Technician Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁⠀⠀⡌⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⣀⣸⣀⣠⣁⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠤⠤⠀⠠⠤⢤⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠙⠢⢄⣠⢴⠓⢒⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⡄⢶⠙⡶⢄⠀⢈⢽⢻⠉⡏⢹⡐⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠈⢺⡀⠘⡄⠙⢇⢸⡋⡇⢸⠀⢇⠀⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⣇⠀⢱⠀⠈⢆⠱⣡⠀⡆⠘⡄⢰⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⠀⢸⡀⠀⠈⢣⢹⠀⢡⠀⢃⢸⡴⡜⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⢸⡄⠀⡇⠀⠀⠀⢻⠀⢸⠀⢸⠀⣇⠣⠼⣫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⢸⡇⠀⡇⠀⠀⠀⠀⠀⢸⣆⡸⠤⠾⢉⠏⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢸⡇⠀⡇⢀⣀⠤⠴⠶⡉⠀⢀⣠⠇⡜⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡶⠤⣀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠧⠤⠼⠥⠤⢀⣀⡠⠓⠺⡁⠈⠉⠙⣦⠞⠀⠀⠀⠀⠀⣀⠤⢶
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠉⠒⠤⢄⡀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⢠⡤⠚⠀⠀⢀⡠⠤⠒⠉⠀⢀⠇
⠀⠀⠀⠀⠀⠀⣀⠅⠀⠀⣀⣀⣀⣨⣑⣦⡄⢣⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⡴⠒⠊⠁⡇⡸⡇⢠⣴⣊⣁⣀⣀⡀⠀⠀⠨⣀
⠀⠀⠀⠴⣊⠉⠀⠀⠀⣠⠃⠈⠙⠻⠿⣿⡁⠀⠣⠤⠔⠒⠂⠉⠉⠀⠀⠀⡇⠀⠀⣜⢠⠃⡇⢈⣿⡿⠟⠋⠁⠘⡆⠀⠀⠀⠉⣑⠦
⠀⠀⠀⠀⠀⠉⠒⠢⢤⠃⠀⠀⠀⠀⠀⠀⠈⠁⠛⠂⠤⠤⠤⠄⣀⣀⣀⣀⠥⠤⠤⠬⠛⠒⠉⠉⠀⠀⠀⠀⠀⠀⠘⣤⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠄⠒⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⢸⣠⣏⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡕⠈⢀⠠⣿⡄⠀⠐⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠔⢊⠉⢻⢦⡴⢀⡔⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣄⡸⡅⢸⠾⠤⣼⢁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠘⣄⠀⢸⡟⠃⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣝⡳⠇⠀⠀⠙⠊⢈⡁⠛⠐⠂⠠⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⠀⠐⡆⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢳⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠙⠢⣌⣐⣐⣂⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡇⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡿⢱⠖⠀⠐⠤⡀⠀⠈⡿⢿⡀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢀⢀⢀⢤⠞⠋⠁⠰⡇⠀⠀⠀⠆
⠡⠼⠓⡆⢲⠁⠣⠀⠀⣹⢷⣝⠶⣄⡀⠀⠀⠀⠈⠓⠀⠀⡼⠀⠀⢀⡤⠒⡈⢷⡻⠊⠀⠀⠀⠀⣼⡇
⠀⠀⣜⣱⠉⠳⡀⠀⣠⡿⡀⡭⠻⣮⣹⢗⠦⣤⣀⣀⠀⢰⣇⠀⢼⡁⠒⠃⡠⠘⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠠⠂
⠀⠀⢳⠀⡝⢀⡿⠉⢁⢠⣿⠀⠤⡤⢝⢷⣄⣊⣴⣏⣵⣯⣿⣧⣀⡍⢠⠂⢽⠠⡆⠀⠀⠀⠀⠀⠂⠭⢡⠀⠈⠠⢤⠀⠀⣀⡀
⠀⠀⠸⠉⠏⠻⠧⠴⢾⡏⢆⡅⣐⡷⣼⣬⡿⣯⣴⡟⢮⣻⣯⣷⣄⢘⡼⠆⠊⠋⡱⠀⠀⠀⡠⠃⠂⠠⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠉⢑⠱⣷⣓⣏⢿⠋⢻⠛⣷⢿⣿⣿⣿⠉⠣⠄⠲⠾⠀⠀⢀⣸⣥⣉⣉⠙⠛⠒⠀⠀⠀⡀⠠⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣒⠐⠡⢮⣉⠍⢘⠳⣯⠖⠿⣟⣿⢞⢿⠀⠀⠀⠀⠛⢰⠖⠂⠤⠄⣈⣉⡉⠛⠓⠖⢀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡻⠶⠶⠶⣶⣤⣥⣴⣤⣤⣿⢾⡏⠀⠀⠀⠀⠄⠸⠄⠀⠀⠀⣀⡇⠀⠈⠙⠙⠸⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠚⣷⡶⣭⣭⣯⣤⣤⣼⠧⣭⣒⣛⡓⢢⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠹⠷⠶⠲⠆⠐⠶⣤⣀⡀
⠀⠀⠀⠀⠀⢠⡴⠋⠁⠀⠀⠀⠀⠀⠠⠤⡀⣀⡀⠘⢣⡄
⠀⠀⠀⠀⠀⢸⠀⠀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠄⠠⠐⣀⢳
⠀⠀⠀⠀⠀⢸⡀⢊⣤⣶⠚⢻⡟⠓⡶⠲⢶⠖⢛⠛⢻⡇
⠀⠀⠀⠀⢀⡼⠿⠉⠀⠙⢦⣤⣤⠾⠁⠀⠈⠲⣤⡤⢻⡇
⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⣀⠀⠉⠛⠋⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠈⠘⣧⠀⠀⠀⠈⣦⣤⣤⣠⣤⣤⡶⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠈⠛⠀⠀⠀⠉⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠢⣀⠄⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠦⣾⠀⠀⠀⠀⡋⠙⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⢇⡀⢀⣈⣦⣀⡀⢩⠶⠾⠀⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠤⠶⠶⠻
⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⠁⠈⠙⣧⣼⠁⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⢠⠾⠛⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢀⣀⡄⠀⣣⠀⠀⠀⠀⠀⠀⠀⢼⠘
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⢰⡆⠀⢻⠀⡗⠘⢇⠀⠀⠀⠀⠀⠀⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⡄⢀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠀⠘⠀⡇⠀⢸⣀⣀⡠⠴⠶⠞⢻⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⡇⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⠠⠟⣰⠟⠑⠒⠶⠗⠂⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡏⢰⠀⠀⠀⠈⠀⠀⠀⠁⠈⠁⠀⠠⠈⠛⠢⠤⣤⣀⡉⠁⠺⠛⠀⢀⡴⠛⠏⠛⠋⠛⣶⠟
⠀⠀⠀⠀⠀⡆⠀⠀⢸⠁⠘⡄⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠰⠶⠶⠤⠶⠦⠼⠿⠟⠛⠃⠘⠛⠴⠶⠶⠖⠛⠋
⠀⠀⠀⠀⠀⡇⠀⠀⡘⠀⠀⡇⠀⠀⠀⠀⠆⠀⢠
⠀⠀⠀⠀⢀⠀⠀⠀⡇⠀⠀⣇⡀⠀⠀⠀⠀⠘⠋⠀⠀⠀⣀
⠀⠀⠀⠀⠸⠀⠀⠀⢷⠀⢀⠋⠉⠛⠛⠒⠂⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⢸⠇⠀⠠⠦⢀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⢠⡶⠒⢾⣄⠠⠀⠒⠈⢷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⣸⡁⠐⣄⠉⠳⠤⠝⠠⠘⠻⣄⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠻
⡇⠀⢸⢌⡓⠠⡀⠀⢀⡀⠀⠀⢳⡄⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆
⠛⢦⣜⠂⠁⠠⡀⢁⠀⠀⠀⢀⣸⡇⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢇
⠀⠀⠉⠳⣄⠀⠀⢻⠀⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠛⢦⣤⣠⠤⣤⡤⠼⠇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠉⠙⠛⠓⠲⠖⠻⠦⠤⠤⠤⠴⠶⠿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⢀⡠⢿⠴⠤⣤⣤⡄⠀⣀⣀⣀⡠⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠦⣀⣀⣀⣀⣠⠔⠃
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠷⣉⡷⠀⠀⠀⠀⠀⠀⠘⢿⣯⣙⠛⠛⠒⠶⠦⣤⣀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣡⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣿⢛⠲⠶⠶⠤⠽⢷⣄⢻
⣰⠖⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠉⠀⠘⠀⠀⠀⠀⠿⣷
⠓⠶⠧⣜⡢⢄⡀⠀⠀⠀⠀⠀⣀⠀⡴⢋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⢀⣀⠏
⠀⠀⠀⠀⠉⠓⠮⣕⠢⢀⣠⠞⠉⢝⣥⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠽⠋⠀⠀⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣍⣚⡉⠙⡝⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⠤⠶⠋⣸⢃⣠⣶⣾⣿⠿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠤⠌⣳⢬⡛⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⣶⡞⠁⠀⠀⢀⣿⡿⢿⣥⠊⣩⠶⠷⢼⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣸⣏⠁⠐⣏⠓⢭⡳⢤⣀⣀⠀⠀⠀⠀⢀⡴⠞⣱⠋⢰⠁⠳⠦⢶⣾⡿⢯⣤⠊⢠⠟⠓⣲⣾⠿⠛⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⡞⠁⠈⡇⠀⠹⡄⠀⠈⠳⣄⣩⠷⡄⠀⣴⠟⠁⣼⠃⠀⡏⠀⠀⢠⣾⡏⣠⡞⠀⡽⠛⣲⣿⠟⠁⢀⣤⣤
⠀⠀⠀⠀⠀⠀⢀⡼⣡⠏⠀⠀⠀⣿⠀⠀⠹⣆⠀⠀⠈⢛⣾⠟⠛⠉⠀⢰⠇⠀⠀⣧⣀⣠⣿⣃⣠⠋⢙⠾⠒⣶⡿⠃⢀⡴⠋⠹⣿⡇⢀
⠀⠀⠀⠀⢀⣴⠞⢰⠋⠀⠀⠀⠀⢹⡀⠀⠀⠹⣆⡴⠶⢿⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢠⣿⢃⣨⠏⢲⡮⠤⣼⡿⠁⢀⡞⠁⠀⠀⠹⡇⣸
⠀⠀⠀⠀⢿⠢⣍⡎⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⢻⠀⠀⠈⣧⠀⠀⠀⢰⡇⠀⠀⠀⠀⣾⡿⣯⣉⣉⠛⠓⢾⣿⠁⢠⡟⠀⠀⠀⠀⠀⢳⡏
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸⡆⠀⢀⣼⡇⠀⠀⠀⢰⣿⢿⣤⣤⣄⡈⠑⠀⠹⣆⣾⡄⠀⠀⠀⢀⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⣀⣼⠷⠞⠋⢹⡇⠀⠀⠀⣼⣧⣽⣤⠤⠤⠀⠀⠀⠀⢻⡟⣿⣄⣴⠞⠋⠉⠉⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠖⠚⠋⠉⠀⠀⠀⠀⠈⣇⠀⠀⠀⣿⣇⣸⠒⣶⠶⣄⡀⠀⠀⠈⠉⠉⠛⠛⠲⠤⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣿⡇⡌⠑⡿⢰⣿⡟⠛⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⣿⣏⡟⢦⣇⣸⣿⠀⠀⠘⠿⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠘⢿⣧⣰⡇⢸⣿⠀⠀⠀⠀⢀⣼⠛⠻⢶⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠙⠻⣷⣼⣿⣆⣀⣠⡴⠟⠁⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⢀⠀⠀⠉⠛⢻⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⢸⡄⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣧⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢹⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠘⣇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡆⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⣇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢻⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⣿⣿⣿⣶⣦⣄
⣶⣶⣤⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢲⠉⠋⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⣀⣀⣀⣃⣀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠠⠤⠤⠤⢴⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡁⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⡀⠀⣠⠧⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⢤⠳⣄⡀⢉⡾⡆⢛⠳⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⢻⡀⢡⠈⢣⠸⠙⠈⡄⢃⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣇⠀⡄⠀⠡⠱⡇⢁⠈⠀⡆⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⢻⠀⠀⠀⠀⠱⡇⠸⠀⡇⢻⢉⣡⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⢸⠀⢠⠀⠀⠀⡇⢀⣀⡠⢼⠓⠖⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⢸⠀⢰⣀⠠⠄⠒⠏⢀⡀⣜⣸⠀⢸
⠀⠀⠀⠀⠀⠇⠂⠄⣀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠃⠘⠓⠐⠂⠒⠈⠙⡀⠀⢀⠗⠁⠀⠀⣀⠤⠐⠸
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠉⠒⠠⣀⡘⡀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠐⠇⡿⢀⣀⠄⠂⠉⠀⠀⠀⡃
⠀⠀⠠⡔⠊⠁⠀⢀⠊⠙⠻⢿⣿⠃⠱⡤⠤⠀⠐⠂⠉⠁⢈⠀⠀⡘⠰⢡⠘⣿⡿⠟⠋⠑⡀⠀⠈⠑⢢⠄
⠀⠀⠀⠀⠉⠒⢠⠎⠀⠀⠀⠀⠀⠈⠈⠓⠂⠀⠤⠤⠤⠤⠼⠤⠀⠘⠓⠈⠁⠀⠀⠀⠀⠀⠱⡄⠒⠉
⠀⠀⠀⠀⠀⠀⠈⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠠⠤⠄⢀⣀⣀⡀⠀⠀⣀⣀⣀⡀⠤⠄⠄⠒⠊⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣤⠔⠀⠀⠀⠀⠀⢐⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⠛⢄⠑⠄⠀⠀⠀⠠⠋⡠⠊⢪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⢀⡀⠤⠤⢤⣤⣤⣀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠴⠚⠁⡀⠀⠀⠀⠠⠤⢤⣈⡑⠢⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⣁⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡮⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⢿⠀⠀⣭⣴⡦⠀⠀⠀⣠⣴⣦⡀⢸⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠈⠀⠀⠈⠉⠀⠀⠀⢸⡏⠉⠁⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠢⠄⠀⠀⠀⠠⠆⠀⢀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡎⠳⣄⠀⠈⠉⠭⠉⠁⠀⣠⣿⡗⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⡊⠰⡇⠀⢈⠳⡀⠀⠀⣀⣀⣴⣿⣿⠁⠀⢣⡀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠈⠀⠀⠰⠀⠀⠀⠀⢸⡀⠀⠀⠀⠈⢹⣿⣿⡿⠀⠀⠀⢳⡀⠀⠀⠁⠂⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠧⠐⠉⠁⠚⢮⣁⠀⠀⠀⠀⠈⠉⣠⠁⡴⠊⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⣷⡀⠀⠀⠀⠀⠀⠀⡤⠒⠘⣢⣤⣶⡿⣆⠈⠣⣀⠀⠀⠀⢠⠿⣄⠀⠀⢀⡤⠒⢄⠀⠀⡀⠀⠀⠀⠀⢀⣾⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠎⠀⠈⠳⡀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⡇⢻⣆⠀⠀⠑⢄⣀⠞⢀⠤⠋⢀⠎⠀⣠⠃⠀⡰⠉⢆⠀⠀⢀⠞⠁⠀⠑
