Appeal Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⠉⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⡟⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⣄⣀⣀⡾⠀⠟⣴⣿⣯⠤⢤⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠖⠚⠛⠛⠛⢻⣄⠀⢉⣡⠀⢀⡶⢸⠏⢀⡶⢀⡟⠻⣦
⠀⠀⠀⠀⠀⢰⡏⠀⠀⣠⣶⣿⠇⣼⠟⠛⠛⡇⠀⠚⠁⡄⠀⢨⣴⠿⢃⣴⣿⠁
⠀⠀⠀⠀⠀⢸⣇⢠⣾⣟⣡⡏⣸⠏⢀⣴⣿⣷⣤⣴⣾⣿⣤⣤⣴⣾⣿⣟⣡⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⣻⠟⠋⠉⠉⣀⠉⣀⠛⢁⣴⠀⣸⠃⢈⡤⢀⡉⠀⣡⠄⢈⠁⢀⡟⠀⣾⣇⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣤⣴⠞⠀⣿⠀⡟⠀⠰⠃⢰⡏⢠⡿⠃⢨⠁⣼⠟⠀⡍⠀⠞⠀⠘⣉⠀⠉⣿⡄
⠀⠀⠀⠀⣿⠙⠶⠟⠋⠀⣸⡇⠀⢿⣤⣴⣦⡔⠁⣦⣤⣶⠇⢰⣤⣴⣾⣧⣴⠎⣰⣿⠟⠻⢷⣿⠇
⠀⠀⠀⠀⠹⣷⣄⣀⣠⣾⣿⣧⣀⣀⣿⡍⡿⠀⢸⣿⢉⡏⠀⣿⠃⠘⠷⠟⠃⣰⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠙⠛⠛⠋⠘⣷⣤⣿⠇⣿⣄⣰⣿⣆⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠛⠛⠋⠙⠛⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠀⠀⠀⠄⠀⠀⠀⠀⣠⣄⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡄⠀⠉⠁⠀⠀⠀⠀⠀⣘⣃⠄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⡆⠀⢏⣀⢠⡤⠀⠀⠀⠀⣌⣠⠀⢀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣶⣶⣲⣄⠠⣿⠿⠶⠀⠀⠀⠀⠀⠀⠼⠿⣿⡿⣀⠀⢦⢽⣾⡇⠀⠀⠀⠾⣍⣿⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠶⣤⣶⣷⡞⢃⣝⠁⠻⠟⠃⠀⣟⡃⠺⡄⠀⠀⠀⠀⠀⣆⠸⠿⠿⢻⣿⣧⣾⣿⡉⠀⠲⠶⣴⣿⣷⣇⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣿⣿⡎⢻⣿⠋⣾⡾⠿⠇⠈⡇⠀⠛⠀⠀⠀⢠⣴⣿⡇⢰⡀⠋⠻⣯⢀⣙⠁⠀⠀⠀⢈⣹⣿⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠑⠂⠀⠺⠁⠀⠀⠀⠀⠀⠈⠁⢸⣧⣼⣷⡀⠼⠁⠰⠇⠈⠀⠀⠀⢾⠏⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⠀⠀⠀⠲⠟⠓⠘⠷⠚⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠖⠀⣠⠄⠀⢀⢸⣿⣶⣿⢿⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣁⣴⣷⠻⣆⠰⣾⣶⢻⣿⡧⢼⣿⢀⣿⠀⠀⠘⣳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠀⢽⣿⣿⣾⣿⡷⢿⣿⣿⣇⠀⣸⣿⢉⣿⣶⣤⣴⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⣿⡹⣵⣿⣿⣿⣿⡼⢁⠈⣿⣿⣿⣷⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢺⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⢸⣷⢀⠘⠛⢿⣨⣾⣿⠐⢻⡄⣿⡝⢳⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣿⣼⣿⣿⡄⠆⢸⣿⣿⡇⢰⣿⣷⣿⡇⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠤⠠⢾⣿⣿⣧⠸⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⢿⣧⣿⣿⣿⣹⣁⣟⣿⠋⢉⣷⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠋⠙⠙⠙⠋⠀⠈⠉⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠸⢋⣹⠉⣛⣋⣙⣛⠿⠿⠶⠁⢼⣯⡉⣤⡄⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠶⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⣧⡤⠀⠰⠟⢃⡀⠀⠀⠀⠉⠀⣸⣇⡀⣀⢿⣇⠚⢠⣶⣧⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠉⡷⢠⣾⣿⣆⠀⠀⠀⠀⠀⠀⠀⢠⣷⣿⠀⠀⠀⢶⣴⣿⡿⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⣽⣻⣿⣿⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣼⡿⢿⡅⠀⢸⡿⣻⣿⣷⠄⠀⠀⠀⠀⠀⣼⠏⣿⠀⠀⠀⠀⢙⣿⣿⡆⠀⠈⠛⢹⣿⠛⣏⠙⠛⡆⠀⢸⡏⠛⠛⠻⠿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⡔⠉⢿⡟⠋⣭⡀⠂⠀⠀⠀⠀⢿⠀⣿⠀⠀⠀⠀⠀⢹⣿⣆⣀⣀⠀⡌⠋⠀⢸⣞⡇⡇⠀⠄⢠⣿⡆⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣽⡇⢀⣿⡄⠚⢁⣾⣟⣷⣀⠀⠀⠀⠀⠸⢦⡿⠠⣶⡆⠀⠀⣾⠃⠈⠁⢡⣾⣀⡤⢾⡛⣻⣷⡇⠰⣶⡿⠻⣏⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟⠿⠟⠟⠓⠀⣾⣿⣿⠟⠛⠁⠀⠀⠀⠀⠘⠛⠃⠛⠃⠀⠀⠀⠀⠀⠀⠀⠅⠈⠀⠘⠉⠀⠘⠁⠠⠿⠓⠚⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣿⡏⠀⠈⣿⣷⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⡶⠀⣿⡇⠀⠀⠘⠿⣶⡾⠟⠀⠀⠀⢼⡇⠰⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣶⡄⢀⣴⡿⠃⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠹⣷⣿⠏⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠁⠀⢀⣤⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣶⡄⢀⣴⡿⠋⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠹⣷⣾⠏⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠙⠁⠀⣠⣦⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣷⡄⢀⣼⡟⠁⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠸⣿⡿⠋⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠘⠓⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀⠀⠛⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣱⡄⠀⠀⠘⠆⠙⢿⣦⡀
⠀⠀⠀⣀⣀⡤⠤⠤⠄⠐⠒⠒⠒⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⢹⣄⠀⠂⠀⠀⠀⠸⣿⣦⡀
⠀⠰⣯⣀⣀⣠⣄⣀⣀⣤⣤⣀⣀⣀⣠⠀⠀⠂⠀⠀⢂⠀⠀⠸⠘⠀⠀⠀⠀⠀⠀⠈⠉⡻⣆⠀⠀⣀⣀⢀⣤⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠉⠁⠀⡀⠀⠤⠤⠄⠐⠂⠙⠉⠁⠀⠀⣄⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠉⠻⠌⠉⢁⠂⠀⠀⠀⠈⠉⠉
⣤⠒⠈⠉⠀⠀⠀⢀⢀⠀⡀⢀⣀⡀⢐⣤⠀⠀⢸⡆⠀⠁⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡄
⠙⢻⠗⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⢱⡀⠀⠀⠀⠙⢦⣤⣀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠇⠀⠀⢹⢣
⠀⠘⠒⠦⠤⠤⠶⣤⣄⣀⣠⣀⡀⢀⣨⣷⠀⠀⠀⢸⠀⠀⠐⠀⠀⠀⠈⠛⠂⠛⠒⠒⠓⠋⢨⢀⠀⠂⡆⠀⠀⠐⡜
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠴⠂⠠⠄⠀⠈⠉⠀⠀⠀⠀⡆⡀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡀⠀⠀⠀⡏⡅
⠀⠀⠀⠀⠀⠀⠙⠲⠶⠴⣦⣤⣤⣀⣠⣼⣤⣤⣧⣶⣯⣿⣧⣼⣷⣄⡄⢄⣀⣀⣀⣤⣈⡶⢤⠠⣤⣤⡇⠀⠀⢸⡇⣧⡀⠀⠐⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠉⠉⠉⠀⠀⠀⠀⠀⢤⣬⣠⣼⡇⡃⣄⡄⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⣯⣁⣁⣃⠆
⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠿⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⢀⠀⠀⣸⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣺⣿⡷⠀⠀⣤⣤⡠⠀⠉⠀⠀⢀⠾⣿⣿⣧⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⠉⢀⡀⢀⣰⣿⠀⠀⠀⠀⢴⡆⠚⢿⣷⣿⣿⣿⡏⣹⣿⣧⠀⠀⢰⣦⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣏⠛⣤⣬⣷⡎⠿⢿⡷⠀⢀⡶⠟⢻⡶⡞⢽⣿⣿⣧⣀⣸⣿⣿⢤⠀⣼⡿⠀⠀⢀⡔⠙⠛⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢟⣸⣾⣿⣿⣿⣭⢁⣰⣿⣄⣼⣸⣿⣿⣿⣶⣄⣿⡿⠛⠉⢹⣿⣿⢶⠶⢿⣓⢠⡀⠈⠀⢀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⢡⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⡧⠀⠀⣼⣿⣿⠇⠀⣼⣿⣿⣿⢆⣶⣦⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡈⠉⣿⣿⣯⣿⡟⣿⠏⣸⣯⢽⣶⣾⣽⣿⣿⠋⢳⣶⠖⠛⢿⡿⣷⣾⣿⣿⣿⣱⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢠⡟⠀⣿⣿⣤⡛⣴⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢁⡄⠀⢴⣷⣜⣩⡾⣻⣿⣿⠛⡟⠿⡇⢻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡜⢃⠀⢹⡟⣷⣤⢉⠁⣾⣿⣿⣿⡏⠈⠻⣥⣏⣼⡇⠀⣿⣿⣏⠉⣿⣯⠟⣽⣧⣠⠀⣧⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠈⣆⠸⣧⣹⣷⢸⣤⠇⠀⠀⠙⣷⣶⣤⣿⢳⣿⡗⠈⠉⢹⣿⡄⢸⣿⣷⠀⢘⣿⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣸⣿⣥⣶⣿⠄⢻⣿⣿⢸⣿⡀⢰⡄⠀⢹⣿⣿⣿⣿⣿⠁⠀⣾⣿⣿⣧⣾⣿⣿⠀⣸⣿⣇⣼⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣷⡀⣸⣿⣿⣿⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿⠀⣿⣏⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⡉⠻⠿⠟⣻⣿⣿⣿⣧⠸⣿⣷⣿⣿⡿⠛⢻⡇⢰⣿⠻⣿⢿⡿⠈⠿⠻⣇⢹⣿⣽⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⠟⢿⣿⣿⣿⣿⣿⣿⣿⡏⠻⣾⣿⣤⡿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠘⠟⠛⠉⠁⠛⠛⠛⠋⠛⠃⠀⠀⠀⠈⠉⠈⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠘⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠠⠤⠄⠒⠒⠳⡀⢀⠁⠈⠛⣦⡀
⠀⠀⠀⣔⣈⢉⣁⣀⣀⣀⣀⣀⡀⠀⠀⠀⠠⠀⠀⠇⠀⠀⠀⠀⠘⠛⣦⡀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠈⠉⠈⠀⠀⠀⠀⠂⠉⠉⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠉⠋⠉⢡⡀⠀⠈⠉⠁
⠀⠐⢧⣤⣤⣤⣤⣤⠤⠤⠤⠤⠴⠂⠀⠹⡀⠀⠀⢠⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⣇
⠀⠀⠐⠦⠤⡄⣤⣄⣀⣀⣀⢀⣰⡄⠀⠀⡇⠀⠀⠀⠈⠛⠐⠲⠴⠒⠂⣁⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠒⠀⠉⡀⢀⡀⣣⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠚⠒⠛⠛⠛⠛⠛⠛⠛⠻⠷⠷⠶⠶⠾⠖⠋⠋⠛⢓⣀⢀⣷⠟⡄⠀⡸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣁⣉⡰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⠟⠉⠑⢆⠀⠀⠀⢠⣾⡟⠑⡄⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠘⠀⠀⠀⣿⣿⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⣾⠠⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⢈⠀⠀⢀⣿⣿⠚⠁⠀⠀⠀⣠⣶⠖⣴⣶⠈⠟⢻⣷⡆⢠⣿⡿⢁⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠈⣠⠴⢺⣿⡏⢠⣾⣿⠀⣸⣿⠃⠀⣿⡿⠰⠀⣿⣿⢁⢼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠘⢿⣿⣮⠀⠀⣸⣿⣧⠇⣿⣿⠀⣿⡿⢀⠼⣿⡷⠃⠀⠻⠟⠋⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠙⣦⡞⢹⣿⣧⠀⣿⣿⡏⠠⣿⣿⠎⠙⠛⠋⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠿⠀⢸⣿⡟⠀⠿⠿⠁⠀⠈⠁⠀⠀⠀⢀⠀⠀⠠⠰⠀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⣠⣾⠟⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠉⠉⠉⠁⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠋⠉⠀⠀⢠⣾⡏⠉⠄⠀⢰⣿⠂⠀⠀⠀⠀⢀⡞⠡⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠊⠀⠀⣾⣿⣀⠀⢀⡤⠤⡀⠳⠦⣤⣤⠀⢰⣿⡗⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢻⣿⣷⠀⠀⠰⣿⣿⠀⠀⣿⡶⢀⠜⡁⢠⣿⣿⠀⣼⣿⡇⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠘⣿⣿⡤⠤⢚⣿⡇⠀⣼⣿⠁⠀⠀⠃⢸⣿⡇⢀⣿⣿⠃⡠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢺⣿⣧⠀⢸⣿⡇⠀⣻⣧⠀⠀⡸⠀⠘⠻⠓⠉⠈⠛⢉⠀⣿⣿⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⢸⣿⡏⠀⠈⠻⠧⠊⠀⠉⠁⠉⠀⠀⠀⠀⠀⠀⢀⠔⠁⢠⣿⡿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣼⠟⠁⠀⠀⠀⢀⡀⠤⠒⠀⠁⠀⠀⠀⠀⠀⢠⠏⠀⠰⣸⣿⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠋

⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢹⣿⡟⣻⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣟⣿⣿⡟⣿⡟⢹⣿⢸⡇⣿⣿⣿⡇⢿⣿⣿⠟⣿⣿⡟⠉⢿⢹⣿⣿⣿⡇⠀⠀⠀⢸⣷⣿⠈⠛⣿⠹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠁⠀⠉⠘⠇⠈⠉⠉⠀⠈⠋⣿⠀⠙⣿⠁⠀⠀⠈⢿⢻⡟⠀⠀⠀⠀⢸⠉⣿⠀⢸⢻⠀⣿⢡⣿⣿⡟⣿⣏
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⡏⣿⡟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠿⠂⠀⠉⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠋⠐⢸⣼⠀⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⠀⣿⠘⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⡿⢻⠃⠀⠀⠀⠀⠀⠀⠀⠘⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⡟⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣇
⠀⠀⠀⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢴⣤⠄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⠀⠀⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠄⠀⡀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⡀⠀⠀⠂⠀⢠⠋⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠛⠃⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⣿⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿
⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠉⠃⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡟⠀⠀⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⠇⠀⠀⠀⠈⠀⠉⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⡀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣽⣿⡇⠀⠀⠀⠀⠀⠉⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⣷⡄⠀⠀⠀⠀⠀⠈⠛⢷⣦⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠶⠶⣤⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠛⠶⠤⠴⠼⠦⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠀⠀⠀⠀⠀⢰⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣀⢀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⢀⣠⠴⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠀⠀⠀⠻⠀⠀⠀⣡⡾⠃⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠸⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡞⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠖⠲⣶⡄⠀⠀⠀⠃
⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⡀⠀⣰⡟⢀⡀
⠀⠀⠀⢨⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⣠⣄⣀⡀⢀⣤⣀⣤⠓⠊⠀⠀⢈⣉⣯⣭⠛⢳⣮⣭⠛⣻⣟⡭⢏⠉
⠀⠀⠀⢸⡀⠀⣀⣀⠀⢀⣀⡀⣸⡋⣒⣦⣤⣤⡴⠶⠙⣿⢹⣷⡄⠀⠦⣴⢟⣥⣖⡛⠛⠷⢶⣍⣡⣠⣭⡅⠛⢶⣉⠉⣻⡿⢻⣦⠀⠁⠀⠀⠾⠁
⠀⠀⠀⣼⡇⠀⠈⢁⣀⣤⣥⠴⢿⡟⢉⣩⣤⠿⠛⠶⠶⡾⢦⡼⣿⣤⣾⣷⣯⣤⣤⣤⢴⠞⠛⠉⠙⠛⠶⠶⠦⠦⠿⡦⣿⡛⠫⠀⠑⠀⠃
⠀⠀⠀⢹⣧⠀⠀⠀⠀⠁⣀⣠⠼⡟⠋⠁⠀⠀⠀⣀⣠⣾⠟⣲⠿⣿⡉⠉⠙⠆⢠⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠸⠉⠀⣠⠴⠒⠋⠁⠀⢀⣿⡤⠶⠒⠟⢉⡽⢋⣽⠞⠁⠀⢿⡇⠁⠀⠀⢀⡁⠀⠂⠀⠀⢀⠀⠀⠘⠃⠀⠀⢸⡏⠀⠁⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢹⡇⠂⢀⡴⠋⣠⠾⠁⠀⠀⠀⠘⣿⡀⠘⠀⢸⠁⢠⠀⠀⠰⡤⠁⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⡄⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠆⠀⠀⠸⣇⡴⠋⠀⠰⠇⣼⠃⡀⠀⠀⠀⢹⣧⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣶⡀
⠀⠀⠀⢠⣀⠀⢠⡮⢁⢸⣿⣶⠗⠺⣿⠀⠀⠀⠀⢸⡟⠾⠃⠀⠀⠀⠀⣿⡆⠀⢸⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠈⠁⠀⠈⠤⠨⠇⠙⠉⠀⠀⣿⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠸⣿⡄⢸⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣸⢠⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠔⠒⠒⠒⠒⠒⠒⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⡀⠤⠄⠀⢀⡀⠀⠀⠀⠉⠓⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣖⠉⡳⠀⡷⠒⠂⣠⠏⣧⠀⡴⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠠⣎⢉⠇⠘⠇⠀⠀⠉⠉⠁⠁⠀⠘⠰⢅⡀⠀⠀⠀⠀⠀⠀⠱⡄⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢠⡤⢀⣈⠧⠀⠀⢀⡠⠴⠒⠒⠒⠒⠒⠦⢄⡀⠀⠀⠀⣈⣠⣤⣴⣶⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠱⡎⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⢀⣀⣤⣤⣾⣷⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⡴⢁⣀⣀⣤⣴⣶⣾⣿⣿⣿⣿⡿⣿⣿⣿⠋⠻⣿⣿⣷⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣉⣤⣤⣶⣾⣿⣿⣿⣿⣿⠿⠿⠿⣿⣯⢡⣶⣾⣿⣿⣿⢸⣷⠜⢿⣿⡇⢿⣿⡿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣤⣴⣿⣾⣿⣿⣿⣿⣿⡿⢟⣛⣛⠻⣿⡆⢿⣿⡇⣹⣿⡌⣩⣥⣾⣿⣿⢠⣶⣾⣎⣻⣿⣬⣶⣶⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⢋⡹⣿⣿⣿⠸⡿⠿⣠⣿⣷⠨⣴⣾⣿⣿⣧⠘⣛⣋⣽⣷⣾⣿⣿⣿⣿⣿⣿⠿⠟⢿⠋⠉⠁
