Seasonally Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⢠⣾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣷⠀⠀⠀⠀⣸⣷⠋⢀⠀⠀⠐⠒⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢀⣤⠶⣄⡼⣿⢿⡟⠀⣠⠀⠀⠀⡹⣶⡁⠀⠀⠀⠀⠀⣀⣀⠀⠀⢰⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⣿⠋⠰⠗⠀⠀⠀⠀⢀⣀⠀⠀⢸⡿⠁⣾⠋⢒⡏⢰⠏⠈⣧⡴⢁⣶⡄⠀⠷⡿⠃⣠⣤⠀⠀⣼⣿⣿⣷⣄⠀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣴⣿⡆⢠⡿⢻⡇⢀⡞⢧⠀⣿⣠⡞⢠⣬⣄⣀⣈⠀⠘⠿⠟⣿⣦⠀⣠⣿⣿⠇⢸⣿⡏⢸⣿⡏⣸⣧⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣼⣏⡜⢀⣿⠃⣼⣧⡾⣥⡼⠁⢈⣁⣀⣠⣿⣿⣿⡿⢲⣶⣄⠀⣿⣿⣴⢻⣿⡟⠀⢿⣿⠃⢸⣿⠇⣿⣏⣽⣿
⠀⠀⠀⠀⠀⢠⣿⠀⣿⡇⢿⣉⣠⠎⠙⣛⣥⡼⠶⠚⠛⣻⣿⡛⠛⢻⣿⡿⠋⢀⣿⣿⠏⢀⣿⣿⠃⢸⣿⣃⣠⡾⠻⢷⣿⣿⠟⠉⠛⠿⠟
⠀⢀⣤⣤⣄⠈⠻⣾⠿⠃⠀⠉⠁⠀⠟⣋⣁⡀⠀⠀⣴⣿⣿⣿⡆⢸⣿⡇⠀⢸⣿⣿⣠⣾⣿⡏⠀⠀⠙⠛⠁⣀⣴⠞⣿⣿
⢀⣾⣿⠿⠿⡇⠀⠀⠀⡀⢠⣤⡀⠀⣸⣿⣿⣿⡄⢀⣿⣿⣿⡿⢁⡿⣿⣧⣴⡿⠿⣿⡿⠋⠛⠃⠀⠀⣀⣴⠞⠋⠀⠀⣿⡿
⣼⣿⠃⠀⠀⠛⠀⠀⠸⣿⣿⣿⡿⠀⣿⣿⣿⠟⠀⣸⣿⠛⠁⣠⠟⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⣸⣿⡇
⣿⡏⠀⠀⣠⣶⣶⣦⡀⡿⢹⣿⠇⠀⣿⣿⠁⣀⣴⠟⠻⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⣰⣿⡿
⣿⡇⠀⠸⡿⣿⣿⠧⡼⠃⠸⣿⣦⣾⠋⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀⠀⢀⣀⣤⣾⣿⠟⠁
⠸⣷⡀⢀⣠⣿⣿⡆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣦⠽⠟⣇⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠋⠁
⠀⠈⠙⠛⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠸⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣀⣠⠴⠾⠯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⢡⡄⠀⠀⠆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣀⠠⠶⠛⠉⢤⡇⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠳⣄⣀⣚⣈⡴⠟⠀⠀⣠⠞⠫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⣦⡴⠋⠀⠉⠉⠁⠀⢲⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠀⠀⠀⣀⠀⠚⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⣉⡀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⣉⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠉⠀⠀⢀⣴⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣿⣶⣶⣶⣟⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⡿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣵⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠐
⣿⠰⡇⠀⠀⠀⠀⠀⠀⠀⠠⡷⢀⡖⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠈⠙⢿⣿⣷⣶⣶⠄
⣿⠀⠀⠀⠀⠀⠀⠀⠮⢹⡇⠀⢿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢹⣿⠿⣿⣿⡿
⣿⠀⠀⠀⠀⠀⢔⡆⠀⢸⠇⠀⢸⣧⣤⣄⣀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠟
⣿⠀⠀⡀⠀⠄⠊⠀⠀⣾⡾⠟⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣶⣶⣶⣶⣦⡀
⣿⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣼⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣇⢠⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⢯⢻⢿⣾⢿⣧⣷⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣤⡀⠀⠀⠀⠤⠄⠀⠾⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇
⣿⡀⢀⣀⣾⣿⡿⢰⢃⢸⡈⠆⣾⡿⠐⣻⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣟⣿⡛⠋⠀⠀⠈⣀⣀⣀⠀⠀⠐⠋⠀⢀⣀⣀⣀
⣿⠋⣹⣿⣿⣿⣷⣸⡞⠇⣇⢸⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣇⣆⣷⣿⡋⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉
⣿⠀⠉⠻⣿⣿⣿⣷⣽⣶⣼⡿⠻⣿⠇⠘⠏⠙⠻⠻⠿⠿⠿⠿⠛⠛⠛⠉⠁⠀⣀⣤⡄⠀⠠⠇⠀⠀⠀⠲⢶⣦⡀
⣿⠀⠀⠀⢀⠻⠿⠿⣿⣧⡉⢀⣄⠈⣼⣷⠀⢶⣶⣤⠀⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⠀⠀⢠⣾⡀⠀⠀⠻⣿⠃⠘⣿⡧⠙⠋⠀⠀⠉⠉⠁⢀⣴⠆⠀⠸⠿⠀⠀⠀⠀⣰⣶⡶⠃⠀⠀⠀⡀⠀⠀⡀
⣿⠀⠀⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⡀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠰⠁⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠒⠀⠀⠀⢀⠀⠤⡄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠘⡆⠀⠁⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣁⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⢀⡰⠋⠀⠀⠔⠉⠁⠈⠃⢀⠀⠀⢃⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡶⢚⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⣀⡀⠀⢠⠀⠀⠀⠀⣸⡠⢆⠁⣥⣜⢬⢷⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢇⠀⠀⢀⠄⠴⠁⠄⢀⠂⢠⠃⣅⠸⢁⡠⢢⡄⠀⠀⢠⠇⠀⠜⠉⠉⠨⡓⣛⡣⣀⠐⡄
⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠈⡀⠀⠀⠈⠁⠀⠀⠘⠦⣠⡤⠊⠀⠄⠀⠀⠀⠀⠐⠀⣘⣄⣂⠐⠀⠄
⠀⠀⠀⠀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠘⢢⠀⠀⠈⠚⠒⠀⠀⠀⠀⠀⠀⠁⠀⢀⢄⠈⡢⠤⠖⠨⠀⠂⠒⠛⠚⠻⠁⠁⠈
⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠠⡜⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡦⠆⠀⠉⠠⠀⠀⠀⠀⡀⠀⠁⠀⠁
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⢠⠅⠃⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⠈⠀⠀⠁⠀⠀⠒⠁
⠀⠀⠀⠀⠀⢦⠀⣀⠔⠁⠀⠈⠁⡀⠁⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⠣⡀⠱⡄
⠀⠀⠀⠀⠁⠢⠖⠜⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠐⡤⠀⠀⠇
⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠤⢤⣤⣴⡾⠛⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠚⠛⠻⣧⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠉⠀⠀⣀⣀⣀⣀⣼⡷⠶⠶⠤⢤⣾⣧⣀⣀⣀⣀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡾⠋⠉⠉⠉⠛⢶⡖⠋⠉⠉⠉⠉⠻⣇⠀⠀⠀⠀⠈⣿⠉⠀⠉⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠉⠙⢷⡀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏⠁⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢀⣿⡤⠖⠚⠛⠲⢶⣿⠟⠛⠛⠲⢾⣟⠛⠛⠳⣦⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢹⣇⣀⣀⣀⣀⠀⣠⣿⣀⣤⣤⣤⣤⣾⡇⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠹⡆⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⣧⣀⣠⠟⠉⠀⠀⠈⠉⠻⣿⠋⠁⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣰⣿⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠹⡏⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡿⠗⠚⠛⠛⠛⠻⣆⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣼⣷⣶⣶⣦⣤⣀⡀⢀⣼⡿⠖⠓⠚⠛⠻⣆⠀⠀⠀⠀⠀⠀⠘⣦⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⢀⣀⡀⠀⠀⣼⣷⣤⣤⠤⠤⣤⣤⣾⡏⠁⠀⠀⠀⠀⠉⠙⢻⡟⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠁⣠⠟⠛⠉⠉⠉⠛⠿⣿⠉⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⣰⠟⠛⠻⡇⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢠⣧⣀⣀⠀⠀⡀⠀⣠⣿⣦⠤⠤⢤⣴⠏
⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣠⣤⢤⣤⣤⣿⡷⠖⠛⠛⠒⠂⢤⣾⡟⠉⠀⠀⠀⠈⠹⣿⠁⠀⠀⠀⢀⡟
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢸⣧⣤⣤⡤⠤⠶⠶⠾⣿⡟⠋⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⡾⠁
⠀⠀⠀⣰⠿⠒⠒⠒⠒⠒⠻⢿⡟⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣸⠁⠀⢀⡾⠁
⠀⢀⣼⣧⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣆⣠⣤⠤⠤⠤⢤⣼⣷⡤⠤⣤⣀⣴⠏⠀⣠⠞
⢠⠟⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣄⣀⣀⣀⣿⣤⠤⠖⠒⠒⠺⢿⣏⠁⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠈⣿⣦⡶⠋
⣿⣀⣀⠀⠀⣽⣦⡤⠤⠤⠤⣤⣼⡟⠉⠀⠀⠀⠀⠉⠻⡏⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠃
⣿⠋⠉⠉⠛⠿⣇⠀⠀⠀⠀⠀⠉⢷⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⡰⠃
⣿⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⣿⣄⣀⣀⣀⡀⠀⠀⣸⣃⣀⠀⣀⣀⢀⣾⡿⠶⣶⠞⠁
⢿⠀⠀⠀⠀⠀⠀⠸⣷⣤⠤⣤⣤⣼⣿⠟⠋⠉⠛⠓⠾⣿⠉⠁⠀⠀⠉⠙⠻⣿⠋⠀⠀⠀⠉⠻⣧⢀⡾⠁
⠘⣇⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠉⢳⡄⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⢀⡿⠋
⠀⠹⣆⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⣠⠞⠁
⠀⠀⠙⠦⣤⡼⠁⠀⠀⠀⠀⠀⠀⠀⣴⠿⠛⠒⠲⢦⣄⣠⣿⣄⣀⡀⠀⠀⢀⡞⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠈⢿⡏⠁⠀⠀⠙⢷⣿⣤⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣤⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⣀⣀⣀⣀⣠⣾⣧⣀⣀⣠⡤⠟
⣶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣷⣭⣷⣶⣶⣶⣶⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⠹⣧⠀⠀⠀⠀⠀⣾⠋⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⠁⣶⣶⣶⣿⣶⣶⣶⣶⣶⠉⣶⣶⣶⣿⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣦⣉⣉⣻⣿⣿⣿⣿⣿⣿⣧⣈⣉⣻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣯⠙⢿⠟⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢡⣥⣤⣤⣤⣬⣥⠉⣿⣿⣿⣿⣿⠉⣤⣤⣤⣤⣤⣌⣹⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠈⢛⣛⡙⣋⣛⠋⢀⣿⣿⣿⣿⣿⣶⣿⣿⣿⡿⠟⠛⣉⣠⣴⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿⣿⠿⢁⣴⣶⣶⣶⣦⠀⣾⣿⣿⣿⣿⣿⣿⡇⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣉⣉⣉⣉⣉⣁⣼⡏⢠⣦⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣿⣿⠿⢿⣿⡄⠻⡿⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢠⣶⣦⡈⣠⣶⣦⠈⣿⣦⣄⠘⣿⣿⣿⣿⡿⠠⣿⣿⣿⣿⣿⣿⣿⡇⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣷⣄⠠⣤⣤⠀⣤⣌⡙⠛⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣆⠙⢿⣿⣿⠟⢁⣾⣿⣿⣿⣿⣿⡄⢻⣿⣇⠸⣿⣿⣿⣶⣤⡌⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⣌⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⡆⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠙⠛⠛⠛⠛⠛⠃⠈⠛⠋⠀⠛⠛⠋⠉






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⢠⣾⣿⡿⠿⢋⣶⣶⡌⠿⠿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⡙⣿⡇⣿⡿⣹⣿⣿⣷⡄
⠀⠀⠀⢀⣿⣿⢃⣾⠿⣾⡿⠸⣷⠾⣿⡎⣿⣿⡆⠀⠀⢰⣿⣿⣭⡛⢋⣤⣶⣶⣄⠻⣋⣽⣿⣿⡀
⠀⠀⠀⢸⣿⣿⡧⢙⣿⡆⣾⣷⠄⣾⣟⢱⣿⣿⡇⠀⠀⢸⣿⣟⣛⡃⣾⣿⣿⣿⣿⡇⣛⣛⣻⣿⡇
⠀⠀⠀⠘⣿⣿⡑⣿⣭⣶⣬⢩⣶⣬⣿⡇⣿⣿⡇⠀⠀⢸⣿⣿⠿⣃⡘⢿⣿⣿⠿⣁⡛⢿⣿⣿⠃
⠀⠀⠀⠀⠹⣿⣷⣮⣭⡘⣿⣾⠇⣬⣥⣾⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⢏⣶⡆⣶⣦⠻⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣶⣾⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣷⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠻⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡟⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣿⡿⣡⡚⢘⣈⠻⣿⣿⣿⣧⠀⠀⠀⠀⣼⣿⣿⢿⢿⣟⠆⠖⣻⢿⠿⣿⣿⣆
⠀⠀⠀⢠⣿⣿⣿⣿⠰⣮⣙⢘⣭⡦⢹⣿⣿⣿⡇⠀⠀⢸⣿⣿⣢⡌⠁⢿⡇⣿⠟⠘⢤⣸⣿⣿⡄
⠀⠀⠀⢸⣿⣿⣿⣿⠸⢦⣍⢈⣥⠖⣸⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⡺⠿⢗⡐⣀⠻⠿⣻⣿⣿⣿⡇
⠀⠀⠀⠈⣿⣿⣿⣿⣷⡐⢭⢨⢔⣴⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣂⢡⢀⣿⠇⠻⣷⢰⡉⣸⣿⣿⠁
⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣦⢰⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠙⣿⣿⣿⣿⣷⡃⣙⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣾⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⢀⠀⠀⠀⠀⡀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⢠⣤⠂⣔⣒⣯⠀⣽⠀⣄⡐⠂⢀⣷⢤⡈⠀⢰⣶⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣳⡷⠇⠐⣿⡧⡛⠠⢽⣄⢱⣄⠱⢇⣽⣉⡡⠄⣸⡿⡷⠀⠀⢤⣦⠒⢀⢀
⠀⠀⠀⠀⠀⠀⠀⡴⣿⣡⣎⡗⠦⠀⢿⣾⠛⢤⠸⣿⠛⣿⣷⠀⣾⣿⡕⠐⢾⣿⡧⠀⣄⢸⣷⠊⡁⠈⠅
⠀⢲⢄⡐⣚⣠⣦⣌⣽⢿⣿⡇⠀⠀⣸⣿⣿⡇⠀⣿⠀⣿⣿⠀⣿⣿⡇⣀⣾⣿⡇⠀⠀⢸⣿⠀⢸⡶
⠀⠃⣜⡷⢿⣏⣿⣮⡽⣿⣿⢿⢶⠂⢹⡿⣯⡇⢶⣿⣦⠿⣩⠀⢿⡿⠟⠛⣿⣿⡇⠀⡀⢸⣿⣤⢸⣿⢁
⠀⠀⢿⡄⢺⣿⣿⣥⡿⣦⣽⡷⠯⠎⢹⡆⢺⡡⣸⣿⣤⡤⠈⠚⠛⠆⠀⠀⣼⠻⡇⠈⠙⠿⣿⣉⣻⣿⠌⠁⠤⠐
⠀⠉⢨⡑⢈⣧⣦⣬⡹⣽⣿⠿⣲⡟⣺⣿⣿⠖⠅⣷⣶⠉⠀⣀⣀⡀⠀⢀⠿⣼⡇⠀⠋⢸⣷⠿⢻⣿⣆⠂
⠀⢀⣼⣷⠀⣿⣿⡿⠗⣻⣿⣤⣕⠄⡀⣿⣿⠰⠸⡅⣤⣀⣠⡿⣿⣄⠴⠼⠃⢸⡇⠀⠀⢸⣿⠀⢸⣿⣿⡄⠠⡇
⠀⠰⣺⣿⠀⢿⣻⣠⡏⡟⠛⠿⠙⠂⠂⢿⣿⠀⢠⠿⠿⢧⠀⠀⠘⠲⡀⠀⠠⣿⡇⠀⠀⢸⣿⠀⢸⣿⣿⡇⠀⣿⡆
⠀⠈⠉⠁⠀⠈⠉⣽⠇⠷⣶⣤⡆⠀⠀⢸⡇⠀⢸⠄⠀⠀⠀⠀⠀⠀⠙⢾⣶⣿⡇⠀⠀⣼⣿⠇⢸⣿⣧⣇⠀⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢸⣿⣇⠀⡀⠸⣿⡃⣼⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⡟⠀⠀⠀⣿⣿⠷⠹⣿⣻⠛⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠩⢾⣿⣿⡿⣦⠿⢿⠿⠷⡴⣶⣀⣀⣤⠀⠀⠀⠀⠐⠚⠳⠀⠗⠋⠀⠀⢘⣿⠀⢻⢶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣫⣿⣿⣳⠇⠜⣿⣿⣻⣷⣶⣤⣙⢻⡦⣤⣀⣀⡀⠀⠀⠀⣀⣀⢀⣴⣾⡿⢠⣿⣹⡾⢶⡤
⠀⠀⠀⠀⠶⢿⣧⣬⣤⣶⣿⣿⣿⠻⠠⠐⠃⢁⣾⣿⣿⣿⣿⣿⣿⡿⠼⣿⣭⣭⠀⠠⢽⣿⠧⣴⣦⣤⣵⣾⣍⣿⣷⣶⡶⠖
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⣿⣿⣿⢯⠀⣼⣧⡙⢻⣿⣿⣿⣿⣿⣿⡷⠄⣿⣿⣿⡄⡀⢸⣿⡖⢽⣿⣿⡿⣰⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣥⣄⣼⣿⠤⢻⣿⣿⣿⣿⣼⡿⣷⢠⣼⣿⣿⣇⢠⣿⣿⣏⣿⣿⣷⠗⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢹⣿⣿⣥⣖⣿⣿⣧⣤⣿⣿⣿⠿⠟⠛⠿⠤⠿⠿⠿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠚⠻⠿⠛⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠐⠐⠀⠀⠐⠂⠂⠂⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢈⠀⠀⠀⢠⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠘⢀⠀⠀⠀⠁⡦⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⡄⡈⠍⠀⠀⠌⠀⠠⠁⠄⠀⠀⠁⢀⡀⠀⠀⠀⠂⠀⠀⡅⠀⠀⠀⠖⠀⠀⠡⡆⠀⠀⠐⠖⠂
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⠳⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠑⠀⠀⠄⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠖⠀⢠⣤⠛⠀⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠂⠀⠀⠀⠀⢀⡀⠀⠀⠁⠀⠀⣻⠄⠀⠴⠒⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠐⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠐⠆⠄⡀
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⠂⠈⠁⢀⡀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠠⠀⠀⠂⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⢀⡀⠀⠤⠀⠀⠀⢰⠅⠀⠀⡀⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡴⠇⠀⠀⠠⠀⠀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⠐⢠⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠐⠀⠂⠐⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡀⠚⠃⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠊⠉⢠⠐⠀⠀⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⡄⢠⠖⠀⢢⠔⠐⠄⠀⡦⠂⠐⠄⠀⠀⠀⠀⠂⢀⡠⠀⣀
⠀⠀⠈⣡⠀⠀⠀⠀⠟⠂⢀⠀⠀⠘⠀⠸⠀⠀⠸⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢰⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁
⠀⠀⠀⠐⢂⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠄⠂⠠⠃⠀⠀⠀⠀⠀⠀⠈⠄⠔⠀⠃⠀⠀⠠⠐⠀⡀⠈⠀⠂⠀⠀⠀⠀⠀⠀⡀⠔⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠩⠔⡀⠀⠑⣀⠰⠀⣄⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣠⠀⠀⠀⡀⠀⠘⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⡘⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠐⡵⠀⠀⠀⠀⢀⢀⡏⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠠⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠆⠀⠀⠀⠀⠀⠀⡦⠀⠈⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠄⡊⡆⡓⢆⠀⠀⢷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠂⠠⠀⠀⠀⠁⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢧⡀⠤⠏⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⢀⡇⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⡄⠈⣁⣰⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠠⠦⠤⠬⢀⡀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠙⠂⣉⡀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⠤⡀⠀⠀⢠⠄⠀⠈⠷⠀⠀⠁⠀⠃⠆⠀⠉⢡⠀⣄⠀⠃⢤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠃⠀⠀⠐⠀⠀⠀⠈⠉⠁⠀⢀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠌⢣⠤⠄⠀⠀⠀⡤⠀⠀⠀⠀⠀⠁⠄⠀⠀⠤⠀⠀⠃⠀⠀⠉⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠁⢷⡀⠀⠐⠀⡀⠀⣀⡀⠀⠐⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⢰⡃⠈⡀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠶⣄⠀⠀⠀⠀⢠⡶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣿⣀⣀⣹⣀⣀⣀⣀⣿⣀⣀⣸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⢿⠉⠉⠉⠉⠉⠉⠉⠻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⠓⠒⠀⠀⠀⠀⠀⠈⠛⠒⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⣛⠛⢛⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡤⠽⠷⠿⠤⣤⠀⠀⠀⠀⣠⡤⠤⠤⠤⢤⡄⠀⠀⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⣤⣤⣤⣤⣤⣾⠀⠀⠀⠀⣙⣀⣀⣀⣀⣤⠴⠞⠋⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⡇⠀⣀⣴⠏⠉⠉⠉⢻⠉⠀⠀⠀⠀⠀⢸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠛⠛⠛⠛⠛⠃⢸⠋⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⠶⠶⣤⡴⠶⣤⠘⠷⣼⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠀⠀⠈⠀⠀⣸⠇⠀⠈⠳⣶⠶⢶⡿⠶⣤⣀⡀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⣄⠀⣀⡴⠋⠀⠀⠀⠀⠹⡆⠀⢿⡀⠀⠈⠙⠛⠶⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠠⠂⡆⡴⠐⠄⠀⠰⠀⠐⡀⠂⠠⠠⠀⠠⠄⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣴⣿⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠛⢿⣆⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢋⣴⡿⠋⠀⠀⠈⢧⡻⡄⠀⠀⣤⡆⠀⠀⢀⣴⠟⠁⠀⠀⣠⣴⠿⡟⠛⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⡾⠋⠀⠀⠀⠀⠀⠈⢧⢱⡀⠀⠛⠃⠀⣠⡾⠋⠀⢀⣴⡾⠋⠁⠀⠁⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠞⣡⣤⣤⣤⣄⡀⠀⠀⠈⣧⢧⠀⢀⣠⣾⣿⣤⣤⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢠⡾⠛⠋⠉⠉⠛⠻⣷⣄⠀⠈⢿⣷⢟⣛⣩⣭⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠙⠇
⠀⠀⠀⢀⣀⣤⣤⣶⣶⠿⠟⣧⡈⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⢸⠁⠿⠛⢛⣻⡷⠀⠀⣰⣶⣶⣤⣤⣴⣶⣶⣶⣶⣾⠿⣦⣄⣀
⠀⠀⠀⣿⠛⠻⠿⠿⠷⣶⣶⣾⣿⣷⣶⣶⣤⣤⣤⣤⣤⣴⣾⣿⣾⣿⣿⠟⣿⠛⠁⠀⠿⠿⠿⠛⠻⢿⣿⣍⣩⣭⣭⣥⣤⣶⣿⣿⣿⣿⣷
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠉⢉⣽⠿⠋⣡⡿⢫⡿⠋⠁⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡟⠛⠛⠛⠋⠉⠉⠉⠉⡏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⣼⠋⢠⡟⠀⠀⠀⠀⠀⣶⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⡼⠃⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠘⠻⠿⣿⠿⣿⣿⣿⣶⣶⣶⠟⠁⠀⠀⢸⠁⠀⡼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⣾⣿⣿⣿⣿⠇
⠀⠀⠀⣶⠀⢰⡇⠀⠀⠀⠀⠈⣹⠏⢀⣠⡆⠀⠀⠀⣸⢷⣶⣶⣦⣤⣄⣀⡀⢸⡇⢀⣠⣤⣾⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⢠⣇⣴⠿⠋⠁⠀⠀⢀⡏⠀⠀⠉⠉⠛⠛⠿⢿⣿⣿⠿⠟⠛⠉⢹⠀⠀⢠⣿⣦⠀⠀⡸⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡟⠁⠀⠀⢸⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡆⢀⡟⠀⠘⣇⣰⠇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢡⡾⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⢿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⣿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⣶⣦⣤⣄⣀⣀⠀⠀⠀⣸⡇⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⣶⣾⣧⣶⣶⡿⠿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⣀⢀⣐⡍⠑⣠⢐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢄⡠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢤⠛⠶⢤⠷⢸⡷⠞⠂⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⣀⣾⢦⣷⠠⠂⢉⡳
⠀⠀⠀⠀⠀⠀⠓⡐⢲⡎⠋⣍⠉⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⢸⡧⠓⢁⠋⢉⠉⠉⠉⠉
⠀⠀⠀⡀⠀⠀⠠⠀⢨⣇⢠⠈⡑⠀⠄⠀⣀⠀⠀⠀⠀⠀⠀⡂⡒⢄⢀⡠⠐⠂⣻⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⢰⣁⡀⢸⢀⡀⢡⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠀⡌⠈⠁⠀⠈⠃⠀⡁⠀⠀⠀⠀⠐⠀⠀⠊⢉⡹⢢⣍⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⠀⠀⠀⠈⠀⠈⣡⣬⠈⠁⠉⠈⢃⣀⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢁⡈⢈⠉⡇⠀⡦⠉⢀⡲⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠐⠀⠈⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠈⠀⠀⢡⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡇⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⣤⣀⠅⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⠂⠚⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⡁⠀⠀⣀⡸⠀⢸⣀⣀⣀⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⣄⠀⠦⡀⠀⢀⠀⠟⡀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⢤⣀⠀⢀⣀⣀
⠀⠐⢄⠂⠈⠒⠁⠀⠀⣻⡆⠀⠞⠤⠊⠠⠖⠡⠂⠴⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠦⠶⣛⢉⡩⢒⡛⠛⠻⣟⡓⠁⡀⠀⠑
⠴⠂⠀⠒⠒⠰⠷⠾⠲⠿⠿⠿⠒⠒⠢⠔⢲⠄⠈⠰⠐⡆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⡇⠀⠈⡀⠀⠀⠀⢆⠀⠀⡄⢠⠊⠉⠀⠁⠈⢛⠂⠀⠀⢀⡀
⠐⠀⠂⠉⠀⡀⠀⠀⠀⠰⠀⡎⠐⠂⠀⠀⠀⠂⠲⠘⢳⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠜⠈⠉⣷⡋⠀⠻⠈⠁⠉⠈⠾⡅⠈⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠀⣠⣄⡀⠀⠀⡀⠄⠀⠘⡆⠀⠀⠣⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡧⠸⣷⠟⠃⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠉⡻⡆⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠩⠁⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⣠⡆⠐⠃⠇⠤⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢸⡇⠀⢰⡶⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⢀⢤⠤⠤⢄
⠀⠀⠀⠀⠀⢰⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣑⣦⡋⣐⡠⠂
⠀⠀⠀⠀⡈⠉⠘⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⡀⢀⠀⡀⠐⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠉⠉⠛⡿⠞⡋⠀⠀⠀⢀
⠀⠀⠀⠠⠇⠀⠇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠒⠲⠶⠶⢧⡀⣅⡞⢁⠴⠃⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⡀⠀⠠⠖⠠⠄⣀⠀⢰⠂⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠴⠂⢤⠀⠀⠉⠁⠀⠀⢠⠄⠙⣀⠄⠀⠀⠈⠉⠈⠁⠓⠀⠤⠆⣀⠀⠇⠀⡨⠦⠛⠉⠁
⠀⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⣀⠀⠈⠀⠐⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢰
⠀⠀⢠⠹⣧⡴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⢲⠖⠊⠀⡇⠀⣀⠂⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⠀⠀⠈⠐⠁⢲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠸⠀⠀⠀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⠀⢠⢀⠀⠴
⠀⠈⠀⠂⠀⠄⢻⣷⡀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠀⠈⠙⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠘⠻⣶⣦⣄⡀⠀⢻⣷⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣄⢻⣷⡶⠿⠿⠿⠿⠿⢿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣤⣀⣀⣀⣀⣀⣀⣀⡙⣿⣧⣿⣷⣶⣶⣤⣤⡀⠀⢀⣨⣼⡿⠟⠻⠿⣶⣦⣤⣀⡀
⠀⠀⠀⣀⣀⣠⣤⣴⣶⣾⡿⠿⠟⠛⠛⠿⠿⣿⣿⣿⣿⠿⠿⣿⣿⣿⡿⠛⠉⠉⠙⠛⢿⣿⡟⠋⠁⠀⠀⠀⠀⠀⠈⠙⠛⣿⣷⣦⣤⣀
⣶⣿⣿⣿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣻⣿⣦⣬⣿⣿⡀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣶⣶⣶⣶⣶⣶⣶⣾⣿⣯⣁⡈⠉⠛⠿⣷⣦⣄⡀
⣿⣯⠙⠿⣷⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⡾⠿⠛⠛⠉⠉⠉⢛⣿⣷⣄⠀⠀⠀⠀⠀⢀⣿⣯⣀⣤⣤⣄⣀⠀⠀⠈⠉⠛⠿⣿⣦⡀⠀⠀⣉⣹⣿
⣿⣿⠀⠀⠈⠙⢿⣿⣶⣶⣶⠿⠻⣿⣉⡁⠀⠀⠀⠀⠀⠀⣀⣤⣾⠏⣹⣿⢿⣷⣶⣶⣾⣿⣿⡿⠛⠋⠉⠛⠻⣿⣦⡀⠀⠀⠀⠈⣿⣿⣾⠿⠛⠋⢹
⢿⣿⠀⠀⠀⠀⠀⢽⡿⣿⣶⣄⢸⣿⠿⢿⣿⣿⣿⣿⣿⠿⠿⠛⢁⣴⠟⠁⠀⣠⣿⣏⠉⢹⣿⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡷⠀⠙⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⢰⣿⠃⠈⠳⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⠏⠀⠀⠀⣿⣿⣴⣶⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⠀⠀⠀⠀⢸
⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⣿⣦⣤⣶⣶⣿⠟⢋⣁⣼⣿⠀⠀⠀⢸⣿⡟⠛⠉⠀⢸⣿⠻⣿⣦⣄⡀⠀⠀⣀⣤⣶⣿⠟⠁⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡿⠟⠛⠋⠙⢿⣿⡿⠿⣿⣿⠀⠀⠀⠘⣿⣇⠀⠀⠀⢸⣿⠀⠉⠻⠿⣿⣿⢿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢿⣿⡀⠀⠀⠀⠹⣿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⣧⡀⢀⣀⣀⣬⣿⣦⡄⠐⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠘⢿⣿⡿⠿⠛⠛⠉⠉⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣧⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⣿⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢻⣿⡀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⢿⣧⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠈⢻⣷⡀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿
⠀⠀⠀⠀⠻⣿⣄⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⢀⣠⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣦⠀⠀⢸⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⢀⣠⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢸⡇⠀⠀⠀⠀⢸⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣠⣴⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣄⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄⠀⢸⣿⡏⠀⠀⣀⣠⣴⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⣿⣷⣶⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡿⠟⠋⠁
⣿⣿⣿⠿⢿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠤⠤⣄⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠛⢄⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⡆⠈⢳⡀⠀⢹⡤⠴⠖⠒⠒⠒⠂⠂⢄⡜⠁⠀⠀⣀⣠⣦⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠱⡚⠉⠙⣄⠀⢇⠀⠀⠀⠀⠀⠀⣠⠋⠀⢀⣴⠋⠁⠀⠘⣆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠊⠁⠀⠹⣦⡱⣄⠀⠘⣆⠘⡄⠀⠀⠀⠀⡴⠁⣠⠔⠃⣀⡽⢦⣀⠀⢸⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⡳⣄⠘⡆⢹⣦⡀⢀⠞⣡⣞⡥⠖⠋⠁⠀⢀⣩⣷⣄⡇⠀⢰⠇
⠀⠀⠀⠀⠀⠀⢻⡲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣿⡞⠉⢩⡿⠛⠉⢁⣀⡤⠴⠚⣋⣉⣀⣀⣼⠇⣠⠞
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⣿⡀⠀⢸⠉⠒⠚⠛⠛⠛⠛⣉⣉⣩⡭⠥⠖⠋⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡄⠀⠀⠀⠉⠢⣄⣀⡤⠴⠖⠋⠉⠀⠀⢀⣨⠽⠻⡏⠁⠉⠉⠙⠯⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣩⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⢨⡇⠀⠀⠀⣀⣠⠴⠚⠋⠁⠀⠀⠈⢢⡀⠀⠀⠀⠈⠳⢦⡀⠀⢀⣀⠤⠒⠋⠉⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠢⣀⠀⢸⡇⠀⠀⣾⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⣀⡤⠖⢻⡟⠉⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠑⠼⣇⠀⠀⣿⠈⠑⢦⡀⠀⠀⠀⠀⠀⢀⠠⠄⠊⣯⠁⠀⠀⢸⡇⠀⠀⠀⠀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡏⠻⣤⡇⠀⠀⠀⠈⠢⢀⠠⠐⠈⠀⠀⠀⠀⢸⠄⠀⢀⣸⡧⠤⠶⠚⠉⠁⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⢈⡗⢄⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⣀⡼⠓⠛⢹⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣸⠀⠀⣼⠁⠀⠈⠒⢦⡀⣿⠀⣀⣠⠴⠚⠉⠀⢻⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⡟⠀⢰⠇⠀⠀⠀⠀⠀⠉⡟⠉⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣸⠇⠀⣾⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢸⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀⡇⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠸⣇⠀⢀⣀⣠⣤⠼⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⡀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⡤⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢸⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢸⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠚⠁
⣿⣏⢹⣟⠻⣿⢛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⢀⠈⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⢶⡄⠀⠀⠱⡀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⣀⣀⣀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡷⠤⣤⣙⣦⣄⠀⠙⣆⢣⠀⠀⠀⠀⠀⠀⡠⠃⢀⠔⠉⠀⠀⡘⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡈⠙⠻⠷⣤⣈⢙⣠⣤⣤⡀⡠⠜⡡⠔⢁⣀⣀⣀⣔⣁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠠⠀⡀⠀⠉⠛⠒⠒⠒⠊⢹⣀⠀⢡⡏⣐⠛⣉⣉⣁⣀⣀⣀⠀⠉⠳⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠤⣄⣀⠠⢤⣤⡄⠁⠒⠒⠒⢒⠶⠾⣿⣆⣸⡇⠀⠙⣦⠨⢭⣉⣀⠉⠀⠁⣰⣧⠴⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠈⠁⠈⠒⢶⡎⣧⢀⣈⣉⣉⣁⠰⡇
⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠠⠤⠤⢄⣀⣀⡤⠖⠇⢸⣦⡀⠀⠀⠉⢻⡇
⠀⠀⠀⠀⠀⢀⠁⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⡸⠀⠸⡇⠉⠛⠲⠒⠋
⠀⠀⠀⠀⠈⠓⠒⠛⡇⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣸⠃⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠃⠀⠀⢸⡃⠀⢠⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⡾⠉⠢⣸⠀⠀⢨⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠠⠷⠒⠛⢆⢠⠇⠀⠀⢹⠀⠀⠸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⠀⠀⠀⢸⠀⠀⠀⠈⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⠈⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡰⠀⡆⠀⠀⢸⠀⠀⠀⠘⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡜⠀⠀⡇⠀⠀⢸⠀⠀⢀⠠⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⣠⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠢⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⢤⣄⡀⠀⡇⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠁
⣿⣏⢹⣟⠻⣿⣟⠟⡻⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠂⢤⠔⠂⠉⠉⠉⠉⠀⡢⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢀⠔⢈⠤⢁⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠴⠶⠒⡄⠀⠐⠆⠠⠁⠀⠀⠀⢢⡁⠀⠇⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠔⠉⢉⠀⠀⣄⡀⠀⠈⠂⣤⠀⠃⢸⡄⡀⢰⣧⠐⡄⢀⣤⣔⠚⠛⠧⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⠦⠀⠙⠓⠁⠀⠀⠀⢸⠀⠀⠁⠀⠈⠓⠀⠐⠈⠀⠁⠂⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⡆⠀⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⣠⡆⠀⢀⠀⢆⠀⢠⣀⣀⣠⡄⠀⠇⠠⡀⢠⣄⣠⡴⢂⠟⠀⠀⠀⠀⠀⠀⠰⠀⡤⠁⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠱⣶⠎⠩⠃⢀⡼⠀⠈⠢⣀⠉⠁⠀⠀⠀⠀⠀⠈⠢⢌⠉⠁⠀⠀⠀⣨⠀⠀⠀⡴⠁⠘⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⢀⠈⠀⡀⠐⠊⠃⢀⠐⠚⡈⠑⠂⠀⠖⢒⡇⠀⠀⢀⠔⠉⠘⠃⣾⠀⢁⣀⠴⠊⡀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⣀⠀⡀⠀⠀⣄⠀⠈⢢⡇⠀⢀⡌⠁⠈⠁⠉⢩⠀⠀⠈⢢⡄⠀⠀⠀⡀⠀⠄⢁⡠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡀⠀⢇⠀⠀⠿⡈⠇⢀⠀⠘⠣⢀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⣡⠐⠄⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⡠⠐⠒⠒⠒⠀⣀⠠⡀⡌⠀⠐⠀⡝⢀⡂⠀⠀⢂⠩⠀⠀⢠⠀⠀⠀⠀⢀⠀⡈⠀⠀⠠⠛⠢⢰⡆⣀⠀⡀⠀⠈
⠀⠀⠐⠀⠀⠠⠁⢃⠈⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠑⠸⠀⠀⠀⠈⠁⠀⡁⠂⠐⠀⠒⢈⠡⠔⠁⠀⡑⠢⠤⣄⡌⢉⣡⡀⠌⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡰⠀⠁⡀⠀⡀⠉⠀⠀⠀⠂⠈⠀⠀⢄⡀⠀⠀⠈⣀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⠘⡁⠀⠁⠀⠀⠀⠀⠀⠀⡀⣈⠅⠀⠀⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠰⠀⠀⠀⠀⠑⠀⠀⠀⠀⠂⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⠠⠐⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠑⠢⠄⣀⡄⡀⡤⠒⢈⠃⠀⠀⠀⠀⣀⣀⡀⠠⠀⠠⠀⢀⠀⣀⠀⠄⠀⠀⠀⠀⠆
⠌⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠋⠙⠉⠉⠉⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠇⠀⠘⠀⠀⠀⢠⠀⠀⠀⠀⠂⡄⠀⣀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡆⠀⠀⠀⢸⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢀⡆⠀⠀⠀⢸⠘⠀⠀⡇⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⢠⣅⡀⠀⠠⠜⠀⠀⠀⡇⠀⡆⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢉⡀⠀⠀⠀⠀⣀⠒⠠⠤⠆⠰⠀⠀⢀⠆⠀⣠⠀⠀⢰⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠈⠁⠄⠊⠀⢠⡁⠠⠀⠁⠀⢄⣁⡢⠑⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣀⣀⣀⣿⠀⣀⣤⣦⣤⡄⢠⣄⣤⣦⡄⠀⣤⣠⣤⣤⠀⣤⡄⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠋⢻⡟⢰⡟⠁⢀⣿⠁⣼⠯⣦⢸⡟⢸⡟⢶⢄⣿⢀⣿⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠿⠇⠘⠿⠶⠟⠿⢀⣿⠶⠶⠟⠁⣼⠷⠶⠾⣋⡈⠻⠶⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⣦⡀⠰⠷⠀⠀⠀⠀⠀⣠⡀⠀⢀⣿⠸⠏⠀⠀⠀⠀⠿⢰⡗⠀⠙⠻⠶⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣤⣴⡟⠁⣾⡆⢰⣷⡶⠗⠺⣿⠿⠇⢸⣧⠶⢿⡆⢀⣴⠾⠻⣿⠇⣠⡶⠟⢷⡶⢀⣾⠀⠀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃⠀⣹⣿⢀⣿⠀⣼⡏⠀⠀⢸⣏⠀⠀⣾⠇⠀⣸⡇⢸⣇⠀⣰⣿⠀⣿⡁⢀⣾⡇⢸⣿⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⠘⠋⠀⠛⠀⠀⠀⠈⠛⠛⠀⠛⠀⠀⠛⠁⠈⠛⠛⠙⠋⠀⠙⠛⠋⠛⠁⢈⠛⢋⣽⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⠁⠉⢉⡉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⠀⠀⠀⠀⠐⠂⠀⠐⠀⠀⠄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠀⢀⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⣀⠄⠀⢠⠀⠀⡠⠀⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠠⠀⠂⢀