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠑⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠿⢋⣨⣿⠂⠀⠀⣀⠽⠊⠁⠘⡉⠑⠂⠤⢴⠀⠘⠁⠀⡸⠀⠀⠊⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢰⠏⠉⣹⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡿⢟⣩⡴⠞⢻⡉⠀⠀⠀⣎⠁⠀⠀⠀⠀⡃⠰⠉⠰⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⣏⠁⠹⡄
⠀⠀⠀⢠⠏⠀⢠⠏⠀⠀⠀⠀⠀⠺⣿⣿⣯⠵⠚⢻⠁⠀⠀⠀⠁⠀⠳⡀⢿⠀⢀⠎⠀⠀⠀⠀⠀⢰⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠹⡀
⠀⠀⢀⡞⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠰⠉⠉⠁⠀⠀⠀⠀⠀⠋⢸⠀⠛⠀⠀⠀⠀⠀⠈⠉⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢳⡀
⠀⠀⠸⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠈⠇
⠀⠀⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⢀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡄⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠠⡀
⠀⡜⠁⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢱⡀⠀⠀⢰⡀⢀⡆⠀⠀⢀⡎⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠈⢣
⠐⠀⡖⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠀⠀⠀⠀⢣⠀⠀⠈⢧⡼⠁⠀⠀⡜⠀⠀⠀⠀⠙⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢲
⡇⢸⠀⠀⠀⣀⡤⠞⠁⠀⠀⠀⠀⠀⣠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠸⠇⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠈⠙⢲⣄⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⡇
⡇⠘⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⣠⠞⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠳⣄⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠃
⠘⡄⠀⠀⠈⠦⡀⠀⠀⠀⠔⠊⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠑⠢⠀⠀⠀⢀⠴⠁⠀⠀⢠
⠀⠈⠢⢴⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡦⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣖⣒⠚⠂⠤⠤⣀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⠑⠲⠒⠚⡿⢶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⣦⣤⣤⠄⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⣠⣧⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⡞⠉⡿⠄⣀⣾⠁⠄⢹⢿⠁⠄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠸⠷⠲⢠⡞⠯⣭⢲⡶⢋⣌⢳⣶⢋⡉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠈⠄⠀⠿⢻⡟⢻⣟⢻⣟⠻⠇⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⡧⣶⣶⠄⠀⠀⠀⠀⡠⠞⠉⠁⠤⣽⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⣼⠟⠓⠒⢤⣄⣠⠵⠋⠀⠀⠀⠀⠒⣾⡇⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠉⠙⠟⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⣠⣤⣉⣿⣧⣀⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠶⠖⣻⠉⡽⠉⢹⠟⠍⠙⠳⢦⣼
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣀⣤⣴⠾⠛⢿⣿⣿⣶⣇⣰⠇⢀⡟⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠛⠛⢻⡏⠁⠀⠀⢠⡄⣹⠟⣿⣿⣦⣞⠁⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣰⣏⣰⣋⣠⣟⠛⢻⣷⣶⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


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

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠿⠖⠒⠶⠐⠒⢶⠤⣄
⠀⠀⠀⠀⠀⢰⡾⠉⠁⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠳⡆
⠀⠀⠀⠀⠀⣾⠀⢀⠔⠂⠁⠒⠀⠀⠀⠀⠀⠀⠁⢂⣩⠂
⠀⠀⠀⠀⠀⢿⣀⣠⠴⡟⠙⢻⠋⣻⠊⠳⡛⢻⠋⢹⡏
⠀⠀⠀⠀⢠⠞⠋⠁⠀⠙⠶⠤⠶⠋⣀⣀⡙⠲⠶⠚⡟
⠀⠀⠀⠀⠈⢦⢀⠀⠀⠀⠀⠀⠀⠀⢉⡉⠀⠀⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠸⣶⡶⠶⠶⢶⡾⠃⡼
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀⠀⠀⠑⠒⠚⠊⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡐⠂⠀⠂⢲⡏⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠛⠧⣀⠀⠀⢸⠀⣹⡶⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠢⠤⠴⢾⡧⠴⡤⠛⠙⠃⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠖⠓⠚⠉⠉⠙⡇
⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣸⠘⢇⠀⠀⢀⠀⡆⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⢹⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡝⠀⠘⣀⡀⢴⡔⢧⠀⣼⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿⠀⠀⡇⢸⠀⢹⡄⠀⠀⢀⣀⣠⣴⡏⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢹⠂⠉⠀⣰⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⣸⠀⣼⠷⢈⣭⡛⠧⠌⠸⣇⠁⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⢸⠀⠀⢠⡟⡄⠀⠀⠀⢰⠀⠀⠁⠐⠤⠀⢹⠘⢧⣄⣀⠈⠙⠠⠄⢴⣽⠀⠀⢠⣤⣤⣤⣀⣀⣸⡇
⠀⠀⠀⠀⠀⡇⠀⠀⣼⠀⢳⠀⠀⠀⠈⡀⠀⢰⠀⠀⠀⢸⣀⣀⣀⣈⣉⣓⣶⣤⣼⠆⠤⠾⣁⣄⣀⡦⠤⠚⠁
⠀⠀⠀⠀⠀⠃⠀⠀⡟⠀⢸⠀⠀⠀⠀⢇⠀⠀⡀⠀⠀⢸⠉⠉⠉⠉⠉
⠀⠀⠀⠀⢰⠀⠀⢰⡇⠀⢘⡀⠀⠀⠀⠈⠀⠾⠃⠀⠀⢘
⠀⠀⠀⠀⡘⠀⠀⠸⣇⠀⢸⠛⠲⠦⠤⠤⠤⠤⠤⠄⠐⠛⡁
⠀⠀⠀⢰⠃⠀⢠⡄⢨⠄⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢀⡤⠤⢾⠀⠄⠀⠄⠙⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⣸⡁⠢⣈⠙⢦⣤⠥⠠⠿⢇⡀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⢸
⡏⠈⠲⣉⠃⢄⠀⠀⠀⣀⠀⠙⢦⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⡆
⠻⢦⡀⠌⠁⢄⠈⡉⠁⠀⢀⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠙⠳⣄⠀⠈⠛⠈⠁⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠈⠳⣤⣤⣤⠤⣤⠶⠿⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣄⣀⣀⣀⣀⡀⣸⡇⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⢋⠉⠁⠓⠒⠒⠂⠉⠉⠐⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠈⣧⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠔⣹⠋⠛⠛⠛⠛⠒⠒⠒⠒⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⠉⠀⠈⠉⠉⠛⠛⠉⠉⢹⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠂⠀⠂⠀⠀⠒⠐⠂⠀⠀⡇⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠀⠀⠀⣄⡠⠀⢀⣧⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠌⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠣⠤⠔⠀⠀⠀⠀⠀⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠠⠤⠤⠄⡂⠀⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢬⣈⣆⣙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠂⢼⡤⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣲⣄⣀⣀⣀⣤⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠛⡹⠀⠀⠀⠸⠢⡀⠀⠀⠀⠀⠀⠀⠀⣠⡜⠁⡃⠀⠀⠀⡘⠀⢱⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠒⢒⠦⡠⠁⠀⠘⢄⠀⠀⠀⠀⡠⠊⠀⠀⠀⣞⣆⠀⣐⡨⠇⠀⠀⢈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢊⠀⠀⠀⠀⠁⠑⠄⣠⠊⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠐⠒⣅⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⠒⠂⠀⠀⠲⠀⠀⠀⠀⡀⠀⠀⢸⠀⠄⢀⠀⠠⠀⠀⣧⢀⠀⢠⣄⠀⢳⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢹⡇⠀⠀⠀⠀⡀⠉⠁⢀⠀⠀⣇⠉⠋⠁⣿⠀⠀⠙⠒⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⢸⡇⠀⠀⡣⠀⠽⠶⢶⢎⠀⢰⢙⣲⣶⣖⡁⠦⠤⢄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡴⠃⠀⠀⢸⡇⠀⠀⢸⠐⢢⠖⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡐⠁⠀⢸⣷⠀⠀⢸⠀⠘⠶⠀⠀⠀⣠⢼⠀⠀⠀⠀⠀⣀⡴⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⢸⣻⠀⠀⢸⠀⠀⠑⠆⠚⠁⢷⢸⡀⠔⠒⠛⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⡼⢻⠀⠀⠘⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠄⠄⡀⠀⠀⠈⠖⠀⠒⠀⠈⠉⠉⠀⠀⠐⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠡⠠⠤⠤⠤⠀⣀⣀⣀⣀⣀⡀⡈⠉⠛⠛⠛⠛⠛⠛⠉⠉⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠄⠒⠈⠁⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁⠀⠀⠒⠒⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠋⠉⠉⠉⠙⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠀⠠⠶⠆⠀⢒⠚⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣏⠀⠀⠀⠈⠁⠀⠈⠀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⢀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⠀⠀⠀⠶⠿⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠀⠄⠚⠋⠛⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⡴⠶⠾⣛⡁⠸⡟⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢛⣋⣭⣭⣿⣷⣶⣶⣿⣿⣿⣿⣿⣇⣈⠻⢶⡶⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢋⣴⣾⣟⣡⡤⠍⠉⣈⣙⣛⣿⣿⣿⣿⣿⣿⣿⣷⠆⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢫⣾⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣧⠀⣉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢣⣿⣿⣿⡿⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡎⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⣁⣧⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⣿⡇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⢿⣿⣿⣆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣄⣀⣀⠀⠀⢠⣿⣿⣿⠇⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣌⢿⣿⣿⡇⠀⢠⡶⠟⠛⠁⠀⠀⠀⠀⠀⠉⠉⠙⠛⠂⠈⢿⣿⢋⣘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⡫⠒⠈⠻⡟⠀⠀⠀⡀⢈⣑⠆⠀⠀⠀⠀⠀⠌⣥⡄⠀⠀⠀⠈⠃⡀⠀⠂⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡃⠀⠐⠀⠁⠀⠀⢰⡀⠿⠟⠀⠀⠀⠀⠀⠀⠘⠛⢃⡼⠀⠀⠀⠀⡀⢀⢲⡘⣝⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢃⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⡗⣌⣼⣧⢡⣌⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⢧⡛⢼⣇⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠁⠀⣠⡶⠀⠀⠀⢰⢡⢃⣿⣿⡼⠄⢳⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠊⣸⡅⢆⣉⠀⠀⠀⠒⢤⣤⠤⠤⠤⠀⠬⠤⠒⠊⣡⣾⠃⠀⠀⠀⠈⣵⡾⠃⠘⣷⡡⡌⢶⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⡌⢰⡟⠻⢶⣬⠇⠀⠀⠀⠀⠹⣿⣶⣶⣶⣶⣶⣶⣿⡿⠃⠀⠀⠀⠀⢢⡿⠀⠀⠀⠘⣷⡘⡌⢣⢻⣆⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⢟⣰⢀⣿⠁⠀⠀⢻⡞⢄⠀⠀⠀⠀⠙⠿⣿⣿⣋⡹⠅⢊⠄⠀⠀⠀⢀⢆⣾⠃⠀⠀⠀⠀⠘⣷⡰⣂⣦⠭⢍⡻⣧
⠀⠀⠀⠀⢀⣴⢶⠟⡑⠂⠤⢸⡇⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⢄⠀⠀⢀⡤⠊⠀⠀⠀⡠⢫⡾⠃⠀⠀⠀⠀⠀⣴⡟⣐⡁⠀⠀⣼⠱⠿⣦⡀
⠀⠀⠀⠀⣼⢧⡂⠈⠑⢒⠶⡘⣷⡀⠀⠀⠀⠀⠈⠛⢶⣭⡲⢄⠀⠀⠀⠉⠁⠀⠀⢀⡴⣫⣴⠟⠁⠀⠀⠀⠀⠀⠀⠸⢷⣬⠹⠚⣍⡁⠉⠱⡸⣧
⠀⠀⠀⣸⠇⠆⠀⠀⣈⡏⡠⠘⣼⣧⠀⠀⠀⠀⠀⠀⣀⣼⢏⠜⢳⠂⠤⠠⠤⠐⢺⢻⡜⣿⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠇⢢⣸⡄⠀⠀⢃⢻⡆
⠀⠀⢀⣿⠸⠀⠀⠀⢉⠟⠁⢠⢣⡿⠀⠀⠀⣠⡶⢟⡛⠥⢊⡤⣄⠳⠀⠀⠀⠰⡁⢀⣙⠒⠭⢛⡻⣦⡀⠀⠀⠀⠀⠀⢻⣇⠂⠈⠨⣀⠀⢀⢸⣹⣿
⠀⠀⠀⣿⡐⡤⠳⠤⠇⠀⠀⠈⡸⣧⠀⣠⡾⣫⠞⠁⠀⠀⠋⠀⠀⠁⠀⣠⠔⠉⠀⠀⠉⠃⠀⠀⠙⢮⡻⣦⡀⠀⠀⠀⣾⠇⡃⠀⠀⠒⣾⠀⠀⠀⢻⣆
⠀⠀⠀⣼⢣⡀⡄⢰⡀⣀⡠⢆⢱⣿⡾⢋⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠙⢎⠻⣦⠀⢀⣿⠇⠝⠢⢄⡄⢁⠲⢌⠛⡸⣯
⠀⠀⠀⠻⣮⣑⣒⣣⡌⠀⠀⠀⢢⠻⢕⣇⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⢸⠃⠀⠀⢀⣇⢹⡷⢛⠕⠀⠀⠀⡄⣶⠟⠻⣮⡀⠐⡹⣧
