Cat symbol Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢠⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⣾⠁⠉⠛⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠘⣧
⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠨⡙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣫⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠈⢢⡙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢁⡜⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢳⡈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⣧⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⡞⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠘⣧⠰⡀⠀⠀⠀⠀⠀⣸⠀⠻⠶⠚⣛⠋⠉⡟⢻⠉⢛⡛⠓⠶⠏⠠⣇⠀⠀⠀⠀⠀⢠⠇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣇⠹⠖⠒⡶⣠⣞⡥⠶⢚⣋⡩⢅⠑⠸⡁⢸⠀⠊⡬⢍⣙⡒⠦⣌⣳⡄⢶⠒⠲⠋⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⢻⡇⠱⢟⣡⠖⢊⠉⠀⠀⠀⠁⠀⠁⠈⠀⠈⠀⠀⠀⠉⡓⠦⣍⠻⡌⢺⡟⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣴⢏⡤⢊⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡑⢌⠻⣦⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡞⠀⣹⠙⠛⠻⣷⡀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠛⠋⡏⠈⢧⣴⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠳⠶⠖⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠲⠶⠞⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⡀⢀⡴⠀⠀⠀⠀⠀⠀⢦⡀⢀⠀⠀⠀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⠔⢉⣴⠏⠀⣠⠤⠄⠠⢤⡄⠀⠹⣦⣙⢦⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⠁⠀⠀⠈⠻⡆⢰⠟⠁⠀⠀⠈⣿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣶⠶⠚⠁⠉⠓⠶⣶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠻⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠿⠋⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣤⣄⠙⢿⣷⣄⡀⣀⣀⣠⣤⣤⣀⣀⡀⣀⣴⣿⠟⢁⣤⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣷⣤⡈⠻⠿⠟⠛⠛⠛⠛⠛⠛⠿⠿⠋⣀⣴⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠘⠛⠛⢛⣛⣛⠛⠃⠘⠿⣿⡿⠟⠀⠙⠛⣛⣛⡛⠛⠛⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠘⠛⠉⣉⣥⠶⠛⠀⠀⣼⡄⠀⠘⠳⢦⣌⡉⠙⠛⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠚⠉⢠⣤⡤⠶⠚⠁⠙⠲⠦⣤⡤⠈⠙⢃⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣶⣶⣶⣾⡿⠿⠟⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⣿⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠐⠷⠦⣤⣀⣴⠿⠻⡆⠀⠀⠀⠀⠀⠀⣿⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⣀⡀⠀⠀⠀⠉⠻⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡆⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠠⠿⡵⠀⠀⠀⣀⡀⠀⢴⠁⠀⣀⣀⣀⣀⣀⣘⣿⣄⠀⠀⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠰⣦⣄⠻⠼⠃⠁⢸⣿⠛⠛⠋⠙⠛⠛⠛⠿⢿⣷⣄⡀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣀⠀⠤⠔⣇⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣆⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠓⠦⠤⣤⣤⣤⠴⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠒⠒⠒⠀⠀⠀⠀⢹⣿⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠄⣠⡾⠁⠀⢀⣠⣄⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢁⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⢋⣤⡶⠿⠛⠋⠉⠀⠀⣀⣀⣽⣿⣦⣤⣤⣤⣄⣀⣴⣿⣿⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠟⣩⠞⠋⠁⠀⠀⢀⣤⣶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠁⠀⠀⠀⠀⠰⠋⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⡀⠀⠀⡘⠀⠀⣈⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠁⢀⠄⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⡡⠚⢁⣀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⡐⠁⠀⠁⠀⠑⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣃⠀⠀⠀⠀⠴⣅⠘⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠁⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠟⠛⠿⢿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠉⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣧⣀⣀⣠⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⣠⢾⣿⣿⣿⣿⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣵⣿⣿⣿⡇⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣠⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⣼⣿⠟⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⠀⣠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⠻⣿⣿⣷⣄⣤⣤⣤⣶⣶⣤⣤⣤⣠⣾⣿⣿⠟⠋⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠛⠛⠛⠛⠛⢛⣧⠀⠀⠀⠀⣽⡟⠛⠛⠛⠛⠛⠛⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠛⣛⣉⣩⡿⠟⣻⣷⡆⢰⣾⣿⠻⢿⣍⣉⣛⠛⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⢋⣭⣶⣿⡿⠟⣡⣌⠻⢿⣿⣶⣬⣙⣻⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠲⣄⡀⠀⠀⠀⠀⠀⣼⠁⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⢿⣦⣤⣤⣶⣾⣿⣦⣄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠋⠈⠻⠟⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⢠⣤⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠉⠁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣄⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠹⣿⣿⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⣿⣿⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⡇⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠈⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢸⡀⠀⠀⣤⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⡇⠀⠀⣿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⡠⠀⢱⠀⠀⢉⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⠟⠛⠉⠀⠀⢊⠁⠀⠘⢀⠀⠈⢦⣀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠏⠉⠁⠈⠢⠀⠀⠈⣂⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠀⠀⠀⠀⠒⠲⠤⠂⠀⠀⠀⠀⠀⠁⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⣿⣿⣶⣦⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⡿⠋⠀⠀⢸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠙⢿⣿⣿⠿⠛⠋⠉⠁⠀⠀⢸⠋⠀⠀⠀⠀⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠈⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⠀⠀⠀⠀⢠⡿⢿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣀⣤⣴⣾⠿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣠⡤⠤⠤⠤⠤⠀⠀⠀⢠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠉⠉⠀⠀⠈⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⢢⡀⠀⠙⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠰⠋⠁⠀⠀⠀⠈⢷⡀⠀⠈⠙⠻⠛⠁⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢶⣤⣀⣀⣀⣀⣤⣴⣾⣿⣿⡿⠿⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠋⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠀⠀⠉⠀⠀⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡅⠉⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠋⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡀⠀⢾⣷⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⣠⠞⠋⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⠀⢀⠰⠖⠛⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠉⠛⠻⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣌⠙⠁⠀⠀⠀⠀⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⢀⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⢿⣿⣿⣆⠀⢀⣠⣶⡿⠿⠿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠀⠀⠉⠛⠿⣿⣿⣿
⢠⣶⣶⣄⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢀⣀⡀⠈⠻⣿⠀⠛⠉⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣤⣀⣀⣀⣀⣀⣠⣤⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠈⢿⣷⣄⣀⡀⣀⣹⣿⠆⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠹⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠶⢶⣴⣶⣶⣾⣿⣿⠟⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿
⠀⠙⢿⣿⣿⣿⣿⣷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⣿⣿⠿⠋
⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠦⣄⠀⠀⠀⢣⡀⠀⠀⠀⢰⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠙⢦⡀⠀⠸⡀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠙⢦⡀⠈⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⣀⡀⠀⠙⢦⡀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡴⠋⠉⠉⠉⠙⠀⠀⠱⡄⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠞⠀⠀⡹⠄⠀⠀⠀⠀⠀⠀⠀⠈⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢥⣤⡤⠀⠸⣤⣤⣤⣀⣠⣤⠤⠤⠖⠚⠁




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⣧⡀⠀⠀⠀⠀⠀⠀⢀⣾⠟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠹⣧⠀⠀⠀⠀⠀⠀⣾⠃⠀⠸⣷
⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⢿⣶⠶⠶⠶⢶⣾⡏⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⡀⠀⠀⠀⠀⢀⣄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠿⠃⠀⠀⠀⠀⠹⠿⠁⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠠⠤⣄⣀⡀⠀⣀⣀⠀⠀⣀⣀⡤⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⣾⡇⠸⣅⣸⠂⣽⠷⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣀⡸⣇⣀⠈⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⢷⡄⠀⠀⣼⡆⠘⠛⠛⡟⠛⠛⠀⢀⣴⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠃⠀⢠⡏⣷⠀⠀⠀⡇⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⣥⣼⡆⠀⠀⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⢀⡆⢠⡏⠀⠈⣧⠀⠀⡇⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠞⠁⠼⠁⠀⠀⠹⠀⠘⠇⠀⠀⠀⠈⠙⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠿⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣀⣀⣀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠙⣿⡿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢻⣿⢋⢡⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⡆⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⢿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠄⣀⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠔⢂⡥⠂⠀⠀⠀⠀⠀⠀⢈⡁⠀⠀⠀⠀⠀⠀⠐⢍⡒⠤⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠔⠁⠀⠀⠀⠀⠀⠀⠐⠚⠋⠉⠓⠂⠀⠀⠀⠀⠀⠀⠙⢦⡈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁




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

⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢠⣿⠛⠿⣦⡀
⠀⠀⠀⠀⠀⢀⣿⠁⢀⡀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴
⠀⠀⠀⠀⠀⢸⣿⡀⢀⣧⠀⠘⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋⢠⠇
⠀⠀⠀⠀⠀⣼⣿⡁⡈⢁⣠⣤⣽⣿⣿⣿⣿⣿⣶⣦⣤⣤⣴⣾⣿⣿⣿⣯⡁⠀⢰⠏
⠀⠀⠀⠀⠀⣿⣿⣷⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⠻⡿⣿⣿⣟⢛⣿⣷⣤⠏
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⢳⡽⣴⣽⢹⡿⣿⣿⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡿⡿⣟⢫⡛⠿⠟⠛⠻⠜⣿⠘⠷⠿⡾⠏⣷⠹⠘⢿
