Jumpy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣦
⠀⠀⣰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠈⠙⢿⣿⣿⣇
⠀⣼⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠻⣿⣿⡄
⢸⣿⢃⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠹⣿⣧
⣿⣿⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡀⠀⠹⣿⡀
⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠻⣧⠀⠛⣛⢳⣄
⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⠀⠀⠀⣤⡟
⠸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢰⠿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠘⢦⣼⡇⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠤⠋⠀⠀⠀⢰⡄⠀⠀⢸⣇⣴⣾⣻⡾⠛⢳⡦⣤
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢸⣿⠁⠀⢨⡇⠀⣸⠃⢸⡇
⠀⠀⠀⠀⣼⠋⡤⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⣀⣀⣀⣠⠤⠖⢺⡏⠀⠀⠀⠀⠀⣸⠿⣇⠉⠙⠿⠒⠛⠲⣾⠁
⠀⠀⠀⠀⢿⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⢈⣷⠀⠀⠀⠀⢠⡿⠀⠙⣷⠀⠀⠀⣴⠾⠃
⠀⠀⠀⠀⠈⠙⠲⠶⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠉⠀⠀⠀⠀⣠⣿⣁⡀⢀⡿⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠋⠈⠉⢻⡟⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⢿⣿⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⢞⣿⠟⠀⠀⠢⣄⣾⠁⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠿⣄⣠⡆⠙⣧⠈⠉⠛⢻⡿⠛⠛⠛⠿⠿⠿⠷⠟⠋⠁⠀⠀⠀⠀⠉⢿⡷⠶⠶⠞⠃
⠀⠀⠀⠀⠀⣼⠳⢤⣀⣸⠿⠀⠀⢿⣄⡀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠻⣦⣀⡽⠁⢀⡀⠀⠀⠈⠙⠛⠛⠷⣤⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠙⠷⣄⡀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢤⣤⠾⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠉⠁⠀⠀⠀⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⢀⣠⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋⠉⣿⠃⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⣴⠶⠟⠛⠉⠀⠀⠀⢠⡟⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢀⣾⠳⠶⢶⡾⢟⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⣼⠧⠤⣀⡀⠒⠻⢴⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠙⠳⠶⢦⣤⣉⣑⠚⠀⢄⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⣠⡾⠙⢻⣦⣄⠀⠀⠀⠀⠀⠀⠉⠉⠛⠳⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠶⣏⠀⢀⠎⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⢲⠆⢠⢃⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠉⡶⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠜⣠⡟
⣤⣤⣤⣠⣤⣤⣄⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣹⣷⣏⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠙⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠾⠟⠛⠛⠷⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⢠⣾⣿⣿⣶⣦⠀⠈⢻⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⢸⣿⣿⣿⣟⠛⠀⠀⠀⣷⠀⠀⠀⠀⣠⠞⠉⣩⣤⣈⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠈⠻⢿⣿⠿⠃⣤⣤⣤⣯⣀⠀⠀⢰⡏⢰⣿⣿⣿⣿⣧⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠾⣇⡘⣿⣿⣯⣙⠛⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠛⠿⠛⠀⢀⣴⠏
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣾⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠖⠛⠋
⠀⠀⠀⠀⡟⠉⢹⣧⣀⣰⡟⠙⣧⠀⠀⡾⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠰⡿⠂⠀⠀⠀⠈⣷
⣠⡶⢦⣾⠛⣶⠘⢿⣍⡽⠟⣰⠏⠀⢸⣇⣀⣀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⣯⠀⢀⣙⣛⠛⠇⠀⠀⠀⣾⠃⠀⠀⢸⣿⣿⣿⡷⠀⠸⢦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣻⡀
⠈⠛⠋⠉⠉⠛⢶⣄⠀⢰⡏⠀⠀⠀⣿⠈⠉⠉⠀⠀⠀⠀⢸⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠘⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢸⣿⣿⣶⡆⠀⠀⠀⢸⡇⠀⠙⠳⢦⣤⣄⣀⣠⣤⡾⠋⠀⠀⠀⠀⠀⠉⢹⡇⣴⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠀⣿⢴⣾⣇⠀⠀⠀⠀⢿⡙⠛⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⢨⡟⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣼⠀⢿⣄⠠⣿⣄⣠⡶⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢿⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⢻⡾⠛⠻⣦⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡏⠀⠀⣿⠀⢿⠉⣹⡇⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠘⢧⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠈⢷⡀⢀⣼⠏⠀⠀⠀⠀⠀⠀⣴⣶⣦⣀⣠⡟⠀⠀⠀⣿⠀⠸⠟⣉⣴⠛⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣶⣿⣷⣄⠀⠀⠀⠀⠙⢶⣤⡀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠿⠋⠀⠀⢀⣴⠇⠀⣠⣤⣌⣉⣳⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠉⠁⠈⠛⠶⢦⡴⠶⠛⠋⠁⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⡴⠶⠛⠋⠁⠀⠀⣠⣴⡏⠁⣠⠞⠉⠀⠉⠉⠙⢧⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⠶⣤⣤⣀⣀⣠⣤⠶⠖⠛⠉⢛⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠹⣿⡿⠀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣧⡾⠃
⠀⠀⠀⠀⣠⠾⠋⠙⠿⣿⡿⠀⠀⣠⡴⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉
⠀⠀⠀⠰⡏⠀⠀⠀⣀⣠⣤⠶⠛⠁⠀⠀⠙⠶⠶⠛⠛⠋⢠⡀⠀⠀⠀⠀⢶⡀
⠀⠀⠀⠀⠻⣄⠀⠀⠻⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⣠⣴⣬⣧
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠙⠛⠛⠛⣧
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠿⠆⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠄⠈⠛⠛⢶⣼⠟⠓⠶⠞⠋⠉⢀⣀⣤⣤⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⣀⣴⠞⠙⣧⠀⠀⣴⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⡾⠋⠀⠀⠀⠘⢦⠀⢻⡀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢸⠀⠘⣧
⠀⠀⠀⢀⣤⠞⡀⢠⡄⠀⡿⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄
⠀⠀⠀⢿⣁⣴⠇⢸⡇⠸⣇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⢷⣤⣄⡀
⠀⠀⠀⠀⠹⢧⣠⣾⡁⠀⢹⡆⠀⠀⠀⠀⠀⣿⠀⢠⡀⠠⣤⡈⢳
⠀⠀⠀⠀⠀⠀⠁⠈⠙⠛⠛⠀⠀⠀⠀⠀⠀⢻⠀⢀⣽⢄⣨⡟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠉⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠒⢉⡉⠉⠛⠛⢓⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⣀⡠⠔⠒⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠤⠤⠤⠤⠠⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢀⠔⠊⠁⠀⠀⠀⣀⣠⠤⠖⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠈⢧⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠎⠀⠀⢀⡠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠤⠖⠤⣤⡤⠶⣤⠒⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠃⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢠⣀⠋⠈⠀⠀⠂⠒⠒⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠶⠀⠀⠀⢀⣀⠀⠀⢀⡞⠋⠛⢦⠐⠀⠰⠀⢀⠀⠐⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡶⡐⠀⠀⠋⠑⠄⠀⠄⠃⠀⠋⠀⢧⣀⠂⠶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⠁⠸⡇⢙⢦⣈⡅⠤⠊⠓⠀⠂⠀⠀⠀⠀⣁⣀⢠⠤⠀⢀⢀⡀⠀⠐⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠠⠛⠁⠀⠉⠀⡀⠄⠀⣀⣤⠀⠋⠀⢀⢁⠠⠤⠤⠀⡈⡀⠤⠐⠚⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠍⠀⠀⠀⠀⠀⠀⡔⠉⠐⠞⠛⠉⠠⢶⣚⣃⠈⠤⠑⠒⢚⡋⠁⣀⠠⠐⡉⠀⢀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢈⠉⠉⠈⠉⠀⠀⢀⡀⣀⣀⡠⠤⠒⠛⠋⠉⠁⡀⢄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⣀⡠⠔⠪⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⣆⠠⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠤⠄⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠙⠉⠑⠆⠀⠘⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠈⠁⠐⣀⠀⠂⠤⠤⠤⠤⠤⠐⣊⠉⠉⠉⠉⠉⡩⠟⠧⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠬⠁⠀⠁⠠⠈⢢⠴⠄⠐⠺⠃⠀⠠⣁⠠⠟⠀⠀⢜⠗⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠀⠑⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⣀⡀⠀⠀⠀⠉⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⡀⠀⠈⠉⠙⠤⠖⠀⢀⠀⠀⠀⠘⣤⠀⠀⠈⣡⠄⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠃⠑⠢⠦⠤⠀⣤⣄⣠⣱⣬⡍⠬⠥⠐⠊⠉⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠈⠣⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠀⠈⠙⠲⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡦⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠶⠖⠛⠛⠉⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣤⣝⣥⣼⣭⣿⣡⣿⣭⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡴⠚⠓⢤⣀⡤⠖⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⣼⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠀⠀⠀⢰⡀⠀⠀⠀⡆⠀⠀⠀⢀⠀⠀⠀⠹⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠋⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⠀⠀⠚⠀⠀⠀⣤⠀⢔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⡶⢄⡀⣀⡤⠶⢄⣀⡤⠶⢦⣄⣀⣠⠤⢤⣀⠘⠁⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣠⠾⠤⢤⣉⡉⢀⠴⠚⣻⣤⠲⣤⣞⠛⢦⣀⡤⠬⠓⠤⣄⣀⠈⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣽⠞⠁⠀⠀⠀⠀⠈⣏⠀⠈⣡⢈⣾⠀⣌⠁⠀⡣⠀⠀⠀⠀⠀⠈⠳⣼⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡏⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢠⡾⠴⢷⢤⡠⠴⠃⠀⠀⠀⠀⠀⠀⠀⠈⡏⠆
⠀⠀⠀⠀⠀⠀⠀⠘⢦⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠃
⠀⠀⣀⢤⣀⡴⣄⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⢠⠾⡝⠃⠙⠁⢻⣀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡜⠙⡤⢛⡶⣄
⣞⠃⠀⠀⠀⣠⠏⠈⠓⢦⣘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣀⠴⣇⠀⠀⠈⠸⢋⡇
⠈⠑⢦⣤⠖⠁⠀⠀⠀⠀⠈⠓⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⠵⠊⠁⠀⠘⢦⡀⠀⠀⣠⠟
⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠈⠻⢷⡦⣄⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢒⡽⠋⠁⠀⠀⠀⠀⠀⠀⣉⠷⠋
⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠉⡿⣄⣉⢉⡉⢉⣉⣉⡭⠤⠚⡟⠁⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⠀⠀⠀⠀⢰⡇⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢀⣠⠖⠉⠀⠀⠀⣏⠀⠀⠀⠉⢦
⢀⡤⠦⡤⣦⢀⡀⠀⠀⠀⠀⠈⠓⢦⡀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠈⠦⣤⣀⣀⣼
⠸⡄⠀⠀⠙⠋⣇⣠⡄⠀⠀⠀⠀⠀⠈⠳⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⠁
⠘⢲⡦⠀⠀⠀⠉⣱⠃⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠘⠒⣲⠆⣀⣾⡁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⣀⣠⣀⠀⠀⠀⣠⣄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠛⠓⠒⠒⠒⡶⠒⠒⠒⠚⠛⠛⠋⢹⠗⠛⠛⡇⠈⠹⡷⣄⣰⠃⠻⠋⠘⠞⠻⣄
⠀⠀⠀⠀⠀⠀⣠⠴⣆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢹⠀⠀⢹⣹⢹⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⣼⠁⠀⠘⡆⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⢸⠀⠀⢸⢿⠸⣄⡟⢦⣰⠦⡜⠉
⠀⠀⠀⠀⠀⢻⠀⠀⠀⡇⠀⠀⠀⠀⠈⠳⢄⣀⣀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣒⢶⣾⡞⠀⠈
⠀⠀⠀⠀⠀⠈⢧⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠒⠒⢶⣖⣶⡶⠊⠉⠉⠁⠙⠓⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣀⣀⣄⠀⠀⠀⠀⠀⠀⢠⠤⢦⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⢳⣰⠓⠿⠈⠀⠀⡇⡠⠶⢤⣤⣄⡾⠀⠀⠀⣿⣤⣶⠦⠤⢤⡀⠀⣠⣀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⣤⠖⠛⢻⠃⠀⠐⣏⠿⠁⠀⠀⢀⣤⣿⠇⠀⠀⠐⢿⣿⣦⡖⢦⡷⠙⣧⡄⠀⠀⠙⢶⡗⠁⠀⠈⡇
⠀⠀⠀⠀⢀⣠⣾⠁⠀⠀⠈⡆⠀⠀⢘⣦⡀⠀⣤⢠⡾⠿⠷⣆⠀⠀⢀⡿⠉⠁⠀⠀⠀⠘⡿⢤⣀⣀⡼⠀⠀⠀⣴⠛
⠀⠀⠀⢶⣼⡇⣉⠉⠙⢉⣿⠿⢆⣠⡎⠀⠉⠓⣪⠟⠀⠀⠀⠈⡧⠀⣸⣀⡀⠀⣀⢀⣶⣆⡇⠀⣘⣦⣀⠀⢠⡔⠚⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣇⠈⢿⡿⠛⠻⣿⡏⢀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡆⢀⣤⣤⡀⢠⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⢇⡀⣀⠟⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣿⣜⢿⣿⢧⣾⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠈⠉⠉⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣶⣿⠏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣯
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣴⣿⣆⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⣾⣿⡟⠁⠙⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⣹⡟⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡧⢀⣾⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⠖⠉⠁⢰⠧⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣷⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⢰⢧⠆⠀⠀⠀⠀⠀⣠⢀⡜⠛⠒⠤⢄⡏⢲⡖⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠊⢛⠁⠀⠠⡏⠀⠀⠀⡔⠓⠊⠈⠋⠀⠀⠀⠀⣀⡀⠀⡇⠀⠀⣤⢦⣀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡶⠲⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠘⢆⠀⢠⡤⠚⠓⠊⠀⠀⢀⣿⣆⠀⠀⠀⠀⡿⠛⠂⠇⠀⠀⡇⠈⡯⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢢⡀⠀⡱⠚⡴⣛⡆⠀⠀⠀⢡⣶⣍⣁⣀⠀⠀⣀⣀⡀⢰⠖⠺⡁⠀⣻⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣴⠃⠀⠑⠬⣄⠀⠀⠀⠈⢻⡄⠉⠉⠉⠉⢉⡝⠀⡼⠀⠀⢱⠎⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠀⠈⢣⡀⠀⠀⠀⠙⢦⣀⣀⡴⠋⣠⠞⠁⠀⢀⠼⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠑⠦⠤⠤⡄⠀⠀⢀⠖⠉⠀⢀⠤⠚⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠦⠤⠜⠀⠀⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠩⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡴⠛⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⣏⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣁⢀⡋⠉⠉⠉⠈⢦⡀⠀⠀⠀⣀⡍⠑⠒⠒⠒⠒⠒⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢳⠿⠋⠑⠢⠤⢀⣀⠠⠜⠓⠒⠊⢱⠃⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠳⢤⣀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢲⠊⠁⠀⠀⠀⣀⠝⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⠟⠒⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠶⠛⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠒⠒⠲⠦⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠒⠒⠒⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⠒⠒⠒⠚⠉⠉⠉


⣿⠿⠿⠿⡿⣟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣛⣻⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⠲⢄⠀⠀⠀⠀⠀⠀⠀⣀⣤⠠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⢸⠁⠀⣱⠀⢀⠖⠉⠀⠀⠀⠀⠘⡦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠘⢤⣀⣀⣠⣴⡁⠴⠮⡅⠀⠰⣄⣄⠀⠀⠀⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠎⠉⠀⠀⠀⠀⠇⠀⠀⠈⡱⠒⠉⠉⠁⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠠⠀⣀⣀⣰⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡀⠄⠊⠉⠉⠀⠈⣇⠀⠀⠀⠀⠦⠄⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡘⠓⠀⠀⡀⠀⠂⠀⠻⡆⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣽⣃⣀⠀⢀⡐⠄⠀⠀⢱⠀⠀⠀⠀⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠁⠀⠀⠈⡆⠀⠀⠈⡇⠀⠀⠀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣀⠀⠂⠀⠀⠇⠀⠀⠀⡇⠀⠀⠀⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣁⣀⡄⠀⠀⠀⠀⠀⣦⠀⡇⠀⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠠⠄⠊⠀⣠⠇⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢹⠟⢒⣒⣋⣩⠞⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠚⠒⢄⣂⣠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⡄⠀⠀⢰⠀⠀⠀⠘⡆⠀⠀⠀⠀⡿⣤⡴⠊⠁⠀⠉⢮⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⡀⠈⠦⣀⡠⠞⠁⠀⠀⢀⣼⠏⠁⡹⢄⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢙⠲⠤⣤⣤⣤⠤⠶⠞⠛⠁⠠⠊⠁⠀⢱⡀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠣⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠤⢄⣀⣀⣀⡠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⢐⡤⠰⠂⠉⠉⠁⠂⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠞⠁⠀⢀⢀⣀⣀⠠⠠⡎
