Panicky Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠤⠐⠒⠒⠒⠂⠒⠒⠒⠒⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠠⣄⠀⠐⠀⠀⠀⠀⠀⠠⠤⣀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡬⡁⣀⣀⣄⣀⡀⠀⠄⠀⠀⡀⣠⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢁⡠⠜⠲⠿⢇⢡⡐⠀⠀⠀⠈⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠀⠒⠀⠀⠀⠐⢠⠉⣈⠀⠀⠀⠀⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠾⠐⡄⠀⠀⡬⠂⠀⠾⠉⠂⠀⠀⠀⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⠁⢀⡀⢳⢤⣴⡀⠄⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠉⠀⠁⠀⠘⡄⢉⣀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠠⡁⡠⠞⢀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠄⠀⠤⠅⠠⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡣⠤⣀⣀⡉⠃⠀⠀⠈⣓⣂⡠⠄⡣⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⠭⣐⣒⣒⣚⣒⣒⣒⡨⠭⠞⠁

⠀⢰⡀⠀⣴⣶⡆⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢷⣾⣿⣿⣄
⢀⣾⠁⢀⣻⣿⣇⡀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⢰⣿⣿⠆⠀⣾⠀⠀⠀⠀⠀⢰⣿⣟⡋⢼⣿⣿⠿⣷⣦⡀⠀⠀⠀⠀⣴⣿⡋⠀⠸⣿⡿⠀⠙⢳⣄
⣼⣿⣾⣿⣿⣿⣿⢿⣾⣷⠀⠀⠀⠀⠀⠀⣾⣿⣥⣄⣸⣿⣏⣀⠀⣼⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣷⣶⣾⣿⠟⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣷⣶⣶⡿
⠙⠉⠀⣿⣿⣿⣿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣿⣿⣿⠿⢿⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⠉
⠀⠀⠐⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⣼⣿⡿⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⢸⣿⠏⠀⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣿⡇
⠀⠸⣿⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⠋⠀⠀⢙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⢸⣿⡇
⠀⠀⣿⡇⠀⠀⣾⡇⠀⠀⠀⠀⢠⡶⠿⠛⠛⠉⠁⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣿⣧
⠀⠰⠟⠁⠀⠀⠙⠷⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣯⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠃⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⠀⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉
⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⢠⣤⣀
⠀⠀⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣼⣿⡟
⠀⠀⠀⣾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣶⡀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣴⠟⠃⢠⣾⣿⣦⡄⠀⠀⠀⠀⠀⣼⣇⣤⣶⣶⣿⣿⣤⡀
⠀⢀⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⣀⣴⡿⠋⠀⠀⠀⣴⣿⣃⣀⣀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠙⠟⠛⣻⣿⣿⣿⣿⣿⡄
⠀⠈⠉⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⠻⠛⠁⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⠘⣿
⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠐⠿⢿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⣿
⠀⠀⣰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠛⢿⣿⣇⠀⠙
⠀⠀⠻⣿⡿⣿⣿⡟⠀⠀⠀⢠⠟⠳⢶⣶⣶⣿⡿⠁⠉⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠹⣷⠈⠹⣷⡄⠀⠀⠉⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⣿⠀⠀⠈⣻⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⣤⡄⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣯⠀⠀⠀⠀⠀⠀⣹⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⠶⠖⠚⠉⠉⠉⠉⠉⠉⠉⠙⠒⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠥⠀⠀⠀⠀⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠱⠕⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣸
⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠀⠀⠀⠀⠶⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⡀⣠⠞⠀⠀⠀⠀⡀
⠀⣀⣀⡀⠀⣀⠀⠀⠀⠀⣾⣄⠀⠀⠀⠀⠀⠀⠀⠠⠌⠉⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⡼⢫⣤⠶⣆
⣴⡿⠭⢉⠻⠏⢳⣄⡀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣇⣀⣠⣤⠶⢾⠃⠋⠰⢚⣩
⣽⠓⠒⠂⠀⢄⠘⠉⠉⠛⠒⠒⠶⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡏⠉⠉⠀⠀⠀⠈⣇⠄⠀⠤⣶
⠈⢿⣏⣠⣤⠶⣧⣀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠀⠀⠀⠀⠀⣀⣤⠴⠚⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣀⡀⠀⠙⠳⣤⣄⢀⡀⠀⠀⠀⠀⢠⣶⠟⠀⠀⢀⣠⡤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢶⡄⠈⠛⠙⠚⠑⠚⠓⠋⠀⢠⡶⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡤⢤⣤⣤⣤⣤⣤⣤⣤⡤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠁⠀⣠⠞⠋⠀⠀⠀⠙⣆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢶⡆⠉⠈⣷⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠉⠉⠉⠀⠀⢸⡄⠀⠀⠀⠀⠀⢸⣇⠤⠤⠤⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣀⣀⣤⡤⠴⠞⠁⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠙⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠤⠤⠄⣀⣸
⣿⣿⣿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣠⣄⣤⣤⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠈⣁⡐⠠⠄⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠶⢍⠀⠀⠉⠲⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡴⠀⠀⠀⠒⠤⣤⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣠⠊⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣶⣶⣶⣶⢶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠘⠉⠑⠒⠤⢄⣾⣧⠀⠀⠀⠀⢀⣾⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣾⢱⡠⠤⠒⠊⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣄⡀⢀⣟⣿⠳⡀⠀⠀⢸⣿⣿⣿⣫⠋⠀⠈⠁⠀⢻⣿⣿⣿⣿⡇⠀⠀⣠⣾⡿⢸⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠇⠈⢿⣷⣬⣓⣦⣌⡉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⢉⣡⡴⢚⣵⡟⡵⠁⠸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⠟⠛⣉⣙⠻⠷⣄⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠈⣰⠟⠋⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠚⠉⠁⠀⠀⠀⠙⠀⠀⠀⠀⠰⠒⠒⢆⠀⠀⠀⠀⠀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⣈⡣⣄⣠⢜⣉⠤⠄⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⢸⡀⠀⢠⠄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣻⣵⡀⠀⣀⡅⢠⠔⡮⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠻⠯⠼⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠂⠦⠔⠀⠀⠒⠒⠒⠒⢢⠆⠒⠒⠲⠖⠒⠒⠢⡴⠒⠒⠒⠒⠒⠒⠒⡤⠒⠒⠒⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣒⣒⣒⣒⠒⠒⡒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⢛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⡇⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣇⣀⠍⢊⢔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠃⣄⠠⡁⢸⡄⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⢣⢸
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣣⠊⠀⠆⠈⠁⡸⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀⢉⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠂⠉⠉⠑⢪⡀⠀⠀⠀⢰⣷⣶⣶⣾⣿⣿⣦⣄⠙⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⢳⠤⠚⠋⢱⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⢻⣿⠋⠉⠁⠀⠹⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠐⠉⠀⠀⠖⠀⡠⠚⡟⠈⠂⠀⠂⠂⠓⢄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠃⠈⠁⠀⠉⠀⡇⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⢀⠴⣄⠀⠈⠙⠀⣠⠷⡄⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⠎⠀⠨⣯⡉⢉⡿⠀⠀⠐⡀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⠀⠀⡌⠀⢀⣀⡈⣛⠉⡁⣀⣀⠀⢰⣀⣠⡼⢇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠉⠛⢻⠁⢀⣿⡿⠿⢟⠌⠵⢻⠿⡄⠀⠏⠉⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⢀⠎⠀⣀⡠⠔⠒⠒⢄⠀⠘⣄⠐⠀⠀⠀⢰⠀⠀⢀⣀⡠⠂⠤⠄⣀⣀
⠀⠀⠀⣀⠤⠒⠉⠀⢀⠬⠂⠐⠠⣀⡀⠀⠀⣀⠔⠁⢀⣎⠀⠀⠀⠀⠀⠀⠳⡀⠈⠢⣄⠀⠀⢈⠀⠀⡧⠊⠀⠀⠀⠀⠀⠄⠑
⠀⠰⢿⡲⣄⣀⠤⠊⠁⠀⠀⠀⣀⡼⠉⠒⠉⠁⠀⠀⢇⠈⢆⠀⠀⠀⠀⠀⢀⣨⡆⠀⠀⠉⠒⠁⢠⣊⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠈⠢⠽⢝⠳⣄⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⢾⣕⠢⣣⠀⣔⠢⢍⣂⣷⣷⡄⠀⡂⠀⠀⠠⣷⣮⣥⣐⢢⡀⢀⠜⡁
⠀⠀⠀⠀⠀⠀⠀⠑⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣬⡛⢗⣛⣽⡿⣟⡼⠃⠀⠀⠀⠀⠀⠀⢹⢙⣿⣿⣔⣁⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⣘⢭⣭⡭⠭⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠉⠉⠉⠉⠴⠆⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢾⡠⠖⠒⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⢼⠃⠉⠙⠉⡉⢉⠉⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⠔⠋⠀⠀⠀⠀⠀⠀⢠⠆⠀⣸⣁⡰⠉⠱⠄⢺⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠏⠀⢱⠀⢀⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢼⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⡈⠀⢸⠄⠴⢁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠘⠤⠼⠧⠤⠃⠀⠀⠀⠀⠀⢀⣉⡩⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣶⡇⠀⠀⠀⠐⢺⠱⡀⠀⠀⠀⠀⠀⣀⠠⠤⠒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⣀⡠⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⡀⠀⠀⠀⠀⠉⣹⠀⠀⠀⠘⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠃⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠉⢹⠀⢀⣀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠧⠭⣈⣀⣈⠈⠉⠉⠓⡶⠂⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣮⠤⢭⠶⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠗⠶⠤⣤⣤⣭⣉⡉⠉⠱⠶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡬⠥⢾⠻⠿⣿⠴⡩⣍⢝⠛⡛⢦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⢳⢾⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣟⣘⣛⠙⠚⠻⠋⠃⢸⠇⠀⢸⠜⠙⢉⣟⣋⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠓⠃⡌⠀⠀⠀⠀⠀⠀⠀⠀⠸⣙⣟⠞⢹⡟⡴⢨⡖⠢⠿⠤⠤⠤⢤⡄⣾⠤⠝⢉⠫⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢀⡤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣍⣀⠆⣄⣏⣛⣏⡃⠶⠚⠚⠀⠋⠀⢰⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀⠀⢀⠒⣲⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣽⠾⣛⣷⣟⠃⠼⠛⡷⡀⠈⠠⣠⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠝⠤⡝⡙⠒⠚⣠⠻⡞⠵⠢⢦⣀⣀⢀⠀⠀⢀⠴⠊⠈⢔⡅⠀⠀⠀⢸⠀⠀⢘⡳⠃⠀⠀⠄⠀⠀⣨
⠀⠀⠀⠀⠀⢜⠁⠱⣀⡜⡌⠀⠀⠲⠧⣄⠀⠀⠀⠁⢦⣴⡘⠐⠐⢂⡀⡾⠁⠤⠤⠄⠀⠘⠦⣤⣤⠞⠁⠈⠉⠁⠒⠒⠒⠒⠒⠊⠁
⠀⠀⠀⠀⡠⠈⠳⠆⠁⢀⣭⣟⣉⡆⣰⠆⠀⠈⠙⠛⠁⠈⠁⠀⠀⣠⠠⣉⢉⣏⡁⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⢲⠒⠂⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⡤
⠀⠀⠀⠀⠙⠒⣒⠟⢁⣎⣀⠨⢽⡾⢥⣀⣀⡤⠾⠀⠀⠀⠸⢍⡍⢤⠀⡸⢧⡤⠸⠃⠀⠀⠀⠀⠀⠀⠀⡠⢄⢸⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠣⠖⠃⠘⠤⢶⣿⡃⢱⡀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠸⠈⢿⢦⣉⡳⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⢸⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠎⠁⠱⡀⠰⠀⠀⠠⠀⠉⠢⣄⠀⠀⠀⠣⠤⠃⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢄⡀⠀⡇⠘⠦⠤⠴⠧⠤⠖⠐⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡖⠐⣀⣀⣠⣶⡶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⠿⠋⢠⢩⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⣀⠤⠤⠤⠤⠤⠤⡤⠄⠀⠀⠄⡔⢁⡴⠅⠀⢀⣀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠉⠁⠀⠀⠐⠒⠀⠘⠖⠒⠢⠖⠷⠿⠶⠶⠲⠮⠤⠤⠤⠄⠈⠳⠆

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⣠⣶⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣷⠀⣠⠤⠄
⠀⢰⣿⣿⣿⣿⠟⠙⢿⡿⢿⣦⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣏⠟⠋⠀⠠⠄
⠀⢸⣿⡟⠈⠁⠀⠀⠀⢀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠙⣇⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⠀⢀⠒⠂⠹⣿⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣾⣦⣀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⣀⠀⠀⠐
⠀⠀⠀⠀⠠⡏⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⢀⡄⡀
⠀⠀⠀⣇⠀⠀⠀⠐⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢄⠀⠈⠐⡄⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⠀⠀⠀⠁⢈⡀⣀⠀⠀⠀⣦⣤⣤⣤⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣿⣿⣿⣿⣿⣿⣦⡀⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠈⠻⣿⠟⠁⠀⠂⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠈⢀⠈⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⠉⢀⣿⠁⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣤⠾⠋⠁⠀⠀⠀⢀⣾⣧⣴⠶⠿⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢀⣾⢿⣄⣴⠟⠁⠀⠀⠀⠀⠀⠀⠾⠛⠉⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⣿⠀⢠⡿⠁⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠿⣦⣴⠟⣠⣶⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣼⠏⣤⣴⠟⠁⢀⣉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⢿⡄⣼⠏⢀⡾⠋⠉⠀⠙⢷⡀⠀⠀⠀⠀⠀⣀⣤⣤⣄⠀⢻⡇⠀⠀⠀⠀⠀⣼⠏⠀⠾⢋⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠀⢸⡇⠀⠟⠀⠀⣸⡇⠀⠀⠀⢠⡞⠉⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⢦⣤⣤⡴⠟⣠⡄⠀⠀⢸⣇⠈⠛⠀⠀⣼⠇⠀⠀⠀⠀⠀⠸⣿⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢾⡃⠀⠀⠀⠀⠙⠶⣤⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⣿⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣀⣠⣤⣈⣉⣀⣀⣀⣠⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⡾⣋⣿⣄⣉⠉⠉⠉⠉⢁⣀⣴⡏⠈⢻⡆⠀⠀⠀⠀⠀⠀⢸⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⢸⡿⠋⠀⠀⠉⠉⠛⠛⠛⠉⠁⠉⠻⣦⣘⣷⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⢶⣤⣀⡀⠀⠀⠀⠀⠀⢀⣠⡼⠏⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣈⠉⠛⠛⠻⠶⠟⠛⠛⠉⠀⢀⣀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠤⣤⠤⠶⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠑⠄⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣠⠿⢿⠟⠉⠉⠁⠉⠉⠙⠒⠦⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠈⠉⠓⢯⡀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠈⠑⣄⠀⢠⣊⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠉⣼⣀⠀⠀⠀⠀⠀⠘⠻⣿⡇⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⣷⠞⠉⠉⠙⢦⡀⠈⠁⠀⠀⠀⢃⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠠⠄⣷⠀⠀⠀⠀⠀⡘⠀⠐⢄⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣠⣏⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠁⠀⠙⣾⣷⣶⣴⡄
⠀⠀⠀⠀⡤⠤⡀⠀⣀⠀⠈⣆⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿
⠀⣴⠉⠙⢇⠀⢸⠋⠉⣳⠀⠈⣦⣀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⡠⠚⠦⣀⡠⢃⠞⠀⠀⣇⠀⠀⡇⠀⠑⢢⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣄⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⠇
⠳⢄⣀⠈⠀⠀⠀⠀⡠⣏⠀⢸⡇⠀⠀⠀⣿⣿⣿⣷⣶⣦⣤⣴⣶⣿⣿⣿⣿⣿⣿⠀⠀⠀⠛⠿⠛⠛⠿⣿⣿⠏
⠀⠀⠈⠙⠢⠤⢴⠋⠀⠈⢦⡈⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡀⢠⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠱⣵⠀⠀⠾⠿⣿⣿⣿⣿⡿⠟⠛⢿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⡀⠁⠉⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠈⢧⡀⠀⠀⠀⠀⠈⠉⠒⠤⣀⠀⠋⠉⠻⣿⠏⠀⠀⠀⠀⡴⠁⠈⠉⢉⡥⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⡰⠋⠀⠀⠀⣠⠞⠀⠀⠀⣠⠏⠀⢸⠦⠒⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⡆⠀⢹⣳⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⣁⣠⣤⡤⢼⠁⠀⢰⣯⣄⣤⢾⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢀⡠⠔⠀⡌⡠⢦⡤⢭⣉⠗⠚⠂⠀⠀⢀⡄⠈⠉⠉⠁⠀⠀⠀⢸⡆⠀⠚⢹⣇⣀⣀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⢇⠀⠀⠀⡄⠁⠀⠖⠊⣠⠀⣀⣀⣀⣠⢤⡞⠂⠀⠀⠀⠀⠐⠛⠛⠷⣤⣄⣈⡉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⣰⠃⠈⠠⡀⠀⠈⢆⠀⠜⠁⠀⠀⠈⠲⢅⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠋
⠀⠀⣀⣤⣤⣶⣾⡄⠠⠀⠈⠠⠀⢈⠞⠀⠲⡀⠀⠀⠀⠀⠩⠀⠀⠀⠀⡎
⠀⣸⣿⣿⣿⣿⣿⣿⣆⠀⠁⠀⠠⠂⠀⠀⠀⠑⣀⡀⠀⠤⠔⠀⠀⠀⠀⡇
⠀⢹⣿⣿⣿⣿⡇⠀⠈⠳⢄⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢰⠁
⠀⠘⣿⣿⣿⣿⣿⡀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠄⠀⢸⠀⢀
⢀⣾⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⢶⣶⣶⣶⢛⣩⣵⣾⣿⡇
⠘⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⡏⡙⠁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⠃⠀⠀⠀⠘⡟⠦⠤⠎⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠘⣌⢦⣀⠀⣀⠜⠀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠀⢀⣀⣀⣀⣀⠀⠀⠀⠐⠀⠀⣠⠈⣆⠀⠁⠀⠀⠀⠀⢠⠃⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⡜⠑⠒⡡⠒⢦⠀⠀⠀⠀⠀⠑⠚⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⢳⡀⠀⢧⡴⠾⠣⠤⣄⡀⠀⠀⠘⠦⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠿⠛⢹⣿⣿⠀⠀⠀⠁⡰⢃⠀⣄⠀⠐⠁⠉⢦⣠⠒⢢⠁⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢉⢤⡀⠀⢠⡄⠈⣿⡏⠀⠀⠀⠀⡇⠀⠀⠙⠥⠤⠛⠟⠁⢻⢆⡼⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⡇⠀⠆⠀⠀⠠⠄⠀⢯⡼⠀⠀⠀⠀⠱⡀⠀⡀⠀⠀⠀⠠⠀⡜⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠁⠀⠀⢀⠤⡄⠀⠀⢸⣄⠀⠀⠀⠀⠀⠙⠢⢄⣀⣀⣀⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⣀⣀⣀⠤⠤⢶⡿⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡇⢸⡄⣀⢸⠃⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⡎⠀⠈⡣⠀⠙⡆⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠘⠒⠊⠁⠀⠀⢰⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣨⡆⢀⣠⠼⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣄⠀⠀⠀⣠⣤⡀⠀⢀⣤⣄⢟⠁⠀⠀⠘⡄
⠀⠀⠀⠀⠀⢀⠀⠀⠰⠀⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠛⢦⠞⠁⠈⠙⠶⢟⡴⡍⢻⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠒⢤⡀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠰⠚⠓⠛⠛⠛⠛⠛⠋⢷⢰⡀⠁⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⣏⠠⠀⠀⠀⠀⠀⠉⠺⣉⠉⠉⡇⠀⠀⠀⠐⠒⠒⠀⡤⠤⠤⠤⠤⠤⠤⢏⢧⡐⣄⠀⠀⣠⠞⠀⠀⢠⠔⠒⠓⠒⠒⠒⢄
⠀⠀⠀⠀⠀⠰⣏⠳⣄⠀⠀⠀⠀⠀⣀⣈⢵⣄⡁⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠒⣳⡈⠁⠩⡀⠀⠀⠀⠈⢆⠁⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠹⣆⠈⠢⡀⠴⢉⣭⡤⣶⠿⠛⢇⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣀⣀⡤⠤⠤⠴⣾⠀⠀⠀⠙⢄⠀⠀⠀⠀⢳⡀⠐⠀⠀⠀⠀⣈⣣⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⠉⠉⠀⣀⡈⢦⠀⠀⠑⠒⠀⠐⠒⠒⠈⠉⠉⠀⠀⠀⠀⠀⢣⠀⠒⣒⣀⣀⠀⠌⢣⡀⠀⠀⠀⠱⣄⢫⡭⣭⣅⣶⣞⠷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠩⣭⣍⣁⣴⠮⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠄⠒⠒⠒⠒⠃⠑⣄⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠒⠒⠒⠂⠉⠉⠁