⠀⠀⠀⠀⣰⣿⣿⣿⣵⢬⣴⣴⣷⠶⠶⣶⠟⢻⠻⣶⡌⢻⣧⠀⠀⢰⣧⠶⣶⣦⡇
⠀⠀⠀⠀⣿⣿⣛⣷⣿⣿⣿⣿⣿⣿⣦⣝⡆⠈⢀⣼⣿⣌⣇⠀⠀⠉⡈⢀⠜⣴⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣭⣉⠙⣿⣿⣿⣿⡷⡦⠀⢿⣿⣿⣷⣀⣀⣻⡕⢺⣿⣿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠡⠤⠮⠯⠉⠙⢿⣏⡝⠀⢿⣿⡟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⡀⠂⠀⠀⢀⣠⣏⠀⠀⡜⠹⠃
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⡶⠚⢻⠋⢹⣷⠞
⠀⠀⠀⠠⠿⠿⣿⠿⠿⠟⠿⠿⢿⢿⡿⣿⣿⣿⣿⣿⣅⣀⣀⣀⣠⠟⠁
⠀⠀⠀⠀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠂⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡌⠑⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢷⣤⡉⢶⢄⡀⠀⠀⠀⠀⠀⠀⢸⣧⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⡄⢹⣿⠆⣼⠏⠀⠄⠀⠀⠀⠙⠚⢿⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡌⢸⣯⡴⠁⠀⠀⡀⠀⠀⠠⣄⠀⠀⠙⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⠟⠀⠀⠀⠀⣀⣤⣜⠃⢸⠰⠇⠰⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠄⢈⡥⠒⡈⠻⠿⠿⣟⠛⠀⠀⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡞⠃⠀⣠⠍⡷⠂⠀⠀⠀⠀⠀⠀⢈⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠑⣖⡀⣾⣻⠔⠁⠀⠀⠧⠤⠀⠀⠀⢠⣴⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⡁⣀⣉⠸⢧⣄⡀⠀⠀⠀⠀⢀⣘⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣧⣾⡟⠻⣿⣍⣛⣶⣤⣬⣬⣴⣷⡿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠾⠋⣀⣾⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻⠀⢀⡜⠈⣼⣿⣿⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡮⠀⠩⠀⢀⣸⣻⣾⡟⢙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣃⠀⠀⠈⠏⣽⠟⠏⣸⡟⠏⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⢱⠆⠀⢀⣼⡷⠀⢸⣏⠀⢀⣿⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠘⣷⣀⠀⠀⣠⣾⠃⠀⢀⡟⠀⢠⠿⣁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠈⠛⠧⠶⠛⠀⠦⠀⣾⢀⠠⠾⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠈⠻⣿⣿⣿⣦⡀⠀⠀⣠⣿⡿⠋⠁⠀⠀⠐⢶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢠⣶⣶⡶⠂⠈⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣤⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣤⣄⣀⣴⣿⠟⠉⠀⠀⠀⠀⠙⣿⣿⣷⣄⠀⠀⣠⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣾⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠋⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠈⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠙⢿⣿⣆⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⣴⣿⡿⠋
⠀⠀⠀⠈⢿⣿⣆⠀⠀⠀⠀⠀⣠⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⢀⣤⣶⣿⡿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣶⣄⡀⣠⣾⣿⠏
⠀⠀⠀⠀⠈⠻⣿⣿⣷⣦⣤⣴⢿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠛⢿⠿⠘⠗⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠟⠁⠁⠙⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣌⠹⠟⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢾⣿⢿⡿⣶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣊⣁⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡙⠛⠀⢘⣿⣿⣤⠄⠄⣄⠀⠀⡀⠀⢀⣔⣿⣿⠿⣿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡶⠀⢨⣿⡿⠁⠀⠀⠉⠀⠀⠀⠠⠀⠀⠙⢿⠀⣤⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⣼⡿⠃⠀⠀⠀⠀⠀⠰⡷⢠⠀⠀⠀⣤⠱⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣅⠀⣄⣠⡤⠶⣿⣿⣦⡤⠀⠀⠰⣿⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠗⠉⠁⠠⣤⡀⠀⠀⠁⠀⠀⠀⠈⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡀⠀⣰⣆⠈⠁⠀⠀⢀⠀⠀⠀⠀⢈⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢧⣼⣿⡿⠀⣂⠀⠠⠶⠆⠀⠀⠀⠟⣭⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠘⠳⠰⣶⣴⣿⠀⡀⠀⠀⠀⠀⢀⢘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠀⠀⡴⠋⣹⣿⣿⣿⣿⣶⣶⣶⣶⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠁⢒⠀⠁⠐⣿⠁⢈⡟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢸⡌⠀⠋⠈⠆⢀⠋⢱⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢺⠁⢸⡶⢻⢠⠀⢠⡀⢠⣾⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢐⡀⠀⠀⢿⢹⠘⣷⡟⢧⡌⣿⣿⡟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠋⠿⡀⠂⠁⠺⡂⠼⢃⡘⢠⣿⡟⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣄⠀⢁⠙⡆⠸⣀⠠⢦⢸⡟⠁⠄⢠⣻⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡝⡆⠀⢼⣧⣦⠈⢤⠈⡇⢀⡄⣠⣾⣟⠊⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⠀⠀⡤⣤⡀⠘⠻⢷⡄⢾⣼⠏⣴⢋⠿⠋⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠃⠀⠀⠀⠻⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⠈⠉⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣷⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠻⠿⣿⠇⣼⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⡿⠃⢻⣿⠋⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣀⣠⡴⠋⣼⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣴⣿⣿⣿⡿⠁⢻⣇⠀⠉⠁⠀⠈⠉⠉⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠋⠉⠁⠀⢀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⠈⡿⣇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠟⠉
⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⢸⠇⠘⢦⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⢀⣠⡤⠶⢯⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠈⠙⠶⣤⡀
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⡞⠀⠀⠈⠳⣤⣀⣤⠖⠋⠀⠀⣀⣤⠞⠋⠁⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠉⠳⠦⣀
⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠈⠙⢲⡦⣴⢾⣉⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⢀⡴⠚⡿⠁⠀⠙⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠈⣇⠹⡆⠈⢳⠀⠀⠀⢀⡀⠀⠀⠀⠈⠙⠶⠚⠁
⡎⠀⠀⡇⠀⠀⠀⠸⣆⠙⠿⣿⣿⣿⣿⣿⣿⡄⠀⣿⠀⢻⡀⢸⠂⠀⣴⢫⣇
⣇⠀⠀⣧⠀⠀⠀⠀⠹⣆⠀⠘⣿⣿⣿⣿⣿⡇⠀⡿⠀⠈⢧⣸⠀⢰⠃⠘⠉⡆
⢸⡀⠀⣿⠀⠀⠀⠀⠀⠈⠳⢤⣿⣿⣿⣿⣿⣃⡼⠃⠀⠀⠸⡏⠀⢸⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰
⠈⣇⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⠉⠀⠀⠀⠀⠀⢹⠀⢸⡇⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⣼
⠀⢹⡀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⣰⠋⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡏
⠀⠈⣇⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⠞⠁⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢻⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⡼⣇⣷⡤⠞⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠈⣧⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⢠⠞⣠⠟⠉⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢹⡄⠙⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠛⠉⠁⢀⣠⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠂⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠶⣶⠖⠒⠛⠛⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠘⠦⢤⡤⠶⠒⠒⠚⠋⠉⠉⠁⠀⠀⣼⡧⠾⠁⠀⠀⠀⠀⠀⢠⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⡀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⢀⣾⣿⣶⣶⣤⣤⣤⣤⣤⣤⣴⣾⣿⠷⣄⡀⠀⠀⢀⣠⡴⠋⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠙⢦⣄⣀⣀⣀⣤⠞⠁
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠉⠙⢻⠏⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡴⠟⠉⢻⡉
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⠏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⢧
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢠⠏⠀⠀⠀⠀⠀⢸⣀⣤⣤⣤⡴⠋⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⡏⠀⠀⠀⢀⡠⠞⠋⠁⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢀⡤⠿⠴⠚⠁⠀⢀⣀⣤⣶⣯⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣤⠞⠁
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠲⠶⠶⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣶⣶⣶⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠡⡰⠋⠓⢤⠁⠀⠈⠦⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⢄⡠⠀⡜⠉⠁⠀⡀⡀⠀⠀⠀⠀⠀⠈⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠁⠢⢁⠄⠊⠊⠄⠀⠀⠀⠀⠀⠀⠁⠀⠐⡨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠉⠒⠒⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠙⠢⡀⠀⢀⡄⠚⢉⠄⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠈⢦⡂⠌⣤⠃⢀⠔⠁⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠽⡾⣔⡾⠥⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣇⠙⡄⠠⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⣤⣾⣷⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⡗⡹⢁⣂⠇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⢀⡀⠈⡁⠈⠉⠢⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⡄⢀⠆⠘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⣁⣲⠁⠀⠀⠀⣀⠀⠀⠠⠆⠀⠀⠀⠀⢂⠀⠠⠀⢄⠠⠀⠐⠀⠘⠃⢍⡲⠤⡀⠀⠀⠠⠀⢼⠀⡈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠸⣇⣾⠀⠀⡆⠀⠢⡙⠋⠀⠀⠀⠀⠀⢀⠄⠰⠐⠀⠄⡢⠈⢡⠊⠠⠃⠰⠁⢁⡸⠦⢢⠀⡄⠙⠾⠇⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠙⢾⣦⠀⡸⡀⠶⠃⠘⡂⠄⠠⠀⠠⠁⡄⠃⢈⠢⠈⢂⠔⡕⠀⠀⠀⠀⣰⠇⡥⢀⣇⠶⢧⣵⠞⠀⠀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢙⠷⠀⣵⡀⠀⠐⠈⠢⣐⠓⢄⠀⢉⡒⠊⠀⠰⣷⢿⠤⠴⠶⣤⠾⠃⠀⠀⣡⡾⠟⡃⠀⠀⣤⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠀⠀⠀⠀⠌⢳⡄⠀⠀⠀⠈⠁⠒⠒⢯⣄⠔⣁⣦⠿⠀⠀⠀⠀⠀⠀⠀⣀⡴⠽⠘⠃⡀⣤⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠈⢳⣾⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣤⣄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠐⠤⢀⢄⠀⠀⠀⠀⠀⠀⢂⠁⡾⢛⠀⠀⠀⢀⠀⣠⡀⡠⠠⠀⠂⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢄⡈⠐⠄⠀⠀⠨⠐⢀⠀⠛⠀⠀⠀⠐⠜⢡⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢲⡀⠀⠐⡈⠄⢸⠀⠀⡄⠀⠀⠐⡀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠠⠐⠀⠈⠀⠀⡇⠀⠀⡨⢂⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⠀⢰⠀⠀⠁⠀⠀⢊⠅⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠈⠀⢠⠀⠀⢈⠔⡊⠄⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠂⠀⠂⠐⠀⠀⠐⡨⢂⡡⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢒⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠰⢴⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⢀⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⠀⠀⠂⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⢀⠁⠀⠀⠉⠉⠉⠳⣄⠀⠀⠀⠀⣄⠀⠀⠉⠀⠀⡀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⢠⠇⠒⠀⠀⠀⠛⠀⠀⠐⡆⠄⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠸⠃⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠳⠄⡀⠊⠀⠀⠀⢀⣀⡀⠀⠃⠀⡀⠀⠀⡇⠀⠀⠀⠆⠀⠀⠀⠁⢰⡄
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠸⠆⠀⠂⣄⠀⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⣰⠀⠘⠀⠀⠀⠀⠆⠂
⠀⠀⠀⠆⠀⠀⠀⢀⠀⠙⠒⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⡟⢀⠀⠄⠀⠀⠀⠀⡔⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠴⠋⠀⠐⠀⠐⠀⠀⠀⠉⠀⠄⢀⠰⠐⠀⠚⠀⠀⢀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠉⠉⢳⡀⢀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣄⠀⠁⠀⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⣶⠀⠀⢹⠀⠀⠀⠀⠀⡀⠀⠀⢰
