Giraffe Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣄⠀⠀⠀⠀⠀⢠⣾⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣤⠀⠀⢻⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⢼⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠈⣿⣿⡟⠣⠀⠀⠀⢻⣏⢻⣿⣿⡆⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢿⣇⠀⢠⣿⣿⣇⠀⣀⡀⠀⢸⣿⣼⣿⣿⣯⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢀⡿⣿⣿⣭⢿⣿⣻⡾⠛⣦⣾⡿⢿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⣵⣿⣿⣿⡀⣁⡤⠂⢸⡿⢏⣡⣤⣹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢾⣿⣿⣿⣿⣿⢶⣄⠀⢸⡿⢻⣿⣿⣿⣿⡷⢠⣤⣶⣾⣷⣿⣿⣯⣥⣿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⠙⢻⣿⣿⣿⣾⣿⣶⣿⣷⣿⣿⣿⣿⣿⣁⠙⣛⣿⣿⡿⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣈⠙⠻⣿⣿⣿⣿⡇⢈⠉⣻⣿⣿⣿⡄⠈⠙⣻⣿⡅⢀⠨⣭⣽⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠛⠻⣿⣿⣿⣯⡿⠻⢿⣷⠀⢠⠻⣯⣋⡙⠇⠀⠀⢀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣰⡌⣿⡇⠈⠀⠸⣿⣿⡄⠀⠛⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢻⣿⣿⡌⠁⢸⣧⣀⠀⠀⠈⢙⠓⠰⡆⠈⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⠀⣸⣿⠋⠀⠀⠀⠸⣷⡄⢰⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⡿⣿⣦⣀⠀⣸⠿⢿⣌⠋⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣾⣿⠛⠻⣷⡀⣰⡆⢹⣷⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣯⢻⣶⣿⣿⡟⢡⣿⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⣿⣽⣿⣾⣿⢿⣿⠀⠻⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⢿⣿⣿⣧⡚⣿⣬⣿⣷⣄⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠈⠻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⣿⠿⣿⣿⣿⣿⣿⣿⣹⡧⠀⠀⣾⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣤⣾⢸⣿⣿⣿⡿⢹⣿⣀⠀⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡘⣿⣿⡿⢧⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣧⠸⢿⣧⣬⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣟⣨⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⠃⠼⠟⢻⣿⠋⠙⠏⠀⡇⠀⠀⠀⠀⢈⣭⣍⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⣃⠙⠿⣿⣿⣿⡟⠀⠀⠀⢸⣿⠀⠀⣼⢀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣆⠀⠘⠿⠋⠀⠀⠀⠀⠘⠏⠀⠰⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠚⠿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⡄
⠀⠀⠀⠀⠀⠀⢀⡀⠀⣭⣀⣀
⠀⠀⠀⠀⢀⡼⠃⠀⠀⠘⣿⣸⡆
⠀⠀⠀⠠⠛⠁⠘⠳⠄⠀⠟⠙⣿⣶
⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⣴⢀⡈⢿⣦
⠠⠦⣀⣠⡶⠴⠿⠶⠟⠻⣿⣌⡁⠸⢿⣇
⠣⠞⠋⠁⠀⠀⠀⠀⠀⠀⢻⡼⣿⠆⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡌⠸⣧⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣶⡀⢾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣥⣌⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⢸⡿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣥⠈⢁⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⢻⡟⠿⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣰⣤⠁⣴⡿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢋⠘⠿⢰⣆⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢧⡀⢿⣿⡆⢹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠆⠛⢡⣿⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⣴⣿⢸⣿⡟⢀⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠈⣉⡀⠸⣿⠏⣈⢻⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠿⠿⠁⣼⣿⣌⡛⢩⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣶⣦⣬⠛⠻⠷⠈⣿⡈⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣯⡙⠛⠀⠺⣷⣤⠉⣉⣂⣈⣻⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⡦⠀⠙⠋⣤⠙⣿⣙⠛⠺⣿⣿⠷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣋⡉⠐⠿⣿⣾⣿⠀⢿⣿⣧⢴⣬⡛⢠⣿⣏⠻⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⡏⢰⣶⣌⠛⢛⣤⡄⢉⠁⣌⠻⠿⢆⠀⠉⣄⣠⣤⣍⢲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠋⢰⣿⣿⡇⢸⣿⣿⠃⣿⡇⢰⣿⣷⡘⣷⣆⠹⣿⣷⣸⣧⠉⠉⣤⡌⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠞⡋⢠⡄⢨⣝⠛⠀⣿⣷⠀⣿⣿⡇⣿⡿⢢⡌⢻⡿⠻⠘⠃⢉⡁⢺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣬⣯⣾⡿⠘⠿⠿⣶⠀⠛⠀⠋⢉⣁⠉⣠⣿⠇⣨⢱⣦⢸⡇⣸⠇⣤⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⣴⣶⣆⢀⣤⣴⠾⠋⢰⣿⣧⡄⠙⠛⢼⣿⢸⣿⣠⡿⠋⠳⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣍⠙⠃⠙⣡⡾⣡⣿⣿⣿⣿⠘⣷⡌⢫⣄⣽⣿⣷⡆⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢁⣼⡇⢰⠟⣴⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⠷⡌⣀⠘⠃⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠤⣿⣿⠈⢛⣋⠈⠈⠛⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠴⣿⠰⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⣿⣿⣷⡌⣩⣀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠹⣦⠰⠟⠘⣦⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡿⣿⠃⣿⡸⡿⠉⠁⠀⠀⠀⠀⠈⢻⡄⠙⣿⣿⡇⠀⢿⣆⠶⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⣿⠏⠀⢻⣇⡀⢰⠀⠀⠀⠀⠀⠀⠀⢹⡆⠻⣿⣟⠀⠘⢿⣇⢘⡟⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢰⠏⠀⠀⢸⡟⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⡆⠀⠈⣿⠈⢠⡆⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠃⠀⠀⠀⢸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⠀⠀⢀⠀⢘⣇⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡗⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⡀⠀⠸⣷⡼⣟⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠈⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⣿⠘⡟⠈⢿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠏⠀⠀⠀⢸⡄⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⡇⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢀⠀⠀⠀⠀⠀⠀⢠⣤⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠀⠀⠀⠀⣠⣄⣸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠇⠀⠀⠰⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣿⣷⠀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⢸⠿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠻⣧⡀⠠⠋⢀⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠙⢿⣿⣿⣭⣲⡀⠀⡇⡤⠀⠀⣾⠂⠀⠠⣿⡀⠀⠀⢠⣾⣽⣿⣿⡿⠋⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢙⣿⣿⡿⠋⣴⡇⠀⢠⣿⣿⢇⡴⠋⠈⠱⡄⣠⠂⣿⣿⣿⡟⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⣿⣏⠀⠰⡟⠀⣴⣿⡿⠛⠸⠱⣦⣒⡒⠆⣡⣴⣿⣿⣯⡷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⡟⢀⡾⠋⢉⡤⠈⠀⠾⣟⢿⡿⣧⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠞⡱⠶⠊⡀⢿⠆⢀⡙⣛⢱⡿⣿⡙⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⢀⣴⣦⢃⠾⢃⣍⠐⠶⠘⠛⣩⡸⢦⡙⢿⣟⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠚⠟⠫⠾⡔⠲⢀⡺⡧⣾⣇⠿⢳⣼⠱⡆⢿⡟⢿⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠶⣂⢤⣿⣥⣉⢡⣦⢘⣣⣤⣾⣭⢸⡆⣴⣜⣿⣿⣗⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢤⣴⣾⣿⣿⠟⠉⠙⠛⠚⠛⢿⣿⣿⣿⠿⠘⢣⡜⢿⡎⢋⢿⣗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⢿⣿⢏⣼⣇⢻⣿⡌⣁⢻⣯⢿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⣾⣿⣿⠸⠟⠁⢻⣷⣝⠞⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⢿⡟⠀⢸⣿⣆⢻⣿⡷⢹⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣴⣷⣌⢻⣿⡆⢉⣠⣾⢿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡘⣿⣿⣿⣦⠉⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠟⣿⣿⣿⠀⠻⢿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⢻⣿⣿⢀⣾⣶⣌⠙⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢸⣿⡇⣸⣿⣿⣿⣟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡸⣿⡇⢻⠻⠿⠟⠯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣷⠆⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⢸⠏⠀⠀⠀⡀⢈⠃⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣄⠀⠀⠐⠂⠀⠀⠀⠈⢻⣦⠀⠀⠀⣰⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⡀⠀⠀⣿⣾⣄⠀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⡿⠳⣄⠃⠀⠀⣿⣨⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠴⠋⣤⡄⠀⢸⡿⠇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣦⡸⢟⡄⠀⢸⣧⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⢹⡏⡘⠁⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⣹⣦⡶⠁⢶⢄⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⣿⣧⠀⠀⠀⠈⢩⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣿⣿⠀⠀⠀⢰⣎⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣤⠈⠃⢀⣀⡀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣿⣷⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⢸⡿⢫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠃⠉⢁⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢡⣾⣿⣿⣷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢃⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠘⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⡿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠛⣿⡟⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣀⠀⠈⠃⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢡⣿⣦⡀⠀⣈⣉⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⢸⣿⣿⣧⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣼⣿⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣿⣿⣿⣿⠀⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡝⣿⢿⣿⣿⣿⣿⠀⠋⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡏⢿⠉⠛⠛⣿⠇⣰⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣽⣷⠀⣄⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣏⣿⡿⢸⣿⣷⣶⣶⣇⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⡉⣻⣿⣇⣾⣿⣿⣿⣿⣿⡀⠉
⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⠿⠿⠿⢿⣿⣿⢿⠿⠿⣿⠿⣿⢿⣿
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠰⠂⠂⠘
⠀⠀⠀⠠⠔⣩⡌⣃⣠⡐⢲⡄
⠀⠠⢘⡂⠀⠷⠯⠬⠞⣼⣋⠿⣆
⠀⠀⠈⠁⠀⠀⠀⠀⠀⢱⠏⣖⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣹⡝⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣣⡜⢓⡙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣵⡌⣯⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣾⢣⡿⢘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡞⡷⢹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣷⡜⢿⠗⣽⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣶⠻⣟⣼⡶⢽⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣍⣷⣜⡋⣾⣷⢺⣎⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠾⡍⢿⣮⣷⣿⣟⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣗⣼⣾⣿⣿⣎⣻⣿⣻⡟⣷⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡟⣽⣿⣿⣴⢿⣿⣦⢻⣷⣿⣻⣿⣻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⣿⢹⣿⣿⣏⢿⠟⣵⣧⣷⣼⣿⣿⣿⣿⣿⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⡝⣷⣾⣷⣽⢻⣿⣿⣾⣿⣽⣯⢿⣻⣿⣾⡛⣿⣟⢫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⡿⣾⣿⣽⣿⣿⣻⣟⣛⢿⣫⢾⡿⣇⣾⣻⢫⣟⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣷⢿⣷⡝⣟⢾⣽⣿⣩⣟⡿⣿⡏⣟⣸⡟⣷⠹⢏⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡶⢾⣭⣿⡏⠙⠻⠻⠟⠙⠋⠀⠑⠱⠎⡿⣷⣛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡅⠙⣇⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣼⠳⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⠘⡷⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠴⣍⠠⣄⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠄⢲⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡃⢆⡱⠒⠆⠀⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢬⣜⡅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡅⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠦⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠆⠀⢀⣤⣀⠀⠀⠀⠀⣠⣄⠉⠀⠀⠀⠀⢴⣶⡄
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⣿⣻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠘⢃⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠔⠋⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⡁⠀⠈⠙⠛⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢡⡿⠃⣼⣭⣦⣅⠤⠄⠒⠂⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⣾⣄⠐⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣶⡆⠘⠓⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⣨⡁⣾⠇⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠘⢿⣇⣶⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⣾⣷⠆⣅⣨⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣰⣤⡉⠐⠛⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠻⠟⢠⠟⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⢁⣷⢰⣶⡄⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⢁⣴⡘⠛⠘⠋⠀⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣋⡉⠠⢾⡿⠁⣼⣿⠃⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠁⢾⠿⢃⣤⣀⠀⢦⣤⣉⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠶⠛⠛⢋⠀⣾⣦⡀⢸⣿⣿⠇⠈⠋⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠋⣴⣾⡿⠂⣴⣿⡷⠀⢉⣄⠘⠟⠁⣾⣿⣦⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠈⠙⠋⠤⣶⠄⢀⣤⣀⠀⠀⢀⡀⠺⣿⡿⠆⠀⠻⠿⠟⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠉⣴⡷⠀⣤⣤⠀⠀⢻⣿⡿⠀⣴⣿⣷⡀⠀⣰⣶⣶⡀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢛⣵⣾⠃⣀⠤⢀⡙⠃⣰⡇⢈⣉⠁⣤⣈⠉⠋⣠⣦⡿⠟⠋⢤⣬⣽⠆
⠀⠀⠀⠀⠀⠀⠀⣘⡧⠈⣉⡁⢸⣿⠇⢸⣿⡀⠛⣷⣿⣿⠀⢿⣿⡧⠀⠙⣻⠋⢤⠀⠈⠙⠏
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⣿⡧⠀⢠⣆⠈⠛⢡⣾⣟⠛⢁⣶⣄⠀⠀⢀⣰⣿⣤⠀⠛⠂⣰
⠀⠀⠀⠀⠀⠀⠈⠸⣿⡀⣉⣥⠐⣿⠟⢠⣾⣀⠉⣹⡄⠀⡀⠠⠄⠂⠉⠀⠀⣿⢀⣴⠀⡿
⠀⠀⠀⠀⠀⠀⠸⠄⢻⠿⡽⣿⡆⢠⣤⡈⠛⠛⡐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡄⣩⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠰⠈⣤⠘⠛⠁⡄⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠏⢰
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣠⠘⠃⢶⡆⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠐⠁⠀⣦⠠⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠃⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣤⢀⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠑⠂⡇
⠀⠀⠀⠀⠀⠀⢀⠘⣿⣿⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⢦⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠳⡜⠓⠄
⠀⠀⠀⠀⠀⠀⢀⢀⣇⠀⠀⠀⠀⠀⠀⠁⠢⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠖⠆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀


⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⢀⣘⡛⠉⠠⣀
⠀⠀⠀⠀⢈⡈⣍⠂⠠⣬⢝⣦⣀
⠀⠀⠀⢀⣌⣄⡻⢀⠈⣷⠿⠗⣛⠒⣀⣀
⠀⠀⠀⡰⢞⡍⠛⠉⠀⠈⢺⡞⣛⡛⣯⡌⣿⢳⣦⣄⣀⡀
⠀⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⠀⠿⠏⣽⣧⣾⢇⣾⣮⣿⣭⣟⣗⢖⣶⡒⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢅⣭⠸⣿⢧⣶⣶⣎⣵⢞⣫⣿⡷⢶⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢰⣾⠻⣿⣾⣷⣿⣟⣶⣾⣿⡿⣺⣷⣦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡮⣿⣿⢣⣿⢣⣬⣴⣿⣹⠋⣾⣫⣽⢷⣆⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣾⣿⣛⣿⣿⡾⣦⡹⢿⣼⣻⣧⣟⣿⣵⠿⣿⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣛⣣⣿⣿⣼⣷⡿⣣⣝⣋⠿⣿⣭⡝⣿⡞⣿⡻⣿⣻⣄⣲⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢋⣿⣯⡹⣿⢾⣿⣿⣿⡿⢿⡾⣿⠿⣿⡏⣿⣻⡓⣾⠛⣄⠀⠉⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣈⣿⣛⡸⠿⣟⣼⣵⣿⣾⣵⡿⢷⢿⢿⣾⢿⢟⣼⣏⣿⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢽⣭⠦⣿⣼⡻⢫⣿⠙⡻⣷⣾⡿⠿⣇⡋⠯⣻⡏⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⣾⠿⢽⠋⠉⠛⠷⠤⣀⣸⡇⢾⡜⠿⢩⡝⣛⣿⠃⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣽⠶⣻⣧⠀⠀⠀⠀⠈⠈⠳⢷⣞⢛⣮⣷⠍⡿⠀⠀⠀⠀⢰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⣿⠻⣟⢪⡆⠀⠀⠀⠀⠀⠀⠀⠙⠻⣧⢘⠷⣤⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣽⡏⠀⢻⣥⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠊⣤⡌⢻⡄⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣈⡾⠀⠀⠀⣿⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⢠⡦⣄⢹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣹⡇⠀⠀⠀⠸⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢼⡷⢙⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠹⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⠁⠈⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⡟⢧⠀⠀⠀⠀⠀⠀⠀⠀⢥⡟⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⢻⢸⠀⠀⠀⠀⠀⠀⠀⢰⣸⠇⠀⠀⢇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⡏⡆⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⢸⡄⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣼⡿⠃⠀⠀⠀⠀⠙⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⢻⡆⠀⠀⠀⠾⠟⠗⠀⠀⠀⠀⠀⠀⠻⠋

⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⣸⣿⡄⣸⣿⠁
⠀⠀⠀⠀⢸⣟⡛⣃⢠⣿⡽⣿⣦⣄
⠀⠀⠀⢠⣾⣭⣕⢻⠷⠛⣁⣛⡟⣾⣿⣦⣄
⠀⠀⣰⣿⣿⣿⣷⡾⠛⠁⠋⢿⡿⢈⣛⣛⣛⡻⣦⣄⡀
⠀⢼⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢾⣿⣿⠟⣱⣷⣻⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⣸⣿⣿⣧⢿⣿⣿⣿⢟⣷⣶⣤⣤⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⡿⢏⣛⡿⠿⢼⣿⡿⣣⣾⡿⣙⣟⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢿⣿⣿⡷⢹⣿⣷⣿⢋⣴⣿⣿⣜⣿⣿⣟⣛⣯⣗⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⣶⣭⣻⡿⣸⣿⣿⣿⢻⣶⠻⣿⣿⣿⣟⣻⣿⣰⠶⣤⣠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢱⣦⢻⡿⣣⣿⣿⣷⣩⣿⣿⣿⣻⣿⣿⢿⣿⣿⣿⢻⣿⡷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⣿⣿⡿⠿⢿⣫⣯⣽⣷⣭⡛⣵⣾⡝⢟⢿⣿⣿⡟⣽⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣾⣿⡎⣾⣿⡏⣿⡝⣿⣿⣿⢸⣯⣿⡟⣿⣷⣿⡽⣿⣿⣿⡿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣿⣿⣷⡘⢿⣧⣿⣷⢿⣮⡇⣾⣿⣿⣿⡿⣿⣿⡿⢩⣶⣶⣾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣶⠘⠙⢿⡜⣿⣧⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⢛⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⡿⢿⣦⢸⣶⢶⣯⠘⢿⠊⠹⣶⣽⠿⣩⣿⣽⣿⣿⣿⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⢿⡿⣬⠵⡞⠛⠓⠳⠶⠶⠶⠄⠚⠻⢧⣽⣬⣻⣾⣿⣾⠁⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣝⣿⡿⡶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⢟⣳⣾⡀⠀⠈⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠈⢳⣿⣝⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡿⣻⣷⢧⣄⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠹⣽⢟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣹⡏⣬⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠉⣛⠖⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⡙⢣⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠁⠀⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠸⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠆⠀⠀⢀⣴⣿⡿⠉⠀⠀⠀⠀⠀⣿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢲⣦⠀⢰⠻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⢤⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⢦⠀⢠⣶⠖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠐⠯⢽⣳⡀⠠⢒⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⣼⣿⣀⣤⡴⠺⠿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⡀⠈⠐⠢⠦⠀⠲⢞⣡⡶⠋⠀⠁⠘⣿⣻⣿⡟⠀⠀⠀⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣩⣉⠀⠲⣤⣄⣈⣿⣿⠁⠀⠀⠀⢠⣿⡟⣿⡇⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡄⢪⣛⣉⣽⣏⠀⠀⠀⠀⣸⡿⠀⠈⠳⣦⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡄⠀⠀⠀⠹⢤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⡿⠡⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⢠⠺⣶⣄⢠⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣯⣐⠀⠀⠀⠀⠄⠀⠀⢀⣀⡀⠈⠳⠦⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣷⡤⠀⣀⣀⠀⠃⠀⠙⠧⠦⣄⠀⠀⠂⠀⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⢴⣿⠻⣿⣦⠀⠀⠀⠀⠈⠙⠂⠠⠤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢨⠻⢜⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢬⠷⠀⠀⠀⠀⠀⠀⠀⠁⢽⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⡜⣿⢦⠀⠀⠀⠀⠀⠀⠀⡲⣷⣻⣿⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡱⣵⣭⣾⢣⠀⠀⠀⠀⠀⠀⣾⣳⣽⠿⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⣌⠻⣬⣍⣴⡀⣀⣀⡔⣮⣿⡿⣓⡙⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⡶⢮⣕⣽⣮⣿⠿⢿⣃⢓⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠪⡗⠵⡒⣥⠳⢣⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠛⠉⠉⠀⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠆
⠀⠀⠀⠀⠇⠀⠀⠰⣦⣤⣤⡀⠰⡀⠀⠀⠀⠀⠀⢠⠀⠀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡄⠁⣀⠀⠀⠀⠰⡆⠟⣰⣿⣿⣿⣷⡄⠀⠸⠛⠓⠸⠂⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣼⣷⠀⠈⠢⠀⢀⠖⢤⠀⣿⣿⣿⣿⣿⠀⢿⣿⠀⣀⠃⢠⣦⠀⠀⠀⠁⠀⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣽⣿⣆⠀⠀⠄⡀⠰⠆⠀⣿⠿⠏⢰⣍⠀⠈⠻⣷⡈⠄⢮⠈⣤⠀⠀⠀⢀⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠃⠀⠀⠃⡀⣾⡄⢁⡤⣿⣦⠻⣷⠀⠘⠋⠀⢠⠀⡀⠘⠁⣀⣤⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢤⠀⢠⣿⢣⣾⣷⣿⣿⣇⣰⣷⣴⣄⡀⠆⠙⣀⡀⠀⠛⠋⡁⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡵⠀⣤⠸⠟⣿⣿⣿⣿⣟⠻⣿⢇⣼⣿⠇⢀⣸⣬⣴⣄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠂⢻⣷⠀⣿⣿⣿⣿⣿⠘⢃⣿⣿⠁⠀⠺⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣾⣿⢻⣿⣿⣿⣿⠋⢠⣾⣿⣿⣷⡀⠀⠻⠶⢆⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⢇⣼⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⠿⠀⠀⢀⠛⡄⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣇⠀⢿⣿⣿⣿⣿⡆⢰⡆⢈⡸⣿⢹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⢟⣼⣿⣿⣿⠃⣾⣨⣿⣿⣿⣿⣧⣿⠇⣾⣇⢹⠄⢿⣿⣭⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠿⣿⣿⣿⣿⣷⡽⣧⣶⣮⢿⠏⢸⣿⠆⠛⢏⠀⣴⡌⠻⣿⣿⣯⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣿⣿⠏⢀⡙⠃⣸⣿⣿⡆⢰⣿⣿⢶⡀⢈⠀⣿⣿⡀⠀⢻⡿⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡘⣿⠀⣿⡿⢀⣿⠇⣼⣿⣿⡿⠇⠸⣿⠀⣿⢣⣿⠘⣿⣿⣷⠀⣈⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣮⣴⣿⣷⡈⢉⣴⣿⣿⣍⣀⣠⣤⣀⠜⠁⢾⡟⠀⠻⣿⠏⢀⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⢟⣵⡶⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠶⠿⠿⠃⠀⢀⡀⢀⣼⣷⣦⠀⣿⣿⣿⣿⣿⣿⡷⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠀⠀⠀⢀⣶⣦⣼⣷⠘⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣷⡾⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣤⣬⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡇⠹⣿⣿⣿⡆⠘⢿⣿⣿⣿⡿⣿⣿⣿⣋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠇⠀⠘⠙⠋⠁⠀⠀⠉⠛⠋⠀⢹⣿⣿⣿⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⣦⡀⢰⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠃⠀⠀⠀⣴⣿⣶⡄⢸⣿⣿⣿⣷⡈⣿⣿⣿⣿⣿⡟⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠀⣿⣿⣿⣷⠀⢿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣷⡀⢻⣿⣿⣷⡀⢿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣩⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠈⠛⢿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠳⠀⠀⠀⣠⣶⣾⣶⣄⠈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠛⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣆⠀⠉⠙⠛⠛⠉⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣠⣦⣈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⣤⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠲⣄⠀⢿⣿⣀⣿⣇⡴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢤⣼⠃⠿⠿⣿⣿⣿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣍⠃⣠⣦⠙⣯⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢁⣤⣤⡄⠀⠀⠈⡿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡄⢠⡙⣶⠆⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣦⢼⣀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠦⣯⡼⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠶⣾⡟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⡜⢻⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠛⣁⡿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⢿⠃⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⢿⡿⢏⣵⡄⣠⣬⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⣿⠆⡀⢿⡿⠙⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠛⡿⢠⣿⣿⡇⣴⣿⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠛⡋⣼⣿⣠⣌⠉⣦⣬⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠛⣥⠸⣿⢨⣥⠛⣻⡄⠛⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣍⢙⣮⣫⣴⣤⠻⢟⣜⠛⠃⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠺⡟⠾⢿⣌⢻⡿⢾⡿⢿⣸⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠰⣮⣉⠹⠟⠀⣿⠆⠻⠿⠃⣴⣄⢰⣿⣿⢷⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡖⣬⠀⠚⠃⠐⣻⡁⠻⠇⣠⡄⢻⣷⠀⣿⡟⢀⣬⣭⣾⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡯⠡⡖⢡⣼⠃⣿⠆⠋⣼⠷⠸⣿⠗⣀⠀⡘⠛⣥⣬⡝⠛⣤⡘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢰⡿⢀⣤⠓⣴⡇⢘⡇⠰⠆⢀⣠⠈⠛⡖⠛⠚⠛⣉⣥⣄⠙⣥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⠀⠞⠰⣶⡎⠁⣴⠧⡀⠛⠃⠾⠃⣾⣿⡇⢸⣿⣗⡸⢂⡉⢻⠟⣀⣛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾⢀⡌⢱⣦⢴⢠⡴⢆⠹⣟⣤⠾⣷⠘⠿⢁⠙⡏⣬⡁⢈⣡⣤⡘⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢳⡿⡉⣥⠈⠟⠠⢶⣂⡈⠥⣩⢩⣄⣀⡶⢶⡟⠛⢱⢀⡘⠉⠛⠉⣥⣿⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢁⡼⠣⢇⡈⣤⠸⣇⢨⡙⠃⣤⣿⠀⠠⠏⠠⠄⣙⡤⢼⠾⣉⠗⠿⣀⠙⣛⣤⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠣⡜⠀⡸⠛⣷⠁⡐⣨⢰⡆⢠⣄⣸⠛⠓⠂⠓⠈⠁⠀⡘⢶⠍⠷⢤⡹⢋⡟⠃⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⢁⠐⠀⣈⡷⢈⣠⢻⣁⠶⠲⠟⠄⣽⠁⠀⠀⠀⠀⠀⠀⠀⣏⠃⠰⢂⠞⠉⠛⠶⢛⣄⠋⣤⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠂⠁⡀⠔⠀⠀⢀⡿⠄⠙⢃⡼⣯⢐⡄⠰⠆⠹⡇⠀⠀⠀⠀⠀⠀⠀⠏⢠⡾⡎⠀⠀⠀⠀⠀⠀⠁⠊⣠⡌⠉⡆
⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣷⠖⠉⠀⠀⢀⡴⠊⡴⠂⡴⠋⠀⠁⡌⠑⣒⠂⣔⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡈⢰⡇
⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡏⢠⡞⢀⡾⠀⠀⠀⠀⣞⣤⠍⠀⡋⠀⠀⠀⠀⠀⠀⢀⠞⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢠⡞
⠀⠀⠀⢼⣿⡿⣿⠋⠁⠀⠀⠀⠀⠀⢠⠐⠁⣀⠞⠁⠀⠀⠀⠰⣭⠧⣶⠀⣧⠀⠀⠀⠀⠀⠀⠰⠃⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢀⡞⢈⠃
⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠀⡰⠁⠀⠀⠀⠀⠀⠀⠣⣰⠇⠀⣷⠀⠀⠀⠀⠀⠀⢀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⢸⡄⠀⠀⠀⠀⠀⠘⠀⡘⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠱⡀⠀⠀⠀⠀⡆⠀⠃⠀⠀⠀⠀⠀⣀⠜⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢣⡀⠀⠀⢠⠀⢸⠀⠀⠀⠀⣠⡎⠁⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠙⡄⠀⡜⠀⢻⠀⠀⠀⠀⢹⣿⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡄⠀⠈⠀⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⠈⣄⠈⡄⠀⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢰⣧⣠⣤⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣿⡿⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⣶⣴⣦⡀⠀⠀⢀⣸⣧⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢐⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣷⣶⣦⣤⣤⣄⠀⢈⣿⣿⣿⠀⠀⠀⣿⣿⡿⢹⡇⠀⠀⠀⠀⠀⢀⣠⣤⠶⠾⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⠀⠀⢰⣿⣿⡇⣼⡆⠀⠀⠀⠀⣰⣿⣉⡀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣦⣠⣿⣿⣿⡁⢸⣇⠀⠀⠀⢰⣿⣿⣿⣿⣿⢠⡏
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠛⠀⠀⠀⣿⣿⣿⣿⣿⣃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⡄⣿⣾⣿⣿⣿⡿⠿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣿⣿⣿⣿⣿⠟⢋⣉⣉⡉⠸⣧⠘⠻⢿⣿⣿⠋⠐⠈
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠀⠙⠀⠀⠀⠈⠁⠀⢿⣿⣶⡦
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⢻⣿⣦⣤⣄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣶⣿⣿⣶⣴⠄⠀⠀⠀⢿⣷⡀⠹⣿⣿⣏⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⡿⠻⢿⣿⣿⣿⣿⡿⠋⠁⠀⣼⣷⡄⠸⠏⠀⣀⠙⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⣠⣄⠀⣙⠿⠿⠟⠃⠀⠀⠀⣿⣿⣿⡀⠀⢰⣿⣷⡈⢿⣿⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣾⣿⠋⠐⠿⠋⢀⡈⠛⠶⠶⠂⣰⡄⠀⢿⣿⣿⡇⠀⢹⣿⣿⡗⠀⢻⣿⣿⣏⣀
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣹⡟⢁⣤⣦⠀⢰⣿⣿⣷⡆⠀⢸⣿⣧⠀⠈⠻⣿⡿⠀⠘⢿⡿⠁⣠⣄⠙⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⢠⣿⣿⠟⠀⠘⠻⠿⠋⠀⢀⣀⣉⠁⠀⣀⡀⠈⠀⣀⣀⠀⠀⢠⣿⣿⣧⡈⢿⣿⣷⣤⣄
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡇⠘⠟⠁⣠⣶⣷⡆⢀⣤⡀⢸⣿⣿⡇⢸⣿⡿⢀⣾⣿⣿⣷⠀⢸⣿⣿⣿⣷⡄⢻⣿⣿⣟⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⠟⠀⢿⣿⡇⠀⠙⢋⣠⣿⠟⢡⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⡿⠀⠀⠙⣿⣿⣿⣷⠆
⠀⣠⣶⣿⣿⣿⡿⣛⣉⣉⠁⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠈⠁⢸⣿⡆⠘⢿⣿⣿⣿⣿⠁⠀⢿⣿⡿⠀⠀⣰⣦⠈⢿⣿⣿⣤⣀
⢸⣿⣿⣿⣿⣯⣾⣿⣿⡿⠀⠀⣠⡴⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣿⣿⣦⡈⠻⣿⣿⣿⠀⠀⠀⠉⠁⠀⢠⣿⣿⣷⡀⠹⣿⣿⣿⠃
⣸⣿⣿⣿⣿⣾⣿⡿⠋⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣦⡀⠉⠁⢀⣴⣶⣦⡀⠀⢸⣿⣿⣿⣿⣆⠈⢻⣿⣶
⣿⣿⣿⣿⣿⣿⣟⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡗⠀⢀⣾⣿⣿⣿⣿⡄⠸⣿⣿⣿⣿⣿⣷⡀⠙⣿
⠻⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⢀⣾⣿⣿⣿⣿⣿⣇⠀⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⠹⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠉⠛⠿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠉⠛⠻⠿⠿⠿⠋⠀⣠⣄⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣤⡀⠀⠀⢰⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠇⠀⣰⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡟⠀⣰⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠏⠀⣰⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠁⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡃⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢀⣀⠀⠀⠀⠉⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⣷⣶⣤⡀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⢸⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣾⣾⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣯⣰⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢠⣿⣿⡿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⣿⡷⠋⠉⠁⠀⠸⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣏⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣏⢾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢫⡥⢾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠲⡶⣴⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡰⡿⣈⢿⡿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡻⠟⢠⣾⠞⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠟⠈⢷⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⣾⣿⣦⣿⣿⠿⢨⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⣬⠉⣼⡟⢫⣶⣶⡍⢶⣿⠟⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⡴⣿⡿⣿⠷⣿⣻⣥⣾⣷⡌⣛⡛⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇⢰⡾⣿⣷⣯⣩⣭⣼⢟⣼⢿⠿⠂⣭⣦⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢶⣿⣿⣷⣾⣭⣿⣧⣿⣰⣿⣿⠜⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠺⣧⠀⠉⠛⠿⡿⠿⡋⣛⣃⣜⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⢾⣿⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⣿⡿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⢨⣧⠿⣷⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠉⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⡀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠉⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣸⠇⠈⠀⠀⠀⢸⡀⠂⠀⢀⣀⣄⢠⡏⠠⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⡤⠶⠶⠶⢇⣸⠟⠛⠛⠉⠁⠀⣼⣀⣀⣀⣤⣤⣤⠈⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣥⡤⠶⠶⢶⣶⣿⡿⠿⠿⠿⠛⠛⠛⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠛⠛⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠶⠶⠖⠋⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⢶⣶⣼⣆⠀⠀⠀⢘⣧⣤⡶⠶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣠⠶⠛⠉⣩⡽⠛⠋⠉⠉⠛⠻⣭⡉⠙⠳⢦⣈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣠⠟⠁⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠙⢷⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⡼⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠹⣆⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⡼⠁⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠹⣆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣸⠃⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢹⡄⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢀⡏⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠈⣧⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢸⠇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢹⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡆⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⣸⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢀⡤⠶⠿⢦⡦⡼⠗⢦⡀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠋⠛⣲⣦⣀⣟⠀⣠⢼⠷⠲⣄
⠀⠀⠀⠀⢰⣋⣀⣀⣾⠻⣣⠖⠻⣼⣁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣸⡝⢦⣰⠃⠧⠟⠹⠿⢻⡀⠀⠀⢿⡀
⠀⠀⠀⣾⠉⠁⣸⡽⠿⣤⡇⠀⠀⠸⣏⣠⣴⡒⠻⢦⣀⣀⣀⣀⣀⠀⢀⠀⢰⡓⠟⠀⠉⠀⠘⣿⣀⣀⣀⡀⠀⣠⣙⣦⢴⡞⠓⠲⢤
⠀⠀⣀⣼⣦⣞⣡⡤⠤⠤⣧⠀⡶⢶⠇⠉⠀⢹⠆⡴⠃⠀⢰⡏⠀⠙⢹⣷⣏⠀⣀⣀⢸⡳⣤⢻⣞⠁⠈⠉⠳⣿⠈⠀⠀⣿⠀⠀⣀
⢤⣬⡿⠉⠸⡅⠀⠀⠀⠀⢾⠳⠟⠀⠀⠀⢀⣼⡿⣥⣀⣀⣾⣅⡀⠀⠀⠁⣿⡶⠿⠈⠉⠁⠘⣿⡿⣄⡀⠀⠀⢸⠀⠀⣠⡾⠷⠯⣅
⠀⠈⢉⣙⣿⡳⣄⡀⠀⠀⠈⣳⣄⠀⢀⡤⠬⠗⠀⣸⠋⠉⠉⠛⢦⡀⠀⢠⡟⠀⠀⠀⠀⢀⣀⢸⡀⠀⠉⠿⣖⠚⠶⡏⠁⠀⠀⠀⢈
⠀⠀⠈⠙⠻⠶⢶⠭⠭⠿⠿⠏⠈⠓⢮⣽⣄⣠⠾⣅⠀⠀⣀⡤⣾⣥⠴⠾⠳⠶⣄⡴⣆⡞⠙⠞⠓⠒⠚⠏⠻⠛⠲⠿⠶⠤⠤⠖⠋