⠀⠀⠀⠀⢀⡄⠠⠤⡀⠙⠀⠀⠊⠁⠀⡖⠑⡆⠀⢠⠀⢀⡔⡇⠀⠀⠀⠀⠠⠤⠀⡀
⠀⠀⠀⠀⠀⠑⢤⡴⢇⠀⠀⠀⠀⠀⠀⠹⡒⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⢀⠷⣤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠢⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀⠁⡉⠂⠀⠀⠀⠠⠂⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠑⠄⠀⠠⠄⡴⠚⠣⠤⠛⠲⠤⠄⠀⠠⠔⠁⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠊
⠀⠀⠀⡴⠂⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠤⠤⠤⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠉⡇⠀⠡⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠀⠡⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⠀⠐⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢱⣄⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⢼⠀⠀⠀⠀⠀⠀⢣⣬⣿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣥⣼
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣠⣀⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢄⣀⣀⣤⠀⢀⣀
⠀⢀⠤⠀⠀⠉⠑⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠊⠉⠀⠂⠠⡀
⠀⠳⢄⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⡀⠴
⠀⠀⠀⠈⠕⠂⠉⠒⠒⠢⠤⠤⠤⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⡀⠤⠤⠤⠔⠒⠒⠉⠑⢊⠁
⠘⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢠⠂⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀
⡆⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢠⣄⣀⣀⣀
⠀⠀⠀⠀⠸⣄⣀⣀⢘⡦⠴⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠖⠚⠓⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢤⣀⢀⡠⠤⢤⡤⢾
⠀⠀⠀⠀⠀⠀⠀⡼⠉⠀⠀⢸⠷⢤⢷⣄⣠⡀⠀⠀⠀⠀⢠⣶⢄⡞⠳⠶⠤⠶⣄⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠁⠀⠀⠀⠈⠻⡖⠦⣤⣤⠞
⠀⠀⠀⢠⣤⠀⣼⠀⢀⣤⠟⠋⠀⠀⡄⢹⡿⠏⠀⠀⠀⠀⠈⠈⢾⠰⠆⠀⠀⣤⠈⡇⠀⡿⠀⢴⣦⠀⠀⢼⣿⡀⠀⢨⠟⡓⠲⠦⠴⢷⡀⠀⣻⠀⣠⡄
⠀⠀⠀⠘⠿⢦⣹⡴⠋⠁⠐⠂⡀⢀⡄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠢⣤⠄⠀⠀⢘⡶⣣⠔⠁⠉⠀⠀⠉⠀⠙⢦⡸⡄⣅⠀⡐⠂⠀⠙⢦⡏⣠⠟⠯
⠀⠀⠀⠀⠀⠀⠙⢿⣦⣄⣀⣀⣉⡭⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡓⠒⠖⠒⠚⢙⡮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣟⣮⣍⣁⣀⣠⣴⡿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⠀⠉⠀⢰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠈⣧⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⣀⣀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢀⣀⣀⡀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢘⣆⡀⠀⠀⠀⠀⠀⠀⠀⢰⠛⢦⡤⡌⣉⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢯⠉⠉⠉⢻⡁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⠶⠒⠋⠉⢨⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠉⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢦⡀⢻⢻⡍⠓
⠀⠀⠀⠀⠀⠀⢠⠶⠦⢤⡼⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠟⠦⠞
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡂⠍⠉⠁⡢⡀⢀⡴⠂⠀⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠌⠀⠀⠀⠀⠈⢪⡏⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⠀⠀⠀⠀⠀⠀⠀⣇⢀⣄⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢸⢸⠀⠀⢰⣻⡆⠀⠀⣿⠰⣿⠀⠀⠀⢰⠓⠚⠋⠉⠛⠛⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⢿⣿⠇⠶⠀⢣⡀⠀⠛⠃⠀⢠⠋⠢⣀⠀⢀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠙⢦⢤⢀⣀⣀⣀⠀⠀⠀⠀⣤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡞⠁⢆⠀⠀⡠⠒⢳
⠀⠀⠀⠀⠀⠀⢀⠔⢦⡀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢑⠀⠸⠠⠊⠀⠔
⠀⢏⠉⠒⢄⡀⢸⠀⠠⠃⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠠⠀⠀⠂⣩⠝⠹⠂⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠎⠀⠀⠀⠀⠓⠀⠒⠒
⠀⠀⠑⠂⣄⠉⠀⠀⠘⢆⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣀⠄⠊⣁⣄⠀⢀⡀⠃⢅⡀⠈
⡖⠉⠉⢁⢀⠀⠀⠀⠀⠐⠢⠤⢀⣀⡀⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔⣊⠥⠖⣊⡡⠤⠒⠉⠀⠀⠈⠁⠘⢆⣀⡅
⠈⠁⠉⡵⠊⢠⠦⠤⠊⠉⠑⠒⠢⠤⢬⣍⣁⣒⠚⡽⠓⠒⠒⠤⠤⠤⠔⠒⠒⠊⠉⠀⢈⢧⠐⠂⠉
⠀⠀⠀⢄⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠱⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣴⠞⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠨⠀⠒⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⣠⠇
⠀⠀⢰⠊⠛⠳⠆⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠪⠄⣐⠠⣀⡀⠀⠀⠰⠁⠀⠀⣠⠎
⠀⠀⠈⠓⡖⠉⠐⠂⠀⠑⢄⠀⠀⠀⠀⠀⠀⡀⢀⡪⠙⠒⠤⢄⣀⣀⣀⡀⣀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠉⠒⢌⡒⢤⡅⠀⠀⡴⠁
⠀⠀⠀⠀⠘⠦⢄⡀⠀⠀⠀⠱⡀⠀⢀⡴⣪⠖⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠱⡴⢋⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠀⠀⣠⠄
⠀⠀⠀⠠⠀⠄⠆⠖⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠆⠀⠀⠀⡀⢀⠚⠁⠀⢀⡀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠏⠸⠀⠀⠀⠀⠀⠈⠀⣠⢖⡜⡐⠁⢀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⠀⠀⢀⠀⣤⠔⠀⢺⠏⡄⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠂⣶⣟⠈⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⡠⢁⠞⣡⠂⢠⠏⠀⢃⠀⢰⡀⠀⠀⠀⢀⡀⠀⠀⠀⣙⠻⢶⣤⠄⠉⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡃⢸⠁⢠⣴⠀⠀⣬⠀⠈⠳⢤⡀⠀⠸⡅⢀⡀⠀⠈⠃⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡜⢀⡔⠀⡁⡃⣸⠀⢀⠻⡄⢠⡈⠃⠰⠄⢸⣷⠤⠄⠀⠺⠗⠒⠾⠂⠀⠀⠀⠀⠀⠈⢲⣰
⠀⠀⠀⠀⠀⠀⠀⠀⣸⢸⠀⠃⡜⣇⠜⠀⢓⣹⡤⠼⢦⣵⣬⣙⠒⠶⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠉⠉⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⢀⡀⢹⡸⣴⣦⣿⠿⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠐⠘⢦⠁⣿⠯⠒
⠀⠀⠀⠀⠀⢀⠀⠀⠁⠠⡈⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁



⣦⣴⣶⣤⣶⣴⣦⣴⣴⣶⣶⣶⣶⣶⣶⣶⣶⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣴⣴⣤⣶⣦⣤⣤⣤⣶⣶⣤⣴⣴⣤⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠛⠛⠛⠻⣿⣿⣿⣿⡆⠀⠀⠠⠤⡀⠀⣠⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣇⠀⠀⠸⠀⠇⢠⡁⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢀⠤⠤⠄⠀⠀⠀⠁⢠
⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠀⠑⢀⠀⠀⠀⠀⠀⡰
⠀⠸⣁⠐⢶⠈⠆⠀⢀⡀⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡀⠀⠀⠆⠈⠀⠀⠀⠁⡂⠀⠀⢸
⠘⢄⠉⠢⠄⠁⠈⠖⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁
⠣⠈⡙⠁⠀⠀⠄⠀⠀⠀⠀⠹⣿⣿⡿⠟⠿⣿⡀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⠀⠀⠀⠀⡇
⠀⠀⠉⠢⠀⠀⠀⠀⠁⠀⠀⠀⠘⠏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠄⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣤⣴⣾⣿⡟⠀⠀⠂⣠⠎⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⢙⣿⣿⠟⠀⠀⣴⠀⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠀⣉⠒⢠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠰⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠛⠓⠓⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠈⠒⠤⣀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠐⠂⠂⡙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⢀⣠⠠⠊⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⣀⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⣀⣀⣀⠠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡤⢤⠞⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⢋⠴⠤⠶⢦⣀⣀⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡟⠿⠀⠀⠀⠰⠄⢀⡟⢸⡷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⠀⠀⠀⢀⡴⠛⠁⣼⠐⠇⡴⠁⠘⠳⠀⠸⣿⣾⠃⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⡇⠀⠀⣴⠏⠀⠀⠀⣿⣠⣈⣃⣀⣤⡄⠀⠈⠛⣯⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠷⣌⡳⣄⣼⠃⠀⠀⠀⠀⠹⣏⠛⢷⡖⠋⠁⢀⣴⠚⠁⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠴⠶⢤⡗⢸⣟⠒⠶⠶⠶⠶⠖⠻⡶⣤⣀⡀⣐⡟⢉⡟⠶⣄⡀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠖⠒⠶⣟⣺⡿⠀⠀⠀⠀⢀⠀⠀⢳⠈⠉⢋⣁⡴⠞⠀⠀⠀⠙⢦⡀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⣥⣤⠤⠴⠖⢻⠀⠀⣼⠀⠀⡟⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⢀⡇⠀⠀⠀⠙⢲⠦⣄⠀⠀⠈⣷⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⢸⡄⠀⡇⠀⢸⡇⠀⠀⠀⠀⡿⣰⠃⠀⠀⣰⠇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠘⡟⠲⠇⠀⠘⣇⣀⣠⡤⠴⣿⡃⠀⠀⣰⠏⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⣧⣀⡀⠀⠀⠀⠀⠀⢀⣠⣿⠙⢳⣶⡃⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢻⠀⠉⠉⠛⠛⠛⠉⠉⠁⢻⣤⢌⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠦⣤⣀⣀⡀⢀⣀⣀⣠⣤⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⡀⠈⠛⠀⣼⣿⣦⠈⠛⠁⣠⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⣀⣸⣿⡿⣿⣧⣠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠙⢿⣿⡟⠀⢹⣿⡿⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣇⣀⡉⠁⢀⣀⣉⣀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⡿⠿⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠛⠋⠙⠋⠉⠙⠛⠉⠛⠛⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣧⣤⣤⣤⣤⣤⠄⠀⠠⣤⣤⣤⣤⣤⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡟⠛⠋⠋⢁⣤⣶⣦⣄⠈⠙⣿⡟⠛⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣿⣅⣀⣀⠀⣿⣿⠟⠻⣿⣧⡀⣿⡗⢀⣼⡿⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃⠘⣿⣿⠛⠛⠁⢀⣿⣿⠀⠀⢹⣿⣿⣿⡿⠛⠻⣿⣶⣽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⣸⣿⠟⣿⡇⠀⢀⣼⣿⣿⡿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠃⣀⢀⡀⢀⣿⡟⠀⠀⠀⠈⢿⣿⣧⣾⣿⡿⠀⣿⣇⣠⣾⣿⡿⠋⠀⠹⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣾⣷⣾⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢿⣶⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠒⢀⣀⣤⣤⢼
⣿⢻⣿⣿⠇⠀⠀⠀⢀⣀⣠⣤⣦⣤⣤⣤⡄⠀⣀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⡀⠀⠀⠀⣀⣤⣦⣤⣿⣷⣶⣾⣿⣿⣿⣿⣿
⡿⣿⣿⣿⠞⠁⠀⠀⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⠀⣀⣤⣶⣶⣦⣼⣆⣶⣿⣿⣿⣿⣯⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣿⣿⣿⣿
⡀⣿⣿⠇⠀⣠⣤⣄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿
⠀⣿⠀⣸⣿⠟⢻⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⣹⣿⣿⣿⡟⢿⢻⣿⣿⣿⣿
⣰⣿⣿⣿⣿⣿⠿⠋⠀⣸⡿⠓⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣀⠉⠛⠀⠈⠸⣿⣿⣿⣿
⠟⢻⣉⣿⣿⠟⣴⢦⣼⠋⣠⣤⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠛⢿⣯⠀⢶⣿⣿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣹
⠟⢠⣼⣿⣿⣦⣸⣎⠁⢸⣿⣿⣁⡄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣤⣶⣄⠀⠈⠿⠀⠈⣿⢹⣿⣿⡇⠀⠀⠀⠈⣿⣿⣿
⢰⣦⢸⣿⣿⣍⣽⢋⣀⠘⣇⠈⢛⠁⠀⠸⣿⡟⢿⣿⣿⡿⠋⠁⢀⣀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⡁⠘⡿⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣶⣤⠀⠀⣿⣿⣽
⢂⣱⠎⢙⣿⣿⣿⣿⣿⣿⣿⣶⠿⣷⣶⣦⣸⣷⣾⣉⣷⠀⠀⠀⠘⢛⡇⠀⢸⣿⣿⣿⣿⣿⣿⠟⠁⠀⣴⣿⣶⣿⣄⣴⣶⠿⣷⣾⣿⣿⣿⣿⣿⣿⡄⠀⢿⣿⣟
⣿⣽⣦⣬⠻⢀⣿⣿⣿⣿⣿⣿⣦⣿⣿⣿⣧⢿⣏⣻⠃⣀⣴⣾⣻⣟⠀⠀⡿⠻⡿⣿⠿⠟⠋⠀⠀⣼⣿⣿⣿⣿⡏⢙⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿
⠻⣿⠇⢿⣴⠾⣿⣹⣿⣿⣿⣿⣿⠿⣿⣿⣿⠀⣿⠟⣿⡿⣿⣿⡿⣿⡆⠀⠀⣴⡄⢀⣰⡆⠀⠀⢸⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡸⠀⣿
⣾⡿⣦⠸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣾⠋⢹⣿⣿⣿⣿⣧⣸⡿⠻⠿⣿⠿⣿⣿⣷⠀⢀⣿⡇⣤⣄⣈⣿⣿⣿⣿⠀⠈⣿⣷⣶⠾⠛⣿⣿⣿⢸⣷⣤⣿
⣿⣷⣿⢹⣇⣿⣿⠿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⠀⠘⢿⣿⣿⣿⡿⣿⡅⢀⠀⠀⠀⣤⠈⣿⣦⢾⡿⢁⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣧⣀⣼⣿⣿⣯⢠⣽⣿⣿
⣿⣿⣿⣬⣿⣿⣿⣤⣿⣿⣿⣥⣬⣭⣿⡏⣿⣿⣀⣤⣼⣿⣿⣯⣀⣿⢉⣭⡕⢦⣤⣴⢈⣿⢿⣿⠧⢼⣿⣿⣿⣿⣿⡇⠲⠾⠟⣿⣏⢹⣿⠋⢻⣿⣫⣼⠛⠛⠛
⠛⠛⠛⠛⠛⠛⠋⠋⢻⣿⣿⣿⣿⣿⣿⣿⠿⠙⠁⠈⢿⣿⣿⣿⣿⡁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣤⠀⠀⢻⣿⣿⣿⣿⣿⡏⠙⠃
⠀⠀⠀⠀⢀⣤⠀⠀⢹⣿⣿⡟⠙⣿⣿⣿⠀⠀⠀⠀⠘⣿⣿⣿⣿⡗⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠿⠋⠀⠀⢸⣿⣿⣿⣿⣿⠇⠀⢀⣠⣤⣤
⣀⠀⣀⣴⣿⣿⡇⠀⠸⣿⣿⡇⠀⠹⣿⣿⡆⠀⠀⠀⠀⣿⢿⣿⣿⡇⠀⣰⣿⣽⣧⣶⣿⣿⣦⣴⡆⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡄⠀⠛⢿⣿⣿
⣿⣷⣿⣿⣿⣿⡃⠀⢀⣿⣿⠃⠀⠀⢻⣿⡟⠀⢀⠀⢰⡿⠟⢉⡛⠁⢰⡟⢡⣿⣿⣿⣿⣿⣿⣹⡇⠀⢸⣿⡿⣿⣧⠀⠀⠀⠀⠀⢹⣿⣿⢻⣿⣷⠀⠀⣼⣿⣧
⣿⣿⣿⣿⣿⣿⣷⠀⣸⣿⡏⠀⠀⠀⢸⣿⣿⠀⠨⠀⢸⡇⣠⠟⠁⠀⠀⠀⠈⠉⠉⠉⠻⣿⣿⣿⡇⠀⠀⣿⣧⣿⣿⠀⠀⣾⠀⠀⢸⣿⣿⠀⣿⣿⠀⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⣸⣿⡇⠀⠀⠀⠘⢿⣿⠆⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠘⡇⠛⠇⠀⠀⣿⣿⣿⡟⠀⠀⡙⠀⠀⠸⣿⣿⠀⣿⣿⡆⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣇⡀⠹⣿⡃⠀⠀⠀⠀⣸⣿⠃⠀⠀⠙⠋⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⣤⡾⠏⠰⢄⠀⣴⡿⣿⣿⡧⠀⠰⡏⠀⣀⣴⣿⣿⣀⣿⣿⣇⠀⣿⣿⣿
⣿⣿⣿⣿⣿⡏⢸⡇⠀⣿⡇⠀⠀⠀⠀⠻⠿⢷⠀⠀⢀⣤⣶⣤⡄⢀⣰⣿⣿⠛⣄⠀⢈⣄⠀⠀⠈⠀⠀⠰⣍⣭⣀⢀⣠⣀⣘⢛⣍⣉⡹⠿⠟⠋⠉⠀⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⠦⢀⣀⣠⣿⣷⣶⣾⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⠃⢠⣼⡀⠸⣿⣷⠀⠀⣦⣄⣀⠙⠉⣯⣿⣿⡟⠛⣿⣿⣬⣹⣷⣶⠟⠋⢀⣼⣿⣿
⣿⣿⣿⣿⣻⣿⣇⣤⠾⠋⢀⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⣴⣿⣿⣦⠴⢿⣿⣿⣿⢲⣿⣻⠛⣛⣿⣿⣿⣿⣿⡟⠛⠀⣴⣿⣿⣿⣿
⣿⡿⢻⣿⣽⣯⣮⣡⣦⣶⣿⣿⣿⠉⣿⣿⡏⢻⣿⠿⢿⡿⠶⢿⣿⣿⣶⣾⣷⣾⡗⠉⠘⠟⢃⣰⣦⣤⠘⣿⠟⣹⣿⣭⡟⠉⠁⠀⠂⠈⠉⠉⠀⣾⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⡏⢹⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢛⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠴⠶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⣶⣶⣆
⠀⠀⠀⠀⠸⣿⣶⣦⣭⣉⣩⣥⣶⣿⣿⣿⣷⣦⣭⣉⣉⣭⣴⣾⣿⣿⣿⣶⣬⣍⣉⣭⣴⣾⣿⣿⣿⣶⣬⣭⣉⣩⣥⣶⣿⣿⣿⣷⣦⣬⡍
⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⢻⣿⣿⣿⣿⣿⣿⣿⡟⢉⣉⣉⣉⡉⠉
⠀⠀⠀⠀⠀⣠⣾⣿⡟⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡟⢿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⡃⢙⣻⣿⣿⠃⣬⣭⣩⣵⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣙⣛⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣋⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣶⣶⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀
⠠⠤⢄⣰⠆⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡀⢸⡇⠀⠀⠟
⠁⠀⠈⣿⠀⠀⢠⡟⡆⠀⠀⠀⠀⠀⡄⡰⠒⢲⡀⠀⠀⠸⡇⠱⣄⠀⠀⡏⢻⡆⠀⡴⢦⡇⠀⡇
⠃⠀⠀⢻⠀⠀⣸⠀⡇⠀⠀⠀⠀⢠⠀⡅⠀⠀⣱⠀⠀⠀⡇⠀⢹⡆⠀⠃⠘⣷⣼⡏⠀⠗⠀⡇
⠀⠀⠠⡀⠀⠐⣿⠤⣧⠀⠀⠀⠀⠰⠾⠛⠋⠉⠉⠀⠀⠀⡇⠀⣸⠃⠀⠂⠀⢨⣿⡇⠀⠀⠀⣷
⠀⠀⠀⠈⠢⠀⡏⠀⢻⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠠⠃⠀⠀⡄⠀⣼⣿⡇⠀⠀⠀⢹⠃
⣴⠀⠀⠀⠀⣸⡇⠀⢸⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⢇⠀⡿⢹⡇⠀⣠⠀⣸
⣿⠀⠀⠀⢠⣿⡇⠀⠸⡆⠀⢠⡎⠀⢸⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⡀⢸⣠⡇⢸⣇⣰⠏⠀⡇⠀⠀⠀⡀
⣿⣠⣴⠾⠟⣿⡃⠀⠀⡇⠀⣸⡇⠀⢸⡇⠀⢀⠇⠀⠀⠀⠻⠆⠀⠀⠸⡾⠟⠀⠘⠿⠃⠀⣸⠃⠀⠀⠀⢹⣆
⢻⠋⠁⠀⠀⣿⠀⠀⢰⣿⠀⢹⡇⠀⠈⠷⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠛⠀⠀⠈⣿⣿⣿⣿⣤⡀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠅⠼⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣤⣨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⣤⣶⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣀⣼⣿⣷⣦⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⣀⣀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣼⣿⣿⡆⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣨⣭⣤⡼⣷⠀⠀⠀⠀⠀⠻⢿⣭⣽⣯⣭⣻⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⣄⣀⣴⣿⣿⣦⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⡇⠀⠀⠛⢿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣛⡛⠻⠿⠿⠟⣛⠓⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣀⠸⣿⣿⣿⠇⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠆⢀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣦⡉⠋⣡⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣷⣼⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⢸⡿⠛⠻⣿⣇⣀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⠀⠀⠀⠀⠘⠁⠀⠀⠙⡉⠉⠀⠀⠀⠀⠀⣿⣿⡟⠀⣿⣿⣿