⠀⠀⠀⠀⠈⠉⠉⢻⣇⡆⠀⠀⠀⢳⠂⠉⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠇⠸⡆⢀⡴⠋⠀⢱⠞⠁⠀⠀⠀⡜⣼⡏⠀⠀⠹⣷⡙⠁⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡰⠀⠀⠀⢠⠅⠀⠀⣨⠗⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⣟⠻⣄⠀⠀⠙⠀⠀⠀⢠⡞⣼⠏⠀⠀⠀⠀⣸⡇⡄⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠻⡀⠀⠈⠀⡠⢚⣵⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣾⢻⣮⡳⣄⠀⠀⢀⡴⣫⡾⠃⠀⠀⠀⠀⢠⡿⣰⢃⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡰⡤⢄⣠⡾⠛⣹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠿⣿⣦⣕⠶⣫⡶⠋⠀⠀⠀⠀⠀⠀⣾⢣⡏⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠟⣉⣀⣀⣿⠝⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⠒⢈⢻⣿⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡏⠸⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⢙⠫⢵⠒⣏⢱⠋⢳⠄⠤⠤⠀⡀⠀⠀⠀⠐⢐⣊⠁⢀⣨⣽⡔⠈⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣜⡒⢆⠸⠉⠹⠒⠺⡄⠀⢀⠀⣀⣀⣀⡀⠀⠁⠀⡖⠚⢫⠷⣈⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⣮⡰⢄⡈⠀⢸⠀⠀⢸⠖⢿⢼⡇⡆⠀⠀⠀⣇⣀⡸⢂⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⢸⡒⣾⢃⣤⣌⣤⣭⣭⣭⣭⣥⣶⣶⡎⠀⠐⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠙⢡⣿⠉⠉⠉⠉⠉⠁⠀⠀⠀⢸⡇⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠆⡆⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣅⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣾⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠶⢞⣻⣿⢦⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠙⠻⣿⣿⣿⡿⢿⣯⣀⠈⣛⣫⣴⣶⣿⣿⠀⡏⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠙⠻⣧⣀⠈⢻⢛⣿⣿⣿⣿⣿⣿⣶⣇⢸⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⡟⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣄⡤⣴⣄⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⡀⢸⢀⡾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣦⣾⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣬⡇⣦⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⢀⣴⣿⣿⣿⣷⣶⣤⣤⣀⣀⣀⣀⣤⣤⣿⣿⡿⠟⠛⠶⣌⠛⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡿⢿⣿⣿⣏⠈⠉⠛⠛⠛⠛⠛⠻⣿⡏⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢦⠹⠿⠃⠀⠀⠲⢿⡷⠂⠀⣾⣿⡁⠀⠀⠀⠙⠳⣦⡄⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠚⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣆⠀⠀⠀⠀⠀⠸⠋⠐⠲⣾⣿⠁⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢀⣴⡟⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠹⣄⠀⠀⠀⠀⠀⠀⣉⣉⣿⣿⠀⠀⠀⠀⠀⠀⠀⢨⣿⡄⣠⣾⠏⠀⠈⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⠀⠙⣷⣦⣀⡀⠀⠀⠀⠈⣹⡿⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⠃⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⢿⡍⠳⣦⣈⢻⣿⣿⣿⣿⣿⣟⣋⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⢿⡇⠀⠈⢿⡄⠀⠙⠛⣿⣿⡋⣹⡏⢻⡛⠛⠷⢶⣤⣄⡀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⣠⣴⡾⠛⠁⠀⢸⣇⠀⠀⠈⢿⡄⠀⣼⣿⣿⣷⣿⡇⠀⢷⡀⠀⠀⠈⠙⠛⢶⣤⣀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢿⣾⠃⢹⣿⡇⢹⡇⠀⠈⢷⡀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣿⣷⢀⣠⣤⣶⣆⠀⠀⠀⢻⣇
⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⢿⡄⢸⣿⣿⣾⡇⠀⠀⢘⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⣹⠟⣫⡿⠟⠋⠁⠀⠀⠀⠀⢻⣆
⢠⣿⠃⠀⠀⠀⠀⠻⣦⠀⠀⠀⠻⠷⢶⣶⡆⠀⠀⢻⣾⣿⣿⣿⡇⠀⢺⡏⠁⠀⠀⢸⣧⠀⠀⠀⣠⢟⣵⠟⢿⣧⠀⠀⠀⠀⠀⠀⠀⠈⢿
⣼⡇⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⣿⣿⣿⡇⠀⠀⣿⠀⠀⠀⠘⣿⣇⡴⠺⣿⣿⠃⠀⠀⠙⠇⠀⠀⠀⠀⠀⠀⠀⠸
⣿⠃⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠹⣿⣿⡇⠀⢰⠏⠀⠀⠀⢠⡟⠛⢷⡄⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠈⠛⠦⡀⠀⠙⣿⡇⠀⠎⠀⠀⠀⢀⣼⣧⣀⣾⣿⠿⣷⣤⣄⣀⣀⠀⠀⠀⠀⠀⢀⣠⣾⠟
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⠀⢀⣠⡶⠟⠛⠛⠛⠛⠓⠾⣿⡀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠉
⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉⠀⠀⠀⠀⠰⠶⠦⣤⣀⠙⣷
⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⠀⣀⣀⣤⡤⠶⣾⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠦⠤⠤⣼⠏⠛⠉
⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⠟⠛⠋⠉⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⢶⣟
⠀⠀⢻⣇⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣀⣤⣤⣤⣴⣶⣶⣶⣶⡿⠃
⠀⠀⠀⢻⡆⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣾⠛⠉⠁⢠⡏⠁⣼⣿⠏⣿⡇
⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠇⠀⠀⠀⡾⠀⣼⣿⡟⠀⣿⡇
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡾⠟⠛⠉⠀⠀⠀⠘⡿⣿⡿⠟⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣤⣤⣤⣴⣶⣾⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣼⣧⣿⣥⣤⣤⣤⣿⡇
⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⣀⠀⠀⠀⠤
⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⠈⠉⠁⠰⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠠⡤⠀⠀⠠⠄⠀⠀⠉⠀⠘⠀⠀⠀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠒⠿⣉⡉⡄⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠠⡀⠀⢀⡄⠀⠀⠁⠀⠰⠁⠀⢀⠖⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠂⡀⢠⠃⢰⠁⠀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⢆⠀⠀⠁⢀⠆⠐⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣤⠄⠐⡉
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣀⡀⠆⡀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠌⠀⠈⠈⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢄⡀⠐⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⡇⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣣⠄⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠜⠀⠀⠀⠀⠀⡠⢃⠀⠀⠀⠀⠀⠃⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠆⣰⠟⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠎⠀⠈⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠔⠂⠀⠘⠀⠀⠀⠠⠤⠤⠤⠤⠄⠄⢠⠄⠀⠀⠀⠀⡠⠀⠀⢠⡰⠇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆⠀⠄⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠠⠇⠀⠀⠀⠣⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⢀⣰⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠳⢤⡄⠰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠴⠴⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⠠⠄
⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣷⣶⣶⣶
⣿⣿⣏⣀⣀⣀⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡱⠀⠀⠀⠀⢶⣖⣚⠡⠤⣄⠙⡄
⠀⠀⠀⠀⠀⠀⡤⣄⠀⠀⠀⠀⠀⠀⢠⠪⠈⠀⠀⠀⠀⠀⠀⢨⠃⠉⠉⠉⠷⡇
⠀⠀⠀⠀⠀⠀⠉⠁⠳⢄⡀⢀⡤⢴⡑⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢺⠉⣙⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⠖⠚⣽⣴⡾⢟⡷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠶⡁⢗⠻⢦⠄⡀⠀⢀⠔⡝⡱⠑⢠⡾⠫⢌⣕⣣⡾⠛⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢮⠊⠀⡇⠈⢆⠀⢑⡽⠖⠃⡼⠀⢧⣠⡟⡑⠣⢎⡾⠋⡔⠻⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣣⠃⠀⠀⢸⠀⠈⠞⠉⡆⠀⢀⡇⠀⠀⣿⣔⡶⠯⣿⢁⡞⠀⠀⢳⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢧⠀⠀⠀⣸⡠⢼⡇⠀⢸⣿⣒⡀⠀⠘⡾⣄⣀⠤⠾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠉⠁⠀⠈⡇⠀⣿⢼⣓⢲⣄⣀⠉⠉⠛⠂⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢿⡒⠬⢼⠃⠘⠗⢦⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠈⠻⢦⣼⣆⣀⡴⠃⠈⠙⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢠⠀⠈⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠈⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢱⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠈⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠃⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠿⠿⠿⠶⠄

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀
⠀⠀⠀⠀⣠⣶⣿⣤⣤⣴⠟⢛⣴⣷⣶⣶⣶⣶⠖⢀⠔⢈⣤⣾⣷⣦⣀⠈⠛⢷⣌⢳⣄⡵⠶⠷⢤⡙⠿⡛⠿⢏⡑⢦⡄
⠀⠀⠀⠀⣿⣿⣿⣤⠞⠁⠀⠛⠋⠉⠉⠛⠛⠁⡴⠋⠔⣩⡤⠤⣬⣍⣙⡛⢲⣤⠙⢿⣧⡀⢶⣶⣤⣈⠂⠈⠢⡀⠹⣦⡀
⠀⠀⠀⠀⣿⣿⣿⠁⠴⠿⠿⠿⢿⣿⣿⠋⣠⠎⠀⠀⣩⣴⣶⣦⣀⠈⢹⡇⣾⣿⠀⠀⠹⣿⡶⣍⡻⢿⣷⣄⠀⢌⠳⣌⠁
⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⣦⣤⣤⠁⣰⠏⣴⣆⢾⡁⢾⣉⡷⠙⣷⡄⣇⢸⣿⡆⣦⣀⣘⣧⡈⠙⠲⠉⣿⣷⡌⢳⡜⠃
⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⡿⠃⣼⠃⣼⣿⣿⡆⠈⠳⣷⠶⢛⡿⠁⢸⡀⣿⣧⠛⡉⠉⢹⡟⠳⣄⠀⣿⣿⣿⡀⠟⠂
⠀⠀⠀⠀⣿⣿⣿⣦⣤⣤⣤⣤⠀⣼⠃⣼⣿⣿⣟⢿⡄⠀⠀⢀⣿⠁⢀⡈⣇⠸⣿⢀⡽⣗⢾⣧⣄⡀⢸⡿⠿⣿⡇⢸⡄
⠀⠀⠀⠀⣿⣿⣿⠿⠿⠿⠿⠇⢰⡏⢰⣿⣿⣷⠉⢉⣷⠀⠀⣼⣇⣻⣨⠣⠿⠀⣿⡌⣳⣯⣼⡧⢤⠄⣼⢡⠄⣿⡇⢠
⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⠀⢹⡇⢻⣿⣿⣿⣧⠸⡜⣇⠀⣿⣿⡇⣿⢰⣶⣶⣿⣷⣿⣾⡏⢰⣶⡦⣽⡏⢠⣿⡇⢸⠇
⠀⠀⠀⠀⣿⣿⣿⠿⠟⠛⠛⠃⠘⢷⣈⠻⢿⣿⣿⣷⣤⣿⡀⢿⣿⡇⣯⣬⣍⣭⣭⢹⣿⣿⠁⣿⣿⡇⣧⢀⣾⣿⠁⡿⠄
⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣾⣿⡷⠄⠙⠷⣦⣉⠛⠿⣿⣿⣧⠸⣿⠇⠰⣞⠳⠶⠖⣾⣿⠃⣼⣿⣿⡇⣿⣿⡿⠃⣸⢧⡀
⠀⠀⠀⠀⣿⣿⣿⢋⣉⣉⣤⣤⣤⣤⣶⣦⣌⡉⠛⠶⣤⣄⣹⡆⠙⠛⡛⢮⡙⢶⣴⣿⠁⣾⣿⣿⣿⢧⡿⠛⢁⡜⠳⣄⠁
⠀⠀⠀⠀⣿⣿⣿⠘⠻⠿⠿⠿⠟⠛⠛⠋⢉⣀⠀⢰⣶⣄⡉⠙⠒⠤⠉⠼⠇⠀⣿⣿⣿⣶⣶⣿⠏⢠⡄⠺⣏⠹⣆⠙⠇
⠀⠀⠀⠀⣿⣿⣿⡀⢰⣦⣤⣴⣶⣶⣶⣿⣿⡿⠀⣿⣿⣿⣿⣿⣶⡶⢀⣤⠀⣾⠋⣉⡛⠛⢛⣡⡆⢸⡄⢢⡘⢧⡘⣧
⠀⠀⠀⠀⣿⣿⣿⣧⠸⣿⣿⡿⠿⠟⠛⠋⠁⡀⠀⣿⣿⣿⣿⣿⡿⢁⣾⣿⡆⢹⡆⢻⣿⣿⣿⣿⡇⢸⡙⢄⠹⡌⢳⣌⠃
⠀⠀⠀⠠⣿⣿⣿⣿⠀⠀⣀⣀⣠⣤⣴⣶⣿⡇⠀⣿⣿⣿⣿⡿⢁⣾⣿⣿⡇⠈⣧⠘⣿⣿⣿⣿⡇⢸⠷⡈⢢⠘⢆⢹⡇
⠀⠀⠀⢸⣿⣿⣿⠿⣇⢸⣿⣿⣿⣿⡿⠟⠋⢀⡄⠹⣿⣿⣿⠁⣾⣿⣿⣿⣷⠀⢹⣷⣄⣉⣉⣉⠀⢺⣇⠙⣄⠁⡄⢻⡇
⠀⠀⠀⠀⣿⣿⣿⠀⣿⡀⠛⠋⠉⣀⣠⣤⣶⡿⠉⠀⣀⣉⡛⢳⡜⢿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⡄⢸⡍⢳⡈⢢⠈⢾⡇
⠀⠀⠀⢸⣿⣿⣿⠿⠛⡇⢰⣾⣿⣿⣿⡟⠉⠁⠀⣴⣿⣿⣿⣿⣿⣆⠻⠿⠟⣀⠾⢿⣿⣿⣿⣿⡇⢸⡟⢦⠹⣄⠱⣼⡇
⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⠀⣿⡿⠟⠁⣠⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣴⣶⣚⣁⣀⣀⣀⣀⣀⣀⠁⢸⣷⡌⢣⡘⣿⡏⠁
⠀⠀⠀⢸⣿⣿⣿⣦⡼⠿⡇⢀⣠⣴⣾⠃⢀⡶⢤⣉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣹⠇⢰⣧⡙⣆⢻⡿⣧⡀
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣧⠈⣿⣿⠋⢀⣾⣿⣶⣬⣙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⢸⣏⠳⣈⣿⢧⣀⡁
⠀⠀⠀⢸⣿⣿⣿⠷⠾⠟⠻⡆⢸⠇⢀⣾⣿⣿⣿⣿⣿⣷⣦⣍⡻⢶⣤⡀⠀⠀⠀⢀⡼⠋⠀⠀⠀⢸⡟⣆⣻⣧⠀⠁
⠀⠀⠀⢸⣿⣿⣿⠀⠀⣠⠴⠇⢤⣀⣼⣿⣿⣿⣿⣿⣿⣿⡿⠛⢡⡏⠈⠙⠳⢤⣴⠏⠀⠀⠀⠀⠀⢸⣿⣾⣧⣬⣍⣁⡀
⠀⠀⠀⢸⣿⣿⣿⢠⡞⠁⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣾⣿⣿⡀⠀⠀⠉⠁
⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠃⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠿⠟⠀⠀⠀⣸⣿⣿⣿⣇