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⢸⣷⠜⢿⣿⡇⢲⣿⣿⣿⣿⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠟⠛⠉⠉⡄⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⢰⣶⣾⣦⣻⣿⣼⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠋⠉⠁⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡿⠿⠛⢟⡉⠉⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⢤⡀⠀⣀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠿⢿⠛⠋⠉⠁⠄⠀⠀⠀⠀⠈⠙⠒⠒⠒⠒⠒⠋⠁⠀⠀⢰⡦⣄⡉⠓⠁⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⣠⠖⡆⢀⡤⢄⡀⡠⠤⡄⢖⢉⡠⠀⢷⠁⠈⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠘⠉⠙⡇⢸⠢⠔⠃⡷⠔⠋⠘⢧⠴⠂⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠈⠀⠀⠀⠈⠀⠀⠀⢀⣀⡤⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠒⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠒⣂⣉⣽⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠁⢀⣠⣴⠾⠿⠛⠋⠉⠁⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⢀⣤⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⢹⣿⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠛⠉⠓⠐⠒⠠⢉⠲⣄⠀⠀⣿⡏⠉⠛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⢁⣀⡤⠤⠤⠤⣀⡀⠀⠈⣇⢹⡆⢠⣿⠁⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠑⠉⠁⢀⠀⡀⠀⠀⠀⢹⠀⠀⠹⣸⡇⡾⠃⠀⠀⠀⣀⣀⣠⣤⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠁⣀⠄⠒⠀⠉⠀⠉⠀⠀⠀⠸⠀⠀⠑⠛⠋⠀⠀⠉⠁⠀⠀⠀⠀⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⡀⢀⠂⠀⠀⠀⠀⢀⠔⠁⡠⠊⣀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠊⠀⣪⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠂⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠐⡀⠘⢷⣄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢂⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠈⣄⠀⠙⢷⠀⠸⣿
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢸⡇⠀⡄⠀⠀⠀⠀⣴⣿⣃⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠘⣦⠀⠈⢷⡄⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⣼⡇⠀⠀⠀⠰⠖⣾⡿⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⢶⣦⣤⣀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠘⣧⠀⠈⣷⣿
⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⣿⡇⢀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣮⡙⣷⡀⠀⠀⠀⠀⠹⣧⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⣿⡇⠈⠀⠀⠀⣾⣷⡶⠷⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⢮⠻⣆⠀⠀⠀⠀⣿⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⢸⡇⣠⣀⣠⣾⣿⠏⢠⠶⠿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⢶⣦⣀⠀⠀⠀⢧⡀⠀⠈⢷⡀⠀⠀⣿⡇⢸⡟
⠀⠀⠀⠀⠀⢀⣤⣤⣽⣷⣸⣧⠈⠙⢿⣿⡇⢠⣏⠀⠀⣸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢻⣿⣿⣆⠈⠻⣷⣄⠀⠑⣑⢄⠀⠈⣷⡀⠀⣿⣇⣾⠃
⠀⠀⠀⢀⡴⠋⠁⠀⠈⠙⢿⣿⡆⠀⠀⢹⡇⢸⡟⣶⣾⣿⣿⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⢸⣿⡏⢿⡄⠀⠙⣿⣷⣶⣤⡛⠉⠑⢻⣧⠀⣿⣿⡏
⠀⠀⠴⠋⢀⣴⠶⠟⠳⣦⠈⢿⡏⠀⠀⢸⠄⠀⠳⡙⠿⣿⠟⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢻⣿⣿⣿⡗⠸⡇⠀⠀⢹⣿⡿⠛⠁⡀⠀⢘⣿⠀⣻⡏
⢀⠀⠀⢠⡟⢠⠀⠁⠒⠈⣇⢸⡇⠀⠀⣼⠀⠀⠀⠉⠑⠒⠀⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢄⡙⠛⠛⠁⢠⠃⠀⠀⠸⠟⠀⠴⠋⠁⠀⢸⣿⢠⣿
⡆⠀⠀⣾⠀⠰⡀⠐⠊⠀⡟⢸⠁⠀⢠⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⣈⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⠟⠃
⢷⡀⠀⢿⠀⠀⠈⠂⠀⠊⣠⠃⠀⢀⣾⠉⢿⡄⠀⠀⠀⠀⠀⠀⢋⠀⠀⠀⠈⠉⠐⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣤⡴⢛⣿⡿⠃
⠈⢷⣤⣈⠳⠄⠀⠀⠀⠀⠀⣀⣴⡟⠁⠀⣿⠻⣦⣄⠀⠀⠀⠀⠸⣀⣤⣤⣤⣴⡶⠶⣾⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⢠⣿⠟
⠀⠀⠉⠻⢿⣶⣤⣤⣤⣶⡾⠋⣹⠃⠀⢠⡏⠀⠀⣿⠳⣦⣀⢀⡴⠋⠀⠀⠀⠀⢙⠄⠀⠀⠈⠹⣷⠀⠀⠀⣀⡴⠾⢫⠟⠁⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠈⠁⠀⢠⠋⠀⠀⣿⠀⠀⠘⣧⠀⠀⢿⡀⢠⡟⢿⡇⠀⠀⠰⠀⠀⠀⢠⠀⠀⠀⠀⡿⠶⣾⡋⢁⣀⣤⣤⡶⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢰⠀⠀⣿⡀⠀⠀⢻⡄⠀⠈⠳⣿⡄⠈⢳⡀⠀⠀⠀⡄⠀⠚⠀⠀⠀⡰⠀⠀⢸⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠣⡀⠘⢧⠀⠀⠀⠻⢄⡀⠀⢹⣇⠀⠀⢻⣄⠀⠀⠀⢹⠀⠀⠀⠴⡄⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠉⠠⠈⠁⠀⠀⠀⠀⠉⠑⠶⣿⡀⠀⡌⠀⠈⠁⠘⣿⡖⠉⠀⠀⠘⢄⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠐⢄⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠁⠀⠀⠀⠀⡟⢿⡀⠀⠀⠀⠀⠀⣸⣧⡀
⠀⠀⠀⠀⠀⣴⡏⠀⠀⠀⠀⢹⣿⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠈⢦⡀⠀⠀⠀⢰⡇⠈⠻⣄⠀⠀⠀⢀⡏⠈⠻⣦
⠀⠐⠀⠀⢸⠀⢻⣶⣤⣤⡴⠞⠋⠀⠔⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠙⣦⣤⣴⡿⡇⠀⠀⠈⠓⠀⠄⠊
⠀⠠⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠄⠀⣰⡟⠉⠁⠀⠛
⠀⠘⠀⠀⠀⠀⠙⠶⣤⣄⣀⣀⠀⠀⠀⠀⣀⣠⣔⢁⣀⣤⣾⡶⠞⠉

⠀⠀⠀⠀⡀⢀⢀⠀⣀⣀⣀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠠⣉⠤⠀⣏⣹⡆⡀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠀⡄⠀⠀⡀⠀⠀⠀⠄⠀⡄⡀⣶⢠⣄⠀⠠
⠀⠀⠀⠀⠀⠈⠂⠀⠉⠈⠀⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠂⠑⡆⠀⢃⣁⢾⣕⣤⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⢠⣛⣄⠀⣤⣀⠄⠀⠀⠉⡀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠋⠈⠀⠉⠁⠀⠉⠁⠀⠀⠛⠀⠀⠀⠀⠀⠀⠄⠐⠒
⠀⠀⠀⠀⠛⠉⠉⠀⠨⠩⠇⠀⡺⢶⠱⣼⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⡀⠀⠈⠈⡀
⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠈⡁⣸⡆⢀⣀⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⢀⡄
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⡧⠿⠀⢸⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠁⢺⠣⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡘⠩⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢡⠈⠁⠘⢐⡄
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠂
⠀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠀⠸⠀⢸⡇⡃⣀⡆
⠀⠀⠀⠀⡀⠀⣿⢁⠀⠠⠖⠀⡶⢺⡇⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠄⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠝⠀⠀⠈⠉⠀⢨⡇⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣵⣋⠠⠀⠀⠀⠀⣾⡿⠟⠛⠿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠒⢰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠙⠙⠢⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠿⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠘⠂⠀⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣿⡄⠘⡛⠂⢠⡏⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣶⣶⣶⠟⠀⠀⠀⠁⠐⠂⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⢀⡆⠀⠀⠅⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠰⢏⡳⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠝⢆⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢸⡇⠀⠀⠀⢷⠂⠘⠀⠂
⠀⠀⠀⠀⠈⠋⠑⠁⠀⢐⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢈⡈⠌⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣋⢉⡔⠆⠀⠀⠠⢠⠀⣟⢉⡄⠢
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠤⠀⠄⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠂⡮⡀⡤⠖⠀⠀⠀⠀⠀⠀⠄⠀⠉⠉⠁⠀⠄⠀⠀⠠⠀⠈⠉⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠂⠰
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⡿⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀⢸⣿⠁⣴⣶⡆⢠⣴⣤⠀⣿⣿⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⠇⢸⣟⠀⢻⣿⡇⢸⣿⡟⠀⢻⣿⠸⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⡆⢸⣿⣿⣿⣿⢡⡄⢹⣿⣿⣿⣿⢠⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣿⣯⣽⡇⢸⣿⣏⣉⣉⣈⣁⣈⣉⣙⣿⣿⢸⣿⣭⣿⣭⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣿⣏⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣹⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠃⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠘⠛⠛⠛⠛⠛



⠀⠀⠀⠀⠀⡴⠉⠁⡠⢤⡀⡄⢠⢀⡤⢰⠄⠀⡴⠉⠱⡄⡯⠀⠀⡞⡄⢠⡤⢄⢠⠤⣄⢠⠤⡀⠤⣄⢸
⠀⠀⠀⠀⠀⠱⠤⠄⠣⠴⠃⠧⠼⠀⠇⠘⠄⠀⠙⠤⠜⠁⠇⠀⠸⠉⠹⢸⠤⠜⢸⠤⠞⠹⠬⠑⠧⠟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈


⠀⢀⡀⠀⡀⡀⠀⠀⠀⡀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠆⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣐⠀⣀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢀⠀⠆⠀⠀⠌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠡⠐⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠰⠁⠀⠀⠀⠀⠙⢢⡀⠀⠀⠀⠀⠀⠠⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠉⠒⠢⣄⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣀⠖⠒⠉⠉⠉⠁⠒⠢⢌⡄⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠔⠁⠈⢀⡀⠀⠁⠀⠀⠀⠀⠈⠢⡀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣽⠢⠀⠑⢄⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡇⠀⠀⡤⠀⠀⠤⠄⢀⣠⣾⡇⠀⠀⠀⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠻⣿⣶⡶⠿⠛⢻⡟⠀⠀⠀⢰⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢄⠀⠉⠻⠄⠀⠔⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠶⠤⠤⠤⡤⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⢠⠃
⡄⠀⠀⠀⠀⠀⠀⡀⢀⡀⠠⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⢀⡎
⠱⡀⠀⠀⠢⡀⠀⠀⠀⠀⠑⠀⠀⢄⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠘⢎⠀⠀⠈⠂⡤⢆⠀⠀⣀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠑⠦⡀⠄⠀⠀⡀⠀⠀⠁⠂⠤⣀⠈⠢⢄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖
⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠦⠤⡄⠀⠀⠀⠀⠀⠀⠈⠶⠥⣄⠀⠀⠀⠢⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠑⢄⠀⠀⠈⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⡸⠁⣀⠀⠀⣀⠜⠀⠀⠈⠢⣀⠌⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⢠⠤⡀⢠⠙⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡀⣼⡀⠀⣬⠀⣴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢩⠀⣠⠚⣣
⡇⠘⡁⠀⢷⠄⠘⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣈⠷⠁⡰⠉
⠃⡀⠀⠀⠠⠠⣤⣄⣡⠄⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣴⡇⠀⠛⠀⠼⠁⠀⢀
⠀⢡⠀⠀⠀⠀⠀⠈⠁⢠⠄⠼⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠈⠡⠀⢀⣀⣠⡀⣠⡀⠀⠀⠀⠀⠀⠙⠶⣀⠈⠃⠀⠀⠀⠀⠐⠂⡀
⠀⠀⠳⣀⠀⠀⠀⣀⡴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠈⠛⠀⠀⠋⠀⠀⠠⠟⠀⢹⠀⠀⠀⠀⠀⠀⠀⣹⠀⠘⠳⠄⠀⢀⡔⠦⢀
⠀⠀⠀⠈⠛⡦⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⣄⡀⠀⠀⠘⢦⡀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⢰⡄⠸⣧⠀⠀⠀⠀⠀⠀⠀⢴⣆⢀⣀⠠⡤⣤⢤⡄⣀⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠀⠀⠀⢀⠌⠁⠀⠖⠋
⠀⠀⠀⠀⠀⠘⡇⠀⢹⡆⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠀⠀⠻⣄⠀⠀⠀⣰⠋⠀⢀⡿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣷⠀⠀⢀⡤⠊⠁⠀⢀⣄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠙⠀⠀⣴⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠈⣦⠀⣀⣧⣠⠛⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠱⡆⠀⠀⣄⢠⡜⠁⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⢸⠛⠉⠉⢨⡏⠀⠀⠀⠀⠀⠀⠐⠀⠈⠃⠀⠀⣀⡤⠞⣋⠀⠀⠀⢰⠀⠀⣿⡏⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⡘⠁⠀⠀⠀⣤⣄⠤⡤⢤⣤⡤⠚⠛⠉⢀⣴⡿⠀⠀⠀⠘⠓⣤⣿⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⡇⠀⠀⠀⠀⣿⣿⡀⠀⠀⢀⣀⣤⡴⠾⢿⣿⠃⠀⠀⢠⣄⢀⡼⠋⠙⢶⡋
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢠⡄⠀⠀⠀⢹⣿⣿⣿⣿⣿⢋⣀⣀⣀⣼⠃⠀⠀⠀⢠⡟⠁⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⢣⠀⠀⠀⠀⠙⢿⣟⠉⠀⠀⣀⡤⠏⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠷⣀⠀⠀⠀⠀⠀⠈⠛⠁⠁⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠙⠳⢤⣄⣀⣀⣀⣀⣀⣀⣠⣤⡶⠛⠉⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠻⡏⠉⠉⠉⠉⠉⠀⠲⠶⠒⠦⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠈⠉⠉⠉⠓⠒⠶⡶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠈⠉⠠⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢣⡀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠀⢀⣀⣀⣀⣀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠉⠉⠉⢛⠉⠉⠀⠘⠒⠒⣶⠶⠤⣤⣤⣤⡤⠔⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⣿⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠸⠃⠀⠀⠀⠀⠀⠘⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣀⡆⠀⠀⠀⠀⠀⠀⢀⡦⠀⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⠉⠉⡷⠀⠀⠀⠀⣀⡤⠋⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⣀⣠⠔⠊⠁⠀⠀⠀⠀⢿⡀⢠⡄⠒⠉⠀⠀⠀⢀⢼⡇
⠀⠐⠋⣀⠀⠁⠙⣀⠀⠀⠀⠀⢀⡼⠶⠊⠙⠂⡀⠀⠀⠀⠀⡰⠃⢠⠃
⠀⠈⠲⠄⣀⣉⣉⣉⣉⣉⠉⣰⠛⠤⢀⡀⠀⠀⢡⡄⢀⣀⠐⠷⢀⡙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣠⣤⣤⣤⣬⣥⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⠻⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣟⣀⣀⣀⣀⣀⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⡠⠖⢚⣉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⡴⠀⢠⡄⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⢀⠖⠉⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠰⠁⢀⣼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⡆⠀⣀⣤⡀⠀⠈⢦⠈⢢⡀
⠀⠁⠀⠀⠀⢿⡉⢢⡀⠀⣠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠷⠄⠀⠠⡶⠿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣿⢠⣇⡼⠋⣰⠃⠀⠀⠀⢧⠀⢳
⠀⠀⠰⣟⠓⢮⣳⣄⠱⣼⡇⣠⠏⠀⠀⠀⠀⠀⠀⢀⡾⣫⡴⠆⠀⠀⠀⠀⠀⠀⠀⣀⡉⠻⣆⠀⠀⠀⠀⠀⣞⢉⣉⡙⢿⣄⠀⠀⢾⡥⠶⣦⠀⠀⢸
⠀⠀⠀⠈⠓⢄⡙⠁⣠⠟⠁⢳⡀⠀⠀⠀⠀⠀⠀⣸⠇⢁⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡦⠹⣧⠀⠀⠀⠀⠉⠁⠀⡝⠀⠈⢷⠀⠀⡤⠖⠋⠀⠀⢸
⠀⠀⠰⣏⣉⣈⡁⠰⠃⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠙⢦⣄⡀⠀⠀⢀⣠⡴⠛⡇⠀⢹⡆⠀⠀⠀⠀⠀⠀⢷⡀⠀⠈⠁⣤⣉⣱⡄⠀⠀⠏
⠀⠀⠀⠀⢠⠞⣡⣄⣀⣀⣄⠙⢦⡀⠀⠀⠀⠀⣠⣿⡀⢸⣇⠀⠀⠈⠉⠛⠋⠉⠁⠀⠀⣿⠀⢸⡷⣤⡀⠀⠀⠀⣰⠋⢠⡷⠒⠋⠁⠈⠉
