Friendship love Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡤⠞⢋⣩⡥⠤⢤⣌⡉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢉⣩⡤⠤⣭⣍⡙⠲⣤⡀
⠀⠀⠀⠀⣠⠋⢀⡴⠋⠀⣀⣤⣤⡀⢹⡄⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢠⡏⢀⣤⣤⣀⠈⠙⢦⡀⠹⣄
⠀⠀⠀⢀⡏⠀⡾⠀⢠⡞⣡⠶⣶⠿⠾⢷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡾⠷⠿⣶⠶⣌⢳⡄⠀⠷⠀⢹
⠀⠀⠀⠀⣇⠀⣧⠀⣾⢠⣏⡤⠞⠛⠛⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠟⠛⠛⠿⢤⣹⡄⣧⠀⣴⠀⣸
⠀⠀⠀⠀⢹⡄⢹⡄⢹⣼⠋⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⠀⠀⠀⠀⠙⣧⡇⢠⡏⢠⡟
⠀⠀⠀⠀⠀⢻⡄⢷⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⡾⢀⡞
⠀⠀⠀⠀⠀⠀⢻⡈⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢃⡟
⠀⠀⠀⠀⠀⠀⠀⡇⣿⢻⡀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⢀⡟⣿⢸
⠀⠀⠀⠸⡗⢒⣿⡵⠃⠀⡿⣄⡀⠀⠀⠀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⠀⠀⢀⣠⣾⠀⠘⢮⣓⡒⢺⠇
⠀⠀⠀⠀⠙⠶⠤⠤⠶⠛⠁⠈⣿⠛⠛⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⠛⠛⣿⠁⠈⠓⠶⠤⠤⠄⠋
⠀⠀⠀⠀⣤⣄⡴⠖⠒⠒⠒⠒⠛⠒⠒⠛⠒⠒⠒⠒⠶⣄⡀⠀⣠⠶⠒⠒⠒⠒⠛⠒⠒⠛⠒⠒⠒⠒⠲⢤⣠⢤
⠀⠀⠀⢸⡁⠈⢷⠟⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⠹⡾⠁⢈⡇
⠀⠀⠀⢸⠳⣦⡏⠀⠀⠀⠀⠈⠉⠛⠆⢠⣄⣠⡤⠶⠋⠁⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠀⠀⢹⣴⠛⡇
⠀⠀⠀⢸⡆⣿⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⣿⢰⡇
⠀⠀⠀⠀⠙⣿⠀⠀⠀⠉⠛⠲⢦⣄⣀⠀⠀⠀⠀⠀⣀⣠⡤⢶⣏⣀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠀⠀⠀⠀⣿⠋
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠚⠋⠉⠀⡇⢸⠀⠉⠙⠓⠒⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠛⠦⣤⣄⡀⣀⣀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠞⠃⠘⠷⢤⣤⣀⣀⣀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⠛⣻⣿⣿⣿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⢞⣻⣿⡙⠳⣦⡀⠀⠀⠀⠀⠀⠈⠉⠋⢁⣼⠏⠻⣟⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⢛⣟⠁⣸⣧⣤⣬⣉⣙⠛⢦⡀⠀⠀⢠⣴⡿⠿⣶⣾⣿⣷⣾⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠻⠞⠋⣹⡷⠟⠀⠀⠀⠈⠉⠳⣦⠹⣆⣰⡿⠉⠀⠀⣠⡀⠉⠉⠉⠉⣿⣿⠛⠛⣻⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣤⡀⠀⠈⢻⣿⡟⠀⠀⠀⠀⠙⠁⠀⣤⡄⠀⣿⣿⣷⢴⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠏⠀⠀⢾⠆⠀⠉⠁⠀⠀⠀⣿⡇⠀⣼⠃⠀⠀⠀⠀⠉⠁⠀⣿⣿⣅⣠⣿⠟⢻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢀⡀⠀⠀⠀⢸⡆⠀⣹⣧⡀⠘⣦⣄⣀⣠⡾⠃⠀⠀⠘⢿⣿⠇⠀⣴⣾⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠘⢷⣄⡀⠀⠘⢷⣤⣤⣤⠾⠁⣠⣿⠉⠻⣦⣀⡉⠉⠉⠀⠀⠀⠀⣠⡾⠋⠀⠀⢿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠉⢻⣦⣀⡀⢀⣀⣀⣤⡾⠋⣿⠀⠀⠀⠉⢻⠛⢷⡶⢶⡾⠿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⡟⠋⠛⠻⢿⣯⣍⣁⠀⢠⡿⠷⠶⠶⠶⢾⣇⣼⣷⣫⢆⡄⠘⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠉⠛⠻⠿⢷⣦⣤⣤⣶⠿⠛⠉⠀⠑⠣⠾⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⡾⠏⠀⠀⢘⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣤⣀⣀⣀⣀⣤⡄⠚⠉⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⢤⣴⠞⠻⢭⠽⢿⣦⡀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠈⠉⠉⢀⣀⣀⣤⣀⠀⢤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠠⡭⠏⠀⠹⢿⣄⡀⠀⠀⢀⣼⡿⠛⢿⡛⠛⠛⢋⣿⠟⠁⠈⢻⡇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠻⢧⡀⢀⣀⣤⣽⠿⢶⣴⡾⠟⠁⣀⣤⡿⢦⡶⠟⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⡁⠀⠀⠈⠉⠉⣀⣀⣀⣤⣤⣴⡶⠛⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠦⠴⠚⠋⠙⠛⠋⠉⠉⠉⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀
⠀⠀⠀⢀⣤⡶⠛⠉⠈⣷⠀⠀⠀⠀⢀⣠⣤⠶⠛⠁⠀⠀⠈⠙⠻⢶⣦⣤⣀⣀⠀⠀⠀⣿⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⢠⡶⠛⠁⣀⣀⣀⣠⠾⣧⣤⣤⣾⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣾⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄
⢰⣏⣀⣀⣠⣿⣿⣤⠿⠲⢤⣀⠀⠉⠙⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⠙⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⣤⡿⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⠛⠷⣦⣤⣀⡀⠀⠀⢸⡇⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣴⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⣿⠉⠉⠛⠛⠿⠿⠇⠀⠀⠘⠛⠿⠿⢿⡿⠿⢿⣿⡿⠛⢻⡟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠸⣿⣷⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡿⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣀⣯⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⣠⣷⣿⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠈⠹⢇⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣉⣭⣼⣿⣧⣴⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⢤⣄⡉⠉⠙⢦⡀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠁⠈⣉⣥⡶⢿⣤⣶⠾⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠦⣤⣠⣿⠀⠀⠀⠀⠘⢷⣤⣀⣠⡾⠛⠉⠁⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⢦⣤⣤⣤⠶⠞⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⠀⢀⠄⠎⠀⠈⠁⡆
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠀⠀⠀⠉⠢⢣⠀⠀⠀⠀⠐⠉⡄⠀⠀⠀⠀⠀⠀⢸⣦⣀
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠰⠤⣀⣀⣀⣀⠅⡜⠀⡀⠀⡄⠐⢹⠀⠀⠀⠀⠀⠀⢠⠿⣿⡏⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⢠⠀⠀⢂⠀⡀⠀⢀⡔⠀⠓⢤⣼⣯⣄⠤⠊⠀⠀⠀⠀⠀⡠⡃⠰⠉⢧⣴⣶⣦⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠃⢀⡞⠀⠀⣀⢀⠄⠀⠀⠰⣿⣿⡟⠀⢀⣤⡀⠀⠀⢰⠁⠀⠘⢶⡜⣿⣿⣿⡿⠹
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡶⣶⣿⡄⠀⠀⡀⣿⡿⠀⠀⢈⣿⡇⠀⠀⠀⢄⠀⠐⠁⡇⠘⢿⠟⡡⠁
⠀⠀⠀⠀⠀⠀⢠⠓⢢⠉⠉⠀⠅⠁⠳⠠⠎⠀⢽⠇⠀⠀⣼⣿⣇⠀⠀⠀⠀⠈⣤⣴⣷⠒⠈⠉
⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠀⠀⠀⠀⠈⡀⠀⡠⠃⠀⠀⢸⠛⠿⢛⠄⠀⠀⣠⡟⠋⠁⠸
⠀⠀⠀⠀⠀⠆⠀⠀⠎⠀⠀⢀⡄⠀⠀⢓⠈⠀⠀⠀⠀⢸⣄⣀⡸⠤⣠⠊⠀⠈⡆⠀⠀⡇
⠀⠀⠀⠀⡜⠀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠘⡀⠀⠀⠀⠀⠀⠙⢻⡁⠀⠉⠀⠀⢀⠇⠀⠀⢰
⠀⠀⠀⣤⠉⠀⠴⠁⠀⠀⠀⠀⢸⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡀⠤⠊⠀⠀⠀⠀⢆
⠀⠀⠀⠛⠶⠎⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⠷⠶⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠘⠤⠤⢀⣀⣀⣀⡀⣸⣀⠠⠤⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣠⣤⣤⡤⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⢉⡁⢠⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⢷⣶⡜⠻⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣽⣿⣽⡿⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠫⠳⢖⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣀⢡⣔⣤⣔⠀⢀⡱⢤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⡰⣡⠏⠐⠉⠉⢦⢀⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠊⠀⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠘⡀⠈⣷⡀⠀⣀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠙⢦⣾⣷⠞⠱⣶⣿⡹⣿⣿⣿⣿⣿⣿⣿⣿⠏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠈⠩⠙⠂⢐⡉⡽⡋⠉⠛⠿⠻⠛⠛⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⠀⠀⠈⣧⠠⠠⠄⠀⠠⠇⠤⠀⢠⠨⠠⠄⠤⠄⠀⠀⠀⠀⢵⠀⠀⠠⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠈⠉⠉⠁⠀⠀⣀⡠⠛⢇⠀⠀⠀⠀⠀⠀⣀⣀⠚⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠤⣀⠴⠤⠚⠉⢂⠀⠈⠉⠉⠉⠉⠉⠉⠅⠈⡆⢐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠈⠤⢤⡤⠤⠤⠤⠒⠒⠑⠈⠁⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⢀⣀⣀⣀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠈⠁⠐⠀⠤⠠⠤⠔⢋⡄⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠈⠀⠀⠀⠠⠬⢁⠒⠒⠒⠉⡒⠈⠉⠡⠉⠐⠒⡒⠒⠒⠒⠒⠚⠋⠉⠉⠉⢉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⣮⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣀⡒⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠘⠍⠀⠀⠢⠘⠤⠘⠐⠕
⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢀⣀⠻⠟⣀⡀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⢠⡄⣴⣶⣶⡄⢠⠀⠀⠀⠀⠀⣶⠀⠀⡆
⠀⠀⠠⣿⣿⣷⣞⣻⣿⡆⠀⠀⠀⢀⣨⣿⣅⡀⠀⠀⠀⢸⣇⣈⠻⠋⣀⣺⠀⠀⠀⠀⣄⣿⡷⣿⡇
⠀⠀⠀⠹⣿⣇⠀⠀⠈⠀⠀⠀⠀⣾⠋⠛⢙⡇⠀⠀⠀⠀⠙⣿⣷⣿⡿⠁⠀⠀⠀⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠈⠛⠀⠛⠁⠀⠀⠀⠀⠀⠘⠋⠛⠃⠀⠀⠀⠀⠀⠀⠙⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⣾⣦⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣆
⠀⠀⠀⠀⠀⠀⣤⣤⣻⠀⠀⠀⢠⣿⣧⡀⣶⣷⡆⠀⠀⠀⣴⣦⠀⣴⣶⢦⠀⠀⠀⠀⢠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⡇⢿⣿⠇⠀⠀⠈⣻⣿⣁⢻⣿⡟⡀⠀⠀⢀⡟⣙⣋⢿
⠀⠀⠀⠒⠛⠊⠉⠁⠀⠀⠀⠀⠈⠹⠋⠀⠘⠟⠀⠀⠀⠺⠿⠿⠟⠿⠿⠿⠀⠀⠀⠈⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⣶⠠⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢐⡷⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⣴⣷⣶⣾⣾⡷⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⢾⣿⠆⠰⣿⡷⠀⠀⠀⠀⢳⣼⠟⠛⣿
⠀⠀⠀⠈⠻⠽⠿⠟⠁⠀⠀⠀⠀⠀⣿⢿⣿⠀⠀⠀⠀⢰⣿⣿⡿⣶⣿⣷⠄⠀⠀⠀⠀⠻⣷⣾⠏

⠀⠀⢀⣶⣶⢦⣴⢾⣶⡀⠀⠀⠲⣬⣟⣀⣻⣤⠖⠀⠀⢀⣴⣶⡄⣠⣤⡆⠀⠀⠀⠀⢰⣶⣶⡆
⠀⠀⠀⢿⣿⣿⣿⣷⡿⠁⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⢸⣿⡿⣧⢾⣿⠆⠀⠀⠀⢠⣀⠙⣋⣠⣀⡀
⠀⠀⠀⠀⠙⠻⠟⠋⠀⠀⠀⠀⠀⠀⠿⠿⠿⠀⠀⠀⠀⠰⠿⠿⠿⠷⠷⠿⠀⠀⠀⠀⠉⠻⠿⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣧⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⣠⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠹⣿⣿⣿⣿⠁⠙⢿⣿⣿⡟⢿⣿⡇⠀⣾⣿⡿⠉⠉⠛⠻⠿⠿⠿⠟⠛⠛⢻⣿⣿⡆
⠀⠀⡤⠤⡄⠀⠀⣤⢿⠁⢀⡤⠄⡀⠀⠀⠀⢀⡀⠉⠀⢸⣿⠁⠀⢿⣿⡇⠀⣀⣄⠀⠀⠀⠀⢀⣀⠀⠀⣿⣿⠇
⠀⠀⢃⢠⠀⠀⠀⡈⢹⠆⢿⣿⡆⠀⠀⠀⠠⣿⣦⠀⠀⢠⡫⢛⠀⣼⣿⠃⣈⡅⢀⠀⠀⠀⠀⠈⣉⠁⠀⣿⡏⠀⠀⠀⣄⣄⡀
⠀⠀⠸⠀⠄⠀⠀⠀⠒⠂⠀⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⢨⡀⠌⠀⡉⢻⠀⣿⣷⠀⠀⠀⠀⣼⣷⡀⠀⠀⡿⠼⡆⠀⢤⡍⠛⡷⠃
⠀⠀⠀⡀⠠⠀⠀⠀⠀⠁⠀⠀⠀⡄⠤⠤⢤⠀⠀⠀⢀⠃⡄⠀⠀⠑⠚⠀⠈⠉⠀⠀⠀⠀⠈⠙⠁⠀⢨⣁⠠⠀⠀⠀⡁⡆
⠀⠀⠀⢁⠀⢃⠀⠀⠀⠀⠢⢄⠀⠀⠁⠈⠁⠀⠀⡠⠊⡇⠰⠀⠀⠀⠀⠇⠀⠀⢠⣄⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠁
⠀⠀⠀⠈⢀⢀⠜⠂⠂⢄⡠⠠⠈⠶⠠⠤⠠⠆⢁⡀⠀⢠⠀⢃⠀⠀⠀⠈⠢⣀⠀⠙⠛⠊⠁⠀⡀⠄⠀⠀⠀⠀⠠⠁⡘
⠀⠀⠀⠀⠀⠣⠀⡀⠀⠀⠋⣽⠉⠑⠠⠄⠊⠑⠂⡤⣤⠀⠆⣀⠔⠀⠢⢠⠐⡀⢱⠒⠀⠐⠦⠉⣀⠀⢀⣀⡀⠔⠁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⠘⠁⠀⠀⠀⠀⠀⢘⠀⡃⠀⠉⠐⡧⢄⡀⠀⢠⠃⡌⠊⠔⠒⠪⠛⡋⠶⠊⠁⠀⢁⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⣄⠀⢀⠛⠠⠈⢱⠂⡜⠀⠀⠀⠀⠀⠀⢣⡀⢀⡀⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠉⠢⠸⠗⠚⠂⠀⣁⠄⡐⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⠎⢰⠆⠀⠈⠁⠘⣐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡈⠳⡄⠀⠀⠀⢦⠉⠁⠀⠀⠐⠒⠀⠀⠐⠀⠃⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡇⠀⠀⠀⠘⠀⠀⠀⠀⢀⡀⠀⣀⣀⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠁⠐⠒⠒⠀⠐⠒⠒⠂⠈⠉⠀⠀⠀⠀⡓⠠⠄⢀⣀⠀⠀⠀⠀⢀⣀⡀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⣀⠂⡀⠀⠀⠀⠀⠀⡆⠀⠠⡀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠑⠦⠄⢀⣀⡠⠀⠀⠰⣀⣀⣀⢠⠔⠁⠀⠰⠁⠢⢄⣀⡴⠁⠀⠐⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢠⠀⠀⢀⠃⠀⡠⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠀⠈⢄⡐⡙⠂⢄⠤⠤⣄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠃⠀⠀⠀⠀⠀⠀⠱⠀⠸⠀⠀⠀⠀⠉⠉⠀⢢⡀⠀⠸⡆⠀⠸⠅⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⠀⠀⠀⠀⠀⠀⠀⢐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⡃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠈⠀⠀⠀⠀⠀⠀⠀⠌⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠋⠸
⠀⠀⠀⠀⠀⢠⠧⢄⣀⡴⠃⠀⠀⠀⠀⠀⠐⢦⡀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠃
⠀⠀⠀⠀⠀⠈⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠫⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢂⡸

⣿⠿⡿⢿⠿⡿⠟⠿⢿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⠛⠿⠿⠻⣿⣿⠿⠿⠿⠿⠿
⣿⣾⣿⣿⣾⣿⣿⣾⣾⣷⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣶⣄⠀⠀⠀⠀⢰⣷⣶⡄⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀
⠀⠀⢀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠔⠛⠓⠀⠀⠀⢀⡿⢿⣿⣿⠀⠀⠀⣴⣿⣿⣿⠁⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⡆
⠀⢠⣼⣿⣿⣿⣶⣄⠀⠀⢀⣄⣠⣾⣷⣦⣄⣴⣾⣿⣿⣿⣷⡄⠀⣄⢿⣿⣿⡿⣧⣀⠀⠀⣴⣿⠷⢶⣄⠀⠀⠀⢀⠒⠛⠛⣛⣿⡇
⢀⣿⠻⡿⠿⢿⣿⣿⡆⠀⢾⢛⣿⣛⡿⡛⢻⣿⣿⣿⣿⣿⣿⡿⢠⣿⣞⣿⣿⣻⣿⣿⡀⣬⣿⣿⣿⣿⠉⢧⠀⠀⣰⠖⢦⣶⣽⣿⡇
⣾⣿⡀⠒⢂⣘⣿⣿⣿⢀⢾⡇⣾⣿⣷⠀⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣮⠽⡻⣿⣿⣯⣿⣾⣿⣿⣿⣷⣾⡂⠀⢸⡇⢠⣿⣿⣿⣿
⢿⣯⡗⡖⠒⠰⣿⡏⠁⢘⣿⣐⣛⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⠋⠈⠻⡟⠲⠒⠂⣿⡟⠋⣿⣿⣿⣿⣿⣿⠉⠁⢤⡾⢧⣿⣿⣿⣿⣿
⠘⣿⣅⣒⣟⣺⣿⡇⠀⢸⡈⠵⡶⠿⠿⠟⠁⣿⣿⣿⣿⣿⣿⠀⠀⢰⡅⠀⠀⠀⣿⣿⠀⠈⣿⣿⣿⣿⣿⠃⠀⠘⠃⠸⣿⣿⣿⣿⡿
⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟⠀⠀⠘⣃⡀⠀⠀⢸⣿⡄⠀⢻⣿⣿⣷⡟⠀⠀⠀⠄⠀⠈⠹⠟⠁
⠀⠀⣿⣿⠟⣿⣿⡄⠀⠀⠰⣶⣶⣶⣶⡆⠀⠛⣿⣿⣿⣿⠉⠀⠀⠀⣻⣿⡗⢚⣸⠛⠁⠀⢘⣿⢻⣿⡄⠀⠀⠰⣄⣀⣈⣁⣈⡀
⠀⠀⣹⣽⠀⢸⣿⡇⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⠃⡿⣿⠀⠀⠀⢸⣿⠘⣿⠃⠀⠀⠀⣿⣿⠁⢿⣿⡇
⠀⠀⣾⣿⠀⢻⣿⠁⠀⠀⠀⢻⣿⣿⣿⠀⠀⠀⠸⡿⢿⡟⠀⠀⠀⠀⢸⣿⡇⣿⡟⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠏⠁⠀⢸⠛⠃
⠀⠀⢻⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣼⠿⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⣧⡀⢿⡆⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⡆⡄⠀⣼⣀
⠀⠀⢸⣿⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⢿⡅⣏⣄⠀⠀⠀⠐⠛⢰⣿⠀⠀⠀⣶⣿⡄⠀⢘⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⣿⣿⣿⣷⣶⣶⣶⡶⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠈⠙⣿⣿⣿⣿⣧⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢹⡿⠡⠐⠂⠀⠀⠀⠘⠻⣿⣿⣿⡀⠀⠈⢛⣿⣿⣿⠟⠻⠟⠻⢿⡿⠿⠿⣿⣿⣟⠁⢸⣿⣿⣿⣿⡿⠛⠀⠈⠿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⢼⠅⠂⠀⣰⡄⠀⠀⠐⣨⣿⠟⣻⠵⠀⠀⢸⣿⡟⠡⠒⠀⠀⠀⠀⠲⠦⠀⢿⣿⡿⠃⣸⡿⠙⡛⡉⠀⠀⠀⠀⡛⠛⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠘⠛⠁⠀⠀⠀⠛⠋⠀⣇⠀⠀⠀⠸⢻⡇⠀⢉⣶⡀⠀⠀⢈⣽⡆⢸⠿⡂⢘⠙⡇⢻⣷⡄⠀⠀⠀⣾⣄⠀⠁⡯⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠰⠠⢀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠈⡇⠀⠘⠛⠀⠀⠀⠘⠛⠁⠬⠄⠁⠀⢱⣾⠀⠉⠁⠀⠀⠀⠈⠁⠀⢰⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⠀⠀⠀⠀⠀⠀⣴⣷⠀⠀⠀⠀⠀⠀⠤⠀⠀⠠⣄⢀⣠⠄⠀⠠⠀⠀⠀⣴⣿⣿⡆⠀⠀⠀⠐⠀⠀⠀⠀⣼⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⢂⡘⠇⠐⠀⡏⣀⣙⣿⣦⠀⠀⠀⠀⠀⠃⠃⠄⠀⠈⠉⠁⠀⠄⠀⠀⠀⠀⠈⢙⠛⠿⠶⣄⣀⠀⠀⣀⣤⣾⡿⠿⠗
⠀⠀⠀⠰⢾⡗⢾⠿⠛⠉⡅⠁⠀⠁⠁⠀⠈⡀⠟⠻⢿⡇⠿⠿⠋⣰⢀⣀⠸⠁⠀⠹⠂⣀⡦⠂⠉⠁⡤⠂⠀⠤⠄⠂⠃⠈⠀⠃⠄⠤⢄
⠀⠀⠀⠀⠀⡀⢂⠀⠀⡀⠇⠀⠀⠀⠀⠀⠀⠡⠐⡀⢀⠁⠀⠀⣰⠝⣿⠇⠀⠀⠀⠀⠀⠻⣧⡠⠄⠂⡐⠈⠀⠀⡆⠀⠀⠀⠀⠀⠆⡀⠀⠈⢀⡀
⠀⠀⠀⠀⢰⠀⠒⠒⠫⢤⠃⢢⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⢨⡁⣸⡏⠀⠀⠀⠀⠀⠀⠀⢿⡆⢀⠊⢠⣄⢠⠆⡅⡀⠀⠀⠀⠀⡥⡅⠦⣄⣤⠀⢂
⠀⠀⠀⠀⠀⠈⠁⠀⠶⠸⠒⠊⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⠠⠄⠀⣌⠩⣀⡀⠀⠀⠀⠐⣀⡨⢁⡀⠠⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣞⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠸⠂⠐⠀⠀⠀⠀⠀⠀⠀⠪⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠀⢀⠀⠀⠀⢀⡀⠠⠐⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣧⣦⠀⣿⣷⣶⣶⣦⣤⣴⣶⣶⣾⡇⠀⠀⠀⠀⢰⠀⠉⠀⠒⠒⠒⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⣶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⣦⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠪⠍⠛⡛⠛⠚⠛⠛⣛⠋⢭⠘⠀⠀⠀⠀⠸⣿⣿⣿⣿⠏⠈⣿⣿⣿⣿⡿⠀⠀⠀⠐⠉⡛⠿⠿⠿⠿⠿⠿⠿⠟⣋⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠩⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠉⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⣰⠀⠀⠀⠀⠀⠰⡀⣺⠀⠀⠀⠀⠀⠀⢠⡀⡄⠀⠀⠀⠀⠀⣄⢠⠀⠀⠀⠀⠀⠀⡆⢤⠀⠀⠀⠀⠀⢠⠄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠢⠀⠀⠀⠀⠀⠀⠃⠈⠄⠀⠀⠀⠀⠀⡰⠠⠀⠀⠀⠀⠀⠀⠠⠰⡀⠀⠀⠀⠀⢀⠒⠢⠀⠀⠀⠀⠀⠀⠒⠣⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠀⢀⡴⠂⠀⠀⠀⠀⠰⣤⡀⠀⠄⠀⠀⠀⡈⠀⢀⡤⠀⠀⠀⠀⠀⣅⠀⠈⡄⠀⠀⡠⠁⢀⣠⠆⠀⠀⠀⠀⠰⣄⠀⡨⡄
⠀⠀⠀⠀⠀⠀⠀⠑⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠀⠀⠀⠠⠛⠋⠀⠀⠀⠀⠀⠀⠈⠓⠣⠘⠀⠀⠑⠐⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠹⠄⠀⠀⠀⠉⠀⠀⡇⠀⠀⠃⠀⠂⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠈⠀⢀⡄⠀⠀⣠⠀⠀⠀⠀⠀⣀⡀⣀⠀⠀⠁⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⠁⣿⠀⠀⠀⠀⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣼⣧⣤⣤⣿⠀⣠⠞⣷⠀⣿⠀⣿⠀⣴⠦⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣿⠀⣿⠶⠋⠀⡟⢰⡟⢸⡇⠀⡿⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⡿⠀⠙⠒⠋⠀⠛⠀⠟⠈⠳⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠛⢳⡄⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⡉⠀⠀⣠⣤⢤⡄⢸⣿⣤⣤⣄⠀⢀⣤⡶⢦⡄⠀⣠⡴⢦⣄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⡍⠻⣿⣦⢠⣿⡇⠈⠁⢸⣿⠁⢸⣿⠀⣸⣿⠀⢸⡿⢺⣿⠃⠘⣿⠚⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⣽⣿⠈⣿⣧⣀⣠⢾⣿⠀⠸⣿⣄⡽⣿⣄⣴⠇⠘⣿⣧⣴⡟⠀⣿⣧⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠉⠉⠁⠈⠉⠀⠀⠈⠉⠀⠈⠉⢁⣀⣀⣀⣭⣥⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢠⣤⣤⣤⡴⢶⣶⣶⣾⣿⣿⡿⠇⣿⠛⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢘⣛⢹⢻⠏⡟⢡⡽⢰⡇⣏⣺⣐⣧⣭⣬⣾⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏⡤⢹⣷⡾⠶⠾⠟⠿⠛⠛⠛⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣾⣿⣿⣶⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⣦⣋⣀
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣾⣿⡿⠿⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣼⣿⡿⣋⡉⠉⠙⠛⣋⠉⢿⣿⣧⠀⠀⠴⠀⣿⡿⢿⣿⣿⡏⠹⣿⣿⡟⣿⡏⠀⢸⣿⡿⠀⠀⠀⠹⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣸⣿⠀⠁⣄⠀⠀⠀⠭⣅⢸⣿⡋⠀⢲⠐⢲⢾⠁⣴⡀⠄⠀⠀⣤⠀⠀⢿⣇⠀⢸⣿⠃⠒⠂⠀⠀⠘⢻⣿⣿⡟
⠀⠀⠀⠀⠀⢍⢻⠀⠰⠿⠁⠀⠀⠺⠟⢸⡡⠀⠁⢛⠄⠘⠨⡆⠙⠃⠀⠀⠘⠻⠃⠀⡏⡩⠀⠛⢧⢹⣷⠀⠀⠀⣾⡆⠈⢻⠛
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⣄⣀⣤⠀⠀⠂⠀⠀⢸⠘⡀⠀⠀⢀⠀⠀⠄⠀⠠⠀⠀⠰⠌⡄⠀⠈⢿⠀⠀⠀⠀⠀⡀⠀⠀⣾⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠉⠉⢁⠠⠊⠀⢀⡠⠀⣀⠐⣄⢀⠀⠑⠀⣀⠀⢀⡀⠔⠁⢀⠐⡀⠀⢹⣧⡀⠀⠙⠛⠁⢀⣴⣿
⠀⠀⠀⠀⠀⠀⠀⣔⡤⠙⠍⠁⠟⢨⣕⠈⠀⢑⠄⠀⠲⡀⠀⠈⢣⠐⠣⠤⠘⠵⠁⣄⠀⠢⡠⠁⠙⠿⡛⡳⠂⠀⠺⣛⣻⣿
⠀⠀⠀⠀⠀⢀⠄⣽⡇⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⣄⠀⠁⢂⠀⠂⢤⡀⠐⠀⠀⠀⠀⠀⠉⠻⡀
⠀⠀⠀⠀⠀⠔⢰⡿⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠆⢎⣲⣤⠄⠐⠀⠘⢁⠀⠀⠀⠀⠀⠀⡄⢀⢆
⠀⠀⠀⠀⡈⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠲⢞⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣧⡀⠂
⠀⠀⠀⠀⡇⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠢⢙⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⢸
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⣴⡦⢸⣿⣷⣶⣶⣶⣶⣶⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠂⠂⠤⠄⠤⠤⠀⠂⠃⢰⣄
⠀⠀⠀⠀⠈⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠄⠀⠀⠀⢸⠀⠀⠀⠀⣤⣄⣀⠀⠀⠀⠀⣀⣀⣴
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠟⠀⠀⠛⠟⠛⠋⠀⠀⠀⠀⠀⠀⠑⠀⠠⠐⠀⠈⠠⠄⠠⠈⠀⠀⠀⠀⠫⠝⠛⡻⠿⠿⠟⡛⠛⡍
⠀⠀⠀⠀⠀⠀⠀⡀⢰⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⢀⡂⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡃⡋⠀⠀⠀⠀⠘⠐⠃⠀⠀⠀⠀⠀⠀⠘⠘⠀⠀⠀⠀⠀⠃⢹⠀⠀⠀⠀⠀⠀⠗⠞⠀⠀⠀⠀⠸⠒⠇
⠀⠀⠀⠀⠀⠀⠀⠃⠇⠀⠀⠀⠀⠠⠐⢄⠀⠀⠀⠀⠀⢀⠔⠐⠀⠀⠀⠀⠀⡤⢄⠀⠀⠀⠀⠀⢀⠆⠄⠀⠀⠀⠀⢀⠤⡄
⠀⠀⠀⠀⠀⠠⢅⣀⡼⠀⠀⠀⠀⠰⣄⡀⠆⠀⠀⠀⢠⠆⣀⡴⠂⠀⠀⠀⠰⣄⠀⣡⠀⠀⠀⢠⠂⢀⡰⠀⠀⠀⠀⢠⡀⠈⣆
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠉⠒⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠑⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣶⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⠿⣿⣿⣹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠘⠻⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠛⠋⠙⠋⠙⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⠀⢸⣯⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠠⠀⠀⠀⠀⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡀⠈⠀⠀⠀⠀⢸⣿⣄⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠉⢇⠀⢠⣠⡄⠀⡜⠻⣿⣧⠀⠀⠀⠀⣴⡟⣽⣿⣦⡈⠐⢁⣴⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠈⠻⡟⠁⠀⠈⢣⣀⠀⣀⡜⠀⠀⠟⠋⠁⠀⠀⠀⠉⣹⠿⣏⣹⠋⠉⠉⠈⡯⡶⠗⢲⡄
⠀⠀⠀⠀⠀⠀⠀⡰⡒⠈⢳⣍⠉⠉⢡⣷⠆⠠⣄⠀⠀⠀⠀⠀⢠⡏⢸⡽⠢⠅⣒⣒⣋⡼⢹⡇⢸⢑
⠀⠀⠀⠀⠀⢀⠞⠀⡇⠀⠀⠙⠛⠛⠛⠁⠀⢠⠛⡀⠀⠀⠀⢠⠛⡇⢸⡇⠀⠀⠀⠀⠀⠀⠘⡇⠈⠀⢆
⠀⠀⠀⠀⠀⢨⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢣⠀⠀⢠⠃⠀⢳⢸⡇⠀⠀⠀⠀⠀⠀⠀⢃⢸⠀⠘⡄
⠀⠀⠀⠀⠀⠇⠀⠀⣆⠀⢰⠶⣤⡾⠓⣀⢀⡇⠀⠈⡆⠀⡇⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⠀⠀⢡
⠀⠀⠀⠀⠸⠀⠀⢰⡿⠘⠋⠋⣿⣟⠈⠉⠙⣇⠀⢀⡇⢸⡀⠀⢰⠁⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢀⣤⠂
⠀⠀⠀⠀⠘⠓⠲⣿⢀⣀⡀⠀⣿⣿⠀⠀⠀⠻⢶⠋⠸⡎⠀⠑⡏⠀⡸⠛⠿⡶⠶⡶⠶⠶⠾⠟⠻⠘⠃⠀⡇
⠀⠀⠀⣀⣬⠀⠀⡟⠉⠉⣷⢸⡧⢸⠀⠀⠀⢰⢰⠀⠀⡃⠀⢰⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠀⠀⡇
⠀⠀⢸⠀⢹⠀⠀⡇⠀⠀⢿⡘⠇⠘⠀⠀⠀⠘⠀⢇⢠⠁⠀⡌⠀⠀⡅⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⡏⠐⡄
⠀⠀⢸⠀⠘⡄⢀⡃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⠂⠈⢿⠀⢰⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠈⡇⢠⠁⢸
⠀⠀⠸⠀⠀⡇⠀⢑⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⡇⢀⠇⢀⣐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠻⣅⣀⠲⠎⠁⠘⠀⡇
⠀⠀⠀⠛⠈⡳⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⠯⠦⠿⠂⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢶⠐⠛⠚
⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢬⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠸⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼
⠀⠀⠀⠀⠀⠀⠈⠉⢲⡶⠞⠟⠳⡶⠶⢺⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠃⣿⠛⠛⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣿⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⣣⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⠀⠀⣨
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⢹⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢀⠃⠀⢸⡄⢸⠢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⢿⡗⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⣺⠀⠀⢸⣿⣿⣎⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠒⣿⣗⠺⢆
⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠛⡆⠀⢸⠋⠈⠛⡯⢧⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⢸⡄⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⣷⣤⣴⣶⠧⠀⢿⠶⠶⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⣾⣶⣶⠶⢾⠷⠶⠾⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠁⠀⠀⠉⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠉⠀⠀⠈⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠀⠀⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣷⡿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣾⠿⠟⠋
⠀⠀⠀⠀⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣄⡀
⠀⠀⠀⣠⣾⠿⠛⠙⠻⢷⣶⣶⣶⣾⠿⠛⠙⠻⢿⣦⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⢴⡿⠟⠋⠛⠿⣷⣶⣶⣶⡾⠟⠋⠛⠿⣷⣄
⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⢀⣴⡿⠛⠉⠉⠛⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢸⣿⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣿⡇
⠀⠀⢸⣿⠀⠘⠛⣿⡇⠀⠀⠀⠀⠀⠀⣿⡟⠛⠀⠀⢸⣿⡀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠻⢻⣿⠀⠀⠀⠀⠀⠀⢸⣿⠻⠃⠀⣿⡇
⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠙⢿⣶⣤⣤⣶⡿⠋⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⢸⣿⣀⣀⣀⣿⣧⣀⣀⣀⣀⣀⣠⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣀⣀⣀⣼⣿⣀⣀⣀⣿⡇
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⣠⣶⡿⠿⢿⣦⣄⣀⣀⣠⣴⡿⠿⢿⣶⣄⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⠀⠉⠙⠛⠛⠋⠉⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠠⣶⣶⣦⠀⠀⠀⠀⠀⠀⣴⣶⣶⠄⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⣶⣠⣴⣶⡆
⠀⠀⠀⠀⠀⠀⢀⠠⠠⣀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⢡⠀⠀⡈⠀⠀⢠⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⢸⣿⣿⡇⣀⡀⠀⠉⠀⠈⣁⣈⠉⢻⣿⠀⠀⠀⠀⢿⣿⠟⠉⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠡⠀⠘⣿⡿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢄⠀⠀⠀⢸⡏⠀⠤⠄⠀⠀⠀⠀⣤⡀⢈⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠐⣜⠻⡅⠀⠔⠒⠀⠀⠀⠀⠐⠂⠀⢰⠋⡈⠀⠀⠲⢿⠀⣶⡶⠂⠀⠀⠀⣠⡄⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠘⠦⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⡀⠀⠀⠂⢸⠀⠻⠟⠀⠀⠀⠸⢿⠇⠀⢸⠗⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡨⠤⣄⠀⠀⠰⣶⣶⣾⠇⠀⢀⡾⠛⣃⡀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠘⠆⢄⡀⠀⠈⢀⡀⠐⠁⠠⠠⠈⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠈⠉⠁⠈⠀⠓⠀⠄⡀⠄⠑⠀⠀⢀⣀⣀⣀⡁⢂⣀⠀⠄⠐⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⡂⠄⠀⣽⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠈⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠠⠶⠀⠈⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠿⠿⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠐⠒⠂⠒⠀⡤⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⠂⢌⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠑⠌⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠚⠹⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠤⠄⠀⠀⠀⠀⠤⠄⠐⡎⠀⠀⠀⠀⠀⢸⠻⠿⢿⣿⣿⣿⣿⡿⠿⠿⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⡀⠌⠀⠀⠂⣀⠀⠀⡄⠀⠀⠀⠀⠀⠃⡄⠀⣀⠔⠀⠈⠄⡀⠀⢠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠠⡆⠀⠀⠀⠀⠀⠀⢰⠠⠃⠀⠀⠀⠀⠀⠠⠦⢴⠀⠀⠀⠀⠀⠀⡄⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠆⠲⠀⠀⠀⠀⠀⡀⠒⢂⠀⠀⠀⠀⠀⠀⠐⠚⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠈⣢⠀⢀⡌⠀⢀⡤⠀⠀⠀⠀⠀⠀⢦⣀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠂⠀⠀⠀⠂⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠐⠁

⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡖⠺⠞⠁⠰⠿⠧⠤⡤
⠀⠀⠀⢀⣀⡰⢲⠿⡴⠻⡷⡆⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢔⠖⠂⠀⠀⠠⠀⠰⠀⠀⠀⠻⣶⠄
⠀⠀⢠⡞⣋⠕⠁⠊⠉⢀⡉⠙⠦⣹⢿⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⠴⠚⠙⢦⠤⡎⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⠾⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⣿⡀⠀⢀⡴⣺⠋⠀⠱⣤⠟⠁⣀⣀⣠⠞⠛⡇⠀⠀⠰⣀⠀⠀⢀⡰⠃⠀⠀⣸⠽⠀⣴⡾⣹⢫⠾⡏⢹⡴⡶⠶⡄
⠀⢇⡇⠀⠀⠠⡀⠀⠀⠀⠀⢀⡞⠀⠀⣿⠞⢀⠞⢰⣃⣠⢤⠖⠛⠛⠉⠁⠀⢳⡀⠀⠱⡀⠀⠀⠀⠉⠉⠁⠀⠀⢀⡰⠃⠀⢰⡏⠴⠃⠀⠀⠉⢀⠁⠘⠜⣷⡄
⠀⠀⢳⠀⠀⠀⠳⣄⣀⣀⣠⠞⠀⠀⢠⣇⠀⢸⠋⠉⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⣧⡀⠀⠈⠢⢄⡀⣀⣀⣀⡠⠴⠋⠀⠀⣔⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⣧
⠀⠀⢀⡷⣄⡀⠀⠀⠀⠀⠀⠀⣠⣴⢏⠈⢣⢾⡀⠀⠘⣆⠀⠀⠀⠀⡼⠁⠀⢀⡟⠃⠀⣠⣞⣹⣦⣥⣼⡏⢑⣦⣄⠀⠀⢸⡇⠀⠀⢢⠀⠀⠀⠀⠀⡰⠃⠀⣯
⠀⠀⠀⠐⠡⢯⠙⢲⠒⠒⢲⡋⠁⠈⠃⠀⠈⠉⢧⡀⠀⠈⠓⠒⠒⠋⠀⠀⣀⠞⠀⠐⡏⢷⣈⡋⣉⣀⣀⣹⣾⠏⡼⠁⣴⠁⠳⡀⠀⠀⠙⠒⠂⠒⠊⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣾⣀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⣀⣤⣤⠖⠊⠁⠀⠀⠀⢸⢦⣽⠿⠿⠿⠿⠿⡿⠴⢧⠀⠧⣠⠴⢻⡶⢄⣀⣀⣀⣀⡤⣶⢯⠸⠄
⠀⠀⠀⠀⠀⢠⠎⡟⠋⠀⢶⡿⡟⢆⠀⠀⠀⠀⠀⢀⡠⠖⡽⠀⠘⣇⠙⠲⡄⠀⠀⢀⠏⢠⣧⣤⣤⣴⣶⣶⣿⡀⠈⢧⠐⠃⠀⠘⣁⣼⣿⣿⣿⡈⠱⢼⠬⣓⠦
⠀⠀⠀⠀⢀⡏⢀⡟⠀⠀⠀⠀⢹⡈⢧⠀⠀⠀⠀⢸⡀⢠⠃⠀⠀⠸⣄⢠⡿⣄⣠⠎⠀⣞⣉⣉⣭⠭⠭⣥⢤⠷⠀⢸⡀⢀⡴⠚⢹⣿⣿⣿⣿⡟⣄⠈⠂
⠀⠀⠀⢀⡜⠀⢸⠁⠀⠘⠟⠀⠀⢧⣈⣧⣀⣄⡴⠋⠷⡏⠀⠀⠀⠀⢻⠉⠀⢸⣿⠯⠁⢸⣥⠋⠀⠀⠀⠘⣾⠀⠀⠛⢻⡇⠀⢠⣿⣿⣿⣿⣿⡇⠈⣦⣀⣀
⠀⠀⢠⠾⠳⠄⡾⠿⠀⠀⠰⠷⠀⣘⣾⠟⣷⡇⠀⠀⡰⠁⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠈⠿⠟⠆
⠀⠀⠀⠀⠀⠀⡇⠐⢷⣀⣀⣠⡼⠿⠛⠈⠙⠀⠀⠀⡽⠋⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠉⠉⣯⠁⠀⠘⡆⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢠⠀⠀⠀⣧⠀⠀⠀⠀⢰⠇⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠸⠿⠿⢿⠛⢿⡿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠘⠒⢻⠊⠙⠓⡞⠉⠁⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠘⠒⠲⡖⠚⠦⠲⡖⠋⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⠢⢄⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠹⣀⣀⠀⠀⠀⠀⠀⣀⡤⠃⠀⠀⠀⠓⠒⠤⠀⠀⠀⠀⠀⢀⡜⠀⠀⢸⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣀⡤⠤⣄
⠀⠀⠀⠀⢀⡴⠒⠓⢶⣴⠶⣄⠀⢀⠀⠀⠀⠀⢤⣀⠜⠉⢿⡿⢳⢀⣼⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢦⠀⠀⣠⣤⠶⠤⣄
⠀⠀⠀⢀⣸⢸⡷⣯⣙⡧⠀⢘⣷⠭⠆⠒⠒⠛⠛⠛⠚⠳⠾⣾⠛⢋⣿⣓⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⣀⣀⣿⣀⡊⢳⣾⣷⠶⢶⡦⡈⠳⡄
⠀⠀⠀⢩⣿⣿⣇⣠⠽⠿⠷⢿⣶⠶⠶⠶⠶⠦⣤⣀⡀⠀⠀⠈⠻⣿⣤⣽⢯⡋⠀⠀⠀⠀⠀⣠⡴⠞⣻⡿⢿⠟⠛⢟⠛⢍⡉⠉⢭⡙⠻⣥⠀⢠⠇⠘⠀⡇
⠀⠀⠀⣞⠧⢄⡟⠁⣠⠖⠊⣿⠟⢍⡉⠛⠓⠲⢤⣈⠙⠲⣄⠀⠀⠹⣆⠘⠚⠁⠀⠀⠀⢠⡞⠁⡔⡹⠋⢠⠏⠀⠀⠀⠱⡀⠉⢳⡀⠘⢆⠈⣷⠊⠀⠰⠀⡇
⠀⠀⠀⡹⡖⢻⠁⣼⠃⠀⡔⣿⡇⠀⠙⠲⣄⠀⠀⠈⠓⢦⡈⠳⣄⠀⣿⠀⣠⣶⡀⠀⢀⡟⢀⠌⡐⠁⠀⡞⠀⠀⠀⠀⠀⢡⠀⠀⢱⠀⠈⡀⢸⡇⠀⠃⣰⠃
⠀⠀⠸⣿⡅⢸⢸⠇⢀⡞⡰⠁⠀⠀⠀⠀⠀⠙⠢⣄⡀⠀⠉⢦⠘⣆⣿⠔⡁⠀⣧⠀⢸⠇⡌⢰⠁⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠇⠘⡇⠀⢠⠃
⠀⠀⣴⡅⠙⣾⣼⣀⣾⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢷⣾⣿⡾⠾⠿⣥⠀⢸⠠⠁⣟⠀⠀⠀⣧⠀⠀⠀⠀⠀⣸⠀⢀⣀⡇⠀⡇⣸⠃⢀⠂
⠀⣤⠍⠶⠶⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠈⡟⠻⣆⠀⢀⡟⠀⣸⣷⠞⣉⠛⠛⠓⠛⠓⠂⠐⠒⠛⠛⠛⠋⠉⠈⢲⣶⠏⢠⡿⠦⣄
⠀⢣⡤⢠⣴⡇⢸⡏⠁⠀⣠⡆⠀⠀⠀⠰⠆⠟⠀⠀⡼⠉⠁⠀⢠⠃⣰⠟⢿⡞⠁⢸⠁⡇⠀⠀⠉⠁⠲⠢⣴⢋⣹⣴⣯⡙⡦⠐⠖⠒⢙⡿⡆⢸⡇⡀⠈⢷
⠀⠈⣿⣣⠜⣧⣄⣿⡄⠀⠀⠙⢄⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣠⠷⢾⡏⠛⢸⣇⡀⠘⠧⣼⣄⠀⠐⠺⡀⠀⠙⠤⠞⠉⠓⠚⣁⠼⠓⠀⣼⣼⠃⢸⣏⣘⡆⣸⣄
⠀⠀⣿⡘⣳⠏⠈⠁⠙⠶⣄⡀⠀⠉⠒⠒⠒⠉⠀⠀⣀⣴⣞⠁⠀⠘⠙⢲⣿⡽⣧⠀⠀⠀⠘⠦⡀⠀⠈⠒⠢⠠⠐⠒⠚⠉⠁⠀⣀⣼⠉⣿⣄⠘⠛⠛⢀⡿⡇
⠀⠙⡿⣿⢽⡆⠀⠀⠀⡠⠛⣻⢶⡖⢦⡤⠄⠒⢚⠟⡟⢷⠈⠳⡀⠀⠀⠘⢧⣤⣿⡀⠀⠀⠀⢀⣼⣷⠶⣤⣤⡤⢤⠤⣤⣴⠶⠻⡟⢿⡀⢳⠙⢶⣤⡤⠞⢠⠃
⠀⢨⣧⡤⠞⠀⢀⡴⠋⠀⠀⣏⠈⢣⠈⠙⠓⠛⠏⡼⠁⠘⡆⠀⠙⣆⠀⠀⠰⠟⣇⢹⠀⢀⣴⠟⢩⡏⠈⡇⠱⣕⣒⣚⠕⡿⠀⠀⢸⡀⠙⢮⣣⡀⠀⠀⣠⠎⢠
⣰⢉⡿⠁⠀⣠⠎⠀⠀⠀⡸⠀⠀⠀⠳⣤⣀⣠⣾⠁⠁⠀⢱⠀⠀⢸⡧⣤⠀⠀⠈⢋⡴⠏⠀⠀⡸⠐⠀⠻⡀⠀⠀⠀⢠⠃⠀⠀⠀⣇⠀⠀⠳⡝⠛⠛⢁⡴⠃
⠘⠋⠀⢴⣚⡛⢦⡀⠀⢠⠁⠀⠀⠀⠰⡁⠙⢉⡿⠀⠀⠀⠀⣧⠖⣫⠼⠙⢦⡀⢀⡼⠷⣀⠀⢠⠇⠀⠀⠀⢙⡶⣤⠖⡏⠀⠀⠀⠀⢻⣀⣀⣴⡟⠒⠛⠉
⠀⠀⠀⠀⠈⣧⣤⣍⣝⡟⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⠀⠸⣷⠁⠀⠀⠀⢱⠎⠀⠀⠈⢹⡿⠀⠀⠀⠀⠈⢧⡀⣰⠁⠀⠀⠀⠀⠘⡋⠀⠀⢷
⠀⠀⠀⠀⠀⣸⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢷⡀⠀⢠⠎⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡆
⠀⠀⠀⠀⣰⠋⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡌⠻⣄⡟⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⢷
⠀⠀⠀⣰⠃⠀⠀⣬⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢹⡇⠀⠀⢀⡞⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠘⣇
⠀⠀⡴⠃⠀⠀⣠⢿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠷⢬⣷⣤⡤⠾⠞⣹⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⣈⠇
⣴⠋⠀⠀⢀⡴⠁⠀⠀⠀⠉⠙⠓⠲⠲⡶⠶⠶⠶⠶⠶⠲⠚⠋⠁⣀⣠⡤⣬⡷⠆⠀⢠⣟⡙⠓⣶⠦⠤⢤⣤⣄⢀⠀⣠⣤⡴⠶⢾⠋⠁⢻⠒⠒⠋⠁
⠈⠓⠶⣤⣤⣱⣄⣀⣠⠴⣆⡴⣆⠀⠀⠀⠀⣠⣄⣠⣄⠀⠀⠀⣼⡟⠉⠀⠀⠀⠀⠀⣼⠀⠀⠀⡎⠀⠀⠀⣿⠁⠀⠐⠘⡇⠀⠀⠈⡇⠀⢨⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠘⠦⣤⡤⠖⣟⠈⠁⠈⠛⠒⠊⢀⠇⠀⠀⠀⠀⠀⢠⣿⠀⠀⣰⠄⠐⠀⠀⣿⡀⠁⠀⠀⣧⠐⠀⠄⣸⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⡟⡳⡄⠤⠴⠿⠷⢤⣤⠾⠛⠚⠶⠚⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠈⠀⠀⠀⠀⢸⡇⠐⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⣰⣇⠀⠂⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡉⠒⠒⠒⠒⢉⣩⣏⣉⣉⣉⣉⣉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽⡀⠀⢀⡀⠀⠀⢸⠀⠀⣠⡄⠀⡼⣀⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⢼⡟⠛⠚⠻⢿⣅⢨⡧⠀⣠⠴⠒⠒⢛⡶⠶⢤⣀⠀⠀⠀⠀⠀⣀⣯⡙⣧⣴⡏⢹⣄⡀⢸⠀⠀⣷⡀⢻⣿⡃⠺⡟⠒⢶⡄
⠀⠀⠀⠀⠀⢀⡴⢫⠶⠀⠙⠢⠤⠒⠋⠈⢿⣧⠋⠉⠶⠶⠶⠋⠠⠶⠄⠈⢷⡀⠀⣰⠟⠁⠻⠚⠋⠉⠛⠛⠀⠙⣾⣤⠏⠹⠶⠋⠀⠙⠛⠁⠀⠀⣻
⠀⠀⠀⠀⣠⣟⠀⠀⠐⠞⠀⢺⡆⠀⠀⠀⣸⣿⣤⣀⣴⠃⠀⠿⠀⠀⠀⣀⡾⣿⣴⠿⣦⣄⣀⡀⠀⠀⠀⠀⣀⣠⣿⣿⣄⣀⣀⠀⠀⠀⣀⣀⣠⡴⣿
⠀⠀⠀⢸⡏⡍⠛⠒⠶⠦⠤⠤⠤⠴⠖⠋⢙⣇⡠⠉⠙⠛⠛⠓⠒⠚⠋⠉⢠⡟⣿⣐⣉⣉⣉⣙⡛⠛⠛⠉⠉⠁⣹⠆⠌⠙⠛⠛⠛⠛⠉⠉⠁⢀⣷
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠙⠛⠛⠉⠉⠋⠉⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠋⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⣛⣡⣤⣴⢶

⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⢄⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢠⣤⣴⣦⣴⣆⣀
⠀⠀⠀⠀⠀⢀⡟⣵⣾⣿⣿⣿⣿⣆⠀⠀⣶⣿⣿⣿⣿⣿⣿⣷⠂⠀⣠⣾⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣀⠀⣀⣼⣿⡿⢿⠿⠟⠻⠟⢻⠀⢰⣿⢋⠉⠛⠻⠛⢿⠋⠀⠀⣻⠃⠄⠉⠛⡛⢻⣿⠂
⠀⠘⠉⠋⣿⣿⡟⡟⠀⠀⠀⠀⠀⢘⠁⢸⡇⢸⡄⠀⣰⡆⢸⠆⠀⠈⡃⢾⠃⠀⢴⡆⢸⠏
⠀⠀⠀⣀⣙⢿⣿⣇⠀⠠⣤⡤⠀⠀⠀⠀⠃⠈⢁⣀⡈⠁⠸⠛⠄
⠀⠀⣼⣿⣿⣦⠈⠙⠀⠠⠀⠤⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⢀⣤⣀⣠⣄⢀⣄⡀
⠀⠀⢘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣷⣤
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⣸⣿⣿⣿⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠛⠛⠿⠿⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠃⠀⠀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠐⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠐⠤⠤⡂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣎⡠⠂⠀⠀⠐⢤⠐
⠀⠀⠀⠀⠀⠀⠘⠐⠊⠀⠀⠀⠉⠒⠁⠀⠐⠋


⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⠶⢶⣦⣄⠀⠀⣠⣶⠾⠛⠛⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⠁⠀⠀⠀⠀⠀⠙⣿⣿⣾⡟⠁⠀⠀⠤⢄⡀⠀⠈⠛⠷⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⢻⣿⣿⠃⠀⠀⠀⠒⠲⠬⣑⠠⠀⠀⠀⠉⠛⠲⢤⣄⣀⣀⣀⡀
⠀⠀⣴⡖⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⢸⣿⣿⠀⠀⠠⠤⠬⣑⡢⢄⡉⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⣿⡏
⢶⡛⠛⣿⠀⠀⠀⠠⣔⡤⠂⠀⠀⠀⠀⠀⠀⡈⠉⠁⠀⣼⣿⣿⠀⠀⠀⠀⢀⠤⢉⠓⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⣸⡟⢰
⠸⣧⠀⢹⣇⠀⠐⢝⠣⠕⠊⢀⠀⠀⠀⠀⠤⠀⠒⠂⠀⣿⣿⣿⠀⠀⠀⠀⠄⢉⠓⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣿⡁⡾
⠀⢹⡇⠀⣿⡀⠄⠀⠤⢐⡠⠄⠀⠀⠀⠀⠐⠂⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠐⠠⠄⣐⠀⣸⠟⣹⠃
⠀⠀⣿⡄⢿⣇⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠂⠒⠂⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢢⡏
⠀⠀⠘⣷⠀⣿⡀⠀⢀⡠⢄⡢⣅⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣼⠁
⠀⠀⠀⢹⡆⢸⣇⠀⠀⠘⣊⠭⠔⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢱⡏
⠀⠀⠀⠀⢿⣦⢿⡀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠤⠀⣀⠀⠀⢸⡇⣼⠃
⠀⠀⠀⠀⠘⣇⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠄⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡈⠀⣾⢠⡟
⠀⠀⠀⠀⠀⢻⡀⣿⡀⠀⢀⣠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠈⠀⡏⣼⠃
⠀⠀⠀⠀⠀⢸⡇⠸⡇⠀⠁⢒⡨⠄⠀⠀⠀⠀⠈⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠇⣿
⠀⠀⠀⠀⠀⠀⣿⡄⣷⠀⠀⠐⠂⠠⠀⠀⠀⠀⠀⣀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣢⣬⠁⣸⢰⡇
⠀⠀⠀⠀⠀⠀⢹⡇⢸⠀⠀⢐⣒⠉⠀⠀⠀⠐⠒⠒⣂⣼⣿⣿⣧⣄⣀⣀⣀⣀⣀⣀⣀⣈⡛⠶⢯⠃⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣷⢼⣧⠴⠖⠚⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠈⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠲⠤⠧⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠒⡫⠶⡖⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⢈⣄⣣⣠⠜⣀⠈⢰⢄⣀⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠞⡱⠋⠹⠹⠓⣆⠈⣷⣾⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⡰⠁⢸⠃⠀⠀⠀⠀⠈⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⡁⠀⢿⣄⠀⠀⠀⠀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠴⢄⢸⣿⣠⡶⣎⣀⣴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠘⠿⠿⢧⠀⠨⠿⣽⡮⠿⢿⣿⣿⣿⣿⠿⠟⠋⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠈⠁⠀⠀⠐⢂⡏⠂⠀⠀⠉⠈⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠐⠀⢀⣶⡂⣂⣒⣀⠀⠚⠂⠂⠀⣰⠐⠀⠐⠐⠒⠀⠀⠀⠂⠐⣓⡀⠀⠐⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠉⢣⡀⠀⠀⠀⠀⣀⣀⣀⣴⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⠤⠤⠤⠶⠔⠋⠀⢇⠀⠀⠀⠈⠈⠉⠉⠉⠙⠄⢸⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠐⠀⠀⠀⠀⠀⠀⠀⠘⠤⠤⡤⠤⠤⠤⠒⠒⠉⠉⠉⠈⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠈⠁⠒⠠⠤⠤⠄⠤⠔⠋⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠄⠀⠒⠉⠉⠀⠀⣉⣒⠂⠒⠠⠔⠒⠂⠒⠒⢡⠑⠂⠤⠤⠤⠤⠤⠤⠤⠒⠚⠒⠋⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠤⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⠀⠒⠀⢀⠉⠉⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣭⡉⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣈⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦
⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⢠⣿⣿⣿⣿⣿⣿⡄⠀⠀⣠⢤⢤⡀⠀⠀⢀⣾⣿⣿⣆⠀⠀⠀⣴⣶⣶⡄⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⣠⣶⣿⣷⣦⡀
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⠃⠀⣰⣿⢿⣿⣧⠀⠀⠘⢿⣿⣿⠏⠀⠀⢰⣿⣿⣿⣿⡄⠀⠀⣿⣿⣿⡿⠀⠀⢰⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠺⣿⣿⣿⠛⠁⠀⠈⠛⣿⣿⠛⠁⠀⠀⣿⣿⣽⡟⠋⠀⠀⠀⠈⣿⡏⠀⠀⠀⠈⣿⣿⣿⣿⣿⡄⠀⠛⢻⣿⠃⠀⠀⠘⠿⢿⣿⣿⠿⠟
⠀⠀⠀⢀⣠⣤⣾⣿⣿⣦⣄⣀⣤⣶⣿⣿⣷⣤⣀⣠⣴⣿⣿⣷⣦⣄⣀⣤⣾⣿⣿⣦⣄⣀⣴⣿⣿⣿⣿⣿⣿⣄⣤⣾⣿⣷⣤⣀⣀⣤⣾⣿⣿⣦⣄⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⠋⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠐⠂
⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣆⠀⠀⣀⣀⣀⠀⠀⠀⠀⣸⣿⣿⡧
⠀⠀⠀⠀⢄⠀⠀⢠⣶⣶⣦⠀⠀⠀⠀⠸⣿⣿⣿⠛⠀⠘⣿⣿⣿⡇⠀⠀⠀⣍⣟⠿⠃
⠀⠀⠈⢆⡄⠀⠀⢹⣿⣿⣿⠇⠀⠀⠀⡀⢘⡉⠁⠀⠀⠀⠀⠙⠿⠇⠀⠐⣤⣨⣿⣿⣻⣶⡀
⠀⠀⠀⠀⠱⡁⠈⢂⠙⠋⠉⠀⣀⠀⠀⢸⠘⠀⠐⠚⣋⠓⠲⠦⢶⡤⢤⡤⢼⣿⣿⣿⠱⣿⢿
⠀⠀⠀⠀⠀⢹⡀⠈⠓⠚⠛⠛⡙⢳⠇⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠠⠋⠀⢀⠙⣿⣿⠟⣽⡇
⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠟⢈⠳⢦⡀⣠⠀⢰⠀⠙⢿⣾⡿⡆
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠠⣶⣷⠀⠈⠤⠤⠴⢂⣠⠁⠀⡼⢃⠄⠀⠀⢀⣸⣎⣀⢀⠙⠢⠇
⠀⠀⠀⠀⠀⠋⠃⠀⠀⠀⠘⣽⠀⠷⣤⣤⠶⠚⢉⣳⣤⠞⠁⠥⠀⠀⠀⠅⢹⣿⣿⣿⣿⣶⣦⠠⠂
⠀⠀⠀⠀⢰⢔⣠⣤⣶⣦⣤⣈⠀⢀⠀⣠⣒⣛⣉⣀⡒⣈⣙⣁⣀⣀⣀⣀⣸⣭⠝⣋⢹⣶⡏
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⢿⣿⠀⠰⡟⠛⠋⡉⠉⠟⡇⠈⣿⣿⣽⣿⣟⢫⡙⣷⢚⣡⡿⠏
⠀⠀⠀⠀⠰⣿⣿⣿⣛⠀⢈⠿⠀⡆⠀⠀⢠⡁⠠⢞⠁⠸⣿⢿⣶⣦⣶⢿⡄⣿⣿⣯⡡⢖⠂
⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⠌⠇⠀⢡⠀⠰⣸⣏⡀⠊⠀⢲⢟⣚⢃⢼⣟⡛⢩⣿⡿⢙⣀⡘
⠀⠀⠀⠀⠀⢸⣿⣿⠁⣿⡀⠁⠀⠰⡴⠦⠉⣻⣀⡀⠀⠐⢙⣽⣽⠸⠟⣁⣸⠟⠁⠈⠋
⠀⠀⠀⠀⠀⢸⣿⣯⠀⢸⣷⢨⠀⠀⠁⠀⠀⠄⠉⠁⠀⠀⠸⢀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⣿⢩⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⣿⠀⠀⠀⠡⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⢻⠀⠀⠀⠀⣴⢀⡈⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢀⣴⢶⢤
⠀⠀⠀⠀⠀⢰⣏⠠⡄⠀⢨⣤⠔⢶⣨⣿⢸⣿⣆⡀⢠⣤⣄⢀⡀⠀⣀⣴⣤⠿⣿⣿⢛⣆
⠀⠀⠀⠀⠀⠠⠿⠗⠁⠀⠘⣿⣦⣷⠏⠁⢯⣭⠥⠀⠚⠿⢻⡿⠁⠀⠘⠍⠉⠙⠘⠙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣬⣤⣤⣷⣵⣾⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡀⠀⠀⠀⠀⠀⠀⠠⠶⣷⣦⠀⠀⠀⠀⠀⠀⢀⠉⠁⠄⡀⡈⠐⠀⠀⠀⠀⠀⠀⠀⠞⢃⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢀⠁⠀⡀⠱⡐⠀⠀⠀⠀⠀⠈⢀⢁⠃⡅⠇⠀⢀⠀⠀⠀⠀⠀⠀⡆⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠄⠀⠀⠀⠀⠈⠀⠁⠀⠀⢀⢠⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⣀⠐⠀⠀⠀⠸⢸⠀⠀⠀⡀⢆⠀⠀⠆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠉⣇⠁⠀⠀⢡
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡀⢀⠔⡀⠀⠰⠀⠰⠰⢸⠠⢀⠌⠀⠀⡁⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⡠⠶⢤⠤⡔⠀⠀⠀⡀⢃⣀
⠀⠀⠀⠈⠢⡀⠀⠃⠸⢠⢠⠰⠤⠖⠛⠁⠨⣴⣄⡆⢀⠀⠀⠀⠀⠘⡀⢞⠀⠀⠈⠁⠐⠀⠀⠀⠀⠈⠀⠀⠀⣠
⠀⠀⠀⠀⢀⢤⠀⠀⠀⠀⠈⠀⠀⠀⢀⠄⠀⠀⠀⠀⠘⠀⡆⡆⠀⠀⠃⠀⡉⠐⠠⠀⠀⠀⠀⠀⠀⢰⠀⠀⠐⠁⢠⠄⡀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠗⠲⠶⡄⠀⠀⠀⠃⠃⠀⠀⢘⠰⠀⠀⠰⠀⠀⠀⠀⠀⡄⠈⠈⠀⡀⠀⠆⠀⢠⠃⠀⠀⠄⠤⠔⡔
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡰⢍⠀⠀⠀⠂⠀⠉⠁⠀⠀⠀⠀⠃⠀⣇⡬⠁⠈⠉⠁⠈⠁⠀⠀⠀⣠⡄⣀⣀⣀⣀⠀⠀⢸⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⣀⣀⣀⡀⠔⡀⠀⠀⠀⠀⠉⠀⠈⠉⠙⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠁⠀⠀⠖⠀⢂⢠⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠈⠀⢘⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢡⣶⣿⣿⣦⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣶⣿⣷⣦⡈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⣤⣬⣿⠿⠟⣿⣤⣤⣍⠻⠟⢉⣤⣍⠻⣿⡿⠋⣡⣼⡿⠿⠿⣿⣤⣤⣌
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⢿⣄⣴⠟⠀⠀⠙⣷⡀⣿⡁⠙⢷⣄⣴⠟⠁⠘⢷⣄⣴⠟⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⢸⡇⠈⠻⣦⡀⠉⠁⣠⡄⠀⠀⠙⠁⠀⢀⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣄⠀⠀⠀⠀⠀⠀⣤⠀⢸⡇⢸⣦⡈⠻⣦⠾⢻⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢠⡿⠶⠾⣧⡈⠛⠛⠂⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⣿⠶⠶⠶⢀⣶⣾⣷⣶⣶⣾⣷⣦⠀⠀⠀⠀⢸⡷⠶⠶⠶⠶⠶⢾⡷⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢸⣿⣿⣉⣉⣉⣩⣿⣿⠂⠀⠀⠀⠈⣿⠀⠀⣶⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢻⡆⠀⣿⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⡆⢉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠸⣇⠀⣿⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣆⠀⠀⠀⢠⣴⣾⠿⠟⠻⠿⢿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⢠⣿⠿⠿⣿⣿⣿⣿⠿⠛⠙⣿⡆⠀⢠⣿⣟⣴⣾⣿⣿⣿⣶⣌⢻⣷⡀
⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠈⠻⣿⣿⣿⣿⡀⠀⠀⠀⠘⣿⠀⣀⡀⠀⠀⠀⢀⡀⠀⣿⡇⠀⣸⣿⣿⣿⣿⡿⠉⢻⣿⣿⣷⣿⡇
⠀⠀⠀⠀⠀⠸⡿⠁⠐⠂⠀⠀⠀⠘⣻⣿⣿⣇⠀⠀⢀⣀⡿⠀⠀⢄⠀⠀⠀⠠⢨⣄⢹⣥⡀⢹⣿⣿⠟⠋⠀⠀⠀⠽⢿⣿⣿⠃⠀⢀⣀
⠀⠀⠀⠀⠀⠫⣇⠉⢀⣼⡆⠀⠀⠁⣾⡟⢹⠛⠀⠀⠘⠉⠇⠀⠰⡿⠇⠀⠀⠀⡿⠟⢸⢁⢸⠻⡏⣿⣧⠀⠀⠀⣴⡆⠀⠻⣿⡄⠀⡼⠛⡷
⠀⠀⠀⠀⠀⠁⢼⠀⠈⠉⠀⠀⠀⠀⠉⠀⣶⠂⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢰⡇⠈⠉⠀⠀⠀⠉⠋⠀⢸⡡⠂⠀⠨⠂
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠉⠀⠂⠀⠀⣠⡃⠀⠀⠀⠀⠀⢠⡲⡀⠀⠀⠀⠀⠀⢀⠴⠈⡀⢀⣾⣿⡀⠀⠀⢦⡴⠂⠀⢀⣾⡇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣸⣿⡇⠤⡄⠀⢠⡤⠸⣿⣇⠀⠀⠀⠀⠀⢀⠀⠀⢐⡆⠀⠰⡆⠀⠄⠄⠐⠬⠉⠛⠛⠦⣄⠀⠀⣤⣴⠿⠿⢟⠢⠀⡐
⠀⠀⣤⣤⢤⣴⡿⠋⠈⠈⠁⠀⠈⠁⢂⠺⠿⣦⡤⣤⣤⡔⠰⠓⠂⠋⠪⠐⠈⠉⠁⠛⠀⢰⣀⠀⠈⠖⡑⠑⠀⠐⠁⠂⠔⠈⠀⠁
⠀⠀⠀⠉⡨⠁⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⡀⠀⠓⠀⠈⠠⠀⡠⠆⠀⠀⠀⠀⠀⠀⠆⠀⠀⠘⢀⡖⡐⠀⠀⠀⠀⠐⠀⢡⠔⠈
⠀⠀⠀⡈⠠⠷⠶⢇⠖⢢⠀⠀⠀⠀⠀⠀⠐⠚⠁⡆⠘⠣⠾⠥⠂⠃⠀⠀⠀⢠⠀⠢⠟⠒⠓⠀⡴⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⢲⠈⠢⠊⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠚⡈⠂⠀⠀⠀⠀⠀⢓⠃⡙⠀⠀⣩⠀⠀⠀⠀⠀⠀⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠀⢘⠠⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠁⢌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠌⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠈⠈⠀⠂⠀⠀⠀⠀⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⣦⣄⡀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠛⠿⠿⢶⣶⣶⣶⣶⠾⠿⠛⠂⠀⠀⠀⠀⢄⠀⠀⠀⡐⠀⠆⠀⠀⠀⡀⠀⠘⠈⢉⡛⠛⠿⠷⠾⠛⠛⡩⠃
⠀⠀⠀⠀⠀⠀⠀⡁⠀⠆⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⡇⠀⠆
⠀⠀⠀⠀⠀⠀⠀⡅⢠⠀⠀⠀⠀⠀⡥⠤⡄⠀⠀⠀⠀⠀⢀⡄⢠⠀⠀⠀⠀⠀⡆⠠⠀⠀⠠⠥⢁⡒⠀⢲⡀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠅⠈⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⢈⠈⠀⠀⠀⠀⠀⠐⢄⡀⠇⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠰⠚⠀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⠀⡘⠐⢄⠀⠀⠀⠀⠀⠀⠁⠁⠐⠒⠂
⠀⠀⠀⠀⠀⠰⠀⣀⡴⠂⠀⠀⠀⠀⠣⣄⡀⠦⠀⠀⠀⡠⠁⢀⡴⠀⠀⠀⠀⠀⢦⡀⠀⢤⠀⠀⠀⠀⠀⠀⠘⢧⡀⢠
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠚⠉⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣤⣾⣿⣿⣿⣿⣠⣤⣤⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣿⣿⠃⠉⠙⠛⠛⠿⠟⠛⠋⠉⠀⠀⠀⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠏⠈⠉⠉⠈⠙⠻⠿⠋⠛⠉⢿⣿⣿⣿⣄⠀⣿⣿⡆⢀⣀⣀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠋⠱⠞⠓⠀⠀⠀⠀⠀⠰⠶⠦⠀⢸⣿⣿⣿⠟⠀⢹⣿⠀⠉⠉⠁⠀⠀⠀⠀⠀⠈⢉⡉⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⠀⠐⠒⢂⣄⠀⠀⠀⠀⠀⠒⣶⣄⠀⣿⠻⡇⠰⢇⡾⠃⠀⠀⢀⣸⣦⠀⠀⠀⠐⠉⣠⣿⣧⠀⣯⠵⠃
⠀⠀⠀⠀⠀⠀⠀⠰⡏⠳⡟⠀⠀⢠⣾⣿⠄⠀⠀⠀⠀⣾⣿⡿⠀⡟⠀⡅⠠⡀⠘⡇⠀⠀⠸⠿⠟⠀⠀⠀⠀⠀⠿⠿⠃⢀⠇⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠶⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠉⠉⠀⢸⠶⠂⠀⠀⠈⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⢠⡀⠀⠀⢠⡀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠆⢰⡄⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠈⠛⢚⡿⠟⠁⠀⢀⡤⠁⠀⠀⠀⠀⡘⠀⡄⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⣀⠀⠀⠀⣀⡀⠒⠁⠀⠀⢀⢀⣠⠆⠀⢠⠁⠀⠀⠈⠡⢦⢀⣤⡄⡴⠌⣡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣂⡀⠼⡉⠉⠉⣹⠆⢉⣉⡶⠒⠁⠀⠑⢤⡄⠂⢀⡐⣩⠐⠴⢓⠄⢀⡀⢜⠢⡔⢠⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣿⡏⠀⠀⠀⠘⠂⠀⠀⠸⣿⣇⠀⠀⢀⠠⠌⡁⠄⠊⠁⢸⠀⠀⠀⠀⠀⠁⠈⠀⡇⠀⠀⠈⠂⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣸⡿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡆⠉⠀⢀⠴⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠄⠀⠀⠀⢸⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢰⣿⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⡐⠁⢀⠦⡀⢴⠂⢀⣀⣀⠀⠀⠀⠀⠀⠀⡇⢄⠈⣶⠄⣀⢦⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⣀⠃⠁⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⢃⡀⠈⠉⠙⠛⠑⡇⠀⢉⠀⠀⠀⠀⠀⠀⠁⠀⢛⠉⠉⠉⠈⣁⡀⠎
⠀⠀⠀⠀⠀⠀⢠⠁⠀⡼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠈⠉⠉⡿⢉⠅⡐⠉⠀⠀⠀⠀⠀⠀⠹⠙⢇⠸⠋
⠀⠀⠀⠀⠀⠀⠈⠀⢰⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢀⠟⠉⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠆⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡄⠀⠃⠘⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠘⠦⠤⠀⣀⣀⡀⠀⠀⠀⠀⢀⣀⡀⠠⠤⠂
⠀⠀⠀⠀⠀⠀⡇⣿⣾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠐⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⠋⠀⠸⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠢⣀⠀⠀⠀⠀⢠⠃⠀⢠⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⠉⠛⠛⠋⠀⠀⠀⠙⠛⠛⠛⠉⡇⠀⠀⠀⠀⠀⠀⠈⠁⠚⢓⠚⠀⠀⠀⠀⠛⠒⡖⠚⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡦⠀⢤⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠇⠀⠀⠀⠀⠀⠀⠰⠦⠤⡆⠀⠀⠀⠀⠀⠀⠀⢰⠤⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠐⠛⠀⠀⠀⠀⠀⠀⠀⠘⠂⠘⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠃⠀⠀⠀⠀⠀⠀⠘⡁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠈⠆⠀⠀⠀⠀⠀⠀⠀⢠⠉⠈⠣⡀⠀⠀⠀⠀⢀⠄⠁⠈⡄⠀⠀⠀⠀⠀⠀⠀⢠⠉⠈⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠆⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠑⡄⠀⠀⢀⠊⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢉⠱⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⡔⢉⣩⠀⠀⢈⣉⢢⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⡔⣉⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣠⣤⣤⣤⣤⣄⣤⣤⣠⣤⣤⣀⣤⣤⣠⣤⣤⣤⣤⣠⣤⣤⣄⣠⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣄⣤⣤⣤⣤⣤⣤
⣿⠁⠀⠈⣿⠫⡉⠛⢻⠙⠛⠉⠟⣏⢩⠛⠋⠻⠛⡿⠛⠛⠛⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣛⣛⣛⣛⣛⣿⣿⣛⣛⣛⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠈⢷⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⡆⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠟⠋⠛⠿⣿⣿⣿⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡟⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣴⠓⠐⠀⠀⠀⠀⠀⠠⢁⠂⠀⠀⠀⠀⠀⠀⢀⠴⠀⠡⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣮⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠃⠔⢢⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⣀⢀⠱⠖
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢹⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢶⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⢠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠓⠛⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣾⣿⣿⣿⣿⣿⣽⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⡤⠀⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠟⢻⣿⡿⠻⢿⣯⣼⠿⣿⡿⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠐⠀⢸⣿⠇⠀⠀⠀⡠⠚⡁⢀⡜⠋⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⠀⠀⠰⣿⢟⣿⣿⣧⡀
⠀⠀⠀⠀⠀⢀⡆⠀⠈⣿⠀⣠⣴⠎⣠⠞⣡⡞⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠛⠒⠲⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⣼⣷⡀⠀⠀⠋⣤⣤⣾⢋⣴⠟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢏⣀⡀⢰⣏⣿⠿⠋⢁⣽⣷⣦⡀
⠀⠀⠀⠀⢠⣿⣿⡷⢤⣀⠀⣻⠉⢠⡾⠃⠘⡄⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⠿⠃⣾⡾⠁⣷⠀⢸⠋⠀⠘⣇
⠀⠀⠀⠀⣿⣿⣧⡹⢄⣉⣛⣉⡴⠋⠀⠀⠈⡇⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣯⣦⣴⡿⠀⠒⣿⢀⡆⠀⠀⠀⢸
⠀⠀⢰⡟⠙⣿⣿⣧⠀⠈⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣷⢻⣿⠀⠀⣿⣿⠀⠀⠀⠀⢸⣧
⠀⠀⣸⡇⢸⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣏⣿⣿⠿⠏⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⡄
⠀⣴⡟⡇⠀⣿⣿⣿⣿⠛⠳⠶⠄⠀⣀⣀⣄⣽⡇⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⢸⣧⠀⠀⠀⣿⡇⠀⡀⠀⠀⠸⣿⣿⡶
⢰⡿⠀⠀⠀⣿⣿⣿⡟⠓⠲⠤⠤⠤⠄⠀⠀⠠⣷⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⣾⠛⣿⣿⣤⣾⣿⢦⣤⠄⣿⡷⣬⣍⠀⠀⠀⣿⣿⢿⡄
⢸⡇⢸⠀⢸⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡿⠚⠋⣈⣵⣿⠇⠀⣶⣶⠏⠀⠀⠀⠀⠀⣸⣿⣿⣎⣷⡀
⠈⠓⠿⢶⣾⣿⣿⠀⠀⢰⠀⠀⢀⠀⠀⠀⠀⠀⡄⢿⡀⠀⢸⠀⠀⠀⠀⠀⠀⡾⠃⣴⠞⣽⣿⣾⠃⢠⠏⢈⣤⣀⡀⠀⢠⣾⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢰⠈⠷⣶⣾⠀⠀⠀⠀⠀⠀⢻⡗⢈⡾⠋⢹⡟⠀⢋⠘⣯⠟⢉⣁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⡇⠤⣄⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⠆⠀⠀⠀⠀⠀⠈⢻⡀⠀⠀⣿⠿⠗⠛⣤⣄⡀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⡇⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢐⡿⠀⠀⠈⠋⠉⣄⠀⢻⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠾⠁⡆⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⠀⠀⠀⢻⣦⣄⠁⢠⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⠁⢸⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢡⠀⠀⠀⠀⠀⠀⢰⡇⠀⣼⢹⠇⠀⠀⠀⠀⠀⠀⠀⢰⠀⣼⡇
⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⢠⡿⠋⠀⠀⠀⡀⠀⠀⠀⠀⠸⢀⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡆⠀⠀⣿⠀⠀⠀⠀⠀⣾⠃⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠰⢸⣿⡇
⠀⠀⠀⠀⠀⠹⣿⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⣀⣸⣇⣠⠤⠛⠀⠀⠀⠀⠀⠛⠦⣿⣷⣀⣀⣀⣀⣁⣀⣀⣀⣀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠁⠈⣿⠛⠛⢻⡷⠶⠟⠛⠛⣻⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡰⠛⠒⢦⣄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣾⣄⠀⠘⠲⣌⣻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⢻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠈⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠻⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠹⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠔⠚⠛⠀⠑⠀⠒⢁⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇
⠀⠀⠀⢀⡾⢢⡀⢀⠀⢰⣧⠀⠊⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠁⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠸⠶⠚⡁⠘⡿⠁⣿⠆⢠⡀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠠⡘⢇⠀⠁⠀⣯⣀⡀⠈⡌⠳⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⢿⣦⡄⠀⠘⣄⡸⣷⣧⣤⣤⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠋⠀⠀⠀⠀⣾⣿⣿⡿⠿⠟⠉⠛⠋

⠀⠀⠀⠀⠀⢀⣠⣤⣴⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⡀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣤⣤⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣼⡟⣩⣶⣶⣶⣮⣙⣿⡄⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⠋⠀⠀⢻⣿⡇⠀⠀⠀⣿⣾⣿⡿⠋⠻⣿⣿⣿⡇⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠄⠈⠻⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠒⠀⢿⣧⣤⠀⠈⣿⣿⠟⠀⠀⠀⠈⣻⣿⣧⠀⢀⣾⣿⣿⠀⠀⣀⠀⠀⠀⣀⠀⢹⣿⣿⡇
⠀⠀⠀⠀⠛⠟⠁⢰⡷⠀⠀⢰⡿⢸⠻⢌⠂⢸⠛⡅⠀⠿⠂⠀⠐⠿⢀⡁⠀⠈⣾⣿⣿⡦⣤⡀⠀⠀⣠⠀⠄⣾⣿⡿
⠀⠀⠀⠀⠀⣿⠀⠀⢠⣀⡀⠀⠀⣾⠀⠀⠲⠀⢨⣧⠀⠀⣤⣤⠀⠀⣸⣧⠀⠁⢼⢿⣧⡅⠙⠃⠀⠀⠙⠃⠀⣥⡾⠃
⠀⠀⠀⠀⠀⣿⣷⣄⡈⠉⢀⣠⣾⣿⢀⠰⠀⠃⡾⠿⠷⣤⡀⠀⣤⠾⠛⠫⡅⢁⠈⠀⠈⠙⠀⠀⠐⠶⠃⠀⠘⠋
⠀⠀⠀⠀⠀⣿⠯⠙⠋⠄⠘⠓⠋⠁⠀⠑⠈⠰⡀⠈⠳⠀⠀⠂⠉⠘⠇⡀⠘⠁⠀⡀⠂⠠⡄⡰⡆⠀⠶⠁⡀
⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⠀⠈⣾⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⢀⠉⠀⢄
⠀⠀⠀⠀⠈⣰⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⣱⡤⠆⢀
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⠒⢑⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢠⣤⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣈
⠀⠀⠀⠀⠀⣴⣤⣀⣀⣀⣀⣀⣤⣼⠀⠀⠀⠀⠀⠸⡻⠷⠶⠶⠶⠾⡛⠛⠇⠀⠀⠀⠀⣄⣀⠀⠀⠀⠀⢀⣀⡄
⠀⠀⠀⠀⠀⠑⠉⠩⠙⠛⠛⠍⠉⠂⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠩⠛⡛⠛⠟⠛⡛⠫⠑
⠀⠀⠀⠀⠀⠠⡄⡄⠀⠀⠀⢠⢀⡄⠀⠀⠀⠀⠀⠀⢠⡰⠀⢰⠋⠁⠅⠒⠊⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠔⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠀⠀⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠉⡂⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⡀⠀⠀⠀⠐⠘
⠀⠀⠀⠀⠈⠱⠚⠁⠀⠀⠀⠈⠓⠆⠁⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠞⠁⠀⠀⠀⠙⠦⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠀⣀⠀⡈⠢⡀⠀⠀⢠⠊⠁⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠉⠉⡛⢷⣴⡀⠁⠀⠀⠊⣀⡤⠴⠐⠒⠰⣦⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠆⠀⠀⠀⠀⡇⠀⠐⠓⣾⣿⠈⠁⠀⠈⠁⡿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡄⠀⢀⢰⣀⡇⠂⠀⠀⢽⣃⡁⠀⠀⢀⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡜⠁⠀⠙⠗⠚⢉⠙⢆⠀⢀⡜⠂⠈⠐⣍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢀⠀⠀⠀⠀⡐⠁⠀⠈⠀⠜⠀⡆⠀⠀⠘⡆⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠙⠂⠀⠈⠛⢀⠀⢀⠀⠀⡅⠀⠘⡄⢀⡤⠴⠀⢀⠀⠀⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠷⣄⡀⠀⠀⠀⠙⠂⠼⠃⠌⠀⠀⠀⠀⠀⠐⠮⡀⠀⠘⠁⡀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠎⠀⠈⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⡆⠀⠀⠀⠀⠁⠛⢶⡋⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⡐⢂⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡐⠃⠀⠀⢸⡄⠀⠀⠀⠀⢠⠈⡀⠀⢠⠀⠀⠀⠀⠀⢡⢞⠃⠀⡜⠁⠀⠂⠀⠀⠀⡳
⠀⠀⠀⠀⢀⣴⢇⠒⣀⠌⠀⠀⠀⠀⢸⠙⠀⠀⠀⢀⡴⠇⠀⠀⠀⠇⠀⠔⠠⠚⢁⠂⡠⠎⠀⠀⠣⢀⠤⡀⢀⡀⢣⣄⡀
⠀⠀⢠⡎⢙⠿⣋⠌⠀⠀⠀⢚⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠓⠾⠆⠀⠤⠀⠁⠽
⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⢸⣦⡄⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠧⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣀⠀⠀⠀⠀⠠⣄⣀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠠⠀⠀⠀⠄⠀⠀⠠⠀⠀⢀⠀⠀⠀⠄⣿⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⡤⠀⠀⠀⡀⠀⠀⠄⠀⠀⠠⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠛⠀⠀⠀⠀⠀⢀⣿⣿⡗⠀⠀⢁⠀⠀⠀⡀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠁⢧⠀⠀⢦⣤⣶⣿⣿⡿⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢄⠀⠐⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠆⢉⡀⢤⣤⣤⣤⣤⣤⣀⣤⠀⠠⢄⠐⠀⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⢐⠄⠂⠀⠀⠉⠀⠻⣿⣿⣿⣿⣿⠃⠀⠀⠀⠈⠑⡀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠲⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⣿⣿⣿⢃⠀⠀⠀⠀⠀⠀⠀⠈⠓⢀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢸⣿⣿⣿⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⡀
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠢
⠀⠀⠀⢁⠀⠐⠂⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠂
⠄⠀⠈⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠑⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠘⣿⣿⡏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⡟⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠆

⠀⢎⠍⡉⠻⠛⡛⠩⡁⠀⠀⠀⠀⠰⣉⠂⠀⢰⣉⠆⠀⠀⠀⠀⢀⡀⣀⣀⢀⣀⠀⣀⡀⠀⠀⠀⢰⡁⠀⠀⠀⢈⡱⠀⠀⠀⠀⠈⡽⠨⠭⢈⠉⠭⢭
⠀⠀⢢⡐⣄⠀⠨⠀⡇⠀⠀⠀⢀⡄⠒⠢⣀⠔⠒⢢⡀⠀⠀⠀⠀⡀⠀⣼⢠⡄⠀⡀⠀⠀⠀⠀⢐⠑⣄⡀⠀⡔⢑⠀⠀⠀⠀⠀⡆⡒⢂⠣⠒⡄⢸
⠀⢘⠀⠉⡉⢉⠍⡂⡇⠀⠀⠀⢸⠀⠀⢰⠭⡁⠀⠈⡁⠀⠀⠀⡠⠍⠅⢸⠀⠠⠍⠤⡄⠀⠀⠀⢸⣕⡇⠀⠆⢇⣿⠀⠀⠀⠀⠀⡀⠉⢁⠀⡀⠀⢸
⠀⢀⠀⠠⠀⠀⠉⠀⡇⠀⠀⠀⠈⠁⠀⠠⠈⡜⠀⠈⠁⠀⠀⠀⢀⣙⢃⢸⠀⢀⣋⡁⠁⠀⠀⠀⠀⢸⠀⠀⠀⠈⢸⠀⠀⠀⠀⠀⡇⠀⠣⠤⢤⠝⢸
⠀⠄⠀⠐⠒⠒⠂⠀⠡⠀⠀⠀⠀⠠⣀⠸⠀⠇⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠁⠀⠀⠀⠠⠈⢦⠀⢠⠎⠸⠀⠀⠀⠀⠀⠁⠀⠲⠲⠒⠀⠩

⠀⠀⠀⠀⠀⠠⠀⣒⠠⡀⠀⢠⠈⠭⠉⠉⠩⠑⠀⠀⠀⠀⠀⢀⢀⠀⡠⠀⠀⠠⡀⡀⡀⠀⠀⠤⠤⠤⠤⠀⠤⠤⠤⠤⠄⠀⠀⠀⠀⢀⡤⠤⣄⡀
⠀⠐⠁⢑⠀⠅⠣⣁⠜⡀⠀⠰⠀⠒⠂⣒⠐⠀⠀⠤⠤⡀⠀⢸⠀⠸⢅⠠⠬⣀⠷⠀⠀⠀⠀⠁⠠⡢⠀⠀⠀⠠⡢⠀⠀⠀⠀⠀⣰⡁⠀⠀⠈⢊⣀
⡠⠐⠐⠓⠐⡜⠂⠁⠈⠀⠀⠈⠲⢄⠴⠀⢤⢔⡀⠛⠀⡇⠀⢸⠀⢀⡠⠓⠚⠣⣀⠀⠀⠀⠀⠠⠥⠤⠦⠀⠀⠧⠤⠇⠀⠀⠀⡘⠤⢂⠀⠀⢀⠣⠜
⠇⠂⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠐⣀⠀⣀⠇⠀⢸⠠⡌⡁⠀⢀⠀⣉⠤⠘⠀⠀⡀⠐⡕⠀⠀⡀⠐⡕⠀⡀⠀⠀⣰⠀⡀⡇⠀⢁⡆⢀
⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⢠⠒⠒⠒⠒⢀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢀⡀⠁⣀⠀⠀⠀⠀⢘⠤⣌⠀⠰⠁⠀⠦⠀⠀⠀⡄⠀⠀⠀⠀⠐⣨⢤⡀⠀⠀⠀⠀⠘⠀⠀⡃⠀⠀⡀
⢸⠈⠁⠁⡁⡤⠀⠀⠠⠀⠀⠀⢐⠐⠰⠀⠀⠠⠴⠀⠁⠀⠀⠘⢀⠀⠀⠁⠘⠆⠂⠈⠀⠀⠀⠃⠈⠩⠬⡤⢄⣀⣁⣄⠆⠀⠀⠀⢠⡀⠀⡇⠀⠀⠇
⠰⠊⠀⠀⡄⠐⠐⠐⠂⠀⠀⠀⠈⠀⠀⡀⠐⣌⠁⠠⠃⠀⠀⠀⠁⠐⠈⠀⠀⠨⣀⠁⠀⠀⠀⠀⠆⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢨⠁⠣⣀⡀⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠢⠄⠀⠀⢸⠀⠁⠀⠀⠀⡂⠀⠀⠀⠀⠀⢅⢨⠀⠅⠀⠉⠉⠉⡀⠀⠀⡄⡁⠈⠁⣐⢂⡒⠒⢀⠇⠀⠀⠀⠀⠘⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠒⠒⠃⠀⠀⠉⠉⠉⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠠⢀⢴⢪⢖⠂⠄⢄⣠⡀⠀⠀⠁⠶⣐⠒⠀⠂⢂⠲⠈⠀⠀⢘⠊⡉⠫⢉⠐⠂⠂⠒⠃⠀⠀⠀⠀⠀⣿⠉⢹⠀⠀⠀⠀⠀⠀⠰⢉⠂⠀⠀⠀⢎⡁
⡇⠁⠰⠀⡁⡆⡸⡐⠳⠜⠀⠀⠀⡇⠀⡱⠤⢆⠁⠀⠀⠀⠀⠠⡊⢀⣒⠀⠡⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋⠩⠉⠂⠀⠀⠀⠀⠀⠊⠒⠀⠠⠂⠀⡐⠂
⠐⠔⡀⡚⠕⠑⠊⠀⠀⠀⠀⠀⡀⢧⢚⡀⠂⢀⠁⢤⠀⠀⠀⢸⠘⠒⠂⢒⠡⠢⢠⠑⡄⠀⠀⢶⠖⢖⡆⢈⠀⡖⠶⢖⠂⠀⠀⠚⠒⠣⠬⠭⠤⠓⠚
⠀⠐⢯⠗⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡠⠒⠉⠉⠈⠉⠁⠀⠀⢸⠀⠀⠀⠈⠂⠌⠥⠆⡀⠀⠀⣾⠤⠼⡏⠉⠉⡧⠤⠼⡀⠀⠀⠀⠀⠠⠡⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢠⠀⠀⠀⡔⠠⠀⠀⠀⠀⠀⠄⠀⠀⢀⠀⠐⡄⠀⠀⠀⠀⠠⠀⡄⠀⠀⡔⢠⠀⠀⠀⠀⣀⣀⣀⣀⠀⣀⣀⣀
⠨⠠⠛⠓⠵⡦⢠⠛⠂⠇⠀⠀⡠⠑⠈⢄⠀⡠⠈⠊⠄⡀⠀⠀⢠⡅⠂⠐⢄⠔⠁⣇⠀⠀⠀⢀⡌⠂⢄⠀⢀⠀⠈⣀⠀⠀⠀⠈⢁⢈⠀⣺⢈⠁⡈
⢠⡈⢡⢐⠨⣧⡈⢢⡀⡄⠀⠀⠀⠀⠀⠣⣛⠜⡆⠀⠆⡇⠀⠀⠃⢆⠠⠀⠀⠀⢀⠗⡇⠀⠀⠈⠂⠀⠘⠁⠁⠀⠀⡄⠄⠀⠀⠀⣈⢎⠀⣽⢈⠶⡁
⠀⢠⠤⡀⢸⡇⢩⠥⡀⠁⠀⠀⠸⠀⠀⠀⠀⠀⡃⠀⡇⠁⠀⠀⠀⠀⡱⠦⡠⢆⡁⠀⡇⠀⠀⠈⡃⡀⠘⠁⡙⠀⠀⢹⠀⠀⠀⠀⡘⢿⡄⠺⢸⠟⡇
⠐⠀⠉⠀⠊⠁⠀⠉⠀⠃⠀⠀⢘⢀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⣄⠊⠀⠀⠀⠀⠙⢢⠇⠀⠀⠀⠆⡇⠠⠀⠉⢉⡉⠏⠀⠀⠀⠊⠉⠉⠒⠐⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠨⠋⠛⠛⠛⡒⠲⢄⠀⠀⠀⠀⠀⣀⣀⠀⠀⣀⡦⡄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠐⠀⡀⠀⠄⢹⠀⠈⠀⠀⠀⠀⡨⠰⣀⡄⢀⠀⠀⡇⠀⠀⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠖⠈⢹⠀⠁⠀⣀⠀⠀⢀⣠⠾⠃⠁⢀⠿⡞⠀⠀⠇⠉⠠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠳⣄⣀⡀⠀⠀⠐⠀⢤⣀⡄⠀⠁⢰⠤⡀⠟⠀⠘⢿⠟⠘⠀⠀⠐⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢒⣩⡩⠭⠉⠉⣋⠴⠂⠀⢌⣛⣃⣞⡀⠐⠒⠉⠀⠀⠀⠠⠄⠀⡠⠊⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠋⠀⢀⣀⠀⢼⠀⠀⢀⠅⠀⠀⠈⠑⠀⠀⣀⠀⠀⢳⣤⣄⠊⠀⢦⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠀⢤⡄⠈⠋⠀⢂⠉⠀⠸⣇⠀⠁⠄⠀⣀⠀⣀⣀⣀⠀⠀⠉⢲⠉⠂⠠⠠⠀⠑⣬⣦⡀⡀⠀⢄⣠⡎
⠀⠀⠀⠀⠀⠀⠈⠀⠈⡁⢁⠀⠠⣤⣴⢀⠁⡄⠀⠀⠀⠈⠄⠄⢌⠀⠤⠴⣦⣦⠄⣈⡀⠠⣠⠀⠀⠀⠀⠈⠙⠁⣴⡿⠋
⠀⠀⠀⠀⠀⡀⢁⠀⠉⣥⢠⠑⠦⣤⠡⠎⡄⢰⠀⠀⠐⢄⣶⣾⡇⡇⠀⠀⠈⠈⠓⢀⡙⠰⠉⠂⠶⠞⣠⠜⠁⢪⠁⣀⡀
⠀⠀⠀⠀⠀⡄⠘⢄⡉⠀⠁⣀⠀⠀⢀⣀⣁⡤⠖⢶⠖⡎⠪⠍⠴⠃⠘⠓⠈⠛⠂⢈⠃⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⣸⠀⠀⠀⢀⠀⠀⠈⠗⠉⠀⠀⠀⠀⠀⢸⠀⠘⠦⣀⡀⠀⢄⢀⣀⣡⠀⢸⢄⠀⠀⣰⡄⠀⠀⠀⠀⢀⠆⢀⠀⠀⠳
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠉⠓⠒⠀⠠⠤⢤⡤⠤⠴⡆⠀⠀⡀⠀⠀⠐⠈⠻⡿⠟⠀⠊⠈⠀⣺⣛⡀⢢⠀⠀⠀⠎⠀⠘⡄
⣴⣴⣦⣶⣤⣴⣴⣦⣶⣴⣴⣦⣤⣤⣤⣤⣤⣤⣴⣷⣤⣾⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣷⣷⣶⣴⣶⣶⣾⣴⣶⣴⣧⣴⣴
⣿⣿⣴⣭⣦⣤⣽⣵⣿⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣒⣐⣋⣀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣦⣤⣄⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣷⣄⡀⢀⣰⣿⣿⣿⠿⠛⠋⠙⠛⠿⣿⣿⣷⡄⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠠⡈⢿⣿⣿⡆⠀⣿⣿⡟⠙⠛⠛⠿⠿⠟⠋⣿⣿
⠀⠀⠀⠀⠸⣿⠿⣿⡿⠋⠹⣿⣿⠟⢿⡇⢽⣿⣿⡇⠀⠤⠀⠀⠀⢠⡄⠀⢸⣿⣿⡆⠀⢸⣿⠃⡒⠃⠀⠀⠀⠓⠂⢿⡟
⠀⠀⠀⠀⢀⣿⠀⠀⣠⡀⠀⠀⣰⡆⢸⠶⠘⣿⡟⡇⠀⢀⣦⠀⠀⠐⣰⡆⠸⢿⣿⡗⠀⠈⠻⠸⣷⠄⠀⠀⢾⡆⠀⠸⠛
⠀⠀⠀⠀⠠⠈⠀⠀⠛⠁⠀⠀⠙⠁⠐⠈⠀⠻⣧⣼⠀⠘⠉⠀⠀⠀⠋⠁⣧⣾⠟⡀⠀⠀⠈⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠁⠀⠀⠄⠈⠙⢂⠀⠘⠶⠆⠀⠀⠈⠉⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢤⠈⢀⠀⠀⢎⡠⠀⠀⠀⢀⢦⠄⠠⡖⢀⡀⠘⠚⠁⠀⠀⠀⠀⢠⢀⣸⠃⠀⣹⡔⣀
⠀⠀⠀⠀⠀⠀⠔⠂⠉⠁⠀⠀⠨⠄⣀⠀⠀⠀⠀⠀⠉⠁⠀⠁⠀⠀⠣⢀⠀⠐⠀⢄⡀⠄⠀⠉⡀⠀⠀⠈⠀⠀⡀⠁⠀⡀
⠀⠀⠀⢀⠐⡀⢀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣃⣀⡔⣁⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠲⠤⠀⠀⠀⠀⠀⠀⠤⡁⣦⣤⠈
⠀⠀⠀⠠⠈⠉⣓⢎⢈⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⢤⠭⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠦⠶⡶
⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢐⠠⠂⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠂⢡
⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣀⣀⣀⣀⣤⣤⣆⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠤⠠⠤⠀⠀⠠⠄⠰⠂⠀⠀⠀⠀⠂⠉⠉⠛⠛⠩⠉⠰⠀⠀⠀⠀⠀⠀⠀⠃⠀⠆⠀⠀⠀⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⠀⢄⢀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠘⠀⠀⠀⠀⢈⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠠⠊⠂⠀⠀⠀⡀⠁⢂⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠤⢀⡴⠂⠀⠀⠀⠘⠦⡄⠂⠀⠐⠠⠴⠋⠀⠀⠀⠀⠑⠦⠈⠀⠀⠀⠀⠀⠃⠤⠞⠀⠀⠀⠀⠘⠢⠄⠃


⣿⣟⢹⣟⠻⣿⣛⠟⣻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠘⠆⢷⣄⡀⠀⠀⠀⠀⠀⣠⠞⢋⠉⠀⠀⠉⠙⢯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⢀⡙⢦⠀⠀⠀⢰⠇⣴⠋⠀⢣⡀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⣀⣼⠁⠀⠀⠀⠐⠊⠁⠀⢻⢸⡇⠀⠀⢸⡄⢷⡀⠀⠀⠀⠀⠀⠀⠀⡿⣦
⠀⠀⠀⠀⠀⢸⣁⣿⠀⠀⠀⣦⠀⠀⠀⣠⠟⣼⠁⣀⣀⣀⣹⣤⡙⠒⠒⠋⠀⠀⣠⡾⢿⣍⡀
⠀⠀⠀⠀⠀⠀⠉⢹⡄⠀⠀⠈⠛⠒⠛⢁⡼⠻⣍⣭⠵⠖⠚⢻⡏⠙⠓⠒⠛⠋⠙⠷⢾⡀⠙⣆
⠀⠀⠀⠀⠀⠀⢀⣴⡿⣦⣄⣀⣀⡤⠖⠋⠀⠀⠹⣆⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⣿
⠀⠀⠀⠀⢀⡴⠋⢸⣥⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣀
⠀⠀⠀⢀⡞⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⡇
⠀⠀⠀⠘⣧⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⣾
⠀⠀⠀⠀⣨⢽⡏⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⠴⠖⢻⡇⢹⠋⠛⠛⠛⠋⣉⣉⣉⠉⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠻⠞⢻⠛⠛⠛⠋⠉⣉⣉⣉⣤⠀⠀⠀⠀⢸⠁⡿⠀⠀⠀⢻⡉⠉⠁⢸⡄⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⠉⠉⠁⠀⢸⠀⠀⠀⠀⣿⢠⡇⠀⠀⠀⠈⡇⠀⠀⠘⠧⠤⢶⠖⠋
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢸⡀⠀⠀⠀⠸⠶⠶⡶⠚⠃⠀⠛⠛⢻⠛⠉⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠉⣽⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⠤⢤
⡴⢻⠟⠦⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏
⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢶⠀⣶⠲⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠠⠤⣤⣀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠄⣀⠀⠀⢠⣾⣿⠟⠁⠀⠈⠙⠲⢮⣟⠎⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⡵⠒⡁⠀⠀⠒⢮⣝⣦⡏⢸⡏⠀⠀⠀⠀⠀⠀⠀⠈⠳⠄⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣫⣶⡿⠂⠀⠀⠀⠀⠘⢿⣧⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠟⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⡇⠀⠀⠠⠤⠭⠄⠀⠀⠀⢀⡏⢹
⠀⠀⠀⠀⢀⣠⠄⠀⣀⠤⠒⠀⠀⠀⠈⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣼⣂⠰⣄⡀⠀⠀⢀⣠⠔⠁⢸⡇⢸⡇
⠀⠀⢀⡴⠋⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠸⡇⣿⠠⠀⠀⠀⠢⣒⡒⣐⡰⠂⢠⢣⡏⣿⢳⡀⠉⠛⠛⠉⠁⢠⠂⣿⡇⠈⣧⠀⠀⠀⠀⠠⡀
⠀⢠⠏⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⠀⡇⣏⠀⠱⣄⠀⠀⠈⠉⠉⠀⣠⢏⡾⢠⡿⢸⠇⠀⠀⠀⠀⠀⡞⡼⣿⡇⠀⢻⡀⠀⠀⠀⠀⠐⡄
⢀⡏⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⢸⡆⡇⢳⠳⣄⠈⠓⠦⢤⣤⠴⠚⡁⣼⢁⡾⢡⡟⠀⠀⠀⠀⠀⢸⡇⣇⣿⢷⠀⠈⢧⠀⠀⠀⠀⠀⠘⢦
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠈⣧⠘⠄⠀⠘⡇⠀⠀⠀⠀⠀⣰⢱⠇⣼⠃⡟⠠⡀⠀⠀⠀⠀⢸⠀⢿⢸⡞⣆⠀⠈⢦⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠉⠀⠀⠀⢀⣦⡏⣾⠀⣿⠀⠁⠀⢳⡀⠀⠀⠀⢸⠀⢸⡇⢳⡘⢆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠢⡀⠀⠀⠀⠀⢸⣿⠃⡇⠀⠋⠃⠀⠀⠀⠳⢤⡤⠔⠘⠆⠀⡇⠀⠹⡌⠂⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢱⠀⠀⡞⠀⣿⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⠀⠀⢸⡄⠀⠈⠳⡄⠀⠀⠀⠘
⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⡼⠀⠀⢹⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠁⠀⠀⠀⢀
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠔⠋⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠲⠤⠤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠒⠉⠉⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏

⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣦⡀
⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠙⣿⡀
⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⠀⣸⣧
⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⢀⣤⣤⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⠀⠀⠀⠀⠀⠀⣴⣿⠃⠀⠀⠀⠘⣿⣇
⠀⠀⠘⣿⣿⣄⠀⠀⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⣀
⠀⠀⠀⠹⣿⣿⡆⠀⠀⢾⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣀⣴⣿⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⢀⣾⣿⣿⣿⡆
⠀⠀⠀⠀⢻⣿⣿⣦⣤⣼⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣷⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⠈⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣼⣿⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢠⣿⠇⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠀⣼⠏⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣷⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢹⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⣿⡿⠀⠀⠀⠉⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⠀⠘⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠀⢿⣿⣿⣿⠀⠀⠘
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠘⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣯⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣼⣿⠀⢴⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⢸⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠀⠀⠈⠻⠇
⠀⠀⡀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠙⠏⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⢸⣿⡋⣀
⣿⣿⠿⠻⡿⠛⠿⢿⡿⢿⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⢀⣠⡴⠶
⡈⢷⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣼⣿⠆⠶⠒⠛⠟⠁
⠈⠋⠻⡀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢘⠀⠖⠚⢛⠻⠿⠂⠀⠀⡄
⠀⠀⠀⠱⣧⡀⠃⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣰⣾⠏⠀⠠⠶⠞⠛⠀⠴⠄
⢰⡲⢼⣿⣿⣿⠀⠀⡘⠂⠀⠁⠒⠀⠀⠀⠄⢀⢄⠀⣠⡯⠀⠋⠋
⠀⠀⠀⢀⢼⣷⣤⣴⣿⡄⣠⠀⠀⠀⠀⡦⠐⠆⠑⡿⠉⠳⠒⠈⠁
⠀⠀⠙⠁⠀⠀⠩⠀⠀⠀⠿⠃⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠐⠀⠱⠀⠀⠀⡀⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠑⡄⠄⢀⠐⡲⠀⠀⠀⠖⠁⡠⠀⢀⣤⣶⡄⠀⠀⠀⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡈⠁⠀⠀⠀⠀⠀⢁⡆⠘⠙⠻⢧⡄⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠶⣤⣀⡀⠀⡠⠀⠏⠀⠀⠀⠈⠁⠈⠙⠀⠀⠀⢠⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠠⢏⣀⡘⢤⡈⠙⠃⠈⠀⠄⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⡠⠞⠀⠀⠀⠙⠦⠤⠈⠂⠀⠀⠺⠀⠀⠀⠀⠀⠘⠀⠂⠀⡬⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⢀⢰⠀⣼⠇
⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⡆⣒⣁⣸⣿⡋⠀⡔⢁⣨⡄
⠀⠀⠀⠀⠁⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣥⣶⠗⠛⠛⠉⠉⣇⠀⢢⠋⡀⠀⠀⢰⡄
⠀⠀⠀⡼⠀⠀⠀⠀⢊⣠⠀⠀⠀⠒⠈⣁⠀⠀⠀⠀⣘⡿⣿⠃⠀⠀⢀⣀⠠⠼⣤⠼⣾⣏⣁⡀⠞
⠀⠀⠀⡀⠀⠀⠀⢀⠋⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠚⣡⠿⠃⠀⠀⡔⠋⠀⠀⠀⡏⠀⠻⣿⡟⠃
⠀⢠⠀⠀⠀⠀⢀⠈⠀⢀⣉⠁⠀⠀⠀⠀⠀⠀⢀⡞⠙⠀⠀⠀⠀⡀⠀⠀⠀⠀⡥⠀⣧⣧⠀⣀
⢠⠟⠀⠀⠀⠀⠘⡗⠶⢫⠀⠀⠀⠁⠀⠀⠀⠀⡟⠀⠟⠁⠐⠀⠀⠹⠀⠀⣀⠀⠀⠀⠈⠙⠋⠁
⠀⠳⣄⠀⡴⢢⠀⠧⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡆⠀⢠⣀⣠⣴⠃⠀⠋
⠀⠀⠈⠀⠀⠀⠑⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠯⡹⠋⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠘⢿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢰⠶⡀⡂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣠⣤⣤⢤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⢠⣴⣿⣿⣿⣿⣿⣷⣾⣿⣦⡀
⠀⠀⠀⠀⢰⣿⣿⣿⡏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢸⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠉⠛⠃⠀⠀⠀⠀⠀⣿⠀⠈⠙⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡇⣿⡟⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠈⠉⠉⠉⠁⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣼⣿⣿⣿⠿⠼⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠀⡴⣯⠀⠀⠀⠀⠀⠀⠀⠏⢹⣿⣿⡇
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠔⠁⢻⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠍⠐⣆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣿⣗⣾⣀⡤⠠⣔⣁⣀⡀⣈⠢⠀⠀⠀⠀⠀⢀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠐⡀⠀⠈⠑⣤⠀⠀⡀⠀⠀⡀⠀⠆⠉⠀⠐⠁⠀⠀⡠⠀⠈⠂⠀⠈⠁⠎⣀⠠⠚⡛⠛⡀⠜⠚⠷⠺⢆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠐⠊⠀⡆⠑⢦⠤⣚⣁⣠⠤⠞⢦⣤⡆⠀⠀⠀⠀⠀⠀⢰⠀⣤⠔⠉⠀⠀⠌⠀⠀⠀⠀⠀⠀⠡⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠖⠀⠀⠀⠀⠀⠀⠌⠀⠄⠨⠄⠀⠂⠀⠀⠚⠋⠁⠀⠀⠀⠀⠀⠀⡎⠀⠀⠑⢠⣀⢰⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠠⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡀⠀⠀⠒⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠐⢦⠒⠂⠒⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠦⠦⠶⠤⠶⢸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⡆⢠⠂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠀⢲⡀⠠⠂⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠢⢨⡀⠀⠀⠀⠀⠀⣦⣤⣤⣤⣤⡤⠤⢴⡇⢡⡎
⠀⠀⠀⠀⠀⠀⠀⠐⣶⣮⠁⠀⠀⠐⡆⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠨⡇⠀⠀⠀⠀⠀⠇⠀⠀⠀⡹⣄⡀⣾⣷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠚⡖⠒⠞⠳⠒⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢰⡇⠀⠀⠀⠀⠀⠃⠀⠀⢈⡃⠈⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⠁⠀⠀⠀⠀⠀⢹⠀⠀⠸⠃⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢸⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣄⠀⢨⣀⣨⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡁⠀⢈⢈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⢰⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋⠀⠘⠛⠛⠛⠀⠀⠀⠀⠀⠛⠛⠛⠋⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣾⣶⣥⣤⣀
⠀⠀⠀⠀⠀⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣾⣿⣿⠿⣁⡀⠀⠀⠈⠉⣉⡈⠀⢻⣿⣿⣷⠀⠀⠠⠦⠈⣿⡿⠙⣿⠿⠿⠿⠀⠙⠿⣿⣿⠛⣿⣿⠁
⠀⠀⠀⠀⠀⣿⣿⡇⢈⠁⣀⠀⠀⠀⠀⣉⢉⡀⠸⣿⣟⠛⠒⢠⡃⠶⡶⢻⡇⣠⣔⠂⠄⠀⠀⠀⣠⠄⠤⠀⢿⣏⡀
⠀⠀⠀⠀⠘⠃⢽⠃⠀⢠⣾⡇⠀⠀⠀⢠⣾⣿⢀⡟⠹⠐⠀⡊⠁⡘⢁⠘⡇⠻⣿⡇⠀⠀⠀⠸⣿⣆⠀⠀⡼⠓⠇
⠀⠀⠀⠀⠀⠁⠨⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠁⠘⠇⠂⠀⡇⠹⢰⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠇⠐
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⣤⣀⣀⣴⠀⠀⠀⠃⠀⠀⠀⢰⠃⡈⠀⠀⠀⠀⡄⠀⠀⠺⠀⠀⠕⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠛⠁⢀⡠⠊⠀⠀⠀⢀⠆⢀⠋⢀⡀⡀⠀⠈⠂⠄⠀⠀⠀⠀⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⢘⡖⠂⠐⢪⠀⠀⣄⠤⠂⠐⢅⢀⠚⢠⠊⠀⠈⠐⢦⢁⠠⡏⠀⠀⣹⣀⡒
⠀⠀⠀⠀⠀⠀⠀⡬⣾⡏⠀⠈⠒⠒⠁⠈⢿⣧⠀⠀⡀⠌⠁⠀⠈⠐⠠⠀⡀⢀⡟⠀⠀⠉⠉⠀⠘⢈⠓⠣⢄⡀
⠀⠀⠀⠀⠀⢀⠊⢰⡟⠇⠀⠀⠀⠀⠀⠀⠈⣿⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠅⡀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠰⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⡤⠸⡘⣦⣀⣆⡈⢣
⠀⠀⠀⠀⠐⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⢄⣠⠇⣩⠀⠁⠀⠘
⠀⠀⠀⠀⠀⢀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠬
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢘⣀⡀⢸⣶⣦⣤⣤⣀⣀⣀⣀⣠⣤⣴⣶⠁⠀⠀⠀⠀⠀⠀⠀⡋⠀⠒⠠⠤⠤⠤⠤⠤⠤⠀⠒⠚
⠀⠀⠀⠸⡿⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠐⢆⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠏⠀⠘⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠘⢤⠀⢀⣀⣠⠁⠀⢂⣀⠀⡀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣀⡀⡈⠀⠀⠀⠀⠀⠀⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡀⠀⠀⠀⠀⠀⠀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠒⡆⠀⠀⠀⠀⠀⠀⠓⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠃⠀⠀⠀⠀⠀⠀⠓⠐⡆
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠁⠀⠀⠀⠀⠀⠀⢘⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠃⠀⠀⠀⠀⠀⠀⢀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢃⠀⠀⠀⠀⠀⠀⠎⠀⠑⠄⠀⠀⠀⠀⠀⢀⠊⠀⠀⣁⠀⠀⠀⠀⠀⢠⠊⠀⠑
⠀⠀⠀⠀⠀⠰⠢⣀⣤⠞⠀⠀⠀⠀⠀⠈⠳⣄⡀⠐⠄⠀⠀⠀⠀⡋⢠⣤⠶⠋⠀⠀⠀⠀⠀⠘⢦⣀⡠⠰
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⢾⣿⢦⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢰⠞⠉⣀⣴⠿⠿⠿⣷⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⠛⠁⠀⠀⠀⠹⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠋⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⠀⢈⣽⠃⠀⠀⠈⡥⠹⠿⣿⣿⣿⠀⠀⠀⠀⠀⠁⠀⣿⠁⠐⢒⠶⠀⠀⠘⢋⡉⠀⠲⢤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⡷⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⣼⣰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣇⠀⠀⠀⠀⠀⠀⠀⣯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⡀⠀⠀⢸⣿⡟⠃⠀⠀⠀⠀⠀⠐⣿⠍⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠈⠿⡿⠋⠉⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢻⣆⢠⣿⡆⢠⡄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠂⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡢
⠀⠀⠀⠀⡂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠒⠒⠀⠄⠠⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠




⡀⣀⡀⢀⡀⣀⣀⣀⣀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⣄⣤⣶⣶⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣷⣿⣦
⠀⠀⠀⠀⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⣠⣤⣤⣶⣶⣶⣿⣿⣿⣶⡄
⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⣿⣿⣿⠿⠿⣿⣿⣿⣿⡿⢻⣿⣿⠄⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠘⢻⣿⣿⡿⠿⠿⢿⣿⡿⠿⣿⣿⣿⠁⠀⠀⢻⣿⡇⠠⠤⠀⠀⠀⠠⠤⠀⣿⡿⠀⠀⢠⣿⡿⠛⠻⠿⣿⡿⠿⠿⠿⣿⣷
⠀⠀⠀⠀⣿⣿⠋⠶⠂⠀⠀⠀⠶⠄⢸⣿⡿⠀⠀⢊⡸⢿⠁⣶⡀⠀⠀⢀⣴⡀⠀⢻⡧⡄⠀⠸⣿⡇⢀⣤⠀⠀⠀⢀⣀⠀⣿⡿
⠀⠀⠀⠀⡟⡿⠀⢈⣶⡀⠀⠀⣩⣷⢈⠟⡆⠀⠠⢀⠃⠜⡀⠛⠃⠀⠀⠈⠛⠃⠀⢨⡐⡧⠀⡴⣿⢡⣶⡂⠀⠀⢀⡤⠀⠀⣿⣃⠀⠀⣶⣶⡀
⠀⠀⠀⠀⠀⢽⠀⠈⠉⠀⠀⠀⠉⠁⢰⢄⠀⠀⠀⠃⠀⠀⢡⠀⠀⠆⠀⠀⠀⠀⢀⠀⠈⡀⠀⠁⠼⠈⠛⠃⠀⠀⠘⠿⠀⢀⡎⠹⠀⠘⡃⠌⠁
⠀⠀⠀⠀⠀⠀⢆⠀⠐⠶⣤⠖⠀⠀⠂⡈⠡⠀⢠⠀⠀⠀⠀⠑⠠⣀⠀⢀⡀⠐⠁⠀⠀⠠⠀⠀⠀⠁⠀⢠⣄⣀⣀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠤⠀⢀⡀⠰⠐⡁⠀⠀⠀⠀⢠⠤⠤⢇⡀⡀⠷⠩⢤⡀⠐⠀⠢⣀⠀⠈⠂⢀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⢀⢶⡆⠀⠀⠀⠉⠰⣯⠀⢀⢡⠘⠄⠠⠆⠀⡀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠑⣀⡆⠀⠀⠲⠔⢾⢁⢁⠩⠄⡢⠄⠀⠆⢀⠌
⠀⠀⠀⠀⠐⢀⣿⠃⠀⠀⠀⠀⠀⢹⡇⠀⠀⠁⠀⠀⠒⢴⢀⠀⠀⠀⠀⠀⢠⠤⣑⣦⡼⠀⠀⠀⠰⠂⠎⠀⠀⠀⠀⢧⠠⣀⠠⠜
⠀⠀⠀⠀⠉⣾⠏⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠘⠀⡷⠶⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠂⢧
⠀⠀⠀⡆⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢘⠀⠁⠐⠂⠐⠀⠒⠀⠌⡃⠀⠀⠀⢰⡴⠀⠀⠀⠀⠈⠁⠀⠈⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠄⢀⡀⠀⠀⠀⢀⡀⠞
⠀⠀⠀⠀⠀⢇⣀⡀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠈
⠀⠀⠀⣾⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠟⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⡀⠀⢀⡄⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠉⢛⠋⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠀⠀⠀⠀⠀⠀⡇⠀⠀⠌⠐⠠⠀⠊⠀⠈⢀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡀⡈⠀⠀⠀⠀⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢰⢠⠄⠀⠀⠀⠀⢤⢠⠂⠀⠠⠀⢰⠀⠀⠀⠀⠀⠀⡅⠀⠁
⠀⠀⠀⠀⠀⠀⡃⠃⠀⠀⠀⠀⠘⠁⡇⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠈⠒⠒⢊⠀⠙⡆⠀⡗⡰
⠀⠀⠀⠀⠀⡠⠃⠄⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⡠⠊⠉⡀⠀⠀⠀⠀⠈⠉⠢⠀⠀⠀⠀⠀⠀⠠⢆⠃⢀⢁
⠀⠀⠀⠀⠰⢀⣠⠞⠀⠀⠀⠀⠲⣄⡀⢦⠀⠀⠀⠀⠀⠓⠤⠞⠁⠀⠀⠀⠀⠙⠦⡔⠃⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠲

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣦⠟⢻⠛⣷⣶⠦⡀⠀⠀⣴⣿⣿⣿⣿⡟⢢⣄
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣻⠟⠉⠦⠜⠃⢀⠙⣤⣾⣿⡝⣿⣿⠟⢰⣓⣺⣿⡄
⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠆⢸⡟⠛⢻⡗⠘⠀⣽⣿⣿⣿⣷
⠀⠀⠀⠀⣸⣿⣿⣿⣿⡝⣿⣿⣿⣻⣿⣿⣦⠀⠀⠀⠀⠀⣰⣿⠛⠲⠄⠀⠀⠀⠰⠀⠀⠁⣠⢀⡄⠀⠈⠉⠛⠫⣿⡟
⠀⠀⠀⠀⡿⣽⣿⣿⣿⣿⣿⣼⣿⠻⣿⠹⣿⡗⠀⠀⠀⢰⡟⡟⢠⣤⣤⣬⣭⣙⠃⠄⠀⠀⠀⠀⠛⠀⡤⠻⣤⢸⡟
⠀⠀⠀⠀⢿⣿⣿⣿⣿⠿⠿⣿⣿⡆⣿⢳⡼⠙⢀⠀⠀⡿⢸⡇⢸⣿⣟⣿⣿⠀⢀⡴⢢⠀⠀⠀⠛⠂⠀⠀⢰⠋
⠀⠀⠀⠀⠀⠉⠉⢀⠀⠀⠀⢹⣿⣿⣿⠃⣿⡀⠀⠀⠘⢀⢸⠁⢸⣭⣉⠛⠿⠀⢈⡿⠃⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⢿⣿⣧⣉⠃⠀⠀⠀⠈⢸⠚⢪⣍⣙⠻⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠈⠋⠈⠃⠀⠀⠀⠀⠀⢸⠀⠼⠛⠛⠛⠀⠀⠀⡶⠀⠀⣀⡀⠀⣤⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⢣⣤⠀⠀⠀⠀⠀⣶⣄⠀⠀⠸⠀⠀⢀⣼⣿⡀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣾⠟⠀⠀⠀⠀⠀⣿⣿⣿⣶⣾⣶⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣤⣄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠙⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣠⣤⡄⠀⠀⠀⠀⠀⠈⠛⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢰⣿⣷⣶⣤⣄⣀⣀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠋⠑⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣦⠀⠀⠀⣀⡀⠀⢀⣾⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⣿⣿⣿⣿⣿⣿⡆⢰⣿⣿⣿⣷⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⢿⡿⠿⠿⠿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⠿⢿⡿⠛⠛⠋⠻⣿⣿⡁⠀⣠⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣿⡟⠀⠀⠐⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇⠀⠀⡈⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⣀⠸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢀⣿⡏⠀⢈⣉⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠿⣿⣿⣿⠃⠀⣼⣶⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠶⢦⣤⣤⠀⣠⣿⣤⣀⣀⣠⣤⣶⣶⡄⠀⠀⠀⠀⢼⣿⡿⠛⣶⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀⣽⣇⢸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠸⣿⡿⠈⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠻⠁⠀⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡤⡆⢲⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⠈⠛⠛⣿⡿⣿⢿⣿⠿⠿⠟⢿⠀⠀⠘⠁⢻⠻⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠉⠙⠟⠀⠀⠀⠫⠀⠀⠀⠏⠀⠀⠀⠘⠀⠀⠀⠀⡞⠀⠈⠙⠿⠟⠹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢠⣾⣏⠀⠀⠀⠀⠀⠀⢀⠀⣄⠀⠀⠸⣧⡀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⢠⡆⠀⢠⣾⡟⠀⠀⢰⣿⣟⣄⠀⠈⣁⣠⣤⣿⣾⠉⣴⢃⣾⣉⣛⡀⣧⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⡀⠀⣀⣿⡀⠾⢛⡇⣀⣀⡻⣃⠀⡄⣽⣿⠻⠘⡟⠆⠙⠛⢻⡾⠉⠿⠟⠋⠛⠉⠉⠛⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠒⠢⠤⠴⠏⡉⠋⠐⢩⠂⠾⠛⠟⠛⠻⠟⠛⠋⠁⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⡀⠀⠀⠀⠔⠁⠀⠀⠀⠉⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⡀⠀⠠⠀⢀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠆⠀⠀⠠⠀⠊⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠋⠀⠀⠈⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣄⣄⡀⠀⡅⠀⠀⠀⠈⠀⠉⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⣖⡷⠃⠀⠛⠚⠃⢀⠆⠀⠐⠀⠈⡁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠈⠠⠈⠁⠈⠱⠃⠀⠀⠀⠀⠀⢦⠀⢴⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠌⠀⠀⡆⠀⣠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠛⠀⠀⠀⠀⠈⠀⠀⠀⢠⠂⠀⠀⣠⠀⠀⠊⠀⠀⠀⠠⡀⠀⢎⣀⠠⠀⡰⡁⠀⠀⠀⠑⡉⠨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠐⠚⠀⠀⠀⠀⠀⠀⠀⠀⡙⠒⢆⠤⠀⠊⠀⠙⠢⡀⠂⠂⣀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠒⢈⠐⠀⠐⠊⠁⠀⠀⠀⡀⠀⠀⠀⠆⢈⠒⢄⡈
⠀⠀⠀⠀⠀⢀⣠⠀⠀⠰⠀⠀⠠⣄⡀⠀⠈⠀⠀⠐⠆⠀⠀⠀⠀⠀⠊⠀⠀⡀⠀⠀⠦⣀⣀⢰⠁⣀⡀⠄⠂⠂⠀⠀⠁⠰⡀⠁⠡⠂
⠀⠀⠀⠀⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⢨⠃⠁⠀⠀⠉⠈⠑⡀
⠀⠀⠀⠀⠀⡀⠀⢄⠀⠀⢠⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠐⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠄⡃⢀⣀⠀⠈⠂⠓⠀⠀⢀⠄⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⣀⠤⠃⠀⠈⠁⠂⠀⠀⠂⠀⠈⠉⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⢶⡉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⡆⢀⡔⠀⠀⠀⠀⠀⠐⠂⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⡈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠎⠀⠀⠀⠀⠄⠂⢠⠀⠀⠀⠐⠀⢀⡀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠒⡠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠐⠀⠀⠱⠐⠐⢈⠀⠀⠀⠠⠀⢨⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⡀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠠⡇⠀⠀⠀⠀⠀⢠⡀⠠
⠀⠐⠀⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡁⠀⠀⠈⡀⠀⠔⠀⠀⠀⡠⠇⠀⠀⠀⠀⠀⠈⠀⠂
⣀⠄⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣀⣀⠠⠤⠀⠠⠤⡄⡗⠀⠀⠀⠀⠀⠈⠀⠀⠀⡃⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠒⠒⠓⠒⠾⠤⠄⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠈⠛⠉⠙⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠛⣅⠀⠀⠀⠀⠀⣿⡿⠛⠿⠿⠟⠛⣿⡇⠀⠀⠀⣠⡴⢮⠁⠙⢿⡟⢻⣿⣿⡇
⠀⠀⠀⣸⣿⣿⣿⣿⣿⠃⠈⠃⢰⠒⡄⠀⠀⣿⠇⣀⣀⠀⢀⣀⣸⡇⠀⠀⢠⣣⠃⢀⠀⠀⠀⠀⠸⣿⣿⠇
⠀⠀⣰⣿⣿⣿⣿⣿⡏⠀⠀⣤⣈⠘⣁⣦⠆⠛⠈⠳⠂⠀⠐⠓⠈⠙⠰⣰⣀⡁⠀⣠⡀⠀⠀⠀⠂⠀⠀⡀
⠀⣴⣿⣿⣿⣿⣿⣿⡟⠀⠀⠁⠈⠉⣍⡋⢠⠴⠀⠀⠀⠄⠀⠀⠀⢠⣘⠛⠉⡙⡉⠁⢢⠀⠄⠀⠠⢀⡥⠷⡀
⠨⢿⣿⠟⠁⠘⢿⣿⣷⠀⠀⢀⡎⠀⠀⠒⠂⠀⠀⡄⠀⠻⠟⢀⠆⠀⠦⠤⠉⠀⠈⣂⣨⡄⣸⣧⠞⠅⠒⠤⡈⢢
⠐⠛⣿⠀⠀⠀⡌⢿⣿⡘⠀⢠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡇⠀⠱⡜⠱⠁⠀⠀⠀⠈⣄⠆
⠀⠀⠘⠀⠀⠀⡧⠀⠈⠀⢀⠄⠀⠀⡀⠀⠀⠊⠡⠀⠀⡀⠀⠀⡈⠐⠀⠄⢀⠀⠀⣧⠀⠀⡇⢧⠀⠀⠀⠀⡄⣿⢰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠲⣦⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠲⣦⡀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⢸⡇⠀⠀⠀⢠⡟⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⣸⡇⠀⠀⠀⠀⢿⡄⠀⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠈⠉⠉⢤⣄⠀⠀⠀⠀⠀⣠⣤⡌⠛⠛⠋⣠⣄⡀⠀⠀⠈⠻⠶⠞⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⣠⠀⠀⠀⣿⠀⠀⠀⠀⣾⠋⠀⠀⠀⡤⠀⠀⠈⢻⡆⠀⣾⠃⠀⠀⠉⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢿⠀⠀⠀⣿⠀⠀⠀⣼⠇⠀⠀⠀⠀⠦⠀⠀⠀⢸⡇⠀⣿⠀⠀⢰⡆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⠀⠀⢰⡟⠀⢰⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢀⡄⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⣿⠀⣿⠀⠀⣿⠁⢠⡿⣿⠀⠀⠀⠀⠀⣦⢸⡇⠀⣿⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢙⣀⣿⠀⣿⣀⡙⠀⣼⠇⢀⡿⠁⣿⠀⠀⠀⠀⠀⣿⢸⡇⠀⣿⢸⡇⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢻⣤⡿⠉⣿⠀⠿⠶⠞⠁⠀⣿⠀⠀⣄⠀⠀⣿⣼⡇⠀⣿⠛⢧⣼⠟⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⢰⡏⠀⠀⠀⣿⠀⢠⡄⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠿⠷⠶⠟⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢰⡇⠀⠀⠀⠀⠀⠀⢹⡄⠀⣿⠀⢸⡇⠀⠀⠀⢹⡀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⠃⠀⠀⠀⢸⡇⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠿⠶⠾⠃⠀⠀⠀⠀⠀⠀⠘⠷⠶⠿⠶⠟⠀⠀⠀⠀⠘⠷⠾⠷⠶⠏




