King Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣐⣯⣀⠀⠀⣼⢷⣾⠟⠲⠶⣤⡀⠀⠀⠀⠀⢀⡤⠄⠀⢦⡀
⠀⠀⠀⠀⠀⠀⣠⣦⠤⣄⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠁⣠⠟⠋⠉⠀⠀⠀⠀⠉⠛⢷⡀⠀⠀⠙⢷⣄⣀⡀⠜⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠈⠓⣠⠶⠲⢦⡴⠋⠀⠀⠀⠀⢿⠀⣼⠟⢷⠀⠀⡾⢻⡆⢸⠇⣀⠀⠀⡾⠉⠈⠙⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠉⢀⠤⠌⠁⢀⣀⡔⠂⣤⡀⠑⠊⠀⠀⠶⠶⠒⠀⠑⢉⡼⠤⠐⡦⠀⠈⠙⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⢀⣀⠀⠀⠀⠸⠇⠉⢁⣘⣃⣀⣤⣶⡿⠀⠀⠀⠳⣦⣤⡥⢀⣀⡀⠀⢀⣀⡛⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⠉⠐⠶⠶⠶⠖⠋⠀⠉⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣀⣀⣀⠒⠒⠠⠤⢤⣄⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣀⣀⣤⡤⣤⡶⠒⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠂⣤⠉⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣴⠖⠛⢉⣵⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣿⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠸⡏⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⢀⡀⠈⠁⢹⡀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⡇⠀⠠⠞⠋⠉⣀⣉⡙⠻⣷⣦⠄⠀⠀⢀⡾⣛⣛⡋⠀⠈⠻⣦⡈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠀⠀⠀⠀⢀⡿⠀⠀⣀⡴⣫⣭⣭⡍⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⠙⠆⢠⠆⠈⢻⡄⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡳⠠⠞⠀⠀⠀⠴⠋⠁⠀⠀⠉⠀⠀⠽⢯⣤⡶⠀⠀⠀⠀⠀⢸⣧⣭⡥⣶⠀⠈⠀⠀⠀⠁⣰⠟
⠀⠀⠀⠀⠀⠀⢀⣀⣸⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠛⠁⠀⠀⠀⠀⠀⠘⣇⠀⠀⠉⠀⣿⠀⠀⠀⣄
⠀⠀⠀⠀⠀⣼⠛⠉⠉⠡⣄⡀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⠄⠀⠀⠀⢿⠀⠀⠀⢰⠏⣦⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⣿⠀⢠⡶⠒⠈⠻⣀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⢠⣤⡄⠀⣸⠇⠀⠀⠋⠀⠈⠳⢦⣄⠀⣿⠂
⠀⠀⠀⠀⠀⠹⣄⢿⠁⠀⠀⢀⣈⡓⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠟⠛⣀⠀⡘⢧⡀⠀⠀⡇⠀⠀⠀⠀⠙⢀⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠢⠀⢰⡟⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⡴⠒⠛⠁⠀⠋⢒⢿⠛⢳⣄⠀⠀⠀⠀⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⢠⠷⣦⣀⣀⠀⠠⠟⠀⣤⠞⠩⣍⣁⠒⠛⠈⣉⣽⡇⠘⣿⡀⠀⠀⠀⣸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠚⠁⠀⠈⠉⠻⣏⠀⠀⢀⡟⠀⠀⠈⠛⠿⣿⠿⠛⠙⣧⠀⠹⣷⣖⠦⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⠋⢀⡴⠋⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠈⠳⢶⡌⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣤⡄⠀⠀⠀⠀⠉⠀⠀⣠⠀⠀⠀⠀⠀⢀⣴⡿⠀⠀⠀⢰⡟⠿⠿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠾⠛⠿⠿⢻⡇⠀⢷⣄⣀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢰⡟⠉⠀⠀⠀⠀⠈⢿⠗⠀⠀⠈⠻⣶⣦⣤⣀⡀
⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⠀⠀⢠⡘⠃⠀⠀⠉⢿⣦⡀⠀⠀⠈⣧⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠦⣄
⠀⢀⣴⠾⠋⠁⠀⢀⣠⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠙⠿⣶⣆⢸⡃⠀⠀⠀⠀⠀⢸⡀⢀⣠⣾⡏⠀⠀⠀⠈⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠐⠋⠁⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠈⠻⣷⣤⡀⠀⠀⠀⠘⣿⣮⠳⠀⠀⠀⠀⣠⣾⣿⠿⠛⠋⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡶⠀⠈⠻⢿⣷⣤⡀⠀⠈⢻⣇⠀⠀⠀⢰⣿⠁⠀⠀⠀⢲⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣠⣆⣠⠌⠉⠛⠳⠄⠀⠙⠳⣆⠀⠘⣿⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢉⡴⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡄⠀⢀⡜⢦⣤⠆⡧⠀⠠⡄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣉⢠⣞⣿⣶⣶⡾⣇⣐⣚⣵⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⠟⠋⠉⠉⠉⠙⣛⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡒⣶⡄⠲⣾⡓⢹⣷⣦⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠁⠀⠀⠀⠀⠀⢻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣾⡿⣷⣄⠀⠀⢸⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⣾⣿⡅⠀⠀⠙⠓⠐⠉⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⣿⣷⡖⠀⠀⠀⠻⣿⡿⠐⠂⡀⠀⠀⢀⣤⣿⡿⣿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣧⢠⣤⣦⣶⣿⢁⢀⡀⠿⠹⣦⣤⣾⠟⠁⣴⠿⠿⠛⠛⠟⠉⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠿⢿⣿⣿⣿⢸⡟⣰⠇⠀⢰⣿⡏⢠⣾⡇⠀⠀⠀⠀⠀⠰⠿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⠉⠀⠀⠀⣿⣿⠟⣽⠀⡿⠆⠀⢻⣋⠀⣿⠟⡥⠤⢠⠀⠀⢠⠃⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣄⣤⣾⣿⣿⣿⣻⣇⡖⠁⣼⣟⣉⣼⣿⠀⠃⠀⡞⠀⠀⡾⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⢻⣿⣿⣿⡟⠉⣛⠻⠟⠦⣛⡝⠙⣿⡿⠀⠀⢠⡄⡴⢸⡇⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡇⢸⣿⣿⣿⣿⡷⠹⣷⡇⢀⣿⣿⣿⡟⠁⡄⢠⢿⣽⢇⣸⡇⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢀⣾⣽⠉⠉⠂⠻⣿⣿⣿⣿⢓⡿⠪⠗⠘⠛⢻⣿⡯⠀⢠⡞⢸⠇⠘⠿⣿⣀⣀⠀⠈⢷⣆
⠀⠀⠀⠀⢰⣿⣿⡏⠀⠀⠀⠀⢾⣿⣿⣷⡀⡿⠸⠆⠸⣷⣿⣿⣾⣡⣄⣸⡾⣿⣿⢃⣿⣋⣥⣤⠀⠀⢹
⠀⠀⠀⠀⠘⣿⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⠉⠉⠀⠀⠀⠈⠙⢿⠉⠉⠉⠑⠚⠉⠙⠻⠯⠤⠤⠤⠀⢀⣸⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣷⣶⠀⢺⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⢀⣾⣾⣧
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠀⢸⣿⣿⣿⣿⡄⠀⠀⠀⢀⣀⣀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡇⣽⣿⡿⠿⣿⢿⣿⣧⠄⢠⣭⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠉⠉⠀⠀⣿⣿⣷⣿⠀⠈⣿⡿⣿⣥⠀⠀⠀⠀⢀⣀⠀⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⢿⠀⠀⡇⠀⢻⣏⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢁⢸⠀⠀⡁⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡚⠀⠘⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⢻⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢰⡟⣳⠀⠸⣯⣽⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣽⣇⠀⣽⢿⣄⢠⡿⢿⡄⢀⣷⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠷⡟⠀⠙⠟⠁⠈⠻⠟⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣷⣀⣀⠀⠀⠀⠀⣀⣀⣴⡟⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⡿⠀⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⣶⣶⣤⣤⣶⣶⠶⠶⢾⡆⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⢸⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⡤⠀⠀⠀⠀⠀⠀⠉⠁⠸⣿⠘⡀⠀⡆⠈⣇⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⢀⡀⠀⠀⠀⠀⠀⣤⣄⠀⣿⠀⢣⠀⠘⡄⠹⣄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠸⣷⠀⠿⠟⢰⠋⠀⢀⣠⣝⡋⠀⢻⡇⠀⠀⠀⠈⠀⠈⣮⣿⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠀⣀⠹⣧⣠⠞⠻⣤⡴⠟⠁⠈⠛⣦⣬⣿⡀⠀⣴⢟⠛⢿⡯⣿⡟⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣧⣤⣟⣿⣶⣿⠴⠶⠤⠞⠒⠚⠉⠁⣸⣷⠶⠟⠿⣧⣼⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡞⢦⡈⠈⣹⠋⠉⠛⠿⢶⣦⣤⣤⣶⡶⠟⠋⠁⠀⠀⠀⣸⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣍⣹⡗⠘⠟⠛⢻⡀⠀⢀⡄⠚⢿⣙⣻⠟⠃⠀⠀⠀⠀⠀⢰⡏⠈⠛⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣥⡌⠁⠀⢀⣀⣸⣇⠀⣸⡇⠀⣼⢻⠇⠀⠀⠀⠀⠠⣄⠀⣸⣷⣄⠀⠀⠀⣽⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⣤⣴⠟⠉⠉⣽⢟⣿⠁⠀⡏⢸⠀⠀⠀⠀⠀⠀⠉⢿⡙⣷⣿⢠⣦⣘⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⣾⠷⣦⣬⣧⣸⣇⣀⣀⣠⣤⡴⠶⠟⢷⡘⣿⡿⠁⠹⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢸⡇⠀⠀⠀⠀⣉⣉⡉⠁⠀⠀⠀⠀⠀⢸⡇⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠿⠋⠀⢸⣇⣀⣼⠷⠛⠉⠻⣄⠀⠀⠀⠀⠀⢀⡾⠁⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⢹⡆⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⠈⠛⢷⡾⠋⠀⠀⠀⢀⣴⠛⠛⣿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠉⠻⣧⡶⢿⣅⠀⠀⢀⣤⡾⣧⠀⠀⠹⢦⡀⠀⠀⢀⣴⡾⠛⠛⠻⠶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠈⠀⠀⠙⢿⣶⠛⠉⠛⠛⢷⡀⠀⠈⢿⡶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⡀⠀⣠⡶⠛⠁⠀⠀⠀⠀⣼⣷⠶⠶⢾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠾⠋⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠟⠋⠉⠁⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⣠⣤⣤⣤⡶⠶⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡔⠂⣄⠀⢠⠖⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠸⢇⠀⠏⠀⠘⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡅⠀⢸⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠆⡼⠂⠀⠃⡴⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡄⠀⠀⠀⡈⠀⡀⠀⠀⢨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⣤⣀⣀⣉⣀⣉⣠⣴⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠎⠙⠋⠉⠉⠉⠉⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢸⡆⠀⠀⠀⠠⣄⠀⠀⠀⠘⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⡈⠁⠀⠀⢨⡅⠐⠀⠀⠀⢀⡀⢀⡀⣀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋⠀⠀⠀⠀⡸⠃⠀⠘⢻⡿⠃⠀⠘⢿⣟⠀⢼⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢧⣤⣤⣤⠾⠁⠀⠀⠀⠀⠀⣀⡼⠒⠀⡙⠧⣼⡇⢠⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠘⠷⠤⠀⠈⠉⠙⠛⠷⠤⠤⢄⣀⠈⠃⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⢀⡘⠻⢀⠀⠀⠀⠀⢀⣀⡀⠒⠶⠒⢶⡄⠉⠻⣦⣄⡈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢎⣁⡀⠀⠀⠀⠀⠛⠃⠀⠀⠉⠛⠋⠀⠈⠀⠀⠘⠓⠈⠻⣦⣀⠀⠉⠛⣷⣶⡄⠀⠀⠀⢠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢴⡆⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠐⠿⠏⠀⠈⢻⡅⠀⡄⠀⠀⠀⠂⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡀⠀⠀⠀⠠⠾⠟⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢀⡀⠛⠃⠀⠀⠀⢀⣉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠛⠁⠀⣠⢤⣤⣄⠀⠀⠀⣤⡈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⢠⡶⠋⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠞⣁⡴⠏⠀⠙⠓⠤⢀⠀⠀⠈⠛⠳⣶⡦⢤⡤⠤⠤⢤⣤⡀⠀⠈⠀⠀⠏⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠈⠁⠂⡀⠀⠀⠀⠀⠀⠁⢠⣤⠚⠀⠉⠐⣶⡆⡸⢷⣄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠠⠆⣀⠋⠁⠀⠀⠀⠀⠹⠃⢰⣦⣈⠉⠀⠄⠠⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⠀⠀⠀⠀⠀⡏⠐⠆⠀⠀⠀⠀⠰⣄⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠰⣟⠉⠁⠒⠒⠒⠐⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⣠⠀⠀⠰⠄⠀⠀⠀⠷⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠘⢷⣄⣀⣀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⣰⠏⠀⠀⣠⠀⠿⠂⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡘⠃⠀⡴⠋⠀⠀⠀⠀⠀⠀⠳⠄⠀⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠀⡄⠃⡄⠿⠀⡀⣀⠀⡀⠀⢀⡀⠀⠸⠇⠀⠀⢸⡇⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠸⠉⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⣴⡧⢸⡃⢰⡆⠀⢿⠀⢠⡸⠃⡴⠀⢠⡄⠀⠀⠈⠃⠀⠀⡈⠀⠀⡇⠀⠀⠀⠀⣀⣀⣤⣈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠈⠓⢤⣅⠀⢸⠀⢸⠇⢀⣤⠒⠋⠙⠀⠀⠀⡀⠀⠠⡇⠀⣴⡟⠀⠀⠀⠸⣯⣀⣠⣄⣁⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣀⣀⣤⡴⠤⠶⠤⠿⠦⠼⠆⣤⣀⣘⣁⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠈⠁⣀⣀⠠⠴⢯⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢰⣷⠶⠐⠒⠚⠉⠁⠀⠀⢨⣥⣄⡀⣀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠉⠁⠒⠒⠒⠶⢤⣤⣤⣄⣀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⠒⠒⠒⠐⠒⠚⠛⠓⠉⠉⠉⠐⠆
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⢦⣤⣤⣤⣤⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⣶⡶⠾⠷⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣟⠛⠻⠿⠶⠞⠻⠟⢻⡟⠉⠉⠛⠛⠻⣿⣿⣷⣿⠛⠛⠋⠀⠀⠀⠈⠙⠻⠶⠶⠶⠶⢦⣴⣶⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠻⠷⠿⠷⣤⣤⣤⣤⣤⡴⠿⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣦⣀⠀⠀⢠⡄⠀⢀⣀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡿⣿⠛⢇⡰⠿⡟⠇⣌⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⡀⢀⣽⣿⠇⣶⣅⣹⣧⣾⣿⣾⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣲⣧⣿⣿⣿⣿⣿⣿⣿⣽⣿⣾⠿⠷⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⡇⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢿⣾⠟⠐⠛⣽⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣶⣿⣧⣀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣟⡀⠀⣈⣩⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⣙⠛⠻⠿⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣿⣿⣿⣿⣿⡆⢠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠈⠉⠛⠻⠿⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡮⠴⠚⢉⡐⢂⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠙⠦⣤⣞⡵⠛⣟⣵⣯⣾⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱⡄⠀⠸⠀⢇⠀⢠⢏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢵⠈⠲⠇⠀⠸⠞⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⢀⣀⡀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠊⢁⠀⠈⠙⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠀⠁⠀⠙⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡔⠡⡀⠀⠀⠀⠀⣼⡀⠀⠀⢀⢿⠉⢦⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠛⠉⠀⡔⠑⠦⢤⡴⠎⠤⠉⢒⠒⠋⠀⢣⠀⠉⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠆⠀⡰⠁⠀⠀⠀⠀⠀⠈⢆⠀⠸⡀⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢀⠀⠀⠞⠒⠚⡇⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠳⠀⢠⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣽⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⣀⣠⠔⠃⠀⠀⠀⠀⠸⠀⠀⠀⠐⢯⢄⡀⠀⠀⠀⠀⠀⠀⡘⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠰⠀⠀⠀⠀⠀⠑⠂⣤⣶⠀⠀⣀⣆⠀⠀⠀⣤⡀⠀⢀⣶⣤⠐⠁⠀⠀⠀⠀⠈⡆⠈⢢⠀⠀⠀⠀⠴⠧⢤⠈⢘⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣀⠇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣷⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⡆⠀⠀⠈⢄⠀⠉⠐⢃⡤
⠀⠀⠀⠀⠀⠀⢀⡠⣤⠔⣀⠺⡀⠀⠈⠢⢤⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⡤⠴⠁⠀⠀⡗⡐⢦⡤⠊⠀⢠⠒⠁
⠀⠀⠀⠀⣠⠞⠁⠀⡇⠒⠀⢸⠳⡀⠘⠀⠀⠆⠀⠀⠀⠀⠀⠈⠛⠿⠟⠁⠀⠀⠀⠈⠿⠿⠛⠁⠀⠀⠀⠀⠀⢠⠁⠐⠃⢀⠜⣧⠐⠂⢱⠀⡠⠃
⡠⠔⠂⠉⠁⠀⡔⠈⢣⡉⠀⠇⠐⠈⠢⣄⠐⠊⠒⠄⠚⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠤⠒⠑⠂⣠⠔⠁⠂⠘⡊⢉⠎⠉
⠳⣔⡛⠗⠀⢰⠃⠀⠀⠈⡙⠀⠀⠀⠀⠀⠉⠲⠤⣀⠛⠈⠓⠠⣄⠒⡛⢤⡀⠀⠀⡠⠟⠒⣠⠴⠊⠁⠛⣀⠤⠒⠉⠀⠀⠀⠀⠀⢿⠁
⠀⠀⠀⠈⠢⠜⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢲⠀⠈⠀⡇⠀⠈⠛⠉⠀⠀⠀⠉⠀⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠀⡃⠀⠀⠀⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⣰⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠴⠀⡇⠠⠀⠀⠀⠀⠀⠀⠤⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠤⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠚⠀⣧⡀⣀⣉⣀⢀⣤⠀⠒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡇⣬⣽⣿⢸⣿⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⣿⣷⣶⣶⣶⣾⣿⠀⠉⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡨⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⣠⠀⣿⣿⣿⣿⣿⣿⣿⠀⣀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⢇
⠀⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⣀⠀⠉⠉⠉⠀⠒⠒⠒⠒⠂⠘⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠓⠐⠒⠒⠒⠒⠂⠈⡉⠉⠁⣀⠀⠀⠤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠓⠀⠦⠤⠄⢠⣄⡈⢀⣀⣘⣃⡀⠀⠂⠀⠀⠲⠀⡏⠀⠀⠀⠀⠀⢀⠀⠖⠀⠀⠐⠀⠀⣘⢃⣀⣀⠁⣠⡄⠠⠤⠴⠄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣉⣼⣿⣿⣿⠁⠈⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⣿⣿⣯⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡆
⣶⣶⣶⠶⡶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⣤⣤⠤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣿⡿⠿⢿⣿⡿⣷⣿⣿⣿⣿⣇⡀⡗⡲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣾⣷⣿⣷⣿⠿⢆⢸⡾⣧⣀⣿⣿⣿⣿⣿⠁⣮⡹⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠉⣾⠛⣿⣿⣿⣾⣄⣨⡟⠂⢘⠿⠿⣋⡍⠻⣿⣟⣡⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣵⣤⣖⣼⣿⣿⣟⣛⣛⠵⠒⠿⠉⠀⢘⢋⡁⠲⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡿⠾⠟⣁⠿⠄⠐⠒⢀⣄⠈⢰⣆⣀⣘⣂⣀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠖⠈⡰⣠⣨⣥⣶⡶⠶⠗⠘⠋⠉⠉⠉⠉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣻⣽⣷⣿⣿⡿⢣⡷⡀⠆⠀⠀⣠⡤⣦⠀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡟⣉⠋⠀⠁⡀⠀⢸⣱⣼⡿⠀⠻⡀
⠀⠀⠀⠀⣠⠞⠛⣶⡄⠀⠀⠀⠀⢸⡿⠛⢿⣼⣿⣿⣧⢾⣡⠄⠀⠈⠀⠉⠀⠀⠀⠹⡄
⠀⠀⠀⠀⢿⣶⣼⣿⣷⠀⠀⠀⢀⣼⣿⣿⣾⣿⢻⣷⡇⣠⣯⡀⠀⠀⠀⠀⠀⢀⢀⡀⠱⡀
⠀⠀⠀⠀⠈⠹⠿⠯⣿⣠⢄⠠⣿⣿⣿⣿⣿⣼⣿⡿⠘⠞⡎⣑⠀⠀⠀⠀⢀⣼⣟⣻⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⢿⡿⣿⣏⠀⠙⣿⣿⣿⣿⣿⠟⠁⠀⢰⣶⡎⠀⠀⠠⣤⣿⣼⢟⠁
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⢿⣦⡀⠙⣿⣯⣍⡈⠀⠀⠀⠈⢿⣻⣧⡆⢆⡐⠋⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠑⣝⢄⣿⣿⣿⣿⣷⢦⣄⡀⢸⣿⣿⣿⣶⣐⠆⠀⣌⡀⢄
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣴⣶⣾⣦⡙⢿⠟⠿⠛⠛⠓⠛⠿⣿⣿⣿⣿⣳⣄⠂⠀⠀⡀⠁
⠀⠀⡤⠀⣠⠀⠀⠀⣴⣿⣿⣿⣿⣯⠿⣦⠁⠄⠀⠀⠀⠀⠀⠀⠹⣿⡿⢿⣿⣣⠄⠁⠬⠀⢀⡄⢀⣀⣀⡀
⠀⠀⢉⣩⡟⢅⣫⣾⣿⣿⡿⠻⠛⢀⢄⡸⣷⣄⠢⡀⠀⠀⠀⠀⠤⠌⠻⣦⣿⣯⣾⣻⣷⣤⣲⣿⣿⣿⣿⣛⠧⢀
⠀⠀⠠⠾⢒⡠⣾⣿⣿⣿⣷⣶⣿⣿⣾⣽⣿⠿⣇⢌⠢⡀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣿⣶⣺⣟⣻⣯⣛⣛⣲⡥⠆
⠀⠀⠀⠢⣑⢼⡿⣿⣻⣿⣿⣿⣶⣿⡭⡨⠏⠛⠚⠷⣤⣈⠢⡀⠀⢔⠛⠁⣀⠉⠻⢿⣿⣿⣧⣻⠿⣷⣍⡙⠀⠁
⠀⠀⠀⠀⠑⢋⠭⠏⠕⢛⡯⠛⣋⠘⠃⢀⠄⠀⠀⠀⠙⢯⣆⡘⢆⠈⠤⠦⡄⠀⠀⠀⠈⠻⢿⣿⣧⣮⣄⡈⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠙⢿⣄⣃⠁⠶⣄⡀⠀⠀⠀⠀⠀⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠓⠘⠂⢀⣀⠌

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠲⡀
⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⡴⠉⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⢈⡟⢯⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢈⠟⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈
⢳⣀⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣠⠜⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⡴
⠀⠈⠉⡟⢄⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⢀⡔⠁⠑⡄⠀⠀⠀⠀⠀⢀⢎⠄⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⠞⡏
⠀⠀⠀⢰⠈⠳⡀⠀⠀⠀⡰⡽⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢠⣮⠎⠀⠀⠈⢦⠀⠀⠀⢀⢮⡟⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠔⠁⢸
⠀⠀⠀⠀⡇⠀⠘⢆⠀⢰⠻⠁⠀⠀⠀⠀⠀⠀⠘⡄⠀⣴⡿⠃⠀⠀⠀⠀⠀⠳⡀⢀⢮⠟⠀⠀⠀⠀⠀⠀⠀⠱⡀⡠⠋⠀⠀⡇
⠀⠀⠀⠀⢸⡀⠀⠀⠱⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⡸
⠀⠀⠀⠀⠀⠀⠀⢰⣓⣒⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣳
⠀⠀⠀⠀⠀⠀⠀⠀⣩⠄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣆⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⣤⠊⢁⢂⣠⡔⠀⣂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣈⣟⡇⠘⠬⣿⠏⠠⣹⣿⠃⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢠⠀⠀⠠⣲⢀⡀⣀⡀⠈⠁⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⠀⠀⣁⠂⠑⠈⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡈⠇⠤⠤⠭⢤⠤⣼⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣔⠂⠂⠀⣘⣤⡈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠆⠓⠃⠀⠀⠓⠃⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠤⠀⠤⡀⢀⠀⠘⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠈⢠⠀⡤⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠈⢀⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡚⠀⠀⢡⠈⠁⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡏⠘⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡀⠨⢇⡀⠀⠀⠁⠀⣀⠀⣰⡀⢠⢶⡀⢀⠀⠀⣷⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⣄⠀⠈⠀⢄⡠⠾⢷⣿⠊⣹⣻⢮⡾⠚⢘⡇⠸⡌⢀⣍⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⢄⠀⠈⠑⠠⢤⣄⡀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠈⣠⠵⠋⠁⠠⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠢⣁⡀⠀⠀⠀⠈⠡⡄⠀⠀⠀⠀⠀⠀⢀⢀⠔⠈⠀⠀⠀⢀⢀⠔⠉⡀
⠀⠀⠀⠀⠀⠀⢀⠂⠉⠀⠀⢀⠁⠀⠀⠀⠀⠀⢳⠀⣐⣤⣤⡀⠀⢀⢸⠀⠐⠀⠀⠀⠈⡗⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢩⠁⣉⠁⠘⠠⢐⠈⠀⠀⠀⡇⠅⠸⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠁⠀⠀⠂⠀⡈⢀⡄⠳⠙⠻⠃⠴⠃⡏⠀⠀⠀⢐⠀⠀⡧⠐⠀⠰
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠀⠀⠀⠨⠀⠁⡀⢐⠂⡠⣀⣀⣀⠤⡀⣂⠥⠐⠈⢀⠀⠀⡅⡀⠅⢐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠛⠛⠚⠃⠐⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢘⡃⠀⢰⡆⠀⣨⠁⠀⣠⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣇⡀⣿⣧⢸⣿⣀⣸⣿⣶⣿⣦⣰⡿
⠀⠀⠀⠀⠀⠀⠀⢦⣦⣘⣿⣼⣿⣷⠾⢟⡋⣝⡿⣆⣠⡠⡬⣽⠉⡉⣿⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣧⣴⣄⣿⡷⠌⢓⣛⣛⣓⣂⣀⣀⡴⢾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣼⢷⠾⣛⣭⣶⣾⡏⠉⠉⠀⠀⠀⠀⠀⠈⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡿⠃⠰⠚⠛⣷⣶⠤⠀⣰⣶⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⠁⠀⠰⠶⠖⠚⠟⠁⠀⢿⠟⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⢿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣌⣉⣿⣿⣧⣀⠀⠀⠀⢀⣴⣂⡀⢀⣴⡄⣺⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣾⣿⣿⣿⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⡅