⠀⠀⠀⢈⠊⠀⠀⠈⠀⠀⢀⠀⠀⡼⢐⠘⠀⠀⡇⠀⢹⠀⠄⠀⠀⠈⠀⠀⡼⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠐⠀⠠⠀⠀⠀⡏⠀⠈⠁⠈⠀⠀⠀⠁⠀⠘⡇⠀⠀⠀⠀⠀⠘⣇⠀⢰⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢷⠀⠀⢤⣄⠀⠀⠀⣤⠀⠀⢻⠀⠀⠀⢠⡄⠀⢻⡀⣦⠀⠀⣴⠠⢀⡠⠇⠀⠈
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠃⠀⠀⢸⠆⠀⠀⡇⠁⠀⠘⡆⠀⠀⢸⠁⠀⠈⣇⠻⠀⠀⠓⠀⠁⣀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠐⠁⠔⡂⠂⠁⠀⢄⠈⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⣾⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⡌⢀⡀⠀⡁⢀⡀⠄⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠈⠅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠐⠁⠂⠀⠒⠀⠁⠃⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⡀⠀⠁⠀⠁⠤⠀⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⢙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠠⠀⢙⢾⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠀⠀⠀⠀⠈⠻⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣞⡀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡎⠓⠄⠀⠀⠀⠀⠘⢦⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣴⣾⡟⡡⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡱⠌⠀⠀⠀⠀⠀⠀⠐⢝⠢⣍⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⣾⣿⣿⣿⣠⠆⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣇⠠⠀⠀⠀⠀⠀⡀⠈⢢⣱⣮⣦⡀⢿⣤⣤⣤⣤⣤⣤⣤⣄⣤⣼⠃⢸⣽⣿⣿⣿⡏⠀⠀⢀⣎⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⠀⠀⠉⠒⠄⠠⠁⣴⣼⢿⣿⣿⠇⠜⠘⠈⠠⠀⠀⠀⠀⠀⠈⠁⠀⠈⢿⣿⣿⡟⣡⠔⠀⠸⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠦⠀⣴⣿⡿⢿⣿⠁⠀⣀⠀⠀⠓⠁⠀⠀⠀⠀⠀⠀⠑⠀⡀⢉⡻⢿⣏⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢡⠘⠂⠀⠀⢀⠳⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠉⢀⣌⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢶⣴⡄⠏⠐⡠⠀⢀⠀⠀⠀⣀⡤⠴⣦⣤⡀⢼⠀⠘⠀⠀⠀⠀⠠⣯⠖⣾⡶⡄⢹⡇⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣈⡀⠀⠀⢨⡆⠀⢸⡀⠠⠀⡏⠀⢶⣿⠇⠘⡆⢃⠠⠀⠀⠀⠀⢰⡇⠸⣿⠇⡟⠈⣷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⡉⢉⡀⠂⠀⢧⠀⠀⠙⠄⠈⠋⠀⣼⠟⠀⠀⠀⠀⠀⠀⠈⣷⠠⠥⠞⢁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠀⣤⡄⠀⠈⢧⠀⠀⠀⠀⠁⠀⠠⠞⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠤⢺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠠⠐⢄⠀⠀⢝⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⡌⠂⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⠂⠀⠈⢕⡢⠤⠉⠉⠈⠀⠀⠀⠀⡀⠀⠈⠉⠻⣿⣿⠋⠁⢀⢰⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡜⢧⡀⠀⠀⠍⡁⣒⡀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⢨⠁⠀⢐⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣀⡈⢀⣀⣠⠴⠞⠳⢤⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠑⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢍⡉⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣠⠝⠛⠃⠙⠓⢒⠠⡴⢢⠟⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠐⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⡀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠠⠐⡮⠀⠈⢢⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡠⠲⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢽⡀⠐⠉⠀⠄
⠀⠀⠀⠀⡀⠓⢰⠁⠘⠓⠨⡀⡎⢉⠆⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡼⠄⠀⠀⠕⠀⣀⠀⢆
⠀⠀⠀⠀⠀⠰⠳⢊⠇⠹⠌⢋⠘⡈⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠢⠌⠂⣀
⠀⠀⠀⠀⠀⠰⣡⠂⠀⠀⠈⠤⠀⠁⠀⠀⠀⠐⠲⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢄⡀⡀⠀⢉⡇
⠀⠀⠀⠀⠀⠀⠠⡤⠁⠋⢢⠄⠀⠈⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣤⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡋⠀⠰⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠘⠰⠗⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠲⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡴⠀⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⠧⠀⠀⣄⠀⢠⡄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡂⠀⠀⠀⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠐⢠⠀⠀⠀⠀⠀⠀⠆⠀⠃⠀⠂⠐⠐⠀⠈⠀⠈⠂⠀⡹⠮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡄⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠘⠀⠀⠈⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀
⠀⠀⠠⣂⢹⡀⠀⢠⠅⠀⠂⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠉⠁⠐
⠀⠀⠀⠙⠀⠡⠀⠀⡑⠂⠔⠣⣤⢤⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠒⠊⢀⠀⠀⡐⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⡘⡆⠀⠀⠖⠼⢿⠁⠀⡴⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠅⠀⠓⠤⠀⠚⠁
⠀⠀⠀⠀⠀⠀⢨⡄⠃⠀⠀⣂⠀⡀⠀⢈⡝⠂⠀⠚⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣘⠊⢤⠓⠀⠀⠀⢾⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⣸⠀⠒⠃⠃⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠁⠀⠀⠁⢠⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡁⢁⡁⠀⠀⠐⠒⣤⢤⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⣆⠀⠀⠠⠠⡿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠓⠘⠁⠀⠀⠀⠘⠉⠋⠀⠀⠀⠀⠠⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣦⣠⣤⣤⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⣀⣀⣀⣾⣦⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⡿⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣼⣬⣿⣧⣿⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣟⣽⣿⣹⣦⣿⡇⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣀⣀⣀⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⠛⢻⣿⡍⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢿⣧⣿⣹⣿⣙⣧⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠈⢿⣿⣎⢿⣿⣧
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⡇⣿⣿⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡟⠁⢀⣴⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⡟⠀⠀⠀⢸⣿⡟⢸⣿⡟
⠀⠀⠀⠀⠀⣾⣿⡉⠉⠁⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⠁⠀⠀⠀⠀⠀⠘⠛⠀⠛⠃⠀⠀⠀⠘⠛⠁⠘⠛⠁
⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣶⣶⣾⣿⣆⠀⠀⠀⠀⠀⠙⢿⢸⣿⢿⣿⡟⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⣰⣷⣄⣤⣠⣾⣦
⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣿⣏⣿⢻⣿⢻⣍⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢻⣿⣿⣿⡟⣿⣧⣤⣤⣄⣀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣽⣻⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣮⣴⣿⣾⣷⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣼⡻⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⡿⢿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠻⣿⣦⣭⣭⣉⣭⡁⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣿⣿⣿⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠀⠀⢀⣼⣿⠀⠀⠀⠴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⣤⣤⣤⣼⣿⣿⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢹⣿⣿⣿⡏⠿⢹⣿⣿⡟⠀⢠⣀⣠⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢿⣍⠻⣿⣿⣿⢛⣹⠃⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢿⣿⣿⣿⣿⣿⠟⠀⢀⣤⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⣼⣿⣿⣿⡅⠀⠀⣾⡏⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⢸⣿⠿⡿⢿⣷⠀⠀⣿⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⢸⣿⡄⠀⢻⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⣿⣿⡇⠀⢸⣿⡄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⢸⣿⣿⣿⣼⣿⣿⡇⠀⠀⣿⡇⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣖⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡖⠓⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⡇⠀⠀⠀⠀⠀⠸⡄⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠊⠀⠀⠀⠀⠀⠀⠀⠑⠒⠚⠁⠀⠀⠀⠀⡴⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠑⡄⠀⠀⠀⢸⡁⠈⠓⢤⡀⠀⠀⠀⣀⣀⣠⠏⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠀⠀⠀⡧⠀⠀⠀⠀⣇⠀⠀⠀⠙⠂⠈⣉⣍⠀⠀⠀⠈⠉⢁⡘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠤⠤⠃⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠈⢁⣀⡀⠀⠀⠀⠀⠿⢿⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡙⢾⣿⠄⠀⠀⣠⣤⡀⠀⠄⢿⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠒⠊⠀⠄⠀⠙⢛⠀⡀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠉⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⣀⡤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠣⣀⠀⠀⠉⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠈⢹⠄⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⢀⠞⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢠⠏⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⡾⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⢳⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠸⡄⠘⣇⠀⠀⠀⠀⠀⡔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⢰⠀⢰⠁⠀⠀⡇
⠀⠀⠀⠀⠱⡄⠈⠢⡀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⣰⠃⡆⠀⠀⣇
⠀⠀⠀⠀⠀⠙⢦⡀⠈⠓⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⡟⠁⠀⡇⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠤⠤⠜⢦⣀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠲⢴⢻⡄⠀⠀⠹⣄⠀⢱⡀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⢤⣄⣀⣤⡏⠀⣄⣰⣨⠇⢳⠀⠀⠀⡈⡆⠀⠙⠒⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠓⠀⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⢸⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣄⢀⣀⣀⣀⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⠿⡀⠻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣯⡀⠀⠀⠙⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⡿⠛⠓⠃⠀⠀⠈⠛⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠘⢿⠁⠀⠀⣀⡀⠀⢀⠀⡀⠀⠀⢊⡙⠛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠒⡀⠀⣀⠠⢄⠀⠀⠤⢀⣾⣿⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠁⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⣺⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⣴⣿⣿⣴⣶⣾⣿⣿⣿⣤⡀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠉⠀⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣤⣄⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠙⠋⠙⠛⠛⠛⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠟⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢄⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠗⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢹⣿⣧⣠⠀⠀⠀⠀⠀⣤⣿⣿⡔⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢰⣿⠟⠁⠀⠀⠸⡖⠀⠀⠙⢿⡟⢘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣬⠋⠀⠀⠀⡀⠀⠀⢠⣀⡀⠀⢳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡤⡾⢹⡄⣤⠀⡫⠖⣦⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡉⠳⣦⠌⠀⠀⠀⠑⠶⢃⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄⠈⠠⣖⣐⡄⢳⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⣿⣇⣁⡀⣿⣿⡀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⣿⣿⡿⢿⣷⣶⣶⣿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⠀⢸⣿⣇⢀⡉⢡⡿⣇⠆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡤⠈⡿⠃⣿⡇⠛⣿⠇⠀⠁⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⡾⠁⡄⠀⣷⣴⡿⠀⠀⠀⠇⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢠⠏⢀⠃⠀⠀⣿⠁⠀⠀⡛⠀⣿⢰⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣾⠀⣼⡀⠀⢀⠀⠀⠈⠀⡟⣼⡇⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠄⣠⠟⢩⡄⡉⢳⣀⠀⠄⠀⢀⣼⣰⣿⡀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣿⣶⣿⡇⡿⣦⣿⣆⡀⠀⢱⣿⣿⣶⡆⠀⠚⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⣿⡏⠉⠳⢷⣮⣹⣿⡆⠀⢸⡁⠈⢿⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠃⠀⠀⠀⠀⠀⢹⡇⢠⡎⠛⣿⣿⡘⣄⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⡟⠀⠀⠀⠀⠀⠀⢸⠁⢻⠀⢀⣾⡿⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⡇⠀⠀⠀⠀⠀⠀⣾⢰⣧⣶⣿⣿⠃⠀⠀⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⣿⠂⡏⠛⠋⠁⠀⠀⠀⡏⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇⠀⠀⠀⠀⠀⠀⡟⠠⡇⠀⠀⠀⠀⠀⢰⣷⣖⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⣸⠇⠀⠀⠀⠀⠀⢸⠁⠀⡇⠀⠀⠀⠀⠀⠀⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⠇⠃⠀⠀⠀⠀⠀⠀⠻⠿⠘⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⣦⣴⡄⣤⠄⠀⠀⠀⠀⠀⠀⠻⡟⠀⠻⠂⠰⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠈⠁⠀⠻⠋⣀⣀⠀⢴⣦⣶⠈⠿⠃⠀⠀⠀⠸⡾⠂⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠘⣿⠟⠺⠂⠀⠙⠋⢀⣈⠛⢃⣀⡀⢲⣶⠆⢀⢠⣤⣤⡌⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠠⣶⠄⣇⠘⠿⠃⠈⠋⠀⠀⠉⠀⢠⣤⠀⣼⠈⠻⠟⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢴⣴⠄⠀⠸⣆⠀⢶⠇⠀⠀⣇⠲⡶⠀⠁⢠⡏⢤⡄⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣍⣀⣙⣃⣘⣷⣄⡘⠿⠀⢸⡄⠀⠀⣠⣾⣥⠤⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⢿⣦⣄⠘⣿⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠿⠿⠦⠄⡀


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠤⠤⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⢀⣤⣄⠀⢀⣤⡀⢀⢠⡀⠀⠀⠀⠀⣀⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⡞⠁⠀⠓⠀⠀⠒⠀⠃⠁⣈⡳⠀⠀⣴⣦⣄⠀⢤⣶⣶⠀⠉⠙⢻⣇⣽⠉⠑⠀⣸⢁⠑⢋⢸⡇⠀⠀⣠⠞⠁⢀⡈⠁⣤⠈⠢⡀
⠀⠀⠀⠀⠀⡈⠀⠋⠀⢠⠀⠀⠄⠀⡄⠀⢩⠃⠀⠀⡏⢁⡥⠠⠼⡀⡸⠀⠀⠀⠸⠀⠀⠀⠀⠀⡿⢀⠐⠀⠜⡇⠀⣰⠁⢀⣄⠛⡹⠀⠫⣄⠀⠙⡀
⠀⠀⠀⠀⠜⢠⠀⠀⠂⠀⠠⠄⠀⠀⢀⠀⠇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠆⠀⠂⠀⡇⠀⠐⠀⠘⡇⠀⠑⠛⠁⠸⠶⠶⠶⠶⠈⠑⠮
⠀⠀⠀⠈⠓⠄⡠⠤⠄⠴⠖⠒⠢⠤⠬⡼⠀⠀⠀⠀⠀⠀⢀⡤⣶⠛⠋⠉⢦⣀⠀⠀⠀⠀⠀⠀⡇⠄⢸⣀⠐⣇⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢤⣀⡀⠠⢤⣄⠀⠀⠀⠀⠀⠀⠀⢀⡾⣵⠟⠣⠬⠶⣄⠈⠳⣄⡀⠀⠀⠀⡗⠀⣼⡇⠐⣿⠀⠀⠀⠀⢀⡏⠉⠉⠉⢷
⠀⠀⠀⠀⠀⢠⠞⡅⠀⠀⠀⠀⠐⠀⢙⢦⠀⠀⠀⢠⠶⡿⡛⠛⢆⠀⠀⠀⢸⣷⡀⠀⢿⡀⠀⠀⣷⠂⢻⣷⠁⣾⠀⠀⠀⠀⡸⠁⠀⠀⠀⠠⡆
⠀⠀⠀⠀⡴⡁⡀⢴⠀⡀⠀⠀⢰⡓⠮⡼⠁⠀⢠⣿⠾⣎⠀⠀⣈⢤⠀⠀⣼⠋⠝⢦⡀⢧⡀⠀⠈⠙⠒⠛⠋⠁⠀⠀⠀⠀⡇⠀⠀⠖⠀⢀⡃
⠀⠀⠀⠀⠑⠋⠀⢸⠀⠁⠀⠀⠀⠇⠀⠀⠀⠀⠈⢷⡀⠈⡷⢦⣘⠁⠈⠳⣿⠐⠀⠌⣷⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠢⠞⠀⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣴⣂⠈⢠⣴⡇⠀⠀⠀⠀⠀⢈⡷⣿⡀⠀⠈⠽⠲⢤⣽⡄⠂⠠⢸⡏⣷⠀⠀⠀⣀⣤⣶⣶⣶⠤⣄⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⣠⣤⠽⠃⠀⠀⠀⢀⡀⠀⣷⠈⡻⡶⢧⣀⠀⠀⠐⠁⠉⠀⢬⣷⠻⡄⠀⢰⣍⡍⠀⠀⠈⣷⢏⠀⠀⡞⡹⢠⠂⠀⠢
⠀⠀⠀⠀⣠⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠛⣦⠈⢧⡀⢨⡷⠌⠛⡷⠬⠤⣤⡤⠞⢩⣧⢧⡀⠈⢿⣻⣤⣤⣴⠿⠁⠀⠀⢂⠰⠰⡀⠀⢠⠠
⠀⠀⠀⠘⣥⠙⣎⠠⠀⠐⠀⠀⠀⠀⠀⠀⠈⠳⠴⣿⠀⠀⠙⡾⠗⠲⢦⣅⣠⣤⠏⠀⠄⠀⢹⡾⡇⠀⠀⠈⠃⠿⡉⠀⠀⠀⠀⠀⠈⠒⠊⠿⠑⠁
⠀⠀⠀⠀⠙⠤⠃⠀⠀⠀⠀⠈⢰⠋⠙⠀⠀⠀⠀⠈⢷⣄⠀⢻⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠂⣿⣇⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⢰⣉⡀⠀⠀⠀⠀⠈⠓⠊⠀⠀⠀⠀⠀⠈⠁⠀⢸⡄⢀⡀⠀⠀⢀⠀⠀⠀⠆⠀⠸⡇⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⣎⠉⢭⡄⠀⠀⢠⠶⠶⢦⠀⠀⡤⠶⢦⡀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠃⢷⠀⠀⡿⠛⠛⠟⠛⣟⠀⠀⣑⡀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠈⡟⠓⡎⠀⠀⢱⠓⢻⠁⠀⠀⢸⠇⠠⠀⠈⠀⠀⡄⠀⠀⠐⠀⢀⣸⠀⣿⣿⡀⠀⠀⢠⣿⡇⠀⡇⠘⡄⠀⠀⡰⠀⡇
⠀⠀⠀⠀⠀⡇⠠⠀⡇⠀⠀⠀⢧⠀⢃⠀⠀⡸⠀⡼⠀⠀⠀⠈⢧⣀⣀⠠⠀⠀⣀⡀⠐⢀⣠⠴⠚⠀⠙⠿⠥⢦⡤⠼⠿⠃⠀⠘⠤⠅⠀⠀⠥⠜
⠀⠀⠀⠀⠀⣇⠀⢀⡇⠀⠀⠀⠀⠑⠋⠀⠀⠓⠊⠁⠀⠀⠀⠀⠀⠀⢹⠙⠉⢻⠉⣿⠉⠉⡇
⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⠀⠀⠀⠀⠀⠶⢾⣿⣿⠷⠆⠀⠀⠀⠀⠘⡆⠀⡼⠀⢹⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⣀⣤⣀⠀⠀⠀⢀⡠⡄
⠀⠀⠀⠀⣠⡶⡿⠟⠿⠉⠉⠝⣄⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⣷⠀⡇⠀⢸⡇⠀⡇⠀⠐⠉⠦⠟⠂⠀⠀⢸⣿⣧⠀⠸⣿⡟⠉⠙⠷⡇
⠀⠀⠀⠸⣏⡄⡆⠀⣻⠀⠀⠘⢾⠀⠀⠘⣿⣧⠀⠀⢰⣿⠇⠀⠀⠀⠀⡿⠀⣇⠀⠀⣇⢰⡇⠀⢸⠂⠲⠈⠃⠀⠀⢸⠀⠀⢀⡀⠀⠀⢴⣆⠀⡇
⠀⠀⠀⠀⣿⣰⢃⢺⡟⡅⠀⠀⢨⠀⠀⠀⠈⠛⠀⠀⠘⠋⠀⠀⠀⣠⠞⢁⣠⠼⠀⢠⠏⠸⡇⠀⠈⠇⠘⠀⠃⠀⠀⣾⣄⠀⠻⠗⠀⠀⠈⠁⢀⢱⡄
⠀⠀⠀⠀⢹⣗⣯⣼⣯⣧⡀⣀⠛⣒⣶⣶⣶⡒⠶⠶⢦⠀⠀⠀⠀⠈⠉⠁⠀⣠⣖⣁⡤⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⡗⠛⢀⡀⠀⢰⣶⡄⠀⠸⢿⡇
⠀⠀⠀⠀⠈⢿⠀⣀⠀⠉⡍⠉⢠⠄⠀⢠⠀⠀⠠⠄⢸⡀⢠⢀⡄⠀⠀⣔⣤⡄⠉⢁⡀⣀⡀⠀⠀⣠⠀⠀⣀⠀⠀⣱⠀⢮⡷⠀⠀⠉⠀⠠⣶⣶
⠀⠀⠀⠀⠀⠘⢧⣀⣐⠀⠀⠆⠀⠀⠄⠀⠀⢤⣀⣐⡞⠀⡎⠹⢓⠀⠀⠉⠋⢑⠀⠈⡖⠘⠻⢦⠀⢘⡉⠙⠛⣧⠀⠻⠔⢤⣀⠀⣰⣿⡦⠤⠤⠽
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠙⠒⠋⠀⠀⠉⠋⠁⠀⠀⠉⠙⠛⠋⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣶⣤⡀⠘⢿⣿⡿⠃⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠁⠀⣴⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠙⢿⣬⣭⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⡿⠿⠛⠉⢠⣤⣤⣀⣀⣀⣴⣶⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⣷⣦⠀⠈⠉⠛⠛⠛⢿⠿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⣷⣮⡻⡇⠀⠀⠀⠀⣀⣴⣿⣿⡆⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⣝⠿⣿⣿⣦⡀⢀⣠⣾⣿⣿⢟⣭⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣮⡻⣿⣿⣿⣿⡿⣫⣶⣿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠈⠙⠛⠁⠀⢿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⣤⠀⠀⢀⣄⠀⢀⢤⠀⠀⢠⢄⠀⢠⢄⠀⠀⡠⢄⠀⣠⣄⠀⠀⣠⡄⠀⢀⢤⡀⠀⢀⢤⠀⠀⠀⠀⡤⡀⠀⠀⣠
⠀⢰⣿⣿⣿⣿⣿⡆⡌⠀⠉⠉⠁⠈⡄⡇⣀⠉⠉⡁⢸⠀⣿⢿⣿⣿⠿⣷⠀⡼⠘⠉⠉⠉⠀⡇⠀⠀⢰⠑⠋⠀⠙⠂⡇
⠀⠘⣿⣿⣿⣿⣿⠃⢣⡈⠁⠤⠉⣠⠃⢇⠚⠣⠌⠓⡹⠀⢷⣿⣿⣿⣷⡿⠀⢳⡈⠁⠀⠘⢠⠇⠀⠀⢸⡀⠛⠠⠘⢃⡇
⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠐⠒⠈⠁⠀⠀⠉⠐⠒⠉⠀⠀⠀⠉⠛⠋⠉⠀⠀⠀⠈⠑⠒⠈⠁⠀⠀⠀⠀⠈⠒⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠈⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠙⣮⡛⢦⣀⠀⠀⠀⣀⣠⣤⣤⣤⡀⢀⡴⠟⠁⢀⡄⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠈⢷⠀⠈⠁⠀⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠘⠳⠄⠀⣠⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡇⢴⣶⣶⡄⠀⠀⠀⠀⠀⠀⢀⡾⣯⣿⡟⠂⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣋⠁⠀⠀⠀⠀⠀⠀⠀⠠⣽⣯⡶⠃⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⠛⠉⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⢁⡀⠤⢄⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⠙⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣄⣀⠈⠙⣛⣻⠏⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢒⡲⣷⠀⣻⠆⠹⠙⠛⠐⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⢠⡜⣧⠙⢿⢸⢇⡏⣦⠟⠰⢷⣾⣴⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣥⡿⢛⡛⠙⠿⠿⣧⣦⠀⠤
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣆⣸⣧⡾⠋⠀⠐⠒⡻⠛⠛⠛⠛⠟⠃⠀⠀⠀⠀⠀⠀⣼⡏⣠⣤⣤⣤⣭⣽⣷⣴⣦⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠘⡏⠈⠉⠉⠀⠀⠀⠀⠀⠀⠻⣷⣶⣤⣦⣴⣦⠀⠀⠀⠀⠀⢀⣿⣧⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⣀⣀⣀⣀⠠⠀⢈⣭⡛⠍⠻⣿⠀⠀⠀⠀⠀⢸⡿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⡔
⠀⠀⠀⠀⠀⠀⠀⠁⠹⡇⠀⠈⢉⢩⡭⠉⠃⠀⣾⡿⢃⠈⢃⠀⠀⠀⠀⠀⠀⢸⡷⣫⢿⣿⠀⠀⢂⡉⠒⠂⠀⢀⣴⣀⡜
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⢣⠀⠈⠀⠀⠀⠀⠀⠀⢸⣧⠉⠀⠋⠀⠀⠉⠛⠀⠀⠀⠸⠋⠉
⠀⠀⠀⠀⠀⠀⠀⡀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢉⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠰⡈⡃⠀⠀⢀⡉⠲⠀⠀⠀⠀⡴⠞⠁⠀⠀⢀⠀⠚⠀⠀⣻⣿⣿⣧⡀⠀⠀⠀⠀⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⣉⣉⠉⠀⠀⠐⠁⠀⡐⠀⠀⠀⠀⢠⣡⢂⣀⣠⣄⣿⣿⣿⠃⠳⣦⡀⠀⠀⠈⠙⠛⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠉⠀⠀⠀⠀⢀⣤⠾⠁⠀⠀⠀⠀⢈⠙⠛⠻⣿⣿⣿⣿⡿⠀⠀⠘⠿⣷⣦⣤⣤⣤⢆⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣴⣶⡾⠟⠁⠀⠀⠀⡐⠀⠀⠀⠙⠳⠾⠿⠿⠛⠙⠄⠀⠀⠀⠀⠈⠉⠸⣿⣿⣯⢀⠇
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⡛⠉⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠑⠢⠔⠊⢂⠀⡐⣈⡻⠿⣏⢀⡀
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠂⠀⠡⠀⢀⡴⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⢀⠠⠉⠉⠉⠀⠀⣹⠀⠀⠉⡒⠤⡀
⠀⠀⠀⠀⢀⢂⡎⠀⠀⠀⠀⣠⣤⣗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠚⠁⠀⠀⠀⠈⠑⠊⠁⠀⠀⠀⠓
⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⣼⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡔⡰⢁⠞⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠉⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣦⣤⣤⣤⣴⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣍⣻⣿⣏⣽⣿⣿⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠈⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠹⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⢻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡇⢀⣿⣿⠀⠀⠀⠀⠀⠀⠘⠿⠿⠃⠀⢠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⣴⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠙⢻⣦⣀⠀⠀⠀⠀⠀⢀⣴⣿⠅⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣮⠀⠀⠹⣿⣷⣾⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠐⠀⣀⣽⡿⢿⣿⣿⣿⣿⣿⣤⡀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠿⠛⠋⠉⠈⠁⠋⠈⠻⠛⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢈⠀⠀⠀⠀⠀⣶⣶⣦⠀⠀⠀⣴⣿⡿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠹⣭⣤⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⠀⠈⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡀⠀⠀⠀⣜⣂⣀⠠⠥⢀⣴⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠂⠀⠀⠈⠉⠳⠤⣴⣾⣥⣴⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡟⢿⢿⣿⣽⣿⡿⢻⣿⣷⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡿⣿⣿⣽⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣧⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣯⢻⣭⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢈⡿⡿⠇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡎⢷⡆⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢀⡀⠊
⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⢰
⠀⣾⣿⣿⣧⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠙⠛⠛⢳⠀⠀⠀⠘⢧⣰
⠀⣹⣿⣳⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠈⠀⠄
⢰⣿⣿⣿⣿⣿⣿⣿⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⠀⠀⠀⢀⣀⣠⣼⣷⣄⣀⡀⣀⣀⣴⣶⠶⠆
⠀⠉⠙⠛⠛⠛⠋⠻⠛⠛⠓⠂⠤⠤⠤⠤⠄⠐⠖⠚⠛⠋⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⠀⠀⠀⠀⠀⠘⠫⠄⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠙⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡀⠀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⡀⠀⢐⠾⡅⠀⠀⠃⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⣌⡠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⢨⡃⠀⠀⠀⠀⠀⡀⠐⠄⠀⠀⢀⠀⣀⠀⠁⠀⠀⠀⠁⢤⠋⠀⠠