⣿⣿⣿⣿⣿⡿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠛⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠟⡿⣿⣿⣿⣿⣿⣿
⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣥⣴⡶⠶⠶⠶⠶⠶⠶⢦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿
⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣄⣤⡶⠾⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣁⣶⡾⡟⠉⢁⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⣎⣼⣾⣿⡿⠶⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠶⢦⣴⣄⣤⡂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠠⠀⣤⣤⣸⣵⣾⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠶⣦⣉⡆
⠀⠀⠀⠀⠀⠀⣰⡨⡒⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣴⣷⣶⣿⣿⣿⣿⣫⣤⣤⣀⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢷⣤⣤⡀
⠀⠀⡀⣤⣦⣽⣯⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣶
⣁⣰⠽⣹⣿⠟⠉⠀⠀⠀⠀⠀⠀⢀⣤⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣠⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣟⣯⡿⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⡀
⠟⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⣀⣀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣤⣠⡀
⠀⠀⠀⠀⠀⢄⣶⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢠⣿⣿⣿⣿⡆⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⢀⣀⣀⣾⣾⣿⣿⣿⣿⣿⣿⣿⠿⡟⠙⠀⠀⠈⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠸⣿⣿⣟⣿⠏⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡬
⣿⣿⣿⣿⣿⣟⠻⣿⣿⣿⣿⣯⠁⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠉⠉⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⠃⠘⣿⣿⣿⣿⣿⠿⠛⣉⣽⣿⣿⣿⠿
⣿⣿⣿⣷⣭⣉⠉⠛⠿⣿⣿⣿⣤⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣆⣄⠀⢀⣠⣠⣴⣾⣿⣿⣿⣿⣿⣿⡟⠀⠀⢹⣿⣿⣿⣧⣴⣾⣿⣿⣿⣿⡹⠃
⣿⣿⣿⣿⣷⣭⣛⡶⣤⣌⣙⠛⠶⢭⣠⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⠀⢀⣼⣿⣿⣿⣽⣿⣿⣿⢿⠉⠙⠌
⡾⠟⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣶⣬⣄⣀⡀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣀⣤⣴⡾⢿⣿⣿⣿⣟⠿⣿⡃⠁
⠀⠀⠀⣰⠋⠸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣷⣶⣤⣤⣤⣄⣀⣈⣉⣛⣛⣿⣿⣿⣿⣿⣿⣿⣶⣞⣻⣭⣽⣶⣿⣿⣿⣿⡿⣿⣇⠀⠙⢦⡀
⠀⢀⠜⠁⠀⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣯⣭⣽⣿⣿⣿⣛⡙⠋⣿⣛⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⡟⠌⠃⠀⠙⠀⠀⠀⠙⢆
⠀⠀⠀⢀⣴⣦⠀⠀⠀⡸⠃⠋⠸⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⡟⢻⠋⣼⠟⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢠⣿⣿⣿⣇⠀⠐⠁⠀⠀⠀⠀⢨⠃⠈⠉⠙⠛⢩⠟⣿⠟⠟⠻⠿⠿⢿⠟⠁⡟⠉⠉⠁⠁⣨⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾
⠀⠀⢸⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠈⠁⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⣰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿
⠀⠀⠀⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠈⠀⠀⠀⠀⠀⣿⣽⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿
⠀⠀⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡮⡡⠀⠀⠀⠀⠀⠀⠙⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⣿⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣗⢼⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡷⣿⣿⣦⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣶⠄⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢀⣴⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠉⡀⠀⠉⠀⠹⣿⣷⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⢀⣴⣿⢇⠀⠀⠀⠀⠆⢀⡅⠰⠀⠀⠀⡀⠹⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢉⠜⠁⠈⣦⣀⡀⣀⣠⠞⠙⠦⢄⡠⢴⠁⠀⠈⠣⡉⡆⠀⠀⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⣠⠁⢀⠭⠋⠉⠉⠉⠁⠒⢤⡀⢠⠄⠀⠀⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⡠⠤⣀⡤⢤⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⠄⡠⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡤⢤⣀⠤⢄
⢠⠖⠧⡀⢸⡀⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠⠀⢀⡅⡀⠼⠠
⣸⠦⢀⡈⠈⠀⠀⣿⠄⣀⠀⠀⠀⠀⣼⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⢧⡀⠀⠀⠀⣀⠠⣯⠀⠀⠁⢁⡀⠤
⠣⠤⠤⢄⣀⣀⠾⣇⠀⣼⣿⡞⠂⣾⣿⣷⣄⠀⠀⠀⢸⣿⣿⣿⣿⣿⠟⠛⠻⠛⠛⠻⣿⣿⣿⣿⣿⡇⠀⠀⠀⣠⣾⣿⣷⠒⢳⣿⣦⠀⣸⠷⣄⣀⡀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣄⡀⠀⠀⠈⠉⠳⢤⣀⠘⠟⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠛⠛⠛⠛⠻⠃⣀⡤⠞⠉⠉⠀⠀⢠⣠⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣦⣬⣭⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣭⣥⣴⣶⣶⣶⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠉⠉⠉⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣴⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠙⠛⠛⠛⠛⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠋⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠉⠉⠙⠛⠛⠛⠛⠋⠉⠉⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠿⣿⣿⣿⣿⡿
⣿⣿⡟⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠇⠀⠀⠀⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡄⠀⠀⠀⠀⠀⠈⢱⣄⡀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠀⠠⠞⠙⢦⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢠⠷⠀⠀⠀⠀⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠿⠶⠦⢤⣄⡀⣸⠖⣒⠲⢤⡼⠦⠒⠒⠒⠒⠒⠻⢤⣀⡀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠈⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠦⣠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣴⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠖⠚⠓⠲⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⣠⠞⠒⠒⠢⡄⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢀⡀⠀⢀⡏⠀⢀⡴⠒⠾⠁⠀⠀⠀⠀⠘⠋⠉⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠊⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢬⡥⠴⠛⠦⣄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒
⠀⠀⠀⠀⡠⠶⢄⠀⢀⡀⠀⢳⡀⠀⠀⠈⠳⢦⣀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠟⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣴⠋⠙⢧⠀⠈⡟⠁⢹⡆⠀⠑⣄⡀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⡠⠾⠦⡀⢠⠟⡴⠃⠀⢸⠀⠀⢸⠀⠈⠒⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⡀⠀⣀⣀⣀⡼⠁
⠳⣄⡀⠈⠁⠈⠁⠀⢀⣼⠀⠀⢸⠀⠀⠀⢸⣯⣈⠉⣶⣦⣤⣤⣤⣶⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣀⣀⡀⣀⡀⠀⠉⢠⠟⠢⠞⠉⠉⠁
⠀⠀⠈⠓⠦⣄⣤⠖⠁⠈⠳⡀⢸⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠈⠀⡀⢄⠉⡖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠙⢌⡆⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⠒⠉⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⢻⡄⠀⠀⠀⠈⠉⠙⠛⠤⣄⡀⠈⠟⠛⢿⣿⡿⠁⠀⠀⠀⢀⡞⠉⠒⠒⢚⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢹⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢄⡀⢀⠟⠁⠀⠀⠀⣠⠎⠀⠀⠀⢠⠎⠀⢹⣀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣠⠞⠁⠀⢀⣀⡜⠁⠀⢠⠊⠀⢀⣼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⢈⡇⠀⠏⢹⡉⠙⣓⠒⢤⠤⢤⣀⣠⠤⢤⠶⠛⠓⠊⠋⠉⠉⠁⢇⠀⠀⠜⢸⠉⠉⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⢰⠊⠁⠀⢸⡰⠁⠈⣁⠜⠊⠀⠀⠀⠀⠀⡠⣧⠤⠤⠤⠤⠤⠤⠤⠤⠼⣄⠀⠀⠘⠒⠒⢶⡋
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠤⡀⠀⠀⢣⠀⠀⡸⡇⠀⠀⠣⣀⠞⠉⠁⠐⠒⠊⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠦⠤⠼⠃
⠀⠀⠀⠀⢀⣀⣸⡇⢃⠀⠀⠄⠀⠀⢣⡘⣀⡃⠀⠀⠀⠈⠑⠦⠀⠀⠀⠀⢸
⠀⢠⣶⣿⣿⣿⣿⣿⡄⠑⠂⠐⠀⢀⠎⠉⠀⠈⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⡟
⠀⢸⣿⣿⣿⣿⡿⠿⠛⢦⡀⠀⡠⠃⠀⠀⠀⠀⠈⠂⠐⠒⠀⠉⠀⠀⠀⠀⡇
⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠙⠲⣅⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡰⢸⠇
⠀⢀⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⣀⣀⣀⣀⣁⣒⣂⣈⣀⠼⢀⣤⣄
⢠⣿⣿⣿⡿⠿⠻⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣤⣶⣾⣿⣿⣿⡀
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠿⢿⣿⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠣⠀⠀⠠⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⢿⡿⠿⢿⣿⡇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⡄⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⠁⢠⠀⠀⠀⣠⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠄⠀⠀⠀⠀⠄⣀⣀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠘
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢷⣤⠀⢀⣴⡾⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⡹⠧⠼⣇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠂⠈⠁⠀⡀⠀⠑⣤⡿⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡠⢆⠀⠀⠐⢀⡅⠂⠀⡀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠑⢲⣶⠧⠬⣴⡊⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠊⠀⠀⠀⠀⢰⣿⣷⣤⣤⣾⣿⡆⠀⠀⠀⠈⠑⠤⢀
⠀⠀⡀⢀⡀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⢀⡀⢀
⠠⠺⢈⠃⠘⠀⠀⠀⢪⠁⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣶⣶⣦⣄⠀⠀⠀⠀⠀⠈⡕⠀⠀⠀⠃⠘⡁⠗
⡜⠀⠂⠀⠀⠑⠂⠤⢀⡣⡀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢀⢎⡀⠤⠐⠊⠀⠀⠐
⠀⠀⠈⠉⠀⠑⠢⢄⡀⠀⠈⠒⢄⡀⠿⠿⠿⠛⠀⠀⠀⠀⠛⠿⠿⠿⢀⡠⠒⠁⠀⢀⡠⠔⠊⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠁⠀⠐⠒⠒⠒⠒⠒⠒⠂⠂⠉⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⠀⠀⠀⢀⠤⠒⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⡔⠁⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⡎⠀⠀⠀⠀⠀⠀⠈⡣⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁
⠀⡁⠀⠀⠀⠀⢀⠠⠊⠀⠀⠈⠁⡖⠂⢤⠤⠀⠀⠀⠀⠀⠀⠤⡤⠐⢲⠈⠁
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣷⣶⣾⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣾⣀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣟⣀⣀⣀⣀⣀⣀⣀⣀⣻⣿⣿⣿⣿⣷⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢷⣾⡀⣠⣠⠄⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣠⣆⣠⣏⣁⣦⣷⣦⡄⣤⢀⠄⠀⠔⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⠛⠉⠋⢩⣯⣅⡁⢻⣧⡙⠘⠢⢆⠄⡀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠛⠁⠈⠀⠀⠉⠛⢿⣷⠀⠀⠀⣼⣃⠤⠀⠙⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣿⣆⠀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣶⣶⣆⠀⠀⠀⢸⣿⣷⠀⠀⢸⣿⡆⠀⣻⣦⣀⡀⡘⣐⣄⡀⠀⣡⣅⣀⡀⣬⣀⣀⣀⠀⣿⠇⢸⣿⣿⣿⣿⠇⢼⣇⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⢸⣿⠇⠀⠛⠏⠉⠉⠛⠛⠛⠛⣛⣛⣉⣉⣛⣿⡿⠛⠻⠶⠦⣤⣤⣼⣿⣿⣧⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠘⠛⠛⠀⠀⢀⣠⣤⣶⡶⠿⠛⠂⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⢀⡀⠈⠉⠉⣿⣿⡏⠉⠛⠛⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⣤⣤⡶⢶⣿⠟⠛⠛⠀⠀⢀⣠⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠟⠁⠀⠀⠀⣿⣿⣿⠀⠀⠐⠲⠦
⣤⣤⣤⡄⠀⠀⠀⣴⣶⣶⣾⡿⠿⠉⢋⣀⡀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣾⡿⠁⠀⠀⠀⠀⠀⠲⠿⠿⠿⠖⠂
⣿⣿⣿⣿⣤⣤⣤⣿⣿⡟⠉⠀⠀⠀⠉⣉⡉⠛⠛⠋⠁⠀⠀⠀⠈⠙⠻⠿⣶⣶⣶⣶⣿⣿⡿⠁⠀⠀⠀⢠⣤⠄⠀⠀⠀⠐⠿⠶⢦⣤⣴⠶⠂
⣿⣿⣿⡿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠘⠛⠻⠷⠆⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣶⣶⣀⡀⠀⠀⠈⠉
⣿⣿⣿⡃⠀⠀⠀⠀⠀⠠⠶⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠙⠻⣿⡆
⣹⣿⡟⠂⠀⠀⠀⠀⢤⡶⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠃⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣇⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣶⣵⣶⣤⣿⣵⣿⣼⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠂⠀⣴⣿⣿⣿⣿⣿⣿⣷⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⣠⡿⠙⠋⣿⢻⣿⣿⡿⣿⢻⣿⣷⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⣸⡏⠈⠁⢹⡇⠀⠀⠁⠘⣷⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢠⣿⠀⠀⠀⠘⠃⠀⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⢀⢠⣿⣿⣿⣿⣇⣠⣤⣤⡀⠀⠀⢀⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣴⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣦⡀⠀⣀⣀⠀⠀⢀⣀⣀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⠿⢿⣿⣿⠙⣿⣿⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣇⡀⠘⢻⠃⢠⣿⣿⠏⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠋⠂⠘⠿⠏⣠⣿⣿⣿⡄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣛⣚⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠇⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣤⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡅⠀⠀⠁⣀⣀⡉⠀⠀⠀⠔⢉⣄⣘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢛⡤⣹⠀⠀⠰⠛⠗⠀⠀⠀⠀⠀⠉⠊⠋⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣼⡄⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⠋⠻⣿⣿⣿⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣤⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⣿⣶⣿⣵⣿⣿⢿⣆⠀⠀⠀⠀⠀⠀⠀⢰⣏⣩⣬⣹⡇⠀⢸⠟⠛⠁⠁⠀⢠
⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⣴⣿⣷⣶⠀⠀⠀⠀⠀⣿⡟⠛⠻⣿⠁⢀⡿⠀⠀⠀⠙⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠁⠘⠋⠳⡤⠀⠀⠹⣇⠀⠀⣿⠀⠋⠀⠀⠂⠀⠑⠆⠐⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⢻⣿⣿⣿⣿⣿⣿⡏⠀⠈⠀⠀⠀⠀⡧⢤⡙⠳⠞⠋⣆⡐⠀⠀⠀⠀⠀⠀⣈⡴⠃⢀
⠀⠀⠀⠀⣀⣤⠾⠃⠀⠀⣰⡏⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣠⠞⠀⠀⠀⠀⠀⣠⡞⠁⠈⠈
⠀⠐⢀⣼⠋⠀⠀⠀⠀⠀⠘⣷⣾⣿⣿⣿⣿⣿⡷⣶⣀⣀⠀⠀⠀⠀⡙⠀⠙⠏⠀⠀⠀⠀⡠⠚⠙⢿⣆
⢂⣠⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⢻⣇⢰⠏⠈⠉⠁⠀⢀⠞⠀⠀⠀⠠⠀⠀⠀⠈⢣⠀⠀⠀⠙⣧
⢾⡃⠀⠀⣠⠶⠛⠛⠛⠛⠲⠤⣀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⣧⠀⠀⠀⠘⡆⠀⠀⠀⠈⣧
⠈⢷⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡜⠀⠀⠀⠀⠀⡘⢣⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠹⡄⠀⠀⢠⡟
⠀⠀⠳⡄⢿⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⡐⠀⠀⠀⠀⠀⢠⠁⠀⢣⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠘⣆⠀⡾⣷
⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⡀⡜⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢳⡄⠀⠀⣸⢻⠀⠀⠀⠀⠀⠘⢶⠃⠸⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠹⡆⢠⠃⠸⡆⠀⠀⠀⠀⠀⠈⢳⡀⠛
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⣀⠠⠤⢶⣿⠀⢹⠃⠈⠺⡇⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢀⡟⠀⠘⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠁⠀⠀⠀⠈⢧
⡀⡀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁

⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⣿⣿⣿⣿⣷⣦⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣭⣶⣷⠿⠿⠟⠛⠛⠻⠿⢿⣿⣿⣷⢦⡀⠀⠀⠔⠁
⠀⠀⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣉⣽⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⡇⢿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⢁⡤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡜⣷⡀
⠀⠠⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠄⢠⣿⣿⣿⣿⡏⠉⣉⣄⡚⣤⣀⣀⠶⠶⠶⠶⢄⡀⢸⣿⣧⣿⠧
⠀⠀⠀⠀⠀⠀⠀⠈⡎⠀⢁⠀⠀⠀⣼⣿⣿⣿⣿⠀⠘⠛⠉⢃⡞⠉⡏⠀⠾⠛⠦⠀⢹⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⣀⡀⢠⣿⣿⣿⣿⡏⠓⠲⠤⠐⠋⠀⠀⠘⢄⣀⣀⣀⡠⠊⠘⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠰⠁⣿⣿⡞⣿⠁⠀⠀⠀⠀⠈⠁⠙⠀⠀⠈⠉⠁⠀⠀⢸⣿⣿⡏⠼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡿⢠⠀⠀⠀⢠⣶⠶⣦⠀⠀⠀⠀⠀⠀⠀⢘⡿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡏⠀⠈⠀⠀⠀⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠇⠈⢙⣿⣗⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣰⣾⣿⣿⣿⡇⠀⠀⠻⠀⠀⢿⣿⣿⣿⠃⠀⠀⢶⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⢿⡯⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣻⣿⣿⡏
⢀⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣌⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⢡⣿⣿⣿⣿⣷⣶
⠀⡅⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⢀⣾⣿⣿⣿⣿⣿⣿
⠐⡇⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⣵⣣⣿⣿⣿⣿⣿⣿⣿⣿
⠀⡅⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿
⠀⠇⠀⠀⠀⢈⣿⣿⣿⣿⣿⢿⣿⣿⣷⡸⡏⣿⡟⠀⠀⠀⠀⡀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⣡⣾⣿⣿⣿⣿⣿
⠠⡆⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⢠⡿⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠃⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡾⠁⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⣿⣿⡻⣿⢻⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠐⢌⠊⠂⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡘⡀⠀⠀⡆⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠁⠀⠀⠇⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⢀⠄⠀⠌⠀⠀⣁⠄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠠⠁⠈⠀⠀⠀⠀⠁⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠅⠐⠀⠀⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⢀⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠈⡄⠀⠀⠀⠀⠀⢀⠔⠈⠉⠉⠐⢄⠐⠂⠐⠀⠀⠀⠀⠀⡐⠀⢀⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠠⠀⠀⢀⠤⠀⠆⡀⠀⠀⠖⠀⠀⢂⠲⠤⠀⠀⠀⡠⠊⠀⢠⠈⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⡀⠈⢆⠀⠀⠀⠀⠇⠀⠐⠈⠉⠀⠢⡈⡄⠀⢀⠔⠈⠀⢀⠄⠆⠐⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⡁⠄⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠊⠀⠀⠑⠂⠠⠀⣀⡘⠤⠂⠀⠐⠒⠁⠐⠊⠀⠀⢀⠆⠌⡠⠊⠀⠀⢀⠀⠀⠐⡈⠅⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡠⠂⠀⠀⠐⠈⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⣁⠀⠉⠐⠂⠀⢀⠀⠀⢀⠄⡀⠀⠀⠀⠀⢀⠄⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠠⢊⠔⡡⢂⠔⡠⠁⠈⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠁⠈⠄⠡⢊⡔⡁⠀⠂⢀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢊⠄⡰⠉⠀⠈⠉⠑⠄⡁⡀⠱⡀⠢⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢊⡇⠀⠀⠀⠀⢀⠀⠀⠐⠀⠱⡀⠐⠈⠀⠀⠀⠀⠀⣀⠀⢀⡠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠡⡇⠀⠀⠊⠔⠡⡠⡂⢀⠀⠀⠈⡀⠀⠀⠀⢀⡀⠅⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠹⠀⠀⠐⠀⠠⡬⣀⣁⣁⣀⠴⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡆⠀⠀⠊⢀⠙⢎⠀⠉⠢⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠄⠈⢧⠂⠀⠈⠒⢄⡀⠀⠐⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⠀⠀⠀⠀⠀⠈⢆⠔⡑⠠⢀⠈⠐⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠂⠒⠒⠘⢅⠀⠊⠔⠀⡡⠊⢀⠄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣈⠈⠀⠔⡡⢂⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠄⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠐⠀⠂

⠀⠀⡠⠐⠀⠀⠉⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⣷⣖⢹⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢾⡙⢦⣄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢒⣶⣶⡟⠁⠀⢛⣿⣆⣽⣷⡄⢸⣧⡀⠀⠀⠀⠀⠀⠈⠳⡄⠛⠷⣦⡀
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠈⢿⣿⣿⣆⠀⢿⣿⣿⡌⢻⣿⡌⢋⢳⡤⢀⡀⠀⠀⠀⠀⠙⢆⠀⠉⢳⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠷⠤⠛⠟⠛⠛⠃⡘⠋⠁⡀⠈⠻⠟⡸⡟⠗⠘⠷⠀⠀⠀⠀⠀⢸⡁⠐⢻⡁
⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⡱⠒⠒⠀⠀⠀⠀⠀⠀⠀⡈⠀⡀⠀⠀⠀⠰⠀⠁⠀⠀⢌⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠠⢾⡄
⠀⠀⠀⠒⢦⠀⠀⠀⢀⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⠀⠀⠀⠄⠁⠃⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣉⣿⠁
⠀⠀⠀⠀⠀⠁⠈⠈⠁⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⢀⡶⠋⢀⣀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠠⠀⠀⠀⠀⠁⠂⠈⠀⠁⠀⠀⠀⢠⠟⢀⣰⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣴⠀⠀⠀⠀⠀⠀⣿⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⣦⢀⡀⢀⢀⣜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣿⣿⠀⠀⠀⠀⠀⠀⢿⠀⣿⠆
⠀⠀⠀⠀⠀⢀⣤⣾⣧⡞⣻⣼⠷⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⡿⣻⣿⠀⠀⢀⣠⣞⡿⠒⢇⢸⡤⠞⠛⠶⣶⠟
⠀⠀⠀⠀⡠⠣⣪⢿⠋⠀⢹⡷⠀⠹⠌⠻⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡋⠲⣾⣿⣿⠀⠀⠘⠿⠿⠤⠦⠾⠿⠧⠴⠦⠴⠦⠒
⠀⠀⡠⠊⠐⠟⠁⠀⠀⠀⠸⡇⠀⠀⠀⢌⠀⠙⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⢱⣾⣿⣿⣿
⢠⢂⣀⠀⠀⠀⠀⣀⡀⣀⡀⢀⡀⡀⣀⣀⠀⢀⡀⢀⣑⡄⠀⠀⠀⠀⠀⠠⣠⢙⣛⣚⣫⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣨⣬⣽⣿⣿⣿⣿⣿
⠀⢀⣤⣴⣦⣄⣰⣿⣿⣿⣿⣿⣿⣷⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠄⠂⠁⠀⠀⠒⠒⠒⠒⠒⠀⠈⠀⠀⢀
⠘⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠋⠉⠁⠀⠀⠀⢤⣀⠀⠠⠤⢤⣤⣤⣀⣀⣀⣀⣀⣀⡠⠜
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠢⢍⣀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠭⠭⣛⣓⣒⡒⠒⠒⠂⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠐⠲⠶⠤⠤⠤⠤⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶
⠀⠀⣄⠀⠀⢠⣶⣿⡶⠀⠀⣀⣰⣄⡀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠘⡈⠉⠉⠭⠤⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠘⢹⣴⣸⣿⣿⢿⠛⢾⡴⣿⠟⠹⣿⣖⣶⣷⣿⡟⣹⠄⢀⡄⠀⠀⠀⠈⠅⠀⠀⢤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡂
⠀⣰⡌⣹⣿⡟⣶⠘⣴⡄⢹⣿⣆⣶⡈⣿⣿⢳⡎⢡⣦⠀⣼⣿⠀⠀⠀⠀⢦⠴⠒⠖⠀⠀⠀⠀⠀⢰⣀⠀⠀⠀⣀⣁⣀⣠⠟⠁
⢠⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣷⣿⣿⣧⠀⠀⠀⠈⠥⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⠴⠖⠋⠁⠀⠀⠀⠀⠘
⠘⠻⠟⠛⡟⣛⡛⣛⣛⣛⣛⣿⣛⣿⣛⣻⣟⣻⣟⣛⣿⠛⠛⠟⠛⠀⠀⠀⢀⣀⣀⣀⣉⣉⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⢀⣀⣈⣉⣉⣉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⠠⡄⠀⠀⠀⠀⠈⠢⣄⠀⠑⠂⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠉⢉⣤⠰⡤⠤⠤⠤⠤⠤⢼⣀⡓⠐⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⢠⠀⠀⢀⣤⠒⠉⠁⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣠⠴⠄⠳⣴⠉⠀⡂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡆⠀⡠⢄⡀⠀⠙⢦⡀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡔⠋⠀⠀⠀⠑⢀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠱⠀⢆⡠⠋⠲⡄⠀⢙⡤⠚⠁⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢢⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠁⠔⠀⠀⠀⠀⠀⠀⠀⣁⠁⢀⡀⡰⠚⠁⠀⠘⠷⢾⠉⠑⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠈⢂⠀⢀⡌⠀⠀⠀⠀⠀⠺⣦⡀⠲⣄⠀⠀⠀⣀⡤⠀⠀⣈⡼⠇⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠤⠄⣭⠀⠁⠀⠏⠀⠀⠀⡸⠙⢦⣌⠳⠤⠞⠁⣠⠴⠚⠉⢡⠀⡀⠀⢀⡠⠄⣄⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠎⠀⠀⡁⠀⡶⠠⡀⠠⠋⠀⡇⠀⣠⣌⡓⢤⠴⠋⢁⣀⠀⠀⡼⢇⡀⠀⠎⠀⠰⡃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⡆⠀⠡⣀⠌⠁⠀⠀⠸⣼⣿⣏⣹⡾⢀⣾⣿⠉⣷⡰⠁⠀⠓⠀⠀⠀⡔⠀⢀⡶⠤⠧⣀⠴⢀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠙⠿⣿⡿⠛⠁⠀⠀⠀⠒⠤⠴⠁⠀⠘⠓⢲⣤⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣌⣁⢿⡀⠀⠀⠀⠀⠤⡐⢶⠀⣹⡟⠈⠉⣶⠂⠶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⡀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⠀⢷⡀⠀⡔⠀⢠⠁⣀⡛⠁⠰⣄⡰⠉⣄⡔⢫⡈⠙⣦⣄⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢅⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⢳⡀⠉⠚⢦⣷⣁⡇⠀⡰⢎⠁⡀⣈⡤⠈⢣⡔⢹⡈⠷⣄⠀⠀⢀⣴⣴⠏⠀⢀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠹⣦⡀⠀⠀⠉⠉⠛⠳⢼⢏⣭⡇⢘⣤⠚⠃⢀⡛⣤⣈⣧⠀⢈⡷⢄⠑⠊⢁⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⢰⠈⠙⢦⣄⠀⠀⠀⠀⢸⠸⠀⠈⠉⠉⠉⠉⠉⠉⠀⢀⣠⠜⠋⠀⣈⠁⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠸⠀⠀⠀⠈⠛⢶⣤⣄⣘⢸⡀⠀⠀⠀⣀⣀⣠⡴⠖⠋⠁⠀⠀⠐⠤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⡀⠒⠀⠁⢄⠀⠉⠌⠋⠉⠉⠉⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⣀⡀⠃⠀⢸⠘⡀⢀⠀⠀⠀⢀⡠⠄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠂⠜⠀⡇⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠓⠤⣀⡇⠀⠀⠀⣀⠀⢀⠀⢠⠀⠀⠀⡀⠀⠀⠀⢀⡞⢀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢸⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠀⠀⠀⠀⢈⡉⠠⠄⢦⠄⠀⠀⠀⠀⢠⣤⣄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢈⡁⠀⢸⠀⠠⠆⡂⠀⠀⠀⠀⢸⡄⡟⢻⡇⣠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⠐⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⠠⠤⢀⠀⠀⠀⠋⠀⣠⣴⡁⢸⠀⠀⠀⠃⠀⠀⠀⠀⣠⢄⠀⢈⠰⠀⠀⠀⡀
⠀⠀⠄⠀⠀⡉⢀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⢠⣈⠿⢻⠏⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠘⡇⠀⠀⡠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢠⡀⠉⠀⠈⠻⡀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠠⢤⠆⠀⡀
⠀⠑⠒⠂⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⠃⡠⠛⠒⠶⡶⠚⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡜⢂⣦⡒⠲⡇⠀⡜⠁⠀⠀⠀⠀⠀⠢⠀⠀⢠⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⢲⡙⢠⣿⣿⣿⢰⠁⢰⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡂⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⢄⡀⠀⠀⠸⡅⠸⢿⣿⢳⡎⢠⠃⠉⠠⠀⠀⠀⠀⠀⠀⣀⠤⠤⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠡⠀⠀⡈⢸⡀⢳⠀⠱⡖⠀⠀⣰⠏⢀⠂⡀⠰⠀⠄⢄⠀⠐⠚⠉⠀⠀⠰⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⡀⢁⠀⠃⢸⠃⡄⠀⠀⠷⠄⠴⠁⣠⠋⠌⢀⡾⠁⢠⠂⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡁⠳⠀⠰⠀⠏⢠⠀⠀⠐⠊⠩⠄⠚⢁⡘⠤⠯⣀⡴⠋⡠⠂⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡆⢀⠀⢀⢀⠀⠀⠄⠀⠀⠀⠀⠈⠰⠞⠀⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠀⠰⠛⠀⡐⠀⠀⠉⠄⠘⠓⠂⠀⠀⢀⠂⠀⠈⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⣀⣠⠞⠀⠀⠀⠀⠢⣀⠀⠀⠀⣀⠞⠀⠀⠀⠀⠀⠀⠀⠰⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⡀⠄⠀⢠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⢀⣔⡀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⡠⢋⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⠀⠀⡌⠁⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⢃⠔⠣⡀⠊⠐⠀⠀⠀⡐⢘⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠔⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⠖⠀⣤⠠⠐⠀⡌⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠁⠀⡔⠀⠀⠄⠈⠀⡀⠀⠀⠀⠀⢀⠆⢀⠈⠀⠱⡀⠰⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠐⠈⣇⠀⠀⠀⢠⠊⠀⠀⠀⡶⠖⠃⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⢠
⠀⢀⠀⢠⡄⠋⠣⠟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠐⡁⠀⠑⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⡀⠀⠀⠀⠆
⠀⡆⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠴⠀⠀⠀⠀⢀
⠀⠐⠤⡀⠀⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠈⡄⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠄⠀⠀⠀⠈⡀
⠀⠀⠀⠈⢂⠀⡆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠋⠀⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡰⠁⡀⢀⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⡐⠒⠒⠀⡐⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠊⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣒⠤⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢀⠠⠀⠠⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⢳⠀⠂⠀⠀⠠⠄⠠⠤⠄
⠠⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢆⠠⡀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⠋⡠⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⢼⣀⠀⠀⠀⠀⡃
⠀⠈⠢⣈⠁⢀⣁⢄⠀⠀⠀⠀⠀⡠⠁⠈⠥⠤⠄⢀⡄⢀⠀⠀⠁⠀⢀⠀⠀⠉⠈⠉⣉⡉⡅
⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠅⠀⠤⠄⠒⠒⢂⠤⠊⠁⠀⠈⠀⠀⠀⠠⠥⠤⠤⠤⠠⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⡠⢶⣟⡡⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣽⠞⡰⢋⡵⢟⣺⠿⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠰⠀⠃⠀⠀⠴⢾⣷⠶⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⠀⠀⠀⠄⠂⠁⠄⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡒⣶⠹⠀⠀⠠⠆⡈⢆⣴⠒⠂⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⠽⣒⠀⠀⠀⠀⠘⢆⠹⠀⠀⠀⠁⠰⠀⡸⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣉⠕⣋⠉⠀⠀⠀⠀⠀⠀⢡⠀⠂⠀⠔⠀⠐⠧⠒⠒⠉⠉⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠾⢓⣿⢵⢋⡤⡶⠀⠀⠎⠥⢃⠀⠀⠀⠀⠀⠈⠁⠒⠒⡖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠟⠁⢰⣱⠀⠀⠉⠋⠁⠀⠀⡀⠔⠒⡢⠤⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣸⢆⠐⠀⠀⠀⠀⣡⡉⡑⠁⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠑⠀⠄⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠄⠀⣀⣠⠀⢃⣠⣂⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡆⠚⣆⡄⠤⣬⠃⢘⡬⣁⣼⡥⠤⠤⠶⠖⣖⡠⢶⡬⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠽⡄⢗⡆⠀⢠⠃⠀⢴⠃⠀⠉⠀⠀⠀⠄⠀⠈⠁⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡄⠠⡀⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⢸⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠰⡏⣀⣽⣠⡧⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠓⠒⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣀⣀⣾⣿⣷⣶⢦⣴⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣿⠿⣿⣦⣄⡀
⠀⠀⠻⠿⠛⣹⣶⣾⣿⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣝⣿⣿⣿⣿⣿⡿⣿⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢿⠧⣾⣿⣿⢿⣿⣿⠆
⠀⠀⠀⠀⠀⠻⠟⠛⣶⣍⣯⢿⣯⣤⣾⣇⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⢭⣻⣿⣿⣴⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⢀⠀⠈⠉⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠁⠀⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⡿⠟⠁⠴⠒⠒⠀⠀⠀⠀⠀⠀⠐⢀⣁⣴⣶⣦⣌⣐
⠀⠀⠀⠀⠀⠀⠀⣈⠀⢼⣇⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣀⣠⣿⣿⡿⠛⠁⠄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠤⣿⣿⣿⣿⣟⣹⣿⡇
⠀⠀⠀⠀⠀⣀⣼⣿⣦⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣯⣭⡼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⢁⣻⣿⠿⣇
⠀⠀⠠⠞⢛⣻⠿⠋⠀⣠⣶⣹⡟⠻⢶⣄⡀⠀⠀⠀⠈⠀⢿⣄⣻⣷⣿⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠨⡥⠌⠹⠿⠛⠿⠿⠻⠤⠟⠨⠥
⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠀⠈⠁⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠀⠀⠉⠁⠀⠃⠀⠀⠈⠒⠊⠈⠒⠊⠀⠒⠒⠀⠓⠒⠁⠑⠒⠁⠑⠒⠁


