Fri Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠙⠀⣴⡶⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⣠⣿⠋⠀⠲⠀⠀⢻⡇⠈⠀⢁⠀⢠⢄⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣧⠀⠀⠀⣠⢼⣿⡇⠀⠀⢠⡄⠀⣿⠁⠀⢀⣿⠃⢻⡈⠀⣤⢴⣞⣻⠂⠀⠆⠀⠀⢀⣠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠴⠒⢁⡼⣿⡀⣠⣾⡁⠈⢟⡵⠄⠀⠐⠣⠀⡟⠀⢖⣨⡿⢁⡀⠀⡁⠈⠁⢾⣇⠀⠀⠀⠀⠀⠉⢳⡀
⠀⠀⠀⢀⢀⣡⡶⢿⣿⡂⠀⠉⠀⠊⠀⢰⡟⢀⠀⠀⠀⠀⣠⠆⣰⣿⡿⠃⠀⠡⠀⡁⠀⠰⠀⠹⣆⠀⣰⠀⠈
⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁⢤⣄⠀⠀⠀⠀⠀⢿⣿⠀⠀⠰⠿⠞⣡⣿⣇⠀⠀⠀⠀⠳⣄⠀⠁⠀⠹⡄⠸⣆⠀⢀⡄⣼⣆⣀
⠀⠀⠀⠀⢰⡷⠘⠓⠒⠂⠚⠻⣦⡀⢀⣤⠀⣀⣿⠃⠀⠀⠀⠈⣿⠃⠈⠂⠀⠀⠀⠀⠈⠙⠦⠀⠀⠉⠀⠈⢲⣾⢿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣀⡠⠟⠛⠀⠉⣽⠠⢤⣄⡀⠀⠘⢷⡄⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⣤⡰⠗⢚⣧⠞
⠀⠀⠀⠀⠀⠀⣴⣆⡠⠼⠛⠒⠉⢀⡀⢀⠀⠀⠉⠰⢆⠀⠀⠙⠈⠛⠿⡀⠀⠀⠀⠀⠀⠀⣀⣤⡚⢉⣠⠞⠉
⠀⠀⠀⠀⠀⠐⠋⠁⠀⡀⠀⠀⠀⣿⣇⣾⣇⣾⣄⠀⠀⠀⠰⢦⣀⠀⠚⠚⠀⠄⣘⣢⡼⢋⣥⠞⢛⡉⠀⠀⠀⣀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣧⡀⢰⣿⣿⣿⣿⣿⣿⣿⣴⣧⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⠳⠞⠉⠁⢠⣿⣿⡄⢠⣴⠿⠖⠛⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠟⠋⠉⠉⠉⠉⠉⠉⠛⠿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⢀⣀⣠⡼⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣼⠬⠼⣘⣸⢻⠹⡜⣎⢧⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠭⢭⣭⣓⡒⠭⣒⢿⡘⣎⢧⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣦⣬⡉⠳⣮⡑⣜⢞⡎⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣧⠈⣷⡘⣎⢦⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⣼⣿⡆⠘⢷⡌⡧⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⢰⣿⣿⡇⢀⣞⡱⣣⠻⡜⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣸⣿⣿⡄⢸⡸⣦⡙⢷⣅⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣴⣿⣿⣿⣿⠀⢳⢱⡙⣆⠙⢷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣼⣿⠟⠛⠛⠉⠀⣸⡇⣇⣟⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠿⠃⠀⠀⠀⢀⣾⡟⢠⠇⣸⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡰⠃⠀⠀⠀⠀⢀⣾⠟⢠⡟⡄⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢳⢣⣾⠿⠿⠶⠾⠿⢋⣠⡟⣰⡇⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡇⡇⣼⠃⡞⣩⢟⡭⡋⡍⡅⢺⣫⣴⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡇⡇⣿⢸⡇⣷⢸⡸⢡⡇⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠀⠀⠀⣶⠀⢰⡆⠀⠀⣶⡆⠀⢶⡶⠶⣄⠀⣶⠶⢦⡀⢰⣄⠀⢀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣄⣀⣠⣤⣤⣤⣤⣤⣾⣿⣿⠀⠀⠀⣿⠀⢸⡇⠀⢰⢿⡇⠀⢸⠃⠀⣿⠀⣿⠀⢠⡇⠀⠙⣧⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⣿⠖⢾⡇⢀⣿⣨⣧⠀⣿⠶⠞⠁⠰⣿⠶⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⠀⢸⡇⢸⡏⠀⣿⠀⣿⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡇⠉⠀⠈⠁⠈⠁⠀⠻⠂⠙⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢘⣿⣿⣿⣯⣽⣿⣿⣿⡀⠀⢀⣿⠁⢀⣀⣤⡀⢀⣤⢤⡄⠺⠆⢰⡶⠶⣄⠀⠀⣾⠀⠲⣄⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡿⠃⠀⢸⡇⣀⠀⢸⣇⡴⠃⢸⡇⢸⡇⠀⢸⡆⢰⠿⡇⠀⠘⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⣿⣿⡿⠿⠛⠁⠀⠀⠀⢸⡏⠉⠀⢸⠿⣄⠀⢸⡇⢸⡇⠀⣾⠀⣾⠚⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⠆⠈⠓⠈⠃⠘⠷⠞⠁⠀⠟⠀⠹⠄⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣤⠴⠀⡄⢠⡄⢠⡤⠤
⠀⠀⠀⠀⢀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢠⣶⣿⣧⠀⠀⠀⣠⣤⣶⣶⣶⣄⠀⠀⢸⠀⠀⡇⢸⡇⣼⣀⡀
⠀⠀⠀⣼⠋⣡⡈⠹⣧⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡆⠀⠀⢠⣿⣿⣿⡿⢀⣴⣿⡿⠛⠉⠉⣹⣿⡇⠀⢸⠀⠀⡏⢹⡇⣿⠁⣀
⠀⠀⠘⢿⣤⡿⠃⠀⢻⡆⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⢿⡿⢻⣿⡇⠸⠿⠟⠀⠀⠀⣰⣿⡟⠀⠀⠘⠃⠀⠃⠈⠃⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⢼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢸⣿⡇⠀⠀⣠⣶⣶⣶⣶⣶⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠛⠿⠛⠉⠉⠉⠙⣿⣷⠐⠖⣾⠛⠃⣶⠀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⣿⠀⠀⣿⢀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠃⠀⠀⣿⠀⠀⣿⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⣿⣧⠀⠀⢸⣿⣷⠀⢠⣶⣶⣶⣾⡿⠟⠋⠀⠀⠀⠀⢼⠄⠀⠿⠀⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠛⢿⣿⠌⠁⠀⠀⠈⠉⠉⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣥⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⢱⡼⣿⣿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣾⣿⢟⣡⠟⠁⠀⠈⠛⢳⣤⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠴⠳⠞⠀⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣪⡴⠟⡏⢈⠉⠙⠛⠁⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⡾⠋⢀⣄⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⣠⣾⡿⠛⠻⣦⡀⠀⠀⠀⢀⡴⠦⣦⠞⠉⠀⣠⠀⠈⠃⠈⠉⢰⣷⠀⠀⠀⠀⠀⠀⣀⡜⠋⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⠀⠈⠻⣷⡀⠒⠠⡀⢾⣡⠤⢀⠀⠁⠈⣠⣦⠀⠀⣼⣿⡇⠀⠀⠀⣠⣴⡿⠞⠊⠀⠀⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠁⠈⠙⢶⡄⠀⢀⣠⠔⠀⣀⡀⠀⣻⣿⡇⢠⣿⣿⣷⣠⠴⠊⠴⠊⠀⠐⠁⣠⣏⠐⣻⣿⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⠎⢀⡴⢀⠄⠀⠀⠀⠀⠀⠀⢠⣌⣿⡷⠊⠀⠀⠜⠉⠀⠀⣼⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⠀⢀⠤⠁⣁⣴⣿⡿⠋⠁
⠀⠀⠀⠀⠀⣴⣿⣤⣀⣀⠐⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⢀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⡖⠀⠀⠀⣡⣤⣾⠟⢁⠀⣀
⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣦⣜⣱⣿⣤⣀⠀⠀⠀⠀⠀⠀⢀⣼⣫⡄⠀⠀⠀⠘⠄⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⣴⣷⠀⣠⣦⡀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠛⠿⣿⠟⠉⠉⠛⣷⣄⠀⡀⣴⣿⡿⠋⠀⠀⣴⣧⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⡄⣿⣿⣿⣿⡿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣀⠀⠀⠀⠀⠀⠿⠿⣿⣿⡋⣀⡀⢠⣄⣿⣿⣷⣄⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⡀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠈⠛⠲⣤⡄⠀⠀⠀⠈⠉⠉⠉⠻⢿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠁⠉⠀⡀⠄⠐⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⡏⣀⠀⠀⠀⠘⠳⣄⣀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣠⣶⣶⠶⠊⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠈⠿⠷⢤⣀⡀⠀⠀⠉⢻⣷⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠈⠀⠋⠁⠀⠀⠀⠀⠀⢀⣴⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠰⠄⠈⠳⢾⣦⣠⣦⠈⡛⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠯⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠉⠻⣿⡿⢿⣿⣿⣿⣿⣿⣿⣟⠛⠉⠋⠀⠈⠁⠀⠀⠀⢀⣠⣾⣧⣤⡶⠃⠀⠀⢀⠜⠋⠁⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢁⣾⣿⣿⡿⠟⠛⠛⠋⠀⠀⠀⠀⠀⠐⠀⠀⣴⠿⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣄⣤⣤⣤⣤⣤⣄⣠⣴⣿⣿⠀⠀⠀⠠⣦⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⠀⣀⣼⡇⢠⣴⣦⣄⠀⣼⣇⣤⣄⠀⢸⣟⣠⣄⢀⣤⡄⠀⣀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠛⡿⣿⣿⣿⣿⣿⣿⡟⢻⣿⣿⣦⠀⣠⣴⣿⠟⠛⣿⢡⣿⠁⢸⣿⢀⣿⡟⠉⣿⡆⣿⡿⠋⣻⣿⣿⡇⣰⣿⡆
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣇⠁⡿⠈⢻⣿⣿⠋⢸⠎⢀⣿⣿⣿⡆⠉⠁⣿⡆⠀⣿⠸⣿⣴⡿⠿⢿⣿⢠⣴⡿⢠⣿⢡⣤⡿⠁⠙⠿⠋⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠛⠀⠈⠁⠀⠀⢸⡏⠀⠉⠀⢸⣿⠀⠉⠀⠀⢀⣶⣿⡿⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣤⡀⣀⣠⣤⣤⣤⣤⠀⠀⠀⠀⠘⠇⠀⢀⣀⠘⠟⠀⠀⠀⠀⠘⠿⠟⠁⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠘⠛⠛⣿⡏⡁⠀⣀⠀⣀⣀⣠⡄⠀⠀⡀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣿⡏⠉⠁⠀⠀⠀⠀⠀⠠⣶⣿⠟⠛⢸⣿⡿⠋⠉⣬⢀⣴⠟⠛⣿⡇⣴⡟⢻⣷⡄⢀⣾⡗⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⣿⠁⠀⢠⣿⢸⡏⠀⢠⣿⢰⣿⠀⣸⣿⣠⣿⢻⣧⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⠀⠀⢠⡄⢀⡄⠀⠀⠿⠀⠀⠀⠸⠿⠀⠀⠈⣿⠘⠿⡶⠿⣿⡈⠿⠾⠟⠛⠛⠁⠀⠉⣁⣼⣏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⣀⣼⣧⢸⡇⣀⠀⣠⣶⡦⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢉⣁⣀⣀⠀⠀⠀⠀⠀⣾⣻⡿⠋⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠈⣿⠀⢸⡿⢹⣷⣿⠟⠁⣀⠀⢀⣴⣿⣿⣿⠀⠀⢀⣾⣿⣿⣿⣿⣷⣦⠀⠀⠀⠛⠛⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡄⢿⡀⠺⠃⢸⡇⠛⠶⠟⢁⣾⣿⣿⣿⣿⣿⠀⠀⠀⠛⠛⠋⠉⠻⣿⣿⡇⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣦⠙⠀⠀⠀⠛⠀⠀⠀⠀⠙⠛⠁⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡇⠀⢴⠆⢰⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⢰⣿⣿⡇⠀⠀⠀⠀⢐⣛⣻⣻⣿⠟⠁⢀⣤⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠛⢋⣀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⠀⠈⠛⢻⣿⣛⣃⠀⢸⣏⡀⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠉⠈⠋⠀⠀⠀⣿⣿⣿⠀⠀⣾⣿⣧⠀⠀⠀⢹⣿⣿⠟⢿⡟⠉⣿⣵⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⣿⣿⣿⠀⠘⣿⣿⣷⣄⣀⣠⣿⣿⡿⠀⢺⡇⠀⣿⠇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠉⠻⢶⣄⣼⣿⣿⣿⣷⣦⡀⠘⠻⣿⣿⣿⣿⣿⠟⠁⠀⠸⣧⡀⠛⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⣵⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠀⠀⠀⢀⣤⣝⣿⠛⠛⠻⠿⠿⠃⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⠏⠍⠁⠀⠠⡄⠀⠈⠘⠟⢝⣲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⡿⣿⣧⠴⠒⠁⠀⠀⠈⠉⠀⠀⠀⠓⠒⠾⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⢿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⢏⣿⡧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⢰⣿⣮⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡏⠜⠁⠀⠄⠀⠀⠀⠀⢠⠀⠀⡀⢸⣼⣿⣿⣿⣿⣿⡽⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠟⡏⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣼⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣼⣧⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⣸⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⢿⢟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⡿⢿⡿⢛⣻⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠘⣿⣿⣿⣿⣿⡟⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⢀⡞⣽⡿⡏⠉⣰⠦⣿⣿⣿⣿⣿⣿⣷⢼⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣷⣶⡀⢐⣿⣶⡟⠘⠀⠀⠙⢸⣿⣧⣿⣿⣿⣿⣿⢼⣿⣿⣿⣿⣿
⠀⠀⠀⢻⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣽⣿⠃⠀⠀⠀⠀⣿⣿⣿⣻⣿⣿⣿⣿⣸⣿⣿⣿⣿⡟
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⠁⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⣿⣿⣽⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠘⣿⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⢯⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡝⣿⣿⣿⣟⢛⣯⣝⣿⣿⣶⠆⠀⠀⠀⠐⠿⣯⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠰⠻⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⢏⣁⠐⠲⠆⡤⢀⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠰⠊⢻⣿⣿⣿⣿⣽⣿⣿⣧⣤⠍⢀⣏⠈⠧⣌⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣴⣶⣶⣄⣉⡁⠌⠭⣿⣿⣿⣿⣻⣿⣾⡿⣿⣿⡿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠙⢿⠿⠿⠿⠿⣿⣿⣿⣭⡽⠛⠛⠛⢃⣒⢀⣹⣿⣿⣿⣗⣿⣻⣽⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠻⣿⣿⣿⣿⣿⣧⠉⠈⠈⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁

⣿⣿⣷⣄⠀⠀⠙⣧⡻⣴⠟⠒⢺⣠⡆⠍⣓⢭⢿⣯⣼⣻⣿⣿⣿⣷⢙⣿⣯⣭⡝⠃⠀⠀⣿⠋⢰⡆⠉⣿⡄⠙⠿⠈⠟⠹⣧⢶⣄⣾⠃⠈⠴⡟⠀⠈⡛
⣿⣿⣿⣿⣿⣶⣯⣅⢀⠽⠿⠁⠄⠽⢦⡄⠈⣷⣆⣿⡟⣭⣯⣿⣾⣿⡾⣟⣋⡮⣥⣠⠖⠖⢨⡆⠀⠠⠀⣈⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⢠⡀⠙⠄⠀⡸⢧
⣿⣿⣿⣿⣿⣿⠻⢿⣏⡙⣥⣤⠀⠀⠀⠉⠀⠏⠈⡍⢲⣭⣦⣿⣿⣥⣽⣬⣻⡹⢯⡝⠀⣀⣸⣷⢶⣆⣦⠈⠀⠀⠀⠀⠀⠀⠈⠇⠀⢀⡄⠁⢳⡘⠋⢠⣬
⣿⣿⡿⠛⠛⢛⣷⢾⣿⡟⢸⠟⠻⠟⠃⠀⠈⠀⢨⡟⠛⠦⠀⠉⠉⠀⠁⠈⠙⠛⠀⠀⠐⢯⠙⠙⣿⢏⠟⣄⢀⡸⢣⡄⠀⠘⠦⠀⠄⠀⠀⣠⣼⡷⠖⠟⠋
⣿⣿⣿⣷⣶⣿⡇⣸⣿⣧⣿⠿⢃⣠⢤⡃⠀⡴⠮⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⠀⣠⠆⠰⠟⠚⠀⢀⣌⠁⠀⠀⠁⣄⣠⣀⠀⣀⠀⢹⠟⠀⠂⣀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣾⠷⠟⣭⢹⡶⠛⠛⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠈⠉⡰⠿⠤⣠⠄⢹⡇⠀⠁⠀⠈⠀⠀⠀⠀⠸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣻⠚⠀⠀⠀⠀⠀⠀⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣱⣴⡇⢾⣇⣻⠁⠀⠀⠀⣤⠀⠀⠀⠀⢐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣠⣿⣰⣧⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡔⢟⢭⣥⡿⣯⣯⠇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣠⣤⣀⣴⡄⠀⠀⠀⠀⠰⠒⠛⠛⠂⠀⠀⠀⠀⠀⠘⠂⠀⢠⡩⠶⠎⢾⡉⠝⠓⡾⠧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⠀⠀⠀⠀⠀⠘⠖⠠⠄⠀⠀⠀⠉⠀⢀⠀⠘⠦⠀⠀⠀⠘⣃⣠⢸⡇⠀⠰⠂⠀⠀⠀⠰⣿⡄⣄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣭⣉⣥⣬⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠒⠀⠀⢀⣤⡰⣎⠉⠀⠘⠀⠀⢀⠀⠀⠀⠀⠠⣼⠷⢋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣡⠖⣠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢻⣠⡄⢀⡴⠂⠀⠀⠀⠀⠀⠀⠴⠿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠼⣋⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠈⠁⢠⣄⠀⠲⠧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣾⣵⠂⠀⠀⠀⠀⢻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⢠⣷⣿⡅⣘⣣
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢿⣹⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⡎⢸⡿⠋⢁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠰⢯
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⣦⠴
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⣶⡟⠟⠢⢠⢁⠀⠀⠀⠀⢀⣦⡆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠾⢫⣷⢥⣀⡀⡈⣇⣛⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣂⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠿⠛⠋⣉⡁⣿⢡⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣽⠇⣠⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠛⢠⣆⣠⠀⠈⢿⠀⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠙⠓⠓⢹⣰⡟⢷⣽⣆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡉⢳⣶⣿⣯⣷⣶⡾⠿⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠀⠀⠀⠀⠀⠀⠀⠀⠐⣱⠺⠿⣍⢣⣴⣿⢧⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢀⢌⡻⠸⡿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣩⠉⠀⡀⠀⠀⠀⠀⠀⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣒⠆⣀⣤⡶⠾⡇⠉⠀⣠⣶⣴⣄⠀⡉⠹⠷⢶⣦⣄⠖⣒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⣧⣮⣳⣌⣛⣛⣟⣟⣀⠀⣼⡏⢰⡆⢹⣧⠀⣀⣻⣿⣛⣛⣣⣟⣽⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠩⣼⣿⣿⣿⣿⣿⣿⣿⠛⠛⠻⣷⣿⣿⣾⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣧⠌⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣠⠀⢐⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠄⠛⣻⡟⠛⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⡀⠀⣀⣤⣠⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡏⠉⠙⢻⣿⠛⠉⠻⢿⣫⣷⣶⣷⣶⡮⣣⣶⣿⣿⣷⣜⢷⣶⣶⣶⣾⣽⡿⠟⠉⠛⣿⡿⠋⡉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⠀⠘⣷⣀⣰⣾⣯⡀⢠⣰⣿⣿⣿⣿⣿⣿⢱⣿⠋⡰⢆⠙⣿⡎⣿⣿⣿⣿⣿⣿⣆⡄⢀⣼⣷⣆⣀⣾⠃⠀⠃⣤
⠀⠀⠀⠀⠀⣠⠶⣾⣃⣦⣠⣴⣿⠛⠋⠻⠁⣠⣴⣶⣮⣝⠻⣷⣟⡿⠖⠛⠉⣁⣈⠉⠛⠺⢽⣻⡾⠟⣫⣽⣶⣤⣄⠈⠿⠙⠛⣿⣦⣄⣴⣌⣷⠶⣄
⠀⠀⠀⠀⢰⢣⣮⣻⣿⣿⣿⣿⣧⡀⢀⣺⢰⣿⠁⣤⢈⡽⠟⢋⣁⣤⣶⣾⣿⣛⣛⡿⣷⣦⣤⣈⡙⠲⢯⣀⣤⠈⣿⡆⣒⡀⢀⣼⣿⣿⣿⣿⣟⣵⡬⡆
⠀⠀⠀⠀⠘⣡⣦⢹⣿⣿⣿⣿⡿⣳⣿⣿⣎⢿⣦⡼⢋⣴⣾⣿⣿⣿⣿⠃⠀⣽⣧⠀⠘⢿⣿⣿⣿⣷⣤⠙⢦⣠⡿⣱⣿⣿⣞⢿⣿⣿⣿⣿⡏⣶⣅⠃
⠀⠀⠀⠀⢠⣿⣿⢸⣿⣿⣿⣿⢹⣿⣿⣿⢿⣖⡾⢁⣾⣿⢴⣷⣄⠀⣿⣶⣴⣙⢉⣦⣶⣿⠀⣠⣶⡦⣿⣷⡈⢷⣰⡿⣿⣿⣿⡏⣿⣿⣿⣿⡇⣿⢿⡆
⠀⠀⠀⠀⠾⠇⠛⢻⣿⠛⠉⠃⢸⣿⣿⣿⣿⡽⠁⣼⣿⣿⠐⠙⡿⢋⢿⣿⣿⢷⡾⣿⣿⡿⡑⢿⠛⠀⣿⣿⣧⠈⢯⢿⣿⣿⣿⡇⠚⠉⠛⣿⡏⠛⠸⢷
⠀⠀⠀⠀⠃⠀⣀⣤⣿⡀⠀⠀⠾⣫⣷⣶⣼⠁⣰⣿⠿⠋⢻⣿⣿⣷⡻⣮⠟⣛⣛⠻⣷⣟⣾⣿⣿⡟⠛⠿⣿⣇⠈⣯⣶⣮⣽⠷⠀⠀⢀⣿⣥⣀⠀⠘
⠀⠀⠀⠀⢠⣾⠋⣉⣽⣿⣟⣤⣾⣿⢋⡠⠇⢠⣿⢫⣀⣤⡔⠻⢿⣟⣿⢣⣾⢛⡟⣷⡌⣿⣻⡿⠟⣲⣠⣰⡜⣿⡄⠸⢄⡙⢿⣷⣀⣻⣿⣯⣉⠙⣷⡄
⠀⠀⠀⠀⠐⢿⣌⣉⣻⣿⣿⠍⢿⣿⣈⠑⡆⠘⣿⡬⠋⠛⠩⣴⣾⣯⣿⡘⢿⣍⣧⣿⢃⣿⣽⣿⣧⠝⠙⠙⢡⣿⠃⢰⠊⣁⣾⡿⠁⣾⣿⣟⣉⣡⡿⠒
⠀⠀⠀⠀⡄⠀⠉⠛⣿⠁⠀⠀⢶⣟⡻⠿⢻⡀⠹⣿⣶⣄⣼⣿⣿⡿⣵⢿⣦⣭⣭⣤⡿⣮⢿⣿⣿⣧⣤⣶⣿⡏⢀⡟⠿⢿⣻⡲⠀⠀⠀⣿⡛⠉⠀⢠
⠀⠀⠀⠀⢷⡆⣤⣸⣿⣤⣀⣄⢸⣿⣿⣿⣿⣳⡀⢻⣿⣿⠀⣠⣵⣎⣾⣿⣿⡾⢷⣿⣿⣷⣠⣾⣄⠀⣿⣿⡟⢀⣞⣿⣿⣿⣿⡇⢤⣀⣤⣿⣧⣤⢰⡾
⠀⠀⠀⠀⠸⣿⣿⢸⣿⣿⣿⣿⣸⣿⣿⣿⣾⠏⢷⠈⢿⣿⠻⡿⠛⠀⣿⠿⢻⢡⡍⠟⠿⣿⠀⠙⢿⠗⣿⡿⠁⣾⠹⣿⣿⣿⣿⣇⣿⣿⣿⣿⡟⣿⣾⠇
⠀⠀⠀⠀⢠⠜⠟⣼⣿⣿⣿⣿⣷⡽⣿⣿⣏⣾⠋⢳⣄⠻⢿⣿⣿⣿⣷⡄⠀⢺⡗⠀⢠⣾⣿⣿⣿⡿⠛⣠⡞⠛⣷⡹⣿⣿⢿⣾⣿⣿⣿⣿⣇⠻⢋⡄
⠀⠀⠀⠀⠸⡞⢟⣽⣿⣿⣿⣿⡟⠁⠀⠽⠘⣿⡀⠛⠈⣳⣦⣌⡉⠛⠿⢿⣶⣭⣥⣷⡿⠿⠛⢉⡡⣴⣞⠉⠛⢀⣿⠇⠯⠁⠈⢻⣿⣿⣿⣿⣯⣻⢳⠇
⠀⠀⠀⠀⠀⠙⠶⢿⡍⠟⢙⠻⣿⣤⣄⣴⡀⠙⠻⠿⢟⣫⣴⡾⣯⣗⠦⣤⣀⡉⢉⣁⣤⠴⣾⣿⣷⣦⣟⡻⠿⠟⠋⢀⣦⣀⣤⡿⠟⠋⠻⢋⡿⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢠⠀⢠⣿⠋⠻⣿⡟⠁⠘⠹⣿⣿⣿⣿⣿⣿⡸⣿⣄⠱⠎⢠⣿⢇⣿⣿⣿⣿⣿⣿⠏⠋⠈⣻⣿⠟⠉⣿⡄⠄⠄⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⣈⣠⣼⣿⣬⣀⣴⣾⣟⡿⠿⠿⠿⢷⡝⢿⣿⣾⡿⢫⡺⠿⢿⠿⢿⣻⣷⣦⣀⣤⣿⣧⣄⣁⣸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠋⠛⠉⠀⠈⢿⣿⣿⣿⣿⣿⣿⡞⠀⠀⠀⣤⣼⣥⣤⠀⠀⠀⢲⣿⣿⣿⣿⣿⣿⡿⠁⠀⠍⠛⠙⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣤⡔⢻⣿⣿⣿⣿⣿⣿⣿⣤⣤⣴⡿⣿⣿⢿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⡿⢂⣤⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡛⣟⣽⢻⣭⣭⣭⣿⠉⠀⢻⣇⠘⠇⣸⡟⠀⠉⣿⣭⣭⣭⡙⣯⣻⢻⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠯⠦⠝⠻⠷⢶⣆⡀⠀⠙⠿⢿⠋⠀⢀⣰⡶⠾⠛⠋⠤⠽⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠄⣹⣿⡄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⢛⢢⡀⠀⠀⠀⠀⢰⣿⠘⢻⣆
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣲⣦⡀⠙⢄⠈⠈⠛⡇⠀⠀⣿⠟⠁⡴⠀⣿⣦⡀⠀⡀⢄
⠀⠀⠀⠀⠀⠀⢠⣇⡀⠀⠀⠀⣿⣷⡀⠈⠁⠀⠘⢇⠀⢠⠟⢀⡄⠁⣼⢹⢿⡙⢧⠀⠈⠆
⠀⠀⠀⠀⠀⠀⣸⣿⣿⡀⠀⠀⠀⠹⢽⢦⣆⠀⠀⣄⡰⠉⠀⡼⠋⠀⢘⣾⠀⣷⠈⠣⡀⠠⠈⠳⠄
⠀⠀⠀⡀⠀⠨⣿⣿⣿⣧⠀⠀⡀⢸⣆⢸⣿⡄⠘⠁⠀⠀⠈⠀⠀⠀⠀⠹⣿⣙⣧⠀⠈⠠⣄
⠀⠀⠀⠁⠀⣠⣿⣿⣿⣿⡄⠀⣿⣼⣿⣾⣿⣗⢀⡄⠀⣄⠀⠀⠀⠀⠀⠀⠈⠿⢦⡀⠀⠀⠈⢃⠀⠠⡠⡀
⠀⠀⠀⠀⠐⠿⣿⣿⣿⣿⣿⣰⣿⣿⣿⣿⣿⣿⣸⣿⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠉⠲⣤⣀⠀⠀⠀⠀⠉⠷⡀
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠃⠀⠈⠉⠉⠉⠉⠛⠛⠿⢇⠀⠀⠈⠣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⠈⡻⠚⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⠟⠑⡧⢲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⠀⠀⠈⣙⠁⡖⣸⡅⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢤⠀⠀⡧⠀⣛⣵⣯⣏⢤⣀⡃⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡤⣲⢶⣷⣿⣿⣿⣿⣌⡹⠴⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠌⠀⠀⢸⡠⢾⡟⠉⣿⣯⢻⣿⣿⣷⢴⣃⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⠥⣃⠘⣽⣾⣧⡀⠘⢿⣿⣿⣿⣿⣳⡗⠂⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠚⠑⠂⡄⠀⠰⢿⢒⣭⣽⣬⣿⠟⠁⠀⠈⠫⣿⣿⣿⣿⣷⡆⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⠀⢀⣀⡁⠐⡿⠘⢷⣿⣿⠿⠛⠀⠀⠔⠀⠠⢹⣿⣿⣿⣷⣴⠆⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠈⢀⡀⢸⣤⠯⣯⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠐⢨⣿⣿⣿⣧⣿⠄⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣿⣷⣦⡿⣧⡟⠀⠀⢀⠀⣀⡀⠀⠀⠄⠀⠀⠈⣹⡖⢸⣿⣿⣿⣿⣾
⠀⠀⠀⠀⠀⠀⣽⣭⣾⣿⣷⣿⣿⡿⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⢀⠀⢺⡯⣿⣿⡿⢿⣏⣿
⠀⠀⠀⠀⠀⡀⣾⣿⣿⣯⣼⣿⠞⠁⣀⣴⣤⠡⡀⠄⠐⠀⠠⢀⠀⠸⣀⡸⡷⣿⠿⢳⣦⣽⣿
⠀⠀⠀⠀⠀⣡⣾⣿⣿⢿⠿⠃⠀⢰⣿⣿⢾⣆⠀⠃⠄⡀⠀⠂⢀⣊⣼⡯⣰⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠀⠀⠈⠋⠹⠼⠟⠾⣿⣿⠿⣿⢻⣿⣿⣯⣿⣧⣈⣠⣤⣨⣤⣿⣿⣿⣿⣿⣿⡙⣿⢼⠛⠓⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠲⠏⠋⣿⣾⣟⠻⣿⣿⠟⠻⠿⠟⠋⠉⠃⠈⠓⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠏⠉⠹⠟⣀⠐⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⣦⣴⠾⠻⠷⠦⠄⠀⠀⠤⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠖⢰⠆⠻⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⢀⣀
⠀⠀⠀⢐⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠂⠠⠀⢸⡇⠀⠀⢀⣀⠀⠀⠰⠿⠟⠿⣿⣿⠟⠛⠛
⠀⠀⠀⠈⠉⠠⠀⠀⠀⠀⠀⠀⣀⠀⠴⢶⠞⠂⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠿⠿⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣺⣿⢿⡷⠀⠀⠀⠀⠀⠰⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⢲⣃⣽⣽⣷⣾⣷⣄⣀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠠⠔⠂⠠⠤⠼⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣃⣀⣋⠀⠀⠀⣼⢯⣿⣾⣿⢸⣿⣟⡿⠂
⠀⠀⠀⠀⢐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠉⠘⠉⠉⢉⡈⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣷⣤⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⣀
⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⡀⠈⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠰⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠂⠊⠀⠂⠀⠀⠠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠚⠿⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠓⠗⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⠄⠘⠀⠀⠀⢀⣀⣀⣀⣄⠀⠀⠀⠁⠀⠀⠀⠀⡀⡉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠋⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⣁⠀⢀⣀⠀⠀⠀⢤⣁⠀⠈⣡⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣼⠿⡆⠀⣀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⢀⡀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⣿⣾⣇⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠉⠃
⠀⠀⠀⠀⢾⣿⣿⣿⠀⠀⠀⡄⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣯
⠀⠀⣀⣀⣼⣿⣿⣿⣦⣄⣀⣀⡀⠀⠀⠀⠀⠤⠤⠶⠶⠶⠦⠴⠀⠀⠀⠀⢴⡾⣿⠋⠹⠿⠶⣦⡄⠀⠀⠀⠀⠀⢠⣤⣤⣄⡀⠠⢠⣉⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢦⠀⠀⠀⠀⢀⣀⡀⠀⢀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠲⣽⣶⣇⠀⢠⠄⠀⠀⠀⡟⠁⢷⢜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠷⡀⠀⢧⠁⣄⠙⢿⡁⠀⡾⠀⠀⠀⣸⠁⠀⢘⡏
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣀⣤⣰⣇⡀⣠⢳⣌⠀⠀⠁⣼⡇⣄⣠⣤⡟⠀⠀⢸⡇⠀⠀⠀⢀⡀
⠀⠀⠀⢸⢀⡔⠛⠙⢯⠄⠀⠉⢉⠙⣷⣯⠀⠉⢤⡀⠀⣏⠙⠻⢿⣄⠁⠀⠀⣾⠀⠀⠀⢠⡾⠛⢆
⠀⠀⠀⠀⣿⣇⠀⠀⠀⢴⣶⢆⣴⠀⠸⣿⠀⠀⢺⠟⢤⠿⣧⣀⡀⢸⠇⠀⢀⡿⠀⠀⣴⠋⠀⡀⣸⣆
⠀⠀⠀⠀⢹⣿⣧⡀⠀⠀⠳⣼⣿⠀⠀⣿⣆⠀⢸⡀⣶⡄⠈⢻⣿⣌⠁⢷⠼⠋⣠⠞⠁⠠⠊⣴⣏⡟
⠀⠀⠀⠀⠘⣷⠛⣷⣄⠀⠑⣾⠿⠦⠀⠛⠛⠷⣼⣿⣿⣿⠀⠀⢻⡀⠀⠸⣇⠀⠉⠐⠛⠷⠾⡿⢻⠇
⠀⠀⠀⠀⠀⣿⡆⠀⢹⣧⡀⢯⣀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣧⠀⢘⡇⠀⠀⡏⢲⣶⢤⣤⠄⠀⠀⣼
⠀⠀⠀⠀⠀⢻⣯⠀⢸⣿⢻⣮⠹⠿⠿⠿⠷⠂⠴⣤⣄⡀⠻⣇⠀⣧⠀⠀⢿⠀⠛⠉⠀⠀⠀⣤⠟⠁
⠀⠀⠀⠀⠀⠧⠹⣦⢸⣿⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⡈⠉⠀⠻⣄⡟⠈⠀⠀⠀⠀⠀⠀⣠⠖⡫⠞
⠀⠀⠀⠀⠀⠀⠱⣬⣙⠿⢶⣿⣀⣀⣠⣤⣶⣤⣄⡀⠀⠉⠻⣷⣿⣡⡀⢀⣠⠤⣖⣚⣯⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⡍⠋⠉⠐⠀⠨⠍⢉⣙⣓⣙⣛⣀⣀⣀⣨⡉⠉⠁⠐⠂⠀⠈⠉⠀⢂⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣆⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⡏⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⢄⣈⡥⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠍
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠾⢠⠆⠰⠀⢸⠳⠀⠀⠀⠇⢰⠄⣤⡄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⣄⣀⣀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⢶⡏⢿⣿⣿⢿⠛⠋⣥⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⣶⡦⢯⣿⣷⣄⡀⠚⠟⠀⣰⣿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡌⠹⣿⣿⣿⣶⣶⣾⣿⣿⣿⠿⠯⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⣆⠈⠙⢿⣿⣿⣿⣿⣿⣿⡲⣄⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⣿⠏⠙⢻⣿⣶⣀⣀⣉⠙⢛⣻⣷⣹⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠟⠀⠠⠘⢋⣿⢿⣿⣿⣷⣿⣿⣳⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⡿⢻⣿⣿⣽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⠸⠛⢛⣿⠟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢿⣧⣀⣐⣾⣂⠉⠀⠀⠀⠀⠀⠶⣤⣤
⠀⠀⠀⠀⠀⠀⡰⠂⣀⠀⠀⠀⠀⠀⠀⠰⠁⣼⣿⣯⣙⣛⣿⣬⣵⣦⣤⠀⠀⠀⠹⡇
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⣼⣿⣿⣿⣿⣿⠿⠋⠁⠹⠡⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣨⣌⡽⠏⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⣿⠋
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢸⢀⡛⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣾⣟⣀⣤⠄⣀⣀⠀⠀⠀⣀⣀⠽⣧
⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣅⣀⠀⠀⢩⡏⣁⠀⢠⣿⣿⣿⣧⡈⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣦⡀⠀⠀⢀⡘⠛⠛⠛⠛⠛⠛⠒⠀⠀⠸⠷⠾⠿⠿⠿⠿⣷⡘⣆⠀⠀⠀⠀⣡⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⠖⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⠄⠀⠼⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠠⡄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢦⡉⠒⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠳⢄⠀⠙⠳⠄⡀⠀⠢⣄⡀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⠘⢆⠀⠑⠦⡀⠀⠈⢤⡀⠈⠑⢤⡁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠳⡄⠀⠱⡄⠀⠘⣦⡀⠀⠻⢆⠀⠀⠹⣄⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡿⣆⠀⠈⢦⠀⠈⠻⣄⠀⠈⠓⢀⠀⠘⠣⡌⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢷⢸⣧⡀⠀⠐⣄⠀⠈⠷⡄⠀⠈⠁⠀⠀⠈⠑⢾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣿⣆⠉⢷⡀⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢡⡀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠈⢟⢦⡘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⢳⡄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢱⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⡈⠘⢿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣸⡄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⢠⠿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣌⣷⣴⠖⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣬⣉⡻⢋⠴⠊⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠸⠹⣿⡷⢤⣀⠀⠀⠀⠀⠀⠀⠐⣤⠋⣱⠏⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⡀⠛⠀⠚⠉⡷⠦⣄⡀⠀⡠⠂⢀⡼⠋⢰⣿⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡿⠋⠻⣄⡀⠀⠀⢀⣿⡿⠋⢀⣴⣿⡇⠀⣾⠁⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠑⠒⠈⠙⠢⣴⣿⢏⣴⣾⡿⠋⠁⣀⡴⢃⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠸⣄⠀⠀⡀⠀⡿⢣⡾⣽⣿⣤⣶⡿⠋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⠳⠠⠁⠀⠈⣹⣣⣾⣴⣿⣿⣿⣿⣶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢠⣿⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠰⠛⠋⠉

⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠰⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠾⠿⠿⠄⠀⠀⠀⠀⠀⣠⣴⣦⣆⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⢰⣿⣻⣿⣿⡗⠀⠀⠀⠶⢿⠉⠁⠉⢻⠿⠂⠀⠀⠀⡿⠉⠀⢿⠀⠀⠀⢠⣿⣋⠉⣉⣿⡆⠀⠀⠀⣬⠔⠿⢣⡄
⠀⠀⠀⠀⠀⠀⠑⠈⠀⠉⠃⠀⠀⠀⠀⠀⢀⠚⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⡁⠀⠀⠘⣿⡇⠀⠀⣿⡇⠀⠀⠀⠛⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣈⣀⣁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠰⣛⣳⣮⣟⡇⠀⠀⠀⠈⠻⠀⠘⠋⠀⠀⠀⠀⠀⡐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠃⠀⠁⢰⠀⠀⠀⠃⠀⠀⠀⠁⠈⣦⢹⡀⠀⠀⠀⠔⠧⡀⢠⠗⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⢰⠰⠀⠐⠁⠀⢠⠀⠀⠀⠀⠀⢠⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠿⢬⠁⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄⠀⠀⠀⠀⢀⣤⠤⣤⡀⠀⠀⠀⠀⣠⣶⣶⣦⠀⠀⠀⠀⠀⣴⠿⠿⣿⡆⠀⠀⠀⠀⢠⣴⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠸⠿⠟⠋⢿⠀⠀⠀⠀⡸⢏⠀⠙⠇⠀⠀⠀⠀⠿⠉⠉⢹⡄⠀⠀⠀⠀⢛⠀⠀⠈⡇⠀⠀⠀⠀⢸⠛⠉⠁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣶⡀⠀⠀⠀⠱⣤⣦⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡈⠁⠈⡀⠀⠀⠀⠀⠀⡈⠀⠹⣿⠃⠀⠀⠀⣀⠍⠉⠁⣀⠀⠀⠀⠀⠠⠆⣠⣰⠦⠀⠀⠀⠀⢰⠈⠀⠈⠴⡄
⠀⠀⠀⠀⠀⣴⣿⣿⣾⣾⣿⣶⡄⠀⠐⠀⠇⠤⠴⠀⠛⠁⠀⠀⡻⠄⢀⠰⠇⠀⠀⠀⠀⠘⠦⠠⠤⠇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣤⠀⠀⠀⣤⠀⠀⠀⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⡇⢀⠀⢀⠀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⣴⣿⡿⢿⣦⠀⠀⠀⠀⠠⠾⠟⠽⠦⠀⠀⠀⠀⢠⣶⣿⣷⣆⠀⠀⠀⠀⢀⣤⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠘⠤⠄⠆⠛⠀⠀⠀⠀⡟⠛⠀⠚⢻⠄⠀⠀⠀⢰⠖⠛⠓⢶⠀⠀⠀⠀⢸⣿⣿⣯⣿⠀⠀⠀⠀⣿⣿⠉⡻⢿⡄
⠀⠀⠀⠀⠀⠀⠈⢀⠀⠀⠈⠀⠀⠀⠀⣿⣄⠀⢠⣿⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⠘⣧⣴⣤⡏⠀⠀⠀⠀⣿⡇⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠔⢀⠐⠄⠀⠀⠀⠀⣿⠋⠀⠈⠿⡀⠀⠀⠀⢀⠉⠐⠈⠁⠀⠀⠀⠀⠀⢙⠛⢛⠀⢀⠀⠀⠀⠙⡻⠀⠘⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣿⣦⣄⡀⢀⣠⣾⡄⠀⡀⠀⢀⣄⣠⡀⠀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⠓⢺⣿⣿⡇⠀⠀⠀⠀⠸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠠⡞⠀⠀⢱⣾⡟⠀⡄⠄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⠁⠀⢀⣠⢞⣵⣤⣾⠃⠀⠀⠐⡽⣷⣜⣿⠀⠀⠈⣶⠂
⠀⠀⠀⠀⠀⠀⢀⠾⣧⡀⠀⠢⠀⠉⠛⠿⣿⣧⡀⠀⠀⠀⠘⠈⣙⣿⠁⠀⠀⠸⣆
⠀⠀⠀⣠⢴⡯⠁⠀⠉⠓⢠⣄⢤⠀⠀⠀⠈⠓⢷⡀⠀⠠⠀⠀⠙⣿⡶⠀⠀⠀⠉⠂⠀⠀⠀⠠⣤⠞⠀⣠⡎
⠀⠀⠈⠛⠻⢧⡀⠀⠀⠀⠀⠈⠙⢶⣤⣀⠀⠀⠀⠀⠀⠀⢀⡀⠘⠣⠷⣿⣦⣀⠀⠀⠀⠘⢷⣤⣄⠀⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠹⣶⡄⠀⠀⠀⢀⡀⠉⠙⠃⠀⠀⠀⠀⠀⡼⠃⠀⠀⠘⠉⢿⡟⡠⡀⠀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠶⠚⠀⠀⠀⠀⡆⠀⠋⣠⣾⠇⠀⠀⠁⠀⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⣴⡀⠀⠀⡔⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⣤⠀⠀⠀⣰⡀⠀⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⡠⢺⠏⢀⣭⡾⠃⠀⠀⠀⠀⠀⢠⣿⣿⡀⠀⢠⣿⡇⠀⠀⣿⡇⠀⠀⡀⠈⠙⠛⠒
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠾⢿⣧⢀⢠⣿⣇⠀⣸⡗⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠚⠩⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠾⣿⡯⠀⣿⣿⣸⣿⡆
⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠻⠟⢼⣿⡷

⠀⠀⠀⠀⢠⠆⠀⢀⠠⠤⢤⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠠⢀⠤⢲
⠀⢀⠀⠀⠀⠀⠀⢘⠉⠉⠉⡁⠀⠀⠀⠀⠀⢠⠕⠠⢀⡀⠀⠀⠀⠀⢠⠖⠉⠉⣐⣂⠤⠚⠚⠉⣰⠔⡇
⡠⠫⣻⠫⣀⠼⢟⣟⣀⣀⠀⠅⠀⠀⠀⠀⠀⠸⠀⠄⠀⠇⠀⠀⠀⠀⡄⠿⣴⡪⡽⢁⡠⢂⣲⡮⢔⡾⢃
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⠧⠄⠀⠀⠀⠀⠀⠈⠲⢤⣉⣙⠚⠛⠕⢊⣉⣡⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⢀⣤⣄⡠⠊⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠰⢂
⠀⠀⠈⣷⠋⠀⢀⠀⠋⢂⠀⠀⠀⠀⢀⣀⠀⣦⣴⣋⣉⠖⢫⠀⢀⡀⠀⠀⠀⠀⢨⠛⠓⠚⠓⡀
⠀⠀⠐⡯⠒⠒⢒⠚⢓⢿⠇⠀⠀⠘⣄⣦⣏⣏⠁⠀⡘⡾⢞⢳⣿⡌⠀⠀⠀⠀⠸⣶⣦⣤⣾⠇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⣽⢧⢭⣋⠽⠶⢺⣭⣠⠵⢊⡇⠀⠀⠀⠀⠀⡆⠀⠀⢠
⠀⠀⠀⠱⣄⡀⢀⠀⣀⡜⠀⠀⠀⠈⢫⡑⠒⠦⠵⡀⢀⠾⠗⢚⡽⠁⠀⠀⠀⠀⠀⠇⠀⠀⠆
⠀⠀⠀⠀⠘⠛⠋⠛⠛⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠿⠿⠚⠛⠁⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⡆
⠀⠀⣠⢴⠒⠒⢲⣤⠆⣰⡶⣢⠀⠀⠀⠀⠘⠣⣄⣀
⠀⡘⠁⡐⣰⡾⠋⡉⣞⠋⢞⠉⠀⠀⠀⠀⠀⡖⠋⠀⠉⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠢⢄
⢰⣷⣚⢿⣽⠀⠮⣀⠴⢀⣼⡆⠀⠀⠀⠀⠰⠛⠒⠒⠒⠚⢢⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡁
⠘⣟⠻⠿⢶⣿⣶⡶⠿⠟⢻⠇⠀⠀⠀⠀⢸⠸⣄⠀⣀⠆⡀⠀⠀⠀⡤⠎⢠⡀⠁⠀⠀⠀⠀⢁⠀⠘⠄
⠀⠘⢿⣒⠤⠄⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠑⢲⠛⡗⠊⠁⠀⠀⠀⠀⢇⠻⠿⠟⠀⠀⠀⠀⠰⠀⡠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⡿⢁⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣄⡈⠻⢿⣶⣴⣾⣿⣿⣿⣄⣰⣿⣷⣤⣄⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⣹⣿⠁⠐⢛⣿⢿⣿⣿⣿⣿⡙⠻⢛⡛⠿⠿⣿⣴⣿⣿⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣾⡿⠷⢶⡄⠘⠋⠘⢿⣿⣿⣿⣇⢀⡻⣓⠾⡷⠛⢹⡟⢿⣒⣐
⠀⠀⠀⠀⠀⣴⣶⠿⢿⣷⣧⣤⠰⠶⢿⣾⣿⡟⠀⠀⠀⠉⠉⠈⠿⣿⡷⣾⡿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⢻⣥⣶⡿⠟⠠⣴⣿⣶⠊⠉⠉⠁⢀⡀⢀⡀⠀⣤⣀⣀⣀⡛⠡⠀⢍⣷⡿⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠫⣿⢗⣼⣶⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿⡿⡇⠀⠀⣾⠿⢀⠀⣀⠏
⠀⠀⠀⠀⢀⣠⣴⣿⣾⣛⠉⠈⠻⣿⡀⠀⠀⠀⠀⠀⠀⠋⠀⢀⡀⣿⣧⣶⣶⣦⣿⠇
⠀⠀⠀⠀⠘⠟⠉⠻⣿⡟⠀⢀⣠⣿⣧⠀⢠⣶⣿⣶⣀⡀⠁⠀⠀⠉⣏⡈⢉⠉⠷⣄⣀⣠
⠀⠀⠀⠀⠀⠠⠈⠁⠙⠇⢸⣿⣿⡏⠉⣲⢾⠁⠀⠀⢹⣷⠾⣿⣿⣷⣧⣴⣦⡴⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠈⠈⠀⠀⠈⠻⣿⣿⠁⠀⠬⠭⣤⣦⣤⣴⣿⠀⠸⣿⣿⠿⠙⠛⡇⠀⠀⠀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⠀⢘⣛⣛⡛⠛⠛⣾⣿⣷⡄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⠀⢹⡃⢀⣴⣿⣿⣶⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡄⠀⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢗⢀⠃⠁⢐⠸⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣴⡰⣲⡄⠀⣤⡟⣥⡏⠐⢀⠀⠄⠂⣺⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢹⠐⣿⠁⣼⡟⠫⣴⠀⢃⠄⠐⠈⡌⠎⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⢀⡜⢳⡆⠀⡖⣿⠀⣞⣀⠢⠉⢐⠒⠠⠁⢀⡂⠠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⢳⡄⢰⡀⡇⢸⠀⣿⠂⡎⠃⠐⠂⠠⠀⠠⢂⣠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠻⡄⢳⣼⢸⡄⣸⣆⢣⠁⠀⠀⠄⠀⠄⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠹⡄⠹⣶⠇⠘⣷⡾⡊⠲⠆⢀⠰⢠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⡄⢿⣷⠀⠘⡿⣵⣬⢁⣂⡲⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡿⣎⠌⢦⠀⠈⢮⠉⢉⠽⠊
⠀⠀⠀⠀⠀⠀⡇⠘⣇⠈⢳⡀⢸⠀⡌
⠀⠀⠀⠀⠀⠀⡇⠀⠻⣆⠀⠙⣯⠀⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⡀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠒⠒⠖⠒⠤⣄
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠷⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⡰⠋⠑⠀⠀⠀⡀⠂⠀⠀⠀⠈⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠃⠀⠀⡁⡄⣷⡀⠀⠀⠈⢦⠀⠀⠀⢀⡞⠀⠀⠀⠈⡶⠊⠀⠀⠀⠀⢀⡀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡇⣿⣧⠀⠀⠀⠀⢣⠀⠀⡸⠀⠀⠠⠐⠈⠀⠀⠀⠀⡠⠂⠁⠀⠀⣀⡤⠞⠓⠒⠒⠒⡆⢄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⣧⣿⣿⢧⡀⠀⠀⠀⢳⡖⠉⠚⣅⠀⠀⠀⠀⠀⢀⠚⢄⠀⠀⣠⠞⠁⢣⡀⠀⠀⠀⠈⠙⠢⡈⢢⡁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⣏⠁⠘⢷⡄⠀⠀⠀⠱⡀⠀⠘⣦⠄⠀⠀⠀⠂⠀⠀⠑⠴⠁⠀⢠⣿⣿⣿⡇⠀⠀⠀⠀⠈⣿⣷
⠀⠀⠀⠀⠀⠀⢸⡄⢀⡔⠛⠃⠷⣄⠀⠻⣄⠀⠀⠀⠰⡄⠀⠘⡄⠀⠀⡆⠀⠀⠀⢀⣧⡠⠔⠈⠻⣿⠿⠗⢀⠀⠀⡀⠄⠛⠿
⠀⠀⣠⠒⠉⠁⠙⣇⠬⠷⢀⠀⠀⠈⠳⠤⠈⢂⠀⠀⠀⠘⢄⡰⠃⡏⠈⠁⠂⠀⠀⠸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⡇⠀⠀⠀⠈⢧⡀⠀⠀⠉⠒⠤⠄⠀⠀⠘⢧⡀⠀⠀⠈⢣⣀⠘⢆⠀⠀⠐⠀⠀⠷⢄⠀⢀⣀⣄⠀⠀⠀⠀⡀
⠀⠀⢣⡀⠀⠀⠀⠎⠙⡇⢄⠀⠀⠀⠀⠀⠀⠀⠈⠑⣄⠀⠀⠀⠹⡑⡄⠉⠈⠹⠤⠤⠒⡀⠙⣿⣿⣿⣷⡄⠀⠈⠁⠀⢄⣀⡰
⠀⠀⠀⠉⠀⠀⠒⠛⢄⣘⣦⡽⠂⠀⠀⠀⠀⠀⠀⠀⣈⣦⠀⢀⠀⣧⣸⠀⠔⠃⠀⠀⠀⠑⢤⠘⠻⠿⢏⠀⠀⠀⠀⠀⣸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠊⠑⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠤⠈⠧⠄⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠖⠢⡀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⢤⡀
⠀⠀⠀⢀⡴⠒⢋⠉⠓⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠐⠀⠀⠠⢹⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢹⠄
⠀⠀⠀⡎⠤⠠⠂⠲⠅⠀⣧⠀⠀⠀⠀⠐⣿⠽⠯⠿⠽⢿⡇⠀⠀⠀⠀⠀⠐⣏⠴⣠⣄⣰⡿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸
⠀⠀⠀⢿⠀⠘⠢⢴⡁⢀⡏⠀⠀⠀⠀⠀⢸⠀⠀⠀⠒⢸⠀⠀⠀⠀⠀⠀⠀⠸⡆⠡⠈⣼⠁⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⣰⠁
⠀⠀⠀⠈⠓⠧⣤⠤⠴⠋⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢻⡐⣱⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠾⠃
⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢠⠖⡛⠉⠉⢉⡉⡙⢦⠀⠀⠀⠀⠀⠀⢀⣖⢒⣆⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠚⠓⠲⣄⠀⠀⠀⠀⠀⠀⣴⣞⣿⣿⣿⡶⡄
⠀⠀⠘⣎⡔⠜⠓⢼⠵⢈⠟⠀⠀⠀⠀⠀⢰⡉⠤⡠⠭⣿⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣘⡇⠀⠀⠀⠀⠀⢿⣯⢿⢹⢻⣽⠇
⠀⠀⠀⠘⣄⣸⡃⣭⣰⠏⠀⠀⠀⠀⠀⠀⢸⠄⣀⡼⡀⣼⠂⠀⠀⠀⠀⠀⣷⣦⣤⣼⣧⣤⣴⡇⠀⠀⠀⠀⠀⢻⠛⠛⠛⠛⡗
⠀⠀⠀⠀⠘⣏⡵⡴⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡤⠜⠁⠀⠀⠀⠀⠀⠈⠲⣀⣓⠉⢀⢛⡸⠃⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠂⠀⠀⠀⠀⢠⠋⣿⢁⣾⡇⢳⠀⠀⠀⠀⠀⡶⢷⣾⡶⣄
⠀⠀⢠⢴⠞⠛⠛⠛⠻⢦⣄⠀⠀⠀⠀⠀⠀⢹⣿⣿⠠⠀⠀⠀⠀⠀⠀⠀⡜⠀⡟⠻⣴⡇⠘⡇⠀⠀⠀⠀⡏⢦⢻⣿⣿⣆
⠀⢀⡿⠁⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⢴⡖⠚⣛⣛⡒⢲⡆⠀⠀⠀⠀⠀⣧⠀⣫⡴⢛⡆⠀⡇⠀⠀⠀⠀⡇⠀⠱⡻⣿⣿⣄
⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣿⠀⠀⠀⢸⣤⣀⣀⡀⣀⣸⠃⠀⠀⠀⠀⠀⢣⠀⢿⠙⢺⡇⢠⠃⠀⠀⠀⠀⡇⠀⠀⠙⢝⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡏⠛⠛⠻⠟⢻⠀⠀⠀⠀⠀⠀⠘⣄⣸⣞⢋⣧⡼⠀⠀⠀⠀⠀⠧⠤⣤⣄⣈⣮⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⡏⠉⢳⠏⠉⢷⠀⠀⠀⠀⠀⠀⠰⠀⠀⣀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠉⢱
⠀⠀⠀⢠⣇⡀⣀⢆⣀⣾⠀⠀⠀⠀⢀⡤⠚⠉⠁⠆⠉⠑⢦⡀⠀⠀⠀⠀⣀⢔⠚⠿⠟⢷⢦⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⡾⡛⠁⠀⠀⠀⠀⠙⣿⡀⠀⠀⠀⠀⠀⢸⡀⠀⣸
⠀⠀⠀⢸⡀⠀⠀⠀⠀⡞⠀⠀⠀⠸⣽⡀⠀⠀⠀⠀⠀⠀⢀⣞⡶⠀⠰⡟⠀⠀⠀⠀⠀⠀⠀⢈⣷⠀⠀⠀⠀⠀⠸⡀⠀⡸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠙⠻⣖⡒⠒⣲⠾⠛⠁⠀⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠈⡏
⠀⠀⠀⠀⠓⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤
⠀⠀⠀⠀⠀⠀⡔⠶⢖⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡿⢿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠐⠈⠱⡤⢈⣏⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⡆⢠⣻⣓⢲⣹⡇⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠘⠙⢻⠶⠊⡁⠀⠙⠃⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⣟⠁⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⣠⠄⣿⠡⣤⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⡀⠴⠆
⠀⠀⠀⠀⠀⡿⠿⣶⣷⣾⠀⠀⠀⠀⠰⣀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠿⠜⣿⣧
⠀⠀⠀⠀⠀⠌⠂⠐⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠚⠻⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠀⠀⠀⢀⠀⠺
⠀⠀⠀⠀⠀⠞⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⣭⣧⣀⠀⣀
⠀⠀⠀⢰⣤⣀⡈⣯⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢰⣖⣐⢁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⡽⢉
⠀⠀⠀⠀⠉⠙⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣰⣾⣇⣹⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠡⢤⣶⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠙⠋⠉⢉⣿⣷⣴⣿⣦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢴⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢻⣿⣿⠁⠉⠉⠢⠀⢀⡐⢀⠀⠀⠀⠀⠀⠀⢠⡟⠘⠛⣿
⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠋⠁⠀⠀⠀⠀⠘⠂⠠⠎⠀⠀⠀⠀⠀⠀⠀⠀⣠⣢⠆
⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠑⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢠⢤⣦⡀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⢀⣴⣥⠞⠀⠀⡠⢴⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠚⠶⡀⠀⠀⠀⠀⠠⢿⣀⠸⣄⣁
⠀⠀⠀⣀⠀⡷⠟⠁⠀⠀⠀⠹⠯⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡁⠀⠁⠠⣤⡀⠀⠀⠀⢀
⠀⠀⠀⠈⠳⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣆⡬⠀⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠀⠈⠹⠾⠇
⠀⠀⠀⠀⠀⢻⣿⣶⣤⣀⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣭⣿⣇⡀⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣮⠁⠀⡀
⠀⠀⠀⠀⠀⠈⣿⣿⡟⠛⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣽⠟⢋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⢿⠉⠉⠁⠛
⠀⠀⠀⠀⠀⠀⠈⡹⣿⣴⠻⢿⣆⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣟⣤⣧⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠉⢰⣥⣤⠆⠈⠙⠂⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣿⠶⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠘⠀⠜⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠉⠂⠀⡛⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠈⠒⢤⡀⠘⣜⣷⣦⣭⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⡿⢶⣄⠀⠀⠀⠉⢦⡈⠛⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠀⣀⣼⣿⡻⢇⢤⡀⠀⠀⠀⠹⣄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣧⠀⠀⠀⠀⠀⠀⠀⢀⣤⣯⣛⠇⠫⠻⠏⠁⠀⠹⢿⣶⣄⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣧⡀⠀⠀⠠⡤⣤⣸⡟⠋⠉⠛⢯⠈⠳⣄⡄⡀⣼⣿⣿⡆⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⠀⠀⠈⣡⣶⣺⡟⢶⣜⠦⣴⣿⡝⢯⣾⣺⣿⠇⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⣴⡾⠟⣿⣿⠷⢋⣙⣷⠉⠿⠻⡾⠉⠉⠀⠀⢲⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⠀⠀⠀⢀⣨⡍⠀⠜⠁⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⡀⠀⣤⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡳⣄⠀⠈⢛⡞⠁⠀⠀⠀⠀⢠⠾⠋⡀⠄⣠⣾⣿⣬⣾⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⠀⠀⢠⡛⢦⣄⡀⠀⠏⠀⠀⠉⠀⣘⣿⡿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⡀⢷⣄⠈⠙⠛⠓⠒⠚⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⣿⠈⢷⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⢟⣴⣦⣀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⣀⣀⣀⡁⢤⠀⡄⢠⠀⠀⠒⢠⣄⢠⠀⠂⠀⠀⠀⠀⣿⠀⢹⡆⠘⢧⡀⠀⠠⠛⠶⠀⠀⠀⠂⠀⠀⠀⣠⠶⣥⣛⠶⣮⠃⠀⠀⠀⠀⠀⠀⢀⣰⢶⣄
⠀⠀⠉⠙⠉⠁⠘⣄⠏⠸⠀⠀⠰⠶⠶⠶⠶⠛⠀⠀⠀⠀⠻⠇⠈⠁⠀⣰⠇⠀⠀⠤⠰⠶⠔⢠⠀⠀⠀⠰⢫⣾⢱⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⣾⡗⣬⠟
⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠦⠶⠞⠉⠀⠀⠀⠀⠀⠀⢄⠶⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠔⠢⡀
⠀⠀⠀⠸⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣠⣄⠾⠂⢠⠀⠀⠀⠷⣿⣷⣿⠿⠃⠀⠀⠀⠴⠀⠉⠒⢄⢀⡀⠀⠀⠀⢰⣶⣶⣶⣾⣧
⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⢠⠖⠐⣮⠙⠃⠀⠀⠀⠀⢷⡆⠉⠀⡠⠋⠀⠀⠀⡀⠀⠈⠁⠀⠀⠀⠀⠀⣀⣾⠃⣀⡀⠙⠷⠀⠀⠀⠈⠉⣙⠿⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⢰⠈⠁⠈⢁⡀⠐⠀⠀⠀⢀⡀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠶⠀⠰⠏⠁⣀⣉⣁⡀⠀⠀⠀⠀⠀⠀⠉⠀⢀
⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠉⠛⠉⠉⠉
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠰⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠋⠉⠁⠂⠀⠀⠀⢠⡀⠀⢛⠿⠀⣀⠀⠀⠀⠀⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇⠀⠀⠀⠀⠀⢀⠁⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⣠⣄⢀⡀⠀⠀⢴⠁⠐⠀⠀⠀⠀⢸⡇⠀⠀⠙⠂⠀⠐⠀⠈⠀⠀⢀⡀⠀⠀⠀⢀⣀⠀⠀⠀⢉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠞⠉⠁
⠀⠀⠀⠀⠀⢨⣯⡹⡭⠀⠀⠀⠁⠀⠰⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠟⡷
⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⢐⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⣦⠀⠀⠀⠀⠀⢀⡀⠲⡀⠚⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠛⠃⠀⡘⠛⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠉⠀⠆⠀⠀⢁⡇⠀⠀⠀⠄⠘⠃⢀⡵⠚⠃⠀⠤⠀⠀⠈⠂⠀⠃⠀⠀⠀⡟⠻⠿⠿⠛⢋⢳⡆
⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⡄⠀⠀⣤⠀⠀⠀⠀⠀⢤⡄⠀⠠⢦⣤⠇⠀⠐⠲⠖⠲⠟⠁⠀⠀⠀⠀⠲⠐⠆⠀⢠⡤⠄⠀⠀⠀⠈⠢⠄⠀⠴⠋⠉
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠑⣮⠀⠀⠀⠀⠀⠈⢀⠀⠐⠀⠀⠀⠀⠀⢀⠄⠴⣿⡟⠀⠀⢀⠋⠉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠒⠒⢠⡆⢠⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠠⠀⢀⡤⠈⠀⠀⠀⠈⠀⣄⠁⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⣄
⠀⠀⠀⠀⠀⠠⠀⠀⡀⠠⠀⠀⠠⢾⣥⣾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠊⠀⠀⠀⠀⠀⠀⠉⠉⡆⠀⠀⠉⠀⠀⠀⠈⠀⠂⠰⣆⣶⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⠘⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠊
⠀⠀⠀⠀⠸⠷⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢟⡻⢿⣧⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠶⠶⠀⠀⠰⠾⠦⠀⠀⠀⢰⠆⢀⡛⡄⠀⡀⠀⠀⠀⣠⣶⣶⣴⡆⠐⠓
⠀⠘⢶⣶⣄⣤⣴⣶⡞⠀⠀⠈⠛⢧⣼⣿⡶⠟⠀⠀⠀⡀⠈⠁⠈⠀⡀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠂⠀⠰⠃⣠⢻⡿⠓⢾⠉⢓⡄⠘⠿⠶⣤⡞
⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠶⠞⠃⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⢾⣤⠴⠛⠓⠋