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠈⠙⣤⠃⠀⠀⠀⢸⠖⠋⠉⠖⠂⠂⠀⠈⠈⠉⠴⠢⡤⠀⠀⠀⢾⡀⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⣷⠀⠀⠀⠀⡏⠀⠀⠀⢀⠠⠄⠡⠈⠢⠀⠀⢀⣰⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⢿⠀⠀⠠⠐⣇⠀⡇⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⢂⠀⢸⡆⠀⠀⡀⠀⠐⠈
⠀⠀⠀⠀⠈⠀⠀⠈⣠⠧⠀⢀⢀⢸⡆⠀⠀⠀⣻⡆⠀⠁⠀⠀⠀⠀⠀⠰⠁⠀⣀⡝⠈⠁⠀⠀⢸⣇⡀⢀⠥⢀
⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠠⠀⠙⣧⠄⠀⠀⠉⢳⣄⠄⠀⣄⠀⠀⠀⠀⡠⢀⡼⠁⠀⠀⠀⠐⢸⠇⠀⠈⣄⠤
⠀⠀⠀⠀⠀⠀⢀⠀⢱⡃⠀⠀⠀⠦⣿⠀⠀⠀⠠⠀⠙⠳⣤⡇⠀⠀⠀⢨⡴⠏⠀⠀⡀⠀⡀⠠⣼⠛⠀⣠⠏⠀⠀⠤
⠀⠀⠀⠀⠀⡀⠂⣀⠉⠉⠖⠲⢤⣀⣿⠀⠀⡀⠐⠀⠀⠇⠉⠛⢶⣤⣷⠋⡄⠀⠀⢀⡈⠀⠁⠀⣿⣠⡾⠋⠰⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢻⡏⠀⠀⠄⠀⠀⡀⢀⣰⡶⠛⠉⠉⠛⠷⢶⣤⣍⠀⢀⠀⢾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣿⠀⠀⢰⠀⢀⣼⠿⣥⠀⡈⡠⠀⠂⠀⠈⠉⢙⣷⣷⣄⣸⣿⠀⠀⠀⠐
⠀⠀⠀⠀⠐⠄⠀⢀⠀⠀⠀⠀⠄⠀⢹⣇⠠⣰⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢨⠗⠙⠂⠈⠻⣿⣿⡆⠀⠈⡁⡀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠁⠀⠈⣿⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⣿⣿⡗⠀⠈⠀⣧⢤
⠀⠀⠀⠀⠁⠔⠒⢬⡂⠀⠀⢀⠇⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⢀⣿⣿⠗⢀⡀⣨⠇
⠀⠀⠀⠀⠀⠀⠀⠞⢯⡀⢠⠎⢁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⡴⣟⠑⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠤⠈⠛⠛⠷⠾⢶⣄⣸⣿⡇⠁⠀⠀⠀⠀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⡇⠙⣿⣧⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⢹⣿⡇⢸⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣼⣿⣷⣸⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⠻⣿⣿⠃⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠆⠀⠀⠈⣻⣿⡏⠀⢿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⣿⣿⠃⠀⠘⣿⣆⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣿⣿⣿⣇⣀⠀⢰⣧⣸⡄⠀⢀⣿⣿⠀⠀⠀⠀⣸⣿⣿⣿⣇⡀
⣶⣶⣶⣶⣶⣶⣴⣶⣶⣾⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠻⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠒⠶⠤⢤⣤
⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠠⠋⠉⠳⡀⠀⠀⠀⠀⠈⠛⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠀⠀⠢⡀⠀⠀⡠⠁⠀⠀⠀⠀⠀⣸⣿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣶⡆⠀⢀⣼⣿⡿⣿⣿⠏⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⠛⢿⣿⣷⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣷⣿⡿⠟⠃⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢺⣿⣿⣿⠀⠚⠀⠀⢱
⠀⠀⠀⢀⣀⣀⣀⣤⣼⠿⠟⠉⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠈⣻⣿⡟⠀⠀⠀⠠⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠛⢲⡀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣶⣄⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠘⠀⠀⠀⠰⣿⣿⣿⣿⣿⡆⢿⣿⢿⣷⣾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠈⠀⠀⠈⠙⠓⠊⠉⠀⠀⠉⠓⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⣠⠔⠒⢤⠀⣴⣾⣿⣿⣿⣷⣀⣀⣀
⠀⠀⠀⠀⠀⡃⠀⠀⠀⢻⣿⣿⣿⠿⢿⣏⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⢷⣦⡀⣠⣶⣀⣀
⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⢻⣿⡇⢀⣼⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠟⣿⣿⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⢀⣿⠀⣸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠁⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢦⡠⠃⠀⠀⠈⢿⡄⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⡁⡉

⠀⠈⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣀⡀
⠁⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠈⢹
⠀⠀⡀⠀⠀⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠰⠋
⡀⢀⡀⠀⠀⠃⠀⠀⠋⠈⠛⠂⠀⠑⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢘⣦⣀⠀⠀⠀⢠⡈
⠀⠈⠱⠃⠙⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣴⡈⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠴⠀⢺⡄⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣲⢠⡄⢀⢠
⠀⠀⢠⠀⠀⢸⠏⡂⠀⠀⠀⢀⡔⣦⠀⠀⢀⠈⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠋⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠤⡄
⠀⠀⠈⡄⠀⠀⠈⡁⠀⠀⠀⠈⠃⠈⠀⠀⠨⠞⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡠⠁
⠀⠀⠀⢊⡀⠄⠀⠠⠀⢠⠀⠀⠀⢀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⡁⠀⠀⠀⢀⡎⠦⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢋⡰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠋
⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⡄⠀⠈⠉⣤⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⣤⢰⠷
⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⢀⢠⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢀
⠀⠀⠘⡁⠂⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸
⠀⠀⠀⠐⠆⠀⠀⠰⠀⠀⣡⠀⠀⠀⠀⢀⠐⠆⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡜⠃⠘
⠀⠀⠀⠀⠀⠘⠆⠀⠁⠀⠁⠀⠀⠀⠀⠈⠑⠂⠀⡐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣀⠠⠰⠀⠀⢀⡄⠀⠀⠀⠀⢶⠀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠐⠀⢠⡄⠀⠀⠠⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⣮⠑⠀⠀⠀⠀⠀⠻⠆⠰⡜⠃
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⣄⡀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢘⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠀⠐


⠀⠀⠀⠀⠀⢀⡴⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦
⠀⠀⠀⠀⠀⠂⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢃⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⢀⠀⠠⡄⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠠⡠⠰⠞⠀⡆⠀⠀⠀⠀⠀⠀⣀⡤⠂⠀⠉⠙⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⡄⠀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠰⠆⠀⠀⣠⡆
⠀⠀⠀⠋⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠰⠈⢁⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡂⢦⣰⡾⠁⠀⠘⠀⣀⡀⠀⠀⠀⠀⠙⠁⠀⢀⢠⡟
⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⡀⢤⡀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⠉⠀⠀⠐⠀⠈⢠⡍⡀⠀⠀⠀⠀⠠⢐⡤⡎
⠀⠀⢀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⣀⠀⠀⠀⠐⠇⠀⠀⠀⠠⠗⠈⢀⠁
⠀⠀⠀⠁⠒⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠒⡄⠀⠀⠀⠀⠀⠀⠙⠄⠘⠃
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢾⡐⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⡈⢃⠀⠀⠀⠐⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣈⠁⢀⣀⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠐⡉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠟⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣄⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣹⡿⠟⢿⡜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣰⠏⠀⠀⠀⣿⢳
⠀⠀⠀⠀⢀⠤⣒⣶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢇⡟⠀⠀⠀⠀⠻⢸
⠀⠀⠀⠀⠁⣾⡟⠛⠻⢦⣄⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⢸⠁⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠈⠳⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢸⠀⠈⠀⠀⠀⠀⢠⣞⣡⣆
⠀⠀⠀⠀⢸⡇⢠⠀⠀⠀⠀⠰⣌⠢⠙⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⣆⠀⠀⡶⢠⣤⠞⠈⠉⣿
⠀⠀⠀⣶⠸⡇⠸⡄⠀⠀⠀⠀⠘⣧⠀⠈⢿⣷⡀⣀⣤⡤⣔⢻⡛⣏⡁⣰⣿⣿⠀⠀⣧⣸⠇⣀⡜⢠⣿
⠀⠀⠀⢻⡄⠁⠀⢧⠀⠀⠀⢀⣦⢸⠀⠀⠀⠙⣿⣏⡓⠣⣌⣿⡧⢻⡵⢌⡉⠻⣦⡘⠉⠁⢸⡏⢠⣾⡿
⠀⠀⠀⠈⢿⢄⠀⠘⣆⠀⠀⣤⠀⠈⠀⠀⠀⢀⣞⣹⡏⡄⡈⡏⢻⠀⠙⣓⠲⠤⠈⠛⠿⣖⢦⡛⢶⣿⡇
⠀⠀⠀⠀⠈⢧⠳⡆⠘⢷⣼⠿⢋⣳⣴⠀⡴⢻⠟⠋⣨⠖⡡⣟⡻⠳⡁⢸⠉⠛⠒⠂⠀⠈⠁⠑⠀⣿⠃
⠀⠀⠀⠀⠀⠘⡇⢧⡀⠀⠀⠒⠻⠉⠀⠀⠀⠀⠀⠈⠀⠉⠀⠜⢳⠀⠙⠘⠀⠀⠈⢷⡈⡙⠶⢤⣸⡇
⠀⠀⠀⠀⠀⠀⠁⢸⣿⣦⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡄⠀⠀⠱⠄⠀⠀⡜⠁⣴⣾⣿⠿⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣿⠶⢠⡇⠀⣧⠄⢠⡾⠵⠒⡒⠈⠳⣀⢰⡀⠀⠈⣾⡇⣸⠝⢿⣿⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣧⢸⡇⢠⡟⠀⠋⢴⢾⣿⡿⣿⣦⡈⠻⣇⠀⠀⠹⡆⢹⣦⣀⡠⠜⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠙⢧⠃⠸⣧⠀⠀⢇⠈⠻⠏⠀⣸⡷⣄⠹⡆⠀⠀⢷⣸⡆⠠⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⠀⢿⡄⠀⢀⠑⠂⠐⠚⠁⣠⣾⣦⡇⠀⠀⠀⢹⣧⠀⠈⣹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠳⡈⠻⣦⡈⠙⠿⣿⣿⣿⡿⠟⠹⡇⠀⠀⠀⠀⢻⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠘⢦⡈⠛⠲⢤⠀⠀⠀⠀⠀⠀⢹⣀⣤⣤⣤⣾⡗⠀⢙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣄⣀⠀⠠⠄⠤⠠⠈⠿⣿⡍⢡⡜⠀⢂⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⢹⠿⠷⣦⣄⣀⠀⠁⠀⢀⣹⠿⢤⡶⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣩⣴⠟⢀⡟⠀⠀⠀⠉⠛⠳⠶⠾⠏⠥⠔⠋⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣤⠶⠋⡽⠋⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠉⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠋⠀⠀⣸⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠾⠟⠛⠛⠉⠉⠀⠀⠀⣀⣤⡾⠋⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣌⠙⠳⡄⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠁⠀⠀⠀⢀⣀⣤⣤⣴⠶⠛⠋⠁⣀⣤⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠀⠀⠀⢀⡴⠿⠛⠛⠉⠀⠀⠀⠀⠀⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⡴⢦⡀⠀⠀⠀⠀⢀⡴⢦⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣼⠃⠈⢷⠤⠤⠤⠤⡾⠁⠘⣧⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⡇⠀⠘⠳⠶⠛⠀⠀⠛⠶⠞⠃⠀⢸⡄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣸⡇⠠⠤⣤⣤⡀⢠⡄⢀⣤⣤⠤⠄⢸⣇⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⠴⠶⠶⠆⠈⠀⠰⠶⠶⠶⣀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣀⣀⣤⡤⠶⠚⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣼⠟⠻⠿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⣿⡄
⠀⠀⠀⠀⣿⠀⢀⡀⠀⠉⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠉⠀⠀⣀⠀⢸⡇
⠀⠀⠀⠀⣿⡇⢸⣿⣷⣦⣤⣀⣈⡙⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⣉⣀⣠⣤⣶⣿⣿⠀⣿⡇
⠀⠀⠀⠀⢹⣧⢈⡻⣿⣿⣿⣿⣿⣿⣷⡌⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⣴⣿⣿⣿⣿⣿⣿⡿⣋⢠⣿
⠀⠀⠀⠀⠈⣿⡄⠻⣦⣍⡻⣿⣿⣿⣿⣿⠀⣈⣿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⣏⡀⢸⣿⣿⣿⣿⡿⣋⣥⡾⠃⣼⡏
⠀⠀⠀⠀⠀⠸⣿⣤⡈⠻⣷⣿⣿⣿⣿⣧⡀⠉⠩⣿⡿⣿⣟⡿⣿⡿⣟⣯⡿⣿⡯⠉⠁⣠⣿⣿⣿⣿⣿⡿⠋⣠⣼⡿
⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠻⡟⣿⣿⠛⠿⠃⣀⣾⡽⢞⣽⠟⠁⣿⡇⠙⢿⣝⠾⣽⣆⡀⠻⠟⢻⣿⡟⡿⠃⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡇⣾⠃⣸⡏⠀⣠⡶⠛⠉⠀⠈⢉⡀⠀⠀⠹⠁⠀⠀⣈⠉⠀⠈⠙⠳⣦⡀⠘⣿⡀⢻⡄⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣝⠷⠟⠁⣼⣋⣀⣀⣐⠒⢦⣌⢻⣦⠀⠀⠀⢠⣾⢋⣤⠖⢂⣀⣀⣈⣻⡄⠙⠷⢟⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠘⠁⢸⡏⣹⠿⣿⣦⡻⣆⢻⡆⠀⠀⣿⢃⡾⣣⣾⡿⢿⡉⣿⠀⠙⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡜⣧⡹⠶⠟⠘⣧⢹⡜⡇⠀⠀⡏⣼⢡⡟⠘⠷⠾⣡⡟⣼⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠈⢿⣮⣛⣛⣛⡛⢻⣆⠁⠀⠀⠀⠀⢁⣾⠛⣛⣛⣛⣫⣾⠏⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢤⣈⣉⣉⣉⡤⠂⠙⣦⠀⠀⠀⢠⡞⠁⠢⣌⣉⠉⣉⣠⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠉⠛⠋⣽⠞⠋⢁⣤⣤⣄⣤⣤⣄⠉⠛⢾⡉⠛⠋⢁⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⡄⠁⠀⡀⠘⠿⣶⣽⣵⡾⠟⠀⡀⠀⠁⣤⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠈⠀⠄⢙⡀⢈⣿⣏⠀⣘⠁⠄⠉⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣭⣐⣢⣤⣾⠃⢻⣶⣄⣒⣨⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠛⠛⠉⠁

⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠴⠶⠦⡀⠀⠀⡞⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢀⣀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠬⠿⠿⠷⠶⠶⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠒⠐⠆⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢣⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠹⣯⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢉⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣣⡞⠀⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠙⢿⣗⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠈⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣧⠀⠀⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢠⠀⢨⣿⠉⠙⠀⣀⣠⡤⣤⣠⡤⢤⡄⠈⠋⢉⣿⡄⠠⠃⢨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⣾⡟⠁⠀⠀⠀⠄⠀⠈⣧⠀⠀⠀⠀⣆⠀⠙⢦⠀⠰⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠃⡼⠃⢀⡴⠁⢀⣀⠀⠀⡀⠃⠰⠀⢀⡀⠀⢱⣄⠀⠁⢤⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⡼⠗⢀⣠⡻⠗⠛⠃⠀⢰⡄⠘⣏⣤⡀⠻⡆⢰⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣼⡇⣴⣿⣽⣽⡆⣿⠀⢠⠈⢧⢠⣟⠿⣻⠆⣵⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⢷⣌⠁⠈⢌⡙⠁⠀⠀⠀⠈⠃⠁⠈⢁⣴⠟⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⣀⠙⣷⠷⠋⠄⠀⠀⠒⠀⠀⠰⣗⢿⢿⡉⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠘⣶⢠⡆⡼⠃⢠⣶⡛⠛⣲⡄⠘⣶⢠⡷⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣽⣴⠛⡇⠀⠸⣿⣿⣿⣿⠏⠀⢷⢞⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡅⠸⢇⠀⠀⡼⢻⡟⢫⡀⠠⣾⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢻⡿⣿⣶⣟⣾⢾⡟⠛⠛⡻⣧⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡝⠀⢾⣿⣿⣿⣶⣥⣤⣤⣶⣿⡿⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠄⠈⡉⣿⣿⣏⣿⣿⡟⢹⡿⠀⣺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠹⡷⠀⠀⠑⣿⣿⢟⠟⢛⠁⢷⡖⠤⠟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠻⢛⡇⠀⠘⣿⡃⠀⠐⠜⠀⣘⠃⠀⢈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠀⠠⢹⣇⠀⠀⠳⡦⠀⠀⠀⠀⢾⠈⠀⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣇⠀⠀⠀⢹⡄⠀⢸⡷⢆⡀⣠⣾⠏⠀⡰⠇⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣇⠀⠀⠀⠙⢷⡀⠼⣡⣼⢧⡄⡟⠀⢸⠁⠀⠀⡀⢜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠛⣧⠀⠀⠀⢸⣧⠀⠙⠀⠆⠁⠁⢠⡟⠀⠀⢶⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠙⣟⡀⠀⠈⠛⣀⡀⠀⠀⡀⢀⡿⠇⠀⡀⠀⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠔⠀⠀⠀⠀⠞⣡⣦⠀⡪⢿⢁⠆⢴⡃⠺⠰⠀⢰⡿⡶⢬⠀⢠⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠛⠀⠐⠒⠁⠈⠁⠐⠁⠂⠐⠀⠀⠘⠃⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠿⢿⣿⣿⡟⠛⠛⠛⠛⠻⠻⣿⣿⣿⡿⠿⠿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⣾⣿⡋⠀⠀⠀⠀⠀⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⡹⠃⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⢸⡇








⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠄⠁⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠈⠙⡕⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣴⡅⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡏⢜⠠⠀⠀⠀⠀⠈⢦⡙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣡⣾⣿⡕⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠘⡟⡄⢀⠀⠀⠀⠀⠀⢌⠪⡳⡈⢳⣀⣀⣀⣀⣀⣀⣀⢀⡼⢰⣿⣿⣿⣿⠁⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠂⡀⠈⢠⣜⢿⣿⡿⠨⠉⠉⠁⠉⠈⠉⠉⠉⠁⠈⢿⣿⣿⢋⡄⠀⡘⣹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡀⠀⠠⢀⠾⣿⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⡻⢿⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠢⣀⠀⠀⡤⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⣀⠀⢦⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠏⠂⠀⢔⠀⢠⠀⠀⢰⠊⣉⣿⠓⣌⠆⠃⠀⠀⠀⢸⡏⣽⡯⢳⠸⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣶⠂⠋⠀⠈⡆⠀⠘⠄⠘⠟⢀⡾⠈⠀⠀⠀⠀⠘⣆⠈⡡⠃⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢠⡤⠀⠘⢄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠱⡀⠀⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡀⠀⢷⣤⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣶⠈⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢤⠀⠀⠪⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠪⡁⠀⠀⠀⠈⠉⠀⠤⣀⠀⠠⠀⠀⢀⣠⢆⡀⣂⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠁⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⠄⠛⠃⠘⠂⠠⠰⢲⠟⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣧⢰⢟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠰⠈⠀⠈⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠀⣀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⣀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠖⠉⢩⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣿⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⡏⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣧⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⠛⠿⢶⣦⣤⣤⣀⣀⣤⠤⠖⠛⠉⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠶⣤⣤⣀⣀⣀⣀⣀⣀⣤⣤⠶⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣦⣀⣀⣀⣀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢛⡻⣿⣿⣿⡿⣛⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠸⢟⣽⣿⣿⣇⠿⢃⣽⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠟⠿⠿⠿⠛⠛⠛⠿⠛⠛⠛⠛⠛⠛⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋


⣿⣿⠛⢿⢛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⡀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠂⠀⠀⠀⠀⠀⢴⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣃⣾⠇⠁⠀⠀⠁⠸⣦⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⠐⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⡴⠂⠀⣄⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠲⠂⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠭⢶⠞⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠶⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⣄⣀⣀⣠⠀⠐⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⢠⠘⢋⠉⠙⠋⠀⠁⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⣡⡴⠄⣸⢃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣇⠸⣿⣏⠠⠿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⠃⢽⢛⣤⡵⠶⣛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠋⠶⠶⠉

⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⢀⣴⣦⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⢳⣄⠀⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠀⠀⢠⣾⣿
⠘⠂⣼⡇⠃⠀⠀⠓⢸⣆⠸⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣶⣶⣶⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣶⣶⣶⣿⣿⣿
⠀⠹⣯⠐⣰⠄⠠⣄⠈⡿⠆⠀⠀⠀⠀⠀⠀⠈⢻⣏⢙⣿⣿⣿⡋⣹⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠘⣦⠀⠶⠆⠈⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿
⠀⠀⠀⢸⢶⣖⣦⡾⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧
⠀⠀⠀⡧⠀⠁⠈⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⠁⠀⠀⠀⠀⠀⡸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⠈⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⡆⢸⡀⣀⡀⣠⡆⠰⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣠⠃⢃⠸⠟⠛⠛⠟⠀⠃⢸⠄⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠁⠀⠾⠀⢸⣂⣬⠄⣸⠖⣸⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⢶⣶⣇⠸⣿⡟⢠⡿⣾⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣄⣀⡍⣹⢞⣈⣫⣤⢖⣻⠀⠀⠀⠀⠀⠀⠠⣾⣿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⣿⣿⣿⣿⡏
⠈⠁⠀⢬⣭⠿⣥⡼⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⣿⡿⢿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⣿⣿⢿⣿⡏⠉⠁
⣿⣿⡿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⢁⣀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡈⡂⠠⠄⠈⠁⠒⠂⠀⠀⠀⠀⡀⠀⠠⠄⢤⠈⠁⠒⠨⢁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⠈⠑⠀⠀⠀⣠⠀⢀⠠⡠⢤⡄⢂⠑⢄⠰⣠⠊⠀⠈⠠⠀⠀⠹⡶⢍⠋⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠎⠁⠀⠋⠐⡀⠲⠂⢤⡱⠀⠑⠁⠀⠀⠀⠀⠀⠀⢣⠨⡆⠑⡄⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⠀⢠⠀⠅⡀⢠⠚⠁⢀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠚⠁⠣⢀⠈⣎⠀⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠠⠀⡌⠨⠄⠂⠀⠁⢱⠀⠀⠀⢀⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠂⡸⡄⠈⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⡀⢠⠀⠨⠀⠩⠀⠀⠀⠐⢡⠎⠉⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠈⠰⢄⢠⣡⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⡀⠀⣤⠀⠨⠀⠀⠀⠀⢠⠊⠀⠀⠀⢀⠀⡠⠂⢔⠅⢀⠀⠀⡀⠀⢠⠏⡰⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠙⡇⠅⢰⠀⠀⠙⠞⠀⡠⢀⠔⡡⡊⠀⠀⠁⠀⡠⠊⠀⢀⠄⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⠀⠀⠈⡄⢘⠦⠒⠀⠠⡊⢀⠀⡠⠂⠄⢀⠀⡠⠊⢀⠔⠀⡠⠂⠓⢄⡀⠘⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠨⡄⢡⠀⠀⡈⠓⠲⡀⡉⣀⠂⢐⣠⣂⠀⠕⠁⡀⢀⠔⡡⠊⠀⢀⠀⠁⠀⣅⠀⠀⢠⠋⠼⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢⡀⠀⠣⠀⠀⢀⠀⠂⢘⠉⡉⢀⡀⠀⠉⠓⠮⠄⢥⠠⠀⠀⠐⠁⠀⠀⡀⠈⠀⢹⠹⣤⠱⠆
⠀⠀⠀⠰⠛⢷⠈⡀⡀⠄⠀⠀⠐⠕⠑⢄⠀⠄⡀⢀⠀⢣⡀⠲⠄⠀⠀⠀⠀⠀⠅⡾⠚⠒⠓⠿⠛⠶⠺⣾⣱⣿⣶⡶⣾⣷⣶⣶
⠀⠀⠀⠀⢄⠀⠓⠱⣴⡰⡀⠀⠀⡄⠒⠢⠁⠤⣀⢐⠀⠂⡈⠆⠡⠀⠀⠀⠀⡌⠀⣟⣘⠋⣈⡜⠓⠒⣙⣋⣙⡋⣹⣷⡿⠃⢸⡏
⠀⠀⠀⠀⠈⠑⠤⡀⠈⠑⠳⠄⠀⢱⣄⠀⠀⠐⠠⠅⡒⢪⣄⡁⠀⠁⠀⠀⢬⡁⠈⠉⠗⠋⠁⠀⢀⠔⠉⣈⢆⠤⠋⠁⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠑⠉⠑⢐⠊⠀⠀⠀⠀⠀⠀⢀⠖⠉⠍⠀⠁⠀⠀⡀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠤⠀⣐⠀⠀⠀⣀⠀⠀⠀⢀⠀⠁⠀⠎⠀⠀⠀⠀⢀⢀⠀⡀⢀⠀⠀⠀⢠⠴⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠂⠤⡀⠀⠀⠀⠀⠀⡀⠀⠀⠡⠀⡑⠁⠠⠄⠂⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡀⠀⠀⠁⠀⠀⡂⠈⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠄⠁⠀⠀⠀⠂⠁⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣷⣦⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⢁⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠾⠿⠓⠀⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⣤⣆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠈⠉⠙⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⢀⡀⠀⠸⣿⣶⠄⠀⢴⣶⣤⡀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠘⠷⣦⡾⣿⡀⠀⣀⠀⠀⠉⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣟⣉⣌⡙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡉⠻⠿⢿⣿⡿⠇⢠⣶⣦⡈⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣈⠛⣻⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⣄⡀⠀⠸⡿⠿⠟⠛⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⣿⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠈⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠊⠀⠀⠑⠀⠀⠠
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⣴⡁⠀⠄⢀⠈⡀⡀⠀⠀⠀⡀⡀⡐⠂⠉⠐⠀⠢⣀⣀⠀⠀⠀⠠
⠀⠀⠀⠸⠀⠀⠠⠀⠀⠒⢠⣶⠁⡀⠐⡿⡇⠨⠼⠲⡉⠀⠀⡷⠖⠀⠂⣵⣦⡀⠀⠀⢰⣶⡄⠀⠐⠲⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠸⠟⠀⠀⠀⠠⠁⡸⠠⠥⠃⠚⠚⠠⠢⡀⠀⠨⠛⠀⠀⠁⡀⠙⠃⠀⠀⠊