⠀⠀⠀⠀⠀⠄⣈⣁⠠⠀⠐⠀⠫⠛⠀⠒⠂⠀⠀⠴⢀⠀⣁⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⢘⡃
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠂⠠⠀⠀⢁⠀⠀⡾⠀⠀⠈⠀⠀⠈⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠐⢲⡄⠀⠀⢤⠄⠀⠐⠀⠁⠀⠀⠀⠄⠀⠂⠈⠀⠘⠀⠀⠁⠀⠀⠂⠀⢀⠀⠂
⠀⠀⠀⡀⠃⠀⠠⠀⠀⡀⠀⠠⠀⠀⠈⠀⢠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠐
⠀⠀⢠⣻⠀⠀⠀⠀⠀⠁⠀⠑⡂⠀⠁⠀⠘⠀⠀⠀⡆⠀⠉⠀⠀⠀⠀⠀⡀⠢⡀⠴
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠄⠀⠉⠀⠀⡇⠀⠰⠀⠀⠀⠁⠵⠶⠐⢠⠘⠀⠀⠀⡀⢠⠀⠀⠐⠁⢀⡀
⠀⠀⠀⠀⠀⠁⠀⢃⠉⠁⠀⠈⠀⠌⢀⡀⠁⠀⠀⠐⠁⠀⠀⠀⠊⠀⠀⠰⢤⣀⠈⢁⡤⠞⠁
⠀⠀⠀⠀⠰⠆⠀⠂⠀⠐⠀⠘⠀⠀⠀⠀⠀⢀⠀⠀⢀⠄⠐⠈⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠠⠧⠀⠀⠀⠀⣀⠀⠀⠂⠀⠀⠚⢃⠀⠈⠓⠀⠠⡤⠒⠔⠀⠛⠒⠀⠋
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⣀⢀⡀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠒⠀⢀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠐⠒⠛⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠡⠀⠀⠐⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣤⠄⠈⠀⢀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⣀⡀⠀⠀⠐⡀⢀⣠⠀⠀⠀⠰⡷⡄⠀⡀⢦⡀⠆⠠⠆⢈⣧⢸⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣷⡄⣠⣾⣿⣿⣷⣄⠀⠀⠀⠀⠁⠀⠀⠘⠃⢤⠍⢧⣿⣧⠉⢻⡄⠀⠶⣿⡿⠈⢣⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠒⠲⢛⡃⠒⠠⣄⠀⣺⡿⠂⠘⢧⣼⠃⠀⢀
⠀⠀⠀⠀⠀⠀⠘⢯⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⡄⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣳⣄⠈⠙⣿⣧⠱⣄⠈⣿⣀⠀⢠
⠀⠀⠀⠀⢀⣤⣾⡄⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⢈⠁⠀⣀⡀⠈⠂⠂⠖⠈⠉⠑⢻⣿⣄⣿⡀⢹⡶⢠⡯
⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣪⣌⣑⣈⣃⡀⠀⠀⠀⠀⢀⣀⣴⣿⣿⡙⢷⣈⣿⣼⣧⠖
⠀⠀⠀⢹⣿⡿⠟⠁⠐⠿⢷⣮⡉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⠀⢀⡽⠛⣶⡶⠾⣿⣿⣄⣿⣿⠁⢀⣿⣿⣿⠗⠂
⠀⠀⠀⢸⡟⠀⢀⠀⠀⠀⢠⣿⣷⡀⢸⣽⡿⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡟⡄⡩⠴⡞⠋⡀⠀⡀⠀⠈⢹⣿⣧⠈⣿⣿⡟
⠀⠀⠀⠿⠶⠒⣿⣧⣾⣿⣿⣿⠿⠇⠜⠻⡄⠀⠿⣿⠟⢿⡿⠿⣿⠿⢛⠛⠻⠋⠊⠟⠀⠂⠒⢿⡐⠛⣺⡟⢿⢀⡏⠑
⠀⠀⠀⠀⠀⠀⢀⣿⣶⣦⣬⢀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠛⠛⠚⠛⠛⠛⠉⠋⠉⠁⠀⠀⠀⢀⣸⣿⣿⣿⡧⠀⢻⠶
⠀⠀⠀⠀⠀⣀⣄⣤⠛⢉⣈⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣠⣄⣀⣶⣿⣿⣿⠿⠟⣿⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠛⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠈⠉⣉⢉⢛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠀⠀⠀⠀⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠘⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣉⡁⠀⠀⠀⠀⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⣠⣾⣿⣿⣷⣄⣀⣀⣠⣠⣀⣀⣠⣤⣤⣾⣿⣿⣆⣠⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣉⣙⡛⠻⠏⢉⣁⣀⠹⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠈⠙⠛⠉⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣴⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⢒⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶
⠀⠀⠀⠀⠀⣼⣟⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢒
⠀⠀⠀⠀⢐⣻⠵⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣛⣭⡿⠿⣟⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⡭⠗⠂
⠀⠀⠀⣠⣿⣓⣫⣭⠥⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢛⣋⣛⣉⣍⣁⡀⠀⠀⠀⠀⠀⠀⠀⠠⠖⣊⣭⢄⣀
⠀⠀⠀⠴⠛⠉⠁⠀⠀⠀⠀⠉⠙⠒⠒⠂⠀⠀⠀⠀⠈⢋⡉⠉⠭⠤⠖⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠤⠤⠤



⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢤⠙⠂
⠀⠀⠀⠀⠀⠀⣨⣠⣶⣾⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄
⠀⠀⠀⠀⠀⣘⣿⣿⣟⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠒⠢⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⣠⠴⠂
⠀⠀⣀⣶⠶⠟⠋⠋⠉⠉⠉⠉⠁⠐⠒⠦⠤⠀⠀⠀⢠⠴⠞⠩⢭⣉⠈⠉⠓⠄⠀⠀⠀⠀⠀⠀⠀⢀⡁⠒⡈⠀⡀⢠⣀
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠐⠂⠙⠂
⠀⠀⠀⠀⠀⠀⠒⠒⠻⠼⡁
⠀⠀⠀⠀⠀⠘⡂⢀⡄⠈⠉⠁
⠀⠀⠀⠀⠀⠘⠛⠻⠤⠄⢶⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠠⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⡨⠆
⠀⠀⠀⡀⠀⣠⠂⠀⠶⢷⡝⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⢀⣀⠴⠶⢤
⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣵⣄⣤⣶
⠀⠀⠀⠶⠠⣰⠆⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡁⠀⢄
⠀⠀⠀⠀⠔⠁⠀⠒⣀⠀⠶⣀⡀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠛⠁⢀
⠀⠐⠧⠄⡤⠀⠀⢀⠙⣀⡀⢀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣘⢀⠤⠚⠐⠾
⠀⠀⠀⠀⠸⠇⠂⠉⡀⢀⡉⠹⠄⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠐⠪⣭⣋⠠⣤⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣤⣀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣶⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠢⡀⠀⠈⠪⡉⠐⠂⠔⠁⠀⠀⡠⠊⡹
⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠔⡲⠏⢉⡒⠈⢐⣄⣀⡈⣦⠎⣀⣀⡔⠪⠔⣊⠉⠲⢄⣀
⠀⠀⠀⡤⣒⠊⠉⠀⠀⠀⠀⠄⢀⣀⡀⠁⣂⣬⠭⡿⣤⠼⡯⣥⣄⠈⢁⣀⡀⠀⠜⠀⠉⠉⠒⡢⡄
⠀⠀⠀⡆⠀⠉⠑⠢⠤⣠⡄⣒⡩⠥⠒⠉⠁⠂⠉⡠⠋⠢⣀⠁⠀⠁⠐⠢⠍⡒⢤⡤⠔⠊⠁⠀⡆
⠀⠀⠀⡇⠀⠀⠀⠀⢀⠗⠋⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠈⠣⡇⠀⠀⠀⠀⡇
⠀⠀⠀⠢⣄⣀⢀⡴⠁⠀⠀⠀⠀⢀⠔⠉⠉⠒⠢⢄⣀⣀⡤⠐⠊⠉⠢⡀⠀⠀⠀⠈⠂⠀⣀⡤⠂
⠀⠀⠀⠀⡇⠀⢋⠀⠀⠀⠀⠀⡠⡇⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⠀⠀⠀⡗⢄⠀⠀⢠⠛⠉⠀⡇
⠀⠀⠀⠀⡇⠀⠈⢆⠀⠀⢀⠜⠓⡧⢄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡠⡷⠊⠢⡴⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠈⠣⡤⠃⠀⠀⡇⠀⠈⠉⠒⠤⢄⣀⣃⡠⠔⠊⠁⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠧⢄⡀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⢀⡠⠇
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠧⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠧⢄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡠⠗⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⣀⢸⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁


⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠓⢤⠀⠀⠀⠀⢀⠴⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠔⢲⣄⢳⠀⠀⣰⠃⢀⠴⠒⠦⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠙⢎⢧⣠⠇⢰⣋⠴⠊⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⠛⠒⠚⠛⠒⢺⠆⠀⠀⢯⠤⠤⠶⠥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠛⠛⢦⠀⠀⠉⠀⠀⡴⠉⠉⠳⣆⠀⢸⠁⠀⠀⣀⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠤⣄⡀⠀⠀⠀⡼⢁⡔⡂⠀⡧⠀⠄⠀⠀⡇⠠⠙⣦⡘⣆⡼⠀⡴⠋⠀⣀⣀⠙⡆
⠀⠀⠀⠀⠀⡜⢁⣠⣄⡀⠉⢦⡀⠀⡇⣾⠃⠀⡰⠃⣀⣄⣀⣀⢱⢄⡀⠘⢧⢳⣇⡜⢀⡴⢋⡠⠊⢀⡇
⠀⠀⠀⠀⠀⢣⡈⠑⠤⣉⣳⡀⠱⠶⢣⣡⠴⠞⠓⠉⠁⠈⣇⠈⠉⠉⠙⠓⠲⠆⠍⠁⣜⣉⣁⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠙⠢⢤⣤⠤⢼⠂⠈⠈⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠉⠉⠙⠓⢢
⠀⠀⠀⠀⠀⢰⠒⠉⠁⠀⠀⠀⠆⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⢀⡀⣇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⢡⠀⠀⠀⠆⠂⠈⠀⠀⠀⢩⠀⠀⠀⠙⠀⠀⠀⠗⠀⠠⠄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣄⠤⠀⠀⠈⠈⠄⠀⠀⠰⡀⠀⠀⠀⠈⠙⡆⠀⠀⡄⠀⠀⢀⣀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⠠⠀⠚⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⡈⠀⠀⠁⠀⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⣆⣀⣀⠴⠴⠒⠉⠉⠒⠒⠲⠤⠤⠴⣄⣀⣀⡀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡤⠴⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⢀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⡎⢇⠀⠀⢒⠄
⠀⠀⠀⡠⠔⠒⠒⢢⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠡⡀⠀⠀⢀⠜⠀⠈⢆⠀⠀⠀⣀⠤⣀
⠀⠀⠀⡱⠀⠀⠀⠘⢲⠶⡏⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠂⠀⠀⢊⠀⠐⢳⠂⠁⠐⠲⡛⠀⠰⠁⠀⠈
⠀⠀⠈⢳⠀⠀⠀⠀⠀⡆⢣⣴⠃⠀⡄⠀⡠⠀⠀⠀⠀⠀⠀⠀⢐⠇⠀⠀⠀⠐⣷⣖⡁⠀⠀⠀⠀⣑⣤⠀⠁⠒⠈
⠀⢠⠤⠶⠀⠈⠁⠈⠄⠉⠁⠈⡆⠰⠀⠀⠀⠂⠀⠀⠀⠀⠀⠒⡡⠤⠀⠄⣀⣀⡤⠋⠀⠀⠀⠀⠀⠈⠢⢄⡀
⠀⢇⠀⢀⡁⠀⠀⠀⠐⠀⠀⢐⡍⡂⠀⠐⠀⣊⠀⠀⠀⠀⠀⢊⠤⠀⠀⠀⠀⢈⠽⠋⠙⠀⠂⠠⠐⠀⠛⢍
⠀⢰⠁⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠠⠀⠀⠒⠀⠀⠀⠒⠁⠀⠀⠠⠀⢌⢑⡶⠀⠀⠀⠀⠀⠀⠀⠴⣖⠉
⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠄⠀⠠⠤⠤⢤⠤⠀⠀⠤⠵⠂
⢹⠀⠀⠒⠒⠒⠁⠀⠈⠉⠀⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠤⠀⢀⣀⡀⠀⢀⣀⠀⠀⠀⠀⠚⠀⠀⠀⡇⠀⡀⢀
⠀⠀⠀⠄⠀⠀⡇⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⠀⠃⠀⠀⠀⢀⠀⠀⣀⣀⠤⠀⢀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⡄⠀⠀⠀⠁⠀⠀⢀⠀⠀⠀⢠⠀⠀⠀⠀⢸⠀⠀⠁
⠀⢧⣀⠄⠀⠀⢣⣀⠄⠀⠀⠀⠳⠤⠂⠘⠦⠄⠀⠘⠦⠔⠀⠈⠳⠤⠤⠂⠈⠳⠤⠤⠔⠊⠀⠳⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠐⠘⠒⠈⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠡⠐
⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⡀⠐⠸⡂⠀⠀⠀⠠⠈⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠃⠀⠀⠄⠀⠆⠀⠀⠂⠀⠀⢀⠒⠒
⠀⢀⠀⠀⠀⠄⠀⠀⠄⠐⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⡄⠀⡤⠀⠀⠶⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⡀
⠀⠀⠈⠀⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣠⡆⢀⠉⠀⠀⠀⠩⠀⢂⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠂⠐⠂⠀⠠⠎⠋⠁⠀⠀⠄⠀⠀⠀⠀⠘⠐⠀⠀⠄
⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠁⠡⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠒⠠⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠁⠀⠃⠐⠀⠘⢀⠀⠀⠀⠰⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⠉⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠄⠤⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⠀⢠⠐
⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀⡀⣀⣀⣀⣀⢀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⠀⠀⠀⠒⠀⠀⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠉⢻⠿⡟⡿⠿⡿⠿⠿⡿⢿⣿⠿⢿⠿⡿⠿⠟⣻⠿⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣾⣷⣶⣇⣷⣶⣶⣶⣶⣶⣾⣿⣠⣶⣶⣷⣾⣷⣿⣤⠀⠀⠂⠀⠀⠩⠀⢀⠀⠀⡀⠀⠀⠀⢠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⡉⠉⡍⠉⠉⢉⠩⡉⠀⠀⠀⠀⠀⠀⠀⠉⠔⠀⠀⠀⠉⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠙⠌⠬⠇⠆⠀⠀⠐⠀⠉⠀⠀⠠⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⣄⠄⠀⢀⠠⡆⢠⣤⢀⠀⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⢀⠰⡈⠀⠈⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠈⠀⠀⠀⠐⠒⠐⠬⠥⠌⠐⠄⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢐⠀⠀⠀⠁⠀⠉⠈⠈⠁⠀⠀⠀⠀⠈⠉
⠀⠰⠈⠠⠈⠀⠉⠐⠀⠁⠀⠀⠀⠀⠂⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⡤⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠀⠐
⠀⠀⠄⠠⠁⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠁⡐⠢⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡄⠀⠀⠀⠀⢠
⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⠤⠄⠀⠠⠤⠤⠤⠶⢏⣩⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⢀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⡀⠀⣇⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠁⠀⢀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠠⠦⠄⣀⣀⠀⠀⣴⠁⠀⠀⡠⠈⠁⠠⠄⠀⠀⢰⡆⠀⠀⠀⠀⠀⢠⡀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⠻⢤⣄⣀⠀⠀⠀⠀⠀⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠱⡀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⢀⣤⠄⠀⠀⡸⠇⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠤⠤⠤⠤⠤⠄⠀⠀⠁⡈⠛⠚⠷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠂⠉⠉⠁⡤⠀⠀⣠⠀⡀⠀⡀⠀⠀⠈⠉⠁⠴⠶⣦⣄⣁⠴⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠴⠦⢠⣴⠞⣉⠀⢀⡴⠋⠂⠠⠞⠠⠤⠚⠁⠜⠋⠈⠉⠀⠠⠤⠈⠁⠀⠀⠀⠈⠉⠰⠈⠉⠰⠞⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣿⡄⠋⠀⠀⠀⠀⣤⡶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠸⠀⢀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⠈⠁⠀⠀⠀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠁⢺⣧⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡞⢻⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣴⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢳⣾⠁
⢀⣤⠀⣼⠃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠘⠁
⠞⠋⠐⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣤⡀⠀⠀⠀⠀⠐⠠⣤⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠃
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⣤⣄⣀⣀⡀⠀⠀⣀⣀⣀⡤⠤⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢺⡇⠈⠉⠉⠉⠁⠉⠉⡟⢹⡗
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠃⡀⠀⠀⠀⠀⠀⢀⡠⠄⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠘⣃⠰⣇⠀⠀⠀⠀⣴⠁⣸⠃⠀⠀⠈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠘⠓⠒⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠈⠛⠢⠤⠚⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠶⢦⣠⣤⠶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⢀⣀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⢹⡆⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⠁⠈⠁⠚⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠉⠁⣉⠀⠊⠒⠀⢢⡘⠃⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠦⢀⣠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⡘⠀⠀⠐⠒⠰⠆⠀⠀⣸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠛⠒⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡗⠉⠁⠀⠀⠉⠁⠘⢂⡀⠀⠀⢉⣁⠀⠈⣹⡀⠀⠀⠀⠀⠀⠈⠀⠀⡴⠀⠀⢀⡀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠙⠉⠙⠛⣶⠶⠶⠊⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⣤⠋⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⣀⣀⡀⠀⡀⠀⠀⠀⡀⢀⡀⢀⡀⢶⡉⠦⡀⠀⠈⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠛⠋⠠⠷⣴⢦⣤⣄⡀⠄⠁⢻⠀⠀⠁⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⢀⣤⣀⠀⠁⠐⠂⢀⣀⠀⣀⣀⣀⣀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠸⠃⠀⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠤⠤⣀⡀⠀⠙⠛⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠂⠲⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⣀⣀⣀⣀⡀⢀⠀⠰⢇⠀⠀⠀⢀⣴⠀⣀⣀⠀⣀⣀⣀⠐⠶
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡀⣁⣀⣀⣀⣀⣀⣀⡼⠃⠀⠀⠸⢁⣀⣀⠀⣀⣀⠀⢀⣤⠴
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠁⠉⠈⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⡏⠀⠈⣧⠀⠀⠃
⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢹⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠁⢤⣤⣤⣤⣤⡤⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣦⡀⣤⡆⠀⣀⠀⠀⠀⠀⢀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠑⠠⡀⢵⠀⠀⡤⠀⢀⠘⠿⣾⡌⢰⠃⡽⠿⠋⢰⠀⢸⡄⠀⢰⢁⡠⠐⠁
⠀⠀⠀⠀⠀⣀⡄⠀⠑⠢⣨⣸⠄⠀⠠⢃⢲⠀⠘⠿⢃⡸⡈⠻⡟⠀⡔⢂⠤⠀⠀⣆⣠⡄⠢⠡⠀⣄⡀
⠀⠀⠀⠐⠁⠀⠀⠀⢀⠐⠛⠛⣓⣐⢤⣄⠈⢳⡟⣺⣤⡞⣥⣸⡚⡶⠁⢀⣤⢔⣐⡛⠛⠓⠀⠀⠀⠀⠀⠃
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇
⡇
⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⡄⠄⣔⢎⠀⠰⡁⡄⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠏⠔⠰⣱⠣⢣⠎⠆⠞⡜
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠏⠀⠀⠜⠁
⡇
⡇
⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠒⠒⠶⢖⣶⠖⠒⣲⠲⣶⣶⢶⣶⡶⢲⡖⠒⢶⣖⠶⠶⠒⠂⠀⠆
