Standard language Emojis & Symbols

⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⡿⠿⠿⡿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⡿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣻⣟⣛⣻⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠸⡆⣸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⠐⣱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠠⠄⠀⠀⠀⠁⠀⣾⣿⣿⠟⠋⠀⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣟⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⠰⠒⠊⠀⠀⠙⠛⠛⠀⠀⠀⠀⠚⠛
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠺⠃
⣶⣿⡆
⣿⡿⠁
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⢀
⣯⡤⠴⠄⠀⠀⠀⠀⠀⠀⠀⢰⡆⢠⣤⣤⣴⠏⠀⠀⣶⣶⣤⣤
⠀⠀⢤⣤⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠀⠀⠀⠘⠛⠋⠛⠛⠇
⠀⠀⠘⣧⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣤
⠄⠄⠈⠻⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢿⠀⠀⠲⠚
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠶⠶⠶⢶⠶⠶⠶⠶⠶⠶⢶⠶⠶⠶⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣀⡀⠀⣰⠃⠘⣧⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⡉⠛⠃⠀⠀⠘⠛⠋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⣤⡀⠸⡷⠶⠶⠶⠶⠶⠾⠶⢶⣶⠶⠀⠘⡇⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢿⣆⠀⠀⠘⣷⠲⠦⢾⠉⠓⢦⣄⣿⠀⠀⢸⡇⠀⠀⢸⣀⡴⠞⠉⢿⠴⠶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣳⡄⠀⠀⠀⠀⠈⠉⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⢀⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢯⣀⠀⠀⠀⠒⠦⣄⡀⠀⢸⡇⠀⢀⣠⡴⠖⠀⠀⠀⣀⡽⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠙⠳⢼⣧⠶⠋⠁⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢛⡛⠛⠛⠛⢻⡟⠛⡟⠛⠛⠛⢛⣛⠛⢦⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠓⠆⠀⢸⡇⠀⣟⠀⠰⠞⠋⠁⠀⠸⠶⢲⡟⢀⡀⢸⡟⠛⠛⠓⠲⠦⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⠛⠁⠀⠙⠛⠛⠛⠛⠛⠛⢛⡿⠋⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠒⠒⠒⠒⣒⠒⢲⡆⠀⡶⠒⢒⡒⠒⠒⠒⢿⠁⠀⠀⢸⠇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠠⠖⠛⠁⠀⢸⡇⠀⡇⠀⠈⠙⠳⠦⠀⣸⠀⠀⠀⢼⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠑⠀⠀⠒⠂⠂⠀⠂⠀⠒⠂⠐⠀⠒⠀⠀⠐⠀⠀⠒⠀⠂⠐⠒⠂⠐⠒⠂⠂⠒⠐⠂⠐⠚⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠠⠠⠠⠄⡤⠀⣀⠠⠀⢄⠀⠠⡀⠠⠀⠄⠀⠄⠀⢄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠶⠒⠒⠛⠋⠉⠉⠉⠉⠛⠛⠒⠲⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⠤⠤⠤⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠈⠉⠙⠲⠦⣄⡀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⡀⠀⠀⠀⠀⠉⢳⣄
⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣆⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠹⣄
⠀⠀⠀⢠⡞⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠘⣆⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠘⣦
⠀⠀⢠⡟⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠹⡆⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠘⣧
⠀⢀⡞⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠹⡄⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠘⣆
⠀⣼⠁⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⢹⡄
⢠⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣥⣤⣄⣤⣠⣀⣄⣀⣷⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣧
⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢷⡀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢹
⡿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠸
⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣏⣁⣀⣀⣉⣳⣶⡆⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣋⣀⣀⣀⣀⣀⣙⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸
⢸⡀⠀⠀⠀⢻⡀⠀⠀⠀⢀⣠⣤⡤⠤⠤⠤⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⠤⠤⢤⣤⣄⡀⠀⠀⠀⣸⠃⠀⠀⠀⣸
⠈⣇⠀⠀⠀⠘⣇⠀⣠⠞⠉⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⢸⡇⠀⠉⢳⡄⢀⡟⠀⠀⠀⢀⡏
⠀⢹⡀⠀⠀⠀⠹⣴⠏⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⡾⠁⠀⠀⠀⣸⠁
⠀⠀⢷⡀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⣰⠇
⠀⠀⠈⢧⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⢠⠏
⠀⠀⠀⠈⢷⡀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⣰⠏
⠀⠀⠀⠀⠀⠻⣄⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣇⡼⠃
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣄⡀⢸⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⢸⡇⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠉⠉⠙⣆⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⣻⢟⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⡤⠶⠖⠛⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣀⡤⢶⣫⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⢤⣶⣒⠛⠛⠲⢒⣚⣫⠵⠞⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣀⣠⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⠛⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀⣀⣀⣀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠛⠛⠛⠛⠛⠛⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡃
⠀⠀⠀⠀⠀⠈⠻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⠻⠿⠷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣦⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⣀⣤⣶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠾⠟⠋⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤
⠀⠀⠀⠀⠀⢠⣿⢃⣠⣴⡿⠛⠉⠀⠈⠉⠙⠛⠛⢻⣿⠿⠿⠛⠛⠛⠛⠉⠁⢶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⡶⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⠀⠿⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠟⠋⠀⠀⠀⠉⠻⢿⣶⣤⣄⠀⠀⠀⠀⠀⢠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠙⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⡀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⠶⣶⣶⣶⡶⠶⠾⠿⠟⠛⠉⠙⠻⢷⣦⣀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀⣀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⡇⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⣿⣿⣿⣿⣿⢸⣿⣿⣇⠀⣤⣠⣿⣿⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⣇⠛⠛⠛⠛⠛⠘⠛⠛⠃⠀⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⣤⣬⣷⣦⣀⢸⣿⣿⢸⣿⣿⢀⣤⣾⣧⣤⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⡿⢿⣿⣿⣿⢸⣿⣿⡿⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣷⣮⣝⠻⠸⣛⣭⣶⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣤⣤⣤⢠⣄⢠⣤⣤⡤⢤⡙⣿⣿⣿⣿⣸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠷⠮⠽⠿⢸⣿⠸⠿⠥⠶⠿⠧⠭⠉⠀⢸⣿⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠿⠿⢿⣿⠿⠿⠿⠿⠿⠟⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣿⢟⣫⣽⣿⢸⣿⢸⣿⣭⣛⢿⡇⠀⠀⠀⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠈⠁⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣶⢾⢠⣤
⠀⠀⠀⠀⠀⠀⣴⣿⣷⣿⣿⡀⠀⠀⢼⣿⣿⣿⡇⣴⡯⠽⣷⡄⠀⠀⢀⠀⠈⣿⣿⠁⢀⣀⠀⠀⢸⡇⣿⣿⢸⣿⡟⡇
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⠇⠀⠀⠸⣿⣿⣿⠏⠼⠿⠶⠞⠁⠀⢠⣏⣧⢸⡧⢼⡰⡇⠈⡁⠀⣼⠇⣿⣻⢸⣿⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠈⠉⠋⠁⠉⠉⠀⠀⠉⠀⠉⠉⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢰⣶⢶⣶⣶⠶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠌⠁⠀⡀⠀⠀⢀⠄⠀⠀⠀⠀⠛⣀
⠀⠀⠀⠀⠀⢸⣧⣭⣿⣯⣽⣟⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⣶⡄⠈⠋⠀⠀⠀⠀⠀⠠⡶⢩⡆
⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠻⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⢀⣬⣿⣥⡀⠀⠀⠀⠀⠀⠀⠾⠿⠷⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⡷

⠀⠀⠀⠀⠀⢀⣀⣴⣶⣦⣴⣦⠀⠀⠀⠀⠀⠀⠀⣴⡯⠻⣶⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢚⡭⣝⠢
⠀⠀⠀⠀⠀⠘⠟⠆⡗⡷⠘⠁⠀⠀⢀⡴⠓⠒⠳⣿⠿⠶⠟⠀⠀⠀⠺⣿⣝⣈⣽⣿⣿⣄⠀⠀⠠⣾⢒⡛⣒⢲⣧
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠟⠁⠀⠀⠀⠀⠙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠿⠿⠿⠁⠀⠀⠀⠀⠀⠉⠷⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⢦⡀⠀⠀⠀⠀⠀⣈⣿⣏⡀⠀⠀⠀⠀⠀⠀⣷⣄⣶⣶⠀⠀⠀⠀⠀⠀⢠⠖⠉⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢯⠘⠢⢆⡇⠀⠀⠀⠀⣾⡟⣯⣿⣿⡆⠀⠀⠀⠀⠀⣿⣿⠟⣛⢢⠀⠀⠀⠀⠀⢉⠚⠊⠒⢅⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⠛⠦⠀⠀⠀⠻⠾⠾⠷⠾⠃⠀⠀⠀⠀⠀⠉⠉⠑⠨⠺⠄⠀⠀⠀⠀⠈⠣⠤⠴⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⢀⠔⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠤⢸⠔⢱⠄⠀⣀⠀⡾⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢸⡖⢺⠀⢸⠽⢠⡇⡜⢘⢹⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠈⣧⠘⣦⠞⠓⠉⠓⠙⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠻⠦⠀⠀⣤⠠⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢂⡄⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠶
⠀⠀⠀⠀⠰⣦⡸⣼⠃⡄⣤⣠⠄⡄⢠⡄⢠⢦⡀⠸⡼⣠⡴⠀⠀⠀⢘⡀⠿⣿⡇⠀⠂⠀⠀⠀⡀⠀⠀⠀⠰⣶⡗⠀⠁
⠀⠀⠀⠀⢤⡶⣝⠁⡄⣇⣿⣿⡀⡇⢸⡇⣾⠀⡇⣄⢑⡵⣦⠄⠀⠀⠉⠐⠃⠈⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠻⠀⠸
⠀⠀⠀⠰⠮⡿⡄⢸⠃⣿⣿⣿⠁⣿⢸⡇⢿⠀⡇⢛⠀⣼⡯⠶⠀⠀⠀⠀⠀⠀⠄⠐⠈⢠⠀⡁⠀⠀⠀⠈
⠀⠀⠀⠰⠛⠥⠖⠉⠂⠙⠟⠻⠶⠻⠊⠷⠘⠶⠃⠊⠑⠶⠝⠳⠀⠀⠄⠀⢄⠀⠠⠀⠀⠀⠔⠁⠀⡀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⢹⡇⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠
⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢉⠉⠀⢀⠀⠀⠀⠀⢀⣸⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠐⠀⠙⠈⠀⠀⠀⣆⢰⠀⣶⠀⢠⠀⠀⠀⠀⣼⢀⢰⡏⡇⠘⢠⡏⢸⠒⡀⠠⠄⠀⠄⠀⠀⠀⠀⠀⢸⡧⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⠀⣿⠀⠀⠀⠀⠀⠀⠈⠉⠈⠓⣃⣀⠛⠀⠈⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠚⠛⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡞⣻⣗⢳⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⡀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠑⢽⡿⠋⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢠⡜⣇⣠⡴⠀⠀⠀⠀⣤⠀⢀⣤⠐⢠⡄⠂⣠⠂⣤⣶⠀⠀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⢀⣌⠻⣬⣵⠶⠀⠀⠀⠀⣿⠠⠀⣿⠀⢸⣷⠀⣯⠀⡿⣿⢶⣰⡿⠀⠀⠀⠀⠀⠀⠀⢠⡄⠙⡞⣦⢀⡟⢼⡇⣀⢀
⠀⠀⠀⢀⠻⣆⣀⣴⡶⠀⠀⠀⠀⠿⠀⠂⠛⠂⠈⠿⠀⠈⠈⠁⠛⠀⠙⠃⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠁⠀⠁⠉⣿⠰
⠀⠀⠀⠹⣷⡈⣉⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡄⠀⠀⠀⠀⣥⡿
⠀⠀⠀⢴⣤⣉⣩⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣶⣤⣄⣀
⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠈⠉⠛⠿⢿⣿⣿⣶⣦⣤⣀
⣾⣿⣿⣿⠿⠟⠛⡉⢡⣤⡄⢶⠾⠛⠃⢀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠰⠆⣀⠀⠀⠉⠛⠻⠿⣿⣿⣷⣶⣤⣀⡀
⠋⠉⠁⡆⢰⠆⠻⠓⠀⣁⣄⢰⡠⠶⠃⠙⢁⣀⡸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⢰⣒⣤⣈⠁⠸⠿⠀⣖⣄⡀⠀⠉⠙⠻⠿⣿⣿⣿
⡄⠀⢀⠄⣤⡴⠶⠛⠛⠋⢁⣠⣤⣦⠴⠿⠻⠏⠁⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⠀⠉⠉⠛⠳⢦⣄⡄⢁⡘⠷⠶⡤⠀⠀⠀⣸⣿
⣷⠀⡈⠉⣉⣤⣤⡶⠆⠑⠊⡉⣀⠉⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠁⠀⠀⠀⣀⠀⠀⢰⣇⡄⢈⡁⠳⠰⢶⠤⣀⠀⠀⢰⣿⣿
⣿⣟⠇⠋⠛⠋⠀⠀⠸⠗⠋⠑⢉⣀⢠⣴⡄⠀⠀⢀⣸⣿⣷⠀⠀⠀⠀⠀⠀⣰⣿⣿⡃⠀⠀⠀⠘⠛⠧⣶⣤⣀⣉⠉⠃⠀⠀⠀⠀⠉⠀⢠⣿⣿⠇
⣿⣿⡀⠀⠀⠀⢠⣤⢠⣤⠶⠿⣟⣋⣈⣥⣴⣶⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠉⠛⠛⢠⣶⡄⠀⠀⠀⠀⣾⣿⡏
⢿⣿⣧⠀⠀⠀⠈⣋⣡⣤⣶⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠛⠻⠿⣿⣿⣿⣶⣦⣤⣀⠀⠀⠀⠁⠀⠀⠀⣼⣿⡿
⠘⣿⣿⣶⣶⣾⣿⣿⣿⠿⠟⠛⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣷⣶⣤⣀⡀⢰⣿⣿⠁
⠀⢻⡿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠸⠷⠖⠶⠶⠷⠖⠶⡶⠀⠀⠀⠀⣿⣿⠀⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡆
⠀⣼⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠐⠠⠀⠴⠀⠦⠀⠤⠼⠏⠛⠋⠀⣿⣿⡇
⣼⣿⡟⠈⠙⠻⢿⣶⣤⣀⠀⠀⣿⣿⡇⠀⠀⠀⠰⠠⠶⠖⠶⠶⠶⠶⣦⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠐⠖⠒⠒⠶⠖⠀⠄⠖⠂⠆⠸⠀⠀⠲⣿⣿⡇
⣿⡟⠀⠄⢁⠒⠀⣈⠙⠻⢿⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⠀⠀⠀⠛⠒⠖⠒⠒⠒⠐⠓⠀⠀⠀⢀⣿⣿⡇
⣿⣀⠀⠠⠄⡈⠢⢤⡉⠲⠀⠈⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠀⠿⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⠉⠛⠿⣶⣤⣈⠐⠢⣄⠀⠠⢄⡛⠻⠿⠟⢻⣿⣿⠟⠛⠛⠛⣛⣿⣿⣿⣿⡿⠿⣿⣿⠛⠀⠀⠀⢰⣿⡟⠙⠛⠻⠿⠿⣿⣿⣶⣶⣦⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠙⠻⢿⣶⣤⣈⠃⠰⠍⠓⠦⣴⣿⣿⣯⣴⣶⣿⠿⠟⠛⠉⢉⠀⠤⠄⠸⣿⡄⠀⠀⠀⣼⣿⠇⠠⠤⢌⢀⡒⠒⠂⠬⢭⣉⠛⠛⠻⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣶⣤⡀⣰⣿⣿⣿⡏⢉⣀⠤⠐⠀⡂⠡⠀⠐⢂⣈⠄⢿⣷⠀⠀⢀⣿⣿⠈⣁⠒⠂⠀⠤⠌⢁⡐⠒⠐⠤⠤⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⠟⣿⣷⠠⠀⢐⣈⠁⠤⠒⣀⡈⠭⠄⠀⠘⣿⡇⠀⢸⣿⡗⠒⠠⠨⢈⣁⡒⠂⠀⠤⠌⣀⣀⠒⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠈⠭⠄⠖⡀⠡⠄⠐⠂⠀⠀⢀⣹⣿⡀⣿⣿⠁⠀⣀⠂⠲⠀⠠⣀⣁⣐⠒⠢⠄⠤⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⠿⠟⠛⠱⣿⣿⣿⣶⣶⣯⣤⣍⣐⣒⠂⠨⢌⣁⠀⠀⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠸⠀⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣄⣀⢠⡟⠈⢧⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⡈⠉⠀⠀⠈⠉⢱⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢦⣘⣏⠉⠉⢉⣉⣉⠉⡿⢯⡀⠘⡇⠀⠀⡄⠀⠀⣼⠀⣀⡴⣷⢀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠆⠀⠀⠻⣍⠉⠁⠀⠉⠳⠇⠀⠀⡇⠀⠀⠿⠚⠁⠀⠉⠉⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣟⠀⠀⠀⠠⣄⡀⠀⠀⡇⠀⠀⣀⡤⠄⠀⠀⢈⣳⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠉⠳⢦⣧⠴⠚⠁⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢛⣛⠛⠛⠛⣿⠛⣿⠛⠛⢛⣛⠛⢳⠀⠀⡀⠀⠇⣀⣀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣈⣙⣃⣀⣿⠀⣿⣀⣘⣉⣀⣀⣘⣛⡿⠃⣤⠘⡏⠉⠉⠙⠓⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣠⠟⠀⠀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣠⠴⠂⠀⣿⠀⣿⠀⠰⢦⣄⠀⢸⠀⠀⢀⣇⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠶⠶⠶⠿⠶⠿⠶⠶⠶⠾⠶⠞⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄⠐⠀⢂⠠⠆⠀⠠⠀⠐⠀⠂⠠⠀⠀⠂⠄⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢟⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣫⣾⣟⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢟⣾⢿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢰⣶⣶⣶⡶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿⡿⣫⣾⣿⣿⣿⢟⣵⣿⣿⣝⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⢸⣿⣿⣮⡻⣿⢟⣵⣿⣿⣿⡿⣫⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣽⣿⣽⣿⣽⣽⣯⣯⣽⣽⣯⣿⣯⣾⣽⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⣯⢿⢟⣼⣧⣝⠿⣫⢸⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⡿⣵⣜⢿⣿⣿⣿⣿⢸⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣾⣧⣭⣭⣽⣿⠇