⠀⢀⡖⠐⠧⠄⠠⠤⠖⠀⢀⣀⠀⢤⠂⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠙⠀⣀⡀⠀⠀⢀⣠⠊⠀⠀⡐⠒⢲⡄
⠀⠈⠁⠘⠠⠀⡀⢀⠀⠀⠀⠀⠀⢠⠑⠀⣐⠀⠀⠀⠀⠀⠀⠀⠀⣘⢀⣀⠀⠚⠁⠉⠈⠁⠀⠂⠀⠀⣊⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠸⠠⠀⢀⡱⢄⠀⠀⠀⠀⢀⣴⡡⠀⠠⠄⡀⠀⠀⠀⠀⠀⢐⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⡀⠠⠁⠈⠁⠀⠀⠄⠀⠀⠀⠐⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠐⠁⠀⠀⠄⠤⠀⠀⠔⠈⠈⢌⠀⡀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢈⠀⠀⠀⠠⠂⠀⡀⣰⠀⠠⡂⢤⠀⠎⠀⠈⡆⠀⠀⡄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠆⠀⠂⠂⠀⠉⢤⠀⠀⠔⠂⢈⢭⡒⠀⠊⠀⠁⠀⠀⠂⠈⠄⠀⠈
⠀⠀⠀⠀⢸⡀⠰⠆⡀⠐⠒⠐⠦⡰⢀⠀⠙⣀⡘⠢⠀⡀⠀⠹⢀⠆⢒⢄⣠⠄⠀⡄⠀⠀⠨⠖⢀⡆
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠁⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠙⣧⣀⣀⣀⣀⣴⠏⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣄⠀⠀⣀⠀⠀⠀⠀⣀⠀⢀⣄⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠉⠛⠛⠁⠀⣤⣤⠀⠙⠛⠛⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⢠⣄⣀⡼⣧⣀⣤⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠶⠶⠚⠛⠛⠛⠛⠛⠶⠷⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⠴⠶⠶⢤⣤⣀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⠞⠋


⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠸⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣶⡆⠈⠛⠁⣿⣿⣿⣿⣿⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣧⠀⠀⣸⠟⠛⠉⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠿⣿⣿⣿⣿⣿⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢀⣾⣿⡿⠛⠻⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⡟⠀⠀⠀⠀⠉⠒⠢⢤⠀⡀⢠⠤⠔⠊⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⢬⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣴⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣿⣿⠁⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣿⣿⠀⠀⠀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⢿⡇⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡧⠖⠁⠀⡰⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣸⣠⠜⠁⠀⣀⠔⠁⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠍⣁⣀⣸⣀⡠⠔⠊⠀⠀⠀⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⣿⣷⣦⣤⣀⣀⣀⣀⣤⣴⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⣾⣿⣦⠀⠀⠀⠀⠀⠀⠁⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠀⠠⣔⠤⢤⠀⠀⠢⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠟⠉⠀⠀⠀⠈⡀⠀⠐⠠⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣷⣶⣦⠀⠠⠄⠀⠘⠈⢄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠟⠹⡄⠀⢀⣠⣤⣶⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠁⠉⢰⠁⠀⢻⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠡⠄⠀⠊⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⣿⣿⣿⠿⠿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢸⣿⣿⠃⠀⠀⠀⢹⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠿⢿⣿⣿⣿⣿⣷⣴⣿⣿⣿⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠁⡤⠠⢀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⢶⣀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠃⠀⠀⠀⠀⣾⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠛⠁⢸⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣴⣦⣄⣀⣁⣤⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠠⣤⣤⡀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇
⠀⠀⠀⠀⠈⠙⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣄⠀⢀⣠⣦⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⣤⣤⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⠫⣷⣄⡀⠀⠀⢠⣿⠇⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⣿⡟⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⣀⠆⠀⠀⠀⠀⢰⣿⠇⠀⠠⣄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⡀
⠀⣠⠴⠊⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⠀⠈⠙⢿⣦⣄⣀⡀⢀⣀⡀⠀⣀⣤⣶⠟⠛⠉⠉⠉⠛⠻⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠉⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢠⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠀⠀⠀⢀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⡛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠐⠈⠁⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠁⠂
⠀⠀⠰⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠆
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠨
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⢆⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡰⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠸⡀⠀⠀⡂⠀⠀⠀⠀⠈⢦⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡴⠃⠀⠀⠀⠀⠐⠀⠀⢀⠇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⣼⠁⠀⠀⠀⠀⡀⠀⠀⠀⢸
⠀⠀⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢂⢀⡀⠀⡀⠄⣀⣀⠠⢀⠀⠀⡀⡐⠀⠀⡏⠀⡀⠀⠀⠀⠀⠀⠀⠀⡒
⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠚⠃⠈⢧⠀⠘⠮⡄⠄⠀⠀⢀⡀⠀⠀⠠⢠⠵⠃⠀⡼⠁⠘⠓⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠈⢇⠀⠀⠀⠠⣄⡴⢤⡇⠀⡾⠃⠀⠐⠁⢀⡀⠀⠀⠀⠀⢀⠀⠈⠂⠀⠘⢿⠀⢸⡤⢦⣠⠄⠀⠀⠀⡸⠁
⠀⠀⠀⠀⠱⡀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⢀⠎
⠀⠀⠀⠀⠀⠡⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠌
⠀⠀⠀⠀⠀⠀⠈⢤⠀⠀⢀⣄⣤⠤⣤⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣤⠤⣤⣀⡀⠀⠀⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠘⣿⠀⢰⡆⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢰⡆⠀⣿⠃⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠓⠢⠄⠠⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠧⠄⠠⠔⠊⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⠀⠀⠀⠈⠲⢬⠑⠊⡥⠔⠁⠀⠀⠀⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢠⡄⠁⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢭⣉⣉⡭⠟⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠈⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠈⠻⣦⡀⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢀⣴⠟⠁⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢠⣀⡀⠡⠈⢿⣌⢻⡷⢦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣿⡟⣡⡿⠁⠎⢀⣀⡄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⣙⣯⠛⠇⠀⣻⣆⢻⣆⢈⠻⠶⠶⣶⣶⠶⠶⠟⡁⣰⠟⣰⣟⠀⠸⠛⣽⣋⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢈⣿⢍⣡⡶⠟⣋⣥⠶⢛⣤⠷⢀⡾⠃⠙⢷⡀⠾⣤⡛⠶⣬⣙⠻⢶⣌⡩⣿⡁⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣰⣿⡷⠟⡩⠔⠋⠁⠀⠀⢿⡁⠀⠸⡇⠀⠀⢸⠇⠀⢨⡿⠀⠀⠈⠙⠢⢍⠻⣾⣿⣆⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⢰⡿⠋⠀⢀⣀⠀⠀⢀⡀⠀⢸⣇⠀⠀⢷⠀⠀⡾⠀⠀⣸⡇⠀⢀⡀⠀⠀⣀⡀⠀⠙⢿⡆⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠟⠀⣠⡾⠛⠁⠀⠀⠀⢹⣄⠀⢻⡆⠀⠸⠀⠀⠇⠀⢰⡏⠀⣠⡏⠀⠀⠀⠈⠛⢷⣄⠀⠻⠀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⢠⣾⠋⠀⠀⠀⣀⣠⣤⣄⣿⡀⠀⢿⠀⠀⠀⠀⠀⠀⡿⠀⢀⣿⣠⣤⣀⣀⠀⠀⠀⠙⣷⡄⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⢀⣿⠃⠀⠀⢠⠞⠋⢉⣀⣈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⣁⣀⡉⠙⠳⠄⠀⠀⠘⣿⡀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢿⡆⢸⣿⣤⡀⠀⠀⣠⠞⠉⠉⠉⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠉⠉⠉⠳⣄⠀⠀⢀⣤⣿⡇⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣿⣶⣄⠀⠀⠀⡄⠀⠀⠠⠀⠀⠀⣠⣾⣿⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⣚⠍⠂⠀⠸⡆⠀⣿⣿⣿⣿⣿⠀⠀⢀⡼⠀⠀⢧⡀⠀⠀⣿⣿⣿⣿⣿⠀⢰⠇⠀⠀⠩⣓⢶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⣆⠀⠀⠙⠦⠈⠛⠿⠛⠁⡠⣠⠋⠀⠀⠀⠀⠹⣄⢄⠈⠛⠿⠛⢁⠔⠋⠀⠀⣰⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢀⠘⣦⠀⣀⡤⠄⠀⠀⢀⡞⢱⡃⠀⠀⠀⠀⠀⠀⢸⡎⢳⡀⠀⠀⠀⢠⣄⣀⣼⠏⡼⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⣦⠈⠛⠁⠀⣠⡴⠞⠛⠁⠀⠙⠖⣶⣶⣶⣶⠲⠋⠀⠈⠙⠛⢤⣄⡀⠉⠛⢁⣸⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡌⢷⣀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⣠⡾⢡⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠁⠈⠙⠛⠓⠒⠂⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠛⢶⣄⡀⠀⢠⡄⠀⢀⣠⡶⠛⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠙⠚⠋⠁

⠀⠀⠀⠀⠀⢠⠖⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠢⡄
⠀⠀⠀⠀⠀⡇⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢀⠀⢸
⠀⠀⠀⠀⠰⡀⠀⠹⡄⠸⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⡽⠏⣠⠏⠀⢀⠆
⠀⠀⠀⠀⠀⠁⠀⠀⠉⠂⢈⠻⣦⡂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢒⣴⡟⠁⠐⠋⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠀⠉⠲⣤⣠⠠⠀⠀⠄⣄⣤⠖⠉⠀⠰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢀⢸⡿⠀⣀⡀⠀⠀⠛⠀⠀⠀⠀⠻⠀⠀⢀⣀⠀⢿⡇⡀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⠾⠁⠚⠁⠀⢀⣀⠄⠀⠠⠄⠀⡠⣀⡀⠀⠈⠃⠈⠷⠁⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣤⡟⠋⣀⣀⣀⠀⠨⣧⡄⠀⠀⢀⣴⠅⠀⣀⣀⣀⠙⢻⣤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣹⣻⡿⡟⣿⣿⣷⡎⣀⠀⠀⣀⢱⣾⣿⣿⣻⢿⣟⣏⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡙⠦⣙⣩⣿⢿⣿⡟⠀⠀⢻⣿⡿⣿⣍⣋⠴⢋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠀⢻⣿⡇⠀⠀⢸⣿⡟⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠃⠀⠘⣿⠋⠀⠀⠙⣿⠃⠀⠘⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠈⠀⠈⠲⣶⣶⠖⠁⠀⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣄⣀⣠⡠⣨⣅⢤⣄⣀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠶⠶⠞⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣐⠢⡀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⢀⡤⣴⣶⣦⢢
⠀⠀⠀⠀⠀⠘⡿⡣⠀⠈⣙⠊⠢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡽⡋⠛⠻⡿⢷⠃
⠀⠀⠀⠀⠀⣄⠁⠷⠀⠀⠸⠀⠀⢌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⡄⠀⠀⠀⠁⡘⡞
⠀⠀⠀⠀⠀⣿⢀⡖⠀⠀⠀⠀⠀⠈⢷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣟⢁⡿⠁⠀⠀⠀⢀⡃⣷⡁
⠀⠀⠀⠀⢀⣿⢸⣿⠀⠀⠀⠀⠀⢠⡘⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⠁⠚⠁⠀⠀⠀⢀⣼⠇⣟⠁
⠀⠀⠀⠀⢘⣿⡏⠻⡆⠀⠀⠀⠀⠀⢻⣯⣿⡟⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⡟⣴⠀⡶⠀⠀⠀⠀⠻⠟⢸⢉
⠀⠀⠀⠀⠀⡏⠷⠀⠀⠀⠐⠀⠀⠀⣎⣾⣿⣇⣿⢿⡿⣿⠟⡟⠻⡟⢿⠻⠿⣿⣿⢏⡆⣿⣷⢻⡎⠁⠀⠀⠀⠀⠀⠴⢳⣾
⠀⠀⠀⠀⠀⣿⣶⣤⢢⣄⣁⠀⠀⢰⣿⣎⢻⣿⣯⣻⡼⠇⡈⣹⡆⣷⠊⡰⠆⣽⣿⣿⣿⣿⣇⣿⣿⣄⠀⢀⡔⠀⣠⣶⣿⣿
⠀⠀⠀⠀⠀⢲⡟⠁⠘⣿⣿⣰⣦⣺⣿⣿⣾⣿⣿⢿⣂⢠⡖⠙⣶⡇⣬⢠⠰⣿⣿⣶⣿⡿⣿⣿⣻⠿⢠⣿⣄⡾⣱⣶⠹⢟
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠙⣿⣿⣿⣿⣿⠿⣏⣻⣷⣮⣩⣶⡇⠀⠉⢹⠣⣞⢳⣦⢈⣽⣿⣟⣿⣿⣿⣿⣿⣿⠟⣱⣏⠄⢀⣾⡀
