Rol Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⠤⠔⡈⠉⢭⠅⠀⠀⠉⠉⠙⠒⠒⠂⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⠁⠀⢀⠀⠠⠀⣈⣉⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠉⢀⠀⠤⠀⠀⠀⠉⠙⠓⠒⠒⠚⠒⠒⠒⠒⠈⠉⠀⠀⣀⡠⠔⠁⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠉⠀⢀⠀⠈⠓⠲⠤⢤⣤⣀⣒⣒⠤⠤⠤⠀⠀⠖⠒⠒⠂⠀⠈⠁⠀⠀⠀⣀⡤⠂⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢉⠑⠒⠤⢤⣄⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠠⠤⠐⠚⠉⠀⠀⠀⠀⠀⠀⣀⢬
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠲⣀⠀⠀⠉⠓⠒⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠛⢷⡀⠠⠃⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠗⠶⠶⣶⣤⡼⠁⠀⠀⠈⢷⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠘⡆⢀⣄⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⢀⡴⠚⠋⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰⠃⠘⢻⠁⠀⡞⠉⠉⠀⠁⠙⠓⠤⠤⠖⠉⠀⠀⠀⠀⠀⠀⠀⢠⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠸⡆⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠰⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠦⠀⠀⢀⣤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠑⠂⠚⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠰⣯⣰⣶⣶⠀⠉⢻⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠻⢿⣧⣀⡀⠀⣀⢀⣀⣤⡾⠃⠀⢸⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⣼⣿⠀⢻⣷⢀⠀⠈⠉⠉⠉⠉⠉⠁⠀⡀⢀⣼⡾⠛⠃⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⣿⣟⠀⠀⠘⠳⠶⠴⢤⠤⠤⠴⠶⠶⠚⠛⠛⠉⠀⠀⠀⢸⣿⡇⢆
⠀⠀⠀⠀⠀⠀⠈⣇⠈⠀⢿⣿⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⡿⠃⣾
⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⠀⠙⠿⣶⣤⣠⠀⠀⠀⠀⠀⢀⢀⣀⣠⣤⣤⣴⢶⠶⠟⠋⣠⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠉⢻⡆⢀⣀⠀⠈⠙⠛⠳⠲⠚⠛⠛⠛⠋⠉⠉⠁⠈⠀⠀⠀⠀⣶⠟⠔⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡟⠛⢷⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣷⣦⠀⡏⠈⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⢸⠀⠀⢸⡀⣿⠋⣇⠀⣸⣿⡆⠀⣼⡿⠛⠒⠋⠀⠸⡆⢷⢸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣾⠀⠀⢸⡇⡟⠀⢸⢠⡟⠉⢻⣠⡏⠀⠀⠘⠀⠀⣤⣷⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣮⣽⣠⣧⣠⣼⣿⠡⣤⣼⣯⣅⠂⠠⣴⣾⣽⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⡟⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢋⡁⠀⠀⡇⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢚⡭⢀⣼⣿⡏⠀⢀⣇⠀⠀⠀⠾⣥⣌⣙⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠁⠐⢋⣀⣤⠭⠷⠒⠋⣽⠻⡏⠛⠒⠶⢤⣌⣉⡀⠀⠈⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣚⣉⣤⠤⠖⠚⠋⠉⠁⠀⠀⠀⠀⡼⠁⠀⠹⣆⠀⠀⠀⠀⠈⠉⠛⠒⠶⠤⣤⣙⡳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠉⠙⠛⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⢠⣤⡼⠟⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣇⠀⠀⠀⣼⠋⢹⡆⣴⠃⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠛⠀⢀⣀⣠⡀⠀⠀⠀⡾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⢹⡀⠀⠀⠈⠀⠸⡿⠁⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠘⣯⣤⠀⠀⠀⢰⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣿⡇⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣠⡤⣤⡀⠀⠀⣠⣤⣄⠀⠈⢷⡀⠀⠀⠈⠙⠇⠀⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⣹⠀⢸⡄⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠋⢀⣼⠇⠀⣸⠃⠀⠹⡇⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⣸⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢻⣿⣿⣙⡆⢷⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⣴⠛⠁⠀⠀⢻⡄⠀⢠⡇⠀⠀⠀⢻⡄⠀⠀⠀⠀⢀⡏⠀⠀⢸
⢴⣶⣾⡖⢲⣤⣀⡴⠋⠙⢶⣿⠟⠉⠛⣾⣆⣤⣤⣤⣴⣧⣤⣄⣀⣀⠈⠛⠛⠛⠃⠀⠈⠛⠛⠛⠁⠀⠀⠀⠀⠹⡄⠀⠀⠀⣼⠁⣴⢶⢸
⠈⢳⣿⣿⣶⣿⣿⡄⠀⢀⠈⣿⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⢰⠇⠀⠛⣿⢸⠀⠀⠀⢀⡄
⠀⠀⠈⠛⢦⠖⠛⠻⠷⣬⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣾⠀⠀⠀⠉⢸⣀⣠⣴⡿⠁
⠀⠀⠀⠀⢸⠀⠀⠀⢲⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⢚⠚⠓⠓⠒⠖⠖⣲⣿⣷⣤⡴⠶⠶⠛⠋⠁⣼⠃
⠀⠀⠀⠀⠈⠓⠶⣲⠟⢷⣄⠀⠀⠀⠀⠀⢀⣀⣤⡤⣤⡤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢠⣀⠀⢀⡘⠋⣛⠀⠀⠀⠀⠀⠀⠀⠠⡟
⠀⠀⠀⠀⠀⠀⠰⡇⣀⣤⠼⡆⠀⢀⣴⡞⠉⣟⡛⣦⠈⠳⣄⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠯⣅⡘⣿⠛⠶⠶⢤⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⣀⣠⠖⠋⠉⠀⠀⢷⣠⠏⠀⠹⣷⢮⣉⢉⣄⣤⡌⠳⣄⠀⠀⠀⠀⢹⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠶⠶⡏⠀⠀⠀⠀⣉⡿⢶
⠀⠀⠀⠀⠉⠉⠛⢦⡀⠀⠀⠀⠙⠲⠶⠶⢼⡆⠈⠙⠻⣤⡁⠀⠈⠳⣄⠀⠀⠈⠛⠚⠛⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⣶⢾⣛⣁⣠⠼
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠉⠓⢦⣄⠈⠳⣄⠀⠀⠀⠀⠀⣠⠟⢳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡾⠭⠉⠉⢲
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢀⣠⠤⠶⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣧⡀⠀⣠⣾⡵⠞⠉⠀⠉⠛⠶⢤⣀⣀⠀⠀⠀⠀⣀⣴⠛⠃⢻⣦⡀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⣸⡥⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⣇⠀⠀⢨⠇⠉⠉
⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢉⡏⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁⠀⠀⢭⡯⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉⠀⣀⣠⡤⠤⠒⡺⢷⡒⠦⠤⣤⣀⢀⡋⢓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣖⣿⠥⠶⠒⠛⠉⠁⠀⢘⣋⡽⠁⠈⢫⣙⣃⠀⠾⠉⠙⠚⠷⠮⣽⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠉⠀⠀⠀⠀⠀⠐⠂⠀⣈⣉⡼⠁⠀⠀⠀⢣⣀⣄⠠⠄⠀⠀⠀⠀⠀⠀⠉⠙⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡝⠀⠀⠀⠀⠀⠀⢳⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡄⠀⠀⠀⠀⠀⠀⣀⣀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⣀⠀⠀⠀⠀⠀⠀⢀⣧⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠀⢧⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣅⠄⠀⠀⠀⠀⠀⡸⠆⢺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢯⣿⠐⠘⡆⠀⡈⠁⢄⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡶⢋⠀⠀⠰⢠⡟⠐⢾⡽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣿⠀⠀⢻⠀⠡⡄⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡞⢠⠀⡾⠁⠀⢿⣠⣿
⠀⠀⣀⣤⣤⣄⣀⠀⠀⠀⣀⣷⣈⣿⠀⠀⠈⡇⢦⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡜⢰⠓⠀⠀⢼⢁⣿⣀⠀⠀⠀⣀⣠⣤⣤⣀
⠀⠀⢻⣙⣾⣇⣹⡟⢶⠋⠁⠀⣯⣿⢧⠀⠐⢻⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡞⠂⠀⠀⢸⠏⠀⠈⠙⡶⢺⣏⣸⣷⣫⡟
⠀⠀⠀⠙⢦⣤⠟⢳⣼⣆⡀⢸⠿⣧⣼⣤⡶⠚⠹⣏⠫⡉⠋⠉⢯⠉⠋⠉⡏⠙⠋⠹⢉⡝⢉⡽⢉⡽⣫⠟⢹⠧⣴⡀⢸⢀⡄⢀⣰⣷⡞⠻⣤⡴⠋
⠀⠀⠀⠀⢀⣬⣿⠯⣭⣙⡻⠞⠣⣽⡿⠓⢬⣁⣀⣀⣷⣝⣂⣁⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠀⣩⠾⠁⠀⠀⠀⢉⣩⣿⡻⣴⢛⠋⠉⠉⢻⠁
⠀⠀⠀⠀⠀⠳⣿⣷⡾⣿⣄⠀⣀⢻⡤⡖⠏⠛⠊⠋⠁⠚⠀⠚⠉⠻⢷⣦⣤⣀⠀⠀⠀⠀⣠⡞⠁⠀⠀⣀⠴⣚⣹⣞⣻⣣⣿⣉⣡⡄⠀⣸
⠀⠀⠀⠀⠀⠀⠈⠛⡖⠊⠛⣳⡾⠋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⢕⠦⣀⠾⠁⢀⣤⣖⡟⠛⡿⠋⠰⠏⣽⣙⣧⢽⡿⡍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣽⣏⢈⠗⠀⠀⡀⠀⠠⠤⠤⢤⠀⢀⠀⠀⠀⠀⠀⠀⠉⠈⡳⢾⣿⡎⡯⠴⢾⡵⠒⢒⡺⠍⢉⠀⡼⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡏⢀⣹⣧⢂⠔⣩⠴⠞⠻⢋⡿⣿⡟⠛⠦⣕⡢⢄⠀⠀⠀⠀⠀⠀⢽⣿⣝⡖⠀⣿⣵⣢⡀⠡⠀⠀⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠛⠙⢫⣳⣂⡮⠙⣆⣀⡴⡿⠀⠛⣆⡀⠀⠲⣽⢷⣬⡢⣄⠀⠀⠀⠀⠈⠙⠉⠓⡳⠴⠶⢼⡧⠐⢀⠈⢓⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⡀⠻⢢⠽⡻⠖⢾⡇⠀⣇⠀⢀⡤⠟⠛⣛⣿⡻⢿⣽⣾⣷⣦⡀⠀⠀⠀⠀⠐⠐⠈⠉⠹⣯⣥⣶⣟⣉⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠒⠀⣒⣏⣨⠏⠀⠈⠙⢻⡀⠀⠀⢉⣼⢻⡏⢷⡉⠀⠉⠺⣵⡢⠄⠀⠀⠀⠠⠬⠀⣺⣯⢿⠣⠉⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣁⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠹⠦⠴⢻⡇⠀⠀⠸⡟⠦⠤⠞⠁⠈⠛⠲⠦⢤⣭⡭⠵⡏⠁⠘⡗⠤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣤⣤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠔⠁



⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠲⣤⡭⠿⠛⢻⡾⠛⠋⢭⣤⠶⠂⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢋⡥⠒⣈⣥⠤⠒⠒⠀⢸⡏⠐⠒⠒⠢⠬⣁⠒⠬⣙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⠞⣡⠔⢋⣁⠤⠖⠐⢛⣋⣹⣏⣙⡛⠓⠲⠦⣈⡙⠲⣌⠳⢌⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡧⠚⡡⠊⡡⠖⣩⡤⠒⠉⠉⠀⠀⢸⡇⠀⠀⠉⠉⠒⠦⣌⠓⢤⡙⢦⡑⢾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⣀⡴⠋⡴⠊⡴⠚⢁⡠⠔⠒⣉⡡⠤⢼⡇⠤⢌⣉⠒⠢⢄⡀⠙⢦⡙⢦⡙⠦⣀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠗⠋⢉⡤⠊⣠⠞⢁⡴⠋⢀⡤⠚⢉⡴⠒⢛⡟⠓⢦⡈⠓⢦⡀⠙⠲⡄⠹⣄⠙⢦⡈⠙⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡴⠊⠉⣠⠞⠁⣰⠿⠶⠊⠁⣀⠴⠋⢀⡴⠊⠙⢦⡀⠈⠲⣄⡈⠉⠲⠻⢦⠈⠳⣄⠈⠙⠢⣈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢀⡴⠚⣁⣤⣞⣁⠤⠤⠒⢋⣁⠤⠒⠉⠀⠀⠀⠀⠈⢓⣤⣌⡙⠒⠠⠤⠀⣑⣤⣈⠙⠦⣄⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⢁⡴⠎⣈⣁⣀⣀⠤⠶⠚⠋⠀⠀⢠⣾⠂⠀⢀⣠⣶⡿⠛⠉⠘⠓⠶⠤⣀⣠⣍⣉⠸⠶⡌⢳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⣯⣼⡑⠒⠊⣡⠼⡇⢀⢠⣤⣤⣆⣠⣻⣏⣠⣴⣿⠟⠉⠀⠀⠶⢤⣤⣄⡀⠸⢷⣌⠑⠒⠊⣹⣮⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢠⠎⢠⠞⠁⡤⣛⠶⠛⢁⣾⣠⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢳⡈⠙⠷⣟⢤⡈⠙⣄⠹⡄
⠀⠀⠀⠀⠀⠀⢸⠀⡏⠀⡞⠱⠉⠀⣠⢾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢹⢄⠀⠈⢦⠙⡀⢸⠀⢱
⠀⠀⠀⠀⠀⠀⢸⠀⡇⢸⠀⡇⠀⢸⡇⢸⠆⠘⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⣤⡀⡀⢀⠀⣀⣠⡞⠑⠀⠁⢸⠗⡀⢸⠀⡇⢸⠀⡸
⠀⠀⠀⠀⠀⠀⠈⢧⣘⣴⡄⢣⣀⠿⢡⣾⠀⣼⠗⠋⢹⢻⡟⠛⠉⠀⠀⠀⠀⠀⠉⠀⠋⠹⠇⠙⠀⠈⠀⠀⢸⡄⠻⣄⡸⢀⣧⣎⡴⠃
⠀⠀⠀⠀⠀⠀⠀⢠⡟⢠⠛⡾⢁⡴⡡⢶⡀⠀⠀⠀⢇⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢌⠳⡌⢳⠞⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⢸⣿⠇⡘⣧⠀⠀⣼⠎⠀⠀⠀⠀⠉⢂⡒⠒⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⡸⢋⡰⣿⡷⢸⠀⡄⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢻⡘⢄⠳⡑⢄⣙⣛⢣⣿⠀⠀⠉⠀⠀⠀⠀⣴⡾⠉⢉⣉⠉⢳⣶⡀⠀⠀⠀⠀⠀⠀⣵⣎⣛⣋⡡⢎⡴⣁⢎⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠙⢌⠳⣦⡬⣟⠹⣧⠀⠀⠀⠀⠀⠰⢦⣷⠚⠛⠛⠛⢾⣵⠧⠀⠀⠀⠀⠀⣰⠟⢹⠧⣴⡞⢩⠎⡰⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢘⠀⣨⣷⠘⡇⢋⠳⣄⠀⠀⠀⠀⠈⣫⠳⠶⠦⠖⢹⠇⠀⠀⠀⠀⢀⡞⢹⠰⠇⣼⣥⠆⡇⢼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⡾⠞⠋⣸⢀⡄⡸⠀⠝⠢⣀⠀⠀⠀⠈⣦⣤⣤⣤⡃⠀⠀⠀⣀⡴⢻⡀⢰⠰⡆⢣⠘⠳⠷⠤⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣇⣬⠴⢃⣼⣣⣶⢟⣷⣄⡀⠀⠁⠀⠀⠈⠀⢀⣠⣞⣻⣷⣬⣵⡈⢧⣵⡘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣼⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣼⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⢃⣨⣿⣿⣏⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⣾⡆⠛⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⢛⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡈⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣴⡆⣻⣿⢋⣴⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢻⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢫⣴⣿⣿⠸⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⢼⣏⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣦⣾⣿⣷⣬⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢻⡧⢺⡇
⢴⣶⣲⡶⢶⣄⡀⣀⡴⠞⢧⣼⣿⡏⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢸⣧⠼⠓⢦⣀⢀⣠⡶⢶⣖⣶
⠈⢷⡟⠳⣦⣯⣹⡇⠀⠀⠀⢹⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⢹⣏⣼⣴⠞⢻⡾
⠀⠀⠙⠶⣎⣩⠿⣿⣦⡀⢶⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡶⢀⣴⡿⠿⣍⣱⠶⠋⠁
⠀⠀⠀⠀⣾⠁⠀⠀⣨⣽⡟⣽⣿⣿⣿⣿⡋⣯⢻⣿⣿⣿⣿⣿⣿⣿⣟⣋⣙⣻⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣾⣿⣿⣿⣯⢿⣏⣅⠀⠀⠈⣷
⠀⠀⠀⠀⠻⣤⣀⣠⣤⠾⣷⣯⣵⣿⣦⣽⠿⠶⠛⠛⠉⠉⠉⠉⠉⠉⠀⠈⠁⠀⠀⠉⠉⠉⠉⠉⠛⠓⠺⠿⢯⣵⣿⣯⣭⣿⠷⣤⣄⣀⣤⠟
⠀⠀⠀⠀⠀⠀⢠⡟⠁⠐⠋⣼⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧⡘⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⣸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣄⠠⠤⢀⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢧⠀⠀⣀⣤⠴⢶⠋⣽⠛⣿⠋⠀⠀⠙⢷⢀⡆⣸⠟⠁⠀⠈⢿⠛⣯⠉⣳⠶⢤⣄⡀⠀⣰⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⣁⣀⠤⠤⢿⡏⠀⠙⢦⣀⣀⣒⣾⣟⣟⢷⡔⠂⡀⣠⠞⠀⢹⣿⠥⢤⠀⣈⣱⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⠉⠀⠀⠀⢀⣿⡀⣠⠖⠛⠋⣩⢷⣿⣟⣽⣏⡙⠛⠳⣄⠀⣸⡇⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢃⣠⣤⠀⠀⢠⠞⠁⠉⢻⡄⠀⠀⢠⡾⠹⡏⢻⣄⠀⠀⢀⣼⠉⠁⠙⣆⠀⠀⣠⣄⣈⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⡇⠰⠏⠀⠀⠀⠀⠛⠲⠒⢋⠃⠀⠀⠈⡟⠓⠦⠞⠁⠀⠀⠀⠈⡧⢀⠇⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠹⣗⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⣙⣦⣸⠃⣐⢡⣇⡇⠀⠈⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡤⣼⣤⣠⠞⠛⠁⠘⠛⠓⣾⣾⠟⠀⠀⠀⠀⠈⠳⣦⢄⡀
⠀⠀⠀⠀⣠⢶⠋⠉⠁⢀⠀⣸⠀⣀⣈⣁⠐⣾⠯⣀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡹⣟⠓⠶⣤⣀
⠀⠀⢀⣴⡇⠀⠹⣆⣀⠈⠀⣺⢟⠉⠀⠀⣿⢁⡤⢼⠷⣄⡀⠀⠀⠀⠀⠀⢸⠇⠈⠳⡀⠀⠉⠻⡲⣄
⢀⡜⡹⠉⢳⡀⠀⠀⠸⠒⠋⢻⣀⠀⠂⠀⣬⣿⡊⣋⠀⢺⠟⠢⣄⣀⣀⣠⠎⠀⠀⠀⠘⢦⠀⠀⠙⡌⢣
⡞⠀⡇⠀⠀⠱⣄⠀⠀⠀⠀⠈⠉⢷⣶⡦⢾⣤⣉⣤⡴⠋⠀⠀⢀⣙⣍⠀⠀⢀⡤⢴⣒⠾⣧⠀⠀⡷⠀⡇
⣇⠀⢧⠀⠀⠀⠈⢦⠀⢀⣴⣺⣯⠯⢛⣇⣠⣞⣿⡫⢳⡄⢠⢾⠟⠷⡚⢦⣰⠛⣏⣍⠄⢀⡾⠀⢠⠇⢀⡇
⠘⢦⠀⠳⢤⡀⠀⠀⠳⣼⠈⠳⠒⠀⣸⢿⡇⠳⠦⠊⢀⡿⢿⣈⣉⢉⣀⡼⠛⠷⢦⣤⠴⠛⣡⡔⠁⣠⠞⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠑⠦⣄⡈⠑⠂⠤⠬⣙⣛⣛⣋⠁⠀⠙⠛⠒⠛⠉⠀⠀⠉⠉⢉⣁⣀⣀⠠⠤⠔⠒⠉⣀⡤⠚⠁⠀⠀⠀⢀⣀⠤⠒⠋⣁⠴⠃
⠀⠀⠀⠀⠀⠉⠻⢖⣒⠢⠤⠤⣬⣍⣉⣉⣉⣉⣍⣉⣉⣉⣉⣉⣉⣉⣠⠤⠤⠄⣐⣲⠞⠋⠁⠀⠀⣀⡠⠔⠚⢉⣠⠴⠚⠛⠋⣉⣩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠂⠠⠤⠤⠤⠤⠤⠤⠤⠤⠴⠦⠴⠒⠒⠊⠉⠁⣀⣀⣤⠤⠖⠋⢁⣠⣔⣚⡭⠤⠖⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠴⠒⢒⣩⠭⠛⢉⣀⠤⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠥⠤⢒⡲⠞⢋⣁⠤⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣋⠥⠖⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠆⠙⠉⠝⡒⢄⣀⠀⡠⠤⢤⡤⠤⢤⠀⣀⡠⢒⠪⠉⠋⠰⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠙⠀⠀⢀⠀⠀⠀⠀⡼⢐⠠⠀⢸⡇⠀⠅⡀⠿⠀⠀⠀⠀⠀⠀⠀⠏⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⣀⣀⠀⡀⡪⢂⠁⠀⢰⡇⢀⠨⡐⠡⡈⠢⣀⣀⠀⠀⠀⠐⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢁⡋⠄⡨⢂⠌⠐⡨⢀⠉⠋⡐⢅⠂⠱⡄⢅⠀⢉⡌⠆
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠃⠠⣔⡤⣒⣓⠂⠅⠂⠁⠀⠀⠈⠐⠨⠀⣊⣒⢬⣢⡅⠈⠐⠀⠠
⡀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⡪⢛⡤⠅⠃⠐⠀⠀⠀⠀⠀⠀⠀⠐⠠⠘⠢⢤⡛⢁⠀⣔⡆
⠹⣤⠂⠀⠀⠀⠀⢒⡥⠹⠀⠀⠉⢠⠚⠠⣀⠀⠀⣀⠀⠀⡀⠀⠀⣀⠄⠓⡄⠩⠄⢀⢏⢮⡒⠀⠀⠀⠀⢐⣤
⠀⠀⠛⢦⡀⠀⠀⠁⡠⠁⢀⡞⠴⢊⢼⠀⠈⠉⠉⠁⠀⠀⠁⠉⠉⠀⠀⡷⣕⢦⠷⡀⠘⢄⠈⠀⠀⣀⡴⠛
⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⢘⠄⣇⡻⢮⣄⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⢀⡷⢟⡴⢀⠇⠀⠀⠀⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡁⢻⡝⠉⢃⠀⠀⠲⣀⡀⠖⠀⠀⡰⠙⢉⡟⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠉⢀⠄⡬⠱⢄⡀⠁⠉⢀⣠⠮⢥⡠⡀⠉⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠊⠂⠀⠈⢈⠋⠙⡀⠁⠀⠐⠁⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠛⡔⣄⠀⢀⠂⠀⠀⠐⠀⠐⢋⡀⠚⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠈⠀⠀⠁⠠⠂⠠⡄⢀⠤⠈⠄⠈⠃⠀⠂⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣽⡴⢂⠀⠠⠁⠀⠾⠅⠸⠤⠁⠈⢄⠀⡐⣀⣯⡆
⠀⠀⠀⠀⠀⠀⠀⢿⠧⣲⠔⢊⣾⠐⠁⣄⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣢⠀⠉⡗⠑⠦⣖⠿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⣙⡟⠛⣾⣷⣖⠈⣈⠢⡀⠀⠀⡀⣀⠀⠀⢀⠔⠡⠅⣢⣷⣴⠛⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⢯⣀⡿⠐⠉⠉⠁⠉⠀⠂⠿⣟⠀⠀⢁⣯⢶⠞⠿⡿⠓⣦⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⢎⠀⠀⠀⡀⣀⣀⡀⠀⠀⠀⠉⠺⣿⡿⠛⠤⠐⠓⢈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⢃⡐⢟⠐⢏⠈⡦⢵⣦⣀⠀⠀⠈⠛⠶⠇⢢⡄⠈⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡸⠀⠀⠙⢄⣠⠏⠙⡗⣤⡀⠀⠀⠀⠀⢲⡾⠵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠁⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠁⠒⠒⠒⢉⠙⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⡖⠊⠉⠁⠀⠀⠈⢧⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⢻⢀⢀⣀⣀⣤⣤⣜⣧⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠛⠛⠉⠉⠉⠀⠀⠀⠀⢀⡽⡏⠛⠢
⠀⠀⠀⠀⠀⠂⠀⠀⠒⠆⠤⢀⡀⠀⠀⠀⠀⠀⢀⡞⠥⠴⠲⣤⠀⠀⠀⠀⠀⢀⡠⠖⢯⠀⠘⡄⣠⡇
⠀⠀⠀⠀⠒⠂⠓⠒⠒⠤⢄⣀⠀⠁⠒⠠⢄⢀⣀⣇⣀⣀⣀⣤⣆⠤⠤⠒⡏⠁⠀⠀⠘⣆⣤⠾⠋
⠀⠀⠀⠀⡑⡄⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⣄⣀⠀⡇⢸⠉⠀⠀⡇⠀⠀⢀⣱⣠⡤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠙⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢉⣏⠙⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⠀⣩⡯⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡛⠁⢀⣀⣠⣤⠤⠶⣺⢷⠲⠦⣤⣠⡀⠀⠈⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣶⣩⡴⠦⠖⠛⠋⠩⠀⠐⠒⣻⠃⠈⢿⡓⠒⠀⠩⠍⠛⠳⠿⢶⣌⣳⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠛⠋⠉⠀⠀⠀⠀⠖⠀⠀⠛⠛⡿⠁⠀⠀⠀⢻⡉⠁⡀⢀⠀⠀⠀⠀⠀⠉⠉⠛⠛⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡾⠁⠀⠀⠀⠀⠀⢻⣉⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡞⠁⠀⠀⠀⠀⠀⠀⠀⠹⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣛⡄⠀⠀⠀⠀⠀⠀⠘⢲⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡴⠂⠀⠀⠀⠀⠀⠀⢀⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣿⠿⣷⠀⠀⠠⠄⠀⠀⢤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⢁⡄⢀⠀⠀⠀⣸⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣿⣴⡿⣇⠀⠄⠀⠐⣦⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⡤⢠⠀⠘⢠⣧⡀⢸⡇
⠀⠀⠀⣠⣤⣤⢤⣀⠀⣠⠼⠦⣽⣿⠗⠿⣄⠀⣦⣠⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠏⢰⠃⣾⠀⠁⢸⣷⡄
⠀⠀⠀⠹⣽⣿⣤⣿⣿⡁⠀⠀⢹⣇⡤⣖⠯⠭⠝⠓⠂⠺⠉⠩⠉⠻⢷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡎⢸⡋⠀⠀⢸⣇⣵⠀⠀⠀⣠
⠀⠀⠀⣤⡬⡿⢯⡥⠟⠿⢦⣹⠞⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠪⢝⡲⣄⡀⣀⣀⣀⣀⡀⣈⣧⡟⣃⠀⠀⢸⣧⠾⣄⣠⢾⡧⢤⣤⣤
⠀⠀⠀⠘⢾⡛⣾⡀⠀⠀⢲⡷⠡⢚⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠙⠯⣭⠟⢩⠟⣩⠯⢹⣷⠤⠦⠼⡛⠚⠉⢀⣿⣷⠛⡳⠃
⠀⠀⠀⠀⠀⠉⢳⠛⠒⡾⠛⠷⣖⡡⠈⡠⢒⣂⡤⠽⢶⡶⠾⠧⢴⣆⡠⢄⡀⠀⠀⠀⠀⠀⠉⠈⢳⢧⣞⠁⠀⢸⣗⣢⠬⠑⠂⡀⠁⢸⠓⢶⠋
⠀⠀⠀⠀⠀⠀⢫⡀⢘⣇⡤⣶⢿⡁⢊⠴⣯⡁⠀⣀⡐⡟⠦⣄⠀⠀⣉⠷⢮⣑⢤⡀⠀⠀⠀⠀⠈⠑⠯⣓⠦⣬⣏⣩⣛⡓⣶⢀⡀⠈⠧⣼⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣏⡁⠉⣸⠟⣧⣿⣒⠸⢿⣟⣹⣅⣳⠀⣄⣹⠞⠁⠀⠀⠈⠓⢮⣳⢄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠭⠭⠭⡿⠄⣀⣢⡴⠞⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢽⡆⠈⠁⠚⠭⢛⡍⣻⣿⡿⠿⠛⠳⣈⠏⡄⢀⣀⠀⠀⠀⢀⣈⠓⣯⡲⢄⡀⠀⠀⠀⠀⠨⠭⠤⠥⢻⣿⣿⣿⠶⠶⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢁⣈⣽⣤⣾⠛⠦⣄⣙⣛⣻⡆⢰⣿⣿⣿⠀⢼⣿⣿⣷⠀⣘⣓⣮⣕⡂⠀⠄⠀⡐⡒⡤⢽⡿⡟⣯⠁⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣣⡴⠚⠉⠁⠈⠀⢣⣀⣀⡸⠁⣹⢫⢷⣈⠛⠛⠋⠾⠏⠛⠛⠋⣠⢏⢻⣧⣀⣉⡿⠒⠲⠶⠖⠚⡏⠀⠀⢹⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢠⣧⠴⠚⠉⠻⣷⣶⢦⢶⣶⡾⠋⠙⠲⢬⡇⠈⠁⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁



⣿⡟⣛⡿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢠⢆⠀⡴⢦⠀⡴⡄⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠃⠈⠛⠀⠈⠛⠁⠙⠋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣺⠉⣰⣿⡇⠀⢸⡇⠀⠀⢰⣏⣍⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠚⣁⣠⡭⠷⠒⢊⡿⢻⡑⠒⠦⠤⣌⣉⡀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣚⣥⠤⠶⠒⠛⠉⠁⠀⠀⠀⢀⡞⠀⠀⢳⡀⠀⠀⠀⠈⠉⠛⠒⠶⠤⣬⣛⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠈⠉⠉⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⢀⡴⣄⠀⢀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠰⣶⠞⠋⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢻⠀⠀⠀⠿⠀⣿⣴⠋⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠀⣤⠴⠖⠀⠀⠀⡞⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⡇⠀⠀⠀⠀⠘⠃⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠼⢷⠃⠀⠀⢸⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣽⡿⣿⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⡾⠋⢻⠀⠀⣴⠛⠳⣆⠀⠘⣆⠀⠀⠀⠈⠀⠀⢀⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣥⣾⣇⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⣠⡶⠋⠀⠀⡇⠀⠀⣿⠀⠀⠘⢧⠀⠀⠀⠀⠀⣸⠀⠀⢸
⠀⠀⠀⣀⣤⣤⣤⣀⠀⢀⡤⠷⣸⣿⡿⠾⣄⠀⠀⢀⣼⠁⠀⠀⠀⠀⠿⠶⠶⠂⠀⠙⠶⠶⠃⠀⠀⠀⠈⢧⠀⠀⠀⢠⠇⢠⣤⢸⢦⡀
⠀⠀⠀⠹⣽⣿⣤⣽⣿⡏⠀⠀⠸⣟⣀⡤⠞⠛⠋⠉⠉⠉⠉⠉⠛⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⡾⠀⠸⢾⢸⠋⡇⠀⠀⠀⣀
⠀⠀⠀⣀⣈⣛⣿⣩⠽⠿⣦⣘⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⢳⣰⠇⠀⠀⠈⢸⠋⣇⣀⣠⣾⢃⣀⣀⣀
⠀⠀⠀⢿⡷⢧⣹⡇⠀⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⡓⠒⠒⠒⠒⣺⠛⢦⣄⣠⣤⠼⠿⠛⠋⢱⢟⣋⡼⢞⡿
⠀⠀⠀⠀⠑⠶⣋⡱⢦⣶⠞⠳⣄⠀⠀⠀⠀⠀⣀⣀⣤⡤⠤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⣠⠞⠁⢀⣞⠁⠀⠀⠀⠀⠀⠀⣾⢏⣙⠶⠊
⠀⠀⠀⠀⠀⠀⡏⠀⠀⣏⣀⡤⠾⡆⠀⣠⣴⢯⣭⡇⠈⣳⣄⠀⠀⠀⠉⠳⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢬⣉⣻⠓⠲⠶⢤⡀⠀⠀⠹⣄⣹
⠀⠀⠀⠀⠀⠀⢙⣶⣖⠋⠁⠀⠀⢳⣴⠃⠈⢳⣦⣴⡶⢿⡈⠙⠲⣄⠀⠀⣼⢯⠿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⢺⡇⠀⠀⣀⣨⠽⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠙⣆⠀⠀⠀⠉⠛⠛⠒⢿⣿⣟⣶⣧⡈⠳⡈⣳⠶⢇⠀⢠⠀⣸⠿⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣿⡯⣤⣤⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢻⠀⠀⠀⠀⢀⣀⣠⣾⡟⠋⢩⣭⣱⣤⣼⠋⠀⣌⡗⠛⠳⣇⠀⠘⣿⣹⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⡙⠀⠀⢙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣠⠴⠒⠛⠯⠤⢻⠃⠈⠓⣶⠤⣤⠤⣾⣤⣤⠞⣧⣀⣴⠳⢦⡤⣿⠤⣤⣤⠼⠛⠲⡦⠤⣤⣤⠤⢶⠋⠈⠀⣟⠢⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠘⠦⠤⠴⠃⢠⠏⠞⣁⡴⣇⠈⠁⣾⡈⠁⣠⠧⣌⡓⠘⣟⠦⠤⠼⠁⠀⠀⠀⠀⠈⠳⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠒⠉⠁⠀⠈⠑⠚⠁⠑⠚⠁⠀⠈⠉⠓⠿


⠀⠀⢀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⡀
⣿⡟⡉⣻⠟⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣰⠦⣀⠜⢦⣀⠶⣀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠁⠀⠉⠀⠈⠉⠀⠙⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠾⠂⠀⣤⠀⠀⠀⢠⡄⠀⠸⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠡⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠙⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠚⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⢖⣀⣀⡀⠀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⠀⣀⣀⣀⠀⠾⣧
⠀⠀⠀⠀⠀⠀⠠⡏⠙⠛⠃⠸⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⡗
⠀⠀⠀⠀⠀⠀⣠⠗⣶⢤⡤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡠⢤⠴⡖⢻⡀
⠀⠀⠀⠀⠀⣠⣆⣰⢷⡞⠀⡌⢸⡅⠀⢀⣠⣤⠀⡄⢸⡇⠀⠀⠀⠀⢹⠀⣤⣀⣤⡀⠀⠀⣽⠀⡇⠀⠷⠿⣄⣷⡀
⠀⠀⠀⢠⢫⢏⡟⢁⢾⡃⣰⠁⡜⠑⠋⠉⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠓⠘⡄⢱⡀⣻⢦⠙⣟⢯⢣
⠀⠀⢀⡇⡎⢸⠁⡎⡌⣹⠇⣰⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢻⡉⡌⡆⢹⠈⡆⡇
⠀⠀⠀⡇⡇⣿⠈⣧⠜⢁⡖⢹⠀⠚⢦⣄⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⣠⠞⠂⢸⠙⣦⠙⢥⡇⢸⡇⡇⡇
⠀⠀⠀⢣⠱⣧⡄⠈⡶⡛⢀⡜⠀⠀⠁⠈⠋⢙⡋⠻⢸⡇⢀⣄⣀⠀⢸⠸⠋⣛⠈⠋⠈⠁⠀⢸⣄⢸⡷⣎⠀⣤⡶⢁⠇
⠀⠀⠀⠈⡷⢩⡷⡞⣼⠷⢉⣼⠀⠀⠀⠀⠀⠈⠁⠀⢸⡇⠘⢤⠼⠀⢸⠀⠀⠉⠀⠀⠀⠀⠀⢸⣌⠓⣾⡜⡶⣯⠱⡏
⠀⠀⠀⢰⠁⢻⡆⡇⠻⣄⡿⢟⢣⡀⠀⠀⢠⡄⠀⠀⢈⡉⠛⠚⠛⠉⣉⠀⠀⠀⠀⠀⠀⠀⣠⢻⠾⣇⡼⠃⡇⣿⢁⢱
⠀⠀⠀⠘⡄⢦⠳⡜⠦⣌⣩⠜⢨⣇⠀⣤⣼⠃⠀⠀⣠⡈⣉⡙⣉⡉⣠⡄⠀⠀⠀⠀⠀⢀⣾⠘⢬⣉⡤⢞⡴⣣⠏⡜
⠀⠀⠀⠀⠙⢦⡱⣌⢳⣶⡴⣖⢫⡉⠷⣏⣹⠀⠀⠰⣯⣿⣍⠭⢍⣽⣯⡷⠀⠀⠀⠀⣠⠟⣩⢛⡶⣴⣶⢋⡔⣡⠞
⠀⠀⠀⠀⠀⠀⣷⠘⡄⣻⠿⡘⡄⢳⠀⣿⡁⠀⠀⠀⠹⢦⣄⣤⣤⣤⠞⠁⠀⠀⠀⣠⡇⢸⠃⡜⡰⢿⡃⡜⢰⡇
⠀⠀⠀⠀⠀⠸⠧⠴⠟⠁⣠⢡⠃⣼⠀⡇⣙⠦⣀⠀⠀⠀⠉⠀⠈⠁⠀⠀⢀⡠⢞⡁⡇⢸⡄⢣⢱⠀⠙⠷⠤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⢎⡴⢃⡼⠛⣀⣀⣬⡽⢦⣄⣀⠀⢀⣀⣤⠶⣯⢀⣀⡈⣻⣄⠳⣌⢆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣫⠶⠟⠛⠂⠉⠁⠐⣀⣀⣩⠟⠑⠉⢯⣩⡀⠐⠋⠉⠁⠛⠻⠷⢮⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⣿⢹⠀⠀⠀⠀⠘⠁⢠⣭⣵⡏⠀⠐⠀⠘⣷⣴⠆⠲⠂⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢿⣸⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⠈⢷⡞⠂⠀⠀⠀⠀⠀⣼⣽⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠁⠀⠀⠀⠀⠀⢀⡈⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⡁⠀⠀⠀⠀⠀⠉⠁⣾⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⢀⠀⠀⢠⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⢂⠀⠀⠀⠀⢸⡟⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠈⡆⢠⠀⠀⢦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⡄⠀⠀⠃⣾⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⢸⠀⠰⣆⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣴⠃⡶⢰⠃⠀⠐⣷⡄
⠀⠀⠀⠀⠀⠀⠠⣍⡿⠀⠀⠀⡆⢦⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠃⣏⠁⠀⠐⣏⡥
⣴⣶⡦⢤⣀⡠⠔⢦⡟⠀⠀⠈⢹⣎⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣾⣩⠀⠀⠀⣧⠖⠤⣀⣠⠄⣴⣦
⠈⢿⣉⡾⣼⣅⠀⡀⣿⣏⠴⣯⠁⠙⢯⡳⣌⠁⠈⠣⠈⠁⠘⠀⠉⠀⠋⠰⠋⡼⠋⡴⣫⠟⠁⠟⠳⣦⣐⡇⡀⢀⣽⡼⢎⣹⠟
⠀⠀⠈⡏⠉⠉⣳⣿⣿⣿⣆⡀⠀⠀⠀⠙⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⣠⣜⠃⠀⠀⠀⣀⣤⣿⣿⣷⡋⠉⠉⡏
⠀⠀⠀⠣⠀⣴⠿⡿⡿⡟⢦⣌⠳⢦⣀⠀⢫⠉⠲⣄⠀⠀⢀⣤⢤⣄⠀⠀⣠⡾⠋⢹⣃⣠⠶⢋⣤⠞⡿⡿⡟⢶⡄⠰⠃
⠀⠀⠀⠀⢸⡀⠀⢠⢗⠀⠀⣱⠬⣿⣿⣿⣾⣆⠰⣄⢱⠴⣇⠀⡄⢀⡷⢶⢋⡴⢀⣾⣿⣿⡯⢴⡁⠀⢀⢧⠀⠀⣸
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠙⡏⠀⠘⣽⢏⠉⠠⠌⠷⣼⡏⠀⣬⠛⠛⢻⡀⠀⣿⡴⠎⠥⠈⢙⡿⠀⠈⡛⠁⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⡃⠀⠉⡗⢲⡖⣺⠗⡶⢋⠷⡴⢏⠳⡖⠿⡒⢶⡖⠈⠉⡙⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣮⣠⠵⠚⠳⣄⡠⢷⣄⡴⠛⠲⢅⣨⡏⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⠿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⡰⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⣅⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣎⢛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⢥⠈⠬⠦⠀⠀⠀⠀⠀⠀⣤⡀⠀⣀⣤⣾⣽⣿⣿⣿⣿⠿⠋⠉⠡⡳⣝⠿⣿⣦⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣧⣼⣦⣴⠖⢦⠀⠀⠀⢀⣀⣗⣿⣿⣿⣿⣿⣯⣿⣿⣿⣷⣶⡆⠀⣸⣿⣜⢿⣮⡻⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣈⣧⣶⣿⣿⣿⠈⠳⡿⣿⣿⠈⠉⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⡻⣿⣦⡁
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣜⢿⠁
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣟⣵⣿⣿⣿⣿⣿⡯⣿⣿⣿⡿⠦⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣾⣽⣟⣛⣻⣭⣾⡟⣡⣶⣾⣿⣷⣶⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣷⣍⣛⡿⠿⢟⣛⣥⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⡤⠮⠽⢦⠄⣀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠁⠀⠀⠁⠀⠀⢨⠀⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠑⡄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅
⠀⠀⠀⢀⢆⠀⠀⢠⠞⠫⠭⠤⠆⠰⠄⠀⠀⠀⠤⠀⠀⢤⠀⠀⠀⠀⠆⠰⠤⠤⠕⠛⠄⠀⠀⢰⡄
⠀⠀⠀⠀⢢⡱⢄⡸⢲⢤⠤⣀⣀⡀⡀⢀⣀⣀⡀⣀⣀⣀⣀⣀⣀⠀⢀⢀⣀⡠⠤⢖⢣⡠⠔⡱⠁
⠀⠀⠀⠀⠀⢉⣦⣑⢿⡇⢠⠀⠇⠴⠒⠈⠉⢸⡇⠀⠀⠀⠀⠉⠉⠒⠦⠐⡀⠇⢘⡽⢊⣴⡎
⠀⠀⠀⠀⢀⢣⢋⡉⢂⢳⠃⡘⢀⡦⢒⠾⢖⢾⡃⠀⠀⠀⡴⡲⠷⡒⢤⣀⢡⡘⡾⠱⠋⡟⡌⡆
⠀⠀⠀⠀⠸⠸⡸⣇⠘⣨⠎⡃⢨⡀⢧⣿⣿⢹⡇⠀⠀⠀⠸⢼⣿⡿⢀⡥⢰⠙⣇⠳⢀⡇⡇⡇
⠀⠀⠀⠀⠀⢢⡷⣎⠞⣍⠴⡅⠀⠉⠓⠱⠳⢺⡆⡔⢲⠀⠻⠒⠎⠒⠉⠀⢀⠶⢼⡱⣁⡾⡴⠁
⠀⠀⠀⠀⠀⠘⢸⠉⠸⣇⡾⡷⡀⠀⢀⡄⠀⠈⡓⠒⠓⢒⡃⠀⠀⠀⠀⠀⠼⢷⣨⠇⡈⢳⢑
⠀⠀⠀⠀⠀⠠⡰⡀⣑⠈⠑⣁⢧⣰⡼⠇⠀⠀⣉⢌⡫⢉⠀⠀⠀⠀⠀⡸⣆⠂⠡⣊⢔⢌⡌
⠀⠀⠀⠀⠀⠀⠙⡌⠌⢻⣟⠍⢂⢹⡖⠀⠀⠙⢇⠂⡁⣺⠏⠀⠀⢀⡜⠰⠡⢻⡟⢡⢢⠊
⠀⠀⠀⠀⠀⠀⠐⠇⠶⠊⠐⡐⡀⢸⠙⢢⣀⠀⠀⠉⠉⠁⠀⢀⣤⠊⢇⢀⢠⢠⠘⠲⠨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⢔⣡⠖⠊⠁⣈⠱⠦⣄⣠⠴⠎⣁⡀⠑⠺⢮⡢⣧⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⡿⡯⠐⠂⠉⠁⠀⠀⡰⠁⠈⢦⠀⠀⠈⠉⠐⠂⠸⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢣⡅⠀⠀⠀⠀⠀⡰⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⢸⡼⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠰⢹
⠀⠀⠀⠀⠀⠀⠀⣠⣷⡈⡄⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⢀⠇⠸⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣏⡅⢱⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡌⠀⢸⢹⡄
⣀⣴⣶⡖⢤⣤⠜⠣⣿⠟⢰⣆⢠⣎⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⢠⠁⠀⢸⣹⣇⡀⠀⣀⣠⣄
⠹⡟⠻⣿⡛⠿⣄⣠⠿⠊⠉⠀⠀⠀⠀⠀⠉⠉⠓⠢⢤⣀⠀⠀⠀⠀⠀⣹⣆⠀⠀⢸⠁⠀⢙⣋⣯⠾⢻
⠀⠈⠛⡴⣇⠀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⢀⡴⠋⠀⠉⠂⣼⣾⣴⠾⡟⢤⠛⠁
⠀⠀⠀⣇⣀⣙⣁⢱⡄⠀⣠⣴⣔⠒⠒⠶⠠⢤⡀⠀⠀⠀⠀⠀⠑⢮⡀⣀⣠⡊⠉⠉⠀⠀⢸⣀⡸
⠀⠀⠀⠠⢾⣉⠀⠀⢣⡎⠙⣷⡉⢷⣦⣄⡙⢦⣨⠗⢤⡀⠀⠀⠀⠀⠈⠛⠻⠏⣉⣲⡄⠀⠘⢍⣀
⠀⠀⠀⠀⠈⠺⡇⠀⠀⠀⠉⣹⠉⠽⣿⣿⠛⠓⢿⠀⡀⣸⠶⢄⡀⠀⠀⠀⠀⠀⠀⠈⣦⣴⣖⣈⡹
⠀⠀⠀⠀⠀⢠⣣⠔⠂⠈⠻⣅⡠⡾⠀⠳⣄⣀⣪⣷⣷⣧⢂⣀⡽⠲⣤⣀⠀⠀⠀⢀⡼⢿⡁⠈⣰
⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠳⢄⡴⠉⠉⢜⣿⣿⡿⠍⠉⢢⡤⠎⠀⠉⠉⠉⠙⣀⡠⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡤⡞⠀⠁⣻⢤⢤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⡇⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⡷⢆⡴⣛⣯⡛⣿⢣⣿⢿⣿⣛⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣯⣴⣏⢼⣿⡿⣣⣿⢸⣿⣌⢿⣿⢸⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣶⣿⣭⣭⣽⣛⣛⣻⠿⠿⢶⣶⣾⣿⡏⣿⣿⣿⣦⣻⣬⣛⣛⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣿⣿⣿⢟⣛⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⡿⣁⣭⣝⣛⣛⡿⠿⠿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡜⣿⣇⠿⠿⠇⠿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⡹⣿⣿⣿⣿⣿⣿⣷⣶⣶⣯⣭⣽⣛⣛⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣹⣿⣷⣶⢰⣿⣦⢻⣿⣿⣿⢟⣵⣿⣿⣿⣷⢻⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⡿⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⢿⣿⣿⣦⣭⣵⣿⣿⡿⣣⠾⡻⣿⡿⠿⢿⣏⢿⣿⣿⣿⣿⣿⡿⢈⣿⣿⣿⣿⢟⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⣭⣻⡟⣿⡞⣿⣿⣿⣿⣿⡿⢫⣾⣿⣿⡇⡿⣱⣿⡆⣿⡞⣿⣷⡰⣶⣶⣶⣿⣿⣿⡟⣵⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣜⡛⢡⢣⣿⣿⡹⣿⣿⣿⢟⣵⣿⣿⣿⢟⣼⡇⣿⣿⡇⣿⣿⡹⣿⣷⣽⣿⣿⣿⣿⣫⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣵⡟⣾⣿⣿⣷⢻⡟⣱⣿⣿⣿⣯⣑⣛⣻⣿⠻⡿⣣⣿⣿⣷⢻⣿⣿⣿⣿⢟⡽⠻⣏⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⣭⡿⢿⣛⣅⣼⣛⣛⣿⠿⠿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣇⢿⣿⡿⣡⣿⢸⡇⢿⣦⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢟⣛⣭⣵⣾⣿⣿⣿⡜⣿⣿⣿⣿⣿⣷⣶⣶⣭⣭⣭⣛⣛⣛⠿⠿⠜⢫⣾⣿⣿⣶⣜⢇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢾⣿⣿⣿⣿⢫⣾⣿⣿⣷⣹⣿⣿⣿⠏⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢟⣴⣮⡻⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⠘⣭⣮⡹⣿⣧⢿⣿⣷⢠⡮⣹⡿⣸⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣮⣝⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣭⣉⣴⣿⣿⡎⣿⣿⠘⣵⣿⡇⡿⣿⣿⡿⣫⣾⣿⡏⣙⠿⣿⣿⣿⣷⣝⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡜⣿⣾⣿⣿⣤⣾⣿⢟⣵⣿⣿⣿⡇⡿⠟⢂⣹⣿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⢹⣿⣿⣿⣿⡿⣡⣿⣿⣿⣯⣶⣶⠾⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣇⢿⣿⡿⣫⣾⣿⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠟⠐⠛⠉



⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠉⠉⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢣⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⣧⡀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⢌⢣⠔⣨⠙⠛⠳⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡼⢉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠑⠤⠀⠀⠠⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⣀⠠⢤⣄⣠⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣤⡤⠄⣀⣀
⠀⢀⡠⠖⠘⠉⠀⠀⠉⠀⣤⡿⠤⠀⠀⠀⣀⣀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢠⠿⣤⠀⠈⠀⠀⠙⠓⠲⢄⡀
⣎⠡⠄⣀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠈
⠈⠑⠂⠭⠴⣍⠣⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⠌⡿⠲⠤⠐⠊
⠀⠀⠀⠀⠀⠈⠳⣌⠻⣤⣾⡅⣬⣹⠗⠀⠒⠶⠶⠒⠀⠀⠀⠀⠀⠐⠒⠶⠶⠂⠐⠺⣏⣥⣨⣧⡼⠋⡡⠎
⠀⠀⠀⠀⠀⠀⡠⣾⢷⣌⠻⣀⠄⠘⠀⠤⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠤⠀⡆⢃⡈⢋⡴⣿⢧⡄
⠀⠀⠀⠀⠀⢰⠱⠘⣷⠀⢃⠛⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣸⠃⡎⠀⢻⠌⡞⡀
⠀⠀⠀⠀⠀⢸⠀⡆⠰⡄⢋⡴⠻⠀⠶⣄⡀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣤⠆⢠⠛⣌⠃⣸⠀⠀⡇⡁
⠀⠀⠀⠀⠀⠨⡘⣧⡀⢹⠺⣁⣼⠀⠀⠁⠉⠸⠇⠋⠉⠀⠀⠀⠈⠉⠏⠚⠈⠃⠀⠀⠈⣦⣘⡿⡃⢠⣧⢡
⠀⠀⠀⠀⠀⠀⡝⣹⠟⢣⡟⢁⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡉⣷⢰⢻⡌⠇
⠀⠀⠀⠀⠀⠀⡇⡸⢀⢆⠛⠮⢹⢱⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠴⠂⠀⠀⠀⠀⠀⢀⠏⡩⠷⢋⡘⣄⢻⠰
⠀⠀⠀⠀⠀⠀⠰⡑⢎⠢⣍⣊⣡⡞⠳⡀⠀⠀⠀⢀⣠⠒⠠⠄⢠⣄⡀⠀⠀⠀⣠⠞⡧⣈⣈⣥⢜⠌⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢡⢩⣷⡜⡈⠆⢸⣄⠀⠀⠀⠙⣍⣠⣀⡽⠃⠀⠀⠀⣠⠁⠴⢰⠱⣾⠁⠆⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠧⠞⠋⣈⢣⠃⡆⢰⣈⠶⣄⠀⠀⠈⠁⠈⠀⠀⢀⡠⠞⣉⡀⢠⠸⠰⠀⠙⠦⠼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣤⢞⣠⡛⣠⡤⠶⠛⢶⣤⣄⣠⡴⡞⠳⢀⣤⡁⣻⣌⡢⣣⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣿⣷⠋⠉⠁⠀⠀⠀⠄⢲⠆⠀⠀⢶⡒⠀⠁⠀⠀⠉⠉⠛⣾⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡿⢸⠀⠀⠀⠀⠀⠐⢛⠋⠀⠀⠀⠀⠻⣀⠀⠀⠀⠀⠀⢀⢩⠇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠉⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠀⠱⣆⡠⠀⠀⠀⠈⠉⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠩⡤⢀⠀⠀⠀⢠⡟⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⠐⢇⠀⠀⠐⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢀⠀⠀⣼⠁⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣋⡅⠸⣀⠁⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⡠⢠⡅⠀⢸⣷
⠀⠀⢠⣤⡄⣀⠀⣀⠾⣙⣿⠇⣠⢇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡼⠀⠀⢸⢾⣁⠀⠀⢀⣀⡀
⠀⠀⠈⢾⣷⡾⣿⡀⠀⣸⣧⡠⠶⠍⠙⠛⠛⠋⠹⠛⢿⣹⡏⠙⠙⠏⡝⢋⠟⡩⣻⠟⠳⢦⣄⣸⠃⠀⢙⣹⡔⠿⠃
⠀⠀⠀⠀⠉⡒⠈⠙⣶⡋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠗⣠⣄⠀⠀⢀⣟⠁⠀⠀⢀⣠⣿⣾⣖⠛⢁⡿⠁
⠀⠀⠀⠀⠀⠳⢠⡴⢿⣀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉⠳⢶⡋⢹⣤⡴⠞⣴⣼⡿⠿⠷⠛⡻⠃
⠀⠀⠀⠀⠀⠀⣸⣤⣤⣿⡁⢨⣔⠺⣿⣿⠿⠿⡙⢷⡮⢔⡠⠀⠀⠀⠀⠀⠹⣿⣿⣿⣺⣦⣀⡘⠃⠀⠀⡇
⠀⠀⠀⠀⠐⠿⢦⡀⠠⡛⢾⣇⣸⣶⠛⠢⢄⣂⣛⣻⣀⡸⣟⣴⣄⢀⠀⠀⠀⠀⠉⠙⠻⠮⣭⣽⡗⠀⠀⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡌⠀⢍⠈⣻⠆⠀⠂⣈⠃⣡⠼⡀⣁⣇⡉⡹⠷⣄⠀⠀⠀⠀⠀⠴⠁⠺⣯⣴⣶⢎⣁⠙
⠀⠀⠀⠀⠀⠀⢀⣇⡠⠴⠒⠚⠉⠁⠀⠀⠀⠉⠉⠀⠀⠈⠉⠀⠉⠀⠀⠉⠛⠴⣂⠀⠀⠀⡀⠐⣾⡿⣟⠉⠉⠣
⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠈⠛⠁⠀⠹⠁⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡔⢀⠐⡀⠀⠀⢀⠀⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢄⣤⠡⠄⢘⣣⡜⡃⠰⠄⣡⣠⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⠤⠓⠀⣠⡒⢂⣀⠴⡧⢁⡠⢒⣨⡠⠒⠤⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣚⠥⢒⡀⣿⠠⣀⡍⠀⢜⡧⠀⢈⣁⠀⢹⢈⡒⠩⢒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡉⠘⣋⣁⠉⠲⡑⢌⠄⢔⡧⠤⡉⢈⡖⠉⣌⣝⠃⢉⣓⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢖⡁⡆⢔⠂⠰⠇⠂⡙⠠⢘⡃⠄⠋⠀⠜⠇⡐⣂⢰⢀⣲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢈⢸⣿⣿⣿⣷⣾⣷⣷⣷⣿⣿⣾⣾⣶⣿⣿⣿⣿⣿⡧⡁⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣱⣿⢞⡇⡈⢹⠩⠀⣸⠀⠀⠀⠀⠀⠀⠀⠠⠍⣏⢠⢀⡓⣼⢆⠔⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣉⡌⠀⡌⡗⢀⠃⠀⢠⠙⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⡄⢺⠡⠀⢡⣈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢃⡄⢫⡔⢡⠈⠑⢶⠦⠤⠀⠀⠀⠀⠠⠤⠤⠊⠁⢌⠢⡝⢀⡐⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢏⡟⢮⡆⣼⣠⡆⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⢱⡴⢻⡙⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠠⠙⠝⣸⡀⠊⠀⠀⠀⡐⢂⠀⠀⠀⠀⠀⣤⠫⠋⢄⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢊⠊⢳⡌⠁⠁⡀⠀⠀⠻⣐⣀⠚⡀⢰⣀⠈⠌⢕⣞⠁⡁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠁⠣⠀⢀⣉⡦⢄⠀⠁⠈⠀⣽⡼⢉⡀⠄⢄⠈⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⠩⠀⠀⠈⠉⠖⠒⡉⠁⠀⠀⠉⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡙⣙⢀⡀⠀⠀⠀⠈⠀⠀⠱⡀⠀⣀⠦⠀⣉⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠉⠰⠀⢀⠌⠀⠀⠀⠀⠑⡀⠀⢰⡆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠘⡀⠀⠀⢀⠊⠀⢊⠆⢠⠒⡆⠐⡄⠀⠁⠀⠂⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢼⠚⢣⠀⠀⠀⠀⠀⠓⠂⠘⠒⠃⠀⠈⠄⠀⡘⢠⡥⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠯⣰⠤⠂⠉⣶⡜⠀⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠁⠈⠷⠉⠐⠤⣆⠽⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡭⠛⠦⣼⢿⣵⠈⣉⠡⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠔⢉⡀⠂⣤⣆⠴⠛⠭⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⣞⠉⢺⡉⠰⠓⠓⠘⠂⠀⠀⣷⣾⠀⠀⠀⠡⠖⣤⣴⢿⣿⡿⣥⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⡵⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢮⣴⣾⡯⢼⡀⢀⣷⡀⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡉⣄⠀⣀⠀⣰⠖⡆⠀⢄⠀⠀⠀⠀⠙⠻⣅⠰⣉⡁⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢍⠀⠀⠦⢈⡏⠱⣄⡤⠛⠺⣿⣦⢄⠀⠀⠀⠀⠉⠑⠐⠃⢀⡈⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠠⠀⠊⠀⠀⠱⠤⢼⠁⠉⠧⠬⠦⢄⡀⠀⠀⠀⢸⢿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠁⠀⠀⠀⠀⠉⠀⠈⠁⡀⠁