⣏⣈⣯⣍⣟⣯⣿⣽⣽⣽⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣶⢀⣤⣤⣠⢤⣰⢠⡄⣶⣄⠀⣠⡤⠘⢿⠉⠹⣦⠀⠀⠠⠤⠤⠤⠄⠤⢄⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠄⠀⠀⠁⠀⠉⠈⠈⠈⠈⢈⣉⠉⢯⠘⠧⠴⢦⠈⢧⣼⢟⡰⢤⡀⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢲⡋⠈⢧⠈⢧⠀⠶⠋⠈⠛⠁⠸⢯⣿⢃⣶⢦⡄⠀⠈⠀⠀⠀⠂⠀⣀
⠀⢀⣀⠀⠀⢀⠤⣄⠈⣡⠔⢳⡀⢳⠀⠈⠁⠀⠀⠀⠀⣤⠤⡶⠒⡆⠡⡾⠳⠾⢃⣼⣲⡄⠀⠀⠀⠈⠈⢀⢀
⢰⡋⢁⣀⠈⢧⠀⢘⡇⠙⠶⠫⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠛⠀⠀⡀⠀⠘⠳⠿⣡⡟⢳⡆⠀⠀⠀⠀⠀⠃⠴⡄
⠈⠉⠉⢉⡷⠈⣿⠋⢀⠔⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠙⠋⠉⠹⠄⠀⠀⠐⠀⠀⠀⠀⠈⡀⠟⢱⡿⢽⡆⣀⡀⠀⠈
⠀⠀⠛⠉⠀⠀⠈⠔⢁⠼⣉⡇⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡴⠻⡛⣦⠀⣠⠄⠄⢐⡀⢀⠀⣀⡀⡀⠙⢂⢤⣻⡌⣡⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⡔⠁⡄⠀⠀⠀⠀⠈⢐⠀⢣⠹⠀⠓⢃⠀⠣⠀⠇⠳⠈⠸⠀⠇⠅⠸⠰⠸⠀⠍⠇⠻⣝⠛
⠀⠀⠀⠀⣀⣀⣤⡀⠤⠤⠤⠕⠎⡀⢏⠃⣀⡀⠀⠀⠒⠄⠀⠛⠉⠙⡇⠀⠀⠀⡀⠀⠀⠀⠀⢀⣿⢐⣯⢸⣆⠀⡀
⠀⠀⠠⠘⠛⣯⡿⠁⠀⠀⠀⡀⠀⠈⠉⢄⡳⢀⠆⢀⡀⢁⡠⠒⠒⣋⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠠⡅⡆⢱⠱⠘
⠀⠀⠀⠀⠀⠛⣴⠾⢦⠀⠀⠀⠁⠢⠤⢄⡀⠁⢞⣲⠇⣠⣄⠰⡏⠉⠙⣆⡝⢡⡀⠤⡀⡆⠆⠧⠕⠑⠊⢸⠂⡞⠆
⠀⠀⠀⠀⠀⠀⠀⣰⠦⣹⡄⠀⠀⠀⡀⠀⠉⠁⠀⠈⠘⢁⡜⠓⢇⣠⣾⡥⠷⠊⣣⠓⣇⢀⢀⢐⠰⠸⠨⠀⠊⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢋⣤⣾⡃⠈⠁⠂⠂⠀⠼⠐⠃⠃⠃⠚⠈⣘⣀⡼⠀⠀⢇⣀⠐⠈⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡼⢇⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⣘⣧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠢⣄⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⢄⣒⣒⡂⡭⠭⠿⠭⡍⢭⢭⢭⢥⣶⣒⣢⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠩⢴⣞⡛⠉⠍⢜⡇⡆⠐⠐⠀⡃⣿⢡⡪⢝⡵⢞⡩⢒⡱⠏⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣡⣖⣶⠤⣀⠈⠉⠛⠾⢮⡇⡇⢈⢎⠄⡄⣻⣯⢟⡡⢒⡡⠖⠉⢀⣴⣟⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠼⠯⠽⠿⠟⠶⠝⠲⠀⠀⠀⠀⣂⠰⢁⢆⢐⠉⠐⠋⢚⣋⣀⣀⣈⣋⡉⠭⠭⢤⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠒⠒⠒⠒⠚⠂⠈⠉⢉⢉⠉⠉⠁⡠⡣⡢⢊⢉⢁⠀⠀⠀⡄⡠⠀⢀⢔⡄⣐⡀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠐⠔⠠⠂⠂⠈⠔⠰⠡⠊⠊⠌⢜⢔⠔⡐⠑⣑⣁⡁⡂⠈⠈⢤⡥⠥⠨⠤⠬⠴⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣉⣉⣉⣉⣉⡉⠉⠉⠉⣉⢉⡉⢹⠠⢊⢊⠘⢁⣀⠀⠰⢖⡲⢤⣚⢿⣽⣿⣿⢗⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣭⡥⠞⠉⠀⣀⠔⢪⣡⣾⢻⢸⠠⡗⡠⢸⡄⣿⠳⢦⡀⠉⠓⢮⣑⣫⠗⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠄⣀⣀⣀⣀⠀⠀⠘⢯⡀⢠⠔⣩⣰⣾⢟⢫⡔⣰⠎⠀⢌⢐⠄⡄⢿⠡⡡⣫⣳⡄⠀⣈⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡯⣷⠶⡶⠶⠳⠳⠖⠶⣯⣭⣗⢢⢍⡚⠃⠭⠽⠟⢓⡛⠓⠻⠃⢧⣤⡤⠴⠧⠬⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣡⠎⠁⠀⠀⢀⢀⡄⡠⠀⠀⠁⠙⢯⡳⢌⢢⣀⠀⢀⠔⠁⠀⣴⡊
⠀⠀⠀⠀⠀⠀⠀⢰⣡⡏⢰⠃⠀⣤⠴⣲⠕⣩⢚⢳⣵⡀⣄⢔⠄⠙⡄⢫⢹⡆⠈⠓⠢⠤⢤⡀⠉⢓⡆
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⡏⢀⡪⢝⣫⡔⠋⠀⠀⠈⠘⠻⢿⣷⣥⠀⠸⠀⢹⣷⠀⠀⢀⠔⠊⡰⠖⠉
⠀⠀⠀⠀⠀⠀⢸⢹⣁⢤⢧⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠖⢤⣠⣇⡀⡟⡆⠀⠙⠒⣦⠈⠑⡦
⠀⠀⠀⠀⠀⠀⣮⠜⠁⢸⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⠉⢣⡇⢀⣴⣊⠥⠖⠋
⠀⠀⠀⠀⠀⢸⣝⢀⡀⢸⣽⠀⠀⠙⠒⠚⠀⠀⠀⠀⠓⠤⠔⠀⠀⢸⢽⠀⠀⢌⣿⠈
⠀⠀⠀⠀⠀⠀⠘⢎⡈⢸⣿⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠏⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠈⠒⠒⠒⠊⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⣤⡀⠀⠀⠀⠀⠀⣠⡾⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⢰⠪⢓⢖⠖⠚⢛⢋⣏⠕⢊⠝⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢐⠨⢊⠡⢾⣷⠤⣌⣈⣈⣈⠼⡧⢊⠥⢒⡡⠄⠙⠢⡀
⠀⠀⠀⠀⠀⢀⡴⠁⠀⡊⠔⣊⠄⣊⢹⣍⠓⠢⠤⠤⠒⢚⣋⠥⢊⡡⠔⣊⠀⠀⠘⢆
⠀⠀⠀⠀⠀⡜⠀⢐⡩⠒⡩⠔⢈⠄⢺⣗⡀⠀⠀⠀⠀⢸⡥⢒⡩⠔⣉⠄⢊⡡⠀⠈⣆
⠀⠀⠀⠀⠘⠀⠐⠉⠒⠁⠐⠊⠔⠨⠚⠷⠀⠀⠀⠀⠀⠸⠖⡡⠔⢈⡠⢚⡡⢔⡩⣀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠘⠁⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⡀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⠀⠀⠀⠀⢠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⡇⠀⣦⣤⣠⣿⣿⣿⣦⣠⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢿⣿⣿⠸⠿⠿⠿⠿⠿⠿⠃⠿⠿⠿⠿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣀⠀⠀⠈⣿⣿⣿⡏⣿⣿⣿⡇⠀⠀⢀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⣿⣷⣦⣄⣿⣿⣿⡇⣿⣿⣿⣇⣤⣶⣿⣿⣷⣶⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣦⣌⡛⠿⣿⡇⣿⡿⠟⣋⣴⣾⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣷⣦⡁⣠⣶⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠶⣶⣶⣶⣶⢠⣶⡆⣴⣶⣶⣶⡶⢶⣶⡌⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣤⣹⣿⣿⢸⣿⡇⣿⣿⣏⣥⣶⣿⣿⣇⣉⡭⠁⠀⠀⣿⡇⠀⠀⠀⠀⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣼⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⠄⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⣉⣉⣉⣉⣉⢹⣿⡏⣉⣉⣉⣍⣉⣉⣉⡁⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢛⣩⣴⣿⣿⢸⣿⡇⣿⣿⣧⣌⡙⢻⣿⡇⠀⠀⠀⠀⠸⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠈⠉⠁⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠔⠀⠀⠀⠀⣿⠃⢰
⠀⠀⠀⠀⠀⡄⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⠈
⠀⠀⠀⠀⠾⣁⡷⠒⡌⢑⠄⢀⠀⠀⠀⠀⣸⣤⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠐⠒⣥⡄⣼⡟⢺⠟⣡⠂⠀⢠⠛⣿⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⠇⢘⡛⠉⢡⣀⡨⠀⠀⠀⠸⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠻⣧⠀⡐⢠⠀⠀⠀⣾⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⡇⢀⠔⠁⠀⠀⠃⠀⣸⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠀⠀⢹⡄⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⢧⠸⡷⡔⠟⠳⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠃⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢸⡇⠀⡈
⠀⢀⢀⢖⡀⠠⡁⠰⣶⣦⣤⣤⣤⡴⣾⠀⠀⠠⠠⠁⡀⢀⣴⣧⡌⠀⠀⠊⠑⠀⠋⠠⠀⠀⠀⠀⡰⡄⠀⠀⠀⠆⠀⠀⠀⣸⡇⠀⠃
⠀⣿⣷⣦⠹⡄⢙⠶⣍⡳⣀⠀⠀⠀⠘⣄⣤⠖⡠⢊⡙⢛⠀⢀⣙⣀⡈⠀⠀⣎⠀⠘⠆⢀⠖⠀⠐⣄⡠⠀⠀⠣⡀⢠⣴⡿⣧⢦⡇
⠀⣿⣿⡟⠀⠃⢸⠀⠈⠑⢤⣳⠀⠀⠀⢿⣧⠂⠀⠀⢁⢡⣸⣿⣿⣿⣿⡆⠀⢻⠀⠀⠀⢆⣠⡀⠀⢸⣧⡄⠀⠀⡆⣸⣿⣇⠀⠀⠰⡀
⠀⢹⣿⡟⠛⢶⡟⠀⠀⠀⠘⠉⠀⠀⠀⢸⡜⣧⠀⠀⡸⠀⠸⣿⣿⣿⣿⠃⠀⣿⠀⠀⡀⠘⠆⠓⠀⠘⠁⣄⡀⠐⠁⠓⢿⣿⣯⡀⠰
⠀⠘⣏⠁⠧⢾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⣷⣶⡾⣦⡄⣠⠄⠉⠛⠶⠤⡏⠘⠦⠀⠱⣔⠙⠂⠠⢼⡇⠀⠁⠀⡇⢐⠀⠀⠈⠙⠻
⠀⢀⠛⠲⢾⡜⢣⠀⠀⠀⠀⠀⠀⠀⠀⣽⠯⣉⣽⡷⣿⣇⣧⠀⠀⠀⢠⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠐⠀⠀⠀⡹
⠀⠈⠀⠀⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⠛⢿⣫⠭⣿⣿⠀⠀⠀⠀⠖⢚⠀⠀⠀⠀⠿⠄⠀⠀⠀⠐⠐⠀⠀⠈⡗⠂⠂
⠀⢀⠆⠀⠀⠀⢸⠛⢳⣶⡤⢴⣶⠒⠋⢱⡗⠀⣿⣿⣿⣿⣿⣤⣀⣀⣀⣀⢨⠀⠀⠀⣰⠈⠁⢰⢂⠂⣠⡦⣤⠴⠏⢰⡄⠀⠐⠒⠞
⠀⠘⠓⣶⠶⢶⠁⠀⢰⣴⠀⠀⢐⠀⠀⢾⣿⣶⣿⣿⣿⠇⠈⡀⢀⡀⠀⠹⠞⠓⠶⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣼⣄⢸⡀⠶⢛⣿⡀⣷⣿⣀⡀⠀⢹⣩⣋⡏⠀⠀⠀⠁⠈⠃⠀⠸⠢⠦⠤⠖⠃⣀⣄⣀⢀⣄⠀⠠⣤⢄⢠⣄⡀⡀
⠀⠀⠀⠛⠛⠉⠛⠁⠘⠻⢿⣿⡟⠻⠿⠂⢸⡿⣇⡷⢤⡀⠀⡆⠀⡆⠀⢲⣤⡖⠐⣲⡄⠡⠽⠿⠶⠦⠥⠶⠙⠋⠉⠋⠁⠁⠈⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠭⣉⡻⠿⠀⠙⠒⠻⠔⠋⠁
⠀⢀⠔⠁
⠔⠁
⣾⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣾⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⣤⣤⣤⠤⠤⠤⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠩⠀⠘⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⢐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⣫⢟⢴⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣿⡍⠽⢛⡛⠻⠷⣶⣦⣤⡀⠖⣰⣪⣴⣾⣟⡫⡕⢕⣋⠾⣛⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣯⡈⠁⠀⠀⠀⠀⠁⠀⠀⠈⠉⢹⠟⡻⣫⢟⢵⣊⢞⡫⠕⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠹⠿⣖⣤⢄⣐⡒⠲⠀⣀⠀⠀⡯⣐⢝⣫⣶⣿⡟⠫⡓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠻⣦⡀⠀⠈⠉⠋⠚⠻⠷⠶⣤⣤⣿⣾⡿⠟⠉⠮⠀⠀⠁⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣷⡄⠙⢷⣔⣂⣀⣀⣀⣠⣤⣤⣤⣤⣬⣥⣬⣥⡤⡤⣤⢤⢴⣶⣶⡶⡶⣭⢢
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠀⠈⢮⣦⠀⠈⢻⡏⡛⡫⡸⡽⢝⢕⢕⣩⡪⡪⡊⡾⡁⢕⢕⢕⢕⠽⡪⡪⣺⠇⣦⣀⡀
⠀⠀⠀⠀⢀⣤⣲⠱⠤⠀⠀⠀⠀⠀⠀⠑⣿⣗⣸⢣⡎⡪⡪⣊⢔⢕⡕⡡⣪⣪⣊⣪⢐⣕⣵⡕⡵⠡⠪⢾⣞⣼⣽⡿⠃
⠀⠀⠀⠀⠀⢻⠶⠬⢥⣄⣐⣠⠄⣀⡀⠀⣈⡛⠯⠿⣾⣯⠭⠭⡭⠼⠷⠶⠶⠶⠶⠰⢒⡾⠛⠓⠪⠽⠛⠋⣩⣿⣻⡇
⠀⠀⠀⠀⢠⣮⣄⡀⠀⠀⠌⠹⠓⠾⠬⠅⠲⡝⡕⠈⢞⣭⣶⣿⣯⢿⣥⣤⣔⣒⣠⡀⠤⠤⠤⣀⡀⠀⣠⣪⢎⣺⣷⡇
⠀⠀⠀⠀⠈⠒⠂⠭⠽⣓⣶⡤⢤⣤⣀⣂⣀⠇⣏⣞⢾⡇⢨⢩⠍⠁⠈⠉⠙⠛⠓⠛⠉⠉⠉⠑⣺⣛⣋⣿⣯⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠉⠉⠑⠒⠶⠶⠴⠾⡿⠟⣾⣻⣤⢄⣀⡀⠀⠤⠤⠶⠤⣄⠀⠀⠀⠠⡟⠿⣿⢾⣿⡿⠛⣦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⡼⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⡉⠛⠹⠖⢭⣭⣽⣒⣒⡤⠬⠭⢀⣀⣷⢞⣵⣿⣾⣶⣖⣾⡮
⠀⠀⠀⠀⠀⠀⡸⢺⠿⠿⠿⠭⠽⠿⠿⠍⠉⠛⠒⠲⠶⠒⠛⠓⠒⠐⠚⠛⠛⠛⠉⠩⠍⠉⣙⠓⡾⣫⣁⠉⠀⠉⢡⡫⢂⠅
⠀⠀⠀⠀⠋⢕⡇⣿⢰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠲⣤⠉⠀⠀⠀⠀⠘⣎⠕⡢
⠀⠀⠀⠀⠀⠀⠣⡻⣧⣐⣒⣒⣒⢂⣀⣀⣀⣀⣀⣀⣀⠤⠤⢤⣤⣤⣤⣀⣀⣒⣒⢒⠒⢒⣐⣀⠀⠠⠴⠶⢶⣶⣶⣿⣿⡪⣔⢄
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⡻⢛⠾⡫⢏⠽⡻⠿⡿⢗⠶⡲⢖⠖⠢⠖⠰⠂⠖⠩⠟⠛⠋⠟⠝⠩⠋⠝⠱⠋⠋⠈⠈⠁⠁⠈⠀⠁