⠀⠀⠀⠀⠘⠋⠀⠈⠉⠉⠈⠳⣄⠙⢦⡀⠀⠀⣧⣂⡀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⢀⣀⣸⠇⠀⢠⡞⢁⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠙⢦⠀⠈⣿⠀⠀⢿⣿⣿⣿⣷⣶⣶⣶⣾⣿⣿⣿⠀⠀⣿⠁⢀⡴⠋⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠳⢄⣿⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⣿⣠⠞⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⣿⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠁⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠈⢧⡈⠉⠙⠛⠛⢛⡿⠃⠀⠀⠀⣿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⡀⠈⠻⢦⣤⣤⠶⠋⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠙⢦⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⠓⠦⣄⣀⠉⠛⠛⠛⠋⠁⠀⣀⣠⡴⠞⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⣀⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢀⣤⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⢦⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⢿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⠾⣦⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠸⣇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⣀⣼⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⢷⡀⠀⠀⠘⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡿⠀⠀⢀⣠⠶⠛⠙⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠙⢦⡀⠀⣠⠏⣹⠛⠻⠶⠶⠶⠶⠶⠶⠶⠾⠟⠛⢻⡙⢦⣀⡴⠋⠁⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⠲⠋⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡉⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠀⠀⠀⢱⣠⠔⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⠙⡄⠀⠀⠀⢀⠀⠀⠀⠀⢠⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣄⠐⢀⡜⠁⠄⠀⢀⠀⠀⡇⠀⠀⠀⠘⢆⠀⠀⠀⢸⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⣻⠟⡝⡟⢷⣶⣤⣀⣀⠞⠁⠀⢀⠀⠀⠈⠧⠀⠀⢈⣀⡀⠈⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⠋⢀⡬⠐⠋⠘⠁⠁⠀⠐⠦⠙⢷⣦⣀⠀⠈⠙⠂⠀⡠⠚⠉⠀⠀⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⣡⡟⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢿⣯⡣⡀⠒⠀⡼⠁⣰⠃⠀⠁⠀⠀⠀⢳⠐⠚
⠀⠀⠀⠀⠀⠀⠐⠟⡴⢡⣇⠛⠀⡀⠀⠀⠀⠐⠂⠀⠈⠀⢀⠀⠛⢸⣻⡷⣵⡀⠀⠱⡀⠹⣄⣀⣀⡤⠀⢠⠏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠹⡄⠀⢣⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢀⠏⠀⠇⠻⠁⠀⠀⣹⢶⠤⢍⡭⠤⠖⠁⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠑⠦⢄⣀⣀⡠⠴⠊⢀⡴⠋⠀⠀⠀⠀⠀⠀⣠⠜⠃⠀⣀⠀⢰⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣲⠦⠤⠤⠤⠤⠔⠚⢿⢄⡀⠀⠀⣀⢀⡠⠊⠁⠀⠀⠀⡇⠀⠀⢇
⠀⢠⠊⢉⣚⠉⢹⠄⠀⢀⣀⣠⠔⠋⡽⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠑⣾⡷⠏⠀⠀⠀⠀⠀⠀⣃⣀⠀⠈
⠀⢿⢠⠋⢀⢱⠈⠓⡄⠀⠞⠿⠁⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢱⣠⠞⠁⠀⠀⠀⠀⠀⣀⣴⡉⠵⠮⣗⠒⣄
⠀⣰⠛⢻⠾⠊⡤⠞⠁⠀⠀⠀⠀⠘⠓⠋⢹⡏⠉⠉⠉⡟⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⣅⢎⣥⡼⢴⠃
⠀⠈⠋⢉⣓⣚⣆⡴⣦⣀⡠⠤⠤⠒⠒⢃⣸⠇⠉⠉⠁⣇⡀⠉⠙⠓⠒⠒⠦⠤⢄⣀⠉⢱⣌⡹⠯⣀⡼
⠀⠀⠀⠈⠓⣮⠿⠛⠉⠁⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⡀
⠀⠀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳
⠀⠀⠁
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢘⣀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠋⠔⠋⠄⡰⠀⠀⠀⠂⠠⠀⠀⠂⠱⠘⠀⠶⠂⠋⠇⠃⡇⠱⠀⠀⠰⠂⠁⠀⠀⠴⠀⠆⠀⠸⠉⠱⠋⠄
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⠿⢿⢻⠿⡿⢻⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣴⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠿⠿⠛⠻⠟⠛⠿⠿⢿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠋⡥⠄⠀⠀⠀⠀⠀⢤⡀⠈⢻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠘⣿⠋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⠀⠀⣾⣷⠀⠀⠀⠀⠀⣿⠟⠀⠀⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠉⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡐⠀⠀⢀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣤⡤⢴⠆⠀⠀⠠⠂⠀⠀⡠⠀⠁⣮⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⢌⣀⣠⡔⢈⡀⠠⣤⠐⠈⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⢹⣧⣼⡷⠀⠀⠀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠠⠐⢻⡀⠀⠀⠀⠙⠈⠁⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⠁⠀⢀⠠⠀⠐⠛⠊⡇⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠑⠂⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠇⠀⢉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠰⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡻⣽⣷⠇⠀⠀⠀⣷⣿⣫⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠈⡻⠃⠀⠀⠀⠀⠘⠿⠿⠛⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐
⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣟⣻⣿⣿⣿⣻⣿⣛⣟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠀⠀⡄
⠀⠀⠀⠀⠀⠠⡄⠀⡖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠺⠳⢾⡆⠰⠛⡠
⠀⠀⠀⠀⠲⡀⠘⡀⠠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⣴⣀⢀⡀⠀⠰⠈
⠀⢰⠀⠀⡀⠑⠄⠀⠄⣀⠀⠀⠀⢀⠄⢀⠔⠀⠀⠈⠑⠒⠀⡀⠉⠉⠉⠆⠀⡀
⠀⠀⠑⢄⠈⠢⣈⡄⠘⠋⠀⢰⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠙⠶⠈⠀⢠⠄⠉⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⢦⡀⠂⡀
⠀⠀⠘⠀⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠰
⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⢀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⢙⡷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠘⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠖⠉⠀⠀⠊⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢂⠀⠀⠀⠀⠀⠀⠀⡰⠋⠑⠶⠀⠉⠉⠉⠛⠻⠶⣿⡄⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠈⢠⠀⠀⢀⡠⠆⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⢦⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⡄⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠔⠋⠀⣼⣷⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⢰⠀⠘⢷⠒⠂⠐⠒⠒⠊⠉⠁⠀⢀⣤⣾⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⢻⠀⠀⠀⢧⣤⣄⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌
⡷⠀⣀⡀⠀⠀⣀⡀⠐⠈⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁
⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠙⢿⣿⡿⠋⠀⠀⡀⠠⠿⢻⡟⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃
⠀⠀⠙⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠈⠓⠴⠂⣀⣀⣀⠴⠋⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⠀⠀⠐⣂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠷⣤⣤⣀⣀⣠⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠦⣤⣤⣤⣠⣤⣤⠴⡾⠋⣀⠀⠷⠈⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⢻⡅⠀⠀⡇⠀⠈⠁⠶⠂⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⢄⣀⣀⠀⠀⠀⠀⠀⢀⣀⠴⠛⠁⠀⠘⠀⢆⣤⠁⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠁⠈⠁⠀⠀⠀⠀⠀⣦⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⡀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠈⢛⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠘⠃⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣤⡀⠀⠀⠀⠀⣠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠶⠛⠁⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠸⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⢠⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⣁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣄⠀⠀⠀⡸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⣤⡞⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣌⠳⠶⠞⠁⡇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠴⠞⠉⠀⠻⣆⠀⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⡴⢾⠁⢠⣼⣁⣀⣀⣀⣀⠀⡴⢶⡞⠉⠁⠀⠀⠀⠀⠀⠀⠘⣧⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⢇⠀⠁⠀⠀⠀⠀⠀⠀⠄⢐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠡⣀⠀⠀⠀⠀⠀⢠⠂⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠓⢦⣤⣤⡴⠋⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠤⣤⣤⣤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣀⠀⣀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠋⠁⣰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⣤⣿⣷⣾⣷⣶⣶⣄
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣦⣼⣿⡏⠀⠀⠀⠉⠉⠈⠙⡇⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⠿⣿⣿⣿⣆
⠀⠀⠀⠀⣿⣿⠟⠋⠛⠛⠿⢿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠏⠋⠉⠈⠁⠀⠈⢻⣿⣿
⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠘⣿⡇
⠀⠀⠀⠈⣜⠀⠀⠀⠁⡀⢀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸⠗⠎⠀⠄⠀⠁⠄⠠⠀⠀⠀⠀⠐
⠀⠀⠀⢠⣿⣦⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠤⠠⡉⠈⠀⠀⠀⠀⠀⠀⠀⢤⡀⣀
⠀⠀⠀⣸⣿⣿⣿⣿⠗⠐⢿⣿⣷⠀⠀⠀⠂⢀⠄⠹⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡛⠃⠓
⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⠀⠈⡀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎
⠀⠀⠀⠈⠉⠉⢱⠀⠀⠀⠀⢈⠁⠀⢠⠀⠀⠀⠀⠀⠈⠩⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠑⠁
⠀⠀⠀⠀⠀⠦⠲⠀⠀⠀⠀⠸⠲⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢒⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠥⡄⠈⢴⠀⠀⠉⡙⠀⠀⠀⠀⠈⡤⠏⠘⠀⠀⠀⠀⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠠⢺⠇⠀⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⣠⠴⠒⠲⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣶⣶⣶⣦⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢸⠃⡄⠉⠐⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣤⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠁
⠀⠀⠸⣄⠡⠀⠄⢀⡼⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⢠⣶⡟⠁⠀⠈⠀⠀⠀⠀⠀⣀⣴⡿⠛⢁⣀⣤⣦⠶⠆
⠀⠀⠀⠈⠓⠲⠒⠋⠳⣦⡀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢍⡅⠆⠳⠰⠀⠀⠀⠀⠙⠻⣵⢼⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠚⠒⠒⠒⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⢀⠀⠀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢅⠳⢔⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣷⡤
⠀⠀⢀⣼⠿⢷⡀⣿⠉⠈⢹⡆⠀⠀⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⠅⢄⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⡄
⠀⠀⢸⡅⠀⠀⣿⠙⢦⣤⠾⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠠⠢⡡⠁⠀⠀⠀⠀⢀⣠⣤⡿⢿⡛⣁⡆
⠀⠀⠀⠙⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠟⠊⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠀⠈⠃


⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⡀⠀⠀⠀⠀⠀⠀⠀⣴⣒⣒⠢⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠾⠿⣆⠀⠀⠀⠀⠀⠀⠀⢴⢰⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢠⣾⡿⠗⠀⠀⠀⠀⠀⠀⢠⣤⢧⣄⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣟⣿⣿⣿
⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠚⣿⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠙⡏⣀⡀⠀⠀⠀⠀⠀⠀⢼⢸⣭⡿⠿⣯⣿
⠀⠀⢠⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⣛⣿⠀⠉⠉⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⣺⢸⣿⣷⣶⣿⣿
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠘⠘⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⣯⢪⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⣶⣓⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⣠⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣾⡿⢦⣶⣤⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⣠⡾⠟⡛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⣷⣾⣿⣶⡄⠀⠀⠀⠀⠀⢨⣿⣿⡅⣾⣶⢹⣿⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠏
⠀⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⠀⠀⠸⠿⠿⠷⠮⠴⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⢀⠀⡀⡀⡠⢸⠠⡂⢠⠠⡀⡄⡂⠄⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢖⠂⣆⠗⠁⠇⠀⠣⠸⠀⠚⠘⠀⠃⠃⠁⠀⠀⡏⢀⠇⠄⠐⢰⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣀⠤⠄⠤⢄⠈⠁⠀⠈⢁⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠎⠉⠀⠁⠈⠀⢀⣀⠀⠈⠑⠒⠖⠢⠤⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⠔⠉⠁⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠠⡉⡅
⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⡀
⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠀⠐⠢⠭⢥⡝⠀⠀⠁⠀⣀⠖⠄⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢒⠄
⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⠷⣮⡍⠁⠈⢣⠀⢀⣀⠆⠁⠀⠈⠙⠔⠆⠤⠤⠦⠀⠀⢀⡀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠘⠀⠀⢸⢸⢀⠆⠋⠁⠀⠀⠀⢹⠉⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠉⠁⠀⢀⡏
⠀⠀⠀⠀⠀⠀⢇⠀⢁⠏⡼⠀⠀⠀⠀⠀⠠⡘⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠃⢣⠀⠀⠀⠀⠀⢠⠕
⠀⠀⠀⠀⠀⠀⠈⢢⣀⠜⠀⠀⠶⠀⠀⠀⡀⡄⢂⠀⠀⠀⠰⡇⠀⠀⠀⠠⠦⠀⠀⠣⣤⢠⡀⡰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡙⣦⣀⠀⠀⠀⠀⠀⢀⣠⠞⠓⢤⣀⠀⠐⠐⠀⠀⠀⠀⢀⡠⠔⠁⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠙⣿⠐⠒⢲⣙⠷⡀⠈⠑⠀⠓⠋⠁⠀⠒⠒⠂⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣀⣠⡿⠃⠀⠸⠄⢠⣆⠀⠀⠀⣀⠀⣀⠐⠤⣦⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⠀⣈⠀⠛⠂⢸⠀⠉⠉⠂⠴⠗⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠁⢀⠄⠉⠁⠰⠦⢼⡀⢀⢀⠀⡀⡀⢄⡒⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠐⠃⠴⠻⠦⠐⠛⠒⠀⠀⢘⠁⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠠⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡃⠀⠐⠀⠀⢀⠀⡀⠐⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢁⡁⠀⢈⢩⡀⠀⠀⡀⣄⡄⠀⢈⡁⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⡿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣟⢁⠉⠑⢀⡀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣹⣿⣿⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⣎⣁⣰⣷⣿⡆⠀⠀⠀⠀⠀⣽⣿⣿⣛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠀⢠⣆⠻⢿⡌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⢿⣿⢿⠀⠀⠀⠀⠀⠀⠉⣿⣟⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣤⡉⠣⣤⠂⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣼⣿⣿⣿⣧⠁⠀⠀⠀⠀⠀⣀⠀⠈⠉⠉⠁
⠀⠀⠠⡎⠈⠂⠀⢸⣿⡿⣿⡻⢿⠉⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠨⠁
⠀⠀⠈⡇⠀⠀⠀⢸⣾⣟⢎⢱⡿⠓⡶⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠐⠀⠠⠞⠉⠙⠛⠉⢲⡔⠀⡑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠄⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠛⠻⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⣿⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠂⡞⠉⢄⣀⣤⣼⡿⠟⠋⠻⣽⣷⡀⠀⠀⠘⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠢⠑⠜⠿⠷⠤⡶⠀⠀⠀⠀⠀⠀⠀⠈⠒⢲⣿⣿⣿⡿⠁⠀⠀⠀⠀⠘⣿⡇⡀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠙⠀⠀⠀⠀⠀⠀⠁⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠋⠀⣠⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⠀⠀⠀⠛⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣻⠄⠀⠀⠀⢻⠼⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠳⠶⣷⠄⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣀⣼⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣠⣸⣿⣤⡈⣯⣀⣀⣀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡛⢛⡃⠀⠈⢛⣻⣿⣿⠛⠙⠛⠛⣿⠿⢿⠿⣶⣦⣀⣀
⠀⠀⠀⠀⠀⠀⣿⣍⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣿⣿⣥⣤⣤⣤⣤⣭⣽⣶⣶⣿⣿⣏⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠲⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡚⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣻⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣽⣿⣿⣿⣾⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣟⣿⡛⣿⠿⠿⣿⣿⡿⠿⢿⣿⣿⠿⠿⠿⣿⣟⠉⠙⣯⡤⣧
⠀⠀⠀⠀⠀⠀⠀⠀⡸⣿⣿⣷⣷⡰⣄⣾⡏⣶⣤⡄⣿⣅⢲⣤⠀⣸⣿⣤⣴⣿⢿⡋⠑⣄
⠀⠀⠀⠀⠀⠀⠀⣼⠻⣿⡈⠉⣿⣿⣿⢿⣿⣾⣿⣾⣿⣿⣶⣧⣶⣿⣿⠿⠿⣅⠀⣳⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠻⢶⣿⣷⣾⣿⣦⣄⡀⣿⣩⣏⠉⠀⠀⣹⡅⠀⣸⣇⣀⣀⣿⣿⣿⣠⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣷⣾⣿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣐⣂⣀⠀⠀⠀⠀⠀⠀⣒⣒⠂⠂⠀⢒⠐⠰⢂⣀⣀⣠⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣴⣦⣶⣶⣾⣾⣦⣤⣄⣠⣴⣾⣿⣽⣶⣶⣤⣭⣷⣶⣾⣷⣯⣽⣶⣶⣶⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⠈⠑⠐⠋⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣿⣿⣿⣧⣶⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣧⣀⣀⠀⠀⠀⠀⠀⠠⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠲⠀⠠⢀⣀⡠⠤⠤⠰⠦⣤⠤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⡷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⣿⣿⣿⣯⣤⣯⣽⣿⣿⣗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣲⣖⣒⢚⣸⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⠿⠛⠿⠿⠿⠿⠟⠛⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣿⣹⣻⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣯⣿⣿⣿⣽⣿⣽⣿⣿⣿⣿⣭⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠖⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣼⠏⠀⣠⣤⡀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣿⣧⣴⣦⠀⠀⢀⣀⣀⣰⣶⣶⣼⣷⣿⡇⠀⠀⢸⡏⠀⠀⠈⢻⣿⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣿⣯⣤⠹⣿⣿⣾⣯⣿⢿⣿⢿⣁⣛⣿⡟⢻⡆⠸⣷⣄⣀⣀⡼⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⡴⠿⠛⢻⣿⠁⢸⡿⠈⢛⠋⢹⣷⣸⠃⠀⠈⠉⠉⠁⠀⡿⣴⣄⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣌⠁⠀⠀⠀⠀⠈⢉⣾⡆⠀⢰⣿⠀⠀⠉⠀⠀⠀⡀⢀⣤⣤⣶⡀⢹⠻⣿⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⣿⢀⡀⠀⠀⢠⣄⢸⣿⡇⠀⢸⣿⡄⣤⣤⣶⣴⣿⢹⠟⣿⡿⣿⡇⠌⣜⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⢀⣴⠃⢹⣿⠀⠀⢈⣩⢸⣿⠁⣰⡿⣿⡇⣿⡏⠉⠙⠿⠥⠔⠛⠁⠙⠋
⠀⠀⠀⠀⣷⣶⣦⣤⣾⣿⡿⠛⠁⠀⢸⣿⣾⣷⢸⣿⠈⢿⣆⠿⣿⠟⢷⣛⡁⠀⠀⠀⠀⠀⠀⠠⣤⡀⢰⣿
⠀⠀⠀⠀⠉⠛⠛⠉⢿⣿⣧⠀⠀⠀⠘⣿⠙⠿⠜⠛⠃⠀⠁⣠⠶⣛⣋⠙⢿⣿⣦⠀⣠⢤⣦⡀⢹⣷⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⡀⣀⡜⠁⢀⣤⣄⡀⠀⢀⣾⠃⠐⣿⣿⠀⠀⣿⣿⡇⢻⣿⣿⣷⢸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣛⣋⠉⠀⠀⣾⠁⢹⣷⠀⢸⣿⣶⣦⢻⣿⡆⠀⣿⣿⠇⢿⣿⠛⠿⠊⣙⣽⣿⡟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠛⠻⣶⣤⣿⠀⢀⣿⡇⠀⠙⠛⠋⢸⣿⡇⣠⡿⠏⠀⠀⠀⣠⡴⠚⠁⠘⣿⡇⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⠀⠀⠙⠟⠀⣸⣿⠁⠀⠀⣠⡖⢺⣿⠋⠉⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠿⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⣀⣀⣴⡿⠃⠀⠀⠈⠻⠟⠛⠁⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣬⣉⡉⠀⠀⠀⠀⣀⣀⣤⣶⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⣥⣤⣤⣤⣤⣤⣤⣤⣀⣀⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⡟⠀⠈⠿⠋⠹⣿⣿⡿⠿⣿⣿⣿⣿⡿⠀⢀⣴⣶⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⠀⠀⠀⠀⠀⠀⠁⠀⠀⢻⠀⠀⠀⠀⣰⣿⡿⠁⣸⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⠀⠀⠀⠀⠀⠀⠀⢀⣀⡶⠾⠂⠀⠀⣰⣿⠟⠀⢠⡟⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣿⠀⠹⣆⠀⠈⢷⣄⡀⠀⢀⡽⠁⠀⠀⠀⢰⣿⡟⠀⢀⡾⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣇⠀⠘⢧⣀⣼⠿⠛⣿⣿⣿⣶⣤⣀⠀⣾⣿⠀⣠⢟⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣠⣿⣿⣿⡀⢠⣿⣿⣿⣿⣿⣿⣿⡾⢿⡮⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢸⣿⡅⠀⣸⣿⣿⣿⠟⠋⠉⠉⠁⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣿⣧⣠⣿⡿⠋⠁⠀⠀⠀⠀⠐⣆⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠁⠀⠀⠀⠀⠀⢠⡈⢳⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠹⣄⢀⣴⣶⡶⢶⣤⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁





⡿⠿⠿⠿⡿⢻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣻⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠉⠉⠉⠉⠉⠙⠓⠦⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠁⠀⠀⠀⣀⠤⢂⠀⠀⠀⠀⠀⠀⠉⠙⠒⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣀⠴⢃⡄⠀⠀⠀⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡞⠀⠀⢀⡤⢴⠾⠦⠒⠫⠿⠴⠒⠉⣘⠋⠯⠔⣏⡀⠀⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢾⠁⣠⠴⠒⠓⡄⠀⠀⠀⠀⣋⣉⡉⠓⢦⡉⣿⡄⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⢸⣸⡡⠒⠋⠙⠁⠀⠀⠀⠀⠀⠀⠈⠙⢦⡇⠸⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣿⣬⠀⡴⠚⠛⠂⠀⠀⠀⠀⠐⠚⠛⠲⡄⠤⠠⡷⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⣿⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⣧⣖⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⣿⠁⠀⠀⠀⠀⠠⣄⣀⣀⣤⠤⠀⠀⢀⡀⠀⠀⡟⣯⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣹⠀⠀⣰⡦⣤⣄⣀⣈⣁⣤⣤⠴⠖⣛⡿⠀⠀⣏⣡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣇⠀⠈⣷⠦⠤⣤⣤⣤⡤⠴⢶⠛⢹⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠸⣦⣼⣁⣀⣁⣀⣠⣼⢷⡟⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠙⠲⠶⠿⠿⠶⠶⠖⠋⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣦⣀⡀⠀⠀⠀⠀⣀⣠⠴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡌⠉⠙⠛⠛⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢺⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⢺⣻⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢸⣯⡏⠉⠛⠒⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠁⠀⠀⠀⠈⣎⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡾⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠘⢦⡑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢭⣓⡲⠦⠤⠤⠤⠶⣚⣩⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣉⣀⡐⠀⠐⠒⠮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⢟⣛⣩⣭⣭⣛⣓⡀⠀⠀⠀⢮⣷⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣋⠵⣾⢿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⢿⡲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡼⠁⠈⠀⠀⠀⠀⠐⠒⠲⠄⠀⠀⠀⠀⠀⠀⠸⣷⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠠⠚⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠁⠀⠀⢀⠤⡀⠀⠔⡡⠃⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⡤⣄⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣧⠃⠀⠈⢀⠡⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠀⠀⠀⠀⠁⠀⢰⠚⠆⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠐⡇⠀⠀⠀⠀⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢸⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⢙⠂⠤⠞⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣏⠀⠀⠀⠀⠀⠈⣏⠉⠁⠀⣠⠔⣻⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢠⢂⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠘⢇⡉⣉⡠⣺⠏⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⡠⠚⡡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠉⠒⠚⠁⠀⠀⠀⠀⣠⠞⠁⠀⣀⠴⣌⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⡀⠀⠀⠀⣀⣠⡴⢾⠁⠒⠒⠉⣁⡠⠟
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣉⠝⠋⢋⡀⢀⡞⢰⠒⠊⠉⠁
⠀⠀⠀⠀⠀⡠⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⢀⣀⣀⣰⠒⠒⠒⠉⣀⣤⡆⠀⠉⠁⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣀⡴⠂⠤⠤⠤⠤⠖⠚⠦⠔⠒⠋⠁⡼⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡒⠒⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⠃⠈⠙⠲⠤⣀⡉⠑⠲⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠒⠒⠒⠒⠲⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠘⣦⠤⣄⢀⣀⣠⠋⠑⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⡭⢷⡄⠀⢀⣀⣀⣀⣈⠑⠲⠦⢤⣤⣀⣀⡀⠀⣀⡁⠀⠀⠀⠀⠀⢸⠀⡟⠈⠀⠀⠈⠉⠉⠉
⠀⠀⠒⠄⠀⠀⠀⠀⠀⡰⢮⡠⠚⠁⠀⠀⠛⠉⠉⠉⠁⠈⠉⠁⠢⢤⣀⡀⢀⠉⠉⠉⠉⠴⡦⠀⠀⠀⢠⠋⡰⠃
⠘⢌⠢⡀⠈⠑⠢⢄⡘⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⡀⠀⠀⠀⣸⠣⠤⠔⣊⡭⠛⠁
⠀⠈⠱⣈⠢⡀⠀⠀⠈⢳⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣤⠏⠙⠋⠉⠁
⠀⠀⠀⠈⠑⢌⠳⢄⢀⡞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠾⠿⠷⠶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣴⣶⣶⣶⣶⣶⣶⣤⣉⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠉⠙⠋⠉⠛⠋⠛⠛⠛⣿⡇⠀⠀⠀⠀⠀⢻⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠻⡿⣛⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢛⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢀⠀⠀⠰⠀⠀⠀⠀⠀⡇⠙
⠀⠀⠀⠀⠀⡀⠀⢈⠉⠇⠀⠀⠀⠀⠐⠂⠀⠀⣀⠀⠀⠀⠀⠠⣴⡾⠦⠆⠀⢘⠀⠀⠀⠐⣲⢶⠆⠀⠀⠀⠀⠀⢀⣿⡀⠰⠘⢻⡵
⠀⠀⠀⢤⡷⠓⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠑⠠⣔⠳⢞⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⠃⠁⠀⠈⠁
⠀⠀⠀⠈⠁⠐⠪⡀⠄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⢀⠤⠑⠠⡀⠚⢷⣶⣶⡄⢠⣶⣾⠿⠟⠋⣠⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢄⠁⠨⢀⠀⠀⠄⠀⠀⠠⠤⠀⠐⠄⠂⠀⠐⠁⠀⠀⠀⠀⠐⣆⠀⠉⠁⠈⠘⠁⣀⣴⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡄⠀⠐⠆⠂⠀⠑⠂⠉⡀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠁⠀⠀⠀⠈⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⠂⠀⠀⠀⠀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡻⠷⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⡀⠄⠐⡀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⣡⣀⢡
⠀⠀⠀⠀⠀⠀⠠⢂⠒⠦⠀⡘⠂⠀⠀⠀⣀⣀⡀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠆⠂⠀⠀⠑⠒
⠀⠀⠀⠀⠀⠀⡃⠲⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠈⠈⠁⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢀⠡⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠛⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡛⠛⠿⢿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣭⣍⠙⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠈⠙⣿⣿⣦⣴⣿⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠈⠙⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣶⣿⣿⣿⣿⠿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⢛⣿⡟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠿⠛⠛⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⡄⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠛⠛⠷⢶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉⠛⠿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣦⣄⣀⠀⠀⠀⠀⢀⣤⣾⣿⣦⣀⣴⣿⡿⠃⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣶⣿⠿⠋⠙⠻⠟⠋⢹⣆⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣃⣤⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢉⣩⣿⠟⠻⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣰⣿⠋⠀⠀⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⢀⣾⣿⡟
⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡏⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⣼⣿⡿⠁
⠀⠀⢀⣴⣿⣿⣿⠟⣿⡿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⢠⣿⣿⠁
⢀⣴⣿⣿⠿⠋⠁⣸⣿⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⢸⣿⠇
⣾⣿⡟⠁⠀⠀⣰⣿⡿⠀⠀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡿⠛⠀⠀⠀⠀⢀⣿⣿
⣿⠏⠀⠀⣠⣾⠛⠋⠀⠀⠀⠀⠹⢿⣷⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣴⣶⣾⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠃⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇
⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠇
⠀⠀⠛⠁
⠀⠀⢠⣦⣤⡠⡆⣶⣶⢠⣶⣦⣴⣶⢠⣄⣤⢠⡄⢰⣄⣴⡄⣰⣶⡀⠀⠀⣴⣄⢰⣶⣠⣶⣄⣴⣶⢀⣶⡄⢰⡀⢰⣶⠀⠀⢶⣦⣰⣶⡄⣾⡆⠲⣦
⠀⠀⠀⠁⠈⠁⠁⠉⠙⠊⠛⠉⠸⠿⠙⠋⠛⠘⠁⠛⠉⠉⠁⠟⠛⠷⠀⠸⠛⠻⠸⠋⠸⠟⠉⠿⠯⠚⠛⠛⠘⠷⠀⠿⠃⠀⠾⠇⠘⠛⠁⠙⠃⠒⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⣿⠛⠻⣦⡄⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠈⣿⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡿⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣾⠃⠀⠀⠀⢸⡇⠸⠶⠶⠶⠶⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠞⠋⣠⡾⠃⠀⠀⠀⠀⠸⠷⢶⣶⡶⠶⠶⠶⠶⢶⣦⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⣤⣤⣤⣤⣤⣤⣤⣤⣄⢠⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⠈⣿⡀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠹⣦⣀⣀⣀⣀⣀⣀⣴⡿⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣠⡾⠛⠛⠛⠙⠛⠛⠛⠛⢷⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢴⠾⠋⠀⠀⠈⣻⣶⣶⣶⣶⣶⣶⣶⡶⠟⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⠀⠈⣿⡀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⠀⠀⠀⠀⢀⣠⡿⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠛⠛⠛⠛⢿⡏⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⣤⣤⣤⣤⣤⣤⣼⡿⠀⠙⠻⠷⢶⣦⣤⣤⣤⣤⣤⣤⣿⣶⣤⣤⣤⣴⠟⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⣤⣤⣤⣤⣤⣤⠤⠞⠛⠓⠶⠦⢤⣤⣤⣤⡤⠤⠤⣤⣤⣤⣤⡤⠤⠶⠛⠁





