Terminus Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠂⢀⡄⡀⣮⠬⠄⠀⠋⢀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠀⢠⡿⢶⡀⠀⠗⠉⠀⠀⠀⠀⠠⠂⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠘⣶⠸⠀⡰⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠠⢈
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢉⠀⠀⠁⠀⠃⢐⡀⠀⠀⠴⠄⢠⠀⣠⣀⡀⣰⠀⠈⠽⠄⠀⠀⠀⠉⣾⠀⢠⠀⠀⠀⢀⢹
⠀⢀⠄⣠⡀⣀⡿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡩⢈⣹⠄⣀⢄⠸⠟⠀⠀⠀⢀⡤⡀⠈⣧⣉⠴⡧⠸⣂⡀⠀⠀⠀⠀⢩⠒⢲⡾⡿⠿⠥⠤
⣴⡮⣿⣿⣿⣏⢳⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠋⠈⠳⠐⠗⠀⠄⢀⠀⠀⢀⣠⣼⡇⣠⠬⣳⢶⡤⠀⠈⠉⠀⠀⠩⠃⢸⠁⣾⡃⡿⣿⡖⣾
⣿⣿⣿⣿⣿⣏⡀⣹⡉⢏⡯⠀⠠⠃⠀⡀⠀⢀⣠⡄⠂⠀⠀⠉⢔⣀⣀⠁⠉⣅⣈⣽⣋⣌⣧⣈⣉⠄⠃⣈⠀⡔⠂⠀⠀⡠⣠⣸⡆⢐⡇⠛⠻⠟⣿
⣿⣿⣿⣿⣿⡷⠍⣬⢹⣿⣿⣫⣶⠁⠠⣡⠴⠿⠙⢠⣤⣀⠀⢀⣠⣷⡦⠄⡌⡞⢻⣽⣿⣍⠉⡟⠛⠛⠉⣛⣙⡓⠒⠃⠀⣝⡝⢻⣇⣿⢁⣴⣇⣸⣿
⣿⣿⣿⣿⣿⡿⢢⡟⠶⣍⡀⢾⣿⣶⣄⣿⣶⣶⣿⠋⣽⠟⣻⣾⣿⡟⠛⠋⡉⡀⢾⣿⣿⠿⡿⣷⡔⠻⢀⣬⣬⣤⣀⢂⣼⣿⣴⠶⠃⢁⣼⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣉⢀⣿⣿⣿⡿⣾⣯⣅⣫⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⡇⢰⣿⣾⣿⣿⣿⣧⣾⣿⣿⣿⣷⣿⠿⣿⣿⣿⠿⠿⢿⣿⣿⠀⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣿⣉⣾⣿⣿⣿⠿⣿⢹⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⠏⢻⣿⣿⠿⢿⣿⣿⣟⣃⣘⣋⣄⣀⣀⣠⣤⣤⣤⣴⣮⣀⣷⡎⣛⣿⣿⡿⣿⣿
⣿⣿⡿⣿⣯⣿⣿⣿⣿⣞⣥⣿⣼⣛⣻⠿⠿⠿⠟⢻⢿⡟⠛⣛⠟⠧⣟⣬⠳⢛⣩⣧⡄⠝⡏⠭⠟⢻⣍⢙⡟⣹⣿⢏⣿⡻⣿⣿⣿⣿⣿⣿⡗⣿⣿
⣿⣿⣿⣿⣿⡿⠿⢿⠿⣟⠯⣽⣿⡟⢽⣋⠔⣃⣭⡶⠈⠙⠉⠁⠀⠀⠀⠀⠀⠒⠃⠽⠯⢧⣠⣶⣔⣶⡮⠛⢛⣻⡀⠛⣛⡑⠾⣿⡶⠾⠿⡿⠛⢿⣿
⣿⣿⣿⣿⠿⠛⠀⠀⠀⣹⣧⠛⢻⣿⣿⣤⣿⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠈⠀⠀⠀⠀⠋⠉⠉⠱⣦⠀⠀⢀⣄⣀⣤⣤⣽
⠿⠿⠿⠿⠷⠦⠀⠀⠀⠈⠉⠁⠈⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠓⠒⠚⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢄⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⢷⡀⠘⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⣀⣴⡟⠛⠀⠀⣠⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⣶⣿⠟⠛⢷⡆⠀⠀⠈⠀⣴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢓⠀⢸⣿⠁⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⣈⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⠀⢀⣿⠿⠗⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠿⠶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠘⠋⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⣨⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⣤⡾⠛⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⣀⣀⣀⣠⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡀⠀⣀⠀⠀⠀⢠⣶⡿⠛⠋⠀⣠⠄⠈⠉⠛⠛⠛⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⡀⢠⣿⣷⣴⠟⠀⢠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⣿⣿⠿⠇⠀⠀⠀⠀⣠⣶⣤⣀⡀⠀⣠⣴⣶⡟⠋⠉⠉⠛⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⣿⠆⠀⠀⠀⠀⠀⠈⠉⠙⠋⠁⠻⠿⠛⠉⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠈⢿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠀⠘⠀⠀⠀⠀⠀⢴⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠂⠀⠀⠀⠀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠐⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢶⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢺⣯⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣸⣷⠀⢠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⢸⡿⠋⠀⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠸⠆⠀⠀⢀⣴⣇⡀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠸⠟⢻⣿⡄⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⠛⠋⠙⠉⠁⣀⡀⠀⠀⠀⠀⢠⡞⠉⢹⡆⠀⠀⠈⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠈⠀⡈⢷⣤⡄⢠⣄⠀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣿⡏⢀⣤⠀⠀⠀⠘⠿⣇⣾⣿⣧⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠁⠸⣿⡆⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣾⠆⠀⢠⡄⢀⣀⠀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣀⠈⢡⣾⣿⣇⠀⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⢾⣿⠀⢸⣿⣿⣿⡀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠸⣿⣿⣿⣷⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠐⡄⠀⢘⣿⣿⣿⠇⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣶⣿⡇⠀⠀⠀⣿⡟⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠉⣷⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠇⠀⠀⠀⢀⣀⣀⠀⠀⣀⠀⠀⠻⠋⠀⠀⠙⣷⡄⠀⠀⠀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⠁⠀⠀⠀⠀⠀⠙⠋⠉⠉⠉⠛⠃⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣸⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡏⠀⠀⠀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠈⠛⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣶⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⠿⠟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠇⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠗⠂⠐⠒⠒⠀⠐⠒⠒⠀⠀⠘⠻⢿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡄⠀⠀⢠⣶⣦⣸⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣦⣾⣿⣿⣿⣷⣽⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁





⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠈⠛⠉⠉⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣶⣾⣿⣿⣿⣿⣧⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠁⠀⢜⠋⣠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣾⠟⠁⠀⠀⠀⠁⠼⡏⢴⡶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠈⠀⠀⠀⠀⠜⡠⠃⠀⡿⠈⢁⣠⡶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⢠⡤⢶⡆⠀⢈⡾⠁⠰⣿⣿⡀⡘⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣕⢠⣿⣷⠻⠀⠀⠘⡇⠀⠀⢍⠹⣷⣇⢰⣶⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣿⣿⣿⣛⣀⡀⠄⠀⠀⠀⠀⠙⢻⣿⡄⣿⡇⢲⡆⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣘⣿⣿⠀⢀⡠⠶⣷⠀⠀⠀⢸⣿⣏⢠⢙⣿⣇⡒⣒⠀⢄⢀⣀⣀⣀⡀⠀⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⡀⠀⠈⠛⠛⠻⡿⢇⣠⣾⣿⡇⠀⣀⣿⣷⣇⠈⣟⣉⡉⡑⠂⠠⣎⣿⠿⢿⣿⣿⡄⠀⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢀⢹⡆⠀⠀⠀⠀⠀⠈⠙⠟⠿⠥⠾⠿⠿⠧⠘⠀⠸⢏⠁⠉⠀⠀⢚⠻⣴⣦⡙⣿⣷⣇⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠛⠚⠂⠛⠿⠷⢿⣿⡧⢹⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⠂⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⠒⣆⠀⠀⠰⢦⡤⠩⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠈⡙⠒⠌⢿⡇⠀⠀⠀⠀⣽⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠀⠈⠙⢶⡀⠐⠁⠘⠾⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⠿⡀⠀⠘⢳⣤⣀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠆⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠠⠌⣀⠘⣷⣌⡳⣦⣴⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣿⣄⠈⠳⣌⠙⢻⣿⠤⠀⡏⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⢿⢷⣶⣾⣁⣸⡃⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠣⡞⠉⠁⢀⠀⠎⠉⠀⠀⣴⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠉⠛⠛⠃⠀⠀⠀⣠⡚⠿⢿⠟⠻⠶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢹⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠦⡀⢀⠟⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣤⣴⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠁⠨⠀⡟⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠀⡀⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⣀⡒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⠆⢰⠭⠉⠾⠈⢡⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢠⠤⡄⠠⠀⠀⠀⠀⠈⢇⠸⠅⠀⠥⠂⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⢀⡤⠄⢄⢰⠏⠁⠱⢀⡤⠀⢄
⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⠉⠈⠉⠉⠉⠉⠈⠑⠢⠤⠐⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠀⠀⠀⠸⣁⢀⠈⠈⠀⡄⠀⠸⡀⢀⡘
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢢⠀⠀⠀⢸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢺⢸⠀⠀⠀⠀⠀⠣⣀⣀⣨⣇⣀⣀⡠⠃
⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⠀⠰⣊⠔⠀⠀⠀⠀⠀⠠⢚⡡⠀⢸⠈⠀⠈⠤⠠⢹⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⢸⠀⠀⡠⢊⠄⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⢀⣀⣀⣂⣐⢸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⢢⠄⠀⠄⠀⠀⠀⠀⠀⣠⠇⠀⢸⠀⠀⠋⠈⠁⢸⠁⢨⠁⠈⠁⢸⠁⢱⠀⢸⠨⠭⠤⠉⠉⢸⢸⢸⠀⠤⠤⢤⠀⠀⠀⠀⠨⢇
⠀⠀⠀⠀⠢⢀⣸⢳⠤⠤⠴⠄⠃⠀⠀⢸⠀⠀⡃⢀⠀⢀⠀⢀⠀⢀⠀⢸⠀⢸⠀⢸⠀⠀⠀⠀⠀⢸⢸⢸⠀⡄⠐⢸⠲⠀⠀⠀⡅⠀⡇
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠫⣤⠄⠀⠀⠀⠠⠀⠀⠄⠈⢠⣬⠇⠘⠒⠒⠒⠒⠒⠚⢸⢸⠀⡷⠀⢸⢀⠀⠀⠀⡄⢀⠆
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⢸⠁⠁⣷⠉⠉⠉⠉⠉⠉⠉⠉⠉⢸⣞⠀⠉⠉⠉⠉⠉⠉⠉⢹⠸⠀⠀⠀⠀⢠⠀⠀⠀⠇⠸⠁
⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠈⠉⠁⠁⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠈⠉⠉⠉⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠁⠁⠀⠁


⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⠂⣀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣴⣋⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣿⣿
⠀⢀⡾⡿⣿⢿⣿⣿⡷⣿⣴⡅⠀⠀⠀⠀⢠⣾⣿⣷⣷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⣴⣿⠿⠾⣧⣽⡿⣿⣿⣿⣿⣥⣦⢀⣄⡰⠿⣛⠻⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣧⣠⠀⠀⣀⠀⠀⠀⣸⣿⣿⣿
⣿⣿⢿⣿⣯⣿⣿⣿⣮⣿⣿⠿⣿⣿⣿⣽⡔⣇⣤⣭⣤⡄⣀⣀⣠⣶⣶⣶⣸⣿⣯⣏⣿⠀⠐⣋⠃⠀⠀⣿⣿⣿⣿
⠈⠉⠙⠿⠯⢭⣿⣿⡿⢿⣿⣿⣿⡿⠻⣦⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣽⣿⣦⣼⢧⡟⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠸⠿⣯⣵⣾⣦⣤⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⢽⢛⣿⡿⠹⠇⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⣶⣄⡀⠀⠀⠀⠀⠈⠉⢈⡛⠲⠾⡍⣛⣻⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣷⣿⠀⠀⠀⣿⣿⣿⣿⣿
⣤⣀⡀⠀⠀⠀⠙⠿⣿⣶⣄⡀⠀⠀⠀⠸⣿⣷⣄⠀⠈⠙⠋⠱⠂⢾⣿⣍⢛⣿⣿⣿⣾⣿⡃⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣦⣤⣀⡈⠙⠿⣿⣶⣄⡀⠀⠈⠻⣿⣷⣄⠰⣾⣶⣤⡀⠀⠈⠻⠿⢿⣷⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣮⣽⣿⣿⣶⣄⡀⠈⠻⣿⣷⣌⠙⠿⣿⣶⣄⠀⠀⠀⠈⠁⠃⠀⠀⢠⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣿⣏⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀⣀⠀⣀⡀⣀⣄⣀⡀⠀⢀⣀⡀⡀⢀⣀⣀⢀⠀⠀⢀⣀⠀⡀⡄⢀⠀⢀⢀⣀⡀⣀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣿⡄⡄⠀⠀⠀⠀⣴⡆⠀⠀⠀⠀⡄⣸⣿⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⢀⣠⣶⣿⣿⣦⣄⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁⠊⠂⠘⠛⠘⠚⠙⠘⠂⠀⠀⠃⠐⠂⠈⠐⠑⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⠈⠂⠃⠀⠘⠘⠙⠉⠙⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⠀⠀⣀⠀⠀⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣩⣌⣁⣩⣭⣭⣏⣀⣨⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠿⠏⠏⠉⠈⠃⢉⣹⣋⣈⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⠀⠀⠠⠤⠤⠖⠺⠛⠛⠛⠛⠛⠛⠛⢛⣿⠛⡟⠛⠛⠛⠛⠛⠛⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠠⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⠶⠶⠶⠶⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠁⠀⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠤⠄⠀⠀⢀⣀⣀⣀⣤⣀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠻⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠟⠛⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣻⡏⣘⡿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⢽⣯⣩⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣯⣿⣿⣿⣯⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣯⣿⣿⣝⣟⣿⠁
⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠰⠄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⣿⣿⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⡶⣀⠰⡆⠀⠀⠰⠆⠀⠀⢰⠄⡀⢰⡀⢱⠀⠀⠀⠀⠀⠀⢈⣻⣿⣯⣿⣟⣖⣿⣿⡃⠀⠀⠀⡀⠀⠀⠀⠀⢼⣿⣠⣠⣠⡀⣀⣀⣿⣀⣀⣀⣿⣇⣿⡄
⠸⢷⣿⢾⣿⡩⠋⣠⡀⠙⢿⣿⣾⣷⣾⡟⠘⠇⠀⠀⠀⠀⠘⢻⣿⣿⣽⣿⢿⣿⣿⣿⡫⠀⠀⠀⠀⣀⡀⠸⣗⣻⡽⣿⡷⣿⡵⡏⢹⣧⠀⠿⣾⣿⣿⣿⡅
⢈⡦⣶⡖⡇⠀⣄⣙⣃⣠⠀⣿⢻⣿⢻⣧⣴⣇⣾⣦⣀⢀⣾⣶⣻⣿⣉⣾⣧⣙⣿⣿⣴⣿⡷⢀⣤⡿⢷⣶⣿⣿⣛⣿⣿⣟⣀⣶⣾⣷⣶⣶⣨⣿⣿⣿⡇
⠘⠓⡋⡐⡗⠒⣭⣭⣭⣽⠒⣿⢛⣿⣻⣏⣉⣿⣿⣶⠽⣿⣿⣷⣿⣿⣵⣽⣤⣧⣿⣿⣿⣿⣿⠮⣿⣿⣿⣿⣡⣹⣿⣿⡯⣿⣿⣷⣶⣶⣶⣿⣽⣿⣿⣟⠃
⠀⠀⡆⣥⣿⣇⣾⣿⣿⣿⣷⡏⢹⣿⡇⣹⣯⣿⣿⣿⡇⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡿⠄⣿⡇⣼⣿⣿⣿⣿⣿⣻⣿⡿⠿⢿⡆
⠀⠒⠐⠀⠝⠣⠿⠿⠿⠿⠟⠓⠚⠏⠲⠟⠛⣿⣿⣿⣷⢿⠷⡟⣿⡿⣿⣟⣟⣷⡻⣿⣿⣿⣿⣿⢿⣟⢛⢺⡟⠓⡶⣲⣖⡟⠛⢟⡛⢛⠛⡻⣛⣿⡶⣾⡇
⠀⠸⠂⢹⣶⡶⣴⡒⢲⣴⣖⣧⡔⣛⡷⠷⡗⣿⣿⣿⣾⣾⣿⣶⣿⡟⢿⣿⣿⣿⠛⣿⣿⣿⣿⣯⣿⣿⣿⡯⣿⣾⢿⣿⣿⣿⣿⣾⣿⣿⣿⢳⣿⣿⠋⠻⡇
⣤⠀⠀⢨⣿⣿⣼⣷⣿⡤⣿⣟⡉⣛⣀⠙⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⡿⢾⡿⠿⣿⣿⢿⣿⡗⢿⣿⢿⣿⣿⣿⠋⡿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣽⣾⣧
⣦⡇⡀⢰⣟⣿⣿⣿⣿⣷⡿⠷⠁⠀⠘⢣⣤⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣾⣷⣿⣶⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣷⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣁⣁⣁⡞⣹⣿⣿⣿⣿⣿⣿⢨⡄⠀⣉⣘⣉⢻⠿⣿⣷⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⢟⠛⣻⣿⣿⢿⢾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⣈⠀⠀⠁⠉⠉⠁⣉⡁⢉⢈⣀⠉⠀⠉⠈⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠉⠉⠉⠉⠉⠉⠉⠛
⣠⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢀⠀⠀⣀⣀⣀⡀⠀⣄⣄⣠⣤⣀⣠⣤⣀⣈⣤⣤⣄⣠⣠⣉⣡⣀⣄⣠⣀⣀⣠⣠⣠⣄⣈⣀⣁⣀⣀⣀⣀⣀⣀⣠⣴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣰⣿⣿⣿⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⢘⣦⡀⠀⠀⠀⠀⠀⠀⢸⣻⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣐⣷⣇⠀⠀⠀⣠⠀⠀⠀⠀⣂
⠀⠀⠀⠀⢸⣿⣿⣶⡶⢀⣅⠰⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⠶⣾⡶⢶⣶⣾⣿⡇
⠀⠀⠀⠀⢸⣿⡟⡗⢀⣘⣟⣠⡀⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣜⣿⣃⡌⢹⣿⣟⡇
⠀⠀⠀⠀⠘⡟⡟⣿⡿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠁
⠀⠀⠀⠀⠀⡶⣶⣻⣟⣭⣿⣿⣿⣗⣿⣟⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾
⠀⠀⢠⣶⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤
⠀⠀⢸⣿⣷⠇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⣿
⠀⠀⠈⠉⠉⠀⠈⠁⠈⠈⠉⠉⠉⠁⠙⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠁⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣷⣾⣷⣾⣤
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢽⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠛⠛⠛⠛⠛⠋⠛⠛⠛⠙⠋⠛⠋⠛⠛⠛⠛⠛⠉⠙⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠋⠋⠛⠛⠛⠛⠙⠋

⣿⡟⠁⠐⢿⣿⣿⣿⣿⡇⢾⣥⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠟⣿⠿⢁⣤⣄⣤⣤⣬⣤⣾
⡟⠁⠀⠀⠈⣿⠿⣿⣿⢇⣾⠟⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡏⠁⠋⠀⠀⠘⣿⣿⣿⠀⣾⠀⠋⠉⠻⠟⢛⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠋⣿⣏⠀⡤⠂⠀⠀⠀⠀⠀⣿⠏⠀⠀⠉⠈⠀⠀⠀⠀⠀⡻⠛⠛⠁⠈⠿⠁⠀⠀⠀⠀⠙⠋⠏⢻
⡀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠁⠚⠀⠀⠀⠀⠀⢠⠴⢃⡠⠂⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣀⠀⠀⠠⠄⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣡⣖⣺⣷⣾⣿⣷⡶⢀⣠⠄⠀⠀⡄⣀⣴⠟⠉⠛⠐⠲⢄⢠⡃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⠀⠀⠀⢰⣦⣤⠉⢹⡀⣿⣿⠟⠉⠛⠉⠉⠻⠟⠃⠀⠀⠰⠷⡿⠇⠀⠀⠀⠀⠾⢦⢿⡅⣿
⠀⠀⠀⠀⢀⣠⡐⠛⢿⡤⣾⣦⣠⠇⠀⠘⠙⣟⠞⠁⢿⡅⠀⠀⠀⢲⣶⣂⣰⡄⣀⠀⢴⠀⠃⢀⣀⣦⠀⠀⠀⠈⢺⡷⣏
⠀⣀⣠⡖⣻⣠⣤⣤⣾⡽⣿⣯⣽⠛⠃⣨⣴⠛⠀⣠⠟⡁⠀⠀⠀⠈⠉⠉⠻⣇⠙⠀⠀⠑⠾⠿⠈⢻⡄⠀⠀⠀⠈⠳⣜
⣤⡿⠁⠀⠈⠻⣿⣟⠁⠀⠀⠀⢈⠆⠀⠀⠉⠁⠀⡁⠀⠀⠀⠀⠀⡀⠀⢰⣆⢻⠀⡀⠂⠀⠀⠀⠀⢠⣧⡀⠀⠀⠀⠀⢹
⡻⠇⠀⣰⠶⠴⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠁⠀⠀⠀⠀⣼⢿⡶⣞⣿⣷⣿⣧⡀⠀⠀⠀⠀⣾⣿⣿⠀⠀⠀⠀⠈
⠧⠈⠛⣿⣿⡖⣷⡶⣀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠀⠀⠀⡀⢠⠃⠍⠀⠀⠘⣿⣿⠿⠿⠿⡄⠀⠀⢰⣞⣿⠋⠀⠀⠀⠀⠠
⠀⠀⢀⡀⠹⢗⣬⣧⡿⠀⠠⠀⠀⠢⣀⠀⠻⠇⠀⢀⠐⠁⣴⠋⠀⠀⠐⢿⣿⣷⣶⣶⣄⠀⢳⠖⠙⣻⠃⠀⠀⠀⠀⠀⢐
⠀⠀⠈⣷⣴⣿⡛⣋⣿⢀⣿⣦⠀⠀⢹⡆⠃⠉⠀⠀⢀⢻⣷⢂⣠⣾⡟⠙⢿⣉⠻⣿⣿⠟⠛⢠⣾⣿⠀⠀⠀⠀⠀⠀⢌
⠀⠀⠀⣀⣸⣿⣿⣿⣿⢻⣿⣧⠀⢂⣼⣻⣟⣡⣤⠄⠙⢷⣾⣿⣿⣯⡥⠘⢇⣼⣷⣌⡿⠀⢠⡿⠃⠋⢈⠻⣄⣄⠀⡀⠈
⠀⠀⠀⠈⢻⠟⢻⣿⣿⣿⣿⣿⣿⡞⢡⡟⢹⡀⢳⡶⣆⡙⢷⣿⣿⣷⣦⠀⡤⠛⠀⠈⠀⠀⠉⠀⠀⠐⠿⢆⡈⠿⡆⣠⢤
⠃⠀⢴⡄⠛⠁⢈⡛⣿⡛⣿⣿⣿⣷⣄⠱⢈⣃⠈⠳⢾⡅⢨⡉⠟⢧⢠⡄⢀⠀⢀⣄⣠⣴⢾⣄⠀⣴⢶⣾⡆⠠⣿⢫⣎
⣤⠆⠘⠛⠀⠀⢬⠃⢈⠿⠏⠻⣿⣯⣼⣷⣾⣿⣃⡀⠈⢧⠀⠀⠀⠀⠀⠘⠿⠶⠎⠙⢛⢋⣀⣌⣹⣷⣄⢿⣿⣴⢃⠛⠋
⣧⣯⡄⢀⠀⣦⡀⠀⣾⡿⢶⡶⢿⣿⣿⣿⣿⣷⣿⡿⣦⣼⣷⡵⠀⠀⠀⠀⢰⣷⣏⣠⣿⣿⣿⠿⣿⣿⣝⡿⢿⣿⠃⠀⠠
⣿⡟⢡⣿⡇⢨⣿⣦⡿⠿⣾⣿⡺⢿⣿⣛⣿⣿⣏⣀⣿⠇⢈⣵⣶⣀⡠⣿⡧⠶⣹⣿⣛⠀⠀⢠⠛⠀⠻⣇⠸⣿⣦⠀⡔
⡇⣀⣿⣿⡿⠀⠘⢿⣵⣶⣼⣏⢻⣿⣭⣿⣯⣤⣬⣍⠿⣴⣾⣹⠿⠛⠷⢉⣤⣶⡿⠋⠛⠛⠶⣤⡆⠀⠀⣸⣷⡀⢘⠞⠠
⣿⣿⣿⢁⣴⣿⣿⣄⡛⠀⣿⡿⢛⡿⣿⣿⣿⡟⢉⣿⢃⣿⣿⣷⣶⣶⣾⣿⡷⠋⢀⡔⠀⠀⠀⠈⢷⣾⡿⣿⣿⣷⠊⡲⠃
⣟⣿⣿⣦⣄⠐⠾⣴⡇⠸⡏⠃⣹⣿⠿⠿⣿⣷⣟⡽⣟⢿⡻⢿⣿⡏⠉⢻⣷⢦⡾⠀⡀⡰⠶⢀⡀⢻⡟⢙⡾⠉⡻⠁⠂
⠬⠭⡿⠉⢉⡀⢸⡋⡀⠀⠿⠀⠀⠿⣇⣴⣿⣿⣏⠀⠀⠈⣷⣾⣿⣷⠀⠈⢹⡟⠁⠀⢸⠆⠀⠄⢱⣤⣽⡄⣤⠞⠀⡴⣷
⠛⠋⢁⣾⣿⡥⠐⠈⠉⠛⠆⠴⣶⠆⠈⠁⠉⠁⠁⠀⠀⠀⠉⠡⠿⣶⠂⠀⢸⣶⠀⠠⠟⢘⠓⠒⠺⢿⠛⣷⠸⣄⡔⢰⣿
⠀⠀⡛⣴⣧⣀⠀⠤⠴⡄⠒⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣁⣼⣟⣵⢦⣤⠴⠃⠀⠀⠁⠀⣿⣏⣷⡟⠔⢋⠁
⠈⠉⠁⢀⠔⠀⠀⢀⠜⠁⠰⠀⠀⣤⠤⠐⣧⠠⠄⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟⠙⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⡋⡄⠆⣯⡄
⠀⠀⠀⢲⠖⠒⠒⠛⢀⠘⠀⠐⠐⠿⠆⢶⣾⠿⠓⠶⠀⡄⠀⠀⠀⠈⢿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣁⠀⠂⠀⠈⠀⠇
⣤⣦⡀⠀⡠⠆⠁⠰⠀⣤⠈⡀⠃⣷⠰⠾⠻⢷⠄⣀⣀⠀⠀⠀⠀⠀⣿⣧⣀⡄⠀⠀⠀⠀⣠⣶⣠⠂⠀⢱⠄⠀⠠⠀⠘
⣀⡀⠁⠀⣠⣐⡜⢁⠰⠀⠆⠀⠈⣩⡓⠈⠀⠀⠀⠉⠉⠉⠁⣤⣤⣔⡿⠛⠿⠦⣀⣠⡶⢋⣵⡿⢧⡄⠀⠈⠀⠀⠀⢳⠠
⠈⠃⢀⣼⣿⣯⢷⠈⣄⠃⣀⠄⠠⠏⠀⠠⠄⢰⣦⠀⢀⠀⣴⣿⣿⠟⠀⢀⡀⠀⠀⣍⢀⣮⣷⣁⡀⣣⡤⠶⠗
⣶⢶⡿⠿⣿⣯⣿⣾⣿⣤⣁⠀⠀⠀⠀⠀⠀⣸⣏⡠⣮⣶⡾⠿⠃⠀⠐⢿⣿⣦⣤⣄⣿⣿⠋⠁⠰⢻⣶⣀⠗
⠁⠀⠀⢠⡙⠻⠾⣿⡳⢿⡝⠓⠀⠀⠀⠘⠛⠻⣿⣴⣿⣿⡿⠂⠁⠀⣠⣜⡯⣿⢽⡿⠃⠐⣿⣦⠆⠈⠙⠿⠀⠃⠽⠀⢶
⠀⠀⠀⢸⣇⠀⠀⠊⢙⢲⣤⣄⠀⠀⠀⢠⡶⡆⠉⠙⠀⠂⠀⠀⠀⠀⡈⣛⢣⠘⣏⠛⠱⢾⣿⠁⠄⠀⠀⡐⠁⠀⠠⢤⣾
⠀⠀⠀⢹⠛⡇⠀⢠⣅⠈⣿⣿⠀⠀⠈⠹⠀⠁⠀⠀⠐⣂⠀⡀⠀⠤⠁⠈⠁⡈⠙⠀⣇⡀⠃⠀⠀⠀⠀⠀⠄⠶⣶⡾⠿
⡄⡵⠀⣼⠀⠁⠀⠀⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠂⠴⢾⡧⠐⢲⡄⠀⣠⣇⠒⠺⣈⡵⠄⠀⠀⠀⡆⠀⢀⡀⠲