⠀⠀⠀⠀⠘⠠⢤⠀⡀⣀⣩⠿⢿⠁⠁⣀⠄⣰⠉⢫⡃⠴⡫⠄⣫⠛⠙⡆⠄⡀⠁⢙⡿⢿⣁⡀⡀⢠⠤⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡠⠄⣌⠀⢠⡀⠈⡅⠀⣛⣿⠂⡟⢸⣻⣃⠀⠬⠘⣠⠀⠀⣅⠢⡈⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡀⠐⠁⠄⠀⠃⠀⠁⠾⠛⢳⡼⠠⣦⠋⠛⠇⠀⠀⠀⠀⠠⠐⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣥⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⢷⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠂⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⣀⠐⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⣟⣘⡾⠾⣼⣼⣰⢢⢄⠀⠀⠀⠀⠀⠈⡈⠁⠀⠄⠀⠠⠀⠁⠀⠐⢀
⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⠀⠀⠀⢀⠈⠻⡇⢠⣴⠀⠈⠉⠛⠺⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠶⠷⠶⢤⣄⣑⡆⠙⠂⠕⠚⣣⡰⠟⣠⠋⠀⠀⠀⠀⠰⠁⠀⠄⠀⠀⡀⡐⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠉⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠊⠀⠀⠐⢠⡄⠀⠀⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⢀⠐⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠄⠀⠀⢸⠃⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠜⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡇⠀⠘⠆
⠀⠀⠀⠀⠀⢈⠀⠀⠀⡄⠓⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⡈
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠈⡐⠒⠅⠀⠀⡒⠲⡁⠈⠀⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠴⠦⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠜⠒⠶⠶⠶⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⣀⣻⣿⣴⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠿⣻⣿⣿⣟⣳⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⠀⢀⣽⣿⣿⣿⣯⣥⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⣯⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡇⢀⣿⣯⣬⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢠⡈⢡⣴⣾⣿⣿⣿⣿⠦⡙⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣭⣿⣿⣿⣿⣿⣿⣽⣿⡛⣿⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢋⣭⡉⠉⢼⣯⣽⣿⣿⣿⣴⣾⣿⣿⣯⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⢻⣿⣬⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⢶⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣟⠛⢻⣽⣿⢻⣿⣿⣿⣿⣿⣯⣛⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⠾⢟⣱⣤⣼⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣷⣦⠤⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠙⢿⣦⣤⣿⡛⣿⣿⣿⣿⣿⣿⣿⠻⣿⣙⠿⢿⣿⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⣼⣿⣿⣿⣿⡿⢻⣿⠛⠿⠯⠙⠆
⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⡿⠿⣿⣿⣹⣿⣿⣿⣿⣿⣿⠛⠏⠋⣻⣿⣿⣶⣶⣶⡶
⠀⠀⠀⠀⠀⠀⠉⠁⠀⢻⣿⣦⣼⣾⣿⣿⣏⣹⣿⣿⣿⣿⣁⡆⣀⣿⣿⣿⣍⠻⠿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠙⣿⡏⠋⠿⠇⠘⠿⠻⣿⡌⠉⠻⠏⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠁⠀⠀⠘⠐⠀⠃⠘

⠛⢿⣿⣷⢄⣿⢦⠀⠀⣀⣀⣤⠊⠁⢾⡁⠀⠈⢢⠴⡜⠀⢀⡜⠁⠙⢶⡋⠒⢶⣶⡌⣦⣀⢲⡲⣤⡀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠘⢿⠂⣸⡰⣯⠚⢓⠤⢭⠝⠉⠉⣡⡀⢜⠉⠙⢎⠁⢀⣴⢟⣄⠈⠻⠿⠹⣷⣵⠙⢺⠗⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠰⣷⣶⡀⣿⣧⠈⠀⠢⢤⣸⠂⠀⠊⠀⡀⠀⠁⠀⠘⠶⠻⣄⡁⠛⠂⣀⣀⡀⠀⠉⠁⢸⣶⣴⣦⣀⣀⣦
⠀⠀⠀⠀⠀⠈⠛⠓⠈⠿⡷⣷⣦⡀⠾⠤⠄⡶⠊⡏⠒⣄⡀⠀⢷⣶⢿⠿⢣⣿⣿⢟⣁⡻⣶⡀⠈⡿⢿⡏⠻⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣧⣝⣛⣛⠢⠤⠀⣹⣦⡀⣠⣥⣯⣹⣿⢈⣏⠙⠲⣶⣾⣿⣿⢧⣿⡇⣠⣴⣿⠿⢗⡒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢁⣴⢶⣾⠃⢰⡿⠃⠈⣼⣿⡎⠻⠿⣿⡗⢿⣷⡿⣿⣇⡀⠸⡿⣩⣽⡏⠀⠘⠛⣝⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⠀⠀⠀⢠⣼⡟⠃⢠⡔⣢⠙⠃⠀⠉⠛⠀⢹⣻⡆⡇⣿⠟⠁⣰⠁⢢⠋
⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠉⠁⠀⠸⣙⡬⠀⠀⠀⠀⠀⠀⠀⠛⠧⢰⠁⢘⠓⠓⠖⠺⣸⠜
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣄⡢⠨⡀⠀⢄⠀⢸
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣶⣀⠀⢉⠟⠁⠤⢵⠄⠀⠗⠈⣲
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢿⣶⣷⠠⡤⠃⠀⠠⠊⢲⠉
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣲⠶⡀⢋⣵⣿⣹⠀⠀⠀⣀⡠⢞⠦
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠓⣡⣼⣿⡛⢛⣾⡏⢘⣧⡀⠈
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⠙⡆⢇⠈⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⣭⠋⢽⢺⣿⠇⠠⣻⣦⠎
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠃⠀⠀⠈⠁⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣯⣴⡇⣶⣭⠀⠟⠋⢀⣠
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⣿⣷⡻⣿⡇⠀⢰⣿⠽
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⢇⣬⠁⢀⣴⡾⡟
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠃⠀⠘⠛⣫⢖
⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠖⠲⡷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⡋
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⡀⠀⢀⣸⣆⣳
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠀⠀⠀⠰⠀⠀⠀⠀⢏⡀⠀⠀⠀⠀⠠⡆⡂⠀⣿⡇⠀⢀⣼⣿⠏⣩⢷⡄
⠀⠀⠀⠀⠀⠀⢈⡇⡆⠀⠀⠀⠀⠀⢠⣭⡄⠀⠀⠀⠐⣆⡄⠀⠀⠼⠃⢈⣅⠀⠀⠀⢰⡏⣵⠀⠈⠠⠀⣁⣾⣿⠿⠁⠈⣉⠄
⠀⠀⠀⠀⠀⢀⣴⡏⠃⠀⠀⠀⠀⢠⠟⠛⠀⠀⣀⠀⢐⣍⣡⡄⢀⢸⠇⠈⠋⠠⡆⠀⣀⡗⢻⣀⠀⠁⢰⢹⣿⠁⣀⡤⠞⠻⣇⠀⣖
⠀⠀⠀⠔⢠⣿⣯⠦⠄⠀⠾⠂⠀⣤⣶⠀⠀⢠⣙⡂⢸⣇⣼⠃⢿⠀⠀⣰⣦⣀⡽⠚⠉⠉⠈⠉⢑⣦⡔⠈⣤⡾⡉⠀⠀⡀⠃⠀⠞⢆
⠀⠀⡌⢠⣿⠟⢏⡔⠒⢠⠋⠳⣀⠹⠍⣴⣤⠚⠿⠀⡾⠉⢹⡆⠀⠀⡿⢻⡇⣹⠛⠛⠛⠛⠛⠋⠉⣩⣤⡈⢇⢸⣥⠔⠊⠁⠀⡾⠃⠙⠧
⠀⠀⢳⡌⣷⢆⢹⡅⠢⠆⡌⠘⢘⠁⠀⠉⠉⠁⠀⠘⠁⠀⠀⢿⠄⠀⠁⠀⠀⢿⣿⢳⡖⢲⣶⣋⢾⣽⣿⠃⠀⠻⠃⡠⠎⠀⠀⡖⠀⠒⢳⡀
⠀⠀⠈⠣⣿⣿⣧⡜⠒⠢⢞⡳⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢨⣧⠆⠀⠀⠀⠈⢿⣛⣛⣛⣛⣉⣈⣭⠇⠀⠀⠀⠙⠃⠀⠀⣴⠷⠒⠒⠚⢷⡄
⠀⠀⠀⠀⠈⠙⠋⠉⠁⠀⠈⠀⠀⠀⣤⣾⣿⣷⡶⠶⠀⠀⠀⢠⠇⠐⠤⠤⠔⠁⠀⠈⠛⠍⠭⠄⠚⠁⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⢿⣿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠻⣇⠀⢧⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡆⡖⢀⠀⢀⡀⣀⠀⠀⠀⢀⠀⠀⣀⡀⢀⠀⠀⠄⡀⠀⡀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠡⠁⠇⠸⠁⠟⠀⢄⡃⠀⢈⠀⠐⠁⠐⠁⠈⠀⠇⠀⠏⠃⠃⠄⡇

⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠗⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⡇
⠈⠟⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡲⡄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⡊⢵
⢈⡼⣋⣁⠄⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⢀⡙⠆⣃⠞⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⡲⢬⣾
⢸⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣽⡇⣿⣯⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢸⡇⠀⠀⠀⣁⣀⣀⣤⡄⢠⡤⢤⣤⣠⣾⣿⣿⣿⡇⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢸⣷⣶⣿⣿⣿⣿⣿⣿⡇⢸⣷⣶⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣆⣀⣀⣀⠀⣀⣤⣤⣴⣶⣶⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⠀⡤⠀⣀⣀⠀⡀⣤⠀⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢻⡀⣀⠄⢻⡟⡼⠀⠀⢠⢸⠁⠘⠁⢀⠀⠀⠐⠌⠀⣷⠀⡏⢆⠰⡅⣿⠘⠇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢣⢈⡎⣧⠇⠀⢸⣧⠀⡐⠋⠀⡘⠀⠀⢠⠎⠀⠀⠀⠘⣀⣷⡟⡁⠁⠀⢠⣿⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠀⢹⠀⠀⢸⣿⠀⠁⠀⠀⣧⡤⠰⠋⠀⠀⠀⠀⠀⠀⣷⠰⠀⠀⠀⠀⢹⠆⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⣆⠀⢸⣿⠠⠈⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⠀⠀⠘⣋⣠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢸⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢽⠀⠀⡀⠀⠀⠀⢸⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⣿⡷⣴⠒⠛⠁⠀⠀⠀⠀⢀⣠⣶⣶⣄⣸⣠⡾⣶⣦⡀⢰⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣹⠇⠀⠀⠀⠀⢀⣠⡶⢿⣿⣿⣿⣿⣿⡛⣅⠉⠙⠃⢸⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⣿⡇⠀⠂⢀⣠⡶⠟⠉⠀⠀⠙⠻⣿⣿⣿⡇⢸⣿⣦⡀⠘⡇⢏⣀⣤⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢾⠉⠷⣿⣿⣇⣤⠞⠋⠅⠈⠀⢀⠀⠀⢠⠠⡄⡙⢿⡟⢸⣿⣿⣿⣷⣧⡇⠸⡿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠌⢀⣠⢼⣿⠛⡇⠀⢀⡶⠒⠲⠓⠐⠁⠀⠀⡗⣱⢾⣏⡼⣿⣿⣿⣿⣿⣿⠦⣀⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠊⢉⣠⠖⠋⠡⢼⡿⠀⢃⡴⠻⣤⠀⠀⢠⠀⠀⠀⠀⠀⢹⣾⡇⠃⠀⢙⣿⡟⣿⢶⣟⣿⡄⠆
⠀⠀⠀⠀⠀⠀⣤⡴⢞⠋⠀⠠⡄⡀⢸⣷⠓⡟⠃⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣿⡿⢀⣴⣿⣿⣿⡟⢸⣿⡟⡃⡄
⠀⠀⠀⠀⠀⠀⠈⠀⢻⠀⠂⠐⠛⠙⢾⣿⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡇⢾⣿⣿⣿⡏⡧⢸⡽⡆⢳⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠀⠀⢸⣿⠇⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⡟⢹⣿⣿⣿⡇⣷⢸⠇⣣⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠈⠆⠀⠀⠀⠀⢠⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⣠⣾⣿⣷⡺⢿⣿⣿⣧⡏⠘⡇⣏⢸
⠀⠀⣤⣄⠴⣶⣼⣿⣾⠀⠀⠀⠂⡀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠁⠀⠐⢀⣾⣿⣿⣿⡟⡁⣾⣿⣿⢿⡇⣠⠃⣿⢸⣦
⠀⢸⣿⣿⡿⣿⣟⣿⣿⠀⠀⠀⠀⠁⢸⣿⣿⣅⠀⠀⠀⠀⠀⡄⠀⣰⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⢸⡗⣿⠀⡏⢸⡷
⠀⢸⣿⠿⠷⡯⠿⡿⠿⠀⠀⠀⠀⠀⢸⣿⣇⠋⠀⠀⠀⠀⠀⠁⠚⠻⡿⡻⠛⠟⢿⣧⠊⠟⠿⠿⢸⣷⡿⠟⠿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠀⠀⠘⠛⠁⠀⠀⠀⢸⠋⠁⠀⠀⢰⡇
⠀⢀⠀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣄⡤⠼⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠀⠀⠀⠀⠚⡇
⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣀⣿⣏⠀⡀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⡈⢿⣿⣿⣶⣤⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣹⣷⣽⣿⣯⣴⢛⢀⡀⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣴⡾⠿⣿⣿⣗⠉⠉⠛⠛⠉⢩⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣍⣯⠀⢀⢽⣿⡿⠦⠄⣁⣀⠀⠛⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⣤⡄⠽⢻⣿⡷⣾⠟⠑⢰⣼⣿⣛⣤⡶⠯⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢙⠓⠓⢓⡋⠰⢿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⣠⣄⠐⠏⣿⣿⡅⠀⠙⣿⣍⢁⣁⣈⣀⣤⢶⡾⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠒⢿⡿⠿⢷⣶⣽⡳⢦⠾⠿⢏⣿⣿⣭⣻⠍⠁⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢂⣀⣩⣿⠀⠀⢰⣟⣿⣿⣿⣿⣷⣿⡛⠙⠻⠏⠈⣤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠙⠛⠿⠿⠲⣿⣭⣿⣿⣿⢿⣝⣿⣟⣶⡴⠾⣻⠿⢷⠶⡦
⠀⠀⠀⠀⠀⠀⠰⢽⣦⣶⣶⣾⣴⣶⡿⠟⢿⣿⢀⣟⣶⣿⣿⣿⠁⠀⠀⡸⣦⠴
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡤⡀⣉⡐⣤⣥⡼⣟⣯⣿⣏⣳⣮⣿⣷⣀⣈⣽⣧⣶⣯⣭⣀
⠀⠀⠀⠀⠙⠗⣶⣦⣄⣉⡉⢁⣭⣍⣛⣷⣿⣿⣿⣬⣋⠻⡟⣿⣯⣹⣿⡷⣶⣤⣬⣍⠙⠓
⠀⠀⠀⠀⠈⠀⠀⣈⠹⠙⢙⣿⢿⣿⠋⠉⠉⢿⣿⣷⣿⣟⣧⣸⣽⣿⠿⢿⣞⣂⣉⣉⡀
⠀⠀⠀⠀⢀⣠⣤⡤⠶⠚⠋⢀⣽⣿⠀⢠⢄⣹⣿⣿⣿⣿⣯⣀⣼⡁⣀⠀⣿⣿⠿⢿⣢⣦⣤⡀
⠀⠀⠀⠈⠉⠀⠀⠀⢀⣠⣶⣿⣍⣿⡿⣛⣿⠻⣿⡿⡏⣿⣏⠿⣼⡿⢶⠾⠻⡷⠄⡀⠈⠉⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠙⠋⠉⠀⣉⣽⠿⠛⠁⣰⠟⠈⣷⡜⢿⣆⠀⠛⢷⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⢹⡧⠀⠈⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠊⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠀⣈⣏⣹⣋⣏⣫⣙⣏⣙⣉⣋⣟⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣭⣿⣿⣯⣿⣟⣿

⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠲⡄
⠀⠀⠀⠀⠐⠀⣁⠃⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢀⠉⠡⡀⡄⠠⠐⢀⠀⣀⠀⠀⠠⣄⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣿
⠀⠀⠨⠀⠁⠀⠀⠄⠈⠃⠋⠀⠀⠀⠀⠐⠀⠀⢒⠀⢢⣦⣄⠱⣤⣠⠊⢁⣨⠴⠠⠀⠂⠀⠀⠀⠈⠉⠉⠉⠉⠋⠉⠙⠋⠁⠉⢉
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠀⠀⠀⠀⠐⠓⠂⠀⠙⡿⠛⠓⢾⣃⣒⣚⣃⣀⣀⣀⠀⠀⠌⠀⡠⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⢉⠀⠈⠉⠉⠀⠀⠀⠀⠸⠁⠀⠀⠄⠀⠀⠿⠠⠀⠀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠈⠀⠀⠐⠢⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠊⠐⡄⠀⠀⠀⠀⠀⠀⡀⠸
⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠀⠰⠦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠀⠠⠃⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⠀⠀⢹⠂⡀⠀⡀⠀⠀⠄⠀⠀⠀⠀⠀⠔⠀⠀⠀⠜⠐⡇⠂⠀⠀⢰⠀⠀⠈⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠒⠒⠓⠀⠘⠒⠂⠀⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠁⠀⠸⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠠⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠘⢾⢀⡠⠞⠀⠃
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡶⠀⠀⠀⠀⠀⠀⠀⢀⡁⠂⠀⣳⡤⠨⡭⠖⠐⠠⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠭⠅⠀⠄
⠀⠀⠀⢀⡠⠄⢲⠁⠀⣀⢄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠤⠖⠁⣀⠀⢄⣀⠠⣀⠀⠀⢠⣠
⠀⠀⠀⠀⠤⠠⡰⡀⡠⢊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠲⡈⠳⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⣿⡁⠁⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢨⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠂⠙⠀⠘
⠀⠀⠀⢀⣀⠀⢀⣿⢋⡀
⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡀⠈⠠⡀⣜⠄⢸⣀⣄⢬⣄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢨⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣄⠽⡶⢃⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⡄⠀⠤⠄⠀⠢⢼⠄⠸⠴⠤⠴⠶⠦⠤⠤⠄⠀⠀⠀⠀⡀⠐⠰⠁⠀⠀⣀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠠⠤⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⢠⡾⠋⠀⠀⠉⣩⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣰⣿⣥⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤
⠿⠷⠶⠶⠀⣴⣏⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣀⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢛⣿⠇⠀⠀⠀⠀⠀⠀⢠⣎⡀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⣻⡿⢉⣿⡋⢽⡃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠶⠶⠆⠀⢉⡽⠋
⣶⡆⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣶⠆⠰⠿⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠋⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢁⠀⠠⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠛⠈⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠐⢒⡶⠆⢀⡩⠗⠀⣬⠓⠀⠆⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⢠⣎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠟⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⣩⡟⠁⠀⠰⣯⣤⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣶⡶⠛⣻⣿⠖
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠆⠘⠁⠈⠀⠀⠀⠀⣀⠚⣿⡠⠿⠗⠚⠿⠷
⣤⠄⠛⠛⣹⠟⠁⠀⠀⠀⠀⠀⠰⠶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡟⠀⠉⠁⠀⠀⠀⠀⠀⠀⠘⠹⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠋⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠴⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢒⡖
⠒⠃⠸⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠛⠛⠓⢈⣽⠋⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⣴⣟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠇⠀⠾⠶⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠂⠘⠛⠛⠀⠀⠀⠀⠀⠈⣹⡟⠁⠀⠈⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢶⠄⠀⠀⠀⠀⠈⢋⣿⠇⠀⠀⠀⢶⡦
⠀⠀⠶⠶⢶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣥⡄⢩⡿⠡⠿⠤⢈⣝⡁
⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠂⠀⠀⠀⠉⠀⠀⠀⠀⠚⢳⣶⠆
⠀⠠⣾⣧⣤⣤⠄⠈⢋⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠁⣀⣀⣄
⠀⠀⠀⣤⣤⣤⠀⢰⣯⣤⣄⠀⠀⠀⠀⠀⠀⠠⠶⣶⡦⠸⠿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⣙⡋⢀⣼⣃
⠀⠀⠀⣠⠞⠁⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⣾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠉⣿⡅⠛⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⣽⠋⠈⠉⠉
⠀⠀⠀⠛⠚⠛⠀⣠⠞⠁⠀⠀⠀⠀⠀⢩⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⡿⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢐⡗⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⡄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠒⠈⠀⠀⠀⠀⠀⠀⠉⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢛⡿⠃⠀⢀⡴⠋⠁⢠⣞⡉⠠⠖⠉⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠧⠤⠄⢀⣀⣀⣀⡀⠀⠀⠀⠀⣴⣋⣀⡀⠀⠉⠙⠛