⠀⠀⠀⠀⠀⠀⣿⣷⡈⣧⠨⣽⠟⠋⢳⣾⣿⣿⣿⣿⣿⣆⣼⡄⢿⣿⣠⣿⢸⣷⣾⣿⣿⣿⣿⣿⣿⡟⢛⡋⠚⠹⣣⣾⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣧⣀⣴⢃⡄⢳⣝⠻⢿⣿⣿⣿⡍⢵⣾⣿⣿⣿⣄⣽⣿⣿⣿⣿⠟⡱⠋⠿⣾⣷⣦⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣇⢸⡇⠀⠙⠿⠿⠋⠙⠿⠃⣬⡟⣟⣿⠃⣿⢿⠋⢿⠛⠛⠛⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡾⡿⢤⡉⠀⠤⣶⠀⠀⠸⣿⣤⣯⣭⣤⣌⠂⠀⠀⠀⢀⠜⢁⣤⢲⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣿⡈⢿⣿⣾⡇⣀⡀⢀⣼⣿⣿⣿⣟⡝⠀⠠⠀⢸⣿⡿⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣽⣿⣇⢨⡤⢨⣯⣿⣿⣿⠧⣁⠢⠤⣾⣠⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣒⣫⣾⡿⠟⠿⢻⣦⣗⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⡿⣿⣿⣿⣯⣿⣧⣶⣆⣠⣟⣻⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⡌⢻⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣾⣿⡏⣽⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠻⠘⣿⢲⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠸⣿⣿⢇⣿⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢻⡟⠀⣤⢸⣼⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣹⣧⣿⣿⡘⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠘⢁⠀⣾⣧⡟⢀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣇⣼⢺⣿⣿⠃⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⣆⢠⣀⠈⠛⠁⣾⣿⢹⣿⣿⣿⣏⢿⣿⡿⣿⣿⠀⣼⣿⠏⢰⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡝⢿⢿⣦⣃⣢⣿⣯⠜⣿⣿⣿⣯⡜⣿⣷⣿⣟⣸⣿⣿⣢⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡄⣼⡿⢻⡏⢰⣿⣿⣿⣿⡹⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠤⣿⠁⠸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⢄⣀⣭⣑⠢⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⢄⣒⣥⣬⠢
⢀⢻⣿⡇⠀⠉⡙⠮⡢⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⢻⠿⢿⣿⣷⢧
⢸⣸⢰⡇⠀⠀⢙⡇⠐⢈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡁⠈⠀⠀⢈⠆⠁⣾
⢸⣟⢠⡜⠀⠀⠀⠀⠀⠀⠳⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⢏⣴⡿⠀⠀⠀⠈⠀⡐⣿
⢘⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣥⣿⠯⢻⠟⠀⠀⠀⠀⠀⢰⡇⣿
⣿⣯⢸⣿⠀⠀⠀⠀⠀⠀⠳⣄⢿⣿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⠁⢀⠀⠀⠀⠀⠀⣰⣿⢃⡏
⣿⣿⠀⠹⢇⠀⠀⠀⠀⠀⢀⢹⣻⣿⡟⣽⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⡇⣿⢀⡟⠀⠀⠀⠀⠉⠩⠉⠸⡇
⢸⡘⠧⠀⠈⠀⠈⠀⠀⠀⢸⣆⣿⣿⣧⣿⣿⣫⢟⡿⠛⡏⡉⣯⠙⣟⠻⡻⣿⣯⢇⣿⢸⣿⡇⣿⣼⠃⠀⠀⠀⠀⠀⠀⠜⣣⣿
⢸⢿⣦⣌⢲⣀⢁⡀⠀⠀⣸⣿⡌⣻⣿⣿⣝⣡⡾⠃⡈⣹⡇⢸⡇⢋⡴⡄⣽⣿⣿⣿⣿⣿⣡⣿⣿⣄⠀⠀⣴⠊⠀⣠⣼⣿⣿
⠘⣿⠟⠁⠘⣿⣿⢇⣶⡄⣿⣿⣿⣾⣿⣿⣿⡿⣁⢌⡥⠛⣧⣾⢠⡋⣦⢰⣿⣿⣷⣿⣿⢿⣿⣿⣻⣿⢂⣾⣧⣡⡞⣱⣦⢹⡟
⠀⢻⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⢟⢿⣿⣭⡛⣏⣬⠇⡀⠙⠃⡾⢣⠿⣬⣉⡝⢻⣿⣿⠷⣿⣿⣿⣶⣾⣿⣿⠟⣰⢟⡡⠀⣸
⠀⢸⣷⣄⠀⣄⢙⡻⣿⣿⠿⣿⣱⣷⣾⣿⣿⣿⣮⡛⣰⠀⢲⣶⡆⢸⡇⢺⣋⣴⣯⣿⣿⣿⣾⣿⣿⡿⠿⣿⢋⡾⣱⢟⣤⣾⣿
⠀⢸⣿⣿⣦⣿⣎⠛⠋⣠⠎⢱⣯⢿⣿⣿⣿⣿⣿⣖⠿⢇⣸⣿⣿⣿⡏⠘⢟⣽⣿⣿⣿⣿⣿⢛⡿⣧⣴⣦⣈⡀⣫⣿⣿⣿⣿
⠀⣾⣿⣿⣿⣿⣽⣶⣷⡇⢸⠀⠹⣷⣬⣝⣿⣿⣿⣿⡆⠹⣿⣿⣿⣿⠿⣷⣿⣿⣿⣿⣿⣋⣥⠊⡀⠏⠙⣿⣿⣾⣿⣿⣿⣿⣿
⠀⢹⣿⣿⣿⣿⣿⣿⣿⣇⣸⣄⠐⢌⠛⠛⢂⡀⠉⡏⠹⣶⡏⣿⣽⡏⠀⠿⠹⠀⢛⡏⠉⠉⢠⠖⠀⢠⢐⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠹⡛⢦⣄⠀⠰⣾⡇⠀⠀⡸⣿⣶⣶⣶⣶⣶⡦⠂⠀⢈⠀⢀⠴⠋⣰⣧⢣⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡐⢿⣿⣿⣿⠀⣀⠀⣀⣼⣿⣿⣿⣿⣿⡓⠀⢠⡄⠀⣿⣿⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⣿⣷⡀⣬⣦⡡⣯⡿⣿⣿⠿⣄⡑⢦⣤⣴⢇⣿⣧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠩⠴⢛⣵⣿⣿⣿⡿⣶⣍⠲⠶⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣛⣿⣟⣀⣆⡀⢸⡟⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡜⢿⡿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣿⣿⣯⣵⣿⢿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⠙⣿⣆⢻⡟⠸⣿⣿⣿⣿⣿⣿⣿⣿⣮⡙⢻⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠘⠃⠘⡏⢿⠃⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢿⣿⡿⣼⣿⡿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⢿⡝⢂⠰⣄⢳⡿⠀⣼⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⡆⢸⡟⣼⣿⣿⡇⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡇⡈⠇⠀⢲⣿⣼⠇⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣷⢸⡿⣹⣿⣿⠁⢸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠛⣧⠰⣄⠀⠙⠛⠤⣿⣿⣽⣿⣿⣿⣿⠞⢿⣿⡿⣫⣿⡇⢀⣿⣿⠏⢀⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡻⢶⣿⣧⡒⠬⢢⣿⣿⠘⣿⣿⣿⣿⡗⢘⣿⣷⣿⣿⣇⣾⣿⣿⢦⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⡈⣙⠙⣳⡾⢿⡏⠀⣿⣿⣿⣿⡟⢾⣿⣿⣿⣷⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⣋⢁⣾⠇⢰⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠤⠄⢿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀
⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⡐⠁⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡀⠈⢂⢀⠌⠀⠀⠀⢀⠠⠂⠁
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠱⡄⠀⠀⠈⠖⢁⠀⣀⡀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠰⠀⠀⠀⠀⠢⠀⠀⠙⠢⡀⠀⠈⠄⠈⠀⡁⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠁⠐⠄⠀⠀⢁⠀⠀⣠⠙⠄⠀⠐⠘⣀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⡄⠀⠘⠆⠀⠀⢆⡀⡄⠐⡀⠈⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠂⠠⠀⠀⠄⠀⠀⠀⢩⠙⢇⣈⠀⠒⠄⣀⡁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡔⠀⠀⠀⠀⣀⠡⡀⠀⠀⡀⠔⠀⠁⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢄⣀⠃⠲⠀⠱⢐⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠻⣿⣿⣷⡄⠀⣀⡀⣠⣾⣿⡿⠋⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠘⣿⣾⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠉⠈⣽⣷⣤⡀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⠻⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣿⣷⣼⡟⠿⠿⠟⠛⠁⢀⠺⡽⢀⠈⠙⠻⠿⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⠙⣦⡀⠀⠀⠀⠈⠉⠉⠉⠀⠀⢠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⠆⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠁⠀⣿⣿⣿⠀⠀⣄⠀⠀⣰⠏⠀⢸⣿⣿⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡟⠀⠀⠀⢹⡿⠃⠀⠀⣿⣠⡾⠃⠀⠀⠘⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⠀⠀⠈⣧⠀⠀⠀⢿⠋⢠⡇⠀⠀⠀⣿⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠘⣇⠀⠀⢸⡇⢸⠇⠀⠀⣸⢇⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⢻⡀⠀⠀⣧⣾⠀⠀⠀⣿⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠌⢩⠉⣤⣾⡇⠀⢠⡟⠻⣄⣄⢠⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠚⠁⠙⠷⠷⠞⠀⠀⠈⠙⠋⠁
⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⣻⣿⣿⣿⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⠛⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢀⣶⣄⠙⢿⣷⣄⣤⣤⣤⣶⣶⣤⣤⣤⣠⣾⡿⠋⣠⣶⡆⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⣿⣿⣷⡦⠈⠛⠋⠉⠉⠉⠉⠉⠉⠙⠛⠁⢠⣾⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⣤⣤⣤⣤⣤⣤⣤⠸⣿⣿⣿⣿⠇⢠⣤⣤⣤⣤⣤⣤⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⣤⠤⠴⠖⢂⣠⠄⠈⢹⡏⠁⠠⣤⡀⠲⠶⠤⣤⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⡴⠞⠋⠁⣀⣠⠾⠷⣄⣀⠀⠉⠓⢦⡄⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣄⠀⠙⠉⠉⠁⠀⠀⠈⠉⠙⠋⠀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⣿⠿⢿⣎⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣰⡟⠁⠀⠀⢻⠸⡄
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢆⡿⠀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠠⢁⣶⣶⣶⣬⣑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣸⠃⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠄⣼⡟⠀⠀⠀⠙⠳⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣿⠀⠟⠀⠀⠀⠀⠀⢠⡎⢀
⠀⠀⠀⠀⠀⣿⠃⡀⠀⠀⠀⠀⠈⠳⡈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⣿⠀⠀⠀⣀⠀⠀⣰⠟⠶⢿⡄
⠀⠀⠀⢠⠀⣿⠀⣇⠀⠀⠀⠀⠈⢳⡌⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⠃⢸⣿⣦⠀⠰⡏⢰⡶⠋⠀⡀⢸⣇
⠀⠀⠀⢸⠀⢿⠀⢻⡀⠀⠀⠀⠀⠀⢹⡄⠀⠈⢿⣷⣄⣤⡶⠞⢯⡹⣎⢱⣦⠠⣿⣿⣿⠀⢀⣿⡾⢃⣤⠞⢀⣿⡟
⠀⠀⠀⠘⣧⡀⠀⠈⣧⠀⠀⠀⠠⠽⢾⡇⠀⠀⠀⠙⣿⣮⣉⠓⢦⣽⣯⠘⣯⣓⠦⠈⠻⣦⣈⡀⠀⠘⣧⡀⣾⣿⠃
⠀⠀⠀⠀⠸⡷⣀⠀⠸⣆⠀⢀⣴⣀⡀⠁⠀⠀⢀⣴⣫⣿⢃⠎⡆⢃⡹⡇⡈⢫⣛⡓⠶⠈⠛⠻⣟⠳⡌⠻⣿⣿
⠀⠀⠀⠀⠀⠹⣎⢳⡄⠙⢷⡼⠟⢋⣹⣴⠆⢠⠞⡽⠛⢁⡤⢞⡡⡻⡱⠝⣆⠀⡏⠉⠉⢑⠂⠀⠀⠁⠙⠀⣹⡏
⠀⠀⠀⠀⠀⠀⢡⠘⣅⠀⠀⠀⠒⠻⠉⠀⠀⠈⠀⠀⠀⠊⠀⠋⠀⡘⢻⡀⠈⠃⠡⠀⠀⠘⢧⡀⣙⠲⢤⣄⣿
⠀⠀⠀⠀⠀⠀⠈⠀⣿⣷⡄⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠘⢦⠀⠀⢀⡜⠁⣠⣶⣿⣿⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⣷⣤⠀⡇⠀⢰⣠⠀⢠⣾⡡⠤⠤⠀⢿⡀⠀⡄⠀⠀⢰⣼⠇⣸⠿⣿⣿⡇⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣆⠀⡇⠀⣾⡇⠰⠋⣡⣴⣶⣶⣦⡄⠈⠂⣿⠀⠀⠈⢿⡀⢿⣄⠈⠉⣠⠆⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⠻⣆⢧⠀⣿⡄⠀⢨⡉⠸⣿⣷⠂⢹⣶⣄⠘⣧⠀⠀⠘⣇⢸⡍⢛⣉⣡⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢂⠀⢹⣇⠀⠀⠑⢤⠈⢀⣠⠞⠋⢹⣆⢹⠆⠀⠀⠙⢾⣇⠈⠛⠻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢦⠈⢿⡄⠀⠲⣤⣤⣤⣤⣤⣴⣿⡿⣿⠀⠀⠀⠀⠘⣿⡆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠱⣄⠙⢷⣄⡀⠉⠛⠛⠟⠛⠋⠀⠹⡄⠀⠀⠀⠀⢸⣧⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠉⠳⠄⠀⠀⠀⠀⠀⠀⣿⠴⡶⠶⠶⢿⠇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣶⣤⡀⠀⠂⠐⠀⢂⠈⠛⢿⣆⣤⠇⠀⣁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣿⠇⣹⠛⠻⠷⣦⣄⣀⠈⠀⠄⣀⡼⠛⣲⠞⠻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⣈⣴⣾⠋⢰⡏⠀⠀⠀⠀⠉⠛⠛⠛⠿⠁⠰⠞⠁⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⡴⠞⢩⠟⠁⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠏⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡿⠟⠃⠀⠀⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⠶⠾⠟⠛⠛⠉⠁⠀⠀⢀⣠⡾⠃⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡘⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⣀⣀⣤⣴⠾⠛⠉⠀⢀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠉⠀⠀⠀⠀⠀⠀⢠⡿⠛⠁⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠉⠉⠀⠀⠴⠾⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⢠⠟⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣦⣤⣤⣶⣶⣶⢶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣯⢾⡟⣾⣯⣾⣇⢿⣧⣻⣷⣾⣿⣝⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⣫⠟⢳⣝⣧⣟⣻⡷⢶⣿⣭⣻⣿⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠁⠀⠀⠉⠛⠛⠛⠻⣷⡝⣿⣇⣿⣯⢿⣮⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⠉⠀⠀⠀⠀⠉⠉⠑⠂⠘⣷⡻⢿⣮⣝⡿⣮⣟⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣄⣀⡄⡸⠀⢤⠀⠀⠀⢀⡄⠹⣟⡻⣾⡻⣿⣮⢿⣧⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⠃⠀⠀⠙⠒⠚⠉⠀⠀⠁⠀⢸⣷⣿⣿⡾⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠈⡁⠀⠀⠀⠀⠀⣴⡄⠀⢀⣀⣤⣿⡟⣿⣿⣧⣿⣇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠉⠉⠁⠀⠀⠀⠉⠁⠀⣸⣿⣯⣽⣾⡿⣻⣾⡟⣼⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠷⠶⢿⣿⣿⣿⣯⣾⣟⣵⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⣀⡀⠀⣴⣿⣻⡇⣰⣾⠏⢸⣿⢹⡿⢛⠛⡛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠁⣩⣿⣿⣷⣿⠇⣿⠏⣴⢸⡇⢈⣴⡿⠸⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣥⡾⣫⣴⢶⡌⢿⣰⡟⣴⣦⢸⢃⡌⢿⡇⣾⣟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢾⡟⣰⢯⣾⣇⣿⡼⠟⠻⢿⡟⢻⣾⡿⣾⣤⣍⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣿⠁⣿⣿⣿⣿⠟⣰⣿⡟⣰⡟⣤⠄⣰⠟⣩⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣯⣒⣥⣿⡄⢿⣾⡾⢋⠰⡿⠟⢠⡟⣰⢋⣼⡏⣼⠏⣴⣯⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⢹⣾⡿⠿⣯⣿⡶⣤⣶⣿⡷⣶⣷⣤⣾⠇⣾⣿⠇⢡⡾⢋⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⠘⢋⣠⣾⡟⢁⣴⣄⠹⣯⡞⢻⣷⢻⣿⣷⣬⣴⣟⣶⣾⣟⣵⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⣿⣿⠀⣿⢷⣿⠀⣿⣿⡿⢀⣿⠁⣾⣿⣼⠇⢸⡏⢈⣉⠙⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣇⣿⣇⣸⣿⡸⣧⠐⣿⡿⠃⣼⡏⢰⣿⣷⣿⠀⣿⠃⠸⠿⠇⢸⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⢿⣮⣻⣿⡿⠿⣿⣦⣄⣤⣾⣻⣧⠘⠿⠿⠃⣼⡟⢠⡄⢰⣾⣟⣵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡽⣷⣽⣟⣿⢰⣷⣾⡟⣩⣍⣻⡿⣿⣷⣶⣶⣾⣽⣇⣸⣿⡀⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣾⣽⣿⣿⣎⢿⣿⠃⠻⢿⣿⢰⣿⣸⡟⢰⣤⣽⣿⡟⣿⣷⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣏⠻⠟⣸⡿⢸⣿⣿⡇⣼⡏⣿⠃⣙⣿⣯⣭⡾⣯⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣧⣌⣛⣿⠑⠿⢿⡿⢸⣿⣿⣻⣽⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣾⣷⠾⠿⠛⠉