⠀⠀⠀⠀⠄⠀⠀⠀⠰⡀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡯⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠄⠀⠀⠀⢸⣣⠚⢀
⠀⠀⠀⠀⠀⠀⠀⠡⣼⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆
⠀⠀⡴⠀⠀⠀⠀⠠⢻⠐
⠀⠀⠂⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⣰⠃⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⠤⠤⡤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣶⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡶⣶⡶⣦⣤⣀
⠀⠀⠀⢀⡔⠋⠀⠀⠀⡇⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢠⡶⠋⠁⠀⠀⣿⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⢀⣶⠟⠉⠀⠀⣿⡆⠀⠈⠙⢷⣄
⠀⠀⢠⠏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡀⠀⠀⢰⠏⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢻⡄⠀⠀⢠⡿⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣇
⠀⠀⡞⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣇⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣧⠀⠀⣾⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡄
⠀⠀⢇⠀⠀⠀⠀⢀⡔⡿⢦⡀⠀⠀⠀⠀⡟⠀⠀⣷⠀⠀⠀⠀⢀⡴⣿⢦⡀⠀⠀⠀⠀⡿⠀⠀⣿⡄⠀⠀⠀⢀⣴⣿⣷⣄⠀⠀⠀⠀⣿⠃
⠀⠀⠘⡄⠀⢀⡔⠋⠀⡇⠀⠙⢦⡀⠀⣰⠃⠀⠀⠹⣆⠀⢀⡴⠋⠀⣿⠀⠙⢦⡀⠀⣸⠃⠀⠀⠸⣧⠀⢀⣴⠟⠁⣿⡇⠙⢷⣄⠀⣰⡟
⠀⠀⠀⠘⢶⡋⠀⠀⠀⡇⠀⠀⠀⢙⠖⠁⠀⠀⠀⠀⠙⢶⣋⠀⠀⠀⣿⠀⠀⠀⣹⡾⠃⠀⠀⠀⠀⠙⢿⣟⡁⠀⠀⣿⡇⠀⠀⣹⣿⠟
⠀⠀⠀⠀⠀⠈⠓⠢⠤⠷⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠿⠴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⣿⡷⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠶⠶⠦⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠦⣤⣀
⠀⠀⠀⢀⣴⡿⠟⠋⢹⣿⠉⠙⠻⢷⣦⡀⠀⠀⠀⠀⡠⠚⣡⠴⠒⢳⠀⡗⠒⢤⡉⠢⡀⠀⠀⠀⠀⣠⠞⣋⡴⠖⢻⡆⣿⠓⠦⣍⠳⣆
⠀⠀⣰⣿⠋⠀⠀⠀⢸⣿⠀⠀⠀⠀⠹⣿⡄⠀⠀⡜⢡⠞⠁⠀⠀⢸⠀⡇⠀⠀⠈⢢⠘⣆⠀⠀⣰⢋⡼⠋⠀⠀⢸⡇⣿⠀⠀⠈⠳⡌⢷⡀
⠀⢀⣿⠇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢹⣿⠀⢸⠁⡏⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⢧⠸⡄⢰⡇⣼⠁⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⢻⡈⣇
⠀⢸⣿⠀⠀⠀⠀⢀⣼⣿⣦⡀⠀⠀⠀⢀⣿⠀⢸⠸⡇⠀⠀⠀⣠⠞⠀⠣⡀⠀⠀⠀⢸⠀⡇⢸⡇⣏⠀⠀⠀⣠⡾⠁⠻⣄⠀⠀⠀⢸⡇⣿
⠀⠀⣿⣆⠀⢀⣴⡿⢻⣿⠛⢿⣦⡀⠀⣸⡿⠀⠸⡀⢧⠀⣠⠞⣡⢶⠀⡦⡈⠢⡀⢀⠏⢸⠁⠘⣇⢻⡄⣠⡾⣫⣶⡆⣶⣌⠳⣄⠀⡼⢠⡏
⠀⠀⠘⢿⣶⡿⠋⠀⢸⣿⠀⠀⠙⣿⣾⡿⠁⠀⠀⠱⡌⠛⣡⠞⠁⢸⠀⡇⠈⢢⡈⠋⡰⠃⠀⠀⠙⣦⠛⢫⡶⠋⢸⡇⣿⠈⠳⣌⠛⣡⠟
⠀⠀⠀⠀⠙⠿⣶⣦⣼⣿⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠈⠢⣌⡑⠒⠞⠀⠷⠒⣉⡤⠚⠁⠀⠀⠀⠀⠈⠳⢮⣙⠳⠾⠃⠿⠖⣛⣡⠞⠋

⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡀⠀⢹⠆⠀⠀⢀⣤⣠⡏⠉⠃⠀⠈⡗⠒⠀⠐
⠀⠀⠀⠒⠶⡶⢼⡶⣶⣦⡀⠀⠀⠀⠂⠀⠀⠀⠂⢤⠤⣘⣤⡟⠲⡁⠠⠒⠱⠋⡇⠀⠀⠀⠀⠀⠀⠀⢀⡼⢅
⠀⠀⠀⠀⠀⠀⠀⡇⠈⡌⢿⠄⠈⠀⠀⠀⠀⠀⠀⡸⢐⠛⠀⠀⢀⣄⠀⠀⣤⠐⠉⠁⠀⢠⣾⢻⣻⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣣⠀⡇⢸⠞⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠈⠀⠀⠀⠏⠈⠲⠒⠊⠉⠙⠋⠁⠀⠀⠀⠈
⠈⠺⡄⠀⠀⠀⠜⡿⠾⡟⠦⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⠀⠄⠀⠀⠉
⠀⠀⠈⢩⠖⠀⠀⡅⠀⢀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠀⠀⠀⠋⠀⠀⣠
⠀⠀⠀⠈⠀⠀⠤⢰⡤⠼⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⠶⠀⠃⠆⠀⠇⠶⠘⠰⠆⠖⠒⠓⠀⠀⠀⠀⠀⠀⠀⠤
⠀⢀⢠⡴⠶⣶⠉⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠄
⢀⠔⠚⠂⠀⠈⠀⠀⠜⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣄⣤⣾⡛⠛⠯⣃⢄
⢸⠄⠀⠀⠀⢀⣤⣤⡄⠀⠐⠋⠃⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⣀⣴⡜⠛⠦⠀⠙⠦⡈⢳⣃
⢀⠀⠀⢀⣨⣿⣭⣷⠁⠀⠀⠠⢄⠀⠀⠂⠐⢀⡀⠉⠀⠀⠀⠀⠆⠠⢤⣿⣤⣔⠉⠀⠇⠀⠀⠑⠄⠀⠱⠄⣿
⠈⠟⠉⠁⠀⠈⠉⠀⠀⠀⢀⣀⠈⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⢡⠈⢇⠀⠀⠀⠀⠀⠈⠆⠀⣧⡟
⠀⠀⡀⢀⠄⠀⠀⢠⠀⠀⠀⠁⣠⠎⠀⠀⠀⠠⠀⠀⠀⠀⠢⣀⡀⢈⠀⠀⡰⢀⣿⠄⠀⠐⠀⠀⠀⢸⡴⠋
⣤⣤⣥⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣥⣤⣽⣶⣶⣿⣬⣥⣤⣵⣤⣤⣵⣦⣥⣤⣤⣤
⣿⣯⣭⣭⣭⣽⣭⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠈⠁⠀⠠⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⡇
⡇
⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠉⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠲⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠄⠀⢀⠀⠀⠐⠁⠤⠤⠤⢀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⣀⠂⠈⠁⠀⠈⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠙⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠂⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠄⢨⣿⣤⠀⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⢻⠀⠀⠀⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢰⠀⠀⠀⢸⠇⠀⠀⠃⡀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣺⢣⠀⠀⡄⢡⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⠰⠆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠠⠀⠀⠀⠀⠀⢶⠘⡇⠀⠀⠸⡈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠐⠀⠀⠀⢀⠂⠈⢃⠀⠀⠀⢃⠡⡀⠀⠉⠦⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⢥⣈⡄⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡁⠀⢹⠀⠀⠀⠸⠀⠈⠢⣀⠀⠈⢓⣴⡀⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠉⢉⠐⠂⢤⠈⠀⠀⠀⢰⠀⠀⠀⠀⠈⢳⡀⠆⠀⠀⠀⢠⠀⠀⠀⠀⠱⠄⠀⢠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⠀⠀⠠⢕⡠⢀⠀⠀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣈⠀⠠⠄⠀⠔⠂⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠈⠁⠞⡴⠋⠛⠿⠿⠿⠿⢃⠿⠿⠛⠋⡙⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠈⠁⠐⠃⢠⡄⠠⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠈⠲⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⣿⣷⣶⣿⣍⠹⣦⡀⠀⠀⡀⠠⠐⠈⢀⣐⡳⣶⣦⡉⠀⠆
⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣵⣴⣧⣬⣉⣤⣤⣤⣤⣄⣀⣠⣌⣉⣤⣤⣤⣤⣬⣿⣿⣷⣌⣠⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡀⠀⠀⠀⣤⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠅⣰⠀⠀⣤⠀⠀⢳⣀⠀⡆⠈⠁⡀⢶⠂⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠶⡘⢃⣤⡘⠊⠷⢿⣽⢶⡗⡈⢠⣼⡇⠉⣿⡆⢆⡀⣴⠆⠀⢸⡀⠀⠊⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡀⠀⠤⢻⠇⠀⢀⠰⣿⡁⠐⣴⣇⣈⡈⢸⡯⣠⣿⣠⣾⣿⡵⣾⢷⠿⢃⡆⣈⡅⠰⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠸⠉⡀⠀⠈⠁⡼⣄⠀⢾⡄⢰⣾⣽⣿⣧⣾⣷⢙⡿⢿⣿⣿⣿⣷⡛⡷⣾⠁⢻⡅⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢰⠛⢙⣻⣯⣿⣸⣷⣎⣿⣿⣿⣿⣿⢿⣿⣿⣾⢿⣿⣾⣿⣿⣴⢄⣴⡮⣱⣦⢀⡔⠀⠈⠁⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⢀⡀⢀⣿⣿⢾⣿⣿⣿⣿⣿⣿⡿⢿⢿⣿⣿⠿⢿⣾⣿⣷⣾⣿⣿⣿⣿⣷⣏⠈⠁⠀⣀⠘⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⡟⢻⠀⡉⢽⣀⣽⢛⣼⣿⣿⣿⠿⣿⠇⠀⠠⠆⠀⠀⠀⠙⢿⣻⣿⣿⣿⣿⣿⣿⡿⣿⣿⠀⠾⠋⠲⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⠾⣣⣴⡞⢿⣩⣿⢻⣿⣿⣏⣶⡀⠀⠀⠀⠀⠀⠀⣀⣴⡾⢿⣿⣿⣿⣿⣿⣿⣿⣼⣯⡘⣷⢻⡿⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢹⡍⠳⠛⣏⢰⡟⣻⣷⣿⣿⣾⣿⣁⣀⣀⣀⠀⠀⢠⣾⣿⣤⣶⣊⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢰⣆⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢢⠓⢶⠲⣷⣸⣽⣿⣿⣿⣿⣿⣿⢿⣯⣿⣿⣿⡆⠀⡿⠿⢿⡿⡿⠟⢿⣿⣿⣿⣿⣿⣿⣭⣿⡿⣠⣤⣀⣔⡂⠰
⠀⠀⠀⠀⠀⠀⠠⠀⠠⠄⠀⠐⣤⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣥⣿⡿⠋⠀⠀⠀⠀⠀⠁⠀⠀⠾⣿⣿⣿⣿⣿⣽⣟⣹⠏⢿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡦⣄⣶⣎⣿⣿⣷⣿⣿⣿⣿⠆⠉⠛⠉⠀⠀⠀⣤⣴⣶⡶⠆⠀⠀⠀⠀⢻⣿⣿⣿⢿⣿⡟⣥⢸⡿⠁⠈⠢⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⡏⠩⠞⠙⠶⢿⣿⣷⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠈⣉⣹⣤⣤⣤⡔⠁⠞⢻⣿⣿⣿⣿⣿⣿⡟⠟⠓⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠘⠂⣠⡼⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣧⣂⡄⢠⠀⠀⢀⣾⣟⣛⣩⣤⡶⢀⠂⣤⣼⣿⣿⣿⣿⣿⣿⢉⡆⠊⠀⠀⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠠⢄⠀⢡⣼⡶⣽⢲⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⠀⠴⠛⠿⠟⠛⠲⡾⠇⣸⣿⣿⣿⣿⣧⠙⠷⠼⣧⡾⠃⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢿⡷⣟⣹⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡄⠀⠀⠀⠀⠀⠲⣾⣤⣾⣿⣿⣿⣿⣹⣿⣷⣠⡄⠈⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢞⡉⢛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣞⣶⡴⠿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣛⣉⣡⠾⣴⠦⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠒⠈⣡⠿⠏⠉⠙⠉⠙⣛⣛⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⠛⠻⠆⠀⠄⠈⠈⠉⢻⣿⣿⡉⠛⠿⠋⣁⠘⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠉⠉⣙⣿⣿⣿⢿⣿⣿⣇⣀⠘⡷⠄⠀⠀⠀⠀⠀⠀⣹⣿⣇⠙⠀⣤⠀⠉⢠⠆⠀⠀⠘⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢛⣿⣿⣿⢎⠛⠛⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⣸⠧⡎⠁⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣭⣿⣷⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠉⠉⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢡⠏⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⢠⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⡤⣀⢀⣠⡄
⠀⠀⠀⠀⠀⢠⣿⣟⢻⣿⣿⣯⣹⣿⣿⣿⣾⣯⡿⣖⡿⣿⢿⣿⣏⣿⡷⣯⣯⣹⣿⣻⣿⢿⣿⡷⣟⠻⣿⣷⣦⣤⡄
⠀⠀⠀⠀⠀⢘⡛⢯⣽⣿⣿⣯⢣⢇⣦⣳⣸⣯⣿⣯⣟⣿⡮⣿⣻⣯⣿⡮⣻⠛⢫⣿⣓⣱⢇⡨⢙⢯⣞⢿⣾⠃
⠀⠀⠀⠀⠀⢘⡹⡟⣉⣟⣿⣿⣏⣸⣿⣿⡿⣜⠋⣿⣿⣾⡷⢹⣿⣿⢟⣿⢃⣼⣛⢿⣿⡉⣿⢿⢻⣷⣿⡿⣾⣷
⠀⠀⠀⠀⠀⠸⣏⣻⣿⢉⣿⢃⣹⣽⡿⠙⢃⡠⠚⣹⣿⣿⣥⣈⣯⣵⣯⣧⢟⢶⡣⢴⠊⣴⢟⣿⣾⡾⣷⣿⢖⣭⠁
⠀⠀⠀⠀⠀⢸⣿⣿⢯⣿⣯⣿⡷⣃⣤⣾⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣿⣿⣿⣿⣿⣷⣚⣿⣯⡷⢿⣱⡾⠅
⠀⠀⠀⠀⠀⠀⠙⡘⠐⣷⠻⣿⣴⣿⣿⣛⢛⡙⣿⣿⣿⣿⣿⡟⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⠻⣯⡝⣫⣿⡿⣏
⠀⠀⠀⠀⠀⠀⡀⠅⠀⣿⠀⠇⣿⣿⣿⠈⠉⠛⣿⣿⣿⣿⡟⠉⠋⢹⡎⣿⣿⣿⣷⣦⣿⣿⣿⣿⣾⡋⠉⠁
⠀⠀⠀⠀⠀⠸⡄⣠⠀⠀⠄⢘⣿⣿⠿⠦⢐⠐⠿⢿⣿⣿⣷⣄⡀⠘⠿⠟⣿⡿⠛⣿⡿⣽⠿⢟⡽⠗
⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⢀⠘⠉⠀⠀⠀⠈⠂⠀⠀⣿⣿⡉⠀⠘⠃⠀⡀⠈⠁⠰⣿⢃⣾⠿⠛
⠀⠀⠀⠀⠀⠸⠉⠑⠂⠠⠸⣷⢤⣀⣂⠀⠀⠀⠀⡀⢽⣿⡷⠀⠀⠀⠀⡟⠀⠠⣷⣽⡸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣿⠏⠛⠙⠛⠉⠁⣸⡿⠁⠀⠀⠀⠀⠁⠀⠸⣿⣿⠧⠉⣀⣄⠀⡀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⣸⡄⠀⢰⣥⣞⡀⡠⠤⣀⠠⢄⣤⡒⠉⠙⠿⠰⡿⠁⠐⡼⣀⠛⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠘⠿⠿⠿⠃⠀⠛⠄⠈⠿⢿⣷⣄⠀⠀⣇⠀⠀⠀⠈⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠋⠑⡀⠸⢿⣳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡆⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⣷⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢀⣾⣷⣆⠀⡃⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢠⠠⢻⡿⣿⣿⣁⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⢛⣾⣿⣏⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠙⠻⣯⡟⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠚⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⠀⢀⡀⠀⠀⠀⠠⡀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⠀⢀⠂⣄⠀⠀⠰⠀⠀⠀⡄⢀⠀⢠⡀⠀⠀⠀⠄⢠⠄⠀⢀
⠀⡮⠷⠀⣠⠈⠀⠀⢀⡀⠀⠙⢆⠀⠈⠁⠀⡛⢌⡀⠀⠤⠀⠀⢻⡁⢲⠀⠈⠱⠀⢁⡁⠀⠺⠛⠁⢀⠖⠀⠘⠀⠀⠫⢠⣫⠂
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠈⠺⠀⠠⠀⠀⣴⠼⠧⢹⠀⣁⠁⡀⠐⠆⠀⠂⠀⠀⠉⠐⠀⠀⠀⡀⡂
⠀⡐⢖⠩⢀⠀⠀⢀⠴⠒⠈⠉⠉⠓⠂⠀⠆⠀⣄⣀⣠⡄⠀⣈⠉⠉⠁⠀⣤⠀⠃⡀⠀⠀⠀⠀⠀⠀⠊⠀⠠⣠⣀⠈⢉⡆
⠀⣚⣅⠤⠑⠀⣀⣀⡀⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢾⣿⠄⠀⠀⠄⠀⠀⠘⠀⠀⠇⠸⠀⡀⠉⠀⠄⠀⠠⡀⢉⠁⠀⠁
⠀⠠⡈⠁⡐⠀⡉⠀⠉⠉⠁⢻⠂⠉⠉⢁⣍⡁⠘⠒⠊⠀⢀⠁⠀⠤⠄⠀⡐⠂⣂⠀⠂⠁⠀⠀⠈⡀⡋⠁⠈⠏⣈⣁⠄⢶⢀
⠀⠠⣈⡁⠀⠀⢹⠃⠉⠀⠀⢸⠀⢤⠀⠨⢱⠀⠀⢰⢈⣄⠊⠀⠀⠀⠀⠀⠂⠀⠉⣉⠰⢢⡄⠀⢀⠁⠁⠀⠈⠱⠛⠓⠀⢠
⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⠀⠀⢪⡇⠀⠀⠀⠀⢈⠀⠀⢀⠀⢀⠄⣴⡀⠀⡀⡀⠀⡀⠘⠛⠃⠀⠀⡀⠠⣀⠂⠄
⠀⠠⠁⠀⡄⠂⠁⠀⢀⡀⢴⡸⠀⣀⠀⠠⠈⡇⠀⠁⠀⠀⠘⠀⠤⠚⠃⢀⣸⠁⡱⠀⢠⣴⠀⠹⢄⠀⠀⢀⠀⠀⠠⠐⠐⡀
⠀⢐⡂⠀⠈⠈⠀⠀⡉⠀⠀⡀⠘⠀⠄⠀⠠⠐⠄⠄⡀⠀⠀⠀⠒⠀⠀⠀⠉⠀⠀⢀⣤⣌⡄⠀⠒⠋⠀⢀⣄⠀⠀⠀⠀⠇
⠀⠈⠃⠀⠁⠐⠀⠺⠗⢀⢀⠁⠀⢀⠀⠠⠂⠀⢈⠂⠡⠀⠀⠀⠀⢠⠄⠀⢶⣦⣤⡄⠀⠀⠁⠀⢀⣶⡤⠐⢀⣉⢓⠄⠀⢀⠄
⠀⠤⠀⠀⠐⠀⠂⠀⠀⠀⠀⢀⠀⠈⠂⠀⠁⠐⠀⠤⠐⠂⠀⠠⠀⠢⠄⠀⡈⠀⠀⠉⠀⢰⡇⠀⣄⢻⣺⣟⣃⣞⣿⣯⡄⢀⡀
⠀⠀⠀⠀⢂⡁⡀⠀⢦⠀⠀⠢⠘⠀⠀⡅⠀⠀⠂⠀⢈⠁⠀⢀⠢⠀⢐⡀⠂⠀⠀⠀⠀⠈⠁⠘⠉⠃⠈⡏⡍⠋⠈⠁⠑⠀⠁
⠀⢌⠤⠀⠈⣄⠁⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢜⠀⠈⠀⠀⠀⠑⠀⡀⠂⠀⠀⡀⠀⣤⣄⡄⠀⠀⠃⠀⠀⠠⠀⠀⠦
⠀⠠⠎⡄⠀⠘⠀⠀⠒⠋⢀⠀⠃⠉⠘⠀⠃⠉⠐⠁⡀⠈⠁⠀⠀⠀⠀⡀⠈⢴⡥⠀⠀⠃⠀⠁⢛⠃⠀⠀⠀⠀⠂⡀⢠
⠀⢁⢚⡠⢡⠀⠀⠀⢀⠀⠉⢀⡁⠀⠁⠉⠁⢈⡁⠈⠁⣠⠘⠠⣀⣀⠶⠃⡀⣈⠈⠁⠀⠀⠀⡀⠘⠶⡄⠓⠃⠠⠀⠉⠠⠠⠆
⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⢀⠃⢀⠀⠀⠀⠀⠀⠀⢀⠈⢠⣀⠀⠀⠀⢠⠅⡊⠁⠀⠀⠀⠀⢉⡁⢁⠁⠀⠂⠀⠄⠠⣌⡀⠂
⠀⡎⠀⠀⠀⠆⠀⠰⣶⡴⣧⡄⠩⠤⠀⠀⠐⠋⢱⠀⠀⠈⢐⠃⠀⣀⠀⠐⠒⠁⠀⠒⠂⢠⠀⠀⢂⡊⠒⠀⠀⠀⡐⠀⠉⣀⡀
⠀⠀⠘⠈⠣⠀⠀⠀⠋⡏⠉⠇⠀⠀⠁⠄⠀⠀⢊⢄⠀⠁⠁⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⡐⠀⠀⠇⠐⡁⠀⢠⠀⠀⠀⠩⠐
⠀⠀⠦⠀⠄⠀⠲⣒⠚⢁⣐⡇⠀⠀⠤⠀⠀⠄⠀⠽⠇⠀⠠⠄⠁⢀⠀⢀⡂⠈⡆⠙⠂⠁⢰⠘⠀⢩⢨⠅⠀⠒⠀⠀⠀⠀⠅
⠀⠀⣀⢠⠀⠀⢠⡌⠉⠉⠉⠁⠀⠄⣷⠲⠀⣠⢶⣖⠤⣄⠀⠲⠆⠀⠀⠶⢄⠀⠀⠀⠀⢤⠖⣆⠀⠒⠰⠂⠀⠀⠀⠐⠔⠁
⠀⠀⢘⠬⠆⠀⢈⠉⠢⠀⢀⠈⠀⠀⠀⠀⢨⣶⠋⠉⠙⢶⠆⠀⠄⢀⠘⠈⠁⠀⠀⠀⠀⢊⡗⢤⠀⠀⠂⠀⠤⣥⢀⢤⠀⠀⢄
⠀⠀⠈⠓⡯⠂⠈⠠⢀⠀⠀⠉⠀⠗⠀⠀⠈⢭⣷⣠⣶⡿⠂⠀⠂⢁⠀⢀⠀⠘⠉⠀⠀⠼⠧⣐⠀⠂⠀⢤⠀⠁⠀⠀⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣈⣁⣈⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠀⠀⠐⠀⠀⠀⢰⡇⡀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠂⠀⠀⠀⠟⠁⠀⠀⡄⠀⠀⠀⠀⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⢹⠂⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⡇⠀⠀⡔⠀⢸⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣴⣄⠀⠈⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⢸⠇⠀⠈⠀⠀⠘⠃⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣍⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠸⣄⠐⠁⠀⢠⣶⠶⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣆⠀⠀⡱⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⡀⣰⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣷⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠛⠡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⣠⣤⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠀⢠⡄⠀⠀⠨⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣛⣿⣆⠰⣄⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠙⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠋⠀⠀⠀⠀⠋⠀⠀⠀⠈⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠓⠀⠀⣠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⡄⠀⠐⡀⠀⠀⠀⠀⠁⠀⠀⠀⡛⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢁⡄⢸⡇⠀⠀⠀⠀⠀⢀⣽⣆⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣼⡇⢈⣧⠰⡀⡄⢀⠐⢸⣿⠿⣸⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠏⠈⠓⠀⠁⠈⠀⠉⠀⠀⣿⢸⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⢠⠀⠀⠀⠀⠀⠀⠀⣰⠏⠸⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣤⣀⠀⠀⣀⣠⠾⠏⠀⠀⠁⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⢹⡝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⣟⡒⢦⣤⡀⠀⠀⠁⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠀⠀⠙⠛⠛⠂⠉⠁⢀⣦⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣀⠀⠀⠀⠀⠀⣠⣿⣿⣧⠀⢻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⠀⠀⣶⢺⣿⡟⣿⣿⡇⠀⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠻⠯⠿⠟⠿⠿⠆⠀⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠁⠁⠂⠀⠀⠀⠀⠂⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠘⠀⠁⠀⠀⠐⠀⠀⠂⠂⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣿⣿⣿⣿⣿⣾⡘⡟⡷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡇⣿⣿⣿⣿⣿⣿⣿⢹⣧⣳⢃⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⢣⢿⣿⣿⣿⣿⡿⢃⣾⢧⣏⢸⣿⡿⣿⣿⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⣸⡇⢸⣾⣿⣿⣿⢏⣶⣫⣿⣿⣿⢸⣿⣽⣿⣿⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢰⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣷⣿⣿⣿⣿⢹⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣼⣧⣸⣿⣿⣿⣏⡿⣿⣿⣿⣿⣿⣿⢹⣿⢿⣿⣿⣼⣿⣈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣧⣻⡌⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⢿⣿⣿⡻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⡟⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⣿⣷⣾⣿⣿⣿⣟⣿⣿⣯⢻⣿⣿⣿⠟⢳⣿⣿⣿⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣿⡿⣿⣿⣿⣿⣿⡟⣿⣿⡿⠁⣼⣿⣿⣿⣿⣣⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠛⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣥⣾⣟⣿⣿⣿⢿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⠻⣿⢿⣿⡿⢛⠯⠋⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⣿⣷⡾⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣻⣿⢿⠿⠟⠛⠉⠉⢡⣴⣿⣯⡀⠀⠈⠉⠉⠛⠿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷
⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⡿⠋⠁⣿⣿⣿⣿⠀⠙⠻⠿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣸⣄
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⣀⣤⣼⣿⣿⠏⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠹⢿⣿⣿⣷⣦⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⣴⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣄⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣀⣀⠀⠀⢻⣿⠻⡿⠀⠀⣀⣠⣴⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠂
⣀⠤⠊⠉⠹⣍⡓⠦⣄⣀
⠀⠀⠀⠀⠀⠙⠮⠑⠢⠉⠙⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠤⠂⠀⠀⠀⠀⢀⣀⡠⠤⣤⠒⠒⠀⠀⠉
⠄⠒⡈⠀⠀⠀⠈⡐⠄⠀⠀⠀⠀⠀⠉⠻⣷⢶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠶⠒⣚⠉⠉⡧⠀⠀⠀⠈⠳⣀⡀⡠⠤⠀⠄⠁
⠀⢀⣇⠀⠀⠀⠀⠘⠤⠀⠀⠀⠘⣄⣀⠀⢹⣍⡪⠀⠉⠉⠓⠲⢶⣞⣋⣩⠭⠥⠖⣒⣚⣉⠡⠤⠄⡘⠀⠀⠀⠀⠀⠀⠈⠑⠆⠈⠉
⠐⢻⣿⠷⠂⠀⠀⠀⢠⣆⡀⢀⡆⠈⠙⠧⠈⡗⠨⡂⠀⠀⠀⠀⠀⠈⢿⠛⢛⠿⢭⣥⣒⣒⣂⠀⠀⠁⠀⠀⠀⠠⢄⡀
⣤⠾⣟⡛⠴⣧⠄⡀⠼⣿⣤⠾⡷⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⡀⠀⠀⠒⠀⠀⠀⠀⠉⠙⠛⠤⢤⣀⡀⠀⠀⠀⠀⠙⠢⠀⠀⠀⢰
⣤⣾⣿⣿⣾⣿⢛⣿⣶⣿⠶⢟⣿⠂⠀⠀⠀⠀⠀⠀⠑⠢⠀⠀⠈⠑⠒⠂⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠉⠉⠐⠂⠀⠀⠀⠀⠀⠔⣿⡃⠀⠀⠠
⣠⣾⣷⣿⣿⣿⣿⣿⢛⣿⢾⢟⣿⠶⠤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣠⡤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⣿⣿⣧⣼⣶
⢉⣿⣿⣿⣿⣿⣥⣿⣟⣿⣛⣻⣟⠗⣚⣳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢟⣿⠻⡿⢾⡷
⣋⣡⠿⠦⠽⠧⢭⢿⣿⣿⣉⣻⡛⠛⣿⣿⠛⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠒⠒⠒⠀⠤⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠲⠟⣻⡟⣿⣿⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠋⠒⢄⠀⠉⠙⢲⡄⠀⠀⠀⠁⠀⠀⠘⠛⠉⠙⠋⠉
⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢚⠀⠀⠀⢀⡼⠔⡊⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⡄⣠⠆⢀⣀⠀⠉⠒⠢⠤⣀⣌⣀⣒⠂⠤⠤⣤⣤⣦⣀⣠⣤⣤⡤⠼⠛⠛⠁⠀⠀⡀⠘⠀⠀⢰⠀⡀⣀⠀⡀
⠀⠀⠀⠀⣀⠒⣾⡿⠛⠒⠾⠿⠷⢤⣤⣀⡀⠰⠀⣈⣽⣿⣿⣟⡛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⣀⠤⢶⠚⣿⣋⠏⡜⠁
⠀⠀⠀⠀⣿⢰⣟⣵⡞⠀⢀⡆⠀⠀⠀⠏⠉⣿⣾⠟⠉⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⣤⣀⠀⣂⣹⣟⡼⢁⠂⠀⠀⣀
⠀⠀⣦⢸⣧⣿⠟⣉⠀⠀⣿⠗⠠⠤⣤⣼⣸⡙⣿⡛⠛⢲⡖⠒⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⢀⠈⠻⣯⣄⠀⠀⠉⡈⠉⠉⣸⣿⣆
⣷⠀⣿⢸⡟⣡⡾⠋⠀⠀⠏⠀⣠⣾⠟⠈⡟⠉⠉⠛⢶⣽⣉⡉⣩⠋⠀⠀⠠⠤⠤⠤⠄⠀⠀⠀⠀⠀⠈⠉⠽⠿⠿⢿⣦⣾⣿⠀⢀⣷⠈⣿
⣿⣴⣿⣿⣿⠋⡀⣀⠀⡶⢀⣶⣿⠋⠀⠀⠀⢒⠦⣄⣸⡟⣯⡽⠯⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⡀⡀⠄⠋⠑⡮⠙⢿⣷⣮⣿⣆⣻⡀
⢿⣿⣿⣟⣵⡿⠁⣿⢀⣿⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⣇⡼⣷⣦⣤⣄⣀⣀⡀⠀⠀⣀⢀⡀⠓⠛⠀⠳⠀⠘⢀⡀⢸⡆⠒⠒⢻⠛⢻⣿⣿⣷
⣸⣿⣿⠟⣁⣴⠶⣿⣜⣿⡟⣯⢿⠦⣤⠶⢦⣤⡥⣄⣠⡞⠀⣿⡗⠢⠄⠨⣍⣁⠲⠶⠶⠛⠀⠀⠀⠀⠀⠉⠉⢉⠽⢿⣿⣄⠘⠙⣏⣹⣉⣩⣿
⣿⣿⣿⡾⠟⠁⠀⣿⣿⠟⢻⡟⠘⡇⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣶⣞⣢⣴⣯⣀⣉⡱⠈⠉⠉⠉⠉⠉⠁⠀⠈⠀⠤⠌⠹⡏⠈⠉⢻⡹⡏⠉⢹
⣿⣿⡿⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⢠⣧⠀⠀⢠⣀⠠⣿⣿⠀⠀⢨⣶⣦⣀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⢶⡿⠀⠀⢘⢿⣷⣿⣯⣶⣶⠀⠛⢿⣯⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠘⣾⡷⠂⠀⠛⣻⡿⣿⣿⣯⡀⠰⢶⣿⠿⣇⣴⣤⡦⠹⣿⠇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣤⠚
⠀⠀⠀⡀⠀⠀⣠⣾⠀⠈⠀⠘⠃⢩⣤⣤⣴⡄⠀⠉⣁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠃⠀⢢⠇⠀⢀⠄⠀⢀
⠀⠀⠀⣿⣧⣰⣿⠃⠀⠀⠀⣤⣾⣿⣿⣿⠟⢀⣴⣾⣯⡀⢰⣿⣧⠀⠀⠀⠠⡀⢰⠁⠈⠣⣀⡼⠒⠋⠁⡠⠏⠁
⠀⠠⣤⣼⣿⣿⣷⣶⡷⠂⠀⠛⠛⠟⠛⠉⢠⣿⣿⣿⣿⣿⠀⠻⠿⠀⠀⠀⠀⠀⠘⠢⣀⡀⢸⡇⣀⠤⠯⣤⠤⠤⠄
⠀⠀⠀⠀⠉⠻⣯⡀⠀⠀⢼⣿⣦⣶⣿⠀⠀⠉⠙⠻⠿⠛⣀⣧⣀⠄⠀⠀⠀⠀⠉⠙⠉⠙⣾⡿⠁
⠀⠀⠀⠀⣠⣤⣄⡁⠀⢀⣼⣿⣿⣿⣿⣿⠆⠀⠀⢀⣴⠀⣺⣷⣻⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣆⠈⠛⣿⣿⠟⠉⠀⠀⠀⠀⢿⡿⠀⢺⣿⣺⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣠⡉⠉⠉⠉⠀⠀⠙⠁⢰⣤⣄⠀⠀⣤⣬⣶⣆⠀⠛⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠾⠟⠣⣦⡿⢿⣠⣤⡤⠀⣾⣿⣿⡇⢸⣿⠇⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠈⠛⠋⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠚⠛⠛⠉⠉⠁⠀⠈⠉⠉⠉⠛⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⣤⣄⡀⠀⠀⣠⡴⠚⠉⠉⠲⠀⠀⠀⠀⣀⠀⠐⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⡏⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠀⡉⠿⠿⠟⠛⠿⡲⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⣠⣤⣭⣀⠀⠀⠀⠀⠀⠀⠘⣏⢿⡈⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠉⠻⢿⣿⣷⣦⡀⠀⡀⠀⠀⠙⡎⢳⡘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⢿⡍⢻⣿⡄⣿⡖⠆⠀⠃⠸⠁⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⣋⡇⠸⠁⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣿⡇⢆⠀⢀⠀⣠⡄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠚⠉⠨⠀⠈⢻⣄
⠀⠀⠀⠀⢀⡼⠋⢀⣶⣿⣿⣿⡆⢹⣤⣿⣿⡄⠀⠀⠀⢠⡾⠻⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⡞⠁⠀⢈⣿⣿⣿⢿⣷⡌⠉⠛⠻⠃⠀⠀⠀⠋⠀⠊⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣇⠈⢿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠘⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⣾⠀⡘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣧⡀⠙⠿⣷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⣉⣠⠄⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡗⠀⠀⠀⠈⠙⠏⠑⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠹⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠑⠓⠒⠦⠤⣤⣤⡤⠞⠁
⠀⠀⠀⠀⠀⠘⢷⣆⠐⠀⠀⢀⢿⣿⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠈⠀⡷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠠⠀⠀⢰⣦⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐⣿⡿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⣿⣏⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠈⢷⣼⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⣿⣧⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣿⣷⣴⣿⣶⣶⣤⣤⣴⣾⣿⠋⢻⣿⣶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠿⢿⣿⣿⡀⣹⣿⣿⣿⣿⣷⣿⣿⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣾⣿⣿⣿⢿⣿⣿⣿⠿⠿⢿⠛⢿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠛⠋⠀⠀⠁⠈⠀⠀⠀⠀⠙⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⢀⡀⠀⣀⣀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡦⣶⣾⣿⣶⣶⣤⣤⡀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠶⠦⠤⣤⣀
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢻⣧
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡀⠙⣿⣿⣿⣿⣶⡄⠀⠀⢿⣿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠑
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⢻⣦⡢⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⣿⠠⡆⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⡿⠁⢰⠙⠻⣧⡀⠙⠳⢦⡀⠀⣠⣤⣶⣶⣿⣶⣶⣶⣶⣾⠆
⠀⠀⠀⣿⢰⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣾⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⠈⠉⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣤⣤⣤⣤⣤⣤⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⣾⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠇⠀⠉⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣇⠘⢀⡀⠀⠀⢐⣦⡄⠀⠀⢣⠀⠀⠀⡜⢁⡾⠿⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡷⠊⠁⠀⠀⠀⠀⢑⣤⡜⠁⠉⠛⢿⣤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⢀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠈⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⠷⠄⠀⠀⠀⢰⠀⢠⣾⣿⣶⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⡄⠀⠀⠘⠀⠀⠈⠉⠋⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢺⢀⠀⢀⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣷⣴⡆⣲⣷⠉⠀⡈⢲⣿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡙⢘⢿⣿⣯⣿⡄⢀⣤⣿⠏⣸⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣅⡀⠀⣀⣽⣿⣷⣯⢿⠇⠸⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⣩⣿⣿⣿⠈⣤⣀⠈⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⡟⠀⣿⡿⢀⣬⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠁⣤⣬⡀⣾⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢻⣿⣧⠘⢿⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣷⠀⠈⠁⠀⢈⠀⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠸⣿⣾⠀⠸⣷⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠄⠻⣿⠂⠀⠻⠿⣿⡁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠁⢘⣀⡠⠤⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠊⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⠊⠀⢀⢤⡤⠆⢁⠠⢄⣰⡭⡄⣤⡤⣌⢑⡄⠀⢠⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⣸⠉⠈⣖⠁⢹⢊⠇⢸⢪⡀⠀⣇⠀⢰⡏⠀⠘⠁⠘⠆⠉⡶⠂⢻⠤⢖⢃⠠
⠀⠀⠀⠀⠀⠀⠘⡤⠶⠀⢿⣧⠀⣽⠀⣜⢸⠀⡏⠁⠇⢠⠇⡇⢸⢹⠀⡴⠀⢀⡆⠀⡟⡄⢠⡄⠸⠂
⠀⠀⠀⠀⠀⠉⢨⠸⠸⠀⢸⡏⠀⣿⠀⣇⣾⠀⢧⠀⡇⢸⠀⡇⢸⢸⠀⣧⠃⠸⡟⠀⢷⠃⢸⠃⢈⠄
⠀⠀⠀⠀⠀⠀⠀⠑⠊⠓⠤⠣⡀⢀⡀⠈⣿⠀⠈⠲⡇⠀⠉⠀⠈⡎⠀⢹⡀⢀⡷⠤⠺⢤⠎⢉⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⢂⠉⠙⠈⠢⠤⠔⢉⠒⠊⠒⠊⠑⠒⠁⠍⠉⠂⠀⠀⠀⡐
⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⠀⠀⠰⠀⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⠀⠂⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⡵⠀⠀⠀⠀⢀⣼⠋⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢃⠞⠀⠀⠀⠀⠀⣾⡏⢸⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⠀⠀⠀⠀⢸⣿⢠⠃⣰⡟⡸⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⣶⠆
⠀⠀⠀⠀⠀⠀⢠⠜⣿⡇⡠⣾⡧⠀⠀⠀⢸⣿⠏⢠⣿⣳⠃⠀⠀⡤⣤⡠⠊⠀⣠⠔⠊⢉⡴⠋⡾
⠰⡿⡢⠖⣄⡀⠀⠀⢿⠟⣰⣿⢃⣴⢦⠀⢸⣿⠀⢸⣿⠁⠀⢸⠃⢩⣿⠃⠀⢰⠃⢀⡴⠋⠀⢐⡇
⠰⣉⡘⠢⣁⢾⠖⢦⠀⠀⣿⡏⢸⣯⠜⢠⠏⠘⠛⢹⣿⠀⣠⠋⠑⠚⠋⠀⢀⡤⠛⢯⣀⣀⡤⠎
⠀⠺⠄⡤⠈⠷⣄⣚⠆⠀⠻⠧⠜⠷⠔⠁⠀⠀⠀⠘⠿⠚⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⠴⠂⢤
⠀⠀⠘⠦⡤⠛⠃⠉⠲⠄⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⣀⠠⠀⢀⣀⠤⣤⡤⢲⣦⠀⠘⢄⣀⡤⣿⡆
⠀⠀⣠⡤⡀⠀⠀⠀⡠⠒⠒⠛⠒⠶⢶⣿⡷⠴⠴⠶⠚⠋⠁⠀⠀⣿⡏⠀⣿⠃⣼⡟
⢀⣾⠋⣠⣿⠀⠀⣀⡀⠀⢀⣀⠀⠀⢸⣿⠀⠀⣤⡄⠀⠀⠀⠀⠀⣿⠁⠀⠛⠀⣿⡇⠀⣸⣷⠴⠲⣦
⢸⣯⠔⣿⡏⠀⣰⣿⠃⢀⣾⣯⠀⢀⣾⣿⠀⣰⣿⠀⢠⣿⡇⠀⡸⠁⠀⠀⠀⠀⣿⡇⠀⣿⡏⠀⢀⣿
⠀⠀⠀⣿⣇⡴⢻⣏⡠⢛⣿⡇⢀⠞⠀⠙⠋⠘⣿⣀⢼⣿⠀⣰⠃⠀⠀⠀⠀⠀⢽⡇⢠⣿⠇⠀⣾⡟
⠀⠀⠀⠈⠉⠀⠀⠉⠀⠘⣿⣧⠏⠀⠀⠀⠀⠀⠈⠁⠀⠻⠞⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⠓⢄⣀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢇⠉⣐⠀⠈⠘⡄⡴⠊⠗⡙⢉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣏⠂⢄⠉⠰⠃⡰⢧⢇⡁⢸⠆⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡕⠒⠒⠪⢕⠤⡀⠀⢿⡟⡇⠀⠀⠀⠈⠫⣍⠉⡴⠊⢨⠄⠀⡵⠂⠸⠥⢲⡒⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⡎⠙⠻⣶⡀⠀⠉⠪⡳⢌⢯⣿⠀⠀⣀⡤⠴⠞⡗⢳⡒⠛⠚⠳⠠⠆⣀⣤⣊⠀⠠⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⡇⡁⠀⠀⠘⣵⠀⠀⠀⠈⠳⣽⣝⣷⠋⠁⠀⡠⠊⠁⠘⡀⠁⣬⣀⠃⠠⠀⡶⢭⣀⣐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠱⡳⣄⣀⢴⡝⠀⠀⠀⠀⠀⠈⣯⠞⣆⠀⠀⣅⠄⠈⠐⢙⠀⢧⢸⡀⢒⠀⠈⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⣸⣝⣛⡟⣆⠀⠈⢆⣰⡀⠚⢢⣈⠎⢦⠮⣡⣹⠁
⡰⠋⠑⠂⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⠫⡏⣀⠹⠊⢧⡀⠀⠁⠙⠢⠇
⣇⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⣠⣤⡶⠶⠶⣤⣴⠾⣾⣷⢦⣤⠿⠛⠷⠻⠿⣦⣤⣤⣤⣤⣀
⠘⢦⣀⣀⡤⠞⠁⠀⠀⠀⢀⡴⣛⡉⢢⡀⠄⢡⢬⡸⣌⣛⡿⠀⠁⠔⡠⣒⡪⡔⠘⡆⢉⢰⠋⢨⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⡏⣿⣷⠄⡇⣰⠿⢀⠅⣂⢀⢀⡈⢿⡜⣇⠳⠟⢣⣴⠣⠘⣾⡀⠻⡿⢃⡿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⢄⠑⠬⠵⠚⣰⠯⡡⢀⡈⠄⠀⠃⠀⢸⡇⡈⠙⠋⠉⡲⠓⠓⢾⡝⠲⠒⠋⠂⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣧⡈⠂⡬⠀⡀⣾⡟⢢⠐⠀⡠⠚⠉⢉⢉⢺⡇⢀⢄⠰⠶⢇⠿⣿⠀⣷⡠⠖⢛⠓⣅⠾⣇
⠀⠀⠀⠀⠀⠀⠀⣼⠋⢅⣁⡤⢴⡀⣿⠿⠠⠀⠀⡇⢴⣫⢯⢣⢸⡇⠁⠣⡠⠄⡀⢝⠩⡀⣿⠡⢸⣟⡿⢸⢆⢻
⠀⠀⠀⠀⠀⠀⠀⢻⠇⡔⠁⣴⠲⡌⣿⠦⠴⣥⠤⡱⣤⣉⣩⢼⢸⣇⠔⠋⡉⠑⣔⣘⠀⢈⣿⠀⠘⠲⡖⠁⠜⢻⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢧⠀⡳⢭⠖⣿⣤⣦⡄⢱⠈⠂⢠⣴⡅⣿⡇⠀⡎⣟⡇⣸⣀⣪⣦⣿⡢⠂⡻⠋⢋⡄⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢙⠲⠤⠤⢿⣜⣚⣃⠜⢀⠄⢀⢠⠡⣺⠇⡁⡑⣢⢒⡏⣡⣤⣨⡏⣴⣕⠐⢈⠚⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣅⠖⡡⢌⡘⡇⢩⠝⡋⠉⠑⢦⡌⡄⣿⢀⠂⠐⠙⠉⢣⣙⣛⣼⠏⣤⣌⠑⣬⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣴⣶⣁⣹⢏⡀⡔⡭⣷⢈⡗⡈⠃⡸⣋⡙⢦⡕⣠⠀⡸⠁⡧⣝⣝⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠻⢤⣧⣉⣡⣞⣀⣤⡾⢧⣿⣿⣀⣏⣥⡶⠶⠶⠶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡩⡴⠖⠚⠓⢲⠞⠛⠨⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠁⢩⠃⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⣠⠚⣄⠀⠀⠀⠀⠀⠀⠈⢂⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠃⠀⠐⠒⠒⠈⠤⣄⣈⢀⣀⣀⣀⣠⡴⠞⠁⠀⠈⠙⢄⠀⠀⠀⠀⠀⠀⠂⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠖⠀⣠⠎⠀⣈⠉⣩⡿⠋⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠀⢀⠔⠠⢰⡞⠀⣴⡏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠀⢖⠐⠁⡘⣼⠀⠉⠋⠈⠀⠀⠀⠀⠀⠹⠿⠛⣟⡀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡤⠀⡈⢀⢄⠁⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡀⡀⠆⠁⡘⢀⠈⠸⠀⠀⣅⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⢰⠃⠘⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠋⠁⠀⠐⠁⠋⠀⠈⠀⠂⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡆⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⠰⠀⠉⠀⡀⠀⣠⠐⠀⠀⢀⠞⢀⠄⢀⠈⢦⡀⠀⠀⠀⠀⢀⡴⠁⠀⠰⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠐⠀⠀⠁⠁⠀⠁⠀⠀⠀⠀⠀⠀⠁⢐⢀⠐⠁⠀⠈⠀⠁⠀⠀⠉⠒⠀⠒⠈⠀⠀⠀⢠⠋⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠁⠀⠀⣠⠘⡁⠀⠀⢀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢠⠃⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠠⣀⣄⠀⢈⠅⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⡃⠀⠀⠀⠀⢀⡴⡁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢠⡤⠀⠀⠈⢅⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠰⠁⠀⠀⠀⡠⠊⠀⢣⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠰⠟⠀⠀⠀⠀⠈⢀⡔⠁⠀⠀⡠⠊⠀⠀⠀⠈⢳⡀
⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⡀⢡⠀⠆⠀⠀⠋⠀⠀⠀⠀⠐⣠⠜⠉⠀⠀⢀⢜⡡⠤⠤⠤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠁⠤⠀⡀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣴⠏⠁⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⢰⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠐⠒⠀⠀⠀⠒⠂⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⠁⠀⣀⠤⠄⠄⠀⠠⠄⠤⢀⡀⠀⠁⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠁⢸⡿⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⠔⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠲⡀⠀⠣⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⡜⠀⢀⡊⠀⠴⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠈⢄⠀⠱⡀
⠀⠀⠀⠀⠀⢺⣿⡆⢰⢀⢠⠏⢴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢅⠀⢠⣄⠀⠈⡄⠀⢣
⠀⠀⠀⠀⠀⠈⠻⠇⡖⠁⠸⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠦⠢⣸⠟⠀⠀⢰⡄⠘
⠀⠀⠀⠐⢿⣿⠦⡘⡇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠈⠢⡀⠀⠀⣠⣤⣤⠐⢄⠀⣿⡟⠀⡂
⠀⠀⠀⠀⢠⣴⡦⠠⡇⠀⢠⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⡀⠀⠀⠀⠀⠑⠄⡉⠉⠁⠀⠀⠀⢺⠀⣾⠟
⠀⠀⠀⠀⠈⠁⠀⠀⢳⠀⠈⡄⠀⠀⢀⠔⠉⠀⢀⠔⠊⠁⠀⠀⠉⠢⣄⠀⠈⠢⢄⠀⠀⢀⠆⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠐⡤⠊⠁⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠑⠤⠜⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⢀⡔⠉⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠑⠬⡂⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⢤⣿⣶⡤⠤⠄⠀⠠⠤⠤⠒⠊⠁⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⠄⣤⣀⣀⣀⣀⡀⠤⢔⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⠟⠁⢂⣤⡶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠈⠓⢀⠔⠊⠉⠉⠉⠉⠓⠦⣔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢰⠁⣀⡀⠀⠀⠀⠀⠀⠀⣠⡂⠸⡇⢀⡀
⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⢀⡇
⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠉⠁
⠀⠀⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠂⠑⢄⡀⠀⠀⠀⠀⣀⡴⠟⠁⠦⡄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠈⠉⣉⠉⠉⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄
⠀⠀⠀⠀⠀⠂⠀⠀⢀⡴⠋⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠛⡩⠢⠈⠀⣰⠀⣠⠀⡀⠈⣆⠀⠂⠀⠀⠒⢄⠀⠀⠀⠀⠀⠢⢄⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣶⠀⠈⠀⢁⡀⢰⠃⠀⠀⠀⠀⠀⢨⢃⠀⠀⠀⣀⢀⠈⠀⠀⠀⠐⠠⠚⠀⠒⠤⠀⠠⢇
⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⠃⠀⠀⠀⠀⠀⠠⢞⠁⠄⠁⠉⠉⠠⢈⠁⢀⠁⠈⠉⠀⢈⡂⢰⡄⠁⠀⠀⠀⠉⢠⠀⠀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⡀⠀⠀⠆⠰⠀⡀⢸⠀⠀⠀⠀⡀⠀⠜⢢⡀⠀⠀⡀⠠⠘⠁⠀⠀⠀⢤⣀⠀⠀⠀⠀⢺⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⠀⠀⠀⠀⠃⢠⡆⠐⠂⠀⠁⠐⠀⠴⠓⣶⠀⠙⠖⠀⣥⡀⠀⠀⠀⠀⠠⠀⠀⣀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠀⠀⠀⠀⠁⠸⡇⢰⠀⠀⠀⠀⣗⠉⠒⠟⠀⠐⢨⡊⠁⠀⠘⠀⠀⠀⠰⡀⠀⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠈⠀⣇⢤⠀⠀⠀⠀⠼⣄⡄⠀⠀⠀⠀⠰⡤⠤⠤⠀⠀⠀⢀⠇⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢹⠁⠀⠀⠠⠒⠀⠠⠃⠀⠀⠴⠒⠀⠄⠁⠀⠀⠴⠂⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣆⠤⣤⣄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠻⡿⠛⠉⠀⠈⠀⢄⠀⠀⠘⠛⠛⠛⢲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠛⠀⠀⠀⢦⠀⠀⠀⢿⠁⠀⠀⠀⣀⠀⣀⡉⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⢀⡀⠀⠀⠈⠀⠀⡀⠀⢀⡀⠀⠀⠋⠈⠛⠃⠈⠁⠱⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⡹⠋⠀⠀⠈⠁⠞⠢⡦⠀⡀⠁⢤⠀⠁⡀⠀⢠⡀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⠀⠀⠀⣄⠀⣅⢠⡀⠰⠆⣩⠀⠀⠁⠀⠀⠘⠃⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠹⠂⠠⠄⠀⠈⠁⠰⠆⠀⠲⢾⣼⡥⣆⠭⠄⠀⠟⠀⠀⡀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⡀⠀⠀⠀⠀⢠⠄⢀⡀⢋⠐⢊⠙⠛⣭⡶⢂⠌⠁⡀⠄⠉⠀⠀⠘⠁⠀⠚⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠰⠠⠌⠼⠽⠠⠓⠀⠁⠀⠄⠚⠂⢀⠀⠈⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠀⠀⠐⠋⠁⠀⠒⠀⠐⠂⠀⠆⠐⠂⠰⠀⠖⡀⠃⠀⠏⠀⠀⠀⠈⠉⠂⠀⢣⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠰⠄⠁⢀⠒⠀⣅⠘⡇⠈⠙⠁⠀⠀⠀⠀⢶⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣶⠀⠀⠐⠖⠀⠀⠁⠀⠀⠀⢀⡄⠈⠀⠰⡀⠀⠑⡄⠠⠖⠢⡄⠂⠀⢜⣥⠂⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣷⣄⠀⠀⠀⠀⠀⣀⠰⠆⠈⠁⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⣴⡟⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣿⣤⡀⠀⣀⠉⠀⠀⠀⠀⢰⡆⠀⠀⠀⡛⠀⠀⠀⠐⣤⠞⠛⠀⠀⠀⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⡦⠀⠀⠀⡄⠀⠀⠉⢻⠾⢿⡠⠶⠄⠀⠀⠀⠀⠀⢀⡀⠀⣀⣤⡶⠉⠀⠸⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⣿⢿⡿⠷⢾⡟⠋⠉⠀⢀⡠⠄⠘⠆⠀⠀⠀⠀⠀⠿⠛⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⡀⢀⣠⠠⠄⠐⠀⠀⠀⠂⠀⠀⠀⢀⣤⠀⠲⠀⠀⠐⠀⠀⠀⢛⠵⠦⠁⡀⠀⠀⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠁⢸⠰⣤⣄⠀⠀⠀⠃⠀⠀⠀⠰⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠐⢮⣑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣶⡟⢻⠀⠀⠁⠀⠀⠠⠀⣠⡔⠀⠺⠁⢀⡀⠀⠀⠓⠀⠀⠰⠆⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢶⡠⠀⡀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢱⣆⠀⠁⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠉⠀⠀⠇⠀⠀⠀⠀⠀⠀⣿⡟⠛⠸⢯⣷⡆⠀⢀⡜⠀⠀⢐⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠉⠀⠀⠀⠐⠀⠟⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠘⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⠀⠀⠀⠀⢰⡄⠀⣠⠄⡄⠀⢀⠀⠀⠸⣷⠀⠀⠰⠄⠀⠛⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠃⠀⠀⠘⠂⠀⠀⠀⢣⠀⠸⠁⠰⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠺⠁⠠⠄⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆

⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣷⣄⢠⣿⢡⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠿⠟⠻⣿⣶⣦⣴⣶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⢾⣧⣀⣾⣁⣰⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⠊⠛⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢠⣼⣟⢁⣶⣿⣿⡿⠟⠛⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⠛⠉⠉⠁⠀⠀⠀⢹⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣤⣷⣿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣿⠄⣾⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣩⠐⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠻⢛⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠾⠟⠉⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢰⣿⡷⣼⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⣴⣷⠙⢿⣭⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠈⢛⣯⣾⡌⠻⠛⡛⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠛⠿⠿⠋⣴⣿⣿⠗⣹⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⣶⣾⠇⠘⢫⣼⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢁⣤⣄⡙⠿⣿⣿⣿⣿⡟⠿⢟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠠⣿⣿⣿⣧⣴⣿⣿⡿⠿⠁⢰⣿⣿⡽⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⡹⠛⠛⣸⣿⣿⡿⢇⣤⣶⣤⡉⢉⣤⣾⣿⡿⢶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣈⠻⣿⠿⣾⡛⣯⣥⣄⠺⢿⣿⣿⡧⠈⣛⣛⠛⠁⣴⣿⣿⣿⢛⣿⠷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⢃⡄⢾⣿⣧⢹⣿⣿⠲⣶⣌⢡⣴⣌⢿⣿⣿⠷⢀⣍⣉⣁⠻⣿⣿⣦⢻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣇⠈⢉⣁⢠⣿⣿⣤⣍⡛⠃⣉⣋⣉⣿⠿⢠⣿⣿⣿⣿⠦⠆⣩⣤⣄⢻⣿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⣸⠿⠂⢹⣿⣿⡟⠃⠺⠿⠟⣿⣯⡌⢿⡄⢉⣭⣤⣦⣌⢻⣿⣿⣀⣿⣧⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢷⣮⣿⣿⢿⣿⣿⣶⣼⣿⣿⣧⣼⣧⢸⣿⣿⣿⠿⠀⣙⣯⣼⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡮⢠⣿⣏⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⡃⠛⠛⠛⢀⣚⠿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣶⠄⣽⣿⡀⠀⠀⠀⠀⠀⠉⠋⠁⠙⠣⡜⠿⣿⣿⣿⣶⣿⣿⣿⠟⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣼⡛⢿⣿⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠐⠾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣝⢻⣸⣿⣿⣇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢻⣿⡇⢰⠀⢲⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⢢⡟⢿⣦⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠀⠈⡆⠘⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠙⠀⠙⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⣿⠄⠙⡄⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠀⠀⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠀⣿⣿⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠘⠃⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢀⣿⠂⠀⠈⠀⠀⠀⠀⠀⢸⣅⠀⠀⠀⠀⠀⣀⣀⠀⢀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⠛⠉⠀⢀⣀⣀⣀⣤⣤⡆⠈⣿⠶⠶⠶⠟⠛⠉⠁⠀⠀⢰⡿⠀⠀⠀⠀⣀⣀⣀⣠⣤⡄⠘⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠶⠛⠛⠛⠛⠋⠉⠉⠨⣦⡶⠿⠀⠀⠀⠀⠀⣀⡀⢰⣤⣾⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⡇⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣤⣤⣶⡶⠿⠿⠛⢿⣦⣿⣷⡾⠿⠿⠟⠛⠛⠋⠉⠉⢀⡀⢸⣟⣁⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠾⠿⠿⠟⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⡄⢲⡾⣧⡠⢀⣀⠐⣀
⠀⠀⠀⠀⠐⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠢⢤⣁⡀⢂⣴⣂⣄⣇⢦⣯⣷⣿⡗⣣⡍⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢤⡄⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣵⢿⣿⣿⣿⣿⣿⣷⣿⣷⣿⣿⣿⣶⣿⣾⣶⡿⣶⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠡⠀⠀⠀⠀⠀⠀⠂⠀⡒⣴⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⣐⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⠃⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣷⡦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⠿⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⡄⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⡇
⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠄⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⠟⠀⠀⠀⢠⡒⠶⣮⢺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠡
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣧⣄⡀⠀⠀⠈⠀⠀⡾⠻⠝⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⢰⡟⠉⢩⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣤⣶⣤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⢾⡯⣿⣿⣿⣿⣿⢿⠿⣿⠋⣽⣾⣿⣿⢿⣿⣿⣿⣿⠻⡞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠷⢿⠿⣟⡿⠿⣿⣿⣿⣿⣿⣍⡬⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠁⠀⠀⠄⠱⠏⠉⠀⠂⠀⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⡀⠀⠀⡠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠃⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⣀⡤⢤⣴⣶⣷⣿⣿⣶⣶⣿⣶⣖⣶⣶⣶⣾⣶⣶⣶⣤⣦⣤⣦⣄⣀⡀⠀⠀⠀⡀⣿⣿⡇
⠀⠀⠀⠀⢤⣠⣦⣴⣶⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣟⣋⣇
⢀⡤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣄⣀
⠘⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⢃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠠⠀⠀⡏⠐⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢋⠀⠀⠀⢀⣾⠀⠀⠀⠀⢠⠔⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠘⠁⣀⣿⣄⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢄⠁⣠⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡐⣴⣦⢐⣒⣶⣯⣤⣶⣾⣿⣿⣦⣽⣷⣧⣠⣧⡀⠀⠀⠀⢀⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢼⣿⣿⣷⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡘⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⢡⣤⡄⢚⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣋⣯⣿⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣜⣿⡿⣀⣻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠐⠶⣞⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠠⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⠟⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠚⢻⡿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣀⣠⣿⣆⠀⠈⢻⣿⣛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣏⢻⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣌⠙⠛⠳⠄⠀⢿⣿⠀⠈⠻⠟⠟⢿⣿⣟⢻⣿⣄⠁⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠻⢿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣧⡀⠀⡀⠀⠈⣿⠛⠀⠀⠀⠐⠿⣿⣿⣀⠈⠙⠿⠋⠀⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⡏⠻⣿⠶⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠒⣤⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠸⣿⡇⠻⠿⠇⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⠷⣶⡆⠀⠀⠈⢿⣿⣿⣦⣄⠀⠀⠲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠠⣿⣿⣦⣿⣿⣿⣿⣷⣄⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣿⣿⣿⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠿⠿⠟⠿⠁⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠁⠀⠀⠀⢠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢴⡷⣶⠺⣧⡴⣏⣠⣾⣤⣤⣄⠀⠀⠀⢀⡔⠛⢰⠌
⠀⠀⠀⡀⢀⡄⢠⠀⠸⣤⣂⣶⢶⢂⣰⣷⣶⣿⣿⣿⣿⣾⣻⣿⣿⣤⢀⡴⢫⢈⡰⠂
⠀⠐⢻⣿⣾⣿⣶⣴⣿⣾⣿⣾⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⢿⣿⡿⠟⣋⠷⠀⡶
⠀⠀⠀⠀⠘⠛⠂⠨⠀⠐⣩⠯⢟⠿⣿⡷⡺⠈⠉⠀⠈⢉⣄⢷⢂⡒⠘⣁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠷⠊⠉⠀⠁⠁⠀⠊⠀⢀⢨⠠⢩⣵⣮⠛⢠⠋
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠂⠉⡀⡀⠀⠀⠴⠐⠆⠰⠆⠙⡘⠊⠍⣥⢙⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠇⠀⠀⠀⠠⠀⠐⠀⠀⠁⠀⠀⠀⠂⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠃⠀⡄⠀⠀⠀⠰⣤⣄⣀⡀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡀⣄⣸⡇⠀⢻⡆⠀⠀⠈⠁⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠙⡻⠁⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢉⣆⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠉⠀⢿⣿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢃⢀⢴⠸⣿⠿⢿⠟⢠⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠈⠛

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡄⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⠴⣈⠀⢈⣭⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡻⣽⣿⣿⣿⣿⣿⣿⣶⣮⣷⣾⣿⣮⣿⣿⠶⠚⠋⠶⣯⣴⣵⣣⣪⣦⣸⣶⣾⣷⡻⠁⣌⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣦⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⣀⣄⣀⡀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣡⠞⣼⢻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣶⣶⣶⣻⣿⣿⣿⣉⣉⠛⢯⣐⣤⣤⣬⣮⣽⣋⣛⢊⠙⠋⠙⠙⢻⣿⣿⠯⣤⣤⡾⣫⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣽⣧⣾⣿⣿⣿⣿⣝⠶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣷⣾⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⠟⠈⠉⠛⠛⠋⠉⠙⠛⣛⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣀⣀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣸⣿⣿⡇⠀⠀⠀⠀⠈⠉⠙⠛⠋⠀⠈⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⢸⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠻⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⡶⠂⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠉⠳⢶⣶⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⡿⢟⡉⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣭⣉⣛⡛⠿⠿⠿⢿⣿⡿⠿⠿⣿⠿⠛⣛⣳⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⢤⣴⣀⣀⠀⣤⣄⠀⠀⢀⣶⡀⢠⡤⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠈⠹⠿⣃⡈⢹⠁⠀⠈⡟⢀⣔⣄⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⢻⣆⠇⠀⠠⠁⠟⠃⠀⠀⠀⠀⠐⠚⠒⠂⠠⠶⢦⡀
⠀⢀⠀⣤⡀⠀⠀⠀⠀⠀⠈⣦⡟⠀⡰⠁⠀⠀⢀⡄⠀⠒⠀⠀⠐⠀⠄⠀⠀⠀⠀⣶⠀⢀⡀⠀⠀⢠⠄
⠀⢾⠃⠉⠀⢠⡄⠀⠀⠀⠀⠘⣧⣰⠇⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠈⠙⠆
⠀⢀⠀⢀⠂⣀⡄⠀⠀⠀⠀⠀⢻⡿⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠒⠀⠐⣷
⠀⠉⠠⡄⠼⣯⠇⠀⠀⠀⠀⠀⠸⡇⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠃⢙⢀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠰⡄⠘
⢀⣦⠀⠀⠙⠲⣶⠀⠀⠀⠀⠀⠀⣿⡇
⠈⠁⠠⢶⠇⠀⣤⠄⢶⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⣠⠄⠀⠀⠀⠀⠀⠛⢁⠀⠀⠀⣿⠃⣀⠀⠀⣀⡀⠀⢀⠀⣀⣀⡀⣀⣀⣀⣤⣠⣔⡚⢠⠦⣴⣶⣄⣵⣤⣿⠁⣀⠀⡴⠂⠤⢡⣤⣴⡦⣀⡀
⠀⠉⠀⠀⠘⠓⠐⢶⠄⠈⣦⠀⢀⡿⠠⠀⢠⣀⠀⠐⠢⡀⠀⠉⠉⠉⠹⠛⠋⠉⡱⠉⠿⠛⠶⠀⠀⠊⠈⢠⣉⠙⠁⠁⠀⡀⠀⢠⣤⣒⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⡀⢂⠀⢸⣦⡾⠃⠁⠀⠀⠈⠀⠀⠁⠉⠀⠀⠐⠤⠀⡄⠀⠂⠀⠀⢀⠀⢤⠀⢀⠀⢠⠤⠠⣶⠉⠀⠀⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⣹⠈⢳⣸⡿⠳⡆⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⢿⣶⠈⣿⠃⢠⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆
⠀⠀⠀⠀⠀⣄⠛⠀⠁⣰⡏⢀⡈⢾⡤⠀⠀⠀⢀⠀⠀⣀⣀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠉⢡⡀⢰⣿⠃⣘⣇⡭⠥⠖⠄⠘⠋⠉⠀⠀⠉⠉⢻⡷⣦⣄⡀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠤⡀⠺⡜⡇⢼⣿⠈⢸⠃⠀⣀⠀⢀⠠⠆⢠⡤⠀⠀⠀⠀⠹⣆⠙⠻⠶⣦⣤⣀⣠⣤⡖⠉⠀⢀⣀⠁⠄⠀⠀⠀⠀⠢
⠀⠀⠶⠤⠛⠀⡠⠐⣾⣻⠀⢸⡗⢀⠫⢐⡛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈
⠀⠀⠀⠤⠀⠀⠀⠀⢻⣿⣷⣼⡰⠋⠀⠉⡀⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢠⡙
⠀⠀⠀⠀⠀⠀⢈⣉⡂⠻⣿⣸⣥⣂⣘⡿⠇⠀⢲⣶⠂⠀⣀⡗⣰⡶⢀⠐⢞⣠⣼⡦⠆⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠿⠟⢲⠀⠈⠋⠱⠿⠈⠈⠁⠀⠀⠀⡀⠂⣟⡱⢿⢐⡧⠋⢶⢏⡙⠀⡅⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠙⠘⢻⣔⠫⢴⡽⠰⢋⣌⢁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓


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

⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿
⠀⠀⠀⠀⠀⠘⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠷⠙⢿⣿⣿⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⢛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡸⣿⣿⣿⣿⣇⣀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣤⠄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⢿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣀⣉⣻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⡟⢰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⡏⠘⢿⣿⣿⡟⣋⣉⣥⣼⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣙⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡿⠿⠿⠿⢿⣿⣿⣿⡿⠿⣛⣫⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡿⣋⣥⣴⣶⣦⠀⢠⣤⣤⠄⠀⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣾⡿⠟⠉
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣷⣿⣿⠿⢟⣉⡄⠀⠁
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⡀⠀⠺⠿⢶⣶⣤⣤⣤⣤⡶⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠛⣿⣿⣿⣿⣿⠟⠁⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡧⣶⣤⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⢸⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⢶⠞⠋⠀⠀⣸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠋⠀⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣀⣀⣀⡀⢹⣿
⠀⠀⠀⠀⠀⠀⣀⣴⣛⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠛
⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⠀⢠⠆
⠀⠀⠀⠀⠀⠀⢀⣨⠁⣌⠀⠀⠈⠂
⠀⠀⠀⠀⠠⠴⢈⢡⣄⠈⣀⣠⣤⠙⣳⣆
⠀⠀⡀⠒⠦⠠⠘⣷⣼⣁⣩⣽⢂⡾⢻⣽⣦
⠀⠀⠀⠚⠖⠀⠈⠉⠀⠀⠀⠀⢼⡙⠿⣉⡸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣭⡍⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢹⣅⠿⢽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣈⢳⡮⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠇⢶⣷⠜⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠆⣿⣦⣘⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣥⣾⢓⣴⣿⢘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣬⣍⣷⣜⢫⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⣻⠛⣠⣭⣄⠯⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⠙⠿⢟⣲⣾⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣵⣔⢿⣿⢇⣼⣶⣉⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣛⠿⣤⡙⣿⡟⣩⣤⡙⣯⣬⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡼⢿⣷⣤⣜⠻⣿⠇⣩⣷⣮⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠾⡧⠸⣿⣷⣾⣿⣭⣟⢻⣿⣿⢿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣿⣿⡦⣹⣦⡽⣿⣿⣧⠛⢷⣿⡟⢿⡷⢶⡦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢿⣷⢻⣿⣷⣝⠛⣵⣿⣷⡟⢳⣾⣇⠻⣿⣿⣿⣿⢶⣿⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡿⠿⣋⢾⣿⡿⣻⣿⡝⣿⣿⣿⣎⢿⣿⢻⡧⣿⣿⣭⡿⣯⣶⣿⣾⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣶⣿⣷⡄⣾⣿⣿⢏⡘⢿⡿⢫⣶⣦⣿⣦⡻⣿⠻⣿⣿⣯⣿⣯⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣙⠿⠛⣿⣯⣴⣷⣝⢿⣏⣾⣿⣶⣴⣾⣽⢿⣽⢻⣽⠿⣿⣶⣸⡿⣿⣿⡿⢟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣟⣼⣾⢻⣿⣿⠿⣬⣿⣿⣿⢻⣿⣿⣿⣿⣏⣹⡦⣿⠋⠿⣧⣾⡿⣧⣜⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⠿⣿⣿⡿⣼⣿⣿⣧⣿⣿⣭⣍⣫⣵⣞⣹⡆⣛⡸⢿⣬⣋⡴⣋⡹⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣙⣷⢼⣷⣶⠙⣿⠲⣏⣵⣿⣍⣹⠿⣯⣹⢻⡏⣻⡃⢻⡿⢳⣶⢹⡏⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⣍⣸⠻⢏⣷⡿⠱⠿⣭⣿⣾⡿⣿⡿⠈⠘⢧⠙⣧⡝⣛⣾⡻⢶⡦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢏⡍⢿⣿⡿⣧⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠙⣡⠏⣼⢃⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡅⠀⢹⣭⢹⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢙⡶⣛⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢡⡄⢻⣶⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣛⡛⢀⣀⠀⠀⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢎⡁⠈⣥⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡶⣉⡱⢆⡉⣤⠀⠘⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡑⠀⠛⠂⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⢩⡴⢎⡁⠉⠀⠀⠀⠈⠛⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠦⢴⣊⣅⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠐⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡋⠀⠂⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡍⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠁⠀⠀⠠⡆⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢘⠃⠀⠀⠀⠀⡃⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠊⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠈⠃⠀⠀⠀⠀⠀⠀⢀⡘⢁⠀⠀⠀⠀⠀⠀⠀⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣶⣦⠈⠀⠀⠀⠀⠀⠀⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⠶⠄⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⣠⣄⠈⠉⠀⠀⠀⠀⠀⢠⣾⣶⣄
⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣧⣤⣤⣤⣤⣬⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢾⣣⣼⣯⣹⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⢉⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇⢀⣤⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⣿⣿⡌⠹⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⢤⣤⠄⡆⠠⢤⠤⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁⣠⣿⣿⣿⣿⣧⣿⣿⡆
⠀⠀⠀⣼⣿⣷⣿⣿⣿⣇⠀⠀⠀⠃⢀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣦⣀⠈⠉⠉⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣯⣈⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⠋⢀⣀⠹⣋⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⠟⢻⣿⣿⣿⡏
⠀⠀⠹⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠉⠙⠻⣿⣿⣷⣾⣿⣔⣢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟
⠀⠀⠀⠙⠿⢿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿⠿⠟⠋

⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⡐⠈⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⡀⠂⠁⠀⠁⠙⠄⠀⠄⢀⠀⠀⠌⠀⠈⠈⠉⡀
⠀⠀⠡⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠐⠄
⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠄⠀⠀⠣⠀⠀⠀⠈⠀⠀⡄⠀⠐⡀⠀⠂⠐⠀⠀⠈⠈⠀⠀⢀⡠
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠎⠐⠠⠄⡀
⠀⠀⠔⠀⠙⠄⠀⠐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠂⡤⠘⠁⠲⠈⠠⢄
⠀⠀⠠⠄⠠⠈⠈⢡⠀⠤⠄⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⡅⠰⠀⡄⢀⠀⢈⠁⡃
⠀⠀⢄⠀⠀⠜⠀⢃⠀⠀⡠⠀⠀⠀⡠⠈⠀⠀⠀⠀⠄⡄⠀⠀⢠⠠⠀⠀⠀⠀⠑⢀⠀⠀⠀⠐⣀⠃⢤⠈⠤⠈⠢⠑
⠀⠀⠈⠀⠸⠀⡀⢈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡇⠀⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠐⠐⡠⠄⠔⠌⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠁⠀⡀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⠐
⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠡
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠐⠀⠀⠀⢀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠐⢠⡂
⠀⠀⡔⠂⣁⠈⠄⠁⠃⡀⠀⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⣡⢀⠐⠁
⠀⠠⠅⠔⠁⠀⠀⠈⣄⠠⢄⠀⠀⠀⠀⡀⠀⠀⠁⠀⠠⠇⠀⠀⠸⠄⠀⠈⠀⠀⢀⠔⠀⠀⠀⠀⠉⠁⠘⡀⢀⠆⠤⢤⠘
⠀⠰⠀⠲⡀⠒⠘⢀⡄⣈⠅⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠊⠀⠩⡀⠀⡀⠁
⠀⠀⠡⠤⢈⠀⡚⢀⡰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠄⠀⠁⠒
⠀⠠⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠒⠄⠀⠠⠀⠀⠠⠀⠀⠀⠄⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠆⠀⠠⠀⠀⡀⠀⡀⠈⡀⠠⠀⠀⢀⠂⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠰⠀⠁⠀⠀⠈⠀⠈⠀⢀⠀⠀⢀⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⢄⠀⡀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⠄⠀⠀⠂⠀⣀⠀⠁⠀⠀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠
⠀⠐⠀⠀⠈⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⠟⢿⠛⠿⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠈⠢⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣤⢄
⠀⠀⠀⠀⠈⠢⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⠀⢹⠀⡀⣀⠀⠀⠀⠀⠆⣊⠑⣢⢉⢛⠐⢺
⠀⠀⠀⠀⠀⠀⠈⢳⣿⡒⠂⠀⠁⠀⠀⠀⠀⠀⣄⣀⡈⠢⢄⣿⣷⠲⠋⠀⠀⠀⠈⠸⠬⣱⣇⠸⢸⣂⠉
⠀⠀⠀⠀⠀⠀⠠⠊⠃⠈⠑⠤⠀⠀⠀⠀⠀⠀⠀⠙⠒⢷⣀⣡⡦⠤⣀⣄⡄⠀⠀⠘⠒⠋⠘⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⢼⠖⠀⣄⣬⣄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠘⡳⠤⣤⡀⠪⠑⠂⠦⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠢⣿⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠠⣨⠆⠁⠄
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢠⡀⠈⠑⣃⠴⢁⠂⠐⣢
⠀⠀⠀⠀⣐⠀⠀⠑⢀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡛⣫⣕⣶⡆⠀⠀⠀⠀⠀⠀⠈⠲⢤⣞⣀⣣⡊⠑
⠀⠀⠀⢐⣀⡀⠠⠑⠖⢨⠁⠀⠀⠀⠀⠀⢀⣀⢉⡚⢬⣭⡼⢿⣜⣇⠀⠀⠀⠀⡠⡀⠓⣤⢵⣿⡿⢇
⠀⠀⠀⠠⠋⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠐⢴⣿⠟⡁⡏⢩⡟⠈⠀⠀⠀⠀⠀⠜⠐⢋⣱⠗⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠩⠀⠖⡀⠚⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢀⠔⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⢀⠀⠀⢀⠂⢠⡎⣰⣾⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠻⣷⣲⡦⣤⣿⢉⡻⣿⠏⣁⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠤⣥⣿⠑⢶⡿⣸⠷⡋⡉⢐⣤⣆⠁
⠀⠀⠀⠀⢀⣀⢠⠾⣍⣄⣿⣺⣠⡳⣿⣧⣿⡾⠿⠓⠈⣀⣀⣀
⠀⠀⠀⢴⣎⢿⣯⣠⡀⢿⣿⢟⢷⣗⣽⣛⢿⣣⢖⣶⠟⠁⠀⠈
⠀⢀⣴⣷⣿⣿⣿⣿⠖⠂⢹⡗⢌⢻⣿⣿⢛⠻⡅⣃
⠀⠈⠁⣻⣿⣿⣿⢉⡈⠙⠃⠳⣾⣷⢼⡿⣿⡴⢛⠳⠔⠂
⠀⠀⠀⢭⣯⣝⣿⣵⠄⠀⠀⢲⢱⣹⣿⣷⢲⣾⣿⢀⡀
⠀⠀⠐⣾⣼⣿⣿⢟⡁⠀⠀⠙⣬⣿⣏⠹⣿⣿⣿⡉⠏
⠀⢀⣼⣿⣿⣿⣿⣿⣇⠄⠴⣏⣾⡻⣿⡽⣿⢟⣻⡟
⠀⠛⠉⠛⠛⠛⠟⠛⠙⣁⢠⣠⢿⣿⣿⣧⠻⢩⢰⣷⢖⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣽⢻⣿⢻⣍⠟⡆⢹⣆⣿⡈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⣿⢾⢏⢸⠿⣰⣧⣘⣿⣿⣷⣷⣇⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢁⣻⣟⣿⡿⣷⡐⣿⣿⣿⣿⣿⣿⣇⣿⣧
⠀⠀⠰⣵⠞⢈⠭⡻⢿⣿⢿⢿⣷⢿⣿⡞⣿⣿⣿⣿⣾⣿⣿⣯⡤⢤⢤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⣷⠃⡎⠀⢠⡏⣆⡆⣿⠙⢿⣷⣾⣯⣺⣿⣿⣿⡿⢁⠄⠠⠈⠉⡟⡍⠛⣆⠀⠀⠀⢀⠀⢠⠊⠁
⠀⠀⠐⠀⡙⣽⡅⣴⢿⣛⠋⠛⠑⢶⣾⣬⣿⡿⣿⣿⣿⡿⣷⣸⡤⡇⣀⣼⣡⣇⣷⣿⣰⣷⠶⣗⡘⠾⠂
⠀⠀⠀⠰⣟⡿⠋⡙⠋⠀⢲⡈⣦⣿⣍⣛⡉⡑⢘⠿⢫⣵⠚⠉⠦⣿⠉⣴⣿⡿⣿⣧⣤⣌⠑⠾⠃
⠀⠀⠀⠀⠀⢁⢡⣹⡘⢦⡤⢿⣙⣷⣤⣤⠾⠵⠿⠛⠚⠉⠀⠀⠀⠈⠉⠁⠩⢭⡍