⠀⠀⠀⢰⣿⡟⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⢾⡶⠂⠀⠀⠀⠀⠁⣠⡦⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⣿
⠀⠀⠀⠘⠟⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢸⣿⣿⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡅⠀⠀⢰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣾⣧⠀⠸⣿⣷⠀⠀⢸⣿⣇
⠀⠐⠒⢶⣶⣶⣶⢶⣴⣿⣿⣶⠶⠶⢾⣿⣷⣤⢤⡤⠄⠀⠤⠀⠀⠀⣠⣿⣿⣤⣴⣿⣿⡷⠶⣿⣿⣿⠿⠟⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢀⡀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠤⠀⠁⢠⡀⠈⠁⠠⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠢⠂⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠀⠤⠀⢠⠄
⠀⠀⠀⠀⠀⠀⡋⠀⠐⠂⠀⠀⠠⠠⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠈⠣⠤⠀⠀⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⡀⠠⠄⠀⠐⡀⠄
⠀⠀⠀⠀⠀⠀⢠⠃⠏⠩⠴⠤⠤⣄⣀⣀⣠⣠⡍⣍⡍⣩⢩⢀⣀⣄⡄⠲⠤⠤⠤⠤⠄⠤⢛
⠀⠀⠀⠀⠀⠀⢦⢀⡀⠀⠂⠀⠀⢀⡄⠀⠀⢀⡄⠀⠀⠀⡀⠀⠀⠀⡤⠀⠀⠠⠄⠀⠀⢀⡠⢤
⠀⠀⠀⠀⠀⠀⠳⢅⣄⢪⠪⡣⡑⢞⠆⠂⠀⠐⠐⠒⠐⠀⠁⠒⠀⠒⠒⠂⠀⠀⠈⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠐⠒⠃⠀⠉⠱⣘⡉⠒⠓⠒⠒⠂⠐⠒⠒⠒⠒⠐⠒⠒⠀⠒⠂⣀⠐⠐⣛⣉⡤⠊
⠀⠀⠀⠀⠀⠒⠂⠀⡀⡐⠀⠄⠀⠀⡑⠀⠄⣀⢀⠀⠀⣠⠄⠀⡄⠐⠒⢛⡋⠀⠈⠃⠉⡤⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡊⠉⢁⠀⠀⢀⠮⠒⣄⡴⠐⠒⠀⣰⠧⠃⡈⠐⠳⡄⠘⠪⠧⠀⠀⣀⠐⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠀⠈⠉⡀⠀⠄⠀⢀⠀⠀⠀⣀⠐⠀⠀⠀⠉⠀⠒⠀⠐⠀⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠁⢠⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⡀⢀⡀⠀⢀⠀⠀⢁⡀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠄⠒⠊⠛⠀⠒⠒⠒⠢⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠄⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠆⠀⠐⠠⠀⠀⢀⣀⠀⠀⠀⠓⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⡠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠞⢀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠙⠒⠒⠶⢶⣒⠒⠛⠛⠛⠛⠛⢛⣛⣻⠿⠿⠟⠛⠋⠉⣉⠙⠋⠉⠉⠉⠀⡀⠄⠲⠀⠀⠡
⠀⠀⠀⠀⠀⠘⠁⠹⡀⠀⠀⠀⢀⡤⠽⠗⠐⠶⢶⢞⠛⠋⠁⠀⠀⠀⢀⣀⠴⠺⠳⠄⡀⢀⠀⣀⡄⠀⠀⢀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⠖⠛⠓⠤⣄⣀⣀⡀⠀⠀⡀⣀⣤⠴⠶⢿⣯⣼⣤⣤⣤⣀⣥⠼⠤⠤⠔⠒⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⢸⠀⠀⠈⠑⠂⠀⠈⠉⣉⣉⣉⡁⠠⠤⠒⠉⠀⠀⠐⠢⠀⠀⠙⠂⠀⡠⠊⡀⠀⠋⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⣀⣤⣶⠿⡋⠁⠁⠀⠒⣤⠀⣁⠀⠀⠀⠀⣀⣀⣀⠔⠀⢠⡆⢀⠄⣠⣠⠌⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠈⢲⠀⠀⠀⠁⣠⠞⠋⠙⠓⠓⠒⠛⠛⠓⠂⠤⠥⠬⠁⠦⡀⠈⠡⠦⠤⠀⠀⠀⠶⠶⠟⠁⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠈⢡⠾⢿⠆⠀⠀⢀⣴⣶⣶⣶⠀⠀⠀⠀⠀⣠⡴⠶⣄⠀⠀⠀⡴⠶⣆⠀⢰⠶⡖⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⠂⠀⠀⠀⢄⣀⠠⠊⠁⠀⠈⠻⠄⠤⠀⠀⠈⠁⠀⠀⠀⠁⠄⠜⠁⠀⠀⠉⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠔⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⠶⣖⣶⣶⣦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢖⣻⣯⣽⣿⣖⣶⣿⣿⣭⡍⠻⢛⣛⠙⠫⠑⠉⠓⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡿⣿⣻⣷⣟⣿⡷⣿⣿⣷⡊⠁⠔⠀⠀⠁⠁⢠⣀⠀⠀⠀⠤⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⣿⣫⣿⣿⣿⡿⢟⢉⡉⠁⠀⠀⠀⠀⠀⠠⠀⠈⠀⠠⠀⡀⢤⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⠏⠙⣋⡀⠉⠀⠀⠀⠇⠀⠀⠖⠀⠀⠦⠀⠀⠀⣄⠀⠢⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠎⢸⣿⣿⣿⣿⣿⣿⡿⠛⠃⣀⡀⠀⠀⠀⠀⠐⠀⠀⠠⡄⢲⡀⠀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠹⡆
⠀⠀⠀⠀⠀⢸⠀⣾⣿⣿⣿⣿⣿⣿⣏⠁⠶⠕⢠⡀⠀⠀⠈⠓⠀⢠⠆⠉⠀⠀⠈⠃⠀⠘⠀⠈⠀⡀⠹⠉⠀⣷
⠀⠀⠀⠀⠀⠸⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣥⡤⠀⠁⣤⡀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⡀⠀⠀⠘⠀⠉⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠈⢢⣤⣌⣉⣙⣉⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣇⣀⣠⣤⣤⡶⠖⠈⠀⠀⢠
⠀⠀⠀⠀⢠⠊⠈⢛⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⣿⣿⣟⣛⣋⠛⠛⠿⠿⣿⠿⢿⠛⠓⠚
⠀⠀⠀⠀⠈⠉⣘⣧⣄⣤⣾⣿⣷⣶⣽⣌⠈⠙⠟⠉⣨⣾⡿⢿⣶⣤⣄⣈⠉⠛⠻⠿⢿⠛⣻⣶⣶⣾⡓⠦⣌⠐⢤
⠀⠀⠀⢀⠎⠁⠀⠑⠋⠉⠻⢿⣿⣿⡛⠿⣆⣀⣤⠞⡉⠀⠊⠁⠛⠛⣻⣿⣿⣷⠶⣤⣀⣴⣯⣭⣤⣉⡉⠀⢻⡒⠲⠃
⠀⠀⠀⠘⠖⢲⠞⠋⠉⠉⠙⠛⠛⠋⢹⣷⣌⠁⠀⠁⠀⠀⣀⡤⠶⠛⠿⠿⢝⠓⠀⠈⠉⠉⡉⣀⠠⠈⠈⠛⢋⣻
⠀⠀⠀⠀⠀⠡⣀⣄⢀⠀⣀⠀⣄⣤⣔⣥⣛⠳⣦⣴⡶⠛⣿⣴⣉⢀⢀⣠⣤⣤⣴⣴⣶⣿⣿⣿⣿⣦⣴⣴⣾⠟
⠀⠀⠀⠀⠀⢀⡿⢛⣿⣿⣿⣿⠿⠿⣿⢿⣿⣿⣿⣶⣶⣶⣿⡶⠿⢿⣿⣛⣛⣋⣭⣥⣶⣿⣿⣿⣷⣾⣶⡴⠿⣄
⠀⠀⠀⢐⠦⠼⣆⢈⠛⠛⠿⣿⣷⣤⡀⠀⠀⠀⢀⡤⠐⢀⣠⣴⠾⠿⠿⠿⠿⠿⠋⠉⠉⠉⠉⠁⠀⠀⢀⣠⣤⣿⣛⠋⣱
⠀⠀⠀⢨⣁⣰⣿⣿⣶⣶⣶⣿⣿⣿⣷⣄⠀⠘⣡⣴⣾⣏⣭⣥⢀⡀⠀⠀⠆⢀⣀⠀⢀⣠⣷⣿⣧⣾⣿⡿⡿⢉⡙⣾⠁
⠀⠀⠀⠀⠉⠙⣧⢙⣿⣿⢿⣿⣟⠙⠛⣿⡷⣿⣿⡿⠿⢿⣿⣿⣿⠿⠿⣷⣶⣿⠿⠿⠿⣛⣻⣿⣿⠏⣉⣤⢦⠏⠙⠁
⠀⠀⠀⠀⠀⠀⠸⡌⢿⣿⣿⣿⣟⣴⣿⣿⣿⣿⣿⣷⡶⠶⠶⠶⠶⠶⠶⠿⠾⠷⠶⠛⠳⠼⠋⠛⠿⠚⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣿⣿⣿⡟⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⠀⠠⠀⠆⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣷⣾⣧⣦⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣍⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠀⠀⠀⠈⠉⠓⢦⡀⠀⠀⣀⠤⠶⠚⡋⠉⠋⠉⠛⠓⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣇⡴⠋⠁⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠐⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠱⠮⣝⣒⢲⠲⠲⠒⠒⣚⣫⡴⠋⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢀⠤⡄⢠⣯⢡⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠒⠲⡎⣷⣈⢷⡼⡆⢾⣷⣌⡓⠦⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠴⣚⣡⡾⡀
⠀⠀⠀⠀⠀⠀⡄⢤⣚⣅⠳⣷⠸⣟⡃⡑⢻⡈⠀⢟⠻⣷⣶⣦⣍⣉⣉⣉⣉⣉⣉⣉⣩⣴⣾⣿⠿⢛⣡⣉⡀
⠀⠀⠀⠀⡠⢄⣝⡀⠹⣿⡄⡙⢇⢻⣧⢳⡜⠷⠀⠈⣇⠤⠬⣉⠿⠛⠿⠿⠿⠿⡛⡛⢛⣩⠭⠒⣀⣥⡾⢀⠈⡇
⠀⠀⠀⠀⠳⢦⣙⢿⣄⠘⢷⠹⣎⠀⢹⡄⠙⠁⠀⠀⠸⣷⣶⣶⣭⣲⣤⣀⢀⣠⣤⣖⣭⣶⣾⣿⠿⣋⡠⠁⠀⡇
⠀⠀⠀⠀⠀⠀⡹⠷⣝⣧⡸⣇⢾⣷⠙⠋⠀⠀⠀⠀⠀⢳⠬⠩⠝⠛⠻⠿⠿⠿⣛⡛⠋⠍⠁⢂⡰⠋⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠙⠢⡀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢗⠲⠤⣤⣀⣀⣀⣀⣀⣠⠤⠴⠚⠉⠀⠀⢠⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠉⠁⠀⠉⠒⠤⠤⠔⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⣀⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣦⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⠿⠿⠛⣛⡛⢻⣛⠿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣯⠅⠤⠀⠀⢀⠈⠀⠀⠘⠈⠀⣨⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠍⢠⡀⠂⠐⠦⠀⠋⠀⠀⠀⠄⠀⠘⡾⠽⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡯⠀⠀⠈⠁⠀⡀⠀⠰⠄⠀⠀⠀⡀⢠⡄⣒⣤⢾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣄⣀⣁⡀⠐⠃⠀⠲⠀⢠⠆⠈⠉⣤⠈⢁⣸⣿⣷⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣴⣷⣴⣶⣶⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠤⠽⢿⣿⣿⠿⠛⣛⣻⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⡿⠿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠒⠋⠋⠉⢉⣉⣉⣬⠤⠾⠿⠃⠀⢐⣉⣭⡿⢛⡉⢁⡰⣬⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣠⣶⣶⣶⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡑⣶⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⣿⣟⠉⢀⣁⣨⣄⣿⣋⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡉⠛⠻⢋⣿⣿⣿⣿⠿⠻⠿⢿⣿⣿⡿⠟⠛⢻⣿⣯⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⣦⣈⠏⢉⣉⣉⣻⣃⡑⠢⠤⢿⣋⣭⣽⢋⣍⣉⣉⣩⣿⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣯⣘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡖⡈⣲⣿⣿⣿⣿⣿⡿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣦⣤⣤⣤⣤⣠⣤⣤⣤⣤⣥⣴⣶⣾⣿⣿⣿⣿⣿⣿⣾⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣾⣧⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⣿⣿⣿⣿⣛⣋⠉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠉⠉⢉⣀⣀⣹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠐⣛⣛⡃⠀⣉⡭⢭⢹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⣿⠄⠉⠉⠀⠀⢧⡀⠀⠀⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⠀⠀⠀⢀⠀⣨⢇⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠸⣿⠟⢿⠟⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⠀⠀⡠⠔⠁⠀⠀⠀⡤⠐⠀⢆⣀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⠁⢱⠐⢄⠀⣀⠄⠘⢆⠀⠈⡄⣀⡀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡠⠌⢀⡆⠀⠀⠀⣱⠀⣹⠇⠀⠀⠀⠣⠀⢛⠀⠈⠂⢄⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠉⠀⡗⠐⠒⠂⢀⠛⠂⢬⣿⣶⣶⣿⣿⠊⠁⠀⠀⣀⠔⠊⠙⠁⠀⠀⠀⠀⠙⠢⣀⠈⠢⡀
⠀⠀⠀⠀⠀⡠⠂⠁⠉⠀⠀⡇⠀⠀⠀⠈⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠈⠈⠀⣤⠦
⠀⠀⠀⠀⠰⠀⠀⠐⠒⠃⣀⠇⠒⠒⠀⠹⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡖⢤⣼⣗⠳
⠀⠀⠀⠀⠇⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⠛
⠀⠀⠀⢰⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢻⣿⣿⣿⣇⠀⠀⠀⡄
⠀⠀⠀⠰⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⡇
⠀⠀⠀⡤⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⣿⣿⣿⣿
⠀⢀⠌⠀⢀⣵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸
⣴⢟⣷⠢⣼⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣿⣿⣿⣇⠀⠀⠈
⠀⠞⢀⡇⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠉⠀⠀⠀⠘⠀⢀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠤⠀⠀⠀⠀⠂⠁⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡌⠀⠆⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⡙⠒⠤⢀⡤⡄⠀⢰⠀⠀⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⢠⣸⣿⣿⠉⠢⢀⡘⠀⠋⠐⡆⠀⠀⠀⠀⢰⠉⠗⠈⠣⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠸⡀⣰⠀⠀⡇⠀⠀⠀⠀⣸⠀⢰⠀⠀⢠⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠁⠑⠠⣃⠀⠀⠀⣠⠉⠁⠚⠋⠁⠀⠓⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⡶⠊⠁
⣶⣶⣶⠶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠉⠀⠀⠀⠄⠀⠀⠈⠉⠋⠚⢿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢹⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠀⢠⣝⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⡧⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣤⣼⢿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣈⣤⣤⣴⣾⣿⣿⢿⠟⠚⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠻⠿⠿⣷⣶⣶⣶⣤⣤⣤⣤⣤⣤⣴⣶⣾⣶⠿⠿⠛⠋⠋⠁⠀⠀⠀⣠⡿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⠀⠠⠠⠀⠛⠛⠛⠻⠿⠟⢋⣉⣀⣹⣝⣋⣃⣀⡀⠀⠀⢀⡰⠷⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠣⠈⠙⠓⠦⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠠⠀⣂⠧⠛⡆⣠⡀⠉⠹⣿⣦⡀⠀⠀⠀⠀⢀⣠⣴⠶⠟⣿⣋⣛⣿⡷⣺⣿⣿⣿⣇⣯⡄
⠀⠀⠀⠀⠀⠀⡀⠘⢸⣄⣤⣸⡅⠈⢲⣿⠉⡝⣆⠀⣠⣴⠟⢛⣇⣀⣼⡏⣹⠟⠓⠚⠃⡀⠉⠙⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠁⠐⠀⠜⠁⠘⠟⠀⠹⠛⠿⠇⠿⠓⠋⠹⠷⡟⣉⢩⠉⡛⡉⠀⠄⠚⠈⣙⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⢀⣸⣀⢀⣠⣶⣮⣤⣤⣤⣤⣴⢶⣶⣶⠦⠔⠄⠔⠶⣶⣾⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠥⣤⣿⣥⣄⡀⠀⠀⠀⠀⠀⠀⠀⠿⣁⣀⣤⣾⣷⣾⣷⡾⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠛⠙⠛⠃⠀⠙⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⠉⠀⠶⠆⠀⠀⠀⠀⠴⠆⠈⠉⠑⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢉⡀⠀⠀⠀⡤⣤⠀⠀⠀⠀⢰⡦⠀⠀⠀⠀⠚⠓⠀⠀⠀⠀⠈⠑⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣜⠕⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⣤⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⡇
⠀⠀⠀⠀⠀⠀⢀⠼⢇⡀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠈⠓⠁⠀⠀⠀⠀⣀⣾⠁
⠀⠀⠀⠀⠀⠀⠓⠦⡄⠉⠒⠪⠶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡒⠤⠔⠚⠉⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡤⢔⢢⠀⠀⣈⡉⠑⠒⣒⣿⠃⠤⠤⠤⠞⠓⠒⣂⣉⣩⠤⠴⠒⠒⠚⢩⡍⠻⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠻⠉⢫⣍⣥⠱⠖⠚⢧⠀⠀⠀⢀⣠⠔⢚⡉⢡⣴⡈⠀⡭⠝⠲⣀⣀⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠻⢶⣼⣧⣄⣈⣓⡛⠀⣧⡴⢊⣁⠊⣧⣐⣻⣤⣤⣤⣥⡶⠶⠞⢛⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⢐⣪⣭⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠲⠶⠶⠦⠤⠤⠤⠤⠤⠶⠖⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡃⠀⠀⠄⠐⢰⠀⢉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠂⠀⠨⠂⠀⠀⠀⠁⢀⠔
⠀⠀⠀⢀⡀⠀⢁⠀⠒⠀⠀⠈⠁⠀⠀⡀⠀⠀⡅⢀⠀⠀⠀⠠⠀⠀⠢⡀⠀⠀⠀⠠⠀⠀⠀⠲⠁
⠀⠀⠀⢙⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡖⠀⠀⠈⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠒
⠀⠀⡠⠂⠀⠒⠒⠠⠀⠀⠀⠈⠀⠀⢂⠀⠀⠀⡃⠠⠀⠀⠈⠀⡄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠨⠧⠀⠀⠀⠀⠀⠀⠐⢠⠇⠀⠀⠀⠀⣀⡀⠈⠙⠃⠠
⠀⠀⠀⠀⠀⢠⠀⢠⠈⠆⠀⠀⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠁⢀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠃⠀⡀⠀⠀⠈⠀⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠃⠐⠠⠀⣆⠀⠐⠀⠀⠙⠀⠀⠀⡄⠂⠀⠀⠀⡖
⠀⠀⠠⠀⠁⠀⠀⠠⠐⠁⠀⠀⠑⠠⠆⠀⠀⡀⠀⠀⠐⠤⠀⠐⠂⠅⠀⠀⠀⠀⠀⠡⠄
⠀⠠⡀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠠⡈⠀⠄⡀⠀⠀⠀⠀⠀⠐⣀⠈
⠀⠀⠁⠄⡀⠀⠀⠀⠀⡈⠀⠀⠆⠀⢀⠰⠀⠀⠃⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠄
⠀⠀⠀⠀⠈⠐⠀⠀⠔⠀⠀⠀⠐⡀⠀⠁⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠁⠄⡘⠂⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠐⠠⢀⠀⠀⠀⡀⠄⠚⠀⠀⢀⡀⠀⠐⠓⠀⢀⠀⠀⠀⡀⠢⡀⠀⠀⠀⠀⡌⠀⠘⠀⢀
⠀⠀⠀⠄⠐⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⠁⡤⠀⡄⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⢠⠀⠀⢠⡄⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠃⠁⢠⠀⠐⠀⡀⢀⠔⠀⠀⠒
⠀⠀⠀⠀⠠⠂⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠈⠘⡆⠀⢀⠜⠂⠀⠀⠒
⠀⠀⡀⢀⡠⠂⢂⣀⠀⠠⠠⠀⠀⠰⠀⡀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣀⡀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⡗⠠⠀⠀⠀⠀⢀⡠⢀⠀⠄⠀⠀⠣⡀
⠀⢀⠀⠈⠀⠀⠉⠀⠈⠐⠄⠀⠀⠀⠀⠒⠆⠈⠀⠀⢠⡄⠀⠀⠀⡄⢤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠐⢄⠐⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠁⢀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⠂⠀⠀⠄⡀⡀
⠀⠀⠀⠁⠀⣰⣤⠀⠔⠀⠀⠀⠙⡀⠂⠀⠀⠀⠲⠀⠀⠰⠁⢈⠐⡄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠑⠄⠔⠀⠀⠀⠀⠀⠀⠀⠂⠁⠊⠀⠒⠈⠀⢂⣥⠀⠀⠂⠈
⠀⠀⠠⠲⠀⠀⠀⠀⠈⡀⠀⠀⠂⡀⠀⢔⠀⠁⢀⡀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠈⠡⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠈⠀⢸⡀⠰⠆⠁⡀⠀⢠⡁⠀⠈⠁⠀⡠⠂⠓⠀⠀⠀⠰⠄⠰⠇⠈⠃
⠀⠀⠀⠈⠁⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⡁⢤⠄⠘⠀⠀⠀⠈⠐⡠⠈⠀⢣⡌⠌⠀⠀⠂⠐⣼⠆⠒⠢
⠀⠀⠄⠀⠀⠠⠀⠀⠀⠀⠀⡋⣤⠑⡀⠀⣄⠐⠀⢀⡀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⢀⣤⠀⠌⢀
⠀⠀⠀⠀⠤⡀⠀⠓⠀⠀⠀⠁⠤⠝⢢⡀⠈⠀⡰⢀⠄⢩⠀⢀⠀⠈⠀⠀⠈⠀⠁⣀⡈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⢲⠀⠀⠀⠀⠀⠂⢠⡜⠳⠒⠀⠀⠨⠀⢀⠚⠀⠀⣀⡀⠙⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⢀⠀⠁⠀⢀⠀⠀⠀⠒⠀⠀⠀⠤⠀⠐⠀⠀⠀⡄
⠀⠀⠀⠀⠈⠁⠂⠀⠒⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠠⠀⡀⡀⠄⠀⠆⠀⠀⠀⠀⠀⠀⠲⡈⠀⠀⢀⠀⡈⠄⠠⠀⠁⠀⢀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠈⠰⡀
⠀⠬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠷⠀⠂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠄⠀⢀⠄⠀⠀⠐⠂⠤⡀⠀⠀⠂⠀⠀⢤
⠀⠀⡈⠁⠑⠂⠀⠒⠐⠊⠈⣁⠠⠀⠂⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠂⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠈⠊⠁⠀⠀⠀⡠
⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠒⠒⠀⠀⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⣛⢿⠛⡻⠻⣟⠟⣿⢟⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣄⣀⢤⠀⢀⣴⣶⡼⢶⠦⣄⠀⠀⠔
⠀⠀⠀⠀⠛⠛⠉⠂⠁⠚⠿⠿⢧⡈⣀⣀⡀⢉⣈⣡⡄
⠀⠀⠀⢐⣷⣏⣙⣛⣭⣛⡝⣫⣠⣕⣱⢉⠿⣋⠸⣟⠀⠀⠀⡠⠦⠠⠀⣀⢀⠀⠀⠀⢀⠀⠀⡀⢀⣀⠀⣀⠄
⠀⠀⠀⠘⠷⢽⣭⣭⣽⣿⣛⣙⣽⣿⣯⣶⠴⠿⠿⠋⠀⠀⠀⠈⠐⢢⣶⢉⡁⡚⢁⠉⢈⠸⢂⡋⠀⢉⣰⡇⠈
⠀⠀⠀⠂⣄⣀⡀⠀⠈⠙⠛⠉⠁⠉⠁⠠⡂⢁⡀⠠⠀⠀⠀⠀⠚⠋⠀⠀⠁⠀⠉⠑⠰⠾⠶⠛⠏⠥⠀⣀⡀
⠀⠀⠀⠀⠈⠉⠁⠀⠼⠁⠐⠁⠦⢒⡐⢐⠠⠀⠀⠠⠀⠀⠀⠸⢿⣿⣿⣶⣤⣤⣴⣶⣟⡾⣻⣻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠢⢠⡀⠀⠀⣠⠖⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢠⡀⡄⣀⢠⡄⣀⣀⣀⡄⣠⠀⡄⣠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠐⣠⠂⠀⠀⠀⠀⠀⠀⠀⢠⠏⣧⢻⣧⡇⡇⢿⡇⢹⢿⢻⡆⣷⢲⡿⡿⠀⠀⠀⠀⠈⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢰⠆⠀⠀⠀⠰⠂⠀⠀⢠⡄⠀⠀⠐⠇⢀⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠛⠓⠀⠀⢈⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠷⠀⣀⣤⡤⠴