⣿⢿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣿⣵⣶⣌⠀⢀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣟⣻⣿⣿⣿⣿⣿⣷⡀⢙⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⣿⣽⣿⣿⡗⡸⡿⡇
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡯⠉⠀⠀⠀⠀⠀⠉⠻⣿⣿⠇⢹⢳⣇
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠐⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡷⣤⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⣠⣤⣤⡀⠘⣿⣯⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⣟⡂⠀⠀⠉⠙⠛⠷⠆⠀⢹⣿⣿⡒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣣⣄⠠
⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣯⠳
⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⢤⠀⠀⠀⢰⣴⣿⡿⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠠⠹⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⡠⠐⢹⣿⣿⣿⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠈⢫⡑⠒⠒⠈⠁⠀⠀⢸⣿⣿⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣏⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢮⣀
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠨⠂⠠⠀⠬⣔⡀⡀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠍⠳⣦⡄
⠀⠀⢔⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠂⠉⠀⠀⠘⡆⠀⠀⠀⢾⣿⠿⣿⡀⠀⠀⠹⣿⡀
⠀⠀⠀⢄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⢸⡟⣀⡘⣿⠠⠴⠂⠙⠣
⠀⠀⢧⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⠀⠟⠋⠋⠀⠈
⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠘⠀⠘⠀⠀⠀⠰⠶⠄
⣶⣶⡶⣶⠶⣶⣶⡶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀
⠀⠀⠀⠴⡤⢄⠀⠀⠀⠁⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⢹⠯⡀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣠⠊⠀⠀⠉⢢⠀⠈⠁
⠀⠀⠀⢣⡠⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠶⠄⠀⠀⣼⠋⠽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣀⣀⠀⠀⠀⠀⠀⣇⢀⠄⠒⠉⣁⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⡊⠭⠄⢳⡀⠀⠀⠀⠀⡠⢄⢤⠀⠀⠀⠀⠀⠘⠃⠀⠃⠀⠀⠀⠀⠹⣀⠠⠒⠊⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⡴⠆⠒⠈⣉⣡⡽⠀⠀⠀⠀⠘⢬⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠢⢄⣀⣠⠃
⠀⠀⠾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠉⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣱
⠀⠀⠀⠀⠀⢀⡏⠀⢠⠁⢀⡄⠀⢹⡀⢈⡆⠀⠀⠀⣆⠀⢀⠀⠀⣶⠒⢢⠀⠈⡟⠀⣤⠄⢄⠐⢲⡖⠒⡀⠀⠀⣾⠑⢂⠀⠀⠀⣄⠀⠀⡀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⡇⠐⠚⠀⣾⢙⡀⢸⡇⠉⢰⠋⠲⡀⠸⣄⠎⠀⠀⣿⠤⢎⡀⢰⡇⠀⣿⡄⠋⠀⢸⡇⢠⡇⠀⡇⢸⠀⠈⠀⢠⡄⠘⣆⠸⠀⠀⠀⡇
⠀⠀⠶⠀⠀⠸⠀⠀⠆⣸⠃⠀⢃⠀⠇⠀⢸⠤⠴⠃⠀⡏⠀⠀⠀⢸⡀⢀⠅⠘⠃⠀⣿⠈⠂⠀⠘⠃⢸⡇⠈⠁⠸⡧⠊⢠⣏⣄⠀⢹⡇⠀⠀⢰
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠿⠀⠀⠀⠀⠁⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠁⠀⡤⡀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠏⠀⠀⠀⠸⠃⠀⠀⡍
⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⢅⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢦⢤⠀⠀⠀⢇
⠀⢀⠀⠀⠀⠀⢘⠒⡤⢄⠀⠰⠋⠀⠤⠆⠀⠀⠀⠀⠀⠀⠘⠶⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⢸⡀⣀⣠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠈⠦⠀⠀⠀⢠⡀⡃⠈⣍⡵⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣂⡈⡭⠭⠖⡒⣇⠀⠀⠀⠀⢠⠤⣄⡠⠤⠄
⠀⠀⠀⠀⠀⠀⠘⠃⢰⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⠀⠀⠀⠀⠀⠀⠀⠀⡯⠤⠐⠒⠈⠁⠀⢸⡀⠀⠀⠀⠰⡀⠈⠀⠀⡌⠀⠀⠀⠀⠀⣀
⢀⡠⠤⡀⠀⠀⢀⠖⠉⢀⣤⣀⠀⠀⠀⡤⡀⠀⠀⠀⠑⠢⠼⠮⣀⡄⠀⠀⠀⠀⠀⠀⣰⣤⣤⣖⢮⠽⠮⠛⠋⠀⠀⠀⠀⠑⢄⡠⠎⠀⠀⢀⠀⠀⠎
⠉⠀⠀⠈⠒⠒⠉⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠉⠉⠀⠶⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣡
⠀⠀⠀⠀⠀⠀⠀⡜⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠈⠁⠁⠉⠄
⠀⠀⠀⠀⠀⢀⣴⡁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠤⣤⠤⠐⠛⠒⠂⠤⣄⡀
⠀⢀⡤⠒⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⣶⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡐⠶⠶⠶⠶⠶⠖⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴
⠀⠉⠒⠒⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⠐⡍⠓⠈⠉⠀⠀⠀⠀⠀⠈⠉⠐⢋⡴⢄⡀⢀⣤⣀⡤⠰⢶⠀⠙⣍
⠀⠀⠀⠸⣯⡜⠁⠀⠉⠉⠉⠉⢁⢾⡵⠛⠀⠐⠒⠂⢠⠀⠀⢀⠀⠀⠀⠀⠉⠉⠲⡹⠄⠀⠀⠀⠀⠀⠑⢲⡟⢠
⠀⠀⢰⠇⠛⠀⠀⠀⠀⠀⠀⠠⣎⠃⠀⠀⠀⠀⠀⠀⠉⠁⠀⠙⠀⠀⠀⠀⠀⠀⠀⠱⠟⡄⠀⠀⠀⠀⠀⠀⢁⠾⡀
⠀⠀⡃⢠⠀⠀⠀⠀⠀⠀⠀⢷⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢁⡀⠀⠀⠀⠀⠀⢸⠄⠱
⠀⠀⠃⣼⡀⠀⠀⠀⠀⠀⢸⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⣤⠇
⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡆⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠈⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠉⡾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡷⠁⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠘⢽⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠛⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠈⠉⠛⠛⢷⡶⠶⠾⡟⠛⠋⠁⠀⠀⠀⢀⣠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠖⠠⣤⡀⠀⢸⡃⠀⠀⣿⠀⣠⣤⠄⠲⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⢙⡇⠀⠀⣟⠀⠀⠀⠀⢸⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⢸⣏⠀⠀⠉⠉⠉⢹⡇⠀⠀⣿⣉⠉⠉⠁⠀⢙⣄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⢠⡏⠙⢦⡀⠀⠀⠈⣅⡀⠀⠇⠈⣄⠀⣠⠞⠛⡀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⡤⠀⠀⠀⣙⠆⢀⣞⠉⠓⠶⠀⠀⠙⠟⠳⠄⣠⣇⠀⠀⠀⠀⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⢀⠀⠰⠷⠴⠆⠘⠛⠛⣏⠉⠛⠀⠀⠀⢀⠟⠀⠀⠀⠀⠈⠉⠑⠦⠄⠀⠈⢣⡀
⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡛⠃⠀⠀⠀⣈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠛⠓⠒⢿⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁
⠀⠀⠀⠘⠓⠒⠀⠀⠠⠤⣤⣤⢤⣤⣤⣤⣤⢤⣤⣤⣶⣤⣤⣤⠤⠄⢋⡤⣈⣁⡀⢲⡖⢶⡦⠤⠤⠤⠤⠤⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣇⣰⠤⠤⢠⡇⢠⣉⣀⢐⠞⠈⠑⠶⠂⢙⡄⢻⠉⠁⢨⣅⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠁⢀⡀⢠⣦⡀⠑⠋⠉⠉⠙⢷⢤⣤⡟⠉⠉⠓⠛⠀⠀⠀⠈⠉⠁⡆
⠀⠀⠀⠀⠀⠀⢀⣶⣤⣿⡖⠻⠿⠈⢻⠟⢦⠀⠀⠀⣾⠇⣸⣇⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢈⣥⠄⠐⠺⠗⠀⣬⡄⠀⠀⡇⢰⠋⠛⡆⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⢿⣍⣤⠄⠶⠒⡿⠒⠲⣾⠀⠀⡾⠀⣼⣀⡂⠘⠇⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⠴⠆⣸⣁⣀⣠⡿⠶⡈⠲⣤⣅⣸⡇⠀⠘⢋⡡⢀⣀⣀⡀⠀⠀⠀⣀⣀⣽⡀
⠀⠀⠀⠀⠀⠀⢠⠙⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢃
⠀⠀⠀⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢀⡇⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠘⠂
⠀⠀⠀⠀⠀⡘⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢸⡇⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠻⡀
⠀⠀⠀⠀⠀⠇⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⢸⡇⠀⠀⠐⢆⠀⠀⠀⠀⠀⠀⠀⢀⡰⠃⠀⠀⠇
⠀⠀⠀⠀⣶⠀⠀⠀⠀⠙⠓⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⢈⡇⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠒⠋⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲
⠀⠀⠀⠘⠤⠤⠄⢀⡀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣠⡅⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⠒⣭⡿⠿⠿⠏⢩⣭⣿⣶⠄⣿⡇⣴⠿⣯⣭⡍⠛⠛⠛⣿⡅⠲⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⢠⡔⣿⡟⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠞⣋⣤⣤⣤⣤⣄⠛⠲⠉⠀⠈⠑⠞⢃⣤⣤⣤⣤⣜⠛⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠒⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣀⣀⣀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠋⠉⠉⠉⠉⠉⠛⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⠆⠀⠀⣿⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢀⡴⠋⠀⣀⠜⠉⢧⡀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⠴⠚⣉⣠⠴⠚⠁⠀⠀⠀⠙⢦⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣞⡥⠖⣋⣉⣀⠀⠀⠀⠀⣀⣀⡀⠀⠙⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⠁⠀⢚⣡⣤⣀⠀⠀⠀⢈⣡⣄⡙⠂⠀⠸⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⣄⠀⠀⠘⠛⠁⠀⠀⠀⠀⠙⠛⠀⠀⠀⡇⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣿⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢸⡧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⢀⣀⣭⣅⣀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⠀⠘⢿⣛⣻⠟⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠳⣄⡀⠀⠀⣀⡴⠋⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠙⠒⠋⠁⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢶
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀⡀⡐⡿⠻⢿⣿⣿⣿⠀⠀⠀⠂⠀⠀⢀⠀⠀⣐⣤⡀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⢶⣷⣶⣶⣿⣿⡿⠏⣳⠀⢿⠟⠉⠄⠀⣠⣾⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⡶⠂⢸⡆
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠙⣿⣿⣿⣷⣿⠷⠀⠀⠀⠀⣳⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠀⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠛⠁⠀⠀⠀⠀⣠⣿⣶⣬⣽⣿⡿⢿⣿⣿⣿⡿⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⣶⣦⣀⠀⠀⠛⠿⢿⣿⣿⡿⠁⠀⠋⠈⡙⣀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠇⠀⠀⠀⠸⣿⡿⢤⠀⠀⠀⠀⠀⢀⣤⣬⡁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠘⠋⠛⠏⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠒⠒⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⣛⣛⣛⣛⣩⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⢶⣶⣖⣒⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣰⠶⠾⠋⠉⣁⣀⣴⣤⣭⣽⣿⣿⠀⠆⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠐⣿⣿⣿⣷⣶⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣻⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣿⣿⠰⠙⣛⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⢻⣏⣀⠍⢡⠌⡑⠚⠃⣾⡿⠽⢟⢿⣿⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⠐⢾⣁⠖⠀⢀⣩⣥⠀⣿⠀⠖⠘⣸⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡎⢄⠘⠂⠁⣘⠷⢶⣫⣽⡦⢜⣴⣏
⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣇⠙⣖⠠⠀⠰⠶⠶⢶⣦⣌⢜⢺⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⢪⡄⠛⣀⠛⡻⠦⠝⣰⣿⣿⣿⣿⣧⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠉⠢⣄⣘⣘⣤⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠿⠛⠛⠛⠷⢶⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢀⣼⣿⠉⠉⠀⣸⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣾⣿⣿⡇⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⠶⠄⠀⠀⠀⢀⡀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⡏⢡⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⠦⢞⠭⣏⠛⠈⣆⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⡽⢊⢂⢸⡕⣀⣰⡎⡾⢆⣠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣹⡧⡉⠴⢹⡇⠀⠌⣷⢆⢠⢿⣆⠨⢿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⣼⣸⣧⣥⢒⡎⣿⡄⠴⢹⡎⡰⡎⣷⢱⡼⠋⠛⠿⣿⣦⣄⡀⠀⠀⠀⠀⣀⣠⣾⠟⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠹⣿⠿⣻⣶⣜⣸⣷⣵⣽⣿⣮⣿⠟⠁⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠟⠋⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⡈⠳⠟⣻⣿⣿⣿⣿⣿⣿⣷⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠔⣟⣋⣛⣛⣛⣙⣋⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⣀⠂⠀⠒⠲⢭⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⣛⣫⣭⣭⣭⣙⣂⣀⣀⣀⡐⣿⣾⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣫⢵⡿⢿⠿⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣗⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⡸⠁⠈⠀⠀⠀⠀⠘⠛⠓⠀⠀⠀⠀⠀⠀⠀⣿⡄⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠊⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠁⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡞⠁⠀⠀⠰⡲⡀⠠⢂⠈⠠⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠖⠆⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡗⠃⠀⠀⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⠉⠁⠀⠀⢀⡤⡆⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠈⢆⠀⠀⠀⠐⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣁⠀⠀⠀⠀⠀⢀⣀⡤⠴⠚⠉⠀⡇⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠘⠀⠊⠉⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⡄⠀⠀⠀⠀⠀⢧⣀⣀⠤⠖⣡⡇⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⢀⠔⡱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠳⢤⣐⣪⠟⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠒⡡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⣀⡠⠴⠋⢷⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠤⠤⣤⢤⠖⠛⢉⡏⡉⣀⠤⠔⠒⠁
⠀⠀⠀⠀⠀⠀⠂⣀⣀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢀⡀⣀⡠⠴⠊⢁⠀⠓⠤⠚⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠤⠇⠀⠀⠀⣀⣀⣀⣒⣒⣒⣸⡀⣀⣠⠴⠚⡹⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠈⠂⡀⠠⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠖⠒⠦⣌⡉⠑⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠄⣤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠸⣄⣀⡀⠀⠈⠱⠒⠤⢄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣚⣳⡄⠀⠀⠀⠀⠈⠓⠦⢤⣄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢹⠀⡟⠘⠉⠉⠐⠒⠒⠊
⠀⠠⠤⠀⠀⠀⠀⠀⠀⣤⡁⡴⠊⠁⠈⠳⠒⠚⠉⠉⠉⠙⠂⢄⣀⠈⠉⠉⠛⠛⠛⠥⣤⠀⠀⠀⢀⠞⣰⠃
⠐⡀⢄⠀⠐⠢⢄⣀⠎⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣇⠀⠀⠀⢠⡏⡀⠤⠔⣨⠖⠁
⠀⠈⠢⡙⢄⠀⠀⠈⠓⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣄⣀⡼⠓⠒⠒⠉⠁
⠀⠀⠀⠈⠢⡑⠦⡀⢠⠃
