Baseball Game Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⢀⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣠⠞⠁⣀⣠⣤⡤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣀⡴⠋⣠⠶⠛⠉⠁⠀⠀⠀⠀⠈⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⣡⠞⠋⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣡⠞⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡤⠀⠰⠏⠰⠃⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣶⡶⠺⢻⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⡀⠀⠀⣀⢀⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠳⣄⠀⠐⠃⢠⠀⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡊⠤⡀⠀⠀⠙⡸⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⣣⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡲⠶⠶⠶⠶⠋⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠈⠉⢤⣀⠊⠈⠢⣀⣀⣀⣀⣀⣤⣤⠶⠛⡇⠀⠀⠀⠀⠠⣀⡀⠀⢀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⡠⠀⠈⣢⣄⠀⠈⠉⠍⠉⠉⠁⠀⠀⢠⡀⠀⠀⠀⠀⢀⠼⢋⠁⠁⠀⠉⠉⠓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢧⣀⠀⢀⠞⣀⣠⣴⠟⠈⠣⡀⠀⣠⠜⠢⣄⠀⠀⠸⡏⠑⣒⠶⠊⠁⡴⣣⣖⡥⠒⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠈⢉⡿⠛⠉⠁⠀⠀⠀⠀⠙⢞⠁⠀⠀⠀⠙⠢⣀⠓⠉⠀⠀⠀⣰⠿⠋⠁⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⡼⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠈⠑⠦⠤⠒⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠜⠀⠈⠳⣄⢀⠳⣄⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⢀⣀⣠⣤⣴⣿⡞⣷⠿⠷⣄⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⣀⣀⣀⣤⣾⠀⠀⠀⠀⠀⣠⢾⠇
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠉⠁⠀⠀⠙⠛⠋⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠙⢿⣆⢀⣠⠴⠋⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⢤⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠲⠦⠿⠶⢤⣄⣀⣀⣀⣀⣀⣤⣤⡤⠤⠤⢤⣤⣤⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠁⠀⠙⠲⠤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣉⣿⠀⠀⢸⡇⠀⠀⠀⠀⢀⣴⣶⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣠⢂⡆⠀⠀⢀⡘⠛⣿⡆⠀⢸⠃⢀⠀⠀⠀⠀⠀⠘⠁⠛⠂⡐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⢀⣴⠗⠻⣿⡄⠀⠸⢷⣶⣿⠷⣄⣼⡆⠉⠀⠀⠀⠀⢄⠠⣤⢒⡴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠁⠀⠀⣣⠁⠄⠀⠀⠀⠀⠀⠹⡇⠳⠀⠀⠀⠀⠁⠘⢾⠗⠈⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⢠⡀⠀⠀⠀⠀⠇⡀⠀⠀⠀⠀⠀⠀⠸⠖⠂⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠂⠚⠻⠶⠉⠀⣤⣄⣀⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠉⠉⢹⡄⠀⠀⠀⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠂⠀⠀⠀⠀⡐⠀⠀⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣼⠷⣶⣤⣥⣿⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⡀⠀⠀⠀⣻⠀⠩⠟⢻⣇⣀⣀⠀⠀⣰⠏⠀⠀⠤⠤⠤⠤⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢽⡃⠄⠀⡾⠀⠀⠀⠠⠀⠀⠤⠀⠤⠤⠄⠀⠀⠀⠙⠀⢤⡄⢠⠄⢠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⣿⣄⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢸⣿⡏⢲⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⢸⣯⣧⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣠⣼⢻⣟⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠁⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣦⣤⣴⣶⣴⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠟⣧⠀⠈⠁⠈⣠⣴⣶⠶⢾⢿⠿⠟⠙⠛⠛⠛⠛⠋⠉⠉⠉⠉⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠶⠦⠬⠿⠛⠛⠛⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣤⣴⣤⣤⣤⣤⢤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣟⣿⣿⠻⠛⠛⠑⠛⠙⢷⡾⣿⡟⠛⠛⠻⣟⢦⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⡿⠿⠛⠃⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⡆⠀⢠⣦⣷⣿⣏⠉⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⡚⠛⠿⠛⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⢄⠀⠀⠀⠀⡇⠹⠇⠀⢀⣍⣹⣿⠋⠀⢴⣬⣧⠟⠛⢿⠲⣜⣷
⠀⠀⠀⠀⠠⡶⢻⠏⣶⠀⠀⠀⠀⢀⢠⠔⢚⣏⢠⣇⣀⣼⠀⠀⠀⠀⡇⢀⠀⠀⠈⠙⣿⣿⠁⠀⣤⣉⡟⠁⣤⠘⣇⣿⣽⣷⡀
⠀⠀⠀⠀⢀⡷⣿⠀⠘⣦⠀⡠⠞⡧⢸⡉⣹⣆⣠⡇⠐⢾⠠⠀⢀⣀⣇⣿⣃⠀⠀⠀⣿⡟⠀⠀⠈⢻⡟⢀⡙⢻⡿⠉⢿⣽⣿⡄
⠀⠀⠀⠀⢸⡇⢹⡀⠀⠸⣇⣷⢈⡷⢚⣉⣸⡇⠤⣧⣤⡾⠞⠛⠉⠉⠉⠉⠉⠉⠓⠶⢿⡇⠀⠀⠀⢸⡇⠈⠻⣾⠄⠀⢸⡆⣿⣷
⠀⠀⠀⠀⠘⡇⠸⣇⠀⠀⢹⣿⣁⡞⠈⣇⠠⣇⣴⣿⣿⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⢸⡇⠀⠀⣿⠀⠀⢸⡏⣿⣽⣧
⠀⠀⠀⠀⠀⢿⠀⢻⡄⠈⢦⢿⣇⠶⠊⣇⣠⡿⢟⣿⢟⣾⢣⡆⣤⣤⠀⡄⢀⠀⡀⣀⢠⡰⣾⣿⣦⣾⠁⠀⢠⡿⠀⠀⢸⠇⣿⡟⠁
⠀⠀⠀⠀⠀⠈⣧⠈⣷⠀⠘⡉⢻⣾⡊⢹⡟⠀⠀⠀⠛⢡⠟⣱⠟⣿⠂⣿⢌⡷⢻⡞⣧⢻⡽⣿⡿⢿⠀⠀⢸⣧⡄⠀⣸⠀⣿
⠀⠀⠀⠀⠀⠀⣼⡀⠸⣇⠀⢹⡆⢻⣗⢺⠁⠀⠀⠀⠀⠀⠀⠁⠀⠃⠘⠇⠸⠃⠸⠀⠛⠘⠃⠁⠀⠘⡆⠀⣾⣿⡇⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢧⠀⢻⡀⠀⢃⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡏⣻⠀⠀⣿⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠈⣷⠀⠘⣇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡾⢱⡏⠀⢰⡇⣼⣿⠁
⠀⠀⠀⠀⠀⠀⢠⣄⢿⡄⠸⣇⠀⠋⣄⠹⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⣸⠇⠀⣼⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣧⢳⠀⠹⡆⠀⠹⡆⢻⣿⣿⣾⣇⣼⢁⡿⢠⡶⣠⡤⠀⡀⠀⠀⠀⠀⠀⠀⣠⣿⠇⢸⡿⠀⢠⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⢻⣄⡀⠀⠈⢻⣦⡀⠉⠙⠘⠇⢿⠹⡟⢠⠞⣡⡖⣠⣄⣀⢀⣴⣿⡇⢀⣼⠁⠀⣼⠁⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢹⣿⡄⢾⣥⣽⡆⠀⠀⡙⢿⣦⣀⠀⠀⠀⠀⠀⠉⠘⠳⣿⣿⣿⣿⣿⠟⠉⠀⣾⠃⠀⢀⡏⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡶⢿⣷⡘⡿⠶⢿⠀⣰⠏⢀⠉⠻⠿⣶⣶⣶⣤⣶⣶⣶⣿⡿⠿⠋⣡⠖⠀⠀⠉⠀⠀⡼⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢿⣭⣿⡛⣶⠾⣧⠀⢠⡟⠀⠀⠀⠀⠉⠉⠛⠉⠉⠛⠓⠖⠀⠀⢀⡀⠀⠀⢀⣀⡴⣧⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⢳⣽⣶⣼⣦⠀⠀⠀⠀⠛⠻⢷⣤⣤⣄⣀⣀⣀⠀⠘⣧⠀⢻⠀⠀⣻⣯⣤⣾⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡈⢿⡙⢦⣼⠷⣆⣀⡀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠙⠃⠀⢀⣿⣇⢠⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢸⣧⣾⣿⣶⣾⣿⣛⣒⣴⡦⢄⣠⣀⠀⢀⠀⢀⡀⣀⣤⠴⣿⣀⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠉⠙⠻⠿⠿⠿⣶⣾⣿⣭⣽⣿⣉⣿⣍⣹⣿⣤⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠿⠋⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢟⣿⣗⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢁⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡀⣘⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⣲⣿⣷⣌⠢⣴⣿⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠏⠀⠀⠀⣁⣾⣿⠧⠘⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢀⣴⣶⣿⣷⠦⣤⠔⠶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠏⠉⠀⡾⠟⠃⠀⠈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠛⠄⠀⠀⠀⠀⠀⠸⡇⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣧⡀⠐⠚⠁⠀⠀⠀⢸⡇⢠⠏⠙⠒⠒⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⢩⡟⠁⠺⠉⠐⠀⢀⣴⣿⣧⠞⣠⡴⠂⠀⠀⠀⠈⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⠏⠀⠀⠀⠀⠑⠛⠻⣿⣻⠿⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⣇⠺⠁⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⣀⣿⠀⡄⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⡀⣼⠲⣜⠞⠃⠀⠀⠀⠀⠀⢀⡐⠓⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠞⠁⠈⠛⠛⢶⠆⠀⠀⢄⠀⡤⠊⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⣘⢀⠒⠀⠀⠰⠈⠧⠀⠠⠤⠋⠀⠀⠀⠀⠀⠀⠀⢈⡳⣄⠀⠀⣠⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠠⡘⠦⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣷⣾⡿⣻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠈⠃⠀⠈⢣⠀⠀⠀⡤⠀⠀⠀⠀⣠⣴⠞⠛⢩⡛⢹⢿⡿⢿⣿⢇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⣶⣴⠦⠞⠁⠀⠀⢀⣶⠿⣦⡠⠰⡆⠰⢸⣇⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠈⣩⡵⠛⠉⠁⠀⢣⠀⠸⣿⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡤⠖⠋⠁⠀⠀⠀⠀⠀⠈⢷⡀⠙⢣⡷⠀⠀⠀⠀⠀⠀⠀⡀⣵⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⡀⠉⠀⠀⡀⣤⣴⣾⡿⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⡿⠇⠿⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠂⠉⠉⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣶⣤⢀
⠀⠀⠀⠀⠀⢠⠞⠑⢫⡀⠀⠀⠀⠀⢀⣝⠃⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡖⢻⣿⣿⣿⣿⣿⠃⣾⣷⡄
⠀⠀⠀⠀⢠⠃⠀⠀⠨⡗⠀⠀⠀⠀⢰⠅⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣨⣿⣿⣿⣿⡏⣸⣿⣿⣿⡄
⠀⠀⠀⠀⡎⠀⠀⠀⠀⣻⡀⠀⠀⠀⣹⡀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠆⢿⣿⣿⣿⠆⢾⣿⣿⣿⣧
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠾⠀⠀⠀⠀⢼⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣃⣿⣿⣿⣿⡃⣻⣿⣿⣿⡏
⠀⠀⠀⠀⠈⢆⠀⠀⢨⡟⠀⠀⠀⠀⢘⣅⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⢨⣿⣿⣿⣿⣧⠸⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠳⣔⡾⠀⠀⠀⠀⠀⠀⠵⣆⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣓⣾⣿⣿⣿⣿⣿⣆⢻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⢼⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⣀⣠⡴⠅⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡏⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢛⣿⣷⠃⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠒⢤⣀⢀⡗⠟⣫⣽⣿⠀⠀⠀⣤⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⠢⠰⡄⠀⠉⠓⢬⣙⠛⠿⠆⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⡱⣆⢸⠀⠀⠀⠀⠈⠓⢄⠀⠀⢸⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣷⡎⠻⣄⠀⠀⠀⠀⠀⠀⠑⢶⣿⣟⡁⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⢣⡞⠀⠀⠈⢲⡄⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠓⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣣⠎⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⢙⡟⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣤⡀⠤⠘⡽⠁
⠀⢠⡖⠐⠚⢃⠚⣤⣼⣧⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠊⠉⠛⠁⢰
⠀⠀⠉⠻⠶⠶⠶⠶⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠙⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⣶⡀⠘⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣽⣶⣿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣏⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠙⠃⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣿⣯⢶⣖⣃⣨⣤⣭⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠙⠋⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣠⢤⣤⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤
⠀⠀⠘⠛⠓⠛⠛⠿⠶⠾⠷⠤⠄⠐⠠⣀⠀⢀⣤⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⣠⣤⣦⣤⣤⣾⣿⣷⣦⣀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠚⠿⠿⢿⣿⠳⠋⠀⠉⢉⠛⣛⠿⢾⣷⣦⡀
⠀⠀⠀⣀⣠⣄⣀⢀⠀⠀⡀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠟
⠀⠀⣄⡙⣋⢿⣇⡺⢐⠓⣀⡒⣢⠂⠿⢄⣰⡆
⠀⠀⠘⠛⠛⠛⠛⠛⠛⠀⠉⠉⠙⠀⣀⣨⡏⠉⠉⠀⠠⠤⠄⠂⠀⠀⢠⣄⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣅
⠀⠀⠀⠀⠀⠂⠒⠒⠀⠐⠲⠖⢳⣾⣶⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⢉⣟⣉⣁⡶⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢙⣧⡄⡀⠀⠀⠦⠸
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠌⠀⠀⠹⢯⣴⣾⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⢉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠛⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣆⡼⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡱
⠀⠀⠀⣄⡀⣄⠤⠤⠤⡀
⠀⠀⠈⠛⠛⢦⠤⠀⠲⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣟⠛⢻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣡⣾⣿⣿⣿⣷⡷⠝⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣶⠿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣧⣨⣿⣿⣿⣿⡿⣿⡉⠙⠛⠿⠗⠿⢧⣤⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣻⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⣵⣿⠿⠛⠀⠀⠷⢠⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣟⣿⣦⣀⣀⠨⠏⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠉⠻⣿⣿⣿⣿⣿⡟⣋⡴⣻⣹⣏⣿⣻⡿⢛⣒⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⣤⣾⣿⣿⡿⣿⣫⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⠯⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢃⡀⣳⣮⣿⣿⣿⣿⣿⣿⣿⡿⣛⣷⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⡃⡠⡄⠸⢾⣯⣼⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣷⡞⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⣤⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠽⢿⣛⣟⣻⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣟⡶⠤⣝⡻⠿⠿⠿⠿⣿⣧⠶⠤⣤⣀⠉⠙⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣅⣀⡀⢄⣲⣦⣀⡤⠀⠙⣿⣿⣶⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣆⣠⢋⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⡶⣩⣧⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⣾⣿⠑⠻⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠈⠑⠒⣞⣿⡟⠛⠛⣶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄⠀⠀⠀⡀⠻⡘⡁⢒⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢶⣷⠀⠀⠀⠸⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠈⢻⣧⠀⠀⠀⠈⠰⠛⠛⣻⣽⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠻⣷⣦⣀⠀⠀⠀⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠙⠓⢆⣄⡈⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣧⢠⣥⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡧⠂⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡏⠀⠀⣤⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣾⣿⠿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⡄⣦⣤⡀⠀⠀⣈⣾⣿⣿⣿⣷⣦⣼⣅⣁⣀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⣬⣴⣣⡶⣟
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠈⢠⣟⣯⡀⠈⠉⠉⠉⠉⠉⠛⠙⠿⣿⡿⢿⣆⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡭⡜⠁⠀⢀⠦
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣦⣤⠀⡀⣀⢀⠉⠀⠶⢴⡴⢪⠇⠀⠑
⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⣺⠟⠒⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢿⣾⣯⡀⠀⣀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⡈⠀⠀⠹⣉⡉⠛⠿⠋⠀⠒⠒⠃⠤⢀
