Nerdy Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡦⢤⡴⣄⣠⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠛⠉⠉⠉⣁⣴⣤⣵⣄⡛⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⢀⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠇⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⡼⢳⡄⠀⡠⠋⠓⢄⣀⡴⠋⢷⡀⠀⠀⠀⠀⣿⠀⣀
⠀⠀⠀⠀⠀⠀⢸⠀⢀⣠⡄⠀⠀⢀⡏⠙⠧⠞⠀⣀⣙⣶⡁⠀⠀⠀⣉⣀⣀⠀⠻⢤⣶⠀⠀⣿⣏⠹⣧⠴⡄
⠀⠀⠀⠀⠀⠀⠘⠟⠉⣼⠀⠀⢀⣸⡷⢒⣚⣛⠭⠭⠷⣤⢸⣉⣉⣹⢇⡭⠭⠭⢝⣛⡛⢶⣀⡟⠉⡀⠀⣤⠚
⠀⠀⠀⠀⠀⠀⠀⢠⢀⣿⣾⣿⣿⣿⡄⡇⠀⠀⠀⠀⠀⢹⠀⡇⠀⢸⢸⡇⠀⠀⠀⢸⡇⢸⢿⣟⠋⢻⡤⠞⠂
⠀⠀⠀⠀⠀⠀⠂⢹⠋⣠⠤⡄⢻⠼⣇⢣⠐⠷⠄⠀⠀⣾⠀⡇⠀⢸⡌⡇⠾⠄⠀⣼⡇⡾⢲⡜⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⠀⠀⠀⠀⠘⢦⣓⠶⠶⠶⠚⣁⠜⠛⠛⠳⣷⣌⡓⠶⢛⣫⡼⠁⢸⠀⡟
⠀⠀⠀⠀⡇⠀⠀⠘⣆⠈⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⢹⠄⠉⠉⠉⠁⠀⠀⠀⢠⠇⠀⠀⠀⠀⢰
⠀⠀⠀⠀⡇⠀⢖⣀⠈⠓⠶⠶⠚⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⡟⠒⠚⠉⠀⠀⠀⠴⠀⢸
⠀⠀⠀⠀⡇⠀⢀⡉⣶⠒⣶⠀⠀⢸⡀⠀⠀⠀⠐⠧⣀⠀⠀⠀⠀⠀⢀⣠⠞⠂⠀⢠⡇⢀⠤⡀⠀⠀⠀⣀⠀⢸
⠀⠀⠀⠀⠇⠀⢸⠉⠙⠀⠛⢲⡆⠈⢷⡀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠉⠉⠀⠀⠀⠀⣼⢵⠚⠓⠉⡃⠀⠘⠤⠁⠸
⠀⠀⠀⠀⡄⠀⠈⢩⣏⡀⡖⠚⠃⠀⠈⠻⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠃⠈⠉⠨⠽⢀⡤⠐⠒⡆
⠀⠀⠀⠀⡇⠀⢀⠀⠉⢉⣁⡀⠀⠀⠀⠀⠀⠈⠛⢦⡄⠀⠀⠀⠰⠂⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⣷⠖⠛⢇
⠀⠀⠀⠀⡄⠀⠈⠀⠐⣏⣈⣽⣤⡄⠀⣀⡂⠀⢠⡞⠙⢦⣀⣀⣀⣀⣴⠏⠙⣶⠼⠉⣧⡀⢄⣁⠀⠈⠀⠛⠋⢀
⠀⠀⠀⠀⡇⠀⠐⣾⠋⠉⠁⢀⣀⣷⠀⠁⠘⠀⣸⠁⠀⠀⠈⠉⡿⣍⠁⠀⢀⣿⣶⣀⡶⠇⠀⠈⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠘⠛⠛⣿⠛⣿⠀⠀⠀⣰⠋⠉⠓⠶⠤⠤⡼⡇⠙⡟⠖⠋⠁⠀⠉⢳⡀⠀⢀⡀⢄⠀⠀⠀⢸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⡟⠀⠀⠀⠄⠀⠄⠀⡇⣚⡇⡆⠀⠠⠀⠀⢸⡇⠀⠘⣖⣻⠀⠀⠀⢸
⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠧⠤⠤⠤⠤⠤⠤⠤⠷⠾⠧⠦⠤⠤⠤⠤⠤⠧⠠⠤⠤⠤⠤⠤⠤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠛⠛⠳⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⣼⠋⠀⠀⣀⡀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠉⠙⠷⣶⣏⡀⠀⠀⣀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⣄⠀⠀⢀⣀⣀⣠⡶⢤⣴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠶⠖⠒⠶⠶⠦⡴⢹⠃⠀⠙⣿⠉⣉⣉⣹⢿⠃⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢛⣻⠛⠒⣶⠶⢶⠁⡇⠀⠀⠀⢸⣛⣉⣉⣽⡟⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣬⠹⡆⢰⠇⠀⡈⢰⡇⠀⠀⠀⢸⠙⠶⠼⠿⠷⠤⠤⢾⣶⣶⠄⢠⡶⣶⢻⡟⣧
⠀⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢾⡀⠀⡇⠸⡇⠀⠀⠀⡆⠀⠀⢀⣠⣤⠶⢲⡟⠋⠁⢠⡾⣧⢻⠘⠇⠟
⣸⡇⢿⢸⠇⣽⡶⣦⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠸⡄⢧⠀⢀⡜⠀⠀⠀⠈⠙⢾⣶⠋⠀⠀⠀⠘⣧⡀⠀⠀⣀⣠
⢿⡛⠀⠀⠀⢋⣴⠏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⣰⣒⣲⣦⠀⠀⢐⡇⠀⠀⠀⠀⠀⢸⡏⠉⠉⠉⠸
⢀⣿⠦⠶⠚⡿⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣭⡭⠏⠀⠀⠀⣧⠀⠀⠀⠀⠀⣸
⠈⡇⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀
⠀⣷⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣠⡞⠁⠀⠀⠀⠀⣸
⠀⢹⡄⠀⠀⠀⢻⣄⠀⠀⠀⠀⣀⣽⣤⣤⣤⣤⣤⣀⣀⣤⣤⣤⣤⣤⣤⡤⠤⠶⠶⠞⠛⣶⠾⠋⠀⠀⠀⠀⠀⢠⡟
⠀⠘⣧⠀⠀⠀⠀⠉⠛⠶⠶⠿⠯⣤⣤⣀⣀⣀⣀⣀⣀⣀⡤⠞⡏⠙⠲⠦⣄⣀⢀⣠⠖⠃⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⡇⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⡞⠁⠀⠹⣤⣀⣤⣤⣤⣤⣤⣤⣶⠶⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⣀⣤⣤⣬⣭⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⡀⣿⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣧⣤⠶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠈⠉⠛⠶⠲⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠋⠀⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣷⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣭⣭⣭⡭⢍⣙⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⢿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠙⠛⠁⠀⠛⠁⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠟⠛⠉⠠⠶⠿⠿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣘⣃⣀⣒⡀⠀⢐⣒⣀⣙⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣃⣐⣒⠀⠀⢐⣂⣀⣻⡁
⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⡤⢹⠻⡏⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠠⡌⢹⢺⡡⡄⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠋⠀⠙⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠁⠀⠑⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠠⠄⣶⢴⢸⠁⣾⡄⡏⡆⠀⠀⠀⠀⠀⠀⡀⠂⠀⠄⠠⠀⡦⣶⣾⢀⣶⣰⣓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠆⣠⣤⣶⣷⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣶⠿⣿⣯⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⣶⣶⣿⣿⣷⣶⠤⢀⣀⡀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⣶⣶⣿⠿⠿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⠿⣿⣿⣿⡻⠿⠿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠤⠀⠀⠀⠤⠀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠠⠀⠀⠀⠤⠄⠀⠿
⠀⠀⠀⠀⠀⠀⠀⢺⠛⠛⣙⢲⢶⡞⡋⠋⢻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠐⡟⠋⢛⠓⡶⣶⣋⠋⠛⡿
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠭⠞⠀⠳⠥⠀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠬⠴⠁⠙⠬⠤⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⢰⠀⣤⡀⡄⡄⠀⠀⠀⠀⢀⠀⠀⡀⢠⠀⣀⢠⣀⣤⡆⢠⣆⠰⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠘⠀⠉⠁⠓⠁⠀⠀⠀⠀⠀⠀⠁⠁⠈⠀⠀⠀⠁⠉⠃⠈⠉⠘⠈