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

⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⡀
⠀⠀⠀⢀⣾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣿⣿⣿⣧
⠀⠀⠀⠾⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠿⠿⠛
⠀⠀⠀⠀⠈⠹⢿⣿⣿⣿⣷⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⡟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣶⣴⣶⣶⣦⡀⠀⠀⠀⠀⢀⣴⠟⠙⠻⠿⠿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⣿⣿⣿⡿⡿⢶⡿⡿⡿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⢞⣿⣿⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⡜⠉⡵⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣩⠀⣿⣿⣀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣟⢁⣴⣿⡷⣶⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠟⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⣿⠉⣝⣿⣧⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⢿⣿⣿⣿⣿⠿⣿⣾⣿⣿⡇⢸⠟⣿⣾⡿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢇⠰⣯⣿⣯⠛⠀⣿⣿⣿⣾⣷⠈⣶⣽⣻⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⡄⠘⣟⡟⠀⣼⣿⣿⣿⣿⣿⠀⠛⠛⠋⠈⢀⣡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡾⣭⣿⣿⣿⣿⠟⣷⣿⣿⡶⠶⠶⠆⠀⢀⠉⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠈⠛⠷⠾⠿⠿⠿⠷⠟⠛⠉⠀⣤⣤⣴⡂⠘⠻⠿⣿⣾⣦⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⡦⠀⠀⠀⠁⠀⠀⠘⢷⣄⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢀⣀⣶⣿⣿⣿⣿⠿⠛⠽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣿⣿⣋⡆⡀
⠀⠀⠰⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢾⢯⣟⣿⡖
⠀⠀⠀⠙⠿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢞⡳⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣶⣦⡀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠛⠂⠀⠀⠠⠀⠀⠒⠀⠐⠂⠄⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢾⣿⣿⣿⡿⢿⣿⣶⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢀⠀⠀⢰⣤⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⢉⣡⠄⢠⡽⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⡩⠀⠉⡁⣀⠴⣿⣿⣿⠃⡀⠁⠘⠉⠉⢉⠀⠸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠳⠆⠁⠸⣻⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⡘⡠⠂⠉⡀⡋⠁⠀⢸⣿⡇⢠⣄⠀⠀⠀⢀⠀⠁⠀⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢂⡀⠁⣠⠊⠀⠻⠿⠟⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⢀⡰⠁⠿⠃⠘⠁⠑⠠⡤⠀⠽⡇⠀⠉⠰⠃⠀⠛⠁⠀⠰⢿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⢠⣠⡆⠴⠷⠰⠏⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⠀⡘⠩⠋⠀⠀⢤⡀⠀⠀⢠⠴⠋⠰⠢⣑⠐⢄⠘⠛⣫⠆⡔⠀⢀⠌⠛⠿⢿⠿⠛⠛⠁
⠀⠀⢠⣾⡄⢈⢻⠔⢃⣰⠀⠈⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠘⣧⠀⠢⠞⣔⠀⠈⠀⠈⡅⠈⠁⠚⣁⠜⡅⠐⠁
⠀⠀⢦⣎⠛⢃⡀⢀⣿⠇⣀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⣀⣀⠀⠀⠘⠃⢀⣀⠀⠀⠙⢂⠆⠐⠈⢁⠏⠁⠀⣧⠀⠀⠀⠑⣸⠋⠀⠀⠀⢂
⠀⠀⠀⠙⢄⣀⠈⣁⣀⣀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⠒⠆⠀⠌⡠⠓⠲⠶⠂⠀⠀⢀⠌⠀⢀⣢⠟⠀⠀⣴⢊⣹⠆⠀⢀⡈⠩⡄
⠀⠀⠀⠀⠀⠙⠗⣛⣋⠠⠋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠈⠁⡀⠐⢸⡁⠀⠀⠀⠀⠀⡶⠓⠂⣀⡼⠃⠀⢀⡎⠉⠁⠀⠀⠤⠤⠐⠂⣃
⠀⠀⠀⠀⠀⢠⡀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠐⣴⠛⠙⠃⠀⢸⠀⠀⡷⢶⠖⠋⠀⠀⡰⡋⠤⢀⢀⠐⡂⠈⣉⡇⠀⠘⠆
⠀⠀⠀⠀⠀⠀⡃⠀⠈⠣⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⢄⣸⠀⠀⡇⣩⠞⢀⣠⠞⡥⠛⠒⠤⠀⠀⣆⠀⣾⡀⠠⢤⣽
⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⣿⠀⠠⠀⠡⠒⢉⣩⠂⣽⣛⣋⠀⠀⠀⠈⠉⠛⠀⠀⠀⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⡰⠀⠆⠀⠀⠀⠀⠀⠀⠀⡎⠀⠠⡠⠀⠀⠀⠀⠀⠀⠀⢢⣇⣤⠁⠀⠠⡆⠀⠈⣁⡤⠓⠐⢧⣤⡶⠖⠢⣤⣀⡀⠀⠀⠀⠄
⠀⠀⠀⠀⢀⡁⠀⠀⢠⠃⠀⠈⢁⠀⠀⠀⠀⠀⢠⠁⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠔⠹⠑⠀⣤⠞⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡏
⠀⠀⠀⠀⡘⠁⠀⠀⠇⠠⡀⠀⠀⢀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⢦⣀⠀⠠⠐⠀⣠⠟⠤⠊⠀⠀⠀⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣧
⠀⠀⠀⣰⠇⠀⠀⠀⠁⠀⢣⣀⣀⠘⢀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡶⠚⠁⠀⠀⠀⡠⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣽
⠮⣀⠳⢿⣶⣄⣠⡜⢀⠴⠿⠿⢋⡤⠏⠺⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⢦⡀⠠⠀⠀⠀⠀⠠⢁⣀⡀⠤⠄⠀⠀⠒⠒⠶⠂⠤⣤⣸
⠀⠀⠳⢤⣈⠉⢉⡁⠛⠶⠶⠚⠉⠀⢀⣴⣿⣿⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡖⠁⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠘⠛⠓⠒⠓⠂

⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⢷⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠙⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣀⣀⣠⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣤⣀⣀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉
⠀⠀⠀⠀⣠⣴⣶⣶⣤⣀⠀⠀⢀⣠⣶⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣶⣦⣀⡀⠀⢀⣠⣴⣶⣶⣤⡀
⠀⠀⢀⣾⠟⠁⠀⠈⠙⠛⠿⠿⠟⠛⠁⠀⠈⠙⠟⠀⢀⣴⣶⡶⣶⣶⣄⠀⠘⠛⠉⠀⠀⠙⠛⠻⠿⠟⠛⠉⠀⠈⠙⢿⣆
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⣿⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⣠⣶⣶⠀⠀⢸⣯⠀⠀⠀⠀⠀⢨⣿⠀⠀⢰⣶⣦⡀⠀⠀⠀⠀⠀⢠⣶⣶⡄⢸⣿
⠀⠀⢸⣿⠀⠀⢸⣿⡁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⢿⣦⣀⠀⢀⣠⣾⠟⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⠛⠿⠟⠛⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿
⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢀⣠⣴⣶⣦⣀⠀⠀⠀⢀⣠⣶⣶⣤⣀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠉⠀⠉⠛⠿⠿⠿⠟⠋⠁⠀⠙⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⣤⣤⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⢹⣿⠀⠀⠀⠀⠀⢸⣿⠉⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠾⠷⠿⠿⠾⠾⠿⠿⠿⠾⠿⠷⠿⠶⠿⠇



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡄⠀⣴⣿⣿⣿⣷⠀⣴⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣏⠉⠀⣠⣿⣿⣿⣿⣿⣃⠀⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⠷⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢹⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣤⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠇⠘⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢰⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣷⣶⣶⣤⣶⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠙⢻⣿⣿⠟⣿⣿⠀⠀⠙⠛⢻⣿⣿⠛⠋⠉⠛⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⢀⣾⣿⠁⢰⣿⡟⠀⠀⠀⠀⠸⣿⣿⡆⠀⠀⠀⠀⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⣽⣿⠃⠀⢸⣿⠁⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⢿⣿⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣀⠀⠁⠀⠀⣿⣿⡀⠀⠀⠀⢀⣠⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠗⠀⠀⠘⠿⠗⠀⠀⠚⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠟⠿⠿⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢀⣾⣿⠟⣁⣤⣶⣾⣿⣶⣶⣄⡙⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣸⣿⣯⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣦⣹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠈⢿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⠀⠙⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⡁⠀⠀⠀⠉⠻⣿⣿⣿⢿⣿⡇⠀⠀⠀⣿⣿⣿⣿⠿⠏⠀⠀⠀⠀⠠⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢋⡉⠀⢀⠀⠀⠀⠀⠀⡀⢈⣁⡸⣿⣧⠄⠀⠴⣛⡿⢻⣷⡒⠢⠀⠀⠀⢀⡤⠀⠈⣿⣿⡿⡀⠀⠀⢹⡾⣾⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠳⢽⠈⠁⠀⣼⣷⠀⠀⠀⠀⢠⣾⣿⠁⡗⠊⠁⠀⢀⡀⢇⠘⢿⣿⠀⠀⠀⠀⢺⣿⡆⠀⠀⡼⠒⠻⠀⠘⢎⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠈⠦⣨⡀⠀⠀⠛⠉⠀⠀⠀⠀⠘⠋⠁⢠⣥⡌⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⠐⠀⠀⠀⡈⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⣸⣿⣿⡀⠀⢀⣾⣿⡆⠀⠀⠀⢰⣤⣴⠞⠀⠀⠀⣼⣿⡀⠀⠀⠰⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⡃⠛⠻⠿⢿⣦⣄⠀⠀⠀⠀⠀⠀⣠⣼⣿⣿⣷⡄⠠⠁⠰
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⠶⣤⠀⣀⠠⣤⠒⠿⠛⠋⠉⠡⡀⠠⠋⠑⠢⣔⡀⢀⢻⠒⠂⠒⣚⣛⠫⢉⣉⠥⠰⠞⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⠆⢲⠙⠠⠤⠄⠋⠶⠀⠼⣀⠀⠀⠀⠣⢀⠀⠀⠸⠁⠆⠀⠉⠀⠁⠀⡁⢸⡏⠀⠀⠀⣰⠊
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⡀⠸⠀⠀⠀⠀⠀⠂⢀⠀⠀⠈⠐⡄⠀⠀⠈⣶⠃⠌⠀⠀⠀⠀⠀⠀⡇⠀⢡⡤⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠄⠙⡀⠀⣀⠜⠁⢠⠀⠀⠀⠀⠀⠀⠘⠡⣀⠀⢀⠛⠢⡀⠀⠇⡘⠀⠀⠀⠀⠀⠀⠀⠂⡄⢰⠃
⠀⠀⠀⠀⠀⢈⠀⠐⠱⠼⠥⡴⠃⠚⠀⠀⠀⠀⠀⠒⠚⢠⠯⠥⠮⠆⠀⠀⠄⠠⠁⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈⢰
⠀⠀⠀⠀⠀⠀⠉⠀⠀⢲⠖⠳⠄⠆⠀⠀⠀⠀⠀⠲⠠⡞⢲⠖⠒⠀⠈⠁⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⡀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⢁⠔⠀⠀⠀⠀⠀⠀⠀⠠⠈⠺⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⢠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠤⠀⠀⣀⡀⠀⠀⠠⠤⠐⢺⠀⠀⠀⠀⠀⠈⠀⠀⠉⠁⠐⠒⠒⠒⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠾⣿⣶⣦⣤⣄⣀⣀⣀⣀⣠⣤⣴⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠿⢿⣿⣶⣶⣶⣶⣶⣶⣶⣿⡿⠟⢳⠀⠀⠠⠃⠂⠌⡉⠛⠓⠛⠿⠿⠟⢛⠛⠉⣭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠢⡄⠉⠉⠉⡉⠉⡥⠰⠀⡆⠀⠀⢀⠃⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠸⡀⠠⠚⠆⣀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⣀⣰⠀⠀⠀⠀⠀⠀⠠⣆⢐⡃⠀⠀⠀⠉⠓⠚⠤⢤⠁⠀⠈⢷⠀⠀⣘⠤⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠼⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠠⡄⠀⠀⠀⠀⠀⠀⠀⠤⠠⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠀⢀⠆⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⢈⡀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠂⣀⡴⠛⠀⠀⠀⠀⠀⠀⠈⠳⢤⡠⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠠⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊

⣿⣿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣝⣹⣷⣦⣤⣶⣦⣶⣤⣶⣴⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠁⡰⡾⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⢠⠀⢩⠇⠁⠀⠀⠀⠀⠀⠀⠈⠉⠻⠷⠦⣾⣿⡿⠿⠻⣟⠛⠷
⠀⠀⠐⠀⠀⡀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠈⠁
⠀⠀⠀⠠⠀⢷⠆⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠐⢌⠀⠐⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⣾⠟⠁⡉⠑⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣶⣾⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠟⠋⢰⣋⣱⡎⠀⣎⣈⣈⣢⠄
⠀⠀⠀⠀⠀⠀⠒⣾⣿⣿⣿⣿⣏⡃⠀⠃⠀⡀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣽⣿⢋⡰⠺⣥⣘⣿⡄
⠀⠀⠀⠀⠀⣀⣤⠈⠙⠛⠿⠂⠀⢀⣀⣼⡄⣸⠀⠀⠀⠀⢀⣾⣿⡿⠿⢭⣿⣿⣿⣻⣷⣾⣿⣷⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣦⣤⠀⠀⠀⢀⣀⣶⣎⠀⡖⠤⠀⠀⠀⠀⠀⠀⠀⣴⡾⠟⠛⢋⣿⣿⣿⣿⣿⣿⠷
⠀⠀⠀⠀⢀⣿⣀⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣎⠀⠀⠀⠀⢇⢀⣬⣥⣴⠛⠁⣾⣿⣿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠐⠋⠁⠘⠛⠃⠀⣿⡿⢿⣻⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⣠⢯⠭⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠊⡇⠈⡿⠟⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣟⣿⣿⢟⢿⣿⣿⣿⡿⣻⡿⢻⡿⡿⣿⡿⣿⣟⣻⣿⣿⣿⣿⣿⠿⠋⠉⠀⠀⠉⠛⠟⠋⠉⠀⠈⠉⠻⣿⣿⣿
⣿⣿⣇⣭⣿⡸⣸⣎⡌⣿⡗⣿⢟⣺⣷⡓⣿⡷⣼⡼⢿⡿⢯⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣷⡏⡯⣿⢫⡟⣭⣿⣽⣿⢹⢹⣽⡯⣽⢽⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⣿⣿⣧⣜⣗⣿⣜⣿⣻⣿⣣⣿⣿⣹⣾⣹⣿⣺⣿⣿⣿⣿⣿⣿⣿⡆⠀⢾⠀⠀⠀⠸⠇⠀⠀⠛⠀⠀⠀⠃⠀⣾⣿
⣿⣿⣿⣿⣿⡿⠻⡟⣛⢛⠉⠉⠉⠙⣿⡟⡉⢻⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⣿⣿
⣿⣿⣿⣿⣿⡇⠀⡇⡤⠸⠇⡇⢅⠀⣿⢿⣤⣾⢿⣿⣿⣯⡉⣽⣿⣿⣿⠂⠀⠀⠀⠀⠘⠿⠉⠀⠀⠀⠀⠙⢿⣿⣿
⣿⣿⣿⣿⣿⣷⣤⡶⠶⠶⠶⠶⠶⠚⠛⠓⠻⠿⢿⣿⣏⢉⣿⡙⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢻⣿
⣿⣿⠀⡂⡔⡄⢔⠂⡄⠀⠦⢠⠂⣀⠀⢰⡄⡄⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⣿
⣿⣿⠀⠃⠙⠀⠸⠀⠘⠆⠖⠠⠗⠉⠀⠊⠀⠂⣸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⣿⣿
⣿⣿⣧⣴⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿

⠀⠀⠠⠀⠀⠄⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢠⠀⠀⠀⢀⡄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡷⠀⢠⡀⣠⣆⣠⡄
⠀⠀⠀⢀⣠⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠘⠟⣿⣿⣽⡇
⠀⠀⠀⣸⠃⠀⠀⡌⡆⢀⡿⠀⢠⡆⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣰⠀⢠⡄⠀⠀⠀⠀⢰⣓⠄⠀⠀⠁
⠀⠀⠀⠛⠀⠀⠀⠃⠋⠈⠉⠁⠁⠃⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⢰⠀⣿⠄⣸⡇
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⣁⠀⠀⠘⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⣰⠀⠀⣿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈
⠀⠀⠀⢻⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⣀⣾⡄⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢲⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢏⠉⣹⠃
⠀⠀⠀⠀⠀⠘⡒⠀⠀⠀⠀⠀⠀⠀⢀⣎⡀⠀⡦⡆⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠘⠻⠿⠓⠀⠈⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠃⠀⠇⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠤⠤
⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⢰⣖⡄⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠐⣿⠀⠈⠴⠀⡏⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠴⠒⠀⠀⡟⠀⠰⡇⠀⠀⠀⠀⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠗⠀⢠⠀⠀⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂
⢞⡣⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⡄⢀⣴⣿⣷⣤⡄⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣼⣿⣿⣿⣿⣯⣀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠟⠛⠛⠛⠓⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⠘⣿⣿⣿⣿⠁⢸⣿⣿⠋⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢹⣿⣿⡏⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⣿⡿⠶⢿⣿⣿⣿⡿⠿⢿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠛⠋⠉⠁⢸⣿⡟⣠⣶⣿⣿⠋⠛⠁⠀⠀⢻⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⣿⣿⣦⠿⡿⠿⠃⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⡛⠋⠀⠀⠀⠀⠀⠀⠹⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠟⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠴⠂⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣦⣀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣿⡆⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⠛⠉⠀⠀⠈⠉⠻⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠯⠀⠀⠈⠙⠿⢿⠿⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠐⠈⠀⠀⠈⠀⠢⡈⢿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⡁⠆⠐⣤⠀⠀⠀⢀⣲⡆⢻⢷⠀⠀⠀⣿⠃⠻⠟⠛⠃⠀⠛⠿⠏⢻⣿⣿⣿⡐⠀⡠⠤⠀⠀⠀⠠⠤⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣈⡃⠀⠸⠟⠁⠀⠀⠘⠟⠁⣇⡘⠅⠀⢘⢻⠀⣿⡌⠀⠀⠀⣶⡉⠀⢸⡿⣿⡟⡇⢴⣖⠂⠀⠀⢠⣄⠂⠆⢼⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⠄⠀⠀⠀⢰⣿⣧⣄⠀⠈⠘⡄⠈⠁⠀⠀⠀⠉⠃⠀⢮⠠⣿⣧⣡⠈⠻⠂⠀⠀⠘⠛⠀⠀⣏⣸⡿
⠀⠀⠀⠀⠀⠀⠨⣿⣿⣷⣄⡀⠀⠀⠀⣀⣴⣿⠿⠟⠉⠀⠀⠀⠠⠀⠀⠒⠀⠐⠀⠀⢠⠄⢂⠈⠙⠻⡄⠀⠀⢠⣀⣠⠀⠀⠸⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠩⠹⠅⣂⠹⠥⢁⡂⠀⠈⠄⢑⠄⢄⡀⠁⢂⡄⣀⣀⡀⠐⠁⠡⠀⠀⣀⠀⠈⠂⢀⡀⠈⢁⡀⠔⠁
⠀⠀⠀⠀⠀⠀⣠⠂⠁⢀⠀⠀⠀⠀⠀⠀⢈⠀⠁⠂⡈⠁⠀⠀⠈⢲⠂⠃⠀⠘⠒⢨⢤⡀⠑⡎⠀⠈⠚⢠⠘⠩⠉⠜⠥⢡⣤
⠀⠀⠀⠀⠀⠀⢩⣀⡤⢊⣀⡀⠀⠀⠀⢀⣈⢠⡄⣠⡉⢂⠀⢉⠃⠉⠀⠀⠀⠀⠀⢠⠀⠈⢐⡀⠁⠂⠆⠊⠀⠀⠀⠀⠀⢀⠀⠉⢂⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣉⢇⣀⠄⠀⠀⠀⢅⣀⢏⡁⠉⠅⠐⠀⠀⠌⠀⠀⠀⠀⠀⡤⢆⣱⣤⣤⠈⠂⠘⠀⠀⠀⠀⠀⠀⡠⢄⢰⣤⣄⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⡥⢂⠄⠀⠀⠀⠀⠀⢂⠢⠅⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠣⠼⠠⠄⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠣⠴⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠢⢘⡀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠠⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠉⠀⠀⡄⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠰⠀⠠⠄⠀⠀⠀⠠⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣄⣀⣀⣀⣀⣀⣤⣴⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠨⠉⢙⠛⠛⠛⠛⡛⠉⢡⠈⠀⠀⠀⠐⣄⣀⠀⢀⠁⠐⡀⠀⢀⡠⠀⠀⠀⠀⠟⠿⠷⣶⣶⣶⣶⣶⠾⠿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⠀⠀⠀⠀⢰⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠇⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠀⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆⠀⠀⠀⠀⠐⠄⠼⠀⠀⠀⠀⠀⡧⢴⠀⠀⠀⠀⠀⣇⣀⠄⠀⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⣄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠁⠀⠀⠀⠀⠀⣃⡀⠀⠀⠀⠀⠀⡁⠠⠀⠀⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡅⢐
⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣡⠀⠀⠀⠀⢠⠁⠈⠢⠀⠀⠀⠔⠉⠑⠀⠀⠀⠀⠀⠌⠈⢄⠀⠀⠀⠀⡠⠃⠣⠀⠀⠀⠀⠀⠐⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠅⠠⠞⠁⠀⠀⠀⠀⠈⠓⠦⠊⠇⠀⢜⣢⠴⠚⠀⠀⠀⠀⠀⠳⣄⠐⡆⠀⠀⠰⠀⣠⠞⠀⠀⠀⠀⠈⠣⣄⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣿⣿⣿⣿⡿⣧⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣷⣶⣌⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣀⡀⠀⠀⢠⡀⢸⡆⠀⠀⠀⠀⢰⡿⢿⣿⣿⣿⢿⣿⣿⣎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⢀⡀⠀⠀⢀⡀⠘⠃⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⠈⠉⣿⡿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⠐⠀⠀⣸⣿⠟⢁⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠴⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁










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

⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⠿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⢿⣷⣦⣄
⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠈⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⠀⠙⣿⣧
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⣤⣶⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣦⣤⣤⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⣠⣴⣶⣾⣷⣄⡀⠀⠀⠀⢀⣠⣶⣷⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣷⣦⣀⠀⠀⠀⠀⣀⣴⣿⣶⣶⣤⡀
⢀⣾⡿⠋⠁⠀⠉⠻⠿⠿⣿⡿⠿⠟⠋⠀⠈⠙⢿⡷⠀⠀⢀⣤⣶⣶⣶⣦⣄⠀⠀⠐⠿⠟⠉⠀⠈⠛⠿⠿⣿⡿⠿⠟⠋⠁⠀⠉⠻⣿⣆
⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠛⠉⠁⠉⠙⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣿⡇⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿
⣿⡇⠀⠺⠿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⠟⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠘⠿⢿⣿⡆⠀⠀⠀⠀⠀⠀⢰⣿⠿⠿⠂⠀⣿
⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠻⣿⣦⣄⣀⣀⣤⣾⡿⠃⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠛⠛⠉⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿
⢿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠿⠛⠉⠛⠿⣿⣶⣶⣶⣾⡿⠟⠋⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢰⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣾⣿⡇
⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⢂⡾⣿⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠏⢀⢺⣦⣑⡾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⠀⣿⢸⡯⡟⣳⣿⢹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣴⡿⣿⠏⣻⣟⢮⡹⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⣸⣼⡀⡟⣿⣿⡾⣿⡇
⠀⠀⠀⠀⠀⠀⣾⠙⢉⡵⢻⣿⣿⣷⣝⢌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⡞⣿⣧⡇⣿⢹⣧⣿⡇
⠀⠀⠀⠀⠀⢸⣟⡴⠿⠀⠈⣷⢹⡎⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠭⠉⢻⣿⡇⣏⢸⡏⣅⡇
⠀⠀⠀⠀⢀⠛⣿⠃⢂⣒⠀⠙⠎⠛⢐⡞⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⣿⣇⣿⢸⣃⡟⡿
⠀⠀⠀⠀⡼⢀⡿⠂⠀⠈⠀⠀⠀⠉⠁⣿⣿⡄⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠒⠀⠀⢀⣿⣿⣿⢸⣿⡇⡏⢠
⠀⠀⠀⡰⠃⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⡧⠞⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣗⠀⠀⠀⠜⢹⣿⣿⣾⣧⢡⡇⢸
⠀⠀⠠⢁⣀⢽⢄⡀⠀⠐⠀⠀⠀⠁⢸⡧⣿⣵⡀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣿⡏⢀⠀⠀⠀⢸⣿⡟⣿⡏⢸⡇⠈
⠀⢠⡴⠛⠶⢬⣿⣷⡄⠀⠀⠁⠀⠀⣸⡷⢻⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢢⢻⡇⣀⠉⠀⠁⢸⣿⣷⡟⡇⢸⠇
⢀⣿⠁⠀⠀⠀⢸⡜⠁⠀⠐⠀⠀⠀⠘⠇⠀⣿⡇⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠃⠸⠇⠀⠀⠀⠀⡜⣿⣿⡇⡇⣸
⢸⣿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⣄⠀⠀⠀⢻⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢸⣷⣽⡇⡏⠀⠀⠀⠠⢤
⣾⡟⡀⠄⠀⡀⠀⠀⠀⢰⠀⠀⠨⠤⠀⣰⠀⡘⢨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⡿⠉⠁
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⣾⠀⠀⠀⠀⣯⠀⡇⠸⡧⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⠁⠳⠃⠀⠀⠀⠀⠀⠀⡆
⢹⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢚⣲⡀
⠘⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⡄
⠀⠀⠀⡐⣂⠀⠄⠀⠀⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠶⢶⣶⣤⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢋⣵⣶⣶⣶⣦⣙⣿⡆⠀⠀⣿⡿⠿⣿⣿⣿⠿⠟⠙⣿⡆⠀⢰⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⡟⠙⢿⣿⣿⣿⣿⠀⠀⣿⡇⣠⠄⠀⠀⠀⣄⡀⢺⡇⠀⣿⣿⡏⠀⠀⠀⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣀⡀⠀⢹⣿⡿⠛⠀⠀⠀⠙⢻⣿⣿⠀⡤⡿⠀⠀⣢⡀⠀⠀⢂⣦⢘⡷⠀⣹⡟⣀⡀⠀⠀⠀⠘⢿⣿⣿⠃
⠀⠀⠀⠀⠻⠉⠧⠀⠞⣟⠁⢠⣷⠀⠀⢠⣿⠋⠟⠁⠠⢸⠀⠘⠛⠁⠀⠀⠙⠋⠸⠐⠀⠙⡇⢿⡦⠀⠀⠰⣷⠀⠙⠟⠁
⠀⠀⠀⠀⠀⠀⠂⠀⠀⣶⠀⠀⢀⠀⠀⠀⠀⢸⣇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠘⡀⠀⠈⣧⠀⠀⠀⡀⢀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⣀⠈⠙⠉⠀⣠⣾⠿⠆⠀⢀⠀⠐⢀⡀⠀⡠⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠉⠁⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢩⡉⠽⠀⠀⠯⠠⡔⠀⠉⡄⠀⠄⠄⠚⠢⠤⠚⠒⠰⠀⠂⡌⠀⠙⠫⠭⠿⠀⠈⠋⠙⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣠⠁⠁⠀⠀⠀⠀⠡⠄⠈⠇⠀⢀⠆⠀⠀⠀⠀⠀⠆⠀⠀⠯⠀⣤⠂⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢲⠀⠚⠒⢡⠂⠂⠀⠀⠀⡐⢡⠛⠒⠂⢈⠏⠈⠀⠀⠀⠀⠀⠰⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠁⠀⠀⠀⠀⠀⢢⠀⠀⢘⠁⠀⠀⠀⠀⠀⠘⠉⢙⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠂⠀⠀⠀⠀⠀⠀⠨⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠈⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠀⢠⢀⠀⠀⠀⠀⠀⠀⢀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠐⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⣶⣶⣶⣶⡶⠾⠛⠇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⢦⣤⣤⣄⣀⣀⣤⣤⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠐⠀⠀⠀⠀⠉⠀⠀⠄⠀⠁⠐⠂⠀⠀⠁⠂⠀⠁⠀⠀⠀⠐⠈⠩⠉⠉⠉⠩⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠄⠀⡀⠩⠤⠀⠀⡄⣀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⡀⡄⠀⠀⠀⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠟⢀⠔⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢂⠈⠀⠀⠀⠀⠀⠀⡀⠁⠁⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠞⠀⠀⠀⠀⠀⠀⠐⠂⡤⠚⠀⠀⠀⠀⠘⠦⡄⠂⠀⠘⢠⠴⠃⠀⠀⠀⠀⠳⢤⠐

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡋⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠒⠒⣤⠴⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⢠⠎⠉⢼⡵⠀⠐⠧⠃⠀⡠⢽⠤⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⡎⠀⠸⡄⠀⢀⢾⣢⢀⡤⠒⠊⠉⠉⠲⢤⡙⠁⠀⠈⡆
⠀⠀⠀⠀⠀⡏⠉⠉⣲⢤⣠⠖⠋⡹⠛⠦⢤⣈⣹⢯⡹⠉⠉⠁⠀⣇⡀⢡⠏⠀⠀⠀⠀⢀⡀⠀⠙⣄⠺⠝⠙⣆
⠀⠀⠀⠀⢸⣁⠔⠚⠣⡼⠓⠒⢩⡄⠀⢠⠀⠀⠀⠸⡄⠀⠀⠀⡞⠣⠽⡎⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⡼
⠀⠀⠀⠀⠈⠁⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠳⠤⠄⢧⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⡕⣨⠟⣒⣒⡲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠳⣄⣀⣀⣀⡤⠞⢁⡼⠀⠀⠀⠀⠀⠀⠀⠈⢦⠐⠤⣤⣤⠤⠔⠊⠁⢠⢇⡇⣾⠡⢦⠙⣞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⣦⠤⢤⠤⠤⢶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⠤⣀⣀⣀⡤⠖⠁⠀⢧⡈⠋⣡⢞⠜
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⢸⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⠏⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⢹⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⢸⠀⠀⠀⠁⠀⢫⠳⡀⠀⠀⠀⠀⠀⠀⡰⠋⡼⠀⠀⠀⠀⠀⢽⢆⠀⠀⠀⢀⡘⡷⠒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⢦⡀⡎⠀⠀⠀⣀⠀⠘⡄⠈⢦⡀⠀⠀⢀⡜⠁⢠⠃⠀⠀⠀⠀⠀⢸⠸⡄⠀⠀⢏⠙⠃⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣭⠍⢳⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⣦⣠⡎⠀⠀⣮⡤⠤⠤⠤⠤⣄⡘⡆⠙⢦⣀⠼⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⣠⢾⠀⠀⠀⠀⠰⠀⠀⡇⠀⠀⣾⣤⡀⢹⠀⢰⠃⠀⠀⢀⠀⠀⠀⠉⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣮⣀⡀⠀⠀⠀⠀⠀⣷⠀⠀⠙⢤⡽⠋⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⠒⠮⠭⣉⣙⣛⣋⣻⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⠋⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⣀⣀⣀⣠⡤⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢹⠉⠉⠉⢻⠉⠒⠚⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢹⠉⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡓⠒⠒⠲⡄⢸⡥⠤⠒⠲⡄⠀⠀⠀⠀⢠⠔⠒⢤⣀⡇⠀⣠⠔⠒⠦⢵⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠁⠈⠓⠒⠒⠒⠁⠀⠀⠀⠀⠈⠳⠤⠤⠞⠃⠀⠘⠒⠒⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠈⡔⡀⠘⠛⠓⢃⠀⡅⠀⠀⠀⠀⠨⠤⠇⠀⠨⠠⠃⠀⠀⠀⢠⠠⣀⡠⢠⠠⢤⣀⡤⢄⠀⠀⠀⠸⡤⠃⠀⠀⠸⡬⠂⠀⠀⠀⠈⡄⠀⠀⢀⢄⠀⠀⠄
⠀⠀⠀⡇⠧⡠⠆⠠⠜⠀⡇⠀⠀⠀⢠⠂⠉⢰⣤⠂⠉⢲⡄⠀⠀⢸⠀⣂⣀⢾⠰⣆⣀⡆⢸⠀⠀⠀⢸⣈⡂⡤⠀⢘⣈⡆⠀⠀⠀⠀⡇⢆⣀⣘⢊⡱⠀⡇
⠀⠀⠀⡇⠀⠨⠀⢄⣈⠇⡇⠀⠀⠀⢀⡀⠀⡰⠲⡄⠀⢀⡁⠀⠀⢸⠔⠉⡆⢸⠀⢠⠈⣒⣼⠀⠀⠀⢸⠿⡃⠀⠒⢸⢿⠇⠀⠀⠀⠀⡇⠀⠁⢄⣦⢀⠀⡄
⠀⠀⠀⡇⠠⠦⠤⠤⠥⠀⡇⠀⠀⠀⠀⠁⡀⠡⠥⠆⡄⠀⠀⠀⠀⢸⠀⠈⠡⢸⢀⠀⠉⠀⡸⠀⠀⠀⢸⠀⡅⠀⠀⢨⠀⠀⠀⠀⠀⠀⡇⠀⠐⠐⠢⡖⢀⠇
⠀⠀⠐⠤⠥⠩⠭⠭⠥⠥⠄⠀⠀⠀⠀⠢⠥⠜⠀⠠⠥⠄⠀⠀⠀⠀⠉⠉⠉⠁⠀⠉⠉⠉⠀⠀⠀⠀⠸⠄⠧⠤⠤⠎⠤⠀⠀⠀⠀⠐⠬⠥⠭⠭⠭⠤⠥⠔

⠀⠀⠀⢀⣀⠀⠀⠄⢄⡀⠠⡀⠀⢰⠀⡀⡀⠀⡀⢙⠀⠀⠀⠀⠀⠠⢀⢀⡄⠀⠠⠐⡀⠀⢀⠀⠀⠔⠦⠤⠖⠄⡔⠢⠤⠖⠆⠀⠀⠀⠀⢠⡤⠤⢤⣄
⠀⠀⠀⢆⠀⡰⠘⡈⠠⡠⢃⠇⠀⢘⠀⠀⠀⠀⠀⢘⠀⠀⠤⠀⠀⠀⠈⠺⢇⡬⠭⡠⠳⢘⢸⠀⠀⡇⠐⠴⠀⠇⡇⠀⠖⠀⠀⠀⠀⠀⡰⢉⠀⠀⠀⢹⠅⡀
⠀⠀⠔⠊⠈⠑⠢⠘⠂⠀⠁⠀⠀⠈⠒⠤⠞⠠⠤⠬⠠⠭⠀⡀⠀⠀⠀⢠⠐⠉⠊⠣⣸⢸⢸⠀⠀⡐⠃⠀⠗⡄⡇⠂⠀⠒⠀⠀⠀⡨⠂⠪⡀⠀⢀⠔⠐⢅
⠀⠀⡀⡆⠀⣰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⡉⡍⢉⣀⠂⠀⢀⠰⠬⠤⠥⠬⠤⠭⠼⢸⠀⠀⡇⠘⠄⠀⡄⠇⠀⠼⠀⡀⠀⠀⢁⡀⣰⡸⠀⢈⣰⠀⣆
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⠀⣀
⠀⠀⠤⢀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠐⠂⠒⠂⠀⠁⢄⡀⠀⠀⠀⡘⣄⠆⠀⠐⠍⢨⢱⠀⠀⠀⡖⢀⣀⣀⣀⠐⢸⣠⢆⠀⠀⠀⠀⢸⠀⠀⢉⠀⠀⢄
⠀⠀⠀⠈⠈⢈⡅⠤⠀⠀⢤⠀⠀⠀⠀⠈⡹⠀⠆⠠⠌⠀⠄⠀⠀⢈⢠⠀⠀⠁⠈⠆⠃⠈⠀⠀⠀⠃⠀⠠⠄⡠⢼⣠⣀⣄⠇⠀⠀⠀⠸⡀⠀⡘⠀⠈⢹
⠀⠀⠒⠈⠀⠀⡄⠂⠒⠒⢸⠀⠀⠀⠀⠀⠣⠀⠁⡅⠈⠀⠀⠀⠀⠀⡈⡀⣉⡀⠀⠸⠀⠃⠀⠀⠀⠀⢆⡁⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠈⡇⠠⣀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠲⠄⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⢠⢀⠅⢐⢁⠉⠈⠂⠀⠀⡄⡉⠉⡱⣀⠒⣂⣐⣀⠇⠀⠀⠀⠀⠐⠓⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠚⠀⠓⠃⠀⠀⠉⠉⠈⠉⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⣀⣀⣀⠀⠀⠀⢀⢀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⡠⠂⡖⡽⠆⠊⠄⡄⣀⠀⠀⠀⠁⠶⡐⠒⠒⠒⢂⠶⠈⠀⠀⢸⠊⡛⠛⢁⠒⠒⠒⠒⢺⠀⠀⠀⠀⠀⢰⠁⠙⢰⠀⠀⠀⠀⠀⠀⢌⠱⠀⠀⠀⠰⠉⠆
⠀⠀⡆⠀⢇⢄⠨⠀⠈⡜⠧⠃⠀⠀⠀⠀⠈⡰⠤⣄⠀⠀⠀⠀⠀⢸⡈⢄⢒⡀⠱⢀⡀⠀⢸⠀⠀⠀⠀⠀⠘⢳⠛⠂⠀⠀⠀⠀⠀⢌⠐⢈⠀⠒⡄⡁⠂⣁
⠀⠀⠢⢂⠤⠫⠞⠅⠊⠀⠀⠀⠀⠀⠀⡤⠊⠀⠄⠀⠑⣤⠀⠀⠀⢸⠘⠂⠤⢐⢡⠠⢤⠑⣸⠀⠀⢤⠴⠖⡄⢸⠀⡔⡶⢦⡆⠀⠀⠒⠂⠘⠤⠅⠤⡘⠀⠃
⠀⠀⠀⠲⣭⠆⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠠⠐⠉⠉⠀⠀⠁⠀⠀⢸⠀⠀⠀⠈⠢⠉⠥⠜⢸⠀⠀⣸⠄⠤⡟⠉⠉⡧⠄⠸⡇⠀⠀⠀⠀⠀⡸⢀⣆⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢄⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⡔⠒⠒⡐⣒⠒⠀⠀⠀⠀⠀⢠⠐⠄⠀⠀⡠⠢
⠀⠀⠯⠹⠿⠭⢴⠯⠽⠿⠭⡆⠀⠀⡀⠒⠊⣀⠀⡀⠑⠊⣀⠀⠀⠀⢠⡇⠅⠠⣀⡠⠀⣄⠀⠀⠀⠀⡈⠒⢀⠀⢀⡀⠒⣀⠀⠀⠀⠉⠛⣛⠉⢰⠉⣛⡛⠉
⠀⠀⡄⠛⢀⡄⢸⡀⠛⠀⡄⠄⠀⠀⠀⠀⢰⠂⢫⠰⠀⠀⡄⠁⠀⠀⠧⡄⠀⠀⠀⠀⢀⡤⠆⠀⠀⠘⡇⠀⠈⠇⡆⡄⠀⠈⠀⠀⠀⡂⢈⠶⡁⢸⠀⡱⠂
⠀⠀⠍⣩⢥⡁⢸⠉⣩⣥⠁⠀⠀⠀⠲⢀⢸⠀⠉⢸⠀⠀⠗⠀⠀⢀⠀⠀⠱⢌⡁⢔⠁⠀⡁⠀⠀⠐⡇⠀⠐⠃⠳⠁⠀⠹⠂⠀⠀⡗⢺⠾⣧⢸⣠⡷⠿⠂
⠀⠀⠂⠑⠒⠀⠈⠂⠑⠀⠀⠂⠀⠀⠀⠐⢸⠀⠀⢸⠀⠀⡇⠀⠀⠈⣠⠂⠁⠀⠀⠀⠑⢄⡅⠀⠀⠀⠇⠀⠀⠀⠙⡃⡂⡚⠀⠀⠀⠂⠉⠉⠥⠘⠠⠁⠍⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