⠀⠀⠁⣠⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠨⠶⠎⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠶⠊⠑⠖⠖⠒⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣾⢿⡿⢿⡿⢿⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠈⠃⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡀⢸⠇⠘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⡿⡀⡧⣏⠸⢎⠀⠀⠸⠇⠀⠀⠀⠐⠒⠀⠀⢰⣶⢠⣶⣰⢶⠀⡇⠀⠀⠀⠀⠆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠽⠋⠿⠘⠟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⡖⢰⡆⢰⠂⣶⠐⠟⠃⠀⠘⠃⠀⠚⠀⠀⠀⠀⠀⣤⢀⡁⠀⠂⠀⠠⡀
⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣐⣘⠃⠀⠀⢃⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⢠⡟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠖⠒⠒⠒⠒⠒⠶⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⢋⡁⠀⠚⠀⠠⠀⠉⠀⠤⠀⠃⡀⠰⠌⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠠⠀⠈⡀⠀⠉⠀⠀⠈⠀⠀⢠⠀⠀⠁⠀⠀⠘⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣟⣉⣙⡒⠦⠤⣤⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠴⢖⣚⣉⣠⣈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠯⣍⣓⣲⠦⢤⣈⡉⠉⠉⠉⠉⠉⠉⠉⢉⣩⠤⠶⠮⠭⠅⢊⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠿⣷⣶⣶⣶⣶⣾⣿⣿⣷⣦⣐⠢⠖⣡⣶⣿⣿⣿⣶⣶⣾⣿⣿⡿⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣄⠀⠛⠉⠋⠛⠛⠻⠟⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠋⠉⠀⢀⡉⠀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣶⣶⣦⣄⣠⣤⣄⠀⠤⢀⣀⣀⡀⠀⢀⣴⣶⣶⣭⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠒⠛⠋⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⣠⠁⠈⠀⠘⠀⣨⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣆⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢰⠓⣆⠀⠀⠀⠀⠡⠘⠀⠑⠴⠃⢀⠝⢁⠜⠀⠀⠀⠀⠠⠂⡆⢀⠀⠀⠀⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠆⢤⠈⠝⠁⠀⠀⠀⠀⠀⢡⠀⠀⠀⠂⠡⠊⠀⠀⠀⠀⠀⠔⠘⢄⢣⡀⠰⡀⠀⠪⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠈⢁⠞⡠⠐⡲⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⡤⠐⠒⡚⠲⣄⠙⢄⠀⠈⠠⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠠⠊⠀⠐⠁⠀⠀⠘⡀⠀⡠⠄⠜⠀⠀⢀⡐⠂⠂⠙⠀⠈⠆⠀⠀⣀⠴⠋⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠩⢧⠑⠀⠂⠀⢀⠐⠀⠀⠉⠀⠀⠸⠷⠿⠁
⠀⠀⠀⠂⠀⠂⢁⡀⠀⠀⠀⠈⠹⣿⣒⡀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠁⠀⠀⠈⠒⠉⠀⠀⠀⠠⡔⠁⡀⠀⢀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⠌⠁⠘⠃⠀⠀⢄⡀⠀⣀⣀⣀⣈⣤⣄⣉⣁⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠈⠃⠀⢀⢀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡄⠘⠀⡀⠁⢠⣾⣿⠭⠯⠿⠥⠿⠧⠷⠧⠽⠯⠽⠯⠿⠭⢷⣭⣭⡟⠻⢖⡶⠂⠀⠀⠤⠔⠁⠐⠒⡄
⠀⠀⠀⠀⠀⢀⣠⡴⢚⣋⣽⡵⡄⡘⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢤⠀⠀⠀⠀⣤⣠⣶⣄⡀
⠀⠀⠀⢀⡴⢫⡴⠖⠛⠉⡀⠐⠀⠀⠀⠀⢸⠤⠀⠀⠀⠀⠀⡀⠼⠟⠏⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡄⠈⠙⢮⣽⣦
⠀⠀⠀⡼⣶⠋⠀⠀⠀⠀⠀⠀⠀⠤⠤⠰⠀⠄⠰⠀⠀⠀⠞⠀⢀⠀⠤⢠⣵⡰⠂⠢⠒⣉⠆⠀⠀⢀⢀⠀⠀⠀⠠⠂⠁⠤⢀⠈⣯⢘⡇
⠀⠀⢰⡗⡇⠀⠀⠀⠀⡴⢁⡀⢸⣄⣀⢴⠂⠀⠐⡄⠀⠈⠃⠠⡤⠤⠀⢋⡽⠃⠀⠀⠉⢀⢄⡁⠀⠘⠄⡠⠐⠃⠒⠂⠰⠤⢁⡴⢉⡝⡇
⠀⠀⢸⠻⣿⣦⣄⡀⠀⠁⠐⠒⠀⠀⠀⠀⠀⠚⠁⠀⠀⡀⡠⠌⣡⢩⠕⠊⡠⠦⠚⠴⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢶⣫⡴⢋⡔⠁
⠀⠀⠸⣄⡈⠉⠛⠻⠿⢶⣶⣤⣤⣤⣀⣀⣀⣀⠀⠀⠀⠧⠤⠊⡔⢉⠔⠊⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡶⡶⣾⣻⡯⠟⢛⣡⡶⠋
⠀⠀⠀⣸⠉⠉⣶⠶⠦⠤⠤⢉⣉⡛⠛⠓⠛⠻⠿⠿⠿⢷⣶⣶⣿⣿⣖⣲⣒⣖⣲⣻⡿⠿⠿⠿⠷⠒⢛⣛⣋⣩⡥⠴⢺⡏⠀⢹⡀
⠀⠀⡼⠁⠀⣸⠃⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠛⠓⠒⡶⠒⢶⡖⠒⠒⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⢳⠀⠀⣇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⢉⣉⣉⡉⢉⢉⠉⠉⢉⡉⠉⡉⢩⠅⣠⡤⠀⠄⣠⡀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⢠⣿⠋⠉⢻⡆⠅⠀⠀⣩⡙⠘⠛⢠⠐⠤⠀⠻⡭⢹⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠈⢿⣦⣤⣾⢏⡁⠀⠐⠛⠑⠀⢀⠖⣢⢤⠙⠓⠃⢸⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠈⠉⠒⠂⠀⠁⠀⠀⠈⠀⢏⠁⣥⡸⠂⠰⡇⣰⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠆⡀⢂⠀⠇⠀⢆⣐⣀⡒⢀⠈⠉⠘⢠⡔⢀⡓⢨⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠐⠀⠀⡀⠀⠀⢀⠉⠁⠉⠀⠈⠇⣸⠉⠰⢆⠀⣀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠰⠔⣬⣴⠆⣀⠠⣂⡀⠀⠨⠆⠘⢠⡔⢃⡀⠀⠄⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⢠⠄⣤⢠⢠⠀⠀⣴⡤⣤⢰⠀⠠⡦⢰⠠⡆⢰⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠰⢸⠇⣿⢸⢸⠀⠀⣿⠀⣿⢸⠀⠈⣇⣸⠀⡇⢨⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠈⣁⠉⠉⠉⡉⢀⠈⢀⠉⠈⠀⠀⢁⣬⡄⢁⡈⡀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢀⠀⡠⠈⣀⠂⠅⠀⠀⠄⠀⠠⠔⠀⣿⣟⣧⠐⢂⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢠⠐⠂⢀⡀⠁⢀⣀⣀⣀⡇⢀⣀⡀⣿⣷⢻⠆⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠐⠂⢰⠦⡤⢠⠎⣉⣉⠁⠁⠀⠀⠀⣈⡙⠛⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠘⠃⠹⠁⠀⠀⠀⠛⠉⠇⠀⠀⠉⠀⠉⠉⠀⠀⠈⠁⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠐⠀⢁⣀⣠⠃⡀⠊⠀⣀⠀⣤⣤⡀⣤⡄⢠⣄⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠁⢀⣈⣙⣛⣁⣉⣁⣈⣁⣀⣉⣉⣁⣉⣁⣈⣉⣀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡄⠀⢠⡀⢹⠀⣀⠀⠀⠰⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡀⢻⡦⠀⢸⡇⢸⠀⡏⢠⢶⠀⣿⠁⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠳⣘⡁⠈⡟⠃⣸⡀⠃⣼⠟⢠⡟⢠⠀⣃⡀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠈⠷⢄⢹⡆⠀⣦⠀⣾⡇⠈⠙⠀⡜⠀⡴⢰⡏⠀⡈⠀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡄⠈⠀⢧⠀⢹⡖⠙⣧⠀⠀⣰⠁⠀⡴⠛⡀⣴⡇⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣧⠀⡄⠈⡆⢘⡇⠀⢡⠀⡤⢯⡄⣰⣿⢠⠋⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠁⠀⠀⠀⢻⡀⠥⠘⠛⠀⢸⠀⠀⡞⠀⣿⡿⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠁⠚⠧⠀⣀⣸⣄⢰⡯⠘⠋⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡃⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣻⠟⣯⡍⢛⠟⠉⣭⣟⣹⣿⡟⣶⠶⢦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣯⣔⣸⣿⣿⣿⣶⢶⠛⠛⠛⣦⣄⡈⠉⠛⠚⠛⣛⣥⣯⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠤⠀⣆⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠾⠿⠛⣛⣯⣿⣿⣿⣿⣿⣶⠿⠿⠿⠶⠶⢿⣿⣿⣛⣋⡛⠃
⠀⠀⠀⠠⠟⠉⠉⠛⠛⠳⠶⣤⣤⣄⣀⡈⠉⠑⠂⠠⠀⣀⠀⠀⠀⠀⢀⣁⣀⢤⣤⣤⠤⣤⠤⠶⠯⠭⢿⣾⣉⡉⠉⠉⠉⠉⠛⣻⠁
⠀⠀⠘⣿⠛⠓⠒⠦⢤⣌⣀⠚⢋⡩⠟⠛⠻⠷⣦⣤⣄⣀⡀⠀⠒⠺⠿⢿⣧⡀⠀⠀⠀⠀⠀⣠⣶⣞⣋⣉⣉⣩⡟⣛⣿⣶⣤⡅
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠦⡈⠢⠐⠊⠉⠙⠛⠿⢶⣶⣤⣠⡈⠙⠂⠀⢀⣴⣿⣿⣿⣯⣝⣻⣯⣭⣽⣽⣿⣥⣬⣥⡄
⠀⠀⠀⠈⣿⣧⡑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢄⣈⠃⠠⢭⠙⠻⢶⣦⡄⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⠇
⠀⠀⠀⠀⠈⠹⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠂⠰⣁⠊⠍⠁⠺⠷⠄⠤⠀⠀⠀⠀⠤⠀⠀⠐⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠧⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠙⣦⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⣤⣄⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠓⠶⠤⠤⠤⣤⣀⡀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠴⠶⠶⠶⠦⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⣀⣴⠦⢄⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⢹⣛⣦⣾⠀⠀⠀⠀⠀⠀⠀⡟⣣⣤⡟⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁⠀⢰⡖⠶⢶⡆⠀⠙⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢛⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⢯⣟⣛⣛⣓⣚⣻⡿⢿⣟⠿⣿⣛⣛⣛⣛⣛⣻⠿⣿⣿⣛⠛⠛⠛⠋⠉⠉⣉⡭⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠉⠓⠲⠶⠶⠷⠶⠖⠚⠉⠈⠙⠓⠶⠶⠿⠶⠶⠒⠋⠉⠓⠮⢭⣝⣛⣛⣫⡭⠵⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣌⣳⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⡤⠤⠤⠤⠤⢤⣤⡤⠤⠤⠤⢤⣤⠤⠤⠤⠞⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢶⠬⠽⢶⣶⣶⣿⣿⣷⣶⣶⣶⣯⣭⣭⣭⣽⣿⣶⣾⣭⣭⣭⣽⣿⣿⡭⠭⠽⠿⠿⠿⡿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠐⠀⠀⠑⠁⠐⠀⠀⠀⣀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⡀⠀⠂⠀⠀⠙⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡄⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠉⠁⢀⠀⠀⡑⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠠⠂⠀⠀⠀⠀⠈⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⣄⣀⠀⠀⠄⠀⠠⠀⠀⠀⠠⣤⣤⠤⡤⠤⠄⠀⠠⠤⢀⠀⠀⠤⠤⠀⠀⠀⢨⡅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠑⠀⠐⠒⠒⠒⠦⠴⠶⠀⠶⠤⠴⠂⠔⠀⠼⠂⠐⠦⠀⠌⠱⠒⠒⠒⠒⠂⠒⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⢤⠀⣀⢂⠀⡀⠀⡦⠀⠀⠀⠤⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠐⢂⣀⡀⠠⠄⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠢⡑⣕⢄⠪⡑⡍⠾⡳⠀⠁⠉⠒⠑⠙⠈⠈⠁⠁⠀⠊⠓⠑⠈⠈⠀⠀⠀⠀⣀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠛⠀⠈⠉⠡⣜⠉⠓⠛⠂⠐⠒⠀⠒⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠂⠁⠀⠐⠛⣋⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⠀⠀⢀⠠⠄⠄⠀⠈⢐⠂⠠⢀⡀⠀⠀⠀⣀⠀⠀⣂⠀⠒⠚⣛⠉⠀⠈⠇⠉⡡⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠚⠋⠁⠄⠀⠀⣴⠩⠄⢠⡊⠥⠸⠀⢰⣉⠆⠀⠃⠠⢄⠀⠧⣝⣄⠀⠀⠀⠡⠛⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠙⠀⠐⠺⠀⠀⣈⠉⠁⠀⠀⠀⠉⠀⠠⠁⠀⠀⠘⠁⠀⠄⠀⢀⠀⠛⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠐⠂⠀⠀⠈⠀⠀⠀⠀⠋⠀⠠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢤⠀⠀⠀⠃⠀⠄⠀⠀⢠⠀⠀⠀⠒⠀⠀⠁⠀⠀⠰⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠒⢀⣀⠉⠉⠉⠉⠉⠉⠉⢁⡀⠒⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠑
⠀⠀⠀⠀⠀⠀⠔⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢂
⠀⠀⠀⠀⢀⠊⡠⠁⠀⠀⠀⡠⠀⠀⠀⠂⠀⢀⡀⠤⠚⠛⠉⠁⠀⡀⠀⠀⠀⣀⠐⠂⠄⣀⠀⠀⠀⠀⠀⠀⢄⠀⠁⡀⠡⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⢺⡆⠀⡀⠀⠀⠀⠀⠀⠁⠢⣄⡀⠁⠂⢄⡀⠁⠢⢤⠄⡀⠀⠀⠐⠀⠐⡀⠡
⠀⠀⠀⠀⢠⠁⠀⠀⠠⠀⠀⠀⠀⠀⠔⠁⠀⠀⠹⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠈⠁⠀⠀⠀⠀⣀⠀⠀⠠⠀⢡
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⠀⠠⠀⠀⢈⠳⢄⡀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠐⠀⠠⠴⠞⢛⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠔⠀⢀⠔⠁⠀⠀⠐⠡⠀⡐⠀⠠⠤⠠⠤⠤⠉⠙⠛⠋⣠⣶⣊⣽⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠇⠀⠀⠀⠀⠀⠀⠄⠀⠦⣠⡀⠊⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⡀⢈⡛⠀⠀⠀⠂⠀⠀⠐⠋⠀⠤⢠⡤⢠⠃⠀⠀⡀⠀⠀⠀⠀⢁
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘
⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠘⠄
⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠁⠄⠶
⡇⠀⠀⠀⠐⠒⠂⠈⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠚
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠸⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⢀⠖⠓⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣀⠀⠈
⠈⡄⠀⠈⠁⡀⠁⠀⠉⠈⠁⠒⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠠⠀⠀⠉⠀⠈⠉⠉⠀⢀⠈⢂⢀
⠘⠵⠀⠁⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠾
⠀⢖⣫⠀⠀⠰⠀⠀⠀⣀⣀⡀⠀⠤⢐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠠⠄⢀⣀⣀⣀⡀⠀⡈⠀⠀⠜⣘
⠀⠈⠒⠒⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡰⠒
⠀⠀⠀⠀⠐⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠾⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠰⢏⣉⣉⣩⣉⠉⣑⠒⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢃⠘⠃⠀⠀⠀⠀⠈⠉⠁⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⢒⣀⣤⣤⣀⡐⠢

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣴⠟⠃
⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⡶⠋
⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⢈⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠞⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡔⠂⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠐
⢴⠖⠖⠤⠠⠠⠀⠦⠤⠄⠠⠠⠤⠄⠤⠤⠄⠶⣤⣬⠆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⠀⣄
⣼⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⠄⠉⠋⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢹⠁⠀⠀⠀⠀⠀⠀⢀⡈
⣿⠀⠀⠀⠀⠀⡏⠉⠁⠲⠀⠀⠀⠀⠀⠀⡀⠀⠸⢋⢲⣤⣤⣀⠀⠀⠀⢀⣰⣄⣀⠀⠀⠀⠀⣀⠀⣷⠾⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⠀⠀⢀⡀
⢸⠀⠀⠀⠀⡀⢠⡀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⢸⣰⣿⣿⣿⣿⠃⠀⢰⣿⣿⣿⣏⠁⠈⠁⠉⠁⠀⡠
⢸⠀⠀⢄⡄⢱⣾⠓⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⡟⠀⠛⣿⣿⡿⡇⠀⠀⠀⠀⠰⠀⠁⠚⣄
⢸⠀⠘⠀⠈⠉⠉⠙⢀⠁⢀⡤⠀⠹⠤⡆⠀⠀⢸⠁⠛⠻⣿⣿⣿⣠⣿⡿⠋⠀⠁⠈⡄⠀⠀⠀⠀⠤⠐⠛⠀⠒
⢸⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠀⠀⠠⠐⠀⠀⠀⢸⡟⠀⠀⠈⠛⢿⣿⠏⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠀⠀⠠⣁⡀
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢠⠀⢈⣀⣀⡁⠀⠀⣠⣴⣇⠀⠀⠀⠀⠀⡐⢃⣀⣀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢶⡂
⢸⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣻⠇⡱⣆⢸⣿⡯⡇⠀⠀⣿⣾⣿⠆⠀⢰⣤⣀⠃⠉⠀⢁⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠂
⢸⢓⣶⣤⡀⠀⠀⠀⢀⠈⠉⠃⠀⠀⠀⠀⠀⠀⢻⡿⢀⣨⡞⣿⣿⣷⣶⣶⣿⣿⢿⠃⠀⣹⣿⣿⠀⠀⠰⡈⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒
⢘⡁⣶⣄⣀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠸⠯⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡼⠀⠙⠂⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠸⡇⠙⠙⣿⠟⠛⠛⠻⠃⠀⠂⠀⠤⡖⠒⠋⠀⠙⠁⠀⠀⠀⠉⠛⠛⡟⢀⣽⣿⣿⣿⡟⠟⢿⣿⢰⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠉⠙⣧⣬⡶⠉⣆⣠⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⢰⠃⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⣀⠠⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠛⠀⠐⠁⠈⠁⠀⠈⠁⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠉
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠇⢀⢿⣀⣀⣀⣉⣀⢀⣀⣀⣉⡀⢀⠆⠀⠀⠀⠀⠠⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠆
⢸⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠃⠖⠤⢤⣤⡤⣤⠄⠶⠊⠁⠀⠀⠀⢀⠀⠆⢀
⠐⡀⠀⠀⠀⠀⠀⠀⢴⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⣀⢀⡀⠀⠀⠀⠀⠀⢠⣤⢄⠀⠀⠀⠀⠈⠂⠄⡈⠳⣀
⠘⡇⠀⠀⠀⠀⠀⠀⠈⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⡈⠡⠀⠀⡁⠘⠇⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠠⠀⠡⠄⡈⠚⡀⠉
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠀⠃⡴⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢲⡄⠀⠠⠀⠀⠀⠀⠀⠀⠁⢀⡀⠐
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⡿⠀⡜⠀⠀⠀⠀⠀⠀⠀⠄⠐⠓⠶⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠙⡂⠀⠈⠱⢄⠀⠁⠀⠀⠈⠙⢤
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠕⡦⢀⠀⠁⠀⠀⠀⠓⠀⠠⠆⠀⠨⢀⠀⢌⠳⢄⢀⣠⠄⡀
⠘⠟⠒⠒⠐⠂⠐⠀⠀⠒⠒⠒⠲⠢⡖⠆⠦⢶⡿⠀⠀⢀⠀⠀⠈⠀⣀⡀⠀⠘⠁⠸⣿⠆⠀⠀⠀⠀⠀⠁⠀⠆⠀⠀⠀⠈⠱⠀⣳⡼⠛⠁⠀⠁⢀
⠀⠤⠛⠀⠀⢀⡠⣦⠀⠀⣀⣀⠀⠀⠀⣘⣶⠋⠀⠀⠀⠈⠀⠂⠀⠀⠈⠁⠄⢀⠀⠀⠁⠀⣤⡀⠀⠀⢢⣾⣷⠀⠈⠀⠀⠣⢀⡞⠁⠐⡆⠀⠀⣀⠞
⠀⠀⠀⢠⡄⠀⠁⠀⣀⠀⠉⠁⢰⡃⡂⡠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠄⢀⠀⠈⠀⠀⠀⢸⣿⡟⠲⠤⣀⢿⠿⠀⣠⠂⡤⠀⠘⣧⡈⢢⣶⣶⡖⠁
⠀⠀⠐⠦⠁⠀⠀⣄⠙⠃⠰⠙⠘⠻⣿⠁⠀⠀⠀⠂⡀⠀⠀⠀⠁⠔⢠⠀⠀⠀⢀⠀⠀⡀⠙⠁⠓⠰⠧⠀⠖⢂⠁⠀⠐⠛⢀⣼⣿⣿⣿⣿⠃
⠀⢷⣦⣄⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡇⠀⣠⡦⠀⠶⡿⠀⠐⠁⠄⢀⠀⠀⠈⠀⠆⠁⠀⠀⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣧⠖
⠀⢸⡏⣿⣤⣾⣳⠆⢀⠆⢙⠃⢠⡼⣇⠀⠉⠀⢶⡶⠀⠓⠞⢁⡀⠀⠈⠐⢆⠓⠀⠀⠀⠀⠀⠀⠀⠉⠀⢶⠀⠄⠀⠀⠀⠀⠀⠀⣬⣿⣿⡟⠁
⠀⠛⠛⠋⠈⠻⠶⠂⠘⠓⠚⠓⠻⠶⠛⠃⣀⢶⡄⠐⡾⢶⠚⠛⠛⠳⢶⣶⣾⡦⠁⠉⠛⠛⠛⠳⣶⣿⠤⠤⡄⢀⣤⡀⠀⠐⠠⠄⠙⣿⠏
⢰⣿⠀⠀⢠⣤⣶⣶⣆⠀⠀⠀⠀⠲⠟⠃⠉⠃⢠⡌⠇⡄⣠⢈⡁⠉⠀⠈⠛⠻⣿⣧⡈⠂⠈⢀⣸⠇⠀⠀⠖⠁⠀⠙⢠⡇
⠸⣿⣀⡘⢠⣿⠉⠉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⢸⡿⠃⠐⠀⢀⣠⠿⠿⠃⠂⠀⠀⠁⠀⠀⠀⢀⣀⣀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⢀⣀