⠛⠛⣻⡿⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠉⠉⠉⠁
⣿⣿⠿⠻⡿⢛⠿⢿⠟⣿⢿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⢇⢻⠀⢹⡞⢸⡼⢪⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠀⠃⠈⠈⠁⠀⠀⠈⠁⠋⠀⠀⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣀⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁
⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⡹⢀⣤⡀⠀⠉⠀⠀⠠⢀⣀⠀⠀⢠⡀⠀⠘⠁⠀⢀⠇
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠐⠋⣰⠋⠀⠀⣀⠀⠀⠀⠰⠂⡀⠉⢱⠟⠲⠦⠴⠶⠚⠁⠸⠀⠀⠀⡀
⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⣰⠃⢠⡆⠀⢑⡦⢀⡼⠀⢸⡋⢠⠋⣴⠀⣴⢂⡄⢀⡴⢚⡁⠀⢸⡃
⠀⠀⠀⠸⣇⠀⠀⡀⠀⠀⠏⠀⠘⠀⠀⠘⠀⠘⠁⠐⠾⠃⠇⠘⠁⠈⠁⠺⠁⠘⠁⡘⠃⠐⠼⠃
⠀⠀⠀⠀⠈⠉⠁⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⡀⠀⠀⠀⠀⠈⠙⠋⠛⠛⠀⠀⠀⠀⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢥⡶⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⡷⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠸⠚⡖⠀⣶⢰⠀⣾⡀⠈⠎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⣤⡶⠄⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡆⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡷⠖⠒⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢪⢀⡇⣀⣀⣄⡈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡞⠠⣔⠀⡀⢲⠊⠎⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢰⡅⠀⠈⠀⠱⠀⠀⠜⡆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠈⣦⠀⠀⠤⠤⠔⠀⡌⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⢦⣀⣀⣀⣀⡔⠀⣆⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠣⠄⢸⠉⠉⠉⠀⢤⠠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡖⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠴⡸⠀⠀⠀⠀⠸⣐⠒⡦⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠢⣀⠆⡌⣘⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠌⠳⣄⡀⢀⢀⣠⠎⢠⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠌⠀⡏⠀⠀⠘⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠅⢀⣄⠈⠉⢹⣿⠆⡄⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠘⡄⠀⠱⡀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡁⠘⠛⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢠⠃⠀⠀⢁⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⣀⠤⡟⠀⠀⢠⠃⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣻⠀⠀⢠⠀⠀⢰⣶⡄⠀⡌⠀⠀⠀⣤⠴⠊⠀⠀⢱⠀⡠⠁⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡟⠀⠀⠈⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⢩⠀⠀⠀⠀⠀⡷⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⠀⠀⠀⠘⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠇⠿⠗⠀⠀⢸⡇⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠀⠀⢀⠀⡆⠀⠀⠀⠀⣾⠀⠀⠠⡀⠀⠀⠄⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢸⠀⢀⠌⠀⢡⡀⠤⠠⠶⢿⠀⠀⠀⠑⢄⡀⠈⣄⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢘⡠⠊⠀⠀⢸⡄⠠⠀⠀⠘⠂⠀⠀⠀⠀⠙⠶⣼⣦⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣶⠉⠀⠀⠀⠀⠘⡇⠀⢠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆⠀⠀⠀⠀⠀⣇⠈⢺⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠉⠀⣀⣀⣀⣀⣀⣿⠀⠀⡀⠀⢸⣀⣀⣀⣀⣀⣀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠙⢻⠉⠉⠉⠉⠉⠁⠀⠀⡇⠀⠀⠁⠀⠀⠉⠉⠻⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠀⣀⡀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢃⡈⢉⠈⠀⠀⢄⠀⢱⣆⣐⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⢢⣌⠀⠀⠀⢋⣁⠼⠒⡉⠤⠖⠨⡑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⢁⠀⠀⢰⡈⠠⡴⠂⠉⢀⠀⠀⠈⠙⠳⡤⠇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠂⠀⠀⢠⣔⣢⡄⢈⠁⢩⠀⠀⠀⢆⠀⠅
⠀⠀⠀⠀⠀⠔⠀⡀⠀⠀⠀⠀⠀⣄⠀⠠⠀⠀⠐⠁⠈⡀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠂⠐⠒⠒⠂⠒⠀⠂
⠀⠀⠀⠀⠐⠠⠀⢀⠀⠀⣀⡔⠀⢈⠂⠀⠀⡠⠈⠁⠀⠀⠈⠠⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠈⣀⠉⠀⣈⠈⠹⠀⡴⠁⡂⠀⠀⠀⢀⡀⠘⡄
⠀⠀⠀⠀⠀⢣⡈⠀⠀⠉⡀⡀⠉⠀⠉⠀⠆⢊⠐⠋⡂⠀⠈⠒⠀⠅⠀⠀⢀⠄⠀⠤⡀
⠀⠀⠀⠀⠀⡀⢱⢔⣄⣀⣀⡀⠀⠀⠀⠀⢃⠀⠀⠈⠃⠀⠰⠆⠀⠀⠀⣰⠃⠀⠀⠀⢈⡄⡀
⠀⠀⠀⠾⠀⠀⠤⠱⠤⡄⢼⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠍⠀⠀⡰⢃⠊⢐⡀⠠⠁⠂⡍⠂⠃
⠀⠀⠀⠀⠀⠀⠀⡀⠐⠂⢋⣖⠀⠀⠀⠀⠀⠰⠨⠬⠉⣂⢝⠡⡄⠸⣄⡐⠭⠔⢀⣀⠂⠀⢀⠀⠀⠄
⠀⠀⠀⣠⢀⠅⠄⠉⠈⠗⡧⡅⠀⠀⠀⠀⢀⣣⠁⠀⠤⢄⢀⢑⣷⡴⠤⠈⠀⢠⠤⠤⢆⡠⠶⠤⠘
⠀⠀⠰⠧⠀⠀⠑⠂⠈⠠⠁⣈⣄⠀⠀⠀⠀⠉⡄⠀⠀⠊⡀⠀⠈⠀⠀⠐⣠⢦⡤⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠂⠀⠀⠀⠀⠀⢨⡉⣩⣉⡅⠀⠠⡀⠀⠁⠀⠉⣕⣾⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠿⠄⠇⠀⠀⢀⢠⠀⠀⠀⠀⠴⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢀⡀⠔⢢⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⢀⠀⠀⢀⠎⠀⠀⢀⠌⠀⣠⠀⠀⡀⠀⠔⠉⠀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠚⠤⡀⠂⠀⠘⠀⠀⠀⠨⡠⠊⠀⡥⠊⢅⡘⠀⠀⠀⠂⠀⠠⠂⠀⠀⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠃⠀⠀⠄⠀⠀⠀⠀⠀⢄⠀⠀⠈⠀⠀⠀⠀⠰⠔⠁⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢂⠀⠀⠈⠁⠒⣣⠀⠒⢂⣀⣀⣀⣀⡐⠒⠲⡈⠉⠈⠉⠁⠉⠃⠐⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⠄⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠑⢄⠐⢄⠀⠘⡆⣾⣿⣿⣿⣿⣿⣿⣿⡆⠘⠐⢊⣤⣤⣤⣭⣀⣀⠐⠠⠄⡁⠐⠄⡀⠀⠀⠉⠬⠠⠂⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠁⠀⡠⠳⣸⣿⣿⣿⣿⣿⣿⣿⢇⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⢠⠗⠠⠈⡒⢄⠀⠀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⢤⠤⢈⠢⡄⠀⠀⡠⠊⠀⠀⠱⣝⣿⣿⣿⡿⠛⠋⠉⠙⠢⢿⣿⣿⣿⣿⣿⣿⣿⡿⢡⠟⠶⣤⡀⠀⠉⠓⠠⢄⡀⠉⠀⠈⠢⢰⠂
⠀⠀⠀⠀⢀⠄⠀⠀⠸⠄⠀⠀⢀⠔⠀⠀⠀⠀⠀⠈⠉⠛⠺⡀⠀⠀⠀⠀⠀⠀⢹⣛⠿⠿⠿⠿⣋⡔⠋⠀⠀⠀⠉⠓⢤⡀⠀⠀⠈⡆
⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠀⡀⠀⢀⠠⠊⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠲⠚⡁⠀⠠⠆⠠⠀⣠
⠀⠀⠀⠀⠀⡀⠈⠆⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡀⠔⠁
⠀⠀⠀⠴⡁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠁⠀⠢⡀⢀⡀
⠀⠀⢀⠀⠀⠡⠄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡰⠁
⡆⠐⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣀⠔
⠈⠂⠀⠀⠂⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠢⣀
⠀⠀⠀⠀⡠⠈⠶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠮⠀⠀⣄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠔⠁⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⡠
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣀⠀⠀⠔⠁⠈
⠀⠀⠀⣀⠀⠊⠀⠈⡰⠂⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈
⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⡄⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠄⠂⠉⠒⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠠⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠁⠀⠀⣀⠤⡤⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠂⠐⠈⠀⢠⠀⠀⢀⣈⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠉⠀⠀⠀⡙⠢⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄⠈⢀⠂⠀⠀⠀⠁⠒⠠⠄⣀⡀⠀⠀⠀⠀⠀⣀⣀⠠⠄⠒⠈⠀⠀⠀⠀⠠⠁⠢⠌⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⠄⢂⠜⠀⠀⢠⠀⠀⠀⣠⠀⠀⠀⠀⢠⡄⠀⠜⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠎⠀⠀⡌⠀⠠⠐⢋⠀⠀⡌⠀⠑⠄⠀⠀⠃⠐⡤⠀⠐⠄⡀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠃⠀⠀⢸⠀⡠⠀⠀⠀⠀⠀⠀⠈⠑⢄⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⣢⣼⣶⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⠈⠁⠀⠀⠀⢠⣦⣤⣶⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⡦⠖⠁⠀⠀⠀⠀⠈⣩⣴⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠿⠶⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⢀⡀⠐⠀⠀⢀⡤⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣾⠿⢛⣿⣶⣾⣿⣯⣴⣦⣴⣴⣿⣾⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣄⣴⡿⠋⠀⣀⣼⡿⠟⠛⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣯⣤⣶⣿⣿⣧⣄⢀⣀⣠⣶⣶⣾⣿⣿⣿⣿⠿⠋
⠀⠀⠀⣀⣤⣄⡴⣲⠟⣫⣿⠟⠋⠁⠈⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠠⣾⣿⣿⣿⣿⣷⣿⣋⣀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣯⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠉⠉⠉⠛⠛⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⣤⣦⡀
⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠀⠙⣦⡀
⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠠⠬⠷⣄
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠞⠻⢷⣶⣶⣾⠂
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣠⠤⠞⠁⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⣼⡟
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⢀⣴⠶⠋⠀⠀⠀⠀⠲⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⣠⠞⠉⠙⢦⣠⠞⠉⠑⢦⡀⠀⣀⣀⣤⣽⣷
⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠠⠀⠀⠀⠀⣠⠟⠁⠀⠀⡼⠁⢠⣶⣤⠈⠁⠀⣤⣶⡄⢳⡀⠘⠻⠿⢿⣥⡀
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡟⢶⡇⠀⠀⣴⠏⠀⠀⠀⢸⠇⠀⠈⠛⠁⠀⠀⠀⠈⠋⠀⠀⢷⠀⠀⠀⠀⠙⢿⣦
⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠻⢦⣴⣿⣶⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠘⢧⣀⠀⠀⠀⣀⡴⠛⠉⠀⣾⣿⠿⠾⠟⠛⠁⠀⠀⠀⠈⢧⡀⠀⠀⠀⣠⢦⡀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠻⢿⣿⣿
⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⠀⢀⣈⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠚⠁⠀⠙⠲⠶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⡿⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⡿⠃
⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠟⣻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄
⠀⠀⠀⠀⢿⣿⣛⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠈⠓⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⡀
⠀⣠⣤⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢫⡧⢤⠄⠀⠀⠀⠀⠀⠀⠀⣤⢬⣏⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⡆
⠀⠘⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢧⡀⠀⠀⠀⠀⠀⠀⠀⡨⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣽⡿⠃
⠀⠀⠀⠈⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠙⡄⠀⠀⠀⠀⠀⡞⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇
⠀⢠⣶⠿⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⠶⣸⠉⠙⠒⠃⠀⠀⠀⠀⠀⠓⠊⠉⣇⡴⡆⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄
⠀⠙⢿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠶⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣻⣿⣿
⠀⢠⣿⣋⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠋⠉
⠀⠈⠛⠛⠛⠛⠛⠛⠛⣿⣷⠆⢀⣠⣤⣤⣶⠶⠶⠶⣶⣤⣀⣀⣀⡀⠀⢀⡀⠀⢀⣤⣶⣶⣦⣄⠀⠀⢀⣀⣀⣠⣤⣦⣤⣶⣶⣶⣶⣶⣶⡶⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠿⠛⠋⠉⠛⠁⠀⠀⣀⣽⠉⠉⠉⠙⠿⣦⣿⡿⣿⠿⠃⠀⠀⠈⠻⣷⣶⡟⠋⠙⠛⠃⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⠀⠀⠀⠀⠈⢹⡇⣿⠀⠀⠀⠀⠀⠀⠈⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠓⠒⠒⠚⠃⠛⠒⠒⠒⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣀
⠀⠀⠀⡐⠒⣲⡔⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⣧⣧⢜⡆
⠀⠀⠀⣘⣬⡼⢦⠾⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⣀⠎⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⠉⡝⢉⠟⡇⠀⠀⠀⣀⣴⢶⠚⢒⣊⣳
⠀⠀⢀⠀⠀⡄⢰⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⠚⡀⡎⠀⡇⠀⠀⠈⡷⢧⡬⢶⣶⣮⣅⣄
⠀⠀⢸⡄⣠⠇⠸⠤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠉⠉⠉⠋⠉⠉⣉⠉⠉⠍⡇⠀⠀⠀⡄⠀⣠⠃⡇⡴⡇⠀⠀⠀⣠⠊⢠⡎⢇⡱⣘⣾
⠀⠀⢸⠤⠤⡀⠀⡤⠤⠤⠤⠤⢼⠀⠀⠀⢸⠀⠁⠀⠈⢀⠀⠀⠀⠄⠀⢁⠀⠀⠀⡗⠊⠁⡠⢻⠀⡇⠀⠀⢸⣁⠜⠁⡳⠚⢀⡴⢻
⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⠀⠐⠂⠀⡄⠀⠐⠀⠀⠆⣸⠀⠀⠀⡗⠒⠉⠀⡼⢀⡇⠀⠀⠀⡁⡠⠊⠀⡴⠋⡐⡻
⠀⠀⠈⠉⠀⠓⠚⠓⠒⠒⠒⠈⠋⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⢀⣀
⠀⠀⣀⣫⣄⣦⣠⣯⣖⡀⠀⠀⠀⠀⠠⡀⠐⡆⠀⣁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠤⢤⣤⣿⣧⣤⣤⠜⠀⠀⠀⠰⣒⢞⣛⣛⣯⡭⢀⣁⣱⠀⠀⠀⠀⠀⠀⠠⣈⣇⢆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠐⢄
⠀⠀⠀⠘⡀⢸⡇⢀⠇⠀⠀⠀⠀⢠⣡⠁⠀⠀⡀⠀⠗⠾⠂⠀⠀⠸⡁⠩⠍⠤⠤⠭⠯⠯⠭⠭⠭⠭⢽⠀⠀⠀⡏⠉⠉⠉⠉⠉⠷⡀⠀⠡⡀
⠀⠀⠀⠀⣇⢸⡇⣸⠀⠀⠀⢠⣔⠟⢹⢠⠀⠀⠇⠀⡄⠀⡅⠀⠀⠀⡄⢒⢐⡒⡂⢒⢖⠒⡒⣒⣒⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⣤⠊
⠀⠀⠀⠀⠘⠾⠷⠃⠀⠀⠀⠀⠈⠀⠸⠤⠤⠤⠤⠤⠤⠤⠇⠀⠀⠀⠃⠠⠥⠬⠬⠤⠥⠤⠤⠤⠥⠤⠞⠀⠀⠀⠣⠠⠤⠤⠤⠤⠤⠦⠼

⠀⠀⠀⠀⢀⢆⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⡠⠉⢢⡀
⠀⠀⠀⢠⢸⠴⣟⢰⣦⠀⠀⠀⠀⠀⠀⠀⢀⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠸⠂⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣋⣄⣊⣁⡀
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⣤⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠂⠄⠰⠛⠛⠂⠀⠤⠂⠀⠀⠀⡆⠀⠙⠛⠛⠛⠛⠋⠀⡂⠀⠀⠀⠀⠈⠀⠠⠀⠀⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠄⠀⡃
⠀⠀⠀⠸⣀⢀⢘⠀⡸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣃⠀⠀⠀⣀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⡀⡀⣈⣈⣀⣀⡀⢀⡁
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣤⣶⣦⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⣿⣿⣿⢹⣏⢹⣿⣾⣿⣇⣽⣿⢻⣿⡿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⠀⠀⠸⣿⣿⣶⣿⣿⣿⢸⣯⡏⣿⣥⣿⡏⣼⣿⣯⣿⣿⣶⣿⠃⠀⠀⢤⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠿⠛⠛⠻⠛⠛⠟⠛⠛⠋⠁⠀⠀⠀⠘⠋⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⠞⢓⠀⠀⠀⢀⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⡠⠚⣽⢇⠔⢡⣿⠀⡰⢫⡿⠀⠀⠙⠃⠀⠀⣠⣿⠏⠀⠀⣴⣿⠋⢀⠎⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠜⠁⣼⣿⠋⢠⣿⠃⡼⢡⣿⠇⠀⠀⠀⠀⠀⣰⣿⣏⡄⠀⣼⣿⢃⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠀⣸⣿⠇⢀⣿⡏⡼⠁⣾⡟⢀⣴⠏⡤⠀⣠⣿⠏⠁⠀⣸⣿⢗⢗⣶⠀⣴⠞⡄⢠⡏⠀⣀⠀⢸⠃⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⣼⠃⢸⣿⠃⢸⡟⢈⣷⠋⢹⡿⠀⢀⣴⣿⣷⢣⣾⠃⣼⡟⡰⢁⠜⠓⣹⡟⠀⠀⠀⠀⣹⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠻⠇⠀⠘⠿⠃⠀⢸⡟⠀⠈⢳⠞⠁⠀⠈⠓⠚⠁⠸⡿⢡⣿⠃⠀⢻⣿⠗⠁⠀⠀⢿⣤⠖⠀⠰⣯⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣠⠎⠀⠀⢠⣤⠀⠀⠀⠀⠀⣿⡏⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣿⣠⠜⠁⠀⠀⠀⠀⠀⠼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⣶⣶⣦⣤⡀⠀⠀⠀⣤⣤⣄⠈⣥⣤⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡆⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣦⠀⢸⣿⣿⣿⡄⢿⣿⣆⠀⣼⣿⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⡀⠀⠀⠀⠀⣿⣯⠉⢩⣿⣿⠀⣿⡿⠻⣿⣇⠈⢿⡟⠙⣿⠏⠀⠀⠀⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⣿⣿⣷⣿⣿⡿⢰⣿⣿⠿⣿⣿⠀⠀⢹⣿⡇⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠟⠁⢸⣿⡇⠀⢸⣿⡇⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⣠⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣄⣠
⠀⠀⢄⢰⠁⠀⢫⡩⢩⡍⢩⠁⡄⠠⣎⢀⡄⠰⠉⠀⣀⣤⠆⣀⣀⠀⢀⢿⡍⠂⠀⠄⠀⠠⠀⠌⢠⡈⡄⠙⠋⠃
⠀⠀⠀⠂⠀⠀⠚⠂⠶⠂⠐⠈⠀⠀⠀⠉⠀⠘⠒⠛⢻⠀⠀⠀⠀⠀⣤⡼⠆⡅⠈⠈⠐⠀⠀⢸⠋⠙⡆⡐⣠⡈⡄
⠀⠀⣰⢞⠒⣄⠉⠀⠼⠄⠀⢠⣾⣯⢳⡅⠐⠽⠀⡀⣈⣇⣀⡀⠀⠵⠃⠶⠸⠫⠢⠀⠀⠛⠀⠀⠆⠀⣐⠋⢹⡖
⠀⠸⣅⠄⡌⠰⠁⡠⡀⠀⠀⢀⢯⣽⡾⠃⠀⠀⠁⠀⠈⠀⠀⠀⠰⠰⠄⠀⠀⡀⠤⠄⡀⠀⢂⡦⡌⠈⠁⢀⠜
⠀⠀⢈⠳⣄⠃⡘⣰⠏⠘⠃⠀⠁⠈⢀⠀⠀⠐⠿⡁⠄⠄⠽⠀⠀⠀⡄⠀⠄⠀⠀⠀⠈⡀⠈⠘⣃⡀⣀⢁⠸⠍
⠀⠈⠥⠀⠈⠙⠞⠁⠀⠀⣸⠴⢦⠄⠈⠀⡧⡠⠶⢶⠷⢳⡃⠿⠀⠀⠂⠀⠀⠂⡤⠤⠚⠀⢠⠚⠀⠀⠀⠀⠓⠄
⠀⠰⣲⠂⢀⠀⠠⠈⠈⢠⣾⡿⣿⡆⠨⠓⠄⠀⢀⣤⠀⠦⠔⠆⠀⠀⠛⡄⠠⠀⠀⠀⢴⠆⠨⠀⠄⠁⢌⡠⠰⠁
⠀⢠⡶⡰⡑⡞⠒⡅⠀⢸⣵⢿⣮⡇⢀⠊⢀⣳⠀⠁⢀⡶⢒⡛⠒⢢⡌⡵⠀⠀⠊⠑⠄⠀⠔⠀⠀⣠⡄⠀⣤⣆
⠀⠀⡀⠀⠀⠀⠀⡀⠀⠈⠻⠿⠟⠁⠀⠀⠐⢮⠀⠰⠇⠰⠋⠈⡓⠉⠀⠀⠀⠰⣀⣄⣬⠀⠑⠐⠘⣏⠏⠀⠈
⠀⠐⡇⠀⢀⠀⠀⢀⡈⡢⡄⠀⡶⠀⣇⣀⠀⣸⠀⠀⣄⣀⠀⢜⢇⠀⢀⢄⢀⣀⠐⣇⢫⠀⢠⡀⠈⡗⠃⠀⠛⠛
⠀⢘⡇⠀⡖⢶⢩⠦⣌⢁⠌⠀⠀⠀⠈⣫⡇⠛⠀⢠⢿⢧⠀⠈⠒⠉⠑⣁⢠⠉⠀⠐⠉⠀⠙⡇⢀⡇⢀⡾⠂⠒
⠀⠠⠇⠀⠑⠤⠖⠀⡀⠀⣤⡞⡉⣧⠀⢛⡀⠀⠀⣸⠀⠀⣇⠀⠙⠊⠀⡼⣿⣷⠋⠀⠘⠇⠀⠈⠢⠌⠋⠀⠀⠠
⠀⠀⢀⣀⣢⣀⡀⠒⣻⡆⢠⠞⣵⠃⠸⠭⠛⠀⣰⢛⣢⢰⣧⠗⠒⢄⡀⠀⠓⠈⢡⠞⠓⣬⢗⣄⠀⠒⠀⠀⡀⣤
⠀⠀⡠⣞⣙⡷⡂⢀⡀⢀⣬⡣⣹⡀⠀⣀⠒⠈⠈⠈⠁⠀⠀⢀⣴⡀⢠⠀⢠⠀⣄⠀⠀⠸⠙⡈⠐⣆⠂⠀⢀⡌