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⡇⠱⡄⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄⡙⠫⣀⠈⠙⠂⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⡇⠀⠈⣦⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠑⠢⢠⣤⢴⠃⠄⠀⠀⣀⠀⣀⡄⠀⠘⢶⡆⠀⠀⠈⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⡇⠀⠀⠀⣷⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡸⣶⡆⠀⠀⠉⠐⠨⠶⠀⠀⠈⠛⢟⠋⠙⢉⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⢀⠀⡇⠀⠀⠀⢙⡿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠀⢸⡍⠢⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⡟⠓⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠁⠀⢀⠖⠀⢉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡇⠀⠀⠉⠒⠒⠋⠁⠀⠁⠀⢀⢳⡦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⣠⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠥⢀⣠⠖⣲⣶⣶⣂⣠⡤⠤⠤⢀⣛⣿⣟⡓⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠁⠀⢸⡆⠴⡞⠫⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠉⠉⠀⠀⠉⠉⠉⠁⠀⠐⠒⠒⠛⠉⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⠶⠶⠶⠶⠤⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣶⣶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣂⢐⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣗⠀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣗⢀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠕⢐⣺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⠏⢡⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡯⠁⣲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢍⢠⣲⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠁⣒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡓⠠⣲⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⠟⠠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠗⡠⣽⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠟⠏⢊⣲⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⣺⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡈⣄⣮⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅⢐⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⢐⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡃⠨⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠈⢽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠩⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣞⠛⢿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁⣤⡉⠻⣿⠷⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⢠⣌⠙⣿⠷⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⣦⡈⣿⠿⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢶⡌⣻⠿⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⡌⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢳⣄⣹⠿⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⢶⣄⣙⣦⣤⣿⣌⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⡀⠻⣄⣽⠿⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡈⠛⣶⡶⢿⡏⢹⣇⡿⢉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣀⠹⣦⣽⠛⠛⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣈⣿⣿⡉⣿⠸⠇⠈⠉⠁⠾⠁⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣤⠹⣦⣿⠛⠛⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣤⣼⠿⣧⠘⠇⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣶⠹⣷⣟⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣶⣾⠻⣧⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣷⣸⣿⡋⠉⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣶⣏⠹⣦⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡿⣯⡈⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⣾⣯⡘⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⣶⣯⡈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⢶⣿⣌⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣧⣌⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣤⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣮⣝⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣸⣷⠮⢥⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣀⠀⠀⠀⠀⠀⠟⢁⣽⣶⡶⣶⣤⣄⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠶⠶⠶⠶⠿⠷⠾⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⢂⡀
⠀⠀⢤⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⣠⣿⡝⠳⢿⠟
⠀⠀⠀⠈⠻⢿⡳⣄⡀⠀⠀⠀⠀⠀⢠⠖⠉⠀⣄⠀⡸⢿⠋⢙⣳⡆
⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣀⠀⠀⠀⢀⣠⣖⣈⠅⠁⠠⣾⣷⠾⠟⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣶⠰⡁⢁⠋⠀⠀⠀⠀⠀⠁⠑⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠭⠀⠀⠤⠤⢄⣀⣠⣤⣤⣶⣶⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢀⠿⠿⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠉⠙⠛⠿⢦⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠠⠀⠀⢚⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣽⣿⣛⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠰⡀⠈⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣯⢉⣵⣾⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⢹⣿⣿⡟⢹⣿⢮⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡏⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⢡⡄⠈⢛⠉⣠⠊⠝⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⢸⣇⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡄⣀⢻⡄⣸⡜⠁⠠⠤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣾⠘⢇⠀⢐⡇⠀⠀⠀⠀⠐⢄⣀⠀⠙⡙⠊⡇⠘⠀⠜⢉⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢸⡟⠀⢸⣷⣾⡇⠀⠀⠀⠀⠀⠲⠐⡂⠀⢷⡀⡀⠠⣶⣾⣄⣁⣁⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢸⠀⢀⣾⣿⡟⠀⠀⠀⠀⠀⠘⢎⠋⠙⠶⡇⠈⣷⣶⣿⣿⣿⣷⣭⢁⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠈⢳⣶⣿⠈⠘⠛⠙⠛⠋⠁⠀⣯⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡆⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⣀⣠⠃⣊⠢⠀⠀⠀⠀⠀⠀⠀⢦⡵⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⡀⣷⠒⠒⠻⠙⠁⠀⠀⠀⠒⠚⠂⠙⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣴⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⢿⣿⣿⠿⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣶⣶⣶⣿⣿⣿⣿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠉⠻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡟⠟⣽⠟⠁⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣶⣶⣶⣤⡆⢠⣤⣶⣶⣶⡄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠉⠟⢫⣿⡇⠈⠉⢽⠏⠉⠀⣾⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⣴⠟⢧⣤⠔⠳⠦⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠞⠡⠄⠀⠄⠂⠀⠀⠀⣸⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣆⠀⠚⠛⠛⠃⠀⠀⣴⠋⠸⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡏⢿⣦⣀⠀⠀⠀⣀⠜⠁⠀⠀⢹⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣆⠙⠻⠿⢿⠛⠁⠀⢀⣄⠀⠈⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣿⣿⡆⠀⢻⣿⣿⡿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄⠀⠀⠀⠉⢿⣿⣶⣾⣿⣿⣧⣹⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠩⣶⡝⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠲⢯⡉⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣷⣤⡞⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⢠⣼⣿⡌⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢻⣿⣯⣶⢆⣼⣿⣿⣿⣷⡀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣈⣽⣧⢸⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⣿⣿⡇⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠋⠉⠀⣀⣨⡉⣿⣿⣿⣿⣿⣿⣆⠀⠀⣿⣿⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣹⡏⠈⠉⠉⢀⣀⣄⣽⣿⣿⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⡇⠉⣼⡆⠘⠛⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇⠃⠘⢋⣠⣠⣤⣴⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⠘⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣇⠀⢐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⢿⣿⣿⣿⣿⠇⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡄⠀⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡘⠏⠉⣿⣿⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⢿⣿⠻⣿⠀⢸⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡏⢻⣿⣿⣿⣿⣿⣦⠙⠷⡘⠇⠈⠎⢿⠀⠙⠀⢸⠻⡟⢿⣿⣿⡟⠀⢀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⢷⡄⢻⣿⣿⣿⣟⣿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⣇⠀⠉⠿⠇⠀⢿⡄⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡟⡏⢷⡀⢻⣿⡻⣿⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⠀⠀⠀⠀⠀⠀⠑⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⠸⠈⠳⠀⢿⣷⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣼⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡄⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⣾⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡿⣿⡇⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢻⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢟⣛⣻⡿⠿⣿⣿⣿⠿⣿⠿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣬⣉⣿⣶⣶⣴⣶⣶⣶⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢠⣴⢻⣦⣄⣀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣾⠁⠀⠀⠃⠈⠻⠛⠋⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠘⣿⣿⠿⠀⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠉⠀⠀⠠⣀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣿⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠰⣄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣅⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠙⣷⠊
⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⠀⠀⠀⢀⡴⡏
⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⡠⠀⠉⠿⣦⣀⣠⡟⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠹⢦⡀⠀⢤⠜⠁⠀⠀⢀⣨⢿⡿⣡⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⡰⠋⠽⠗⣾⠧⣤⠴⠛⠁⠸⣿⣯⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⣦⣞⣓⣐⣀⣀⣀⡼⠃⠀⠀⠀⠀⠻⣯⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣴⣿⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠿⢿⣿⣿⠏⠀⠀⠈⠃⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠀⠀⠛⠁⠀⠀⠀⠀⣾⠇⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠠⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⢻⣿⣿⣿⣿⣷⣦⣀
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣙⠿⣿⣿⣿⣿⣿⣷⣄
⠀⢨⣿⠃⢠⣤⣀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣮⣝⠻⢿⣿⣿⣿⣦
⠀⢸⡏⠀⠈⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠛⢿⣦⡙⣿⣿⣿⡄
⠀⢸⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣿⣷⠸⣿⣿⣇
⠀⢸⠁⠐⠐⠛⠋⠙⠛⠻⢿⣿⣿⣿⣿⣿⠏⠀⠀⠀⢹⣿⡇⢻⣿⣿⡀
⢀⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⢿⣿⠸⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⣿⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠉⠀⠀⠀⠀⠀⠀⣸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⡤⠶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠄⠠⠤⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠒⠲⠤⠤⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠉⠑⠒⢤⡀
⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠈⣖⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠲⢄⣀⠀⠀⢀⡠⠤⠤⠄⢀⣀⡔⠲⡎⠁⠀⠀⠀⢀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⡉⠓⢺⠀⠀⠀⠀⠀⡀⠷⢾⠛⢄⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠚⢦⣀⡠⠔⠊⠀⠇⢠⠡⡀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡸⠀⠙⠒⠠⠼⠆⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠈⠑⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⠊⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠐⠒⠒⠒⠈⠉⠁

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠄⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠈⢇⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠐⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤⢄⡀⠀⠂⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡐⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⡤⢀⡀⠔⠆⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⢀⡄⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠤⢠⠀⠈⠁⠀⠐⢃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠄⣀⠀⠈⠁
⠀⠀⠤⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠲⣀⠀⠀⠀⠠⠀⠀⣀⣀⣀⣀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠃⠔⢠⠂⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠓⠲⠦⢤⣤⣀⣀⣀⣀⣀⣀⡠⠤⠴⠒⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢈⣉⣁⣀⣤⠤⠤⠒⠒⠒⠈⠉⠉
⠀⠈⠓⠲⠶⠤⣤⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠲⠶⠤⠤

⣶⡶⠲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣿⣶⣴⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡄⠈⠩⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣯⣾⣾⣿⡿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡏⠻⠟⠻⠿⠿⠋⠁⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡎⠿⠀⢠⣾⣿⢿⣶⡀⢠⣶⣿⢏⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⠀⠀⠀⣦⣀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢠⠀⠀⠀⠀⢀⣀⣉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⣿⣎⢷⣄⠀⠀⠀⠀⠤⠤⠂⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠶⠖⠚⠹⣿⣿⣿⣿⣿⡄⠻⣷⣶⡀⠀⠀⠀⣠⡟⢦
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠈⠻⢏⣛⠛⢀⣴⠋⣿⣷⡶⠞⠁⠀⢸⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠈⠉⡴⠀⢈⣵⠏⠉⠀⣴⣆⠀⡄⠀⠀⠈⡢⠉⡏
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢱⡄⠀⡴⠋⠀⢰⠞⢁⠂⢠⠘⢻⣿⡗⠀⠀⠀⠺⠄⡀
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⡱⠀⠋⠋⠛⠟⠙⠚⠃⠀⣼⣇⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⢖⣶⡶⠟⠋⠀⣼⡁⠀⠀⢶⠶⡶⠶⠤⠀⠀⣿⣿⡄⠀⠀⠀⠀⢸⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠧⣿⠗⡿⠃⠀⠀⡀⢸⣿⡇⠀⠀⣀⣨⣧⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠸⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣠⠃⡼⠃⠀⠀⠀⢱⣿⣟⣿⡂⠀⠁⢸⣇⡀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠸⢿⣆⠀⠀⠀⠀⡠⣼⣿⡙⣾⣿⣶⣿⠛⠋⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⠀⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⢀⡀⢳⣝⠻⠿⠞⠛⠻⠿⠇⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⡇⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣿⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣿⡇⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏⢻⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣧⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣠⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣛⣛⣛⣛⣛⣙⣻⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣭⣭⣿⣯⣿⣯⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⢸⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠄⠀⡉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣠⣿⡟⠀⠀⢸⡧⠉⠀⠈⢁⠀⠀⠀⠀⠀⠀⠀⠁⡞⠁⠀⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⣀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠠⠀⠐⠒⡄⢀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡿⠃⠀⢄⡀⠀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⣂⡔⠁⠐⠤⠸⠁⠀⠀⠀⠈⠈⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⡤⠀⠀⠀⢠⠖⡉⠂⠠⠀⠀⢀⠀⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠛⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠄⡽⠀⡀⠀⢀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠉⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠍⠀⠀⠀⠙⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⢀⠒⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢄⡀⠁⠀⠀⠀⠀⠀⢀⠄⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢀⠚⠣⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⢀
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⣄⠐⠓⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠃
⠀⠀⠀⠀⡄⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠄⢀⡀
⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⡴⠶⠶⠶⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣛⣉⣭⣭⣤⠤⠤⢤⣤⣤⠀⠀⠀⢤⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠕⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠋⡩⠥⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⠤⠶⠒⠉⠀⠀⢠⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡌⠉⠙⢏⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣷⠶⠶⠶⠶⠶⢶⣶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠁⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⡥⠈⣷⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⡤⠸⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠛⠛⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡶⠟⠋⠁
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⠶⠞⠛⠋⠉
⠀⠀⠀⠙⠛⠲⠶⠦⣤⣤⣤⣤⣤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠒⠛⠛⠛⠛⠉⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠉⠉⠑⢦⡀
⠀⠀⢀⣀⣠⣤⢤⣤⣾⣷⣤⣀⠀⠀⠀⠀⠑⢤⡀
⢀⡜⠁⠀⠀⠀⢸⠀⡤⢦⠈⠹⣷⣀⠤⠖⠒⠦⢷⣀
⢸⠀⠀⠀⠀⠀⡟⠀⠀⡇⣧⠴⠋⠁⣀⡤⠴⠶⢶⣬⡇
⣸⠀⠀⠀⠀⠀⡇⠀⢴⡇⡧⣤⡶⠋⠁⠀⠀⠀⠀⣳⡀
⡇⠀⠀⠀⢀⡼⠁⠀⢀⠇⢷⣿⡁⠀⢀⣠⠴⠚⠋⠉⠉⢓⣦⡀
⡇⠀⢀⡴⠋⠀⠀⠀⣸⢸⡤⠽⣿⠟⠋⢀⣠⠴⠖⢋⣉⡭⠭⠷⢄⡀
⢹⣶⣏⣠⡤⠴⠒⠋⡇⢸⡀⢸⣷⣶⠊⡿⣷⡤⠖⢉⣁⡤⢤⣤⣤⣷⡀
⢸⠻⣿⣷⡀⠀⠀⠀⡇⣯⠟⠀⣉⡴⠊⢰⣣⣴⣾⣩⠔⠊⣉⡤⠤⣶⣿⣷⣄⡀
⢸⠀⠈⢿⣿⡀⠀⠀⢹⡰⡖⠋⠁⢦⠴⣄⠀⠈⣏⣀⣴⣏⡴⠚⣉⡭⠴⠶⠶⢭⣆
⢸⠀⠀⠈⢿⣷⠴⠚⠉⠱⣜⢦⡀⠀⠀⠘⠖⢄⠈⠁⢸⣃⠴⢿⣷⣤⡴⠒⠋⠉⣉⣉⡳⡄
⢸⠀⠀⠀⠸⣿⡇⠀⠀⠀⠹⡆⢳⡀⠀⠀⠀⠀⠑⠀⠀⣠⣾⡿⢿⣁⣠⡴⠚⠉⢉⣉⣭⣿⣦⡀
⢸⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⣿⠈⡇⠀⠀⠀⠀⢀⡤⣺⠟⠁⠀⢀⣽⣿⣷⣾⣿⣿⡿⠿⠛⠛⠙⠲⢄⡀
⣾⣤⣀⡀⠀⢹⣿⠀⠀⠀⠀⣿⠀⡇⠀⠀⢀⢔⡽⠊⠁⠀⠀⢠⡿⠻⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄
⡟⠲⠬⣍⣛⠺⠿⢧⣤⣄⣀⣿⢠⡇⢀⣴⡷⠋⠀⠀⠀⠀⠀⢻⡣⣀⣀⣀⣠⣤⣴⢐⡀⣄⣠⣀⣆⡰⣴⡳⠽⠊⠱⡄
⠹⣄⡀⠀⠈⠉⠙⠒⠒⠒⠒⢧⠘⣷⢿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠛⠉⠉⠁⠀⠉⠑⠒⠒⠚⠋⠉⢁⣀⣠⠴⢚⣽⠦
⠀⠈⠛⠷⢶⣦⣤⣤⣤⣤⣤⣌⣧⣈⡓⠮⣍⡓⠒⠦⠤⠤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠒⠒⣛⣋⣩⡥⠶⠛⣉⣠⣴
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠻⠿⣿⣽⣲⠭⣟⣯⣛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⢛⣋⣉⣉⣥⣤⣶⣶⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣾⣭⣤⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⣿⡿⠿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣴⡶⠶⠿⠟⠛⠛⠻⠿⢶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⣠⣤⣤⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⢿⣿⣿⣶⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⣀⣤⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⣠⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣄⡀
⠀⣠⣾⠟⠁⠀⠀⠀⢠⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⡇⠈⠙⠻⢷⣤⡀
⣰⡿⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠉⠻⢷⣄
⣿⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠈⠛⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠷⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠶⠶⣶⣶⣶⣤⣤⣤⣤⣤⣶⣶⣶⡶⠶⠶⠿⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⡟⣡⣾⣿⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠈⠉⠁⠉⠉⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⠀⠲⡿⠿⠭⠀⠻⢫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⠉⠁⠀⡀⠀⠀⠀⠄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠢⠴⠊⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⢶⡃⠐⠋⠀⠀⣼⣿⢳⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠱⡀⠉⠀⠀⣠⡾⡿⠃⢸⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠉⣑⣽⠗⢢⣼⠛⡟⢛⡭⠊⠀⢀⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⣩⡝⣢⣽⣯⣴⣟⡍⠀⣠⠖⢰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠉⠉⣹⣷⣹⣿⡿⢙⣿⡞⠐⠁⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⠀⠉⣵⣎⣿⣿⡇⢸⣷⡷⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏⠳⡒⢻⣿⣿⣿⡇⣟⣥⡾⠀⣼⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣧⠳⣄⠀⢾⣿⣿⣿⣷⡿⢋⣄⣼⣿⣿⣿⣿⣿⠿⣛⣩⣤⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⡈⠛⢘⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⢋⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⠿⠛⠟⠁⠀⠙⠋⠙⠃⠀⠀⠛⠋⠈⠟⠻⢿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⢀⠇⡇⠀⡠⠴⡄⢀⢠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢦⠀⡇⠘⠾⠀⠘⠊⠀⠀⠙⠛⢤⠀⠙⢆⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢇⠓⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠈⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⣙⠚⠁⠀⠀⠀⢀⡠⠔⠒⠉⠑⠒⡶⠀⠀⠀⠀⢨⡆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡗⡒⠉⠁⢀⣠⣢⠀⠀⡸⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠜⢢⠀⠀⣞⠀⠈⡇⢀⢧⡆⠀⣀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡆⡜⠀⠀⢿⢀⡴⠃⣸⠁⠹⠏⣅⢸⡄⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠙⠊⠐⠉⠀⠀⠉⢀⣴⣿⡄⠀⢀⠡⢞⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠐⠢⣤⣀⣠⣤⣾⡟⠋⢹⠇⠀⢀⣤⠞⢛⣣⠤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠈⢿⡿⠛⡟⣀⡴⠋⠀⡴⠋⣟⠉⢹⠈⢩⠃⠁⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢯⠟⠒⡄⠈⠓⠤⣍⣛⣉⣁⣠⠤⠞⠁⢀⣸⠤⠞⠀⢇⣠⠆⢠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢇⡔⠰⡇⠀⠀⠀⢀⣀⡴⠋⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠑⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⠍⠊⠳⣌⡑⠒⣒⣳⠎⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢡⣏⣀⣀⣀⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⢫⡉⠉⠉⠁⠀⠀⠀⠀⠈⣹⠃⠀⠀⠀⣠⠦⠤⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢨⢗⢳⠀⣀⡤⡀⠀⢀⠀⠀⠉⠒⣦⠀⡠⢷⡒⢦⢀⠀⡝⢝⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣾⠀⠙⠊⠁⠀⠈⠉⠻⣄⠀⢀⡴⣃⡰⢋⡵⠟⠳⢮⣎⡠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠾⢛⣛⡿⢝⡢⢼⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠉⠲⣬⠖⠁
⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠈⠑⠪⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⠤⠤⠴⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⢳⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢁⡴⢢⠞⡴⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡴⢋⡴⣣⠞⠁⠀⣀⣀⠀⠀⠀⣠⠞⣉⢀⡼⢱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡞⢡⠞⡴⢋⣠⠴⠛⠁⣹⡄⢀⡾⣡⠞⣡⢞⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡼⢠⢏⡼⠗⠋⠁⣠⡴⢚⣥⣇⡞⡴⠃⡴⣫⠞⠀⣠⠴⠚⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⣷⡟⡼⠁⢀⡴⠞⣡⠞⠋⣼⡟⣰⢁⡾⣱⣧⠖⠋⠁⣀⡴⢛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⣼⢰⣇⡴⢋⣠⣏⠀⠀⢰⡿⢡⣧⡾⣱⠋⠀⣀⡴⢛⣡⠞⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⣴⠇⠘⢋⡴⠋⠁⣿⠀⢀⡿⠁⠀⣿⢁⡇⣠⠞⢉⠔⠋⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⣠⠎⡟⠀⢠⠎⠀⠀⠀⣇⣠⠏⠀⠀⢠⠇⢸⠟⣁⡴⠛⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⢀⡴⠃⣼⠁⢠⠏⠀⠀⢀⡤⠛⠁⠀⢀⡔⡿⠀⢀⡴⠋⠀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⣇⣠⣤⠴⠶⠛⠉⢀⡼⠃⢀⡟⠀⢀⠔⠉⠀⠀⠀⣠⠞⣰⠃⢀⡞⠀⠀⠀⢠⡇⢀⡟
⠀⠀⠀⠀⠀⡀⠄⠂⠁⠀⠀⠻⣄⣀⣀⣀⣠⠴⠋⠀⠀⣞⣠⡎⠀⢀⣀⣠⡤⠞⠁⣠⠏⠀⡾⠀⠀⠀⠀⢸⡇⣸⠁
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⣀⡠⠞⠉⠘⣷⠛⠉⠉⠀⠀⣀⡴⠋⠀⡼⠁⠀⠀⠀⠀⢸⢀⡏
⠀⠀⠀⠢⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠁⠀⠀⠀⠀⠈⠙⠒⠒⠚⠋⠁⠀⠀⡸⠁⠀⠀⠀⠀⠀⢸⣼
⠀⠀⠀⠀⠀⠈⠉⠉⠛⠒⠒⠒⠒⠚⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠲⠤⠤⠤⣤⣤⣤⡤⠤⠴⠒⠋⠁



⣤⣤⣤⣴⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣴⣬⣴⣬⣽⣴⣾⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣙⣿⣷⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣩⠾⠛⠛⠛⠑⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠉⠀⠀⠀⠀⢀⣠⣼⣧⡤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⡾⠛⣷⠈⣡⠾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢼⡿⠀⢴⣏⣡⡄⠻⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⣇⠀⠈⠉⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⡄⣠⣤⣶⣯⣤⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠛⠁⠀⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⡿⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⢁⣴⡿⠋⠟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⣬⠞⠁⠀⢀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠀⣱⠋⠀⠀⠀⠀⠆⠀⣼⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡘⠯⠃⠀⠀⠀⠀⠀⠈⡔⠻⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠈⢲⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣜⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣌⡉⢛⠻⠛⠛⠛⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠘⡆⠀⢲⠀⢸⡇⠀⡶⠀⢠⠇⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣠⠞⡟⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⢧⠀⢹⡄⠸⠆⠘⠇⠠⠇⢀⡏⠀⣰⠃⠀⡴⠋⠀⠀⠀⠀⢀⣀⠀⣀⡼⡿
⠀⠀⠀⠀⠀⠀⠀⠙⠫⣍⠁⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠈⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⣠⠞⠁⠀⠀⠀⠀⠀⠙⠻⣍⠉⠀⠳⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣠⠖⠒⠻⠗⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⠴⠒⠛⠛⠛⠓⠶⣤⣄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⠖⠒⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢀⣠⢶⣿⣤⠞⠋⠳⣤⡀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⢀⣴⡞⣿⣹⣭⢯⣿⣉⡛⠶⣄⠀⠀⠀⠀⢸⡇⠀⣰⢏⣡⣬⡁⠀⠀⢀⣤⣄⡙⢦⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⠶⠞⢛⣛⡛⠳⠦⣄
⠀⠀⠀⢀⣴⣏⡶⠋⠁⢰⣶⣀⣀⠀⠉⠳⣌⠳⣄⠀⠀⠸⡇⠸⠃⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠈⠃⢠⡇⠀⠀⠀⠀⣠⠟⠡⠶⠛⠉⠉⠉⠛⠂⠈⠳⡄
⠀⠀⠀⣾⣳⠏⠀⢠⣾⠛⣿⡏⠙⣿⠄⠀⠈⢧⠘⣆⠀⣼⢳⠀⠀⠀⣠⣄⠀⠀⠀⠀⣠⣄⠀⠀⠀⠚⣧⠀⠀⠀⢰⡏⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⢹⡄
⠀⠀⢸⢯⡟⠀⠀⢸⣿⣷⣿⣷⣤⣄⡀⠀⠀⠈⡇⢹⠀⣧⠀⣤⠀⠀⠻⠟⠀⢠⠀⠀⠻⠟⠀⠀⣠⠀⣼⠀⠀⠀⢸⠁⠀⢀⡴⠛⠁⠀⠀⠀⠰⡇⢹⢘⡇
⠀⠀⢸⠒⡇⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣦⠀⠀⡇⢸⠃⠘⠳⠿⡄⠀⠀⠀⡶⠶⠶⢶⠀⠀⠀⢠⠿⠞⠃⠀⠀⠀⢸⡇⠀⠘⣳⠀⠀⢀⡾⠇⠀⠉⠉⣸⠇
⠀⠀⠘⣯⢿⡀⠀⠀⢰⣦⠀⣿⡇⢹⣿⡟⠀⣸⢃⡾⠀⠀⠀⠀⠙⢦⣀⠀⠙⠛⠛⠋⠀⣀⡴⠋⠀⠀⠀⠀⠀⢠⡟⠀⢠⠞⣡⠎⠳⠋⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠹⣆⠳⣄⠀⠈⢻⡷⢿⣿⣟⡉⢀⡴⢃⡾⠁⠀⠀⠀⠀⠀⠀⣨⣿⠃⠀⠀⠐⣿⣍⠀⠀⠀⠀⠀⠀⢀⣾⣇⠀⠁⢞⣡⣀⡴⢦⣄⣤⠴⠋⠁
⠀⠀⠀⠀⠈⠳⣌⡙⢶⠏⠀⣸⠋⠀⣿⠯⣴⠋⠀⠀⠀⠀⢀⣠⠶⠛⢧⣙⣳⠖⠲⣞⣋⡼⠛⠶⣄⡀⠀⢠⣿⣿⣿⣷⠶⣾⠀⠹⣆⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠈⠙⡿⠀⢰⡇⢀⡾⠁⢠⠿⣄⠀⠀⠀⣴⠋⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠀⠀⠀⠈⠙⣦⠘⢿⣿⣿⢿⡄⠈⢷⠀⢸⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠺⠀⢸⠁⢠⡏⢀⡟⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢳⡄⢹⡀⠈⡇⠀⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠀⠸⡇⠀⣸⠃⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⣇⠀⢸⠇⠀⠋⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⡿⢠⡏⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⣿⠀⠀⠀⠹⡄⢿⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⢤⣤⠴⠞⢻⡟⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢻⡟⠳⠦⣤⡤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⣧⠀⠀⢀⡞⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢳⡀⠀⠀⡼⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠸⠆⠀⡾⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢷⠀⠰⠇⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⡼⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢧⡀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠞⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⠦⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⠀⠀⠀⢈⡷⣄⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣷⡀⠀⠀⠀⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⡤⠒⠁⠀⠀⠉⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⡠⠐⠛⠢⡺⣯⡆⠀⠀⠀⢰⣀⣀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⠀⢠⠉⠈⣀⠙⣷⡀⠀⠀⢸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡄
⠀⠀⠀⠀⠀⠐⢹⠀⣿⠀⢸⣿⡑⢾⡇⠀⠀⠀⣿⠃⠀⠐⠁⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣍⣿⣿⣿⣿⣿⣿⡟⠉⠛⠃
⠀⠀⠀⠀⠀⠀⢸⠀⢿⣀⣠⠄⠐⢸⣧⣄⣀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡿⠛⢛⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠨⠀⠀⢸⡟⠉⠉⠉⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡿⠛⠛⠒⠚⠀⠀⠘⡇⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠰⠁⠀⠀⢠⣙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⢀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠻⣧⡶⣛⠶⠥⠾⠶⢦⡴⡟⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣥⣼⣿⣿⡿⣿⣿⣷
⠀⢸⢹⢳⡒⣿⣥⣄⡴⠛⣁⠀⠀⠄⠤⠊⠀⡇⠀⡇⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠁⣸⣿⡇
⠀⢸⣏⠀⠀⢸⡟⣡⠂⢩⠃⠀⠀⠀⠀⠀⠀⢇⠀⡇⠀⠀⠀⠀⠀⣿⡟⠁⢸⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠘⠋⠀⠀⢸⢸⡇⡐⠁⠀⠀⠀⠀⠀⠀⠀⠈⢦⠰⠀⠀⠀⠀⠀⠛⠀⠀⠸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠿⢯⣑⣠⡄⠀⠀⠀⠀⠀⠀⠀⢸⡧⠓⢒⡲⠀⠀⠀⠀⠀⠀⣽⣿⣿⣦⡤⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣴⠶⠶⠶⠶⠶⠶⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⣀⣴⠶⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⢀⣴⠟⠋⠀⠀⣼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⡟⠷⣦⣄
⠀⠀⣰⠟⠁⠀⠀⠀⠀⡇⠀⠉⠓⠢⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠤⠖⠊⠁⠀⡇⠀⠀⠉⠻⣦⣄
⠀⢠⡿⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠓⠒⠒⠒⠒⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⠜⠁⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠒⠒⠚⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿
⠀⠀⠀⠈⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠞⠁⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢐⢦⡐⠉⠉⠉⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⡆⠹⠆⠁⠘⠒⣂⡤⠔⠀⠈⠓⠀⡀⠀⠀⠀⢀⣴⠟⣉⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡤⠆⠰⣊⡥⠀⣀⠤⠖⠉⠁⠀⠀⢤⣀⠀⠀⠈⠑⠲⠶⠿⠛⠉⢁⡔⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠘⠁⠃⠀⣈⠥⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⠀⠀⠀⠀⠀⢀⠴⠋⠀⠈
⠀⠀⠀⠀⠀⠠⠤⠤⠒⠂⠀⡠⠂⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⢀⡠⠚⠁⢀⠴⠚⠉⡇
⠀⠀⠀⠀⣈⣉⣓⣒⣒⠂⠀⠐⢦⡀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠤⠚⠁⠀⠀⠐⣁⣀⠤⢒⠀⡄
⠀⠀⡇⠀⢀⣉⠉⠉⠛⠉⠂⠀⠀⠙⢂⣙⠲⠄⠀⠀⢀⣀⠤⠤⠤⠀⠒⢒⡲⠒⠒⠒⠒⣒⠤⠦⠒⠊⠅⠐⠂⠉⠁⠀⠸⡄
⠀⠀⢧⡀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠋⠉⠀⠈⠉⠀⠉⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠐⠒⠒⠈⠉⠈⠃
⠀⠀⠀⠈⠑⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢦⣄⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠛⠿⠿⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠋⠉⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣦⡀⠀⠀⢀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⢠⡤⠀⠀⠀⠀⠀⠀⠛⠋⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⢻⣿⣿⣿⣿⣿⣿⣤⣶⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⢠⣿⠇⠀⠀⠀⢀⣶⣄⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⣾⣿⣿⣿⡿⠙⢿⣿⠟⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⡄⠀⠀⠀⠘⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣿⣿⣧⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠶⣤⣀⡀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣙⠛⠿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⡏⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠀⣸⣿⣿⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⢀⣼⣿⠁⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃⢀⣿⣿⡏
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⢀⣴⣿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠿⠋⠀⠀⠀⠛⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠉⠉⠉⠛⠛⠒⠶⠶⠤⢴⣶⡶⠾⠛⠛⠛⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠑⠦⠤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⣶⣶⣒⠲⣤
