Satisfied Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⡋⡴⣶⣀⣨⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡮⠒⠴⣦⣀⠙⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠓⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠦⢄⡀⠀⠀⠀⢀⠔⠈⠡⣀⡠⠊⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣤⠀⠀⠀⠀⠈⠑⠲⢤⠋⠀⠀⠀⠀⠀⠀⠀⠐⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠄⣾⠟⡀⢱⠤⠤⠄⠀⠀⣀⠆⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠐⠓⢅⠀⠀⠀⠈⠑⢆⢄⡀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⡈⠓⢄⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠰⣯⡁⠀⠀⠀⠀⠀⢀⠜⠢⡀⠀⠀⠀⠀⢸⠂⠈⠑⣤⢶⠇⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⢳⠀⠀⢘
⠀⠀⠀⠀⠀⠀⢸⡟⢦⣄⠀⠀⠀⠀⠐⡄⠑⠬⠦⡒⠕⢊⣠⣶⣋⠡⢎⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⢀⡂⠀⠀⡌
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠳⣄⠀⢦⠀⠈⠢⠊⣀⡤⠖⠋⠁⠀⢀⡤⠈⠑⠦⣄⡀⠀⠀⢀⠀⠀⣠⢃⠀⢀⠜
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠙⣦⡀⣠⣴⠚⠉⠀⠀⠀⣠⠔⠋⠀⠀⠀⠀⠀⠈⢁⡠⠛⢺⡱⠀⡘⠊⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠋⠁⢸⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⣀⣤⣐⣊⣉⣀⣠⡞⠦
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠲⡞⠉⣿⠀⠀⠉⢹⡇⠸⣧
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆⠀⠀⡘⢇⠀⠀⠀⠘⡀⣿⠀⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠈⢦⠀⠀⠀⢣⣿⠀⠀⠀⢀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢸⡄⢀⣰⠧⠤⣤⠃⠀⠈⢆⠀⠀⠈⠇⠀⠀⠀⠘⠃⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⢸⡏⢉⣀⣀⣤⡼⠀⠀⠀⠈⢆⠀⠀⠘⡄⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⢸⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠤⠖⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⣠⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⣁⣤⠞⠁

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

⠀⠀⠀⢀⣤⣶⣶⣶⣤⡀⠀⠀⠀⢀⣤⣴⣶⣶⣤⡀⠀⠀⠀⢀⣠⣴⣶⣶⣤⣀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⣴⡿⠛⣿⣿⣿⠛⣿⣆⠀⣰⣿⠛⣿⣿⣿⠛⢻⣦⠀⢠⣿⠛⢻⣿⣿⠛⢻⣧⠀⢠⣾⠛⢻⣿⣿⡟⢻⣷⡀⢀⣾⡟⢻⣿⣿⡟⠻⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣷⣿⣿⡆⣿⣿⣷⣿⣿⣿⣷⣿⣿⡇⣾⣿⣿⣿⣿⣿⣷⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⡟⣿⡇
⠀⠀⢿⣿⣾⣿⣿⣿⣷⣿⡟⠀⢻⣷⡻⣿⣿⣿⣿⣽⡿⠀⠹⣿⣿⣿⣿⣿⣿⣿⡿⠁⠸⣿⣟⣿⣿⣿⣝⣿⣿⠃⠘⣿⣿⣫⣿⣿⣽⣿⣿⠇
⠀⠀⠀⠙⠿⣿⣿⣿⠿⠋⠀⠀⠀⠙⠿⢾⣿⣷⠿⠋⠀⠀⠀⠙⠻⢿⣿⣿⠿⠛⠁⠀⠀⠈⠻⢿⣿⣿⠿⠟⠁⠀⠀⠈⠻⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⣆⠀⣼⣿⡄⣠⣾⡆⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⣷⣼⡿⠿⠿⢿⣿⠿⠿⢿⣷⣴⣿⡇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⢟⣫⣵⠶⡟⣻⣏⣿⡛⣟⡷⢶⣭⡻⢷⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⣶⣿⢟⣵⣿⢯⣯⠶⠛⠛⠋⠉⠛⠛⠷⣯⡾⣿⣷⣝⢿⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⡿⢣⣿⣿⣿⠋⠁⠀⣀⠀⠀⣀⠀⠀⡀⠀⠙⢿⣿⣿⣦⢻⣿⣁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⢣⣿⣿⡿⠁⣾⡇⢰⠟⣧⢸⠟⡇⣼⢻⢀⡏⠀⠹⣿⣿⣧⢻⣿⡿⠋
⠀⠀⠀⠀⠀⠀⣀⣤⣾⡏⣾⣿⣿⠁⠀⢸⡇⢸⠀⣿⣿⠀⣷⢹⣼⣸⢱⢶⠀⢿⣿⣿⡸⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠙⠻⣿⡇⣿⣤⣿⠀⠀⣸⣇⢸⣦⡟⢹⣤⡏⠀⢀⡏⢹⣸⠀⢸⣦⣼⡇⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⣠⣶⣿⣧⢿⣿⣿⡄⢀⡉⡉⣀⣉⡁⡈⡉⢁⣀⡈⣁⣈⠉⠀⣾⣿⣿⢹⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⢛⣿⡜⣿⣿⣷⡈⠷⠛⠀⠫⠷⠃⠷⠢⠀⠀⠣⠏⠃⣼⣿⣿⡏⣼⣟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡜⢿⣿⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⣿⢟⣼⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣮⡻⢮⣿⢻⡷⣦⣤⣤⣤⣤⣤⡶⢛⣽⣯⠟⣡⣾⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⣹⣿⣶⣝⣛⠶⣧⣧⣽⣎⣿⣤⡷⢞⣫⣵⣿⣿⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠉⣿⣿⡿⣶⣶⣶⣶⣶⣶⢿⣿⣿⠏⠙⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠸⣿⠋⠈⢿⡟⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠉⠈⠁⠀⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠋⠉⣅⣠⠄⠀⠀⠀⠠⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⡉⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⢴⠗⠛⠀⠀⠀⣤⡀⣼⠿⠶⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⠠⠠⢤⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠁⠀⠀⠹⠏⡄⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠢⡀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣏⣀⣧⠧⠄⠒⠀⠀⠀⢆⡀⠀⠀⠀⢱⠀⠀⠀⣿⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠜⢹⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠥⣄⠀⢸
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⡀⢀⡀⠀⠀⠀⠀⢤⣿⣄⣠⣿⡄⠀⢸⡇⠀⠀⢀⡀⠀⠀⢀⡤⣞⡛⣉⠀⢲⡀⠀⠀⠀⣀⣤⣾⣧⣾
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠁⠈⣀⣀⠀⠀⣷⣦⣰⣿⣿⣿⣿⣿⣿⣆⠀⢻⡄⠀⠀⠀⠀⠀⣿⣷⣶⡞⠉⠛⠑⡶⠊⠀⠚⠫⠽⠿⣿⠏
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⠘⠁⠀⠀⠀⠙⢿⢃⣀⢹⣿⣿⣿⣿⣿⡆⠀⠻⣄⠀⣀⠄⠀⣿⣿⣿⠁⠀⢀⡜⠁⠀⠴⠒⠉⠁⣴⠋
⢰⣿⣿⣿⣿⣿⢻⣿⠟⣻⣿⣿⣿⣿⣀⡄⠀⠀⠉⠁⠀⠀⠀⠀⢃⣾⣿⣿⣿⣿⣿⣿⡀⠀⠙⢷⣄⠀⢠⣿⣿⠉⠀⠀⠈⠀⠀⠀⠀⢀⣤⠞⠁
⣾⣿⣿⣿⣿⠧⣼⡁⢼⠛⠛⢿⣿⣿⠀⠤⠐⠀⠀⠄⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠛⠾⣿⣇⣀⣀⣀⣀⣀⣠⡤⠞⠋⠁
⣿⣿⣿⣿⣿⡈⠤⠇⠈⠀⣿⣖⢻⣿⡄⠀⠀⡀⠁⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣿⣿⣿⣿⣿⣷⠀⢀⠀⢀⠈⠋⢸⣿⣿⣄⣀⡀⣀⠆⣩⣾⣿⠿⠛⠉⠨⠿⣿⣿⣿⣿⡇
⢹⣿⣿⣿⣿⣿⡄⠀⡱⠁⠀⣠⣿⣿⢿⣿⣿⣦⣤⣾⡚⢋⣰⠆⠀⠀⠴⠒⠚⠛⢿⣿⠇
⠘⣿⣿⣿⣿⡿⠟⠂⠀⠀⠚⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠘⣿⣿⡿⠁⠈⠉⠀⠀⠘⠿⠋⠻⠋⠁⠀⠌⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢷⣶⣦⣤⣤⣤⣤⣤⣴⣶⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⢀⣠⣤⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠈⠀⠀⠀⣠⣯⣯⣿⣫⣽⣿⣿⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠟⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣉⡀⠀⠈⠀⠀⠉⢿⣿⠀⠀⠀⠐⢦⡄⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡥⣬⣭⣧⡀⠀⣀⣿⣷⣈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣾⣿⣿⡟⢻⣿⣶⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⣼⠀⠉⠉⣉⣁⡀⠀⠛⡛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⢲⡊⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠇⠀⠀⠡⣄⣀⠀⣛⣦⣶⣿⣿⣿⣿⣶⣿⣿⣿⣶⣞⠀⠀⢀⡿⠀⠀⠀⠀⢀⠴⡆⠀⠀⠀⠉⠛⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⣏⠙⢿⣍⣉⣉⣉⣽⡟⣿⣿⠛⠋⠉⠀⠀⠀⠀⠀⢂⣀⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠛⠿⢿⣿⢟⣰⣿⣿⣆⠀⠀⠀⠀⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣦⣦⣾⣿⣿⣿⣿⣿⣿⣷⣀⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠗⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣟⠛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠛⠛⠛⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠿⣿⣿⣿⣿⣿⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣬⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠋⠻⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣀⠀⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈

⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀⠀⠀⣠⣾⣿⣿⣿⣶⣄⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀⠀⢀⣴⣾⣿⣿⣿⣦⡀⠀⠀⣠⣶⣿⣿⣿⣶⣄
⠀⢀⣿⣇⣨⣿⣿⣅⣸⣿⡀⣼⣿⣀⣿⣿⣯⣨⣿⡆⢰⣿⣅⣸⣿⣿⣀⣽⣷⠀⣾⣯⣨⣿⣿⣇⣸⣿⡄⣸⣿⣄⣽⣿⣯⣀⣿⣧
⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⠃⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇⢻⣿⣿⢿⣿⡿⣯⣽⡷
⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠈⢿⣮⣿⣿⣿⣿⠟⠁⠀⠻⣿⣿⣿⣿⣿⡿⠃⠀⠙⢿⣾⣿⣿⣿⣿⠟⠀⠈⠻⣷⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢴⠦⡀⠀⠀⠀⠀⠀⠀⣀⢴
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠻⠦⠤⠤⠤⠤⡾⠃⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⢀⡀⠀⠀⠀⠀⠀⡼⢄⣀⣀⡠⠤⠔⠒⠋⠉⠉⠉⠉⠑⡒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠽⣧⢀⠈⡄⢉⠁⢤⡼⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠊⣹⢦⡉⠉⣁⡴⡛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⢣
⠀⠀⠐⢏⠉⠉⠀⠀⠉⢻⠀⠈⠉⠁⠀⠁⠀⢠⠊⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠈⡆⠀⠀⠀⣠⠬⢧⡀⠀⠀⠀⠀⠀⢎⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇
⠀⠀⠀⠀⢡⠀⠀⠀⣏⡉⠉⠉⠉⠑⠒⠒⠂⠘⢄⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠳⡄
⠀⠀⠀⠀⠀⢣⡀⠀⠉⠓⠉⠉⠀⠀⠈⠉⠉⠉⠙⠓⠤⣀⠀⠉⠑⠒⠒⠀⠀⠀⠀⠀⠀⠀⣀⠚⠁⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠤⠤⠤⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢀⣀⣀⣀⣀⣀⣀⠠⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠤⠄⠒⠒⠒⠂⠤⠤⠄⠀⠠⠤⠤⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢁⠄⠈⠀⠀⠀⠉⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣶⠀⠀⣶⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠁⠀⠀⢳⡀⠀⠀⠀⢤⣀⣀⣀⠄⠊⠀⢀⠌⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠉⠒⠒⣒⢒⠈⠁⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠌⢰⣿⣦⠱⡄⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠌⠀⠈⠻⠃⠀⠘⢆⢀⡀⢄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡠⠲⡆⠠⠤⠤⠸⣆⡠⠂⠈⠆⠦⡀⠙⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢃⠀⠐⠤⢸⠁⠀⠠⠐⢸⣷⡈⠉⠁⠉⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸⠀⠀⠀⠈⠁⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠬⢄⡒⠥⡂⢄⡀
⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⣁⣀⣀⠀⡠⠀⠀⠀⠀⠀⣀⠀⠸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰⠈⠑⢬⡢⡙⢦
⠀⠀⠀⣇⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢜⠆⠱⡀
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⢒⡀⣡
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠻⣿⣿⣿⠃⣀⡀⠀⠀⠀⠀⠐⠂⠀⠉⠁
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢀⣀⢀⠀⠀⠤⠤⠐⠒⠀⠀⠉⠁⠈⠑⠒⠤⣀⡀
⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣠⠃⠀⢀⠠⡉⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠱⡕⡄⠀⠀⠈⠉⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢜⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣄⡀⠀⠀⢀⣀⠀⠤⠔⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣴⠆⠰⠟⣶⠒⡠⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠁⠀⠀⠀⠀⠠⢢⣠⣸⣧⣾⣀⣁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡀⢹⣿⣟⣍⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡠⠤⠤⠌⣉⠐⠀⣀⢺⡿⣻⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠊⠁⠁⣴⡶⠶⡶⠶⠲⠤⠄⠸⠿⢡⡿⠋⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⢀⠀⠀⢷⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡊⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠈⢧⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⠁⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣵⣾⠀⢸⣏⣑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣿⣇⣸⠃⠸⠿⢼⠀⠀⠀⠰⠶⠒⠒⠂⠂⠀⠀⠀⠀⠀⡆⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⢸⣿⢿⣿⣷⠀⠀⠀⠘⠀⠀⠐⠚⠓⠀⣶⡶⣶⡄⢀⠀⠀⠀⣵⡀⠀⢹⠐⡀⠀⠀⠀⠀⠀⢠⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠑⡛⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠀⠀⡾⣾⣷⡄⠸⣷⣀⠀⠀⢢⣤⣦⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣇⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⣳⣤⡈⠀⠀⠀⠘⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣿⣿⠀⠀⠀⣈⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⡀⠸⠸⡿⠀⢰⡇⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣆⠀⠀⠀⠉⠉⠓⠒⠶⠂⠄⠀⠀⠀⠀⠀⢀⣾⡿⣿⠃⠀⠀⠃⠀⠀⢱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣿⣿⣦⠀⠐⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢞⣥⣶⠏⠈⠀⠀⠀⡀⠀⡸⡀⡀⠀⡀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠴⠿⠟⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⣴⡿⢋⡥⠊⠀⠀⠀⣰⡇⠀⢠⡇⣿⡶⠈⠢⢺⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⢨⣶⣶⡖⠒⠖⠒⠛⠉⠀⠀⠡⠖⠊⠁⢀⡤⠊⣀⢴⠟⠀⠰⠟⢰⡯⡧⠐⡦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠅⠀⠂⠀⠀⠀⠀⠀⠶⣦⣶⠛⢃⣀⣈⣁⢀⠁⠄⠀⠐⠋⠸⢀⡼⢃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡿⠛⠉⠀⠀⠀⠠⠀⠀⠈⠙⢾⣍⡁⠁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠥⠉⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢴⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⠀⠀⢈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⣀⣠⠴⠊⠁⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⡞⡄⠀⡰⡄⠀⣴⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠢⡀⢸⠈⢦⡜⠀⠸⡔⠁⠳⠜⠰⣆⠌⣽⠀⣠⢶⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⠢⣄⣸⠀⠈⠃⠀⢀⡀⣀⣤⢤⠤⡄⣤⣀⣉⠰⠸⢞⢵⢙⣀⢔⣹⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢤⢄⣀⢘⡀⠀⠉⣀⢤⣲⠬⠗⠓⠛⠊⠉⠉⠉⠉⠛⠛⠿⢵⣳⢥⣁⠕⡍⣺⠤⡪⣭⠀⢀⡀
⠀⠀⠀⠀⠀⣀⡀⠈⣆⠀⠉⠁⣠⣺⠝⠋⢀⠀⠀⠴⠒⠀⠰⡀⢰⠙⠂⠒⡦⠄⠈⠙⠺⢽⣲⢬⣎⠆⢷⢊⢵⠃⠀⡀
⠀⠀⠀⠀⠀⠩⡈⠉⠉⠀⣠⣺⠞⠁⢠⡀⠛⠤⢄⠘⡎⠀⠏⠙⠘⣤⠄⠀⡁⠀⢀⠁⡠⣄⠈⠑⢮⣗⢭⡑⡿⡒⡹⠃
⠀⠀⠐⢖⢖⠒⠚⠀⢀⣞⠞⠁⡠⡂⠀⠑⢄⠀⠚⠀⣐⣠⡤⡤⡴⡶⡤⣤⣀⠀⠁⠸⡀⠎⣠⡆⠀⠈⠻⣟⣮⢚⢗⠒⣨⠃
⠀⣀⣀⣠⣱⠄⠀⢠⣻⠊⠠⣄⣀⡈⠳⠄⢠⢠⢵⣻⠝⣚⢛⢉⢉⢍⢉⢉⠛⠿⣶⣄⡀⠐⢀⡧⠔⠁⠀⢊⢾⢧⡣⠺⠧⠤⢤⠄
⠀⠉⠢⡂⡀⠀⢠⣷⠃⡠⠀⠘⢮⡉⠁⢀⡶⡧⠋⠀⠀⠀⠈⠀⠈⠈⠀⠈⠈⠑⢌⡙⢽⣦⠀⠀⠀⠀⠀⠀⠈⢯⢧⢕⢵⣟⣁⡀
⢀⡤⠄⠞⠂⠀⣞⠃⠀⠁⠈⠀⢹⠀⠨⣸⡿⠁⢀⣀⠀⠀⣠⠤⣀⠀⠀⡴⡖⣄⠀⠀⠈⢻⣷⠀⠀⠀⡶⡀⠀⠼⡝⡆⢺⢀⡶⠞⠁
⠀⠑⠪⣀⠄⢰⢾⠀⠀⠀⠀⠀⠀⠀⢀⢯⠃⡴⢩⢤⡇⢸⢳⣶⢎⢧⠸⢉⡾⡮⣇⠀⣀⠀⢻⡇⠻⡉⢠⣿⠝⠁⣯⢵⢐⢝⣑⠢⣤
⠰⢖⣊⡉⠀⠸⠿⠀⠀⠀⢠⠀⠀⠐⢸⢿⠀⠓⢳⡇⡇⢸⢺⡇⢳⣻⡀⡞⡇⢸⣽⠘⢾⠃⢸⢿⠀⣯⠝⢾⡄⠀⢸⢽⠀⡻⣟⡋⠁
⠀⢀⣉⡦⠀⢸⣷⠀⣀⡠⠃⠣⢤⡄⠺⣿⠀⠀⢀⡣⣛⠸⡮⣧⣸⡺⡇⢡⢽⣆⢼⡇⡜⠯⢺⣿⠀⠉⠀⠀⠀⠀⢸⢽⠎⣴⢧⣮⠵
⠛⠫⢔⣒⠀⢸⣻⠀⠉⡆⢰⡿⡏⠀⠁⢯⣆⠀⠈⠚⣽⡀⠻⣕⣫⡾⠁⠈⠣⠥⠟⠁⠀⡀⣾⠇⠀⣠⠤⣄⠀⠂⣎⡿⢺⢪⠫⣀
⠀⢀⡤⠊⠁⠀⠿⡆⠀⠷⠁⠉⠓⠀⠈⢌⣿⣆⠀⠓⠋⠁⠀⠀⠀⠀⠀⠂⢀⠠⢀⠀⢂⣼⡟⠀⡀⠑⠢⡬⢀⢵⣶⡵⣡⡥⠷⠿⠓
⠀⠉⠀⠒⡲⠁⠘⡽⣄⠀⠀⠀⠀⠀⠀⠀⠩⠱⣷⣤⣀⡀⠀⠀⠄⠌⠄⠌⠆⠂⣀⢴⡿⠋⠀⠐⢧⡆⣠⠀⢊⢯⡇⣴⣑⣍⢦⡀
⠀⠀⢠⣎⣀⣴⡀⠘⢯⢆⠀⠀⠀⠀⠔⠉⠃⡀⠀⠉⠛⠻⠽⣿⣖⣖⣶⣲⣾⠯⠟⠉⠀⠀⢮⣠⠀⡈⢁⣀⣯⣏⣬⣌⣯⡉⠉⠁
⠀⠀⠀⠀⢀⠞⡀⠀⠈⠻⡵⢄⠀⠸⠔⠃⡜⢀⠎⢀⡆⠀⡤⢄⠀⢀⡀⠀⡀⠀⠀⠐⠤⠀⠀⠐⠊⣰⣺⢯⢟⣾⢷⠻⠿⠟
⠀⠀⠀⠀⠚⠒⠓⣹⠃⢀⠈⠫⢗⡦⣀⠀⠓⠃⡴⠛⡆⢀⠓⡎⠀⣨⢷⠀⠋⠲⣧⠀⠈⠃⠀⠠⣫⢾⢿⢇⢾⣝⣏⣇
⠀⠀⠀⠀⠀⠀⠰⠧⠐⠊⡇⠀⠀⠉⠫⠽⣲⣤⣀⡀⠀⠀⠀⠁⠀⠃⠈⠂⠘⠀⡈⠀⢐⣀⢴⡻⢾⢡⣎⡻⠌⡆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣠⠔⢻⠀⣠⣀⠀⢝⣞⠻⠯⠧⣖⣶⣤⢤⢤⡤⣶⣲⡖⠿⠽⠺⢁⢂⣥⡑⢹⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠉⢸⠐⡣⢶⡨⢊⣢⣖⢔⢄⡌⡘⣙⡱⡰⣡⣴⡱⣱⠳⡗⠃⠈⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠎⠁⢸⡀⠏⠁⢳⢃⠎⢫⠊⡘⠱⡜⡰⠈⢣⡝⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠋⠀⠀⠛⠀⠀⠘⠃⠀⠀⠁
⠀⠀⠀⠀⢤⢔⠔⡠⢂⠔⡠⢄⡠⢂⠔⢂⠄⡠⠀⡠⢀⠀⢀⠄⣀⠀⡀⢀⠀⡠⠀⡠⠀⡀⡠⢀⠄⡠⢀⠄⡠⢀⠀⡀⢀⢀⢀
⠀⠀⠀⠀⠀⠀⠈⠈⠁⠊⠐⠁⠒⠁⠊⠡⠊⡠⢊⠔⡡⢊⡡⢊⡴⣈⢔⡡⢊⡠⢊⠔⡨⠊⠔⠡⢊⠔⡡⢊⠔⡡⢊⠞⠕⠱⠡⠊⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠂⠀⠀⠀⠀⠀⠉⠀⠀⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠀⠀⠀⠉⠐⢳⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠰⠁⠀⠀⢈⠂⠀⠀⠀⠀⠴⠒⠀⠀⠀⠒⠀⢳⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠰⠆⠠⠀⠂⠒⠉⢸⠀⠀⠀⠀⠀⢈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⡠⢊⡠⠊⠁⠒⠤⠤⠤⠤⠤⠤⠤⠤⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠆⠀⠀⠀⠀⠀⠐⠀⠠⠁⠀⡇⠀⠄⠈⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠬⠙⡆⠀⠀⠀⠠⣤⣤⠤⠴⠋⡼⢄⡀⠀⡀⣀
⠀⠀⠀⠉⠉⠙⠁⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠰⠀⠆⢀⠏⠀⠈⠓⠄
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠕⡦⢀⠀⢀⣠⠆⠀⠀⢸⠀⠀⠀⡰⠈
⠀⠀⡇⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡌⠀⣠⡈⠓⠢⢸⡀⠀⠰⠁⢱
⠀⠀⠌⠀⠀⠑⡄⠀⠀⠀⠜⠀⠀⠀⠀⠠⣀⠘⠈⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⡸⠀⠀⠀⢦⡇⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠑⠢⢄⢰⡀⣀⠄⠀⣸
⠀⠁⠀⠀⠀⠤⣙⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢦⠀⠀⣀⣀⡴⠃⠀⢠⠃
⡄⠀⠀⠀⠀⢀⠘⠐⠂⠤⠀⠀⠀⠀⠀⠉⠁⢠⠀⠘⡀⠀⠀⠀⠀⢸⠀⢸
⢷⡀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃⠀⠀⠀⢀⡇⠀⢸
⠀⠳⡇⠀⠀⢸⡤⠤⠤⠤⢀⡀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⡾⠀⠀⢸
⠀⠀⠹⠄⠠⠒⠀⠐⡄⠀⠀⠈⠑⠢⠤⠤⠤⠜⠀⠦⡀⠀⠀⢠⠇⠤⠚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠁
⠀⣁⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠦⠲⠶⠳⠦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠙⠃⣠⠾⡾⣋⢨⢈⣤⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⣘⣡⣄⣠⡄⠉⠉⠻⡅⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠻⢃⠀⠈⠙⠳⠃⠦⡀⠄⣀⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠰⠛⢳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡤⠀⠀⢠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡤⠤⠒⠁⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠙⠛⠲⢦⣄⡀⠀⠀⠀⠀⠀⣠⠞⠉⡀⠸⢄⣀⡴⠋⠁⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀⡴⠃⠀⠀⠐⠀⠀⠘⠁⠀⠀⠀⡠⠜⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⣀⡴⣶⠟⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢁⠃⠀⠀⠀⠀⢀⠈⠀⠀⠀⠘⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠁⢀⣠⠆⠑⣻⡿⠡⡀⠀⣜⠳⠒⠶⠶⠦⠤⠀⢀⣰⠇⢰⠀⠀⠀⠂⠀⠀⠈⠀⠀⠁⢠⡀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⣶⢡⡁⠀⠀⠁⠀⠀⠀⠙⢣⡤⣀⠀⠀⠁⠀⠀⠀⠉⢠⡇⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠟⢦⡀⠄⠀⠠⠀⢳
⣀⣤⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⢠⡉⠢⡀⠀⠀⠀⠀⠀⠀⠘⣆⠉⠲⢤⡀⠄⠁⣠⡾⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⢀⡏⠀⠀⠑⢄
⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠊⠉⢦⡈⢂⡀⠀⠀⠀⣀⠐⡏⠀⠀⠀⢈⣵⠿⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠀⠀⠀⠀⠈⡇⠀⠀⠐
⠀⢸⡟⠿⣦⡀⠀⠀⠀⢀⠀⠀⠀⠱⡄⠀⠱⣄⣳⣄⠴⣋⡤⠖⠁⣀⣤⠾⠋⣁⣴⠋⠈⠁⠀⠈⠀⠀⠈⠀⠀⠤⡴⠀⠀⠀⠀⠀⡼⠃
⠀⢸⡇⠀⠈⠛⢷⣄⠀⠘⡄⡆⠀⠀⠘⣆⠀⣀⠤⠒⠉⠁⣠⣴⠞⠛⠛⠉⠉⠉⠈⠳⣄⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⢀⠀⢈⡼
⠀⢸⡇⠀⠀⠀⠀⠙⠳⣦⡁⠸⠆⠀⠀⠈⠋⠁⢀⣠⡶⠛⠉⠀⠀⠀⠀⠀⣠⠶⠃⠀⠀⠙⠓⠦⣄⣀⡀⠀⠀⢀⡇⠀⡀⠠⢏⣀⠀⠀⣠⠞
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣀⣴⠾⠋⠁⠀⠀⡀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⣸⠟⠳⣏⢰⠆⠀⡸⠖⠋⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠟⠋⢹⠀⠀⠀⠀⠲⣼⣶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠋⠁⠀⢀⣿⠁⢀⡴⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⡀⠀⢀⣀⣤⣶⢮⣭⣤⣤⣤⣤⡤⢶⣿⠈⠉⠉
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠒⠼⡟⠋⠀⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣸⡆⠀⠀⠀⠀⠀⠸⡄⠀⣿⡇⠀⠀⠀⠀⠀⣿⠃⠀⣿⡇
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢀⡏⠹⡄⠀⠀⠀⠀⠀⢳⠀⣿⡇⠀⠀⠀⠀⠀⣿⡄⠀⢸⡇
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠈⣧⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢀⣸⡇⠀⠀⠹⡄⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⢀⣿⡀⠀⠘⣿
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⣿⣀⣤⡴⠿⠛⠉⠉⢻⡏⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣼⠇⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⣿⠟⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⣀⣀⣤⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⢀⣠⣤⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⢀⡼⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⡠⠒⠀⠁⠀⠀⠀⠈⠙⠳⣦⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠄⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠀⡐⠀⠀⠀⠂⠐⠂⠀⠀⠀⢄⣀⣀⠀⠀⣻⠆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⢀⠤⠆⠶⠤⣀
⠀⠀⠀⠀⠀⢀⠀⠀⠘⡇⠀⠂⠀⠀⠀⠈⠂⠁⠀⠀⠀⡀⠀⠀⣎⠇⠀⠀⠐⠁⠐⠄⠂⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠘⢄⡀⠀⠀⠀⠀⠀⢀⠐⠉⠀⠼⠟⠀⠐⠀⠂⠀⠀⢀⠠⠈⡀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠉⠒⠂⠀⠐⠂⠁⠀⢀⣼⠝⠁⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣤⣀⣀⣀⣀⣤⡖⠟⠁⠀⡀⠀⠀⠀⠀⠀⠂⠄⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠁⠈⠀⠀⠢⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠡⠀⠄⠂⠀⡀⡔⠔⡨⡤⢄⠀⢉⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤⣫⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠔⠈⠠⠄⢁⠀⢠⠆⠪⢔⠊⡢⠈⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡅⢎⣁⡀⠈⠢⣄⠀⠀⠀⢀⠆⣀⠐⠈⠊⠐⠀⡊⠔⠪⠒⠡⢺⠀⠀⠀⠉⠢⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⡜⠂⡙⢤⡈⠣⡀⢀⠎⢈⠡⣰⠀⡀⠀⠠⢀⠅⡢⠐⣀⡄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠇⠀⠘⡄⠦⡢⠑⠶⠀⠐⠀⢸⠀⢒⠁⡠⢀⠡⠀⠈⠂⠅⠀⠀⢀⠊⠈⠔⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠂⠕⢀⠌⠀⢸⠀⠁⠁⡊⠐⡈⠐⠈⡠⠙⣄⡴⠕⠊⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⢀⠀⢁⠀⠀⠀⠈⡀⠅⠠⢀⢀⡀⢈⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢀⠄⡂⢐⠠⢂⠡⠂⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⠀⠀⢀⠦⠈⠁⡀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠅⠈⠘⠀⡄⠠⠆⡈⠐⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡂⠀⠀⠂⡇⠀⠁⠀⠠⢀⠀⠂⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠂⠀⠀⠠⠀⢊⠄⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⢸⠀⠀⠈⠀⠀⠀⠊⠄⠄⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⠄⠄⠸⠀⠀⠀⠇⠀⠊⢀⠄⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⣔⠁⠀⠀⠀⠐⠃⡖⢀⠄⠠⢠⠀⠐⠁⠊⠂⠂⠈⣓⠀⡀⢄⠠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠃⠒⠐⠂⠊⠁⠈⠐⠈⠘⠁⠚⠛⠋⠘⠁⠊⠀⠈⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣾⣿⣿⠿⢿⣿⣿⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡀
⠀⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀
⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄
⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⢠⣶⣿⠿⠿⠿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠿⠿⠿⣿⣶⣤⠀⠀⠀⠀⠀⠹⣿⣆
⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄
⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣷⣶⣤⣤⣤⣤⣤⣴⣶⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣦⣤⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣶⣿⠿⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣙⣛⣛⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡾⠋⠉⠉⠉⠉⠻⣿⣿⣿⣇⠀⠀⠀⠀⣾⡟⠋⠉⠉⠉⠙⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠻⢿⡿⠛⢿⣿⠟⠛⠻⠿⣿⠀⠀⠀⠀⠀⠠⠴⠄⠠⠦⠤⠀⠸⣿⣿⣿⣧⠀⠀⠀⠻⣤⣤⠀⣤⣤⠀⠀⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠃⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠙⠀⠀⠙⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠘⠃⠀⠘⠃⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣿⣷⣶⣶⡄
⠀⠀⠀⠀⠀⢠⣿⣿⡿⠿⠿⠿⠿⢿⣿⣿⣆⠀⠀⠀⠀⢸⣿⠿⠛⠛⠛⠛⠛⣿⣿⣷⠀⠀⠀⠀⢈⣿⡿⠟⠛⠛⠛⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣯⣤⡄⠀⣤⣄⠀⢻⣿⣿⠀⠀⠀⠀⠈⢃⣀⣀⠀⣀⣀⠀⢻⣿⣿⠀⠀⠀⠀⠸⣿⠤⠴⠄⠠⠤⡤⠀⠿⠿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⠀⣀⣀⡴⠶⢶⣄⣀⣀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⢛⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠲⣾⣿⡶⢂⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠀⠟⠛⠃⠨⠿⠏
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠰⣷⣄⣸⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠻⢿⢿⡿⣿⢿⣿⠟⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣧⣤⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⠶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣤⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣿⣶⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡁⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⣴⣯⣿⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣽⡿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣿⣧⣤⣤⣾⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠀⠈



⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢿⣿⣿⢿⣿⣆
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⢿⡏⠏⡿⠉⣿⠹⢻⡟⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣟⡻⡝⢎⣣⣶⡵⠷⠶⠿⠶⠷⣾⣵⡏⣴⡿⣻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⣿⣮⣣⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣾⣿⣿⡿⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⠀⠀⣰⣾⠿⣷⣄⠀⣰⣾⢿⣷⡄⠀⣴⡿⢿⣆⠀⢠⣾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠀⠀⣿⡏⠀⢻⣿⠀⣿⡏⠀⣿⣿⠀⣿⡅⢸⣿⢀⣾⠃⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⡇⠀⣸⣿⠀⣿⡇⠀⣿⣿⠀⠙⠿⠿⠃⣾⠇⣼⡟⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣿⣿⣤⡄⢿⣷⣤⣿⡟⠀⢿⣷⣤⣿⡟⠀⠀⠀⠀⣼⠏⠀⢻⣧⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⡃⢀⢷⡈⣻⠃⡖⢾⣓⢰⣟⡃⣸⣆⢰⠛⠒⠘⡟⢰⣦⡞⢳⡔⣧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢮⠟⠼⠛⠧⠸⠀⠇⠢⡽⠳⠏⠠⠟⠻⠜⠦⠖⠀⠇⠸⠟⠧⠼⠹⠏⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣤⣿⣿⣿⢳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡟⣭⢻⣧⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣥⣿⢃⣿⣻⡟⡷⠶⡶⠶⡿⣛⠛⣥⢷⣹⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣿⣴⣼⣷⣰⣯⣤⣽⣷⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣰⣦⡶⣠⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣬⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⢀⣠⡴⢾⣿⣿⣿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠈⠉⠻⠖⢳⡄
⠀⠀⠀⠀⠀⠀⠈⣥⠞⣿⡿⠋⠀⢀⡀⠀⠀⣀⠀⠚⡉⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣠⣸⠏⠀⠀⠀⠋⡉⣠⡘⠉⠀⣠⠏⠁⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠸⣧⡈⠀⠀⠀⠀⠘⠉⠛⠓⠒⠋⠁⠀⢀⣠⠾⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣅⠀⠀⠀⢀⡴⠋⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⣷⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣌⠀⢔⣫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⡄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠹⣷⡈⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠳⠶⠞⢩⡟⠀⠀⠀⠀⠀⠀⠀⠸⣷⠘⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⢀⠢⣝⣲⠄⠀⠀⠀⠀⠀⠀⣿⠀⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣶⠖⠶⢽⠿⠃⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠋⠐⢮⣓⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣆⠀⠀⣀⡀⢠⡾⠋⠁⣀⣀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠀⢠⣿⠇⠈⠉⠉⠉⠉⣹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠿⠿⠶⠶⠶⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠆⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⠀⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠆⠀⠀⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠉⠒⠢⣄⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣀⠖⠒⠉⠉⠉⠁⠒⠢⢌⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠔⠁⠈⢀⡀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣽⠢⠀⠑⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡇⠀⠀⡤⠀⠀⠤⠄⢀⣠⣾⡇⠀⠀⠀⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠻⣿⣶⡶⠿⠛⢻⡟⠀⠀⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠉⠻⠄⠀⠔⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⠤⠤⠤⡤⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⢠⠃
⡄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⡎
⠱⡀⠀⠀⠢⡀⠀⠀⠀⠀⠑⠀⠀⢄⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠘⢎⠀⠀⠈⠂⡤⢆⠀⠀⣀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠑⠦⡀⠄⠀⠀⡀⠀⠀⠁⠂⠤⣀⠈⠢⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠦⠤⡄⠀⠀⠀⠀⠀⠀⠈⠶⠥⣄⠀⠀⠀⠢⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠀⠀⠈⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡸⠁⣀⠀⠀⣀⠜⠀⠀⠈⠢⣀⠌⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⢠⠤⡀⢠⠙⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣬⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢩⠀⣠⠚⣣
⡇⠘⡁⠀⢷⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⠷⠁⡰⠉
⠃⡀⠀⠀⠠⠠⣤⣄⣡⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣴⡇⠀⠛⠀⠼⠁⠀⢀
⠀⢡⠀⠀⠀⠀⠀⠈⠁⢠⠄⠼⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠈⠡⠀⢀⣀⣠⡀⣠⡀⠀⠀⠀⠀⠀⠙⠶⣀⠈⠃⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠳⣀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠛⠀⠀⠋⠀⠀⠠⠟⠀⢹⠀⠀⠀⠀⠀⠀⠀⣹⠀⠘⠳⠄⠀⢀⡔⠦⢀
⠀⠀⠀⠈⠛⡦⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⣄⡀⠀⠀⠘⢦⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢰⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⢴⣆⢀⣀⠠⡤⣤⢤⡄⣀⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⢀⠌⠁⠀⠖⠋
⠀⠀⠀⠀⠀⠘⡇⠀⢹⡆⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠻⣄⠀⠀⠀⣰⠋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⢀⡤⠊⠁⠀⢀⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⠀⠀⣴⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠈⣦⠀⣀⣧⣠⠛⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⡆⠀⠀⣄⢠⡜⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⢨⡏⠀⠀⠀⠀⠀⠀⠐⠀⠈⠃⠀⠀⣀⡤⠞⣋⠀⠀⠀⢰⠀⠀⣿⡏⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡘⠁⠀⠀⠀⣤⣄⠤⡤⢤⣤⡤⠚⠛⠉⢀⣴⡿⠀⠀⠀⠘⠓⣤⣿⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣀⣤⡴⠾⢿⣿⠃⠀⠀⢠⣄⢀⡼⠋⠙⢶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⢋⣀⣀⣀⣼⠃⠀⠀⠀⢠⡟⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢣⠀⠀⠀⠀⠙⢿⣟⠉⠀⠀⣀⡤⠏⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠷⣀⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⠳⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡶⠛⠉⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⡏⠉⠉⠉⠉⠉⠀⠲⠶⠒⠦⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠶⡶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠉⠠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉⠉⠉⢛⠉⠉⠀⠘⠒⠒⣶⠶⠤⣤⣤⣤⡤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠘⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡆⠀⠀⠀⠀⠀⠀⢀⡦⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⡷⠀⠀⠀⠀⣀⡤⠋⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⠀⠀⢿⡀⢠⡄⠒⠉⠀⠀⠀⢀⢼⡇
⠀⠐⠋⣀⠀⠁⠙⣀⠀⠀⠀⠀⢀⡼⠶⠊⠙⠂⡀⠀⠀⠀⠀⡰⠃⢠⠃
⠀⠈⠲⠄⣀⣉⣉⣉⣉⣉⠉⣰⠛⠤⢀⡀⠀⠀⢡⡄⢀⣀⠐⠷⢀⡙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣠⣤⣤⣤⣬⣥⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠻⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡟⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⣀⣀⡾⠀⠟⣴⣿⣯⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⢻⣄⠀⢉⣡⠀⢀⡶⢸⠏⢀⡶⢀⡟⠻⣦
⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣶⣿⠇⣼⠟⠛⠛⡇⠀⠚⠁⡄⠀⢨⣴⠿⢃⣴⣿⠁
⠀⠀⠀⠀⠀⢸⣇⢠⣾⣟⣡⡏⣸⠏⢀⣴⣿⣷⣤⣴⣾⣿⣤⣤⣴⣾⣿⣟⣡⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣻⠟⠋⠉⠉⣀⠉⣀⠛⢁⣴⠀⣸⠃⢈⡤⢀⡉⠀⣡⠄⢈⠁⢀⡟⠀⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣤⣴⠞⠀⣿⠀⡟⠀⠰⠃⢰⡏⢠⡿⠃⢨⠁⣼⠟⠀⡍⠀⠞⠀⠘⣉⠀⠉⣿⡄
⠀⠀⠀⠀⣿⠙⠶⠟⠋⠀⣸⡇⠀⢿⣤⣴⣦⡔⠁⣦⣤⣶⠇⢰⣤⣴⣾⣧⣴⠎⣰⣿⠟⠻⢷⣿⠇
⠀⠀⠀⠀⠹⣷⣄⣀⣠⣾⣿⣧⣀⣀⣿⡍⡿⠀⢸⣿⢉⡏⠀⣿⠃⠘⠷⠟⠃⣰⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠙⠛⠛⠋⠘⣷⣤⣿⠇⣿⣄⣰⣿⣆⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠋⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⡠⠖⢚⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⢠⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⣀⣤⡀⠀⠈⢦⠈⢢⡀
⠀⠁⠀⠀⠀⢿⡉⢢⡀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠄⠀⠠⡶⠿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢠⣇⡼⠋⣰⠃⠀⠀⠀⢧⠀⢳
⠀⠀⠰⣟⠓⢮⣳⣄⠱⣼⡇⣠⠏⠀⠀⠀⠀⠀⠀⢀⡾⣫⡴⠆⠀⠀⠀⠀⠀⠀⠀⣀⡉⠻⣆⠀⠀⠀⠀⠀⣞⢉⣉⡙⢿⣄⠀⠀⢾⡥⠶⣦⠀⠀⢸
⠀⠀⠀⠈⠓⢄⡙⠁⣠⠟⠁⢳⡀⠀⠀⠀⠀⠀⠀⣸⠇⢁⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡦⠹⣧⠀⠀⠀⠀⠉⠁⠀⡝⠀⠈⢷⠀⠀⡤⠖⠋⠀⠀⢸
⠀⠀⠰⣏⣉⣈⡁⠰⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠙⢦⣄⡀⠀⠀⢀⣠⡴⠛⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠁⣤⣉⣱⡄⠀⠀⠏
⠀⠀⠀⠀⢠⠞⣡⣄⣀⣀⣄⠙⢦⡀⠀⠀⠀⠀⣠⣿⡀⢸⣇⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⣿⠀⢸⡷⣤⡀⠀⠀⠀⣰⠋⢠⡷⠒⠋⠁⠈⠉
⠀⠀⠀⠀⠘⠋⠀⠈⠉⠉⠈⠳⣄⠙⢦⡀⠀⠀⣧⣂⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢀⣀⣸⠇⠀⢠⡞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢦⠀⠈⣿⠀⠀⢿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠀⠀⣿⠁⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⢄⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣠⠞⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠈⢧⡈⠉⠙⠛⠛⢛⡿⠃⠀⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⡀⠈⠻⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⣄⣀⠉⠛⠛⠛⠋⠁⠀⣀⣠⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣀⣼⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⠀⢀⣠⠶⠛⠙⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠙⢦⡀⠀⣠⠏⣹⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⢻⡙⢦⣀⡴⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠲⠋⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠠⣄⣠⣟⠀⠀⠀⠀⠀⣀⣀⣺⣦⠀⠀⠀⠀⢀⣼⣃⠀⠀⠀⠀⢀⣟⣀⠀⠀⠀⠀⠘⢶⣷⠞
⠀⠀⠈⠉⡿⣍⠁⠀⠀⠈⢿⣿⣟⠁⠀⠀⠀⠠⣤⡿⠷⢤⡄⠀⠀⠀⢸⡯⡅⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠼⠑⠃⠀⠀⠀⠀⠚⠑⠋⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠏⠁⠀⠀⠀⠀⠀⠞⠘⠇
⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠻⣮⡉⣫⡾⠃⠀⠀⣠⣶⢿⣦⡄⠀⠀⢠⢴⣿⣿⡦⡄⠀⠀⠀⣀⣴⣀⡀
⠀⠀⠀⠙⣿⡛⠁⠀⠀⠀⣿⣿⣈⣿⡿⠀⠀⠈⠲⢿⡿⠗⠁⠀⠀⠀⢹⠿⣯⠁⠀⠀⢠⣤⣿⣽⣇⢀
⠀⠀⠀⠈⠳⠃⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠚⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠚
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⣤
⠀⠀⢉⣶⡟⡛⢦⡀⠀⠀⢰⢿⡆⠀⣄⠀⠀⣤⠠⡆⢰⠄⣤⠀⠀⠀⣀⠚⣿⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠸⣇⠘⠃⣸⠃⠀⢠⡟⠛⣿⡈⠛⠁⠀⠈⢷⣿⣿⡾⠁⠀⠀⠘⠻⢀⣿⠀⠀⠀⠀⠈⢠⣿⡄
⠀⠀⠀⠈⠙⠋⠁⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁
⠀⠀⢀⣤⠶⣤⡀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⣠⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⢠⣄⣀⣀⡀
⠀⠀⢸⣿⣴⣿⣷⠀⠀⠀⣠⣾⣿⣦⡦⠀⠀⠀⣿⡀⣠⣿⡀⠀⠀⠰⠐⠖⡶⠟⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⠀⢹⣷⣿⣿⠁⠀⠀⢸⣏⣛⣹⡇⠀⠀⠀⢈⡉⠉⠉⠉
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣠⡀⠀⠀⢀⡀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⢀⡶⠆⠀⠀⢠⡀⠠⣶⠄
⠀⠀⠀⠀⠘⣿⡿⠂⠀⢸⡇⠀⢀⡼⠻⠀⠀⢀⣠⡍⣤⡉⣷⠀⠀⠀⠐⣡⣰⣶⠀⠀⠀⠓⠦⠥⠤⡶
⠀⠀⠀⣀⣴⠟⠁⠀⠀⢸⣇⣞⣉⣀⣀⠀⠀⠋⠈⢦⣍⣤⠟⠀⠀⣺⣇⣿⣟⣿⠀⠀⠀⠀⠰⣷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠂⢊⠀⢫⣟⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢻⣿⣷⡔⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⢈⣶⣿⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣀⣲⡶⠒⠒⠒⠒⠀⠙⢻⠇⠀⠀⠀⠀⠀⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠀⠀⠹⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣼⠀⡸⢠⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠨⠛⣹⣿⡆⡄⠀⠁⠇⠀⣰⣿⠃⣀⣀⡀⠀⠇⢸⠇⠀⢰⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠘⣿⣆⠐⣯⡃⠀⢠⣤⢾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⣾⡅⠤⠈⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠛⣷⣬⠀⢀⠘⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠬⠄⢀⡀⠀⠀⠀⠔⠀⢀⣿⣿⢰⠃⠏⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⡀⠠⢤⠀⠀⠀⠀⠀⠀⠀⣠⣿⡷⠃⡸⠀⠀⡀⠈⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢋⠿⠇⠀⠀⠀⠀⠀⢀⣠⣴⠾⠿⠋⠠⠞⣁⠀⣰⠁⢀⠄⠁⠂⠀⠸⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⡟⢻⡟⠫⠆⠀⢀⠉⠀⠀⠊⠁⠊⠑⠀⠈⠜⡸⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠁⣰⣶⠖⠂⠉⢀⡀⠀⠀⠀⢅⠋⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠹⠋⠀⠀⠀⠀⢀⡠⠋⠁⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠔⠋⠀⠀⠀⡀⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠉⠁⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠀⠀⠀⢱⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⠙⡄⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣄⠐⢀⡜⠁⠄⠀⢀⠀⠀⡇⠀⠀⠀⠘⢆⠀⠀⠀⢸⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⣻⠟⡝⡟⢷⣶⣤⣀⣀⠞⠁⠀⢀⠀⠀⠈⠧⠀⠀⢈⣀⡀⠈⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠋⢀⡬⠐⠋⠘⠁⠁⠀⠐⠦⠙⢷⣦⣀⠀⠈⠙⠂⠀⡠⠚⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣡⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⣯⡣⡀⠒⠀⡼⠁⣰⠃⠀⠁⠀⠀⠀⢳⠐⠚
⠀⠀⠀⠀⠀⠀⠐⠟⡴⢡⣇⠛⠀⡀⠀⠀⠀⠐⠂⠀⠈⠀⢀⠀⠛⢸⣻⡷⣵⡀⠀⠱⡀⠹⣄⣀⣀⡤⠀⢠⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠹⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⠏⠀⠇⠻⠁⠀⠀⣹⢶⠤⢍⡭⠤⠖⠁⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⠦⢄⣀⣀⡠⠴⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⣀⠀⢰⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⠤⠤⠤⠤⠔⠚⢿⢄⡀⠀⠀⣀⢀⡠⠊⠁⠀⠀⠀⡇⠀⠀⢇
⠀⢠⠊⢉⣚⠉⢹⠄⠀⢀⣀⣠⠔⠋⡽⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠑⣾⡷⠏⠀⠀⠀⠀⠀⠀⣃⣀⠀⠈
⠀⢿⢠⠋⢀⢱⠈⠓⡄⠀⠞⠿⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠞⠁⠀⠀⠀⠀⠀⣀⣴⡉⠵⠮⣗⠒⣄
⠀⣰⠛⢻⠾⠊⡤⠞⠁⠀⠀⠀⠀⠘⠓⠋⢹⡏⠉⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣅⢎⣥⡼⢴⠃
⠀⠈⠋⢉⣓⣚⣆⡴⣦⣀⡠⠤⠤⠒⠒⢃⣸⠇⠉⠉⠁⣇⡀⠉⠙⠓⠒⠒⠦⠤⢄⣀⠉⢱⣌⡹⠯⣀⡼
⠀⠀⠀⠈⠓⣮⠿⠛⠉⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⡀
⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢘⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠋⠔⠋⠄⡰⠀⠀⠀⠂⠠⠀⠀⠂⠱⠘⠀⠶⠂⠋⠇⠃⡇⠱⠀⠀⠰⠂⠁⠀⠀⠴⠀⠆⠀⠸⠉⠱⠋⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣖⠛⠭⠉⠒⠀⠀⠀⠀⠚⠉⠍⠛⣲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠯⠂⠉⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠀⠈⠐⠙⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠊⠁⣀⣀⣾⣦⣀⡀⠀⠈⢿⣿⣿⡿⠃⠀⢀⣀⣼⣧⣀⡀⠀⠐⢝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠫⠊⠀⠀⠀⠈⢻⣿⣿⡏⠀⠀⠀⠼⠛⠛⠇⠀⠀⠀⢹⣿⣿⡟⠁⠀⠀⠀⠀⠝⣆
⠀⠀⠀⠀⠀⢀⡜⡑⠀⠀⣀⣶⡀⠀⠋⠁⠉⠃⢀⡀⠄⠀⠀⠀⠀⠠⠀⠀⠘⠉⠈⠙⠀⣀⣶⡀⠀⠈⢈⢧⡀
⠀⠀⠀⠀⠀⡞⡌⠀⠀⠙⣿⣿⣿⠋⠀⠀⡠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠙⣿⣿⣿⠋⠀⠀⠡⢳
⠀⠀⠀⠀⣼⠑⠀⠀⠀⠐⠛⠉⠛⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠋⠉⠛⠀⠀⠀⠀⠋⣧
⠀⠀⠀⢀⡇⠂⠀⠀⣠⠤⣄⢠⡄⠀⣤⢀⡤⢤⡀⠤⣤⠤⠀⣠⢤⡄⢠⣄⠀⣠⡄⣤⠤⠤⠀⣤⠤⣄⠀⠀⠐⢸⡄
⠀⠀⠀⢸⠹⠀⠀⠘⡇⠀⠀⢸⡇⠀⣿⠈⠓⠲⡄⠀⣿⠀⢸⡇⠀⣿⢸⡟⣦⢿⡇⢿⠒⠀⠀⣿⠶⣞⠀⠀⠀⠆⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠙⠛⠉⠀⠙⠛⠁⠈⠛⠛⠁⠀⠙⠀⠀⠙⠛⠁⠈⠁⠉⠈⠃⠙⠛⠛⠀⠋⠀⠙⠀⠀⠀⠀⡇
⠀⠀⠀⢸⣰⠀⣞⡓⠀⢠⢶⡀⠐⢻⠃⢰⠀⣞⡒⠀⣞⠛⠂⢀⣶⡀⢠⡞⠓⠀⢻⡟⠀⡆⣴⠛⣦⢰⣆⢰⠀⣄⡇
⠀⠀⠀⠈⡇⠀⠤⠼⠂⠟⠛⠷⠀⠸⠀⠸⠀⠤⠽⠂⠏⠀⠀⠞⠛⠳⠘⠦⠤⠀⠸⠇⠀⠇⠻⠤⠟⠸⠈⠻⠠⢸⠃
⠀⠀⠀⠀⢹⡠⠀⠀⠀⠀⣤⣀⣤⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⣀⣤⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⢧⢂⠀⠀⣠⣿⣿⣿⣄⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣠⣿⣿⣿⣄⠀⠀⡐⡼
⠀⠀⠀⠀⠀⠈⢳⡡⡀⠀⠉⠿⠉⠀⣀⡀⣀⡄⠈⠁⠀⠀⠀⠀⠀⠒⠈⠀⢀⣀⢀⣠⠀⠉⠿⠋⠀⢀⢜⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣔⢄⠀⠀⠀⢀⣼⣿⣿⣇⠀⠀⠀⢰⣤⣤⡆⠀⠀⠀⣸⣿⣿⣧⡀⠀⠀⠀⡠⣢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣕⠄⠀⠉⠉⢿⠟⠉⠁⠀⢠⣾⣿⣿⣷⡄⠀⠈⠉⢻⡿⠉⠉⢀⠠⣪⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠄⡀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⣀⠠⣢⠵⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠮⢤⣐⣀⣠⠤⠀⠀⠠⢄⣀⣀⡠⠵⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⠛⠻⠟⠛⠿⠿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⡥⠄⠀⠀⠀⠀⠀⢤⡀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠘⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⣾⣷⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡐⠀⠀⢀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⡤⢴⠆⠀⠀⠠⠂⠀⠀⡠⠀⠁⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢌⣀⣠⡔⢈⡀⠠⣤⠐⠈⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⢹⣧⣼⡷⠀⠀⠀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠠⠐⢻⡀⠀⠀⠀⠙⠈⠁⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⠁⠀⢀⠠⠀⠐⠛⠊⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠑⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⢉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠰⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣽⣷⠇⠀⠀⠀⣷⣿⣫⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⡻⠃⠀⠀⠀⠀⠘⠿⠿⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣶⠛⠛⠛⠛⠓⠒⠶⠶⠶⠶⠴⠿⠭⣿⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⢲⡄
⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢛⣉⣩⣭⣭⣭⣍⣉⠉⠓⠲⣄
⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⣖⠆⢠⣞⣁⣠⣤⠶⠛⢉⣠⣤⣤⡹⠋⢿⠟⣿⡿⡿⠆⢀⡼⠃
⠀⠀⠀⠀⠀⠘⣿⡤⠴⠒⢚⣉⣩⡏⠁⠀⠀⠀⣴⣿⠿⢛⡉⠀⠀⠀⢠⣶⣤⣼⡖⠋
⠀⠀⠀⠀⠀⢀⣿⣷⣿⣟⣛⣻⣿⣷⡆⠀⠀⠀⠋⠡⠞⣩⠭⣧⠀⠀⢸⣟⢝⢿⠁
⠀⠀⠀⠀⣼⠋⢉⣍⣉⠛⠻⣷⣾⡿⠀⠀⠀⠀⠀⠀⠚⠛⣀⠀⠀⠀⡰⠿⢦⠈⡆⠀⠀⢀⣤⠶⢓⣛⣶⣄
⠀⠀⠀⠀⠙⢦⣌⠁⠉⢷⡄⠨⣿⣯⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⡖⢊⡀⠀⣸⠇⢸⣤⠞⢉⡤⠞⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠙⡗⠠⣿⣇⠀⠉⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠁⠉⠀⢀⣁⣤⠴⠟⠚⠫⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣦⣉⣉⣀⡧⠀⠀⠀⠀⠀⠀⡞⠓⠒⠒⠒⠒⠛⠉⠉⠀⠀⣀⣀⣀⡐⠂⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠉⢻⠀⠀⠐⠀⠀⠀⠀⢧⠀⠀⠀⢀⠀⠀⠀⠦⠀⠈⠁⣤⠀⠙⢶⡀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⡇⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⠀⠀⠤⠀⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⡄⠀⠀⠀⠀⠐⠂⠀⠄⠀⠐⠀⢀⡀⠀⠀⠀⠀⡀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⢻⣶⠛⢻⣦⡀⠐⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠈⠀⣀⣡⣴⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣹⣿⠉⣿⣿⠛⢷⣶⠦⠤⣤⣤⣤⣤⣤⡤⠴⢶⣾⣿⠋⢁⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣏⠙⣿⣷⠾⢿⣦⣼⢿⣷⡞⢻⣿⠀⠘⣿⣷⠀⣿⣿⣇⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣽⣿⣦⡬⠿⠿⠼⠿⠷⠼⠿⢧⣀⣿⣟⣀⣿⣿⣿⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣤⣿⣷⣾⣷⣶⣶⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣼⣿⡏⠀⠀⠀⠉⠉⠈⠙⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⠟⠋⠛⠛⠿⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠏⠋⠉⠈⠁⠀⠈⢻⣿⣿
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠘⣿⡇
⠀⠀⠀⠈⣜⠀⠀⠀⠁⡀⢀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠗⠎⠀⠄⠀⠁⠄⠠⠀⠀⠀⠀⠐
⠀⠀⠀⢠⣿⣦⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠤⠠⡉⠈⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀
⠀⠀⠀⣸⣿⣿⣿⣿⠗⠐⢿⣿⣷⠀⠀⠀⠂⢀⠄⠹⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡛⠃⠓
⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠈⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎
⠀⠀⠀⠈⠉⠉⢱⠀⠀⠀⠀⢈⠁⠀⢠⠀⠀⠀⠀⠀⠈⠩⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠦⠲⠀⠀⠀⠀⠸⠲⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢒⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠥⡄⠈⢴⠀⠀⠉⡙⠀⠀⠀⠀⠈⡤⠏⠘⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠠⢺⠇⠀⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⠀⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡟⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆⠀⠀⢻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⣿⣿⣿⣶⡄⠀⣀⣉⠛⠛⣉⣀⠀⢠⣴⣿⣿⣿⣶⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⣶⣿⣷⣦⡶⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⢶⣶⣾⣿⣶⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣇⠰⣶⣶⣿⣿⣶⡶⠆⠀⢰⡆⠀⠰⢶⣶⣿⣿⣶⣶⠆⣹⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠙⠀⠀⣽⣿⣿⣍⠰⣶⣶⣿⣿⣶⣶⠆⣩⣿⣿⣯⠀⠀⠋⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠈⠙⠂⠈⣽⣿⣿⣿⠁⠐⠋⠁⠈⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠛⠁⠈⠛⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠠⡄⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠺⠳⢾⡆⠰⠛⡠
⠀⠀⠀⠀⠲⡀⠘⡀⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⣴⣀⢀⡀⠀⠰⠈
⠀⢰⠀⠀⡀⠑⠄⠀⠄⣀⠀⠀⠀⢀⠄⢀⠔⠀⠀⠈⠑⠒⠀⡀⠉⠉⠉⠆⠀⡀
⠀⠀⠑⢄⠈⠢⣈⡄⠘⠋⠀⢰⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠙⠶⠈⠀⢠⠄⠉⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⢦⡀⠂⡀
⠀⠀⠘⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⢀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢙⡷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠘⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠉⠀⠀⠊⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢂⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⠶⠀⠉⠉⠉⠛⠻⠶⣿⡄⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢠⠀⠀⢀⡠⠆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⢦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⡄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠋⠀⣼⣷⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢰⠀⠘⢷⠒⠂⠐⠒⠒⠊⠉⠁⠀⢀⣤⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢻⠀⠀⠀⢧⣤⣄⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⡷⠀⣀⡀⠀⠀⣀⡀⠐⠈⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⡀⠠⠿⢻⡟⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠓⠴⠂⣀⣀⣀⠴⠋⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠐⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠷⣤⣤⣀⣀⣠⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣤⣤⣤⣠⣤⣤⠴⡾⠋⣀⠀⠷⠈⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⡅⠀⠀⡇⠀⠈⠁⠶⠂⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⢄⣀⣀⠀⠀⠀⠀⠀⢀⣀⠴⠛⠁⠀⠘⠀⢆⣤⠁⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠈⠁⠀⠀⠀⠀⠀⣦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠈⢛⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⠃⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀⠀⠀⠀⠀⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠛⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠸⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣄⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⣤⡞⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠳⠶⠞⠁⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠴⠞⠉⠀⠻⣆⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⡴⢾⠁⢠⣼⣁⣀⣀⣀⣀⠀⡴⢶⡞⠉⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢇⠀⠁⠀⠀⠀⠀⠀⠀⠄⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠡⣀⠀⠀⠀⠀⠀⢠⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠓⢦⣤⣤⡴⠋⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠋⠁⣰