⠀⠀⠑⠹⠩⠃⠁⠘⠃⠘⠮⠤⠞⢁⡀⠋⠁⠰⡿⠷⡲⡒⣫⠝⠁⠀⠀⠂⠄⠀⠐⠄⣠⣁⡰⠋⢀⡀⡁⡛⣉
⠀⠀⡤⢴⠶⣆⡀⢰⢎⠂⠀⠀⣀⢷⡝⠀⠀⢀⠠⡀⠈⠢⠎⠀⠀⠀⢀⠀⠂⠀⠀⠠⠀⠀⠤⠀⢈⠉⠉⠋⠙⠯⠁
⠀⠀⡀⠸⠀⡇⢈⠀⠱⣟⡾⠬⠿⡼⣀⠀⠰⠈⠀⠀⠰⡀⠠⢔⡢⢄⢀⡀⠀⣀⠀⠐⠠⢀⠂⠀⡶⠀⠀⠓⠀⠠
⠀⠀⠉⠛⠒⠓⠋⠀⠨⢿⣁⠒⣊⣼⠗⠀⠐⠀⢀⠀⠐⠁⠨⢵⡪⠅⢸⠀⢀⠘⣆⠀⠂⠀⠉⣁⠑⠃⠠⢄⠘⡀
⠀⠉⠙⠉⠑⠉⠁⠉⠀⠘⠉⠁⠈⠈⠀⠀⠀⠈⠁⠋⠁⠀⠀⠀⠀⠀⠈⠃⠁⠁⠁⠀⠀⠀⠈⠁⠀⠐⠁
⣿⣿⠛⢛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⠀⢀⣠⡀⠀⣠⣤⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣿⣿⣿⣤⣶⣾⣽⣿⣶⣾⠅⣂⡠⠼⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠟⠉⢶⣿⣷⣿⣟⣿⡿⢿⣯⣿⣿⣥⡟⢺⡤⣗⠒⠒⠐⠂⠛⣤
⠀⠀⠀⠀⠀⠀⠀⢰⢃⣜⣢⣼⡿⣯⢿⣿⣏⣹⣇⣿⡿⣟⢯⠻⣿⣿⣤⣿⣷⣶⣀⣤⣞⢿⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠸⣯⣼⣿⣿⣿⣿⣿⡿⠿⠉⠙⣿⣿⣜⡎⢃⠣⢸⣻⣿⢿⣽⣿⣷⣼⠘⡏⣷⡄
⠀⠀⠀⠀⠀⣴⣦⣜⣿⣿⣿⣿⣿⢛⣂⣇⠀⢀⠼⠛⢿⣿⣿⠛⠒⠓⢿⣿⣿⣿⣿⣻⣿⣧⣇⣲⣿
⠀⠀⠀⠀⠀⡯⣷⢻⢷⣿⣿⣿⡟⡛⠉⢙⠖⠉⠀⢀⠔⠉⢳⣄⠀⠀⠀⣿⣿⠏⣸⣿⣿⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⡗⣿⢼⢊⣿⣅⣝⢳⠄⠀⣠⠃⠀⠀⠸⠀⠀⢸⡿⠀⠀⠀⢿⡿⣾⣏⣿⣿⣯⣿⣿⣿⣿⠷
⠀⠀⠀⠀⣰⠖⠛⢾⣸⣿⣿⣯⣨⣶⣾⣯⠀⠀⣠⡃⠀⠀⠘⣇⢣⡀⠀⠀⠛⠻⣽⣷⢻⣯⣅⡼⣌⠙⡄
⠀⠀⠀⠀⣽⢂⠀⠀⠈⢯⣿⣿⣿⣽⡟⠁⠀⢰⣯⣤⣤⠄⣭⣿⣯⡵⠀⠀⠀⢰⡟⠃⠁⢺⡤⢶⠃⠀⠨⡂
⠀⠀⠀⠀⠘⡄⢰⡀⡀⠸⣷⣿⣿⡿⢟⣧⡀⠀⠀⠀⠈⠒⠋⠀⠀⠀⠀⣀⣤⣾⣟⠀⢀⣉⣷⣤⣁⠀⢀⠇
⠀⠀⠀⠀⣤⠛⠋⠉⠃⣢⣼⣿⣿⣏⡙⠲⢷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣽⣿⣿⢿⣿⣿⣽⡏⠃
⠀⠀⠀⠀⠘⢶⡂⢀⣀⠯⣨⣿⣹⣿⣿⣏⣺⣿⣻⣷⣦⣀⠀⢀⣤⣺⣿⣟⢙⣿⡿⣿⣿⡿⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠈⠱⣿⣾⣿⣿⣷⣿⢿⣿⣽⣁⣽⣿⣿⣿⣭⣵⣶⣾⣿⣿⣾⣯⣿⣿⣛⠄⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⡟⠋⠉⠙⢿⣽⣻⣿⣿⣿⣿⡟⣠⠿⣿⣾⣿⣿⣿⣿⣿⡆⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⣄⣘⢼⣹⣍⣿⣿⣿⣿⣿⣿⡽⡋⣟⣺⣿⣾⣽⡗⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⢿⣿⣿⣿⣝⡻⢭⣩⠿⠬⠤⣥⡽⢻⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠉⠹⡿⠿⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡲⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢰⠏⠂
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠂⠀⠀⠠⠒⠀⠚⠀⠀⢀⡠⠀⠂⠃⠀⠀⠀⠀⠠⠀⠀⠀⡄⠀⠀⢸⡆⣰⣧⣆⠒
⣴⣧⣿⣇⡆⠀⠀⠘⣯⠐⠂⠀⠀⢲⡄⠀⠀⠂⠻⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⢰⢷⠀⢠⣾⢷⣭⠌⠛⢶⠒⢠⡆
⣻⡟⢫⣿⣧⣷⣏⡠⠀⠔⠐⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⡔⢯⡀⢘⡁⢂⠪⡦⠴⡘⢦⣝⢳
⠚⣯⡶⢹⣿⣿⠋⠂⠀⠂⠐⡈⢺⣶⣆⣤⣀⣤⢀⠀⠀⠀⢀⡀⣤⢀⣠⡊⠃⣴⠞⠃⢜⡺⠤⠉⠱⣿⠀⠈⠀⠙⢿
⢈⠁⠖⢸⣉⡤⠎⣠⠆⠀⠀⠐⠲⣿⣿⡿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠈⠈⠺⣂⠘⣯⡤⠒⣀⡠⣜⡀⠀⠈⠶⢾
⠸⠄⢠⣟⡉⠠⠠⠶⢀⣰⢿⠑⢠⣙⣯⢓⣹⣭⠟⣋⡁⠀⠀⠀⠙⢿⠀⠈⠙⠳⣎⢢⡟⠒⠉⠈⠣⠌⡂⠀⠠⡀
⠀⠀⢠⡏⠠⢰⠂⢠⠊⠙⢄⣯⡀⠀⠉⠁⠀⢄⣪⣷⣻⣓⣦⣴⡶⠂⠀⡀⠁⣌⠚⢿⣗⠄⣇⠀⢰⡆⠈⠲⡀⡀⠑
⠀⡰⠋⠀⢀⡬⣟⠀⠀⠀⢣⣀⠧⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠉⡋⠀⠀⠉⠐⠉⠄⠀⠁⢿⡁⣀
⠀⠀⠀⠈⠀⠀⠀⠙⠊⠀⠊⠁⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠄⠘⠉⠀⠄⠐⠤⠦⠀⠽⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣶⣶⣾⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⣠⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡈⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⣿⡇
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢿⣿⣶⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣷⣰⡿
⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⣠⣴⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣿⣿⣿⣻⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡰⠋
⠀⣀⣀⡀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠛⠋⠉⠛⠲⢶⣾⣅⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣠⠖⣫⣽⣯⣭⡉⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠉⣶⣶⣿⣆⠀⢸⣿⣿⡇⠀⢀⡾⠁⠼⠿⠿⠟⠛⠁⠀⢀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣆⠀⣿⣿⣧⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣴⠟⢉⣿
⠀⠀⠀⠀⠀⢴⣷⢸⣿⣿⠙⣿⣆⢹⣿⣿⠀⠸⣧⠀⠀⠀⢀⣠⣾⠟⠁⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠘⣿⣿⣿⣿⡇⠀⠙⢷⣤⣴⣾⠟⣁⣀⣴⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠘⣿⣿⣿⣷⠀⣀⠀⠈⠙⠛⠛⠛⠋⠁⠀⢀⡀⢀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠈⠀⠀⠀⣸⠤⡇⡀⢠⢒⢄⡆⠐⡄⠥⢦⣮⠜⠾⠽⣀⣤⣶⣤⣷⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠝⠁⠀⠈⠒⠁⠈⠉⠡⠀⠀⠀⠄⠘⠠⠿⢷⠿⠿⠿⠿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠀⡲⠚⠻⣿⣟⢛⣶⣂⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⠀⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⢀⣾⣧⣠⣠⣿⠿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣦⠛⢿⣿⣿⣷⠀⠀⠀⠀⢰⣅⣹⠿⠛⢿⣿⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠋⠀⠀⠈⠉⠀⠀⠀⠀⠀⠘⢿⣿⣄⣀⡼⠋

⠀⠀⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠃⠘⠀⠃⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⡇⠀⢰⣶⣲⣶⡲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢖⣶⣒⣶⡆⠀⢸
⠀⠀⡇⠀⢸⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡇⠀⢸
⠀⠀⡇⠀⢼⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⢦⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠍⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠁⠒⢆⡒⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣐⣊⠝⣂⠁⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠻⣽⣦⡁⢦⠀⠀⠀⠀⠀⢀⡴⠛⠂⣠⠀⠀⠀⠀⠀⢀⡖⣀⠾⡽⠒⢬⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡹⣄⠈⠰⡁⠀⠀⣷⡄⠘⠇⠀⠀⠛⠃⣴⢷⠀⠀⣡⠈⠀⣴⣋⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠂⠀⠀⠙⡭⠛⢦⡀⠘⢧⣀⢣⣋⣱⣤⡀⣀⣶⣇⣹⣆⣤⠿⠁⣠⠞⡫⣿⠁⠀⠀⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡖⠒⣛⡦⠉⠛⣀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣮⣄⡼⠉⠠⢶⡍⢉⣤⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣊⠶⠄⣀⠌⠁⠂⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡗⠈⠠⢤⠴⢚⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣳⠖⢂⣨⡀⠀⠀⠘⠻⣿⣿⣿⣏⣿⣿⣷⣬⡉⠲⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⢀⡠⣾⣷⣶⣶⣶⣦⣤⣈⣿⣿⣿⣇⢀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠸⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⢻⡇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠡⠖⣀⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣨⠑⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⠞⣠⠈⠄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠲⢁⣄⠳⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣩⠆⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠳⡄⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⠔⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠣⣿⣑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣷⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⣿⣿⣿⣿⠟⡜⢿⣿⡆⠀⠀⠀⠀⠀⠉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⣻⠃⠈⣿⣿⠏⢧⣴⢸⡙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⡇⠀⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣇⡼⠁⠀⠀⠘⠛⠁⠈⠻⠸⠇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠀⢸
⠀⠀⡇⠀⡹⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡇⠀⢸
⠀⠀⡇⠀⠼⠿⡹⠿⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣞⠿⢯⠿⠇⠀⢸
⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⢀⠀⡀⢀⠀⢀⠀⠀⣄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣇⣯⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡭⢫⠄
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⢿⣷⣿⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡓⠾⢿⡖
⠀⠀⠀⠀⠀⡒⠢⠔⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣿⣿⣿⣛⣞⠀⠀⠀⠀⠀⠀⠀⠀⣸⣮⠑⠦⣽⠅
⠀⠀⠀⠀⠲⠌⠁⠂⢁⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣷⣿⡿⢿⡥⠀⠀⠀⠀⠀⠀⠀⠐⠒⣻⠴⣄⢯⡥
⠀⠀⠀⢄⣁⠀⠁⢀⠉⠀⠀⠀⠀⠀⠀⠀⠠⢤⡼⣿⣾⣿⣿⣃⠄⠀⠀⠀⠀⠀⠀⣉⡭⢩⠤⢄⣈⣟⢉
⠀⠀⢰⡆⡀⢩⠁⡀⠀⠠⠀⠀⠀⠀⠀⢀⣀⣸⣷⠟⣻⢿⡟⣿⡷⠂⠀⠀⠀⠀⠠⠠⣧⣶⢴⡍⢬⡭⢐
⠀⠀⢄⠠⢠⡀⠀⠀⠀⠂⠈⠀⠀⠀⠀⣀⠙⢛⣿⣶⣾⣼⣷⣿⡿⠋⠀⠀⠀⣀⠮⢙⢁⣋⣬⠉⠔⣊⣅⢉
⠀⡘⠀⡀⠀⢀⠁⠀⠄⠀⠀⠐⡄⠀⠀⠘⠺⣫⣿⣥⣾⣧⠿⣿⣿⣿⡞⠁⠀⣸⠿⠷⢦⠀⣩⢁⣈⡉⢛⣀⢀
⠀⠈⠁⠀⠂⠀⠂⠐⠐⠀⠀⠀⠀⠀⠈⠙⠛⣻⣿⡭⣿⡻⣿⣭⡤⠿⠟⠂⠀⠉⠛⠑⠙⠿⠛⠀⠉⠁⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠑⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤
⠀⠀⠐⢠⠂⢀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠀⠀⠒⠬⢘⢭⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠉
⠀⠄⠐⠂⠀⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠄⠀⠀⠀⠀⠁⢀⠐⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢰⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘
⠠⠀⠌⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⡤⠓⠀⡀⠀⠀⠀⠀⠐
⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠰⠋⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤

⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⠄⣖⡶⢔⡠⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠌⢘⣦⡀⠄⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠋⠔⠚⠀⠀⠀⠀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠈⠀⠈⠄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣩⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⢀⣨⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣲⣫⠄
⠀⠀⠀⠀⢠⣄⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⣩⢧⠶⠂
⠀⠀⠀⠀⣄⣈⣱⣆⣠⠤⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣤⣩⣿⣦⣶⠛
⠀⠀⠀⢠⡌⠉⢡⣎⣁⣀⡄⠀⣠⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣉⣤⣾⣥⡼⠳⠶⠾⠃⢀⠀⣠⠄
⠀⠀⠀⠀⠓⠚⣯⠉⢉⢁⣠⠀⠋⡀⠀⠀⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⠀⡞⠀⠀⠀⢰⠃⠁⢺⣤⠿⢶⣧⡾⢿⣶⠶⠾⠋⠊⠀⠟
⠀⠦⠐⠶⡴⠾⢛⡓⠟⠉⠁⠀⠀⣧⠀⡀⢀⡀⠀⠈⢧⡄⠠⢠⠆⠀⠐⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⣟⣫⡿⠶⢟⡿⠶⠶⠂
⠀⠀⠶⣦⣀⣴⠞⠛⣶⣄⣼⠻⠇⠘⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣳⣫⣿⣭⣿⣟⣻⡷⢶⡾⠇
⠀⢀⣤⣬⣡⣂⣴⣾⣇⠈⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣙⣋⣽⠶⠖⠺⢛⣿⣷⣤⣤⣶⠄
⠀⠉⠉⠉⠙⡉⠉⣹⡋⠛⠋⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠛⠛⠶⠼⠿⠿⣿⣧⣤⣤⡤⠆
⢀⡝⠛⢿⣍⣠⠾⠻⡿⢷⠆⣦⠆
⠈⣿⠏⣉⣈⠈⣀⢾⣧⣀⠀⡾⠟
⠀⠀⠀⠀⠿⠋⠏⠈⠙⠛⠋⠁
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⣿⠙⢦⣴⠲⢤⡀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣫⠇⣠⠞⠛⠁⢀⠀⠀⣀⣀⣙⣛⡁⠳⠶⠊⢩⣏⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢰⢟⣛⡵⠞⠛⠛⠿⠷⠞⠛⠩⣍⠉⠙⠛⣶⠻⣟⣿⠹⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠞⠀⠀⣿⠷⠖⠓⣆⠀⠀⡟⠢⢄⠀⠉⢀⣀⣠⣾⠀⠈⠉⠀⠀⠉⠢⣀
⠀⠀⢀⡀⢀⡴⠋⠀⠀⠀⠀⠁⠀⠀⠀⠀⡹⠋⠀⠀⠈⠳⠶⠛⣁⠀⠀⠈⠀⠘⢇⡈⠆⠀⠈⢷⣄
⠀⠀⢸⢋⣿⠐⠁⠀⢀⠀⠀⢀⣴⣦⡀⠀⠀⠀⠀⠀⠲⡄⠀⠄⣈⠀⠀⢀⣴⣤⠀⠈⠃⠀⠀⠀⢯⢧⡀
⠀⣿⣾⡟⠛⠀⠀⠀⠈⢢⡀⣾⣿⣿⣷⢀⠀⠀⠀⠀⠀⢋⡀⠂⣹⠂⠠⣿⣿⣿⣧⠀⡀⠀⢀⠀⠀⠀⢻⡄
⢰⢧⡏⣰⠃⠀⠀⡀⠀⠀⢱⡝⣿⡿⣱⡇⠀⢰⡀⠀⠀⠀⢨⠟⠀⠀⢰⡝⢿⡿⣻⡏⠀⠀⠘⢆⠀⠀⠀⢹⡄
⢸⠇⠉⡟⠀⠀⢰⠃⠀⠀⢸⣿⢎⠸⣿⡇⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣾⠟⣀⢺⣿⡇⠀⠀⠀⡘⢱⢸⡄⠀⢻
⡾⠀⠀⠀⢠⠀⣿⢄⠀⠀⠘⣵⣿⣷⣝⠇⠀⠀⢈⣇⠀⠀⠀⠀⠀⢆⣴⣾⣿⣷⡽⠇⠀⡀⠀⠰⣹⠰⠄⡄⠘
⡇⠀⠀⠀⢈⡴⢁⣀⠀⣠⠞⣿⣿⣿⣿⠦⡀⠀⢈⠈⠳⡄⠀⣠⠖⢯⠸⣿⣿⣿⡿⠢⠀⠹⡄⠀⠙⡸⠂⠆
⡇⠀⠀⢾⢨⡴⠊⠈⠔⠁⠀⠘⢿⡿⠋⠀⠙⢆⠀⠁⡌⢻⠀⡟⢆⠘⠆⠹⣿⡿⠁⠀⠀⠀⡇
⣧⠀⠈⠲⣈⠀⠀⠀⠀⠀⠀⢲⣀⠀⣠⠂⠀⠀⠑⠄⠙⢺⡇⡧⠚⠂⠀⠀⢠⡤⠞⠀⠀⠀⠉
⢸⡄⠀⠀⠁⠀⠀⣴⡒⠂⠀⠀⠈⠉⠁⠈⣦⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⡾⠁⠀⠀⠐⢲⢦⠀⠀⠀⠀⠀⣸
⠀⢷⡀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠘⣤⡀⠀⠀⠀⠙⠀⠀⠀⠀⢰⠁⠀⠀⢀⡴⠃⠀⠀⢀⡄⠀⠀⣿
⠀⠈⢷⡀⠀⠀⣆⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣴⣶⠋⠀⠀⠀⢀⠾⠀⠀⠀⣏
⠀⠀⠈⣷⠀⠀⠈⡗⡄⠀⠀⠀⠈⠻⢶⣤⣄⣰⣄⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋⠀⠋⠀⠄⠀⡴⢣⠤⠀⠀⢀⡇
⠀⠀⠀⢸⡆⠀⠸⠀⢷⠀⠀⠀⠀⠀⠈⢿⠀⠉⠉⠛⠛⠛⢿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢄⡷⠁⠀⣴⡀⢸⠃
⠀⠀⠀⢸⡇⣀⠀⠐⠬⢛⠀⢻⠀⢀⠀⠈⠀⠠⣄⠀⠀⠀⠈⠀⢀⠊⠀⢀⡔⠀⠀⠀⠀⠈⠀⢀⡞⠉⣇⡞
⠀⠀⠀⢸⣰⡟⡆⠀⠀⠠⡀⢏⡇⠘⡇⠀⠀⢀⡜⠳⠀⣦⠀⠀⣼⠀⢠⡾⠀⡀⠀⠀⠰⠹⠂⡜⠀⠀⠻⠇
⠀⠀⠀⠈⠛⠁⣧⠀⣴⠋⢻⡄⠁⠀⣻⠀⠀⠈⢧⠀⡄⠹⡇⠀⠁⠀⡾⢧⡀⠻⣴⡀⠰⠂⠀⡇
⠀⠀⠀⠀⠀⠀⢹⢠⡇⠀⠘⢧⡀⠀⣿⡀⠀⠀⠈⣄⠷⣠⠃⠀⠀⣠⡇⣼⠿⣄⠘⠇⣰⢻⡆⡇
⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⢱⢸⡇⠳⣄⡐⣄⠈⠒⠃⠀⢠⠞⢻⠁⡏⠀⠈⢧⠀⡇⢸⣇⡇
⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⢸⢸⡇⠀⠈⣿⣿⣆⠀⠀⢠⠏⠀⠀⡇⠃⠀⠀⢸⡆⡇⠀⠉
⠀⠀⠀⠀⠀⠀⣾⢸⡇⠀⠀⠀⢸⠀⡇⠀⠀⢿⡘⢹⡀⠀⡏⠀⠀⠀⠃⠀⠀⠀⠈⡇⡇
⠀⠀⠀⠀⠀⠀⢻⠀⡇⠀⠀⠀⢸⢀⡇⠀⠀⠀⡇⠀⡇⠀⡇⠀⠀⢸⠀⡀⠀⠀⠀⡇⢀
⠀⠀⠀⠀⠀⠀⠘⠾⠃⠀⠀⠀⠈⠛⠀⠀⠀⣾⠀⠀⣷⠀⡇⠀⠀⠘⣦⠃⠀⠀⠀⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⡟⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣠⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠇⠀⠀⠀⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠠⠀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⡀⠀⢹⠀⡀⠀⠀⢁
⠀⠀⠀⠀⠀⢀⡤⠒⠉⠁⢱⠀⠐⠟⠀⢠⠀⠋⠀⠀⠄⠀⠉⠑⠦⡀
⠀⠀⠀⠀⠀⢸⠀⠠⡖⠀⠀⠑⠄⠀⠠⠂⠑⠀⠀⠈⠀⠀⢀⠆⠀⢳
⠀⠀⠀⠀⠀⢸⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⡘⠠
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠏⠀⠀⠂⠀⠀⠀⡰⠂⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⡉⠁⠦⠁⠘⢀⠄⠂
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠧⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢠⠂⠤
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠂⠁⠉⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⣄⠲⠖⠁
⠀⠀⠀⠀⠀⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡗⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⢓⡖⠒⠒⠒⢲⠒⠒⠒⠒⠒⠒⠒⢲⠖⠒⠒⠒⠒⠒⣶⡿⠞⠉
⠀⠀⠀⣠⡶⠛⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡂⠀⠀⠀⠀⠀⣿
⠀⣴⠞⠉⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⣿
⠀⢻⣄⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢻⠦⢤⣀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻
⢀⡤⠞⠀⠀⠈⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣭
⢇⠹⡀⠀⠀⢀⡇⠀⠀⣀⠼⢄⠀⠀⠀⠀⠀⢀⠼⢇⠀⠀⠀⠀⠀⣿
⠀⠑⠢⠴⠒⠊⣧⠒⠉⠀⠀⠀⠙⢦⣀⡤⠚⠁⠀⠀⠙⢦⣀⡠⢔⡿
⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠶⣶⣶⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⠟⠁

⠀⠀⣰⠞⠓⢦⡤⣄⣠⠞⠉⡲⠲⢤⡀⢀⣴⡦⠀⣴⣶⠂
⣠⡼⢯⣀⣄⣾⡀⢹⣏⠁⣾⡷⣄⠸⠛⠀⠑⠒⢨⣄
⢿⣤⡀⠘⡿⠾⠓⡶⠛⢲⣧⣤⠏⠀⠀⠀⠀⠀⠀⠙⠁
⠀⡹⠷⠾⣤⣠⣾⣧⠌⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣴⡆⠀⠀⣤
⠸⢅⡛⡢⠉⠉⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢣⡁⢾⡿
⠀⠀⠙⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⢧⠀⠂
⠀⠀⠲⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠿⠿⠘
⢠⡾⡂⢱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡁⠀⠀⠀⡀⢠⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣵⣿⡄⠀⢇⡰⣤
⠀⢀⢱⣎⠙⠁⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠁⠁⠃⠇⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣈⡀⣼⣿⠟
⣠⡆⢁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣇⢈⢡
⠈⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⡴⡆⡄⢠⠀⠀⠀⠀⠀⠰⣿⣾⡆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠣⠀⠂⠘⠀⠀⠀⠀⠀⠀⠉⡛⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡋⣀⡛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠀⢀⠀⣠⣤⣴⣾⠀⠘⡿⠓⢤⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠆⠾⠇⣴⣏⠧⠙⠟⣻⣞⣧⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣿⠆⠇⠀⢸⠿⠁⠀⠀⢀⡟⠉⠨⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠙⠋⣷⣦⣤⡴⠾⠷⣦⣾⢿⣿⣶⠄⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣿⡄⠐⠀⠀⠀⠀⠀⠀⢀⣀⣏⣀⣤⣴⡁⠀⠻⣿⣆⠀⢽⢿⣶⢿⡟⠦⣤⣀⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⡤⠶⠶⠄⠠⡤⣼⡍⠀⠀⠻⣯⢿⡧⢤⣀⣿⣿⠀⠀⣼⠃⠀⠁⠀⣬⣻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣷⡎⠀⢀⣘⢿⣿⠀⠀⠙⢯⣥⡄⢠⡏⠀⠀⣠⣿⠟⢿⣦⡾⢿⡀⡀⢠⠠⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠋⠀⠀⠀⠙⠃⠁⠀⠀⠀⠀⠈⠛⠋⠳⣶⣵⠟⠁⠀⠀⠁⠀⠈⠓⠿⠷⠟⠋
⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡤⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡠⡄⠀⠀⣠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⣡⢿⡈⠀⠀⠀⠀⠀⢀⡀⠀⠀⢸⠃⠀⠀⠀⢩⠄⡠⠹⠄⠀⠀⠀⠀⠀⠨⠥⠀⠠⠬⠀⠀⠈⠁
