Fit Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣼⣿⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢽⣿⣿⢺⣿⡏⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠫⡕⠛⠻⠇⠀⠀⠳⡂⠀⠀⠀⠀⠀⠀⠀⣴⣾⣷⣿⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠘⢦⡀⢀⠀⠀⠀⠘⣿⣿⣾⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠘⣷⣤⡀⠀⣠⣄⣸⣷⡀⠀⠀⠀⠀⢉⡿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠐⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⢠⣀⠴⠏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣷⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠘⣿⠟⠋⠀⠈⠋⠉⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⢀⣴⣾⠟⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢠⣿⣿⠁⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢸⣿⣿⡄⠀⠀⠀⠀⢀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⢸⣿⣿⣷⡀⠀⢠⣤⣿⣄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣷⣄⢻⣿⣿⣧⠀⠀⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣽⣿⡏⣿⣿⣿⡆⠀⡇⠈⠻⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⠿⠟⠀⢻⣿⣿⡇⠀⠃⠀⠀⠀⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠞⠓⠐⣿⠖⠚⢳⣾⠛⠂⣾⣶⡀⠀⣶⠀⢰⡞⠛⠂⣰⡞⠛⠓⢠⣶⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠓⠀⣿⠃⠀⢸⣿⠀⠀⣿⡏⠻⣦⣿⠀⢸⡟⠛⠂⠈⠛⢿⣦⠀⠙⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠘⠛⠃⠀⠘⠛⠀⠀⠛⠃⠀⠙⠛⠀⠘⠛⠛⠃⠘⠛⠛⠋⠘⠛⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⡷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢀⡀⣴⢾⣷⠿⣿⣿⠿⣿⣄⡰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠛⢻⣿⣿⣿⣶⣿⣿⣿⣿⣿⡛⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢤⣧⣠⡠⣶⣷⣿⣦⠉⢿⣿⣿⡿⠉⣾⣿⣯⣠⣬⣤⣼⣥⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣿⣿⠛⠛⠂⢫⠀⠈⠁⠀⠸⠿⢉⠇⠀⠈⠁⠈⣿⡿⠛⠛⣿⣿⡄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢲⣿⡟⠀⠀⠐⢾⣷⡿⠀⠀⠀⡀⢈⠀⠀⠀⢾⣾⣿⠃⠀⠈⢻⣿⣧⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⢿⣷⡀⢀⡀⢠⣻⣿⡀⣿⣀⣶⣾⣠⣷⢠⣿⡟⡁⢀⠀⢠⣿⡿⠋⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠹⣷⣄⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣴⡿⠋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠸⣿⣟⣛⣽⠿⠷⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢟⡉⠀⠀⢿⣀⣠⣬⣿⣿⣿⣶⣿⣿⣿⣿⣿⠇⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠈⠷⠖⠛⠉⠁⠀⠀⢀⡀⠀⠀⠈⠉⠛⠿⠏⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⢠⠀⠤⠄⣠⠀⢀⢠⠀⠒⣮⣷⠒⠀⡀⡠⢀⣠⡀⠤⠀⡆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡢⠤⠠⠹⣄⠂⣆⢀⠂⣽⣿⠐⠀⢠⠂⣠⢇⠑⠠⠴⡓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⣀⠠⣕⢆⣁⣘⣷⠞⣿⣿⠳⣾⣇⣀⣐⣺⣀⡀⠀⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠈⣡⠞⣫⠴⠚⣹⣟⠷⡦⡙⠂⠈⠁⠤⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠄⠛⣡⠎⠅⢀⠠⠟⠻⣄⡀⠢⠰⣜⠋⠲⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⠀⠈⠁⠂⡀⢁⠐⠀⢡⠀⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠂⠉⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠤⢀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠔⢤⠊⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡠⠃⠌⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⢒⠄⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠁⠀⠀⠀⠀⠀⡆⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⠀⠐⠀⠀⠀⢰⡀⠀⠀⠀⠀⡠⠊⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠋⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠰⠀⢀⠤⠊⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠓⠁⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠁⠀⠢⡀⠀⠀⠀⠀⡐⠀⠈⠢⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⡄⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠈⠁⠀⠤⢀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⢐⠂⠂⠠⢀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢆
⠀⠀⠤⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠑⠠⠀⠀⡔⠁⠀⠀⠀⠀⠈⠐⠂⠄⡀⢀⡌⠀⠀⠀⠘
⠀⠀⠌⠀⠀⣖⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄
⢀⠀⠀⠀⠴⠁⠘⠂⠀⠀⠀⠀⠀⠀⠠⠄⢀⣀⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃
⡄⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠰
⠷⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠂⠀⠠⠛
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠈⠉⠉⠁⠰⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣠⠞⠁⠀⡏⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢸⡦⠋⠀⠀⠙⠷⡁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⢣⣤⡟⠤⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⠘⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⣀⡀⢀⣄⣠⠃⠀⠀⠀⠀⡴⠆⠠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⠒⠉⠀⠀⠀⠸⠉⢆⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⢛⠢⠤⠼⢦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⣀⢠⠀⠀⡰⠃⠀⠀⠈⣄⠀⠀⢰⠀⢡⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠖⠲⡉⠁⠀⢸⠗⠊⠁⠀⠀⠀⠀⠈⠑⠒⠂⠀⠀⡆⠸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈⣄⠀⠱⣄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢰⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠸⠀⠀⠀⠑⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡆⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⢡⠀⠀⠀⠀⠀⣠⠞⢀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠇⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢋⠁⠀⠀⠘⡀⠀⢀⡠⠚⠁⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠘⢄⠀⠀⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠈⠆⠀⠀⠀⢷⠔⠋⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣓⢠⠁⠀⠀⠉⠁⠒⠒⠂⠀⢠⡔⠁⠀⠀⠘⣄⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢻⡌⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠈⠣⣄⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣱⡾⠃⠠⠤⠤⠤⠀⢀⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠈⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠉⠒⡦⠤⠄⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠐⠒⣳⠆⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡚⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⢀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠐⠀⠉⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⡀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠙⢄⡀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⣤⠃⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⡌
⠀⠀⠀⠀⠀⡘⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠽⣤⠾⡆
⠀⠀⢀⡤⣾⠀⢀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⢆
⠀⡠⠛⠐⠚⠻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠑⣠⡄⠀⠀⠈⠳⣄
⢎⠳⣄⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠩⠔⣂⣤⣈⠑⡄
⠀⠙⠬⡑⣄⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠀⠀⠈⠻⢧⣄⠘⢆
⣤⣤⣤⣤⣤⣤⣭⣷⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠉⠛⠛⠛⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⣀⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⣼⣿⣿⣿⣿⡿⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠒⢸⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁⠀⠀⣿⣿⢻⡿⠋⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠘⠊⠀⠈⠙⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⣶⣿⣿⠁⠀⠀⠀⠹⡟⠂⠀⠀⠀⣀⡸⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡇⠀⠀⠀⠐⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⡙⠁⠀⠀⠀⠀⠀⢀⣴⠅⠂⠐⠶⣄⠀⠀⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠔⠉⠀⠈⣿⡦⢀⠀⡃⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⢀⠆⠰⠀⠙⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠈⠀⠀⠀⠀⠀⢠⡛⢱⠀⠈⢱⡶⠢⡀⠀⢁⠀⢀⠤⡄⠀⢡⠄⠊⠀⠀⠑⠤⠄⠈⠈⠄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡤⠀⠂⠈⣥⣴⠗⠀⠀⠓⢀⡸⠃⠁⠠⠀⠀⠀⠀⠆⠐⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⢰⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢃⠀⠀⠀⠏⡇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆⠀⠡⡀⠀⢠⠊⠀⠑⠄⠀⠀⠀⠀⠀⠀⡰⠀⠃⠀⠀⢀⠜⢢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠸⠀⠀⠀⢰⢱⠀⠀⠀⠀⠀⠀⠀⢀⠆⠘⠀⠀⠱⣀⡬⢠⣤⠊⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⣀⠔⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡆⠀⠀⣼⡎⠀⠀⠀⠀⠀⠀⠔⡁⠀⠀⠃⢀⡴⠋⢠⡎⠈⠁⠒⠀⠄⠀⡤⠁⠀⠁⠀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠂⠰⠙⠂⠤⠀⣀⡀⢀⠌⠀⠐⡀⠀⠈⠀⢀⠴⢤⣿⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢘⡂⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠈⠢⣀⠔⠁⠰⠋⠀⠀⠈⠈⠓⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠟⡒⠂⠠⡤⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌⠀⠀⠁⠀⠀⠈⠒⠣⠀⢀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠐⡀⠀⢅⡂⠤⢀⠀⠀⠀⠈⠁⠁⠂⠄⣀⢀⠎⠀⠀⠀⢀⠎⠈⠉⠀⠒⠒⠒⠒⠂⠄⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡰⡙⠃⠂⢌⣀⡀⠀⠁⠀⢧⠀⠀⠀⠀⠀⠉⠂⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢐⠕⠀⠀⡠⠃⠀⠀⠀⠉⠉⠉⠉⠁⡐⠀⠀⠀⡆⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠉⠛⠤⣀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⢰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠂
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⢀⠔⠃⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠷⠺⡀
⠀⠀⠀⠀⢀⠂⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠹⠶⠁⢠⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⣀⠀⠈⢄
⠀⠀⠀⢀⠂⠀⡐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡠⡀⠀⣠⢾⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠭⠂
⠀⢀⣔⣇⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⢦⡍⠀⠀⠈⢄
⠴⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⢐⡤⣀⠑⢄
⠈⠃⢑⡄⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⣀⣀⣀⣙⣲⣌⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣠⣾⣗⣦⡀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡺⣷⣄⣀
⠀⠀⠀⠀⣀⢞⣶⣾⣿⣿⣿⣿⣆⠀⠀⠀⢠⢏⠀⠼⡇⠀⢸⡏⠀⢸⡟⢹⣦⠀⠀⠀⠀⣰⣯⣿⣿⣿⣷⣶⡳⣀
⠀⠀⠀⡼⢿⠚⠿⠉⢹⣿⣿⣿⡽⡆⠀⡴⠛⠉⠔⣻⣷⠀⢸⡇⠀⣿⣀⣾⣿⡆⠀⠀⢰⢿⣿⣿⣿⡏⠉⡟⠓⡿⢧
⠀⠀⠀⡇⢸⠀⢰⠀⢸⣿⣿⢽⠿⠿⠒⡇⠀⠴⣾⠉⣿⡀⣼⡇⢤⣿⣼⣟⣉⣓⠒⠒⠿⠿⡏⣿⣿⡇⠀⡇⠀⡇⢸
⠀⠀⢠⢇⠸⡀⢸⠀⠘⣿⣿⣸⣶⣾⡿⣧⠀⠀⠛⠛⠻⣀⣻⣇⣸⠟⠿⢿⣿⣿⠿⢿⣷⣶⣇⣿⣿⠃⠀⡇⢠⠃⡸⡄
⠀⠀⠸⣿⠀⢇⠀⡇⠀⢻⣿⣷⣿⣧⡇⠸⣦⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣿⡏⠀⣸⣾⣿⣾⣿⡟⠀⠸⠀⡸⠀⣿⠇
⠀⠀⠀⠈⢇⠘⡄⣸⣀⣬⣿⣿⣿⣿⡇⠀⠈⠻⣶⣤⣀⣀⣀⣀⣀⣀⣤⣾⣿⠁⠀⢻⣿⣿⣿⣿⣥⣀⣇⢠⠃⡸⠁
⠀⠀⠀⠀⠈⢿⣿⡿⣷⣽⣿⣿⣿⣿⠇⠀⠀⠀⠈⡏⠙⠛⠛⠟⣿⠛⣿⣿⡇⠀⠀⠸⣿⣿⣿⣽⣯⣾⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠺⠿⠞⠿⠿⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡏⠀⣿⣿⡇⠀⠀⠀⠙⠿⠿⠳⠿⠗⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠒⠶⠶⠶⠞⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢷⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠁⢢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⣾⣿⣿⣿⡿⠋⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠄⡠⠔⠠⣬⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣠⣿⡿⠀⠀⠀⢻⣿⣘⠉⠀⠈⠛⡀⠀⠀⠀⠀⠀⠀⠀⡠⠈⢤⠓⠁⠀⢠⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠁⠀⠀⠀⠈⠁⠀⠀⠀⠐⡞⠀⠀⠀⠀⠀⠀⡠⠊⠤⣄⡌⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠐⣦⣀⠀⠀⠈⠀⠁⠀⠀⠀⡀⠄⠊⠁⠀⠀⠀⠘⡁⠰⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⡟⡡⠀⠘⣦⡤⠀⠀⠐⠁⢀⠀⡀⠀⢆⠠⠁⠀⠐⠄⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠤⠀⠂⢹⣤⠘⠀⠈⢇⠄⡾⠐⠈⡄⠀⠀⠈⡄⠱⢀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡔⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢡⠰⠀⠀⠀⠀⠐⠋⠈⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡀⠦⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣸⠀⠀⠀⠀⠀⠀⠀⡰⠀⠆⠀⠀⢨⠔⢢⡔⠁⠀⠀⠀⠀⠠⠙⡀⠀⢀⠀⠊⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⡀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠫⠀⠀⠐⡀⠜⠁⠰⡄⠀⠈⠁⠒⠂⡴⠁⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠉⠁⠀⠀⠀⠀⡄⠀⠀⠡⡀⠀⠁⠠⠚⡴⠗⠠⠄⢀⡀⣠⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣀⡀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠂⠊⠀⢢⠁⠀⠀⠀⠈⠂⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⠀⠉⠒⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⡀⢄⡀⠀⠀⠀⠀⠲⠀⠄⡀⠀⠀⠐⠀⠀⠀⢀⠆⠀⠤⠄⠀⠀⠀⢀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢀⠐⠀⠄⠐⠀⠤⠄⠀⠀⠀⠀⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢠⠁⠀⢀⠌⠁⠀⠀⠀⠀⠙⠀⠀⠄⠀⠀⠄⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠐⠣⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⡀⠋⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠘⠺⠀⡁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄⣀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⣄⠀⠞⠒⠺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠊
⠀⠀⠀⠀⠀⣀⠚⠐⢲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢀⡀⠀⠈⠄⡀
⠀⠀⠀⠀⠀⠃⠂⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠴
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣶⣶⣶⣶⣶⡷⣿⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣿⣶⣿⢿⡷⠾⠷⠶⢶⣶⡷⠶⠶⠶⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⢤⠀⠀⠔⠊⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⣠⠄⠠⠀⠈⠀⠀⡀⠛⢄⠀⠀⠈⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⣀⡀⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀⡤⠐⠰⠋⠀⠀⠉⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢀⠜⠣⠠⠀⠀⠐⡆⠀⠀⠀⣀⠀⠠⠋⠀⠀⠁⠀⠠⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠚⠁⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢠⠀⠀⠀⠂⠀⠀⠀⡧⠠⠤⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⠄⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡀⢀⡀⠀⠒⠂⠀⠀⠀⠤⠄⣀⠀⠀⠀⠀⠀⢑⡄⠀⡠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣐⣀⣀⡀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠴⠋⠀⢦⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠈⠂⠀⠳⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⡀⠬⢽⣦⠄⣰⢷⠆⠄⠀⠀⠀⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠊⠀⠀⠀⠀⠘⡆⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⡼⠧⠤⠚⠀⢀⣈⣩⡭⠤⠤⠤⠀⠀⠈⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⢠⠃⠀⠀⣠⠴⠛⠁⠀⠀⡀⡀⠤⣐⡤⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠐⠀⠀⡰⢃⡞⠀⠀⡠⠤⣀⣠⠤⠀⠀⠐⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⢀⣠⠤⠤⣄⣀⠀⡇⢠⠃⢀⡜⢁⡞⠡⡀⠀⢠⡞⠉
⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠑⠒⠂⠀⠈⠀⠀⠀⠀⠀⠠⣼⠷⠃⣠⠎⢠⠞⠀⠀⠈⠄⠘⠁
⠀⠀⢀⠜⠁⠀⠀⠀⢀⣀⣒⠀⠤⠤⠤⢤⣀⣀⣀⣀⣀⠠⠤⠤⣥⡴⠁⠀⠀⠀⠀⢈⣄⣀⣂⡀
⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⡁⠀⠀⢾⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣆⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣼⣿⣿⣿⣿⠏⣿⣿⣿⡏⠀⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣧⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⡿⠿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⢀⣿⠋⠁⠙⢿⣿⣟⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠛⢷⣌⡙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠘⠿⠿⠶⠀⠹⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⣀⣈⣀⢀⣀⠀⢀⠀⣀⣀⡀⢀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠒⠂⢸⡇⠀⣿⠀⢸⠙⢦⣾⢀⣟⠒⠂⠙⢲⡄⡙⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠁⠀⠈⠀⠀⠁⠀⠉⠉⠁⠈⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⢟⢿⣿⣿⡿⣠⠖⠾⣷⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡿⣡⢀⣿⡆⠙⢿⡇⢧⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠁⠿⠂⡍⠀⠀⠀⠑⠮⣵⣦⡀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠐⡆⠀⠛⠀⣸⢻⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠐⠒⠃⠀⠀⢠⠷⢋⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣽⣦⡐⠛⠛⠷⠆⢀⣴⣿⣿⠟⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣳⣶⣶⣾⣿⣿⣿⣏⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣇⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣟⣻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠋⠻⡿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡒⣦⣾⡷⢾⣷⣔⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⡿⣿⣻⡳⢞⣟⡿⢿⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⢷⡄⡴⣤⠖⢁⠌⣹⣿⢷⠀⢀⢼⣿⣍⡡⡈⠲⣤⢆⢠⡾⡇⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢌⣦⠦⠼⠶⠷⣥⣜⣰⢳⠋⠉⢻⣤⣧⣾⣤⡟⠉⠙⣎⣆⣣⣬⠾⠶⠧⠤⣜⢡⡽
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⣴⣺⣿⣿⣿⣿⣿⡟⠀⢀⠼⢾⣿⣿⡷⠧⡀⠀⢻⣿⣿⣿⣿⣿⣗⣮⣿⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⣿⡏⠀⡦⠟⡏⢿⣇⡀⠙⠾⣶⣿⣿⡶⠟⠃⣀⣸⡿⢸⠻⢴⠀⢹⣿⣿⡿⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢮⣙⠻⣧⢰⣗⢂⣿⣾⣏⢸⠀⠀⠷⣿⣿⠾⠀⠸⡇⣹⣷⣏⡐⣺⠆⣼⠟⣛⡵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⣳⢮⣳⡉⢉⣝⣻⣿⣏⠸⡵⢰⣟⣿⡆⢮⠇⣹⣿⣟⣩⡉⢉⣞⠵⣞⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⡀⢙⣿⢨⣿⢿⣿⣿⣦⣖⣿⡿⢿⣿⣲⣴⣿⣿⣿⣿⡅⣿⡃⢀⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠛⢿⡿⢿⣿⣭⣿⡶⠿⠿⠷⠶⠿⠿⢾⣿⣭⣽⣿⢿⡿⠚⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣫⢭⣔⣲⣶⣶⣾⣤⣵⣷⣷⣶⣖⣢⡭⣝⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠻⠟⠿⠿⠿⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣤⡄⠀⠀⠀⠀⠀⢀⣴⡶⢶⣄⠀⠀⠀⠀⠀⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠸⠫⠀⠌⠛⠀⠀⠀⠀⠀⣞⠡⠀⠄⢻⡆⠀⠀⠀⢰⡟⠩⠉⠙⢻⠀⠀⠀⠀⠀⠀⠟⡛⠉⠿
⠀⠀⠀⠀⠀⠀⠀⢠⣦⣀⣴⡄⠀⠀⠀⠀⠀⠘⠂⠀⠀⠋⠀⠀⠀⠀⠀⢛⣦⣠⣴⡋⠀⠀⠀⠀⠀⠀⢠⡀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣹⡏⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠐⠞⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠠⣤⠀⠀⠀⠀⠀⠀⠈⢋⡥⠀⠀⠀⠀⠀⠀⠀⢠⡄⠿⠀⠀⠀⠀⠀⠀⠋⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⢷⡄⠀⠀⠀⠀⠀⣴⣿⠷⣶⡀⠀⠀⠀⠀⠀⣾⣿⡿⣿⡄⠀⠀⠀⠀⢠⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⠀⡍⠀⠐⢈⡅⠀⠀⠀⠀⠀⠉⠀⠐⠈⠁⠀⠀⠀⠀⣟⠡⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠸⣷⣴⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⣶⣴⣦⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣴⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠙⠿⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠂⠈⠁⠀⠀⠀⠀⠀⠠⠖⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⠶⠀⠀⠀⠀⠀⠲⠖⠁⠀⠀⠀⠺⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣧⣤⠾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠙⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣰⠋⠉⠛⠓⠶⠶⠦⣤⣤⣬⣙⣛⣛⣛⣛⣂⣀⣀⣀⣀⣀⣀⣀⣀⣠⣾⠃⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⡏⠀⠀⢀⠀⢤⣄⣈⠙⠯⡷⠀⠿⠀⣖⣲⠶⠆⠀⠀⠀⠀⠀⠀⠀⢠⡏⢹⡰⠃⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⣴⠟⠉⢻⠁⠀⠀⠈⣭⣭⣭⣿⡿⠶⠀⠀⠀⠀⠠⢴⣶⣶⣿⡿⢿⣯⣥⠀⠀⣸⠀⠀⣧⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⣿⣀⣘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⠄⠀⠀⣏⠀⠀⣿⠀⡰⢚⠋⠻
⠀⠀⠀⠀⠀⠀⠈⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⠶⠛⠚⠙⠏⠀⣠
⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡿⠟⢿⣄⣠⣄⢀⣼⠗⠒⣲⠟⠻⢯⣉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠓⠒⠛⠋⠁⠀⠀⠀⠀⠉⠛⠛⠿⠶⠶⠚⠉⠀⠀⠀⠈⠙⠳⢦⣄⡀
⠀⠀⠀⣀⡀⠀⠀⢀⣸⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⡦
⢀⡤⣤⣯⠹⠶⠞⠋⠁⢘⣇⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡼⠓⢦⡀⠀⢀⣀
⣹⡶⠄⠀⠀⠀⣀⣤⠶⠛⢻⣆⣀⣤⡴⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⡼⣧⡀⠀⠉⠲⠋⢽⠦
⠛⢲⡾⢃⡶⠛⠉⠀⠀⠀⠀⠉⠉⠁⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠳⣤⠾⠁⠀⠙⠳⢤⡀⠀⠀⠴
⠀⠈⠛⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠘⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⡶⠤⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⢀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣠⣀⣀⣈⣉⣉⣉⣉⡉⠉⠙⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢰⡏⠈⠉⠉⠉⠉⠉⠙⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⢸⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠉⠉⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠚⠋⠉⢹⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣴⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⢿⣿⣿⠏⠀⠀⠈⠃⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⣾⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢻⣿⣿⣿⣿⣷⣦⣀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣷⣄
⠀⢨⣿⠃⢠⣤⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣮⣝⠻⢿⣿⣿⣿⣦
⠀⢸⡏⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠛⢿⣦⡙⣿⣿⣿⡄
⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⣷⠸⣿⣿⣇
⠀⢸⠁⠐⠐⠛⠋⠙⠛⠻⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⣿⡇⢻⣿⣿⡀
⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢿⣿⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⣄⣀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⣾⣿⡤⢛⣉⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣵⣿⣺⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⣿⠟⡿⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠋⠉⠻⣷⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣼⣿⣿⠀⣁⣠⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠠⠀⠻⢻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⣿⣿⣿⠁⠈⠁⣸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⡞⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠔⠉⠉⠀⠀⠀⠙⠩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠠⠤⠔⢋⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡖⡉⠀⠀⠀⠀⠀⠀⠀⢐⠀⠙⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢴⡆⢰⣦⣤⣴⠂⢀⢔⡶⠦⢀
⠀⠀⠀⠀⠀⡴⡃⠔⠈⢀⡆⡀⢀⡀⠄⡄⠈⢀⡀⣀⢬⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⢸⣷⠀⠘⠃⠀⣠⢫⡾⢠⡞⠋⠛⢦
⠀⠀⠀⠀⡜⠘⡄⠰⠁⠘⠁⡄⠘⢠⠀⠁⠰⠀⠀⠘⠀⡈⠄⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠈⢿⣿⠀⠀⠀⣰⢣⡟⣿⠏⠀⠀⠀⠀⢢
⠀⠀⠀⠈⣷⣦⣤⣀⡀⠃⠀⠀⡆⠈⠀⠄⠀⠀⠇⠃⣀⣡⡞⠀⢠⣄⣏⡀⠀⠀⡰⠣⢄⠀⠐⡋⠀⠀⢠⢃⡿⢸⣿⠄⠀⠀⠀⠀⠀⢃
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢲⣶⣶⠒⡒⠂⠉⠁⠙⣿⣿⣿⣤⣤⠴⠧⡁⠀⢺⣤⡀⢠⠑⣄⣧⠀⢀⡟⣼⡇⣿⢫⡤⠤⠠⠀⠀⠠⠤⡄
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⡟⠀⠃⠀⠀⠀⢣⠈⢿⣿⢿⣫⣀⣀⣥⠀⠈⡅⠈⢢⡁⣼⣹⡄⡼⣷⠛⣿⣿⡷⢒⣀⣉⣉⣉⣉⡒⠇
⠀⠀⠀⣿⣿⣿⣿⠟⠛⣻⠟⠁⠻⡇⢰⣄⣠⣤⣤⣼⣧⠈⠿⠛⠻⠻⢹⠃⠀⠀⠇⠀⠀⣹⣿⣟⢷⡿⣱⣾⣿⣿⣿⠋⠀⠀⠈⡹⠿⠿⠃
⠀⠀⠀⣿⣿⣿⡇⠀⢰⡇⠀⠀⣸⠃⠘⠛⠛⠛⠛⠋⢿⣧⠀⢸⣿⡏⡉⠀⠀⠀⢸⠀⢸⣿⣿⣷⣤⣾⣿⣿⣿⡿⠁⠀⠀⣠⠊