⠀⠀⠀⠸⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⠿⢿⣷⣄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡌⠉⢀⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣶⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣶⣿⣿⣿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡇⠙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂⠀⠛⠛⠛⠛⣛⣛⣛⢁⡀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⢤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣼⣿⣿⣿⠿⠀⠀⠀⠈⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⠀⢤⡖⠀⠀⠀⠀⠀⠀⠀⠠⠀⡴⠫⠟⣭⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠀⠨⠀⣠⡿⢄⡀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠸⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠋⢉⠀⠉⠈⠀⢨⠊⠁⠀⠀⠙⢆⢠⣀⣄⠀⠀⠀⠁⠒⠊⠻⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⢷⣤⡞⠋⠙⢧⡀⢀⣼⣷⣶⣷⣶⣄⡼⠘⠿⠏⠀⠴⠦⠤⠀⠀⠀⠙⢿⣦⠀⣶⡆⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣄⣠⡾⠉⢧⣄⣠⣞⡁⠈⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠐⠢⣤⣀⠀⠀⢠⡄⣀⠉⡀⢉⠐⠂⠀⠀⢀⣼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡌⢁⣀⠀⣀⠈⠁⠈⠀⠀⠙⠻⣿⣿⣿⣧⢀⡄⠒⣄⡄⠒⣤⠙⢷⣄⠘⠁⠉⠀⠃⠈⠀⠀⣤⠔⠋⠉⠉⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣍⣉⣱⠏⠀⠀⠀⠀⠀⠀⠀⠛⡛⠛⠛⠓⠗⠒⠈⠑⠒⠟⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠩⠙⢦⠀⠀⢰⣶⠀⠀⠀⣯⡽⡀⠀⠀⠀⠀⠀⠀⣀⣀⡞⠃⠀⠘⣧⠀⢀⣀⠀⠀⣠⣴⠟⢋⣡⠄⠀⠀⢀⠀⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⠚⠂⠘⠃⣴⣤⣤⣦⣤⣤⢸⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⣿⠆⠀⠀⢻⡆⠹⠟⣠⣾⠋⠁⠀⣾⣿⠇⣀⣤⡴⢄⡀
⠀⠀⠀⠀⠀⠀⠰⣶⠀⠀⢨⡗⠒⢦⡀⠀⠙⣿⣿⣿⣿⣿⣄⣿⣿⡆⠀⣤⣄⠀⠀⠙⠛⠋⠀⠀⠀⢸⣇⠀⣰⡿⠁⠀⠀⠀⠀⠁⠈⠉⣴⡤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠠⠤⠄⠻⠀⠘⠿⠿⠿⠿⠿⠏⠸⠿⠗⠀⠙⠁⠀⠀⡀⠛⠀⠘⢀⡀⢸⣯⢀⣿⠃⠀⣿⣿⡇⠀⠀⣀⠀⠁⠀⠀⠀⠈⠂⣀
⠀⠀⢀⠂⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠦⣤⡀⠀⠁⠀⠀⠀⠈⠁⣿⡇⣼⡿⠀⠀⣿⣿⡇⠀⡘⠟⠀⠀⠀⠀⠀⠀⠀⠟
⠀⣰⣯⣀⣀⠀⠛⠃⢠⡀⠴⢀⡴⠀⠀⠀⠈⢷⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠛⠀⠘⠀⢰⣿⢁⣿⡇⠀⠀⢉⣛⣁⣸⣿⠀⠐⣄⡀⠀⢀⣠⠁
⠀⠀⠉⢩⡟⠁⢀⠓⠄⠀⠀⠀⠀⠚⣁⠀⠀⢸⡇⢀⣬⣤⣤⠀⠀⠀⠀⠂⠀⠀⠀⢻⣧⠀⠀⢀⣿⡏⣼⣿⠀⠀⠀⢘⣿⣿⣷⢿⡇⠀⠛⠁⠀⠈⠋
⠀⠀⢀⠏⠀⠀⢉⡁⠀⠀⠀⠀⠀⢋⣁⠀⠀⢸⡇⣾⣿⣿⣿⡧⠀⠀⡀⠃⠄⠀⠀⠈⣿⣇⣠⣾⡿⢱⣿⡇⣰⡗⢀⣀⠉⠉⠉⢈⣁⣤⣶⣷⣶⣶⣶⡦
⠀⠀⠈⣀⡀⠀⠈⠁⠀⢀⠀⠀⠀⢠⠵⣄⠀⣿⠇⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣿⡿⠀⠿⠛⢻⡼⡀⠀⠐⠙⠛⠛⠛⠛⠛⠋⢁⠒
⠀⢠⣾⠟⠁⢰⣤⣄⠀⠸⣿⡗⠀⠀⢰⠋⠀⣿⠀⠚⠿⢿⣿⣿⣿⣿⢿⠶⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠀⢠⣤⣠⠞⣰⣅⣀⠀⠉⠀⠀⠒⠒⠀⠉
⠀⠌⠁⣀⠀⠘⢿⡟⠀⠀⠀⠀⢀⣤⠀⠀⠀⣿⡆⠀⠀⠘⠿⣿⠿⢟⡀⠀⠀⠀⢠⣿⣿⣿⡿⠟⠁⠀⠀⠀⠼⠿⠦⠀⢉⠟⡀⠀⠀⣀⠀⠀⠀⠀⢠⠄
⠀⠀⠻⠿⣶⠄⠘⢁⡀⠀⡀⠀⣾⠷⢶⣙⠄⠸⣿⣦⡀⠀⠀⠀⣴⣿⡆⠀⠀⣠⣿⣿⡿⠋⢠⠴⠎⠀⠀⠀⠀⣀⠄⠀⠈⠀⠀⠰⠀⣀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⡀⢀⣀⠀⣼⠋⠀⠀⠈⠻⣿⣷⣦⣄⡙⠛⠁⠀⣴⣿⡿⠋⠀⠀⠀⠀⠀⢀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠐⡀⠉⠐⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⢹⣿⣅⠀⠀⠀⠠⠀⠈⠙⢿⣿⣿⣷⣤⣾⣿⣿⠁⣄⣉⢈⡌⣁⣠⠏⠀⣄⠈⠉⠀⣤⡄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠈⠀⠉⢠⡤⠀⠀⠀⡄⠀⠀⠈⢿⣿⣿⣿⣿⣧⠘⠄⠐⣦⠄⠐⠃⠀⠀⣿⠀⠀⠀⣴⣄⠀⠀⢰⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠁⠀⠀⣘⣿⣿⣦⠀⠀⠀⢸⡀⠀⣠⣾⣿⣿⣿⣿⣿⣦⡀⠠⠿⠄⠀⠀⠀⣼⡟⠰⠖⠀⠉⠀⠲⠦⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠇⠀⠀⠀⠈⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠄⣰⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⢀⣀⣤⠤⠴⠶⠶⠶⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⣰⣏⣤⠶⠞⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⢦⣄⡀⢸⡇⠀⠀⢀⣠⣤⠴⠶⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠶⠶⢦⣤⣄⣻⡄
⠀⠀⠀⠀⣰⡿⠧⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⣤⣤⣄⣈⣙⣾⣧⣶⣾⣭⡥⠤⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠶⠶⠾⠿⣿
⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⢀⣜⣼⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⣿⣿⣿⣿⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣡⡀
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠛⠛⠛⠛⠛⠁
⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠿⣿⠿⠿⣿⣿⣿⠿⠿⠿⣿
⣿⣶⣶⣶⣶⣶⣶⣷⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠋⠙⠋⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣾⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣛⡛⠀⠀⢀⢀⣛⡛⠿⣿⣿⡇⠀⠀⢀⡀⠀⠀⠀⠀⣀⡀⠤⠴⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⢿⣿⣿⣿⣿⣿
⣋⠉⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⢉