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

⢀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⣸⣿⡅⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢸⣿
⣿⣿⡇⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣼⡆⠀⠀⠀⠀⢰⣧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⣿⣿⡇⠀⢸⣿⣧⠀⠀⣀⠀⠀⠀⠀⡀⠰⣆⢠⡆⢸⣿⣄⣤⣴⣿⣧⣠⣤⣤⣄⣼⣿⣤⣤⣠⣿⡇⢰⡄⣰⠆⢀⠀⠀⠀⠀⣀⠀⠀⣿⣿⡇⠀⣸⣿
⢻⣿⣿⣦⣼⣿⣿⡀⣼⣿⠀⠀⠀⠈⣿⣶⠋⠻⣿⠋⠙⣯⣠⣿⣿⣿⡿⠉⠉⢻⣿⣿⣿⣤⣽⠋⠙⣿⠟⠙⣶⣿⠁⠀⠀⠀⣿⣇⢀⣿⣿⣧⣴⣿⣿
⠈⠻⣿⣿⣿⣿⣿⣷⣿⣿⣶⣆⠺⠃⠈⢿⡀⠀⣿⠤⡶⢿⡍⡿⠁⠈⠻⣀⣀⡾⠉⠈⢿⢹⡟⢶⣤⣟⠀⢠⡿⠁⠘⠗⣰⣷⣿⣿⣾⣿⣿⣿⣿⣿⠟
⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⡳⡿⠃⢲⣄⣠⡾⠛⣦⣤⣽⣿⣿⣾⣤⣴⠛⢷⡄⣠⡶⠚⠷⢺⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⠟⢿⡁⢹⣷⣤⣼⣿⣿⣷⡎⠀⠀⣩⠿⠿⣅⠀⠀⢱⣴⣿⣿⣦⣤⣾⡏⠈⡿⠿⣿⣿⡿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠧⣤⣸⣤⢼⣿⣿⣇⠀⣨⠟⠻⣇⠈⣳⠊⣍⠀⠀⣯⠒⣶⠁⣸⠟⢻⣅⠀⣸⣿⣿⣧⣤⣜⣤⠼⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⣩⡄⣼⠛⢻⣄⣹⠫⣄⣠⡬⣿⣵⠋⠉⣰⣆⠈⠳⣼⡯⢥⣄⢴⠋⣯⢠⡟⠛⣧⢀⣍⣓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢉⡤⠿⢛⡄⠀⣯⢼⡾⠋⣩⡴⠋⣁⠴⠊⢹⡍⠓⢤⣈⠓⢤⣉⠚⢷⡥⣬⠃⢴⡻⠿⣄⡙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⣀⠴⠋⣀⡜⢁⣠⠴⠋⣡⡔⠊⠁⣀⡴⠋⠙⢦⣀⠉⠛⠦⣉⠓⢦⣀⡉⢦⡀⠙⢦⡀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⢁⣤⠚⠉⠁⣉⠡⠔⠋⣁⠠⠤⠚⠉⠀⠀⠀⠀⠉⠓⠤⠄⣈⡑⠦⢌⣁⠈⠙⠓⣄⠉⢂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡍⢀⡾⣷⠓⡿⠋⡟⠓⠂⢈⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣉⠁⠒⠚⢿⠙⠗⠚⣿⢧⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢔⠯⡻⠃⡴⡇⢠⠋⢸⠀⠞⠋⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⠈⡄⠻⡀⣿⢆⠹⡟⡽⢢
⠀⠀⠀⠀⠀⠀⠀⠺⠘⠀⠃⣴⠹⢻⠎⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢳⠛⡎⣁⢸⠀⠇⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠇⣰⠀⣯⡸⠏⡰⠛⣄⡴⢦⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣠⠶⣆⢨⠛⣄⠻⣇⡿⠀⢀⡂⡇
⠀⠀⠀⠀⠀⠀⠀⢲⠰⣇⡄⠈⡧⣞⠃⣰⠃⠀⠊⢩⡟⢳⡖⢻⠛⠏⠀⠀⠀⠀⠙⢻⠗⢲⠛⠟⠉⠃⠀⠸⣄⠸⢧⡜⠁⢰⡜⠇⡗
⠀⠀⠀⠀⠀⠀⠀⠈⡳⢟⣷⣦⢱⣾⠊⣩⡀⠀⠀⢸⡇⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢰⣍⠳⣯⡜⣴⢎⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⡇⣀⢹⣇⣸⣾⢧⠀⠀⢸⡇⢸⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢾⡷⣆⣸⠇⣿⠸⡇⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠱⡉⢦⣈⢋⡤⢁⣤⠀⢸⡇⢰⡇⠀⠀⠉⠛⠓⠚⠙⠁⠀⠀⢸⠀⡇⠀⠀⣿⠐⣄⣋⣁⠴⡩⢆⠷⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠳⣌⢶⣤⡄⣴⢾⡙⢧⠘⡇⢨⡇⠀⢀⣼⣉⣉⣉⣙⣤⠀⠀⢈⠀⠇⢀⠞⢉⠷⣤⢠⣴⠞⣡⢊⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠈⡆⢹⡷⠀⡆⢓⡀⢾⡇⠰⡇⠀⠈⠣⣄⣀⣀⣤⠏⠁⠀⢠⠀⣢⡟⢠⠋⡘⣠⢷⡉⠸⢠⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣮⣤⠷⠋⢠⠀⡇⢈⠃⣟⠑⢶⡇⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⣠⡶⠋⣥⠸⠀⢃⠙⡀⠹⠶⣄⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠖⣠⠏⣰⢁⣴⠎⠙⠶⣤⣀⠀⠀⠀⢀⣀⡴⠞⠉⢳⢦⣈⣄⠳⡌⣤⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢮⡵⠞⠋⢠⡜⣷⠰⠒⠊⠉⢙⠗⠻⡉⠉⠐⠒⢶⡛⣧⠈⠙⠷⣬⣳⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⡿⠖⠈⠉⠁⠉⠁⠀⠀⠀⢠⠏⠀⠀⠱⡄⠀⠀⠀⠉⠁⠈⠉⠁⢺⠛⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⢱⣀⡆⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⢸⣀⠇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣧⠀⠉⠁⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⡄⠀⠀⠀⠀⠀⠀⠀⣰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡇⠀⢣⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⡜⠀⢸⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠻⡇⠀⠘⡀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⣠⠃⠀⢸⠟⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢶⡇⠀⠀⢛⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠟⠀⠀⢸⡖⢻
⠀⣶⣖⣶⠖⣤⣄⡀⣠⠜⠻⣤⡇⠀⠀⠈⡇⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠸⠀⠀⠀⢸⣤⠟⠣⣤⣀⣠⣤⠒⣶⣲⣶
⠀⠙⣾⠛⠳⣬⣍⣿⠁⠀⠀⢸⡇⠀⠀⠀⣸⣏⠀⣀⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⣀⠀⢀⡀⠀⢀⠀⣾⣇⠀⠀⠀⢸⡇⠀⠀⠀⣿⣩⣤⠞⠛⣷⠋
⠀⠀⠈⠛⠷⣍⡽⠿⣷⣴⢿⣦⣇⠠⠔⠊⠉⠙⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠉⠓⠢⠄⣸⣴⡶⢷⣾⠿⢯⣩⠾⠛⠁
⠀⠀⠀⠀⢰⡏⠀⠀⣠⣧⣀⠭⢙⣧⡔⠻⠂⠀⣀⣉⣦⣀⡀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣀⣴⣋⣀⠀⠀⡾⠢⣼⡛⠩⢀⣨⣧⠀⠀⢹⡆
⠀⠀⠀⠀⠈⠳⣄⣠⣿⣟⠁⠀⣈⡽⠿⠛⠛⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠾⢿⣁⠀⠀⣹⣿⣇⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠉⢨⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢾⣯⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠻⣄⣈⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴⠶⣤⡤⠤⠤⢤⣤⡴⠦⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⣄⠀⣀⠴⠖⠻⣏⠉⣽⠟⣿⠃⠀⠀⠈⠷⣠⣆⣸⠏⠀⠀⠈⣿⠻⣯⠉⣹⠛⠲⢦⣀⠀⣠⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣅⣀⠤⠤⠼⣿⡟⠀⠈⠳⣤⣄⣙⣿⣻⣿⣳⣞⣀⣀⡴⠏⠀⢹⣿⠧⠤⠤⢀⣩⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣸⠷⣀⣠⠞⠉⠉⡭⢿⣿⣿⡾⢧⠉⠉⠳⣄⣀⠾⣇⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⣀⣠⡀⠀⠀⡴⠃⠀⠀⠹⣆⠀⠀⣠⡟⠹⠇⢻⣆⠀⠀⢠⡟⠀⠀⠘⣦⠀⠀⢠⣄⣀⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢹⠀⡜⠁⠀⠀⠀⠀⠈⠛⠛⢹⡇⠀⠀⢸⡇⠙⠙⠋⠀⠀⠀⠀⠈⢣⠀⡏⠀⠀⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⢀⡀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⣉⡠⠤⠔⠒⢋⡝⢫⠉⠒⠢⠤⢄⣉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠛⠉⠉⠀⠀⠀⠀⢀⠞⠀⠀⢣⡀⠀⠀⠀⠀⠉⠉⠛⠓⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡇⠸⡀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⠀⢠⠃⢸⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣌⡇⠀⢣⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡜⠀⢸⣡⡇
⠀⢠⣶⣶⠤⣄⣀⡤⠾⢬⣇⠀⠘⡄⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢰⠁⠀⢸⡥⠷⢤⣀⣠⠤⣖⣶⡄
⠀⠀⠳⣋⣳⠧⣹⡀⣀⣿⡟⡆⣀⠽⡥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢼⠧⣀⠀⢸⢃⣀⢀⣏⠼⣞⣙⠞
⠀⠀⠀⢈⣱⡎⣉⢻⡭⢏⣿⠯⢤⡀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⣤⣙⡾⠞⠋⡟⠉⠑⡏⠁
⠀⠀⠀⠀⠻⣷⠾⣿⣀⢀⡿⠔⠊⠉⠁⠀⠀⠈⠉⠉⠒⠢⢄⡀⠀⢀⣴⣷⠀⣠⣴⣷⣾⡋⠉⠉⣿⣇⡠⠇
⠀⠀⠀⠀⠀⠀⡏⠀⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢟⡵⣿⣿⣋⣟⡻⠮⠷⠖⠋⡏⣧
⠀⠀⠀⠀⠀⠀⠈⢳⠋⢳⡄⠀⣀⠤⠶⢲⢶⡶⠢⠤⣀⠀⠀⠀⠀⠀⠙⠣⣶⠏⢯⣀⣀⠀⠀⠸⡖⠁
⠀⠀⠀⠀⠀⠀⣠⣜⠋⠁⢳⡜⠙⢦⠴⡟⠈⠳⣄⣀⣺⣿⣦⣀⠀⠀⠀⠀⠀⠉⠙⠒⠀⡗⠀⣀⡩⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⠈⠉⢙⠆⠳⠤⡞⠉⠁⢞⡿⡿⡛⠗⠢⣀⠀⠀⠀⠀⠀⠀⠘⣿⡿⡒⠪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣜⡠⠔⠒⠉⠉⠀⠀⠀⠙⠤⠤⡟⠀⠁⢻⠦⠤⠞⠉⠒⠦⠤⠤⠤⠔⠉⢹⠤⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠶⠶⠶⠖⠒⠶⠶⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⣄⣤⠞⠋⠉⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⣠⣾⠟⠁⠀⠀⠀⣁⣠⡤⠶⠖⢚⡛⠛⠲⢦⣄⣀⠈⠁⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡼⣻⠋⠀⠀⠀⣴⢞⣭⠶⠚⠛⠉⠉⢉⡍⠛⠓⠶⢯⣝⡓⠶⣄⡀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⠟⣰⠃⠀⠀⠀⣾⡷⠋⠀⠀⠀⠀⢀⣀⣈⣀⡀⠀⠀⠀⠈⠙⠳⣌⡻⣦⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⣼⠃⠀⠀⠀⢠⡟⢰⡏⠀⠀⠀⣸⡟⠁⠀⠀⢠⡴⢛⣉⣭⣭⣭⣉⣛⠲⢤⡀⠠⣄⠈⠳⣌⢳⡀⠀⠀⡀⠀⢻⡀
⠀⠀⠀⢠⡇⠀⠀⣦⠀⣾⠀⣼⠀⡾⠃⢰⣿⠀⠀⠀⢠⡟⣰⠋⠁⠀⠀⠀⠈⠉⠻⢦⡙⣦⠀⠀⠀⠹⡎⣇⠀⠘⠃⠀⠈⣧
⠀⠀⠀⢸⠃⠀⠀⠈⠁⡇⠀⣿⠀⠀⠀⣿⣧⠀⠐⠋⠸⡇⢿⣀⣀⣠⣿⠀⠀⣠⠀⠈⢷⠸⡆⠀⠀⠀⣿⢿⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⢿⠹⡄⠀⠀⠀⠙⠶⢭⣭⠵⠋⠀⠀⠁⠀⠀⢸⠇⡇⠀⠀⢀⡟⢸⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠸⡇⠀⠀⠀⠀⢿⠀⢹⡄⠀⠀⠘⣇⠙⢦⣀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⢀⣠⠟⢠⡇⠀⠀⡼⢡⠟⠀⠶⠂⠀⠀⣿⡇
⠀⠀⠀⠀⢿⡀⠀⠀⠀⠘⣇⠀⠻⣄⠀⢿⠈⠳⢦⣉⡛⠶⠦⢤⣤⣤⡤⠴⠶⠚⣋⣁⡴⠋⠀⢠⡞⣱⠏⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠙⣦⠀⠈⠳⣬⡀⠀⠀⠈⠉⠛⠓⠒⠶⠶⠶⠶⠶⠛⠉⠁⢀⣠⠾⢋⡼⠃⠀⡆⠀⠀⠀⢀⡟⣸
⠀⠀⠀⠀⠀⠘⣿⣄⠀⠴⠋⠈⠻⢦⣀⡀⠙⠓⠶⣤⣄⣀⣀⣀⣛⣀⣀⣀⣠⣤⠴⠞⣋⣡⡴⠋⠀⠀⠀⠁⠀⠀⢀⡾⢁⡟
⠀⠀⠀⠀⠀⠀⠸⣏⠳⣄⠀⠀⠀⠀⠈⠙⠓⠶⠶⠤⣤⣬⣭⣭⣭⣭⣭⠥⠴⠶⠚⠛⠉⠁⠀⠼⠀⠀⠀⠀⠀⣠⠞⠁⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⠳⢦⣀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣀⣠⡴⠞⠋⠁⣀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠉⠛⠲⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠶⠶⠒⠛⠋⠉⠀⢀⣠⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠶⠶⠶⠶⠤⠴⠶⠶⠶⠒⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣨⡷⢊⣁⣀⡉⠒⠶⣶⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠇⠀⠻⠷⠬⠯⠿⠟⠀⠀⢈⣿⡆⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⠉⠻⢿⣧⣀⣀⡀⠀⣀⡀⣀⣤⣴⡾⠋⠀⢀⢈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠂⣹⣿⡇⠈⢿⣿⠂⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢀⣠⣿⠿⠛⠉⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡃⢸⣿⣿⡇⠀⠈⠹⣧⣵⣄⣀⣀⣀⢀⣀⣀⣀⣠⣤⣴⡷⠾⠿⠿⠁⠀⠀⡀⠨⣷⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⡇⠸⣿⣿⣦⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡀⠁⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣴⣿⡿⠇⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠉⢻⢷⣗⣠⣄⡄⠀⠀⠀⠀⡀⠀⢀⠀⢀⣀⣠⣤⣤⣦⣶⣶⡿⠿⠟⠋⢀⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠀⠀⠀⠀⠀⠉⠻⠿⣿⡶⢴⠶⠷⠷⠾⠟⠿⠛⠛⠋⠹⠉⠂⠁⠀⠀⠀⢀⣴⣼⢣⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⢰⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⢸⡟⠁⢮⣴⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⡆⢸⡏⠁⠀⢻⡄⢠⣶⣤⡄⠀⠀⣀⣦⠄⠀⠀⢀⣄⣀⣀⠀⣰⠟⠛⣇⠘⣇⢈⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣼⡇⢸⠇⠀⠀⢸⡇⣼⡟⠁⣧⠀⢠⣿⣿⡗⠀⢰⣿⠟⠋⠉⠉⠁⠀⠁⢻⡀⣿⠠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣾⠀⠀⠀⢸⡇⣿⠃⠀⢸⠀⣼⠟⠉⢻⡄⣸⠏⠀⠀⠀⠍⠀⠀⢀⣸⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣮⣀⠲⣾⠁⣿⠀⠀⡹⣶⣟⣀⡀⢈⣿⢟⢀⠀⠀⢀⣤⣾⢧⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⣽⣿⣾⣿⣷⣧⣽⣯⣿⣿⣿⣮⣦⣤⣼⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⣀⡤⢴⣾⣿⠶⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠀⡠⢊⡗⡡⠂⣠⠔⢛⣏⣛⡒⠈⢽⡳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⡟⠀⠀⢰⠁⣿⠰⡇⠀⡇⠀⠈⠉⠉⠙⡧⠀⠹⣦⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⣿⡇⠀⠀⢸⡀⢿⡀⠻⣄⠈⠀⠰⠴⠶⠊⢀⣀⣠⡿⠀⢳⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⢧⠀⠀⠘⣷⡈⠯⣰⡈⠉⠒⠂⠀⠤⠾⠿⠟⠉⠀⠀⣿⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠹⣿⣦⠀⠀⠹⣿⡄⠈⠛⠶⣤⡀⠀⠀⠀⠀⠀⣀⣠⠖⠋⢰⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢷⡄⠀⠙⠿⣷⣤⠀⠙⢷⣷⣄⠀⠀⠉⠛⠛⠛⠛⠛⠋⠁⠀⣶⢏⡴⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠅⡄⠀⠀⠈⠛⠷⣄⡀⠙⠿⢽⣷⣄⣤⣲⣶⣤⣦⡤⠴⣫⡴⠛⣄⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠒⠀⠀⠀⠀⠀⠉⠐⠲⠤⡤⢤⣭⣭⡭⠥⠴⠖⠋⣥⠖⡃⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠⡄⠴⠖⠀⠀⠀⠀⠁⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠐⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢠⢶⠄⠀⢰⣄⢀⣶⣦⠀⡦⠀⠀⣄⠠⣆⠀⣰⠀⡴⠒⡆⣴⡀⢰⡄⠀⠀⢠⠒⢦⢀⡔⠲⢰⡄⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⠘⢶⡟⡇⠳⡇⠀⠀⡏⠀⠈⠶⢹⢠⡅⠀⡇⣿⠘⢾⠃⠀⠀⢸⠀⢸⢨⠀⠀⡇⡇⠀⢸
⠀⠀⠀⠀⠘⠇⠀⡇⠀⠀⡀⢸⠀⠈⡧⡇⠀⡇⠀⡀⠀⠀⠀⠀⢸⢸⡇⠀⡇⣸⠀⢸⠀⠀⠀⢸⢀⠌⢸⠀⢸⢠⠃⠀⢸⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⢸⠀⠀⡇⠁⠀⠀⢠⠃⠆⢸⠀⠀⢸⠈⡇⠀⡇⢸⠀⢸⠀⠀⠀⣼⢧⠀⢸⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠚⠀⠃⠘⠀⠀⠃⠀⠀⠉⠀⠀⠁⠈⠀⠀⠘⠀⠃⠘⠁⠘⠀⠈⠁⠀⠀⠛⠀⠃⠈⠂⠊⠀⠂⠀⠘


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠒⢒⠒⠒⠒⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠉⢀⣴⠾⠛⠋⢉⣉⣉⡙⢉⠙⠲⢦⣄⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⡵⠃⠀⣶⡟⠀⢠⡾⠛⠛⣉⣉⣙⡛⠻⣶⣀⠈⢷⣄⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢸⡇⠀⢰⣿⡁⠀⣿⣁⡀⠈⠛⠿⣿⣷⡄⠈⣿⡀⠸⣿⣿⡆⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢸⣇⠀⢸⣿⡅⠀⠘⠿⢿⣿⣶⣶⣾⠟⠐⢀⣾⡇⠀⠈⣿⣿⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣿⡤⠀⢻⣿⡄⡂⠀⠀⠀⠉⠉⠀⠀⢰⣿⣿⠃⢀⠈⢧⣿⠀⠠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠈⢿⣾⡁⠀⠻⢿⣷⣷⣶⣤⣤⣤⣶⣶⣾⡿⠃⢠⠞⣼⣿⡟⠀⡂⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⡈⢿⣷⣬⡂⢄⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠰⣯⣾⣿⢟⠂⡰⢃⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠈⠀⠙⠿⣿⣶⣦⣌⣢⣤⣀⣰⣤⣤⣦⣴⣿⣿⠟⠋⣠⣾⢇⣽⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠈⠛⠿⠻⠿⠿⠿⠿⠿⠛⠛⠛⣉⠤⠒⠛⠱⣭⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠁⠀⠀⣀⢤⡴⣞⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣠⣀⡤⠤⠤⠤⡄⢠⣄⣀⡄⣠⣥⣾⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢠⡖⠉⠀⢠⡟⠀⠀⠀⢀⡧⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠑⠀⠀⢻⡀⠀⠀⠙⠢⠄⠒⠋⠀⠀⠀⢸⡏⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠀⠘⢧⡀⠀⠀⠈⠒⠦⣤⢄⣀⣀⣀⣀⡤⠀⠊⠀⠀⠀⡝⠈⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠈⢆⠀⠀⠀⠘⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠀⢀⡿⠈⠢
⠀⠀⠀⠀⠀⠀⠀⡰⢻⡀⠀⠈⠳⣀⠀⠀⠀⠀⠉⠋⠀⠠⠦⠤⠤⠴⠤⠀⠒⠂⠀⠀⠀⠀⠈⠀⠈⣴⢣
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠣⡀⠀⠀⠀⠉⠒⠦⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠂⠁⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⡞⣦⡀⠈⠢⢀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⠀⡠⠲⡄⡀
⠀⠀⠀⠀⠀⠀⢰⡇⠈⠉⠳⣄⠀⠉⠑⠒⠄⠀⠀⢀⢀⣀⣀⡀⣀⡀⠤⢀⡀⠀⠒⠘⠋⠃⠀⢠⠃⠀⢃⡃
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠘⠷⠖⠂⠉⠀⠲⣄⠈⠉⠁⠀⡀⠀⡠⠔⣄⠀⢀⣀⠀⠀⠀⠀⣼⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠘⠉⠋⠀⠀⠸⠛⠋⠀⠻⣦⣴⢶⡏⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠐⠐⠹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢂⣀⣩⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣴⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠒⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⢉⣥⢖⠪⠛⡿⠿⢛⣛⢓⡲⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠅⠀⣰⢋⡴⢣⡞⠁⡴⠚⠉⣭⣍⣩⡁⠒⢮⣽⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠁⠀⠀⢰⡅⣾⡇⢾⠀⠸⡁⠀⠀⠉⠈⠉⢛⡆⠀⠙⣷⡈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣼⣿⠀⠀⠀⢸⡇⢿⣇⠘⣧⡀⠈⠀⠲⠦⡶⠶⠋⠀⠀⢀⣿⠇⠈⡎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⣿⡀⠀⠀⠸⣧⡈⢿⡁⣄⠙⠲⠤⢀⡀⠀⢠⣶⣿⡿⠟⠁⠀⢰⣿⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣧⣟⡀⠀⠀⢻⣿⣆⠙⠲⢦⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⣼⠏⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣧⠀⠀⠻⣿⣿⣄⠀⠀⢻⣿⣀⡀⠀⠉⠛⠦⢤⣤⣤⣀⣤⣤⣼⠶⠋⠀⣠⡾⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⢻⣷⠀⠀⠈⠛⢿⣷⣄⠀⠙⢿⣿⣦⣤⡀⠀⠀⠀⠉⠁⠀⡁⠀⠀⢀⡼⢟⣵⠏⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡦⠰⠀⠀⠀⠀⠉⠛⠷⣤⡀⠉⠛⠿⠿⣧⣶⣬⣿⣯⣴⠷⠾⢚⣭⡶⢛⣁⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠄⠀⠐⠃⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠤⠴⠦⡤⠤⠤⠶⠶⠛⠉⠠⣴⢚⠁⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣴⠶⡦⠖⠀⠀⠀⠀⠩⠀⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠐⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣐⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⠤⠄⠀⠀⠀⠀⠤⠤⠼⠛⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⡦⠀⠀⣶⡄⠀⡦⣶⡀⢰⡆⠀⠀⢰⠀⣶⡀⠀⣶⠀⣴⠚⢲⢰⣦⡀⢰⠀⠀⠀⠀⡖⠒⡆⢠⡖⠒⡆⣶⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⠘⢦⡇⡇⠙⣬⡇⠀⠀⣿⠀⡇⢱⡌⢸⢀⡁⠀⢸⢸⡇⠳⣼⠀⠀⠀⠀⡇⠀⣿⣘⠁⠀⡇⣿⠀⠀⣿
⠀⠀⠀⠀⠀⢶⠆⠀⡟⠀⠀⠀⠀⣿⠀⠈⡇⡇⠀⠸⡇⠀⠈⢸⠀⡇⠀⠀⢸⢸⡇⠀⢸⢸⡇⠀⢹⠀⠀⠀⠀⡇⢀⡇⣿⠀⠀⡇⢿⠀⠀⣿⠀⠰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⣿⠀⠀⡇⡇⠀⠀⡇⢀⡁⣸⠀⡇⠀⠀⣼⠘⡇⠀⣾⢸⡇⠀⢸⠀⠀⠀⢰⣷⡋⠀⣿⠀⠀⡇⠈⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡆⢸⠀⣿⠀⠀⡧⡇⠀⠀⡇⠎⠀⢸⠀⡇⠀⠀⣿⠀⢇⢀⡿⢸⡇⠀⢸⠀⠀⠀⢸⡅⠳⡄⠹⡀⢠⠇⢠⢀⠀⣧⢀