⠀⣴⣦⠀⠀⣀⣀⣄⢀⣀⣠⣄⠀⣂⠀⣀⠀⠀⣀⠀⢀⣶⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⠁
⠀⣾⣭⣧⠀⢹⡏⠉⠈⣿⡇⠉⠈⣟⠀⠙⣧⣸⢏⡴⠋⢹⣀⢸⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠈⠉⠛⢿⣿⣿⣿⣿⡀
⠀⠀⣴⣿⡟⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣧
⠀⣼⣿⠿⣷⣾⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡆
⢸⣿⡟⢠⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠈⢿⣿⣷
⣿⣿⣣⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠈⢿⣿⡄
⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⡀⠈⢿⡿⢶⣤
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢹⣷⠀⠀⠒⠀⢻⣧
⢻⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⢶⣾⠋
⠘⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠹⣦⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⣄⡀⠀⠀⠀⢿⡇⢀⣠⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢸⣷⡾⠛⠳⢶⣯⠀⠀⣿⠋⢻⣧
⠀⠀⠀⠀⠈⠻⣿⡿⠿⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⠀⠀⠀⢸⣿⡇⢀⣀⣰⣯⠀⢀⡟⠀⢰⣿
⠀⠀⠀⠀⠀⣼⡏⢠⡤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⣀⣀⣀⣀⣠⣤⡴⠚⢻⣏⠀⠀⠀⠀⠀⠀⠀⣼⡿⣷⠀⠉⠙⠻⠟⠛⠛⠓⣿⠇
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⣏⡉⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⣰⡿⠀⠙⢷⣤⠀⠀⠀⢀⣤⡾⠏
⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⣤⠶⠖⠋⠁⠀⠀⠀⠀⣠⡿⠁⠀⠀⢸⡏⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣠⣾⠿⠷⣶⣤⣤⣾⠇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠁⠀⠀⠀⢸⡿⠉⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⠟⠛⢿⣿⡿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠟⣻⣿⠋⠀⠀⠐⣤⡀⣾⡇⠀⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⣇⠀⢀⡄⠙⢷⡄⠉⠉⠛⠿⢶⣶⠶⠶⠶⢾⣿⣿⣿⣭⣵⠶⠞⠋⠁⠀⠀⠀⠀⠀⠙⣿⣷⣶⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠙⢿⣿⠁⠀⢸⡇⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢾⡏⠈⠛⢦⣴⠟⠁⠀⠀⠈⠛⠷⣦⣠⣿⣇⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣾⠁⢀⣀⣄⡀⠀⠀⠀⠈⠙⠁⠉⠻⢷⣤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠟⠋⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⣰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣤⣀⣠⣼⠟⢨⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠶⠟⠛⠉⠉⣿⠛⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢀⣀⣤⣴⡾⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠋⠉⠉⣹⡏⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠟⠋⠁⠀⠀⠀⢠⣿⠁⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣤⣤⣤⣤⣤⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣀⣀⣀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠉⠈⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠉⠉⠻⣟⣡⠟⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⢰⣟⠒⠖⢠⣤⡀⠉⠙⠲⡿⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠈⠛⠻⠶⣶⣤⣄⣉⠁⠚⠷⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣼⠟⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠶⣦⣤⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⣰⡿⠁⠀⢀⡞⢹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠙⡆⠀⣠⠏⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣏⣉⣿⠁⣰⠃⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠈⠡⡾⢁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⢀⡞⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣯⣴⡿⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⠋⠉⠋⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⢿⢿⠍⠃⠚⠫⠛⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⢿⡟⠁⠀⠀⠀⠀⠘⠀⠀⠀⠉⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣃⡄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢐⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣆⢊⣠⡄⠀⠠⣄⢰⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣼⣷⣶⣴⣶⣿⠿⡷⢿⣏⡙⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣿⣿⣿⠟⠛⠿⠉⠉⠠⢀⠹⠟⠛⢻⣿⣿⣧⣶⣧⣬⣿⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠟⠷⠶⠴⠦⢤⠤⠄⠀⠤⠤⠾⠿⠿⠟⣱⣿⣿⣭⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢛⣋⣤⣴⣶⣌⠛⠛⠋⠙⠉⠉⢉⣥⣶⣤⣄⣼⣿⣿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⠯⢉⡛⣿⣿⣿⣧⡀⠀⢀⡴⣿⢿⣟⣩⣿⣿⣿⢿⠿⣿⡳⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠗⣤⠞⡯⡣⠎⣠⣝⢶⠏⠇⢀⡐⠟⠉⠑⢉⠅⣠⣴⢇⡦⢻⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⣰⣩⣤⣶⣌⣛⠀⣤⡭⢀⣉⣀⡀⢠⠾⠿⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠯⣼⣄⣉⢁⣀⠀⠀⠁⢉⡉⠪⠴⠷⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣤⠖⢀⡀⠓⠈⢲⠞⠛⠻⠿⠓⠂⠲⢠⣀⣢⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⠄⠘⠀⢀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠓⠀⠀⢾⡍⠻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠟⠉⠀⠠⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⢻⣿⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⠁⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⡀⠁⠸⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣾⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡷⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢠⡓⠀⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢷⣥⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣿⣿⡿⠿⠟⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣭⣟⠛⠷⠶⢶⣶⣾⣶⣶⣶⣿⣷⣦⣤⣴⣶⡶⣶⡶⠶⠿⠶⣾⢿⡻⠿⠟⠛⣻⡿⢏⣽⢶⣛⡵⠖⠋⠃
⠀⠀⠀⠀⠀⠀⢀⡤⠶⣽⣿⡷⠶⢶⣶⣶⣶⣴⣾⣷⣄⠈⠈⣉⣭⣟⣻⡿⣿⣷⣿⣿⣡⢬⣝⠦⣤⣶⣿⡤⠊⠀⠈⠂⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠖⠈⠉⠉⠵⠾⣿⡿⣯⣬⣿⣿⣿⣿⣿⠿⠭⠿⠭⠋⢁⡀⠩⠟⠁⠀⠈⠐⠉⠁⠉⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣙⣷⣦⣄⣠⠴⠖⠭⠄⠨⠭⠘⠫⠋⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣴⣶⣲⣶⣶⡶⣿⣿⠙⡟⢁⣿⠛⠇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣈⠻⢿⣋⠹⣷⣦⣀⠀⠀⠀⢀⣠⣶⡟⢋⣩⣿⠟⣩⣿⢋⣽⠟⢉⣽⣿⡿⣶⡾⠛⢿⣾⢣⣿⢋⡼⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠤⠟⠻⠽⢷⡻⢿⡷⢶⠿⣿⣧⣀⣴⣿⡿⠟⢻⣿⣟⣹⣿⠟⣥⣾⣿⣴⣾⠟⣸⡏⢻⣷⣶⣿⣯⢟⠯⡌⠙⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠐⠐⢦⣔⣁⠀⠀⠉⠀⠉⠉⠻⠛⠛⣋⣀⡰⠋⠭⢍⠙⡟⠩⠡⠌⠀⠑⠛⢙⣇⢉⡉⣹⠇⢸⣾⣤⣬⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠱⠦⠾⠗⠺⠿⢻⠟⡿⡷⠖⠷⢶⡦⠶⠿⠾⠿⠿⠿⠿⠛⠉⠙⠙⠛⣿⠏⠀⢀⣨⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⠀⠀⡄⢠⣦⣀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠤⠤⣤⣄⣀⣤⣀⣠⣀⣀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣮⡾⠷⠤⠶⠾⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠤⡤⡦⡴⣶⣠⢤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⣆⠙⠳⢇⣋⠀⢵⠲⠪⣸⠗⢳⠛⠾⠷⠒⣻⠉⢛⠛⠀⢮⡭⡳⠿⣶⣆⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⢋⡉⠙⠉⠁⠈⠁⠀⠀⠈⠈⠈⠀⡀⠀⠂⠠⡄⠂⠀⠐⠢⠨⠀⠐⠒⠀⡉⠛⠸⢿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⡁⠀⡀⠘⣯⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠁⠠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠈⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡀⠐⠀⠀⣀⡤⢀⡾⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂⠁⠀⠀⠀⠀⠀⠊⡽
⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣷⣶⣤⣀⣀⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣁
⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⡟⢿⡟⠛⠛⠿⢿⣿⡿⠿⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⢾⣿⣿⣿⣿⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠯⣠⣴⡷⣦⣄⡀⣀⣈⣷⣤⣀⡀⠀⠀⠉⠉⠉⠉⠁⠈⢉⣿⣿⣿⣿⣿⣭⣤⣀⣀⣀⡼⠟⠿⠿⠛⠿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠀⠐⠁⢉⠑⠻⠿⠿⠿⢿⣟⣿⣿⣶⣶⣶⣶⣾⣿⠿⠿⠿⠛⠛⢙⡋⠉⠉⠙⠛⣟⠳⣆⢠⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⣦⠁⠀⠀⠀⠀⠀⠖⠀⠘⠛⢿⣿⠿⣿⣷⢦⣄⣀⡀⠀⠀⠀⣴⡟⠻⠳⡞⠁⣰⡄⣠⣾⣿⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠖⣜⠛⢾⣤⣰⣤⣠⠀⠀⠀⠐⢠⢤⡄⠀⠀⠉⡁⠈⠉⠉⢳⣄⡴⠾⠏⣠⡶⣀⣑⣾⣥⣿⣿⡿⠟⢉⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⠀⠙⠛⠟⠿⢿⣶⣶⣤⣷⣾⣥⣼⣦⣴⣿⣷⣠⣾⣖⣋⣥⣤⣶⣷⣿⡿⣿⠿⠛⠉⠉⣠⣠⡾⠿⣷
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠓⢤⣀⡀⠀⠀⠉⠀⠉⠉⠉⠉⠋⠉⠙⠟⠋⠈⠙⠛⠛⠙⠉⠉⡈⠉⢀⡀⡀⢰⣯⣴⠿⠋⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠈⠙⠓⠲⠦⢤⣄⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣀⣈⣡⣤⠶⠞⠋⠉⠀⠀⣠⣴⣿⡶⣄
⠀⠀⠀⠀⠀⠀⢠⡈⠉⠻⡆⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⢀⣀⣤⣶⣿⡿⠋⠀⣼⡟⠓
⠀⠀⠀⠀⠀⠀⢸⡷⠆⢤⣶⡄⠲⠿⣿⣯⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣶⣶⣿⣯⣥⣾⠛⠉⠀⣠⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⠛⢦⣄⡀⠈⠉⠛⠛⣶⡶⠶⠦⠄⣄⡴⠿⠿⠿⠿⠿⣿⠿⠿⠟⠛⠋⠉⠉⠻⠿⠁⢀⣤⠾⠋⠉⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣀⠀⠀⠈⠟⠳⠤⣤⡄⣈⣀⣤⠾⠷⢾⣭⣤⣀⣀⡀⠀⠉⢀⣀⣠⣤⣶⣶⣶⠦⠤⠾⡏⠀⢀⣠⣼⡟
⠀⠀⠀⠀⠀⠀⢀⢴⡻⣟⡈⠛⠶⣦⣀⠀⠀⠀⠀⠉⢹⠏⠀⠀⠀⠀⣾⠋⠉⠉⠉⠉⠉⠁⠙⠙⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢰⣞⢳⣌⠛⢦⡞⡈⠟⠻⡿⣶⣶⣶⣶⣤⣄⣀⣀⣀⣬⣄⣀⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⡿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣢⣄⡙⠲⠿⣧⣶⣾⣿⣿⣿⣥⣭⣿⣿⠿⠛⣛⠛⠛⠛⠛⢛⣏⣿⣯⣾⣷⣿⡷⠿⠛⢁⡼⢋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡓⣶⣤⣄⣀⠉⠉⠉⠛⠋⠙⠛⠻⠾⠶⣶⣷⣾⠿⠿⠿⠟⠛⠛⠉⢁⣀⢤⡞⣉⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠲⠿⢭⣯⣷⠒⠦⠤⣤⢤⣤⣄⣀⣀⣀⣀⣠⣤⡤⠶⡶⠚⢋⣥⡶⠶⠋⠉⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⢷⣼⣶⣴⠿⠛⠛⠻⠾⠶⠾⠿⠟⠛⠛⠉⠁⠀⠀⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠲⠶⠶⠶⠶⠶⠖⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⡦⣤⠄⠊⠁⠀⠀⠀⠀⠀⠀⡖⠊⠉⠁⠀⠈⠉⠉⠘⠣⣀⠀⠀⢀⡆⢑⣦
⠀⠀⠀⠀⠀⠀⢀⢳⣬⡷⠧⣀⡗⠒⠒⠒⠒⠒⠒⡖⠲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠷⠺⠧⠺⣤⣟
⠀⠀⠀⠀⠀⠀⠈⠛⢇⠀⠀⠉⠄⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠀⠀⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠴⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁⠘⢄⢀⠀⠰⠆⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠆⠀⠀⠀⠀⠀⠀⢰⠀⠀⡀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⡄⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⡄⠌⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢖⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢸⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠓⡀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⡀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⣅⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠈⠑⠤⣀⣀⠤⠊⠀⠀⠀⠀⠀⠃⠀⠀⢰
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢨⠀⠀⠉⠉⠘⡆⠐⠒⠒⢅⠀⠀⠀⠀⠈⡢⠤⠤⠆⠒⠂⢙⠂⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠊⠁⠀⠀⠀⠀⠀⢸⡄⣀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠀⠀⣀⣀⠐⠂⣤⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠠⠆
⠀⢀⠆⠈⠒⢤⣀⠀⠀⣠⢄⠀⠀⠀⠘⡦⣀⠀⠀⠀⢠⢃⠀⠀⠀⠀⠀⠀⠐
⠀⡌⢀⠋⣦⣁⡈⠍⠃⠉⠒⠈⠲⠦⢠⠀⠀⠉⢰⠂⠂⠈⠒⠒⠒⠒⠲⠉
⠰⢀⡴⠋⠀⠈⠉⠓⠤⠀⡀⠀⠀⠀⠀⠀⠀⢀⠊⠘⠂⠐⠃⠀⠀⢀⠃
⠁⠉⠀⠀⠀⠀⠀⢐⡀⠔⠉⠈⠐⠠⠄⣀⣀⠬⢀⡀⠀⠀⠀⠀⠀⠎
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣒⣤⣤⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢧⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣘⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠀⠑⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡆⠀⠀⠀⠀⠀⠙⠳⠦⢄⡀
⠀⠀⠀⠀⠐⣀⠀⢀⣀⠀⠀⠀⠈⠃⡀⠁⠀⠀⠀⠀⠁⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠘⡇⢸⠉⠸⠂⠀⠀⠀⠫⠄⠀⠠⠤⡄⣄⠀⠀⡉⢳⡀⠀⠀⠀⠀⠀⠀⠰⡞⢙⠄⣀⣤⠤⡄
⠀⠀⠀⠀⠀⢇⠘⡄⠀⠘⠀⡠⠀⠀⠀⠀⠐⠀⠃⠉⠀⠈⠉⠉⠑⣄⡀⠀⠀⠀⠀⡜⠙⠛⠂⢀⠀⠂⠘⠲⣌
⠀⠀⠀⠀⠀⢸⡀⠇⠀⡀⠒⠇⠀⠀⠀⠀⠀⠀⣀⣴⡆⠀⠀⠀⠀⡿⠃⠀⠀⠀⡔⠁⠒⠀⠀⠀⣇⠉⠲⡟⣤⡑⠂
⠀⠀⠀⠀⠀⠀⣇⢸⠀⠀⠀⢀⠀⠀⠀⢀⣤⡞⡩⠋⠀⠀⠀⠀⣰⠇⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⠁⠖⠻⠍⠙⠃
⠀⠀⠀⠀⠀⠀⠹⢬⣦⣄⡀⢴⠀⣠⠞⢉⢌⠊⠀⠀⠀⠀⠀⡠⠃⠀⢀⠀⠈⡐⠀⠀⠀⠀⠀⠀⠲⠀⠒⠋⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠐⡑⠁⠀⠀⠀⠀⢀⠜⠀⠀⠀⢸⠀⢰⠂⠀⠀⠀⠀⠀⠀⡕⢾⣍⠷⠆⠀⠀⢀⠀⣷⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠊⠀⠀⠀⠀⠀⠀⡎⠀⢂⠀⠀⠈⠀⠀⠑⠄⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠞⠉⠀⠈⠁⠠⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠁⠚⠒⠀⠀⠀⠀⠀⠀⠂⠡⡀⠄⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣤⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠀⠠⠀⠀⠂⠁⠀⠀⠘⡀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⣴⠉⢇⢀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣰⢷⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠉⠣⠀⢀⣠⣔⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠢⡍⠃⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣬⣭⣥⣬⣩⣯⣡⣿⣩⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠈
⠀⠀⠀⠀⠀⠀⠀⢳⠈⢲⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠠
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣤⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠈⢿⡻⣿⣦⣙⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠋⢐⣀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⢈⣷⣠⣏⣿⣷⣤⣌⣛⠛⠒⢦⣤⡈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⢲⠀⠠⣦⠀⣦⢈⠉⢈⣠⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠙⠻⣾⣿⣟⣿⠉⠩⠛⢶⣾⣧⣤⣽⢿⣏⠻⠂⡴⢦⡜⡐⠆⢀⣰⠞⣰⢺⣖⠶⠃⣁⣄⠈⢁⣼⢿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠂⠀⠉⠙⢷⣧⣤⣈⡈⠝⠛⠛⠿⣿⡶⣾⢷⣶⣆⣤⢶⣀⣠⣴⣿⣈⣀⣶⠾⢟⠥⠞⢋⡰⠛⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠣⢀⠀⠀⠀⠀⠀⠨⣍⢹⣿⡿⠶⢾⣤⣿⣤⣿⣾⣿⣿⣧⣿⣾⠶⠟⠛⠛⠉⠁⠀⠀⠀⠈⠁⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀⣠⠖⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠙⠛⠛⠛⠐⠃⠒⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠀⠒⠢⠤⠤⠤⠄⣄⣀⣤⣀⡠⠤⠤⠤⠔⠒⠚⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠒⢛⠛⠛⣛⠛⠛⠓⠒⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢚⡉⠀⠀⠓⠀⠀⠈⠀⠀⠈⠀⠐⠂⠀⠀⠲⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠈⠁⠠⠆⠀⠀⣤⠀⠀⠘⠃⠀⠀⠀⠀⠁⠀⠀⠀⠙⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠈⠃⠀⠀⢤⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⣤⠀⣴⢶⠀⣠⣄⠀⠀⣀⣀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⡤⠶⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣾⠀⢸⠀⡇⠸⣤⠇⠈⠉⠋⠉⢈⡇⢾⠉⠉⠙⠛⠹⣆⢀⡟⠛⣋⣿⠰⣞⣁⡀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣁⣤⠼⠟⠛⠛⠛⠒⠶⢤⣀⠀⢸⡇⢸⡆⠀⠀⠀⠀⢈⣩⠖⠋⠉⢿⣤⠏⠉⠉⣿⠲⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⡛⣦⡉⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠉⢉⣉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠀⢀⡼⠉⠉⠉⠉⢩⣭⡉⠉⠉⢉⣉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠖⢷⣼⠟⢦⡴⠞⠶⠾⡇⠀⣰⠞⠶⣷⣄⡞⢳⣀⡼⠳⠶⠋⠈⢳⣄⣠⠞⢹⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⣀⣀⣀⣀⣀⣀⣀⣙⣚⣋⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⡀⣀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠌⠀⠀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⢠⠄⠀⠠⠴⠀⠄⠈⠉⠀⠀⠈⠉⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣄⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠦⠀⠀⠀⠐⠤⡀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠂⠀⠁⠀⠀⠢⠂⠀⠀⠀⠠⡠⠀⠀⠈⠁⠀⠀⠀⠀⠚⠂⠀⠀⠀⠄⠀⠀⠀⣰⠀⢀⣠⠀⠀⠈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠴⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠤⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠡⡤⢤⠀⢐⠒⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠔⠀⠀⠒⠂⣀⠀⠐⠂⠀⠀⠀⠀⢐⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠂⠉⠉⠑⢒⠓⠚⠧⠼⠀⠩⠤⠶⠀⠂⠀⠃⠈⠃⠘⠀⠼⠁⠓⠚⠛⠛⠋⠉⠁⠈⠃⠤