⠀⠀⠀⠀⠀⠀⢀⡏⠈⠑⠦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠖⠚⣧⠀⠀⠈⣷⠘
⠀⠀⠀⠀⠀⣀⡞⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠚⠛⢏⠉⠁⠀⠈⡇⠀⠀⠀⢸⡄⢀⡴⠃⢠
⠀⠀⣠⠶⠛⠉⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⣀⡷⠋⠀⠀⣼
⢠⠞⠁⠀⠀⠀⠀⠉⠓⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⣀⣆⣴⠞⠉⠀⠀⠀⣠⠇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠓⠒⢋⣭⠟⠋⠀⠀⠀⢀⣠⠞⠃
⠙⢧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠞⠋⠀⠀⣀⣠⠴⠚⠋
⠀⠀⠉⠛⠲⠦⣤⣄⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⠶⢟⣛⣋⣥⡤⠴⠖⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⢐⣿⣷⣦⣠⣄⡀⣤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⣠⣴⣦⡀
⠀⠀⠀⠀⠀⠀⡔⢠⣿⣟⣽⣿⣿⡵⣶⣾⡗⢻⣆⠀⠀⠀⠀⠀⠀⣰⡿⣷⡆⣼⣿⡿⣿⣿⣷
⠀⠀⠀⠀⠰⣿⣤⣿⣿⣿⠟⠁⠀⡀⠈⠋⠁⢸⠿⠳⡀⠀⠀⣠⢾⣿⣿⠦⠰⢿⣿⡛⠋⠻⣿⡆
⠀⠀⠀⠀⣾⢩⣿⣿⣿⣿⡄⡉⢻⣿⠂⠀⢀⡄⠀⠀⢸⣠⡞⣵⣿⠿⠧⣤⣦⡑⡘⣿⣷⡄⢸⣿⡀
⠀⠀⢠⠀⣭⡟⠿⠇⠹⡟⢳⣀⠀⠀⠁⢠⡾⠁⠀⢀⣾⣿⡧⣿⡇⢇⠀⠘⠛⠀⢠⣿⣿⡛⢾⣿⣧
⠀⠀⠈⠷⣟⣻⡶⣰⠿⠀⠘⢿⣿⣿⣿⡿⠁⠀⢠⡾⠻⣿⣧⠈⠻⣄⠁⠀⠤⠞⠓⣹⡿⣻⡆⢻⣿
⠀⠀⠀⠀⠈⣽⣿⣤⠀⣰⠓⢴⠞⢻⠟⡸⠃⣠⠟⠀⠀⢻⣿⡆⠀⣇⠀⠀⢀⣴⡊⣹⢿⣻⣧⠾⠋
⠀⠀⠀⠀⣰⣿⣿⡿⣧⠈⢷⣄⢄⢬⣼⠇⣰⡟⠀⠀⠀⠘⣿⡇⠀⠃⢳⡄⢧⡾⣣⣭⣼⣯⠁
⠀⠀⠀⢀⡿⢹⣿⣷⣌⣛⡟⣿⣿⣦⣌⡑⠻⣀⠀⠀⠀⠀⣿⡇⠀⣦⣼⠻⠾⠟⣿⣿⣼⢿⡄
⠀⠀⠀⣼⠇⢿⣿⡐⠛⠛⢛⣿⡍⢿⡏⢹⠇⠈⣤⠀⠀⠀⢿⡇⠀⢹⣿⣋⣁⣿⡶⢻⣿⡛⣧
⠀⠀⢠⡿⠶⢿⠿⣿⠻⣿⣿⡿⣿⣿⣦⣎⡀⠀⠛⠀⠀⠀⣿⡄⠀⣾⡿⡿⢛⣁⣴⣿⣿⡄⣿
⠀⢀⣼⡇⠀⠅⣶⠛⢻⣿⠃⢴⣯⣿⠙⣏⠙⠷⣄⠀⠀⣰⣿⣇⡴⠟⠋⣠⣿⠟⢩⣹⣿⡟⣾
⠀⣸⣿⠀⣴⣦⠬⠭⣿⣿⠟⠚⡋⡟⢀⣿⡀⠀⠈⠱⣴⣿⣿⠥⣶⣾⡿⠶⠿⠗⢻⣻⡟⢡⡟
⠀⣿⣿⣄⠈⠁⣠⣾⣿⣶⣦⠴⡿⡇⣸⠛⠙⠶⠤⢀⡼⢋⣠⠶⠿⢛⡓⠚⠛⠒⠛⠳⠇⢼⠃
⠀⣿⢹⣯⠷⠾⠿⠿⢿⣿⣿⣷⣄⢶⡟⠀⠀⠀⢰⡾⣷⡟⣛⣉⣩⣤⣿⣿⣿⣿⣟⣶⣶⡤⣤⢄⣀
⠀⣿⡎⣿⠀⠀⠀⠀⠀⠀⠀⠈⢉⡿⠀⢀⡀⢰⢿⣿⣯⠄⣤⣝⠛⠋⠉⠉⠑⠚⠛⢿⡿⠀⠃⠑⢿⡆
⠀⢿⣷⡘⣧⡀⠀⠀⠀⠀⠀⢠⠞⠁⢠⣾⣿⣷⣾⣿⡏⣴⣾⣿⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠁
⠀⠘⣿⣷⡌⠳⢦⣄⣀⣠⡶⠋⡴⠀⠻⣯⠥⠋⠠⠙⣿⣿⣿⡏⠠⣤⣀⣀⣀⣴⣟⡄
⠀⠀⠸⢿⣿⣶⣦⣤⣥⣤⣴⠞⠁⠀⠀⠀⠀⣠⣤⠀⠙⢿⣿⣷⣤⣾⣿⣿⣿⡿⠛

⠀⠀⠀⠈⠉⠙⠛⠉⠛⠉⠁⠀⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠈⢻⣟⠛⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠛⢶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠦⠶⠶⠶⠶⠶⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠚⠉⠉⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠤⠤⣄⡀
⠀⠀⠀⠀⢀⠴⠊⠉⠀⠀⠀⢾⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⢖⡡⠃⠀⠀⠀⠉⠢⡀
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠓⠤⣈⡉⠓⠒⠒⠒⠚⢛⣉⣉⡥⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⣹
⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠲⠦⠤⠤⠤⢤⣤⣤⣀⣀⣠⡤⠤⠤⠤⠶⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⣛⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⢿⣿⣧⣿⠾⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⢸⣿⣿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠩⠿⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⣴⣿⣵⡀⠀⢺⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢶⡟⠉⠁⠀⠀⠙⣷
⠀⠀⠀⠀⠀⢀⠴⠋⠉⠁⠈⠀⡝⢯⢿⣾⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢾⡇⣈⣹⣷⣶⣶⣶⢾⣿
⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠸⣧⣠⠿⠋⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠶⡟⠁⢈⣿⣩⣬⣯⡙⠻⣿⣶⡿
⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠈⠛⠲⠶⣄⡀⠀⠀⠀⣠⡾⠃⠀⢸⣄⣼⣏⡰⢿⣷⣦⣤⣄⣹⠇
⠀⠀⢀⡇⠀⣘⣷⣄⡀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣽⡿⣿⣿⣿⣄⡠⠘⠛⠀⠈⣹⡿⣦⣄⣈⣉⡽⠃
⠀⠀⣠⠁⠀⠀⠀⠙⣿⣤⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡟⠋⠻⣿⣿⠛⠀⢀⣠⣾⣿⣿⣮⡛⠛⠋⠁⣠⣤
⠀⣠⡟⠀⠀⠀⠀⠀⢻⡿⠀⠀⠀⠀⠰⣆⠸⣧⣀⣄⣴⣿⣿⠏⠀⣤⣾⠏⠁⣤⡄⠀⠀⣿⣿⠟⠋⠁⠀⠀⠘⠉⣿
⠀⡿⠀⠀⠀⠀⢠⣴⡿⠁⠀⠀⠀⠀⠀⠈⣷⣾⣿⣿⣿⣿⣏⢠⣾⡟⠁⣠⣾⣿⣿⣦⣴⠟⠁⠀⠀⠀⠀⣰⠃⢰⣿
⠀⡇⠀⠀⠀⢠⣤⣶⠃⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣯⡙⢉⣿⣿⣠⣼⣿⣿⣿⣿⠛⢳⣤⡀⢀⣶⣶⠾⠋⢠⣼⣧
⢈⣷⠀⠀⢀⣠⣴⠃⠀⠀⠀⠀⠀⠀⢠⣀⣾⡟⣿⠿⠋⠻⢿⣶⣭⡾⢿⣿⣿⣿⣿⡔⠀⠀⠉⠙⢃⣽⡆⢠⣼⣿⠋
⠀⣿⣤⡾⣋⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠉⠀⠈⠙⠿⣏⠻⢶⣤⡄⣀⣸⣿⣀⣾⣿⠇
⠀⣿⣿⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠄⣉⠛⢿⢻⣿⣿⠿⡄
⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣾⣿⣯⡞⠁
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠃⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⢸⣿⣄⣀⣀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⣼⣿⣿⡄
⠈⣿⡿⣿⣿⣶⣤⣤⣤⣤⣴⡿⠃⢀⣼⣿⠀⣿⣷
⠀⠹⣿⣄⡀⠈⠙⠻⠿⠛⠉⢀⣴⣿⣿⣿⣷⣿⣿⡆
⠀⠀⠙⢿⣿⣶⣤⣤⣤⣴⣶⣿⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡇⠀⢠⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣠⡿⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢀⡄⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣏⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡀⢀⣀⢠⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣉⠁⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⠿⣿⣿⣷⣶⣭⣭⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣦⣭⣽⣭⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣤⣉⡙⢛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⢿⣿⣿⠟⣻⡏
⠀⣀⣤⣶⣾⣷⣶⣶⣦⣬⣿⣿⡆⣸⢟⡱⣾⣯⣿⣿⣿⣶⣾⣄⣀⣀
⠚⠛⠿⣿⠿⠿⠿⠿⣿⣿⣿⡟⢣⣿⠄⠀⠛⣋⣙⠿⡿⠿⠿⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠶⠿⣿⣿⠋⠸⣿⣷⣴⣾⣿⡿⠶⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣤⣼⣿⣿⣿⣿⣤⣤⡍⠙⢋⢙⣶
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡿⠟⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣠⣀⣠⣤⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠛⠛⠛⠛⠛⠛⠛⠓⠲⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⡻⢁⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠢⣔⢎⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣴⡟⡻⢛⡿⢿⣴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⠙⢳⣈⣴⢿⡛⣟⢶⡄
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠡⢏⡴⠋⣹⣶⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⣀⠈⢳⣴⣛⠛⢶⡙⡎⠋⠹⣄
⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠨⠇⡾⠋⢈⣷⠿⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣼⣋⡉⠻⡄⠁⠀⠀⠀⠘⢧
⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠇⣴⠟⠉⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣄⣼⠋⠉⠻⡆⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣴⡿⠛⡏⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⢹⡿⠿⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⣴⣷⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣟⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⣸⠟⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣤⣼⣷⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣴⣶⡶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠋⠀⢾⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣀⢸⠏⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⢁⣼⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣾⠉⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⠾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⡏⠈⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⣿⠀⣀⣴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⢻⡿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⢿⡟⠛⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣠⣾⠀⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢀⣴⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⠟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⢿⡟⠋⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢶⠶⣿⡀⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢸⣦⣴⠿⢁⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠈⢷⣤⣤⣾⠉⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡁⣠⡾⠁⡆⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⡀⠷⣤⣠⣴⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣤⣿⡋⢀⣴⠃⡆⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠈⢷⡤⡜⣆⡳⠦⣤⡞⠻⣄⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣿⣟⣁⡴⢃⡆⡆⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠾⣶⣽⠟⢏⠹⢦⣈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣠⡽⢿⣶⣫⣜⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣔⠪⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠋⢜⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⠶⠤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠖⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠠⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣇⣀⠐⣻⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⠿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠹⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣌⢿⣿⠀⠀⢀⣤⣤⣀⣀⠀⠀⠀⢀⣠⣄⠈⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢹⠆⠛⠀⠀⠿⠿⠿⢿⣿⡇⠀⣾⠿⠿⠢⢤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢃⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⢀⣶⡀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⣆⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠸⣦⠀⠀⡀⠀⠛⡒⠒⠒⢛⡊⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣶⢿⣿⢻⣿⡆⠹⣷⣄⠀⡀⠀⠈⠙⠛⠁⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠚⠛⠛⠉⠀⠑⠏⢸⣶⡆⣿⠇⣆⠀⠹⠓⣷⡀⠀⠀⠀⠀⠘⠃⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢈⡀⠚⠁⣠⠊⠉⢫⢾⡆⠚⠃⠀⠀⠸⡆⠀⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠉⠀⣀⠀⠀⠀⣴⠃⠘⠀⢠⣦⠀⠀⠀⠁⠀⠀⠙⣌⠉⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⣠⠞⠁⠢⣴⡟⡁⠄⢀⠐⢿⣿⣿⡆⠁⠀⠀⢤⣖⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⡁⠀⢦⣼⣧⣶⣶⣤⣼⣆⠀⠙⢿⡀⠀⠀⠀⠀⠈⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣀⣠⣼⠃⠀⣰⠇⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⠀⠀⠀⢡⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡴⢚⣻⡿⠟⠛⠁⢀⣼⣧⡀⠀⠀⠻⠿⣿⡟⠓⠒⠀⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⢸⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⣿⡾⣻⠟⠀⠀⠀⠀⢞⣿⣿⠃⠀⠀⣀⣀⣽⣄⠀⠀⠂⠀⢸⣿⣿⣷⠀⠀⠀⠀⠀⢸⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠏⣰⠏⠀⠀⠀⠀⢱⣾⣿⣻⣷⠀⠀⠉⠉⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⠀⠀⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠉⠀⠇⠀⠀⢀⣸⣿⡿⣿⣿⣶⣤⣴⣶⡿⠟⠁⠀⠀⠀⠈⣿⣿⣿⣷⡀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣷⠀⠀⠀⠀⠘⣌⢿⣿⣷⢬⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣽⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⠇⠀⠘⢦⡖⣽⢧⡍⠙⠊⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⡇⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⡀⠀⠀⠀⠃⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⡏⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠠⡐⠒⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢨⢠⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⢀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣧⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⢨⠁⠩⡤⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠣⣀⡀⠀⡼⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⢚⡰⠾
⠀⠀⠀⠀⠀⠀⠀⠰⢻⠋⠹⠂⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⣿⢡⠋⣉⣉⢉⡉⠁⡀
⣦⡖⠊⠂⠀⠀⢠⠂⠐⠀⠐⠀⠀⢤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣤⠀⠀⠀⠈⡏⠁⡏⠀⡇
⢿⢷⡇⡞⠁⠒⠁⠀⣇⠀⠀⠀⠀⢠⣇⣀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣯⠔⡖⠒⠊⠈⣮⡄⠀⠀⠘⣯⠉⠙⠃
⠈⠉⠉⠓⢤⣀⣀⠾⡏⡄⠀⠀⡼⣁⠀⡹⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠒⠓⠤⣀⠔⠁⣿⠀⠀⠀⠻⢷⠞
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣆⠀⠈⠛⠙⠁⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡄⠄⠐⠊⠁⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠘⠓⠒⠀⠒⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡞⠁⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⣀⠔⠈⠀⠑⠦⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣠⠞⠫⡀⠀⠀⠀⠉⠉⠉⠑⣲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠋⠈⠢⣀⠀⠀⠀⠀⠀⠈⠑⣤⣠⣀⡀
⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡾⠁⠀⠀⠈⠲⢄⣀⡠⠤⠶⣄⣏⠠⢛⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⡜⠀⠀⠀⠀⠈⠉⠒⠒⠲⠦⠤⠴⣧⡇⢠⢿
⠀⠀⠀⠀⢀⡼⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⠇⠀⠀⢀⠀⠀⠀⠀⡼⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣮⠏
⠀⣀⣀⣾⣉⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠝⠀⠀⠀⠸⣏⠳⠆⢰⣙⡶⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⠏
⠑⠪⢿⣍⣉⠙⡷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡢⢄⣨⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⠊⠀⠐⢌⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⣠⠖⠁⠀⠈⢳⡄⠉⠀⠀⠀⠀⠈⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠳⡈⠀⡀⠤⣀⣀⠀⠛⠀⠀⠀⠀⢀⡴⠁⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢻⣄⣉⡙⠛⠒⠀⠀⠀⠉⠑⠂⠤⠤⠒⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠨⠦⠬⠍⠉⠀⣀⠀⠤⠤⠒⣁⠀⠀⠀⢀⣀⡤⠊⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢀⣋⠙⠂⠀⣠⠔⠉⠀⠀⠀⠀⠀⠀⠉⠉⣩⠔⢋⡤⠊⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⠹⣮⠈⠉⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⡫⠔⠊⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⡠⠔⢊⠁⠀⠈⠆⢠⠈⠀⠀⡜⠁⡀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠉⠀⣀⠤⠒⠉
⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⢤⡀⠀⠊⠀⠀⡸⠀⠀⠈⠑⠦⡀⢠⡺⠛⠁⢀⡠⠔⠊
⠀⠀⠀⡀⠐⠤⣀⡀⠀⠀⠀⠈⢀⣀⠤⠚⠁⠀⠀⠀⠀⣠⠔⠉⠀⡠⠒⠉
⠀⠀⠰⡙⠦⣀⡀⠀⠉⠉⠉⠉⠀⠀⠀⠀⢀⣀⡠⠔⠊⢁⡠⠖⠉
⠀⠀⠀⠙⠦⣀⡈⠉⠉⠒⠒⠒⠀⠐⠊⠉⣁⣀⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠠⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢽⡀⠀⠀⠀⠀⡀⠦⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠙⠛⠲⠶⠚⠃⠀⠀⠀⠀⠀⠀⠉⠒⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⢿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠯⠸⠟⠋
⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣷⡆⠀⠀⠀⠈⠻⣿⡭⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡀
⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣦⣀⡐⠦⠄⠀⠰⠟
⠀⠀⣶⣿⣧⣈⣉⣉⣀⣤⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠄⠀⠐
⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠈⢙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠊⠀⠀⠀⠱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⣀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣀⣠⠀⠀⠉⣠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⢁⣠⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⢷⣶⣶⣿⣿⡧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⣁⡤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣿⣯⣦⣉⠀⠉⡧⠀⣿⡍⠐⠢⠤⣀⠀⠀⠀⠀⠀⠀⠀⡀⢔⣩⠴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠻⠦⢙⣿⣿⣿⡘⢦⣿⣦⣄⣀⣀⣀⡛⡙⠂⠠⢠⡔⢦⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⠀⠀⠹⣜⣯⠙⠛⠛⠛⠛⣿⡀⢪⣹⣎⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⢦⣄⡀⣀⢀⡠⠟⠛⠉⠁⣠⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⣓⣄⢀⣀⣀⣠⡤⢞⣡⡎⣸⣀⣠⣄⣸⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⣶⣶⣶⣿⣿⣿⣶⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣶⣾⣛⡿⢿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠘⠋⠰⡿⠉⠿⠛⠋⠱⡘⣿⢛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠊⣵⣽⠗⠉⠘⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣩⠶⠒⠉⠀⠀⠀⠀⣠⣾⣿⠋⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠄⠤⠤⠤⠔⡂⣴⣾⣿⣿⣷⣶⣿⠄⠀⠀⠀⠀⠈⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠘⢹⣿⡿⠛⠿⣿⣿⣿⣷⣦⣀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⣠⣿⡟⠀⠀⠀⠈⠙⠻⢿⣿⣷⣦⡀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠤⣤⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣯⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡤⡶⠃⠀⠀⠀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣵⣢⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡇⠀⠀⢣⣦⣀⣤⣭⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣻⣿⣿⣄⣷⣴⣿⣿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢋⣠⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡟⠻⣇