⣤⣤⣤⣤⣤⣬⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣬⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⡶⠾⠿⠿⠛⢛⣿⠿⠷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠉⠀⠀⠀⠀⠀⣰⣿⣿⣿⣷⣶⣤⣉⠛⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⢰⣷⣶⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠹⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠈⠉⠛⣠⣾⠟⠳⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣼⡿⠁⠀⠀⡇⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠙⠁⠀⠀⢰⣿⠃⠀⠀⢠⠇⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠻⢿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⠀⠀⣠⠏⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⠷⠞⠁⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣦⣀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠟⠀⠀⢰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣠⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⢀⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣤⣶⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠋⠛⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠙⣿⡇⠀⠀⣼⡟⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⣸⡿⠀⠀⣼⡟⠀⠀⡀⠀⠀⠀⢠⣾⠂⠀⠀⢀⣶⠆⠀⠁⢀⣀⣤⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⠷⠶⣾⡿⠀⣠⡾⠛⣿⠂⢠⣿⡗⢩⣿⠀⣾⡧⠊⣿⠂⢻⣿⠃⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⣿⠇⢰⡟⢁⣾⡇⢀⣾⣯⢤⣾⠏⣼⣿⢤⣼⠟⢀⣿⠃⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢿⣄⠈⠛⠁⠘⠛⣽⡏⠙⠋⠁⢰⡿⠙⠛⠉⠀⠀⠛⢋⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣿⠃⠀⠀⠀⣴⠞⣻⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⡀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⢀⣴⡆⢀⣴⡷⠀⠀⢰⣄⠀⠀⣶⠄⠀⠀⠀⠀⠀⠐⠛⠟⠁
⠀⠀⠀⠀⠀⢠⣶⠿⣿⡟⠉⠙⢦⠀⠁⢀⣤⠉⠉⠉⠛⠛⢻⣾⢿⣶⣿⣯⣤⣀⣠⣼⠟⠀⣼⡟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠹⠅⣰⣿⣀⣀⣤⠏⢀⣠⣌⠉⢰⣛⡆⠀⠀⣼⠏⢠⡿⢁⣭⣭⡍⢉⣡⣤⣴⡟⠀⠀⢠⣴⠶⣤⠀⢀⣴⠆⢠⣶
⠀⠀⠀⠀⠀⠀⣰⡿⠉⠉⠉⢹⡄⢹⡿⠁⢀⣼⢛⡿⠀⣸⡏⠀⣾⡷⢋⣿⠋⣴⠟⠉⢠⡟⠀⠀⣴⡿⢁⣼⠇⢀⣾⠏⣠⣿⠃
⠀⠀⠀⠀⠀⢠⣿⡃⢀⣀⣤⠟⠁⣿⣧⠶⠋⢰⣿⣥⢾⣿⠀⢸⡟⠁⢼⣇⡤⢿⣤⡶⠻⣧⣴⠞⢿⡥⠊⣿⡤⢺⣿⠔⣿⠏
⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⣾⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⣴⡟⠀⠐⢶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠿⠶⠶⠶⢶⣶⣦⣤⣤⣤⣤⣤⣤⣾⣯⣤⣤⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣬⣿⠟⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⡾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠰⣦⣤⣀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⠤⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣧⣄⡀⠀⠀⠁⠀⢨
⠀⠀⠀⠀⣠⣴⣆⣤⣤⣤⣤⣬⣭⣍⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠉⠁⠒⠠⠀⢨⡀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠓⠒⠒⠲⠶⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⣿⢿⣿⣿⣿⣿⣿⡿⢋⠡⢤⠙⣿⣿⣿⠏⠀⠀⠀⠠⠴⠿⢷⣦⣄⣀⣿⣿⣿⣿⣿⣿⣾⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠇⡠⢀⡄⠀⣿⣿⣿⠀⠀⠀⠀⠀⠠⣠⡄⠉⢻⠿⠿⠿⠿⢿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡀⠻⡿⠃⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠻⢶⣤⣿⣿⣿⣿⣿⣧⠀⠁⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠙⠻⣿⡿⠿⠟⠑⢻⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠳⣄⡀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣀⠀⠀⠀⠀⠙⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣶⠆⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⣰⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⣍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠙⠷⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣄⡀⠀⠀⠀⠀⠰⢶⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠈⠁⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣋⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢿⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡀⢀⡠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠁⢈⠉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣧⠾⠻⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡴⣾⡃⠀⠀⠸⢻⣦⡀⠀⠀⠀⠀⠀⡴⡈⡲⠚⠋⠀⠀⢉⣻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠘⣹⣿⣭⣿⡟⣿⠟⠖⣾⢺⣿⢿⣿⣿⣦⡳⡀⠠⢿⣇⣀⠀⡆⡀⠈⣿⡍⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠡⡁⣐⣸⣿⣾⣿⣿⣏⡀⢀⣿⣌⣿⣾⣿⣿⣿⣧⡁⢠⣨⣽⡜⢀⣤⠀⣾⡇⠙⣷⠶⣶⠂
⠀⠀⠀⠀⠀⠀⢠⣤⣐⣥⣁⣴⡟⣿⣯⣿⣿⣿⠋⠙⠀⣿⠟⢻⡇⣿⣿⣿⣿⣿⣾⣿⣻⣩⡀⠉⠀⢀⡱⠾⢿⡄⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⣿⣯⣭⣿⣿⣹⣿⣿⡿⠈⣿⡏⠀⠀⠠⠛⠀⣿⣷⣿⣿⣿⣿⣿⣿⣿⠟⠛⣻⡖⢿⡏⠁⠀⠈⢻⣄⠀⠀⢇⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣷⡌⢻⣿⣿⣿⣧⣤⣿⠀⠀⠀⠠⡄⠀⠙⢻⣿⣿⢯⣿⢿⣿⣿⠀⠀⡏⠉⠉⠃⠀⠀⠘⠋⢹⣦⠀⣸⣧⣄⡀⣰⡆⠀⠁⣄
⠀⠀⠀⠀⠀⠀⠠⢠⣻⣴⣿⣭⣿⣿⢻⣿⣿⡗⠁⠀⠀⢀⠀⠰⣿⣿⠿⠞⢿⣿⣿⠃⠀⢀⡆⠀⠀⠀⠀⠀⠊⠀⠀⠈⣷⣾⡿⠛⠋⠁⠀⠀⢀⣿
⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⣿⣿⣿⣿⡏⢀⣦⣤⣀⠀⢀⣧⣀⣊⣿⣅⠀⠀⣼⣿⠃⢠⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠚⡹⠋⠀⢀⡀⠀⣠⣶⠟⠋
⠀⠀⠀⠀⣢⠰⠋⠉⠻⣿⡿⣿⡋⠉⠀⠀⢈⣿⠇⠈⠉⠄⠈⠀⠀⠘⣷⣾⠟⠁⠀⢸⡀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⣾⢡⠀⠀⣼⣿⣿⣿⣧
⠀⠀⠀⣠⠁⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣻⡯⡄⠀⠀⢂⣀⡀⠡⢽⡟⠁⠀⠀⠀⣼⡇⢘⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢢⣴⣦⣿⠉⠁⡘⢿⡄
⠀⠀⠘⠋⠀⠀⠀⠀⠀⣸⠋⠈⠀⠀⠸⣿⠍⢿⣦⢰⣀⣴⠟⣥⣤⣀⢀⣰⢤⣶⣿⣿⡄⣿⡁⠀⠀⠁⠂⠀⠀⠀⣀⠤⣀⣾⢼⣿⡔⠀⡇⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠂⠀⠀⠀⠀⠠⢤⡉⡸⢿⣧⠼⢿⡿⠷⣿⣿⣿⣿⣿⣿⡇⣿⣷⣤⣀⠀⣤⣀⣩⣭⠶⠚⠩⠀⠀⡝⣿⠀⡇⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠈⣾⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣿⡇⠀⠄⠀⢲⣾⣿⣿⣿⣿⣿⣿⠛⣿⣿⣿⡆⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⢷⠀⢧
⠀⠀⠀⠀⠂⠀⠀⠀⠹⣧⠀⠀⠀⣥⣤⡄⠀⣺⣿⣿⣇⡀⠀⠐⢄⣈⡉⢻⣿⣿⣿⣿⠂⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢹⠀⢸
⠀⠀⠀⠀⢰⠀⠀⠰⣀⣿⣆⠀⠀⠈⢧⠃⠀⠀⢿⣿⣏⡀⠀⠀⠀⢉⠙⢾⣿⣿⣿⣿⢼⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⣼⠀⢸
⠀⠀⠀⢀⢘⣦⠀⠀⠘⢿⣿⣦⡀⠀⠀⠀⣀⣲⠟⠉⠁⠀⠀⠀⠲⢆⠀⣼⠿⠿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⣴⣼⡆⠀⠀⠀⠀⠀⣀⣻⠀⠸⠆
⠀⠀⠀⠈⠰⣿⡇⠀⠀⠀⠻⣿⣿⣤⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠱⠷⢾⣿⡤⢤⣀⠙⢿⣿⣿⣿⠃⠀⠀⠀⠀⡌⣸⣧⠀⠀⠀⠀⠸⣿⣦⠀⠤⠂
⠀⠀⠀⠀⢸⣿⣿⢠⠀⠀⠀⠈⠛⠉⢹⡿⠒⠐⠒⠳⠂⠦⣀⠀⠈⡘⠀⠘⠒⡀⠀⠀⠀⠉⢿⣿⡀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⣿⣿⠔
⠀⠀⠀⠀⠀⠈⠉⠸⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠒⢦⣄⠠⠄⢹⠞⠁⣠⣤⣄⠀⠀⠀⠀⠀⠰⣿⣄⠀⠀⠀⠀⢘⣿⢧⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡦⠀⠀⠀⠀⠀⠀⠀⠈⣶⡎⠍⠁⠀⠀⠀⠀⠑⣄⠀⠀⠀⢢⠻⡛⢆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣮
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢏⠃⠀⠀⠀⠀⠀⠀⠠⠻⠇⠀⠀⠀⠐⠃⠌⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠉⣉⠉⠀⠀⠀⠉⠉⠙⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⡣⠔⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢾⣵⢟⣥⠶⢚⣛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠙⢶⡄
⠀⠀⠀⠀⢀⣴⠟⠀⢸⡵⣫⠖⢋⣩⣤⣤⡤⠤⠤⣀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣭⣭⣍⢙⠓⢦⣻⣆
⠀⠀⠀⢀⣾⢏⡔⠀⠘⡎⣀⣭⣴⣶⣬⠁⠒⠒⠢⠤⣉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣀⣀⣀⣹⡿⠷⠲⠂
⠀⠀⠀⣼⡟⡞⢀⢀⠀⠈⠛⠛⠓⢦⣍⠳⣄⠒⠦⠤⢬⠙⠒⠤⠤⠤⠀⠀⠀⠉⠁⠀⣀⣀⣀⣉⢻⣦
⠀⠀⠀⣿⣿⠁⣆⡎⠀⠀⠀⠀⠀⠀⠙⢷⣜⠳⣶⠦⠤⠤⠤⠤⠤⠤⠀⠀⣀⣤⠶⠋⠉⠀⢠⡈⡌⣇
⠀⠀⠀⣿⣿⢸⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣦⣤⣤⣶⣶⡶⠾⠟⠋⠁⠀⠀⠀⠀⠀⢷⣻⣿
⠀⠀⠀⢸⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠈⣿⣿⠇⠀⠀⠀⠀⣠⣤⡾⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⠀⠀⢼⣿⠁
⠀⠀⠀⣠⢽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠀⠀⠀⣼⣇
⠀⠀⢸⠁⠛⣿⡆⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⣴⠟⠊⡇
⠀⠀⠘⢷⡀⠘⣷⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⢰⡟⠀⣼⠃
⠀⠀⠀⠈⠻⣶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡾⠃
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠤⠶⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠆
⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠉⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢶⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠙⠿⣍⡛⠓⠒⠒⠒⠒⢒⣻⠟⠋⠁⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠈⠛⢷⣤⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠷⠶⠚⠛⠛⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠀⠀⠀⢀⠆⠸
⠀⠀⠀⠀⠀⠀⠠⠄⠠⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣄⠀⠀⠀⣀⠤⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠘⠉⠁⠒⢄⠀⠀⡀⠀⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠈⠳⠶⠚⠁⠀⣡⣀⠔⠁⠠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠀⠒⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢠⠟⢉⡍⠀⠀⣇⠀⠉⠉⠓⠛⠒⠒⠒⠒⠒⡟⠀⠀⠰⡟⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⡠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⢈⡶⢹⠀⠀⠀⡇⠈⠐⠒⠀⠀⠤⠄⠀⠒⡾⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠉⢻⠀⠀⠀⠸⡆⠀⠀⠤⠄⠀⢀⣠⠞⠀⠀⠀⠀⠀⠰⢂
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠳⠶⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⠀⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠃⠀⢀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣀⡀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠒
⠐⠂⢉⡆⠰⠒⠈⠀⠀⠀⠀⠁⠒⠂⠤⢄⣉⡑⠢⠤⣀⠈⠙⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⢀⣀⠤⠐⣀⠔⠉⢦⡀⠀⠀⢀⠀⠠⣔
⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠤⣌⡉⠒⠺⠛⣿⠶⠶⠶⣶⠶⠶⠟⠉⠀⢒⣂⣈⣭⠤⠔⠒⠉⠁⠀⠀⠀⠉⠓⠀⢧⠘⠆⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⡰⠁⠘⠒⠛⠁⡶⠂⠂⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠐⠉⠑⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠤⡴⠀⢈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢄⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⢦⣤⣤⣄⣀⣀⣠⣤⣴⠞⢉⠄⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠀⠀⠀⠀⠀⠀⠢⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⠄⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⢀⣀⠇
⣿⣿⣿⠿⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣯⡿⣿⣿⣿⣿⣿⣿⡇⡿⣿⣿⣿⣿⡿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⠀⠉⠉⢡⢸⡇⡏⣬⠉⠉⠁⣴⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⡌⠀⠀⠀⠈⣻⣶⣞⠋⠀⡀⠀⢩⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣈⣽⣿⣿⣿⣿⣷⡀⠘⠙⢛⣛⠿⣛⡛⠃⠃⢀⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠠⢶⣿⣿⣿⣿⣿⣿⣿⣿⠹⣷⣄⢀⠀⠹⠛⠋⢀⣀⣴⡟⢸⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣏⠀⠈⠻⢿⣷⣤⣤⣤⡾⠯⠋⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣽⣿⣦⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠙⢆⠀⠀⣼⣿⣿⣿⣧⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⡆⠀⠓⠊⠹⣿⣿⡟⠁⠀⣹⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠉⠻⠟⠛⠛⠛⠛⠃⠀⠀⠀⠠⣿⣿⡇⠀⠠⠿⡿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⠤⣀⣀⡀
⠀⠀⠀⢠⠊⠁⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠙⢦
⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⢀⡴⠒⠓⠲⣄⠀⠀⠀⣀⣤⣄⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸
⠀⠀⠀⠀⢣⣀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⡞⠀⢀⣀⡀⠈⢧⠀⡼⠁⢀⣀⠈⢢⡀⠀⠙⣄⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠈⠑⠢⣀⠀⠀⡜⠀⠀⠀⢸⠁⠸⣶⣿⡷⠀⢸⠀⠁⠸⣾⣿⠂⠀⡇⠀⠀⠘⡄⠀⠀⡠⠔⠋⠀⠀⢠⠞⠁
⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠈⢆⢸⠁⠀⠀⠀⠈⢧⠀⠉⠉⠁⠀⡼⠀⢆⠀⠀⠀⠀⣠⠃⣀⠀⠀⢹⢀⠎⠀⠀⠀⠀⠀⠘⠓⠦⣄
⠀⠀⠰⠁⠀⠀⠘⠁⠈⢣⠀⠀⠀⢸⣾⢠⢶⣄⠀⠀⠀⠉⠒⠀⠒⠊⠀⠀⠀⠉⠒⠒⠊⢁⣠⠞⠳⠀⢸⣼⠀⠀⠀⢠⠎⠉⠑⠒⠀⠀⢱
⠀⠀⠘⢄⡀⠀⠀⠀⣠⠞⢄⠀⠀⢺⢸⠀⠀⠈⠛⢶⠤⢤⣀⣀⣀⣀⣀⣀⣀⣤⡤⠶⠚⣩⠃⠀⠀⠀⢸⢹⠄⠀⢀⠼⢆⠀⠀⠀⠀⢀⡔
⠀⠀⠀⠸⡉⠉⠉⠉⠁⢀⡜⢧⠀⡠⠸⡄⠀⠀⠀⠀⠱⣦⣤⣉⣉⡉⠉⠉⠉⣀⣠⣴⣾⠇⠀⠀⠀⠀⡾⢠⠀⢀⠾⡄⠀⠉⠒⠒⠊⣹
⠀⠀⠀⠀⠈⡒⠂⠐⠒⠉⣠⠏⡟⠁⠀⢱⡀⠀⠀⠀⠀⠈⢿⡟⠋⠉⠉⠉⠉⠉⠛⣿⠏⠀⠀⠀⠀⡰⠁⠀⠑⡏⢇⠈⠓⠦⠤⠄⢞⠁
⠀⠀⠀⠀⠀⠑⠲⣶⣒⣊⡡⠔⠁⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠙⢶⡄⠀⠀⠠⢔⡶⠋⠀⠀⠀⢀⡼⠁⠀⠀⠀⠑⢄⣙⠲⢤⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠉⠓⠒⠚⠉⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣀⣀⣀⣀⣀⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠲⠂⠈⠉⠉⠉⠉⠉⠉⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡁⠀⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠸⠇⢀⠀⢀⡠⠄⢲⡾⠉⠉⠉⠉⠉⠉⠉⠀⠒⠒⠂⠠⠤⢈⣳⠀⠀⠀⠀⢄⠀⠀⠃
⠀⠀⠀⠀⠆⠀⣾⠞⠁⠀⠀⢈⡴⢂⣈⠁⠢⡀⠀⠀⠀⠀⡠⢊⣡⡌⠑⢤⡈⠑⠤⣀⠀⠀⠀⢸
⠀⢀⡤⠞⠂⢴⡇⠀⠀⠀⠰⠋⣰⣿⣯⣴⣦⣸⡆⠀⠀⢠⣧⠼⠿⠷⠷⠤⣵⠀⠀⠘⡷⠀⠀⢸
⠀⢸⡇⠀⠀⢾⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠙⡗⠒⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⠸
⠀⠀⠀⢇⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠀⠀⠀⠀⠀⢀⡏⠀⠉⢿
⠀⠀⠀⠘⢧⡀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠘⣏⠃⠠⠴⠦⠖⠘⢋⡼⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠈⠳⢤⣀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠑⠦⣤⣤⣤⠴⠋⠁⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⣐⠂⠲⡄⠀⢀⠔⠁⣠⠞⣠⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠳⢦⣀⡀⠀⠀⠀⠀⠀⠴⠄⠀⠀⠀⢀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⢈⣤⡁⠀⠘⠣⠚⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⡶⠶⠶⣶⠶⠶⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠓⠀⠀⠰⣉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⠇⠀⠀⢷⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠈⠯⣀⠀⠀⠀⢀⣨⠋⠈⠁⠀⠙⠛⠒⢶⡆⠠⠤⠴⠶⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢤⣤⡤⣤⠞⠓⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣯⠤⠤⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣀⣀⣸⡧⣤⣤⠤⠤⠴⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⢀⠘⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠘⢄⢸⣧⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⢶⡿
⠀⠀⠀⠀⣌⠁⠀⠀⠀⠀⠈⠻⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⠀⢃⠀⠀⠀⠀⠀⢀⡴⣦⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⢤⣀⡘⠀⣴⠀⠘⣧⣈⡄⠀⠀⠀⢀⡏⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠉⠛⠦⠻⡤⠚⠀⠈⠀⠀⠀⠀⣘⠁⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠶⠒⠀⠀⠀⡟⠀⠀⠀⢸⡄⠀⠀⠠⠴⠶⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣀⣀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣰⠇⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣇
⠀⢀⠴⠚⠓⠠⣄⠀⠀⠀⠀⠀⠀⠀⡌⠉⢻⡄⠀⠀⠀⠀⠀⢸⠇⠀⣠⣤⠤⠴⠦⠄⢈⣦
⡰⠃⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠙⢒⠦⣷⠀⠀⠀⠀⠀⢸⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⣧⣀⣀⣀⣀⣀⣀⣀⣘⣁⢀⠀⠀⠀⢘⠋⠀⣼⠀⠀⠀⠀⠀⠸⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣽⣿⣷⣶⣶⣶⣶⣶⣾⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠛⠙⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠛⠛⠻⣿⣏⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣿⠋⢀⠀⢀⣀⡈⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⡄⠁⠀⠀⢀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣻⠿⢏⠙⣆⢀⡠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠁⠀⣛⣀⢤⣀⢛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡡⢋⡁⠐⣎⠛⠶⠤⠜⠛⢆⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣅⠘⠃⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⣱⡏⠀⠀⠀⠀⠀⠀⠀⠗⢸⠀⡅
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⢀⠀⠀⠀⢸⠀⢨⠑⢹⣿⡇⡴⣿⣤⠀⣆⢸⠀⠁
⡉⠢⣀⠀⠀⠉⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⣀⡀⢀⠎⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠅⠔⢉⡀⠀⠀⡼⠀⡶⠀⠀⠐⠅⠃⠙⠉⠲⠁⠇⠄
⢈⡓⠦⠟⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡦⣬⠁⠀⢦⡬⢥⡄⢀⠀⠀⠀⠀⠀⡆⠈⠁⠀⠈⡠⢠⠀⢀⡼⢁⠃⠁⠀⠀⠐⠀⠀⠀⡀⢁⠰⠠
⠀⠙⠚⢉⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⣼⠉⠳⢄⡐⠦⣀⢮⣶⣯⠜⢻⠀⠀⠀⠀⠀⣃⠀⢀⠀⠀⠈⠉⠀⠈⠂⠛⠛⠛⠃⠀⢀⣀⠀⠐⠖⠲⠤⠠
⠀⠀⠀⠀⠀⠈⠀⠰⣮⠗⢄⠀⠀⠀⢠⡇⠀⠀⠈⠳⣞⢛⡻⠋⠀⠀⠀⡇⠀⠀⠀⣰⠯⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⢳⡀⠀⠘⡇⠘⢇⠀⠀⠘⣿⡁⠀⠀⠀⡄⢿⠀⠀⢀⡏⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⡇⠀⢠⣀⡴⠀⢠⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠹⢆⣸⠀⡇⢸⡆⠀⠈⣿⠁⠀⠀⣼⢇⠸⡆⠀⡈⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⡏⠀⢤⠙⡄⠀⡎⣷⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⢰⠇⠀⣿⠀⠀⢸⠀⠀⠀⣿⠘⡄⢃⡜⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠘⣆⠀⠉⠀⢰⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⢿⠀⠀⢷⡆⠀⢠⡇⠀⢷⣈⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⢿⠀⠀⠀⢸⠀⠸⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠸⡇⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⢸⡄⠀⠀⢸⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⢠⡅⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⠀⢸⠁⠀⠀⢸⡇⠀⢠⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣤⡀⠰⡟⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠀⢸⠀⠀⠀⠀⡇⠀⠘⡄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡼⠁⠀⣇⠀⢹⡉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⠀⠀⠀⠀⢳⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣀⠀⣟⠀⠀⣉⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⢀⡟⠀⠀⠀⠀⠘⣇⠀⠸⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠙⣛⠛⠛⠻⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢙⠁⢸⠇⠀⠀⠀⠀⠀⢻⡀⠀⠁⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿⠉⢳⡀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠷⢢⠄⠁⠀⡟⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠈⠁⠛⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⣾⠇⠀⠸⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠁⠀⠁⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⣿⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⢀⡟⠀⠀⠀⠸⡇⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢸⡇⠀⠀⠀⠀⣧⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⠀⠀⠀⡆⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠀⢸⡇⠀⠀⠀⣸⠃⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢸⡇⠀⠀⠀⣭⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠸⠁⠀⠀⣘⡇⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢾⡆⠀⠀⣿⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⡇⠀⢸⡏⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡆⠀⣈⡇⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⢼⡇⠀⡿⠃⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣽⠟⠛⠉⠉⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⢸⡇⠀⣷⣄⢀⡀⣸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⣼⡇⠀⠙⠿⠿⣿⡿⠿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⢦⠀⠀⡠⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡿⠽⠗⠀⠀⠀⠀⠈⠳⣄⡉⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠒⠀⢀⢀⠄⢐⡀⠄⠊⠀⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡈⢲⡄⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠀⠀⠘⠷⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢻⠿⠻⣿⣿⣷⣶⣶⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠻⠿⠿⠿⠿⠛⠋⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⣟⣿⣶⣶⣦⣤⡀⠀⠀⠀⠀⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⢾⣛⣿⣭⠽⠿⡟⠚⢟⢷⣄⠄⠊⠁⠀⡀⠁⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢽⡿⠛⣿⠉⣷⣿⣀⣁⢹⡤⠈⢧⢿⣧⠤⣤⡀⠀⣤⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⣁⢠⢿⢈⣿⣿⣧⡁⢣⣿⣰⣸⣬⣿⣄⣾⣧⠴⣆⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⡇⢸⣇⢸⣙⣿⣿⡿⣿⡟⣿⣻⣯⣵⣶⣷⣿⠿⠘⡘⢸⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⣾⣟⣿⣯⣽⣿⣶⣿⣿⢿⢿⣿⣿⡿⡟⣿⡀⡆⠃⢁⣗⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⡟⣿⣧⣘⢻⡏⣿⣿⣿⣾⣿⣿⡟⠁⢧⢹⣧⣰⡰⠐⢸⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢳⠈⢻⠿⣿⢿⠛⢿⣿⡏⢀⣀⣷⣤⣼⣼⢿⣿⡷⠷⠳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡼⡆⠘⣆⣠⣼⣦⣾⣿⡿⢿⣿⣛⣿⣿⡷⢿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣷⢻⣿⣛⣿⣩⣿⣤⠿⠳⢻⡟⠛⠛⠁⠀⢸⣧⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣏⠙⠛⠛⠉⠈⢶⣶⣾⣿⣥⣀⡀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣽⣦⠀⠀⠀⣴⣾⣿⣿⣿⣿⡿⠿⠂⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠛⠛⠿⣿⣷⠟⠀⠀⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣄⣠⣄⣤⣤⣴⣿⣿⣿⠀⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠋⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠶⠀⠘⠃⠀⠸⠂⠀⣤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⣀⣠⣤⣶⡶⠶⢶⣶⣦⣤⣄⡀⠈⠃⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⣤⡾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄⠈⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⢠⣶⠟⠁⠀⠀⣰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⢀⣴⠟⠁⠀⠀⠺⣿⡟⠙⣻⡿⠂⠀⣰⡾⠻⢷⡄⠀⠀⠀⠀⠀⠙⢷⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢀⣾⠏⠀⠀⠀⠀⠀⣼⣷⠶⣿⡇⠀⠀⣿⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠈⢿⡆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠐⠶⠶⠶⠶⠶⢶⣶⠀⠀⣰⡿⠁⠀⠀⢰⣿⣤⣤⣤⣤⣤⣄⠀⠀⢸⣧⠀⢴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢸⡇⠀⠀⠀⠀⢠⣶⣶⡄⢸⣿⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠈⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢸⡇⠀⠀⠀⠀⠸⣷⡾⠇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⠀⢀⣿⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⡟⠀⠺⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡿⠁⠀⢀⣿⠁⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⢻⣆⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣦⣄⣀⣀⣀⣀⣀⣀⣀⣠⣼⠇⠀⢀⣾⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠻⣧⡀⠀⠀⠘⠛⠛⠛⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⣰⡿⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠈⠛⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠉⣀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣀⠉⠙⠛⠛⠻⠿⠿⠛⠛⠛⠉⠁⢠⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠖⠀⠰⠆⠀⠰⠄⠀⠛





⣿⠿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣿⣿⣿⣛⣿⣟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣺⡽⠶⠖⠛⠛⠒⠶⢯⣗⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⣫⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢮⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⠵⠋⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀⡀⠀⠈⢧⢳
⠀⠀⠀⠀⠀⠀⣠⢞⣭⣾⠿⠿⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⡿⠿⠛⣿⣷⣦⣸⡟⡆
⠀⠀⠀⠀⠀⢸⢱⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⠀⠀⠸⣟⢹⣿⢻⣃
⠀⠀⠀⠀⠀⠘⢮⣻⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡿⢿⡋⠉⠀⠀⠀⢻⣟⣻⡬⠾⣝⢦
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣶⣿⠀⢀⣀⡀⠀⠀⠈⠉⠀⠿⠇⠀⠀⠀⠀⢸⣿⣿⢆⠠⢸⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⡼⣿⠐⠿⠿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⡄⠋⢸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⣯⢸⠀⠀⠀⢋⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠐⠀⢠⣟⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡝⣾⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢻⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⡄⠀⠀⠀⡀⠼⠀⠀⠤⠤⠠⣄⠀⠀⠀⠀⠀⠀⣇⠳⡤⠤⣤⡀
⠀⠀⠀⠀⢀⣠⠤⠤⠤⣬⡟⣇⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠌⠣⡀⢀⣠⠒⠂⠉⠉⠉⠙⣷⡷
⠀⠀⠀⣰⣫⢞⠟⠋⠙⢳⣕⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡄⠙⢧⢘⡇⠈⠉⠉⠁⢸⡏⡇
⠀⠀⢠⣷⡇⠈⠀⠀⠈⡄⢻⣼⣮⡳⣄⠀⠀⠀⠀⠀⢖⠉⠀⠈⢦⡀⠀⠳⡀⠀⠀⠀⡼⣧⣳⡀
⠀⠀⠘⣟⢇⠈⢄⣀⡠⢀⡾⣹⠁⡵⣸⣷⣤⣀⠀⠀⠈⢣⡀⠘⡄⠉⠒⠠⠤⠤⠴⠊⡠⢾⡹⡅
⠀⠀⠀⠈⢫⡝⢲⣊⣶⢿⢱⠁⠈⠳⢽⣛⡿⢿⣿⣿⣶⣾⠃⠀⡟⠢⠀⠀⠀⠀⠀⢀⡁⠀⠻⡽⡄
⠀⠀⠀⢰⢫⡞⢉⡇⡇⠸⣜⢄⣀⡤⠴⣫⣵⣿⣿⣿⣿⣿⣾⠀⠸⣆⠀⠀⠀⠀⠀⢘⢇⠀⠀⢹⡽⡄
⠀⠀⠀⢸⡳⣏⡡⡇⠀⠀⢘⡎⣵⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢀⠀⠙⢤⣀⠀⠀⠀⠈⠈⡀⠀⠀⢷⣻
⠀⠀⠀⠀⣇⢿⡔⢻⡦⠀⠈⠙⠛⠿⠏⠉⠛⠿⠿⠿⠛⠁⠘⢄⡀⣀⠀⠉⠉⠒⠒⠤⠤⡇⠀⠀⣾⣇⣧
⠀⠀⠀⠀⠈⢮⣓⡛⡟⠁⠒⠶⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⡸⠀⠀⡼⣿⣿⡼⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢀⠆⠀⠀⠀⠀⡴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠞⠤⠶⠋⠀⣿⣿⣷⢣
⠀⠀⠀⠀⠀⠀⠀⠘⣞⠷⢶⠦⣤⣤⡔⠊⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢰⣿⣿⣿⡸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⢣⣿⣿⡇⢀⠀⠀⠀⡐⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢸⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⣠⢖⣭⣶⣿⣿⣿⢃⡌⠀⠀⠸⠁⠀⠀⠀⠀⠰⡄⠀⠀⠄⠀⠠⠤⠀⠀⠈⣿⣿⣿⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡞⣿⣿⣿⢟⣩⣹⡁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⣿⣿⣿⣿⢣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠭⠗⠋⠙⠮⣗⠦⣄⣀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⣀⠤⠀⠀⠀⣿⣿⣿⡟⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠯⠯⣟⣓⣶⠶⠤⠤⠤⠤⠭⠥⠤⠶⢶⣒⡛⣿⣿⢟⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠘⠮⠵⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣀⣀⣉⣀⡀⠀⠐⠒⠯⣑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⢟⣛⣭⣭⣭⣝⣓⣀⣀⣀⣐⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣛⡵⡾⢟⠿⠛⠉⠉⠉⠉⠉⠉⠈⠉⠉⠙⣟⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢰⠋⠀⠀⠀⠀⠀⠘⠛⠓⠆⠀⠀⠀⠀⠀⠀⢹⡇⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠐⠉⣧⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠁⠀⠀⠀⠀⠀⢀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⠁⠀⠀⠀⡖⢆⠀⠔⠄⡁⠄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⡔⠢⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡸⠈⠀⠀⠠⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠰⠋⠉⠀⠀⢀⡠⢦⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢣⡀⠀⢀⡀⠒⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⢁⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠉⠀⣸⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢀⢃⠒⠉⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣄⠀⠀⠀⠀⠀⢳⣀⣀⠤⠖⢋⡞⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢀⡠⢢⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠑⠦⣐⣪⠝⠁⠀⠀⠀⠀⣰⠏⠀⠀⡀⠔⣉⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢞⣁⣀⠤⠚⢹⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠦⠤⢤⡴⡖⠚⠉⡏⣈⣁⠤⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠒⠁⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠠⠤⠒⢁⡀⠑⠦⠞⠁⢡
⠀⠀⠀⠀⠀⠃⠀⠀⠺⠀⠀⠀⣤⣀⣀⣐⣒⣚⣻⣇⣀⣠⠴⠖⢻⠃⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠈⠀⠀⡠⠒⠒⠊⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⡔⠓⠲⠤⣈⠉⠒⠢⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⢤⡒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢧⣀⣀⠀⠀⠉⠗⠲⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⣊⣻⡄⠀⠀⠀⠀⠀⠑⠲⠤⣤⣀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇⢸⠋⠉⠉⠁⠒⠒⠊
⠀⡀⠤⠀⠀⠀⠀⠀⠀⣤⣉⡤⠊⠁⠀⠙⠒⠛⠉⠉⠉⠉⠐⠤⣀⡀⠈⠉⠙⠛⠛⠩⢤⡄⠀⠀⠀⡸⢁⡏
⠐⡌⢆⠀⠐⠲⢤⣀⠞⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢳⠀⠀⠀⠀⡾⠀⠠⠔⣊⡵⠊
⠀⠈⠢⡙⢄⠀⠀⠈⠑⣮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣸⠓⠒⠒⠉⠁
