Temperature Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠶⠲⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠴⠤⠴⠴⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠶⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡅⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣀⣀⣀⣀⣀⣀⠀⣿⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣿⣿⠀⣿⠐⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⣿⣿⣿⡇⢸⡇⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⣠⣾⣿⣿⡿⣿⣿⣿⣦⣌⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⣡⣴⣿⣿⣿⢿⣿⣿⣷⣤⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣴⣿⣿⣿⣿⣍⡁⢉⣽⣿⡿⣿⣷⡌⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢡⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣦⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢸⠇⣼⣿⣏⠃⠸⣿⣿⡇⣿⣿⣿⠆⢋⣽⣿⡄⢻⡄⠀⠀⠀⠀⠀⠀⢀⡟⢠⣿⣯⡙⠻⣿⠟⠛⠛⠻⢿⡿⢛⣹⣿⣧⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣧⣶⣶⣬⡙⠃⠛⣉⣴⣶⣦⣽⣿⣷⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⣾⣿⣿⣿⣷⠃⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⡛⠻⠟⢋⣡⡄⣤⣍⠛⠿⠟⣻⣿⡿⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢿⣿⣿⣿⡿⣆⠀⠀⠀⠀⢀⡿⣿⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠹⣿⣏⡄⢸⣿⣿⡇⣿⣿⣿⡆⣬⣽⣿⠃⣾⠁⠀⠀⠀⠀⠀⠀⠈⣷⠘⣿⣯⣥⣶⣿⣶⣤⢤⣴⣿⣷⣦⣹⣿⠏⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠙⢿⣿⣿⣿⣋⡄⣈⣹⣿⣿⣿⡿⢁⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡈⢿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⡿⠋⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡙⠻⠿⣿⣷⣿⡿⠿⠛⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣈⠛⠿⢿⣿⣾⡿⠿⠟⣋⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⠶⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠛⠉

⠀⠀⠀⣠⠞⣉⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣉⡙⢦⡀
⠀⠀⠀⣿⢸⠁⢹⠀⡇⡒⠒⠀⠀⠀⠀⠀⢸⠁⡏⠈⡇⢹⠠⠤⠄⠀⠀⠀⠀⠀⡇⢸⠁⣿⢸⡇⡒⠒
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠂⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⢨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣄⣿⢸⡇⠂
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠥⠤⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⢐⣀⡀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢨⡇⠥⠤
⠀⠀⠀⡟⢸⠀⢸⠀⡇⡁⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⠐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢰⡇⡁
⠀⠀⠀⣷⢸⠀⢸⠀⡇⣂⣀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⢸⠨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢰⡇⣂⣀
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠆⠀⠀⠀⠀⠀⠀⢸⠀⣷⣾⡇⢸⢈⠉⠁⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠆
⠀⠀⠀⣿⢸⠀⢸⠀⡇⠅⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢨⡇⠅
⠀⠀⠀⣿⢸⠀⢸⠀⡇⡋⠉⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠰⠒⠂⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⡋⠉
⠀⠀⠀⣿⢸⠀⢸⠀⡇⡂⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠨⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⡂
⠀⠀⠀⣿⢸⣷⣿⠀⡇⠍⠉⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢘⠒⠂⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠍⠉
⠀⠀⠀⣿⢸⣿⣿⠀⡇⠅⠀⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⢐⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⢸⡇⠅
⠀⠀⠀⡏⢸⣿⣿⠀⡇⠒⠒⠀⠀⠀⠀⠀⢸⠀⣿⣿⡇⢸⠨⠤⠄⠀⠀⠀⠀⠀⡇⢸⣿⣿⠐⡇⠒⠒
⠀⠀⡴⢃⣼⣿⣿⣄⠳⡄⠀⠀⠀⠀⠀⢠⠞⣠⣿⣿⣧⡘⢦⠀⠀⠀⠀⠀⠀⡴⢃⣼⣿⣿⣄⠳⡄
⠀⢸⡇⣾⣿⣿⣿⣿⡆⢻⠀⠀⠀⠀⠀⡟⢰⣿⣿⣿⣿⣷⢘⡇⠀⠀⠀⠀⢸⠃⣿⣿⣿⣿⣿⡆⣿
⠀⠈⢧⠙⢿⣿⣿⠟⢡⠟⠀⠀⠀⠀⠀⠹⡌⠻⣿⣿⡿⠋⡼⠁⠀⠀⠀⠀⠈⢧⠙⢿⣿⣿⠟⢡⠏
⠀⠀⠀⠙⠶⠤⠴⠖⠉⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⠤⠶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠑⠶⠤⠴⠚⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡗⠒⠀⠀⠀⠀⢰⡋⠉⡆⠀⣠⠞⠋⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠚⠁⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⢸⠀⢠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣤⣤⣤
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡏⠉⠉⠉
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⣇⣀⣀⣀⣀
⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢈⠀⢸⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡧⠤⠤⠤
⠀⠃⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠀⢸⠀⠀⠧⣄
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢀⣴⣶⣿⣷⣶⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢰⣿⡟⠉⢻⣿⣿⣿⣷⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢺⣿⣿⣶⣿⣿⣿⣿⣿⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠘⢿⣿⣿⣿⣿⣿⣿⠟⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠙⠛⠿⠟⠛⠁⢀⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣤⡤⠔⠚⠁
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡏⠛⠛⠉⠉⠉⠉⠉⢉⡉⢉⠩⠉⠉⠉⠉⠉⠉⠉⠉⠉⠃
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⡇⠀⠀⡀⣀⠀⠀⠀⠀⢰⡆⢰⠀⠀⠀⠀⠀⠀⠀⣀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢄
⡇⠈⠒⠃⠚⠀⠈⠉⠉⢸⡄⢸⠀⠉⠉⠁⠀⠐⠂⠛⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠈
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⠀⠁⠀⠀⠉⠒⢦⡀
⡇⠐⠂⠃⠜⠀⠁⠈⠉⢰⠆⢸⠀⠉⠉⠀⠀⠐⠀⠟⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡐⣁⡀⣠⢤⠤⡄⡄⣤⢠⣄⢙⣆
⡅⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡔⠋⠈⠉⠁⠀⢀⣀⣀⣀⡀⣀⣈⣀⡉⣣
⡇⠘⠓⠃⠛⠀⠀⠀⠀⢸⠃⢸⠀⠀⠀⠀⠀⠘⠃⠛⠀⡇⠀⠀⠀⠀⠀⠀⣥⠐⠐⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⡇⠀⢀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⢠⣌⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐
⡇⠀⠈⠿⠃⠀⠀⠐⠀⢨⠀⢸⠀⠐⠒⠀⠀⠻⠌⠂⠀⡇⠀⠀⠀⠀⠀⡀⢰⠀⠈⠛⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠈⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢳⠀⠳⡄⠀⢀⣀⠠⠤⠐⠀⠀⠀⠀⢀
⡇⠀⠼⠣⠇⠀⠒⠒⠒⢈⠀⢸⠀⠒⠒⠂⠀⠿⠧⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠈⠓⢤⣀⣀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠢
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⠤⠄⣤
⡇⠀⠀⠛⠃⠀⠒⠒⠒⠀⠀⢸⠀⠒⠒⠂⠀⠙⠛⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⣀⣤⣲⣄⣭⡟⠁⠉⠻⠀⠀⠁⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠄⢸⢀⡤⢄⠀⡇⠀⠀⢸
⡇⠀⢴⠴⠇⠀⠒⠒⠒⢈⠀⢸⠐⠒⠒⠂⠀⣤⠼⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠌⠳⣄⡼⠉⠐⢦⠇⡇⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡘⠀⠘⡇⠀⣄⠀⠀⢣⠀⣀⣴⣷⡄⠂
⡇⠀⠸⠿⠇⠀⠒⠒⠒⢀⠀⢸⠐⠒⠒⠂⠀⠷⠽⠂⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠀⠈⠻⡀⠏⠀⠀⠈⠻⣿⣿⠋
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢈⠀⢱⣎⠀⠰⠀⠀⠈
⡇⠀⢤⣴⠇⠀⠒⠒⠒⠀⣶⣾⠐⠒⠒⠂⠠⣤⡄⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⠀⠀⢻⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⠀⠀⠀⠁
⡇⠀⢰⢴⠆⠀⠒⠒⠒⠀⣿⣿⠀⠒⠒⠂⠀⡤⠤⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠃
⡇⠀⣰⠄⠆⠀⠀⠀⠒⠀⣿⣿⠀⠂⠐⠀⢀⡆⡖⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠰⢄⠆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠠⠀⡶⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⢀⠆⠀⠤⠤⠤⠀⣿⣿⠀⠤⠤⠀⠀⢰⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⢸⡅⠐
⡇⠀⠀⢴⠆⠀⠤⠤⠤⠀⣿⣿⠀⠤⠤⠄⠀⠀⣤⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⡁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⡇⠀⠀⠀⠀⠀⠀⢀⣨⠟⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⡇⠀⠀⠀⠀⠀⠀⣤⣟⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⡇⠀⠀⠀⠀⠀⠀⠀⠙⠋⠿⠭⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⣇⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⡄⠴
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣭⣥⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣤⣾⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣠⣾⠟⢹⣿⠙⣿⣦⡀⠀⠀⠀⣀⣀⠀⠀⠀⣠⣾⠟⢹⣿⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠿⣷⡄⠘⠛⠁⠀⢸⣿⠀⠀⠙⠃⢀⣴⡿⠛⠛⢿⣶⡘⠛⠁⠀⢸⣿⠀⠀⠙⠛
⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢹⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢠⠞⠓⣦⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣰⣾⣷⣆⢸⣿
⠀⠀⠀⠀⠀⠀⣿⡇⢸⠀⠀⣿⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⢀⣿⡇⢸⠀⠀⣿⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⣿⣿⣿⢸⣿⡀
⠀⠀⠀⠀⣴⡿⠋⠁⢸⠀⠀⣿⠀⠙⢿⣧⡀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣿⣿⣿⣿⠀⠙⢿⣦
⠀⠀⠀⣸⡿⠁⠀⣠⠞⠀⠀⠛⢦⠀⠀⢻⣷⠀⠀⠀⠀⣼⡟⠀⠀⣠⣿⣿⣿⣿⣦⠀⠀⢻⣧
⠀⠀⠀⣿⡇⠀⢰⡇⠀⠀⠀⠀⠘⡇⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿
⠀⠀⠀⢹⣷⠀⠀⢳⣄⠀⠀⢀⡼⠃⠀⣸⡿⠀⠀⠀⠀⢻⣧⠀⠈⢿⣿⣿⣿⣿⡿⠁⠀⣼⡿
⠀⠀⠀⠀⠻⣷⣄⠀⠈⠉⠋⠉⠀⢀⣴⡿⠁⠀⠀⠀⠀⠈⢻⣧⣀⠀⠉⠙⠋⠉⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠈⠻⢿⣶⣶⣴⣶⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣶⣦⣴⣶⡾⠟⠋



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠑⠢⣄⢀⡤⠖⠚⠋⠉⠙⠓⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠒⠾⠄⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣀⣀⡤⢾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣆⡀⠀⠀⣀⠼⣄⣀⠀⠀⠀⠀⢀⣼⡏⣹⡅⠈⢹⡿⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠚⠉⠁⠀⠀⠉⠉⣭⡟⢦⣬⣽⠝⢯⡿⢯⣿⠛⣾⡭⠶⠤⠞⠵⠯⠴⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⠞⣹⢟⡴⠋⢳⣞⢟⣴⡻⣽⡾⠉⢧⢯⣉⠙⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡾⢧⣀⣨⢗⠼⠒⠷⡼⣅⣀⣼⣧⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⢻⢧⡼⣄⣀⡰⣣⣔⣖⠦⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡒⠫⡟⠧⠼⢻⡤⠗⢘⡎⢳⡶

⠀⠀⣀⡸⠓⣧⣎⠀⡀⢀⠁⡠⠈⠀⣨⠻⢈⣠⣀⡐⠫⠓⡦⢲⡖⣔⡀⠀⠀⡧⡤⢀⠪⢇⡠⠽⣿⢸⡸⠈
⠀⠀⢣⠠⢀⡇⠛⠠⢯⣧⣟⣉⣶⣴⡐⠚⡩⢤⠈⣙⡏⠙⢏⠁⠠⣬⠛⣫⢬⠅⠆⢹⣡⠀⢠⢂⡈⠧⢀⠉
⠀⠀⠠⠂⢸⠓⣌⠘⠳⡞⣽⡟⢒⠯⡇⠀⢠⡀⠈⠋⢹⠼⠘⡠⠔⠃⠀⠘⠰⣎⠀⠘⠂⠂⢉⣁⣀⡠⠴⠦⢴
⠀⠀⣠⡀⠆⠈⢀⠄⠀⠁⠉⠙⠚⠦⡼⠀⠲⡷⡤⢤⠬⢶⠛⠀⠀⠀⠀⠀⠀⠈⠡⣿⣿⡿⣿⡛⣥⣀⡂⠐⠨
⠀⡆⢩⠘⢠⢚⠀⢂⣐⡂⣐⠤⣄⠀⡘⡄⣟⣍⢍⣿⣿⡏⠀⡀⠴⠀⠁⠈⠀⠐⠀⢻⣈⣹⡇⣷⢏⠁⠸⡈
⠀⠓⠐⠋⣶⠀⣤⣭⣶⠀⠀⠀⣤⡑⡆⢹⢤⣿⠀⣿⣿⠱⠀⠿⠿⠇⠀⠀⠿⠉⠆⢸⡿⡿⣿⣝⠀⡁⠈⠀⠢
⠀⠀⠰⣦⢸⠟⠙⠘⠋⠃⠀⠺⠯⠿⠇⢫⠒⠛⠃⡿⠿⠀⣇⠠⠄⠀⠀⠀⠀⠠⢀⣬⣼⣥⣬⣧⣼⠁⢸⢅
⠀⢐⢾⣷⠀⠀⠓⢄⠀⠀⠀⠀⡠⠒⠀⠀⢀⠐⠀⢠⠄⠉⢘⠀⠀⠐⠀⠀⡞⠀⢽⢯⣤⣤⣀⡀⢉⣠⣨⡀
⠀⠘⢰⢸⡉⠀⠀⢀⠍⣲⢦⡀⠀⠀⠀⢸⠊⢀⣂⡉⢸⠀⣇⠤⢀⣠⣀⡀⠘⠄⡍⠐⠈⡗⠮⢤⢠⠀⢠⠀⢀
⠀⠀⠰⢸⠀⢠⡪⠑⠦⣤⣜⠟⠳⠀⢠⠆⢰⠀⢽⠏⢠⠊⠁⣦⠉⠀⠈⣰⠆⠀⠘⡆⣇⣁⣿⠀⢸⠀⠰⠤⠉
⠀⠀⢥⠿⡄⢠⠀⠸⡶⢣⢝⣲⢾⢰⡇⠘⠘⠂⠙⠀⢨⢠⠀⠁⠀⡀⠀⠘⠀⢠⠄⢿⡍⣉⣉⣉⣙⡂⡤
⢀⡀⢠⡄⠀⢺⠀⠱⢨⢋⠦⢱⡾⡟⠀⡀⠀⠀⠀⠀⠙⢿⢘⠀⠀⠀⠀⢀⠀⠈⠄⠗⠒⢰⣐⠀⠈⠃⠰
⠙⠋⠶⡄⢻⠸⠀⡖⠢⠍⠖⠸⠀⠓⡄⢠⢀⠰⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠀⠀⢀⠀⣀⣀⣈⣂⠂
⠀⠀⢬⣀⡌⠁⡰⠀⠀⠈⡁⠀⠉⣁⢱⢸⡂⠀⠀⠀⠀⡸⠀⣀⠀⠀⠀⠈⠀⠀⠀⢀⠁⠲⠤⠽⣽⠀⠈
⠀⠀⠛⢉⡖⠐⠀⠀⠠⠀⠀⠸⠀⠘⡂⠃⠃⠀⠀⢀⣖⡃⠐⠁⠁⢀⡀⡀⠀⠤⠄⠈⣉⠛⢛⢛⢃⠀⠈⠉
⠀⠆⠽⢸⠀⡤⠀⠀⠀⠂⠀⠀⠀⠀⠇⠸⠷⠶⠦⠉⢯⡄⢀⠒⠄⠉⠀⠓⠀⢀⡄⠀⠂⠀⠀⠻⠾⠓⢾⢾⡀
⠀⠆⡽⢸⢧⡀⠀⠀⠀⠀⠀⣄⠀⠢⠄⢖⡁⠀⠀⠀⠀⡃⠉⠀⠀⠃⠀⠰⠠⡄⠀⣴⠠⠀⡀⠀⠘⠂⢨⠀⠈
⠀⡄⢠⡼⡾⠡⠄⣀⠀⠀⡔⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⡀⠀⠀⠘⠀⠈⡀⠙⠙⠁⠀⠆⠀⠀⢀
⠀⠉⢠⣐⠢⠲⠀⠀⣀⠈⠀⠀⠀⠀⠀⠀⠿⢠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠄⠒⠀⠃⢀
⠀⠀⠘⢉⠈⠐⠲⣦⠀⠀⣠⠐⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠾⠀⢐⠀⢸
⠀⠀⠀⠈⠀⠀⠀⠉⡀⠈⢹⠘⡆⠀⠀⠀⠀⠘⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⡄⠀⠀⠀⠉⢆⡀

⠀⠀⣤⡀⢻⢀⡤⠄⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⢀⣩⢝⠯⣤⡀⠀⠀⠀⢀⠐⠳⠄⠀⠀⠀⠁⠀⠀⠀⠀⠙⠇⠀⠀⠀⠂⠀⠀⠀⡀
⠀⠀⠉⠁⢼⠀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠂⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⡍⠀⠀⡀⣀⠀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠘⠳⠀⠀⠀⠀⠀⡇⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠈⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠐⢗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⠀⣃⣁⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡈⠃⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠃⠓⠀⠀⠀⠉⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⡤⢤⢤⡀⠀⠀⠁⠐⠃⠀⠁⠀⠀⠸⡿⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⢂⢄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀
⠀⢀⡡⣆⣾⣶⠬⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢈⠊⠀⠀⠀⠀⣠⢀⡄⠀⠀⠀⠰⢂⠚⢛⢂⡄⠀⠀⠀⡘⡈⠀⠐⠀⣀⠀⠃
⠀⠀⠝⠽⣿⡛⡏⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⡛⠤⠀⡃⠆⠀⠀⠘⠀⠀⠀⠠⠂⢀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡞⠁⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢻⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠔⣪⠿⠋⠀⠀⠀⠀⠀⠀⣠⠄⠂⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⣉⡼⢋
⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣨⡇⠀⠀⠀⠠⠚⠁⠈⡙⠀⠀⠀⢀⡤⠔⠊⠡⠉⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⢰⡆⠀⠀⠀⢼⡇⢨⣡⣶⣮⠈⠒
⢠⣾⠴⣻⠇⣸⠀⠀⠀⠀⠚⣍⡄⢦⠀⠀⠀⠄⣦⠐⠀⠀⠐⠚⢻⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⣾⣧⠀⠀⣠⣿⣿⣈⣋⡼⠟⢶⠶⣶⢀⣆
⣿⡿⣊⢃⣲⣼⣇⣤⡶⠚⢈⠉⢰⢌⣁⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⢸⡆⠀⠀⣫⣜⠅⠀⡜⢛⡎⢉⣻⣀⣤⠈⢃⡀⠙⢿⣀
⣿⠞⣹⣈⡽⢌⣿⣿⡷⠂⠀⠈⡃⠀⠀⢣⠀⠈⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡷⠂⠘⣁⡙⢶⡀⠟⠁⡴⣄⡘⣿⡏⠳⡌⠙⠷⣾⣿
⠿⠓⣅⡟⢘⣦⣹⡿⠗⠚⠶⠖⠉⠛⠧⣌⠳⣦⣦⣴⣀⣴⣠⡠⢼⡿⢄⣀⣰⣄⣦⣦⣾⣷⣶⣏⢜⡁⢸⠞⢉⡅⢸⡡⠒⠉⠻⡀⠻⣌⣗⣦⣤⣀⠈⠻⣿
⣰⡎⠀⣿⢺⣏⣠⡤⢾⣀⣤⠾⢶⢸⡆⠘⠷⣿⣿⣿⣿⣿⣿⡗⡹⡟⢺⣿⣿⣿⡟⢹⣿⣿⣿⡥⢟⠙⠓⠛⣩⢸⢛⣀⠀⠂⣧⠘⣤⣼⣿⣇⠀⠙⠷⢴⣿
⠴⠳⣀⣿⣶⠟⠉⣀⢀⣉⣇⠀⢨⡽⡟⢦⣀⠸⣿⡟⡟⡿⡿⣉⣠⠉⢈⢻⣿⣿⡴⢊⣻⣿⠋⡠⠴⠗⠶⣦⣽⣇⣼⠡⠤⠖⠛⢿⡀⠘⢯⠉⠁⣄⠀⠀⢹
⢀⣶⡟⣹⣷⡞⠋⢀⣬⠋⠀⡖⠈⠀⢀⡀⠙⠳⣾⣿⡏⣿⠯⠎⣀⢧⡼⠨⢿⡵⡁⠊⢉⡟⢋⠀⣠⢤⠦⣀⢹⣿⣅⠀⣠⡀⠀⢹⣉⠉⠺⣆⡀⠈⠂⢤⣀
⣾⣁⣼⣿⡯⠀⠚⢻⣁⣠⡞⠀⢠⡀⠘⢳⡄⣀⠈⡿⢿⡅⠀⠈⡇⢸⠀⠀⠠⡾⠁⠲⢎⠁⣣⣤⡇⢀⡷⣄⣉⣿⣿⠜⠛⣧⠀⢠⡏⠶⣄⡘⢧⣠⠂⠀⢻
⠀⠀⡽⠃⠀⠀⣀⡤⠞⠉⠀⠀⠀⠛⣆⢨⡗⠛⠓⠇⠘⠓⠐⠋⠙⠛⠒⠀⠀⠈⠀⠀⠀⠁⠀⠀⠑⠁⠀⠘⠛⣁⠆⠀⢈⠉⠓⠃⠀⠀⠀⠙⠻⣯⠁⢀⢈
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠲⠀⠀⠀⠙⠻⠇⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠁⠀⠠⠾⠴⣄⡀⠀⢠⣀⣣⡀⠈⣻⠿⠿