⠀⠀⠀⠀⠀⠀⢀⣶⢿⣋⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣰⣇⣸⢡⣿⣿⣿⣿⣿⣿⡿⠃⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⠛⠘⣿⡏⣿⣏⠟⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⢸⣿⢻⠘⣿⣷⣷⣿⡿⠀⠀⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠘⢼⣿⣱⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⢿⡿⡿⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠈⢻⣿⣿⡿⠛⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡾⢷⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⢿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣀⣼⣧⣼⣧⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢤⣶⣿⡿⣧⣼⢿⣿⣶⡴⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⣀⢀⡤⠄⣓⢷⣿⣏⠋⠙⣹⣿⡞⣚⠠⢄⡀⣀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⢦⢀⡀⣏⣹⡞⢉⡏⢠⢪⠶⠺⣿⠙⣄⣰⢉⡿⠗⠶⡕⡄⢹⣉⢢⣏⣹⢀⡀⡴⡒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡯⣾⡟⣩⡭⣭⣽⣿⣷⣿⡏⠀⠀⢺⣻⣿⣿⣟⡇⠀⠀⢹⣿⣾⣿⣯⣭⣭⣍⣿⣷⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣷⠟⢛⢻⣿⣿⣿⡇⠀⢞⣭⣛⣿⣿⣋⣭⡧⠀⢸⣿⢿⣿⡛⡛⠻⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡌⢿⣿⣿⠀⢸⣛⢨⣇⣿⡷⢦⠈⢉⣿⣿⣿⢯⡉⠀⡴⢾⣿⣸⠅⣙⡇⠀⣿⣿⡿⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢲⣬⡙⢧⡻⢮⣼⣿⣿⣧⡾⣀⢄⢙⣿⣿⡋⡠⣀⢷⣾⣿⣿⣦⠥⢏⡴⢋⣥⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⠟⠺⣷⡀⢶⣦⣿⣿⣦⣘⣋⣼⣷⣿⣧⣑⣃⣶⣿⣿⣴⡶⢠⣾⠞⠹⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⣼⣯⣾⣟⡿⢿⣿⠿⣿⣿⣟⢻⣿⣿⠿⣿⣿⣿⣿⣷⣽⣧⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠙⣿⣍⣿⡷⠾⠿⠛⣛⣛⣛⣛⣛⣛⡛⠿⠷⢾⣿⣟⣿⠃⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣷⣲⣭⣿⣿⣷⣿⣶⣶⣿⣿⣿⣯⣭⣖⣾⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⣦⠿⠉⡀⠹⢥⠞⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡷⢒⠒⠀⣓⠂⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⠉⣩⢀⠉⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠁⠦⠘⠃⡄⠀⡼⠃⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡦⠤⠄⠰⣏⠀⠀⠀⠀⡜⠙⢛⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠊⣹⡁⠀⠐⢀⣹⡍⠉⠓⢤⢳⣦⣴⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⡰⢁⠜⠻⠿⠣⠇⠹⠛⠉⡄⠀⠀⠙⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⣀⡐⢠⠎⠀⠀⠀⡶⢾⠀⠀⠀⠘⡀⢡⠣⠉⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠶⠤⣠⣼⣶⣷⠀⠀⠀⠀⠷⠿⠀⠀⠀⠀⣷⢂⢇⡀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣧⠀⠀⢹⠀⠻⢿⡄⠀⠀⠀⠛⢿⠀⠀⠀⢠⡏⣬⠈⠈⢣⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡟⢤⠊⠹⣀⡴⣌⠳⣤⣀⣀⡛⠛⣀⣀⣤⡟⣰⣿⠀⠀⡾⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇⡚⣦⡈⠛⠀⢫⡆⠸⠛⠉⣳⡈⠉⠉⢨⠥⠇⠃⣤⡾⠃⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇⠀⠀⣰⠦⢠⣜⠿⠃⠀⠀⣯⣄⠀⢰⣧⣄⠀⠈⠑⠧⡄⠀⢸⣧⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣠⡴⢿⣧⠒⠀⣾⠁⠀⠀⠈⠢⡄⠁⡈⠁⡀⠠⠀⠉⠃⠀⠀⠀⠀⡑⣰⣟⣉⣭⣿⣶
⠀⠀⠀⠀⠀⠀⠀⣿⠹⣷⣦⠇⠀⠀⣿⠀⡄⠀⠀⠀⢰⠠⠈⠣⡀⠀⢠⠀⠀⠀⠀⠀⠀⠗⣿⣿⣿⠿⠋⣽
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢻⡀⣤⢀⡿⠀⠃⠀⠀⠀⠀⠃⢡⠀⠈⠁⢻⠀⡆⠀⠀⠀⠀⢰⣾⡁⠀⣀⣤⣿
⠀⠀⠀⠀⠀⠀⠀⣿⢀⡤⢴⠁⣿⢹⡇⠀⢠⠀⠀⠸⡀⢸⠀⠇⠀⢀⠺⡰⠀⠀⠀⠀⠀⢸⡞⢧⡾⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⢸⠀⢸⢸⡇⠀⠈⠀⠀⠀⢧⠈⡆⠀⠀⢼⣶⠁⠀⠀⠀⠀⣀⡈⡇⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣸⠇⠀⠀⠀⠀⠀⠈⢣⣁⠀⠘⣾⠏⠀⠀⠀⢀⡾⠁⣩⣷⢠⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⢻⠖⢶⡀⠀⠀⠀⠀⠀⠻⡏⢠⠏⠀⠀⣀⡤⠾⠸⢛⠁⠀⠀⢃⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠰⠘⡄⠀⠙⣄⠀⠀⠀⠀⢀⣹⣯⠶⠞⠉⠹⠀⡀⢠⠶⠀⠀⠀⠈⢆⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠘⠀⠘⣴⠻⡍⣗⢠⣤⠞⣫⢅⣿⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠳⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⣄⣀⠠⠖⠁⠀⠀⠈⠁⣸⠋⠉⡸⣿⠉⠷⠻⡐⠒⠲⠋⠳⡀⠀⠀⠀⠀⠀⠀⠈⠂⠈⣓⣦
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢠⠾⠐⢢⠊⠁⠈⠳⣆⠠⠽⢄⡀⠀⠀⠙⠳⢤⣄⣀⠀⠀⢀⣀⣀⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣤⠃⠀⠀⠀⠀⠈⠳⣀⣋⣹⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⡀⠀⠀⠈⠠⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠀⠈⠀⠀⠀⣤⣤⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⡀⠄⠐⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠄⠲⠆⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂⠀⢀⠐⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢡⠂⠀⠈⠁⠈⠐⠠⡀⠀⠀⠀⠀⠀⠉⡀⠄⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠀⠉⠒⠠⠼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⠀⢀⡄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⢄⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⢸⡢⠀⣛⣁⡀⡄
⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣈⡷⠖⠶⠈
⠀⠀⢠⠎⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠢⣠⣀⠀⠀⠀⠈⠢⠄⡀⣀⠤⠊⠀⢀⠜⠀⡀⠀⠀⠀⠀⠰⠋⠉⠙⢦⡄⢀
⠀⠀⢯⠀⠀⠀⠀⠀⡀⡤⣠⠀⠀⡠⠀⠀⠀⠑⠦⣄⣉⣁⣠⠤⠞⠓⠶⠤⠤⠤⠴⠊⠑⠄⣀⣠⠁⠀⠦⠤⠄⠀⢀⣀⣠⠏⢀⡇
⠀⠀⠈⠳⣄⡐⠐⠞⠀⠋⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠠⢲⣦⣄⣀⣀⠀⠀⠀⠠⡀⠘⣀⠀⠀⠀⠀⠀⠠⢺⠉⠤⠔⢋⠄
⠀⠀⠀⠑⠄⡁⠀⠀⠀⠀⣀⡾⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠸⣿⣿⣿⣿⠃⠀⠀⠀⠈⢦⢄⠀⠀⡰⠂⠳⣄⡀⠀⢤⠠⠔
⠀⠀⠀⡀⠀⠮⡒⠦⠀⠀⠱⡇⠀⠀⠀⠀⠀⠀⠀⠘⣶⣦⣤⣤⣴⠟⠛⠛⡉⠀⠀⠀⠀⠀⢸⠏⠄⠊
⠀⠀⠀⡁⠀⠀⠈⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠀⢀⠔⠀⠀⠀⠀⠀⡀⠞⠰⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠈⠂⢤⠠⠄⠀⡤⠚⠁⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠛⠿⠀⠁⠀⡀⠠⠀⠀⠈⠀⠀⢠⠃⠀⠀⠀⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠐⠤⣀⠀⠀⠀⠀⠈⠀⠀⠐⠒⠒⠒⠀⠀⠈⠀⠀⠀⠀⠀⣀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠉⢳⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣠⣤⣤⣴⣶⠞⠋⠁⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠘
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠛⠉⠉⠉⢛⠏⠀⠀⢿⠛⠉⠀⠀⡘⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡆
⠀⠀⠀⣰⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡌⠀⣤⠀⣨⠀⠀⠀⠀⡇⠀⢀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡴⠁⠀⠀⠀⠀⡏⠀⠀⠀⠠⡄⢄⣐⠀⠀⠀⢀⡅⢀⡼⢧⠀⠱⡄⢀⠀⠃⢠⠊⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣠⠞⠀⠀⠀⠀⠀⡼⠁⠀⠐⠀⠀⠘⣆⠈⠲⣄⡴⠁⢀⡞⠀⠘⡆⠀⠈⠁⠀⣰⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠙⠢⢄⣀⠀⠀⠔⠀⠀⢀⣁⣀⣀⣀⣈⣆⠀⠈⠁⠀⣸⠒⠂⢈⡇⠀⠀⠀⢠⡁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢳⡀⣠⠀⠀⠀⠀⠀⠉⠀⠁⠐⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⢙⡎⠁⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠤⣀
⠀⠀⠀⠀⠀⣠⠄⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣸⣀⣀⠀⠀⢀⠴⠊⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠞⠁⠀⣿⣿⣿⣿⣭⣥⣤⣀⡀⢄⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣥⢀⠆⠀⠀⣿⣿⣿⣿⣯⠄⢀⡠⡤⠀⠉⣀⠦⡤⠔⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠁⠀⠀⠀⠘⢏⠉⡉⠿⠛⠁⢰⠁⠀⠀⠈⠛⢶⣤⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠦⣄⡀⠀⢐⣿⣿⣦⡁⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠔⠓⣄⠀⠀⠀⠀⣠⠄⠠⠀⠀⠀⣅⠀⠀⢠⣽⣿⣿⣿⡿⢛⣿⣎⣂⣈⣩⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⠀⠀⠀⣀⡤⠂⠀⠈⠢⡀⠀⠰⠛⠦⢀⠃⠀⠀⠀⠑⠢⠄⣈⠩⠛⠉⠀⢸⣿⣿⣿⣿⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠘⡝⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠷⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⢀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⢶⠀⠢⠀⠀⠉⠛⠛⠋⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⢠⠏⠀⠀⠀⠀⠀⡴⢂⠀⠀⠀⠀⠠⢒⠟⠋⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⡼⠅⠉
⠀⠀⠀⠀⠀⠀⠀⡜⠀⣰⡟⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⣀⠼⠃⠂⠀⠠⠄⣀⣀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⣸⠃⠀⢡⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⣀⠀⠀⠀⠀⠀⠀⠈⠉⢿⢹
⠀⠀⠀⠀⠀⢀⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠜⣡⣴⠦⣥⣀⡉⠢⣀⣰⠐⠇⠠⢒⣾⢸
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠊⠁⢀⡮⢵⢒⣯⡆⠀⠋⠀⢈⡾⣳⢽⣼
⠀⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣠⠆⠀⠠⠐⠉⠂⠼⠾⢿⠀⠈⠃⠀⠀⠱⡏⠀⠀⠄
⠀⠀⠄⠀⣀⣴⠖⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⣰⡗⠁⡄⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠘⡆⢀⢰⡀
⠚⠓⠈⢉⣌⠁⢴⣆⡤⠂⠀⠀⡀⠴⠞⠁⠀⠀⠀⠀⠰⠏⠀⠀⢇⣠⠀⠀⠀⠀⠀⠀⠈⠀⡞⠉⠀⠀⠀⠀⢸⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢁⡞⠀⠀⠀⠀⠀⠀⠀⠖⢀⠔⠀⠀⠀⠀⠀⠀⠘⠋⡀⠀⠀⠀⣠⠴⣲⠾⠁⠊⠻⠄⠀⠴⡃⠀⢠⡄
⠀⠀⠀⠀⠀⠀⢠⢀⠀⠀⠐⠛⠉⢠⠶⠪⠅⠊⠀⢠⠎⠀⠀⠀⠀⠀⠀⣰⠖⣋⡀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠁⠠⣺⠇⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⢨⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⣀⠴⠊⠁⠈⠁⠀⠀⠀⠀⡀⣠⣴⣶⣶⣀⣀⠖⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠘⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠃⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⢀⣴⠞⢾⠻⢿⢿⠟⣿⣄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠁⠐⠀⣈⠉⠀⠚⠋⠀⠈⠛⠂⠰⠞⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠈⠑⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠈⠳⠀⠀⣳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣀⠀⠀⡠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠚⠆⠀⠀⠤⢀⣀⣀⠀⠀⠀⢤⣤⡤⠴⠖⠚⠛⠑⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⣰⣛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣠⣿⣤⣤⣤⣤⣤⣤⣤⡴⠋⠘⠂⠀⠀⠀⠙⣷⣴⠶⠖⠛⠛⠛⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡁⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⢢⡀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣿⣿⡛⠛⠷⠦⣤⣤⡀⠀⠀⠈⠉⠛⠷⢦⣍⠉⠙⠛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠻⡇⠀⣸⡇⠈⠛⠿⢶⣤⣀⠀⠈⠛⠛⠶⠶⠦⣀⣀⠈⠛⠶⣄⠈⣿⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣸⠁⠀⣿⠃⠀⠀⠀⠀⠈⠉⠛⠓⠶⢤⣀⣀⠀⠀⠈⠛⠳⠶⣌⣳⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⠶⢶⣶⣬⣿⣿⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢿⠁⠀⠀⠀⣀⣠⡄⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⢸⡟⠛⠂⠀⢸⠁⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠉⠙⣄⠀⠀⠀⠘⠛⠀⠛⠁⠀⠀⢠⡞⠁⠀⠀⣰⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣱⣄⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣶⣄⡀⠀⠀⠀⠉⠳⠶⠤⠤⠶⠶⠋⠁⠀⣠⣴⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⡿⠛⣷⣶⣶⣤⣤⡴⠶⠶⢶⡶⠶⣿⡿⡟⠻⢯⠻⣌⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢡⡿⠳⣁⣿⢣⡾⣆⠙⠳⡶⢶⠋⠀⣴⣿⠃⢸⡇⠈⣧⠘⣧⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⡴⠀⣾⠛⠀⢸⡏⠄⠙⣿⣷⣄⣈⣀⣠⣟⣿⠇⠒⠀⢿⠀⠈⣷⡈⢷⡀⠙⣆⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡜⠀⣸⡏⠀⠀⣾⠇⠀⠀⠈⠱⣿⠏⠉⠉⢻⡇⠀⠀⠀⠈⡇⠀⠘⣧⠈⢱⡄⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡾⠁⣼⠇⢸⡟⠀⠀⢀⣿⠀⠀⠀⠀⠀⣿⣆⠀⠀⢸⡿⠀⠀⠀⠀⢹⡄⠀⠹⣆⠀⢻⡄⠀⢳⡄
⠀⠀⠀⠀⠀⢠⡟⠀⢀⠋⠀⣿⠁⠀⠀⢸⡿⠀⠀⠀⠀⠀⠈⠻⠷⠶⠛⠁⠀⠀⠀⠀⢸⣧⠀⠀⢻⡄⠈⢱⡄⠀⠻⣄
⠀⠀⠀⠀⢠⡞⠁⠀⠺⠀⣸⠃⠀⠀⠀⣿⡟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠀⠈⢳⡄⠀⢿⡀⠀⢹⡄
⠀⠀⠀⠀⣾⠁⠀⠸⠇⢠⡏⠀⠀⠀⠀⣿⡷⢦⣤⣤⣄⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠛⠁⣸⣷⠐⠀⠈⣿⠀⠀⢣⡀⠀⢻⡀
⠀⠀⠀⢠⡟⠀⠀⡆⠀⣼⠁⠀⠀⠀⢰⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣀⣤⡾⠟⣿⡆⠀⠀⠸⣆⠀⠘⣇⠀⠈⣷
⠀⠀⠀⢸⡇⠀⣠⠀⢰⡟⠀⠀⠀⠀⢸⡿⠿⠶⠶⣦⣤⣤⣤⢤⣤⡴⠶⠶⠚⠛⠉⠀⠀⠀⠘⣷⠀⠀⠀⠹⡆⠀⠹⡄⠀⠸⡇
⠀⠀⢀⡟⠀⢰⡇⢀⣿⡄⠀⠀⠀⠀⣿⡇⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⢀⠀⠀⢿⡀⠀⢿⠀⠀⢹
⠀⠀⢸⠇⠀⠸⠀⣾⣧⡄⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠤⠀⣸⡇⠀⢸⡇⠀⠘⡆
⠀⠀⣿⠀⠀⡄⠀⣿⢛⠻⠶⣶⣶⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠟⠉⢿⡆⠘⣧⠀⠀⣿
⠀⠀⡿⠀⠀⡇⠀⡏⢸⠀⠀⠀⠈⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢀⡿⣿⠀⢿
⠀⢰⠇⠀⠀⠁⠀⣷⡼⣦⡀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣴⠟⠁⣿⠀⠘⡆⠀⠠⡇
⠀⢸⠀⠀⢰⠀⠀⢸⣇⠈⠙⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⠂⠀⣿⠀⠀⡿⠀⠀⢻⡀
⠀⢸⠀⠀⣾⠀⠀⠘⣿⡖⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣸⠇⠀⠀⠑⠀⠀⢸⡇
⠀⡟⠀⠀⣿⠀⠀⠀⠙⢿⣆⢹⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⠃⠀⠀⠀⠠⡇⠀⠈⣧
⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⠙⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⡇⠀⠀⣿
⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣇⠀⠀⢻
⢸⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⠀⠀⠈⡇
⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢸⠀⠀⠀⣧
⢸⠇⠀⢸⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢸⡇⠀⠀⢿
⡆⠀⠀⢸⡁⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⡇⠀⠀⢸
⡇⠀⠀⢸⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠘⡇⠀⠀⠸
⣿⠀⠀⢸⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⡇
⣿⠀⠀⢸⡄⠀⠀⠀⠀⣸⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⣇
⣿⠀⠀⠰⣅⠀⠀⢀⡀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣤⡏
⢻⡄⠀⠀⠙⠛⠳⠾⣾⣿⣦⣀⣀⣀⡴⠞⠛⢛⠳⢦⣀⡀⠀⠀⢀⣀⠴⠶⣛⡳⢦⣀⣀⠀⣀⣀⡴⠶⠛⠷⠦⠤⠦⠾⣿⣿⠁
⠈⠻⣦⣀⠀⠀⠀⠀⣻⣯⡈⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⠉⠛⠛⠉⠁⠀⠀⠀⠀⣀⣉⣉⣉⣉⡁⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣤⣠⣄⠠⠴
⠀⠀⠈⠙⠛⠓⠲⠶⠟⠻⠷⠶⠶⠶⠞⠛⢻⣏⣉⣉⡛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣛⣋⣉⣉⣉⣙⣛⣛⣛⣻⣿⡏⠀⠀⢀⣀⢠⣄⣀⣤⣤

⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡀⠀⠀⠀⠠⠄⣀⣀⣀⡀⠠⠀⠀⠄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⡀⠀⠀⠀⠀⡤⠂⠄⠀⠀⠀⠀⢀⠀⠀⠐⠋⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡂⠄⠀⠀⢀⠂⠊⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠇⠈⠐⠀⠀⢔⡀⣀⠄⠊⡀⠀⠀⠀⢇⠀⠀⠀⠀⠈⢧⠀⣠⡀⠀⠀⠀⡘⠁⠀⢀⣠⠀⢁⣀⢀⣀⢀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⢸⡅⠀⠀⡄⠀⠀⠁⢂⢀⠏⠀⢀⣀⠤⢀⠒⢉⠁⠀⢸⠂⠀⠁⠀⠀⠀⢻⡉⠟⠤⡀⠀⠃⠀⠰⠪⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢈
⠀⠀⠀⠘⡅⢣⠀⣡⡀⠉⠀⢸⠀⠤⠐⠋⠀⠀⠳⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⢸⠁⠀⣥⠟⢁⠅⠲⠚⢁⠁⠂⠉⠀⠀⠀⠀⡺⠒⠀⠀⠀⡀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠁⠁⠀⠐⠸⠯⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⢸
⠀⠀⠀⢀⡂⢀⣧⠐⡠⠐⠀⠘⠱⠀⠂⠀⠴⣌⣐⢊⠀⠠⠀⠂⠉⠀⠀⠐⠒⠄⠀⠀⠀⠀⠀⠈⡄⠂⠄⡀⠂⡀⠀⠀⠀⠄⠀⠀⠈⠉⠁⠀⢸
⠀⠀⠀⠸⣧⠦⡰⢤⢷⠀⠐⢐⡀⠀⠠⠐⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢰⠂⡐⠓⢧⣠⠀⡀⠂⠀⠀⠀⠊⢌⠀⢸
⠀⠀⠀⠰⡄⢅⣴⠋⢉⣱⣰⣻⠉⠦⡁⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠀⠁⣠⠐⠀⠀⠀⠀⠆⠒⠘⡇⠃⠀⣹⡢⠠⣐⠀⢄⡂⠀⠘⠒⠀⠀⠀⢸
⠀⠀⠀⠀⣗⣼⠑⡐⠂⠀⠍⠑⡀⠂⠡⡀⡠⠤⠒⠂⠒⠛⠚⠊⠉⠂⠀⠉⠀⠀⠈⢁⡀⠂⠀⠼⠁⠘⢹⣄⠁⠈⠀⠈⠂⠈⠐⠠⢐⠠⠀⠀⠈
⠀⠀⠀⠀⢛⣯⠊⠀⠀⢬⠀⡀⠹⡀⢂⠡⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡀⠐⠈⠀⠀⠀⠀⢀⠀⠀⢿⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐
⠀⠀⠀⠀⣾⡹⠀⠀⢀⠂⡳⠒⡁⢀⠀⠀⠷⠗⠀⠀⠀⠀⠠⠈⣀⡀⠀⠑⠆⠀⠈⡁⠀⠀⠀⠀⠉⠁⠛⢤⣁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⡯⣧⠀⠀⠉⠟⢁⠤⠠⠉⣃⢠⢭⠀⠀⡄⠀⣐⡠⠉⠁⠈⢂⠖⠻⠇⢀⡈⠀⠀⠀⠀⠐⠒⠂⠜⠁⠂⡌⠀⠀⠀⠀⠃⠐⠀⠀⠀⢨
⠀⠀⠀⢀⡹⣁⡈⠑⠶⢀⠀⠠⠄⠀⠂⠀⠈⠃⠀⠘⣛⣁⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠒⠈⢀⡠⠄⠀⠜⠩⠀⠁⠀⠀⠔⡩⠀⠀⢸
⠀⠀⠀⠰⠇⠀⢠⠃⠈⡅⠀⡂⠀⠘⡩⢠⠃⢻⢄⠄⠀⠈⠊⠀⠀⠀⠠⠤⠐⢄⠀⠀⠀⢰⠤⠀⠘⠉⠉⠀⠂⠁⠀⠀⣷⠖⠀⢀⠙⠁⠀⠀⢸
⠀⠀⠀⠰⠂⡜⠸⠀⠀⠀⠀⠀⠀⠀⠠⠈⠆⠈⠀⠀⠀⠀⠀⡤⠀⠀⢀⠀⠀⠠⡉⠂⠀⠏⠀⠀⠀⠀⠀⢠⠀⠠⠄⠈⠏⠀⡠⠂⠀⠄⠀⠀⠰
⠀⠀⠀⢠⡋⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠠⠤⠴⢥⡈⡦⡎⠀⠀⢀⠀⠁⠘⠆⠀⠀⠀⠀⠀⠆⡀⠀⢁⡴⣀⠀⠀⠀⡌⠒⠤⠤⠐
⠀⠀⠀⠘⢁⠸⢀⠀⠀⠘⡀⠰⠀⠐⠀⢠⡀⠀⣀⡡⡄⠀⠈⣆⡙⠣⡐⢲⠖⠴⠆⡀⠠⠤⠤⠀⡄⠀⠀⠉⢈⠀⢰⠀⠀⠑⠦⠂⠀⠄⠀⠀⠀⡇
⠀⠀⠀⠀⡄⠠⠄⠀⠐⠄⢰⠀⠀⠨⠰⠀⠁⠀⠰⠕⡄⠐⠀⢘⠑⡠⢬⠲⢄⠑⢄⡀⠀⠀⠀⠀⠈⠒⠒⠐⠠⠀⠀⠀⠀⠀⠒⠀⢀⠀⠁⠀⠀⡅
⠀⠀⠀⠀⠂⠀⠂⠀⠀⣀⣐⠀⠀⢀⣐⣀⠈⠀⠀⠈⠸⢦⡠⠰⠶⠀⠀⠑⢄⡙⢦⣈⠂⠀⣀⡀⠆⠰⠀⡈⠀⠀⠀⠆⠀⠀⠀⡀⣀⠁
⠀⠀⠀⠀⠗⣀⠂⠒⠁⠸⠟⠁⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠲⠄⣈⠒⠊⠁⢁⡀⠤⠄⠠⠄⠀⠀⠀⠠⠠⠄⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢆⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠙⢿⣦⡿
⠀⠀⠀⠀⠀⠐⠂⠀⠀⣴⡏⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠠⣷⡄⢠⣿⣯⣤⡀⠉
⠀⠀⠀⠲⠔⠋⠉⠉⣾⣿⣧⣠⣴⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠁⠀⠀⠀⠀⢠⣷⠀⠀⠀⠀⠀⠸⣾⡀⠀⠀⠀⠀⠀⣠⣸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢦⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⢀⣼⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡈⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠸⣷⣤⠖⣱⡿⠋⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⢐⠀⠈⢂⡀⠀⢹⡇⠀⠀⠀⠀⠀⠀⣸⣿⣇⣾⣯⢀⣠⠆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣯⠈⢳⣄⢺⣧⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⢃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⣿⣥⡈⠙⠯⠙⠹⢿⠛⣿⠟⣻⣿⣿⠋⠥⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠛⠀⢀⣀⠱⣾⣾⣿⣿⣩⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⢹⣿⣿⣿⣟⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣜⠼⢿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⣴⣿⣿⣶⣿⣿⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⠈⠂⠹⠿⣿⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⢿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠚⠳⠟⣩⣆⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣄⣀⣀⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠒⢿⠛⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⡀⠀⠘⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⠿⣟⢫⠟⠋⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣀⠱⣤⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢠⠀⠀⠈⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠊⠁⠀⠀⣂⠀⠀⠁⢘⡻⠿⠟⢿⣿⣟⠍⠉⢿⠿⠋⠀⠀⠀⠀⠸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠈⠵⠒⣩⣿⣯⣱⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡿⠻⣿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉

⢸⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⢶⣿⠛⠉⢸⡗⠓⠀⣿
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡤⠄⠒⢲⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣶⡾⠿⠛⣿⡇⠀⢈⣿⢄⠀⢸⡧⣭⠀⠻
⠀⡇⢀⡄⠀⠀⠀⢀⠀⠀⣀⡀⢠⢴⡲⠊⠀⠀⠀⠀⢂⠁⠀⠀⠀⠓⢲⡖⠒⠒⠒⣞⠟⠉⠁⢹⣧⣤⢶⡾⠉⠀⠀⠀⣀⠱⠘⣿⠁⣠⣀
⣤⡇⣜⣁⠀⠄⠅⠀⢿⠠⡏⠄⢘⢸⣧⣶⣶⡆⠀⠀⠿⠷⠀⠀⠀⠐⠀⣷⠀⠀⠀⢻⠲⣄⣀⣸⡗⢿⠏⣀⠀⣛⠂⠀⠄⢷⡄⡇⢺⡏⡍
⠁⠙⣿⣷⠀⠀⠀⠀⠌⣆⣀⠁⠘⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡖⢀⢻⡴⠖⠚⠛⣯⠟⠏⠈⠁⡾⢀⣿⠀⠀⠀⠀⠐⢠⣄⢸⣿⠀⣄
⠃⠒⣿⣾⣄⢙⡆⡿⣳⢿⡀⠀⣾⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠳⣦⠈⡷⣆⠀⠀⠀⡈⠀⠀⠀⢻⣸⣿⠀⠃⠒⠦⠀⣸⣿⡾⢏⣀⠤
⠚⠃⠀⠜⢌⣉⣽⡽⠿⠻⣠⢾⠘⠈⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⣷⠀⠀⠁⢰⠁⠀⠁⠀⠁⣿⢿⠃⣄⣀⣀⢴⣿⡗⠉⢈⣁⢰
⠀⠀⠀⠀⢀⠀⢛⠀⠀⠀⡈⢾⢨⠶⣿⠉⢹⡇⠶⢤⣄⣀⠀⢀⣰⠀⠂⠀⠩⠐⠃⠒⢰⡇⢗⣠⠦⣅⠀⣠⡷⡔⣾⠕⠋⣡⣿⣿⠟⠀⠐
⠀⢀⣦⢠⠁⠊⠭⠀⠀⡀⡁⢸⣿⠀⣿⠀⠈⠁⠀⠈⡁⢠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠘⠉⣠⡏⠁⡴⣿⡟⠁⠀⠀⡏⡞⣼⡟⡿⠁⠀⠀⠈
⠀⢼⣿⢿⡇⢠⠀⠀⣘⡃⡿⠻⠛⠀⣿⠀⠀⠀⠀⠀⠇⠸⠀⠀⠀⠀⠀⣤⠾⡷⠶⣆⠜⢁⣐⣾⠷⢟⠠⠆⠀⡀⡇⣧⣿⠀⠀⢀⡀
⠉⠁⢻⠀⣱⢊⣤⡔⠋⠹⣷⠆⠀⠀⢿⠃⠂⠐⠂⠀⠁⠀⠙⠐⠖⠶⠾⣧⡀⢰⡞⠉⠙⣍⠃⠀⠀⠀⠡⣴⡞⡯⠦⣯⡿⣴⡿⢟⣻⠿
⡀⠠⢈⣺⣿⣽⣤⠤⠀⢀⡿⣿⣤⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠛⢻⠆⠀⠀⠁⠀⠀⠀⠀⠠⠼⢃⡅⣷⡸⡇⠀⢠⡤⠄⣄
⠤⢖⡗⣉⣿⣮⡉⠗⠀⠈⡏⢺⣿⠧⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠔⠹⢄⣀⡀⠀⣀⠤⠶⣏⠀⠀⢜⢧⡹⡇⠁⠀⠈⣀⣤⠀⠈
⠀⠀⡇⡴⠋⠀⠀⠀⠀⠀⠁⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⡇⢀⣰⣇⠀⣤⣄⠓⠱⣤⠈⠀⠀⠀⠉⠀⡀
⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠈⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⡗⠉⠀⠀⠀⡀⠈⠹⢦⡈⣎⠈⠀⠀⠀⠈⠉
⠀⠀⢇⣣⡖⢀⠤⠤⣤⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠘⠐⠵⠋⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣾⣆⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢶⣼⣿⣽⣧⣶⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣿⣾⠆⠀⠐⣞⣿⡏⢹⣿⣣⠆⠀⠰⣿⣿⠂⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢀⡀⠀⠀⢲⣶⣿⣶⠇⠀⣨⣿⣿⣿⣿⣇⠀⠘⣷⣿⣦⡖⠀⠀⢀⣀⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣇⠀⠀⠀⣼⣿⣿⣶⣾⣡⣿⣧⣼⣷⣌⣿⣶⣿⣿⣿⡀⠀⠀⣸⣿⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⣖⣺⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣟⣒⣮⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣩⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⣾⣿⣿⡿⣿⣿⣋⣿⣿⢿⡽⢿⣿⣯⣻⣿⡟⣿⣿⣷⣼⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣻⣽⣿⠛⠉⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢻⢿⣏⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣹⡿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⣙⣵⣶⣾⣿⣷⣷⡔⣰⣾⣶⣾⣿⣶⣮⣿⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⢿⡿⣿⣿⣿⡿⣳⣼⠻⣿⣿⣿⠿⣿⢹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢡⡊⢓⡉⣉⡁⢰⣿⣿⣧⢀⣉⠩⡓⢃⣦⢑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣶⣤⣶⣶⣹⠇⢿⡿⢿⠿⠹⢟⣴⣶⣤⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏⣇⣀⣀⡀⡀⣀⣀⣬⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣶⡦⢤⠥⠤⢶⣿⣯⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⠁⠉⠛⠛⠛⠛⠛⠛⠋⠀⢈⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⢀⣈⣀⣀⣠⣀⢠⠞⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⢀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣦⣽⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⡸⣹⡿⠛⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⡠⣜⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠻⣾⣷⠏⠀⠀⠀⣿⠀⠀⢰⠇⠀⠀⠀⠀⠀⢀⣴⣿⡿⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣬⣿⣷⡄⠀⠀⢸⣧⣠⣾⣤⣤⣤⣴⣶⠾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢮⣭⡙⢻⢿⣶⣤⡾⣿⣿⠛⣫⣭⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣥⣞⣿⡿⠁⠘⠿⡿⢧⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠿⠟⠛⠋⠜⠛⣛⣦⣾⡋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⢷⠔⣛⠉⣁⠀⠈⠉⠈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠘⠦⠬⣿⣿⣾⣿⣾⣿⣴⣤⣿⣧⣀⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣇⠀⠉⠲⡂⢤⡤⠄⠀⢀⣀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠀⢠⠀⠀⢠⣾⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠐⠄⢰⡀⠀⠀⢠⣷⠈⣤⣤⣾⡾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣇⠀⠀⢰⡄⠄⠀⠈⠻⠤⣿⣿⣿⣿⡻⠂