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠈⠘⠀⠀⠀⠐⠀⠂⠘⠀⠀⠀⠀⠂⠀⠀⠀⠠⠈⠁⠄⠀⠀⠀⠘⠠⠀⠘⠂⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣶⣾⢖⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣞⣼⣿⣿⣿⣿⣿⣿⢿⣷⣽⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣆⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣼⣿⠁⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣧
⠀⠀⠀⠀⠀⠀⢠⢏⣿⣿⣿⣿⣿⣿⣿⣿⣞⣯⣿⣽⣾⣿⣿⣷⣤⣄⣀⠀⢀⣠⣴⣬⣽⣿⣿⣷⣿⣿⣴⣿⣿⣿⡿⢉⣿⠛⣿⣿⣿⣧⣾⡆
⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣿⣿⡿⣿⣿⡿⣿⢸⣿⣿⣿⣿⣿⣾⣯⣿⣿⣿⣟⣭⣷⣿⣟⣯⣭⣭⣭⣿⣿⣿⣿⣷⣶⣿⡿⣡⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣟⢿⡝⡝⣿⣿⣷⣶⠷⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⢿⣿⣿⢻⣿⣿⣿⡵⢻⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣮⣻⣶⣿⠟⠉⣶⣮⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣆⣾⣿⣿⣧⠻⣿⣧⣔⣛⣵⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣿⣅⠀⣿⣿⣿⣿⣿⣿⣿⢟⡯⣙⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⡇⡙⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣿⣿⣿⣿⣿⣯⡿⣫⣾⣿⣿⣿⠿⠛⠻⢿⣿⣿⣿⣿⣿⡻⣷⣝⢿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢫⣾⣿⡿⣻⠟⠁⠀⠀⠀⠀⠙⢿⡻⣿⣿⣿⣌⠻⣎⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣳⣿⣿⢟⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡝⣿⣿⣿⣧⣹⡎⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣻⣿⣿⣿⣿⣯⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡜⣿⣿⣿⣿⣿⢸⣿⣿⣿⢫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⣿⣿⣿⣿⣾⣿⣿⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣷⣿⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⣿⣿⡏⡽⠙⠛⠿⢿⣿⣷⡀⠀⠀⠀⠀⣰⣾⣿⡿⠟⠻⣟⣿⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⢻⣿⣿⣇⡇⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠹⣿⣧⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⢻⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⣿⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣤⣤⣴⣾⣿⠓⠀⠀⠀⠀⠀⠐⢻⣿⣶⣦⣤⣴⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠻⠟⠻⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠛⠟⠟⠉⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣯⠉⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⡀⢀⣶⣶⣶⣶⣶⣶⠒⠒⠒⠒⠒⠋⠀⠀⠙⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⣠⣀⡀⢸⣿⣿⣿⣿⣿⣧⣀⣀⣼⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⡇
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⣿⣷⡀⠀⠀⢠⡞⠉⠙⠓⠚⠁
⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⡿⠁⠀⠀⠘⢧⣀⣠⠴⢦⡄
⠀⠀⠀⠀⠈⠉⠉⠁⢸⣿⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⢠⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠃⠀⠀⠘⠛⠛⠛⠛⠛⣤⣤⣤⣤⣤⡄⠀⠀⢠⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⢀⣿⣿⣿⣿⣿⡇⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠖⠋⠉⢳⡄⠀⠀⢀⣾⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣄⣀⡼⠃⠀⠀⠈⢿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⣿⣿⣿⡟⠉⠉⢻⣿⣿⣿⣿⣿⡇⠈⠉⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧⠀⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠤⠤⣄⠀⠀⣠⠤⠤⠤⠤⠤⠿⠿⠿⠿⠿⠿⠁⠈⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⢿⢿⣿⣿⢷⣶⣶⢶⢶⣶⣶⢶⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢈⣿⣘⣻⣭⣧⣮⡧⠧⠿⠼⠾⠾⠾⠾⠾⠾⠼⠿⠴⣼⣼⣼⣼⣸⣝⣿⡁
⠀⠀⠀⠀⠀⣀⣬⣭⣁⣀⡀⠀⠀⠀⠀⠀⠀⠠⠄⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣌⣍⣡⠄⢀⡀
⠀⠀⣶⢱⡿⢯⣝⣫⡿⣹⣿⣤⡀⠀⢀⠐⣧⡈⠦⠀⠐⠡⠀⡸⢀⠀⠀⠀⣴⣿⡅⠟⢨⢁⡺⢷⢁⣠
⠀⡆⢧⡈⠙⠒⠲⠶⠞⠛⠉⢠⡇⠀⢘⡧⡈⠓⠤⠀⠀⠠⢆⡀⣈⠀⠀⢸⣧⡌⠜⣀⠋⠞⣱⠀⢿⠷⡇
⠀⡇⢨⠙⠓⠶⠤⠤⠴⠶⠛⠫⠆⠀⢸⣿⠈⢀⣄⠀⣠⣄⣉⣼⠻⡇⠀⣟⢿⢉⣬⣻⣘⡿⣩⣰⣿⠀⣿
⠀⠙⢮⡀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠘⢯⡓⠻⣟⠾⠃⠀⣸⣧⣤⠇⠀⠹⣾⣸⠋⢻⠟⢳⡛⠛⣿⣸⡿
⠀⠀⠀⠈⠑⠒⠒⠒⠒⠒⠈⠀⠀⠀⠀⠀⠉⠛⠻⠷⠶⠿⠟⠋⠁⠀⠀⠀⠈⠙⠳⠿⠷⠿⠷⠞⠛⠉
⠀⠀⢀⡴⠞⢉⣉⡉⠈⢑⣦⡀⠀⠀⠀⣠⠖⠉⠁⣀⣀⣈⠉⠖⢄⠀⠀⠀⢀⡄⠀⣙⣛⣋
⠀⠠⠊⡠⢺⡓⠂⠁⠓⣹⣿⡜⢶⡀⢸⠁⢀⡾⣫⣽⣿⣭⣝⢶⠀⣧⠀⣰⢏⠀⠈⠁⠀⠀⠁⢀
⠀⣂⠀⣧⣿⠡⢟⠗⢀⡷⡏⢃⠀⡇⠀⠀⡞⣼⣿⣿⢫⢻⣽⡾⡇⢸⠀⣧⡎⠀⢠⠞⢉⡉⢢⠀⠃⢀⣆
⠀⢸⡆⠆⢬⠐⣷⡤⡾⠕⡡⠛⢠⠃⢀⠀⣷⢻⡿⣿⣿⣿⠿⣹⠇⠸⠀⠻⠇⡀⢸⠀⠘⠀⠂⢀⡎⢸⡇
⠀⠀⠻⣆⡃⠷⠀⡄⠴⠞⢋⡤⠃⠀⠘⣖⠘⢧⡙⠷⠶⠶⠞⠋⢀⠆⠀⠀⠂⠙⠄⠁⠀⠤⠴⣫⢴⡞⠁
⠀⠀⠀⠈⠙⠣⠶⠦⠤⠒⠋⠀⠀⠀⠀⠈⠓⠢⠭⠗⠒⠂⠐⠊⠁⠀⠀⠀⠀⠀⠚⠦⠤⠾⠇⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣴⠞⢋⣩⡴⠶⢖⠶⠶⣭⡙⠻⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠋⣡⣴⠖⢛⣩⣭⣍⣙⠓⢦⣌⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⣰⣿⣏⠀⢾⣿⡀⠺⣿⣥⣴⠿⠋⣀⣿⣿⣷⡄⠀⠀⠀⢸⣿⣿⣿⠏⠀⢸⣿⡁⠐⣿⣅⡐⠛⢛⣀⣼⡿⠀⡀⠉⢻⣿⣿⣿
⠀⠀⠀⣿⡏⠻⣦⣄⡙⠛⠷⠶⠶⠶⠶⠟⢛⣡⡾⢻⣿⠀⠀⠀⢸⣿⣿⣿⠀⣶⣄⠙⠿⣶⣤⣍⣉⣉⣉⣉⣥⡴⠞⢁⡄⠈⣿⣿⣿
⠀⠀⠀⢻⣧⠀⠈⠙⠛⠿⠶⠶⠶⠶⠶⠿⠛⠋⠀⣼⡏⠀⠀⠀⢸⣿⣿⣿⣆⠸⣿⣷⣦⣤⣀⣉⣉⣉⣉⣉⣁⣤⣶⣿⠃⣸⣿⣿⣿
⠀⠀⠀⠀⠙⢷⣦⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡾⠋⠀⠀⠀⠀⢸⣿⣿⣿⣿⣦⠈⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣁⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠷⠶⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣤⣤⣬⣉⣉⣉⣩⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⠤⠤⣤⠤⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢖⠒⠒⠉⠉⢉⡡⡠⡄⡄⢀⠀⠠⠄⠘⡈⢆⢠⠀⡈⠉⡒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⡠⠋⡔⡠⠀⠀⠀⠃⠂⠀⠈⠈⠀⠀⠀⠀⠁⠘⠀⢇⠰⡀⢱⡀⡌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢰⠁⠈⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⡀⠀⠁⠀⠁⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⣀⣥⡴⠶⠞⣛⡻⠋⠉⠋⠉⢛⡁⠈⠁⠀⠉⠁⠈⠋⠁⠙⠛⠉⠓⠒⢤⡀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⡞⠉⠀⠁⠀⠐⠋⠉⠀⠀⢠⠊⠉⠉⠀⠀⡀⠀⠀⢰⠋⡧⠀⠀⠀⢀⣀⠀⠉⢳⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⡴⠄⠀⣠⠄⠀⠀⢠⢦⠀⠀⢀⣀⠀⠀⡔⠋⠉⠀⠀⠀⠀⠀⢀⣀⠀⠀⠸⣄⠀⠀⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠠⡇⣼⠃⠀⣴⠃⠀⠀⠀⠀⠀⢀⠀⠈⡈⠁⠀⠀⠀⠸⣄⣤⣀⠀⠀⠈⠨⠇⠀⠀⠉⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⠀⠀⢀⡖⠃⠀⢠⠶⠒⠋⠉⡿⠙⢢⣤⣤⣤⣀⡀⠀⠁⠀⠳⠖⠀⡀⠀⠀⠀⠐⠳⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢈⡷⠀⢀⡀⢿⠀⠀⣹⠉⠀⠀⢀⡼⢡⠞⠋⠁⠀⠀⡿⠉⢲⣤⣴⣆⠀⠀⠑⢦⡄⠀⠀⠀⠀⠀⣴⡃
⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⠙⠂⠀⠀⠈⢧⢀⣠⡴⠋⡰⠋⠀⠀⠀⠀⡼⠁⣰⠋⣰⠏⣠⡿⢇⠀⠀⠀⠀⠀⠀⠶⡄⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⠀⠀⠀⠀⢠⡟⠋⠁⣠⠞⠁⠀⠀⠀⣠⠞⢀⠜⢁⡴⠁⡴⠋⠀⢼⠁⠀⠚⣦⠀⠀⠠⠟⠀⢾⠁
⠀⠀⠀⠀⠀⠀⠀⡱⠂⠀⠀⠀⣞⠂⠀⣹⡴⠊⠁⠀⠀⣀⠴⢊⡡⠞⣁⠴⠋⣠⠞⠁⠀⠀⢈⠇⠀⠀⠘⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⣏⠀⠘⢯⣤⠶⣒⣮⠭⠒⠋⣉⠴⠚⢁⡤⠞⠁⠀⠀⠀⠀⠘⣆⠀⠴⠄⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⣨⡷⠟⠉⠀⢀⡤⠚⢡⡴⠚⢁⣀⣀⣀⣀⡀⠀⠀⠰⡃⠀⠀⢀⠀⠀⠈⣳⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢀⡟⠁⡠⢤⠀⢠⣄⠀⠻⡄⠀⢠⠖⠉⠀⠀⠘⠛⠉⢉⣀⣀⣀⣀⣉⠙⠲⣄⡹⠂⠀⡇⢀⠀⠀⡀⠀⢺⡀
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠾⡆⠀⠀⠀⡰⠁⠀⠙⠦⠤⠤⠤⠒⠒⠋⠉⠁⠈⠉⠉⢉⣙⣲⣬⣇⠀⠀⠉⠁⠀⠘⠙⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠘⡦⠀⠀⠈⣁⡀⠀⠀⠙⣶⠶⠶⠦⣤⣀⣀⣀⣀⣤⠤⠶⠒⠛⣋⣉⡉⠛⢿⠋⠀⠘⠙⠃⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠘⡧⠀⠀⠀⠧⠄⠀⠀⠸⠧⣤⣴⣶⣒⣒⣒⠲⠶⠶⠶⠒⠚⠉⠉⢹⣿⠖⠛⠀⠀⠀⠀⠀⣴⠂⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠀⠈⠙⠉⠑⠒⠉⠙⠒⠊⠙⠃⠀⠀⠀⠀⢠⠤⠄⠈⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠸⣤⡄⠀⠀⠔⠒⣄⠀⠀⠀⠀⠈⠹⠆⠀⠀⠀⠀⢀⠴⢦⡀⠀⠀⠋⡁⠀⠀⣴⠛⠀⠀⢠⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣄⠀⠀⠀⠈⠙⠀⠀⠀⠀⠀⢠⣆⡀⠀⠀⠃⠀⠀⠙⠀⠀⠐⠧⢦⡤⠁⠀⣀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠴⢦⣀⣤⣄⣀⣄⠀⢀⣍⠀⣠⡀⣠⣄⣠⠦⣤⣠⣴⣄⣠⡤⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠙⠛⠋⠉⠉⠁⠈⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠉⠀⢀⣀⣀⣀⣀⣀⣀⡀⠀⣀⣈⡙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⣠⣴⣾⠿⠟⠛⣛⣛⡛⠻⠿⣿⣿⣍⠙⠛⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠏⠀⣠⣾⡿⠋⠀⣀⣴⣿⠿⠛⠋⠀⠀⠀⢹⣿⡆⠀⠀⣿⠹⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣿⡟⣰⡟⣿⡿⠀⢠⡾⠙⣿⣧⣄⣀⣀⣀⣀⣤⣾⡿⢁⣴⠟⠁⠀⣿⣿⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⢰⣿⡇⢿⡀⢿⣿⠀⠈⠛⠷⣌⠙⠛⠿⠿⠿⠛⠛⠉⣠⠟⠁⠀⠀⢠⣿⣿⡀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⢀⣴⠛⣿⡇⠘⣧⠈⢿⣷⣄⠀⠀⠘⠷⣤⣤⠶⠶⠶⠖⠛⠉⠀⢀⣠⣴⡿⠋⢸⡇⠀⠀⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠸⣧⠀⢻⣿⡄⠙⢶⣤⡙⠻⢿⣶⣤⣤⣀⣀⣀⣀⣀⣠⣤⣴⣾⡿⠟⠋⢀⣠⠟⠁⠀⣼⣿⠛⣿⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠘⣷⠀⠹⣿⣦⣀⠈⠻⣦⡀⠈⠉⠛⠛⠛⠛⠛⠛⢛⣫⡿⠶⠶⠶⠛⠋⠁⢀⣠⣾⡟⠁⣰⠟⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⢹⡆⠀⠈⠙⢿⣷⣦⣌⡛⠛⠛⠛⠛⠳⠶⠶⠶⠛⠁⠀⠀⣀⣠⣤⣶⣾⠿⠛⠁⠀⣼⠏⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠙⠛⠿⠿⣷⣶⣶⣶⣶⣶⣶⣶⣶⡿⠿⠿⠛⠋⠉⠀⣀⣀⣀⣴⠏⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠈⠛⠛⠛⠛⠷⣤⡀⠀⠀⠀⠀⠀⣉⡉⠉⠀⠀⠀⠀⠀⠀⣠⡶⠛⠋⠉⠉⠉⠀⠀⣀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣤⡶⠟⠛⠉⠙⠛⠶⢦⣤⣤⣤⠾⠋⠀⠀⠀⠀⢀⣠⣴⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⢿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⠿⠿⠟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⠤⠴⠶⠶⠶⠦⠤⢤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⠶⠶⠶⠶⠶⠤⢤⣀⡀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⡸⠁⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠘⣆⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠙⢦⡀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⣰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢳⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⣀⡼⠋⣧
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣸⢠⠞⠓⢦⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⠖⠋⠁⠀⠀⠀⠀⢀⣤⠞⠋⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢸⠃⢸⠄⠀⡼⢀⣠⠤⠤⣄⡀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⢀⣠⣤⣄⠀⣴⠋⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠓⠛⠀⡼⠁⣾⠁⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢸⡇⢹⡆⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠀⠀⠀⠀⢀⡿⠀⣰⠞⠛⠛⠒⢦⡀⠀⣾⠀⠀⠀⠀⠈⠳⠞⠃⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠸⣧⣀⡿⠀⠀⠀⠀⠀⢻⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣀⣤⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠒⠶⠶⠶⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠶⠶⠖⠛⠋⠉