⠀⠀⠀⠀⠀⠀⣼⠋⣠⠷⢿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⡸⢦
⠀⠀⠀⠀⠀⣴⠏⠀⡹⣻⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣇⣮⣷⣄
⠀⠀⠀⠀⢸⡏⠀⠊⢷⠃⢀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⣸⣵⣦⣤⣾⠆⠙⠷⢄⡀
⠀⣀⣀⣀⣸⣷⣀⣠⣾⣬⣉⣉⣉⣉⣉⣉⣉⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⢉⣉⣉⣉⣉⣁⣀⣉⣫⣽⣿⣿⣦⣤⣤⣀⣉⣦⣀

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

⠀⠀⢰⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣦⣄
⠀⠀⠀⠉⠻⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣻⣿⣿⣟⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣀⣀⡀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣷⣶⣿⣿⣿⠿⠿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⠟⠁⠸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠟⠋⠀⠀⠀⢻⣿⣿⣿⣿⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⠛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⡿⠿⠟⠛⠛⠛⠛⠛⠿⠶⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⣳⠄⠈⠉⠑⠲⠤⠤⢤⣤⣤⡤⠤⠤⠶⠛⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⣓⠲⠿⡧⢤⣄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠦⣌⣹⣖⠂⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⣝⡲⠭⠍⠙⠒⠒⠀⠠⣤⣤⣤⣀⣀⣀⣠⣄⣀⠤⠴⠚⠀⠀⢻⠀⠀⠀⠀⢀⠴⠛⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣛⠲⢭⡙⠀⠰⠤⠀⠀⢀⣀⣀⣀⡀⠀⢀⠀⠀⠀⣴⠶⠦⣄⠀⠸⡇⠀⢀⡾⠁⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠮⣍⡓⠠⢤⣀⣀⢀⠀⠀⠈⠁⠀⠈⠉⠉⠀⠀⣼⠃⠀⠀⠈⠳⣄⣷⣰⠏⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣗⡶⢤⣉⡁⠀⠀⠀⠉⠉⠋⠉⠙⠒⠒⠒⠒⠂⠀⢿⣄⣀⠀⠀⠀⢈⣻⣯⣀⣤⣤⠴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⣟⣡⡈⠙⠳⠮⣭⣛⣒⠶⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣉⣛⣩⣿⣿⣿⡿⢦⣄⣀
⠀⠀⢀⣀⣀⣤⣤⡶⠶⡟⠻⣝⠯⣕⠮⠉⠙⠓⠦⣤⣀⡀⠉⠉⠛⠓⠾⠿⠿⣿⣿⣿⣶⣶⣿⣟⡋⠩⠽⠛⠛⠉⢸⡏⠙⢦⡀⠉⠛⠳⠶⢤⣄⣀⡀
⣶⢿⡉⠯⣉⠙⠦⣙⠢⣌⠓⠎⠉⠲⢤⣀⣀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠲⢦⣤⣤⡤⠤⣤⣠⡤⢤⣤⡤⠴⠒⠃⠀⢸⡇⠀⠀⢳⣠⠞⠀⠀⠀⠀⢩⡍⠙⢻
⠙⣦⡙⠢⠈⠑⠀⠀⠀⠈⠓⠦⢤⣀⣀⣀⠀⠀⠀⠀⠙⠲⠤⠶⠶⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⠀⢈⡗⠚⠁⠸⠏⠀⠀⠀⣠⠖⠋⢀⣴⡟
⠀⠈⠛⠶⣦⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠈⠙⠛⠒⠒⠦⠤⢀⡀⠀⠀⠈⠉⠀⠀⠈⠙⠋⠁⠈⠉⠉⠁⠀⠁⠀⠀⠾⠃⠀⠀⢀⣠⡤⠖⠋⠀⣀⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠶⠶⠶⠶⠶⠶⠶⢶⣮⣝⡛⠛⠛⠒⠲⠤⠤⠄⠠⢤⣤⠤⠴⠤⠤⠂⠀⠀⠒⠚⢉⣀⣀⣤⣴⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⣄⣀⣀⣀⣀⣠⣤⣤⣴⣶⣶⠶⠶⠾⠿⠛⠛⠛⠉⠉⠁
⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⢂
⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠑⣦⠙⡄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠀⣄⣀⣀⣤⣴⣿⣇⡈⣄⠀⠀⠰⠾⠿⠛⢻⠟⠛⢻⣿⣿⣶⣤⡀
⣼⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣿⠿⣿⣿⣶⣿⣿⡿⣿⣿⠛⣿⣿⣷⣦⡬⠿⣿⢛⢿⣿⡇⠀⣈⠁⠀⢴⡄⣶⢀⠄⠀⢀⡼⠃⠉⠻⢿⣿⣷⣦⣄⡀
⢻⣿⣿⣿⣿⡏⠉⣿⣿⣿⣿⣿⣴⣿⡿⠻⣿⣿⣤⣾⣿⣶⣿⠟⢿⣿⢷⣶⣿⣇⠀⢀⣴⢰⣿⡆⠐⡼⣷⣿⡟⠀⣴⣿⣿⣦⡀⠀⠀⠙⠿⣿⣿⣿⣷
⠀⢻⣿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣶⣿⣿⣿⣿⠛⣿⣿⣶⣿⡇⠀⠤⣠⣌⠙⢛⠋⢸⣿⡏⠀⣿⣿⣿⣿⠞⠉⠉⠉⠻⣿⣷⣦⣄⡀⠀⠉⠛⠛
⠀⠀⠱⣼⣿⣿⣟⠉⣿⣿⣿⣧⣤⣿⡿⣿⣿⣁⣽⣿⣿⣿⣿⠋⢻⣷⡈⠉⣛⣿⣿⣿⣷⣿⣿⡿⣴⣿⣿⣿⣿⣶⣶⣶⣦⠀⠈⠛⠿⣿⣿⣷⣶⣶⣶
⠀⠀⠀⠈⢿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣏⣩⣿⣿⣿⣿⣿⣷⣄⠀⢘⣿⣿⢿⣿⣿⣷⡿⠟⠻⣿⣿⣿⠉⠛⣿⣷⣤⡀⠀⠀⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠈⢣⣽⣿⠋⢻⣿⣯⣈⣿⣿⣿⣇⣰⣿⡿⣿⣿⣏⣨⣿⣿⠇⠉⠒⣻⣌⣁⠀⠀⠀⢀⣠⣴⣾⣭⣤⣤⣄⡀⠈⠻⣿⣿⣿⣶⣶⣶⣶⣦⣼
⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⣿⣿⣿⣿⣅⣸⣿⣿⣿⣤⣼⣿⣿⣿⣿⣿⣦⠀⠀⣽⣩⣿⣿⣶⠶⣟⣿⣿⠟⠛⠛⠻⢿⣿⣶⣄⠀⠉⠙⠛⠛⠛⠻⢿⠏
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠿⠟⠋⢋⣹⣿⣿⣧⣤⣿⠿⢿⣿⣤⣿⣿⠛⠉⠀⢀⡙⣿⣿⣿⠷⠿⠟⠋⠀⢦⣀⣀⠀⠀⠻⣿⣿⣿⣶⣶⣦⣴⣶⣦⡌
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠈⣿⣇⣼⣿⣿⣿⣶⣾⡿⠛⣿⠏⠀⠀⠀⠘⢿⣿⣿⣿⣶⣶⣦⡶⠂⢠⣿⣿⣿⣆⠀⠀⠉⠛⠛⠛⠿⠛⢉⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣾⣿⡉⠀⠀⠀⠀⠉⠉⠛⠻⠃⠀⠀⠀⠤⢤⣼⣿⡿⢿⣿⣿⣤⡀⠀⠀⠈⠉⢻⣿⣷⣦⣤⣀⣀⣀⣠⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⠋⢱⣾⣿⣿⣿⣿⣷⡶⠀⠀⠀⢲⣶⣾⠟⢻⣡⣶⠿⢿⣿⣿⠦⠄⠀⢰⣶⡀⠙⠿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢀⣀⣤⣿⠉⠉⠀⡉⠉⣠⣶⣶⣾⣿⣯⣤⣶⡟⠉⠀⢀⣀⣻⣿⡄⠀⠀⠨⠻⣿⣷⣄⣀⠀⠉⠁⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⡾⠡⡄⣾⣿⣿⣿⠛⠛⠿⠛⢋⡍⠉⠉⠉⠉⠉⢻⣿⣤⡖⢀⡀⠈⠙⠛⠿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⠁⠀⠘⢀⣴⣿⡿⠛⠻⣿⣶⣤⣤⣤⣾⠁⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠉⠛⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣠⣴⣶⣶⣿⣿⣿⣷⣦⣀⠀⠉⠛⠛⠛⢉⡄⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣉⠙⠿⣿⣿⣶⣦⣴⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡇⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣂⣼⣿⡿⠿⠛⠉⠉⠃⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡚⠉⠙⡀
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⣷⣤⣤⣶⠶⠶⠞⠋⣉⣉⢭⡉⢉⡭⣉⣉⠙⠳⠶⠶⣶⣤⣤⣾⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⢭⣽⣶⣦⣤⡈⢉⣓⣬⡳⠄⠠⢞⣥⣛⣉⢁⣤⣴⣶⣮⡭⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢟⡥⠒⠭⠙⠻⢦⠙⢿⢳⡼⣧⡞⡿⠋⡴⠟⠋⠍⠒⢬⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡵⠋⠀⢠⣵⣤⣀⠈⣇⣧⣼⣱⣎⣇⣼⣸⠁⣀⣤⣮⡄⠀⠙⢮⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⢺⡏⣠⣤⡌⢳⢸⡼⣿⣿⣿⣿⢧⡇⡞⢡⣴⣄⢹⡓⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢁⠀⠀⢀⣄⢧⡙⠛⣃⡾⠀⢷⣽⣿⣿⣯⡾⠀⢷⣘⠛⢃⡼⣠⡀⠀⠀⡈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣷⠏⣀⠀⠘⠿⠷⣍⠿⠋⠀⡀⢸⡿⠉⠉⢿⡇⢀⠀⠙⠿⡩⠾⠿⠃⠀⣈⠹⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢯⠻⡜⣴⠋⣠⡀⠀⠈⠀⢀⡄⢀⣯⣾⡇⠀⠀⢸⣷⣿⡀⢠⡀⠀⠁⠀⢀⣄⠙⣶⢣⠟⡽⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣦⡑⠝⠜⣦⡏⠀⢀⡶⢀⣼⣷⣿⣿⣿⣷⢠⡄⣾⣿⣿⣿⣾⣧⡀⢶⡀⠀⢹⣴⠣⠫⢊⣴⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣶⣤⡈⠻⠶⠿⣿⣿⣿⣿⣿⣿⣿⣿⡾⢷⣿⣿⣿⣿⣿⣿⣿⣿⠿⠶⠟⣁⣤⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣄⡙⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠹⢋⣠⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠈⠉⠉⠉⠁⠒⢤⡤⠒⠒⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⣠⣤⣶⠶⠶⠶⠶⣶⣤⡈⢿⡆⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠋⡁⠤⠤⢀⡀⠀⠀⠈⢻⣌⠻⠦⡀⠀⠀⠑⢄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡞⣁⡀⠀⠀⠤⣄⠀⠈⢆⠂⠀⡆⠆⠈⠉⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠰⣦⠀⠀⠶⠀⠀⠀⠘⢤⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡠⡤⠄⠊⠅⠀⠀⠀⢀⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠬⡀⠉⠀⠀⣀⠔⠋⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⣩⠀⠀⠀⣒⠖⠂⠁⠀⠸⠉⠉⠀⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⠀⠀⠊⢠⠀⠀⠀⠀⡧⠀⠀⠠⢄⠀⠈⠢⡀
⡔⠀⠚⢱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⠀⠀⠀⠀⠀⠀⠁⠂⠀⡞⠀⠀⠀⠀⠀⠑⢄⠀⠑⡀
⠁⠀⢠⣿⠉⠐⠀⠠⢀⡀⠀⠀⢀⠄⠈⠑⢪⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⢆⠀⠐⡀
⠀⠀⠀⠀⠁⠢⢄⡀⠀⠀⠈⠉⠀⠀⡠⠔⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠐⠂⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠀⠀⠀⠀⠀⠀⡰⠉⠆⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠒⢄⠀⠀⣀⠤⠊⠀⠀⠈⢆⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⢏⠁⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠎⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⣱⣄⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⣿⠃⠀⣱⠆⠀⠀⠀⠀⠀⠀⢂⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣶⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⢣⠀⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣟⠁⠈⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢏⡴⡚⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢖⠻⠊
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣪⠗⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⣾⣿⣿⡌⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣿⣿⡽⣿⣆⠘⡦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⣿⣿⠟⠀⢰⢀⣿⣿⣶⢄⡀⠀⠒⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⢃⣿⣿⡏⠀⠀⠀⠸⢸⣿⣿⣿⠀⠙⢾⣾⣷⣤⡈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⣉⣡⣠⣶⣿⣿⠟⠀⠀⠀⢠⢃⣿⣿⣿⣿⠀⠀⠘⣿⣿⣿⡿⠼⠦⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠃⠚⠋⣩⡭⠿⠛⠛⠋⠁⠀⠀⠀⢠⠃⣼⣿⣿⣿⠃⠀⠀⢸⣿⠋⠁⠀⠀⠀⠀⠀⡱⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⢤⢠⡞⢁⣾⠁⠀⠀⠀⠀⠀⠀⢀⢃⣾⣿⣿⡿⠃⠀⠀⢠⣿⢻⣇⢀⣄⣀⣠⣤⡴⢁⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣱⡟⠺⠷⣯⠟⠀⠀⠀⠀⠀⠀⠀⣸⣾⣿⣿⠏⠀⠀⢀⣴⣿⡫⣜⠛⣨⢜⡭⡼⢶⣖⠀⡀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡏⢨⢿⠀⣗⣦⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠟⠁⠀⢀⣴⡾⢻⠊⠀⢹⠓⢡⡏⢀⡠⠼⢿⣇⠀⠀⡠⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣸⠇⢐⠦⢇⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⢀⣴⠟⡉⠀⡇⡆⠀⢀⣴⢯⠁⡴⠄⠚⢿⣟⣆⠚⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣘⠖⢁⣴⡧⣾⢒⠆⠀⠀⠀⠀⠀⠀⣸⠇⠀⣰⡿⢻⣆⠹⣦⡙⠿⠓⠁⣹⠟⣰⠷⠖⠈⣛⣻⣿⣧⡒⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢁⣴⢫⣌⣣⡼⠾⠀⠀⠀⠀⠀⠀⢰⡟⠀⣼⡟⠀⠀⠙⢦⣅⢈⡢⠔⠊⡡⢾⣿⡶⠞⠓⠈⣛⣻⣏⢷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⢮⣀⠈⢄⣹⣿⠂⠀⠀⠀⠀⠀⢀⡿⠀⣼⡏⠀⠀⠀⠀⠀⡙⣋⡤⡶⠉⠀⠀⠿⣿⡟⢋⡁⠈⣭⣿⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⢀⠸⢋⣛⢳⠋⠀⠀⠀⠀⢀⠀⣾⠃⣰⡟⠀⠀⠀⠀⠀⢠⠏⢘⣷⠸⡆⠀⠀⠈⢪⡫⢸⠗⠶⣿⠍⠀⢳⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣷⡾⣋⣿⠽⠉⠀⣀⠀⠀⢀⠎⣼⠃⣴⠋⠀⠀⠀⠀⠀⠀⠈⢰⣼⣿⠀⣧⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠈⢼⣿⣧
⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠈⣀⠔⠊⣹⣿⣧⣬⣷⡾⣡⡞⠁⠀⠀⡀⠀⠀⠀⠈⠁⣿⣿⣿⡀⣿⣿⣷⢮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⣿⡆
⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⣠⢞⣡⣾⣿⣿⣿⣿⡿⣋⣾⢟⣡⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇⢸⣿⣿⣿⣿⣦⡢⡀⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠀⠀⣾⣏⠣⣀⣀⠤⢊⣵⣿⣿⣿⣿⣿⢟⣡⡾⠛⢋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣄⢻⣿⣿⣿⣷⣿⣮⣵⣢⡤⠄⣂⣰⣿⣿⣿
⠀⠀⠀⠀⠹⣿⣷⣶⣶⣿⣿⣿⣿⡿⣟⣽⣾⣉⣭⣶⠊⠉⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡏
⠀⠀⠀⠀⠀⠉⠛⢿⣿⣯⣿⣿⣿⣯⣿⣷⣾⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢷⣍⡛⠿⠿⠿⠿⢿⣿⣿⣟⣿⣽⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠋⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠾⠿⠿⠿⠛⠛⠋⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡏⠉⠉⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣉⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⢀⣶⡄⡐⠋⠁⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣾⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠸⠯⡀⢀⡄⠀⣰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣷⣄⣀⣤⣤⠛⢀⠀⠉
⠀⠀⠀⠀⠀⠀⢀⠎⠀⠱⡀⣀⣴⣿⡯⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠈⠋⠈⠆⢴⡀
⠀⠀⠀⠀⠀⢀⠎⠀⣳⢄⠈⠉⠀⠙⢷⣆⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠽⠶⡟
⠀⠀⠀⠀⢀⡎⠀⡰⠁⠀⠳⢄⠀⠀⠀⠙⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡖
⠀⠀⠀⠀⢼⡄⣨⠃⠀⠀⠀⠀⢱⣦⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠄⠀⠐
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢸⠻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⠘⠋⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⢽⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣤⣶⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⢠⣶⣿⠿⢿⠿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣐⡅⠀⠀⠀⠀⠀⠙⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡁⣉⣉⡇
⠀⠀⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠒
⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤
⠀⠈⠙⠷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠐⠞⠉⠃⣀⠀⠐⠦⠴⠖⠚⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠉⠛⢦⣄⣀⠀⠀⣀⣀⣤⣤⣤⣶⣀⣿⣿⡷⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡴⠚⠋⡉⠀⢟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⡈⠛⠛⠛⠉⠩⠿⠟⠛⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣻⡟⠁⠀⢐⣿⣿⡀⢙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡋⠀⠀⢈⣙⡧⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡋⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡴⠶⠈⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢷⢀⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⣿⡿⠛⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠈⢿⣄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⡝⠁⠀⠀⢹⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⣄⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠘⣄⣀⠈⠁⠁⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣀⣀⣀⡇
⠀⠀⠀⠀⠀⡰⢅⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠒⠤⣴⠇⠀⠀⠀⠀⠀⢀⢒⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢂⡄
⠀⠀⠀⠀⠐⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠆⠀⠉⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠋⠀⠀⠀⠀⠀⠙⢦⡀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣊⡽⠏⡉⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⡉⠉⠉⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⣸⡷⠀⠀⠀⠀⠀⠀⠁⠀⢨⢺⡇⠀⠀⢐⢂⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⡿⣱⡀⢀⣴⣖⠒⠒⠦⣄⠈⣷⣿⠀⠀⠐⠐⣅⠀⠀⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢺⡇⠀⠀⠀⣷⣿⠇⢿⡽⠿⠶⠶⠤⢬⣧⡘⣿⠀⠀⠀⠀⠙⠀⠀⢠⡎⠓⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠞⠀⠀⠸⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢹⠀⠀⠀⠀⠀⠀⠀⢔⡴⠁⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠰⡀⠂⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠠⡃⠐⠀⠀⣿⣦⡀⠀⠀⠀⠀⡄⣢⡿⠃⠀⠀⠀⠀⠀⣧⠀⠀⠰⠁⠀⡄⠀⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠐⠃⠠⡀⠶⣼⣷⢩⣗⣶⣖⠺⠋⠁⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣰⡿⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠐⢦⡰⡄⠀⣶⣾⣿⢺⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⡴⠁⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠇⠀⠀⠀⠀⠀⠤⣌⣻⣟⣏⣨⣭⣽⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠇⠀⠀⠀⣸⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠐⠒⠮⠝⡏⠛⢡⢿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⠀⢠⡏⠀⠀⠀⣠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⡠⠤⠶⠦⣷⣠⡀⠉⠈⠓⠆⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⡼⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⡴⠋⠀⠀⠀⠀⠹⣿⣿⡂⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⣸⠃⠀⣤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣻⠃⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠌⠻⣮⣭⠫⠃⢀⠢⢀⣀⣠⡼⠃⠀⠀⠀⣰⠟⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠢⠀⣤⠀⣴⢆⢠⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣴⡖⠋⠋⠁⠀⠀⠀⠀⣰⠋⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡘⠁⠼⢿⣦⡏⣠⡿⠒⠦⣄⡀⠀⠀⠀⢷⠀⠀⠙⣦⣄⣐⠄⠀⢀⠔⠁⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠓⠚⠻⣤⣀⠀⠀⡉⠓⢦⠀⢸⡄⠀⠀⠀⠙⠯⣙⣛⣅⣀⡠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡐⠀⢸⠂⡾⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢁⡈⠀⠀⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠁⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠁⢀⡦⠀⠀⠀⠀⠀⠀⠀⣀⠔⢁⠄⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠁⠀⠠⠀⠀⠀⠀⠀⠀⢠⠞⠡⠖⠁⠀⠀⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠘⠀⢠⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠘⠃⠀⠀⠀⠄⢐⠈⡃