⠀⠀⠀⢀⣤⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⢿⠆
⠀⠀⠀⠀⠉⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠤⣴⡄⠀⠀⠀⠀⠘⠿⣿⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢷⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠖⢠⡀⠀⠀⠀⠀⠀⠂⠀⡈⠀⠀⠀⠀⠀⠀⠀⠹⠿⠀⠀⠀⠀⠀⠄⢀⡤⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢲⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠓⠇⠀⠀⠄⠀⠀⠀⠙⠷⣶⣶⠶⠟⠁
⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦
⠀⠀⠀⠆⠈⠄⠀⠑⠄⠀⠀⠀⠀⠀⠀⣀⠀⠠⠔⠀⠀⠀⣤⣙⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣰⣧⡄⠀⢀
⠀⠀⠀⠴⠆⠀⣤⣾⠃⠀⠀⠤⠤⠤⠌⠉⢫⠀⠀⠀⠀⣤⣹⣿⣧⣄⠉⠀⠀⠀⠀⠀⢠⡄⠐⢳⡄⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⢠⡤⠲⡦⠴⠶⠀⠀⠂
⠀⠀⠀⠤⠄⠻⣄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⠋⣻⡑⢫⣧⠀⠀⠀⠀⠤⠸⠃⠶⠊⠀⠀⠀⠀⠿⠟⠀⣀⢠⣦⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠂⠐⠛⠀⠀⠀⠀⠀⠀⠀⠻⠋⠀⠉⠀⠀⠀⠈⠤⢀⣀⣀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠉⠉⠉
⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⣀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡦⠄⠀⠀⠀⣀⠐⠂⠀⢠⡤⠀⠀⠀⠀⠠⠶⠶⠦⠀⢀⠀⠀⠀⠀⠛⠏⠀⠀⣤⢶⠒⣄
⠀⠀⠀⣤⣴⣶⡆⠀⠀⠀⠀⠰⠀⠀⠀⠹⠀⠀⠀⠀⣴⠿⠛⠉⠀⠀⠀⠘⠀⠀⠁⠀⣄⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⡌⠄⠀⠀⠀⠁⠂⠂⠀⠐⠫⠜⠋
⠀⠀⠀⢉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠠⠟⠃⠀⠀⠀⢈⣿⡷⠲⠶⠿⠓⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⡿⢿⣿⡿⠛⢿⡿⣿⡿⣿⠿⢿⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣥⣤⣼⣷⣠⣾⣠⣴⣷⣬⣴⣿⣥⣾⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⢄⡀⠀⠐⢄⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣀⠀⠀⠒⢦⣛⢦⡈⢾⣷⠀⠈⢀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠠⠜⠂⣠⠙⣿⣶⣦⣸⣿⡂⣿⡤⣿⡆⢀⣸⠀⡀⣸⠞⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠜⠳⣾⣿⣿⣿⣿⣾⣷⣽⣿⢸⣿⣾⣽⣽⡾⢁⠄
⠀⠀⠀⠀⠀⠈⠀⠚⠉⢀⡓⣤⣤⣙⣧⣹⡟⠁⠀⠈⠙⠺⢿⣿⣿⣿⣽⣉⣤⣴⠖⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡄⣤⠀⠀⢈⣛⣛⣻⣷⠀⠀⠀⠀⠀⠀⠀⠈⠭⠛⠛⢿⣿⣯⣉⣀⡤⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠓⠿⣾⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⣥⣄⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢣⣼⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣾⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠸⢛⣯⣿⣷⠀⠀⠀⠀⠀⠀⢴⣆⠀⠀⠾⠛⠀⠀⠀⠀⠀⣠⣼⣿⠷⣬⠄
⠀⠀⠀⠀⠀⠀⠀⠄⠚⠉⠀⠡⠛⠀⠀⡠⠀⠀⡄⢱⠚⠛⢣⠀⠂⠀⠀⢀⣾⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⢳⣤⣶⣿⣿⠳⢿⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⡟⢻⡁⠙⠂⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⣿⣿⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠁⠀⠠⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠃⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣦⡀⠀⢀⡴⢦⠤⢤⢴⠟⢡⠶⢒⣻⡶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣯⣷⣴⠋⠙⠂⠀⢀⠂⠀⣻⣶⡿⠿⠿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠉⠠⢣⣌⠀⠐⡃⢠⣻⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡶⠳⢄⣟⡡⠂⠀⠀⠀⠰⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠰⠋⠀⠀⠀⢀⠠⠔⣰⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢰⣏⠀⠄⢀⠁⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡩⠿⠋⠀⠀⠈⠀⠀⣼⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠅⠴⠒⠠⠀⠠⠀⠂⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢣⠃⡄⠈⠂⠀⢀⠊⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠀⢷⡀⠂⢀⠐⠀⢠⣿⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠸⠇⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⢀⡀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣀⡘⣁⡀⢀⣾⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⠖⠾⠟⠀⠀⣠⣾⣿⣿⣿⠟⣿⠿⢿⣿⣿⣧⣌⣉⣩⠿⠿⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡎⠉⣩⣤⣔⠀⠈⠀⢸⣿⣿⡟⠉⠀⠀⠀⠀⠈⠿⠿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣿⣿⣿⡇⣿⣿⣩⣽⣦⠑⢠⣾⣿⣿⣷⠀⠀⠀⠀⠀⠦⠀⠀⣨⣿⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣮⣿⣿⠿⠿⣲⣄⠸⣿⣿⣿⡖⣀⡀⠀⢠⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢲⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣩⡽⠛⣋⣵⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣿⣿⣿⣿⢻⣿⣿⠿⣿⣿⣿⣿⠋⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣿⣿⡿⠋⣿⣿⠋⠵⠚⠉⠁⠈⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠻⠿⠧⠈⠩⠀⠀⢀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⢿⣽⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣿⣅⠀⡰⠀⠀⢠⣠⣾⣿⣄⣴⠾⢋⠁⣁⣀⠀⠤⠛⣁⢀⢰⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣧⣸⣤⣾⣿⣿⣿⢏⣁⠴⠊⠀⢹⡄⠡⠠⢾⣟⢢⣸⣿⠀⠀⢀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢘⣭⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⡿⠉⠂⢀⣐⣢⡶⣾⡇⢸⣿⣨⣿⣿⠿⠛⠛⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⣿⣿⣿⣿⠏⠁⢸⡏⠙⠛⠟⠛⠋⢀⣉⣠⣽⣿⣷⣦⣍⡤⠿⢿⣿⠟⠁⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⢤⣷⣤⡀⠀⠀⠀⠈⡻⢿⣿⣿⣿⣿⣿⡇⢠⣿⣬⠀⠄⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⣿⣭⣶⣾⣾⡇⣤⠀⠈⠋⠁⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿⠀⢾⣿⠇⣀⣴⢀⡀⠀⠁⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡛⠿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⢿⣦⣉⣀⣿⣿⣿⣷⣤⣶⣶⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠒⠠⠘⠟⠀⠀⣸⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠠⠹⠿⠿⠿⠿⠒⠾⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⡛⠻⠿⢿⡿⠿⠷⠄⠜⠛⠛⣿⣿⣶⡞⠃⠀⠘⣿⠿⣿⣋⣉⠚⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢦⣿⣿⣦⡜⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⣄⠀⡀⢭⠔⠀⠀⠉⠁⠀⢹⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡄⠀⠀⠀⠀⠀⢀⠻⣿⣿⣿⣿⣿⡀⠀⢀⠴⢤⠄⢀⣴⣋⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠿⢿⣿⣶⣄⣤⠀⢀⣀⣼⣿⣿⡿⠛⠁⣠⢄⠱⣦⣄⣴⡿⢟⣛⣑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⣶⣆⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣾⣿⣾⡿⢟⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀⠀⠙⠿⣿⣧⠉⠙⠛⠋⠺⠷⠿⠛⠻⣽⣴⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄⣀⡀⠤






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣤⣴⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠉⠉⠉⠀⠀⠀⠀⠁⠄⠠⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠄⠀⠀⠀⠀⠐⠆⠀⠀⢣⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⡆⠀⠀⠀⠐⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠄⠀⠀⠀⢀⠄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠄⠄⠈
⠀⠀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠂⠀⠀⠐⠖⠀⠀⣀⡀⢀⠠⠐⠋⠀⠀⠀⠀⠀⠠⠀⠀⡔⠁⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⡠⠤⣀⠂
⠀⠀⡐⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢐⡰⠒⠀⠁⠀⠄⠀⠀⠀⠢⢄⡈⠀⢀⣀⡠⠊⠀⠀⠀⢀⠈⠄⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠐⠒⡒⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠋⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠀⠀⠀⠀⡠⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠈⠉⠑⠒⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⡤⠊⠀⠠⠄⠐⠂⢐⠆⢀⠠⠗⠀⠐⠒⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠄⠀⠀⠀⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠙⠓⠂⠀⠀⠀⢀⡄⠀⠒⠛⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠉⠩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⠀⠀⠀⠤⠤⠤⠌⠀⠀⠤⠄⠀⠤⠤⠀⠂⠂⠩
⠀⠀⠸⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠙⠲⠶⠒⠋⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠲⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣯⣓⢦⡀
⠀⠀⠀⢀⡴⢋⠡⠈⠄⡀⠍⠙⣦⠀⠀⠀⠀⠀⣰⣒⣚⣛⣛⣶⡀⠀⠀⠀⠀⠀⢸⡇⢸⡀⠀⠀⠀⠀⠀⣠⢞⣩⠴⣿⣼⣚⣍⣳⣽
⠀⠀⠀⢸⡧⠬⠤⠤⠤⠤⠤⠤⣼⠆⠀⠀⠀⠘⣧⡤⠤⠤⠤⣤⠇⠀⠀⠀⠀⠀⣞⡉⠉⣧⠀⠀⠀⠀⣼⢡⢏⠤⡅⡏⠿⢟⣫⡵⣆
⠀⠀⠀⢸⡗⠛⢒⣒⣛⣒⠒⠓⢺⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡟⠀⠀⠀⠀⠀⠸⡇⢹⡇⣸⠀⠀⠀⠀⣇⢸⠘⢒⠋⠷⢚⣭⡀⡇⣿
⠀⠀⠀⠀⢿⣙⣯⡉⠉⢉⣽⣋⡟⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⡟⠉⠀⣿⠀⠀⠀⠀⠹⡜⢧⡻⠗⣡⣌⣋⡼⣡⠏
⠀⠀⠀⠀⠀⠉⠉⠙⠶⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣸⠃⠀⠀⠀⠀⠀⢠⣇⣀⣀⣻⠀⠀⠀⠀⠀⠙⠦⣍⣛⣓⣛⣩⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⣠⣞⢻⡏⣿⢓⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢿⡳⣄
⠀⠀⠀⢸⠷⠾⠮⠷⠧⠾⠶⣿⠀⠀⠀⠀⠀⠀⣤⣤⣯⣯⣤⣄⠀⠀⠀⠀⠀⣠⠖⠋⡀⢀⠐⡈⠓⢦⡀⠀⠀⠀⠀⣰⢋⠀⡁⢈⠙⣿
⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⢛⡧⠀⠀⠀⠙⣶⠓⡶⠚⠲⡖⢳⡞⠁⠀⠀⠀⠀⣏⠠⠀⠄⣠⡼⠁
⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣀⣀⣿⣀⣀⣀⣀⣟⣋⣀⠀⠀⠀⠀⢸⡄⢹⠀⢸⠃⣼⠀⠀⠀⢀⣤⠞⣩⠷⠖⠛⠁
⠀⠀⠀⠀⠸⢤⡤⠤⠤⢤⠼⠀⠀⠀⠀⠈⠧⣄⣀⣀⣀⣀⣀⣀⣤⠟⠀⠀⠀⠀⠀⠳⣼⣧⡾⡴⠃⠀⠀⠀⠘⢦⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣉⣳⣆
⠀⠀⠀⠀⠀⢀⣀⡤⠞⣧⠀⠀⠀⠀⠀⠀⢀⣠⣀⣀⣀⣄⣀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⡄⠀⠀⠀⣸⣥⣤⣽⣧⣬⣧
⠀⠀⠀⠀⣰⠯⠤⠤⠤⠬⢷⡀⠀⠀⠀⣰⠋⠀⠀⠏⠀⠀⠟⠁⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⣠⡏⠙⠓⢿⠀⠀⠀⠹⡞⠛⢚⣷⢶⡏
⠀⠀⠀⠀⡷⠶⣤⠴⢤⡴⢶⡇⠀⠀⠀⣟⣶⣒⣲⣒⣲⣒⣒⣖⣒⣶⣻⠀⠀⠀⠀⠀⢀⡞⠉⠙⠦⣄⣸⠁⠀⠀⠀⢻⣚⣩⡴⡾
⠀⠀⠀⠀⢹⡀⢸⠀⢸⠀⣸⠁⠀⠀⠀⣿⠦⠭⡽⠯⢥⡬⠵⠮⡭⠴⣻⠀⠀⢀⣠⢾⡉⠙⢦⡀⠀⢈⡗⠀⠀⠀⠀⠈⣏⣡⢴⠃
⠀⠀⠀⠀⠈⣧⣸⣄⣼⣠⡇⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠒⠚⠛⠛⠉⠁⠀⠀⠘⠲⢤⣷⣀⣠⠿⠖⠋⠀⠀⠀⠀⠀⠀⠹⣄⡟
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⢶⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⢀⣤⡀⠀⠀⠀⠀⢠⡤⠤⣤⢀⣾⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡔⡿⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠛⣧⡾⡶⣴⢧⡀⠀⠀⣴⠋⣄⠙⣶⠋⣄⠙⣆⠀⠀⠀⠈⡏⠉⡇⠸⡷⢾⡇⠀⠀⠀⠀⠀⠀⣠⢞⡼⣳⠋⠈⢹⡄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠶⢿⣹⠋⠀⠀⠰⡇⢀⡼⠀⢸⢀⡞⠀⢸⠀⠀⠀⢀⡇⠸⣧⡞⠁⠈⢳⡆⠀⠀⢀⡴⢞⡵⣫⠟⠁⠀⠀⣸⠃
⠀⠀⠀⣴⠋⠀⠀⢀⡴⠃⣠⠞⠁⠀⠀⠀⠀⢳⡈⠃⣠⢾⡈⠇⣠⠟⠀⠀⠀⡏⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢿⢚⡿⠞⠁⠀⠀⣠⠞⠁
⠀⠀⠀⠘⢦⣀⡴⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠙⠶⠃⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠈⠻⣄⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠙⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠲⠚⠓⠒⠶⠚⠃⠀⠀⠀⠀⠈⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠠⠀⠀⠀⠄⠄⠀⠀⠀⡄
⠀⠀⠀⠐⠂⡾⠁⠀⠀⠀⠀⠀⠀⣀⠀⠈⠀⢀⡠⣄⡀⠀⠀⠀⠀⢟⣂⣠⡔⠦⣀⠀⢀⡤⠀⠀⠀⠁⠲⢖⢛⢄
⠀⠈⠢⣤⡈⢇⠀⠀⠀⢀⠂⠀⠀⣇⠀⠄⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠻⢻⣝⢟⢛⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠈⠁


⠀⠀⠀⠀⣴⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⡒⣄⡀
⠀⠀⠀⠀⠁⠁⠀⠒⠋⠀⠀⠀⠀⡀⠀⠀⡀⢠⠀⡠⠄⡀⠀⠀⠀⡄⠰⠀⣢⣄⠀⠀⠀⠱⠀⠀⠀⠀⠀⠉⠜⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠀⠁⠀⠠⠀⡈⠀⠀⠀⠱⢤⠝⠁⠘⠄⠀⠀⠈⠀⠀⢀⠀⠀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⣳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠒⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡂⠹⠆
⠀⠳⠦⠀⠀⠒⠢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠉⠂⡄⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠒⣀⣠⡀⠀⠀⠀⠀⠉⠃⠊⡆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈

⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⡟⢀⣴⠦⡄⠀⢄⢠⣤⠈⠀⠀⠀⠀⡀⠄⠀⠀⠀⢀⡠
⠀⠀⠈⢀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠀⠀⢸⢹⠃⠀⠀⠔⠋⠀⠰⠂⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠁⡀⠀⠀⠑⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠑⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⢶⣿⣿⡈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠟⠳⠀⠉⠛⣿⡄⠀⢤⣄⣤⡄⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⣠⣠⣴⣿⣿⣷⣾⣿⠳⢄⡀⠀⠘⠛⠀⠈⠙⣿⣌⠾⠛⠿⠦⣄⠀⠀⢀⣴⣿⣧⣀⠀⠀⠀⠀⣰⣿⣷⡀
⠀⠀⠀⢴⣶⢾⠻⢿⣿⣿⠿⢻⣿⣿⡟⠂⠀⠙⠀⠀⠀⠀⠀⡀⠈⠛⠳⠀⢀⠀⠈⠻⠛⠿⠛⠋⠈⠙⣻⣶⣤⡞⠁⠀⠈⠛⣠⣄⠀⠀⡀⢀
⠀⠀⠀⠀⠁⠘⣁⣀⡙⠃⢴⣿⣿⡿⠟⠀⠀⠀⠀⢠⣶⣶⣿⣿⡔⢦⣄⠀⠀⠀⠱⣶⣤⣤⣀⣀⡴⠟⠛⢟⡇⠉⠀⡀⠀⠀⠀⠈⠛⠘⠉⠉⠑⠦
⠀⠀⠀⠀⠀⠘⠿⠛⠻⣄⠈⠛⠿⣃⣄⠀⠀⣤⣾⣿⣿⣿⣿⣿⣯⠀⣉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠻⠿⣿⣿⣿⠿⡿⢋⠜⠛⣿⣿⣻⣷⣦⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⢀⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⡜⠀⠐⢋⣾⡿⠛⢻⠀⠀⠀⠀⠀⠀⢶⣿⣿⣏⠁⠀⠀⠀⠀⣼⣿⣿⡇⠀⣀⠐⡋
⠀⠀⠀⣶⣶⣦⡀⣀⠀⠀⠈⠀⠀⠒⠀⠀⠀⠀⠀⠀⠰⠀⣀⢠⠟⠀⢠⣤⣠⣰⣄⠀⠀⠀⠀⠀⠈⠻⣽⣦⣦⡀⠀⢠⣿⣿⣿⣿⠟⠁
⠀⠀⠀⢹⣿⠿⠿⢾⣷⣶⣆⣠⣄⠀⠀⠀⣼⣧⠀⠀⠀⠀⠁⠀⠴⢪⣿⣿⣿⣿⡿⠷⡀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠠⣶⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠈⠠⡀⢸⣿⣿⣿⣿⣿⣶⣾⣆⣿⣿⡇⣼⣷⡀⠀⠀⠀⢀⡉⠛⠿⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣿⣿⣿⣿⣿⣿⠆⢀
⠀⠀⠀⢠⣄⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⢠⣿⣇⣴⡀⠀⠈⠙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣤⣶⠄
⠀⠀⠀⠀⠉⠉⠛⠋⠁⠈⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣸⣿⣿⣿⣷⣴⡄⠀⢀⣄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⠙⠛⠈⠋⠛⠛⠛⠉⠙⠿⠿⠿⢿⣤⣾⣿⡂⠀⠀⢀⣴⣄⣼⣿⠿⠛⠿⠿⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⢀⠤⡂⠀⠉⠙⡌⠒⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠐⠉⠙⡆
⢀⠔⠁⠀⠈⠓⣄⠀⠸⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠈⠙⢗⠢⢌⣒⠒⠁⠀⣀⣀
⡄⠀⠀⠀⠀⠀⠈⣇⠀⡇⡼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⢠⣚⡉⣉⣥⣴⢾
⡧⢄⣀⠀⠀⠀⠀⢸⠀⣧⣃⡠⠞⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⡴⣩⠖⠊⠉
⢱⡀⠀⠈⠉⠐⠒⡞⢺⠛⡅⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡞⡴⠁
⠀⠑⢤⣀⠀⣠⠞⠀⡞⢀⡹⠖⠁⠀⠀⠀⠀⣠⠔⠐⠒⢔⠒⠦⢤⡀⠀⠘⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣫⠟
⠀⠀⠀⠀⠉⠉⠒⠋⠉⠉⠀⠀⠀⠀⠀⢠⠎⢀⡉⢦⡀⠈⢂⠀⢡⠈⢣⡀⠘⠢⢝⡒⠶⠤⠤⠴⣒⡪⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⡔⠁⢀⡠⢷⠀⢸⠀⠘⠀⢸⢳⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⢀⣀⠤⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠀⡠⠃⠀⣀⣄⢸⠀⢠⣀⣸⠼⡇⠀⠀⠀⠀⠀⠀⠀⡠⢚⣅⡢⠾⣿⠷⢦⡑⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠸⡇⠀⠁⢀⠾⢄⣀⠈⠉⠉⢀⣀⢴⠇⠀⠀⠀⠀⠀⢠⢊⣴⢷⡏⢨⠞⢄⡜⠃⢼⠸⡆
⠀⠀⠀⣠⠔⠛⢧⡔⠋⠀⠈⢟⠦⡀⠀⠱⣄⠀⢸⠤⣀⠈⠉⠉⢉⣁⣠⠞⠀⠀⠀⠀⠀⠠⢡⣳⠗⢅⡜⠑⠦⣎⠈⡰⢺⠀⡇
⠀⠀⡌⠀⠀⠀⢠⡀⠀⠀⠀⡘⠲⣝⣆⠀⠈⠑⠮⢷⣤⣬⣭⡭⠽⠛⠁⠀⠀⠀⠀⠀⢠⢣⡿⠋⡦⡂⢈⠖⢄⡠⠋⡲⡞⢰⡇
⠀⡜⣸⠦⠤⡖⠁⠙⠦⡠⠴⣄⠀⣸⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣼⢉⠶⣀⡼⠓⣤⡊⢩⠾⡕⡡⢣⠁
⠀⣏⠀⠀⠀⠱⣄⣀⡜⠀⠀⢠⠋⠈⢦⡇⠀⠀⠀⠀⠀⢀⠤⠀⠲⠤⡀⠀⠀⠀⠀⠀⡇⡧⠉⣦⡎⠑⠴⣁⡜⡳⢊⠔⡱⠃
⠀⠘⣄⢀⣀⠊⠀⠀⠑⢢⠖⠣⣀⠀⡼⠁⠀⠀⠀⠀⢠⠊⠐⠢⡀⠀⢈⣆⠀⠀⠀⢐⢠⡘⠶⣁⣨⡳⠤⢛⡩⢔⡡⠊
⠀⠀⠈⠳⢄⣇⠀⢀⠠⠯⣄⣰⠧⠋⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠆⠊⢸⠀⠀⠀⠌⣬⠶⢋⡠⠴⡒⠉⠁⠂⠁
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣠⠔⠋⢉⡷⠃⠀⣀⡌⢈⡵⢊⠕⠊⠁⠀⡠⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠒⠒⠲⢤⠤⣀⡀⠀⠀⠈⠉⠉⠁⠀⢀⠾⠉⢍⢫⠈⠀⠀⠀⠀⢸⡀⣀⣸
⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⣀⣠⢤⠴⠤⢦⠈⡍⡗⠀⠀⠀⠀⠀⡰⡉⢦⢱⢸⠎⠀⠀⠀⠀⢀⣼⣖⣗⣮⣄
⠀⠀⠀⠀⢀⠜⢄⠀⡠⠴⠎⠙⠉⠀⠀⠀⠀⠀⠇⢸⠃⠀⠀⠀⠀⡾⠤⡙⡼⡤⠃⠀⠀⠀⠀⠀⣾⢏⡏⢿⢙⢿⣄
⠀⠀⠀⠀⠊⠡⡀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠑⢄⣈⡷⠁⠀⠀⠀⠀⢀⣎⡟⣹⠓⢼⠉⣍⣷⢆
⠀⠀⠀⠀⠀⠀⢱⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢧⠾⢁⣼⠑⣾⠁⣤⢹⠛⠧⡀
⠀⠀⠀⠈⠢⢄⣼⣠⠁⠀⠀⠀⠀⠀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠘⣅⠘⡹⠃⠫⣀⣨

⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠛⢣⡙⢄⠀⠈⢻⡛⠛⠛⠛⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠈⡀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠠⠿⣿⡃⣇⠀⠀⠘⣧⠀⣀⠄⠀⠀⠀⠀⠀⠀⠁⠑⠀⠤⠤⣤⣤⣴⣶⡿⠉⠀⣹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢌⣿⡅⡇⣠⣔⠒⠛⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠋⠀⣠⣾⡿⠉⠋
⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⡆⠀⢩⣞⣅⡀⡀⢤⠠⠀⠀⡀⢀⣴⣶⡿⠟⠁⠀⣠⣾⣿⣿⢣
⠀⠀⠀⠀⣤⣿⣿⣿⣿⣿⣿⣿⠹⣿⠯⠐⠚⠛⠛⠟⠛⡷⠿⢷⣾⣿⡿⠛⠁⠀⠀⣠⣾⣿⣿⣿⡟⣾
⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣾⣏⠀⠀⣀⡀⣔⠀⠀⠀⠀⠀⠁⢀⢀⣀⢀⣴⣶⣿⣿⣿⣿⣿⠃⠨
⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣟⣒⠀⢀⠀⠀⢠⣯⣷⣿⣾⣿⣿⣿⣿⣿⣿⢏⢄⡇
⠀⠀⠈⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢛⣉⣙⠛⠛⠛⣛⠋⠉⠉⠉⠉⠉⠉⠙⢿⣿⣿⣿⢿⡼⠁
⠀⠀⠀⠼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣷⡶⠦⠾⠿⠶⠷⠳⠤⠴⠤⢤⣤⣴⠷⠋⢵⡟
⠀⠀⠀⠘⠻⠻⠿⣿⣿⡟⣿⣿⣿⣿⣷⣄⡀⠹⣶⣤⣴⣤⣤⣄⣄⣠⣤⣤⣤⣤⠤⢤⣀⡀⣼⠛⠓⠚⠓⠒⠛⠛⠃⠑
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⢿⣿⣿⣦⣌⠛⠛⣟⠿⠿⠛⠿⠻⠿⢿⣶⣶⣤⣬⣿⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠤⢠⠄⢠⣄⣀⣀⠀⢀⡁⣁⡀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠉⠀⠁⠀⠀⠀⠉⠁⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢤⣆⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠀⣤⠉⠁⢠⣰⣾⣯⡄⢸⣿⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⠀⢻⣴⣦⣜⣻⣯⣿⣯⠈⣫⣾⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠍⡀⣬⣥⢠⣜⣿⣿⣿⣿⣿⣿⡿⡃⣠⣄⡝⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣿⢋⣯⡻⠈⣉⣭⣭⣥⣭⣙⠛⠿⣽⣏⣷⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣋⣭⣿⣿⣿⠟⢛⣛⣛⠛⠻⣿⣶⣤⣬⣭⣻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⡿⠐⠋⠙⣿⠤⣦⣿⠿⠿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⢉⣷⣦⣄⣀⣸⣙⣻⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⡘⠉⢯⣙⠻⣷⣿⣿⣶⣤⣤⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣻⠿⣇⡼⠇⢸⣾⣿⣦⣞⠯⣙⢏⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠓⠀⠀⠀⢠⢰⣼⣿⣿⣿⠝⠵⣞⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣤⣄⣀⡀⠀⠈⠀⠀⠈⠉⠀⠀⠀⢀⣀⣠⡤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢾⣿⣽⣿⣛⣛⣛⣛⣛⣛⣛⣿⣿⣭⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡀⣰⠋⢳⡄
⠀⠀⠀⠀⢀⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⣀⣩⡇⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣠⠤⣄
⠀⠀⣤⠐⠿⡞⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡞⠛⣡⢼⡛⠷⢯⣷⡀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠉⠒⠚⠉⠀⠀⢸⡆
⢀⡞⣹⠷⠒⠿⣄⠀⠀⠈⠓⠦⢤⣄⣀⡄⠀⠀⠀⣸⣴⣣⢾⠷⠶⢿⢻⡞⠁⢧⠀⠀⠀⠀⠀⠀⠳⣄⠀⢀⡴⠛⠛⠳⣄⠀⠳⣄
⠉⠉⢻⠀⠀⠀⠈⠳⢦⣄⣀⣀⣀⣀⠿⣽⠀⠀⠀⢻⠙⠛⠻⠶⠶⢾⠏⠀⠀⠘⡆⠀⠀⠀⠀⠀⢀⡼⠀⢸⠀⠀⠀⠀⣹⠀⠀⠈⢳⡀
⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠉⠛⠛⢦⣴⡆⠀⠀⠀⠈⢦⠀⠈⠁⣰⠋⠈⠀⠀⠀⢻⡀⠀⠀⠀⣰⠋⠀⠀⠈⠳⢤⣤⠴⠋⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠈⠙⠲⠦⢤⣤⣤⣤⠤⢿⣸⠇⠀⠀⠀⠀⠈⢧⡀⣰⠃⠀⠀⠘⠀⠠⠈⣇⠀⠀⠀⢿⡀⠀⠀⢀⣠⣤⣤⣄⣀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⢹⠇⠀⠀⠀⢀⠀⠀⠀⢸⡆⠀⠀⠀⠙⠓⠚⠉⠀⠀⠀⠀⠈⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠤⠀⠈⠀⠀⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣦⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠙⢦⣀⣀⣀⣤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⢰⢡⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠲⣾⡀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⣠⠤⡎⢉⡆
⠀⠀⠀⠀⠀⠀⠀⡟⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⣀⠀⠀⢀⡞⣠⠀⠀⠀⠀⢀⣴⡀⠀⠀⠀⠙⣦⠟⠳⣦⡟⣏⠹⣞⣦
⠀⢠⠶⠶⠶⠶⠚⠓⠓⠒⠒⠒⢲⡄⠀⠀⠀⠀⠀⣞⣛⣛⣛⣛⡇⠀⠈⢇⢹⢀⣤⣄⣠⡿⠋⠀⠀⠀⣏⣛⠛⢦⣞⢹⢠⡿⣆⠘⢧
⠀⠉⢻⠉⠉⠉⠉⠉⠉⠉⠉⢉⡏⠁⠀⠀⠀⠀⡴⠛⠉⠉⠉⠉⠑⢦⠀⠞⢀⢸⣄⠈⢹⡇⠀⠀⠀⠀⢀⣈⡟⣆⠈⢻⣸⠃⣼⣄⣨⣷
⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⣞⠉⠙⢿⡳⠞⠁⠀⠀⠀⠀⠀⢻⠉⠉⠙⠛⠛⠉⠉⠁⢠⡇
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡼⠶⢿⡶⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣤⣀⣷⣠⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⣧⠀⠀⢰⡏⠐⠈⠙⣏⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠶⠶⠒⠒⠒⠶⠟
⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠻⠶⠶⣶⡿⢦⣧⡴⠋
⠀⠀⠀⠀⢀⣀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠓⠲⠶⢤⣄⡀
⠀⠀⠀⣰⠿⢞⠳⠮⣍⡹⣯⣽⣇⣤⣀⠀⠀⠀⠀⠀⠁⠀⢀⣤⠶⠒⠒⣖⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣾⣀⡀⣠⠀⠈⠉⠳⢤⡀
⠀⢀⡞⠁⠀⠈⠀⡄⠀⠙⢦⡸⢭⣳⣈⠱⡄⠀⠀⠀⠀⣴⢋⡀⠳⠀⠶⠀⠡⠆⢈⠳⡄⠀⠀⠀⠀⠀⢸⡷⣤⡈⠉⠓⠦⣤⡄⠀⠀⠙⢦
⠀⠸⡆⠠⠀⠀⠀⠀⠀⠂⠀⠹⡄⢿⣽⣇⢹⠀⠀⠀⠀⢷⣌⣁⣀⣀⣀⣖⣀⣈⣋⣁⣹⠀⠀⠀⠀⠀⢸⡷⢈⡇⢀⣩⡤⣤⣙⠶⣞⠀⣈
⠀⠀⠘⠦⣀⠈⠁⠀⢀⠀⠈⠃⠹⡄⣹⢾⣼⠀⠀⠀⠀⣸⣭⣭⠭⣍⣉⣩⠽⢯⣍⣩⠿⡄⠀⠀⠀⠀⢸⠟⠋⢠⣯⡴⡖⣟⣮⠷⠞⠋⠉
⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠠⠄⠀⢷⡷⢺⠃⠀⠀⠀⠀⠹⣦⠤⠶⠶⠿⢶⣶⠶⠶⠖⢺⠁⠀⠀⠀⠀⢸⠶⣄⢘⣦⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣄⣸⡴⠃⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⣀⣈⣹⣦⡴⠋⠀⠀⠀⠀⠀⢾⠴⠛⠉
⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⣷⡄⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢠⣾⡗⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠿⢦⣽⠁⢴⡞⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠰⠆⠀⣿⠛⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢣⣤⢹⣿⡧⠀⠠⡾⢂⠀⢀⡀⠀⠀⠀⠠⣤⡄⠀⡀⠀⢀⢤⠀⠀⠀⠀⠀⢸⣿⣿⢠⡘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠆⣾⢱⡇⠌⠉⠁⠀⠀⢪⠀⠉⣿⠇⠀⠄⠁⢄⠛⠀⠀⣿⡗⠀⡐⠀⠀⠘⣴⣶⠮⠙⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⣦⡀⠈⢀⠁⠂⠘⠃⠀⠀⣀⠉⠀⠀⠁⠀⠀⠀⠉⣀⣠⣽⣭⣤⣤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⢀⠀⠀⠀⠀⣄⣰⡔⣿⡇⠀⠀⠀⠠⠀⣘⠀⠀⢉⠀⢠⣦⣤⣤⡀⠀⠀⠘⡛⢃⠈⠛⠁⢠⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠙⡇⠀⠘⠀⢀⡀⣲⡶⠋⠁⠁⠈⠁⠈⠀⠀⠀⠠⠀⠀⠄⠸⠠⠀⠀⠉⣥⣤⡔⠀⠒⠀⠀⠐⠊⠀⠀⠀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠏⢶⣶⣄⠶⠋⠁⣀⠀⢂⡀⠂⠀⠘⠀⠀⠔⠀⠄⡀⠀⠤⠀⠀⠣⣄⣩⣭⣽⣦⣄⠀⠀⠀⠀⠀⠀⠐⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⠇⠀⡙⠛⠂⠀⠀⠀⠐⠻⣄⠀⢀⣴⣦⣦⣄⣀⡀⣀⣤⣶⣷⠀⠀⠹⠿⠿⠿⠛⠉⠁⠀⠰⠒⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠐⠒⠀⠀⠐⣀⠀⠄⠀⠀⠀⠀⣶⣿⣿⣿⣿⣯⣭⡛⢿⣿⣿⣿⡄⠀⠀⣤⣤⣤⣴⣶⣶⡆⠀⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠢⢱⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣀⣰⡿⣿⣿⣿⣿⣿⣿⡿⠈⠉⠛⠿⣷⣶⣶⣿⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠐⠂⠁⠀⠀⡈⢿⣿⣿⡟⠁⠰⠿⠿⠿⠿⠛⢋⣀⣀⠀⠀⠀⢈⠛⢻⣿⠃⠀⠀⢀⡀⠀⠐⠂⠂⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣈⣠⠴⣢⣿⡟⠀⠀⠀⠀⢠⣄⠀⠀⠈⠻⠿⠟⢃⢀⡀⠀⠁⠡⠀⠀⠈⠈⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⢠⣾⣿⣿⠁⠀⠀⠀⣰⣿⠟⠀⠀⣀⣤⣴⢾⣿⣸⣧⠀⠀⢁⣀⡀⠀⠀⠀⠀⠀⠀⠁⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢶⣶⣦⠀⠙⠛⣿⠀⢄⠀⢀⣤⡄⣶⢺⡇⠀⠘⢿⣸⣇⡿⠃⠀⢠⣾⠟⡁⠐⠀⠄⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⣿⠿⠧⠀⠀⠎⠁⠀⠀⠀⣼⣿⠀⠀⠢⠘⢿⣸⡏⡿⠇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠡⣠⣿⠖⣀
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠘⠹⠛⡀⠀⠀⠀⠀⠩⠍⠀⠄⠀⠄⠀⠉⠀⣤⣶⣕⠂⠀⠀⠀⠈⠁⠛⠁⣴⣿⠓⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢂⠠⠀⠀⣢⣤⣤⣤⡄⠀⠀⠀⠠⡀⠘⣿⡿⠀⠀⠀⠂⡀⠀⠀⠄⠈⠁⣴⣿⠆
⠀⠀⠀⠀⠀⢠⣶⣶⣾⡆⠀⠀⠂⠀⠀⠠⢀⡀⠀⢠⠄⠀⠀⠀⠈⠿⠛⠁⠀⣘⠀⠤⠷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⢀⠐⠒⢦⡀⠠⠀⠀⢀⠀⠀⠠⠀⠀⠀⠈⠁⠀⠀⠀⠀⢈⡁⢀⣀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢈⡇⠀⠀⠂⢀⡆⡀⢀⠀⢄⠀⣘⡂⡡⠄⠀⣼⣿⠋⢈⡻⣷⠀⠀⠀⠐⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠁⠤⠀⠀⠸⢓⣵⣦⠠⣻⡿⢓⠂⠃⠥⠁⢉⠄⠀⠘⡿⠆⠀⠀⠀⠄⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠉⠀⠈⠒⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⠀⠀⠂⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡤⣤
⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠙⠲⡟⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡀⣤⣋⡷⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣤⠤⠞⣻⡥⣤⡷⣄⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⠦⣤⠴⠚⠉⠉⡇
⠀⠀⢀⡼⢣⡤⣼⣆⠀⠀⠙⠢⣄⣀⠀⠀⣀⠀⠀⠀⢸⣡⢇⣴⣏⣭⣷⢯⡽⠛⡆⠀⠀⠀⠀⠀⠸⣅⠀⠀⢀⡤⠶⠦⣄⠀⢳⣀
⠀⠐⠛⠲⡏⠀⠀⠉⠳⣄⡀⠀⠀⠈⠉⣻⣹⠀⠀⠀⢾⠻⠿⢾⣁⣄⣽⡞⠁⠀⢻⠀⠀⠀⠀⠀⠀⠈⡷⠀⡏⠀⠀⠀⢸⡆⠀⠈⢳⡀
⠀⠀⠀⠀⠹⣄⠀⠀⠀⠈⠙⠛⠻⠿⣏⣁⡍⠀⠀⠀⠘⣆⠀⠐⠂⢠⠏⠐⠀⠀⠈⣇⠀⠀⠀⠀⣠⠞⠁⠀⠳⣄⣀⣀⡼⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠙⠦⢤⣀⣀⣀⣀⣀⣼⣁⡇⠀⠀⠀⠀⠘⢦⠀⢠⠏⠀⠀⠰⠀⠀⠸⡄⠀⠀⠀⡇⠀⠀⠀⠀⣀⣉⣁⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⢳⠇⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠙⠲⠶⠒⠋⠉⠁⠉⠉⠓⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⡀⠀⠘⠀⠀⢂⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠹⣄⡀⠀⣄⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡽⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⠧⣽⡀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢦⢠⡤⣼⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢳⡀⠀⠀⢠⡞⣠⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⢳⡼⠛⢦⡼⢾⠙⢶⣳
⠀⠀⠀⣠⠶⠶⠶⠖⠛⠛⠒⠒⠒⠒⣦⠀⠀⠀⠀⠀⣾⣛⣚⣛⣛⡇⠀⠘⣇⢳⢀⣤⣄⣴⡿⠋⠀⠀⢸⣛⡛⠳⣴⡋⡇⣾⢧⠈⢻⡀
⠀⠀⠀⠙⣟⠉⠉⠉⠉⠉⠉⠉⠉⡿⠉⠀⠀⠀⠀⣠⠟⠉⠉⠉⠉⠓⢦⠀⠟⣀⣻⡀⠈⢿⠄⠀⠀⠀⠈⣠⣸⣳⡀⠙⣧⡏⣸⣇⣠⣷⠆
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⡞⠉⠙⢿⠳⠛⠀⠀⠀⠀⠀⠘⣏⠉⠉⠛⠋⠉⠉⠁⠀⡟
⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡾⠶⢿⡶⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⣠⡤⣤⣷⣠⣁⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⣹⠀⠀⠰⣇⠐⠀⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠚⠛⠛⠓⠛
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⢛⣿⡿⠲⠵⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⢀⡤⣤⠤⣀⣠⣀⣠⣀⠀⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡉⠉⠛⠲⠦⣄⡀
⠀⠀⠀⠀⣠⠟⠛⠉⠛⠲⢤⡿⠶⣿⣶⠦⣄⠀⠀⠀⠀⠀⠀⣠⠶⡛⠉⠉⠯⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⢸⠾⠧⣤⣠⠆⠀⠀⠉⠳⣄
⠀⠀⠀⣼⠁⠀⠀⠀⠰⠀⠀⠙⢾⠓⣼⡧⣌⢧⠀⠀⠀⠀⡾⢡⠆⠙⠀⠛⠀⠘⢁⣤⠹⡄⠀⠀⠀⠀⠀⣼⠹⢦⠀⢈⡙⠲⢦⡀⢀⡀⠳⣄
⠀⠀⠀⠹⣄⠂⠀⣤⠀⠀⠈⠀⣌⢳⡘⢾⣻⣸⠄⠀⠀⠀⢹⡦⠤⠤⢤⣼⣭⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⣿⣃⡾⢀⡴⡒⣒⡶⣝⢻⣄⡤⠿
⠀⠀⠀⠀⠈⠳⢤⣀⠀⠠⠆⠀⠀⠀⢧⣸⣹⣾⠀⠀⠀⠀⢼⡳⠒⠛⣲⣶⣖⣛⣓⣶⣞⣹⠆⠀⠀⠀⠀⣟⠡⠀⣟⢾⣹⣽⠽⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠃⠀⢸⡏⡽⠃⠀⠀⠀⠀⠈⢿⡛⠉⠉⠉⠛⠯⣍⠉⢉⡟⠀⠀⠀⠀⠀⡏⣙⡦⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠚⠛⠋⠀⠀⠀⠀⠀⠀⠛⠉

⠀⠀⠀⠀⠀⢀⡴⠂⠁⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡀
⠀⠀⠀⠀⠠⠈⠁⢀⠀⠀⠀⠀⠀⠄⠠⡤⠄⢤⡤⢠⡄⠶⠀⠛⠂⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠡⣤⠀⠈⡄⠀⠀⠀⡧⠔⠒⠗⠒⠒⠳⠼⠧⠤⢀⣥⣿⣦⠀⢴⠆⠀⠀⢠⣶⣄⠀⠀⠿⠀⠀⠀⠀⠐⠠⠄⡀
⠀⠀⠀⠀⠀⡩⠋⣑⣷⡀⠀⠀⠁⠤⡴⠃⠂⠂⠁⠀⠀⠀⠀⠀⠉⠫⡀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠀⢠⣿⣷⠀⣶⠀⠈⠐⢄
⠀⠀⠀⠀⠠⠀⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⢶⠄⠀⠀⢰⡦⠀⠀⠈⠛⠋⠀⠀⠀⣠⣤⠀⠐⢄
⠀⠀⠀⠀⠸⠟⠐⣶⣾⠁⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢿⣿⠃⠀⡀⠁
⠀⠀⠀⠀⣀⣀⡠⠛⠋⠀⠑⠄⠀⠀⠀⠀⠀⠀⠐⠠⢀⠀⠀⠀⢀⣠⡜⠀⣿⣿⠂⢀⣄⠀⢠⣾⣦⠀⠀⠛⠁⠀⠀⢀⣄⠀⠀⠘⠋
⠀⢀⣒⠁⣶⠀⠀⠀⠀⠀⠀⣈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠉⠀⠈⣥⢀⡈⠁⠀⢈⡉⢀⠈⠿⠏⠀⣀⠀⠀⢀⣤⠀⠉⠐⠀⣤⠈⡀⠀⡀⢀
⠀⢿⣿⡇⠀⠀⠄⠀⠀⠀⠛⡤⠇⣀⣀⣀⠀⣀⣀⣀⡀⢀⠀⠀⢀⠀⠰⡎⠉⠁⠉⠀⠀⡀⠝⠒⠄⠀⠛⠀⠀⢿⣿⠇⣴⠁⠀⢈⡤⠚⢈⣵⣦⠀⣨⡄⢀
⡆⠈⠛⠡⣦⠀⠐⡀⠲⠀⠀⠈⠐⡆⠠⠄⢀⠀⠀⠀⠀⠀⠀⢠⠋⠉⠒⠲⣤⠦⠀⢠⣾⠀⠀⢀⢀⡠⠀⠀⠠⢤⣥⣴⠁⠀⢀⠋⢴⠆⠘⢿⠟⠀⠈
⠇⠿⠂⢰⣿⣧⢠⡀⠀⠁⠀⠀⠀⠈⢠⠀⢠⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡰⠉⠀⠀⠀⢠⠟⠁⠰⢲⠟⠉⠁⣿⡄⠀⠀⢠⣤⡀⠀⣠⡄⠀⣰⣶⡄
⠀⢄⠀⠈⠛⠃⠈⢁⡀⢀⠀⠀⠀⠀⣨⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⣬⠏⠀⠀⠀⣿⣧⢀⠀⢻⣿⠇⠈⠀⡀⠀⠹⠿⠃
⠀⠀⠈⠐⠀⡄⠠⣿⣿⡆⠀⢀⡈⢁⣸⠠⢥⢀⠑⠢⣀⠀⠀⠀⠀⢠⠠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢩⣿⣾⢰⡆⠀⠀⠀⣿⣿⡄⠀⣦⡠
⠀⠀⠀⠀⠀⢘⠀⣉⠉⠙⣦⡙⠁⠀⠊⠀⠉⢚⣴⣶⡄⠱⠆⠀⠀⢲⡗⢦⠅⣀⠀⠀⠀⠀⠀⣀⣀⡠⡧⠴⠐⠂⠈⣿⣿⣍⠀⣠⣤⠀⠀⠙⠋⡶⠈⠉
⠀⠀⠀⠀⠀⢸⠀⠛⠀⢸⣿⡇⠀⣶⢼⣶⠂⠀⠹⠿⠇⢠⣾⣷⣀⣀⡀⣰⣶⣦⠉⠿⠀⠉⠈⢿⡿⣀⠀⢰⣿⣷⠀⠛⠋⣶⠀⢿⣿⠇⠀⢶
⠀⠀⠀⠀⠀⠀⠀⠉⣁⠚⠛⠃⠠⠤⠊⠈⠂⢺⡇⠀⠀⣴⣦⡀⠀⠟⠀⠘⠿⠋⠀⠀⠰⡷⠀⠀⠀⠛⠁⠈⠻⠋⠀⠀⠀⠀⠀⠀⠠⠄⠤⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⢀⠄⠒⣁⠀⠀⢸⡏⣿⡀⡄⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠆⠜⠛⠛⠿⠥⠅