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠠⠖⠀⠀⢰⡰⣦⣛⣛⣥⠶⠛⢀⣠⢇⡄⠀⠐⣶⠀⠀⡖⢀⡤⠐⠀⢀⢠⣶⠀⠀⠐⠢⠄⠦⠤⡀⣀⡠⠔⠀⠀⢠⡂⣦⣛⣛
⠀⠀⠀⠀⢜⢧⣄⣀⣀⣀⣠⠴⠛⠁⠀⢰⠂⠀⠛⢠⡈⢇⠀⠁⠈⢀⠎⢈⡇⠀⠀⠀⢀⡠⠄⠤⠄⢀⠀⠀⠀⠀⢀⢷⣄⣀⣀
⣽⢂⡀⠀⠈⡄⠈⠉⠉⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢑⡀⠀⢐⣒⢹⣒⠟⠀⠀⢀⠔⢉⣤⣒⠒⢀⠠⣿⡂⣀⠀⠈⠄⠨⠉⠉
⠸⡇⠙⡆⠀⠘⠶⠦⠤⠤⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠈⠉⠒⠋⠛⠈⠀⠀⠀⠀⠀⢰⢡⡟⢠⣤⠀⠞⡻⡇⡉⠇⠀⠈⠳⠦⠤
⠀⢴⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠖⣲⢔⣤⠄⢀⠀⠀⠀⣦⢘⠛⠇⡉⡊⢠⡎⠇⢰⠀⡄⠀⠀⠀⠀⣀
⠨⢀⠜⠀⠀⠀⣶⠀⣄⠐⠋⠆⡀⢀⠀⠀⠀⠀⣼⣯⢘⡧⢁⡴⣀⠻⡯⡁⠀⠀⢹⣎⠨⢣⡘⠛⢉⣠⡬⠀⠐⠀⠀⠀⣴⠀⢇
⠴⠋⠀⠀⠰⢇⠙⢦⡈⠀⠀⠀⡀⠊⠴⠀⠀⣸⢻⠋⡂⢾⣆⣞⣹⠆⠹⣿⡇⠀⠀⠙⠳⢬⣡⣄⢈⣁⠴⠋⠀⠀⠠⢧⠘⢷⡌
⠀⠀⠀⠀⢸⣷⠑⠀⠀⠉⠀⠠⠔⢠⣶⠀⠀⠸⣞⡸⠻⣿⣮⢫⣤⣼⠗⢰⡇⠀⠀⠀⢀⣀⡀⠀⡀⠀⠀⠀⠀⠀⢰⣿⡐⠂
⠀⡢⡀⠀⠸⣟⠦⢾⣷⣰⠟⠑⢲⡏⢸⡄⠀⠀⠙⠷⣴⣅⣼⡏⠑⣿⣑⣿⠀⠀⠀⡔⠋⢀⣠⣤⣄⣀⠀⡢⡀⠀⠘⣟⠧⢾⣿
⠻⡆⢸⡀⠀⠈⠳⠶⣶⣤⣤⣴⣿⠿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⠘⠀⣰⢫⣷⣶⣶⣮⡹⣆⠈⡆⠀⠈⠓⠶⣷
⡧⡇⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⠖⠶⠀⠀⠀⠀⠀⠀⠀⢠⡇⣿⡯⣎⢎⡷⣿⣿⠀⡇
⣰⠃⠘⠀⠀⠀⠀⣠⣴⡶⡖⣷⣦⢤⠀⠀⠀⠀⣰⠋⠀⠚⠉⠉⠒⠀⠀⠀⠀⠀⡇⠘⣇⢿⣿⠿⠿⠿⣣⠇⠀⠀⠀⠀⠀⣠⡴
⢠⡤⠁⠀⢀⢠⡼⠧⠭⢶⣣⡿⠋⢀⡷⠀⠀⢰⣧⠂⠀⡠⠒⠒⢢⠀⠳⠀⣤⠀⠘⠦⡘⠦⣉⠛⠛⠛⠉⣠⠂⠀⠀⠄⣴⠷⠭
⠈⠁⠀⠀⡸⣆⠛⠒⠚⠋⠁⣀⡴⠟⠑⢣⡀⠀⢿⢀⠐⡇⠀⠋⠀⠀⡸⠸⡟⠀⠀⠀⠀⠉⠉⠉⠀⠈⠉⠁⠀⠀⠘⣇⠙⠓⠚
⣰⡀⠀⠀⠰⠙⠓⠒⠛⠛⠋⠁⠀⠀⢀⠜⠀⠀⠐⠈⠆⠁⠀⠀⡤⢔⡥⡴⠃⠀⠀⣠⠶⠉⡁⣉⠀⢑⣶⡄⠀⠀⠘⠈⢛⠒⠚
⢻⡞⠶⡀⠀⠣⣔⡀⠀⠀⠀⣀⡤⠐⠀⠀⠀⠀⠀⠀⠐⠦⢤⠴⠷⠘⠊⠀⠀⠀⠐⢁⠔⣟⠂⠈⠘⡨⣿⡇⠷⡄⠀⠱⣄⣀
⠈⣇⠀⠃⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠠⠀⠀⡀⠀⠀⠀⠀⡀⠰⣾⡇⠾⠛⠀⡵⠏⢃⠀⠁⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣆⣀⡈⣀⣠⣿⠖⣲⣀⣀⡀
⠀⠀⠀⠀⢀⣤⠖⣨⡕⢹⣏⠡⠶⣏⠙⢯⡙⠲⣶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⡀⠀⠉⠉⠉⢉⣤⣼⣿⡿⢃⣽⡄
⠀⠀⠀⣠⣼⣿⡀⢿⣷⣈⠙⠛⠻⠛⠀⣸⡿⠀⢈⣏⢷⢠⡀⠀⠀⠀⠀⠀⠀⢀⣀⢴⣯⣈⠀⠉⠛⠷⠶⠾⠿⠛⠛⣛⣉⣀⣾⣿⡇
⠀⠀⢠⣿⣦⡙⠳⣤⡙⠛⠿⠶⠶⠶⠚⠋⠀⢀⣾⠋⣼⠈⣿⣆⠀⠀⠀⠀⣰⡟⣿⢦⠈⠙⠷⣿⣧⣴⣦⣤⣶⣿⣿⣽⣿⣿⣿⠿⢩⣦⡀
⠀⢀⣼⣟⠿⣧⣄⡈⠉⠛⠒⠒⠤⠤⠤⠴⠚⢋⣠⣾⠏⠀⣿⣿⠀⠀⠀⢠⣿⣯⣝⠢⡛⣷⣦⡤⣈⠉⠉⠛⠛⠛⠋⠙⠉⣉⣠⣼⣿⣿⡇
⠀⢸⣷⡉⢷⣄⣉⠛⠻⠷⣶⣴⣶⣦⡴⠶⠶⠟⠉⠀⣀⣼⢿⡿⠀⠀⠀⠘⣿⣿⣉⡠⠐⢌⠻⣿⣦⡜⢋⡉⢸⠁⠈⠏⠛⠓⠠⢠⣿⣿⡇
⠀⠸⣿⡏⠠⠟⠛⠻⠷⢤⣤⣀⣀⡀⣀⣀⣠⣤⠴⠞⣋⣥⡾⠃⠀⠀⠀⠀⠙⢿⣿⣧⣷⣄⠙⠢⡙⣾⣟⣷⢋⡀⠄⢠⣤⣴⣼⣿⣿⡿⠁
⠀⠀⠈⠛⠿⣿⣷⣦⣤⣤⡀⡀⠈⠉⠉⢉⢀⣀⣶⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣷⣤⣌⡪⣟⠿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢀⡾⠀⠘⠃⠀⠀⢀⢀⠀⡴⣦⠀⡾⢻⢠⠏⡟⣸⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡾⠀⠀⠀⢀⠆⣼⣾⢁⡿⡟⢸⣡⣇⢸⡥⠏⠘⠞⠁⠁⠘⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠐⠋⠀⠠⠴⢺⡇⠀⣠⠃⠼⠐⠇⢧⠘⠁⠓⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢂⡛⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡦⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣛⣾⣲⣴⡦⣶⣴⠄⠀⠀⣀⣀⠴⠶⠟⠛⡂
⠀⠀⠀⠀⠀⠀⠀⣀⡨⢄⠙⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⠛⠛⢉⣁⣀⠴⠾⠛⣋⣉⣀⣀⣠⣴⡥⠟⠁
⠀⠀⠀⢀⠄⡂⣤⡶⠅⠬⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣀⡴⠞⠚⢋⠉⣠⣠⣴⡿⠿⠿⣿⣭⣛⠿⣶⣄
⠀⠀⠠⠀⠶⣟⡊⠀⠠⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⣿⣿⣿⣉⢁⣢⡽⠶⠚⠋⠉⠙⠻⣿⣲⣄⣴⣿⣟⠳⣶⣭⡛⢦⣌⣠⡄
⠀⠀⠀⠉⠁⠋⠨⠤⠀⠀⠀⠀⣀⠀⠠⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⡄⠙⠳⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡐⠰⣶⣦⠠⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡁⠀⠙⠛⠷⠶⠤⠤⠴⠟⠋⠀⢈⣤⢢⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢧⣌⣙⠛⠶⠶⠴⠤⠤⠤⠤⠴⠶⠚⣋⣡⠜⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⢒⣒⣒⣒⣛⣛⡯⠭⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢤⣄⣀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠂⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢠⣥⣼⣿⣢⡄⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⣤⠄⠠⠀⠀⠒⠀⠉⠀⠀⠀⠀⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠒⠂
⠀⠀⠀⠀⠀⠀⡔⠉⢴⣮⣽⡷⡄⠀⠂⠺⡟⢉⣻⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⣽⠉⠏⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠣⠤⠈⣹⣿⣷⡷⢾⠀⣠⠖⠈⠙⠦⢳⠀⠀⠀⠀⠀⠀⢀⡘⣱⣏⣸⠸⠿⣾⠒⠂⠉⠉⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠒⣡⣶⠶⣤⡀⠑⠀⠐⣆⡠⣾⣟⣀⣠⣤⡴⠖⢐⡯⠀⠉⢉⡇⠂⢀⡇⠀⠀⡀⠄⣀⡤⠴⠒⠚⡉⠑
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣏⣶⣜⣛⠻⡄⢀⠞⣉⠠⢞⣥⣍⡁⠀⠀⠀⠀⠀⠀⢰⡏⠃⠀⡂⠿⠷⠂⠈⢁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠸⡷⢬⣽⡟⢿⣗⡷⣻⠸⢟⡇⠛⠉⣹⡟⡄⡤⠠⠄⠀⢀⡠⠥⠖⠋⠁⠀⠀⠀⠀⠘⢀⣀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠈⠀⣃⣘⣿⡄⡲⢟⣱⣾⢮⣓⠟⣤⣉⣩⠾⣁⡠⠴⠖⠊⣉⡀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠆⠂⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⢅⣼⡿⣫⣭⡿⢎⠋⠚⠚⠛⠉⢀⠠⠀⠀⠉⠀⡀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛⢭⡟⢷⣼⣐⡠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠁⠀⠇⣤⡴⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡸⢯⣵⢻⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠘⣹⣴⠎⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠈⠉⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠿⢿⣿⣿⣿⣿⣿⣿⣿⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⠋⠉⠉⠉⠉⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢾⣿⠏⠁⠉⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠀⣼⡟⠀⠈⠀⠀⠀⠉⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⢰⡏⣿⠀⠀⠀⠀⣀⡀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⣼⠁⢻⠀⠀⠀⠀⣿⣿⣷⣄⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠸⡇⠠⠀⠀⢻⣿⣿⡟⣷⣀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠈⢿⡿⢡⢣⠛⢦⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢧⠀⠀⠀⠈⢧⡎⡜⢰⠈⠳⡄⠈⢷⡄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢧⠀⠀⠀⠈⢷⡀⠃⠀⠀⠙⣄⠈⢻⡄
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⢷⡀⠀⠀⠀⠘⣆⠈⢷⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠻⣄⠀⠀⠀⠘⡆⠈⣷
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠈⢷⡀⠀⠀⢹⠀⠸⣇
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⠳⠤⠾⠀⠀⢻
⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡐⢤⡀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣭⣅⡤⠋
⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⢀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⣯⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣤⡤⢤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡤⡤⠤⣀
⠀⠀⠀⠀⠀⡀⢤⠒⠋⢹⡖⠂⠈⢳⡌⢂⠙⣿⠣⡀⠀⠀⠐⡟⠡⠔⣛⣫⣶⣦⠀⠀⢀⠎⠀⣀⠤⠜⠀⠧⣀⠀⠡⡀
⠀⠀⡠⠒⣭⢄⠸⡍⠳⡈⢳⡀⠀⠀⢻⡌⡄⠘⣄⡇⠀⢸⠀⠃⢠⠊⠉⠉⢉⣈⣁⠀⣸⡀⠚⠀⠀⡄⢠⠀⠀⠱⠀⡧
⠀⠰⠁⠀⢿⠀⠂⠹⣆⢱⡈⢷⡀⠀⠘⡇⣧⣶⠟⠀⠀⠘⢧⡀⠐⠔⠒⠉⠉⢉⡟⠀⢿⣧⡀⠀⠀⠸⠂⠀⠀⣀⣴⠇
⠀⢠⡀⠀⠈⠓⠄⣦⣘⢦⣧⣀⣷⣶⡶⠟⠋⠁⠀⠀⠀⠀⠀⠙⠲⠤⣤⣤⠴⠚⠁⠀⠈⠻⣿⣷⣶⣶⣶⣾⣿⡿⠋
⠀⠀⠉⠛⠓⠒⠛⠛⠛⠛⠛⠋⠉⠀⡠⠒⠀⠁⠀⠠⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢉⡩⠭⠟⠫⢤⣀⣀
⠀⠀⣠⠒⠫⣅⠉⠉⠱⣶⢤⠀⠀⡌⠀⠀⠀⠀⠀⠀⠳⠀⢀⠄⠀⠀⠙⡿⠉⠱⡀⠀⠀⢀⣴⡙⠀⠀⠀⠀⢨⠀⢨⣉⠵
⢀⣞⠣⠄⢈⢮⡀⠀⣠⡇⢠⠳⡄⡇⠁⠀⠀⠀⠀⠀⣸⠀⣸⠶⢦⣄⡰⠁⠀⠀⣷⠀⣰⡋⠀⠘⢎⢂⠀⠀⣸⠀⠊⠁
⠠⠁⠀⠉⣷⣿⣿⡾⠛⠶⠾⢤⠁⠘⢄⣀⣀⣀⣠⠴⠃⠀⡇⠀⠀⠘⡇⠀⠤⠞⡿⠠⠀⠬⢷⣤⠼⠿⠶⠛⠁
⠘⠦⠒⠛⣉⣁⠀⠄⡠⠠⣀⡀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠰⣤⡀⠀⠰⣄⠀⠚⠀⠀⠀⠀⠤⠶⠾⠥⣄⡀⠈⠙⠲⣄
⠀⠀⣴⠍⠁⠀⠀⠀⠀⠀⠛⣯⠀⠀⠀⠀⠀⠀⠀⠤⡤⢤⣤⣤⣭⣉⠉⠁⠀⠀⠀⢠⠚⠀⠀⠀⠀⠀⠀⠙⢲⡀⠀⠈⡆
⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣤⠈⠢⡀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⢲⣧⡤⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⡴⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⡱⠀⠀⠠⠁⠀⢀⠀⠀⠀⠀⠀⢹⠁⠀⢸
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠑⠦⠄⣀⡀⠀⠀⠀⠀⠀⠀⠈⣿⡾⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣸
⠀⠀⠀⠙⠢⠤⠴⠄⠂⠒⠚⠉⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⠒⠒⠒⠒⠋⠀⠀⠀⠀⠘⠓⠢⠤⠤⣄⣀⡀⠀⡼⠖⠉
⡶⠒⢶⢶⡶⠶⠶⣶⢶⠶⠶⡶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⠶⠶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠀⠈⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀
⠀⠀⠀⠀⠀⢀⡤⠒⠉⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠽⢦⣄
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⡡⠤⢶⡒⠒⠦⣌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢀⠞⢁⡔⠉⠀⠀⠀⠈⢲⣄⠈⠱⡄⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡎⢀⡎⣱⣄⠀⠀⣠⠊⢁⣈⠙⣆⠸⡄⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⢸⠀⢸⠈⠳⡄⠱⡄⡇⠀⣯⣸⠀⣸⢀⡇⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠘⢦⣠⠇⢀⡟⢿⡀⠙⢌⠒⢃⡜⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡌⠶⢤⣤⡖⠋⠀⠀⠙⠢⢄⣉⣉⣤⣞⣙⣢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⡠⠖⢉⡡⠤⠲⢶⠢⢤⡉⠳⣄⠀⠀⣠⠖⢉⡡⢤⡖⠤⢌⡉⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⢠⠎⢠⠾⣍⢦⡀⣰⠋⡳⠀⠈⡆⠘⣆⠞⢁⣴⢏⠀⠀⠙⠢⡀⠈⠳⣄⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⡇⢠⠛⣄⡨⢦⡙⢇⠀⢧⡤⠔⢃⣰⡏⢠⠿⣌⠳⣕⢄⡀⠀⣨⠶⡄⠈⢧⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠘⣄⠈⠳⡄⠹⡌⠑⠲⠶⠒⠉⠙⢧⡈⠣⣌⣳⠉⢳⠙⢦⡇⠰⣇⣀⠜⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣈⣉⠉⣁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠴⠋⠀⠀⠙⠢⠤⠤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⡿⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⣾⠁⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠹⣆⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣿⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⣼⠁⠀⠀⡾⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠛⢉⡽⠟⠉⠙⠓⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⣠⡶⠋⠀⠀⠀⠀⢀⣠⠞⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⣶⡦⠾⣧⣄⣀⠀⠀⢼⡁⢀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣠⡴⠚⠉⠁⢀⣤⠞⠉⠀⠀⠀⠀⢉⡿⠶⣬⡉⠉⠁⠀⠀⠀⠀⢀⣠⡴⠶⠚⢛⣻⠟⠛⠛⠓⠶⢤⣀
⠀⠀⣠⠞⠁⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣠⡶⠋⠀⠀⠈⠙⣦⡀⠀⢀⣠⠞⠋⠀⠀⢀⡴⠛⠁⠀⠀⠀⠀⣠⡾⠋⠛⢦⣄
⠀⣴⠋⠀⠀⠀⠀⡾⠁⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠈⢳⣤⠞⠁⠀⠀⠀⣴⠏⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠙⢧⡀
⢰⠃⠀⠀⠀⠀⠈⠷⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠀⠀⠀⠀⢸⡃⠀⣀⣀⠤⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠙⠳⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠞⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣯⣭⣥⣤
⣿⣿⣁⣩⣭⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠰⠂⠀⢀⡔⢮⣻⠭⠶⠛⣠⡼⡢⠀⠐⣶⠀⣰⠀⡤⠐⠀⡄⣶⠀⠀⠐⠀⠶⠤⠀⠤⠔⠀⠀⢰⠰⣾⣻
⣀⠀⠀⠠⠳⠦⠤⠤⠔⠚⠉⠀⢠⠇⠀⠁⠠⠘⠀⠀⣀⣔⣀⡟⠀⠀⢀⠔⠒⠒⠀⢤⡀⠀⠀⠈⠳⠤⠤
⢻⠰⠄⠀⠡⣀⡀⠀⠀⣀⠠⠀⠁⠀⠀⠀⠀⠴⠄⡴⠦⠛⠊⠀⠀⠐⠡⢲⠗⠀⠒⢽⣧⠲⡄⠀⢦⣀
⠈⡄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⣠⣀⠀⠀⠀⠀⡄⠲⠿⠘⠛⢠⢞⢉⡄⠀⠀⠀⠀⠉
⠀⠀⠂⠀⠀⡄⢨⡀⠒⠆⠀⠀⠀⠀⠀⣼⣋⣻⠈⡨⣁⢶⠕⠀⠀⢻⡍⢖⣘⠓⢛⣤⠂⠠⠁⠀⢠⡄⢨
⠐⠁⠀⠀⣧⠙⠦⣀⠀⢀⠀⠁⠂⠀⢸⣻⢡⡰⣆⣼⠽⢀⠿⡇⠀⠀⠙⠲⢥⢄⠈⠤⠚⠁⠀⢰⡆⠻⢆
⢀⠀⠀⢘⣿⠈⣠⡀⣀⣔⣊⡞⡇⠀⠘⣯⡏⣻⠷⡷⣶⣿⢀⡇⠀⠀⢀⡤⠄⠀⠀⠀⠀⠀⠀⢸⣿⢁⣄
⣤⠑⡄⠀⠻⣍⣻⣓⠃⢀⣽⣤⠗⠀⠀⠀⠙⠻⠾⠧⠴⠷⠞⠀⠀⠰⠁⣠⣖⣲⣖⡲⣌⠑⡀⠈⠿⣍⣛
⣾⡇⠃⠀⠀⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⡤⠀⠀⠀⠀⠀⠀⢸⣱⣿⡿⣿⣿⣾⠀⡇⠀⠀⠈⠉
⣸⠁⠂⠀⠀⠀⣤⣤⣶⣶⣤⣄⠀⠀⠀⣴⠃⠐⠒⠐⠂⠀⠀⠀⠀⡆⢸⡸⣿⠿⡿⢟⡽⠀⠀⠀⠀⢀⣤
⢀⠄⠀⠀⠠⣾⠿⣖⣯⠾⠋⣸⢇⠀⠰⣷⠀⢠⠒⣁⠢⠈⠂⣤⠀⠘⠠⢑⣬⣙⠛⢉⡠⠂⠀⢠⢰⡿⢾
⠀⠀⠀⢈⢧⣉⣉⣁⣠⠴⠚⠁⠉⡆⠀⠛⢀⠹⠀⠁⠀⣠⠁⡿⠀⠀⠀⣀⠄⠠⠄⣀⠀⠀⠀⠘⣦⣉⣉
⣿⢠⡀⠀⢄⠈⠉⠀⠀⠀⢀⠀⠊⠀⠀⠀⠀⣀⡀⣀⡎⣷⠝⠁⠀⠠⠊⣡⡆⠒⠤⣺⣇⣤⡀⠀⣄⠉⠁
⠸⠀⠑⠀⠈⠙⠒⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⣀⢈⠁⠁⠀⠀⠀⠀⠀⣤⣾⢠⠶⠀⣭⠛⠀⠃⠀⠈⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡔⣲⣶⣿⣿⣿⣟⣿⣿⣶⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡢⠀⠀⠲⠀⠠⢌⡻⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣄⣶⣤⣤⣬⠀⢁⡊⣁⠀⠀⠀⢀⡲⣍⠇⢨⢭⢭⢄⣄⣄⡀⢄
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣢⢒⣾⠳⠉⣖⠂⠁⠠⠠⠦⠵⡬⡦⠤⠤⠤⠤⠽⠮⠦⠦⠠⡁⠃⡒⠲⠭⠚
⡄⠠⠔⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠰⠁⢟⠐⠂⠀⠀⠄⡠⠀⠀⠀⠉⠉⡁⡠⢀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⣂⣰
⡇⠀⠠⠀⣀⡀⠀⠀⠂⠄⠄⠀⠉⠁⠘⡀⢀⠀⡚⠪⠄⠄⠁⣠⣠⣀⢅⡄⡠⣨⢀⣔⡅⣠⣡⣪⠆⠴⠵⠖⠂⠒⠚⠉⣁
⡇⠀⢀⠀⡇⠀⠀⡏⠁⠀⢠⠐⠀⠀⢠⡀⠡⠊⢧⣯⣥⢀⣒⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢨⣤⣤⣶⠖⠛⠛
⡇⣄⢀⠀⠀⠀⠀⠣⠄⠀⡘⡀⡀⢐⡾⠇⢊⠀⠋⠈⠁⠒⠛⠻⠧⠯⠯⠯⠿⠯⠮⠮⠮⢉⡴⠿⠕⠓⠛⠉⠁
⡇⡺⢛⠽⣀⠀⠀⠀⠀⠀⢂⠀⠒⣡⠀⠁⠠⠂⠄⠀⣤⠀⡀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠞⡡⢊⠵⠀⠀⠀⠀⠀⡠⠔⣩⠔⡡⠆⠐⠡⠆⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠉⠀⠀⠀⣻⡇⠀⠀⠲⠀⠘⠃⠋
⢰⣊⠔⣡⠔⠀⠀⠀⠀⣠⠔⣋⠔⣋⠔⠃⢈⠔⢪⠀⠀⠀⠂⠀⠐⠀⠀⠈⢀⡀⠀⠀⠀⣻⡇⠀⠐⠀⣈⠠⠄⠐⠀⠀⢀
⠸⠁⠊⡵⢊⠀⠀⠀⣀⠀⢊⡡⠞⣡⠖⠃⠀⠀⠹⠀⠀⠒⠒⠒⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⢇⠀⠀⠀⠀⠀⠁⠀⠀⠀⢈
⠀⠂⠀⠘⠅⠀⠀⠀⡠⠖⣉⠴⢊⡥⢊⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢹⠀⠀⠀⠀⠀⠀⢤⠪⡊⢸
⠀⠀⠑⠄⠀⠀⠀⠀⠴⢊⡡⠔⡡⠖⡡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠍⠀⠀⠀⠀⢀⠀⡠⠈⠀⢸
⠀⠀⠀⠈⠀⠀⠀⠀⠆⣈⠔⢉⠴⢋⡤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢰⠎⡄⠀⠀⠂⠁⣀⡴⠑⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢊⡥⠚⣡⠔⡩⠖⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⢸⠌⡇⠀⠀⠀⠄⠈⢐⠡⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠴⣊⠔⢋⡠⢒⡥⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⢸⠀⡇⠀⠀⠐⡡⡀⢔⠀⠀⢨
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠚⠁⠔⡩⠖⡡⠚⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⢀⠀⢠⠀⡠⠀⢻⢔⣧⠀⠀⠀⠀⠀⠀⠠⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠔⢊⠴⣊⠔⠚⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠛⢉⡷⣿⡗⠿⠇⢸⡇⣿⠀⠀⠀⠈⠡⠐⠀⢁⣰
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠠⢊⡥⢊⡡⡎⠀⠀⠀⠀⣦⢤⣤⣄⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⣜⠗⠉⠆⡤⡤⡴⠰⣐⡺⠟⠙
⠀⠀⠀⠀⠀⠀⠃⢰⠀⠀⠀⠀⠄⠈⠂⠀⡁⠀⠀⠋⢉⠛⠚⠛⠲⠶⠾⠰⠱⠟⠫⠃⠲⠢⠲⠮⠆⠒⠚⠘⠉⠉
⠀⠀⠀⠀⠀⠀⢢⣐⡤⢠⠄⡀⡀⠀⠀⠀⡇⠀⠀⢀⠄⡠⢊⠄⠀⡠⢂⠔⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠄⢀
⠀⠀⢀⢄⠀⢈⡀⢀⠉⢙⠃⠨⢄⡴⢀⣂⡇⠀⢀⠄⡈⢀⠀⢀⠄⠐⢁⠈⡠⠊⡐⡡⠂⠀⠀⠀⠀⠀⠀⢀⠀⢈⠄⠁⠆
⠀⠀⠀⠁⠂⠡⢂⠅⠊⠄⢊⠔⠀⠔⡡⢊⡅⠀⠁⠦⠀⡁⠚⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠁⠐⡈⣠⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠠⠊⠠⢊⠔⢁⠆⠀⠨⠿⣦⣀⣠⣀⡀⡀⠂⡀⠀⠀⠀⠀⢀⠀⠀⣈⠀⢀⣠⣆⣴⠾⠞⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠄⢁⠜⢉⠄⢫⢄⡉⠙⠏⡉⠙⠛⠻⡻⠋⠛⠃⢛⢃⢟⢛⠋⢉⠉⠈⠀⢁⣠⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡁⠂⡡⠊⡡⠊⢁⠔⢀⠄⢠⡤⠀⡀⢀⡀⢀⡀⢀⡠⢠⠄⠠⠆⠖⡡⠫⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠊⡠⠊⢀⠔⢁⠔⢁⠄⠀⡠⠊⡠⠊⠀⠈⡰⢂⠔⣡⠞⡠⢂⠔⡀⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡀⠔⢁⠔⠁⠔⠀⡠⠊⡠⠊⡠⠀⡠⠊⡠⢁⠔⢁⠊⠠⠊⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠔⠁⠀⠁⠠⠊⡠⠊⡀⠊⠠⠊⡠⠊⠐⠁⠐⠁⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣄⣽⣷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣇⣀⣷⣶⣿⣿⠛⠛⣻⣿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣿⣿⣿⡙⠀⠀⠈⠁⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠉⠽⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠅⠸⠿⠿⣿⣇⠀⣤⡄⠀⢠⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣦⣾⣿⡇⠀⠀⠀⠀⠀⢻⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⡟⠃⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣶⣿⣀⣾⣶⣿⣿⠿⡟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⠀⠀⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣿⡟⠛⠻⠿⠿⠉⠿⠿⣿⡏⠁⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣇⣠⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣿⠷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⡷⠀⣀⡀⠀⠀⣤⣿⣿⣿⣿⣿⡏⠉⠉⣿⠏⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠇⠈⢱⣶⣿⣷⣶⣶⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠀⠐⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣶⡆⠀⣸⣾⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠚⠛⠉⠉⠉⠉⠙⠛⠲⠦⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⠋⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠈⠉⠛⠓⠲⠶⠶⠶⠶⠶⠚⠛⠛⠛⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠶⠚⠋⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⢀⣴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⢀⣀⣠⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢸⡷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠉⠁⠀⠀⠀⢠⣀⣠⠤⠤⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠉⠛⠓⠶⠦⢤⣤⡤⠤⠶⠞⠋⠁
⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠹⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠁
⠀⠀⠀⠈⠳⣦⣄⣀⣀⣠⣤⡴⠶⠛⠉⠀⠀⠀⠀⠈⠙⠲⠶⠤⣤⣤⣤⣤⡤⠤⠤⠶⠶⠖⠚⠛⠙⣯⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣰⠋⢠⡄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠿⠃⢠⣀⣻⠀⠀⡟⠀⣀⡀⠀⠀⠀⠀⠈⣧⠐⡄⠀⣿
⠀⠀⠀⠀⠀⠀⣾⠉⠭⠭⠭⠭⢭⣭⣿⣗⣒⣒⣒⣲⣷⡶⢤⡀⠀⠉⠁⠀⠀⠉⠉⠉⠀⠀⢀⡴⠶⣆⣿⠴⠃⠀⣿
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣧⢻⣀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⣿⣇⣀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠉⠉⠉⠛⠛⠓⠲⠶⠶⠾⣦⣀⣴⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣴⣾⣧⠀⠀⠀⠀⠀⠀⢸⡜⡗⠒⠒⠲⠶⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⣠⣞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢻⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢶⣒⣲⡶⠦⠤⠤⣤⣤⣄⣀⣀⣀⣀⣙⣼⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠓⠒⠒⠒⠶⠭⠭⠽⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢒⣋⣩⡭⠭⠭⣍⢁⣂⡒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⣴⠋⢉⣤⠶⠶⠶⠶⠦⣌⠙⢲⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⣏⠀⢼⡇⠀⣿⣁⠀⠻⢿⣿⡆⠈⣷⠀⢿⣷⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢘⣇⠀⢸⣇⠀⠘⠻⠿⣷⣶⡿⠃⢀⣿⠀⠀⣿⣧⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣿⣂⠀⢿⣧⣄⣀⡀⠀⠀⣀⣠⣿⡟⢀⡄⣵⡿⠀⢺⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠘⣿⣄⠀⠉⠻⠿⠿⣿⢿⠿⠟⠋⣠⣦⣼⣿⠃⡰⢸⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⠻⣿⣮⣤⣐⣤⢀⣠⣤⣄⣤⣿⡿⠟⢀⠴⢑⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠙⠟⠛⠻⠛⠛⠛⠋⢉⠡⠔⠈⢑⢵⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢔⣆⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠾⣭⣥⣽⣷⣾⣶⣷⣿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠉⣉⣉⣉⣉⣉⣉⡉⠉⠉⠙⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠟⠁⢀⣶⠟⠋⢉⣩⣭⣭⣍⡉⠛⠿⣶⣄⡀⠈⠳⣖⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢩⣿⠄⠀⠀⣾⠛⠀⠠⣿⣀⣦⣄⠈⣿⣦⠀⠹⢿⣿⡄⠀⠘⣷⠄⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⢸⣿⠀⠀⠘⢿⣗⣶⡄⢈⠛⠛⠋⢀⣴⣿⠀⢀⣼⣽⣇⠀⠌⡹⣧⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢻⣇⡂⠘⣿⣧⣢⠀⠀⠙⠻⢿⣿⣬⣿⣾⣷⣿⠟⠀⢴⣷⣿⡇⠀⢆⣶⣿⡆⠀⡔⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢺⣿⣆⡀⠈⠻⣿⣶⣶⣄⠀⠀⢀⡉⢉⠋⢉⠀⣤⣠⣜⣿⠟⠀⠀⣞⣿⣿⡇⠀⠀⠐⠜⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀⠀⠻⣿⣦⣵⠄⠀⠉⠙⠻⠿⢿⣿⣶⣿⣿⣿⣿⣿⣿⡿⠃⡀⠴⣿⣿⣿⣿⠇⠀⠠⡄⡀⢻⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢔⠄⠀⠙⣿⣿⣷⣾⣧⣀⡀⠀⠀⠀⠀⠈⠈⠉⢀⠀⠠⢠⠤⢫⣶⣿⣿⣿⠋⠀⠀⣙⢷⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡑⠀⢨⡑⠠⣀⠈⢻⢿⡿⣿⣿⣷⣷⣼⡦⣤⣼⣁⣤⣪⣾⣷⣿⣿⣿⡿⠋⠀⠐⠀⠀⣞⠌⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠁⠀⠀⠀⠈⠑⠂⠤⢀⠉⠙⠻⠛⠿⠿⢿⠿⠿⠿⠟⠛⠋⠁⠈⠀⠀⠀⢀⡀⡴⢿⠿⣁⠤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠀⠤⠄⠀⠀⠀⠀⠀⠤⠄⡀⠡⠄⠀⢅⡬⢚⡅⠶⡠⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣔⡂⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⡁⡅⣊⣟⠿⢖⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣾⣗⣢⣀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣣⣠⣶⣻⣫⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢾⣥⣿⣣⢞⡂⠀⠀⢀⢀⣀⣀⡠⢀⠔⡘⣳⣟⣵⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠷⢶⣶⣾⣥⣬⣥⡾⠿⠟⠋⠉






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

⠀⠀⠀⠀⠠⢴⣇⠀⠀⠘⠠⡄⠀⠀⠀⠀⠀⠀⣠⡶⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠂
⠀⠀⠀⢄⡀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠁⠀⠀⠀⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⠉⠉⠛⣿⡆⠀⠀⠀⠁⠀⠂⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣤⣤⡄⠀⠀⠀⠛⠃⠀⠀⢀⣠⣤⠀⠀⠀⠀⠀⠈⠀⣀⣀⠀⠀⠈⢧⡀⠀⠀⣀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⣶⢲⢶⣢⠀⠀⠀⠀⢀⡴⣄⡔⠆⠀⠀⠀⠐⠛⠛⠁⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢊⠀⠋⠉⠀⠀⠀⠀⠀⢀⣄⠀⠀⡟⠜⠓⢻⠀⠀⠀⠀⣾⠧⡤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣦⣤⡶
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠒⣄⠘⠂⠀⠀⠀⠀⠀⠀⢤⣴⣶⣾⣿⡆⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠛⢛⢻⣧⣀⣀⠄⠀⠀⠀⠀⠀⠈⠻⢿⡿⠿⠁
⠀⠀⠀⠀⠀⠉⠛⠖⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠿⠟⠀⠀⠀⠀⠀⠀⠠⢤⣤⡄⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠄⠀⠀⠀⠄⠈⠛⠀⠴⠂⠛⠐⠠
⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⣶⡦⠄⠒⠀⠓⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠄⢐⡀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠢⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⢄⠀⠀⠀⠀⠀⠀⡔⠉⢣⠀⢠⠇⠀⡤⠚⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⡎⠀⠀⠀⠀⠀⠀⠂⠄⢤⠙⠁⠀⠘⢆⠀⠀⡨⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠂⠒⠋⠀⠀⠑⠒⠒⠊⢳⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠉⠉⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⣀⡀⠀⠀⠀⠀⢀⠤⢄⠈⡆⠀⡠⠊⠀⢀⠤⢤⡀⠀⠀⠀⠀⠘⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⡆⠀⠀⠀⡇⠀⠀⠉⠀⠀⠳⣄⠀⡊⠀⠀⢘⠆⠀⣀⠤⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠢⣀⡰⠀⠀⠀⠀⠑⠤⠒⠉⡆⠀⠀⢀⣉⡁⢠⡒⠉⠀⡰⠁⠀⠙⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣅⠀⡸⠀⠈⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡖⠉⠉⡇⠀⠀⢏⠉⠉⢉⠧⣀⣀⣀⡸⠐⣅⣀⣀⠠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⡀⠀⠀⣀⠇⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⠖⠢⠤⠞⠀⠀⠈⠉⠉⠀⠀⠀⠸⢄⠠⠲⡄⠀⠀⠀⠰⠁⠉⠒⠎
⠀⠀⠀⠀⠀⠀⠸⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⢀⡇⠀⠀⠀⠰⡀⢀⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢱⠀⠀⡴⠊⠉⠐⠄⠀⢈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⠀⢱⠄⠀⢠⠃⠀⢸⡤⠤⠒⠦⠀⠀⡤⠒⠒⠤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠉⠉⠁⠀⠀⠀⢠⠃⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠤⠔⠁



⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠐⢟⢢⡀⠒⣂⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢓⣒⠃⠀⠀⠀⠀⠀⠀⠀⠐⢿⣲⡈⠐⢒⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠉⠉⠀⠀⠀⠀⠀⢀⡾⠁⠀⠉⠉⠀⠉⠁⢀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⣼⠉
⠘⠆⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣵⠁⠀⡠⠖⢉⡓⡄⠈⠇⢀⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣵⠃
⢀⡇⢺⡇⠀⠀⢀⠀⣇⠀⠋⠹⢀⠀⢸⠀⡄⠀⠀⠠⡿⢀⠀⡇⠀⠋⠡⠀⠀⡆⢸⡏⠀⠀⢀⠀⢃⠀⠋⠹⠂⠄⢸⠀⡄⠀⠀⠘⣿
⢚⣠⡼⠁⠀⠀⠻⣄⡈⠓⠀⠀⠀⡠⢀⠀⣠⡀⠀⠀⠠⠘⢆⠙⠀⢀⣀⣠⢞⣠⡼⠃⠀⠀⠸⣦⣈⠑⠀⠀⠀⣠⠄⡀⣠⡀⠀⠀⠠⠈⢆
⠽⠊⠀⠀⠀⠸⠦⢈⠙⠓⠒⠂⠀⠐⠋⣰⡏⢻⡀⠀⠀⠈⠠⢤⣀⣠⣶⡄⠿⠊⠁⠀⠀⠸⢦⢈⡉⠓⠒⢂⠀⠀⠊⢠⡏⢹⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣿⣧⠉⢠⣦⡀⣾⠋⠉⢹⣧⣼⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠉⢠⣤⡀⢾⠏⠉⢹⣧⣤⡏
⠀⠀⠀⠀⠀⠀⠸⣿⡳⠛⠻⣛⣉⣀⣠⣿⠟⠋⠀⠀⠀⠀⠠⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣳⠚⠻⣛⣋⣀⣠⣿⠿⠋⠀⠀⠀⠀⠠
⢰⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⡀⢾⡀⠸⣄⠈⠀⢃⠄⢀⠂⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⡀⢸⡇⠘
⠈⠘⠃⠀⠀⠀⠀⠀⢀⢀⡤⢤⣤⣀⠀⠀⠀⠀⠸⢷⠈⠻⢦⣀⠀⠀⢀⡄⠈⠐⠃⠀⠀⠀⠀⠀⠀⣀⡤⢤⣤⣄⠀⠀⠀⠀⠐⠷⠀⠛⢦
⢠⡾⡆⠀⠀⠀⠀⠀⠛⠋⠉⠑⠀⢈⠉⠀⠀⠀⢸⣿⡎⠂⠀⠀⠄⠀⠠⠖⢁⡾⣦⠀⠀⠀⠀⠀⠛⠛⠉⠑⠀⠈⠉⠀⠀⠀⢸⣿⡎⠓
⢻⠀⣧⠀⠀⠀⢡⠞⣡⡤⠤⢄⡀⠀⢳⡸⣧⠀⠸⣟⢧⣴⣿⣇⣰⠟⠙⢲⣿⠀⣯⠀⠀⠀⢁⠞⢁⡤⠤⢤⡀⠀⠱⡰⣷⠀⠈⣿⢧⣴⣿
⣿⠿⠃⠀⠀⡟⠀⠀⠁⠀⢤⡄⢱⠀⠀⡇⠉⠀⠀⠘⠳⣤⣼⣮⣁⣀⣠⣾⣿⠿⠋⠀⠀⡟⠀⠀⠁⠀⡤⡄⢹⠀⠀⡇⠈⠀⠀⠈⠳⢦⣼
⠀⠀⠀⠀⠀⢳⣤⢰⡀⠀⠂⠠⠊⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⢳⣤⢰⡀⠀⠢⠠⠚⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠈⢩⡴⣝⠒⠤⠤⠐⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⢶⠶⠆⠀⠀⠀⠀⠀⠀⠈⠩⡴⢝⠒⠤⠤⠐⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠚⠷⠶⠾⠟⠓⠀⠀⠀⠀⠀⠀⣴⠛⠀⠒⠒⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠷⠶⠾⠟⠓⠀⠀⠀⠀⠀⠀⣰⠚
⠰⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢫⠂⠀⢀⡤⠴⠤⡀⠐⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⠆
⠀⡆⢼⡇⠀⠀⠀⠀⡆⠀⠖⢲⠀⠀⢰⠀⡀⠀⠀⢈⣿⠀⠀⡏⠀⢾⢂⡁⠀⡄⢸⡿⠀⠀⠀⠀⡆⠀⠔⢲⠄⡀⢰⠀⡀⠀⠀⢸⣿
⠴⠁⣸⠃⠀⠀⢾⡀⠙⠦⠀⠀⠈⢀⠈⠀⢁⠀⠀⠀⠉⠰⡄⠳⠀⠀⠀⢀⡴⠁⣸⠇⠀⠀⢰⡄⠙⠦⠀⠀⠀⢀⡈⠀⢃⠀⠀⠀⢉⠐⡄
⢺⠽⠃⠀⠀⢴⡌⠙⠳⠦⠤⠄⠀⠉⠔⠠⠟⠧⠀⠀⠀⠐⠈⡀⠀⠀⣭⠉⣾⠽⠋⠀⠀⢰⡄⠙⠳⠦⠤⠄⠀⠉⠔⠁⡟⠷⠀⠀⠀⠐⠈