⠀⠀⢠⡶⠛⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢦⠸⡀⠇⢸⢃⡶⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⣴⣿⡁⠀⠀⠡⢀⠁⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠠⣌⠑⠌⠃⠁⠀⠀⠈⠰⢋⡤⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣿⣿⣷⣄
⠀⣿⣙⣷⠀⠴⠄⢡⢺⡉⠁⢠⣤⠀⠀⠀⠀⠀⠀⣼⣿⡌⡆⠀⠁⢀⡔⢹⣿⡝⣦⠀⠡⠴⠖⠀⠀⠀⠀⠀⠀⠘⡄⠻⣿⣟⣿⣿⡿⠋⠉⠉⢃
⠀⣿⣏⠁⠀⠀⠀⢺⡺⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⣿⣿⠉⠁⠀⠀⢸⡇⣿⣿⡇⢸⠆⠀⣉⡉⠀⠀⠀⠀⠀⠀⠀⠀⢄⠈⢉⠈⠛⣣⣾⣿⡷⣾
⠀⣿⣿⣷⣄⠀⠀⠀⠙⢢⣄⠀⠀⣄⡀⠀⠀⠀⠀⢹⣿⣆⠀⠀⢀⣸⣇⣿⣿⣧⣼⣄⠀⠀⠉⠀⠀⠀⠀⠾⠆⢀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⡏
⠀⣿⣿⣿⠟⠀⠀⠀⢰⣿⣿⣿⡾⣿⣿⣶⣄⠀⠀⠀⠹⣿⣷⣤⣮⣙⢿⡛⠛⠛⠛⠋⠀⠀⠀⠀⠀⢀⣴⣶⣦⣄⢱⡀⠀⢀⣾⣿⣿⣿⣿⣟
⠀⠿⠟⠋⠀⠀⠀⠀⠸⠿⠧⢿⣿⣿⣿⣟⣻⠀⠀⠀⠀⠈⠙⠻⠿⠿⠏⠀⠦⠶⠶⠚⠂⠀⠀⠀⠀⠀⣻⣾⣏⣿⣠⣇⣴⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⢠⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣠⣴⣶⣤⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣚⠛⣟⠛⢀
⠀⠀⠀⠀⠀⣶⡆⣿⡿⣿⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣴⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣦⠀⠀⠀⠴⣿⠦⢠⣶⠛⠻⣿⡙⢿⣦⣶⡄
⠀⠀⠀⠀⠀⣯⣧⣿⡿⢿⣿⣯⣾⣅⣠⡄⠀⠀⠀⠀⠀⠈⠻⠿⠟⠛⠛⠛⠿⣿⢿⣿⣿⣿⠟⠋⠀⠀⠐⠶⣽⣻⣾⣿⠶⠂⣿⣽⢶⣿⣿⣿⡧
⠀⠀⠀⣶⢲⣿⣿⣿⣿⣿⣿⣯⣿⠻⡦⣇⣀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⡀⠀⢈⡉⠀⠀⠀⠀⠀⠀⠀⢷⣭⣿⣿⣿⣇⡿⣶⣾⠻⣿⠿⣂
⠀⠀⠀⣻⡞⢹⣧⠟⢙⣿⡿⢿⡃⢠⣿⠟⢹⡆⠀⠀⠀⠀⢸⣿⡇⠀⠀⢀⡞⠛⣾⠛⢻⣧⠀⠀⠀⠀⠀⣠⡾⢹⣧⠼⠃⣿⡿⢻⡏⠀⣿⠾⢹⡇
⠀⠀⣾⣋⣀⣉⣀⣀⣋⣀⣀⢈⣁⣈⣁⢀⣻⡆⠀⠀⠀⢤⢼⠿⠿⢿⠿⠿⠧⢦⠿⠶⣼⠿⠀⠀⠀⠀⣼⣋⣀⣈⣀⣀⣋⣁⣀⣀⣁⣀⣁⣀⢸⣧

⠀⡉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠛⢻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠛⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠙⠢⡀⠈⠻⣷⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣹⢃⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢾⣿⡿⡽⢾⣷⣄⠈⠳⣄⠈⠻⣿⣿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠙⣧⣠⣦⠀⠀⠀⠀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡟⢫⡳⡿⣧⣑⣜⣦⣳⣽⡕⣄⠈⢣⣀⡈⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣯⠀⠀⠈⠁⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠴⢶⣖⣛⣻⣿⣶⣽⣦⣷⣿⠶⠶⡶⠶⠶⠶⠶⢾⠟⢹⠋⢉⣿⠿⣿⠿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡞⠳⣬⣇⠀⠀⠀⣞⣷⡶⠾⣛⣭⠵⠖⣛⣋⣩⣭⣥⠤⠶⣾⣷⣧⣧⣇⣀⣀⠀⠀⠀⠀⢠⠃⢠⠇⢠⣾⠋⠘⢛⣣⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠁⣴⢶⣷⣿⣿⠿⣛⡭⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠿⠿⠿⣿⣤⣋⣠⡿⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣋⣬⢟⣫⠶⠋⠁⠰⣦⣀⣠⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢚⣭⢞⣫⠷⣋⠁⠀⠀⠀⠀⠹⡍⢉⣤⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⣋⣥⠞⣋⡴⠛⠁⢰⣿⡀⠀⠀⠀⠀⠀⠛⠋⠈⠳⣼⠁
⠀⠀⢀⣀⡤⠶⠚⣋⣡⡴⠞⣉⡴⠞⠉⠀⠀⢀⣟⣈⠙⠛⣶⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢩⣥⠴⠞⢛⣩⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠉⣽⠁⠀⡟
⠀⠀⠀⠰⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠉




⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠈⠀⠒⠀⠀⠀⠐⠂

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠛⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢌⠀⠀⡧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠈⠀⠈⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢄⠀⠀⠀⠀⢠⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⡇⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⣀⠀⠀⠀⠙⠆⠖⠃⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⣶⡦⠛⠀⠀⣄⣀⣠⣴⣾⣿⣿⣿⣿⣷⣼⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠛⢠⠎⠂⠀⠿⠿⠿⠛⠋⠉⠈⠉⠻⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠈⠠⠀⠀⠀⠀⠀⠀⠀⣠⡂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠹⠷⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⠀⠀⠈⠀⠀⠁⠁⠀⠉⠈⠁⠀⠠⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⣢⠂⠐⠂⠀⠀⠒⢠⡆⢀⡖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠁⠀⠀⠀⠀⠀⢈⠁⠈⠉⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⠛⠋⠉⠉⠁⠉⠉⠉⠙⠛⠿⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠉⠉⠉⠉⠉⠉⠉⠛⠿⢷⣤⣀
⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣄⠀⠀⠀⢿⣇⠀⢀⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠀⣈⣿⣷⡀⠀⢸⣿⢰⣇⢀⣼⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣁⣹⣿⡿⠁⠀⠀⣿⠘⢧⣤⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠈⠙⠿⣿⣶⣤⣤⣀⣀⣀⣀⣠⣤⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣄⣀⣀⠀⠀⠀⠀⣀⣀⣤⣴⣾⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠛⠛⠋⠉





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⢠⣶⣿⠟⠛⠛⠻⢿⣷⣦⣄⣀⣀⣤⣶⡿⠛⠉⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠋⠉⠉⠙⠿⣿⣶⣤⣤⣤⣾⣿⠟⠁⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⡀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣾⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⣛⠉⠐⣶⠀⠈⠿⠢⠙⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠂⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢴⡆⠀⠀⢀⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢒⡁⢤⠗⢲⠛⠛⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⠒⠢⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠹⠀⠈⠀⠀⠀⠀⢠⠱⡀
⠀⠀⠀⠀⢀⡤⠚⡀⠀⠀⠀⢀⠟⠳⣟⠄⠀⠀⠀⠀⠀⢀⠎⠁⢰⠀⠀⠀⠀⠀⠀⢨⠀⢣
⠀⠀⠀⢠⠊⠀⢠⠁⠄⠀⠠⠄⣧⠀⠈⢊⡱⠂⠄⡀⢠⠂⠀⢰⠀⢃⢀⠀⠀⠀⠀⢸⠀⠀⠃
⠀⠀⠀⡌⠀⠀⠸⣆⠀⣤⡄⢠⡏⠀⠀⠈⡦⡀⠀⠀⠉⢆⠀⡸⡄⠀⠓⢀⣤⢀⣀⣸⠀⠀⠘⡀
⠀⠀⠀⢣⠀⠀⢸⠎⠣⣈⡰⠁⢹⠀⠀⢠⠁⠈⠲⢄⠀⢀⡕⠁⢡⢀⠔⠁⠀⠀⢀⡔⠀⠀⠀⠇
⠀⠀⠀⠀⠑⠒⠺⠀⠀⠀⠀⠀⠈⠒⢖⠁⠀⣀⣀⣀⣉⣁⣀⡀⠜⢅⠀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡏⠩⢀⣤⣭⠋⠀⡠⠚⡅⠀⠀⡎⠉⠁⠉⠉⡇⢀
⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠱⣜⣈⣉⣿⣶⡞⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢸⠀⠀⡇⢀⡀⠀⠳⡀⢸⠀⠀⠀⠘⢘⠁⠀⠀⠱⡀⠀⠀⡠⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠧⡀⠀⠀⢸⠀⠀⠃⢸⠀⠀⠀⠁⢸⠀⠀⠀⡆⠸⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⡇⠉⣦⡀⠘⠀⢠⠀⠈⢆⠀⠀⡄⢸⠀⠀⢀⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠜⠒⠂⠈⠁⠐⠒⠦⢠
⠀⠀⠀⠀⠀⠀⠀⠃⠀⡝⠜⢄⡀⠸⠀⠀⠈⡄⠀⡃⡸⢔⡺⡛⠘⠤⠤⠀⠀⠠⣤⠄⠒⠂⠁⠀⠀⠀⠀⠀⢀⡀⠄⡷⢺
⠀⠀⠀⠀⠀⠀⢀⠆⠀⣷⢃⡸⠃⢰⠆⠀⡶⠗⢀⣛⣈⣏⣉⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠒⠒⠂⠈⠉⠁⠀⠀⠀⡄⢸
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠉⠀⠁⠁⠉⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠉⢩⣒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⢀⠀⢀⠴⠻⠔⠋⠀⠀⣀⠬⠥⢬⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣠⠔⠹⠖⠁⢀⡀⠀⢀⡔⠋⠀⠀⠀⠀⢱⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠠⢖⠉⠀⠀⠀⠀⡠⢋⣀⡀⡞⠀⠀⠀⠀⠀⠀⢸⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣥⠀⠀⠀⠀⠘⠧⡀⠀⠀⠀⢠⠏⠀⠉⣳⣼⡇⠀⠀⠀⢀⠎⣠⠤⣄⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⢷⠀⠀⠀⠀⠀⠀⣼⠆⠀⠀⠘⡄⠀⠀⣻⣧⣀⣀⣀⠴⠋⠸⠁⢀⡸⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢉⣓⣄⣀⢱⠀⠀⠀⠀⠈⠉⠉⠋⠀⠀⠀⠀⠒⠀⠀⠒⠉⢠⠇⠀⠀⠀⠀⠀⠀⢠⠖⠉⠉⠉⠙⢶⢦⡀
⠀⠀⠀⠀⠀⠀⠰⡓⡔⡹⠂⠀⡔⠉⠤⣶⠦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣒⣲⣒⠖⠒⠲⠤⠴⠃⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢳⢳⡀
⠀⠀⠀⠀⠀⠘⠉⠁⡀⠀⢤⢠⣿⡦⠄⢤⡤⠄⠲⢄⠀⠀⠀⣰⠋⣉⣙⣂⢠⣤⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠲⣄⣤⣀⢀⠄⢠⠄⡇⡇
⠀⠀⠒⣠⡀⠀⠐⠦⠜⠀⠀⠘⣿⠤⠠⠀⠀⠀⠀⠀⠉⠒⠤⢝⣋⣉⣁⠤⡴⠿⠀⠀⡀⡘⠈⢀⣛⡒⣲⠀⠀⠀⡇⠶⢁⠖⣿⣿⣿⡇⠀⡇⡇
⠀⠘⣃⠤⠄⠀⠀⠀⠀⢰⡮⣁⡀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⠤⠤⣀⣀⣿⠋⠉⠉⠉⠑⣄⡜⠀⠈⠁⠀⠀⠀⣇⠠⠈⠙⠛⠸⠉⠈⢁⣇⡇
⠀⠀⢴⣺⣽⠶⢚⢋⠟⡏⠀⠀⠉⢀⡤⠄⠀⢹⣀⣀⣠⠔⠊⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⢀⣜⡼
⠀⠀⡿⠋⠣⠘⠙⢼⡀⠙⠒⢒⡶⢁⣠⡴⣿⠻⡟⢿⠻⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⣀⣤⡾⠛
⠀⠀⠘⠦⣤⡤⠜⠋⠈⠒⠋⡸⢿⠛⡅⠱⡈⢆⠹⡈⢣⠈⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠳⠸⠦⠗⠘⠒⠛⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠉⠀⠉⠡⠞⠧⠤⠦⠤⠒⠲⠶⠄⠒⠶⠒⠶⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠞⠛⠛⠛⠛⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡾⠁⠀⢠⣂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⣸⠇⠀⢈⠏⢹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⣿⠀⠀⠺⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⣾⠀⠀⠀⠀⣴⠞⠋⠉⠁⠀⠀⠈⠉⠉⠛⠶⣤⣀
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣧⡴⠇⠀⣀⣀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡉⠳⢼⡇⠀⠀⠺⠤⠜⠘⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠹⣆
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠳⣼⠁⢀⣤⣤⣄⣀⣀⣹⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣦⢄⢻⡄
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⠋⣿⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠢⠸⡇
⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⢀⡏⠓⢾⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢸⠷⢤⡾⠀⠀⢀⡔⣶⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⡾⠀⠀⢀⠀⠀⣸⣵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠇⠀⠐⠭⠇⠀⣿⠃⠀⠀⠀⠀⣀⣠⣤⣤⠶⠶⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢠⣯⠤⠶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣌⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠢⣄⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣷⡀
⠀⠀⠀⣒⣶⠀⠠⡃⡄⣤⠀⢀⠀⢈⡄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇
⠀⠀⠀⢹⣥⣤⣶⣿⣶⣿⣿⣗⡀⣽⣷⣦⣶⡄⠀⠀⠀⣴⠀⠀⠀⠀⣤⡆⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣸⠟⢁⠀⡠⢄
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣧⣽⣿⣿⣿⠇⠀⠀⠀⡽⣆⠀⣠⣾⣿⣿⣿⣿⣦⣀⣀⡀⠀⠀⢰⣇⠈⠛⠃⠀⣾⠁⠰⠟⠈⣤⠠⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠈⠉⠉⠛⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁⠀⠰⠟⠍⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⢀⣤⣄⣾⣷⣦⣿⣿⣿⡆⣶⣦⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⢡⠀⠀⡀⢀⠄
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠻⣿⣿⡷⣿⣿⡇⠀⠀⠀⢠⣿⣿⣿⠀⢘⡿⠋⠎⣳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠃⣠⣾⠀⣷⠅
⠀⠠⣤⠀⢾⣿⣿⠿⢿⠛⢡⣾⣣⠷⢿⣿⠇⠀⠀⠀⠀⢿⣿⣿⣦⣉⣂⣂⣚⢠⡌⠃⠠⣠⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣰⡉⣻
⠀⠈⡽⡀⠀⢸⣏⣷⣬⣫⣾⣶⣿⣷⣾⣽⣟⣆⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⡄⠀⠐⢻⣧⣀⣠⣄⣻⣿⣃⣀⣀⣀⣀
⠀⠀⠁⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠉⠀⠈⠙⠛⠛⠉⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠸⠙⠂⠉⠛⠛⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⢦⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⡾⡤⣾
⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⢸⣿⣮⡙⠛⣃⣜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⢴⣿⣷⠸⣁
⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⢸⡇⠈⠀⢰⡆⠀⠀⢸⣿⣿⣿⣶⣤⠉⠀⠄⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣎⣹⣩⠴⠏⠁
⠀⡀⠀⠀⠀⠘⠛⣛⣛⣿⣛⣓⠛⣦⣼⣽⣄⠀⢸⣿⣿⣿⣿⣿⡗⠃⢐⢦⣥⣤⣴⣤⠀⠀⠀⠀⠀⠀⠘⠉⠁⠈⠀⠀⠀⣧⡀
⠀⠀⢠⣤⣤⣤⣤⣽⣽⣿⣮⣿⣤⣶⣶⣶⣶⡆⢸⣿⣿⣿⣿⣿⣷⣄⣸⣟⣶⣦⢷⡞⠀⠀⠀⣀⣀⣠⣄⣤⣠⣀⣤⣤⣤⣿⣤⣤⣀⡀
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠈⠛⠛⠛⠛⠛⠛⠛⠂⠒⠒⠈⠙⠃⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⡿⠿⠿⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢹⣟⡻⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⠛⢿⡄⠀⠀⠀⠀⠀⢸⣿⣿⣦⡙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⣸⡿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⡸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠛⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠗⠀⠀⠀⠀⣠⣾⣿⣿⢿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⡀⢀⣠⣴⣶⣿⣿⣿⡿⣿⣷⣶⣤⣙⠿⢋⣴⣶⡝⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢋⣴⣿⠿⣋⣭⣶⣶⡆⣶⣶⣦⣍⡛⢿⣷⣌⠻⣫⣾⣿⡟⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡙⢿⣷⡙⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⣼⣿⣿⣿⣿⣿⣿⡏⣿⣿⡿⣫⣾⣿⣿⡎⢿⣷⡸⣿⣷⡀⠀⠀⠀⠀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⡇⠿⣫⣾⣿⣿⣿⣿⣿⠸⣿⡇⣿⣿⣿⣷⣶⣶⣿⣿⣿⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢨⣭⣽⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣯⣭⠀⣿⡇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣿⡟⣸⣿⣿⣿⣿⣿⣿⣿⡟⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣌⠻⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⢛⣴⣿⠟⣴⣿⣿⣿⣿⣿⣿⠿⣫⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⣭⣛⣛⣃⣛⣛⣫⣥⣶⣿⠟⣣⣾⣿⡿⠿⠿⣛⣫⣵⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠿⠿⠛⠉⠀⠘⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⣾⠀⣠⠔⡿⢋⣀⣠⡅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣧⠞⢡⡾⢛⣥⣾⠥⢶⣦⠤⠀⠁
⠀⠀⠀⠀⠀⠐⠀⢠⡀⢰⠃⣾⠋⠀⠋⠰⣛⡭⠖⣛⣉⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠁⠀⣼⣧⡟⠰⠃⠀⠀⠀⠚⣛⡭⠿⠛⠋⠉⠀⠁
⠀⠀⠀⠀⠀⠂⠀⢻⠘⠇⠀⠀⠀⣠⡖⠋⠁⣠⠀⠀⠐⠀⠈⠲⡄⠄
⠀⠀⠈⠀⠐⠶⣤⠬⣧⠀⠀⠀⣜⠁⠙⣄⢠⡔⠲⢤⡀⠀⣠⠔⠒⢦⠀⠁
⠀⠀⠠⠀⠚⠿⢭⣇⣀⠀⠀⡄⠁⠈⢰⡌⠻⡄⠀⠀⠈⢾⢁⣤⠀⢸⡇⢈⠀⠀⠀⠀⠠
⠀⠀⠀⠠⠴⠾⢭⣥⠤⠀⠀⠀⠀⠀⠀⠑⣄⠱⣀⡠⠔⠚⡇⠉⠀⡼⠀⠀⣂⣀⡤⠤⠖⢒⡆⠠
⠀⠐⠀⢀⡴⢚⣩⠄⠀⠀⠀⠀⠀⠀⠀⢀⣈⡄⠈⠓⠶⣺⠋⠒⢚⠛⠋⠉⠉⠀⠀⣀⡴⠛⠀⠂
⠠⠀⡰⠿⢺⣟⢵⡞⢁⡤⣶⠂⠀⣰⠊⡅⠶⣀⠀⠀⠀⠁⠀⠀⠉⠁⠒⠒⠒⡺⠋⠁⠀⠀⠂
⠀⠀⠂⠀⠟⢁⣾⠞⠉⣰⢃⠀⠀⠘⠲⠶⠚⠙⡄⠀⢀⣠⡤⡄⣀⣀⡀⠀⡴⠁
⠀⠀⠀⠒⠀⠌⢁⠀⠀⡏⣿⠀⡆⠀⢀⠀⠀⠀⠁⢀⡯⡾⢴⢥⠏⠈⣡⠞⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⣼⡟⣆⢸⠳⢄⡀⠀⠸⠴⠦⠧⡏⣠⠞⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⠈⠁⠈⠛⠀⡶⢭⡇⠀⠀⠀⡠⠞⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⢀⣀⣐⡀⠈⠀⡟⢸⠀⣠⠔⠋⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⠿⡇⠀⠉⠉⠛⢧⠀⢟⠻⠛⢛⠿⠥⠄⠀⠀⠀⢀⣤⠤⣲⣓⣢⣤⠀⠀⠀⢀
⠀⠀⠀⠠⠀⠢⣖⣶⠾⠇⣀⠓⠚⠉⠉⡟⠀⠀⢸⠉⠒⢼⡟⠉⠉⠉⠉⠉⠉⠳⠴⢭⣻⠉⠍
⠀⠀⠀⠀⠀⠀⠯⢶⢫⠒⢢⠕⠀⠀⢸⠁⠀⠐⣏⠁⠀⠀⠀⠈⠐⠁⠐⠀⠁⠈⠀⠀⠁⢈⡃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢈⠀⡏⠀⠀⠀⠸⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣷⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠳⢴⡚⠋⢁⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣇⠀⣹⠄⣻⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢽⠠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢸⠏⠁⢨⣗⣐⣯⠤⣄⠐⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣃⣽⣿⣉⡭⠗⠒⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠀⠀⠀⠀⠀⠄
⠀⢀⠠⠀⠀⠀⢀⡤⠠⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⡀⡀⡀⢀
⠔⠁⡨⢀⠄⠊⣀⣠⠠⠠⠐⠂⣁⣁⣁⡅⠭⠭⠬⠤⠤⠤⠄⠀⠀⠀⠀⠠⢀⠀⠀⠐⠈⠂⠀⠀⡀
⡈⠀⠐⠁⡴⠊⢁⠀⣢⠖⠊⢉⡡⢔⣂⣉⣉⡉⠉⠉⠉⠁⠂⠀⠈⠀⠀⠀⠈⢠⠔⡡⠀⠀⡂
⠀⠀⢇⡀⠘⢤⠀⡀⠃⠀⠀⠊⠰⠫⠟⢩⣭⡁⢂⢀⠄⢀⠄⢂⠠⠌⠔⢚⠉⠀⠀⠜⠀⡶
⠀⠀⠈⠫⢅⡠⠀⡈⠂⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠀⠀⠐⠒⠀⡀⣀⡤⠤⠄⢊⢁⠠⠖⠁
⠀⠀⠀⠀⢀⠘⠛⠒⠺⠤⠤⠤⠀⣀⣀⣀⣤⡤⠤⠤⠐⠀⠀⠂⠈⠁⠀⢀⠤⢸⠊
⠀⠀⠀⡈⠘⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡀⠀⠄⠂⢀⡂⠀⠈⠀⡰⡏
⠀⠀⠀⠀⠀⠠⠀⠤⡉⠐⠒⠒⠒⠒⠒⠒⠀⠁⠀⠀⠀⡀⠀⠈⡀⠰⠐⣠⡖⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠒⠋⢀⣠⣴⢃⡄⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⠀⢦⣀⡀⠀⠀⠀⠀⠠⠤⠴⠒⠓⡻⠋⠡⠾⠄⠀⠀⠤⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⣦⣤⠶⠖⠒⣘⣩⠀⣀⠔⣣⠶⡀⠀⠀⠚⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⡙⢿⣟⣋⣩⠍⠁⠞⢫⢃⡄⠀⠀⠀⣤⠈⠁⡤⣦⠀⠐⠂⡀⠀⠚⠉⠀⠳⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠊⠀⠈⢻⡷⠞⠛⡿⠷⡊⡁⡀⠀⠀⣈⠀⠀⣤⠉⢀⣠⣦⣾⡀⠀⣡⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⢰⢤⡀⠀⠈⠁⢀⠶⠛⠒⣂⡺⢽⣻⡃⢀⠀⢀⡉⠀⢀⣀⡀⠈⢠⡀⢀⡈⠉⠀⣀⢉⠁⢽⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠛⠉⠁⠀⠑⠀⢳⣿⣶⣶⡢⠝⣫⠃⠘⠀⠚⣥⣄⣌⠁⣠⡄⣀⣄⣤⠉⠁⣈⣉⣈⡤⣤⣶⠊⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⠀⡀⠐⠂⠄⠀⣠⣷⣰⣬⣾⣶⢡⠎⠀⠀⢀⡈⠁⢠⠁⠀⠉⡤⠜⠛⠙⡀⡄⡌⠀⠘⡄⣀⠈⠙⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⢀⠁⠶⠁⠈⠸⣬⣂⣈⣁⠲⢼⡢⠃⠀⠀⠃⠀⡚⠀⢊⡒⢔⠂⣰⠂⠠⠋⠂⢄⡀⠀⣑⠂⡀⠀⡁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠰⠆⠠⢆⡀⣠⢐⡂⣿⣶⡰⢶⣛⣟⢇⣤⣘⠀⠉⠀⠂⠐⠠⠀⠀⠊⠫⢄⠀⠀⠀⠀⠀⡂⠈⠡⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢠⢞⣿⣶⠌⠱⠝⢛⣽⣾⠿⢱⠉⡵⠄⠈⠁⠀⠀⢐⣁⡀⣀⣠⣀⣠⡼⠒⠙⠘⠉⢁⣄⡷⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢺⣾⣽⣎⣁⣐⠀⠁⠂⠽⠛⢂⢀⠠⠀⠀⠀⠀⡈⣁⡀⠘⡃⢈⠀⠠⠀⠄⠄⠈⠀⠀⣁⡼⠁⢔⣢⠥⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠉⠻⡆⣀⣌⢋⠉⠸⢅⠢⣄⢜⣙⠍⠁⠉⠁⢘⠩⠳⠦⠂⠁⠛⠋⠙⠁⠁⠚⠉⠁⠀⠀⠉⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⣤⡄⣀⣀⠀⣤⠀⣠⡄⢀⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠻⠆⣯⡅⢿⡼⠃⣿⠸⢧⡄⡾⠿⠼⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⣀⣀⢀⣠⡀⣀⣀⢀⡀⣀⢀⣠⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⠇⡗⢯⠸⣔⡶⢸⣏⠸⠟⠿⠸⣄⠄⢹⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⡤⡤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⣰⠒⣆⢸⢰⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠹⠤⢸⠯⠭⠭⠭⠭⢿⠭⡿⢿⢸⠉⠉⢹⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠁⠀⠀⠸⣶⡦⣏⣳⠤⠬⠥⠯⠼⠤⣎⣳⡺⠁⠀⠈⠓⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⢬⢧⡤⣤⣤⣤⣤⡤⡬⣄⠀⠀⠀⠀⣀⣤⣬⣀⠀⠀⢀⣠⣼⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡏⠉⠉⠉⠉⠉⠉⠉⠉⢹⢸⠀⠀⢀⢞⠕⠋⠉⠲⣕⢖⡵⠊⠉⠓⣝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⢀⡤⣄⠀⠭⣒⢤⠀⢸⢸⠀⠀⢸⢼⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⣎⠀⣼⠁⠙⡎⣇⢧⢸⢸⠀⠀⠘⣾⢶⣦⠴⠭⠭⠭⠭⠷⢤⣶⢮⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠈⢦⡈⠳⣄⠤⣄⠀⢸⢸⠀⠀⠀⡇⠀⣀⣤⣀⠀⠶⠲⠶⠶⠶⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠙⠦⣄⡠⠞⠀⢸⢸⠀⠀⠀⡇⢸⣏⡀⣹⡇⠀⠤⠿⠤⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⢸⠙⢦⡤⡇⠀⠉⠛⠉⠀⠀⠩⠥⠍⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠋⠀⠀⠀⠙⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⣠⡴⠶⠚⠛⠛⠛⠒⠲⢤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢷⠀⡞⠁⠀⠤⠶⢤⡀⠀⠀⠀⠀⠈⠳⣄⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠧⣾⠸⠟⠃⠀⢠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⣎⣉⠀⠀⠀⠀⢯⣴⣶⡿⠀⠀⠀⢀⣈⡙⠆⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡇⠀⢹⡄⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣾⣤⡟⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣇⣀⡼⣇⠀⠀⠀⢀⠀⠀⢤⣀⣀⡆⠈⢉⡾⠁
⠀⠀⠀⠀⠀⣀⣠⣴⣶⠟⠛⣻⡇⠏⣽⠃⠘⣦⡀⠀⠈⠑⠒⠤⠍⠉⠀⣠⡞⠁
⠀⠀⠀⢠⡾⠋⠉⠉⢿⡀⣰⠏⢷⠸⣿⡀⠀⠈⠻⢦⣤⣀⣀⣀⣤⡴⢺⠏
⠀⠀⠀⣿⠃⠀⠀⠀⢸⣿⠋⠀⠈⢧⡙⢷⡄⠀⠀⠀⠀⣼⡿⢉⡿⡡⠋⠙⢦⣀
⠀⠀⢀⡏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠙⣯⣙⠓⠶⠶⣿⠽⢟⣩⠞⢷⡀⢀⣠⠿⠷⠤⣤⣀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⣦⣄⡀⠀⠀⢹⡈⠙⢦⠀⢀⣴⠋⢀⡤⡾⠛⠁⠀⠀⠀⠀⠀⢈⣳⣄
⠀⠀⢸⠁⠀⠀⠀⠀⠀⡇⠀⠉⢻⠆⠀⠸⡇⠀⠈⣿⢿⢸⠆⠎⡜⡀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠙⢦⡤⢤⣀
⠀⠀⣼⠀⠀⠀⠀⠀⠀⣿⠀⣰⠋⠀⠀⠀⣇⠀⠀⣿⢸⢸⠂⡞⠀⠁⠀⠀⠘⠛⠀⠀⠘⡇⠀⠀⠀⢱⡈⠏⠱
⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡀⣇⠀⠀⠀⠀⣿⠀⠀⣿⢻⢨⣏⣀⡀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⣷⠾⢤⣇
⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⡇⠹⣷⡀⠀⠀⣿⠀⠀⣿⢺⠸⣿⡿⠟⠀⠀⠀⠀⠀⢀⡞⢸⣇⠀⠀⠀⢠⠏⠓⠦⣼⡇
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⣷⠀⠈⢿⡄⠀⢹⠀⠀⢿⢸⠀⣿⠳⣤⣄⣀⣀⣤⢴⠟⡰⠃⠘⠦⣄⣠⠞⠀⠀⠀⠈⠙⢦⡀
⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠻⡄⢺⠀⠀⢸⡘⡆⢸⢠⠇⠀⣠⠟⠁⢶⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡤⠤⠤⠤⠤⠤⠤⢤
⠀⠀⢸⠀⠀⠀⠀⠀⠀⣠⠾⢻⣆⣀⠀⠹⣾⠀⠀⠈⣇⠻⣼⣿⠴⢋⡴⢻⣄⣨⢷⡀⠀⡇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⣧⠀⢠⢶⡄⠀⠀⢸
⠀⠀⢸⡇⠀⠀⠀⠀⠊⠁⠀⠈⠀⠉⠓⠦⢿⣄⣀⠀⠘⠶⢤⡴⠚⠁⣀⡼⠅⠤⣤⣷⣸⠃⠀⢀⡆⠀⡜⠁⠀⠀⠀⠀⠀⢻⣤⠏⣸⠇⠀⠀⢸
⠀⠀⠀⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢷⡶⠞⠉⠀⠴⠖⠦⣌⣹⠆⠀⢸⠁⢸⠁⠀⠀⠀⠀⠀⠀⠸⣧⠾⠋⠀⠀⠀⢸
⠀⠀⠀⢸⡏⠙⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠚⠐⠒⢤⣸⡆⠀⢸⣀⣸⡀⠀⠀⠀⠀⠀⢀⣸⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠇⠀⠀⠀⠀⣉⣳⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣏⣀⣀⣀⠐⠾⣻⢗⡶⠃⠀⠀⡎⠉⠉⣹⠶⠂⠀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⣸⣤⡴⠶⠚⠛⠉⠁⠀⠀⠉⠛⠷⢶⣤⣄⣀⠀⠀⣿⢁⡼⠯⣌⣙⣶⠬⣹⠶⠀⠀⢀⡇⣰⠟⠁⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠺⠷⠊⠀⠀⠀⠀⠀⠀⠳⣤⣤⣤⠾⠑⠧⣄⣀⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⢤⣤⣤⠤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠰⡂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣹⣤⢧⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣀⡸⠲⢀⣤⣄⡲⡀⢢⠠⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡏⠀⣴⣿⠟⠛⠻⢾⣾⣾⣏⣎⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠗⢠⣿⡇⠒⠀⢀⢄⠙⠿⣽⡿⠗⣃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣠⠾⢿⠃⠀⠈⠈⢳⠀⣠⠤⡍⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⢰⣦⣠⣿⣷⡈⠈⠁⢠⠒⢢⠈⣈⡃⢀⠇⠀⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⣿⢏⠃⣠⣄⠀⠀⠀⠈⠨⢀⡵⠊⢀⠜⠁⠀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⢋⡕⠉⠑⢉⡭⠗⠤⢩⣤⣴⢮⢄⡀⠈⠀⠘⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣉⣴⣿⣿⣶⣿⣿⣄⣀⠐⢒⡛⢡⡶⢾⣀⡅⠀⠤⠄⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣯⠻⢿⣿⣿⣿⠃⢀⠄⠉⠁⠘⠱⡀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆⠀⢈⠙⠋⠀⠐⠀⠀⢡⠐⢤⡇⠀⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠿⠟⠛⠀⠈⠀⠀⠀⠋⠀⠀⠰⣖⠠⠙⠋⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠤⠔⠂⠁⠀⠀⠀⠀⠀⠒⠒⠈⠁