⠀⠀⠀⠘⣿⣿⡇⠀⢸⡇⢸⣿⡿⠀⠀⠄⠀⠀⠀⠀⠸⣿⡆⠸⣿⠃⠇⠀⠀⠀⢸⡀⠈⢿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣴⣾⣿⣧
⠀⠀⠀⢰⣿⣿⣧⠀⠸⠁⣼⣿⠇⡄⠀⠀⠀⠀⣠⠀⠀⢻⣷⠰⠘⠚⠀⠀⠀⠀⣮⡄⠀⠙⣿⣿⣿⡿⠃⠀⠀⣴⣿⣿⣿⣿⣿⣆
⠀⠀⠀⣾⣿⣿⣿⣷⣶⣾⣿⡏⠐⠀⠀⠀⠀⡀⠂⡀⡀⢸⣿⡄⠀⠀⠀⠀⠀⣼⣿⣧⢠⣼⣿⣿⣿⡁⠀⣠⣷⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠄⠀⠿⠿⠿⠿⠿⠿⠿⠿⠤⠾⠤⠤⠤⠤⠤⠤⠤⠴⠷⠿⠧⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⣤⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⡔⢆
⠀⠀⠀⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⡇⣿⠇⠐⠒⠒⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠀⠾⢿⣿⣿⣿⣿⣿⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣾⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢻⣿⣿⣿⣿⣰⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⠿⠛⠈⠻⣷⣔⣄⠀⠀⠀⠀⠀⠀⣠⣴⣶⣄⠀⠀⣤⠋⢀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠦⣄⠀⠀⠀⠀⣿⣿⣿⣿⡇⠈⠁⠐⠋⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣌⡂⢰⣤⣠⣿⣿⣿⣿⢀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣹⣿⣿⣿⣧⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣒⠈⠙⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⡿⣋⣽⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠛⠛⠛⠛⠛⠋⢉⣼⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣴⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠛⠻⠿⠿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠛⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡈⣀⡈⡸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣦⡛⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⢟⠀⢘⠧⠤⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⡇⢹⠀⢈⠑⠒⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠠⢰⠈⠀⠀⠀⠀⠀⠀⠄⠀⢲⡧⠬⠈⠁⠀⢰⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⡈⡀⠐⢉⠧⠰⠀⠀⠀⠈⠧⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⢠⠁⠀⡠⢃⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠅⠀⣀⡀⠠⡆⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⡰⡀⢠⠀⡀⣘⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢈⠿⣁⠘⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⢠⠀⠀⠀⣰⣤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡥⠜⠒⠒⠴⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢰⣿⡇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⠀⠀⠀⢠⠟⣽⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠀⠀⢧⠂⠀⣿⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠓⠀⠀⠀⠐⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡁⠀⠐⠉⠓⠉⠈⡈⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠀⠀⠀⠀⠀⠰⠀⢀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠀⢀⠀⢸⡥⠤⠥⡴⣀⡀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠃⠁⠃⠀⠀⠀⠀⠀⠸⣇⠀⠆⠀⠀⠆⡉⠉⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⢸⠀⠀⠈⠉⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠘⢩⠁⠀⠀⠀⠀⠀⢸⡾⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⢈⣷⠋⠁⠀⠀⠀⠑⠄⠀⠀⠀⡍⢶⡇
⠀⠀⠀⠀⠀⠔⠋⢡⠀⠀⡼⢹⠑⠠⠆⡱⠀⣠⣛⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⣆⢀⡜⠛⢀⡀⠀⠀⠀⠀⠈⠢⠀⠀⡁⠘⠙⣄
⠀⠀⢀⠔⠁⠀⠁⠁⠀⠀⠈⠈⠀⠈⠐⠀⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠈⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⠀⠉⠁⠀⠉
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠭⡓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠈⠈⢳⡀
⠀⣀⡤⠤⣤⣀⠀⠀⠀⠉⠓⠦⣄⠈⣇
⡞⠁⠒⠒⠂⠉⠓⢦⡀⠀⠀⠀⠈⢧⣿
⢧⡀⠀⠀⠀⣀⡀⠀⠙⣆⠀⠀⠀⠸⡏⠀⠀⠀⢀⣠⣀⣠⣤⣤⣀⡀
⠀⠉⠙⠋⠉⠉⠉⠙⠲⣌⣧⠀⠀⠀⠀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡆⠀⠀⠀⢠⡏⠀⠀⠀⣀⣠⣄⠀⠀⠀⠀⠙⣆
⠀⠀⣠⢚⣛⠛⠒⠶⢤⡀⠀⠀⠀⠀⠀⢸⡇⣠⠴⠛⠉⠀⠈⠳⣤⣀⠀⠀⣹
⠀⠘⣇⠁⠀⢀⣤⠴⠦⢭⣷⡄⠀⠀⠀⣼⠛⠃⠀⠈⠓⠚⡁⠀⠀⢉⠛⢶⠏
⠀⠀⠈⠓⠛⠉⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣦⠀⠀⠀⠉⠉⠀⠀⠘⠬⢶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠶⠦⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣟⠻⢦⣀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠉⠉⠁⠀⠉⠙⠛⠛⠛⠛⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣤⡛⠶⠦⢤⣤⣀⡀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣾⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠈⠉⣻⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠘⣧⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⡀⣠⡏⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⢸⠿⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⡿⣋⡴⠳⠦⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⣀⠀⠀⠀⠈⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⢀⣤⣄⣉⣙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢲⡏⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠓⠶⢤⣤⣀⣀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠉⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⣾⣻⣭⣥⠴⠶⣦⣹⣦⣄
⠀⠀⠀⠀⠀⢰⡿⢃⡀⠠⠤⠶⠾⣿⠇⣹⡀
⠀⠀⠀⠀⠀⣸⠃⢏⣀⠀⠤⠤⣤⣾⣽⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠶⠶⠒⠒⠒⠒⠀⠀⣀⡤
⠀⠀⠀⠀⠀⣿⠀⠈⠁⠐⣦⣤⡤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠁⠀⠀⠐⠦⣤⣤⣤⠶⠟⠉
⠀⠀⠀⠀⠀⡿⠀⠀⠀⠈⠙⣧⣴⢟⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⣷⠀⣶⠀⠀⠀⢸⣧⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⣿⠀⠛⠶⠶⠶⢋⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⡴⢒⡦⠖⣻⠓⠆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠻⡻⣌⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣴⣿⣤⣘⣵⣋⡉⠉
⠀⠀⠀⠀⠀⠀⢻⡀⢢⡀⠀⠀⠙⣦⠙⣧⡀⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⠀⠈⠉⠛⢿⣟⢷⣄⠐⢶⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢷⡀⠀⠀⠈⠃⠈⢷⣄⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠹⣆⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⢷⠀⠀⠀⠀⠀⠀⠙⢷⣄⣾⠃⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⣧⣄⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠙⢿⣸⡄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠉⠛⠦⣌⠙⠻⠷⢶⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⢠⡿⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⢸⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⠸⣄⠀⣿⣿⢧⣀⣀⡀⠀⠀⠀⠀⠀⣿⡀⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⢸⡇⣸⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡎⠒⠀⢹⡄⠑⠚⠒⠊⠀⠀⠀⠀⠀⣿⣿⢯⠐⠦⣤⣤⣤⣴⠾⠋⠀⡠⣞⠄⠀⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠸⣧⡀⠀⠀⠀⠀⠀⢠⣤⠾⠃⢹⡄⠀⠀⠀⠀⠀⠀⢄⠀⡼⡼⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠈⠛⠷⠶⢶⣤⡤⠤⠄⠀⠀⠸⠃⢰⡄⠀⠀⠀⠉⠢⣭⣥⣷⠄⠀⢀⣾⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⣾⠋⠀⠙⢷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣀⠙⠦⣤⣤⠤⠀⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠉⠛⠛⠿⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⡀⣀⣠⣴⠾⠛⠛⠋⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠍⣿⣿⣿⣿⠟⢿⣇⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⠟⣡⣶⣌⠻⢿⠃⣾⡆⢻⣷⣤⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⣸⣿⣿⣿⣷⡆⢸⣿⣿⠄⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠟⢛⠻⣿⣿⣿⣿⣦⣉⣩⣴⣦⣬⢹⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⠛⣋⣼⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣤⣤⡙⢿⣧⡘⢿⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⢸⣿⣿⠆⣿⣿⣿⣿⣿⠃⣼⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⢾⠟⣡⣾⡟⠛⢛⣋⣤⣾⣿⣿⢃⣶⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⢸⡟⣛⣉⣉⣉⣉⣉⣉⣉⣙⡛⢿⡇⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠘⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢀⠇⡇⠀⡠⠴⡄⢀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⡇⠘⠾⠀⠘⠊⠀⠀⠙⠛⢤⠀⠙⢆⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢇⠓⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣙⠚⠁⠀⠀⠀⢀⡠⠔⠒⠉⠑⠒⡶⠀⠀⠀⠀⢨⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡗⡒⠉⠁⢀⣠⣢⠀⠀⡸⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜⢢⠀⠀⣞⠀⠈⡇⢀⢧⡆⠀⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⡜⠀⠀⢿⢀⡴⠃⣸⠁⠹⠏⣅⢸⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠙⠊⠐⠉⠀⠀⠉⢀⣴⣿⡄⠀⢀⠡⢞⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠢⣤⣀⣠⣤⣾⡟⠋⢹⠇⠀⢀⣤⠞⢛⣣⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠈⢿⡿⠛⡟⣀⡴⠋⠀⡴⠋⣟⠉⢹⠈⢩⠃⠁⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢯⠟⠒⡄⠈⠓⠤⣍⣛⣉⣁⣠⠤⠞⠁⢀⣸⠤⠞⠀⢇⣠⠆⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢇⡔⠰⡇⠀⠀⠀⢀⣀⡴⠋⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠑⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠍⠊⠳⣌⡑⠒⣒⣳⠎⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣏⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢫⡉⠉⠉⠁⠀⠀⠀⠀⠈⣹⠃⠀⠀⠀⣠⠦⠤⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢨⢗⢳⠀⣀⡤⡀⠀⢀⠀⠀⠉⠒⣦⠀⡠⢷⡒⢦⢀⠀⡝⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣾⠀⠙⠊⠁⠀⠈⠉⠻⣄⠀⢀⡴⣃⡰⢋⡵⠟⠳⢮⣎⡠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠾⢛⣛⡿⢝⡢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠲⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠈⠑⠪⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢈⡷⣄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠒⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠛⠢⡺⣯⡆⠀⠀⠀⢰⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢠⠉⠈⣀⠙⣷⡀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠐⢹⠀⣿⠀⢸⣿⡑⢾⡇⠀⠀⠀⣿⠃⠀⠐⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⡟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢿⣀⣠⠄⠐⢸⣧⣄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠨⠀⠀⢸⡟⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠒⠚⠀⠀⠘⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡶⣛⠶⠥⠾⠶⢦⡴⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣥⣼⣿⣿⡿⣿⣿⣷
⠀⢸⢹⢳⡒⣿⣥⣄⡴⠛⣁⠀⠀⠄⠤⠊⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁⣸⣿⡇
⠀⢸⣏⠀⠀⢸⡟⣡⠂⢩⠃⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⣿⡟⠁⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠘⠋⠀⠀⢸⢸⡇⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠰⠀⠀⠀⠀⠀⠛⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠿⢯⣑⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⡧⠓⢒⡲⠀⠀⠀⠀⠀⠀⣽⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠏⣩⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⠐
⠀⠀⠀⠀⠠⠒⣠⠈⠿⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡽⠊⡄⠂⡄
⠀⠀⠀⠀⠐⢾⡿⠆⠀⢦⡙⢄⠀⠰⠒⣄⠀⠀⠀⠀⣼⣿⣿⣷⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⠀⠀⠀⢠⠐⠢⠀⣠⣾⡿⠀⠈⢿⠶⠃
⠀⠀⠀⠀⠀⠀⠘⡜⢆⠘⣷⠈⢣⡰⢾⣗⠀⠀⠀⢠⣿⣿⣿⣿⣿⣄⠀⠀⠀⣦⣀⠀⠀⠀⠴⡿⣿⣿⣷⣷⡀⠀⠀⠀⣺⡷⢦⠖⠈⣿⡇⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣮⣦⣿⡀⢸⣷⣾⣿⣧⡀⣠⡿⣹⣿⣿⣿⣿⣮⡳⣤⣀⢿⡿⠄⠂⠀⠀⠀⠈⢹⣿⣟⣷⣄⢀⣼⠿⠛⠁⠀⣴⢿⣴⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢹⡇⠸⣿⣮⣻⠿⠾⣋⣼⣿⣿⣿⣿⣿⣿⣿⣾⣧⠤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣮⡳⠶⠶⣿⠏⠀⣼⡏⢸⢟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣰⠁⠀⢻⣿⣿⡟⠋⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡏⠀⢀⣿⠟⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⣻⣯⣥⠤⣐⣢⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⢉⣉⣉⣛⣛⣻⣿⡻⠥⠤⣾⣟⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠛⣭⣭⣿⠁⠀⠈⠹⣿⣿⣿⣿⣿⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣾⣶⣶⣯⣭⠛⠒⢒⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡇⣹⣿⣿⡀⠀⠀⣠⣿⣿⠿⠿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠾⣿⣿⣿⣿⣿⣤⣤⣾⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢷⣛⣯⣭⣿⣿⣿⣿⣿⡿⣶⠶⠶⠶⠶⠶⠤⠤⠄⠀⢀⣀⣀⣀⣀⣀⡤⡤⣶⣒⡚⠛⠻⠿⠿⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣾⣿⣷⣾⣿⣿⣿⣿⡿⠿⠶⠶⠶⠶⠶⠀⠠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢷⣀⡜⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠿⠿⣷⣤⣭⣭⣭⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣩⣽⣿⣿⠷⢿⠿⠿⠿⠿⠋⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠙⠃⠠⠀⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠠⢼⡇⠀⠀⡀⠉⠓⠦⣤⣀⡀⣀⣀⣀⣀⣤⡤⠖⠒⠈⡍⢩⠁⡏⠈⠉⠛⠦⣄
⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⠀⡀⠀⣴⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠸⣦⣇⠀⠀⠀⠀⠀⠙⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣯⢰⡇⠈⡉⠰⠾⠛⠶⠃⠤⠀⠀⢉⣥⠖⠀⠐⠀⢠⠈⠁⣴⣏⣿⠢⣄⣉⠀⠀⠀⠀⠈⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⣟⠁⠀⠀⠀⢀⡀⣤⠀⠀⠀⠁⢀⣠⠄⠀⠀⠀⡜⠁⠋⣸⣏⣿
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣤⣤⣶⡌⣿⣤⣦⣄⠀⠈⠁⢠⠁⣀⠀⠀⣀⠁⠒⠀⠀⠀⠁⡀⠀⣹⣯⣿⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢈⣀⣤⣴⣿⣿⣿⣷⢾⣿⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⣠⣴⣿⣿⣿⣿⣅⣷⣤⣤⣀⠀⠐
⠀⠀⠀⢠⣶⣶⣿⣿⡏⢹⣿⣿⠏⣿⣼⣿⠙⣿⣿⣿⣿⣷⣄⠀⠀⠂⠚⠃⣀⣴⣿⣿⣿⣏⣿⣿⣿⣾⣿⣻⣿⣿⣷⣶⣄
⠀⠀⠀⢸⣿⣿⣿⠟⠁⢸⡿⠇⠀⣹⣿⠶⡖⢋⡍⠛⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣹⣿⣿⣿⠟⣿⣿⣿⠟⣻⣿⣿⢻⣿⣿⣷⣤⣀
⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠠⠈⣿⡄⢀⠀⠀⠀⠉⠉⠙⠛⠉⣿⣿⣿⠋⠉⢹⠿⠧⠈⣴⣿⡈⠁⠉⠻⢿⡟⠀⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣉⣠⡶⣿⠀⠀⠀⠀⠀⠈⠛⠛⣛⡆⠙⠛⢉⣈⡁⣤⣀⣤⣾⣿⣻⣷⡘⠛⢠⡄⡈⠁⠠⢿⣿⣿⣿⣿
⠀⠀⠀⢀⣀⣠⣤⣶⣾⠿⠛⠛⠛⣧⣿⡟⠛⠿⠿⢿⣿⣾⣟⡿⠿⠟⠛⠟⠛⠛⠈⣡⣬⠀⣾⡿⠯⡤⠄⠀⠄⠉⠀⠀⠐⠉⠉⠙⠿
⠀⠀⠀⠺⡿⠿⣿⣬⣤⣤⣀⣀⣀⣼⣼⣿⣠⣤⣤⣤⣤⣄⡉⠀⠀⣤⣴⣾⣷⣿⠿⠿⠷⠶⢿⣿⡿⠿⡶⠶⣾⣶⡆⢰⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠛⠛⠛⣿⣯⡉⠈⠉⠙⠁⠈⠋⣻⣿⣿⣯⣅⣩⣭⣶⣶⠛⠓⢾⣾⡇⠀⠀⣀⣉⢀⣀⣌⣉⣤⣴⣿⣿
⠀⠀⠀⠿⠛⠒⠿⠿⠿⢿⣿⣷⣤⣀⣿⢿⡧⠤⣴⠒⣿⣿⣿⡿⠋⠉⠭⠉⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⠛⠋⠁⠀⠈
⠀⠀⠀⣤⣤⣤⣤⣴⣶⣶⣿⣿⠿⣿⣿⣿⣧⣤⣴⣦⣽⣿⣿⣿⣶⠾⠿⢿⣿⣿⣿⣿⣿⡿⣿⢻⣿⣟⣛⣿⡿⣿⣻⣿⣿⣿⣦⣤⣴
⠀⠀⠀⣿⣿⣍⣉⣉⣉⣁⡀⠀⠀⢀⣿⣿⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⠀⣤⣤⣸⣿⡏⠀⠀⢀⠀⠈⢀⣤⣀⣸⣿⣿
⠀⠀⠀⠈⠉⠛⠛⠻⠿⡿⢿⣿⣶⣾⣾⣿⣞⡛⠛⠛⠒⠤⣤⠄⢿⡛⠛⠛⠻⠿⠿⣿⣿⣿⣾⣿⣿⣏⣥⣴⣶⣶⣶⣿⣿⡿⠛⠹⠿
⠀⠀⠀⣀⣀⣀⣦⣤⣶⣶⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣤⣤⣶⣿⣿⡿⠙⣿⣿⣏⣉⣉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠈⣿⣿⣿⣥⣬⣤⣉⣠⣄⣀⡀⣿⡟⠉⠃⣀⣠⣬⣿⣿⣿⠿⠟⠿⡿⣿⣿⣯⣥⣤⢾⣿⣯⢀⣙⠃⠀⡀⠀⢀⣀⣠⣤⣾⣿⣿
⠀⠀⠀⠀⠈⠉⢋⣻⣟⣻⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣄⣁⣶⣦⣤⠤⠿⠿⣿⣿⣾⣿⣿⣿⣿⣿⣿⢿⣿⠿⠟⠛⠉⠁⠈⠋
⠀⠀⢀⡀⠀⢀⣬⣉⣉⣀⣤⣤⣶⣶⣿⣿⣿⣷⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠛⣿⣿⠉⣉⡉⣤⠔⠂⠀⠀⠉
⠀⠀⢻⣷⣿⣿⣟⣛⣛⠉⠉⠉⣉⣹⣽⣿⠻⠿⢿⣶⣾⣶⣦⣤⣤⣤⡀⣀⣈⣽⣿⣿⣿⣶⣿⣯⣸⣿⣿⣦⣶⣦⣤⣴⣤⣤⣀
⠀⠀⠻⠻⠿⠿⢿⣿⣿⣿⣶⣶⣥⣼⣿⣷⣶⣶⣿⣿⣿⠿⠿⠛⢿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢛⡛⣉⣉⣍⣉⣤⣄⡁⢘⡛
⠀⠀⠀⠀⠀⠺⠆⢉⣁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀⠀⣀⠀⠀⠉⠘⠓⣿⣿⣿⣿⣿⣿⣿⢿⣿⠿⡿⣿⣿⣿⣿⣿⣿⣷⡼⠃
⠀⠀⡀⣀⣤⣴⣾⣿⣿⣿⡟⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣍⣀⣄⣤⣴⣦⣾⣿⣿⠿⠟⠉⠁
⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣏⡙⠛⠿⠿⣿⣿⣶⡷⣤⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⢶⣀⣀⣄
⠀⢸⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣯⡿⠿⠛⣛⠋⣉⣉⣉⣀⣀⣀⣀⣀⣉
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣯⣽⣿⣾⣿⣾⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⠻⠋
⠀⠀⣼⣟⣹⣟⣿⣿⣿⣿⣿⣿⣷⣶⣾⡿⣿⣙⣿⣿⣿⣿⣿⢿⣿⠿⣟⠿⣾⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⣼⢟⣿⡛⣿⡇
⠀⠀⢸⣾⢻⣟⣻⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣴⣿⡿⣛⡛⣿⣿⣿⢛⣜⣿⣽⣿⣿⣷⣿⣧⣿⡹⣿⡄
⠀⠀⣌⢙⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣧⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡝⢟⣱⣿⣿⣿⣹⣿⣿⣇⣿⣿⣿⣿⣿⣿⣷⣿
⠀⠰⣿⢎⣽⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠻⣙⣿⣿⡟⣿⠏⣼⣿⣿⣿⣿⣿⣿⡿⣿⠁
⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡟⢻⣿⣿⣿⣿⣾⣿⣟⣿⡟⣼⢿⣛⣿⣽⣿⣿⢻⡌⣀
⠘⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣧⣴⣤⣼⣿⣿⡟⢿⣄⢻⡅⢥⡄
⢀⣄⡸⣻⣿⣿⣿⣿⠿⣿⢿⣧⠿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣿⣯⠿⢛⣿⣿⣿⣿⣿⣿⣹⣯⠙⠿⠋⢳⠈⢻⡀
⠀⠏⣰⠻⣿⣿⣿⣿⣷⡞⢿⣏⣦⣹⣿⣷⠆⣽⣿⣿⠟⣿⣿⣿⣿⡟⣿⡍⢁⠁⣠⢿⠛⡿⢿⣿⣿⡟⣧⢈⣿⡞⢀⣤⢿⡆
⠀⣿⠛⢀⣿⣿⣿⣿⣿⣿⣿⣽⣯⣬⣿⣽⣾⣿⡿⢿⣿⢿⣧⣿⡄⢻⣿⣤⣟⠠⣧⡼⢧⡰⣿⣿⣿⣿⠋⣼⡟⣿⣏⠛⠛⠁
⠀⠈⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⡎⣩⡟⠻⣿⡴⢾⢋⣶⣉⣱⣦⠰⢞⠶⣿⡿⢻⣿⣿⣿⣾⣿⣿⣿⣟
⡀⢠⣼⣿⠿⣛⣿⣿⣥⣴⢾⣿⡿⣾⣿⣿⣿⠷⣯⣿⢿⣴⠿⠛⢀⠶⣤⢿⡛⡉⠁⣴⣛⣏⣹⣶⣿⣿⣿⣿⣏⣙⡿⣏⡋
⠙⣻⠧⣄⣳⢮⣿⣿⣿⣧⣺⠿⣿⢫⣿⣿⣿⡊⣿⣿⣿⣬⡟⢳⣘⡖⢻⣴⠏⢠⠦⢈⣿⣿⣿⣿⢿⡤⣥⡼⢿⡬⠃⣬
⠀⣽⣶⣟⣱⠾⣿⠛⣿⠟⢿⡶⢿⣉⠿⣿⣿⡍⠿⣶⣿⠷⣍⠉⣁⡽⢮⣥⡾⠃⣠⣾⣿⣿⠟⣿⣿⣵⣽⠶⠏⡤⠺⠋
⠀⣼⣽⣿⣿⣿⣾⣷⣽⣶⢾⡿⠖⣉⣿⣾⣿⣿⣤⡈⠉⠶⢈⡀⠠⠶⠂⣉⣷⣿⣿⣿⣿⣿⣷⣿⠇⣻⡁⠋⣶⠄
⢄⡉⠙⠛⢿⡿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣧⣁⣰⡏
⠀⠛⠂⠳⡞⠒⠇⠈⢉⡍⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⢙⣫⣿⠿⣿⠗
⠀⢲⣦⠆⠐⠀⢛⠁⢠⠏⠸⣇⠹⣿⣿⣿⣿⣿⣿⣿⠿⢿⣉⡙⠫⠤⠒⠀⠀⠈⠛⠋⠀⠈⠛⠋
⠀⠀⠀⠀⠘⠟⢀⡇⢠⡶⠛⣃⠀⠸⣿⠘⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣀⠁⡀⠀⢀⣤⠿⠛
⣤⣤⣤⣄⣠⣤⣤⣤⣤⣤⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢦⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠛⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⡤⠀⠀⠀⠀⠀⠀⠛⠋⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢻⣿⣿⣿⣿⣿⣿⣤⣶⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣿⠇⠀⠀⠀⢀⣶⣄⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⣾⣿⣿⣿⡿⠙⢿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡄⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⣿⣧⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣤⣀⡀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⠛⠿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⣸⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣼⣿⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃⢀⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣴⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠀⠀⠀⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⡎⠉⠉⠉⠉⠉⢳⣶⣶⣶⣶⣿⣿⣿⡉⠉⠉
⠀⠀⠀⢀⣠⣶⣶⣶⡖⠀⠀⠀⠀⠈⠛⠛⢻⣶⣶⣶⣶⣶⣶⣾⠛⠛⠛⠋⠉⠉⠉⠓⣶⣶⣶⣶⣄
⠀⠀⠶⠿⠿⠿⢿⣯⣤⣤⣤⣤⣤⣤⣤⡤⠿⠿⠿⠿⠿⠿⠿⠿⣤⣤⣤⣥⣤⣤⣤⣤⣼⡿⠿⠿⠿⠷
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠠⡐⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢨⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣧⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢨⠁⠩⡤⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⡼⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢚⡰⠾
⠀⠀⠀⠀⠀⠀⠀⠰⢻⠋⠹⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢡⠋⣉⣉⢉⡉⠁⡀
⣦⡖⠊⠂⠀⠀⢠⠂⠐⠀⠐⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⠈⡏⠁⡏⠀⡇
⢿⢷⡇⡞⠁⠒⠁⠀⣇⠀⠀⠀⠀⢠⣇⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⠔⡖⠒⠊⠈⣮⡄⠀⠀⠘⣯⠉⠙⠃
⠈⠉⠉⠓⢤⣀⣀⠾⡏⡄⠀⠀⡼⣁⠀⡹⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠒⠓⠤⣀⠔⠁⣿⠀⠀⠀⠻⢷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣆⠀⠈⠛⠙⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡄⠄⠐⠊⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⠓⠒⠀⠒⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡞⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣀⠔⠈⠀⠑⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠞⠫⡀⠀⠀⠀⠉⠉⠉⠑⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠋⠈⠢⣀⠀⠀⠀⠀⠀⠈⠑⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡾⠁⠀⠀⠈⠲⢄⣀⡠⠤⠶⣄⣏⠠⢛⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡜⠀⠀⠀⠀⠈⠉⠒⠒⠲⠦⠤⠴⣧⡇⢠⢿
⠀⠀⠀⠀⢀⡼⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠇⠀⠀⢀⠀⠀⠀⠀⡼⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣮⠏
⠀⣀⣀⣾⣉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠝⠀⠀⠀⠸⣏⠳⠆⢰⣙⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠏
⠑⠪⢿⣍⣉⠙⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡢⢄⣨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⠀⠀⠀⠈⠉⠉⠐⠀⢀⠄⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡇⠀⠀⠀⠄⢈⠁⠄⠀⠀⡝⢧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⡽⠁⠘⠦⣀⣀⣤⣊⡑⠤⠤⠞⠀⠀⠹⣍⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⡔⠉⠀⠀⠀⠈⠐⡄⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⣄
⠀⠀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⣀⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀
⡠⠸⢄⠘⠀⠸⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠀⠸⠀⡙⢀⠼⠠
⡰⠂⠤⠀⠀⠀⠃⠤⠀⠀⠀⠏⢇⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢀⠏⢃⠀⠀⠠⠄⠃⠀⠀⠀⠄⠲
⠑⠒⠒⠄⠔⠓⠤⣀⠀⠀⠉⣇⠈⠢⡀⠀⠀⣿⣿⣿⣿⣏⣁⣉⣁⣉⣿⣿⣿⣿⡇⠀⠀⡠⠊⠀⡜⠁⠀⣀⡠⠔⠓⠄⠔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠈⢢⠀⠈⠓⢦⡥⣄⣀⣀⣀⠀⠀⠀⢀⣀⣀⣀⡤⢥⠶⠋⠀⢀⠎⠀⢰⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠑⠤⡄⠀⠙⠢⡀⣿⣿⣿⣿⣿⣿⣿⡇⡠⠒⠁⠀⡤⠴⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠙⠢⠄⣀⠈⠙⢍⡀⠉⠁⢈⠝⠋⢀⣀⠤⠒⠁⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠭⢦⡾⠕⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢍⡉⠙⠃⠒⠐⠒⠒⠒⠲⠊⠳⠒⠒⠒⠒⠐⠂⠛⠈⣉⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠒⠒⠀⢤⠤⠤⠤⠄⠤⠤⠤⢤⠄⠐⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣤⣤⣤⣤⡼⠀⠀⠀⠀⠀⠀⠀⠸⢤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⠋⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢱
⠀⢠⠒⢺⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡗⠒
⠀⢼⠀⠸⡇⠀⢸⠀⠀⠀⠀⠀⢀⣠⢤⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡤⢤⢄⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡶
⣤⢸⡀⠀⡇⠀⠸⣒⡒⢒⣒⣒⣇⢠⡠⠀⢗⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡙⠀⠂⡤⢠⡁⠀⠀⠀⢈⡇⠀⢸⡇⠀⡇
⠛⠀⡇⠀⡇⠀⠀⠁⠀⠀⠀⠀⠙⣆⣀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⣴⠋⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⡇
⠀⠒⡇⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⣀⡤⢤⣤⡄⠀⢀⢤⡖⠲⠤⣀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠘⡇⠀⠉
⠀⠀⠣⠤⡇⠀⠀⡇⠀⠀⠀⠀⠀⢸⡴⠋⠀⠀⠘⠉⠀⠈⢑⠈⠉⠀⠀⠈⠳⣼⠃⠀⠀⠀⠀⠀⠀⡇⠀⠈⠓⠒
⠀⠀⠀⠀⠣⠄⠤⠃⠀⠀⠀⠀⠀⡸⣇⠀⠀⠀⠀⠀⣀⠀⠎⣤⠀⠀⠀⠀⠀⢉⡀⠀⠀⠀⠀⠀⠀⠓⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠈⠀⠀⠀⡄⠘⠛⠀⡆⠉⠁⢀⠃⠀⠀⠈⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⠀⡠⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⠀⠑⠠⣀⡀⠀⠀⠀⡀⡀⠐⠀⠀⠀⠀⠀⡀⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠀⠀⠀⠀⠀⠀⠖⠂⠀⠃⠠⠃⠀⠔⠀⠀⠀⠀⠀⠀⠼⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠂⠀⠀⠀⠀⠠⠀⣀⡉⠦⠀⠀⠀⠐⠤⠀⠀⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⣀⢀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⢀⡤⡀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠑⠒⠒⠒⠉⣿⠀⢹⡏⠉⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣿⠀⢸⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠉⢂⠀⢀⡹⠂⢎⡀⠈⡐⠉⠈⠲⡀

⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⡀⠀⢀⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢤⣼⣧⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣶⣾⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃
⠀⠀⠀⠀⠀⠀⠘⣽⡏⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⡵⠃
⠀⠀⠀⠀⠀⠀⢀⡟⢿⠀⠀⠀⠀⠀⠀⠀⠀⢿⣋⡉⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠱⡏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡆⠀⢠⡟⠿⠂⠀⠀⠀⠀⠀⠀⢸⠃⢻⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠀⢧⠘⣆⣀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠚⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡦⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⢇⡀
⠀⠀⠀⠀⠀⠠⣤⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⢿⣿⡀
⠀⠀⠀⠀⠀⠀⢹⣾⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡀⠈⣿⣷
⠀⠀⠀⠀⠀⢀⣼⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⡀⠀⠀⠀⠀⠀⠀⢰⡋⠀⣰⡋⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⣦⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⢾⣷⠄⠀⠀⠀⠀⠀⠀⠀⠘⠻⡄
⠀⠀⠀⠀⠀⠀⣀⣼⣿⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⣿⠉⠁⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣿⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠋⢛⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠛⠂⠀⠀⠀⠀⠀⠀⠘⠟⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠙⠿⠟⠃⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠄

⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣆
⠀⠀⠀⠀⠙⢣⣁⣸⣇⣹⡜⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠾⢿⡿⠷⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠹⠏⡇
⠀⠀⠀⠀⠀⠀⠀⡏⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠉⢷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠀⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠤⠐⠒⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠄⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣯⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⣤⠤⠔⠒⠒⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣀⣤⠤⠐⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦⠙⠂⠀⠀⠀⠀⠘⣿⡄⠀⢀⣿⣇⡀⠀⢰⣶⠀⡠⠀⠀⠀⠀⢸
⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣷⠆⠀⠀⠀⠀⠀⣍⣿⣦⣌⡙⠉⠀⠈⠻⠯⢀⡟⠀⠀⠀⠀⢸
⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣛⡳⣆⠘⢷⣶⣆⠀⠀⠀⡉⠄⠀⠀⠀⠀⠉⠕⠓⢶⣾⡇
⠀⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢞⣫⡶⠟⢋⡙⢿⡇⣼⠟⢿⡆⠀⠈⢷⣶⣦⠀⠰⣶⣾⣶⠈⣷⣬⡁
⠀⣿⡿⠧⠀⠀⠀⠀⠀⠀⢀⡰⣿⡵⠛⣁⠶⠛⣉⣡⣼⣿⣿⣁⠈⠀⠀⡀⠺⠟⣷⡄⠀⠛⠿⠛⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡿⢋⡰⡟⠍⠒⢉⣁⣠⣤⣤⣌⣉⠁⠀⠀⠡⢤⠘⠛⠛⠀⢀⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠋⠃⣠⣴⣾⣿⠟⠉⠀⠀⠈⠉⠀⠀⠀⠀⢰⣶⠶⢦⣤⣤⣴⣿⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡟⠉⢻⡿⠁⠀⠠⠶⢦⣄⠀⠀⠀⠀⠀⠘⠛⠛⠛⠿⢟⡿⠟⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢤⣶⠀⣶⠂⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⠾⠃⠒⢖⣊⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣤⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⢸⠿⠛⠉⠁⠀⢀⠀⣜⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡀⠀⠀⠀⠀⠀⢀⠀⠀⣆⣠⠄⠒⠈⠀⠀⣀⣤⣴⢎⠔⣁⠸⠿⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⢿⡆⠀⠀⠀⠻⠿⠁⠀⡏⢀⠀⠀⣠⣶⡿⠿⢟⠕⢁⡀⠿⠇⠀⢠⡞⢁⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠊⠀⠑⢤⣀⣀⣤⡄⠀⣿⣿⠃⠀⢸⣿⣀⡴⠁⢀⣈⠿⠀⠀⣴⠏⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⢀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡀⡏⣀⣠⣶⣿⢏⠎⠀⣤⣼⠛⠀⢀⡾⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠙⢿⣧⣤⣤⣤⣤⣤⣤⣤⣾⣿⣿⣿⡇⣿⠟⠋⠁⣦⠃⠀⣀⡘⠛⠀⣰⠏⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠙⣿⡈⠉⠉⠉⠁⠀⣿⣿⣿⣿⠇⠀⠀⠀⠐⠋⠀⣀⡙⠋⢀⡾⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣀⠀⠀⠀⡸⣿⣿⣿⠏⢀⠀⠀⠀⠀⠀⣠⡝⠃⣠⠟⠁⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠷⣦⣴⡿⣿⣿⠏⠀⠸⠄⣀⠆⠀⠀⠙⠋⣴⠏⢠⠃⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⣿⠟⠀⠀⠀⠒⠀⠀⠀⠀⠀⣼⠏⣰⡇⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⣤⣀⠃⠀⠀⠀⠀⠻⡐⠂⡴⣦⡸⠋⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠄⠀⠉⢻⣿⣿⡦⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠂⣠⣶⠾⠿⠋⠀⡠⠀⠐⠆⢠⡄⠀⣼⡏⣿⠀⠉⠀⢴⣆⠠⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣾⠰⡆⣿⡇⣴⠆⠀⣰⠁⠀⠈⠀⠀⠻⠿⠟⢰⠏⠀⠀⠀⠀⠉⠁⢦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠁⢸⡇⠻⠷⠿⢰⣾⠏⠀⠀⠀⠀⠀⠀⠀⣶⠋⠀⠀⠀⠀⠀⠀⠘⠆⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠂⠸⠧⠾⠿⠶⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠁⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⢿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⡇⣏⣿⣯⣿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠛⠉⠉⣝⠛⠻⣿⠇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠘⢷⣧⠈⢠⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠀⣾⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢈⣿⡿⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢰⣶⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⡀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣆⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠿⠿⠓⠀⠲⠶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⣀⣤⣶⣿⢿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢾⣟⣿⢮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸
⠀⠘⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠸⣄
⠐⠦⣽⣉⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⢤⣄⣀⣙⣂
⣿⣿⡿⠻⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠰⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢰⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⣠⠀⠀⠘⠻⠿⠃⣼⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠐⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢸⡏⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣴⠟⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠈⢿⡄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⢿⣆⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠋⠁⠀⠀⠀⠀⢀⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢿⣶⣶⣾⠇⣀⡀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⠿⣷⣦⣤⣤⣤⣾⣿⣿⣿⣷⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠁⢴⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢠⣌⠙⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⡄⠹⣿⣿⣿⠟⢋⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢠⣿⣿⠃⣼⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⡁⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⡀⢿⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⡷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⢠⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠛⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠀⠛⠛⠛⠁

⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⠀⢀⣀⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢀⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠿⠿⠿⠿⠿⠟⠁⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣤⣀⣬⣶⣿⣷⣄
⣸⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠇⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⢻⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠈⢿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠸⣿⣿⣿⣿⣿⣿⣦⣄⠀⠸⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣶⣤⠀⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠈⠉⠻⣿⡏⠉⠁⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡏⠘⣿⣿⣿⠿⠃⠐⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠘⠋⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠋⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠀⣹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣦⣤⣾⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣠⣴⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡒⢢⣤⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠉⠉⠁⠒⢤⡤⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣠⣤⣶⠶⠶⠶⠶⣶⣤⡈⢿⡆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⡁⠤⠤⢀⡀⠀⠀⠈⢻⣌⠻⠦⡀⠀⠀⠑⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⣁⡀⠀⠀⠤⣄⠀⠈⢆⠂⠀⡆⠆⠈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⣦⠀⠀⠶⠀⠀⠀⠘⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡤⠄⠊⠅⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡀⠉⠀⠀⣀⠔⠋⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣩⠀⠀⠀⣒⠖⠂⠁⠀⠸⠉⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠀⠀⠊⢠⠀⠀⠀⠀⡧⠀⠀⠠⢄⠀⠈⠢⡀
⡔⠀⠚⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠁⠂⠀⡞⠀⠀⠀⠀⠀⠑⢄⠀⠑⡀
⠁⠀⢠⣿⠉⠐⠀⠠⢀⡀⠀⠀⢀⠄⠈⠑⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠈⠉⠀⠀⡠⠔⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠐⠂⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⡰⠉⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠒⢄⠀⠀⣀⠤⠊⠀⠀⠈⢆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢏⠁⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠎⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣿⠃⠀⣱⠆⠀⠀⠀⠀⠀⠀⢂⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠁⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢏⡴⡚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢖⠻⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣄⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⡷⢼⠂⠀⠀⠀⠀⠀⣸⠃⢣⠀⠀⠀⠀⠀⢰⣤⡷
⠀⠀⠀⠀⢀⠀⣨⣧⡀⠰⣆⣰⣶⣦⣀⣺⣄⣰⣖⣶⣀⣺⣆⢠⣩⠀⢀⠀⠀⢠⣔⣤⢈⡲⣀⠀⠀⢀⠞⡁⠀⢀⠳⡀⠀⠀⣠⢞⠐⣤
⠀⠀⠈⣷⢼⡷⣿⠈⠋⠛⠋⠋⡿⠿⠉⠉⠉⣫⣾⣯⠛⠛⠉⣿⣿⣿⣯⠆⠃⠈⠉⣿⢸⡇⠈⢷⣖⣩⠊⠀⠀⠀⣷⣍⠒⢚⣵⣿⢨⠏
⠀⠀⠀⢸⣦⣤⣿⡄⠀⠀⠀⠀⠀⠀⡀⠀⣼⡿⠿⠧⠀⢀⣸⣿⣿⣯⡙⠐⠀⠀⠀⠱⣾⣧⣄⠀⠉⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣧⠌
⠀⠀⠀⠈⢛⠉⣿⣟⠉⠉⠀⠀⠀⠈⢹⣿⡟⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡆⠀⠀⠀⠀⣿⡯⠉⢩⡔⠀⠀⠈⠙⣿⣦⣤⠉⣽⢿⣿
⠀⠀⠀⠀⠋⠼⠿⠿⠷⠶⠶⠶⠂⠉⠉⠀⢀⣐⣶⣶⠶⠶⠶⠶⠾⠿⠟⠃⠀⠀⠀⠠⣿⣷⣶⣿⣭⣤⡀⠒⠺⣭⣭⣭⠀⣻⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⢉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠺⠇⡀⠀⢴⡄⠀⠀⠀⠀⣶⠈⡄⠀⠀⠀⢠⡄⠀⠐⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⠀⢠⡇⠀⠀⢀⢃
⠀⠀⠀⠀⠀⠀⠀⠳⣤⡄⠱⡄⠀⠀⢸⡇⠀⣷⠀⠀⠂⠈⠄⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣸⣯⠄⢸⡇⠀⠀⡌⢸⡀⠈⢃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡜⣿⣶⡤⣼⡇⠀⣿⡶⠀⣰⣴⡴⠁⠀⠀⠀⠀⠀⠀⠘⢥⡈⠀⣧⣿⣿⡰⣼⣇⠀⢴⠀⠸⣧⠇⢘⣀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡻⢣⣿⠃⠀⣿⣧⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⢰⠘⠃⠀⣾⣿⣿⣇⣿⠟⠀⠀⢀⣿⡟⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠃⠹⣿⠀⠀⠿⡿⢠⣿⡇⠄⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⠀⣿⢣⡌⣿⡇⣾⡆⠀⠈⣥⡛⠀⢸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⡀⠁⠀⠀⢀⣠⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣦⠀⠈⣿⣎⣿⣿⡧⠙⠛⠀⠠⠝⠂⠀⣉⣥⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠀⠉⠻⠶⠶⠶⠿⠇⠀⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠙⠯⠀⠀⣠⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⢠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠃
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⢨⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣲⡆
⠀⠀⠀⠀⠀⣽⡆⠀⠀⠀⠀⣿⣆⠀⠀⠀⠀⢀⡇⢁⠀⠀⠀⠀⣾⢫⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠁⡀
⠀⠀⠀⠀⠀⠸⣿⣄⠀⠀⣸⣯⢹⣦⣀⢀⣠⣾⡇⠈⣵⠀⠀⣼⣿⡗⠀⠀⠀⢲⠀⠀⠀⠽⡆⠀⢀⠀⠀⠀⠐⣷⡄⠀⠀⠽⠇⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⣧⣻⣦⣴⣿⡟⠈⣿⣿⣿⣿⣏⠀⣬⠈⢻⣿⣿⣳⠇⠀⠀⠀⠙⢠⡀⣀⣦⣷⡄⡀⠀⠀⠀⠀⢹⣿⡔⣾⣤⣆⠀⢀⡌⠁
⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⠇⠀⠘⢿⣿⣿⡿⢳⣷⠐⠿⠿⣣⣿⠀⠀⠀⠀⠀⠈⢿⡃⠙⠛⠊⠀⠀⠀⠀⢐⣿⣿⣿⠮⣭⣆⣴⣿
⠀⠀⠀⠀⠀⠀⠸⣤⣷⣾⠁⠀⠀⠀⠀⠛⠛⠁⣼⠟⢂⠐⣺⣿⠟⠀⠀⠀⠀⠀⠀⠉⣿⠒⠀⠀⢈⣿⣿⣿⣿⣿⠃⠀⠀⠘⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠙⠾⣧⡀⠘⠶⠟⠀⣰⠶⠆⠀⠰⠟⣡⠿⠏⠀⠀⠀⠀⠀⠀⠀⠺⠿⠶⠶⠶⣶⣶⣶⣶⡶⣾⠶⠶⠶⠶⠾⠍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠛⠀⠀⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠠


⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⣹⠳⡀
⠀⠀⠀⠀⠀⢀⡞⠁⡟⠀⢠⠃⠀⡇
⠀⠀⠀⠀⠀⡼⠀⢸⠃⠀⡎⠀⠀⠃
⠀⠀⠀⠀⢰⠇⠀⡟⠀⢰⠃⢀⢰
⠀⠀⠀⠀⣸⠀⢰⡇⠀⡾⠀⣾⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⠤⣀
⠀⠘⠀⠠⡿⠀⣸⠀⢀⡇⢠⡿⠧⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡸⠀⢸
⠀⠀⠀⠀⡇⠀⡿⠀⢸⠁⢸⣿⠃⠐⠂⠀⠨⢍⠋⣰⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⡇⠀⣸⠑⣆
⠀⠀⠀⠀⡇⠀⡟⠀⢸⠀⢸⢋⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠐⠚⠀⠬⢐⡂⠀⠄⡀⣀⠀⠀⠀⠀⠀⠀⠀⠜⡤⠀⡼⠀⢠⡇⠀⡏
⠀⠀⠀⠀⡇⠀⡇⠀⣾⠀⠀⠆⠀⠀⠀⠀⠀⠸⡄⠀⠀⠤⣤⡉⣿⠆⠀⠀⠀⠁⠒⠇⡀⡐⠀⠄⠀⠀⡜⣼⠇⢰⠃⠀⡾⠀⢰⠃
⠀⠀⠀⠀⢃⠀⡇⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠙⢄⡉⠈⠛⠑⠀⠈⠑⡄⠀⠀⢀⡴⠀⠀⠈⠑⠂⠨⣙⡟⢀⠏⠀⣸⠃⢀⡟
⠀⠀⠀⠀⠈⠉⢧⠀⢧⢠⠃⠀⠀⠀⠀⠀⠠⡀⠀⠀⠙⢄⠀⠀⠀⠀⠠⣼⡤⠔⠋⠀⠀⠀⠀⠀⢀⢱⡿⢀⡞⠀⢰⠏⠀⡼⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠙⢶⠢⢲⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠞⠁⡜⠀⢠⠏⠀⡼⠁⠂⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠖⠉⠉⣽⣦⣄⣀⣠⣴⣄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⡼⠁⢠⠏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠻⠿⣿⣿⠿⡇⠑⣄⠀⠀⡀⠈⠀⡎⠀⡜⠀⣠⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡿⣿⣦⣤⣀⣀⣴⡇⠀⠈⡠⠊⠀⠀⠀⣇⠎⢀⡴⢇⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⡈⠙⠻⠿⢻⠏⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠙⠷⠶⠞⠁⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠂⠀⠁⠀⠀⠀⠀⠀⠁⠀⠐⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠛⠻⠳⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⣶⢴⣦⠀⠀⠀⠀⠀⠀⠀⢸⣵⡾⣷⣶⣶⢿⠿⣷⡷⠀⠀⠀⠀⠀⠀⠀⣴⡤⣶⡶⢶⡄
⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢰⣻⡅⣴⣿⡟⣿⣁⣤⡀⠀⠀⠀⣯⣷⣷⣅⠀⠉⠉⠀⣰⣿⣿⣷⠀⠀⠀⢀⣤⣀⡿⢿⣿⣄⢈⣿⡆⠀⠀⠀⠀⠀⣀⡀
⠀⢠⣶⣿⣿⣷⣠⣄⠀⠀⠠⣿⣷⣯⠀⠀⣼⢱⡿⣇⠀⠀⢰⣇⢿⡏⠉⠁⠀⢠⡈⠉⢱⣿⣽⡆⠀⠀⣸⢿⡎⣧⠀⠀⣹⣿⣿⠅⠀⠀⣠⣄⣾⣿⣿⣶⡄
⠀⠈⠉⠙⢧⢹⡿⢿⣶⣄⠀⣽⢧⢹⣯⠉⠑⡘⡇⢹⡇⢀⢸⣷⣼⡇⠀⠀⠷⠾⠃⠀⢿⣇⣼⡇⡀⢸⡏⣸⢇⠊⠉⣽⡏⡼⣯⠀⣠⣶⡿⢿⡏⡼⠋⠉⠁
⠀⠀⠀⠀⣸⢾⠇⡀⠈⢻⡆⢻⣆⠋⢛⣟⣣⣁⢙⣼⣿⣁⣿⣿⣿⡶⣠⣶⡟⢻⣗⣄⢠⣿⣿⣯⣈⣿⣿⡋⣈⣌⡻⣛⠙⢰⡟⢰⡟⠁⠀⠼⡷⡇
⠀⠀⠀⠀⣇⣼⣿⢨⣤⣾⣻⣉⣽⣷⣾⣿⣏⣃⣿⡿⣫⠏⡙⠷⣷⣚⣡⠗⢰⡟⠻⣅⣙⣿⠾⢋⡹⣝⢿⣿⣘⣽⣿⣷⣾⣫⣉⣞⣷⣤⡄⡿⣧⣸
⠀⠀⠀⠀⠈⠙⠛⡷⢿⣾⣿⢟⣿⣛⣺⡿⠿⠿⠟⠿⠓⠳⠿⠯⠶⣾⣯⣽⣿⣿⣮⠽⠷⠶⠿⠿⠮⠺⠿⠻⠿⢿⣿⣗⣛⣿⠻⣿⣷⡿⢾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣾⣾⣛⣀⡤⠤⠤⢄⣀⣀⣀⣀⣠⣀⣼⡿⠻⡦⢤⣤⣀⣀⢐⣒⣒⣒⣤⠤⠤⣤⣬⣽⣶⣷⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠘⢾⡿⣿⢫⣭⣶⣟⠋⠩⣷⣶⡌⢻⣯⠻⠟⠙⡟⢫⣷⠻⠛⣿⡍⢻⣿⡸⠟⣸⣿⢣⣶⣿⠟⠉⡺⣶⢹⣽⣿⠻⡯⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢿⢻⣽⣮⣡⣿⣜⠷⠶⠿⣿⢅⣼⣏⢾⡷⢸⣷⣦⣭⠛⣛⣯⣴⣿⡿⢾⡷⣹⣯⡺⣻⡿⠶⠶⢫⣿⣉⣵⣿⡟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣥⣯⡟⠳⠿⠻⠟⠻⠶⠾⣟⣋⣈⣭⣭⣭⣉⣭⣭⣭⣭⣭⣉⡁⣈⣁⣀⣉⣥⣔⣚⠻⠧⠾⠿⠻⠿⢮⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿⠿⠿⠿⠖⠚⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⠻⠽⠷⠶⠶⣶⣾⣟⣻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣦⣶⣶⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢿⣿⠁⠀⠀⠙⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⢸⡿⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠢⢄⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠚⠃⠤⣀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠐⠈⠀⠀⠑⢌⢣⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠆⠂⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢀⣀⠠⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠒⠒⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⠰⠁⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢐⣀⡠⠄⠂⠉⢀⠠⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠑⠠⠀⣀⠀⠀⠀⠀⠀⠈⠀⠀⣀⠤⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⢄⣀⠀⠀⠉⠑⠂⠀⠚⠃⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠉⠀⠀⠐⣄⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣶⣤⣤⣤⣤⣶⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⠿⣿⣿⣿⣿⣿⣿⠋⠛⠋⠑⠢⡀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠜⠀⠀⠈⢻⣿⣿⠇⠀⠀⠀⠀⠀⠈⠑⢄⡀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠁⠀⠀⠀⠀⢠⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⢡
⠀⠀⢀⣤⠒⠉⠀⠀⢀⡀⠤⠐⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠆
⣾⠉⢧⣠⠦⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠸
⠘⣧⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⢃
⠀⠘⢧⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠘⡀
⠀⠀⠈⢷⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠚⣛⣤⡶⠊⢉⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⢁⣠⡴⠞⠋⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶
⣿⣿⣇⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⣴⣴⠀⠀⠀⠀⢠⣤⣄
⠀⠀⣠⣿⣿⣄⣄⣼⣿⣧⡄⠀⠀⠀⣰⠛⢻⣷⣶⣾⡇⠀⠀⠀⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠀⡏⠁
⠀⠀⠀⠿⣿⠁⠀⢸⡿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢠⣾⣿⣷⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠿⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠚⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣶⣄⣀⣀⣰⣶⠀⠀⠀⠀⡷⠶⠶⠶⠶⠶⡇⠀⠀⠀⠀⠀⠀⢠⣥⡄⠀⠀⠀⠀⠀⢰⣿⣿⡆
⠀⠀⣤⡄⠿⠏⢉⣯⡹⠿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⢀⣹⣏⣀⣀⣀⣀⣀⣀
⠀⠀⣿⡟⠛⠛⠻⣿⠅⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠈⢹⡏⠉⠉⠉⠉⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠘⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠘⢷⠂⠀⠀⢠⡾⠏⠀⠀⠀⠀⣀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣀⠄⢀⠤⣀⢰⣄
⠀⠀⣤⣾⣶⣶⣶⠾⢧⡀⠀⠀⠀⠈⣿⠀⠉⠉⠍⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣈⡀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⢿⣯⣼⣿⣿⣄⣠⠇⠀⠀⠀⠸⠿⠸⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⡇⠀⠀⠙⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠉⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⢠⣄⢀⣤⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣠⣤⣤⣤⣤⣿⣇⠀⠀⠀⠀⣿⣿⣤⣄⣾⣿⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣄⠀⠀⠀⠀⠙⢻⠉⢹⣿⡍⠉⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣄⣅⣸⣿⠀⠀⠀⠀⠀⢸⠀⢸⣿⡇⠀⡇
⠀⠀⢶⣶⣶⡶⡶⣶⣿⡇⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣭⣭⣭⣭⣍⠀⠀⠀⠀⠀⠼⠄⠨⠿⠄⠤⠧

⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⡄⠀⠀⣠⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⣠⣾⣿⣿⣿⣶⣿⣿⣿⣷⣄⣀⣼⣿⣿⣦⣴⣦⡀
⠀⠀⣠⠶⠚⣿⣿⣿⠉⠙⠳⢦⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⡼⢃⡴⠛⠛⠛⠛⠛⠳⣄⠈⢳⡄⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢰⡇⡾⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⡇⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢸⡇⣷⠀⢀⠀⢀⠀⣀⣀⣀⡿⢀⡇⠀⠀⢠⣿⡿⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠉⢉⢁⠀⠀⠘⣿⣿⣿⡆⠀⠀⠀⣀⣤⣤⣀
⠀⢳⡌⢻⣋⣿⣋⣿⣉⣿⣹⣇⡼⠃⠀⠀⢸⣿⠃⢸⣼⠆⠀⠀⢠⣆⡄⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠉⠳⣄⢈⣷
⠀⠀⠙⠻⢯⣄⣉⣰⠿⢤⡴⢿⠀⠀⠀⠀⢸⣿⠀⠀⠁⣀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⣠⠶⠖⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⢀⣶⡞⣧⡉⠙⣷⠀⠁⢸⠀⠀⠀⠀⢸⡏⠀⠀⠀⠉⠛⠶⣄⡀⢠⣄⠀⣠⡴⠞⠉⠁⠀⠀⠀⠀⣄⡶⠸⣿⣿⡀
⠀⠀⠈⠛⠁⠈⠙⣆⠉⠀⣀⣾⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣿⠟⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠙⣧⢠⡟⠁⢹⡇⠀⠀⠀⠐⠦⣤⣄⡀⠀⠀⠀⠀⢤⣶⣖⣛⡀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢻⣄⣀⣤⣤⠴⢿⠈⢧⡀⢸⡇⢀⠀⡀⠀⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣧⡴⠞⠀⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠘⣧⠀⠙⢻⡆⠘⠟⠁⠀⠀⠀⠀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⠀⠀⣧⠀⠀⠀⠀⠈⢿⣝⣷
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⣿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠙⢦⣀⡇⠀⣆⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠈⠁⠀⣿
⠀⠀⠀⠀⣠⢴⡞⠁⠀⠘⢧⡀⠀⠀⢀⠀⠉⠹⡄⠈⠁⠀⠐⠛⠳⠤⠖⠲⠴⠞⠲⠞⠉⠛⠉⠛⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠸⣧⠞⠀⠀⠀⠀⠀⠳⣄⠀⢻⡾⠧⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⢀⣰⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⠐⣶⠆⠀⠀⠀⠀⠀⠀⠀⣉⣤⠶⠋⠁⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⠦⣤⣄⡀⠀⠈⠉⠛⣶⠶⢦⣤⡤⠤⠶⢶⠚⠋⠁⠀⠀⠀⡀⠀⠈⢷⠀⠀⠀⢶⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠻⠚⢻⣿⣿⡷⠦⡾⠀⠀⠀⠀⢀⠀⠻⠟⠀⠈⣇⠀⠀⠀⠻⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⢸⣿⣿⣧⠀⠸⣦⠶⡄⣠⠀⢸⡆⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⣿⣿⣿⡀⠀⠀⠀⠙⠁⠀⠈⣿⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠹⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢸⡏⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣶⣶⣤⣬⣥⣄⣀⣀⣀⣤⣤⣤⣴⣾⡇⢿⠶⠚⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣄⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⠇⣸⠶⠶⠶⠚⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣏⣠⡀⠀⠀⢀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⠷⠶⠒⢛⣁⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⠀⢿⣿⣿⣿⣿⣿⣿⠿⢿⠿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⠏⣠⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⠛⠁⣼⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣴⣤⣤⣤⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠶⠀⢸⠇⠀⠀⠀⠀⢸⡇⣐⠖⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣩⣿⡇⣼⣿⣿⣿⣿⣇⢸⣿⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠃⠸⡄⠀⠀⠀⠀⣠⠇⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣾⣇⢻⣿⣿⣿⣿⠟⣸⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡻⠿⠿⢋⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⡤⣶⡞⠁⠀⠈⠁⠀⠘⣳⡶⢤⣀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⣛⡉⢡⣾⣿⣿⣿⣿⣷⡌⢙⡛⠿⣿⣿⣿⣿⣿
⠀⠀⠀⢀⡞⠁⠀⢽⣿⡀⠀⠀⠀⠀⢀⣿⡇⠀⠈⢳⠀⠀⠀⠀⢸⣿⣿⣿⣿⢱⢚⣛⠃⠘⢿⣿⣿⣿⣿⡿⠀⢸⣿⣷⡜⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⠀⣽⣿⣿⣶⣤⣴⣶⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⣿⠅⠀⠈⠙⠛⠛⠋⠁⠀⢸⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⢸⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢸⡇⠀⠀⠀⢸⣿⣿⣿⡇⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠈⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⣠⢀⣆⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡦⠀⠀⠀⠀⠀⠀⠀⣰⣀⣀⣀⣀⣀⣶⡀
⠀⠀⠀⠀⠀⠙⠶⣾⣷⡶⠞⠁⠀⠀⠀⠀⠀⢰⣿⠻⠦⠄⣴⠂⠀⠀⠀⠀⠀⠠⠞⣿⢦⣠⢀⡿⠀⠀⠀⠀⠘⠐⣆⡶⣠⠆⠛
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣄⡀⠀⠀⠀⠀⠀⠀⢶⣭⣷⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣸⠟⢶⡀⣸⠇⠀⠀⠀⠀⠀⠀⢘⣿⡏
⠀⠀⠀⠀⠀⠀⢻⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠶⠿⠻⣗⠀⠀⠀⠀⠀⠀⢠⣭⣥⣴⣾⣶⡿⠀⠀⠀⠀⠀⠀⠀⣼⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢉⡳⢄⠀⠀⠀⠀⠀⠀⠀⠐⢲⡂⢶⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⣸⣃⣀⡽⠂⠀⠀⠀⠀⠀⠀⠀⠹⣶⣦⡄⠀⠀⠀⠀⠀⠀⢈⣭⣀⣠⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣥⡀
⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠟⠂⠀⠀⠀⠀⠀⠀⠻⣿⣥⣄⠈⣻⡇⠀⠀⠀⠀⠀⣠⠿⣿⣿⢷⣄
⠀⠀⠀⠀⠀⣤⣬⣿⣏⣿⣤⣄⠀⠀⠀⠀⠀⠀⣾⠁⢻⡀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡟⣷⣿⡇⠀⠀⠀⠀⠀⠘⣿⠿⢿⣿⠃
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠂⠀⠀⠀⡀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⢠⡴⣿⣿⠦⡄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣷⣆⠀⠀⠀⠀⠀⠀⠠⡾⣿⢧⡀⢀⡾⠃⠀⠀⠀⠀⢶⠎⣷⣦⣤⣄
⠀⠀⠀⠀⠀⢰⠀⣿⣿⠀⡰⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⣮⡅⠀⠀⠀⠀⠀⠀⢹⡟⢳⡆⢸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣮⡙⣷
⠀⠀⠀⠀⠀⠈⠢⡌⣿⠴⠁⠀⠀⠀⠀⠀⠠⠴⠚⠉⠀⠸⠁⠀⠀⠀⠀⠀⠀⢸⠃⠼⢤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠻⣿⡿⠃⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠹⣄⡀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣪⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⡇⠀⠀⠀⠀⠀⠀⠰⡶⣻⠄⢀⣄⡀⠀⠀⠀⠀⠀⠀⠈⢉⣹⣷⡘⠟
⠀⠀⠀⠀⠀⠀⠸⢿⣟⢻⠆⠀⠀⠀⠀⠀⠀⠀⠸⡶⢛⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⣷⣾⡿⢻⡄⠀⠀⠀⠀⠠⣔⣛⡯⠽⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠒⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠷⡴⣇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⡆⢠⣿⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⣽⡁⠈⣿⣭⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠗⣛⣛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁⠀⠁⢉⠁⠀⠀⠀⠀⠀⣿⣿⣿⡟⢛⠛⠾⠛⠿⠿⠻⠾⠛⠛⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣾⠛⠃⠀⠀⠀⠀⠘⠛⢷⡆⠀⠀⠀⣿⣿⣿⣇⠹⣿⣶⣿⣷⣾⣷⣶⣿⠇⣸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⢏⣼⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢀⣀⡀⠀⠈⣀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣦⠀⠀⠀⠀⣴⣏⣀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠆⠹⣿⣿⣿⣿⠋⠸⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠋⠀⠀⠀⠀⠘⢻⣿⡇⠀⠀⠀⣿⣿⣿⡇⠀⢤⣾⣿⣿⣿⣿⣦⡄⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠆⠀⠀⠠⠞⠛⠛⠁⠀⠀⠀⣿⣿⣿⣷⣦⣤⣌⣿⣿⣿⣟⣥⣤⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⢸⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⣿⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⣷⡀⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⡁⠀⠀⠀⣿⣿⣿⡿⠀⠘⣿⣿⣿⣿⣿⣿⠁⠀⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⡿⠟⠋⠀⠀⠉⠉⠀⠀⠈⠛⢿⣷⡄⠀⣿⡿⠁⣀⣤⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀⠙⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⣿⣇⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⣿⡏⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠳⣦⣤⣀⣀⣈⣉⣉⣉⣁⣀⣠⣴⠾⠃⠀⣿⣷⣌⡛⠻⠿⠿⢿⣿⣿⣿⠿⠿⠟⢋⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿

⢰⡲⡄⠀⠀⠀⢀⡠⠖⠉⠉⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⢤⠀⠀⠀⠀⢀⠔⠊⠉⠉⠑⢄
⢀⣈⠁⠀⠀⠀⡞⠀⡠⣄⣀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠁⠀⠀⡰⠁⠀⡤⣀⠀⠀⠀⢣
⠓⠁⠀⠀⠀⢀⢧⡞⠡⠄⠈⠍⢹⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢠⡧⣤⠊⠔⠀⠩⠙⡆⠀⢣
⠀⠀⠀⠀⡰⠃⠀⡇⠀⠁⠈⠀⢸⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⢸⠀⠈⠀⠁⠀⣏⡇⠘⡄
⠀⠀⠀⡴⠁⠀⠀⣧⡀⠰⠆⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢸⢄⠐⠾⢂⡰⠁⠀⠀⡇
⠀⠀⠈⣳⣄⢤⣼⣧⡿⣗⣛⢿⢢⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣧⠔⡶⠾⣾⡏⠉⢙⠳⡒⠤⢞⡀
⠀⠀⡜⠈⠉⢺⠄⡧⣧⣎⣱⣼⢰⡃⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⣠⠁⠈⠉⢳⢸⠉⠗⠒⢺⠀⡇⠀⠀⠹
⠀⡼⠀⠀⠀⡼⠸⡅⢸⠇⠈⡏⢀⡇⠀⠀⠈⡇⠀⠀⠀⠀⠀⢠⠃⠀⠀⢀⠇⢸⠀⡆⠀⢸⠀⡇
⢠⠃⠀⠀⢠⠃⠀⢣⠘⡄⢀⡇⡼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡼⠀⠀⠀⠘⠀⠈⡆⡇⠀⢸⢸
⠘⡄⠀⠀⢀⡆⠀⠀⠓⢳⡸⠘⠁⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⢣⠀⠀⠀⣴⠀⠀⠓⡇⠀⣾⠊⠀⠀⡇
⠀⠈⠒⠒⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠒⠊⢸⠀⠀⠀⣇⠀⡟⠀⠀⢀⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢹⠀⡇⠀⠀⠸⠀⠠
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢰⠹⡀⠀⣀⢤⡄⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠣⡟⠀⠳⠞⠀⠀⠳⠞⣄⡇⠙⢦⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⡟⠓⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠤⠼⠆⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⢀⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢤⣀⠤⠔⠚⠁⠀⡀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠀⠀⠀⡴⠛⢲⡀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠈⡇⠀⢰⣧⠀⢀⡇⢀⡸⢏⣀⠤⠶⡄⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠟⡰⢁⡄⠘⢧⡤⠞⠀⣼⠀⠀⠋⢴⠄⡿⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠉⠀⠁⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠐⠚⣅⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠑⠲⣤⣤⣴⣶⣿⠟⠁⢨⠇⠀⢀⣤⠖⣉⣓⣒⡢⢄⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠹⣿⠟⠫⡇⣠⡴⠋⠀⣴⠋⢸⠉⠉⡇⠀⡸⠃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣯⠝⠓⡆⠀⠙⠲⢬⣉⣉⣉⣁⣀⡤⠞⠁⠀⣀⡧⠴⠃⠘⣅⣰⠆⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢎⡠⠀⠇⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠣⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣐⠪⠔⠲⣄⠑⠦⠤⠼⣯⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⢹⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢖⠒⠒⠶⠖⠋⠉⠉⠉⠉⠑⢶⡇⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⣒⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⢸⡩⣉⠉⠉⠓⣯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠙⢬⣧⠴⠊⠙⠤⠤⣖⠀⠀⠀⢀⡽⠀⣰⢭⣳⣤⣇⠄⡠⠃⣱⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣴⡓⢺⡀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠴⣋⠳⢞⡡⠚⠉⠀⠉⠛⠒⠊⠁
⠀⠀⠀⠀⠀⠀⢾⠛⠒⠒⠭⢵⣒⠭⣚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠈⠙⠪⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣒⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⡀⠤⠒⢢⠀⠀⠀⢸⣏⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠢⢄
⠀⠀⠀⢀⡤⠊⡶⠶⠂⠒⠁⠀⠀⢾⡏⢉⡇⠉⠀⠃⠀⠀⠀⠀⠀⣴⡿⠛⢻⣷⡀⠀⠀⠈⠧⠾⢤⡑⡄
⠀⢀⠔⣪⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⢿⣜⣛⣣⡶⠁⠀⠀⠀⠀⣸⣿⠶⡄⠠⣿⡔⠀⠀⠀⠀⠀⠀⢱⠈⢆
⢠⢫⣾⣧⣨⠠⠤⢀⣀⣤⣤⣤⣄⣀⣬⠻⡟⠋⢣⣤⣀⠀⠀⢀⣿⣿⢼⡇⠤⣸⣇⡀⠀⠀⠀⠀⠀⠀⣧⠀⢣
⢃⠻⠿⣿⣿⣷⣦⡈⢿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣿⣿⣿⣿⣷⡾⣿⣿⣷⠶⢶⣿⣦⣀⣀⣀⢀⡀⠠⠤⠽⠇
⠀⠈⠁⠒⠨⠍⠉⣉⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⡿⠀⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⣿⣿⣿⠂⠀⣿⣿⣿⣿⠃⡠⠀⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠹⣧⣼⣿⣿⣿⣧⣤⣀⣾⣿⣿⡿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢏⣴⠂⣻⣿⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⢃⣞⢡⣾⣿⣿⡆⠘⡿⢟⠛⠛⠉⠉⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⢿⣇⠀⣾⣿⣇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⡄⠻⣿⣿⣿⣿⣿⣿⣄⡄⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢛⡛⠿⠿⠿⣿⣿⣿⣄⠘⠛⠋⡻⣿⣿⠀⠇⠀⠀⣀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣾⣿⣿⣿⡁⣰⣿⣿⣿⣿⣷⣄⢦⣪⣿⣶⣶⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣨⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣿⣿⣿⣿⡿⠏⡔⠑⡝⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⠇⠌⠀⠀⠈⠌⢿⣿⣿⣿⣿⠿⠀⠀⠀⠈⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣌⣛⣉⣀⠤⠀⠀⠀⠀⠘⢌⣿⣿⣿⣿⣄⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⡠⠀⠀⠀⠀⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠐⣼⣿⡿⠆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣯⣿⣿⣷⠀⠀⠀⠀⠈⢻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠄⣿⣿⢡⠂⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠇⣿⣇⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠘⡙⣿⣧⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠌⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⠀⠀⠀⠐⡹⡿⡀⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣦⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⢴⣴⣷⡄⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⣰⣿⣿⠿⡆⠀⠀⠀⠀⠀⠀⠀⢠⠛⠛⡇⠀⠀⠀⠀⠀⠾⣿⣿⣷⡀⠀⠀⠀⠀⠈⠋⠉⠱⡀
⠀⠀⠀⠀⢸⡒⠿⣿⡶⠃⠀⠀⠀⠀⠀⠀⢠⠃⠀⠈⡇⠀⠀⠀⠀⠀⠷⣞⣛⣓⣳⠀⠀⠀⠀⢀⡀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠈⠓⠤⠤⠔

⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⣤⣤⠶⠶⠶⠛⠛⠛⠛⠛⠛⠷⠶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⣰⠟⢿⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⣤⣀
⠀⠀⠀⠀⢀⡾⠃⠀⠈⠻⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⣾⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⢸⡏⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⢸⡇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠷⣦
⠀⠀⠀⠀⣿⡀⠀⡴⠋⠓⠲⠤⠤⣤⣄⣀⣀⡐⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠈⢷⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡼⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣾⠃⠀⠀⠀⢀⣀⠀⠙⢿⣒⡆⠀⡇⠀⡤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⡄⡎⠀⠀⠀⢀⣿⠁
⢀⣴⠶⠶⡟⠀⠀⠐⠒⠒⠿⠿⣷⣦⣍⠁⠀⠀⠀⢉⣉⣡⣤⣤⣤⣶⣦⠄⠀⠀⢰⡇⠀⢳⠀⠀⠀⠀⣼⠇
⢼⡇⠠⢤⠇⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠭⣍⡉⠁⠀⢸⠀⠀⢸⡆⠀⣀⠴⠿⣦⡀
⠘⠷⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣼⠧⠜⣴⠒⠀⢸⡇
⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⢠⣤⣤⣴⠟
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠉⠁
⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠙⠻⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁
⠀⢀⣠⣴⣛⣳⣄⠉⠛⠳⠶⣤⣤⡄⠀⠀⠀⠀⢠⣀⣀⣀⣠⣤⡴⠶⠚⠛⠙⠛⠳⢦⣀
⠀⢿⢻⡀⣿⠄⠉⣷⠀⠀⠀⠀⠘⢳⣶⣶⠦⠤⠟⠉⢉⣡⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠈⠻⣏⣡⡤⠶⠛⣦⣀⣀⣤⠾⠋⠀⠘⠦⠴⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⣿⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⣿⠶⠶⣤⣄⢀⣠⣤⠶⢯⣀
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠙⡏⠁⠀⠀⢀⣉⡇
⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⢘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠻⢶⣀⡶⠶⠋
⠀⠀⠀⠀⠀⠈⠛⠷⢶⣤⠴⠶⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣬⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⣾⡁⠀⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠛⠋⠉⠁⠀⠻⣤⣤⣤⡤⠴⠶⠚⠛⠉⠁⠀⠀⠈⠛⠶⠶⠶⠛⣷⠒⠋⠁⠻⣄
⣴⠟⠛⠶⣤⡀⣠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣴⣄⠀⠙⢷
⠘⢷⡄⠀⠈⢻⠋⠻⣷⡀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠀⠀⠀⠘⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣯⠀⠈⠛⠶⠶
⠀⠀⠻⣄⠀⠀⣠⡴⠞⢳⣄⠀⢠⡞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣤⣤⣤⡤⠶⠛
⣀⣀⣀⣈⣛⣛⣉⣀⣀⣀⣙⣷⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠂⠐⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠁⠀⠀⠀⠀⣤⡢⠀⠀⠀⣱⡀⠀⠀⠲⠀⠀⠐⡄⠀⠀⠀⠀⠛⠀⠀⠀⠀⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠉⠀⠠⢐⡣⢞⠅⠀⠀⠀⠀⠀⡜⠁⠀⠊⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢶⣶⠆⠀⠳⠀⠀⠀⠀⣀⠤⠒⠍⠂⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠠⠄⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⡀⠀⠐⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⢲⡤⠀⣾⠁⠀⠀⠀⠀⠈⠒⠄⡀⠠⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⢠⡀⢰⡟⠂⠀⠀⠀⠀⠉⠒⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃⠀⠀⠀⠀⠀⢸⡇⢸⡟⢸⠇⢹⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠛⠀⠈⠁⠘⠶⠌⠉⠀⠀⠀⠰⠆⠀⠀⠀⠸⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⣤⠤⠤⠴⠶⠶⠶⠦⠤⠤⣤⣀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⣿⣰⠏⢘⠀⣤⢠⡄⢀⣤⠀⠀⠀⠀⠈⠉⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢻⠉⡆⢸⡔⠿⢸⡇⠻⢰⡇⠀⠀⠀⠀⠀⠀⢠⠇⢀
⠀⠀⠀⠀⠀⠒⠪⡁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠐⠈⠀⠉⠀⠀⠀⠀⠀⣦⠟⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠈⠁⡒⠶
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠦⠀⠰⡂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠒⠲⠦⠞⠁⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⡐⣀⡠⠤⠴⠒⠒⠉⠁⠀⠀⠀⢠⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠲⠤⣀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⣶⠀⢀⠀⡀⠀⠀⡷⠀⠀⣾⠋⢰⢀⡀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠈⠁⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⡆⢸⡄⣿⣸⣰⡇⠀⠐⣇⠀⡘⢸⡁⣏⢸⡇⢠⡟⠁⣼⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠟⠀⠘⠊⠁⠉⠀⠀⠉⠀⠀⠉⠛⠣⠀⠀⠉⠀⠉⠀⠙⠁⠋⠐⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠑⢄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⡒⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⣴⠒⠀⠀⠀⢤⠀⡄⠻⢷⠴⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠁⠈⠁
⠀⠀⠀⠀⠀⠁⡆⠀⠀⠀⡀⢴⣂⠘⠠⠂⠊⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⡠⠔⠁⠠⡀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠄⠀⠋⠙⠓⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠰⢀⡠⠂⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠄⢀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠄⠈⠁⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⡚⠀⠀⢦⠀⠑⠀⠸
⠀⢀⠀⠀⠀⠁⡀⠀⠶⡄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠘⠐⢋⣰
⠀⠋⠐⠊⠕⠈⠈⠁⠰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠈⠒⡄⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡖⠢⠤⠀⢰⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠁⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⣄⠀⢀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠰⠂⠀⠀⠀⠀⣀⣠⢒⡨⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠛⠖⠁⠪⡀⠀⠀⠀⡀⠐⠀⠀⣄⠂⢤
⠀⠀⠀⠀⠢⠀⠔⠁⠀⠅⢠⠐⣄⠀⠈⠄⠐⠂⠊⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⢸⠁⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠄⠀⢐⡴⠃⠈⢄⣀⡄⢀⠈⠢⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠘
⠀⠀⠀⠀⠈⠀⠀⠛⠅⠠⡤⠐⠅⠁⠀⢹⠀⠈⡂⠀⠀⠪⠂⡀⠄⠒⠈⠋⠑⠠⡀⡀⠀⠰⠘
⠀⠀⠀⠀⠤⠀⠀⠀⠀⢀⠖⠀⠀⣂⡖⠁⠀⠀⠀⠀⠈⡱⠊⣀⠠⠐⠈⣄⠀⠀⠘⡤⠂⠈⢉⡐⢖⡒⠀⠲⡁⢢⢄
⠀⠀⠀⠀⠀⠁⠴⡆⢄⣀⢀⣀⠴⠁⣠⠆⠀⠀⠀⠀⠐⠀⢶⡁⠀⠀⠀⡈⠄⡀⠀⠂⠑⡀⠀⠀⠀⠀⢱⠂⠉⠓⢄⠈⠂
⠀⠀⠀⠀⠀⠀⠔⠁⠈⠃⠘⠁⠈⠈⡠⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠁⠘⢃⠀⠰⠀⢰⠀⠀⠀⢀⠸⠀⠀⠀⠈⠑⠈
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⡄⠁⠀⠀⠀⠀⠀⣜⡇⠠⣀⡀⠀⠈⠀⢀⣠⡦⠖⠂⠈⠉⠉⠉
⠀⠀⠀⠀⠈⠀⠖⠂⠀⠀⠀⠀⠃⡄⠀⠀⠀⠀⠀⠀⠉⠢⠖⠘⢱⢫⢉⡅⣌⠄
⠀⠀⠀⠀⠀⠐⠂⠒⠀⠠⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠈

⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢀⠀⡠⠤⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠈⠀⡀⠀⡇⠀⠀⠀⢀⠠⠠⠀⠀⠀⠀⠀⠀⡤⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⢸⠀⢢⢄
⠀⡄⠁⠀⡇⠀⡇⠐⠂⠰⡃⠀⢀⡘⢂⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠘⠒⠀⠀⠀⠰⠁⣀⠀⠀⠡⠄⢸⠀⢸⠸
⠀⠀⠁⠀⡇⠀⡇⢸⠀⠀⠹⡁⠀⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠎⠀⠧⠤⠡⠆⠈⠀⠸⠀⣩
⠀⠀⠘⠀⠃⠀⡇⢸⠀⠀⠀⡁⠀⡜⠀⠀⠀⠀⠀⣀⠀⠀⠢⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⡆⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠐⢤⠀⠐⠀⠄⠈⠁⠀⡄⠄⠀⠀⠁⠀⠀⠀⠁⠀⢠⣠⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠱⡀⠀⠁⠀⠀⠀⢆⠀⠀⠐⠀⠀⡰⠀⠀⠤⠀⠀⠠⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠐⠂⠀⢠⠤⡕⠢⠀⠠⡊⠀⡤⡀⡀⠄⠂⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠄⣀⢀⠎⠀⡇⠀⠀⠀⠂⠈⡅⢩⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠰⢄⡀⠀⡀⠎⢠⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠀⠀⡆⠀⠀⠈⡀⠠⠀⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣤⣶⣶⣷⣆⠀⠀⠃⠀⣃⣉⢡⠀⢃⢀⣠⣼⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⢻⣿⣿⣿⣧⣀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⠋⠁⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠖⢪⠉⠻⣿⣿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣿⣿⠇⠀⡠⠀⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠑⢄⣹⡿⠿⠟⠛⠛⠛⠛⠿⠿⠿⣿⡿⠦⠊⠀⠀⢠⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⠤⡌
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡯⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢶⡀
⠀⠀⠀⠀⠀⢀⠊⠁⠈⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣀⠈⠈⠉⠒⠄
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⠀⠀⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⡄⠀⢠⡄⢠⡄⢠⡄⠀⢠⠠⡄⠀⢠⣤⡀
⠀⠀⠀⣄⡀⠀⢨⣧⣤⣄⠀⢠⡺⢗⣄⠀⢠⣤⣴⠇⠀⢀⣠⠄⢸⢂⠈⢳⠀⠀⢻⣾⣧⣼⡃⡀⢼⡇⡆⠠⢠⡿
⠀⠀⠀⠈⢿⣶⣾⠿⣹⠙⡴⠋⣹⣎⠘⠧⠊⣹⠿⣯⣶⡾⠃⠀⠈⡇⡇⠈⣏⢸⠫⣿⣿⡜⣿⣧⣰⣿⢿⡅⡼
⠀⠀⠀⠀⠀⣿⣯⣔⣋⢉⣉⣙⣾⣃⣁⣈⣈⣉⣧⣿⣿⡀⠀⠀⠘⡿⢃⣀⠀⠠⣶⣿⣿⣀⠀⠈⠉⣉⣰⠗⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⡾⣿⣿⣶⣟⣻⠿⠺⢟⣷⠾⢽⡟⠀⠀⠀⠘⠷⠘⠓⠀⠀⣼⣿⣿⣀⣀⡀⠐⠛⢛⠤⠿⠋
⠀⠀⠀⠀⠀⠈⠉⣩⡄⠉⠉⠉⠉⠈⠁⠈⠉⠁⢀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⡠⡀⠀⠀⠈⣷⠄⠀⠀⠀⠀⠀⠀⠀⢠⣞⠃⠀⠀⢀⢤⠀⠀⠸⣮⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⡋⢇
⠀⠀⠀⠉⠻⣦⠀⠀⢁⠈⡄⠀⠀⣶⠀⠀⢠⠃⠘⠀⠀⣠⡟⠁⠀⠀⠈⢢⠣⠀⠰⡆⠀⣇⠀⠰⡶⠀⠀⣿⠂
⠀⠀⠀⠀⠀⠑⡱⡀⠘⠀⣟⠄⢀⣿⡆⢀⣶⣴⡇⠀⢌⠎⠀⠀⠀⠀⠀⠈⠇⡄⠀⢁⠀⣾⡇⢰⡇⢀⠃⡇
⠀⠀⠀⠀⠀⠀⠰⠐⡄⣿⣿⣆⢾⣿⣷⢾⣿⣿⣧⠈⡌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣦⡿⠃⣿⡇⠈⢠⠁
⠀⠀⠀⠀⠀⠀⠀⡅⡐⣿⠟⢋⣾⣿⣿⡼⠿⣿⠏⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⡅⣠⣻⡿⠀⣇
⠀⠀⠀⠀⠀⠀⠐⡴⢷⣌⠀⠈⠀⠉⠀⠀⠀⢨⣾⣴⡆⠀⠀⠀⠀⠀⠀⠀⢠⡟⠻⣯⣿⣿⣿⣿⡧⠐⣿
⠀⠀⠀⠀⠀⠀⠀⣇⣻⣇⠀⠀⠀⢀⠀⠀⠀⣼⡿⠞⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⢸⣿⣿⣿⣷⡖⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠿⢊⠀⢀⣈⣉⣉⣀⣠⡤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢠⠀⠀⢠⡆
⠀⠀⠀⠀⠰⣶⠀⢀⢀⢸⠩⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⣶⠆⠀⠀⠀⠀⢠⢣⢠⣿⠄⣼⠀⡀⡆⢠⠰⢃
⠀⠀⠀⠀⠀⠸⡗⢌⠻⡀⠀⠑⢄⠻⠄⠸⣿⣄⡘⣾⣾⠇⠀⠀⠀⠀⠻⡌⠘⣾⣿⡴⣿⡄⠸⠀⣸⡇⢸⣎⡅
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣤⡀⠀⠀⠀⠀⣠⣿⣿⣶⣿⡟⠀⠀⠀⠀⠀⣇⠁⠸⣾⠿⣧⡟⡁⠀⠸⢿⠀⢸⣽⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣤⠀⠀⠀⠀⢹⣿⣤⠀⠀⣠⣿⡇⠀⠀⠀⠀⠘⣟⡀⠰⣧⢻⣿⡘⠿⠀⠀⠿⠀⠘⣻⡗
⠀⠀⠀⠀⠀⠠⣿⣻⣥⣤⣤⣤⡶⠶⠶⠶⣤⣴⣿⣿⡇⠀⠀⠀⠀⠘⠿⣀⠀⣹⣿⣿⣄⠀⠀⠀⠀⠀⢝⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠂⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣞⠳⢦⣄⣀⣴⡶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⠀⠈⠉⣠⠔⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣁⣤⣆⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠹⣆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢦⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣿⠀⠈⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⢻⡿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⠛⠀⢀⣀⡷⣤⣤⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠇⠀⠈⠁⠰⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡾⠉⠚⠋⠻⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠖⡹⠟⠒⢦⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠸⣇⠐⢛⡄⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠐⠟⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡅⣀⠈⠛⠋⠀⡄⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠠⣄⠀⠀⠀⣸⠇⠀⠀⠀⠀⢠⠖⠲⡆⠀⣿⠀⠿⠀⠋⠀⢸⠃⣿⠀⢰⠞⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠙⠶⠾⢳⡟⠀⠀⠀⠀⠀⠸⣆⠀⠀⢀⡿⣧⣤⣀⡴⢠⠏⠀⣷⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠶⠒⢻⡀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠉⠀⠘⠷⣤⡼⣏⠀⠀⠙⠓⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⠀⠀⠀⠀⢹⡘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠘⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⡾⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠃⠀⠀⠀⠀⢹⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢧⣤⡟⠀⠀⢲⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠉⠀⠀⠀⠀⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⣤⠏⠀⠀⠐⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡸⠁⠈⠁⠀⠀⠀⠀⠈⠇⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢹⡋⠉⢹⡏⠙⣿⠛⠿⠦⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣼⡇⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⢶⡶⠶⣦⣦⣤⣤⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣿⣁⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠟⠀⠿⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⠃⠀⠀⠿⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⠾⠿⠿⠿⠿⠿⠿⠶⠶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣦⣀⠀⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⠁⠘⢿⣆
⠀⠀⠀⠀⠀⢀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠖⠋⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢻⡆
⠀⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠘⣿
⠀⣠⣴⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⢀⣿
⢰⡿⠁⠃⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠶⣏⠀⠀⠀⢸⡟
⣿⡇⠀⠀⠀⠀⠀⠘⣿⠦⣤⣤⣠⣤⣤⣤⣤⣤⣠⣤⣤⣭⣭⣥⣤⣤⣤⡤⠤⠶⠶⠶⠛⠛⠛⠉⠉⠁⠀⠀⠈⢷⡀⢠⡿⠁
⣿⡆⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⢻⣇⠀⠀⠀⠀⠀⢀⠀⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆
⠸⣿⡀⠀⠀⠀⠀⢸⠀⣼⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⢸⡇⠀⣤⠖⣾⠿⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⢻⣧⠀⠀⠀⠀⠈⢰⡏⠀⢿⡄⠀⠀⠀⣀⣀⣀⣀⣀⣀⡈⠙⠻⠶⠿⠀⠘⠁⠀⠙⠛⣁⣤⣶⣾⣿⣯⣤⣤⠀⠀⠀⢹⣧⣤⣤⣄
⠀⢈⣿⡆⠀⠀⠀⠀⣿⠀⠀⠸⣇⠀⠀⠤⣤⡿⠿⠿⢿⣿⣿⣿⡿⠶⠂⠀⠀⠀⠀⠀⠛⠻⠿⠯⣥⣤⣤⡤⠄⠀⠀⠀⠈⡏⠁⠉⠻
⣰⡟⠉⢙⠳⢦⡀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠠⠶⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⠁⢀
⢻⣇⠀⠈⢙⡷⠛⠄⠿⣆⣀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟
⠈⠻⣶⣤⣀⣥⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠈⠉⠉⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⡟⠛⠷⢶⣤⣄⣀⣀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢠⣀⣀⣤⣴⡶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⢧⣀⣀⣤⣤⣀⣼⡿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠙⠸⡆⠀⠀⠀⠀⠀⠀⢻⠀⣇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢸⡆⡏⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠈⣿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⢻⣷⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⣀⣽⡟⠛⠛⢿⣶⣦⣄⡀⠀⠀⠀⢀⣀⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡞⠋⠉⣤⠿⣦⣤⣠⣤⣤⣽⣷⣤⣀⣰⡟⢻⣷⡟⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⢀⣀⣤⣤⡶⠶⠟⠛⠁⠹⣦⣄⣀⣀⣀⣨⣿⣀⣀⣀⡉⠉⢩⡟⠀⣸⡏⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠉⠉⠉⠛⠛⠛⠀⠀⣿⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠶⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠛⠻⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠘⣿⡳⣦⡀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡿⠉⠙⠷⢦⣤⣀⣀⠀⠈⠻⣦⡙⢷⣄⠀⠀⠀⢸⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⠀⢸⡇⠀⠀⣀⣀⣀⠉⠉⠛⠛⢻⣿⣿⡿⠛⣦⠀⠀⣿⠃⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠉⠉⠻⣿⠀⠐⠛⠉⠉⠙⠃⠀⠀⠀⠀⠀⠈⠙⠷⠈⢷⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠳⣦⡶⠞⢻⣇⠀⠀⠀⠀⣿⠀⠀⠀⢾⣿⠆⠀⠀⠀⠀⠀⢿⣿⠆⠀⠀⢨⡟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢸⣧⡀⠀⣿⣦⣤⣤⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣠⡾⠋⠙⢶⣄⠙⠳⣦⣈⠉⠻⣄⠀⠀⠀⠀⢦⣤⣤⣤⡦⠀⠀⠀⣠⡾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠙⢷⣄⠀⠙⠳⣤⡉⠳⣦⣄⣀⠀⠀⠀⠀⣀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⡼⠛⠳⣤⣬⣽⡟⠛⠛⠛⣿⣭⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠈⢷⣀⣀⣠⡟⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴⠏⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠻⣦⡀⠈⢷⣄⠀⠀⣀⡴⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠻⣦⡀⠙⣦⣸⡏⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇⣠⡶⠶⠾⣿⡆⠈⠹⣧⡀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡏⠀⠀⠀⠈⣷⣴⠾⠋⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡾⠛⠛⠛⣷⠀⠀⢸⡟⢧⣀⠀⢀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⢺⡇⠀⠀⠀⣿⠛⠒⢻⡇⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠾⠷⠶⢾⡇⠀⠀⠀⣿⠶⠶⠾⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠶⠶⠶⠾⠇⠀⠀⠀⠻⠶⠶⠶⠶⠶⠿