⠀⠀⠀⠀⠀⠀⢀⠈⠒⠈⠁⠠⡍⠀⠀⠀⠀⡠⠋⣠⠆⠀⠀⠀⠈⡀⠀⠂⠀⠂⢉⢑
⠀⠀⠀⠀⠀⠐⠀⠀⠉⡁⠀⡈⠀⠀⠀⢀⠜⢁⠔⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠂⢩⣀
⠀⠀⠀⠠⠤⠀⠤⢀⡀⠀⠉⠁⠀⣀⠔⠁⢠⠏⠀⡁⠢⣀⠀⠀⠀⠀⢀⠀⠈⢉⠁⠂⠀⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡤⠔⠊⠀⠀⢰⠃⠀⠀⠑⠀⠠⠑⢄⠀⠀⠰⠄⢀⣉⠀⠤⠤⠈⠀⢥⣄⡀
⠀⠘⠤⠄⠠⠤⠤⠒⠊⠁⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠸⠑⠤⢀⡔⢉⠀⠀⢀⣀⠠⠄⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⡀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠐⠁



⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠱⡀⠂
⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠈⣦⠘⢄⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣙⣿⣿⣿⡛⢻⣿⣿⣿⣿⣷⣦⣤⣀⠀⢦⣤⣴⣶⣾⣿⡧⠤⢦⠐⠀⠚⠛⠉⠉⠟⠉⠉⣿⢿⣿⣷⣤⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣉⣿⣿⡟⢻⣿⣿⣿⣿⡟⢻⣿⣉⣹⣿⣿⣿⣖⢛⣿⡉⠻⠟⡇⠀⣦⠀⡀⢾⣀⣇⡖⠀⢀⣼⡃⠀⠈⠛⢿⣿⣿⣶⣤⣀
⠹⣿⣿⣿⣿⣅⣸⣿⣿⠻⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⣿⣉⣹⡿⠛⠻⠿⠧⣄⣤⡟⣴⣿⡀⢸⣜⣿⣿⣄⡴⠿⠿⣿⣶⣄⡀⠀⠉⠻⢿⣿⣿
⠀⠘⡟⢻⣿⣿⡿⣿⣿⣶⣿⡟⢻⣿⣿⣿⡿⠻⣿⣅⣹⣿⣿⣿⣇⠠⢵⣦⣶⣄⣭⣄⣿⣹⣇⢸⣿⣿⣿⣏⣀⣀⣀⠈⠻⢿⣿⣶⣤⣄⣀⣀⣀
⠀⠀⠈⢿⣿⣿⣧⣤⣿⣿⣿⣷⣾⣟⠙⣿⣷⣾⡿⠿⣿⣿⣧⣼⣿⣆⠐⠚⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣦⡀⠀⠉⠛⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠹⣉⣿⣿⠿⣿⣿⠟⢻⣿⣿⣿⠛⣿⣿⣾⣿⡿⠛⣿⣿⡟⠷⢀⣔⠻⠟⠙⠛⠛⠛⠉⣀⡼⠟⠛⠁⠀⠙⢿⣿⣷⣦⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⣿⣷⣤⣼⣿⣷⣾⣟⠙⣿⣿⣿⡟⢹⣿⣿⣶⣿⣿⣅⠐⠀⠾⠿⣶⣤⣀⣤⣴⣿⣿⣿⣿⣿⣿⣷⣤⡀⠙⠛⠿⠿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡿⠻⠛⣿⣿⣿⠉⣿⣿⣿⣿⠋⣻⣿⡿⠿⠁⡈⢻⣿⣿⣿⣯⣵⡾⠟⠉⠀⠀⠀⠈⠻⣿⣿⣶⣤⣀⣀⣀⣀⣀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣏⠁⠀⠀⢻⣿⠉⣿⣿⣿⣇⣸⣿⠿⣿⡿⠀⠀⠀⠨⣷⣿⣻⣿⣭⣤⣀⣠⠀⣻⣿⣶⣦⠀⠈⠛⠻⠿⢿⣿⡿⠟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣠⣾⡟⠛⠉⠉⠉⠛⠛⠻⢶⠟⠀⠀⠀⢀⣀⣼⣿⣿⣿⣿⣟⡋⠁⠀⠙⠛⠻⣿⣷⣄⣀⠀⠀⠀⢀⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⣤⣶⣶⣶⣶⣶⣤⠄⠀⠀⠰⣤⣶⡾⢻⣋⣤⣾⣿⣿⡯⣅⠀⠀⣤⣄⠙⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⣼⡛⠉⠉⠉⠛⢁⣤⣤⣴⣾⣟⣡⣤⡾⠛⠁⠀⠈⣿⣷⠀⠀⠀⡹⣿⣷⣄⡀⠉⠉⠉⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣿⣿⣿⣷⣶⠆⣇⢸⣿⣿⣿⠟⠿⠿⠟⠋⠏⠉⠉⠉⠉⠛⢿⣧⣤⠆⢀⠀⠙⠛⠿⢿⣷⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠉⠉⠉⠃⣠⣾⣾⠟⠻⣿⣦⣤⣤⣤⣾⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠘⠛⠓⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣤⣶⣶⣾⣿⣿⣷⣶⣄⡀⠉⠛⠛⠛⢹⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣯⠍⠿⠿⣿⣶⣶⣶⣶⠿⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣿⣿⠿⠛⠉⠁⠑⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣥⣀⡁
⣿⢿⢿⠿⡿⡻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣿⣿⢿⣿⡿⢿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣽⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⠋⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⡛⠛⠛⠋⠁⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠆⠀⠈⠙⠳⠬⢿⠿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⠀⡀⠀⠀⠀⠀⠀⠐⣄⠉⠛⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣦⡝⢋⣀⠀⠀⠀⠀⠘⠇⠀⠀⠈⢿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⢚⣳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣻⣿⣿⣿⡿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣉⣼⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⠻⠽⠻⣷⣿⣿⣿⣧⢄⣀⣀⣀⣐⣾⣏⢥⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉⠙⠋⠉⡟⠁⠉⠉⠀⠉⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⠀⠀⠀⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠠⠄⠀⠀⠤⠄⠀⠀⡼⢿⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡄⠀⠀⠀⠀⠀⠀⠸⢁⠈⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣷⠀⢰⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⠤⠂⠀⠀⠀⠈⢿⣿⡿⠃⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡄⠀⠀⠀⠀⠀⣀⣤⠂⠀⠀⠀⠀⠈⠴⣄⠔⠀⣠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⡀⠀⠀⠲⣾⡉⠋⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⡀⣀⣀⣀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠷⠖⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢰⢳⣏⣿⠿⠿⣿⣿⠿⠟⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣿⣿⣿⣦⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠋⣀⣀⡀
⠀⠀⠀⠀⣾⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⣯⣿⣿⣿⣄⣀
⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⡿⣿⣿⣷
⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣷⣶⣶⣤⠺⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⠿⢷⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢻⡆⣤⠙⠷⣀⠙⠿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣿⣿⠛⠻⠿⠛⠻⢶⣿⣿⣿⣿⣿⠇⠀⠀⠀⢀⠀⠀⠻⣷⠘⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣋⡛⠀⠀⠀⠀⢀⣴⣿⣿⣿⡻⠏⠀⠀⠀⠀⠘⣁⠀⠀⠸⣷⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣹⡿⢠⣾⣿⣿⣿⣿⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠛⠲⠶⠾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⠿⣿⣿⣯⣈⣉⣠⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣤⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠀⠒⡀⠂⡀⠀⠀⠀⠀⠀⢻⣿⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢿⡆⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠈⢿⡄⢰⠀⠀⠀⡇⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠈⣅⡀⠀⠀⢻⣿⣿⣿⣿⠶⠦⠀⠴⢤⣾⡇⢸⣿⣦⣶⠀⠀⡧⠀⢸⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠃⠂⠀⠀⢻⣿⠏⠀⠀⠀⠀⠀⠀⠘⣿⠈⣿⣿⡇⠀⠠⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢠⡶⢶⣶⣦⣶⣄⠀⠛⠀⠀⠀⠀⣴⠶⠦⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣻⣿⡇⠀⠀⣨⢦⣈⣿⣧
⠀⠀⠀⠀⠀⣠⠋⣰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⣤⣶⡿⠋⠀⠀⠀⠀⠀⠀⣀⣀⣀⣽⣿⣿⣷⢀⡀⠹⡆⠙⢿⣿
⠀⠀⠀⠀⢠⠉⢸⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠛⢉⡽⠋⢀⡠⠀⠀⣀⣤⣶⣿⡟⢻⡏⣿⣿⣿⢹⣿⡇⠀⣷⡈⣤⣹
⠀⠀⠀⠀⢸⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⣤⡶⠋⠀⣠⣿⣡⣴⣾⢟⣭⣿⣿⠀⢸⡇⣿⠟⢈⣿⣿⣇⠀⢻⣧⣼⠏
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⢀⣠⣾⣿⠿⣛⡿⠉⠙⠿⢯⡌⠀⢠⠀⣹⠶⢿⣿⣿⣿⣿⠿⠛⠁⢀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠉⢀⣤⣴⣾⣿⣟⣯⣵⣆⠘⢿⣦⣀⠀⢤⣷⡔⣾⣶⠀⠀⠀⠀⠀⠉⠛⠳⠶⠛⠋
⠀⠀⠀⠀⠀⠀⠘⣧⡉⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⠿⣿⣿⣿⣿⣦⡈⠛⣻⠷⣦⣍⠓⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣟⠻⠿⢿⣿⣿⣿⣼⠿⣿⣿⣝⢿⣄⠀⢶⡈⠛⣷⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠛⠿⢿⣿⣾⠿⣿⣿⣿⣷⣄⠙⢣⣀⠙⢷⣄⡀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠉⠀⠻⣿⣿⣯⠙⢷⣤⡙⠶⠀⠉⠻⣶⣄⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠳⠀⠀⠀⠀⠀⠙⠿⣿⣶⣄⠘⢿⣶⣄⠀⠀⠀⡉⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠤⢤⣈⣠⣄⣉⠛⠿⢶⣼⣿⣿⣤⠠⢾⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠠⣤⣤⣠⣤⣬⣤⠬⠿⠿⣶⠚⠛⠋⠉⠀⠀⢺⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣀⣠⣤⣬⣿⣟⠙⠁⢿⣤⣶⣦⡸⣯⣤⡴⢶⣶⡶⠾⠋⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⢿⣤⣶⣾⣿⠿⠛⠁⠀⠀⢠⡀⢼⣤⣄⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣥⣾⢟⣿⠟⠋⠀⠀⠀⠀⠀⠀⠠⣽⣾⡏⠩⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡰⠻⡌⠻⣄⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠉⠀⠀⠀⠀⠛⡀⠁⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⣠⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢶⣄⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⣿⣟⢀⡄⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠈⢷⣿⢿⣼⣷⣰⣶⣶⣾⣿⣭⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠄⠀⠀⠀⠀⠀⣉⠞⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠉⠂⣿⣿⣿⣿⡟⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠹⡀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⢿⣿⣿⣂⣤⡄⠁⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡆⠀⠀⣼⣿⣿⡟⠁⠀⠑⠀⠄⠀⠀⠀⠀⢶⣿⣿⡝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠆⠀⠀⠀⠹⢿⣿⣅⣀⣴⠆⠀⣈⠙⢶⣀⠀⠈⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣦⣽⣧⠀⢹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⣠⡶⠶⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣠⣾⣿⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠿⠟⠻⠿⠀⢀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣏⠀⢠⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣆⠈⢤⣤⡸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⣿⣿⣿⣿⣿⣀⣿⡇
⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣤⣈⢻⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⠟
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⢿⣿⡄
⢠⣾⣿⣿⣿⣿⣟⠁⠀⣉⣴⣿⠟
⣿⣿⣿⣿⣿⣿⣯⣴⣾⡿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣴⢲⠸⠏⠏⠻⠹⠛⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢴⠸⠇⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠹⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⣠⡡⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠻⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢦⣹⣾⣷⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣹⣮⡳⠍⠋⠁⠀⠀⠈⠙⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣙⢳⣞⡳⠎⠁⠀⠀⠀⠀⠀⠀⠀⡈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣙⣯⣭⠉⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠦⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡽⠶⠆⠀⠀⠀⠀⡄⠀⢀⠄⢰⣄⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢶⣟⠛⠂⠀⠀⠀⢈⣧⠐⢞⢇⣨⡟⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢀⠠⠄⠀⠀⠀⠀⡄⠀⠀⣙⠶⣏⣉⠀⠀⠀⠀⠀⢇⠐⠂⢦⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢼⣛⠯⠦⠀⠀⢰⣿⢀⠀⣰⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠩⣤⡦⢐⣀⡦⠾⢂⡷⣽⣛⠓⠀⣈⣜⣿⣧⣼⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣠⡋⠉⠁⣁⣠⡾⣝⠦⠯⠠⢈⣿⡎⣥⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣿⡞⢉⣴⣦⣿⣿⣭⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣋⡦⢀⡀⠀⣠⠀⣀⢀⡞⠀⣀⠀⢠⠆⡞⠀⠀⢠⠏⠀⣀⠀⣰⢢⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢈⠇⠾⣶⣄⡳⢾⠻⡾⣽⠳⢷⣦⡏⡞⠀⠀⢠⢟⡼⠾⢷⢰⣣⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣴⣴⣶⣿⣿⣧⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣭⣒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡻⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⣿⡿⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠯⣩⠟⠛⣥⣄⣀⣈⠙⠳⢦⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⣁⣠⡔⢻⠿⠇⠀⠱⣄⠀⠹⣶⡶⡆⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢁⣀⣠⣤⣤⣈⣳⡤⠏⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⠀⠀⠀⠀⠉⠛⡛⠉⣁⠠⣠⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢚⣷⣦⣤⣤⣿⢷⢾⡯⠥⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠌⠩⠿⠵⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡛⠁⠀⠀⠠⠴⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠻⢢⣤⢦⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⢈⠀⠀⠈⣴⠅⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⣀⣰⣷⢀⣿⣿⡁⠺⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⢈⣽⠛⠉⢿⣧⣄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⡾⠃⠀⠀⠀⠹⣼⢶⡀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⣀⣴⡟⠁⠀⠀⠀⠀⠀⠈⠳⢭⣦⡀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⣸⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠀⣤⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⡀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⢠⠞⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⢧
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢈⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠆⣀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠃⠀⣘⣱⣶⠤⢤⠤⠤⣶⠦⠤⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢉⠻⣶⠈⠂⢀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠋⠉⠉⠉⠉

⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⢟⣵⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠛⠿⠇⠸⠛⠋⠁⢹⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⢀⣠⣤⠤⠈⠹⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⠿⣷⡀⠉⠟⠙⠛⠂⠀⠋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠲⠶⠛⠁⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠐⢶⡋⠕⠛⠁⠀⢀⣿⣿⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢣⡀⠉⠁⠀⢀⣴⣿⡿⠁⣸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠚⢉⣷⠶⢄⣱⠶⣶⠶⢛⡫⠚⠀⢀⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠴⠞⣛⠿⡀⣻⣿⢀⣽⣖⠋⠀⢀⠔⢸⣿⣿⣿⣿⡿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠲⠤⠼⣿⡖⢿⣿⣿⢿⣿⢫⡄⡰⠁⢀⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠤⠴⢻⠷⣾⣿⡟⢠⣟⣻⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⣇⠠⢴⣿⣶⣿⣿⡇⢸⡿⢛⡀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⣋⠈⠩⠁⣾⣿⣿⣿⡇⣾⣾⡿⠃⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡘⢧⣄⠈⣿⣿⣿⣿⣿⠟⣡⣴⣰⣿⣿⣿⣿⣿⣻⣟⣫⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⣄⠈⠃⣸⣿⣿⣿⣿⣾⡿⣿⣿⣿⣷⣿⣟⣽⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣟⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⢟⣱⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⢿⡿⠉⠁⠻⠟⠻⠟⠁⠀⠙⠿⠟⠹⡿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠛⠀⠀⠀⠀⠉⠉⠉⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⣿⣟⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣻⣿⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠀⢰⠹⡀⠀⣀⢤⡄⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢸⠀⠣⡟⠀⠳⠞⠀⠀⠳⠞⣄⡇⠙⢦⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠣⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⡟⠓⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠤⠼⠆⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⢄⣀⡀⠀⠀⠀⠀⢀⡐⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠴⢤⣀⠤⠔⠚⠁⠀⡀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣄⠀⠀⠀⡴⠛⢲⡀⠀⠈⢀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠈⡇⠀⢰⣧⠀⢀⡇⢀⡸⢏⣀⠤⠶⡄⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⠟⡰⢁⡄⠘⢧⡤⠞⠀⣼⠀⠀⠋⢴⠄⡿⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡀⠉⠀⠁⠀⠀⠀⢀⣠⣾⣿⡆⠀⠀⠐⠚⣅⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⠑⠲⣤⣤⣴⣶⣿⠟⠁⢨⠇⠀⢀⣤⠖⣉⣓⣒⡢⢄⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠹⣿⠟⠫⡇⣠⡴⠋⠀⣴⠋⢸⠉⠉⡇⠀⡸⠃⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣯⠝⠓⡆⠀⠙⠲⢬⣉⣉⣉⣁⣀⡤⠞⠁⠀⣀⡧⠴⠃⠘⣅⣰⠆⢠⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢎⡠⠀⠇⠀⠀⠀⠀⠀⢀⡽⠋⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠣⣤⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣐⠪⠔⠲⣄⠑⠦⠤⠼⣯⠏⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠋⢹⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢖⠒⠒⠶⠖⠋⠉⠉⠉⠉⠑⢶⡇⠀⠀⠀⠀⣠⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⣒⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⢸⡩⣉⠉⠉⠓⣯⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠙⢬⣧⠴⠊⠙⠤⠤⣖⠀⠀⠀⢀⡽⠀⣰⢭⣳⣤⣇⠄⡠⠃⣱⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⣴⡓⢺⡀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⠴⣋⠳⢞⡡⠚⠉⠀⠉⠛⠒⠊⠁
