⣷⠀⠀⢳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⣸⣿⠃⠋⣀⡁⣯⠀⠀⠀⠀⠀⠀⡿⠀⠀⣸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣔⣷⡿⠃⠀⢰⣿⠃⢀⡰⣯⢙⣈⡳⠤⣤⣤⠶⡀⠣⠀⢠⣿⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⡿⠋⠀⠀⢠⡏⠋⢀⣴⣿⣦⡸⠥⠍⣶⠓⠖⢿⡉⢀⣀⣿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠟⠁⠀⠀⠀⡾⠀⠀⢴⡿⠋⠀⠙⠷⠶⣮⣥⣤⣶⠾⠛⠁⠈⠻⢿⡇⢻⡄
⠀⠀⠀⠀⠀⣜⣿⠿⠋⠀⠀⠀⠀⠀⢸⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣁⣜⣷
⠀⠀⠀⠠⣬⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠁
⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣯⣿⣿⣿⣿⣶⣤⣄⣠⡀⡀
⠀⠀⠈⣷⡾⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠛⠿⢷⡆⡠
⠀⠀⠀⠸⣿⣬⡳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣮⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠈⠻⣿⣷⣭⣕⣦⢤⢤⣤⣐⣒⣦⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠢⡈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠓⠲⠦⠤⠤⠄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠤⣤⣤⣄⣀⣀⣀⡀⠀⠀⡇⢰
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠓⠒⠂⠀⠀⠀⠀⢠⡇⡦⠀⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡇⣧
⠀⠀⠀⠀⠀⢠⢸⠃⠀⠀⠀⠀⠀⠀⡐⣼⠁⢸⠁⣿
⠀⠀⠀⠀⠀⢂⡞⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢸⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣁⣒⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠉⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⢦⡀⠀⢀⠻⡄⠒⣞⣛⣉⣉⠉⠉⠉⠉⠙⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⠦⠶⠊⠉⠉⠉⠉⠉⠉⠛⠛⠶⣤⣀⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣸⠽⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠴⠚⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⠤⢽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠴⠒⠒⠂⠀⠀⠀⠀⡠⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⣾⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⡀⠤⢤⡀⠀⠀⢸⠀⠠⣿⠆⣿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡎⠀⣶⡆⢻⠀⠀⠘⢷⣄⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠘⢦⣀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠳⣄⣀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⠀⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠳⢦⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠖⠛⣇⠀⠘⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣀⣀⡿⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠙⣧⠀⠀⠰⠖⢚⣿⣭⠴⠾⡶⣄⠘⢧⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠉⠙⢳⡄⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠘⢷⡄⠀⠰⣟⡛⣉⣡⣼⣧⡈⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⠀⠴⠶⣏⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠶⠶⠒⣛⡛⢿⣿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢀⣀⡼⣃⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠶⠶⠟⠛⠛⠋⠉

⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠙⢦
⡎⠀⠀⣠⠤⡤⠴⠶⢶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠘
⠁⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣇
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⡆
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⡄⠀⠀⠀⡴⠋⠁⠀⠀⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠀⠀⠀⠀⠀⠀⢸⣿⣷
⣤⣤⣤⣤⣤⣤⣤⣼⣿⣧⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣈⣁⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠈⠉⠀⠀⠒⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⣀⣠⣶⢾⡧⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣠⣼⡿⡟⠹⠃⣾⣷⣧⡀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣫⡅⠀⠠⣖⡒⠂⠈⢻⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡏⠉⡇⠀⠀⠈⠉⠀⠀⢸⡀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡏⣇⠐⠇⠄⠀⠀⠀⠀⠀⠈⣹⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡫⣻⠿⡝⡄⠶⠶⠶⠀⠀⠀⠀⠀⠈⣶⠷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣼⠀⠀⢣⣽⡈⠉⠀⣀⡤⠚⠀⢀⢞⡟⢠⠏⠑⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠳⣄⡬⠆⢈⠏⠉⡗⣄⣠⠔⢁⠎⠀⡞⠀⠀⠀⠈⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡼⠀⠀⢞⠀⢸⣤⠹⡁⢠⠎⠀⣸⠁⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠠⢠⠞⠁⠀⢰⠛⠃⡏⡽⡴⢻⠃⠲⡴⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡉⠒⠒⢻⡇⠀⡆⢸⠀⠀⡶⠁⣷⠃⠀⡴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠳⣄⡀⢸⡇⠀⡇⢸⠀⢸⠇⢠⡏⢀⡞⢱⡗⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠓⣮⢷⠀⡇⢸⠀⡾⠀⡸⢀⠎⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢰⠁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠠⠃⢸⠀⡇⢸⠀⠇⢠⢃⠏⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⠀⣧⠘⣦⠀⣼⡞⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡴⠛⠳⢼⡇⣿⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⣀⡼⠁⠀⠀⠀⠙⠳⢤⣴⢁⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⢼⡁⠀⡀⠀⠀⠀⠀⠄⣿⠀⢣⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⢰⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠉⠢⢄⠀⠀⠉⠢⢜⢦⡀⠀⠀⠀⡏⡇⠀⠀⠀⠀⠀⡰⢇⡀⢀⠔⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠑⠻⢦⡀⠀⠙⠃⠀⠀⠀⢀⠜⠀⠀⠈⠓⠤⣀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠈⠒⢄⡀⢀⣀⠔⠋⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⢠
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠙⠳⢄⡀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⢀⡝⠒⠒⠒⠆⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⡀⠄⢊⡿⠥⣀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⢀⠔⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠀⠀⠐⠀⠱⣄⣴⠏⣀⡀⠀⠀⢀⡴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠬⠭⢒⡯⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡟⠉⠉⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢿⣿⣿⣧⠀⣀⣤⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣫⣶⡿⠛⠻⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⣵⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⢀⣴⣿⡿⠀⠀⠀⠀⠀⣀⠤⠤⡀⢠⡔⢶⠆⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⣿⡿⠋⠀⠀⠀⠀⢀⣴⢃⣶⣦⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⣾⣿⣿⣿⣿⣿⣿⣿⣯⠟⠧⠭⠶⠛⣝⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⠶⢌⠂⠀⠀⠀⠈⢴⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⠀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣄⠟⠄⠢⣄⣀⣠⡾⡤⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡆⢀⡤⠖⠒⠉⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣷⣜⣉⠖⠻⢹⣅⣠⣬⣭⣧⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠟⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣷⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡇⠙⠛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣿⢿⣿⣽⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣩⣶⣦⣭⡑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⣉⠙⡿⣷⣼⣯⣄⣯⣝⣿⣷⣠⣶⣶⣦⡄⠀⠀⠀⣾⠗⠤⣄⡙⠋⠉⠉⠳⣌⠳⡄⠀⠀⠀⣀⣀⣀⡤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢀⣩⣿⣿⣿⡾⣿⣼⣳⣿⣿⣿⣿⣿⣿⠟⣛⣿⣿⣿⣿⡎⠀⠀⠘⡟⠉⠳⣄⠙⣆⠀⣄⠀⠈⠀⡸⠖⠀⠉
⠀⠀⠀⢀⣴⣶⣿⣿⣿⣍⣩⣿⣧⣿⠘⠻⠃⡟⠉⠉⢿⣿⠸⣛⢻⣿⣿⢸⡏⠀⠀⠀⠹⣄⠀⠈⡏⢺⣀⠞⠳⡤⠋⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢰⣿⣿⣿⣿⣿⣀⠁⠾⣿⢋⣿⣶⠖⠀⠁⠀⠀⢸⣿⣶⣿⣸⣇⣿⠾⠿⣶⡀⠀⠀⠈⠳⢶⣧⠜⠉⢀⠎⢀⣴⡗⠀⠈⡀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⣿⣿⣿⣿⣿⣯⣿⢶⡿⢻⢿⣿⡟⠁⠀⠀⠀⠀⠀⠛⠟⠑⠋⠸⠇⢿⠟⠁⠈⠓⣄⠀⠀⠀⠀⠀⠀⣸⢀⠋⡟⢀⡄⠀⠀⡄⠀⢀⡤⠖⠋⠁
⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⢀⡀⣞⣿⣷⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⡇⡎⠀⠑⡞⠀⡔⢠⠀⡔⢫⡀⠀⠀⠀⠀⡤
⠀⠀⣿⣿⣿⣿⣿⣿⣀⠙⢧⣷⣜⢿⣿⣧⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡀⠀⠀⠘⡄⠀⠀⠀⢸⡸⠁⠀⠀⠹⣌⣦⣸⠀⠇⢹⡝⣄⠀⠀⢀⡉
⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⣙⣿⣿⣿⣿⡎⠀⢠⠂⣠⠴⠂⠀⠀⠀⠀⠀⠀⢀⠘⠂⠀⠀⠐⠀⠀⠀⠛⠁⠀⠀⠀⣠⣜⡒⠻⡄⢢⠀⢷⢸⣄⠀⠉⣻
⠀⠀⣿⣿⡏⣿⣿⣿⣿⡆⠀⠙⣷⠈⢻⣿⡇⠀⠀⣼⡇⢿⡗⠀⠀⠀⠀⠀⠈⠀⠀⣀⡶⠄⠀⢡⡀⠀⠀⠀⠀⠀⢸⠁⠄⠍⢳⡘⢮⡇⠸⣳⠻⠋⠉
⠀⠀⣿⣿⢱⣿⣿⣿⣿⣷⣴⡆⢸⡀⠘⠘⠁⠀⠀⠹⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣀⣰⠆⠀⢱⠀⠀⠀⠀⠀⠈⢧⠠⠀⠆⢻⡈⣇⡴⠁
⠀⠀⣿⢃⣾⡿⣽⣿⣿⡟⠘⣦⣠⣧⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠳⣌⣀⣾⡴⠁
⠀⠀⢣⣾⣿⣱⣿⣿⡿⢁⢰⣿⣿⣧⡉⣿⡸⡘⣆⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴
⠀⠀⣿⣿⣧⣿⣿⣿⢣⠇⣼⣿⣿⣿⣿⣿⠇⣇⢻⣄⣦⣾⠢⣐⠦⠤⠤⠤⠖⠒⠚⠉⠁⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⣿⣿⣼⣿⣿⣧⡟⢰⣿⣿⣿⣿⢧⡟⠀⣿⢸⣷⡻⠹⣇⠈⠑⢦⡻⣖⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉⣀⣀⣀⣰⠓
⠀⠀⣿⡇⣿⣿⣿⣿⠇⣿⣿⣿⣿⡟⣾⣠⡇⣿⣼⣿⣿⡇⣶⣶⣦⠀⠙⣆⢀⣠⠶⠚⠋⠉⠉⠙⠳⣄⠀⠀⠘⠷⣶⠶⠿⠭⠵⠶⠒⠛⠉⠉⠹⣏⣨⠟
⠀⠀⢿⡇⣿⣿⣿⣿⡜⣿⣿⣿⡿⣸⣿⣿⣷⢹⡏⣿⣿⣇⠛⣹⣿⡆⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣶⣤⣌⣙⠀⠀⠀⣄⡀⠀⠀⠀⠐⢿⡧
⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⢇⣿⣿⣿⡟⠘⣿⣾⣿⣿⣿⣿⣿⠇⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠉⠻⢿⠿⣦⣀⣀⣽⠛⣿⣭⠿⠿⠷⣄
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣱⣷⣜⠿⣿⣿⣿⣿⠟⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣩⡿⣨⠕⠋⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡟⣿⣿⡇⣿⣿⠟⢷⣦⣭⠉⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡞⠁⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⢻⣿⣿⣜⣛⣃⣼⣿⣿⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠰⡇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡿⠃⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣀⠀⠀⡴⣻⠇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣁⣠⣤⡄⠀⠀⠈⠳⣤⢟⡿⠀⠀⢨⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⡋⠁⠀⠀⠀⠀⠀⠀⣰⣉⡿⠁⢀⣠⣾⢴⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠹⣆⠀⠀⠀⠀⠀⢸⣉⡿⡀⠠⢤⣙⣹⣾⣞⣷⣤⣴⣦⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠘⠇⠀⠀⠀⠀⠞⠹⠃⠙⠆⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀
⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⢀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄
⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠀⠀⠀⢠⡟⠁⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⣴⠟⠁⠀⠀⠀⣠⡴⠶⠂⢀⡟⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠸⣿
⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⡾⠃⠀⠀⠀⣴⠟⠉⠀⠀⠀⠘⢷⣀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣼⠃⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠉⣻⠗⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢿⡀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⢿⣿⡀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠈⠁⠀⠀⠈⠁⢠⡴⠶⢶⣄⣰⠏⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠈⣿⣷⡀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠈⠋⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠈⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣤⣤⣴⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⢀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠻⣿⡆⠀⠀⣼⣿⠛⣿⡇⠀⠀⢰⣿⠛⣿⣷⠀⠀⢰⣿⠟⢻⣷⠀⠀⢀⣿⡟⢻⣿⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣿⡇⠀⣿⣧⣤⣤⣿⣿⠀⣿⣷⣤⣤⣾⣿⠀⣿⣿⣤⣤⣼⣿⠀⢸⣿⣤⣤⣼⣿⡇⢸⣿⣦⣤⡀
⠀⠀⠀⠀⣴⣿⠟⠉⣿⡇⠀⣿⡏⠉⠉⣿⣿⠀⣿⡟⠉⠉⢹⣿⠀⣿⣿⠉⠉⢹⣿⡀⢸⣿⠉⠉⢹⣿⡇⢸⣿⡋⠛⢿⣷⡀
⠀⠀⠀⢸⣿⠃⠀⠀⠙⠿⠿⠟⠁⠀⠀⠘⠿⠿⠿⠃⠀⠀⠈⠿⠿⠿⠋⠀⠀⠈⠻⠿⠿⠏⠀⠀⠀⠻⠿⠿⠟⠀⠀⠈⣿⡇
⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⢸⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⢠⣤⣤⣄⠀⠀⠀⣤⣤⣤⡀⠀⠀⢠⣤⣤⣄⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠀⠙⠛⠛⠁⠀⠀⠈⠛⠛⠋⠀⠀⠈⠙⠛⠋⠁⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠿⠿⠿⠃⠀⠀⠸⠿⠿⠟⠀⠀⠀⠿⠿⠿⠃⠀⠀⠺⠿⠿⠟⠀⠀⠐⠿⠿⠿⠂⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣾⣿⣶⣶⣦⣿⣇
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣶⣶⣶⡄⠀⠀⢰⣶⣶⣦⠀⠀⠀⣶⣶⣶⡄⠀⠀⢴⣶⣶⣶⣾⠿⠛⠉⠁⠀⣴⣦⠀⠈⠉⠙⠿⣿⣦⣀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⢀⣽⣿⠟⠁⠀⠀⠀⠀⠀⠺⠿⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⠀⠀⠀⣀⣀⣀⡀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠛⠛⠛⠃⠀⠀⠘⠛⠛⠋⠀⠀⠀⠛⠛⠛⠁⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢀⣀⡀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⣀⠸⣿⡇
⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠿⠿⠿⠇⠀⠀⠸⠿⠿⠷⠀⠀⠀⠿⠿⠿⠆⠸⣿⡇⠻⠿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣤⡀⠀⠀⠀⠀⠻⠿⠟⢀⣿⡇
⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡷⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣦⣤⣬⣥⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠉⠲⣄⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢀⡄⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠸⠵⠋⢧⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⢄⣀⣀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠑⢄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢀⠔⠁⠒⠉⠁⠀⠀⠀⠀⠀⠉⠙⠂⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠿⠇⠀⠀⠀⠀⠐⠿⠇⠀⠀⠀⠳⣴⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠁⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣠⠃⠀⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡄⠀⠀⠀⠀⣷⣬⣭⣭⣥⣼⠇⠀⠀⠀⠀⣇⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣄⠀⠀⠀⠈⠻⠿⠿⠿⠋⠀⠀⠀⢀⡜⣏⠙⡟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠤⣀⡀⠀⠀⠀⠀⠀⣀⡠⠖⠉⠉⠁⠀⢇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣉⡏⠉⢹⣉⠀⢀⡤⠦⣠⠤⠒⢚⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢄⠀⢀⡠⠔⠊⠉⠻⠤⠤⠤⠟⠉⠉⠀⠀⢳⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢨⠇⢎⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⢇⡎⢨⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⣠⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢷⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿
⣿⣿⣿⣿⡁⠰⣨⠃⢠⣄⡟⠀⠛⠀⡏⠀⡄⠘⠃⢠⡄⢹⠀⣾⣿⣿⣿⡦⠀⣶⠇⢸⠀⠘⠏⠀⡟⠀⠲⣾⣦⠀⣶⣿⠃⠀⢻⡇⠀⠂⣽⠀⣾⣿⠀⠐⢶⣿⣿
⣿⣿⣿⣯⣉⣀⣸⣆⣈⣩⣇⣀⣧⣀⣧⣀⣁⣼⣄⣈⣠⣾⣀⣉⣹⣿⣿⣧⣀⣿⣄⣸⣀⣶⣦⣀⣇⣀⣉⣹⣿⣀⣿⣇⣠⣄⣈⣇⣘⣀⣼⣀⣉⣹⣀⣈⣙⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣿⣿⣿⣿⣯⣭⣭⣽⣭⣿⣿⣿⣿⣿⣯⣭⣯⣭⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣿⣿⣿⣭⣯⣭⣭⣯⣽⣿⣿⣿⣿⣭⣯⣭⣽⣿⣿⣿⣿
⣿⣟⣟⡿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠟⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿
⣿⣿⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣯⣿⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣏⣿⣿⣧⣀⡀⣀⣀⣀⣀⣀⣀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣭⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣷⣽⣶⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁⠉⠀⠀⠻⠟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠤⠄⠀⢶⣽⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⣠⡅⢰⣶⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡦⠀⠀⢰⣶⣿⡇⣺⣿⣙⡻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢃⣤⣤⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢸⣟⡇⠢⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠀⠀⠀⠀⣴⣧⠀⠀⠀⠔⠀⠀⠀⢀⡄⢸⢯⣇⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⡀⠀⠀⣴⢧⣼⡄⢀⣡⣴⡞⠛⣷⠿⠂⡻⠿⠫⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⠿⠇⠀⠠⠧⣼⣿⣷⣿⣿⠟⢛⠞⢁⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡷⠀⠀⠰⣴⣇⣸⠃⡽⠋⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠐⠆⠠⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⣧⡆⠀⢠⣏⣠⡋⣹⠟⠡⠈⠠⠀⠀⣠⣶⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢰⣯⠏⠀⠀⠀⠀⠀⣠⡄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⢸⣇⡻⠹⠃⠀⢀⣀⢠⣼⣿⣿⣿⡇⠀⢸⡄
⠀⠀⠀⠀⣠⡀⠀⠀⠄⠔⠀⠀⠀⣴⠾⠿⠻⠈⠰⠶⠾⠿⠿⠛⠛⠀⠁⠀⠀⠀⠐⠤⡄⡘⢋⡟⠁⠀⠀⠙⣿⣶⣿⡿⢛⣿⠃⠀⢸⡇
⠀⠀⠀⠀⠀⢻⡄⠀⣠⢠⣶⣠⣤⡄⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠴⠆⠀⠱⠇⡝⠀⣠⣴⣶⣿⣿⡿⠋⠀⢰⡏⠀⠀⠀⡇
⠀⠀⣠⠀⠀⠀⠱⣾⠀⠀⠈⠉⠛⣷⡶⠄⡀⠐⠷⠀⠤⠀⠀⠀⢤⡾⠟⠋⠀⠀⠀⣠⡄⠀⠀⠀⠉⠉⠙⠛⠋⠀⠀⠀⢸⡅⠀⠀⢀⡇
⠀⠀⠁⠀⠀⠀⠀⠡⠤⠴⠶⣣⡴⣿⣵⣿⡀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⡠⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠘⢿⣶⣶⡿⠁
⠀⠀⠀⣿⠁⠐⠒⠛⠛⢿⣿⣟⡇⠛⠳⣾⡉⠛⠒⠀⠐⣿⣿⡏⠀⠀⠀⠐⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠑⢄⠀⠀⠉⠉
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢿⣿⡇⠀⠀⢸⣧⡀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠿⠄⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⢺⣿⣿⠀⠀⠀⣿⡿⠗⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠰⢷⣦⣀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢀⡀⠀⠈⠻⣷⡄⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠈⢿⣆⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠘⣿⡄⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⢿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢰⣿⠁⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⢿⠿⠿⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠋⠀⠹⢿⣿⣿⣿⡛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠉⠁⠈⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠁⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠿⡞⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠆⠢⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢠⠋⣟⣿⣿⠟⠑⢀⣀⡠⠔⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⠃⣀⣤⣴⣧⠀⠀⠀⠀⠀⢰⣿⡿⠃⠀⠀⠀⣀⣤⣹⣧⡀
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠿⣿⠇⢰⣿⣿⣿⣿⣧⣀⡄⠀⢠⣿⣿⠃⠀⠀⢠⣐⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣷⠁⠀⠀⠂⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠉⠛⠻⠿⢿⡟⠀⠀⣿⣿⣿⠠⣜⢦⣷⡿⠏⡉⠂⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⣿⡝⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣯⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣽⣿⢻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠉⠀⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠒
⠀⠀⠀⠀⠀⣀⣀⣀⣤⣄⠀⢀⣤⡤⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⡀⠈⠋⣀
⠀⠀⠀⠈⣀⡙⠛⠻⠛⠛⠀⠒⠒⠒⠊⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣤⣥⣭⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⢠⡄⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢿⡿⣿⣿⡏⠙⢿⡄
⠀⠀⠀⠀⠀⣧⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⡇⢿⡟⡇⠀⠈⠇
⠀⠀⠀⠀⠀⢻⠘⣟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠘⠃⠘⠀⠁
⠀⠀⠀⠀⠀⠈⠛⠣⠤⠤⣤⠤⠤⠔⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣀⣁⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠀⠉⠉⠉⠉⠐⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠀⠠
⠀⠀⠈⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⢡⠀⠀⠀⠈⡄⠀⠀⠂⠀⠀⠀⢀⣀⠀⠀⢀⣀⡐⠁⠀⠀⠀⠀⢰⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⢱
⠀⠀⠀⠈⠀⠀⠀⡠⠠⠤⡀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡔⠀⢀⡠⢤⠤⠒⠉⠀⠀⠀⠀⠀⠀⠉⠐⠠⡀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠈⠀⢰⠈⠀⠀⠀⠈⠀⠦⡀⠀⠀⡇⢸⠀⠀⠀⠀⠉⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⢘⠀⠀⠀⠀⠀⡠⢀⡤⠴⢎⠑⢄⠀⢸⢸
⠀⡠⠄⠀⠀⠈⠳⣖⡒⠂⢐⠚⠀⡄⠀⠀⡅⢸⠀⠀⠀⠀⠀⠀⡆⠀⠀⠐⠒⠂⠐⠒⠒⠒⠒⠒⠂⠀⠀⢘⣠⠼⠀⠀⠀⠀⢎⠎⠁⠀⠀⠀⠱⡸⠀⢸⢸
⠀⡄⠀⠐⢀⣀⢸⡁⠀⡀⡂⢠⣴⠃⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⡘⢰⠠⠀⣀⣀⠀⠗⠤⠰⠆⠢⠀⢀⠟⠀⢸⢸
⠀⢠⠔⠚⢋⠨⠿⡉⢉⣉⠝⠁⡏⠀⠀⠀⡇⢘⡠⣤⠀⠀⠀⠀⠀⠑⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⡴⠚⢋⡡⣿⠀⠀⠀⠀⠀⣶⡟⠀⠀⢸⢸⣀⢤
⠀⡃⠀⢀⣴⡪⠄⠐⣆⠀⢀⡂⠠⠤⣀⡀⡇⢘⢰⢠⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠀⠤⠤⠀⠐⠂⠀⠤⠀⠵⠎⠀⠀⡠⠊⠀⢇⠋⠀⢁⡠⠊⠀⠀⠀⢸⢸⠁⡂
⢀⠆⠒⠙⣉⢹⠆⠀⠈⠋⠋⠀⠀⠀⠀⠀⡇⢸⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠒⠘⣉⠩⣷⣤⠟⠉⠉⠙⡦⠠⢀⠀⢸⢸⠀⢸
⠈⠀⠀⣀⣤⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣤⣎⡁⠇⠀⠀⠀⠀⡀⠀⠀⠑⢾⢸⠀⠈
⠀⢀⠞⠛⣍⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼⠛⢋⡈⡿⠀⠀⠀⠀⠀⠐⡀⠀⠀⢸⢸
⡜⠁⠀⠀⡠⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⡠⠐⡇⠀⠀⠀⠀⠀⠀⠐⡄⠀⢸⢸⠀⢰
⠇⠀⢀⢾⡄⠀⠀⣀⠤⠒⢁⡠⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠼⠀⢠⠳⠀⢀⣀⣀⡀⠀⢀⠁⠄⢸⢸⠀⢸
⠘⡄⢸⠘⠃⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠒⠃⢸⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠒⠃⠐⠂⠀⠀⠀⠀⠀⠐⠂⠚⠂⢸
⠀⠐⣼⠀⠀⠀⠘⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⢸⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⡄
⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⢀⠠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡂⢄
⠀⠠⠁⢔⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠆⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠰⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠜⠈
⠀⢠⠀⠀⠈⠑⠂⠠⠤⠤⠤⠄⠐⠈⠁⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠈⠁⠒⠠⠤⠤⠤⠄⠐⠂⠉⠀⠀⠰
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰
⣿⣿⡟⠛⠛⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣼⠃⠀⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿
⠀⠀⠀⢀⣀⢿⠏⢠⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡋⠉⠀⠀⢹⣿
⠀⠀⢠⣎⠿⠀⢠⣬⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠻⠺⠃⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡀⠸⣭⡍⢹
⠀⢰⠋⢩⡖⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠁⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇
⠀⠈⠉⠀⠛⠀⠀⣔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠰⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠻⣆⠴⠀⢨
⠀⠀⠂⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⣠⣄⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⡀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⢁⠙⠳⡄⠀⠀⠀⠀⠀⠀⠐⠛⢻⣷⣄
⠀⠀⠠⠀⠀⣠⣤⣰⠠⢀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠛⠁⠀⠀⢠⡄⠀⠀⠀⠀⠀⢋⣿⡀
⠀⠀⠃⠈⠠⣯⡏⠘⠻⠇⠀⢀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⢘⣧⠀⠀⠀⠂⡀⠈⣿⣧
⠀⠀⠉⣙⡦⢄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣩⣿⠟⠃⠀⠀⡄⠀⠀⠀⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⢻
⠀⠀⠙⠓⢠⠈⠀⠤⠇⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⣀⢐⣼⢁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣀⣿⡸⠋⠉⠉⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⢿⣿⡇
⠀⠀⢰⣿⡍⣷⠀⣁⣤⣸⠿⠟⡒⠀⠀⠀⠀⠀⠀⠀⣠⣤⣥⢠⡧⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠑⠂⠸⣿⠃
⠀⣤⠈⠉⣀⣀⣤⣾⢿⣾⠀⠃⠘⠀⠀⠀⠀⠀⢀⣄⠛⣃⢀⣤⡙⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣾⣇⡀⠀⠰⠁
⠀⣿⡆⠛⠋⢉⣼⡿⣼⣿⣃⡀⠐⠃⠀⠀⠀⠀⠈⠈⠀⠈⢹⠙⡁⠀⠀⠀⠀⠀⠀⢦⣸⣿⣯⣭⣭⣿⣷⣰⠂
⠀⣿⠁⠀⠰⠞⣻⡗⠐⠹⣿⠀⠀⠀⠀⠀⠀⠘⠛⠒⠳⣾⠿⠃⠐⠀⠀⠀⠀⠀⢀⣾⠋⣿⣟⣛⣛⣿⣿⡟⠤⢤⠄
⠀⠁⠀⠙⠛⠛⠉⠙⠋⠚⠛⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠛⠛⠛⠛⠛⠋⠀⠀⠐⠂

⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡇
⣿⠛⠛⠛⠿⡿⠟⠟⠛⠛⠟⢿⡿⠻⠟⠛⠻⠻⣿⠟⠛⠛⠟⠟⠻⡿⠻⠿⠛⠛⠻⢿⡿⠛⠛⠛⣿⡆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⢀⣤⣤⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⣤⡄⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⣿⡇
⠛⠀⠀⠀⢸⡏⠀⠉⠀⢹⣿⠈⣿⠁⠉⠉⠈⣿⠀⣿⡏⠈⠉⠁⢹⡇⢸⡏⠀⠀⠈⢸⣿⠀⠀⠀⣿⡇
⣀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⣿⠀⠀⠀⠸⠿⠿⠿⠿⠾⠟⠐⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠇⠸⠿⠿⠿⠿⠿⠟⠀⠀⠀⣿⡇
⣿⠀⠀⠀⢰⣶⣶⣶⣶⣶⣦⠠⣶⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⣶⡆⢰⣶⣶⡆⢀⣀⣤⣤⣴⣶⣿⣧⣤⣄⡀
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⢸⡇⢸⣇⣤⡶⠟⠋⠉⢀⣀⣀⣀⣀⠈⠉⠛⠷⣦⣀
⣿⠀⠀⠀⢸⣇⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠘⠁⣼⡿⠋⢀⣤⡶⠟⠛⠋⠿⠋⠛⠻⠷⣦⣄⠈⠻⣷⡄
⣿⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠛⠛⠀⠛⠛⠛⠛⠃⢀⣾⠋⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠈⢿⣆
⣿⠀⠀⠀⢰⡶⠶⠶⠶⢶⣶⠀⣶⠶⠶⠶⠶⣶⠀⣶⡶⠶⠶⠶⣾⠇⢀⣾⠏⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⠀⠈⢿⡆⠀⢿
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿⠈⣿⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⢸⡿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸
⣿⠀⠀⠀⢸⣇⣀⣀⣀⣸⣿⠀⣿⣀⣀⣀⣀⣿⠀⣿⣇⣀⣀⣸⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠀⠀⠲⣿⡇
⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⢹⣷⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⠀⢠⣿⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢀⣾⠃⢀⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⢀⣾⠏
⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⣷⣄⠈⠙⠿⣶⣤⣤⣶⣦⣤⣴⡾⠟⠉⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣤⣀⡈⠉⠉⠉⠁⣀⣠⣤⡾⠗
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤
⣿⣿⣀⣩⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠿⠟⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠍⠀⠀⣫⣭⡀⠹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀⠀⠀⠀⠀⠈⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣇⢠⣀⢠⠀⠀⠀⢀⣤⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣆⠁⠀⠀⠀⢀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡏⠁⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠞⡼⣿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⢠⣿⣿⠋⠀⠀⠀⠀⣸⣤⠿⣆⠀⠀⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢇⣶⣿⡟⠙⡆⠀⠀⠀⢴⠇⢸⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⠋⢚⠃⡀⢙⡦⠒⢡⡞⠀⡇⠈⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠙⢻⡟⠛⠛⠛⠛⠃⠠⡇⠀⡆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⢀⠱⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠤⠀⠀⠀⠀⠀⠈⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⠀⡀⠀⠁⠀⢠⡀⠀⠆⠀⠈⠄⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⢠⠄⣀⠤⢀⡀⠤⠐⣢⢄⡸⠯⢄⣐⣄⣠⠤⢄⠀⢠⠀⠙⢶⡞⠸⠇⣸⣇⣧⠀⠐⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣥⣽⣆⣛⢱⣶⣛⣡⣶⣏⣉⣶⠚⠓⢻⣤⢤⡽⣷⣾⢃⡘⠂⢀⡠⠀⣲⡾⠓⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠠⡀⠈⠈⠀⠉⠋⠀⠹⠿⠟⠙⠟⠛⠀⠙⠳⠝⠏⠩⠇⠙⠛⡁⠉⢉⡀⠌⠤⠎⠉⠉⠀⠈⠓⢦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⣀⡀⠀⠀⠀⠂⠐⣀⣤⡴⠖⠋⠁⠀⠀⠀⠀⠤⢴⡶⣶⠟⠐⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡷⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠋⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠁⠳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⣿⠂⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣠⠏⡴⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢦⡙⢆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢰⠃⡼⢡⠞⣠⠆⠀⠀⠀⠀⠀⠀⠐⢆⠙⡄⢳⡈⣇⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡟⢸⠃⡞⢰⠇⠀⢰⠒⠂⡠⠒⠂⠀⠈⣇⢹⠀⣇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⢸⠀⠀⠘⠛⡖⢧⡐⣲⠀⠀⡿⢸⠇⡿⢸⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢻⠘⣇⢹⡈⢧⠀⠈⠉⠁⠀⠉⠁⠀⣰⠃⡼⢠⠇⡼⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⢧⠘⣆⠱⠄⠁⠀⠀⣸⣷⠀⠀⠈⠁⠞⣡⠎⣰⢃⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠳⠌⠁⠀⠀⠀⢠⡔⢆⣄⠀⠀⠀⠀⠁⠜⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢞⣵⣷⣝⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⢯⣭⢏⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢟⣣⣤⣛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢒⣛⣛⣛⣛⣓⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣮⡝⠛⣭⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢟⣫⣵⣾⣿⣶⣭⣛⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣽⣿⣿⣿⣽⣿⣿⣿⣬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⡆

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢸⣙⣉⣉⣉⣙⣳⠀⠀⠀⠀⢸⣙⣉⣉⣉⣛⡃⠀⠀⠀⠀⢘⣛⣉⣉⣉⣋⡇⠀⠀⠀⠀⣞⣋⣉⣉⣉⣋⡆
⠀⠀⠀⢸⠀⠴⡆⣲⠂⢸⠀⠀⠀⠀⢸⢰⣲⣖⣖⡆⠁⠀⠀⠀⠀⠈⠠⠄⠤⠠⠄⡇⠀⠀⠀⠀⡇⠀⠠⠤⠤⠀⡇
⠀⠀⠀⠸⣀⣀⣁⣉⣀⡼⠀⠀⠀⠀⠸⣘⣚⣓⣓⣃⡄⠀⠀⠀⠀⢠⣘⣁⣃⣘⣀⠇⠀⠀⠀⠀⢇⣀⣈⣉⣉⣀⠇


⠀⠀⠀⢠⠴⠤⠤⠤⠴⢤⠀⠀⠀⠀⢠⠴⠤⠤⠴⢤⠀⠀⠀⠀⠀⢠⠤⠤⠤⠴⢤⠀⠀⠀⠀⠀⡤⠦⠤⠤⠤⠦⡄
⠀⠀⠀⢸⠒⣒⣒⣒⠒⢺⠀⠀⠀⠀⢸⠒⠒⠒⣒⣺⠀⠀⠀⠀⠀⠘⠒⠒⢒⣒⣻⠀⠀⠀⠀⠀⡗⢒⠒⢒⠒⠒⡇
⠀⠀⠀⢸⠀⠣⢌⠴⠁⢸⠀⠀⠀⠀⢸⣀⣀⣸⠰⠆⡃⠀⠀⠀⠀⢂⣀⣀⣎⠀⠀⡇⠀⠀⠀⠀⡇⠐⠐⠐⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⢸⣉⣉⣉⣉⣉⣽⠀⠀⠀⠀⢸⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⢨⠭⠭⠭⠭⢽⠀⠀⠀⠀⠀⣯⠭⠭⠭⠭⣽
⠀⠀⠀⢸⠀⠀⠆⡖⢲⠞⠀⠀⠀⠀⢸⠀⠀⢠⠊⡙⠆⠀⠀⠀⠀⡀⠀⠀⡰⠊⡙⡄⠀⠀⠀⠀⡇⠀⠀⡴⢊⡛⡄
⠀⠀⠀⠸⠤⠥⠥⠗⠁⠀⠀⠀⠀⠀⠘⠒⠒⠚⠤⠥⠃⠀⠀⠀⠀⠘⠒⠒⠓⠬⠵⠃⠀⠀⠀⠀⠑⠒⠒⠳⠬⠴⠁