⠀⠀⠀⠀⡄⠣⣴⢀⣀⢠⡀⢠⣷⡄⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠀⠂⠸⡇⠀⠀⠀⠀⠀⠀⣻⣿⣿⡅
⠀⠀⠀⣌⣶⣿⣿⣿⣏⠉⡤⣿⣿⣷⡼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⢀⡄⢰⠈⢀⠀⠀⠀⠀⠁⠈⣿⣿⡷⠂
⠀⠀⠀⡿⣿⣿⣿⣿⣿⣖⣿⣿⠿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢀⡤⠀⣿⠀⠀⠀⠀⢸⣆⣿⡿⠇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣏⣿⣿⣷⣶⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣥⢀⣧⡴⢂⢀⣤⣸⣿⣟
⠀⠀⠀⠀⠀⣈⣽⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠛⣻⡟⢙⣿⣿⣿⣿⣿⣿⡷⢠⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣿⡿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠁⠀⠰⠂⠈⠛⠻⡿⠯⠁⠚⡿⢿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠁⠈⠁⠀⠙⠛⠛⠛⠛⠁⠁⠀⠀⠀⠈⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⣴⢫⠻⣄⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡆⠀⢀⣯⣝⣒⣋⣵⡾⢷⣬⣓⣚⣉⢧⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡿⠟⠉⣉⣰⠤⠤⠤⠤⣀⣉⠙⠻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣷⣯⣥⣶⣶⣶⣶⣶⣶⣬⣙⣾⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏⠉⠛⠿⣿⣝⠃⠨⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠉⠛⢷⣿⣿⣿⣿⣗⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⢴⣿⣿⣇⣉⣙⡒⠀⠀⠀⠐⣚⣩⣹⣿⣿⣿⡆⣢⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⡇⠑⠈⠁⠀⠀⠀⠀⠈⠒⠁⢸⢡⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡘⣿⣿⣇⠀⠀⣀⣀⣀⡀⠀⠀⢠⣿⣿⣧⣻⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠒⣺⠇⣿⢻⣿⣄⠀⠙⠲⠖⠋⠀⣠⣾⣿⣿⣏⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣰⢞⣭⡆⠀⣼⣿⣿⣿⣦⣤⣤⣴⡾⠟⣿⣿⣿⣯⢿⣎⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡿⣡⣾⣿⣿⣿⣿⡟⠿⠿⠋⠀⠀⣿⣿⣿⣿⣿⣷⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣻⣿⡿⠇⠀⠀⠀⠀⠀⢹⣿⡿⠻⢿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣇⣈⡿⠯⠄⠀⠀⠀⠀⢀⣠⣤⣿⣿⣦⠀⣵⢻⡄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⢿⡳⣄⡀⠀⠀⠁⠀⢀⣠⠴⣫⣿⣿⣿⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠂⠀⠙⠲⠭⣟⣓⣒⣻⡭⠗⠋⠀⣸⠟⠉⠈

⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠩⡜⠻⠆⠒⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢌⠃⠳⠶⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⢀⣤⣶⣕⠲⡈⢳⡄⠀⠀⠀⢀⣠⠶⢾⡛⡻⣿⠟⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⣏⡬⣯⣇⠇⢻⡔⠹⡒⡇⠀⠀⣰⢋⡤⣸⣾⠷⣣⣰⠿⣇⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣐⣨⢟⣢⠽⢿⠿⢿⣳⢁⣟⣿⣿⣃⣘⡷⣿⢿⠀⢙⣯⣢⣯⡊⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣙⠦⠶⣾⡤⡄⣛⡿⢉⣿⡷⣁⣽⡇⠋⠨⠌⡟⡋⣿⣿⣿⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠉⣹⣾⣭⣞⣯⠹⢟⠊⠞⢮⡏⣛⠛⣒⣒⣺⣷⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠋⠉⡍⣉⠉⣩⣿⣿⠿⣷⣉⢍⠢⡈⢆⢧⢨⠝⡛⠫⡋⣽⢻⢈⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠁⠀⠀⠀⢂⣠⣿⣿⠟⠁⠄⣿⣈⠣⡆⢁⠸⣎⠧⢤⠬⠒⢩⣭⣭⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣿⠔⣀⠂⠔⣤⣼⣿⠟⢡⠄⡒⢠⠆⠨⠀⠟⢆⢤⠿⡍⢠⣼⠒⣁⣥⣸⣘⡼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢿⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⠃⡴⠠⠘⢊⣽⡿⠋⠈⠊⠀⠀⠈⠁⢀⠀⢆⠸⠘⢩⡐⢻⡄⠀⠉⠀⣠⣤⣯⣛⣍⡉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡐⣄⠀⠛⠃⢀⣴⡿⠋⢀⡀⠈⡀⡠⠀⡠⠀⠀⠀⠸⡤⢳⣣⡩⠲⣧⠀⢀⣴⣿⡻⠿⠿⢉⠉⢸⠁⠀⠀⠀⠀⣀⣀⠀⠀⠀⢸
⢸⡾⠟⠲⠶⣄⠀⠔⢁⡿⢀⡔⢸⣿⣏⣳⣭⣭⣉⡛⠛⠚⠚⠳⠶⠷⠚⢓⣛⣊⣉⣳⣧⠘⠇⠉⣉⣠⣴⣶⣦⠀⡇⣎⣀⣤⣴⣶⡿⡟⠛⠳⣄⠀⢸
⠈⡟⠂⠀⠉⠀⡟⠀⣺⠁⡾⡇⢵⡟⠛⠓⠂⠁⠈⠩⠛⠿⣿⣿⣿⣿⠛⢻⠀⠛⠿⢭⡻⣆⣰⣿⣿⢯⢅⣼⠇⡜⣠⣿⡿⢿⢿⠟⢂⣠⣦⡰⠾⣆⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣬⣷⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡔⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠟⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣾⣿⠟⢈⣿⣿⣾⣿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⠀⠹⣿⣶⣿⣿⣿⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣤⣶⣿⠀⠀⣠⣿⣿⣿⣴⡿⣹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣁⣰⣿⣿⣿⡆⢸⣿⣻⣿⣯⣿⠁⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣿⣿⣿⣿⣿⡿⢋⣠⣾⣿⣿⣿⣿⣿⣷⢀⣹⣿⣿⡿⣣⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⣿⣿⡛⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣴⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣿⡿⢋⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⡟⠘⣫⣿⣿⣿⡿⠿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣿⣿⣇⠸⣿⣿⣏⡻⣿⣷⣶⣄⡉⢉⣉⣽⣼⡟⣿⣿⣏⠳⣀⠟⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⡌⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠊⠝⢍⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢐⢀⡫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡓⠼⢚⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠂⠉⠉⠛⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠯⠭⠭⠭⠭⠭⠭⠭⠭⠭⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠿⢸⠸⠇⡇⠹⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠁⠀⢠
⠀⠀⠀⠀⠀⡠⠔⠒⠸⠤⠤⣀⡀⠀⠀⠀⠀⢀⢀⠤⠤⠗⠒⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠜⠀⠀⠀⠀⡠⠚⠁⠀⠉⠉⠉⠁⠈⠀⠑⢄⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡠⠂⢀⠜⠁
⠀⠀⠀⠀⠑⡤⠤⠤⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠤⢤⠊⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⠜⠁⡴⠁
⠀⠀⠀⠀⢀⠃⢀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⡄⠘⡄⠀⠀⠀⠀⠀⢀⠜⢁⡴⠃⣠⠊
⠀⠀⠀⢀⣜⣒⠚⣄⣁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⡷⢒⣳⣄⠀⠀⠀⡴⠁⡠⠊⢀⠜⠁
⣿⠛⠛⠛⠛⢻⠋⠉⠉⠉⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢳⠨⠽⠋⠀⡠⠊⠀⠀⠀⡰⠃
⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⣀⣀⣀⡀⡇⠈⠀⢰⢤⠜⠀⠀⠀⣠⠊
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢨⣿⡅⠀⠀⠀⠀⠀⡃⡗⠒⢺⢶⣕⢄⢀⠔⠁
⣿⣀⣀⣀⡀⠠⡧⠤⠤⠤⠄⣿⠤⠤⣿⠿⡿⠤⠤⠤⢤⠴⠁⢣⠔⣋⢶⣛⡟⠛⠂
⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⡇⠀⠃⠀⠀⠀⢸⠀⠀⠣⠬⠼⠒⠁
⠈⣿⡀⠀⠀⠀⠃⠀⠀⠀⢸⡿⠤⣀⡇⠀⢠⠠⠤⢄⡸
⠀⠘⣷⡀⠀⠀⢠⠀⠀⢠⣿⠛⠛⢂⠇⠀⢇⡚⠛⢓⡝
⠀⠀⠘⢿⡢⢀⣸⣀⢔⡿⢫⠉⠉⢸⠀⠀⢸⠈⠉⢸
⠀⠀⠀⠀⠉⠳⠯⠖⢋⡠⠚⠁⠈⠁⡧⠀⡇⠉⠀⠉⠢⣀
⠀⠀⠀⠀⠀⢠⡔⠊⠥⠤⠐⠒⠂⠉⠀⠀⠈⠁⠐⠒⠀⠤⠭⠲⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⣰⡤⠀⠀⣤⣖⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⡤⠀⢈⠛⣯⠃⣝⣶⣽⣈⣡⡶⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣽⣵⣠⣳⣶⣶⣶⣷⣿⣿⣿⣿⣿⣷⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠈⢻⣯⣿⠟⠟⣽⣿⣿⣿⣿⡿⢿⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⢊⣥⡾⣷⣿⣿⣿⣿⣶⣶⣿⣾⣿⣿⣿⣷⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢉⡎⠉⣠⣯⣻⣿⣿⣿⣿⡿⣿⣿⣿⣏⠙⠉⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠙⠡⣽⡿⣯⣽⡏⢻⣻⠞⠉⠉⠀⠀⠀⢀⣆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠙⠀⠀⠘⢈⠀⣷⠊⠀⠀⠀⠐⠀⠀⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠁⣀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⠉⠈⠉⠻⠛⠿⠷⠴⠇⠾⠴⣾⣷⣀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡖⢶⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⣘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣤⣤⣀⠀⠀⠀⠀⢀⣴⠛⣳⣄⣤⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣞⠉⠻⢿⣿⣿⣮⣿⠻⠯⢻⣆⠀⡰⠻⣿⡟⠋⠛⡝⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⣀⣈⣷⣀⣀⠸⣿⡜⠁⠈⣿⣶⠄⠉⠻⢡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠈⠻⢏⣹⡏⠉⢀⡿⠀⠀⠀⣿⠧⠖⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡁⠄⠀⠘⡏⠣⠀⣺⡇⠀⠀⢠⣿⡀⣀⣠⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠉⠁⠀⡠⠖⢚⠉⠀⠈⢿⡇⠀⠀⠸⣿⣿⡿⠛⠉⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣾⡿⢿⣿⣿⡿⠇⣀⠙⢆⠀⠀⢻⣿⣿⠷⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣅⣿⡷⠊⣼⠋⣠⣿⡿⠠⠀⠑⠠⣴⣿⣿⣤⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣻⡿⠁⣠⢆⡼⣹⡧⠀⠀⠀⠒⣤⠚⢉⡿⣏⢾⣴⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⣴⣿⢣⣟⣽⡇⣰⢟⣿⣇⠝⠀⠠⠔⣲⣯⡀⠀⢰⣿⣿⢤⠹⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣿⠈⠰⢾⡇⣼⣏⣿⢀⣽⠋⣰⡏⠀⠀⣴⡖⠎⠀⠀⢤⡿⠃⠈⢿⡆⠑⢶⣜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡈⣡⠛⢰⣿⢻⣯⣜⠃⢀⡏⠇⢠⡾⠋⡀⠠⣠⡶⠛⠁⠀⠀⠀⠙⠳⣶⣤⣭⣛⣴⡶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠿⡏⠀⣼⣟⣿⣿⠋⠂⠈⣦⠄⣼⡶⠾⡷⣺⡟⠁⠀⠀⠀⠀⠀⢠⠖⠈⠩⡀⠀⣿⣿⣦⡛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣅⡈⠁⣚⣿⢻⣿⡿⠐⢄⡄⠱⢏⡿⢰⡿⠛⢁⠌⣠⡴⠆⠀⠀⠀⣠⠊⠔⢳⠍⢰⣿⡯⡿⠙⣴⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⠛⣿⣿⣻⡟⠀⠢⠀⡀⠀⣼⡷⢻⠁⠀⢁⠩⣠⣿⡆⠀⢠⡀⡁⠀⣰⡏⠀⣾⣿⣟⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⢈⢫⠀⡀⣿⡟⣿⡇⠀⠀⢠⣷⠔⠹⡀⢿⣎⢀⣠⡾⠋⢀⣠⣄⠀⢿⣿⣿⡿⠃⠀⣿⡹⡏⠀⣶⠿⢄
⠀⣀⡀⠀⣀⠤⠴⠿⠁⡀⡱⣿⣿⣿⡇⠀⠀⢸⣿⠃⢀⡇⠈⢻⣿⣿⣿⣿⣿⣿⡟⠀⠈⢿⡿⠀⠀⣰⠿⣧⠁⠀⠉⠀⠘
⠰⢿⡄⠋⠀⠀⠀⠀⢚⡽⣱⠿⣿⣿⡇⠀⢀⢻⣷⣶⣾⣿⣶⣿⢟⣹⣿⣿⣿⣿⣅⡀⠀⢤⠀⢀⣞⠿⠚⠛⠃⣀⠃⢀⣼
⠀⠀⢡⡀⢀⡄⠀⢲⣿⠿⠃⢠⣿⣿⡧⠀⠘⡚⠛⣩⣿⣿⡿⠿⣫⣿⣿⣿⣿⣿⠏⠁⠀⠀⠰⠿⠁⠀⠀⠀⠤⠿⠦⠜⠁
⠀⠀⠈⣿⣍⣤⣞⡃⢀⣀⡴⣿⣿⣿⠀⢀⡔⠀⣹⣿⠈⠿⠓⠈⣥⣤⠴⣿⢿⣿⣄⠀⠉
⠀⠀⠀⠀⠀⠉⠉⠉⠙⠬⣾⣛⣿⣿⣿⣿⣷⣾⡅⠘⡆⠀⠀⠀⠀⠉⠀⠁⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠙⠷⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⢥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣸⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢤⠀⠈⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠉⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠠⠔⢖⠒⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢻⢄⠀⠀⠑⢁⡀⢢⠀⠀⠀⠀⠑⠆⡠⠠⣶⡦⠀⠀⠀⠠⠀⠠⢻⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡌⢢⡀⠘⠁⠀⠀⠆⠀⠀⠉⠈⠙⣄⡘⠁⠐⠃⢠⠐⠀⠀⠀⢹⣧⠀⠀⠀⠀⠠⣢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡜⠄⠑⡄⠀⠂⠀⠀⡄⠀⠠⠤⠓⣂⣕⠀⡀⠐⡡⠂⠀⠀⠀⠀⢩⢃⠀⠀⠀⠀⠑⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⠀⠞⠂⠂⢀⡈⠳⡀⠂⠀⠀⠤⣼⣿⣥⣯⢤⣀⡀⣀⠀⠀⠀⢆⢣⠀⠀⢀⣠⠘⣐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢉⠄⠐⠂⣼⠀⢆⣨⡔⠿⠓⣀⣷⣁⣈⣀⣉⣈⣀⣈⣈⣆⣩⡉⠁⠁⠀⣽⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠠⠀⠂⠈⠈⡀⠡⡘⠓⠋⠉⠉⠉⠛⠃⡉⠉⠩⢭⠉⠉⠩⠟⣿⠆⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⠀⠚⠙⡅⠀⠀⠀⠈⠀⢀⢤⠀⡀⠀⠉⠧⠴⠶⠶⠂⠀⠀⢰⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠀⠄⠂⣀⡇⠀⠀⠀⠀⠀⡆⠠⠰⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⡞⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⢸⡿⡝⠀⠀⠀⠀⠀⠀⢡⠀⡷⠀⢤⡖⢠⠀⠀⠀⠑⠀⡰⠁⠀⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠋⡘⣞⡄⠁⠀⠋⡃⢇⣠⡀⢀⠠⡇⣀⢸⠀⢤⠀⠸⠣⡌⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡃⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠀⢡⡄⠀⡆⠁⢸⡿⠛⢻⡀⠄⡽⢙⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⡞⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣖⠠⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⡀⠀⠃⠀⠃⠀⠀⠀⠀⣾⠉⠃⠀⠸⡀⠀⠆⠀⠀⠀⠀⠑⠄⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀⠀⠀⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠀⠘⠄⠀⣰⣏⠀⢠⠀⠀⠘⢄⠰⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⢀⡤⢠⢀⣿⣿⣄⢀⡀⢀⠀⠘⠀⠀⠐⣤⣾⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄⢁⠀⠀⠀⠐⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣴⣿⣧⡈⣾⣿⣿⣿⡏⡅⡿⠀⢰⠗⠠⣶⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢠⠃⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣵⣿⢿⠉⣱⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⡟⠁⠀⢠⣿⣿⡟⠁⢻⣿⣿⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠠⠀⡀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠃⠀⢠⣿⡿⠋⠀⠀⠈⣿⣿⡀⠀⠀⠀⣾⣿⡁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⣰⣿⡿⠁⠀⠀⠀⠀⢘⣿⣇⠀⠀⠀⠛⠿⠿⣷⣄⠀⠀⠀⠂⠀⠔⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⠀⠀⠀⣀⣴⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⢸⣿⡄⢻⡆⠀⠀⠀⠀⢰⣿⠏⣸⣿⠃⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠘⣿⣷⠈⣿⠀⠀⠀⠀⣿⡿⢠⣿⡿⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⢹⣿⡆⣿⡇⠀⠀⢰⣿⠇⣼⣿⠇⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⣿⣇⢸⣷⡀⢀⣼⣿⢠⣿⣿⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⣟⠛⠻⢿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⢻⣿⣀⠀⣠⣀⣀⡀⢀⣿⡋⠁⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣾⡿⠛⠀⣸⣿⣿⡿⠿⠛⠛⠻⢿⣿⣷⣄⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣾⣿⠥⠤⣴⠏⠉⠙⣧⠶⠲⠶⢤⠀⠀⠌⠻⣆⣒⣻⣧⣀⣀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⠀⠀⠀⠂⢀⠉⠄⠐⣿⣿⣿⣿⠃⠀⠀⠀⣽⣷⣦⣶⡏⠀⠀⠀⠉⢹⣬⣭⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠙⣿⣿⣿⠀⠀⠀⠀⢾⣿⣿⣿⣆⠀⠀⠀⠀⣸⣿⡿⠧⢈⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⡦⢴⠶⠾⣿⣿⣿⠿⠶⢶⢤⣺⣿⣿⣷⣤⣀⠈

⠀⠀⠀⠀⠀⢸⣿⠀⣼⣁⣴⠞⢁⡆⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢹⡀⣿⠀⠀⣷⣿⠀⡄
⠀⠀⠀⠀⠀⠀⣿⣷⣿⠟⠁⠀⡟⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡎⣷⣿⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣧⡀⠀⣼⠃⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣷⢸⣿⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣸⣾⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣶⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣀⣤⣀⣽⡿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣟⣿⣿⣿⣦⣀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣟⣫⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣠⣄⣸⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⢟⣿⠿⣷⢀⣤⣤⣤⠘⠿⢿⣯⣾⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣾⣿⡀⠀⠘⠛⠛⠋⠁⢀⣹⣿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠼⢻⣿⣿⣆⠀⠀⠀⠀⣠⣾⣿⠏
⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⢈⣷⣼⣿⣷⠚⠛⢾⣿⣯⣿⠀⠀⠀⠀⠶
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⣿⡿⣿⠀⠀⢸⢿⣿⣿
⢸⡇⠀⠀⠀⣠⡄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡷⢡⣤⣤⣤⢾⣿⣿⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢸⡇⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣿⣷⣿⣿⣿⣿⣼⣿⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠘⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⡿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠏⡈⠁⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡖⣾⣹⠃⠀⠀⠀⠀⢀⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⠀⠀⢀⡴⠋⡒⡁⣉⡒⠂⠄⡒⠚⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠓⠒⣾⣥⠜⠣⠚⣓⣘⣛⡛⠛⠏⠴⢋⣬⣧⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣯⡾⠒⡺⠕⠒⣉⣁⣤⣤⣤⣤⠍⣉⣐⠂⠭⡙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⡫⣐⣬⣷⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣶⡪⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣾⣿⡿⠛⢁⣶⣿⡿⢋⣾⣿⣝⢿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠟⡋⠀⠀⠁⣡⢟⣴⣿⡏⠹⣽⣦⡙⠿⣿⣟⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣤⡾⢀⣴⣶⣿⣷⣿⣿⡟⠀⠀⠈⠻⣿⣶⣜⣮⣷⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣿⣿⣷⣿⣿⣿⣿⢟⡵⠋⣀⣠⣤⠶⠦⠌⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠻⠿⡄⠀⠘⠋⡡⢴⣶⣶⡶⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡝⠿⠗⠁⠀⠀⠀⠁⠈⠋⠉⠀⠀⢸⣿⡛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢴⣿⣿⣿⣿⣧⡀⠀⠈⠛⠢⠖⠘⠀⠀⠀⠀⠀⠉⣽⢀⣿⠟⣿⢸⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣴⣯⣿⣿⣿⣿⣿⣷⡀⢠⣴⣶⠶⠶⣤⡤⠀⠀⠀⣠⣿⣽⠏⣶⠇⣼⣿⣎⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡙⠷⠶⠖⠁⠀⢀⣤⣾⣿⣿⠃⣸⣯⣶⣿⣿⣿⢳⡹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣠⣴⣿⣿⣿⣿⣧⢠⣿⣿⣿⣿⣿⠇⣸⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠹⣿⣿⡝⣿⢻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⢹⣿⣿⣿⣿⣿⡙⠷⠃⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣴⠀⣿⣿⣧⡡⢁⡿⢱⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠸⣿⣿⣷⡞⡕⣰⣿⣿⣿⣿⣿⣿⣯⠀⠀⠀⠀⢀⣠⣽⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⡻⣿⣼⣇⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⠴⠚⢋⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⠃⠸⠟⠁⠀⣠⣴⣿⣿⣿⣿⠋⢩⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡀⠀⢀⣤⣾⣿⠿⠟⠋⠁⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⠀⣀⣠⣤⣤⣤⠄⢹⠿⢻⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⠃⣔⡒⡷⣆⣹⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⣿⡟⣿⣿⠄⠙⣁⣱⣿⣇⣿⢿⣷⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⡴⠀⠀⣼⣿⣯⣿⣿⣿⣷⣹⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣱⣿⣿⣿⣿⡗⠖⠂⢸⣟⣿⣿⣷⣯⣽⣯⡻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢠⣿⢹⣿⣿⣿⣿⢰⣷⠸⣿⡏⠴⡌⣿⣻⡿⣇⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⠸⣿⠘⣿⣻⣿⡏⣾⣿⣧⡙⢿⣿⣵⡿⣿⣿⣮⢻⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢿⢠⡘⢿⣿⣻⣿⣾⣿⣿⣧⠀⠀⠛⠿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣾⢆⠀⣯⣬⡻⣿⠶⠿⣿⣷⡄⠀⠀⠀⠈⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣌⡉⠙⢦⡘⢿⣷⡹⣸⡇⣬⣙⣿⣆⠀⠀⠀⠘⠿⠿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣶⣿⣦⣙⢿⣶⡁⠻⣌⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠻⢶⣶⣮⣭⣷⣯⡙⠻⣷⣌⠑⢝⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⣶⣄⠙⢷⡀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⢳⡀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠛⢻⡟⠛⠛⠛⢶⡶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⡏⠀⠀⣠⡴⠟⠁⠀⠀⠀⢸⡇⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⢸⡇⠀⠘⠃⠀⠀⠀⢠⡶⠛⠛⠃⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢁⣴⠞⠋⠀⠀⠀⠀⠀⠀⢠⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠳⣤⡙⢳⡄⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⡀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠃⢸⡧⠤⠼⠿⠦⠁⠀⠀⣾⠛⠷⠶⠞⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠙⣧⠀⠀⠀⠀⠀⣼⠃⠀⣠⡾⢷⣄⠀⠀⠀⠠⣟⠛⠛⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⣩⡴⠟⠁⠀⠀⠈⠻⢦⣀⠀⠙⢳⣦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠳⣄⠰⠶⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠷⠶⠆⠁⢀⡀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⣿⠀⠰⢶⡶⣶⠶⢶⡶⢶⡶⠶⣶⢶⡶⠆⠀⠀⣸⠇⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢰⡏⠀⠀⢸⡇⣿⠀⢸⡇⢸⡇⠀⣿⢸⡇⠀⠀⠘⠉⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠃⠀⠀⢸⡇⣿⠀⢸⡇⢸⡇⠀⣿⢸⡇⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠆⢀⣸⣇⣿⣀⣸⣇⣸⣇⣀⣿⣸⣧⡀⢰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⡄




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠿⠿⠿⠿⠿⠦⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⡀
⠀⠀⠀⠀⠀⣠⠔⠊⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠤⠄⠀⠂⠑⢄⠙⠀⠀⠋⠀⠐
⠀⠀⠀⢀⠴⠧⢄⡀⠀⠀⠀⣼⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⢀⠠⠀⠀⠀⠀⢀⠀⠘⡄⠈⠢⡀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⣄
⠀⠀⠐⠁⠀⠀⢀⣠⠴⠋⠓⢦⣐⣀⠀⠻⡒⢄⠀⢀⡠⠊⠀⠀⢰
⠀⡶⢒⠒⠒⠋⠉⠀⠀⢀⠀⠀⠈⠉⠛⠒⢛⡲⡎⠁⠀⠀⠀⠀⣠⢔⡂⢄⠀⠀⠀⠀⠀⠀⢤⡀
⠀⡇⠉⠀⢠⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⠀⠈⠁⣇⠀⠀⠀⠁⠀⢬⠉⠉⡥⡀⠊⠀⠀⡀⠀⠀⠱⡀
⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⡟⣆⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁
⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠈⢣⡀⠀⡐
⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡟⢦⡀⠑⢄⠈⣀⡀⣀⣀⠄⠠⢀⣀⠀⣀⡰⠋
⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠑⠄⠀⠑⠦⠀⡀⠔
⠀⢣⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⡏⠉⢳⠀⠛⠂⠀⠘
⠀⠘⣆⠛⠀⠘⣄⠀⠀⠀⠀⠀⢀⡼⠁⠀⠛⡼⠒⠒⠚⠀⠀⠀⠀⠘⡀
⠀⠀⠘⢆⠀⠀⠈⢦⡀⠀⠀⢠⠞⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠈⠳⣄⠀⠀⠙⠦⠔⠃⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃
⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⡀⠀⣠⡞⠉⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⠉⠁⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠬⠴⠊⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠙⢟⠁⠀⠀⣻⠀⢸⡄⠀⠀⡾⠂⢨⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣏⠀⢿⡀⠀⠀⢸⣆⠀⠀⣿⣆⣾⣿⡄⣾⠃⣠⣟
⠀⠀⠀⠀⠀⢀⣤⡀⢠⣄⠀⠀⣿⣧⡸⣷⣄⢀⢸⡿⣷⣄⣹⣿⡿⣽⣿⣾⣾⣿⣏
⠀⠀⠀⠀⠀⠈⠛⠣⣄⠉⢳⣦⣽⣿⢿⣿⣽⣵⣿⣿⣯⣿⣿⣿⣷⡿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣾⣾⣿⣿⣿⣿⣿⣿⣿⡽⠿⠿⠿⣿⣿⣿⣿⣟⣛⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⢿⣻⣭⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣛⣽⣷⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠛⠛⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣾⣿⠟⡿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⣼⠁⠀⠀⠀⠀⠀⡠⢰⣿⣿⠋⠙⢅⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿⠀⠀⠀⠀⢀⣀⣠⣿⡿⠁⠀⠀⠈⢶⣤⣀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⢫⣽⣷⣶⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡿⠻⣿⣿⣿⣷⣀⣀⣶⣀⣴⣿⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡛⠛⠛⢿⣦⣿⣿⣿⣻⣿⢿⡻⢿⡿⠿⡏⢻⠛⠉⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡛⣿⣿⣥⣤⣸⣿⣿⣏⡝⣿⠿⠋⠿⠜⠿⠿⢾⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠁⣹⠛⠃⠀⠙⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⣷⠀⠀⠀⠀⣿⠉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠀⠀⠀⠀⠸⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⢻⠀⠀⠀⠀⠀⣿⠀⠀⣠⣀⣄⣠⢀⡀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⣸⣾⣆⣀⣀⣴⣤⣿⣴⣶⣿⣿⣿⣿⣾⣧⣆⠀⢀⡀⣇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠻⡝⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿⣿⣷⣶⣿⣾⣿⣿⣿⣷⣽⣛⣾⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣾⣶⣶⣾⣿⣶⣤⣀⣀
⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⠿⢟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀
⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⢠⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠂⢻⢆⣸⡀⠀⡜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠹⡾⠀⠁⢡⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣲⠬⠡⠤⣤⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠟⠀⠀⡠⠲⠈⢧⡠⠆⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣶⣶⢤⡄⠀⠀⠀⠀⢾⠄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣝⣿⣟⣿⣆⠀⠀⠀⢾⠆⠀⠀⠀⠀⠀⣿⡀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣷⣤⣀⣸⠀⠀⠀⠀⢀⡞⠉⠉⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⡠⢣
⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠟⠛⠉⠁⠀⡄⠀⣽⡙⠓⠆⠀⣾⣀⠀⠀⠀⢀⣈⣵⡄⠀⠀⠀⠀⠀⢸⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡱⠀⢻⣇⡀⠧⡜⣿⣈⠉⠉⠉⠉⠉⢀⣹⣄⠀⠀⠀⢠⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠿⣷⣾⡇⠘⠗⠫⣅⣒⣚⣛⣛⣉⠽⣍⠓⠒⠦⢤⣀⡇⠑⢄
⠀⠀⠀⠀⠀⠀⢀⣀⠈⢲⣶⣄⠀⠲⢄⡀⠀⠀⠀⠰⠘⢍⠣⠀⠀⠀⠈⠳⢝⡺⢭⣉⣉⠙⣆⠀⠀⢠⡞⢁⣀⣬⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢈⠙⢻⡛⠛⠷⢤⣬⣝⡄⢀⠀⠀⠀⢠⠑⢄⠀⠀⠀⠀⠀⠈⠑⢾⣯⣽⣿⣷⡄⠛⠉⠁⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠠⠤⠤⠤⠿⠏⣛⢁⡀⠤⣀⣍⠙⠌⣧⡀⠀⡀⠀⠀⠑⠦⣀⡀⠀⠀⠀⠀⠙⢦⡀⠀⠙⢆⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⢋⣑⡦⠤⠁⢍⣂⡙⠦⠿⣅⠀⢀⣈⣷⠟⠋⠉⢃⠀⢀⣔⣹⠦⡀⠀⠱⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠟⠒⠒⢤⣽⣁⢀⣵⡀⠀⠀⡝⡗⠃⣿⠟⢤⡘⣶⣄⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠷⣖⣨⡟⠉⠹⡀⠀⠘⠀⠀⠙⣿⡄⠙⢾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠙⣝⣟⡈⠧⣀⣀⠀⢰⡀⠈⠃⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⢌⠉⠉⢣⡀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡀⠈⠑⠢⢤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⢱⣾⢍⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⣾⣶⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⣦⣭⣽⣏⣥⣤⠼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⢨⣿⣏⠁⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⡆
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⣤⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⡀⢠⡒⠂⣀⠀⠀⠀⠀⣄⣧⣿⣶⠶⣶⡄⠀⠀⠄⢠⣜⢛⣓⣤⣠⠄⠀⠀⣴⠶⢶⣿⣼⣧⠄⠀⠀⠀⢀⠀⠒⣦⠀⡀
⠀⠀⠀⡹⣿⣿⡀⠀⠀⠀⠀⣴⠿⠿⣿⡇⠀⠸⠃⠀⠀⡠⢛⠛⣼⡟⣏⠢⡀⠀⠀⠻⠀⢈⣿⠿⠿⢶⡀⠀⠀⠀⠀⣸⣿⢟⡁
⠀⠀⠀⠛⠉⠙⣷⡄⠀⡀⠈⠿⠀⢀⢈⣌⣦⡤⠤⠴⠾⠖⠛⠛⠛⠛⠻⠷⢾⣶⣤⣤⣤⣊⡍⢀⠀⠼⠃⠀⠆⢀⣼⡋⠉⠹⠃
⠀⠀⠀⠀⠀⠈⠻⠦⣾⣅⣀⡠⠔⠛⠋⣙⣦⣤⡤⠤⠖⠒⢒⡖⠒⠲⡖⠒⠛⠿⢭⣭⣀⣀⠉⠙⠲⢄⣀⣀⣵⠰⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠑⢼⣿⡋⣠⡴⠚⠻⠋⠁⠀⣠⡤⠀⠐⣽⣷⠤⢴⣿⠇⠀⠠⣤⡀⠀⠙⢿⠓⠶⣤⣾⣿⣿⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡟⢩⡄⠀⢠⡶⣬⣾⣿⣿⡷⠾⠴⠿⠿⠿⠧⠯⣾⣿⣿⣿⣧⡴⣆⠀⢀⡬⠹⡯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣎⣷⢾⠏⠋⠉⠁⣀⣀⣀⣤⣤⣤⣤⣤⣤⣶⣟⣁⣀⡀⠉⠉⠙⠿⣼⣉⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣁⣸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠻⠿⢿⠛⠋⠉⠛⠻⢿⣛⡛⠛⠛⠛⠉⠉⠉⠛⢻⠿⠿⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠑⠀⠀⣀⡀⠀⠀⠈⢧⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣼⠵⡒⠢⣄⠀⢺⡵⠋⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠰⣿⣮⠾⣷⠈⡄⢸⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣿⡏⢰⡟⢀⢀⡾⠁⠀⠀⠀⠈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠼⢉⠹⠿⣦⣮⣼⣅⣱⡀⠀⠀⠀⠙⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⣀⠔⠩⠑⣄⠀⠙⢿⠻⠿⠁⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡴⠞⠁⠀⠀⠀⠈⠑⠦⣤⣷⠀⠀⠀⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣀⡴⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠠⣶⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⡦⠠⢿⠀⣤⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠙⠛⠛⠶⠶⠶⠚⠛⠛⠉⠀⣠⣾⣤⠼⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⡂⠄⡀⠈⠲⠂⡀⡀⠄⡂⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠂⡁⠆⣈⠀⠀⡁⠆⡁⢂⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠃⠤⠀⠀⠄⠃⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠿⠶⢄⢀⢀⠤⠊⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⢿⡁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠝⠛⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣟⣻⣿⣿⣻⣻⣟⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠂⠄⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡤⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢀⡇⢀⠀⠀⠀⠀⠈⣷⠴⡞⠀⠀⢠⡎⠑⠀⣧⣀⣠⠇
⠀⠀⠐⠀⠈⡆⣰⠒⢆⠀⠀⢻⠀⢳⠈⠃⢁⠀⠀⠀⣿⠃⡇⠀⠀⢀⠉⠋⢠⣿⢹⠁⠀⠀⠀⠀⠀⢠
⠀⠀⠘⠒⢖⢳⡘⠲⡊⠄⠀⠈⠀⠈⢲⠀⠈⢆⠀⣰⠏⠀⠱⡄⢀⡎⠀⡴⠻⠃⠈⡀⣀⠔⡱⠒⣠⡾⣡
⠀⠀⠀⠀⠈⢧⠙⠦⣠⣀⡤⡈⠀⣀⠀⠑⣂⣘⣫⠟⢀⢤⡀⠘⣟⣒⣂⠐⠀⢀⠀⢽⠁⣸⣡⠴⠋⣼⠃
⠀⠀⠀⠀⠀⠈⣧⠀⠀⠒⠊⠁⠰⠉⢧⠀⠀⠈⠁⠀⡎⠀⢳⠀⠀⠁⠀⢀⡞⠁⢂⠈⠙⠚⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⠑⡀⠸⠀⠀⠀⠀⠀⠱⣄⠊⠀⠀⠀⠀⠀⢇⠀⠈⠀⠀⠀⠀⡸⢹
⠀⠀⠀⠀⠀⠀⠣⡘⠀⠀⠀⠀⠀⠈⠁⣀⣀⣀⣠⣤⣶⣤⣤⣄⣀⣀⣀⣀⠈⠀⠀⠀⠀⠀⠀⢧⠎
⠀⠀⠀⠀⠀⠀⠀⣿⠴⢖⠒⠛⠏⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠓⠤⢲⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⠀⠀⠀⠀⢄⡠⠀⠀⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⣷⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠸⠀⠀⣠⣤⡤⠤⠔⢿⣿⣿⡄⠀⣿⣿⣦⡾⠁⣀⣀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⣧⠀⠰⡔⠢⡄⠀⠀⠀⣠⣿⣧⣀⣀⣠⣼⣿⣿⠇⠀⠀⢉⣉⣀⡼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠈⠉⠹⠜⣶⣿⣄⢠⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⠟⠁⢹⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢰⡊⠰⡿⣿⣟⣿⣥⠀⢋⠙⢯⣿⠿⠿⣿⡟⠀⠀⢸⠀⢴⣷⣠⣤⣴⣶⣒⣂⡴⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⢸⣿⣿⣷⠀⠀⠀⠉⠻⠿⠋⠀⠀⠀⣾⠀⠈⢯
⠀⠀⠀⠀⠀⠀⠀⠙⠒⢤⡀⢠⣿⣿⣿⡆⠀⡠⢀⣠⣤⣀⣰⣤⣶⣿⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠿⢿⣿⠁⠀⠀⠀⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠰⡎⠉⠻⠿⢿⠟⠀⣴⠶⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣴⡆⠀⠀⠀⠈⡃⢩⠙⠄⠀⠀⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⢬⡷⢤⡄⠁⢸⠃⠀⠀⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⠀⠀⠀⢀⢔⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠲⠖⠁⠉⠑⠕⠚⠕⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⠭⣟⣛⣛⣛⡛⣛⣟⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⠀⢿⠁⠀⢀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣠⠶⠀⠀⡞⠠⠀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⠐⢮⣁⣼⣧⣀⠸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⢀⠀⢀⡀⠹⣯⡙⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣸⣿⣿⣿⡉⠀⠉⠀⠀⣾⣷⣼⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⡄⠀⢻⡆⠀⣴⣿⣿⢿⣿⣷⣀⠄⠀⠀⢸⣿⣿⣿⣿⡆⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⡦⠞⠀⣸⣿⣿⣿⣿⣿⣿⣿⡶⠆⠀⠈⢿⣿⣿⡿⣿⣄⠋⠱⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠛⠀⢠⣿⣿⣿⡿⠛⠉⠈⠀⠁⠀⠀⠀⠀⠉⠙⢿⣮⡛⠃⠀⢿⣤⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⣾⣿⣿⣿⣿⠀⠀⠲⣶⣠⣀⣀⣠⣤⣦⣁⠈⢿⣿⣦⠀⢿⣿⣾⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢿⣿⣿⠟⠁⠀⠀⠀⠀⠘⡃⣸⡇⠀⠀⢹⡟⠿⣿⣿⠂⢿⣿⣿⣹⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⠀⠘⣿⠁⠀⠀⣰⠀⠀⠀⢰⡇⢹⢰⠀⠀⠀⣧⠀⢻⡟⠀⢸⣿⣏⣿⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡀⠀⠻⠀⠀⣴⣿⣴⠆⣀⣼⡇⠀⣿⣆⠀⠀⢹⠀⠘⠀⠀⣸⣿⣽⣿⣿⢟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣇⠀⠀⢀⣾⣿⣿⠏⣼⣿⡿⠃⠀⣿⣿⠀⣿⠘⣧⠀⠀⢀⣿⣿⣿⣿⡿⣈⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⣿⣿⠃⣴⣿⣿⣿⡏⢰⣿⠟⠁⠀⢀⡟⣿⡇⢸⣇⠘⣷⣦⣾⣿⣿⣿⣿⣿⣼⣿⣷⡇⣤
⠀⠀⠀⠀⠀⠀⠠⣬⣿⣿⣿⡿⡋⣸⣽⣿⣿⡟⢠⠟⠁⠀⠀⢀⡾⠁⢹⣧⠘⣏⠛⢉⡴⠀⠉⠛⠛⠿⢿⣿⣿⣁⣾⣿⡇
⠀⡀⠀⠀⠀⣼⣿⣿⣿⣿⠏⠀⣠⣿⣿⣿⡏⣴⠋⠀⠀⠀⣠⠈⠀⠀⠀⠿⡆⢻⣰⢣⠀⠀⠀⠀⢀⠀⠀⠉⠻⣿⣿⣿⡟⢠⣀⡀
⠀⣿⣦⣤⣼⣿⣷⣿⣿⣿⣴⣿⣿⡏⣽⢏⠜⠁⠀⡀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠀⣀⣤⡴⠋⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⡿⣯⡿⠻⠕⢸⡿⢰⡟⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠛⢭⠴⠞⠃⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷
⠀⣿⣿⣿⠏⢽⣴⠏⠀⠀⢀⣿⠇⠘⣧⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⣿⡏⠠⢄⡈⡻⣦⣾⣆⠾⠉⠀⠀⢿⡄⢸⣆⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢿⣿
⠀⢿⡤⣤⣴⣿⣛⣮⣿⣿⡄⠀⠀⠀⢸⣿⡌⢿⣿⣴⡀⢿⣄⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣿⣿⣿
⠀⣇⣻⣶⣤⣤⣭⣝⣿⡿⢿⣆⠀⠀⢸⣿⣿⡜⣿⣦⣦⡈⢿⣅⠀⠀⢾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣉⣿⣿⣿
⠀⠉⢱⣬⣿⠿⠛⠋⠁⠶⣾⣿⣶⠀⠀⣿⠉⣿⣸⣿⣿⣿⣾⡻⣷⣔⢤⠛⢷⣄⠀⠴⢦⣤⣀⣀⠀⢦⣄⠀⠀⠀⣈⠸⡅⠻⣿⣿⣿
⠀⢠⣿⠟⠷⠿⠶⣦⠤⡤⣠⣾⣿⣦⠀⠉⢠⠘⣿⠉⠟⢿⣯⠑⢘⣿⣿⣷⣠⣙⠻⣷⣶⡤⣤⣉⠉⡛⠻⠷⣦⣠⣇⠀⠁⢺⣮⡻⣿
⠀⣴⠟⠀⠀⠀⠊⣤⠀⠀⠈⠀⠙⣿⣧⡀⠈⢄⣸⡦⠌⠰⣤⡖⠉⠴⠛⠉⠁⠈⠑⠀⠉⠃⠈⠉⠀⠈⠀⠀⠀⣠⡴⠞⠻⣿⣿⣿⠎
⠀⣿⣶⣤⡀⠀⠀⣯⠀⠀⠀⠀⠀⢸⢿⣷⣤⣀⣹⢸⣆⣴⠇⠙⠶⠂⣄⣠⣤⠀⠠⢀⡀⠀⠀⣀⣀⠀⢤⡞⠙⠁⠀⠀⠀⠈⠢⠐⠛
⠀⠿⠻⣿⣷⣄⠀⢿⡄⠀⠀⠀⠀⠀⠾⣏⡉⣍⠝⢾⣿⣦⣄⣠⣴⣶⣿⠟⠁⠀⣴⠀⣅⣠⣿⣥⣤⣤⣽⣿⣂⡀⠀⠀⠰⣂⡴⣄⠐
⠀⠀⠀⢻⣿⣿⣅⠘⣿⣄⠀⠀⠀⠀⠀⠘⢿⡄⠦⡄⢛⢀⣿⠋⠉⢹⣿⠀⠀⢰⡟⠀⣇⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠻⠿⣭⡿⠟
⠀⠀⠀⠈⢿⣿⣿⣆⢻⡙⣦⠀⠀⠀⠀⠀⢸⠇⢆⣿⢀⣿⣿⡀⠀⢼⣿⡆⠀⣿⠁⠈⠉⠛⠲⠶⣶⣤⣤⣤⣤⣤⣤⣤⡴⠂⠉⠄⣁
⠀⡀⠁⣄⠼⣿⣽⡿⢷⣤⣈⠳⣄⠀⠐⢤⣐⣶⣸⣿⡿⠋⠠⣿⡆⢨⡤⠷⡀⢞⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⡞⠁⠀⠀⣠⣿
⠀⣁⠀⠙⠳⣞⣿⣿⠈⠙⢿⣶⣌⡳⢄⡀⠀⣽⠇⢻⣱⣦⣶⠶⢁⠛⠀⠀⡇⢸⠀⠀⠙⢿⡷⠒⠠⢤⣤⡤⢤⣤⠬⣅⣤⣴⡟⠿⠿
⠀⠀⠣⠴⠀⢬⠙⣯⣦⡀⠀⠉⠉⠻⠦⠙⢲⡏⣠⠟⠁⠀⠀⢠⡀⣠⠀⠀⠁⣸⣄⣀⣀⣠⡹⣆⠀⠀⠀⠀⠸⠃⠀⠀⠙⠟⠁⢀⣴
⠀⠀⠁⠀⠀⠉⣝⣤⡹⠟⠷⠦⣤⣤⣤⢤⣼⠟⣯⠙⠷⠶⣦⠼⠃⠉⠀⠰⠚⠋⠩⠭⣉⡉⣁⣸⣷⣶⡄⠀⡀⠀⠀⢀⣀⣤⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠘⢧⣝⠦⠑⠀⠐⠋⠙⠇⠀⠃⠀⠙⠶⣷⣭⣤⣤⡠⢴⣤⣤⣀⡀⠀⠀⠀⠉⠉⠙⠋⠑⠊⣀⣤⣶⣿⣿⡿⢿⡟⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠠⠄⠈⠀⠀⠀⠀⠄⠀⠀⠀⠈⠻⢷⠀⡁⠀⠀⠀⠉⠙⠛⠻⠿⠳⠞⠾⠛⠿⠿⠿⠛⠡⠈⠀⠁⠈⠉⠿
⠀⠀⠀⠀⠀⠈⠀⠐⠀⠒⠂⠀⠀⠀⠚⠛⠒⠀⠀⠀⠀⠀⠀⠐⠑⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣤⣤⣀⣀⣳⣼⣁⣠⣤⣤⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⣼⣿⣿⣿⡇⠉⣩⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣿⣿⢿⢿⣷⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣤⣴⣶⣄⠀⠀⠀⠀⣀⣴⣾⣿⡿⣫⣿⣿⣝⢿⣿⣿⣦⣀⠀⠀⠀⠀⣀⣴⣶⣤⡀
⠀⠀⠀⢠⣾⣿⣿⣿⣿⣧⣴⣶⣿⣿⠿⢛⣡⣾⣿⣿⣿⣿⣷⣬⡛⠿⢿⣿⣷⣦⣼⣿⣿⣿⣿⣷⡆
⠀⠀⠀⠈⠻⣿⣿⢿⣿⣿⣯⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣭⣿⣿⡿⣿⣿⠟⠁
⠀⠀⠀⠀⢸⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢹⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿
⠀⠀⠀⠀⣠⣿⡇⢸⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⠀⢸⣿⣅
⠀⠀⠀⠀⣷⣾⡇⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢸⣿⣾
⠀⠀⠀⠀⢿⣿⣿⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣶⣿⣿⣿
⠀⠀⠀⠀⣾⣯⣿⡏⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠉⣿⣽⣿
⠀⠀⠀⠀⠙⣿⣿⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣿⣿⠊
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡟⠉⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠈⢿⣿⣧
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⡏⠈⠙⠛⠛⠿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠁⠀⢸⠀⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠦⠤⠤⠤⠤⠤⠤⠤⠒⠒⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⣽⠆⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣉⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣠⣷⣯⠶⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣨⠟⣿⠀⠐⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡎⠁⣀⣽⡆⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⣾⡇⣼⣿⣿⡆⠀⠀⠀⢻⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⢀⣿⣿⣷⣾⣿⡇⣤⣴⡶⢟⣵⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢹⣏⠛⠻⢿⣿⠛⠓⠁⣉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠘⣿⣿⣿⣿⡿⢰⣾⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢉⠿⣿⣿⡇⢸⣿⠿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⣬⣷⣦⣽⡇⢈⣠⣾⣯⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⢰⣿⣿⣿⣿⠋⠋⠉⢙⣿⣿⢿⠌⠀⠀⢀⡤⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣿⡇⣤⠄⠀⠙⢿⣿⠟⠀⠀⠀⠈⢛⠕⠁⠀⢠⡾⠁⣀⡤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠶⢤⣀⢀⣿⣿⡇⡄⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣍⡀⣠⣤⠴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⠀⠐⠀⠀⠀⠀⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣄⣀⣿⣿⣿⣿⡇⣿⡄⠀⢣⡀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣀
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⡏⣿⡇⣿⡇⢀⣀⣳⣀⣀⣸⣿⣇⣀⣀⣀⣀⡀⠀⢠⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠠⠀⣼⣿⣿⣿⣿⡿⠁⣽⡇⠹⣗⠻⠿⠿⠿⠿⢿⣿⣿⠿⣿⠿⠿⠿⠂⣼⣿⣿⡿⣯⣿⡟
⠀⠀⠀⠀⠀⣠⣦⣀⣄⠀⠸⣧⣄⣿⣇⣀⣈⣤⡄⠀⠀⠀⢸⣿⡇⠀⠘⣦⠀⠀⢠⣿⣿⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⣻⣿⣿⣿⢃⠀⠁⠈⠉⢉⣉⣉⣹⠀⠀⠀⠀⢸⣿⣿⣄⠀⠸⣧⠀⠛⠉⠻⣄⡀⠙⡿
⠀⠀⠀⠀⠀⠻⣿⣿⠃⣿⣿⡟⣤⠁⢈⡉⡡⡴⠀⠀⠀⠀⢸⡿⠏⡙⠷⠀⠀⠀⠀⢀⡀⣬⣍⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡿⡟⠃⣯⠀⠠⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⠙⣄⣀⣤⣰⣶⣿⣿⢸⣿⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠰⠌⢿⠆⠐⠀⠒⠉⠀⠀⠀⢀⣠⣤⢶⣷⢻⡟⣛⣡⣤⠤⠴⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠴⢿⠄⠀⠀⠀⠰⡴⠿⠛⠋⠈⠁⠀⢠⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⣀⡷⠒⠈⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠾⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣴⣿⡇⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣛⠻⠿⠿⢿⣿⣿⣷⣦⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣬⣛⣛⠛⢛⠿⠷⠶⣈⣭⣥⣭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣌⣿⣿⣛⢛⡻⢿⣿⣷⣎⢻⣷⣾⣿⣿⡏⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣵⣿⣿⣿⣿⣿⣿⣿⣿⣌⣩⣬⣿⣿⣿⣇⢿⣿⡝⢿⣵⡟⣱⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⠛⠁⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡌⣥⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡄⠀⠀⠀⠀⢹⣿⣿⣿⣿⡿⢋⣿⡿⣸⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣷⣦⣄⣀⣤⣾⣿⣿⡿⣋⣴⣿⡿⣡⣿⣿⣿⢱⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠿⠿⠿⠟⣋⣵⣾⣿⡿⣫⣴⣿⣿⣿⢣⣿⣿⣿⣿⣿⡏⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⣠⣴⣿⣿⡿⣋⣵⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⡇⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠡⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⣿⣿⡿⣰⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠙




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⢀⣼⠙⣐⠠⣶⣿⣦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣤⣯⣕⠆⠉⢛⠑⣁⡭⣫⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣽⣧⣷⣮⣬⣥⣽⠾⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡿⠛⠛⠛⠻⣿⣿⣿⣿⣿⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⡤⣄⠀⠐⢪⣥⡄⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣿⣿⣿⣿⠟⠋⡃⠀⠀⠉⠀⠀⠈⣁⣽⣿⣿⣿⣷⣾⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⢿⣿⣿⡀⠐⠅⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⣥⣾⣿⣿⣿⣤⣾⣛⠟⠀⠀⣠⠞⣿⢟⣿⣿⣿⡻⡟⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣅⣻⣿⣿⣿⣯⣄⣠⡴⠚⠁⠀⢿⣿⣿⣿⣏⢿⡈⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢿⣿⣾⣿⣿⣿⣿⣿⠛⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⣿⡿⠟⠛⠛⣿⠿⠟⣋⣁⣁⠀⠀⢀⣀⠀⠀⠸⣿⡿⠛⠻⠿⢿⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣏⠀⠀⠀⢺⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢹⡿⠛⠀⠀⠐⢿⣏⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⢛⣿⣧⡀⡹⠓⢀⠇⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢸⠀⠀⠀⠀⠀⠛⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⡯⠓⣄⠀⢸⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣇⠀⠀⣶⠂⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⣽⣻⣿⡇⠀⢻⡶⠛⠳⣄⡀⠀⠀⣀⢀⡀⠀⠀⣀⡴⠋⠉⠳⣶⣿⠀⠀⢰⣿⣫⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢠⠏⠀⠀⠀⠀⠙⠶⣤⣿⣼⡧⡴⠞⠉⠀⠀⠀⠀⠸⣿⡀⠀⠁⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⡄⠀⠀⠀⠀⠀⠀⠈⠂⠐⠈⠀⠀⠀⠀⠀⠀⠀⣰⢿⡄⠄⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠤⣤⡤⠤⠄⠂⠁⠈⠐⠢⠤⢤⣤⣤⠤⠞⠁⠘⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣯⣀⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⡟⣤⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣮⢻⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠛⣿⣿⣿⣿⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⣿⠟⢾⣿⢶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⣇⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢀⣀⣰⣿⣿⣿⠷⠀⣠⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠋⢻⣿⣏⣿⡄⣰⣿⣄⠀⠀⠀⢠⣿⠀⠀⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠛⠁⢀⣼⠏⠉⠀⠀⠈⢿⣿⣷⡀⢀⣼⣿⣎⡽⠃⡈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⢀⣀⣨⣤⠀⠀⠀⠀⠀⠈⣿⣿⡿⠁⣿⣿⣿⡀⣿⡷⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⡟⢻⣿⣿⣦⡀⠀⣀⠀⣀⣿⡿⢡⣿⣿⣿⣿⣿⡗⠰⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⢹⡇⢸⣿⣿⣾⣿⣶⣿⣿⣿⡿⢡⣾⡟⣿⣿⣿⣟⠀⠀⢰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡾⣿⣿⡿⢻⣿⣿⣿⠟⢰⣿⡿⠀⣿⣿⣿⣿⣦⠀⠈⠷⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⠠⣿⡿⣀⣿⣿⣿⣿⣴⣿⣿⠇⠀⢿⣿⣿⣿⣿⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣧⡼⣶⡿⢣⣿⣿⣿⣿⣿⣿⣿⡋⠀⠀⠸⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠉⣻⣿⠟⣷⡟⣱⣿⣿⣿⣽⣿⣿⣻⣿⠳⣆⣀⣠⣾⣿⣿⣿⠛⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠀⠀⣀⣠⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠈⠉⠉⠉⠙⣿⣿⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣙⡛⠻⢿⣮⣻⡶⢶⢶⣦⢤⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠛⠋⣿⢛⣿⣿⡟⠃⣽⣿⣿⣿⣿⣶⣦⣽⣿⣷⣿⣶⣷⣶⣯⣽⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠄⠀⠀⠀⣦⢸⣿⣿⡧⢾⣿⣿⣿⣿⣿⡉⠀⠀⠈⠙⠙⠛⢻⣿⣿⡏⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠀⢸⣿⣿⣿⣧⡀⠀⠀⣀⡿⣾⣿⣿⣧⣿⣿⣿⣿⣷⣿⠟⣿⣄⠀⠀⣀⣴⡟⢹⡉⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⣹⠉⠀⣸⣿⣿⡿⣿⣟⠛⢿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠛⠿⣶⣿⣿⣥⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣶⣿⣿⣿⣿⡇⠘⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠹⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠙⢿⣿⣿⣾⣿⡿⠛⠀⠙⢿⣿⣿⣿⣿⡿⠻⣿⣇⠀⠘⡋⠁⠈⠛⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠹⠿⢿⣿⡇⠀⠀⠀⠀⣻⣿⣿⡟⠁⠀⠈⠻⣷⣄⠱⡄⠀⠀⠀⠀⠉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⠀⠀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⠙⠻⣷⣾⣦⡀⠀⠀⠀⠀⠀⠹⣷⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣆⠀⠀⢸⣿⣿⡃⡀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠀⠀⠀⠀⠀⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠙⠀⠀⠀⠙⠻⣿⣧⡄⠀⠀⠀⠀⠀⠀⠈⢻⣿⠇⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠈⠙⢿⠖⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⣸