⠀⠀⠀⠀⠀⠀⢾⠛⠒⠒⠭⢵⣒⠭⣚⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠈⠙⠪⣿
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣓⣒⣒⣚⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠔⠂⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠑⠂⠤⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⡄⠀⠂⠐⠛⠀⠐⠚⠓⢗
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊⣃⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢲⠒⠀⠤⢄⣀
⠀⠀⠀⠀⠀⢰⠁⠀⠬⢙⠄⠀⠀⠀⠓⠤⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡠⠤⠤⠧⢍⢐⣲⡄
⠀⠀⠀⠀⠀⠈⡄⠬⣑⣲⠁⠀⠀⠀⠀⠀⠀⠀⡷⡄⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠈⠫⣐⡯⠇⠀⠀⠀⠀⠀⠀⠀⡄⠸⠄⠀⠀⡢⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠉⠈⠉⠒⠨⠬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠀⣀⠀⠀⠀⠀⠀⣀⠠⢊⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⣀⣀⣒⠀⢈⠭⡗⠈⠁⢀⢴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠔⠉⠀⠀⠈⢆⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠑⢦⡢⣀⠑⠤⡀
⠀⠀⠀⢀⡀⣀⣀⡔⠁⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠲⢤
⠀⠀⠀⠈⠻⣦⣄⠑⢄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠘⠒⠒⠓⠊
⠀⠀⠀⠀⠀⠀⠉⠛⠶⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣯⣴⣶⣤⣠⣤⣬⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⢻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣏⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠑⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣠⣶⣦⣤⣀⠀⢀⣀⣤⣶⣾⡶⠀⢸⣡⠟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠘⠩⠙⣟⣿⡏⠘⠿⠯⠝⠝⠉⠀⠘⢼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠛⢻⠇⠀⠀⠀⠀⠀⠀⡀⠀⣦⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣧⠀⠀⣰⣷⣄⢀⣤⠶⣄⠀⠀⠁⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠂⠸⠀⠈⠙⠋⣁⢤⡘⡄⢀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⣧⠀⠶⠶⠾⠯⠵⠾⠃⠀⢸⣾⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⣤⢸⡆⠀⠀⠒⠒⠂⠀⠀⠀⣼⠏⠀⣹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣛⣿⠀⣷⣀⣀⣀⡀⠀⠀⢀⡴⠁⠀⠀⢸⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢟⣿⣦⡍⠛⣿⣿⠛⠛⠛⠁⠀⡀⠀⠀⢸⣿⣿⣿⣿⡛⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⡦⢾⣿⠿⡇⠀⠈⠛⢳⣤⣤⣤⠉⠀⠀⠀⣾⣿⣿⣿⣿⣿⣷⣮⣳⢶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠁⣤⣤⣾⣿⣤⡿⢤⠄⢠⢾⣿⣿⣿⡆⠀⢀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣭⣝⠛⢶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠶⠴⣾⣿⣿⣿⣿⠋⠀⠉⣹⠏⠉⢀⣠⣴⠀⠾⢠⣿⣿⡟⠀⠄⠀⠀⣿⣿⣿⣟⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣇⣤⣿⣦
⠀⠀⠀⠀⠀⡼⢣⣿⣿⣿⣿⣿⣿⣿⣶⡶⢶⣾⣶⣶⡿⠛⠁⠀⣀⡀⠈⣿⣿⠀⠀⠀⢀⣿⣿⣿⣿⣿⢫⣿⣿⣿⣿⣿⣿⢟⣥⣾⣿⣿⣿⡏
⠀⠀⠀⠀⣼⠡⣾⣿⣿⣿⣿⣿⣿⣿⡙⠳⢦⣈⠙⠻⠿⠶⠖⠋⠁⠀⠀⠈⢿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⠙⠷⢤⣄⣀⡀⠀⠀⠂⠀⠀⠈⠛⠦⢤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢸⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⣿⣷⣶⣌⣻⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢦⣤⣀⠉⠁⢀⡀⠀⠀⠀⠀⢠⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣝⠦⢙⠻⢶⣯⣭⣭⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⠀⢉⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠉⠻⠤⢤⠼⠶⠛⢻⡟⠻⠿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣦⣌⠛⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⣤⠀⠀⣿⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣝⣷⣶⣿⣿⣿⣿
⠀⠀⠀⢰⡇⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣰⣿⡆⠀⣿⣿⡆⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣾⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣾⣿⡇⠀⣿⣿⣧⣀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⢀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣾⣿⣿⣿⠀⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢿⣄⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣾⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠙⢷⣮⣿⣿⣿⣿⣿⣿⡿⢟⣭⣾⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⢿⣿⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⠿⠻⣿⣿⣿⣿⣿⣿⣿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣶⣶⣶⣶⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠛⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠉⢻⡇⠀⠀⠀⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠂⠸⠃⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠧⡀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⡀⠈⠁⠒⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡮⠀⠂⠐⠢⡑⠄⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠄⠠⢗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣀⠠⠊⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡠⠂⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠴⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠀⡀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠀⠂⠄⢀⠀⠀⠀⢀⠀⢀⠠⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠤⠀⢀⠀⠈⠁⠈⠁⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢈⠦⢀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣀⣀⡀⠀⣀⣠⣮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⢿⣿⣿⣿⣿⣿⠛⠻⠋⠐⢄⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠃⠀⠀⠙⣿⣿⠏⠀⠀⠀⠀⠀⠉⠢⣀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠊⠁⠀⠀⠀⠀⠌⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈
⠀⠀⠀⠀⢀⡠⠐⠈⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⢀⣀⠄⢆⠁⠀⣀⠠⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠘⡀
⢻⡀⢲⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⢡
⠀⢳⡈⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠆
⠀⠀⠻⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⢼⣄⣀⡤⠒
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠛⣋⣤⠖⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣯⣭⣤⣤⣤
⣿⣿⣏⣈⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢠⣶⣦⣄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣦⣴⣤⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄
⠀⠀⣾⣿⣿⣿⣷⣶⣿⣿⣿⣶⣄⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⣼⣿⣇⠙⠛⠿⠛⠛⠁⣸⣿⣦⢠⣴⣶⣶⣿⣛⣛⣛⣛⣫⡅⣶⣦⣄⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠙⠻⠿⠿⠿⠿⣿⣿⣿⣧⡀⠀⠙⠻⣿⣷⣦⣤⣴⣶⣿⣿⠟⠃⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣹⣿⠃⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣦⡄⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣴⣶⣿⣿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣴⣿⠇
⠀⠀⠀⣀⠿⣿⣿⣿⣿⣿⣿⠿⣀⠀⠀⠀⢀⣠⣌⠙⠛⠛⠛⠛⣠⣀⡀⠀⢀⣴⣶⣶⣦⣤⣤⣀⣉⣁⣀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢾⣿⣦⣄⣉⣉⣉⣉⣠⣴⣿⡷⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠸⠿⠋⠁⠀⠀⠙⠻⣿⣿⣿⣿⣷⠄⢀⣴⣾⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠛⠉⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⠋⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⡀
⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡄⠀⠀⠠⣄⡀⠀⣠⣿⣿⣿⣶⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⢀⣙⠿⠿⠿⠿⢟⣃⡀⠀⠀⠈⠻⣿⣯⣛⢿⣿⣿⣿⠂
⠀⠀⣠⣄⠛⠻⠿⠿⠿⠿⠛⢉⣿⠀⠀⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡆⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⠻⣿⣷⣮⣟⠋
⠀⠈⠻⢿⣿⣶⣶⣶⣶⣶⠾⠛⠁⠀⠀⠀⠀⠉⠙⠛⠉⠉⠛⠻⠿⠿⠿⠋⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢦⡀
⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⢰⣶⣾⣿⣿⣶⣶⡀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠷
⠀⠀⣠⣾⠛⠛⠿⠿⠿⢿⣻⣤⡄⠀⠀⢀⣸⣿⡿⠿⠿⠿⠛⠛⠛⢉⣀⡀⠀⢠⣿⣇⠀⠀⠀⠀⠀⠀⣾⣶⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣄
⠀⠐⣿⣿⣶⣤⣤⣴⡾⠿⠛⠋⠀⠀⠀⠘⠻⠿⠿⠟⠚⠛⠛⠛⠛⠉⠉⠉⠀⠀⠻⢿⣿⣿⣿⣿⣿⣿⠿⠏⠀⠙⠿⠿⠿⠿⠛⠛⠛⠻⠿⠿⠛⠁
⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡄⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣀⣀⣀⣀⣾⣿⣿⣿⣿⣦
⠀⠀⢀⣀⣸⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠐⢤⣠⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠉⠛⠶⣍⡻⢿⣿⣿⡿⠁
⠀⠠⣿⣿⣧⣄⣈⣉⣉⣉⣠⣿⣿⡇⠀⠀⠀⠀⠉⠙⠛⠻⠿⣿⣟⣄⣀⠀⠀⠻⣷⣬⣭⣭⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠈⠛⢶⣬⣉⡁
⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠶⠂
⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⣿⣿⠿⠻⡿⠛⡿⢿⡿⢻⡿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡈⠁⠀⠀⠀⣈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡏⣿⠋⠀⠛⠃⢼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣻⣿⠄⣠⣤⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣾⣩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⢿⣿⣿⣿⣿⣿⣿⠟⣿⠁⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡆⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠉⠀⢢⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠙⠂⠀⠀⣀⣀⣴⡶⠶⠶⠶⠿⠟⢋⣁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣶⣶⣶⢿⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠻⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠁⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⣴⣶⣦⡈⢳⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠉⠁⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⠆⠿⣿⣿⣿⣷⡸⣿⣿⣿⣿⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣦⠀⠠⣴⣦⣶⣿⣿⣿⡿⠛⠋⢁⡘⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⢿⣿⣿⣿⣿⣿⣷⡄⠄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⡿⠛⠁⠀⠀⠀⠂⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢿⣿⢟⠉⣰⡿⣿⠿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣉⠑⢊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⢸⠿⣬⣿⡆⠀⣴⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⣀⠀⡡⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⡀⠀⠈⢳⣭⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢁⡠⠄⠔⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠈⠻⣶⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠖⠐⠋⣡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢋⡈⠠⠞⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢁⡴⠀⠋⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠐⡡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢈⠀⡴⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢠⠄⠊⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⡰⠘⣁⠰⠏⢀⠖⢀⠄⡠⢀⡄⠀⢀⠀⡰⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠠⠛⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠰⠃⢀⡞⠀⡴⢡⡄⢡⠊⣠⠈⠀⠂⠃⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠞⠀⠁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠐⠋⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⡈⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢥⣨⡅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢃⣾⡟⣠⡴⠾⣛⣋⣉⡉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢡⣾⠏⣰⢽⣿⣿⣿⣿⣿⣿⣦⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣱⣿⠃⠀⣿⣼⣿⣿⣿⣿⣿⣿⣿⣷⣧⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⠀⠀⠸⡟⡛⢻⠟⠛⠻⠝⠿⠛⣏⠙⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠟⠀⠀⠀⠀⠀⢳⣈⠀⠁⠀⠀⠀⠰⢤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠋⠀⠀⠀⠀⠀⠀⠘⢹⢳⣄⠀⠀⠀⠐⢺⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⢾⣟⡁⠀⠀⠀⠀⠀⣀⣤⡆⠘⠃⠛⠳⣦⡤⣠⠔⢲⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣞⡛⢦⢃⡉⣸⠦⠒⠒⠒⣿⠀⠀⣷⡀⠀⠀⠀⢀⡴⠃⢀⡎⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣤⡌⢦⣼⣿⠇⠀⠀⠈⠀⠸⡀⠀⠻⣿⣷⣶⡾⠋⠀⣠⣋⠔⠁⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢀⣶⡃⢌⠛⣾⠟⠉⠀⠀⠀⠀⠀⠀⠑⢄⠀⢹⡏⠀⠀⣠⢴⡿⠃⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⢀⣾⠿⠉⠂⠀⠃⣀⣠⣀⠀⢠⠀⠀⠀⠀⠈⣧⠘⠀⣠⠞⢡⠞⠁⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠸⣿⣶⣤⠔⠁⣷⣿⣽⣿⣰⠏⠀⠀⠀⠀⠀⣿⠀⢰⢇⣴⢯⣉⠙⠶⣄⠀⠀⠀⠠⡾⠁
⠀⡄⠀⠀⠀⠈⠹⣯⠀⠀⠁⠈⢹⠿⣿⢀⠀⠀⠀⠀⢀⡇⠀⣿⡾⠁⠀⠀⠑⠠⡈⢷⡀⢠⡞
⠀⠁⠀⠀⠀⠀⠀⣸⢻⣷⣄⢀⣾⣶⡆⡀⠀⠀⠀⠀⠈⠁⠈⠉⠓⠄⠀⠀⠀⢀⣱⡄⣿⣿⠁
⠀⠀⡀⠀⢀⣀⡼⠛⢿⣿⣸⠟⠛⢿⣿⣷⣷⣶⣠⡀⣤⢀⡄⠀⠀⠀⡀⢀⣴⣿⣿⣿⣿⠇
⠀⠀⠀⠁⠉⠀⠀⠀⠀⠉⠁⠀⠀⠈⡏⠙⠻⢿⣿⣿⣿⣿⣿⣤⣷⣾⣷⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢸⠋⠙⣿⣿⡿⣿⠻⠿⠛⠛⢩⠏⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠒⣀⡈⠁⠀⠀⠉⡒⣀⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⣂⡡⠄⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢍⠒⠩⢐⡢⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢐⡬⠔⠂⠉⠀⠀⠀⣀⣀⠤⢴⣖⣒⣖⣖⣒⣲⣶⢤⢄⣈⠈⠀⠢⣬⠑⠺⢷⣦⣄
⠀⠀⠀⠀⠀⠀⡀⣐⠔⠊⠁⠀⠀⣀⡠⡔⣖⡽⢜⣚⣮⣽⣶⣶⣾⣿⣷⣶⣯⣭⣙⣻⠿⣳⣦⣀⠈⠒⠀⠹⣟⢧
⠀⠀⠀⠀⡠⡪⠊⠁⠀⠀⡠⡴⣛⡬⣚⣭⣷⣿⣿⣿⠿⢿⢿⢛⣉⣉⣉⣿⣛⡛⠛⠿⢯⣶⣭⡛⣗⠄⠉⢄⠘⣿⢧
⠀⠀⣠⡮⠊⠀⠁⢀⡔⢝⠽⣪⡵⣞⣿⢿⢿⡟⣉⣀⡤⠾⠛⢋⣩⠴⠊⢓⢤⣉⣽⣖⢾⡉⠻⡮⣺⡼⠀⢰⠀⣟⣿⡦
⠀⠴⠋⡀⠁⠀⡰⣿⢞⣵⣿⣿⣾⠏⠋⣁⣴⠟⠋⢁⡠⠔⠊⠁⠀⠀⠰⣅⠀⠫⣑⡿⣇⠙⣆⢘⡿⠹⠀⢬⢀⡯⢺⡏⣃
⢈⠃⢨⠅⠀⣼⡞⣱⣿⣻⠿⠛⢼⣾⠾⠉⣁⠤⠒⠉⠲⢄⡀⣀⣀⠀⠀⠑⠭⠶⢄⡴⠊⠑⣾⡟⡵⠁⢐⠃⣼⡪⣼⡯⢔
⡇⡀⠂⠀⠀⣿⡧⣿⣿⢎⡠⣤⢞⡠⠔⠊⠀⡀⠀⠀⠀⠀⢸⠳⢄⡙⡆⠀⠀⠤⠤⠒⡠⢞⠕⠁⠀⠔⢁⣾⡯⡾⣿⡛⠕
⢧⣣⠐⠠⠀⠹⣟⢿⣿⠁⢟⣿⠙⠦⡠⠔⠉⠉⠲⣦⢄⡀⠀⠑⡒⡛⠣⣈⣀⢤⡲⠝⠊⠁⠀⠀⣂⡴⣑⢵⡞⣏⡯⣒⠥
⠘⣶⣳⣄⠑⠄⡀⠉⠒⠭⠞⣻⣦⣄⣈⣲⣮⣢⣄⣀⣑⣷⣀⣤⣄⡲⠾⠛⠋⠁⠀⠀⠀⠀⣠⡾⡑⢌⣶⡇⡇⡿⡭⢕⡪
⠀⠱⡱⢿⡝⠶⢤⣁⠐⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⡴⠻⡳⣜⣾⣿⣿⣷⣿⣿⣲⢵⡺
⠀⠀⣷⣎⡉⡾⣮⣌⢋⢳⠲⢤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣴⣖⢋⢃⢡⣙⣹⣛⣛⣛⣹⣿⣿⣿⣽⣺⢽⡿
⠀⠀⠨⡻⣷⣰⢃⠋⡏⠗⡦⣧⣡⣘⡍⡇⡏⡋⢹⢉⢹⢉⢋⢋⢱⣘⣈⣬⡼⡞⡟⠹⢹⣏⣘⢼⣿⡿⢟⠭⡒⢍⡪⠅
⠀⠀⠱⣵⣌⢿⢿⣜⣰⢸⢡⢰⢡⢉⢋⠋⠟⡙⢻⢸⢻⠛⢫⠏⡏⡟⡘⡰⢀⠀⠁⣤⡄⣿⢕⠽⣭⡊⢕⡩⡪⠗
⠀⠀⠀⠙⢿⡻⣷⣿⣝⢧⣎⡄⡎⡌⡼⠘⣸⣧⣮⡌⡘⠸⠸⢸⢰⢡⠇⢷⣨⣸⡴⢾⡻⡣⢕⠭⡪⣝⢷⠯⠂
⠀⠀⠀⠀⠀⠙⠜⢿⣟⢗⢕⠫⡫⡻⢗⠧⡿⡿⢮⣻⣵⡧⢦⢷⠾⡺⣛⢝⣭⡪⡪⢕⢌⠪⣓⢕⠬⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⠑⠌⠪⠕⢕⢬⡺⣦⡙⠿⢿⢕⢵⡪⣛⢾⠵⢍⠪⣪⠑⠑⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠿⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠀⢀⣶⠀⢠⣦⠀⢀⡈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣿⠁⢠⣾⠃⢰⡿⠃⢠⣿⠃⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣧⣤⣿⠁⢠⣿⠃⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠏⣠⣶⣶⣦⡙⣿⣿⣿⣿⣿⣿⣿⣷⣿⠃⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢿⣿⣿⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣷⣌⣙⣛⣁⣴⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠿⠿⣿⣿⡿⠿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠓⠶⠦⢤⣤⣤⣄⣀⣀⣠⣤⡤⠤⠶⠚⠃⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⡟⠶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠛⡷⠦⢤⣄⣀⡀