⠀⠀⠀⠀⠀⣠⣤⣤⣄⡀⠀⠀⠀⠀⢀⣤⡤⠄⠀⠀⠀⠀⠀⠀⡠⠄⢶⣤⠀⠀⠀⠀⢀⡤⠤⠤⢀
⠀⠀⠀⠀⣾⣿⣿⣿⣯⠙⠀⠀⢀⣆⣿⣿⣿⣿⣦⠀⠀⠀⠀⠺⣴⣶⣿⡷⠀⠀⠀⠀⠸⣿⣳⣶⣾⠃
⠀⠀⠀⡀⢿⣿⣤⣿⣿⠀⠀⠀⠀⢿⣿⣷⣼⣿⡟⡇⠀⠀⠀⠀⢘⣿⠿⢁⠀⠀⠀⠀⢀⠹⣿⣿⠁⠶
⠀⠀⠀⠙⠾⠿⠿⠟⠃⠀⠀⠀⠀⠀⠩⠽⠿⠋⠀⠀⠀⠀⠀⠀⠸⡏⠐⡔⠀⠀⠀⠀⠐⠀⠙⠿

⠀⠀⠀⠀⠠⣔⣂⣢⠄⠀⠀⠀⠀⢀⡤⣒⡒⠦⡀⠀⠀⠀⢐⡦⣠⡀⣀⠠⠀⠀⠀⠀⠀⣰⣦⠀⠤⠠⠂
⠀⠀⠀⠀⣀⢺⣟⡟⠔⠀⠀⠀⢠⣟⣒⣳⣛⣆⡘⡆⠀⠀⠠⢦⠒⣁⣏⠐⠀⠀⠀⠀⢛⣆⡴⢿⡇⣫⡄
⠀⠀⠀⠀⡈⢹⣷⡆⢠⠀⠀⠀⠈⣗⠒⡒⣦⠰⣳⠃⠀⠀⠀⠩⣮⢀⡀⢱⠦⠀⠀⠀⠠⠘⡠⢟⠻⡹⠇
⠀⠀⠀⠀⠾⠿⠿⠿⠷⠀⠀⠀⠀⠈⠓⠺⠵⠚⠁⠀⠀⠀⠰⠀⠷⠾⠿⠾⠆⠀⠀⠀⠆⠰⠷⠶⠶⠶⠄

⠀⠀⠀⠀⠀⣠⣖⢦⡀⠀⠀⠀⠀⠀⢀⣴⣖⣄⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣖⣖⣒⣲⣒⡶⡆
⠀⠀⠀⠰⣿⣿⣿⣿⣿⠇⠀⠀⠀⢾⣿⣿⣿⣿⡿⠀⠀⠀⡇⣤⠀⢠⣖⣷⢄⡇⠀⠀⣿⣿⣿⠿⡿⠛⡇
⠀⠀⠀⠀⣶⢠⣶⡀⣶⠀⠀⠀⠀⠀⢴⢰⡆⠶⠀⠀⠀⠀⡇⠿⠟⠁⠉⠁⢈⡇⠀⠀⡏⣿⡾⠿⢣⡀⡇
⠀⠀⠀⠀⠈⠀⠋⠀⠁⠀⠀⠀⠀⠤⠲⠴⠦⠴⠦⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠷⠷⠶⠶⠶⠶⠇

⠀⠀⠀⠀⡞⠉⣛⠉⢳⠀⠀⠀⠀⣗⣴⢶⡤⢤⣤⡀⠀⠀⠀⡤⢾⠶⡶⢦⠀⠀⠀⠀⢠⠔⠚⡙⢲⣄
⠀⠀⠀⠠⡙⢜⠿⣣⢊⠄⠀⠀⠀⡿⣿⣸⣧⣼⠧⠇⠀⠀⠔⠇⢬⠅⡭⢸⢠⠀⠀⠀⡇⢀⣴⣧⣍⠈⡇
⠀⠀⠀⢀⣩⣤⣍⣁⡌⠀⠀⠀⠀⡇⠠⢤⣴⣽⣀⠀⠀⠀⣐⣧⣸⣀⣟⣼⣒⠀⠀⠀⢧⡽⢩⡧⠉⣱⠃
⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠃⠀⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠉⠒⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠟⠛⠙⠛⠻⢿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⣠⠀⠀⠀⢨⡿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣤⣤⣶⣾⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⠄⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⣀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡆⠐⢿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠸⣿⣆⠀⢹⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠙⣿⣿⣿⣿⣿⣷⡀⣿⣿⠆⠘⠛⢿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠘⢿⣿⣿⣿⣿⣿⣿⡯⠀⠀⠀⠀⠉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣇⡀⠀⠀⢀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠙⠛⢿⣿⣿⣿⣿⣿⣿⡃⠈⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠈⢻⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄⠀⠀⠀⠀⣾⠉⣷⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⡀⠀⢰⡏⠀⢻⡆⠀⠀⠀⣠⡾⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⡀⢾⠁⠀⠘⣷⠀⣠⡾⠋⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣤⣀⣀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢹⡿⠋⠀⠀⢸⡏⠀⠀⠀⠀⣀⣀⣤⣴⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⡙⠛⠳⠶⡆⠻⣿⣿⣿⣿⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣥⣴⠶⠞⠛⢋⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠹⣿⡯⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⢶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠐⠻⠿⢿⣭⣥⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣬⣭⡿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⣶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⣀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⡴⠶⠟⠛⠉⢉⡿⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⢿⡍⠉⠛⠳⠶⢦⣼⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⡄⢸⣷⠀⠀⠀⣼⡇⣤⡀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢀⣴⠟⠁⢸⣿⡄⠀⢠⣿⡇⠈⠻⣦⡀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⠟⠁⠀⠀⢸⣿⣧⠀⣾⣿⡇⠀⠀⠈⠛⠊⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢸⣿⣿⣤⣿⣿⡇⢠⣾⣿⣿⣷⣆⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⣿⣷⠀⣠⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⡿⠿⠇⠿⠟⠛⠛⠛⠛⢀⠛⡛⢛⠛⠛⠻⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣴⣶⣿⠆⣠⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢟⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢋⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣴⣿⣿⣿⣿⣿⣿⣿⡿⢋⣴⠆⠀⢴⣶⣶⣶⣶⡦⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣯⣀⠉⢥⣾⣷⣌⠛⠿⠿⠛⣡⡄⠻⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡙⢿⣿⣿⣷⣾⣿⣿⣿⣦⣌⣉⣉⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⡉⠈⠀⠐⢣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠒⢄⠀⠀⠈⢑⠆⡠⠄⠀⢜⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠈⡀⠘⠄⠈⢐⢀⠀⠡⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⠀⣀⡀⠀⠣⡀⠳⠎⠁⣈⠆⢀⡷⠁⠀⠀⠀⠀⠀⠀⠤⣐⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡈⠧⠤⠚⠀⠘⣓⢈⠑⡆⠀⠀⠉⠈⠐⠂⠒⠒⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⢠⠀⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⡖⠀⠉⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⢢⡄⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢄⠈⡌⠀⠁⠢⠜⡄⠐
⠀⠀⠀⠀⠀⠀⢀⠠⠆⠀⠄⣈⠢⡁⠺⡄⠀⠀⠀⢁⢧⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⠄⠐⠂⢈⢠⣈⠆⠈⠐⠂⠉⠁⡀⠀⠀⡞⡩⠃
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⠝⠁⠀⠀⠀⠀⠀⢀⠃⠮⠄⠁⢅⢄⠒⡄
⠀⠀⣾⠉⠑⣄⠃⠈⠐⠄⣴⠁⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠈⠣⠕⠂
⠀⠀⠈⠛⢶⣿⡣⡀⠠⠑⠨⢃⣀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠀⠀⠙⠣⣒⠒⠤⠴⢏⡰⠳⠤⣦⣄⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⡀⠙⢄⠀⣿⡥⠛⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⠼⠀⠁⠒⠊
⠀⠀⢀⣀⡀⣀⠀⣀⠀⠀⠀⣀⡀⠀⢀⡀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣉⣀⣈⡉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣰⣿⣿⣿⣿⣿⣦⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⣿⣿⣿⣿⣿⣿⣿⠆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣿⣿⣿⣿⣿⡿⠋⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠹⣿⡿⢀⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⣶⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⣻⣿⣿⣿⣿⠟⠋⠉⠉⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠿⠋⣠⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣤⣶⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡤⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢄⣐⣒⣒⣒⣒⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠉⠔⣩⢎⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⣰⠇⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣄⣀⠀⠀⠀⠀⡠⡪⢄
⠀⠀⠀⠀⠀⠀⢰⡱⡏⡄⠙⢆⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⡔⠘⠦⣤⡠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣒⢄⠀⠀⠀⢱⡕⠚
⠀⠀⠀⠀⠀⠀⢰⢃⡉⣥⢀⢞⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠉⣇⠜⢀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡗⣐⣋⣡⠃
⠀⠀⠀⠀⠀⠀⣠⢕⠊⠩⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠥⡘⠁⡴⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⠠⠊⡔⠊⠉
⠀⠀⠀⠀⠠⡤⢧⣤⢥⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠭⣤⠭⣤⣭⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⢤⠯⣤⢧⣬⠦⠧⡤⣤⣤