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⣼⣟⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣀⡀⠀⠀⠀⠀⠈⣯⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⡿⠆⠀⠀⠀⠀⠻⣷⣤⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣶⣿⡄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠸⣷⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠛⠻⠟⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣁⣀⣀⣀⣀⣠⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⡒⠶⠎⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⠡⠆⡆⡴⣤⡙⢁⡴⡾⣭⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⢿⣏⡠⢔⣒⣒⣦⢼⡟⢠⢏⣒⠂⡤⢿⡄⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠐⢻⣣⡞⣉⠨⠦⣮⣝⢠⢿⣦⡷⣬⠽⣟⠷⡜⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣳⡘⠺⠗⢸⣱⡋⠙⣙⢷⡱⠆⢒⣽⣛⠟⠟⣓⣒⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⣇⣼⡗⠢⡤⠤⣐⣾⢹⣦⣣⡈⠘⠕⢋⠙⠉⠁⠾⣹⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⡜⢿⣷⣄⠈⠲⢬⢝⣛⣳⢍⢿⠏⢏⢸⣭⠑⠺⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣮⡳⣑⠠⢠⡱⣯⣿⡄⡿⣆⣲⢣⢋⡴⠂⠁⡹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣫⣙⢦⣓⠛⣛⣴⠷⢾⠏⣹⣿⣷⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣯⣭⣏⠂⢡⡟⠼⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⣚⠞⠠⠷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣍⣲⢋⡆⠸⠴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡋⡠⠺⢁⠒⡒⣸⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣏⠈⠞⣯⠘⣷⡑⢦⠃⠂⠆⠌⡜⣎⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢻⠐⡈⡜⣶⠊⠻⣷⣧⣉⡹⢍⣉⣒⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠈⢇⢡⠄⢍⠷⣝⢠⣍⢛⣛⠛⠿⣛⣻⡝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠈⢦⠲⢈⠰⣈⠳⠦⣬⣙⣛⣠⣭⠵⠞⡩⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠑⢞⡀⡄⢉⡈⠢⠤⠼⠤⠔⠚⣍⠀⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠓⠮⣥⣘⣂⡸⠀⠻⠀⠙⢁⣘⡯⠜⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⢠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣰⣿⣿⣷⣤⣤⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠿⠟⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠟⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣋⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣭⣛⣿⣻⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⡀⠀⠀⢿⠧⡀⡴⡄⢸⡟⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⢿⣄⠀⡏⣑⠄⡄⠀⢮⡳⢹⠀⣡⡚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⢶⡀⢤⢰⢋⠘⠯⠉⠼⢁⣉⣠⣭⣥⢩⣙⢫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⠈⠛⣿⠈⣁⠴⣺⠟⣩⣾⡛⢅⣧⡾⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣔⠀⣠⠞⣱⠎⢠⠞⠁⠀⠀⠀⠁⠀⣯⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⢾⣯⣈⣋⡡⡒⠁⠀⠐⠒⢤⡄⠀⣠⢼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠻⢶⣬⣭⣽⠆⠀⠀⠻⠿⠏⠀⢰⠷⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣷⣠⢫⢯⡓⠀⠀⠀⠀⠀⠀⠀⣀⡆⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣝⣿⣌⣿⠑⡀⠀⠀⠀⠀⠤⠾⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⣌⣥⣤⠿⠀⠈⠲⣤⣀⡀⠈⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⣿⣥⢾⠀⠀⠀⠀⠙⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠎⠀⢰⠀⠀⠀⣼⢻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠲⠶⠊⠁⠀⠀⠈⣇⠀⢠⠋⣀⣑⣌⠿⠋⠉⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠀⠈⠀⠀⠀⠉⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠊⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⣞⠀⠀⡄⠀⠀⠀⠀⠀⠀⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠾⢰⡋⢐⡄⠀⠀⣿⠂⡞⠀⠀⢀⠀⠀⠀⢸⠋⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠸⡀⠹⠈⣄⠀⢀⠃⠀⠃⠘⠃⡎⠀⠀⠀⣸⣦⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠙⢦⡀⢀⠄⠊⠀⠀⢦⠀⠀⡇⠀⠀⢰⡿⢠⡇⠀⠀⠀⡴⠙⡄⢠⠏⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠐⠋⠑⠂⠀⣤⠀⠸⡆⢀⣧⣀⣴⠟⠀⠸⠄⠀⢀⡄⠙⠖⢁⡼⣧⣤⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡃⠀⠀⠀⠀⠠⠀⠸⡇⠀⠐⠎⠹⠞⠁⣠⠀⠀⢇⡴⠊⠀⣨⡴⡿⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⠁⠀⠀⠀⠀⠘⣀⠴⠁⠀⠀⠀⠀⠐⠋⠀⡇⠀⠻⣕⣶⠬⠀⠈⠀⠀⣇⣀⠤⠲⣀⠁⢀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠷⠤⠶⠦⢤⣀⣀⡁⠀⠀⠀⠀⠀⠀⢰⢀⡴⠃⠀⠀⠈⠀⢀⡤⠶⠀⢰⡍⢁⠔⠃⣀⣴⠞⣣⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠁⠐⠀⠋⠉⠐⠚⠣⠴⣶⣬⣁⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠓⠟⠉⠋⠉⣰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣷⣶⣤⣄⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠦⣤⣀⠀⠀⠷⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣽⣿⣿⣿⣿⡏⠘⠀⠀⠀⡴⢶⣶⣶⣶⣶⣶⣤⣀⠀⠀⠀⢠⣌⠙⠖⣠⠀⠀⠀⠀⠀⢠⠉⣠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⢻⣭⣴⣄⢴⠀⠀⠈⣡⣼⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⢸⣷⡀⠀⠀⠈⠔⣢⢄⡀⡸⠂⠁
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⢟⣽⣿⣿⣿⣿⠘⢠⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣇⠀⠀⠀⠀⠀⠉⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣾⣿⣿⣿⣿⣿⣿⠃⠈⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠠⠀⠀⠀⠀⠈⣿⣿⣿⠿⣿⣿⣿⡿⠁⠀⣠⣤⣤⡀⠀⠈⠉⠙⠛⠛⠛⠁⠀⠀⣸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠠⠀⢀⠄⠀⠀⠈⠛⠁⠃⠸⠿⠋⠀⠀⠊⠉⠩⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⡿⢶⠦⣠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠃
⠀⠀⠀⠾⣀⣄⡀⠉⠀⠘⠋⣱⣧⠴⣦⠀⢄⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣄⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠰⡶⠡⣘⢙⡷⣾⣦⣤⣾⠃⢠⠟⢳⣦⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣶⣶⣶⣦⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠐⢲⡄⣁⠀⠋⣹⣿⣿⣶⣮⣀⡤⠀⡙⠚⣄⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⡇⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⡇⢹⣷⣶⣿⣿⣿⣿⣿⣿⣷⣄⡁⠰⠙⣷⣰⣿⣿⣿⣿⣿⣿⣿⢛⣻⣿⣿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣿⣿⣿⣿⣿⣿⣿⠇⠀⣿⣿⣿⣿⠁⣀⠂⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠈⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣿⠛⢁⡼⠁⠀⠀⠀⡐
⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⠏⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⣀⣠⣜⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⣛⣀⡀⣀⠔⠁
⠀⠀⠀⠀⢼⠷⠐⠆⠾⢿⡿⢿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠛⣿⡿⠟⠉⠀⠀⠀⠀⠀⣸⠛⠋⠉
⠀⠀⠀⠀⢿⣴⡀⡀⢠⢸⡇⢰⠀⡌⠈⡅⠈⠁⠀⠉⠀⠀⢹⠀⠀⠂⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠛⠷⢾⠎⢁⠸⠧⠟⣠⡧⣌⠣⠴⣶⠟⠛⠉
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠑⠒⠀⠠⣤⠄⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣷⣖⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣯⣵⣮⣉⣿⣶⣦⡀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣿⣿⡿⠛⠉⠉⠙⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢠⣼⣿⣿⣿⣿⣿⡀⠀⠛⠛⢚⣽⣿⣿⣿⣿⣿⣟⠛⠳⠤⣄⣠⠟⣽
⠀⠀⠀⠀⠀⠀⠀⢀⢯⢲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣴⣆⡴⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠉⢸⣼⣿⣿⣿⣿⣿⡿⣿⣿⡏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣯⠿⣿⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣬⢿⣾⣻⣿⣿⣿⣿⣿⣮⢹⣤⣿⣿⣿⣿⣿⣿⣿⣿⣟⣇⠲⣿⣟⠛⠙⠻⢷⣄⡀
⠀⠀⠀⠀⠀⢨⠀⠀⠈⢷⢻⡉⠻⢮⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡒⠿⢿⡦⡀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠈⣏⠀⣄⢸⠃⠛⢦⣄⡙⣿⣿⣿⣿⣿⣿⣿⣿⠿⣯⣥⣿⣿⡿⢿⡳⣄⢼⡇⠈⡆⠘⣿⣿
⠀⠀⠀⠀⠀⠀⢹⣷⣨⢨⣰⡄⠀⠉⢿⡿⣸⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣛⣄⠘⣧⠙⢻⡇⠀⢸⠀⢹⣿
⠀⠀⠀⠀⢠⡀⠀⢻⣷⣿⣿⣿⣧⡄⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣸⣿⣗⠀⣿⢀⡾⡁⢸⡌⢧⢸⡏
⠀⠀⠀⠀⠀⠳⣄⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⡟⠀⢹⣧⣿⣿⠰⣿⠚⠀⠀⢰⡆⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠘⡷⢀⣿⣿⠏⠻⣿⣿⣿⣿⣿⣿⠃⣿⣿⠃⠀⠀⠀⢻⣿⣿⢡⣿⠀⠀⠀⢸⠇⠀⢸⢧
⠀⠀⠀⠀⠀⠀⠀⢷⢸⣿⡿⠀⠀⢸⣿⣿⣿⣿⠋⢸⡿⢁⠆⠀⠀⠀⠸⣿⣿⡾⡏⠀⠀⠀⡞⠀⠀⡾⠈
⠀⠀⠀⠀⠀⠀⠀⡼⣸⡟⠀⠀⢀⣾⣿⣿⡛⠏⢠⡟⠀⡜⠀⠀⠀⠀⠀⣿⣿⡅⣷⠀⠀⠜⠁⠀⢰⠃⠘
⠀⠀⠀⠀⠀⠀⡴⢱⣿⣧⠀⢀⣾⣿⣽⣿⡃⠀⣸⠇⠀⢳⠀⠀⠀⠀⠀⣿⡟⢠⡇⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⣴⣃⣾⠿⣟⠀⢸⡟⣿⠟⠋⡁⠐⠹⠀⠀⠘⠀⠀⠀⠀⠀⢻⠀⣸⡇
⠀⠀⠀⢀⠜⠁⣹⠃⢀⣣⠜⠉⠑⠋⠀⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠇
⠀⠀⠀⠂⠠⠴⠓⠐⠊⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠄
⠀⠀⠀⠀⠀⠀⡄⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠢⠄
⠀⠀⠀⠀⠄⢠⠀⠿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠗⡁⠂⠄
⠀⠀⠀⠀⠲⢿⠇⠀⢢⡑⡄⠀⠖⣠⠀⠀⠀⠀⣾⣿⣿⣆⠀⠀⠀⠀⣠⡀⠀⠀⠀⢠⣾⣿⣿⠀⠀⠀⠀⠀⠐⠢⠀⣰⣿⠇⠀⠿⠶⠂
⠀⠀⠀⠀⠀⠈⢂⢂⠘⣇⠈⢂⠲⣿⡀⠀⠀⣰⣿⣿⣿⣿⣧⡀⠀⠸⡦⡈⠀⠀⠐⠿⢿⣿⣿⣧⠀⠀⠀⣸⣟⡢⠃⢘⣿⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⢳⣷⣿⠀⢹⣿⣿⣷⢄⣴⢟⣿⣿⣿⣿⣿⣽⣶⡤⠛⠓⠀⠀⠀⠀⠀⢹⣿⡞⣧⡀⠴⢟⣋⠀⢠⡜⢷⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠟⢸⠀⢸⣿⣾⣟⣯⣵⣿⣿⣿⣿⣿⣿⣿⣿⡎⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣮⣝⣛⣽⠃⢀⣿⡀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⠀⢀⣿⡿⠏⠀⠈⠻⣛⣛⣛⠛⠛⠛⡛⠓⠀⠀⠀⠀⠀⠰⠾⠿⠿⣿⣿⣿⣿⡇⠀⣸⡿⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠾⣿⣛⠏⠉⠒⢻⣿⣿⣿⣿⣿⣿⠿⠟⠉⠉⠉⠀⠀⠐⠂⠐⣦⣖⣒⣛⣶⡥⠬⣙⣷⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡔⠘⣿⣿⡀⠀⠀⢨⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣿⣿⣿⣄⣀⣼⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⢼⣛⣻⣿⣶⣶⣿⣿⣷⣤⡤⠤⠤⠤⠤⠀⠀⠀⣀⣀⠀⠀⠀⣀⣩⠿⠿⠽⠿⠻⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⣿⣾⣿⣿⣿⠿⠶⠶⠶⠶⠦⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⢶⢶⣀⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠷⠶⢤⣬⣭⣭⣤⣤⣤⣤⣄⣀⣀⣨⣭⣭⣭⣭⣽⡿⠿⠷⠿⠿⠻⠟⠛








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⢩⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢀⠀⠒⢢⣽⠃⠀⠉
⢀⡠⣀⣠⣦⠀⠀⢀⣴⣿⣆
⣷⣿⣿⡿⠿⠗⠒⠈⠁⢨⣿
⠈⠙⠉⠀⠀⠀⠀⠀⠀⡂⢸⠇
⠀⠀⠀⠀⠀⠀⢰⠶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠸⠒⠆⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠚⠷⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠸⣿⠃⠀⠀⠀⠀⠙⠋⠉⠛⠓⠠⠤⣤⣀⣀⠀⠉⠉⢻⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠻⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢰⡏⢄⠀⠀⠐⠀⠀⠀⡼⣿⠉⠉⠉⣻⠁⠐⠂⢄⠀⠀⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠢⢤⡤⠀⠀⠀⠀⠀⢠⡇⠀⠘⣷⣌⣓⠦⠴⠦⠖⢋⡴⠏⠀⠀⢰⡇⠀⠀⠀⠈⡻⠞⢀⣤⠀⡒⠠⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⣤⡀⠀⣼⠃⠀⠀⠸⡍⢛⡑⠋⠙⠋⠉⠀⠀⠀⠀⢸⣇⡇⢀⠐⠈⢀⡴⠋⠀⠀⠁⠐⠊⠀⢠⠝⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣠⡟⠀⠄⠀⠀⠹⣦⠈⠑⠢⠄⢀⠀⠀⠀⠀⣼⠉⣁⣠⡵⠚⢉⡀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠿⠀⡰⠀⠀⠀⠀⠘⢷⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⠀⠀⠉⠘⠢⠄⠀⡇⠀⠀⠀⠘⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣰⣃⠄⠀⠀⠀⠀⠀⢂⠙⠦⣄⠀⠀⠀⠀⣿⠐⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢤⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⠀⡠⠀⠀⠀⠀⠀⠳⣄⠈⠉⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣠⡄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢧⠈⣷⠠⠀⠀⠀⠀⢠⠀⠀⠿⠀⠋⢠⡇⠀⣀⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⢈⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢺⡄⢘⡆⢀⠄⠀⣀⣠⡀⠀⢠⡖⠒⠀⠹⠆⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠉⠭⣀⡙⠛⠂⠒⠒⠛⠛⢛⣛⠿⢳⡀⠹⡸⠔⠋⠉⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢋⣼⠶⠀⠀⣀⡉⠛⠛⠛⠛⠛⠋⠉⣀⣀⣼⣷⣀⢷⡀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⠀⠀⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠚⠁⠀⠀⠀⠀⠃⠈⣩⡟⠁⢰⠟⠉⠉⠁⠀⠀⠉⠒⢷⡀⠀⠀⠀⠁⠀⢹⡄⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠈⠀⠀⠀⠀⠀⢀⡠⠞⠁⢀⡴⠃⠘⢦⣀⡀⠰⠶⠤⠤⣄⣈⠀⢀⡀⠀⠀⠈⣧⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣊⠀⠀⠀⠀⡀⢤⡐⢊⠁⣀⡤⠖⠁⠀⠀⠀⠀⠉⡀⠀⠉⠉⠉⠉⠉⠒⢦⣌⢲⣄⠀⠘⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⢀⡤⣔⡪⠽⠓⠋⢉⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠉⢦⡀⠀⠙⢷⡝⢦⠀⢱⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢠⠖⠃⢚⡀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⡀⠀⠀⠑⠄⠀⠈⢿⠈⠃⠠⠦⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣴⣫⠏⠀⠺⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢟⠁⠀⠀⠀⠀⢨⡆⢰⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡐⡦⠻⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠈⠀⠀⡿⢸⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠹⣠⡆⠁⠀⠀⢐⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⠶⠆⠰⣇⢸⡆⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⢰⡏⠑⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠀⠀⠀⠙⣾⣷⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠞⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⠀⠀⠀⠀⠘⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡄⠀⠀⢀⡴⠋⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠛⠋⢙⡷⠀⠈⢣⡀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣀⠀⠀⣰⠉⣴⠟⠛⠉⠉⠉⠙⠒⠒⠤⠤⠤⠤⠤⠔⠒⠋⠉⠀⠀⢀⣀⡤⠚⠁⠀⣀⣤⠟⣄⡠⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢶⣈⠀⢀⣅⠰⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉⠀⠈⠋⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⢩⡋⠈⠙⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠁⠀⠘⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⠈⠃⠀⠳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣢⣄⣂
⣿⣿⣿⠿⠻⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠘⣿⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⢀⠀⣀⠀⠸⣷⠠⡀⢿⣷⡡⡛⢔⣶⡷⢀⠄⣴⠃⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⢢⡳⣿⡟⢮⣤⡌⢋⡀⡀⣀⣃⡈⠁⠀⠓⢂⣉⡎⣶⣿⣰⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡿⠰⢦⣬⠹⣷⠬⡻⢃⡈⡰⣒⣛⣞⣛⠲⠍⢁⠻⠿⢀⣿⠛⣊⣂⣀⢄⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠈⣬⡽⢻⣔⣮⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣥⣜⣅⢸⣿⠄⠠⢋⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⢛⣧⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠿⠛⠈⠉⠉⠉⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⣿⣿⣿⠀⢀⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⣿⣿⠀⢀⣤⣤⣌⡙⠛⠀⠀⠀⠰⠶⠛⢋⣉⠉⠂⠀⣿⣿⡿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡟⠀⠙⠿⠿⠿⠋⠂⠀⠀⠀⠀⠠⠺⣿⣿⣿⡶⠀⣿⣿⣵⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠠⠄⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⠂⠀⠀⠀⠀⠀⠂⢺⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠆⠲⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡐⣁⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠦⠀⡙⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⡴⠛⠛⢻⡇⡠⢎⡀⠀⠀⠀⠀⠀⠀⠈⣹⡷⠀⠈⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⡀⢾⡟⠃⢀⡶⠾⠋⣠⢮⣿⡀⠀⠀⠐⢄⠀⠀⢈⣷⢘⡇⠐⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣠⡷⠀⣰⣿⡤⠛⣡⠷⣤⣿⣿⣏⣵⢟⣆⡀⠀⠀⠉⠘⠋⣿⠸⢇⡤⢠⠇⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣴⠏⠀⣸⡿⢃⡰⣮⢀⣼⣿⣿⣍⣩⡾⢶⣙⠛⠛⢠⡶⣶⠟⠁⠀⠉⠀⠁⠀⠀⠒⠈⠀⠀⠀⠤⣀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣟⠃⠀⠀⣿⠦⠻⠇⣠⣶⣤⠟⣁⣾⠷⣶⢎⣭⣙⣿⣤⣾⣧⢀⠀⠀⠈⠀⠀⢠⣀⡴⠂⢡⡀⣠⡆⣬⡀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣳⣮⠀⠀⠐⢋⠀⠈⢶⣿⣿⣶⣾⣿⣷⣯⡷⣿⣥⣮⠑⢋⣼⠶⠋⠀⠀⠀⠀⠀⠟⣹⢰⡶⢿⡏⠠⡄⠻⢃⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢿⠏⠀⢠⣿⠋⢠⣴⣿⡿⢻⡟⢻⣯⡟⠉⣽⣿⡟⣡⠛⣱⡄⠀⠀⠀⠀⠀⣠⣾⠷⡛⠿⠃⢸⡀⣦⢹⡀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣴⠂⢠⣿⢃⡔⣧⣹⣯⣴⣿⣿⣿⣿⡷⠰⣧⣼⣿⠁⣾⠋⡀⠀⠀⠀⣠⠖⢩⡟⢠⠁⠆⢸⣼⣷⡏⢸⣿⡎⢿⣼
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠇⠀⢸⡟⠏⢰⣿⠟⣼⠟⠛⠉⠁⠈⠳⠀⣥⠘⣧⠒⣡⠖⢀⣤⠰⢾⡷⢾⣏⣰⣿⡜⠆⠁⢁⡟⢡⡜⠘⠁⠘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡟⢀⣼⠇⠈⠀⠀⠀⡏⠀⠀⠀⠀⠀⢀⣠⡶⢪⠀⠀⣿⠀⢡⣶⣿⣵⠔⠋⠀⠲⣾⣧⢀⣤⠘⢣⣾⣠⣿⣿⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠠⠀⠰⠀⢠⠁⠀⠀⠀⠀⣰⣿⠋⣼⠃⠀⣴⠃⣀⣷⡍⠀⠀⠀⠀⠀⠀⠸⠏⢹⣇⡶⢻⢻⠏⢹⣿⡄⢠⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣱⠄⠀⢰⠀⢸⠀⠀⠀⠀⣠⣿⣏⣴⢟⣤⡽⠞⢻⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⢇⡀⠀⠀⣼⠉⣧⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⠀⢀⣿⣦⠈⠀⠀⠀⠀⣿⣿⢏⠲⣟⣥⣴⢖⡹⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⣾⢷⣤⠀⡟⠀⠉⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⢸⣿⡆⢸⣿⣿⣆⡆⠀⠀⠀⣿⣿⣼⣷⣿⣇⣱⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡡⠀⣤⢸⣿⢰⣇⢀⡄⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢟⠈⣿⠁⢸⣿⣿⠿⣧⠀⠀⢀⣿⣿⣫⣿⡟⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠘⣿⣿⣿⠀⡇⢸⢩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣿⡀⢸⡟⣻⠄⠘⠂⠀⢚⣽⡾⢛⣿⣷⡸⣿⣿⣀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣿⣿⢿⣿⣿⣾⣟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠇⢸⣏⡷⠀⠀⠀⣱⡟⣥⠛⡿⠉⣽⣷⡌⢻⣯⣁⣀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⠈⢋⠹⡇⣟⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⢳⠀⠀⢀⢿⣿⠁⠀⠠⣼⠉⠛⡷⣾⡍⠉⠉⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠈⠀⠹⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠛⢸⠀⠀⢸⠈⠙⠷⠀⡷⢈⣦⡀⠲⣿⣧⠛⠿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠂⠸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠈⠀⠀⠈⠂⠀⠠⠆⠀⠀⠀⣠⠞⢩⣿⢰⣿⢻⣿⠷⣄⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢻⠶⡖⠀⠒⢺⠇⣰⡏⣹⣿⣏⣽⠿⠀⠻⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠆⠀⠀⠀⡤⠙⠶⣤⡄⣾⣟⣤⡋⣴⣻⣿⢇⡀⠀⠀⠀⠸⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⣇⠀⠠⡄⠀⠉⠻⡿⢹⡏⠻⢿⣼⣷⣄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠀⠀⠀⢻⡄⠀⢸⢶⡆⠀⠀⠹⡗⠀⠘⠙⡏⠻⣿⡀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠈⢻⣦⡀⠸⣷⠱⣄⡂⠈⠠⠔⠚⠀⠰⠞⣷⡀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡻⠆⠈⠢⡙⢷⣌⡀⠀⠀⣀⣴⣶⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣧⣿⡿⢣⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣥⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⢀⣿⠿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠐⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢸⣦⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⢙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣌⣃⣠⣀⢛⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣰⣬⣵⡦⠀⠈⠉⠻⣿⡿⠟⠛⠀⠀⢴⣬⣁⡄⠤
⠀⠀⠀⠀⠀⠀⢠⢀⡀⠸⠛⠿⣿⣫⠄⠀⠀⠀⢷⣿⣷⡏⠀⠀⠠⢤⣽⣿⠿⠛⠃⢀⡀⡄⠴⠃