⠐⠀⠀⠀⠀⠀⠀⠀⠀⠷⡠⡤⢀⣀⣂⠀⠀⠀⣰⠄⠀⠀⠠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠐⢂⣀⡀⠠⠤⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠊⡐⡁⢮⡢⡙⢮⡳⡐⠊⠉⠑⠂⠒⠒⠒⠀⠁⠐⠂⠀⠒⠂⠒⠈⠈⡈⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠭⢈⠈⠳⠑⠚⠲⠽⠤⠠⠠⠤⠤⠤⠀⠤⠄⠄⠄⠄⠠⠀⠀⠠⠄⠒⠒⠂⠐⠶⠚⢉⣡⠖
⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⢀⠈⡉⠁⠐⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⢤⡶⠶⠂⠀⠰⡆⠒⠏⠉⠀⠠⠤⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⡼⠀⠀⠅⠀⢀⠔⡆⠀⢁⠜⢒⡂⠀⢀⡜⡅⠀⣼⣂⠀⠀⢀⡐⢤⠀⠀⠀⠀⠸⣰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⡀⠠⠤⠁⢀⣠⠛⠊⠙⠦⠗⠈⠀⠀⠄⠼⠈⠉⠤⠀⠀⠉⣦⠀⠈⠛⠓⠀⠀⠨⠀⠀⠺⠀⢀
⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠁⠀⣀⡀⠀⠈⢡⠀⠐⠀⠀⢀⡀⠀⠀⢀⡤⠀⠁⠠⠀⠈⠀⠀⠘⠂⠀⠚⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢀⡀⠁⢤⡄⠀⠀⠀⠀⢀⡀⠀⠀⠚⠀⠀⠀⢀⠀⢀⠀⠀⢀⡀⠀⠀⠁⡀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠈⠀⠘⠂⠀⠀⠀⠀⠐⠂⠀⠀⠐⠂⠀⠀⠁⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⠈⡇⠀⠀⠀⠀⠀⠀⣀⣿⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⣠⠞⠀⠀⠀⠀⠀⠀⣸⣿⠃⢸⠁⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠔⠁⠀⠀⠀⠀⠀⠀⢠⣿⡟⢀⡏⠀⣰⡟⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡜⠀⣰⣿⢁⡇⠀⠀⠀⠀⠀⠀⠀⣀⡤⠄⠀⠀⢀⣀⠠⢴⣾⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⣿⣿⠀⣠⢾⣿⡇⠀⠀⠀⠀⣿⣿⡿⠀⢠⣿⣇⡞⠀⠀⠀⢠⢶⣦⣠⠞⠁⠀⢀⡴⠊⠉⠀⣠⠟⠁⣿
⠀⡶⡲⣤⡶⡄⠀⠀⠁⠀⢹⣿⡞⢡⣿⣿⠁⣠⣤⡀⠀⢸⣿⡇⠀⢸⣿⠏⠀⠀⢠⡇⠈⢻⣿⡇⠀⠀⢰⠏⠀⢀⣠⠞⠁⠀⢰⡇
⢀⡵⠝⢮⣁⠛⣈⡆⣀⡀⠀⠉⠀⣾⣿⠇⣸⣿⢁⡟⢀⡜⠻⣧⡤⣿⣿⠀⠀⠀⡞⢧⣀⣾⠟⠀⠀⠀⠘⣶⣶⠋⠀⠀⠀⢀⡾
⠘⠓⡶⠢⣙⢦⣄⠛⠉⢘⠆⠀⠀⣿⣿⠀⣿⣷⠋⢀⠞⠀⠀⠀⠀⢿⣿⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⡴⠋⠁⠘⠢⠤⠴⠞⠋
⠀⠘⠢⠀⡦⠄⡨⠳⣦⣈⠵⠀⠀⠙⠿⠖⠛⠿⠖⠋⠀⡀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠤⣤⠶⠏⠀⢦
⠀⠀⠀⠸⣄⣀⡼⠣⠇⠉⠳⢤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⣀⣤⣄⡠⠴⣦⡄⠀⠹⣄⡀⠀⣨⡟⣷⠄
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣄⣼⣿⣏⣀⣀⣀⣠⣤⠴⠖⠃⠀⠀⢠⣾⡟⠁⢸⣿⠁⢸⣿⡇⠀⠀⠀⠉⠉⠁⠙⠛
⠀⢀⣴⡿⠋⢳⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠉⠉⣿⣿⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⣾⡿⠀⣾⣿⠀⠀⢀⣀
⢀⣾⡟⠀⣠⣿⡇⠀⠀⣠⣤⠀⠀⢀⣤⡄⠀⠀⠀⣿⡟⠀⠀⢰⣷⠀⠀⢀⡀⠀⠀⠀⢹⡿⠀⠀⠙⠁⠀⣿⣟⠀⠀⣼⣿⡗⠋⠉⢿⡆
⢸⣿⣥⠞⣿⣿⠀⠀⣰⣿⡏⠀⢀⣼⣿⡅⠀⠀⣴⣿⣇⠀⣠⣿⡿⠀⠀⣾⣿⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢀⣿⡇
⠀⠉⠀⠀⣿⡏⠀⣴⢿⣿⠁⣠⠾⣿⣿⠁⠀⡼⠃⠹⠿⠞⠋⣿⡇⠀⣨⣿⡟⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⢰⣿⡟⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠿⣧⠞⠁⠘⠿⠞⠁⢸⣿⣿⢀⡾⠁⠀⠀⠀⠀⠀⠘⠷⠞⠉⣿⣇⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠤⠞⠙⠁⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠈⠐⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢶⠞⣛⣉⣿⣿⣼⣿⣏⣛⠒⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⡐⠀⠠⠤⠤⠤⠤⠤⠀⠐⢲⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣀⠠⠄⠂⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⣀⢤⠇⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠐⠒⠀⠒⠒⠒⠭⣀⣀⣀⠀⡈⢗⢄⡉⠢⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⡫⢋⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡑⢄⠂⠑⢜⠠⣈⠳⢄⠉⠫⡷⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⡜⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠑⢄⠄⠓⠊⡑⢄⠑⠢⢈⢎⠒⠒⢆
⠀⠀⠀⠀⠀⢠⠇⠀⡰⠇⠑⡤⠄⣀⡀⠀⠀⠀⠀⢀⣀⣀⣀⡤⢤⡶⠝⠆⠀⠀⠉⠲⢄⠙⠚⢓⠢⣀⠉⠣⡀⠈⢣
⠀⠀⠀⠀⢠⠏⠀⡰⡝⠀⣼⠹⠶⢶⡛⠛⠛⠉⠡⠘⠯⠍⣭⣄⣐⣀⢹⠀⠀⠀⠀⠀⠀⠑⠢⣄⠉⠩⠁⣄⢱⠀⠈⢣
⠀⠀⠀⠀⠈⢱⡴⡑⠀⡫⠑⡄⠀⠀⠈⠢⡀⢀⡤⠒⠉⠁⣀⡠⠄⢀⠟⠦⡀⠀⠀⠀⠀⠀⠀⠈⠲⡀⠀⢸⡌⢣⣀⠜⠁
⠀⠀⠀⠀⢠⡿⠡⠁⠀⠁⢰⢌⠘⠛⢓⣒⡚⣓⣒⠶⢐⡉⡠⠤⠄⠤⡼⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⢱⠲⠞⠁⠀⠹⣧
⠀⠀⠀⠀⡋⣅⠁⠀⠀⠀⠈⠂⠥⣈⣁⣀⣀⣀⣀⣀⣀⣀⠠⠤⠐⠊⠁⠀⠀⠀⠀⠈⠑⠒⠂⠂⠒⠁⠀⠀⠀⠀⢀⢝⠄
⠀⠀⠀⠀⠑⠒⠾⠲⠲⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⠿⠿⠶⠾⠿⠿⠿⠿⠿⠿⠛⠛⠟⠻⠟⠿⠿⠿⠿⠿⠿⠶⠾⠒⠊

⢠⠄⠀⠀⠀⠀⠀⠐⠂⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄
⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠐
⠀⠀⠆⠀⠀⣀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠬⠃⠀⠐⠁⠐⠀⠄
⠀⠀⠈⠀⠠⠀⠂⢀⠀⢰⡄⠀⠀⠀⠀⣀⢀⠒⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠐⢈⡂⠀⠀⠀⠀⠀⢁⡀
⠀⠀⣀⣄⠀⢢⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠀⠸⢐⠂⠀⠀⠀⠀⠀⠀⠈⠤⠐⡀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢶⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠑⠂⠓⠀⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠂⠠⠌⡈⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠙⠒⠐⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠛⠻⠿⠿⠿⠟⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠀⠀⠀⡠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠑⣤⠒⠉⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠉⠉⠑⡄⣻⡴⠚⠲⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢤⠀⠀⠀⠀⢹⣿⣀⡁⠀⠀⢀⣀⣀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠷⠴⠚⠛⢿⣿⡟⠛⠛⠺⠏⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇
⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣴⣦⣤⣤⣤⣴⣾⣿⣿⣿⣦⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⡀
⠀⠀⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢾⡿⠀⠀⣼⡄⠀⢸⣿⡧⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⣼⣿⣷⠀⠀⠀⠀⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⡀⠀⣸⣿⣿⣿⣇⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣦⣿⡿⠈⢻⣿⣴⣿⠋⠞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠑⠉⠻⣿⠁⠀⠀⠉⠛⠁⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⣤⣤⣤⣶⣦⣤⣤⣤⣤⣤⣤⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠒⠒⠒⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⢒⠠⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢄⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠒⠲⠦⣤⠀⠀⠀⠀⠀⠀⠰⢡⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠚⠠⠆⢠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⠀⠀⠀⢀⠀⠆⢠⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⢠⣴⣶⣥⣒⠠⢀⠀⠀⠀⠀⠀⠀⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠉⠛⢿⣿⣿⣿⣶⣕⠂⠀⠆⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⣉⠛⠿⠿⣷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⡉⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⡄⠀⢀⠀⠀⠑⠄⡀⢴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢰⡆⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠈⡇⠀⢈⠀⠈⠁⠉⠀⠉⠉⠐⣶⣤⠀⡀⠀⠀⠀⠀⠀⠙⢋⠀⠀⠀⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠨⠄⠁⠀⠀⠀⠐⡤⠁⠀⠀⢀⡀⠀⠂⠀⠀⠉⠂⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⡠⠛⠀⠀⠛⢄⠀⠀⠀⠰⠀⠀⠀⠀⠈⠁⡀⠀⠴⠄⠀⠀⠀⠘⠃⠀⠈⠢⡀⠀⢠⠁⠀⠀⠀⠀⠀⠰⠈
⠀⣀⠁⠑⢤⣤⡔⠀⠘⠀⠀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠐⠀⠀⠀⠀⠀⠀⠀⠄⠇
⠰⡉⠂⠠⡴⠂⠀⢀⡘⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠆⠀⠁⢐⡀⠀⠀⠇⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⠁⢂
⠀⠉⠀⢁⡀⠀⠀⠀⢆⠀⡀⠀⠀⠀⠑⠠⣀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠙⡎
⡆⠀⠀⠀⠁⠀⠀⠀⢀⠞⠉⠂⠀⠐⠀⢺⠀⠀⠂⠀⠀⠀⠀⠢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠳⢄⣀⢄
⠀⠁⠈⠀⠀⠐⡀⢰⠏⠀⠀⠀⠀⣴⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠄⠀⠀⠧⡀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⡦⠀⠊⠀⢠⠐⠣⠄⠀⠅
⠀⠀⠀⠀⠀⠀⠘⠂⡀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠠⠐⡇⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢈⣿⣿⡿⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠷⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⠶⠶⠶⠒⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠓⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⠀⠀⢀⠊⠁⠀⠲
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠩⠁⢃⠀⣠⡀⠀⠀⠆
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠀⠀⠈⠢⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢳⡄⠒⠀⠀⠘⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢠⣤⣄⣀⣀⣤⡄⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢰⠘⠊⠔⠀⠈⠀⠀⠚⠱⠤⠦⢴⣆⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠠⠇⠀⠀⠀⠀⠈⡀⠀⠁⠀⠀⠐⠀⠠⠀⠠⠰⠆⢀⠕⠠⡀
⠀⠀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠐⠂⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠁⢀⡙⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⣿⣿⣧
⠀⠀⠀⠀⢀⠐⣄⠀⠀⠀⠀⠀⠀⠁⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⣿⠟⣼
⠀⠀⠀⠀⣤⠄⢸⢻⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⣷⣟⣻⣷⣟⠻⢶
⠀⠀⠀⠀⠫⠀⣀⣤⣯⣴⠻⠟⠿⠷⢷⠾⠶⠶⣶⠶⡶⢶⠿⣿⣶⣾⢖⣿⣿⣿⣷⣿⣿⡶⠂
⠀⠀⠀⠀⠀⢀⣿⣟⠻⠿⢷⡁⢀⣠⣞⣠⣾⣏⡽⢿⣴⢾⣄⣿⣏⣾⣃⣸⣿⠛⠛⣉⡼⣷⡀
⠀⠀⠀⠀⠀⠸⣿⠟⠿⣶⣦⣬⣭⠿⠿⢃⠠⣾⡛⠓⠒⠲⠖⠚⠛⣩⣿⣽⣿⣿⣷⣤⣴⣿⠧
⠀⠀⠀⠀⠐⠀⠙⢷⣼⣧⣿⠋⠙⣗⢶⣿⡿⢿⣿⣆⠀⠀⢀⣴⡾⠟⠛⣿⢋⣩⣥⣿⣿⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣤⡿⠃⠉⣙⣛⠻⠿⠾⠿⢷⣶⣷⣾⣷⣶⣿⣿⣶⠾⠿⠿⠛⣿⡋⠛⣫⣤⡤
⠀⠀⠀⠀⠀⠀⠹⣗⠛⠂⠿⢼⠂⠉⠂⠐⠊⢉⣭⡿⠋⠀⣭⠁⢼⣶⡌⠉⢀⣼⣇⣠⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠛⠂⠒⠚⠋⠙⠳⠄⡠⠿⢠⡼⠟⠗⠀⠋⠉⣙⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⠤⠤⣤⠀⠀⢠⠀⠤⠤⠤⠐⠒⠊⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠒⠒⠛⠋⠉⠉⠉⠉⠉⠙⠓⠒⠲⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿
⠀⠀⢀⡴⠻⣑⠒⠶⠤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠤⣶⠶⠮⠄⠛⢦⡀
⠀⢰⠛⠒⠲⠦⠭⠭⣿⣭⣩⡆⠀⣴⠶⢄⣀⣀⡀⠀⢀⣀⡀⢀⣀⣤⠤⡤⡀⠀⣮⣉⣉⣩⡿⠿⠷⠒⣚⣋⣉⣉⡇
⠀⠈⣉⣽⠙⠓⠒⠒⠶⣤⣬⣙⡔⠳⠤⣀⡀⠑⢺⠀⢹⣈⣉⡩⠠⠔⠋⢀⣸⠀⠸⣋⣩⠤⢶⠚⠋⠉⠉⠀⡏⠉⡓⠦⣄
⢀⡞⠁⣸⡄⠀⠀⠀⠘⠁⠀⠀⠉⠙⠒⢦⣍⡓⠧⣤⡜⠀⠀⣀⡤⠔⣊⡭⡏⠀⢀⡇⠀⠀⠀⠃⠀⠀⢀⠜⠣⡀⠘⠄⠈
⢘⣦⣾⣁⣈⣳⣄⣀⡀⠀⢠⠀⠀⠀⠀⠘⠀⠉⠓⠦⣍⣙⣋⡥⠖⠋⠉⠇⠙⠒⠚⠀⢀⡆⢀⣀⣴⣷⠥⠤⠤⠿⡄⢠⠞
⡎⠛⠶⠤⠤⠬⣍⣉⣙⡛⠛⠷⢶⣒⡒⠒⠒⠂⠀⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⢒⣲⠶⠟⣛⣉⡭⠤⠶⠒⢒⣚⠛⠃⠸⡄
⠳⡐⠒⠂⢀⣠⢤⡀⠈⠉⠉⠓⠲⠤⣍⡓⠦⣄⡀⠀⠀⠀⠀⠀⠀⣀⠤⠖⣋⡥⠴⠚⠉⠁⠀⡀⠀⣠⠤⢤⣈⣁⣠⠴⠃
⠀⠈⠉⠉⢻⠀⠀⢳⡀⠈⠲⠤⠔⠀⠀⣉⡑⠲⢬⣑⠲⠤⠤⢒⣩⠴⠚⠉⠁⠀⠀⠀⠒⠒⠊⠁⡼⠁⠀⣸⠁⠁
⠀⠀⠀⠀⠈⢣⡀⠀⠑⠦⣤⣠⡤⠖⠉⠉⠉⠉⠳⣌⠙⠒⠚⠉⠖⢁⡴⠚⠉⠉⠙⠲⠤⠤⠴⠋⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠶⠒⠚⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠒⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣓⣒⣶⣶⣶⣦⣤⣤⣴⣶⣶⣶⣶⣖⣒⣚⣋⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠘⣦⣶⣷⣀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡗⠀⠀⣤⠀⣸⡓⠼⡁⠀⡄⠀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣤⠐⣿⢲⣿⣿⣿⣵⢰⡷⠀⣄⣖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⣽⣿⣦⣿⣿⢯⣿⣿⣿⣿⣷⣼⣿⣯⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣽⣾⣿⡿⣿⠟⠉⠉⠻⣿⣿⣿⣾⣇⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣱⣾⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣷⣙⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣷⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠿⢿⣿⡇⠀⠀⠀⠀⣿⣿⡿⠏⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⣆⡖⠋⢹⣧⠀⠀⠀⠀⣿⡏⠛⢶⣰⣾⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣹⣿⣶⣰⣿⣿⣶⣤⣦⣴⣿⣷⣦⣼⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣿⠟⠋⡁⠀⠀⠈⠀⠉⠙⠻⣿⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣟⣿⣿⣿⣀⣄⣾⣿⣿⣥⣤⣸⣿⣿⣿⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢻⠋⣿⠉⠛⠻⢿⡿⠟⠛⢙⡏⠙⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣼⡏⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢀⢠⡄⠀⠠⣦⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠄⣀⣙⠫⠟⠃⡱⢙⢓⣶⠄⡚⠁⢤⡄⣀⣶⣆⠠⡶⣶⢂
⠀⠀⠀⠀⢀⠀⡔⡷⠝⡸⠂⢽⡄⠀⠈⠔⢷⡍⠃⢀⣓⣆⡎⢩⣣⠮⡭⣿⡗⠢⣟⣻⡃⠀⠀⠴
⠀⠀⠀⠀⠘⠖⠒⠃⠄⠀⢀⡾⡓⠀⠀⡸⠃⠀⠦⢖⡛⠘⠁⢡⠄⠀⠓⠦⠡⠀⠠⠭⡥⠄⠀⠀⠀⢀
⠀⠀⠀⠀⣐⢺⡄⠲⠄⢐⠈⣷⣏⠀⠉⠀⠀⡀⠘⠻⠇⠀⠈⠈⠁⠐⡀⠀⠈⠁⡁⢍⣫⠁⠐⠂
⠀⠀⠀⠀⢤⡕⠳⠀⠀⠀⠀⠃⠀⠀⠑⠄⠀⠀⠀⠀⠀⠁⠀⠀⡠⠂⠀⠀⠀⠀⠙⢿⠏⠀⠀⢀⡀⠀⠲
⠀⠀⠀⠀⠘⠋⠉⠁⠀⠀⠀⠀⠀⠄⡀⠀⠠⠀⠀⠀⢀⠀⠐⠊⠀⠀⠀⠀⣨⣚⡖⢢⣗
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠠⠀⠀⠀⠀⠀⠁⠀⠂⠁⠀⠀⠀⠀⠀⣤⠀⢨⡈⠑⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⢀⠀⠀⢲⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⣀⣠⡀⡀⠄⠀⠀⠰⡤⣄⡀⡀⠀⣀⠀⠀⢄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠛⠛⠗⠞⠷⠟⠖⠘⠦⢿⢪⡗⡿⠥⡞⠷⠓⠿⠶⠣⡞⠫⠟⠿⠾⠓⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠀⠉⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⢠⣤⠀⠁⠈⠁⠒⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠁⠀⠀⠴⠰⠀⠀⠀⡀⣀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠀⠀⠀⠆⠆⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⣄⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢔⡠⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡔⡄⠀⠀⠀⠘⠘⠁⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠙⠉⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠰⠄⠦⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢐⠈⠁⠄⠬⠊⣥⢌⠁⡉⠉⠉⠉⠉⠉⠉⢉⠁⠀⠀⠀⠀⠒⠒⢒⠒⢲⠒⢒⠒⠒⢒⠀⣀⣐⠂⠈⠁⡉⠹⡄⣀
⠀⠀⠀⠀⠀⠀⡀⠅⠀⡃⢠⢴⠄⠀⠓⠚⡀⠄⠠⠀⣤⠌⢁⢱⡀⠀⠀⠀⢠⣄⣊⡒⠉⠀⠁⠈⣢⣄⠠⡊⣉⣢⡀⣈⡀⣱⡍⢢⠇
⠀⠀⠀⠀⠀⠀⢇⢐⠉⠀⡧⠤⡴⠄⠀⡸⠀⠀⠀⠀⢀⡀⠀⠀⠈⢉⢀⣈⢁⡀⡀⢀⡀⠀⠈⠀⠀⢩⠐⢧⣀⢀⣀⢀⣀⣈⠈⠁
⠀⠀⠀⠀⠀⠀⡾⠀⡇⢀⠃⠈⠒⠂⠁⠀⠄⠀⠀⠠⠀⠁⠂⠀⡄⠐⠊⠀⠀⠀⡀⠀⠀⠠⠀⠀⠀⠀⠉⠁⠘⠲⠀⠓⠃⠘⢂
⠀⠀⠀⠀⠀⠀⣟⠚⠂⠁⠀⠀⠀⠐⠂⠀⠀⠤⠀⠀⠀⠀⡈⠈⠀⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠈⠠⠄⠀⢸
⠀⠀⠀⠀⠀⣘⢀⣑⠤⠠⠄⡠⠔⠢⠄⠀⠄⠀⠤⢀⣤⠤⣀⠀⣀⠀⢀⣀⣃⡀⣀⣀⠠⠄⢀⣀⠀⢄⣄⣀⠀⠀⠀⠀⢠⠤⠋
⠀⠀⠀⠀⠀⠉⠙⡤⠐⢤⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡈⠁⠒⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⢀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢆⡰⠉⡆⢀⠰⡉⠫⢄⠀⡰⠁⠘⢀⠠⡀⢰⠓⣀⠥⣐⠤⡀⢀⣔⡁⠤⡄⠀⡎⠑⠨⡱⢈⠆⠑
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠈⠀⠙⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠤⠄⠀⠓⠃⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





⣿⠿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠰⠖⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠚⠛⠓⠲⠤⣀⣘⣦⠀⠀⢤⣀⣀⠤⠒⠚⠉⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣤⣶⣶⣶⣶⣤⣬⣽⠸⠇⣯⣤⣤⣶⣶⣶⣦⣄⡀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⣿⣿⣿⣿⣿⣿⣿⣿⣟⢠⡤⣻⣿⣿⣿⣿⣿⣿⣿⣿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡸⣿⣿⣻⣿⣿⡿⣟⠛⢈⡁⢛⣿⣿⣿⣿⣛⣿⣿⣇⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢿⠯⠭⢫⡙⡇⢨⠓⣠⡄⢛⡁⣿⣫⡛⠭⠻⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⢰⢇⡇⣨⡴⢈⡁⢦⡅⣿⡻⠀⠀⠞⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢳⣦⠬⣛⣛⣉⣀⣈⣁⣀⣉⣛⣻⡤⣤⡼⢥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢐⠆⠀⠠⡄⠀⠀⣾⡇⠀⢀⣤⡀⠀⢰⠂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢈⠉⠓⠒⠒⠧⠤⠤⠽⠥⠤⠤⠼⠒⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⡀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠉⠁⠀⠀⠒⠂⠈⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⣶⠒⠋⠩⠋⠉⡉⠉⡉⠉⢉⠉⠉⢙⠛⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⣠⠴⣚⡉⠀⠀⠀⠁⠈⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠃⠠⠄⠀⠐⠀⠐⠋⠓⢦⡀
⠀⠀⠀⡼⢋⠀⠀⠀⠀⠀⠃⠀⠈⠁⡀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠐⠀⠀⠄⠀⠈⠀⣙⢦
⠀⠀⡾⠁⠁⠀⠈⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⡀⠈⢧