⠀⠀⠀⠀⠀⠀⠀⢠⢂⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠤⣀
⠀⠂⠀⠀⠀⠀⡰⠉⢥⣇⠀⢙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠒⠒⠒⠒⠒⢄⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⠀⢸⡀
⠀⠀⠀⠀⠀⠀⢹⠀⡏⢃⠹⡈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡀⣀⢠⡄⢀⠆⡔⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡇⠀⢸⡉⠑⣆
⠀⠀⠀⠀⠩⠨⢆⢆⣭⠨⠈⠥⠭⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡊⠀⠀⠁⠀⡠⠢⡉⠁⠀⠀⠀⠀⠀⠀⠀⠐⡡⢌⣓⠁⢏⠒⢹⠈⠒⡇⠈⠂
⠀⠀⠀⠀⠀⠛⣸⡠⠊⠨⠽⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠚⠒⠒⠒⠚⠇⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡜⢤⠋⠆⡠⠃
⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⠘⣿⡟⠀⠀⠀⣠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⣿⡇⠀⠀⢰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡀⠀⠀⠈⠻⠃⠀⠀⠈⠀⠀⠀⠻⠋⠀⠀⠀⣠⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⢿⣷⡄⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀⠀⠀⣴⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣼⡟⠉⣠⣤⣴⣶⣶⣿⣿⣿⡄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣦⡄⠀⣿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣤⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⢠⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣛⣉⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣙⣉⣛⣛⣋⣙⣉⣙⣙⣉⣉⣛⣙⣁⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠆⠂⠀⠀⢀⠚⠁⠀⠀⠀⠀⠀⠀⠀⢠⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠟⠚⠛⠀⠀⣄⣿⡄⠀⠀⠀⠀⠀⠀⠀⣤⣿⣿⣿⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢈⡰⠁⠀⠀⠉⠈⠂⣄⠀⠀⠀⠀⢀⡺⠟⠋⣉⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣸⢈⠀⠀⢀⠀⠀⠘⠆⠐⠠⠀⢀⣶⣿⣣⠀⠀⠉⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⢈⠈⠒⠘⠀⣉⣤⣶⣿⡿⠿⠟⠀⡇⠀⠀⠀⡀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢀⡁⠈⠀⡀⣸⣴⣿⠛⠈⠉⠀⠀⠀⠀⣇⠀⠀⢀⣇⠀⠀⠁⠀⠀⠀⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠓⡃⠁⠀⠀⠀⠀⣀⣿⠇⠄⠀⢚⣂⠁⠀⠀⠀⠃⠀⢀⡞⢰⠀⠠⢀⣀⡀⠀⠘⠀⢠⣺⠂
⠀⠀⠀⣆⣤⡤⠠⠀⠺⠀⢿⣤⣤⣠⣤⣴⡟⣅⡀⠃⠀⠘⣷⠄⠀⠀⢄⣤⡔⣾⡶⣾⡗⣶⣟⡄⡗⠀⠀⢠⠫⣿⣸
⠀⠀⢠⣷⡏⠀⠀⠀⣱⣷⣞⣋⣩⣭⣉⣈⣭⣻⣿⣷⢦⣗⢿⣿⣿⣷⣿⣏⣿⠶⠲⡞⠃⠊⢨⠙⠷⣰⣂⡘⠇⣾⡏
⠀⠀⠙⠈⠓⠛⠀⠁⠐⠈⠀⠀⠀⠰⠛⠻⠓⠊⠉⠸⠛⠃⠘⠖⠋⠀⠈⠉⠀⠁⠐⠉⠀⠙⠉⠃⠀⠀⠈⠉⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⡄⠀⠀⢠⢀⢈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⡀⠒⡄⠉⠀⠈⠁⠉⢹⣧⣴⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⢠⡄⠀⠀⠰⠀⠀⠀⠸⡛⢉⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⢀⠠⠒⠚⠋⠁⠀⠀⠀⠀⠈⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⠆⠀⠰⡀
⠀⠀⠀⣆⠀⠀⠀⠀⠀⡇⢀⡄⠀⠀⠀⠀⢀⡤⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠘⠢⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡀⢀
⠀⠀⠀⢸⡄⣀⠀⠀⠀⣇⣼⠀⠀⠀⠀⠀⣾⢣⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠈⢠⠀⠀⠀⠀⢣⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⡠⠀⠀⠀⠈⠀⠠⠊
⠀⠀⠀⠀⢷⣿⡀⠀⠀⡿⢹⠀⠀⠀⠀⣾⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⠀⠀⠀⠀⢀⠈⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡠⠊
⠀⠀⠀⠀⠈⠁⢳⠀⠀⠀⠘⠀⠀⠀⡼⠉⠁⢀⣤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠓⠑⣄⠄⡠⠊⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠸⢁⡠⠚⣽⡧⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠔⠏⠈⢀⠴⠁
⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⣠⠔⠉⠀⠈⠁⠀⢀⠠⠔⣊⠷⠔⠒⠁⠀⠀⢺⡄⢀⡀⠀⠀⠀⡘⠃⠀⠐⠁⠰⢧⣆⡡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠐⠀⠀⠀⡀⠀⠀⠒⢆⠁⠀⠀⠀⠀⠀⡐⠒⠤⠐⠄⢳⠈⠙⠢⠤⠔⢁⡆⠁⠀⠀⡀⡀⠛⣄
⠀⠀⠀⠀⡆⠀⠀⠀⢠⣄⣠⡄⠉⢹⠀⠚⠀⠂⠈⠃⠀⠉⣉⣀⢸⡇⠸⡀⣀⠜⢦⡀⠡⡀⠀⠀⠁⡆⠀⠀⠀⠀⠀⠘⡈⢆
⠀⠀⠀⠀⡇⠀⠰⠂⠓⠋⠈⠁⡄⠀⠓⠛⠒⠲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⠆⠛⠓⠑⢄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⣀⠀⠀⠒⠒⠢⠦⢤⣁⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠇
⠀⠀⠀⠀⠀⠉⠉⠁⠉⠉⠁⠘⡄⠀⠀⠠⢄⣀⡀⠀⠀⠀⠘⠛⠛⠲⠦⢤⣄⣀⠀⠸⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⢹⡄⣀⠀⠀⠀⢭⣽⣶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢠⠀⢀⣀⣀⣤⣶⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠐⡀⠀⢳⣹⡄⠀⠀⠀⠀⠉⠙⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠨⠠⠐⣦⣴⡿⠟⠛⠋⢉⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢧⠀⠀⠻⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⢀⣾⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢸⣀⡄⠀⠀⠙⡄⠀⠀⢀⣠⡤⠤⠤⠤⢄⣀⠀⠀⠀⠀⣼⣿⠿⠁⠀⠀⠀⢀⣾⣿⠏⣁⣠⠤⠐⠂⠀⠒⠶
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⣿⢷⡀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠒⠭⣒⠦⣼⣿⠃⠀⠀⠀⠀⣠⣼⡿⠟⣋⠥⠒⠈⠁⠀⡀⠔
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⢀⠀⢀⡈⠉⠂⢽⡀⠀⠀⠀⠀⡹⣫⠔⠉⠀⠀⣀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠐⠊⠁⠀⠀⠀⢸⡟⢁⠀⠔⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀

⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀⣤
⠀⠀⠀⢀⣤⣶⠏⢿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢻⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣧⣿⣿⣋⣷⠀⠀⠀⠀⠀⢠⣴
⠦⢂⣶⣾⣿⣿⣿⣿⣿⣟⣿⠃⠀⢿⢀⣀⣀⣠⣀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⢟⣿⣿⠛⣿⣀⣠⣠⣴⣶⣿⣿
⣿⣿⣿⣿⣿⣯⣍⣩⣍⣿⣿⣿⣶⣤⣼⣿⣿⣿⣿⠛⢢⣤⣤⣄⣀⣀⠰⣿⣽⣿⣿⣿⣿⣾⣿⣿⣿⣯⣿⣽⣽⣿⢿⣿⣿
⡄⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿
⠶⠶⠖⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠐⠢⠀⠐⠒⠐⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠒⠛⠆⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⣿⠿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣾⣿⡏⢹⣿⣴⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠙⢶⣈⣡⣶⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⣿⢿⠛⠛⠈⢿⣿⣿⣿⣿⡿
⣁⣀⣀⠠⠤⠀⠀⡀⠀⠀⠒⢁⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⡿⠿⣿⣶⣤⣀⣠⡇⠀⢀⣾⣿⠿⠛
⣿⣦⣤⣮⠀⠀⠀⠬⡀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣬⣿⣿⣿⡟⠛⠛⣿⣏⣵⣤⡀
⠟⣻⣿⣴⣿⡿⠚⢀⣶⣾⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⢿⣿⣿⠃⢀⡀⡌⣻⣿⣿⣿⣶⣾
⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠛⢛⣟⠄⢹⣿⣿⣿⣿⣾⣿⠹⣿⣿⣿⣿⣿⣿⣿⣷⣜⢾⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣏⣽⣧⣾⣿⣷⣿⣿⣷⣦⣹⣿⡻⣿⣿⣿⣿⣿⣶⠘⣽⣿⣿⣿⣿⣿⣷⣍⠛⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣾⣟⡛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⡿⣷⣽⣿⣴⣿⣹⣿⣿⣿⣿⣿⡟⣇⣀⠀⣼⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣷⣤⣽⣿⣿⣿⣿⣿⣿
⣿⣿⣉⣉⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⣀⣆⣈⣿⣿⣿⣿⣿⣖⡀⠀⠀⠀⠀⡠⠚⢹⣿⣿⣿⣿⣿⣥⣬⠤⡄⠀⠀⢀⣔⣀⣽⣍⡀⠀⠀⠀⠀⣠⣤⣴⣿⣿⣿⣿⣿⣯⡭⠍⠩⠈⠁
⠀⠀⢿⢿⣿⣿⣿⣿⡿⣿⠿⠿⠛⠀⠀⣶⣿⣷⣿⣿⣿⣿⣿⣿⣿⠷⠖⠲⠀⠀⠾⡿⣿⣿⣿⣿⣭⣁⠚⠁⠀⢿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠉⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠉⠑⠊⠉⠉⠉⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋

⠀⠀⠀⠠⠀⢤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣿⠂
⠀⠀⠀⣀⣜⢂⣀⣏⣀⣹⡄⠀⠀⠀⠀⠒⢶⣶⣶⣶⣶⣶⣶⢰⣶⡒⠂⠀⠀⠀⠀⣀⣀⣾⣷⣤⡀⠈⣧⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣾⣯⡝⢦⡀
⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⠏⠀⠀⠒⠀⠾⠿⣿⣿⣿⣿⣿⣁⣠⣿⣷⣆⠀⠉⠉⢽⢿⡾⢿⣿⣿⣼⣁⣾⡷⢦
⠀⠀⠈⠛⠾⠋⠉⠉⠙⠷⠟⠁⠀⠀⠙⠼⠋⠉⠉⠉⠙⠧⠋⠈⠁⠀⠀⠀⠀⠀⠀⠿⠦⠟⠛⠉⠉⠉⠀⠧⠞⠀⠀⠀⠀⠁⠑⠚⠉⠉⠉⠉⠈⠓⠋

⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣦⣤⣤⣤⣄⡀
⠀⠀⣐⣉⢹⣧⣿⠛⣷⠶⣤⣤⢀⡀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡂⠀⠠⠤⣤⣬⣭⣿⣿⣿⣿⠛⠛⣿⣷⠄
⠀⢰⣿⣿⣿⡹⣿⡛⢻⣦⣠⣷⠴⠛⢋⣀⣤⣤⣼⣿⣿⣿⣿⣿⣿⡅⢀⣀⢂⣰⣶⣿⣾⣿⣿⡟⠂⢀⣿⣄⡀⠐⠶⠾⠿⢿⣿⣿⣿⣶⣾⣿⡟
⠀⠈⠻⣉⡟⠃⠛⠛⠉⠁⠸⣉⡗⠀⠀⠠⠴⢾⣿⣟⣻⠛⠛⣟⣻⠃⠀⠀⠀⠀⢩⣿⣛⣿⠿⠿⠶⠟⢹⣛⡧⠀⠀⠀⠀⠀⠙⠿⡿⠿⠿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⡀⢀⣄⡀⠀⠀⠀⡠⠀⢄
⠀⠀⣀⣀⣀⣠⡞⠛⠛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡄⣶⠖⠀⠀⢠⣘⣿⣿⣿⣿⣿⣷⣿⣿⣿⠂⠀⣾⣶⣶⣶⣧⢀⣀⡀
⠐⣒⡿⢿⣿⣿⠀⠐⠆⠀⣿⣶⣶⠄⠀⠀⣀⣠⣴⣾⣿⣯⡤⣯⣤⠟⠁⠀⠀⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⢸⣉⣹⣄⣀
⠀⠀⠀⠀⠛⣹⣦⣄⣀⣴⣿⡿⠏⢀⡀⠬⢷⡾⣿⣿⣿⣿⣤⠟⠁⠀⠀⠒⠚⠛⠻⣿⣿⣿⡿⠿⣿⣿⣿⡿⠀⠀⣰⠿⣿⡿⢿⣿⣾⣿⣿⣿⢿⡿
⠀⠀⠀⠀⠀⠧⠊⠈⠉⠸⠤⠁⠀⠈⠀⠁⠐⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⠶⠛⠛⠋⠁⠀⠀⠘⠒⠉⠓⠋⠀⠀⠀⠀⠙⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣭⣭⣭⣿⣵⣭⣯⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣾⣿⣿⣽⣿⣷⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠉⠉⠠⣀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠤⣀⠀⠀⠐⠦⡀⠈⠁⡀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⢒⣂⠤⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠄⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠠⠀⠀⠀⠀⠘⠀⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠀⣠⡴⠚⠉⠉⠀⠈⡁⢀⣀⠂⠠⠖⢀⠀⠀⠀⠀⠀⢤⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⢺⠁⠴⠋⠀⢠⠂⠀⠀⠉⠉⠑⠀⠀⠀⠱⠀⠀⠀⢀⡈⠄⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠴⠃⠀⠀⡆⠠⠐⠒⠀⠀⠀⠈⠉⠑⠄⠀⠀⣤⠣⠀⠀⡴⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⡾⢠⠀⢰⣇⡀⠀⠀⠀⠀⣠⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠜⠉⡘⠘⠿⢠⠀⠒⠀⠘⠃⠤⠄⠴⠀⠀⠀⡐⢀⡄⠴⣼⠀⡠⠤⢄⢀⡔⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡘⠁⠀⠀⠀⠀⠂⠀⠀⡤⡀⠀⠀⠀⠠⠞⠟⠉⠀⠀⢀⠋⠀⣠⠋⠀⢀⣗⣈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⠀⠀⠈⠁⣨⠀⢠⠇⢀⡔⠋⠀⢀⠗⠂⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⢀⠀⠀⢈⠐⠀⠀⡀⠀⠀⠀⣠⢊⠁⠉⠉⢡⠀⠀⠀⠈⠀⠴⠚⠡⡤⠤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⠜⠉⠉⠉⠙⢷⠀⢿⡀⠁⡒⣉⣋⡁⣸⠓⠂⠀⠙⠀⠀⠀⠀⡠⠀⣤⡂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⠒⢴⠒⠊⠉
⠀⠀⠀⠀⡀⠄⢈⠀⠀⠀⠀⠾⠰⡀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠢⢤⠀⣤⠃
⠀⠀⡠⠊⠀⠐⠠⠀⠀⡀⡠⠂⠰⡇⠀⠀⠀⠀⡀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠌⠀⠀⠀⠀⠀⠘⢠⠞⠁⠀⠀⠈⢢⡀⢰⠈⢀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠊
⡎⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠈⡱⠇⠀⡠⠦⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣀
⠳⠤⣄⣀⡠⠔⠥⡎⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠉⠐⠂⠤⠤⠄⡀⠀⠀⠀⡜⠀⠀⠀⠉⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠋⠉⠐⠒⠶⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡀⠄⠒⠒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⠠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠱⣁⣔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⢀⡤⡄⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⣀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡄⠀⠀⠀⠀⠀⡔⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⣀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠄⠀⠀⠀⠠⠄⠦⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢀⠠⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠓⢶⡤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠔⢚⡇⠀⠀⠀⠈⠁⠂⠹⠂⠠⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢰⠁⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠆⠀⠂⢸⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢸⠀⢸⠀⢀⠀⢸⣤⡇⠀⠀⠀⡀⠀⣠⠀⠄⠄⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⣦⣜⠦⠿⢷⠋⡠⠶⠆⠀⠐⠁⠘⠡⠾⡦⢀⠜⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢶⡛⠂⠀⠀⠘⣤⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⠀⣀⣀⠈⡁⠀⠀⠀⢀⣄⣀⠀⠀⠀⢈⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠘⢉⠉⠻⠧⠀⠀⠀⠟⠋⢉⡈⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠕⠒⠲⡄⠀⠀⢀⠴⠚⠓⢦⡀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢻⠀⠁⠀⠶⠀⠸⣵⠀⣮⠀⠴⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⢳⠇⠢⣀⠀⢀⠼⡏⠀⢻⣆⣀⣀⠴⡄⢠⣧⠄⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠈⠓⡾⠀⠘⠓⠒⠀⠈⠀⠀⠀⠙⠀⠀⠀⠀⠀⢳⠊⡁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠉⡇⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣷⠀⠀⠀⢠⠊⠀⢄⠀⠀⠀⠲⡄⠀⠀⠀⡞⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⠀⠿⠄⠀⢴⠿⠃⠀⠀⠀⢀⡟⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⣯⣉⣉⣉⣛⣋⣉⣉⣉⣭⣭⣇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⣏⣉⣉⠉⢉⣉⠉⠉⠉⢙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⣆⠀⠤⠤⠤⠤⠄⣰⠶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⢀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠀⢸⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡼⠁⠀⠀⠀⠸⢇⡴⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠈⠑⠤⡄⢀⣠⡴⠊⠀⠀⠈⠁⠒


⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⡶
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣛⣿⣻⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠔⠒⠊⠉⠉⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠤⢤⠔⠉⠠⠔⠂⠘⢦⣀⡴⠛⢦⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⡖⢩⡅⠀⠀⠾⠂⠀⠀⠀⠁⣠⡾⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣌⡎⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣬⡁⠀⠀⠀⢀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⢄⣤⠀⡙⠀⠀⣀⠴⠛⡄⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠳⢍⡠⠴⡊⠁⣀⣷⢵⣤⣤⣄⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⡟⣻⠋⢠⠎⠀⠀⠈⢷⡄⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢀⡞⠳⠃⠀⢸⠀⠀⠀⠀⢸⢡⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢰⠋⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠈⡆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⢡⠀⠘⡄
⠀⠀⠀⠀⢀⠴⠊⢩⡯⠉⠓⢀⡀⠀⠀⠀⠀⠀⢀⢷⡙⠢⡄⠀⠀⠈⠀⠀⠀⠀⢸⠀⠘⡄⠀⡇
⠀⠀⣀⠔⠉⣆⡀⡼⢸⠀⣀⡞⠑⢄⡀⠀⠀⣀⣾⢄⣙⠒⡇⠀⠀⢸⡀⠀⠀⠀⣸⠀⠀⢧⠒⠁
⢀⠞⠁⠀⠀⡏⣹⠃⠈⣆⠉⣷⠀⠀⠙⢯⠉⠀⠀⠀⠀⢹⠁⠀⣀⠜⠋⠉⠉⢉⡏⠀⠀⢸⡄
⡎⠀⠀⠀⢸⢠⠏⠀⠀⠘⢆⢸⠀⠀⠀⠈⣧⠤⠐⠒⠊⣏⠴⠊⠁⠀⢀⡠⠔⠋⠀⠀⠀⠀⠳⡀
⡇⠀⠀⠀⢸⡇⠀⣶⣶⣦⠈⢿⡀⠀⠀⠀⣇⢀⣀⣠⠾⣅⠀⢀⡠⠖⠉⢀⣀⣀⣀⣀⣀⣀⡀⠱⡀
⡇⠀⠀⠀⢸⣇⠀⠭⡿⠅⢀⣿⡀⠀⠀⢈⣷⠷⠀⠀⢀⡨⠟⠉⠀⠀⠀⣏⣀⣀⣀⣀⣀⣀⣹⠀⠹⡄
⢱⠀⠀⠀⣸⡌⠓⠒⠒⠒⢺⠹⡇⠀⠘⣷⣾⢄⠀⣰⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⡀
⠘⢄⡀⢀⡟⡇⠀⠀⠀⠀⡜⠀⢇⠀⣀⠽⣛⣿⡋⠈⢢⡀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠙⡆
⠀⠀⠉⠉⡏⠀⠀⠀⠀⠀⣇⠀⠈⢹⠁⠀⢀⠤⠵⣄⠀⡟⢦⡀⠀⠀⠀⠈⢦⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⢹⠀⠀⡜⠀⣰⠁⠀⠀⠘⣆⡇⠀⠙⠢⣄⠀⠀⠀⠉⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢸⢣⡀⠀⠀⠀⢸⢀⡴⡇⠀⢻⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⣀⣀
⠀⠀⠀⠀⢸⠀⠱⣄⠀⠀⢸⠁⠀⡇⠀⠈⣇⠀⠀⠀⣿⠀⠀⠀⣀⣤⠀⠀⢨⡍⠓⠲⠤⢤⣤⣤⣤⣤⠤⠴⠒⠊⠁⠀⠀⠀⠉⢹⠒
⠀⠀⠀⠀⢸⠀⠀⡟⡗⡄⢸⠀⠀⡇⠀⠀⠘⡄⠀⢀⡯⠤⢶⣏⣁⠞⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢼⡤
⠀⠀⠀⠀⢀⣀⣀⣇⣸⠘⣼⣀⡀⣧⡀⠀⠀⢱⣀⣸⠦⠐⠒⠉⠁⠀⠀⠀⠀⠙⠢⠤⢀⣤⣀⣀⣀⣀⣤⡀⠤⠤⠔⠒⠊⠉⠀⡾
⠀⠀⠀⠠⠯⠥⠬⢿⠤⠿⠭⠤⠭⠷⠿⠶⠏⠭⠭⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡶⢦⣰⠇⠀⠘⣧⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣟⣰⡇⠀⠛⠀⠀⠀⣹⠶⣄⣿⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠀⠀⠹⠋⠀⠀⠀⣀⡀⠀⠉⠀⠹⠃⠀⢸⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠈⠀⠀⠀⠀⠀⢠⡾⠋⢳⡄⠀⠀⠀⠀⠀⠈⠈⠇
⠀⠀⠀⠀⠀⠀⠀⣾⣛⣛⣛⣛⣛⣛⠛⢃⡴⠋⠀⢀⣿⣠⡶⣄⠛⢛⣛⣛⣛⣛⣛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠉⣟⢉⣉⣉⣉⢉⣴⣋⣀⣀⣀⣈⣉⣁⣀⣈⣷⣌⢉⣉⣉⡉⢙⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡏⠉⢹⡆⢹⡏⣿⠉⠉⠉⠉⠉⢹⡏⣿⠁⣎⠉⠉⣿⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡷⢈⣉⣉⣉⡁⢸⡇⣿⢀⡾⠋⠻⣆⢸⡇⣿⠀⣉⣉⣉⡁⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⢸⡇⢸⡇⣿⢸⡇⠀⠀⣿⢸⡇⣿⠀⣏⠀⠈⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣧⣈⣉⣉⣉⣁⣸⣇⣿⣸⣇⣀⣀⣿⣸⣇⣿⣀⣉⣉⣉⣁⣨⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣦⣤⡀⠀⢠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠛⠻⢿⣿⡿⠛⠉⠙⠛⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠁⢠⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣿⣿⢁⡀⠀⠀⠀⠀⠙⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣇⣤⡄⢀⣾⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠼⠿⣿⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⣠⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣟⣀⣉⢩⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠰⠿⠶⡾⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣧⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⡼⠁⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠶⠒⠚⠒⠶⠶⠦⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣟⣉⣀⣤⡤⠤⠴⠶⠶⠶⠤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠗⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠿⠖⠒⠒⠋⠙⠓⠒⠲⠶⢤⣌⣉⠛⠳⢦⣄⠀⠀⠀⢠⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢛⣉⣠⣄⣀⣀⣀⣀⣀⣠⣀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣍⠛⢶⣄⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣫⣵⣾⣷⣌⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⣔⣠⢀⡀⠉⠛⢶⣍⠻⢦⡀⠀⠀⠀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣙⠸⣿⡿⣿⣿⣿⣿⣿⣻⣋⡀⠿⢿⡿⢪⣥⣢⣄⠉⠳⣦⣟⣦⡀⠀⠀⢈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⡈⠁⠀⠇⠈⠁⠈⢙⢧⠘⠷⢿⣿⣿⣷⣷⡢⡈⠻⣿⣿⣧⡄⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⣍⣻⠇⠀⢁⠀⠀⠀⠀⠀⠆⠀⠃⠙⣶⣿⣿⣿⣿⣿⣿⣮⡢⡈⠛⠿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣭⡀⠀⢿⡑⠄⣀⡰⠆⢀⠀⠠⠀⠏⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢻⣿⣾⣮⡗
⠀⠀⠀⠀⠀⠀⠀⣾⠿⠿⠟⠛⠛⡛⣛⣉⣭⣭⣭⣷⣶⡶⣆⢈⣇⠀⠀⠁⢀⢤⣄⡀⠀⢒⣉⣀⣲⡶⣿⣿⣿⣿⣿⡆⠀⠜⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⣶⣾⣿⡿⢋⣿⢻⣿⣿⣿⣿⣿⢿⢟⣿⡁⣙⣶⣄⣀⣀⣠⡴⢛⡀⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡇⡀⢹⡟⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⢾⣿⣿⣿⠿⣤⣴⣿⢯⣿⣿⣿⡿⣹⣾⣶⡟⣿⡝⢿⣿⣿⣾⣭⣛⠿⣿⣿⣿⡇⠇⡼⣰⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢀⣿⡿⠏⣸⣷⣿⡛⠿⠻⠋⢹⣶⣿⡿⢣⣿⣿⣿⣿⣧⣿⣽⣿⣿⡘⣿⣶⣽⣿⣿⣟⢿⣿⣶⣭⣛⡇⢀⣷⣿⢛⣿⣾⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡆⣿⣿⣿⣤⣶⣴⣷⣿⣿⡟⣱⣿⣿⣿⣿⣿⠛⣿⣹⣷⣿⣷⡘⣿⣿⣿⣿⣿⣿⣯⣻⢿⠏⢠⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡾⣿⣿⠇⣯⠉⠁⠀⠀⠉⢘⣛⠿⢿⡛⠻⠿⠷⣼⣿⠿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⡿⡣⠁⣸⣿⣿⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠋⠿⣿⢻⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⢲⣿⣿⢠⡾⣿⣿⣧⠘⣿⣿⣿⣿⣟⠝⠀⣰⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣤⣈⡌⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⣠⡞⣻⣿⡞⢛⠛⠛⠛⠧⣹⣿⡿⠟⠁⣠⣾⣿⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡁⣢⣤⣀⣀⠀⠀⠀⠒⠆⠻⣦⣼⣭⣿⣿⣷⠀⡀⠀⠅⠈⣀⣴⣾⣿⠿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣏⡛⢿⡟⠋⣿⠷⣆⣤⠾⠌⢀⣀⣤⣀⣈⣉⣉⣉⣋⣁⣩⣤⣴⣶⣿⣿⡿⠟⢡⣾⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣽⣻⢿⣵⡞⢳⣿⠟⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⢁⣄⣶⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣾⣏⣍⣉⡉⣉⣉⡉⣉⣁⣀⣀⣤⣤⣶⣾⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣶⣿⣭⣭⣉⣛⣛⣓⣛⣛⣻⣛⣛⣻⣿⣭⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣠⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⢀⣠⣴⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣣⣴⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣄⡀⢸⣿⡟⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣾⣿⣶⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡿⠃⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡿⠛⠉
⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣄⣠⣿⣿⣿⣿⠋
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣠⣴⣾⣿⣷⣶⣄
⠀⠀⠀⠙⠻⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠛⠛⠛⢻⣿⣿⣿⣿⠁⠈⠻⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠛⠛⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏⠀⠀⠀⠈⠉⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋


⣿⠿⠿⡿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠲⠌⠘⠃⠉⣩⢛⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡃⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠪⠅⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠉⠑⣄⠀⠀⠀⠀⢸⠋⠁⠒⠀⠀⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠀⠈⠇⠀⠀⠀⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣠⠜⠃⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠐⠛⠓⠒⠒⠦⠀⠀⠘⢦⡀⣇⠀⠀⠀⠀⠀⢀⣂⣀⣀⣀⠀⠀⠀⢀⣤⡀⠀⠀⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⠀⠀⠀⢤⣒⠄⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡀⡀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⡠⠠⠤⠤⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⢤⡤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣀⣠⠁⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢄⡀⠹⡈⠀⠈⠑⠣⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⠉⠢⠱⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠁⠊⠀⠀⠀⡈⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⢈⣁⣰⢤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠘⠤⠔⠀⠢⡀⢄⡘⢲⡄
⠀⣴⡋⠙⠲⡀⠘⠀⠉⠂⢄⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⢦⣱⠤⠃
⠀⠈⢷⠀⢀⣰⣧⡀⠀⠀⡦⠘⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠐⠝⢣⣕⣄⡰⠁⠀⠙⠂⢠⢤⡀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⠂⢈⠑⠒⠀⠚⠇⠰⠛⠛⠀⢵⡆⢀⠂⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢄⡀⠀⠈⢦⠀⠀⣿⣓⡰⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡑⣄⠃⠙⠤⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋
⠀⢀⣤⣤⡤⠤⢀⣰⣤⡀⠠⠄⢀⣤⡤⠄⣀⣤⣄⡀⠀⠀⠀⢠⣀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠀⠀⠀⢀⣠⡤⠖⠚⠛⠉⠉⠙⠛⠒⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⣀⣠⡤⠤⡤⠤⣤⣀⡀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⡼⠁⠀⢀⡴⠛⠉⠀⠀⠀⡇⠀⠀⠈⠙⠳⣄⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⢠⠟⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠳⣄⠀⠘⣧
⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⡞⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡴⠋⠀⠀⠀⠹⡆⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢸⠃⠀⠀⠀⠀⠀⠀⢀⣤⠧⣴⠋⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠸⣅⢀⡽⠒⠒⠒⠒⠒⠒⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠸⡇⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⣠⠤⠶⠶⡶⠶⠶⠶⣶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⢳⡀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣰
⢠⠏⠁⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⣰⠋
⡏⠀⠀⡄⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠙⠲⢤⣀⣀⡀⠀⣀⣀⣠⠴⠚⠁⠀⣠⠞⠁
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉⠉⠉⠉⠛⠦⣄⡀⠀⠀⠉⠉⠉⠉⠁⠀⠀⣀⡤⠞⠁
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠉⠛⠒⠶⠶⠶⠶⠖⠚⠋⠁⠀⠀⠀⢠
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⡴⠃⠀⠀⣠⣄⠀⣏⠀⠀⢹⠀⣠⣄⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⣰⠃⠈⠙⠃⠀⠀⠘⠋⠁⠘⣆⠀⠀⠘⡇⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⣀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⣤⠖⠋⠀⠀⠀⢿⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⢿⠀⠀⠀⣶⠋⠁⠀⠀⠀⠀⠀⠀⠈⠙⣶⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠘⣆⠀⠀⠘⣦⣠⠶⡇⠀⠀⣸⠶⣄⣴⠃⠀⠀⣰⠃⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠘⢦⠀⠀⠈⠀⠀⣧⣤⣤⣼⠀⠀⠁⠀⠀⣰⠃⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠁⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⡇⠀⠀⡇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢸
⠻⡄⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⢠
⠀⠙⠲⣤⣤⣤⣧⣄⣀⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣀⣄⣼⣤⣤⣤⠖⠋
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣷⡀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠙⢿⣿⣿⣧⠀⢀⠞⣁⣠⠏
⠀⠀⠀⠀⠀⠀⣰⣿⣧⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⠈⠋⠉
⠀⠀⠀⠀⠀⢠⣿⠇⣼⣿⣿⣿⡿⠟⠉⠁⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠀⠘⣿⣷⠀⢠⣴⠛⠛⢳⡄
⠀⠀⠀⠀⠀⣼⣿⣼⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣿⡆⠀⠈⠙⠓⠛
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢳⡀⠘⢛⠻⣆
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⣤⡿⠀⠀⠀⠀⠀⢀⣀
⣠⣤⣤⣄⡀⠘⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢰⡟⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⣰⠏⠉⢻⡶⢶⣄
⠳⣤⠤⠞⠋⠀⠹⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⢿⣾⡟⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢻⣶⠟⠛⢻⡏⠀⢠⠏⠀⢀⣿⣄
⠀⠀⠀⢀⡠⢤⠀⠙⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠞⠀⠀⠀⠀⠀⠀⠀⠸⣿⢷⡀⠀⠳⡖⢧⡀⣠⠊⠁⢉⡇
⠀⠀⣴⠁⣰⠇⠀⠀⠈⠻⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⣿⠘⣧⡀⠀⠁⠀⠙⠀⣠⡶⠋
⠀⠀⠈⠛⠁⠀⠀⠀⠀⢰⠏⡤⠦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⢰⡏⣠⠟⠁⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠿⢿⣿⡿⠆⠀⢀⣾⠟⠁⠀⠀⢠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⠞⠉⠁⠀⠀⣠⣾⡁⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢾⡷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣠⡾⠋⠀⠻⣆⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⠶⣴⣿⠀⠈⢷⡈⠙⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⢞⣻⠟⠀⠈⠲⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢘⣧⣀⣀⣀⣸⣇⠀⠀⢿⣤⣤⣤⡼⠿⣿⡛⠛⠛⠛⠻⠿⠿⠶⠛⠋⠁⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⠀⠠⡟⠁⠈⠉⠉⣿⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠛⣶⠖⠒⠚⠛⠀⠀⢀⣤⣤⣤⣤⣤⡤⠶⠿⠖⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠿⣤⣤⠶⠞⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠾⠛⠛⣯⠉⠉⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣀⣠⣴⠶⠛⠋⠉⠉⠀⠀⠀⠀⣿⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣍⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠟⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠉⠛⢶⡶⠶⠶⠶⢾⠋⠉⠀⠀⠀⣿⠀⠀⠀⢸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠏⠛⢷⠶⠶⠶⠾⠿⠒⠶⠶⣿⠀⠀⠀⠀⠀⢸⠟⠛⠛⢻⣡⣏⢼⠟⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢰⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣿⡄⠠⠄⠤⠠⠤⠶⠶⠄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢀⣇⢹⠶⢦⣤⣤⣤⣤⡴⠟⠀⠀⠀⠀⠀⠀⠙⠶⠶⠖⠚⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⡺⠼⣦⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⣠⡿⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣾⣿⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠛⠛⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣾⣶⣿⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠄⠤⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠞⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⢀⣀⠀⠠⠀⠂⠉⠀⠀⠀⠈⠂⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⢀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⢰⠀⠀⠀⠀⠀⠀⠀⣴⣶⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⡄⡨⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣈⠳⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠀⠈⠛⠃⠀⠀⠀⠀⠀⣀⠰⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⢄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡠⠂⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠊⠁⠀⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠔⠈⠴⠁⠀⠀⠀⠀⠀⠁⠀⠀⠰⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠐⢶⢰⣶⠖⠠⣂⠀⠐⠀⠀⠘⢏⠁⠀⠀⡲⠊⠉⠁⢀⡀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡿⠀⠀⠈⢠⠁⠀⠀⠀⠀⠑⠤⠊⠀⠀⠀⠀⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀⠀⢰⡷⠀⠀⠀⢰⡷⠀⠀⠀⠀⠠⠁⠀⡤⠑⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠰⡷⠀⠀⠰⡷⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠃⠀⠀⠀⠀⢾⠆⠀⠀⠀⠀⣇⠀⠰⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠂⠀⠀⠀⡸⠅⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠃⠀⠀⠀⠀⡈⠠⢄⠀⠀⠈⠀⠀⢀⣠⡀⠀⠐⠀⠨⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⡀⣀⡀⡤⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠉⠛⠙⠓⠋⠁⠀⠈⠀⠤⣀⣀⠄⠁
⠀⠀⠀⠀⠀⢀⢠⠄⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛
⠀⠀⠀⠀⠂⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⡀
⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢠⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⢣⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠶⢀
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠸⠃⠀⠸⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⡸⠀⠀⠀⠃⠀⠀⢰⠀⠀⠀⠀⠁⠀⠁
⠄⠀⠀⠀⠐⠂⠁⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠁⢥⢠⡄⠠⠇⠀⠀⠈⠑⠾⠖⠂⠀⠀⠠⡀
⠀⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣿⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⠿⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⡄⠀⠀⣀⣠⣤⣤⣶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣈⣉⣉⠉⠉⠉⢁⣴⠾⠋⣉⣤⣤⣤⣤⣤⣉⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⡀⠀⣰⡟⢁⣴⠿⠋⠉⠀⣿⠀⠉⠙⠻⣦⡈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⠉⠁⣸⡟⢠⡿⠃⠀⠀⠀⠀⣿⠀⢀⣴⠄⠈⢿⡄⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⣿⠁⣾⠃⠀⠀⠀⠀⣠⣿⣴⠟⠁⠀⠀⠘⣿⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⠀⣿⠀⣿⠀⠀⠀⠀⠀⢿⣤⡿⠛⠛⠛⠛⠛⣿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⢿⡆⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⣿⠉⠉⢹⡏⠉⠉⠉⠉⠁⠀⠈⢿⣄⠹⣧⣄⠀⠀⠀⠀⠀⠀⠀⣀⣼⠟⢠⡾⠃
⠀⠀⠀⠀⠀⠀⢰⣿⠀⣶⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⢀⡀⠀⠈⠻⣦⣌⠙⠿⢶⣤⣤⣤⡶⠾⠋⣁⣴⠟⠁⣶⡆
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⣠⡾⠟⠁⢀⣀⣀⡈⠙⠻⢶⣦⣤⣤⣤⣴⡶⠟⠋⠁⣤⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⣼⠟⠀⣴⣤⣿⡏⢹⣿⣤⣦⠀⠀⢀⠀⢀⡄⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⢸⡟⠀⣼⣏⡈⠛⠃⠘⠛⢁⣹⣧⠀⢻⡇⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⢸⡇⠀⣤⡿⠟⠃⠀⠀⠘⠻⢿⣤⠀⢸⡇⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠈⣿⡀⠘⢷⣶⣿⡇⢸⣿⣶⡾⠃⢀⣿⠁⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⠈⢿⣄⡈⠀⠻⠿⠿⠟⠀⢁⣠⡿⠃⠀⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⣿⠀⠀⢸⡇⠀⠀⠀⠙⠻⠶⣦⣤⣤⣴⠶⠟⠉⠀⠀⠀⢸⡇⠀⠀⣿⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣄⠉⠀⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠉⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃
⠀⠀⠀⢠⡀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢰⠒
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠂⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠈⠓⠦⡀⠠⠀⠀⠀⠀⠒⠤⠌⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⣀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠁⠀⠀⢘⠀⠀⠂⠈⠄⠄⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠰⠀⠀⠰⢦⡀⠀⠠⠀⠀⠑⠤⢠⠁⡀⠃⠀⠘⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠑⠄⠈⢦
⠀⠀⠀⠀⠂⠀⠀⠀⢘⠀⠀⠈⠒⠀⠂⠀⠉⠀⠀⠑⢄⡀
⠾⠇⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⢠⠬⠲⣤⠢⠦⢂⣤⣄⠀⠀⠀⠈⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡟⠿⠃⠀⠀⠠⠏⠀⠀⠈⠛⠃⠈⠀⠀⠀⠀⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢆⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠢⠄⠀⠒⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢠⠀⠀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⢃⠀⠀⠸⠀⠀⠀⠈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠢⡑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡐⠢⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡁⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣊⣂⣀⣀⣀⣀⣀⣀⣀⣆⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣂⣀⣀
⣿⣿⡟⠙⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡠⡀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠽⢠⡆⠀⠀⠊⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡸⠊⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⡀⠀⠀⠀⠀⠀⢀⡀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⣰⠀⠀⠀⠀⣠⡊⠭⠊⠀⡆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡌⠀⡰⠉⢆⠈⢦⠊⠁⠀⠀⠀⢠⠃
⠀⠀⠀⠪⡐⡡⠀⢆⠀⠀⠀⠈⠈⠀⠀⠈⠢⡀⠀⠀⠀⣀⢴⣉⣈⡈⠐⠂⢄⡀
⠀⠀⠀⠡⠱⢴⡖⣸⡑⠒⠀⠰⠄⠀⡀⢀⡰⠚⠉⠉⠁⠀⠀⢃⠀⠀⠈⠑⠠⣈⠢⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠈⠉⠉⠁⠀⡄⠉⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠑⢤⠃⠀⠀⠀⢄⠙⠄⠀⠀⠀⢀⣀⠠⠄⠂⠒⠒⠂⢄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⢸⠐⢍⢧⡈⡢⠝⠀⠀⠀⠉⠉
⠦⡀⠀⠀⠀⢸⠁⢠⠀⠑⠤⣀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠨⣀⡄⠀⠹⡄
⠐⡌⠠⡀⠀⡜⠀⠘⠣⡀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢄⠀⠸⠤⠤⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠈⠢⣈⠲⠀⢀⠃⠀⠈⠑⠄⣀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢣⠠⡇⠀⠀⠀⠈⠢⢄⡀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⢑⠑⡼⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⠠⠀⢀⢀⣀⣀⣀⠔⠠⡀⠀⠀⠄⡈⢳⡈⠢⡀⠀⠀⠀⠀⠀⠉⠁⠐⠒⠀⠉
⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠢⣤⢀⠀⠀⢣⡀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠊⠀⠀⠀⠱⣀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡀
⡀⠀⡆⠐⠀⢤⡀⠀⠠⠐⠀⢄⠁⠀⠀⢀⠀⣀⣀⠀⠈⠀⠐⡀⢤⠀⠀⠂⣀⠀⠠⡆⢀⣀⠦⣀⢀⠠⠐⠀⠤⠁⠀⠀⢀⠀⣀⣢⠀⢎⠀⡶⣔⢧⠀⠀⠂⢄⠠
⣆⡀⠠⡀⠉⣶⠬⢀⠀⠰⢃⣤⠗⠺⠘⣀⠘⣏⣩⣀⢡⠤⡌⢠⡞⡒⠄⠨⡍⠃⡰⠠⠀⡙⣒⠥⢂⠂⢰⠆⢰⠟⠲⠀⠀⠚⣏⣭⡀⢡⣴⡯⢰⣎⣐⠄⡠⠅⢰
⢿⡁⢠⣎⡳⠎⠠⠟⠵⣎⠀⣛⠶⢌⠲⠚⢀⠷⢠⡕⠲⡖⣓⠸⢿⡄⢨⡅⠄⢶⡀⠀⣌⡩⠞⠰⠞⡵⣊⠄⣋⠄⢙⠢⡜⠒⠦⠠⣕⠲⣿⣇⠀⠿⡏⢩⡴⠦⢤
⠏⠁⢛⣁⠁⢥⣜⠂⠰⢾⡻⢟⡀⠑⠔⣀⡭⠄⡫⠬⡁⢲⡌⢳⡬⢶⡉⠛⣽⠆⠁⣚⣅⠢⠅⣘⠃⠀⢬⠭⠗⣠⠉⠄⣀⡼⠄⡋⠬⣝⢒⡍⣳⡌⢶⡉⠛⣤⡻
⣥⣄⠛⣈⢷⡀⠀⠿⣄⠈⣝⠂⣠⠌⢩⣄⠘⣦⢈⡐⠃⠈⢶⣯⡑⠢⢌⡦⢘⣣⣄⠙⣌⢷⣆⠐⠋⠐⢂⣹⠃⡠⠍⠠⡄⠘⣧⠈⠐⠓⢈⣰⣋⡙⠲⢀⣶⠌⠙
⣳⣤⠀⠩⢄⠑⢾⠀⠸⣏⠁⢀⡎⠁⠙⢢⡄⠩⢉⡈⠠⣄⡉⠶⠞⢒⡒⣉⡘⣷⣀⡀⠬⠄⠑⢦⡔⠢⣌⠁⢰⣎⡉⠉⢲⡄⠩⠉⡛⡁⠻⢦⡀⠊⢁⣀⠈⠴⠆
⡀⠾⢧⡀⢀⡀⢹⠧⢤⡉⡛⢠⣄⠛⠳⠍⡁⠓⣎⣱⡂⠀⡁⠴⠫⡤⠉⠻⣀⡼⠋⢡⣀⠀⠀⢙⠷⢤⡉⠛⢢⣄⠙⠫⠉⢉⠛⣏⠉⢀⢠⡴⠛⠃⠶⣪⡉⣡⡤
⣶⡂⢻⠖⢠⡄⠘⢷⣄⠀⠠⣴⣛⠒⢤⠠⢮⣉⡡⠤⠎⣉⢷⡖⠲⢈⣃⠀⢨⣿⠋⣹⣇⣀⡄⠘⢳⣤⠀⠠⢴⡚⢓⣦⡰⠮⣉⡰⠦⠤⣈⢳⡆⠐⢨⡉⣉⣱⣦
⠴⣷⡐⡮⢅⠈⠥⢼⣉⠀⠼⣦⠀⢼⡷⠶⠀⣿⠀⠀⢶⡀⠈⠉⡷⠚⠻⣃⣀⠠⣷⣀⡈⢥⡁⠴⠼⣍⠄⠼⣧⠠⠼⠷⠒⠀⣿⠀⠀⢴⣈⠈⢉⡽⠚⠓⠂⢲⣉
⠾⠫⢄⣀⣠⠙⠶⠢⣾⡀⠰⠷⢠⡴⠖⢂⠞⠓⠰⣶⠚⢁⠈⠙⣳⠶⣌⣉⣁⠀⠾⢄⣀⣠⠙⠓⠲⣾⡀⠐⠿⢠⡴⠒⢲⠚⠛⠰⣶⡊⣍⠈⠙⣳⠶⣬⣀⣀⠿
⡶⢂⣁⣳⣄⡄⠒⠚⠉⢡⠓⠒⢒⠃⠀⡻⠶⠒⡲⠞⠡⣜⠿⡈⠀⠐⣀⠍⠩⣷⡀⠀⢁⠤⢀⣀⠛⠉⢀⠵⠆⠤⠷⣈⠻⠒⣾⠀⣈⠉⠘⠋⣬⠡⣀⣤⢼⢆⡰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⠀⣴⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⡄⣰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣬⣹⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠈⠉⠉⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣶⠄⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⣿⣿⣏⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣏⣀⣀⣀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠉⠉⠉⣿⠋⠉⠉⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡀⠀⠀⠈⣹⠋⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⠐⢛⠇⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣄⠀⣠⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⡖⠆⣠⠀⠤⠀⠂⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠘⠀⠘⠀⠀⢉⡓⠓⠛⠚⠒⠐⠒⠀⠀⢀⡁⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢙⢋⡉⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⣐⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⢼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⣀⣀⠀⡀⡀⠀⠐⣐⣬⠽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⡀⠟⡉⠋⠌⠀⠀⠀⠴⣆⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠻⣶⣳⠂⠀⠸⣿⡇⠀⠀⠐⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⡀⠀⢀⣿⣬⡴⠔⠒⣶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠀⠈⠀⠂⠀⠂⡀⠄⢄⢀⠀⣢⣤⣭⠵⠦⠐⡭⠀⠀⡐⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⠂⠀⢱⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠇⠀⠀⡀⢀⠀⠀⢈⡄⠠⡘⠐⣼⣭⢽⠓⢠⡬⣒⡄⠀⠠⠈⠄⡄⣀⠠⠤⠔⠠⠄⡀⠀⠀⠀⠄⠀⠀⢀⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠁⠃⠀⢠⠘⠀⠀⠀⠀⣤⠔⠂⠐⠤⣿⣿⡄⠀⠀⣄⠀⢂⡄⠀⠀⠀⠁⠀⠜⠀⠀⢴⢆⠤⡀⠈
⠀⠀⠀⠀⠠⠐⠀⠀⠁⠀⣀⠀⠀⠈⣐⡈⠪⠀⠢⠐⠂⠠⠀⠨⢓⣤⠤⠲⣿⢟⠘⣦⢀⠀⢀⠄⠑⠀⠀⠀⢐⠀⠀⠀⠀⡌⢀⡄⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢄⠔⠡⠀⡢⣀⠳⡀⠒⢬⠤⢜⣯⣀⠸⣴⠯⠷⢻⠄⠂⠔⠀⠀⠀⠀⠀⠀⡁⠀⠀⡒⢴⠂⠴⠉⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠶⠤⡠⠊⡀⠀⡀⠀⠋⠐⠨⠛⢶⣶⣬⠻⢯⣞⣭⣑⢈⠔⠧⠄⠀⠀⠰⠆⠀⠀⠀⢙⡄⡐⣣⠈⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢇⡀⢀⠀⣤⣤⠀⠁⠀⠀⠀⢀⠀⠀⠉⠱⡤⠈⠹⢷⣶⣾⠠⠂⡴⠀⠀⠂⠀⣠⢨⠤⠣⠐⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢤⠀⠀⠦⠘⠀⠆⡀⠀⠀⠈⠉⠙⠛⠁⠢⠊⠁⠀⠐⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣿⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⣠⣿⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⣰⣿⣿⣱⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠈⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡄⠀⢸⣿⣿⡏
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠸⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢰⡟⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⢻⡄⠀⣿⣿⣀
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢻⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣾⠇⠀⢉⣍⢻
⠀⠀⠀⠀⢠⣾⠟⠛⠛⠿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠛⠋⠀⠀⢀⡀⣠
⠀⠀⠀⢠⣿⠁⠰⠛⠷⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛
⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⡼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠻⣦⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠟⠛⣿⠿⢶⣄⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⢹⡆⠀⢻⡄⠈⣿⡄⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡀⠸⣇⠀⢻⡇⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠈⠁⠀⠀⠀⢸⡿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠉⠉⠛⠛⠿⣷⣶⣶⣶⣾⡟⠀⠀⠀⠀⠀⣠⡿⠃⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢻⣦⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣷⣄⠀⠀⠀⠸⣿⠁⣀⣤⡾⠟⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⢿⡆⠀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠈⠻⣷⡀⠀⠀⠙⢿⣏⠁⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠃⠀⠀⠀⠀⣸⣿⣾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠙⠧⠀⠀⠀⠸⣷
⢀⣤⣤⣤⣤⣤⣤⣶⡾⠋⠀⠀⠀⠀⣠⡾⠋⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠋⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢀⣿⠃
⠻⣦⣀⣀⣀⣀⣀⣀⠀⠀⠀⠸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣤⣤⣤⠾⠁
⠀⠸⣿⡉⠉⢉⣽⡿⣧⡀⠀⠀⢹⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⠀⠈⠛⠛⠛⠉⠀⠹⢷⣤⣶⠿⠃⠀⠀⢸⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠋⢙⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⣿⢻⣿⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⣤⣤⣤⣤⣴⣿⠸⣿⣤⣤⣤⣤⣤⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢨⣿⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡚⠻⡏⢻⡿⠿⠿⠿⢿⡏⠀⣿⡿⠿⠿⠿⣾⠏⣹⠟⢲⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠈⠱⡛⠖⠛⠚⠁⠀⠀⠀⠘⣿⢀⣿⠀⠀⠀⠀⠘⠓⠛⠒⢿⠋⠉⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠓⠒⠒⠒⠛⠒⠒⠒⠒⠒⠒⠒⠚⣿⢸⣿⠒⠒⠒⠒⠒⠒⠒⠒⠛⠒⠒⠒⠚⣿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤
⣿⣿⣿⡿⠟⠻⠿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣏⣠⣄⣠⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⠿⠛⠛⠛⡛⢛⠟⠿⠿⢿⣿⣷⣦⣤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣯⣽⠿⢛⣻⣯⣷⠾⠗⢛⡛⡛⠛⠛⠛⠛⠛⠷⠶⣦⣌⠉⠛⠷⣧⣗⣠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⣫⣴⠞⢛⣉⣥⡶⠖⠛⠛⠉⠋⠀⠀⠀⣛⠒⠒⠶⢤⣌⡙⠻⢦⣤⣉⠻⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢆⣽⡿⢋⣴⠟⣋⣤⣶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠑⠲⠿⢶⣦⡈⠙⠷⣀⡙⠿⢇⣄
⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⠟⣁⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠈⠻⣦⡈⢻⣧⡀
⠀⠀⠀⠀⣀⣾⠟⣰⡟⢁⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠈⢣⣄⠈⢿⣆
⠀⠀⢀⣾⡿⠋⣼⠏⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⠙⢦⠀⢹⣦
⠀⠀⣼⣿⢃⣾⢋⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠘⣷⡀⢿⣆
⠀⠐⣿⠇⣸⡇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠘⣧⠈⣻⡆
⠀⣴⡟⢸⡟⡸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⡄⢿⡆⠈
⢰⣿⡇⣼⢃⡇⢠⣤⣤⢠⣤⣤⢀⣤⣤⣤⣤⡀⠐⠀⣀⣤⣤⣀⠀⢀⣀⣀⣀⠀⠀⣀⡀⢠⣤⡄⣤⣤⣤⣤⣬⡀⣤⠤⡄⢸⣇⠀⣶⠀⢦
⣶⠟⠀⣿⣸⠁⢸⣿⣿⢸⣿⣿⢸⣿⣿⠛⣿⣿⠀⠘⣿⡟⣿⣿⡇⢸⣿⣿⠿⠂⣤⣿⣧⢸⣿⡇⠛⢿⣿⣿⠛⠃⢸⣯⡅⠀⣿⠀⣿⠀⢺
⣿⡆⠸⣯⢹⠀⢸⣿⣿⢸⣿⣿⠀⣿⣿⣶⣿⡟⠀⣿⣿⡇⣛⣛⡃⢸⣿⣿⣤⠀⣿⣿⣿⣿⣿⣇⠀⢰⣾⣿⠀⠀⢸⣿⠀⠀⣿⠀⣿⡀⢸
⢻⠟⠀⣿⢸⠀⢸⣿⣿⢸⣿⣿⡄⣿⠿⡯⣿⢿⡆⣿⣿⡟⣿⣿⡇⢸⣿⣿⠉⠀⣿⣿⡿⣿⣿⡇⠀⢸⣿⣏⠀⠀⢸⣿⠀⠀⣿⠀⣿⠁⣸
⠈⢶⡆⣿⡘⡇⠘⢿⣿⣾⣿⡿⠁⣿⣷⡄⣿⣿⡀⢿⣿⣧⣿⣿⡇⢸⣿⣿⣶⡆⣿⣿⠇⢿⣿⡇⠀⢰⣿⣿⠀⠀⢰⣶⡆⠀⣿⢀⡿⠀⣿
⠀⠀⣄⢸⣧⢻⡀⠀⠉⠉⠉⠀⠀⠉⠈⠁⠉⠁⠀⠀⠉⠉⠁⠉⠀⠈⠉⠉⠉⠁⠉⠉⠀⠈⠉⠁⠀⠈⠉⠉⠀⠀⠀⠉⠁⢸⡏⣸⠃⢰⣿
⠀⠀⠈⡄⢿⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢠⡏⠀⣾⡏
⠀⠀⠀⣭⡈⣷⡌⠳⠈⠁⠠⠄⡥⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡾⠁⣼⡿
⠀⠀⠀⠈⣷⡈⢻⡄⠀⠀⠘⠀⠀⠀⠐⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⡾⠃⣰⠿⠁
⠀⠀⠀⠀⠹⣷⣄⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣰⠟⠁⣼⠃
⠀⠀⠀⠀⠀⠙⣻⣧⡘⢳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⠊⣠⡾⠉⣰⣾
⠀⠀⠀⠀⠀⠀⠈⠉⠻⢦⡙⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⠀⠀⠉⠚⢉⡤⠾⠋⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣄⠘⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠀⠀⣀⣤⠾⠋⢁⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣬⣙⠛⠿⢶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣤⣴⠶⠞⠛⢋⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣿⣷⣶⣤⣬⣍⣉⣉⣉⣉⣋⣉⣉⣥⣤⠶⠖⠛⠉⠁