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⣉⣤⠒⠒⢲⠒⠠⢌⡑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡰⠋⠐⣎⢦⠀⠘⡀⠀⣠⠜⣦⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢠⠃⣼⣁⠀⡀⠈⣆⠻⠀⡦⢊⠴⠊⠀⠱⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠒⠂⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡖⠂⠠⢤⣄⡉⠀⡀⠀⡋⠒⠒⠀⢠⠇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠲⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣁⠀⠀⣠⣞⡵⠞⣳⡌⢫⠉⠁⠀⢐⠂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠻⠙⢫⠋⠁⠈⠀⡇⢿⠉⢦⡱⡀⢀⠎⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠉⠀⡀⠀⢀⡸⠀⢀⡱⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠈⠈⣿⡉⢤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⠀⠀⠀⢀⣀⡀⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⠁⠀⠉⠉⠉⠁⠀⡀⠄⠂⠙⡗⠲⣤⣀⣀⣀⣤⠔⠚⠉⠀⠀⠀⠀⠀⠙⢆⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⢀⠔⠈⠁⠀⡄⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⡈⠀⠠⠂⠁⠑⢄⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢣⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⠀⠀⣇⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢸⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢀⡔⢃⠀⠀⣀⣀⣀⣀⠀⢦⠀⠀⠀⠀⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠇⠀⠀⠀⠈⢠⡡⣀⠤⠒⠉⠀⡀⠀⠀⡏⠉⠉⠉⠀⠀⢱⠀⠀⠀⠿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠂⠀⠀⠇⠀⠀⠀⠀⠀⢈⠀⠀⢀⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢿⠀⠀⠀⠀⠀⠀⡠⠉⠀⠉⠓⠴⢦⡂⠀⠀⠀⠀⠀⡕⠸⠀⠀⢸⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢸⡄⠀⠀⠀⠀⡰⠁⠀⢠⠧⢄⡀⠀⢩⠁⠒⠒⠂⠐⠈⠁⠀⠀⡶⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⣁⠀⠰⠃⠀⠀⠁⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⠀⠀⠀⠀⢋⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⢸⢂⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡀⠀⠀⠀⠈⠢⢀⡀⠀⢀⠄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⡎⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⠀⠀⠀⠀⠀⠀⠈⠑⠂⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠋⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⣄⠙⠶⢤⣀⣀⣀⣀⡀⢀⣀⣀⡠⠴⠚⠋⣠⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡈⠙⠲⠤⣤⣀⣀⣈⣉⣀⣀⣤⠤⠖⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠊⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣟⣿⣻⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⢀⠀⢘⡅⠀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣆⢸⣿⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡆
⠀⠞⡞⢻⡇⠰⣾⣷⢖⡶⡖⣀⣤⣤⡌⢓⣂⣄⢘⠇⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡀⢠⣿⠯⢿⠓⠒⣦⣠⣤⡔⢂⣤⣼⣰⣇⢰⡆
⠀⣛⣥⠘⠉⢛⠿⡟⢙⡄⠁⡩⢨⣗⢩⡍⢹⡟⢊⣩⣬⠇⣀⠀⠀⢀⣤⠀⠀⣀⣀⣀⠾⢠⡮⢳⠈⣼⠨⣯⣳⡍⢿⣿⠯⠀⠛⡚⢇⠤⢳⠼⣧
⠀⡯⠘⢶⣸⢓⡖⢹⣅⣷⠚⡳⢘⣵⠈⣓⡭⠶⣤⠒⣶⡂⢿⣿⢻⡏⠛⢿⣿⣯⣿⡿⣺⣙⢈⢫⠀⣶⡠⢿⢧⠱⠶⣿⣆⠩⢠⡈⢻⠒⠸⢱⡼
⠀⠈⢀⣯⡜⠸⣆⡠⠿⣠⠋⣡⡬⡀⢛⡿⢖⡱⢉⣞⠉⡍⠺⣻⡤⠁⠡⠈⢿⢁⡹⡂⡩⠌⢹⡏⣹⠁⢩⡘⢄⠱⠦⣬⡁⠲⡆⠓⠘⣥⡋⡜⠳
⠀⠀⠀⠙⠀⠺⠤⡘⠀⡀⠀⠉⠐⠉⠛⢡⣄⢱⣏⠀⢢⣸⠓⠈⠁⠐⠒⠀⠈⠀⠀⠈⠁⠉⢚⡙⢉⠑⠏⠉⠒⢇⣠⡁⠉⠠⠆⠨⠂⡌⠔⢠⡉
⠀⠀⠀⠃⠐⠁⠄⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡇⢈⠧⠀⠘⡟⠠⠩⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣭⣯⣭⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠀⠀⠀⢀⡠⡆⠀⢀⠛⠃⠀⠀⠀⠐⡀⠂⢄⠈⡆⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠉⠀⣠⣤⣤⡄⠀⠀⠀⠡⠄⠤⡔⠀⠀⠀⣀⠹⢼⣦⡔⣤⡌⠳⣌⡇⠈⠁⠒⢼⣆⡿⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣆⢰⣾⣿⣿⣿⣷⣄⡀⢀⠀⠠⠀⠀⠀⠀⠀⠙⡀⠶⠦⠿⣾⣿⠆⠈⠻⡄⠀⠛⠿⣿⡇⠈⢣⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠒⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠐⠲⢞⣫⠉⠃⠤⣄⠀⢰⣿⠟⠁⠈⢳⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣴⣤⣤⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠐⢳⡀⢈⠙⣿⣿⡄⢣⡀⠀⣿⡇⡀⠀⢅
⠀⠀⠀⠀⠀⠀⣀⣴⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣀⠉⠓⠀⢀⠀⠀⢅⠠⠦⢀⡭⠙⠒⠲⢼⣿⣇⠀⣿⡀⣘⣧⡅⢠⡏
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠒⠶⢄⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⡷⠿⣧⠀⢿⡆⣸⣇⡠
⠀⠀⠀⠠⣼⣿⣿⣿⡿⠛⡻⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡿⠚⠶⠒⠲⢦⣄⣀⣀⡀⣠⣴⢶⡞⢻⣿⣿⠄⠈⣷⣾⣿⣿⡟⣀
⠀⠀⠀⠀⣻⣿⠟⠋⠀⠀⠉⠙⢻⣷⡆⠈⠙⣿⣿⢿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⡶⣆⢀⣠⠜⠉⣹⡿⠛⠛⠻⠿⠷⣾⣿⣿⡀⢠⣿⣿⣿⡿⠁
⠀⠀⠀⠀⣾⠃⠀⣀⡀⠀⠀⠀⣧⣿⣿⣦⠀⣷⣿⠿⠻⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣏⡄⣔⠒⣚⡙⢀⠀⢀⣄⠀⠀⠀⣿⢿⣷⠈⢸⣿⡿⠁
⠀⠀⠀⠘⠷⠶⠛⣷⣯⣴⣿⣿⣿⣿⡿⠟⠠⠛⢟⡄⠁⢹⣻⣿⠛⢿⡿⠿⢿⠿⠟⠛⠛⠻⠊⠒⠾⠓⠐⠂⠒⢿⣋⠘⢷⣿⡾⢿⠀⡷⠈⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣶⣦⣤⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠃⠒⠂⠀⠀⠀⣸⣽⣷⣿⣟⡇⠀⢿⣧⠄
⠀⠀⠀⠀⠀⠀⠀⡀⢙⡹⠟⠏⠙⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢰⣴⣶⣶⣶⣾⣿⣿⡿⠋⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠳⠿⢿⡷⠶⣶⣶⣶⣶⣶⣤⣤⣤⣴⣦⣤⣤⣤⣶⣬⣾⣿⣿⣿⣾⣷⣿⣿⣿⣿⣿⣿⠟⠉⣉⡥⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠊⠻⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠄⠄⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢰⣿⣿⣾⠟⠉⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠃⢿⡿⠛⠁⢠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠼⠡⠙⠿⣦⣶⣤⡀⣠⡀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣤⠀⠀⢀⣹⣿⣿⣿⡿⣿⡿⠆⠀⠤⣄⠀⠤⡀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⠼⠟⠋⠉⠉⠉⠀⠀⠀⠀⠈⠁⠀⢤⣀⣤⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣿⡃⢠⣤⣤⣤⡄⠀⠀⠶⣲⣷⠒⠒⠾⣿⡏⠀⢠⣿⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡀⢿⣿⡿⠑⡄⠀⠀⠀⠿⠿⠀⠀⠀⠉⠃⢠⣿⡟⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠇⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠋⠏⠀⠀⡴⠁
⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠾⣷⣦⡀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⣴⣮⡄⠈⠁⠀⡸⠁
⠉⠁⠀⣀⠿⡇⠀⠐⣧⡀⠀⠀⠀⠀⠙⠿⠧⢀⠸⣿⣿⣦⣀⠀⠀⠶⠖⠀⠀⡠⣺⣿⣿⡇
⠀⠀⠀⠡⠤⠀⠀⠀⢈⣽⡗⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡆⠙⣶⠦⠄⠀⠀⢰⡈⠳⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠋⠀⠀⠘⠋⠀⠀⣰⣆⣀⠀⠈⡡⠊⠉⠉⠉⠛⠛⠅⠀⠀⠀⠈⣿⡀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠊
⠀⠀⠀⠀⠀⢤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣶⣤⣄⡄⠀⢀⡀⣤⣴⣶⣤⣶⣄⣐⣥⡘⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠷⢶⡶⠾⠿⠿⠛⠋⠻⢿⣿⣿⣿⣷⣾⣿⣶⣿⣿⣸⣿⣿⣿⣿⣿⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠻⠛⢛⡷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣬⣉⠙⢀⡄⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡧⠥⡬⠀⠥⠬⠼⢿⣄⡀
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁⠊⠀⠰⠀⠀⠀⠈⠀⠀⠀⠙⢷⣄
⠀⠀⠀⢰⠞⠉⠉⠉⢳⡀⠀⠀⢀⡞⣁⣀⣀⡠⠤⠤⠖⠒⠀⠒⠢⠐⠄⠳⢤⣙⣧
⠀⡴⠖⠛⠂⠀⠀⠀⣸⡃⠀⣴⠛⠏⠠⠀⠀⡆⠀⠈⠀⠀⢘⠀⢰⠀⢠⠀⠘⡀⠉⣧
⢸⡇⠀⠀⠀⠈⢢⠀⠈⢿⡀⡏⢠⠀⢈⠀⠀⠁⠀⠀⠂⠀⠈⠀⠀⡄⠀⠀⠀⡇⠘⣸
⠀⣿⠂⠀⠀⡀⠈⠀⠀⣸⡿⠛⠀⢤⣚⣩⣭⣩⣒⡉⠉⠀⠴⠖⠒⢆⣀⡐⣖⣳⠂⠉⢳⡄
⢀⣸⡇⠀⠀⠁⠀⣤⠞⢹⠃⠀⠀⢰⠀⢀⠊⣴⡿⢷⠀⠀⠀⡴⢷⣦⠑⡄⠈⡇⠀⠀⠈⣷
⡟⠀⠀⠀⣆⠀⠀⡹⠶⠛⠠⡀⢀⡸⢠⣬⣄⣛⣿⠞⣠⣤⣄⢻⣿⠟⡠⣁⢀⡃⠀⠀⠀⣿
⠻⢤⣠⣤⠁⣀⢀⣠⡶⠓⠲⣦⣠⣾⣻⠟⠟⠩⢋⠕⠛⣏⠙⢌⠫⠑⠟⢿⢙⣷⣤⡤⠞⠁
⠀⠀⠀⠈⠉⠉⠉⠁⢀⣴⠞⠋⠉⠙⡇⠀⠀⠀⠀⠀⠚⡍⠂⠀⠀⠀⠀⠀⡋⠉⠉⠙⠦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⠤⠔⠊⠘⠀⠀⠀⠀⠀⠀⠜⠙⢆⠀⠀⠀⠀⠀⡈⠉⠑⠤⢤⡙⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⣀⣀⡀⢳⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠀⣈⣢⡀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⡸⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣦⠄⠊⠁⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢸⠒⠂⢲⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠤⢽⡇⠀⠐⠒⠀⠤⢼⠀⠀⠀⠀⠀⠀⡸⠄⠒⠊⣿⣤⡴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠁⠀⠐⠒⠂⢎⣰⣣⣆⣴⣰⣠⠗⠒⠈⠉⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣅⠀⡽⠃⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣛⣛⣦⣤⣀⣀⣀⣤⣐⣿⣾⣓⣠⣄⣀⣀⣠⣴⣞⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠉⠉⠀⠀⠀⠈⠀⣼⣿⡀⠈⠀⠀⠀⠈⠉⠁⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⣿⢷⡳⢄⡀⠀⠀⠀⠀⠀⠀⢀⡼⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠿⡕⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⠐⠀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠀⠀⠲⣧⡟⠛⠛⠾⠼⣵⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠁⢄
⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢧⣀⡀⠀⠦⠁⢣⡐⠿⣀⡀⢀⣭⢙⡇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠆
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠛⠓⢲⡝⠀⠈⠭⠅⠚⠪⠥⠎⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠀⠌⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢨⠏⠉⠠⠂⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⢑⠬⢖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠁⠀⠀⣸⠁⠰⡄⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢣⠈⠀⢀⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠐⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⣀⣀⠀⠀⠀⠐⠀⢁⠒⢪⠀⠈⠒⠲⡁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠶⠶⠤⠤⠤⠤⠤⠤⠴⠤⠤⠤⠤⠦⠴⠛⠒⠲⠶⠶⢦⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠀⠀⡉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⢶⡀⢀⡼⣿⣬⣀⠀⣤⣈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠘⣷⡟⠸⠋⢨⠉⠿⠈⠉⠙⠿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠛⢻⣤⣾⢋⡀⢡⡆⠁⠀⢀⠈⢀⡔⣰⠃⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠊⠂⣬⠟⡇⢠⢊⠉⠀⡀⠀⢡⣭⡎⢳⣟⠀⠀⢷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⣤⠞⢿⣏⠀⢹⣠⣖⡀⡢⢸⡼⢏⣿⡇⠾⣧⢤⣤⣼⠷⣧
⠀⠀⠀⠀⠀⠀⢠⣤⣿⡶⣾⢹⡆⣸⣇⣀⣸⡇⢻⣷⢃⣼⠁⣾⣿⣷⡶⠟⠻⠉⠛⢶⣏⣷⣰⡆
⠀⠀⠀⠀⠀⡰⠿⢸⢏⣤⠿⠿⠛⠿⠾⢷⣿⣷⡿⠙⣦⢿⣶⡿⠋⠁⠀⠠⠖⠃⠀⢀⠹⣮⣟⢳⡄
⠀⠀⢀⣠⣼⣅⣠⡾⠛⠰⠋⠀⠠⠄⠈⠑⠂⢄⠙⠻⣿⣯⣉⣀⠁⠘⠀⠀⠤⣄⡀⠀⠁⠬⠻⢦⣷⣄⣀
⠀⠀⠀⠀⠈⠉⠁⠀⠒⠀⠁⠀⠀⠀⠂⠔⠀⠀⠀⠀⠀⠀⠙⠉⠀⠈⠈⠀⠀⠀⠈

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⣍⢺⣷⣶⣖⢲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⣼⠆⢼⡃⠀⠇⢙⣷⠈⢱⣤⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⣿⠻⣾⣷⣆⣷⠟⠈⠓⢶⡒⢧⠀⠀⠀⠀⠐⣦⡜⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢿⡶⢻⣻⠉⠉⠀⠰⢣⣾⢏⡏⢡⠀⠀⠀⠀⢾⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠺⡿⣶⣯⠓⣒⣄⠀⠉⠁⢸⡏⡈⠀⠀⠀⠀⢀⣹⣿⣷⣜⢿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠈⠛⠻⢮⢿⣿⣟⠁⠁⢀⣠⠀⣧⠏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⣝⢿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠻⣿⢿⠷⢦⣬⣤⠂⠈⠄⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣝⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢰⡆⢚⣏⠐⣾⣟⠛⣷⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⡝
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢻⣿⣷⠀⣿⡧⠤⠿⢧⣌⣿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠈⠯⣀⣽⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⢷⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠉⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣮⠄⢀⠠⡵⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣠⠴⠇⠀⠀⠀⢟⠎⠙⢆⣀⠔⠒⠒⠒⠲⢮⣀⢀⢽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢈⡲⣄⠀⠀⠀⢉⡹⢇⠀⠀⠠⠤⠤⠀⠤⠋⠀⠀⢀⠠⢀⠶⠒⠠⠔⣉⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⠞⠁⠀⣈⠧⢤⡀⠚⢓⡶⠋⠀⠤⠪⠅⠐⠀⢶⡀⢀⡄⠀⠀⠀⠘⡀⠀⠰⣈⣓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡷⠈⠀⠀⠀⠙⣯⡤⣶⠉⠀⠀⠤⢤⣄⣠⠀⠀⢠⡙⣺⢦⣀⣀⡠⢼⡂⠀⠠⠈⠳⢤⡀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⢘⣯⠶⢃⣀⣀⢀⠂⠀⡠⠀⠊⠁⠐⠀⠤⠈⠄⢠⠄⠀⠀⠽⠿⡤⠤⠆⢶⠣⠴⠄⠨⠙⠷⠁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠊⠀⡀⠂⠀⢀⣵⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠦
⠀⠀⠀⠀⠈⢁⠀⠀⠈⢇⠀⠀⠀⠀⠄⠐⠐⠇⢀⡈⠃⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠀⢀⠄⠀⠀⠀⣿⠸⢀⢀⠆
⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠳⠂⠐⠒⠋⠉⠉⠀⠀⠁⠀⢀⠭⠐⠠⣄⠀⠀⠀⠀⡰⠆⠗⣇⠀⡀⡀⠀⠩⢅
⠀⠀⠀⡐⠀⢀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⢄⠨⣀⢘⠀⠀⡰⠃⠀⠀⠉⠂⠀⢄⠀⠀⠀⠡
⠀⠀⠀⠃⠐⠈⠰⠚⠂⡀⠀⣀⠀⠀⠀⠀⠀⠈⡅⠢⠎⢰⠤⠔⠂⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⣀⣸⣀⣠⠴⠒⠊⠁⠀⠀⡀⠀⠰⠄⡀⠀⠐⣤⠀⠀⠺⠀⡀⠀⠀⠂
⠀⠀⢐⡒⠀⠀⠠⠤⠨⠓⠈⠁⠉⠀⠀⠀⠟⠦⠓⠂⠉⠀⠑⠊⠈⠀⠀⠀⠀⠀⠀⠒⠒⠒⠠⠸⠱⠈⠑⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⢒⡾⠟⢋⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⢁⢤⣾⡯⠖⢛⣩⠝⠛⠉⠉⠉⠉⠉⠉⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣾⠟⠁⠀⠀⠁⠾⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⠛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣀⡾⢋⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡽⠶⠾⠝⠻⠷⠶⠦⠽⣶⡀
⠀⠀⠀⠀⠀⠀⣴⣿⠋⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⢰⡏⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣷⡄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⢀⢀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠘⣧⠀⠀⡆⢿⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⣨⠄⠒⠒⠒⠒⠢⢬⣁⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⡗
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠡⠈⢧⡸⡀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠈⠉⠉⠉⠲⣄⣂⠤⢀⠀⠀⠈⠙⠲⣄⡀⠀⠀⠀⠀⢀⡼⣣⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠁⠈⠳⣷⡀⠀⠀⠀⠀⢸⡇⠀⣠⠖⠒⠲⢤⡀⢸⡇⠙⢦⡀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠶⠟⠛⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⣤⡀⡞⠳⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢱⢠⠇⠀⠀⠙⢦⡀⠀⠀⠀⠀⣀⠀⠀⠀⡀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡎⣹⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⡀⠀⠀⠀⠈⣝⠳⣄⠀⣀⣈⠁⠈⠉⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣧⣻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁⠈⢀⠀⠀⡀⠀⠀⠸⡆⢠⠞⢉⡉⢳⣀⣠⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣷⠒⠒⠶⢦⣤⠀⠀⠰⠞⠋⠀⢄⣴⣾⣿⠛⣟⣿⠋⠀⠀⠁⠈⢀⠊⡀⢸⣏⢢⡘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣹⣶⣯⣴⣀⠀⠀⠀⠀⠀⠀⣸⣏⢹⣿⣿⠇⡾⠃⠀⠀⠀⠀⣠⠀⡠⠁⡼⢻⡄⠱⡀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣏⠈⡉⣿⣷⣄⠀⠀⠀⠀⠛⢈⣲⣬⡵⠎⠀⠀⠀⠀⠀⠀⢿⠀⢀⡼⠁⠀⠹⣆⠁⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⣵⣬⠿⠛⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⠙⠀⠀⠀⠘⢦⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⠀⢣⡀⠀⠀⠀⠀⠳⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⡜⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣧⡀⠱⣄⠀⠀⠀⠀⢡⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠀⠀⠀⡀⠠⡶⠃⠀⠀⠀⠀⢀⣤⠞⠁⠟⢻⡏⠙⠷⣤⣅⣀⠀⠀⠀⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣍⠑⠆⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠈⣿⠀⠀⠈⠻⡙⢿⠶⣤⣸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⣀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⠀⢳⡀⠹⡇⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⠀⠀⠀⠀⠀⠀⢇⠀⣷⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⣤⠤⠀⠀⠀⠀⢰⡇⠘⣷⡀⠀⠀⠀⠀⢸⠀⢹⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⣾⠁⢰⢸⠛⢦⣄⠀⠀⢸⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢸⠈⠀⠀⠙⣧⠀⠀⠀⣟⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⣧⠸⠀⠀⠀⠀⢸⣆⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⡠⢒⡽⢷⡀⠀⠀⠀⢸⣏⡴⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⢠⣿⢙⣁⣏⣹⠂⣐⠠⣀⣠⣴⡟⠀⠀⠐⠉⠀⠈⢷⠀⠀⠀⡼⠋⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⣿⠁⣀⠀⠀⢉⣼⠾⠛⠉⢁⣿⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⡜⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠋⠐⠐⠐⢺⡏⠀⠦⣠⡶⠋⠁⠀⠔⠃⣸⡇⠀⠀⠀⠀⠀⠀⠀⢠⣷⣮⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⢀⣾⣤⠶⢻⡿⠁⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡀⡀⠀⢀