⠀⠀⠀⠀⠀⢨⣷⣿⣡⣄⠋⠹⣿⣿⣄⠘⣆⢀⣾⣿⣿⣇⣀⣶⠁⣠⣿⣿⠏⠁⠀⣈⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⣷⣴⣿⡿⢿⣻⣛⣟⣻⣏⣻⣛⣟⣻⣻⠿⢿⣿⣦⡾⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣏⣿⠶⠛⠛⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠛⠻⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣇⣀⣀⣀⣀⣠⠤⠤⠤⠤⠤⠤⣄⣀⣀⣀⣸⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣬⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠙⠛⠿⠿⠿⠿⠿⠻⠻⠟⠛⠛⠉⠉⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⢀⣀⡀⠀⠀⢀⣀⡀⠤⠤⠔⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢴⡄⠀⡀⣤⡄⡀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣋⣾⡇⢸⠝⠁⠀⣁⠀⣀⡀⣀⡀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢩⠽⣏⡗⠉⡶⡄⠈⣿⠀⢻⡏⢻⡇⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠴⣷⣦⠖⣧⡷⠀⣿⠄⢼⡇⣼⡧⢸⣧⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠒⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡃⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠠⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠐⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⡟⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣶⣀⡠⡄⢠⣤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣷⣄⠹⠿⠋⣘⣿⣿⡄⠀⡀⠤⠂⠈⠛⠄⠆⣠⠹⠃⠐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠛⢋⣠⡐⢿⣿⣿⣿⣿⡴⢖⢆⠀⠀⠠⠀⠀⣉⠀⠀⠊⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣿⣿⣇⣠⣽⡿⠛⠁⠀⠈⢎⣷⡄⠀⢀⠄⠀⠀⠀⠀⠐⠶⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠛⣁⣦⠀⠀⢀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡋⠁⠀⠀⠈⢻⣦⡴⠿⠒⠀⠜⣿⣦⣀⢀⢀⢀⣠⣴⠛⣆⡀⠀⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠰⣶⠟⠙⣧⣀⠀⠀⢈⠌⠻⠏⠛⣉⡉⠉⠀⢿⣿⣟⡢⢴⠞⠂⣀⣀⠀⠀⢀⣀⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠁⠀⠀⠸⠉⡀⠀⢽⡀⠀⠀⠀⠙⠀⠁⠀⢡⣝⢿⡟⠛⠓⠐⠏⠉⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠐⠁⠀⠀⢸⠇⠀⠀⠀⠀⠘⡄⠀⠀⣿⣿⡇⠀⠀⢠⡆⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⣿⡇⠀⠶⢺⡗⠲⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⠃⠀⠀⣿⣿⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠆⢀⠀⠀⢀⠸⣧⣀⣀⣀⣀⣀⣀⣿⣿⣶⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠀⠀⠘⠀⣿⠛⠛⠛⠛⠛⠛⢻⣿⠉⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢠⡴⠿⢻⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠶⢶⠂⠐⠀⠀⠞⣧⢱⠃⠘⡄⠀⠀⠀⠀⠀⣼⣿⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡆⠀⠀⠀⡇⡘⣧⠀⠀⣇⠀⠀⠀⠀⠀⣿⣿⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠆⢰⠁⠀⠀⠀⢣⠁⡘⠣⢀⠸⡀⠀⠀⠀⠀⣿⣿⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠂⡏⠀⠀⠀⠀⠘⠈⡇⠀⢡⠀⢷⡀⠀⠀⠀⣿⡿⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠈⣇⢡⡆⠀⠂⠀⠀⣘⡣⢰⡃⠠⢀⡁⠳⡀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸⡗⠀⠀⠀⠈⢹⣓⣃⣰⠢⢀⢑⠀⠘⠦⣀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠒⣬⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⠋⠁⠰⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⡿⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣾⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠆⠀⠀⠀⠀⣄⡼⣧⣠⠀⠀⠀⠀⠠⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠶⣤⣬⢙⣃⠀⠀⣀⡽⣰⣎⣯⣄⠀⢀⣰⣋⣇⣠⠴⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀⠠⠊⣻⣗⠭⢛⡋⢭⠞⣫⣏⠛⠿⠿⠿⠿⣿⣿⣧⣤⣤⣴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡏⠐⠉⣿⣷⣤⣀⣤⠤⢿⢿⣦⣤⣤⣴⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣷⠚⠯⠉⢉⣹⠿⠀⣈⣥⡀⢾⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣿⣷⡶⠞⠛⠛⠉⠉⠻⠟⠉⠙⠿⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⣿⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢢⣿⣿⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣳⡿⠞⠁⠁⠠⡀⠀⡀⢀⠀⠀⠠⣈⡂⠛⢿⣾⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢳⣿⣿⣿⣷⣾⣿⡿⢿⣾⣷⣿⣿⣿⣿⡖⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⢿⣽⣿⡿⢿⠟⣴⣦⡻⠿⢿⣟⣿⡹⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣄⡠⢀⣀⣰⣶⢾⣿⣻⣷⣿⣶⣄⣀⠀⣤⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡽⠉⠙⠋⡙⠋⠘⢿⣽⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⢿⣿⢮⣭⣱⣆⣧⣽⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠻⣿⠷⠶⠷⠶⣿⣿⣟⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠣⡐⠈⠀⠀⠀⠁⠞⣿⠷⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠉⠋⠉⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢤⡀⠀⢠⡶⠉⠀⠀⡀⠀⠀⢾⡁⠤⠒⠀⠈⠀⠀⠀⢀⡄⠁⠀⠀⠀⠈⠀⠀⠀⡀⠀⡠⠖⠒⢠
⣴⠟⠛⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⢸⣵⣠⣦⢀⡀⠀⠀⠀⠀⠑⢄⠀⣀⠀⢀⢌⠅⢀⡀⠀⢰⡄⡀⠀⠀⠰⡀⠘⣁⣠⡴⣤⢣⡀
⣿⣆⡶⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠉⢿⣦⣄⡉⠀⠀⠀⠀⠀⠀⠀⢀⢫⣴⣛⢀⡶⣻⣦⣶⠞⠉⠀⠀⠀⠀⣰⣾⢿⣡⡼⢩⣿⣧
⠉⢷⣴⣶⠻⣀⡀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠔⢠⠀⢨⠏⠚⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⣹⢟⣿⣦⡬⡈⠁⠀⠀⠀⠀⠀⡜⢿⡿⠚⡻⠶⠁⠀⠈⠛⡄
⡾⣿⡿⢾⣷⣛⣳⡦⠀⠀⢿⡞⠠⠐⠚⠁⢸⣿⣷⣾⣤⣤⡖⡆⠀⠀⠀⠀⠀⠀⡰⠻⠉⠻⠉⣩⠼⠇⠀⠀⠀⠀⠀⣸⠇⢸⡷⢤⣤⠀⢸⣆⠀⡆⠐
⠇⡌⠈⠀⡄⣹⣏⡐⢨⡒⢄⠀⠀⠀⠀⠀⠈⢻⡽⠉⢿⠟⠁⠀⠀⠀⠀⠀⢀⡄⠂⠀⠀⠂⠒⠁⠀⠘⠀⠀⠀⠀⢠⡿⠀⡀⣿⣯⡈⢀⣰⣟⠀⢩
⢰⣷⣀⡴⡁⠘⣾⣷⡮⠥⠤⠀⠀⠀⠀⠐⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⣀⣖⠞⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣹⣷⡄⢉⠿⠀⣉⣴
⠞⢋⣁⣾⠃⠀⡆⣏⠀⠀⠀⠀⠀⠀⠀⠒⢸⠗⠲⠀⠀⠀⠀⠡⠀⠀⢻⣏⣻⠉⡋⡙⠡⠖⠀⠀⣯⠀⠐⠒⠀⠀⣿⠁⠀⡄⣿⣇⣈⣷⠸⠀⠘⢃⠜
⠈⢏⢱⣿⠰⡀⢠⠘⠈⢄⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠸⣴⠆⠠⢉⣉⠓⡙⣉⣱⡀⢀⡠⠒⠀⠀⠀⠀⠀⢸⠀⠀⡇⠡⡿⠟⢀⡀⣈⡦⣬⠈
⣄⠘⡸⢿⡀⢰⣞⠀⣜⡀⣱⡄⠀⠀⠠⢶⢤⡟⠣⠄⠠⠄⠀⠀⠀⠀⠀⠀⠈⠉⠟⠾⠋⣲⠉⢀⠀⣀⡀⢀⢀⡀⠈⡆⢹⣧⢀⣅⡰⣏⠀⢎⢠⣄
⣷⡀⢣⠸⣟⢸⣁⣾⣿⠻⠉⠀⢰⠣⠊⢛⣻⢛⣛⠛⠒⠀⠀⠀⢀⣀⣤⠄⠐⠲⣷⣶⠞⠾⡇⠁⠀⠀⠀⠀⠀⠀⠀⣇⠾⠋⢨⣹⡿⡇⠀⢰⠞⠀⠜
⣿⣧⠀⡀⢸⠈⢸⡇⡿⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠙⠙⡇⠀⢤⣍⠙⠀⠀⠀⠘⢳⣆⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⡠⠋⡀⢶⣿⠉⢡⠃⢀⠇
⣿⣄⠣⠙⠻⣠⠘⡃⠁⠁⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⣈⠈⠻⠶⠀⠀⠀⠈⠟⠀⠀⠀⠈⠵⠁⠀⠀⠀⠐⡀⠣⠻⢾⣫⠃⡸⠀⡘⠀⠀⠁
⠘⣝⣳⣢⣴⡙⣾⢣⣄⡷⠐⠿⠇⠤⠀⠀⠀⣈⠭⠀⠀⢠⣦⡉⠉⠀⠂⠀⠰⠂⠀⠀⠀⡠⠁⠀⠀⠀⠀⠰⠀⠀⠙⡄⠀⠠⡀⢀⢃⠞
⠀⡼⣹⣿⢀⣨⠟⢂⡁⠀⠀⠀⠀⠀⠀⠀⢠⡠⠀⠀⠀⠸⠙⠁⠐⠀⠀⢀⠀⠀⢀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠄⠀⠈⢇⠸⣦⣹⣆
⡼⠷⠞⠋⠀⠘⠶⢟⢮⡐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⡠⠂⠀⠁⠀⢀⡀⠈⠈⠀⠀⠀⠀⢀⣤⠐⠀⠀⠀⡀⢠⠀⠀⠀⠣⡀⢋⠘⣧
⢀⠀⢀⢴⠀⠀⣴⡀⠣⡀⣄⠀⠀⠀⠐⠂⢀⠀⠀⠔⠀⠀⠈⢨⠁⠀⠂⠀⠀⠳⡲⠄⠀⠀⠀⠀⣠⡿⢤⣠⡀⠠⡰⠛⠤⠸⢀⠀⠀⠀⡽⠊⢏⡂
⢨⠀⢁⡄⠸⠻⣷⠳⣶⠙⣟⢳⣄⠈⠓⠀⠀⠀⠀⠀⡰⠠⠆⠀⠀⠀⣰⡖⠀⠀⠈⠞⠀⠀⣠⣾⢻⣿⣷⡐⣿⣧⡶⠋⠀⠀⠀⠀⠀⠀⣠⠖⠀⢠⢰
⡄⠀⠉⠃⠂⠀⠈⠳⡝⠳⣄⡙⢌⠳⣬⡐⣤⡘⢓⣾⠿⢗⡄⢀⡴⢊⡼⠁⠀⠠⢤⣄⡤⠊⠁⢠⠎⣿⠋⠉⠍⠁⠆⢁⣀⠀⣀⣀⠀⢀⡈⢤⢁⣴⣷
⣄⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠙⠢⣌⠀⠉⠓⠮⣍⡉⢩⣿⢟⠁⣠⠋⠀⢠⡇⢸⢈⣿⣇⠀⢀⠄⣸⣿⡀⢀⣦⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠛
⣿⡗⠂⣠⡐⠀⣀⢤⣽⣿⣷⣦⡀⠈⠙⠶⣄⡀⢀⡉⢕⡚⣶⡽⠃⠀⠀⡀⣫⣿⡿⢿⣿⠏⠁⢠⣿⣿⡟⠋⣹⣿⠈⠼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⣸
⣿⣵⣶⢋⡵⣛⣿⣿⣿⣿⣿⣿⣿⣦⣀⣴⠛⠟⠷⠦⣼⣭⣽⣶⠶⢁⣌⡙⠳⢾⡃⠧⠍⠀⣠⡿⠿⡏⣰⣶⣿⡟⢳⣄⣿⣿⣿⣿⣿⣿⣿⢉⣁⣴⣿

⠀⠀⠀⡆
⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣶⣿⢃⣤
⠀⠀⠀⡇⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⡇⢀⣴⣿⣿⣿⣿⣿⣿⡁⠘⣿⣿⣿⠿⣿⣿⣿⣿⣿⣏⠂
⠀⠀⠀⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⢆⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡾⣿⣿⣿⠿⢿⣿⣿⡅⡀
⠀⠀⠀⡟⢻⣮⣿⡟⠁⠈⠙⣿⣿⡇⠙⠿⣿⣿⣿⣮⣭⣵⣿⡈⢿⣿⣄
⠀⠀⠀⡇⠀⠻⣿⣧⢀⣤⣾⣶⣿⣿⣆⠀⠻⣿⣿⣿⣿⣿⣿⣿⣦⡳⠙⠃
⠀⠀⠀⡇⠀⠀⢫⣿⣿⡿⣿⣿⣿⡿⣿⢿⢞⢻⡿⣿⣿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⡇⢰⣶⣾⣿⣿⣥⣻⣿⣿⠁⠙⠘⠄⠁⠃⠀⠙⣿⣿⣿⣿⣿⠁
⠀⠀⠀⡇⣾⡟⠋⢻⣿⣿⡿⣻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⣷⡟⠁⠀⠈⢿⣿⡟⢻⣿⣿⣿⣧⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠤⠖⠁⠀⠈
⠀⠀⠀⡟⠀⠀⠀⠀⠀⠉⠃⣸⡿⣿⣿⣮⢻⣷⠀⠀⠀⠀⠀⠀⠘⣁⣤⡔⠁⠀⠀⠀⢀⠀⢀⠔⠀⠠⠊
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡟⣷⠀⠹⣿⣧⠹⣄⡀⣀⣤⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣿⣞⠥⠒⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣷⣄⣼⣧⣠⣞⣡⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠋⡟⠉⠁⠉⠉⠻⣯⡉⢛⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣾⡟⠁⠀⠀⢈⠀⠀⠀⠀⠀⢹⣿⠀⠈⠻⣷⡀
⠀⠀⠀⠀⠀⢠⡿⢻⡅⠀⠀⠀⠺⣤⣀⣤⡴⢂⣾⡿⠀⠉⠛⠻⣿⡆⠀⠀⠀⡀⠀⢀⡄⠀⢀⡄⢀
⠀⠀⠀⠀⠀⢸⠃⠈⠛⠃⠀⢉⣒⣺⣿⣿⠾⠟⠋⠀⠀⠀⠀⠀⡿⠁⠀⠉⠲⣼⣶⣼⣷⣴⣿⡶⠋
⠀⠀⠀⠀⣠⠜⠀⠀⣠⡶⡟⠛⠉⠉⠀⠀⠐⣒⡦⠄⠀⠀⢤⡂⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣥⣤⣤⣄⡀
⠀⠀⠀⣼⡏⠀⢠⣾⢫⡾⠁⠀⠀⠀⠀⠙⠛⠛⠉⠀⠀⠐⠋⠈⠀⠀⢀⣴⠟⠋⢡⣾⠟⠉⠀⠀⠀⠀⠙⠻⣷⣄
⠀⠀⢸⣿⠀⠀⣿⡇⡟⣇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣧⣤⠄⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀
⠀⠀⢸⣿⣦⡟⣿⣧⠃⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠟⠁⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠠⣼⠏⠹⣇
⠀⠀⠘⣿⣿⡇⢹⣿⣷⣤⣀⣴⡇⠀⠀⠀⠀⠀⠰⠾⠿⠃⠀⠀⠀⠻⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⣦⣤⣄⣀⠀⠀⠀⠿
⠀⠀⠀⠘⠿⣿⣄⠙⠿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡀⠀⠐⣲⣭⣁⣀⠀⠀⠀⣌⡙⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠉⠉⠁⠀⠀⠀⠈⣿⣆⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⢹⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠋⠀⠀⠀⠀⠀⠸⣷⣤⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⣰⣬⣫⣀⣴⣀⡜⣣⣤⣠⠀⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣧⣈⡏⣠⣤⠀⠀⠘⠛⠙⢿⣿⣿⣿⣿⠿⠛⠯⠀⠀⢀⣤⣄⠻⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣦⣍⣱⣿⣿⣿⡿⠉⠀⠀⠠⠂⠒⣦⡹⣿⡟⣡⣆⢀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣎⣭⣴⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⡆⣤⡄⠀⠘⠛⠙⠻⣿⡟⣴⠅⠀⢀⠀⠀⠀⢹⣷⣿⣷⣿⠁⠀⠀⠀⢠⠖⢺⣦⣻⣿⠟⠋⠙⠃⠀⣠⣤⢲⠄⠺⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣘⣷⣿⣿⢁⣀⠀⡎⠙⢷⣾⣿⣿⡀⠀⢸⣼⠀⠀⣼⣿⣿⣿⣿⡀⠀⢐⣶⠃⠀⢠⣿⣿⣷⡾⠛⠅⠀⠀⠉⣿⣶⡖⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⢿⣷⣾⠉⠳⣆⠀⠘⣿⣿⣿⣿⣦⣴⣿⣦⣾⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣶⣿⣿⣿⣿⢁⠀⣠⠀⠀⣇⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣤⣻⣶⣿⣿⣿⠿⢛⣟⣫⢏⣹⢯⣠⣿⣅⣼⣭⣼⢻⣩⢯⣛⡟⠻⢿⣿⣿⣿⣼⡏⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡟⢻⣽⣤⣶⠿⠷⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠿⢿⣶⣼⣿⠻⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣯⣼⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣧⣄⣀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠤⢤⣤⣤⣀⣀⣠⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣭⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣭⣽⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠿⠿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⠁⠉⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠇⠃⠁⠀⠀⠀⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢻⣷⡀⠂⢠⡄⣴⣿⡀⡰⠀⠠⣴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠻⠓⣵⣿⡇⠘⣏⡿⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠿⠿⣿⢻⡃⣼⣏⡀⠀⠀⠰⣿⡖⢠⣾⣳⡤⣾⣳⠆⢠⣴⡞⢿⢶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢰⡖⠻⣧⣼⠃⠀⢸⡿⡆⠀⠀⣿⡇⠀⢸⣿⠁⢸⣿⠀⢠⣿⡇⢸⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣡⣤⣭⣤⣄⣤⢾⣄⢷⡄⢀⣿⣇⡀⣸⣿⡆⣸⣿⣀⢸⣿⣇⣸⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠉⠉⠻⠿⠁⠌⠻⠟⠀⠈⠙⠋⠀⠙⠟⠁⠉⠿⠁⠊⠙⠋⢹⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣶⣶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀
⠀⠀⠀⠠⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢠⠴⠀⡤⠔⠉
⠀⢰⠤⠀⠙⣆⡀⠀⠈⠉⢁⡄⠀⠀⠀⠀⢠⢀⣀⣀⠀⠀⣠⢶⣂⢤⡀⠉⢁⣠⡴⡄⠀⠀⢀⡌⢀⡌⠀⣠⠖⠚⠁
⠀⢈⠀⠈⠠⡄⠰⠤⡄⠀⠀⠀⠶⠀⡀⠐⠀⠋⠙⠿⠁⠀⠈⠙⢿⣰⡏⠐⣋⡿⠑⠉⢉⣠⡈⢀⠂⠑⠄⠀⠸⢎⡁
⠀⠘⠀⠀⠀⢀⠤⠀⠛⠦⠀⠀⠀⠀⢑⣲⡀⠃⠀⠀⠀⠀⠀⢰⠈⠁⠀⠀⠉⠀⠀⢙⡍⡼⠁⠀⠀⠀⠀⠀⠄⠘⠃
⠀⠠⠀⠀⠠⠋⠀⢠⡀⠈⠁⠀⠀⠈⠻⣿⠉⠠⣄⠀⠀⠀⡀⠈⠀⠀⠀⠁⠈⠉⠀⠋⢿⡤⠃⢀⡀⠄⡀⠀⢀⡀⠄
⠀⢀⠀⠀⠀⠀⢀⡈⠁⠀⠀⠀⣈⠉⠳⠎⠀⠀⠈⠖⣺⡟⠙⣶⡀⠀⠀⠀⢀⣛⠉⢀⠄⠐⠀⢀⣁⠤⡅⠠⠃⡈⠃
⠀⢸⠀⠀⠀⣀⠈⠁⠀⠀⣈⡀⠉⠀⣤⡶⠦⣼⣿⣞⡛⢿⡾⣧⠘⣧⡠⠴⠛⢛⣻⣻⣦⡀⠀⢻⣫⡜⠻⠁⠔⢀⡄
⠀⠰⠀⠀⣤⡉⢉⠀⠀⣼⠻⠮⡙⢄⠉⠁⠀⠻⠏⠈⢉⠿⣤⣿⡆⣩⣧⠤⢾⡭⢥⣬⣉⣿⣶⠀⠉⠀⠋⠈⢐⢋⡁
⠀⠰⠰⠦⠀⠀⠀⠠⠶⣄⡱⡿⢡⡄⠀⠀⠀⠒⠒⣒⡟⣺⠏⠃⠉⣉⣿⣦⡜⡖⡚⣃⣬⠿⣦⡄⢀⠀⠀⠀⠀⣬⠂
⠀⠰⠀⠀⠀⠠⠄⠀⠀⡛⠭⣃⡘⠙⣼⣤⠠⠾⠁⠀⠀⡎⠀⠀⠴⠀⡺⣹⠶⠦⡄⠤⠤⠁⢾⠗⠀⣆⣀⠋⠐⠀⡇
⠀⠘⠀⠀⣐⠚⠃⢀⡘⠙⢲⠏⣙⠛⠅⠉⣗⣴⡶⢚⠽⡆⠠⠄⢀⣴⡷⣳⣖⣒⡀⠰⠊⡙⢢⠤⢤⢁⠲⣤⡈⠹⠇
⠀⢰⢀⣐⠋⢡⣄⡈⠉⠁⢺⣃⣤⡘⢉⣴⣿⣿⣿⣤⢊⡌⣰⣾⣇⠠⡘⣿⣯⣿⣏⣤⠞⠀⠂⢸⠀⢾⡂⣹⣆⠑⡄
⠀⢨⠍⠀⣀⠈⠉⠉⠉⠉⢹⠁⠀⠀⣼⣏⣿⣿⠀⡙⠉⡱⠟⢘⠟⠶⢻⣯⣿⠁⠈⢹⣇⠐⠀⠀⢈⠀⣻⡇⠈⠆
⠀⢈⠀⠘⠛⠓⠒⠀⠒⠐⢾⡀⠀⠀⡾⠉⢿⣿⠊⡀⠨⠐⣥⣸⠐⠀⢸⣿⡯⠚⠠⣿⣿⠀⢠⠲⢾⣁⣉⣤⠂⠁
⠀⠸⠄⠀⢠⡀⠀⡀⠀⠀⣤⣧⡀⡼⠛⢤⡺⣿⠀⠀⠀⠑⡀⡉⢀⡔⢢⡿⠁⠀⠈⠿⢿⣷⣿⡄⠀⠀⢃⡠
⠀⢠⣄⡐⡜⡓⣀⠀⠀⣀⡄⠈⠉⠉⠓⠉⠀⢹⡇⠀⠀⠁⠀⠐⠁⠀⣿⠙⣦⣄⡐⠒⠉⠀⠀⢘⣦⡀⠈⡔
⠀⢈⣿⠷⣤⢺⣧⢽⡿⢢⡀⠀⢤⡄⠀⠀⣀⢀⣿⣄⠀⣀⣵⠦⠀⠐⢿⣆⣀⣙⠛⠺⣭⠦⠚⠋⠀⠓⠋⠁
⠀⢰⣸⢋⡿⢿⣧⣴⠋⠻⠇⠀⠠⣠⡷⠯⣽⣥⡈⠉⠉⠹⠿⢾⣀⡴⠶⠏⠀⣈⣤⣬⡯⢿⠭⢀⠀⠀⠀⠀⠀⠀⠁
⠀⢸⢛⢿⣔⣙⢣⣠⣤⣤⣀⡀⢴⡏⠀⠀⠀⡉⠉⠲⢤⣜⡶⠀⠁⠀⣠⣦⡾⠋⣠⠀⠀⠀⣸⡟⢀⡄⡀⠀⠀⠀⡄
⠀⢘⡙⢻⢯⣭⢿⠏⠁⠜⢰⠉⢈⣿⣶⡇⠀⠀⢀⡀⢀⠹⣴⣤⢦⣤⣞⣻⠀⠀⠀⢦⣤⠖⠹⢇⠀⠄⠐⠖⣶⠂⠂
⠀⠘⠁⣬⣲⣓⡾⠒⠒⣶⢃⠴⠋⠀⠉⠙⠗⠲⣦⡀⠀⠠⣙⢷⣞⠋⠀⠀⣦⡤⠶⠟⠑⣆⠀⢰⡔⠀⠰⠀⠀⠀⡄
⠀⠈⣰⢏⠀⠸⠓⠀⢀⣆⠙⠶⣈⠀⠂⠀⠀⠀⠀⡰⢳⠾⣅⠀⠉⡓⢾⡽⢯⡀⠀⠀⠀⠈⣆⠈⠓⠀⠀⠀⠀⡏
⠀⠈⠹⡀⠀⠀⠀⠀⡘⠉⠀⠉⢻⠑⢦⢄⣀⣀⢶⠠⣟⡢⠊⠛⠶⢍⡀⠻⣄⠻⣄⣰⣄⠿⠝⠀⠀⢹⠀⡞⠈⢱⡆
⠀⢠⣀⣿⠀⢠⠀⢘⠀⠀⠀⢀⣬⠉⠉⢻⠉⠿⡌⢻⣙⣯⣐⢰⡀⠀⠉⣷⣟⠱⠈⢿⠈⠁⢷⠴⠀⠺⠀⡇⠀⠀⡇
⠀⠈⢹⣯⣀⣳⠀⢹⠓⠒⠚⠋⡙⢚⢛⠏⣹⠒⣽⣻⠟⠛⠫⣟⠝⠻⢾⣷⢾⡽⠳⣼⠀⠀⠀⡀⠀⠧⢤⡷⠂⠈⠁
⠀⠐⡼⠂⠛⢽⡇⢸⠙⠁⠀⠀⠀⠌⣤⡵⢢⡄⠀⠘⣆⠀⠀⣼⠆⠀⠚⣄⣀⠤⠒⢚⣒⣉⣉⣉⡉⠐⠂⢸
⠀⢸⠁⠀⠀⡰⢢⡟⠠⣌⠄⠛⠀⣎⠹⣏⣓⠋⠀⢀⡼⢣⠉⢻⠆⠀⠐⡋⠀⠀⣋⠩⣁⢍⠹⣀⣀⣻⠁⠘⣷⠄
⠀⢀⠀⢀⣊⠀⠀⠈⠙⠟⠚⠛⠛⠛⠲⠤⠥⠴⠖⠋⠙⠊⠃⢸⡆⠀⠀⣱⢦⡀⠀⠈⣉⣈⣉⣁⣈⣀⠲⠁⠙⡆
⠀⠈⠉⢹⠟⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣄⣶⣞⣡⠞⡟⠛⣿⠉⠙⡆⠀⠈⣿⣁⣠⣀⣈⡟⢳⣀⠀⠀⢁
⠀⠀⠀⣼⠦⠄⠀⢈⣹⠗⠲⠦⠤⣶⣶⣾⣿⠛⣽⠁⠀⠀⠓⣶⣶⡿⠀⠰⣋⣀⣤⠀⠽⡇⠈⠉⠑⣙⣶⣶⠆⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠋
⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣙⣿⣿⣽⡍⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⡘⢻⡿⢂⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡴⠋⠈⠉⣻⣟⠋⠉⠈⣧⠀⠀⠀⠀⠀⢠
⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⠀⠀⠀⠁⠀⠀⢠⣤⣽⡀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢀⣀⣆⠀⠄⠀⠀⠀⢠⣷⣿⡶⢲⡆⠀⠀⠀⣴⠙⣛⣦⠤⠀⠀⢰⠖⢶⣿⣿⡄⠀⠀⠀⠠⠀⣢⣀⡄
⠀⠀⢠⡿⡿⡄⠀⠀⠀⣴⠛⠛⠿⠄⠀⠃⠀⣀⢂⣭⣸⣏⣭⡀⣀⠀⠘⠀⡰⠿⠛⠛⣦⠀⠀⠀⣠⢿⢯⡄
⠀⠀⠈⠀⠀⣾⡂⡠⠀⠈⢀⣤⠴⢖⠓⠋⢉⣉⣉⣉⣉⣉⣉⣉⣿⡿⠿⠛⠲⠦⣄⡈⠁⠀⣅⢰⡿
⠀⠀⠀⠀⠀⠈⠅⣻⡷⠋⣁⡤⢶⠞⠛⠛⣉⠀⢠⣾⣀⣀⣷⡄⠀⣉⠛⠛⠲⡦⢤⣈⣱⣾⣟⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠙⣿⠛⡁⠀⢠⣤⣴⣾⣷⣶⣶⣭⣷⣾⣭⣶⣶⣾⣷⣦⣤⡄⠀⣈⠻⣟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢞⣹⢾⠋⠋⠉⣀⣀⣠⣤⣤⣤⣤⣴⣶⣏⣁⣉⠉⠙⠙⠷⣏⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⣼⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠁⠀⠈⠉⠧⠄⠀⠀⠀⠀⠀⠈⠙⠛⠙⠉⠁

⡏⠭⠭⠍⠥⠅⠤⠬⠭⠌⠤⠤⠥⠤⠭⠬⠭⠭⠬⠭⠌⠍⠭⠩⠩⠩⠥⠥⠤⠤⠤⠤⠤⠠⠥⠨⠥⠡⠁⢀⡉⠉⠉⠭⠉
⡇⣠⠶⠞⠁⢈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⠓⠢
⡇⡇⢀⠈⠁⢠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡧⠶⢤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⡐⡂⠀⡀⢰
⠀⢇⣙⣣⡼⣚⣽⣷⣦⣤⡤⠤⠤⠰⠴⠒⠂⠈⠀⠐⠋⢰⣁⠀⠀⠀⠁⠁⠀⠀⠐⠂⠀⠀⣀⣠⡿⣿⣦⣙⠦⣈⣃⠟
⠀⠀⠙⣷⢾⢿⡋⠛⠯⣝⡙⠛⠛⠂⣠⢰⣤⢊⣇⣶⣰⠷⣶⠟⣦⡖⠶⠖⣆⣤⡉⠁⠀⠀⣉⠥⠞⢋⣿⡿⣿⡏⠁
⡀⠀⠀⡟⢸⢿⠛⠢⣄⣤⡈⠀⠀⠠⡷⣅⣹⡰⢦⣤⠶⣄⡷⣠⠄⣤⣜⣈⣧⡨⢐⠀⠀⠉⠀⣀⠴⠊⠀⣍⠻⣧
⠀⠀⠀⡗⢸⣿⠀⠀⠀⢉⡑⠒⠒⠒⢳⡽⢿⣶⢬⣹⠶⠧⢾⠼⣓⣿⠤⢰⣿⢁⠚⠒⠒⠉⢁⡄⠀⠀⢠⡋⡄⣿
⠀⠀⢸⡏⢸⡿⡄⠀⠀⠘⠇⠀⠀⣦⠀⠉⢻⣥⣭⠒⣂⣻⣽⣏⣱⡖⡵⢚⡏⠀⠀⡆⠀⠀⠘⠀⠀⠀⢨⡅⡇⣟⡆
⠀⠀⠸⣇⠀⣿⡇⣠⠀⠀⠀⣤⢤⡤⡤⠤⣤⠿⠶⠿⠯⠿⢽⡽⢿⣿⣶⣿⣤⣤⣤⣤⣤⣄⠀⠀⠀⣴⠸⠃⡇⣏
⠀⠀⡴⣿⢠⢻⡇⠛⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠃⣸⠀⡇⣿⠀⡀
⠀⢀⡇⠂⠈⢸⣷⠀⠀⢰⡆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠃⠀⠀⡧⢸⠇⢿⠀⡇
⠀⢸⡇⡀⠘⢸⣿⠀⠀⠀⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⡇⠘⠀⠀⢰⡇
⠀⢸⠁⠇⠀⠀⣶⡀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣤⢠⣇⠀⠀⠀⠀⠇
⠀⢸⣀⣤⡤⡾⢟⡆⣻⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠻⢨⣍⠛⠲⣄⠃⡄
⠀⢸⠋⠃⠀⢐⡹⠃⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⢬⣙⠲⢼⣈⡃⢸
⠀⣎⣤⠶⠛⠛⡇⠀⠀⢀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡀⠀⠀⢸⡛⠛⡳⣮⣳⣸
⠀⣿⣒⣒⣀⣀⡇⠀⠀⠘⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⠇⠀⠀⢸⣥⣤⣄⣈⠝⢹
⠸⢈⠻⠉⡍⠉⡃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⠀⢸⡇⠄⢈⣀⡀⣸
⠀⠈⡀⠀⣡⡾⠁⠀⣶⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡁⠀⣴⠀⠀⠙⢶⣬⡛⣇⣾
⠀⠘⣣⠞⠋⣀⠀⠀⠈⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠹⠄⠀⢠⠀⠙⠿⡿⢿
⠀⢰⡏⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⡀
⠀⠈⠻⣦⣀⠁⠀⠀⠀⠈⡇⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡴⠀⠀⠀⠀⢀⣠⡞⠃
⠀⠀⠀⠈⠙⠻⠶⣦⣤⣀⠁⠘⢧⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡴⠋⠀⢁⣠⣤⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⡀⠀⠈⠉⠉⠉⡍⠉⠙⢦⣀⡴⠋⠉⠉⠉⡅⠀⠀⠚⠁⣠⡴⠟⠋⠉
⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⢶⠀⠙⠷⢦⣤⣤⣤⣤⣤⣄⣀⣙⣀⣀⣠⣤⣤⣥⣤⣤⠶⠞⠃⢰⠃⠈⠛⠚⠛⠛⠛⠓⠓⡆
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⢤⠸⠤⠤⠤⠤⠤⠤⠤⠤⠾⠀⠂⠀⠀⠄⠂⠀⠀⠐⠀⠂⠂⠐⠐⠀⠀⠀⠂⠀⠀⠀⠀⠒⠘⠒⠒⠒⠒⠒⢒⣒⣒⣒⣃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠙⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣦⣄⣀⣤⣤⣼⠶⠞⠋⣠⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⡛⠉⠉⠁⣠⣤⡿⠋⣱⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡀⠀⠀⠛⠛⠛⠛⠋⠉⠀⢀⣤⠿⠿⠃⣴⠿⠿⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠆⠉⢣⡼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⣰⣿⡇⠘⠀⠀⠀⠀⠀⠀⣤⠾⠉⠀⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⢀⣀⣹⠇⠀⠀⠈⠛⠾⠷⠘⠋⠀⠀⠀⠀⣿⠟⠉⠙⠳⠄⠀⠀⠀⠉⠉⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠈⣿⡟⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⢰⡇⠀⡀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⣿⡃⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⣰⡿⠀⠀⢀⠀⠀⠻⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠰⣿⠁⠀⣠⠞⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣟⣠⠀⠀⠀⠀⠠⢤⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⢴⣿⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣉⣡⣼⠖⠈⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠋⠉⠙⣿⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⡦⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠶⣶⢿⣧⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠈⣶⡀⠀⠀⠀⡰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣹⣷⣦⣶⣦⣤⣀⣤⣄⣤⣤⣠⣤⣤⣤⣄⣀⣠⣤⣼⠏⠀⠀⣿⡃⠀⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡄⠀⣼⠏⠩⢴⡶⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢻⣿⢷⡄⠀⠀⠘⢻⠀⠠⠞⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣠⣼⡟⠉⡟⠁⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣠⣎⣵⡄⠀⠀⠃⠀⣀⡀⠀⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡉⠐⠒⠙⣿⡿⠀⣿⠛⠒⠀⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠒⠀⠈⠉⠉⢉⣟⣧⡀⢰⠟⠀⠉⠻⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⢶⣆⠀⣿⠃⢰⡟⠀⠛⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠓⠺⠋⢻⣧⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⡀⢿⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠖⠲⢿⣿⣿⡇⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⠀⠀⢀⣤⡀⠀⠓⠀⠀⠀⢀⡀⠀⢀⣀⠀⠀⠀⠒⠒⠺⢇⣿⣇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠁⠀⢠⠉⠀⠀⠀⠀⠀⢠⣶⠄⠀⠀⠈⠀⠻⠿⠛⠲⣶⣾⣿⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⡾⠉⠁⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⡇⠀⠀⠀⠈⠀⠀⢠⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⢀⣸⣿⡇⠀⠀⠀⠀⢀⣾⡿⣍⠁⢹⡷⣄⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣶⠟⣦⣀⠰⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠋⠻⢯⣽⠃⠀⠀⠀⣰⠀⣿⣧⢻⣷⠎⠀⠘⣠⠆⠀⠀⠀⢸⡄⠀⢀⡼⠋⠀⠘⠿⠷⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠣⣤⣼⣿⠀⠀⠀⡸⠃⢠⡟⠁⢸⡟⠀⠀⠀⠙⣀⠀⠀⠀⣼⣅⣴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡦⠠⠼⠃⠀⣾⠇⢠⣿⠀⠀⠀⠀⠀⠻⠟⢻⡦⠘⠋⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠄⢀⡀⠀⢠⡆⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⣼⣇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠰⠲⠶⠶⠰⠶⠿⢣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣤⣤⣤⣤⣤⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢠⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣤⠄⠀⡀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣶⡶⠎⢹⡇⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⢠⣶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡶⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡷⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠋⠁⠠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡠⢬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢶⠄⠉⠡⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⢶⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠃⠀⢰⣏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠘⠛⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⡿⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠶⣏⣁⣀⣀⠀⣤⠤⠀⠀⠰⠆⢿⡁⣹⡿⠿⠃⠀⠀⠀⠀⠀⠀⠰⠉⠀⠋⢁⣤⣴⠶⠂⠉
⠀⠀⠀⠀⠀⣠⠐⠾⡍⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠛⠉⠁
⠀⠀⢠⡄⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡶
⡠⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡟⠛⠉⡛⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣀⣤⣀⣠⣤⣤⣄⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣼⠟⢻⣆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠻⢷⣶⣶⠿⠏⣼⣧⠹⢷⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⢴⣾⣿⣿⠸⣗⡤⠔⣒⡛⠻⠿⣟⣒⣮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣾⠷⣢⠴⣋⣠⣥⣶⣶⣶⣿⣿⣷⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⢿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣸⣿⣿⣿⣿⣟⡥⣢⣾⣿⣿⣿⣿⣿⠏⠀⠈⢿⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣿⣿⣾⣿⡿⠿⣟⣛⣉⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡾⠀⠀⢀⣀⣈⣙⠛⠃⠀⠴⠞⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢿⣧⠀⠀⠈⠛⠘⠟⠁⠀⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡟⠁⠈⠈⠓⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⠟⣿⣿⣿⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹⡇⣤⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠐⢶⣟⣻⣇⣾⣝⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⢸⣿⣿⣿⣿⣿⠉⠳⣦⣄⣀⠀⠀⠀⠀⢁⣾⡿⠿⣯⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣧⡻⣿⣿⣿⣿⠀⠀⠀⠙⢿⣿⣷⣶⣶⣾⣿⣷⣄⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⢹⣿⣿⡀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣆⠻⣿⡇⠀⠀⠀⠈⠛⠛⠿⣿⣿⣿⡛⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡝⣿⣍⡛⠻⠷⣌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠟⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠤⢠⣸⡗⠀⠂⠀⠀⠁
⠀⣀⣀⣀⣠⡀⠀⠀⢸⣾⣇
⣾⣵⣿⣿⡿⠷⠤⠜⠋⠉⣿⡄
⠙⠛⠟⠁⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⢀⣤⣴⠿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢨⡅⢤⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⡤⢧⣀