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⡟⠛⠁⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢀⣛⣶
⠀⠀⢀⡄⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣾⣿⣿⢻
⠀⠀⠀⠳⡀⠀⠘⠛⠋⠀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣷⣦⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠈⠟⠋⠀⢘⡆⠀⠀⠀⢀⡀
⠀⠠⡈⢆⠱⠴⣒⣒⡚⢊⠴⢒⡈⠥⠴⠀⠀⠀⠀⠀⣀⣤⣤⣠⣤⣤⡀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠠⣄⠀⠀⠒⠉⠀⠀⠀⠀⣸⣿⣷⣶⣶⣤⢗⣤
⠀⠀⠱⣈⣻⠭⠥⠤⠭⣥⠞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⡿⠿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣛⢯⠅⣠⡄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠙⠛⠉⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣘⠷⢟⡻⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠐⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡏⣿⢛⢛⣯⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡧⠀⠀⠀⠀⠀⠀⠸⡿⠿⠧⠉⠿⠃⣿⣋⡬⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⠀⠀⢀⣥⡀⠀⢸⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡀⢺⢻⣿⡿⡗⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠋⢻⢸⣿⣇⡇⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣩⣔⣚⣿⣭⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠈⠉⠉⠉⠉⠉⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠁⠀⠀⠀⣀⣀⣀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣿⣿⡷⠶
⢰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⢟⣛⣛⠿⢛⣛⡻⠿⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⣿⣿⣿⡇⠸⠄
⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⡟⠛⡟⠛⢻⠛⢻⠛⠛⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣦⣖⣤⣭⣉⣥⣴⣶⣷
⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠒⢳⠒⡗⠒⢺⠒⢺⠓⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣼⣿⣯⣽⣿⣋⣭⣴⣿
⠐⠐⠒⠒⢲⣿⣿⠒⠒⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⡚⢗⢻⠒⡟⡲⢛⡲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠘⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠛⠛⠛⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠋⠉⠙⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠀⠄⠤⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣤⣴⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⢠⣾⣿⣿⠿⠛⠛⠛⠻⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠿⠿⠛⠻⠿⣿⣿⣿⣆
⠀⠀⢀⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄
⠀⠀⢸⣿⣿⡀⠀⠀⠀⡀⠀⠀⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⠀⠀⠸⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃
⠀⠀⠀⠹⣿⣿⣿⣷⣦⣤⣤⣀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠏⠀⢀⣤⣤⣤⣴⣾⣿⣿⠏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣫⣴⣾⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⡿⠿⠿⠿⠿⢿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⣿⡿⠗⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣷⣄⠀⠀⠀⣤⣀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⣶⠀⠘⣿⠀⠀⠀⠈⠻⣦⣠⣾⠏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢀⣤⡿⠿⣷⣄⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⠃⢿⡆⠀⠀⠀⠀⣽⠀⠀⣿⠀⠰⠾⠟⠉⠀⠀⠈⠙⠻⠶⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣾⣯⣀⣨⣿⡄⠀⠀⠀⢻⡇⠀⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⣸⡿⠉⠉⠉⠹⣷⠀⠀⠀⠀⠀⠀⣿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠻⣶⣤⡀⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡧⠀⣀⣴⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠛⢉⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡵⣾⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠾⠱⠿⠏⢺⣿⣏⣼⠋⢹⣯⡉⣯⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⢀⠴⠒⠿⠛⢁⣽⣯⠀⠀⠀⡛⠟⠚⢆⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡼⣶⢁⠀⠄⡤⡸⠈⣻⣦⡀⢄⠹⠆⣀⣀⡹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠿⡀⢻⠀⢶⢋⣽⡿⠯⢭⣟⠿⠖⡺⢿⡝⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠠⣿⣿⣤⠾⣸⠏⠀⠐⡭⠛⠛⠋⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⠁⠙⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣔⡐⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠁
⠀⠀⠀⠀⠀⢀⡴⠀⠀⠈⠐⠂⠀⠀⠈⠂⢤
⠀⠀⠀⠀⡴⠡⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⡀
⠀⠀⢀⠎⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⢀⠔⠀⠀⠂
⠀⠀⡎⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠰⠃⠐⠀⠀⠀⠀⣠⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠓⢠⠀⠀⠀⠀⠀⠀⠀⠐⡀⡄⠘⠀⠀⠀⠀⠐⠀⢀⠔⠁
⠀⡆⡠⠀⠀⠀⠀⠀⠀⠠⠼⠀⠈⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠢⠤⠄⠀⠀⠀⠈⢁⠂⠀⠀⠀⠴⢀⡔⠁
⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠠⠁⠀⠀⠀⠀⠀⠄
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠂⠠⠘⠂⡀⠁⢦⣀⠴⠤⣀⡀⠀⠀⠈⠀⠀⠀⠀⠨⠡⠀⠀⠀⠀⢀
⡄⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠀⡀⠕⡊⠁⠁⠀⠀⠀⢼⠀⠀⠁⠀⠀⠂⠀⠉⠁⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠈⠀⠀⠀⠄⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠠⢤
⠘⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠊⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐
⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⡠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠘⠒⠀⠀⠄⠀⠀⡀⢀⡠⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠲⠤⠄⢀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⢀⣀⣠⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠂
⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢤⣄⡠⠄⠀⡤⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆⣴⣰⣆⣦⣰⣄⡀
⠀⠀⠀⠀⢠⠖⠒⣒⢒⢒⠒⡒⡒⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠒⠒⠒⠒⠒⠒⠲⡄
⠀⠀⠀⠀⢸⠀⠀⢻⣏⡙⠋⣿⡏⠀⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⣗⢘⡃⣚⠐⡗⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡟⢻⣆⢻⣧⣿⠃⠀⡇
⠀⠀⠀⠀⢸⠀⠀⣼⣯⠁⠀⣿⣇⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⣹⡁⣭⢨⡅⣭⢈⣍⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠸⠷⠼⠋⠼⠏⠿⠦⠀⡇
⠀⠀⠀⠀⠈⠳⠶⠶⠶⠶⠶⠶⠶⠶⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠴⠄⠶⠰⠆⠴⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠶⠶⠶⠶⠶⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⣾⠆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠈⠻⠆⣨⡤⢤⡀⢋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢠⣤⣤⣤⢠⣤⣤⣤⠙⡆⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⢿⠿⣿⡿⠿⣿⠿⣿⡆
⠀⠀⠀⠀⠀⠈⣓⠠⡏⢰⠀⣹⠈⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢙⠿⢶⣭⢸⣿⠶⡁⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⣸⡷⢠⡧⠌⢻⡇
⠀⠀⠀⠀⠀⠉⠋⣤⠙⠒⠚⠁⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠛⠛⠛⠋⠘⠛⠛⠛⣠⠇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣶⣿⣿⣷⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠀⢸⠇⠈⠇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠟⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠠⠀⣀⢀⡀⡀
⠀⠀⠀⠀⢀⡴⠶⠶⠶⠶⠶⠶⠶⠶⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠉⠉⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣦⣤⣩⣤⡆⠉⣠⠄
⠀⠀⠀⠀⢸⠀⢲⣦⣰⡖⣰⡖⢶⡄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣾⢿⣿⣿⣿⢿⣇⠠⠄
⠀⠀⠀⠀⢸⠀⣼⡝⢿⡇⠻⣧⣼⠇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠙⢿⣶⣶⣶⣶⣾⡿⠐
⠀⠀⠀⠀⠘⠦⠤⠤⠤⠤⠤⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡌⠿⠿⠿⠻⠏⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠘⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀⢀⠀⠀⡄
⠀⠀⠀⠠⠀⠀⠦⠀⠀⠠⠀⠢⠌⠀⠈⠉⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠉⠈⠁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⠀⠁⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⠠⠁⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣤⠀⣿⠀⠀⠠⣀
⠀⠀⠀⠀⠰⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠁⠀⠀⠰⠀⠀⠀⠀⠀⣿⡇⠀⣿⠄⣿⠀⣶⠄⣿
⠀⠁⠀⠀⠀⠀⠀⣠⡄⠀⠑⢀⠔⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠉⡉⠁⡉⠉⡍⠉⡋⠐⠂⠂
⠀⠰⠤⠀⠀⠀⠈⠁⠀⢀⠐⢁⠔⠠⠀⡄⠀⠀⠀⠀⠀⠀⠐⣉⠀⠢⠘⠃⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠤⡀⠀⠀⠉⠄⠀⠀⠂⠁⠀⠀⠀⠐⠒⠀⠉⠁⠀⠠⣄⠀⢠⠄⠉⠂⠈
⠀⠀⣀⠀⠀⠐⠐⠀⠀⠀⠀⠀⡀⠀⡀⣢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⡀⠐⡀⠀⠈⠁⠀⠀⠁⢦⣤⣄
⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠅⠀⠄⠂⠀⠀⠀⠀⠀⠀⢠⠂⠐⠀⠢⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣟
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠷⢴⡤⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⠘⠂⢀⣨⣥
⠀⠀⠀⠀⠀⠀⠄⠈⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⠀⠀⠀⠀⠘⡃⠵⠀⠀⠀⠀⠀⠣⠀⠔⠉⠙⠉
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⢨⠌⠀⠀⠀⠠⢄⠀⡆⢠⣄⠀⡄⠀⡀⡄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠈⠀⠀⠁⠀⢀⡁⡙⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠣⠀⠀⠀⠀⠀⡄⢀⡄⠀⡆⠓⡀⢀⠔⠐⣠⠀⠀⠀⡡⠊⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⡀⠀⠀⠀⠈⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠓⠺⠁⠀⠃⠀⠀⠘⠛⠛⠋⡀⠀⠊⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⢀⢡⠀⠀⠀⠀⢰⣤⣼⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡀⠀⣀⠠⠀⣤⠠⠀⢠⠀⠄⠄⠰⠀⠆⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠐⠉⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄
⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⡠⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⡇⠀⠀⠀⠀⢀⠀⠁⠉⠉⠉⠉⢉⡉⠉⠉⠁⢀⠁⠉⠉⠉⠁⠐⢤⠀⢸⢠⠀⠀⣀⡀⠀⠀⠈⠁⡈⠀⡀⠀⠈⠉⠁⢀⠀⠀⠉⠈⠉⠑⡀
⡇⠀⠀⠀⢣⠈⠁⡀⠁⠁⠀⢈⡁⡈⢈⣁⡀⠈⡁⠀⢉⠀⠁⠀⠈⠀⢸⠸⠀⠈⣀⢁⡈⠀⠁⠀⢀⣀⣉⢀⡀⠉⣀⠀⡀⠀⣈⡀⣀⠠⠁
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⢄⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⡎⢉⢁⡀⣉⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠃⣀⡠⣤⣄⡄⠀⠀⠀⠀⠀⠂⠀⠠⠄⠀⠀⠀⠇
⡆⠐⢀⠤⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢸⠀⠀⠠⢾⢔⡚⠋⢈⣠⠀⠀⠀⠀⠀⠄⠀⠐⠂⠀⠀⠀⡀
⡆⠀⠛⢀⢤⡤⣴⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⢾⣹⡾⠇⠀⠘⠋⡗⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⠉
⡇⠀⣾⣿⣨⣙⣿⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣿⠓⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠼⡿⣿⣿⣿⣦⡀
⡇⠨⣿⡏⣷⠋⠁⠺⠛⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣽⣷⣦⣶⠒⠀⠀⠀⠀⠀⠀⢠⡀⠙⠃⢸⣿⣿⣿⣧
⡇⠀⠙⡧⠙⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣶⣶⣤⡀⠀⠀⢸⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢠⣿⣿⣿⣿⡟
⡇⠀⢀⣿⣦⣀⠠⠦⠤⠀⠀⠀⠀⠀⠀⠀⠐⠾⠁⢹⣿⣿⣿⣿⡆⠀⢸⠀⠀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠑⠺⣍⣻⣟⡉
⡇⢠⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣀⠀⢸⣿⣿⣿⣿⡇⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄
⣧⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡿⣿⣿⣿⠿⠁⠀⢸⠀⣸⣿⣿⣷⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣆⠀⠀⢸⠀⣿⣿⣽⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇
⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆⠀⢸⠀⣸⣿⣿⣿⣿⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣽⣿⣿⡀
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡆⢸⠀⣿⣿⣿⣻⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣼⠂⣿⣿⣿⣿⣿⣿⣿⣏⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣧⣿⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⢸⣿⣿
⠿⠿⠿⠿⠿⠛⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠸⠿⠿⠿⠟⠿⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠘⠿⠿⠿⠇⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇

⠀⠀⠀⠀⠀⠀⠀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⢀⣤⠀⠀⠀⡋⣰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣿⡿⠠⣤⣀
⡤⠚⠉⠀⠀⠀⣾⣿⣟⠛⣶⡀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣇⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⣸⣿⠿⠋⠛⡛⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠢⣀⠀⡀⣀⣀⣀⣀⡀⢀⣾⣿⣿⣿⣿⡗⠀⠀⡀⠀⠀⠀⡇
⠀⣄⠀⠀⠀⣈⣁⣤⣾⣷⡀⠞⣶⢿⡗⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣿⣿⣿⡽⣿⣿⣦⢛⣿⣿⣿⣿⣥⢀⣼⡻⣦⡀⠀⡇
⣀⣼⣷⣶⣿⣱⣿⣯⣿⣿⣷⣿⣛⣳⣧⣯⣿⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣿⣾⣿⣽⣿⣣⣟⣻⣿⢿⣿⣷⣿⡶⢿⣿⣦⡇
⣿⣿⣅⣽⣭⣿⣏⣻⣭⣩⣿⣿⠿⡿⣿⣿⣿⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⢸⣯⣩⣿⣭⣭⢼⣿⠿⣿⣛⣷⣴⣟⡿⣾⡙⠻⣿⣿⡇
⣽⣿⣯⣿⣛⣿⣝⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣧⣽⣿⣞⣛⣺⣿⣿⣝⣛⣿⣿⣻⣿⣝⣹⣟⣿⣿⠇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠉⠉⠉⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⣤⣾⣿⣿⣷⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣦⣄⡀
⠀⠀⣰⠟⠁⠀⠀⠈⠉⠙⠻⠿⣿⣷⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠛⠉⠉⠈⠉⠙⠳⣄
⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣾⣿⠋⠉⠉⡉⠛⠾⣓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣆⣠⡤⠶⣶⣷⣦⣀⡀
⠀⠀⠀⠀⠿⡃⠀⣾⣿⣿⣷⡄⢈⣄⠑⠔⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡷⢢⡿⠃⢠⣴⣶⣤⡀⠙⡙⣿⣄
⠀⠀⠀⠀⠸⢗⠢⡟⠟⠿⠿⠴⡾⠿⠓⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣶⡿⢇⡀⢻⣿⣿⣿⠇⣀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠉⠉⠉⠛⠚⢿⠻⠺⠚⠁⠨⠂
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠒⠒⠦⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⡄⠀⠀⠩⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣲⠟⢻⡆⠀⡴⠶⠶⡷⠶⠶⠖⠋⠁⠸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠋⠀⠀⠻⠀⡇⠀⠀⡇⢀⣀⣰⣄⣀⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢀⣤⠖⠛⠛⠛⡟⠛⠛⠁⠀⠸⣄⣰⠏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⠛⠙⠒⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠫⣄⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡞⡨⡽⡸⢹⠸⡅⡎⡬⡇⡏⡮⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⠁⠀⠉⠁⠁⠉⠁⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⢀⡀⠀⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣤⣤⣤
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡿⠛⠉⠉⠻⣿⠄⠀⠀⣾⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠⣶⣤⠀⠙⠀⠀⠀⠀⠀⣿⣈⣿
⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠰⡇⠀⠰⠀⠀⣹⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠐⠃⠈⠛⠋⠀⣤⡆⠀⠀⣴⠶⣿⣿⣿⢶⣦
⠀⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⣱⢤⣀⣀⢴⡇⠀⠀⠀⢻⣻⣟⣻⣟⡟⠀⠀⠀⠀⢴⣦⡄⠀⠛⠁⠀⠀⣿⣤⣿⣿⣯⣿⡿

⠀⣠⠴⣶⢲⡶⣄⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⡖⠒⠒⠒⠒⢲⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⢀⣶⣶⣆⠰⡆⠶
⢰⣛⣶⣿⣿⣷⢛⡇⡇⠀⠀⠀⠀⡟⢏⠀⠀⠀⠀⠀⡧⢤⡤⢤⡬⣼⠀⠀⠀⠀⡎⠙⠧⠀⠀⠀⠀⠀⢰⣿⣟⣿⡀⢠⠂
⠈⠻⣿⠛⣿⣏⡿⢰⠃⠀⠀⠀⣸⠁⠸⡄⠀⠀⠀⠀⡷⣿⡷⢾⡷⢾⠀⠀⠀⠀⠘⠢⡤⠀⠀⠀⠀⠀⣺⣟⣖⣿⡆⢸
⠀⠀⠙⠶⠛⠋⠀⠘⠀⠀⠀⠀⠃⠀⠀⠃⠀⠀⠀⠀⠳⠿⠷⠾⠷⠞⠀⠀⠀⠀⠲⠾⠿⠷⠀⠀⠀⠀⠛⠓⠒⠚⠓⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠰⡄⠠⠀⠀⠀⠀⠀⠀⢷⡀⣠⠏⠀⠀⠀⠀⡥⠀⣤⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⢀⠃⠙⠆⠀⠀⠀⠀⠀⠀⠀⣻⣇⠀⠀⠀⠀⡜⡥⡔⠓⢃⡀⣌⡅⠀⠀⠀⠀⠿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡇⣏⢱⠀⠀⠀⠀⡐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⣠⡤⢤⡀⠀⠀⠀⠀⠀⠀⡀⠴⢦⡄⠀⠀⠀⢀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⣤⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡃⣴⡄⢹⠀⠀⠀⠀⢠⣾⣿⢿⣦⡁⠀⠀⠀⢸⢻⡿⡄⠀⠀⠀⠀⠀⢀⢈⣱⣿⡿⠸⡄
⠀⠀⠀⣼⣿⣷⡀⠀⠀⠀⠀⠈⣳⣦⣴⡋⠀⠀⠀⠀⠘⢿⣯⣿⣿⠃⠀⠀⠀⢸⢸⡇⠹⣆⠀⠀⠀⠀⠘⣿⡙⢻⣷⣦⠁
⠀⠀⠼⠿⠿⠿⠷⠀⠀⠀⠀⠀⠛⠏⠻⠓⠀⠀⠀⠀⠘⠶⠬⠉⠁⠀⠀⠀⠀⠸⠽⠷⠤⠼⠧⠀⠀⠀⠀⠀⠱⠟⠋⠁

⠀⢀⣤⣄⢰⣿⣿⣆⠀⠀⠖⠂⠀⠒⠒⠒⠒⠂⠀⠀⠀⠀⢰⠀⠁⠀⠀⠀⢠⣤⣤⣦⣤⣦⣤⣤⠀⢀⣠⣴⣾⣿⣶⣦⣄
⠀⣾⣿⣿⣿⣿⣿⡿⠀⠀⣋⣉⠉⢁⣀⣉⣉⡁⠀⠀⠀⢀⡇⢸⠃⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠈⣿⠟⠛⠻⡟⠁
⠀⠹⣿⣿⣿⣿⡿⠁⠀⠀⠅⠀⠤⠤⠤⠤⠤⡅⠀⠀⠀⠘⠣⠾⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠈⠙⠛⠋⠁⠀⠀⠘⠓⠒⠒⠒⠒⠒⠒⠛⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⠀⠀⠸⣿⣿⠀⠀⢰⣿⡆⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⣿⣿⣦⡀⢸⢋⣥⣤⣬⣭⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣾⣿⣿⣿⡇⢸⣆⠛⠿⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣶⣤⣶⡿⠷⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠇⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⠿⠛⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣤⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣼⠇