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⠶⠶⠶⠶⠶⠶⠶⠿⠿⠛⠛⠛⠛⠛⠛⢛⣋⣉⣉⣉⣉⣉⣉⣉⣥⡌
⠀⠀⠀⠀⠀⢸⣦⣀⡀⡈⠙⢿⣿⣯⣭⣤⣤⣤⣤⣤⣤⡴⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠛⠛⠛⢛⣛⣛⣛⣛⣉⣉⣉⣉⣉⣭⣭⣭⣭⠁
⠀⠀⠀⠀⠀⠰⠆⠀⠀⠇⣤⠀⠙⢿⣿⣷⡶⠶⠶⠶⠶⠶⠶⠖⠒⢛⣛⣛⣛⣛⣛⣛⣋⣉⣉⣩⣭⣭⣭⣭⣭⣭⡤⠤⠤⠤⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣛⣛⣛⣛⣩⣭⣭⣭⣭⣭⣭⡭⡭⠭⠤⠤⠶⠶⠶⣶⣶⣶⣶⣒⣒⣒⣒⣛⣛⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⠄⢉⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣭⣭⣭⣭⣭⣽⣿⣿⣷⣾⣶⣶⣶⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠭⠭⠿⠭⠭⠭⠭⠭⠭⠭⠽⠿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠘⠽⢿⣿⣿⣿⣿⣿⣿⣭⣭⣿⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣭⢭⣭⠭⣭⣭⣭⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠿⣿⣿⠿⡿⠭⠭⣭⠍⠉⣉⣉⣉⣉⣙⣛⣛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⢿⣿⣿⣿⣿⣿⣿⣭⣽⣿⣿⣷⣿⣿⡟⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⢠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠸⠿⠭⠭⣩⠿⠉⠉⠿⣿⡅⠀⡀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡤⢢⠜⠁⢠⠄⣤⠈⣙⡆⠀⡇⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣸⠸⡧⠀⢀⣀⠀⣹⡇⠀⠄⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠠⠎⠉⢀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⢶⣿⣭⡥⢭⡭⠉⠉⠉⣿⣛⣛⣶⠀⡀⠀⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣠⣀⣀⣤⠰⠶⠾⠛⠛⠛⠓⠲⠶⠶⠶⠶⠂⠀⠀⠀⠀⠸⣯⣠⠉⢈⢹⡇⢀⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣓⣒⣛⣧⡁⠘⠈⠙⠉⠉⠉⠙⠛⠋⠉⠋⠉⡋⠋⠀⠀⠀⠀⠀⠀⠀⡏⢁⡜⢁⣸⠁⠀⢙⠀⢀
⠀⠀⠀⠀⠀⠠⠓⠂⠀⠀⠐⡤⠤⠭⠭⠭⠭⠭⠭⠭⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⢄⠀⠂
⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠑⠛⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠄

⠀⠀⠀⠀⢀⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠒⠒⠒⢤⡀
⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠈⠆
⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠠⣇⡀⠀⠀⠀⠀⠀⠀⢀⣰
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢀⠉⠉⠉⠉⠓⡀⠘⠉⠹⣧⡀
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⣀⣴⢇⠀⠀⠀⠀⢀⡁⠀⠀⠀⡌⠻⣆⣀
⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⢸⡔⠁⠀⠑⠤⢤⡴⠂⠈⡒⠒⠊⠀⠀⠈⢦⡇
⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡠⠊⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⡜⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠔⠲⠖⠒⠒⠒⠚⠀⠀⠀⠘⠀⠀⠀⠡⢀⠀⠀⠈⠒⠤⠤⠤⠄⠚⠁⠀⠀⡠⠘⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡰⠢⡄⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠤⠄⠂⠁⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠴⢤⠀⢸⠄⠠⠤⠤⠤⠴⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠦⠔⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢺⡀⠞⠀⠠⠤⢄⡀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠁⢀⠀⢀⠠⢄⡀⡱⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠀⠀⠀⠀⠀⢀⢃⠔⠁⠀⠀⠀⠉⠒⠲⣆⠠⠤⠤⠤⠤⠤⠤⠄⣰⠖⠒⠉⠀⠀⠀⠀⠢⡘⡄⠀⢏⠀⢠⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠰⣀⠌⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡉⠑⠒⠒⠠⠤⠤⠄⣀⣀⣀⣀⣀⡰⢆⣀⣀⣀⣀⡀⠠⠠⠤⠄⠒⠒⠈⢉⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠉⠛⡇⠈⠀⠀⠁⠀⠁⠈⠁⢸⠋⠉⢈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠈⠁⡃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠑⠢⢄⡀
⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣯⣀⣄⣠⢦⣀⣠⣃⣀⣀⣀⣀⣀⣀⣀⡀⣘⣄⣀⡴⣄⣀⣀⣽⡀⣀⣀⡀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡶⠶⠀⢤⠀⡤⠀⣶⠀⢦⡶⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡧⠶⠀⢈⣿⠁⠀⣿⠀⢸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠧⠶⠀⠞⠀⠳⠀⠿⠀⠘⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡖⠋⠓⢦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⢀⡼⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠉⠙⠲⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⡴⢒⠆⠀⠀⠈⢳⣀⡤⢦⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⢠⠏⠀⠀⢠⢦⡀⠀⣀⡤⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢠⠏⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢀⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠚⠀⢠⠎⠙⢦⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠶⠏⠀⠀⠀⢷⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠼⠇⠀⣤⣤⣤⣤⣠⣤⣼




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠚⠛⠋⠛⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣰⡆⠀⢀⠀⠀⠀⢀⡇⢻⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⣤⣀⡀
⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⣠⠞⠓⢦⣰⠃⠀⠀⠀⠀⠀⢀⣴⣿⢁⣠⣯⣤⣤⠴⠚⣁⡀⠀⠀⠀⠀⠀⠀⠀⡠⠖⣉⠤⠒⠒⠒⠒⠒⠬⣉⠲⣄
⢠⠎⠁⠀⠀⠀⠉⠛⠲⢤⣠⢤⡿⠀⠀⠀⠃⠀⠀⠀⠀⣀⡴⠞⠁⢉⣉⠀⠀⠀⢿⣠⡴⠶⢬⣄⠀⠀⠀⠀⢠⠎⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢎⠳⡄
⢸⡀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⡇⠀⠀⠀⠀⠙⢻⠏⠉⠁⠀⠀⣈⣤⢤⣄⠀⢠⠟⠁⠀⠀⠀⠙⣧⠀⠀⢰⠃⡜⠀⢠⣴⡦⣶⣶⢦⣶⣿⣦⡶⣶⠆⢣⠹
⠈⢷⡀⠀⠀⢠⡞⠉⠉⠉⠻⢤⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⡼⠁⠀⠀⠈⠻⡏⣤⠀⠀⠀⠀⠀⢻⠀⠀⡎⢰⠁⠰⠿⢶⡆⢸⡇⣿⠁⣸⣿⠟⠁⠀⠀⡇
⠀⠈⢷⡀⠀⠸⠻⣄⠀⠀⠀⠀⢻⠀⠀⠀⠉⢳⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⠀⡇⠈⠀⠀⠀⠀⠀⣾⠀⠀⡇⢸⠀⠀⠟⠋⠀⠛⠁⠻⠟⠁⠛⠀⠀⠀⠀⡇
⠀⠀⠀⠹⡄⠀⠀⠈⢷⡄⠀⠀⠘⡆⠀⠀⠀⢸⠀⠀⠀⠀⢈⣧⣉⠁⠀⠀⢸⠷⣄⡀⠀⢀⣠⠞⢧⡀⠀⢱⡈⢆⠀⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡇⡰⢠
⠀⣠⣤⡤⠗⠀⠀⠀⠀⢹⡄⠀⠀⢹⣄⡀⠀⣼⠀⠀⠀⠀⡀⠀⠉⠉⠓⢶⡋⠀⠈⠉⠉⠉⣁⠤⣄⢳⡄⠀⠱⡈⢦⡙⠛⠉⠉⠉⠉⠉⠉⠉⠉⢉⠔⣡⠏
⢸⡇⣠⠶⠶⠶⠶⠶⠖⠋⠀⣠⠞⠋⣉⠙⢷⡏⠀⠀⢠⠿⡁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠁⠀⣸⠂⣷⠀⠀⠘⠢⡉⠲⠤⣀⣀⣀⣀⣀⠤⢒⣩⠞⠁
⠈⢷⡏⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠛⠙⠳⠀⠁⠀⠀⠀⠀⠈⠓⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⡿⠀⠀⠀⠀⠈⠙⠒⠦⢤⣤⡤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⢖⣊⣙⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⢀⡤⣼⡷⠓⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣍⠙⢦⠉⠹⣷⣶⠶⠶⠤⠤⣤⠶⣶⣖⣚⣛⡭⠵⣶⡆⠀⠀⠀⠀⠀⣀⣀⣸⠀⡀⡷⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠈⢷⡈⢧⠀⠹⡍⠓⢦⣀⠀⡿⢆⢹⡍⠁⠀⠀⠀⡟⠛⠛⠛⣉⣉⣉⣉⣉⣉⠀⢿⡵⣾⣃⡈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢻⡈⡇⠀⢹⡄⠀⠈⠻⡇⠸⡄⢣⠀⠀⠀⠀⣟⢹⠉⠉⠁⠀⠀⠀⠀⢹⣤⠼⡇⡇⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢸⡇⢸⠀⠀⢷⠀⠀⠀⢷⠀⢷⠀⢿⠒⠲⠤⣿⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⣇⠈⡂⠀⠸⡆⠀⠀⢸⡄⠸⡆⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⢶⣶⣶⠚⠋⢀⣇⣀⣠⣧⠤⠤⠴⢷⠶⠓⠒⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⢻⡆⣿⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣸⣀⣀⣀⣠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣷⣌⠙⠁⢸⣶⡶⢶⡖⠛⠛⠉⠉⠉⠉⠙⣧⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡶⠷⠀⣀⣤⣶⣲⡞⠃⠀⢀⣀⣀⣀⣠⡼⠁⠳⠶⠚⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠚⠋⢻⡀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⡄⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣁⣤⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⠀⠀⠀⠀⠀⠨⠭⠟⣻⠋⢓⣂⣠⡼⠦⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡁⠀⠉⢙⣛⠷⢒⡶⠒⠒⠚⠋⠉⠛⠯⢍⣃⣠⠤⠤⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠭⠭⠭⠭⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⣰⠿⢳⣷⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⣿⢲⣬⠧⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡰⢰⠆⠀⠠⣌⡈⠙⠋⠀⣹⣾⣦⣶⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣿⣷⠛⠀⠀⠀⠈⠛⠷⠂⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢟⠟⠀⠀⠀⠀⠀⠀⢠⣿⡌⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⣍⡀⠊⠀⠀⠀⠀⠀⢸⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢨⣿⣿⠉⠀⠀⠀⠀⣶⣾⣶⣾⣿⠿
⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⠉⠀⠀⠀⠀⣴⢿⠟⢛⣿⡷⢸⡋⠾⠆
⠀⠀⠀⠀⠀⠀⠀⣸⢀⣿⠏⡄⠀⠀⠀⣰⣟⣈⣠⠜⢋⣴⣾⣄
⠀⠀⠀⠀⠀⠀⢀⡏⣸⠿⢺⡇⢀⣴⠾⣿⡇⠀⣀⣀⠈⢤⠾⢿⣦
⠀⠀⠀⠀⠀⠀⢸⣷⡇⢰⣄⢿⠟⠁⢰⠟⣀⣠⣤⡠⠤⠒⠶⣛⣟⠳⣤
⠀⠀⠀⠀⠀⠀⢾⣿⣷⣿⣿⠀⣀⡴⢷⣾⠀⢉⣠⠀⢀⡤⠞⢋⣥⣁⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⡝⠉⠀⠘⣿⣦⠉⢠⡾⢭⡴⠞⢋⣵⡌⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢢⡜⢦⣀⠚⢥⡴⣾⠟⣥⠤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠁⠖⢛⣵⣿⣿⣧⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡯⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠠⠀⠀⠘⢧⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣶⡦⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⡿⢙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⠓⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠟⡩⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠡⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⣴⣾⣿⣌⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣸⣿⡇⠘⠁⢘⣶⣾⣳⣾⣷⣾⣿⣭⣭⣭⣭⣭⣥⣶⣿⣿⣿⣿⣯⣿⣷⡆
⠀⠀⠀⠀⠀⠀⣶⣾⣷⣎⠉⠛⠻⠿⠿⢷⣶⣿⣤⠭⠏⠚⠛⠛⠛⠉⠉⠉⠉⠉⠐⠶⠤⠀⠙⢿⣿⢿⣿⣦⣀⠠⠄
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⠐⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣒⣀⣀⣀⣀⣃⣀⣉⣛⣃⡀⠀⠀⠀⠈⠓⣹⣿⣿⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⣤⣤⣴⣤⣦⣄⣰⣶⣶⣶⣛⣦⣤⣤⣤⣦⣤⣤⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣒⣒⣒⣻⣻⣦⣀⣀⣰⣦⣤⣶⣀⡀
⠀⠀⠀⢠⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⣿⣿⡷
⠀⠀⠀⠸⣿⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡄⠐⣶⣶⣦⠀⠀⣴⣶⣶⠀⢰⣶⣶⣶⣶⡦⢠⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⣿⣿
⠀⠀⠀⢠⣿⠀⠀⠀⢸⣿⣿⡏⠉⠉⠉⠁⠀⠘⣿⣿⣧⣸⣿⣿⠃⠀⠀⢸⣿⣿⡇⠀⠈⠉⠉⢻⣿⣿⠋⠉⠉⠀⠀⣿⡏
⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣧⣤⣤⡀⠀⠀⠀⠘⣿⣿⣿⡿⠃⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⣸⣿⠀⠀⠀⢸⣿⣿⡿⠿⠿⠃⠀⠀⠀⢀⣾⣿⣿⣷⡀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⡟⢿⣿⣷⡀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⣻⣿⠀⠀⠀⢸⣿⢿⣷⣶⣶⣶⡆⢠⣿⣿⡏⠀⠈⢿⣿⣿⡄⢠⣼⣿⣿⣧⡄⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠁⠈⠉⠉⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⡶⢶⣶⣶⡶⣶⡶⠶⣶⣶⠿⠋
⠀⠀⠀⠀⠁⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠃⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠶⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠴⠶⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠴⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠴⠖⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡶⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠶⠶⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠟⢿⡋⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠴⠶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠶⠶⠛⠋⠉⠁⠀⠀⠀⠀⢻⡇⢸⡇
⠀⠀⣀⣠⣤⣴⠶⠶⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢴⡶⠞⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠶⠶⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠶⠛⠛⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠞⠛⠋⠉⠁⠀⢀⣠⣤⣤⠤⣿⣤⣤⡤⣤⠤⠤⠤⢤⠴⠿⠦⠤⠤⠤⠴⠶⠶⠶⠶⠖⠒⠒⠒⠒⠛⢒⣻⢿⡆⣿
⠀⢤⣤⠶⠟⠛⠉⠉⠁⠀⠀⠀⠀⢀⣠⣴⠞⠋⠉⠀⠀⠀⠀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣀⣀⠀⠀⠀⢀⣀⡀⠀⠀⠀⢀⣠⠔⠋⠁⠰⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⠉⢉⣉⣉⣉⣉⣉⣉⣉⠈⡇⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣶⠃⠉⠉⠉⣿⠀⢺⡋⠙⠷⢀⠾⠛⢙⡟⠀⣟⠉⠙⡇⢸⡅⣀⠀⠀⢀⣀⣽⠀⠁⠀⠀⠀⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠀⢰⣛⣛⠋⠀⠀⣿⠀⠰⡟⠀⢠⡞⠀⠀⣿⠀⠀⣇⠈⠁⢹⡇⠀⣹⠁⠀⠀⢸⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⢉⣉⡇⠀⠀⢸⡇⠀⠀⠀⡞⠁⠀⠀⣿⠀⠀⣿⠀⠀⢸⡇⠀⢼⠀⠀⠀⢸⡇⠀⠀⠀⢸⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⠀⠀⣏⣉⣀⠀⠀⣾⠁⢀⡀⠀⢻⡄⠀⠀⡿⠀⠀⣿⠀⠀⢸⡇⠀⢸⠀⠀⠀⠸⠇⠀⠀⣠⡟⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣀⣀⣉⣉⣹⠀⣼⣁⣀⡞⢣⣀⡀⣷⠀⠀⣷⣤⣤⡿⠀⠀⠘⠷⠶⠿⠀⠀⠀⠀⠃⣠⡾⢙⣇⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣈⣉⣉⣉⣉⣩⣤⣍⣭⣭⣤⣤⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⠶⠶⠶⠶⠶⠾⠋⠀⠀⣏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⢿⡏⠁⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⣿⠀⢶⡴⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣄⣨⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢀⣤⡴⠾⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠶⠟⠋⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⢦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠶⠶⣦⣤⣀⡀⠀⢿⠀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠿⠷⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠰⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⢁⣠⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡻⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⣤⡀⠀⠀⠀⠀⢀⣤⡶⠟⠋⣀⣴⠞⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⢀⣠⣤⣀⠀⠀⠀⣀⣤⠞⠋⠁⢀⣠⣴⠾⠋⢁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⠟⠉⠁⠀⠙⠳⡶⠛⠉⠀⣀⣤⡶⠛⠉⣀⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⢁⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣠⡴⠞⠋⢁⣠⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠾⠛⠉⣀⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣠⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠁
⠀⠀⠀⣀⣴⠾⠛⠉
⠀⠴⠟⠋⠁