⣿⣿⠛⢿⠛⠿⣿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠳⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⠒⠚⠛⠛⠒⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⢸⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡾⠀⠀⠀⠀⡀⠠⠖⢛⠒⠀⠁⠈⠐⠂⠀⠀⢀⣄⡀⠉⠂⢸⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⢸⠇⠀⠀⠀⠀⠣⠀⠀⠈⠀⠀⡀⠀⠀⠀⠄⠀⠰⠆⠘⠀⡄⢸⡇⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠚⠀⠀⠀⠀⠱⠄⣀⠀⠀⡖⠀⢸⡇⠀⠀⠀⢨⠟⠀⣀⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⡇⠀⠀⠀⠀⠘⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡇⠀⢸⡇⠀⠀⠀⠘⣆⠀⣇⠀⠀⠉⠙⠛⠒⠶⠶⠶⠶⠶⠚⠛⠉
⠀⠀⠀⡇⡆⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠐⠂⠀⡜⠀⠀⣾⠁⠀⠀⠀⠀⡸⢃⡼
⠀⠀⠀⣧⡅⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⠀⠀⠀⣿⠀⠀⠐⠚⠋⠉⠉
⠀⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢸⠁⠀⡀⠀⠀⠀⠀⠀⢴⠃⠀⠀⠀⢰⡏
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠉⠀⠀⠀⠀⣼⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⣿
⠀⠀⠀⣠⡴⠞⠳⠀⠀⠀⠀⠀⡏⢄⠀⠀⠀⠀⠀⡰⠁⠀⠀⠠⠀⣿⠳⠦⣄⡀
⠀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠢⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⠁⠀⠀⠀⠉⠀⡀
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢻⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠈⣧⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡄⠀⡇⠀⠀⠀⠀⠀⢠⠃⠀⡿
⠀⠀⢹⡄⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⡇
⣤⣤⣤⣿⣤⣼⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣦⣤⣤⣦⣤⣤⣤⣤⣤⣤⣧⣤⣼⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣿⣮⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣷⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣧⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠋⠉⠙⠛⠉⠉⠛⠛⠛⠃⢠⡀⢀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⢀⡀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⠚⠛⠿⣿⣷⣶⣄⠀⠀⠀⠀⠀⢀⡴⠛⠂⠀⠀⠀⠀⣼⣿⣿⠗⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠾⠛⢿⣿⠃⠀⠀⠀⠀⠀⠀⠈⢁⡈⠁⢀⡀⠁⠀⠀⢈⣷⣦⣄⠀⠀⢠⡀⠈⣿⡇⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠹⠀⠀⠀⠀⣿⣿⠀⣿⡇⠀⠀⣾⣿⡿⠟⠃⠀⠀⠀⠀⠀⠙⠁⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⢤⣤⣴⣾⠥⠼⠿⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡄⠀⠛⠂⢀⡀⠀⠀⠀⠀⠚⠁⠀⠀⢾⣿⠟⢦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⡟⠉⠛⢃⡀⠀⠀⠀⠀⠲⠶⠈⠁⠀⠀⠀⠀⠈⠀⠀⠈⠉⠛⢿⣶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠿⠿⠋⠁⠀⠀⠀⠀⠘⠃⠀⠀⠀⡆⠀⠀⠀⠀⣠⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⢀⣤
⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟
⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣶⣄
⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠘⢻⣦⡀
⠀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⢻
⣀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠶⠶⠀⠰⠶⠶⠶⠶⠿⠿⠇⠀⠀⠀⠠⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠿⠿⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣀⣤⣤⣀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠉⡿⠛⠛⠛⠀⠀⠠⢤⣶⣶⣶⣦⣤⣤⡤⢤⣴⣦⣤⡤⠤⣤⣤⣶⣶⣶⠶⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠤⢤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠆⠀⠀⠈⠙⠛⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁⠀⠀⠻⡿⠟⠃⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡤⠀⣠⣤⣤⣤⣶⣶⠄⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⢿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⣠⠀⠀⣤⠄⠀⠀⠀⢴⣿⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠇⠀⠀⠘⠷⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⡼⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⠟⠛⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣿⣿⣿⣿⣿⣿⣟⠋⠉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡿⠿⠛⠉⠻⣿⣿⠿⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⡏⠉⠉⠉⠉⠉⠉⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠀⠀⢹⣷⣤⣄⣀⣤⣤⣤⣄⣀⣠⣤⣤⣤⣄⣀⣠⣬⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣷⣿⣿⣾⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡊⠉⠁⠐⠂⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠠⠀⠀⡔⠁⠀⠀⠉⠒⠤⡀⠀⠀⢥⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠛⠉⠙⠳⣄⢀⣤⠶⢦⣤⡀⡇⠀⠀⠀⠀⠀⣀⠡⠀⠐⢦⡀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡀⣿⡟⠀⠀⠀⠈⢻⣇⠀⠀⠀⠀⡐⠁⠀⠀⠐⡦⡙⡑⢴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠉⠉⠉⠉⠉⠀⠘⢿⡀⠀⠀⠀⣡⣿⡇⠚⠀⠀⠀⢀⣿⣄⡀⠀⣰⠃⡔⠂⠒⠐⠁⠈⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠛⠉⠀⠻⣤⣀⣀⡤⠚⠓⠭⠙⠻⣿⠚⠉⠉⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⢒⡰⠖⠂⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⡐⡦⠀⠀⠀⠘⠉⠁⢠⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⠊⢱⠁⠀⠠⠤⠀⠤⠊⠁⠈⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡉⠉⠉⢉⠉⠉⠉⡁⢀⣨⡄⠎⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢔⠉⠁⠀⢺⠀⠀⠈⡋⠀⠀⠘⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠉⠂⠄⣨⣀⣀⣀⡥⠄⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢉⠙⠉⠀⠀⠰⠁⡸⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⢠⡥⢄
⠀⠀⠀⠀⠀⢀⠄⢸⡀⠂⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠈⠈⢢⢄⡀⠀⣠⠖⢩⠊
⠀⠀⠀⠀⠀⠸⣠⠁⢰⠋⠙⠄⠀⠀⡠⠃⠌⠀⠀⠀⠀⢀⡰⠟⠀⡇⢰⡅⠀⢀⠟⡩⢭⡆
⠀⠀⠀⠀⠀⠀⠘⠄⣈⠀⠘⢁⠈⠩⠔⢂⠀⠀⢠⡶⡈⠁⡔⡗⠈⣉⢹⠄⣀⢪⡊⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢣⢀⠴⣿⠷⣰⡈⠀⢎⠣⡀⠀⠀⠉⠸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠉⢰⠉⣅⠀⠀⠘⡆⠀⠀⠀⠸⡊⠀⣤⣶⣶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠁⠉⢰⠒⢓⢢⠤⠔⠐⠋⠫⢁⡁⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠈⠈⠀⠀⠀⠢⠀⠀⢀⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣄⣄⡀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⢄⣀⣉⠉⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢭⣽⡅⢰⣶⣶⡄⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠤⡤⡤⡶⣎⣣⣾⣴⠀⣞⡹⡧⠐⣤⢠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢀⣀⣀⣈⡉⠉⠉⠡⡀⠀⢈⡆⣄⣉⠛⡉⠓⠓⠾⣮⠁⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠃⠁⠂⠐⠀⠀⠀⠀⠠⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡒⢒⣢⣭⣒⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡦⢽⣿⣿⣿⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⣿⣿⣿⣿⣿⡿⣿⣿⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣟⢁⣤⣀⠀⠀⠀⢿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡇⠙⠉⡟⢿⠛⡷⣴⣿⠶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡔⡂⠠⠘⠒⠊⠀⠲⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢫⡇⠀⣀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣖⣿⣿⠀⠀⠀⠈⠳⣄⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⣹⣿⣿⡆⢠⣆⠀⠀⡟⠉⠁⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⢃⣿⣿⣿⣇⠈⠱⣻⡁⡰⣾⣷⣾⣷⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢿⢿⣿⣶⣶⡶⣮⣿⡀⠀⠸⡀⠀⢿⣿⣯⡲⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⡿⡌⣾⣿⣿⣏⢾⣿⣿⣧⠀⠀⠀⠰⣾⣿⣿⣿⢆⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡈⣿⣴⣿⣿⣿⣿⣆⠻⣿⣿⡆⠀⠀⢀⣿⣿⣿⢃⡽⣛⣒⣒⣶⣶⣶⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⡇⢻⣿⣿⣿⣿⣿⣿⣧⡹⣿⣿⡄⡀⣾⣿⣿⣡⢯⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣾⢋⣥⣾⣧⣬⣿⣿⣿⣿⣾⣿⣿⣿⡾⢿⣷⡀⣿⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⠭⣛⣭⡅⣰⣯⣽⣿⣿⠿⠋⠀⢔⠨⡙⣿⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣫⣾⣿⣿⠏⠀⠀⠀⠀⣈⠂⢁⣴⣿⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⣶⣾⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠉⠙⠛⠿⡿⠿⠿⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣠⠴⠒⠚⠛⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡍⣧⢤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣬⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⡀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠈⡄⠀⢰⣄⣀⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢄⠀⢀⠧⣤⡜⠳⠤⠜⠑⠦⠜⢉⡶⠻⡦⣄⡜⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢀⢀⡜⣉⣩⠋⠢⡀⠙⢆⠀⣀⣠⣠⠧⡈⠳⡈⠢⣻⠈⢦⠞⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠓⢖⠚⠋⢿⡳⣄⠈⠢⣸⡟⠊⠉⠉⢇⠈⠢⡈⢢⡜⠀⠀⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⣄⠀⠀⠀⠑⢬⣦⡤⠞⠠⡤⣤⢴⠀⠉⠒⠛⠉⠀⠀⠀⢸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣬⣽⣷⣤⣤⣴⣟⠛⣛⣖⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⠔⠒⢒⠀⢤⣤⣤⡀⠐⠂⣈⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢛⣚⣋⣛⣛⣛⣒⣻⠿⠿⡿⡷⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢼⣁⣀⣀⣀⣀⣀⣀⣀⣐⣒⣒⣒⣻⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠂⠹⠛⠒⢒⣚⣛⣛⣛⠩⠭⠭⠭⠭⡽⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⢳⣤⣀⠒⠀⠀⠀⠀⠀⠈⠉⢀⣀⣾⣧⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⠘⣹⠧⢤⣤⣤⣒⣒⣲⣶⣾⡝⠃⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⡗⠒⠒⢲⠒⠒⢲⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣼⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢿⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠮⣭⣭⣽⠿⠶⠯⠭⠽⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠉⠉⠙⢦⡀⢰⡎⠤⠒⠈⠁⠐⠢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⡤⠔⢻⠁⠀⠀⢀⣀⣀⣀⡀⠐⢮⡐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢚⡁⠀⠀⣨⠀⢠⠞⠋⠉⠉⠉⠙⠳⣄⠁⠀⠁⠠⡀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⣠⠊⠀⠈⠁⠀⣰⢇⣴⠁⢀⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠤⢤⠄⠀⠁⠀⡃⠐⠯⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠋⠉⠉⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⢀⡼⢷⣄⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢦⣀⠀⠇⠀⠀⠀⠛⢦⣀⣀⣀⣠⡴⠋⣀⠀⠉⠳⣮⡀⠀⠀⢀⣀⠄⠔⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢀⡤⡾⠁⠀⠀⠀⠉⢉⣉⡉⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠄⣀⣀⣀⣀⡀⢤⠖⠚⢅⡜⠁⠀⠀⠀⠀⠉⠓⠂⠛⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠹⡄⠀⠓⡀⠋⠀⡠⠔⠊⠉⠀⠠⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠹⠖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠚⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣠⠛⣒⡈⠙⣁⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠊⠒⡫⣀⢠⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⣀⡙⡉⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠀⠀⠛⠓⠘⡇⠀⠀⣫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠁⠞⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠀⡖⢲⠆⠀⠤⢤⠠⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡊⠈⢙⣦⠀⡣⢦⠀⠀⠐⠂⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠜⡆⠀⠀⠀⠀⠀⠐⢀⣴⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣉⣉⣉⣩⣦⠀⠀⠀⢀⡠⠖⡩⠀⢹⣆⢢⠐⠀⢠⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⠒⠂⢴⡇⣀⠤⠶⠭⡤⠞⣁⢾⠀⣿⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⢀⠀⠀⣄⠀⢠⣤⡤⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢖⣫⣩⠭⠉⡽⠿⠿⠓⠒⢾⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣾⣿⣿⣿⣀⣤⣶⣿⣿⣿⣶⣄⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠈⠛⠻⢿⣿⠿⠛⠉⠛⠻⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣶⣦⣸⡇⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡋⠁⠀⢀⣴⣶⣤⣤⡀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡶⣍⢿⣿⠿⢿⣷⣿⣷⣿⣿⣿⣿⡶⣾⣿⡿⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⣏⠈⠋⠠⣄⡈⠙⣿⠶⠟⢉⡿⠀⢻⣿⣅⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣴⡄⠨⣿⡧⠀⠈⣿⠿⠛⠁⠀⠘⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠼⠟⢳⡀⠀⠁⠀⠀⠀⠀⠀⠉⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣰⣿⣧⡱⣄⠳⣄⠀⠀⠀⠈⠁⢻⣛⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣤⣼⣿⡽⣿⣷⡈⠳⣜⢿⣦⣄⡀⠀⠀⢉⣽⡏⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⡝⣿⣿⣄⠀⠙⢿⣿⣿⣷⣾⣿⣯⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠛⢿⡿⢿⣿⣿⣿⣶⣦⣄⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣧⢠⣤⢀⣾⣿⢰⣿⣿⣻⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢿⣿⣿⣧⠀⣼⣿⣿⣾⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠐⠒⢶⡒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣀⠤⠒⠉⠉⠉⠉⠀⠀⠈⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⢉⠼⠃⠼⠿⠦⠤⠤⠤⣀⡀
⠀⠀⣠⢎⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⣀⠲⣒⠠⠤⢤⣤⣀⠀⠀⠀⡠⠤⠰⠆⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠞⠓⠄
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠚⠛⠃⠀⠀⠉⠉⠀⠀⠉⠀⠙⠐⠊⠀⠉⠉⠀⠀⠉⠳⠾⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⣇⠄⠀⠀⠀⠀⠀⠀⠀⢴⡮⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣕⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢂
⠀⠰⠃⠀⠀⠀⠀⠀⢀⣴⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠃
⢠⢡⠆⠀⠀⠀⠀⠀⠈⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⡿⣹⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣀⣠⣤⣀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⢻⡆⠀⠀⠀⠀⠀⢸⣱⠂⢀⣴⠟⢋⡉⠉⠉⠻⢷⣶⠚⠷⠙⣄⢷⡀⠀⠀⣀⣈⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠻⡇⠀⠀⣀⢀⠄⣸⣿⢀⡾⠡⣄⡔⠀⠀⠀⠀⠀⢻⣧⠀⠀⠈⢛⣼⠖⠛⠉⠉⠛⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸
⠀⠀⢇⠀⢠⢫⡆⠛⠁⠸⣸⡇⠈⠉⡄⠀⠀⠀⠠⡶⢸⣿⠶⠶⣦⡟⠁⠀⠀⠀⠀⢖⠙⡜⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣦⢰⡀⠀⠀⠀⠀⠀⠀⠀⡿⡇
⠀⠀⠘⡄⣻⠘⠁⠀⠀⠀⠙⢿⣌⠉⠁⠀⠀⠀⠀⢀⣼⠇⢤⡀⣿⠁⠸⠃⠀⠀⠀⠘⡒⡋⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⣿⡎⠉⠐⣴⣰⡇⠀⠀⠀⣸⠛⠃
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠈⠻⢶⣤⣤⣤⣤⡶⠟⠱⠆⡶⠙⢿⣇⠀⠀⠀⠀⢀⣜⣩⢃⣾⣅⠀⠀⠀⠀⠀⠀⣰⢀⡎⠛⠀⠀⠀⠀⠀⠋⡀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡉⠙⢃⠀⠀⠀⠀⠀⠀⠀⠹⢷⣄⣀⠀⢀⣀⣴⠞⠁⠛⢉⣉⠑⠲⣄⠔⣽⠞⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠈⢧⣤⣀⠀⠀⠀⠀⠀⠀⠉⠛⠛⡛⠉⠁⠀⠀⠀⠉⠉⠙⠦⠈⡆⠁⠀⠀⠀⠀⠀⠀⠀⠰⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠑⠚⢻⠒⣢⡤⠤⣤⣤⣤⠄⠘⠆⠀⢀⠤⠊⠐⠤⠀⠤⠜⠻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠸⠿⠿⠛⠉⠉⠀⣀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⢽⣶⣶⡖⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡋⣻⣄⡼⠧⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣠⣔⡀⠉⠀⠀⠱⢤⠞⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠴⠥⣄⣉⡉⠓⠲⠤⣀⢸⢀⣠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡠⢄⣀⠀⠀⠀⠀⠀⠉⠙⠒⠤⣈⠙⡧⠶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣉⣶⣦⣄⡀⠀⠀⠀⠀⠀⠙⢆⡀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⠻⠿⣿⣦⣄⡀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠢⠤⢷⠬⢙⡿⣗⠢⣀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠚⢫⣙⡜⢦⣀⣸⡀⠀⠀⢣⣄⣈⠳⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠂⠀⠀⠀⠀⠯⢕⣲⠋⠎⠭⠛⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⠷⠶⣦⣤⣤⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢛⣫⣴⣿⣿⣿⣧⣬⣵⣶⣾⣿⣷⣶⣍⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡶⠖⣛⣡⣾⣿⣟⣽⣿⣿⡿⠋⠙⢿⣷⣦⣨⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⢛⣩⣥⣶⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⢣⣶⣿⠟⢡⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢋⣥⣶⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢻⣷⢹⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⣡⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠹⠀⠀⣀⣀⣀⡀⠀⠀⠀⢰⣿⣤⣼⣿⣤⣵⣦⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣰⣿⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢘⣡⣶⣿⠟⠛⢋⣉⣥⣴⠶⠿⠻⣿⣉⠁⢩⡄⠈⢿⡌⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣴⣿⠟⢋⣠⣶⠿⠟⠋⠁⠹⣦⠀⠀⢻⡿⠃⡀⠹⣄⢸⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡟⠁⠈⠋⣠⡶⢿⣿⣄⣀⠀⠀⠀⠀⢸⡄⠀⠀⠻⣏⠀⠀⢈⡿⢷⣌⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⣼⣿⣿⣿⣏⣰⣾⣿⣿⣿⠀⠀⣀⣀⣿⡅⠘⣿⣿⠿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢻⡛⠉⠀⠀⠙⢷⣌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠹⣿⣿⣿⡿⠿⢿⣿⣿⣿⣾⣿⠿⠿⠛⢳⡀⠙⠦⠄⠀⠀⢀⣰⠏⢀⡀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠙⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢸⡿⠁⣠⠤⣄⠻⣿⣿⠁⠀⠀⠀⠀⠈⠳⢦⣤⠴⠶⠛⠛⠁⠀⢾⡁⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠘⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠸⠁⠀⢸⡆⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠼⡇⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢼⣧⠀⠘⠀⠈⠑⠆⠀⠀⠀⠀⠀⢾⡇⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠄⠀⠒⡿⠀⠀⠀⢠⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢏⣼⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⣾⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡸⠿⢃⣤⣍⡻⠶⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠑⠢⡄⠀⠈⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⠞⠁⠀⠀⠀⣰⡟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠈⠉⠛⠶⢮⡙⢷⣄⠀⡄⠀⠀⠀⠀⠀⠀⠁⠐⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡹⣷⣌⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠊⠁⠀⠀⠀⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢻⣦⠀⠉⠙⠳⠶⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⠷⣶⣦⣤⣀⣀⣤⠶⢟⣋⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢋⣼⠛⠇⠀⠀⠀⠀⠀⠴⣿⡏⢱⡶⠶⢬⣍⣛⣩⡴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⣛⣋⣭⣴⣾⣿⡅⠀⠈⠶⣄⠀⠀⠀⠀⡌⠹⣦⠻⣦⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢈⣙⣢⣤⠞⣃⣤⣼⢷⣬⣙⠳⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢘⢿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣦⣝⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢰⣿⣿⣿⠿⣿⣿⠎⢻⣿⣿⣿⣿⣧⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠙⠋⠡⠀⢈⠁⠂⠘⣿⣿⣿⣿⣿⣧⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⢀⠀⢀⠀⠀⢻⣿⣿⣿⣿⣿⣧⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⢃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⡀⠈⡀⢠⢸⣿⣿⣿⣿⣿⣿⣇⢻⡄
⠀⠀⠀⠀⠀⠀⢀⣼⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣎⠀⠃⠀⡇⢸⠈⣿⣿⣹⣿⣿⣿⣿⣇⢻⡄
⠀⠀⠀⠀⠀⢀⣾⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢠⠀⢠⡀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣎⢻⡀
⠀⠀⠀⠀⢀⡼⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⡾⢷⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠸⡇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢿⡀
⠀⠀⠀⠀⣸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣄⠙⢷⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡼⠀⣳⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢷⡀
⠀⠀⠀⠀⢿⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣦⠻⣎⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⣷
⠀⠀⠀⠀⠈⠻⣦⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠳⣜⣧⡹⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡂⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡘⣇
⠀⠀⠀⠀⠀⠀⠈⢳⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢷⡹⣾⢳⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣌⠻⣿⣿⣿⣿⣿⣿⣿⡟⠋⢤⣼⣃⣛⡚⢛⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣮⡙⢿⣿⣿⣿⠏⡀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣬⡙⢿⣤⡾⢷⣄⠀⠀⠀⠀⠀⠀⠀⠠⢈⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⡟⣱⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠾⢦⣝⠷⣤⣀⡀⠀⠀⣀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡿⢋⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣍⡛⢷⣆⢸⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢎⣱⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⡻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⡄⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣬⣍⣛⡛⠿⠿⠿⣿⠟⣡⠟⢿⡘⠿⠿⣛⣛⣛⣭⣥⡴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠲⠶⠞⠋⠀⠈⠙⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⢈⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠐⢠⠀⡀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠹⢿⡿⠟⠛⠉⠀⠉⠻⣿⣿⡿⠋⠀⠀⠀⠤⢠⠀⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⡀⢀⠈⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⡀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠶⠞⠛⠛⠛⣉⣉⣉⣁⣤⣤⡤⠶⠶⠛⠃⠀⠀⠀⠀⠀⠁⠈⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠐⠛⠛⠛⠛⠛⠋⠉⢉⣉⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣜⣛⣛⡛⠷⢦⠀⠀⠘⣫⣭⡶⠶⠶⢶⣤⣄⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠁⠉⠉⠙⠻⣦⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠉⠻⣆⣠⣴⠟⠁⢠⡶⠻⣷⠀⢰⡟
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⣠⡄⠀⢸⡇⣿⡇⢰⡆⠀⠀⠀⠀⠀⠀⢀⣿⠟⠁⠀⠀⢺⣄⠀⠉⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠈⠁⢀⣼⡇⠙⢷⣄⣀⡀⠀⢀⣀⣠⣴⠟⠁⠀⠀⠀⠀⠀⠉⣋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢦⣤⣤⡴⠶⠛⣿⠃⠀⢀⡈⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢰⡟⠀⣴⠏⣻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣼⡇⠀⠀⢠⡟⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠹⢷⣤⣴⡟⠁⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⢰⣿⣿⣿⣿⣿⣶⣶⣶⣿⠀⠀⠀⣠⣴⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣧⣀⠈⠉⠉⠉⠉⠉⠉⠉⢀⣠⣴⡿⠟⠋⠁⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠉⠛⢷⣤⣀⣀⣀⣠⣴⣾⠟⠋⠁⠀⠀⠀⢠⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⢶⣤⣤⣤⣿⡿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⢈⣿⡁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣧⡀⣸⡿⣧⠀⢀⣀⣠⣤⣶⣿⡟⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁⢤⡀⢉⣿⣿⣶⣿⣿⠿⠟⠛⠉⠈⣿⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣠⣾⣿⣿⠃⠀⠲⠾⣿⣾⠃⠀⠀⣿⣥⣤⠴⠄⠀⢰⡟⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣠⣾⣿⣿⣿⣿⣄⣀⣀⣠⣾⣏⡀⠀⣸⡿⢦⣤⣀⠀⢀⣿⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠉⠙⠛⠿⠿⣷⣤⣤⣁⣠⡾⠃⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣰⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠶⠶⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⣴⣶⣦⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣛⣛⣫⣭⣭⣷⣿⣿⣿⣿⣿⣿⠿⠿⣿⣯⣛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣶⣦⣍⡳⣌⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⢮⣭⣛⣛⣻⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣶⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⠾⠿⠟⠛⠉⠉⠉⠉⠉⠛⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣼⣷⣤⣴⣿⣿⣭⣙⡛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠒⢛⣛⣭⣿⣿⣦⣤⣼⣧⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⡿⣻⡏⢉⣩⣿⣭⣽⣟⡿⣷⣦⣀⣀⠀⠀⣀⣀⣴⣾⢿⣻⣯⣭⣿⣏⡉⢹⣟⢿⣿⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡟⣿⠉⣽⢡⣶⢿⣟⢮⡻⣿⣿⣿⣿⣿⣿⣟⡵⣻⡿⣶⡄⡿⠉⣿⢻⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⢿⡇⠸⣆⠘⠈⠻⠿⠏⣠⣷⣿⣟⠀⠀⢻⣿⣾⣅⠙⠿⠿⠁⠋⣰⠇⢸⡿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣼⣷⠀⠙⠶⣤⣀⣨⡽⠟⣱⣿⢻⡆⠀⡟⣿⣎⠻⢯⣥⣀⣤⡶⠋⠀⣼⣧⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡀⠀⠀⠀⠀⢀⣠⣾⡿⠃⣸⠁⠀⣇⠘⢿⣷⣄⡀⠀⠀⠀⠀⢀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⢻⣿⠿⣿⣿⠿⠿⠛⠉⢠⡏⣇⠀⠀⣸⢹⡆⠉⠛⠿⠿⢿⣿⠿⣿⡟⢋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⣷⣾⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⡠⠀⠀⢀⣰⣧⣼⣀⡀⠀⠀⢠⡀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⠟⠉⠉⠛⠛⠛⠛⠛⠛⠋⠙⠻⢷⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠁⠀⠀⠀⠑⠶⢦⠶⠶⠊⠀⠀⠀⠀⠁⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠻⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠐⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣀⣤⣴⣶⣾⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣴⣿⣿⣿⠿⠿⠿⢿⣿⣿⣯⣝⠋⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣎⣿⣿⣿⣿⣷⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠛⠛⠛⠻⠿⠋⠁⠀⠀⠀⠙⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣮⣿⣷⡟⣻⣿⣷⣦⣠⣤⣠⣶⣿⣿⡛⢿⣶⣯⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠈⠫⠉⠹⢿⡟⠛⣿⠛⠉⠉⠋⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣏⠳⣤⣤⣤⢾⡟⠀⠙⠷⣤⣤⡴⢏⣿⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣤⠀⠀⣾⣷⣤⣴⠆⠀⠀⠰⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⢠⣈⣽⣟⣥⣄⣀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣿⣦⠀⠰⠶⠶⠦⠀⠀⢀⣴⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⢻⣿⣿⣧⣄⣀⣠⣀⣀⣴⣿⡿⢠⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⣷⡉⠻⣿⣿⣿⣿⣿⣿⠿⠋⢠⡿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⢾⣿⣿⣿⣿⣿⣿⣿⡇⠁⠁⠀⣠⡉⢻⣿⠋⣡⡀⠀⠁⢠⣿⣿⣿⣿⣿⣿⣿⡷⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢿⣜⡿⣿⣿⣿⣷⠀⠀⣼⣿⣿⣶⣿⣾⣿⣷⡄⠀⣸⣿⣿⣿⣿⣫⡿⣫⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡛⢿⣿⣿⣿⣆⡄⠀⠘⣿⣿⣿⠏⠀⠈⣿⣄⣿⣿⣿⡿⢟⡭⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠛⠿⠷⠀⠀⣾⣿⣿⣇⠀⠀⠼⠿⠛⠋⠁⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠤⠤⠐⠒⠂

⠀⠀⣰⣶⣶⣦⣠⣤⣀⣤⣤⣾⣆⣤⣤⣤⡄⣷⢤⣤⣤⣤⣀⣤⣄⣠⣤⡄⣠⣤⣀⣤⣤
⠀⠀⣿⣿⣿⣿⣽⣿⡇⣿⣇⣸⣇⣿⣿⢺⡇⣿⢸⣏⣿⢺⣅⣿⣿⣿⣿⣷⣿⣻⣏⣻⣷
⠀⠀⠈⠉⠁⠁⠈⠉⠉⠁⠉⠉⠁⠁⠉⠈⠁⠉⠈⠁⠉⠉⠀⠉⠉⠘⠛⠃⠈⠉⠁⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⢄⣠⣤⣄⡴⢲⣆⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠐⠘⠃⠀⠸⠇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠈⠁




⢠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣤
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠐⠦⠤⡀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡘⣧⣅⣀⡨⠍⣁⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠻⢿⠞⠋⠙⠛⠉⠻⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡷⣌⣀⣤⣤⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠃⠀⠹⣏⣩⠏⣟⢛⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣤⡴⢹⠛⡀⠀⠀⠀⡲⠈⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠀⠠⣵⣦⡌⣄⠻⣦⣄⡀⢉⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⢠⡀⡘⡇⢇⣨⣷⣼⣿⠛⡙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⡄⠀⠀⠀⢷⡰⠹⡿⠻⡿⣿⣿⡾⠃⠀⠀⡄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠺⣿⣄⠀⢠⣸⡇⠀⠀⠀⠈⢹⣟⠁⠀⠀⢀⣧⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣵⣧⠙⠦⠁⠀⠹⣷⡀⠀⠀⠀⣼⡿⠰⠷⠤⣼⣿⢷⢇⣀⣰⠆⠀⠀⠀⠀⢀⣀⣠⣤⣤⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣋⣤⠶⢦⣽⡻⣿⣿⣦⣤⠞⠉⣵⠁⠀⡰⢿⡷⢖⣫⠞⠉⠐⢔⣴⣶⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⣾⠟⠛⠛⢻⣿⣿⣿⣯⣾⣼⣧⣾⣿⣷⠀⠈⠉⠑⢄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠠⣤⣳⣼⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠋⢉⣿⣿⣇⢧⣄⣀⣀⣠⠀⢀⣀⣄⡀⣀⣄⡀⠀⠀⠤⣝⢲⣽⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢳⣿⣾⣿⣿⣿⣿⣿⠭⡁⠀⠙⠓⠶⣿⣛⡛⠿⢿⣿⣿⣶⣬⣶⣿⣿⣿⣿⣿⣿⣿⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⡿⣫⡥⣀⡀⠀⠀⠈⣿⣿⣿⣷⣦⣭⣍⢟⣾⣿⣿⣿⣿⡿⡍⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣯⣾⣿⠁⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⢡⣆⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣾⣿⣷⡿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⢷⣿⣧⣴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣆⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀⠀⠀⣿⣿⡇⠀⠀⢀⣀⣤⣤⣤⣄⣀⣀⣼⣿⣿⣦⣤⣤⣤⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⢿⣿⡇⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⡿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⢸⣿⠁⣼⣿⣿⠋⠀⠀⠈⣉⡉⠀⢸⣿⡟⠀⣀⣤⣾⣿⣿⣿⡇⠈⠙⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣄⠈⣿⣠⣿⣿⣧⣤⣴⣶⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⡿⠛⣿⣿⡇⠀⠀⠈⠻⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠻⣿⣿⣶⣿⣿⣿⣿⡿⠿⠟⠛⠋⠙⢿⣿⣿⣿⣿⠛⠉⠁⠀⠀⣿⣿⣧⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠙⣿⣿⣿⣿⣿⣿⡀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣷⣤⣀⣀⣀⣿⣿⣿⠀⠀⠀⠀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⡿⠋⠀⠙⠻⣧⣭⣭⣽⣭⣿⣿⣶⣶⣶⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠈⠉⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠓⠋⠛⠚⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠒⠻⠿⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠸⠿⠿⡀⠀⠀⣶⡆⢀⡴⢺⣧⣭⡁⢳⣿⡏⠀⠀⠿⠀⠀⠀⠈⠛⠛⠉⠀⠙⠛⠋⠁⠈⠛⠒⠋⠀⠉⠛⠚⠃⠈⠓⠒⠂
⠀⣠⣤⢸⣶⣦⢀⡄⢸⣇⣨⣏⣈⣁⣈⣉⣁⡀⠀⠀⠀⠀⠰⠃⠘⣦⡞⠁⢱⡴⠁⠙⣦⡞⠀⢳⡞⠀⠙⣶⠋⠀⢷⠎⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠈⢫⡌⠛⠋⠀⠀⣼⣿⣿⣿⣾⣿⣷⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⢸⣷⠀⠀⣿⡇⠀⢸⣧⠀⠀⣿⡄⠀⣸⣇⠀⢀⢿⡄⠀⡀⢠⣤⣀⢀⡖
⠾⡿⠈⢱⣿⡆⠀⠘⣏⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠋⠈⠀⠘⡅⠙⢀⠋⠈⠀⠈⠀⠙⠀⠁⠈⠀⠈⠀⠙⠈⠁⠸⣿⣿⣿⣿
⠐⠀⠀⠀⠈⠀⠀⠀⢈⠀⠀⠀⣀⣀⣀⢈⣁⠀⠀⠀⠀⠀⢀⠀⠘⠃⠀⠘⠁⠠⡄⢠⣵⣤⣼⣤⣄⡖⠀⠀⠈⠀⠀⠀⠀⠀⠐⠒⡋⠀⠠⠛⠿⡿⠟
⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⢿⣧⠀⠀⠀⠀⠠⠤⡀⠀⠈⠃⠰⢤⣠⣾⡿⡿⠻⢿⣷⣼⣷⣴⢂⣀⠀⣴⣶⣧⡀⢠⡆⢠⡀⢰⢰⠂⠠⠰
⣯⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣯⢿⣿⢾⠀⠀⠀⠀⠀⢀⡁⠀⠃⠀⢀⢰⣿⣿⣿⡆⣰⣿⣮⢛⢿⣿⡇⠀⠀⠹⠿⠟⠀⠘⠇⠈⠁⠾⠿⢷⣶
⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⣉⠀⢀⠴⠶⠟⠫⡅⠀⢘⡷⠈⢭⣟⣻⣿⡟⢻⣿⣟⣶⣾⣿⡗⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠰⢶
⣭⣿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡆⠠⣀⡀⢀⣀⡴⠀⠈⠀⠐⠊⣻⣿⣿⠾⣿⣿⣿⣿⣥⡼⠧⠀⠀⠈⠀⠀⠀⠀⠁⠀⠐⡞⠆⠀⠀⣈
⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣀⡜⠃⢠⠾⠛⠛⠓⠄⠀⠉⣂⠀⠈⠁⡹⠿⠿⣿⣿⠿⡟⠫⠀⠀⠀⠐⡂⠐⠦⠶⠛⠛⠂⠀⠉⠓⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠈⢳⡄⠠⣀⠀⣀⣀⡤⠀⠀⠀⠂⠀⢀⡁⠀⠀⠘⠈⠀⠘⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠆⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠹⣄⠀⠀⣸⠇⠠⠞⠛⠋⠓⠦⠀⠂⠀⣴⡶⢀⠀⠀⠀⣰⣶⣶⡀⠀⠀⢐⣾⠿⣿⣿⠿⠟⠋⠉⠀⣀⠛⠛⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠟⠓⢧⡀⠐⢦⣄⣠⡤⠖⠀⠀⠀⠠⠀⠘⠁⠀⠂⠙⠿⠟⠀⠀⠀⠉⠉⣉⣉⣀⡀⠀⣀⡄⠀⠙⢻⡷⠸⣿⣿
⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀⠀⠀⢸⡇⠰⠚⠋⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠂⢰⣦⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠁⠀⠀⠺⠇⠀⠗⠀⣁
⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⠀⣀⣈⠀⠐⠤⠄⠀⠀⠀⠚⠃⠀⠀⡤⠀⠀⣴⣶⠆⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⢀⣄⢀⠀⠄⠀⠐⠐⠂⠸⣿
⠀⠉⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠘⠛⠐⢿⣿⠲⠿⠀⣤⣤⡀⢘⣿⣿⣿⣇⣶⡆⠀⠉⠉⠁⢨⣿⣿⣿⣿⢶⠀⠀⠀⠰⡀⠀⠀⠑
⠀⠀⠂⠤⠄⣚⠳⣼⡑⢞⣿⠀⠀⠀⠀⠀⣀⠸⠷⠀⠛⣿⣇⠘⠿⠿⠇⠛⣿⠄⠘⠛⢀⡀⠹⣿⣿⣿⡿⡟⠀⠠⠀⠀⠈⠀⠀⠈⠀⠠⠴⠤⣄⠐
⠀⣴⡄⠀⠀⠀⢠⡿⢙⣫⡞⠀⢀⠀⠀⣿⣿⡗⠀⠀⣴⣿⣟⣤⣤⣄⡀⠀⢀⣰⢤⠀⠈⠁⣠⡈⠉⠛⠀⠀⠀⠀⠁⠀⠐⠀⠀⠀⢐⡀⠐⠀⠀⠇
⠄⢸⡇⠀⠀⢀⡘⣰⡾⣼⠇⠀⢈⠀⠿⠇⣽⣄⢸⣟⡸⠋⢿⠿⣯⡟⢤⠀⠀⢼⣻⠸⠀⠀⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠳⢽⠽⠃⠈⠉⠁⠘⠟⠀⣾⠁⠸⣿⣿⡀⠀⠁⠀⠙⠀⠀⠀⢀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠒⠀⠀⠀⢸⡇⠻
⠑⡀⠀⣠⣄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⡆⠀⠀⢻⣿⣷⡀⣶⠦⣾⡿⠋⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠠⠀⠄⠠⠀⠀⠺⠇
⠀⡆⠀⣉⠙⠛⠀⣀⠀⠀⢀⣾⢯⣿⣿⣷⣿⣧⠐⠈⢿⣿⣿⣟⢛⣁⡀⠀⠀⣴⡀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠸⣿⠇⠀⠄⣡⣴⣶⣷⣶⣤⡀⠆
⠘⡇⠈⠙⠃⢰⡆⠙⠀⠀⢸⣿⣿⣅⡸⢻⡿⣿⠠⠀⢠⣿⣏⣿⣽⡍⣿⣀⡀⠀⢀⠈⠀⠠⡆⠀⣾⣿⠆⢀⣤⡀⠠⠀⣼⣟⣹⣿⣿⣯⣽⣷⡌⠄
⠰⡇⠸⠿⠀⠘⠃⠘⣇⠀⠀⠛⢿⣭⣿⣿⣿⠟⠄⠀⢾⣿⣿⣿⣿⣟⠛⠻⠿⣀⡀⠀⣠⡀⠀⠀⣨⣇⠀⢀⣉⠀⠐⠰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐
⠀⣁⡠⠤⢄⣀⡤⢤⣍⠀⠀⠀⠀⠀⠙⠉⠈⠀⢠⣄⠀⣬⣉⡛⠻⠿⣿⠇⠸⣿⣿⡦⢈⣹⠿⠚⢿⡿⠃⠺⠿⠇⠈⠁⠻⣿⣽⣿⣿⣿⣵⡿⠁⠁
⣼⣿⠾⣧⣤⢩⡞⡯⢴⢩⠽⣖⠀⠀⠀⠀⢠⣤⠀⠙⠀⢀⡈⠉⠀⠀⠀⠀⠀⠈⠉⠐⠀⠉⠀⠀⣾⡆⠀⠘⠃⠸⠿⠀⠂⡈⠛⠛⡟⠛⠋⡀⠂⣠⣶
⡱⣿⣮⣗⡣⠒⠽⠣⠘⢜⣗⢾⠜⢀⣄⣤⣿⠃⠰⠇⢠⡟⠁⠘⠉⠉⢳⣤⠛⠉⠙⢲⡴⠛⠉⢳⣤⠞⠉⠙⠳⣴⠛⠋⠃⠀⠈⠀⠁⠈⠀⠀⠀⣯⠿
⢠⣦⣄⠀⠠⠶⠶⠆⠀⠀⠀⢠⣄⣼⣿⣿⣿⠀⠀⣄⠀⣤⠆⠀⠀⠀⣠⣏⡀⠀⢀⣼⣃⠀⠀⢀⣿⡀⠀⠀⣨⣇⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⡤⠔⠉⠁⠉⠀⠈⠐⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢭⣀⠀⠀⡠⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠉⠀⠀⠀⢈⢳⡔⠃⠁⠀⠀⠀⠀⢱⡠⡀⠀⠀⠀⠀⠀⠀⠁⠒⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠔⣊⠁⢀⡧⣠⠴⠚⠛⠛⠳⢦⡹⠈⠐⢄⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠘⢆⡔⠁⠈⠁⢀⡾⣰⠃⣀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠙⢒⣶⣞⠁⠀⢿⠘⠓⠀⠀⠀⠀⠀⣼⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣇⠀⠘⠀⠀⠘⢦⡀⠀⠀⠀⢀⡴⠋⠳⣄⡐⠁⠀⠀⠀⠀⣀⡨
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠈⠐⠀⠀⠀⠀⠀⠉⠓⠒⠋⣁⡼⠁⠀⠈⠛⠒⠒⠢⣍⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⣀⢀⣀⡀⣤⠐⢎⡝⠁⠀⠀⠀⠘⠯⠝⠂⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠙⡄⢈⡶⠋⢀⠔⠊⠉⠐⠒⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠉⠉⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⡧⢠⠀⠴⢃⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠌⠃⠚⡂⡘⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⣀⣍⣍⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠉⠉⢨⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡅⢄⣀⣀⣀⡀⡸⣹⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠲⣄⣐⣰⡁⠀⠠⠥⠉⢩⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠁⢹⡏⠀⠀⠀⠀⠐⣠⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠩⠭⢽⡇⠀⠀⡀⠔⣩⢂⠘⡇⠆⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡭⢭⡽⠇⠒⠉⢑⠞⢭⠹⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠳⠤⠿⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣻⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠟⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⣿⣿⣿⡿⠋⠀⠀⠀⠈⠹⠿⠿⠿⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣰⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣤⣶⣶⡿⠿⠟
⠀⠀⢠⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠿⢿⣛⣫⣭⣷⠶⠛⠁
⠀⠀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠚⠛⠛⠛⠋⠉⠁
⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⢠⣤⣤⡄⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡤⠟⠉⠉⠁⠓⠶⠶⠦⠤⠴⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠂
⠀⢸⣿⣿⣿⣿⡿⠋⠁⠀⠈⠉⠻⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿
⠀⠈⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠂⢠⠆
⠀⠀⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠙⣿⣿⣷⣄⡀⠀⠀⣀⣴⣿⣿⡇⠀⠈⠋⠉⠉⠉⠉⠉⠉⠋⠙⠁
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠢⠤
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣠⣤⣤⡄⠀⣶⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠾⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⣤⣾⠟⠁⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⣠⡴⠋⠀⠀⠓
⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⢀⣴⠟⢉⡿⠀⠀⠀⠄⠋⢁⣠⠀⣤⡴⠛⣡⣤⣤⣤⣤⠚⠛⠁
⠀⠀⠀⠀⠀⣸⠋⠀⠀⢀⣤⣾⣿⣧⣴⣿⣦⣶⣾⣦⣤⣾⣿⣿⠀⠀⠀⠘⠛⠉⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⣿⣀⣤⡞⠛⣿⣿⡿⠛⠋⠉⠉⠉⠁⠉⠉⠉⠉⠛⣷⣆⠀⠀⣠⣶⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⣤
⠀⠀⠀⠀⠀⣿⠟⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣥⣤⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠘⣷⣦⣶⣶⣶⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣿⡇
⠀⠀⠀⣦⠀⢿⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠃
⠀⠀⠀⠙⣶⣦⡀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⣹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⠟
⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠻⣷⡟⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣿⠛⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠉⠻⠟⠛⠶⠶⠆⠀⠠⣤⡤⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠴⠤⠤⢤⣤⣤⣤⣤⡄⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⣦⣤⣤⣀⠀⠀⠀⠀⠀⠠⣤⡤⠀⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡿⠀⠀⠀⢰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⠀⠀⠀⠀⡀⠀⠀⠙⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⣀⡀⠈⠓⣤⣀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⠀⠀⠘⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⢀⣾⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⣠⡟⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣴⠟⠀⠀⣿⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠹⣶
⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⣴⠋⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡴⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠿⡀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⢠⣷⠀⠀⢀⠿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠻⣄⠀⢠⣴⡆
⠀⠀⠀⠀⢀⡤⠉⠁⠺⠟⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢹⠃⠀⠀⠛⢦⣤⣀
⠀⢠⣴⣶⠋⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⣠⣿⣿⡿
⠀⣿⣿⣿⣦⣤⣀⣤⣤⣍⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⠀⠀⠀⢠⣤⡾⠛⣦⠘⣿⣿⣿⣿
⠚⣿⣿⣿⣿⣿⠏⠀⠈⠙⠛⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠻⠿
⠀⠛⠋⠀⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⡀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⣿⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣶⡤⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⣿⡀⠀⣿⣿⣤⠄⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠈⡇⠀⢨⡁⠀⠀⠀⠀⠸⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⣧⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣠⣤⡤⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠂⢴⣶⣤⢤⣴⣶⣤⣤⣤⣤⣤⣤⣿⣦⠘⠛⠛⠛⠉⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠈⣏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠘⠛⠛⠢⣤⣀⠀⠀⠀⢀⣴⡄⠀⠀⠛⠢⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠙⠇⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣾⣷⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣶⣿⣿⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡇⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣦⣴⣤⣤⣤⣶⣶⣶⣄⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣗⠀⢠⠆
⠀⠀⠀⠀⠀⣀⣴⣿⡟⠁⠀⠀⠈⠻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡛⠻⢻⠛⠟⠛⢻⡟⠀⠀⠀⠀⠀⠀⠐⢄⢀⡡⠔⠒⠶⢍⡀⡤⠂
⠀⠀⠀⢀⡞⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⡿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⠀⢰⠀⢸⡇⠀⠀⠀⠀⠀⠤⢤⢀⡞⠀⠀⠀⠀⠀⢳⡀⡤⠤
⠀⠀⠀⠸⡅⢿⣿⣿⠀⠀⠀⢀⣀⡤⠞⣡⠏⠳⣄⠀⠀⠀⠀⠀⠀⠀⣿⡇⡁⢀⢸⠀⠀⣄⢸⡇⠀⠀⠀⠀⠀⣀⣀⠸⡇⠀⡤⠶⢤⠀⢸⠇⣀⣀
⠀⠀⠀⠀⠙⠦⣬⣉⣉⣉⣉⣉⡤⣴⠞⠁⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⣿⡗⠟⠛⠛⠃⠀⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⣠⠹⣄⢃⠀⡸⣠⠏⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠶⣞⠻⣶⣄⡀⣀⣾⡇⠀⠀⠀⠀⢠⡞⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢣⠀⠀⠀⠀⠀⠉⠀⣰⢸⣾⣤⣷⡗⢆⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠚⠛⠛⠛⠋⠀⠀⠀⠀⠀⠸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢻⡗⢲⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣏⠉⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⢀⣤⠴⠶⣾⠇⣠⣤⠙⠿⠶⠶⣤⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡄⠃⢸⡇
⠀⠀⠀⠀⣠⢤⣰⡏⠁⠀⢹⡄⠀⠀⠀⢀⣹⣿⡧⠀⠀⠀⠀⢸⣇⠀⣠⡿⢁⡠⣤⡈⠐⣦⡀⢸⡿⠀⠀⠀⠀⢸⡇⢠⠄⢀⡀⣶⡷⠾⠤⠀⢸⡇
⠀⠀⠀⠀⠸⡌⣿⢷⣀⣤⣴⣷⡴⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⣴⠻⣯⣀⡇⢸⣤⣾⡿⠀⣄⣼⢿⡁⠀⠀⠀⠀⢸⡇⠘⠥⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⠁⠀⠀⢸⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠁⠙⣷⣦⣉⣉⣤⠶⣛⣁⢈⣷⠀⠀⠀⠀⠸⣷⣤⣤⣤⣤⣤⣤⣾⣿⣷⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣻⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠶⢿⡟⠛⠙⠛⡾⢯⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣿⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢁⡀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠁⠈⠛⠉⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠯⠼⠂⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⢀⡆⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⣤⣠⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⢸⣷⡖⠀⡟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠼⠿⢴⣶⠿⠷⢤⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠋⢹⠇⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⡿⢩⡍⠉⠉⠉⠉⠉⠉⣿⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣶⠖⢾⡷⠲⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠈⠀⢰⠇⢀⡇
⠀⠀⠀⠀⠀⢀⣷⣬⣥⣤⣤⣤⣤⣤⣴⣿⣾⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⡏⣶⢹⡆⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣦⡤⠖⠋⢀⡞
⠀⠀⠀⠀⠀⠀⣟⣻⣋⡉⣩⣽⣯⣭⣭⣿⣹⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠁⠀⠀⠀⠀⠙⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⠧⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⡀⠀⠀⣀⡀⡀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠰⡂⠀⠐⠚⡠⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⡈⠀⠀⠠⠀⠀⣀⣱⠮⢀⠀⠓⠘⠓⠃⠀⠀⡀⢌⡃⠠⡆
⠀⠀⠀⠓⠓⠀⠀⠀⠠⠀⠢⢨⠀⠀⠀⢀⠀⠉⠀⠐⠸⡁⢐⠀⠁⠊⠀⠀⠀⠁⠀⠀⡁⠀⠀⠀⠀⠀⠀⠄⠌⠀⢁⠂⠃⢸⡓⠃⠰⣀⠇
⠀⠀⠘⠀⠇⠀⠐⢀⠶⠀⠀⠈⠀⠀⢀⠀⠀⠀⠄⢀⠠⡤⠀⠠⠀⠀⠄⠀⡆⠀⢴⠀⢉⠀⠐⠀⠀⠀⠀⢄⠠⠁⠈⠄⠠⡀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠠⠀⠉⠔⡀⠢⠀⠂⠀⠀⠄⠀⠠⠀⠀⠄⠈⠈⣠⣀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢀⣀⠀⢀⠠⡁⢈⠆⢄⠀⠤⣄⠀⠀⢔⣀
⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⢴⠆⡄⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠤⠦⠘⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⣱⠀⠀⠀⠀⢀⠈⠠⠄⠈⢷⠀⠢⡤⠃
⠀⠀⠀⢾⡐⠁⠙⢀⡇⠀⠀⠖⠂⡀⠀⢐⡛⠆⠐⠶⣆⣶⡄⠀⠮⠆⠀⠌⣀⢂⡂⠀⠔⠢⠀⠸⠭⠭⠿⠿⠽⠇⠀⠌⠉⠈⢀⠅⠀⠀⠀⠐
⠀⠀⡀⠀⣅⡋⣀⠁⠁⠁⠀⠈⠀⠀⣲⢢⡀⠀⠀⠧⣍⠴⠀⠠⣄⠄⠑⠀⢻⠀⠈⠀⠀⠀⠀⠀⠉⠈⠚⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠉
⠀⢀⠤⢌⠉⢠⠈⠉⠀⠠⠀⠀⠀⠀⠠⣉⠀⢻⠒⡄⢠⠀⠈⣡⢀⡐⠂⠠⠃⠀⠀⠀⠄⠀⡄⠁⠀⣤⠀⣴⡶⢿⠂⡀⢠⡀⠄⡀⠈⠉⣂⠈
⠀⡇⣀⠈⠀⠈⡀⠀⠀⠈⠁⠀⠀⣦⡄⠉⡀⠈⠒⠁⠀⠈⠃⠀⠀⠁⠤⠆⠀⠠⠀⠂⠀⠂⡄⠀⠘⠓⠛⣂⡅⠂⠀⠀⢀⢌⠉⢋⡀⠁⠄⠈
⠀⠑⠤⠜⠀⠀⠀⡀⠐⠀⠀⢀⠤⠤⠤⠀⠫⠀⠀⠀⠀⠰⢆⠀⢀⠀⠘⠂⠀⠜⣁⠀⢿⠆⠠⣛⠃⠰⠊⠁⣀⠋⡁⢀⣠⡃⣀⢀⠋⠤⢰⡄⠆
⠀⠀⣰⡆⠀⢀⠀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠀⠠⡦⠀⠐⠀⠀⠀⢠⠀⠁⡠⠒⠢⡀⢀⠠⠢⡀⠀⠀⠀⠤⠀⣿⠀⠶⠀⡀⠀⣀⢄
⠀⠀⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠚⠃⠄⠀⠀⠀⠈⠀⠒⠉⠀⠀⠀⠀⠱⠄⢤⠃⠘⢈⠌⠀⡡⠀⢀⠀⠀⢰⠀⠀⠀⢠⠀⠀⠃
⠀⢀⡀⣠⠀⠀⠄⠄⠂⠀⠠⠴⡏⠲⢀⠔⠁⠐⠄⠀⢠⠒⢸⣿⠷⠾⡀⠀⠀⠀⠀⠂⠀⠀⠠⠀⠐⠀⠂⠀⢀⡞⠀⠀⡀⠀⠀⠀⠰⠁
⠀⠑⠸⠈⠈⠆⠀⠀⠀⠠⠀⠀⠇⠀⠈⠑⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢟⡠⠀⠂⡄⠀⢬⣽⠀⣄⡠⠨⠀⡀⠐⢀⣁⠀⠂⠀⠂⠀⠁⡀
⠀⠒⢤⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠈⠀⠄⠀⢨⠾⡀⠀⠀⠀⡀⠀⠀⢸⠀⠐⠉⠁⠀⠀⣉⢀⠀⢈⡉⣍⠀⠂⠘⠄⡃⠉⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠈⠀⠐⠊⠀⠀⠂⠐⠀⠀⠀⠈⠀⠀⠑⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣠⣤⣀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣲⣬⣭⣤⣤⣶⣶⣿⣿⣿⣿⡿⢿⣿⣿⠀⢹⣿⣿⣿⣿⣿⣛⣻⡶⠶⣖⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣻⡿⣟⣉⣉⣭⣤⣐⣒⣒⣒⡒⠒⠒⣚⣛⣛⣭⠀⢸⣿⣷⣾⣶⣿⣿⣿⣷⣶⣯⣭⣞⣻⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⣵⡿⢽⣷⣿⡿⠿⠛⢛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣉⣧⠀⢺⣿⣿⣿⣿⣿⣿⣟⣿⠿⢿⣿⣿⣷⣶⣜⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣳⡿⣫⣾⠿⣛⣥⣶⠾⢟⣛⡿⠿⠿⠿⠛⠛⠛⠻⠿⠿⣿⠀⣸⣛⣉⣉⣉⣉⡛⠿⠿⣻⣿⣶⣾⣳⢌⣻⢿⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⠟⣡⣾⡿⣯⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⢮⣵⠾⣿⣯⢿⣷⡹⣿⢿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⢫⣾⣟⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣝⢷⣍⢷⣜⢯⢿⡆
⠀⠀⠀⠀⠀⡴⠟⢿⣯⣿⣯⣼⣥⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣾⣶⣿⣧⣿⣯⣷⠖⠉⠳⣆
⠀⠀⠀⠀⢸⠃⣤⣄⢿⣿⣿⣾⣷⣦⣤⣤⣤⣭⣤⣬⣿⣯⣤⣽⣿⣿⣿⣿⡆⠀⠀⣼⡟⢉⣍⣩⣭⣥⣤⣤⣤⣤⡶⡶⠶⠶⣷⡆⣿⣯⡿⢿⣴⣿⠄⡿
⠀⠀⠀⠀⠘⣿⣿⣿⡇⢹⡟⣿⡏⠀⠉⠉⢉⣉⣤⣐⣠⣀⠀⠀⠀⢸⡿⣿⠇⠀⠀⣿⣷⣿⠉⠉⣡⣼⣯⣭⣤⣒⠀⠀⠀⠀⢸⡃⣿⡟⠁⣾⣿⣿⡴⠃
⠀⠀⠀⠀⠀⠙⣯⠉⠀⢸⣷⣿⡄⠀⢀⣴⡟⠉⢹⣍⡹⢿⣆⠀⠀⢸⣿⣻⡄⠀⠀⣿⡿⣿⠀⣼⣿⣁⣨⣿⣉⣿⣷⡀⠀⠀⣸⠁⢸⠀⢀⠈⣻⠏
⠀⠀⠀⠀⠀⠀⠹⣦⣴⢸⣿⣿⡀⠀⡎⢿⡿⣾⣿⣿⡋⢸⣿⢆⠀⢸⡇⠿⠷⣶⣶⣿⡇⣿⢰⡟⣿⡛⣿⣿⡏⣸⣿⣹⠀⠀⣿⢁⣿⠀⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⣿⣿⡇⠘⠷⣼⣿⣌⣉⣉⣠⡾⠏⠉⠀⢸⣧⣶⠾⠿⠻⣿⣿⡇⠈⠉⠻⢿⣤⣤⣶⠿⠟⠁⠀⠀⣿⢶⡿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠈⣿⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⣀⣸⣿⣿⠶⣤⡶⢿⣿⣷⣤⣴⣶⠶⢶⡶⠶⠶⠶⠶⠶⠾⡟⠸⣷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢀⠙⠛⠛⠛⠛⠛⠛⠛⠛⢛⣛⣛⣛⣟⣹⡧⠀⠀⠀⢸⣧⣤⣤⣦⣤⣤⣤⣤⣤⣦⣦⣤⣶⣿⣴⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢶⣿⠿⠿⠿⠿⠟⢿⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⢾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠶⠛⠙⢿⠛⣿⠶⣶⣤⣤⣤⣀⣀⣀⣀⣠⣤⡴⢶⡾⠛⣙⠿⣦⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⢹⡇⢻⡄⠀⢸⡏⣻⣿⣿⣿⣟⢹⡿⠀⣸⡇⢰⡟⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠘⣧⢸⡇⠀⠸⠿⠋⢉⡟⠀⠉⠻⠇⠀⣾⠁⣼⠃⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⡟⢸⡇⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⢡⣿⡀⣿⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⣾⣿⣧⣾⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⢻⡇⣿⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⡇⠀⠀⠀⠀⢸⠉⠀⠀⢻⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣧⣿⣿⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⢸⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⣿⠀⠀⠀⠢⠀⠀⢸⣿⣿⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠘⢦⣄⠀⠀⢿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣖⠀⠀⠀⠀⠉⠙⢶⣿⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠘⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠉⠳⣄⠀⠀⠀⣀⢿⢻⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣯⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡤⠤⠼⠷⣄⣠⢹⡆⢀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⣿⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢿⣉⣉⣀⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⡇⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠈⠉⠈⠉⠙⠛⠛⠛⠛⠶⠶⠶⠶⠶⠚⠋⠁⣸⣿⣷⣜⣛⣿⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡋⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠐⢶⣤⡶⠞⠀⠀⠀⠀⠀⣰⠏⠀⠙⠓⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⡏⡀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡇⡇⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣀⣀⣀⣼⠁⣷⣀⣀⣠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⠿⠿⠿⠿⡿⠀⢿⣿⣿⣧⣿⣷⡄