⠀⠀⠀⠸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠈⠉⠙⠀⠀⠀⠀⠀⠀⠒⠈⠉⠉⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠀⠐⠊⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣵⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⢿⣿⡎⢻⣿⣿⣿⣿⣿⣿⣟⣿⣟⣿⣿⡿⠁⠀⠀⠀⠀⠀⡄⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣾⡾⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣛⣾⣿⠁⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣷⢻⠀⠀⠀⠀⡀⠀⠀⢶⡦⠬⠉⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⢻⢐⣋⠸⣿⣿⣿⣿⣿⢿⣿⣿⣿⣼⡀⠀⠀⠀⠄⠀⡄⠀⠀⢀⣀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠘⠈⠠⢐⡏⢿⣟⣿⣾⢿⣿⣿⣿⡔⠃⠀⠀⠀⠀⠢⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠃⠀⣠⠀⠀⠈⡀⠈⠀⢸⢰⠾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠛⠀⠀⡀⠀⢀⠀⢟⠀⠀⢿⠃⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠇⠔⠘⡆⣿⡇⡴⣾⣁⢻⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⣰⠀⠀⠀⣦⠁⠀⣿⡁⠀⢻⡟⡾⣗⢁⠀⣄⠀⠀⠀⠀⠀⠀⠈⠀⢢⣤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡆⣄⢿⠆⣀⠀⣇⠆⢰⣿⠓⣶⣾⣇⡕⣺⡟⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠹⢾⡆⡁⡐⡯⠀⠈⢸⠆⠛⠙⠛⠠⣿⢹⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠟⣆⣳⡰⠁⠳⠜⠃⠻⠇⠀⠀⠎⠄⠌⠁⡆⠀⣠⣄⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢀⣿⣟⣇⢀⣀⠀⣀⡄⢀⣀⠀⠀⢀⣠⣶⣺⣨⣯⣻⣳⡄⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠐⣶⣶⡶⠒⢠⣎⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠹⣿⡟⡿⣝⣛⢛⣂⡀
⠀⠀⠀⠀⠀⠀⢄⡲⢦⣬⣂⡘⢿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡏⠉⠀⠀⠲⠈⠀⢀⠤⠉⠜⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠐⢛⠉⣉⠈⠰⠢⠬⢕⣀⣈⣙⢯⠉⠉⠻⣿⣿⠟⡖⣥⣂⡀⠠⠰⠺⣿⣻⣿⣿⣷⡶
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⠶⠞⠒⠾⠛⠛⠛⡛⠛⠿⠷⢦⣥⡴⣶⣿⣷⣦⣀⣤⣤⣴⣮⣽⣙⠒⠲⠶⠌⠧⠝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠛⡛⠻⠛⠛⠻⢿⣷⡶⣶⣶⣶⣶⣧⣄⢀⣈⠛⠛⠛⠛⠛⠛⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠂⠉⠈⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣴⠞⠋⠉⠉⢉⡙⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣾⠃⠈⠑⠀⢀⣾⡶⠆⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣴⢿⣧⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣆⢻⣄⠛⣻⠁⠀⠀⠃⣤⢠⡿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⡯⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣿⣦⣄⣚⣚⣛⣠⣼⠟⡡⣫⢼⠿⠿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣭⣥⣶⣭⣍⠛⡉⠩⠄⠢⠡⠘⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣡⣶⣶⡶⣶⣶⣶⡶⠶⠶⠶⣶⣶⣶⠶⠶⢶⣶⣦⡙⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣿⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⢀⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇⣿⣿⣿⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⢀⣿⣿⡇⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠥⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠿⠛⠛⠉⠉⣿⣿⣿⣎⡙⠿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠿⠷⠶⠒⠒⠒⠒⠀⢀⣾⣟⠻⢍⠛⢿⣦⡈⠻⣿⣷⣄
⠀⠀⢀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⡿⠶⠤⠤⠀⠀⠀⠀⣠⣾⣿⣿⣷⣍⠢⢄⡀⠈⠛⢦⣈⠻⣿⣷⡀
⠀⢠⣿⣿⡟⠉⠀⣽⣿⡟⠉⢈⣿⣿⡿⠛⠉⣿⣿⠟⢹⣿⡟⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⢿⣟⠿⣯⣙⠦⡙⠢⣄⠀⠉⠃⠈⠻⣿⣦
⠀⣾⣿⡏⠀⠀⣼⣿⡏⠀⠀⣼⣿⡿⠁⠀⣰⣿⠁⠀⣾⠀⣶⣾⣿⣿⡇⠀⣼⣿⣿⣿⢷⣦⡉⠳⣤⡙⠷⣄⠠⣌⠓⢄⡀⠀⠀⠈⣿
⢰⣿⣿⠀⠀⢠⣿⣿⠀⠀⣸⣿⣿⠃⠀⢸⣿⡏⠀⣸⣿⢠⣿⣿⣿⣿⡇⠀⢸⣿⣿⡟⢷⡉⠻⣦⣈⠛⢦⣌⠑⠄⠑⢦⡁⠀⠀⠀⣿
⣼⣿⣿⠀⠀⣾⣿⡿⠀⢠⣿⣿⣿⠀⢀⣿⣿⠃⢠⣿⣿⢸⣿⣿⣿⣿⡇⠀⢸⣿⣿⢿⣦⡈⠲⣌⠻⢷⣄⠉⠳⣄⠀⠀⠙⢦⡀⠀⣿
⣿⣿⣿⠀⠀⣿⣿⣿⠀⣸⣿⣿⣿⠀⢸⣿⣿⠀⣸⣿⣿⣾⣿⣿⣿⣿⡇⠀⢸⣿⣿⡶⣌⠛⢦⡈⠳⣄⡙⠷⣤⡈⠳⣄⠀⠀⠙⢦⣿
⢹⣿⣿⠀⢀⣿⣿⣿⠀⣿⣿⣿⣿⠀⣾⣿⣿⢀⣿⣿⣿⣿⣅⣈⣁⣀⣀⣀⠀⠙⠻⢷⣌⠳⣄⠙⢦⡀⠙⢦⡈⠻⢦⡈⠳⣤⣶⣀⣿
⠀⠹⣿⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠈⠛⢦⡙⠢⡄⠙⢷⣌⠻⣦⣼⣿⡿⠋
⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠙⠂⠘⢦⡈⣿⣷⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣀⣤⣄⣀⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣬⣷⣦⣼⣮⣾⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣄⣀⣠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠙⠋⠉⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⣶⣾⣿⡿⢷⣤⣤⣄⣀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⠀⣶⣶⠆⣠⣶⣶⣶⠀⣿⡇⣿⣍⣿⡇⠀⢈⣉⡛⠛⠛⠻⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢸⣿⠀⣿⣿⣴⠆⠀⣿⣿⠀⣿⡇⣿⠛⣿⡧⢌⠛⠛⠻⠶⣶⣶⣶⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⡀
⠀⠀⠀⠀⢸⣿⠀⠿⠿⠋⠀⠀⠿⠟⠀⣿⡇⣿⣿⣿⣇⠘⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠘⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠇⣷⠦⠾⠋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠰⠶⠦⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⣻⡟
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣫⣾⣿⠁
⠀⠀⠀⠀⢸⣿⡟⠛⠛⠛⢛⣛⣛⠛⠛⠛⠛⠛⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢫⣾⣿⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠁⠘⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠘⣿⣿⣿⣇
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠰⠶⠶⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⠀⠀⠀⠹⣿⣿⣿⡀
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⢻⣿⣿⡇
⠀⠀⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⠿⠇⠀⠀⠀⠀⠀⠀⠠⠴⠾⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠚⠛⠋⠉⠉⠉⠉⠙⠛⠓⠶⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⢴⣞⡛⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡇⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠰⣷⢦⣤⡾⠁⠀⠀⡻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣀⣠⣀⢾⣙⠾⣝⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣭⡳⣌⡻⢮⡳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣝⢦⣝⠳⣝⠶⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢮⡳⠟⣰⠖⠶⣎⢳⡀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⢷⣀⣀⡽⣸⠇⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⣭⣭⠶⠋⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⢤⣤⣄⣀⣀⣀⣀⣠⣤⡤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠿⠟⠛⠉⠉⠁⣸⣿⣿⣦⣉⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣶⣿⠛⠋⠉⠉⠉⠉⠁⣠⣾⣿⠭⡓⠉⠛⢷⣌⠻⣿⣧⡀
⠀⢠⣶⣶⡶⠒⢶⣶⣶⠞⢿⣿⣿⠿⢿⣿⣿⡿⣿⣿⠋⠉⠉⠀⠀⠀⣠⣾⣿⣿⣿⣷⢮⡑⢤⡀⠈⠳⠄⠛⣿⣦⡀
⠀⣾⣿⠏⠀⢠⣿⡟⠁⢀⣾⣿⠏⠀⢰⡿⠁⢰⠛⣣⣤⣴⣶⠀⢠⣾⣿⣷⣬⡙⢦⣙⠳⣌⠀⢈⠳⣄⠀⠀⠈⢻⣿
⢰⣿⡟⠀⠀⣿⣿⠀⠀⣼⣿⡏⠀⢰⣿⡇⢀⣿⠀⣿⣿⣿⣿⠀⢸⣿⣿⡷⣌⠻⣦⡈⠳⣬⡑⢦⡑⢬⡑⠀⠀⠀⣿
⣾⣿⡇⠀⢸⣿⣿⠀⢸⣿⣿⠃⠀⣿⣿⠀⣸⣿⠀⣿⣿⣿⣿⠀⠘⣿⣿⢿⣌⠓⢌⠻⢷⣌⠙⢦⡀⠀⠙⢦⡀⠀⣿
⣿⣿⡇⠀⢸⣿⣿⠀⣿⣿⣿⡄⢸⣿⣿⠀⣿⣿⣿⣿⣿⣿⡿⠀⠀⣿⣿⡶⣌⠳⣄⠙⢦⡙⠳⣄⡙⢦⡀⠀⠙⢾⣿
⠹⣿⣇⠀⣼⣿⣿⣀⣿⣿⣿⡇⣸⣿⣇⣸⣿⣿⣿⣦⣤⣤⣤⣤⣄⠈⠙⠻⠌⠳⣌⠳⢦⡙⢶⣌⠻⣦⡙⢶⣷⣼⡿
⠀⠘⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠈⠑⢦⡙⢦⠙⢷⣌⣳⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠁⠈⣷⣼⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣿⡿⣿⣿⡿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⡞⠋⠀⠈⢹⣿⢹⢹⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣾⡿⠋⠀⠀⠙⠆⢸⣿⡿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠙⠛⠛⠓⠚⣧⠀⠐⠂⠢⡀⠀⠀⠀⠀⠀⠀⢸⠀⠙⠀⠀⢠⡄⠀⢀⠠⠛⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢹⠀⡀⠀⠀⠉⠛⠓⠶⢦⣄⡀⠈⣵⠀⠠⠆⠀⠀⣸⠀⢴⢀⣞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢸⡉⢉⡿⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠙⠳⠁⠀⠀⠐⢄⠀⠀⠀⣠⡪
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢧⠀⠈⠁⠀⠊⠑⡦⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⡆⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠤⠠⡤⣴⡦⠁⡲⠚⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠘⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠜⠇⠜⠁⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢣⡀⠘⣆⠀⠀⠀⣀⠤⢤⣴⠿⠛⢻⠇⠀⠀⠀⠀⠀⢀⠃⠀⢣⡀⠀⠀⠀⠀⣀⢀⢤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠙⢦⡈⠒⢂⣩⣶⣿⣿⣮⡓⠦⣂⡀⠀⠀⠀⠀⢀⣈⡀⠤⠤⠓⠒⠒⢲⠈⢲⣡⣀⡄⠤⠧⠄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠃⠀⠀⣀⡀⠀⠉⠛⠛⢿⠿⠿⢿⣿⣿⣶⣿⣿⠷⢤⣴⡟⠓⠶⠶⠦⠤⢤⣤⣤⣀⣀⣀⣀⣀⡤⠤⠶⠒⠛⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣼⣷⣄⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⣰⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣷⣦⣤⣄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⣼⡗⡢⣇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⡟⠹⡏⢱⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⣸⢀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⡇⢸⡦⠈⣆⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⡤⠄⠀⠂⠀⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣿⢨⠁⣀⠘⢦⡀⢹⣿⣿⣿⣿⡏⠉⠻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⢠⠠⠄⢸⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠞
⠀⠀⠀⠀⢹⡾⠤⢼⡳⣾⣷⡌⢿⣿⣿⣿⣷⠖⣲⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢀⢩⡳
⠀⠀⠀⠀⠈⢧⠀⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⡀⠙⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⢧⣤⠇⠻⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠘⠳⢶⣴⣦⣤⣤⣤⣤⣤⣤⣀⣇⣀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⠤⣞⢄⠈⡉
⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⠠⡄⠀⠀⠀⠙⠻⣿⣿⣿⣿⣧⠀⠀⠌⣩⠟⠛⠛⠒⢲⠖⠶⠤⠤⣤⣯⠴⠒⠉⢡⠖⡺⠙⠶⠃⡆
⠀⢀⣀⠀⡔⡊⠀⠘⣄⡥⠋⢰⢉⡿⣿⣿⣿⡟⠐⠑⠒⠉⡴⠓⣆⡜⣻⠿⠟⠁⣀⠈⠙⠑⠺⠙⠀⠮⠯⠇⢰⠛⣃⣈⡈⣦⡀⠀⢉⡄⠳⠒⠀⠂
⠀⠉⠦⠍⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⢙⡩⠀⣀⠖⠀⠾⢴⡈⠋⢁⠃⠐⠀⠊⠘⠁⠚⠁⠀⠀⠀⠀⠒⠀⠀⠉⠀⡈⢍⡽⠀⣀⢉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⠟⠀⠀⠀⠀⠐⡄⠀⠱
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⡟⠁⠀⣀⡀⠀⠀⢢⠀⠈⡀⠀⢃⠀⠀⠀⣠⢴⣶⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡅⠀⠰⣿⣿⠂⠀⠀⠀⠀⡁⠀⢸⠀⢠⣾⣿⡿⣿⣿⣿⣾⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠌⠀⢀⠀⠀⡌⢠⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠐⠁⠀⡐⠀⠘⠛⣻⣟⣻⣿⣿⣿⡛⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠟⠿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢴⣯⣯⣿⣿⣿⣿⣯⣯⣵⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⣿⣿⡿⡟⠁
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠄⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⣿⣿⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣯⣶⣾⣾⣿⣟⢿⣿⣿⣿⢿⣽⣿⣾⣷⣦⣯⣟⢿⣿⣿⣿⢿⣛⣟⣻⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣼⣿⣿⢯⣿⣿⣿⢻⣿⣧⣿⣿⣷⣿⣿⣷⣿⣿⡏⣿⣿⣿⣿⣿⣯⢿⢿⣾⣿⣿⣿⣿⣿⣮⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣷⣺⣿⣿⣿⢘⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⢻⣿⣿⣿⡇⣿⣿⠿⢽⣷⡯⢿⣿⡷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣛⣛⣛⣻⣿⣛⣛⣛⣹⣏⣛⣹⣿⣿⢿⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉


⣀⡀⣀⢀⣀⣀⣀⣀⠀⣀⣀⣀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⠾⠿⡾⠿⣿⡷⣶⣦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠛⢉⡙⠋⢠⣶⣿⠷⣶⣶⣶⣤⡀⠀⣶⠙⠻⣿⣶⣤⣤⡀