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

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠉⠉⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣤⣄⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠘⠛⠛⠛⠉⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠗⠐⢶⣤⣀⣄⡈⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠈⠙⢩⠙⠛⠛⠛⢨⣿⣷⣄⣤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡅⡏⠀⣀⡛⠀⠀⠀⠀⣌⢙⠛⠉⠉⠀⢠⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠻⢠⠋⠀⠀⠀⠀⣴⣿⠹⣷⣆⠀⠀⠀⣿⣷⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⣾⢀⣴⠀⠀⢼⣿⠏⠀⠈⣻⣿⣶⣶⣿⠿⡷⢴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠁⢸⣿⠀⠀⠀⣀⠀⣠⣾⣿⣿⠟⠿⠀⠀⠀⠀⣝⠢
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⡀⠈⠻⠄⠀⠀⠹⣿⣿⣿⣧⡡⣔⣶⣶⣤⣄⠀⠘⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣄⣇⠀⠀⠀⠀⢈⣿⣿⣿⣿⣮⣽⢥⣄⠉⠓⠐⠈⠻⠷⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠛⢿⣿⣆⠀⣤⣶⣿⢟⣿⠁⠈⠻⢷⣶⣏⣷⡄⣴⠆⠀⢦⣀⣈⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠗⣤⡻⡝⢿⣧⠛⠋⢁⣾⣟⠀⠀⠀⠀⠙⣿⣄⡙⠶⣗⢦⣈⠙⠢⣾⣍⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⠘⢿⣶⣿⣦⣤⠖⣾⣼⡇⠀⠀⠀⠀⠀⠘⣿⣻⣷⣤⣑⣌⣿⣷⡌⠙⢦⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋⠵⠾⠍⠿⣆⢸⣿⡀⠀⡄⢔⠂⠀⠀⠈⠻⣿⣯⣻⣿⣤⣀⠹⡄⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠹⣧⣾⣦⣤⣄⢀⢻⣷⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣿⣷⣬⣄⣤⡆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⣙⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠛⠉⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣦⣀⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⢿⣿⠿⣿⠿⠛⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠙⠿⣿⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⡛⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠄⠀⠀⠀⠀⠀⡌⡁⠈⢠⠀⠀⠀⠀⠀⠀⠠⡤⣁⣻⣈⣩⢄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢈⣀⠤⡀⠀⠂⠀⠀⠀⠠⡁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⣶⣾⣦⣿⡉⣹⡜⢀
⠀⠀⠀⠀⠀⠁⠐⠋⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠣⡙⢯⠤⠀⠀⠃⠿⠍⡟
⠀⠀⠀⠀⠄⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⢿⢲⡇⡀⠀⠀⠲⣦⡐⡄
⠀⠀⠀⠀⠀⠠⡤⠒⠋⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠘⡁⠁⠠⡀⠀⠀⠀⢈⠻⣇⣹⡿⢿⠭⣧⣤⠠
⠀⠀⠀⠠⡤⠀⡄⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⣤⠐⠀⠀⠀⠀⠀⠙⠋⢺⣉⢸⡏⡘⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠠⢀⡂⠀⢩⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣴⠞⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⣿
⠀⠀⠀⢸⡇⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⡇⢸⣿
⠀⠀⠀⢨⣿⣷⣯⡀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⡟⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠾⠿⠶⢿⣿
⠀⠀⢠⣿⣿⣿⠿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⡀⠀⠀⠀⢻⣿⣸⣿⡇⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⢸⣿
⠀⠀⣿⡟⠻⣿⣦⣙⢷⠀⠀⠀⠀⢀⣴⣿⠟⠻⢿⣿⡆⠀⠀⠀⠀⢠⣿⠟⠛⠛⣿⢣⣾⣶⠀⠈⠛⠛⣿⡇⠀⠀⠀⠀⢀⣼⡿⠻⣿⣦⠀⠀⠀⢸⣿
⠀⠀⣿⡇⠀⠀⠙⠻⣷⣄⠀⠀⣴⣿⠟⠁⠀⠀⣼⡟⠀⠀⠀⣀⣴⣿⠏⠀⠀⠀⠹⣯⠛⠋⠀⠀⠀⠀⣿⡇⠀⠀⢀⣴⣿⠏⠀⣴⡿⠃⠀⠀⠀⢸⣿
⠀⠀⠘⠋⠀⠀⠀⠀⠈⠋⠀⠈⠉⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠙⠃⠀⠀⠈⠋⠁⠀⠀⠛⠁⠀⠀⠀⠀⠘⠛

⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢾⣿⡇⠀⠀⠀⠐⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⣴⣿⠄⠀⠀⠀⠀⢀⣤⣶⣿⡆⠉⠀⠀⠀⠀⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⢿⡿⠃
⠀⠀⢰⣿⣿⣶⣦⡀⠀⠰⠟⠁⣿⣿⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣅⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠈⢿⣿⢹⣿⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠘⣿⣿⣶⣤⣄⡀⠀⠀⠀⠘⣿⡟⠛⠛⣛⣻⣿⣇⣀⡀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⢸⣿⡆⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣿⡇⠈⠉⠛⢿⡗⠀⠀⠀⣿⣷⠀⠀⣿⣿⣿⣿⣿⡇⢀⣤⣄⠀⠀⠀⢿⡇
⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⡇⠀⠀⣿⡏⠀⠀⣿⣿⣿⣿⣿⡇⠘⠿⠿⢳⣶⣶⣿⣿⣶⣶⣿⣿⣶⣦⡄
⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠘⠛⠛⠁⠀⠈⠛⠁⠀⠀⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣤⣀⠀⠀⠀⠀⢀⣾⠃⠘⢷⡀
⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣶⣮⢻⡄⠀⠀⠀⠀⠀⠀⣰⡏⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣴⣶⡝⣧⠀⠀⠀⢸⣧⣾⣷⣼⣷
⠀⠀⢿⣿⡗⣀⠀⠀⠀⠀⠀⠘⢛⣯⣾⡇⠀⠀⠀⠀⠀⠀⣿⣼⣿⣷⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣝⣻⣵⣿⠀⠀⠀⠘⣧⣽⣯⣼⠇
⠀⠀⠀⠀⣸⣿⣷⣄⠀⠀⠀⢠⡿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠘⣷⣿⣷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣧⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠹⣯⣿⣿⡆⠀⠀⠈⢷⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⡀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⣸⣿⢹⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⣧⡀⠀⠀⠀⠀⣠⣶⣞⡻⠟⠁⠀⠀⠀⠀⢀⣿⡏⣿⡇⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣰⣿⠃⢸⣿⠀⠀⠀⣠⣾⠟⠉⠀⠈⠻⣿⣦⠀⠀⣾⣿⣟⣉⣛⣷⣶⠄⠀⠀⠀⣾⡟⠀⣿⣇⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠿⠁⠀⠀⠀⠿⠋⠀⠘⠿⠀⠀⠐⠿⠃⠀⠀⠀⠀⠀⠈⠻⠇⠈⠻⠿⠛⠛⠛⠛⠿⠷⠀⠀⠸⠟⠁⠀⠻⠗⠀⠀⠸⠿⠻⠇

⠀⡁⠉⠅

⠀⠀⠀⠀⢀⢰⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣆⣀⡆⠃⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠐⢻⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⣦⡀⠀⠀⠀⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢾⡷⡴⠂
⠀⠀⠀⠀⠀⣞⡷⠀⠀⠀⠀⠀⢀⣯⣍⠁⠀⣻⣇⠀⠀⠀⠀⠀⢰⢿⠀⠀⠀⠀⠀⠀⢀⣀⡿⣇⢣
⠀⠀⠀⠀⡀⠘⠃⢀⠀⠀⠀⠀⠘⢿⡿⠃⠀⢏⠉⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⢠⡄⢠⣼⣼⣦⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⣻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣷⡟⠀⠀⠀⠀⠀⠀⡞⣿⡦⠓⠀⠀⠀⠀⠀⠀⣠⡾⣷⡄
⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⣰⠿⣬⠁⠀⠀⠀⠀⠀⠀⠓⣿⡄⠀⠀⠀⠀⠀⢀⡼⢻⡄⠸⣿⡀
⠀⠀⠀⠀⠰⠁⢸⠀⠀⠀⠀⠀⠀⠞⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠰⠁⡇⠀⠀⠀⠀⠀⠏⠀⠞⠀⠀⠻⠇

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢈⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⠀⠀⠀⠀⠐⣿⠂⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⣤⣾⣿⡇⠀⠀⠀⠀⠀⣀⣤⠟⠻⢿⠀⠀⠀⠀⠀⠀⠼⣿⠄⠀⠀⠀⠀⠀⣶⣾⣷⣶⣶⣦
⠀⠀⠀⠀⠘⠿⠿⠁⠀⠀⠀⠀⠾⠉⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢶⠠⢠⡄⡄⡦⠀⠀⠀⠀⠀⠀⣃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣀
⠀⠀⠀⠀⠙⢾⡗⠋⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠀⠀⠁
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⢀⣠⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡄⠀⠀⠀⠀⠀⠀⠀⡎⠋⡇
⠀⠀⠀⠀⠀⡃⠃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠓⠂⡄⠀⠀⠀⠀⠀⡞⠀⣇⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢰⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠹⣿⣿⣷⣤⣴⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠙⢿⣿⣿⣿⠟⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃⢻⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⣽⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠋⠀⠈⣿⣿⣿⣿⣿⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣟⣉⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣀⣀⠀⠀⠀⢿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣬⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⢿⣿⡿⠟⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢲⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣷⣦⣵⠀⠱⡀
⠀⠀⠀⠀⢀⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠙⡦⠀⠀⠀⢠⣿⣿⣿⣿⠿⠟⠙⣇⠀⣣
⠀⠀⠀⠀⠈⠳⢴⣯⣑⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡾⠀⠀⠀⠀⢿⣿⣿⠋⠀⠀⠀⠀⣿⠉⠉⠱⡀
⠀⠀⠀⠀⣠⣶⣿⣿⠟⠲⢄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡃⢣⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⢀⡏⠀⠀⠀⢡
⠀⠀⠀⢀⣿⣿⣿⠁⠀⠀⠀⠱⣄⡨⠓⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠀⠀⠀⠹⣿⣿⡿⠉⠉⠙⡆⠲⣄⢸
⠀⠀⠀⠈⠻⣿⣿⠓⠀⠀⠀⠀⢈⠇⢀⣀⡈⢢⡀⠀⠀⠀⠀⠀⠀⢱⠀⢸⠖⠒⠛⠂⠈⠉⡹⠣⠤⠶⠊⠀⠀⠈⢫
⠀⠀⠀⠀⠀⠈⠛⠿⠦⠤⢶⠊⠉⠉⡧⠀⠀⠉⠙⠢⣄⠀⠀⠀⠀⠀⠳⣸⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣠⠞⠁⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠈⠑⠒⠒⠊⠁⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⢀⡠⠄⠒⢲
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠱⣤⡞⠁⠀⠀⠀⢘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠱⡄⠀⠀⠀⠀⠀⠀⠀⣨⠞⠉⠉⠉⠉⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡟⠈⠁⠘⠦⡀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠁⠀⠀⠘⡀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠀⠀⠈⠲⢄⣀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⢈⡄⠀⠀⢀⠇⠀⠀⠀⠀⡠⠴⢄⡀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡄⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⢀⠜⠁⠀⠳⡄⠀⠀⠀⡇⠀⠀⡘⠀⠀⢀⡴⠊⠀⠀⠀⠙⠦⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢇⡀⠀⠀⠀⠀⡜⠀⠀⢀⠔⠁⠀⠀⠀⠀⠘⢆⠀⠀⡇⠀⠀⡇⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢀⡏⠀⠀⠀⡜⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡇⠀⢰⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⣸⣷⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠗⣒⣛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⣒⡲⣤