⣠⠾⠛⠁⠀⠀⣾⠁⠀⠀⠀⠀⣠⡟⠁⠀⢀⣀⣤⡴⠞⢋⡼⠃⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠁⢠⣴⠟⠁⠀⠀⠀⠀⣼⠇⠀⠀⠀⣠⠾⢋⣠⣶⣞⣫⣭⡴⠶⣿⠟⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣤⡞⢁⣀⣀⣀⣠⣤⣞⣁⣀⡠⠶⠛⢛⣋⡽⠟⠉⠉⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠙⠶⢶⣄⣀⠀⠀⠀⠀⣀⡶⠛⢶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠀⠀⠀⠀⠀⠉⠙⠷⠲⢦⡼⢧⡀⠦⢄⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⡀⠀⠀⠀⠔⠲⡤⢄⢀⠀⠀⢀⣈⢥⠀⢈⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠈⠉⠃⠀⠈⠀⠀⢀⠉⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⢠⠒⠃⠀⠀⢠⠀⠀⠀⠀⢀⣄⣼⠃⠰⠚⠁⠁⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠀⠀⣀⡴⢿⣰⠀⢀⣘⣿⣄⣠⣄⠀⣀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣴⠗⠚⡅⠐⠂⢹⡆⣸⠋⠁⠀⠀⣬⡟⠉⣟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠀⠀⠀⠓⠋⠀⠀⢠⠄⠀⠀⠀⢻⣧⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢀⠀⠀⠀⠀⠀⠫⠇⠀⠀⠀⠀⠀⠀⡀⢺⡧⠤⢤⡈⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⣸⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⡆⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⣿⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠄⣿⡗⠒⠚⢁⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣟⠒⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠋⡦⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣑⣑⣚⣘⣃⣄⣤⣤⣤⠤⠤⠬⠶⠮⢕⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⡄⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡠⠂⠀⠀⠣⠖⠚⠋⠁⠀⠀⠀⠀⠀⠈⠳⡄⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡰⠀⠘⣄⠀⠀⡀⢊⢻⡅⣀⣠⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠀⠀⠀⠆⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠐⡄⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⢀⡐⠖⠒⠀⠀⠈⠀⠀⠀⠀⠈⠙⢧⠀⡄⠀⡠⠐⠉⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠉⠑⠖⠒⠆⠙⠁⠀⠀⠐⠒⠢⢤⣀⠀⠀⠀⠀⠀⠀⠲⠂⠁⠀⠀⠀⠀⠀⠒⠐⠒⠒⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⣀⡀⠀⠀⠀⣅⠀⠈⠀⠀⠀⠠⣶⡀⠀⠀⠀⠉⠡⠘⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⡘⠂⠀⢀⡀⠀⠀⠙⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⣌⣻⣶⣶⡄⠀⠀⠀⠀⣀⠚⠤⣀⠇⠀⠀⠈⠀⣀⣀⠀⠁⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⡃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣔⣋⣀⣀⡄⠋⡀⢲⡶⠛⠈⠉⠉⠀⢂⠄⡠⠞⠀⠀⠀⢈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠐⠠⠒⠀⠤⠐⠂⠀⠀⠀⡀⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⣛⠛⠉⠀⠀⠀⢀⡀⠀⢠⡄⠀⠀⠐⢆⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢀⠀⠀⠀⠀⠀⡠⠄⠂⠠⠀⢲⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠁⠀⠀⠈⠂⠉⠁⠀⠀⣴⠀⠀⠀⠈⡄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠄⠀⠀⠀⠀⡾⠀⠀⢀⠂⠀⠀⠳⡀⠀⠀⠘⠷⠄⣀⡐⠠⠀⠀⠴⠀⠇⠀⠀⠀⠀⠀⠈⠲⡀⣼⠇⠀⠀⠀⠀⡇⠀⠀⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠐⠄⠤⢶⠀⠀⠠⡏⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢉⡅⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⡿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⣀⠠⠀⠀⢰⡇⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣄⠀⠀⠈⡄⠀⠓⠀⠐⠀⠀⢀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⡠⠂⣽⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠄⠒⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠦⠀⠀⢁⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠉⡔⠀⠀⡠⠊⠐⢀⡆⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠤⠶⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠶⠄⢀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠘⠆⢸⡅⠀⠄⣸⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⠀⠀⠀⢀⡠⠉⠀⠓
⠀⠀⠀⠀⠀⠀⠀⡆⠀⣼⠁⠀⡶⢋⠀⠀⠀⠀⠀⠀⠒⠀⠀⠤⣦⠤⣤⢤⣤⣤⣤⠀⠐⠛⠒⠈⠉⠉⠉⠉⠉⠉⠳⢾⡃⡟⠀⠉⠁⠀⠀⠀⢀⠦⠠
⠀⠀⠀⠀⠀⠀⠀⠐⡀⢈⣠⠀⡃⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⡀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣙⠷⣏⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠸⠄⠈⠂⠈⠠⣤⢵⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⡀⠈⠂⠤⠀⠀⠀⡄⠀⠈⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠹⣦⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⣦⠀⠑⠄⠀⢰⡃⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡉⠀⠀⢀⠀⠀⠀⣄⡀⠠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠠⠒⠀⠘⢦⡀⢀⠞⠉⠉⢰⠶⠆⠠⣤⢄⣰⡒⠀⢹⣄⠈⢢⠈⠓⠤⠤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡕⢤⡀⠀⠆⠀⠀⠉⠐⢄
⠀⠀⠀⠀⠀⠀⢀⡆⢄⠁⣀⠴⠀⠄⠀⠹⡄⢀⡀⢀⡀⠛⠀⠀⠀⢦⠁⠁⠤⠉⠙⣦⠀⠱⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠬⢒⠠⠦⠄⠀⠀⠀⢠
⠀⠀⠀⠀⡐⠒⠣⠠⠔⠂⠀⠀⠀⠀⠀⠀⠈⠲⡙⠂⠑⠂⠀⢀⢀⠀⠱⣄⠀⠤⠀⠀⢳⣄⠀⠢⡀⠙⠦⠠⠐⣄⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄⠐⠆⠤⠀⢀⡼
⠀⡀⠀⠀⠁⠉⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢢⡀⠀⣀⠺⠄⠀⠀⠈⢢⡀⠀⣀⣸⡜⢣⡀⠀⠄⡀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⢰⠀⠀⠀⢀⠈⠙
⢈⡁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠙⢶⡁⠀⠀⠁⠔⠒⠀⠙⠦⠀⠬⠥⠽⠽⠶⠠⠘⠖⠐⠚⠓⠒⠋⠉⠁⠀⠀⠀⠀⠀⠸⠀⠐⠀⠀⠙⢤
⢸⠇⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡁⠂⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⠄⠀⠀⡀⠀⠀⢀⡀⠀⠀⠃⢠⡀⠀⠀⠀⢰
⠰⠃⠶⡤⠀⠀⠈⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⡕⢶⡀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠒⠀⠁⠐⠁⠀⢀⣂⡈⠓⠺⠛⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⣳⠄⠁⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⣆⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⣴⠶⠚⠛⠉⡉⢹⡇⡀⠀⢐⢀⡴⢀⠴⠛⢆⠀⠀⠀⠘
⠠⠀⠀⠀⢂⡑⣾⠃⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡃⠀⠀⢘⣆⢠⡙⣢⣤⣴⣴⠖⠒⠉⢉⢛⣄⡀⢠⣀⡈⠀⢰⡂⣈⠙⣻⣰⡀⠁⠀⠀⠀⠈⢂⡀
⠀⠈⠀⠁⠈⠉⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢐⣿⠆⠀⠀⠀⣠⣴⣾⠏⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡏⣇⠀⣸⣶⣝⣠⢴⣾⡿⠛⠁⠀⠀⠀⠀⠀⣀⢠⠖⠈⠁⠀⠀⠲⣻⡶⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⡰⢧⠀⢠⡀⠀⣄
⠀⠀⠀⡀⠀⠀⠀⣦⡀⣀⠀⢞⠛⠋⡼⢡⠿⣞⣏⠌⢉⡅⠀⠀⠀⢀⣤⠆⠉⠁⠂⠀⢸⡧⠀⠀⣿⢉⣦⡄⠀⠀⠰⢿⠆⠀⠀⠀⣝⡄⠶⣷⡀⡶⣷⢴⣿
⢠⣦⣸⣷⠀⣶⣜⣍⣁⣿⣶⣯⠼⣷⢶⢿⡀⢈⠛⠁⠈⠀⠠⠔⠪⢿⠇⠀⠀⠀⠀⠀⣻⡟⠄⠘⣻⣾⢧⣀⠀⠀⠶⢻⣼⢠⡄⢸⣏⣹⢼⡀⢻⢧⣼⣌⣿
⢸⣿⣵⢾⣿⣯⠼⡯⣤⢿⣃⡞⠀⣿⠂⡸⣧⣰⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⢤⡇⠿⠠⠶⣿⣌⡟⠀⠰⡶⣿⣄⠚⠄⠚⣿⣥⡾⠷⠾⣧⣏⢸⣹
⢸⣿⢿⡤⣿⣁⡠⢆⣦⣤⣿⣧⣾⣿⡿⢀⣾⣷⡀⢦⣠⣀⡆⠀⡀⡀⠀⠀⠀⠀⠐⣸⠟⣿⢼⣤⠚⢻⠉⠉⠀⠠⠖⠻⠎⢃⡄⠘⠻⣯⡴⣦⡀⢹⣽⢻⣿
⢸⣷⣿⣿⡀⣠⠀⣈⣧⣀⠙⣿⣿⣿⣽⣿⡏⣿⣷⣾⣿⣿⣿⣿⣿⡿⣿⠶⣾⣶⣋⢼⡴⢿⠶⣴⣦⣼⣤⣤⢺⣧⣤⡖⣲⠃⠑⣊⠻⣿⣾⣿⣿⣿⣷⣼⣿
⢸⣿⣿⡿⠉⢉⡶⠏⠈⢻⡟⢻⣿⢿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⢷⣷⣯⣿⣿⣿⡵⢚⣇⢘⠧⣜⣦⣿⡿⡄⣩⡐⢿⣋⠉⡇⢸⡅⢙⣻⣿⣿⣿⣿⣿⣿⣿
⢸⡿⣿⡿⠑⢺⡀⠈⡧⣀⣷⡖⠟⠘⠁⠹⡇⠁⠈⠋⠛⠁⠀⠀⠀⠉⠙⠛⢻⡡⠒⠻⠋⠀⠴⠗⠀⣿⠁⢯⠡⠅⠐⠵⡎⢁⣏⠙⠦⢭⢹⠟⠛⠉⠀⠙⠻
⠀⠀⡈⢛⠚⠉⡉⢁⣄⠀⠈⠿⠒⠂⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠃⠘⠃⠀⠚⠂⠉⠁⠀⠀⠀⠐⠈⠘⠃⠋⠸⠀⠆⠀⠦⠴
⠀⠀⠉⠉⠀⠈⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠙⠢⡄⢀⡤⠔⠒⠚⠛⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⠀⡰⠒⠋⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆⢀⣀
⠀⠀⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠓⢤
⠀⠀⠀⣀⣤⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠙⠢⡀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⣄
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠒⠒⠒⢲
⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠫⠽⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⡇⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠁
⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠈⠉⠉⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠃
⠀⠀⠀⠀⠀⠀⠉⠢⣄⣀⡀⠀⠀⠀⣀⣠⠞⠁⠙⠲⢤⣤⣀⣀⣀⡤⠴⠊⠁⠀⠈⠉⠙⠛⠒⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⡼⢧⠀⠀⠀⠀⠀⠀⠀⡸⢳⠀⠀⡀
⠀⠀⠀⠀⢀⣴⡀⠀⠀⢯⣸⠇⠀⠀⠀⠀⠀⠀⡏⠸⡆⠀⠀⠀⠀⠀⣇⢸⡆⠀⠀⠀⡀⠀⠀⢇⣸⠇⢰⠻⡄
⠀⠀⠀⠀⢸⠀⣷⠀⠀⠀⠁⠀⠀⠀⢰⢧⠀⠀⠳⠼⠁⠀⠀⣠⡀⠀⠘⠛⠀⠀⠀⢰⠻⡄⠀⠈⠉⠀⢸⣀⡇
⠀⠀⠀⠀⠀⠙⠋⠀⠀⢠⡄⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⢠⡇⢧⠀⠀⠀⠀⣤⠀⢸⡀⡇⠀⢀⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢻⠀⠀⠀⠙⠚⠀⠀⣠⡀⠀⠀⠘⢇⡼⠀⠀⠀⢸⠀⡇⠀⠉⠁⠀⢸⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⠀⠀⠀⠀⠀⠀⢠⡇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⠏⠀⠀⠀⠀⠘⠲⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣼
⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠤⠤⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢂⠩⠤⠰⢒⣲⣲⣟⣉⣡⣆⣖⣶⡶⡤⢄⣉⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠐⣿⡷⠴⠛⠻⠿⠿⢿⢟⢻⢛⡛⠛⠃⢠⣤⣾⠃⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠃⠂⠈⠑⢾⣿⢿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣶⡤⠊⠀⡠⣿⡟⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠁⠕⠶⣤⣀⡀⠙⠻⣿⣿⣿⣹⣿⣿⣿⣿⠿⠋⠀⣀⣬⣤⢟⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣛⡿⣿⣶⣤⣀⡉⠛⢷⣆⡈⠛⢿⣿⣽⡿⠛⢡⠴⠶⠛⢉⣩⣴⠞⠋⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢡⣴⣬⡿⠀⠀⠉⠛⠛⠿⢷⣬⣥⡶⢤⣽⢫⣴⢶⣀⣤⡴⠶⠛⠉⠀⠀⠈⠁⠈⠉⠛⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣞⣱⠅⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⣻⡷⠊⠋⠉⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠠⠟⠻⢏⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠲⢎⢽⡄
⠀⠀⠀⠀⠀⠀⣗⠰⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠻⡀
⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⡰⠀⠀⠈⠥⠀⣧
⠀⠀⠀⠀⠀⣿⠘⠄⠄⠈⠳⢘⠲⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⠋⢻⡤⠂⠐⠠⢰⣽
⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⠐⢮⢀⣦⢠⡀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠺⠁⠀⠀⠀⠀⠀⠀⣀⢰⣿⣿⠞⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠀⡿⠃⠁⠀⠀⠀⠀⣴⣆⣿⡜⠇⢠⡐⡠⣠⠀⠘⢁⡤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢻⣾⡿⠄⡖⡶⠆⠀⠀⠐⣿
⠀⠀⠀⠀⠀⣯⡆⠀⠀⠀⠀⠚⠟⢻⡿⣟⣿⣒⣛⣛⣈⣀⢀⠏⢐⢄⡟⠁⡽⠋⠉⠱⣖⠶⠺⡗⡻⢿⡇⡇⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢇⡗⠀⠐⠠⠶⢻⣿⡝⠫⠍⠀⢹⠟⠛⠇⡮⠀⠴⠮⠒⠀⠀⠀⠀⠈⠁⠀⠀⠫⠞⢹⡗⠦⠄⠀⠀⠀⠀⢰⡾
⠀⠀⠀⠀⠀⢸⠜⠉⠀⠀⠀⠀⠸⠋⠙⠀⠁⠀⠀⠀⠀⠀⢳⣦⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠀⠐⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣥⣽⢿⣦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢺
⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡑⢀⡻⣿⠛⣛⠩⣶⢤⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⠆
⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡄⡀⠀⠀⡀⠄⠀⢨⡴⣟⣫⣻⡿⡖⠱⠘⠈⠁⠹⠃⢽⡠⣢⣠⣔⠀⢀⡀⠀⠀⠀⠀⣧⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⡲⡶⣿⡳⠛⠟⡿⠃⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢧⡙⣲⠤⢆⣠⣤⣶⡔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢩⠏⠁⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠉⠉⢡⢁⣞⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠋⠄⠀⠀⠀⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⠀⠀⠀⠈⣹⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠀⠀⠈⠀⠚⠛⠛⠒⠒⠒⠠⠒⠚⠟⠉⠉⠛⠊⠉⠁⠀⠀⠀⠀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⠶⠄⠄⠖⢦⠐⢶⡶⠶⣶⣠⢶⣶⣶⡶⠲⠹⠟⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒

⠀⠀⠀⠀⡅⠢⠄⠀⠔⠁⠀⠀⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⡀⢤⡝⣦⠉⠁⠀⠀⢀⡘⢧⡛⠂⠀⠀⠀⠀⠀⠈⠑⠒⠂
⠀⠀⠀⣸⡱⡀⣀⠎⠀⠀⠀⠀⠀⠀⠢⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠃⠃⠺⣂⢠⡄⢨⣥⠤
⠀⠀⡀⠹⣱⠡⠃⠀⠀⠀⠀⢀⠀⠀⠀⠨⣄⣷⠄⠒⠀⢲⢏⣄⠀⠀⠀⠀⠀⠀⣽⣍⠦⠀⠮⣶⠻⠀⠀⠀⣀⠀⣄⠀⠀⠀⠠
⠀⠀⡇⠀⠉⠀⠀⠀⠀⢀⠔⠁⠀⢀⠔⠉⠁⠀⠀⠀⢠⠊⢃⠿⠀⠀⠀⠀⠀⠀⠆⠉⠀⠀⠀⠀⠈⠀⠀⢀⡻⠋⢛⡶⠂⠀⡆⠀⢈
⠀⠀⠁⡀⠀⠀⠠⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⢀⣾⠛⡆⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⣶⠺⠧⠀⡘⠀⠀⣈
⠀⠀⠀⠩⡀⠀⠀⠠⠂⠀⠀⢠⢃⠤⠔⠒⠒⠁⠀⠀⠘⠿⠞⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⣼
⠀⠀⠀⡇⠉⠁⠀⠀⠀⠀⣠⠋⣁⣴⣾⡟⠋⠲⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠠⠛
⠀⠀⠀⠙⣄⠀⠀⣀⣠⠞⠉⢀⠉⠁⠚⠒⠊⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠈⠑⠺⠇⠀⠠⢀⡈⣁⠀⠀⠀⠀⠀⠀⠀⠤⠔⠪⠁⠓⣒⠒⠢⡄⠀⠤⡀⠀⠀⠀⠀⢀⣀⠤⠐⠊⢁⣀⠀⣠⠀⡴⢩⠁⠀⢀⠀⢤
⠀⠀⠀⠀⠀⠀⠈⢄⡀⠆⠀⠀⠀⠉⠲⣤⣀⣀⡀⠀⣀⣀⡤⠂⠀⠀⠀⠀⠘⠀⠀⠘⢄⠔⠒⠽⠷⠃⠀⠀⠀⢠⠎⢂⣁⠴⠛⠁⠀⠀⡴⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠑⠤⣀⣀⠠⢤⠀⠐⠀⠀⠸⡛⢻⡋⠀⠐⠒⢊⣠⠅⠓⠘⠤⠤⠀⠀⠊⠴⢔⢋⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡀⠀⠀⡀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠦⢤⠤⠘⠀⠀⢃⣈⠭⠛⠁⠀⠀⠀⠄⠀⠀⠀⢀⢀⡀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⣄⣀⣘⠦⠄⠀⡇⠀⠀⠀⢀⠄⠂⠀⠀⠠⠀⠀⢸⠀⢧⠤⠤⡬⡥⠢⡀⠀⠀⠀⠀⡄⠀⠀⡨⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠀⠀⠱⡄⢲⡆⠀⢸⠀⠀⠀⡰⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⡴⠊⠉⠀⠀⠀⠀⠀⠹⣻⣇⠀⡜⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣠⣔⠰⢠⠏⠀⠀⠀⠙⡄⠀⠀⠀⠀⢻⠈⢦⢀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠄⠪⡄⠉⣰⡟⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠈⡹⠁⠀⢠⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡚⠂⠀⡰⢡⣅⠀⠀⠑⠲⠤⣀⠀⠀⠀⠀⢸⣧⠎⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⠢⣀⠁⡠⣉⠀⢢⠀⢄⠀⠀⠉⠒⠠⠀⡸⠁⠀⠀⠐⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠙⠲⡥⣀⠀⠑⠀⠑⠀⡁⠤⠄⠀⢁⡀⠀⠠⢚⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡤⢀⠴⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⡾⠛⠉⠠⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⠳⣤⣤⣤⡤⠴⠚⢁
⠀⠀⠀⢠⠁⠀⠈⠁⠀⠀⢀⣀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠟⠛⠀⠑⠒⠒⠒⠛⠋⠉⡠⣄
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠠⠴⠶⠶⠒⠚⠉⠁⢈⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡤⠶⠶⠖⠒⠒⠒⠒⠶⠤⢬⣽⠓⠦⢤⣤⣀⣀⣀⣀⣀⣤⠴⠶⠛⡏
⠀⠀⠀⢰⠀⠀⠀⠀⠈⠁⠀⣤⠀⠹⠓⠒⠶⠦⠤⠤⠤⠤⠶⠒⠚⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢠⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠈⠄⠀⠀⠀⠶⠀⠀⠀⠀⠿⠀⠀⠀⠘⠃⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠇⠀⠀⠀⠀⠰⠆⠀⠠⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠛⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣷⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢚⡽⠛⠀⠀⠀⠀⠀⠀⢀⠠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠒⡿⠒⠂
⠀⠀⠀⣠⣼⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⣠⠖⠁⠴⡏⠀⠀⠀⠀⢀⠀⠔⠠⠏⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠀⠀⢸⣧⠀⡴⢏⣤⠒⢦⠄
⠀⣴⢠⣿⡿⢀⡀⠀⠀⠀⣠⡶⠂⣀⠀⠀⠀⢁⣀⠀⡐⠀⠀⣠⢤⣾⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⠀⠀⠀⣸⣷⡄⢿⠛⢿⣣⣀⣀⡄⢀⡀
⣾⣿⣿⣿⣇⣾⠁⡀⢀⢠⠠⠶⢂⠈⠀⠀⠀⠁⢸⠇⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡴⠂⠀⠀⠀⠀⢠⡀⠀⠀⠸⢻⡄⠀⢸⣿⠟⠷⣦⠒⠀⠈⣯⠉⢉⣸⡇
⢺⣿⠛⢨⣿⣿⣼⣷⣟⣠⠰⠀⡼⠁⠲⡴⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⡀⢀⠞⠿⣁⠀⢙⡀⠚⡀⠲⣧⠔⢦⡘⢶⣤⡙⢻⣴
⢸⢿⣃⣤⢹⣿⣿⣿⠟⣡⡄⠀⡤⢀⡠⠀⢲⣶⣴⡄⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡬⢓⠇⢠⣠⡼⠆⠙⡧⣎⣀⠚⠙⢢⣻⠆⠀⠑⠂⠘⠿⣿⣿