⠀⠀⠀⠈⠢⣝⠢⣀⢠⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠋⠉⠙⠛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠋⠁⠀⠀⠀⠉⠛⢿⣦⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣴⣾⣿⣿⣿⣷⣦⣄⠙⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⣠⣤⣄⠀⢠⣶⣦⠹⣷⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣴⣿⣿⠿⠛⠛⣿⡿⠋⠉⢧⠈⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣾⣿⣿⠟⣠⣿⣿⡿⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢰⣿⣿⠋⠀⠀⣠⡿⠁⠀⢀⣼⡇⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣮⡙⠋⢠⣶⣮⠛⠉⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⡿⠉⠢⣴⣾⠋⠙⢦⣴⣿⣿⡇⢰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⠛⠁⠀⠀⠉⠁⠀⠀⠀⣼⡟⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠻⣦⣾⣿⣿⣷⣶⣿⣿⣿⠟⢀⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣶⣄⣀⠀⠀⢀⣀⣴⡾⠋⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠈⠻⠿⣿⣿⣿⠿⠟⠁⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠈⠙⠛⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠃⠀⠀⠀⢠⣾⣶⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠁⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠈⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣄⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠋⠩⠉⣉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠉⢻⡿⠛⠉⠁⠀⠀⠀⠀⠒⠠⢀⡀⠀⠙⠂⠄⡀⠀⠉⠛⢿⡉⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⢀⣀⣀⡠⠔⠙⣷⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀⠀⠈⠑⠢⢄⠀⠈⠑⢠⣤⣤⣀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢠⣾⣿⣿⠏⠁⠀⠀⠘⣧⡀⠈⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⠀⠀⠁⠠⢄⠀⠈⠙⢿⣷⡄⠀⠀⠘⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⣼⣿⡿⠃⠀⠀⠀⠛⠀⠈⢳⣄⡀⠀⠀⠈⠑⠂⠤⣀⠀⠀⠀⠀⠀⠈⠁⠒⠠⠄⣀⣈⠑⠂⣠⣽⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡿⠁⠀⠀⢀⠌⠀⠀⢀⠜⠉⠻⡂⠀⣀⠀⠀⠀⠀⠀⠓⠠⠤⢀⣀⣀⣀⠀⣀⣠⡤⠤⠛⠉⢡⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⡿⠃⠀⢀⡠⠊⠀⠀⡠⠊⠀⠀⠀⠈⠑⠢⠌⣁⠒⠒⠀⠤⠤⠤⠤⠤⠄⠐⠒⠛⠛⢒⣲⣖⣋⣠⡖⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠁⢀⡤⠋⠀⡀⠔⠋⠀⣀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠙⠗⠀⠠⠤⠤⠠⠠⠤⠄⠶⢈⠃⠈⠉⠀⠈⢉⣹⠂⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢹⣿⣶⠒⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⢴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣷⡀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⡿⠛⠀⠀⠀⣼⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣧⠀⠀⠀⠻⣿⣿⣿
⣏⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⠀⠉⠉
⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣴⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⣀⣀⠀⠀⠠⠤⠠⠀⠀⢀⣀⠠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⡿⠂⠀⢈⠉⠉⠉⠉⠉⠁⠀⠤⠀⠿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠣⡀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⡨⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠈⠑⠤⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢀⡠⠔⠉⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡤⠀⡀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⣄⣨⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡾⠁⣴⠀⣤⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⣠⣄⠠⡄⠘⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣾⣿⣿⣿⣿⠁⠐⠒⠒⠂⠀⠤⠤⠤⢤⠤⠤⠤⠀⠐⠒⠒⠀⠉⢹⣷⣶⣿⣿⣦⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⣀⣀⣀⣀⣀⣀⡀⢸⡀⣀⣀⣤⣤⣤⣀⣀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⣛⠻⠶⠶⠶⠶⣾⡁⣹⠁⣙⡿⠶⠶⠶⠿⢍⡋⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠉⠁⠀⠀⠀⠀⠀⠀⣮⡣⠌⠀⠀⠀⠈⠉⠉⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⢷⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⢿⣿⡄⠀⠀⠀⢀⡞⠛⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣼⣿⣿⠀⠀⠀⢸⣷⣴⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⠀⢸⣷
⠀⠀⣿⡇⠀⠀⢠⣴⣶⠶⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⠘⢿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⠉⠛⠆⠀⠀⣿⡇
⠀⢰⣿⠀⠀⠀⠋⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢸⣧
⠀⢸⣿⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⣿
⠀⢸⡟⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣷⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟
⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⠶⠶⠴⠶⠶⠶⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣤⣤⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢶⣶⣶⣦⣤⣤⣤⣤⣶⣶⡶⠾⠿⠛⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⡿⠿⣿⠻⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡀⡔⠉⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠐⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⢀⠀⠀⠂⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃
⠁⠀⠀⠀⠀⣄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠙⠙⢦⠀⠀⠀⠀⢀⡀⢀⠀⠀⠘⠛⠦
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⠀⠀⠀⢈⡁⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⡀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠸⢄⠀⠀⠀⠈⠁
⠀⢀⡤⣄⡀⠀⠀⠀⠰⠀⡀⠶⠀⠀⠀⠀⠀⠀⠀⣴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣉⣷
⠓⢦⣄⠀⡔⠈⠒⢄⡀⠀⠀⠀⠤⢀⣀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⠚⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠛⠛⠋⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠘⠉⠙⣄⠠⠀⠀⠀⠀⠀⢀⡔⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⢀⡼⠶⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⣱⠀⠀⠀⠀⠀⢠⠊⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠋
⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠠⠀⠀⠀⡀⠔⠃⠀⠀⡠⠚⠀⣠⠞⣹⡇⠀⠀⠀⢠⡀⠏⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢀⡀
⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠺⣟⣛⡿⠛⠛⠋⠀⠀⠀⠀⠀⠈⠒⠠⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⠄⠈⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠐⠂⠁⠀⠀⠠⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠠
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⠶⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠞⠁⠀⡄⢤⡦
⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠀⠀⠀⠤⢀⣠⠖⠀⠀⠀⠀⠀⠀⠀⠠⡏⠀⠀⠈⢢⠀⠀⠀⢺
⠀⠀⠁⠔⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠠⣀⠀⠀⢠⡀⠀⠀⠀⠀⣀⠀⠤⠀⠀⠀⠀⠀⠄⠈⠁⠙⣄⠠⠀⠀⠀⠀⠀⢀⠄⠀⣠⡴⣾⠀⠀⠀⠈⡆
⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⡀⠀⢀⣾⣥⣤⡼⣯⡴⠏⠀⠀⠀⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠃⠀⠀⠀⠀⠀⠀⢳⣤⣤⣀⠀⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡠⠜⠁⠀⠈⠛⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠃⠀⠀⠀⠀⠀⠀⠀⠻⣦⡽⠷⣶⣛⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⠀⠂⠀⠀⠰⠄
⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⡠⢀⠔⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠈
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⠀⠀⠀⠀⠀⠀⠀⠲⠐⠀⠀⠀⠀⠀⠀⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈
⢀⣤⣄⣠⣞⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⣿⣴⠟⠉⠉⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠤⠀⠀⠀⢀⠘⢣⠀⢀⡀⠀⠀⠀⡎⠙⠳⠶⠾⢷⣶⢶⡄
⠈⠉⠉⠉⢣⠀⠀⠀⠈⠒⠒⠀⠙⠂⠀⠀⠀⠀⢀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠑⠒⠊⠀⠀⠀⠀⠠⠤⣙⣶⡇⠀⠀⠀⠀⠐⢀

⠀⠀⠀⠀⠀⢀⣴⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⠀⠀⠀⢀⣤⣶⣶⣶⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣧
⠀⠀⣀⣀⣠⣤⣭⣛⠋⠉⠀⠀⠀⠛⠛⣿⠆⢸⣿⣿⣿⠏⢠⡀⠀⢸⣿⣿⣿⣿⠀⠀⠀⢸⣿⡿⠦⢤⣤⣤⣀⣘⠿⢿⣿⠃
⠀⠀⣿⠉⠉⣽⣿⡟⣷⣀⣠⣄⠾⠷⠀⣿⡇⠈⠁⢠⣶⡆⢀⣽⡆⠀⠙⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⣦⣀⠉⠉⢁⣾⣿⣇
⠀⠀⣟⠀⢰⣿⡿⠀⠘⠋⠉⠀⠀⣀⠀⣿⡇⠀⠀⢸⣿⣷⠟⠉⠀⠀⠀⠀⠈⢿⣿⣷⣤⣤⣀⣠⡄⠀⠉⠉⠛⠷⣾⢻⣿⣿⡀
⠀⠐⠿⠂⣼⣿⣷⣶⣶⣶⡆⠀⡴⢁⠀⣿⡇⠀⣠⣾⣿⡇⠀⠀⠀⠀⣀⣤⣤⡀⠹⣿⣿⣄⣬⣽⣅⣀⠀⠀⢀⡀⠀⠀⢿⣿⣃
⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣧⣾⠟⢩⣿⡇⠀⠀⠠⣾⣿⣿⣿⣿⠀⠈⢿⡏⠉⠁⠉⠙⠓⠶⡾⠁⢀⣴⡿⠙⣿⡄
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠘⠛⠒⠀⣠⣾⡿⠟⠁⠀⠈⣿⡇⠀⠀⠀⠈⠻⠿⠿⢋⡀⠀⠈⣿⡄⠀⠀⢠⠀⠀⠀⣴⡿⠋⠀⠀⢹⣧
⠀⠀⠀⠀⢿⠇⠀⠀⣠⣶⣶⣤⣾⡿⠋⠁⠀⠀⠀⠀⣿⡇⠀⢀⣤⣦⡄⠀⢹⣿⣷⠀⠀⠘⣿⡄⠀⠸⠀⠀⠀⣿⠁⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⢿⠷⡄⠰⣿⣿⣿⣿⠋⢻⣆⠀⠀⠀⠀⢀⣾⡆⠀⠘⢿⣿⠿⠶⢶⣿⣿⡄⠀⠀⠈⣷⣤⣤⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⣷
⠀⠀⣀⣀⡀⠀⠀⠀⠉⠻⣿⠛⠀⠀⠉⠛⠛⠛⣆⠀⠀⠀⣤⠀⠀⠀⣀⣀⡀⢹⣿⣷⠀⠀⢀⣈⡁⠀⠀⠚⠛⠛⠀⠀⠀⠀⠐⠚⠛
⠀⣾⣿⣿⣿⣦⠀⠀⠀⠀⠘⢷⡀⠀⢀⣤⣤⣌⠉⠀⠀⠀⠿⠾⠚⠛⠋⠙⠛⠻⠿⣿⠀⣴⣿⣿⣿⣧⡄⠀⠀⠀⢠⣾⣿⣦
⠀⠻⣿⣿⠿⠁⠀⠀⠀⠀⠙⠞⠀⠶⣿⣿⣿⣿⡇⠀⣤⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⣿⠀⠹⣿⣿⡿⢋⣁⣤⣴⠞⠛⠻⠿⠋
⠀⠀⢰⣶⣷⡀⠀⣤⣶⣤⡀⠀⠀⠀⠙⠻⡿⠛⠁⠀⠿⠳⠀⠀⣿⣿⣿⣿⠟⠀⠀⣿⠀⠀⢸⣿⠴⠛⠉⠁⠀⠀⠀⢀⣴⣿⣿⣦
⠀⠀⣿⡿⣿⣿⡄⢿⣿⡿⠛⠛⠛⠛⠛⠛⣿⣿⡄⠀⠀⠀⣀⣀⣈⣛⣛⡋⠀⠀⠀⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⠃
⠀⠀⣿⡇⠙⣿⣿⣆⣠⣴⡾⣦⣀⠀⠀⠀⢹⣿⣷⡀⠀⠀⣿⠉⠉⢹⣿⡇⠀⠀⠀⣿⠀⠀⢸⣿⣤⣴⡾⠿⣷⡄⠀⠀⠈⢹⣯⡁
⠀⠀⢿⠇⠀⠈⣿⡟⠋⠁⠀⠈⠻⢧⣄⣴⠇⢿⣿⣧⠀⠀⣿⠀⠀⢸⣿⡇⠀⠛⠛⠛⠀⠀⢸⡟⠉⠁⠀⠀⠈⢻⣆⢀⣄⢸⣿⡇
⠀⠀⠛⠀⠀⠀⠘⣿⡄⠀⡇⠀⠀⠀⠙⢃⣤⡾⠿⣿⡆⠀⠉⠀⠀⢸⣿⣃⣀⣀⡀⠀⠀⠀⢸⡇⠀⠀⠹⣄⠀⠀⠛⠋⠁⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠁⠀⠀⣠⣴⠟⠋⠀⠀⢸⣧⠀⠀⠀⠀⢸⡟⠉⠉⢻⡇⠀⠀⣤⡾⠇⠀⠀⠀⠀⠀⠀⢀⣀⣤⡜⣿⠃
⠀⠀⠀⠀⠀⠀⠠⡿⠃⠀⠀⠀⠀⣿⠁⢰⣶⣶⣶⡄⢿⣄⠀⠀⠀⢸⡇⠀⠀⢸⣇⠀⢰⣷⠾⠂⠸⣦⣤⠴⠾⠟⠛⠉⠁⠀⣿⡆
⠀⠀⠀⠀⠀⢰⣿⣥⡄⠀⠀⠀⠐⣿⠀⢸⣿⣿⣿⣿⠀⠹⣷⡀⠀⢸⡇⠀⠀⠈⠉⠉⠀⠀⢠⡾⠃⠉⠀⠀⠀⠀⠀⡄⠀⠀⣿⡇
⠀⠀⠀⣄⠀⠀⢠⣄⠀⠀⠀⠐⠚⠛⠀⣀⣙⣛⣛⡁⠀⠴⠾⠿⠆⢸⣧⡀⠀⠀⠀⠀⠀⣠⡿⠁⢀⣤⡾⠓⠀⠀⠀⠇⠀⠀⠿⠇
⠀⠀⣤⠘⣶⣤⣿⠛⠿⣶⣄⡀⠀⠀⣼⣿⡿⠉⢻⡇⠀⠀⡄⠀⠀⠈⢩⣽⣶⣤⠀⠀⣰⡟⣠⡾⠛⠁⠀⠀⠀⠀⠀⡀⣠⣴⠿
⠀⠀⣿⣄⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⡟⠁⠀⢸⡇⠀⠀⠇⠀⠀⢠⣿⣿⣿⣿⡇⢿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠇⢿⡿
⠀⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠛⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠺⠿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⢰⣳⣿⡄⣀⠀⠗⠀⣈⣉⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠂⠁⢤⣾⣼⣻⣿⡿⡶⡀⢠⠀⠀⠀⠀⠈⠪⡶⢇⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠖⢀⣶⣬⡈⢉⣿⣿⣯⣽⡟⡇⠘⠠⢹⡲⣷⠀⠠⡴⠁⠔⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡾⢿⣾⠏⢻⢛⠰⡿⠈⢦⣀⡀⣡⠈⠀⢀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⡿⣿⣿⡶⣸⠗⡠⠾⢑⣂⠐⢄⡈⠒⢭⣹⣦⠀⢆⡷⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣹⣽⢿⣿⣿⣧⣪⣴⠊⠁⠀⠈⠻⣧⣰⠳⣾⣉⢥⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⣿⠮⣿⢿⣿⡟⠀⠀⠀⠀⠀⡿⣣⡑⣞⣝⡻⢏⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣴⢷⡝⢿⡛⣏⠭⢱⡄⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⡉⠙⠿⢼⡟⠀⠀⠀⠀⠀⠛⢋⣴⢱⣧⣹⡯⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠻⡗⢤⣸⡇⠀⠀⠀⠀⠀⠀⢛⣿⣷⣟⠍⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠡⠙⠀⠀⠀⠀⠀⠀⠈⡟⠹⡟⡡⠉⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢒⡻⠣⣀⡀⠀⠀⣰⣄⣠⠘⢿⠥⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡤⣿⠬⢿⣭⠫⢽⡗⢒⡲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠇⡔⠁⣝⠑⢦⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠈⠁⠒⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⡤⠤⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣴⣶⡷⠶⢶⣶⣦⣤⣤⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⢿⣿⣿⠓⠒⠀⠀⠈⠉⠛⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠻⠃⠀⠨⠽⢶⡆⣤⠠⠿⠿⠿⢓⣴⡮⠞⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠆⠀⠀⠀⠀⠀⠬⠇⠂⠀⠀⠀⠠⡫⠋⠀⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⡞⠉⠁⢿⣶⠀⠀⠀⠠⠠⠔⠡⠀⠀⠀⣿⠃⠀⠀⢀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠚⠉⠀⠀⢰⠁⠀⡆⠈⢿⣦⠀⠀⠀⣹⣦⣄⡀⠀⠀⢩⣘⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⡀⠀⠘⣤⠀⢰⠀⢠⣿⣿⢿⣿⣿⣿⡿⠋⠉⢉⡉⠛⡙⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢱⠀⠸⡁⠀⠘⣆⠏⣼⣿⣤⣿⣿⠟⠀⠀⠀⢄⡈⢢⠇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⡇⠀⠙⣄⠀⢻⣼⣿⣿⣿⠟⠁⠀⠀⠀⠐⢤⣹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⠆⢸⡿⠛⠋⢁⠀⠀⢀⣀⣤⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⠴⠊⠁⠀⠀⢠⠈⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠾⢷⣦⠀⠀⣰⠟⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⣾⠋⠀⣼⠏⠀⠀⠀⠀⠀⣠⡦⠀⠀⢀⣤⠄⠐⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⠿⠶⢶⡿⢀⣴⠟⣳⠂⢠⣿⠖⣿⠆⣾⡷⢺⡇⢘⣿⠃⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⣿⠃⢼⣃⢼⣏⣠⣿⣗⣾⠏⣸⣿⣲⡿⠁⣾⣃⢼⡟
⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠘⠧⠀⠀⠻⠦⠈⠁⠀⠉⣼⠇⠈⠀⢠⡟⠈⠁⠀⣀⣈⣡⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠿⠀⠀⠀⠸⠃⠀⠀⠸⠯⠴⠟⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⣠⡄⠀⠠⣄⠀⢀⣄⠀⠀⠀⠀⠀⠸⢿⠆
⠀⠀⠀⠀⢀⣴⠶⣾⠟⠓⢦⠀⠉⣉⡉⠙⠛⠓⠶⢶⣿⣥⣾⣏⣀⣀⣠⡿⠀⣼⠏⠀⠀⠐⠂
⠀⠀⠀⠀⠘⠇⣼⣯⣤⣤⠞⣀⣤⡍⠡⣞⡆⠀⢠⡿⢁⡾⢁⣭⡭⢉⣡⡤⣴⠏⠀⢀⣴⠶⡤⠀⣠⡖⢀⡶
⠀⠀⠀⠀⠀⣸⡏⠈⠀⢈⡇⣸⠏⢀⡴⣫⡟⢀⣾⠁⣸⡷⢫⡿⢡⡟⠁⣰⡏⠀⣠⡿⢁⣾⠃⣰⡟⢠⣿⠃
⠀⠀⠀⠀⠰⢿⣤⣤⠶⠋⠀⠛⠚⠉⠀⠻⠞⣻⡇⠀⠟⠀⠘⠓⠚⠛⢛⡉⠛⠛⠉⠓⠁⠛⠂⠙⠓⣽⠇
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⣰⠏⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠓⠲⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣾⣏⣀⣠⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢉⣩⣿⠟⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠾⠿⠛⠋⠁⡀

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠔⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⣾⣿⡀⢰⢁⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⡆⠀⣰⣾⣿⣳⣿⣿⣿⣿⣿⣿⣻⣿⣉⣻⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⣿⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣚⡂⠠⠊⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠘⠉⢹⣿⣿⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⣈⡉⠁⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠉⠉⢡⡄⠀⢀⣀⣠⣤⣶⣾⣿⡿⢱⠄⢀⠀⣾⠠⠄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡄⠀⠀⢀⡇⠀⠰⠀⣀⠀⠀⠒⣀⠲⢶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠛⠀⣿⡇⠀⢠⣿⡀⠚⡍⠛⠢
⠀⠀⠀⠀⠀⠀⠀⣾⡇⢠⣴⣷⠔⠉⢛⡥⡤⣀⠀⣛⠒⠋⠉⡜⠛⣒⠿⠛⠋⠉⠌⡁⠩⠩⣠⠄⡈⠉⡓⠴⠸⢿⡇⠛⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⣿⠧⠘⠉⠁⢀⠔⡅⢀⠈⠁⠀⠁⠀⠸⠓⠒⠉⠀⢀⠄⠘⠆⢀⣿⠀⠀⡀⠉⠈⠑⡷⢤⠀⠀⠈⡖⠉
⠀⠀⠀⠀⠀⠄⠀⠙⠤⠊⠁⠀⡀⠂⠀⣼⢰⡄⠘⠃⠀⠀⠀⢠⡀⠂⢀⠀⠀⣿⢸⣿⡄⣀⣀⠀⢀⠀⠂⠀⡆⠀⠀⠇
⠀⠀⡆⠀⠂⠀⠀⠠⢀⣀⠀⢀⣤⣤⡤⣭⡁⠀⠰⠆⠀⣠⡄⣾⡇⠲⠵⠄⠜⠛⠛⠉⠉⠀⠄⠈⠀⠀⠈⠀⡇
⠀⠀⠁⠨⠁⠠⢈⠀⢀⠀⣀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠀⠒⠠⠅⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠻⠿⡷⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⠃⢂⡆⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