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣍⣿⣋⣙⣋⣛⣉⣤⣉⣙⣟⣋⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣭⣯⣭⣬⣭⣿⣯⣭⣭⣭⣯⣭⣭⣭⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⠀⠀⠀⣀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡗⠲⡀⠈⡀⢀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣇⠀⠻⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⣀⣀⣠⣔⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢠⣤⣴⣿⣿⠃⠀⠀⠀⠈⢰⡇⣄⣀⢀⣀⣤⡄
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣠⣤⠴⢿⣈⣿⡟⠉⠻⢿⡄⠀⠀⠠⡆⢸⣷⣿⠉⠈⣿⣿⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠻⣯⠑⢛⡁⠂⠈⢻⣿⣷⠀⠀⡀⠀⢀⣤⡀⣷⣿⠿⣛⡆⣸⡿⢿⣿⣿⡿⠿
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠁⠀⠀⢿⣿⣧⢠⣿⠀⠀⡿⠀⢉⡟⠀⠏⣸⣿⣄⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢠⠡⠹⠆⠀⠀⠈⠀⠀⠘⢻⣿⣾⣧⠀⠰⢷⡄⠸⠷⠠⠂⢁⣾⣯⡿⠛⠿⠿⠿⢇⣤⣄
⠀⠀⠀⠀⠀⣀⣸⣤⣤⣄⡀⣠⠂⠀⠘⢿⣭⢿⣿⣷⠀⠀⣿⠀⠀⠀⠀⠀⠘⡿⠛⠁⠀⠀⠂⢀⣿⠹⠙⢷⡀
⠀⠀⠀⠰⠋⠁⠀⠀⠀⠀⠀⠀⠁⠈⣀⠉⢿⣾⣿⣿⡀⠀⡽⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⢀⡴⠻⠁⠀⠀⠀⠁
⠀⠀⠀⠘⢷⠶⠦⠤⠂⠄⠀⠀⢂⣄⡈⠻⣾⣿⣿⣿⡇⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠪
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣌⣻⣿⣿⣧⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢦⡈⠻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠰⣶⣬⣁⡈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣷⢆⠄⠀⠀⠀⠀⠀⡐⢋
⠀⠀⠀⠀⠀⠀⠈⠙⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⣤⣄⠀⠀⢠⣦⣰⣾⣴⢂⡤⠶⠦
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣄⣀⣤⣀⣀⣀⣤⡶⠶⠖⠛⠛⠻⣿⣿⣧⣄⡀⠉⠉⠻⣿⠿⢶⣄
⠀⠀⠀⠀⠀⠀⣾⠙⠛⠻⠿⠿⠿⠿⠛⠉⠀⠐⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⡄⠀⣶⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠸⣧⡈⠀⠀⠀⠀⠤⠐⠂⠀⠀⠀⠀⢀⣤⡴⢊⡿⢳⣼⡟⣹⠟⡀⢹⡄⠀⢼⣁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⣦⣤⣠⠤⠄⠀⠀⢀⣍⣸⡿⠋⢀⣬⣿⣷⡿⠟⡃⢸⣷⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠟⠉⠀⠀⣶⠋⠋⠀⣠⡾⠃⢸⡟⣆⠀⠀⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠇⠀⠀⠀⡰⠃⠀⠀⡄⠉⠀⠀⣿⠀⠘⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⣼⠁⠀⠀⠀⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⢸⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠈

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠟⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣄
⠀⠀⠀⢸⠃⠀⠘⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠟⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢿
⠀⠀⠀⠸⡄⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣤⣤⠎
⠀⠀⣀⣴⠫⣀⠝⣦⣀⠀⠀⠀⠀⠀⢀⣀⣤⣦⠉⣌⢡⣦⣄⣀⠀⠀⠀⠀⠀⣀⡠⡿⢕⡪⢷⢄⣀⠀⠀⠀⠀⠀⣀⣤⣴⡌⣻⡁⣴⣤⣄⡀
⡔⠉⠀⠈⠋⣿⠛⠁⠀⠉⢢⠀⠀⠀⣾⣿⣿⣿⣆⣿⣸⣿⣿⣿⡇⠀⠀⢠⠋⠀⠀⠳⢺⡗⠇⠀⠀⠑⡄⠀⠀⣼⣿⣿⣿⡇⣼⣀⣿⣿⣿⣿
⡇⠀⠀⠀⠠⣿⠆⠀⠀⠀⠸⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠇⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⠿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣦
⠀⠀⠀⣸⠛⠛⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⣹⣃⣀⣀⡹⡇⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢸
⠀⠀⠀⠙⣄⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⡍⠉⠙⡱⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⢀⡞
⠀⢀⣀⣰⣯⠉⣹⣆⣄⡀⠀⠀⠀⠀⢀⢠⣤⠋⠒⠊⠣⣤⢀⡀⠀⠀⠀⠀⠀⣀⣰⣏⠐⢉⣧⣀⡀⠀⠀⠀⠀⠀⢀⣠⣴⠃⠉⠁⢳⣤⣀
⢠⣿⣿⣿⣾⣿⣷⣿⣿⣿⡆⠀⠀⢰⠁⢸⣿⣄⣀⣀⣼⣿⠀⢹⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⡜⠁⢸⣿⣷⣶⣶⣿⣿⠀⠑
⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠸⠀⠸⠿⠿⠿⠿⠿⠿⠄⠸⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠇⠀⠾⠿⠿⠿⠿⠿⠿⠆

⠀⠀⠀⢠⣶⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠐⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣷⣦⡀
⠀⠀⠀⢸⠉⠛⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠛⠉⢻⡇
⠀⠀⠀⠸⡄⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠒⠁⡸⠂⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⣰⠇
⠀⢀⣠⣴⣏⠒⣫⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⣷⠄⢴⡁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣍⣩⣿⣶⣄⣀⠀⠀⠀⠀⠀⣀⡤⡾⣷⣦⣾⣿⣦⣄⡀
⣰⣿⣿⣿⣿⣷⣿⣿⣿⣿⣧⠀⠀⢠⠔⠊⠉⣧⢿⣥⠋⠙⠒⢀⠀⠀⠘⢻⣿⠛⠿⠟⠿⠿⠿⡿⢿⣧⠀⠀⢠⡞⠉⠀⠹⡼⣿⣿⡼⠁⠈⠙⣆
⠹⠛⢻⣿⣿⣿⣿⣿⡟⠛⠻⠀⠀⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⢾⠛⢻⣿⣿⣿⣿⣿⣿⡟⠛⠿⠀⠀⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸

⠀⠀⢀⣠⠖⠒⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⣶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠂⠒⢦
⠀⠀⢸⣧⡀⡀⡀⣸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠉⢿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⡇
⠀⠀⠀⠳⣀⣉⡀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠋⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⡀⢰⠃
⠀⠀⣀⣠⡟⠿⢛⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣑⣤⣚⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⡻⣿⣿⢟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⠿⢿⣿⣦⣄⡀
⣴⣿⣿⣿⣿⣶⣿⣿⣿⣿⣦⠀⠀⢰⣿⣿⣿⣟⠉⢹⡿⠿⢿⣧⠀⠀⢠⣴⣾⣷⡻⣮⡴⢏⣿⣷⣦⡀⠀⠀⢠⣾⡿⠿⠿⣿⡆⣿⠿⠿⠿⣿⣦
⣿⣷⣶⣶⣶⣿⣶⣿⣿⣿⣿⠀⠀⣼⣿⣿⣿⣿⡄⣾⣿⣿⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣧⣽⣯⣼⡇⣿⣬⣿⣥⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡌⠀⠀⠀⠀⠀⠀⠀⠄⠀⡦
⠀⠀⠀⠀⠀⠀⠀⢀⠒⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠌⠀⠜
⠀⠀⠀⠀⠀⠀⠐⡁⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠃⠀⠀⠀⠀⠀⠀⠌⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⢀⠖⠀⡰⠆
⢀⠒⢄⠀⠀⠀⠀⠀⠀⠀⠘⢄⠃⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⢀⠔⢁⠄⠊
⠢⡀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠈⠂⡄⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⡠⣃⠔⠁
⠀⠀⠑⢄⠈⠢⡀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⠬⠊
⠀⠀⠀⠀⠈⠂⢈⠢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠙⠲⢤⡀⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠱⣄⠀⠀⠀⠀⣠⠔⠁⠀⣀⣤⣴⣦⡀⠀⠀⠀⠴⠿⢿⣿⣦⡀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⢀⠜⠁⠀⢀⣾⠟⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢿⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠼⠁⣠⠀⠒⠒⠤⡀⠀⡠⠐⠂⠐⠠⡀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠖⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢀⣀⡈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡃⠀⠀⡴⣦⠀⠀⢸⠀⠀⣴⣶⠀⠀⠘⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⡜⠉⠉⠉⢢⡤⠎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠉⠉⠀⠀⡘⠀⠀⠈⠉⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⠀⠀⠀⡇⠀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠾⠛⠁⠀⠈⢄⠀⠀⠀⠀⠠⠉⢄⠀⠀⠀⠀⠐⠁⠀⠲⠶⠄⠀⠀⢸⠀⠀⠀⠘⡅⠀⠑⠤⠐⢇⢀⡰⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡃⠀⠀⠀⠁⠂⠀⠈⠀⠀⠀⠁⠒⠂⠈⠀⠀⠄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠉⠣⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣤⠆⠀⠀⠀⡏⠀⠀⠀⠀⡰⢁⠞⠢⠤⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢹⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⡟⠀⠀⠀⣼⣁⣀⣀⠴⠊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⣧⡀⠀⢿⣿⣿⣷⣶⣶⣶⣤⣤⣤⣴⣶⣶⣾⣿⣿⣿⡿⠁⠀⢀⣼⣿⣄⠀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⠟⠳⣄⠈⢻⡿⠋⠉⠉⠉⠛⠉⠁⠈⠉⠙⢿⣿⣿⡿⠁⠀⡠⠊⠀⠈⠉⠉
⠀⠀⠀⡀⠠⠀⣀⡜⠁⡴⠋⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⢠⠎
⠀⠰⠉⣀⡀⠀⠈⠁⠎⢀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⢀⠗⠉⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠎⠀⠀⠀⣠⡖⠀⡄⠀⠰⡀⠀⠀⠀⠀⠀⢠⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀
⠘⠦⠤⠾⠇⠆⠀⠱⣄⠀⣈⠀⠀⠀⠀⠀⢸⠀⠀⠉⠁⠀⠒⠀⠀⠀⠀⠀⣀⡀⠀⠬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣀⠀⠤⠄⠀⠒⠂⠀⠈⠉⣀⣀⠠⠤⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠤⠤⠀⠒⠒⠀⠉⠉⣀⣀⡀⠤⠤⠤⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠆⠒⠒⠂⠉⠉⠀⠀⠀⠀⠀⡴⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠶⢄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢷⣶⣶⣾⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠉⢀⠐⠂⠒⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠟⣆⠀⠬⡁⠄⢉⡟⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠁⢈⡃⠠⠆⠀⠀⠀⠀⠀⠀⣡⠀⠀⡀⡙⠇⡐⠀⢨⠄⠈⠵⠊⠀⠀⠘⠛⠀⠠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⣿⠀⠀⠀⠀⠀⣸⣿⣄⠀⠆⠀⠀⠈⠈⠉⢰⠆⠀⠀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⢄⠀⠀⠈⠀⠀⠀⠀⠈⠈⠻⠛⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠃⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣿⡀⢀⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣁⡆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⣤⡀⢸⠿⡷⢦⣻⢲⢲⡖⣾⡗⡦⡏⡷⡟⡟⢹⣶⣠⡇⠈⠉⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡠⠄⠀⠀⠁⠀⡀⠀⠉⠉⠈⠈⠀⠉⠀⠀⠁⢉⠁⠁⠀⠈⠀⢉⡇⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⢒⠇⠀⠙⠛⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⢀⡁⠐⠀⣠⠀⢤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⡖⠀⠀⠀⠂⢀⡀⣙⠕⠀⠀⠘⠦⠤⠀⢀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⣿⡟⠚⠃⠀⡆⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⣉⠑⡋⠐⢠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⣄⣀⠀⡀⠀⠈⠛⣷⣶⡄⣰⠇⠀⠰⣄⠀⠀⠀⠀⠀⠁⢀⠨⢁⠊⠁⣄⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⣼⡟⣻⡟⠀⠀⠀⠘⠛⠛⠥⠂⠀⠀⠹⣷⡆⠀⠀⠀⠀⠀⠀⠋⣀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠉⠀⠀⠀⠐⠀⠉⠐⠋⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡀⠉⠀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠰⠇⠀⠈⠁⠀⠀⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠀⠀⠁⡀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠹⠀⠀⠉⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠠⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢢⢀⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣀⠰
⠀⠀⠀⠀⠀⠀⠀⡄⠉⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠀⠀⠀⢄⡀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⠟⠁⠀⠀⠀⠀⠀⢑⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠄⠀⠀⢀
⠀⠀⠀⠀⢀⢀⠀⠈⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣼⡇⢿⣷⣾⣿⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣾⠁⠀⠀⢀⠘
⠀⠀⠀⠰⡀⠆⠀⠀⠁⠀⠀⠘⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⣤⣤⣤⣤⣾⣿⣿⡷⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠸⠿⠿⠿⠛⣿⣿⠟⠁⠠⠀⠛⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠸
⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐
⠀⠀⠀⠐⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⠆
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠘⠀⠀⡀⠰⠀⠀⠀⠈⢿⣿⣶⣤⣤⣀⣀⣀⣀⢀⣀⣀⣀⣀⣠⣤⣴⣶⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢂⡄⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠐⠀⠀⠀⠇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⡀⠂⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠈⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠉⢉⠙⠛⠛⢛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠤⠐⠁⠀⠂⠠⠐⠀⠂⠀⠀⠀⠀⠈