⠠⡾⠋⢃⢸⣿⠟⠋⢁⠇⠈⣀⠀⠀⠀⠓⡀⣻⣿⣿⣿⣷⣶⣼⣴⣶⣠⣄⢀⣠⣴⣶⣼⣷⣶⣾⡶⠣⠤⣘⣇⡠⢢⣈⡶⠉⢀⡄⠀⠀⢻⡃⠀⠀⠠⣄⠀⢈⣿
⢀⣆⠀⠋⢘⣥⡴⠚⠛⠠⠖⠃⠀⠀⣀⠀⠉⢺⣿⢿⣿⢻⣟⣿⣻⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⡿⠂⢀⡀⠈⠡⣴⠀⢹⡴⠞⢉⡴⣦⠜⢶⡤⠀⠀⠀⠘⠛⠿⣿
⠸⠧⠆⢁⡞⠋⠀⠰⠂⡴⠦⠀⠀⡾⣯⠉⠢⣄⡛⣏⠍⣋⣽⣥⡿⠟⢁⡀⠀⠀⠀⠀⠈⠛⣿⡀⠄⠈⠉⠻⢷⣎⢢⡼⠷⠒⠉⠀⠘⠦⢄⡙⠀⠀⠐⣆⠀⠀⢈
⠀⠀⠀⠈⡽⠋⡀⣀⡔⠀⢀⠌⠛⢇⠘⡶⠀⠈⠉⠉⠉⠉⠏⢀⠤⢾⣧⣽⣷⣤⣀⣀⣠⣤⠌⡀⠀⠐⠂⡐⠦⣽⣿⣯⠔⡠⣆⠀⠀⢦⡄⠈⠛⣤⠀⠈⠁⠢⣌
⠀⠀⢀⡼⠓⠀⠁⠰⢤⡤⠟⠀⠀⢀⡓⠙⡇⢠⣄⢨⠀⠀⠀⠙⠺⠿⠛⠒⠓⠚⠛⠛⠚⠡⠀⠤⠐⠃⠠⠟⢦⣤⡹⠟⠊⠁⡼⡀⢠⠿⠟⠀⢠⢈⣦⣾⠀⠀⢻
⠀⠠⠞⠡⠀⣀⣤⠖⠾⡄⠀⠀⠀⠈⢳⣠⠝⢦⡉⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡀⡀⠙⠈⢢⠀⢄⠀⠻⣆⡀⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠚⠃⠐⠊⠉⠂⠘⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⣤⠄⠞⠉⠣⠠⡀⠀⢦⡨⠦⠀⢨⣿⢯⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠰
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣽⣭⣿⣭⣯⣽⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣾⣶⣾⣿⣿⣶⣾⣶⣶⣶⣿

⠀⢘⣛⡛⠯⣪⢿⣿⣿⡿⡦⠹⠇⠺⣿⣐⣦⣼⡊⠞⡉⠊⡠⠐⠈⠈⡸⠃⠀⣶⠋⠀⠂⠀⠀⢈⠀⠲⠌⢀⢭⣴⣂⣿⠂⠻⢏⡼⣿⢿⣯⡯⣴⠟⣛⣛⡛
⠀⢸⡏⢩⡆⠰⣿⠿⠛⢻⡽⠎⠲⠶⣰⠈⡻⠒⡀⠁⠄⣠⠤⠴⠀⣊⣤⠴⡾⢬⣙⣒⣒⠀⠔⠤⣍⣑⠒⠶⣈⠓⠂⡒⢂⣒⣽⢿⡟⠛⠿⣿⢁⢮⣌⠈⡟
⠀⠘⠣⢙⠑⣡⡇⣬⡿⠋⢐⡿⡙⡢⢃⠔⠠⠂⣠⠖⢋⡤⣶⠛⣿⡩⡓⣏⠘⡾⡅⠿⣹⠏⣻⡶⢦⣬⣙⠲⣄⠑⢄⠑⠴⣥⡲⢦⢱⡿⣗⡨⣄⠀⠙⠐⠷
⠀⢸⠏⢸⣿⡿⠷⠭⣍⡛⡮⠍⠊⠔⣁⡴⢞⣷⢋⡾⠋⠻⣠⠂⣸⠆⡇⣘⣼⠶⠛⠲⠿⢾⣭⣧⡸⠰⠛⣳⣌⠁⠀⠀⠤⣤⣭⣭⢩⢉⠭⠶⠯⣽⣧⠙⡿
⠀⢸⢇⢘⠏⡶⣿⠗⠈⠃⡥⢋⠔⡁⢊⡴⢻⢛⡼⡈⡆⢆⢹⠰⡄⣱⠞⢋⠠⣤⠊⠉⡀⠀⣠⠌⠹⣼⢶⠋⡈⡷⣄⠀⠀⠈⠻⣿⣇⠘⠼⣛⠷⡌⡏⣰⣧
⠀⢸⡟⣽⠄⣒⠒⣪⢂⠞⠁⠀⠔⣠⣿⡘⣸⢻⡳⢁⠋⠜⠄⢎⣾⣇⠴⠃⣰⠃⡠⠊⡠⠃⠑⣔⢀⡟⡌⡎⣱⢁⠽⣶⡄⠀⠀⠈⢿⡌⢦⢛⢛⣀⣏⣿⣟
⠀⢸⣭⡟⠀⢤⠒⡴⠏⠠⠀⠌⣰⣿⣰⢶⣛⣿⣖⠵⠓⠋⣿⣛⣉⣁⣀⣠⡇⣊⣥⡬⠀⠰⠆⣌⣾⣃⡹⡸⠁⣁⣂⣯⡼⣄⠠⠀⠄⢻⡄⠲⡚⠉⠛⣽⡿
⠀⢸⢋⢔⣦⠆⠟⡐⠀⠁⢀⣴⡇⡿⣅⢀⣨⢣⣿⠀⢀⣔⡘⣋⣿⡍⠡⠈⢼⣿⠇⠀⠀⡐⣾⣹⣗⣿⡃⣷⣉⠀⣩⡎⣵⠜⣦⠀⡈⠀⢻⡄⠡⢷⢢⡜⢿
⠀⢠⠱⣷⣖⡡⠀⠀⠀⢀⣿⠃⡇⣹⡟⢫⣉⡎⢹⡀⢪⠳⠉⡜⢻⡘⠢⣄⠀⠉⠈⠀⠀⠁⠀⣰⢧⢡⢋⣩⠿⡜⢋⡜⣛⠁⠻⣧⠀⠀⠀⠛⢌⢲⡧⡸⣦
⠀⢐⠒⠆⡊⠀⠀⠐⠀⣎⠌⢢⡜⡇⣇⣼⠙⢘⣆⡌⡆⣆⡇⢳⠌⣷⣼⡈⠳⣤⢀⠐⢒⡠⣔⢧⣸⢞⡈⢋⢀⣿⠎⢞⢩⡄⢢⣽⢆⠀⠀⠀⠈⠔⢷⢬⡟
⠀⢸⣟⡄⠃⠀⢀⠁⣸⣯⠂⠈⣁⣿⢻⣇⣨⣣⣏⢌⣴⣭⣹⡌⢾⠟⢼⣛⣦⡈⢢⡼⠘⣧⡙⠘⠇⢜⣹⠀⢟⢀⣡⣽⡿⡠⠟⠍⣻⡄⠂⠀⠐⠘⣿⢸⣿
⠀⢰⠍⡔⣠⠁⠀⠀⡸⢻⣶⡷⠿⠿⠈⠻⣇⠝⢹⠯⠀⠀⣰⡿⢸⡞⢓⡏⠡⠀⣾⠇⠂⢀⠙⠲⣄⠠⢩⡼⠣⣡⣿⢿⡷⠿⣷⣻⠋⡇⠀⠀⠀⠃⢸⡷⣿
⠀⢸⠧⡔⠹⠀⠀⠀⠀⡾⣬⣝⣃⠀⠀⠀⢯⡞⠟⠛⣁⠛⢟⣀⣿⠁⣟⠀⠂⢸⡏⠀⢊⠁⡠⠂⠈⢷⣾⢂⢖⣿⠀⠀⠀⢛⣫⣥⣧⠀⠀⠄⠀⠀⠈⢣⠙
⠀⠀⠉⠁⠀⠄⠀⢀⢰⠟⡯⠥⢧⢤⡤⣤⣌⠳⣇⠨⠧⢊⠆⣿⠇⢸⠁⠈⢀⣿⡍⠀⠀⠀⢳⡄⠀⠀⠻⣆⣾⣁⣀⣠⣤⠬⠥⢊⢺⡄⣄⠂⡀⠀⠀⡈
⠀⢐⠒⠒⠒⠒⠒⠚⢸⡇⠉⠒⣸⢉⡅⠂⠙⢷⡌⠙⠲⢦⣔⡾⠀⠘⣇⠀⠈⣿⠇⠀⠀⠀⣼⢛⢦⡀⠀⠈⠻⡟⠉⣿⣫⠐⢧⣛⣿⡇⢙⢀⠈⠁⠉⠄⠤
⠀⢘⡀⢀⣀⣀⡀⢸⢸⣏⡁⢠⡟⢸⢾⠶⠲⠋⡽⢦⡀⠀⠈⠳⠀⠀⡼⣇⠀⢸⠀⠀⠀⢰⣏⡟⣸⠝⣦⡀⢸⠀⠀⣸⢟⠂⠀⡺⢏⡄⠀⢰⠀⠀⠀⠀⠥
⠀⢨⢶⠂⢕⡄⣩⢀⢸⡟⣡⠟⡘⡃⣎⠐⣡⠈⡷⣻⣼⠗⣦⣄⣡⡾⣡⢿⡄⠀⠀⠀⠀⣾⡟⣴⡅⣾⢹⣷⣬⣀⣴⢫⣎⢣⠱⠘⡈⡇⠀⢈⢩⡝⠉⣌⠛
⠀⢰⠂⢠⣸⢁⠛⢸⢸⡦⠳⠀⢇⠜⢈⡞⡠⡏⠁⡇⡝⠠⠷⣀⣭⢺⣼⠾⡇⢀⠀⠀⡀⢿⣇⢡⠋⡌⣧⣸⣮⢋⡸⢫⠟⡘⢸⢇⢣⡇⢠⢸⣆⢳⢀⠊⣶
⠀⠈⠀⣸⢃⠆⠀⢸⢸⡶⠖⠛⣌⣋⠒⠤⠤⣵⡳⠊⢲⠤⢲⣹⠾⣻⡇⢨⠀⠄⡠⠊⡠⠚⢿⠻⣤⡠⢷⡷⡥⠬⢴⡧⠾⠶⠓⠚⢢⡇⢰⢸⠈⠄⢿⠀⢹
⠀⢠⠀⣽⠘⡀⠀⠘⢸⡧⢀⣴⠏⣼⣅⡀⣀⣾⢡⢞⣦⣠⠞⠁⢰⠃⢡⢠⠰⢊⠔⠃⠀⠀⠘⡆⠈⠳⣼⢏⡀⢈⣍⣴⣯⡂⢀⠐⢋⡅⢸⠈⠀⡘⣸⠀⣨
⠀⠀⢠⠁⠱⠈⠳⠀⢸⢡⠡⢡⠶⠌⣸⢇⠹⣠⢏⣦⡾⠋⠀⢠⠏⠀⠈⡤⠞⠁⠀⠀⠀⠀⠀⢹⠀⠀⠈⢳⡟⡙⢇⡾⢫⠅⡰⠃⢣⡆⢸⠸⠃⠐⠁⢁⠈
⠀⠀⠃⡀⡱⡡⠀⢠⢸⡄⠇⠎⡤⣄⡣⢸⠗⣵⣿⠋⠀⠀⠀⠞⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⣷⡎⣷⣿⠀⡇⢃⠈⡃⢸⠀⠸⡕⠁⢈⠿
⠀⠀⠠⠋⠈⠁⠀⠈⠀⡵⢈⠂⠑⡄⠠⡟⣴⡟⠁⠀⠀⠀⠌⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢷⣷⠱⢠⢃⠼⡄⡆⠰⠀⠀⠁⠈⢣
⠀⠀⡆⢀⢂⠅⡆⠀⠀⣷⠘⢪⠖⢫⢋⣴⠏⠀⠀⠀⠀⠈⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠈⢻⡄⠱⡄⠒⣼⡆⢸⠀⡀⠀⣆⠀⡄
⠀⢰⢇⠸⣤⠀⠀⠀⠀⣿⡯⠴⢲⣿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢻⣇⠈⢩⡽⡇⢸⠀⠀⢀⡿⠰⢰
⠀⢠⡸⢆⠹⣆⡶⠀⠀⣿⢀⣎⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⠈⡰⡇⢸⢠⡄⡾⢠⢧⠈
⠀⢠⡁⠘⣇⠹⣀⠀⠀⣿⣷⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠈⠌⢷⣱⡇⢸⢈⡿⠰⠇⠀⡆
⠀⢨⠱⠾⠄⠲⢻⠀⢨⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢸⢸⡥⠁⠀⠎⣠
⠀⣠⣀⣀⣀⣀⡘⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡞⢀⡀⣀⣀⣁⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣹⣷⣤⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠉⠀⠙⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠞⣶⡞⠛⠛⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠠⣄⣀⣉⠒⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠆⠀⠀⠀⠈⠓⠉⣙⣍⡉⠧⠄
⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⣠⠇⠀⠀⠈⠑⢠⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣁⣶⢠⡄⢰⡌⠳⢟⡀⠀⠀⠀⠐⠒⠙⠀⠐⠖⠀⠈⠲⢄⡀
⠀⠀⢠⠀⠀⠀⢘⣁⡤⢄⡐⠚⢁⠀⠀⠠⠀⠀⠀⠘⠉⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠩⡥⠾⠳⣦⠙⢛⡋⢁⣠⣀⠀⠀⣠⡤⠤⠤⠄⠀⠡⣀⠀⠙⣦
⠀⠀⠿⠀⠀⣀⠀⡡⠐⠀⠀⠀⠀⠈⠢⡀⠀⢆⠀⠀⡇⠀⠘⡀⠀⠀⠀⠀⠀⣀⠴⣤⠶⠶⠀⠀⠀⠙⠃⣼⡙⣯⡁⠂⠀⠀⠀⠀⠀⠰⠄⣀⣀⠀⠋
⠀⣤⡆⠈⠁⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢰⠀⡠⠁⠀⠰⠃⠀⠀⠀⠀⠄⣹⡿⣹⢇⣠⠀⠀⠀⠰⠿⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘
⢠⣿⡒⣤⣤⠂⠀⠀⠂⠈⠉⠀⠀⠈⠐⠄⠀⣼⡂⠁⠀⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⣨⣯⣤⠰⠷⣰⠶⣟⠂
⠃⢻⣝⠄⠀⠀⠁⠀⢀⠄⣀⠤⠀⠀⠉⢺⣦⣿⠗⠂⠀⠘⢻⢧⡴⠀⠀⠀⠀⠀⠀⠀⢴⢶⣼⠇⠀⠹⠿⡏⢋⠀⠀⠀⣄⡀⣠⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠜⠁⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⣠⠊⢃⣹⠀⠀⠀⠀⠀⠀⠀⠰⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠟⠙⢷⣶⠄⠀⡀⠀⠀⣙
⡄⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠠⡞⠁⠀⠀⠀⠀⠀⠀⠠⢀⣴⠿⡇⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣧⡤⢾⡆⠀⢠⠁⠀⠀⠙
⠘⣶⠒⠈⠁⠀⣠⠖⠀⠀⠀⠀⠀⢀⣀⣠⠴⠎⠀⠀⠀⡾⣿⡶⠋⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⡌⠀⠀⠀⣿
⠀⠉⠤⠠⠄⠂⠁⠀⠀⠀⣼⠶⠋⢉⣠⣦⡤⣄⠀⠀⠈⡁⠋⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⡟
⢸⡆⠀⠀⠀⠀⠀⡐⢀⡾⠁⢤⣾⠟⣿⣟⣀⠈⠃⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠹⣦⣀⣀⣠⣈⡴⠛⡠⢣⠀⠀⠘⠉⠋⠀⠀⠀⠀⠀⠠⠀⢈⡁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠘⠀⠀⠀⢀⣰⠄
⠀⠀⠀⠉⠺⠷⠀⠀⢀⣀⠀⢉⡀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢦⡻⠀⡰⢦⣌⠑⠶⣄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠉⠁⢀⣀⠀⣠⠆⢠⠋⡉⠀⠀⢀
⠀⠀⠀⠀⠀⣄⠐⢠⠄⠀⠋⠁⠘⠂⢤⡀⠀⠀⠀⠀⠀⠀⣀⢀⠔⠁⠀⠀⠀⠀⠈⡀⠀⠀⠳⣄⡤⠲⠲⣿⠷⠁⠀⠀⠀⠄⣨⡍⣶⠋⣠⠿⠊⠁⠀⠐⣼
⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⡄⠀⠀⡀⠀⠉⣈⠉⠁⠀⠀⠉⠉⠣⣄⠀⠀⠀⠀⣀⡀⠠⢄⠀⠀⢹⡦⣶⡷⠂⠀⠤⠤⠴⠄⠚⠩⡆⠻⡉⢀⣀⠀⠠⠄⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⣧⠀⠀⠀⠇⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠒⠉⠀⠉⢄⡀⠀⠀⠀⣸⠇⠈⢧⠀⠤⢤⣴⠶⠛⠁⠀⢸⠀⠀⠁⠀⠀⠀⠀⠈⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡆⠀⠀⠠⡄⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠘⣟⠙⡀⠀⠀⠈⣶⢖⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣐⣦⠴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣤⣤⡘⠒⠒⠂⠳⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⡟⠐⢷⡒⠟⠶⠾⡌⠐⢄⠀⠀⠀⠀⢀⠖⠀⠁⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢀⡠⠤⠃⠀⠀⠙⣧⢸⣷⠁⠀⢸⡀⠀⠀⠀⠌⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡠⠊⣀⡤⠒⠉⠁⠀⠀⠀⠀⠀⠹⣿⣿⡀⠀⡜⠀⠀⢀⠊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣰⡔⣺⠋⠀⠃⢠⡘⠇⠀⠀⠀⠀⠀⢹⡇⠻⡄⠁⠀⠠⠃⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣴⠛⠿⠉⣴⡇⠀⠀⠀⠁⠀⠻⠀⠀⠀⠀⠀⠈⣷⠀⠻⢞⠄⠁⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡖⢊⡄⠙⢃⠀⡐⢻⠍⠑⠢⣄⡀⠀⠀⠂⠀⠀⠀⠀⠀⣿⠀⡠⠊⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡉⠀⠀⡔⠀⢷⡀⢀⠀⠀⠉⠐⠢⠆⣀⠀⠀⠀⠀⣿⠞⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠢⢼⠀⠰⣾⠀⠈⢢⡀⢤⡀⠀⠀⠈⠑⠂⠤⢰⠏⠀⠀⠀⠈⣁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠘⠳⣼⣄⠀⠀⠱⠀⠘⠂⠀⣁⡠⠤⠀⠃⠀⡀⠀⠠⢲⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠳⡉⠑⠲⠦⣠⣄⣁⡀⠠⠴⠖⠛⠁⠀⣀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⣼⠉⠁⠒⠒⠠⠤⣤⣥⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠲⣿⠀⠀⠀⠀⠀⠀⠸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡌⠀⢰⡇⠀⢀⠀⠀⢸⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠙⠿⣀⣠⣿⡇⠀⣸⠀⠀⣻⠀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⢈⠉⠉⠉⠉⠈⠓⠂⢿⣇⡀
⠀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠁
⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠈⠁⠀⠀⠰⠄⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠆⠀⠀⡤
⠀⢀⡄⠀⢀⣄⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠉⠀⡠⠀⠠⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢀⡀⠉⣄⠑⠀⠁⢀⠀⠀⠀⠀⢀⣿⠀⠐⠀⠀⠀⠺⠀⠀⠐⠀⠀⢢⠀⣰⡄⠊⠀⠠⠀⠈
⠀⠀⠀⠈⢀⢃⠈⠀⣠⣤⣤⣵⠀⠀⠖⠢⠂⠒⠆⠀⡌⣂⠀⣐⠩⠁⢀⢠⣆⠰⢃⠱⡀⠤⣼⡇
⠠⣾⡀⠀⠐⠐⠰⠛⠛⠛⠃⠌⠶⠒⠀⠓⠐⠃⠘⠘⡖⠿⠀⠿⠀⠗⠀⣤⠈⠻⠫⠛⢋⠈⢀⠃⣠⣤⡀
⠈⣾⡅⣠⠨⠺⠀⠰⠂⠗⠀⡂⡄⠚⢀⠄⠄⠄⠘⠃⡇⠀⣏⡅⠀⠀⣶⠀⣶⠀⠿⠿⠀⢠⡄⢘⢁⣤⠉
⠐⠛⠂⢸⠐⠾⠀⠐⠂⠂⢸⠽⠃⠀⠀⠂⡀⠂⠀⠰⡇⠰⠃⢧⠈⠘⠛⠀⠛⠂⠿⠽⡄⠘⠃⢸⠀⠅⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⠆⣀⡀
⠀⠀⠀⣠⣄⠀⠀⠀⠀⠠⢤⢀⢬⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡸⠄⡀
⠀⣠⡾⠟⠁⠀⠀⢀⠀⠐⣊⣠⢤⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⣠⠀⡾⠃⠠⣄⠒⠆
⠠⠿⠿⠿⠄⠀⠀⠀⠀⡀⠈⣤⠀⢀⠰⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠀⠛⠂⣮⠅⣈⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠁⢨⠀⡠⠀⠺⠇⠀⠀⠀⠀⠀⠂⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠵⠂⠨⡡
⡀⠀⠀⠀⠀⠀⠀⠀⡘⠂⠉⠙⠒⠋⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠆⠀⠰⠀⠀⠀⢠⣅⣋⣭⡟⠒
⠈⠀⠀⠀⠀⠀⠀⢐⡤⠾⠀⠼⠀⢳⡤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠑⠢
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠄⠿⢶⣶⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⢀⡀⠀⠀⣄⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢘⣿⣻⡆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡃⠀⡄⠀⢠⣯⠄⠀⠀⠀⠀⣸⠀⠀⢀⠀⠀⠀⠀⣨⡆⠀⠀⠀⠀⢀⡀
⠀⠀⠠⣿⣿⠿⣿⣄⡀⠀⠀⠀⢰⣧⡀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢢⢠⣇⠀⣰⣿⡇⢰⠀⠀⣴⣿⡷⠀⢀⣆⢰⣿⣇⠀⣾⠀⠀⡀⠀⣼⡆⢀⠀⠀⢀⣼⣧
⠀⠢⢼⣿⣿⣶⣉⡶⢿⣤⠀⠀⣼⣿⣇⠀⢰⠀⠠⣿⠀⠨⠀⢠⣿⣾⣿⣤⣼⣿⣿⣿⠀⢴⣾⣿⡟⠂⢼⣿⣿⣿⣿⣿⣿⣦⣠⡆⠀⣼⣧⣾⡆⠀⢠⣿⣿⠄⢰
⢠⣾⣾⣿⡿⢿⠯⢹⣿⣿⣷⣾⣿⣿⣿⣶⣿⣶⣼⣿⣦⣾⣄⣸⣿⣿⣿⣾⣿⣿⣿⣿⣀⣿⣿⣏⣼⣷⣿⣿⡿⠟⢟⡛⢿⣿⢿⣿⢈⣿⣿⣿⡇⠀⣼⣿⣿⣇⣾
⠘⢿⣿⣿⡿⣌⣴⣿⠿⠉⣿⡿⠿⠿⠟⠛⠙⠛⠛⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⢿⢿⠟⠓⠚⠛⣵⡎⢿⣾⣿⢾⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿
⠀⠛⠻⠿⣤⣿⣿⢡⠤⡤⠠⠀⡦⠰⠆⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠀⠉⠀⠂⠀⠀⠀⠀⠀⠉⠋⠉⠉⠙⠛⠛⠿
⠀⢺⠙⠇⠉⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣀⣤⣤⣀⣀⣠⣠⣤⣤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⡄⠤⠀⠀⠦⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣶⣶⣶⣤⣄⣀⣀⣀⣀⠀⠀⢀⣀⣀

⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠉⠉⢡⠈⠉⠉⠈⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠐⠂⡇⡇⠒⠂⠐⠒⢸⠀⠀⠀⠀⠀⠀⠀⠠⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⡇⡇⠀⠀⠀⠰⢸⠀⠀⠀⠀⢀⢀⣀⣀⠀⢀⣑⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠤⠄⡇⡇⠤⠄⢠⠤⢸⠀⠀⠀⢰⠁⠡⠀⠀⠀⠐⠒⠺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⣀⡀⡇⡇⣀⡀⢠⡀⢸⠀⠀⠀⠈⠀⣦⠰⠅⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡇⡇⠀⠀⢀⠀⢸⠀⠀⠀⠆⠣⡀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡇⡇⠀⠀⠉⠁⢸⠀⠀⠀⠀⠀⠈⠓⠂⠀⠀⢀⣀⡀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⡇⡇⠉⠁⠈⠁⢸⠀⠀⠀⠀⠀⠀⠁⠐⠒⠓⠋⣁⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠂⡇⡇⠀⠀⠒⠂⢸⠀⠀⠀⠀⠀⠀⠀⠄⠘⢠⠙⠑⢲⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠒⠂⡇⡇⠒⠂⠰⠂⢸⠀⠀⠀⠀⠀⠀⠀⢀⠄⠘⢆⡆⠀⠲⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠄⣧⡇⠤⠄⢤⠄⢸⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣿⡇⠀⠀⣠⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⣿⡇⣀⡀⣀⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⣿⡇⠀⠀⠈⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢠⠀⠉⠀⠀⠀⣿⡇⠀⠀⠈⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⣿⡇⠀⠀⠘⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠐⠀⠒⠂⣿⡇⠒⠀⠐⠂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⣿⣿⣦⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢿⣿⣿⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⡀⢀⣀⣀⣀⣉⣁⣀⣀⣀⣀⣸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡜⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⡜⡰⠃⡠⠊⠁⠀⠀⠀⣈⠩⠝⠉⢆⠀⠙⢎⠳⣥⠀⠀⢀⣠⢏⢦⠀⢠⡔⣞⣠⡤⡄⡠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡜⡰⢁⡜⣠⠔⠊⠉⠉⠁⠀⠀⠀⠀⠈⢆⠀⠀⢣⠸⡆⠀⠀⠀⠈⠈⠀⠀⠀⠉⠈⠀⠈⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢰⢠⠃⠎⡴⠷⠦⣤⡀⠀⠀⠀⠀⣠⠤⠶⠬⣄⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠇⡇⣰⢮⣴⣶⣶⣤⡉⠀⠀⠀⠀⣡⣴⢶⣦⣬⢆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⢸⢠⡑⡄⠈⣄⣻⣿⠛⠀⠀⠀⢀⠻⣿⢂⡽⠁⠠⢣⡀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡌⠂⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡆⡸⠹⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢸⠈⢧⠀⠀⠠⣄⡀⠑⠀⠘⠀⣠⠄⠀⠀⡸⢠⠃⢸⠃
⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⡂⠀⡗⢄⠀⠈⠣⣁⠀⢂⠜⠁⠀⢀⠜⠗⠁⢀⠎
⠶⠈⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢣⠀⠇⠀⣑⢤⡀⠀⠉⠀⢀⡠⢺⠁⠀⢀⠴⠃
⣲⠒⠃⠀⠸⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠚⠣⣴⠊⠀⣰⠊⠉⠀⠈⠉⡅⠸⠒⠂⠁
⠹⠞⢲⠊⠁⠀⠈⠷⢄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠠⣇⠀⠀⡥⠔⣯⣃⠀⠀⠀⠀⠀⠈⡧⠤⠄⣀⣀
⠀⠀⠀⡇⠀⠀⠀⠈⡄⢑⡤⠔⠖⠈⠉⠀⠀⠀⡰⠉⢦⠎⠀⢀⠎⠀⠑⠂⠤⠄⠒⠉⠀⠀⠀⠀⠀⠈⠉⠒⠠⣀
⠀⠀⠀⡇⠀⠀⠀⠀⠸⡀⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⡧⠔⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⢆⠂⠂⢧⡀⠀⠀⠠⠛⠢⡘⠀⢀⠔⠁⡀⠀⠠⠜⣡⠆⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡹
⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⡟⠢⣀⠃⠀⠀⠙⣴⠃⠀⠠⠙⣦⠀⠘⡏⠀⠠⠍⠄⠀⠀⠀⠀⠀⢀⠞⠁⠀⠙⢦⡀
⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠣⠤⠌⣹⠦⠀⢰⢃⠀⣀⡢⣄⠀⠓⡤⠦⡞⠀⢀⣀⡄⠀⠀⠀⢀⠏⠳⡄⠀⠀⠀⠙⢦⡀
⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⢃⢨⠀⠈⠝⠋⠉⠩⠇⠐⢟⠉⠉⠂⠉⠉⠉⠁⡟⠀⠀⠈⢦⡀⠀⠀⠀⠙⢄
⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⡀⠈⠾⡀⠀⠠⠶⠾⠁⢀⣅⠀⢻⡶⡆⠀⠀⠀⢰⠁⠀⠀⠀⠀⠈⠢⣀⢀⣀⠀⢧
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢈⠚⠀⡱⠀⠀⠀⠀⠀⠇⡹⠄⠀⠉⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⡸
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠠⣀⡔⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠔⠁
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣻⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠗⠀⠀⠀⣀⣴⠃⠀⠀⢀⡠⠒⠁
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢺⠁⣤⠏⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠩⡁⣀⠞⠁⢀⡱⠔⠊⠁
⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠋⡄⠘⡏⠹⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠠⡔⠃⠀⢀⠂
⠀⠀⠀⠀⠈⠑⠒⠂⠀⠐⠒⠉⠁⠀⠀⠀⠙⠶⠓⠁⠀⣠⠮⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠫⢄⣂⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠀⠀⠀⠈⠑⠦⡀⠀⠀⡠⠔⠉⠀⠀⠀⠀⠈⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠔⠒⠒⠲⠤⢄⡤⠴⠒⠒⠒⠒⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠊⠉⠉⠉⠉⠓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⡆⠀⠀⠀⠀⠀⣠⢋⡜⠀⠀⢠⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠈⠀⡴⠒⢄⢀⣠⠗⠉⢀⣀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⡄⠑⠒⣺⢡⠃⠀⢀⡎⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠘⢄⣀⡤⠊⠑⠤⠖⠁⠘⠦⠴⠋⠳⠔⠃⠘⠦⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢢⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⣠⠎
⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣀⡀⢰⡟⠀⡠⡄⢠⣤⠀⠀⣠⡉⠀⠀⣠⡦⢦⠀⠈⣿⡔⠒⠆⠀⠀⠀⠀⠈⡇