⡃⣀⢀⣀⣀⣈⣳⣌⣉⣴⣦⣤⣤⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⢿⠺⠟⠛⠻⠛⠺⠛⠻⠿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⢉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠙
⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠟⠋⠉⠉⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠄
⡶⣦⣅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠃⠀⠈⠙⢷⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡤⠤⠔⠒⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⢀
⠤⠒⠊⠉⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢓⠁⠀⠐⠀⢰⣿⡇⠀⠀⠀⠀⠰⠀⠂
⠁⠀⠀⢀⠀⢸⣿⡇⠀⠀⠀⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣄⠀⠈⠀⠀⠈⣿⣇⡨⠀⠁⠀⠀⠀⠠
⡓⢤⡆⠀⠒⠀⢛⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣀
⠀⠀⠁⢘⠃⠀⠘⠀⠈⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠶⠂⠀⠀⠀⠒⠀⠆
⠁⡆⢀⡶⠀⠑⠂⠀⠀⠀⠀⠀⠀⠉⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣨⡁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣈⣃⣀⣀⣐⣀⣀⣀⣀⣀⡀⠀⡀⡀⢀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⣀⠤⠶⠒⠉⠉⠉⠉⠉⠓⠶⢤⡀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠁⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠉⠈⢇⣀⣤⣞⣡⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⣠⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⣸⡇⠀⢠⠀⢸⣧⣠⣿⡀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⢠⡟⣷⢀⡿⣇⣸⡏⠉⠘⢿⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠾⢿⣤⣴⠶⠋⣠⣿⠟⠁⠹⠟⠀⠀⠀⠈⣷⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣠⣶⡿⠿⠟⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠈⣷⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣿⠀⠀⠀⠀⠉⠀⢀⣠⡤⢤⣤⣉⡻⣦⡀⠀⠀⠀⠀⢴⣤⣴⣾⢿⣋⣁⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣆⡴⠋⠀⠀⠀⠀⠀⢠⡞⠋⠀⠀⠀⠀⠈⠙⢮⡻⠆⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠉⠛⢦⡀⠀⠀⠀⠀⠀⠻⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢹⡏⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⣠⣶⣿⡆⠀⠈⢷⠀⠀⠀⠀⠀⣰⠁⠀⣰⣶⣿⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⣿⠻⡆
⠀⠀⠀⠀⠀⠀⢀⣀⣿⣀⠸⡇⠀⢀⡀⠀⠀⠀⢸⡀⠀⠀⠀⢿⣿⣿⣿⡿⠀⢸⠇⠀⠀⠀⠀⣿⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⡿
⠀⠀⠀⣀⡀⣰⠏⠉⠉⠙⢾⣿⠟⠋⠙⢳⡀⠀⠈⠃⠀⠀⠀⠈⠛⠛⠛⠁⠀⡌⠀⠀⠀⠀⠀⠸⣄⠀⠈⠛⠛⠋⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢈⣃⡼⠃
⣴⠟⠉⠉⠘⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣧⠀⠀⠀⠈⠉⠛⠒⠶⠶⠶⠶⠊⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠒⠒⠊⠙⠉⠀⠀⠀⠀⠀⠀⠀⣾⠋
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⡟⠋⠛⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢀⣼⠋
⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⠇⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠛⠻⣯⠀⠀⠀⠀⣠⡾⠁
⣿⠀⠀⠀⠀⢹⡇⠀⠀⠀⠘⠗⠀⠀⠀⠀⢸⠂⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⣿⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟
⢻⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠘⠷⠀⠀⢻⣤⡽⠿⠶⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⡞⠛⠛⠶⣤⡀
⢸⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣾⠁⠉⠉⠉⠻⣿⣟⣛⣉⣽⠟⠁⠀⢸⡇⠀⠀⠀⠈⠙⢧⡀
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣉⣉⡉⠀⠀⠀⠀⠘⣿⢶⣄⠀⠀⠀⠀⢻⣆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⠾⢻⡟⠀⠀⠀⠀⣰⣿⣿⣿⣽⣷⡄⠀⠀⠀⣿⠀⠙⣧⡀⠀⠀⠀⢿⡀
⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⢸⡇⠀⠀⠀⠀⢻⡟⣿⠿⣿⣿⡇⠀⠀⠀⢿⠀⠀⢸⡇⠀⠀⠀⣸⠇
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⢠⣤⣤⣤⠌⠁⠀⠀⠀⢸⣄⡴⠟⠀⠀⠀⢠⡟
⠀⠙⠷⢦⣤⣤⣤⣤⣴⠴⠶⠿⠿⠟⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡿⠶⠶⠶⠶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⢘⣧⠀⣴⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⢿⣄⣨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⠀⠀⠀⠀⠀⣿⠛⠛⠛⠶⢶⡀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠛⠛⠒⠂⠴⠿⣤⡀⠀⠀⣨⣷⣤⡴⠶⠒⠒⠛⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⠖⠲⣶⠶⠾⠏⠀⠀⠀⠘⠷⠶⢶⡶⠶⠲⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣤⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠋⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⣀⣤⣶⣶⣤⣀⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⣿⣯⡀⠀⠀⠀⠀⠀⢹⣿⠀⠀⢀⣾⡟⠋⠁⠈⠙⢻⣷⡀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣽⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣀⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣀⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣦⣼⣿⠀⠀⠈⢿⣦⡀⠀⠀⢀⣴⡿⠁⠀⠀⣿⣧⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣉⣻⣿⠀⠀⠀⠀⠙⠿⠿⠿⠿⠋⠀⠀⠀⠀⣿⣟⣉⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠛⠛⠛⠛⢿⣿⠀⠀⠀⢀⣴⣾⠿⠿⣷⣦⡀⠀⠀⠀⣿⡿⠛⠛⠛⠛⠛⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⡀⠀⠀⠀⢸⣿⠀⠀⢠⣿⠏⠀⠀⠀⠀⠹⣿⡄⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⡀⢸⣿⠀⠀⢸⣿⡀⠀⠀⠀⠀⢀⣿⡇⠀⠀⣿⡇⢀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⠀⠀⠀⠻⣷⣤⣀⣀⣤⣾⠟⠀⠀⠀⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠈⣙⣛⣛⣋⠁⠀⠀⠀⠀⣿⣷⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⣿⣧⡀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣰⣿⠟⠛⠛⠻⣿⣆⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⣀⠀⠀⢸⣿⠀⠀⢰⣿⠁⠀⠀⠀⠀⠈⣿⡆⠀⠀⣿⡇⠀⠀⣀⣴⣾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣾⣿⠀⠀⠸⣿⣄⠀⠀⠀⠀⣠⣿⠇⠀⠀⣿⣷⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠀⠀⠙⠿⣷⣦⣴⣾⠿⠃⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⠉⣿⡏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠀⠀⠈⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⣿⣷⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣿⡀⠀⠀⠀⣰⢿⠉⢳⡄⠀⠉⠉⠉⢻⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠟⢻⡏⠀⢿⣷⡀⢀⡾⢷⡞⠀⢿⡁⠀⠀⠀⢠⠏⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠁⢀⣀⣼⣦⡀⠸⡟⢿⡾⠃⣿⠃⠀⣠⠃⠀⠀⠀⡟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡽⠶⠛⣏⣧⣬⠇⠀⣧⢸⠃⢸⣿⠀⢠⠏⠀⠀⠀⢸⡇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣁⣤⣶⡞⠛⠋⠹⣆⠀⢸⣼⠀⣸⡇⠀⡞⠀⠀⠀⠀⢸⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢿⣿⣿⣿⣿⠏⠀⠸⣧⠀⠀⠀⠹⣆⠀⣿⠀⣿⡇⡸⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣏⣿⠀⠀⠀⢿⡀⠀⠀⠀⠹⣆⢹⡆⣿⣷⠇⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁⠈⢹⠃⠀⠀⠀⢸⣇⠀⠀⠀⠀⠹⣶⣇⣿⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠉⠀⠀⠀⡾⠀⠀⠀⠀⢸⠻⡄⠀⠀⠀⠀⠙⣿⣿⠇⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣟⠁⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⣸⠀⢻⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⢀⡞
⠀⠀⠀⢀⣤⣾⣿⠿⢿⣿⠿⠂⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⢸⠁⠀⢺⡄⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⡾
⠀⠀⣶⣻⠿⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣳⣦⣽⡆⠀⠀⠀⢠⣏⣀⡀⠀⠀⢀⣾⣷
⠀⠀⠈⠀⠀⠀⠈⠉⠙⠛⠷⢤⣄⣀⠀⢸⣿⡿⠛⠉⠉⠉⠛⠻⣶⡶⢦⣄⣀⣬⣭⣭⣍⣿⣿⣿⣷⣤⣤⠾⠛⠻⢿⣦⣀⣾⠋⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢿⡅⢀⣀⡀⣀⣠⣉⣙⣿⣿⣦⣀⣀⣼⣿⣿⠃⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⢿⣿⣯⣉⣁⢿⣤⢧⣦⣡⣽⣿⠟⠛⠉⠉⠁⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠁⠉⠁⠀⢸⠏⠈⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣤⣶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⢰⡷⣶⡶⠾⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⢻⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⢹⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠿⠂⠀⠀⠀⠀⠀⠸⠇⠀⠻⠀⠀⠸⠇⠀⠀⠀⠀⠀⠛⠀⢿⡄⠀⠀⠀⠀⠀⣿

⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⠒⣚⣋⣉⣉⣉⣉⡙⠓⠒⠶⢤⣄⡀
⠀⣀⣤⡴⠶⡏⠁⢹⠶⢦⡀⠀⠀⠀⠀⣠⠔⠚⠉⠁⠀⠀⠀⠻⣭⠄⠐⠒⠒⠦⣽⠟⠀⠀⠀⠈⠙⠳⢦⣄
⢰⠏⢹⠀⠀⡇⠀⢸⠀⠘⡇⠀⠀⠀⠘⠷⢦⣄⠀⠀⠀⣀⡤⠼⣿⡀⠀⠀⠀⠀⣿⠤⠄⣀⠀⠀⠀⠀⠀⣈⣻⡆
⢸⠀⠘⠀⠀⠃⠀⠈⠀⠀⣇⡴⠛⣩⡷⠀⠀⢹⣦⢖⣉⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣉⡳⢦⣠⡾⠋⠁
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⣏⠀⠀⢰⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠿⣧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢰⠋⠀⠀⣿⢠⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣭⣤⠾⠛⢡⡄⣿
⣿⠀⠀⠀⠀⢠⡾⠀⠀⠀⠀⠀⢸⡆⠀⠀⠙⢿⠀⠀⠛⢂⠈⠉⢩⡉⠉⠉⠉⢉⡍⠉⠀⠲⠏⠀⠈⣷⠟
⢿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⠀⢀⣶⠵⠚⠛⠛⠻⣦⣤⡴⠟⠛⠓⠶⣵⣤⡀⠀⢻
⠈⠻⣦⣀⠀⠘⣷⣄⣀⣠⡤⠖⠋⠙⠓⠒⢶⡏⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⢸⠋⠉⠉⠓⠒⠶⠦⢤⣤⣀⡀
⠀⠀⠈⢿⡉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣀
⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣷⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣸⣦⡶⠃⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠶⠶⠿⠿⠿⠶⠶⠒⠚⠉⠁⠀⢠⣿⣿⣿⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣦⠀⠀⠀⠀⢠⣴⣶⣦⡀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠀⠀⠀⠀⠘⢿⣿⠿⠁⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠿⠋⠉⠛⠿⣿⠆⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡆⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠉
⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⠀⠀⢿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⠿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠓⢶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀

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




















































⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣶⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⡿⣿⠿⢿⣿⡿⢿⣿⣿⣿⠿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣦
⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠈⠛⠛⠉⠉⠉⠉⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣧⣤⣴⣶⣦⣤⣴⣤⣤⣤⣶⣶⣄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⢠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡀⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠧⠤⢤
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡆⠀⠂⢀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠁⠈⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⡞⢤⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⡀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡁⠀⢀⡁⠈⠀⠀⠠⠂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠰⠀⠀⠀⠀⠠⠂⠉⠠
⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠑⠀⠈⡀⠠⢀⡀⢠⣄⠄⠐⢁⡀⢈⠇⠀⠠⣃⠀⡀⠀⠐⠀⠀⠀⠰⠄⠀⠀⠀⢶⢂
⠀⠠⠀⠁⠙⠀⡀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠓⠈⠆⠀⠂⠀⠄⠀⠈⠉⠉⢻⠅⠀⠀⠀⠈⠀⠑⠀⠀⠀⠂⢠⠁⠀⠂
⠀⠀⢄⠀⠀⠀⠀⠂⠀⠀⡈⠀⠀⢀⠄⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⠀⠐
⠀⠀⡀⢀⡀⠘⡤⠄⠠⠄⠀⡀⠤⠂⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠖⠔⠐⠀⠀⡡⠈⠀⠀⠀⠦⠀⠀⠒⠀⠀⣀⠏⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠠⠄⡄⠁⠀⠏⠀⠀⠀⠀⠀⠀⠒⠪⠂⠀⠀⠀⠀⢀⠄⢠⡀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⡆⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠨⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢿⠄⠄
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⢠⣶⣤⣿⡀
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢤⠈⠹⠿⣧⣀
⠀⠀⢘⣻⣇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⢰⣬⣾⣦⣄⣨⣿⣟
⠀⠀⠨⣿⣿⡇⢀⣾⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡗⠋⠙⣿⡈⠁⠚⠻
⠀⠀⢸⡇⣿⣿⢴⣿⣍⡇⠀⠀⠀⠀⠀⠀⣀⣤⡤⠐⠀⠐⠒⠒⠀⠀⢠⠤⠌⠐⢀⢐⣿⡆⣐⣿⣶⣷⠎⣳⣴
⠀⢸⣿⣯⣿⣿⣿⣿⢻⡇⠀⠀⣠⣴⡶⢛⡉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⡽⣾⣿⣿⣿⣅⣝⣫⡤⠿⡟
⠐⠛⣿⡗⢿⣿⡿⢿⣿⣧⣶⡿⠟⠋⢉⡉⢺⣷⠤⢄⣠⣤⣄⣀⣀⣀⣀⡀⠀⣾⠱⣿⣿⡛⢿⢿⣭⡼⢿⣾⣷
⢰⢏⣽⣷⣼⣿⣯⣿⣾⡏⣃⣀⣠⠤⢺⣇⣀⣯⣤⣤⣤⣤⣤⣤⣤⣬⣸⣉⡸⢿⣯⣉⠹⣷⣾⣿⣿⠛⢻⣏⢳
⢠⣾⣿⣦⣿⣿⡡⣽⣿⡷⠛⣉⣤⡶⠿⣛⣩⣭⣤⣤⡶⠿⣿⣯⣽⣟⡛⠿⢿⣿⣧⡙⠶⣼⡟⠻⣿⣯⡉⢹⡄
⠸⢿⣽⣥⣽⣿⢓⢉⣠⣿⠾⠋⣡⣴⣿⣿⣿⠏⠒⠀⠀⠀⠀⠀⠈⠉⠛⠿⣦⣟⣿⡛⢷⣿⡙⠲⠜⣿⣿⠟⢀
⠀⠛⢹⠛⢿⣿⢹⣿⠋⠀⢀⣴⣿⡽⣿⣷⣿⣶⣤⣀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⣰⡏⠛⣷⣯⣽⣷⣶⣿
⣿⣶⣼⣄⣸⡏⢹⣧⣆⣴⣿⣿⡿⠆⠠⠀⠀⠀⠄⠉⢛⣿⣶⣦⣤⣤⣄⣀⡀⢈⣿⣿⣇⠀⠉⢿⣯⣭⣏⡬⢿
⢬⣽⣽⣿⣿⣿⣾⣿⢿⣿⣿⣿⣤⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⠿⣿⣿⣿⣦⣀⣿⣿⣙⣃⣀⣀
⣀⣠⣽⣹⣿⠿⠛⣿⣀⢀⡘⣿⡏⠿⠿⠛⢻⣿⣷⠿⣿⠿⢟⣻⣿⣿⣷⣾⠾⣿⣿⣿⣻⣍⢉⣿⡿⡛⠿⠿⡿
⢹⣿⣿⠛⣾⣤⣤⢻⣏⡻⢿⣝⣿⣾⣟⡛⣿⣿⣛⣛⣿⣿⡾⠿⠛⠋⠁⠀⢠⣿⣿⣿⣩⣿⣿⡟⢈⣿⢠⣴⡦
⠰⢿⢻⡐⣽⣯⣿⠀⣯⡟⢿⣿⣿⣿⣿⣽⣿⡿⠷⢚⠩⠀⠀⠀⡀⣀⣤⠶⢛⣡⣿⣿⣿⣿⣿⣄⠀⠀⠀⠙
⢰⣿⢸⣷⣿⣿⢹⡆⣿⠀⠲⠀⢉⣻⢿⣿⣿⣿⣿⡷⠶⠿⠿⠿⣻⣿⣱⣾⣿⣿⣿⡏⠟⢿⣻⣿⠷⣤⣆
⢠⡷⢺⣯⣿⣿⣹⠁⣿⢖⠀⠀⠀⣩⠴⠇⠉⠉⠉⠛⠛⠛⢛⡛⠋⠉⠉⠍⠃⠸⣿⠃⠀⠘⣿⣿⣆⣸⣿⣇
⠘⣗⣻⣧⡟⣿⡟⠀⣿⠈⠃⢒⣋⣠⠐⠁⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄⣠⠸⢻⡏⡿⣿⠿⣶
⠀⣹⣮⣿⡀⢿⣿⠀⢹⣃⠈⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣇⡸⢆⢈⡁⠳⠉⡆
⠀⠀⣾⣿⠀⠘⡏⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣟⡇⠀⠺⣷⡄⢀⠉
⠀⠈⢯⣿⢀⡀⠹⠄⠀⠀⠀⠐⠂⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣤⣄⠿⡙⠛⠁
⠀⠀⢸⡏⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣧⠉⠻⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠙⢻⣛⢿⡀⠘⣿⣾
⠀⢀⣤⡧⠀⠀⠀⠀⠀⣸⣿⣤⣸⣿⣯⡀⠀⠙⢷⡀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣽⣾⣟⣿⣿⡿
⣤⣤⣬⣤⣶⣤⣀⢀⣾⣿⣧⣁⣈⣿⣿⣷⣀⣀⣈⣻⣧⣀⣠⣤⣤⠖⢲⣾⡿⢿⣯⣩⣿⣿⢻⣿⣿⣏⣿⣿⣿
⠈⠈⢻⡿⠘⠻⠿⠿⠿⢿⣿⣟⣛⣛⠻⠿⠿⠛⢛⣻⣯⣭⣥⣤⣽⣿⣽⠿⠷⢼⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣛

⠀⠀⠀⣠⢞⣿⣿⣿⣭⣭⣭⣝⢦⡀
⠀⣠⢞⣵⣿⣿⣿⣿⣿⣿⣿⣿⣷⣝⢦⡀
⡞⣵⡿⠿⣿⠿⠿⢿⣿⠿⢿⣿⠿⠿⣷⢹⡆
⡇⣿⠠⢶⣿⣶⢰⣾⠃⣶⡄⣿⠰⡆⢸⣾⡇
⡇⣿⣷⡄⢹⣿⢸⣿⡀⢿⠇⣾⠀⣤⣾⣿⡇⠀⠀⠀⢀⣠⠤⠖⠒⠒⠋⠉⠀⠒⠒⠒⠤⢄⡀
⣧⢿⣦⣤⣾⣿⣼⣿⣷⣤⣶⣿⣶⣿⣿⢻⡇⠀⡠⠖⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠈⠓⢤⡀
⠈⠳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⡵⠋⣠⠊⠀⠀⠀⠀⠀⠠⠟⠛⠀⠀⠀⠀⠈⠛⠳⠀⠀⠀⠀⠙⢦
⠀⠀⠈⠳⣝⡻⢿⣿⣿⣿⣟⣛⡵⠋⠀⢰⠃⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠔⠁⠀⠀⠑⡀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠈⠉⠉⢹⢹⠉⠉⠉⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢘⠀⠀⢠⡆⢸⠀⠀⣤⠀⠀⡇⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠁⣸⡀⠀⠛⠀⢠⠃⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⢀⢸⠸⠤⡀⠀⠀⠀⠀⠘⣆⠀⢢⠀⠀⠀⠀⠀⠁⠀⠈⠀⠑⠄⠀⠔⠁⠀⠀⢀⠉⠀⢀⠎⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠅⢈⠁⠀⢸⠀⠀⠀⠀⠀⠈⢧⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢠⠟⠉⢻⠁⠀⢹⣤⣀
⠀⠀⠀⠀⠀⠘⡆⠀⢻⢁⡸⠧⣤⣀⠀⠀⠀⠀⢳⡀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⣸⡄⠀⢸⠀⠀⡜⠀⠈
⠀⠀⠀⠀⠀⠀⠉⢸⢹⠁⠀⠀⠀⠉⠙⠛⠲⠶⠤⣷⠀⠀⠀⠀⠀⠀⠉⠰⢁⡉⠁⠀⠀⣶⠴⠾⠧⣇⠀⠘⠀⠀⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⢀⠀⠧⢄⡠⠋⠘⡦⣀⠀⠹⢄⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⡇⠀⠀⠑⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⡀⠀⠧⣀⣀⣀⣤⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⣀⣀⡠⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠀⠀⠀⠀⠀⠀⠀⠰⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠴⠶⠋⠉⠙⠶⠦⠖⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠛⣹⠟⢉⡟⠉⠉⠉⠉⠉⠉⠻⡉⠛⣏⠙⠻⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢰⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣼⣧⣤⣼⡄⠀⠀⠀⠀⠀⠀⢠⣿⣤⣼⣧⡀⢸⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠿⣿⣿⢿⠗⠒⠒⠒⠒⠒⠒⠺⢿⣿⣿⣿⠇⠚⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃

⡶⣶⢶⢶⡶⠶⢶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⡶⠶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡛⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⣠⡴⠚⠛⠛⠛⠲⢦⣀⠀⠀⢹⠛⠛⠛⠛⠛⠛⠛⣛⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⠀⢸⡇⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⢸⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄⣸⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣦⣀⣀⣀⣀⣠⡴⠛⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⢾⡇⠀⠀⠀⣠⣬⠿⠿⠯⣥⣀⠀⠀⠀⢸⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⠀⠀⠀⠀⢸⡇⠀⣴⠟⠁⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⢸⡇⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⠀⢀⣠⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⢸⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣦⣀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠉⢛⣲⣶⣞⣛⠉⠀⠀⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡴⠞⠉⠉⠀⠉⠉⠻⢦⡀⠀⢸⠁⠀⠀⠀⠀⠀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⢸⡇⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⠀⠀⢀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣼⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠳⠦⣤⣤⣤⡴⠞⠋⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⣿⠛⣿⠛⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠲⠀⠂⠰⡀⣀⠘⠀⠀⡀⠐⢧⡀
⠀⠀⠐⠁⠈⠉⠀⠀⠀⠀⠂⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠈⠵⠛
⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠤⠠
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⠀⠀⠀⢀⡄⠀⠠⠀⠀⠒⠊⢀⡀
⠀⠤⢤⣀⣀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠁
⠀⠀⠀⠀⠉⠉⠓⠲⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣠⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓
⠀⠀⠁⠀⠀⠀⢀⡂⠐⠂⠀⠐⡓⠀⠀⠀⣸⠀⠠⡁⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠂⠉⠈⢈⢉⠁⡁⠐⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⠤⠤⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢈⣀⢤⠠⠄⠄⠈⡈⠒⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⢠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⢤⠐⠀⡆⠀⠀⣀⠀⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠂⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠐

⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠖⠒⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠉⠀⣀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠛⠶⠦⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⣈⡢⡀⠀⢀⠤⠔⠊⠉⠀⣄⠤⢴⡷⠆⢯⠌⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢠⡽⡗⠒⠺⡄⠀⢉⡁⢀⣀⡀⠀⠄⡂⠈⠀⠐⠀⠀⠀⢉⡁⢇⣴⣿⣂⢤⣰⣎⠸⠛⠁⠄⣁⠈⠈⠃⠀⠀⢸⣧
⠀⠀⠀⢀⣀⡿⡁⠻⡄⠀⠸⡀⠀⢺⣿⣿⣷⠀⠀⠀⠐⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡼⠉⢈⡀⠈⢊⠩⠄⠠⠄⠀⠀⠀⢸⡏⡆
⠀⠀⠀⢸⡍⢻⢱⣧⢳⠀⠘⣧⠀⠀⢿⣿⣽⣦⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⢿⣿⣿⣿⡿⡄⠨⠒⠉⠀⠴⠃⠀⠒⠀⠀⠀⢸⣿⡘⡄
⠀⠀⠀⠈⣧⠸⣧⣻⣇⢇⠀⢸⣧⠀⠈⠑⢉⠵⡀⠀⢡⡀⠠⠄⠊⠦⠀⠀⠸⣿⣿⠋⡰⣷⠀⣀⠠⠀⠠⠒⠐⠒⠂⠠⣤⠈⣿⣧⠘⡄
⠀⠀⠀⠀⢻⡄⠿⣯⣧⡞⡆⠀⣿⣧⠀⠁⢠⠀⠁⠀⡈⠀⢀⠀⠀⠀⠀⠀⠀⢻⣿⢷⡀⢻⡆⠡⠊⠁⠠⠀⠀⠀⠀⠀⡄⠀⠘⣿⣇⢹
⠀⠀⠀⠀⠈⣷⠀⣾⣾⣿⡸⡄⢸⡽⣆⠀⠛⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣄⣸⣷⠀⠈⣀⠀⠀⠐⠒⠶⠤⣄⠀⠀⠈⣿⢸⡄
⠀⠀⠀⠀⠀⢹⣆⢹⣿⡉⣿⡇⠀⠻⣿⡄⠐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣿⣿⣿⣷⢿⠁⠚⢁⣠⣤⣤⣤⣤⣤⣀⣀⣀⣀⣿⣆⣷⡀
⠀⠀⠀⠀⠀⠈⣿⠀⠿⣷⢸⡹⡆⠀⢁⣿⡠⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣩⣴⣾⣯⣽⣿⣦⣭⣭⣭⣉⣛⠻⢾⣿⣧⣵⡹⡄
⠀⠀⠀⠀⠀⠀⠸⡇⠀⢻⣯⡇⣇⠀⠸⣏⡉⠒⢄⠀⣼⣤⣀⢀⣀⡴⣶⢿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣯⣟⣛⠻⢿⣻⡿⢿⣷⣶⣿⣌⢦
⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⣷⣹⢸⠀⠀⠐⠂⠁⠈⢦⣿⣿⣿⣿⣿⣿⣾⣿⣿⣋⢻⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣶⣭⣿⣶⣮⣭⣭⡤⠷
⠀⠀⠀⠀⠀⠀⠀⠸⣤⠀⢻⣻⡀⣇⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣷⣿⡿⠟⠛⣿⡹⠿⢿⣿⡿⣿⡇⠀⠀⠈⠉⠙⠛⠲⠶⣮⣽⣻⡛⠿⠿⣿⣶⣾⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠘⣯⣧⣩⣭⣽⣿⣿⣿⣿⢽⣿⡿⠿⠋⠉⠀⠀⠀⠈⠙⠻⠷⠷⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣶⣤⣭⣓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠑⠚⠻⠿⢿⣛⣭⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠶⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠒⠉⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢩⠀⢀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡀⡸⠀⠂⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠉⠁⠂⡀
⠀⠀⠀⣰⠋⢹⠀⢠⠞⠉⡆⠀⠀⠀⠀⠀⢸⣀⣀⠤⠤⠒⠒⢊⣀⡚⢥⣒⣄⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⣴⠢⡀⢹⠀⢸⠀⡿⠀⢰⠃⣠⠞⢳⡄⠀⠾⡿⠁⠊⢀⣠⠀⠀⢒⡂⠀⢸⡿⠃⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⢿⠀⢳⢸⠀⢸⣧⡃⠀⡏⣼⠃⢠⡞⠀⠀⠲⢿⡄⠀⢶⢈⠇⠀⠘⠁⠀⠠⣷⠶⣄⠀⠀⠀⠀⢸
⠸⣧⠀⠛⠉⠉⠉⠉⠉⠛⠃⡴⠋⠀⠀⠀⡴⣾⡇⠀⠀⠧⠤⠄⣂⠀⠀⠀⢻⡇⡁⠀⠀⠀⠀⠼
⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠈⢧⡘⣿⠀⠀⢀⣀⣤⣤⣶⡄⠀⢸⡏⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣿⡀⠀⠀⢀⠴⠒⠚⠛⠒⠚⠛⢶⡄⠀⠀⠉⢹⠀⠀⢸⣿⣿⣿⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠀⠃⠀⠀⠀⢸⡀⠀⠈⠛⠛⣉⠥⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊
⠀⠀⢻⣇⡀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣤⣄⣀⣤⣤⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡖⡤⠤⠀⣀⡠⠚⠁
⠀⠀⠀⠛⠿⠷⣤⠬⠘⢧⡀⠀⠀⠀⠀⣀⣀⢀⣤⠤⠶⣦⡿⠀⠀⢸⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠈⠳⠒⠛⡛⠉⠉⢻⡇⠀⠀⠈⠳⡟⣱⠞⠁⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⡀⠀⣇⠀⠀⢸⡇⠀⠀⠀⠠⢼⠇⠀⠀⠀⠀⢸⡏⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⢺⡇⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡿⢀⠠⢾⠀⠀⠀⠀⠀⣠⠴⠲⠤⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠹⡄⠀⠦⠤⣤⠔⠞⠁⢠⡛⠒⠊⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠇⠀⠀⠙⠦⢤⣤⣼⡀⠀⠠⢾⣉⣉⣉⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⠀⠀⢀⣼⠃⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⣼⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠙⠛⠉⠈⠓⠚⠛⠋⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⡴⠚⠓⠶⣄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠈⠳⡄⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⢀⣀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠠⠞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⡤⠖⠲⢦⣯⣄⣀⣀⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⣠⠤⠶⢾⠷⠒⠶⣄
⠀⠀⠀⠀⠀⠀⣜⠤⢄⣀⣀⣁⠀⢈⣉⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢂⣤⣀⠀⣀⣀⡠⠼⡂
⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣤⣬⣿⣶⣶⣾⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣯⣧⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣷⣶⣾⣿⣿⣿⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⣀⣀⡀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠴⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠾⣕⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⢶⠖⠖⠓⠐⡟⠻⠊⠿⠂⠴⣶⠀⣠⣄⠀⣀
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣀⣠⡶⠾⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣇⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⢴⣦⠀⡀
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠛⠉⡏⠉⠉⠉⠉⠙⠛⠓⠲⠶⣤⣤⣀⠀⠀⠈⠛⠷⠀⢀
⠀⠀⠀⠠⠁⠄⠀⠀⢀⢀⠔⠁⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⢀⣀⣀⣧⣤⢤⣤⡤⣤⣄⣀⣀⡀⠀⠉⠙⠺⢧⡀⢄⠀⠀⠙⢅⡀
⠀⠀⢀⠃⠀⠀⠀⠀⢠⠋⠀⠀⠀⢀⠀⡠⡊⠀⠀⠀⠀⢀⠴⠛⠋⠋⢀⣇⠒⠀⠀⢰⣀⣀⡀⠈⠉⠛⠓⠶⣤⡀⠈⠋⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⢀⠀⡀⠀⡼⠤⠒⠒⠂⠀⠔⠁⠀⢤⣔⣋⠉⣏⠱⠴⣶⡾⣤⣬⣍⣙⠓⠒⢤⠄⠀⠀⠀⠀⠇
⠀⠀⠀⢠⡄⢀⠘⠁⠁⠲⠀⣀⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠙⡟⠐⠚⢿⠻⠛⢹⡯⠶⠿⠓⠐⠂⠂⢀⠀⠀⠠
⠀⠀⠘⠈⠗⠈⠀⠀⠂⣀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⠃⠀⠀⣶⣹⠆⡀⠀⣸⣇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠹⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠤⠀⠀⠀⢤⠀⠘⠃⠀⠀⢀⢿⡏⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠄⠀⠐⠠⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⠀⠀⠀⠀⠀⠘⡏⠂⠀⠀⠀⠀⠀⠘⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠓⠶⠔⠂⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠋⠉⠀⠀⠉⠁⠐⠆⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠲⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠓⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠹⣆⡀⠀⠀⡀⢀⢀⣠⡠⠴⠄⠐⠒⠒⠒⠒⠰⢶⣤⡀⠀⢀⠄⣠⠔⠁⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠙⠳⢶⡾⡋⠩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠞⠓⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢀⣤⣤⣄⡀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣠⣿⡿⠿⠃⠉⠉⠀⠀⠙⠋⠙⠻⠿⣤⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣼⣿⡟⠁⠉⢡⣴⣶⢄⠀⠀⠀⣊⢤⣶⡶⢤⠀⠈⣿⠗⠒⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠐⠁⢤⣈⠿⠧⠄⠀⠒⠛⠛⠉⠀⠀⠀⠁⠉⠉⠉⠉⠀⠤⠁⠶⢆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣢⠎⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⡞⢭⠑⠢⡀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⢠⡀⠀⠀⢀⣀⣴⣷⣤⣿⣤⣴⣷⣾⣧⣀⠀⠀⢀⣴⣀⣠⠿⠆⠀⠀⠀⠀⠀⣼⣶⡤⠔⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⠿⠿⡭⢿⣿⣟⣽⣗⡿⢿⣼⣿⣿⣿⣿⣷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⢄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠙⣍⡂⠤⠀⣊⡽⠁⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢀⡤⡄⠩⠀⣀⠀⠉⠙⠳⢄⡀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠋⠀⠀⠀⠀⠀⠈⠑⡄⠀⠈⠳⣦⣀⠀⠀⠀⠘⠦⠀⠤⠒⣠⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠰⣌⠫⡒⠢⠤⠀⠀⣠⡴⠗⣥⣂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡳⠀⠀⠀⢩⠐⠬⠁⠒⠀⢲⣿⣧⣅⣂⠩⢛⡒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠎⠀⠀⡅
⠀⠀⠀⠀⢀⠄⠐⠀⣿⡀⠀⠀⠀⣠⠴⠚⣉⡩⠍⠿⣄⠀⠀⠈⠁⠀⠀⠉⠙⠛⠿⠿⡿⣿⣿⣶⣮⣕⡀⠨⢕⡒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠃⡎⠀⠉⠻⣷⣅⣤⠛⣡⠴⠋⠁⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡽⠁⣾⣝⣿⣿⣿⣶⣤⣁⡂⠬⢐⡠⢄⣠⢀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠳⡔⠢⡀⠀⠀⢹⣁⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠉⠁⠀⠈⠉⠉⠀⠒⠋⠀⠀⠁⠀⠀⠈⣈⢉⡉⠙⠻⣷⣶⣬⣐⠪⢝⡳⢴⣔⠠⠐⠐⣾
⠀⠀⠀⠄⠀⠀⠑⠶⠶⠂⣭⡽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣨⠍⢉⣋⣿⢻⠿⣶⣦⣅⡛⠩⣗⢼⡇
⠀⠀⠀⠢⢀⠠⢤⠖⠙⠋⠀⠉⠊⠝⡢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⠀⠀⠀⠀⢤⣤⣤⡄⠉⣲⠤⢚⠷⠒⠊⡩⠛⠿⢷⣦⣽⣿
⠀⠀⠀⠀⠀⠐⠡⠈⠀⠁⠀⠀⠀⠀⠀⠉⠙⠳⠲⣤⢦⡀⠀⠀⠀⠀⠀⠀⣀⣤⢶⠛⠋⠊⠉⠛⠙⠛⠗⠦⣌⣫⠑⢄⠀⠀⢁⡠⠔⠈⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠙⠓⠺⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⢯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⢄⣀⡀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⣀⣠⣤⣤⢴⢶⠶⣮⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠀⠀⠈⠀⠀⠀⠁⠁⠋⠓⠉⡳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠛⡻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⣿⣶⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⢠⣿⠹⡜⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡴⠁⣿⡿⣿⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⣠⠎⢸⠀⣿⠀⢸⣿⠀⠇
⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⣠⠞⠁⠀⠀⢸⣿⡿⢿⣿⡄⠸⡀⠀⢀⡀
⣤⣤⣼⣦⣶⣅⣀⣀⣀⢳⡞⠁⠀⠀⠀⠐⢸⣿⠀⢸⣿⢹⡀⢣⠀⢐⣂
⠀⠂⡀⠂⡀⠈⠉⠉⡙⢸⣫⣙⣩⣿⣿⣿⣿⣿⣶⣾⡿⣶⣿⣬⣦⣸⢿
⠀⠂⠓⠂⠁⠈⠀⠀⠉⢻⡏⠀⠀⠀⠀⠀⠀⣿⣜⣯⡏⠀⠀⠉⠉⢹⡿
⠀⢀⠔⢚⣫⣐⠢⡀⠀⢸⡇⠀⠀⠀⠀⠀⠐⣿⡳⣏⡷⠀⠀⠀⠀⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⡴⣇⠆⠁⠀⠀⠁⠜⡀⣼⡇⠀⠀⠀⠀⠀⢠⣿⣉⣹⣿
⣠⠃⠀⠀⠀⠀⠀⠈⣘⣾⡇⠀⠀⠀⠈⠉⠉⠉⠉⠙⠋⠁⠈⠁
⠃⠀⠀⠀⠀⠀⠀⠀⠘⠻⠇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⠿⠿⠿⠷⢶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⢀⣀⠀⢀⣀⣀⡀⣾⡆⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠈⣿⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⣧⠈⠛⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⢷⣤⣤⣀⣀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣾⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⢶⣾⣿⣤⣄⣸⣿⡀
⠀⢀⣤⣶⣾⠿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣩⣿⣿⣷⡀
⣰⡿⠟⠛⠛⣿⡟⠛⠻⣶⠶⠶⠶⠶⣶⣦⣤⣤⣤⣤⣀⣀⣀⠀⠀⠀⠀⠘⠃⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⠶⠛⠉⠀⣿⣿⣇
⣿⡇⠀⠀⠀⡿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠉⠉⠉⢻⡟⠷⠶⢦⣤⣀⠀⠀⣀⣀⣤⣤⡶⠾⠟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⣿⢻⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⡇⠀⠀⠀⠈⠹⣿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿
⣿⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿
⣿⡇⠀⠀⠈⣿⠀⠀⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆
⣿⡇⠀⠀⠀⣿⣀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⣿⡇⠀⠀⢠⡿⣻⣿⠛⢷⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇
⣿⡇⠀⠀⢸⡇⢿⣹⡇⢸⡇⠀⠀⠀⠀⠀⠀⢀⣿⣦⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧
⣿⡇⠀⠀⢸⣇⠘⠟⠁⢸⡇⠀⠀⠀⠀⠀⠀⠘⡯⢈⣉⡉⠉⢹⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿
⣿⡇⠀⠀⠀⢻⣶⢶⣶⡾⠃⠀⠀⠀⠀⠀⠀⢸⡇⣿⣉⣿⡄⠘⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢶⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟
⣿⡇⠀⠀⠀⠠⣿⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠸⣧⣬⣍⣉⣀⣀⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠀⠀⠀⠀⢰⣤⣦⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢙⣿⠛⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢹⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡾⠛⠻⣷
⠀⠀⠀⠻⣿⣷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡟⠀⣀⠀⢹
⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⢀⣠⣤⣶⣾⣿⡿⠿⠛⠋⠉⣿⣿⡇⠀⠿⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣶⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣶⣿⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⣤⣠⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣶⣶⣶⣤⣶⣶⣿⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⠠⠤⠤⠒⠒⣂⣉⠉⠉⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢤⠤⠄⠒⠒⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠠⢤⣤⣄⣀⣀⣀⡉⣿⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⡇⣿⠀⠀⠀⠤⠀⠔⣶⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣧⣿⣿⣷⣶⣶⣶⣶⣦⣤⣀⣈⡟⠀⠉⠀⠁⠱⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣭⡍⡉⠉⣉⣛⣛⠛⠒⠂⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⠀⣿⣿⣀⠀⠀⠀⠀⠀⠀⣿⢰⡖⠓⣶⣒⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⠀⢈⡀⠀⠀⠠⠀⠀⠀⠀⣿⣼⣿⣿⣿⣿⠇⠿
⠀⠀⠀⠀⠒⠀⠀⠀⠠⠄⠀⠀⠀⠒⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⠿⠙⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣀⣤⠴⠞⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣩⠽⢿⡇
⠀⠀⣶⠿⠟⠛⠓⠚⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⢻⠁⠀⠀⢸⡇
⠀⠀⣿⠀⢠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⣿⠀⠀⠈⠁⠀⠀⢰⡋⠀⢹⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠳⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⢻⠀⠀⠀⠀⡖⣒⣛⡉⠰⣯⠜⡹⠀⠀⢰⡞⠛⠛⠛⠛⠃⠀⠀⠙⢄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿
⠀⠀⢸⡄⠀⠀⢸⣱⠃⢰⠇⢰⠖⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⠀⠀⠀⠀⡇⠀⠀⠀⣸
⠀⠀⢸⡇⠀⢾⠣⡤⠴⠞⣀⣉⣭⡍⢹⡀⠀⠀⠙⠉⠉⠉⡇⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⡇⠀⠈⠳⡤⠖⠛⠉⠀⠀⠹⡤⠇⠀⠀⠀⠀⠀⠀⣿⡤⠞⠁⠀⠀⠀⠄⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡇⠀⠀⣀⡟
⠀⠀⠀⢿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣧⠶⠛⠉
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠶⢤⣄⡀⠀⠀⠀⠀⠀⣿
⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⣏⠀⠀⠀⣠⡾⠋⢁⣤⡴⠶⠶⠦⣤⡈⠙⢷⣄⠀⠀⠀⣹⠶⠶⠶⠶⠶⠶⠶⠶⠶
⢣⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠋⢠⡾⠋⠀⠀⠀⠲⠶⣤⡙⢷⣄⠙⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⣿⠀⣸⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢹⡆⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⢹⠷⣤⡀⠀⠀⠀⣿⠀⡿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⣿⠀⣿⠀⠀⠀⢀⣤⠾⡏
⠀⠀⠀⠘⡇⠀⠙⣦⠀⠀⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⠀⠀⣴⠋⠀⢸⠇
⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⣏⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⣿⠀⠀⣿⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠳⣿⡀⢀⣿⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⣿⡀⢀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣤⡴⠶⠶⠶⠶⢦⣤⡀⠀⠀⠀⠀⠀⣿
⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣯⠀⠀⠀⣠⠾⠋⣀⣤⠴⠶⠶⠶⣤⡀⠙⠷⣄⠀⠀⠀⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶
⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠋⣠⠞⠉⠀⠀⠀⠒⠶⣤⡙⠷⣄⠙⣧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⣿⠀⣸⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⡆⠘⣇⠀⣿⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⢸⠷⢦⡀⠀⠀⠀⣿⠀⡿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⠀⢿⠀⣿⠀⠀⠀⢀⡴⠾⡇
⠀⠀⠀⠘⡇⠀⠙⣦⠀⠀⣿⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⠀⠀⣴⠋⠀⣸⠃
⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⣇⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⣿⠀⠀⣿⠀⣰⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣀⣀⡿⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⣿⣀⣀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⠶⠶⠦⣤⣀⠀⠀⠀⠀⠀⣿
⡶⠲⠶⠶⠶⠶⠶⠶⠶⠶⣟⠀⠀⠀⣠⠞⠋⡀⠠⠔⠒⠒⠀⠄⣀⠙⠳⣄⠀⠀⠀⣹⠶⠶⠶⠶⠶⠶⠶⠶⠖
⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡼⠃⣠⠞⠁⠀⠀⠐⠒⠶⢤⡈⠳⣄⠘⢧⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣿⠀⣸⠃⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠹⡆⠘⣇⠀⣿⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⢸⠿⢦⡀⠀⠀⠀⣿⠀⠿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⠀⠿⠀⣿⠀⠀⠀⣀⡴⢿⡇
⠀⠀⠀⠘⣇⠀⠙⣶⠀⠀⡿⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣿⠀⠀⣶⠋⠀⣸⠃
⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⣷⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⣿⠀⠀⣿⠀⣰⠏
⠀⠀⠀⠀⠀⠈⠻⢿⣀⣀⣿⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣀⣠⣤⠞⠋⠀⠀⠀⠀⠀⣿⣀⣀⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠹⣦⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣴⠏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣏⢉⣉⡉⠉⠉⢉⠉⠉⣉⠉⣹⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⣿⠉⠉⣿⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣾⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠋⠙⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣴⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣄
⠀⠀⠟⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⣀⠀⠦⠄⠀⠤⠀⠤⠄⢀⡀⣀
⠀⢼⡦⠀⣰⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠗⠒⢂⣀⣉⣀⣙⣓⣒⠒⠂⢄⠒⠀⠂⠄⠠⠤⠤⠄
⠀⣠⣠⣿⣿⣿⣿⣿⣇⠀⢀⡂⠀⢐⠀⠑⣿⠛⡏⢻⡟⢸⣿⠋⣿⡏⢽⡍⠂⡀⠀⣀⠈⢉⠉
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣃⠈⠁⠀⠈⠀⢀⣉⡀⠁⢈⠁⢈⢹⠀⣿⠇⠙⠀⠀⠃⠀⠉⣀⣘⡀⠸⠀⣀⣀⣀⣀⣤
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⠁⠁⢨⠀⢸⠞⠀⡇⠀⢸⠀⠀⣶⠀⣲⣿⣿⣷⣶⣶⣿⣾⣷⠀⣠⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣴⣯⣿⣿⣧⣶⣶⣶⣶⣊⡒⠀⢓⠂⠘⠂⣠⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⢸⣧⣼⡗⢋⡴⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠇
⠀⠀⠀⠀⠀⢤⡤⣤⡤⣶⣄⢤⣤⡶⢶⣷⣾⣿⣛⣛⣛⣷⣿⣿⣿⣿⣿⣈⡉⣹⣟⠿⠛⠋⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠜⣤⠁⠀⢀⢈⡩⠉⠽⠿⣽⣿⣿⣿⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠈⠀⠁⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⢀⠀⡄⡄⠀⠄⠀⠀⠐⠆⠀⢀⡖⡄⠀⠀⢄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⡀⠰⡁⢊⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⢐⠄⠀⠀⠈⠀⠐⠩⠋⠀⠤⡄
⠀⠀⠀⠀⠀⣀⠠⡀⢀⠈⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⢠⠀⡀⡪⠢⢤⡤⡀
⠀⠀⠀⠀⡀⢨⢛⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣾⣾⣿⣿⣯⣅⣦⣀⣡⡙⠉⡊⠀⢼⣿⢆⣠⡄
⠀⠀⠀⠉⡪⠖⠁⡁⠀⠀⠀⠁⡀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣦⣤⡛⠱⠛⢙⢇⡀
⠀⠐⠯⢊⠀⠀⠢⠀⠂⣀⣴⣿⣶⣿⠿⠿⠟⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣦⠐⡖⡻⣛⡂
⠀⣆⠆⠀⠀⠀⠀⢀⣼⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⣿⣿⣿⣿⣿⣿⣿⣿⡇⡍⣭⡰⢤
⢀⣩⠅⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡶⠶⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠠⠀⠦⠼
⠰⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠛⣉⣠⣤⣴⣶⣶⠶⣶⣶⣶⣶⡦⣤⣉⠙⢿⣿⣿⣿⣿⡗⠀⠀⠀⠔⣺
⢀⣈⣂⠀⠀⠀⠀⣀⣤⣤⠄⠒⠋⣁⣤⠶⠛⢉⣿⣿⣿⣿⣧⣀⣿⣿⣿⣿⣷⠀⠙⢿⣦⡉⠻⢿⠿⠁⠀⠀⠀⢀⡉
⠰⢶⣤⠀⠠⠤⠀⠀⠀⠤⠤⣶⠏⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠘⢻⣿⣦⣀⣀⠀⠀⠀⠀⠈⠤
⠀⣜⡒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣿⣧⣼⣿⠇⠀⠤⠀⠀⠢
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣀⠤⠞⢋⣠⡁⠀⠀⠀⠀⠀⡠⠈
⠀⠀⠱⠮⠢⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠿⠟⠛⠓⠒⠋⣉⣤⣴⣾⠟⠋⠁⠀⠀⠀⠠⡠⠐⠃
⠀⠀⠀⠐⢿⠿⢪⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⣶⣶⣶⣿⣿⠟⠟⠋⠉⠀⠀⠀⠀⠀⠀⢀⣄⠥
⠀⠀⠀⠀⠀⠀⠽⡵⠫⣤⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡐⣀⢑⢕⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⡵⠁⡔⢄⢆⠀⠀⠎⠀⠀⠀⠱⠀⠀⠴⢀⡀⢂⢐⣀⡔⡡⢎⢞⡏⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠊⠀⢰⡼⠎⠀⠀⠠⠀⠧⠀⡆⠀⠤⠰⠉⠉⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣤⣴⠶⠶⠾⠷⠶⢶⣤⣄⡀⠀⠀⠀⠀⠀⢸⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⢀⣠⡾⠛⠉⣀⣠⣤⣤⣤⣤⣀⡀⠉⠻⣶⣄⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠘⢿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⢠⡾⠋⣠⣴⡿⠟⠋⠉⠉⠉⠉⠛⠻⣷⣤⡈⠻⣦⡀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠟
⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢰⡿⢁⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠘⣷⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠙⢿⣦⣄⠀⠀⠀⣼⣿⣿⠀⣿⠃⣾⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⢹⡇⢸⣿⣿⡄⠀⠀⠀⣤⣾⠟⠉
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣴⣿⣿⣿⠰⣿⣾⣿⠁⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⡿⢸⣿⣿⣿⡄⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣿⠀⠀⠘⣿⡀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿⣿⣿⣿⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⢻⣧⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠀⠀⠀⠈⢻⣧⡀⠙⠿⣿⣷⣦⣄⣀⣀⠀⠀⣠⣾⠏⠀⠀⠀⢸⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣽⣿⣶⣤⣀⣉⠉⢉⣉⣠⣴⣾⣿⡅⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣤⣴⡶⠶⠾⠶⠶⢶⣤⣄⡀⠀⠀⠀⠀⠀⢸⣿
⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⣠⡾⠛⠉⢀⣀⣤⣤⣤⣄⣀⡈⠙⠻⣶⣄⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠘⢿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⢠⡾⠋⣀⣴⡿⠟⠋⠉⠉⠉⠉⠛⠻⣷⣤⡈⠻⣦⡀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠟
⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⢠⡿⢁⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣆⠘⣷⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠈⠙⢿⣦⣄⠀⠀⠀⣼⣿⣿⠀⣿⠃⣾⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⢹⡇⢸⣿⣿⡄⠀⠀⢀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣰⣿⣿⣿⠰⣿⣾⣿⠁⢰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡿⢸⣿⣿⣷⡄⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣿⠀⠀⠘⣿⡀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿⣿⣿⣿⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⢿⣇⠀⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⠀⠀⠀⠈⢻⣧⡀⠙⠿⣿⣿⣦⣤⣀⣀⠀⠀⣠⣿⠏⠀⠀⠀⢸⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣽⣿⣶⣤⣀⣉⠉⠉⣉⣡⣴⣾⣿⡅⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠶⠿⠷⠶⣶⣤⣄⠀⠀⠀⠀⠀⠀⢸⣿
⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⣠⡾⠛⠉⣀⣀⣤⣤⣤⣄⣀⡈⠙⠻⣦⣀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠘⢿⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⣿⠀⠀⢠⡾⠋⣠⣴⡿⠟⠋⠉⠉⠉⠉⠛⠿⣷⣤⡈⠻⣦⡀⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠟
⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⢠⡿⢁⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄⠙⣷⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠈⠛⢿⣦⣄⠀⠀⠀⣼⣿⣿⠀⣿⠃⣾⡟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⢹⡇⢸⣿⣿⡄⠀⠀⠀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⣴⣿⣿⣿⠠⣿⣼⣿⠁⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⡿⢸⣿⣿⣿⡄⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⣿⣿⣿⣿⠀⠀⠀⣿⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢸⣿⣿⣿⣿⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⢿⣇⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⢸⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠀⠀⠀⠈⢿⣧⡀⠙⠿⣿⣿⣶⣤⣀⡀⠀⠀⣠⣾⠏⠀⠀⠀⢸⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣽⣿⣶⣤⣀⡉⠉⠉⢉⣀⣤⣾⣿⡅⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣠⣾⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀⠀⣀⢀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣭⠀⢻⣿⠋⢸⣿⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠶⠤⠟⠹⠷⠸⠿⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣉⡉⠉⠉⠉⠉⠉⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⣴⣦⣀⣉⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠸⣿⣿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠁⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⣿⠀⠀⠀⣠⣤⡶⢶⣤⣄⠀⠀⠀⣿⡶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⣿⠀⠀⣾⠏⠁⠀⠀⠀⠙⣷⡀⠀⣿⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⣿⠀⢸⣟⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣿⠀⠈⢿⣄⠀⠀⠀⠀⣠⡿⠃⠀⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠙⠻⠶⠶⠟⠋⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⣿⠀⠀⠀⣠⣴⡶⠶⣦⣄⠀⠀⠀⣿⡶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⣿⠀⢀⣾⠋⠀⠀⠀⠀⠙⣷⡀⠀⣿⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⣿⠀⢸⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⠀⠈⢿⣄⠀⠀⠀⠀⣠⡿⠁⠀⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠙⠻⠶⠶⠟⠋⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠶⢶⣿⠀⠀⠀⣠⣴⠶⠶⣦⣄⠀⠀⠀⣿⡶⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⣿⠀⢠⣾⠋⠀⠀⠀⠀⠙⣷⡀⠀⣿⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⠀⠈⢿⣄⠀⠀⠀⢀⣰⡿⠀⠀⣿⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠙⠻⠷⠾⠛⠋⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟

⠀⠤⠀⡄⢶⡒⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣶⣾⡖⠶⠶⠶⢶⣶⣷⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠈⡆⣷⠀⠀⠀⠈⠉⠒⠂⠤⠀⢀⠀⠀⠠⠤⠤⠤⠀⠒⠒⠀⠂⠉⠀⠈⠁⠉⠙⢿⠿⠿⢿⣿⢿⡿⣿⣿⣿⣿⣿⠭⠁
⠀⠀⠀⠀⡇⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠈⢃⠀⠬⠋⢋⡽
⠀⠀⠀⠀⡇⠰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠦⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⣴⠀⠀⠀⠈⠃
⠀⠀⠀⠀⡇⠀⠀⢠⣶⡶⠶⠶⠶⣤⣤⠈⠀⠀⠀⣐⡂⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⡇⢠⠀⠈⠉⠁⠀⠀⠀⠉⠉⡄⠀⠀⠀⡿⠻⠘⣿⣇⠀⠐⠓⠐⠒⠀⠒⠂⢰⠀⠈⠀⢸
⠀⠀⠀⠀⡇⣴⠀⠀⠀⢰⣶⠆⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⢀⠀⢀
⠀⠀⡀⠀⠁⠙⠀⢀⡀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣤⢸⡇⡇⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⣿⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠙⠋⠈⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⠠⠄⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⣿⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠎⠋⠁
⠀⠀⣿⣶⡶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠈
⠀⣴⣿⣿⣷⡇⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⣿⣿⣿⣿⠁⠀⠀⣠⠔⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⢒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠤⣀⠀⠀⠀⠀⠹⠆⡀⠣⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣔⠖⣿⣍⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⡟⠙⠛⣍⢼⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⡀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠀⢀⠁⠈⠛⠙⠻⢗⣄⠀⠀⠀⠀⡀⠀⣄⠈⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠐⠀⠀⠀⠀⠀⠸⣿⣿⠇⠀⡾⠀⠀⠁⣀⢶⡗⠈⠓⢄⠀⠀⠀⠁⢸⡄⡆⠀⡃⠀⠀⠠⢀
⠀⢀⠀⠀⠀⠀⠀⠀⢸⠇⢀⠄⣳⣤⣬⣦⣉⡼⠧⠀⠀⠀⠑⢦⣄⠀⠘⣇⠀⠀⠀⠐⠀⠀⡈⠄
⠀⠀⠀⠀⠉⠀⠀⠀⡿⠀⡸⢀⡟⠁⢌⡋⡇⠣⠀⠀⠀⠀⢠⠀⠀⠀⠨⠘⠀⡦⠀⠈⡁⠍⠁
⠀⡀⠀⠀⠀⠀⠀⠀⢣⠤⠇⢸⡇⠀⠀⡇⠁⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠀⠐
⠀⠃⠂⠀⠈⠀⠂⠀⠀⠀⠀⢸⡇⠀⠀⠐⠀⠀⠀⠀⠨⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣈⡁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠃⣀⡀⢀⣀⡠⣠⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣇⢴⡰⢦⠆
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣆⢻⡄⠈⡟⠐⢣⡔⢦⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣔⠉⢙⣶⣿⣵⢀⡄⢀
⠀⠰⠚⠍⠁⠀⠀⠀⠀⠹⠀⠀⠸⢼⡿⠤⡇⠀⠈⠀⠒⢂⠀⠀⢀⠇⠂⠀⠁⣨⠿⠃⣄⣙⣿⣿⠁⢈⠡⣔⡃
⠀⠀⠂⠀⠀⠊⠀⠀⠀⠀⠁⠀⣧⠈⣧⢀⡀⣀⣠⠄⠀⠨⠛⠀⠀⠀⠂⠀⣄⣄⠀⣒⠐⡩⠁⠪⣶⠀⢀⠰
⠀⠀⠀⠀⠉⠈⠁⢠⠀⠀⠀⢀⣿⡀⣻⡟⢀⠁⠀⠀⠄⠀⠀⠀⠀⠀⢀⣠⣤⣿⣮⠛⠲⠀⠙⠘⠉⡄⠛⣠⠄
⠀⠀⠀⠀⠀⠂⣛⠉⠀⠛⠺⠟⠿⠛⠿⣿⣿⣿⠀⠀⠛⠛⠛⠀⠀⠀⠉⠙⠙⠿⣟⠓⠲⠶⢿⠛⠟⠓⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠐⠈⠀⠐⠐⠀⠋⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⠛⠛⠒⠒⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠀⣠⣤⣀⣀⣀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠉⠛⣻⡇⠀⢠⣧⣤⣄⣀⣉⣉⣙⠛⡦⠀⠀⡾
⠀⠀⠀⠀⠀⢀⡤⠚⠫⠤⢤⡤⠖⠉⣀⣇⣀⣸⠁⠀⣠⠖⠛⠛⣿⠙⣇⠀⢸⡧⣤⣀⣀⣀⡀
⠀⠀⠀⣠⠖⢫⠀⠀⣀⠔⠋⢀⡴⠚⠧⠬⣭⣁⣀⠀⠛⠉⠉⠙⠃⠀⣇⠀⣼⠀⣠⠞⠉⢉⡽⠛⠓⠲⢶⣤⣀
⠀⠀⢻⠛⢲⠶⠷⣾⢤⣤⣞⣁⣀⣀⡀⠀⠀⠀⠈⠉⣩⠿⣖⣒⠒⠶⠬⢭⡵⠊⢀⣠⠞⠉⠀⠀⠀⢀⣸⡶⢿
⠀⠀⢸⡴⠋⠀⠀⡇⠀⢸⠁⠀⠉⠉⠙⠛⠛⠓⠶⢾⠧⣤⣤⣏⣀⣀⡴⠋⢀⡴⠋⠈⢉⣑⣒⣤⡴⠋⠁⠑⣾
⠀⠀⢸⡇⠀⠀⠀⠁⠀⢸⠀⣴⠒⠋⢻⠀⠀⠀⠀⠘⣆⠀⢹⡀⠈⢹⠉⠛⡿⠶⡿⠿⣭⡽⢻⠃⠁⠀⠀⠀⣾
⠀⠀⠘⣧⠤⡄⠀⠀⠀⢸⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠸⠤⠶⠃⠀⢸⠀⠀⡏⠀⠱⡀⠀⣇⡞⠀⢸⠀⠀⠀⣿
⠀⠀⠀⡇⠀⣧⠀⢸⠀⢸⡇⢸⡇⢀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠙⢦⣿⠃⠀⠸⠀⠀⠀⣿
⠀⠀⠀⡇⠀⢸⠀⢸⠀⠀⡇⠀⠛⠉⠁⠀⠀⠀⠀⣠⠶⠒⠦⣀⠀⣸⠀⠀⡇⣤⠖⢲⠀⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣿⠤⠼⠀⢸⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢰⠁⠄⠀⢦⠈⣇⣿⠀⠀⢷⢹⠀⢸⡄⣿⠀⠀⢠⠀⠀⠀⢸⡀
⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⣿⠉⠉⠉⠉⠉⠉⡇⠘⣄⠢⠤⠜⢀⡏⣿⠀⠀⢸⠈⠷⠴⠇⣿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⢸⡀⠀⠀⠀⡇⠀⢹⠒⠒⠒⠒⠒⠚⠁⠀⠈⠓⠒⠚⠋⠀⣿⠀⠀⢸⠑⠲⡄⠀⣿⠀⠀⢸⠀⠀⠀⢸⡇
⠀⠀⠀⠈⣧⡀⠀⠀⣧⠀⢸⠀⠀⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢻⡄⢸⠀⢻⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⣧⠈⢢⡀⢻⠀⢸⡄⠀⠀⡏⠉⠁⠀⢸⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⣀⡤⠋⠀⢸⠀⠀⠀⡇⠀⠀⢠⡇
⠀⠀⠀⠀⠙⠓⠲⠷⢼⣤⣤⣇⣀⠀⠻⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⡄⠀⠀⡇⠀⢀⡞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠲⠶⠤⣤⣀⡀⠀⠀⠀⣾⠀⠀⢸⠀⠀⠀⢀⣼⣇⠀⠀⡇⢀⣾⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠦⣿⣀⡀⢸⡆⢀⡴⠋⢸⡇⢳⣤⠷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠶⠦⠼⠗⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⢉⡁⣀⣀⣀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⢀⢠⡤⢀⠀⣠⡤⢄⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⢙⡣⠤⠭⠥⠠⠀⠐⠉⢀⣀⠚⠂⠐⠀⢁⣄⠀⢀⠠⠀⣤⠰⡄⢀⣶⣼⣯⠙⡆⣄⠞⠉⠢⣷⠸⡼⠓⢺⡇⠏⠀⠈⡇⠛
⠀⠀⠀⠀⠀⠀⠀⠌⡴⠉⠀⠀⢸⠀⡇⠀⡐⠖⢐⠀⠾⡦⣀⠐⡒⣤⢿⡀⢰⠊⠷⠋⠓⣿⠘⡁⠀⠀⢰⡇⢸⢀⠀⠀⣹⠀⡇⠀⠠⡇⢘⠠⠀⠇⠐
⠀⠀⠀⠀⠀⣢⠀⢸⠁⠀⠀⢀⠼⠀⠗⠖⠀⠀⠀⠙⢀⣿⢁⢚⢢⠁⢠⠁⢸⠀⠈⠀⠀⣼⠀⡆⠀⠀⢈⣿⠀⡀⡀⡀⠼⠀⠱⠤⠒⠃⠈⠒⠚⠁
⠀⠀⠀⠀⠀⠏⠂⠸⢄⢀⣀⠀⠂⠁⠈⠀⠀⠀⠀⠀⢠⢬⠈⠈⠸⠀⡸⡄⠘⣄⣀⡀⠠⠘⠀⠑⠒⢃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠈⠒⠆⠠⠀⣄⠀⠀⢀⠀⡀⣠⠀⢀⠀⠀⡨⠸⣄⡁⠀⠒⡒⠣⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠒⠤⠤⠤⠴⠦⠔⠂⣀⣀⣀⣀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠲⣄⠐⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡄⢤⡀⠀⠀⠀⠀⢀⡀⢀⡤⠴⠖⠒⠒⠲⠏⠁⠐⡀⢠⣾⡷⣟⡷⣿⡀⣐⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠤⣤⢀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡞⠘⢳⢡⠀⠀⠀⠨⠀⠀⠈⠁⠀⠐⠀⠀⠠⠴⠲⠺⠧⠞⠯⠷⣬⣿⣿⢿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠘⠤⠤⠸⠀⠠⠄⠀⠀⠀⠀⣄⡁⢈⠀⠀⡐⠀⢰⠅⠘⠈⢠⢐⡀⡀⡈⠐⠶⠲⠛⠛⠋⠛⠛⠋⠛⠑⠑⠊⠒⠐⠋⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠶⠻⠿⣀⣀⣀⣀⣀⣀⣀⢰⣯⠤⠠⠄⠠⠒⠧⣅⣘⡤⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈

⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣴⡆
⠀⠀⠀⢸⣿⣿⣶⣤⣀⠈⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣄⡀⠉⠛⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⠉
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠉⠻⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠇⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠈⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣴⠟⠉⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣿⣷⣦⣀⣀⣉⣛⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠋⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⠉⣿⠉⠛⠿⣿⣿⣯⡳⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⢹⣿⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿⠀⠀⠀⠀⠙⢿⣿⢦⣉⣷⣤⠀⠀⠀⠀⠐⣿⣄⡇⣘⣿⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠻⠀⠀⠀⠀⠀⠀⠆⠀⠀⠙⠙⠀⠀⠀⠀⢀⣃⣸⣇⣿⣿⣐⡂⣀⠀⡄⣸⣧⣸⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠸⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠐⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⠛⠛⠛⠛⠻⠷⠶⠶⢦⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⠈⢸⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠶⠶⠶⠦⠬⠭⠭⠭⠙⠛⣓⣒⣶⣾⠤⢼⣤⣀⣀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢠⠘⡅⠀⠀⠈⠁⠒⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⢤⣤⣤⣤⣶⣖⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠖⠛⠉⠉⣀⣴⡾⢛⣿⣿⡟⠀⢹⡷⣄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢤⠄⣀⠀⠀⠀⠀⠀⢀⣀⠤⡤⣀⡀
⠀⠀⠀⠀⠀⢀⣤⣾⣿⠋⣰⣿⣿⣿⠃⠀⠀⣿⠸⣆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡴⠚⠁⠀⠀⠀⠈⠓⢦⣤⣠⠖⠋⠀⠀⠂⠀⠙⠲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡇⢡⠉⢠⠞⠇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⡇⠀⡠⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⢧⡴⠿⠲⠒⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠃⠀⠀⠀⢠⡎⠘⠀⠉⠈⠇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⡔⠃⡇⢸⠀⠀⢀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⢀⡄⠃⠀⠀⠀⠀⡠⠊⠀⠀⡇⠐⠿⠛⠂⡇⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡇⢸⠉⠉⠉⡇⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠁⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡅⠸⡄⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠘⣤⣤⠄⡇⠀⠱⡀⠀⠈⡄
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠡⠤⠤⡇⠀⠀⢱⡀⠀⠈⡄⠀⠀⠀⠀⢰⠂⢲
⠀⠀⡠⠒⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⢠⠀⠀⠀⡇⠀⠀⠀⠡⡀⠀⠐⡀⠀⠀⠀⠀⡖⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠨⠀⠀⠀⠁⠐⠂⠀⠂⠓⣄⠀⠑⡀⠀⡀⠀⡶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡉⡉⠉⠉⠉⠍⠉⠁⢡⡴⠒⠒⠒⠤⠄⢦⠀⠈⣄⢀⠐⠃⠰⠀⠀⠐⠢⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢤⣤⠶⣶⡶⢲⠀⠀⠀⠀⠀⠀⠒⠒⠒⠲⠤⢬⣄⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡋⠳⣼⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢐⠹⢏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⢿⠀⠀⠠⠀⠀⠀⠈⡀⠀⠀⠀⢀⣐⣃⡨⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠄⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠸⠀⠀⢀⡀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣼⡇⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣴⠋⠀⠀⠀⠈⢳⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣄⠀⠀⠀⢸⡇⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⢀⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⣸⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⣰⠇⠀⠀⠀⣿⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠙⠓⠶⠒⠛⠁⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⢸⠋⠛⠛⠛⠛⠛⠛⢻⡇⠀⠀⠀⢀⣤⠶⠒⠲⢦⣄⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠋⠛⢻
⠀⠀⠀⠀⠀⠀⠘⠷⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡞⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⣿⠀⠀⠀⠀⠀⣠⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⢸⡇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣿⠀⢀⣠⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⡇⠀⠀⠀⠹⢦⣀⡀⣀⣠⠞⠁⠀⠀⠀⣿⠞⠋⠁
⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⣼⡇⠀⠀⠀⠀⠀⢉⣉⡉⠁⠀⠀⠀⠀⠀⣿⡤⠤⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠞⠋⠉⠉⠛⢦⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⢸⡇⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣿⠀⠀⠀⣀⡤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⣸⡇⠀⠀⠈⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣿⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠈⠳⢦⣤⣤⠶⠋⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠚⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠢⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣉⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠔⠃⠀⠀⠠⢤⣌⣻⠽⡯⠟⠋⠉⠉⠉⠉⠉⢉⣉⣤⠾⠒⠁
⠀⣷⡀⠀⠀⠀⠐⡩⠒⠻⢶⣤⣀⠀⣀⣤⣴⠾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠
⠀⠛⠬⠀⠀⠀⠈⠤⣄⣀⣠⣴⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠀⠀⣠⠆
⠀⡇⢰⠀⢀⡀⠄⠐⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠒⠈⠀⠤⠀⠒⢠⣿⣭⡇⢸⠉⠃⢀⠃
⠀⡇⢸⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠁⠀⢸⠈⠉⡇⠸⠀⢄⠀⢹
⠀⡇⢸⢰⡇⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠤⠄⠈⠀⠀⠀⠀⠐⠪⡀⢸
⠀⡇⣸⢸⠠⠒⠀⠀⠀⣀⢀⠀⠀⢀⠠⠀⡀⠀⣨⢇⠀⠀⠀⠀⢀⡀⠀⣠⣀⣰⣄⠀⠃⠀⢰⠀⡃⠀⡮⢹
⠀⡇⣸⠸⠏⠤⠶⠶⣐⠀⢸⠠⠈⠈⡁⡀⠀⠰⡆⠘⡇⠀⠀⠀⠀⠄⢸⣿⣿⢿⣿⠀⡇⠀⠸⠠⣇⣀⡑⠸
⠀⡇⠸⠀⡠⠠⠤⠀⠤⠀⢼⡆⠠⡄⠂⠙⠂⡄⢻⠀⠰⠀⠀⠀⠀⠠⠀⡼⠏⢸⣿⠇⠃⠀⠠⢄⠀⠁⠧⠀⠂⢀⡀
⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⣤⡁⠀⠒⠂⠈⠀⠀⠀⠀⠀⠘⠑⡆⣾⣿⡆⠀⠀⠀⠠⠀⢤⡀⠂⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠈⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠉⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⢸⣿⣷⣄⡀⣀⣤⣤⣼⣿⠀⠂⢠⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣤⠄⢈⣹⣿⣿⣿⣿⣿⣿⣿⡿⣶⣿⣿⣿⣿⣿⣶⡄⢀⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣼⣿⠇⠀⠈⠈⣿⡟⣿⣿⣿⣿⣿⡇⠀⠙⣩⣈⠙⠛⠻⣿⣾⣿⣿⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⠿⠿⢷⡄⠀⠉⠀⠘⢿⣿⣿⣿⣿⡀⢌⡛⣋⠸⣿⠆⡃⢻⡿⢻⣌⣡⡉
⠀⠀⠀⠀⠀⠀⣤⣶⣾⣿⣿⣷⣋⣤⡀⢴⡶⢿⣾⣿⣿⠓⠈⠀⠀⠉⠉⠃⠸⢿⣯⣷⣶⣾⣿⣿⣷⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⢿⣋⣠⣤⣾⣿⠉⢉⡴⢦⣠⠜⠛⠻⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠟⣀⠉⡛⢿⣴⣶⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠈⠿⢟⣿⡍⢸⣠⣿⣿⣿⠃⠀⠀⠀⠀⠀⠘⠀⠀⢂⣴⣶⣿⣷⣾⣿⣷⠀⠀⠀⢹⣿⡏⠁⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣻⡭⣵⣿⡿⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠛⠛⠙⣿⡿⠙⢀⡀⠀⣿⡏⠑⠂⠀⠒⠁
⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣍⠀⠀⠈⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠠⠄⣹⣷⣿⣿⣿⣿⡿⠗
⠀⠀⠀⠀⠀⠘⠟⠉⠙⢿⣿⡟⠀⢀⣀⣼⣿⣧⠀⠀⣾⣿⠿⢷⣀⡀⠉⠀⠀⠀⠈⡟⣀⠈⢉⢸⡿⣦⣀⣀⡤
⠀⠀⠀⠀⠀⠀⢠⠀⠉⠀⠻⠀⢰⣿⣿⣿⠋⠉⡲⠼⡏⠀⠀⠀⢹⣿⣶⣿⣿⣿⣷⣧⣠⢦⣈⡼⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠀⠂⠀⠀⠐⢾⣿⣿⡟⠐⠀⣙⣛⣉⣁⣀⠉⣈⣿⠀⠀⢹⣿⣿⣿⠟⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⣿⡏⠀⢠⣭⣭⣽⣿⣿⣿⣻⣿⣧⡀⠀⢻⡟⠉⠀⠀⠈⠁⠀⠀⠀⠉⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣤⣴⡶⠆⠒⠂⣤⠄⠛⠛⠿⣿⣇⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⣽⡀⢠⣶⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠐⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⠤⠤⠤⢀⡈⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠤⣤⢤⣤⣠⣀⣀⣀⣀⢀⠂⣐⣁⣀⠀⠀⠀⠈⡄⠰
⠀⠀⡠⠒⠚⠛⠙⠛⠒⠚⠛⠻⠯⠿⠲⠶⠤⠤⠤⠤⣛⣲⣿⣃⠉⡉⣉⢉⣉⣷⣼⡷⠓⠒⠒⠒⠶⠴⠀⠒⠖⡦⠤⠤⠤⠤⢤⣤⣀⠀⠀⡀
⠀⠘⠐⠈⠉⠉⠁⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠤⠤⠤⠤⠀⠀⣀⢀⣉⣉⣁⠉⠈⠁⠉⠉⠃⠓⠒⠚⠛⠿⠿⠻⠘⢛⠀⢤⣀⡠⠤⠔⠈⠀⢮⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠂⠐⠒⠒⠀⢄⠀⠈⢢⠈⠆⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⠀⠀⡇
⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃⢸⠀⠀⠀⠀⠀⡇
⠀⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢺⠀⠀⠀⠀⠀⠠
⠀⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀⠀⠴⠀⠀⠀⠀⠀⣷
⠀⠇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⡄⠀⠀⠀⠀⢀⢸
⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⡇⠀⠀⠀⠀⢸⢸
⠀⢇⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⢸⠸
⠀⠀⠁⠐⠨⠭⠍⠅⣐⣒⠂⠤⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠀⠀⠀⣸⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠒⠒⠬⠭⠄⣁⢐⣒⠀⠤⠤⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠸⠀⡇⢀⣠⠴⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠒⠂⠨⠭⠤⢀⣀⣒⠀⠠⠤⠀⠀⠴⠀⢀⣄⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠐⠒⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠴⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠛⠉⠉⠈⡇⢺
⠀⣤⠴⠖⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠖⣶⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⢺⠋⠉⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡄
⠀⠀⣀⣠⡤⠴⠒⠚⠉⠉⢀⡠⠴⠒⠚⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⠊⠈⢉⡹⠻⣾⡇
⠈⠉⠁⠀⠀⠀⠀⠀⠀⡾⠋⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡒⣒⡒⢒⡒⠲⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏⢉⡉⠗⠘⡍⠙⡄⠋⡹⠁⡏⠉⡇⠧⢤⠀⣠⠼⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡇⠘⢒⡆⠀⢱⠀⠁⣰⠁⠀⡇⠀⡇⠀⢸⠀⣿⠀⠀⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡇⠠⠤⣤⢀⠏⢠⡄⠸⡄⠀⡇⠀⡇⠀⢸⡀⣿⠀⠀⠃⢀⣴⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣓⣒⣒⣋⣘⣒⣃⣘⣒⣋⣀⣩⣍⣥⣤⡤⣥⡤⠤⠤⠴⠋⠀⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⢸⡇⠀⠀⠀⠀⠀⣷⠀⠇⠐⠒⠋⠀⠀⠀⠀⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣤⠶⠛⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠓⠲⠦⣤⣀⡀⠀⠀⠀⠀⠸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⢤⣠⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣼⠉⠉⠉⠉⢿⡃⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⠀⢈⡱⠄⠀⣀⣤⠖⢋⣥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠸⠗⠊⠙⠲⠴⠚⢉⣠⡴⠚⣩⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡤⠖⢋⡥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⠴⠚⣡⡴⠚⠉
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⣋⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⢀⣤⠶⠋⠁
⠀⢀⣠⡴⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠸⠭⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⢠⠃⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢔⠂⠁⠉⠒⠢⠄⢀⣈⣀⣀⠦⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⢀⣴⣆⢠⣤⠤⠤⠤⠤⠤⠄⣀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣄⠀⣤⠈⡛⢃⠀⠀⠀⠀⠀⠰⡀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⡄⠀⡿⠆⠉⠱⠂⠀⠀⠀⠀⠀⢱⠀⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⣦⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⢤⠤⢀⠖⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⡇⠀⢈⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡤⠘⠇⡀⣀⢠⠄⢉⠆⠀⠀⡀⠔⠊⢠⠀⢄⣉⠠
⠀⠄⠐⠀⠀⠀⠀⠐⠢⠎⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢦⡤⢠⡀⠂⠈⠉⠁⠖⠁
⡈⠀⠀⠀⠸⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠐⠂⠠⠀⡔⠀⠘⡄⢰⠀⠀⡀⠐⡀
⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠘⠠⠐⠿⣀⠀⠀⠀⡉⠀⠀⠀⠀⠐⠀⠀⠀⢰⠈⠖⡴⠁⠤⠃
⡇⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⣤⠃⠀⠠⠍⢀⡀⠐⠓⢁⡰⠂⠠⠄⠠⠀⠄⢸⠀⡇⡁⠁⠐
⡇⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⠶⠆⣀⠠⠖⠠⡼⠁⠀⠀⠀⠑⠠⠉⠓⠆⣀⡆⠈⠀⠃⡇
⠃⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⢃⠘⣤⠆⠁⠀⠀⠀⣠⠊⠀⣇⠀⠀⠀⠈⠙⢀⠀⢃⡁⠀⢀⠄
⠸⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢇⠀⢀⠀⠀⠸⡐⢇⠀⠀⠀⠀⠀⢏⠀⠀⠉⠤⠤⠲⡁⠀⡈⠀⠀⠉⠀⡇
⠀⢣⡀⠀⠀⢹⠄⠀⠚⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠑⠤⠄⠠⠤⠚⠁⠨⠖⢂⡀⠀⡠⠀⠀⠀⠀⠀⠠⡡⡜⠒⠀⠀⠀⢀⣠
⠀⠀⡙⠀⡄⠁⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠶⠤⣤⣄⣀⣵⣌⠗⢂⠀⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠠⡤⠂⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠀⠠⡅⠂⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢳⡀⠀⠀⢀⡠⠸⡇⠀⠀⠀⠷⠐⠒⢻⣄⢀⣤⣤⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠈⠘⠋⠱⠤⠒⠉⠀⠀⠇⠀⠀⠀⡄⣀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠐⠠⠆⠀⠀⠰⠄⠀⠀⠀⠀⠇⠀⣄⠀⠀⠠⡄⢀⡂⢀⠐⣀⠐⠆⠀⠀⠀⠀⠀⠈⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠁⠈⠀⠉⠀⠀⠀⠃⢻⠀⠀⠀⡒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠄⠘⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠒⠒⠒⠈⠀⠀⠀⠀⠀⣣⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠂⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⣀⣀⡀⠠⠰⠈⠓⡀⠖⠉⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠑⠀⠂⠑⠊⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠤⠀⣀⣤⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⡠⠔⢤⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠠⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⢀⢀⠀⠀⠀⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠏⢨⠷⠹⠹⠸⠾⡥⠀⢍⡶⢹⠸⡹⢸⡹⠀⠶⢾⡅⢏⠗⠏⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠾⠿⠿⢷⣦⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠛⠛⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⢀⣼⡿⣷⡄⠀⠀⠀⢀⣴⠟⠁⠀⠀⡄⣶⣀⠀⠙⢷⣄⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢀⣴⡿⠋⠀⠘⣿⠀⠀⠀⢸⡇⠀⠀⠀⡇⡇⣿⣿⠀⠀⠀⣿⠀⠀⠀⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣧
⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⣴⡿⠋⠀⠀⠀⢀⣿⠃⠀⠀⢸⡇⠀⠀⠀⣷⣿⣿⣿⣴⠆⠀⣿⠀⠀⠀⢿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢿⣆⣴⡿⠋⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠘⣷⡀⠀⠀⢿⣿⣿⣿⠋⠀⣠⡿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣦⣄⣀⣀⣠⣴⡿⠋⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⣉⣉⡀⣠⡾⠋⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢻⣿⣿⣿⣦⠀⠀⠀⠀⠀⢠⡾⠋⠁⠀⢀⡄⠀⢙⣷⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⡿⠹⢸⢸⢸⣿⣿⣿⣧⠀⠀⠀⢰⡏⠀⠀⢰⣾⣿⣿⡇⠋⠀⠹⣆
⠀⠀⠀⠀⠀⢸⣿⣧⡒⢿⡆⣞⢰⡎⡇⣒⣽⣿⠀⠀⠀⢸⣿⣿⣿⣧⠘⠘⠘⠸⠟⣻⣿⣿⠀⠀⠀⢾⠀⠀⠀⢸⣿⣿⣯⣇⣤⠄⠀⣿
⠀⠀⠀⠀⠀⢸⣿⣷⣶⣿⣷⣿⣶⣾⣷⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⣰⣿⣿⡿⠀⠀⠀⢸⡆⠀⢀⣸⣿⣿⣿⣿⠃⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠘⢿⣿⣿⣦⣤⣤⣾⣿⣿⡿⠁⠀⠀⠀⠀⠹⣶⡛⠀⠙⠛⠛⠁⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠶⠶⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠀⠀⠀⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢲⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡶⠾⣿⠙⠻⠇⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠈⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣀⠤⠒⠊⠉⠁⠀⠀⠉⠉⠐⠢⢄⡀⠀⣰⠿⠐⠂⠈⠉⠉⠁⠐⠒⠬⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠗⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⠛⢦⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⣠⡤⠞⠋⠙⠳⣄⡀⠀⠀⠀⠀⠀⢠⣤⠴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⣀⣀⣀⡈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⠀⠀⠙⠳⡄⠀⠀⠀⠀⠀⢀⠖⠉⣁⡀⠈⠙⠲⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣴⠛⣿⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣴⣿⣿⣄⠀⠀⢹⠀⠀⠀⠀⢠⡏⢀⣾⣿⣧⣀⠀⠀⢹⡄⠀⠀⠀⠀⠀⣿⢳⡄
⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠻⣿⣿⠿⠀⠀⢸⠃⠀⠀⠀⠸⡄⠈⠿⣿⡿⠏⠀⠀⢠⠇⠀⠀⠀⠀⠀⣿⠀⡷
⠀⠀⠀⠀⢀⡶⠛⠉⠳⣼⡶⠖⠒⢦⡀⠀⠀⠘⠒⠤⢤⣀⣀⡀⠀⣀⠚⠀⠀⠀⠀⠀⠳⣄⠀⣀⣀⣀⡀⠠⠎⠀⠀⠀⠀⠀⠀⣋⡼⠃
⣠⠞⠋⠉⢿⠁⠀⠀⠀⣿⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⡿⠀⠀⠀⢸⡀⠀⠀⠀⣿⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢻⠀⠀⠀⠀⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠆⠀⠀⠀⣰⠏
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⣿⠁⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠞⠋⠉⠁⠙⠃⠀⢀⡼⠋
⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡄⠀⠀⠀⢸⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁
⣿⠀⠀⠀⠘⠃⠀⠀⠀⠈⠀⠀⠀⠀⢸⡀⠀⢸⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠸⡇⠀⠸⡏⠙⢳⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠾⠿⣥⣀
⢸⠀⠀⠈⠉⠉⠉⠁⠀⠀⣼⠋⠁⠀⠀⠀⠀⠀⣿⠛⠉⠁⢀⡏⠉⠉⠛⠛⢛⡿⣟⠉⠉⠁⣷⠀⠀⠀⠀⠙⠳⣄
⢸⡄⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⡟⠢⡀⢀⠴⢿⣿⣯⠓⢄⡸⢹⠶⠶⣄⠀⠀⠀⠘⣧
⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠶⠛⠉⡇⠀⠘⠁⠀⢸⣿⣿⠀⠀⠀⠸⡇⠀⠈⢷⡀⠀⠀⠸⡇
⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢐⠋⠁⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠙⠂⠀⠀⠀⣇⠀⠀⣘⠃⠀⠀⢰⡇
⠀⢿⡀⠀⠀⠀⠀⠀⢀⣸⣇⣀⣀⣀⣠⣤⡴⠟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⢸⡴⠞⠁⠀⠀⣠⡟
⠀⠀⠉⠙⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠘⡇⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠿⣿⣿⠿⠁⠀⠀⠀⣿⡀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⡶⠶⠶⠶⠶⣄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢠⡇⠀⠀⠀⠀⠹⣆⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣫⣄⣀⣠⣼⣧⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣄⣌⣷⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠺⠯⠭⣥⣀⣀⡀⠀⠤⠄⠒⠂⠈⠉⠉⣹⢄⡉⠉⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⢩⣝⣧⡤⢀⣀⡀⠀⢸⠀⠀⣀⣠⣤⣤⠶⠾⠝⠃⠀⠁⠀⠀⠀⣀⣀⣉⣑⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠒⣒⡻⡥⠤⣄⡀⠀⠀⠀⠈⠉⠙⠒⠪⠭⠭⠭⠖⠊⠉⠁⣀⣀⡤⠤⠖⠒⠛⠋⠉⠉⠉⠉⠉⠁⠈⠉⢣
⠀⠀⠀⠀⠀⢀⠔⠋⣇⠀⠀⠀⠀⠉⠉⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⠉⠁⠀⠀⢀⡀⢤⡀⢀⣀⣠⠤⠤⡄⡔⠒⢲⠸⡄
⠀⠀⠀⠀⢰⠋⠀⠀⠈⠉⠓⠒⠶⠤⢄⣠⡿⠀⠀⠀⠀⢀⡤⠖⠛⠉⠀⠀⠀⠀⣀⢄⢀⣠⣤⣓⣶⣿⣿⠉⢩⣿⠀⠀⡇⡇⠀⢸⠀⡇
⠀⠀⠀⠀⣼⠓⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⣀⡀⢯⣶⡿⠛⣿⠉⠀⢸⣿⡿⠀⢸⣿⣀⣠⡇⠧⠴⠞⢰⡇
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠉⠉⠓⠲⠤⣤⣀⡀⠀⠀⡇⢀⣀⡀⡴⠚⠉⠀⡇⢘⣿⠀⠀⢹⡇⢀⣸⣿⠘⠒⠋⠀⠉⠀⢀⢀⣀⠤⢼⡇
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⡟⠉⠀⡇⢇⠀⠀⠀⢸⠸⣿⣄⢤⣺⣉⡉⠀⣿⣀⣀⠤⠒⢒⣉⡩⠤⠖⠒⢸⠁
⠀⠀⠀⠀⠘⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⢸⡀⣀⠠⠚⠀⣿⠀⢸⢹⣿⡇⠀⣿⡥⠔⡶⠛⠳⣄⣀⠤⠔⠒⣺
⠀⠀⠀⠀⠀⢧⠈⠉⠓⠲⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠉⠁⠀⢀⡠⣿⠀⠈⠋⡇⠀⠀⣿⣀⣼⠠⡞⣦⢸⣤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠸⡭⢷⣄⠀⠀⠀⠈⠉⠛⠒⠦⢤⣄⡀⢸⡄⣀⡸⢆⣠⠔⢊⡡⠔⣿⡇⠀⠀⢠⠀⠀⣻⣥⣼⣄⣑⣋⠾
⠀⠀⠀⠀⠀⠀⣇⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣹⣏⡥⠒⣉⢴⡮⠷⢤⡀⣿⡇⠀⣀⡴⠖⠋⠁⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢹⠟⠥⣀⡴⠤⣀⡀⠀⠀⠀⠀⡞⡩⠻⡇⣿⠊⢠⠋⢠⠶⡄⢹⡽⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠓⠤⢄⣿⣷⡆⢨⣹⠲⣄⠈⣟⢇⣀⡇⣿⢀⠼⡄⠸⣤⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠬⣏⣘⠂⣇⠈⠉⠛⠓⣿⠋⢁⡠⠷⠤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠑⠒⠦⠿⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠀⠀⠀⠀⠀⠈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠶⠾⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⡇⢠⡶⠶⠶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠶⠶⠶⡆⢸⡟⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠷⠾⠿⠿⠿⠿⠿⠿⠷⠿⡇⢸⡇⢸⡇⠀⠀⣠⡶⠟⠛⠛⠛⠛⢷⣦⡀⠀⠀⡇⢸⡇⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡄⠀⠀⠀⠀⡇⢸⡇⢸⡇⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⡇⢸⡇⣿⠀⠀⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⡇⢸⡇⢸⡇⠀⠙⢷⣄⡀⠀⠀⠀⢀⣠⡴⠟⠀⠀⡇⢸⡇⣿⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠻⣇⣸⡇⢸⣇⣀⣀⣀⣉⣙⣛⣛⣛⣛⣉⣀⣀⣀⣀⡇⢸⣇⣿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣽⡇⢠⡭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⡅⢸⣯⣭
⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡇⢸⡇⢸⡇⠀⠀⣀⣴⡶⠾⠿⠿⠶⣦⣄⠀⠀⠀⡇⢸⡇⣿⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⡄
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⡇⢸⡇⣿⠀⠀⠀⠀⣠⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⡇⢸⡇⢸⡇⠀⠹⣦⣀⠀⠀⠀⠀⠀⢀⣠⡾⠁⠀⡇⢸⡇⣿⠀⠀⠀⢰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠶⢾⡇⢸⡇⢸⡇⠀⠀⠈⠙⠻⠶⠶⠶⠞⠛⠁⠀⠀⠀⡇⢸⡇⣿⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⡇⢈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⢸⡟⠛
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡟⢻⡇⢸⡇⠀⠀⠀⣀⣤⣴⣶⣦⣤⣄⡀⠀⠀⠀⡇⢸⡟⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡇⢸⡇⢸⡇⠀⣠⡾⠋⠁⠀⠀⠀⠀⠉⠻⣦⡀⠀⡇⢸⡇⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⡇⢸⡇⢸⡇⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡇⢸⡇⣿⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⡇⢸⡇⢸⡇⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⡇⢸⡇⣿⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣀⣀⣠⡇⢸⡇⢸⡇⠀⠀⠙⠻⢶⣤⣤⣤⣴⠾⠛⠁⠀⠀⡇⢸⡇⣿⣄⣀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣧⣼⡇⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠇⢸⣧⣿⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿⠃⠀⠀⠀⠀⠀⣤⣤⣤⡄⣤⢠⠤⣤⠄⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⣤⡤⠤⠤⠶⢶⣦⣤⣶⣤⣤⣤⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣶⣯⣽⣽⣃⣀⠀⢀⣀⣉⠶⠖⠒⠈⠷⠿⠾⢭⣤⣞⣻⠿⢷⣶⣦⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⢿⠟⠋⣸⡍⠉⢻⣿⣿⠷⠶⢦⢤⡤⢤⣀⣀⠀⠀⠈⠉⠙⠒⠢⢍⣟⣻⣿⠦⢀⡀⠐⠀⠲⠄⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠎⠯⠀⢸⢯⠀⢠⣴⣖⣶⣿⣿⠋⠙⠉⠺⠶⢦⣀⢀⣀⠄⢘⠩⠐⠒⠛⠀⠠⢄⡀⠀⠀⡀⢭⣽⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢹⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⡾⣁⣀⢴⣮⣭⣤⣭⣶⣾⢿⣵⡚⠙⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠄⡐⡬⢖⣦⡀⣼⣬⡀⠈⠁⠀⠀⠀⠀⠀⠀⣴⣿⢯⠡⠀⠸⠿⢧⣿⣿⠈⢿⣿⣿⣿⣿⠃⢸⡿⠃⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣄⢛⣩⣥⠄⠿⢿⠇⠀⠀⠀⠀⠀⢠⡄⠀⠰⣼⣧⠜⠀⢸⠀⠄⠈⠃⠀⠀⢹⢻⣿⡿⠀⢸⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢨⠈⠘⠛⠙⠀⢠⣤⠂⠀⠀⠀⠀⠀⢳⣬⡀⠀⠀⠀⠀⠀⣻⣿⡇⠀⠀⠀⠀⢼⢸⣿⠁⠀⠸⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠸⡟⡳⡄⠀⠀⠀⠀⢈⠙⡃⠀⡶⠟⠀⠘⢸⣷⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢃⠁⠀⢀⣴⡇⠀⣬⠀⠀⠀⠀⠀⠀⠀⢹⡯⠖⠀⠀⠀⠀⢸⣾⠃⠠⢿⢟⠠⠀⢸⡏⠁⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⢐⠞⡿⣿⡇⠀⣟⡇⢠⣶⣄⠀⠀⠀⣸⣔⠑⠾⠆⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⢸⡧⠄⠀⠀⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠾⠿⠙⠗⠀⣏⡷⣿⡬⢉⢿⣦⡀⠉⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⣿⠄⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⡀⠀⠀⠀⣧⡇⠈⠑⠿⣷⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣵⣀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢲⠦⡄⠸⠀⠀⢾⢿⣦⣄⠰⢸⣏⡀⠀⠀⣿⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⣄⣀⠀⠀⠀⠀⠀⣆⣻⡄⣼⠀⢠⠀⠀⡴⣿⣄⣠⠞⠀⠀⣿⣗⠂⠀⣿⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠤⣤⢀⣀⠀⠈⠉⠉⠉⠛⠀⡄⡇⠀⠈⠻⠀⠀⠀⣯⣅⠀⠀⡏⠀⠀⠀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠒⠆⠤⣄⡀⢀⣄⡇⠀⠀⠀⠀⡠⣶⣿⣬⡤⢠⡏⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠢⢤⣠⣎⣀⣿⣿⣿⣷⣼⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄
⠀⢰⡖⣶⢴⢲⢀⣠⣤⡀⠀⠀⠀⠀⢸⢹⡛⡏⡏⣹⡆⠀⠀⠀⠀⢰⣿⢻⡯⡟⡟⡇⠀⠀⠀⠀⠀⣿⢸⠀⠇⡆⣾
⠀⠘⠓⠛⠛⠘⠀⠉⠉⠀⠀⠀⠀⠀⠸⠬⠴⠤⠷⠿⠀⠀⠀⠀⠀⠘⠿⠾⠿⠧⠧⠇⠀⠀⠀⠀⠀⣿⣿⣛⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁


⠀⠀⠠⢶⣶⣶⣶⣶⠆⠀⠀⠀⠀⢠⠶⠶⠶⠶⣆⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⠶⢶⡄⠀⠀⠀⠀⠀⣴⢶⠶⣶⣶⣦
⠀⠀⢸⠺⠼⠷⠇⠇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⣥⣶⡤⠀⠀⠀⠀⠀⣭⣴⣦⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⣿⣟⠋⣻
⠀⠀⠸⣀⣐⠲⡆⣀⡄⠀⠀⠀⠀⢸⣥⣤⣤⣤⡷⠀⠀⠀⠀⠀⠀⠀⢶⣤⣤⣤⣼⡇⠀⠀⠀⠀⠀⢿⣿⣫⣿⣿⣿


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡄
⠀⠀⡟⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⣀⠸⢀⣶⣿⡇⠀⠀⠀⠀⠀⣀⡸⠃⣴⣾⡇⠀⠀⠀⠀⠀⣾⣿⣿⣿⣟⣻
⠀⠀⡗⠿⠿⠏⢒⠺⠗⠀⠀⠀⠀⠀⠘⠛⢛⠀⣿⣿⡇⠀⠀⠀⠀⠈⠙⢫⡀⣿⣿⡇⠀⠀⠀⠀⠀⣿⣅⣹⣿⣿⣿
⠀⠀⠷⠶⠶⠶⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⠿⠃⠀⠀⠀⠀⠀⠀⠘⠁⣿⠿⠃⠀⠀⠀⠀⠀⠻⠿⠿⠿⠷⠾