⠀⠀⠀⢀⣰⣒⣒⣒⣒⣢⠀⠀⠀⠀⢠⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⡆⠀⠀⠀⠀⡴⠶⠶⠶⠶⠦
⠀⠀⠀⢸⢺⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠘⠀⢀⢀⡀⠀⡇⠀⠀⠀⠀⡏⠉⠉⠉⡡⣶⠄
⠀⠀⠀⢸⠘⠤⠤⠤⢤⠜⠀⠀⠀⠀⢸⠀⠘⠲⠃⠀⡀⠀⠀⠀⠀⢀⠀⠰⠛⠄⠀⡇⠀⠀⠀⠀⢧⡀⢠⣪⠖⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⣀⠔⠀⠀⢀⡴⢀⢳⠹⣿⡇⢀⣀⡀
⠀⠀⠀⢀⣴⠁⠠⠀⠀⣿⡂⠀⢸⣿⡿⠃⣼⡇⣿⣦⡀⠀⠀⢀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⢻⡿⠟⠉⠤⠚⠉⠸⠟⠛⠛⣶⣄⢸⡆
⠀⠀⠀⠰⠢⠀⠀⠀⡠⠐⠋⠀⠀⠀⢀⣀⠀⠀⠀⠀⠲⠿⠻⣿⣷⡀
⠀⠀⠀⠀⢣⣄⢀⣠⠀⠀⠀⠀⠀⢠⠟⠀⢀⣠⡀⠀⠀⠀⠀⠹⣿⣷⠄
⠀⠀⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢠⣼⣯⡑⠠⠤⢀⣈⠳⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⠖⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡙⠂⢐⠳⢿⡇⢨⡻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣴⠃⠉⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠙⠿⣯⣙⣿⠻⠄⢠⢹⣿⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⡴⢋⣭⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⣀⣸⣃⣤⣤⣤⠤⢂⢠⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣦⣄⡀⠙⠿⠟⠁⠀⢀⠀⠀⠀⠀⠀⠀⢰⠛⠻⠮⠛⠿⢿⠿⠿⠿⠿⠷⠾⣿⣿⠿⠿⠿⠿⠟⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⢻⡿⣶⣄⣀⣀⣴⠂⠀⠀⠀⠀⠀⠀⣼⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⡆⡾⢀⣿⡿⠟⠉⠣⢄⡀⠀⠀⠀⠀⠀⢸⠁⠈⠙⠛⠋⠉⠛⠟⠋⠉⠉⠛⠛⠿⠋⠉⠙⠋⠁⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⡤⢴⣄⠀⠀⠀⠀⢰⠒⠂⠁⠀⠀⢿⠷⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⠶⠾⡟⠀⠀⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⡏⠀⠀⠹⣗⢦⢀⢸⡈⠀⠐⠶⠲⠀⠺⠂⠀⠀⠄⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠛⣀⣴⣖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠄⠀⠀⠙⠆⠀⣸⠃⠀⠀⠀⠀⠀⠈⠍⠑⠒⢆⣤⣦⣤⣤⣤⡤⠤⢤⣤⣤⣴⣦⠰⠦⡀⠻⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⠀⠐⢈⡄⡞⠀⠀⠀⠀⠀⠀⣁⠔⠚⠓⠋⠀⠁⠀⠉⠉⠀⠀⠀⠉⠂⠚⢿⠀⠀⠀⠀⢠⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠂⠀⠀⢀⠀⢹⠀⠀⠀⢲⡖⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠐⡀⠸⣇⠀⠀⠀⡿⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⣿⢿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢸⡂⠀⡉⢀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢻⣞⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠅⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡟⣇⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡼⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⠶⠚⠛⠃⠀⠀⠀⠀⠀⠇⣼⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⡀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢦⡀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢈⡀⢀⣷⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠐⠛⠀⠀⠀⠀⠀⠀⠐⠛⠉⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠿⢿⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⡗⡲⠿⠟⠉⠉⠉⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⣀⡀⠀⢀⣄⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣿⣿⡿⠀⢫⣥⠦⠀⠈⠋⠈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⠶⠀⠀⠀⠐⠢⠴⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠈⠒⠲⠦⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠈⢳⡀⠀⠀⠀⣿⠛⠢⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠹⠀⠀⠙⢄⠀⠀⠇⠀⢴⣼⡛⠛⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⠀⠀⠀⣧⣤⣤⣤⠀⠰⡂⠄⢀⡾⠀⠁⠀⢰⠀⢨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢰⠘⠋⠉⠉⠀⠈⠐⢌⣠⠋⠀⠀⠀⠀⢸⢀⡼⠃⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢌⡄⠀⠀⠀⠀⢸⢸⠇⠀⠀⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⢀⡜⠈⠀⠀⠀⠀⢀⣠⠂⠤⢀⣣⠀⠀⠀⠀⠘⣸⠀⢀⣰⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⠀⢠⣾⣿⠇⠀⠔⢋⣁⠀⠩⠖⠀⠈⠉⠂⣤⠀⣰⡿⡶⢛⣁⡁⠉⢣⡀⢂
⠀⠀⠀⠀⠀⠀⠀⣶⠾⠭⠤⠄⣠⡟⠉⠻⣶⡤⢀⣀⣘⣆⣀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠙⢶⣭⣥⣤⡄⠁⢸
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠀⣀⡿⠁⠀⢶⠋⠁⢀⣼⣿⣿⣿⣿⣿⣶⣦⣟⣿⣿⡃⣠⢀⡀⠿⠿⠿⠛⠛⠛⡄
⠀⠀⠀⠀⠘⢇⣬⠴⠊⠁⠀⠀⠀⠀⠀⣬⡲⡾⣻⣿⣿⣿⣿⣿⣿⡟⠉⠭⠥⠤⠭⠝⣻⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢀⣀⡤⠖⠋⠁⢈⡉⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣿⡤⠞⠒⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠈⠂⠒⠒⠛⠉⠁⠀⠀⠀⢀⠎⠛⠢⠤⠀⢐⢒⠀⠀⠸⣀⡀⠀⠀⣀⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⢠⣤⢾⣿⣶⡟⣻⣟⠛⢟⢫⣠⡉⢭⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣵⠿⡿⣹⢁⡿⠃⢸⠈⣮⣿⢷⢸⣇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡠⠚⠣⠶⠄⣧⢈⡇⣦⠄⠀⠐⠛⠘⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⣼⡄⠀⠈⠁⠀⢸⢸⠇⣍⡀⠘⠛⠛⠛⠻⠿⠿⠿⠿⠶⠒⢁⡀⠀⣠⠞⣷⡐⠉⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⣓⠀⠀⠀⠀⣿⣿⢰⣿⡙⠛⠛⠲⠶⠶⠶⠦⠤⠶⠶⠖⠛⠁⠀⣧⣤⣤⡷⠶⢶⡇
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣿⣧⣭⠄⠀⠀⢸⣿⣇⣾⣃⡄⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⠀⣁⣀⣀⡇⠀⠀⡇⠀⢸⣇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢠⢀⡾⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠳⠶⠞⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠙⠿⠏⢁⠀⠀⠀⠀⠀⠙⠓⠶⠤⠤⢤⠤⠤⠴⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣁⣀⣉⣉⣉⣉⣉⣉⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠛⠛⠛⢛⡛⠛⠛⠛⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠶⠶⠰⣿⣿⠆⠶⠶⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣶⣶⣤⣶⣶⣤⣶⣶⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣤⣿⣿⣤⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣤⣿⣿⣤⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣀⣉⣉⣀⣉⣉⣀⣉⣉⣀⣾⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣤⣶⣶⣤⣶⣾⣥⡄⢢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⣿⠏⠛⢿⣿⣿⣿⡿⣻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣯⣾⠐⠃⠈⠉⠙⠻⠷⠛⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠚⠉⣁⣀⡞⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣯⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠀⠀⠀⠀⠀⠀⢀⡴⢋⣩⡭⠶⠞⠋⠉⠹⡄⠀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⣉⠉⠛⠦⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⢻⠛⠉⣁⣤⣶⡶⠞⠠⠄⣳⡼⢩⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⣿⢟⣻⠉⠀⠀⠀⠘⠻⠟⠇⠀⠀⣠⣌⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠉⣉⡩⡤⣶⣾⣏⢯⣗⣏⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢿⡿⠳⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⢸⣟⣽⣷⢺⡛⢏⡀⢺⡜⡞⡾⡼⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⡛⡄⠀⠀⠀⠀⣀⣀⡀⠂⠼⢇⡾⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠆⢉⡼⡽⣌⣛⠻⢿⣾⣿⣼⣽⣷⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠁⠀⢀⠀⠀⠀⠙⣶⣯⣽⢣⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⠀⠁⢣⢻⡫⢈⡟⠋⢀⣻⣧⢣⢿⢧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣻⠀⠀⠀⠙⢦⣀⠀⠀⠉⣱⠋⣄⡠⠔⠢⡄⠀⠀⠀⠀⠀⠀⡏⠀⢃⡈⣇⢿⣿⣿⣷⢶⣿⣿⣏⣏⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠘⢧⡀⠀⠀⠀⠀⠈⢻⡗⢚⣱⣶⣆⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⡜⡜⣿⣿⣻⣹⣷⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣀⢠⡇⠀⠀⠙⠢⣄⠀⠀⠀⢸⣈⢢⠙⠌⠛⣆⠀⠈⠣⠤⠤⠀⠀⠀⠀⠀⢳⠀⠰⠃⢹⡿⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠔⠋⠁⠀⢹⡄⣀⣴⡾⣟⢯⣢⡀⠀⠀⠀⠈⠀⢠⣞⣹⣛⡟⠛⠓⠒⠒⢲⡆⠀⠀⢀⡜⠀⠀⠀⠀⣼⠁⠀⢠⠃
⠀⠀⠀⢀⡈⢀⣴⣤⣤⡾⠛⢿⣳⣝⢮⢳⡽⣷⠀⠀⠀⣠⡶⢰⠶⣩⣭⠩⠭⠭⣭⠭⡏⠃⠀⣴⡏⠀⠀⠀⠀⢰⠇⠀⠀⡞
⢀⠔⢋⣡⣶⣿⣿⣿⣿⠃⠀⠀⠛⠻⢿⠳⣽⣎⠀⣠⣾⡋⢀⠴⢊⠴⢳⣶⣶⣶⡆⠀⣷⠀⠘⣹⡇⠀⠀⠀⢀⡏⠀⠀⠀⡇
⡇⠀⢸⣿⣿⣿⣿⣿⣿⣧⣤⣴⣶⣶⣶⡟⢯⣤⣾⡻⠁⠔⣁⠴⠥⠤⢌⣽⣭⡛⡇⢠⣿⢠⡀⣹⣿⠀⠀⢀⡞⠀⠀⠀⢸
⠙⢆⠸⡔⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡻⠻⣿⣵⠠⡏⣁⠔⠊⣡⠤⠖⣯⣶⡇⢸⢻⠈⣷⣿⣿⠇⢀⠞⠀⠀⠀⠀⡌
⠀⠀⠀⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⢟⢿⣧⡉⠀⢰⣫⠔⠒⠀⠲⣧⣀⢸⣬⠀⠘⣿⠊⣱⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢧⣀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣮⣳⣵⠀⠀⠰⡎⠑⠒⠒⠒⠒⠚⠁⢀⣼⡿⣰⠃⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠈⠱⣼⣿⣿⣿⣿⠿⠛⠛⢻⣿⣿⣿⡀⢉⡵⠋⠀⢀⡴⠃⠀⢀⣴⣿⡡⢊⣴⣿⠋⠀⠃⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⢻⣿⠟⢉⡤⠖⠋⠉⠻⣿⣿⣿⡿⠁⠀⠀⢠⠎⠀⢀⠴⣫⡾⠋⣰⠟⠁⠘⢆⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠐⡏⣠⢖⠉⠀⠀⠀⠀⠀⣻⠟⠋⠀⠀⠀⢠⣿⣿⠟⢁⡴⠋⠀⢠⡏⠀⠀⠀⠀⠑⢤⡀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠘⠓⠚⢇⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⡰⠛⠛⠁⠀⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣾⣶⣶⣾⣷⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣤⡄⣿⠀⣤⣤⣤⣤⣤⣤⢸⣿⢠⣄⡀⠀⠀⠀⢸⣿⣿⣿⡿⠟⣿⠀⣿⠛⠛⠛⠛⠛⠛⡇⠀⡟⠻⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣧⣭⣼⣿⣿⣿⣿⣿⣿⣦⣭⣼⣿⡇⠀⠀⠀⢸⣿⣿⣿⡇⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠛⠶⠃⠀⢸⣿⣿⣿
⠀⠀⠀⢸⡏⠉⠉⢩⣭⡍⣭⣭⢩⣭⡍⣭⣭⢹⡇⠀⠀⠀⢸⣿⣿⣿⡇⣾⣿⣿⡟⠛⣻⠛⠛⡟⠛⢻⠛⠛⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⣀⣀⢘⣛⡃⣛⣛⢘⣛⡃⣛⣛⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠿⢿⡷⠤⣿⠦⠴⡷⠤⢾⠦⠴⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⠿⠿⠸⠿⠇⠿⠿⠸⠿⠇⠿⠿⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⣀⣸⣇⣀⣽⣀⣠⣇⣀⣼⣀⣠⡇⢸⣿⣿⣿
⠀⠀⠀⢸⡇⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⣿⠀⢸⡇⠀⣻⠀⢀⡇⠀⣸⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⠀⠀⠀⢸⣿⣿⣿⣧⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣡⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠋⠉⠙⠒⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠔⠉⠀⣀⠤⠤⠒⠒⠒⠤⠤⣀⠀⠙⠢⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡔⠁⡴⠀⡀⠀⢀⠀⢦⠈⢢⠈⢧
⠀⠀⠀⠀⠀⠀⠴⠋⠀⢀⣠⡤⠤⢤⣄⡀⠈⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢸⠁⡜⠀⡞⢀⣤⡀⢳⠀⢣⠈⡇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠴⠊⠁⠀⣀⣀⣀⠀⠈⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⣇⠀⡇⠸⠤⠟⢸⠀⢸⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠉⢀⣀⡀⠉⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠘⣆⠘⣄⠘⠆⠀⠠⠋⣠⠃⣰⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⠂⠈⠀⠀⠀⠀⠀⠁⠐⠁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢠⠎⢠⠆⢀⠀⠀⠀⠀⠀⡀⠰⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠉⢀⣀⡀⠉⠑⠢⣄
⠀⠀⠀⠀⢀⡇⢠⠃⢰⠃⣠⠂⠀⠐⣄⠙⡆⠸⡄⠹⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣠⠖⣉⠭⠤⠭⣉⠲⣄⠈⠳⡀
⠀⠀⠀⠀⢸⠀⢸⠀⡇⠀⡇⢠⣒⡄⢸⠀⢸⠀⡇⠀⡇⠀⠀⠀⠀⠀⠀⢀⡎⠀⡜⢡⢊⢔⠿⠿⠿⣢⡑⡌⢣⠀⢹⡀
⠀⠀⠀⠀⠸⡄⢸⡀⢣⠀⢣⡀⣿⢀⡜⢀⡎⢀⡇⢰⠃⠀⠀⠀⠀⠀⠀⢸⠀⢰⠃⡇⣿⡇⢠⠒⡄⠸⣻⢸⠘⡇⠀⡇
⠀⠀⠀⠀⠀⠱⡀⠳⡀⠓⠀⢠⠸⠀⠀⠊⢀⠞⢀⠎⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⡇⢣⠻⣣⡀⣿⠀⣜⡟⡜⢰⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⢸⠀⡄⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢆⡑⠬⣀⠁⣆⡭⢊⡰⠃⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⠒⢸⠀⡇⠒⠉⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠘⠶⠃⠖⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣜⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠆⠿⠛⠉⠀⢷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢬⠀⡀⠀⠀⠀⢀⡈⢧⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠻⡦⢴⠀⠀⠀⠁⠀⢸⡁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⣷⣄⠀⠀⠀⣄⠀⢸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠛⠹⠃⢀⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠶⢃⣴⠛⡢⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢁⠀⢠⣶⡇⢀⡴⢿⣋⣱⣟⡺⡊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢠⣿⣿⠞⣉⣐⣸⣿⣿⣯⣾⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⣀⣿⡿⡟⣯⣽⠟⢻⡆
⠀⠀⠀⠀⠀⣰⠓⢛⠛⠓⣬⡿⣈⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣾⣫⡘⢣⣿⠆
⠀⠀⠀⠀⣀⠇⠄⠨⣄⣹⣭⢼⣿⢿⣖⢲⣤⠀⢤⣤⣄⣤⣀⣀⣤⡀⠀⠀⠀⠀⠀⣠⣿⡿⣉⡼⠿⣿⢿⣿
⠚⠷⠶⣿⣋⣿⣭⣽⣿⣻⣏⡀⠡⢴⠤⠶⣿⢟⣻⣾⣿⣿⣿⣿⣿⣷⣄⣀⣀⣤⣴⣿⣾⣟⣩⣗⠾⣵⢟⡃
⠀⠀⠀⠀⠈⠉⠙⠉⠉⠙⠛⠛⠓⣲⣂⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⢿⣿⢿⣿⣽⡿⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡿⣴⡿⠛⠁⢽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠛⠿⢿⣿⣿⣿⣟⢿⣿⣷⠏⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠰⢶⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢰⣾⡾⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⢀⣬⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣠⣴⠶⠟⣻⣿⣿⣡⣤⣤⣤⢤⣄
⠀⠀⠀⠀⠀⣀⣰⣯⣿⠿⠋⣀⣴⢛⡿⣿⣿⡟⢿⣿⣿⡂⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠘⣿⣿⣿⡇⢀⣼⣿⡿⠯⠇⠈⠛⠃⢹⣿⣿⣿⣾⠃⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠈⠽⠓⢀⣀⠠⠖
⠀⠀⠀⠀⠐⣿⣿⡏⠀⢺⣿⣿⡿⠰⠶⠦⣄⠀⠈⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠴⡶⠟⣹⠟⠁
⠀⠀⠀⠀⠀⠀⠁⢰⠀⢚⣿⡷⠀⢠⣴⡦⠀⠀⠈⣉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⠑⠂⠤⠴⢶⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠾⡘⠫⠁⠀⠀⠈⠀⠀⠀⠄⠛⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⠀⠀⢀⡀⣀⣀⣀⡘⠁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡂⠀⠀⠈⢷⣄⣨⡿⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣇⠀⠀⠀⠀⠈⠉⠀⡘⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠶⠾⠑⣆⡠⠒⠁⢀⡈⠷⠲⣄⠠⡀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⣤⣄⡙⢷⣦⣀⢻⡄⠀⣰⡇⠀⠀⢀⡟⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢠⡄⠀⠂⠈⠻⡿⣿⣿⣿⣿⣤⡿⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⣷⣄⠀⠀⠈⢧⠀⠀⠀⠀⠈⠢⣝⣿⣿⣿⣧⡀⣀⣧⣿⡷⠶⠶⠶⠶⠶⠚⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⣿⣿⣷⣤⣄⠈⠀⠀⢀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣉⠛⠛
⠀⠀⠀⠀⠀⢀⡔⠯⡀⠀⠀⠀⢹⣿⣿⣿⣿⡄⠀⠀⠀⠙⠶⣄⠀⠀⢨⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢸⡆⠀⠙⣆⠀⠀⠸⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⠁⠀⢀⣈⣧⡀⠀⣿⣿⣿⣿⣿⣇⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⡜⠑⠛⣋⣉⠀⠓⠀⠸⣿⣿⣿⣿⣟⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢀⣠⡟⠉⠉⣡⣴⠾⢓⣴⠆⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⡈⠀⠀⠀⠀⠉⢀⣾⠋⠙⢷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠛⠂⠀⡀⠀⣨⣽⣧⠀⠀⠀⠙⠿⣻⣿⣿⡿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠀⠀⠀⠈⣶⠟⠉⠀⠀⠀⠀⠀⠀⢱⣿⣿⡟⢠⣿⡿⢋⣿⣿⣿⣿⣿⣿⣿
⠀⠠⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠀⣸⡟⣡⣿⠟⣿⣿⣿⣿⣿⣯⡀
⢀⠃⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⠉⠐⢋⣴⣿⣿⣿⣿⠟⢁⡽⢟⡓⣶⠶⣄
⠘⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢠⠿⠿⢟⣛⣭⠶⠓⠞⠁⠀⣿⠃⠀⣼⠃
⡇⠀⠀⠀⠈⠉⠉⠉⠉⠁⠐⠂⠀⠀⠀⠐⠚⠁⠀⠀⢠⣿⣾⣿⣯⣥⣤⣤⣴⣶⠀⠀⠛⠁⠀⠸⠃
⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴⠶⠛⠿⢷⣮⣍⡉⠉⠉⠉
⣬⣭⣩⣽⣾⣿⣿⣿⣿⣿⣿⣿⣟⣋⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠉⠉⢉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸
⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⡀⢰⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⣶⡆⣿⡇⢸
⠀⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣄⢸⣿⢸⠀⠀⠀⠀⠀⠀⣤⡄⣿⡇⣿⡇⠀⢸⠀⠀⢠⣤⢸⣿⠀⣿⡇⣿⡇⢸
⠀⠇⠀⠀⠀⠀⠸⠠⠀⠀⠀⠸⠿⠀⠀⠀⠄⠅⠀⠀⠀⠀⠸⠿⠸⠿⠸⠀⠀⠀⠀⠀⠀⠿⠇⠿⠇⠿⠇⠀⠸⠀⠀⠸⠿⠸⠿⠂⠿⠇⠿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⣛⠿⣿⣷⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠭⠉⣀⣹⣿⣿⡏⠩⣽⡳⣤⠀⢰⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣅⢸⣿⣟⠛⠿⣿⡿⠿⢿⣿⠦⣸⣿⢿⣿⣮⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢯⡋⣹⣧⢿⣿⣥⠴⡀⢻⣷⠶⣿⣿⣿⣿⣧⣼⣋⡛⢿⣿⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣽⡦⠁⠹⠀⠘⢹⣿⠿⢅⣾⣿⡞⠻⠻⢿⢿⣧⡉⣻⣿⣮⡿⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⢀⡀⠓⠂⠋⠀⠠⢟⣿⠻⢿⣿⠶⢿⣿⡇⢿⢻⣿⡇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠷⣄⠀⢀⣀⣬⣿⡴⢈⣿⣷⣮⣿⠁⡎⣼⣿⢡⢆⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠳⠄⠀⠀⠀⠀⠀⠀⣀⣀⣴⣟⣹⡿⣿⣿⣿⣿⡿⠃⣰⣳⣿⠃⣾⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣬⣹⣿⣿⣿⣾⣿⣿⠿⣋⣤⣾⣳⣿⢃⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠟⠛⠛⠋⣩⣔⣫⡽⢛⣿⣿⣷⡿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠁⠸⣿⣿⡁⠀⠀⣠⣾⠟⡉⢀⣴⡻⣻⡿⢫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠀⢰⡿⢡⢊⣴⣿⣿⣿⣽⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣊⠭⣓⢤⡀⠀⠀⠀⠀⠳⣦⣄⣿⢃⢃⣾⣿⡏⣠⡖⣫⣭⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠒⠲⡎⢝⢴⢷⣝⠢⢄⣀⣀⡴⠋⣸⣿⣼⣌⣸⣿⣇⢿⣿⣿⣋⣭⡻⣿⣧
⠀⠀⠀⠀⠀⠀⢀⡞⣰⠏⠀⠀⠁⠀⠨⠳⣟⠄⠂⠀⠀⠀⡈⢿⡿⣿⣿⣿⣿⣟⡛⣿⣿⡿⠛⣷⣿⣿
⠀⠀⠀⠀⠀⢀⣸⠃⡏⠀⠀⠀⠀⠀⠐⡄⠈⠓⠦⣤⣄⣤⡾⠷⠋⢹⣷⢿⣍⠛⢿⣿⣿⣿⣾⠿⣿⡟
⠀⠀⠀⢀⡴⣿⢻⡇⡇⠀⠀⠀⠀⠀⠀⠘⣎⣿⡀⠀⠀⢸⡇⠂⠀⢸⠙⠷⣝⣷⣾⣿⣟⣫⣵⠞⠋
⠀⠀⣰⣫⣶⡏⣨⢧⠀⠀⠀⠀⠀⠀⡄⠀⠈⠸⡇⠀⠀⣸⠰⠀⠀⡏⠀⠀⠀⠉⢹⡉⠉⠁
⠀⠀⢹⣿⣿⠙⠈⠈⣣⡀⠀⠀⢠⠀⡇⢠⢀⢄⣷⠀⢀⡏⠆⠀⡸⠁⠀⠀⠀⠀⣾⠇
⣀⣀⣈⣁⣁⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣈⣈⣁⣀⣈⣁⣀⣀⣁⣀⣀⣀⣈⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢉⡉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⢀⠀⠂⠈⠀⠀⠀⠀⠈⠀⠠⡀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠐⠀⠀⠀⠀⣴⡟⠋⠉⠉⠉⠛⢿⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⠀⢰⣾⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠈⡀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟

⠀⠀⠀⠀⠀⣠⠶⠚⣿⣟⠛⠛⠲⢤⣀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢠⡞⠁⢀⣼⡿⠉⠻⢦⣄⡀⠈⢷⡀⠀⠀⢸⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣋⡉⠉⠁
⠀⠀⢠⡟⠀⣠⢞⡿⠃⠀⣰⠋⠉⠛⡿⢷⣷⣄⡀⠀⠀⠀⢸⡷⢦⡄⠀⠀⢀⡤⠖⠒⠚⠁
⠀⣀⣸⡁⣾⡵⢻⠀⣠⠄⢹⡀⠚⠁⡰⠀⣿⢧⢻⠀⠀⠀⠺⠇⢠⣿⠀⠀⠀⠉⠙⠒⢢
⡜⢩⢭⡻⣿⠴⠚⢃⡀⠀⡠⢩⡒⠚⠁⢀⣿⡋⣸⠀⡐⠉⠦⠠⡄⠈⠀⠒⠛⠉⠉⠉⠁
⡇⣷⠆⣷⣿⣇⠀⠀⠀⠀⣄⠈⢀⠞⠀⣼⠃⠉⠁⠀⣇⡀⢀⢚⡇⠐⡼
⢷⠙⠛⠋⣼⣙⣦⣀⠀⠀⠈⠉⠁⣠⣾⠃⠰⣒⠦⠤⢤⡍⢉⣍⠀⠀⢀⣠⣿⣄
⠈⣑⡾⠟⠉⠁⢀⠉⠳⡖⠒⠒⠋⠉⢫⡙⢦⣸⠆⢸⣿⠂⠈⣯⠳⡴⠟⠓⠒⠺⢧⣴⣶⣶⡄
⢚⣿⠦⣤⣠⡞⣯⠀⠀⠘⣆⣀⣀⣠⠤⠽⢺⡉⠀⢸⠀⠀⠀⣸⡗⠀⠀⠀⠀⠀⠈⠉⣯⠇⡷
⢾⡁⠀⠀⠙⠦⣼⡤⡄⠀⠀⠀⠀⠀⠀⠀⣸⠧⣀⣼⣀⡤⢾⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⡞⠋⢙⣿⣿⠂
⠀⠉⠙⠓⣒⣦⢄⣀⡟⠀⠀⠀⠀⣀⣤⠾⠋⠀⣾⣍⣀⣀⣀⣀⣉⣉⣉⣛⣛⣛⣛⣛⣛⣵⣾⣿⣿⡿
⠀⠀⢠⣾⣿⠉⠉⠋⠀⠀⠈⠁⠙⣿⣿⣦⠀⠀⢻⠀⠠⣶⡄⠀⢴⣦⠀⠠⣶⡆⠀⠈⣿⣿⣿⣿⣿⡇
⠀⠀⣾⣿⣿⣆⠀⠀⠀⠀⠀⢀⣼⣻⣿⣿⡆⠀⢸⡆⠀⣬⠤⠤⠤⠴⠶⠶⠶⠶⣦⠀⢹⣿⣿⣿⣿⡇
⠀⢰⢿⣿⣿⣷⣷⣦⣤⢤⣖⣻⣾⣿⣿⣿⣧⠀⠘⡇⠀⢿⡄⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⣿⣿⣿⣿⠃
⠀⠀⠀⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄⠀⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⡅⠈⠙⣿⡏⠉⠉⣿⠀⠀⠀⠀⠀⣿⠀⠈⠛⠛⠒⠒⠶⠶⠶⠾⠧⠀⢸⣿⣿⠋
⠀⠀⠀⠀⠀⠀⢳⣀⣸⠇⢹⣆⣴⡃⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⡿⠁
⠀⠀⠀⠀⠀⠀⠐⠿⠛⠳⠌⠛⠉⠛⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⣿⡇⠀⠀⠀⢰⣿⡆⠀⠀⠀⢰⣿⡆⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠠⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢀⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⣶⣦⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣶⡆⠀⠀⠀⢠⣶⡆⠀⠀⠀⢠⣶⡄⠀⠀⠀⢰⣶⠄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡇
⠀⠀⠀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣾⣧⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣤⣤⣴⣷⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⡟⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⢾⠷⠶⠶⠶⠶⠶⠶⠶⢾⡷⠶⠶⠶⠶⠶⠶⠶⠶⡿⠶⠶⠶⠶⠶⠶⠶⠶⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⢻⣄⣀⣀⣀⣀⣀⣀⣀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣸⣇⣀⣀⣀⣀⣀⣀⣀⣠⣯⣀⣀⣀⣀⣀⣀⣀⣠⡿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠺⣶⡤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⡏⠀⠀⢨⡂⠀⠀⠀⠀⠀⠇⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⣾⣜⡀⠀⢸⡁⣤⠄⢸⠇⠀⠀⠀⠀⠀⣸⣆⠻⣿⡆⠀⠀⢀⣀
⠀⢠⣤⣤⠀⢀⣀⣤⡌⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠼⠿⠛⠉⠀⠀⢠⠇⠀⡀⠋⠀⠀⠂⠀⠀⠀⢹⣿⣿⣾⠿⠀⠀⠀⠙⡿
⠀⢸⣿⡋⠀⢰⡿⠿⡿⣽⣽⠇⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣰⣄⣤⡁⠀⠀⠀⠀⠀⣹⡿⣿⡀⠀⠀⠀⢠⣶⣤⣤
⠀⠈⠛⠁⠀⣿⢄⠀⠳⠗⠃⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣴⣯⢟⣷⣿⡆⠀⠀⠀⢀⣿⣶⣼⣇⢰⠏⠀⢸⣿⣿⣿⡇
⠀⠀⢀⠀⠀⠉⠙⠃⠀⠀⠀⠀⢉⣠⣴⡖⠂⠀⢠⣶⡀⠀⠀⠀⠀⠀⢻⣯⡏⣟⣿⣿⡇⠀⠀⠀⠸⠿⠿⠿⢿⠿⡀⢀⣬⣶⣿⣿⡇
⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠀⢀⣿⣿⣟⣶⣷⠀⠀⠀⠐⠀⠀⠀⠀⠚⠛⠃⠉⠉⠉⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠇⠀⠀⠀⠀⠀⠙⠛⠿⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠈⠐⠃⠐
⠀⢸⣿⡇⢸⡇⠀⡇⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⢈⠀⣴⡾⠛⠻⣦⠀⠀⠀⠂⠀⠀⠂⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤
⠀⢸⡿⠧⠸⢧⣤⣇⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠰⢈⣿⣷⠦⣚⢹⣷⣄⠀⣁⢈⠀⡁⢈⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⢛⠻⣿⣧
⠀⢠⣤⣴⣶⣿⠟⣋⠀⠀⠒⠿⢿⣠⣤⡈⣛⠂⢘⣠⣿⣿⣆⡛⣹⣿⡁⢄⠄⡡⠈⠄⠡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⢈⠁
⠀⠀⠈⣿⣿⣤⣴⣿⡆⠀⠀⠀⠀⠀⠉⠓⠛⠀⢪⢿⣿⠿⠻⢭⣭⣿⡔⠀⢀⣀⣀⠀⠐⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢊
⠀⢠⣤⡟⢏⠀⠀⢘⣥⣤⡄⠀⠀⠀⠀⠀⠀⠀⣱⣿⣿⣷⣩⢂⡙⣿⣷⣿⣿⣽⣶⣶⣾⠀⠀⠀⢸⠿⣼⡟⠇⠀⠐⠇⠀⠀⠀⠀⠀⠰⠄
⠀⢘⣿⢧⣾⠀⠀⢻⣿⣿⣇⡀⠀⠀⠀⠀⠀⠀⢾⣿⣏⠻⣦⡡⢬⣾⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠈⠉⠰⠆⠀⠀⠀⠀⠂⢄⠀⢠⡅
⠀⢸⣷⡄⠁⢠⡀⠀⠈⠙⢻⡟⣷⢤⣄⠀⠀⠀⣻⣿⣿⣵⣿⣿⣷⣟⢝⡫⣿⣿⡧⢀⠌
⠀⢸⣿⠇⠀⠀⢶⣶⠖⣥⠀⠠⠙⠩⣿⢹⣶⡆⣿⣿⣿⣿⣿⣿⣿⣯⢾⡵⢻⠿⠕⠬⠢⠔
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⡀
⠀⠀⠀⡞⡡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠋⠉⠉⠹⡧
⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⡤⠀⠀⠃⣿
⠀⠀⢀⣤⠀⣄⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠀⠴⠂⡎
⠀⣾⡿⣷⠀⠛⢹⣦⠀⠀⠀⠀⠀⢄⡐⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⡀⢀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠒⣢⡞
⠀⣿⠗⣸⠀⣤⣼⣿⡇⠀⠀⠀⠀⠈⠀⠀⠆⠀⠁⠀⡃⠀⠀⠀⠀⠀⠀⢉⠁⡶⠙⠈⢤⡀⠀⠀⠀⠉⣉⣩⡝⠛⠊⢀⠀⠀⣾⡿⣊⣩⣷
⠀⠀⠺⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⢀⣀⣠⡀⠛⠀⠾⠗⠒⠒⠛⠋⠀⢀⠀⢠⣾⠿⠀⠀⠀⣾⠃⠀⠀⣲⣿⣿⣿
⠀⣴⣾⣯⣭⢹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡰⠋⣩⣿⠣⠯⠁⠀⠀⠀⠀⠀⠀⠀⠂⠐⠓⠒⠶⠀⠠⡀⠹⡀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠒⠂⠘⠗⠀⠈⠈⠻⠛⠛⠛⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⢀⣠⣴⣶⣾⣿⣿⣿⣷⣶⣦⣤⣀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣾⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣠⣾⣿⣿⣿⣿⣿⣿⠛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠻⡝⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠃⠀⣰⣿⣿⣿⣿⣿⣿⡟⢋⣀⣉⣉⣉⣉⣉⠙⣿⣿⣿⣿⣿⣿⣧⠀⠀⢹⡄⠙⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢿⠀⠀⠙⢿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢸⡇⠀⠀⠀⠙
⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣿⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡇⠐⠶⠰⣿⡗⠰⠖⠀⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢸⡇
⣿⣿⣿⣿⣿⣿⣇⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠸⠿⠖⠿⠗⠾⠷⠀⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⣿⠁
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⣿⣿⣿⣿⣿⣿⡇⢘⣛⡛⣛⡋⣻⣛⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⣼⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠈⠻⣿⣿⣿⣿⣧⣈⣉⣉⣉⣋⣉⣉⣠⣿⣿⣿⡿⠏⠀⠀⢀⣼⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⣠⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⣀⣴⡾⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⠳⠦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⡶⠿⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠈⠉⠉⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠄
⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⠁⠈⠉⠉⠉⠉⠉⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠠⠀⠀⠀⠀⠀⠀⠄⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣽⠀⠐⠷⠀⠀⠀⠀⠷⠀⠀⠘⠛⢥⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠄⠀⠀⢀⠀⠀⠺⡁⠀⢹⠀⠀⢀⠤⠄⠠⢤⡀⠀⠀⠀⢰⠃⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠚⠃⠈⢆⠀⠈⠿⣿⣿⠟⠁⠀⠀⢀⠜⠦⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⡀⠀⠀⣀⠠⠔⠁⠀⠀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡇⢀⣷⣶⣴⣶⠐⣂⠡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠒⠆⠀⣀⠌⡐⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠂⠚⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠜⠣⡀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⠊⠀⠀⠑⡄⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⠃⠀⠀⠀⠀⠈⠢⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠀⠌⠀⠀⠀⠀⠀⠀⠀⠈⣢⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡒⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣠⠀⠀⠀⠀⠀⠀⢠⢞⣻⣿⣻⣧⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⢸⠀⠀⠀⠀⠀⠀⠻⡟⢉⠀⡉⠟⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⣖⢰⡋⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡰⠹⡀⠈⠢⡀⠀⢀⣀⠀⣈⢀⢂⡀⣠⠔⠂⠂⠒⠒⠈⠉⠁⣀⠤⠋⢸⠀⢠⠀⢸⠀⠀⠀⠐⣲⠆
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠉⠀⣀⡤⠖⠃⠀⠋⠶⠶⣤⣖⠒⠛⠛⠉⠁⠀⠀⠀⡖⠠⠋⡆⢸⠀⠀⠀⠘⠅
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⣀⠀⠈⠉⠉⣽⣅⣠⠴⢄⡀⠀⠀⢦⡀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠈⠂⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⢀⡌⠀⡛⠁⠤⠀⠑⡆⠀⢨⠑⠦⣀⠀⠉⠒⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⡀⢹⡆⠤⠀⣀⡿⠀⣸⠀⠀⠈⠑⠲⠶⠴⠏⠸⠞⠲⢞⣛⡛⠛⢛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣴⠋⢀⠠⠂⠀⢘⠂⢀⣅⡀⣀⣴⠋⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⡄⠒⠁⠀⠀⠀⠸⠀⠀⠀⠁⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠢⠀⠀⠀⣇⠀⠀⠀⠙⠀⠀⢁⠀⠀⠀⠀⠨⠀⠀⠀⠀⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡈⠀⠀⠡⡀⠀⠈⠒⠒⠂⠀⠠⡀⢈⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⡀
⠀⠀⠀⠀⠀⡠⠀⠀⠂⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠑⠤⠄⠀⣀⠀⠀⠀⣸⡟⠧⣄⠀⠀⠀⢠⢡⣾⣿⣷⡌⡆
⠀⠀⠀⠀⠐⠃⠉⠀⠀⠀⠀⠀⠀⠚⠀⠀⠄⠀⠀⠀⠀⠘⠀⠀⠀⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠉⠀⠀⠀⠈⢄⠻⠿⠿⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⣟⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⣄⣀⣀⣉⣁⣁⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣫⡅⠀⠀⠐⣟⠁⠀⠈⠛⠀⠀⢠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⠀⠸⢤⣤⣠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂⠂⠀⠀⢶⡄
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⢈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡽⠛⡋⠀⠀⠀⠀⣶⣧⡀⠀⠀⠀⠀⠀⠀⣰⠅⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠈⠓⠀⠀⠠⠤⠏⠀⠀⠀⠀⠀⠄⠀⠄⢀⠀⠁⠀⠀⠸⠥⠜⠁⠀⠀⠀⠀⠂⢿⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢠⠀⠀⠀⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢈⣶⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⣋⣻⣄⣀⣠⣞⣉⣳⣄⣀⣀⣴⣛⣶⣀⣀⣠⣖⣛⣲⣀⣀⣠⠖⠒⣆
⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⣏⡉⠉⠉⠉⠹⣍⠉⠉⠉⠉⠙⣯⡉⠉⠉⠉⠹⣍⠉⠉⠉⠉⢻⣛⠛⠛⠛⣷⣦
⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣈⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣬⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣉⣀⣀⡀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⢸⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡤⠒⠒⢢⠠⠒⠒⢢⠠⠒⠒⢢⢰⠒⠒⢢⡔⠒⠒⢢⠄⠀⠒⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣀⣀⣀⢣⣤⣤⡜⢰⢤⣤⡜⢰⣤⣤⡜⢸⣤⣤⣜⢣⣤⣤⡜⣢⣤⣤⡇⠀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⢸⣇⡀⠀⡀⢀⠀⠀⣸⢀⠀⠀⢸⢸⠀⠀⢸⣇⠀⠀⣸⡁⠀⠀⡇⠀⣿⣾⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠒⠋⠱⡔⠒⠒⠂⠐⠒⠚⠂⠐⠓⠒⢢⠐⠛⠒⢢⡔⠒⠒⢢⠒⠒⠚⡄⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⡇⠀⢸⠴⠤⣔⡳⠤⢤⠆⡰⠤⢤⠄⡠⠤⢤⠎⡰⠤⠤⡜⡳⠤⢤⢚⠤⠤⡤⠃⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⣀⣀⡿⣀⣀⢀⠀⡀⣀⣀⠇⣀⣀⢀⠇⡇⠀⣀⢮⣀⣀⣀⢇⡀⠀⡘⠀⣸⣿⣿⡇
⠀⠀⠀⠀⢀⣼⠃⠀⡎⠉⠉⢁⠆⠉⠉⢷⠂⠉⠉⣲⠂⠈⠉⡳⠂⠈⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣿⠃
⠀⠀⠀⠠⠾⣥⣄⣀⣉⣉⣉⠁⠑⠒⠒⠈⠒⠒⠒⠉⠒⠒⠒⠉⠒⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⠉⠉⠛⠛⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠴⠞⠁⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣛⣛⣛⣛⣟⣻⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⢟⣿⣯⣥⣄⡉⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣮⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⣿⠯⠛⠉⠈⠁⠈⠉⠛⣿⠿⠻⣿⣷⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣀⠀⠀⠀⣀⣀⠀⠀⠘⣿⣿⣿⣟⡛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠺⣿⣿⣯⡭⣄⠀⠀⠡⣤⡀⠀⠀⠙⣿⣿⣿⣷⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣿⣏⣿⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠙⠨⣙⣛⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⡀⠦⢀⣀⠀⠀⠀⠀⠀⠀⣤⣴⣯⡿⣧⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⡸⣭⣭⡽⠃⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠷⡈⠁⠀⢀⡠⠄⠀⢰⣿⣟⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠒⠉⠉⠑⢤⡀⢠⠃⠀⠘⢯⡉⠁⠀⠀⠀⢸⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⢢⠙⣏⠀⠀⠀⠈⢻⡀⠀⠀⠀⢸⡇⠈⠢⡤⢀⡀
⠀⠀⠀⠀⠀⡰⠁⠀⠀⡤⠤⠤⠔⣲⠆⢀⠂⠇⢌⡂⠀⢀⣀⣤⣵⣄⡀⠀⠀⡇⠀⠢⣬⣤⠈⠱⣄⠀⠒⡄
⠀⠀⠀⠀⢰⠁⠀⠀⡸⠀⠀⢀⡴⠛⢷⠇⠀⠈⢾⠙⠛⠉⠉⠉⠀⠑⢍⠢⣀⡇⠀⠄⠉⣁⠀⠀⠈⠀⠉⠙⠢⡀
⠀⠀⠀⢠⠃⠀⠀⢀⠁⢀⡴⠋⠀⠀⠀⢣⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡇⡆⠀⠀⠀⠂⢠⠄⠈⠀⠀⠀⠈⠳⢄
⠀⠀⢠⠃⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⡆⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠃⠀⠀⠀⠀⠀⢹⣲⠀⢄⣀⠀⠀⠀⠙⢆
⠀⠀⠂⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⢱⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⢻⣃⠁⢉⢻⣶⠄⠀⢸⡧⣶⣶⡾⣶
⠀⡄⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣴⣷⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀⠀⠀⠀⠀⠀⢈⣩⡉⣉⣩⠁⠀⠀⣾⢷⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡴⣏⠙⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⣇⢠⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⡘⠀⢹⠏⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠃⢀⠃⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⡌⠀⡀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⠁⠀⠀⠀⡘⠀⠀⠃⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⡴⠃