⠀⠀⠀⠉⣺⠈⠒⠀⠀⠀⣠⣾⣿⡀⠀⠠⠂⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠉⢸⠿⣿⣿⠿⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠁⠂⠁⠀⡤⠀⠀⠀⢠⠄⠀⣼⡇⠀⠀⠀⠀⢸⠇⠀⠀⠀⣲⣆
⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⣀⣀⣀⠀⠀⣴⣶⠀⠀⣤⣤⣤⣤⣤⡄⢀⣿⣿⠀⠀⠀⠀⠸⠂⣠⣤⣤⣌⠁
⠀⠀⠀⣴⣿⣿⣧⠀⠀⠀⠈⠉⡧⠀⠀⠈⠀⠀⠀⠻⢿⣿⣿⣿⠀⣸⣿⣿⡇⠐⣞⣳⠂⠘⠋⣉⢉⠙⢧
⠀⠀⠈⣉⡤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠀⠉⠉⠀⠀⠙⣀⠂
⠀⠀⠀⠈⠠⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⣠⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⢿⠿⠆⠀⠀⢠⣶⣶⡄⠀⠀⠀⠀⠴⠿⢿⡇⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣶⠆⠀⠈⠛⠛⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⠃
⠀⠀⠐⠿⠾⠿⠂⠈⠀⣀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠘
⠀⠀⠀⠸⢆⡀⠀⠀⠀⠉⣀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠸⣶⣀⢀⡀⠀⢠⡄⠀⠀⣠⣶⣿⣿⡆
⠀⠀⠀⢰⡌⠃⠀⠀⣠⠀⣭⠀⠀⢀⡀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠉⠀⠀⠘⢷⣤⣴⣿⣿⣿⠟
⠀⠀⠀⠘⠃⠀⠀⠀⠉⠀⠛⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠘⠃⠀⠐⠀⠋⠙⠉⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣾⡟⠋⠉⢿⣴⠶⢤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢣⠀⠀⡼⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣳⡦⣤⡷⢶⣷⠲⢿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣊⡴⠚⢉⣴⠞⢹⡇⢸⠈⠳⣄⠈⠻⣆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣾⣁⠀⣰⠟⠁⠀⢸⡇⢸⠀⠀⠘⢧⡀⣽⡇⠃
⠀⠀⠀⠀⠀⢀⣀⣐⠒⣠⠤⠤⠶⠶⣮⡭⠿⣞⣹⣄⠀⠀⢸⡇⢸⠀⠀⠀⠈⣿⠃
⠀⠀⠀⠀⣠⠟⠡⡍⢻⡇⠀⠀⠀⠀⡿⣸⠃⠈⠛⣷⡖⠒⢾⠀⢸⠲⠶⠶⢶⡟⠀⠀⠀⠀⠀⠀⠒⠄
⠀⠀⠀⠐⣇⠀⠀⠙⣾⣇⠀⠀⠂⢸⣿⠏⠀⠀⣰⣿⠀⠀⢸⠀⢸⠀⢀⠀⢸⠇⠀⣤⠶⠶⠛⠛⠛⢷
⠀⠀⠀⠀⠈⢧⣤⠖⠚⠿⢤⣴⠤⠟⠛⢶⣡⣾⣋⣿⠀⠂⢸⡇⣿⠠⠀⠀⣿⣤⣼⡇⠀⠀⠀⠀⠀⢸⣷⠞⡛⠳⣦⡀
⠀⠀⣠⡤⠶⣾⠃⠀⠀⠀⣴⠛⣆⠀⠀⠈⢿⡉⠛⠛⢻⡆⢸⠃⢻⣠⡴⠾⠋⢰⡂⣇⠀⠀⠀⠛⠀⢸⡇⢠⠇⠀⠈⠙⣷
⠀⠀⡇⠀⢰⣏⣴⣦⠀⣼⠃⠀⢸⡆⠀⠂⠈⣧⠀⠀⢸⡗⢺⠀⢺⣹⡇⠀⠀⠀⢳⡸⡄⠀⠀⠀⠀⢸⣧⠏⠀⠀⠀⢀⡿⢰
⠀⡆⢿⣠⣼⣯⣤⣽⣴⡇⠀⠀⠀⢿⠀⠀⠀⢸⡶⢶⡾⠛⣽⠀⢸⠀⢳⡀⠀⠀⠀⠹⣷⠀⠀⠀⠀⣼⡋⠀⠀⠀⣠⡾⠁⠁
⠀⠘⡄⣿⠀⠀⠁⠀⠉⣧⠀⠀⠀⢸⡇⢰⠷⣬⡇⢸⣇⡀⢸⠀⣸⣀⣀⣿⣦⠔⠒⠒⠚⣳⡤⡖⠋⠉⠉⠳⣶⠾⠿⠶⠶⠶⠶⣦⡈
⠀⠀⠀⢿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣷⡏⠀⠘⠁⢸⡇⠁⢸⣼⡏⠉⠁⢠⠏⠀⠀⠀⡼⠁⠀⠈⢦⡀⠀⠀⠘⢷⣀⣀⣀⣀⣀⣼⠃
⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠁⢸⡏⠀⠀⠀⠀⣸⠄⠀⢸⡝⣷⡤⢤⣿⡴⢶⡄⢸⠗⠖⠛⠛⠻⢷⡀⢠⣄⡈⢿⠉⠉⢉⡏⠁
⠀⠀⠀⢸⡇⠀⠈⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠠⠀⣾⠠⠀⢸⡇⣿⢹⡄⠁⠀⠀⠙⠟⠀⠀⠃⠀⠀⠈⣧⠟⠈⠙⠻⠆⠀⣾⠁
⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⡋⠀⢀⣀⣿⣤⣤⣼⣇⣿⣄⣱⣤⣤⣤⣤⣤⡤⣤⣤⣤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠋
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠶⠶⠶⠾⠋⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢄⠀⠀⠀⠀⠀⠀⠀⡇⠅⠃⠇⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢩⠕⣄⠀⠀⠀⡠⢾⠀⠀⢀⡋⠉⠉⠢⡄⢀⠊⢀⣶⡆⠀⠀⠉⢂⠥⡍⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠁⢀⠈⢸⠀⢠⠎⠸⠀⠻⣀⣉⣀⣈⣉⣉⠺⣤⠀⠀⠩⣶⠦⣾⠇⠀⠬⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢦⣀⠀⠢⠁⣘⢀⠀⡆⠀⠑⠤⣀⡀⣀⣀⡀⢤⣑⡏⠣⡐⠴⡀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣯⠙⢦⡀⠉⠉⠛⡉⢄⢣⣙⣢⣄⣀⣀⣀⠤⠄⢲⠘⠏⠏⠀⠀⠀⠐⠀⠐⢪⠀⠀⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠺⡥⠀⠑⢄⠙⠢⡀⠀⠈⠂⠣⣣⣩⣶⣎⣵⢶⡝⢇⡊⣏⣪⣓⠀⠀⠐⡉⣘⠂⠀⠁⠀⠐⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠀⠙⠦⣈⠑⠤⣀⠑⣘⢸⣿⢿⣿⣛⡱⣾⣷⠼⣿⣭⢆⡀⠀⢉⠤⢐⠲⠚⢭⡽⣒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⣴⢠⢉⡑⠢⢄⣑⠪⣸⣿⣤⣽⡟⠁⢾⣧⡼⣧⣿⢧⠗⢈⠄⠊⠁⠀⡤⣀⡹⡀⠑⡱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠩⠂⢈⡉⠉⠆⠉⠋⠘⣵⡶⣯⠉⠉⢱⡶⣮⠉⠁⠀⣶⣩⣶⠻⡄⠀⡀⠀⠓⠢⠄⣙⡄⠔⡡
⠀⠀⠀⠀⠀⠀⠀⠤⠓⢺⠊⠁⠀⠀⠀⠁⠄⢠⣿⡿⢻⣿⣧⢸⣷⣟⠛⣿⣟⣛⣿⠣⢠⠞⠉⢹⠀⡔⠁⠀⠀⣹⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡆⠀⠄⠠⠈⠀⢁⣿⠛⡝⣾⠷⢿⡟⡣⠝⡓⣿⡧⢜⠃⠀⠀⡓⠚⢇⠀⠈⢀⠡⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠎⠓⠺⡁⠈⠐⠀⡠⢴⡛⠁⢀⠘⢊⠈⠈⠑⠀⠀⠀⠊⠈⠂⢄⠀⡆⠁⠀⠈⢓⡨⠑
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⠁⠀⠁⠀⠙⠡⡀⠻⡀⠈⠓⠢⠤⠤⠄⠀⠀⠀⠀⠀⠙⢄⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡔⠀⠀⠀⠙⠢⠤⠤⠤⠔⠠⡀⠐⣆⠀⠀⠀⠀⢣⢠⣄⠀⢀⡄⢠⡀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠈⠛⠢⡤⠀⡘⠁⠈⠒⠋⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⠤⠚

⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⠠⢟⡏⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡄
⠀⡼⣤⢻⡏⠱⣾⣶⣶⢩⣶⠞⢢⣤⣀⠺⢢⢀⡀⢸⠥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⣾⣷⣯⡷⠶⢠⡄⣀⣀⠬⡀⣀⣰⢸⡇⢰⡀
⠀⠾⣋⠴⡏⠲⣿⣿⠗⢏⠋⢘⡋⣻⠕⣇⠟⣿⠷⠞⣎⣘⡆⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⣼⢆⣸⡳⡔⢊⡅⣳⡓⣜⢸⣿⡿⡟⠠⡧⠽⣏⡙⢧⣽⣇
⠀⣟⠹⣤⣑⡾⣱⠹⡆⣵⡤⠾⡀⢟⡅⢻⢆⣋⣄⣨⢥⡭⠌⣿⣾⢶⡎⠿⣶⣶⡟⣿⣿⢴⡾⢇⢺⡃⢩⡁⣨⣿⡙⢤⣽⣷⠔⡳⢈⠲⣯⠤⣾⣦⡽
⠀⠑⠀⣾⡹⢆⣯⢀⣿⣇⡓⢛⣀⣋⢱⢾⣯⠔⡰⢣⡜⢻⠳⢼⢷⣸⠗⢂⠸⣿⢏⢹⢇⢚⣝⠺⣿⢳⡏⢻⠀⢇⠑⣤⡈⡛⠁⢤⠘⡄⢼⡐⡗⡸⢧
⠈⠀⠈⠻⠁⠴⣏⠹⠬⠘⠁⠛⠋⡔⢦⠟⣃⠹⢤⠎⠱⡌⢳⠦⡛⠇⠨⠤⠀⠘⠉⠋⠑⠃⠔⠬⠿⠞⠳⣼⡓⠢⣥⠆⡸⠙⡆⣨⠀⢰⢀⠛⡥⢁⠲
⠀⠀⢀⡆⠠⡆⡀⠓⢌⡁⠀⠀⠀⠀⠀⠀⠋⠁⠉⠓⠀⠘⠛⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢷⠀⢦⠈⠈⢶⢌⠛⡛⠒⠀⠀⠀⠀⠈⠉⠂⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠉⠉⠐⠈⠂⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⣦⠀⠀⠀⠀⠀⢠⡶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣤⣿⠆⠀⢀⣀⣄⣀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⢀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⣾⠋⠁⠈⠙⢳⣤⣤⣤⡀⠀⠀⠀⠀⠀⠸⣇⠀⠘⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⡇⠀⠀⣀⣠⣴⣿⠞⠋⠀⢠⣟⣻⡆⠀⠀⠻⣦⣀⠀⠉⠙⣛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠾⠿⣿⣛⣉⣡⡾⠃⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⢀⣤⢶⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠿⣧⡀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣸⠇⠀⢰⣶⠄⠀⠀⠀⠸⣿⣤⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠀⠀⠛⠷⠀⣶⡄⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣄⣀⣀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡴⠶⠶⠾⣦⡀⠀⢰⡏⠉⢹⡆⣾⠋⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠞⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣟⠛⣷⡀⣹⡿⠛⠛⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿⢾⡟⠶⠶⠞⠳⠶⠶⠛⠷⠶⠶⠛⠾⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠶⠾⠃




⣏⠉⣻⣛⣛⣛⣛⣻⣛⣛⣛⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣛⣛⣛⣿⣟⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⠿⣷⡿⢋⣤⣤⠄⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠀⠰⣶⣄⠀⣿⡇⠀⠀⢀⠴⠒⠶⣾⠀⢤⣭⣁⠀⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⢀⢉⣴⣾⠿⠀⠙⠻⢗⠉⢀⣤⣤⠾⠀⠀⠀⠈⢧⢨⡟⣻⠧⣄⡿⠋⠀⠀⢀⣤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠻⣭⡷⠶⠶⠶⢦⣼⣄⡉⠉⠁⢰⣷⠀⠀⠀⢸⣤⣿⣶⡄⢈⡇⠐⢦⣠⣿⠏⠀⢀⣾⡆
⠀⠀⣄⡀⠀⠀⢠⣴⣾⡟⠀⠀⣼⠋⠀⠀⠀⢰⡀⣠⠭⠿⡻⣶⣌⠻⠀⠀⢀⡿⠙⠿⠿⢁⣾⣁⡀⠀⡉⢀⣀⣀⢸⡿⠀⣶
⠀⠀⠙⢿⡄⢠⣿⠿⠏⠀⠀⠀⢿⠀⠙⠿⠿⢾⢸⠁⣶⡶⢤⡀⠹⣷⣄⣠⠞⠀⠰⢶⣴⠟⢋⣭⠉⠙⣦⣭⠟⠛⠙⠅⠀⡹⠤⣶⣶⣶⠆
⠀⠀⠀⢠⠁⢀⣀⣀⡀⠀⢠⣴⡞⠳⠤⣤⠤⠊⠸⣦⠀⠀⢸⣿⠀⠘⣿⣏⠀⣀⣴⠟⠁⣦⡿⠃⠀⣾⣿⣿⠄⣀⣤⡓⢈⡤⠀⠈⠉
⢴⣶⣶⡾⡶⠿⠿⠋⠀⠠⠿⠛⠀⣴⠞⠉⠑⠂⠀⠘⣧⣀⣠⣍⣀⠀⢹⣿⣿⡿⠁⣀⣧⡉⢡⣤⣄⠙⠛⢛⠎⠁⠀⣹⣿⣁
⠀⠀⠀⠀⠈⠳⠶⢶⡶⠶⠶⠶⣼⣇⠐⠶⠦⡤⠀⣰⠟⠉⢠⡀⠈⠳⣸⣿⣿⢡⢹⣾⠙⣇⠸⠿⠏⠀⢀⡞⢀⡄⠀⢿⣿⡿
⠀⠀⢀⣴⣾⣿⠦⠾⣴⣤⡀⠀⣈⠻⣦⡀⠰⠆⢠⡟⠀⠀⢻⣷⠀⠀⢻⣿⣿⠀⡞⠛⢀⣟⣀⣠⡤⠞⠋⠀⢾⣿⣰⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠉⠀⢴⣿⣾⠉⠁⢸⡿⠀⡈⠉⣟⠛⠻⢿⣄⠀⠀⠙⠂⠀⣼⣿⣿⠀⣡⡶⠛⠋⢉⣉⡙⠒⠤⣄⣀⠙⢯⣄⠀⠀⠿⣿⠇
⠀⠀⠀⢀⣤⣤⣬⡛⠋⠙⠷⢮⠴⠿⢃⣀⠿⣠⡶⠂⠙⠷⣦⣤⣤⣾⣿⣿⣿⣿⠋⠀⠀⠀⠺⣿⢇⡤⠶⠶⠶⢤⡀⠀⠀⠀⠀⢨⠀⢾⣿
⠀⠀⠀⣾⣿⣟⣿⣿⡀⠀⢀⣘⣆⡀⠻⠿⢀⣇⡀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣄⡀⠀⢀⡠⣱⠟⠀⠀⠀⠀⣀⣝⡓⠶⠶⠚⠓⠲⣌⠉
⠀⠛⠿⠮⠿⣿⠟⣛⣤⡖⠋⠉⠉⠉⣫⠽⠿⣭⡉⠉⠙⠛⠷⣦⣄⣿⣿⣿⣿⠏⠉⠉⠉⠉⣰⠏⠀⣰⣶⣆⠀⠿⡿⠃⠀⠀⢀⠰⡆⡮
⠀⠀⠀⣠⣿⡚⠉⠉⣹⡴⠟⠀⠀⣼⠁⠀⢀⣹⣿⡆⠀⠀⠀⠘⢿⣿⣿⣿⣿⡄⠀⣀⣤⡾⠋⠀⠀⠘⡿⠃⢠⣠⣄⡀⠀⠀⢨⢿⣿⢯⡁
⠀⠀⠀⣿⣿⠃⠀⣾⣿⣿⠀⠀⠀⢿⡀⠀⣿⣿⣿⣿⠀⠀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣶⣞⡋⠀⠠⡻⣿⡷⠄⠀⠀⠀⠛⠜⠂
⠀⠀⠀⠙⠁⠀⠀⠈⠋⠹⣦⡠⠒⠺⢷⣤⣘⣿⣻⣯⣴⡾⠿⢿⣿⣿⣿⣿⣿⡿⠋⠀⣴⣶⡀⠀⢈⡙⠲⣤⠖⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣶⣦⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠛⠛⠀⠠⣿⣿⡳⠈⡇⢰⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⢧⣁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠈⠀⠀⠀⠀⠁⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢧⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣄⣓⣤⣀⢳⣀⡤⠚⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⡤⠠⡄⠀⠀⣿⠀⠀⠈⠉⠙⡟⠺⡷⢮⣭⣉⣀
⠀⠀⠀⠀⠀⠀⠀⣼⡀⣈⠳⣄⣠⣼⣖⣙⣤⣯⣤⠀⠀⠀⢰⠀⠀⠇⠀⠀⠀⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣭⡿⡥⢞⠻⢍⡀⠀⠘⣧⣤⣄⡜⠀⢸⠀⠀⠀⠀⢀⡿⣀
⠀⠀⠀⠀⠀⠀⠀⣏⠉⠰⢯⡴⠇⠀⢳⡄⣾⠷⠶⣏⠀⠀⡇⠀⡼⢍⠙⢯⠁⣹⡇⠀⡱⠂
⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⣸⡥⠾⡀⠀⢏⠃⠀⠀⠹⣆⠠⠇⣸⣀⣀⣙⠦⣧⣼⣴⣋⣠⣤⡀
⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠙⠓⠦⣧⣀⠈⡆⠀⠀⠀⠹⣼⠀⢸⣁⣨⡭⠿⠿⣿⢿⠍⠀⠈⣷⡀
⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢾⣄⠀⢠⡶⠛⠛⠉⠁⠀⠀⣀⣀⣽⡎⣶⠚⠋⢻⡀
⠀⠀⠀⢰⣧⠤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠾⢧⣠⣤⠴⠖⢛⡏⠉⠀⠼⠛⢇⠐⣂⣈⣷⣠⣄
⠀⠀⠀⣰⠃⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠉⠀⠀⢹⡄
⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⡾⠀⠀⠀⡐⡗⣀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⢸⠃⠀⠀⠿⡪⡻⣗⠓⠀⠀⠀⠀⠀⠈⠙⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠚⠋⣧
⠀⣾⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠉⠛⠛⡛⠛⢻⠛⠛⠉⠉⠁⠀⢠⡄⠀⠀⢹
⠀⡏⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⠀⣀⠀⠀⠀⠤⣶⣶⠄⠀⠀⠀⢀⣇⠀⠈⠀⠀⠀⠀⠠⡮⣶⢕⠵⠆⢸
⠀⣇⠀⠀⠀⠀⠀⢀⢄⣹⣳⣿⣼⣏⡠⡀⠀⠀⠘⠀⠀⠀⠀⢰⠁⠀⣱⠔⠢⢄⠀⠀⠀⠈⠡⡇⠁⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠈⠉⣽⠟⣿⡻⣯⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠒⠴⡃⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⣸⠤⠤⠤⢀⣀⠀⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣆⠀⠀⠀⠑⠤⠤⠚⠁⠀⠀⢀⡀⠤⠒⠉⢧
⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈
⠉⠙⠛⠓⠒⠒⠶⠶⠦⠤⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡤⠴⠶⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⡠⠀⠀⠑⠀⠀⠀⠘⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠀⢀⡈⠐⠀⠀⠀⡀⠀⠀⢀⡁⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⠀⠀⢰⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠢⢄⠀⠁⠂⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⠠⠊⠀⠀⠀⠀⠑⠀⠒⠀⡀⠀⠀⠐⠆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠘⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⡁⠀⠀⠀⠀⠑⢒⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠠⠀⡀⠀⠸⠀⠀⠀⣀⠰⠀⠈⠀⠀⠀⡇⠀⠀⠀⠙⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠱⡁⠉⠀⠀⠀⠀⠁⠀⠐⠀⠀⡀⢠⠀⢀⡉⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⢀⢀⠀⠀⠀⣀⠀⢀⠀⠣⡈⠓⠢⠤⠤⠤⢐⡙⠀⠒⠉⢀⡌⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠈⠰⠁⠀⠀⠈⠁⠀⠂⠒⠚⡉⠀⡑⢲⠒⠍⠀⠀⠀⠀⠀⢙⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠠⡀⢈⠢⠄⣀⣀⣰⠀⠀⠀⠈⠣⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠌⢰⡀⢀⣀⡀⠀⣄⡀⠠⡄⠈⠠⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠁⠀⠀⠈⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⠀⠀⠢⡀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠉⠉⠁⠒⠂⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⣀⠤⠤⠄⣀⣀⠀⠀⠀⣀⠤⠄⠀⠤⡀⠀⠀⠀⠀⢃
⠀⠀⠀⢠⠒⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸
⢀⠸⠉⢆⠀⣱⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡤⠤⠄⣰⣧⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⡠⠤⠤⣨
⠇⢹⡆⢸⡄⢹⡀⡇⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠰⠋⠀⠀⠠⠶⣛⠿⠝⢶⣄⢀⣴⠿⠿⣛⠶⠄⠀⠀⠙⠲⠀⣸
⠘⡀⠳⠈⠁⠀⠀⠣⡜⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢱⢸⡇⠀⠀⠀⠀⣾⣿⡇⠀⢨⠦⠑⣦⠀⢸⣿⡇⠀⠀⠀⠀⠀⢿⢸⠳
⠀⢡⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿⣴⡀⠀⠀⠀⠘⠛⢁⣴⠟⠁⠈⠛⢷⣌⠛⠁⠀⠀⠀⠀⣰⣞⠎⠀⠇
⠀⠘⡃⠐⠒⠀⠐⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣀⣻⠈⠛⠶⠶⠦⠴⠖⠋⠁⠀⠀⠀⠀⢸⠟⠓⠶⠦⠴⠶⠞⠉⣏⢀⠔
⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠋⢀⡀⠀⠀⠀⠀⠀⡜
⠀⠀⢠⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠸⡉⠐⠒⠒⠒⠒⠈⣩⠃⠀⠀⠀⢠⠞
⠀⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠈⠒⠢⠤⠤⠔⠊⠁⠀⠀⡠⠖⠁
⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⡵⠖⣤⡤⠀⣀⡀⠠⡤⢤⠖⠪⣄⡀
⠀⠀⠀⢡⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠚⣀⡰⢿⠛⡀⠀⠀⠀⠀⠀⡜⠟⠳⣤⡓⠂⠤⣀
⠀⠀⠀⠈⡀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣼⡟⠀⠈⠀⠈⠐⠒⠒⠒⠉⠀⢸⠀⠘⣷⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⢣⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⣰⠃⠉⠆⢰⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⡎⠉⣇⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠋⠀⠀⠀⢀⡏⠀⡜⠀⠈⡄⠀⠀⠀⠀⠀⠀⢀⠇⠀⢱⠀⢹⠀⠀⢦⡀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠙⠢⠤⣀⣀⣀⠠⠤⠒⠉⠁⠀⠀⠀⠀⠀⢸⠇⠀⠈⢆⠀⠰⡀⠀⠀⠀⠀⠀⡌⠀⢠⠊⠀⢸⡄⠀⠸⠑⡄⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢺⣠⡴⠿⢶⣄⠀⠀⢢⠀⠱⡀⠀⠀⢀⠜⠀⡰⠃⠀⠀⢸⠇⠀⠀⡇⠘⡄⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⢀⣀⠠⠔⠊⠁⠀⣼⠏⠀⠀⠀⠙⣧⠀⠀⠱⡀⠘⢄⡠⠊⠀⡔⠀⠀⠀⠀⢸⠃⠀⠀⢰⠀⢡⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⢄⠈⠁⢀⠎⠀⠀⠀⠀⣰⢋⡦⠀⠀⠸⠀⠈⡀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠈⢆⣠⠃⠀⣀⣀⣰⣿⣟⣻⣿⣄⡀⠀⡆⠀⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠁⠀⠀⢣⠄⠤⠤⠤⠤⠄⢤⠇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⢠⣆⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠖⢢⡄⡔⠐⡟⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠢⠀⠀⡀⠀⠠⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣃⣀⣃⣤⣤⣤⣤
⣿⣿⡟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣿⣧⠄⠠⣼⣿⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣧⣤⣤⣼⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⡀⠀⠀⠀⠀⠀⢨⡿⣇⢀⡀⠀⠀⠀⠀⠀⣤⢸⠟⣡⣷⡉⣟⣿⠉⣾⣌⠹⡇⣤⠀⠀⠀⠀⠀⢀⡀⣸⢿⡅⠀⠀⠀⠀⠀⢀
⠀⠀⢻⢷⣆⢰⣤⡀⠀⠈⢬⣻⡇⡀⠀⠀⣰⡄⢿⣾⣮⠹⣿⢋⣌⣁⡙⣿⠏⣵⢷⡿⢠⣆⠀⠀⢀⢸⡟⡥⠀⠀⢀⣤⡆⣠⡾⡟
⠀⢀⣀⣓⡽⣆⢎⢻⡜⢷⣦⠀⡼⡇⢀⠀⠹⣇⣼⣿⣵⡛⢖⢻⣾⢷⡟⡲⢻⣿⣿⣧⣘⠏⠀⡀⢸⢯⠀⣴⣾⢣⡟⡱⣰⢿⣚⣀⡀
⠐⠻⠭⠉⠁⣹⡿⠛⠵⣴⣽⡃⠀⣭⣮⡇⠀⡼⠫⢵⡦⢠⣤⡜⠃⠼⢣⣤⡌⢴⡮⠝⢧⠀⢸⣵⣥⠀⢘⣯⣦⠮⠛⢿⣏⠈⠉⠭⠟
⠀⠀⠀⠀⠟⠉⢠⡴⠾⢋⣀⣽⠗⠂⢌⣇⠰⠁⠐⠋⣾⢷⠉⠀⠒⠒⠀⠉⡾⣷⠙⠂⠈⠇⣸⡡⠀⠺⣯⣀⡙⠷⢦⡄⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠃⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠘⠀⢰⢰⢂⠸⣠⠰⢐⠀⠀⠀⢰⠃⠠⠀⠀⡀⡘⠀⡀⠇⠐⡆⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠀⡘⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣤⡀⠘⠲⣤⣌⠙⣳⡦⠀⢒⡏⢰⠀⢀⣀⢿⡶⣀⠀⣄⣠⠀⣀⢶⡿⣀⡀⠀⡆⢹⡒⠀⢴⣞⠋⣡⣤⠖⠂⢀⣤
⢠⣤⠀⢄⠈⣻⣧⣠⡶⠿⣷⡏⠀⢚⡟⡇⠀⢣⣐⠾⡧⠘⠛⢂⡖⢠⡐⠛⢃⢸⠷⣂⡜⠀⢸⢻⡓⠀⢹⣷⠿⢶⣄⣼⣟⠁⡠⠀⣤
⠀⠉⠉⡩⣷⠏⢏⣼⢃⡶⠟⠐⢱⡏⠈⠀⣰⡏⢻⣿⠶⣧⠤⣸⢷⡽⣏⠵⣼⠶⣻⡿⢹⣆⠀⠁⢹⡔⠂⠻⣾⡘⣧⡹⠹⣮⢍⠉⠉
⠀⠀⣼⣴⠏⠰⠟⠁⠀⠀⠚⣶⡇⠁⠀⠀⠹⠃⣼⢯⡟⣠⣶⣉⢋⡙⣉⣷⡄⢻⡽⣧⠘⠏⠀⠀⠈⢸⣶⠓⠀⠀⠈⠻⠇⠻⣧⣧
⠀⠀⠉⠀⠀⠀⠀⠀⠘⣥⡏⠈⠁⠀⠀⠀⠀⠀⠛⢰⣆⠹⣟⢉⣿⢿⡉⣻⠏⣰⡇⠛⠀⠀⠀⠀⠀⠈⠁⢹⣬⠃⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⡿⠛⠛⢿⣿⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡟⣿⡟⠆⠰⢻⣿⢻⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠾⠁
⡶⢶⣶⠶⠶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⡶⣶⣶⣶⣶⡶⣶⡶⣶⡶⡶⠶

⠀⠀⠀⠀⠀⠀⣴⠀⢤⣠⠄⠀⠘⠆⣐⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⢤⣠⠄⠀⠈⠇⣰⡂⢠⡆
⠀⠀⠀⠀⠐⠲⢺⡯⠄⢤⣚⢁⡀⠀⠈⠴⠻⢄⠖⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡛⠂⣺⠥⢄⣅⡀⣿⡆⢀⣼⡅⡎⢀⡴⠀⡄
⠀⠐⠆⠴⠏⠀⣀⠒⠂⠠⢍⡹⣅⠌⢠⠔⡗⠂⠒⠊⠙⢤⡤⠀⠀⠀⠀⠀⠘⠴⠟⠁⢀⡅⠒⢢⢈⣭⡷⣄⠈⢨⡵⡗⢢⡐⠒⠁⠣⣤⠄
⠀⠶⠶⠛⠁⡚⣡⢊⠀⢱⠄⠁⢸⡶⠁⠘⣢⡝⠀⣨⣤⠅⠀⠀⠀⠀⠀⠐⠖⢸⠟⠐⡋⡰⢠⣤⣫⠙⠁⢹⡶⢃⠎⣥⠘⠛⢈⣴⣇
⠀⠀⠶⢶⡙⠆⡝⠢⡤⣆⢀⠇⢸⡇⢀⠼⠥⠀⠸⡔⢈⡐⣠⠄⠀⠀⠀⠀⠀⢠⣾⣅⣀⠑⢤⢭⡇⢀⠇⢸⡇⢉⣤⡤⠔⠀⠇⢛⡛⢂⡶
⠀⠀⢰⠞⠇⠉⡔⠂⢳⠈⠓⠶⣿⣿⡁⠀⣠⠂⠀⠬⠀⠉⠠⠆⠀⠀⠀⠀⠀⠈⠛⠁⢀⡚⠈⠒⠈⠒⢶⣿⣿⡁⢿⢦⠔⠀⣠⠉⠀⠀⠴
⠀⠐⢢⡄⠒⣥⠄⢊⣁⢤⣍⢲⣿⣏⣀⣴⠃⠜⠈⠍⠓⡜⠆⠀⠀⠀⠀⠀⠉⡤⠐⣊⠀⠠⠋⢣⡈⠙⢶⣿⣟⣀⣠⠃⠨⠰⣿⡷⢠⣄
⠀⠀⠘⠁⠸⢐⠐⠤⣀⡡⠴⢾⣿⡿⠋⠉⠉⠠⠖⠶⠆⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠉⣄⠰⢄⣙⡯⠴⢾⣿⣿⠛⣿⠍⠁⠖⠼⠗⠘⠋