⠀⢸⡗⣔⠀⡀⠀⠀⠀⢀⠀⣀⠀⡀⣀⠀⣄⣠⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⡇
⠀⢸⣿⡿⠾⢥⣷⣶⣤⣿⣶⣿⠿⠿⠟⠛⢛⡋⣉⣋⣉⠉⠛⢛⣿⣗⡒⠒⣶⠦⠤⢤⣤⣤⣀⣀⣠⠃
⠀⠀⠻⣿⣿⣿⠿⠟⠛⠉⢁⡤⠚⢉⠍⠉⠁⠉⠁⠀⠈⠉⠉⠉⣀⠀⠀⠈⠀⠉⠉⠲⢤⣤⠤⣄⣬⣵⣤
⠀⠴⢺⣵⡾⠛⢉⠉⢙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠠⠀⠀⠀⠀⢀⠍⢸⡄
⠀⠀⠘⣧⣤⣄⠀⢀⢈⣀⣴⣾⣯⠿⠷⠶⠶⠶⠶⣆⣠⡄⠀⣤⣤⠾⠿⠶⢶⡆⠀⡀⣠⣙⣀⣀⣀⡼⠁
⠀⠀⠀⠀⣿⣿⣃⠏⠉⠀⣀⣀⣀⣀⣠⣤⠤⠤⠤⠼⠿⢿⠼⠿⠋⠓⠒⠒⠶⠿⠼⢍⣉⣈⡉⢉⡏
⠀⠀⣤⣞⣻⡟⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣠⣤⣤⣤⣤⣀⡀⠀⠀⠀⠈⠉⠢⡀
⠀⣾⠿⠿⣿⠟⠻⠶⢦⣄⡀⠀⠀⢰⡀⠋⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⡄⠀⠀⠀⠈⠙⠚⢻⡄⢀⣄⡖⢻
⠀⢻⣤⣤⡷⠀⠠⠀⠀⠉⠛⢶⣄⡀⠁⠀⢀⡴⠋⠀⡀⠒⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠻⠼⠈⠄⢠
⠀⠀⣽⣿⡷⠶⣶⣶⣶⠶⣶⣶⣼⣿⣷⣠⡯⣤⣤⡴⠾⠶⢶⡶⢲⠶⢶⣤⣴⠶⣤⡤⣤⢴⡷⣶⣶⣶⠋
⠀⠀⠛⣿⣷⡼⠁⠉⠁⠀⠸⣄⡜⠛⠛⠛⠁⢸⡇⠀⠚⠦⠞⠀⢸⣤⠜⠀⠉⠛⠁⠀⢿⡀⢰⠏⠙⣇
⠀⠀⠀⢻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⢠⡇
⠀⠀⠀⠘⢿⣿⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠳⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣿⣟⣛⣟⣛⣻⣻⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⢶⠒⢚⠛⠛⢫⠛⠓⠒⠶⠶⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢋⠁⠘⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠲⠀⠎⣙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠅⠀⠀⠁⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢚⠽⢀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢳⠒⠦⠤⢤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠤⠴⡶⠒⠛⢋⣒⠮⠕⠊⠁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣜⢬⣒⠠⠀⠒⠋⠯⣬⣍⣭⣿⣿⡿⢿⡛⠋⠉⠉⠉⢹⡙⠋⠚⠛⠛⠛⠛⠛⠂⠉⠉⠁⣀⣀⡤⠴⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣁⡬⠭⠿⠿⣟⣛⣿⣛⠛⠋⠉⠉⠀⠑⠤⣌⣉⣉⣉⣉⣉⣉⡶⠶⠶⠶⠶⠖⠒⠒⣒⣋⣉⠁⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⣤⣀⢹⠁⠀⠀⠀⠀⠀⠈⠙⠲⢍⡒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠭⠭⠿⠶⠲⠶⠶⠴⠯⠭⢤⡼⠃
⠀⠀⠀⠀⠀⠀⢰⡃⠀⠙⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢢⡀⠀⠀⢀⡠⢐⣠⠴⠚⠉⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠒⡆
⠀⠀⠀⠀⠀⠀⢀⡽⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠑⠤⠔⢋⠔⠋⠀⠀⠀⠀⠀⠚⠉⢁⣀⣀⡴⠋⠙⠒⠦⠤⠴⠃⠀⣳
⠀⠀⠀⠀⠀⠀⠼⠥⠤⢤⡀⠀⠙⠦⠶⣦⣤⣤⣤⣀⡀⠀⢀⡈⠢⡤⠞⠁⠀⠀⣀⣠⠤⣀⢀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠉⠉⠉⠉⠉⠓⠚⢉⣡⠶⢄⣈⣉⡀⠀⠀⢀⠴⠲⠦⣄⣀⣠⠤⣤⢤⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠹⢄⣤⠤⠤⢤⡤⠞⠉⠙⠒⠲⠦⠴⠒⠒⠋⠉⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠶⠶⠤⠤⠤⢤⣤⣄⣀⣀⣀⣀⣤⣤⠤⠤⠤⠶⠖⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠀⠐⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⠠⠀⠀⣠⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠈⣤⠀⠀⢀⡄⠀⠀⠀⠆⢀⠀⠻⠂⠈⠀⠈⠁⠀⠋⠀⠀⠀⠀⠀⢀⠈⠁⠀⠠⠄
⠀⠀⠀⠀⠀⡐⠂⡀⠙⠃⠀⠁⠀⠀⠀⠀⠈⠁⠛⠀⠀⠘⠃⢠⠀⠀⢉⠀⠘⠂⢀⠈⢁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠛⢾⢙⠀⠀⠀⠷⠀⠄⠀⠛⠂⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠
⠀⣀⠀⠀⠀⠶⠀⠀⣸⠙⠃⠀⢀⠜⠷⠀⠀⣀⠀⠘⠀⡀⠃⠀⠀⠀⠈⠁⠐⠆⠀⠀⠈⢁⡀
⠀⠀⠀⠘⢤⠀⢀⣴⡉⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⢠⠄⠀⠁⠐⠆⠰⠆⠀⠈⠁⠀⠀⠘
⠀⠀⠀⠀⢰⡴⠃⡨⠥⠞⠧⣺⡴⠅⠀⡿⠄⠀⢀⡀⠀⠛⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⢹⣟⡉⠀⡀⠀⠀⠉⠰⣠⠀⠀⠀⠀⠈⠁⠀⠰⠆⠀⠀⠛⠁
⢸⠁⡀⣠⡂⠀⠉⠀⠀⠔⢴⡦⠀⠙⠃⠀⠀⠀⠶⠂⠀⠶⠀⠀⠠
⠈⠀⣿⣩⠙⠦⢤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄
⣤⣼⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⠃
⠃
⣿⣟⣛⣛⣛⣻⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠒⠈⠉⠉⠁⠁⠀⠀⠀⠉⠉⠒⠦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠦⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢀⣀⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠦
⠀⠀⠀⠀⠀⠀⣀⠀⠙⠒⠢⠶⣖⡒⠖⠒⠒⠒⠖⠖⣾⣿⣿⠿⠿⠛⠛⠐⡰⠛⠋⠉⠉⠉⡠⠀⠤⠀⠀⠄
⠀⠀⠀⠀⠀⠋⠉⠇⠀⠀⠀⢀⡤⠿⠇⠲⠶⠶⠛⠛⠉⠀⠀⠀⣀⣀⠶⠟⠢⢀⠀⡀⢀⡄⠀⠀⢀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠷⠞⠉⠓⠠⠤⣤⣀⣀⣀⣀⣀⡤⠴⠒⣿⡷⠶⢤⣤⢤⣤⡤⠦⠴⠐⠂⠈⣀
⠀⠀⠀⠀⠀⠀⠀⢪⣸⡄⠀⠀⠉⠀⠀⣀⣈⣭⠥⡄⠤⠐⡊⠁⠀⠀⠀⠂⠀⠈⠉⠀⡄⠋⠄⠀⠃⢹
⠀⠀⠀⠀⠀⣀⠀⠀⠈⠃⠀⢀⣤⣶⣟⡻⣀⣀⣀⡈⠰⠀⢤⡀⠀⠀⢠⠤⠤⠀⠁⠠⠆⠀⣂⡴⠒⠀⠀⢠⠤⠆
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠴⢋⣄⠀⠉⠉⠉⠉⠉⡉⠉⠀⠒⠂⠓⠀⠀⠘⠂⠃⠀⠀⣉⠉⠁⠀⣀⣀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠐⠋⠙⠀⠀⢀⡼⠛⠛⣿⡇⠀⠀⠀⠠⠞⠛⠛⠢⠀⠀⡞⠙⠳⠄⠘⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⠀⠀⠀⠐⠒⠁⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠴⠶⠶⠶⠶⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢀⣤⠶⢛⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⣉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠑⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠟⡔⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡸⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠠⠄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⡇⠀⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠄⠀⠀⡀⠀⠀⡀⠀⠀⢢⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣓⡲⠶⠶⠶⠶⠤⠤⠤⠤⠤⢤⣤⡬⠽⠿⠷⠶⠶⠶⠶⠶⠶⠖⠒⢛⣻⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣬⣭⡉⢛⣛⡛⠛⣻⡛⠛⢛⡛⠛⠉⣛⠓⠲⠶⠶⠶⣶⡶⠿⠟⣻⣛⢉⣡⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⠜⣟⢟⣿⣿⣿⣭⣙⣛⠋⠙⠛⠛⠋⠛⠛⠛⠛⣛⣩⣽⡿⡿⡏⠟⡻⣿⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡷⠷⠾⠾⠿⠾⠿⠿⠿⣿⠷⣶⣿⣶⣾⣿⣿⣿⣿⣿⣾⣼⣽⣿⣾⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠙⠛⠛⠓⠒⠒⠒⠶⠒⠖⠒⠚⠋⠙⠓⠲⠶⠶⠶⠶⠶⠶⠖⠒⣛⠋⣉⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡙⠳⠶⠻⠦⠶⠻⠦⠶⠛⠶⠤⠾⠶⣤⡴⠻⠶⠶⠻⠦⠴⠿⠶⠶⠋⠙⢩⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⡰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠳⢦⣀⡀⠤⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣊⡴⠋⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠶⠶⠾⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠉⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠒⢊⡉⠁⣠⡄⠀⠀⣠⡄⠈⢉⣁⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣩⣤⠀⠀⠀⠛⣁⠀⠉⠀⣤⡀⠀⢶⡦⠈⠉⠀⠀⢀⣀⠈⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⢠⡶⠀⠐⠟⣀⣀⠀⠈⣁⣀⠀⠀⣠⡄⠀⠀⠈⠉⢠⡶⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠈⢉⣤⠀⠈⠉⠀⠀⠉⠀⠐⠷⠀⠀⠲⡦⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡈⠁⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣷⣿⠀⠀⠀⠀⠀⠀⣿⣿⣶⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⢀⣤⣴⣦⣤⡀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠢⢄⡀
⠀⠠⡄⠒⠒⠒⠊⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⡹
⠀⠀⠈⠢⣀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠀⣹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠉⠺⡀
⠀⠀⠀⠀⠈⢳⣄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⢢
⠀⠀⠀⠀⢠⠋⠈⠓⢄⠙⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⣇⠀⠀⠀⠀⠑⢦⡀⠈⠉⠑⠒⠒⠢⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⠤⠶⠶⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⣿⠐⠒⠾
⠀⠀⠀⠀⠈⡗⠲⠤⣀⣀⣀⣙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣴⡒⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⢹
⠀⠀⠐⢦⣉⠱⣄⠀⠀⠀⠀⠀⠀⠈⢢⡀⢀⣀⣤⠤⠒⠒⠉⠁⠀⣀⠼⢇⠈⠓⢄⡀⠀⠀⢀⣀⠤⠤⠒⠒⠒⠀⠈⠉⠁⠀⠀⢀⠜⠲⢄
⠀⠀⠀⠀⠈⡝⠀⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡏⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠳⢄⣀⠓⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⣪⠆⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⡏⠑⠒⠒⠒⠒⠒⠒⠶⣄⠒⠒⠒⠒⠂⠉⠉⠉⠉⠁⠒⠢⢄⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠤⠤⠖⠒⠛⢳⠎⠉⠉⠉⠉⠐⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠵⠶⠦⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠦⠤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡠⠤⠴⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠉⠉⠁⠀⠀⠀⢠⠀⠀⠀⠀⢐⠀⠀⢀⡀⠀⠀⠆⠀⠐⠠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⠀⠀⠐⢄⠀⠀⠐⠀⠀⠄⢀⠀⠀⠀⠂⠀⠁
⠀⠀⠀⠀⠀⠄⠀⡄⠀⢀⣀⣀⠖⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠇
⣦⣄⡀⠀⠀⠀⠀⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⣤⡶⠶⠶
⠈⠉⠻⢷⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⢹⣧⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢸⣿⡀⠀⠀⠀⠻
⠀⠀⠀⠰⣾⠇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠠⠀⡀
⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⡦
⠠⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠞
⠀⠀⠒⠠⠶⠄⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⣤⠶⠞⠛⠉⠀⡖⠲⠄⠀⠀⠀⠀⣀⠠⢰⠒⠉⠄⠀⠀⠀⠀⠀⠀⠴⠓⠋⠀⠈⠁
⢁⠀⢀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠂⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠂⠀⡀⠂⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⢀⠀⠀⠆⠁⠀⠀⢀⠀⠈⠀⠀⠀⠸⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⡀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⢀⡠⠂⠀⠀⠀⠀⡀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠘⠈
⠠⠀⠀⢰⣶⡆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠈⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢
⠀⠀⠀⢸⣿⠇⠀⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠄⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠂
⠀⠒⠀⠀⠀⠠⠀⠀⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⢀⠊⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠂⠈⠄⠐⠈⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠄⢠⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠧⡈⢀⠀⠀⠀⠀⠀⠀⠷⡀⡟⠠⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠦⠈⠀⠀⠀⠀⠀⠤⡀⠙⠋⢀⡀
⠀⠀⠁⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⣀⡀⠈⠉⠀⠴⠆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠓⠀⢤⠀⠀⠀⠀⠀⠀⠀⠘⣶⠂⠈⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠠⠀⠀⣼⡟⢿⣦⡀⠀⠀⠀⠉⠁⠀⣴⡆⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠙⠇⠀⠀⠀⠠⡦⠀⠀⢠⠴⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⢀⡀⠀⠈⢲⡀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠂⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠐⠒⠀⠀⠂⠀⠀⡀⢠⠞
⠙⠀⠀⠀⠀⠁⠀⠀⠀⠈⠅⠀⠀⠀⢀⠀⠀⠀⠀⢀⢐⣤⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⢀
⠂⠈⢀⡀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⡎⠁⢀⠥⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠮⠀⠀⠀⠀⠀⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠈⠈⠉⠉⠉⡁⠁⠐⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠚⣄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠀⠁⠈⡰⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠑⠒⠒⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣂⣒⡂⠉⠀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⢘⠹⠭⣙⠒⡲⠖⣒⢀⢊⠀⡬⠖⠤⠔⠒⡾⠔⠂⠲⣽⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠁⠈⠁⠀⠀⠀⠀⠐⠛⠉⠉⠒⠚⠲⠃⠈⠉⠉⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⢀⡀⠀⢀⣀⣀⣀⡀⣐⡀⣀⣀⠂⠀⠀⠠⠴⠬⢚⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⢦⡠⠤⠄⠒⠢⢀⠤⢤⠀⡤⠒⠒⠒⠒⠢⠼⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠠⠄⠀⢀⣀⣀⣀⡀⠀⠀⠳⠃⠀⠤⠤⠄⠐⠒⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠠⠤⠤⠤⣤⣀⡀
⠀⠀⠀⠀⠀⣠⢴⣲⡴⠚⣋⣡⣤⡶⠶⠞⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠷⠶⢶⣭⣗⡢⢄⡀
⠀⠀⠀⠀⣠⠞⢋⣡⡶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣭⡂
⠀⠀⠀⠚⣡⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄
⠀⠀⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡿⢣⡄
⠀⢠⠒⠋⢈⣽⡛⠷⢶⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣶⠶⠿⠛⡛⠉⢙⢾⡀
⠀⠈⢽⣠⣸⡴⠷⠧⣾⡽⠃⣉⡉⠉⣉⣝⠛⡛⠛⠛⠛⠛⠛⠛⡛⠛⠛⠉⣩⣍⢵⢖⡆⢾⡩⢋⢹⡤⢯⣭⡀
⠀⠀⠀⢧⣉⠉⠳⢤⡙⢻⣆⣷⡟⢸⡇⠹⠞⢩⣇⣀⠰⣿⣀⡼⠹⠴⡆⣐⣇⣙⣚⣰⣧⣖⣁⣂⡤⠴⠚⡿⠃
⠀⢰⡇⡏⠈⣿⢙⠯⣹⣄⠀⠀⠉⠉⠀⠀⠀⠈⣉⣻⣴⣃⣀⣀⣤⡶⠛⠭⣄⣀⣉⣉⣉⣁⣀⣀⣠⡤⠚⢹⡄
⠀⠀⢇⢧⠀⢇⡼⢠⠀⢹⡄⠀⠀⣀⣤⣶⣿⡍⠉⠉⠉⠉⠉⠁⠀⣝⡦⠴⠒⢢⡈⢩⠽⠛⠉⠁⠀⠀⢀⣾⡄
⠀⠀⢀⣈⡳⢦⣄⣁⡀⠀⠳⠶⠚⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣇⣸⠀⠀⠀⢀⣠⡴⠻⣏
⠀⠀⠣⢤⠄⢸⡙⢽⣟⣛⠻⠷⠶⠶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠶⠾⠶⠖⢺⡟⠉⠁⠀⢰⡞⠁
⠀⠀⠺⠵⢦⣠⡝⠦⣄⣉⡉⠙⠛⠛⠛⠉⠉⢉⣀⣠⠭⠋⠙⢓⡲⠦⢤⣀⠀⠀⠀⠀⣠⢟⠀⣀⠀⡖⡶⠟
⠀⠀⠀⢦⡹⣧⡷⢻⢠⠦⣽⠛⢉⡏⣹⢯⢩⠟⠶⢋⡟⣀⡘⣏⠻⠞⢻⣫⣝⠒⠶⢺⡏⠀⠋⠈⢛⡽⢣⡗
⠀⠀⠀⠈⠻⣮⡙⠾⣍⣀⠀⠀⠘⠶⠋⠘⠚⠀⠐⠯⠴⠋⠳⠛⠀⠀⠈⠁⠈⠋⠙⠋⠀⠀⢀⣠⠟⣡⡿⠁
⠀⠀⠀⠀⠀⠀⠙⠶⣄⡉⠛⠲⠦⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡤⠶⠛⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠒⠒⠒⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⢀⡇⣤⠀⠀⠀⠀⡀⠀⣷⠀⣬⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⢰⣆⢧⠜⢆⠿⣴⠀⠀⠀⢹⣶⡟⠾⠹⡇⠀⠀⠀⠈⣁⡀⡏⢳⢀⣌⠀⠀⠀⠀⣆⡞⢳⣰⡆
⠀⠀⠀⠀⠀⠀⠠⠦⠤⠤⠬⠵⠂⠀⠀⠀⣧⣤⣶⡾⠿⠛⠁⠀⠀⣇⠉⠀⠀⠈⠸⡆⠀⠀⠐⣈⣃⣈⣃⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⡄⢰⠄⢠⠆⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⣀⣀⠀⠀⠀⠀⠀⢨⡅
⠀⠀⠀⠀⠈⢆⢠⠣⡼⠤⡟⣠⠀⠀⠀⠀⡤⠞⠚⠖⣄⠀⠀⠈⢷⠦⣏⣳⣞⣙⠞⠱⠃⠀⠀⠢⣄⠁⡼⠅⣩⡜⠁
⠀⠀⠀⠀⠀⡌⠩⠤⠤⠀⠈⢧⡴⠋⠀⠿⠡⠄⠀⠠⣌⡭⠆⠀⠀⠋⠁⠀⠙⠓⠶⠈⠀⠀⠀⠀⠌⢎⣃⣋⣸⠀⠚⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢴⡶⠀⢠⣿⡀⢰⢶⠀⠀⠀⠀⣧⠀⣴⢀⣿⠀⠀⠀⠀⢀⠀⢸⣷⢰⣇⠀⠀⠀⠀⡀⠀⣶⠀⠲
⠀⠀⠀⠀⠀⣶⣆⣾⢹⣇⣾⡆⠀⠀⠀⠀⡟⣿⢻⡟⣿⠀⠀⠀⠀⢸⡷⣼⢠⡌⢩⠀⠀⠀⠈⢷⣰⡿⡴⡇
⠀⠀⠀⠀⠀⣿⠙⠟⠀⠿⠙⣧⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠈⠠⢥⣼⠆⠀⠀⠀⢐⣈⣀⣂⡖
⠀⠀⠀⠀⠀⠋⠀⠀⠐⠒⠒⠛⠀⠀⠀⠀⠀⠀⠉⠉⠋⠀⠀⠀⠀⠚⠛⠹⠿⠿⠿⠄⠀⠀⠀⠈⠟⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⡆⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⣀⢰⢳⢠⡄⣀⠀⠀⠀⢤⡌⠛⠇⠹⠧⢸⠃⠀⠀⠀⠀⢀⠳⠀⠀⠀⠀⠀⠀⠠⣄⣸⣇⡤
⠀⠀⠀⠀⠀⢓⢋⣙⣀⣋⢛⡏⠀⠀⠀⠈⢎⣀⠀⠀⠀⠌⠀⠀⠀⠀⡈⠎⠀⠋⢢⠀⠀⠀⠈⠳⡇⠁⠉⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠛⠀⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⠙⠻⣿⣿⣿⣿⣿⣇⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣤⣤⣘⣿⣿⣿⣟⠿⣶⣿⣿⣿⣿⣿⣿⣿⠛⠛⠛⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣭⡉⢉⡉⣿⣿⣿⣿⡿⠾⠿⣿⣿⣿⣿⣿⣿⢿⣶⠠⠆⢼⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⣷⣿⠘⠁⠀⣿⣿⣿⣿⣿⣿⣦⡉⢻⣿⣿⣿⣿⣿⣄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⠀⢀⣾⣿⣿⣿⠿⠿⠿⠿⢿⡎⣿⣿⣿⣿⣓⣀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡧⠄⣼⣿⣿⣿⣿⣷⠶⣶⠶⠶⡝⣿⣿⡻⢿⡏⢀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡀⠚⠂⠀⠇⣿⣿⣷⣦⣤⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠰⠄⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⠂⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⣴⣾⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠉

⣿⡿⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠉⠛⠉⠉⠋⠙⠛⠛⠉⠙⠛⠻⠟⠛⢛⠛⠛⠛⠛⠛⠿⠟⠻⢿⡿⠿⣿⣛⠙⠛⠿⠟⠋⠉⠙⠛⠿⠿⠿⠿⠛⢛⠛⣿⣻
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠐⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⣿
⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⣿
⣿⡄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠀⠘⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠰⠆⠈⠀⠀⣾⣿
⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠎⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⢁⠀⠐⠀⠀⠐⠈⠀⠀⣿⣿
⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠚⠀⠀⠀⠀⠁⠈
⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠄⠀⠤⠁⣈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠰
⣿⠇⠀⠀⠀⠀⠈⠁⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠐⢂⠻⡀⠐⠒⠞⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡪
⣿⣁⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠀⠁
⣿⡿
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⠀⠀⠀⠀⠀⠐⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠤⠀⠀⠀⠀⠀⢰
⣿⠄⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠀⣸
⣿⡆⠀⠀⠀⠀⠠⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⢠⣾
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⣿⣆⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⣤⣤⣶⣶⡄⣀⣴⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡤⠤⠤⠀⠀⠤⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠀⠀⡀⠀⠀⠀⢀⡄⠀⠀⠀⠉⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢦⣄⣀⣀⣀⣀⣤⡤⠤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠤⠤⢤⣤⣤⣤⣤⣠⣤⠴⠚⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣑⣒⡚⣉⠉⠙⠒⠤⠤⠤⠴⠒⠒⠲⢤⣀⣀⣀⡤⠴⠒⠲⠦⢤⣴⣾⠭⠥⠤⠴⠖⠒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⣈⣉⣉⣉⡉⠉⢙⣛⡶⢶⣶⠦⠤⢤⣤⣀⣀⡤⠤⠒⠛⢉⡥⠤⢤⣀⣀⠴⠚⠛⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⠉⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠈⠙⠶⠤⠴⠞⠋⠙⠒⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣱⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠾⠧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣁⠀⠀⠈⠉⠉⠙⠛⠛⠛⠓⠒⠒⠒⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⠀⣠⠴⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠟⢷⡀⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠙⠶⠿⠖⠛⠳⣄⡀⠀⣀⣠⣀⣀⠠⣀⣀⣀⠀⢀⣠⠴⠶⠶⠦⠶⠞⠁⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠉⠀⠀⠉⠓⠦⢤⡤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⠴⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠓⠒⠒⠚⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐⠠⡀
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠢⡀
⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⡇⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸
⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⣤⠒⠒⠃⠈⠁⠀⢀
⠀⠀⠀⠀⠀⠀⢠⠐⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠒⠈⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢐⣠⣴⣾⣷⡄⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣪⣴⣾⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⢆⣴⣿⣿⠿⠛⣋⣁⣤⣤⣤⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠁⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⠿⠿⢿⠿⠟⠛⠉⠙⢿⣷⡄⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠰⠐⢀
⠀⠀⠀⠀⠀⠀⠰⢠⠀⠀⠀⠀⠀⠀⢹⠿⢿⣿⠇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⡷⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⢰⠀⠀⠀⠊⠀⣀⠀⣈⠂⠄
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠠⣀⠀⣠⠀⠀⠀⠀⢀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠸⠀⠀⠀⠆⠀⠉⠭⡀⠠⠘
⠀⠀⠀⠀⠀⠀⠀⢈⡠⠚⠒⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠿⢀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠈⠆⠀⠀⠑⠐⠊
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣤⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠚⠁⠀⠀⣀⠀⠀⠀⠀⠀⡌⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠐⢈⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠚⠛⠒⠀⠉⠱⠀⠀⠀⠄⠀⠳⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠀⣁⠤⢁⠀⠐⠀⢠⣶⣶⠀⠄⠰⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠳⡅⠀⠀⠀⠱⡀⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡀⠀⠀⠉⠉⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⠀⢀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣧⡀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⣠⣬⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣾⣿⣿⡿⠟⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣠⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢦⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠀⠀⢁⠀⣔⢻⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠈⡐⠀⠀⠀⠠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠁⠒⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⣀⣀⣀⠠⠐⠁⠀⠀⠀⠀⠀⠉⠐⠒⠠⠤⠤⠤⠤⠤⠐⠊

⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠴⠆⠠⣀⠤⠀⠀⠀⠀⠀⠀⣠⣴⠶⠒⠒⠦⢄⡀
⠀⠀⠀⢀⠞⠁⣤⡄⢠⣶⡈⢦⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢾⢁⣀⡖⠂⢛⠍⡀⣽⠒⠀⠀⢀⣾⠋⢀⣤⠀⠀⢠⣆⠈⠲⡀
⠀⠀⠀⢸⡀⠀⠛⢃⡘⠛⠁⢸⠇⠀⠀⠀⡠⠚⢉⠁⠀⠀⠀⠙⢦⠀⠀⠀⢿⠉⣨⠡⠤⠬⢥⡄⢹⡇⠀⠀⣾⠃⠀⠈⣿⠀⠀⠸⠟⠀⠀⠘⣆
⠀⠀⠀⠀⠑⠄⠀⠀⢀⡤⠔⠋⠀⠀⠀⡜⠀⠀⠸⠇⠀⢾⠆⠀⠀⡇⠀⠀⠘⣆⠹⢀⠀⣀⠞⢀⡾⠁⠀⠀⡟⠀⢀⡴⠲⠒⠒⠒⠒⣶⠀⠀⣿
⠀⠀⠀⢀⣀⢠⣠⣴⣾⠁⠀⠀⠀⠀⠀⡇⢠⡀⠀⠀⠀⠀⢀⡄⠀⡁⠀⠀⠀⠈⠙⠒⠛⠓⠊⠁⠀⠀⠀⠀⢻⡀⠈⢇⠀⠀⠀⠀⣰⠏⠀⣰⡟
⠀⠀⠀⡧⠻⠤⠤⠤⠤⠤⠞⡇⠀⠀⠀⠰⡀⠙⠢⠄⠀⠀⠋⢀⠔⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⣀⣀⠀⠀⠙⢦⣄⡑⠲⠶⢚⣁⡤⠚⠋
⠀⠀⠀⠳⠞⠋⠉⠉⠉⠉⠳⠇⠀⠀⠀⠀⠈⠓⠦⠤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⢀⡤⠶⠶⠶⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢿⣷⣶⣶⣾⣷
⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⢱⡄⠀⠀⠀⠀⣿⠈⣙⠛⠛⠋⠁⠘⠿⠿⠿⠋⣿⠇
⠀⠀⡾⠀⠒⠐⠶⠀⠒⠚⠳⠆⠘⡆⠀⠀⠀⠀⢀⠀⠀⠀⠀⡼⠃⣄⠀⠀⠀⢹⡀⠙⣦⡀⠀⠀⠀⣠⡆⠀⣰⡟
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠺⣄⠀⠀⠈⠀⣀⠞⢄⠀⠀⠀⠓⠄⠀⠉⠀⠒⠋⠉⣠⡾⠋
⠀⠀⢻⡄⠀⠀⠀⣀⣤⠶⠛⢀⡼⠀⠀⠀⢸⠀⠀⠀⠀⠀⡠⠊⠁⠀⣸⡀⠀⠀⠀⠀⠀⠐⠒⠒⠚⠋⠁
⠀⠀⠀⠙⠲⠤⣤⣀⣤⣤⠶⠋⠀⠀⠀⢠⠈⠓⠶⠀⠀⠀⠀⢀⠴⠊⠁⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣰⠀⠀⠠⣿⠀⠀⢰⡆⠀⠈⢆
⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠈⠉⠲⡄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠁⣤⢤⣀⠀⣠⡴⠶⠶⠶⢦⣴⡃⣿⠀⠀⠀⡇⠀⠀⠀⠛⠀⠀⠸⠃⠀⠀⠘
⠀⠀⠀⢀⡞⠀⠰⣷⠀⠸⣿⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣸⠋⠁⠀⠀⠀⠀⠀⠈⠹⣯⡀⠀⠀⡇⠀⢶⡀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣀⣤⣤⡀⠀⣴⡋⠉⠀⢸⡇⠀⠀⢻⡀⠈⠳⢤⣀⣀⣤⠞⠁⠀⡠
⠀⠀⠀⢸⡄⠐⠏⠳⢄⡞⠉⠦⠾⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠘⠃⠀⠙⠀⣀⠀⢸⡇⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⣀⡠⠖
⠀⠀⠀⠀⠹⢦⣀⠀⠀⠀⢀⣀⡴⠋⠀⢀⣠⡴⠶⠒⠢⠤⣀⠀⠀⠀⢇⠀⠀⢤⡀⠀⠀⢀⡴⠃⣠⡿⠁⠀⠀⠀⢀⣀⣀⣈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠀⠀⣰⡟⠁⢠⡄⠀⠀⣴⡌⠑⢦⠀⠈⠢⢀⡀⠈⠉⠛⡉⠠⠞⠋⠀⠀⠀⣠⠞⢉⠁⠀⠉⠙⠲⡀
⠀⠀⠀⢀⠔⠒⠒⠒⠢⡀⠀⠀⠀⢠⡟⠀⠀⢸⡟⠀⠀⠿⠇⠀⠀⢳⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠿⠂⠀⡿⠀⠀⠙⡀
⠀⠀⠐⠁⠀⠇⠀⠗⠀⠈⠀⠀⠀⢸⡇⠀⣠⠶⠶⠶⠶⠶⢲⡆⠀⢨⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⡤⠠⠆⠤⢠⠀⢀
⠀⠀⠀⠐⢄⠀⠀⠠⠆⠀⠀⠀⠀⠘⣧⠀⠸⡀⠀⠀⠀⢀⡼⠃⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢏⣁⣉⣈⣅⣭⠼⢁⠆
⠀⠀⠀⠑⠤⢤⣤⠤⠊⠀⠀⠀⠀⠀⠈⠳⣤⣈⠒⠶⠖⣋⣡⠴⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⣀⣀⣀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣤⠤⣤⠤⠤⠴⠠⢶⣦⣰⠦⠤⢤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣶⠖⡮⢏⡳⠮⠓⡓⠨⢿⣅⣈⢴⡞⢕⠋⠲⣼⠂⠉⣰⢷⡴⢮⢼⣝⠻⣷⣶⠶⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠳⠗⠀⠀⠀⠀⠀⢁⠰⡋⠰⠥⠀⠀⣢⢄⠤⡀⠄⡉⠈⠡⠉⢡⣶⢣⡜⠃⡤⢽⣿⣲⣶⣏⣒⣤⡀
⠀⠀⠀⠀⠀⣠⣾⠟⠃⣦⠀⠈⠀⠀⠀⢀⠀⠀⠋⠀⠀⡀⠒⠂⠀⠀⠉⠀⠁⠠⢄⠀⠀⠀⠀⠀⢀⡖⠆⣔⠚⠁⠙⠠⠱⠫⠙⣿⡦⡀
⠀⠀⠀⠀⣰⣿⠡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠴⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠘⠀⢀⡀⠂⠀⠁⠈⠠⠄⠀⢀⠈⠀⠀⠀⠈⡝⣿⡄
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠄⠀⢠⡄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢹⣷
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠐⠀⠀⠈⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⣛⣿
⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠃⠀⢸⠀⠀⠀⠀⠠⠃⠈⠀⠀⠇⠀⠀⢸⢀⣼⠃
⠀⠀⠀⢀⣾⣄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡼⠋
⠀⠀⠀⠘⣿⣿⣿⣶⣦⣄⣀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠖⢋⣶⣀
⠀⠀⣠⠜⠉⠻⢿⣟⠉⠀⠉⠛⠿⣿⣶⣶⣤⣤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⣿⢿⠏⢀⣴⠿⠿⠀⠐⣦
⠀⣴⣧⣤⣤⡄⠀⠉⢛⠓⢲⣤⣤⣀⣈⡁⠉⠉⠉⠉⠉⠉⠉⠻⢿⣿⣿⡿⠟⠛⠛⢻⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⠛⠀⠒⠋⢛⣿⣷⠶⠒⠿⠧
⠀⠀⠀⠀⢸⢟⠛⣳⢦⣌⠉⠋⠉⠉⠉⠙⠛⠛⠶⠶⣶⣶⣦⣤⣬⣬⣤⣄⣀⣀⣠⣮⣍⣭⣥⣤⣤⡶⠾⠛⠉⠀⡠⢀⣠⡞⣿⠉⠞⡇
⠀⠀⠀⠀⢸⣏⠷⠀⢈⡏⠳⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠉⠉⠉⠉⠬⠟⠛⠂⠉⠀⠀⠀⢀⠔⣋⣴⠏⠋⢇⡘⠀⣞⠁
⠀⠀⠀⣴⠟⠻⣶⣈⢸⠁⢄⣀⡄⢈⠉⠛⠛⢿⠿⠿⡿⠿⢷⣮⣵⡦⠄⠀⢀⣡⣤⣤⣤⣶⣶⣶⢶⣦⣍⣠⣾⡿⠈⠀⣀⡞⣀⣼⠿⠳⢄⡀
⠀⠀⠀⢹⠂⠀⠈⣙⣿⣄⣈⡀⠁⣴⠟⠃⣤⠚⠶⠂⢀⡌⢡⣈⠰⠋⠀⠰⢫⠁⠀⠀⢠⡌⠁⢈⡆⠉⡻⠗⠛⣱⣖⣴⣿⡾⠟⠁⠀⠀⢀⡿
⠀⠀⣠⡞⠚⠋⠉⠉⠉⠉⠛⠛⠳⠾⢷⣤⣔⣂⣀⠈⣀⡀⡋⢍⠁⣀⣪⣆⢀⣄⣙⣷⣤⣅⣨⣄⣤⣾⣧⣶⣟⡛⠋⠉⠀⠀⠀⠀⣀⣴⣿⣧
⠀⠀⢸⡿⢶⣀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣬⡭⢽⣷⣿⣷⣾⣿⣿⣛⠙⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠈⠁⣠⣶⡾⣿⣿⣿⠿⠟⢻⣿
⠀⠀⠨⣿⣾⡿⣷⡄⠀⢀⣴⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣿⡿⣿⡿⠒⠛⠁⠰⣣⡿⠋
⠀⠀⠀⢹⠚⠟⠛⣧⣠⣿⣿⣯⣦⣹⢿⣦⣀⣀⣀⣤⣤⣦⣄⠀⠀⠀⠀⠀⣸⣿⣿⣦⠀⠀⣀⡤⣶⣿⠿⠟⠿⠥⢵⠋⣤⣴⣦⢴⠞⢻⡇
⠀⠀⠀⠈⢷⣄⠀⠈⠉⠀⠀⠀⠀⢉⡴⣾⣯⠹⣏⢛⣿⣿⣿⣦⠀⠀⣀⣼⣿⣖⣉⣻⡷⠾⠗⠋⠉⠀⠀⠀⢀⣠⣴⠟⠙⡻⢁⣫⣤⣿
⠀⠀⠀⠀⠈⠫⠙⠶⢷⣶⡞⠛⠛⠉⠀⠀⠀⠿⠿⠛⠻⠟⠛⠛⢎⡿⠛⠉⠉⠉⠉⠉⠙⠲⠶⠚⠛⢫⠓⠺⢛⡁⣀⣈⡬⠔⢋⡀⠉⠁
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⠙⠓⠾⠶⣶⣴⣯⣀⣀⣀⢀⣀⣠⣄⣀⣠⣦⣶⣤⣀⠀⣤⣤⣤⣤⣴⡾⠶⣖⣛⣩⠭⠀⠀⠀⡡⢋⣠
⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⢨⠩⠭⠉⠛⠫⠉⠉⠉⠉⠙⠛⠛⠉⠛⠉⠡⠄⠀⠈⠉⠁⠀⣀⡤⠀⠋⣡⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣁⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣶⣤⣀⡠⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢃⢀⣀⣤⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠲⠿⠿⠛⠳⠾⠿⠗⠛⠛⠓⠒⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣽⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡿⣿⣿⣿⣿⡟⣵⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣿⣿⣿⣿⣾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⣿⣿⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⣣⣿⣿⢷⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣴⠿⢋⣴⣿⠯⢠⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠠⢤⣄⣤⣴⡿⢟⣫⣵⣾⣿⢿⣁⣤⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠋⠉⠉⠁


⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣤⡀⠀⠀⠀⠀⠀⣠⡤⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡂⠄⡴⢟⣦⠀⠀⠀⢀⡟⠁⣀⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠰⣇⣴⠟⠃⠀⠀⠈⢆⠀⡀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠦⢤⣾⡛⠀⠀⠀⠀⢨⠀⠑⣇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠛⠁⠀⠚⠟⠓⢲⣴⠚⠃⢀⠴⠧⡁⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⡀⡀⠀⢀⡻⣤⣴⣥⡖⣴⢀⢷⡀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠁⠁⣀⢠⡿⣿⣾⣷⣻⢻⢸⡇⠑⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠓⣷⠀⠀⠀⠀⠉⠀⢉⣻⠙⡿⢩⣇⠎⡿⡏⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠛⡇⠀⠀⠀⠀⠀⡞⢯⠀⡇⡀⢸⠀⠠⡱⠀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡜⢠⡇⠀⠀⠀⠀⠀⣷⢸⣤⣿⠜⡟⢀⡣⢟⢀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⡃⢸⣄⣀⠀⠀⠀⠀⣹⣼⠯⡏⢠⡄⡈⢁⡤⢁⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⠃⠀⣟⠽⢻⡓⠒⠫⠭⣿⣬⢿⣯⠀⠖⠚⠀⣾⡇⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⢸⠋⡁⢐⡀⠘⠂⠉⢱⠾⠊⢿⠂⠀⠀⣼⡏⡉⠛⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢶⠀⠀⡇⣆⠀⠈⠇⣆⠀⢘⣠⠀⠀⡰⡇⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⠀⢠⣦⠏⣆⢀⠊⣯⠀⠈⢧⡀⠀⠀⢰⣤⣌⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠓⠛⠊⠀⠈⠈⠉⠉⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣠⠀⢀⠀⠐⠖⠀⠀⠀⠚⠖⠀⠠⠀⠠⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠈⠉⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⣀⠀⠰⠆⠀⠀⠀⠛⠒⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠁⠀⠀⠀⠀⠰⠼⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠺⠆⠀⠀⠀⠀⠙⠤⡀
⠀⠀⠀⠀⡐⠀⠈⠉⠀⠀⠀⠦⠋⠀⠀⠀⠀⠠⣈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⢠⡄⠀⠀⠀⢀⣰⠀⠀⣐⡄⢅
⠀⠀⠀⢨⠇⠀⠀⢄⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠀⠀⢠⡤⠂⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠉⠙⢒⠢⠄⠀⠀⠀⢀⣀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⣀⡀⠄⠀⠒⠀⠀⡒⣃⡠⠤
⠀⠀⠀⠀⢸⢥⣒⣒⣒⣀⡠⠤⠭⠤⢁⣀⣀⣀⣀⣀⣈⣛⣛⣛⣛⣉⠉⡙⣃⢀⣀⣀⣉⠑⠤⣄⣀⣈⣀⣁⣀⣀⣀⣀⠽⡋
⠀⠀⠀⢀⠝⠠⢀⠀⠀⠍⠉⢉⠉⡙⡓⡒⠒⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⡉⡉⢉⢉⠉⠉⠉⠄⠂⠐⠨⠧
⠀⠀⠀⡼⣶⠠⣄⢀⣀⠙⡋⠀⠀⠀⢣⣧⠔⠐⠸⠀⣥⡅⠀⠀⠀⠀⢀⡥⠁⠀⠘⠀⢠⠆⠃⠂⠀⠘⠖⢁⣀⣀⢀⢤⠴⠒⢢
⠀⠀⠀⠻⣌⠑⣭⢢⠌⢶⡝⢍⠙⣕⠺⣥⠰⠒⠛⢗⠲⡢⠤⠄⠖⠒⠒⠒⠲⠒⢒⠒⠶⠒⠒⠒⠈⠋⠉⠁⠀⠀⠀⢁⠀⠀⡼
⠀⠀⠀⠀⢀⠉⠒⠱⡬⢢⡉⢢⣛⣄⡱⣬⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⣊⡀⠀⢀⣀⡲⡶⠖⠀⣤
⠠⠀⠋⠉⢉⠉⠑⠐⠆⠀⠬⣥⣀⠈⠉⠛⠉⠉⠉⠉⠙⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⢉⣀⣀⣈⣉⣁⣀⣩⣤⠶⠋
⠀⠒⠒⠶⠀⠀⠄⢀⡡⠄⣀⡤⠀⠀⠉⠑⠆⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠐⠐⠂⠶⠶⠫⠉⠀⠀⠹⠄⠈⠉⣀⠀⠀⠰⠛⠉
⠀⠀⠀⠀⢀⣠⣪⠤⠧⠀⠀⠆⠀⢀⡔⢙⡄⠀⢀⠞⠉⣋⡇⠀⢀⡍⢳⠀⠀⣟⣁⢦⡀⠀⢠⠀⠠⡀⠀⠘⠀⠀⠘⠨⣵⣄
⠀⠀⠀⠀⠸⢁⠀⠀⡥⡉⠀⢂⣠⠎⠶⠉⠹⠤⠧⠇⠉⠀⠐⣄⡼⠔⠋⠡⠤⠀⠀⠉⢳⠄⠌⠍⠛⠚⠢⠔⠀⣬⡄⠉⠠⡞
⠀⠀⠀⠀⠀⡈⡃⠀⠀⠀⠀⠨⠉⠃⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⢀⠀⠀⠫⠁⠀⠰⠆⠀⠠⠆⠀⠈⠠⠖⢀
⠀⠀⠀⠀⠀⠐⢦⡀⠘⠓⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠋⠀⠀⣤⠀⠋⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴⠆⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠄⠀⠻⠇⣀⣀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⡤⠦⠀⠀⠰⠂⠀⠀⠀⣤⡄⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣘⣛⣀⣀⣀⣀⣠⣄⣉⣁⣀⣀⣈⣉⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⡛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⢤⡄⠐⠶⢒⣦⠤⠤⡤⠀⡀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⠴⠴⠛⠢⠍⠁⠀⠙⠈⠀⠁⠈⠀⠉⠉⣂⣂⡀⠀⢀⡈⢋⡔⠢⡄⣀
⠀⠀⠀⠀⡀⡚⠁⠀⡀⠁⠴⢒⠦⠛⠼⠑⣓⠂⣨⡈⣏⣉⠋⣅⢚⡒⡓⠦⠌⠡⠀⣘⠈⢱⢄
⠀⠀⢀⣾⣧⢁⠀⠀⠀⠤⡀⠐⠈⡄⡁⠀⣀⠈⡀⡀⠆⢁⡁⠤⠌⢀⠁⡐⠀⠀⠸⠄⠀⠠⠨⣳⡀
⠀⠀⣎⠚⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠃⠈⠉⠐⠀⠠⠄⠀⠀⢋⠀⠀⠀⠉⠀⠀⢀⠀⠠⣠⠈⢿⣧
⠀⠀⣟⠦⡬⠷⠀⠀⠀⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡃⠀⠀⠀⠀⠄⣠⣄⣙⢌⢶⣿
⠀⠀⠉⣃⠉⠐⠐⠤⣅⢄⢐⠁⠤⣢⡀⢤⠆⠏⠒⢄⠲⠴⡄⠀⠌⠩⡦⢨⡛⠐⠤⡆⠈⡌⠘⣹⡃
⠀⠀⣠⣻⣶⣤⣀⠀⠀⠠⢀⣨⡂⠤⢱⡄⢀⠀⣤⣀⠱⠂⢡⠆⠴⠖⣩⢣⠹⠖⠀⠀⠀⣠⣾⡿⠗⢄
⠀⠀⡙⣹⠃⠉⠛⣛⣶⣦⣄⣀⡀⠀⠀⠀⠈⠃⠀⠈⠁⠘⠀⠀⠉⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⢿⡿⠇
⠀⠀⡟⢷⠚⡫⠋⠙⢻⡟⠻⡍⠛⠛⠻⠶⠶⣶⣦⣤⣤⣤⣤⠶⠶⠶⠞⠛⠛⠛⠉⠉⠀⢸⡿⢋⣼⠃
⠀⠈⠉⣿⢦⣑⣔⡀⠀⢈⠓⠱⡀⢀⣠⡴⢾⡙⠋⠻⠿⣿⠟⠛⠷⠶⠦⣄⣀⣰⡶⣄⡀⢸⣿⡿⢡⠆
⠀⠀⠀⢿⣠⠉⣟⠓⠲⠿⢧⣤⣯⣿⣿⠀⣌⡑⢒⡃⣀⣀⣨⡠⠛⣀⣠⣤⣿⣻⣧⣷⡿⣿⡋⠁⣿
⠀⠀⠀⢈⡟⢇⢺⠃⣠⠴⠦⡜⠋⠉⠉⠉⠻⠿⠿⢿⣛⠛⠛⠓⠚⣛⣭⣭⣥⣈⠁⢀⣼⢿⢹⣿⡿⠇
⠀⠀⠀⠻⢿⣮⣝⠲⢬⣘⠑⠛⢆⠰⠠⠀⡀⠀⢠⠀⠃⣑⢤⡶⡝⠁⠉⡀⣋⠬⣱⣾⣿⡼⢟⡽⣿⡀
⠀⠀⠀⢠⣦⢀⠈⠑⠶⣤⣉⠙⠻⠷⢷⠿⢶⣶⣦⣤⣼⣉⣀⣡⣤⣤⣤⣴⠿⠾⣟⣩⣴⠶⠉⢀⣬⡩⠤⣄
⠀⠀⠀⡄⢿⣆⡈⠓⠲⠶⣇⡀⠀⠀⠀⠐⠂⠀⠰⠯⣭⣤⣭⡭⠷⣶⣶⢶⠟⢟⡻⠟⢁⣠⣶⣿⣿⡃
⠀⠀⠀⢣⠀⠙⠻⣷⣦⣤⣌⣉⣁⣀⣀⣀⣀⡀⠀⠀⠉⠉⠓⠀⠈⣚⠛⠋⠉⠀⢀⣴⣿⣿⣿⠟⢹⠃
⠀⠀⠀⠈⢦⠀⠀⠀⠉⠛⠻⠿⢿⣽⣭⣤⣭⣿⣿⣦⡀⠀⠀⢠⣿⣿⣿⣷⠶⠶⠟⠛⠛⠁⢀⣴⡟
⠀⠀⠀⠀⠈⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠉⠑⠀⠀⠋⠉⠁⠁⠀⠀⠀⠄⡀⣰⣾⡾⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣦⣤⣤⡀⢂⣀⣠⣠⣤⣀⡀⣀⡠⢠⣤⣠⣠⡤⢠⢳⣤⣿⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠷⠾⠿⠯⠿⠿⠷⠞⠒⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠸⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼
⠀⠀⠀⠀⠀⡀⠀⠈⠉⠉⠉⠛⠒⠒⠒⠒⠒⠒⢒⠲⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠉⠉⠁⠀⡃
⠀⠀⠀⠀⠀⠈⠁⠒⠒⠒⠀⠀⠐⠲⠖⠒⠈⠉⠀⠀⠀⠂⠀⠤⢤⣄⠤⠀⠀⠀⠠⠄⠐⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⢠⡄⠠⠤⠀⠀⠈⠀⠀⠉⠖⠒⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠁⠤⠠⠀⠐⠂⠀⠀⠀⠆⠀⠀⠀⠠⠀⠀⠀⠄⠀⠠⠤⠠⠀⠠⠀⠀⠀⠀⠀⣠⠂⠄
⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⠀⠀⠁⠀⠈⠀⠀⠈⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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