⠀⠀⠀⠀⢀⣠⠶⠛⠉⠁⠀⠀⢸⡇⠀⠀⠉⠙⠓⠲⠦⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠴⠶⠒⠛⠉⠀⠀⠀⠀⣿⠀⠀⠀⠈⠉⠛⠶⣄
⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠻⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⠴⠖⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠶⠦⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠶⠖⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⠀⣾⣿⣦⠈⠢⣀⣠⡴⣶⡟⠲⣄
⠀⠀⠀⠀⣸⣿⣿⣿⣦⡀⠈⠻⢿⣻⣿⣅⠈⠢⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣳⡀⠀⠀⠙⢿⣿⣆⠀⠘⢦⡀
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠙⢿⡆⠀⠀⠙⣄
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⢔⠠⡀⠈⠳⣄
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⢻⡆⡀⠀⠀⠀⠈⠳⡙⢦⡀⠈⢢⡀
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⡌⢢⡀⠀⠀⠀⠈⠢⡑⠄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣿⣿⣿⣿⣿⣄⠹⣄⠀⠀⠀⠀⠈⠢⣄⣀⣀⣉⣲⣤⣤⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠈⠻⣿⣿⣿⣿⣿⣦⡘⢦⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⣵⣀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠿⠃⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣻⠿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠿⠿⠷⠶⠿⠛⠛⠛⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠒⠋⠉⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠶⠒⠚⠋⠉⠉⠉⠛⢢⡀⠀⣀⣠⢤⣶⠶⠞⠛⠛⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠛⣉⠥⠒⡙⠀⡰⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡤⠚⠁⠀⠀⠀⠀⠀⢀⣀⡤⠴⠒⠋⠉⣀⣠⣴⣾⠟⠁⠀⡰⠁⡐⠁⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡰⠋⠀⠀⠀⢀⣠⠴⠒⠋⢉⣀⣠⣤⣴⣶⣿⣿⠿⠋⠁⢀⡠⠊⣠⠊⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⣠⠖⣋⣠⣴⣶⣿⣿⣿⣿⣿⣿⠿⠋⠁⢀⠠⠐⠁⡠⠊⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢶⣺⣿⣶⣿⣿⣿⣿⣿⣿⠿⠟⠛⠉⣀⠠⠒⢈⣀⠤⠒⠁⠀⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⢀⣠⢖⡭⢖⠉⠀⠉⠉⠉⡉⣉⠡⠤⠐⠒⠂⠈⣁⠠⠐⠊⠁⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⣠⣴⡯⠚⢍⡀⠀⠈⢉⣉⡉⠀⠀⠤⠤⠀⠀⠐⠂⠁⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠾⣋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁
⠀⠀⠀⠀⠈⠓⠦⠤⠤⣤⣤⣄⣠⣤⡤⠤⠶⠒⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠒⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⣧⣄⣀⣀⣤⠞⠁⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⢀⣠⣤⣤⣤⣄⡀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣠⠞⠋⠀⠀⠀⠀⠈⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⣀⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠶⠶⠦⠤⠤⣤⣤⣤⣤⣀⡀⠀⠀⢀⡾⠀⣠⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⣖⡛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣟⢻⣧⡼⢿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⡞⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⣼⡉⠘⠉⣧⡀⠈⠙⠻⣿⣿⠟
⠀⠀⠀⠀⠀⠀⡴⢺⢱⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠈⢿⣦⣿⠃⠀⢠⣟⣿⣶⣤⡀⠈⣯
⠀⠀⠀⠀⠀⢰⠇⣾⢸⣿⠸⡆⠈⠒⠦⣤⣀⡀⠀⠀⠐⣿⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⢀⣤⣽⣏⣿⣿⡟⠳⠿
⠀⠀⠀⠀⠀⡿⢠⡿⠘⣿⣧⢳⡀⠀⢀⣄⣉⣙⣻⣿⣿⣿⣿⣦⠀⠐⣦⡀⠀⠀⠀⠀⠀⠀⢻⣧⢸⣿⣿⡿⠟⠋
⠀⠀⠀⠀⢸⣧⡿⠁⣀⢨⡻⣧⡹⣄⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣷⡄⠈⠷⣄⡙⠲⢦⡀⠐⡌⠿⠛⢛⡋
⠀⠀⠀⠀⢸⣿⠃⠀⢿⣜⢿⣿⣿⣌⣳⣤⣀⡀⠀⠀⠉⠛⠿⣿⣿⣿⣶⣜⣛⣛⣛⣫⣭⣿⣿⣅⠶⠟
⠀⠀⠀⠀⣾⡏⠀⣷⠈⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣯⣭⣤⣶⣶⣶⣶⣶⠶⣄⠀⠀⠀⠀⡴⠲⢶⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⣿⠁⠀⠸⣇⠀⢻⣿⣿⣿⣿⣿⣵⣾⣿⣿⣿⣷⣶⣶⣶⣾⣿⣷⡍⠻⣿⣿⣿⣿⣿⣿⣆⠙⣦⠀⢀⣼⣧⢀⡞⠐⣣⠋⣤⣶⠟
⠀⠀⠀⠀⣿⠀⠀⠀⢻⣦⠘⣿⣿⣿⡿⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠈⠙⢿⣿⣿⣿⣿⣧⡘⣷⣿⣿⠃⢺⣣⣴⣇⣾⡿⠋
⠀⠀⠀⠀⣿⠀⠀⢰⣬⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠈⠙⠿⣿⣿⡿⠟⠛⠻⣿⡆⠀⠀⠀⠙⢿⣿⣿⣿⣷⣌⣙⡛⠿⣿⣿⣿⡿⠋
⠀⠀⠀⢀⡏⠀⢻⣾⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⡶⠀⢀⣴⣿⣇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧⣄⣈⡙⠶⣄
⠀⠀⠀⢸⡃⠀⢠⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠙⢿⣿⢿⡄⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣷⡾⢠⣄⠐⠢⢄⡀
⠀⠀⠀⢸⣇⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣈⣡⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠟⠁⣾⣿⣷⣦⡀⠉⠲⢄⣀
⠀⢀⣀⣼⣿⣷⣦⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣀⣉⣉⣙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣤⣬⣍
⣰⡏⣁⣸⠇⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠿⠻⢿⡝
⢹⡿⣿⣶⣾⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⣿⡇
⠀⡇⣹⡿⣿⣋⣽⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣶⢹⡇
⠀⡇⢻⣿⣸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣼⡇
⠀⢿⣘⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠸⣿⣇
⠀⠀⠙⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡼⠛⣿⣿
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣀⣈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣡⣤⣤⣬⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡽⣿⣻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡛⢿⡯⠽⠉⠒⠂⠤⠤⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠁⠀⠀⠖⣙⣯⣍⣂⠀⠀⠀⠀⠀⠀⠉⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠘⢁⣵⠴⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⡷⡞⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡿⢟⠉⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⢠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⡿⢯⣁⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡿⠣⣤⡴⠖⢀⡀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡿⠑⣴⣞⠁⣰⠀⣀⠀⠀⠀⠀⣱⡀
⠀⠀⠀⢠⠇⠀⡀⣤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⠀⢹⡇⢴⡟⣠⡇⣹⢸⣹⡟
⠀⠀⠀⢼⣨⣸⣄⣧⣸⡇⢠⠇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠸⠇⢸⠄⢻⢹⠁⡟
⠀⠀⠀⢼⢫⢿⢸⡅⡾⢉⡿⠧⣼⡇⠀⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⢰
⠀⠀⠀⠈⢇⠀⠀⠀⠁⠘⠃⠼⠋⢉⡽⠧⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠉⠀⣤⠾⠿⣄⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠞⠻⣄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠋⢓⣤⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣍⣉⣦⡼⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠤⢷⠞⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢵⣚⣿⡥⠆⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⠤⢤⣄⣸⢶⣿⣾⠔⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⢛⢿⢛⠟⣿⢟⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠀⠀⠀⠀⠀⠀⠐⠀⣰⣤⡀⠀⢰⡀⢀⡀⠀⠀⢠⡄⠀⠀⠨⠀⠀⠘⠅⠀⢶⡶⣤⢠⣭⣤⠂⠀⠈⣼⣷⣧⠘⠁⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠰⠿
⢸⣆⠀⠙⣿⡆⢰⣍⠻⣭⣿⣅⠀⣿⣿⡟⣷⡀⠀⠀⠚⠆⠀⠴⡄⠰⠀⠰⢀⣸⣿⣻⢛⡿⠓⠀⠰⡄⣼⣿⢾⡤⠄⣻⣇⣗⢰⣷⠆⠂⢀⣀⣀⡀⠄
⢸⣿⠀⠈⢹⡇⠈⣯⠈⠻⠵⢻⣣⣯⣏⠐⣊⡀⠛⠆⢄⠄⠀⠈⠃⠛⠎⠀⠴⠸⢍⡭⣼⠀⠀⣌⠟⠈⠁⢴⠞⠀⠀⠛⣩⢧⡼⣿⢻⣿⡟⢈⣦⠈⠃
⠘⢋⠀⠀⣿⠧⠰⡧⠀⠀⣾⣶⣶⣿⣦⠺⣿⣧⢈⡠⠄⠠⠀⠀⠀⠀⠀⠀⢈⠁⣀⢀⢙⣤⣩⠅⠀⠜⢒⢴⣦⣒⠀⣠⠀⠀⠀⢄⠙⠋⢀⠄⠅
⢀⣾⡁⠀⢙⠀⠀⠀⠗⠶⢸⣿⡿⢋⢚⠂⢠⠙⣠⡦⠀⡒⠀⠀⠀⠀⠘⠀⡆⠀⣉⡐⢈⡷⠿⠂⠢⣖⠝⢀⠺⡿⠾⠀⠀⠰⠀⠘⠋⢛⡿⠶⠀⠀⢰
⠀⠧⠸⠦⠤⠏⣴⠓⠶⠒⢸⡋⠴⣿⠉⠃⢠⠠⡏⠉⠄⠀⠀⢯⣉⠃⢨⠉⣤⣰⠯⢥⢐⠀⣀⠐⠂⠀⣴⡌⠀⠘⠀⠀⡴⣠⠧⠀⠀⠈⢠⣆⠀⢠⠛
⠀⣶⣶⠸⠀⠄⣰⡟⠂⠊⢹⡿⠻⡇⠀⠓⠶⠘⠃⣠⣴⣥⣴⡞⡛⠀⢶⡞⠀⠹⠇⠽⠅⣀⠈⠀⠄⠀⠐⠿⠀⠀⠀⠌⠄⠟⠓⠆⠀⢋⠫⠁⠼⢇⢀
⠀⠙⠉⠀⢀⣾⣧⠀⡐⠂⠸⠇⠐⠁⠀⠀⠀⠀⠐⠏⠀⠨⠿⠀⠀⠀⢠⠶⢰⡇⠀⠀⡄⠟⢀⡀⠀⣨⡵⣬⢡⡀⠀⠂⠰⠄⠀⠀⠶⢾⢠⠀⠀⣨⢨
⠀⠀⠀⡀⠀⠠⢀⠀⠀⠁⢀⠀⣤⠶⢾⣼⠆⠀⠠⣶⡖⠀⠀⠀⢠⣶⣾⡶⠀⠁⠀⢤⠤⠀⠘⠂⠀⠟⠱⢤⠉⠄⠀⠀⠱⣄⠈⠛⠀⣈⡀
⠀⠀⠘⡇⠀⠀⠉⠏⠀⠁⠀⠘⠏⠀⡀⠇⠀⠀⠀⠈⠡⠄⠀⠀⣼⣿⣿⡃⠀⠀⠀⠀⣖⢀⣴⣓⡀⢠⡶⠒⣀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠉
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠘⠋⠀⠀⠀⠀⠐⠀⠀⠀⠀⢠⣾⠟⢹⣿⡆⠀⠀⠀⢠⣬⣿⣀⠐⠄⣝⠃⠙⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⡛⠁⢀
⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡀⠂⠀⠀⠀⣾⡏⠀⢸⣿⡇⠀⠀⣴⡄⠐⡼⢅⢰⣰⣿⡇⠀⠸⢷⡄⠀⠀⠀⠀⠀⠀⠰⠆⠀⠈⠀⠂
⠈⠃⠀⢠⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠁⣄⢿⡛⢿⣇⠀⠉⠀⠀⠄⠘⠀⠀⢠⣷⣤⣄⣀⣿⣦⠀⠀⠀⠀⠀⠂⢀⣢⣄
⠀⠀⠀⠺⠀⠀⠀⠀⠸⣿⠋⢰⡾⠶⠶⡆⠀⠀⠀⠂⠂⣞⣤⣴⣿⣷⣿⣾⣿⡇⢠⢠⣤⣖⡖⠀⠀⢸⣿⣿⣿⣷⣝⣿⠂⠀⠈⠀⠀⠀⠀⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣻⡇⠘⡦⠗⠛⠳⠄⠀⠸⣿⣿⣿⣿⣿⣮⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠈⠿⠁⠙⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶
⣄⣄⣠⣄⣤⣄⣄⣀⣐⣙⣀⣀⣀⣠⣤⣠⣀⣙⣥⣀⣉⣠⣄⣀⣄⣀⣄⣀⣠⣀⣠⣠⣀⣀⣀⣠⣠⣤⣠⣄⣀⣄⣠⣄⣠⣄⣠⣀⣀⣤⣠⣄⣀⣠⣠
⣿⣻⣛⣟⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣽⣿⣿⣯⣯⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠱⡈
⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢈⣧⠘⠄⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⣤⣄⡀
⢠⣾⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣉⣿⣿⣿⣿⣿⣷⣶⣤⣀⠲⣶⣶⣿⣿⣿⠖⠒⠦⠂⠈⡉⠉⠁⠘⠁⠀⣸⠟⢿⣿⣶⣤⣀
⣿⣿⣿⣿⣿⠿⢿⣷⣼⣿⣿⣉⣿⣿⣿⣿⣟⣙⣿⣧⣼⣿⣿⣿⣏⣩⣿⡄⠙⠛⡆⣘⣆⠀⡈⣷⣼⣴⠃⢀⣴⣯⡀⠀⠈⠻⣿⣿⣿⣶⣤
⠘⣿⣿⣿⣿⣦⣾⣿⡏⢻⣿⣿⣿⣯⣤⣿⣿⣿⡿⢿⣿⣧⣼⡿⠉⠙⠛⠓⢤⡾⠇⣿⣿⠀⢻⣽⣿⣷⡴⠛⠛⠻⣿⣦⣄⡀⠀⠙⠻⠿⠿
⠀⠈⢏⣻⣿⣿⠟⢿⣿⣿⣿⣉⣿⣿⣿⣿⠉⣿⣷⣼⣿⡿⠿⣷⡐⠚⢷⣿⣶⣶⣤⣿⣿⣆⣾⣿⣿⣿⣤⣤⣤⡄⠈⠻⢿⣿⣷⣶⣤⣤⣤
⠀⠀⠀⠻⣿⣿⣷⣾⣿⣿⣿⣿⣿⣇⣹⣿⣿⣿⠛⣻⣿⣷⣾⣿⣷⡀⠉⢛⣿⣿⣿⣿⣿⣽⡿⠿⣿⣿⣿⠛⢻⣿⣦⣀⠀⠈⠙⠛⠻⠿⠟
⠀⠀⠀⠀⠑⣬⣿⡿⠛⣿⣿⣉⣿⣿⣿⣅⣹⣿⣿⣿⣿⣉⣿⣿⡏⠙⠐⣯⣙⡁⠉⠉⠉⢀⣠⣶⣋⣉⣀⣀⡀⠙⢿⣿⣿⣶⣶⣦⣤⣤⣤
⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⣿⣿⣿⣅⣹⣿⣿⣷⣤⣿⣿⣿⣿⣿⣧⠌⠀⣭⣽⣿⣷⡶⢾⣿⣿⠿⠛⠛⠻⢿⣿⣦⣀⠈⠉⠛⠛⠛⠻⢿⡏
⠀⠀⠀⠀⠀⠀⠈⢿⠿⠟⠋⢩⣹⡿⣿⣧⣼⡿⠻⣿⣧⣼⣿⠛⠉⠀⢤⣙⣿⣿⡿⠾⠟⠋⠁⢰⣤⣀⡀⠀⠻⢿⣿⣿⣶⣶⣶⣶⣶⡜
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣿⣦⣼⣿⣿⣷⣾⣿⡉⣻⠏⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⠶⠊⠠⣿⣿⣿⣧⡀⠀⠈⠉⠛⠛⠛⢉⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⣶⣿⣿⡄⢀⣀⣀⣀⣀⡈⠙⠃⠀⠀⠀⢐⣲⣾⡿⢛⣿⣿⣷⡤⠀⠀⠀⠀⠹⣿⣿⣶⣦⣤⣤⣤⣶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⠁⠸⣿⡿⠿⠿⢿⡿⠃⠀⢀⣨⣿⠟⠋⣹⡶⠟⠛⢻⣿⡏⠃⠀⠸⣿⣦⡈⠛⠿⠿⠿⠿⠿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⣤⣶⣾⣷⣀⣀⠰⠀⣤⣿⣿⣿⣿⣿⣾⡿⡧⠤⠤⠤⠤⢿⣷⡀⢀⠐⠉⠻⢿⣷⣦⣤⣄⣤⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠟⠛⠻⠃⢸⣼⣿⣿⣿⣅⡈⠉⢁⣠⠀⠀⠀⠀⠀⠀⢿⣿⣏⠀⣦⣤⠤⠬⠍⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⣀⣠⣀⣼⣿⣿⣅⡀⠙⠻⠿⣿⡿⢿⠀⠀⠀⠀⠀⠀⠈⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠻⢿⣿⣶⣤⣀⣀⣤⣼⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⣿⣿⣿⠿⠛⠻⠁⠀⠉⠉⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠉⠙
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣼⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢟⡿⠋⠁⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡃⠈⠁⠀⠀⠀⠀⠀⠸⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⢓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣶⡇⠰⣶⣿⠿⠇⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠁⢻⡇⠀⠀⠃⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀⢀⡼⢧⣀⠰⢤⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡷⠚⠀⠤⠥⠀⠀⠀⢸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠘⠛⠛⠁⠀⡰⠋⠘⣿⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣌⠻⣦⣤⣤⣤⠄⠁⠀⠀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⣿⣿⣿⣿⣧⡀⠉⠛⠀⠀⢴⣷⡀⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠘⢿⣷⣤⣿⣿⣷⣻⢿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠉⠙⠻⢿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠷⠎⢻⣿⣦⡀⠀⠀⠀⠀⠐⢳⡍⢿⣿⣷
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣿⣿⢏⡀⣹⣿⣷⣄⠀⠀⠀⠈⣾⣿⡎⢻⣿⣇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠛⢛⣟⢱⣿⣿⣿⣿⣦⠀⠀⠀⣿⣿⣇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠚⠋⢉⣘⠻⣿⣿⣿⣿⣧⠀⠀⣿⣿⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣹⡇⠘⠉⣉⣉⣌⣿⣿⣿⣿⣿⣧⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢻⢡⡿⠀⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣇⠈⢈⣴⣶⣶⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠐⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡿⢻⣿⡇⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣄⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣸⣿⣿⣿⣷⣶⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⠿⡿⣿⠿⣿⠀⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡈⢿⣿⣿⣿⣿⡌⠛⠈⠂⠁⠸⠀⠈⠀⡏⣿⠙⢿⣿⠃⢀⡀⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⢷⡈⢿⣿⣿⡟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⠀⠛⠀⠘⢧⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡞⡎⢧⠘⣿⡟⢧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⣌⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡿⣿⡄⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠻⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⡏⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢒⠖⠉⡩⠝⠂⠠⠴⣊⠁⠀⠀⣀⡰⠃⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠂⠀⣘⡁⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣠⠊⠉⢁⣀⡀⠁⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠈⠒⡌⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠞⠁⠉⠉⠀⢀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⢀⠴⠂⠉⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠁⠴⠈⠋⠀⢀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⡴⠛⠂⡀⠀⠀⠀⠀⠈⠱⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢌⡁⡀⠤⠤⠤⠤⠣⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⡠⢊⠌⢠⡤⠐⠚⡉⠁⠉⠁⠒⠀⠂⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⠄⠘⠀⠁⠀⠰⠠⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⡠⠊⠀⠊⠀⠈⠛⠀⠀⠀⠀⠤⠾⠂⣈⡉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⢤⠒⠈⠉⠉⢉⠉⠊⠀⠀⠀⠀⠀⠀⢀⠞⢁⠔⡠⠂⠀⠀⠀⠀⠘⠖⠀⠀⠠⠀⠀⠤⡌⠒⠚⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠗⢀⣀⢤⠤⠄⠀⣸⠇⠀⠀⠀⠀⠀⠀⣰⠋⠠⣢⠞⠀⠀⠀⠀⠀⠀⠀⠰⡧⠤⠀⠠⠤⠄⠀⠑⣯⡵⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠤⣈⠀⠀⢠⡀⠀⠀⠀⠀⠀⢀⠞⠁⣀⡵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⣈⣁⠀⠘⠁⠠⢴⡆⡄
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢨⠈⠙⠁⠁⠀⠀⠀⣀⡔⠁⠀⣨⠏⠀⠙⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠁⠒⢲⣈⣉⣉⣠⡀
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠑⠢⢌⣆⠀⠀⠀⢀⡤⠚⠁⠀⣀⢼⠏⠀⠀⢳⣄⠀⠉⠢⣀⠀⠀⠀⠀⠀⠋⠃⠨⢟⣭⣿⠯⠥⠭⠭⠤⢬⡽⣦
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣦⠴⠚⠉⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠙⠍⡓⠢⢌⠣⣀⠀⠀⠀⠀⢹⠈⢉⣉⣀⣤⣤⣤⣤⣤⣤⣽⣷⣷⣤⣀⡀
⠀⠀⢧⣀⠀⠀⠀⠀⢀⣀⣀⡤⠴⠚⠉⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠁⣽⠣⣄⡀⠀⠀⣠⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠋⠈⠢⡀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠉⠑⠚⠒⠤⠤⠄⠐⠒⠒⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠡⢄⣀⣀⣀⣀⣀⣠⡤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠒⠂⠀⠀⠀⠐⠒⠒⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠛⠋⠉⠉⠉⠉⠛⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠀⠀⠀⠀⠀⣀⡠⠤⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠶⣄⡤⠤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣀⡤⢴⣺⠃⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⣠⢖⠟⠋⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⡏⠓⠢⢤⣀
⠀⢀⠞⠁⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡷⠖⠛⠛⠛⠛⣲⠤⣄
⢀⠏⠄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠚⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱
⡼⠘⠀⠀⠀⠀⠀⠀⠀⣸⢤⣀⣀⡀⠀⠀⠀⠀⢀⡠⢞⠝⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⡇⡀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠬⠭⠭⠭⢭⡭⢋⠔⠁⠀⠀⠀⠀⢀⣀⡤⠴⠒⠚⠛⠋⠉⠉⠉⠉⠉
⣧⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡪⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢔⠝⠊⠀⣀⡤⠖⠊⠉
⠀⠙⢆⡀⠀⠀⠀⠀⠀⢀⠤⢚⣓⡥⠴⠒⠉⠁