⣠⠤⣴⣶⣶⣶⣶⣶⢒⡤⢤⡀⠀⠀⠀⠀⢀⣠⣠⣤⣤⣤⣤⡤⠤⣄⣠⣀⠀⠀⠀⡀⣠⠴⣄⣀⣀⣀⣀⡀⠀⢀⣀⠦⢄
⢧⠀⠘⣿⣿⣿⣿⣿⠼⠂⢀⡇⠀⠀⠀⠀⢎⠀⠉⣿⣿⣿⣿⣿⡇⠍⠁⢨⠀⠀⠀⠉⠣⡀⢹⣿⣿⣿⣿⣿⣿⣷⠹⢀⠜
⠀⠑⠠⣽⣿⣿⣿⣿⡤⠖⠋⠀⠀⠀⠀⠀⠈⠢⣀⠘⣿⣿⣿⣿⡝⢀⠤⠃⠀⠀⠀⠀⠀⠠⡈⠛⠿⣿⣿⣿⠿⠗⢏⠅
⠀⠀⠓⠟⢿⣿⣿⡟⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠃
⠀⠀⠀⠀⠈⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣇⡀
⠀⠀⠀⠀⠰⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣾⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡴⢀
⠀⠐⢶⣬⣉⡋⠠⠓⢒⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢊⠨⣭⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡄⣤⣌
⠀⠀⠀⠹⢉⣾⠘⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢻⡗⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠀⠴⠛⢻⡓⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⣠⣶⣾⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⣴⠿⠿⠿⠿⠿⠷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⣠⠆
⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠛⠓⣦
⠀⠀⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠒⠒⠛⠃
⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⠀⠀⢀⣀⣀⡀
⠀⠀⢠⣾⣽⡿⠿⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⢀⣪⣿⣿⡿⠟
⠀⠀⣽⣿⣿⠆⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣾⣿⣶⡶⢶⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣲⣿⣿⣿⠏
⠀⠀⠈⠿⣿⣿⣿⣯⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣏⠉⣯⣿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠻⣵⣿⣿⡿⠏
⠀⠀⠀⠀⣿⣭⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⣀⣿⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢛⠷⢁
⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⣷⣔
⠀⠀⠀⣾⣿⡟⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⣿⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣿⡏
⣤⣤⣤⣯⣥⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡙⠋⠀⢛⣻⣿⣿⣯⡛⠁⠈⢋⣹⣿⣿⣏⡛⠁⠉⣋⣿⣿⣿⣍⠋⠀⠙⣻⣿⣿⣿⡛⠁⠀⢙⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣠⣤⡸⣿⣿⣿⣿⣏⣠⣄⣹⣿⣿⣿⣿⢃⣤⣄⣿⣿⣿⣿⣿⢁⣤⡘⣿⣿⣿⣿⡟⣠⣤⡸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⣉⣉⣉⣀⣀⣀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⢀⣀⣀⣈⣉⣉⣉⣉⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠈⠛⠿⠿⢿⠿⠿⠟⠋⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⣫⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢀⣀⣀⣴⣤⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠟⠛⠛⠻⢿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⢸⣿⠏⠀⠀⠀⠀⣠⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣿⣿⠀⠲⣦⣠⡾⠁⠀⢨⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀⠈⠋⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡍⢿⣿⣶⣶⣶⣾⣿⠏⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢡⣦⣤⡌⠙⠋⣤⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢿⡟⠀⠀⠀⠈⢿⡿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠶⠦⠉⠉⠟⠛⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠳⠶⠆⠀⠀⠀⠀⠀⠀⢀⣖⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠓⠲⠶⠄⠀⢂⣬⢆⣀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⠰⣝⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⣤⣶⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⠛⠛⠷⠶⠤⣤⣤⣀⠀⠀⠀⣀⣤⡶⣉⡙⠒⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⡀⠀⠀⠀⠂⠄⢀⠈⠉⠙⠓⠶⠻⠭⣬⣉⣛⢳⣦⣄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⣶⣦⣤⣝⣒⠐⠢⠄⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠉⡉⠙⠛⠻⢷⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠉⠛⠻⠿⣿⣿⣷⣶⣦⣤⣂⣄⠀⠐⠂⠤⣄⣀⣀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⣹⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣖⣟⢲⡀⢦⡍⣉⠋⠉⠛⠛⠻⢿⡷⣶⣶⣦⣤⣅⣈⠁⠐⠂⠀⠀⠀⠀⢀⡾⠃⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣎⡲⠿⠌⣹⣞⢷⣿⡀⣞⠳⡄⢠⣄⡀⠙⠓⠿⢿⡿⢿⣶⣶⣤⢤⣴⣿⠃⠀⢤⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⢤⣤⣙⡃⠘⠧⢿⠎⢷⠻⣄⢟⢳⡄⣤⢤⡀⠉⠉⠃⠙⢻⣧⠐⠈⢆⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⢬⣅⣉⣀⠀⠹⠈⠉⢿⡌⣯⢳⡀⠈⢻⣆⠐⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠤⣅⣘⡋⠳⠀⠀⢻⣆⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠲⠿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠄⠀⣀⡄⢀⡉⠉⣉⣉⡁⠉⢐⣒⡒⠒⠒⠒⠒⠂⠤⠤⠄⠠⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢀⣤⠶⢛⣿⡟⢁⣤⠾⠧⠖⠋⣠⡾⠯⠴⠛⣰⣿⣍⣹⠟⢀⣴⡟⠙⣦⠀⢰⣦⠀⣴⣤⢀⣤⡤⠄⠉⢩⣭⣭⣍⡈
⠀⠀⣀⣀⣉⣀⣀⣈⣉⣀⣉⣁⣀⡀⠀⠚⠁⠀⠀⠀⠰⠟⠀⠠⡿⠀⢻⣿⣀⣴⡟⠀⠘⣿⣶⣿⠃⢸⣿⡷⠒⠀⠘⣿⣆⠈⢿⣦⡀
⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠐⠒⠈⠩⠍⠉⠛⠛⠓⠛⠛⠿⠿⠷⠷⢶⣶⣶⣯⣥⣤⣤⣤⣉⣙⣁⣀⣈⣙⣛⣛⣒⡀⠛⠻⠶⠟⠛⠃
⢀⣀⣀⣀⢀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠉⢉⣉⡙⢛⠛⠛⠛⠻⠛⢛⠓⠒⠲⠶⠦⡤
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄⢀⣼⣿⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢺⣿⣿⣾⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠚⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣆⠀⠀⠀⣼⣿⣆⠀⠀⠀⠀⠀⠀⣾⣧⠀⠀⠀⠀⠀⠀⢠⣿⣧⡀⠀⠀⣰⣿⣿⣄
⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⣛⣛⣉⣭⣼⣿⣿⣿⣦⣤⣀⡀⠀⣼⣿⣿⣧⠀⢀⣀⣠⣤⣿⣿⣿⣧⣭⣉⣛⣛⢻⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⢩⣭⣴⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⣭⠝⣿⣿⣿⣿⣿⡿⢋⣼⣿⣿⠁
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠟⠻⢶⠙⣿⣿⣿⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⣾⣿⣿⣿⣿⡋⡴⠟⠻⢿⣿
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣷⠙⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠀⠀⠈⠙⠻⢰⣿⣿⣿⣿⣿⣿⣿⣿⡆⠟⠋⠁⠀⠀⠉⠛⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠟⠋⠀⠈⠙⠻⢿⡇