⠀⠀⠀⠀⠀⠛⠈⠛⠀⠀⠀⢸⣿⡇⠀⠀⠉⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠐⠄⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⠘⠛⡛⠀⢀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⣀⣀⡘⠛⠛⢀
⠀⠀⠀⠀⠈⢓⡄⣧⣼⢣⠛⢻⣀⠓⢻⣸⣂⢰⠶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⡼⣇⢰⡏⠉⡏⠁⣧⢀⡿⢸⠷⢾⡀⠧⣸
⠀⠀⠀⠀⠈⠉⠀⠈⠀⠈⠀⠀⠈⠀⠈⠈⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠛⠋⠀⠀⠃⠀⠙⠛⠁⠘⠀⠈⠛⠀⠙
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⣄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀
⠀⠀⠀⠠⣤⡀⣫⠄⠛⠞⠁⢄⣁⠂⢺⠇⣰⠏⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢀⡈⠛⠁⠀⠉⠀⣠⠀⠀⢠⣦⡄
⠀⢀⢀⣴⠆⠀⢖⣉⠑⠓⠤⣼⠿⢂⠳⣦⡄⠐⢋⣀⡗⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⠈⠁⢀⠀⠰⠆⢀⣉⡀⢤⣀⢁⡀
⢀⡈⢙⣥⢀⡴⠂⡈⠘⣤⣿⡎⢷⣀⡼⢫⠟⣨⡎⠁⢀⡈⠛⠃⠀⠀⠀⠀⠐⠷⠀⠃⢈⡑⠀⠧⣐⠘⢄⣠⠞⠉⣤⠀⠀⢈⠈⠁⠰⣴⡄
⠈⠉⠙⣯⠌⠤⣾⣿⡇⡼⠋⡄⢸⡟⢠⠡⢶⢏⠙⣠⣾⡆⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠈⠁⠀⠳⠈⢳⣼⠃⡀⢀⡄⠀⠄⠘⠃⠀⠀⠉⠁
⠀⠀⠀⣿⠆⠄⣈⢳⠒⢧⡰⠋⣼⣇⡠⣚⠚⠁⣀⠛⢤⠤⠱⠟⠀⠀⠀⠀⢀⠀⠀⠶⠀⠀⣀⣠⡀⠀⣿⠉⠀⠀⢳⣀⣠⠞⠉⠉⠋⠁
⠀⣠⣿⣷⣀⠠⢄⣀⡉⢀⣉⡛⣿⣿⠌⠟⣴⠋⠀⣼⠂⠐⠂⠚⠀⠀⠀⣀⠈⠀⠀⠀⠀⠤⡐⠂⠈⢲⣿⣷⣤⠴⠋⠀⠇⠀⠏⠠⡦⠀⣀
⠀⢠⡎⢉⣵⠀⠀⢀⣽⡀⠈⢹⣿⣿⣤⣴⣁⡉⠸⣿⡇⢰⣷⠀⠀⠀⠈⠟⠀⠀⣤⠀⠀⠦⠘⣄⠐⠛⣿⣿⡤⠒⠂⠀⢐⡀⠁⠢⠄⠀⠛
⠀⠈⠁⠈⠉⢢⡘⠲⠿⠗⠛⢻⣿⣿⠉⣶⡄⠁⠈⡙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠻⣾⣿⣿⣥⠴⠖⠦⢈⠁⠀⠂
⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⢸⣿⡇⠀⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⢠⡤⠀⢠⡤⢄⠀⡤⢤⠀⢤⠀⣀⢀⡄⢀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⡀
⠀⠀⠀⢈⣑⡆⢸⡗⠚⠀⡷⢮⡀⣸⠀⡏⢺⡇⢧⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢦⡏⡟⠰⢤⠉⡏⢹⡍⢸⣥⡛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠁⠈⠉⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠄⢒⠀⠐⠀⡄⢀⡶⠂⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠁⠀⠀⠀⠀⠀⠉⠈⠛⠣⠝⠀⠠⣄⠀⠀⠀⠍⡲⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠀⠁⣀⣤⠎⠁⠀⠴⠃⢀⣀⠀⠙⢤⠀⢠⠇⠀⠀⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⣀⡄⢀⣢⡤⠵⠶⠛⠛⠛⠛⠻⠶⢶⡦⣥⣐⠀⡈⠀⠀⠀⡀⠀⠬⠦⡀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢠⠦⠠⢞⣭⡾⠋⠉⠀⠀⠀⠀⠰⠤⠀⠀⠀⠀⠀⠈⠻⢿⣶⣅⡠⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠠⠂⠒⢂⠒⠈⠁⢂⣴⣿⡟⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣮⡢⡾⠆⠰⠄
⠀⠀⠀⠠⠁⠀⠀⠀⢱⡔⣵⣿⣿⡟⣡⣶⡿⠁⡀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣜⠾⠄⠈
⠀⠀⠠⠃⠀⠀⠀⠀⢌⣾⣿⣿⣿⣷⣿⡁⠀⠀⠙⢁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠸⡆⠀⢿⣿⣿⣿⣆⢀⣠
⠀⠀⡄⠀⠀⠀⠀⡀⣼⣿⣿⡿⣻⣿⣿⣷⣶⣇⢀⣼⡇⠀⢀⠠⠖⠛⠉⠉⠙⠛⠛⣶⣧⣄⣸⣿⣿⣿⣿⡎⢙
⠀⠀⠀⠀⠠⠂⠀⢸⣿⣿⣿⣷⣚⢿⣿⣿⣿⣿⣿⣿⡷⣾⠟⠒⠀⢀⠀⢰⡔⠛⠃⠈⢯⡹⣿⣿⣿⣿⣿⣿⡸
⠀⢸⠀⢀⠀⠀⡟⣾⣿⣿⣿⣿⣿⣼⡿⣿⣿⣿⣿⣯⠀⢹⡤⠖⠋⢻⣦⠀⠁⢀⡀⢀⠖⡂⠈⠉⠹⣿⣿⣿⡇
⠀⠈⠀⠈⠀⢀⠃⣿⣿⣿⣿⣿⠿⡯⢉⠛⣿⣿⡿⢿⣷⣴⡠⠀⠀⢀⡝⢁⣀⠘⠃⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⢰⠀⠀⠀⠎⢀⢿⣿⣿⣧⡀⠌⠀⢀⣴⣿⣿⡇⠀⠈⠙⠁⢀⡚⠉⠈⠁⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⠈⣿⡇
⠀⠀⠈⡀⠀⠀⠀⢸⣿⣿⣣⠏⠀⠠⢿⣿⣿⣿⡷⠀⠀⠀⠀⠈⠀⠀⡀⣄⣤⡤⠐⠚⠛⠙⠁⠀⠀⠀⠀⢹⢡
⠀⠀⠂⠸⣄⠀⢀⣴⣿⡟⠁⠀⢀⠀⠀⡀⣻⣿⣿⣆⡀⠀⡀⢼⠿⡿⡏⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠘⠀⣨⣶⡿⢿⡟⠁⠀⠀⠈⠀⠀⠠⡈⣿⡀⠈⣿⠀⠄⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⢠⣿⡿⣫⢀⠀⡄⠀⠀⠀⠀⠀⠠⡄⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⣰⣿⡟⠰⠁⠄⠀⠀⠀⠀⢀⠐⠀⣴⣶⣾⣿⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢰⣟⡅⢠⠰⠀⠀⠀⠀⣀⡄⠀⠀⠺⣿⣟⣿⣿⣿⣿⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡏⡇⠀⠀⠀⠀⠀⠀⡛⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠉⠀⠲⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠶⣠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠺⠂⠰⠂⠰⠗⠀⠒⠀⠘⠃⠀⠀⠀⠀⠈⠁⠒⠀⣤⠄⢠⠄⠰⠖⠈⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣘⠿⠻⠟⣛⣡⣆⠀⠀⠀⠀⠀⠀⠀⢠⣶⣍⣫⣅⣡⣴⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⣈⡙⠛⠙⠛⢋⠭⢲⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠟⠿⢛⣉⡵⠀⠀⠀⠀⠀⠀⠀⠉⣓⠛⠝⠊⠛⢈⣠⠄⠀⠀⠀⡠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠛⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠾⠿⠛⠁⠀⠀⠀⠀⠄⠠⠀⣄
⠀⠀⠀⠀⠛⠉⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠝⢳⠒⡓⢛⠉⠅⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢦⣤⢤⣤⠶⢚⡉
⠀⠀⠀⠀⠂⠀⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⢻⡿⢿⡿⢿⠟⠓⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠒⠈⠁
⠀⠈⢀⡀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠰⣄⣠⣄⣀⣤⣠⡶
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠽⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣾⠟⠁⠀⠀⠀⠀⢀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠰⢸⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⣠⠖⠀⠺⠁⠀⠀⢀⡀⠔⢂⠋⠀⠀⠀⠀⠀⢤⠀⠀⠀⣄⠀⠀⠀⣿⠀⣚⣡⣔⠲⠄
⢠⣦⠔⡧⢰⡄⠀⠀⠐⢚⡁⢦⠀⠀⠠⢤⡄⠠⠀⠔⠒⡏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⣰⣿⣆⣙⡹⠿⣦⢤⡆⣠
⣿⢗⣍⣰⢾⣧⣤⡶⠂⡉⣰⠄⣁⡀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣸⡄⠀⢸⣼⠇⠠⡟⣱⠈⣹⣄⣤⠘⣇⡈⢻⣆
⣿⣋⢥⡼⡃⢽⣿⠟⣁⠀⣩⣀⡀⢷⠀⠀⠀⠀⠀⠀⠀⣴⡀⠀⠀⠀⠀⠀⣀⠀⣰⢿⠒⣀⣡⣌⠳⡄⢉⣠⡷⢤⢿⡏⠱⠄⠙⣷⣿
⠉⡱⢾⣅⣿⠞⠛⡡⠈⢋⣀⣈⡛⢦⣽⣾⣾⣦⣾⣴⣐⣿⣓⣶⣼⣶⡿⣿⣿⣿⡇⣤⢤⣻⡡⢦⣸⠔⠁⣈⠳⡌⢳⣿⡗⠣⣄⠈⣿
⢻⡆⢸⣇⣷⠾⠋⠙⢶⠋⠹⢶⣧⡈⠙⣿⣿⣿⣿⣿⠄⠹⠈⢿⣿⣿⣡⠻⣿⡿⠋⣱⣄⣈⢰⡄⢱⠒⣀⡸⣧⠛⠻⡿⠷⠀⠈⠙⢿
⠁⣼⡟⣿⣥⠔⠋⣸⠋⢠⠠⠋⠁⠙⠲⣽⣿⡟⣾⣗⠕⢱⣀⠞⣿⡫⠡⠒⢻⠇⠂⢈⣠⡈⠻⣷⡏⠉⢁⠀⠸⡗⠢⢽⡀⠈⠣⢀
⣾⣉⣿⣿⠁⠐⢚⡁⣠⠏⠠⡀⠘⢦⠀⡈⣹⢿⡋⠀⢘⠁⡏⠀⠈⡖⠀⠤⡋⢀⣠⡏⢨⣌⠙⢻⣿⠔⢻⡄⠀⡟⢦⣀⠹⣆⡄⠈⢻
⠈⢉⠟⠀⠀⣠⠤⠋⠉⠀⠀⠙⣆⣸⠟⠛⠚⠀⠓⠐⠈⠛⠓⠂⠀⠈⠀⠀⠈⠀⠀⠁⠁⠀⠛⣋⠄⠀⢈⠙⠚⠁⠀⠈⠙⢿⣅⢀⣀
⠀⠀⠀⠀⠈⠀⠀⠀⠙⠂⠀⠀⠈⠛⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠰⠛⠲⠤⠀⠰⠤⠧⠄⠹⠛⠛



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠠⣶⠀⡷⣀⡀⠀⢠⠄⠀⠀⠀⡀⠠⢀⣀⣤⠀⠰⣄⣰⠀⡄⢰⡆⠀⠀⣐⠷⠄⢰⡄⢠
⠀⠀⠀⠀⠀⠀⠀⠉⢁⡔⢁⡀⠀⡉⢀⡄⣿⢣⢠⣾⡀⢠⡤⣶⠀⣼⠊⡌⣉⠂⢠⢉⢹⣷⣠⡟⠁⠐⠃⠁⢀⠀⢀⡠⠟⢀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡅⠙⠁⠀⠀⠙⠟⣡⡼⠀⠁⣁⢄⣤⣄⢠⠀⠀⣇⠈⠳⣾⣙⠸⣫⡟⢀⢸⡇⠹⣷⣿⡏⠹⠵⡎⢠⣾⣤⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢲⠈⠉⠙⠲⠾⠟⠁⢀⣴⣿⠀⢉⢟⣼⣧⠃⠈⡀⠐⢿⡿⠟⠻⢄⣴⣎⣁⡀⠉⣍⠀⢲⣶⡿⠋⠦⠛
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠒⣄⡨⠒⠀⡀⠀⢀⣀⣀⣹⣿⡷⣿⢸⡇⠹⣿⡟⠁⢠⠀⠀⢶⡄⢼⣇⣠⢸⣄⣴⣿⣦⣼⠏⡼⢻⡆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢢⣈⠶⡇⢰⣾⣇⢀⣼⢿⡏⠹⣟⣡⣿⠚⢃⡗⢹⣷⣶⡟⢳⣤⡟⠀⢠⣙⣧⡜⠁⣿⣿⡿⠃⣰⠆⠋⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠟⠻⠑⢿⡆⢻⢿⣿⣯⡿⠃⠀⠈⠉⠀⠰⣏⠁⢈⣿⣿⢁⡿⣿⠗⢠⣾⠂⢩⠁⣸⣿⡟⢡⠞⠁⣼⣶⢳⣦
⠀⠀⠀⠀⠀⠀⠈⠢⣤⣴⣷⡄⠑⢾⣧⠙⢿⣶⣦⡶⠀⢰⣄⠀⢿⡼⢻⢹⣟⣼⣇⡿⠠⣮⣽⡆⠘⠑⢿⡏⡴⠁⣠⠾⠛⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣤⢀⠹⣅⡛⢦⡀⠙⢷⡀⠙⣿⣿⣶⣿⡏⢀⡼⠁⣿⢸⣷⣿⣁⣀⡀⠀⠀⠀⠰⣶⣿⡟⠀⠀⢁⣤⠞⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⠀⠙⠳⢬⣶⣌⠻⣦⡈⢻⣿⣿⢀⡾⠁⢸⣿⢸⣿⣿⢠⡏⢧⠀⠘⣦⣄⣿⡟⣠⣤⠾⢛⠁⠣⡀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⠀⠀⠹⣿⠿⠿⠿⠾⢿⡿⣿⣷⣶⣿⣿⣾⣿⣿⣾⣷⣾⡿⠿⠿⠿⠟⠛⠛⠋⢩⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠆⠉⡇⠀⠀⠘⣷⡀⣤⡶⢤⡴⣦⠀⣤⣄⣀⠀⠀⠀⠀⠀⢠⣾⠀⠀⢠⡄⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠄⠈⠀⠀⠀⠀⠘⣷⡈⠷⣌⠃⢻⠈⣇⡠⠚⠀⣠⡖⢲⠀⢸⡟⢀⣤⠞⠁⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠓⢾⡇⠉⣠⣤⠀⣿⡀⢸⠁⣸⣿⠟⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⣠⢶⡄⠀⠀⢰⡇⠘⡆⢿⣇⡸⣴⠻⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⣤⠰⠃⣼⣇⡀⠀⢸⡇⠀⡇⠈⢿⣿⣯⡀⣧⢀⡴⣤⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⢰⣿⡀⣰⡟⠉⢻⣦⢸⡇⢰⠁⠀⣸⠻⣎⠀⣿⣸⠀⣸⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠜⢧⣼⠏⠀⠀⠈⣿⣸⣧⠏⠀⢀⡏⠀⠘⣇⣿⣇⣠⡟⢀⡴⣦⡄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠸⠀⢸⡏⠀⠀⠀⠀⣿⢸⣏⠀⠀⣼⣤⡄⠀⣿⠈⠛⠋⠀⠸⣷⡎⢿⡄⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢠⠟⡇⢠⠊⣦⢀⣿⠟⣿⣄⡴⠃⢿⣦⣴⠟⠀⠀⠟⠙⣧⡈⠀⢸⡇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⡏⠀⣷⣿⠀⠁⣼⠃⠀⠈⠉⠀⢄⠀⠀⠀⠀⢠⡶⣮⠠⣘⣷⠀⢸⡇⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⣾⠀⢸⡏⢿⣤⡼⠃⢠⠶⡀⣠⣤⡈⢣⣀⣤⣄⢸⡇⠘⣧⠈⢻⡧⣾⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⣿⣶⡿⠁⠀⠱⣄⠘⣦⠛⡀⣿⠈⢻⠀⢻⣟⠻⣮⣷⡀⢻⡄⠈⣷⣿⡛⣆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⢡⠟⢳⡄⢦⢸⡇⢿⡞⢻⡌⢷⣜⡆⢸⠉⢧⠹⡟⢷⣼⡇⠀⣿⠸⣧⡼⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⢸⡷⡄⢻⡌⢿⡇⢸⡇⡆⣿⠀⠙⣷⣈⡄⠈⢷⣻⡌⢿⣧⣤⣿⠀⠉⢀⡾⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⢷⣽⠈⢿⣼⠷⣼⡇⢹⣼⡇⠀⡏⠻⣷⡀⠘⣿⣷⢸⡟⠛⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⣿⡀⠸⣇⠈⢿⡇⠀⡇⣀⠹⣿⣦⣸⡟⢿⠇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⢛⠻⡄⢿⡑⡘⣿⣀⠙⣯⣀⣽⠆⠉⠁⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠈⠢⣽⠈⠻⠖⠈⠉⠀⠈⠉⠁⠀⠀⠀⠀⢀⡴⠋
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡾⠟⠁
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠐
⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠘⠀⠀⠀⠢⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⣀⠀⠐⠊⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠐⠢⢤⣀⣀⡀⠀⠬⠂⡸⠁⠀⠀⠀⠀⠀⢀⣀⠢⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⣠⠊⠀⢀⠔⠂⠀⠀⡐⠀⠀⢉⣢⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠄⣀⡀⠀⠀⣀⡀⠥⠐⠁⠐⣀⡀⠀⠁⠀⢸⠠⢀⠴⠊⠁⠀⠀⠀⠈⠑⠢⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠑⠄⠀⠤⠇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⢲⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠁⠢⣀⣀⡀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠉⠈⠁⠀⠀⠀⠀⠀⠀⡀⠈⢢⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⡘⢠⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⢀⠀⠀⠀⠀⠀⠀⠘⡀⢀⠜⠄⠀⠀⠀⠀⠠⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⠂⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡰⠀⠢⣀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠃⡀⠊⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠈⠁⠀⠀⠀⠀⠈⠀⠁⠀⠸⠁⠀⠀⠀⠀⠀⠐⡀⠈⢄⠠⠼⢖⠂⠠⠄⠶⢤⡀⠀⠈⠀⠀⠀⠀⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⢀⠀⠀⢀⡰⠅⠐⠁⡀⢄⠀⠀⠀⠀⠀⠀⠀⠻⡃⠀⠀⠈⠒⠦⠀⠤⠄⠘⠃⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⡇⢀⠑⠄⠀⠀⠀⠀⠀⠀⢀⠑⢦⠄⢀⠀⡁⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠁⠉⠂⡉⠁⠒⠂⠂⠐⠊⠀⠛⣇⠈⠠⠆⠤⠄⠖⠉⠀⡅⠀⠀⠐⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⠀⠠⠀⠠⠄⠠⠄⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠠⢄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠀⣨⠓⠂⠒⠋⠀⠀⠀⡀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠄⡀⠀⠀⠀⠀⠀⣀⣀⠠⠴⠋⠘⠃⠈⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡤⠊⣉⡱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⠀⠀⠀⠸⠀⠈⢪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠐⢄⠀⢣⠀⠀⢿⠀⠀⠀⠀⠀⣀⡀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡀⠀⠐⢤⡑⠄⠱⣄⠈⣧⡠⠤⡒⠁⠀⠀⣀⡀⠤⠊⠙⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠖⣌⣉⣁⡀⣉⣉⣐⡊⣑⣦⣬⡗⢻⢡⠀⠘⣍⠉⠁⠀⠀⠀⠀⣀⣨
⠀⠀⠀⠀⠀⢸⠉⠉⠁⠐⠒⢲⠐⠂⢉⠭⠽⠟⠉⣉⢁⡉⣲⠰⠀⡴⠮⢍⠉⠉⠛⠛⠂⠈⠉⠉⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠂⠀⢸⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣤⠄⢀⡀⢀⣸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⠀⢸⠀⠀⢇⡀⢀⡀⠠⡤⠆
⠀⠀⠀⠀⢀⠜⢸⠀⠀⠀⠀⢸⠀⠀⢸⠉⠉⠉⠁⠐⠚⠒⡇⠀⠀⢸⠉⢹⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠓⠂⢹⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢣⠔⠂⠤⠄⠘⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠒⠂⠐⠧⣀⡠⠄⢀⡀⢀⣀⣀⣀⣀⡀⢀⡀⠠⠴⠀⠠⠔⠐⠒⠂⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣋⡹⣷⡈⠏⢀⣴⣿⣉⠉⢉⡻⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣏⣿⣿⠉⣿⣿⣯⣶⣬⣿⠉⣼⠃⣼⣬⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⠙⠛⣧⣤⡾⠛⠉⢁⡀⠉⠙⠹⣧⢘⡿⠲⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣦⣼⠿⣾⡋⠋⠁⣦⣦⡞⡟⠛⠂⣄⠙⠼⣳⠔⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣺⣿⠿⠏⠉⢡⡴⠋⠁⠀⠀⠀⠉⠂⣌⢃⠀⣿⡀⣾⡿⠟⠋⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠻⠿⣿⣿⣿⡍⠀⠀⢤⡘⠀⠀⠀⠀⠀⠀⠀⠀⠐⣨⡀⢹⠇⢸⠙⠆⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠉⠷⢦⣠⣲⠀⠐⠦⠀⠀⠀⠀⣀⣤⠀⡧⠗⣾⠂⠃⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠠⡿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠊⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠶⠶⠀⢀⡀⣰⡟⠂⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠻⠿⣿⠛⠛⠻⣿⣿⣿⣿
⠛⠛⠛⠛⣛⠋⠉⠙⠿⠛⠦⠐⠀⠀⠒⠶⠶⠊⠁⠚⠛⠷⠀⢤⣀⣄⠰⠆⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠩⠋⠀⢄⢈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⠉⠂⡤⠤⣀⠀⠀⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠰⠂⠉⠀⢠⠂⠀⣀⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠣⡀⠠⠀⣰⢨⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡇⠀⡀⠀⠀⠀⠀⠀⠐⠒⣋⡽⠃⠈⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠁⡍⡁⠀⠀⠀⠀⡀⠊⠬⠼⣬⣤⠤⠰⢃⠓⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⢆⠀⠀⠀⠌⢀⡎⠀⣰⣿⠃⢀⢃⡌⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⢂⠀⠌⠀⢿⣧⠄⠻⣿⣆⢊⡷⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠀⠈⣿⠀⠀⠘⣿⣎⠂⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠈⠀⣸⠀⢸⣇⠀⠀⠸⠏⠀⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡰⠋⠀⠀⠹⣆⠀⠀⠀⠀⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠌⠀⡶⠄⠉⢵⠲⠶⠚⠁⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡀⡇⠀⠀⢈⠀⠀⢀⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠁⠀⠀⠀⠒⠛⢻⠋⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠘⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢠⣿⡀⠀⠀⠀⠀⡀⠀⠠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣧⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⢳⡝⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣽⠗⣄⠹⣮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢟⣼⣿⡿⣄⠘⢷⡈⢯⡪⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢡⣾⣿⡿⡇⠈⢷⡈⠻⡦⡙⢮⡺⣗⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢞⡿⢋⣴⢏⣾⡿⢡⡟⢦⡀⠙⣦⣘⢮⡓⢝⣦⡙⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣋⠴⢋⣴⡿⢣⣾⡟⣱⠿⡇⠈⠻⣦⡈⠻⣷⢽⣦⡉⠛⠷⣄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠵⢚⣽⢋⣴⡿⢋⣾⠏⢠⡟⢆⠀⠙⢮⡳⣌⠳⣍⠻⢶⣄⡀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⡵⣿⢟⡵⣻⠏⣠⡾⣧⡈⠳⣦⡀⠙⢄⠙⠪⣝⢦⣈⠙⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠟⢋⣾⡵⢋⡼⢁⣴⠟⢀⡇⠻⣦⣌⠻⣶⣄⡙⢦⡀⠉⠛⠿⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⡴⠋⣠⡿⠋⣰⢫⡟⢆⡈⠻⣷⣌⡳⣝⡲⢭⣲⣄⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⣠⠞⣠⣾⠟⣡⢾⠏⣼⡗⢮⡻⣦⣈⠻⢽⡲⢽⣦⣍⡛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣠⣮⠴⣫⠟⢁⡼⡳⢋⡞⢹⠁⠀⠙⢮⣻⣗⢦⣉⠲⢭⡙⠻⠷⣦⣤⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢉⣠⠞⣡⠔⣫⢞⡴⠋⠀⢸⠀⠀⠀⠀⠈⠛⢷⣮⣉⠓⠻⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡯⠖⠋⣡⠞⣡⠎⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣄⣀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠁⠀⣠⠞⣡⠞⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡥⠞⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢁⡀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠤⠒⠂⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣶⣶⣶⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡄⠀⠀⢻⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣷⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠃⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