⠀⠀⠀⢀⣠⣴⣶⡖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡖⣲⣶⣦⠀⠀⠀⡠⢴⠒⠚⠈⢃⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣤⣶⣶⣄
⠀⠀⠀⢀⣀⠀⣀⡀⠉⢤⠂⢀⠀⠀⠀⠀⠀⢀⡴⠯⣤⣾⡿⠟⢋⡙⠁⠀⠀⣧⣿⣶⠟⠋⣉⣉⡉⢻⡄⠀⠀⠀⠀⣠⡿⠛⢺⣷⡬⠻⢧⡀
⠀⠀⠀⢸⣵⣾⡿⣿⠀⣼⣿⣿⠀⠀⠀⠀⣴⢿⡱⣾⣿⠟⢁⠐⠀⠀⠀⠀⠀⠁⠀⠋⠠⢾⣿⣻⠷⠈⠀⠀⠀⢀⣴⡟⠀⠠⡠⢿⣿⡦⠻⢷⡀
⠀⠀⢠⠘⢿⠿⣶⣿⠀⣿⣿⠟⠀⠀⢀⡾⢯⣶⣿⣿⡟⠔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠋⠀⠀⢠⠀⠀⠀⣾⡟⠑⠓⠖⠹⠒⠿⣿⣾⡿⠁
⠀⠀⠈⠁⠀⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠒⠉⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠐⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⢢⠇⡖
⠀⠀⠀⢀⡠⠠⠾⡛⢉⠁⠠⡀⡀⠀⠀⠀⢸⣯⠱⣶⣶⡖⠀⠀⡄⠀⠀⠀⠀⠀⢠⠴⠟⠛⠺⢴⡟⢓⠀⠀⠀⠀⢠⠀⠏⠐⠀⠀⡙⣿⣥⡖
⠀⠀⠀⠫⡀⣀⠀⠀⢲⡉⣳⠷⡗⠀⠀⠀⠸⠙⣧⣀⠈⠀⠀⡃⠸⢀⠀⠀⠀⠀⠁⢠⣶⣀⠀⠀⣡⡾⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣿⡏⣧
⠀⠀⠀⣂⣈⠉⠩⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠈⠙⢿⠻⡏⠟⢙⡇⠀⠀⠀⠀⠀⣿⣇⢽⣧⠀⢹⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣳⡿
⠀⠀⠀⢩⠄⢀⣀⣠⠐⣀⠤⠀⠀⠀⠀⠀⠀⠈⠿⣶⣴⣄⣠⡄⠾⠁⠀⠀⠀⠀⠀⠻⠿⢿⡿⠃⢸⠏⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠄⠠⠊⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣠⣾⠄⡀
⠀⠀⠀⠤⠒⠋⠛⣿⡟⠛⢣⣴⡢⢾⠇⠀⠀⠀⠀⠀⢀⣀⡴⣦⠀⠁⠀⠀⠀⢀⠀⠀⠐⣯⣀⣀⣚⡵⠀⠀⠀⠀⠀⢀⡔⣠⠟⢢⠟⣠⣥⡟
⠀⠐⢮⣖⣀⡀⢠⡎⣿⣿⡿⠃⠀⠀⠀⠀⠀⠂⠀⠡⣮⢻⡯⠟⠀⢀⠀⠀⠀⠄⠀⡀⠀⠀⢸⠛⢻⡄⠂⠀⡄⠀⠀⢸⠚⣶⣾⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⣧⡿⠀⠄⠀⠀⠀⠀⣼⠀⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠛⠋⠉⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠠⡄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠
⠀⠀⠀⠀⢉⠀⠠⠀⢀⠀⣠⢀⠄⠀⠀⠀⢠⣄⡀⠠⠬⠤⠀⠐⠀⢀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⢀⣐⣦⠶⠆⠀⠀⡌⠀⠐⠀⣀
⠀⠀⠀⠀⢸⡀⠀⠀⠀⡆⢸⣧⡀⠀⠀⠀⠀⡧⠀⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⢀⡐⠉⣀⣤⠾⠞⠉⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⡁⢀⠀⢰
⠀⠀⠀⠀⠈⠁⠸⣿⣿⡇⢸⣿⠇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠸⢧⠜⠐⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⣷⣄⡀⠀⠌⠡⢈⣥⣾
⠀⠀⠀⠀⠈⠀⠘⠛⠛⠁⠜⠃⠀⠀⠀⠀⠀⠙⠲⠤⠤⠤⣿⠿⠿⠃⠀⠀⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢏⡉⠀⢀⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⣾⣧⠀⠘⠢⠄⠀⠀⠐⠂⠀⠀⠀⢀⠏⠁⠀⡀
⠀⠀⠀⠀⠀⣀⡴⣶⢿⢻⡆⠈⠻⠈⠒⡤⠀⠀⠌⠠⡀⠀⠀⠹⠇⢠⠞⠀⢰
⠀⠀⠀⠀⡼⠁⠘⠁⠘⠇⢩⠇⢄⠀⠀⠒⠂⠤⠤⠠⠅⠄⠀⠐⠀⠀⡤⠔⠀⠀⢁⡖⢀
⠀⠀⠀⢸⣣⢶⡀⠀⠀⠀⠚⠅⠀⣉⠈⢆⠔⠂⠐⠒⣆⠄⠉⠂⠹⡜⠀⠀⠀⠀⠊⢀⡼
⠀⠀⠀⢹⡏⠈⢧⡀⢀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠒⠒⠂⠀⠈⠀⠉⠁⠀⠀⢁⡠⡄⡏⠀⢰
⠀⠀⠀⢿⣧⡀⢸⣷⠉⠙⠛⠛⠳⡄⠀⠀⣀⣄⡀⠀⠀⢀⣀⠀⠀⡾⠋⠙⠊⠀⠸⠇⠀⢸⣷⠄
⠀⠀⠀⠈⡿⣇⡀⠉⠀⠀⠀⠀⠀⣿⢠⠎⠀⠀⠙⠓⠈⠀⠀⢳⣀⠁⠀⠀⠀⠀⠀⣀⣴⡟⠁
⠀⠀⠀⠀⠀⠙⠳⢦⣠⡀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣄⣀⠀⠀⠀⢀⣀⣠⣀⣀⣤⣴⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⣀⠀⠀⠀⠀⠤⠀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠒⠉⢀⠀⠀⢀⣐⡚⠒⠂⠀⠀⠀⠀⠀⠀⠀⡁⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⢠⡀⠀⠀⠀⠉⠉⠓⠒⠚⠒⠒⠒⠈⠁⠀⣀⠤⠊⠀⠀⠀⠈⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⢄⡈⠑⠒⠒⠒⠬⠅⠂⠀⠀⠈⠉⠉⠁⠀⠀⠀⣀⠤⠒⠁⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⢀⠠⢄⠀⠀⠢⠤⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠈⠁⠀⠀⠀⠤⣄⠀⠰⠘⡄
⠀⠀⠀⠀⠀⠀⠸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠦⠤⣦⣄⡔⠁⠀⠘⣆
⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⣤⡀⠀⣀⣀⡴⠦⣄⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠀⠹⡄⠘⠁⠀⠀⠀⠈⠑⠒⠉⠀⠀⠀⠀⠀⠀⢰⠡
⠀⠀⠀⠀⠀⠀⠀⠳⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⠀⠀⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠈⠉⠉⠉⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠛⢛⣳⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⢀⣠⠴⠚⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⣋⡁⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⡴⠊⠁⠀⡤⠚⢻⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢞⣭⠞⠉⣠⠏⠀⢀⣤⠾⢩⢧⠀⠀⡴⠃⠀⠀⠀⠸⣄⡤⠊⠀⣀⡤⠤⢤⡇
⠀⠀⠀⠀⢀⡴⠋⠠⣿⣯⠴⠊⠁⣠⠴⠋⣠⠗⠁⡨⢧⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⡠⠔⠋⢹
⠀⠀⢀⡴⠋⠀⠀⠀⠀⢀⣠⢴⡫⢅⣤⠞⠁⡠⠺⡡⢩⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⢸⣁⣀⣠⣤⠶⠚⣋⠴⠑⡨⣚⠥⢒⣥⣴⢝⠕⡜⢠⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠈⠁⠀⠈⢯⣥⠤⠒⢋⠡⢐⣨⡷⠟⠋⡡⣊⢔⡕⢀⢷⡄⠀⠀⠀⠀⠀⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣛⠛⠛⠉⠁⠒⣈⡤⢚⠉⡠⡪⢊⡼⢋⢿⣄⣀⣀⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣈⣙⠫⠅⡊⢥⡠⢔⠅⡪⠔⣩⢗⠧⢊⣿⠈⢋⡷⠀⠀⠀⠙⠒⠚⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡜⠚⠊⠥⢚⣋⠥⢒⢞⣑⡵⡫⣵⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠄⠬⠭⠀⡴⠖⢉⡠⢞⡩⢞⣥⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⡂⠨⠥⣒⣾⡯⢖⣫⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⠈⠛⠛⣉⣭⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠒⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠢⠀⠠⡀⠀⠙⢦⡀⠀⠀⠀⠀⡴⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡉⠳⢄⡀⠀⠀⠀⠀⠙⢆⠀⠀⢰⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠙⠢⣀⠀⠀⠀⢤⡑⢄⠸⡄⢸⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣈⠑⢤⡀⠢⠀⠀⠑⢧⣸⠴⠒⠉⢉⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡉⠢⣄⠀⠀⡴⠛⢏⡋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢌⣑⣾⠀⢀⡤⠛⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⡛⢧⡀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡙⢦⡀⠀⠀⢤⡉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠙⢦⡀⠀⠀⠀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢶⣶⢶⣤⣸⡇⢸⣇⣤⡶⣶⡶⠀⠀⠀⠀⠀⠀⠳⡄⠙⢦⡀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣌⣙⣷⣼⣋⣡⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⢦⡐⠄⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⣉⡿⢿⣍⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠙⣦⣀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢼⡿⠶⠞⢻⡇⢸⡟⠻⠶⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠔⠒⠒⢦⡀⠀⠀⠀⠙⣦⠀⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⡿⠀⠀⠀⠀⣠⠿⠖⠀⠀⢀⣿⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⣻⣃⠀⠀⠈⠐⠦⠤⣀⣀⢀⣠⡦⠞⠛⠉⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠉⠀⢣⣅⡒⠲⠦⢤⣀⣀⣀⡀⢀⣀⣄⣀⣀⣀⣒⣬⣾⣿⠃⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆⠀⠀⠀⠀⠀⠉⠙⠓⠶⠤⣭⣥⣬⣭⣤⣾⣷⠿⠿⠛⠋⠉⠀⠀⣠⢄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢠⡜⠀⠀⠀⠀⠦⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣀⡴⠞⠛
⠀⠀⠀⠀⠀⠀⡇⠀⠠⡀⠀⣴⡽⠷⣬⣵⣄⠀⢀⣄⣠⣄⡀⠀⠀⢀⣠⣀⡀⢁⡀⠀⣀⣀⣤⡴⠟⠋⠀⠀⣴⡤⠠
⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠉⠛⠻⠷⡾⠿⠾⠿⠾⠟⠿⠛⠛⠛⠉⠉⠀⠀⠀⢀⣀⠀⠙⢠⡆
⠀⠀⠀⠀⠀⠀⢳⢾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠶⠒⠋⠁⣀⠀⡹⠃
⠀⠀⠀⠀⠀⠀⠈⠻⣝⠇⢀⣀⡀⠀⠀⠀⠀⣰⣯⣏⠀⠀⠀⠀⠀⠀⠤⠞⠛⠛⠻⢿⣏⢦⠀⠀⢤⣺⣟⣵⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⢀⡟⣱⣄⠀⣶⡋⢽⡇⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⡄⠀⢸⣿⣯⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⣻⣧⣴⣯⡒⢶⣶⣾⠄⠀⠀⠀⠀⣀⣄⣠⣀⣠⣤⡀⠀⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠳⠶⠿⠷⠶⠴⠶⠶⠿⠷⠞⠛⠋⠉⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠛⠛⠛⠛⠛⠛⠓⠒⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡾⠋⢁⣴⠶⠛⢛⣛⣛⣛⠓⠲⣶⣤⡀⠉⠳⣤⠦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⣿⡇⠀⠀⣾⠇⠀⢸⣋⣩⣉⠉⢻⣆⠀⠹⣿⣆⠀⠈⢷⡄⠻⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇⢨⣿⡆⠀⠘⢿⣳⣦⡀⡙⠛⡛⠀⣴⣿⠀⣠⣭⣿⡂⠐⣸⣷⠀⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣿⡔⠀⢻⣷⣲⠀⠀⠙⠻⢿⣷⣿⣷⣿⣿⠏⠀⢼⢾⡿⠀⠰⣼⣿⡇⠀⠃⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠻⣿⡠⠀⠉⠻⣿⣿⣤⣄⢀⡀⣈⡉⢡⣤⣴⣴⣿⡿⠁⢀⣾⣿⣿⡇⠀⠀⠐⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡀⠈⠹⣿⣾⣇⣀⡀⠈⠙⠛⠻⠿⠿⠿⢿⠿⠿⠿⠋⢠⠄⣋⣿⣿⣿⠃⠀⢀⣆⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢋⠊⠦⡀⠙⠿⣿⣿⣿⣤⣤⣀⣀⡄⠀⢀⡤⠀⡠⢀⣒⣥⣾⣿⣿⠟⠁⠀⠀⢮⠿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠄⠐⠀⠑⠦⣀⠋⠟⠿⠿⢿⣿⣿⣿⣿⣶⣾⣾⣿⠿⠟⠛⠋⠁⠀⠀⢀⣤⣷⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠈⠐⠀⠠⠌⠈⢁⢈⠋⠉⠉⠀⠀⡀⠀⠊⢀⠀⠔⢐⡫⡟⣕⠪⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⢄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⣀⠤⢒⢽⣺⣿⣄⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⣁⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢒⣁⣵⢿⣟⣴⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢾⣷⣶⣦⣖⢄⡀⠀⠀⡀⠄⡠⢀⢖⡚⢾⣫⣿⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠾⢶⣾⣭⣬⣷⠶⠾⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣀⡤⠶⣒⣒⣒⣒⣒⠶⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⣛⡭⠭⠭⠭⠭⣍⡛⠿⣿⣿⣿⣿
⠀⠀⢀⣾⡏⠀⣾⡋⢴⣶⣀⣽⠗⢈⣿⣶⣄⠀⠀⠀⣿⣿⣿⠏⠀⣾⡋⢰⣟⡉⠿⠗⣀⣿⠆⠈⠛⣿⣿
⠀⠀⢸⡏⠻⣦⣈⡛⠓⠶⠶⠶⠚⣋⣵⠟⣿⠀⠀⠀⣿⣿⣿⠀⣦⡙⠻⢦⣬⣭⣉⣩⣭⡤⠖⢁⡄⢹⣿
⠀⠀⠈⢷⣄⠀⠉⠛⠛⠛⠛⠛⠛⠉⠁⣰⠟⠀⠀⠀⣿⣿⣿⣆⠹⣿⣷⣶⣤⣤⣤⣤⣤⣴⣾⡟⢁⣿⣿
⠀⠀⠀⠀⠙⠳⢶⣤⣤⣤⣤⣤⣤⠶⠟⠉⠀⠀⠀⠀⣿⣿⣿⣿⣧⣤⣉⡛⠛⠿⠿⠿⠛⣋⣡⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠚⠉⢉⠁⠀⠀⠄⠈⠩⠟⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⡅⠀⠀⣨⠶⠛⠛⠙⠻⢿⣦⡀⠈⠑⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠙⣿⡿⠀⠀⢈⠿⣴⣤⣾⣿⡶⠀⢹⡇⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠠⠀⠂⠀⣰⡿⢁⠀⠃⠀⢀⡗⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⢠⠀⠀⠀⠀⠉⠛⠢⠤⠄⠶⠆⠓⠊⠁⠀⠀⠀⠀⢠⡼⠁⠈⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢿⣿⡗⡁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠀⣰⡟⠀⡀⠀⠰⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠐⠀⠀⣌⠑⠻⠆⣲⡆⣀⠀⠀⠀⢀⠀⢀⠀⣠⡄⣼⠝⠉⠀⠀⠁⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠘⠀⠈⠉⠛⠷⠾⠷⠾⠧⠿⠗⠛⠁⠀⠄⠀⠀⠀⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠙⠀⠀⠀⠘⠀⠀⣄⠒⠀⢀⠀⠀⠀⠊⠀⡀⠀⠀⠀⠀⢀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣷⢶⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣶⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣽⣇⣶⢴⡀⡄⢀⠀⠀⠀⠀⢀⢀⣠⣤⢦⡾⢧⣿⡿⠗⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣾⣟⣱⣾⣾⣷⣢⢳⣾⣿⣷⣽⠿⠕⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⠀⢀⣀⣀⣀⢀⣀⣀⣀⠀⠀⣀⣀⣀⢀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣀⠠⠄⣔⣒⡤⣭⣭⣭⣭⣭⣤⣤⣒⣂⠀⠠⠄⢀⡀
⠀⠀⡀⠤⣒⣬⠵⠒⣊⣿⣭⠖⣻⠿⠟⠛⠓⠚⠒⢮⠓⣤⡉⠁⠒⠤⠤⣉⠒⢤⡀
⣠⠊⡴⢛⡠⠾⡛⠫⠉⠂⢸⢦⣙⠲⠾⠾⠗⠴⠶⠞⣠⡟⠉⠑⠲⠤⢄⣀⠉⠲⡌⠢⡀
⣧⡀⠱⣏⡭⠁⠀⠀⠀⠀⠸⡀⠈⠙⠛⢹⢻⢻⢙⢛⠇⣿⠀⠀⠀⠀⠀⠀⠙⢢⡼⢀⣷⣤⣤⣀⣀⡀
⢧⡙⠦⣈⠙⠒⠤⢤⣀⣀⡀⠉⠒⢦⣤⣼⣬⣨⣼⣿⣾⠛⠁⠀⣀⣀⡠⠤⠒⣉⠴⣿⡏⢹⣾⠓⠿⢫⢿⡖⡦⣤⣀⡀
⠘⡦⣀⠀⠉⠐⠂⢤⢄⣈⣉⣉⡉⠛⠓⠚⠒⠒⠒⠒⡂⣉⣉⣉⣥⢤⠤⡖⡏⢹⣠⣿⢳⣶⢛⢦⡼⢻⣀⣟⡋⢳⣫⠯⡍⢒⡤⣄⡀
⠀⠶⣌⣉⠓⠢⠤⣼⣸⢀⡁⡏⢩⠙⠉⡏⠛⢹⠙⡉⡉⠉⠠⠀⣇⣠⣠⡧⠟⡟⡼⢿⡵⣡⢫⠎⡔⡱⢩⢿⠓⡞⡹⣄⡇⠋⠉⡆⣩⡓⢤
⠀⠂⢀⣉⠙⠓⠒⠶⠽⢯⣽⣳⣿⡶⣦⢷⠾⡼⢼⢧⡷⡾⠾⣛⡟⣍⣼⠼⢞⢋⠀⢚⡔⡱⢣⢊⠎⡔⡡⢣⢊⢌⢖⡀⠀⠙⠒⠃⣇⢸⢸
⠀⣿⡿⣿⣷⠀⠀⠀⣠⡀⠀⠀⠉⠉⠉⠛⠛⠛⠛⢛⠛⠙⣿⢎⣾⠡⢠⢢⡮⠁⠀⣺⡜⡰⡡⢣⢎⠜⡔⡡⡣⣣⢋⠄⠀⠀⠀⠀⠈⠉⢸
⠀⠁⣗⢿⡏⠀⣠⣺⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⣹⢊⡏⠐⡡⠃⠀⠀⠀⣏⢜⠞⡵⡡⢫⢊⢔⡝⡵⡡⢫⠆⠀⠀⠀⠀⠀⠀⢸
⠀⣶⣹⣧⣻⢠⣟⣿⣖⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡡⢋⢘⠔⠀⠀⠀⠀⣯⢊⢎⠜⡵⡡⢫⢪⠞⣔⡱⣡⠆⠀⠀⠀⠀⠀⠀⢸
⠀⡛⠶⠶⠟⠸⣿⡦⣾⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡱⣯⠨⠆⠀⠀⠀⠀⡷⢣⢋⢎⡜⡱⡡⢣⢎⢎⡔⡡⢊⠀⠀⠀⠀⠀⠀⢸
⠀⣧⡘⠯⢶⢠⡌⠙⠋⠁⠀⠀⠀⠰⣒⠆⡠⣒⢄⢠⡦⡄⢸⡔⡇⢰⠁⠀⠀⠀⢀⡷⠡⣣⢊⢜⠔⡱⡡⢣⠊⡔⡱⢣⠀⠀⠀⠀⠀⠀⠸
⠀⡓⠬⣑⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⡉⡈⣶⠭⠺⡿⡍⡆⢸⢜⡅⠠⠀⠀⣠⠴⣫⡗⡵⡱⢣⢊⢎⢜⣵⠥⠮⣼⣴⣡⣂⠀⠀⠀⠀⡂⣼
⠀⡃⠀⠀⠉⠑⠢⠬⣭⣙⣒⡒⠦⠤⢬⣁⣈⣒⣁⣈⣚⣁⣼⢊⢦⠴⢒⣉⡤⢺⢹⢎⠜⡕⡡⣣⠟⠉⠀⠀⠀⠀⠀⠉⠙⠒⠦⣄⡪⠁⣯
⠀⠇⠀⠀⠀⠀⠀⠀⠔⡡⠎⡩⢋⠗⡲⢒⡶⢲⠖⡴⠀⡴⢾⡡⣟⡹⡫⢡⢡⢁⢺⣣⢎⡜⡕⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠌⠀⠀⠀⠀⠀⢊⡴⢊⠔⣡⢞⡵⢋⠔⣡⠊⡠⠊⡠⢛⡱⣻⠔⡡⢣⢣⠃⢸⣡⢮⢎⢜⡇
⠀⠀⡐⠀⠀⠀⠀⠀⠡⢂⡡⠊⡴⢋⠔⣡⠞⣡⠚⡠⠊⡠⣟⢔⣭⠜⡱⢣⢣⠃⢸⣡⢣⢪⢪⡇
⠀⠀⠂⠀⠀⠀⠀⠀⠖⣉⠔⣋⠔⣡⠞⣡⠞⣡⠚⡡⢊⡔⣯⢊⣥⠜⡰⢣⢣⠆⢸⡕⡱⣕⣽⡇
⠀⢰⠨⠀⠀⠀⠀⠀⢊⠥⢚⡥⠊⡠⠊⡡⠊⡠⢊⡤⣪⠎⣥⢪⣇⡔⡱⢡⢂⠆⢸⣎⣼⣫⢞⣧⣀
⠀⢸⠩⡀⠀⠀⠀⠀⠠⠒⢉⠔⢋⡤⢊⠔⢋⠴⣫⠞⡡⢊⡷⣳⣏⠜⡴⢡⢊⠆⢸⣿⣷⢷⣿⡇⠂⠿⠤⢤⠄
⠀⠈⣄⢄⠀⠀⠀⠀⡤⢚⡡⢒⡡⠚⡡⢖⡥⠊⣡⠞⡡⢊⡗⡹⣯⠞⡠⢣⢃⠆⢸⣿⣿⣿⣾⠥⠥⠤⠭⠭⠭⠭⠭⠭⠥⠠⠤⠤
⠀⢰⢯⡣⡀⠀⠀⠀⢒⡡⠖⣁⠔⣋⠔⠉⡠⢊⡠⢊⠔⣡⣟⢼⣥⢞⡴⡡⢃⢎⢸⣓⢚⣛⣛⠒⠀⠠⠤⠭⠭⠍⠉⠉⠉⠩⠭⠭⠭⠭⠭⠤⠄
⠀⠸⡦⠙⠺⣷⢤⣈⡥⠖⣉⠴⢊⡡⠔⣋⠴⢋⠔⢁⢞⡵⣯⢪⡷⢊⠔⡡⢣⣾⣾⠹⣆⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣀⣀⣀⣀⣀⣀⣘⣂⡀
⠀⠀⠑⢄⡀⠀⠹⢻⢷⣻⢥⣖⣩⠔⣊⠥⢚⡡⢞⠵⣫⣞⡷⠣⢷⣡⣼⠴⣟⠼⢻⣼⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣀⣒⠒⠒⠚⠒⠀⠀⠀⠀⠐⠒⢒⣒⣒
⠀⠀⠀⠀⠈⠓⠲⣴⣼⡸⢹⠺⠽⣭⣗⣚⣳⢾⠷⣾⢾⢗⣗⣱⡵⡧⢾⠚⢩⣼⡞⢓⣂⣀⣀⣀⣀⣀⣀⣀⡀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠲⢶⣄⣅⣍⣙⢉⡏⣝⢉⠉⣏⣸⣠⣦⠖⢚⣛⣒⣒⣒⣂⡀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠯⠭⠭⠭⠠⢖⣒⣒⣒⡓⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠐⠠
⠀⠀⠀⢀⣤⣤⠖⠠⠀⠀⠀⠐⠁⠀⡠⠀⠀⠀⢀⣀⢤⠀⡀
⠀⠀⢠⣿⣿⡿⠦⠀⣤⣖⠞⠁⠀⠒⠀⠀⠀⠀⠈⠁⠀⠀⡀
⠀⠀⢈⣿⣿⣂⠆⠘⠁⠈⠃⣀⠀⠀⠀⠀⠀⠰⠚⠁⠀⠂
⠀⢀⣼⣿⡿⢁⠀⠀⠀⠀⠀⠸⡆⠀⢀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⣾⣿⡟⠀⠘⠀⢀⡐⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡄⠐⠀⢀⠠⠐⠒⠙⠉⣀⢠⡆
⢰⣿⡏⠀⠀⠀⠀⠀⠑⠤⠀⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣻⣻⠃
⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⠄⠀⠉⠀⠂⠀⠠⡀⠀⠀⠀⣸⣿⣿
⠀⠉⢺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢀⡄⢒⠆⠁⠀⠀⠀⢉⢽⣿
⠀⠀⠈⠀⢶⠦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠘⠒⢒⡒⠒⠶⠶⠲⣶⠦
⠀⠀⠀⠀⣁⣤⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⡍⠀⢈⡭⠉⠭⠭⠿⠿⣿⣟⠀⠀⠁
⠀⠀⠀⠀⣼⣭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠉⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠸⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠂⠀⢹⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢰⠀⠀⠉⠉⠉⠉⠀⠀⠀⠐⡄⠀⠀⠈⠉⠀⠐⠀⠐⠒⠒⠒⠒⠁⠀⠀⠐⠒⠀⠀⠀⠐⠒⡆
⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢲⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠈⠑
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠣⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠄⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠢
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠸⠀⠐⠒⠒⠒⠒⠄⠀⠀⢀⠤⠤⠤⠤⠄⣀⢐⠀⠀⠠⠤⠤⠤⢄⠀⠀⠀⠠⠄⠄⠤⠤⠄⢘⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⡀⠀⠒⠂⢄⠀⠘⣄⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠉⠒
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠒⠄⠔⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠒⠤⠀⠀⢡
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠣⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠁⠀⠀⠢⠀⠀⠒⠀⠀⠀⠓⠠⠤⠀⣀⡀⠀⠇⠀⠀⠰⠀⣀⣀⣀⣀⠠⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠇
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣤⣤⣄⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠋⠉⠉⠉⠉⠉⠉⢒⠶⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⡡⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⣀⣠⣄⣠⣴⣦⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠙⢄
⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⠁⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⠀⠀⡇⠀⠀⠀⠀⠈⢧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠘⡆⠀⠸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⢁⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠙⠿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣄⣀⣀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠠⣂⣰⠆⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠙⠛⠛⠛⠉⠉⠉⠛⠛⠉⠁⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠈⢲⡤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠤⠤⡖⠋⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢰⠁⠀⠀⢦⠀⠀⠀⡔⠁⠀⠀⠀⠘⡄⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⠬⡘⠢⢄⣀⡡⠀⢌⠀⣀⡠⠀⠀⠀⠇⠀⠀⠀⠈⠳⣄⡀⠀⠀⣀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠙⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠉⠁⠀⠀⠀⠀⢣⠀⠀⢀⣄⠀⠀⠀⠀⠀⣣⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠋⠀⠉⠁⠉⠉⠉⠉⠑⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠋⠀⠂⠙⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠁⠀⠀⠀⠈⠙⢦⡄⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⢰⣿⡿⠀⠀⣀⣤⠖⠋⠛⢇
⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢠⣤⠞⠛⢲⣿⣿⡃⠀⢀⣀⣼⡿⠷⠞⠉⠁⠀⠋⠀⠀⠈⢣⡀
⠀⠀⠀⠀⣼⣿⡿⠛⠿⣿⣷⣶⣄⣀⠀⠀⠀⠀⠀⡟⣰⡟⠁⢄⣰⣿⣿⣽⣿⠿⠟⠋⠁⠀⠀⠀⠀⡀⠀⠀⠀⠴⠄⠴⣿
⠀⠀⠀⠰⣿⣿⣄⠀⠀⠈⠙⠿⣿⣿⣷⣶⣄⡀⠘⣧⣿⣷⣶⣿⣿⠟⠛⠩⠉⠀⠀⠀⣀⡀⢀⣀⣠⣅⣤⣦⣤⡴⠶⠶⠿⢶⣄
⠀⠀⠀⠀⠛⠿⣿⣷⣤⣤⣤⣦⣤⣽⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣶⣧⡴⠾⠟⠛⠛⠛⡉⠩⠍⠀⠉⠙⠓⠒⠀⠉⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⢉⣩⣽⣿⣿⣿⣿⣿⠿⠟⠙⠉⠙⠛⠿⠿⣿⣷⣶⢒⣊⠉⠁⠀⠀⢀⣀⣀⣤⣤⠄⠂⠀⢉⡞
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠟⢻⣿⢿⡯⠿⠛⠛⠋⠉⣁⡀⠀⢀⢀⠀⠬⠘⠛⣿⣷⣦⠀⠀⠤⢭⣴⣆⣈⣥⣤⠴⠖⠒⠛⠁
⠀⠀⠀⠀⢠⡶⣛⣉⣙⡛⢤⠀⠀⠚⢉⣀⡤⠔⠚⠁⠀⠀⠀⠈⠁⠀⠀⠀⣀⣀⣭⣿⣿⣶⣒⠻⢋⠍⡉⠩⠄⢹
⠀⢀⡴⡶⣿⢾⣿⣿⣿⣿⣦⢳⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠾⠛⢛⣽⡿⠟⠙⢻⣿⣿⡆⠊⠀⠀⢀⣨⠟
⣰⢫⣮⣁⠻⣮⣙⣵⡏⣶⣿⡏⣧⠄⠀⣀⣤⣴⣶⣿⠿⠻⢉⠄⣰⣶⡾⠿⠙⠐⢁⡤⢚⣿⣿⣯⣩⠴⠚⠉
⢿⠸⣭⡍⠻⡄⡈⣫⠳⣙⢛⣡⣿⣾⣛⣩⡴⠛⠙⠀⣆⣀⣴⢺⣏⡁⡩⠖⠃⠀⠥⡀⢘⣿⣿⡟
⡼⣃⢨⣴⢷⢃⡤⢄⣐⡭⣿⢏⡵⣶⣶⣝⢷⣅⡴⣾⠿⣛⠿⣿⣯⢻⡗⠒⠂⣀⣤⠶⠛⠉⠁
⢳⣝⠴⣫⣞⡤⠶⠛⠉⠀⣿⢾⢸⣿⣿⡟⣸⢋⢼⠇⡞⣿⠇⣿⡿⢸⣧⡴⠟⠉
⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠘⢮⣓⠶⠶⣚⣵⠷⠛⢧⡳⠶⠞⣋⡴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣦⣤⣬⣅⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣵⡦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣷⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢙⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⠾⣋⢿⣽⣟⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣡⡾⣵⣿⣿⣿⣿⣿⣿⡿⣿⣿⡻⣿⢿⣿⡝⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣵⣿⣽⣿⣿⣿⣿⣿⣿⣿⣸⢸⣿⣷⢹⡇⠃⢹⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⢟⣛⣿⣿⣿⣿⣾⣿⣿⣼⡇⣀⢸⢼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⣫⣴⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣛⣻⣷⣾⣸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢏⣾⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠉⠻⢶⣭⣙⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⣯⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣾⣿⡻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣾⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣌⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣼⠷⠾⢿⣶⣶⣤⣀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡿⠿⠶⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣿⣿⠀⠀⠀⠀⠉⠛⠿⡧⠀⠀⠀⢸⠿⠛⠉⠀⠀⠀⠀⢘⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⣹⣧⢶⣦⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⣠⣶⢸⡏⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⠏⠈⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⠛⠁⡼⡇⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⡟⣫⣝⡿⢻⡟⣟⢻⡏⣯⢹⠿⡉⣿⢹⠉⡏⢹⠹⠍⡯⢹⠍⣿⣿⢀⡃⢿⠛⡟⣿⢉⡏⢹⠿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣬⣭⣿⡼⠷⠿⠾⠷⠿⠾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠾⠷⠾⠷⠶⠿⢼⣧⣾⣆⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠉⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⠀⣈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢬⠍⢩⡄⠀⢀
⠀⠀⠀⣠⠄⢠⢄⣴⠂⣭⣻⠟⣮⢪⡻⡦⡀⠀⠀⠀⠀⠀⠀⣶⠀⠰⡄⠀⠘⠉⡹⠀⡀⠀⡤⠀⠀⠀⠀⠀⠀⣠⡶⡷⠊⣴⢎⠾⠟⣐⣰⣏⡓
⠀⠀⠄⡾⠀⣿⣌⠛⠷⠶⠶⠾⠟⣰⣗⣿⠊⣷⠀⠀⠀⣤⠀⢻⣦⡀⠙⠂⠀⠀⠀⠊⠀⢠⠇⢠⠀⠀⠀⠀⣰⡿⠵⣵⢂⠁⠀⣰⢀⣤⠉⢻⢀⣦⡴⡄
⠀⢸⠂⣿⡄⠈⠙⠓⠲⠶⠶⠿⠿⠛⠋⠁⠀⣿⠘⠀⠀⢛⠧⠀⠙⠻⠦⣀⣀⠀⠀⠤⠖⢀⠀⠉⠀⠀⠀⢠⣿⣥⣦⢠⠏⣀⠘⢣⠾⢃⣷⠀⠸⡙⡧⣷
⠀⢸⠀⢈⠻⢶⣤⣤⣀⡀⠀⢀⣀⣠⣴⣶⠾⢋⠀⠀⠀⣿⣷⠘⠗⠀⠤⠄⠀⠀⠠⠤⠚⠁⣴⠿⣇⠀⠀⢸⢿⣿⠣⢆⠸⢷⡇⢰⣾⠸⠋⢀⡞⣄⠀⢾⡀
⠀⢸⡇⠰⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠈⠙⣀⠀⠀⣿⢿⡇⢀⣶⣷⡄⢠⡶⠟⠲⠶⣾⠋⠀⣇⠀⠀⢸⡄⣿⡀⣾⣷⣶⢣⣬⡍⣴⣾⢼⣿⡄⢠⣼⡇
⠀⠀⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠙⣧⣙⠋⠛⠟⠳⠞⠁⠀⠀⣰⣿⣦⣤⡿⠀⠀⠈⢿⣾⡀⡏⠀⢻⡾⠋⢳⡝⠘⠚⣹⡇⣸⣽⠇
⠀⠀⠀⠀⠉⠑⠲⠤⢤⣤⣤⣄⡤⠤⠄⠒⠉⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⣤⣤⣾⣿⠿⠟⠉⠀⠀⠀⠀⠀⠙⠻⢿⣦⣾⣴⣤⣿⣿⣦⣤⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠭⠍⠥⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠭⣭⣭⠉⠁
⠀⠀⠀⠀⣠⠞⠉⠀⣀⢀⣀⠀⢀⡈⣽⡶⡄⠀⠀⠀⠀⢀⣴⠿⠁⠀⠀⢀⣀⣀⣀⣀⠐⠬⠑⢦⡀⠀⠀⠀⠀⣠⡶⠀⠀⣈⣉⣉⣁⡀
⠀⠀⢠⠞⠁⣠⢶⡿⡤⠒⠒⠐⠶⢉⣿⣿⡀⣶⣄⠀⠀⡼⠁⠂⢀⣴⠟⣋⣩⣭⣭⣉⠻⣷⡄⠀⢻⡀⠀⠀⣼⡏⠀⠀⠉⠁⠀⠀⠀⠉⠂⠀⠀⠀⠀⡀
⠀⢠⠃⠀⠸⠁⣼⡎⢀⡤⣄⠀⠈⡎⣻⢸⡆⢉⠉⡇⠀⠃⠀⢠⡟⣡⣞⣭⠿⠿⢿⡝⢿⡌⢿⡀⠘⡇⠀⢸⢋⡴⠀⠀⠀⢀⡤⠴⠤⣄⠀⠈⢦⠀⠀⢀
⠀⢸⣤⠀⡿⣿⢹⡇⠜⣫⠽⠁⢀⡗⢻⠘⢁⡈⠀⡇⠸⠀⠀⢾⠁⣿⣿⠣⣞⠹⠆⣿⢸⣿⢸⡇⠀⡇⠀⡞⣾⠁⠐⠀⣰⠋⠀⣤⣤⠈⡇⠀⠈⡀⢠⣿
⠀⠈⣿⡆⢳⠀⣍⠃⢢⠳⡄⢀⡟⡡⠋⣀⠘⠃⢠⠃⠈⠀⠀⢾⡄⣿⡹⣧⣙⡷⣚⣫⣿⠃⣸⠃⠀⡇⠀⠠⣿⠃⡀⠀⣿⠀⠀⠑⠄⠔⠁⠀⣸⠁⢸⣼
⠀⠀⠸⣿⡄⠰⡍⢢⡀⠙⠓⠈⡁⣰⣮⠡⠀⢠⠇⠀⠀⢧⢀⠘⣷⡘⢷⣬⣛⣛⣛⣉⣡⣾⠏⠀⢠⠃⠀⠀⢠⠀⠹⡄⠘⠆⠀⠀⠀⠀⢀⡴⠃⢀⣼⠏
⠀⠀⠀⠙⠿⣦⣄⡙⠛⠉⠭⠩⠛⠋⠀⣠⡴⠃⠀⠀⠀⠘⢦⡁⠈⠳⣄⡈⠙⠛⠛⠛⠉⠀⠐⣠⠎⠀⠀⠀⠀⠐⠆⠉⠂⠀⠀⠀⢒⡛⠉⣴⢋⡹⠋
⠀⠀⠀⠀⠀⠈⠙⠺⠭⠽⡦⠤⠤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⠤⠭⠿⠒⠶⠆⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠛⠦⢤⣴⣛⡿⠀⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢶⠋⡁⠤⠤⢤⣀⠀⠉⠑⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡏⠁⠳⣄⠉⠙⠚⠉⣁⡼⠀⢄⡤⠕⢸⠉⢲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠙⣄⠙⠒⠑⠦⣭⠯⣍⣉⠥⠒⠮⢚⣀⠤⠞⠀⡎⠀⣻⢦
⠀⠀⠀⠀⠀⠀⠀⣾⠸⡜⣄⠀⣉⣉⡉⠉⠉⠉⠉⠉⠉⢉⣩⣥⣤⣴⠟⢉⣡⠔⠁⠀⢇
⠀⠀⠀⠀⠀⠀⠀⡏⡄⠘⠪⣍⣉⡉⠓⠒⠒⠉⠉⠑⠒⠉⣀⣠⠤⠖⠊⠉⠀⡔⠒⠒⣻
⠀⠀⠀⠀⠀⠀⠀⢹⠈⠉⠈⢢⡄⢉⣩⣛⠛⠛⠋⠉⣉⡉⠀⠀⠀⡺⠋⠙⠚⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠉⠉⠀⠀⠑⠂⠒⠉⠀⠈⠑⠊⠉⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠒⠒⠒⠤⠤⠤⠤⠤⠖⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠤⠀⢀⠀⠀⠈⠳⣄⡀⠒⠂
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠲⢦⡶⠟⠡⠤⠤⠥⠤⠀⠀⠀⠶⠖⢖⡖⣶⡂
⠀⠀⠀⠀⠐⠒⠒⠬⠤⢤⣄⣀⡀⠀⢀⡶⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣶⢘⡏⢅⣀⡀
⠀⠀⠀⠀⠐⠲⠤⣤⣄⡀⠀⠀⠈⠉⠙⡋⡛⠿⠻⠿⠿⠷⠿⡿⣾⣶⣶⣾⢤⣷⣜⣻⢴⣀⠀⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠒⠧⠂⣦⣤⣄⣠⣄⡀⠀⠀⠀⠀⠀⠉⠉⢫⣝⡧⠇⠛⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠒⠒⠢⠶⢦⣠⣘⡛⠪⠈

⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⣸⠇⠸⠇⢀
⠀⠀⠀⠀⢀⣿⠀⠀⠀⣛⢀⡀⣀⡀⢀⡀⣀⡀⠀⣀⣀⡀⢀⡀⣀⡀⢀⡀⢀⣀⣀⢀⡀⢀⣀
⠀⠀⠀⠀⠸⣿⠀⠀⢸⡏⢸⣿⢩⡇⢸⣿⢩⡇⢸⡇⢹⡇⢸⣷⢻⣷⢻⡇⣾⠉⢻⢺⣷⠊⡿
⠀⠀⠀⠀⠀⢿⣀⣴⠸⣇⣼⡇⠸⣇⢼⡇⠸⣇⣸⣇⢼⣇⢸⡇⢸⡏⢸⣇⢿⡀⡾⢸⡏⢸⣇⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣩⣤⠀⠀⠀⢣⡀⣠⠮⣍⡻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠙⠟⠁⢸⢠⣶⠛⠛⠛⠛⠶⢦⣍⣹⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠹⣧⠀⠀⢸⠀⠉⠛⠶⠶⠂⠀⠀⠈⢻⣿⡟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠁⠀⠀⠈⠳⢦⣼⣇⡀⠀⠀⠀⠀⠀⢀⣠⣿⡿⠀⠈⣿⣧⣄⣀
⠀⠀⠀⠀⢀⣤⠞⠋⢹⣿⢀⡴⣶⡀⣠⣤⣌⣉⠟⠛⠻⠿⠿⠿⠛⠛⠁⣀⣠⡾⠟⢹⡏⠙⠳⣦⣄
⠀⠀⢀⣴⠟⠁⢀⣠⣾⣇⣸⠁⢸⡇⡏⠀⢀⣴⠶⠖⠒⠶⠶⠶⠶⠚⠛⠉⠁⢀⣤⣼⡇⠀⠀⠀⠙⢷⣄
⠀⢠⡿⠁⠀⣠⠟⠁⠸⣿⠃⠀⢸⣧⡇⠀⣿⢁⣶⠷⢦⣄⠀⠀⣠⠖⠛⢻⡇⢸⠃⠘⡇⠀⠀⠀⠀⠀⢻⣆
⠀⢸⡇⠀⢰⡏⠀⠀⠀⠻⣧⡀⠀⠛⠁⢸⡇⠘⡏⠀⠀⠙⠛⠋⠁⠀⠀⠀⣧⣼⠀⢰⡇⠀⠀⠀⠀⠀⠘⣿
⠀⢸⣧⠀⠈⢻⡄⠀⠀⠀⠙⠻⣶⣤⣄⣸⣷⢰⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣴⡿⠁⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠻⣧⡀⠀⠉⠓⠦⠤⣀⠀⠀⠉⠉⠛⢻⣿⣁⡀⠀⠀⠀⣀⣀⣠⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠈⠻⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⠿⠶⣶⣶⣶⣦⣤⣴⣶⣶⣶⡶⠾⠿⠟⠛⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣤⣠⠠⢤⡄⣤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡇⠀⠀⠀⠀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠴⠒⠂⠉⠠⠄⠀⢑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⡚⢀⡃⠄⢀⣀⣀⣮⢉⣤⡆⠘⡆
⠀⢀⠤⠒⠈⠁⠀⠀⠀⠀⠀⠀⡠⠀⠄⠀⠘⠀⠀⢀⠀⢀⣠⢀⣴⣖⡶⣻⠇⠸⠀⢰⡏⠳⠅⡹⠭⠟⠃⢠⡷
⠀⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⡐⠁⣰⠃⣞⢀⣯⣿⣟⡸⠀⠀⠀⢸⣶⢤⣬⡄⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠈⠑⠈⠀⠂⠀⠀⠀⠊⠈⠀⠀⠀⢰⠁⡼⡯⡝⠓⠛⣾⠛⠈⢀⠜⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⢀⠇⢘⣿⡥⠅⠛⠊⣀⠤⠒⠁
⠀⠀⣀⠴⠚⠉⠀⠀⡀⢰⠁⠉⠢⡄⠀⠀⡜⠀⠈⣀⠠⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠐⠀⢖⠋⠀⠀⠐
⢀⠜⠁⠀⠀⠀⠀⠀⠄⢨⣀⠀⠀⠀⠐⠐⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⠀⠲⡚⠀⠀⠀⠛⠀⠀⢀
⣬⠐⠒⠢⡀⠀⢀⡠⠔⠋⠏⠀⡠⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠙⠭⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈
⢀⠀⠀⠠⠾⠩⣉⠀⠀⠀⠂⠀⠀⠀⢀⠀⢤⣴⠀⠀⠙⠇⠀⠀⠀⠀⠀⠄⠂
⠈⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠒⠯⠀⠀⠉⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⠀⣀⠄⠐⢦⢤⡤⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠔⠀⣾⡿⠁⠀⣤⠟⠉⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⡎⠀⠀⠋⠀⠀⠀⠋⠀⠀⠀⣰⠗⠋
⠀⠀⠀⡠⠐⠈⠹⠯⠥⣤⡒⡄⢀⠄⠈⡠⠴⢒⠒⡶⠤⢤⡉⠢⡀⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴
⢀⡔⢋⣀⣈⡛⠓⠶⣄⡬⠞⠁⠈⠀⠘⠃⣻⠒⡓⢀⣖⡮⠋⠀⠀⠀⡐⠉⠑⠒⠒⠂⠀⠒⠒⠒⠊⠉⡁⢠⠃
⡞⣤⣦⣴⡪⡉⡣⠖⠁⠀⠀⠀⠠⡀⠁⠀⠀⠈⠉⠉⠁⠄⠁⡠⠂⠀⠸⣄⡀⠀⠀⠀⠀⠀⡁⠀⠡⠀⣰⠏
⠈⠂⠈⠉⠁⠋⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣄⣀⣀⣠⡤⠔⠊⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠒⠒⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠟⠋⣩⣥⣀⠹⣻⣶⣾⣛⠻⣿⡛⣛⡓⠲⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⡿⣿⣿⣷⣤⣭⣭⡿⢀⣼⣿⣿⣿⡆⠈⣽⣿⡙⢿⣦⡈⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡏⣰⠏⣸⣿⣦⡉⠛⠛⠉⠠⡌⣋⣿⣿⡿⠀⠀⢻⣿⡧⠈⡈⢻⣦⠀⡉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣱⡏⢀⣿⠄⢻⣿⣿⣏⢈⠀⣀⣢⣼⣿⣿⡿⠉⠀⣾⣺⣿⠇⠀⠀⢸⣿⣿⠀⠠⠌⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢠⣿⡀⠘⣿⣵⠈⠻⠿⣿⣿⣿⡿⠿⠟⠉⠉⠀⣄⣾⣿⡿⡏⠀⠀⣴⣾⣿⡿⠆⠀⢹⡼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⣷⠀⠚⣿⣷⣤⣀⡀⢀⡀⠀⣄⣀⣶⣼⣿⡿⠿⠋⠀⠀⠀⣀⣭⣿⣿⠏⠀⠠⣼⠇⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠘⢻⣶⣄⠈⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⣀⣄⣿⣿⡿⠇⢀⠀⢀⣾⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⡣⡀⠀⠙⢿⣷⣦⣠⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣠⢠⢤⣾⣾⣿⠿⠁⠀⡀⢀⠀⣾⡇⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍⠀⢜⢦⡀⠀⠈⠙⠛⣿⣷⣶⣤⣶⣶⣶⣾⣯⣷⣿⡿⠟⠛⠓⠈⠀⠀⠂⠠⣢⢞⣝⣱⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠕⠍⠰⠆⠀⠀⠀⠀⠈⠙⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠐⠈⠅⣪⠔⣃⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣐⠀⠀⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣠⠦⡂⣴⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣄⣠⣦⡼⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠧⣴⣤⣀⣐⣤⣤⣄⣄⣤⣤⣄⣤⣶⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⢶⢶⣶⣶⣶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢹⣏⠲⣾⣹⣏⣏⣇⣧⣿⣧⣧⣧⣦⣼⣬⣾⣼⣼⣿⣴⣼⣏⣫⣇⣇⣼⠹⣿⠁
⠀⠀⠀⠀⠀⠀⣀⣛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⡉⡛⢃
⠀⠀⢠⠀⡴⣶⡻⠿⢟⡿⢿⣶⢀⠀⠀⠀⠀⣶⠀⢢⡀⠉⠹⢂⠀⢠⠀⠀⠀⠀⢀⣴⣿⢩⡞⠜⠃⢴⣮⠗
⠀⠀⣿⠈⠷⣬⣉⣛⣯⣵⣿⠟⠘⡇⠀⢰⣇⠘⢷⣄⠁⠀⠀⢀⡀⠈⠰⠀⠀⠀⣾⢋⠛⡕⠀⡜⣰⢆⠘⣸⢿⣃
⠀⡀⠙⠷⣤⣀⡀⠀⠀⢀⣠⣤⡾⡃⠀⢰⣶⡔⠄⠈⠉⡁⠀⠩⠔⢠⣶⡆⠀⢸⣿⡿⠌⠴⣷⢸⣥⠾⠀⣼⡈⢽
⠀⢷⡀⠁⠀⠉⠉⠉⠉⠉⠉⠀⠈⣡⠄⠸⣿⣇⣠⣾⡆⢰⡞⠒⢶⡟⠀⣇⠀⢸⡌⡇⣸⢷⡞⣮⡅⣶⡾⣷⠑⣼⡇
⠀⠈⠛⠦⣀⡀⠀⠀⠀⠀⠀⡀⠴⠋⠀⠀⠙⢦⣤⣹⣍⣉⢀⣠⣾⣿⡶⠋⠀⠀⠹⢧⣿⣀⣟⡁⣽⣅⢉⣿⣤⠿⠁
⠀⠀⠀⠀⠀⣈⣉⣉⡉⢉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠙⠛⠛⣛⡉⠁⠀⠀⠀⠀⠀⠀⠉⠙⣛⣛⣛⠛⠋⠁
⠀⠀⢀⡴⠋⢁⡠⢤⠀⠀⣹⣷⠄⠀⠀⠀⣰⠞⠉⢀⣀⣠⣤⣀⣐⠉⠢⡀⠀⠀⢀⣴⠀⢀⣉⠭⣁⡀
⠀⠠⠋⢠⢚⣟⠊⠉⠈⢃⢼⣿⡘⠷⡄⢰⠃⠁⡴⢛⣥⣶⣶⣦⣝⢷⡀⢹⠀⢠⡿⡀⠀⠀⠀⢀⠀⠈⠠⡀
⠀⢀⠀⣶⡜⡇⠼⣻⠇⢨⠴⡟⢁⠀⠁⢸⠀⢸⢣⣿⣟⡝⣯⢳⢹⡎⡇⠈⠀⠸⣼⠁⠀⡰⠊⢉⡉⢳⠀⢙⠀⣴
⠀⢸⣇⢢⢠⡁⢮⣢⢠⡟⠜⡡⠛⢀⠇⠘⠀⢸⡘⣿⢧⣟⣒⣻⡿⢡⡇⠀⠀⠰⠿⢀⠀⣷⠀⠙⠁⠊⠀⡼⠈⣿
⠀⠀⢻⣆⠑⢵⡦⠉⠅⣤⠾⠒⣀⠎⠀⠈⢇⠌⢷⡙⠷⣯⣭⣥⠶⠋⠀⡜⠀⠀⠰⠈⠧⡈⠀⠀⣀⡤⢞⡡⣴⠏
⠀⠀⠀⠉⠛⢶⣠⣬⣁⣀⠤⠞⠁⠀⠀⠀⠈⠢⠄⣉⣒⡤⠤⠄⠠⠴⠊⠀⠀⠀⠀⠀⠠⠦⣄⣠⣶⡆⠻⠒⠁