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

⠀⠀⠀⠀⠀⠀⠄⡒⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣇⠀⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⢨⠀⠀⠀⠁⡆⣇⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠠⠁⠡⠤⠀⢀⠎
⠀⠀⠀⠀⠀⠈⠣⠀⡄⢀⡞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢐⠂⣀⣊⣔⣚⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠹⠀⠙⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠴⠁⣄⣀⣀⡀⠁⡄
⣯⡏⡉⢁⠀⠀⡔⢁⡀⠀⠑⠐⠒⢶⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠄⠀⠀⢀⢴⢀⠀⠀⠀⣆⢀⡇⢀⡇
⢻⣻⣅⡏⠉⠉⠀⢈⡇⠀⠀⠀⢀⠼⠠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⡎⡏⠑⠋⠈⡶⣆⠀⠀⠘⣷⠈⢉
⠀⠀⠀⠑⠢⠤⠴⢻⠘⡄⠀⠠⣷⣄⣀⠎⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠉⠒⠤⠘⠀⣿⠀⠀⠀⢙⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⢦⠀⠀⠉⠁⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠆⠀⠈⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⣸⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⢀⣀⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⣠⡼⠁⠀⠙⠦⡤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⣤⢶⠋⠀⠀⠉⠛⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣠⠞⠉⠢⢄⠀⠀⠀⠀⠀⠀⢸⠗⢤⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⢠⠆⠀⠀⠐⠦⢄⣀⠀⠀⠀⠀⢹⢲⠖⢢
⠀⠀⠀⠀⠀⢠⠃⠀⠀⣰⠃⠀⠀⠀⠀⠙⠢⠔⠚⠋⠳⣾⡄⣧⠂⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⢾⢄⣎
⠀⠀⠀⣀⠤⠃⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢠⣤⣄⡀⢀⣾⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟
⢄⠲⣶⠷⠶⣤⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠳⣬⡁⠘⠲⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚
⠀⠉⠛⠳⣦⡤⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠕⡶⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣾⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣏⣻⡟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣵⣾⣟⣿⣷⣮⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⡀⣀⠌⡾⣿⣿⠛⠛⣿⣿⠷⠡⣀⢀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⢠⣖⣦⣀⣫⣽⣞⣹⢡⣮⠞⠻⣏⣧⣼⣹⠟⠲⣵⡌⣏⣳⣯⣝⣀⡴⣲⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣵⣶⣷⣾⣿⣿⡏⠀⡠⢿⣿⣿⡽⣄⠀⢸⣿⣿⣿⣶⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⢀⡘⣿⣿⡏⢰⠾⣿⣻⣧⣀⠛⢳⣿⣿⡞⠛⣠⣼⡟⣟⠷⡆⢹⣿⣿⢃⡀
⠀⠀⠀⠀⠀⠀⠈⠛⢯⣝⣷⡸⣭⣼⣿⣧⡞⣀⡙⣿⣿⢋⣀⢷⣾⣿⣧⣭⢇⡼⣫⡽⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠾⠙⢿⡄⣶⣾⣿⣦⣝⣵⣿⣿⣮⣫⣶⣿⣷⣶⢠⡿⠋⠷⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⢾⣿⣿⣿⣻⣿⣿⣿⣯⣽⣿⣿⣿⣟⣿⣿⣾⡷⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣯⣭⣷⣶⣶⣶⣶⣶⣶⣿⣭⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⣡⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⢀⡀⠄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⡄⠀⠀⡆⠀⠀⠀⣀⠀⡀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠈⠁⠒⣆⠀⠀⡞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢻⣇⠀⠀⠀⠈⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠹⠢⡀⢂⠆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⣧⡀⢈⣷⡦⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠸⣿⣷⣾⣿⣾⣿⣶⡌⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣷⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠀⠙⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣤⣦⣀⣀⣀
⠀⠀⠀⠀⠀⣀⡠⠀⠀⠐⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣿⣿⣿⣿⣷⣦⣄⣀⠠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⡎⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠶⠤⠅⣀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⠹⠂⠀⠠⠖⠉⠀⠀⠀⠐⠂⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔⠊⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡰⠁⠀⠀⠀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠸⠁⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠣⡄⠠⢄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠰⠒⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠺⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢀⡀⠤⠈⠁⠀⠀⠀⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡠⠀⠀⠀⠑⠢⣤⣀⡀⠀⠀⠀⢀⣀⠠⠔⠊⠉⠀⠈⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⢳⠀⢀⡀⢀⠔⠹⢸⠀⠀⠀⠩⠙⡅⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢤⡈⠃⠀⠈⠉⡜⠀⠘⡀⠀⠐⠁⠀⢀⠀⠀⠀⣠⠚⠓⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⡀⠀⠀⡠⠃⠀⠈⠉⠁⠀⠀⠀⠉⠄⣀⠈⢘⠀⠈⠢⠦⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠁⡀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠈⡀⠈⠢⢀⢀⠀⢀⡀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠮⠄⡀⠀⠀⠀⠀⡇⠀⠀⠈⠀⠀⠀⢱⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣠⡂⠀⠀⠈⠢⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠑⠢⡾⠁⠈⠆⠀⠀⠀⢠⢾⡼⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠑⢄⠈⠢⢤⠀⠁⠁⠄⠀⠀⠀⠀⠔⡶⡥⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠀⠀⠂⣠⠄⠀⠀⠀⢣⠀⢀⡠⠴⠊⠀⠂⠈⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠁⠀⠀⠀⠀⠀⠙⠣⢄⠀⡀⠂⢀⡠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⣧⠤⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡂⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⡦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⡷⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣰⣼⣿⣦⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⣧⣼⣿⣼⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣵⣶⣶⣮⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣭⣭⣽⣿⣭⣭⣀
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠛⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠖⠒⠉⠉⠉⠲⠚⠉⠉⠉⠓⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠉⠙⠲⠤⣄⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⠋⢀⡀⠀⠀⣀⣀⠀⠈⠑⢦⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠠⠦⢄⠀⠀⣠⠋⠲⠃⠞⠁⠈⠀⠀⠁⠀⠤⣀⠀⠀⠳⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣧⠸⠀⠙⠶⠁⡗⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⡀⠀⣠⠜⣗⠒⠀⠀⠀⣇⠀⠈⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠳⡴⠁⠀⠸⣄⠀⠀⠀⠘⢆⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠙⠒⠒⠚⠁⠀⠀⠀⠀⢀⠴⠃⠀⠀⠀⠀⠀⡤⢄⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⢤⣀⣠⣠⠤⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⡘⠀⢈⣄⠈⠄⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠤⢤⠶⢦⠐⠀⠀⠗⠦⣤⣄⣠⣀⠀⠀⠀⠀⠀⢸⠀⢠⣀⣠⠀⢰⢸⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠆⠐⠂⡏⠀⠀⠀⠀⠀⠀⠙⢌⠣⡀⠀⠀⢀⠃⠀⠀⠀⠉⠢⢄⠀⠀⠈⠣⣀⠱⢤⠦⠊⠁
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢀⣀⣠⠂⠀⢀⣀⣀⠀⠀⠀⠀⠳⡌⢤⢀⠎⠀⠀⠀⠀⢱⠀⢀⣁⡦⠀⠀⡄⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡎⠀⠼⠖⠋⠁⠀⡇⠀⠀⠀⠀⠀⠈⠦⠂⠀⠀⠀⠀⠀⠀⢷⠉⠙⠦⣠⠎⠀⡸
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠁⠀⢠⠃
⠀⠀⠀⢀⡀⠄⠐⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⠦⢤⠞
⠀⠀⢰⡁⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠴⠖⠀⠀⠀⠀⣤⡀
⠀⠀⠈⠣⡄⢀⠀⠆⣠⠊⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡠⠊⡀⠈
⠀⠀⠀⠀⠈⠙⠚⠳⠃⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⢀⡜⠀⠀⢳⠀⢀
⠀⠀⠀⡠⢊⠔⡉⠐⡄⠀⠀⠀⠀⠀⠀⠀⢺⣤⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⠴⠚⠉⠑⠦⣄⡀⠀⠀⠀⢀⡞⢢⠆⠀⠀⢀⠆
⠀⠀⡔⡱⠁⠘⠤⢀⣼⡀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠈⠉⠉⠉⠉⠉⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠎⠀⡼⠀⠀⠀⡌
⠀⠌⡘⠀⠀⠀⠀⡎⠀⡈⠑⠲⢄⡀⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠀⢀⣠⠤⠀⠦⢄⡀
⡘⡰⡇⠀⠀⠀⣰⠉⠚⢅⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⢒⡧⠊⠁⠐⡂⠀⠀⠂⠙
⣷⠁⠘⢄⣀⡱⠁⠀⠀⠀⠑⢦⡀⠀⠈⢆⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⢊⠀⠀⠀⠈⠀⠀⠀⠀⢀
⠻⣄⣀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠈⣦⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⢀⠔⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠙⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⠁⠀⣀⠠⠤⠒⠊⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣧⣷⣤⣶⣵⣾⣽⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡛⢶⣄
⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣆⢻⣧⡀
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣄
⢶⣆⣚⠛⠛⠻⠉⠙⢻⣿⣿⣿⡇⢸⣿⡦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⣿⣿⣿⣷⡌⣷⡄
⠀⠀⢸⣿⣿⣿⣤⣾⣼⣿⣿⣿⡇⢸⣿⠀⢠⠋⠻⠿⠿⠿⠶⢶⣶⣶⣶⣤⣤⣤⣤⣀⣤⠚⢿⣿⣿⣿⣿⣿⣷⠸⣿
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣧⡀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠻⡁⠀⢯⣀⣈⣿⣿⣿⢀⣿⣦⣤
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣡⣿⠏⠀⡇⠀⢳⡀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⠀⠀⠀⠀⢷⠀⠘⣿⣿⣿⣿⡿⣸⡟⠉⠁
⠀⠀⠀⠀⠈⠛⠿⣿⣯⣵⣾⠟⠁⠀⠀⢸⠀⠀⢳⡄⠀⠀⠀⣾⣿⡟⠁⡤⠙⡄⠀⠀⠀⠸⠁⠀⣿⣿⣿⢟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠻⣄⡀⠀⣿⠋⠁⠀⠁⠈⣧⠀⠀⢀⣇⡀⠀⡝⠻⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠱⣄⣘⡇⠀⣀⡀⢈⣧⣠⡔⠁⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⣀⡀⠉⠑⠲⣤⡙⠛⢷⣄⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠛⠛⠆⠀⠀⠸⣷⡀⠀⠙⣦⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢠⡀⠀⣿⣿⣿⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠈⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⣿⣿⣿⣿⣿⣿⣿⣅⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡿⠟⠉⠀⠀⠀⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⢰⠂⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⣠⣾⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⡦⠰⣤⣀⣀⡀⠀⢠⠛⠙⠒⠉⠉⠀⢸⡀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢧⡸⣇⠀⠀⠉⠗⠁⠀⠀⠀⠀⢀⡴⠻⣓⢀⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⡏⠉⠉⠳⢤⡀⠀⠀⠀⠀⣠⠖⠉⠀⠀⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠻⠿⠿⠟⠉⠉⠉⠉⠉⠓⠶⠶⠊⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠋⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⢸⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠉⢻⣿⣿⠀⣿⣿⡟⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⠁⠀⠀⠀⠉⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠏⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⣿⣿⡿⠃
⠀⠀⠀⠀⢀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⣡⣾⡟⠁⠀⠀⣿⣿⣿⡁
⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣠⣴⣿⠏⠀⠀⠀⠀⠘⣿⣿⣷
⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⢿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠟⠋⠁⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿
⠿⠿⣿⣿⣿⣷⣶⣶⣶⣶⣤⣤⣤⣴⣶⣶⣿⠿⠟⠛⠉⠀⠀⠀⠀⣀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇

⠀⠀⠀⠀⠀⢀⣠⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⡄
⠀⠀⠀⠀⠀⢸⠉⠛⠉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣉⠂⠀⣦⠁⠀⠈⡆⠀⢂⠑⢄
⠀⠀⠀⠀⠀⠻⠈⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⢽⣧⠀⠀⠀⠀⠀⠀⢀⠔⠡⠁⠀⠀⠀⢃⣠⣤⣁⠇⠀⠀⠀⠠⠑⢄
⠀⠀⠀⠀⠀⡀⣵⠀⣊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣤⠶⣰⡿⠀⠀⢀⣀⠀⡐⠁⠰⠀⠠⠤⠀⠀⢀⡷⠔⢇⠀⠀⠀⠤⠄⠥⠀⠑
⠀⠀⢀⠂⠈⠃⠰⠠⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⣼⡿⡛⠈⡛⢿⡀⠀⠈⢴⣧⡈⠂⠠⢀⠀⡀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠀⠂⠀⡀⡎⢲⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠄⢀⡆⠉⠀⢡⠈⠈⢂⠀⠀⢮⠗⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⢀⠂⠉
⠀⢸⠖⠁⣠⠩⠁⠀⠀⠀⠘⢂⡀⠆⠀⢀⠊⠀⠘⠀⠃⠀⢸⠀⠓⢄⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠣⠀⠊⠀⠆⢀⠀⠀⠀⠇⠀⠠⠘⡀⢃⠪⡀⠀⠚⠒⠒⠒⠋⠀⠀⠁⠂
⠀⠀⠀⠀⠘⠀⠊⠀⠀⠀⠀⠀⠀⢂⠠⠀⠀⠪⡔⣳⠆⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠰⠀⠀⠀⢂⡄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠰⠀⠀⠀⠤⠀⠀⠀⠂⠀⠀⢠⣄⡂⠀⠓⠀⢤⢣⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⢀⣿⡏⣇⠀⠀⠀⡐⠈⡀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⢰⠀⡾⢻⡇⢸⡀⠀⠠⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠂⠀⡀⠀⠀⡠
⠀⠀⠀⠇⠀⡐⠀⠀⠀⠁⠀⠈⠎⠀⢸⠀⠀⡰⠀⡀⠀⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⡁⠈⠀⢀⠘⠁⠀⠉⠢⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸
⠀⠀⠰⠠⡠⠀⠀⠀⠀⠀⣄⠄⠟⠒⠒⠚⢀⠀⠈⠀⠀⠁⠀⠀⡀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⡀
⠀⠀⠆⠠⠁⠀⠀⠀⠀⠀⠘⠆⠨⡀⠀⠠⡌⢠⠀⠀⠠⠂⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠐⠄⢠
⠀⣰⡲⠇⠀⠀⠀⠀⠀⠀⠀⢨⢆⣦⠀⠀⣦⡄⠀⠀⠀⠀⠀⠀⠀⣧⣅⠀⠀⠀⠀⠀⠀⣜⡶⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⡆
⢀⠿⠇⠁⠀⠀⠀⠀⠀⠀⠀⠸⠸⠿⡀⢰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠏⢿⡀⠀⠀⠀⠀⠰⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⡀
⣿⣤⣶⣧⣤⣤⣤⣤⣤⣤⣤⣼⣶⣤⣿⣯⣿⣧⣤⣤⣤⣤⣤⣤⣤⣿⣧⣿⣤⣤⣤⣴⣧⣴⣶⣥⣤⣤⣤⣤⣤⣤⣤⣼⣶⣤⣿⣤⣤
⣿⣿⣡⣬⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⣦⠀⣴⡆⠀⠈⠉⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡅⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣷⠖⠲⣶⣿⣧⠀⠀⠀⠀⠀⢀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⡀⠀⠀⠀⠀⢀⣠⣾⣿⡿⣧⠀⠴⡿⢻⣿⡦⠀⠀⠀⠀⠛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⠁⠀⠀⢀⣼⣿⣿⣿⣿⣇⢀⡄⠀⠀⣿⣿⣿⠀⠀⠀⠀⢰⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣟⣿⣿⣯⣿⣿⣎⣉⣠⠾⣿⣿⠟⠀⠀⠀⠀⠲⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⡵⢿⣿⣿⣿⡿⠻⡿⢯⣽⣉⣡⡿⣁⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣣⣿⣿⣿⣿⡿⣹⣤⠉⠀⣻⣿⣟⣟⣿⠵⢶⣶⣀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢡⡞⣵⣿⢿⣿⣿⣧⡻⣽⡄⠧⣿⠁⠁⠈⠡⢀⠀⠈⢻⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⣰⢏⣼⣿⣿⠹⣿⣿⣿⣿⣿⣷⠀⠹⣧⢰⡤⢀⡀⠐⣀⡦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⡏⣼⢿⣿⣿⣦⠻⢿⣿⣿⢷⣼⢃⠀⠹⣿⣿⡀⣿⣝⠃⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⣿⢠⣿⡿⣿⣿⣿⣶⣶⣶⣾⣿⣿⣯⡀⠀⣹⠙⣿⠉⣿⣇⣿⣧⣿⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢻⠼⣿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣟⢿⣷⡠⠞⣡⣿⡇⣿⣿⣿⣯⣹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡻⣿⣽⢿⣿⣿⣿⣿⡟⢿⣿⡿⣿⣿⣔⠀⢿⣧⣿⣿⣿⣿⠃⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⡞⣿⣿⣻⣿⡿⠉⠀⢠⡟⢷⡷⢿⣿⣿⣾⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⠀⢿⣷⣿⡟⠁⠀⣠⣿⡏⠉⠑⠈⢿⣿⣿⣿⣿⣿⢁⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⢸⣇⡸⣷⢿⣿⡇⠀⣠⣾⡟⣿⠁⠀⠐⠃⠘⣿⣿⣿⣿⡇⢀
⠀⠀⠀⠀⠀⠀⠠⣿⣿⡇⢰⣿⣷⠟⠁⠿⠈⣶⣿⣟⣽⡿⣆⠀⠀⠀⠰⡿⣿⣿⣿⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠁⢃⣀⣿⠟⡠⠀⡄⢰⣿⣿⢿⣿⣿⣌⣀⠀⠀⠀⠀⢻⣿⡇⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⡿⢁⣠⠀⢀⣿⡿⣿⣶⣍⠛⣿⣿⣿⣧⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠞⣯⣶⡿⢡⠎⣼⡋⠱⢌⠙⢏⢓⣀⠀⠀⠀⠉⠉⢉⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠰⣙⡷⣊⣰⢿⢝⡦⠀⠑⠠⣉⠉⠓⠲⠿⠶⠟⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠹⢷⢛⡁⢰⠌⢫⡟⡶⢤⣀⢉⠐⠒⠒⠲⠂⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣡⠄⠉⠧⡵⢋⡔⠀⠀⣶⣠⡄⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢩⣿⣴⢾⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡄⠠⠀⢈⣁⠀⠂⠸⣧⣿⣿⣾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠊⣉⠟
⠀⠀⠀⠀⠀⠀⠀⡶⣮⠈⠀⠠⠀⠀⠢⠐⠻⠄⣠⡟⠛⠟⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠆⠀⠀⠀⠀⣤⣬⡔⢉⣤⠀⠛⠷⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡾⠀⠋⠗⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣠⣤⣶⠶⠾⠿⠿⠿⠿⠿⠿⠶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢻⣤⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠈⠻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠘⣦⣄⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠛⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣠⠞⠀⠀⠉⠉⠉⠛⠓⠒⠲⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠖⡟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠶⣶⣤⠀⣷⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⢠⡄⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⢿⠃⠀⠀⠲⠾⠿⠷⣦⣌⠉⠀⠉⠀⢛⣋⣉⣤⣤⣤⣤⡤⠀⠀⢸⠇⠸⡇⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠤⡼⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠛⠿⠯⢭⡉⠁⠀⣾⠀⠀⣿⠀⣀⠴⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣠⠿⠔⠳⠂⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⢀⣤⣤⣤⣄⣀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⣤⡶⠛⢉⣠⣤⣤⡈⠙⠳⣦⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠛⠁
⠙⠛⠛⢻⡙⣧⡟⠉⠀⠀⢈⡿⢦⣀⠀⠀⠉⠛⠶⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⢀⣀⣤⡤⠖⠛⠉⠉⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠘⢧⣽⣷⡾⠛⢳⡞⠁⠀⠉⠛⠲⠶⠦⠶⠶⠞⠛⠻⠧⠤⢦⣤⡼⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠈⠉⣩⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠟⠛⠛⠛⠛⠲⠶⣤⣿⣄⢀⡀⠰⣌⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠶⠶⠚⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠿⣼⠛⣦⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⠀⠀⠀⣀⣀⣤⣤⡤⠶⠖⠒⢻⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠲⣄⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠷⣄⠀⠀⠀⠀⠀⠀⠀⠉⣳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡆⠈⠛⢦⣄⡀⠀⠀⠀⢀⡟⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠙⠳⠶⣤⣼⡇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠶⠶⠶⠶⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣶⣶⣶⣤⣤⣤⣟⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