⢰⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⣠⡤⣤⡤
⠀⢿⣿⡿⠻⠦⠄⠀⠀⠀⢀⡀⠀⢿⣷⣆⠀⠀⢶⣾⣷⣿⣿⣦⣍⡒⠀⠀⠀⠀⣀⣤⡄⠀⠀⢀⠀⣤⡀⠀⠀⠀⣤⣄⡀⣠⣄
⠀⠈⢿⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠘⠏⠁⠁⠀⠀⣽⡟⠉⢙⠉⠻⣿⡗⠀⠀⠀⢻⠟⠀⠀⢰⣾⣼⣿⡿⣶⣤⣈⡙⠛⠿⣿⡿⠃
⠀⠀⠈⠀⠀⠀⠀⠀⣴⣿⢿⣷⡀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠠⢀⠀⠈⠀⠀⣠⣶⡷⠓⠀⠰⠿⣿⡟⠉⠁⠀⠈⠻⢿⣿⣷⣶⡖⠠
⠀⠀⢷⣦⠀⠀⠀⠈⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⢿⣶⠚⠛⠿⣶⣤⣤⡤⠀⠘⠳⣾⣿⡆⠀⢶⣾⡟⠋⠡⣤⣶⡇
⠀⠀⠈⠻⡆⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⡀⠀⣠⣴⣆⠀⠀⠀⠀⢠⢾⡟⠀⢀⣼⣿⠟⢋⣀⣤⣤⠄⠘⢿⣷⠀⠀⣟⠀⠀⢀⣼⣿⡇
⠀⠀⢀⣀⣼⣷⣦⡄⠀⠀⢀⡀⠈⠹⡿⠃⢠⣿⣿⣿⣧⣤⣷⡀⠀⣾⠁⠀⠈⠜⠻⣿⡿⠋⠀⠀⠂⠀⠀⠘⠃⠀⢉⣠⡶⠛⠟⠋⠁
⠀⠀⠘⣿⠟⠁⠀⠀⠀⠀⠘⢿⣆⠀⢀⠀⣾⠏⠘⠀⠹⣿⠏⠀⠀⣽⣧⡀⠀⠀⣠⣼⣇⠀⣠⢶⣤⣄⡀⠀⠀⢀⡿⠋⠀⠀⠀⠀⣀
⠀⣷⣼⣾⠀⠀⣿⡀⡀⠀⠀⠀⣷⣤⡷⢀⡏⠀⠀⠀⠀⠁⠀⠀⠀⠋⠻⢻⣿⠋⠉⠉⢙⣿⠏⠈⢿⡿⣧⢄⠀⠹⡀⠀⠀⠻⠿⠿⠛
⠀⡿⣿⠻⠀⠀⣿⣿⣃⡜⠀⣠⣼⣿⣃⡌⠀⠀⠈⠀⣲⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡤⠂⠈⠃⠀⠈⠐⠀⢤⡀⠀⣠⣶
⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⠀⠀⠈⠉⢙⠁⢀⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠠⢤⣤⣄⣀⣾⣿⣥⡄⠀⠀⠀⠀⠀⠀⠈⣿⢿⣿⡿⠃
⠀⠀⠀⣀⡀⢠⣿⠟⢹⡿⠀⢀⣴⢀⡆⠀⠐⠋⠀⢀⣦⡀⢠⣄⠀⠀⠀⠀⠛⠛⠋⢹⠿⠛⠙⠋⢀⣠⣴⣦⣤⠀⠀⠘⠀⠈
⠀⠀⢀⡾⠟⠉⠁⠀⠀⢃⠀⠈⣿⣿⠇⠀⠀⢀⣠⣾⡿⠿⠾⣿⡇⠀⠀⠀⣀⣴⡆⠀⠀⣀⣤⣾⠏⠁⠉⠙⠙⠶⢶⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⣄⣰⣿⣿⡄⡘⡿⠀⠀⠀⠸⣿⠃⢁⣠⣴⡾⠁⠀⠀⠀⠹⣿⠀⠀⠀⠘⣼⡟⠀⠀⠀⢀⠀⠀⠀⣿⠇⠈⠉⣦⣤
⠀⠀⠀⠀⣠⣤⣤⣽⣿⣍⠙⠿⠛⠁⠀⠀⠀⢢⣯⡀⠀⠝⠫⠀⠀⠀⠀⠀⠉⠛⠷⣤⠀⠀⠈⠀⠀⠀⠀⢸⣷⣆⠀⠁⠀⠀⢠⣿⣿
⠀⠀⠀⣀⣾⠛⠉⢀⠈⠛⠂⢀⣀⠀⠀⠠⠴⢾⣿⣷⣄⣴⣿⠿⠾⠶⠂⠀⠀⠀⢀⣶⠁⠀⠦⣤⣀⡀⠐⠶⠿⠿⣷⡄⠀⠀⠈⠉
⢠⣶⣿⣿⣇⠀⠀⠀⣤⣤⣀⣾⣷⣀⠀⣀⢀⣀⣤⣾⠏⠀⠀⠀⠀⠀⠀⢠⣤⡴⢾⠇⠀⠀⠀⢼⣿⣦⣀⣇⠀⠀⢠⣤⣤⣠
⠀⣿⣿⣿⣿⡀⠀⠀⣿⣿⠏⠈⣩⣽⠿⠟⠛⠋⠿⠟⢀⣼⡤⠀⠀⠀⠀⢸⡏⠀⠈⠀⠀⠀⠀⠚⠉⢉⣽⣆⡠⠀⠀⠈⠙⣧⡀
⠀⠉⠉⠛⠣⠘⠀⠀⣿⠏⠀⠸⡿⠋⠀⠀⠀⠀⠀⠀⢾⡿⠁⠀⠀⢠⣾⣿⣷⡀⢿⣦⡀⡀⢀⣀⡴⠘⣿⡿⠁⠀⠀⠀⠀⠈⠛
⠀⠀⠠⣀⡀⢀⣠⣴⣏⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠨⠀⠀⠘⢴⣿⣿⣿⣿⣷⠀⠀⠁⠈⢻⣿⣁⣀⡟⠁⠀⢠⣿⣿⡀
⠀⠀⠐⠛⠛⢉⣿⠁⠸⣷⣄⠀⠀⠀⣾⣧⡀⠀⠀⣦⣶⠀⠀⠠⠶⠛⠛⠉⢼⡏⠀⠀⠀⠈⠉⢿⠿⠋⠀⠀⠀⣾⣿⣿⣧⡀
⠀⠀⢠⣶⠶⣿⡿⠦⠀⠉⠏⠀⠀⠀⠁⠻⣿⠶⠶⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣼⣶⣄⢶⣷⣶⣶⡶⠔⠂⠈⠉⠉⠉
⠀⠀⣽⠃⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣦⡀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⠀⠀⠀⢀
⠀⣾⣿⡶⠆⠀⠀⠀⣀⣤⣦⠤⠀⠀⠘⣿⣆⠀⠀⣀⢠⣤⣴⠀⠀⠀⠻⠿⡷⠀⠀⠀⠀⢰⣿⣷⡶⠀⠀⠀⣰⣷⣿⣿⡷⠚⠀⢠⣾
⠀⠀⠈⠑⡤⢀⠴⠟⠛⠉⠁⠀⠀⠀⠀⠘⡿⠀⠀⢹⣿⣿⠇⠀⠀⠀⠀⠀⣴⠀⠀⢤⣄⣿⡏⠁⠀⠀⣐⣿⡿⢛⣯⠁⠀⢀⠄⠛⡏
⠀⠀⠀⢨⣴⣿⣷⣶⣤⣤⣴⣦⢀⣶⠀⠀⠀⣠⣄⣙⣿⡿⠀⠀⢠⣦⣏⣶⣿⣦⠀⠀⠉⠛⠧⣤⣶⣿⠏⠉⠀⠸⢿⣄⣴⠋⠀⠠⠁
⠀⠀⠀⠀⣿⣿⠇⣀⠄⠀⠈⢻⣿⣿⠇⠀⠐⣻⡟⠉⠁⠀⠀⠀⠘⣿⣿⠇⠁⠀⠀⣤⡀⠀⠀⠈⠉⠁⣀⡀⠀⠁⢈⣿⠁⡀
⠀⠀⠀⠀⠈⢉⣷⣿⣄⠄⠀⠘⣿⢋⣠⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠘⣿⣶⠂⠀⢤⣾⣿⡿⠙⠛⠛⠿⣧⡀
⠀⠀⢀⣤⣄⣸⠿⠿⠯⢤⣴⣶⡟⠛⢿⣿⣿⣤⣤⣀⠀⠀⣷⣀⣤⠀⠀⠀⠀⠒⠛⠛⠻⣧⡀⠀⠀⠁⠟⠁⠀⠀⠀⠀⠈⣻⡄
⠀⠀⠈⠙⠉⠀⠀⠀⡀⠀⠀⠙⠀⠀⠸⠟⠁⠈⠙⠳⡄⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⢀⡄⠀⠀⣼⣷⠀⢀⣹⠟⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⠦⠀⠀⣴⣷⣂⡀⠀⠀⢈⣁⣼⠏⠀⠠⠾⣿⣶⣶⣦⢤⡗⠀⠀⠀⠀⣛⣿⣦⣀⣙⣿⣠⡴⠗⠂
⠀⠀⠀⠀⢀⣠⡴⡍⠉⠁⠀⠀⠀⠈⠉⠉⠉⠀⠐⠋⣹⣿⠏⠀⠀⣀⣤⣄⠀⠈⠻⣿⡇⠀⠀⠀⠀⢩⣿⣿⣿⠛⠉⠁⠀⠀⠀⣀⣸
⠀⣤⠀⠀⠋⠁⠀⢻⣆⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢼⠃⡄⠘⠛⠻⠿⣅⣀⣀⢘⡇⠀⠀⠀⠀⠁⠀⡾⠻⠀⠀⠀⠀⠀⠘⠿⣿
⢠⡏⠀⠀⠀⣀⣾⡄⠉⢲⣿⡁⠀⠀⠀⣀⣤⣤⣀⠀⠀⣿⡟⠁⠀⠀⣀⠀⠀⠉⠻⣿⠿⣄⠀⢀⣀⡄⠈⠀⠀⠀⠀⠀⠈⠳⣦⣄⢨
⠸⠁⠀⠀⠀⠨⠿⠿⠦⠤⠿⠂⠀⠀⠀⠹⠿⠿⠿⠿⠶⠏⠀⠀⠒⠾⠿⠷⠄⠀⠀⠀⠀⠀⠉⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡾⣻⣿⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠛⠛⠛⢿⣿⣷⣯⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⢛⣟⠳⠶⠶⢶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣯⣍⣤⣶⣿⣿⣿⣷⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⣀⡀⡠⡟⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠉⠈⣻⣿⣏⠉⠉⣽⠋⢹⡿⠋⠉⠛⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⣿⣄
⠀⠀⠀⠀⠀⠀⢀⠶⢿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣹⣿⡇⠾⢋⡀⢀⡇⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷
⠀⠀⠀⠛⠛⠟⠻⣦⣭⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣼⣧⠀⠈⣁⡼⠃⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⣄⠀⠀⠀⢠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⢠⣾⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢹⠁⠀⠀⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⠂⠀⠀⠀⠀⠀⠠⣿⣿⠀⠀⠀⠀⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⣤⣤⣤⡤⣿⣿⡇⠴⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣏⣛⣼⣿⡽⠛⠲⣿⣿⡗⠶⠟⠿⠿⠿⠿⣿⠇⠿⠛⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠉⠛⠙⢛⣿⠁⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡰⠏⠀⠀⠀⡾⠁⠀⠀⠀⢸⡄⡇⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠰⠇⠨⠄⠈⠰⠀⠀⠀⠀⣤⣼⣆⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣿⢿⡟⡏⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⣶⣾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡇⠀⠀⠀⠀⠀⢿⣾⣷⣶⣶⣶⣶⣶⣶⣿⣟⣿⣶⡶⠶⢶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣀⣀⣀⣈⣿⣀⣤⣴⣈⣤⣭⣵⣦⣿⣃⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢻⡿⠟⠉⠉⠍⡿⠛⠉⠉⠻⠫⠁⠈⠉⠛⠧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠤⠆⠀⠀⠀⠀⡠⠚⣶⠋⠉⣿⣷⣥⣄⡀⠀⠀⠀⠀⠉⡓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠈⡏⠀⠀⢩⡯⠀⠛⢻⣧⡀⠀⠀⠀⠀⠀⠀⠉⠑⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡀⠀⠘⠛⠙⠛⠛⠛⠉⠋⠙⠁⣀⠀⠀⠀⢺⠃⠀⠀⢸⡇⠀⠀⠸⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡘⠀⠀⠀⠀⡇⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣷⣶⣦⣤⣤⣤⡄⠀⠀⢀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⣿⣿⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢿⠿⠿⠟⠛⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠁⡈⣉⣀⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣷⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⢟⡉⣿⠟⢛⣭⣥⣿⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⠘⠳⠑⢰⣿⡏⢿⣞⢿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣦⠁⣀⠰⣵⣾⣿⠟⠁⠀⠛⢷⣯⣛⠿⣿⣇⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠁⠀⣼⡟⠀⠀⠀⠀⠀⠀⠈⠙⣿⣌⣿⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡎⢈⡙⠃⠀⠓⠸⠊⠉⠀⢴⡆⣿⡬⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡌⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣦⡀⠀⠀⠀⣀⣀⡀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢹⣿⢿⣦⠀⠀⠙⠏⣳⡾⣿⡿⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠙⠒
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣟⡎⣿⣦⡹⣷⣄⡀⠐⠟⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡾⢁⣿⢹⣷⣌⡙⠿⣷⣄⠀⠤⠀⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⢻⡟⣱⠟⠁⠀⠙⠛⠻⣷⡀⠘⣗⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠀⠀⣀⠀⠀⠀⢸⡇⠀⢸⡆⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠡⢻⣿⡶⠴⠃⠀⠀⠀⣼⠃⠀⢸⣇⠈⠓⠀⠀⠀⠀⠀⠀⣰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠡⡹⣷⡄⢀⠀⠀⣾⡇⠀⠀⠀⢿⣦⣀⠈⠂⡀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢿⣎⠣⡈⢿⣇⠀⠀⠀⠀⠐⢤⣤⣤⣼⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠈⠢⢍⠓⠦⠀⠀⠀⠀⠙⢿⡿⠀⠀⠀⠸⠃⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠑⡑⢠⡀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠸⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠂
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠛⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣦⣤⣤⠄⣠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⢿⣷⣄⠀⢀⡄⣹⠷⣄⡀⢰⣄⡀⠀⠀⢀⣙⣿⡦
⠀⠀⠀⠀⠀⠀⠀⢘⡁⠀⠀⠈⣧⠘⢿⡿⢿⣿⣷⠇⠀⠛⠁⢸⡝⠋⣄⡀⠐⣻⣿⣯⡄⠀⢰⡄
⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⠀⣾⠀⡜⠻⠘⣧⢨⡔⠀⠀⣤⠸⠇⢧⠉⣷⣍⣛⣿⣻⡷⠀⣻⣇
⠀⠀⠀⠀⠆⢈⡱⣿⣷⣄⡀⢠⠏⠀⠻⡀⠠⠾⠋⠀⠀⢸⡏⠀⠀⠀⠈⣿⣿⣿⣿⣿⡟⣲⣿⣿⡂
⠀⠀⠀⠀⠠⠀⠀⠙⡛⢿⣿⡶⣤⠖⠉⠤⠀⠁⠀⠀⠀⠛⣀⠊⠀⠀⢨⣿⣿⣿⣿⣿⣷⣯⣽⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣦⠯⠹⣿⣦⡀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠒⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠻⣿⣿⣿⣷⣄⣀⣀⠀⠀⠀⠐⠀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠉⠛⠛⠿⣿⠿⣷⣦⣀⡠⠿⠛⠛⠟⠛⠛⣿⣿⣿⣿⣷⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⢷⣾⣷⣴⣶⣤⣤⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡉⠠⠜⠛⡏⠀⠀⢀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣤⡆⢀⠀⠲⠒⢲⡶⠶⠶⣶⠶⠖⠠⢀⣀⢀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠓⠀⠈⠀⡀⠈⠀⠀⠀⠈⠀⠀⠀⠤⠀⠀⠰⠀⠈⢩⣟⠻⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⠞⠃⠉⡀⠀⠀⠊⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠅⠀⠐⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠤⠀⠀⠀⠙⠁⣽⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠛⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢤⣿⣿⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣘⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡄⠐⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠘⣟⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣷⣤⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢛⠿⢶⣦⣤⣀⣀⣀⣀⣄⣀⣀⣠⣤⣀⡀⠀⢀⣀⣀⣀⣀⣀⣤⣀⣀⣤⣤⣤⣤⣶⣿⣿⣿⠿⣟⣛⡭⣉⣠⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢈⢿⣷⣦⣈⠉⢉⣛⣛⠛⢛⣻⣟⠛⠟⠻⢿⢿⣿⣯⣿⣯⣍⣭⣭⠽⠈⢦⡉⠁⢀⣰⡟⢉⠾⠋⢗⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠰⠚⣉⡙⠿⠿⠍⣉⣿⣿⣿⡟⢻⣿⣿⣦⣤⣤⣴⠿⣷⠾⡯⠟⠛⠿⢿⡴⠋⠳⢍⣻⡿⠻⠷⠁⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣭⡥⣄⡀⠀⠋⠙⠿⠻⠿⠛⢻⠿⢻⡿⡯⠋⠀⠋⠉⠀⠐⠂⠒⠉⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣶⢶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠧⣿⣿⣷⣶⣏⡋⠉⠀⠀⠒⠓⠀⠊⢀⣠⣤⣴⣶⣶⠶⣶⣶⣶⣾⣿⣿⣿⣿⡿⢛⣽⣿⣇⣼⢧⣼⡏⣨⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣌⢙⣧⣄⡙⢟⣷⣄⠀⠀⢀⣠⣾⣿⣯⣤⣶⣿⣭⣼⡿⢃⣾⠏⢀⣾⠿⣿⢹⣟⠉⣸⣿⣏⣾⡯⣴⡏⢀⡀⠘
⠀⠀⠀⠀⠀⠀⠂⠈⠉⡃⠛⢮⠛⠿⢙⣛⢻⣿⣧⣴⣿⣿⣛⠉⢻⢿⣭⣼⣿⣋⣾⡿⠿⠿⠿⢛⣴⣟⣨⡿⣿⣿⠿⡿⡗⠓⢄⣁⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠙⠶⡥⢄⡠⢀⣀⠀⠀⣈⡉⣉⣉⣤⣤⣞⡀⠒⡢⠈⠟⠀⢐⠊⣀⣀⣉⣡⣽⣄⣤⣤⣏⣀⣿⡿⠶⠾⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠙⠛⠻⠙⠿⠟⠋⠛⠻⠗⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠁⠉⠹⡟⠀⡀⢠⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡲⣄⠀⠀⠀⣰⣀⣼⣦⣼⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⠤⠤⠤⠤⣤⣤⣤⣄⣤⢀⣀⣠⣤⣄⣴⣶⣴⣤⠤⠴⠾⠿⠿⠒⠶⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣦⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣉⠶⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠈⠉⠇⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣭⠒⡋⢱⠀⠀⠀⠀⠘⠁⠀⡙⣥⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠀⢀⠼⠄⠀⠀⠀⠈⢦⡀⠀⠀⣀⣤⠤⠤⠤⣄⣤⠀⠀⠀⢠⠃⠀⢀⣘⠃⡌⢳⠀⠀⠀⡞⠁⠘⠟⠀⢒⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠴⠯⠶⠼⠅⣅⣠⠋⠀⠈⠉⠃⠸⠀⠀⠈⠛⠀⠔⠋⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣀⣀⣀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⣛⢙⠉⠀⢻⡰⠂⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠸⠤⠄⠀⣠⠶⠋
⠀⢀⣀⣤⡼⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⡠⠔⡏⠨⠇⠀⠀⡖⢨⡠⠖⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⢡⠁⡄⠁⠀⠀⠀⠀⠀⠀⡀
⢀⣼⠀⠀⠀⠀⠀⢀⣀⣀⠀⠺⢤⡀⠀⣌⣀⣀⡂⠈⠅⠀⠀⡽⠋⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠉⠀⠀⠀⠀⠀⠀⠀⢣⠀⢨⠆⠀⠀⠀⠀⠀⡼⠁
⠛⠢⠤⠖⠉⠙⠛⠃⠀⡠⠤⢤⠴⠛⠋⠉⠁⠈⠹⡄⢙⠠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠃⠀⠀⠀⢰⠋
⠀⠀⠀⠀⡀⠂⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠆⠀⠀⠀⡜
⠀⠀⠀⢮⡁⠀⠀⠀⠀⠀⢀⣀⢀⣀⣀⣠⠤⠤⣶⠫⠑⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⢐⣊⠤⡀
⠀⠀⠀⠀⠻⠤⠶⠚⠛⠚⠋⠀⠉⠁⠀⠀⠀⠀⠈⠙⠣⢿⣄⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠉⠀⠀⠀⠀⠙⠚⢢⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠲⠖⠓⠲⠶⠤⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢈⣳
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣒⣲⣞⣛⣶⣴⣙⣓⣒⣋⣉⣉⣈⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠁⠀⠀⠀⢰⠂⠀⢠⣤⠶⡄⢘⣥⣤⣘⠚⠑⡻⢂⠀⡃⠀⠀⣥⡀⡀⠀⠐⠊⢡⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⢠⢤
⠀⡴⠲⡄⠀⡀⠀⠀⠐⠀⠀⠜⣨⣾⣅⢸⣅⣘⣿⢇⠴⠶⢦⣤⢀⡴⠎⠸⠁⢴⡞⢓⡆⠘⠦⠾⢟⡤⣄⠀⡤⢤⢄⠀⠇
⣄⠁⠜⠱⠀⠈⠁⠀⠊⠑⡆⠀⡀⠀⢸⡇⣨⠭⣤⣈⠘⠤⠤⣷⠛⢦⣦⣤⣀⣈⠁⠉⠄⠂⠂⢀⠈⠷⠞⠀⠣⢼⣟⣀⡀⠀⢠
⠜⡤⠦⡄⠀⠀⠀⠀⠀⠄⠉⠀⢈⠒⡫⢤⠿⠒⣛⣁⣠⠀⠺⠤⣄⢠⠏⠐⠀⢹⡇⠀⠀⠀⠀⡞⢠⢲⠖⠮⣅⡴⠃⠉⣻⠂⢁
⣄⠣⠀⠋⢠⠎⠉⠙⢢⠀⠈⠀⠀⠃⠑⠐⠁⢈⠁⠀⢉⡄⠘⠀⠈⠈⠳⣄⣤⡾⢡⠄⡀⠠⠶⣄⠘⣿⣀⣤⣿⣀⡀⠉⡱⠒⠓
⠏⣠⣤⣀⢈⠀⠀⠀⣸⠀⠀⠀⠀⢀⡊⠙⠢⠈⠁⡀⠉⠀⠀⠀⠀⠀⣠⡠⠄⠠⠀⢈⣁⣠⠀⢠⡀⠤⡀⢘⠀⠀⠙⣦⢧⠀⣀
⣄⠧⢈⡋⠀⠚⠛⠛⣡⠞⠛⢷⠀⠘⢄⣀⣴⠀⣮⠛⠀⠀⠀⠀⠀⣠⠉⠁⠀⠘⠠⡟⠀⠊⡇⠹⢄⣤⠃⢸⡀⠀⠀⠋⢀⣴⣾
⡾⢴⠋⠙⣿⡶⠤⠀⠻⣮⣴⠿⢰⠊⡁⢸⠉⠉⡦⠀⠚⠛⠁⠄⢰⠁⢀⢀⠀⠀⢰⠛⢲⠒⠓⢀⣴⠏⠙⣯⠀⠀⠀⠀⠰⠀⠠
⠀⠘⠢⠜⠉⠀⢸⣉⠀⣠⠤⢦⣄⠉⢁⣌⣒⠚⠁⠄⠀⢀⡴⠀⠈⠑⠃⠀⠀⠀⠀⠀⣀⡠⠀⠁⠈⠧⡄⠋⠀⠀⠀⠄⠀⠙⠊
⠀⠁⢀⠖⠁⠀⠀⠀⠀⢋⡀⢀⡝⠀⠎⠀⠈⣿⠙⠀⠀⠀⣏⠱⠀⠀⠒⠀⠀⡠⠥⠄⠀⠀⡀⠐⠂⠈⠉⠙⣆⠀⠀⠀⠀⠈⠄
⠀⠀⢠⠀⠀⠀⠀⠀⠀⠈⠉⢉⣤⡌⠠⠀⠀⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠤⠀⠀⠀⡠⠛⠀⣉⣀
⠠⠄⠤⡀⠀⠀⣀⠀⠀⢀⠃⢸⣤⡿⠀⠖⠉⠀⠀⠀⠔⠈⠀⢀⠀⠀⠀⡀⠀⡀⠰⠀⠀⢀⠀⣠⠂⣠⠴⣦⠀⢸⡉⠩⠀⠐⠂
⠀⠠⠀⠁⢰⢙⡉⢳⠀⠈⠑⠀⣠⠖⢦⡀⠐⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⡀⡄⠅⠀⠋⠀⠀⠀⠀⣤⣀
⠀⣀⣤⣄⡈⠃⠖⠚⢀⡨⢤⠰⡅⠐⢻⠃⢠⣤⠇⣈⠤⠀⠀⢌⣆⠀⠒⠐⠁⠀⠈⠀⠠⢤⣤⠗⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠂⠁⠁⠀⡹⠠⢁⡰⠀⠀⢏⡤⣀⠉⢁⣤⢤⣤⠀⠁⠀⣠⠤⠊⠉⠀⣰⠀⠐⠀⡴⠀⠠⡀⢀⡤⣄⠀⠀⠈⠱⠀⠀⣐⠂⡀⠜
⣄⠀⢀⣒⠁⠀⠀⣠⣤⡀⡿⣄⡼⠀⢿⣧⣀⣿⠇⠀⢠⠄⠀⢀⡀⠀⢁⣀⠀⡨⢌⠀⢡⡧⠜⠀⡾⠀⠀⢀⠜⡀⣮⠗⠠⠄⠶
⡿⠀⣁⢀⠇⠀⡏⠈⠀⢹⡇⠀⣎⣱⠀⡙⠛⠁⠄⠢⡀⣄⢻⣛⡿⢠⢿⢙⣷⠘⠘⠒⠋⠀⠀⠀⠀⠀⠀⢀⡏⠉⢘⡆⠂
⠓⢂⠀⢀⠀⠀⠓⠀⠴⠛⢀⡤⢬⣥⡘⢘⣁⡀⠠⠔⢻⣨⣧⡼⢶⡈⠢⣬⠜⣠⠴⢦⡀⡀⠀⠀⠀⠠⢒⡺⠓⠤⠞⢀⠀⡌⢹
⣀⣁⣨⠀⠉⢳⡌⢀⠤⡀⠀⡇⠀⢸⡷⠂⣀⡽⠀⠄⠀⠀⠘⠳⠚⠀⠔⠤⣄⢿⣀⣠⠯⠐⠤⣄⠀⡴⠛⡧⠠⠒⣦⠇⣀⣾⣬
⢷⣹⠏⢢⣆⠾⠁⠘⠴⠃⠀⠛⡛⠉⡔⠲⢦⣄⠄⠀⠀⠀⣼⢉⣆⠀⣆⠀⢹⡇⢈⡀⠘⠀⣾⠙⠧⢈⡶⢧⡈⣹⠗⢄⣠⠤⡈
⠓⠋⠀⠧⠈⠇⢠⡏⠚⢷⡀⠠⠥⣤⣅⠐⣠⣏⣤⣴⡒⡧⠾⠒⢀⣠⣅⣤⠾⣁⡀⠈⠀⡠⠬⠋⠁⠣⣔⣼⣋⡁⠴⣫⣼⠁⠃
⠀⠀⠂⠀⠀⠀⠜⠢⠤⠞⠁⠀⣀⣈⡏⠀⠀⣿⡡⠀⣹⠁⢀⣁⣸⠦⢀⡀⢸⠅⢹⠄⠀⠀⠀⠀⠀⠀⠈⠉⢁⡍⠆⠣⠜⠁
⠀⣀⠀⠂⠀⠀⠀⢰⡋⠑⣄⡠⡌⢉⡴⠶⠶⣚⠒⠋⠁⡞⣁⡀⣸⠇⢞⠉⡋⢉⠄⢤⡀⠐⠀⡲⢤⡀⠀⠀⠈⠈⠀⡄⠒
⣇⣀⣀⣀⣀⣀⣀⡈⢒⣉⣘⡓⠋⣘⣀⡀⣈⣟⣔⣲⣆⣑⡄⢚⣩⣒⣢⣅⠀⣠⣀⣀⣀⣀⢈⣁⣐⣃⣀⣀⣀⢀⣀⣁⣀⣀⢐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣰⣿⣿⣦⢶⣾⣷⣔⢣⠀⠀⠀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣇⠀⠀⠀⠘⢿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡷⠁⠛⠛⠉⠙⠛⠘⠻⠟⣿⣣⣾⣷⣄⠀⠈⠻⣿⣿⣿⣿⣿⣷⡀
⢀⣠⣶⣿⣿⣿⣿⣿⠛⠛⠁⠀⠀⡀⢠⢰⣴⣦⣀⣤⣀⠀⠘⣿⣿⣿⣶⡄⠀⠸⣿⣿⣿⣿⣿⡀
⠈⢿⣿⣿⣿⣿⡿⠃⠀⢀⣴⣶⡆⠻⡎⠙⠋⠉⠛⠻⠿⣿⣦⡄⠻⣟⣽⣶⡄⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠻⣿⣿⡿⠃⠀⣤⣿⣿⣿⠃⠀⢀⣤⣶⣾⠾⢷⣦⡉⠹⣶⠆⠙⣿⣿⣆⠀⠰⣿⣯⡉⠹⣯
⠀⠀⠀⠈⠻⠀⠀⢀⣿⡻⠓⠀⣀⣶⣾⣋⣥⣶⣿⢷⣝⣿⡄⢹⣟⠈⣻⣿⣋⠀⠀⡈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠷⠀⢠⣝⡿⢩⣿⢳⠋⣿⣿⣏⡻⠁⣼⡿⠀⣹⣿⣧⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⢠⣾⡷⠾⣿⡸⢶⣉⣛⣋⣡⣾⡟⢁⣼⣿⡏⠁⢠⣶⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⣫⣿⣆⠹⢷⣴⡋⠛⢻⢛⣁⣶⣿⣿⠃⢀⣰⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡿⢽⣦⡄⠀⠙⢿⣿⣧⣀⡙⠿⢿⣿⡿⠿⠉⠉⢀⡔⣾⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⠘⠋⣿⣿⣿⣤⣠⢀⣄⣤⣄⢴⣾⣻⣬⢿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⣫⣿⣿⣿⣇⡀⠀⠀⠈⠹⠿⣿⣿⣿⣿⣿⣿⡾⡿⠙
⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣾⣿⣿⣧⡆⢀⠀⠀⠀⠈⠉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢫⣿⣿⣿⣵⣿⣦⣠⡄⣀⡀⠀⢀⣀⣀⣀⣀⣠⣴⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⠛⣾⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣷⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠏⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠂⠀⠀⠀⠐⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡔⠀⠀⣤⠀⠀⠀⠀⠀⠯⡆⠀⠀⢸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄⠀⢰⠀⠒⠀⠀⠀⢂⠀⠀⠀⣶⠀⠀⡁⠀⠀⢸⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢆⠂⠀⠈⠉⠇⠀⠀⠀⠀⠀⠀⣌⠞⠀⠀⠁⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⡈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⡀⢳
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠰⡌⠀⠀⠀⠊⠀⠀⠑⠀⠀⠀⡄⠂⣠⣆⣀⣀⠀⠀⢁⡆⠀⠀⠃⠈⡧
⠀⠀⠀⢀⠀⠐⠁⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠋⠉⠁⢀⠀⠀⠈⠁⠀⠀⢱⢐⣿⡄⠀⠀⡟⠗
⠀⠀⠀⡰⣿⡿⢤⢼⡁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠰⡎⠁⠀⠀⠀⠀⠀⠀⠀⢬⢾⡷⠀⠀⢻
⠀⠀⠀⠀⠀⡈⣦⣽⠇⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠠⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⡇⠀⠀⢠⠀⠀⠀⡤⠔
⠀⠀⠀⠀⠀⠐⡻⢿⡇⠀⠀⠀⠀⠀⠠⠑⠁⠀⠀⠀⠀⠀⠀⠄⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠐
⠀⠀⠀⠀⠀⠈⠘⠀⣃⠀⠀⠀⣀⡀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⡿⠄⠀⠀⡁⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣎⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢇⣸⡀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡸⠥⠄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠘⠈⡳⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣼⠀⠥⠀⢂⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⢃⠀⠀⠀⠀⠀⠀⠾⣾⠻⠇⢀⡀⠂⡼⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢻⠀⠒⠀⠀⠀⠣⠀⢑⣄⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠁⡠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡀⢀⡀⠲⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠰⣦⢇⠀⠀⠐⠀⡂⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⢰⡆⠘⢈⠀⠀⢶⡄⠀⣆⢸⡀⠀⠀⠀⠀⣤⠁⠀⢣⡃⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⠀⠀⠀⠘⣀⠋⠀⢀⠀⠀⠙⠲⣜⣦⣱⠀⠀⠀⠀⠈⠉⠀⠀⠹⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡅⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⡠⠀⢻⣦⡄⠀⠉⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⢸⠔⢘⠕⠀⠙⠿⠑⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣳⠀⠀⠀⠀⠀⠒⠠⠀⠀⠀⠀⠀⠈⠀⠀⠁⢀⠀⠀⣰⣾⠤⠀⠀⠀⠀⡠⠄
⠀⠈⢄⠀⢀⣀⢀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⣀⠔⠉
⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢀⢀⠀⠀⠡⣀⠀⠀⠀⠔⠀⠀⠠⢂⠴⠊⠀⠀⠀⣠⠆
⠀⠀⠀⠠⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⡠⠈⠀⢀⠀⠀⠉⠈⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠰⠛⠂⠀⢴⣢⣄⠀⠀⠉⠁⠀⠈⠀⠀⠀⠙⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠒⠀⠁⠀⠁⠀⠀⠀⠈⢪⠘⣆⠀⠀⠀⠀⣦⠀⠀⠀⠠⠀⠀⢀⠔⠉⠀⠀⠀⠞⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⣺⠟⠓⠲⡀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⡀
⠀⠀⠀⠐⠤⣤⠠⠔⠀⠀⠑⠤⠀⠀⢠⠏⠓⣾⠇⠀⢇⠀⠀⢀⡀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢀⡞⠁
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢜⠁⠀⣠⠞⠀⠀⠀⠀⠀⡰⠀⠐⠂⠀⠀⠀⠀⠀⠃⠀⢠⡞⠀⠀⠀⠀⢐
⠀⠀⠀⡯⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠑⠊⠀⠀⢠⣼⡦⠀⠀⢃⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠠
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠉⠃⠀⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⢤⡦⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠐⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠀⢀⠀⠀⠀⠀⠀⠀⠀⡰⠆⠀⠀⠈⠲⢄⡀⠈⠀⠀⠀⣀
⠀⢸⡀⠀⠀⠀⠀⣠⣾⡀⣀⣀⠀⠀⠀⠀⠈⠁⠘⠉⠉⠀⠀⠀⠀⠴⠠⢰⠄⠀⠀⠀⡠⠀⢀⡴⠶⠀⠀⠒⠁⠀⢠⠀⡄
⠀⠀⠳⢤⠀⠐⠛⠛⠻⠻⣛⣙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⢀⡀⠀⠀⡐⠂⠀⢀⣠⠤⢤⠀⢀⠀⠀⢸⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡈⠁⠀⢀⣿⠇⠰⣿⣿⡆⠃⠀⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⣶⣤⠄⡘⠁⠀⠀⠈⡉⣄⣀⠉⠩⣠⡤⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠉⠁⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠹⢿⣿⠂⡴⠿⠁⠀⠀⠀⡇
⠀⠀⠁⠀⣀⠀⢀⠀⠀⡀⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡀⡀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠚⠁⠀⠀⠀⠀⠠⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⠶⠒⠒⠒⠲⠶⠶⢤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠛⢋⣩⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠲⠶⠶⣬⣍⡙⠓⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢋⣡⠶⠛⢋⣡⡤⠶⠶⠒⠛⣛⣛⣛⡛⠛⠛⠒⠶⢤⣌⡙⠻⢶⣤⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣠⠴⠋⣡⡴⠟⣉⣤⠴⠖⢛⣋⣉⣭⣭⣭⣭⣉⣛⡛⠲⠦⣬⣉⠳⢦⣈⠙⢷⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⣾⠋⣠⠞⢁⡴⠛⣡⠶⢋⣡⠴⠒⣋⣩⡭⠤⠤⠤⠶⠦⠤⣌⣉⡙⠲⢦⣍⠛⢦⣈⠳⣦⡈⠳⣦⠹⣦
⠀⠀⠀⠀⠀⣠⡟⢁⡞⢁⡴⠋⣰⠞⢁⠶⠋⣠⠶⠋⣩⠤⠖⠒⠒⠒⠒⠒⠲⠦⢬⣍⡛⠲⢬⡙⢦⣈⠳⣌⠻⣆⠈⢷⡌⢷⡄
⠀⠀⠀⠀⣰⠏⢠⠏⢠⡟⢡⡞⢁⡴⢋⡴⠋⣠⠖⢋⣠⠴⠒⠛⣉⣉⣉⣉⡙⠓⠦⣤⡉⠳⣄⠙⢦⡉⢷⡈⢷⡈⢳⡀⢻⣄⢻⡄
⠀⠀⠀⢠⡟⢠⡏⢀⡟⢠⠏⣠⠏⣠⠏⣠⠞⢁⠴⠋⣠⠴⠚⢉⣉⣁⣀⣈⡉⠙⠲⢄⡙⠧⣀⠳⣄⠙⣆⠹⣆⢳⡄⠻⡄⠹⣆⢻⡄
⠀⠀⢀⡾⠁⣿⠀⣾⠀⡟⢰⠇⣰⠃⡴⢁⡴⠃⡰⠊⣡⠖⠋⣉⣠⠤⠤⢤⣉⠙⠢⣄⠙⢦⠈⢳⡈⢧⡘⢧⠘⣆⠹⡄⢻⡄⢻⡄⢿⡀
⠀⠀⣸⠇⣸⡇⢠⡇⣸⠁⡾⢠⠇⡼⢁⡞⠀⡜⢁⡞⢁⡴⠊⣁⡤⠤⠤⢤⣀⠙⢦⡈⠳⡄⢳⡀⠳⡀⢷⠈⣇⢸⡄⢱⠀⣇⠈⣷⠸⡇
⠀⠀⣿⠀⣿⠁⣼⠁⡟⢸⠃⡾⢠⡇⢸⠁⣼⠀⡞⢀⡞⢀⡞⢁⡤⠒⠒⠤⡈⠳⡄⠹⡄⢹⡀⢳⠀⢷⠘⡇⢸⡄⣧⠸⡇⢹⡀⢹⡄⣿
⠀⠀⣿⠀⣿⠀⣿⢰⡇⢸⠀⡇⢸⠁⢸⠀⡇⠀⡇⢸⠃⢸⠀⡜⠀⠖⠒⠄⠘⡄⢹⠀⢧⠈⡇⢸⡆⢸⠀⣿⢸⡇⢻⠀⡇⢸⡇⢸⡇⣿
⠀⠀⣿⠀⣿⠀⣿⠘⡇⢸⡀⣇⠸⡇⢸⠀⣧⠀⡇⠸⡄⠸⡀⠱⣄⣁⣠⠎⢠⠇⢸⠀⡼⠀⡇⢸⡇⢸⠀⣿⢸⡇⣿⠀⡇⢸⡇⢸⠇⣿
⠀⠀⢻⡄⢻⡀⢹⠀⣧⠘⡇⢹⡀⢧⠘⡆⠸⡄⠹⡄⠱⣄⠙⠦⣤⣉⣠⠴⠋⣠⠏⢠⠃⢰⠃⣸⠀⡾⢀⡏⢸⠃⡟⢠⡇⢸⠃⣿⠀⣿
⠀⠀⠸⣇⠸⡇⠸⡇⠸⡆⢻⡀⢧⠘⣆⠙⣄⠙⢆⠙⠦⣈⠑⠦⠤⠤⠤⠴⠚⣁⡴⠃⣰⠋⣰⠃⣸⠃⡾⠀⡞⢸⠃⣼⠁⡿⢠⡟⢰⡇
⠀⠀⠀⢿⡀⢻⡀⢻⡄⢹⡄⢳⡈⢧⡈⢦⡈⠣⣄⠙⢦⣈⠑⠲⠤⠤⠤⠖⠊⣁⡤⠚⢁⡴⢁⡼⢁⡼⢁⡼⢡⡟⢰⠏⣸⠇⣼⠃⣾
⠀⠀⠀⠘⣧⠈⢷⡀⢳⡄⠹⡄⠳⡄⠳⣄⠙⢦⡈⠓⠦⣌⡙⠒⠲⠶⠒⠒⠋⣁⡤⠞⢁⡴⠋⣠⠞⣠⠞⢠⠞⣠⠏⣰⠏⣰⡏⣸⠇
⠀⠀⠀⠀⠘⣧⡈⢷⡀⠹⣄⠹⣆⠙⢦⡈⠓⢦⣈⠓⠦⣤⣉⡉⠙⠓⠚⣋⣉⣥⠴⠚⢉⡴⠊⣡⠞⢁⡴⢃⡴⢋⣴⠋⣰⠟⣰⠏
⠀⠀⠀⠀⠀⠈⢷⣄⠻⣆⠙⢧⡈⠳⣤⡙⠳⢦⣈⠛⠶⠤⣤⣉⣉⣉⣉⣩⡤⠤⠖⠚⣩⡴⠞⣡⡴⢋⣴⠋⣠⡞⢁⣼⢋⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡈⢳⣄⠙⠶⣄⠙⠳⣤⣈⡉⠓⠶⠤⣤⣤⣠⣤⡤⠴⠖⠚⣉⣡⡴⠞⣁⡴⢋⣠⠞⢋⣴⠟⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⠳⣦⣈⠙⠶⢤⣍⡛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⣉⣥⠶⠛⣡⡴⠛⣡⡴⠟⣡⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣤⡉⠻⢶⣤⣌⣙⠛⠓⠶⠶⠶⠶⠖⠚⠛⠛⣋⣩⡤⠶⠛⣡⡴⠟⣉⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣌⡉⠛⠛⠷⠶⠶⠶⠶⠾⠛⠛⠛⣉⣡⣤⠶⢛⣩⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⠶⢶⣶⡶⠶⠶⠶⠞⠛⠋⠉

⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⡠⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⢀⠠⢐⡠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇
⠀⣀⣢⢠⣈⣀⣥⠆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡊⠀⠀⡿⠂⠀⠀⡐⢁⢲⡀⠀⠸⠀⠀⠀⠀⣠⠤⠶⠶⠿⠿⢶⡄
⠀⠘⠈⠈⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠀⠀⠾⠀⠀⠀⠀⠀⠛⢾⡡⠀⢁⡆⠀⠀⠀⢹⠀⠀⠀⠄
⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠁⠠⠀⠀⠀⠀⣶⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⢠⠀⠀⠀⢨⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠏⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡄⠀⠀⠐⠀⠀⠀⠐⠒⠘


⠀⠀⠀⠀⢤⠉⠀⠀⠀⠀⠀⢀⣠⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢘⠆⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠰⣆⠀⡀⢀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⡄⠄⠀⠀⠀⠀⠸⠁
⠀⠀⠠⠂⠀⠀⠂⠄⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠲⢎⠣⣄⠀⠀⠈⠀⡃⠈⠠⠈⠀⠠⠀⠑⠠⠀⠀⠀⢀⠀⠐
⠀⠀⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⢙⡧⠄⠀⠀⠀⠀⠐⠨⠤⠒⠀⠡⠀⠀⠀⠀⡁⢈⠀⠠⠋
⠀⠀⠈⠀⠈⠁⠒⠂⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⣠⠀⢠⠀⣀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠠⠊⠀⣀⣤⣤⣀⡈⠀⠀⠀⠀⣿⢿⡄⢸⣟⠖⠁⠀⠀⠀⠀⢀⡨⠢⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄
⠐⠀⠂⠀⡰⢛⡅⠈⠀⠁⠀⠀⠰⠀⠀⢨⠉⠃⠀⠀⠀⠀⠔⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡜⢋⣀⣠⡠
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠇⠀⠨⡀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠀⠠⡬⠀⠀⠀⠀⠀⢁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⠉⠹⡁
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⡐⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠚⠀⠀⠀⠀⠀⠀⠀⠀⠑⠰⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢀⠀⠀⠀⠀⠀⢀⣠⠄⣤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠠⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⢀⠂⠈⠂⠀⠀⠀⢠⡏⠃⠀⠀⠀⠙⡆⠀⠀⠀⡀⠔⠒⠂⠀⡀⢀⠠⠀⠀⢀⠔⠀⠀⠀⠑⠄
⠀⣀⠀⠀⠐⠚⠛⠛⢲⠇⠀⠀⢀⠈⠀⠂⠂⠀⠁⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠸⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁⢈⠀⠀⠒⡄⠶⠀⠒⢄⡜⢂
⠀⠀⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢆⠤⠁⠔⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠈⠘

⠀⣀⠀⠀⠀⠀⡀
⢀⠀⠀⣷⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⣰⣤⣤
⠀⠀⡀⠙⡏⠀⠐⠀⠀⠀⠀⠀⠀⠀⠶⠄⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⠃
⠀⠀⡇⠀⢣⠀⠀⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⢀⡜⠁⠀⠀⠀⠀⠀⠇⠀⠀⠀⠠⠦⡀
⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠜⠃⠀⠀⠀⠀⠀⠀⢐⡀⠀⠀⠶⠒⠃
⠀⠀⢀⠀⠸⠀⠀⠠⠀⠘⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠄⠄⠤⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⡶⠿⠿⠿⠿⠿⠿⠿⠷⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⢤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣦⣄⠀⠀⠀⠀⠈⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠈⠹⣿⣦⡀
⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⣴⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠛⠻⢿⣷⡄⠀⣴⣿⡿⠿⠛⠻⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠙⢿⣾⡿⠋⠀⠀⠀⠀⠀⠀⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷
⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢴⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣄⡀⠀⠀⣀⣠⣾⠟⠹⣿⣦⣀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⡿⠟⠁⠀⠀⢹⣿⠿⣿⣷⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⢻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃
⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠘⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟
⠀⠀⠀⠈⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣶⣶⣤⣤⣤⣄⣀⣠⣤⣤⣤⣴⣶⣾⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⡤⠀⠄⠠⠀⠄⠠⠄⠄⡄⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⣒⡦⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣾⠿⡗⣿⡿⣾⣿⣿⣿⣷⣶⣦⣤⣄⣀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣠⣿⣾⣿⡞⠹⠙⠛⠛⠿⠽⣟⣿⢾⣿⣿⣭⣿⣿⣳⣾⣻⣶⡦⠀⠀⠀⠠⠀⠀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣴⣷⣿⣾⣏⡄⠀⡀⠰⠀⠤⠄⠀⠀⠈⠁⠋⠐⠺⠟⠿⣳⣿⣿⣷⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡬⢿⡿⠿⠁⢋⠀⢀⠛⠃⠀⠂⢀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠸⢹⡎⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⠀⠀⠀⠀⠄⢀⢐⠀⠀⢀⠀⠀⠀⢀⠀⠀⠃⠀⠀⢀⠀⣾⠁⠀⠀⠀⠀⠀⢰⡿⣴
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠚⣈⣿⠏⠀⠀⠀⠀⠀⠀⣼⠀⣩⡀
⠀⠀⠀⠈⣇⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠀⡈⢠⡟⠀⠀⠈⠀⢀⡀⣸⣿⣀⣈⣀
⠀⠀⠀⢀⡈⢃⣻⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠰⠠⠐⠦⠄⠀⠀⠀⠰⣿⣿⣿⣿⡿
⠀⠀⠀⠀⢀⡀⡙⠩⠷⢄⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠃⠀⠈⠑⠀⣰⣿⠏⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⣳⡤⢀⠀⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣯⠇⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠧⠔⠄⡄⢸⣁⠀⠀⠀⠀⠠⠀⠀⠒⠛⢻⢨⠽⡿⠟⠁⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠘⠂⠈⠲⠂⢀⢰⢊⣹⣷⡁⠽⠴⠋

⠀⠀⠀⡠⠖⠋⠉⠈⠉⠒⢄⠀⠀⠀⣠⠔⠋⠉⠉⠉⠒⢤⠀⠀⠀⢀⠔⠚⠉⠁⠉⠓⢤⡀⠀⠀⢠⠖⠉⠒⠢⢄
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⡰⠁⠘⢡⡄⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠱⡀⢰⠁⠀⠀⠀⠀⠀⠑⢦
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡇⠴⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⢣⡀⢿⠧⠀⠀⠀⠀⠀⡼⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠸⡀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠑⢄⠀⠀⠂⠀⠀⣀⠇
⠀⠀⠀⠑⠤⣀⣀⣀⣀⡤⠊⠀⠀⠀⠙⠦⣀⣀⣀⣀⡠⠚⠁⠀⠀⠈⠢⣀⣀⣀⣀⡠⠞⠁⠀⠀⠀⠀⠀⠙⠢⠤⠠⠔⠁
⠀⠀⢀⡀⢄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢤⡤⠄⣀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠤⣀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣀
⠀⠀⡇⠀⠀⣠⠄⠉⠲⢄⠀⠀⠀⢀⣴⠁⢀⣴⣶⣄⠀⢱⣄⠀⢀⡔⠋⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠋⠀⠀⠀⠀⡈⠙⢆
⠀⠀⢧⠀⠀⠀⠑⢕⠄⠀⠱⡀⠀⣾⠃⠀⠈⠿⠿⠏⠀⠀⠹⡄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢸⠁⠀⠄⠀⢠⠀⡀⠂⠈⡆
⠀⠀⠈⢆⠀⠀⠀⠀⠈⡦⠀⢱⠀⠇⢠⣤⣄⠀⠀⠀⣠⣤⢀⠇⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢰⠐⠑⠁⠀⠇⠈⠌⠀⢂⡇
⠀⠀⠀⠀⠓⢄⡀⠀⠀⠀⠀⢘⠀⠈⢾⣿⣿⠆⠀⠐⢿⣿⠏⠀⠈⢧⡀⠀⠀⠀⠀⠀⢀⠞⠀⠈⠳⡀⠀⠀⠀⡐⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠊⠀⠀⠀⠉⠒⠢⠤⠔⠊⠁⠀⠀⠀⠀⠉⠒⠶⠤⠔⠚⠁⠀⠀⠀⠀⠉⠓⠲⠦⠖⠚⠁
⠀⠀⠀⢀⠴⠂⠉⠉⠑⠢⣄⠀⠀⠀⢀⡤⢆⣉⠉⠓⠢⣄⠀⠀⠀⢀⡴⠒⠻⠉⠓⠲⢄⠀⠀⠀⠀⠀⠀⡐⠳⢤
⠀⠀⡰⣡⡶⡻⣶⡄⠀⠀⠈⢣⠀⠰⠃⠀⠀⠀⠀⠀⠀⠈⢣⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⢰⠄⠀⠀⠙⢄⡀
⠀⠀⡃⣿⠰⢡⣾⡇⠀⠀⠀⠈⠃⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣰⣿⣦⡀⠀⠀⠀⠑⣄
⠀⠀⢣⠈⠛⠟⠋⠀⠀⠀⠀⣰⠁⢣⠀⠀⢀⠀⠁⠀⠀⠀⣰⠁⢸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢀⢴⣿⣿⣿⣿⣦⣄⠤⠔⠊
⠀⠀⠀⠳⢄⡀⠀⠀⠀⣀⠔⠁⠀⠀⠑⢄⡁⠀⠀⣀⣀⠔⠁⠀⠀⠑⢤⣀⠀⠀⠀⣠⠔⠁⠀⢀⠀⠙⡿⠛⠉
⠀⠀⠀⠀⠀⣉⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠀⠀⣈⣉⣉⣉⡀⠀⠀⠀⠀⠁⠈⢀⣀⣀⣀⡀
⠀⠀⢀⡴⠋⠀⠀⠀⠀⠉⠢⡀⠀⠀⡴⠋⠀⠘⣄⠀⠈⠲⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠓⡄⠀⠀⡠⠊⠁⠀⠀⠀⠈⠑⣄
⠀⠀⡞⢰⡷⠀⠀⠀⠐⣷⠆⢹⠀⡜⠀⠀⠀⠀⠈⠱⢄⡀⢘⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠘⡄⢰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡗⠒⠠⣀⠀⠀⠀⠀⠈⠉⠇⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠘⣄⠀⠀⣠⣀⠀⠀⢀⠎⠀⠘⣄⠀⠈⠳⡄⠀⠀⢀⠜⠀⠘⢄⠀⠀⠀⠀⠀⠀⢀⡜⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠈⠑⠶⠬⠥⠤⠚⠁⠀⠀⠀⠈⠑⠢⠄⠼⠤⠒⠁⠀⠀⠀⠀⠑⠢⠤⠤⠤⠒⠉⠀⠀⠀⠀⠑⠢⠤⠀⠤⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠷⠴⠛⠁⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠁⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠠
⠀⠀⠀⢤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠛⠛⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⡀⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣀⡀⠠⠴⠀⠒⡁⠀⠀⠤⠤⠖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠐⠒
⠀⡀⠀⠀⠀⠈⠤⠠⠴⠒⠈⠀⠀⠀⠀⠀⠀⠄⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡀⠉⠁⠀⠀⠀⣤
⠀⢁⠀⠀⠀⠀⠀⢀⡀⠄⠠⠀⠀⢉⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠔⠈⢀⡂⠀⠀⠀⠀⡅⠀⢀⡀⢀⡤⢠⠠⠀⠁
⠀⠈⠀⠬⠀⠈⠀⠃⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠀⠘⡀⠀⠁⠀⠀⠀⠇⠀⢀⣤⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠐⠛⠁⠀⠀⠄⠀⣙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠂⠀⠀⠀⢛⠉⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⢀⡀⠀⣁⠀⣴
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢨⢀⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁
⠀⠀⠁⠈⠐⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠄⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠲⠠⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠘⠆⠘⠃⠂⢂⠄⠀⠀⠀⠀⠀⠤⠄⢙⣐⣒⣉⠠⠤⡆⠀⠀⠀⠀⠀⠉⠉⠉⠀⠁⠀⠀⡀⠀⠀⠀⠀⣀⠤⠀⠀⠠⠤⢀
⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠸⡀⢆⢀⣄⠲⠀⢂⠀⡇⠀⠀⠀⠀⠰⢠⠂⠈⠲⠂⠀⠀⠀⠀⠀⢀⠈⠀⣀⡀⠁⡁⣀⣀⢱
⠀⠀⠀⠀⠛⠄⠀⠀⠀⠆⠀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⢀⠜
⠀⠀⠀⠀⠁⠀⠈⠠⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣄⠈⠒⠒⠒⠒⠒⠁⣀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣅
⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠉⠁
⠀⠀⣼⠷⢦⣄⡀⠀⠀⠀⠀⡠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢄⡀⠀⠀⠀⠀⠀⢀⢤⡠⠄⠤⢄⠀⠀⢠⢊⠋
⠀⠀⠀⠀⠀⠈⠛⢗⢄⠀⠀⡇⠀⢱⠀⡀⠤⣀⠀⠀⠀⠀⠀⣏⣉⠉⡇⠀⠀⠀⠀⠰⡅⠀⠑⣴⡄⠀⠑⡴⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡕⢤⡃⡠⢼⠉⠀⠀⠀⢑⠖⠢⢄⣸⡀⣩⡧⠧⠒⠊⠉⠒⡎⠀⠀⢀⣩⠉⣶⠎⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠓⢌⡐⠪⢅⣢⡎⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠺⠵⣴⣝⣢⣁⠔⠁⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠈⠑⠤⣈⡏⠉⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢹⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠉⠐⠂⠀⠤⠦⠤⠤⠤⠤⠒⠂⠉⢛⠉⠉⢹⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠂⠤⠤⠤⠄⠐⠂⠉⡠⠃⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣀⣸
⠀⠒⠒⠂⠒⠂⠒⠒⠂⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⢿⡄⠀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⢀⣤⡄⠈⢿⣎⠉⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠘⣿⣷⣠⡞⢻⣷⡀⠚⢷⣦⣤⣤⡴⠶⠒⠉⠁⠀⠙⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠷⠶⢤⣤⣴⣿⣡⡤⠄⠀⣻⣿⣿⣿⡀⠙⠻⠷⣄⡙⠳⢄⡀⠀⢠⣶⡶⢄⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠶⣦⣤⣧⣀⣀⣊⣁⣉⣉⣤⠤⠖⢋⣨⣿⣿⣿⣿⣆⠀⣴⣎⠙⠲⠤⢀⣰⣿⣿⣃⣀⣾⡇⠶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⡠⠘⠹⣟⠛⠛⠛⠉⠰⣷⣶⣤⣼⣿⣿⣿⣿⣿⣿⣆⠸⣿⣧⣤⣶⣴⣿⣿⣿⠏⢸⣿⠆⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠘⣿⣿⣶⣿⣿⣶⣶⣦⣄⣈⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⠀⢲⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠉⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣀⣹⡀⢾⣿⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣠⠁⡼⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣾⣿⠉⢧⠘⡇⠉⠛⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠃⣼⠁⣠⣼⣿⣿⣿⣿⣿⣿⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠘⣄⠀⢀⣀⣀⠀⠉⠓⣦⣄
⠀⠀⠀⠀⠀⢀⣠⠴⠛⠏⣠⡾⠘⠿⠿⠛⣻⣿⠉⣡⠌⣿⣇⣿⣥⣿⣟⣿⡟⣿⡿⣿⣤⣿⣧⣾⣏⣹⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⠟⠇⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⢿⣇⠀⣰⣴⡿⠁⢀⣤⣶⣾⣿⣿⡟⣉⡀⣼⠯⣿⡿⣿⣿⡿⠓⢿⣇⣿⡿⣿⠗⣽⡏⢹⣿⣿⣿⣿⣿⣿⣿⠿⠿⢦⣤⣤⡾⠛⠉
⠀⠀⠀⠀⢀⣠⣴⠿⠻⣥⣤⣴⣿⣿⣿⣿⣿⣿⡀⠛⢁⣿⣃⣿⣿⣿⣯⣧⣀⣾⡇⣿⣧⣿⣯⠽⠛⢻⣿⣿⣿⣿⡿⠟⠋⢀⣴⡿⠟⠉⣶⣦
⠀⠀⢠⣾⣟⠉⠀⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡾⠃⢸⣿⣿⣏⣁⣤⣤⠀⣾⠏⢠⣠⡴⠟⠋
⠀⠀⠘⠻⢿⣀⠀⠈⠛⠛⠉⠉⢣⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⡿⠛⢉⡟⢁⣴⣿⠁
⠀⠀⠀⠀⠀⠈⠙⠲⣤⣀⠀⢡⠀⣧⠀⣾⡿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⡞⢀⠞⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣾⡇⢸⡝⠛⠁⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣀⠈⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠘⣿⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣦⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⢠⣿⠇⣠⣿⣿⣿⠿⣿⠿⢿⣿⡉⢻⣿⣿⣿⣿⣿⣿⡿⠿⢿⣷⠀⠀⣀⣀⣈⣻⡀⡉⠟⠁⠀⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⢼⣿⠚⢻⣿⣿⠟⠒⠤⣄⡈⣿⡧⠀⢻⣿⣿⣿⣿⠟⢁⣠⠴⠾⠒⠛⠛⠛⠟⠛⠿⠷⢶⣤⣴⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⢢⣾⣿⠋⠀⠑⠢⣄⡉⠳⣤⣄⠀⠙⣿⣿⣿⣟⠁⠤⠰⢚⣿⡷⠿⠦⣤⣤⣾⠄⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⡀⠀⠀⢀⣀⣤⡴⠶⠾⢿⣦⣈⠛⣿⣆⣽⠻⣿⣿⡀⠀⢠⡿⠉⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠒⠋⠉⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣿⣧⠀⠹⠟⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠁⠻⣦⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠏
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⢀⡀⢀⠀⣀⢀⣀⡀⣀⣀⣀⣉⡉⣀⡀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣾⣷⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠟⠻⠿⣿⣿⣿⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢟⡁⠀⠀⣤⢦⣤⣝⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣛⣀⣀⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⣻⣿⣟⣁⣤⡤⠄⢈⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣏⣉⣀⣀⣤⣶⣿⣿⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⢨⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢀⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣡⣧⣾⣿⣷⣮⡙⢿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣄⣼⣿⣿⣿⣿⣿⣿⣿⣧⡙⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠏⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⢡⣼⣿⣿⣿⣿⣿⣿⡿⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⣿⣿⠁⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠉⢿⣿⡟⣼⣿⣿⣿⣿⣿⣿⣿⡇⠉⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⡐⣫⣿⣿⣿⣿⣿⣿⣾⡟⢸⡿⣿⣿⣿⣿⣿⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠋⠺⠿⠟⠛⠛⠛⠛⠛⠛⠃⠿⠃⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⣀⠀⡀⠀⡀⢀⣀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠐⠘⡀⠀⠁⠀⠀⠀⠈⠀⠒⠃⠀⠀⠀⠀⠀⠀⠀⠐⠚⠀⠀⠀⠀⡚⠣
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠣⠄⠃⠇⠀⠀⠘⠂⠀⠀⠁⠀⠀⠀⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠀⠂⠀⠀⡆⢰⠀⠀⠀⠀⡤⣠⡀⠀⠄⠀⠀⠀⠈⡁⡠⠀⠀⢀⡀⠀⠝⢃
⠀⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⢀⠠⡄⡀⡀⣄⢀⠀⠀⠀⠀⠀⠆⠀⠒⠒⠀⠀⠆⠀⠀⠀⠀⠊⠀⠀⠀⢐⠀⠀⠕⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠄⠄⠀⠀⢠⠄⠠⠣⠠⠇⠜⠄⠇⠈⠣⠀⡀⠀⡆⢰⠀⣉⠘⠃⠀⠂⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣶
⠀⠈⠈⠁⠀⠀⡀⠀⣦⣧⣷⣴⣾⠀⡦⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⢸⠀⣿⠐⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠠⠀⠀⠀⠀⠀⠃⠀⠈⠛⠙⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢀⠀⠀⡠⠤⢆⠀⠈⠛⠙⠘⠘⠋⠛⠛⠃⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠈⠀⠁⡀⢘⠂⠀⠀⡀⠀⠈⠀⠀⠐⠀⠁⠀⠀⠀⠀⢆⣙⡸⠂⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠂⡀⠀⠀⠀⠂⠀⠀⠀⠐⠂⠐⠀⠐⠂⠐⠂⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉⠀⠀⠸⠇
⠀⠘⠐⠘⠐⠀⠀⠀⠀⢀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⢋⠀⠀⢀⡀⢀⠀⠀⢰⡆
⠀⠈⠛⠛⠃⠀⠀⠀⠀⠉⠀⠁⠈⠁⠃⠀⠀⠀⠀⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⡀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⣿⣧⡄⠀⠀⠀⠀⠀⠻⣶⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⡁⠠⠽⠟⠃⣄⡐⠰⠶⠀⠉⢠⡀⠀⢀⣴⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡁⢋⡳⠀⢀⡤⠀⠈⠠⢦⣤⣲⡖⠀⠀⠘⢿⠯⢚⡽⠿⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠂⠘⣷⣤⣜⠁⠘⠋⠀⠩⣳⣹⡿⡶⠓⡤⡄⢈⢻⣧⡶
⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢾⡆⠹⢏⣿⣷⠂⠀⢀⣀⢸⡿⣿⠏⠘⠛⠻⢻⡟⣳⣄⠢⠐⣀
⠀⠀⠀⠀⠀⠀⢀⣿⡗⠦⣷⠬⣅⡶⠎⠻⢯⡄⠂⠻⠿⠸⢿⠿⠀⣀⠀⠀⠀⢰⣿⣅⣤⠖⡭⠅
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠈⢢⣴⣿⠶⠼⠾⠀⢀⣀⠠⢀⣰⣤⣾⣿⣧⣤⣀⠙⣏⢀⣀⡰⢶⣿⠁
⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣈⠉⠀⠀⠐⠂⠀⠈⠁⣠⠀⠊⠋⡉⠉⠙⣄⠈⠀⢀⡀⠈⠃⠄⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠠⠞⠉⠙⠻⣿⣶⡷⢤⣤⣌⣀⠀⠀⠀⠵⠀⠘⠉⠉⢘⣿⡟⠂⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠐⠀⠀⠀⠀⠙⠴⠛⠧⣾⣻⣾⠃⠐⢒⡀⠀⢐⡛⠘⣿⣁⣀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⢝⡷⢸⡀⠈⢀⠺⢻⡿⠁⣒⠠⣴⣮⠤⠀⣉⣡⠀⠈⢹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣮⣷⣾⣧⣼⣠⣴⢫⠀⠸⠟⢳⠎⢡⣾⣿⠉⠉⠀⣰⠚⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⣿⣟⢊⢸⣿⡒⠀⡀⢀⠍⢶⣦⡉⠹⠷⢘⢢⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣼⣀⡐⠻⠞⠃⠀⠀⠀⢀⣀⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⢛⣿⣿⣿⣿⣿⣿⣯⠻⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⣿⣿⣷⣮⣭⣭⣭⣭⣭⣴⣾⣿⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠭⠯⠭⠯⠏⠛⠛⠉