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⠉⠉⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⢀⡠⠴⡦⢤⣀
⠒⠉⠀⠀⠙⢗⡮⣙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⠐⠒⠒⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠄⠒⠒
⣀⠤⠤⠀⠀⠈⡧⣀⠁⠂⠀⠉⠙⠲⢦⣤⣀⡀⠀⠀⠐⠂⠉⠉⠉⠒⠒⠒⠒⠉⣉⣀⣤⡤⠴⠶⡒⠚⠉⠙⢥⣀⣀⣀⣀⠀⠠⠤
⠀⢠⡆⠀⠀⠀⠘⢆⡁⡀⠀⠰⡀⠀⠀⢿⡻⣯⠛⠲⠦⢤⣤⣤⠤⠶⣒⣚⣫⡭⠭⠶⠒⣂⡉⡜⠀⠀⠀⠀⠀⠙⢳⣒⣾⠭⠭⠅
⠠⣾⣿⣦⠀⠀⠀⢈⣬⡑⠠⣆⠙⠻⣦⠸⠯⡲⢄⠀⠀⠀⠈⠙⢿⣿⡿⣶⣒⣋⠩⠭⠥⠐⢰⠁⠀⠀⠀⣀⠀⠀⠀⠳⣀
⣠⣾⡿⢦⣼⣄⠀⢨⣿⣏⡴⣿⡄⠀⠀⠀⠳⢬⡗⠀⠀⠀⠀⠀⠀⠳⠍⠀⠈⠉⠛⠻⠷⣤⣜⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⣭⢐⣴⢀
⣿⣾⣿⣷⣾⡷⣾⣦⣼⣦⣶⣿⡆⠀⠀⠄⠀⠀⠙⠢⢄⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠢⠤⠀⠀⠀⢁⣼⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣄⣀⣁⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠒⢰⣿⣿⣿⣷⣿
⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣾⠀⠀⣤⡤⠀⠈⣁⠶⠂⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⣫⡽⠿⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣵⣿⣟⣛⠙⠻⢄⣀⣘⡒⢐⣒⣒⣲⠤⠤⣄⣀⡀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⠾⠎⠀⠠⠤⠀⠀⠀⣁⣉⣉⣉⠓⣄⠀⠉⠙⢦⠀⣭⣴⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⡽⠀⡐⢶⣿⠟⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣳⣿⣶⣿⣧⣴⡶⠶⠖⠛⠋⢁⠀⣀⣰⠟⠁⢁⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣫⣿⣿⠿⣿⠿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣴⢾⣻⣏⣿⣿⣿⣿⣽⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣭⣵⣲⣓⣒⠀⠀⠀⠀⣀⣀⣀⣀⣀⣬⣭⣭⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢿⣌⣠⡿⠤⠤⣤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠚⠻⠶⢶⣬⣿⣿⣯⣝⣿⣿⣽⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣀⠈⣿⣷⠶⡶⠧⠤⠀⠉⠉⠉⠉⠡⠬⠤⠤⠤⠤⠍⠩⢭⣽⣿⣿⣿⣿⣯⣺⣿⣼⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠛⠛⠻⣿⢙⣿⣍⣉⣩⠭⠭⠬⠀⠀⠀⠀⠠⣭⠍⢡⣟⣓⠎⠒⢻⣿⣤⣽⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣭⣍⣶⣶⣀⣸⡟⢹⣿⡿⣟⣛⣓⠒⢂⣤⣻⣿⣃⣀⠀⠀⠶⠶⠖⣶⣾⣿⡶⠶⢿⡶⣿⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⠍⠉⢉⡉⠉⠙⢛⣤⣾⣿⣴⣧⣤⣿⣙⣓⣢⠐⠛⢛⡛⠒⠒⠂⠔⢊⣁⣉⣻⡿⠒⢚⣿⣻⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠛⠉⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠉⠛⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣾⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷
⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣧
⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⢸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⣀⣴⣾⣿⣿⣿⣿⠿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⣦⣀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⢹⣿⣿⣿⣶⣿⣿⣿⣿⡿⠟⠋⠁⠀⣿⣿⣿⣿⠀⠈⠙⠻⢿⣿⣿⣿⣿⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣤⣀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣀⣤⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣶⣶⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠠⠴⠶⠶⠶⠾⠱⠶⠾⠾⠾⠿⠊⠒⠒⠒⠒⠂⠒⠒⠒⠂⠀⠀⠀⠀⠀⠒⠐⠒⠒⠒⠓⠛⠚⠛⠒⠒⠒⠂⠂⠂


⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠸⢀⠀⡃⠀⠀⠀⠀⠀⠀⠀⢀⣀⢔⣒⣂⡴⠂
⠀⠀⠸⢠⢠⢠⢠⢀⠀⢀⣠⣤⢒⣁⠒⢛⣲⠖⠩⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⢹⣴⣸⣸⠞⣈⣛⡓⠒⠒⠒⢒⣲⠞⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⡆⠛⡄⠀⠸⠁
⠀⠀⠘⠛⠀⠐⠂⠄⠉⠩⠤⣤⡼⠛⡁⢐⠀⠀⡐⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⢀⠀⡄⠀⡇⠀⣠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠙⠁⠘⠁⠈⠃⠙⠸⠃⠶⠁⠀⠀⠀⠀⠀⠀⠐⠟⠛⠋⠀⠙⠁⠀⠂⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⢀⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⢀⣼⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣷⡀⠀⠀⣀⣀⣀⣀⣀⣀⠀⣀⣀
⠀⢿⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡆
⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣏⠙⠉⠉⠀⠀⠀⢀⣠⣤⣤⣤⡀
⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⠿⠛⠛⢻⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣿⠿⣿⣿⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣦⣤⣼⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠋⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡆⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢿⣿⠀⠀⢰⠟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣤⣤⣤⣀⠀⠀⢀⠁⠈⠛⠦⡀⠁⠀⣿⠀⠀⠀⠀⣀⣤⣄⣀⣠⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠙⢿⣿⣶⣵⡄⢸⠱⠂⢀⣼⣇⠀⣀⠻⣆⠀⠀⣰⣿⣿⣿⡿⠋⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣠⣿⡿⠋⣠⡿⠀⣠⣿⣿⡏⣞⡁⢀⣈⠓⠾⢀⣿⣿⣿⣁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣣⣀⡝⢀⣼⠿⠟⠉⠄⢩⣿⣷⣮⡅⢺⣿⠿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⢁⡾⣁⣀⠞⣠⡄⠈⠳⡿⢏⣿⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣨⣰⢮⣵⠰⠎⣥⡰⣷⠺⢺⣋⢿⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢾⡿⢇⣇⢤⡝⠠⣮⢡⣆⢿⣎⣹⢢⡘⣷⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠋⠀⣠⣤⢀⣨⣖⡐⠗⣈⡸⢟⣀⣼⢋⢰⡁⣍⢱⣿⣿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⣤⣿⣶⣿⠟⠛⠿⠯⠿⣷⣿⣿⣿⡿⠸⢃⡻⣷⡻⠻⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁⠀⠀⠀⠀⠀⠈⢻⣿⡟⣰⡆⣿⣷⠙⡁⣷⣽⣷⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⣼⣿⣿⠸⠟⠁⣿⣮⠿⢻⣷⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⢿⡟⠀⣾⣿⡜⢿⣿⡎⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⣼⣷⡌⢿⣿⠈⣉⣴⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⣿⣿⣿⣆⢡⣾⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⠋⣿⣿⣿⠀⠛⠿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢻⣿⡏⣰⣿⣷⣌⡙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢹⣿⡇⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡜⠿⠇⠻⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣧⠀⠀⠀⠀⢀⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡀⠀⢀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⣿⣟⣛⣟⣛⣻⣛⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣼⣿⠃⢠⡅⠀⠬⢶⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣷⣷⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⢸⢿⠞⢠⠄⠀⠀⠀⢾⣶⣿⣿⣿⣿⣿⣿⣷⠀⠈⠙⠻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⣿⡅⡼⠀⠀⠀⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠙⠛⠉⠁
⣿⠿⠿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣴⣶⣾⣷⣦⢉⣿⡿⢿⠿⠋⠝⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀
⠀⠀⠀⠀⣀⣤⣾⣿⡿⠟⠉⠉⠀⠀⠀⢠⣾⠀⢈⡃⠀⡀⠀⢸⣾⣿⣷⣿⣿⣿⣿⣿⠿⣿⣿⣷⣄
⠀⠀⢠⣾⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢰⣾⡇⢀⣷⡄⠀⠀⢏⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⠙⠿⣿⣧⡀
⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠈⠓⠈⠋⠁⠀⠀⢸⣿⠁⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠈⢿⣷⡄
⣶⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡄
⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⣀⣀⣀⣀⣠⣴⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠻⢄⡀
⠿⠿⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⡀⣨⣬⣤⣤⣤⣤⣤⣁⣙
⠄⢃⠀⠀⠈⠉⠉⠋⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡄
⠈⠙⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣩⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⢀⠠⠄
⠀⠐⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⢃⡀⠚⠭⣽⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠠
⠀⠀⠈⢹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡯⠴⠒⣒⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⡿⠋⢚⠉⣼⠿⠛⠋⠉⠉⠉⢻⣿⣿⣿⣿⡄⠀⣠⠾
⠀⠀⠀⠀⢀⣹⣿⣦⡀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡎

⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠢⠀⠀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠄⠠⠯⠁⠐⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡠
⠝⠁⣲⠀⠒⠈⠀⠀⠉⠁⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠠⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠠⠀⠄⠈⠀⠃⠀⠘⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠊⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠃⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠁⠀⠎⠀⠉⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⠡
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠁⢀⠈⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⡀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠈⠁⠀⠀⠀⠠⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠠
⠀⢀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⡀
⠀⠀⠁⠀⠑⠠⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⢳⠀⠈⠀⠀⠔⠀⠀⠀⠀⠀⠀⢄⠐⠂
⠀⢀⡤⣄⢲⢯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠉
⠀⡀⢈⠙⠙⠃⠀⠀⠀⢀⣀⢀⢀⣀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⢀⡀⢀⣀⡀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⣿⣿⡇⢠⣾⣶⠀⠀⠀⠀⠀⠀⣶⣶⡆⢸⣿⣿⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢀⣿⣿⡇⢸⣿⣿⠀⣠⡄⣠⣄⠀⣿⣿⡇⢸⣿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢸⣿⣿⡇⣿⣿⣿⢸⠿⠇⠿⢿⡆⣿⣿⣿⢸⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡿⢛⣥⣶⣾⣿⣿⣷⣶⣬⡻⢿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢏⣴⣿⡿⢟⢻⣿⣿⠛⡻⣿⣿⣦⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣤⣄⠀⢸⣿⣿⣿⣿⡏⣾⣿⡟⣴⣿⢸⣿⣿⡸⣿⣎⢻⣿⣧⢻⣿⣿⣿⣿⡇⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠘⠿⣿⣿⣿⣾⣿⣿⣿⣿⠠⣿⣿⠰⢟⣵⣿⣿⣿⣷⣌⠻⠈⣿⣿⢸⣿⣿⣿⣿⣷⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆⣿⣿⣶⣿⣿⢿⣿⣿⠿⣿⣷⣼⣿⡿⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡜⢿⣿⣿⣁⢸⣿⣿⠀⣈⣿⣿⡿⢡⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣮⡛⢿⣿⣿⣿⣿⣿⣿⠿⣋⣴⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉

⠋⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠰⠀⠀⠐⠤⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣄⠀⠀⠄⠀⠀⠢⠀⠀⠀⠂⠀⠠⠀⢈⠀⠀⠀⠂⠀⠁
⠒⠀⠀⠀⠀⠠⠀⣀⠀⠀⣀⣀⡰⠚⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣤⣄⡀⠀⠀⠀⠀⢠⣤⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⢀⣤⣴⣶⣶
⠉⠙⠻⣶⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⠟⠉
⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⡟⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢿
⠀⠀⠀⢠⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠿⠃⢀
⠀⠀⠀⠀⠙⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠂⢠⡄⠀⠈
⢀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠡⠤⡦⠀⠂
⠀⠀⠤⠀⣀⣀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠠⠀⢠⠀⠀⠀⠀⢀⣀⣤⠴⠟⠃⣄⣤⡀⠀⠀⠀⠀⠀⠀⣀⣌⠤⢄⠀⠀⠀⢀⠀⠀⠀⣠⢤⠆⠀⠐⠂
⠂⠀⠀⠀⠈⠉⠀⠀⠀⠁⠆⠀⠸⠢⠀⠀⠀⠀⠀⠃⠸⠀⠀⠴⠛⠋⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⡠⠠⠈⠉⠀⡗⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠘⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠆⠀⠀⠀⡆⠀⠀⠀⠈⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠈⠀⡀
⠲⠐⠀⠁⠁⠀⠀⠀⠈⠀⠀⠰⠀⠀⡀⠐⠀⠀⠀⠀⢴⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠘⢠⡀⠀⠘⠢⠄⠈⠀⠀⠀⠀⠀⠀⠀⠃⠰⠀⠄⠤⠠
⠀⠀⠀⠀⣀⡀⠀⠀⠀⠐⠀⠀⠀⠒⣀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠠⠰⠲⠘⠊⠁⠀⠀⠀⠀⠲⠀⡀⠁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠈⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠬⡀
⠀⢀⠀⠘⠻⠉⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡠⠀⠀⠀⠀⠀⠂⣡⠀⠀⠀⠐⠀⠈⠁⠀⠀⠀⠠⠀⠂⢈⠠⠄⠀⡀⠄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠂⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⡒⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡠
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠙⠄⢀⡀⠀⠀⠀⠀⠀⠘⢳⣸⡁⠺⠂
⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⣠⡄⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⠀⠀⠀⠀⠀⠀⠐⠢⡀⠉⠀⣤
⠀⠁⠁⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢁⣤⠀⠈⠀⠀⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠂⠀⠀⠀⠀⠀⠀⠀⠹⠟⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠠⠀⠀⢸⡿⠛⢷⣄⠀⠀⠀⠀⠁⠀⠀⣶⠆⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠛⠀⠀⠀⠀⠰⠆⠀⢀⣰⠲⠤⠤⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢀⡄⠀⠀⠱⡄⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠁⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠐⠒⠀⠘⠃⠀⠀⠀⡰⠋
⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⡄⠀⠀⠀⠀⣀⢐⣼⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⠀⠀⠀⠘⠀⢀
⠂⠈⢀⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠀⡊⠅⠀⠀⠀⠀⠀⠀⠀⠀⣦⣠⡀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠠⠏⡀⠀⠀⠂⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠂⠀⠈⠀⢰⠀⡄⠀⡄⡇⣁⠄⠀⠀⢀⠀⢀⣤⠂⠀⠒⠐⠀⠀⠀⣂
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠆⡈⠀⠀⠀⠀⠀⢤⡀⢰⠲⣗⢉⡦⡆⡶⢄⡆⢒⣀⠔⠁⠖⠋⣠⠄⢈⠟
⠀⠀⠀⠀⠀⠀⠨⢯⡆⠀⠀⠀⠀⠁⠀⠀⢁⠀⠀⠰⠃⠀⠈⢺⠁⡟⠃⡠⠼⢁⠏⠀⢂⠰⡽⠉⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠒⠂⠀⢁⠀⠀⠐⠐⠂⠀⠀⠀⠂⠀⠀⠀⠄⢄⣸⡷⠊
⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠴⠖⠂⡄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠋
⠀⠀⠀⠀⠠⠤⠄⠀⠀⡄⠀⠀⠉⠀⠀⠀⡇⠀⠀⠀⡀⢈⡁⡀⡄⠀⠀⡜⠀⠀⠄⠴⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢈⠁⠁⠁⡀⠀⠁⠀⠀⡆⠀⠁⠀⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⠄⠀⡀⢠⠀⠀⠀⡇⠀⠀⠠⠄⠄⠤⠀⠒⢛
⠀⠀⠀⠀⣽⠚⣾⠀⠀⠀⠀⠀⠃⠀⠇⠀⠀⠀⠙⠂⠀⡈⠀⠈⣿⡇⢈⠀⢀⡎⠀⢨⠀⠈⡇
⠀⠀⠀⠀⠘⠒⠁⠀⠀⠀⠀⠀⠁⠀⠀⠠⠄⠳⠖⠀⠀⠀⠀⠀⠑⠀⠈⠀⠈⠁⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣦⣴⣄⠀⠀⠀⠀⣀⣿⣄⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡇⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⠀⠀⠀⠈⢿⣿⣿⡟⠀⠀⠀⢸⣿⣿⡿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡆⠀⣸⣿⣿⣷⠀⠀⠀⣿⣿⣿⠁⣿⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠉⠉⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⠀⣰⣿⣿⣿⠀⣿⡄⠀⠀⠀⠀⣰⣿⣷⣤⣤⣤⠀⣿
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⠀⢿⣇⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⢰⡏
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠉⠀⣀⠀⣿⣿⣿⣿⣿⣿⣥⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠐⡄⢿⣿⣿⣿⣿⣿⠿⠛⡯⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣉⣉⡉⠁⢸⣇⠘⠻⢿⣿⣿⡟⠁⠒⠉
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣶⣿⣿⣿⣿⡧⠈⠛⠀⠀⠀⠉⠉⠀⠸⣿⣶⣦⣤
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠹⣿⣿⣇⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣠⣤⣤⣤⣀⣀⡀⠀⠀⠀⢐⣿⣄⠘⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢀⣤⡀⠈⣿⡿⠀⠈⢿⣿⣿⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠛⢿⣿⣿⣿⣿⠏⠀⠀⠀⣾⣿⣷⡀⠉⠁⢀⣶⣄⠻⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⣿⣿⡟⢁⣴⣶⠀⢤⣙⡛⠛⠁⡀⠀⠀⠀⣿⣿⣿⣧⠀⠀⣾⣿⣿⣆⠙⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣹⣿⠏⠀⠘⠛⠁⣀⣀⣉⠛⠛⠋⢀⣾⡆⠀⢹⣿⣿⣿⡀⠀⢿⣿⣿⡟⠀⠈⢿⣿⣿⣧⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⠁⣠⣶⣦⠀⣸⣿⣿⣿⡿⠀⠀⢾⣿⣧⠀⠀⠙⢿⣿⠇⠀⠘⠿⡿⠁⢠⣶⡀⠻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁⣾⣿⣿⠏⠀⠘⠻⠿⠟⠁⠀⣀⣀⡉⠁⠀⣀⡀⠀⠁⢀⣀⡀⠀⠀⠀⣾⣿⣿⣄⠙⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡇⠀⢿⠟⠁⣠⣶⣶⣦⠀⣠⡄⠀⣿⣿⣿⡇⢠⣿⣿⠆⣰⣿⣿⣿⣧⠀⢰⣿⣿⣿⣿⣧⠈⢿⣿⣿⣿⠃
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣾⣿⣿⡿⠁⢼⣿⣿⠀⠘⠿⠛⢁⣾⣿⠏⣰⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⡟⠀⠀⠻⣿⣿⣿⣶⣦
⠀⠀⣀⣴⣿⣿⣿⣿⠿⠿⠿⠟⠀⠀⠘⠛⠛⢉⣀⣀⡈⠛⠛⠀⠀⠴⠚⢻⣿⡅⠀⣿⣿⣿⣿⣿⣿⠂⠀⢿⣿⣿⡟⠀⠀⢠⡀⠙⣿⣿⣿⣏
⢠⣾⣿⣿⣿⣿⢟⣵⣶⣿⣷⡆⠀⠀⠀⣠⣶⣿⣿⠿⠿⠛⠀⠀⠀⠀⠀⠈⣿⣿⡄⠈⢻⣿⣿⣿⣿⠀⠀⠘⠿⠟⠁⠀⢠⣿⣿⣆⠈⠻⣿⣿⣿⣷⡄
⢸⣿⣾⣿⣿⡏⠿⣿⣿⣿⠟⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡀⠙⠿⠿⠿⠀⠀⣀⠀⠀⠀⠀⣼⣿⣿⣿⣷⡀⠙⣿⣿⣯⡀
⣼⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦⡀⠀⠀⢠⣾⣿⣿⣷⡄⠀⢻⣿⣿⣿⣿⣿⣄⠈⢻⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣡⣤⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇⠀⢠⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠙⠿
⠹⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡏⠀⢠⣿⣿⣿⣿⣿⣿⣿⡆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠙⠻⢿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠇⠀⢀⡀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣶⣤⣤⣤⣀⠀⠀⠀⠀⢠⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢠⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠏⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠋⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠛⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⡀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢻⣿⣿⣶⣦⣄⡀⠀⠀⠈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡷⠈⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⠀⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠟

⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⣄⣀⡀
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⢸⣿⣿⡆⠀⢠⣾⣿⠟⠃⠉⠀⠀⠀⣻⣿⡟⠉⠙⠿⣿⣷⣄⡀⠀⠀⢀⢀⣠⣶⣶⡆
⠀⠀⠀⠙⣿⣿⣧⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⢈⣿⣿⣿⣿⡟⣹⣿⣿⢿⣿⣷⡀
⠀⠀⠀⠀⠚⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣾⣿⡟⠀⠀⠀⡰⠈⠨⢿⣿⣿⡎⣽⣿⡇⢀⣽⣿⡿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⣤⣤⣤⣴⣾⣿⣿⡿⠁⠀⠀⢞⠀⠀⠘⣿⣿⣿⣷⠙⠻⣿⡿⠛⠛
⠀⠀⠀⠀⠀⢠⣿⣿⣿⠛⠿⠿⠿⠿⠿⠛⠉⠀⠀⡀⢀⣾⣴⣤⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠹⣿⣿⡘⠂⠀⠀⠀⠀⠀⣠⡶⠞⠛⠉⠉⡍⡍⠀⢩⣿⣿⣿⡿⣶⣄⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢻⣿⣧⡄⣠⣤⣤⡀⠾⠉⠀⠀⠀⠀⠘⠁⠁⠀⢸⢹⣿⡟⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣴⣾⣿⠟⠀⠀⠀⠀⠀⠀⠸⣇
⠀⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡉⠉⠛⠉⠉⠁⠀⢀⣴⡖⣿⣿⣿⣿⣷⣦⡀⠀⠀⣿⡄⠀⠀⠐⠁
⠀⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⢠⣾⣿⣿⣏⠛⢿⣿⣿⣿⣗⠀⢀⣿⠃
⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⢸⣿⣿⣟⠁⠚⠛⠉⢻⣿⣿⡀⠚⠁
⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠚⠛⠛⠁⠀⠀⠀⠀⠘⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠀⢸⣿⢆⣼⠇⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡈⠿⠃⠈⠓⠼⣟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣝⠃⢠⣼⡳⣦⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⣰⣤⣯⡿⣬⣶⣄⠀⠹
⠀⠀⠀⠀⠀⣢⣠⡄⠀⠀⠀⣶⡻⠂⠀⠀⠀⠀⠀⢀⣾⣯⡶⢀⡠⢻⣿⡔⢶⠬⣭⣤
⠀⠀⠀⠀⠈⢉⣚⣿⢃⠀⡶⣾⣼⢁⡲⠀⠀⠀⠀⢸⡏⢼⡗⠹⠷⣔⡮⣕⣿⣿⣙⡻⠂
⠀⠀⠀⠀⠀⠀⠫⠽⠏⣰⣿⣿⣡⣮⡞⠀⠀⠀⢀⣾⣷⣆⣴⣿⣷⢙⡟⠀⠀⠉⠈⢋⡀⡰⣦⡄⠀⠀⠀⠀⣀
⠀⠀⢀⣰⡄⠦⣰⡄⣠⣭⡿⠿⠍⠁⠀⠀⠀⠀⣸⣿⣿⡿⠹⠿⠟⣸⠁⠀⠀⠀⠀⠀⢹⣻⣿⣷⠀⠀⢀⣽⡥
⠀⠀⠀⢭⣽⣦⡈⠟⠻⠯⠍⠉⠉⠀⠀⠀⠀⢀⣿⡇⣠⡄⣼⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⡆⣿⣿⡺
⠀⠀⠘⠳⠶⣽⢣⣗⣢⢠⣆⡤⠀⠀⠀⠀⠀⣼⣿⢠⡻⡇⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⢠⣽⠕⠛⣛⢡⢰⣟⡩
⠀⠀⠀⠉⣳⣮⠚⢁⣤⠼⢟⡂⠀⠀⠀⠀⣨⣿⣇⣿⣷⡴⣦⣬⣉⠁⠀⠀⠀⠀⠀⠀⢨⣷⣩⡀⠀⣰⣹⣧⣛⣒⡘⠁
⠀⠀⣠⠀⠈⠁⠀⢸⣛⣭⡭⠄⠀⠀⠀⢀⣿⡟⢸⣿⡿⢁⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣒⡼⣯⡜⣻⢿⣿⠽⠿⠃
⠀⠰⣌⠧⠘⢦⣠⡿⠷⠖⠂⠀⠀⠀⠀⣼⡿⢱⣦⡙⢃⣶⣭⣛⠁⠀⠀⠀⠀⠀⠀⢀⡦⠭⣻⠁⣹⠆⠒
⠀⠀⠤⣑⠦⠓⢯⢳⣤⡀⣄⠀⠀⣀⣼⡟⢡⣽⣛⠋⢸⣿⣿⣿⠸⠀⣠⢀⡄⣖⣄⡀⠉⡛⠿⠚⠁⢀⣖
⠀⠀⠀⠨⣽⣿⣾⣷⣿⣝⢻⣧⣀⣿⠏⣐⣿⣿⣿⣇⠙⠿⡿⠃⠆⡄⣿⣼⠀⠿⣛⡥⣊⣀⣠⢤⣆⣿⠍
⠀⠀⠀⠀⠀⠒⠛⠉⣶⣾⣿⣿⣯⣅⣀⡈⢙⠻⠿⠿⢡⣶⣶⣾⠀⠹⣾⡿⠾⣭⣯⣾⣿⣛⠛⠻⠾⠆
⠀⠀⠀⠀⠀⠀⠀⠙⢛⡉⠉⣉⡡⢭⣭⣿⠟⡿⣷⣄⠚⢻⣛⣭⡴⣚⣷⡛⠼⠬⠅⠈⠺⡯⠩⠉
⠀⠀⠀⠀⠀⣀⣀⣒⣾⣽⡶⢮⡿⣿⡏⢼⡇⡇⠀⢈⡿⢿⡙⠦⣍⡻⢭⣝⠓⠂⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠘⠊⢉⠾⢁⣼⠁⠸⠁⠊⠉⠁⢠⠏⠀⠀⠛⢦⡀⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠈





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠖⡁⠀⠀⠀⠤⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠠⠀⠐⢃⠔⠃⠈⠁⠀⠀⠤⠀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⠁⠀⠜⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢠⠤⠄⠈⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠂⢄⠀⠀⡄⠀⠀⠀⠀⠀⠠⠆⡀⠀⠀⠀⠀⠀⢀⠂⡔⠀⠀⠈⠀⠀⠀⠒⠠⢄⠀⠀⠀⠀⠀⠴⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⢀⠀⠀⠁⠄⠀⠉⠀⠀⠀⠀⢂⠀⠀⠈⠄⠐⠲⠆⠶⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠒⠂⠠⠄⠤⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠁
⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠒⠴⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢇⣀⣺⠶⠈⠃⠈⠀⠀⠈
⠀⠀⠀⠀⠉⠉⠐⠂⠀⠒⠀⠀⠀⠁⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⣤⠄⠀⠀⠘⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠙⠀⣀⠀⢠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⡀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠐⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠄⡐⠚⠶⠒⠤⠀⢀⣀⣀⣀⠤⠄⠐⠖⠊⢹⡇⣀⢀⠀⠠⠤⠤⠄⠀⠀⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣂⣀⣐⣈⣀⣀⣒⣒⣒⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣤⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⢠⣶⡄
⠀⠀⠀⠀⠀⢀⣾⠟⠉⠀⢸⣿⠀⠈⠙⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⢸⣿⡇
⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⢸⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣤⡴⣿⣾⣿⣬⣭⣥⡄
⠀⠀⠀⠀⠘⣿⡀⣠⣾⠟⢹⣿⠙⢿⣦⡀⢰⡿⠀⠀⠀⠀⠀⠀⢻⣿⢻⡿⣿⡽⣹⣿⡇
⠀⠀⠀⠀⠀⠘⢿⣿⠁⠀⢸⣿⠀⠀⢙⣿⡿⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣾⣿⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⠳⡀⢸⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣚⡭⣭⢩⢭⣙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢣⣸⠀⣿⠀⠀⠀⠀⠀⠀⠀⡴⢡⠞⠁⠀⣿⢸⠀⠀⠑⢎⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⣆⠈⣿⠀⣿⠀⠀⠀⠀⠀⠀⢰⢣⡇⠀⠀⠀⣿⢸⡀⠀⠀⠈⡇⢧
⠀⠀⠀⠀⠀⠀⠐⡏⢹⡀⢹⣴⠋⠉⠉⢧⠀⠀⠀⠀⠀⢸⢸⡄⠀⣠⢞⣥⢠⡙⢦⡀⠀⡇⣸
⠀⠀⠀⠀⠀⠀⠀⣧⡨⠧⠼⠈⣹⠗⠀⣸⠀⠀⠀⠀⠀⠘⣆⠣⢞⡵⠋⣽⢸⠙⠦⡙⠶⢣⠇
⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠈⠢⣘⠦⣄⣽⢸⣠⠴⢛⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠶⠶⠶⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠶⠖⠚⠉


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣽⣿⣯⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⡐⠀⠃⠦⠤⠀⠀⠉⠀⣀⠤⠒⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⢀⡂⠐⠀⡠⠂⠀⠀⠀⠀⠀⠂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠆⠀⢀⠢⠀⠊⡀⠐⠀⣁⠒⠀⠒⠂⠀⠀⡠⠉⠀⠀⢠⠶⠈⠁⢆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢤⠀⠀⢈⠉⡀⠀⠠⠕⠂⠀⠀⠀⠀⢐⡀⠑⠉⠑⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠉⠓⠀⠀⠀⠀⠆⠄⠀⠀⠀⠀⠀⠖⠰⠒⠂⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⡄⣈⠲⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⢢⠚⠳⠀⡀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠊⠀⠴⠉⡠⠀⠀⠀⠐⠀⠀⠀⠤⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠖⠠⠀⡴⣂⠀⠁⢼⠶⠆⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⡀⠠⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠁⠂⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣻⣩⣛⣿⣹⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⠻⢦⠜⢷⡖⠺⠇⠀⣴⡶⢦⣴⣄⠀⣤⣀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⠈⠁⠀⠀⠀⣤⡀⡠⣀⣴⣿⠀⠂⣀⠙⠻⠷⣇⣰⣿⣲⣿⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡤⡀⢫⣄⠠⠌⠉⠀⠀⠀⠀⠰⠟⠁⣼⣿⠙⣿⠇⢸⣼⠛⢻⣯⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⣾⣤⣿⡦⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠓⢉⡠⠐⢋⣵⢞⣿⣧⠬⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣦⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⢸⡿⠃⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡤⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⢣⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡞⢿⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⣯⢀⡰⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠻⠿⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⡷⢸⣿⣷⢰⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⢴⣿⠍⠋⢴⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣭⣡⣬⣍⢼⣷⡦⠲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡏⠿⣿⠎⡛⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⣿⣷⡌⢩⣿⡖⣴⣿⣿⣆⢉⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⣻⠇⣠⣥⣈⣛⡟⢡⣦⣙⣃⠹⠿⢟⣩⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⡿⣄⢠⣛⡋⠿⠿⢋⣬⡻⣿⣿⣄⣀⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢰⣄⣶⣤⠘⣟⣁⡿⢂⣿⡟⣧⣿⡟⣿⣿⣆⣭⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣼⣛⢛⣠⣛⣻⠻⠋⠉⠁⠈⠉⠁⣿⡥⢬⣻⣯⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠸⣏⣈⡛⠟⠁⠃⠀⠀⠀⠀⠀⠀⣭⡵⠶⢿⡏⠀⠀⢀⣤⣤⣴⢶⣾⣿⣷⣾⣿⣿⣴⣮⣥⣴⣆⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠎⣀⠶⠍⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠂⠠⠀⠀⠀⠉⠉⠛⠙⠛⠛⠛⠻⠯⠿⠟⣿⢿⠛⣿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⢈⡋⠒⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠂⠀⠂⠀⢤⠶⠶⠴⠾⠿⡿⢶⣤⣤⣄⠀⠀⠙⣧⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⢀⠸⠇⠀⣀⠀⢀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⣠
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣌⣤⠶⠆⠀⠀⠀⠀⣀⣀⣀⣤⣤⡴⠶⣶⣧⠴⠟⠃
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣶⡶⠶⣶⣾⣟⣭⡭⠽⠶⠖⠒⠋⠉⠉
⠀⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉
⣿⣿⣿⡿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣥⣤⣧⣤⣦⣤⣷⣤⣷⣭⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣿⣿⣿⡿⠿⣿⣿⣿⡿⠋⠀⠸⣿⣿⣷⣦⣤⣀⣀
⠀⠀⠀⠀⠀⢸⣿⠟⠋⠿⠿⠿⣛⣛⠿⢿⣏⡛⠙⠋⠉⠙⠊⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⢸⣿⡆⠀⡠⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣰⠀⠆⠀⠺⣦⣢⣤⡀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣄⣀⣼⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣀⡤⢾⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠟⠛⢛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⣾⣶⣬⡆⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢄⡀⢀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣠⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⣦⣤⡄⠀⣯⡀⠀⠈⠙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⠀⣿⡇⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠋⠁⠀⠀⠈⠙⠛⠿⠇

⠀⠀⠀⠀⠀⣿⡀⢠⡄
⠀⠀⠀⢰⠾⢟⢡⣿⣷⣤⡀
⠀⠀⢠⣮⣽⢹⣟⢫⣼⣛⣿⣷⣤⡀
⠀⣴⣿⡿⠿⠛⠁⠀⠘⢛⣼⣿⣷⢛⣷⣤⣀
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠙⠿⢱⣿⣿⢿⣿⣿⡶⣦⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡿⣫⣟⣿⢸⣿⢿⣾⢿⣽⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⣸⠿⣿⣷⣿⣿⣷⣛⣿⣿⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠏⣽⣿⡿⣽⣿⣝⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⣿⣿⣯⣽⣯⣟⣽⣮⣛⡿⣿⣿⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢻⣿⣿⣿⣿⣿⢿⣿⣿⣽⣿⣿⡿⣿⣟⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣭⠿⣿⣹⣿⣻⣿⣿⣿⣿⣿⣷⣿⡿⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⣿⣿⣿⡆⣾⢷⡍⠻⠋⠻⣿⣿⣽⣯⣿⣿⣟⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣾⣛⡉⠉⠛⠒⠂⠈⠉⠳⢷⣯⣿⣟⡇⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡷⡀⠀⠛⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠈⣷⡿⢡⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠈⣽⣧⠀⠀⠀⠀⠀⠀⠀⢻⣿⣬⣱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⢸⡟⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⣿⡇⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢠⣿⡃⠀⠀⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠀⠀⢀⣴⡿⠋⠀⠀⠀⠸⡿⠃