⠀⠀⠀⠀⠀⠀⠀⣶⡟⠉⠁⠈⠉⠑⣤⠰⠶⠦⢤⣀⢀⠀⠀⠀⠙⠿⠿⢿⡿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⢰⣿⠀⠀⠀⢈⠀⠀⠀⢠⣤⡤⠤⠤⢤⡤⠤⢄⡀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢸⣇⠳⣤⣀⣈⣀⣠⡤⢊⡿⠀⠀⠀⣾⠁⠀⠀⠈⢷⣄⠜⠁⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢥⣀⡀⠀⠀⠀⣼⠃⠀⠈⡿⠷⡮⢭⠭⠭⠥⠔⠋⠁⠀⠀⢸⣇⢠⠀⠀⡠⠔⠁⣠⠞⠑⠴⠂⢉⠔⠒⠒⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡆⢠⡟⠀⠀⠀⠘⣦⡑⠢⢀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠀⢠⣀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⡾⠁⠐⠀⠀⠀⠈⢷⣄⠀⠈⠐⠂⠠⠄⠀⠀⣿⠃⣴⠖⠋⠁⠀⠢⡄⣀⠀⠠⢸⡇⠀⠀⠈⢁⡼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣊⠃⢠⠇⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⢻⡀⣹⠀⠀⠀⠀⠀⠀⠈⠙⠀⢸⡇⠀⠀⠀⠈⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣿⠔⠁⢀⠀⠀⠀⠀⠡⡀⠙⠲⠤⠀⠀⠀⢸⡇⠙⡆⠀⠀⠀⠀⠀⠀⠄⠀⢸⡇⠀⠀⠀⠀⠖⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣤⠂⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠈⣷⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⠤⡾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⡄⠸⡗⠂⠀⠀⠀⠀⠘⠀⠈⠉⠁⠀⢾⣇⠈⠂⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⢬⣷⣄⣀⣤⡀⠀⠀⠀⢀⣀⡠⢂⣨⣧⢀⢻⡀⡘⠀⣠⡤⠶⡀⠀⣾⠛⠋⠀⠙⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠈⠑⢦⣄⣉⣉⣉⣉⣉⣉⣩⣭⠗⠙⣇⠈⣴⠃⠉⠀⠀⠀⠇⠀⢻⡆⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣡⠟⠓⠂⠠⠄⣀⢉⣉⡉⠉⠉⠁⢀⣀⣠⡴⠿⣦⡘⢧⠀⠀⠀⠀⢸⠀⠘⣧⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠋⠀⠀⠀⠀⠀⠁⠀⣸⠟⠁⣈⢿⠁⠀⠀⠀⠀⠀⠉⠻⠦⡀⠀⠀⠘⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠁⠀⠀⠀⠀⠀⣀⡴⠛⠁⣀⠔⠁⠀⠳⣄⣀⣙⣛⣦⠤⠶⡤⣈⠠⢄⠀⠀⠀⠘⣧⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠡⠐⠀⠁⢀⣠⡤⢒⣉⢁⣤⠔⠊⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⠈⠉⠉⠉⠲⢦⣝⢦⡀⠀⢹⡆⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠉⠀⣀⠤⣚⡽⠟⠛⠉⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠑⢄⠀⠀⠙⣧⡙⣆⠀⠳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡴⢊⠐⣋⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⠣⡀⠀⠘⣧⠘⡁⠐⠇⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢀⢞⣾⠃⠀⠏⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⡋⠀⠀⢱⠀⠀⢻⡄⡇⠀⠀⣀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⣈⠎⡿⠀⠘⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠃⠀⢸⡇⣿⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠈⣤⠀⠀⠀⠀⣲⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡲⠤⠄⢼⡇⢹⡄⢠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠀⣿⠉⠂⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⢳⣼⣧⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡸⠏⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡞⡀⠀⠀⠀⢹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠞⠛⢳⡄⠀⠳⣄⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⠀⠔⢋⣤⠶⠟⠛⠛⠓⠒⠤⢤⣄⣀⣀⣀⣀⣠⠤⠖⠚⠉⠁⠀⠀⢀⡴⠟⠀⠀⢀⣸⢆⡀⣀⡀⠙⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡉⠛⠀⢘⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠲⠖⠛⠛⠁⠸⠋⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠉⠒⢲⠋⠳⢾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠖⠒⠂⠀⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠏⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣐⣶⣦⣄⡀⣤⣸⣧⣠⠀⣠⣤⣶⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣷⣤⢤⣤⣈⠙⢿⣿⣿⣿⣿⡾⠛⢉⣠⣤⣴⣾⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣞⠉⠉⠉⠉⠙⠳⠈⠻⣿⡄⢿⣿⣿⣿⢁⣾⠟⠉⠐⠋⠉⠉⠉⢙⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠙⢻⣿⣿⣿⣿⣶⣶⣤⡀⠀⠈⢻⣼⣿⣿⡇⠞⠁⠀⠀⣀⣴⣾⣿⣿⣿⣿⡟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣖⠒⢛⣉⣤⡦⠍⠍⡉⢿⣿⣿⣄⢿⣌⣦⣿⣿⣠⢠⡾⣠⣾⣿⡯⠉⡉⠉⢬⣥⣉⡓⢒⢲⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⢿⣿⣿⣯⣴⣦⣴⣶⣶⣤⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⣶⣶⣶⣮⣽⣿⡿⠿⠿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⡲⠄⠔⠒⣚⣿⣿⣿⠋⠙⣻⣿⣿⡿⠿⠿⠻⠿⢻⣿⣿⡟⠿⠿⠿⠿⢿⣿⣿⡟⠋⠹⣿⣿⣿⣟⠒⠦⠤⢖⣲⣶⣦⡀
⠀⠀⠀⠀⠀⢀⣿⣯⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣦⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣼⣿⣿⣿⣿⣟⠿⠿⠿⣿⣿⣽⣿⡀
⠀⠀⠀⠀⠀⢸⣿⠋⠁⠀⢀⣀⣤⣄⣶⣿⣿⣿⣿⣿⣿⡇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣠⣤⣀⣀⡀⠉⠛⣿⡇
⠀⠀⠀⠀⠀⣾⣇⢀⣴⣿⣿⠿⢟⡛⠝⠛⣩⣿⣿⣿⣿⣿⣶⣿⣿⣦⣀⠀⠀⠀⠀⢀⠀⣀⣴⣾⣿⣶⣿⣿⣿⣿⣿⣯⡻⢯⣟⡿⢿⣿⣿⣷⡄⣼⣷
⠀⠀⠀⠀⣸⣿⣿⣿⡿⠛⢁⠉⠁⠄⣾⡿⢟⣽⣿⣿⡇⠙⠿⣿⣾⢽⣿⣿⣿⠠⠤⢿⣿⣿⣿⣧⣿⠿⠃⢸⣿⣿⣮⡙⢿⣷⠠⠀⠁⠈⠛⢿⣿⣿⣿⡆
⠀⠀⠀⣰⣿⣿⡟⠁⠐⠈⠁⣀⣴⡿⠋⠀⢉⣵⣿⣿⣷⣄⠀⠀⠉⠻⣿⣿⠃⠀⠀⠈⢿⣿⡟⠋⠁⠀⣀⣼⣿⣿⣮⡉⠀⠙⢿⣦⣀⠀⠁⠂⠙⢿⣿⣿⣄
⠀⠠⠾⠛⣹⣯⡁⠀⠀⢠⣶⣿⠋⠀⠀⣤⣿⣿⣿⣿⣿⣿⣷⣷⡆⠀⣹⠃⠀⠀⠀⠀⠈⣯⠀⢠⣶⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠹⣿⣷⣄⠀⠀⢈⣿⡏⠛⠦
⠀⠀⠀⢠⣿⣿⣷⣴⣶⣿⡿⠁⢀⣔⣴⡿⢟⣽⠟⣉⣽⣿⣿⣿⡇⠀⢿⣂⠀⠀⠀⠀⣀⣿⠂⢨⣿⣿⣿⣿⣍⡛⢿⡙⢿⣦⣲⡀⠈⢿⣿⣷⣴⣾⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⢀⢮⡿⢋⣴⣞⣥⣾⣿⣿⣿⣿⡿⠁⠀⠀⢫⣿⣶⣶⣿⡿⠃⠀⠀⢻⣿⣿⣿⣿⣿⣦⣹⣦⡙⢿⣷⡀⠈⢿⣿⣿⣿⣿⣿⡂
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡋⣎⣿⣿⠟⣵⣿⣿⢟⣿⣿⣿⣿⣧⠤⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠤⣼⣿⣿⣿⣿⠻⣿⣿⣯⠻⣾⣷⡇⡹⡌⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠸⣿⣿⣿⣿⣿⡏⣑⣸⣿⠃⣸⣿⡟⠁⣾⣿⣿⣿⣯⠉⠀⠀⠀⠀⢹⣿⣿⣿⠀⠀⠀⠀⠈⣟⢿⣿⣿⣷⡈⠻⣿⣧⠸⣿⣿⠉⣻⣼⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⣿⠏⠀⠈⣿⣿⣿⡿⣿⠀⠀⢀⣤⣶⣿⣿⣿⣿⣷⣤⣀⠀⠀⣸⣿⣿⣿⣿⠁⠀⢻⣿⠃⢹⣷⣿⣿⣿⡟⣿⣿⡿
⠀⠀⠀⠀⢹⣿⡟⣾⣿⣿⡟⣿⢳⣿⣿⠀⠀⢰⣿⣿⣿⣿⣿⣤⣾⣿⡿⡟⠻⠿⠿⠛⡿⣿⣿⣷⣦⣿⣿⣿⣿⣿⡇⠀⠘⣿⣿⡎⣿⢻⣿⣿⡇⣿⣿⡇
⠀⠀⠀⠀⠈⣿⣧⢿⣿⣿⣇⣿⢸⣿⡿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡏⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⣧⠀⠀⢹⣿⣷⣿⣿⣿⣿⡇⣿⡿
⠀⠀⠀⠀⠀⠹⣇⠸⣿⣿⣿⣿⣿⣿⡇⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣄⠀⠀⢠⣀⣿⣿⣿⣿⣿⣿⣿⣿⡇⢿⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⣿⠃
⠀⠀⠀⠀⠀⠀⠹⡄⢿⣿⣿⣿⣿⣿⡇⠀⠀⡿⢸⣿⡟⢻⣿⣿⡿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡇⣽⠆⠀⢈⣿⣟⣿⣿⣿⡟⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⣷⠀⠀⠁⢸⣿⡗⢸⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⢸⣿⣿⠛⢀⣿⠇⠸⠀⠀⢸⣿⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡏⣿⣿⣿⣯⣀⣀⠈⣿⣷⡀⠹⣿⣷⢠⣿⣿⣯⢻⡟⢿⣿⣿⠠⣈⣿⡟⠀⣼⡿⠀⣀⡄⣴⣿⣿⣿⢹⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣧⠸⣿⣧⠀⣿⣿⣿⣿⣿⣿⠀⢀⢏⣿⣿⣿⣼⣿⡑⣰⣿⣿⣼⣿⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⣷⣽⣿⣦⢙⣿⣿⣿⣿⣗⠀⠀⣻⣿⣿⣿⣿⡥⣰⣿⣯⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⢿⣿⣿⣿⣿⣿⡛⢿⣿⣿⣿⣷⣾⣿⣿⣿⠿⢋⣾⣿⣿⣿⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠦⣉⠛⠿⢿⣿⣿⣿⣿⣿⣿⡷⠾⠛⣩⠴⠟⠋⠁

⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⡀⡎⢱⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢲⡀⡀⠀⢨⡀⠀⠀⠀⠀⣌⠀⠀⡀⡰⢢⠄
⠀⠀⠐⠇⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣿⣿⣄⣼⠀⣆⠙⠀⠄⣿⣄⣼⣿⣷
⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡿⡄⣷⠀⠀⠀⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣽⠷⠿⣶⡶⠿⠿⣽⣿⣿⡄
⠀⠀⠀⣶⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡞⠁⠀⠀⠀⠀⠀⠀⠙⣶⣿⣿⣧⡀
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢞⣿⣭⣿⣿⣤⣀⠀⠀⢀⠀⠀⢀⣠⣼⣿⣯⣻⣗⠄
⠀⠀⣠⢠⣄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⣿⣧⣿⠁⠈⣽⠟⠁⠻⠁⠙⢿⡍⠀⢿⣯⣷⣯⡃
⠀⠀⠻⠼⠁⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⠀⣼⠇⠀⢄⡀⣠⠄⠀⣿⠀⣸⣿⣿⣿⡂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠻⣯⣽⣷⣸⣧⢀⡈⠙⢁⡀⣰⣟⣼⣿⣹⣟⣿⢷
⠀⠀⡿⢿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣿⣻⠏⢻⡟⣸⠛⠻⠛⢿⠘⡿⠉⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠖⣿⣗⢟⣼⣷⣼⡇⠀⠀⣿⣴⣷⣝⢷⣿⡷⢦⠁
⠀⠀⣤⢤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣻⣿⣍⣰⣿⡋⠛⢶⠞⠉⣿⣧⣉⣽⣟⠓⠄
⠀⠀⠚⠛⠋⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠿⡿⣸⣿⣄⠀⠀⠀⢀⣼⣧⡻⡿⢮⠑
⠀⠀⢀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⣷⣿⠿⠿⢷⡿⠿⣿⣧⡟⠷
⠀⠀⣿⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠜⠄⢀⣞⠀⠆⠼⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⡾⣦⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠋⢩⡦⠀⠀⠀⠀⢀⡀⠀⡇⢸⠉⣻⠹⣇⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⠤⢤⠞⠋⠁⠀⣠⠀⠀⣰⠇⠀⠀⠟⠁⢀⡇⢸⡴⢡⢀⣯⠟⣹⡁
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠀⠀⣰⠇⠀⢀⡄⠀⠘⠀⠀⠀⠀⡀⣄⣤⠠⣼⡔⠉⠱⠙⢘⠅⡼⣉⢳⡀
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⢰⡇⠀⠀⠘⠇⠀⠀⣦⠀⡤⠂⢹⣾⢻⠀⡿⣌⡶⠤⣤⣬⡞⠁⠉⠀⣷
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠷⣦⣄⡇⠀⠀⡀⠀⠀⠀⠁⣼⢧⠀⠘⠛⠀⠀⠻⣯⡝⠻⣚⡟⠀⢰⠇⠀⢸
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠈⣷⠀⠀⣤⣤⡿⢶⠀⠀⠀⠀⠀⠀⠀⠉⢩⣥⠖⡇⠀⢀⠀⡼⠓⢆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣠⣤⠶⠶⠌⣇⠀⠀⠀⠐⡇⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠋⢠⡇⠀⠀⢳
⠀⠀⠀⠈⣇⠀⠀⠀⣠⠟⠉⠀⠆⠀⠀⠘⢦⡀⠓⠆⠙⢮⡆⠀⠀⠙⢶⣆⠀⠀⢀⡴⠟⠂⢸⡇⠀⢀⡼⠀⢀⣀⣸⡄
⠀⠀⠀⠀⢹⣆⠀⣾⠉⠀⠀⣤⡀⠀⠓⠀⣀⣽⣦⣀⣤⠘⡇⠀⠐⢤⣀⡉⠀⠀⠠⣄⣤⠆⣸⢇⣠⠛⣦⠞⠛⠉⢉⡉⠓⢦⡀
⠀⠀⠀⢠⡿⠉⣿⠇⠀⣆⠀⠘⠃⠀⠀⡾⠁⠀⠀⠋⢼⣁⡙⢤⣀⣀⠉⠀⣀⠀⢠⣀⣠⠾⠛⠉⠀⠀⣿⠀⠺⠀⠘⠃⠀⠆⣧
⠀⠀⠀⣸⡇⠀⣯⠀⠀⠈⠁⠀⣠⣤⠀⣧⠀⠀⠀⠀⡆⠀⠉⠙⠺⠋⠉⢻⡙⠦⡼⠋⠀⠀⠀⠀⠀⢰⡏⠴⠆⠀⠰⠂⢠⣄⡿
⠀⠀⠀⢹⡇⠀⠻⣄⠰⠶⠂⠀⠀⠀⠀⠈⠙⢶⣤⣄⠠⣤⠤⠖⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⣧⠘⠉⠉⠙⢢⣠⠞⠁⠀⡀
⠀⠀⠀⠘⣧⠀⠀⠙⠶⣄⣀⠐⠾⠃⠀⢀⣠⠎⠀⠉⠙⠮⡖⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣧⠀⣄⡀⠀⠀⠈⠙⠛⡿⠁
⠀⠀⠀⣼⠟⠣⡀⠀⠀⠀⠉⠛⠲⠶⠚⠋⠀⠀⠀⠀⠀⣠⠶⣦⡀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⣼⡴⠏⠙⠳⠶⠶⠖⠋
⠀⠀⣸⡏⢀⡲⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣿⣷⠟⠁⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠸⣧⡈⠓⠀⢲⡌⠙⠲⢦⣤⣤⣤⣀⣀⣀⣀⣤⣤⣴⠿⠋⠁⠀⠀⢀⣠⣴⡾⠟⠋⠁
⠀⠀⠀⠘⠻⣦⣀⠀⠠⢤⠄⠘⠆⠀⠉⠉⠩⠭⠽⠿⢯⣥⣤⠤⠶⠶⠛⢋⡿⠃
⠀⠀⠀⠀⠀⠀⠙⠛⠲⣦⣤⣤⣀⣀⣙⣛⠀⠀⠶⠖⠀⠀⠟⠃⣀⣨⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣋⡉⠛⣯⣿⠙⠉⠙⣿⡿⠿⡿⢿⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠈⠉⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣾⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣷⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢷⣦⣀⡀⠀⢀⣤⣤⡀⠀⠀⠀⠀⢀⣤⣤⡀⠀⢀⣀⣤⡾⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣴⠟⠉⢻⣶⣿⠋⢸⡟⠀⠀⠀⠀⢺⡇⠙⣿⣶⡟⠉⠻⣦⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣴⣿⣿⣿⣆⠀⢀⣀⣾⣧⣀⡀⠀⣰⣿⣿⣿⣦⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣧⠈⣻⣿⣿⡟⢁⣾⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣋⣀⠈⠙⢿⣿⣿⠻⡆⢉⣹⣏⡁⠸⠻⣿⣿⡿⠋⠁⣀⣙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠈⠛⠛⠋⠀⣾⣿⣿⣷⠈⠛⠛⠛⠁⠀⠀⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠟⣿⡾⠛⠛⢻⡟⢻⡟⠛⠛⢷⣿⠻⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠶⢠⣿⠀⠚⠂⢸⡇⢸⡇⠀⠛⠀⣿⡄⠶⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠶⠘⣿⠀⢛⠀⢸⡇⣼⡇⠀⡛⠀⣿⠃⠶⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⠰⠆⢻⣆⠘⣁⢸⡇⣿⡇⣈⠋⣰⡟⠐⠆⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠙⣷⣜⣃⣿⣦⣉⣸⣷⣿⣄⣉⣴⣿⣈⣣⣾⠋⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⠟⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠏⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡚⠁⠀⠀⠀⠘⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣀⡀⠀⠀⠀⢠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⠀⠀⣀⠤⠀⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡶⠒⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠀⠀⠠⠞⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠶⡤⠀⠀⠤⠄⠂⠈⠉⠝⠦⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁⠀⠀⠀⡄⠤⠀⠀⠀⠀⠀⡀⠔⠋⢩⠉⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢠⠖⠀⠉⠁⠀⠀⠀⠀⠀⠐⠂⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠂⢤⠁⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠘⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⣶⢶⣿⣿⣿⡧⠀⠀⠀⠺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣸⡿⢻⠆⠀⠁⠀⠀⠀⠀⠀⠀⠁⠈⠀⠭⣿⡇⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢽⣥⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢣⣿⠃⠀⠄⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⡟⢁⣤⠀⡀⠀⠀⠀⠀⡀⠀⣀⣀⣰⠿⠇⠹⡆⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⣹⠟⠉⠙⠤⡀⠀⠀⣠⡶⠟⠉⣭⠥⠀⠀⠘⠋⠀⠀⠀⠠⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡃⠀⠀⠠⡴⠉⠀⠀⠀⠀⠐⠱⠚⣡⠖⠉⠀⠀⠆⠀⠈⠁⠊⠙⡅⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠃⠀⣶⣶⠇⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠀⠀⠀⠀⠀⠐⠂⠓⠒⣀⠠⠒⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⢀⡀⠈⢁⣭⠀⠀⠁⠐⠄⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠠⠤⠀⠂⢈⣥⣤⣤⣉⣙⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠰⣮⣀⡀⠚⠀⠀⠐⠂⠀⠀⢀⣼⣧⣞⣷⡄⠀⠀⠀⠀⠀⣄⣠⠞⠉⠀⠀⠀⠀⠀⠉⠱⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣈⠛⣠⡴⠎⠀⠒⠀⠀⠀⢠⡿⠀⠉⠼⢿⡄⠀⠀⠀⢀⡬⠁⠀⠀⠀⠀⠀⢀⢴⡄⠀⠈⢻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣋⣸⣿⠆⡀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⣠⠿⢤⡀⠀⣼⠇⠀⠀⠀⠀⠀⣀⠂⡜⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠔⠃⣀⢰⠠⢠⠌⢩⠏⠠⢀⡾⠋⡄⠀⠉⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠫⠀⠀⢰⠃⣔⢡⡟⠁⠀⠃⠀⠐⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠟⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⡝⠢⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢀⡜⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠈⡀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠀⡀⣠⣤⠀⠘⣤⠤⠤⢤⣤⣿⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠬⠤⡾⠃⠀⠀⠀⠀⠀⠠⠀⠈⢳⡦⠔⠊⠉⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢤⠀⣸⠀⢴⡇⠀⠀⠢⡀⠀⠀⢀⡀⣰⣿⡅⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡠⣾⠀⠀⠀⠀⠐⢄⠀⠈⠁⠈⣣⡷⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠔⠂⢚⡁⣷⠀⠀⠀⠀⠀⠀⠡⠀⢀⣀⡿⣴⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠛⠋⣿⠀⠀⠀⠀⠀⠀⠀⢁⠀⣤⠤⠀⠹⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⢧⠀⠀⠀⠀⠀⠈⡄⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣆⠀⠀⠀⠀⠀⢷⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⡆⠀⠀⢸⡆⠸⡄⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢿⣀⡤⠃⠈⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⡁⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠸⣧⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠢⡀⢨⡷⠺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⣽⡆⠀⠀⠀⠹⣆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠦⠀⣸⡇⠀⠀⠀⠀⠻⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠶⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⢛⣿⣿⣿⣦⣄
⠀⠀⠀⠀⣿⣿⡯⣿⡙⣿⣿⣿⡷⣶⣆⣼⣿⣿⣿⣿⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠉⣿⣧⢻⣿⣿⣿⣿⡟⠻⢘⣿⣿⣿⣿⣿⣿⣿⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠁⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠷⠶⣤⣄⠀⠀⠀⠀⠀⢸⣦⣷⣿⣡⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠳⠙⣽⣿⣸⣿⣿⡀⢀⣿⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⣄⠀⠀⠀⢸⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⢹⣿⣿⡇⢻⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⡆⠀⠀⣿⣿⣿⣿⣧⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣶⣅⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠁⣼⣿⣿⣿⣿⣇⠀⣐⣿⣿⣿⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⣮⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠋⠁⠀⠀⣹⣿⣿⠏⠉⠛⢿⣿⠹⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⢸⣿⣧⣿⠀⠀⠀⣿⣿⡿⠏⠙⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⢰⣿⡟⠀⠀⠀⠀⠀⠈⢷⣿⣿⡀⠀⢠⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠄⠹⣿⡁⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⢀⣾⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⢀⣿⣿⡆⠀⠀⠀⠀⢀⣾⡟⠁⠀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⠆⠀⠀⣠⣾⡿⠁⢠⣾⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠻⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣱⣄⣀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠉⠉⢉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢈⣳⠒⠚⠙⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡉⢉⠖⠛⠿⣲⣶⠿⠋⠈⡝⠉⡹⣫⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣼⠀⠀⠀⠀⠀⠀⠀⢸⡇⢎⢞⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢽⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⡡⢚⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠈⢘⡆⠀⠀⠀⠀⠀⠀⠀⢸⣯⠊⢥⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣤⡴⡿⣄⡀⠀⠀⠀⢀⣀⡞⢿⢶⣯⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⢯⡼⢁⠷⣭⣛⠿⣻⡏⠭⢝⣤⡟⠉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⡼⢹⡇⣿⢹⡶⣕⠲⣸⡽⡇⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢤⣱⠃⡾⢠⡟⢸⡇⢻⡿⣾⣷⡷⠒⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡿⢽⡇⢸⠇⠸⡇⠸⣧⠩⢹⡃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣶⣯⣻⣿⣿⡰⣇⣿⣋⡽⠥⠾⣗⣶⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠁⣻⠋⠈⢻⠀⠀⠀⢻⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⢀⢀⡇⠀⠀⠘⣆⣠⣤⡟⣇⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⠽⣿⠆⠀⠀⠀⡿⠝⠿⡇⢸⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣖⣒⣺⡆⠀⠀⠀⣿⣶⢲⣿⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⠽⢻⠁⠀⠀⠀⢻⠫⠝⢹⠀⢸⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡟⠀⠀⠀⠀⠈⣆⠀⢸⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣗⡠⣷⠀⠀⠀⠀⠀⣰⣒⣮⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⡐⢨⡾⠃⠀⠀⠀⠀⠽⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛⠋⠁⠀⠀⠀⠀⠀⠘⠦⣤⡶⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠁⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⣠⣦⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠛⢻⣿⠛⢀⣀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢄⣾⠟⠛⠛⠹⣿⣿⡿⠟⠛⠋⠻⣤⠀⠀⠀⠀⠀⠀⠀⢀
⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⠐⠛⢿⡆⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⢠⡖⠀⢀⣄⠀⠀⠀⠀⢰⣼⣷⣿⣿⠿⢾⣇⠀⠀⠠⣄⢢⣶⠙⣛⣲⣴⣦⠄⠀⠀⢰⡶⠿⣿⣿⣾⣧⡄⠀⠀⠀⠀⢠⡄⠀⢳⡄⢀⡄
⠀⠀⠀⠹⣿⣿⡃⠀⠀⠁⠀⠀⣠⡾⠿⢿⣿⡇⠀⠸⠏⠀⠀⠀⡠⢚⡟⢤⣿⠛⡋⠤⠀⠀⠀⠹⠇⠀⢸⣿⡿⠿⢷⣄⠀⠀⠈⠀⠀⢘⣿⣿⢏
⠀⠀⠀⠿⠛⠋⢿⣤⠀⠀⠀⠘⢿⡄⠀⠀⢙⡙⣤⣀⣀⣤⡴⠶⠴⠾⠷⠾⠿⠾⠿⡦⢶⣦⣄⣀⣀⣤⠋⡍⠀⠀⢠⡿⠃⠀⠄⠀⣰⡿⠙⠛⠿⠁
⠀⠀⠀⠀⠀⠀⢿⣿⣁⣴⠂⠀⠀⢁⣴⠶⠾⠿⣉⠉⢀⣀⣀⣀⡤⢤⣤⣤⣤⣤⡤⡤⢾⣿⣛⡛⠋⠉⠙⠓⠶⣦⡈⠀⠀⠐⣦⣀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⢿⡁⢻⣿⡾⠛⠁⣀⣠⣴⣶⠟⠛⠛⠉⠁⠀⠀⣴⣆⠀⠀⣰⣦⠀⠀⠀⠉⠛⠛⠳⢶⣦⣄⣀⡈⢙⣶⣿⡿⢁⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠿⣦⣴⠟⠁⠈⠉⠀⠀⣀⣴⣿⣁⣀⣀⠺⠿⣍⣹⠿⠗⣀⣀⣨⣿⣶⣠⠄⠀⠉⠀⠙⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢀⣶⠀⣀⣼⣻⣿⡿⠿⠿⠿⠮⠙⠛⠛⠛⠛⠛⠛⢫⣭⡿⠿⢿⣿⣿⣟⣏⣀⠀⣶⡀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣸⠿⣟⠋⠉⠀⢀⣀⣀⣤⣤⣤⣤⣶⣶⣶⣿⣭⣭⣭⣤⣀⣀⡀⠀⠉⠈⠛⠿⣯⡹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣅⣠⣽⡶⠾⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⢶⣤⡴⣮⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡤⠀⠀⢀⡀⠀⠀⠀⠀⢼⣤⡏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠀⠀⣀⣤⣤⣴⣶⣄⡠⢌⡁⠀⢀⣀⡠⣠⣴⣿⣿⣶⣶⣤⣀⠀⢻⣿⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⠃⢠⣿⣿⣿⣿⣿⣿⣿⣷⡀⠁⠀⠀⢹⠄⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⠈⢷⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠻⣿⣿⣿⡛⢫⡅⠛⢿⣾⣿⣿⣿⢮⠙⠻⢬⣽⣿⣿⣿⡟⠀⣼⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠛⣋⣁⣌⡄⠀⠀⢠⣿⣿⢾⣿⣿⣶⢀⣄⣀⠉⠙⠁⠀⠘⠛⢛⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢞⠿⢟⡛⢶⣸⣧⣼⠙⣿⣸⡟⣘⣿⣿⣷⡾⣿⣶⣴⣧⣴⠿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⠻⣾⣅⠸⣡⠇⠻⣷⡟⢹⠿⣏⣿⣿⣿⣿⣿⣿⣿⣿⡗⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠿⠀⠇⠀⠀⡔⠻⠿⠶⠿⠙⣧⣼⣿⠏⣹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣹⣿⢤⣏⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠸⣿⣷⣌⠀⠂⢠⠀⢲⠒⢀⠰⡆⢹⣽⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⣶⣷⠀⠀⠶⠀⠸⠀⠃⢙⣿⣿⣿⡷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢫⣼⡧⡆⠀⠀⠀⠀⠀⢀⣉⣹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡙⣧⠀⠀⠀⠀⢀⠦⣬⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⣸⠣⠀⠀⡴⡉⣷⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠑⠾⣴⣤⣧⣿⣾⣿⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⢶⡿⠿⠿⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⣷⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣠⡄⣠⣶⡮⠙⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠐⣶⣿⣿⣾⣿⡷⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠉⠀⠀⢿⣿⠙⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⡀⢀⣤⡴⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣇⡀⠀⠉⠂⠀⠀⡀⠀⠀⠀⠐⢡⣴⣿⣷⣽⣿⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⢰⣧⣬⣿⣿⡇⡀⠀⡰⠀⠀⢋⠀⠀⠀⠀⠀⠫⠉⠉⣿⣿⠃⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢿⡛⠋⠹⣿⣇⠑⢴⠃⠀⠀⠀⠈⠀⠠⠤⠀⠀⠀⠀⠸⣿⢀⣲⠊