⠀⡠⠀⣶⣶⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠰⠀⠐⠻⠿⠀⠶⠟⠚⠲⠿⠂⠀⠷⠟⠳⡶⠶⠿⠆⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠂⠀⠀⢀⡴⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠃⠀⢀⣴⠋⢀⣼⠿⣷⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⣙⢻⣿⠀⠀⣾⠁⠐⣾⠟⢶⡿⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⢻⡾⢿⡄⣼⠁⠀⢰⡿⢶⣾⠀⢠⡿⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣿⣃⣸⣷⡿⠀⠀⣾⠧⣾⠃⢀⡾⠁⠀⠀⠀⢀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣦⠀⠀⢸⡍⣻⢿⡌⣿⠉⣿⡄⣾⡿⣶⠇⣸⣜⣣⣤⡴⠖⠒⠉⠥⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢰⠀⠀⢠⣙⢤⡀⠈⣿⡿⣿⣿⣯⣤⣼⣷⣿⣳⣟⡾⠟⠋⠡⠀⠔⠒⠳⡀⢀⡟⠀⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣀⢚⡀⢸⣿⡿⣱⣦⡘⣧⣠⣾⡻⣾⣿⣘⣿⡿⠛⡤⠊⠐⠀⠲⠀⠳⠀⢁⡼⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠻⠎⠙⢿⣹⠶⡿⠏⠙⢯⠛⢿⣓⣿⣿⣻⣿⠑⢲⣦⠀⠀⢦⠄⠈⠁⣈⡿⠁⣠⣾⡟⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠰⣶⡅⠸⢿⣶⣿⣿⣶⣤⠑⡄⠻⣿⣾⣿⠛⢢⡈⢡⡴⠐⠚⠇⢈⣴⣏⣤⣾⡽⠏⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢶⣤⣄⠀⠩⡀⠀⢂⡀⢰⣬⣿⡀⢈⣵⣶⣿⡄⢹⣼⠷⣄⢂⣀⣀⣴⣤⣿⣶⡟⣛⣿⡿⠟⠁⣀⡴⢋⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡿⢶⣿⡈⠹⡿⠊⢙⡉⢳⣾⣿⠧⠾⡇⢸⣯⣴⣿⣯⣿⠿⠿⠿⠿⠿⠟⠉⣡⣤⡆⢈⠇⢰⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠙⢧⣽⠷⣦⠶⣷⠼⠷⡾⢿⣶⣤⣄⣱⣾⡿⢟⣋⡁⠀⠀⠀⠖⠻⣇⠀⢀⠄⠈⠻⣯⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⢄⠈⠻⣆⣸⢷⣽⣀⡀⣥⣸⡟⠛⣿⣿⡋⠀⣼⠋⢁⡄⠀⠌⠀⠀⠘⡷⢎⣀⡰⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣸⣆⡹⢦⠙⣿⠘⣿⣿⠃⠈⢻⣿⣾⠟⠻⡰⠾⠃⠀⠀⠘⣁⣀⣤⠶⠆⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠻⣷⠀⠙⣼⣟⣡⣾⣷⡆⢸⣿⠋⣀⡰⡷⠶⠋⠛⠛⠛⠉⠉⠁⠀⠀⠈⠓⠂⠀⠀⠠⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠘⣯⣀⡀⣿⢠⠞⠙⣿⢀⣾⠛⣶⣿⠿⠛⠉⠀⠀⠀⡀⣀⢀⣀⣀⣀⣠⣤⠄⠀⠀⠀⠀⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⡿⣿⣿⣿⣷⣤⡏⠉⠙⣿⡇⢀⡴⠻⡿⢱⣿⣿⣁⡤⠶⣤⣐⠾⠾⠛⠋⠉⠉⠉⠉⠉⠉⠘⠛⠛⠒⠒⣒⣶⡾⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠘⠿⣿⡿⣿⣽⣿⡤⠤⣸⣇⡺⠀⠀⣷⣾⣿⣻⣭⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣿⡿⠋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠉⠛⢷⡖⣤⢄⡀⠀⢻⣙⢿⣡⣿⡄⠀⠈⠻⡇⢀⣼⣿⣿⣛⣭⣶⣿⣻⣟⢻⣿⠿⣷⡾⣷⣞⡿⢳⣿⣿⣿⣿⣿⡷⠋⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣟⣬⡁⣦⣼⣯⣿⣯⣾⣷⡤⠒⠢⣿⣾⣿⣿⣿⣿⠿⠿⢿⣿⣿⢿⣼⣿⢿⣾⣷⣿⣿⣿⣟⣥⡿⠛⠋⣀⣤⣴⣯⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣉⣿⡅⣿⣿⢯⣿⣿⣿⣧⡀⠀⢠⣿⣿⣿⣅⠀⠀⠀⠀⠉⠙⠛⠽⠿⢾⠶⠽⠛⠛⠋⢉⣠⣤⢶⡾⣭⡀⣯⠛
⠀⠀⠒⠚⠻⠿⢻⣿⠳⣶⣤⣋⣭⣿⡏⣯⣾⣿⣿⣿⣿⣷⣄⣿⣿⣁⣿⣍⣿⡶⠶⡖⢢⣤⣤⣤⣤⣤⣤⣤⠄⠶⢶⣿⣅⠤⠘⢰⡯⠛⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠉⠺⣍⢳⡌⠹⣧⣽⡇⣿⣟⣿⡯⣿⣿⣯⣿⡿⢛⣿⣿⣽⣿⣷⠈⢛⣳⡙⠛⣷⣄⠾⢷⣦⣀⣀⡈⠻⣿⠖⠃⠁⠀⣠⣴⣿⠃
⠀⠀⠀⢰⡄⠀⠀⠀⠉⢾⠳⡾⢿⡞⣿⣯⣿⣿⣿⣿⣿⣿⣿⡗⠋⣁⣠⣄⣤⣤⣬⣭⣩⣄⣴⡚⠛⠒⠒⠛⠋⠉⠉⢀⣠⣤⣴⣾⣿⣯⡿⠁
⠀⢀⠀⠈⡿⢦⠀⠀⠀⠘⣧⠑⢀⣿⢉⣽⣿⣽⠏⠻⣿⣿⣿⣿⣷⠘⠿⢓⣿⣿⣿⣿⣿⣇⣘⠿⢛⣿⣿⣿⣿⣟⢻⡿⢦⣿⣏⣩⣼⠛
⠀⢸⣦⡀⠀⣶⣻⣦⣄⡀⠀⠻⣯⣾⣿⣤⠞⣹⠿⣶⣿⣿⣻⣿⣿⣿⣷⣿⡏⢡⣤⣴⠌⣻⣿⣷⡿⠟⢛⣹⣿⣿⣷⣴⣾⣿⣼⠟⠁
⠀⠀⢿⣿⣦⡘⣿⣌⣱⢾⣉⣴⠄⢉⣻⣿⣿⣷⣾⣡⣿⣿⣫⣾⣿⡿⠟⠛⠛⠋⣉⣉⠉⠛⠛⠿⠿⢿⣿⣯⣥⣿⣿⡿⠿⠋
⠀⠀⠈⢻⣬⠛⣿⣿⣿⣾⣿⣿⣹⣾⣧⢤⡤⣄⠈⢹⣿⡟⡋⠁⠀⣀⣀⣀⠀⠀⠀⠀⠉⠲⠦⠀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠠⣴⣶⠟
⠀⠐⣄⠀⠙⢶⣿⡿⣄⢀⡿⣿⣿⣿⠛⠿⣿⣟⣱⣿⣿⣿⠟⠛⠋⠁⠈⠉⠉⠙⠓⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⡲⠛⠁
⠀⠀⠈⢦⠐⣌⡛⠷⠧⣾⣵⣿⣿⠟⠀⢠⡄⣿⣿⣿⣿⣇⡴⠞⠛⠋⢉⣉⣉⣉⠲⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠉⣀⣀⡴⠞⠁⢠⡆
⠀⠀⠀⠀⠳⡌⠻⣲⢤⣄⣈⡙⠛⠶⣦⣌⣴⣿⡟⣿⣿⠏⣀⣴⢾⣿⣾⣿⣼⣯⣴⣗⢠⣄⠈⠙⠛⠛⠛⠛⠛⠋⠉⠁⠀⡀⠀⡿
⠀⠀⠀⠀⠀⠘⢦⡙⢿⡿⢉⡝⠓⢲⠤⢌⡙⠿⣀⣿⣿⣷⣤⣿⣨⣅⣸⡇⢸⡏⢹⡟⢻⡿⢿⣷⣶⣤⣤⡄⢠⡤⣤⣶⣿⣿⣾⠃
⠀⠀⠀⢲⣤⣀⡀⠙⢷⣝⡿⣷⡤⣧⠀⠈⠙⠲⣌⢻⣿⡏⢙⡋⢹⡟⢻⡿⢿⣿⣾⣷⣼⣧⣸⣏⠹⡟⠻⡷⣾⣷⣼⣇⣸⠟⠁
⠀⠀⠀⠀⠙⣿⣭⡓⠲⠿⠿⠯⠿⣧⣶⣇⣠⡄⢀⡼⣿⣿⣾⠿⠾⢧⣼⣇⣸⡇⢸⡏⠹⡟⠻⣿⢿⣿⣶⣷⣸⣯⣹⠟⠉⠀⠀⣀
⠀⠀⠀⠀⠀⠘⢿⡙⣶⣦⡴⣄⣀⡀⠈⠉⠙⠛⢾⡇⠘⣿⣇⠀⠀⠀⢀⣉⣿⣿⣿⣿⣶⣿⣤⣷⣀⣧⣤⠿⠛⠋⠀⠀⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠻⣧⠐⣾⣍⢻⣄⣶⣆⡀⠀⠀⠀⢿⣿⣦⡴⢾⠋⠉⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠓⠒⠖⢒⣠⡿⠛⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠞⣿⡶⠌⣷⠜⠻⡔⣈⣿⣿⣧⡄⠀⣀⣠⣤⣤⣴⣶⣶⣶⣦⣤⣤⣶⡞⠛⠋⠁⠀⠀⢀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡴⢶⠿⢿⠿⣾⣥⣾⣆⣸⡏⠙⢿⣿⣷⣾⣿⣿⣿⠿⢨⣿⡟⢠⣿⣿⠀⣸⣿⣷⠖⣶⣶⣾⣹⠟
⠀⠀⠀⠀⠀⢀⣴⣾⠛⢛⣳⠶⣄⢳⣬⣷⠀⠯⡉⣉⣹⣿⠶⢾⢿⣿⣿⣿⣟⣿⣷⡖⣴⣶⣧⣤⣴⡷⣉⠛⣉⡴⣿⣿⡿⠋
⠀⠀⠀⠀⣰⣿⣥⠀⠻⣦⣍⡰⢟⣻⠾⣹⣿⡶⣿⣟⣷⣿⠰⠆⢀⣨⣿⣿⣿⣿⣿⣁⡺⢏⠿⠿⠿⢁⠿⣿⣿⣤⠟⠋
⠀⠀⢀⣼⣿⣮⣿⣷⣮⣥⣼⣷⣌⢻⣾⣹⣿⡟⣫⣴⣦⣭⣶⣿⣿⣿⣿⠿⠋⠛⠿⣿⣿⣶⣶⣷⠚⠛⠛⠉⠁
⠀⢀⣾⠟⠋⠁⠀⣀⣼⣿⣽⣿⣿⣿⡟⣟⣻⣛⣟⣿⣿⣥⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣷⣦⣤⣀
⠀⠘⠀⠀⠀⣠⡾⠿⠛⠋⠙⠛⠛⠛⠿⠿⠿⠿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⠛⠃
⠀⠀⠀⠀⠜⠋

⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠲⣄⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⠖⠁⠀⠀⣠⠏⠈⠙⠲⡄⠀⠀⠻⠏
⠀⠀⠀⠀⢠⠇⣀⡤⠖⠋⠁⠀⠀⣀⣀⣼⡀⠀⠀⠀⢀⣀⡀⠀⠀⣀⣀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢠⠴⠟⢤⡀
⠀⠀⠀⠀⢈⠟⠁⢀⣀⣤⡴⠞⠋⠉⠀⡀⣧⢀⣾⡿⠛⠛⠛⠛⠛⠛⠛⠻⠻⠟⠛⠛⠛⠛⠛⢿⣦⠀⣫⡀⠀⣠⢷⡀
⠀⠀⠀⠀⢸⡖⠚⠉⠈⠁⠀⠀⠀⠀⣀⣡⡟⠸⣿⡄⠀⠀⠀⠀⠀⠀⣤⠀⠀⢴⠀⠀⠀⠀⠀⡠⠛⠉⠀⠉⠉⢁⣠⠧⣄⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠳⣄⢀⣀⡠⠤⠖⠚⢩⡁⠐⢳⠀⠈⢿⡆⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⢀⡤⠓⠒⠛⠉⠉⠉⠉⠀⠀⠈⢇⠀⠈⡿⠃
⠀⠀⠀⠀⠀⠀⢈⡽⢇⠀⠛⠀⠀⠈⣥⠷⠞⠓⠛⠛⠓⠒⠲⠤⣄⠀⠣⠈⡈⠠⠁⠀⢀⣏⠤⠤⠤⠤⠤⣀⣀⣀⣀⣀⣠⠾⣦⠴⠤⢤⣀
⠀⠀⠀⠀⠀⠀⡎⢠⢟⣧⡀⢠⣹⠟⠁⢠⠒⠒⠒⠚⠉⠉⠦⠤⣄⠙⢦⡀⠀⠀⠀⠀⡎⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣦⣄⡈⠳⣄
⠀⠀⠀⠀⠀⢸⠀⣾⣼⠀⣟⡞⢁⡤⠖⠃⠀⠰⠆⠀⠀⠗⠀⢀⠸⣆⡀⠙⣆⠀⠀⠀⡷⠂⠚⠉⠙⠋⠉⠑⠋⠉⠲⠉⠳⠲⣞⠛⠿⣿⣿⡄⠸⡄
⠀⠀⠀⠀⠀⠘⡄⢹⡽⣄⡿⠀⣸⠀⠀⠁⠀⠀⢤⣀⣠⠆⠀⠈⠀⠈⡹⠀⢻⡀⠀⠀⡇⠀⠀⠀⠀⡲⠀⠀⠀⣴⠀⠀⣠⡶⡍⠙⡆⢹⣿⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠱⡄⠙⢼⡇⢺⡁⠀⠒⠀⠀⢀⡴⠦⣄⠀⠈⠀⠀⡀⠙⡆⠈⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢯⡉⢡⠟⠋⣩⠉⠛⢯⣼⠁
⠀⠀⠀⠀⠀⠀⣀⠈⠢⣤⡇⢀⡇⠀⠁⠀⠐⠸⣄⠀⣨⠂⢂⣀⣀⣀⡸⠥⠼⠷⠶⠶⢿⣄⣤⣀⣀⡘⠲⠤⠖⠀⠀⠹⢤⣾⠀⢀⠉⠁⠀⡀⠹⡄
⠀⠀⠀⠀⠀⠺⣿⡇⠀⠀⢳⠈⠣⡄⠐⠀⠄⠠⢌⡽⠛⢋⡩⠤⠤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣍⡉⠓⠶⡄⣠⠤⠶⠾⣘⣆⠈⠦⠤⠘⠋⠀⡷⢒⡦
⠀⠀⠀⠀⠀⠀⢀⣀⢀⢤⣈⣧⣀⡣⣄⠀⠀⡄⢸⣦⣀⠳⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢀⡠⣟⡡⠖⢲⠦⡌⢮⠙⢦⣄⣀⣠⣞⠁⢈⡹
⠀⠀⠀⠀⠀⠀⣸⠉⡍⢀⣉⠆⠀⠙⢾⡦⠴⢄⣈⡇⠈⠉⠓⠒⠒⠬⠭⠭⠭⠭⠭⠭⠭⠥⠔⠒⠒⠉⠉⠀⣏⢀⣠⡟⠀⢸⢸⡆⠀⣠⣁⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠸⣅⢄⣠⠏⠀⠀⣄⠀⠀⢳⡦⠤⠭⢿⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⢰⠋⠉⠁⠀⣠⠏⡸⠁⠀⠿⠟
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⠀⠀⠀⡄⠀⣧⠀⠀⠈⢇⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⢀⣮⣤⡤⠴⠚⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠙⠀⠠⠜⠁⠀⣸⠀⠀⠀⠈⢣⡀⠀⠀⠀⠈⠢⣄⣀⣀⡠⠞⠀⠀⠀⠀⣠⠾⠒⠒⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⢤⡤⠤⠤⠖⠁⠀⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠉⠉⠉⠉⠉⠙⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣀⣴⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⣡⠿⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡤⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠁⠀⠀⠀⠉⠙⠳⣄⠀⠀⠉⢽⣒⠦⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⣾⣣⡼⠖⠋⠉⢑⡜⠂⠀⠀⠀⠉⠉⡿⠒⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢙⣲⣤⠤⠤⠤⣿⠄⠀⣀⣀⠞⠁⠀⠀⠀⢀⡀⠀⢧⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠘⠄⠀⠁⠀⠀⠘⠉⠉⠁⠉⠁⠀⠀⠀⣠⣏⠏⠁⠀⣁⡤⣤⣄⠹⡄⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣔⢛⡶⠋⠀⠀⣀⣀⣉⣁⣠⠼⠁⠀⣠⠀⠀⣦⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡀⠀⠉⠒⠦⠤⠤⠤⠤⠴⠒⠋⠉⠀⠘⢉⡴⠚⠑⠺⣄⠀⠀⠈⠑⠢⠤⠴⢻⣀⣠⠿⠴⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣤⣀⣀⣀⣀⣀⣀⣠⠤⠤⠒⠚⠙⠁⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⡖⠒⢜⣯⣍⣠⣟⠔⠊⣉⠆⠀⠀⠀⠀⠀⠀⠀⣠⣞
⠀⠀⠀⠀⢀⣴⣾⠏⠁⠉⠹⣿⣦⣄⠀⠀⠀⠀⠀⠀⠸⣉⣱⢇⡠⠤⣽⠏⠇⠠⢐⠄⠁⠀⠀⠀⠈⣦⡠⢾⡿⢿
⢠⠞⡑⠠⡟⠋⠑⣄⣀⣀⡠⠋⠙⠻⣆⠀⠀⣀⠤⡄⠀⢉⣒⣊⣀⣴⠃⠀⠢⢈⡁⠔⠁⠀⠀⠀⠀⣧⢠⠼⠀⠘⠓⡄
⡄⠀⠇⣶⡅⠀⢠⣿⣿⣿⣇⠀⠀⠀⣻⡀⢈⡟⢠⢞⣆⣸⡄⠀⠀⠈⠙⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⢻⠈⢆⣠⠤⠚⠁
⠃⠀⠀⢻⡟⠀⠙⢿⣿⠿⠛⠒⢤⣴⣿⠁⠈⠻⡈⢧⠚⢇⣻⣄⠀⠀⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠚⠛⠦⡄⢡⣊⠙⢶⡄⠀⠀⢄
⠀⠀⠀⠈⢅⠀⠀⠀⡇⠀⠀⠀⢰⣿⠃⠀⠀⠸⢵⡌⠁⠛⠁⠈⠳⡀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡷⢺⠉⣇⠀⢣⠀⠈⡆⠱⠀⠀⠑
⠀⠀⠀⠀⠀⠑⠲⠾⢿⣶⡶⠖⠋⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠒⠊⠁⠀⠱⣄⠀⠀⠀⠀⢀⣴⣿⣁⡼⢾⡽⣇⢈⡿⠀⠁⠀⠄⠘⠃⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣤⠤⠖⠉⠀⠀⠀⠀⠈⠫⣈⡼⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⢠⠀⡄⢀⠀⡄⢀⠀⡄⠀⡀⠀⡀⠀⡀⢀⠀⡀

⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⡄⢀⠐⠁⠔⢸⡄⠀⠀⠀⠀⠀⠀⠁⠀⣵
⠀⠀⠀⢀⡠⠈⠋⠆⢰⠆⣝⠀⠀⠀⠐⠐⠶⠚⠙⠿⠿⣖⡂⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⢀⡼⠎⠀⠀⡀
⠀⠀⠈⠐⠂⠀⡧⠐⠒⡠⠡⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠚⣿⡷⠀⣤⢠⠌⣍⢀⣀⣈⣉⡀⢀⡀⠀⠀⠈⠙⠒⠒
⠀⠀⠀⠤⡀⢶⠸⠟⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⣴⣿⡇⠀⠈⠻⠴⣧⣰⣄⣤⣠⡷⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠃⠀⢰⡄⠀⠀⠀⣼⣿⣿⠇⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣷⣄⣤⣾⣿⣿⡿⠀⠀⢀⡀⠡⠔⠤⠀⠀⠀⡇⠀⠀⠀⠐⠒⠀⠡⡀
⠀⠀⠀⠀⠀⡠⠤⣄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠙⠁⠀⠀⠀⢀⣀⠀⢐⠇⠀⢠⠃⠀⣄⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠸⠀⠀⠹⡇⠀⠀⠀⠀⢀⡘⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠀⠉⠉⠁⣀⣴⠋⠀⠀⠘⢄⣀⢀⣠⣠⣶⡞
⠀⠀⠀⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⢘⣁⠀⠀⠀⠀⠻⣷⣶⣾⣿⢃⡖⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠨⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠖⠋⠀⠀⠀⠔⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀
⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⡀⠀⠀⡀⡇⠀⠀⠀⠀⢰⣀⠀⠀⠀⠀⠀⠀⣼⠁⠀⣸⡀⠀⠀⠀⠀⠀⠀⢂⣿
⠀⠀⠀⠀⠀⠑⠀⡾⠀⠀⠐⢦⣀⣧⣀⣴⡷⠃⠠⢂⣱⣯⢧⠷⠾⣷⣄⠀⢐⠀⣏⠀⠀⠈⠻⣤⣀⣀⣀⣰⣦⣿⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠃⠁⠈⠉⠆⠀⠀⠈⢻⡆⠀⠀⢹⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⢲⠀⠀⠀⠀⠡⠀⠀⠀⢴⡇⠀⠈⠒⢤⣄⣠⣼⣶⠿⠋⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠑⠒⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣤⣀⣠⡶⡀⠀⠀⠇
⠀⠀⠀⠀⠀⢰⣾⣿⣿⣷⣾⣎⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣶⣄⠀⠀⠏⠀⠀⠀⠀⠠⠟⠃⠙⠁⡧⠠⠄
⠀⠀⠀⠀⠰⡿⢿⣿⣿⣿⣿⣟⠓⠀⠈⠳⣦⣄⣀⣀⣀⣀⣀⣴⣠⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠃⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⠀⠀⡀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣤⡅⣦⡍⠆⠀⠀⠀⢂
⣠⠔⠀⠉⠈⠁⠀⡐⠀⠀⠀⠀⠀⠀⡀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠈⠀⠉⠒⠀⠀⠀⠀⠈
⠣⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣥⣴⡆⠄⠀⠀⡜⠀⠄
⠀⠑⡀⠀⠀⠀⠂⡇⠀⠀⠀⠀⠀⠀⠤⢀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠰⠁⠙⠁⠀⠀⠈⠃
⠀⠀⠘⡄⠀⠀⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠡⢀⠔⠒⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠉⠓⡀⠀⠀⠸⠀⡀⡇⢄⢀
⠀⠀⠀⠠⠀⠰⠀⠀⠀⠀⠀⠄⠁⠈⢑⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠈⠀⠐
⠀⠀⠀⠀⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠤⠀⠀⠠⠄⠀⢁⠀⠀⠀⠀⠀⠀⠀⠈⠐⡄⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⢠⢈⠀⠀⠀⠀⠀⢀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⣀⣀⡤⣊⡁
⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠈⠀⠀⠀⡁⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠈⠉⠉⠁⠈⠐⠂⠤⠤⠼
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣦⣶⣦⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣤⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣀⣤⠶⠶⠖⠶⠶⠦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⣠⡶⠋⣡⠤⠒⠛⠉⠉⠙⠓⠲⢤⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠁⠀⣠⢂⡼⠋⡴⠊⡁⢀⡤⠶⠚⠛⠓⠶⣤⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣼⡾⠀⠀⣰⢃⡞⢠⡞⢡⢎⡞⠉⢀⣤⠶⠶⠶⢦⡀⠙⢧⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢸⢳⠃⠀⢰⠏⡾⢀⡿⢠⢣⡏⠀⣰⠟⠀⠀⢀⣈⠛⣿⡆⠈⡇⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⣾⣼⠀⠀⣿⢠⡇⢸⠃⣾⢸⠀⢰⡟⠀⠀⠠⣿⣿⣷⣿⠇⢀⡇⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⣿⠀⠀⣿⢸⡇⢸⠀⣷⢸⠀⠘⣧⠀⠀⠀⠙⠛⠛⠁⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⢸⡇⠀⢰⠀⣷⢻⠀⠀⣿⠸⣇⠸⡆⢹⡜⣆⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⢹⠈⡇⠀⢸⡄⢸⡸⡇⠀⢸⣇⣿⡄⢻⡀⢳⡘⢦⡀⠈⠛⢦⣤⣀⣀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⢠
⠀⠀⢰⠀⠀⠀⠘⡄⢻⠀⠈⣇⠈⣇⢻⡀⠀⢿⡜⣿⡀⠳⣄⠙⢦⡙⠦⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠋⠀⠀⠀⢀⡜⠀⢀⠎⠀⠀⠀⢠⠇
⠀⠀⠸⡄⠀⠀⠀⢣⠸⡆⠀⠹⡄⠘⣎⢷⡀⠈⢻⣞⢷⡀⠈⠲⣄⡙⠲⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⠖⠋⢀⡴⠊⠀⠀⠀⠀⣠⠊⠀⣠⠋⠀⠀⠀⢠⠏
⠀⠀⠀⢷⠀⠀⠀⠘⣆⢻⡄⠀⢻⡄⠘⣎⢳⡄⠀⠙⣷⣿⣦⡀⠈⠙⠳⠦⣤⣉⣉⣉⣉⣉⣉⣀⡤⠔⠊⠁⠀⠀⠀⠀⣠⠞⠁⢀⠜⠁⠀⠀⠀⡰⠃
⠀⠀⣀⠘⣆⠀⢣⠀⠘⣄⢻⡄⠀⠹⣄⠈⢧⡙⢦⡀⠈⠛⢿⣿⣷⣤⣀⡀⠀⠀⠈⠉⠉⠉⠁⠀⠀⢀⣀⡠⠔⣀⠴⠊⠁⣠⠔⠁⠀⡠⠊⢀⠜⠁
⠀⠀⠸⡄⠘⣄⠀⢳⡀⠘⣆⠻⣦⠀⠘⢦⡀⠙⢮⡛⢦⣀⠀⠉⠛⠿⢿⣯⣿⣓⡒⠶⠶⠒⠒⣚⣋⡩⠴⠒⠉⠁⣀⠴⠊⠁⢀⡴⠊⢀⠔⠁⠀⠀⡀
⠀⠀⠀⠹⡄⠘⣆⠀⠳⡀⠈⢳⡙⢷⣄⠀⠙⢦⡀⠙⠶⣝⠳⣦⣄⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⣀⣀⡤⠴⠒⢋⣡⣔⡾⠖⠉⣠⠞⠁⠀⢀⡠⠊
⠀⠀⠀⠀⠙⣄⢈⢦⠀⠙⢆⠀⠙⢦⡹⣷⣄⡀⢝⢶⣄⡈⠙⠶⣭⣛⠷⢶⣦⣤⣤⣤⣤⣤⣭⣭⣤⣤⣶⣶⠿⠟⠋⣁⡤⠖⠉⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠘⢦⠣⡷⣄⠈⠳⣄⠀⠙⢦⣙⢿⣦⣝⠻⢿⣶⣤⣄⡈⠉⠓⠚⠻⠿⠟⠛⠛⠛⣉⣉⣉⣠⡤⠴⠒⠋⠁⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠱⣌⢿⣢⡀⠈⠳⣄⠀⠉⠳⢬⡻⢽⣲⣭⣛⠻⢿⣿⣷⣦⡄⠀⠀⠀⠀⠀⠉⠁⠀⣀⣀⣠⣤⡶⠾⠛⠉⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⣮⡢⣄⢀⠙⠦⣄⠀⠉⠓⠮⣝⣚⠭⣷⣶⣬⣭⣝⣛⣛⡻⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣮⣕⠷⣢⣌⠙⠲⣬⣐⠢⢭⣙⡒⠦⠭⢍⣀⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣮⣄⠙⠲⢤⣉⠙⠲⠦⣭⣝⣓⡲⠶⠶⠤⣤⣤⣤⣤⡤⠴⠶⠶⠚⣛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣦⣄⡉⠓⠶⠤⣤⣉⣉⡙⠛⠛⠒⠒⠒⠒⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣈⡙⠻⠿⣿⣶⣦⣤⣤⣉⣉⣉⠉⠉⠁⠀⠀⠀⠈⠉⢀⣀⣀⣤⠤⠴⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⣄⣉⡉⠛⠛⠿⠿⠿⠿⠿⢿⡿⠿⠿⠿⠿⠟⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠒⠲⠶⠶⠶⠤⠴⠖⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠠⠤⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠤⠤⠒⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀
⣠⣤⠶⢚⣛⣭⠥⠶⣒⣒⡿⠶⣶⣩⡹⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⣥⠶⢚⣋⣥⠶⢚⣫⡥⠶⣚⣩⠵⢖⣚⡭⠼⣳⡐⣔⢢⡀⣄⠀⠀⢀⣀⡤⠴⣒⣛⣉⣭⣭⣍⣉⣉⣙⠛⠛⠳⠶⠶⠶⠖⠛⣋
⠴⢚⣫⡥⠶⣛⣩⠥⢖⣛⡭⠴⣚⣩⠵⢚⣫⢵⣳⣌⣃⠭⠼⣒⣻⡭⠷⣚⣋⣭⣭⣤⣤⣤⣬⣉⣉⠛⠛⠿⠷⠶⠶⠾⠿⠟⠛
⣫⠭⣖⡺⣭⣵⣒⣫⠭⠶⣚⣫⠵⣖⡫⢵⣚⣭⠥⣶⣒⣛⣭⡭⠶⢚⣋⣭⡤⠴⠶⠶⣤⣤⣌⣉⠛⠿⢷⣶⣤⣤⣤⣤⣤⣶⣶
⠶⠯⠭⠭⠭⠵⠶⢒⣛⣭⠵⢖⣫⢵⣚⡭⠶⢶⣦⣄⠉⣷⠒⣚⣛⣭⠥⠶⢒⣒⡒⠲⠶⣤⣍⠛⠿⣦⣄⣈⠉⠉⠉⠉⠉⠉⠁
⣶⣶⣶⠶⠶⠞⢛⣋⡥⢖⣫⢵⣺⠭⠶⢶⣶⣄⠈⣿⠀⡿⢸⢳⢖⣒⣚⣫⠭⠭⢭⣙⡓⢦⣍⠻⣦⣌⠙⠻⠿⣿⣿⣿⣿⣿⣿
⣤⣤⣤⠶⠶⢛⣩⠵⣚⢽⡺⠵⠶⠶⣶⣄⠈⣿⢀⡿⢸⡇⢿⣘⠲⠭⣗⣒⣛⣛⡓⠦⣍⡳⣌⠳⣌⠙⢷⣦⣄⣀⡀
⣉⣤⡤⠶⣛⡭⣖⣭⣿⣓⣛⣛⠻⣦⢨⡟⢸⡇⢸⣇⠸⢷⣬⣭⡭⠭⠶⠶⠦⣭⣙⠳⣬⡙⣮⠳⡌⢷⣄⠉⠛⠿⠿⣿⣿⣿⣿
⣉⡥⢖⡫⣷⣿⡿⠛⠋⠁⠐⠘⡇⡟⢸⡇⢸⣧⠈⠻⠶⠶⣖⣒⣻⣯⣝⣛⣲⠶⣝⠳⣮⠻⣎⢷⡻⣦⡙⢿⣶⣦⣄⣀⡀⠈⠉
⢵⣪⣵⡿⠛⠁⠀⠀⠀⠀⠀⢰⡇⣇⠸⣷⣀⠙⢛⣓⡻⢭⣓⣶⠶⡶⣶⣒⣿⣗⠦⠳⢤⠷⡜⢦⠳⣜⢿⣦⣈⠛⠻⢿⣿⣿⣶
⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠹⢦⣬⣩⡭⢽⣒⠯⣟⣺⠭⠭⠯⠭⢷⣛⣿⠿⡾⠿⠷⠟⢮⡳⣝⢷⣍⠻⢿⣶⣤⣀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⢟⣒⣒⣒⡯⠽⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉⠈⠳⢭⡳⢦⣌⡙⠻⢿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠪⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣼⠟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠤⠚⣩⣶⣄⠀⠀⠀⠀⠀⠀⣤⣶⣿⠿⣦⡀⠀⠁⠂⠸⣯⠀⠌⢳⡌⢙⣧
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⢠⡟⣽⣿⡤⣿⠧⡀⠀⢀⡼⣻⣤⡠⠴⣟⣀⡞⣒⣲⣤⣀
⠀⠀⠀⠀⠀⣀⣽⣿⣧⣄⡀⣴⠀⠀⡠⠿⣧⣿⣀⣤⣿⣇⠳⣄⢸⣟⠻⠧⠀⠀⣽⣿⣿⣏⠉⣿⡿⣿⡀
⠀⠀⠀⠀⠀⠻⠿⡿⠿⠟⠉⠁⠀⠸⡄⣠⣾⡟⠀⠀⠛⣛⣧⣼⠘⣿⣿⣤⣤⣴⣾⡑⡷⠲⠖⠲⢻⣌⠙⢶⡄
⠀⠀⠀⠀⠀⠘⠉⠋⠉⠉⠀⠀⡀⠀⢹⣿⣿⣿⣿⣶⣿⣿⣿⣿⡀⢹⣿⣿⣿⡿⠋⠀⣇⣂⡂⢀⣠⣿⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡶⢈⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⣎⠻⣿⡿⣿⣤⡶⠏⣽⠃⠀⠀⠹⣶⣶⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣧⢀⣿⠆⠀⣰⣿⣿⣿⡆⢨⡀⢀⣀⣤⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠠⠂⠐⢀⣴⣔⣦⠀⠀⠀⠘⣿⡻⠿⣿⠿⠛⠛⠋⢹⣿⡿⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⠯⣀⡟⣿⣣⠀⠀⢀⡜⣴⣿⠛⢳⣶⢦⠀⠀⠙⣱⣿⣿⠋⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠙⣿⣿⣇
⠀⠀⠀⠀⠀⢀⣏⡶⠀⢹⠙⢧⠀⠀⡜⣼⣿⡚⣀⣶⣿⢎⠛⢳⠄⢹⣿⣿⠀⢄⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⠂⢈⣿⡇
⠀⠀⠀⠀⠀⣷⣠⣼⣧⣼⣆⠘⣳⡀⢹⠟⢸⡧⠿⠟⠈⣸⣆⠘⢤⡈⢿⣿⡇⠀⡘⣿⣿⣿⣿⣿⣿⣿⣿⠛⠧⠂⢹⣿⠇
⠀⠀⢀⡀⠀⠉⣿⣿⣿⣿⣿⣿⣿⣷⡇⢰⢿⣷⡀⢒⣀⢻⣿⠿⣦⣄⠘⣿⡇⢤⡆⣹⣿⣿⣿⣿⣿⣿⣿⠀⡓⡄⢸⡟
⠀⠀⣏⢠⠙⣼⣿⣿⣿⣿⣿⣿⣿⠉⣰⣽⣾⣿⠛⠉⠀⠀⠿⠁⣽⣿⣦⠘⡇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠐⠋⠸⠃
⠀⠀⠘⠀⠠⢸⣿⣿⣿⣿⣿⣿⡛⣸⣿⣿⣿⣿⣄⣀⣀⣀⣠⣶⣿⣿⣿⡇⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠂⢸⣿⣿⣿⡿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⠈⠉⠉⠉⠛⠛⠿⠿⠿⠿⠿⠟⠛⠛⠉⠉⠁⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⢹⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣿⣿⣿⣿⣿⣿
⠀⠸⢿⡟⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿
⠀⠀⠈⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢉⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⣿⣿⣿
⠀⠀⢰⡏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿
⠀⠀⠀⠁⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠘⠛
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠲⠾⠿⠃⠀⠶⡀
⠀⠀⠀⠀⠀⠠⠆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⡘⠂
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠃⠀⠀⢀⠄⢿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⡄⠁⠀⣠⣦⣼⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣥⡄⠙⠻⣯⣿⣿⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣹⣦⡀⠈⢿⣿⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢹⣿⡇⠈⠻⣿⣿⣶⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢠⣿⡇⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣠
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣻⡋⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⠁⠀⠀⣠⣆⡀⢿⣿⣿⣿⣿⡿⠿⠿⠋
⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠉⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢃⣿⡧⣀⠀⠀⠿⠟⠀⠺⣿⣿⣿⣿⣄
⠀⠀⠃⠀⠀⠘⠀⠀⣀⡄⠀⣤⠖⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⢻⡇⣿⣶⠶⠖⣿⣦⡀⢹⣿⣿⣿⣿⣷⡄
⠀⡆⠤⠄⠐⠚⠃⠈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⢹⣥⣿⡀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣠⣼⣿⡿⠃⠀⢠⣿⣠⣼⣯⡙⠿⣿⣿⣿⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⠟⣯⣤⠀⢰⣿⣿⣿⣿⡏⠀⢰⣿⣿⣿⣿⣷⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠙⢾⣿⣿⠀⢾⣿⣿⣿⣿⣿⠿⠋⣿⣿⣿⠟⠃⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣯⣌⣽⣯⣤⣾⣿⣿⣿⡿⢃⣴⣦⠋⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠻⠿⠿⣿⠿⠿⣿⣿⡿⠋⢻⣿⣆⢸⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⣠⣶⣶⡆⠀⠀⠙⠋⠀⣠⣾⣿⣿⣿⠿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣶⣶⣿⣿⣿⣿⡀⠀⠀⠀⠀⠘⢿⣿⢿⣿⡿⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⡿⠿⠟⠻⡿⠗⠀⢰⣶⡆⠀⠀⠀⢸⡟⠀⠀⠀⠈⣿
⠶⠦⢀⡄⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⡟⠻⠃⠀⠀⠀⠀⣠⣶⠀⠀⠀⠀⠀⠀⣼⣷⡄⠠⠄⠀⣿⡀
⠀⠀⠀⠀⠒⠚⠉⠉⠉⠙⠻⣿⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢋⣤⣯⠀⠀⠀⠀⠀⠀⣾⣿⣯⣀⣶⠄⠀⠀⢀⣿⣿⣿⣦⡀⠀⠻⣧
⠐⢢⣴⣎⠀⠀⠀⠀⠀⠀⠀⢿⡀⢠⡀⢻⣿⣿⣿⡿⣿⣿⣿⣿⣷⣀⣀⠈⣻⡟⣀⢰⣿⠇⢀⣴⣮⠛⣿⡟⠁⣀⠀⠀⣸⣿⣿⣿⣿⣿⣷⣄⢀
⠦⠀⠀⠀⠘⠓⠀⠀⠀⢀⡀⠀⠙⠦⠘⣿⣿⣿⡏⠀⠈⠛⢋⣼⣿⣿⣿⣿⡿⠀⠉⣿⣿⣤⣮⣿⡟⠀⠋⠰⠿⠋⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⡀⠀⠀⠀⠀⠰⠉⠃⠀⠀⠀⠀⠙⣿⡟⠀⠀⠀⠈⠈⣿⣿⡿⣿⣿⡀⠀⠀⠉⠙⠿⠿⣿⠇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⠁⠀⠀⠀⢠⡤⠀⠀⠀⠐⣄⠀⠀⠘⠇⠀⠀⠀⠠⠀⠀⠀⠀⠻⠿⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⠇⠀⢀⠀⢀⠀⠀⠱⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⢠⣿⣄⣤⣶⣛⣿⣋⣭⣿⣛⡛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⠓⠀⢸⡟⠀⠀⢠⡄⢸⣧⠀⢀⣈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠈⠀⠀⠀⠀⠸⡀⠀⠀⠀⠿⠄⢈⣤⠈⠉⣩⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠦⢀⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡱⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⢩⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡿⢛⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢋⣿⣿⣿⣿⣿⣿⣿⣿⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⢤⠄⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⡀⠸⢒⢊⡐⣬⣥⢀⢀⡳⣦⣂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣥⠏⣅⡛⠐⣩⡿⣛⡛⠉⠀⠈⢢⣀⣉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⠴⠿⠛⢑⡷⠓⡛⢈⣭⢤⠶⣶⡀⢬⣛⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠇⡀⠀⣸⣬⣦⡀⠄⠋⢸⣣⡌⡉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⢳⣿⣶⣷⣗⠈⢹⢛⣷⣇⢂⡘⠋⣴⣶⠂⠒
⠀⠀⠀⠀⠀⠀⠠⠾⠟⣻⣷⡿⡿⠻⢻⣿⢿⡿⠵⠙⠋⢶⣥⣌⣁⠀⠐⢆⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣯⣿⠕⣟⣿⣴⢮⣾⣿⣿⣿⣶⢼⡏⡿⠛⢛⡛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⢴⣶⣿⡿⣷⣿⠼⠋⠙⣲⠛⣿⣷⣞⣿⣴⣇⠖⣔⡲⡜⠐⠾⡄
⠀⠀⠀⠀⠀⠀⠈⡫⠿⠉⠉⠁⠀⠀⠀⠀⣮⡟⠻⣿⠟⢿⣿⡟⣲⣤⡀⡀⠐⡄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢚⣿⣿⠿⠒⢀⢛⠟⠳⠟⡾⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⠀⣀⠂⠀⠀⠋⢿⡒⠻⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠁⠀⠀⠀⠀⠀⠛⢿⡳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣷⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⠫⠉⢍⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢞⠍⠊⠀⠀⠀⠀⠈⠒⢍⡻⢿⣿⣿⣿⡿⠟⢋⠡⠐⠀⠀⠀⠀⠉⡳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⡫⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠋⡡⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⡎⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄
⠀⠀⠀⠀⠀⣴⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢰⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⠟⡁⠤⠈⢐⡀⠀⢀⡀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⢀⡀⠄⠀⠒⡈⠈⠀⠙⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⠟⡡⠊⠀⠀⠀⠀⠀⠁⠀⠒⠁⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢩⠄⠒⠀⠈⠀⠀⠀⠀⠀⠀⠙⢿⣿⡇
⠀⠀⢸⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠈⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣸⠁
⠀⠀⠀⠸⡔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⠉⠉⠉⠉⠉⠉⠩⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠇
⠀⠀⠀⠀⢳⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡡⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⡞
⠀⠀⠀⠀⠀⠳⡌⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢪⠞
⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⣤⣤⣤⣤⣤⠡⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣤⣤⣤⣤⣶⣶⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣼⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠤⠬⠥⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢁⣀⣈⡉⣀⣶⢤⡐⠦⡤⠤⠤⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡟⠀⣨⣿⢭⠳⣤⡼⠀⢙⣷⠀⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠠⠒⠎⠀⠤⠤⠠⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠞⠁⠈⠐⠳⣌⣑⣦⠪⡾⠀⡆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢸⠀⠂⠀⠀⠀⢠⠀⠄⢸⠀⠫⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠀⠀⠈⡿⠀⡐⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⣁⢸⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠠⠀⠤⠀⠀⠀⡠⡿⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃⢠⣀⣀⣀⣀⣀⣈⢸⠀⠰⢸⡇⠀⠀⠀⠀⢠⡤⣤⣤⠤⠖⠒⠂⠀⠀⠀⠀⠀⠀⢀⢀⡣⠁⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣤⣄⣶⡂⠐⠀⠀⠀⠀⠀⠘⠀⢀⣺⡇⠀⣠⠔⢈⢉⣁⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣠⠿⠵⠶⠶⠇⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠁⠒⠓⠛⠓⠒⣶⡴⢶⣶⠒⠒⠓⠛⠒⠛⠊⠀⠀⠀⠑⡍⠀⠀⠀⠀⠐⠀⠀⠀⢀⠴⠊⠁⠄⡀⠀⠀⠀⠀⡴⡀
⠀⠀⠀⠀⠀⠀⢼⢇⣰⣂⣀⣀⣘⣉⣉⣉⣉⣉⢓⣒⣲⣶⣶⣖⡆⠔⠲⡀⠀⡀⠀⠀⠀⠐⠂⠌⢀⡖⠁⠀⣀⡀⠀⠀⠆⠀⠀⡞⢈⡅
⠀⠀⠀⠀⢀⠀⡈⣆⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢄⡇⠙⢞⠁⠠⢣⡀⠀⠀⠀⣀⣂⣈⡁⠀⣊⡀⣱⠀⠀⠰⢀⠈⠀⠸⠁
⠀⠀⠀⠀⠹⡭⠟⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠔⠁⣦⡀⠀⠀⣇⠀⠉⠁⠈⠀⠀⠀⠈⠣⡀⠀⠉⠁⡀⠠⠚⠁⢀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠀⡇⠘⠛⠉⠉⠑⠒⠴⠤⠤⠤⠤⠤⠤⠾⠢⠤⠤⡶⠥⠟⠒⠋⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠀⠀⠀⢊⡅⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡃⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⣀⣄⣀⣴⣶⣦⣴⣶⠶⠲⡶⢲⣀⣸⣨⡗⣷⣷⣷⠤⢦⣤⠤⠤⢤⡬⢬⣤⣤⣤⣤⣀⣀⣥⣥⣼⠶⠶⢶⠾⠃⠈⠑⠂⠉⠀⠄⠒⠀⠂