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⣸⠁⠈⠀⠈⡏⠀⠀⢀⡷⠁⠀⠀⡸⠀⡼⠀⢀⠇⠘⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠘⠤⠚⠀⠣⠔⠂⠀⠣⠤⠴⠊⠳⠴⠚⠐⠃⠀⠘⠒⠊⠀⠀⠉⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠈⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠊⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠦⠤⠤⠤⠤⠔⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠤⠤⠔⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢸⢂⡬⠴⠀⠞⠂⠀⢠⡒⠒⢐⡂⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠠⡄⠐⡠⠁⠀⠀⡢⣀⠀⡿⠦⣸⡀⠄⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⢑⡔⡅⠀⠀⠀⢑⣄⠤⠛⠊⡁⠄⠺⢐⠌⡑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⠁⡐⠁⠀⣔⣈⢁⣸⠔⢂⣉⣀⠀⠀⠀⣙⠚⠴⣤⡇
⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠂⠠⠠⠎⣀⡀⠀⢨⠠⢤⢸⠈⠐⠐⠚⠀⢀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣁⠀⠊⠁⠀⣈⣀⠐⠠⡀⠀⠀⠀⠀⠀⠀⡠⠀⠠⡀⠀⠀⠁⠉⠉⠚⠚⠴⠥⠀⣈⡂⢈⣈⣈⣀⣀⣁⠀⣀⡀
⠀⠀⠀⠀⠀⢀⠋⢀⠔⠈⠀⠀⠀⠀⠈⠘⢦⠀⠈⠀⠀⠀⡧⠄⠀⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠈⠰⠁⢀⣄⣈⠠⣤⠖⢀⡑⣄⢃⠀⠀⣰⠡⢐⡂⠀⠀⠀⢀⠓⣄⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⡟⠁⢠⣄⠀⠀⢠⣄⠸⡛⠀⡸⡡⢊⠄⠒⠀⠀⠀⣐⢐⠩⡄
⠀⠀⠀⠀⠀⠀⢳⡀⡀⠀⠈⠉⠀⠀⠈⠋⠀⠃⠀⡗⡐⡡⢚⠃⠂⠀⠀⠓⠢⠬⡇⠀⠀⠀⣀⠄⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠑⣔⢄⠀⠀⠁⠈⠀⣀⠔⠀⠀⠶⠐⠀⠈⠸⠇⠀⢀⣴⠂⠀⡇⠀⢀⠼⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⡔⢀⡂⠒⠢⡉⠉⠓⢒⠎⠉⠀⠀⡀⠀⠘⡃⠀⠂⠀⢐⠒⠀⠈⠀⢰⠀⡠⠊⡀⠒⠂⠤⣀⢸⢏⣹
⠀⠀⠀⠀⠁⠀⠀⠉⠈⠉⡉⡏⡽⣀⠀⠀⠀⠀⠀⠀⠡⡀⠷⠤⣄⡈⢀⡤⠐⠁⢶⠀⠀⠲⡤⢉⠀⠀⡛
⠀⠀⠀⠀⠀⠀⡈⠀⢢⡀⡈⡉⡱⣿⠅⠀⠀⠀⠀⠀⠰⢩⠚⠒⠀⠤⠚⣄⢹⡀⢉⠦⢤⠑⠒⠡⠤⠤⡃⠀⠀⠄⠀⢈⠃
⠀⠀⠀⢀⡤⠠⠇⠒⠀⠀⢨⠋⡇⢆⠀⠀⠀⠀⠀⠀⣇⡆⠀⠀⠢⢴⠀⡈⣿⢿⠜⠂⠠⠤⠤⡴⠴⢤⠣⡠⠬⠷⠶⠚
⠀⠀⠀⣮⢅⡀⢀⠨⠒⠐⠛⢤⠁⠀⣣⡀⠀⠀⠀⠀⠉⠡⠀⠐⠚⠒⢀⠀⠀⡆⠀⠀⠐⠂⡰⣥⡇⠈⡀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠡⠂⠀⠀⠀⠀⠀⠈⠃⠀⢈⠉⢻⠀⠀⠇⠀⠀⠐⠈⠁⠒⡧⣤⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⣿⠁⢏⠀⠀⠈⠢⡀⠀⠀⠀⠀⠉⢉⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠋⠉⠈⠀⠀⠀⠐⠘⠃⠈⠁⠀⠈⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡶⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠩⡟⠄
⠀⠀⢀⣤⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠠⠀⢴⠋⠀⠀⠀⠀⠠⠠⠎⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⣼⠀⢴⢎⡠⢤⡀
⢠⣆⣼⡗⣠⠀⠀⠀⠴⠂⠠⠀⠀⠈⢀⡀⠀⠀⠠⠠⡎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⢀⣼⡆⠛⠛⠿⣤⣠⡄⣀
⣿⣿⡟⣷⣿⣠⣠⠔⢐⠁⡄⣀⢀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠐⣆⠀⠀⣯⠃⠀⡜⢛⠉⣻⡁⣀⠹⣀⠠⣿⣀
⣸⡿⢀⢻⣿⣿⣿⢞⠁⠈⡁⠀⠡⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠕⢀⢘⣤⠓⣄⡔⠀⠶⢌⣯⠈⠣⠈⢳⣮⣿
⢨⡿⢛⢰⣿⠟⠃⠌⠁⡀⠀⠀⢂⢘⣿⣷⣿⣶⣼⣤⣄⣀⡀⣠⣤⣤⣷⣤⣼⠴⠅⣘⣋⠀⣀⡥⠋⣁⠀⠈⢿⠀⠀⠠⡀⠈⣿
⢠⡀⠊⢈⣥⠖⠋⠀⠚⠀⠀⣄⡀⠑⢿⣿⣟⣹⣽⣽⣿⡿⠟⠟⠿⠿⣿⣿⠟⠀⡀⣈⡑⡆⢈⡶⢋⡠⢦⠜⢦⡂⠀⠀⠉⠙⢿
⠘⠋⠀⢯⡥⠀⠁⠚⠁⢤⡜⢳⡀⠱⠦⢷⣀⢼⠷⠛⣡⠤⣄⠀⠀⠀⠈⠻⠀⠀⣀⢉⠻⣄⣾⢉⢁⠀⠀⡁⠂⢄⡀⠀⠢⢀⠠
⠀⠀⢠⡎⠀⠂⣏⡀⠔⠀⠈⠤⠷⠀⣀⢀⠀⠈⠰⣭⡼⠏⢿⣽⠶⠾⠟⠁⣨⠈⢁⣁⣉⣻⣿⡷⢛⣧⣀⣿⣷⢄⣻⣇⣧⠈⢿
⠀⠠⠋⠀⣀⡤⠶⣅⠀⠀⠈⢳⡠⠲⡍⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠤⠀⢩⠉⠛⠙⢦⠠⠉⠻⣏⢠⣼
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠈⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠠⠀⠊⠁⠐⠄⠀⠠⠳⠄⠸⠿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣛⣛⣛⡛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣻⡿⣶⣿⣾⣿⣿⣿⣿⣆⢻⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣼⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣝⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⣿⣴⣦⣄⡀
⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣟⣯⣭⣶⣮⣭⣟⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣶⡝⣿⣆
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣸⡿
⠀⠀⠀⢸⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣝⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⢷⣝⢷⣭⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠈⠛⢷⣿⣿⣿⣿⠿⠋⠛⠿⢿⣿⣿⣿⠿⠛⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢆⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠛⠛⠛⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠴⠋⠻⣦⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠄⠀⠀⠈⢃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠀⠀⠀⠀⣸⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⣀⣿⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢹⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⡿⠓⠤⡶⠞⢿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⣧⠀⠀⠀⠀⢸⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⣿⠀⠀⠀⢀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠈⢹⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠉⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢸⡀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠷⠚⠻⣧⠀⠀⠀⠀⠀⠀⠀⣶⡶⠶⠾
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠘⠁⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣧⠀⠀⠀⣾⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⣼⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣤⣤⣤⣶⡶⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢻⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⢺⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠉⠉⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⢀⣸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠉⣻⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⣸⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣰⡄⠀⠀⣸⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁⠀⠸⠇⠘⠛⠛⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⢠⡆⠀⠀⣴⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡁⠀⢀⣀⠀⢰⣄⠀⠀⣭
⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠘⡏⠀⠀⠙⠀⠀⣿
⠀⠀⠀⠀⣀⣤⠞⠛⠉⠁⠀⠀⡇⠀⠀⡆⠀⠀⠛⠻⠶⣤⣄
⠀⠀⢀⡶⠋⠀⠀⠀⠀⢀⣠⣤⡇⠀⠀⠻⣤⣄⣀⠀⠀⠀⠉⠳⣄
⠀⢠⡿⠁⠀⠀⢀⡴⠎⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⠀⠀⠀⠈⢷⡄
⢀⡿⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠈⢳⡄
⢸⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠘⣧
⣿⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿
⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿
⢻⡆⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⢰⡏
⠈⣷⡀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣰⡟
⠀⠈⢷⣄⠀⠀⠀⠈⠳⢦⣤⣄⡀⠀⠀⢀⣠⡴⠞⠁⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⠉⠉⠉⠛⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⠋⠉⠉⠉⠛⠓⢲⢤⣄⡀⠀⠀⠀⠀⠉⠳⢤⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣀⠀⠀⠀⠀⢠⣦⠰⣿⠇⠀⣭⡓⠦⣄⡀⠀⠀⠀⠉⢳⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠥⠼⠃⠀⢸⣻⣮⠟⠛⠙⠲⢼⡟⠁⠀⡀⠉⠓⢦⡀⢀⡟⣦⣼⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡤⡄⢀⣤⢀⣸⠃⠀⠀⠀⠀⠀⠓⢤⡿⠋⠀⠀⠀⣙⡾⠀⣇⣀⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠖⠃⠀⢹⡏⠀⠽⠂⠀⠀⠠⢶⡆⠈⢣⡴⠶⠂⠈⠉⣷⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣯⡿⠁⣠⣄⣠⣄⣀⣠⡀⢀⡀⢸⠀⠀⠀⣄⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠳⡀⠀⠀⠁⠀⠉⠀⠈⠉⠀⢈⡷⠤⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡶⠤⢤⣤⣄⣠⡤⣶⡿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠋⠙⠲⢶⣶⠶⢾⣋⣉⠀⢸⣧⠳⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⣨⣧⣄⣠⣼⣧⣀⣀⣈⣉⣠⣼⡇⢷⡀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠀⣿⢻⠀⢠⡄⠀⣄⠀⠀⠸⠀⠀⠀⣴⢃⡜⠉⠛⠒⣶⠶⢭⡉⠁⠀⢸⡇⠀⢳⡀⢘⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣧⡇⠘⡆⣸⢧⢠⢿⡀⠀⠀⠀⣠⠂⣿⢿⡁⠀⠀⢸⠁⠀⠀⠙⠒⠦⣼⡇⠀⠀⠙⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⣳⠈⢠⡄⠙⠃⠘⠋⣸⠁⠀⠀⠐⠇⠀⢁⣸⠧⠤⠤⣼⡀⠀⠀⢰⣧⡖⠃⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠙⠀⡼⠙⣆⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠙⠒⣾⣲⣶⣏⣙⣛⣻⣛⣲⠭⣟⡷⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡾⠁⠀⠘⡆⠀⢻⠀⠀⠀⠀⠀⠀⠀⠰⣯⣥⣤⠤⣾⠿⠭⢞⡿⠁⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣧⣀⣀⣠⣧⣤⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⢡⢾⠉⡏⢇⠙⣍⠪⣝⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠳⣴⠃⣰⠖⢾⡘⡶⠋⣳⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠃⠀⠉⠒⠋⠀⠉⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⢀⠀⣘⡅⠀⢀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣆⢼⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡆
⠀⢾⣞⢻⡇⠱⣿⣶⣖⣶⣞⣃⣴⣤⡌⣗⣆⣄⢘⠏⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡀⢠⣿⢯⢿⠓⠂⣦⣠⣤⡜⢂⣤⣼⣰⣿⣰⡆
⠀⣻⣥⠚⡍⢛⠿⡟⢙⡄⠁⡭⢨⣗⢩⡍⢹⡟⢋⣩⣬⠇⣀⢀⡀⢀⣤⠀⠀⣀⣠⣀⠾⢠⠯⣳⠈⣼⠪⣯⣳⡍⢿⣿⠯⡁⠛⡚⢧⠤⢳⠾⣧
⠀⡯⠘⢶⣸⢛⣶⢹⣥⣷⠚⡳⢘⣵⠈⣗⡤⠶⣤⠒⣶⡃⢿⣿⢻⡏⠛⢿⣿⣯⣿⡿⣺⣛⢈⢫⠀⣶⡀⢾⢧⠑⠶⣿⣖⠩⢢⡈⢻⠒⡼⢷⡞
⠀⠈⣠⣯⡜⢸⣦⡸⠿⣬⠋⣥⡬⡀⢟⡿⢂⡱⢉⣾⠉⡍⠺⣿⡴⠋⠡⠈⢿⣣⡹⡋⡩⠎⢹⡯⣹⠀⢯⡐⢆⠱⠦⣬⡁⠰⡆⢓⠘⣥⡯⡜⠳
⠀⠀⠀⡙⠐⠺⠤⡘⢀⡀⠀⠉⠘⠉⠛⢡⣄⢱⣏⠀⢳⣸⡛⠂⠁⠒⠒⠀⠈⠀⠁⠉⠁⠉⢚⡙⢉⠑⠟⢉⠒⢯⣠⡃⠛⠠⠆⠨⠢⣬⠔⢠⣍
⠀⠀⠐⠇⠐⠁⠄⠁⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢉⡇⢈⠧⡀⡘⡟⠢⠩⠉⠁

⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⠐⢛⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢚⣊⡤
⠀⠀⠀⣈⣙⣬⣁⠄⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⣿⣤⠖⠂
⠀⠀⠠⡌⣁⠼⣤⠤⠀⣠⠀⠀⠰⠀⢀⠀⠀⠀⠰⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⢀⡤⣵⣶⢦⣞⠳⢚⣃⠀⢀⠆
⠀⡀⣀⣉⣼⣧⣠⠴⠂⠀⡆⠀⠀⠀⠈⢠⠀⠀⢀⠀⢀⠀⠠⠀⠃⠀⠀⠘⠀⢈⣻⣭⢿⣟⣽⣟⣉⡁⠁⠈
⠈⠡⣬⣉⣠⠷⣦⣀⡴⠖⠉⠐⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣞⣷⣻⡶⣾⣧⣽⡆
⠀⣠⣬⣭⣡⣴⣇⣉⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣟⣽⠶⠒⢛⣿⠶⣤⠶
⠀⣄⣀⡩⢀⣠⣧⣉⠑⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠛⠛⠛⠳⠲⠖⠃
⠐⣧⠘⠓⠋⢠⣏⠉⢩⡧
⠀⠀⠈⠿⠲⠃⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠈⠉⠉⠉⠉⠉⠰⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢁⡄⢶⠋⠉⠉⣙⢻⠋⠑⠶⣤⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢀⡾⠋⠹⣄⢳⡀⠀⣿⢸⠀⠀⠀⣠⠟⣦⡈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢰⠏⠀⠀⠀⠙⣆⠙⣦⡁⠸⠀⡴⠚⣡⠞⠁⢻⡄⢹
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡤⠄⠀⠲⠶⠒⠒⠒⠒⠒⠒⠒⠲⠂⠀⠀⠄⢠⡏⠀⠷⠠⠤⢤⣤⣤⣬⣧⠘⠁⠘⢁⣀⠊⠁⠀⠀⠀⢳
⢀⣠⠔⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠐⠗⠒⠲⠢⠤⣤⣤⡀⠀⡀⠀⠈⣉⠉⠐⠒⠒⠂⣾
⣿⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⢀⣴⣋⣤⡾⢻⣆⠙⣏⠉⠉⠉⠉⠀⢶
⣿⠀⠀⠀⠉⠙⠓⠶⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣵⣤⣵⡦⢲⡶⠛⡩⠛⠉⡄⢸⡟⢶⡈⢷⡀⠀⠀⢰⠇⢰
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⢂⡀⠈⠀⠀⠀⡇⢸⡇⠀⠙⣎⡱⣄⣰⠋⢠⠟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠑⠤⣀⡀⠀⣧⢸⡇⠀⣀⣨⡷⠉⢁⡴⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⢈⠉⠁⠘⢻⡟⠈⣉⣁⣤⠖⠉
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⢺⠟⠀⠉⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿
⠀⣷⢠⡞⠙⠛⠦⠴⠦⠤⠖⠛⠛⠉⠉⢉⣛⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠛⠛⠋⠉⠙⠛⠶⣄⠀⢠⡏
⠀⣿⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠚⠁⠀⠈⢣⠉⠛⠳⠶⠶⠶⠶⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⡇
⠀⢻⠀⣧⠀⠀⠀⠀⠀⠀⡞⠋⠀⠀⠀⣠⠂⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⠇
⠀⢸⡆⢿⠀⠀⠀⠀⠀⣾⠀⢀⣀⠴⠚⠉⢣⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡿
⠀⠀⡇⢸⡄⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⡀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⡇
⠀⠀⣷⠈⡇⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡹⠶⠖⠒⠐⠀⠒⠀⠶⣄⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⢹⠀⣧⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡧⠀⠀⢀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⢸⡇⣼
⠀⠀⢸⡇⢻⠀⠀⠀⠀⠀⠈⢷⠘⢦⡀⠀⣀⠴⠞⠁⠀⡧⠀⠀⠀⡟⠉⠉⠉⠁⠀⠀⠈⡇⠀⠀⠀⠀⣼⠁⣿
⠀⠀⠸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠑⠶⠞⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⢠⡏
⠀⠀⠀⣿⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⠀⠀⢟⠀⡀⠀⠁⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⢀⡏⢸⡇
⠀⠀⠀⢻⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠉⠳⢮⣵⣏⠀⠀⠀⠀⠀⠀⠀⢸⠅⡇⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠘⡆⢿⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢐⣇⡀⠀⠀⠀⢹⡗⠶⠠⠤⠤⠀⠀⠀⠶⠁⠀⠀⠀⣼⢠⡇
⠀⠀⠀⠀⣇⠸⡇⠀⠀⠀⠀⠀⢰⠋⠀⠀⢠⠏⠀⠉⠳⢦⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⠇
⠀⠀⠀⠀⢻⠀⣇⠀⠀⠀⠀⠀⢃⣦⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⢸⡇⢹⠀⠀⠀⠀⠀⠘⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟
⠀⠀⠀⠀⠀⣧⢸⡇⠀⠀⠀⠀⠀⠙⣄⡀⠀⠀⠀⠀⠀⡄⢀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢸⠇
⠀⠀⠀⠀⠀⣿⠀⣧⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣄⣀⠖⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼
⠀⠀⠀⠀⠀⢸⡆⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣸⠁
⠀⠀⠀⠀⠀⠀⢻⡀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿
⠀⠀⠀⠀⠀⠀⢸⡇⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠁
⠀⠀⠀⠀⠀⠀⠀⢙⡆⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠞⠋⢀⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⢦⣀⠈⠙⠛⠛⠲⠶⠶⠶⠶⠶⠶⠖⠛⠛⠉⠁⣀⡤⠾⠋⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠉⠛⠳⠶⠦⢤⣤⣀⣀⣀⣀⣤⠤⠴⠶⠛⠛⠉⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠶⠦⢀⣀⣀⣀⣀⣀⣀⣀⡤⠶⠶⠛⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡇⠀⠀⠀⢀⡀⠨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠄⠂⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠢⠄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠡⠀⠄⠀⢄⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠠
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⣈⢀⣀⠔⠒⢢⠖⠒⠴
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠔⠘⠈⡩⠐⠩⢅⠁⠉⠈⠭⣐⠂⠠⢀⡀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠈⠑⢄⠀⠀⠁⠂⢄⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠬⠦⡀⠀⠀⠀⠐⠂