⠀⠀⠀⣀⣤⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⡀⢸⠀⡰
⢀⣴⣾⡟⠁⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠉⠉⠙⠉⢻⡇⠀⠀⠀⠀⠈⠓⡤⠓⠉⠑⠣⡔⠋
⡏⣿⣿⡇⠀⠀⠀⢀⣠⢇⠧⣄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠆⢸⡇⠀⠀⠀⠈⠉⢰⡇⠀⣀⡀⠀⡗⠈
⠙⠬⣙⣓⠒⣒⣋⣩⡴⠉⠀⢸⡀⠀⠀⠀⠀⠀⣿⠷⠶⠾⠀⠺⢿⡇⠀⠀⠀⠈⠉⣀⢣⡸⠀⡸⡰⢃⡉
⠀⠀⠀⠙⠛⠛⠲⣟⢷⣤⣀⣼⡇⠀⠀⠀⢠⠎⠉⠉⠉⠉⠉⠉⠉⠙⡆⠀⠀⠀⠈⠀⠔⣷⣧⣿⠗⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠈⠣⡤⠤⠤⠤⠤⠤⢤⠴⠃⠀⠀⠀⠀⠀⠀⠻⣿⡿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢳⠲⡄⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⢀⣀⠤⣏⠁⠀⠀⠸⣿⣇⠀⠀⠀⢀⡴⠒⢲⡟⠤⠬⠛⠒⠲⣦⡀⠀⠀⠀⡿⠀⠀⠀⠀⡀⡠⠈⢸
⢰⣳⣼⡀⠀⢘⡄⣀⣠⠤⠿⠟⠀⠀⠀⢈⣷⡄⢺⢡⠒⣶⡄⠑⢤⡾⠁⠀⠀⠀⣟⢰⠅⠢⠘⠛⠓⠁⢸
⠀⠳⠃⠛⢹⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢸⡤⠛⢿⣌⡻⢛⣡⡶⠋⢹⡀⠀⠀⠀⣿⣀⣁⣀⣀⣀⣴⣶⣼
⠀⠀⠀⠀⡾⣿⢳⡀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠾⣿⠛⠛⢳⢯⡹⠛⠀⠀⠀⠀⠀⠀⠀⠈⡇⡏
⠀⠀⠀⠸⠃⠿⠈⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿

⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠑⢢⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⡜⠉⠉⠛⠉⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠪⠗⢀⡇⠀⠀⠀⠀⠀⠀⢸⠁⣹⠀⡄⠀⢸⠁⣹
⠀⠀⠀⢳⡀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠦⢴⣮⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢠⣷⡏⢸⠉⢹
⠀⠀⡤⠤⠷⠤⠤⠴⠯⣤⣄⠀⠀⠀⠀⠀⠀⠀⣏⣉⣹⣏⣉⣹⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠃⢸⠀⡸
⠀⢀⣧⣿⣤⣤⣤⣤⣤⣯⣿⠀⠀⠀⠀⠀⠀⢠⣿⠁⠈⠁⢸⣫⢲⠀⠀⠀⠀⠀⠀⢿⣿⣦⣤⠤⠋⣠⠃
⠀⠈⣟⣟⣉⣹⣿⣿⣿⣿⣻⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠈⢲⠋⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠄⠠

⠀⠀⢀⣀⣤⣦⣜⣶⠟⠉⠀⠀⠀⠀⠀⠀⠈⠲⠇⡩⠛⠓⠁⠣⠘⠕⢊⡡⣨⠜⡟⣶⠃⠄⡈⢄⡔⠈⡀⠀⠁⢀⡰⠀⠉⠀⡜⣸⠂⠀⠀⠀⠀⠂⡄
⠀⠘⣯⣿⢽⣟⣿⠃⠀⠀⠀⠀⠀⠀⢀⢀⠤⠀⠌⠂⠑⠙⠊⠔⠡⢊⠄⡁⠠⢑⢸⠇⠀⠈⣀⡭⡀⠈⢀⣼⣷⡞⠁⠀⠠⢰⢁⡟⠀⠐⠀⠀⢊⠀⠀⠐⠃
⠀⠀⢋⡡⢞⡾⠁⠀⠀⠀⠀⠀⠔⠉⠈⢀⠀⠤⠰⠂⠐⠀⠂⠒⠀⠄⡈⠊⣤⣴⣴⣌⣠⠁⠎⠀⠀⣡⣾⣿⣿⡟⠀⢀⢠⡇⡘⠀⠀⠀⠀⠀⠄⠒⣂
⠀⠀⢟⣅⣮⡆⠀⢀⡠⠂⠁⢀⠠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢼⡏⣿⡧⣻⡎⠉⠆⢂⣼⣿⣿⣿⠟⠀⢀⢀⠊⠐⠀⠀⡀⠤⠔⣢⠫⠐⠀⠂⠄
⠀⢀⣄⣿⢿⠀⡐⠂⠀⡠⠘⠀⠀⠀⠀⢀⣠⣴⣶⣞⣛⢿⣿⢿⣶⣶⣤⣈⠻⠿⠿⠟⠀⠀⣰⣿⣿⣿⡿⠋⠀⠀⢀⠎⡴⠀⠀⢨⣴⡮⣜⣖⣫⠜⡄⣇
⠀⢘⣯⠷⣿⠂⠀⡠⠊⠀⠀⠀⣀⣴⣾⣿⣿⠿⠋⠁⠉⠀⠉⠙⣶⣛⣿⣿⣷⡀⠀⠀⠠⣼⣿⣿⣿⡟⠀⠀⠀⠀⢊⠜⡡⢪⡴⣟⡇⣯⢵⡸⠆⡫⠩⠈
⠀⢘⣾⡾⠁⠀⠌⠀⠀⠀⣠⣾⣿⢟⣻⠟⠋⠀⡀⢨⣔⣧⡄⢔⠘⣿⣿⣴⣿⣿⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣴⣻⣃⡇⢻⠜⢃⠐⠂⠀⠆
⠀⠀⡟⠀⠀⠂⠀⠀⣠⣾⣿⠟⡝⡡⠱⠀⣀⢨⡾⣿⣿⣿⣻⠐⡆⢸⣿⣭⣿⣿⡇⢸⣿⣿⣿⡟⠀⠀⠀⠁⠄⠀⢈⣾⡋⢓⠃⠜⠠⠀⠀⠀⢀⣀⣤⢤⠄
⠀⠀⠀⠀⠂⠀⢀⣾⣿⣿⣧⢞⣤⢄⡆⡐⣠⣛⢏⠀⢆⠋⢿⠀⡇⢸⣿⣿⣿⣿⠁⣿⣼⣿⣿⡁⠀⠠⠀⠀⠠⢀⣿⣛⣉⠃⡘⠀⠁⢀⡴⠛⠉
⠀⠀⠀⠌⠀⣰⣿⣿⣫⢻⢿⠻⠍⠿⠀⢤⣯⣦⣷⣥⣂⣳⣻⣿⠇⣼⣿⢟⡧⠜⠀⣿⣿⣿⣿⣧⡀⠀⠀⠀⡠⠚⠋⠉⠉⢀⠀⠀⠐⠋⢀⠀⠤⠀⣠⣤⡀
⠀⠀⠘⠀⣰⡿⣟⣃⠄⠀⠀⠐⠀⡌⢸⠋⠣⠊⢿⣿⣄⣞⣧⠂⣰⣿⢣⡔⠄⠀⠠⠹⣾⣿⣿⣿⣿⣶⡦⡄⠖⠘⠙⡉⠉⠉⠩⠛⠫⠱⡄⡀⠲⢿⡏
⠀⠀⠃⢰⣿⡷⡷⣧⠀⠀⠀⠈⡂⢱⠹⡀⠀⢀⣼⣶⠟⠃⢁⣼⣟⣿⡿⠃⢀⠀⠀⠑⣝⠿⣿⣿⠟⠋⠁⡀⡄⢠⢢⠂⠄⠆⠦⠤⣀⠀⠈⠂⠑⠆⡉⠠
⠀⠀⢠⣿⠯⡜⠉⠍⢐⣁⣀⠀⣑⠇⡈⠀⠀⣾⣿⣏⣠⣶⢿⠷⠷⣋⣀⠀⠰⠀⡆⢤⠀⢠⠈⠃⣀⠔⠈⠀⢀⣀⣤⣤⣤⣤⣤⣀⠀⠉⠂⢄⠀⠀⠘
⠀⠀⣸⡟⠀⢄⠀⠀⣾⣿⠗⡨⡨⠁⠅⠀⢸⣿⣿⣿⠽⢒⣍⣿⣼⣿⣿⣷⣦⡀⠁⠈⠠⠈⢀⠆⠁⢀⣰⣾⠿⠿⠿⠿⠿⠿⠿⠿⣿⣶⣄⠀⠙⢄⡀⠈⠄
⠀⠀⣿⢮⢰⠆⡑⢾⠁⠘⢁⠠⠆⠧⡁⠀⣿⣿⡿⢁⣜⣷⠟⠉⠀⠈⢿⣿⣿⣿⡄⠀⢀⠔⠁⠀⡴⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡈⠙⢷⡄⠈⠳⡀⡁
⠀⠀⣟⠀⢘⠃⢌⡄⠀⣴⢶⢾⣷⡠⠀⢸⣻⣿⠁⣾⣽⠃⠀⠀⣦⣄⣀⠀⠉⠙⡎⡖⠚⠀⢠⠞⠤⠤⠞⢳⡄⠀⠀⠀⠀⠀⠀⢯⠀⠉⠳⣌⢻⡆⠀⠙⡅
⠀⠀⠘⣤⣶⣶⠄⠑⡀⢸⢹⢹⠞⣯⠀⢸⣿⣿⠀⣱⣿⡀⠀⠀⠘⣿⣿⠇⠀⠀⢇⢈⠀⠀⡎⢠⠖⠲⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠁
⠀⠀⡌⠜⠿⣻⣔⡤⠢⡐⠀⠠⡂⢸⡄⠘⣿⣿⡄⠹⢞⣷⣄⠀⢰⣿⡿⠆⠠⠀⣠⠖⠀⢰⠓⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⢺⠂⠀⠁
⠀⠀⣦⠘⠢⡏⠽⢧⠀⢀⠀⡀⢧⣇⣻⡆⠘⢿⣿⣦⣈⠁⠋⠿⠋⠉⡀⣠⠴⠋⡡⠀⠀⠸⣠⣆⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡆
⠀⠠⣏⣆⡁⠀⡌⠌⣴⣖⣦⡘⠀⢙⡯⣿⠀⠀⠈⠛⡡⢍⢁⣀⣐⣉⠍⠥⠚⡡⠐⢀⠀⠀⢻⣿⣄⠀⡏⠀⠙⢆⢠⡀⠀⠀⠀⠀⠀⠀⣴⠋⢳⠆⠀⠀⡇
⠀⠀⢿⡿⠈⠑⢄⡐⢿⢟⣞⠛⢆⠀⢻⣾⣷⡀⠀⠀⠀⠀⢄⠀⠀⢀⣀⡄⠂⠁⣡⣤⣖⠀⠈⠻⣿⣆⠙⠢⠤⢄⡀⠈⠓⢦⢣⠀⠀⠀⠈⣲⠏⠀⢀⡼⡇
⠀⠐⢸⢲⣷⣶⣦⡁⠈⠸⢹⠀⡀⠃⠌⠹⢿⣿⣷⣤⣀⠀⠚⠈⠋⠘⠀⣁⣤⣾⣿⡿⡀⢀⠂⠀⠙⠿⣷⣄⠀⠀⠳⡄⢸⡜⢰⡀⢀⣠⠞⠁⠀⡄⠪⠯⡗
⠀⠀⠈⠨⠨⢩⢿⡿⣔⠐⠀⢀⠁⠣⣖⣀⠀⠙⠟⢿⣿⣿⣿⣷⣷⣿⣿⣿⡿⡿⢋⠀⢀⠀⠠⠁⢀⠀⠈⠙⠳⠤⠄⢌⢼⠡⠤⠖⠉⠀⡀⠔⢉⡒⣝⣝⡃
⠀⠀⣄⠰⣄⢆⡆⠘⠻⠧⠈⠈⢢⡀⢀⠁⠠⣖⡄⡠⠄⠍⠩⢙⢙⢋⡉⠡⡔⠈⠈⠀⠀⠀⠐⢆⣠⣿⣦⠂⠀⠀⠀⠀⠀⠀⠀⠄⠖⠈⢀⠠⢎⢪⡵⢣⠁
⠀⠀⠠⡠⠉⢻⣿⣿⣆⢰⠀⠢⠄⡵⡯⣅⡲⡀⠁⠀⠐⠀⢁⠉⠊⠁⢀⠀⣠⠄⠖⢲⠂⢀⡄⢬⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢎⠝⣪⠟⡨⠃
⠀⠀⢁⢋⠊⠐⠎⢿⣷⣿⣦⣤⡁⠀⡁⠊⣔⢎⡉⠺⢄⡤⣠⢄⡂⠱⠵⣯⣇⣤⣧⡦⣆⣴⣿⣿⣿⣯⡟⠁⠀⠀⠀⠀⠀⠀⢀⡠⠿⣚⡱⠞⣡⠞
⠀⠀⠀⡀⠈⠐⠲⣦⣿⣮⣿⣿⣿⣲⣶⡀⢠⠋⢄⠐⡀⠖⢈⡁⠂⠀⣘⠻⡿⢿⣟⡫⠿⠛⠻⣿⠿⠉⠄⠀⠀⠀⠀⠀⠀⣠⡤⠶⠚⣩⡴⠞⠁
⠀⠀⠐⠊⠵⠒⠦⠄⡛⠿⠿⢿⢟⠹⡛⠛⠻⠶⢆⣻⢷⡨⠀⠹⡗⡶⠿⠻⠏⠹⠈⠁⠁⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠠⡾⠾⠔⠃⠉⠁⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⣾⣶⣾⣷⣶⣶⣿⣶⣶⣶⣶⣷⣾⣶⣷⣶⣷⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡚⡠⠠⠠⡄⠀⠄⠀⠉⠀⠒⠂⠈⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⡀⡀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠘⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⠚⠀⠀⡀⠠⡄⠀⠁⠈⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡃⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠈⠁⠉⠉⠀⠑⠤⠂⠀⠀⠀⠀⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠀⠁⠈⠀⠀⠀⠀⠀⠀⠉⠀⢹⠀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡂⠄⠀⠀⠀⠀⢐⣓⣒⣛⣒⣒⣳⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠀⢄⠀⢀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⡩⠘⠁⠄⡒⠀⠀⠂⠀⠀⠀⠄⠂⠀⠀⠂⠠⠀⠀⠀⠀⠺⠂⠀⠀⠀⠈⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠛⡄⠛⠂
⠀⠀⠀⢠⢅⣠⠀⠀⠀⠀⠛⢠⡶⣆⠀⢶⠆⠀⡴⠶⣦⠀⣴⠀⠀⢀⡴⠲⣦⠀⣠⠖⠶⠄⠐⣆⠀⣴⠀⠀⠈⠢⠤⠚
⠀⠀⠀⠀⠁⠒⠀⠀⠀⠀⠀⢸⡷⣯⠀⢸⠀⢸⡇⠀⢸⠄⣿⠀⠀⢸⡇⠀⣹⠀⡿⠀⠰⡤⠀⠘⣾⠃⠀⠀⠠⠀⠠⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠸⠧⠞⠀⠾⠆⠈⠷⠶⠛⠀⠻⠶⠶⠈⠷⠶⠛⠀⠻⠶⠾⠃⠀⠰⠏⠀⠀⠀⠁⡀⠁⡀⠈⠁
⠀⠀⠀⠀⠐⠠⡁⠀⡀⠈⠀⠠⠂⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒⠐⢀⢀⡘⠀⠃
⠀⠀⠀⠀⠀⢀⠄⠀⢀⠀⠀⢼⠒⠂⠀⠈⡈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⣀⠀⠠⢄⠀⠎⠀⠀⠄⠠
⠀⠀⠀⠀⠈⠠⢀⠀⠂⠀⡀⠀⢀⠔⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠘⢤⠀⠀⠀⠀⠰⠂⠀⠀⠀⡘⠀⠈⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠰⠀⠀⠁⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠒⠆⢠⠐⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠦⠀⠀⠀⢶⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠖⠀⠆⢀⡄⠐⡠⠀⡄⠀⣐⠀⠠⡄⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡈⠉⠐⠀⠀⠰⠒⠀⠀⠀⠰⡐⠒⠀⠰⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⣄⠀⠀⠀⠴⠂⠨⢦⡀⠀⠀⠀⠀⠁⠁⠐⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡄⠀⠀⡤⠒⣆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢠
⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡆⠀⢠⠄
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⣦⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠄⣀⢀⡀⠀⢠⠁⠀⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣁⢀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⣣⠃⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⠁⠈⢀⠀⢀⠎⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠁⠀⠀⠀⠀⢤⡾⠁⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠔⠁⡄⠀⠀⠂⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠀⠀⠈⠢⠤⠤⠐⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⣶⠶⣦⡄⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⢰
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⠟⣿⠖⣦⢶⣄⣴⠆⣶⡶⢼⣆⣴⠶⠆⣶⡶⠄⠀⢠⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠀⠀⡿⠀⡿⠈⣿⠏⠐⢯⡿⢾⡟⢷⣤⡔⢯⡿⠂⢀⡇⠀⠁⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠆⠀⠈⠄
⠀⠀⠀⢀⠀⠀⠀⠠⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠎⠀⠀⣀⠀⠀⠉⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⠠⡄⠀⠀⢘
⠀⠀⠀⠀⠀⠈⠉⠠⠅⢘⠊⠛⠁⠀⠀⠀⠀⠂⠄⣀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣀⣀⣀⣀⣀⣀⣀⣲⠘⡣
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠀⠀⣼⠀⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠳⢤⣄⣀⣇⣀⡤⠔⠻
⠀⠀⠀⠈⠀⠐⠁⢠⠄⠀⠀⠊⠀⠈⠁⠀⠜⠁⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⡇⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠴⣲⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢸⣀⣀⣀⣤⣧⣄⣀⣀⠀⠀⡆
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠔⢛⠉⠀⠀⠀⡇⠀⠀⠉⠙⠲⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢸⡗⠲⠘⡲⠖⠒⠐⡓⠒⠒⠒⠒⠒⣾⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠳⢬⣉⠁⠀⠀⠘⠃⠀⠀⣀⣠⠴⠃⠈⠇
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠒⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣺⣭⣶⣍⣓⣫⣭⣭⣝⡲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣒⣫⣶⡿⣻⣿⡿⠛⠻⢿⣭⡹⣿⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢊⣥⡶⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠉⠻⣿⡇⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣺⣵⣿⡏⣰⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⢀⣾⢿⡼⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⣾⣿⣿⣾⣿⣿⣿⣿⣿⢿⠋⢀⣀⣀⣠⡀⠀⢀⣼⣥⣼⡧⢷⡎⣷
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣾⣡⣿⣿⣿⣿⣿⣿⡿⢁⣴⡾⠟⣉⣥⡴⠾⣟⠉⢹⣧⡄⠰⡀⢻⡼⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠰⣿⣿⣿⣿⣿⣿⣿⡟⠁⠛⣩⡴⣿⣉⠀⠀⠀⢹⡄⠀⠹⣆⠀⠈⡾⣇⠧⣀
⠀⠀⠀⠀⠀⠀⠀⢨⡇⣿⣿⣿⣥⣾⣿⣿⣀⣀⣤⣿⡀⢻⠿⠃⠀⠀⢸⠇⠀⠀⠈⢯⡉⠀⠈⠳⣬⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡝⣿⠟⣉⡙⢿⣿⠟⠛⠉⠉⠳⣀⣉⣀⣤⠴⠋⣰⠀⠀⠀⠀⠙⣄⠀⠀⠈⣷⢹
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⡇⠸⠁⢁⠈⠛⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠒⠀⠀⠀⠀⠼⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣷⡀⠀⠀⠙⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠘⠃⠀⠀⡾⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣏⢾⠟⣙⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⣠⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠋⠉⠙⠓⠾⣝⢷⣄⢄⠀⠀⠀⠀⠈⠂⠒⠂⠀⠀⠀⠀⠀⡀⠀⠀⢠⡟⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡝⢧⡑⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⢻⡄⠀⠀⠉⠉⠛⠓⣶⣤⣄⣀⣀⣀⡤⢖⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⣋⡞⠃⡀⠀⠀⠀⠰⣯⣱⡖⠾⢭⣭⠵⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣭⣷⣾⣿⣿⣦⠀⠙⠦⡄⣀⡤⢈⣳⣝⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⣱⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠺⣿⣿⣶⣿⣯⣽⣿⣮⣝⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢿⣿⡟⠻⠿⠈⣿⣿⣿⣷⡹⡄
⠀⠀⠀⠀⠀⠀⢀⡼⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡹⡄
⠀⠀⠀⠀⠀⢀⡞⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣹⡀
⠀⠀⠀⠀⢀⢎⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣷⣿⣿⣿⣧⢳⡀
⠀⠀⠀⢀⡎⣾⣿⣿⣿⣿⣿⣿⠟⡵⣌⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠁⠰⠸⣿⣿⣿⣿⣿⣿⣧⢳⡀
⠀⠀⠀⡜⣸⣿⣿⣿⣿⣿⣿⣿⣮⡳⣌⠳⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⡀⣿⣿⣿⣿⣿⣿⣿⣧⢳
⠀⠀⠀⢳⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣝⣮⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢁⣿⣿⣿⣿⣿⣿⣿⣿⣎⢧
⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣎⢿⢧⡹⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⢇
⠀⠀⠀⠀⠀⠀⠙⢮⡻⣿⣿⣿⣿⣿⠛⠡⣾⣝⣛⣣⣿⣿⣿⣿⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢞⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣝⡻⣿⢁⣤⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⢿⣿⡿⣫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠭⠶⣝⠦⣄⡀⠀⢀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⢟⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣍⢳⣄⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢌⡻⢿⣿⣿⣿⣿⣿⣿⡿⣣⢹⣿⣿⣿⣿⣿⣿⢎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠭⣝⣛⣛⣛⡼⠙⢧⣛⣯⠭⠷⠒⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠎⠉⠉⠁⢈⣉⠡⣬⣩⠲⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣶⣾⣿⠳⠓⠉⠀⠀⠀⣠⣾⡿⠇⢫⣯⡀⠘⢛⣻⣟⣦⣄
⠀⠀⠀⠀⠀⠀⠀⡴⢋⣶⣽⡯⠊⠀⡄⠀⠀⠀⢀⣼⣿⠋⠀⠀⠀⣮⠠⠖⠋⠁⠉⢻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⡎⣰⢿⣿⠏⠑⠒⢶⢧⣀⠀⣠⣿⡿⠁⠀⠀⣠⠞⢹⡆⠀⠀⠀⠀⣼⠉⠻⣼⣇⡀
⠀⠀⠀⠀⠀⠐⣿⣟⡛⠀⠄⠀⡠⠎⠀⡸⢋⣿⣿⠁⠀⣠⠞⠁⠔⠘⠁⠀⠀⠀⢠⡏⠀⠀⠬⢿⣿⡀
⠀⠀⠀⠀⠀⢸⡏⣟⠀⠠⠼⠊⠀⢠⣟⣀⣼⣿⣅⣠⣼⣁⣠⣎⣀⣾⢀⣀⣀⠀⣎⡠⠊⠀⠀⠸⠁⢷⡄
⠀⠀⠀⠀⢠⣿⠖⣦⣶⣶⡿⢿⣷⠛⠛⢻⣿⣿⡿⠋⢉⠕⠛⠋⠉⠈⠙⠛⢛⡿⠛⠿⠿⣶⣦⣦⡄⠘⣧
⠀⠀⢀⡴⣾⣿⠾⠋⠁⠀⣠⠋⠀⠢⠀⢸⣿⣿⢀⣴⠂⠲⣶⡄⠁⠐⢀⣴⡞⠀⠀⠀⠀⠀⡜⠉⠻⣴⣾⡇⠠
⠀⠀⣿⣷⣿⣿⣇⠀⠀⡼⠁⠀⠀⠀⢐⣿⣿⡯⠊⢠⠀⠀⣸⣿⢠⣀⣈⠿⡀⠀⠀⠀⢀⡼⠀⠀⠀⣿⢻⠗⠒⢂
⠀⠀⠘⣞⠃⠛⢿⣦⣾⣀⡀⠀⠀⢀⣞⣷⣿⣧⠀⠀⠉⠚⠋⠁⢨⠉⠁⠀⠀⠄⡀⠀⡘⢁⣀⣴⣾⠟⠁⠀⣠⠊
⠀⠀⠀⢿⠿⣶⣤⡈⢉⡛⠛⠷⠶⢈⢘⡛⢛⣧⠀⢀⡀⠠⢴⣧⠀⢀⣀⠀⣤⡐⠩⠭⠁⠉⢉⣩⣤⣴⣶⣿⡇
⠀⠀⠀⠈⣦⣹⣿⣿⡤⠚⠋⢀⡖⠀⢲⣶⣤⡄⠀⠀⢀⢤⠤⣤⡄⣤⣴⠂⠀⡲⠂⠀⠊⣀⠴⣶⣽⣿⠉⠋
⠀⠀⠀⠀⠸⡇⠹⣶⣷⡀⠀⣸⠁⠀⣼⣿⣿⡇⢀⡠⠖⠁⠈⠉⠉⠋⣿⣤⡎⠀⠀⣠⠞⢣⣴⣿⣿⠋⡄
⠀⠀⠀⠀⠀⠘⠄⠙⢳⣅⣤⡇⠀⠀⢿⣿⣿⡛⠹⠁⢠⡶⠃⠀⠀⢰⣿⠁⠉⠐⠲⠅⠠⢿⣿⡟⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠢⡈⠻⣿⣿⣶⡖⢻⣿⣿⡇⠁⣠⡟⠁⢀⠄⠚⣼⡇⠀⠀⣀⣼⣶⣷⣿⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣌⠛⢿⣿⣟⣿⣛⣃⣰⡏⠁⠀⠀⠀⠀⣿⣧⣶⣾⣿⡿⣿⣯⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣯⡉⠛⠻⠾⣷⣤⣤⣬⣬⠁⣠⠿⠛⠛⠉⢁⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠲⠶⠶⢞⣿⣷⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢼⣾⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣻⣿⣿⣿⣿⣋⡁⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢞⠘⠿⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⠿⣿⡿⠿⠿⠟⠛⠉⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠷⣶⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⡷⣾⣟⣻⣻⣿⣿⡽⣿⢶⣄⠀⣀⠔⠋⠀⠀⠐⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣻⣿⡿⣶⠗⢿⣿⡏⠡⠎⢳⡀⠈⢦⠹⣿⡀⢀⣀⠀⠚⠁⠀⠈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⣻⠅⠀⢿⣀⣼⣿⣿⠰⡖⡈⢿⡑⡈⣧⢻⣿⠉⢻⣷⠀⢘⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠤⢸⣹⣇⣾⣿⣿⣷⣆⣘⣿⣷⣷⠿⢺⠻⣦⣿⠿⡚⡟⣧⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡀⣿⣖⣸⣆⣿⣿⡿⢿⣿⣏⣿⣭⣿⣴⡾⣿⣿⣿⠛⡄⣃⢡⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢷⣻⣏⣿⣧⣼⣿⣷⣾⣿⣿⢿⡟⣾⣿⣿⢿⡛⣿⡇⢡⠸⠈⡜⡗⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢺⠻⣿⡧⣈⠛⣿⢹⣿⣿⣿⣶⣿⣿⣿⠛⠀⣇⢻⣷⠈⡆⠆⢡⢻⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⡇⠉⢿⣿⣷⣿⡿⢿⣿⣾⡟⠉⠁⢸⣆⣀⣸⣨⣿⣿⣿⣼⡬⡼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢹⠀⠸⡄⠉⠀⣷⣀⣽⣷⣳⣶⣿⡿⠿⣻⢿⣿⣮⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣸⣏⣦⣤⣷⣾⠿⢿⡟⠛⣿⣏⣽⣿⣶⣾⣿⠟⠁⢸⣿⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣿⣽⣿⣷⣿⣷⠾⢿⠛⣉⣉⣥⡇⠀⠀⠀⠀⠀⢸⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣆⠀⠉⠉⠀⠀⠈⣻⣿⣿⣿⣷⣦⣤⣀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢿⣦⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣟⠛⠟⠀⠈⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠘⠛⠻⢯⣽⣷⡾⠁⠀⠀⠀⢀⣿⣯⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣾⣿⡇⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣀⠀⢀⡀⠀⣀⣀⣴⣿⣿⣿⠇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⡿⠀⠈⠙⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠓⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⡿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣆
⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣆⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣯⣭⣭⣭⣭⣭⣭⠍⠙⠻⣷⣦⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⢻⣿⡿⠛⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⣿⣿⠈⠛⠋⠀⠀⠈⠉⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⣿⠟⠛⠉⠀⠀⠖⠛⠛⠛⢿⣿⡿
⠀⠀⠀⠻⠋⠀⠠⠦⠀⠀⠀⠠⠦⠄⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠤⠦⠀⠀⠀⠰⠤⠀⠀⢿⠃
⠀⠀⠐⣾⠛⠛⠛⠛⢶⣤⣤⠖⠛⠛⠛⢻⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠛⠛⠛⠒⢦⣤⣴⠖⠚⠛⠛⢷⡶
⠀⠀⠀⢸⡀⠀⠘⠗⣸⠃⢹⡀⠳⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠸⠇⣸⠁⢻⡘⠗⠀⠀⢸⠇
⠀⠀⠀⠀⠙⠓⠒⠋⠁⠀⠀⠉⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠛⠁⠀⠀⠙⠒⠒⠚⠉
⡤⠒⠀⣖⠀⢐⠀⢠⡀⢰⣆⣰⢰⡒⠀⡴⡀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⡄⠀⢀⠀⢰⣀⡄⢰⡂⢀⢶⠀⡶⠲⡄
⠑⠀⠂⠃⠀⠀⠀⠈⠑⠘⠃⠙⠸⠄⠸⠃⠃⠺⠔⠃⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠁⠀⠀⠀⠸⠁⠃⠸⠄⠘⠙⠇⠳⠴⠃



⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠖
⠀⠀⠀⠀⣤⣀⠀⢰⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⢀⣤⣶⣯⣶⣿⡿⠋⠁
⠀⠀⠀⠀⢿⣿⣿⣶⣿⣿⣿⣎⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⣤⣈⣻⣿⣿⣦⣍⣉⠉⠁⠀⠈⠉⣻⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣽⣿⣯⣯⣽⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⣿⣿⣿⣿⣛⣛⣛⣛⣉⣤⣴⣶⣄⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⠆
⠀⠀⠀⣿⣿⠿⠿⢿⣿⣿⣿⣿⠿⠛⢻⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣭⠽⠿⠿⠿⢿⣿⣟⣛⠛⠛⠛⣿⣿⠃
⠀⠀⠀⢸⡟⠀⢀⣀⠀⠀⠀⢀⣀⡀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢀⣀⠀⠀⠀⠀⣀⡀⠀⠀⡿⠁
⠀⠀⢠⣬⡥⠶⠤⠤⣄⡀⣀⣤⠤⠤⠦⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣥⠤⠤⠤⣤⡀⢀⣠⠤⠴⠶⢤⣥⡄
⠀⠀⠀⢹⠀⠀⢰⡦⢸⡟⢻⠠⣦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠠⣦⠀⡟⢻⡇⢶⡄⠀⠀⡯
⠀⠀⠀⠘⠶⠤⠤⠴⠚⠀⠈⠳⠤⠤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⠞⠁⠀⠓⠤⠤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢠⡤⠀⣠⠀⢠⢤⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⢀⠀⢀⠀⢀⠀⠀⡄⢠⠀⡤⠀⢠⡀⢀⡤⡀
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⢻⢸⣃⢠⠧⢧⢸⣀⠇⠀⠀⠀⠀⠀⠀⠸⣀⡆⢸⡀⢸⣀⠰⠛⠄⡟⢸⠀⣇⣀⠶⢷⢸⣄⡼

⠀⡀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠃⠦⠉⠀⠰⠟⠛⠋⠁⠀⠀⠀⠀⠄⣀⠂⠀⠰⠊⠀⢀⢀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠀⠁⠰⠰⠀⠁⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠰⠀⠤⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠞⠁⠀⠞⠀⠀⠀⠓⡀⠀⠀⠀⠀⠐⢺⡆⠀⠀⠀⠀⠚⠻⠀⠀⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠄⠂⠀⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠀⠀⠀⡀⡀⠀⢠⠆⠀⠀⠘⠀⠀⠄⢀⡀⢄⠀⠀⠀⠀⠀⠀⣴⣃
⠀⠀⣠⠀⠀⠀⢀⣀⠈⠋⠀⠀⡄⠀⠀⠀⢬⡻⣄⠔⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢶⡇⡘⣏
⠀⠀⠃⠀⣀⠈⠐⠀⠀⢀⠀⠉⠀⠀⠀⡐⢉⠥⠂⠠⠆⠀⠀⠀⣠⣀⠀⠀⠀⠀⠆⠀⢠⠀⢈⣩⣵⣵⠇
⠀⣸⠀⠀⠛⠀⠀⣀⠐⠈⠀⠄⠀⠀⢀⡁⠊⢀⡛⡀⠠⠆⠀⠘⠑⠀⠀⠀⡦⠀⠀⠀⠈⠀⠀⠀⠹⡏
⠀⣼⠀⠀⠀⠐⠀⠁⢀⡀⢀⠀⠐⠂⢣⡞⠄⣼⠃⢹⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⢠⡄⠄⡀
⠀⠦⠀⡿⠇⠀⠀⠀⡀⠀⠌⠁⠀⢇⠁⢀⡾⡟⢎⡎⠀⠀⣀⡰⠄⠊⡷⠀⠀⠀⠈⠀⠂⠀⠀⠀⠰⢠⠁
⠀⣴⡔⢳⣦⣀⠀⠀⠐⠤⠀⠀⢀⠈⠣⡾⠈⠠⠌⠀⠀⠁⠻⢯⠆⠀⠁⠈⢀⣀⠈⠓⠒⣂⣸⡧⠄
⠀⠀⠀⠀⢋⠁⠀⢀⡀⠤⢒⡖⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠏⠀⢰⠋⠁⠀⡇
⠲⣄⢠⡞⠀⠀⠀⢀⠖⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠀⠀⢠⠆⠀⠀⣀⣤⣼⡤⠴⠊⠀⠼⣆
⢰⡿⣆⡔⠀⠀⠀⠠⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⢈⡀⠀⠀⠀⠀⠁⡆⠀⠀⠀⣇⠀⣸⠇⠀⢀⠀⠀⠛
⠀⠙⡏⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⡒⠁⠀⠀⠀⡄⡀⠖⠀⠀⣰⠚⢻⡍⠁⠀⢰⣶⠧⠶⣄
⠀⢀⡄⠀⡀⠐⢀⠠⠀⠀⠀⠀⠀⠀⠀⠐⠠⠂⠀⠨⠄⠀⠀⢑⠈⠀⢀⣻⠤⠞⠁⡀⠈⠉⡇⠀⠀⢉⠃
⠀⠘⠏⠀⠱⡀⠀⠀⠖⠀⠀⠒⠚⠉⠉⠉⠁⠀⣀⣀⣴⣶⡶⠆⠀⢀⣯⠟⠀⠀⠢⢏⠀⠀⠑⠧⡐⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⣔⠂⠘⠈⠓⠉⠐⢢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠁⣶⡖⢠⣆⣤⣆⠔⠒⠐⠃
⠀⠠⣎⠁⠀⡄⠀⠀⠀⠀⠐⠒⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⣃⣋⣘⡘⠒⠃
⢠⠊⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠊⠉⠀⠀⠀⠀⠉⠉
⢸⠴⢃⠆⠈⠠⠀⠀⠀⠰⢄⠀⠀⠀⠀⢀⡔
⠀⠀⠃⡀⠀⠀⠀⠄⠀⡢⠔⠛⠒⠒⠒⡍
⠀⠀⠀⡇⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢰⠀⠀⢄⠀⡼
⠀⣀⠠⠄⠤⢄⠀⠀⠀⠀⠀⠀⠘⠽⠆⠀⠂
⠀⠀⢀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠨⠀⠍⠀⢀
⠀⡀⠀⠉⠁⣈⡺⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠉⠀⠀⠀⠁⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠈⠄⢀