⣤⣤⣤⣤⣤
⣿⣽⠭⣭⣿
⣿⣿⠤⣿⣿
⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⢞⣱⠆⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣤⡀⠀⠀⠀⠀⠀⠀⢀⡐⠠⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣦⡀
⠀⠀⠀⠀⠀⣀⣿⣿⣀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠉⣿⣿⡄⠀⠀⠀⠀⢀⡃⡐⠁⢠⡆⠀⢡⠐⡀⠀⠀⠀⠀⠀⣰⣿⣧⢸⣿⢹⡆
⠀⠀⠀⠀⠀⠙⠁⣿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⣿⡿⠁⠀⠀⠀⠀⠐⡎⢷⡀⠠⠇⠀⣸⢅⡖⠀⠀⠀⠀⠀⢿⣿⣿⡾⢋⡼⠃
⠀⠀⠀⠀⠀⢀⣰⣿⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠿⣿⣶⡀⠀⠀⠀⠀⠀⠹⣞⡅⠀⠀⠈⣃⠾⠁⠀⠀⠀⠀⠀⢀⣙⣲⣖⠋⡀
⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣿⠶⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⢤⣀⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡆
⠀⠀⠀⠀⠰⠻⢶⡾⠛⠁⠀⠀⠀⢠⣤⣄⣴⣿⡿⠙⢦⣠⣤⠀⠀⠀⢸⡁⢹⣿⣿⣿⣿⠸⠁⣹⠀⠀⠀⠀⠀⠀⠀⣷⣿⠃
⠀⠀⠀⠀⠀⣠⣬⣧⣤⡀⠀⠀⠀⢀⣿⣿⡞⢋⣰⣿⣯⣿⣿⡀⠀⠀⠀⠑⠶⣿⣿⣿⣿⡧⠖⠁⠀⠀⠀⠀⠀⠀⢀⣸⣟⣀
⠀⠀⠀⠀⢸⣿⠛⡿⢻⣿⠀⠀⠀⠀⠀⠀⠈⠚⠙⠓⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠻⢿⡇
⠀⠀⠀⠀⠸⠿⠾⠷⠾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠾⠇

⠀⠀⠀⠀⢀⢔⣮⣗⡢⡀⠀⠀⠀⠀⠀⠖⢸⣿⣿⣿⡆⠲⠀⠀⠀⠀⠀⢀⠠⠀⢀⡀⠀⠠⢀⠀⠀⠀⠀⠀⣄⣀⣀⣀⡀⣀⣀⣠⡀
⠀⠀⠀⠀⡦⣿⡛⢛⣿⢰⠀⠀⠀⠀⠀⠈⠊⢿⣿⡿⠔⠁⠀⠀⠀⠀⢐⠂⡆⠀⢸⡇⠀⠈⠄⠂⠀⠀⠀⠀⣿⣿⣿⢟⣵⣿⣿⣿⡇
⠀⠀⠀⠀⠰⣝⡿⢿⣫⠖⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠨⡄⢷⡀⠾⠷⠄⡼⠁⡍⠀⠀⠀⠀⣿⡿⣣⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢺⣿⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⣻⣷⡆⠀⠀⠀⠀⠀⠀⠛⣦⡅⢀⣀⠠⣤⠞⠀⠀⠀⠀⠀⠉⠀⠁⠉⠻⣄⣼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⡀
⠀⠀⠀⠀⠀⣿⠛⢛⣛⠇⠀⠀⠀⠀⠀⠀⣔⣉⣯⣉⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⠁⠀⠀⠀⠀⠀⠰⠒⠄⣶⣶⣶⣶⠠⠒⡆
⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⠀⠀⣆⢹⠿⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠿⢧⠀⠀⠀⠀⠀⠀⠀⢂⠀⣿⣿⣿⣿⠀⡰
⠀⠀⠀⠀⢀⣀⣼⣁⣀⡄⠀⠀⠀⠀⠀⠀⠈⢑⣶⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡏⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠏⠔
⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⣀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⣛⣛⣛⡛⢵⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣾⣿⣿⣿⣿⡎⠇⠀⠀⠀⢰⣶⣶⡶⡲⣶⣶⡶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣶⠀⠀⠙⢿⡟⣯⡵⡻⢎⣵⢏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣟⡿⢟⣿⡏⠁⠀⠀⠀⢸⣿⡎⣷⣝⣛⢱⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣛⣟⠋⠀⠀⠀⠀⠀⠸⠿⠧⠵⠶⠮⠾⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣜⠿⢟⣴⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣩⡆⣮⣣⣿⣿⣷⡄⠀⠀⢀⣴⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⡏⣿⡇⠀⢠⣿⡿⡿⣿⡿⡿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠿⣿⠿⠿⡿⠇⣿⣿⣿⡇⣿⡇⠀⢸⣏⢹⣾⣿⣿⣾⢩⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣴⣾⣿⣿⣿⣮⣭⣭⢃⣵⣾⡆⠈⢿⣮⣛⠿⣟⣫⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⠋⠀⠀⠀⠉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠟⠈⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠄
⠀⠀⣠⣾⡿⠋⢁⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠖
⠀⣼⣿⣏⣀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣾⣟⠀⠀⠘⠛⠋⠀⣀⣀⡀⠀⠀⢀⣴⡿⠃
⠀⢈⣛⠻⢿⣿⣷⠀⢀⣴⡾⠿⣿⣿⠏⠘⣿⣿⠟⠛⢃⣾⡿⢃⣴⡿⢛⣿⡿⢀⣠⣾⣟⣀⣤⣾⡟⠁⣰⣿⠏
⣴⣿⣏⣀⣴⣿⣿⢾⣿⣟⢁⣼⣿⣃⣠⣾⣿⣃⣀⣴⣿⡟⣀⣤⠟⢿⣿⣭⡴⣿⣿⣿⠛⢩⣿⣏⣠⣾⡿⠃
⠻⠿⠿⠿⠛⠉⠀⠈⠿⠿⠛⠻⠿⠟⠉⠻⠿⠟⠋⠹⠿⠟⠛⠿⠿⠿⠛⢉⣼⡿⣿⡟⠀⠻⠿⢛⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣰⡿⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣴⠟⠁⠀⠀⠀⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣍⠀⠈⣩⣿⣿⣿⣯⡉⠀⢉⣽⣿⣿⣿⣍⠀⠀⣩⣿⣿⣿⣯⡉⠀⢈⣽⣿⣿⣿⣍⠁⠀⣩⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣇⣴⣦⣹⣿⣿⣿⣿⣤⣶⣌⣿⣿⣿⣿⣏⣴⣦⣹⣿⣿⣿⣿⣡⣶⣤⣿⣿⣿⣿⣏⣴⣦⣸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣤⣀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠛⠉⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄⠀⠙⠿⣿⣿⣿⣿⣿⣿⠿⠋⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣓⣀⡀⢀⡉⣉⢉⡉⣀⣀⣀⣚⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣢⣤⣤⣤⣭⣭⣬⣶⣤⣄⣀⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⠏⣀⣤⣤⣀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣇⠞⠉⠀⠈⠉⠀⠀⠀⠀⢀⡛⠟⠙⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠀⠀⣠⠤⠦⣄⠀⠀⢀⣴⣾⡵⡶⢶⣤⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠃⠾⢷⡾⠀⠀⢸⣿⣿⣔⡚⣧⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡍⠈⠁⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢰⣿⢿⣿⠀⠀⠀⣺⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠀⠀⠀⠀⠈⢹⣿⣿⣧⠀⢤⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠴⠶⠖⢒⡚⠿⡿⠂⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⣿⣰⡆⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠃⣾⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠀⠀⠀⠈⠛⠛⠛⢹⡟⣸⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⡌⣰⣿⣿⣿⣿⡿⡙⣊⣿⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠚⠁⣿⣿⡟⠁⠀⠀⠀⠀⠈⠈⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⡀⠀⠀⠀⠀⢀⣀⡞⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣻⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠞⡿⣷⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠧
⠀⠘⡆
⠀⠀⢃
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠇⠀⠀⠀⠂⠤⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⣽⣿⣿⣷⣿⣶⣶⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉


⠀⢀⣀⠀⢀⡀⠀⢀⣀
⠀⠠⠁⠀⠲⡄⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠈⠉⠀⢫⡀⠀⢀⡈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫
⢀⠀⠀⠀⢀⠀⠉⠀⠂⢠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠙
⡎⡀⠀⢀⠂⡀⠀⠀⠀⡌⠀⢸⡀⢠⠤⡴⡂⠉⠀⠀⠂⠀⠒⠀⢒⣀⣾⣷⡶⢠⣐⡀⠀⠀⠀⠀⠀⠂⠐⢻⡂⠀⠀⠀⠀⢧⠙⡄
⢠⠁⠠⠃⢠⠁⠀⠀⢀⠆⠀⠈⠀⣼⢀⠀⢉⠄⣀⠀⠀⣀⡀⣺⡟⡉⢡⢔⠀⠀⠈⡕⠒⢤⡆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠐
⠇⡰⠃⢠⡇⠀⠊⡁⠀⠀⠀⠀⠀⡄⠘⢀⢸⠂⣿⠴⢋⢹⢭⠉⠉⣡⠱⠛⠓⢤⣤⣄⠠⣠⣶⠀⠃⠀⠀⣴⡈⡀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢡⢀
⠐⠋⣥⡄⠀⠰⠍⠠⢄⠠⠄⠠⢈⣧⠟⣬⡇⠨⠄⠠⢘⣼⠖⣏⠚⣧⡟⢿⡉⢸⡆⡿⠒⢘⣿⣄⠀⡴⠀⠄⡡⠶⠀⠔⠀⡐⠀⠀⠀⠀⠀⡀
⠢⢄⠈⠑⠢⣄⠀⠑⠠⡐⠀⠑⢼⠉⡖⢦⢄⡑⠦⣀⢸⣛⠲⠦⣌⣧⠥⡤⠦⢤⠭⡿⠆⠹⣿⣧⣾⡢⠄⢠⠀⠀⢂⢀⣌⠄⠊⡀⠄⠂⡀
⢸⠀⠹⢢⣄⡀⠙⠢⣀⠈⠐⠄⡆⠀⡇⠸⠨⠘⡑⠠⢙⣎⣂⣁⣚⣛⡛⣶⣯⡴⣊⢀⡤⠤⣭⠿⡏⠀⠰⠈⡄⡈⠌⢋⡠⠐⢀⠠⠂⠁⣀⠴⠞
⠇⢀⠇⡸⢘⡏⢰⠤⠀⠙⠢⢀⠃⢀⠇⡆⠈⠦⠥⠀⠀⠉⠁⠦⠄⣀⠀⠀⠀⠞⣁⣩⠜⠋⠀⠀⠀⠀⠀⠀⠉⢡⠈⢌⡄⠊⢀⣠⠔⠊⢅⠘⡄
⠀⡼⢠⠇⢸⠁⢸⡂⠀⠁⠂⣼⣄⠈⠠⠄⠀⠀⠔⠀⠀⢄⠀⠠⠀⠈⠙⠒⠶⣶⠿⠀⢀⣤⣄⣠⡄⢀⣠⡄⢂⡔⠀⣪⡼⡖⠉⠀⠀⠀⠈⢢⠐
⠀⣇⣞⠀⠛⠀⠈⠁⠐⠂⠈⠉⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢀⣴⡿⢶⠐⢂⡁⠲⠀⠀⠀⠀⠀⣠⡴⠚⠉⠒⠒⠶⠶⠶⠶⠶⠶⠷
⣸⣿⣟⣉⣙⣑⣀⡒⠒⠂⠒⠒⠖⠖⠀⠈⠳⢦⣄⠀⠀⠀⠀⠀⠠⢀⠑⠦⠖⣊⠴⠊⠁⠀⠀⠀⢀⣠⡖⠛⠁
⣍⣁⣀⣀⣀⣉⣉⣉⣉⣙⣛⣛⠛⠒⠒⠒⠒⠖⠚⠻⠦⣄⡀⠁⠂⢀⠉⠒⠋⣀⡤⠊⠁⣀⣴⠾⠉⢁⣀⣀⣀⣀⣀⣀⡀⠀⠈⠁
⣿⣽⣯⣭⣭⣭⣭⣭⣭⣭⣙⣛⣛⣛⣛⡛⠛⠛⠓⠶⠶⠾⠿⢶⣄⡀⠑⠲⠟⠁⢀⣴⠾⠯⠤⠤⠤⠤⠤⠤⠄⠀⠠⠤⠤⠤⠤⠤⠄⠤
⣿⣭⣽⣯⣭⣯⣭⣭⣭⣩⣍⣩⣉⣉⣉⣛⣛⣛⣛⣛⣓⣶⣶⣶⣮⣿⣶⣤⣤⣾⡛⡒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠄
⣿⣯⡛⣿⣯⣝⣯⣿⢿⣭⡋⠩⠻⣿⣍⠉⠙⠻⣯⣍⡉⠛⢿⣍⠉⠉⠉⠙⠛⠉⠉⠉⣉⡿⠛⠉⠙⣻⡿⠛⠛⠛⣓⡿⠛⠒⢒⡶⠛⠒⠒⣲⡶
⣿⡿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⣖⢹⣿⢷⣌⡠⠐⠉⠻⣶⣄⡙⢷⣄⠀⠀⠀⢀⣤⠾⠋⠀⣀⣴⠟⠉⠀⢀⣤⠞⠉⠀⣠⠶⠋⠀⣀⡴⠛⠁
⣿⣿⣿⣽⣿⣿⣾⣿⣿⣿⣽⡯⣿⢷⣧⡶⣉⠻⣦⣔⠉⡈⠙⢿⣦⡙⢷⣶⡾⠛⠁⢀⣤⠞⠋⠀⢀⣤⠾⠋⠁⣀⣴⠟⠁⢀⣤⠞⠉⠀⠀⣠⠴
⣿⣻⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣯⡻⣿⣿⣦⣗⡬⣻⢷⣤⡀⠀⠙⢻⣦⡉⠀⣀⣴⠟⠁⢀⣤⠾⠋⠁⣠⡴⠟⠉⢀⣠⡶⠋⠁⠀⣠⠴⠋⢁⡴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣮⣿⣷⣤⡷⣝⡿⣦⣄⠀⡈⢿⣿⡟⠁⢀⣴⠟⠁⣀⣴⠟⠉⠀⢀⣴⠟⠁⠀⢀⡴⠋⠁⣠⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣄⠀⠀⠀⢠⣤⣶⣶⣿⣿⣿⣷⣶⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠃⠀⠀⠀⠀⠸⠟⠋⠁⠈⠛⠿⠀⠀⠀⠀⠀⣰⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣏⣾⣿⡿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢨⣵⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡼⠿⢿⣛⣻⣭⣿⣶⣷⡄⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣛⣯⣭⣽⣶⣶⣷⠀⠀⠀⠀⢀⣀⣰⣿⣿⣆⣀
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⠿⢟⡋⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠟⠿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⢀⠀⡀⡀⡠⢸⠠⡂⢠⠠⡀⡄⡂⠄⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢖⠂⣆⠗⠁⠇⠀⠣⠸⠀⠚⠘⠀⠃⠃⠁⠀⠀⡏⢀⠇⠄⠐⢰⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠤⠄⠤⢄⠈⠁⠀⠈⢁⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠎⠉⠀⠁⠈⠀⢀⣀⠀⠈⠑⠒⠖⠢⠤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠔⠉⠁⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠠⡉⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠐⠢⠭⢥⡝⠀⠀⠁⠀⣀⠖⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠄
⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠷⣮⡍⠁⠈⢣⠀⢀⣀⠆⠁⠀⠈⠙⠔⠆⠤⠤⠦⠀⠀⢀⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⢸⢸⢀⠆⠋⠁⠀⠀⠀⢹⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢇⠀⢁⠏⡼⠀⠀⠀⠀⠀⠠⡘⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃⢣⠀⠀⠀⠀⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠈⢢⣀⠜⠀⠀⠶⠀⠀⠀⡀⡄⢂⠀⠀⠀⠰⡇⠀⠀⠀⠠⠦⠀⠀⠣⣤⢠⡀⡰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⣦⣀⠀⠀⠀⠀⠀⢀⣠⠞⠓⢤⣀⠀⠐⠐⠀⠀⠀⠀⢀⡠⠔⠁⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⣿⠐⠒⢲⣙⠷⡀⠈⠑⠀⠓⠋⠁⠀⠒⠒⠂⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⡿⠃⠀⠸⠄⢠⣆⠀⠀⠀⣀⠀⣀⠐⠤⣦⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⠀⣈⠀⠛⠂⢸⠀⠉⠉⠂⠴⠗⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⢀⠄⠉⠁⠰⠦⢼⡀⢀⢀⠀⡀⡀⢄⡒⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠃⠴⠻⠦⠐⠛⠒⠀⠀⢘⠁⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡃⠀⠐⠀⠀⢀⠀⡀⠐⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡁⠀⢈⢩⡀⠀⠀⡀⣄⡄⠀⢈⡁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⢴⣦⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⣰⣿⣇⣀⣀⠀⠀⠀⠀⠀⠀⠉⢿⣮⡛⠛⠻⣶⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⠀⠀⠀⢀⡄⠀⢶⠆⠀⣶⠄⢀⣄
⠀⢰⣿⣿⣿⡯⣳⣿⠃⠀⠀⠀⠀⠀⢸⡇⠉⣇⣷⠄⢸⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⢻⣄⣸⣧⣠⣿⣠⡾⠉
⠀⢸⣿⣿⣿⣯⣿⡏⠀⠀⠀⠀⠀⠀⠘⣷⣄⠈⠉⣀⣾⠏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠃
⠀⠈⠉⠛⠛⠓⠛⠂⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠒⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⣤⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠻⣷⣆⠀⠀⠀⠀⠀⠀⠀⢒⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄
⠀⠀⢀⣝⠉⣀⣀⠈⢻⣆⠀⠀⠀⠀⢸⣿⣄⠀⢀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠠⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢹⣿⡄⠀⠀⣴
⣠⠾⠻⠟⠁⠿⠿⠃⢸⡿⠀⠀⠀⠀⠀⢻⣿⣀⣾⣿⣿⡗⠀⠀⠀⠀⠀⠀⠉⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣴⣿⣷⠀⠛⣿⠛
⠃⠀⠀⠺⣶⣤⣤⣴⠿⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⢀⣸⣟⠀⠀⣸⣿⣇⡀⠉
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⣠⣶⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣜⣛⡴⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⣰⡄⠀⠘⢀⠀⠀⠀⠀⣀⣀⣚⣛⣛⣛⣂⣀⡀
⠀⣼⣹⣿⡅⢸⣿⣏⣧⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠀⠲⣶⣿⣿⡶⠂⠀⠄⠀⠀⠀⢻⡈⣯⣿⣿⣿⠏⣼⠃
⠀⢿⣚⣿⡇⢸⣿⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠿⡇⠀⠠⠀⠀⠀⠀⠀⠙⠚⢿⣿⠟⠚⠁
⠀⠈⠻⣾⣿⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⢰⠇⠈⢿⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠈

⠀⠀⠀⠀⠀⣠⡿⡇⠀⠀⠀⠀⠀⠀⣤⡄⠀⣤⣿⣤⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⣘⠿⠃
⠀⠀⠀⠠⠚⠁⢠⣤⣄⠀⠀⠀⠀⠀⠘⢷⣄⣀⣛⣀⣠⣄⠀⠀⠀⠀⠀⠀⠬⠿⣾⣿⢷⡴⠆⠀⠀⠀⠀⠀⠀⢰⡶⢶⣶
⠀⢀⣀⡀⣿⣷⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠙⣛⠛⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⠿⣦⠀⠀⠀⠀⠀⣴⣤⣤⣾⣿⣼⣿⣀⣀⣀
⠀⠸⠿⠇⠿⠿⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢼⠃⠀⠀⠀⠀⠀⣿⡗⣼⣿⣿⣿⣿⣿⢂⣿
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⡄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠌⡀
⠀⠀⠀⠉⠒⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠢⠀⠄
⠀⠀⠀⠀⠀⠀⠈⠂⠀⠑⠄⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠂⠀⠀⠁⠀⠘⠐⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑
⠖⢺⡟⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠠⢀⠆⠀⠀⠠⠀⠈⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠐⠅⠀⡄⠐⠈⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⡀⠤⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢠⡍⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈⠘
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⡤⠶⠒⠒⠶⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠒⠲⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠒⠲⠦⣄⡀
⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⣼⠁⠀⢶⠆⠀⠀⠠⡶⠀⠈⣷⠀⠀⠀⠀⡿⠀⠠⣶⠀⠀⠀⢰⡦⠀⠸⡇⠀⠀⠀⢰⡏⠀⢰⡦⠀⠀⠀⢶⠆⠀⢹⡄
⠀⠀⢿⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⣿⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢸⡇⠀⠀⣀⠤⠤⠤⡀⠀⠀⢸⠇
⠀⠀⠘⢧⡀⠈⠢⠤⠤⠔⠋⢀⣼⠃⠀⠀⠀⠀⠹⣆⠀⠉⠉⠉⠉⠉⠁⢀⡾⠁⠀⠀⠀⠀⢻⣄⠘⠁⠀⠀⠀⠈⠂⣠⠟
⠀⠀⠀⠈⠛⠦⣤⣄⣀⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠈⠳⢦⣤⣀⣠⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⣀⣠⡴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣄⣠⣾⣿⣿⣷⣤⡀⢤⣶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠄⣾⠿⢛⣛⣩⣭⣭⣤⡭⠭⣭⣭⣭⣍⣛⡛⠿⣷⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⣋⡤⠀⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠄⣨⣙⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡖⠼⣋⡴⠟⠉⠀⠀⠀⠀⡀⡄⠀⠴⢲⠀⠀⢀⣴⡀⣤⣄⠀⠀⠀⠀⠉⠻⢦⣝⠻⡶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣁⠆⠀⠀⠀⠀⢠⠒⡌⢲⡇⠻⠔⠘⠊⠀⠀⠊⠉⠓⠋⠉⠾⠽⢢⢤⡀⠀⠀⠈⠳⣄⠺⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⣣⡾⠋⠀⠀⣀⠠⡽⠆⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠟⡵⢒⠄⠀⠀⠈⢳⣌⢿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣴⠏⠀⠀⣠⣜⢾⠁⠁⠀⠀⠀⢀⣠⠴⠖⠛⠛⠛⠛⠛⠛⠲⢦⣄⡀⠀⠀⠀⠉⠉⣼⡡⢄⠀⠀⠙⣦⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⠏⣼⠃⠀⠀⡀⡓⡇⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⢖⠋⠃⠀⠀⠈⣧⡹⣟
⠀⠀⠀⠀⠀⣠⣾⡟⣼⠃⠀⠀⠀⠻⡏⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠈⢲⣉⡇⠀⠀⠘⣧⢹⣷⣄
⠀⠀⠀⠀⢸⡏⢸⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⢿⣿⡇
⠀⠀⠀⠀⠀⢿⡇⣾⠁⠀⠀⣀⡀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⡀⠀⣀⠀⠀⣀⡀⣀⣀⣀⣀⢀⣀⣀⡀⠀⠈⢿⢸⣿⠁
⠀⠀⠀⠀⣠⣿⡇⣿⠀⠀⢠⣿⣷⠀⠀⣿⠛⠛⣷⠀⣿⠛⠛⣿⠀⣿⠛⠻⣿⠀⣾⠟⠛⢿⡄⢿⡆⢀⣿⠃⣿⠟⠛⠋⢸⣿⠛⢿⣧⠀⢻⠈⣿⣆
⠀⠀⠀⠰⣿⣿⠀⣿⠀⠀⣾⣏⣿⣇⠀⣿⡷⠾⠟⠀⣿⡷⠾⠟⠀⣿⡷⣾⡟⢰⣿⠀⠀⣾⡇⠸⣿⣼⡟⠀⣿⡿⠿⠀⢸⣿⠀⢘⣿⠀⢀⡀⣿⣿⡇
⠀⠀⠀⠀⠙⣿⠄⣿⠀⢸⡿⠉⠙⣿⡀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⢹⣧⠀⢿⣦⣠⣿⠃⠀⢿⣿⠃⠀⣿⣇⣤⣤⢸⣿⣤⣼⡏⠀⣼⢡⣿⠏
⠀⠀⠀⠀⠀⣾⠀⢿⡀⠉⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠉⠀⠀⠙⠋⠁⠀⠀⠈⠉⠀⠀⠉⠉⠉⠉⠈⠉⠉⠉⠀⠀⣿⢸⣿
⠀⠀⠀⠀⢸⣿⣶⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⡇
⠀⠀⠀⠀⠀⠛⢿⣇⢻⡄⠀⠀⢰⣋⡷⡀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⢐⣆⠀⠀⠀⢠⡟⣸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣆⢻⡄⠀⠀⢡⣠⡕⠀⠀⠀⠀⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⠀⢐⣍⡉⠁⠀⢀⡈⣱⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣆⠻⣄⠀⠀⠁⠒⡟⣀⣀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣀⣀⣠⣤⠤⠶⠛⠉⠀⠀⠀⠀⠀⡳⡜⠊⠀⠀⣠⡟⣱⡛⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣧⡙⢧⡀⠀⠀⠀⠧⠜⣰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡰⣜⡦⠛⠁⠀⢀⣴⠋⣼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡙⢦⣀⠀⠀⠈⠳⠋⣖⡶⢀⣠⠄⣀⣀⠀⠀⣠⣄⠤⢦⢰⡦⡘⠤⠇⠈⠀⠀⢀⡴⢟⠱⠞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣦⠁⠀⢠⣀⠀⠀⠈⠁⠙⠚⠀⠿⠁⠀⠀⠧⠼⠠⠝⠈⠁⠀⠀⠀⠀⣤⠾⣋⣴⠿⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣦⣝⡳⠶⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⢠⣴⣾⣥⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠛⢷⣆⣠⣍⣛⣛⣛⠲⠶⠶⠖⣛⣛⣛⣩⣭⣶⡾⠛⠓⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠛⠛⢿⣿⣿⡿⠟⠛⠛⠽⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣰⡶⠀⠲⣶⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⢸⢉⣉⢹⠀⠀⠀⠀⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠢⠀⠄
⢽⣇⠀⢀⣿⠅⠀⠀⠀⢈⣿⣻⡅⠀⠀⠀⠀⢸⠐⠒⠀⠀⠀⠀⠀⠀⣹⣿⣿⡏⠀⠀⠀⠀⠀⠄
⠀⠛⠓⠛⠃⠀⠀⠀⠀⠈⠁⠘⠓⠀⠀⠀⠀⠈⠉⠉⠛⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛

⠀⢶⣦⣶⠂⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠀⢠⡿⣥⠀⠀⠀⠀⠸⣧⣾⣷⣴⡇⠀⠀⠀⢲⡞⠰⠠⠤⠀⠀⠀⠀⠢⣿⣿⠕⠀⠀⠀⢰⣶⣾⣿⣷⣶
⠀⠻⣤⡾⠀⠀⠀⠀⠘⠁⠀⠀⠈⠑⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⡜⢣⠀⠀⠀⠀⠀⠀⠈⠛⠁

⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣤⢠⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⡇⢀⣤⣀⠖⠀⠀⠀⠀⠀⠀⢀⣤⣾⠀⠀⠀⠀⠐⠿⠈⠀⠀⠀⠀⠀⣤⣴⣧⣤⠀⠀⠀⠀⣼⣯⡊⠙⡆
⣇⢋⣀⣁⣀⠀⠀⠀⠸⠇⠿⠻⠟⠿⠀⠀⠀⠀⠐⣿⡖⠀⠀⠀⠀⠀⠿⠿⠿⠿⠀⠀⠀⠀⠙⠢⣝⠴⠃
⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⡖⠒⠒⣒⡒⡄⠀⠀⠀⣠⣴⣮⡭⣄⠀⠀⠀⠀⣙⡿⣏⠀⠀⠀⠀⠀⢀⡴⢦⡀⠀⠀⠀⢀⣼⣿⣿⣷⡶
⠧⠭⠤⠭⠤⠇⠀⠀⠘⢔⡻⠿⠃⠁⠀⠀⠀⢸⣿⠴⣻⡇⠀⠀⠀⠰⠮⠽⠯⠽⠆⠀⠀⠀⠻⢿⣿⡿⠃
⣶⣶⠶⠶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣿⣷⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠋⠙⣿⣿⣿⡟⠉⢻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣴⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⢿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⡙⢿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⠿⣫⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣶⣬⣙⡛⠿⠿⠿⠟⣛⣫⣵⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣏⠀⣹⣿⣿⣏⠀⣹⣿⣿⣿⣷⢹⣿⣟⠿⣻⣿⣿⣟⠟⣻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⡿⣸⣿⡿⠟⠋⠉⠉⠉⠻⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢡⣿⣿⣀⣀⣀⣀⣀⣀⣀⣈⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠋⠁


⣀⣀⣀⣀⣀⣀
⣿⡿⡿⢿⢿⣿
⣿⣿⡏⢹⣿⣿
⣿⣿⣳⣚⣿⣿
⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⠿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⣫⣭⣝⡻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢻⣽⣩⣴⡆⡇⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣾⣿⣿⣿⣷⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⡹⠿⠸⣿⣧⡇⣱⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣙⢿⣿⡿⢏⣾⡏⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⢻⡜⣶⣭⣣⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⠾⡫⣶⣶⢰⣾⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣥⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣞⡃⣿⣿⢸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣝⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠃⠐⣦⠀⠉⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⡿⠿⢷⢍⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠿⠥⠤⠄⠘⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡀⣰⣟⠍⣻⣟⠉⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⢀⣤⣤⢰⣿⣕⣸⣿⣅⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢣⣿⣿⣇⣿⡿⣛⣭⣭⡻⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣚⣇⢿⣞⣿⡸⣯⣰⣶⣶⣶⢆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠞⠃⠙⠲⠛⠘⢛⢻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⠀⣩⠏⣿⣭⡷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⡴⠋⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣢⣞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⣿⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣿⣿⣷⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁






⣿⣿⡿⠿⣿⠿⢿⠿⡿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣶⣿⣶⣿⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡴⣖⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣲⢦⡀⠀⠀⠀⢸⣿⣿⣿⡿⢋⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⢍⡛⢿⣿⣿⣿
⠀⠀⠀⡿⡞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⢻⣿⠀⠀⠀⢸⣿⣿⣿⢠⢱⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣷⡽⠈⣿⣿⣿
⠀⠀⠀⡇⡇⠀⠀⣠⣶⣄⠀⢀⣴⣿⣿⠟⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⢸⣿⣿⠟⠉⠻⣿⡿⠋⠀⢀⣼⣿⡇⠀⣿⣿⣿
⠀⠀⠀⡇⡇⠀⠀⠙⢿⣿⣷⣿⣿⠟⠁⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⢸⣿⣿⣦⡀⠀⠈⠀⢀⣴⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⡇⡇⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⢸⣿⣿⣿⣿⣦⣀⣠⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿
⠀⠀⠀⡇⡇⠀⡄⣠⣠⣄⣄⣠⣀⣀⡄⣠⡀⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⣸⣿⢿⠿⠿⠿⠿⠿⣿⢿⣿⠿⣿⣧⡄⣿⣿⣿
⠀⠀⠀⣇⡇⠠⠿⡟⢹⠹⠯⢿⠻⠇⡇⣿⠆⢸⣿⠀⠀⠀⢸⣿⣿⣿⢸⢴⣖⣐⣰⣦⣶⣸⣋⣧⣿⣺⣁⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠹⣾⣢⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣖⣵⠏⠀⠀⠀⢸⣿⣿⣿⣆⠣⠝⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠯⢊⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣉⣀⡐⠀⠐⠒⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢟⣛⣩⣭⣭⣛⣓⡀⠀⠀⠀⢮⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⠵⣾⢿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⢿⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡼⠁⠈⠀⠀⠀⠀⠐⠒⠲⠄⠀⠀⠀⠀⠀⠀⠸⣷⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠠⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠁⠀⠀⢀⠤⡀⠀⠔⡡⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡤⣄⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣧⠃⠀⠈⢀⠡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠁⠀⢰⠚⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⡇⠀⠀⠀⠀⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢙⠂⠤⠞⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣏⠀⠀⠀⠀⠀⠈⣏⠉⠁⠀⣠⠔⣻⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠘⢇⡉⣉⡠⣺⠏⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡠⠚⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⣌⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⡀⠀⠀⠀⣀⣠⡴⢾⠁⠒⠒⠉⣁⡠⠟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢋⡀⢀⡞⢰⠒⠊⠉⠁
⠀⠀⠀⠀⠀⡠⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠒⠒⠒⠉⣀⣤⡆⠀⠉⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡴⠂⠤⠤⠤⠤⠖⠚⠦⠔⠒⠋⠁⡼⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠈⠙⠲⠤⣀⡉⠑⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠲⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣦⠤⣄⢀⣀⣠⠋⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⡭⢷⡄⠀⢀⣀⣀⣀⣈⠑⠲⠦⢤⣤⣀⣀⡀⠀⣀⡁⠀⠀⠀⠀⠀⢸⠀⡟⠈⠀⠀⠈⠉⠉⠉
⠀⠀⠒⠄⠀⠀⠀⠀⠀⡰⢮⡠⠚⠁⠀⠀⠛⠉⠉⠉⠁⠈⠉⠁⠢⢤⣀⡀⢀⠉⠉⠉⠉⠴⡦⠀⠀⠀⢠⠋⡰⠃
⠘⢌⠢⡀⠈⠑⠢⢄⡘⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⣸⠣⠤⠔⣊⡭⠛⠁
⠀⠈⠱⣈⠢⡀⠀⠀⠈⢳⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠏⠙⠋⠉⠁
⠀⠀⠀⠈⠑⢌⠳⢄⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⠉⣁⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠐⠈⠁⠀⠀⠀⠀⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠂⠀⢄⠔⠒⠢⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠈⠉⠂⢀⠀⠀⠈⠂⠀⣀⡀⣀⠤⠄⠊⠀⠀⠀⠀⡀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄
⣤⡒⠁⠁⠀⠈⠈⠝⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠮⠀⠚⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡎
⠑⠈⠦⣄⣀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠐⠠⠁⠀⠈⠀⠀⡤⠄⠚⢡
⠀⠀⠀⠈⠢⡀⠉⡀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠁
⠀⠀⠀⠀⠀⠈⠁⠙⡄⠀⠀⠙⠄⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠒⠄⠠⠀⠰⠒⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠈⠐⢲⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⡀⠁⠒⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⢄⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠁⠺⠀⠀⠱⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠑⢄⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⢀⣀⣄⠐⠂⠀⠠⢄⠀⢤⠃⠀⠀⠀⠀⠠⠄⠒⠂⠒⡠⠴⢤⡀⠀⠀⠂⠀⠀⠁⠢⡀⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⢏⠙⢳⣻⠁⠀⠀⠀⠀⠀⠀⣀⡘⠀⠀⠀⠀⠁⠀⠀⠀⡠⢡⡀⠤⢈⠀⠀⠀⠰⡀⠀⠀⠈⠑⠠⣀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⡋⠀⠄⡀⢐⣠⢜⠤⠉⠀⠀⠀⢄⠀⢠⡄⠀⠀⠁⠀⣀⣀⠚⠁⠀⠀⠀⠈⠐⠢⢀⡀⠀⠀⠈⢛⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠤⠛⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠍⢐⡀⠀⠀⢄⠀⠈⠁⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠁⠁⠀⠄⢛⣒⣐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠀⢀⢀⡀⠀⠀⠘⠁
⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⣴⣿⡁⠀⠀⠀⠀⠈⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠁⠀⠀⠀⠀⢈⣿⣦⡀
⠀⠀⢀⣴⠟⠁⠈⠻⣦⡀⠀⠀⠀⠀⠹⣿⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣿⠏⠀⠀⠀⠀⢀⣴⠟⠁⠈⠻⣦⡀
⠀⠀⠀⠻⣦⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠈⢻⣦⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⡟⠁⠀⠀⠀⢀⣴⡟⠁⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣹⣷⠶⠟⠛⠛⠉⠙⠛⠛⠻⠶⣾⣏⡀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⠀⠙⣷⡶⠟⠉⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠉⠻⢶⣾⠋⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⢠⣾⠋⠀⠀⣠⡶⠟⠋⠉⠁⠀⠀⠈⠉⠙⠻⢶⣄⠀⠀⠙⣷⡄⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡿⠁⠀⣠⡿⠋⠀⠀⠀⠀⠀⣰⣆⠀⠀⠀⠀⠀⠙⢷⣄⠀⠈⢿⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⣼⠏⠹⣧⠀⠀⠀⠀⠀⠀⢻⣧⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⢰⡿⠀⢀⣀⣤⣤⣤⣼⠏⠀⠀⠹⣧⣤⣤⣤⣀⡀⠀⢿⡆⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠈⠻⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠟⠁⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⣿⡀⠀⠀⠀⢸⡟⠀⠀⢀⡀⠀⠀⢻⡇⠀⠀⠀⢀⣿⠁⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⡄⠀⠘⣷⡄⠀⠀⣼⣧⣴⠾⠛⠛⠷⣦⣼⣧⠀⠀⢠⡾⠃⠀⢠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠈⢿⣄⠀⠈⠻⣦⣄⠙⠉⠀⠀⠀⠀⠀⠀⠉⠋⣠⣴⠟⠁⠀⣰⡿⠁⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣹⣷⣄⠀⠀⠙⠻⠶⣦⣤⣤⣤⣤⣴⠶⠟⠋⠀⠀⣠⣾⣏⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⠶⠶⢶⣾⠋⠀⠙⠷⣦⣄⣀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣠⣴⠾⠋⠀⠙⣷⡶⠶⠶⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⣠⡾⠟⠛⠛⠷⠶⠶⠶⠶⠾⠛⠛⠻⣷⣄⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣠⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠿⠿⢷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠈⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠠⣶⡦⠀⢰⣶⡄⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢻⣦⣄⣀⣀⣴⡿⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠈⠙⠛⠛⠉⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⠛⠛⠻⢿⣦⣄⠀⠀⠀⠀⠀⠉⠻⠿⠶⣶⠶⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣿⠇⠀⠀⣶⣦⠀⢠⣶⡆⠀⠀⢿⣇⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⣿⡀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⣸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠛⠛⠛⠛⠛⠛⠛⠃⢠⣿⠃
⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠹⣷⣤⣀⠀⠀⠀⢀⣠⣴⡟⠁
⠀⢀⣴⡿⠛⠉⠉⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠟⠛⠁
⢠⣿⠉⠀⣠⣄⠀⢀⣤⡄⠀⠙⣿⡄
⣿⡇⠀⠀⠛⠛⠀⠈⠛⠋⠀⠀⢹⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀
⣿⡇⠀⠀⣠⣴⣶⣶⣤⣀⠀⠀⣸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠘⣿⣄⠸⠟⠁⠀⠀⠉⠻⠆⣰⣿⠃
⠀⠈⠻⣷⣤⣀⣀⣀⣀⣤⣾⠟⠁
⣤⣤⣤⣤⣭⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠐⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠠⢀⠀⠀⡀⠢⠁⠀⠀⠀⠀⠀⠀⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠢⠄⠀⢀⠄⠂⠀⠀⠈⠉⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠐⠀⡀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⡀⠀⠠⠀⠣⡀⠀⠀⠈⠀⠀⢀⠌⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠁⠀⠀⠂⠈⠉⠁⠠⠀⠀⠑⠄⡈⠐⠂⠤⠀⢒⡡⠊⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁⠀⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⡀⠔⢨⠂⠀⠀⠆
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠄⡐⠀⠀⠀⠐⠒⠁⡀
⠀⠀⠀⠀⠀⠐⢄⢀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠎⠀⠀⠀⠀⠀⢑⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠒⠂⠀⠀⢀⡀⠀⢀⡐⠯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠀⠀⠀⠀⠀⢉⠁⡀⠈⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠃⡋⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠠⠼⢬⣼⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢰⠉⠢⡀⡜⠀⢣⣀⡴⠉⠃⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠆⠀⠀⠈⢀⣀⣀⠉⠀⠀⠠⠔⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠐⠒⠀⢀⣠⣴⣶⡿⡿⡿⢿⢿⣿⡿⣶⣦⣄⡀⠀⠗⠂⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠄⠀⡄⠀⣠⣶⣿⣿⡿⣪⣧⣾⠷⠾⠶⢾⣽⣣⡿⣛⣿⣶⣄⠀⠰⠄⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⢀⣼⣿⣿⣛⡽⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⢯⣞⣽⣿⣷⡀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠃⠀⠈⠉⣉⠉⠉⠀⣀⣀⡀⠀⢀⣀⣀⠀⠀⢀⣀⡀⠉⢉⡉⠉⠁⠀⠘⠂⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠰⢾⣿⠀⠀⣾⡟⠹⣿⠀⣾⡏⢻⣷⢠⣿⠉⣷⠀⣾⠃⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⣿⠃⣿⡁⢸⣿⠀⠿⠾⠏⣼⠃⣾⠛⣧⠀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⢠⣴⣿⣦⡄⢻⣧⣼⡿⠀⢿⣧⣼⡟⠀⠀⠀⣸⠏⠀⣿⣤⡿⠀⠀⠀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⣀⡀⢀⠀⣀⣈⡁⣀⡀⣀⡈⢁⠀⣀⡀⣀⣀⡀⢀⣀⢉⠀⠀⠀⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⡤⠀⠀⠀⣓⡆⡼⣧⢸⠀⡏⣓⡄⡗⢀⡿⡆⣇⣀⢸⡁⡷⣇⣸⢸⢷⡀⠀⢠⠐⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢠⣬⣤⣤⣤⡀⠀⠀⠈⠀⠀⠈⠀⠀⠈⠁⠀⢀⣤⣬⣥⣤⡄⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⢐⠂⠀⠹⣿⣿⢷⣽⢶⣤⣀⡀⠀⠀⠀⣀⣀⣤⢖⣫⣲⣽⣿⠟⠀⠐⡂⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⠀⣀⠈⠛⢿⣧⣾⣡⡏⣿⡻⣟⢻⡟⢹⣅⣷⣿⡿⠛⠁⢀⠀⢀⣱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⢀⣀⠈⠙⠻⠿⠿⢿⣿⡿⠿⠿⠟⠋⠁⢀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠊⠁⢸⠀⢀⠤⡄⠀⠀⣀⠀⠀⢀⠤⡀⠀⡀⠈⠑⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠰⣀⠞⠀⠱⣀⠌⠀⠈⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈



⣿⠿⠿⠿⡿⡻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣵⣶⣌⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣟⣻⣿⣿⣿⣿⣿⣷⡀⢙⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⣿⣽⣿⣿⡗⡸⡿⡇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡯⠉⠀⠀⠀⠀⠀⠉⠻⣿⣿⠇⢹⢳⣇
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⣤⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⣠⣤⣤⡀⠘⣿⣯⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⣟⡂⠀⠀⠉⠙⠛⠷⠆⠀⢹⣿⣿⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣣⣄⠠
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣯⠳
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⢤⠀⠀⠀⢰⣴⣿⡿⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠠⠹⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⡠⠐⢹⣿⣿⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⢫⡑⠒⠒⠈⠁⠀⠀⢸⣿⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣏⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢮⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠨⠂⠠⠀⠬⣔⡀⡀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠍⠳⣦⡄
⠀⠀⢔⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠉⠀⠀⠘⡆⠀⠀⠀⢾⣿⠿⣿⡀⠀⠀⠹⣿⡀
⠀⠀⠀⢄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢸⡟⣀⡘⣿⠠⠴⠂⠙⠣
⠀⠀⢧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⠀⠟⠋⠋⠀⠈
⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠘⠀⠀⠀⠰⠶⠄
⣶⣶⡶⣶⠶⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡿⣿⣿⣿⡿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣟⢱⣿⣿⣿⣧⡀⠉⠉⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠟⣿⡿⠀⠈⣿⡿⢀⣀⣀⡀⣿⣿⣿⣰⣶⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠰⣿⠇⢸⣿⣿⣇⠛⠋⠁⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⣾⣿⣷⡸⠛⠉⢸⠁⢀⣠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠿⠋⢹⡇⠀⣀⡘⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠸⠟⠉⣿⠀⢀⣘⢳⣾⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⢹⡇⠀⢀⠛⣶⣿⣧⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣾⣿⠀⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠃⠘⠟⠿⠀⠻⠿⠗⠘⠿⠛⠃⠻⠿⠟





⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠶⠿⠶⢶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣥⣴⣶⣶⣶⣶⣦⣤⣉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠏⠙⠛⠛⠛⠛⠛⠟⠻⠿⣿⠂⠀⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⠀⠈⠉⠁⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠿⢇
⠀⠀⠀⠀⠀⠀⠀⢀⠿⢿⡿⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⠀⠀⡄⠀⠀⠀⠀⢠⡄⠀⠀⠀⠈⡞⢙⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠊⣇⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣀⡄⠀⠀⠀⠀⠀⠠⢬⣤⡄⠀⠀⠀⠀⠀⠀⣴⡆⢰⡤⢧⣦
⠀⠀⠀⣀⣢⡷⠠⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢸⠛⠤⢀⡠⣦⡶⡀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⢀⣴⣿⠧⠘⠀⠐⠛
⠀⠀⠀⠐⠋⠀⠀⠠⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⠌⣁⠠⡉⠁⢴⣤⣤⣤⡀⠀⣠⣤⣶⣾⠟⠋⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠑⠀⡁⠂⢁⠂⠄⢀⠀⠀⠀⠀⢀⠀⠀⠠⡊⠄⢈⠀⠊⠀⠀⠀⠐⢀⡘⠙⠛⠣⠄⠻⠋⠁⢡⣴⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠂⠀⢀⠤⣀⠨⣀⠄⠆⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠂⠀⠀⠀⠐⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⡀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⣿⣿⢿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠁⠀⠃⠀⠖⢂⠢
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢄⠀⠀⢀⠠⠐⠻⣀⠇
⠀⠀⠀⠀⠀⠀⡔⣀⣡⠉⠑⠐⠀⠀⠀⠀⠐⠒⠒⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠠⠮⡄⢨
⠀⠀⠀⠀⠀⠀⠃⠘⠁⠑⢀⠀⠀⠀⢀⠠⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡤⠀⢣⡄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠂⡡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣴⠟⠋⠉⠉⠙⠻⣦⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⠏⠀⠀⠀⠀⢀⣤⠜⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢿⡎⠻⣦⣠⣴⠟⠁⢀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢿⣄⡈⠉⠀⢀⣠⡾⠃⠀⠀⢸⣿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠘⠟⢠⡾⠋⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠶⠶⠶⠶⠶⢀⣤⡾⠛⠛⢶⣤⡶⠛⢀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣀⣀⣀⣀⣀⡀⠀⣿⠉⠛⢶⣄⠀⠙⢷⣴⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠉⠉⠉⠉⠁⠀⠹⣦⠀⠀⠙⢷⡀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⠘⢷⣄⠀⠈⣷⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠙⠳⠶⠾⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣧⡀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣿⣿⣷⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢠⣆⠀⠐⠒⠒⠒⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⣠⣴⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁⠀⠀⠉⠋⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠉⠙⠛⠀⠀⠀⢶⣿⣿⣿⣿⣿⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣹⣿⣿⣿⣆⣀⣀⡀⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣦⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣾⣿⣿⣿⣿⣿⣿⠿⠿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⣠⠄⢠⢠⣠⡄⢠⡄⡤⢤⢠⣠⡄⣤⢠⣤⠄⢠⠄⣠⡄⣤⢠⢠⡄⢠⡄⣤⣀⡤⠀⣤⢠⣠⠤⣄⣄⡄
⠀⠀⠀⠀⠙⠒⠘⠚⠐⠛⠘⠃⠓⠊⠛⠛⠓⠛⠚⠛⠂⠐⠃⠋⠓⠛⠘⠒⠛⠛⠐⠋⠋⠓⠂⠛⠘⠘⠒⠉⠋⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠫⠉⠁⠀⠂⠐⠀⠈⠉⠝⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⠂⠀⠀⢠⣤⡀⣶⣦⢠⣶⡄⣤⣀⠀⠀⠐⠉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢋⠀⠀⢀⣄⢺⣿⠌⣿⠃⣿⠷⢸⣿⢃⣿⠏⣼⡓⣠⡀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠘⢿⡷⠟⣀⠬⠔⠒⠒⠒⠒⠢⠍⣘⠃⣼⣿⠏⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣏⡄⣀⣀⣀⣀⣀⢈⣴⠊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⠑⣤⣁⣀⣀⣀⣀⣀⢠⣹⣄⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⣤⡀⢠⣤⣤⡀⣤⣤⣄⢠⣤⣤⡀⢀⣤⣤⡀⣤⡀⣠⣄⣤⣤⡀⣤⣤⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣿⣇⢸⣿⣹⡇⣿⣏⣿⢸⣿⣽⡗⢸⡏⣿⡇⣿⣇⣿⡇⣿⣏⠀⣿⢽⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⣿⣾⣿⢸⣿⠉⠁⣿⡏⠉⢸⣿⢹⣇⢸⣇⣿⡇⢸⣿⣿⠀⣿⣏⠀⣿⣾⣿⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠛⠈⠛⠙⠛⠀⠀⠛⠃⠀⠘⠛⠘⠋⠈⠛⠛⠁⠈⠛⠋⠀⠛⠛⠁⠛⠛⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⡄⠄⠂⠀⠀⠀⠀⠀⢤⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡤⠀⠂⠀⠀⠀⠐⠠⢠⠄⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡔⠀⠀⠀⠀⢀⡴⣦⣉⠢⠤⣀⣀⣀⣀⣀⣀⠤⠔⣉⣴⣦⠀⠀⠀⠀⠠⣢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠻⠿⢁⣾⢣⣶⠆⣤⡄⣤⣤⠀⣶⢀⣿⡌⠻⠇⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠉⠁⠛⠿⠸⣿⡇⢽⣿⠘⠿⠇⠙⠉⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⡀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠶⠶⠖⠒⠚⠋⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣴⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠞⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠦⠊⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣄⠅⠉⠀⠀⠀⠠⠙⡄⠀⠀⠖⠙⢴⠈⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠂⢠⣉⡭⠋⠀⠀⢀⡄⠀⠈⠁⠀⠉⠀⠠⡄⠐⠂⠀⠘⠀⢠⡄⠀⠀⠐⢖⠒⠁⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢖⡠⠼⣷⡀⢲⠄⢈⠠⣄⠀⠰⡆⠺⣄⠀⠈⢦⠀⡀⢾⢵⠉⠀⠀⠀⢀⣈⡠⡞⢶⠛⠁⠀⠀⡀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⢠⡥⠯⣒⠁⠀⠰⠕⢵⠋⠁⠀⢸⡨⠚⠑⢦⡇⠀⠀⠀⠈⠁⠀⠈⠾⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⣒⡒⢦⡀⠀⠈
⠀⠀⠀⠁⠀⠀⠞⡱⠒⠢⡙⣆⣀⠊⡉⣉⠉⠉⠉⠀⠉⠀⠀⠀⠀⡈⠈⠀⠁⠠⠠⢠⠈⠄⠑⣀⣠⠿⠉⠀⠀⠈⢆⢳⠀⠀⠉
⠀⠀⠄⠀⠀⢸⣸⠀⠀⠀⠒⠚⠛⠒⠐⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⡓⢒⠒⠂⠀⠂⠐⠐⠒⠚⠉⠉⠁⠀⠀⠀⠀⢻⢸⠁⠀⠒
⠀⠀⠄⠀⠀⢠⢸⠀⠀⠀⠀⠀⠀⡄⢈⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⢊⠔⡡⢀⠔⡡⣂⢸⠀⠀⠀⠀⠀⠀⠀⡌⢸
⠀⠀⠀⠀⠀⠘⢸⡀⠀⠀⠀⠀⠀⠃⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡠⢊⠔⡡⢊⠔⡁⠊⠀⠀⠀⠀⠀⠀⡘⢠⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠡⢇⠀⠀⠀⠀⠀⠸⠀⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢊⠴⢈⠔⡡⢊⠔⠃⠀⠀⠀⠀⢀⠜⣠⠏
⠀⠀⠀⠀⠀⠀⠐⢟⣆⠀⠀⠀⠀⠀⢆⠀⡊⠀⠢⠀⠀⠀⠀⠀⠀⠀⠐⡡⢚⡡⢊⠔⡡⠆⠀⠀⠀⠀⣠⢊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢮⡷⡀⠀⠀⠀⠈⡆⠀⠆⠀⠑⠀⠀⠀⠀⠀⠀⣫⢔⡡⢊⠄⡊⡐⠁⠀⠀⢀⠔⣠⠊⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⡢⣄⠀⠀⠈⢀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠔⡡⢂⠕⣪⠊⠀⢀⡠⢐⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢝⣢⢤⣀⡑⣄⠀⠂⠀⠀⠀⠀⢐⣡⡺⣿⣭⡥⣴⣶⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠷⠦⣄⠀⢀⠬⢓⡥⠚⠛⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠕⢐⡺⢥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⡰⠖⣟⣦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠸⠇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠉⠀⠀⠢⢒⠔⠉⡑⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡁⠀⠀⠀⠀⠀⢀⠕⡈⠢⢀⢀⢈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠺⠿⠧⠄⠾⠋⠭⠤⠿⠿⢿⠿⠿⠿⠿⠿⠧⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠟⠛⠛⠒⠒⠒⠒⠒⠒⠚⠛⠚⠚⠛⢺⠌⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠐⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢌⢼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣤⣿⣯⣺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠄⣀⠸⡀⠀⠈⠀⠀⠀⠰⠛⠛⠃⠚⢒⢂⠶⠖⡷⢋⡞⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠙⠋⠙⠉⠛⠐⠛⠓⠓⠓⠓⠛⠟⠺⠞⠟⠛⠛⠟⠩⠓⠩⠂⠩⠒⠝⠊⠔⠚⠊⠓⠉⠊⠂⠒⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠋⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠙⣿⡇⠀⠀⣼⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣸⡿⠀⠀⣼⡟⠀⠀⡀⠀⠀⠀⢠⣾⠂⠀⠀⢀⣶⠆⠀⠁⢀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠷⠶⣾⡿⠀⣠⡾⠛⣿⠂⢠⣿⡗⢩⣿⠀⣾⡧⠊⣿⠂⢻⣿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⠇⢰⡟⢁⣾⡇⢀⣾⣯⢤⣾⠏⣼⣿⢤⣼⠟⢀⣿⠃⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿⣄⠈⠛⠁⠘⠛⣽⡏⠙⠋⠁⢰⡿⠙⠛⠉⠀⠀⠛⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠃⠀⠀⠀⣴⠞⣻⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣴⡆⢀⣴⡷⠀⠀⢰⣄⠀⠀⣶⠄⠀⠀⠀⠀⠀⠐⠛⠟⠁
⠀⠀⠀⠀⠀⢠⣶⠿⣿⡟⠉⠙⢦⠀⠁⢀⣤⠉⠉⠉⠛⠛⢻⣾⢿⣶⣿⣯⣤⣀⣠⣼⠟⠀⣼⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠹⠅⣰⣿⣀⣀⣤⠏⢀⣠⣌⠉⢰⣛⡆⠀⠀⣼⠏⢠⡿⢁⣭⣭⡍⢉⣡⣤⣴⡟⠀⠀⢠⣴⠶⣤⠀⢀⣴⠆⢠⣶
⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⢹⡄⢹⡿⠁⢀⣼⢛⡿⠀⣸⡏⠀⣾⡷⢋⣿⠋⣴⠟⠉⢠⡟⠀⠀⣴⡿⢁⣼⠇⢀⣾⠏⣠⣿⠃
⠀⠀⠀⠀⠀⢠⣿⡃⢀⣀⣤⠟⠁⣿⣧⠶⠋⢰⣿⣥⢾⣿⠀⢸⡟⠁⢼⣇⡤⢿⣤⡶⠻⣧⣴⠞⢿⡥⠊⣿⡤⢺⣿⠔⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣴⡟⠀⠐⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠶⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣾⣯⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣬⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⡾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢹⡄⠀⠀⡀⠒⠈⣿⣿⠆⢰⣶⣶⣶⣶⣶⣶⣶⣶⣿⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠤⠞⠀⠀⠀⠀⠀⠐⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠤⠀⠄⠀⢠⣤⣤⣤⣤⣤⣤⡦⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠀⢰⣶⣶⣶⣶⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣰⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣼⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣄⣀⣀⣀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⣿⣿⣿⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠰⡦⢰⡆⢴⠆⢶⠄⣿⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠁⠀⠀⠉⠁⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⠉⠉⠀⠀⠀⠀⠉⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠐⣿⣿⡿⠀⠀⢻⣿⣿⠂⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠻⢦⣀⡀⠀⠀⢀⣀⡴⠟⠁⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⣀⣀⣀⣀⣀⣤⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣿⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀
⠀⠀⠀⠀⠀⠀⠰⣾⣿⣿⣿⡶⠂⠀⠀⠀⠀⢠⡿⠟⠛⠛⠿⣿⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⣿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣦⠀⣾⣿⣿⣿⣿⣦⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠘⣿⣿⣦⡀⠀⠘⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⡿⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠈⠛⠿⠿⠋⠀⠀⠈⠛⠿⠿⠛⠁⠀⠈⠛⠿⠿⠛⠁⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣶⣶⣶⣶⣶⣶⣄⠐⣶⣶⣶⣶⣶⣆⠐⣶⣶⣶⣶⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠿⠀⠿⠿⠿⠿⠿⠏

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

⠀⠀⣠⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣷⣄⠀⠀⠀⣠⣶
⠀⠀⣿⡇⠀⠀⢀⠀⠀⢀⣀⡀⠀⣀⣀⡀⠀⣀⣀⡀⠀⠀⣀⡀⠀⡀⠀⠀⡀⣀⣀⣀⠀⣀⣀⠀⠀⢸⣿⠀⠀⢠⣿⡇
⠀⠀⣿⡇⠀⢠⡟⣧⠀⣿⣉⣹⠆⣿⣉⣿⢈⣯⣉⡿⢠⡟⠉⠹⡆⢻⡄⣼⠃⣿⣭⣍⠀⡏⠉⢻⡄⠸⢿⠀⢀⣿⣿⠁
⠀⠀⣿⡇⠀⡾⠛⠻⡇⢿⠁⠀⠀⠿⠀⠀⠨⡇⠈⢷⠈⠳⣤⡼⠃⠀⢿⠏⠀⢿⣤⣤⠀⣷⡤⠾⢡⣶⣦⣀⣼⣿⡇
⠀⠀⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠿⣿⣿⡿
⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠁⠈⠻⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣭⣿⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣭⣿⣿⣯⣿⣽