⠀⠣⠤⠴⡆⠀⠀⣀⢀⡀⠀⠀⠶⠤⠤⠃⠀⠘⠤⠤⢶⠀⠀⢀⡀⣀⠀⠀⣰⠦⠤⠜⠀⠀⠣⠤⠴⡆⠀⠀⣀⢀⡀⠀⠀⡶⠤⠤⠃
⠀⠀⠀⠀⠇⠀⠘⡇⢀⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⣸⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⢀⠇⠀⠀⠇
⢸⡉⠉⢙⠋⠐⠒⠋⠈⠓⠒⠈⢋⠉⠁⡹⠀⢏⠈⠉⠻⠁⠒⠚⠁⠙⠒⠂⠹⠉⠉⢉⠇⠸⡁⠉⢩⠉⠐⠒⠋⠈⠓⠒⠈⢋⠉⠁⡹
⠔⠃⠀⢸⣀⠤⢄⠀⠀⢀⠤⣀⣸⠀⠀⠓⠠⠚⠀⠀⣇⡠⠤⡀⠀⠀⡠⢄⣀⡆⠀⠘⠢⠔⠁⠀⢸⣀⡤⢄⠀⠀⢀⠤⣀⣸⠀⠀⠓
⡀⠀⠀⢠⠤⣄⡰⠀⠀⠸⣀⡤⢤⠀⠀⢀⣀⠀⠀⠀⡤⢤⣀⠅⠀⠀⢄⣠⠤⡀⠀⠀⢀⡀⠀⠀⢠⠤⣄⡸⠀⠀⠠⣀⡤⢤
⢨⠆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢏⠀⡹⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠸⡁⢈⠆⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢏
⠘⠂⠒⠒⡓⠈⠉⡗⠰⡋⠉⠐⡗⠒⠀⠚⠀⠓⠐⠒⢺⠀⠉⢹⠆⢾⠉⠁⢺⠒⠒⠐⠃⠘⠂⠒⠒⡗⠈⠉⡷⠰⡏⠉⠐⡗⠒⠀⠚
⠀⢀⣀⡀⡇⠀⠘⠧⠠⠇⠀⠀⣁⣀⣀⠀⠀⠀⣀⣀⢸⠀⠀⠸⠄⠼⠀⠀⢼⢀⣀⡀⠀⠀⢀⣀⣀⡇⠀⠀⠧⠠⠃⠀⠀⡇⣀⣀
⠀⢇⣀⣨⡅⠀⠀⠀⡀⠀⠀⠀⣭⣀⣀⠇⠀⠸⣀⣀⣭⠀⠀⠀⢀⠀⠀⠀⢨⣅⣀⡸⠀⠀⢇⣀⣨⡅⠀⠀⠀⡀⠀⠀⠀⣭⣀⣀⠇
⠀⠀⠁⠀⡇⠀⠠⡏⠉⡆⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⢸⠉⢱⠀⠀⢙⠀⠈⠀⠀⠀⠀⠁⠀⡇⠀⠀⡏⠉⡇⠀⠀⡇⠀⠁
⢰⠒⠒⠲⠆⠦⠤⠟⠘⠦⠤⠆⠗⠒⠒⢲⠀⡖⠒⠒⠾⠰⠠⠼⠃⠻⠤⠴⢸⠒⠒⠒⡆⢰⠒⠒⠒⠧⠦⠤⠟⠘⠧⠤⠖⠗⠒⠒⢲
⡨⠇⠀⢸⠀⣀⡀⠀⠀⠀⣀⠀⢸⠀⠀⢧⣀⠽⠀⠀⡆⢀⣀⠀⠀⠀⢀⡀⠈⡄⠀⠸⣄⡠⠇⠀⢸⠀⣀⡀⠀⠀⠀⣀⠀⢸⠀⠀⢧
⠀⠀⠀⢈⣉⡁⢘⠀⠀⢸⠀⣉⣉⠀⠀⠀⠀⠀⠀⠀⣉⣉⠈⡇⠀⠀⡁⢈⣉⠁⠀⠀⠀⠀⠀⠀⢈⣉⡁⢸⠀⠀⢌⠀⣉⣉
⢑⡆⠀⢸⠀⠉⠁⠀⠀⠀⠉⠁⢸⠀⠀⡖⠉⢲⠀⠀⠇⠈⠉⠀⠀⠀⠈⠉⠀⠃⠀⢰⠊⠑⡆⠀⢸⠀⠉⠁⠀⠀⠀⠉⠁⢹⠀⠀⡖
⠸⠤⠤⠤⠆⠖⠒⡦⢠⠖⠒⠆⡦⠤⠤⠼⠀⠧⠠⠤⢴⠰⠒⢲⡄⣴⠒⠲⢸⠤⠤⠤⠇⠸⠤⠤⠬⡆⠆⠐⣦⢠⠖⠒⠆⡦⠤⠤⠼
⠀⠀⠀⠀⡇⠀⠰⣇⣀⡇⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⢸⣀⣸⠀⠀⣬⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣇⣀⡇⠀⠀⡇
⠀⡎⠉⢑⠃⠀⠀⠀⠁⠀⠀⠀⣓⠉⠉⡄⠀⢰⠉⠉⣚⠀⠀⠀⠈⠀⠀⠀⢙⡊⠉⢱⠀⠀⡎⠉⢑⡃⠀⠀⠀⠁⠀⠀⠀⣓⠉⠉⡆
⠀⠈⠊⠁⡇⠀⢀⡔⠒⡄⠀⠀⡋⠉⠋⠀⠀⠀⠙⠉⢹⠀⠀⢰⠒⢢⠀⠀⠺⠉⠙⠁⠀⠀⠈⠋⠉⡇⠀⠀⡖⠒⡄⠀⠀⡏⠉⠋
⢠⠤⠤⢤⡅⣀⣀⡧⠰⣅⣀⡀⡧⠤⠤⢤⠀⡤⠤⠤⢬⢀⣀⣸⠆⢾⣀⣀⢸⠤⠤⠤⡄⢠⠤⠤⠤⡇⣀⣀⡷⠰⣇⣀⡀⡧⠤⠤⢤
⢘⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣎⠀⣳⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⡁⢘⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣎
⠉⠀⠀⠘⠒⠊⢑⠀⠀⢰⠉⠒⠚⠀⠀⠈⠉⠀⠀⠀⠓⠒⠉⠆⠀⠀⠊⠑⠒⠃⠀⠀⠈⠉⠀⠀⠘⠒⠊⢱⠀⠀⠐⠉⠒⠚⠀⠀⠈
⠢⡀⠀⢸⠉⠒⠊⠀⠀⠈⠒⠋⢹⠀⠀⡤⠒⢤⠀⠀⡏⠑⠒⠁⠀⠀⠑⠚⠉⡆⠀⢠⠔⠢⡀⠀⢸⠉⠒⠊⠀⠀⠈⠒⠊⢻⠀⠀⡠
⢰⣁⣀⣨⣄⡀⠤⣄⢀⡤⠄⣀⣌⣀⣀⣱⠀⣎⣀⣀⣠⡀⠠⢤⡀⣠⠤⢀⣰⣀⣀⣈⡆⢰⣁⣀⣘⣄⡀⠤⣄⢀⡤⠄⣀⣌⣀⣀⣱
⠀⠁⠁⠀⡆⠀⢠⡇⠈⡆⠀⠀⡇⠀⠀⠁⠀⠈⠀⠀⢸⠀⠀⢸⠁⢹⠀⠀⢸⠀⠈⠈⠀⠀⠁⠁⠀⡇⠀⠀⡏⠈⡆⠀⠀⡇⠀⠈⠁
⠀⡔⠒⠶⠇⠀⠀⠉⠊⠁⠀⠀⠶⠒⠒⠄⠀⠠⠒⠒⠾⠀⠀⠈⠑⠉⠀⠀⠹⠔⠒⠢⠀⠀⠔⠒⠶⠇⠀⠀⠉⠊⠁⠀⠀⠷⠒⠒⠄
⠀⡀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣮⣑⣶⢹⣿⡝⢧⡭⣕⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠏⡎⢿⣦⢉⣀⣾⡿⠃⣼⡇⠈⣷⡈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⣤⣉⣛⣉⣡⣴⠾⠋⠀⣾⡿⠁⡸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠙⢦⣄⣉⠉⠉⠁⣀⣠⣴⡾⠛⠁⢠⢇⡇⠀⠀⠀⣀⠤⣐⠒⠒⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠑⠀⠈⠙⠛⠛⠛⠛⠉⠁⠀⠀⠈⣳⠞⠀⠀⢴⡞⢡⡞⢡⣛⡒⣍⠻⣦⠙⡔⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⣀⣠⡤⠞⠁⠀⡠⣾⠘⣧⡈⠛⠶⣤⣴⠟⢰⡿⠀⣿⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⣇⠘⣦⣄⡙⠓⠶⠶⠶⠾⠟⢁⣾⣿⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢆⣀⣀⣀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⠙⠻⠷⣶⣶⣦⣴⡾⠿⠋⠁⣀⡏⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣶⠋⣠⠞⣩⠴⢖⣒⠶⢬⡙⢢⡉⠢⡀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣣⠃
⠀⠀⠀⠀⠀⠀⢀⠏⡿⠀⣿⠀⢧⣤⣈⣛⣿⠀⢻⠀⢻⣆⢹⡄⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⠀⠀⠀⢀⣤⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠘⠀⣿⡀⢹⣆⡀⠉⠉⠋⠁⣰⡿⠀⢐⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠅⠀⠘⣷⣀⠙⠻⠿⠷⠾⠿⠛⣡⣬⣾⠏⡰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢒⡄
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠛⢷⣦⣴⣤⣤⣶⣴⡾⠟⣡⣖⣽⡟⠀⠀⠀⣀⣤⡤⠤⠤⠤⢤⣒⣚⡓⠿⠏⠩⠄⠚⠛⢣
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠁⠉⠀⠀⠀⢈⢤⣾⠟⠁⠀⠀⢰⣟⡓⠒⡚⢛⣩⣍⣤⡤⠴⠶⠶⠗⠒⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠦⣦⣴⣦⣦⣴⣶⠿⠟⠁⠀⠀⠀⠀⠈⢛⡿⠟⠋⠉⣽⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⡔⢨⡥⣠⠀⡩⣄⣀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡊⠀⢾⣷⣄⣸⠀⠈⠉⠛⠛⠁⢈⣿⣿⠆⠀⠱⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠀⣶⣄⠀⠀⠀⠈⠉⠛⠛⠛⠓⠲⠓⠚⠛⠉⠁⠀⠀⠀⣸⣷⠈⢧⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣿⣀⠈⠙⠻⠿⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠟⠛⢁⣀⡿⠁⣸⠇⢂
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠉⠛⠿⠶⣶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⠶⠟⢋⣁⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⣉⣉⣉⣉⣩⣭⠭⠤⠶⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠤⠤⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠤⠀⢀⠀⠀⠠⠤⠤⠄⠐⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠂⠀⠀⠀⠀⢀⠀⠀⠈⠈⢁⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣉⣉⣁⣬⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠉⠉⠛⡇⠀⠀⠀⠀⠠⢤⣀⡀⠉⠛⠦⡀⠀⠈⠂⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⣤⣀⡀⠀⠈⠓⠦⠀⠀⠀⠀⠀⠀⠀⢰⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠞⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⣀⠤⠀⠠⠠⡄
⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠃⠀⠀⠀⢈⡅
⠀⠀⠀⠀⠈⠙⠻⠷⣠⣤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠐⠢⠀⠀⡀⠀⠀⡐⠉⠁⣴⠀⠀⠀⢸⠃
⠀⠀⠿⣶⣄⡀⣀⠀⢀⣍⣉⣉⣵⠖⠂⠀⠀⠀⠀⠤⡀⠈⠑⠀⠀⠀⡀⠤⠒⠉⠀⡠⠶⢃⡀⡠⠉⠀⠀⢀⡿
⠀⠱⣦⣄⣈⠁⠙⠃⢠⠇⠀⠈⠀⠀⣀⡀⠐⠒⠂⠀⠀⣀⡠⠐⠂⠁⠀⠀⠀⠀⢸⠀⠀⠈⡘⠃⠀⠀⣠⡎
⠀⠀⠀⠉⠉⠈⠛⠃⠉⠀⠀⠀⠂⠀⠀⠀⠀⠀⠔⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⠛⣧⢠⡎⠁⠀⠀⠠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠖⠒⠁⠀⠀⠀⠀⡀⠀⢄⣀⠀⠀⠀⠀⠀⠸⢀⣿⠖⠀⠀⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣔⠁⠀⠀⠀⠀⠈⠉⠐⠄⠀⠀⠀⠙⠿⣤⡤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢻⣿⣿⣶⣦⣀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠉⠛⠿⣶⣬⣛⠶⠂⠈⢀⣠⠞⠀⠀⠀⡀⢀⠚⠉⠉⠉⠙⣄⡀
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠋⠁⠀⠀⠀⠀⡌⠀⠀⠽⠖⣶⠦⡔⠉⠹⡄
⠀⠀⠐⣄⣀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣄⡄⠀⠀⠙⡾⠃⠀⢰⠇
⠀⠀⠀⠈⠙⠛⠻⠥⠬⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⢀⢀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⣠⣀⠶⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠤⠤⠤⠤⡤⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡚⠋⠉⠉⠉⡐⠄⡄⠄⢀⠀⠠⠀⠘⡈⠆⢂⠀⡀⢉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠊⡰⠡⠂⠀⠀⠘⠈⠈⠀⠀⠀⠀⠀⠀⠀⠘⠈⡆⢣⠀⢣⠠⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⠆⠀⠀⣀⡀⣀⣀⣀⣤⣀⡤⢤⣤⢤⣀⡤⣄⣠⣤⣀⣀⣀⠀⠀⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⡬⢴⠚⠛⣩⣝⠁⠀⠉⢀⣉⣀⠀⠀⠀⠀⠀⠀⣁⠀⠈⠉⠉⠛⠒⢦⣀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠾⠃⠀⠀⠀⠀⠁⠈⠀⠀⠰⠋⠀⠀⠀⣠⣄⠀⠀⠋⠈⠇⠀⠀⠀⠤⡄⠀⠈⢧⣸⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⣳⠛⠀⣠⠖⠀⠀⠠⠛⠂⠀⠀⠤⡄⠀⠎⠁⢀⠀⠀⠀⠀⠀⡠⢤⠀⠀⢳⠄⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠧⠀⠐⠇⣀⠀⠀⠀⠀⣠⣄⣀⣤⡀⠀⠀⠀⠘⠒⠢⠔⠀⡀⠀⠈⠀⠀⠀⠀⣀⠀⠀⢹⠄
⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢀⠞⠉⢀⣤⠏⠉⠀⢀⡟⢀⣽⠿⠛⠒⣶⣦⡀⠀⢀⠉⠁⢠⣀⠀⠀⠀⠉⠃⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⣖⠈⠃⠀⣴⠃⠀⠀⣠⠞⡰⠋⠀⠀⠀⢠⠏⢀⡽⢻⡟⠣⣤⡄⠈⠓⠀⠀⠀⢀⡀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⣨⣦⠶⠛⢁⠞⠁⠀⠀⠀⣠⠏⢠⠎⢠⠞⢠⡾⠉⣧⠀⢀⣄⠀⠀⠉⢻⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠸⣅⠀⠀⠀⢀⣄⠀⠻⡀⣠⠖⠁⠀⠀⢀⡠⠞⣡⠔⢁⠔⠁⡴⠋⠀⠀⢣⡀⠀⠘⠇⠀⠀⠉⠀⢘⡇
⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⢸⠆⠀⣴⠟⣁⣀⡠⢤⣒⡭⠖⢊⡡⠖⢁⡠⠞⠁⠀⠀⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢈⡗⠀⠀⠀⠈⠁⠀⠈⣿⣭⠶⠛⠉⠀⣠⠖⣉⡤⠖⠋⠀⠀⠀⠀⠀⠀⢸⠃⠀⠉⠀⠀⠀⢤⡀⠈⢧
⠀⠀⠀⠀⠀⠀⢘⡦⠀⣀⡀⠀⡀⠀⠰⡟⠁⠀⣠⠔⠋⠀⠸⠧⠴⠒⠒⠚⠛⠒⠲⢤⣀⠘⣤⠀⢰⠂⠀⠀⠘⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠘⡆⠘⢉⣧⠀⠙⠀⠀⡟⠀⠸⣅⣀⣀⣀⣠⠤⠴⠒⠒⠛⠛⠛⠒⠦⣌⣻⡁⠀⠘⠒⠂⠀⡖⠆⢀⠇
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠚⠁⠀⠀⠺⢤⣠⣤⣀⣀⠀⠀⠀⠀⢀⣀⣠⡤⠴⠖⠒⠲⠾⣿⠗⠀⠔⠶⠀⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⢰⣉⠀⠀⠠⣾⣁⣠⣤⡭⢭⠭⢭⣭⣭⡥⠤⠴⠒⠛⢻⣦⡤⠇⠀⠀⠀⠀⢀⡤⠀⠐⣏
⠀⠀⠀⠀⠀⠀⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠛⠉⠉⠲⠒⠲⠴⠒⠲⠤⠖⠻⠋⠀⠀⠀⠀⣀⣄⠈⠃⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⢯⣀⠀⠀⢀⠤⢤⠀⠀⠀⠀⠀⠲⡆⠀⠀⠀⠀⢀⡠⢄⠀⠀⠰⠚⠀⠀⢀⡴⠃⠀⠀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⠀⠈⠛⠂⠀⠀⠀⠀⠀⡄⡀⠀⠀⠏⠀⠈⠻⠀⠀⠸⣤⣀⡈⠁⠀⣀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠦⢴⣄⣀⣄⣀⣀⠀⠀⣉⠁⢀⡀⢀⣀⣠⣤⣄⣀⣤⣀⣨⣤⣤⠾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠉⠛⠉⠉⠉⠈⠁⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠉⠉⠀⠀⠀⠈⠉⠙⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠄⠀⠀⠀⠀⠐⠀⠄⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⡀⠀⠐⠄⠀⠀⠀⠀⠐⢀⠀⠀⠐⠀⠀⠀⠀⠀⡦⠀⠈⢆
⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢂⠀⠀⠀⠐⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠠⠀⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠸⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠃⠀⠈⠐⠀⠀⠀⢀⠀⠀⠀⠂⠀⠠⠐⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠁⠀⠠⠄⠀⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⠀⠀⠐⠀⠀⠀⠂⠀⠴⠀⠈⠀⠀⠀⡀⠀⠁⠂⠀⠄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠠⠐⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡎
⠀⠀⠀⠀⠀⠘⣗⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢊⠏
⠀⠀⠀⠀⠀⠀⠈⠢⡈⠑⠒⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠤⠔⠒⠉⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠤⠄⣀⣀⣀⣀⡀⠤⠤⠤⠔⠒⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⢋⠉⢉⡛⢯⠉⠳⠶⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢰⠞⠁⠀⠰⣎⣀⣂⣀⡼⡃⣠⠀⠀⠙⡌⠒⠶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠶⠋⠁⢀⡏⠀⠀⠀⠈⠳⠛⠯⠕⠶⠋⠁⠘⠁⢀⡟⠠⠀⠀⠙⠢⡈⢉⠓⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⣱⠋⠀⠀⡀⠘⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢩⡇⠀⠀⢀⠄⠀⠘⡄⠀⢁⠀⠉⠲⡄
⠀⠀⠀⠀⠀⡠⠞⣱⡟⠁⢰⠿⡄⠀⠄⠉⢀⠀⠙⠻⢯⣝⣓⡒⣒⣒⣛⣉⣡⡤⠔⠋⠀⠀⠀⢀⡀⠀⠀⣧⠀⠀⠀⠀⠀⠘⣦⣀
⠀⠀⠀⢠⠎⠀⢰⠃⡇⠁⠈⢧⠳⣄⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⣠⠞⢻⠄⠀⠈⠀⠀⢀⢿⡎⠳⡀
⠀⠀⢠⡏⠀⠀⢸⠀⡇⠀⠀⠀⠙⢮⡳⢤⣀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠒⠀⢀⣀⣀⡤⠴⠚⢁⡤⠋⠀⠀⠀⠀⣠⠎⣸⠁⠀⡇
⠀⠀⣼⠀⠀⢐⠘⡄⠘⢦⡀⠀⠁⠀⠙⠳⢬⣝⣓⠖⠶⠦⡤⠤⠤⠶⠶⠒⠛⠋⣉⣉⣁⣠⡤⠶⠚⠁⠀⠀⠄⠀⢀⠔⢁⡴⠁⢀⠼⡇
⠀⠀⢿⣧⠀⠈⠱⡘⢦⡀⠙⠢⡀⠀⠂⠀⢠⠀⠉⠉⠛⠓⠚⠒⠒⠚⠒⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⣠⠴⣋⡴⠋⢀⡴⠋⠀⣷
⠀⠀⠈⡏⠳⣄⠀⠀⠀⠙⠲⢤⣌⠑⠠⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⢀⣀⣠⠴⢚⣥⠖⠉⢀⡴⠋⠀⠀⠀⡏
⠀⠀⠀⢣⠀⠈⠓⠦⣀⠀⠀⠀⠀⠉⠙⠓⠲⠬⣍⣉⠋⠐⠒⠒⠒⠒⠒⠖⠒⠒⠒⠚⠛⠉⠉⠉⣁⡤⠞⠋⢀⣠⠖⠉⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠈⠇⠂⠀⠀⠀⠉⠲⣦⣄⡀⠀⠀⠀⠀⠀⠈⠁⠘⠒⠚⠶⠴⠦⠴⠤⠤⠤⠶⠖⠒⠛⣉⣁⣤⠴⠚⠉⠀⠀⠀⠀⠀⠐⡠⠋
⠀⠀⠀⠀⠐⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠂⠤⠄⣀⣀⣀⣀⣀⣀⣀⣀⠀⠠⠤⠤⠒⠒⠛⠋⠉⠁⠀⠀⠀⠀⢀⠠⠂⠀⠀⠊
⠀⠀⠀⠀⠀⠈⠓⢤⡈⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠂⣁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣈⠉⠒⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠴⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⢤⣄⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁




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

⠀⠀⢀⠔⠁⠀⠀⠀⠀⠁
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⠄⠀⢢⣄⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⢠⠠⠀⢀
⠀⠀⠀⠀⠗⡢⠠⠸⠄⠈⠲⠀⠀⠀⠀⢸⠀⠾⠀⡙⢷⣄⠐⠇⢸⠃⠀⠀⠀⠀⠠⠃⠐⠐⠘⠃⠈⠘
⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠇⠀⠊⠀⠀⠉⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⠀⠈⠂⠄⠀⠀⠀⠀⠠⠊


⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡦⢴⠟
⠀⠀⢰⠀⠀⢈⣷⣺⡋⠀⠈⣧⠀⠀⠀⠀⠀⠀⣄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣿⣧
⠀⠀⢻⠀⠀⢺⣿⣿⡗⠀⠀⡼⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷
⠀⠀⠈⠃⠀⢿⡿⢿⡿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⢸⣷⡾⡿⢧⣼⠿⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠘⠛⠒⠚⠛⠚⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠡⠀⠇⠼⠨⠃
⠀⠀⢠⠁⠀⢶⣿⣿⣿⣿⠦⠀⢸⠀⠀⠀⠀⢀⠀⢀⣤⡄⡀⣤⢠⡄⢤⡄⣠⣀⢠⣠⢠⣤⡄⠀⢀
⠀⠀⠀⠃⠀⠂⠀⠀⠄⠦⠄⠀⠚⠀⠀⠀⠙⠛⠋⠛⣿⡧⠆⡏⡍⠧⡜⠇⠾⠯⠼⡇⣿⣷⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠇⠘⠤⠸⠖⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠂⠉⢠⣤⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠤⠒⠉⠁⠀⢀⣠⡴⠞⠋⠁⠀⣠⠔
⠀⠀⠀⠀⠀⠀⠰⠆⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⣾⣟⠁⢀⣀⣤⠖⠛⠉⠀⢀⣠⣤⡶⠖⠃
⠀⠀⠀⠀⠀⢀⠛⠛⠛⠒⠶⠶⠦⢤⣿⡻⣦⣒⠉⠱⠋⣿⢒⣽⡿⡏⢀⣠⣴⣾⠿⠋⠁⠀⠈⠁⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠛⢳⣶⣤⣀⠤⠐⠂⠉⠻⣽⡌⣿⡆⡰⣻⣟⣁⡾⣿⣿⣿⡷⠦⠤⣤⣤⣄⣀⣀⣐⣶⣴⢰⣿⡦
⠀⠀⠀⠀⠀⠰⢤⠄⠉⠉⠀⢠⣶⣾⠿⠷⠾⣷⡝⣿⣶⣻⡿⠿⢭⣥⣍⣉⣠⣀⣀⣀⠀⠀⠈⠉⠉⠉⢋⡽⡝⡇⠠⠀⢀
⠀⠀⢀⡀⠀⠀⠀⠀⢀⣤⠶⠛⠛⣿⡷⣲⣶⡷⡶⣽⣯⡰⣗⣦⣤⣤⠄⠛⠛⠛⠿⠿⠿⢿⣷⣶⣴⣶⣜⠻⢖⡁⠀⢀⣴⣶⣶
⠀⠀⣿⣹⡄⡤⡴⠚⠉⠀⠀⣀⡠⣪⣞⣡⣴⠾⣻⢫⣼⣧⡀⠹⣮⡍⠀⠀⠀⠀⠀⠉⠀⠒⠒⢨⣭⡝⠟⠋⠉⠙⠃⣾⣮⢭⠃
⠀⠀⢻⣿⣼⣷⠠⠀⣨⣄⣩⣀⣸⣿⣿⣷⣦⣤⣿⣼⣿⣏⡙⠦⣝⡻⠟⠲⠶⢦⢤⣤⣤⣤⣤⠿⣿⡇⠙⠓⠒⠂⠀⠙⠋
⠀⠀⠀⠈⠉⠑⠒⠛⠛⠛⠋⠉⠉⠉⠁⠈⠁⠀⠹⣿⣿⣯⠍⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠙⢴⣷⠟⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉







⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⣀⡀⠀⠀⣎⣙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡏⠁⣸⠀⢸⠃⢠⡇⠈⣹⣿⣿⠉⢩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⢸⣤⡿⠛⠋⠉⠙⠛⠛⠳⠶⠿⠾⠿⠖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠇⠀⠀⠀⠀⠀⠀⢸⢿⠀⠀⠀⢀⣠⣄⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⢸⠃⢹⠓⠦⣤⣄⣀⣀⣠⡼⢿⠀⠀⣰⣿⣍⢻⠻⣦⣱⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⣹⠳⢦⡈⠳⢾⣄⣼⣄⢰⡇⠀⣸⣠⠇⠀⢰⣿⣿⡻⠯⠇⠀⠁⠸⣎⢳⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢴⡇⠀⣴⠛⣆⠀⠉⠛⠛⠛⠛⠛⠋⠁⠀⠀⢸⣿⣿⠂⣶⣆⠀⠃⢀⣤⢍⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⡃⢀⡼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡸⣿⣿⣧⣘⣿⣄⣴⣄⠛⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣞⡁⢸⣆⠀⠀⠀⠀⠀⠀⠀⠀⢿⡋⠚⠛⣿⣿⣿⣿⡅⢀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠏⢻⡄⠀⢠⡤⣤⣄⠀⡀⠙⢽⣦⠀⠈⠻⠿⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⡼⣷⡀⣻⢤⣒⣊⠰⠩⠏⡟⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⡸⣷⡔⢼⣀⣹⠉⢹⣷⡖⢲⠶⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡁⡼⢿⣆⠈⠉⠉⠛⠛⠓⠦⠤⢴⣇⡈⡽⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣅⣼⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡟⠈⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣹⠏⠙⣷⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠹⢞⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢶⣧⣀⡽⠙⠓⠶⠶⣶⠶⢶⡶⢚⡏⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠾⠷⣤⣧⣤⣼⣧⠾⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡘⠱⢾⣰⣶⡔⢦⣤⣀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⠂⠀⠛⠿⣶⣧⣤⣄⣀⣀⣀⣼⡿⠟⠀⠀⠡⣄⠠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢢⡇⠘⠷⣤⣤⣄⣀⡀⠀⠀⠉⠉⠉⠁⠀⠀⣀⣀⣠⣾⠟⠀⣷⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠻⠶⣦⣄⣈⡉⡉⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣉⣤⣴⠶⠚⢁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠂⠠⠌⠉⠉⠙⠛⠛⠛⠚⠛⠛⠛⣛⣛⣛⣩⡭⠴⠖⠛⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠱⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠤⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⠀⠀⠒⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠶⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠤⠤⠐⠒⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠂⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⢲⠀⠀⡰⢳⡹⣦⡀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⢀⡴⠊⠁⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠊⠁⠀⠀⠀⠀⡜⠀⡐⠻⣶⣱⡑⡙⣧⠀⠀⠀⠀⠀⠀⠘⡄⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⢀⠞⣆⠀⠀⠀⠀⠀⠀⠀⢇⠈⠩⠭⠭⢻⣷⣠⡟⠀⠀⠀⠀⠀⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣼⢈⠎⢇⠀⠀⠀⠀⠀⠀⠈⠢⢤⠀⠭⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⢀⣀⡼⠁
⠀⠀⠀⠀⠀⢠⡃⠙⣦⡄⣈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡑⡰⠅⠑⠩⢉⢉⡿
⠀⠀⠀⠀⠀⣦⠴⠒⠚⠛⠶⠮⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣞⢓⣧⣷⣷⣾⡇
⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠈⠳⣴⢶⣚⡤⠃
⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡞⡝⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠉⠉⠙⢻⣦⠀⠀⠀⠀⠀⠀⠈⠫⡀
⠀⠀⣤⢧⢹⢦⡀⠀⠀⠀⣀⡴⣏⡎⠘⡜⢳⡀⠀⠀⠀⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣼⡠⣄
⠠⠄⢹⣆⢊⢊⠋⠛⠹⢉⠁⢈⡿⠿⡾⡀⢨⢳⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⢀⣿⠁⠀⠀⢀⣠⣴⠾⠛⠁⠀⠀⠑⢄
⢹⡃⠀⠛⣮⣇⢊⠁⢡⣛⣱⣟⣛⣓⣿⡰⢃⠚⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣤⡀⠀⠘⣿⣤⣶⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀
⠀⠒⠠⠤⠈⠙⠻⠿⠿⠿⠷⠶⡴⠒⢻⡇⠀⠌⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠈⠀⠐⠀⠭⠤⠛⠉⠀⣀⠀⢸⣧⠈⢈⣾⠷⢶⣶⣶⡶⠾⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⢋⠙⡷⠶⠤⡴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣐⣿⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⢾⣮⣎⡣⠀⡑⣼⠃⡴⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠐⣻⡿⣹⡇⠀⠀⠀⠀⠀⢀⡤⢴⠶⢦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢭⣔⣟⣻⣿⣾⡯⣼⠃⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢻⣷⠆⣻⡀⠀⠀⠀⢰⠟⡅⢄⠂⠀⠀⠘⢳⡀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠈⠓⠒⠊⣰⠋⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⢀⠈⣷⡂⢀⠿⣄⠀⠀⣾⠿⠛⣻⣿⣷⣟⢐⢎⡗⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⢂⣘⣷⢃⢈⠊⢳⡾⠉⠉⠛⠉⠍⠭⢿⣎⢄⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠍⠠⠬⠳⣦⠽⠟⠀⡠⠤⠤⠀⣰⣶⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠥⣄⣀⠁⡼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠴⢒⠚⠛⠙⡉⠈⠌⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣸⠄⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠉⠓⠳⠦⣦⣴⣤⠴⠾⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⢻⡆⡔⠱⠳⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠏⢙⡿⠀⠀⠀⠀⠀⠀⠀⠉⢳⣅⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣁⠀⠹⣾⡕⡐⠙⢦⣄⠀⠀⣠⠶⠛⡀⡀⣤⣾⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠬⢿⣄⠄⢌⢋⠛⠛⠅⢀⣰⣴⠛⣛⣻⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠬⢙⣮⣧⣢⣀⣤⣶⠛⣉⢭⠭⠕⣿⠀⡟⣦⣄⣀⣀⣀⣠⡴⣺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢬⡀⢀⠛⠋⠉⠉⠋⠉⠀⠈⠙⢋⡻⣾⣜⣆⢌⡍⡌⢁⣣⡾⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠲⠚⠉⠁⠀⠀⠀⠀⠀⣐⣚⣙⣏⣿⡿⠾⠋⠭⠭⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⢒⣒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣞⢩⣭⣿⣣⢠⣀⣿⣷⣾⢶⣶⣿⠏⠇⢤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣡⣴⣬⡙⠻⢿⢿⡙⠉⠉⠐⡙⠿⣿⠀⠀⠘⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠉⠉⠉⠀⠈⠿⠿⡷⣀⠻⣿⣾⣴⣼⣿⣄⠈⣷⡖⠋⠉⠹⠺⣧
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⠄⠀⠀⠀⠀⠀⢀⡤⠺⠿⣿⣦⡙⢿⣿⡟⠛⠉⠛⠂⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣌⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠶⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠧⢹⢸⢸⢸




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