⣏⣈⣯⣭⣟⣯⣿⣹⣽⣻⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡀⠀⡠⠊⠡⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠒⠂⠂⢡⠀⠈⠁⠀⠀⠀⠈⠁⠁⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠒⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠤⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠂⠀⠃⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣀⣀⠀⠀⠀⠀⡠⠒⢄⡀⠄⠒⠄⡀⣀⠠⠀⠄⡀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⢀⢴⠃⡠⠊⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⠀⠰⠢
⠀⠀⠀⠀⠀⠀⠀⡈⣨⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⣆⠆
⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢀⣀⣀⣀⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠠⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⣀⣀⣀⡀
⠀⠀⠀⠓⠚⠉⠛⠛⠛⠿⠷⠶⠶⢶⣦⣤⣤⣤⣀⣀⣀⡀⢀⣀⣀⣠⣤⣤⣤⣴⣶⠶⠶⠾⠿⠛⠛⠛⠉⠁⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠉⠀⠀⠉⡏⠉
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡐⠈⠐⠃⢁⡼⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡈⠙⠃⠁
⠇⠀⠀⠀⠀⠠⠴⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠦⠄
⠘⢄⠀⠀⠀⠐⣚⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣓⠂⠀⠀⠀⡠
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⠀⠠⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠀⠠⠄⠨⠁⣈⣁⡀⠥⠠⠄⠀⠒⠂⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⠛⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣛⣶
⠀⠀⢀⠄⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣾⣿⡿⢻
⠀⠀⠀⢳⡀⠀⠘⠛⠋⢀⡀⠠⠀⡀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣦⣄⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠟⠋⠀⢸⡄⠀⠀⠀⢀
⠀⠠⡈⢆⠱⣖⣒⣒⣛⣩⠖⣊⠭⠥⠂⠀⠀⠀⠀⢀⣤⣤⣄⣤⣤⣄⠀⠀⠀⠀⠀⠉⠀⠂⠀⠠⣆⠀⠐⠀⠉⠀⠀⠀⠀⣼⣿⣿⣶⣶⣼⣧⡄
⠀⠀⠱⣈⡯⠵⠒⠒⠬⠵⠊⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠟⢿⠟⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠠⢾⣿⡦⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣻⠿⢁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡙⣾⣉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠂⠐⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⣎⡝⣿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠄⠀⡀⠀⠀⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡧⠀⠀⠀⠀⠀⠀⠸⠟⠛⠂⠘⠛⠀⣿⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⣀⣤⣤⣀⠀⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣄⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⣿⡈⡿⣿⣿⣿⣁⣿⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⠟⢁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣿⣿⣿⣽⡂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⣴⣾⣭⣷⣾⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⣠⣤⣤⣄⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡋⣡
⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣻⣯⣭⣯⣝⣩⣿⣭⣽⣋⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⢘⠿⠿⠿⠿⠁⣙⡀
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣸⣀⣸⣀⣈⣇⣀⣇⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢻⣶⡶⣶⣶⣾⠿⢿⡇
⠀⠀⣀⣀⢰⣿⣷⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣈⣅⣸⣀⣨⣃⣸⣁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣶⣿⣿⡇
⠈⠀⠀⠀⢹⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⣪⣇⣼⣔⡡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠻⡿⠿⣿⣿⠿⠟⠛⠁
⣤⣤⣤⣤⣤⣽⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⡠⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⢠
⠀⠀⣧⡀⠀⠀⠀⢀⠜⠛⡒⢴⣶⠄⢸⠀⠀⠀⠘⠀⠀⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠈⠀⠀⠠
⠀⠀⣧⠹⣄⠀⠀⢎⠐⠰⠀⠀⡇⠀⠈⠦⣀⡠⢾⣄⠀⠀⣿⣿⣿⣏⣀⣀⣀⠀⠄⠀⠀⠀⠢⠂⠀⠀⠀⠀⠡
⠀⠀⢿⠀⠈⠂⠀⠚⠢⢤⡤⠊⠀⠀⠀⠀⢀⡀⠀⠙⢷⡄⠉⠉⠛⠿⠿⠿⡏⠀⠀⠀⠀⠀⠀⠁⡀⠰⢀⠀⠐
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⣁⠀⠀⠀⣠⣤⣄⣀⡀⠀⠀⠾⠃⠀⠀⠀⠀⠀⠀⠀⠌
⠀⢀⠀⠀⠀⠀⠀⣴⠛⢲⠀⠀⠀⠀⣾⣿⣿⣿⡇⢀⣾⣿⣯⣿⡟⠁⡀⠀⠀⠀⣠
⠀⠹⣿⣷⣶⣦⠀⠙⠶⢾⡶⠶⣄⠀⠘⠻⠿⠟⠁⠀⢰⣿⣿⡿⠿⠂⠷⠰⡦⠀⠀⠶⠀⠀⠠⠂⠀⠀⠀⠐⠄
⠀⠀⢈⠏⠉⠀⠀⠀⠀⢿⣄⣀⡿⠀⠀⠀⠠⠲⠤⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠈
⠀⠰⢾⡆⠀⢴⠞⢳⡀⠀⠈⠉⠀⢠⣿⣶⣦⠀⠀⠹⡄⠠⠀⠠⣇⣠⠀⢀⠀⠀⠀⠀⠂⠄⠀⠀⠀⠀⠐⢀⠂
⠀⠀⠀⠀⠀⠙⠳⢟⣵⠀⠀⠀⠀⠀⠻⣿⡟⠀⢾⡤⡇⠀⣨⠒⣿⣿⠒⠁⠀⠀⠆⠀⡀⠀⠀⠀⠀⡁
⠀⠻⣷⡄⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠑⢌⣛⣀⡸⠟⠀⠀⠀⠀⠁⠈⠙⠐⠀⠀⠀⠀⢀⠀⠀⠀⠘⠿⣷⣤⡀
⠀⠀⠈⠀⠀⠀⠈⢿⣿⠀⠀⠀⠀⠀⠀⠀⢉⣽⣤⠀⠀⣤⠀⠀⠈⠁⠀⢀⠀⠀⠀⠈⠩⠀⠀⣶⣄⡀⠈⠙
⠀⠀⠀⠀⠀⢀⡄⠀⢀⠤⡤⢠⡀⠀⠀⠀⠀⠀⣀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠉⠻⠂
⠀⠀⡀⠀⠀⠀⠉⠀⠈⠛⠁⠸⡽⠀⠂⠀⠀⠀⠈⠠⢿⡟⠹⠄⠀⠀⠀⣴⣶⡄⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⢲
⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣄⠀⠀⡄⠰⠆⠀⠀⠀⠀⠒⠦⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠳⠤⠤⣀⣀⡀⠀⠀⠀⡾⠀⠀⢀⡀⣀⠀⠀⠀⢀⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠠⣤⣿⣦⠤⠈
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠚⠃⢀⢀⣧⡄⠐⢿⣦⠘⠁⠀⢀⡾⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⢿⡷⠀⣠⠴⣄
⠀⠀⠀⠀⡀⠀⠀⠀⢠⡀⠀⢈⠙⠿⠇⣀⣀⠀⠃⢀⣤⠉⠀⠀⠀⢩⡅⠀⠀⣴⣦⡀⠀⢀⢠⡞⢁⠴⣩⣦⡀
⠀⠀⠀⠈⣀⡄⠘⠁⠈⠀⠠⠈⠶⣄⠀⠿⡿⠊⠀⠈⠀⠀⢠⡄⣄⠶⠀⠀⠀⠉⠛⠁⠀⠀⠀⠙⠹⣾⣿⣿⠛
⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⣴⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀
⠀⠀⠀⠀⠀⣿⠀⡧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⡄⠈⠳⡄
⠀⠀⢠⡴⣄⣿⠀⠇⠸⠉⡟⢦⠀⠀⠀⠀⠀⠀⡇⠀⠙⣆⠀⠀⠀⠀⠀⠀⢸⡇⠐⠲⡗⠒⠀⡿
⠀⠀⠀⠳⣌⠛⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠳⣄⣀⢁⣀⣼⣇
⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡇⢀⡴⠶⠶⠾⠗⠀⠀⠀⠀⠀⠈⠉⠉⠁⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢸⣄⣀⣀⣀⣰⠃⠀⠀⠀⠀⠀⠀⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛



⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣿⢰⡆⣤⢀⡀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠰⠿⣿⡄
⠀⠀⠀⠘⢶⣀⣿⣾⣿⣿⣾⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠘⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⣀⠀⣄⣀⣈⡀⠀⠁⠀⠈⠀⠀⠀⣈⣀⡤⡰⠂⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠈⠳⢦⣤⣭⣍⣛⣢⡄⠀⣐⣙⣥⡍⣴⠚⠁⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⠑⠀⠀⠉⠀⠈⠁⠈⠳⠋⠉⠁⠀⠀⠀⠙⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⡀⠀⠁⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠼⠀⠀⠀⠀⠀⢀⡀⣀⠀⢼⡀⠀⡀⠀⠀⠀⠀⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡄⠠⠡⡀⠠⡏⠠⢄⡿⠚⢿⠈⡄⠑⡄⢀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡀⠚⠦⣌⠊⠴⢓⢴⣛⠡⣀⠿⢿⣶⠚⡁⠁⢠⢈⡮⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠒⠶⢖⡛⢷⣾⣷⣄⠀⡈⠳⡤⠤⡤⠠⠄⠉⠓⠤⠤⠤⠤⣚⢃⣡⠿⠖⠂⠊⣁⠤⠓⠂
⠀⠀⠀⠀⠀⢀⠀⠀⠀⣨⠉⠓⠓⠤⠍⣈⠉⠉⠓⠒⣶⣦⡠⠶⣷⠷⠟⢛⡿⣿⠥⠶⠒⢊⡉⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠐⣚⣭⡍⠀⠀⠀⠀⠀⠀⠀⠉⠉⠺⢭⣀⣁⣁⣠⠐⢉⡡⠁⠀⠀⠀⠀⠀⢛⠒⠂⠠⠀⠁
⠀⠀⠀⡔⠀⢐⡩⢴⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⡐⠒⠆⠀⠂
⠀⠀⠀⠈⠄⢡⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠉⠂⠈
⠀⠀⠀⠀⠀⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠤⠴⢒⡟⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠙⢤⡀⠀⠁⠐⠚⢄⡀⣠⠴⠐⠈⠀⠀⣠⠴⠭⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⣭⣮⢐⣦⣄⢀⢀⣀⠀⡀⣀⣀⣤⣾⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠗⠹⠕⠋⠁⠀⠀⠀⠀⠀⠀⠉⠉⠻⢽⡟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠐⠊⠁⠀⣀⠠⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠊⠁⠀⢀⣠⠤⠒⠉⠀⠀⠴⠂⠀⡀⠀⠈⠒⠤⣀
⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠀⠀⣀⠤⠒⠉⠁⣀⠤⠒⠉⢀⣀⠤⠒⢉⡀⠤⠂⠀⢀⠀⠈⠓⠢⣄
⠀⠀⠀⠀⢘⡯⢄⡀⠀⠀⠀⠒⠉⣀⠠⠔⠊⠉⣀⠤⠒⠊⠁⠠⠔⠊⠁⢀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠸⣟⠒⢍⡒⠤⡀⠀⠀⠀⠀⠠⠒⠉⣀⡀⠐⠊⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠐⠖⠀⠀⠀⠀⠀⠀⠀⠉⠲⠄⣀
⠀⠀⠀⠀⢘⡷⢌⡒⠬⣓⠬⣑⠤⡀⠀⠀⠀⠈⠁⢀⠤⠒⠉⠀⠀⠀⢀⡀⣀⠀⣀⣀⠜⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⢉⣲⠄
⠀⠀⠀⠀⠰⣏⡑⠬⣑⠮⣑⠪⣑⠬⡑⠦⣀⠀⠀⠀⠀⠠⢄⠀⠀⢀⠏⠀⠀⠀⠀⠀⣀⠤⠒⣉⠤⠖⠉⠀⠀⢀⡠⠔⣊⠥⢾⡁
⠀⠀⠀⠀⢨⡷⠬⣑⠪⣕⠢⣕⠪⣑⠢⣑⠢⢝⡢⢄⡀⠀⠀⠉⠒⠾⢄⣀⣀⠤⠖⣋⡥⠒⠋⠀⠀⢀⡠⠔⣊⠥⢒⡩⠔⣊⣽⠂
⠀⠀⠀⠀⢐⡯⣑⠢⣙⠢⢍⠢⢝⠢⣑⠢⣉⠲⢍⡒⢌⡑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣊⠥⠒⡩⠔⣊⠥⢒⡩⠔⣺⠅
⠀⠀⠀⠀⠨⣗⠢⣕⠢⢍⡢⢍⡢⢍⡢⢍⡲⢍⡒⢬⣑⠮⣑⠢⢍⡒⢤⣀⠀⢀⡠⠔⣊⡥⢖⣪⠵⣒⡭⢖⣩⠴⣊⠥⢖⣨⣽⡂
⠀⠀⠀⠀⢘⡯⢍⠢⢍⡒⢌⡒⢍⡢⢍⡲⢍⡒⠬⡑⠤⣑⠢⢍⡒⠌⣑⠬⣉⠤⢒⡩⠔⢂⡩⠔⣈⠤⢚⠡⠐⣊⠔⢊⡥⠐⣺⠅
⠀⠀⠀⠀⠠⣟⡲⢍⡒⠬⡒⢬⡓⢬⡒⢬⡒⠬⣑⠦⣕⠲⢍⡒⠬⡓⠮⣑⠒⣊⠥⢖⡫⠵⢒⡭⠔⣪⠵⢒⡩⠔⣫⠴⢚⡩⢽⡃
⠀⠀⠀⠀⢘⡧⢌⡒⠬⡓⠬⣑⠬⡓⠬⡓⠬⣑⠢⣑⠢⢍⡒⠬⡑⠬⣑⠢⠭⠔⢊⡡⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⠤⢒⣻⠄
⠀⠀⠀⠀⠰⣏⡒⠬⣑⠮⣑⠪⣑⠮⣓⠮⣑⠢⣕⠢⢍⡒⠬⣓⠮⣑⠢⢍⣒⡫⠴⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠖⣊⠥⢾⡁
⠀⠀⠀⠀⢨⡗⠨⣑⠢⢑⠢⢕⠪⣑⠢⣑⠢⢉⠢⢍⡒⠬⡑⠢⣑⠢⢍⡒⠤⠒⣈⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⣽⠂
⠀⠀⠀⠀⢐⣯⣑⠢⣝⠢⢝⡢⢝⠢⢝⠢⢝⠲⢍⡒⠬⣑⠮⣑⠢⢍⡒⠬⣛⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣚⠥⢒⡩⠔⣺⠅
⠀⠀⠀⠀⠨⣗⠢⣍⠢⢍⡲⢍⡢⢍⠢⢍⡢⢌⡒⠬⡑⠠⣑⠢⢍⡒⢬⡑⠤⢒⡩⠔⣊⠡⠒⣉⠔⢊⡡⠐⣉⠔⣊⠥⠒⡉⢽⡂
⠀⠀⠀⠀⢘⡯⢍⡢⢍⡒⢭⡒⢍⡒⢍⡒⢍⡒⠬⣑⠬⢑⠢⢍⡒⠬⣓⠮⣉⠥⢒⡩⠕⣚⠭⠒⡫⠥⢒⡩⠐⡫⠔⣊⠭⢒⣻⠅
⠀⠀⠀⠀⠨⣟⡒⢌⡒⢬⡑⠬⡓⢌⡒⢬⡐⠬⣑⠢⣑⠢⢍⡒⢬⡑⠬⣑⠒⣊⡥⠒⣉⠤⢂⡩⠔⣊⠤⢂⡩⠔⣉⠄⢒⡩⢼⡃
⠀⠀⠀⠀⠀⠀⠉⠒⠬⣑⠬⣑⠬⡓⠬⡑⠬⣑⠢⣕⠢⢍⡒⠬⣑⠪⣕⠢⠭⠔⣊⠭⠐⣊⠥⢒⡩⠔⣊⠥⢒⡩⢐⡫⠥⢒⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠪⣑⠬⣑⠮⣑⠢⢙⠢⢍⡒⠬⣑⠤⣑⠢⢍⣒⣉⠤⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠮⣕⠢⢝⠢⢍⡒⠬⡑⠪⢑⠢⢍⡒⠤⠒⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⣊⠥⢒⡩⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢌⡒⠬⣑⠦⣑⠢⢍⡒⠬⣁⡡⠔⣊⠥⢒⡩⠔⣊⠥⢒⣩⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣝⠢⢍⡒⠬⡑⠒⢚⡩⠔⣊⠥⢒⡩⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢌⣑⠬⣉⠥⢒⣩⠔⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠺⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⠉⢁⣰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠛⡙⢿⣷⣄⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣷⣦⠾⠿⠿⠿⠿⣿⣿⣿⣿⣿⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡤⠀⠀⠀⠀⣠⣤⠀⠀⠈⠀⠀⠀⠠⠂⢠⣀⣸⣶⣿⣿⣿⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡶⢚⣱⣿⠄⠀⠀⠀⡀⠀⠀⠀⣶⣺⠟⠻⠿⢿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡔⢡⣴⣿⠋⢀⠀⠄⠀⣀⣤⣴⣶⣷⣿⣿⣶⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢉⣾⡿⢻⣿⣷⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢶⠍⣡⣿⣿⠶⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠆⢰⣄⡠⣿⡿⣿⣶⣺⣿⣿⣯⣻⣿⡿⢷⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⢃⣰⣿⣷⠿⣿⡗⣿⣿⣿⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣀⣀⣀⣀⣀⣀⣀⣈⣿⣋⣛⣐⣻⣳⣺⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣛⣛⣹⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠈⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠄⠀⠀⠀⠀⠀⠠⠄⠀⠠⠀⠀⡀⠀⠀⠐
⠊⠀⠐⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⢦⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⡊⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠶⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠈⢀
⠀⠀⠀⣀⡀⠀⠀⠀⠠⠀⠉⠀⠁⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠋⠡⠀⡀
⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⠀⠀⠀⠀⠀⠂⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⣀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠐⠀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⡤⠞⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠈
⠀⠐⠀⢦⣀⡀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠁⠐⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠉⠙⠳⠶⡵⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⠒⠢⠤⠤⠢⠆
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠀⠀⢰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠔⠊⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⠄⠀⡠⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⠈⢿⣿⣿⣧⡀
⠀⠀⠀⠀⢀⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⡇⠈⣿⣿⣿⣿⣇⣀⣀⡀⠀⢀⣠⣿⣿⣧
⠀⠀⠀⠀⣼⣿⣿⣿⠀⢸⠟⢻⡟⠉⠛⠃⠀⠛⠙⣿⣿⡿⡋⣿⡏⣆⢸⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠈⠀⠀⠻⠀⣿⣿⠀⢀⣷⡸⡟⣼⣃⠛⣰⣿⢸⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⣿⣦⣼⣿⣿⡄⢸⣿⣷⣤⣽⣿⣿⣿⣿⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠿⠃⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡿⣿⡿⠿⣿⣿⣇⠻⠿⠷⠍⠿⢿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠀⠈⣰⣦⠘⣿⣿⡇⢰⣶⣶⡆⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⢿⣿⣧⠸⣿⡇⠸⣿⣿⡇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣦⡛⣰⣧⣙⣛⣡⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠿⠟⠛⠋⠙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠞⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡢⠂⠀⠀⠴⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⣀⡄
⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⣤⣄⡀⠀⣶⣯⠀⡏⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠙⠛⠁⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⢀⡀⠀⠀⠀⠀⡤⡀⠀⠀⠀⢠⠶⢦⠀⠀⠀⠀⢸⡇⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⢡⣀⠠⠀⠀⠀⠀⠠⠀⠀⣰⢆⠀⠘⢛⣉⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⡡⡴⠖⠀⠀⠐⠀⠀⠰⠛⠙⠆⠀⠀⠀⢀⠤⠤⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢟⠙⠻⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⡀⢸⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠀⢉⠀⠄⠀⡉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢴⣀⣄⣐⣀⣀⣀⡦⠴⢂⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣸⢀⠈
⠀⢀⠀⠴⠀⡄⠀⠀⠀⠀⠀⠠⠁⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠶⠀⣀⠀⠀⢠⠀⢠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣬⣥⣾⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆⠤⠎⠁⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠴⣶⣄⣀⣀⣀⣠⡄⢀⣠⠞⠁⢴⡄⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠉⠛⠛⠉⠁⠚⠉⠀⠀⠀⠺⣧⠀⠀⠀⢠⣤⡆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⢆⣿⠟⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠀⠀⠀⠀⠨⠋⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠶⢄⠀⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡴⡇⠀⠀⠀⠐⠲⠖⠋⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡆⠀⠀⠀⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣧⡀⠀⠀⠀⠀⣰⠁⢐⠢⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⠟⣿⣿⠋⠀⠀⠀⢀⡴⠃⢀⣾⣇⢀⡀⠉⠉⠉⠛⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡎⢧⡀⠀⠀⠀⣀⡴⠋⢀⣴⣿⣿⣿⣿⣗⣴⣦⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣦⣉⠓⠒⠋⠁⠴⠾⠛⠛⠛⠋⠉⠙⣿⣿⣯⣌⡆⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣠⣤⣴⣶⣷⣾⢿⣿⣿⣿⣿⣶⣆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡙⠻⡿⠿⢿⣿⡿⠟⠉⢀⣀⣤⣶⣶⣿⠿⠿⠛⠉⡉⠡⠿⢗⣿⣿⣿⣿⣿⡷⠃⣠⠀⠘⢆
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠀⠀⠀⢎⡉⠑⣸⣿⣿⣿⠟⢋⠉⠀⠀⠀⠈⠁⠀⠀⠉⠁⠸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⢻⣽⣿⣿⣛⠛⠛⠛⠛⠋⠙⠈⣿⣿⡟⠘⣿⣿⣯⡿⢷⣾⣯⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠰⠴⠭⠯⣛⣉⡁⠀⠀⠀⠈⢿⣿⣿⡁⢹⣿⣿⣿⣞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⠀⠙⣿⣿⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⣀⣠⡼⢖⠀⠀⠀⢸⠏⠻⣇⠠⢿⣿⡿⢛⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⣀⣀⣀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⢀⣠⡯⠗⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⢁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⡠⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣴⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⢻⣧⣤⠀⠀⠀⠀⠀⠀⠀⡠⣔⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢿⣿⣿⣿⣿⡿⣻⡏⡀⠀⠀⠀⢀⣴⣺⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣍⣿⣿⣻⣿⣿⣿⣷⣿⣷⡿⠃⠀⢠⣫⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⠟⣿⣿⣿⠟⣡⡄⠀⢠⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠁⣿⣿⠾⣿⣿⣿⣿⠟⠀⢀⣾⣿⣿⡿⠿⢟⠉⠉⠉⢻⣿⣿⣿⡿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠟⠉⢰⡎⠛⠋⠉⠀⢠⣿⣿⣿⣵⣾⡿⠭⣟⣹⣷⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢹⠀⠀⠀⠀⠀⠈⢿⢿⣟⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⠟⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⡿⢿⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣄⠀⠀⠀⠀⠀⠀⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣌⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⢛⣻⡇⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠙⠏⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⢫⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣇⠀⠀⠾⠁⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣱⣾⣿⡿⣿⣿⣿⣷⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣄⣀⣄⣠⣾⣿⣿⣿⣿⡟⠀⣿⣿⣿⣄⣿⣾⣷⣽⣯⡝⢻⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠃⠀⢸⣿⣿⣧⡌⣿⣿⣿⣭⡿⠀⠙⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣿⣿⣿⣿⣶⣿⣾⣏⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⢻⢀⣼⢻⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡗⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣶⣿⣿⣿⢸⣿⣿⣿⡹⠃⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣿⣿⣿⡇
⠀⠀⠀⢸⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⢸⣿⠟⠋⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⡿⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠙⠻⣿⡿⠿⠟⠟⠘⠁⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⢸⣿⣿⣿⣿⡇⠀⢠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⠁⠀⢸⣿⣿⣿⣿⣤⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠿⠟⠛⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⠀⢠⣴⠿⠛⠉⢿⣿⣿⣿⡇⠀⠀⠀⢀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⢰⣟⡁⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠰⣟⠉⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠙⡆⠀⠀⠀⠻⣿⣿⣿⣿⣄⠀⢉⣡⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠾⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣾⡾⠋⠁⠙⠃⠈⡇⢻⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⢻⣿⣿⣿⣿⠁⠀⠀⣼⠀⣸⠀⣣⡜⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣠⢚⣿⣿⣿⣿⣇⡞⠀⣰⠃⣰⠃⣠⢷⣷⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⡞⢡⣿⣿⣿⣿⣿⡿⣁⡼⢃⡼⢃⡴⢯⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡾⢠⣿⣿⢿⣿⣿⣯⣾⡯⠶⠛⣿⠟⠓⢈⡙
⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⠇⣾⢻⡏⣿⣿⣿⣿⡇⠀⠀⠀⣽⠿⠀⠙⡋
⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿⢰⡟⣼⢣⣿⣿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⠳⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⡇⢸⡇⣿⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⢸⠃⣿⠀⣿⢸⣿⣤⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠸⡀⢩⡄⠠⡆⠹⣿⣄⠀⠀⣶⣶⣶⣶⡶⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠈⢳⣄⢇⠈⢷⡄⠘⡀⢈⠹⣿⡆⢻⣿⣿⡇⠀⣠⠇⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠎⠈⣆⠈⢿⣄⣱⣤⣾⠟⠁⠀⠙⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠶⣤⣼⣷⣾⣿⠿⠟⠁⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠰⠯⢇⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠂⠴⣿⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣾⠁⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⣰⢦⢀⡀⠉⠂⠰⡇⣠⢦⢀⠀⠀⠀⠀⣰⠋⣬⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡀⠈⠳⢦⣀⠀⠀⠀⠈⡿⢶⣄⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠙⠋⠯⡼⡦⢢⡄⡀⠀⠘⠋⣾⠃⠀⣼⠃⣾⣿⠃
⠀⠀⠀⠀⠀⣀⣤⡾⢟⡱⣞⣿⣦⣀⠀⠈⠑⠒⠒⠋⠀⣸⡏⣻⣿⣶⣄⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠛⢫⠀⠀⠀⠀⠀⣼⢃⣾⣿⠃
⠀⠀⢀⣴⡟⠛⠿⢶⣧⣿⣿⣾⣿⠿⣿⠒⠀⠤⠤⠤⠴⠛⣧⢸⣾⣽⣿⣧⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⣾⣿⠏
⠀⠀⣿⠋⣰⡆⠙⠄⠹⣿⣿⣿⣟⡷⡘⣷⡀⠀⠀⠀⠀⠀⢻⡏⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢃⡟⣽⠋
⠀⢸⡏⢰⡉⢰⠿⣦⣋⢻⣿⣿⣿⣆⣧⠘⣷⠀⠀⠀⠀⠀⠘⣿⢹⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⡟⣾⠃
⠀⣾⡇⣾⢧⣾⣶⣿⡟⢹⣿⣿⣿⣿⣿⡄⢻⡆⠀⠀⠀⠀⠀⣿⡄⢸⣧⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣰⡟⡵⣸⠃
⠀⣿⣄⣿⢨⣿⣶⣿⣆⡟⢻⣿⣿⣿⣿⠃⠘⣧⠀⠀⠀⠀⠀⢸⡇⣉⣿⡠⠼⣿⡲⢦⣀⠀⠀⠀⠀⠀⣠⠦⣼⠙⢾⡏⢳⣴⡟⡐⢰⠏
⠀⢹⣿⣽⡎⢉⣭⡙⠉⡇⢾⣿⡿⣿⣿⡗⣧⣿⡄⠀⠀⠀⠀⢸⠟⠉⠀⢀⣴⣿⣃⡴⠞⢻⣶⣤⣀⣘⠙⡄⠘⡇⠈⢧⠀⢿⡰⢡⠏
⠀⠀⣿⣿⣿⣦⣽⡏⢹⠇⣼⣿⣧⣿⣿⣷⣿⣾⣇⠀⠀⠀⢠⠏⠀⠀⠀⠉⠉⠉⣁⣠⠖⣻⣇⡈⠿⣿⡆⢱⠀⠱⠀⠀⠀⠸⣆⠊
⠀⢸⣿⢸⣿⣿⣯⡄⢡⣤⣿⣿⣿⣿⡿⠵⢟⣁⣿⣤⠒⠲⡏⠀⠀⠀⠀⠀⠛⠉⣉⡤⠞⣹⢯⣿⣷⣾⡟⠀⠀⠀⠀⠀⠀⠀⡟
⠀⢸⡇⢿⣿⣯⣻⡇⠨⣍⣾⣿⣿⣷⡾⢷⠿⠋⠉⠙⢦⡀⢳⡄⠀⠀⠀⠀⠐⢉⣡⠔⣿⡽⣿⡟⣛⢻⡇⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠸⣧⠸⡉⠉⠹⣣⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⣧⠀⢓⠀⢀⣀⣀⣈⣥⣴⣾⠏⠑⢿⢩⡀⢯⠑⠦⢄⣀⣀⡴⠞⠁
⠀⠀⢹⣿⣿⣆⠸⡿⣿⣿⡿⠉⠀⠀⠀⠀⠀⢀⣠⣄⣀⢸⡆⢸⡟⢉⢉⣉⣽⣿⣿⣽⣇⠚⣍⠳⡔⠎⢳⣤⣤⠤⠋
⠀⠀⠀⣿⢼⣿⣧⣼⣿⣿⣄⣀⣀⣤⣴⡾⣿⣿⣿⣿⣿⢸⣿⣿⡇⠀⠀⠷⠸⣿⠿⣶⣭⣤⣤⣤⣶⡾⠋⠁
⠀⠀⠀⠹⣎⢿⣻⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣏⣩⣤⣾⠏⣽⡇⠤⠰⠆⠀⣿⡆⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠙⠂⠉⠙⠿⠿⡿⢿⣧⡼⠿⠿⠿⠟⠛⠛⠉⠿⠀⢿⠅⠀⠀⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣀⠀⢻⠀⢀⡄⢀⡀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣁⢀⡀⡨⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⠉⢇⠀⣂⡈⢀⠦⠟⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣀⠘⣄⡉⠉⠉⢁⣠⠏⠀⠀⠀⡏⢠⡀⣄⣠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣶⡄⠀⠀⠀⠀⠀⢾⣿⣷⠀⠿⠚⠉⠉⠀⠀⢠⠒⠒⢇⣈⣁⣁⡈⢁⠇
⠀⠀⠀⠀⠀⢴⡦⣰⣿⣿⡆⣶⠆⠀⠀⠀⡎⣶⢯⣥⣶⢹⡀⠀⠀⠀⢠⣼⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⠘⠛⠻⠛⠓⢸⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠻⠞⠿⠟⠵⠃⠀⠀⠀⠀⠀⠿⠦⡥⠼⠁⠀⠀⠀⠀⠺⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⢀⡷⠖⠉⠉⠉⠁

⠀⠀⠀⠀⢀⠔⢂⠉⡉⡀⠂⠄⠀⠀⢀⠀⠐⢄⠒⢒⠐⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠈⠢⣌⢀⣁⢁⡤⠊⠀⠀⠐⠄⢈⠔⠙⠁⢪⡉⠁⠂⠀⠀⡁⠠⠆⢈⣄⢀⣶⣾⣶⣄⠀⠀⠀⠠⠒⠰⠀⠠⣀⣸⣀
⠀⠀⠀⠀⠀⠀⣤⢄⡁⢴⡄⠀⠀⠀⠀⠀⢴⡶⣀⡰⣶⠀⠀⠀⠀⠀⡉⠨⠧⢉⠆⠰⣿⣿⣿⠟⠀⠀⠀⠘⢠⡦⠀⢠⡄⢸⡿
⠀⠀⠀⠀⠀⣰⡿⣻⣿⢸⣧⡀⠀⠀⠀⢀⣾⢇⣿⣇⢿⣦⠀⠀⠀⠀⡉⠡⠉⠈⠶⠀⠻⣿⣿⠀⠀⠀⠀⣸⣇⢇⣿⡸⣺⣇
⠀⠀⠀⠀⠀⠙⠱⣿⣿⡷⠙⠁⠀⠀⠀⠈⠉⠾⠿⠿⠎⠉⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠉⠉

⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⣠⠶⠛⠳⢦⡀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠆⣀⠤⠄⣀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⣾⠙⣧⣼⡱⢿⣷⠀⠀⠀⢰⡏⡼⣆⣰⡆⣷⠀⠀⠀⠀⠀⠈⡇⠀⠀⡞⠠⢰⠤⠊⡣⠀⠀⠀⣿⠈⣡⢤⣄⠉⡏⠈⡇
⠀⠀⠀⠀⠙⠾⢯⣤⡏⢽⣸⠇⠀⠀⠀⠻⣤⣀⣠⣼⣇⡀⠀⠀⠀⠀⠀⢹⠀⠀⠈⣢⡤⠤⠒⠁⠀⠀⠀⣿⠀⠻⣤⠞⠀⡗⠒⡏
⠀⠀⠀⠀⠀⠀⠀⠉⢳⡟⠋⠀⠀⠀⠀⠀⠀⠉⠁⠀⠻⣿⣦⠀⠀⠀⠀⠈⣇⣀⡠⣤⡆⠀⠀⠀⠀⠀⠀⠿⣴⣤⣤⣤⣤⣯⣭⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠘⠛⠃⠉

⠀⠀⠀⠀⠀⢰⣶⡶⢶⣶⣶⠀⠀⠀⠀⠀⠠⣽⣿⡿⢿⣿⠀⠀⠀⠀⠀⠀⣴⣶⣶⡶⣶⣶⠀⠀⠀⠀⠀⡠⠖⢒⡒⠐⡒⠒⡦⡀
⠀⠀⠀⠀⠀⢸⢉⡉⣉⣉⠉⡆⠀⠀⠀⠀⢸⠉⣩⠉⢉⠉⡇⠀⠀⠀⠀⠀⣧⣯⣤⣜⣷⣼⠀⠀⠀⠀⢐⡇⡁⠛⠃⠘⠧⠀⡏⡇
⠀⠀⠀⠀⠀⢸⠈⠡⠉⠍⠀⡇⠀⠀⠀⠀⢸⠀⠩⠁⠍⠀⡇⠀⠀⠀⠀⠀⣿⠷⢶⣶⣶⣿⠀⠀⠀⠀⢰⡇⡂⢿⠧⠀⠧⠇⡇⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡿⠀⠘⠛⠛⢛⠀⠀⠀⠀⠨⣷⣭⣭⣴⣶⣭⣭⣽⡇
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⢈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠠⠀⠈⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠉⠉⠉⠒⢄⠀⠀⠀⠀⠀⢀⠞⢁⡤⠤⠄⠀⣠
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣀⣀⡀⠀⠀⠀⠀⢣⠀⠀⠀⣰⠋⠀⠈⠀⠀⠀⠀⡟
⠠⠊⠉⠀⠀⠉⠉⠒⢄⠀⠀⠀⢣⡀⢀⡠⠴⠒⠒⠚⠧⣌⠉⠓⠠⢀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠒⠦⠀⠀⡆⠀⢰⠃⠀⠀⠈⠀⠀⠀⡰⠁
⢇⠄⡀⠀⠀⠀⠀⠀⠀⠉⠲⠄⠀⢧⠋⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠑⠢⣀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡴⠃⠀⠀⠀⠀⠀⠀⠐⠁
⠈⠢⡈⠑⠀⠀⠀⠀⠀⠀⠀⠈⠓⢢⣇⠐⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠈⠳⡼⠀⠀⠀⠀⠀⠈⠁⠀⢠⢃⠎⠀⠀⠀⠀⠀⠀⡀⡔
⠀⠀⠈⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠢⠀⠀⠀⠀⢈⠀⠀⢸⢧⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠂⠀⠀⠀⡶⠏⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⢂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⠀⠀⠈⠁⠀⠠⠋⠀⢷⣄⣀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠸⠙⣿⠃⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣐⣀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠀⠐⠒⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⡠⠾⡆⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⢀⠚⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⢤⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠠⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠀⠤⠄⣀⠀⠠⠀⠀⠀⡀⢀⣀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠂⠤⠀⠀⠀⢀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢸⣿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠀⠀⠀⢿⠿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠹⣿⣿⣦⠀⠀⠀⠘⣯⠟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⢦⠀⠀⠘⢿⣿⣷⡀⠀⠀⠸⣼⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣨⢷⣄⠀⠈⢿⣿⣷⡄⠀⠀⢳⡴⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⢉⣦⠀⠀⢻⣿⣿⡄⠀⠈⣧⣬⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠘⢳⣄⠀⢻⣷⣿⣄⣀⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠈⢳⣅⢞⣦⣀⣿⣿⣿⡿⣿⣿⡿⢿⣆⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⡀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⠛⢿⣆⠘⣿⣧⣀⠀⠀⢀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣄⡀⠀⢘⣿⣿⡿⢻⣿⣿⡄⠀⠙⢷⣞⣿⣿⣿⣾⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⡿⢉⡉⢻⣄⠻⡿⢷⣠⣷⣾⣿⢿⣿⣿⣿⣿⣿⣟⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⡄⠻⠄⣻⣦⣶⣿⣿⣿⣟⣸⣿⣿⣿⣿⣽⣳⣭⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣶⣿⣿⣿⣿⡿⣾⣿⣿⢿⠁⣙⠛⢛⠉⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⡱⣌⣶⣿⣏⣉⣹⣟⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⡴⣶⣶⡾⣿⣿⣹⣿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣞⡿⣿⣿⣿⣿⣿⢿⣿⣿⣯⣟⠿⣷⣿⣿⣷⣿⣿⣻⢿⣿⣿⣿⣿⣿⡟⠳⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣭⡻⢷⣯⣟⢾⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢟⣛⣻⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣙⠿⣿⣿⣿⣯⣗⢮⣝⠿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⣛⣛⢭⣭⣴⣶⣦⣿⣻⣯⣽⡟
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣮⣝⣿⣿⣿⣿⣾⣝⡶⣍⣯⣭⣭⠶⢶⣚⣿⣿⡿⣿⣻⣯⣽⣿⣿⣿⣿⠿⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣷⣭⣻⢿⣷⣽⣶⣾⣿⣿⣭⢷⣟⣿⣻⣿⡿⢿⣛⡯⢽⣒⣯⣽⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⠿⣟⣻⣭⣵⠶⠛⣩⡶⣾⣻⠯⠓⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣾⣽⡟⣻⣽⣷⣾⣿⢟⣋⣥⣤⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠮⣽⡿⣿⣿⣷⣝⠟⣭⣺⠽⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢾⣿⡿⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢻⠛⡟⣻⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣷⣶⣤⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⣿⣿⣿⣿⣾⣿⣿⣶⣦⣄⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠟⠿⣿⣿⣿⣯⠽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣀⡈⣤⣿⣿⣿⣿⣿⣿⣧⠤⢀⣴⣦⣴⣶⣷⣾⣶⡀⠀⠀⠀⠀⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣰⣿⣿⣿⣶⢿⣿⣿⡾⣿⣿⣿⠟⣿⢤⣄⢠⣛⡿⣿⢿⠹⣿⣿⣿⡇⠀⠀⠀⢠⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⣸⣿⣿⣿⣿⡿⣿⣿⣿⣷⡘⡻⢿⣄⣀⣰⡿⣿⢣⢸⣿⡇⣸⣿⢿⣿⡇⠀⠀⣴⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠫⣿⣿⣿⣿⡗⠀⣿⣞⣿⣻⣿⠿⢀⠉⠩⠁⠂⣿⣧⡸⣿⣷⡸⢿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⣿⣿⣽⢧⣟⣿⣿⣿⢯⣯⠋⠤⢀⡄⣠⣶⣿⡿⣷⡿⢿⣷⣚⣯⣧⡟⠀⠀⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣤⠶⣽⣿⡣⣞⡬⢡⡞⣻⣆⣶⣭⡭⠾⣻⡿⣟⣟⣙⢯⡟⣵⡟⠀⠀⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⣻⣟⣩⣥⣼⣥⣶⢦⣄⠷⢿⣟⣹⣻⢌⣿⢟⣻⣿⡿⢿⣯⣰⣦⡾⡷⠀⠀⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠿⠻⠟⣟⣻⡜⣩⢻⣽⣞⣿⣻⣟⡿⣿⣽⣿⡻⢾⣧⣾⣿⣿⣻⢟⡁⠀⠀⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣠⣿⣿⣿⢿⣾⡇⢿⣻⣾⣋⢾⣿⣷⣼⣻⣿⣿⡝⣿⣿⣿⣿⠎⣃⠀⠀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⣿⣟⣿⣯⣽⣾⣿⣿⣚⣭⣏⢻⡿⣼⣟⣻⢿⣿⣶⣧⣽⣟⣿⢭⡚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣻⣿⣿⣷⠝⣿⣿⣼⣾⣿⣾⣷⣿⣿⣮⣿⣷⣯⣸⣿⣿⣿⣿⡷⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣻⣿⢻⣿⣿⣿⣿⣿⢿⣿⣽⢿⣿⣿⣽⣟⣿⣿⣿⣿⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡋⠉⣩⡔⢊⡷⣯⡹⣧⢸⠹⢿⡟⣽⣿⣿⣾⣿⣻⣿⣿⣿⠻⠿⠘⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⣠⠞⠉⣱⣿⠷⣿⠛⠟⣾⣛⠾⢲⣿⣿⣿⣿⣿⣿⣿⣿⣻⣠⡾⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⠾⣑⣹⣿⣿⣃⣐⣁⣔⢩⣽⣙⣾⣿⣿⣿⣿⣿⣯⡗⣿⠿⣏⣟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣭⣟⣿⣿⣿⣼⣿⣿⣿⣿⣜⣿⣿⢿⣿⣿⣯⣵⣷⢾⡻⡟⠋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣥⣿⣿⣹⡛⣪⣿⣟⣻⣷⡄⣠⣼⠿⠻⠁⣙⢔⢻⡇⠦⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣗⣴⣻⣿⣿⠟⣻⢚⢲⣟⡿⣏⡻⣤⡾⠫⠀⢖⣨⠀⠌⢼⡽⡇⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡾⣸⣿⠛⠉⣤⠚⢐⣂⡵⢙⣾⡛⣿⣿⡇⢻⠠⠟⠋⠘⣉⣈⣑⣃⡃⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣻⡿⢩⢦⡙⠀⢟⣩⢀⣩⣥⣤⣌⣙⠫⣴⡶⢿⣛⣛⣻⣿⣿⣿⣿⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⢾⣍⠛⣵⣟⠃⢸⢿⣿⣿⣿⣿⣿⣷⣌⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠓⠘⠋⠾⠧⠷⠤⢾⠙⢫⣿⣿⣿⣽⣿⡿⢸⣑⣽⡦⡙⢿⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣟⣋⣥⣏⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣹⠿⠿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟

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




⣷⣶⣶⣶⣶⣾⢶⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⡿⣶⠶⡶⠶⠶⣶⡾⠷⠶⢷⠶⢶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⣤⣶⣾⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣤⡀
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠉⠉⠛⠛⢿⣔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠾⠿⠛⠉⠁⠈⠉⠓
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡠⢄
⠀⠀⠀⠀⠀⢠⠿⠉⠀⣀⣄⣉⠓⢬⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠔⠒⠛⠛⠻⠦⣤⡀
⠀⠀⠀⠀⠀⠈⠁⠀⢾⣿⣿⣿⠀⣐⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣃⣿⠁⢰⣿⣿⣷⡀⠀⢘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠍⠉⠉⠠⢬⣛⣿⡟⠀⠠⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠱⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠶⠀⠼⠁⠈⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⢧⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⡄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢰⡋⠂⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠜⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⢀⣯⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡆⠀⠀⣏⣀⡀⠀⠀⠀⣠⡴⠿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣷⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠉⢩⠿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⢀⣒⣚⣯⣷⢶⠶⠞⣻⣿⣟⡉⠉⠁⠀⠈⡄
⠀⠀⠀⣀⣤⣠⢄⣀⠀⢳⣴⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⡀⠀⣀⣴⣁
⠀⠀⠈⠀⠈⠐⠋⠚⠛⠛⠿⢿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡋⣙⣰⡿⠋⠉⠁⠀⠀⠙⠉⠓⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠉⠉
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠀⠀⠀⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢹⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡿⠒⠉⠉⠲⣄⢠⠖⠉⠉⠒⢤⣤⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡴⠀⠀⠱⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠼⡧⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⣠⠊⠀⠑⠦⢰⠀⠀⠀⠀⠀⢸⡇⣀⠀⠀⠀⠀⡤⠤⠊⠀⠑⢄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠇⠀⢠⠀⠀⠈⠂⠀⠀⠀⠋⢰⡇⠈⠀⠀⠀⠐⠁⠀⠀⡀⠀⠘⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⡆⠀⠀⡁⢰⠀⠀⡤⠀⠀⠀⢨⡇⠀⠀⠀⠦⡀⠀⡁⢈⠁⠀⢠⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠙⢤⠊⠀⠈⠀⠀⠀⠀⠖⢀⢼⡧⣀⠲⡀⠀⠀⠀⠃⠀⠑⣤⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠘⢆⠀⡀⠀⠀⠀⠀⠀⢀⡜⢣⡀⠀⠀⠀⠀⠀⢀⠀⣠⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠬⣉⣀⣈⣂⠴⠊⠀⠀⠙⠦⢐⡀⠀⣈⡡⠞⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡇⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣀⡠⠔⠒⢋⣉⣉⣉⣉⡃⢸⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡋⠉⠉⠉⠉⠉⠉⠉⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣤⣖⠭⠔⠒⠉⠉⠉⠀⠀⠀⠀⠉⠛⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣤⣤⣤⣤⣤⣤⣤⣤⣧⣶⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⢧⠀⠀⠀⢀⡜⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠙⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠀⠀⠈⠙⠂⠉⠛⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⠟⠋⠀⠀⠘⠛⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠉⢀⡀⠀⠀⢦⡔⠀⢉⣼⠏⠀⠀⠀⠀⠹⣧⡁⠀⢢⡵⠀⠀⢀⡀⠻⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠑⠷⠀⣀⡄⠁⣠⣾⣿⡀⠀⠀⠀⠀⠀⠈⣓⣄⠈⠀⡀⠀⠾⠊⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⡤⠀⠐⠚⠁⠈⣿⠧⡀⠀⠀⢀⠔⠁⠀⠈⠐⠂⠀⢤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣟⣉⡳⣄⠀⠀⠟⠀⠀⠀⠀⣀⢀⡈⠀⢁⠀⠀⡘⠀⢀⡀⣀⠀⠀⠀⠀⠡⠀⠀⣠⢞⣩⣻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣼⠉⢻⠾⣷⣎⣤⣂⠨⠭⠒⡞⡿⠓⢀⢘⠀⠀⡁⡤⠚⢽⢓⢶⡭⣄⣀⣤⣁⣼⠿⠛⢻⣡⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⣇⠞⡕⢿⣿⠙⠓⠲⠒⠶⡓⢧⣀⣜⡸⠀⠀⠇⣢⣀⡼⠘⠉⠀⠚⢴⡋⣿⡿⡰⠒⢸⠃⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢧⡻⠸⠸⣿⡆⢀⠀⠀⢀⠶⠁⡇⠀⢠⠀⠀⡄⠀⢸⠸⢁⡀⡀⢀⣈⢱⣿⡇⡘⠌⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣷⣶⣿⣷⣂⣤⣵⣶⣾⣿⣿⣶⠌⠀⠀⠡⣶⣿⣿⣶⣶⣦⣭⣌⣸⣿⣴⣾⣉⠁
⠀⠀⠀⠀⠀⠀⠍⠢⣀⡀⡤⠐⠊⠹⣿⣿⠿⠿⠿⠛⠿⣿⡿⠿⣿⠇⠀⠀⠀⠀⠘⣿⠿⠿⣿⠿⠛⠿⠿⠿⣿⣿⡏⠁⠂⢤⢄⣀⠔⠹
⠀⠀⠀⠀⠀⠀⠈⡄⠀⣀⠴⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡉⠀⠀⠀⠀⠀⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠦⢀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠢⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀⣀
⠀⠀⠀⠀⠀⢰⠇⡆⡴⣤⢤⢰⢠⡀⣦⡄⢀⠤⠆⢻⡁⢙⣦⠀⠀⠆⠀⠤⠀⠠⠠⠀⠀⠀⠀⠜⠁
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠘⣇⠙⠒⢲⠄⣣⡾⣩⣖⣦⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠦⡄⠹⡄⠈⣧⠘⠆⠙⠋⠀⠉⠀⢙⠲⠏⣼⠛⣶⢀⡀⠀⠀⠀⠀⣠⡀
⢀⡤⠄⠀⢤⠖⠳⡄⢰⡋⣹⠄⠑⠀⠀⠀⠁⠀⠀⠒⠛⣏⠙⡇⠘⠋⠛⠡⣖⣾⠇⣠⣀⠀⠀⠀⠀⠀⠀⢀
⠸⠦⠴⢦⡈⢧⡠⠟⠀⠉⠀⠀⠀⠠⠀⠀⠀⠄⠀⠠⡤⠤⢆⠀⠀⠀⡀⠀⠀⠉⠸⠋⣼⢣⡤⣄⠀⠀⠈⠈⠅
⠀⠠⠤⠞⠁⠘⢂⠔⢡⠦⡄⠀⠀⠀⠀⠀⠀⠀⢰⡀⢀⢤⢬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠟⠰⣗⠄⢀
⠀⠀⠀⠀⠀⠀⠁⠐⠉⡟⣡⠀⡀⠀⠀⠀⠀⢈⠀⢥⢫⠘⡤⠃⠀⡁⢠⢰⠰⢰⠀⡇⡆⠂⡀⡆⠛⢻⢸⡻⣷
⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⢣⡜⠱⡒⢀⡀⠀⠀⠨⠈⠀⠶⠒⢲⡀⠈⠁⠀⠁⠀⠀⠀⢰⡆⠸⣂⣆⠀⠀⠉⠁
⠀⠀⠀⠚⢿⣹⠏⠀⠀⠀⠀⠀⠑⠣⣘⡌⡠⠀⠀⠊⡀⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠀⢨⠉⢌⠡⠉⡙⠉
⠀⠀⠀⠀⠘⢣⡴⢦⠀⠀⠁⠀⠢⠤⣀⠘⢱⣒⡕⢠⣅⢰⠋⠉⢳⣠⠃⡄⠠⡀⡔⠆⠅⠇⠂⠃⡆⢰⠢
⠀⠀⠀⠀⠀⠀⢠⡶⣽⡆⠀⠀⠀⡀⠀⠉⠀⠈⠐⢁⡜⠓⣇⣴⢾⡼⠖⢑⠐⢎⢀⢀⠆⠢⠣⠁⠘⠁⠉
⠀⠀⠀⠀⠀⠀⠈⠙⢋⣴⢾⡁⠀⠀⠒⠀⠰⠀⠂⠁⠁⠚⢀⣀⣤⠇⠀⠠⣀⠘⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠾⣣⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢩⡶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠺⠇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⡀
⠀⠀⠀⠀⠀⠖⠍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠑⠒⠂⠀⠠⠄⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠤⠶⠾⠿⢿⠃
⠀⠀⠀⠀⠀⠐⠦⣄⡀⠀⠀⠀⠀⠉⠁⠲⣤⣶⣶⠾⠟⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢚⡀
⠀⠀⠀⠀⠀⢠⡀⠀⠈⠙⠲⢤⣀⠀⠀⠀⣿⡅⠀⠀⠀⠀⠀⢀⣀⣀⠤⢤⣴⣶⣲⣾⣯⣿⠏⠁
⠀⠀⠀⠀⠀⠉⠒⠍⠒⠤⡀⠀⠀⠉⠓⠦⢼⣿⣷⣶⣶⣿⠭⠿⠶⠚⠛⠋⠉⠉⠁⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠂⠄⡀⠀⠀⠀⣹⠏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣖⣻⡏
⠀⠀⠀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠈⠲⢤⡁⠂⠄⣿⣦⣀⣤⣤⣴⣶⣾⡿⠿⠾⠛⠛⠉⠉⣸
⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠈⠑⣢⣤⡾⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⢘⣿⡄⠀⠀⠀⣀⣀⡠⠤⠖⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠀⠒⠒⠒⠒⠂⠰⠷⣭⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⢀⠤⠟⠀⡠⠞⠉⠦⡀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠷⠾⢎⣡⠶⢶⡉⠀⠀⠀⢀⣈⠶⢼⡏⠲⠶⡦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠘⠀⠠⠐⣀⡀⠀⠀⠀⢠⡴⠆⠠⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⣰⠇⠀⢠⣿⣿⣿⠀⠀⠀⢸⣿⣿⣦⠀⠀⢿⡋⠈⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⣽⠀⠀⠀⠙⠛⠁⠠⠀⠠⠀⠙⠛⠁⠀⠀⢸⡇⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠔⠷⠄⢠⡄⠀⠀⠀⠰⣦⠴⠂⠂⢠⡷⠀⠀⠀⠀⣴⠀⠴⠿⠴⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠘⠾⡉⢩⠗⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠓⠤⠤⠤⠤⢤⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡴⠶⠆⠀⠶⠆⠀⠙⠛⠁⢦⡴⠘⠛⠃⠀⠲⠦⠀⠰⠖⢦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠁⠉⠈⠉⠉⠁⠀⠒⠒⠶⠶⠶⠶⢶⠖⠒⠀⠀⠈⠉⠉⠉⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⢀⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠤⠤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠰⠒⠒⠺⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢨⡧⢼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣎
⠀⠀⠀⢀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢸⡀⣀⢸⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀
⠀⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠂⠀⠀⠞⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⡀
⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠤⢤⣤⢤⡤⠤⣤⣤⣤⢠⣤⣀⣀⣠⣤⠤⠤⠤⠤⠤⠄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠠⠤⠤⠤⠤⠤⣤⣄⣀⣀⣤⢠⣤⣤⠤⠀⣤⢤⡤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣅⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⢹⡆⠀⠀⠀⠀⠀⢸⠉⠁⠉⠀⠀⡇⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡤⠀⣀⣀⣀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣀⣀⣀⠀⢤⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣀⣴⡆⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢸⣦⣀⣴⠂⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢉⡙⠏⡇⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠻⣏⡉⠷⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⢀⡴⢉⠀⢺⡟⡰⠃⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠉⢆⠹⡗⠀⡈⢦⡀⠀⠀⠀⣧⣤⠀⠀⡇
⠀⠀⠀⠀⠀⡀⠐⠂⠀⠁⠀⠀⠀⠀⢠⡍⠀⠀⠑⠊⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠙⠊⠀⠀⠉⡄⠀⠠⠀⠠⡇⠀⡃
⠀⠀⠀⠀⠀⠟⠛⠆⠀⠉⠀⠁⠀⠀⠘⢇⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⡠⠃⠀⠈⠂⠘⠁⢀⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡅
⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢤⡀⠀⠀⠀⢠⢦⡀⣠⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠑⢦⡀⢰⠃⠀⠙⠦⡀⢠⠏⠀⠙⢧⣀⠀⠈⠉⠒⠲⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠄⡉⠺⣄⠀⠀⠀⠈⠛⠦⡄⠀⠀⠈⠓⡄⠀⠀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⢄⠑⢤⡙⠲⢤⡀⠀⣼⠁⠀⠀⢀⠎⠀⠀⣎⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⣠⠤⠤⠤⠷⡄⠠⣉⠢⢌⠀⣾⡄⠀⢻⡀⠀⠀⣼⠀⠀⢸⡀⠉⠓⠒⠛⠉⠁⠀⠈⠢⢄⣀⡀
⠀⠀⠀⠀⠀⢹⣷⣦⣄⡀⠈⠲⣄⡑⠢⡀⢻⣜⠀⠈⠳⣄⠄⢻⡀⠀⠸⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⠛
⠀⠀⠀⠀⠀⠈⠛⢿⣝⣿⣷⣦⣄⠉⠓⠦⣀⠙⢦⣀⠀⠈⠢⣄⠳⣄⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣮⣤⣽⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢝⡿⡿⣿⣦⣄⠈⠉⠓⠛⠿⣶⣦⣬⣽⣾⣿⣶⣮⣷⣶⣶⣶⣶⣶⣾⣿⣿⣷⠶⠶⢾⣛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⢷⣝⢷⣴⠯⣽⣿⣿⢿⣿⣿⣿⣿⣿⠿⠿⢞⣛⣻⠿⠿⠛⠛⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠿⡷⣋⣍⣴⠶⠾⠓⠛⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉




⣿⣻⣻⣻⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣻⣿⣛⣻⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠒⠲⠤⣄⡀⠀⠉⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣊⠏⠁⠀⠙⢄⠉⠉⠑⠒⠤⣀
⠀⠀⠀⠀⣀⡴⠷⢞⡉⠁⠀⠀⠀⠀⠀⠉⠳⢄⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⡴⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⢮⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠜⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠸⠀⡜⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢃⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⡇⡸⠀⠈⠢⢄⠀⠀⠀⠀⠐⢤⠄⣀⣀⣀⣀⣀⠠⡄⠀⠈⠀⠀⠀⠀⣠⠼⠁⠀⠀⠀⡰⠀⠀⢠⡆⠀⣿⠀⠀⠀⠀⠀⠀⢠⡆⠀⠘⢿
⠀⠀⠀⣇⠃⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠘⡀⢰⠀⠀⢀⠎⠱⣼⠀⠀⡴⠸⠇⣀⣀⢾⠀⠀⣿⡀⠀⠀⠀⣀⣀⡠⠧⢤⠀⢾⣼⠉⠑⢄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠀⢀⠇⡐⠀⣃⠀⢰⠁⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⢸⠀⢸⡈⠎⠀⠈⡆
⠀⠀⡎⠉⠀⠀⠀⠀⠿⠂⠀⠀⠀⠀⡀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠉⠑⡌⢰⠇⡜⠉⡶⠁⠀⠀⠀⠾⠆⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠀⠈⣀⡏⠈⣾⠀⠀⢸
⠀⠀⠑⠂⢄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠚⠁⠸⠀⠘⡆⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠊⠇⢠⠀⢸
⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⡀⣀⠂⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡀⠀⠀⠀⡇⠀⠈⠢⣀⠀⠀⠀⠀⠀⠐⠲⠒⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⡼⠀⣼⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⣀⡀⠀⠀⠀⠀⣀⣀⠠⠔⠊⠁⠀⠀⠀⠀⠠⠤⠀⠀⠤⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠉⠀⠀⠀⠐⠓⠐⠿⠤⠌
⠀⠀⠀⠀⠤⢤⣀⡀⢀⡤⢺⠹⡈⠀⠉⠉⠍⣱⣤⠀⠀⢀⣀⡵⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠉⠉⠉⠉⠓⠦⣄⢀⠀⠀⡀
⠀⠀⠀⠀⡇⠀⠀⠉⠙⠓⠦⢷⣦⡭⠭⣴⣾⡋⠤⠉⠊⠉⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⢨⠉⠛⠒⠤⢮⣀⡀⠀⠀⠀⢀⣠⡛⣨⠤⠖⠚⠉⡁
⠀⠀⠀⠨⠁⡆⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⢰⠌⡇⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠀⠀⠀⠉⠛⠶⠾⠛⠉⠁⠀⠀⠀⠀⢠⠳
⠀⠀⠀⠀⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠀⠀⠀⠀⠀⠀⠀⠰⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡌
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡐⠈⠂⠀⠀⠀⢠⠈⠡⠀⠀⠀⡀
⠀⠀⠀⠀⢠⠓⠒⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠓⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢠⣴⠃⠀⠀⡄⠀⠀⡄⠀⠀⢷⠖⠂⠁
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠌⠑⢶⠤⣦⡖⠉⢰⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⡗⠖⠈⡇⠀⠀⢸⡜
⠀⠀⠀⠀⠀⠙⠢⢄⠀⠀⠀⡅⠀⢸⠉⠣⡜⣄⠀⠂⢀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠾⡀⡀⠀⣃⣄⣤⡇⠀⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠥⠠⠮⠤⠄⠋⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⠒⠀⠐⠢⠤⠒⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡴⠶⠶⢦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠛⠈⠛⠛⠛⠛⠀⠛⠒⠒⠒⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⢀⣤⠴⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢰⡏⠀⠀⠀⣬⢻⡄⠠⣤⣤⣤⣤⣤⣤⣤⠄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣇⠻⣦⡾⠁⣸⠇⠠⡤⠤⠤⠤⠤⠤⣤⠄⢀⣿⠴⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠶⢤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠐⠒⠶⢦⣄⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠷⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⠞⠛⠛⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⣀⣀⣀⣀⢸⡇⠐⠶⠶⠶⠶⠶⠶⠶⠞⢧⣀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢧⡀⠀⠀⢀⡼⠁⠐⠶⠶⠶⠶⠶⠶⠶⠆⠀⣿⠁⠒⠒⠛⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡟⠁⠀⠀⠈⢳⡄⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠛⠛⠛⠛⢸⠇⢀⣤⣤⣤⣤⣤⣤⣤⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠙⠦⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠧⣶⡄⣶⣠⣶⢄⣶⢰⠆⡖⠂⡗⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠉⠀⠉⠀⠉⠈⠉⠈⠀⠉⠁⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠮⠅⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣸⡏⠀⢠⣄
⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠁⠀⣀⠀⣤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠂⣀⠀⠒⠀⠀⠂⠀⠀⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠉⠀⠁⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠃⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣠⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠘⠛⠉⠁⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠐⠠⠿⠿⠛⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⠄⢀⣤⡐⠀⠰⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⡑⠀⠀⠠⢀⣡⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠟⠁⠀⠉⠀⠁⠀⠀⠈⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢀⣠⠂⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⡇⠀⠀⠠⠀⠀⠀⠠⢀⣀⣀⡈⢿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡶⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢱⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠈⠿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⢿⠃
⣾⣿⡄⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡀
⣿⣿⡇⠀⠀⠀⠀⣀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⣿⣧
⣿⣿⡇⣴⣶⡄⢸⣿⡧⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣿⣿⠀⢰⣿⠈⢿⣿⡇
⣿⣿⡇⣿⣿⡇⢸⣏⢰⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⠀⣿⡇⣴⣶⣶⣶⣶⣤⡀
⣿⣿⡇⢿⣿⡇⢸⣿⡆⢙⠛⠛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⣿⣿⠀⣿⣷⣌⡛⠛⠛⢿⣿⣷
⣿⣿⣷⣤⣭⣴⣦⣭⣴⡿⠟⠛⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣥⣶⣬⣥⣾⣿⣿⣿⠿⠟⠃⣸⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣿⣿⣿⡿
⢻⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⡇
⠈⢿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢠⣴⣄⠀⣠⡴⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⠈⠙⣛⣦⡼⠿⠷⣾⢋⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⠉⢱⣶⣶⠻⣯⣉⠉⣹
⠀⠀⠀⢀⣀⠀⢀⣤⡄⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⣿⠀⠀⠙⠛⠋
⢀⣀⠀⣾⣿⡇⢸⣿⠟⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⣦⡀⣾⣿⡇⢸⣿⣿⠀⣤⣶
⣿⣿⡇⣿⣿⡇⢸⣯⠰⣶⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣿⣿⡇⣿⣿⡇⢸⣿⣿⢸⣿⣿
⢿⣿⠇⢿⣿⠇⠸⣿⡆⢉⣛⣛⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣀⠻⠟⢃⠙⠿⢃⣼⣿⡿⢸⣿⣿
⣦⣤⣾⣦⣤⣶⣶⣤⣶⠿⢛⣋⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⣤⣶⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⢡⣶⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿
⢻⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣏⣸⣿⣿⣿⣿⣿⣿⡏
⠈⢻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⡿⠟⠁