⠀⠀⠀⣀⠀⠀⣿⢀⣷⣾⣿⡆⠀⣴⠏⣤⣤⡀⣰⠀⠀⠀⠀⣠⡎⠀⢀⣀⡀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠀⠀⠀⣾
⠀⢠⣿⣾⣷⠀⢻⡾⣿⡏⠙⢁⣾⠋⠀⡩⣻⣷⣿⠀⠀⢀⡴⡟⢠⡘⡧⢳⡽⠆⠀⠀⢠⢏⡾⠉⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣤⣴⡶⠟⠛⠃
⠀⢸⢞⣧⡉⠀⣿⣱⣿⣦⣜⡟⢡⣔⣯⡷⠛⠡⣟⣠⣶⣯⠝⣠⡟⠽⠀⠛⠋⠀⠀⢠⠯⣘⠇⠀⠀⠀⠀⠀⡼⢻⠀⠀⢀⡴⢯⡶⠉
⠀⠈⢏⣼⢛⡶⣾⠏⢿⣿⣺⣼⡷⠛⠁⠀⠀⢰⣟⡙⣭⣡⢾⡿⠥⡇⠀⠀⠀⠀⣰⢯⣓⢾⡀⠀⠀⠀⡠⢜⠣⡝⠀⣠⠻⣽⠋
⠀⢰⡀⢑⣿⣴⣿⠸⢘⣧⡿⠋⠀⠀⠀⡀⢠⣾⡾⠟⠉⠁⢡⣍⣸⣁⣀⡤⢤⣾⣟⡒⠮⣿⡇⠀⣠⠮⣙⠢⣱⠃⡰⠯⣽⡇
⠀⢸⣷⠸⣿⠿⣿⣷⡘⣿⡇⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⢀⡞⠋⠉⠁⠀⠀⢀⣿⠣⣌⣶⣿⣷⡴⢉⠓⣌⢢⡏⣰⣛⡓⡞⠀⢠⣿⢽⣹⢦
⠀⢸⣟⠆⠀⠀⣿⣧⡛⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠀⠀⣠⣿⢣⣹⠞⠉⣱⡿⢍⢢⢱⣼⣾⢡⡧⠤⢬⠃⠀⠘⠷⠂⣳⢺⡇
⠀⠀⢷⣉⢆⡀⠰⣻⡿⣮⣿⡆⡀⡌⠀⠀⠀⠀⠀⢰⡎⠀⠀⠀⠀⢀⣴⣿⠞⠁⠀⠀⠀⣰⣍⣦⠿⣿⣿⣏⣞⣉⣑⡟⠀⠀⠀⠀⣰⠯⣼⣡⣶⡞⠃
⠀⠀⠀⠛⣾⣌⣙⠻⢿⠐⠹⣷⣷⡇⠀⠀⠀⠀⡀⣮⠀⠀⠀⠀⢠⠞⠋⠀⠀⠀⠀⡠⠐⠉⠉⠀⠈⢸⣿⠿⠤⢤⡝⠀⠀⠀⢀⣴⣏⢲⣿⣳⠏
⠀⢰⣿⡶⠘⡿⠛⢷⣿⢦⣀⢿⢿⣇⠀⠀⠀⠀⠁⠿⠀⠀⠀⢀⠏⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠠⢮⣛⣛⠓⠶⠁⢀⣠⡴⠻⢤⣉⣟⠢⣽
⠀⢈⣛⣿⣞⣿⣀⠀⠈⠃⢽⣿⣇⢹⣦⣠⠚⠛⢶⣄⣄⣄⠀⣸⡄⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⢐⣱⠾⠭⣭⣙⣤⠞⢿⣅⠙⢶⣤⡿⢯⡙⠇
⠀⠈⣙⣿⠉⠛⠛⡻⣦⡀⢼⡋⢻⣶⣿⡿⣶⣤⢯⠁⠒⠈⢴⣿⡇⢀⠀⢸⠁⠀⠀⠀⡀⣀⣴⠻⣭⡻⠶⠾⠿⠿⠶⣶⡿⢟⡻⢿⡝⢦⡝
⠀⢾⣿⢟⣧⣄⠀⠀⠉⢿⣋⡿⠋⠉⢻⢷⡀⠙⣷⣤⠀⡰⠧⣜⠋⠉⠙⢻⠮⠉⢲⡮⠉⢹⡞⠃⠀⠀⠀⢀⠠⠀⠉⢀⡰⢯⡙⢦⣹⠎
⠀⠀⠁⠀⠛⣽⣶⡶⢿⡏⠙⢳⣄⣀⣈⣘⠿⢾⣤⣬⣻⣦⣄⡈⢫⣥⠤⠼⢧⣀⠼⠳⣶⠋⡄⠂⠀⠀⠀⠀⢀⣠⡾⣏⠳⣌⠻⠊⠁
⠀⠀⠀⠀⣴⣿⠋⠴⠊⠓⠚⢿⡁⠄⠘⣢⡀⠐⠛⢿⣧⠼⢿⣷⠾⢤⣀⢀⠀⠘⣊⣱⢸⣇⣀⣠⣤⣤⣴⠾⠻⠛⠉⠚⠛⠣⢤⣄
⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⢠⠎⠙⠒⠋⣹⢫⠁⠲⣈⠘⠙⣶⢌⡁⠀⢹⣿⣶⣾⣥⣐⣂⠙⠈⠉⠉⠁⠀⠀⠀⠀⢀⡀⣀⣠⣼⡾⢿⣦⣄
⠀⢠⠼⣿⣧⣤⣴⣤⠖⠚⠁⠀⠀⣀⠔⠁⠀⠁⠃⠋⠁⢠⡇⠀⠈⡙⠳⠮⣷⣭⣛⡿⣙⠳⠖⠦⣤⣀⣤⣤⣴⣯⡶⢻⣿⠠⣿⢰⡟⣩⣴⢦
⠀⠀⠴⢻⣟⢻⡎⢁⣀⣤⣴⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⡞⢴⡸⡿⠿⣷⡄⠀⠉⠙⠳⠿⣷⣈⠂⠈⠙⣷⡜⢿⡘⣿⣄⠿⠗⠋⠉⡉⠙⣿⣻⠂
⠀⠀⢠⡻⣬⠿⢋⡽⢏⣵⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣴⣦⣽⡇⠐⢿⠟⠀⠀⠀⠀⠀⠈⠙⠳⣶⣤⣌⠹⣎⠉⠀⠀⠀⠀⠀⢾⠿⣴⣿⠏
⠀⠀⠼⣿⠁⢀⡾⣑⣾⠃⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠀⢻⣤⣤⠼⢲⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣍⠻⢿⡄⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⠾⣷⡄
⠀⠀⣸⠇⠀⢼⣓⣺⡏⠀⣀⣀⡤⠤⠞⠛⠉⠀⠀⠀⠀⠘⣿⠳⢶⣟⣡⡾⢁⡿⢙⡶⢢⣄⠀⠀⠀⢿⠌⢣⣊⣷⠀⠀⠀⠀⣀⣀⣀⣀⣀⡤⣾⠟
⠀⠀⢁⡀⢠⡗⣲⡿⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⣤⡤⠼⣧⡐⢍⠉⠓⠻⠧⣾⣡⡾⢉⣷⢄⠀⣾⢂⠀⠹⣾⠀⣴⢞⠻⡉⣇⠱⣙⣬⣞⣥⡶
⠀⢠⣿⣷⢞⡾⠋⠀⣀⡀⠀⠀⠀⣀⠰⣶⣆⢙⣿⣆⣹⣿⡀⣿⡟⢢⡁⠠⠀⠀⡀⠉⠱⡞⣡⡾⣧⡏⡄⠀⠂⢹⣟⠉⠉⠉⠙⠛⠛⣛⢉⣩⢿⣄
⠀⠀⠙⠛⠋⠀⠀⠺⣿⡥⡤⢶⡌⣿⠦⠚⠛⠛⠿⣯⣟⠓⠩⠍⠉⠘⠿⣦⠀⠀⠀⠀⠀⠹⣯⡴⢿⠂⠀⠀⣠⣾⣤⣃⠀⠀⠠⢀⣩⡴⢟⠿⣭⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⢀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⢿⣶⡏⢀⡤⢞⣻⡏⢳⣉⠳⡎⠙⠉⠋⠛⢦⣤⣤⣤⡄
⠀⠀⠀⣤⣼⠟⠻⢶⡀⠀⠀⠀⠀⢀⣠⡴⣖⢶⡖⢶⠲⣦⣤⡄⠀⠀⠀⠀⠀⢡⠀⠀⢀⣈⣰⣯⡾⡏⢣⡼⣿⠓⡴⠈⢷⠃⠀⠀⠀⠀⠀⡠⠞⠁
⠀⠀⢰⠶⡇⠀⠘⠛⠁⠀⣀⠠⣆⢻⡌⢷⢹⡆⣷⠘⡇⢻⡜⡷⡀⠀⠀⠀⠀⢈⣧⡖⡡⠒⣿⡏⣧⡼⠊⢀⣋⡴⡇⠁⡼⠒⠤⣄⡀⢄⠈
⠀⠀⠘⣶⠟⣶⢲⣖⢶⢻⡌⣧⢸⣄⡷⠾⠚⠃⠉⠈⠁⠙⠁⠛⠿⢄⠀⠀⠊⡡⠜⣧⡁⢰⣿⠟⠁⡀⠀⣼⠩⠀⠳⠈⡅⠀⠀⠙⢶⢈⠤
⠀⠀⠀⠈⠘⠻⠴⠿⠼⠷⠛⠛⠙⠛⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⢘⠀⣠⠎⣉⣿⡇⠀⠀⠀⢠⡟⢂⣇⡠⢾⣇⡀⠀⠀⠈⣧⠘⡀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡵⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣤⣵⢁⠊⡠⣿⣇⠀⠀⠀⣾⠗⠛⠁⠀⠀⢻⣗⢀⠀⠀⢸⣧⢃⠻⠉⡷
⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⢠⡆⢿⠘⣄⢷⣹⣦⣤⣄⣀⠀⠀⠀⣀⡞⠁⠀⠼⡀⢀⢨⣿⡽⢆⠂⢠⡟⠀⠰⣧⠁⠀⠀⢻⢷⣷⣄⠀⣧⠘⠲⠚⠁
⠀⠀⠀⣾⡏⠁⠀⠀⣀⣤⢻⡆⢳⣌⢳⣼⣟⠉⠀⠀⠈⠉⠙⠛⠋⢩⠁⢄⣀⡔⠹⢦⣼⣏⢷⣋⢧⠰⠇⡀⠀⢘⣧⣄⡀⣾⣶⢩⠛⠿⢾⣧⡀
⠀⠀⠈⢯⢻⠶⡖⣏⢳⡘⢦⣹⣦⠿⠿⠭⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡨⠀⢀⢀⡹⣿⡕⡽⢮⣿⡆⠹⣦⣼⣒⡏⠛⢿⡀⠉⠐⠒⠒⣒⢯⣦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣴⠞⠂⠀⠀⠀⠠⠂⠡⠤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠰⠃⢀⣴⠞⠁⠀⠀⢀⢴⠒⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣡⠰⠛⡀⢤⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠤⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⠍⠉⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠆⠀⠀⠁⠀⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠆⠀⠀⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⠀⠈⠀⠒⢊⠑⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠤⢤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣐⣂⣄
⠀⠀⠀⢠⠄⠂⠁⠀⠀⠀⠀⠀⠀⡀⠄⠀⢆⡀⠀⠘⠀⠀⠀⠀⠀⢀⠀⠐⡀⠨⠤⠤⢠⠀⠘⠑⠃⠀⡤⣄
⠀⢀⡖⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠉⠀⠀⠀⡇⠀⠐⠶⠡⠡⠀⠀⠀⠀⠁⠀⠨⣀⠠⠰⠀⠀⠉⠉⠀⢀⣀⣀⣀⡀⠀⠤⣄⣀
⠀⠘⣇⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⢆⡂⣀⠈⠁⠄⠀⠀⠒⠒⠒⠀⠀⠘⠁
⠀⠀⠈⠑⠂⡵⠒⢚⡯⠟⠁⠈⠉⠒⠤⣀⠀⠀⠀⡸⠀⠀⠀⠀⠠⢤⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠚⠋⠉⠉⠉⠀⠀⠉⠉⠉⠙⠓⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⡴⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⢦⡀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⣴⣾⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⢤⣤⣤⡤⠤⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⢤⣄⣀⣀⣀⠀⠀⣀⣀⣀⣠⡤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⡶⠒⠦⠤⠖⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⣤⠤⠶⠶⣤⣤⣶⠶⣤⡠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣶⣶⡿⠿⣭⣍⡞⢒⣖⠛⢻⣦⣴⣄⣬⣬⠓⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⡾⠏⣉⠋⠉⡁⠐⠇⠝⠛⣟⠉⠙⠃⠛⠛⠫⢦⡼⢜⣿⣦⣽⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣀⣹⠛⠁⣰⠁⠀⠀⡁⠋⠀⠈⢀⡴⠋⠁⠠⡀⠠⠀⠁⠀⢈⠶⠻⡀⡀⠉⠻⣿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢄⡽⠈⠡⣀⡾⡉⠀⠀⠀⣧⠀⠀⠑⠋⠀⠀⠀⢀⡘⠀⣄⢠⡞⢃⣀⡷⢎⠗⠒⠀⡞⢧⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡦⠊⠀⠀⠀⠈⠃⡈⢠⡀⢀⣾⣆⣰⠀⠈⠀⠀⠀⠀⠀⠴⠚⠏⡍⠀⠁⠀⢀⣠⡄⠠⠷⠾⣧⣉⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡞⠁⠀⢀⡀⠠⠆⠣⡌⢣⡈⢚⠀⠠⡀⠀⡀⠀⠘⢂⠀⠀⠀⠀⠸⣅⠰⠀⡀⢠⡆⢦⡄⠰⢀⠘⣏⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⠜⠀⠨⠀⠀⠀⠀⣤⠀⢰⡀⢘⣂⣴⣄⡀⣀⣀⡋⠀⢀⡤⠚⠉⠉⠀⠀⡆⠀⡷⡨⠂⠀⠈⠈⠈⠗⣬⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡸⠀⠀⠀⠐⠄⠀⠐⠀⠘⡟⠁⠉⡍⠈⠈⡧⠀⡄⢈⠘⣓⣠⣎⠀⠉⡃⠘⠒⢴⣷⣶⡆⡁⣤⠀⣀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣸⣥⣫⠒⢃⠀⠀⠒⣋⠈⠀⢡⠜⠈⠸⠀⢀⠗⡼⠿⠃⢀⠼⠉⠉⡱⠠⣄⢀⠠⢮⠛⡮⠁⠸⠇⠰⣅⠀⡸⠉⠈⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣷⠋⡄⢀⠺⢆⠀⠀⢸⠀⠐⠘⠆⠀⠆⠀⢸⠃⠐⠀⠀⠯⠀⠀⠀⠀⡴⠊⠉⢠⡻⠆⢠⠐⣄⡂⠀⢹⠞⢁⢿⡄⢀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢠⠜⠀⡄⠀⠀⠀⡄⠀⠀⢸⡀⣠⠀⠀⢫⠰⡀⢀⠀⠀⣀⠀⠀⠀⠀⢠⡄⠋⠀⠷⡀⡔⠀⣷⠀⢜⠀⢏⣹⡄⠙⡁
⠀⠀⠀⠀⠀⠀⠀⣷⠞⠀⠀⠀⠑⢦⣀⠀⠀⠀⠸⠎⠹⡄⢀⠉⠀⣤⠁⠀⠀⠙⠀⠣⢀⠐⠉⢣⠀⠀⢀⡏⢒⠀⠛⡇⠄⣱⠞⠙⡀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⡄⠀⠀⢃⠀⠐⠸⢅⢰⣴⣀⡌⠐⣠⢻⡀⠀⠀⠀⣱⡞⢀⡠⢄⠖⢶⠀⢸⠀⢁⢈⠄⣻⠻⡏⠀⠠⡁⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⢀⠁⢄⡀⢀⠉⠦⠐⡘⢻⡇⠈⢸⠇⠘⡟⢓⣾⣓⡺⢛⡟⡅⣈⠄⠀⠆⠘⠀⠘⡠⠀⠹⡄⠃⠀⢸⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠂⣎⠀⠀⢹⠂⢀⠀⠀⠁⠬⣷⡀⠈⠃⠀⢀⠀⠰⠈⡉⠂⢦⣴⡀⢁⠀⢐⣭⡀⠈⠀⢠⣸⠷⠀⠀⡔⢹⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢄⠀⠄⠀⠀⠙⠣⢤⣐⣆⢷⡄⢄⠀⢀⡀⠀⠀⠱⠄⠀⠖⡈⡏⠀⠘⡿⠋⠙⠀⠊⠒⠴⢐⣏⡰⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠀⢀⠀⠒⠒⢿⠁⠩⠬⣇⠀⠄⢾⠓⢇⡀⢀⡀⡔⠘⣁⣆⠀⡁⠤⡀⠠⠀⠈⢨⡤⠟⢿⣡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣱⡄⠀⠘⠀⠀⠀⠶⡄⠀⢠⣵⢶⠄⠀⠈⠉⠀⡍⠩⠚⠉⠂⠰⡐⢠⠔⡓⠀⠠⡟⠀⣰⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⣀⠠⡄⠢⡄⠀⠐⢖⡁⠀⢦⠀⠀⢀⡤⠞⠃⠀⠀⢀⠀⢀⡔⠛⢛⢤⣴⣒⠁⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠛⠠⢀⠀⠙⢦⡈⠳⠞⠀⠀⠀⠀⠀⠀⣨⡺⢟⡣⠴⠧⠔⣠⠱⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠶⠚⠋⡙⠲⣶⣴⣶⣶⣦⣟⡼⠛⠉⢁⡠⣋⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠐⠈⠙⠛⠛⠛⠉⠉⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⣼⣿⡿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆
⠀⠀⠀⠀⠀⢰⣿⡟⠀⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣾⠋⠀⠀⢈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⣿⣆⡠⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⠺⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣿⣿⣧⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡋⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⢻⡾⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡄⠈⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣧⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⣿⡿⣿⣆⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⢈⠛⣷⣾⣻⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⡀⠈⠁⠀⠀⠀⠛⣿⣄⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡉⠉⠉⠀⢉⠁⠉⠁⠈⠉⠉⠉⠉⢙⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣗⡤⠀⠀⠒⠀⠒⠀⠀⠀⠀⠀⠚⠙⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣧⠐⠂⠀⠀⠀⠀⠀⠀⠀⠓⠒⠚⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠉⠀⠀⠀⠀⠀⠀⢠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⣷⡀
⠀⣠⣶⣶⣤⣤⣦⣄⢺⣿⣿⣿⣿⣷⣄⠠⣴⣾⣿⡷
⠀⢿⣿⣿⣿⣿⣿⣿⣦⢻⣿⡿⢿⣿⣿⢹⣿⠃⠈⠁
⠀⣼⣿⣿⣿⡅⠈⠻⣿⣿⣿⠁⣸⣿⢣⠟⢉⣡⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⣀
⠀⠻⣿⣿⣿⣿⣶⣄⢹⣿⠇⣰⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⢿⣿⣿⣿⣶⣄⡀
⠀⣀⣤⣤⣭⣭⣽⡿⠇⢴⡿⢟⣛⣉⡉⠉⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣷⣤⡀
⣾⣿⣿⣿⡟⠁⠀⢀⣴⣦⡘⢻⣟⡻⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⢘⣿⣿⣿⣧⣤⣶⣿⡿⣿⣿⠀⠙⣿⣶⡙⠿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆
⣾⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣷⣾⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀
⠈⠻⠛⢻⣿⣿⠟⠁⣿⣿⣿⣿⣿⣿⠿⠏⠀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄
⠀⠀⢠⣿⣿⠏⠀⠀⢿⣿⣿⣿⣿⡿⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷
⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢹⣿⣿⠀⠀⠀⠀⠠⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣷⡄⠀⠀⠀⠀⠀⢀⣿⣿
⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠈⠁⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⢹⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⢻⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟
⠀⠀⠀⠀⢻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣷⣦⣤⣤⣤⣤⣤⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠊⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⡀⠀⠙⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣦⠀⠀⠈⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣷⣤⣄⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣠⣴⣾⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⡸⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⢇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⢸
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣶⣶⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