⠀⠀⢦⡼⢩⣽⣗⣛⡽⠻⡿⣏⡽⣁⣹⣄⠤⡼⠉⡙⡆⣠⡰⣬⣉⠘⡍⣛⣬⣳⢯⢘⣼⢂⡀⠈⣠⡴⣖⣒⡲⢤⡈⠈⠗⣤⠸⠾⠀⣦⡀⠈⣛⢓⠚
⢰⡷⠓⢰⣯⡋⠿⠳⠾⣿⣯⡵⣗⣓⣤⢸⣋⡾⠊⠠⠟⢀⢀⢀⣠⡀⠡⣣⡙⠃⠜⣼⠃⣀⣤⢾⣡⢎⢕⡴⣮⢂⣷⢸⡀⠠⠐⠂⠀⡏⠙⢉⡦⢀⠁
⠈⠟⢮⠿⢍⡉⢶⡆⠙⡳⠿⠟⣴⠥⣎⡭⡀⢟⡛⠲⠼⠀⠎⢋⡰⡷⡦⢩⣿⣿⡛⠉⡏⣍⡼⢞⡵⣻⣋⠀⠻⠼⢏⠼⠀⣑⡀⢀⠈⡄⠰⠀⡀⣭⠅
⠀⠀⣤⠃⢢⠤⣽⠮⠯⠯⣕⠒⠣⠟⢐⠰⠇⢘⠃⢃⠐⢣⡞⢶⣍⢢⢛⡫⡿⣿⠽⢻⠏⡽⡌⠙⢎⠹⡮⠁⣈⡡⠁⣤⣀⠀⠂⢙⠄⢷⠆⠐⡏
⠀⠀⠀⠐⠋⠀⠘⠈⠀⠀⠙⢦⣤⡤⢆⠲⠅⡵⠨⠬⠔⠀⠙⠚⠀⠉⠙⠷⠦⠾⠓⠛⠉⠊⠀⠀⠑⠓⠒⠊⠙⠃⠑⠛⠈⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠛⠺⠛⠟⠛⠺⠊⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋⠾⠺⠳⠗⠗⠺⠛⠃⠀⠀⠀⠀⠀⠘⠾⠺⠸⠓⠛⠺⠸⠓⠗⠟⠛⠃










⠀⠀⠀⠀⠀⠀⠐⠂⠶⠶⠶⠦⠶⠶⠶⠖⠆⠀⠀⠀⠀⠀⠀⠀⠰⠇⠶⠰⠶⠶⠲⠆⠀⠀⠀⠀⠀⠀⠀⠰⠦⠖⠒⠲⠶⠆⠶⠶⠲⠆




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠛⠋⠛⠛⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠂⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠧⠀⠀⡶⠲⡌⠳⠗⠈⠁⠲⣴⠆⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⠒⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠉⢹⠀⠈⠁⠀⣠⡀⠀⠀⠀⡄⠀⠘⠁⠀⠀⠀⢩⠶⠋⡥⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⣼⠀⠀⢩⣿⠿⠄⠀⠀⠀⠞⢿⡿⠂⠀⠀⠀⢸⠀⠈⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠉⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢿⠀⠉⢢⡀⠀⢠⣤⡤⠀⣤⣠⠄⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠄⠀⠄⠃⠀⠀⠀⠈⣁⠀⢊⡟⠢⡀⠙⠓⠒⠛⠁⠀⣀⠠⠀⣀⠠⠀⠀⠀⠀⠀⠠⠀⠈⡆
⠀⠀⠀⢠⠈⠙⡏⠀⠀⠀⠀⠐⠂⠈⢦⠀⠘⢓⠙⠣⠄⣀⣀⠠⠀⠀⠶⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⢲⠁
⠀⠀⠐⠀⠶⡀⠈⠀⠀⠀⠀⠛⠀⠀⠀⢃⠀⢡⠂⣀⠀⣤⢄⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢧⠈⠀⠈⠠⢀⡀⢀⡀⠜⠀⠀⠀⠀⠂⠀⠉⠁⠀⠙⠄⠙⠄⠀⠀⠀⣘⡡⠋⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⢀⡎⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠀⠁⠂⠀⣤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠘⢦⡀⠀⠀⠀⠀⢀⡴⠼⢷⠂⣄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢣⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠉⠲⠤⠀⠴⠟⠁⠀⢠⠆⠀⢀⠉⠢⡀⠀⢰⠇⣤⡀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣄⠉⠀⠀⠁⠀⠛⢿⣷⠧⢰⣁⠄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⠈⢂⠀⠀⠀⠀⠀⠈⣿⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠉⠀⠀⣠⣀⣀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠦⠀⠀⠀⠀⣸⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠃⡠⠋⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠈⠲⠀⠀⠀⠀⠀⠀⠀⣠⡴⠊⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡄⠀⠀⠀⠀⠸⣀⠋⠀⠉⠲⠄⠀⠑⢄⠀⠀⠀⠀⡄⠁⡀⡴⠟⠁⠀⠀⠀⠀⠀⠀⣀⠀⡀⠰⠂⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠐⢦⠀⠀⣀⠂⠀⠀⣾⣆⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢠⡄⠀⡎⠀⠀⠀⠀⠀⠀⠀⠠⠚⠃⠀⠸⣾⣩⣿⡧⣤⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠸⢿⣽⣶⣿⣦⣤⣄⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡰⠀⠰⣀⠀⢃⠀⠠⠤⠀⠀⠀⠀⠀⣠⣤⣼⣍⠛⠸⠗⣿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⠤⠄⠐⠒⡚⠇⠀⣾⠋⣠⠅⠀⣀⠀⠀⠀⠀⣴⠿⣿⣷⣼⣿⣦⣄⠀⣽⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒
⠀⠀⠀⠀⠀⠠⠞⠁⠀⠀⠀⠀⡤⠒⠆⠀⠛⡒⢈⡀⠀⠠⢶⣤⠀⠀⠓⠤⢾⡏⠀⠙⠋⠛⠿⠿⣿⣶⣿⠁⠀⠀⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠⡀⠸⣄⠐⠴⠀⠘⠉⡈⢁⠀⠀⠀⣠⠀⣀⡀⠀⠀⣄⡀⢀⠀⠀⠀⠀⠈⢻⡁⠀⠀⠄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⣀⠁⠰⠶⠂⠀⠀⠀⠀⠀⠉⠀⠈⠶⠈⠁⠀⠉⠀⣃⠘⣢⡀⠀⠀⠀⢻⡄⠀⠀⠛⠂⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣀⣀⣤⣤⣄⠠⠶⠶⠶⠶⠚⠉⠛⠛⠉⠻⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠀⠀⠀⠈⡛⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠈⠓⣤⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠐⠀⠈⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⣀⣀⠤⠖⠋

⠀⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠾⢶⡤⠴⠿⢦⠤⠴⠷⣦⠤⠶⠶⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠲⠚⠀⠀⠲⠊⠀⠐⠖⠃⢀⣲⣞⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⣭⡭⠇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⢠⠤⡄⣤⢤⠀⡖⢦⢰⠒⡄⣖⢲⠀⡖⣆⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠈⣒⡃⢘⣚⠀⣉⢉⢈⡭⡅⢨⠤⡀⡬⢤⠀⣇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⡇⠀⠧⠼⠘⠤⠇⠙⢚⠂⢓⣛⢘⣒⡃⢙⣚⡀⢸
⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣇⣀⣳⠀⢻⣹⠄⣏⣳⠸⠭⠇⠯⠼⠀⠧⠗⠘⠤⠇⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢹⠀⢰⡒⡆⢴⢲⡀⣖⣲⢰⣚⡆⢯⣹⠀⣏⡧⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣩⡥⠬⠭⠤⠬⠤⠤⠤⠤⠤⠶⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠶⠶⠦⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣠⣶⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠐⠛⠛⢷⡄⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠉⠲⠿⠃⠀⠀⠰⢆⣠⣾⠏⠀⠀⠀⠀⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣤⣤⣀⣿⣄⣀⡀⠀⠘⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠚⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠉⠉⠉⠉⠉⠉⠙⢻⣿⡁⠀⠀⠙⢿⣦⡄
⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⠀⠀⠀⠈⠙⠻⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠷⠶⠶⠾⠟⠻⡿⠀⠀⣠⡶⠃
⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⠾⠋
⠀⠀⠀⠀⣠⣿⠃⠀⠀⠀⠀⠴⠛⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠟⠻⠿⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉
⠀⠀⠀⣾⣏⣤⣄⣤⡶⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⢁⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠙⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠛⠻⠆⠀⠀⠀⠘⢢⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠶⠄⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠙⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡷⠦⣤⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆⠀⠀⠹⣷⡀
⠀⠀⠀⠀⣀⣤⣶⣦⠤⠤⠤⠄⢺⣷⣄⣀⣤⣾⡿⠛⠀⠀⠀⠀⠠⠄⠀⠀⠁⠈⠁⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⣼⡏⠙⡏⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⢀⣠⣴⠟⠁
⠀⠀⢸⡟⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠟⠋
⠀⠀⢸⠃⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣰⡶⠛⠉
⠀⠀⣠⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⠋⠁
⠀⠀⢻⡇⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⣾⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⣿⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⢰⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⢸⡇⠀⢸⡟⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⢸⡇⠀⠈⠓⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⡇⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠁⠀⢿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣦⠀⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠉⠀⣿⡟⠻⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠖⠛⠟⠛⣿
⢀⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⣿⣴⣆⣠⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⣿⣿⣿⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠈⠛⠛⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠰⣿⢿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠠⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢠⡇⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⠏⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⣿⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⣿⡆⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣶
⠀⠀⠀⣀⠀⠠⠆⠈⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⡄
⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⢰⣿⠁⠀⠀⠀⠀⠁
⠀⠐⢿⣿⠿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠁⠀⠀⠸⢿⣧⣠⣄⣀⣰⣷
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⡿⠟⠻⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡇⠀⠐⠀⢀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡆⠀⢀⣴⠾⠛⠉⠉⠉⠉⠛⢷⣄⠀⠀⢸⣿⢩⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣄⢸⡇⢸⡇⢸⡇⠀⠈⠁⢠⡶⠟⠛⠛⠷⣦⡀⠉⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⢸⣿⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⣴⠶⢶⡄⠀⠀⠀⠀⠀⢸⣿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠉⠈⠉⠈⠁⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋


⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣯⣽⣷⣦⣶⣶⣦⣶⣷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠉⢘⠛⠁⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⡀⠺⣿⠃⢃⣀⣿⡀⢲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⠀⣴⢿⣷⡶⠘⠛⠿⠛⠛⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠳⣤⣤⣶⠉⠀⠀⠀⠀⠀⠀⠠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣄⠀⢀⢀⣤⠀⠀⣶⠂⠀⠀⢰⣤⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣇⠈⢻⠏⠀⠘⠛⠁⠀⠀⠀⠁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣴⠛⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⣹⣿⣦⠀⠀⠀⠀⢾⠛⡻⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⡇⠀⣤⡀⠀⠈⠁⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠈⠻⣿⣿⠋⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⢀⣠⣤⣶⠊⣡⠀⠀⠀⠀⠀⢿⡇⠀⣴⠞⠿⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡟⠉⣡⠀⠀⠛⠋⠀⠀⠀⢿⡀⡀⠀⠀⠀⠈⠉⢻⣤⡄⠀⠀⣶⠇⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⢀⣶⣿⡿⠀⠀⠙⣦⠀⠙⢡⣶⡦⢤⣀⡀⢠⣿⣷⠆⠀⠈⠀⠀⠉⠱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣴⣿⠋⢋⣀⡀⠀⠀⠀⣷⡀⠈⢩⣀⠀⠈⠛⠁⣀⣷⣶⠓⠀⠀⣄⠀⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠻⣿⣿⢿⣿⡟⢻⣦⣤⠀⠙⣷⠀⠀⠛⢻⣿⣂⣿⣿⢙⣿⣦⡄⠀⠿⠀⠀⠻⣀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣧⡀⠿⢿⣿⡿⣅⠘⠀⠉⣦⠀⠀⠉⠀⠛⠛⠛⠛⠋⠀⠁⠀⠀⠛⠀⢠⡟
⠀⠀⠀⠀⣤⣿⣿⣤⣿⣧⣬⡟⢱⣶⣾⣿⣤⣿⣦⡀⠀⣿⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⣂⠀⠿
⠀⠀⠀⠀⣬⠿⣿⣿⡛⢻⣿⣇⣠⣿⣧⣼⣿⠛⣿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⡀
⠀⠀⢀⡿⠁⠀⠈⢿⣿⣿⣿⣏⠛⣿⣿⣿⣿⣿⡿⠀⠛⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠁⠛
⠀⣤⠛⠀⠀⠀⠀⠀⣿⣧⣼⣿⡿⠀⠹⣤⣽⣏⠀⣾⠀⠀⠀⠀⣤⣤⣤⣤⠀⠀⠀⠀⣴⣿⢯⣤⠌
⣄⠀⠀⠀⠀⠀⢠⣴⡏⠀⠈⠉⠀⠳⠀⢿⣿⣿⣄⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠐⠒⠶⠀⠀⠀⢤⡀⠀⠀⣠
⠁⢀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⠀⠀⠀⢼⣿⣁⣿⠀⠙⠛⠀⠀⠀⣀⣀⣤⣤⡆⢀⣶⠀⠛⠛⠛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠤⠉
⠀⠀⠀⠛⠀⣤⡀⠀⠀⠀⠀⠀⠛⠣⣤⡀⢈⣿⠟⠛⠛⠛⠀⠀⠀⠀⠿⠉⠍⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠚⠛⠃⠀⠀⠀⠀⠀⠀⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠓⠠⣤⡄⠀⠀⠀⠘⠛⠻⣶⡶⠿⣶⣦⣄⣤⣤⣀⣤⣤⡀⠻⠿⠛⠣⣤⣄⠚⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢓⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⢠⣥⡄⢸⣷⠀⠻⣦⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⢀⣴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⣿⠀⠀⢻⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⠀⠛⣿⠁⠀⣴⠁⠀⠚⡀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⢿⣅⠘⠻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣼⠛⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠃⠀⡿⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣾⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⣿⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠟⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠈⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠗⣀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡀⣀⣠⣤⣶⣿⠃⠂⢤⣤⣤⠄⠀⢀⣠⣤⣤⡄⠀⠘⠛⣤⣤⣀⠀⠀⠀⢀⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⢿⣿⣿⠛⣿⣿⣦⡀⠀⠀⢀⣴⣿⣷⣤⡀⠀⢀⣤⣿⣿⣿⣿⣷⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣤⣼⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⢿⣿⣤⣿⣿⣿⣿⣉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣦⣸⣿⣿⣟⡛⢱⣷⣾⣿⣿⣧⣼⣿⣿⣿⣿⣀⣿⣿⣿⠀⣿⣿⣿⣅⣽⣿⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢻⣿⣿⣿⣿⡀⣸⣿⠛⢻⣿⣿⣿⣅⣽⣿⣿⠟⢿⣿⠀⡿⠛⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣼⣿⣏⣸⣿⠟⢿⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣷⣿⣿⡇⢸⣿⣿⣿⣿⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⠟⢿⣿⣶⣿⣿⣯⣤⣿⣿⠙⢛⣧⢸⣿⡟⢹⣿⣿⣿⣧⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣾⣼⣿⣏⣉⣿⣿⢾⣿⣷⣿⣿⡟⠛⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣿⣏⣹⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⡿⢿⣿⣿⣿⣿⣤⣼⣿⣏⣹⣿⣿⣶⣿⣿⠿⣿⣿⣿⣿⣿⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣯⣠⣿⣿⣿⣿⣿⣷⣶⣿⣿⠋⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣤⣿⣿⣤⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣼⣿⡏⢻⣿⣿⣿⣿⣿⣿⣧⣴⣿⣿⣿⣶⣿⣿⠟⣿⣿⣿⣿⣿⣿⡋⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⠉⣿⣿⣿⠛⣿⣿⣷⣿⣿⣧⣼⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠉⠉⣭⠉⠉⠉⠙⠛⠛⠻⠿⠿⢿⣿⣿⣿⠛⠉⠀⠀⠈⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠛⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣷⣄⠀⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡄⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣿⣦⡀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢿⣿⣦⣀⣱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⢿⣿⠟⢻⣿⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣧⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣴⣦⣤⣤⣶⣶⣶⣄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⠉⢻⡅⠁⠀⢂⠀⣁⣾⠋⠀⠉⠑⡍⠈⢹⣦⡾⢛⣿⠛⠛⠋⢿⡀⡸⠏⠛⠛⢻⣟⠿⣖⣾⠉⠉⡕⠉⠁⠈⢹⣆⡁⠀⠐⠀⠁⣽⠋⢁⡀
⠏⠁⠐⠚⠓⠒⠁⠈⢫⠈⠙⠃⠘⠃⠀⢹⠀⠒⠓⠛⠈⠛⢀⡀⠀⠒⠛⠙⠒⠀⢠⠀⠈⠚⠓⠀⠀⣹⠁⠑⠋⠒⠛⠋⢨⠋⠀⠑⠒⠛⠂⠀⠉
⠠⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠦
⠲⡆⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡖
⠑⠃⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⢛⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⢈⡅⠀⠀⠀⠀⠀⠀⠓
⢐⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢒
⣿⠀⠀⠀⠀⠀⠀⠀⡥⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢼
⣾⠃⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⢹
⡶⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠴
⠲⡄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⢝⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡴
⠳⠄⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠴
⢸⡅⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢐⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿
⠒⣳⣒⣶⢲⡶⣒⣶⣶⢆⠐⢲⠐⠒⠀⢛⣷⠂⠒⢂⠀⠀⢺⣅⠒⠒⠀⠐⠒⢀⣼⠂⠀⢀⠒⠂⢲⣞⠃⠐⠒⢰⠒⢀⢶⢶⣖⡲⡶⣲⣖⣲⡛
⠈⠉⠉⠉⠁⠉⠉⠉⠈⠉⠀⠁⠉⠀⠀⠉⠁⠀⠀⠀⠀⠉⠈⠉⠁⠀⠀⠀⠀⠉⠉⠈⠁⠀⠀⠀⠀⠉⠁⠀⠈⠉⠁⠈⠉⠈⠉⠉⠁⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠿⡿⠀⠀⠀⠀⠀⠀⠀⠾⡷⠸⡿⠸⡿


⠀⠀⠀⠀⠀⠀⣲⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢾⣿⠀⠀⠀⣰⣆⡖⡄⢀⡀⢸⣿⠀⠀⠀⠰⣆⡖⣀⠀⣀⢰⣿⡆
⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣿⣿⢸⣿⠀⠀⠀⠀⠁⢉⡁⢸⣿⢸⣿⠀⠀⠀⠈⠁⢈⣁⢸⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⣿⡇⣿⣿⢸⣿⠀⠀⠀⣠⡄⣿⣿⢸⣿⢸⣿⠀⠀⠀⣠⣄⢸⣿⢸⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠃⠙⠋⠘⠋⠀⠀⠀⠙⠃⠙⠋⠘⠋⠈⠛⠀⠀⠀⠙⠋⠘⠋⠈⠛⠈⠛⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⢟⣛⢛⡟⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⢿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠉⠀⠀⠀⠀⠀⠀⠤⢤⡀⠀⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠛⠉⠁⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠉⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⢿⡿⠂⠀⠀⠀⠀⠀⠘⢿⠿⠀⠀⠀⠀⠈⢹⡷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠀⠀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠀⢰⣛⠓⠒⠒⠒⠚⣛⡆⠀⠀⠀⠀⠀⡟⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣰⣿⣷⣶⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⠀⠀⠈⠙⠛⠿⠛⠋⠁⠀⠀⠀⢀⣼⣿⣿⣿⢸⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⣿⣿⣿⠈⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠙⣿⣿⣿⣿⣿⡿⢿⣿⡿⠿⠖⠋⣨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠟⠋⢹⣋⡀⠀⢀⣙⠏⠉⠛⠁⠀⠈⣇⣀⣤⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⣻⠏⠁⢸⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⣸⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⣀⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡼⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⡼⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠳⢄⡀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡙⠋⠳⢤⡀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⣀⡟⠀⠀⠀⠀⠀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠁⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟











⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⠀⠀⠘⠃⠀⠀⠀⢀⣠⣤⣴⣶⣶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠈⠁⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⢻⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⡿⠋⠈⠁⠉⠉⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢁⣀⡀⠀⠀⢀⡤⣄⠀⠀⠀⣹⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠚⠀⠉⣼⠀⠈⠀⠈⠀⠀⣸⣿⡿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠛⣀⣀⠀⠀⠀⢠⣿⣯⣿⡏⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠐⢿⡉⠉⣹⠃⠀⢀⣿⣛⠳⡌⠧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠙⠚⠁⠀⢀⣼⣷⡌⡇⠛⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠿⡖⠒⠚⠛⠉⠙⣿⠇⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠻⣄⡀⠀⣀⡼⢻⠀⠀⠀⠀⢸⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⢷⠶⠒⠛⠙⣇⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠹⡆⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠹⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠈⠁

⠀⡰⠜⣫⡷⠚⠻⢲⡤⡤⠶⢿⣄⠀⠈⣷⢵⡒⠳⣙⣦⠴⠿⠦⠤⠴⠔⢃⢖⣿⠥⠤⠴⣮⡜⣝⠢⠬⢭⡣⢸⡀⣼⠃⠂⠀⣄⢾⣁
⠀⡗⣊⠳⣯⣴⡤⠿⡷⠿⠀⠀⠙⢷⠀⣸⠀⠀⠀⠘⣤⣇⣚⠛⠛⠚⠛⣩⠞⣁⡠⠤⣠⡀⠙⢿⡗⢀⡠⡑⣢⢻⡁⣀⠘⠶⢣⡔⠄
⠀⠓⠞⠠⠈⣼⣤⢶⡁⢘⢷⣶⡀⠀⢨⣿⣦⣀⡀⢤⣤⠽⡈⡉⠱⠾⡻⠟⡋⢥⣤⡦⠸⡇⠀⣸⠀⢌⣿⢛⠼⣊⡱⠌⠑⠣⢤⡙
⠀⠀⠀⠀⢠⣶⠉⠉⠉⠇⠀⠨⠬⠐⠛⢣⠁⠈⢃⣞⠽⣏⠛⠁⠁⠁⠔⠋⠀⠀⠹⣴⠗⠁⣠⠟⠛⠋⠁⢩⣦⠘⡦⣄⢰⠳⠊
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠉⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⣊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⣀
⠀⠀⠀⣇⠀⠀⠲⠶⠖⠶⠲⠶⠖⠆⠀⠀⢈⠀⣇⠀⠀⠒⠰⠲⠲⠖⠖⠖⠒⠀⠀⠈⠀⣇⠐⠶⠲⠲⠒⠖⠆⠶⠶⠶⠒⠂⠈
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣇⠀⠈⠑⠚⠒⠛⠸⠺⠒⠛⠃⠀⣈⠀⣇⠀⠀⠀⠛⠛⠒⠗⠟⠚⠀⠀⠀⢈⠀⣇⠀⠸⠒⠋⠉⠒⠚⠸⠚⠛⠃⠀⢈
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡗⠀⠀⢀⡀⣀⣀⣀⣀⣀⡀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠷⠀⠀⠈⠁⠉⠉⠈⠉⠁⠁⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣠⣾⣿⡿⠿⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣀⡤⢶⡻⠿⣿⣿⣿⠋⠩⠶⢟⣷⣾⣷⡀
⠀⠀⠀⠀⠀⢀⣠⣄⣤⣤⣦⣵⠀⣸⣯⣾⣯⣥⣼⣹⣿⣿⣿⣶⡀⠨⣭⣿⣿⣷
⠀⠀⠀⣠⡾⣿⣿⣿⣿⡟⢻⣿⣯⣽⣿⣏⡉⣹⣿⠏⠀⠰⢽⣿⣆⢿⡿⢿⡿⣿⡇
⠀⠀⣼⡟⣼⣿⢋⣠⣶⣿⢗⣿⣷⣾⣿⣭⡉⣿⡏⠀⣬⣳⣿⣿⠛⠷⣷⠀⠔⣹⡄
⠀⢰⣿⠰⠟⣯⣾⣿⣧⠤⣼⣟⠻⠛⢛⣊⣻⣿⣿⠥⠚⢿⡿⠋⠀⠀⠃⠀⠸⢿⠎⠂
⠀⢸⣿⡇⠀⢿⣿⡇⣿⠰⣿⣿⠗⢾⣻⡿⠋⣻⡟⠦⠀⠀⠃⠒⡉⢀⡀⠀⠀⠀⠀⠀⠈⠂
⠀⢸⣿⡇⢷⠸⣿⣀⢿⠀⢿⣾⣦⣷⣟⠀⣦⣿⣱⣀⠀⠀⠀⣠⢾⣧
⠀⠀⣿⣿⡼⡆⢻⣿⡜⣆⠈⢿⣿⣿⣿⣿⣾⢿⣟⠿⣧⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⣀
⠀⠀⠈⢿⣿⣿⡄⢻⣿⡼⣷⣄⠙⢿⣿⣿⣿⣮⣿⣿⣿⣿⠟⡁⠀⠀⠀⠀⠀⠀⠖⠒⠚⠋
⠀⠀⠀⠀⠻⣿⣿⣦⣽⣿⣞⢿⣟⠶⢌⡛⠿⠿⣿⣿⣿⣿⣿⣵⡆⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠈⠳⣝⢿⣿⣿⡷⣍⠙⠲⢽⣷⣄⠀⠀⠀⠉⢿⣿⣿⠀⠠⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠙⣦⢹⡷⡝⣝⣷⣄⠢⡉⢻⣷⡀⠀⠀⡸⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡿⠾⠿⣿⣿⣧⡘⢆⢻⣷⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣴⣾⣭⣭⣭⡙⣶⡀⢻⣿⢳⠈⡄⣿⣀⣴⡶⠃⠀⠀⠀⠀⠀⡠⢞⠭⢶⣙⡆
⠀⠀⠀⠀⢠⣟⣿⢋⣭⢉⢻⣟⣿⢇⣼⣿⡆⣅⣇⣿⣻⡀⠳⣄⣀⣀⣠⠴⣫⣾⢷⡴⡫⢶⠖⠒⠲⢤⡀
⠀⠀⠀⠀⢸⣿⣇⣿⠿⢿⣿⡿⠷⠿⣿⣿⣿⣿⣿⢿⣿⠇⠀⠀⠉⠁⠠⠀⠾⣣⢋⠊⠀⠀⠀⠀⠹⣧⠙⣆
⠀⠀⠀⠀⠸⣿⡿⣿⣦⣾⣿⣿⣿⣿⠿⢫⣿⣸⡟⠳⢍⣾⣅⡀⠀⢀⣀⡴⠛⠁⣠⠀⠀⠀⠀⠀⠀⠸⡆⢻
⠀⠀⠀⠀⠀⠙⢿⣮⡛⠿⣿⣿⣿⣠⣴⠿⣹⠟⡇⠀⠀⡌⢻⠋⠋⠉⠁⣤⡆⡴⠁⠀⠀⠀⠀⠀⠀⠀⡇⣸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⠿⣿⠿⠷⠛⠁⠀⢳⠀⠀⢀⢸⡄⠀⠀⠀⡿⠸⠃⠀⠠⠀⠀⠀⠀⠀⠀⠁⡯⠈⣿⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠘⡆⠀⠈⠸⡇⠀⠀⢸⠇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⣼⢻⠦⣼⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡇⠀⡀⠀⠀⠀⢹⡀⠀⠃⢿⠀⠀⢾⡆⡇⢰⠀⠇⠀⠀⠀⠀⢀⡼⡄⠀⠀⠁⣿⣿⡟

⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢀⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⢠⠄⡇⠀⠀⠀⠀⠀⠀⢠⠸⠇⡿⠀⠀⠀⠀⠀⠤⢸⠿⡇⠿


⠀⠀⠀⠄⠀⠀⠀⣤⠀⠀⠀⠀⡄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠠⠀⠀⢠⡄⠀⠀⠀⢠⠀⠀⠀⣠⠀⠀⠀⠀⢄⠀⠀⠀⣤
⠀⠀⠀⣀⢰⣼⡇⣿⠀⠀⠀⠀⡀⣤⢸⡇⣿⠀⠀⠀⠀⡀⣤⢸⢸⡇⠀⠀⠀⢀⡀⡆⣿⢸⡇⠀⠀⠀⢀⢠⡄⣿⣻⠀⠀⠀⠀⣀⢠⣼⡇⣿
⠀⠀⠀⠉⠈⠉⠁⠉⠀⠀⠀⠀⠁⠉⠈⠁⠁⠀⠀⠀⠈⠁⠉⠈⠈⠁⠀⠀⠀⠈⠁⠁⠉⠈⠁⠀⠀⠀⠈⠈⠁⠉⠈⠀⠀⠀⠀⠉⠈⠁⠁⠉

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⠿⠿⠿⠿⠿⠿⢿⠿⠿⣿⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢘⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⡿⢿⣿⣶⣄⣤
⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢀⡤⠛⠉⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠼⣿⣧⣤⣤⣤⣴⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣦⣤⡴⠾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡆⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⣠⣿⣇⠀⠀⠀⠀⢀⣴⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢀⣴⠿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠈⢹⣿⣿⣦⣴⣶⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿
⠀⠀⠀⠀⣰⣿⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⢀⣤⣴⠿⠋
⠀⠀⠀⠀⣿⠏⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣿⠿⠋⠉
⠀⠀⠀⣸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠟⠉
⠀⠀⠀⣿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡾⠛⠉
⠀⠀⠀⣸⡀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡾⠟⠁
⠀⠀⠀⣿⡟⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⣿⠇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⢘⡏⠀⠀⠀⣸⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⣾⡇⠀⠀⠀⣿⡇⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⡇⠀⠀⢠⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⣿⡇⠀⠀⢸⡟⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⡇⠀⠀⢸⡇⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⣿⠁⠀⠀⢸⡇⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⣀⣀⣀⣤⣤⣦⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣉
⠀⣴⠀⠀⠀⠀⢰⡄⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⢠⣿⠇⣠⣄⠀⢸⠁⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠁
⢿⣿⣄⣿⣿⡆⣸⡇⠀⢸⣷
⠸⣿⣏⣻⣿⣿⡿⠁⠀⢸⡏
⠀⠻⠿⣿⠟⠉⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣀⠀⢀⣀⡀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠙⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣴⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠿⠄
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⢸⣿⡁
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⣷⡄⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⠃⢺⡿⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⡿⠟⢻⡿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣄⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠊⠀⠠⠄⠀⣤⣦⡄⠀⠀⠀⠀⠀⠀⠀⠰⡾⠿⠿⢷⠄⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⢰⣿⣷⠀⢁⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⢠⣤⡄⠀⢠⡘⠃⠀⠀⠀⠀⠀⠀⣿⣿⣇⣀⠀⠀⣠⣶
⠀⠀⠀⠀⢸⣿⣿⢀⣸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣤⣼⣿⣦⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣴⣦⣿⣿
⠀⠀⠀⠀⠸⠿⠿⠈⠹⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠙⠁⠿⠿


⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠒⠀⠀⢁⡉⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠖⠈⠀⢁⡀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠐⠂⠴⠆
⠀⠀⠀⠀⠀⠀⠾⠀⢸⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⡆⢀
⠀⠀⠀⠀⢰⣾⣿⠅⣾⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⣶⣿⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠠⠂⢿⣿⣿⡧⠌⡐
⠀⠀⠀⠀⠸⠿⠿⠁⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠟⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠗⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⣾⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣿⣇⣀⣀⣀⣿⣇⣀⣀⣀⣿⣿⣀⣀⣀⣸⣿⣀⣀⣀⣠⣿⣇⣀⣀⣀⣿⣇⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣤⡀⠀⠀⠀⣠⡄⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤⠀⠀⠀⠀⣤⡀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠛⠃⠀⠀⠀⠛⠃⠀⠀⠀⠙⠋⠀⠀⠀⠘⠛⠀⠀⠀⠀⠛⠁⠀⠀⠀⠛⠃⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⣿⡇⠀⢠⡤⠦⠤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⢤⡤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠤⠤⠤⠤⠤⢤⡄⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡏⠀⣿⠋⠉⠉⠉⠉⠉⠉⢻⡏⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠙⡟⠉⠉⠉⠉⠉⠉⠙⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡏⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡏⠀⣿⠋⠉⠉⠉⠉⠋⠋⢻⡋⠉⠉⠉⠉⠉⠉⢹⡟⠉⠉⠉⠉⠉⠉⠙⡟⠉⠉⠉⠉⠉⠉⠙⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⣇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠛⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⣿⡇⠀⠘⠓⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠚⠃⠀⢸⣿
⠀⠀⠀⠀⢻⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠀⡤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣀⡀⠈⠒⢶⣤⣬⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⣛⠛⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠰⡿⠟⠁⠀⣒⠲⠀⠀⢤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⡀⠀⠀⠑⠂⠀⠂⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠷
⠀⠀⠀⠀⠀⠀⠀⢀⣔⣂⠙⢶⡆⢸⡄⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠁⠒⠀⠀⠀⢀⡔⢁⡠⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⡴⣇⣈⠁⢦⣀⠀⠘⠟⣰⡇⠀⠀⠀⢠⠀⠀⣤⣀⣀⠉⠀⣭⣴⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣷⡟⠋⠀⠀⢈⣷⣶⣾⣿⣿⠆⠌⠠⠂⢀⣼⣿⣿⣿⣟⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡊⠙⢄⠀⠀⠀⠘⠛⠿⣿⠿⢛⠀⠠⠁⣰⣼⣿⣿⣿⡟⣾⣿⣿⠃
⠀⠀⠀⠀⡖⠺⠗⠚⡉⠘⡑⣄⠀⠑⠢⢤⣀⠀⢠⠈⢀⠄⠀⠄⢰⣿⣿⣿⡿⢏⣼⣿⣿⠃
⠀⠀⠀⢸⡿⣻⣦⠛⢀⡀⠀⡋⠲⣤⠖⠀⠸⠟⠃⢸⣋⡀⠀⢐⣾⣿⣿⡏⠀⠄⣿⠟⠁
⠀⠀⠀⢸⣿⡏⠻⣦⡈⠛⢄⠘⡷⢀⡴⠓⠂⠘⠛⣊⠀⣙⢦⠿⠿⡿⠿⢢⠑⠏⠀⠀⠀⢀⡀
⠀⠀⠀⠸⣮⡁⣶⠈⢑⣶⠘⠆⠘⠉⠀⠀⢀⡴⠁⠹⠆⣻⣀⠛⢴⠃⡀⠀⠌⠀⠀⠀⣴⡯⠢⠄
⠀⠀⠀⠀⠘⣯⢀⣐⢶⣁⠰⠇⠀⢀⠠⠐⠉⢀⡀⢌⠀⢀⡝⠒⢚⡀⠁⡐⠀⠀⠀⠀⠉⠵
⠀⠀⠀⠀⠀⢿⠀⠀⢹⣿⠥⠚⠈⠁⢀⠔⠂⠉⠀⠈⡴⣘⡇⠠⠊⠀⢰⠃⠀⠀⠀⠀⠀⠈⡖⠈⠄
⠀⠀⠀⠀⠀⢸⠀⠀⠉⠀⠀⠀⣠⠖⠁⠀⠀⠀⢠⠊⡅⢃⠹⠒⠟⢀⠂⠀⠀⠀⡄⡄⠀⠈⠑⠊
⠀⠀⠀⠀⠀⠘⣇⠀⢀⣀⡴⣞⠁⣰⠀⠀⠀⠀⢢⡀⢈⡁⢀⡆⡠⠃⠀⠀⠀⢀⠏⢁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⢛⡩⠀⠀⢻⡋⠉⠉⠀⠀⠀⠈⠟⢒⣷⠀⠀⠀⠀⠀⠀⣠⣞⠄⠘⠄
⠀⠀⠀⠀⠀⠀⠠⣇⡤⠞⠁⡼⠃⠀⠀⠐⠒⠂⠀⠁⢸⣷⣂⠀⠀⠀⠀⠀⣿⠀⣤⡄⢘
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⡼⠀⠀⠀⠀⠀⠻⢦⣠⠤⢄⣀⠤⠀⠠⠄⠀⠀⣠⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠘⣧⠀⣄⠀⠀⠀⠀⠀⠀⡀⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢠⠀⠀⠀⠀⠹⣦⡙⠃⠀⠀⠀⢀⣠⠃⣿⠀⣿⡇⠀⠀⢀
⠀⠀⠀⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠾⠶⠶⠶⠦⠤⠼⠷⠷⠦⠤⠤⠽⠤⠤⠿⠤⠿⠧⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠂⡙⢂⠓⣀⠂⠘⣀⠛⣘⠓⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠈⢁⣉⣴⣿⣿⣿⡿⠿⠯⣬⣅⡉⢀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣷⣾⣿⣭⠤⣄⠈⠛⢷⣟⢥⡄⢀⠀⠀⠀⠀⢀⡴⠎⠁⠠
⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⢈⢻⡀⠲⠄⡶⠶⠖⠋⠉
⠀⠀⠩⠙⠲⠦⠤⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠛⠋⠉⠉⠀⠀⠀⢹⣧⠀⡌⢇⣷⡄⠲⣿⡄⠀⠀⠠
⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠉⠒⠶⢤⣤⠴⠖⠋⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡃⢴⣌⣾⣿⣔⣿⡇⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠦⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⣿⠋⡝⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣤⠃⢀⠀⠀⠀⢸⡄⠀⣀⣀⠀⠀⠀⠀⢠⣶⣛⣀⠀⠀⠀⠉⠀⣤⡃
⠀⠐⠐⡐⠐⡀⠀⠀⠀⡐⢂⠀⠀⠀⢂⢲⡉⣱⠀⠀⠀⢃⠀⡨⢿⣿⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⢘⣠⣿⠂⢂⠀⡐⡐⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣦⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣂⣀⣴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠠⣬⣤⣴⣶⡾⠀⠀⠀⢠⣿⣿⣉⣀⣀⣾⡟⡀
⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠁⠀⠀⠀⣀⠠⠀⠘⠛⠿⣦⡀⠈⠛⠛⠋⠀⠀⢀⣴⠟⣿⣿⡂⠀⠾⣿⡟⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠖⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣠⠴⠋⠁⠀⢸⣿⠆⠤⠟⠉⠀⠀⠀⡤⢄
⠀⠀⠄⠄⠀⠀⠤⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⢦⡀⠀⣄⠀⠀⠀⠀⢀⣠⣴⠿⠃⠀⠀⢀⣼⡃⠈⠁
⠀⠤⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢰⣦⠀⠀⢹⡧⢀⣼⣆⣽⣶⠿⠟⠛⠉⠀⠀⠀⠀⠾⠟⢃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠂⠂⠉⠉⠀⠀⠀⠀⠀⠀⠠⠟⠠⢠⣾⣿⣾⣿⡿⠗⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠡⠴⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠀⣠⠤⠄⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡄⠀⠀⠛⠁⣤⣴⣆⠀⠁
⠀⠀⠠⠤⠤⠀⠀⠀⠉⠉⠀⠀⠀⠀⠠⠤⠤⠤⠤⡀⠉⠀⠀⠐⠟⠀⠀⠘⠛⠀⠀⠈⠁⠀⣀⠀⠀⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⢀⣀⠀⠀⠠⣦⠀⠀⠈⠉⠀⣸⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠁⢀⡀⠈⠉⠀⠀⢠⡄⢠⡄⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⡀⠀⢀⡨⠁⠸⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣠⣤⣤⡀⠀⠀⣀⡀⠄⠀⢀⣀⣀⣠⣤⠤⠤⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠶⠶⣤⣈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢁⣤⣔⣷⣽⣧⣬⣦⠤⠶⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠘⡙⠛⠉⠉⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠈⣧
⠀⠀⠀⠀⠀⠀⣀⣴⣿⢋⣤⣻⠃⠀⠨⠀⠠⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢿
⠀⠀⠀⠀⠀⣴⠋⠘⣝⢲⡿⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾
⠀⠀⠀⠠⢸⡇⠀⠀⠈⠛⠁⠁⠀⠀⠙⠬⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏
⠀⠀⠀⠀⢸⡁⣀⣀⣤⣤⠐⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠇
⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⡇⠀⠀⢠⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡿
⠀⠀⢀⠀⠀⠀⠀⢸⣿⣿⡿⠀⢠⣾⣿⣿⣿⣧⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⡼
⠀⠀⠸⡄⠀⠀⠀⣼⣿⡿⠁⢀⣾⣿⣿⣿⣿⣿⣎⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⣼⠁
⠀⠀⠀⡇⠀⠀⠀⣿⠟⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣰⠃
⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⢰⣿⣿⣿⣿⣿⣿⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡏⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡼⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡞⠁
⠀⠀⠀⠀⠈⠧⡀⣠⡾⢿⣿⣿⣿⣿⠿⠿⢇⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠚⠁
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⡇⠀⠀⢰⡏⠙⡇⠀⠀⢰⠋⠙⡆⠀⠀⢸⠋⢹⡆⠀⠀⢸⠉⢹⡄
⠀⠀⠀⠀⢰⠶⠶⠶⢾⡇⠀⡷⠶⠶⢾⡇⠀⡷⠶⠶⢾⠀⠀⡷⠶⠶⢾⠀⢸⡷⠶⠶⢾⠀⢸⡷⠶⠶⠶⡆
⠀⠀⠀⠀⢸⠀⠀⠀⣸⡇⠀⣷⡀⠀⣼⡇⠀⣧⡀⠀⣼⠀⠀⣧⠀⢀⣼⠀⢸⣧⠀⢀⣾⠀⢸⣇⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⢿⡳⠴⣻⠇⠀⢿⡧⠴⣻⠃⠘⣟⠧⠼⣻⠃⠸⣟⠦⢼⡿⠀⠸⣟⠦⢞⡿⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠀⠈⠉⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡇
⠀⠀⠀⠀⢸⠀⠀⣤⣤⣤⣤⡄⠀⣠⣤⣤⣤⣤⠀⢠⣤⣤⣤⣤⡄⠀⣤⣤⣤⣤⣄⠀⢠⣤⣤⣤⣤⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⣧⣤⣤⣤⠇⠀⢿⣤⣤⣤⡼⠀⠸⣧⣤⣤⣼⠇⠀⢧⣤⣤⣤⡿⠀⠸⣤⣤⣤⣼⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡶⠶⠶⠶⡆⠀⣴⠶⠶⠶⢶⠀⢠⡶⠶⠶⢶⡄⠀⡶⠶⠶⠶⣦⠀⢰⠶⠶⠶⢶⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠷⠶⠶⠶⠇⠀⠻⠶⠶⠶⠾⠀⠘⠷⠶⠶⠾⠃⠀⠷⠶⠶⠶⠟⠀⠸⠶⠶⠶⠾⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡟⠛⠛⠛⡇⠀⣾⠛⠛⠛⢻⠀⢰⡟⠛⠛⢻⡆⠀⡟⠛⠛⠛⣷⠀⢸⠛⠛⠛⢻⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿⠀⠀⠀⢸⠀⢸⡇⠀⠀⢸⡇⠀⡇⠀⠀⠀⣿⠀⢸⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⢸⠀⠀⠛⠛⠛⠛⠃⠀⠙⠛⠛⠛⠛⠀⠈⠛⠛⠛⠛⠁⠀⠛⠛⠛⠛⠋⠀⠘⠛⠛⠛⠛⠀⠀⡇
⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢀⣀⡤⠤⠄
⠀⠀⣠⢖⠉⠀⠀⣼⠞⠹⡄
⠀⢛⡽⠋⠀⠀⢴⡁⠀⠀⢳
⣰⡯⢺⠃⢠⡀⠀⢙⡄⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠁⠀⢫⡀⢸⣧⡰⢦⡑⡄⠀⠘⡆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⡠⠤⠖⢢⣄
⠀⠀⠀⠈⠛⠈⠁⠀⢻⡽⡄⠀⠹⣄⡠⠴⠒⠒⠛⠋⣐⣒⣾⠗⠖⠉⠉⣉⡽⢀⣀⣀⡤⠤⠒⠒⠚⠛⠛⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠽⡄⠀⢹⡶⠒⠚⠉⠛⠉⣁⡤⠏⠉⠉⢉⠙⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⢀⡠⠒⠚⠻⢷⠀⠀⠉⠉⠛⠛⠛⠛⠉⠀⠀⠀⠈⠓⠀⠠⠤⢀⣀⣀⣀⣀⣤⠬⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠐⡿⠛⠒⠢⣄⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠃⡴⠋⠀⠈⠳⡄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠘⡆⢇⠀⠀⢀⣀⣹⣰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠤⡿⢵⣿⣿⣷
⠀⠀⠀⠀⠀⢀⡼⠾⠒⢦⡸⡈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣧⣿⠋⢻⡿
⠀⠀⠀⠀⠀⠘⣄⣳⣲⣞⡁⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣄⡿⡷⠿⣷⠟⠁
⠀⠀⠀⠀⠀⠀⢧⣀⣠⠤⠽⣄⢻⡀⣴⢧⡚⠋⠲⡄⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⣃⣠⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢤⣐⣖⢏⠀⠻⣅⠀⠙⢧⡀⠘⣆⠀⠀⠀⣮⠶⢦⠖⠀⠀⢸⢤⡀⠀⠀⠀⠀⠀⢸⠀⡙⢿⡧⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⣹⣮⣣⡀⠀⠀⠀⠈⡇⠀⢹⠶⠦⣄⣉⠋⠁⠀⠀⠀⡼⠒⠾⣇⣀⣀⡀⠀⠈⡆⢱⡏⡟⠦⡜⠃
⠀⠀⠀⠀⠀⠀⢀⡔⢁⡞⠉⠀⠑⢝⡆⠀⠀⢀⡇⠀⣸⠀⠀⠀⢀⡽⠓⢦⡀⣼⡁⠠⡀⠀⢀⣾⡇⣀⠜⡰⠺⡇⢹⡞⠁
⠀⠀⠀⠀⠀⢠⠊⠀⣋⣹⡀⠀⡴⠂⠈⠑⢲⣮⠔⠒⠉⠒⢄⣠⠊⠀⠀⠰⠿⣿⠇⠀⢳⣀⡾⠤⢍⣁⠜⠀⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠙⠘⠉⠉⣤⣇⠎⢀⣤⠀⣠⠞⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⢀⣈⠗⠀⠀⠉⠀⠀⠀⢙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡫⣥⠤⣽⢿⣭⣥⣀⣀⣀⣀⣤⡤⠤⠤⠤⠤⠴⢶⢒⠖⠋⠉⢡⠏⠀⠀⠀⠀⠀⠀⢀⠞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡮⠼⢎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡼⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⢠⣎⣕⠠⠒⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣠⣾⣼⣥⣖⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣎⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠉⠀⣀⣉⡯⠤⢭⡭⣾⠦⠤⢄⠀⠛⠁⣬⠍⢹⢹⠁⢹⡏⡏⢹⠙⠤⡤⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠛⠉⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠀⠀⠀⠉⠉⠉⠉⠀⠈⠓⠓⠚⠒⠚⠓⠓⠒

⣯⡟⠏⠉⠛⠩⠍⠷⢻⣛⣯⡉⢹⡍⡸⢩⡜⠛⠻⣿⣹⣟⣹⢻⣿⣈⣹⣝⢉⡽⠯⢿⡛⢯⣙⠿⠍⣍⠍⣿⣯⣼⣵⠫⡯⢹⡁
⣇⣧⠀⡘⠈⠉⢠⣄⣤⣬⢭⣡⣬⡅⠉⠉⡏⢰⠀⠀⢩⢍⣩⣌⣍⣍⣭⣭⠈⠉⠻⣹⠉⣄⣭⣤⣭⣭⣌⣍⣭⣥⢭⠁⠩⣼⣤
⡷⡔⣲⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠃⠈
⡯⣎⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣬⠄
⣗⡧⠜⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣙
⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⢧
⣷⠎⣙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠳⡄
⣇⠀⣸⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠹
⣏⡞⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⢹
⣥⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢃
⣯⢐⣲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⡓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⣿
⣾⠎⢳⡤⠀⣀⣀⠀⠀⣄⠀⡀⣄⠀⡀⣀⣾⣧⡤⠀⢀⣀⣀⣀⢀⣀⣄⣄⣠⢀⡧⠻⠀⣄⠀⣀⣀⣀⣀⠀⠀⠀⠀⣀⠜⠐⢏
⣧⡂⣺⠟⠀⠩⠌⣄⣀⣉⣀⣡⣩⣠⡀⠁⡆⢹⠉⠁⠈⣩⣡⢉⣌⣉⣌⠉⠁⠉⠢⢹⠀⠈⣅⣍⠭⣉⣩⣠⣩⣡⡀⠁⠲⢄⡨
⣯⠛⢇⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⢻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡄⢳
⡏⠒⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢃⢧
⡇⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠻
⡧⢖⡻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣠⣕
⡁⢸⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⡀⠀⢀⢀⠀⠀⡀⠀⠀⢖⣒⡨
⠧⠌⠑⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣻⠀⠀⠀⠋⠋⠉⠙⠉⠉⠁⠀⠀⣴⣲⠌
⡄⡪⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡾
⠓⠬⠤⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⡭
⣷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠜⠛
⡟⠦⣄⣢⣀⣀⣀⣀⢀⣀⠀⢀⡀⠀⡀⠀⢧⣤⣀⢀⡀⡀⠀⠀⢀⡀⢀⣀⡀⣀⠔⣸⣀⣀⣀⢀⢀⢀⣀⠀⡀⣀⣀⣀⠴⡾⠿
⣏⣁⣂⢀⣀⣀⣾⡟⠂⣈⣒⣀⣠⣄⣀⣀⣁⣀⣿⣸⣄⠑⢄⣀⢊⢀⣠⣏⣀⣸⣀⣩⣿⣇⣏⣾⣰⣌⣓⣠⣃⣿⣖⣒⣒⣂⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⣠⣶⡄⠀⠀⢠⣿⡿⠉⠀⠀⠀⠀⠐⠒⠒⠒⠂⠲⠖⠀⠀⠀⠀⠀⠈⢿⣿⡆⠀⠀⢠⣶⣄
⠀⠀⢀⣾⣿⠟⣡⣴⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⣦⣌⠻⣿⣷⡀
⠀⠀⣼⣿⠇⣰⣿⡟⠁⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠈⢻⣿⣆⠸⣿⣧
⠀⠀⣿⣿⠀⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢀⣿⣿⠀⣿⣿
⠀⠀⢻⣿⣆⠹⣿⣷⡀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣾⣿⠏⣰⣿⡟
⠀⠀⠈⢻⣿⣦⡈⠋⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠙⢁⣴⣿⡟⠁
⠀⠀⠀⠀⠙⠛⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⠛⠛⠛⠛⣛⣿⣟⡛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣼⠋⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠹⢦⣀⣠⠞⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠉⠙⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⡤⠤⠶⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡴⠋⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣸⠁⠴⠚⠒⠂⠀⠀⠀⠀⠀⠒⠛⠒⢧⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⣿⠀⠀⢠⣶⡀⠀⠀⠀⠀⠀⢰⣦⠀⠈⢧⣀⣤⣀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢈⡟⣼⠃⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣱⠏⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢰⢣⡟⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⢳⡀⠀⠰⣟⠛⠛⠒⣲⠆⠀⢀⡞⠉⠓⠲⠖⠚⢳⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢳⣄⠀⠈⠉⠉⠉⠀⣀⡴⠋⠀⠀⠀⠀⠀⢀⡾⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⠙⠒⠒⠖⠒⠋⢹⡏⠛⠛⠛⠛⠛⠉⣹⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠟⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀⠀⠀⠀⢠⠇⠀⡼⠁
⠀⠀⠀⠀⠀⣀⡤⠶⠖⠒⠚⠛⠳⠶⠤⣄⡀⠀⠀⠀⢀⡏⣾⠉⠙⢿⡶⢶⣏⠀⣼⡁⠀⢀⣀⡀
⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⡴⠚⡿⣸⠇⠀⠀⢀⣙⣿⡘⠟⠷⠻⠶⣞⣉⣇⣀
⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢡⡟⠀⠀⢀⡏⠉⠉⠉⣉⣩⣭⣉⡉⠉⠉⠉⣧
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⢹⡗⠶⠶⢺⠁⠀⢠⣾⣿⣿⠿⣿⣿⣦⡀⠀⢸⡀
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢳⡀⠀⡾⠀⠀⣿⣿⡏⠀⠀⠈⣿⡿⡇⠀⠈⡇
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⢳⡇⠀⠀⢻⣿⣷⣄⣀⣴⣏⣠⠇⠀⠀⢿
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠙⠿⣿⣿⡤⢿⡽⠂⠀⠀⢸
⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠶⠶⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣴⣴⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠐⠌⢋⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠇
⠀⠀⠀⢠⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠂
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣣⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⢀⠀⠀⠈
⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣻⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡄⠈⠀⠁
⠀⠀⠀⠸⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⢀⡀⠀⠀⠘⠂⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠈⠁⣠⢴⡶⣶⢤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣤⣼⣤⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣀⠀⡀
⣀⠀⢰⣷⣿⣿⣿⡷⣷⠀⢀⡀⠀⠀⠀⠀⢀⣤⢸⣿⠀⠀⢠⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠠⣤⣀⣿⣿⣿⣿⣿⣿⣷⣠
⠉⠀⠘⣟⢿⡿⣿⢟⡟⠀⠈⠁⠀⠀⠀⠀⠨⠤⠶⠿⠄⠀⠸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢠⡄⠈⠙⠛⠛⠉⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠛⠿⠟⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢽⡯⠉⠉⠉⠉⠉⣿⡏
⠀⠀⠀⠀⠀⠰⠆