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢮⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣸⣿⡟⢀⠐
⠀⠀⠀⠀⠀⠐⠆⠀⠀⠐⠒⠀⠀⠀⠀⢲⡞⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠿⠇⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠸⠁⡜⡀⠀⠀⠀⠀⡠⠔⢀⣀⠀⢀⣄⣠⢄⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⡉⠀⠀⠀⠀⠐⡷⣿⣿⣦⣿⣿⡅⠀⠱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠉⠈⠁⣿⣿⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣿⢃⢸⣠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠃⠀⠀⢀⣄⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠐⠋⣡⠾⡿⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡐⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁

⠟⠒⣢⠤⡄⠴⢶⣶⢤⠶⢶⡷⠀⣸⠇⠀⠀⠀⠀⢄⠀⢸⠤⢤⡖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⡋⢰⠀⡇
⠀⣀⠃⠀⠀⡀⠈⠁⣀⣠⣀⣳⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⣿⣇⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠂⠁⠘⠀⠁
⠎⢙⣇⠀⢦⠿⡀⠀⠀⣮⣽⣷⠄⠀⢐⠀⠀⠈⠁⠈⠑⠀⠀⡙⢫⠀⠀⠀⠁⢐⣀⠀⢰⡀⢈⠀⠀⣀⠀⣀⠀⠀⠀⢰⣶
⠰⠋⠋⡁⠐⢤⣽⢀⠀⡏⢿⣿⣰⠀⠘⠃⢀⡀⡀⣀⢀⠀⡀⢫⡉⡃⠀⠀⣀⢘⣁⣄⡛⢀⠈⡀⠈⢳⣦⢾⡶⠀⠀⠀⠏
⠣⠀⢌⠦⣠⡜⢷⡎⠠⣇⢠⡀⢻⣼⢀⠠⣼⣧⣤⢻⣼⣼⢧⠾⢿⢹⣅⠀⢉⢉⣿⠁⠀⣤⠄⡶⠰⠘⠘⠊⡑⠄
⠀⠐⢿⠀⢂⠶⢈⣻⢂⡘⢇⣭⢉⠅⠈⠶⠿⣿⠙⠙⢿⢃⠘⠀⣾⡍⢻⣧⣌⠩⣟⣾⡖⢻⡇⠃⡇⣄⠐⣰⠁⣐⡚⠈⠰
⡆⠐⡀⢣⣀⠘⣀⠻⠛⠳⡘⠡⣘⣌⣴⠴⡶⣼⣇⠀⣺⢿⠀⠈⡿⠉⠉⡻⠟⣴⠓⢫⠇⠽⣤⢰⣧⣭⠶⣛⣥⣴⡀⢀⡀
⢱⣰⣛⣞⠧⠀⡤⠄⠀⢂⣿⡂⣤⣾⠴⠆⢹⣿⡏⠠⠈⢠⣴⣤⣠⡾⢶⡐⠠⡁⣴⡇⠖⣸⢁⡁⡔⠳⠄⠈⠯⠙⠷⣦⡽
⠛⠉⣆⠀⠀⢘⣧⡦⠀⠉⣭⣅⠃⡘⠂⠠⣬⢙⣿⠴⢴⣘⣦⣝⠨⠄⡰⠀⣈⢁⣭⡁⣬⡄⡀⠁⠀⢠
⣄⣴⡟⠄⠀⣁⡿⠈⢘⣧⡅⣠⣤⣆⣈⠷⠄⠭⠁⢠⣌⣁⣝⣇⠠⣦⢓⣆⣴⣰⣿⡀⣄⢠⣧⣀⣠⡴⣤⣴⣄⣠⡀⡆⠠
⣾⡿⠗⢀⡄⠈⣰⣞⣟⠙⠘⠩⢁⡑⠛⣀⠦⣤⣾⣿⣿⣿⣾⢹⣅⠾⣺⣵⠏⠟⠿⣷⡸⣿⡟⢹⣿⠅⣿⣿⣿⣿⣇⠇
⡩⠃⠀⢺⡤⠀⣸⣾⡻⢟⠙⣠⣬⡷⠞⣟⣂⣩⡛⡿⢡⣿⡏⡌⣿⡐⣄⣉⣴⡗⣀⣿⡾⣿⠀⣿⣿⢰⣿⣿⣿⣿⣿⣾⠻
⢕⣡⠀⢹⣿⡀⢼⡿⣿⠿⣷⣿⣿⢀⣿⣭⣭⣭⡟⠀⣏⣿⠇⡇⣿⡶⢩⡍⠠⠱⢈⡿⣷⣿⢰⣿⣻⢸⣿⣿⣿⣿⣿⣭⣤
⢿⣻⡆⢹⣟⡇⢸⠿⠿⣻⣿⣿⠗⣺⠙⣋⠛⠟⣟⢴⡿⣿⣤⣧⣷⣦⣿⣿⣵⣶⣶⣿⣯⣾⣿⣿⣯⣸⣿⣿⣿⣳⡿⠭⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⠈⠂⠀⡀⢀⣺⡇⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢺⡅⠠⣠⠋⢸⡎⣣⣴⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡄⠐⣸⠧⢘⡀⣤⣀⡿⠵⠎⠠⠂⠀⠠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠬⢃⡰⢛⠐⠉⣀⠠⣿⣛⣥⣀⠯⣀⣠⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢐⠀⠹⡄⣛⢛⣽⣿⡓⢀⣋⣸⣿⣿⣿⡿⠚⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠉⠘⠶⣄⠾⣋⢿⢃⡟⣜⡛⠉⠁⠈⠌⠀⠀⠀⠀⢀⣤⢴⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢹⣦⢡⣸⢯⠼⣿⠀⢢⡄⢀⡤⢀⠀⠀⢀⡋⢻⣿⣿⣿⣿⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡘⡏⢍⣧⡰⠁⠀⣪⣷⠘⠫⠀⠀⠀⡘⠁⠀⢹⣟⣿⣿⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢺⠈⡼⠟⠢⠛⠉⠀⠀⠀⠀⠀⠀⢨⣐⣶⡞⠉⢹⣿⣇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⣤⢤⣴⣤⣦⡞⢈⣙⣽⣤⠶⠾⠂⡂⠀⠓⢂⡲⣦⡵⠔⡧⠀⠀⠁⠛⢶⣤⣴⣤⣴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢧⡟⢻⣿⣿⣎⠙⠂⠉⠻⣿⣏⣴⣷⣤⣾⣷⣮⣯⡿⠁⢨⢀⡴⠊⠀⠀⠀⠈⢿⡟⠛⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣙⡳⢦⣾⣿⣿⡿⠤⣤⣤⣤⣀⠈⠉⠉⠛⠛⠛⠾⢛⡑⠛⠋⠉⠀⠀⠀⠀⠀⠀⠘⣷⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣽⡿⠋⠉⠙⠃⠀⠀⢀⣿⣿⣿⣿⣶⣦⣄⣀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣁⣶⣶⣤⣿⠀⣠⡼⣿⡿⠛⠛⠛⢛⡿⣿⡿⠋⢻⣿⣤⠤⠤⠴⠤⠤⠄⠀⠀⣼⢷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡿⣿⣼⣿⣷⡾⣥⣄⡈⢣⡀⠀⢁⡞⣠⢋⣴⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⢰⣿⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡻⡿⣻⣿⣿⣟⣠⣅⡸⠙⡆⠉⠛⢋⡴⠃⡾⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡟⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣃⣴⣿⣿⠿⠛⠟⠉⠀⣴⠇⢀⠞⠁⠀⣠⢳⣄⠀⠙⠛⠙⠛⠣⢔⠚⠒⠢⣀⠘⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠋⠻⢅⡀⠈⢀⠀⢠⠼⣀⡹⠁⣀⡼⠱⠘⢅⠀⠀⣀⢀⣀⡠⠞⠀⠀⠀⠈⢢⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⠻⣿⣷⠋⡔⠱⠦⣤⠁⠀⡀⠀⣼⡗⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠙⡿⠋⠀⣠⣔⡩⠂⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀⠀⠀⠀⢀⡸⠖⠛⠉⠉⢀⡴⠃⣠⢫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣄⣴⣏⠀⠀⢀⣠⠖⠉⠳⣤⠃⠓⠤⢀⡀⢀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠙⠊⠁⠈⠳⢤⡴⠃⠀⠀⠀⠀⠈⠁⠀⠈

⠀⠀⢠⡆
⠀⠀⢸⡇
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⠒⠐⠶⠆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⢷⣤⡉⠉⠢⠄
⠀⠀⡆⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣤⣀⣀⠀⠈⠛⠢⠄⠈⠦
⠀⠀⠂⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡁⠀⠀⠉⠉⠀⠐⠒⠂⠀⠀⠀⠀⠀⠰⢾
⠀⠀⢣⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠿⠿⠟⠛⠛⠛⠓⠚⠓⠒⠀⠀⣀⠀⠀⣸⡇
⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣚⣋⣛⠋⢻⣶⣴⡾⠿⠂⠛⠀⢀⣀⣿⠇
⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡛⠛⠛⠛⢻⡀⠀⠀⢀⣀⠠⠤⢛⣿⠇
⠀⠀⠀⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣶⠶⠟⠉⣀⣤⣾⡿
⠀⠀⠘⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⢿⣯⣭⣴⣶⣾⣿⡿⠟⠛⠛⢦⡄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⢿⡟⠉⠛⣉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡉⢻⣧⠀⢀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣠⡾⢿⣄⠛⢂⡸⢳⣾⡇⢀⣠⡾⠃
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⣼⡛⢧⣀⠉⠃⠀⢀⠿⠋⠁⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠘
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢰⡟⠃⠤⠈⠉⠀⣠⡞⠀⠐⠀⠘⢿⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿⣶⣤⣤⣄⣀⣼⡇⠰⠆⠀⠀⠸⣷⡄
⠀⠀⠀⢸⣿⡇⠀⠀⠀⢀⣼⡇⠀⠈⠉⠉⢉⡝⠁⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠘⣿⡇⠀⠀⠀⢸⣏⡀⠀⠂⠀⢀⡼⠁⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣉⣤⡶⠞⠛⠻⠶⡆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⠏⠑⠶⣿⠀⣸⠃⠀⠀⠸⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣰⠾⠛⣿⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣿⣇⠀⣀⣴⡇⠀⠀⠀⠀⢻⣿⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁⠀⠀⠈⢿⣆⠀⠀⠀⠒⠀⠁
⠀⠀⠀⠀⣿⣿⢺⣿⣿⣿⡶⠿⠶⢶⣤⣿⣿⣶⣶⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⠀⠀⠈⣿⣆⢠⣴⠁⠀⠀⠀⠐
⠀⠀⠀⠀⣿⣿⠈⢹⣿⠉⢠⡄⠀⢀⣿⣉⠉⠉⠉⠉⠛⠛⠋⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠋⠀⠀⠸⡏⢹⠟⠁
⠀⠀⠀⠀⢸⣿⣷⡞⠃⣀⠾⠿⠿⣿⠉⠙⠿⠿⠶⠶⢶⣶⠶⠴⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠛⡀⠀⠃
⠀⠀⠀⠀⢸⣿⣯⣤⣀⣀⠀⢀⡼⠿⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡤⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣠⡀
⠀⠀⠀⠀⣸⣿⣿⣙⠛⠛⢶⣿⣇⡀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣴⡏⠀⠀⠈⣿⡿⠷⠶⠦⠄
⠀⠀⠀⢸⠛⣿⡇⠉⠈⠳⡞⠉⠙⠿⠷⢶⣤⣤⣄⣠⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠐⢰⡿⠀⣀⡀⠀⣿⣷⢤⣤⡀
⠀⠀⠀⠸⣗⣿⡇⠀⠀⢰⣶⣄⣀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢠⡿⠃⢠⠿⣷⣄⠈⠻⣿⠉⠻⢿⡄
⠀⠀⠀⠀⣸⣿⣏⣀⢰⡏⠈⠛⠿⢿⣶⣶⣦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠠⠞⠀⠈⠻⣷⣄⠈⠁⡀⠘⠁
⠀⠀⢀⣾⠋⣿⣿⣛⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣶⣄⡀
⠀⠀⢸⣇⠀⠉⠻⢿⡟⠙⠿⢷⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠷⣶⣤⣤⣀⡀⠀⠰⣄
⠀⠀⠀⠹⣧⣄⠀⠀⠑⡄⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠂⢀⣴⡾⠛⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⣿⡄
⠀⠀⠀⠀⢈⣿⣿⣶⣾⣷⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⢠⣾⣿⣿⣿⣌⣙⠛⣻⣿⡿⠿⢿⣶⣶⣶⣶⣶⣶⣾⣿⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧
⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⣾⣿⠀⡉⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠘⠛⠛⠉⠉⢹⡏⠀⡄⢀⣽⠇⠀⠀⠀⠀⠀⢠⣾⠏⠁⠘⣧⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿
⠀⠀⠀⣀⣀⣴⣧⣤⣷⣾⡿⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠴⠶⠶⠶⠶⠆⠀⠈⣿⡀
⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⢹⠇⠀⢀⣀⣀⠀⣤⡄⠀⠾⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢻⣿⣿⣿⣿⡧⠄⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⢰⣷⠚⠋⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⣄⣀⣀⠀⣸⣿⣿⣶⠦⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⣿⡿⠛⠀⠈⠉⣉⠰⣿⣀⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠉⠉⢶⡿⠛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⢿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡇
⠀⠀⠀⠀⠀⠀⠈⣷⠀⣀⣀⣀⣀⣠⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠛⠙⠛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣿⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⢻⡧⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⡼⢇⠀⢠⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇⠀⠀⠀⢿⣅⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠸⣇⠀⠄⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⣿⣇⠀⠂⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⢊⠀⠀⠈⠙⣦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣿⠗⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠺⠀⢀⡄⠀⢻⣦⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠷⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⢹⠄⠀⢀⡀⠀⠀⢀⣾⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠀⢰⡆⠀⠀⠀⠘⢷⡀⠀⠀⠀⠘⢄⣠⠟⠳⠄⠀⢸⡿⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣾⣖⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⡇⠀⠀⠀⠀⠈⠻⣆⠀⠀⣠⠟⠁⠀⠀⠀⠀⣬⠃⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠙⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⢿⡇⣶⡄⠀⠀⠀⠀⠙⢷⣄⠈⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⡧⠀⢹⡆⠀⣿⣯⣽⠆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠧⣴⣄⣨⠟⣈⡅⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⣿⠃
⠀⠀⠀⣀⠀⠶⡶⠞⠉⠁⠀⠀⠘⠿⠞⠋⢉⣿⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠘⠉⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⢆⡀⠀⣸⡿
⠀⣶⠀⠀⠀⠀⠀⠀⢀⣄⣠⣀⣀⡀⢀⣠⣾⡇⠀⠀⠀⠀⠀⡐⠛⠉⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠀⡀
⠀⠻⢶⣶⣤⣶⣤⣴⣶⡷⠾⠿⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠘⢿⣶⣶⣦⣶⣶⣶⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦
⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡿⠻⠉⠻⠟⠿⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⢀⣀⣀⣀⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡀⠘⣄⣀⣤⣄⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠟⠻⠿⢿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠩⠁⠓⠉⠉⠚⠉⠀⠉⣽⣿⣶⣴⣶⠆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⠛⠛⠛⠀⢲⣶⣿⣶⣭⣶⣾⣼⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠈⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠁⠀⠀⠀⢻⡇⠈⣻⡇⠀⠀⠀⠀⣿⣿⣿⣿⠃⢰⣿⡇⢰⣿⣿⣿⣷⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠏⢻⠃⠀⠀⠀⠀⣸⡷⠚⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣭⡁⢹⣿⣿⣿⣿⡇⣰⣄⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡆⠀⠀⠀⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⣶⣿⣿⣿⣿⠇⣿⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⠄⠀⠹⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣡⣿⣿⣿⣿⣟⣸⣿⣏⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣤⣤⡤⣤⣤⣤⢤⡤⠤⠤⢤⡤⠤⠤⢤⣤⠤⢄⠄⠠⠤⠀⠠⠀⠀⣠⡤⠄⠀⠀⡀⢀⠤⠀⠀⠀⠀⠀⣤⠠⢤⠠⠀⠤⠀⠀⠀⠠
⠀⡿⠟⠋⢻⣿⣄⠋⠄⠀⠀⠈⠑⠸⠃⠀⡱⠋⠉⠉⠀⠀⠀⣠⠀⣴⡷⠁⠘⢆⠀⠀⡞⠀⠢⠀⠀⠀⢰⢇⡀⠀⠱⡀
⠀⣿⢿⣷⣿⣽⣷⢀⢾⡦⠐⠂⠀⠀⠀⢱⡦⠄⠀⠀⠀⢤⣤⣀⣶⠯⠁⠀⢀⠈⢆⠸⡑⠲⡄⠀⠀⢀⠋⡈⠈⠀⠀⠙⢦⠹⣄⠀⠘
⠀⠏⠳⣹⣿⠉⢰⠟⠸⠁⠀⠀⠀⠀⠀⠸⢧⣐⣀⠈⠀⠰⠮⣿⡣⠁⠀⠐⡟⡀⣨⢦⠀⠀⣘⠠⢰⡎⠦⠀⠀⠄⠀⠀⠀⠑⣌⠢⣼
⠀⠀⠀⠘⣷⢄⣨⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⠀⠀⠿⠿⠷⠤⠴⠌⠋⠹⠷⠄⠳⠐⠿⠋⡁⠿⠔⠤⠀⠠⠀⠀⠀⠢⠸⠧⢸
⠀⣧⣤⣥⣤⢬⣿⠄⠀⢤⣤⢤⡤⣄⠠⠤⠐⠤⠂⠴⢆⠐⣶⠶⠂⠶⠶⠶⠶⢶⣶⠶⠢⡞⡚⢦⣤⣲⣆⠲⡖⢰⣶⣶⣶⡶⠈⢶⡆
⠀⡿⣿⠀⡿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠳⡈⠀⠀⠘⠒⣦⣔⢼⢸⡤⢸⡇⣸⡟⠹⠿⠋⠳⣄⠘⠚⠋⠈⠙⠛
⠠⣧⡾⠛⣡⡇⠀⣷⠀⡄⠙⠦⢀⠀⠀⢃⣽⠶⠒⠀⠂⠈⣽⢶⡂⠄⠀⠠⠀⠸⢚⠀⠀⡇⢈⠄⢀⣰⣞⡀⢁⠀⢀⠤⣤⠤⣤⣤⡼
⠐⣧⡿⠖⣿⠇⠙⠛⢿⠿⠚⠐⠋⠑⢦⡙⠢⠔⠒⠒⠀⠀⣤⠤⠀⠀⠀⠀⠀⢸⢸⠀⠁⠋⠃⣵⣉⠀⠀⠀⠀⢶⡜⠀⠁⡴⠇
⠐⣴⣳⠀⠉⠰⠶⠶⣶⠠⠶⠤⠤⠴⠤⠭⢃⢀⣀⠰⠶⠶⠿⠀⠀⠶⠶⠶⠦⠾⠶⠆⠛⣧⠀⠿⣿⣖⡒⢒⣻⡇⠴⠶⢶⢿⡄⠐
⠐⡿⠋⡜⣡⣰⡶⠶⠷⢤⠀⣒⠀⣀⣐⠒⣉⡩⠃⣀⡀⠀⣀⣀⢀⣀⢀⣀⣀⣀⣀⠀⡂⠑⠲⣠⣤⠀⣀⢠⣄⣉⣩⣥⣬⣿⣦⡸⣡
⠀⣷⡜⠐⠻⣛⠷⠚⠁⡖⠁⠒⠒⠀⡶⠴⠲⡄⢲⢿⣶⢦⡀⠲⠟⣟⢷⠮⡍⢯⡴⢋⣤⣼⡁⠀⠀⢀⣤⣤⣤⣤⣤⣤⡅⢠⣍⠀⣭
⠀⡏⢹⢉⣡⠄⠼⢦⣤⣦⣀⡖⡬⠄⠠⠤⣀⢱⣌⣡⠄⠉⠙⠲⠦⢤⣼⣶⣆⣲⣀⣻⣛⣒⣖⣀⣀⣠⣠⣽⣖⠂⠀⠤⠔⢾⡆⠷⠤
⣤⣭⣤⣭⣭⣭⣭⣥⣭⣭⣭⣭⣤⣤⣤⣤⣬⣭⣥⣬⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣥⣬⣭⣥⣭⣭⣬⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡀⣴⠃⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠹⡄⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⡐⠀⠀⠀⢠⡟⢀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢧⡀⠀⠀⢠⡄⠀⢻⡇
⠀⠀⠀⠣⠀⠀⠀⢻⣤⠜⠁⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠁⢴⣾⡇⠀⠀⢀⡂
⠀⠀⠀⠹⣖⡀⠀⠘⣿⣦⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣾⡿⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠈⠛⠲⠶⣿⣿⣦⣤⡆⠘⣇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡸⠀⢦⣠⣼⣿⡷⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡻⢿⣿⣶⣾⣶⢿⡀⠀⠀⠀⠀⠀⠀⢰⣿⣾⣧⣦⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠛⢿⣿⢳⣆⠀⠀⠀⠀⣴⠰⣿⠟⠋⠁⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢄⣻⣿⡿⠒⢰⠄⠲⣿⣿⣆⡴⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⢧⠀⠏⠀⠀⡔⠁⠀⠀⠀⠈⠉⠠⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⢢⣠⣄⠀⠘⠒⠒⠂⠀⣠⣀⠖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠀⠀⠀⠀⠀⠘⠛⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠠⢋⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠑⠄⣰⣶⣶⡀⣶⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠘⠪⣛⢛⣶⠏⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠻⠟⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⡀⠀⢀⡴⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣌⣱⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⢿⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣾⣷⡶⠀⠀⠀⢠⠖⠚⠩⠿⣧⡤⠾⠉⠙⠒⣄⠀⠀⠀⣀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠔⠛⢸⡿⠋⠛⢆⢰⠟⣶⠀⠀⠀⠀⠀⠀⠀⢰⢶⠈⠔⣺⠉⠈⠉⢧⢨⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠸⠇⢀⢀⣘⡏⠀⢻⡂⢀⠄⢀⣀⠀⢀⢸⡀⢄⢧⠸⡄⠀⠀⠀⢣⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢠⡌⠿⢇⠁⢹⡑⣿⠋⠀⠛⠿⣆⢈⡃⠈⠆⠑⣔⣦⡀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠀⠀⡜⣷⠤⠊⠀⠀⠉⢳⠀⠀⡤⠀⡛⠁⠀⠀⠑⠬⣙⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⣰⡾⠄⠄⠀⣰⣧⣄⠀⠀⠀⠀⢸⠟⢷⠆⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡀⠀⢠⣄⣀⣟⠛⠶⣶⣃⣶⣮⠿⠟⠃⣦⠀⠀⡸⢁⣿⠂⡞⣱⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣤⣿⡿⠿⠳⣦⣬⣭⣁⡀⠀⠀⡰⣾⣷⣿⣠⡼⠃⠀⠬⢯⣽⠅
⠀⠀⠀⠀⠀⠀⢀⡀⣤⠀⠀⠐⢲⣾⡿⣷⣄⠀⠀⠀⠀⠙⡶⣦⡶⠀⠀⠀⠛⠙⢿⣿⣷⡶⠦
⠀⠀⠀⠀⠀⣴⠉⡙⠁⣀⠄⢒⣶⣿⣇⠃⢸⡿⣄⡀⠠⢄⡉⠉⠤⠖⢀⣀⣤⣾⠏⠉⣿⡿⢷⡦⠀⠁⢤⡤⣀
⠀⠀⠀⠀⠀⡏⠀⠀⠀⢣⢶⣿⣿⣽⣿⣧⡈⠈⣙⡿⢷⣦⣬⣭⣻⣉⡬⣋⡵⢻⢆⡼⣿⡧⠚⡿⠆⣆⠈⠋⠻⣥
⠀⠀⠀⢀⡤⠃⠀⠀⠹⣎⠋⡀⢈⣿⣿⣿⣯⠲⣙⣿⣶⣿⣿⣿⣿⣿⠿⠫⢐⡿⠊⢱⣿⠁⢟⡳⢃⠿⠀⠀⠀⢹
⠀⠀⣰⠉⠀⠀⠀⠘⠢⣉⣁⡡⢞⣵⣿⣿⣯⠱⣜⣿⣶⣿⣿⣿⣭⣷⣦⡶⠏⡡⢠⣿⣿⣦⠀⣀⡄⠀⠁⠀⠀⠸⡄
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣽⢿⣿⣿⣿⣿⣿⣿⣿⣷⣛⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣾⢿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⠿⣛⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⢻⣿⡿⢿⣿⣿⣿⣷⣭⣿⣿⣿⣽⣾⣿⣿⣿⡿⠋⠸⠏⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⠛⠉⣸⡎⡿⠛⢿⣿⣏⡉⠛⠛⣻⣿⠛⠹⢿⡇⢠⡅⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣶⠈⠛⠹⠀⠀⠈⠳⢌⣻⣿⣛⣫⠽⠃⠀⠀⢨⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠎⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢇⠱⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣆⠀⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠄⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠐⠋⠃⠛⠛⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠶⠶⣶⡄⢀⣠⣄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⣐⡟⢫⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⠄⠀⠀⠀⢹⣽⣭⣽⣿⡿⠀⠀⢀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⣄⣈⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄⠀⠀⢈⣿⣿⣿⣏⣠⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡇⠀⠀⠉⠁⠈⠉⢁⢹⣿⡟⠛⢿⣿⣿⣿⣿⣿⣿⡧⠀⠉⠋⠻⠿⠟⠛⠓⠙⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⢀⠠⠈⢻⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣿⣿⣃⠀⠀⠀⢀⡐⢐⣿⡟⢙⣿⣿⣿⣿⣿⣿⡿⠁⡆⠀⠀⢀⣼⣿⡂⠀⠀⠀⠀⢻⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣀⣩⣽⣿⣿⣿⣧⠚⣆⡀⡰⣼⢼⣿⠇⣾⣿⣿⣿⣿⣿⣿⠇⠰⠃⢀⡴⠋⠀⠀⠙⡐⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠯⠾⣿⡅⣿⣿⣿⡟⠶⣧⣄⣶⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣈⣁⡀⢠⣿⣶⣧⢠⣀⠀⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠰⡀⠈⠇⢻⣿⣿⣿⣿⣗⢞⣿⣿⣣⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⡏⠛⣛⠿⠿⢿⡈⣿⣼⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠱⣾⣾⣤⣭⣾⣯⣿⢿⣍⡿⢡⡈⠈⢻⣾⠛⠈⠛⠻⢈⡫⠼⠟⠿⠉⣰⣯⣶⡶⠿⠃⠉⠃⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠿⠿⠿⠿⢻⠟⢿⡿⣧⡦⠦⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⣠⢞⠡⠔⠂⠲⣤⣤⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡄⠀⠀⠀⣰⠋⢀⣀⠄⠀⠀⠀⠹⣧⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⠀⠀⡰⠀⣰⣿⠵⠆⠀⠤⣀⣼⣋⣙⣻⡤⠠⠤⠤⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠻⠦⠶⠁⡰⢋⣴⣾⠛⠉⣠⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣰⠗⠉⠉⠉⣩⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣬⣵⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⣼⠟⠂⠀⠀⠀⠀⢀⣠⣶⣴⠾⣿⣿⢿⡿⠿⣿⠿⢷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠰⡆⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠃⠀⠀⠀⢀⣴⣿⠷⠟⠏⠀⠀⠀⠀⠀⠀⣀⠀⣀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢀⡏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⣿⠟⠁⠀⣤⠀⠀⠀⠀⠀⠀⠀⠋⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⣼⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢤⡀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡅⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡇⢸⡇⠀⠀⠀⠀⠀⠀⢠⣎⡀⠚⠻⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠄⠀⠀⠀⠀⠀⡇⠈⠀⠀⢀⣀⣀⣀⣠⠄⠀⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⢘⠟⠛⠻⢷⡒⠤⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠈⢿⠛⠻⠟⠀⠀⣠⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⡀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠊⠀⠀⠺⠟⡇⡿⠀⠀⣠⡤⠖⠂⢠⣴⠛⠐⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⣸⣶⠤⠤⠄⠠⢴⡞⢣⡐⠲⠦⣄⣁⣤⠠⠿⠓⣄⠀⠸⢿⣤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠲⣇⠀⠉⠉⠐⠀⠀⠀⠀⠁⠀⠉⢀⠀⢨⡇⠀⢀⡄⢲⣿⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠰⢿⢯⣀⠀⠀⠀⠈⢶⣄⠀⠀⠀⢀⣄⠀⠀⠀⠠⣿⠀⠀⠇⠀⠀⠄⠘⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠈⠉⠈⠙⠀⠛⠓⠟⠷⢄⠀⡀⠾⠇⠀⢆⠀⠌⢛⡒⠂⢉⣠⢛⡑⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠃⠀⠀⠀⠈⠂⠀⢀⢦⡀⢀⣿⡱⠀⠀⠀⠙⠦⠈⠛⡛⠤⠌⠙⠮⢹⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠁⠀⠀⠀⠀⠀⠁⠀⠘⢸⡿⣿⣟⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡾⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠀⠀⠀⠀⠀⠈⠀⢠⠓⣍⠉⠛⠷⣆⣁⣠⡴⠖⠀⠀⠐⠀⠀⡠⠞⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠀⠀⠀⠀⡐⠀⢀⡇⠀⠀⠀⠀⡽⠛⢿⡟⣀⡀⠀⣀⠀⠀⠞⢿⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢰⡀⠀⠀⣤⠁⢀⡜⣤⠌⠀⠀⣰⡄⠤⣤⣧⠀⠀⠀⠀⠀⠀⠌⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢸⣿⣤⠰⠃⠀⡬⠹⠏⠀⠀⣰⠋⠀⠀⠀⢻⣧⣀⡀⠀⢀⣼⠇⠹⣆⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠚⠁⢹⡸⣿⡟⠁⠀⠺⠖⠁⠀⠀⣼⠇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⠇⠀⠀⢻⡀⠀⠀⠈⡄
⠀⠀⠀⡤⠀⠈⠃⠀⠀⠀⠈⣇⠉⠀⠠⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⣰⠿⠍⡂⣿⡿⣀⠀⠀⠘⡇⠀⠀⠀⠘⡄
⢀⣀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠛⠚⠂⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⣰⣷⠏⠐⠢⠄⠘⡏⠀⠀⣀⠴⢿⠀⠀⠀⠀⠘⡄
⠚⠉⠉⠉⠉⠙⠢⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠠⠁⠀⠘⣯⡠⠉⠁⠈⠀⣷⡀⠀⣻⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⣿⢀⠃⠀⣠⡾⠁⠀⠀⠀⠀⢀⠉⠁⠀⠈⠳⢀⡿⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⢀⠆⠀⠀⠈⠀⠂⠈⢩⠃⠀⠀⠀⠀⠀⢸⡀⠈⠀⠀⠀⢸⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⡸⠤⣄⣀⡀⠀⠀⣿⠠⡀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡤⠄⣀⠀⠀⢠⣴⠁⠀⠀⠈⠉⠁⢰⣿⣿⡇⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠈⢀⣴⠾⠃⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠀⠤⠀⠀⠀⠈⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣀⣤⠎⠁