⠀⠀⠀⡄
⣤⣶⣴⡇⢠⠄
⣿⣿⣿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡴⣦⣅⡀
⣿⣿⣿⣿⣿⣿⣿⣇⣸⣇⣀⣀⠀⣀⠀⠀⠀⠀⠀⣴⣿⡦⠄⠀⠀⠀⠀⠀⢀⡔⠁⣾⢃⠈⠙⢿⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⡀⢀⡶⠀⠈⠉
⣿⣿⣿⡿⢟⣻⠟⠉⠉⠉⠉⡍⠉⠉⠉⠉⠉⠉⣿⣿⣷⡀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠹⣼⡄⠀⠀⠙⢿⣷⣦⣶⣶⣿⣿⣶⣿⣿⣿⣿⡿⡟⠁
⣍⣉⣥⠀⠈⠉⠀⠀⠀⠀⠄⢀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣇⠀⠀⠀⢠⡞⠁⠀⠀⣀⠈⣭⢿⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⡀⠀⠀⠀⠶⠶⠖⠛
⣿⣷⡁⠀⠘⠉⠎⠰⣷⣶⣿⣿⡟⣿⢋⣍⣀⠀⠀⠀⣿⣿⣤⣤⣴⠋⠀⠀⠈⠙⣽⡄⢸⣾⣿⣿⣿⣿⣱⣿⣿⡄⠙⠿⠿⣿⣿⣿⠿⣿⠀⠀⠀⣀⣀⣠⣤⣤⣴
⣿⣯⣥⣰⣿⡾⣾⣿⣶⣿⣿⣿⣿⣼⠟⠻⣿⣧⡟⣧⢻⡿⣿⣿⣿⡀⠀⠀⠀⠀⢷⢷⡈⣿⣿⡫⠿⢿⣿⣿⣿⣿⣴⣧⣄⣸⣶⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⣽⣿⣟⣿⣿⣿⢈⣻⡻⠿⠿⠿⠿⠿⠶⠂⠛⠋⣈⡰⠶⣼⢋⡾⣏⣱⣶⣴⠠⡆⠨⡘⠧⢛⣩⠔⠀⠀⣉⠘⠚⠻⢛⣋⣉⣹⡿⢻⣿⣿⣿⣿⡿⢾⣿⠋
⣿⣿⣿⣛⡛⣫⠏⠉⠻⣿⣿⠿⠿⣫⠈⠁⣿⣻⠿⠉⣠⣿⡿⡽⠙⣿⣿⣿⠛⠿⠿⢿⡞⠗⠬⣷⣾⡏⢻⣗⠀⠀⢛⢛⣯⣿⣿⣿⣿⠟⣡⣾⣿⣆
⢿⢿⣿⣽⣞⣯⡃⠀⠴⠆⢠⣄⣰⡎⠚⢉⡩⢀⣶⡆⠿⠟⠀⠀⠀⠋⠛⠁⠀⠐⠠⠤⢯⡾⣴⣿⣥⣾⣿⣿⣆⣀⣼⣏⠁⣙⡛⠙⢃⣼⣿⣿⣿⣿⣿⣦⣄⡀⢀
⣿⡿⣿⣯⢟⠃⢹⡇⣤⡄⠸⠿⣘⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⢀⣀⡀⢠⣤⡍⣡⢟⠻⣿⡿⣿⡿⠋⢽⣿⡳⢀⣴⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣾⢿⡝⢿⠵⠯⠛⡉⠀⠀⠉⠀⣀⣠⡤⠴⠖⠉⠙⠂⠙⠛⠳⢀⠀⢠⣿⣿⡟⢿⣿⡷⣀⠞⡰⠡⣀⣾⡀⠟⠀⣠⠌⢀⣰⣿⣿⢋⠟⠧⠀⢉⣽⣿⣿⣿⣿⣿
⣧⡀⠖⠃⠀⣈⣀⡤⠤⠒⠒⣈⡉⣥⡤⠖⠒⠚⠁⠀⠀⠐⠂⠀⠀⠀⠀⠻⡿⣿⣦⠙⠰⢋⢒⢖⠕⣛⡛⠁⢀⣶⣾⣟⣿⠿⠟⣱⣿⣿⣔⠦⡬⣿⣿⣿⣿⣿⣿
⣿⣩⠄⠘⠉⠁⠀⠀⢰⣿⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠲⠚⠑⠋⠀⠀⠀⠀⠀⢈⣚⡡⠛⠁⢰⡿⣫⣿⣿⣿⢧⣾⣿⣿⣿⣿⣇⣒⠾⣿⣿⣿⣿⣿
⣌⣿⢓⣖⣦⠔⠂⠀⠿⠿⠿⠟⠋⢉⠄⠒⠀⡀⢀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇⢘⡴⢋⠼⣯⡝⠛⡿⣫⣟⢿⡌⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿
⣿⣧⣼⣿⡖⣛⡻⠀⠀⠀⠀⣨⡅⠀⠀⠀⠀⠀⠀⠀⠒⢸⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣏⠈⣿⣿⠗⡀⣠⠎⠸⠟⠃⠠⢄⡹⢿⡿⣙⢻⣿⣿⣿⣟⠻
⠿⠿⠓⠀⠁⠀⠀⠀⠀⠀⠉⠉⠿⠿⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠿⠀⣿⠿⠛⠉⠁⠀⠀⠀⣀⡐⠌⠁⢤⠙⢿⣆⠻⣿⠿⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⣠⣤⡤⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⣻⣿⣿⠋⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⠛⠃⠀⠀⠀⠀⠀⠀⢰⣶⣶⡆⢠⣿⣿⣿⣿⣿⣿⣶⣤⣤⡄⢀⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣧⣿⣿⣿⣿⣿⣟⠛⠋⠉⠹⠿⠁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣦⠶⠶⠞⠛⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⠏⠁⠀⠀⠀⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣥⠀⠀⠀⣴⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⠗⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣿⣿⣿⣶⡾⠛⠄⡈⢻⣿⠛⠛⠿⣿⣿⣧⣀⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⢻⣿⣿⣿⣿⣶⣄⠙⢺⣿⣀⣀⣀⣠⣿⣿⣿⣦⡀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⣿⣿⣷⣻⣟⡛⣿⣿⣿⣿⠋⢹⣿⣿⣦⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣷⡜⣿⣿⣅⡢⠀⢿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢻⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣷⣤⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠏⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⣿⣦⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⡠⡀
⠀⠀⠀⠀⠀⠀⣴⣶⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠄⠀⣰⣫⡤⠋
⠀⠀⠀⠀⠀⢸⣿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠙⢿⣿⣿⡄⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠙⣿⣿⡀⠀⠙⠲⠤⠞
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠂⠘⣿⣇⡀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠘⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⠄⠀⣋⣿⣿⣿⣿⣀
⣎⣉⣱⠆⠀⢻⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⢰⠓⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⣿⣿⣿⣿⣿⠟⠻⣦⢤⣄
⠀⠀⠀⡀⣄⠈⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠘⠷⡟⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢹⣿⠉⠙⡏⢀⡜⠁⢀⠿⣄
⠀⠰⣏⡠⠃⢀⣬⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⣁⣀⡀⠀⠀⠀⠈⣿⣷⠀⠈⠏⠣⡔⠁⣀⡾
⠀⠀⠀⠀⢀⣿⣿⣿⣿⢛⣉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣷⡀⠀⠀⢠⡟⣻⠇⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠸⣿⣿⣿⣯⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢿⡿⢿⣿⡷⠀⠀⣼⠟⠁⠀⢀⡞⠉⠁
⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠖⠉⠀⠀⣠⡾⣅⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⠋⢻⣿⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⠉⢀⣈⡷⠋
⠀⠀⠀⠀⠀⠀⠰⣏⠀⠉⠙⡆⠀⢳⣄⣈⣙⣷⣶⡦⠤⠴⢶⣶⣾⡯⠷⠛⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣾⠉⠉⠉⢻⠉⠀⠀⠉⠉⠀⠀⠘⡇⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠉⡿⠊⠉⢉⣀⣀⡴⠶⠶⠶⠶⠖⠛⠛⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⠞⠛⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣀⣤⠴⠖⠛⠉⠉⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣥⣀⠀⠀⠀⣀⣀⣤⠴⠚⢻⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⠦⣤⣀⣀⣉⣿⠛⠋⢹⡇⠀⠀⠀⢸⣧⣤⣤⡟⡴⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠘⠀⢸⠅⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢸⡇⣀⠀⣉⣁⠽⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡄⢰⣙⠶⠤⣤⣤⣤⠴⠋⠀⠀⠀⠀⠘⠷⠶⠖⠒⠒⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠓⢳⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⠞⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠙⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠞⠁⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⢠⡶⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣄⡴⠋⠉⠉⠛⢶⣄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠂⠉⠁⠀⣠⡟⠁⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡤⠖⠛⠀⠀⠀⠀⠀⢀⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⢿⡄⣀⠔⠊⠁⠀⠀⠀⠀⣀⣠⡴⠞⠉⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿
⠀⢀⣀⣀⣸⣿⣁⢀⣀⣀⣤⠴⠖⠛⠉⠁⣀⣤⠴⠖⠚⠛⠋⠉⠉⠉⠁⠀⠀⠀⣴⠏
⢰⠟⠁⠀⠈⠉⠉⠉⠉⠀⠀⢀⣠⡴⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⣿⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⣿⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋⠀⠀⠀⠀⢀⣀⡀⠀⢀⣀⣀
⢹⡆⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠶⡟⠋⠁⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣄⡀
⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠞⠛⠉⠙⠈⠁⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⠂⡡
⠀⠀⠀⠉⠛⠛⠓⠚⠛⠛⠛⠉⠁⠀⠀⠀⢀⣀⣀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡟⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⠈⠉⠙⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⢀⣿⣿⣿⣿⠟⠁⣼⠿⠟⣫⡿⠿⠛⠁⠀⠰⡏⠀⠀⠀⠀⠈⠻⣿⣿⡇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠈⠛⠄⠀⠀⢹⣿⠇⠀⠱⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⠀⢠⡀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠏⠀⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣯⠈⢣⠀⠀⠀⠀⢀⣏⠈⠇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠠⡄⠀⠀⠼⠀⢀⣾⣿⣿⣿⣿⣿⠋⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⠀⠀⠀⠣⠤⠃⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⢶⣄⣀⣤⣾⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠟⠛⠋⠉⠀⠀⠤⠭⠋⠀⠀⠀⢻⣇⠀⣀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡆⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠛⢦⡈⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⠀⢀⡼⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⢰⡟⠋⠉⠉⠀⢙⠁⠀⠀⢀⣴⠏⠀⢻⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠉⠻⣦⣀⠀⠀⠀⠀⠦⠄⠒⠀⠀⠈⠀⢀⣴⠟⠁⠀⠀⢀⡙⠓⠰⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠁⠀⠀⠉⠒⢤⡙⠻⠶⣤⣤⣀⣀⣀⣀⣠⣴⠾⢿⠁⠀⠀⠀⠀⣸⠋⠁⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢯⡈⢢⡤⠚⠃⠄⠀⠀⠈⠲⣄⣀⣈⣉⣉⣉⠟⠛⣤⠠⣄⠁⠀⠀⠀⠀⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠀⠀⠀⢹⠃⠀⠀⠀⠀⠈⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⠠⠴⠦⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠤⠞⠚⠀⠀⠀⠀⠀⢸⡇⠀⠀⡀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣠⡶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⢦⠀⠀⠀⠐⣿⠙⠛⠿⠿⠷⣶⣶⡶⢿⡀⠀⢨⠛⣿⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⠈⢳⡄⠀⢠⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣠⡾⠃⠀⠈⠉⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⣄⢀⣤⠿⡶⠋⠀⠀⣠⠶⠷⠶⠶⠶⠶⠶⠟⠛⠉⠙⠛⣶⣖⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠘⢳⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⣴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⢤⣸⡇
⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣀⣀⣀⣀⣀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣶⣶⣶⢶⣶⠶⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣷⣿⣿⣿⣿⣿⣷⣾⣶⣷⣾⣿⣾⣶⣾⣷⣷⣾⣶⣶
⣿⣿⣶⣷⣷⣾⣶⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠖⠲⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢠⢆⡴⣄⣤⠀⢀⣸⣁⢀⢔⣤⠀⠀⠀⠀⣴⡢⡀⣈⣇⡀⠀⣄⣠⢦⡴⡄
⠀⠀⠀⠀⠀⠀⢯⢻⡸⠧⠏⡜⠀⣯⡄⢸⠈⠁⣀⠀⠀⠀⠀⣀⠈⠁⡇⣠⣷⠀⢣⠻⡼⣧⡟⡽
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠚⡩⠞⣇⣻⠸⣄⣉⣁⠐⠂⠐⠀⣈⣉⣠⠃⣟⣸⠲⢝⠓⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⡟⠒⠒⢺⠀⠀⠈⣹⠀⠀⠀⠈⠛⠒⠒⠛⠁⠀⠀⠀⡏⠀⠀⠀⡗⠒⠒⢻
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠦⠤⠤⢾⠀⠀⠀⣶⣿⣶⣶⣿⣦⠀⠀⢀⡷⠤⠤⠴⠇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⢰⠃⠳⣄⡀⠻⠿⠛⠛⠿⠟⢀⣠⠎⠱⡀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠤⠤⠤⡆⠀⠀⠀⠉⠓⠲⠶⠶⠖⠚⠉⠀⠀⠀⢠⠤⠤⠤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠔⠤⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠄⢬⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢔⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠄⢸⡀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠉⠓⠒⠶⠶⠤⠼⠦⠤⠶⠶⠒⠚⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢲⡖⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣀⣀⡀⠀⢸⡇⠀⢀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠉⠉⠉⢹⡏⠉⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⠢⡁⠀⠀⠀⠀⣀⠀⠀⠀⠈⡉⠰⠄⢀⡰⡈⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⡀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⢦⢫⠻
⢀⡀⠄⠂⠀⠀⠀⠀⣀⠐⡄⠀⡅⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠁⠀⠀⠀⠀⢚⣛⡧⢄⠐⢀⠀⢀⠀⠙
⠈⢆⠀⠀⠀⠀⠀⠀⢘⡂⠱⡀⠘⠦⢤⣀⣀⡤⠤⠀⠀⠚⠀⠀⠀⠀⠀⠴⠂⠒⠤⠓⠊
⠀⠘⠀⠀⠀⠀⠀⡀⠤⢈⣂⣱⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⡀⢘⠧⡦⠚⠁⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠂⠀⠀⠀⠣⡡⣀⣠⣄⠀⣀⡤⢠⠎⠀⠀⠀⠀⠀⢀⡌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡦⢃⣀⠉⡩⠙⢧⠀⠀⠀⠀⠀⢠⠮⠀⠈⠠⠃
⠀⠀⡠⡀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠃⡀⠀⢠⠀⠀⠀⠠⠄⠐⠀⠠
⠀⠀⠘⣼⡤⠤⠄⣀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡤⠤⣠⠁⠀⠀⠀⠄
⠀⠀⠀⠈⢷⠀⠀⠀⠈⠊⠑⠲⢤⠀⠀⠈⠢⡀⠀⠀⠘⣣⠀⣀⡤⠤⢄⠐⡤⠆⢀⡡
⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⠀⣃⠢⢤⢤⡜⠂⠀⠀⠁⠘⡟⢶⠓⠐⠊
⠀⠀⠀⠀⠀⠀⠓⠊⠁⠀⠀⠀⠀⠈⠁⢢⠁⡤⠀⠀⠀⠀⠀⠀⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣈⠀⠀⠀⠐⠊⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢤⣄⠀⣀⣀⣀⣀⢈⣉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠉⠉⠉⠁

⠀⡏⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⡀⠀⠀⠀⠀⠉⠉⠀⢀⣀⣀⠠⣾⡆⠈⠁⠈⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⢹
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢴⠀⢠⣶⠊⠈⠀⠀⠀⢾⡐⠖⠚⠈⠁⠀⢈⡤⠥⠄⠀⠈⠀⠺⢡⠀⣠⠶⠢⣄⠀⠀⠀⢸
⠀⣷⡞⠋⣙⡂⠀⠀⠀⠀⠀⠀⢠⠌⠀⠀⢘⣿⠷⠶⡨⠀⠀⠀⠀⠁⠀⣤⠀⡠⠵⣰⡤⢀⣷⠀⠀⠀⢣⢀⣿⡤⣶⣿⡆⠀⠀⢸
⠀⡿⢯⡘⣿⣇⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⠀⠸⣷⣶⠃⠀⠀⠀⠀⠀⡠⠿⣿⡯⡽⠻⠿⠟⠁⠀⠀⠀⣴⣿⣹⣶⡻⠿⢿⣄⠀⢸
⠀⣅⣬⣿⣻⡅⠶⣀⠀⠀⣶⠅⠀⠀⠌⢥⣂⣰⡃⣀⣄⠀⠀⠀⠀⣶⠠⣼⠾⡾⠟⣫⡆⠀⠀⠀⠀⣰⠉⣿⣈⣿⠁⣄⠀⡆⠆⢸
⠀⠛⢉⠁⠈⡙⣷⠉⠆⡄⠈⠁⠀⠀⠀⠸⣿⠟⣿⠟⠈⠀⠀⠀⠀⢠⠞⢀⠠⠡⠞⠁⠀⠀⠀⠀⢠⣿⣄⣿⣿⠛⣀⣿⠀⢻⠀⢸
⠀⡄⣾⣄⡐⠁⢡⣿⡦⠥⠤⠀⠀⠀⢐⢰⢏⢀⠀⠀⠠⠀⠀⣠⣤⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡋⢹⣿⣧⣙⣿⠛⣩⣶⢼
⠀⡍⣋⡡⣾⠀⠈⢮⡀⠀⠀⠀⠀⠀⠁⣸⠀⠃⠀⠀⠀⡀⠀⠻⣨⣏⣩⢙⠓⠒⠀⠽⠠⠾⠃⠀⣿⠐⢰⣾⣿⣿⠟⠧⡘⠳⠣⢚
⠀⡇⠙⢰⣿⠐⠀⡈⠁⠠⡀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠛⠃⠀⠂⠄⢤⣢⢁⣠⠔⠁⠀⠀⠀⠀⢽⣠⣼⠻⡿⢃⣴⢴⠗⠇⠀⢸
⠀⣷⠂⠈⢻⣄⠻⢢⣶⣢⠞⠀⡄⠨⠭⣿⠤⠭⣀⠀⠀⠀⠀⢀⡀⠀⣍⣁⡠⣇⠀⠀⠒⠂⠒⠒⢸⡇⣿⢷⣿⣿⡇⢺⣾⠃⡌⢹
⠀⣿⣆⠀⠀⡣⠙⣿⢿⡆⠀⠀⠁⠀⠈⠁⠒⠒⠠⠀⠀⡐⠚⠉⠀⠀⢶⠉⠀⠘⢀⠀⠀⠀⠀⠀⣸⠟⣡⣾⠿⢫⠄⡼⠁⠀⠠⡘
⠀⡿⢌⠂⠑⢏⡀⡟⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠉⠒⠦⠀⠀⠈⠗⠀⠀⠀⠳⠀⠀⠀⠀⡅⠿⢷⣛⠇⡜⢀⠃⠀⠀⠠⢸
⠀⡇⠫⣡⣤⢌⣳⠛⣄⡇⠸⠾⠀⠒⠀⠀⡽⠀⠀⢠⡄⠉⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣆⠀⠸⡄⠀⢄⡀⠡⢧⡀⠀⠀⠀⢸
⠀⡇⡠⠹⣻⢠⡟⣩⢄⠀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠑⠢⠀⠘⡀⠀⠣⡘⢮⢷⡀⠀⠀⢸
⠀⡵⠛⠉⠀⠀⠉⡉⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢄⠐⠁⠁⠀⢠⣒⠀⠀⠀⠀⠀⣴⡞⠁⠀⢀⣄⠂⠄⠀⠑⠈⣼⣿⠀⠀⢸
⠀⡂⠄⠀⠀⠠⣼⣷⣌⢢⣦⣄⠀⠤⠀⠀⠀⠀⠰⢀⠈⠉⠦⣠⡄⠀⠙⡁⠀⠀⣠⣾⣿⡏⢱⣄⣁⠔⠉⠀⠀⠀⠈⣤⠘⠀⢰⣾
⠀⡇⠀⠐⠇⠀⠈⠳⡟⢦⡑⠜⠳⣄⠠⣀⠲⡰⠿⣤⡀⣀⢔⡽⢀⠀⣄⡄⡠⠞⢛⡾⣿⠛⢊⠋⡅⢃⠀⢀⡀⠀⢈⢃⠂⣸⡌⢹
⠀⣇⠀⠀⠀⠀⠀⢀⢙⣄⠉⠳⣌⡀⠙⠻⢭⣍⣿⡿⢏⣠⠞⠀⢼⣠⣽⣿⣧⡀⣬⣻⣿⣁⣠⣼⠇⢸⣿⣿⣿⣿⣿⣿⣿⠿⠛⠹
⠀⢹⡟⠂⢀⠁⢀⣰⣾⣿⣷⣄⡀⠉⠳⣦⣀⠄⢩⢡⡿⢃⡀⢠⣬⠿⡟⡿⡿⠙⢡⣿⡿⣟⣹⣿⣸⢾⣿⣿⣿⣿⣿⣿⡟⢢⣾⠟
⠀⢸⣷⡖⠀⣪⣽⣿⣿⣿⣿⣿⣿⣶⣆⡀⠈⠙⠛⣚⣛⣋⣤⣶⣌⡙⢃⣁⣆⣴⢟⣻⣧⣿⣿⣏⣹⣶⣿⣿⣿⣿⣿⠿⣓⣺⢿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⢦⡄⠀⢄⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠉⠀⢸⢦⣀⣀⣤⠖⠁⠀⢀⣀⡀⠀⠀⣀⠀⠄⠤⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠈⠀⢯⣀⣀⠈⠉⢉⠠⠂⣏⠔⠚⠈⠑⢦⠑⡄
⠀⠐⠒⠒⢀⣴⣃⡀⢀⣴⠞⠩⠥⢒⢈⡠⠒⠉⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠘⠉⠁⠀⢹⡾⢸⠄⠁⠉⠈⠀⠀⠀⠀⠀⢀⣀⣠⣤⣼⣥⣧⣤⣤⣀⣀
⠀⠀⣠⠞⠀⣴⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠀⠀⠀⠀⠀⠈⠉⢹⡇
⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⣤⣤⣤⣤⣤⣤⣤⣤⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢛⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣠⣮⣍⠛⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡆⠀⠀⠈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡴⠚⠉⠉⠉⠙⠲⣄⠀⠀⣠⠤⠒⠒⠦⢄⡀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢠⠋⠀⠀⠀⣦⣿⡆⠀⠈⢳⠞⢧⣿⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⣠⠞⠳⣄⡀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠒⠤⠤⠤⠶⠚⠁⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠒⠒⠒⠒⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠥⠄⢤⡀⢠⡄⠀⠤⠤⠈⠛⠖⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⢀⠤⣄⢀⣀⡀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣸⠀⠀⠀⣀⠤⠀⢤⠀⠙⡤⠸⠦⣄⡀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⢿⣀⣤⣮⣬⣤⣤⣼⣤⣤⣤⣤⠤⠦⠤⠦⠤⠤⠷⣤⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠦⣤⣤⣤⣤⣤⣤⣤⡴⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢆⣠⡄⢦
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠿⠟⠛⠻⠷⣶⣤⡀⠀⠀⠀⠀⣠⣴⠾⠟⠻⠿⢿⣽⣿⣾⣷⣀⣠⡀
⠀⠀⠀⠀⠀⢀⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⣠⠛⠉⠀⠀⠀⠀⠀⠀⠉⠁⠙⢿⣿⣿⣿⣄
⠀⠀⠀⠀⣄⣾⣿⣿⠉⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡄
⠀⠀⠀⠰⢻⣿⣿⠇⠀⠀⠀⠀⢰⣀⠴⠈⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠃
⠀⠀⠀⢠⡟⣿⣿⡖⠀⠀⠀⠀⠈⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⣩⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⣀⡠⠀⠴⠒⢐⡩⠋⠀⠀⠀⠀⢠⣿⣿⠫⡉
⠀⠀⠀⠀⠈⣻⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⠔⠒⠊⠉⠀⠀⠀⠀⠔⠋⠀⠀⠀⠀⠀⠀⣰⣿⣿⠾⠃
⠀⠀⠀⠀⠀⠿⣿⣿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠎
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⢀⡀⢀⡀⠀⣶⣿⣿⠏⠈
⠀⠀⠀⠀⠀⠀⠀⠺⠿⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠠⣞⣁⡀⠤⠄⠒⠂⠈⠉⠀⡔⢀⣼⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢿⣿⣶⡄⠀⠀⠀⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⣌⣴⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠜⠻⡾⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡶⡀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠀⠀⣠⡼⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠉



⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠂⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⠘⠿⡟⠒⡟⠀⠁⠀⠀⠀⢀⣤⠒⠀⠠⠄⠀⠀⠀⠀⠈⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠁⠸⢀⠀⠀⠀⠀⡊⠜⡶⠠⠞⠆⡀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⠀⠀⡀⡐⠙⢐⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠰⢀⣩⢷⢁⡰⠀⠀⢀⢀⣾⠀⠀⠀⠀⠑⢦⡀⠀⠀⢄⣐⡶⣠⡸⠖⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠉⠛⠖⠺⠤⠐⠀⠉⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠀⠀⡀
⠀⠀⠞⠀⠀⠹⣤⠻⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⢀⣀⡀⠀⠀⠈⠀⠈⠉⠚⡁⠉⠁⠁
⠀⠀⠴⠀⣀⠂⠀⠀⠹⠇⠀⠀⠀⠀⠀⠀⠀⠈⣲⠞⠌⠉⠀⠀⠀⠀⠀⡄⠙⡎⠀⠀⠀⠀⠀⠀⠄⠄⠂⠁
⠀⠀⠈⠂⠙⠃⠠⠖⠂⠀⠀⠘⠙⠘⠀⠀⠀⠀⠓⠀⠂⠀⠀⠀⠀⠠⠒⠛⠈⠱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⢰⢀⠠⠄
⠀⠀⠀⠀⠈⢂⠀⠀⠀⢶⣄⠀⠀⠁⠰⢠⡄⡴⠀⠂⠄⠀⠀⠀⠀⠀⣀⣬⣁⡀⠀⠀⠀⠀⠀⠀⠙⠻⠞⣂
⠀⠐⠀⠀⢀⠀⢷⣀⠀⠠⣬⣄⠀⠀⠀⠀⢽⡇⢤⠇⠀⠀⠀⠀⠀⠈⠀⠀⢇⠀⠠⡀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠠⠼⠶⠿⠟⠓⠐⠒⠉⠀⠀⠀⠀⢨⢱⢇⣠⠀⠀⠀⠐⠒⠂⢀⣀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⣀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⣠⣎⠀⢀⣀⣀⣘⣉⠀⠀⠀⡀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠈⠑⡦⠀⠀⠀⠀⠀⠀⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠾⣾⠅⠂⠠⡀⠀⠈⠁⠀⠁⠀⠀⠀⠊⠉⠁⠀⠀⢠⡄⠈⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠦⠀⠠⠬⠧⠶⠴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠑⠘⢆⣀⢈⡩⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠂⠀⠈⠉⠉⠉⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠔⠀⠂⠄⠀⠀⠠⠄⠐⠂⠄⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠊⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠑⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⢀⠐⡀⠇⠀⠀⠀⢀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠊⡀⠀⠀⠀⢀⠎⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠢⡈⠁⠀⠉⡠⠃⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡒⢄⣀⠀⠁⠀⠁⢀⢠⣴⠨⠄⠀⠤⠤⠤⠤⠤⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠡⠂⡠⠂⢃⠀⠈⠉⠀⢠⡠⠁⢀⠔⣡⣄⠀⠀⠀⣠⣌⠲⡀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡐⠁⡼⠁⡈⠀⠢⠀⠀⡠⠋⠀⠀⠃⠀⢐⡂⠀⠀⠀⢐⡂⠀⠘⡀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠠⡀⠀⠀⠀⠂⠀⠀⢰⡃⠀⠃⠀⠀⢐⠊⠀⠀⠀⢠⠔⠁⠈⠍⠉⠉⠉⠫⠁⠈⠢⡄⠀⠀⠀⠑
⠀⠀⠀⠀⢠⠂⠂⣰⢂⡄⣀⢀⡀⡖⡂⠅⠀⠌⠀⢠⡴⠋⠳⣤⠀⠀⠀⠘⡀⠀⠀⠀⣴⠀⠀⠀⠀⠐⠒⠂⠐⠀⠀⠀⢦
⠀⠀⠀⠀⠬⣨⠄⠨⠁⠃⠏⠂⠡⠁⢅⡠⠬⢀⠀⣸⠁⠀⠀⠘⡄⠀⠀⠀⠀⠢⠤⠔⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠣⠀⠔⠈
⡀⠀⠁⠀⠀⠁⠈⠈⠉⠁⠉⠉⠀⠀⠀⠀⠁⠈⠉⠉⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢾⠟⠀⡀⠀⠀⠈⠀⠀⢀⠀⠳⡀
⡇⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠈⠀⠐⠒⠂⠀⠁⠀⠀⠁
⡇⠀⠀⠀⠀⠀⠄⠀⢀⠂⠀⢂⠀⠑⠄⠀⠀⠀⠀⠀⣆⡄⡄⢾⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⡇⠀⠀⠀⠀⢸⠀⠀⠀⢶⢷⠆⠀⡆⠀⠀⠀⠀⠀⠀⡧⠀⠀⠠⠀⠀⠀⠀⠀⡌⠠⠄⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠠
⡇⠀⠀⠀⠀⠈⠄⠀⠐⠉⠈⡂⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠐⣄⠀⠀⡄⠀⠀⠀⠠⠀⠀⣠⠂
⡇⠀⠀⠀⠀⠀⠈⠂⠄⠉⠉⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠼⠀⡀⡀⠀⠀⠀⠀⢀⠀⠧⡀⠀⠄
⣧⣤⣄⣤⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣀⣂⣀⣀⣘⣂⣀⣀⣀⣐⣃⣀⣀⣐⣀⣂⣀⣀
⣿⣿⠉⡉⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠄⠀⠤⢄⣾⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠑⠁
⠀⠀⠀⠀⠀⠀⠐⠛⠒⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⣦⡚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠄⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⣢⡀⢀⠄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣑⡞⠓⠈⠔⣄⢳⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣄⡄⠰⠐⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠳⠄⢀⣿⡅⡤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢀⡂⠀⠘⠀⠀⠀⠀⢸⡏⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⡰⣄
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⡀⢀⣠⣿⡷⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠃⠘⠁⠁⠈⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠔⡲⢂⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓
⠀⠀⠀⠀⡞⢀⢰⣥⠊⠁⡖⢀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠳⠈⠀⠁⠠⠀⠁⠒⠀⠀⠀⠲⠠⢀⡀
⠀⠀⠀⠀⠀⠀⠑⠢⠁⠠⠤⠶⠄⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠉⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢾⣿⡏⣤⠈⣿⣿⠿⠋⠀⠀⠀⠐⠂⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣗⣂⣽⣿⡶⠞⠋⠁⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠀⠀⠀⠀⠀⢀⣶⡆⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡄⠘⣿⣿⡀⡀⠤⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣶⣶⡶⠿⠟⠛⠙⣿⣷⠠⠙⠉⠁⠀⠀⠐⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣀⣀⣀⣀⣤⣤⣤⣤⣴⣆⠀⠆⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⣶⣾⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⢡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠈⠉⢙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡍⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⢃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠂⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⠀⠑⠠⠀⣀⠀⠀⠀⠀⠀⣀⠠⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠤⠚⠁⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠈⠡⠀⠀⠴⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⢀⠀⠒⡉⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠡⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠔⠈⠀⠀⠀⢡⠀⠀⠀⠀⠀⠘⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠈⠁⠀⠀⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⡠⠔⠊⠐⠒⠀⠠⠃
⠀⠀⠀⠀⣀⠊⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠈⠉⠀⠀⠂⠊⠐⠀⠂

⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⢀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠟⠿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⡟⣿⣿⢿⣿⠿⣿⠿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠶⠶⠶⠦⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⢠⢄⡴⣄⣄⠀⣀⣼⡀⢠⢐⣤⠀⠀⠀⢠⣔⢠⠀⣸⣄⠀⢀⣄⡴⣄⢤
⠀⠀⠀⠀⠀⠀⢏⢿⣼⢧⢏⡎⠀⣧⡀⠀⠉⢁⡀⠀⠀⠀⠀⡀⠉⠁⠀⣠⡇⠈⢎⢧⢿⣴⢏⠇
⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢚⡩⠺⣅⣷⠰⣀⣉⣁⠐⠀⠐⢀⣈⣁⡰⢰⣇⡽⠺⣙⠂⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⡗⠒⠒⢾⠀⠀⠀⡽⠀⠀⠀⠉⠑⠂⠒⠉⠁⠀⠀⢸⡍⠀⠀⢸⠖⠒⠒⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⠤⠤⠴⣧⠀⠀⢠⣾⣿⣶⣾⣿⣦⠀⠀⢠⠷⠤⠤⠼⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠰⠉⠳⣄⠈⠻⠟⠋⠛⠿⠋⣀⡴⠋⢃⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠤⠤⠤⠆⠀⠀⠀⠉⠓⠲⠤⠴⠒⠋⠁⠀⠀⠀⠦⠤⠤⠤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠉⠓⠒⠲⠤⠤⠼⠤⠤⠴⠒⠒⠋⢩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠐⢲⠒⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⠀⠀⣸⠀⠀⢀⣀⣀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠉⠈⠉⠉⢹⠉⠉⠉⠀⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠒⠒⠒⠒⠛⠓⠒⠒⠒⠒⠚⠃

⣿⡿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠹⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣭⣍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⠀⣉⣛⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣿⣷⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢋⣿⣿⣿⣿⣧⣀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⣹⣿⣿⣿⣋⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠀⠐⢿⠟⠁⢸⣿⣿⣿⣿⠿⠿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁⣰⣿⣿⣿⡟⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣿⣿⡏⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣿⣿⠛⠛⣿⣿⡅
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢹⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠏⠀⠀⠸⣿⣿
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡾⠁⠀⢠⡆⠀⠈⢷⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⡇⠀⠒⢺⡗⠒⠂⢸⣂⣀⣀⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣬⣿⠉⠉⠉⠹⣄⡀⠘⠃⢀⣠⠏⠉⠉⠉⣿⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⣄⣀⣀⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠿⠀⠀⠀⠀⠀⠸⢁⡞⠁⠀⠀⠀⠹⡌⢷⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢺⠒⠶⠶⠶⠖⣶⢸⡇⡉⢀⡀⢉⡀⡇⢸⠄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣾⠀⠆⢀⠰⠀⣿⠈⡟⠁⢯⡽⠃⢻⠗⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠘⣦⣈⣉⣁⣴⠋⢠⠟⠲⠚⠙⠶⠛⢧⠀⢤⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢦⠀⣰⠃⠀⠉⠀⠈⣧⠘⠀⣦⠀⡆⢠⡄⠘⣇⢸⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠁⡏⠀⡆⢰⢰⡆⢸⠀⠀⡟⠀⡇⠘⡇⠀⢻⢸⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣄⣧⠀⡇⢸⢸⡇⣸⠀⠀⡇⠀⡇⠀⡇⢀⣾⠟⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⣤⣬⣤⣤⣤⣭⣿⣧⣼⣤⣿⣥⣤⣯⣥⣤⣧⣤⣭⣿⣥⣤⣤⣤⣥⣤⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣶⠒⡆⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⡤⠂⠙⡉⠁⠘⠙⣻⡅⠀⣠⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄⣩⠁⠠⠁⠹⠁⠀⠐⠀⠀⠈⠀⠁⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠴⠄⠀⠉⠰⠟⠓⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⣸⣀⣤⣠⣄⣠⣤⣤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⠛⣿⢹⠛⣟⠛⠋⠙⠛⠋⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣶⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣨⣠⣠⣤⣄⣠⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠐⠂⠀⠀⠐⠹⠿⠟⣛⢿⣭⣿⣿⡿⠿⠿⠿⠿⠛⠛⠿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠥⠄⠠
⠀⠁⠃⠀⠀⠚⠛⠃⣀⣉⣀⣀⣀⠀⠀⠒⢀⡀⠀⠒⠖⠉⢿⣛⡊⠁⠀⠀⠀⣀⣤⣤⣌⣻⣏⣿⢿⣿⡿⠾⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣭⢬⡥⣦⢴⣶⠷⠖⠤⡾⠶⠠⢴⣶⣶⣾⣅⣈⠈⡡⠂⠒⠒⠀⠀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠀⠈⠉⠐⠒⠒⠒⣋⡴⠖⠛⠋⠽⠉⡸⠜⠒⠒⠛⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣤⣴⣶⣴⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⣿⣿⡿⠟⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠶⠶⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⣠⡴⠞⠛⠉⠉⠉⠛⠳⢦⣄
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠲⣦⡄⠀⠀⠀⢸⡇⠀⡠⠲⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⢶⣿⣿⠀⠀⠀⢸⡇⠀⣿⢐⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠛⠁⠀⠀⠀⣾⠃⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⢠⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⡄
⠀⠈⣧⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠉⠻⣦⣄⣀⠀⠀⠀⣀⣠⣴⠟⠁⠀⠀⣼⠁
⠀⠀⠈⠳⣄⠀⠈⠛⠶⢦⣤⣄⣀⣠⣤⣴⠶⠛⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠉⠉⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⡟⠛⠉⠉⢹⠉⠙⠛⢻⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⡇⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠈⢉⣻⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠛⠓⠲⠷⠶⠶⠶⠾⠶⠶⠶⠚⠛⠛⠉⠁⠀⢰⡏⠉⠛
⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⡀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠛⠳⣤⡴⠚⠓⢦⡀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⣀⣾⠤⠤⣤⠲⠶⠶⠶⢦⠤⠤⢷⣄⣀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⣿⠁⠀⠀⠀⣠⠞⢹⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠉⠳⣄⠀⠀⠀⠈⢿⣆
⠀⠀⠀⣿⣤⣤⣤⣤⣾⣦⣼⣧⣤⣤⣤⣿⣤⡤⡶⠶⠾⠶⠶⠶⠶⠿⠶⠶⠟⠛⠛⠛⠛⠛⠛⣻⡆
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡏
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⣶⠓⠓⠒⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠒⠚⠛⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⣿⡴⣾⠁⠀⠀⠉⠳⣄⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⢳⡄⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⢀⡀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢠⡟⠀⣠⠶⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⢀⣾⡴⠞⠁⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠜⠋⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⢒⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡤⠴⠖⠛⠋⠉⠁⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣚⣡⣤⠤⠶⠖⠛⠋⠉⣡⠏⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣁⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠀⠀⠀⠀⢈⡭⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⡀⢀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡶⠞⠛⠛⠛⠻⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣶⠿⡆
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢠⡿⢻⡆⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⠀⢀⡿⣠⡤⠀⠀⠀⠀⣠⡟⠁⠀⢻⣄⠀⣿⠻⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣼⠿⣿⠃⠀⠀⢀⣴⠏⠀⠀⠀⠀⠹⣦⣿⠀⠙⢷⣄⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⣿⠀⠀⣠⡾⠡⠶⠞⠗⠀⠀⠀⠈⠿⠆⠐⠿⠿⢷⣄⠀⠀⠀⢸⡏
⠀⠀⠀⠀⢿⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⣀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣴⢾⡏⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠈⢹⡿⣦⡀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠹⣆⠀⠀⠀⣼⠋⠻⣆
⠀⠀⢰⡏⠀⣷⠀⠀⠀⠀⠀⠈⠁⢠⣄⡀⢀⣠⡤⠀⠉⠀⠀⠀⠀⠀⣸⠇⢸⡇⠀⠀⠀⠀⠀⠀⣾⢃⣤⡀⢿⡄⠀⣸⠇⣠⣄⢻⡄
⠀⠀⠈⢷⣤⣿⡆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠀⠀⠀⠀⠀⠀⠀⢀⣿⣤⣼⠃⠀⠀⠀⠀⠀⣸⠇⣾⠉⣷⠸⣇⢠⡟⣰⡏⢻⡌⣷
⠀⠀⠀⠘⣧⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⡇⠀⠀⠀⠀⠀⢠⡟⢰⡏⠀⢹⡆⣿⣼⢃⡿⠀⢸⡇⢹⡆
⠀⠀⠀⠀⢻⡄⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⣿⠀⠀⠀⠀⠀⠀⣼⠃⣸⠇⠀⠘⣧⢹⡿⣸⠇⠀⠘⣇⠸⣧
⠀⠀⠀⠀⠘⣧⠸⣯⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣹⡇⣸⡇⠀⠀⠀⠀⠀⢠⣿⣠⡿⠶⠶⠞⠛⠻⠷⠿⣤⣄⡀⣿⠀⣿
⠀⠀⠀⠀⠀⢹⡆⢻⡆⠀⠙⠳⢦⣤⣀⣀⣀⣀⣠⣴⠾⠋⠁⢀⡿⢠⡿⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠀⢹⡆
⠀⠀⠀⠀⠀⠈⢿⡀⢿⡀⠀⠀⠀⣿⠉⠉⠉⠉⢹⡇⠀⠀⠀⣾⠁⣾⠁⠀⠀⠀⠀⠀⢸⣏⣀⣀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠘⣷⠘⣷⡀⣀⣤⣿⠀⠀⠀⠀⢸⣧⣀⠀⣼⠇⣼⠇⠀⠀⠀⠀⠀⠀⢸⣿⠉⢹⡇⠀⠀⠀⠀⣿⠉⢹⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⢿⡋⠀⢿⡀⠀⠀⢀⣾⠁⢉⣿⠋⣼⡏⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠁⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣠⡾⠛⢷⡌⠻⣦⣈⠻⠦⠴⠞⢁⣤⠾⢁⣼⠋⠻⣦⡀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⢸⡇
⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠙⢷⣌⣉⠛⠛⠛⠛⢋⣡⡴⠟⠁⠀⠀⠈⠻⣆⠀⠀⢸⡿⠛⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢾⡟⢸⡇
⠀⠀⠀⣰⠟⠀⠀⣄⡀⠀⠀⠀⠈⠙⣿⠀⠀⣺⡋⠉⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⢸⡿⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⣼⠃
⠀⠀⣰⠏⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⣿⣤⣤⣾⠄⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⢷⡜⢷⣿⡀⠀⠀⠀⣀⣠⣤⠶⠶⠛⠛⠋⠁⢀⠀⠀⠀⣿
⠀⢠⡟⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⣿⠈⢿⡀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢿⡀⠉⠛⣿⠛⣛⣩⣅⣀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⣿
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠹⣆⠘⢷⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⣷⠀⠀⣿⠟⣉⣥⣬⣉⠻⣦⠀⠀⢀⡀⣼⠇⢻⡆⢻⡄
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠹⣦⡈⠻⣦⡀⠀⠀⣿⠀⠀⢀⣠⣤⣤⣼⣧⣼⣧⣼⣏⠀⠀⠙⣧⢸⡇⢀⡾⢿⠏⡀⢸⡇⠈⢷⡀
⠀⣿⠀⠀⠀⠀⣀⣀⣀⣀⣹⡆⠀⠀⠀⠀⠀⠈⠳⣦⡈⠛⠶⣤⣿⣀⡴⠛⠁⢀⣤⣤⣤⣤⣤⣤⣽⡃⢀⣰⡟⢰⡿⠟⠁⠀⣸⢿⡿⠀⠀⠈⢻⣄
⠀⣿⣤⡶⠛⠛⠉⠉⠉⠉⠙⠻⣆⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣤⡽⠋⠀⠀⠀⣀⣀⣀⣹⡏⠻⣦⣉⣛⣛⣉⣴⠟⠀⠀⣀⡼⠋⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠈⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠉⠉⠉⢹⡆⠀⠀⠉⠙⠋⠉⠀⠀⠀⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠞⢹⣿⡙⢿⣶⠶⣄⠀⠀⠀⠀⠀⠈⣧⡙⣦⣀
⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣿⣿⣧⠀⢠⣿⣷⡘⢿⣆⢺⣧⡀⠀⠀⠀⠀⠀⠳⡀⠉⢷⣄
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢠⡿⣿⣀⢸⡿⢿⠷⠈⡿⠛⠓⠸⣿⡆⣿⣿⠹⣆⠀⠀⠀⠀⢸⠆⠠⣾
⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠠⠘⠀⠀⠐⠀⠀⠈⠀⠤⡀⠀⠀⠀⠀⠀⣸⠁⠠⣿
⠀⠀⠀⠀⠂⢤⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⣹⡟
⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⢀⣀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡞⠁⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⡏⠀⠀⠀⠀⠸⡗⢰⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣧⣶⢰⣴⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣶⣿⣇⠀⠀⠀⡠⣤⣷⡜⣇⣤⣤⣄⣠⡤⡀
⠀⠀⠀⠀⠀⠠⠊⣻⡿⠃⢸⣯⠈⢏⠻⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢥⣽⣿⡧⠀⠰⣿⣷⣁⣠⣿⣏⣠⣄⣨⠥⠜⠁
⠀⠀⠀⢀⠄⠠⠞⠁⠡⠀⠸⡇⠀⠀⠀⠂⠘⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡣⣶⣿⣿⡇
⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠢⣎⣻⣓⣯⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣴⣶⣿⣿⣿⢿⡧
⠀⠀⠀⣶⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠈
⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠃⠀⢀⠈⠀⣀⡀⠀⠀⠀⠀⠈⠀⠀⠀⢈⠄
⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⢀⠣⡂⠀⠉⠉⠉⠉⠙⠉⣉⣉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠨⠝⢒⣒⡶⠶⠶⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠒⠲⠦⠤⠤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣦
⠀⠀⠀⣠⡀⢀⣴⣿⣷⡃⠀⣤⣴⣤⡀⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⣄⣀⡄⠒⢒⠁⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⡘⣿⣾⣿⠛⡏⠘⣿⣿⠁⡙⢿⣿⣿⣻⠃⠟⠆⣤⠀⠀⠀⢘⠂⠐⢒⠂⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⢉⠇⠀⠀⠈⠁
⠀⠀⣰⣿⣼⣿⣽⣇⣼⣷⣸⣿⣷⣿⣼⣿⣽⣇⣼⣷⣸⣿⣇⠀⠀⠀⠏⠩⠉⠀⠀⠀⠀⠈⠓⠀⠀⠀⣀⣠⠖⠋⠀⠀⠤⠤⡄
⠀⠀⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠒⠒⢒⣛⡉⠁⠀⠀⠀⠀⠀⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠠⢒⣟⠮⠛⠉⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⢀⣊⠬⠚⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡠⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠂⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠚⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⢤⣴⠿⡞⡃⠀⠀⠀⠀⠀⠈⢪⡀⣈⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠤⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡦⢻⣽⣬⣿⣆⠘⡜⡄⠀⠀⠀⠀⠀⠀⠓⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣒⣤⠹⣯⠤⡄⢫⢛⠻⣝⣆⠘⠛⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⢀⣐⠀⡰⠀⢀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⣐⡌⢞⢄⣥⣦⠹⡩⣤⣤⠱⠃⠈⠀⠀⠀⠀⡀⠘
⠀⠀⠀⠀⠀⠀⠈⠑⠳⢊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⢆⢹⢡⠹⡱⡾⣶⡌⢼⢬⣿⠇⠙⠉⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⡀⠀⠀⠀⠀⠀⠀⠈⢮⡆⠱⣣⠹⣷⡘⠾⠀⠉⠀⠀⠀⠀⠀⠀⠄⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠄⡀⠀⠀⠀⠈⠜⠰⠟⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠠⢀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠂⢠⠜⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠤⠀⠀⠉⠓⠶⠶⢦⣤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠿⢿⣷⣶⣄⣀⠀⠀⠀⢠⠈⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠻⢗⢒⠀⠀⠛⠋
⠀⠀⠀⠀⠀⠀⠘⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠅⠓⠁
⠀⠀⠀⠀⠀⠀⠠⠚⡄⠀⠀⠀⠀⠀⢀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠠⠀⠂⠂⠘⠈⠀⠀⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⠠⠊⠁⠀⠄⠀⠀⠀⠈⢀⠀⠄⠠⠀⠰⠀⠂⠐⡀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠅⠰⠂⠀⠐⠈⠀⠉⠀⠀⠀⠀⠀⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠂
⠠⠐⠀⠂⡠⠈⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠀⣼⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⡇⢰⠿⠛⠁