⠀⠀⠄⠀⠀⠀⠀⣀⠀⠠⠀⠠⠀⠀⠂⡄⠀⠤⠀⠀⠈⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠂⠀⠀⠀⢀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢻⠀⠀⡔⢀⠆⢀⠃⠀⢀⠂⠀⠄⠀⡐⠀⡐⢀⢀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠀⠋⣐⠀⠈⠀⠠⠄⠈⢀⠄⡔⠀⠉⠀⠘⠄⢀⠄⠀⡅⠁⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢠⠀⠀⢸⠀⠊⠀⠀⠚⠀⠀⠀⠇⡀⠘⠀⢈⠄⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠐⠀⠃⢀⡔⠠⠂⠰⠀⡰⠀⠈⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠒⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠈⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⢠⡀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠀⢀⣀⡀⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠘⣼⡇⠠⠀⡀⢸⣿⢆⣣⠀⠀⠀⡄⡧⠋⢀⠆⠀⠠⠀⠀⠀⣄⡇⢸⣧⠀⢸⢡⠇⢀⡇⢸⠀⢾⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡄⠡⡄⣿⠀⠂⠀⢸⣿⠈⠉⢀⡴⠔⠃⡧⠐⠁⠀⢀⠆⠀⠀⠀⠈⡆⢸⣏⣤⡏⠠⠓⠀⠇⢸⡇⠸⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡿⢸⡞⠁⠀⠀⣿⣆⠀⠆⠀⠀⢠⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠐⢼⡟⠻⠰⠁⠠⠀⠐⢾⡇⠐⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠣⡄⠀⢷⠀⠀⠀⣿⡿⠐⠀⡀⠀⠸⠴⠶⠚⡋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃⠀⣂⠀⠀⢘⡿⠂⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢾⡄⠀⠀⣿⡟⠤⠉⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⣿⠻⠀⠀⠀⠀⠐⡈⢋⣁⣤⠅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⢰⣿⣇⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣿⠀⡆⢀⢀⠀⠀⠁⠀⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⣿⣯⣄⢀⣀⣥⠤⠤⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⢹⠂⣇⣎⣁⡀⠀⠀⡀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⣭⡿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣷⣼⣄⡟⠻⠿⣷⡄⠀⣇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⣿⠃⠀⠀⠀⠀⠀⣀⣴⡾⠛⢿⣿⣿⣿⣿⣿⣿⡜⢱⣄⡀⠈⠀⠠⣿⠘⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⡀⠀⢸⣿⣿⠂⠀⠒⠀⣀⣴⠾⠛⠁⠀⠀⠀⠈⠻⢿⣿⣿⣿⡏⢸⣿⣿⣦⡀⠀⣿⠸⢃⡀⣴⣄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⠹⣷⣾⣿⣿⢀⣠⡴⠟⢋⠥⠘⠀⠀⡀⠄⠀⢸⢠⢀⠉⠻⣿⡷⠇⣿⣿⣿⣿⣷⣿⣸⠀⠹⣧⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡃⠐⠀⢸⣿⣿⣼⠋⠀⠄⢼⣤⢤⣤⣴⣃⡴⠇⠈⠘⢽⡌⣆⣸⡷⢀⣿⣿⣿⣿⣿⣿⣿⣦⣄⠡⠜⢉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠆⠈⣀⡴⠞⠉⣿⡇⢰⡇⢀⣧⡞⠁⠠⠤⡀⡄⠀⠀⠀⠀⠸⠀⣇⣿⣿⢷⠉⠻⣿⣿⣿⣿⣏⣋⣥⣶⡟⡋⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⠶⠛⠁⠀⠀⠇⣿⢃⡀⣥⡾⠁⠑⠎⠀⠀⡁⡇⠀⠀⠀⠀⠀⠀⢹⣿⣗⡈⠀⢀⣼⣿⣧⣼⡏⣻⣵⣾⣷⢹⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣴⠾⢛⠁⠀⠀⢲⡀⡆⠀⣿⣿⠙⡍⠁⠀⠀⠀⠀⠀⠃⠇⠐⠀⠀⠀⠀⠀⠘⣿⣟⠃⣰⣿⣿⣿⡿⣿⠃⢿⣿⡏⢻⣈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢻⠀⠐⠂⠺⠛⠉⢦⣿⣟⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢻⡏⢰⣿⣿⣿⣿⡇⣿⡄⢺⡻⡇⠈⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⠀⢸⠀⠀⠀⣿⣿⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⣼⣿⢸⣿⣿⣿⣿⡇⣿⡆⣿⡇⣧⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠈⠀⠀⠀⠀⠀⣻⡏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⢀⣾⣿⡇⡄⣿⡿⣿⣿⡇⣿⡇⢿⡇⡿⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄⣸⠃⠀⠀⢰⠀⠀⠀⣿⣧⡔⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⣰⣿⣿⣿⣟⣥⢛⣿⣿⣿⣷⣿⠂⠸⡃⣷⡇⣿⠃
⠀⠀⣰⣦⣔⣃⣹⣷⣾⣿⣿⠀⠀⠀⠈⠀⡄⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾⣿⣿⣿⣿⡏⠅⣸⣿⣿⣿⢿⣿⠀⡆⠁⣿⠇⡿⡇
⠀⢸⣿⣿⣿⣿⢿⣿⣻⣿⣿⠀⠀⠀⠀⠀⠃⠀⣿⣿⢿⡅⠀⠀⠀⠀⠀⠀⣄⠀⠀⣴⣿⣿⣿⣿⣿⣿⣏⠀⣿⣿⣿⣿⢸⣿⠚⣿⠀⣿⠀⣿⡇
⠀⢸⣿⣿⣿⣓⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡃⠏⠀⠀⠀⠀⠀⠀⠃⠠⠾⠿⣿⢿⠿⠿⠿⣿⣯⠴⢻⠿⣿⣿⠑⣿⣆⡿⡿⣯⣿⣿⠁
⠀⠀⠉⠉⠉⠀⠉⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠧⢰⠀⠐⢠⢿⠿⠇⠁⠀⡃⠠⢀⡿⠿⠁⡀⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⢠⠀⠇⠀⠘⠇⠀⠀⠃⠀⢸⣽
⠀⠠⠠⢠⣄⣄⣤⠤⠄⠤⠤⠤⠴⢤⠴⠶⠖⠚⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠘⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠴⠄⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⠀⠂⠀⠀⡀⠀⠂⠀⠀⢄⡴⣤⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠨⠀⣿⣓⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠦⠤⢄⠀⠀⠠⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠐⠠⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠋⠉⠀⠀⠀⠀⠀⠐⠀⠐⠵⠶⢄⠶⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠈⢀⡀⠛⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⢔⡠⠆⡠⣀⢄⣀⠢
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢀⠞⢁⡄⠏⢠⢄⢤⠱⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡆⣠⢠⡀⣉⡠⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⢰⠇⠀⠠⠸⠟⠽⣈⡀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣱⢀⡁⢍⣬⣙⢢
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠽⠡⠠⣤⡀⡀⠀⢀⢀⠀⠀⠑⡴⠊⠐⠔⠂⠀⣷⣭⡥⠄⡛⠚⠿⠻⠤⠐⠂⠀⠂⠐⠂
⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⢀⣀⣒⠓⠧⠗⠀⠾⠿⡇⠀⡧⣿⣏⣷⠁⠈⠃⠀⠿⠓⠚⣇⢀⠠⠀⠦⠔⠐⠒⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠸⠒⠀⠀⠐⠒⠸⠔⠒⠢⣼⠀⠐⡆⡠⠴⠊⠋⠉⠉⠐⠀⠀⠀⠓⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠺⣄⠤⢄⣀⣀⠀⠀⣀⣀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠂⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⠤⣄⡀⢙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⣰⠉⠉⠓⠲⢤⣱⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⣄⡀⠀⠀⠀⢢⠀⠘⢶⡖⠢⢄⣀⠀⣙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⣰⢿⣦⣠⠤⠬⣃⣀⡜⠀⣔⣉⡴⠷⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣈⣳⠦⠤⠬⣿⡆⠀⠀⠀⣀⠝⠀⢰⣷⣏⣀⣀
⠀⠀⠀⠀⡜⣱⠋⣠⠶⠚⠋⠉⠉⠀⠀⠀⠀⠀⠀⣀⠠⡤⠈⠓⠒⠒⠉⠁⠀⠀⡷⢋⣡⡤⡌⢳⠀⡇⢸
⠀⠀⠀⠀⠀⡇⢸⡁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⢀⡴⠋⠠⢁⠃
⠀⠀⠀⠀⠀⠉⠀⠑⠦⣥⣈⣁⣈⣀⣤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⡆⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡆⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠰⠁⠇⠀⣀⠤⠒⠊⣿⠫⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⢹⠛⢲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡠⠖⣉⠀⠄⢀⣀⣸⡯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢥⡰⠧⠤⢄⣀⣙⣢⣾⣠⣏⣙⣆⣀⣀⣠⣤⠤⠤⠤⠞⣆⠿⢋⡀⠒⠒⠢⢄⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⢙⡇⠠⡄⡈⠑⣄⢀⣴⠿⣦
⠀⠀⠀⠀⡰⢃⠖⣠⠞⣁⣸⠧⠤⠿⠭⠭⠭⠭⢉⣉⡉⢉⡩⠟⠉⠀⠀⠀⣠⠞⡇⠰⠁⠇⠀⠘⢟⠗⠕⠀⠁
⠀⠀⠀⠀⠃⣏⢰⡁⢊⡔⠀⣀⣀⣠⣤⠤⠶⠒⠒⣲⡞⠋⠀⠀⠀⠀⣠⠞⠁⠀⢿⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡏⠉⢹⠀⠈⠓⠦⣀⣰⠁⡇⠀⢀⣠⠴⠋⠁⠀⠀⠀⢸⡀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠆⡼⣸⠀⠀⢸⠀⠀⠀⠀⠈⠙⠲⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⣜⠀⢇⠻⣄⣀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠇⢘⡇⢳
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⡿⠿⢷⡶⠒⠒⠒⠒⠲⣶⡶⠶⣶⠟⠋⠉⠉⠁⠀⠈⠉⠠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⠤⢾⠀⠀⠀⠀⠀⠀⣧⡤⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣚⠭⠿⠀⢯⠀⠀⠀⢀⣔⠾⣕⡶⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠁⠀⠀⠛⠒⠩⠥⠤⠬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⣢⣴⣤⣶⣶⣾⣆⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠡⢚⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡛⣥⣤⣤⣤⣄⠙⣿⣿⣿⣿⣿⣽⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠛⠃⠈⠻⣿⣿⣿⣿⣏⢻⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢄⣀⣿⣿⣿⣿⣿⡟⣿⣿⣿⣍⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡦⠿⠋⠉⣿⣿⣿⣿⠻⣿⡃⣿⠻⢛⣹⣿⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣆⣨⢀⣀⣥⣿⣿⣮⣿⣷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡤⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣤⣠⣤⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠇

⠀⠀⠀⢀⡤⠶⢦⡀⠀⠀⠀⠀⠀⢠⡴⠶⣄⠀⠀⠀⠀⠀⠀⣠⠶⢦⡄⠀⠀⠀⠀⠀⢀⡴⠶⢤⡀
⠀⠀⠀⢸⠁⠀⠀⣷⢠⣤⡄⠀⠀⣿⠀⠀⢸⡇⣤⣤⠀⠀⢸⡇⣤⡄⣿⢠⣤⡄⠀⠀⣾⢰⣷⠈⡇⣠⣤
⠀⠀⠀⢸⠀⠀⠀⣿⠠⠤⠄⠀⠀⣿⠀⠀⢸⡇⠤⠤⠀⠀⢸⡇⣿⡇⣿⠠⠤⠄⠀⠀⣿⢸⣿⠀⡇⠤⠤⠄
⠀⠀⠀⢸⠀⠀⠀⣿⠐⠒⠂⠀⠀⣿⠀⠀⢸⡇⠒⠒⠀⠀⢸⡇⣿⡇⣿⠐⠒⠂⠀⠀⣿⢸⣿⠀⡇⠒⠒⠂
⠀⠀⠀⢸⠀⠀⠀⣿⢘⣛⡃⠀⠀⣿⠀⠀⢸⡇⣛⣛⠀⠀⢸⡇⣿⡇⣿⠘⣛⡃⠀⠀⣿⢸⣿⠀⡇⢙⣛
⠀⠀⠀⢸⠀⠀⠀⣿⢈⣉⡁⠀⠀⣿⢸⣿⢸⡇⣉⣉⠀⠀⢸⡇⣿⡇⣿⢈⣉⡁⠀⠀⣿⢸⣿⠀⡇⣉⣉⡁
⠀⠀⠀⢸⠀⠀⠀⣿⠠⠤⠄⠀⠀⣿⢸⣿⢸⡇⠤⠤⠀⠀⢸⡇⣿⡇⣿⠠⠤⠄⠀⠀⣿⢸⣿⠀⡇⠤⠤⠄
⠀⠀⠀⢸⠀⣶⡆⣿⠰⠶⠆⠀⠀⣿⢸⣿⢸⡇⠶⠶⠀⠀⢸⡇⣿⡇⣿⠰⠶⠆⠀⠀⣿⢸⣿⠀⡇⠲⠶
⠀⠀⢠⠞⣰⣿⣧⡻⣆⠀⠀⠀⣴⢋⣼⣿⣌⢳⡀⠀⠀⢀⡾⣡⣿⣧⡙⣦⠀⠀⠀⣠⢟⣼⣿⣆⠳⡄
⠀⠀⣿⢸⣿⣿⣿⡇⣹⠀⠀⠰⡇⢾⣿⣿⣿⠀⡇⠀⠀⢸⡀⣿⣿⣿⡷⢸⠆⠀⠀⣯⢸⣿⣿⣿⡇⣿
⠀⠀⠘⢧⣍⣛⣋⡴⠋⠀⠀⠀⠹⣮⣙⣛⣩⠾⠁⠀⠀⠈⠳⣌⣛⣋⣥⠏⠀⠀⠀⠙⢦⣙⣛⣫⡼⠃
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠿⠟⠛⠋⠉⠉⠉⠙⠛⠿⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠞⠛⠛⠉⠛⠛⢲⣾⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣿⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠛⠉⠉⠀⠀⠈⠉⠙⠿⣦⣄
⠀⠀⠀⠀⠀⠀⣀⣤⡤⠴⠶⠤⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⢠⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣿
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣿
⠀⠀⣿⣇⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢣⣾⠏
⠀⠀⠘⢿⣦⠂⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠤⠀⠄⡀⠂⡀⢂⣥⡾⠋
⠀⠀⠀⠀⠙⢷⣦⣥⣈⣀⠄⠀⢀⣰⣤⡶⠏⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢿⣇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠾⣤⣥⣀⣤⣥⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠙⠲⣥⣬⣈⣠⣂⡄⣐⣀⢀⣅⣈⣤⡬⠿⢷⣬⣄⢄⠀⠄⠤⢀⢠⠠⣠⣴⠟⠋⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠈⠙⠛⠳⠿⠶⠶⠛⠛⠉⠀⠀⠀⡀⢀⠀⠀⢄⢡⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠤⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠨⣣⠆⠀⠪⠃⠊⠠⡆
⠀⠀⠀⠀⠀⠀⢈⣐⡸⠀⠙⠋⠈⢀⢩⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠁⢬⠳⠀⢀⠠⠐⠕⠀⠀⠀⠀⠀⠀⠀⣀⡐⠀⢂⡐⠀⠤⡀⣱⠩⡈⠁
⠀⠀⠀⠀⠀⠀⠁⠀⠒⢌⠂⠐⡀⠭⠌⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡔⠮⠂⠄⠌⠔⠀⠙⠈⠃⠀⠀⠀⠀⠀⠀⢀⢐⠔⡎⠄⠒⠠⢑⢄⡀⢀⠁
⠀⠀⠀⠀⠀⠀⠡⢰⠢⠀⠂⠖⠕⢀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠂⠄⡬⠆⡠⡆⢌⠀⠀⠀⠀⠀⠀⠀⠀⠃⢁⠀⡤⠀⠀⡞⡕⢂
⠀⠀⠀⠀⠀⠀⠈⠤⠁⠠⣦⠦⠀⢁⠅⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡨⢲⠁⢠⣤⠀⠰⡰⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠂⠂⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⡏⠑⠂⠀⠀⠀⠀⠀⠀⡀⡐⠀⠀⠀⠀⠀⠀⠁⠀⠊⠰⠐⠩⠒⠂⠀⠀⠀⠀⠀⡉⠠⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡆⠐⠭⠊⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢄⠑⠨⠃⠀⠀⢼⠡⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠚⢀⢠⠀⡀⢄⢒⠏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠲⣁⠄⠄⠒⡕⠀⢁⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣒⡄⠀⣡⢀⡒⠀⢀⣁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀⣀⠘⠠⠰⠂⠠⠩⡎⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⢂⠜⢀⣀⣄⠀⠐⣑⠊⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣒⡆⠀⠤⡔⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢚⠢⠤⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠚⢈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⣤⣄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠀⠐⠁⠀⠀⠀⠐⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⡇⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠓⢤⡀⠀⠀⠹⡆
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠙⣄⠀⠀⢿
⠀⠀⠀⢰⡟⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⣀⣀⣤⡴⠶⠛⠻⠋⠀⠀⢀⠐⠀⠀⠀⢿⠀⠀⢸⡄
⠀⠀⠀⣿⠁⠀⠊⠀⠀⠀⠀⠐⢀⡤⠖⢋⡙⠉⠉⠉⡉⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⢀⢀⣀⣀⣸⠀⠀⢸⡇
⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⣠⠥⡞⠀⠀⠀⠀⢀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢉⡁⠉⠛⠋⡄⠀⣼⠃
⠀⠀⠀⢻⣦⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠰⠞⠋⠁⢀⡄⠱⡄⠀⠀⠀⠀⠀⡀⠀⠰⠾⠇⠀⠰⠀⡇⠀⣿
⠀⠀⠀⠀⡿⠀⠂⠀⠀⠀⠀⢸⡄⠀⠀⠀⢤⣀⠈⠛⢃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⡇
⠀⠀⠀⠀⢿⣄⠀⠀⠀⠀⡀⠈⠃⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⢸⠋
⠀⠀⠀⠀⠈⣿⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠑⢄⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢤⠂⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠘⢷⡀⢀⠆⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⢆⠀⠀⠈⠉⠙⠓⠶⠤⣄⣀⣀⣀⣴⡟⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠐⠠⠀⠠⠀⠀⢤⠤⠀⣀⡀⠀⠉⠁⠀⡗⠠⢀⡘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢶⡶⠁⠀⠀⠉⠀⠀⠀⡦⠀⠀⠈⠑⠢⡀⠀⢷⠀⠀⠅⠀⠀⢻⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢠⠜⠀⠀⠀⠀⠀⠀⠈⠃⣜⠰⠀⠐⡀⠀⢈⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠂⠄⠀⣀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠃⠠⠀⠈⠆⡈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⢀⠀⠀⠈⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠀⠙⢣⠀⠘⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠈⠐⠀⠄⠀⠈⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠐⠀⠣⡀⠀⠑⠧⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠈⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠳⠠⢤⠄⠀⢸⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢣⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠙⢀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠤⠀⠃⠈⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠑⡦⢀⠀⠠⣽⡄
⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⡀⠀⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠇⠀⠉⠀⠘⣧
⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠈⠄⠀⠀⠀⠀⠀⢀⠈⡑⠊⠀⠀⠀⠀⠀⠀⠀⢈⠢⣖⠀⠈⡒⢄⡀⢀⣼⡂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠸⠉⠀⣷
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢰⠀⠀⠀⠀⠀⠢⠠⠆⠀⠀⠐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠈⠳⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢤⠀⠀⠸⡀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⡸⠀⠀⠀⠀⠃⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠔⠁⠀⠀⠀⢰⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣷⡄⢠⡟
⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇⢹⡞⠁
⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⠟⠿⣿⣆⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾⣿⣿⣿⡟⠱⢖⣞⣿⣿⣷⡀⠁⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠏⠀⠠⠤⢿⣿⣿⣿⣿⡄⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⠏⠈⠀⠘⠚⠛⢻⡿⣿⣿⣇⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⣾⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣡⡀
⠀⠀⠀⠀⠀⠛⠛⠛⠿⠏⠿⢻⣿⣿⣿⠻⢻⢿⣿⣿⣿⡿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣶⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢉⠀⠀⢹⠁⠀⣿⠀⠈⠉⠙⠛⡿⠛⠉⠉⠋⠋⢃⠈⢻⢡⡄
⠀⠀⠀⠀⠀⠀⠄⠀⢸⠀⠀⠀⠀⠀⢘⠀⠀⠸⠀⠀⢻⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠸⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠂⠀⢀⡗⠀⠀⠁⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣤⣤⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣯⣻⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠟⣋⣥⣾⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣤⣶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣇⢻⣿⣿⡟⣿⡿⣹⡇
⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⣿⣿⢱⣿⣿⣿⣿⣿⣿⣿⣎⢿⣿⣇⣟⣼⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⡇⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠋⣾⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢻⣧⠸⣿⣿⣿⣿⣿⣿⣿⡇⢸⠏⣼⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠙⠀⣿⣿⣿⣿⣿⣿⣿⡇⠛⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿⣿⣿⣿⣿⣿⠃⢠⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⠀⣸⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⢛⣾⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⢸⣿⣿⣿⡇⠘⣿⡿⢋⡙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⡟⣿⣿⣿⣇⠀⠟⣴⣿⣿⣦⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣅⣤⣤⣤⣤⠀⣾⣿⣿⣿⣿⣷⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢹⣿⣿⣿⡇⠀⣿⠿⠿⠟⠛⠛⠁
⠀⠀⠀⠉⠉⠉⠉⠛⠛⠿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠈⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠂⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠘⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠌⠒⠒⠒⠒⠲⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢊⢾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣌⢸⡇⢀⣀⠠⠤⠐⠒⠒⠒⠒⠲⠤⢤⣀⠀⣿⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣃⡼⠛⠉⠀⠀⢀⣠⠀⠁⠀⠀⠀⢀⣀⠈⠙⢻⣅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠤⠿⣄⡀⠀⠀⠀⢋⠄⠒⠒⢄⡀⠒⠒⠠⡹⠀⠀⢸⢿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠴⠦⢤⡀⠀⠀⠀⡞⠀⠀⠀⠈⠷⠤⠘⠐⠁⠀⣤⡀⠀⠀⣠⣄⠀⠐⠰⡦⠏⠻⡀
⣦⣄⡀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠉⠢⡀⢀⡇⠀⠀⠀⠀⡼⠀⠀⠀⠐⢀⠉⢀⠀⠃⠚⢅⡠⠂⠀⠀⠄⠀⡇
⢹⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠋⠙⠦⣀⣠⡴⠧⠙⠢⠄⠀⢀⣀⡀⠀⠀⠀⠀⣇⣠⡴⢦⡀⢀⠇
⢸⡆⠀⠀⠀⠀⢀⣠⣴⡾⠟⠛⢛⠛⠛⢿⡀⠀⣀⠤⢾⠁⠀⠀⠐⠠⠀⡈⠙⠻⠶⠶⠖⠊⠀⡀⠠⠀⣻⠁
⠀⣇⠀⣀⡠⣶⡿⠛⠁⡠⠀⠀⠀⠀⠀⣠⠝⠋⠀⠀⣸⣇⠠⠀⡀⠀⠀⠀⠈⠉⠁⠈⠉⠁⢀⠠⠐⠈⣸⠳⢤⡀
⠀⠉⠁⠀⠀⠘⡆⡠⠊⠀⠀⠀⢀⡤⠊⠁⠀⠀⢀⡾⠁⠈⠲⣄⣀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢧⡀⠀⠙⠷⣄
⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⣠⠞⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⡖⠒⠲⠒⠚⠉⠉⠀⠀⠈⣷⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⢀⠤⠶⣿⣷⣄⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣠⡾⠓⠤⣀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠈⠙⠻⠿⠿⢷⡚⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⣧⣤⣤⣶⠿⠋⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢠⡇⠀⠀⠀⢀⣠⠶⣦⠀⠀⠀⣿⣷⡈⠙⠳⠦⣤⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⠋⣷⡞⠉⠀⠀⡶⢤⣀
⠀⠀⠀⠀⠓⠶⠶⠖⠋⠀⠀⢻⡀⢀⣠⠇⠈⢳⣦⣀⡀⠀⠀⠉⠙⠛⠲⠲⠶⠶⠶⠶⠶⠶⠒⠛⠉⠉⠀⠀⣰⠟⢧⠀⠀⢀⡇⠀⠈⠓⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠘⡇⠈⠙⠲⠦⣤⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠚⠋⡇⠀⠈⠙⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠙⡓⠒⠈⠉⠉⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣰⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⢳⠆⠀⠀⠀⠀⠠⠞⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠠⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡘⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠒⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣁⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠚⠃⠂⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠨⠈⠏⠙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡧

⠀⠀⠀⠀⠀⣀⡤⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠦⣄⡀
⠀⠀⠀⠀⣸⠋⣠⡶⣤⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⣠⡶⣤⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢠⡴⢤⡈⢳⡀
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣓⠒⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣛⠓⠓
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⠒
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠭
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣛⠛⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣛⠛⠋
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣒
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠭
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣀⣸⡇⢸⡇⠶⠶⠖⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣉⠉⠉
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣒
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠤
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣭⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣭⠉⠉
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣒
⠀⠀⠀⠀⣿⠀⣿⠀⢸⡇⢸⡇⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠶
⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⡭⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣒⠒⠒⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣭⠉⠉
⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⣗⣛⡂
⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⣋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⣽⣿⣽⡇⢸⡗⠶
⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⣿⣿⡇⢸⡇⠉⠉⠉
⠀⠀⢀⡴⠋⣠⣿⣿⣿⣧⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣿⣿⣿⣧⡈⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⣿⣿⣿⣧⡈⠳⣄
⠀⠀⣼⠁⣸⣿⣿⣿⣿⣿⣿⡀⢻⡀⠀⠀⠀⠀⠀⠀⣼⠁⣸⣿⣿⣿⣿⣿⣿⡀⢻⡀⠀⠀⠀⠀⠀⠀⣸⠃⣸⣿⣿⣿⣿⣿⣿⡀⢹⡄
⠀⠀⢿⡀⢻⣿⣿⣿⣿⣿⣿⠃⣸⠃⠀⠀⠀⠀⠀⠀⢻⡀⢻⣿⣿⣿⣿⣿⣿⠃⣸⠃⠀⠀⠀⠀⠀⠀⢻⡀⢻⣿⣿⣿⣿⣿⣿⠃⣸⠇
⠀⠀⠈⢧⡀⠛⠿⣿⡿⠟⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠛⠿⣿⡿⠟⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠛⠿⣿⡿⠟⠃⣠⠟
⠀⠀⠀⠀⠙⠶⢦⣤⡤⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⡤⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢦⣤⡤⠶⠞⠁

⢀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀⠀⠀⠀⢠⡀⠀⣶⡀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆
⢹⡆⢀⠀⠰⠇⠀⠀⢿⡀⠀⠀⣧⠀⠈⠀⠀⡀⠀⠸⠇⠀⠈⠀⠀⢸⡃⠀⠹⠃⠀⠀⠀⣗⠀⠀⠀⠀⠘⡼⠀⠀⠀⠀⣷⠀⠀⠀⡜⡆
⠈⠀⣿⡄⠀⠀⣄⠀⠙⠁⠀⠀⠙⠀⠀⠀⢸⣣⠀⠀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⢰⢆⠀⠋⠀⠀⢠⡀⠀⠀⠀⢠⣳⠀⠈⠀⠀⠀⠙⠁
⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⣎⡇⠀⠀⠀⠈⠛⠀⠀⠀⠀⢠⡄⠀⠀⢸⣃
⢠⡀⠀⠀⠀⠀⠉⠀⠀⠀⠠⡴⠀⠀⠀⣄⠀⠀⢰⡄⠀⠘⠃⠀⠀⡀⠀⠸⡇⠀⠀⠀⣀⠀⠀⠀⠘⠁⠀⠀⠀⠀⣀⠀⢸⡃⠀⠠⡴⠀⠀⠘⠋
⢰⡇⠀⠠⣻⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⣽⠆⠀⠸⠃⠀⠀⠀⠀⢀⣿⠀⠈⠃⠀⠀⠀⣏⠀⠀⠀⠀⠀⡀⠀⠀⠘⣬⠄⠈⠁⠀⠀⠁⠀⠀⠀⣆
⠀⠀⠀⠀⠉⠀⠀⢸⣥⠀⠀⠸⡧⠀⠀⠀⠀⠀⠀⣀⡀⠤⢤⠤⠼⣿⣄⣀⣀⠀⠀⠀⠉⠀⠀⠀⠀⠘⣵⠀⠀⠀⠈⠀⠀⠀⠀⡄⠀⠀⡀⠀⠉
⠀⠀⢠⡄⠀⣴⡀⠀⠁⠀⠀⣀⡤⠴⣒⡾⣛⠯⣋⠤⣒⠭⠐⠊⠁⠈⣣⢰⠂⡉⠉⠙⠓⠶⢶⣤⡀⣤⣉⣀⠀⠀⠀⠀⠀⠀⢸⣱⡀⢠⢛
⠀⠀⠰⡿⠀⠙⠀⠀⡠⡴⠚⠁⣠⡮⠕⡩⡰⠝⣂⠥⠒⠊⠁⠀⠀⠀⠃⣂⢳⠄⣐⠂⠄⠀⠀⠀⠙⠳⢤⡩⢉⡑⠲⠤⣄⠀⠈⠛⠀⠀⠛
⢰⡄⠀⠀⠀⣠⠔⡭⠭⠁⢀⠞⡩⢒⠕⣊⠔⣊⠅⢂⡁⠀⠀⠀⠀⠀⠀⢸⡀⡡⠄⡒⠨⠥⠀⠀⠀⠈⠁⠙⢧⡊⠕⡂⠄⠉⠲⢄⡀⠀⠀⣿⡀
⠸⠇⠀⣠⢊⡑⠨⠅⠀⢀⡎⡨⢊⠵⣊⠕⡪⠔⣊⠅⠀⠀⠀⠀⠀⠀⠀⠈⡅⣈⠥⠐⡀⠉⠥⠀⠀⠀⠀⠈⠀⠳⣬⠔⡂⠥⢀⠀⠙⢦⡀⠉
⠀⢀⠎⣄⢀⡂⠀⠀⠀⡼⣡⢜⡥⡊⣕⠯⢒⠭⢒⠠⠀⠀⠀⠀⠀⠀⠀⠀⢾⡴⠎⢁⡒⠤⠀⠁⠀⠀⠀⠀⠀⢃⠙⡯⢒⠨⠁⣀⠀⠠⡱⡄
⢠⡯⠾⠼⠯⠤⣔⡠⠀⣏⢔⡁⡢⢛⣥⣊⡴⢪⣤⣊⣡⣐⡠⢀⠀⠀⠐⠀⠨⢒⡩⢔⡂⠬⣀⣒⠂⠲⢀⡀⠀⠀⣂⠹⡥⢒⡡⣔⡈⠥⠽⢼⡆
⠛⠀⠀⠀⠀⠀⠀⠈⢲⣧⠿⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⢺⣴⣀⠀⢸⠄⣪⡵⠾⠛⠒⠚⠐⠒⠒⠒⠒⠪⠽⢆⣷⠟⠋⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠈⠑⢾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⡎⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢿⡀⠀⢀⠇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡉⠛⢁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⣀⠀⠀⡀⢀⠠⢀⠤⢂⠤⢀⠠⢀⠤⢀⠄⡠⠀⡀⢀⢄⡀⢀⠀⢀⢀⡀⣀⠀⡀⠀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠀⠁⠈⠁⠚⠁⠂⠁⠒⠡⠊⠵⠊⠔⠥⠊⠕⢩⠔⡡⠒⠕⡡⢊⠕⢋⠔⣡⢚⠕⡫⢐⠅⡢⢖⠕⡢⢒⡡⣂⠔⣡⢂⠔⡠⢠⠄⡀

⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡈⢛
⣻⣿⡭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠏⢤⣴
⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠰⠂⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡇⠀⠀⠙
⣿⣿⣟⣋⡀⠀⠀⠀⠀⠀⢠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢠⡿⠤⠄⡀
⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠶⣶⣶
⣿⣿⣿⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠁⠀⠀⠀⢐⣛⣿⣿
⣿⣿⣿⣟⣋⠁⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⢩⣽⣿⣿
⣿⣿⣿⠯⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣻⣿
⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠉⠛
⣿⣿⠧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢾
⣭⣇⣐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡛⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠾⠿⠛⠛⠻⠷⢶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠶⢤⣼⡟⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠉⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠉⠉⠙⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣿
⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⣿
⠀⠀⠀⠀⢻⡏⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⣽⡟
⠀⠀⠀⠀⠈⢿⣄⢅⡂⢀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡀⡠⢐⠠⢐⣼⠟
⠀⠀⠀⠀⠀⠈⠻⣷⣬⣄⡂⠀⠀⠄⣀⣤⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡼⢤⣄⣅⣂⣌⣤⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠛⠛⠉⠑⢦⣥⣅⣐⡐⡀⠀⢂⠄⢅⣐⣐⡽⢿⣦⣄⠄⡀⡀⡀⠀⠀⠄⣄⡶⠟⠁⠀⠈⠉⠉⠉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠈⠙⠛⠷⠾⠶⠶⠞⠛⠉⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⢀⣠⠖⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⢤
⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⣰⡟⠁⢰⡄⠀⠀⠀⠀⠀⠀⣠⡾⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⣾⠋⠀⢱⡀⠀⠀⠀⢿⡄⠀⣸⠇⠀⠀⠀⠀⠀⢰⡟⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋⢤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠋⠀⠀⢀
⠀⠀⠀⢿⣄⣀⡼⠁⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⢀⠀⠀⠘⢧⣀⣠⠟⠀⠀⢀⣴⠞⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣠⡞⠋
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠉⠁⠀⢀⣼⠟⠁⠀⠀⠀⠙⠷⠴⠟⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢠⡾⠋⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣄⠀⠀⠀⠀⣿⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⣿⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠻⣄⣠⡾⠁⠀⠀⠀⢀⡀⠀⠀⠀⣿⠀⠀⣸⠀⠀⠀⠀⠙⢦⡴⠟
⠀⠀⠀⠀⠀⢠⡾⠋⡀⠀⠀⠀⠀⠀⠹⢦⡴⠟⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠀⠀⠀⠀⠙⠶⠾⠋
⠀⠀⠀⠀⠀⣿⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⠏⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠻⣄⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠐⣄⠀⠀⠀⠀⠀⠀⢼⡇⠀⢈⡇⠀⠀⠀⠀⠀⠀⢀⣴⠟⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⡿⠀⠀⠀⠀⠀⠀⠈⠷⠤⠾⠁⠀⠀⠀⠀⠀⠀⣼⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢧⣤⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠻⠟⠛⠿⣿⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠕⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠃⠀⠀⠠⣾⣾⣿⣶⣄⠀⠀⠀⠀⠀⠘⢷⣌⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⢀⠸⢿⢙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣧⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⢤⡟⢸⣃⣬⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⢠⡶⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡄⠀⠀⣉⣦⡆⣿⣾⠓⣲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣴⣶⣫⣅⣀⠀⢾⡇⣽⢾⣦⣻⢿⣞⣿⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠟⢛⣼⣋⢠⡀⢰⣛⣸⡇⣉⢋⣹⣻⣿⣍⣭⣦⡀
⠀⠀⠀⠀⠀⣄⠀⡀⠘⠖⣿⢠⡀⠐⠂⣀⡀⠀⠀⣀⡄⣂⡾⣻⢷⢋⣿⣿⣻⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳⠄
⠀⠀⠀⠀⠀⢬⡻⢤⠤⢼⠿⣏⡀⣤⡄⠀⠀⠐⠂⠉⢙⣋⡶⣉⡄⠙⣿⣿⡟⠿⢟⣫⡿⢟⡿⣿⣿⣿⣿⣋⣥⢼⣇⣶⣄
⠀⠀⠀⠀⠀⠀⠠⠈⣤⣤⡤⣿⣧⡄⠳⢄⣤⣤⣀⠲⣾⣿⣴⠏⢠⠟⠹⣿⣤⣦⣟⣉⣷⢟⣿⡟⣿⣷⣾⣿⢻⣿⡿⠿⡝⣄
⠀⠀⠀⠀⠀⠀⠀⢾⣄⠐⠂⠞⢻⣧⣀⢀⣍⠛⢋⣴⣿⣛⣥⡆⠀⣠⣾⠛⢻⣟⣿⣿⣷⢬⣿⣼⣿⣿⣿⢿⡼⣿⡇⣶⣷⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⢄⣠⠼⢿⣿⡿⢃⣥⣿⣁⡿⢧⣿⣀⡾⢻⣇⣵⠟⢉⣽⣿⢿⡿⣿⣿⣿⣾⢛⣼⣇⣻⣿⣿⣧⡿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣮⣳⣶⣊⣿⠿⣿⣾⣯⣿⣟⣷⢿⣯⢥⣿⡟⣻⣆⢸⣯⣴⣎⣾⣯⣿⣻⡿⠿⣿⡛⣿⣣⣼⣽⣷⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣏⣭⣿⠟⣽⣾⡟⣻⡿⠿⢿⣣⡾⣻⡾⢻⣿⡿⣿⣼⣯⡿⢿⣿⠏⣹⡿⣷⠿⣿⡿⢿⣿⣿⡿⠻⣾⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣻⣾⣋⣷⣿⣍⣛⠻⣻⣿⣿⢛⣿⣿⣧⣴⣋⣿⣏⣴⢦⢶⣾⣻⣶⣏⣤⣾⣵⣶⣿⣶⣶⣾⣿⣿⣿⣯⡛⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣟⣻⢶⣿⣿⣹⣷⣾⣏⢛⣿⣿⣩⣿⣿⣻⣿⣽⣯⣶⣿⠛⡿⡟⣿⣿⡿⢿⣿⢿⣏⢉⣿⣽⣿⣿⣿⣿⣽⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣯⡽⣿⣿⣯⠵⣿⣿⣽⣿⣿⣏⣴⣿⣿⣿⣿⣏⣴⣤⣧⡆⢩⣼⣀⠀⣼⢛⢿⠻⢿⡛⠛⣍⢿⣿⣷⡿⣯⠛⠩⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠱⠶⢻⣷⡺⣿⡋⢷⣅⣭⣿⣿⣿⣿⣿⣿⣿⠛⢿⡛⠿⣿⡟⣿⣿⡾⢿⣿⣴⣿⣤⣿⣿⣿⣼⣿⣿⣷⣄⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣿⣿⣾⣿⣮⣿⣽⣿⣿⠿⣟⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣻⣿⣎⣻⣿⣿⣿⣽⣿⣿⣯⣿⣿⣿⠿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣛⣯⣟⣹⣷⣿⣷⣶⣛⣿⡼⢿⣵⣿⣿⣿⣿⣿⣿⣿⣟⣭⣿⣯⣷⣿⡿⣿⣿⣿⣿⣿⣿⣯⡢⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣡⣯⣍⣬⣯⣙⣿⣿⣋⣶⠾⢿⣿⠟⣫⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣟⣭⣿⣟⣿⣾⣷⣤⠶⠶⢿⡾⢫⡜⢻⣿⣿⣯⡀⠙⠻⢿⡿⠿⣿⣿⣷⣿⡿⡏⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠿⡿⠋⠙⢿⣯⣷⣶⠞⣻⡅⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⢇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢀⣾⣿⡟⠀⠀⠈⢀⣀⣀⠀⠀⠀⠀⠈⠛⠟⠀⠀⠉⠉⠹⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⢻⡏⠙⠃⠀⠀⠀⣼⠇⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣻⣷⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⠉⠉⠉⠉⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⢀⠞⠃⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⡀⠀⣰⠇⣰⡇⠀⣰⣆⠀⠀⠀⣠⡶⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⢺⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠰⡏⠀⢠⡇⡜⢱⣣⠎⠿⡟⠀⡠⠊⢰⠇⡴⠚⡇⡴⣿⠀⣠⡆⠀⠀⠀⠀⢳⡀⠘⡦⢘⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠇⠀⢸⠏⠀⠸⠃⠀⠀⣷⠊⠀⠀⢸⠏⠀⠀⠟⠀⣿⠞⢹⢃⣤⠀⠀⠀⠈⡇⠘⣇⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠇⠀⣸⠀⠀⣾⠀⠀⠀⢀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⠏⢹⢀⢼⡇⠀⣿⠀⢇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⡼⠁⠀⢀⣿⣤⣴⠾⠟⠋⠉⠛⣆⠀⠀⠀⠀⣀⣽⣦⣄⡀⠀⠀⠀⠸⠋⢸⠃⠀⣿⠀⠘⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⠟⢸⠁⠀⠀⠀⠀⢀⢸⡆⠀⠀⡜⠁⠀⠈⠳⣝⢦⡀⠀⠀⠀⢸⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⢻⠀⠀⠀⠀⠰⠿⣸⠁⠀⣸⠀⠀⠀⠀⠀⣈⠙⡿⠒⠀⠀⢸⣠⠼⠷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢦⡼⠁⠀⠀⠀⠈⠳⣤⣄⣀⡤⠞⢁⡀⠀⢿⡀⠀⠀⠀⠰⠟⣠⠇⠀⠀⠀⣿⣣⢚⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡤⠋⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠋⣭⠀⠀⠀⠉⠓⢦⣤⡤⠞⠁⠀⠀⠀⢸⠛⢋⡽⠋⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠻⣗⠺⠭⣄⠤⠼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣣⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢳⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠦⣄⡀⠀⠸⣆⠀⣠⣄⣤⡴⠶⢻⠖⢦⠞⣯⠤⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠛⢯⡀⠀⠀⠀⠀⣀⢀⣤⠖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠶⠶⠯⢤⡶⠀⠈⡻⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠷⣦⣤⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢽⣭⢭⡭⠭⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠠⠮⠤⠖⠊⠀⠀⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⣃⣠⠴⠋⠁⠀⣀⠀⢦⣤⣉⣓⣦⣤⠀⣀