⠀⢀⡠⠤⣀⠀⠀⠀⠀⢀⣠⣤⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⣀⣤⣄⡀
⠀⡾⠷⡴⠿⡆⠀⠀⢾⣯⣿⣿⠇⠀⠀⢀⣩⣯⣅⡀⠀⠀⠀⢀⡴⠛⠃⠀⠀⢀⠎⠆⠳⡀⠀⠀⠺⢿⣿⡿⠗
⠀⠑⠼⠻⠔⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠙⠛⠚⠛⠃⠀⠀⠈⠋⠀⠀⠀⠀⠀⠫⠤⠤⠤⠗⠀⠀⠀⠨⠿⠅

⠀⠀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣶⠀⠀⠀⢠⢒⠉⡒⡄⠀⠀⠀⢠⠚⢆⠀⠀⠀⢠⠖⠋⣳⣆
⠀⠈⢫⣬⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠠⡀⠀⢀⡜⠀⠀⣰⠋⠷⡍⢦⠀⠀⠸⣤⠞⢁⡼
⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣄
⠀⣶⣶⣶⣶⣶⠀⠀⠀⣼⣿⣧⠀⠀⠀⠀⡸⣿⢃⠀⠀⠀⢨⣿⣿⣿⠇⠀⠀⢀⡞⣤⠳⡀⠀⠀⢠⣮⣹⡋⠉
⠀⠉⠉⠉⠉⠉⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠽⠒⠯⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠺⠤⠬⠤⠕⠀⠀⣼⣿⣿⡇

⠀⠀⣠⠀⠀⠀⠀⠀⠀⠄⣐⠄⠀⠀⠀⠀⣶⣤⣴⠀⠀⠀⢠⡶⣨⣶⡄⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣄⠒⢢⡀
⠀⠛⠱⣷⣷⣤⠀⠀⠀⠘⢿⠃⠐⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⠁⢾⡧⡀⠀⠀⣤⣄⣠⣍⠉⠀⠀⢀⣀⡀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠱⠵⠳⠻⡿⠀⠀⠰⣿⣿⣿⠆⠀⠀⠀⠃⣶⠀⠀⠀⠀⢠⣶⣿⣤⡀⠀⠀⠀⣠⣤⡉⠀⠀⠀⣾⣧⣤⣤⡤
⠀⠐⠀⠀⠀⠂⠀⠀⠀⢳⣿⡞⠀⠀⠀⠀⢀⣉⡀⠀⠀⠀⠘⠇⠀⠹⠃⠀⠀⠀⢿⡿⠇⠀⠀⠀⠀⠛⠓

⠀⠀⡀⢀⠀⠀⠀⠀⢠⡔⡐⠢⡄⠀⠀⠀⠀⢀⣤⠆⠀⠀⢀⣴⣶⣦⡀⠀⠀⢠⣄⣠⣤⠀⠀⠀⠀⣀⣠⣤⡀
⠀⠀⢸⣿⡇⠐⠀⠀⢺⣿⣦⡀⡀⠀⠀⢠⣤⠟⠁⠀⠀⠀⢸⣿⣿⣽⠇⠀⠀⢸⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠛⠛⠛⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣽⣦⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠛⠛⠛⠛⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡴⠖⠛⠿⣄⣀⢀⡼⠷⠶⣄⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡋⣿⡄⠀⠙⠃⡟⠉⢯⠘⠟⠀⢈⣿⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣿⠉⠓⠒⢾⣀⠀⣘⡷⠶⠆⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣀⣋⣏⣹⣀⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⣈⣝⣛⡋⠁⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⣄⡤⠀⠁⠄⠅⣠⠶⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡇⠈⠙⠒⠚⠒⠋⠁⠀⡧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣞⠁⣧⠀⠀⠀⠀⠀⠀⠀⣠⠇⠈⡷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠚⠁⠀⠀⠀⠈⠙⠒⠒⠒⠒⠚⠉⠁⢀⠜⠁⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣦⣤⣤⠴⠾⠦⠀⢀⡀⠠⡄⠀⠠
⠀⢀⡀⠂⠂⠀⠂⠀⠀⠀⠀⣤⣾⣷⡀⠀⠀⠀⣀⠴⠻⠴⠇⠀⠀⠀⠀⠀⠀⠀⠰⠄⡷⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠉⠁⠀⠀⠚⠁⠀⠀⠘⠺⠁⠀⠀⠉⠀⠀⡛⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠲⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠒⠀⠁⢀⡛⠿⠃⠈⢡⡄⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠘⠟⠀⠀⠀⠀⠀⠀⠄⠀⡤⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠠⠄⣀
⠀⠀⠀⠀⠀⠀⠀⢢⡈⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠃⠈⢀⠀⢀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠸⠂⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠃⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠐⠂⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠇⠀⠀⠃


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⢠⣄⣠⡀⠂⣴⢄⣺⡫⢛⠀⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀⠒⢀⣿⡟⠛⠏⣅⠨⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠂⠡⠃⠘⠾⠀⠃⣀⢌⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠓⠄
⠀⠀⠀⠀⠀⠆⠄⠠⠀⡄⡌⠀⠂⠀⢀⡨⠁⠧⠀⠠⠀⠀⠀⠀⢀⠀⠀⠠⡀⠀⣠⣘⠻⠀⠀⠠
⠀⠀⠀⠀⠀⢷⢈⠂⠠⠈⠀⠀⣤⠐⠌⠱⠆⠀⠀⠀⠀⢄⠀⠀⠈⡠⠈⠃⡅⢷⢠⠈⡄⢁⠢⠂
⠀⠀⠀⠀⠀⡉⠴⢆⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⣄⠈⢷⣄⠀⠘⣅⢎⠛⣿⡠⠖⣖⠂⠜⠁
⠀⠀⠀⠀⢠⠀⠄⡐⢦⠀⠀⠀⠂⠀⡀⠀⠀⠀⠀⠀⠁⠰⡌⠢⣚⠀⡈⠺⢷⡞⠁⠠⠀⠠⠂
⠀⠀⠀⠀⠘⠀⠀⠀⠂⠬⠀⠀⠀⠀⠈⢶⡤⢀⡀⠀⠀⢀⡈⠠⠉⢤⡈⠀⡄⠛⣐⣈⣘⣐⣆⠴⠚⠁
⠀⠀⠀⠂⠀⠉⠀⠀⠀⠚⠀⠀⠄⠀⠀⠀⠱⠏⠲⠂⠀⠀⠘⠄⠘⡄⠀⠨⢠⣌⠀⣈
⠀⠀⠀⠀⠀⢈⠀⠂⡤⠀⡀⠀⠀⠢⠄⠀⢀⠀⠀⠄⠢⢄⠀⢀⠀⠔⢰⠀⠉⠍⠀⡈⠀⢛⠉⠟⠒⠂
⠀⠀⠀⡀⢀⡄⠐⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠐⠤⡈⣀⠏⠀⡀⠀⠀⠒⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⢂⡐⠙⠀⢁⠀⡀⠠⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠂⠁⡋⠀⠄⠄⢀⠀⠠⢠
⠀⠀⠄⠀⢀⠈⠡⠀⢀⡜⢁⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠄⠈⠈⠀⠀⡀⠀⠀⠀⢀⡀⠤⠆⠐
⠀⠀⠠⠀⠠⠐⡂⠐⢀⡰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠃
⠀⠀⠀⠘⠀⠀⠁⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠆
⠀⠀⠀⠀⠀⡀⠁⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡠⠀⠀⠀⠠⠈⠁
⠀⠀⠀⠀⢀⠀⠰⠂⠀⠰⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⠁⠀⡆⠀⠀⠄⠁
⠀⠀⠀⠐⠠⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠠⠐⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⢀⠀⠀⠀⡀⠄⠐⠀⢈
⠀⠀⠀⠠⠀⠀⠀⢸⠀⢇⢀⡀⠠⠔⠒⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠹⣄⠀⠀⠂⠀⡄⠀⠀⠀⠀⠀⠈⠀⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠠⢈⠀⠀⠀⠡⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠁⠘⠀⠀⡀⠑⢀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠁⠢⠄⢠⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡠⠄⠒⠃⠀⠀⠀⡤⠠⠄⠀⠀⠄⢀⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⢀⠍⠊⣁⠤⠀⠚⠠⠤⠄⠐⠁⢀⠠⠀⢀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣊⠥⠂⠉⠀⠀⢀⣀⡀⠤⠐⠂⠈⣁⠠⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡾⠋⠀⠙⣭⠀⠀⠐⠂⠈⠁⢀⣀⠠⠤⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠱⠟⠰⠖⠊⠀⡼⣁⠄⠋⠉⠀⠀⠀⠀⠀⠀⡀⠤⠤⢄
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠉⠀⠈
⠀⠀⠀⠈⡉⡲⠦⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⡂⠀⠀⠀⠀⠀⠀⠀⣀⡠⠠
⠀⠀⠀⠀⢯⣮⢂⠆⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⣾⠃⠀⠀⢀⠤⠆⡴⡋⠁⠌⡈
⠀⠀⠀⠀⠀⠳⡁⠰⡉⠀⠐⢎⡂⢄⡀⠀⠀⠀⠀⠀⠀⢀⠤⢄⠀⠀⢀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠐⠇⠈⠠⠀⡫⢊⠎
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠨⠠⢀⡑⢦⡈⠰⠄⠐⡈⠉⡏⠣⣀⠀⠑⠀⠀⠁⠒⠂⠚⠉⠀⠀⠀⠀⣄⠆⢀⡠⢊⢔⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢊⠢⡀⠠⢂⠜⠆⠄⡀⠀⠐⡼⠀⠐⠢⣵⡠⢄⡀⡀⠀⠀⠀⠀⠀⠀⢀⢞⣈⠖⣊⢔⣁⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡈⠒⠓⡀⠀⠀⠘⢇⠀⠈⠂⢴⠂⢝⠳⣌⠲⣎⠓⠀⠀⠀⠀⠀⠠⠠⠔⠊⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⣠⡀⢀⠲⡂⠀⠀⠈⢧⡀⠧⠄⣤⡞⠁⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠑⠸⠄⠀⠀⠀⠙⠃⠐⠰⠌⠶⠆⠌⣂⡀⠀⠀⠀⠄⣀⡀⠀⡲⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠂⠠⣄⠀⠀⠀⠀⠐⠂⢰⠆⣀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠒⢁⠴⡄⠉⢢⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠂⠀⠀⠀⠀⠑⠕⠀⠀⠀⠀⢬⠆⠀⠙⢀⢈⡁⠀⠀⠀⠀⠀⠀⢤⣀⠉⠠⡤⠣⣤⠀⢰⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠀⠀⠀⠀⢈⠑⠢⠀⠀⠀⠕⣀⡸⠑⠠⠈⠒⢆⠀⠀⠀⠒⢅⣸⡄⠀⠗⠪⠀⠘⢣⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠀⠈⠡⠒⠂⠉⠁⠈⠀⡨⠀⠀⠀⢤⢡⡈⠂⠄⠀⠀⠡⠀⠀⠀⠈⢀⣀⢆⡀⠀⠦⠔⠈
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠉⠀⢺⡆⠒⠒⠀⢲⡈⠀⠀⠀⠀⠀⠐⠴⢄⠀⠀⣠⠞⠃⠀⠀⠀⠀⡁⢄⠉⠐⠂⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠧⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⠚⠀⡀⠀⢀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⣎⣥⣶⣶⣮⣿⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠻⣿⠋⠀⢈⣿⡿⠿⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠚⠉⢁⡀⠀⠀⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⣿⣿⣿⡔⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠊⠀⠈⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢈⠁⠀⠠⠈⠀⣠⣤⣦⣀⠹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣾⣧⠄⠀⠀⠡⠼⣿⣿⣿⣿⣷⣬⣿⣿
⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣽⣿⡍⢀⠀⠀⠀⠠⠄⠀⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⣠⣀⣻⢤⣀⡐⣻⣿⠋⠀⠀⠐⠈⠐⠀⢀⡀⠀⠙⠻⣿⣿⣿
⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢉⡽⢟⡩⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠈⠻⣿
⠀⠲⣴⣤⠀⢄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠄⣀⣠⣤⣿⠿⠿⣾⠶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠉⠒⠶⠦⢔⢀⣂⣀⠄⠈⢁⡈⠉⠩⠻⠃⠀⡠⠒⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢊⡥⢔⡈⠀⠀⠠⡉⠀⡠⠚
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣻⠷⠚⠋⣰⠻⢅⠎⠀⡤⠄⣦
⠀⠀⠀⣀⣠⣾⡿⠟⠋⠁⠀⠀⢰⣟⣿⡇⠀⠀⠑⣖⠌⡆
⣷⣶⣬⡏⠉⠁⠀⠀⠀⠀⠀⠀⣿⣤⠟⠀⠀⠀⠀⠙⠲⢿⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⣿⣶⡄⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡆⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠙⠿⠟⠋⠁⠀⠀⠀⠀⠈⠉⠻⠿⠋⣠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠟⠉⠉⠀⠀⣤⣀⡉⠙⠻⣿⣿⣿⡿⠋⠀⠀⠙⣿⣿⣿⣿⠟⠋⢉⣠⣤⠀⠀⠉⠉⠻⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠙⠿⠇⠀⠀⠀⠀⠸⠿⠋⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⢤⣶⣤⣤⣶⡄⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⣿⣿⣿⣿⠀⠀⢀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡆⢀⣿⣿⣿⣿⡀⢰⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⣼⣿⣿⣿⣿⣧⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠾⠿⠿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠷⠶⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣷⣿⣿⣶⣦⣄⡀⠀⠀⢀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠛⠉⠁⠀⠀⠉⠛⢛⢿⣿⣿⣦
⠀⠀⠀⠀⢠⣧⣶⣾⡟⠀⠀⠀⠀⠀⣀⠠⡜⠁⢾⣿⣿⣿⣧⡀⠀⠀⣀
⠀⢀⣤⣤⣼⣿⣿⣭⡄⠀⠀⠀⠀⠀⢀⣿⡄⣴⡌⣿⣿⣿⣿⣧⣤⢤⡷⣤⣤⣤⣤⡤⡤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠉⠉⢹⣿⣿⣭⠁⠀⠀⠠⣀⣠⣾⡿⣣⡽⢡⣿⣿⣿⣿⡏⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢏⠋⠻⣧⡀⠔⠑⠈⠉⠉⠞⠁⣠⣾⣿⣿⡿⣿⠁⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠻⣦⣤⡀⣿⢢⣴⣾⣿⣿⣿⣿⠁⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠠⣔⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⡆⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠛⠛⠛⠋⠁⠀⠀⡀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡁⠀⡅⠈⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠐⠈⢀⠉⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⢰⠀⠇⠀⠀⢰⠀⡀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⢀⢐⡥⠈⠂⠒⠐⠑⠂⠒⠂⠐⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⣠⠤⠚⠁⠀⠀⠀⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠄⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠊⠀⠀⠀⠀⢂⡁⠤⠀⠀⢀⠀⠈⣀⠦⠁⡀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣠⢄⡀⠀⠉⠈⠁⠀⠀⠀⠀⠐⠆⠲⠋⠀⠀⠀⠀⠄⠀⡀⠅⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠠⠀⠈⠈⠳⣕⣄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠘⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠳⡄⠀⠠⡢⡢⡁⠀⢁⠽⠟⠀⠀⢀⠀⠂⣠⠀⠀⠀⠄⠪⡄⠀⠀⠈⠁⡀⡄⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠸⣗⠄⢀⡌⠔⠊⠁⠀⠀⠀⠀⠀⠜⠠⠄⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠐⠂⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠘⣆⠀⡀⢀⡠⠖⠊⠁⠉⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠖⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠘⢶⠟⠓⠊⠁⠀⠀⠀⡀⠀⠂⠀⠀⢤⠈⠢⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⢀⣀⢐⣈⠵⠼⠛⢕⣉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠀⠸⠈⠰⠀⠀⠂⠀⠀⠀⠀⠀⠦⠄⠀⠀⠄⠀⢘⣀⢔⠄⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠳⠾⠉⠀⠀⠀⠀⣈⣩⡶⢦⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⠀⠀⢤⠄⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⡄⡀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠄⠢⡎⠀⠀⠈⠀⠤⠠⠴⠈⠀⠀⠈⠀⠀⢀⣠⣎⢊⢌⠎⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⡤⠒⠩⠢⠁⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⢸⠀⢡⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠃⠐⡐⠂⠒⠐⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣟⠀⠀⠇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠂⢰⠻⠾⠀⠀⠲⣠
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⣸⠀⠀⠀⠀⠀⢉
⠀⠀⠀⢀⣀⠀⣄⣦⠀⠔⠛⠐⠀⠠⠀⢠⣠⡌⣤⢀⠀⠀⣀⠄⠠⠀⠄⠠⠀⠄⠠⠠

⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⠀⠀⠀⡠⠋⠊⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⡠⠋
⡰⠋⠀⠀⠠⠔⣒⠒⠐⠄⠀⠀⠀⠀⠀⠀⠰⠀⢀⢾⣷⠶⠋⠀⠀⠀⡰⠋⠀⠀⢠⠔⣒⠒⠀⢄⠀⠀⠀⠀⠀⠀⠈⠀⢀⢾⣿⠶⠋
⠀⠀⠀⠀⠁⠠⣴⠄⠀⠀⠑⢄⠀⠀⠀⠀⠠⡠⢄⡀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠐⠁⠀⣴⠄⠀⠀⠁⠀⠀⠀⠀⠀⠘⡠⢄⡀⠀⠀⠀⣀⠴
⠀⠀⠀⠀⡄⠀⠹⡄⠀⢀⠀⠈⠆⠀⠀⠀⠀⠈⠈⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⡄⠀⢀⠀⠈⠆⠀⠀⠀⠀⠈⠈⠀⠀⠀⠉
⠀⠀⠀⠀⠐⡄⠀⠹⣷⣾⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠙⢷⣾⡇⠀⠈
⡀⠀⠀⠀⠀⠈⠳⣄⣀⠐⢾⡰⣼⠀⠀⢀⠤⠒⠒⠉⠉⠉⠁⠀⠐⠤⡀⠀⠀⠀⠀⠀⠲⣄⣀⠐⢾⡴⡱⠀⠀⢀⠠⠂⠒⠉⠉⠉⠉⠀⠐
⠈⠢⠀⠀⠀⠀⠀⠀⠉⠛⠓⠋⠀⠀⠀⠁⠀⠀⢀⡀⠂⠄⠄⠤⢀⠀⠈⠲⡄⠀⠀⠀⠀⠀⠉⠛⠓⠋⠀⠀⢠⠃⠀⡀⢀⡀⠂⠤⠄⠤⢀
⠦⡀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣯⠉⠁⠀⠀⠀⠀⠘⠢⣀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣯⠁⠁
⠲⡘⡂⠐⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠲⣌⢆⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠳⣄⡀
⣿⠟⠛⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠉⠓⠶⠶⣶⣿⠟⠟⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠉⠓⠷⠶
⢀⣀⢌⣳⠋⠀⠀⠀⢀⠀⠠⠤⣤⣀⡀⠀⠈⠢⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⢐⣳⠋⠀⠀⠀⢀⠀⠤⠤⢤⣀⡀⠀⠀⠀⢀
⠭⠞⠋⠁⠀⢀⠔⠊⠀⠀⠀⣀⡀⠍⠻⣦⠀⠀⠀⠉⠐⠢⠤⠬⠬⠬⠭⠖⠋⠁⠀⢀⠄⠊⠀⠀⠀⣀⡀⠍⠻⣦⠀⠀⠀⠈⠐⠢⠤⠬⠤
⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⣹⠏⠂⢈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⢹⠇⠆⠈⡻
⠀⠀⠀⢠⠁⠀⢀⠀⠀⠀⡠⠋⠋⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢀⠀⠀⠀⡠⠫⠊⠀⢰⠏
⠀⠀⠀⠀⠄⢀⢾⣿⠶⠋⠀⠀⢀⡰⠃⠀⠀⡠⠒⠀⠀⠒⢄⡀⠀⠀⠀⠀⠀⢰⠀⢀⢾⣷⠶⠋⠀⠀⠀⡰⠋⠀⠀⡠⠒⠀⠀⠒⠄⡀
⠀⠀⠀⠰⡴⢄⡀⠀⠀⢀⣀⠴⠊⠀⠀⠀⢰⠁⠀⣴⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⡰⢌⡀⠁⠀⢀⣀⠴⠊⠀⠀⠀⠀⠁⠀⣴⠆⠀⠀⠐⢄
⠀⠀⠀⠀⠈⠁⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠈⡄⠀⠸⡄⠀⢠⠀⠈⢂⠀⠀⠀⠀⠈⠁⠒⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⡄⠀⠹⡄⠀⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠙⣿⣶⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠹⣷⣾⡆
⠀⠀⢀⠄⠐⠚⠉⠁⠀⠉⠁⠒⠤⡀⠀⠀⠀⠀⠈⠳⣄⣀⢐⣿⡴⡧⠀⠀⢀⠤⠒⠚⠉⠁⠀⠁⠀⠀⠤⡀⠀⠀⠀⠀⠈⠳⣄⣀⢈⣮⡴
⠀⢠⠃⠀⠀⢀⡂⠒⠠⠄⠄⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠉⠙⠓⠉⠀⠀⢀⠃⠀⠀⢀⡐⠒⠢⠄⠤⠀⠀⠈⠒⡀⠀⠀⠀⠀⠀⠉⠙⠓⠉
⠀⡆⠀⠀⠐⣿⠁⠁⠀⠀⠀⠀⠙⠦⣀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣯⠈⠁⠀⠀⠀⠀⠙⠦⣀⠈⠦
⠀⠠⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠈⠰⣌⢆⠀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠳⣌⢦⠀⣃
⠀⠀⠱⡀⠀⠀⠀⠀⠉⠓⠳⠶⢶⣿⠟⠛⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠉⠛⠳⠶⢶⣿⠟⠓⢀⣸
⣤⡀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⢀⡀⢔⡣⠃⠀⠀⠀⣀⠤⠀⠤⣄⣤⡀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⢀⢀⢰⡳⠋⠀⠀⠀⣀⠤⠀⠤
⡀⢛⡧⠀⠀⠀⠈⠒⠲⠤⠬⠬⠭⠽⠞⠋⠀⠀⢀⠔⠊⠀⠀⠀⣀⡀⠌⢛⣦⠀⠀⠀⠈⠐⠲⠤⠤⠬⠬⠽⠚⠋⠀⠀⢀⠔⠊⠀⠀⠀⣀
⠃⢈⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⣹⣇⠂⢈⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣸
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⢀⠊⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⡄⠀⣄⠀⠀⠀⠀⠀⢸⣍⠋⠀⠀⠀⠱⠀⠀⠀⠀⠀⢀⡴⠋⢊⠔⡡
⠀⠀⠀⠘⡀⠣⢀⣀⣀⠀⠀⠀⠀⠀⠈⠟⠀⣔⠉⣦⠀⠀⠀⠀⠀⠘⡓⠠⢀⡀⢠⠓⡄⠀⠀⣰⣍⡊⠐⡡⢀⠔
⠀⠀⠀⠀⠈⠒⠒⠒⠁⠀⠀⠀⠀⢼⡣⠀⠀⠈⡶⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠊⠉⠉⠁⠀⢸⡌⠈⠉⢀⡴⠃
⠀⠀⠀⠀⣠⡤⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⡀⠀⠀⠀⠀⠀⠀⠑⠚⠊
⠀⠀⠀⠰⣊⠾⣸⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⢤⢀⠔⡭⠁⠀⠀⠀⠀⠙⡗⠯⠳⡀
⠀⠀⠀⠀⠙⡆⠙⢆⠀⠉⠀⠀⠀⠀⠀⣔⢎⡹⢥⣮⡀⠀⠀⠀⠀⠀⠈⢢⡰⣉⢎
⠀⠀⠀⠀⠀⠈⢆⠀⠑⢄⠀⠀⠀⠀⠀⢠⠧⣪⡲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠁⠀⠀⢀⠄⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣀⡴⠢⣄⡀⠀⠀⠀⠀⠀⡃⢊⠥⣰⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡼⡓⠤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠽⠲⠶⠯⣅⠀⠀⠀⠀⠀⠐⠤⠭⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⢷⡦⠀⠀⠀⠀⠀⣇⣰⠶⠲⣆⢨⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠯⠴⡦⢼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠐⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠄⡀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂⢨⠽⠂⠀⠀⠀⠀⡝⠐⠀⠀⣠⠁
⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠁⠈⠂⡀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠈⢁⡀⠀⠀⠀⠀⡀⠀⠀⠠⠨⠈⠀⠀⠀⠀⠀⡀⠡⠮⠠⠀⡀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⣄⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠧⠈⠀⠐⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢸⡀⠊⠀⠂⢸⠀⠀⢄⠠⠀⠀⠀⠀⢠
⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠈⠶⠠⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠐⠀⢈⠠⠠⠘⠈⠀⠀⠀⠀⠀⠈⠂⠀⡆
⠀⠰⠰⠀⠀⠴⡆⣠⣄⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠯⠭⢭⠭⠭⠭⠀⠀⠀⠀⠚⠁⡈⠑⠂⠀⠀⠀⠠⠁⠄
⠀⠀⠀⠀⠀⢆⣘⡠⠄⠀⠀⠐⠁⠐⠀⠀⠀⠈⠀⠀⠈⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠀⠄⠅⠐⠀⢀
⠀⡰⠒⠢⠀⠀⠀⠀⠀⠀⢀⡦⡄⠀⣶⠄⢠⡴⢦⡀⢰⠀⠀⢀⡴⢶⡄⠀⡴⠶⡄⠀⣦⠀⣴⠀⠀⠀⠀⠠⠁⠀⠈⡆⠒
⠀⠀⢀⠀⠆⠆⠸⠆⠃⠀⢸⡷⣇⠀⣿⠀⣿⠀⢸⡇⢸⠀⠀⢸⡇⠀⣿⢸⡇⠀⣦⠄⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠘⠷⠋⠐⠛⠂⠘⠶⠚⠁⠘⠓⠒⠈⠳⠞⠃⠈⠳⠶⠋⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣄⠀⠀⠀⠀⠷
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡠⠀⠀⠀⠀⠀⠀⣼⣃⣘⠿⠃⡋⠀⠀⠀⢀⠁⠀⠂
⠀⠘⠃⠀⠁⠀⠸⠂⠀⠀⠀⠐⠇⠀⠀⠀⢄⠁⠀⠀⠀⠉⠀⠀⠀⠠⠀⠀⠀⡀⡄⠀⠀⢀⡠⠔⠀⣠⠀⠀⠀⠁⠀⠀⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠐⠅⠀⠀⠉⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠃⠂⠇⠁⠑⠀⠀⠐⡂⠈⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠨⠀⠀⠀⠀⢀⢀⠂⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠄⡀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⡐⠀⠃⡀⡀
⠀⠀⠀⠀⠈⠀⠀⠀⢀⠠⠀⠀⠀⢈⠐⠁⠀⠸⣀⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⡀⠀⠂⠀⠄⡐⠀⢂
⠀⠀⠀⠀⠀⡴⠁⠀⠠⠄⠀⠀⠀⠈⠸⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠁⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠹⠃⠀⡐⠄⠈⠁⠀⠄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⣾⣿⣿⡏⠛⠲⢤⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠙⢷
⠀⠀⠀⠀⠀⠙⣿⠛⠀⠀⠀⠀⠈⠛⠦⣄⠀⢸⠃⠀⠀⠀⠀⠀⠀⠸⡇⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠈⠙⣾⣀⠀⠀⠀⠀⠀⠀⣀⣷⠋⠁⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠳⣄⠀⠀⣠⠞⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢈⡵⢯⡁⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⣄⣀⣤⣤⣤⢼⡧⢤⡴⠿⠤⠤⠿⢦⡤⢼⡧⣤⣤⣤⣀⣠⣞⡀⠀⠀⠀⢀⣀
⠀⠀⢀⣠⡤⠶⠒⠛⠉⠉⠉⠙⢧⡀⠀⠀⢸⡷⠋⠀⢀⣀⣄⡀⠀⠙⢾⡇⠀⠀⢀⡼⠋⠉⠉⠉⠛⢲⣿⣿⣷⡀
⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⡴⢻⡇⠀⢰⣿⣿⣿⣿⡆⠀⠸⡟⢦⣴⠋⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠉⠳
⢳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠳⣼⠇⠀⠈⢿⣿⣿⡿⠁⠀⢸⣧⠞⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠈⠓⠶⢤⣄⣀⡀⠀⠀⠀⢠⡞⠁⠀⠀⢸⡷⣄⠀⠀⠀⠀⠀⠀⣠⢾⡇⠀⠀⠈⢳⡄⠀⠀⠀⢀⣀⣠⡤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⡶⠛⠶⠶⠦⠤⢼⡧⣬⣷⣤⣤⣤⣤⣾⣥⢼⡧⠤⠴⠶⠶⠛⢶⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠈⡇⠀⠀⢈⣷⣾⡁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⡴⠋⠀⠀⠙⢦⡀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⢿⠉⠀⠀⠀⠀⠀⠀⠉⡿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⣰⣿⣿⣶⠋⠀⢸⡆⠀⠀⠀⠀⠀⠀⢰⡇⠀⠙⠲⣤⡀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢧⣄⣀⣤⠴⠛⠿⠿⠃⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡾⣿⣽⣷⣟⢋⠓⠠⢀⠀⠂⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡡⡑⠍⢠⡄⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⣟⠻⡥⠀⠁⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⠵⠢⠁⡥⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⢤⠡⠘⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⡀⣤⠠⡀⡰⢠⡶⠶⢠⡄⠀⢀⡴⠶⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠉⠀⠿⠠⠊⠢⠘⠯⠥⠸⠧⠤⠠⠭⠭⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⡛⢡⡊⠀⠲
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡤⠐⣠⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣷⡿⣪⣠⠄⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣿⣿⣯⣾⣟⡩⡒⠃⠐⢀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣵⣿⣯⢶⣽⣯⡽⢂⠄⠴⠂⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣟⣹⡷⢟⢢⡐⢂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠠⠀⠒⠒⠂⠠⢄⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠚⠢⢄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠠⠄
⠀⠀⠀⠀⠀⡄⠀⠠⡁⠚⠀⠆⠀⠀⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⣄⣠⢠⣤⡄⠀⠀⠠⠐
⠀⠀⠀⠀⠀⠘⠦⡀⠈⠂⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⡜⠀⠀⠀⠀⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⠀⡠⠈⢂⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠱⡄⠀⠀⠀⠀⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠈⠢⡀⠀⠀⠀⠀⢨⠒⠆⠤⢤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠈⠀⠀⠈⠢⡀⠀⠀⣜⢉⢉⢉⣘⣻⣻⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠠⢂⠄⠈⠂⢤⣌⣉⣉⣩⣩⣽⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⡂⢀⠘⠳⢛⢗⠿⡯⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠔⠁⡡⠊⠔⡡⡪⢐⠔⠀⡉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⡊⢔⠁⠠⢊⠔⠁⡠⢄⠀⠔⠄⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠔⡡⢂⢄⠁⣠⠞⡇⠀⠈⠒⠢⢄⠐⢀⣤⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠊⠄⠨⢊⠔⡁⠞⠡⠀⡇⠀⠀⠀⢀⠜⡡⠎⠅⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⢊⠔⡡⡪⠢⢪⢌⢄⢧⠀⡠⠊⠔⡼⠉⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠔⡡⡢⢊⢔⠔⠅⠕⠡⣸⠏⠀⠈⢊⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢊⢌⠔⡽⠣⠢⢊⠈⠘⠡⠀⢡⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⡡⠠⢲⢘⡁⡁⡁⠀⠑⣀⠀⠥⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠔⡀⢉⣬⠇⠄⡀⠀⡄⣭⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠎⠔⠁⠹⣇⠌⠔⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠈⠊⢂⢻⠎⡐⠀⢂⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⡡⡁⠘⡆⠈⠀⠣⠀⢰⡤⡀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠃⠈⠈⠄⠀⣴⠀⠀⠠⠡⠘⡇⠀⢀⣀⣀⣀⣀⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢅⣀⣤⠤⠤⠤⠊⠙⠆⠀⠡⡠⡁⢙⠒⠒⠂⢀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣠⠟⠒⠲⠶⠂⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⣀⣤⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⢠⣶⣤⠀⠀⠀⢰⠄⡀⠀⠀⠀⠀⣀⢄⡄⠀⣶⠂⠀⠀⠀⢠⡔⣶⠆⠀⠀⡠⣾⣦
⠀⠀⠀⠀⠀⠀⠀⣿⣾⣷⡀⠀⠀⣤⣈⡃⠐⠛⢘⣣⡌⠀⠀⠙⣦⡀⣿⣿⡍⡤⡟⠀⠀⣜⠦⣹⣿
⠀⠀⠀⠀⠀⠀⠀⢿⢘⠻⣷⣄⣤⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠛⠂⠉⠞⠁⢤⣤⣹⣶⣧⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣰⣾⣥⣾⣤⣀⣀⠀⠀⠀⠀⠀⠠⣤⠤⠄⠀⣀⠠⢦⠐⢺⣶⣽⣿⡟⠏⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠟⢩⣄⠙⢿⣿⣏⣶⣄⠀⠀⠀⠁⠀⠀⠘⢦⣄⣶⣿⡻⠿⠟⡊⠙
⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢀⣰⠂⠉⠉⠉⣉⣉⠁⠀⠠⢤⠤⢤⠄⠀⠀⣀⣀⠘⠻⠿⠛⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⢧⣴⡔⠁⠀⣴⣶⠀⢻⣦⠽⣦⡀⠀⠀⡀⢀⣴⠿⢹⡟⢐⣿⣦⠀⠀⠀⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣽⣆⠀⢿⣮⡑⢪⣆⢠⣴⡑⢈⣰⣿⠀⢸⡿⣿⣇⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⠿⠟⠀⠰⣇⠄⣀⠹⣿⡗⠀⣤⣸⡆⠀⣸⠗⣿⣿⡀⠀⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⡀⢾⣉⡀⠠⢘⡀⠀⠙⣶⡟⠀⠉⠀⢼⣦⠋⠀⠀⣿⢠⣿⡿⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠃⠈⠛⢧⡤⣘⠁⢀⡼⠃⠀⠀⣀⠀⠀⠛⢆⠀⠸⠿⠚⠋⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⣦⠀⠀⠀⠀⠀⣀⣁⠀⠀⠀⠸⣿⣧⣄⣀⠀⠁⣰⣦⣄⡀⠀⠀⢀⣤⣺⣿
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣷⣀⣠⣶⣿⣿⡟⠃⠀⠀⠀⠲⠶⢤⡙⣷⠀⢸⣋⢿⣿⣦⣤⣿⣿⣟⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⠀⢈⣀⡀⡀⠀⠁⠀⠀⣹⣲⣿⣿⢿⢿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡿⠟⠙⠃⣿⣿⣿⠇⠀⣶⣦⣄⠀⠀⠀⠀⠀⢀⡀⠀⠸⢿⣿⠇⠀⠀⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⡈⠋⠁⣀⠀⢸⡾⣿⣷⣤⣄⣲⣿⣛⠏⠀⣶⣀⡀⠶⠠⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠾⠛⠋⠀⣼⣿⣧⠈⠙⠻⠋⠿⠩⢿⡿⠇⣀⣼⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⢤⡾⠉⠈⣀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⣰⡿⢟⡗⣿⠁⢘⠃⠀⠐⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃⠀⠀⢰⣍⣻⣿⣷⣦⠀⠀⢠⣾⢟⡹⣾⣻⣯⡏⠠⠎⠈⠀⢐⣄⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠋⠀⠀⠀⣰⣾⣿⡿⣿⣿⡿⠀⠀⢿⣫⣾⡷⠟⠿⣛⣧⠀⠀⠀⠀⠈⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣶⣶⣾⣿⣿⣿⡙⠳⠄⠀⢀⠀⠤⣤⣤⣤⠄⢈⣉⣿⣿⣯⣳⡶⢦⡤⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⣿⡿⣿⣿⡆⢠⣾⣭⠽⠾⠉⠉⠀⡴⠟⠻⢌⣛⣍⣭⣿⣶⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠿⠛⠁⠀⠀⠀⠀⠀⠉⠛⠿⠿⠋⠀⠉⠉⠁⠀⠉⠁

⠁⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠈⠲⠮⠁⠲⠤⠔⠉⠀⠀⠀⠀⠀⡤⣨⣗⣻⡞⠡⠒⠀⠀⠀⠀⠀⠀⠈⠓⠯⠓⠁⠀⠀⠘
⠀⠀⠀⣀⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠛⠙⠒⠃⠀⠀⠀⠀⣀⣄⠀⠀⠀⣠⣄⠶⠂⠀⠀⠀⡀⠀⠀⠐⠀⠖
⣄⣴⣌⣻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀⠀⠀⠀⠂⠴⠟⠃⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⡴⣖⠅⠀⠀⠀⠀⠊⠙
⠳⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⢉⣛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠣⠚⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⠀⠐⢁⡀⠸⢭⡆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⡅⠀⠉⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠱⡀⠀⢠⠀⠀⠀⠐⠃⠀⢀⡴
⠀⠄⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠉⢁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢈⠄⠀⠀⠀⢡⡀⠀⡀⠀⠛⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⢀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⠶⠀⣠⠾⠋⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠈⢧⡤⠞⠁⠀⠀⣰⠿⣎
⣂⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⢤⣀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠐⠃⠀⠈
⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠠⢀⠉⠹⢦⡀⠀⠘⢦⡘⣎⠷⠀⠀⢀⣤⠽⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠛⣦⠀⠀⠹⠈⣤⢀⣄⣿⠁
⡀⢀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠐⡷⣾⡄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠂⠀⠔⠂⠀⠐⠲⢤⡀⠀⠀⠳⡏⠙⡇⠀⢠⢞⡟⠈⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣀⣀⣄⠀⠀⠀⢠⠖⠿⣆⠀⠀⢳⠀⡿⠀⠸⠈⣷⣀⠀⠀⠀⠀⠀⠸⡍⢀
⠀⠀⠀⠁⠔⠁⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⡤⠀⠀⠀⠀⠙⠘⠇⣿⠀⠀⠘⠶⠃⠀⠀⠀⠻⠽⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠈⠠⡄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠈⠁⠒⠁⠀⡀⠀⠀⠀⠀⠀⢠⠄⠀⠁⠀⠀⢀⣀⣀⣤⣀⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠁⠀⠈⠁⠀⠈⠉⠙⣦
⠒⣤⠂⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⢀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠈⠳⠦⠶⠛⠓⠶⠒⠒⠤⠖⠤⠚
⢉⠈⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢀⠔⢉⠌⢁⠐⢄⢈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠲⡆⠀⠀⠀⠀⠀⠀⢇⠀⠘⢄⡈⠂⠊⣀⠜⠀⣹⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠁⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠈⠉⠉⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢇⠀⢠⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡄
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡚⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠈
⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⡀⠀⠈⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠛⠐
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢀⡠⠂⢰⠀⠀⠀⠀⠀⠄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠠
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠐⠉⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡄⢠⠀⡖⠒⡄⠀⠀⠀⠉⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⡉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⡀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣎⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣆⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣶⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠻⣦⡀⠀⠀⠀⠀⢰⣶⣶⣶⣶⡸⣿⣿⣲⣶⣶⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠈⠻⣷⣄⠀⠀⠘⣿⣿⣿⣿⠏⠀⠀⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠈⠿⣿⠏⠀⠀⠀⠀⢻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⣠⣴⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠙⢷⣦⡀
⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠿⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣹⣛⣩⣛⣟⣽⣻⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