⠀⠀⢀⣤⣴⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶
⠀⢰⣿⣿⣿⣿⣟⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣫⠴⢒⣛⣛⣒⠲⣍⡳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠘⣿⣿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢚⡭⠴⠶⠬⣙⠦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇
⠀⠀⠘⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⣫⣍⡓⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⡂
⠀⠀⠀⣿⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢀⣀⠤⠀⣌⢙⡀⢀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣰⠆⣄⢲⡀⠀⠀⠀⠀⠀⢠⡿⠾⣧⠠⠉⠀⠀⠀⠀⠘⠒⠒⠁⠀⠀⠀⠀⠀⢠⡶⣤⠐
⠀⠀⠀⠀⢹⣿⣛⣿⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⠀⠀⠀⠀⠀⠉⠁⠨⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀
⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠆⠀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠀⠀⠺⠛⠻⠶⠗
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠠⣉⡣⠀⠀⠀⠀⠀⠀⠀⢠⣰⠦⠤⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠠⣦⡤⣦⣰⠄⠀⠀⠀⠀⣿⡿⡇⠀⠀⠀⠀⠀⠀⠀⠆⣀⢤⡀⠀⠀⠀⠀⠀⠀⠐⠷⠾⠆
⠀⠀⠀⠹⠜⠠⠀⠀⠀⠀⠀⠀⠀⢠⡉⠹⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⣿⣟⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⢀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣄⢀⡄⠀⠀⠀⠀⠀⢀⡀⠀⡀⠠⠄⠀⠀⠀⠀⠀⡀⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠴⠉⠁⠙⠄⠀⠀⠀⠀⠀⠀⢠⣥⣔⠀⠀⠀⠀⠀⠰⠿⠓⠒⠷⠇⠀⠀⠀⠀⠀⠀⠟⠒⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡤⠀⠀⠀⠀⠀⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠐⠒⠂⠤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠈⠀⠀⠁⠀⠨⠀⠀⠀⠀⡂⠀⠐⠁⡠⢀⠄⠀⡀⠂⠀⡗⠉⡧
⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⠁⠀⠊⠀⠔⠐⠀⠊⠀⠐⠁⡪⢀⠔⠁⠈⢀⠔⠀⠈⠐⠀⢰⡃⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⠉⠙⠓⠒⠒⠲⠤⠤⠤⢄⣀⣀⣈⡐⠁⠀⠀⠐⠑⠄⢸⠆⠠⡆
⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠠⠀⠀⠉⡀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⢀⣉⠉⠉⠑⠓⠒⢢⢹⣴⣸⠁
⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠘⠀⡄⠀⡀⢠⠀⠀⢀⡀⠀⠀⠀⠀⠀⠸⣨⠞⡁
⠀⠀⠀⠀⠀⠀⢀⠁⡄⠀⠀⢀⠈⠡⠀⣀⠐⠀⠀⠄⠀⠀⠍⠀⠁⣐⡁⠀⠘⠀⠘⠀⠁⡼⢸⠁
⠀⠀⠀⠀⠀⡆⠈⠀⠤⠀⠂⣈⡁⠀⠀⠀⠀⠁⠴⠂⠀⠀⡆⠀⢠⣄⠀⢀⢁⠀⡄⢨⢀⡇⡎
⠀⠀⠀⠀⢠⠀⠆⡖⡀⢠⢠⢀⠀⠄⣈⡉⠀⠐⠒⠂⠀⠤⠤⠀⠀⢉⡀⠘⠙⠰⠀⡄⠠⡑⡇
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠬⢈⠀⠂⠋⠃⠀⠸⠸⠀⠁⣦⡀⠀⡄⠄⠀⠉⠀⠀⢠⠁⠠⢳
⠀⠀⠀⠀⡆⠀⠘⠈⠃⠃⠔⡄⠐⢠⢈⠉⠀⠐⠒⠀⠀⠀⠀⠀⣁⠁⠀⠛⠃⠁⠐⠂⠠⡌
⠀⠀⠀⢠⠀⡄⢈⡈⠠⠐⠂⠄⠀⠈⠈⠐⠀⠢⠅⠀⠠⠰⠀⢠⣄⠀⢈⠀⠘⠀⡎⠀⠐⡃
⠀⠀⠀⠘⠀⠀⠉⠁⠀⠰⠉⢀⠁⡠⡄⠀⢈⣁⠀⠀⠒⠀⠀⠀⠀⠀⠂⠀⡀⢠⡇⠀⢰
⠀⠀⠀⢠⢤⣀⣀⠉⠀⠒⠀⠤⠄⣀⡀⠀⠘⠘⠘⠀⠿⠂⠁⠀⠀⠀⠉⢠⠀⢸⠃⠄⠆
⠀⠀⠀⠈⠳⣶⣬⣭⣉⢓⣒⠤⠤⣀⣈⠉⠐⠢⠠⠄⣀⡀⠀⠀⠀⠀⠀⡄⢀⣿
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠟⡿⣿⣶⣮⣭⣯⣖⣢⠤⣄⣀⡉⠉⠐⠒⠀⠀⡼⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⣷⣯⣿⣭⣹⣃⠐⠊⢄⠰⠀⠆
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠷⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠄⢀⠀⠀⠀⠀⠀⠀⠄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠂⠁⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⠿⠿⠿⠿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠈⠉⠙⠛⠳⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⢠⣤⣤⣀⡀⠀⠉⠙⠻⠷⢦⣤⣀⣀⠀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⣿⠃⠀⠈⡛⠁⠀⢤⣄⣀⠀⠀⠙⠻⠿⣿⣿⣯⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⢸⡏⠀⠀⠀⠁⠀⠀⡀⢨⠭⠝⠛⠶⣤⣄⡀⠉⠛⠿⣿⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣸⡏⠀⣼⣷⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⣷⠀⢀⡿⢀⣾⠇
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⡾⠟⠛⢿⣶⣿⠁⢠⡿⠀⠀⠀⠀⠀⣀⣀⣀⠀⠠⠀⢰⡟⠙⠀⢠⡟⢀⡾⢁⣾⠏
⠀⠀⠀⣠⣾⠿⠛⠛⠉⠉⡽⠀⠀⠀⠀⠹⡿⣷⣾⡇⡀⠦⠀⠀⣰⠏⠈⠉⠛⢦⣴⡟⢀⣠⣶⠿⠶⢿⣴⡿⠃
⠀⠀⣰⡿⠁⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⣰⠇⢸⣿⠈⠻⠶⣤⣴⠁⠀⠀⠀⠀⢴⠏⣿⡟⠉⠀⠀⢀⡴⠛⠻⣦⡀
⠀⠀⣿⡁⠀⠀⠀⠀⠀⠧⣀⣀⠀⣀⡴⠋⠀⢸⣿⠀⡀⢀⡼⠉⠛⠦⠀⠀⠀⠀⢸⣿⡀⠀⠀⢰⡏⠀⠀⠀⢸⣿
⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⣰⡿⠉⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠈⢧⡀⣀⣤⣼⡟
⠀⠀⢸⣇⠀⠀⠀⠀⠲⣦⣤⣤⣤⣶⣶⣿⠏⠀⠀⣀⠸⣤⣀⠀⠀⢀⡀⠀⠀⢀⣿⣿⣿⣄⡀⠀⠀⠉⠉⠀⢸⡇
⠀⠀⠘⣿⠀⠀⠀⠀⠀⣠⣿⣿⠛⢻⣿⠟⠃⣴⣿⣿⠀⠀⠉⠛⣶⣿⣁⠀⣴⠟⠁⠀⠀⣉⣽⢿⣶⡄⠀⠀⢸⣇
⠀⠀⠀⢻⣷⣤⣤⣤⣶⣿⡿⠁⢠⣿⣯⣄⣀⠈⠉⠠⡤⠀⠂⣶⠏⠀⢉⣿⠋⠀⠀⢀⡾⠉⠀⢀⣿⡇⠀⠀⠀⣿⡀
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠁⢠⣿⡿⣿⡟⠉⠀⠀⠀⠀⠀⠸⠃⣄⠀⣸⣿⠀⠀⠀⡼⠁⠀⣀⣼⣿⣧⠀⢀⣠⣿⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣿⠃⢠⣿⡟⠀⠀⢀⡀⣤⠀⠀⠸⠂⠀⠀⢀⣼⣿⣿⡆⠀⠀⠱⠶⠞⠉⠸⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⠙⠳⢾⣸⣿⣧⠀⠀⠀⠀⣄⠀⣠⡾⠟⠛⢻⣿⣄⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠈⠉⠛⠶⢦⣄⡀⠈⢀⡿⠁⠀⢀⡞⠁⠹⣆⡀⠀⠀⢀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠙⠛⢿⡇⠀⠀⠸⡁⠀⣴⢿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠙⠚⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣦⣴⣿⣼⣿⣷⣦⣄⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠉⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⢁⣾⣿⣯⣭⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠈⠛⠛⠉⠉⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣨⣿⡿⠁⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣶⣶⣹⡏⠈⠁⠀⣄⠀⣴⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣀⡤⠀⠐⢧⣬⣿⣷⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠦⠤⣌⣿⣿⣅⠀⠀⣠⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡛⠿⢷⡄⡉⣩⣏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⣿⣷⣾⣷⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⡿⢿⣿⣿⠏⠀⠸⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⡄⠀⢩⣿⣿⣿⣆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣆⠀⢻⣿⣿⣿⡄⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⣼⢿⣿⣿⣧⡀⠙⠻⢿⣷⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣵⣾⡿⣸⡟⣧⠛⢤⣬⣥⣶⣧⣆
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⠧⠿⠧⡿⢆⣍⠀⠀⣺⣧⠹⠤⠤⠤⠤⠤⠆⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠁⠀⠛⠛⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡖⣭⠛⣿⣷⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣧⡼⠇⠉⠀⢠⡟⠀⢰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡅⠀⠀⠠⠤⠸⡶⠎⠹⣧⡇⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠆⠀⠐⠻⠂⠑⢺⡿⣿⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣄⣄⠀⠀⠀⢈⢁⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⢀⠀⠀⣬⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠶⠊⠁⠀⠀⡿⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢰⠀⠀⠀⡴⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣴⡆⠀⡏⠀⠀⠈⠀⣤⣄⣀⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣻⡏⢳⡀⠀⠀⠀⣠⠞⠋⠉⡝⣿⠛⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡿⠀⠀⢷⡄⠀⠞⠁⠀⢀⠶⣨⠃⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⡼⡇⠀⠀⠘⠟⠀⠀⠀⢀⢾⣻⡇⠀⠀⠀⠚⣹⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢟⣻⣟⣝⣋⡹⣙⢍⡫⣝⢻⢎⡽⠀⣠⣷⡀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⢻⣿⡿⣷⣦⡆⠢⠸⠻⠷⢷⣼⣡⣽⣾⣿⣿⡟⢄⣠⡤⡟⢆
⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠁⠀⠀⠀⡩⠁⠈⣿⢯⢘⣈⡁⣃⣘⣉⡕⢾⣿⣿⣿⠀⠀⠹⡼⠀⠈⠳⢤⡀
⠀⠀⠀⢀⡠⠖⠋⠀⠀⠀⢀⠠⠄⠊⠀⠀⠀⢸⡆⡂⡆⠆⠰⠸⠰⠆⣻⣿⣿⠇⠀⠀⠀⠘⢆⠀⠀⠀⠈⠓⢄
⠀⡔⠁⢀⠀⠄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢧⠥⢭⢭⢨⠉⡩⡋⢽⣿⣟⣤⢞⡧⡤⣄⢀⡙⠦⡀⠀⠀⠀⠑⢤⡀
⠀⡄⡐⠻⠚⡆⢤⣀⣠⡤⠔⠀⠀⠀⠀⠂⠀⠀⠀⠓⠒⠒⠲⣶⣦⣍⣾⠿⣿⣿⣤⡶⣞⠛⣁⡏⠀⠀⠙⠒⠤⢄⠀⠉⠠
⠀⣿⢷⣾⣷⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡤⠖⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⡀⠀⢀⠆
⣼⡏⣿⣿⣿⠂⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⣠⣤⠖⢋⡁⠔⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁⠈⢻⣿⡍⢃⡴⠂
⠙⣧⣌⢟⣛⠉⢀⡒⠀⠀⠀⠀⠀⠁⠀⢘⡉⢀⡔⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⣀⠄⠀⠀⠘⢻⢿⣿⠇
⠀⢩⣿⢻⣍⡑⣋⣉⠓⠒⠤⠠⢤⡀⡴⠼⠀⢉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠊⠀⢀⠁⣿⡏
⠀⠈⣿⣧⣿⡠⣡⢽⣖⣿⠻⠦⠰⠐⠒⣶⠷⣬⠾⣭⣾⣭⣯⣽⣏⣱⣉⣾⣐⣲⡞⢺⣂⣞⣳⣏⣲⣏⣿⣽⣿⣿⠃
⠀⠀⢹⣿⡾⣧⢹⠮⣿⠼⣧⠻⡀⣳⠀⣺⠄⣺⡍⣿⢛⡄⢸⠠⢿⡥⣾⢩⡟⢽⠍⢽⠏⣿⣭⣿⢽⡯⣿⣼⣿⡟
⠀⠀⠀⢻⣿⣽⣆⢫⠸⡧⢻⡇⣷⣸⡦⢼⣖⢺⡇⢻⡷⣷⢼⣎⢽⣏⣿⢰⡏⢹⠄⣽⢛⣿⢲⡿⣾⢯⣏⣿⡿
⠀⠀⠀⠀⠫⣷⣨⡍⠡⠯⠄⣿⡸⣷⢷⣚⣷⡐⣿⠘⣇⢸⢘⣀⢸⡇⣿⢸⡏⣽⡆⣿⢺⡿⣾⡇⡿⣾⣽⡿⠁
⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣾⣽⣌⣋⣹⠳⠾⠐⢻⡄⢿⠸⠎⣿⢸⡇⣿⣸⡇⣾⢰⡟⣻⠧⣿⣼⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠿⠷⠾⠷⠾⡶⣿⣾⣿⣿⡿⣷⡿⣾⡿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⢖⠙⠟⠷⢮⣕⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣦⣼⣿⣿⣯⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠟⠛⠿⠿⠛⠛⠻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⣤⣄⠸⣿⣿⣿⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⠋⢉⡳⠀⠀⠀⠉⣀⣀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢶⡋⠏⠑⠀⠀⠀⠐⠚⠁⠈⡿⣻⣿⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢰⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡽⠋⠠⠽⠦⡀⠀⠘⠃⠀⠀⠀⠀⠀⢠⢸⣿⣿⣷⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠂⠀⠀⠀⠈⠵⡄⠠⡒⠀⣂⠄⠀⢠⣾⣿⣿⣿⣿⣿⣿⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⣶⢸⡀⠉⠉⠁⢀⠔⠁⣿⣿⣿⣿⣿⣿⡿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠄⠀⠀⠀⣿⣞⢻⡄⢠⣖⠁⠀⠀⡿⢻⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠁⠀⠀⠀⡠⠻⣿⣿⡟⣿⣿⡇⠀⠀⣇⠀⠻⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⡠⢎⠀⠀⠈⠙⠷⢿⡛⠳⣄⠀⢹⠀⠀⡉⠫⠉⠒⢤⡀
⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⡠⠊⠀⠸⣄⣠⣤⡶⠶⠿⢿⠦⢈⢧⢸⠀⠀⡸⠟⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠙⣿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