⠀⠀⠀⠀⠀⡤⠤⠤⣤⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣤⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⣀⣰⠆⢰⣿⣿⠀⠀⠀⠀⠀⢸⡿⠟⣿⣿⣿⡆⠀⠀⠀⠀⠀⣳⡬⡇⢰⣿⣿⠀⠀⠀⠀⠀⣿⣿⡿⠿⣿⣿
⠀⠀⠀⣱⢟⠓⢸⣿⣿⠀⠀⠀⠀⠀⢸⣟⣔⣿⣧⣼⡇⠀⠀⠀⠀⠈⣹⢷⣀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣧⣄⣠⣼⣿
⠀⠀⠈⠁⠈⠁⢸⠿⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⡠⢔⣢⣭⣷⣶⣶⣮⣬⣔⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⡠⣪⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣕⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢌⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢢⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⡌⣾⣿⠿⢿⣿⠿⠿⠿⢿⣿⠿⠿⣿⣿⠿⠿⢿⣇⠆⠀⠀⠀⢸⣿⡿⠿⢿⡿⠿⠿⠿⣿⡿⠿⢿⣿⡿⠿⠿⣿⡇
⠀⠀⢠⣿⣇⠻⠷⣿⣿⡇⣿⣿⢱⣿⣷⡜⡇⠾⠿⢆⣿⢰⠀⠀⠀⢸⣿⡘⠿⢶⣿⣿⢸⣿⡏⣾⣿⣦⢹⡇⠿⠷⣸⡇
⠀⠀⠈⣿⣝⠿⠗⣸⣿⡇⣿⣿⣜⠿⠟⣱⡇⣺⣿⣿⣿⠘⠀⠀⠀⢸⣯⡻⠿⢂⣿⣿⢸⣿⣧⡻⠿⢋⣼⣇⣿⣿⣿⡇
⠀⠀⠡⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⠇⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠡⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⠊⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠪⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠑⠊⠭⠟⠛⠛⠫⠩⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠀⠀⠉⠉⠉⠉⣭⠺⣶⣶⡤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⠀⣴⠦⠀⠠⡄⣵⠞⠛⠭⣵⣶⣦⣿⣦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⣀⠀⠀⠀⠀⠐⣿⣶⣴⠰⠂⢂⣬⣣⡐⣄⠀⠀⠉⠛⠿⣿⣷⡌⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠯⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢡⣴⣿⣿⣿⣶⣿⣿⣶⣭⣉⣀⣰⣇⣉⠉⢻⣿⠪⡓⣄⠀⠀⠀⠀⠈⠑⠢⢌⣢⡀⠀⠀⠀⠀⢀⠔⠉⠈⠉⠁⢆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠫⠅⠰⢿⣿⠗⠈⢪⡑⣄⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⡠⠊⠁⠀⠀⠀⠀⠀⠸⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⡿⠛⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢼⣷⡆⣁⠀⠀⠈⠻⠦⡱⢄⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢰⣾⣿⣿⣿⣆⠀⠀⠀⠀⠈⠪⡓⣄⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⠉⠹⠄⠀⠀⠀⠀⠀⠈⠪⣁⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⢣
⠀⠀⠀⠀⠀⠀⢸⡠⠔⠂⠐⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠈⢆
⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠘⡄⠘⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢠⠔⠁⠀⠀⠀⠀⠈⢆
⡔⠚⠓⠒⠒⠒⠒⠒⠒⠛⠛⠓⠚⠒⠚⠻⣿⣿⣿⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⣠⣾⣧⡳⣄⠀⠀⠀⠀⠀⠈⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡇⠻⠿⣤⣤⡄⠀⠀⠀⠀⠀⠀⢀⡰⠋⠈⠉⢐⠎⢀⣤⠀⠀⢀⣴⣿⣿⣿⣿⣿⡮⡳⣄⠀⠀⠀⠀⠘⡄
⡇⠀⠀⠀⢀⡴⣶⢻⡟⢶⠦⣄⠀⠀⢸⡇⡇⠀⠀⠀⠈⠙⢧⡀⣀⣄⣠⣤⣋⠀⠀⠀⠠⢃⣴⠟⢁⡠⠊⠁⠈⠻⠿⠛⠛⢿⠃⠈⠪⡳⣄⠀⠀⠀⠘⡄
⡇⠀⠀⣴⠛⢶⠗⢺⡷⠚⣷⠚⢧⡀⢸⡇⡇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣭⣙⠓⣛⠥⠊⠁⢁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠈⠪⡳⣄⠀⠀⠘⡀
⡇⠀⢰⣇⣀⣿⣀⣸⣇⣀⣼⣄⣨⡇⢸⡇⡟⡄⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠈⠪⡳⣄⠀⠰
⡇⠀⠈⣇⠀⢻⠀⢸⡇⠀⣼⠀⢰⠇⢸⡇⡇⠘⡄⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⡀⡠⠊⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣁⠔
⡇⠀⠀⠘⢿⣙⣏⢹⡏⣹⢏⡽⠋⠀⢸⡇⡟⣀⡼⠆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠁⠉⠀⠀⠀⠀⠀⠀⣠⠎
⡇⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⢸⡇⡏⡥⠢⡈⡄⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⡇⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢸⡇⣷⡘⡄⠱⡈⠄⠀⠀⠀⠀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣷⡘⣄⡱⢘⡀⠀⠀⠀⢿⡿⠀⠀⠀⣀⣀⠤⠒⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡿⢛⡡⠔⠊⠁⠈⠉⠁⠀⠈⠉⠈⠉⠁
⢇⣈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣋⣠⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠻⣿⣿⣿⣿⠿⠿⠛⣨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣋⣉⡹⢦⣿⡄⠛⠫⣿⣼⠇⠀⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣾⣿⣿⠽⢿⣿⣿⣶⣆⣀⡁⢀⣼⣿⣿⣿⣷⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠓⢼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⣿⣻⣿⣷⡄⠀⠙⠿⣿⣿⣿⣿⣿⢿⣯⣳⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣿⣿⣿⢋⠘⠹⣿⣿⣦⣠⣀⣠⡉⢉⣉⣿⣏⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠻⠟⠀⠠⠄⠛⣹⡿⣿⣿⣿⣷⣿⣿⣟⣦⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⡝⠿⣿⣿⣯⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠋⠁⢸⠿⠿⢿⣿⠿⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡃⠀⢤⣨⡄⠰⠛⠁⠀⠈⠠⢀⣠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠎⢈⣿⣿⠶⣶⣿⣷⠀⢄⣀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠲⠀⠀⠀⠂⠀⠀⠀⠈⠁⢿⣿⣿⣷⣤⣬⣿⣿⣾⣿⣿⢧⡄⠀⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⣿⢷⣽⡿⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠸⠀⢸⣿⣿⣿⣿⣿⢿⠟⠊⠁⠀⠙⠰⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢨⣤⣈⡿⠁⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣼⡽⠃⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠼⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣷⠂⠀⢀⡑⠀⠀⠀⠄⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡅⡁⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠖⠛⠋⠘⠒⠶⡤⠆⠀⠘⢻⣾⣥⣿⣆⠠⡀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⢀⣐⣄⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⣶⣤⣀⡀⠚⠓⠲⣄⠠⣿⣾⣿⣿⣿⣆⠡⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣤⠀⠀⠀⣤⡄⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠙⠉⠙⠛⠛⠛⢿⣦⡻⠆⠀⠀⠀⢀⣴⣶⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠀⠾⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠭⠭⠭⠭⠭⠭⠭⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢝⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠾⠖⠚⢉⡤⢖⠉⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣐⡮⠽⡛⠯⠉⣉⠀⠀⠀⢀⠘⢉⠅⠀⢀⡞⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⣩⠭⠟⢓⡉⠉⠀⠈⠀⠀⠀⠀⠈⠀⠀⠈⢠⠜⠁⠀⢀⡞⠀⠀⠃
⠀⠀⠀⠀⢀⣀⣠⠤⡤⠴⣤⡀⠤⠔⠒⢉⡉⠤⠐⠂⠉⠀⠀⠀⠀⠀⢀⢂⡄⠣⠀⠖⣡⠖⠅⠀⠀⢀⡾
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠞⡅⠂⣠⠟⠁⠀⠀⠀⢠⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠈⠀⠈⠙⠲⡄⠀⠀⡎⡐⡀⡄⢘⡁⡆⠀⣰⠟⠡⠂⠀⠀⠀⢠⢣⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠂⠀⣧⡗⣃⠀⠈⠀⡡⠞⠡⠂⠀⠀⠀⠀⢠⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⡄⠀⢰⠐⠀⣤⠏⢡⣤⡤⠊⢀⠔⠀⠀⠀⠀⠀⢠⢃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣉⣉⣉⣉⣀⣰⣟⠀⠀⢠⣾⡿⠋⢀⠔⠁⡀⠁⠀⠀⠀⢠⢃⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⣠⡄⠀⠀⢡⢀⣰⠤⢄⡾⠋⢀⣴⠃⠀⠆⠀⠀⠀⠀⢠⠏⡜
⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠈⠉⠛⠙⠋⠁⢀⡴⠋⠀⡠⠋⠀⠔⠁⠀⠀⠀⠀⢠⠏⡘
⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⣠⠎⠀⠠⠊⠀⠀⠀⠀⠀⢠⠏⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⣠⠞⠀⢀⠔⠀⠀⠀⠀⠀⠀⢠⠏⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⠞⠁⠀⡠⠂⠀⠀⠀⠀⠀⠀⢠⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⠔⠋⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⢠⠃⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⢀⢀⢀⣀⣀⡀⢀⡀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠕⠁⠈⠀⠀⠀⠀⣀⣄⣂⡀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠸⠀⢰⠒⠒⠒⠉⠈⢸⡏⠈⠉⠲⣬⠊⠉⠉⠩⠃⣌⠉⠉⠙⢹⡇⠈⠀⠁⠉⠀⠀⢹⠀⡠⠴⠀⠀⠉⠢⣄⡀⠐⡇⢸
⠀⠀⠀⠀⠀⠀⡇⢨⠀⠀⢀⡀⡤⡼⠞⢄⠀⠀⠉⠀⠀⢠⠇⠈⢰⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⣯⠟⢤⡄⠀⠀⠨⡗⠚⠀⡇⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠉⢹⠂⡸⢆⠀⠀⠀⠀⣋⠴⠂⢸⠀⠀⠀⢸⠍⠉⡇⠀⠀⢸⠉⠉⣠⠖⠋⡆⠀⠀⠀⡇⠀⠀⡃
⠀⠀⠀⠀⠀⠀⡇⠈⡆⠀⠀⣤⡴⣞⠉⢀⠟⠀⠀⠀⠀⠘⢆⠀⠸⠀⠀⠀⢸⡂⠀⡇⠀⠀⢸⠀⢈⣁⣤⠠⠁⠀⠀⠀⠇⠀⠀⡁
⠀⠀⠀⠀⠀⠀⣇⡰⠆⠀⠀⠀⠀⠈⢅⠏⠀⠀⢠⠓⡀⠀⠈⠣⡀⡇⠀⠀⠸⠇⠀⡇⠀⠀⢸⠁⡄⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠃⢐⠰⠶⠒⠒⠸⠴⠔⠐⠃⠀⠘⢦⠤⠠⠙⠑⠥⠤⠰⢁⠀⠲⠤⠄⠘⠀⠃⠤⠤⡤⣤⠤⠀⠀⠀⠀⢠⠅⢠
⠀⠀⠀⠀⠀⢀⠐⠠⡀⠠⠠⠀⠴⠄⠅⠄⠄⠄⠄⠠⠤⠤⠄⠄⠤⠤⠤⠤⠄⠀⠤⠤⠀⠀⠐⠒⠂⠒⠐⠒⠒⠒⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⠀⠀⢰⣿⣦⠀⢀⣾⣿⡆⠀⠀⣾⣿⡆⠀⠠⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠁⠀⠀⠈⢿⣿⣧⣾⣿⡟⠀⠀⠀⣿⣿⡇⠀⠀⠈⠉⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣴⣶⣄⠀⠀⠀⠈⢿⣿⣿⡟⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠛⠛⠛⠋⠀⠀⠀⢀⣾⣿⣿⣧⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣀⣀⣀⡀⠀⠀⢀⣾⣿⡟⢿⣿⣧⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠀⠀⠸⣿⡟⠀⠈⢿⣿⠇⠀⠀⢿⣿⠇⠀⠀⠀⠀⢿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿
⣤⣦⣴⣶⣶⣶⣶⣴⣿⣿⠛⢿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿
⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠙⠻⣿⣿⣦⣄⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣷⡆⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠛⠁⠀⢸⣿⣿⠀⠀⢸⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⣠⣶⣿⣿⠟⠉⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⣿⣿⡇⠀⠸⣿⣿
⠈⠉⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣾⣿⣿⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠒⠉⠉⠉⠉⠐⠢⣄⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢤⣀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠓⠤⡀
⠀⢀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣒⣒⣀⣤⣤⣤⣤⣤⣼⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠈
⢸⠉⠸⢹⡏⠛⠻⣦⡤⠔⡻⠑⠉⠁⠀⠀⠀⠀⠀⢁⣠⣴⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⢸⣿⣧⠘⣧⣠⠞⡫⠐⠙⠛⠷⢦⣤⣀⣀⣤⣶⠾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠔⠂⠉⠀⠀⡀
⢸⠑⣿⢀⢛⠵⠋⠤⣤⣀⣀⣤⣴⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠔⠂⠉⢀⡀⣠⢦⡤⡶⡞⡋
⢨⠀⢼⢨⢈⣀⣤⢴⠾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⢈⡀⠀⠤⠐⠂⣰⣯⣭⡆⢸⠛⡿⢯⣿⡇
⢸⠀⢘⠘⠸⠑⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢐⣊⡉⠥⠐⡂⠀⠈⠄⠀⠀⢀⠀⠀⡇⠙⢿⡇⢸⠀⣇⠀⠙⢧
⢸⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠐⠬⠑⠀⠀⠈⠂⢀⠀⠤⡀⠂⠀⠂⠁⠀⠃⣈⡀⠀⣿⢀⡜⡿⢸⠠⣶⡃⠀⢸
⢸⠀⠘⢸⢀⣀⠀⠀⠄⠀⡤⡚⠉⠉⠁⠀⠀⠂⠐⢖⢲⡃⠀⠀⠀⠀⠀⠀⠀⠀⢉⣸⢻⡍⠀⢷⠎⠀⢳⠘⠆⠀⢘⠤⣾
⢸⠀⢺⢸⠸⠄⠀⠀⣀⣒⠂⡇⠀⢈⠹⢋⠃⠐⠒⢀⢛⢱⡄⠀⠀⢀⣀⠀⡂⢠⣾⣷⣾⣿⡆⢸⠀⠀⢸⠀⡆⠀⢸⡉⢹
⢸⠐⢶⢸⠰⠝⠛⠛⠙⠁⠀⢷⡰⠁⠂⡥⢴⡐⠀⢸⣞⢸⢡⠀⠐⠒⠒⠒⠂⠚⢿⣷⡞⣯⣇⢸⢀⠤⠼⢀⠗⠆⠆⣧⡼
⢸⠀⢻⢸⠘⠈⠒⠀⠀⠀⠉⠟⠇⠐⠢⡋⡀⢉⠰⠀⠽⣈⢸⠃⠀⠀⠀⠠⠠⢰⠞⣗⠸⣿⡿⠈⠂⠐⠒⠺⠄⠀⡸⠃⠰⠤⠤
⢸⠀⠈⣨⠀⠀⠀⠀⠀⠀⠀⠈⣄⠈⠀⠈⢪⡌⣰⠚⠛⠃⠘⠀⠀⠀⠀⠀⠈⠉⠉⡿⠿⣿⣿⠀⠒⠐⠂⠀⠀⠴⠆⠘⠂⠙
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠨⠂⠖⠐⠀⠀⠀⢠⠀⠠⠄⠀⢰⡗⠀⢸⢿⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠈⠠⠀⠅⠀⠂⠐⠄⠀⠄⠀⠀⠀⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⣆⠀⠁
⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⡷⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡁
⢀⣤⣎⠀⠀⢸⡋⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠿⣆⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⡀⠀⠀⠴⡇⠀⠈⣿
⠀⠛⠛⠲⠄⠈⠷⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢢⠀⠀⠄⠀⠀⠀⠀⠈⠀⠀⣠⣾⣤⡀⠈⠛⠄⡀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠁⠀⠀⢴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠦⠄⠆⣀⠀⠈⠛⠒⠒⠒⠒⠲⠀⠀⠠⠤⠤⠤⠠⠤⢤⡄⠤⠤⠤⠠⠤⠤⠤⠠⠤⠤⡄⢠⣤⣤⣄⢤⢠⡤⢠⠤⠄⠤⠄⠀⠠⡶⠶⠖⠒⢲⠒⠂⣀
⢦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢀⠀⠀⠦⠀⠀⢠⡄⡀⠀⠀⢸⣇⣰⡆⣤⣾⠁⠀⠀⠀⡀
⠀⠀⠀⡄⡄⠀⠀⠀⠀⣀⣤⣠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠉⠀⠀⣮⠀⠀⠀⠀⠀⣼⣷⣯⠀⠀⠸⠟⠛⠃⠙⠻⠌⠀⠢⠉
⠄⠀⠀⠹⠀⣀⠀⠀⠀⠿⠿⠛⠀⠀⠀⠸⠷⢇⠀⠀⢠⣼⣵⡇⣠⢀⠀⠀⠈⠛⠃⠀⠀⠰⠷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠂⠀⠀⡄⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⢀
⠅⠀⠀⠀⠀⠀⠀⠀⢒⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠄⠀⠀⠂⠀⠡⣶⠂⠉⣁⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠡⢀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣄⠀⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠻⢿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣷⢿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠿⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠽⠿⣿⣿⣷⣶⣦⣄⡀
⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣾⣾⣿⣦⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠛⠋⠉⠛⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠋⠁⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⠀⠀⠈⠙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⣠⡴⠞⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄⠀⢄⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⡾⢁⣴⣶⣶⣶⣤⡀⠀⣤⡀⠀⢀⣀⠀⠀⣀⣤⣤⡌⢷⡀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣼⠃⣾⡏⠀⠀⠀⢈⣿⠀⣿⡇⠀⢸⣿⠀⣼⡟⠉⠉⠁⠈⣷⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⣿⣇⣠⣤⣴⡿⠃⠀⣿⡇⠀⠘⣿⠀⢿⣦⣄⡀⠀⠀⣸⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⡋⠉⠉⠻⣷⠀⠀⣿⠁⠀⠀⣿⠀⠀⠈⠉⢿⡆⠀⣿⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢻⡄⣿⣇⣀⣀⣠⡿⠀⠀⢻⣧⣀⣸⡿⠀⣤⣀⣠⣿⠃⢠⡏⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠻⣌⠙⠛⠛⠋⠀⠀⠀⠀⠉⠛⠋⠀⠀⠉⠉⠉⠁⢀⡾⠁⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠏⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠈⠙⠲⢦⣤⣀⣀⣀⣀⣀⣤⣤⠴⠚⠉⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣐⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣀⣤⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⠷⢶⣶⣶⠶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣄⣤⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⢀⣶⡟⠛⠛⠿⠀⠛⠛⠛⣿⡿⠛⠛⠃⢀⣴⠿⠛⠛⠻⣷⣄⠀⠀⣿⡿⠛⠛⢿⣶⡀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠸⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣿⠏⠀⠀⠀⠀⠈⣿⡇⠀⣿⡇⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠉⠛⠿⣶⣄⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣹⣿⠀⣿⣷⣤⣴⡾⠟⠁⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣿⡇⠀⠀⠘⣿⣆⠀⠀⠀⠀⢀⣿⠇⠀⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠰⣶⣤⣤⣶⠿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠈⠻⣷⣤⣤⣴⠿⠋⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⡜⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⠍⣐⠠⠠⠃⠤⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⠈⡇⠒⡶⢾⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠁⠢⣠⠚⠉⠈⣿⠀⣜⡠⠶⠎⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⣤⠏⠉⠀⢀⡏⢀⣿⡁⠀⠀⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠤⠤⠤⠀⠀⠐⠚⠓⠈⠙⣄⣀⡠⠞⠀⠸⠫⠦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠉⠀⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢺⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⡀⠀⠀⣀⡤⠒⠁⠀⠀⠀⠀⠀⠸⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢠⡰⠄
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣩⠆
⠢⡀⠤⠖⠉⠢⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⢉⠴⠊⠇
⠀⠙⢄⠀⠀⠘⠕⣾⠀⠀⠀⠀⠀⠀⠀⢠⣆⡃⠀⠀⠀⣠⢔⠒⠈⢁⠤⠊⠁⠀⠀⠲⢄⡀
⠀⠀⠀⠑⠚⠛⠁⠫⠣⡀⠀⠀⢴⡄⠒⠈⠀⢡⣀⠴⠊⠀⠀⠢⠀⠁⠀⢀⣄⡀⠀⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⡐⠀⠟⢿⢆⡞⠀⠞⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠓⠊⢈⠇⠀⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⡝⠀⠀⠙⣌⣂⣩⡧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⠄⠀⣀⣀⠀⠀⣀⣀⠀⠠⠤⠐⠊⠁⠀⠀⡇⠀⠀⠀⠋⣧⡵⣚⢕⢖⠒⠀⡰⠶⡶⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠬⢶⣚⢺⠶⠶⠤⠀⠀⠀⢀⣀⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⣿⠥⠀⠢⠴⠮⡀⠀⠈⠢⡈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠶⣔⢦⡦⡿⢉⣹⣭⠁⠀⠀⠀⣿⠯⢷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠈⠢⡀⠀⠈⠫⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢀⠂⢘⣂⣓⣘⣂⣂⡸⠆⠀⠀⡠⢻⡂⡩⠂⠷⣐⠦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠢⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠈⠍⢓⣺⠤⡀⠠⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣤⢤⣀⠀⢀⣠⣚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠀⠀⠀⠄⣀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠄⠂⡏⠁⢀⠘⠋⠉⣉⣠⣤⠤⠤⣄⡁⠠⡀
⠀⠀⠀⠀⠀⢀⠀⠤⠥⣂⠀⠀⠀⠀⢀⠀⢺⠀⠀⠀⡧⠈⠀⢤⣶⠉⢹⠘⢻⠀⣆⠈⡏⠰⣌⠂
⠀⠀⠀⠀⠀⠚⠁⠀⠐⡈⠂⠈⢀⡄⠁⠀⣸⠠⡐⠋⢀⢔⠃⢚⣿⢰⠛⠖⠚⠂⠛⠒⠓⠄⢸⣣⡐
⠀⠀⠀⠀⠀⠀⠐⠐⠚⠀⠁⡀⠈⣗⡐⠋⢀⠆⠀⠔⣿⠀⠩⠞⠉⠀⠤⣶⣶⣿⣽⢿⢷⣶⣄⡙⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠇⢠⡄⡀⢹⡦⠟⠀⡠⡁⡀⢒⠠⣿⠈⢁⡄⠀⠀⠀⣹⢿⡿⣿⣟⣟⣾⣿⣷⡘⡂⠁⠐⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⡈⢄⣘⠓⣇⡺⠀⠀⠀⣿⡐⡇⢈⠙⢀⣴⠟⠷⠿⢿⢦⣿⣫⣀⡀⡲⢦⣌⡿⠚⡁⠀⠀⠀⠴⠧⠠⠰⣄
⠀⠀⠀⠀⠀⠐⠛⠙⠻⠠⠟⠲⠤⠥⠤⣀⣤⡧⠃⣠⣿⢈⣧⠐⠿⢆⠐⢩⣬⠀⠉⠉⠉⢈⠀⠉⠙⠛⠂⣖⣤⣀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠈⠄⠀⠀⠀⣀⡠⠡⠼⠭⠿⠀⠛⠉⠀⠈⠀⠀⢬⠤⠀⠸⠄⠉⠀⠀⠢⠤⠀⠀⠀⠀⠭⠙⠁⣭⠙⡢⡄⠐⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣄⡀⠈⠃⣀⡀⠀⢉⠁⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⢅⠀⠀⠀⠀⠄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠊⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠀⣄⡂⠀⠀⠀⠉⠙⠋⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣯⣥⣤⣤⣤⣤⣤⣤⣤⣽⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢫⠉⠉⠉⠉⠉⣿⠀⠀⠀⣀⢤⣲⣤⣴⣖⡤⡀⠀⠀⠀⢸⠉⠉⠉⠉⠉⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⣿⠀⠀⣜⣵⠟⠉⠉⠉⠉⠛⢮⣢⠀⠀⢸⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣿⠀⠸⣼⠁⠀⠀⠀⠀⠀⠀⠈⣧⡇⠀⢸⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⣿⠀⢰⢻⠀⠀⠀⠀⠀⠀⠀⠀⡘⠇⠀⢸⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣿⠀⠀⠣⠦⡀⠀⠀⠀⠀⢀⠴⠟⠀⠀⢸⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⠤⠤⠤⣿⠀⠀⠀⠀⠀⣁⡀⠀⣈⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣿⠀⠀⢀⢔⣽⠶⠿⠿⠷⣯⡢⡀⠀⠀⢸⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⣿⠀⢠⢣⡟⠁⠀⠀⠀⠀⠀⠹⣞⡄⠀⢸⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⣿⠀⢸⢾⠀⠀⠀⠀⠀⠀⠀⠀⢹⠆⠀⢸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⣿⠀⠘⣜⡀⠀⠀⠀⠀⠀⠀⢀⣬⠃⠀⢸⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⠀⠀⠈⠉⠂⠄⠀⠀⠀⠐⠉⠁⠀⠀⢸⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡤⣔⣒⣒⣢⠤⡀⠀⠀⠀⢸⠒⠒⠒⠒⠒⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⣿⠀⠀⡔⣵⠟⠋⠉⠉⠉⠻⣮⣢⠀⠀⢸⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣿⠀⢰⣸⠃⠀⠀⠀⠀⠀⠀⠈⢧⡇⠀⢸⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⣿⠀⢰⢹⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⢸⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣿⠀⠀⠣⠦⡀⠀⠀⠀⠀⠀⠰⠟⠀⠀⢸⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣐⣀⣀⣁⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠁

⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡎⠀⠄⠄⠤⠄⠀⠀⠤⠤⠤⠤⢤⠄⠤⠤⠤⠤⠤⠤⡄⠀⠠⠠⠄⠤⠄⢄⢠⠤⠤⠤⠤⢄⣱⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⡇⠀⣀⣀⣀⣄⣀⣀⣀⡀⠀⠀⢨⠀⠀⠀⣀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⡸⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⢠
⠀⠀⡇⢀⡀⠀⠀⠀⢀⣴⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⢀⠀⠀⠀⠩
⠀⠀⠣⠤⠤⠤⠤⠤⣿⡁⠀⣹⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⣿⠁⢈⣿⠤⠤⠤⠤⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣴⣤⣼⠏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠈⠠⡀⠶⠀⠀⡄⠀⠀⠀⠀⠀⠀⠘⢷⣴⡤⠀⠀⢠⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡔⣤⠀⠆⢠⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠢⠀⣒⠀⠀⠑⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠙⡇⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢂⣰⣶⡿⠃⢾⣈⠉⠀⠀⠀⠀⠀⠻⠀⠀⠀⡄⠀⢹⣇⠔⠡⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡟⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣂⣐⠁⠸⡉⢀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠈⠀⠀⡐⢁⣤⢝⣷⡀⡤⣌⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡼⠫⡆⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⢾⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠿⠿⠛⠛⠛⠻⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣠⠀⠀⠀⠀⠀⠰⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⠤⠀⠄⠀⣤⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣸⠏⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⣭⣄⣤⣤⣼⣶⣿⣾⣷⡯⠄⢐⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡾⢠⣤⠁⠀⠀⠀⠀⠀⣈⡾⠿⡃⠀⠀⠀⠈⠋⠼⠍
⠀⠀⠀⠀⠀⠀⢀⣐⢮⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠙⠀⠀⣠⡀⠀⠀⠈⠃⠀⠀⠀⠀⠹⡧⠉⠿⠄⠀⠀⠀⠀⠘⠁⢀⣠⣅
⠀⠀⠀⠀⠀⠀⠰⣿⠿⠟⠋⠙⠏⠉⣹⣇⡀⠀⣤⠀⠀⠀⠻⡇⠁⠀⠀⠀⠀⠀⢸⡶⢀⡏⠰⠄⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⣄⠀⠀⠀⢰⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠛⣯⡿⡀⠈⠀⠀⠀⠀⢳⡀⠀⠀⠀⢄⠀⠘⡆⢸⠷⣄⠀⠀⠀⠐⠀⠀⠀⠞⠛⠉⠁⠈⣿⡷⣶⣷⢻⣷⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠇⢸⡏⠑⠀⠀⠀⠀⡠⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠾⠀⠘⢣⡀⠀⠀⠀⣤⠀⠀⢀⣠⣶⣿⣿⣇⣿⣿⢀⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢧⠀⡀⠀⢠⠀⠀⠀⠀⢳⣦⡄⢀⣀⡀⠀⠀⡷⠄⠀⠀⢱⡄⠀⣠⡜⠃⠴⠿⠋⠉⢹⡾⣸⣿⡟⢸⣷⡄
⠀⠀⠀⠀⠀⠀⢀⠤⠶⠀⠀⢸⡁⠀⠁⢈⡾⠃⠀⢀⠀⠀⠀⠀⠈⢿⣆⠒⠉⠛⠂⠀⠇⢀⠈⡀⠇⢷⡀⢻⠀⠀⢀⣤⣶⣿⣽⢧⣿⣄⠀⠾⢿⣿⡆
⠀⠀⠀⠀⠀⠰⢿⣿⣦⣵⠛⠉⣿⣶⡅⠘⣆⠄⠀⠈⢽⠗⢀⠀⠀⠈⢯⠆⠀⢿⣶⡄⠥⠀⠁⠁⡀⠈⣷⠀⠀⠐⣿⠉⠉⢹⡿⣸⢀⡻⢐⣆⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠴⠀⠃⠀⢩⠀⣯⣄⣤⣴⣏⣀⣤⣤⠀⢀⠸⢻⡴⣐⠂⡄⢠⣤⡀⡜⡠⠀⣿⡇⠀⠀⣩⣴⣿⣿⢡⣿⡄⠁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡌⠾⣶⣍⡻⡿⢧⠉⠀⠿⠿⣮⡳⣿⢸⣋⣁⣌⡠⠆⢹⣷⠀⠛⡏⠀⣼⢃⣿⠿⡟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆⠀⡄⢻⠀⡴⠈⢉⣝⢷⠦⣴⣆⢠⡄⠈⢳⣮⡀⣿⣿⠟⠟⠋⢸⣿⣀⣴⣷⣿⢏⣾⣿⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⡁⣰⠎⢿⣷⡵⣬⣿⣾⠟⠁⣿⣿⡇⢻⣿⣶⣄⣄⣸⣿⣿⣧⣿⢯⣿⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⠀⠇⢹⣿⡻⣧⣄⠸⣿⠋⠀⠸⡹⣿⣿⣿⣿⢸⣿⣿⣫⣿⡇⠈⠡⠇⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣳⣿⠉⢿⣾⣿⣷⣿⣶⠦⠂⣷⢻⣿⣿⣏⣿⡿⣹⡟⠉⢁⠐⠁⠀⠀⠀⢀⠀⣨⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠙⠃⠀⠸⣿⣷⣝⣿⡀⠀⠈⢸⡏⣿⡿⣸⣟⣿⠋⠀⡠⠊⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⣿⠏⢻⣎⢿⣟⡀⠈⣿⢸⣿⣿⠟⡃⠔⠁⠀⠀⠀⠀⠀⠀⠀⢼⣿
⢰⣷⡄⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⡀⠸⠿⣷⡽⣧⠀⣿⠜⣫⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠈⠀⠀⠀⠀⠀⠀⠄⢸⠦⣀⣀⣠⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣉⢢⣄⣈⣿⣝⡼⣣⣾⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣠⡀⠘⠛⣰⡀⠀⠀⠀⠀⠀⢲⣶⣬⣿⣿⣿⡿⠿⠋⠉⠉⡷⠞⡾⡿⠁⠀⠈⠻⠿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠰⣦⡄⠀⣾⠟⠁⠄⠉⠙⠀⠀⠀⢾⣷⣾⡿⣿⣿⠉⠟⠀⠀⠀⠀⢠⣇⠀⢻⡧⠀⠀⠀⠀⠀⠤⠈⡉⠳⠶⣾⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠙⣤⠀⣧⡆⠈⣆⠀⠁⠀⢰⣶⣿⡟⠻⡧⠀⠈⠀⠀⠀⠀⠀⠀⠸⠀⢀⣾⠇⠀⠀⠀⣰⣄⠀⠀⠘⠀⠀⣈⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢺⠁⠀⠀⠀⠀⠀⢸⡿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣻⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⠃

⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿
⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣿⣿⡆
⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⠉⠉⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⡿⣿⣄⣀⠀⠀⣤⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣧⣤⣿⢠⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⣼⣿⣿⣿⡟⠻⠿⣷⣿⣾⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠈⠢⣿⠀⢣⡠⢻⣄⣾⡄⠀⠀⠀⠀⠀⢀⣀⣀⣄⣀⣠⠔⢻⣷⣾⣷⡇⠀⠀⠀⢹⣛⣟⡃
⠀⠀⠀⣠⠀⠀⢀⡀⠀⣠⣤⣤⣤⣤⣿⣶⡀⠀⣿⠁⠈⠙⣿⣯⣼⠀⠀⠀⠠⠴⠶⠶⣶⣶⣶⣦⣤⣤⣙⣀⡻⣿⣇⠀⠀⠀⢘⣹⣯⣿
⠀⠀⠀⢹⡆⠀⠐⡇⠀⠀⠉⠉⠛⠛⣿⠿⣿⣿⣿⣦⣤⣀⡛⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣿⣿⣿⣿⡀⠀⡷⢶⣤⣼⣽⠿⢇⡐⣄⡀
⠀⠀⠀⣸⠓⡀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠉⠉⢉⣿⠈⠉⠙⢻⠿⢾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣶⡄⣆⣼⣿⣿⣷⣬⢋⠛⠉
⠀⠀⢀⣙⣀⣀⣐⣃⠀⠀⠀⠀⠀⠀⠿⠉⠉⠉⠿⠀⠀⠀⠸⠒⠚
⠀⠀⠨⡏⣿⠿⡟⣿⣷⣦⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠄⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⡀⠀⠀⠐⠢⣤⣀⡀
⠀⠀⠀⡇⠀⠀⡧⢸⠋⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⠀⠀⣿⢦⣄⣀⡀⠀⡀⠀⠀⠀⠀⢠⣄⠻⠄⣈⣹⣗⣲⣶⣿⣥⣉⣳⡦⢤⣀
⠀⠀⠀⠃⠀⠀⠛⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠆⣿⢶⣴⠿⢿⣷⣦⣄⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠉⠙⣻⣶⣧⣽⣗⣆⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼⠁⠀⣿⡌⠀⢠⡀⠙⣿⠟⠛⡃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡴⠛⣩⣤⣶⣤⣙⢯⠙⢳
⠀⠀⠀⠀⠀⠀⣠⣶⠾⡿⣦⣾⢿⠶⣄⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⣿⠁⠀⠀⡇⠀⠸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡾⢡⠞⣿⣿⡿⠟⠙⣷⡣⣸
⠀⠀⠀⠀⢀⣼⡏⢸⡀⣼⠋⣿⢻⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⡏⠀⠘⣯⠀⠀⠀⠘⣧⣿
⠀⠀⠀⠀⠾⠛⠇⠘⣼⣿⡀⣿⢸⢻⠀⢠⣷⠦⣠⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁⠀⠀⠛⠒⠒⠒⠒⠛⠚
⠀⠀⠐⢲⡶⠶⠶⢦⡿⣭⣤⣉⣘⣘⠃⡿⢸⣆⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠈⠉⢹⠋⠛⣻⠛⠖⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢦⡄⠀⠀⠀⢼⠷⣄
⠀⠀⠀⠙⠁⠀⠀⠙⠁⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠟⢦⣀⠀⣛⣉⢸⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢄⡄⠰⠀⡟⢶⣷⣶⡾⠤⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢫⡧⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣁⣀⣉⣀⣁⣁⣙⠻⢹⡿⣯⣛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⣠⣦⣤⣤⡔⠀⠀⠀⡇⠀⠀⢸⡄⠀⠀⠀⢠⠀⡀⠀⠀⠈⢏⠉⠛⠛⠛⠛⠻⠿⣿⣯⣁⢿⢻⣿⣿
⠀⠀⠀⢐⣀⣀⣈⣁⡈⣀⣁⣐⠄⡀⠀⣻⠀⠈⡇⠀⠀⠀⠉⠙⠛⠻⠿⣶⣦⣥⣀⡀⠀⡇⠀⠀⠀⡀⠀⡁⠀⠀⠀⠀⢹⠛⠉⠉⠉⠛⡻⠿⣿⣿⣷⣮⣽⢿⡄
⠀⠀⠀⠀⢏⡙⠛⠛⠛⠛⠻⠿⣿⣯⣐⠿⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡏⠛⠻⠇⣷⣦⣤⣀⡃⠀⢅⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⡇⠀⠀⠉⠉⠉⠻⠛⠃
⠀⠀⠀⠀⠀⢸⡟⡇⠀⠉⣿⠻⠿⣿⣿⣿⣦⣄⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠄⢿⠀⠉⠙⡿⠻⢸⣤⣄⠀⠀⢸⣀⠀⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⣿⠁⠀⠀⠉⢩⠉⢹⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠀⠀⠀⠃⠀⠘
⠀⠀⠀⠀⢀⡼⠧⠇⠠⠀⠿⠶⡄⠀⠀⡸⠀⠸⡀
⣄⠀⠀⠀⠈⠷⠒⠒⠒⠒⠒⠒⠃
⣿⣷⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⣿⣿⠿⠿⣿⣿⡿⠿⠿⠿⣿⣿⠿⠿⢿⣿⣿⡿⠿⠿⣿⣿⣿⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠁⣴⣦⠈⣿⣷⡆⠀⣶⣿⠃⢰⣶⡀⢹⣿⡇⢰⣦⡀⢻⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣄⡈⠛⢿⣿⣿⡇⠀⣿⣿⠀⢸⣿⡇⢸⣿⡇⠸⠿⠃⣸⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⠛⣿⣷⠀⢻⣿⡇⠀⣿⣿⠀⢸⣿⠇⢸⣿⡇⢠⣶⣾⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣄⡉⢉⣠⣿⣿⣇⣀⣿⣿⣧⣈⢉⣀⣾⣿⣇⣸⣿⣿⣿⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⠤⠤⠤⢄⣈⢑
⠀⠀⠀⠀⠀⣀⢀⣤⣤⢶⡤⢔⣠⣶⠤⢤⣼⣧⠀⢀⣀⣀⣨⣧⣀⣤⣄⣀⣤⣤⡤⣤⣤⣤⠄⡀
⠀⠀⠀⠀⣎⠀⢀⡠⠊⠁⠀⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⣴⠋⠁⣄⠀⠀⠀⠆⡇⢱
⠀⠀⠀⠀⡁⡔⠁⠀⠀⣂⣀⣓⠀⠀⠈⢄⠁⠀⠀⠀⠀⠀⠀⠰⣂⡀⣹⠀⠀⠈⠑⢔⠁⢸⡇⢛
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡿⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠨⡂⠊⡇⢸
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠓⠀⠘⠀⢸
⠀⠀⠀⠀⡇⠀⠈⠚⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⣇⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠩⢄⠄⠐⠞⠀⠀⢸
⠀⠀⠀⠀⠸⢄⠀⠀⢂⠀⠀⢸⠀⠀⠀⠀⠒⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠑⠄⠀⠈⠚⠀⠀⠀⣆
⠀⠀⠀⠀⡆⢀⠠⡀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠄⠀⢠⡀⢠⡄⠀⢲⠀⢠⠀⠀⠀⠀⠀⠂⢱⡄⣾
⠀⠀⠀⠀⠑⠒⠒⠞⠒⠒⠀⠘⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢸⣀⠀⡀⠀⠀⠄⣨⠷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠁⠉⠐⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⡠⠀⠀⠀⠀⠀⢀⠀⠀⣀⡠⠀⠀⠀⢀⠀⠀⢀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃⣀⠔⠉⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⠀⠀⠀⠀⢸⠀⠀⡸⠔⠉⠀⠀⡀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠟⢁⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢸⡠⠋⠀⢠⠀⣠⠤⠓⠢⢤
⠀⠀⠀⠸⠏⠀⡀⠀⠀⠀⠀⠀⠐⠐⡷⠣⠄⣨⣄⣀⡀⠀⠀⣠⢼⡡⠦⠄⠀⠀⠀⠐⠀⠀⣻⠤⠀⠀⠀⢸⠃⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⣁⣀⣀⣀⣀⣀⣀⣘⣇⣀⣀⣀⣀⣀⣀⣀⢀⢀⢸⣀⢀⣀⡁⠀⠀⠀⠀⣀⣽⣀⣀⣀⣀⣹⣄⠀⠀⠀⢰⣃⣀
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⡏⠁⠁⠀⣁⣀⡀⢀⣀⣀⢸⣁⣀⣀⣀⣈⡧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢉⣀⣀⣸⣀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠈⡇⠀⠀⠀⠉⠉⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⣤⣤⠴⠶⣿⠚⠛⡛⠛⠛⠉⠉⠉⠉⢹⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠀⢀⡀⠀⠠⠄⠀⠀⢸⠂⠀⠘⢉⠀⠀⠀⠀⠀⣧⠀⢀⡇⢠⣴⠆⠀⠂⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠐⠒⡇⠀⠁⠀⠀⠀⠀⠀⠀⣀⣸⣀⠠⠤⠁⠤⠀⠀⠀⠀⡏⠀⠸⠃⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠤⠒⠂⠉⠁⠀⠀⠀⠀⢀⡇⠀⣀⠠⠐⠂⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⣻⡇⠀⡀⠠⠠⠀⠁
⠀⠈⠀⠀⠀⠀⠀⡀⠠⢤⣧⡗⠊⢹⠇⢠⣿⠇⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⢀⢸⠉⠂⠁⠀⠀⠀⢠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢈⠁⠀⠘⠄⣠⠋⠀⠀⠀⠀⠀⠀⠛⠐⠒⠂⠀⠀⠀⠈⠈⣄⡀⠐⠈⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡀⠀⠁⠚⠀⠄⠅⠀⠀⠀⠂⡀⠀⠀⡸⠄⠐⠀⢀⠑⠠⠂⠀⠀⠀⠀⠀⠈⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⠁⠐⠀⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠈⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡀⠀⠀⠀⣈⠡⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣠⠤⠖⠊⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⠀⢶⡄⢀⣴⠀⣴⡄⣶⣶⣶⣶⠄⡶⠒⠒⠒⠒⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⠀⠈⣻⣿⠁⠀⣿⡇⠀⢸⡇⠀⠀⡇⢀⣀⣻⠇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⡀⣰⡟⠹⣧⡀⣿⡇⠀⢸⡇⠀⠀⡇⠚⣸⡋⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣀⣉⣀⣀⣈⣀⣈⣤⣤⣈⡁⠀⠀⣧⣴⠋⠙⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠿⠟⠛⠉⠀⣦⣤⣤⣤⣭⣼⠇

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⡖⠒⠒⠒⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠶⠶⠶⡶⠂
⠀⠀⠀⠀⠔⠁⢸⠀⠸⠉⠁⠀⠀⠈⠉⢙⠀⢁⠀⠀⠂⠀⠀⠀⠀⡰⡆⠀⠒⠒⠛⠓⠒⠛⠛⠛⠛⠃⠘⠓
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢒⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡇⠀⠈⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢽⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⡆⠀⠀⠀⠉⢢⣠⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣀⠄⠁
⠀⠀⠀⠀⠀⠁⠓⠒⠀⠀⠐⠂⠐⠀⠀⠀⠒⠚⠉⠈⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉
⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⣾⡿⡿⣷⣆⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣴⣶⣶⣶⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠊⠉⠁⠉⠐⠉⠈⠁⠈⠁⠈⠀⠈⠫⡉⢻⡇⠀⠀⠀⢠⡟⠉⠉⠉⣿⠛⠉⠉⠉⠙⠋⣿⠋⠋⠉⠂
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⡇⠀⠀⢀⣸⡗⠈⠀⠈⣉⠀⠀⠀⠀⠀⠀⣉⡁⠀⠀⠙
⠀⠀⠀⠀⡇⠀⠀⡟⡷⠀⠀⠀⠀⠀⢟⢻⠀⠀⠀⢠⠄⠀⠀⢸⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⣤⣤⠀⢠⣤⣤⣤⣤⣤⡄⢀⣤⣤⣤⢠⠀⠀⠀⠘⢿⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⡿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢨⡆⠀⠀⠀⠘⡧⠁⠀⠀⣉⡀⠀⠀⠀⠀⠀⣉⡁⠀⠠⢤
⠀⠀⠀⠀⠄⠙⠿⠇⠸⠿⠿⠿⠿⠿⠇⢸⠿⢟⣁⠝⠁⠀⠀⠀⠀⠑⠀⠀⠀⠿⠒⠂⠀⠀⠀⠀⠿⠗⠂

⠀⠀⠀⠀⠀⠀⢀⣀⣤⠔⢒⡼⠛⢯⣉⣶⠖⠒⢲⢶⣶⣶⣒⣢⠤⠤⢄⣀⣀⡀
⣶⣶⣤⣤⣶⣾⣭⣭⣾⣞⣡⣔⣚⡉⠳⣼⡤⠚⠁⢸⠈⠳⣄⢹⠉⡳⠺⣿⣌⠉⡟⠒⠢⣤⣀
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠻⠿⠿⠷⣶⣾⣶⣤⣬⣿⣏⣀⣀⡇⠈⢳⣇⡠⠊⡝⢯⡑⣦⡀
⣶⣶⣶⣶⣶⣦⣤⣤⣄⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠻⠿⢿⣿⣷⣤⣧⣤⣱⣧⠞⢦⡀
⠉⢻⠟⠉⣠⠞⠁⠙⢯⡉⢢⡔⠋⣠⠞⠁⠀⠉⠹⣟⠶⣖⠒⠶⠦⣤⣤⣤⣀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣷⣿⣶⣤⣤⣀⡀
⠳⣼⣠⠞⠁⠀⠀⠀⠀⠑⢬⣧⠞⠁⠀⠀⠀⠀⠀⠈⠣⡀⢱⡖⢉⡼⠋⠙⢯⡑⢤⠔⢲⠟⠒⠒⢶⣤⣤⣨⣉⣙⠛⠿⢿⣿⣶⣦⣤⣀⡀
⠀⢸⡋⠉⠉⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣷⣋⣀⣀⣀⣀⣑⣾⠞⠁⠀⠀⠀⠀⠳⡝⢦⠴⣻⠟⢿⡒⢤⠭⣽⢿⣿⣿⣿⣶⣦⣄⣀
⠀⠘⢧⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⢾⣞⣁⣀⣀⣙⣾⠞⠁⠀⠙⣿⣍⣛⣿⣿⡿⢿
⠀⠀⠀⡏⠉⠉⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠈⢿⣞⣁⣈⣿⠟
⢀⣀⣠⡇⠀⠀⠀⠀⢸⣿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠄⠀⠀⠀⣿⣿⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣿⠉⠉⢹⣿⠀⠀⠀⠀⠀⠀⠘⣤⠤⠤⡼
⠉⠀⠀⣇⠀⠀⠀⠀⢸⣿⠀⢴⡒⠒⠒⠤⠤⠤⠤⠤⠤⠤⣄⢐⡀⠀⠀⠀⣿⣿⢀⡤⠴⠄⠀⠐⠒⡶⠀⠀⣿⠀⠀⢈⣿⠉⠉⠉⠉⠉⠉⠉⣇⠀⠀⡗⠒
⣄⡀⠀⢀⣈⣈⣀⣈⣈⣩⣤⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⣅⣠⡤⠤⠭⠥⠊⠀⠀⠀⠀⠀⠀⠑⠒⠒⠚⠒⠛⠒⠓⠒⣤⠀⠀⠀⠀⠀⠉⠀⠁⠁
⣠⠇⠀⠛⠒⠠⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣖⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠲⠦⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠐⠯⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⠀⠀⠀⣀⠤⠤⠔⠒⠒⠒⠒⠒⠒⠒⠢⠤⠤⣀⠀⠀⠀⠀⡰⠋⠁
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠈⠒⠲⠦⣄
⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⣀⣀⣀⣀⣠⡤⠴⠋
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡈⠉⠉⠉⣉⠈⠀⠀⠂⢀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡹⠁⠀⣠⠞⣏⣉⣉⣉⠳⠦⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣴⢢⠔⠇⠀⣼⣽⡿⠟⢟⠟⡿⣷⠷⣝⣧⡀⢸⡌⢳⠄⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣴⠢⣶⠀⢸⡆⢨⡟⠎⠠⠡⠁⢊⠌⠔⡝⣾⣿⡇⢸⡇⢸⡦⠤⠦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢸⡇⠰⡇⠀⠀⠀⠀⡠⠆⢄⡫⢨⣿⠀⢸⡥⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⢠⠇⠀⠻⣦⡀⠀⢐⡒⣈⡅⣃⡞⠉⢠⠸⡂⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⡄⠀⠀⠈⠉⠑⢲⣶⣾⣛⠉⠀⠀⠘⠀⡖⢽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⡇⠀⢀⡠⠞⣋⣍⣭⣭⣍⣟⠷⣄⠀⠀⡷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⢼⠀⢸⠇⠀⡟⣤⣿⠟⠛⠛⠳⠦⣍⢻⣿⡧⠀⣇⢽⠂⠐⠂⢂⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠾⠭⣼⠁⠘⢰⠀⡾⠛⠀⠃⠀⠁⢎⢔⠜⣧⣻⡿⠀⣟⢼⣮⠒⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢸⠀⢹⡀⠀⠀⠀⠀⢖⠅⡪⢎⣾⡇⠀⣗⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠘⠀⠈⢿⣤⣀⠀⠨⠔⠚⣫⡾⠋⠁⠀⣗⢽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠙⡛⡛⠚⠉⠀⠀⠀⠀⣗⢸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡇⠀⡀⠀⢀⣴⣶⢿⣿⣯⡯⠭⣓⣦⡀⠀⠀⣟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢶⣒⣇⠀⡇⢠⣍⠞⠿⠛⠉⠁⡩⡙⢮⡻⣿⡄⠀⡧⣸⢣⡉⡉⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⡀⠐⢻⠚⠀⠁⠀⠀⢁⣈⠔⣷⣿⡟⠀⡿⢹⡟⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⡆⠀⠨⣆⠀⠀⠀⢀⣺⠷⠶⠛⢿⠇⠀⠿⢿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⡇⡀⠀⠙⠷⢥⣀⣐⣫⠈⠤⠄⠀⠀⠀⠦⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⣂⣀⣀⣀⣀⣉⣉⣁⣀⣀⣀⠀⠴⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠺⠷⠖⠶⠾⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⡊⠊⢔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢑⢔⠥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢕⠥⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠕⡨⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠦⢊⠄⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣧⠀⠀⠂
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⣴⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⣀⡔⠀⠀⢀⠟⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠁⣭⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⢀⣴⡀⠀⠘⣿
⠈⠿⠻⢤⡀⠈⢣⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡨⡀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣴⡀⠀⠀⠭⡁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣄⡀⠘⠄⠀⢀⣠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠉
⡀⠀⡄⠀⠀⠀⠤⠤⠄⠤⠤⣠⡀⠀⠀⣀⣀⣀⡀⣀⡀⣀⠀⣀⢀⣀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⣀⢀⣀⠀⠀⣀⣀⣀⡄⡤⣤⠤⠄
⣁⠀⠀⠋⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⠀⠩⡌⠉⠈⠁⠉⠉⠈⠉⠉⠉⠈⣉⠉⠉⠈⢉⠉⠁⠉⣉⠈⠁⠀⠀⢸⡄⣠⠀⡀⣼⠀⠀⠁
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠈⠁⢐⠀⠀⠀⠁⠀⠀⣟⣰⠄⠀⠀⣿⣷⣿⡄⠿⢏⡡⠀⡀⠔⠂
⠀⠀⠀⢳⠃⠀⠀⠀⠀⣰⣾⣶⡇⠀⠀⠀⣶⣠⠀⠀⠀⠀⣰⢀⡀⠀⠀⠀⠀⠰⣿⡇⠀⠀⠀⣤⣿⣇⠀⠀⠀⠀⠘⠛⠋⠃
⡁⠀⠀⠀⠀⠒⡀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠂⠀⠀⣿⣿⣿⣥⣧⣖⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠄
⠄⠆⠀⠃⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠖⢒⡒⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠁⡗⠀⠀⡤⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣶⠛⠋⠉⠉⠁⠀⠀⠀⠀⠈⠓⠀⠀⠐⠂⠀⡠⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠀⠀⡀
⠰⡇⠀⠐⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠘⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠻⠯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣟⣂⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣏⠻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣿⣿⣄⣀⣀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣞⣛⡛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣿⣿⡿⢷⢦⣀