⠀⠀⠀⠀⠰⡇⠀⠀⣻⠄⠀⠀⠀⠀⠀⣀⣀⣀⡈⠳⣄⠀⠉⠉⠛⠓⠒⠒⠒⠒⠒⢒⣒⡿⠿⣍⠹⡄⢠⣄⠀⢹⠓⢬⣦⡀⠉⣩⠗⢴⠋⢠⡄
⠀⠀⢀⠀⠀⡇⠀⠀⠙⣦⠀⠀⠀⠀⠚⠯⢤⣉⡉⠙⠛⠒⢶⣶⡴⢲⣄⡤⣄⠀⣜⠉⠀⠀⠀⠈⠳⡝⣼⠀⠳⣼⡀⠀⠈⠉⠀⣳⢀⡼⠁⣾
⠀⠀⠈⢣⡀⠈⣢⠀⠀⠷⢤⡄⠀⠀⠀⠀⠀⠈⠙⠢⡀⠀⠈⡇⢧⠀⠀⠀⠙⠉⠙⠖⠚⠳⠒⠲⠖⠿⠈⠀⠀⠈⠁⠀⠀⠀⢼⠁⣨⠇⢶⠁
⠀⠀⠀⠀⡇⠀⠙⢦⠀⠀⠈⠳⢦⣤⡤⠶⡄⠀⢀⡀⠈⢦⣤⣇⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠈⣧⠀⠼
⠀⠀⠀⠀⠘⡆⠀⢎⠀⠀⠀⠀⠘⣧⠀⠀⠙⠋⠉⠓⠒⠋⠉⠉⠓⢦⢤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠋⠀⡴⠁
⠀⠀⠀⠀⢀⠃⠀⠈⣷⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠙⠚⠙⠲⠛⠶⠶⡤⢦⠖⢶⡿⠙⠁
⠀⠀⠀⠀⠘⠃⠀⠀⠛⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⡄⠠
⠀⠀⠀⠀⠀⠦⠲⠴⠂
⠀⠀⠀⠀⢘⣀⡄⠈⠉
⠀⠀⠀⡀⠈⠉⠁⠃⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠦
⠀⠀⠀⠉⢒⠀⢠⣤⡦⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣉⣁
⠀⠀⠈⠘⠃⠀⢀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣭⣅⣠⣬
⠀⠀⠐⠀⡴⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣅⠀⢤
⠀⢀⠀⠈⠀⠀⢢⡄⠛⠦⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⢈⣀⡠
⠀⠈⠁⢨⠄⠀⠂⠐⠢⣆⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⢴⣲⠊⣀⠀⠥
⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢲⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣆
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣶⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⢺⡛⠉⠉⠉⠉⠉⠈⠉⠁⠀⠀⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⢲⣟⠛⠙⠋⠛⠛⠉⠉⠀⢀⠀⠁⠐⡇⠀⠀⠀⢠⣤⢠⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⡴⠖⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⢀⣶⡀⠀⠀⢀⡇⠀⠀⠀⢸⣿⡜⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢰⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⡜⢀⡀⠀⣶⠀⢠⣇⠀⠀⣸⠹⣧⠀⠀⣸⣷⠀⠀⠀⡞⣿⣧⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⣠⡄⠀⣸⣇⠀⢠⢿⣷⢠⡇⢸⣇⠀⣿⣆⢸⣿⡄⠀⡅⠀⣿⡇⢀⠅⣿⠀⠀⠀⡇⣿⣿⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣿⣧⡀⡿⣿⡀⣼⠈⣿⣼⣷⣼⣿⢸⢻⣿⣾⣿⣇⠀⡇⠀⡿⣿⢸⠀⣿⡇⠀⢰⠃⢻⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⣿⠀⡇⢻⣷⡏⠀⢻⣿⡿⠛⣿⡿⠀⢻⠁⣿⣿⢀⠇⠀⡇⠹⠟⠀⢹⡇⠀⢸⠀⠀⠀⣇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⣿⣿⡇⠘⢷⡇⠀⠀⠿⡇⠀⠙⡇⠀⢸⡆⣿⠹⣾⠀⠀⡇⠀⠀⠀⣼⡇⠀⣼⠀⠀⢀⡿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⡏⣿⠋⣷⠀⠀⠁⠀⣰⠀⡇⠀⠀⡇⠀⢸⠇⠻⠰⢹⠀⠀⠁⠀⠀⢰⣿⡇⠀⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠸⡀⣸⡆⠀⠀⠀⣿⠀⠟⠀⡆⢹⠀⢸⠀⠀⠀⣼⡇⠀⠀⠀⢀⣿⢻⡇⠀⣿⠀⠀⡌⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡄⠀⣿⣿⠀⠀⠈⣿⡆⠀⠀⠀⣿⡆⢸⠀⠀⠀⣿⣧⠀⠀⠀⣼⢹⢸⣿⢀⢿⡄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣧⠀⣿⢹⡇⠀⠀⣿⡇⠀⠀⡀⣿⣇⣸⡄⠀⠀⣻⣿⠀⠀⠀⡇⢸⠸⡏⢸⠘⡅⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣿⡘⣿⠀⠘⡟⣿⡀⠀⠀⡏⢹⣿⡇⠀⠀⡏⣿⠀⠀⢰⠃⢸⢰⣷⡜⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⡏⣿⢿⠀⢰⡷⣿⠀⠀⢠⡇⣼⢹⡇⠀⠀⡇⣿⠀⠀⢸⠀⢸⣸⠻⠇⠀⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉⢸⡆⢸⣟⢻⠀⠀⢸⣇⣿⠸⡇⠀⢸⠁⣿⠀⠀⡏⠀⢸⣿⠀⠀⠀⣧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣸⢸⣾⠄⠀⢸⢿⠟⡀⡧⠀⢸⠀⣿⡄⢠⠇⠀⠈⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⢿⡇⠀⡾⠈⠀⠃⣟⠀⢸⠀⣿⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⢸⣿⠀⢸⡄⡇⢯⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡇⠀⠀⠀⢹⠀⣿⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁⠀⠀⠀⠘⣦⢿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡎⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠿⠀⠀⠇⠻⠼⠃⠇⠀⠇⠿⠸⠀⠿⣸⢯⡄⠀⠿⠀⠘⠧⠟⢸⢯⡍⠷⠼⠃
⢀⢀⣠⠫⢂⠢⠍⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⢁⠀⡼⠠⠐⠒⠻⠄⡀
⡴⠂⠀⠀⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊
⠂⠀⠀⢀⠀⠀⠀⠰⠶⡄⠀⠀⢬⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠁
⠀⠀⣀⣤⡄⠀⠀⢰⣿⣾⠁⠀⠈⠀⠀⠀⠀⢀⠀⡔⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠁⠀⠀⠴⠂⠀⠀⠀⠖⢖
⠀⠀⠘⠿⠃⠀⠀⠀⠉⠁⠀⠀⡀⠀⠀⠀⠀⠤⠎⠑⠙⣫⠀⠀⠠⡀⠠⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⢤⣀⣤⡎⠀⠐⠋⠁⠀⠀⠀⠀⡨⠆⠀⠀⠄⠃⠆⠣⠐⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠀⠀⠀⠺⠃
⢀⡄⠀⠀⠀⢀⠀⠀⠄⠀⣄⠀⢡⠀⠀⢀⠜⠀⣠⠊⠀⠀⠈⠈⠀⣀⢀⡁⠀⠀⠀⠀⢁⠀⠀⠀⡀
⠀⠁⠀⠠⠒⠂⠀⠀⢀⠁⠅⠀⠆⠒⠀⠈⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠂⠀⠀⠈⠂⠤⠌⠉⠀⠐
⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡄⠀⠂⢒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⠀⠄⠠⠀⠇⠀⠁⠀⠀⠀⠀⠂⠠⡀⠀⠀⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⢄
⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠒⠈⠈⠀⢄⠰⠀⠀⠉⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀
⠀⠀⠂⡀⠢⣀⠀⠀⠀⢀⠤⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⡀⢀⢀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⠠⢠⣤⠀⣀⡀⠀⣠⠴⠀⠀⡉⠻⣀⠯⠤⠶
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠒⠃⠀⠘⠤⠧⠽⠀⠀⡁⠣⡀⠀⡁⠜⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠃⠀⠀⠀⢰⡀⠀⠀⠀⠖⠒⠒⡀⠀⢀⡐⠀⠈⠑⢂⠂⠁⣸⠷⠀⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠄⠠⠄⡀⠢⠅⠀⠀⠀⢀⠤⠀⠘⡄⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠚⠁⠀⠑⠬⢀⡀⠀⠀⢠⠃⠀⣀⣤⣒⡉⠥⠀⠐⠸⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⠋⠙⡗⠛⠥⣌⠀⠄⣸⠃⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠈⠠⠀⠀⠑⠢⡉⠢⣄⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡀⠤⠤⠤⠤⠴⠟⠂⠀⠈⠑⠠⠈⣈⠁⠉⠀⠀⠀⠀⠀⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⡀⠄⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠰⡀⢤⣶⣾⣿⣿⣿⣿⣿⣶⣴⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡤⠀⠂⠀⠀⠀⢀⡽⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⣿⣿⣿⣿⠃⠀⠀⠀⠠⣶⠋⠀⠀⠉⠉⠉⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⠀⠀⠀⠀⠀⠐⢕⢄⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠿⠛⠁⠀⡄⠀⠀⠀⠀⠀⡟⢻⠶⣄⡀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣃⣠⣾⣦⠀⡀⣻⡇⠀⠀⠀⢸⡇⣾⠀⠈⠻⠃
⠀⠀⠀⠀⣠⠀⠀⠀⣴⣿⠀⠀⠀⠀⣰⣀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⣿⡇⠀⠉⠛⠛⠋⠀⠀⣾⣿⣿⡆⠀⠀⠀⠀⢀⣠
⠀⠀⠀⣼⣿⡆⠀⢀⣿⣿⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣿⠀⠀⠀⠀⠀⠀⡀⠀⢻⣿⡿⠀⠀⠀⠀⠀⠐⠻⢧
⠀⠀⢰⣿⣿⣷⠀⣸⣿⣿⡇⣠⡄⣿⣿⣿⠀⢠⣾⡅⠀⠀⠀⠀⣿⠋⠁⠉⠀⢀⠀⠤⠤⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⢀⣿⢽⣧⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣏⣇⣼⣿⡃⠀⠀⠀⠀⠉
⠀⠀⡗⡐⠻⣿⠟⠉⠉⢉⣿⣿⣟⠁⠉⠉⣿⡿⣿⡇⢀⣤⣤⠠⠀⠰⡄⢶⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡀⢠⣀⡀
⠀⠈⠀⠀⠀⠀⠀⠄⠀⢼⡡⣿⣿⠀⠀⠀⠈⠀⠁⠀⠙⠚⢻⣿⣿⣿⣿⣿⣿⡟⠋⠋⠁⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠿⠻⠋⠉⠉⠀⠀⠀⠀⠀⠀⢀⣽⡿⠁⠀⠀⠀⠠⠤
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣮⣵⣦⣤⣮⣴⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡠⠴⠒⠒⠶⠖⠒⢤⠄⠐⠒⠒⠒⠂⠀⠀⠐⠒⣠⣄⢀⡤⠖⠒⠒⠒⠒⠒⠒⠒⢦⣤⡀
⠀⠀⠀⠀⠀⠻⠤⡀⣀⡀⣐⠤⠖⠚⠛⠋⠀⠀⠠⣀⡀⢀⡀⠀⠀⠀⠈⠉⠉⠉⠩⠽⢯⣄⣀⣀⣀⠀⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⢃⡏⢙⡧⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⣻⣦⠶⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠘⠦⢤⠀⣠⣄⠀⢀⠀⠀⠀⠑⠦⢤⣄⢠⣶⠶⣸⢟⠀⠀⢲⢸⣿⡞⡏⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣶⢏⠘⡇⡟⠀⠀⠀⠀⠀⠀⠹⢸⡇⠀⣏⣌⠀⢠⡿⣼⣿⡇⣁⣸⠀⣇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣿⣷⡆⢠⠖⠀⠀⠀⠀⣾⡁⠀⣼⣿⠀⢸⣇⢻⢸⠀⢻⡜⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⡏⣿⡇⢸⡆⠀⢰⡄⠀⣿⡇⠀⡿⢹⠀⣿⣿⠀⣾⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣷⢸⣿⠀⣼⡇⢠⡇⡇⢰⡇⢸⢘⣿⣿⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣲⢿⢸⢸⣿⠀⣿⠇⢸⡇⡇⣸⡇⣷⢸⣿⣿⡀⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⢸⣿⡇⣿⢰⢸⣧⡇⠏⡇⣇⢸⣿⣿⡇⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⢸⡇⡇⣿⢸⠸⣷⡷⡆⡃⡟⢸⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⣇⢹⣼⠀⣿⣿⡄⠀⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢸⡇⣿⣾⣿⡆⣿⣿⡇⠀⡿⢻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢸⡇⣿⢻⢿⣧⣿⣿⡇⠰⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢃⡟⣼⣿⠷⠿⣿⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠛⢻⠀⠀⣿⠇⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡆⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⢠⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠠⡄⢣⢠⠁⠀⠀⠀⠀⠀⠠⢬⡄⣰⣾⠁⠀⠀⠀⠀⠀⠀⡄⠀⡨
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠋⠀⠠⠀⠀⠀⠀⡂⢤⡄⢄⡦⢄⠀⠀⠀⠀⠈⠀⠘⣔⣅⠤⠆
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⠀⠬⠭⠕⢘⢛⢻⠅⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢃⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠂⠴⢶⡋⠁⠀⠀⠀⡠⠀⠀⠀⠀⠀⡠⠌⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢀⠔⢏⠀⠀⠈⡼⠑⡜⠀⢀⡠⠊⠁⠀⠀⠀⠀⠁⠀⠱⠚⠒⠎⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠢⠀⠀⠐⠀⠀⠠⢀⡡⠀⠈⡇⠀⠉⠐⠉⠀⣴⠊⠁⠀⠁⠀⢄⣀⢀⢀⡂⠐⠶⠚⠲
⠀⠀⠀⠀⠀⢴⣤⣨⡦⠐⠶⠜⠶⣖⠂⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⣈⣉⢁⣉⡑⠶⠶⠴⣴⡄
⠀⠀⠀⠀⠠⣖⢤⠤⣴⠲⠶⠻⠔⠴⠄⠀⠐⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠈⠈⢉⠉⠉⣶⣄⢠⣤⣄⠴
⠀⠀⠀⠀⢐⣠⣀⣤⣬⢖⢰⣠⢤⠠⣄⠐⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠐⡋⠉⠉⠉⠙⠮⣌⠠⡀⢈⠁⠀⠀⠄
⠀⠀⠀⠀⢄⣠⡠⣄⣬⢅⡀⢀⠀⡀⣀⡠⠀⠀⠈⠂⢄⠈⠁⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠛⠓⠓⠘⠓⠓⠦⣿⡆
⠀⠀⠀⠄⠄⢀⣀⢀⠍⡉⠛⠙⠛⠉⠛⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⡆⠀⠀⠀⡾⡇⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠠⣇⠀⠈⠠⠁⢺⡆⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠠⠀⠀⡁⠀⠀⠀⠀⡌⠉
⠀⠀⠀⠀⠀⠠⣁⠀⠀⠀⠤⠆⡗⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠬⠀⠀⠀⠀⠉⠤
⠀⠀⠀⠀⠀⠀⠉⠓⠶⠀⠐⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠄⠤⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠁⠒⠁⠛⠀⠀⠀⠐⠀⠠⠀⠀⠈⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠟⠿⠶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠟⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⢀⣤⡶⣶⣦⡾⠟⢿⣦⣀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⣾⠟⠿⣦⣴⠶⣦⣄⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⢠⣿⠁⠀⠀⣉⣤⣶⠾⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⢷⣦⣌⡁⠀⠈⢻⣇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⢿⣤⣴⠿⠛⢋⣠⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣤⡈⠛⠻⢶⣤⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⣄⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠘⢷⣄⠀⠀⣿⠁⠀⠀⠀⠀⢀⡀⠀⢾⡇⠀⠀⠀⠀⠀⠀⠙⠃⢀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⣿⠀⠀⡀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣴⠟
⠀⠀⣸⡟⠀⢠⣿⠀⠀⠀⠀⠀⣾⠃⠀⠘⠧⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠠⠟⠀⠀⢻⡆⠀⠀⠀⠀⢸⡗⠀⠘⣿⡀
⢠⣿⠋⠀⠀⢨⣿⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡗⠀⠀⠈⢻⣧
⠈⠻⣦⡀⠀⠈⣿⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠀⢹⡄⠀⢸⡇⠀⢸⡇⠐⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⢸⡇⠀⠀⣠⡾⠋
⠀⢀⣾⠃⠀⠀⣻⣧⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⣤⣤⡴⠾⠋⠀⠀⢸⡇⠀⢸⡇⠀⢸⡇⠀⠈⠻⢶⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⠀⣠⣠⣿⡃⠀⠀⢻⣆
⣴⡟⠁⠀⠀⣼⠟⠉⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠸⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠙⢿⡄⠀⠀⠙⣷
⠙⢷⣄⠀⠀⣿⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢸⡿⠀⠀⠀⣸⡇⠀⠀⣴⠟
⠀⣸⡟⠀⠀⢻⣆⠀⠀⠀⢻⡇⠀⣠⠾⠛⢛⣿⠇⠰⣶⣶⣾⣿⠷⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣶⣶⡶⠀⢿⣟⠛⠻⢦⡄⠀⣾⠇⠀⠀⢀⣿⠃⠀⠘⣿⡀
⠀⣿⡁⠀⠀⠀⠻⣦⡀⠀⢸⣧⢸⣇⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢀⣿⠀⣿⠀⠀⣠⡿⠁⠀⠀⠀⣹⡇
⠀⠈⠻⢷⣦⠀⠀⠈⠻⠷⢾⣧⠀⠙⠷⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⠟⠁⢈⣿⠾⠟⠋⠁⠀⢠⣶⠟⠛
⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠈⣿⠀⠀⠀⢠⡶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣦⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⣿⡁
⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⢻⣧⡀⠀⣿⣀⣿⠀⠀⣠⡶⢦⣄⣀⣤⣄⡀⠀⠀⣀⣤⣄⣀⣤⠶⣦⡀⠀⢸⣇⣸⡇⠀⣀⣿⠁⠀⠀⠀⠀⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣦⣌⡉⠁⠀⠀⠉⠀⠀⠙⠋⠁⠈⠛⠛⠛⠋⠀⠉⠙⠁⠀⠈⠁⠀⠀⠉⣉⣤⡾⢿⣥⡀
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠉⠛⣷⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡿⠋⠁⠀⠀⢹⡇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠈⠛⢿⣦⠀⠀⢀⣿⠀⠀⠀⢀⣾⣿⡛⠿⠟⠉⢻⡿⠛⠿⠛⢷⣶⣶⠿⠿⠛⠛⢛⣿⠋⠉⠀⠛⣿⣦⠀⠀⠀⢿⣇⠀⠀⢠⡾⠛⠋
⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢸⣏⠀⠀⠀⣿⣯⠉⠁⠀⢀⣴⡾⠿⠿⠶⠾⣿⣯⣄⣀⠀⠀⠀⣼⡟⠀⠀⠀⠸⣿⡀⠀⠀⠀⢸⣿⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠛⣷⡀⠘⣿⣉⠀⢠⣶⠟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣿⣧⣄⣀⠀⣾⡟⠁⠀⠀⣰⡟⠁⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠈⠻⣶⡄⠀⠀⣿⠁⠀⢼⣯⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠃⠀⠀⠀⣿⠇⠀⠀⣴⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠹⣧⠀⠀⠉⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠉⠛⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣶⣦⡀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣠⣴⠾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡧⠀⠀⠀⠈⠙⠻⠷⢶⣦⣦⣴⣶⠾⠛⠉⠙⠻⠿⠶⣶⣤⣤⣄⣀⣀⣀⣠⡾⠋⠀⠀⠀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⢸⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣀⠀⠀⡰⠾⠀⢺⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢬⢺⣁⠀⠀⣰⠀⠀⠀⣂⢠⣭⢀⣶⣠
⣤⣤⢧⢰⣆⡋⣼⣶⠦⢶⡶⢧⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⠺⣾⢅⡀⠀⢺⡦⢀⢐⣏⡷⡎⢻⢼⣞
⣿⣿⠛⡿⠊⡓⠚⣧⡇⣼⣡⠚⣶⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⣣⣇⡃⢺⡿⠬⠁⠘⣻⡧⠑⢈⣿⠞⡓⠛⡿⣴
⣿⣥⣷⠓⠋⢸⡏⠙⣿⣿⣧⣾⣿⣦⣿⣷⣷⣴⣶⣦⢤⣤⠒⣧⠯⢝⣄⡀⣀⢠⣌⡠⢥⠘⢄⢴⣯⣿⣦⣿⣈
⢸⣿⣟⠉⡱⢋⠙⣟⢻⡿⡿⣿⢻⢿⣿⣿⠿⠿⢶⣿⣿⡯⠔⣣⠘⢦⠷⣿⢇⡨⡚⣧⢉⠄⣍⢸⣿⣿⣿⠿⢿
⠘⠹⢏⡨⠦⠈⠓⠛⣍⡁⠀⠸⠀⠀⠈⠀⠀⠀⠀⠀⠩⠃⠁⠭⠀⠉⠠⠓⠐⠀⠀⠠⡇⠘⠀⠓⢊⣃
⠀⠐⠒⠀⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠂⠂⠠⠤⠤⠤⠤⠀⠄⠀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣌⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣩⣶⣤⠠⠄
⠀⠀⠀⠀⠀⣀⣰⡀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠀⠈⠡⠔⠂⠀⡴⠆⢞⠿⡿⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡠⠤⠤⠤⠤⠷⠾⠳⠲⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠠⠴⠶⠛⠡⠈⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