⠀⠀⠀⢀⣤⣶⣿⣿⣷⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣾⣶⣿⣿⣿⣿⣶⣶⣦⡀
⠀⠀⠀⢸⣯⣤⣤⣤⣄⢰⣶⠄⣠⣤⣤⣤⣤⣤⡀⢴⣦⢀⣤⣤⣤⣤⣤⣄⢰⣶⠄⣀⣤⣤⣤⣽⡇
⠀⠀⠀⠀⠸⣿⠉⠙⣿⣶⣶⣶⡿⠉⠉⠉⠉⠻⣷⣶⣶⣾⠟⠉⠉⠉⠉⢿⣶⣶⣶⣿⠋⠉⣿⠏
⠀⠀⠀⠀⠀⢿⣇⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⣸⡿
⠀⠀⠀⠀⠀⠘⣿⡀⢀⣠⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⣤⣠⣤⣄⣠⣤⣤⣤⣄⣤⡀⢀⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣧⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⣸⡟
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠋





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣒⣀⣴⣦⣴⣾⣅⢢⠀⠀⠀⣠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢻⣾⣿⣿⣿⣿⣿⣿⣿⣷⠆⠀⠀⣿⣿⣿⠟⡻⢏⣴⣆⠻⢛⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣫⣶⣶⣶⣦⣭⣙⣛⣛⣅⠀⠀⠀⣿⣿⣷⢸⡿⢛⣉⣉⠻⣿⠡⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡏⢴⣿⠁⣿⣿⣿⡇⢸⣷⠎⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣷⢸⣧⣈⠛⢛⣡⣾⡱⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡿⠿⠋⠀⠀⠀⠀⠀⣿⣿⣷⣌⣩⣍⢿⠟⣭⣍⣵⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣶⡇⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⡸⣌⠿⣿⠿⣣⡞⣰⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣷⣬⣓⡒⣚⣫⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⣀⠀⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣥⣤⡀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⣠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢟⣛⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣸⠟⠉⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣙⠻⢛⣠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠈⢻⣄⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠘⠉⠻⠋⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠄





⡿⠿⠿⠿⡿⡻⢿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⢛⣛⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢩⠎⠀⣠⠾⠋⠁⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣀⡏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⠉⠉⠈⠉⠙⠛⠶⠾⣧⣄⣀⠀⠀⠀⠀⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢤⠴⢶⠶⠶⣶⠶⢟⡛⠛⣿⠛⠉⠉⠻⣥⢄
⠀⠀⢠⠋⠙⢳⡤⠤⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⠏⠀⠸⢿⠤⢋⡘⠿⣦⠴⠿⠂⠀⠀⠀⢻⣤⠇
⠀⠀⣸⣄⢀⢠⠿⣆⠀⢻⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⣰⠚⠉⠀⠀⠀⠀⣻⠃⠀⠀⠀⢸⡇
⠀⠀⣿⠘⢯⣉⣠⠟⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠸⣦⣀⣤⡖⠀⡰⠃⠀⠀⠀⢀⣿⠁
⠀⠀⠘⢧⡈⠉⠀⠀⠀⣞⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠉⠁⠀⠠⠴⠂⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠙⠓⠚⠻⡄⠘⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⣄⣀⠀⠀⠀⠀⠀⣀⣠⡶⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⡼⠓⠚⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢾⠀⢰⡏⠉⠙⠛⠛⠛⠛⠉⡇⠀⣿⣌⠳⣄
⠀⠀⠀⢀⣤⠖⢦⣄⠀⠘⢎⣦⠤⠶⣎⢳⡄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⣾⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⠙⢦⡈⢷⡀
⠀⠀⠀⢸⡄⠀⠀⢹⠀⢰⣋⣀⠤⠚⢩⡇⢻⠙⠲⠦⣤⣤⡴⠋⠀⢀⣴⡟⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡄⠀⠹⣄⠹⡄
⠀⠀⠀⠀⠓⢦⠴⠋⠀⠀⠈⠳⢶⣈⣥⢿⣾⣤⣄⣀⣀⣀⣤⣴⠞⠋⢰⡇⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇⠀⠀⠹⡄⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠤⠾⡄⠹⡄⠀⠈⠉⠉⠀⠀⠀⠀⠸⡇⠀⡿⠶⠶⠶⠶⠞⠟⠛⠛⣫⢄⢨⡇⠀⠀⠀⣷⠀⢻⡄⠀⠀⠀⠀⠀⢰⢋⡤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⣦⠘⠋⠙⠓⣦⠀⠀⠀⠀⠀⣷⣎⣹⠆⠀⠀⠀⠀⠀⠀⠀⣇⡼⠃⣇⠀⠀⠀⠸⣄⠀⣻⣄⡀⠀⢀⡤⠞⣀⠳⠴⢚
⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢸⠀⠀⢰⡇⢀⡴⠛⢣⢨⡆⠀⠀⠀⢠⡇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠰⡥⠚⠁⠈⠙⢶⣥⠴⠋⠉⠓⠚⠉
⠀⠀⠀⢸⡋⠀⠈⢳⠀⠀⠀⠀⢸⠀⠀⠈⢧⣀⠳⣄⠀⠛⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⡼⢁⣴⡄⠀⠀⠀⠙⢦
⠀⠀⠀⠘⢧⣀⣠⠞⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣤⣀⣠⠟⢉⣠⣿⠀⠀⣠⡤⠚
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⡟⣁⣄⠉⣡⠖⠋⠑⠦⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠘⠋⠁⣸⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢰⡿⡇⠀⠀⠀⠀⠀⠀⢎⣁⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠇⣿⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣸⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⢀⣠⣿
⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠟⠶⠶⢤⣤⣤⣴⠿⠂⠘⣷⢤⣤⣄⣠⣭⣭⠴⢛⣶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠛⠛⠀⠀⠀⠀⠀⣀⣤⣼⠀⠀⢰⣇⠀⠀⣄⠀⠀⠀⠈⠁⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⣀⡀⠀⠀⢀⣀⣠⣴⠿⠥⠤⠾⠀⠀⠸⣍⣉⣋⣹⠗⠶⣤⣤⡤⠴⠶⠶⢚⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⠭⠭⠭⠽⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡖⠢⠔⢢⣄⠀⢀⡤⠐⠒⠒⠒⠒⠒⠒⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠯⢉⣀⡀⠀⠀⠀⢙⠇⢸⠀⠀⡤⠤⠞⠢⠤⣄⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠁⠀⠀⠉⠓⠒⢹⠀⢸⠀⡠⠇⡔⠉⠐⢆⠫⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⢀⠏⠀⢸⠀⠑⡆⠣⣀⡠⠎⣔⠃⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⢄⡠⡴⠋⠀⠀⢸⠀⠀⠓⠒⢆⡔⠒⠋⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⡖⣇⠀⢀⡗⡶⠲⠄⡈⠲⠤⠤⠤⠤⠤⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠈⠪⠭⠍⠒⠀⠀⠀⠀⢆⠀⠀⠀⢀⡀⢏⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣸⡀⢠⡦⠺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⣠⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠖⠦⡀⠀⠀⠀⠀⠀⢸⠀⢠⠇⡐⠲⡄⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠓⠊⠁⠀⠀⠀⠀⠀⢸⠀⠈⣇⣝⢊⣄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠝


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠂⠐⠀⠄⠀⠠⠄⠰


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣈⣉⠉⠓⠦⢬⣙⠻⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⣀⠤⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠤⣈⠙⠲⣌⡛⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠀⠀⣠⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢀⠙⢦⣿⣧
⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⢿⣷⡀
⠀⠀⠀⠀⠀⢠⣾⠋⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣦⣤⣄⡐⠢⢄⠀⠀⠀⠀⠀⠀⠡⡀⢻⣷⡄
⠀⠀⠀⠀⢠⡿⠃⣠⠋⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⠿⠛⠛⠉⠉⠛⠛⠿⣿⣿⣷⣄⠑⢦⠀⠀⠀⠀⠀⠘⡄⢹⣿⡄
⠀⠀⠀⣰⡿⠁⢠⠃⠀⠀⠀⠀⠀⠀⣰⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡄⠳⡄⠀⠀⠀⠀⠀⠀⢻⣿⣆
⠀⠀⣴⡟⠁⠀⡎⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⢹⡄⠀⠀⠀⠀⠀⠈⣿⣿⣆
⢀⣼⡟⠀⠀⢸⠁⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠀⢷⠀⠀⠀⠀⠀⠀⢹⡇⣿⣷
⣾⡟⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⡇⣿⣿
⢿⣧⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠈⢻⣧⡀⠀⢸⡄⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⡞⠀⠀⠀⠀⠀⠀⣿⣏⣿⡟
⠀⠀⢻⣷⡀⠀⣧⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⣰⠁⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠹⣷⡄⠸⡆⠀⠀⠀⠀⠀⢆⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃⢀⡴⠃⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠙⣿⡄⠙⣄⠀⠀⠀⠀⠀⠱⣄⠙⠿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣶⡿⠟⠋⠀⣠⠞⠁⠀⠀⠀⠀⠀⢀⣾⣿⠋
⠀⠀⠀⠀⠀⠘⣿⣆⠈⢦⠀⠀⠀⠀⠀⠀⠙⠢⢄⡉⠙⠉⠉⠉⠉⠉⠉⠀⣀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠒⠒⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣯⠢⣄⠁⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣮⡙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⢟⣿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣮⣝⡛⢶⣦⣤⣤⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⢿⣻⣽⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠂⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣷⣿⣿⣶⣦⣄⡀⠀⠀⢀⠤⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠺⣿⡿⠛⠉⠁⠀⠀⠉⠛⢛⢿⣿⣿⣦
⠀⠀⠀⠀⢠⣧⣶⣾⡟⠀⠀⠀⠀⠀⣀⠠⡜⠁⢾⣿⣿⣿⣧⡀⠀⠀⣀
⠀⢀⣤⣤⣼⣿⣿⣭⡄⠀⠀⠀⠀⠀⢀⣿⡄⣴⡌⣿⣿⣿⣿⣧⣤⢤⡷⣤⣤⣤⣤⡤⡤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠉⠉⢹⣿⣿⣭⠁⠀⠀⠠⣀⣠⣾⡿⣣⡽⢡⣿⣿⣿⣿⡏⠉⠉⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢏⠋⠻⣧⡀⠔⠑⠈⠉⠉⠞⠁⣠⣾⣿⣿⡿⣿⠁⢰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠻⣦⣤⡀⣿⢢⣴⣾⣿⣿⣿⣿⠁⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠠⣔⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⡆⢀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⠛⠛⠛⠋⠁⠀⠀⡀⠀⠇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡁⠀⡅⠈⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠐⠈⢀⠉⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⢰⠀⠇⠀⠀⢰⠀⡀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠔

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢴⠒⢲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⢸⠀⢸⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠤⠾⠶⠾⠦⢤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠙⠒⠲⠒⠒⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⠴⠧⠤⣤⠤⠤⠷⢤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢳⠀⠘⡆⠀⠀⠀⠀⠀⠀⡞⠀⢰⠓⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⣇⠀⠘⠦⠤⠤⠤⠤⠞⠁⢀⡏⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡏⠠⣞⡦⠀⠀⠀⠀⣞⡦⠈⣇⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣄⡀⡼⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠸⡄⣀⡤⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠲⠦⠤⠤⠤⠤⠤⠤⠖⠒⠚⠉⠁




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠉⠁⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠀⣀⠔⠁⠀⠀⠀⠀⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠠⢤⡪⠒⠈⠉⠁⠈⠉⠁⠒⠢⠤⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⡰⠆⠀⠀⠀⠀⢤⠤⢀⡀⢠⣄⡀⠀⠀⠀⠀⠈⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢗⠋⠀⠀⠀⠀⡞⠁⠀⠀⠉⠓⠈⠑⠢⢄⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠠⣄⢸⠀⠀⠀⠀⠠⠀⠄⠁⠀⠉⠢⣀⠀⢀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠀⠀⠀⡠⠠⠇⢀⠐⠀⢢⠀⠀⠑⢄⠻⠢⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠃⠀⠀⠀⠐⠀⢀⡬⢀⣀⠄⠀⠀⠀⠀⠑⠧⠈⠂⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠒⡒⠀⠀⠉⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠤⣀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢁⡱⡀⠀⠁⠀⠸⠁⠀⡀⠀⠀⡸⠄⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠐⠦⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⠀⡄⢀⣧⣴⠀⠀⠁⠀⠉⠀⢀⠜⠀⠈⢢⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⠀⢋⡡⢴⣛⡀⠀⠀⠀⢀⠴⠅⠢⢤⠀⢸⠑⠠⣀⠀⢀⡠⠊⠀⠀⢸⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⣀⡠⡄⢰⡊⠉⠉⠁⠀⠀⠀⠐⠀⡹⠐⣬⠀⠈⠢⠄⠂⠉⡁⠠⣚⠮⡀⣼⠀⢀⡀⣉⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡈⠈⡠⠚⠁⠱⡀⠀⠀⠀⠀⠐⠦⡘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠌⢑⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠃
⠀⠀⢀⢀⠁⣤⠇⠀⠀⣀⡨⠢⢀⠀⠀⠀⠀⠛⢦⠛⢄⡀⡀⠀⠀⠀⠀⡄⢐⡠⠄⠒⠨⡀⠘⠋⠉⠉⠉⠉⠉⠀⡇⠀⠔⠀⠘
⠀⠀⠘⠄⠜⠀⠀⠀⠀⠀⠀⠀⠀⠁⢲⢤⠀⠀⣀⢟⡂⡠⢮⡁⢠⠒⢀⠜⠠⠀⠁⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡘⠀⠃⠘⠂⢬⠁⠀⠀⠈⢆⠀⠀⢀⣖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⡰⠀⠀⠀⠀⠀⢇⡀⠀⢀⡘⢡⡿⣮⣿⣛⡃⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⢰
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠐⠷⠂⠀⠀⠁⠀⠀⠀⠀⠀⠂⢝⠁⠊⠀⠈⠉⡑⠠⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡀⢄⡸
⠀⠀⡀⠀⠰⣤⡀⡀⠀⠀⠀⠀⡘⠀⠀⠀⢠⠀⠀⠀⠀⠈⠱⠇⠈⠀⠀⠀⠀⠀⢀⠀⣠⣇⠀⠠⡀⠀⢄⠀⠀⠀⢠⢠⠶⠋
⠀⠉⠀⠀⠀⠘⠃⠈⠉⠁⠀⠈⠉⠐⠒⠢⢟⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢤⣤⣤⢰⡞⢤⣤⡤⢤⢠⣤⣤⡄⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠈⢈⣻⣿⢸⣧⠚⠛⠋⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣘⣿⣿⣸⡇⢸⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠖⠀⠠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠘⠐⣒⣀⣀⣉⡀⠓⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠜⠉⠉⠀⠀⠀⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡾⣿⣽⣷⣟⢋⠓⠠⢀⠀⠂⠀⠐
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡡⡑⠍⢠⡄⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡿⣟⠻⡥⠀⠁⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢧⠵⠢⠁⡥⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣀⠠
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠾⢤⠡⠘⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⢆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⡀⣤⠠⡀⡰⢠⡶⠶⢠⡄⠀⢀⡴⠶⠆
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠉⠀⠿⠠⠊⠢⠘⠯⠥⠸⠧⠤⠠⠭⠭⠂
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣟⡛⢡⡊⠀⠲
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡤⠐⣠⠀⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣟⣷⡿⣪⣠⠄⢀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣷⣿⣿⣯⣾⣟⡩⡒⠃⠐⢀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⣵⣿⣯⢶⣽⣯⡽⢂⠄⠴⠂⠀⠄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡯⣿⣿⣟⣹⡷⢟⢢⡐⢂⠀⠐

⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⡉⠉⠉⠙⠛⠿⢭⣍⡛⠻⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⣀⣤⠴⠖⠛⠛⠉⠉⠉⠉⠉⠙⠛⠓⠶⢤⣄⡉⠛⠶⣌⡛⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⢀⣤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣈⠙⢶⣝⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠙⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠃⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣈⠑⢦⡀⠀⠀⠀⠀⠀⠈⢢⠈⢿⣿⡄
⠀⠀⠀⠀⠀⢀⣾⡿⠁⣰⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠛⠋⠉⠉⠀⠈⠉⠉⠛⠿⣿⣿⣷⣄⠙⢦⡀⠀⠀⠀⠀⠀⢱⠈⢿⣿⣄
⠀⠀⠀⠀⢀⣾⡟⠁⣰⠇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⡀⠱⡄⠀⠀⠀⠀⠀⠡⠘⣿⣿⣆
⠀⠀⠀⢠⣿⠟⠀⢀⡟⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡀⢹⡄⠀⠀⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⣠⣿⠏⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣣⠀⣧⠀⠀⠀⠀⠀⠀⠘⣿⣼⣿⣧⡀
⠀⢰⣿⠃⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣷
⠀⢸⣿⡄⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿
⠀⠀⠻⣿⡄⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⣿⣷⣿⡿⠃
⠀⠀⠀⠹⣿⣆⠀⠘⣇⠀⠀⠀⠀⠀⢀⠘⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⣰⠇⠀⠀⠀⠀⠀⠀⣸⣿⣿⡿⠁
⠀⠀⠀⠀⠙⣿⣆⠀⢹⡀⠀⠀⠀⠀⠀⢣⠈⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⣠⠏⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠘⢿⣧⠀⢳⡀⠀⠀⠀⠀⠀⠳⣄⠙⢿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⠟⠁⢀⡴⠋⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣧⡀⠳⡄⠀⠀⠀⠀⠀⠈⠳⣄⡉⠻⠿⣿⡿⢿⣿⣶⣶⡿⠿⠟⠛⠉⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣈⣁⠀⠀⠀⢀⣀⣀⣤⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣝⠦⣄⠙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣌⡓⢦⣄⡈⠑⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⣟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣶⣬⣙⡻⢶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣀⣤⣤⣴⣶⡿⣟⣫⣵⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⣀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣍⡉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⡀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠿
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣎⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣆⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣆⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣧⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣧⡀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠉⠙⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⢀⠠⠔⠀⠀⠀⣀⠄⠤⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠒⢄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠊⠀⠀⠀⠈⡧⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠬⠤⢄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⣠⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠦⢄⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠁⠀⠰⠀⠀⠀⣀⠀⠀⡀⠀⢀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡠⠊⠁⠀⠀⠀⠀⠉⠢⡀⠀⠀⡄
⠀⠀⢀⠂⣠⢀⠘⠀⠀⠀⠁⠀⠀⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣝⣁⡀⢀⠀⠀⠀⠀⣀⠀⠨⠢⣠
⠀⠀⠀⠀⠃⡟⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠊⡌⠉⠁⠈⠀⠀⠈⠁⠀⠙⠀⠀⠉
⠀⠀⠀⠀⠀⠱⠀⠀⠀⢱⠢⠤⠉⠂⠉⡤⢊⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⣆⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⢸⡀⢀⡀⠂⠀⠀⠁⡠⠂⠀⠀⡏⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠘⠀⠀⠀⡄⢠⠀⠀⠀⠀⠀⡆⠀⢣⠀⠑⠀⠀⠀⠀⠁⠀⡀⠀⠁⠀⢀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠸⢵⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠸⣇⠀⠀⠑⢕⠂⠀⢀⣀⣤⠊⠀⠀⢀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⣦⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠑⢄⡀⠀⠈⠉⡙⠉⢡⠃⠀⠀⠀⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⠀⠈⢒⡤⠊⠐⣠⠟⠀⠀⠀⢀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⢀⠠⠐⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁

⠀⠀⢀⡀⠀⢀⠎⠉⠁⡀⠀⢀⡀⠀⠀⠀⠀⠤⠔⠐⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠰⡁⠀⠁⠄⠐⠒⠂⠠⢈⠀⢀⠆⠀⠀⢔⠁⡉⢈⠆⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢈⠀⠀⠀⠀⠀⠀⠀⢰⠉⣈⠄⠈⡀⠱
⠀⠀⠁⠈⠀⢀⠀⢀⠦⣀⠀⠡⠸⡀⠀⠀⠀⠃⢀⡠⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠁⠀⠀⢁⠈⢡
⡆⠀⠠⠀⢈⢎⠀⢞⠂⣹⢍⠆⣣⢌⠒⠒⠦⠒⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠐⠀⠂⠀⡀⠀⠀⡠⠒⡻⠹⠹⠒⢄⠁⢢
⠑⠢⡀⡄⠀⠀⠸⠂⠀⠈⠀⠎⢀⡀⣀⡀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⠀⡇⠀⡰⠡⠀⠧⠀⠴⠤⠤⢇⠁
⠀⢀⠃⠈⠢⣀⠀⠀⠀⠀⠄⢶⠁⠀⠀⠀⠉⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠇⠀⠈⠀⢠⠀⢹⠆⠀⠀⠀⠈⡀⠱⡏
⠀⠀⠑⠔⠂⠒⡈⠉⢠⠂⠐⠘⡄⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢈⢀⠁⡀⠀⠂⠸⣀⢀⣀⠦⣀⢕⣠⠃
⠀⠀⡠⠤⠢⠤⡁⠀⠉⠀⠀⠀⠈⢢⠄⠀⢠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⠉⠇⠈⢰⠉⠀⡆⠀⠈⢢⠄⠀⢠⠔⠁
⠀⢰⠁⡊⠁⡄⢣⠀⠀⠀⣀⠄⢰⠺⠀⠀⢀⠖⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠀⠑⢊⡠⠰⡒⢾⠀⠀⢠⠖⢒⠂
⠀⠀⠆⢌⣈⠀⠇⠀⠀⠎⠀⠀⠀⠣⢁⣒⠁⠌⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⡸⠊⠁⠀⢩⠰⡁
⠀⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠑⠠⡅⠀⣤⠐⠁








⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠆⠤⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠁⣀⣀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡀⠍⠱⠄⡁⢘⠀⠀⠀⠀⠀⡎⢉⠡⠤⢉⠉⠂
⠀⠰⢄⡀⠀⡰⡀⠸⡎⠀⠀⠀⠉⠀⠈⠀⢀⢆⠔⢊⠡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⠠⡀⢤⠄⠐⣀⡱⠀⠀⠀⢖⠁⠆⠀⠀⢈⠀⢐
⠀⠀⠀⠓⠨⠦⡈⠀⠰⡀⠀⠀⠀⢀⠄⠀⢁⠴⠅⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⢋⠲⢄⡛⢹⠀⠀⠀⠴⠞⠓⠐⠊⠐⢢⠐⡅
⠀⠀⠀⠈⠉⠀⠀⠀⠀⢸⡒⠀⢒⡃⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠊⠁⠀⠀⠀⠃⡀⠤⠄⣀⠀⠀⣨
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⠁⠈⠀⠀⠀⠀⠀⠀⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡍⠉⠉⡽⠀⠀⠀⠟⠀⠀⠂⠀⢈⠉⠴
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠤⠒⢤⠘⠂⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀⠀⢄⠀⠀⠀⢣⠀⠀⠒⠓⠀⠠⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠀⣀⠆⠕⢀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠃⠈⠆⠀⠀⠀⠑⣄⣀⣀⣠⠊
⠀⡀⠀⠀⠀⠀⠀⠀⠐⠈⠓⠖⣘⠠⠃⠀⠀⠀⠀⠀⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠁⠀⠐⠀⠘⢂⠠⠤⠰⣆⠀⠀⣸⠀⡤⠄
⠀⠀⢰⠄⠤⠤⠤⠤⠤⠤⠤⠄⠠⠤⠄⠤⠤⠤⠤⢤⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄⠰⠷⠀⠃⡐⠁⠀⠀⡖⠉⠉⠉⡅⢎
⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠀⢠⠤⠊
⣿⣿⣿⠿⡿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣾⣶⣾⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⢀⣀⣀⡀⠀⠀⠀⡀⠀⠀⠀⣀
⠀⠈⡛⠛⣿⡏⠀⠸⣦⡿⠀⣸⣧⣿⠇⠀⠀⠀⠀⠀⠀⠀⠠⢉⣉⣐⣒⣢⡤⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠿⠿⠿⠿⠿⠿⠇
⠀⠀⠉⣿⣿⣿⠀⠀⢹⠁⠀⠹⣿⡏⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⠀⠒⠀⠘⡧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠤⢴⣶⣶⣿⣿⣿⡆
⠀⠀⢰⣿⣿⡟⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⣸⣏⣻⣆⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣈⣉⣛⣉⡁
⠀⠀⢸⣿⣿⡇⠀⠀⢸⠀⠀⠀⣿⡇⠀⣴⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⢸⣿⠉⠁⠀⠀⢸⠀⠀⢠⣿⡇⠀⢺⣿⣽⣿⡟⠀⠀⣾⣷⠀⠀⠀⢰⣏⣷⠀⣶⣴⣦⡤⠀⣤⣤⣤⣤⣤⡄⣤⣤⣤⣤⣤⡀
⠀⠀⢸⣿⠀⠀⠀⠀⣸⠀⠀⢸⣿⡇⠀⠘⣿⣿⣿⡇⠀⢀⣿⣿⡀⠀⠀⠘⣿⡟⠀⠀⣿⠇⠀⠐⠿⢿⣿⡿⠿⠇⠉⠀⢹⣿⠉⠑
⠀⠀⢸⣿⠀⠀⠀⠀⠉⠁⠀⠸⠛⡇⠀⢠⣿⠛⣿⡇⠀⢰⡏⢹⣆⠀⠀⢸⡿⣧⠀⠀⣿⡆⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣿
⠀⠀⢸⠿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⣾⠀⢹⡆⠀⣿⡇⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠘⠶⠁⢠⣤⣤⣤⣤⣤⣤⣤⣄⠀⠠⠓⠀⣿⠃⠀⣿⠀⠀⣿⠀⠀⣇⠀⢸⡇⠀⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⡆⠀⠀⠀⠿⠀⠀⡿⠀⠀⢿⠀⠀⢻⠀⠸⠁⠀⣿⡇⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⢸⣿
⠀⣤⣤⣤⣤⣤⣤⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⣦⠀⠀⠀⠀⣶⣶⠀⠀⡆⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣿⣿⣿⣿⠀⠀⣿⠂⠀⠀⠀⠀⠀⢠⠿⡀⠀⠀⠟⠿⠀⠀⠀⠀⢿⡟⠀⠀⡇⠀⠀⡇⠀⢰⡇⠀⠀⢀⠀⠀⠀⠉⠉
⠀⠀⠙⢿⣿⠋⠁⠀⠀⣿⡇⠀⠀⠀⠀⠀⠜⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡇⠀⠀⡇⠀⢸⡇⠀⠀⢸⠀⠀⢸⠀⣠⣀⣄
⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⣿⣷⠀⠀⣇⠀⠀⢾⠀⢸⡇⠀⠀⢸⠀⠀⢸⠀⢸⠘⡿
⠀⠀⠀⣾⣿⠀⠀⠀⠀⠛⠃⠀⠀⠈⠙⠛⠷⣶⣦⣄⣤⠞⡛⠋⠀⠀⠀⣿⡏⠀⢀⣿⠀⠀⣿⠀⢸⡇⠀⣴⣿⣦⡀⢸⠀⢸⠀⣇
⠀⠀⠀⢻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠋⠉⠉⠉⠙⠌⠉⠹⠂⠀⣿⡇⠀⠸⣿⠀⠀⣿⠀⢸⡇⠈⢿⣿⣿⠇⠘⠀⠘⠀⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠉⠒⠒⢢⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠔⣲⠊⠉⠸⡇⠀⢀⢐⡮⢥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠣⣄⠀⠉⠁⠈⠗⠒⠉⠉⢙
⠀⠀⠀⠀⠀⠀⠾⡥⢀⠉⡆⠀⠀⢧⣩⠖⣎⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠏⠑⠘⢢⠀⠀⢲⡍
⠀⠀⠀⠀⠀⠀⠐⠛⠙⠒⣞⣄⣀⠤⢧⠀⣸⡀⠀⣸⠁⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡤⠒⡞⠀⠀⢀⡟⣀⠽⠁
⠀⠀⠀⠀⠀⠀⢠⠊⡭⣍⡛⠏⠀⠀⠀⠉⠁⠈⠉⠉⠉⠉⠉⠉⠁⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠖⠒⠒⠒⠒⠒⠒⠚⠓⠄⠜⠳⣄⠀⣼⡼⠁
⠀⠀⠀⠀⢠⠤⣀⠓⢨⠅⠀⠀⠀⠀⠀⠀⠀⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢹⠿⠥⠤⡄
⠀⠀⠀⠀⢈⡦⢧⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢔⣐⣀⠤⠞⠁⠀⠀⠀⠀⠀⠊⠐⣠⢤⡈⠳⢤⣀⣂⣀⠤⠖⠋⠀⠀⠀⠀⠀⠀⠀⠶⠒⢆⡽
⠀⠀⠀⠀⠈⣧⡈⠣⡀⠀⠀⣠⠄⡖⠘⡟⠛⡶⠢⢤⠄⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡄⠄⣸⢁⠞⠐⠄⠀⡔⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠒⠒⠋
⠀⠀⠀⢀⠎⠀⠑⢄⠘⢶⣿⣾⠒⢻⠋⠉⠉⠁⠀⠀⠀⣠⠎⠀⠀⠀⢀⣀⠀⠀⠰⡛⢇⡴⠐⠛⡦⠀⠀⠐⠀⠘⢇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⢠⠋⠀⠀⠀⠈⢢⡀⠙⠻⡄⣸⡤⢤⠀⠀⠀⢀⡴⠁⠀⣀⣀⣴⡿⠀⠙⢦⡀⠑⢾⢉⡥⠊⠀⠀⢀⡄⠀⠳⣎⠆⣀⣤⡤⣀⠀⠀⠀⣞⡂⣄
⠀⢠⠏⠀⢀⠔⠈⠑⠢⡙⢄⠀⠈⠳⢄⠀⠀⢀⡴⠋⠀⠀⠨⣠⠼⡏⠀⡀⢠⠞⠀⠀⠘⢠⠤⣤⠤⠴⣷⢆⣠⣤⠭⢎⣤⠜⢈⣿⡍⣒⢀⡩⠿⠁
⠀⡞⠀⢠⠃⠀⠀⠀⠠⣱⠈⠳⣀⣴⠂⣧⢔⡯⠧⢤⠀⠤⠘⠣⣒⣪⡛⣗⠑⢄⠀⠀⠯⣀⠀⣸⢀⠜⠁⣾⠅⠈⠒⠞⠃⢴⠣⡈⡋⠉⢸
⢸⠁⠀⣿⠀⠀⠀⠀⠀⠈⢲⣄⡀⠀⠳⡥⢈⡇⢤⢨⠀⣀⠄⠈⢮⣀⠗⠋⠑⠉⠀⠀⠀⠀⠀⡹⠋⠀⠀⠀⠀⠀⠘⢄⣠⡞⡀⠏⠀⢀⠏
⡟⠀⠀⠘⠦⠀⠩⠷⠤⣤⡋⠀⠉⠓⠒⠤⢼⠇⠜⣈⡩⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢡⠸⠑⢢⣾
⠣⡤⣤⣀⣀⡀⠀⠀⠀⠀⠈⠉⠉⠀⠒⠂⣶⡀⢀⡶⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⠎⢸
⠀⢣⠀⠀⠈⠉⠉⠉⠉⠉⠉⠘⢟⢛⠉⠉⢇⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠙⠲⡤⣤⣀⣀⠀⠀⠀⠀⠉⠁⠀⢸
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⣸⡀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⢣⠱⡀⠉⠉⠉⠓⠒⠒⠒⢠⡞
⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠈⠹⡍⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⣇⣱⡀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⢀⠿⠑⠄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠈⠙⡍⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢾⣀⠀⠀⠀⠀⠀⠀⣀⣰⣁⣀⣀⣼⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⠉⠭⠽⠥⠤⠚⠚⠂⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠛⣒⡒⠶⠚⠁⠒⠀⠒⠲⢺⡟⠁⠐⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⣅⢠⠕⢀⠀⠀⢀⠀⢀⣀⣀⣀⣀⠀⠀⠤⠄⠀⠀⠀⢒⠀⡆⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠙⡏⡔⠠⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⡁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⣻⠁⠀⠀⠀⢸⡟⠘⣿⠀⠀⠀⠀⠀⡔⠁⠀⡜⣼⡷⠀⡇
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠁⠀⠀⠀⠀⠈⠉⠙⠋⠀⠀⠀⠀⠀⢆⠀⢸⠁⢀⠂⠀⡇
⠀⠀⠀⠀⠤⠤⠤⠤⡀⠀⠀⠀⠀⡀⠄⠀⠰⢒⣾⣿⡿⠛⡇⠀⠀⠀⠀⠀⠈⠁⢀⣀⣀⠀⠀⠀⠀⠑⠚⠂⠣⠀⠀⡇
⠀⠀⠀⢰⠀⠀⠁⡤⢇⠀⠀⢠⠊⠁⠀⢠⣴⣿⣿⣷⡄⢀⠇⠀⠀⠀⠀⣀⢀⣀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠶⠀⠃
⠀⠀⠀⠸⠈⠀⠁⢈⠇⢣⡀⠆⠀⠀⣠⣹⣿⣿⣿⠿⠁⣸⢀⠀⠀⠀⠀⢀⣌⣆⠀⠀⠈⠉⠁⠉⠁⣶⣿⣷⣦⠤⢀⠇
⠀⠀⠀⠈⠉⠉⠉⠙⠎⠁⠈⢆⠀⡠⠚⠛⠛⠿⠉⠐⠊⡍⡄⠑⠀⠀⢀⣌⠈⠻⠁⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠋⠀⠀⠀⠀⢒⠃⠀⠀⢀⠀⠀⠠⣀⠃⠈⢑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⣀⡴⠃⠀⠈⠀⠀⠀⡈⠀⡀⠀⠋⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⢸⢨⠀⠀⠄⠀⢀⠁⢀⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⢀⡤⠔⡩⡄⡘⠀⠀⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢸⠀⠀⠀⠰⠀⠈⠀⣼⣿⣿⡟⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠘⡄⢃⠄⠌⠀⢀⡷⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣂⠀⣠⣿⣿⣿⣷⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠘⢄⠆⡀⠠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣀⣅⣀⣈⣓⣂⣠⣤⣼⣿⣿⣿⣟⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠖⣑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⡏⠉⠉⠙⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣍⠉⠉⠉⠉⠉⠙⠉⢻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠂⠤⢙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢸⠀⢠⠀⠀⡠⠆⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⠀⡏⢂⡘⠀⠘⡠⠉
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠃⠀⠀⠀⠸⣿⡿⠟⣿⣿⠀⠀⠀⠈⡁⠉⠉⠣⢿⡉⠀⠑⠀⢄⠀⠘⠀⢡⠀⠑⠄⠠⠁⢠
⠀⠀⠀⠀⠀⠀⠈⠂⠈⠢⠀⠀⠀⠀⠉⠀⠀⠈⠁⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢠⣷⣶⣾⣷⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⡀
⠀⠀⡄⠀⠐⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢁⠄⣴⡼
⠀⠀⠧⣀⢠⣀⢼⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡴⠾⠛⠉
⠀⠀⠀⠀⠈⠉⠀⠉⠉⠙⠒⠤⢄⣀⠀⠀⠀⠀⠀⡀⠔⠀⣀⠔⠉
⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⡲⠖⠉⢠⡔⠛⠁⠀⠀⠀⠀⠀⠀⢀⡐⢒
⠀⠀⠀⠀⠉⠙⠒⠀⠤⢀⠀⠀⠀⡀⠔⠀⠀⣠⠔⠁⠈⠑⠒⠢⠤⣀⡈⠀⠐⠒⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢈⠀⢀⣠⣞⠁⠀⠀⠀⠀⠀⠀⠀⡀⠝⠛⠒⠦⢤⣀⠈⢰⠀⠀⠈⠉⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠐⣞⡷⠊⠀⠀⠀⠀⣂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣄⡆⠀⠀⠀⢀⣻
⠀⠀⠀⠀⠀⢠⠶⢤⣆⠀⠀⢠⡽⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠉⠒⠄⡤⠤⣀⠀⠀⠀⠀⠈⠓⠶⣦⣠⣸⠏
⠀⠀⠀⠀⠀⠰⡞⠉⣹⣆⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠻⢿⣿⣶⣦⣤⣀⣀⠈⠉
⠀⠀⠀⠀⠀⠀⠈⠘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠼⠿⠿⠷⠶⠿⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⣰⣾⠿⠿⠛⠛⠛⠛⠛⠛⠉⠉⠉⠙⠛⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦
⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠿⠛⠁
⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣦⡀⠀⠀⠀⣠⣾⠟⠁
⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣦⣠⣾⠟⠁
⠀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣧⡀
⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠈⠻⣿⣦⡀
⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⠙⢿⣦⡀
⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣤⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⢠⡞⡟⣿⢻⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢻⡟⡏⣿⣷⠀⠀⠀⠀⠀⢰⣿⣀⠀⠀⠀⠀⠀⢸⡇⡇⣿⠸⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⢸⢸⡇⣷⢻⢸⡄⠀⠀⠀⠀⠺⣯⣭⠂⠀⠀⠀⠀⢸⡇⣇⣻⣇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣄⠀⠀⠀⠀⣸⣸⣧⣿⣼⣿⡧⠀⠀⠀⠀⢰⡏⠹⣆⠀⠀⠀⠀⠈⠉⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡄⠀⢀⣴⣿⠏⠀⠈⠻⣄⠀⠀⠀⠀⣠⡟⠀⠀⠘⢷⣄⣀⢀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢉⣴⣿⣯⣿⡏⠀⠀⠀⠀⠉⣳⣾⣶⡾⠋⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⠟⢋⣠⣄⣠⣤⣶⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣯⣛⡛⠿⠿⠟⠛⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣍⡛⠿⣿⣿⣿⣿⣿⣿⣇⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣶⣬⣭⣭⣉⣿⣿⣷⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⠀⢻⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠈⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠀⣠⣿⣷⣤⣤⣤⣤⣤⣤⣤⣽⣿⣦⣤⡀
⠀⠀⠀⠀⣸⡏⠉⠉⠉⢿⡍⠉⠉⠉⣿⠋⠉⠉⠙⣿⡀
⠀⠀⠀⢠⣿⣤⣤⣤⣤⣬⣷⣤⣤⣼⠏⠀⠀⠀⠀⠹⣧
⠀⠀⢀⣾⠉⠉⠉⠉⠉⠉⠹⣿⢿⡿⠷⠶⠶⠶⠶⠶⢿⡆
⠀⠀⣼⣇⣀⣀⣀⣀⣀⣀⣀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⢰⡟⠉⠉⠉⠉⠉⠉⠉⠉⣽⣿⡶⠶⠶⠶⠶⠶⠶⠶⠾⣷
⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⣾⠃⠀⠀⠀⠀⠀⠀⠀⠠⡿⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠉⠀⠉⠙⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣤⡄⠀⣀⠆⠀⠀⠀⠀⠉⠉⠛⠛⠻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣦⣁⠀⢀⣤⣶⣿⣿⣿⠟⠢⣄⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⠿⠿⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⡰⠁⠀⠀⠈⣿⡀
⠀⠀⠀⣄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠿⠋⠁⠀⠀⠀⡜⠀⠀⠀⠀⠀⡟⢣
⠀⠀⠀⠈⢪⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡜⠀⠀⣠⠊⠀⣸⣤⣾⡆
⠀⠀⠀⠀⠀⠱⡳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⢀⣼⣿⠦⠶⠛⠋⠁⣡
⠀⠀⠀⠀⠀⠀⠑⡕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⣠⣊⣀⣀⣤⣤⣶⣶⡿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣜⠄⠀⠀⠀⠀⠀⠴⠶⠖⣉⢱⣖⣒⠶⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⡰⠉⠙⡛⠛⠉⠉⢁⣀⣤⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⠆⠀⠀⣤⡤⠖⠁⠒⠭⠍⣄⣀⣀⣀⣀⣉⡀⠶⠤⠤⠤⠐⠒⠚⠀⠀⡔⠁⠀⠀⣷⣶⣾⠿⠿⠛⠛⠉⡇
⠀⠀⠐⠒⠒⠒⠒⠒⠒⠚⠓⠒⠒⠃⠒⠑⠁⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠒⠂⣠⣧⣤⣤⣤⣤⣤⣤⣶⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠶⢶⡶⣾⣷⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁⠀⣀⣄⠀⠀⠈⠉⠉⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠁⠛⠀⣈⣋⣀⣀⡀⠀⠀⠀⠘⠻⠎⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣀⠰⠟⠉⢩⡄⠠⡦⠰⠀⠁⠰⠃⣀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣅⠛⠀⡀⠀⠰⠦⠤⠤⠖⠀⠤⠴⢆⣀⣀⣁⣿⡇
⠀⠀⠀⠀⠀⠀⣼⠋⠁⠘⣦⠉⠀⠀⣔⠒⠤⠴⠦⠄⠀⠀⠀⠀⠤⠀⣿
⠀⠀⠀⠀⠀⠀⢹⣤⡤⠞⠛⣷⠀⠀⠹⣿⠿⠟⠓⠒⠒⠶⠶⣶⣶⢀⡟
⠀⠀⠀⠀⠀⢠⡾⠉⠀⢘⠃⢜⣧⠀⠀⠘⢢⡀⠀⠀⠀⠀⢀⠜⠁⣼⠃
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠐⢢⠾⡃⠀⠀⠀⠈⠛⠀⠀⠚⠁⢀⡾⠁
⠀⠀⠀⢠⡿⠀⠀⠀⣤⢶⣀⠀⣴⠟⣦⠀⠀⠀⡀⠀⢀⣠⠶⠋
⠀⠀⢀⣿⠁⠀⢀⡼⠁⠀⠙⠛⠉⠀⢸⡇⠀⠀⠀⢸⣏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠶⠦⢄
⠀⠀⣼⠇⠀⠀⣼⠃⠀⠀⠀⠀⠀⣀⣾⠙⠢⢄⣀⣸⡉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠉⠅⠀⠠⠤⠄⠀⠒⠂⠒⠛
⠀⣠⠟⠀⠀⢀⠏⢀⣠⣤⡴⢾⠋⠁⠙⢷⣄⡸⠋⠙⣷⣠⡟⠓⢦⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇
⢀⡿⠀⠀⠀⠸⠛⠛⠁⠀⠀⠘⡇⠀⠀⠈⢿⣷⢦⡴⢿⣹⠃⠀⢸⡀⠉⠳⣄⡀⠀⠀⣀⣠⣾⡇
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣿⠀⠀⠀⠘⢧⠀⠁⢈⣿⠀⠀⠀⡇⠀⠀⣈⣱⡶⠛⠯⡉⠋⠁
⣿⠀⠀⠀⠀⣀⣀⣤⣤⣆⡤⠀⣿⠀⠀⠀⠀⠘⣇⠀⣼⠇⠀⠀⠀⡷⣴⠾⠍⣁⠘⠥⠒⣡⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤
⠙⠓⠒⠒⠊⠉⠉⠀⠙⡏⠀⠀⣿⠀⠀⠀⠀⠀⠻⣀⣿⠀⠀⠀⠀⢹⣯⣶⣦⣌⡁⠒⠪⡭⢀⡤⠀⠀⠀⠀⣠⠏⢠⠆⡄⢠⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣄⢰⡏⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠘⣧⠈⠉⠛⠿⣶⣦⣄⣈⡁⢀⣤⣤⣶⣇⣀⣀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠋⠁⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⢰⢲⣿⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⢹⡆⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢡⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⠀⠀⠀⢀⣀⣀⣰⠿⠶⠾⣦⣤⠤⢶⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡟⠙⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⡟⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣷⡤⠤⠤⠼⠧⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠓⠒⠒⠒⠚⠛⠒⠙⠓⢤⣀⠀⠀⠈⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠤⠤⠀⣀⣀⠀⠀⠀⠀⠀⠈⢳⣤⠀⠠⠤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠮⣭⣭⣥⣴⣦⣄⣀⣀⣠⡿⠤⠤⠶⠋
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡟⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣾⣷⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠤⠤⠤⠤⠒⠒⠒⠒⢦⡀
⠀⠀⠀⢀⣠⣤⡤⠤⠤⠤⠤⠶⠶⠖⠒⠒⠒⠒⠛⠋⠉⠉⣉⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⣭⡥⠤⠤⠤⠤⠤⠤⠤⠴⠖⠒⠒⠒⠒⠚⠛⢳⡆⠀⠀⣿
⠀⢀⣿⠀⠀⠀⡤⠐⠊⠉⠉⠉⠉⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⣿
⠀⠈⣿⠀⠀⠀⠃⠀⠀⠀⠀⢀⡴⠃⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿
⠀⠀⣿⠀⠀⠀⢠⠀⠀⠀⣠⠚⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡀
⠀⠀⣿⠀⠀⠀⢸⠀⢀⡔⠁⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⣿⡇
⠀⠀⣿⡄⠀⠀⢸⠐⠋⠀⠀⣴⠋⠀⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⣿⡇
⠀⠀⣿⠇⠀⠀⠈⠀⠀⢠⠞⠁⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢿⡇
⠀⠀⣿⠀⠀⠀⠀⡆⠰⠋⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢀⣶⣤⣀⣀⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⡇⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⢠⠈⡋⠠⡄⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⣤⣴⣷⣤⣤⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⣇
⠀⠀⠀⢻⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣀⣀⣀⣠⣤⠤⠴⠋⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠈⠧⠠⠤⠤⠤⠤⠶⠶⠒⠒⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡤⠤⠤⠴⠶⠒⠚⠛⠛⠛⠉⠉⠙⣦⡀
⠀⠀⠀⠀⣿⠛⠲⠶⠶⠾⠟⠛⠛⠛⠛⠛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠶⠶⡶⠶⠒⢶⠚⠛⢿⠉⠙⢦⠈⠳⣄
⠀⠀⢀⡞⠁⠀⡴⠚⢻⡟⠉⣹⠋⠉⠉⣿⢉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠐⠒⣻⠒⠒⠛⣗⠂⠘⣯⣉⣉⣱⡄⠈⠳⣄
⠀⢠⡟⠀⢀⡞⠃⢐⡟⠁⢰⡏⠀⣈⣹⣏⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠓⢀⣹⣏⣉⣉⣹⣍⣁⣹⣇⣀⣠⣼⠂⠀⠈⣧
⣰⡟⠀⢠⣞⣁⣉⣽⣉⣩⣿⠤⠀⠤⠿⠴⠶⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣋⣉⣉⣉⣉⣩⣭⣭⣭⣭⣭⠤⠤⠴⠦⢐⣲⡟
⢻⣦⣤⣤⣭⣭⣤⠤⠶⠶⠶⠶⠶⠶⠖⠒⠚⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣋⣩⣭⣤⣤⣤⠤⠴⠶⠶⠶⠶⠶⠶⠒⠚⠛⠛⠛⠉⠉
⠀⠙⠲⠶⠶⠶⠶⠒⠒⠒⠛⠛⠛⠛⠉⠉⠉⠉⢉⣽⣿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡀
⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡀⠀⠀⠀⠀⠀⠀⢀⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⣷⣦⣾⢴⣻⡦⣶⠋⢺⣯⣁⠀⠀⢠⣟⡷⣰⠛⡇⠀⠀⢰⣟⣠⣞⣳⢰⢿⡾⣆⢾⣻⠆⣗⣿⠘⣿⠛⣿⡗⣠⣶⠏⠸⣯⣄
⠉⠀⠀⠀⠉⠀⠉⠈⠉⠀⠉⠀⠈⠀⠈⠀⠀⠀⠉⠁⠈⠀⠁⠀⠀⠀⠉⠁⠉⠁⠈⠀⠁⠉⢺⠁⠀⠉⠈⠁⠉⠀⠈⠉⠁⠉⠀⠈⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠛⠉⠉⠉⠀⠀⣄⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠁⠀⣀⣤⠴⠶⠒⠀⠀⠀⠈⠳⡄⡄⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⣠⠞⣉⡤⠖⣛⣻⣿⣂⠙⢦⠀⠙⣿⡄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⢀⡾⢁⡾⢉⡴⣫⣽⣿⣿⣧⠄⠈⣧⠀⠸⡿⡄⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⡾⢡⠏⡴⢋⡾⡽⣫⣿⠛⠁⠀⠀⢸⡄⠀⠇⠙⠆⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠘⠁⠈⣼⢁⣞⡞⣽⡿⠁⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠰⢤⣀⢣
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠇⠈⣼⠹⣿⠁⠀⠀⠀⠀⠀⡿⠀⡴⢠⠆⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠃⠀⡇⠀⠀⠀⠀⠀⡼⠁⣼⣳⠃⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢳⡀⠀⠀⠀⠀⠃⠀⠀⠀⣠⠞⢁⡼⣳⠏⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠙⢦⣄⣀⠀⢀⣀⡤⠞⠁⣠⠞⢡⡏⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢤⡀⠀⠈⠉⠉⠉⠁⢀⣠⠞⠋⢠⡟⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠉⢛⣒⣤⣤⠤⠄⠁⠀⠀⠠⠞⢁⣀⣀⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡭⠭⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⣀⡀⢀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⣷⠀⠀⠀⣠⡄⠾⠂⠙⠁⠉⠀⠉⠀⠉⠁⠛⠂⢴⠀⣾⡋⠉⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⢹⣇⡀⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⢻⣦⣼⠇⣠⡶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⣶⡜⠿⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠉⢹⣿⠉⢹⣿⡋⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣾⠛⣿⣄⢈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡶⠟⣹⡿⢿⡍⢛⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣼⡇⠀⢈⣿⣾⠇⠀⠀⠀⣀⡀⠀⢰⡟⠻⣷⠀⠀⣀⠀⠀⠀⠀⣿⡀⠈⣻⡎⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⠿⠛⠁⠀⠀⢀⣾⠋⠻⠾⠛⠁⠀⠛⠻⠾⠛⢳⡄⠀⠀⠈⠛⠛⠋⠀⠈⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⣠⣤⣤⣄⡀⠀⠰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⣴⠾⠟⠀⠀⣾⠋⠀⠀⠙⣷⠀⠀⠻⠷⣶⠀⠀⠀⠀⠀⠀⠀⢈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠻⣶⣦⠀⠀⢿⣄⠀⠀⣠⡿⠁⠀⣠⣶⡿⠀⠀⠀⠀⠀⠀⠀⢈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠙⠛⠛⠋⠁⠀⠰⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⣀⢰⡄⣀⠀⠀⠘⢿⣄⣴⣦⣤⡀⠀⣠⣴⣶⣄⣼⠇⠀⠀⠀⠀⠀⣴⡄⢀⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⡀⠻⣿⡿⠋⢠⣶⣀⠀⠉⠁⠀⠹⣧⣤⡿⠀⠀⠉⠁⠀⠀⠀⠀⠀⣼⠟⣷⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⣿⣷⣄⣸⣇⣠⣾⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⣿⣿⣯⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⣦⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⡟⠁⣸⡇⠈⢻⣿⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣽⠇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⣾⢿⡿⣦⠀⠋⠀⠤⠀⣤⠀⣀⠀⣀⠀⣀⠀⣤⠀⠶⠀⠀⠻⣧⣄⣀⣈⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉




⡿⠿⡿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⡿⢿⢿⡿⢿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠛⠛⠛⠛⠛⠛⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠉⠀⢀⣀⡀⠐⠶⠶⣦⣴⡖⣻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠀⠀⡔⠁⠜⢋⣀⡠⠶⠾⠛⠛⠉⠉⠀⠙⠛⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⢁⣴⡷⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⣠⡿⠉⣀⣴⠾⠛⠛⠛⠛⠛⠻⢶⣦⣀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⠋⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡄⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⡸⢃⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣴⡷⠻⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠴⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⠉⢡⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠆⠉⠀⠀⠀⠀⣠⡀⠀⠀⠀⠰⠶⠶⠆⠀⢀⡀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⣴⡄⠀⠀⠀⠀⣶⣤⣀⣀⠉⠀⠈⠀⠀⢁⣸⠁⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⡄⠀⠀⠀⠈⢿⡉⠉⠉⠁⠐⠛⢿⠿⠃⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣛⣦⣀⠀⠀⠀⠘⠓⠀⠤⠄⠚⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢦⡙⠿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⠟⢣⠘⣿⠀⠀⠀⠀⠀⢴⠟⠫⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠈⠿⣟⠀⠀⠀⠀⠀⢠⣤⡀⠀⠉⠲⣤⡀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠃⠀⠙⢳⡄⠀⠀⢀⣎⠉⢻⣦⡀⠀⠀⠹⠿⠁⢹⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠀⠀⠀⠀⣄⣹⡆⠀⣼⠟⠀⠀⣿⣍⠳⣤⣿⠀⠀⠈⠀⠉⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠛⠁⠀⠙⣦⡀⠀⣴⠉⠛⢳⣶⠋⢠⡞⠳⣾⠛⢆⣼⠟⢤⡄⣠⡛⠄⣹⡿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⠻⢾⠇⠀⠀⠀⠉⣦⠀⠀⠀⠀⠀⠈⣻⠀⣤⣈⠙⠿⢾⡉⠀⠀⠉⠳⣦⣀⣴⠶⠶⠶⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠠⡤⠀⠘⣧⡔⠉⠛⣼⡅⠀⠀⢰⡿⢧⣤⣀⠀⠈⠉⠁⠀⠀⠀⠀⠉⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢄⣀⣀⣀⠀⠀⠀⢹⡇⠀⠀⠘⣷⠀⠈⠙⢷⠀⠀⠀⠀⡰⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⠀⠈⡄⠀⠀⢸⡇⠀⠀⠀⢻⡆⠀⠀⢸⡇⠀⠀⠸⣧⣄⠈⠙⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⣀⣀⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠈⢷⡀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠻⣤⠘⠃⠀⠀⠀⢸⡇⠀⠀⠀⠈⠻⣦⣀⣀⣠⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⢷⡀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠉⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠙⠻⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⣷⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠏⠀⣹⡧⠤⡤⠀⠤⠤⢤⡄⠀⠴⢆⡀⢀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣶⠀⠀⢻⡇⠄⠀⠀⠰⠠⣼⡅⠀⠀⠀⠄⠼⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢀⠉⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⣼⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣰⡞⡿⠟⠁⠀⠀⠀⠀⢿⡆⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠀⠀⠀⠀⢸
⠀⣤⡄⠀⡀⣰⡧⣛⠀⠀⠀⠀⢀⣀⣀⡀⣷⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⢸
⠀⡏⠙⣤⠀⠸⡇⠈⣷⣄⠀⠀⣿⠉⠉⢻⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡆
⠀⠃⠀⠀⠑⢄⠀⠀⠀⠘⠷⠶⡟⠀⠀⣿⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⢠⣄⣤⡀⠀⠀⠑⢤⠀⠀⠀⠀⣤⣤⣴⠇⠉⠳⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⢸⡇
⡄⠘⢧⡈⠀⠀⠀⠀⠙⢤⡀⠀⣀⣀⣀⣀⣀⣀⣀⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⣧⠐⠟⠁⠀⢠⡶⠀⠀⡈⠁⠀⠀⠀⠉⠁⠉⠉⠉⠉⠈⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡇
⠉⢦⡀⠀⠀⠀⣸⡆⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠙⣦⡀⠀⠛⠀⠈⠟⠀⠒⠂⠀⠂⠀⠀⠀⠉⠀⢿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠛⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠋⠉⠉⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠇⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠶⠶⠶⠶⠶⠶⠶⠒⠲⠿⢦⣤⣤⣠⣤⣤⣤⡴⠿⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⣸⣿⣥⠒⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⢀⣠⣤⣤⣴⣷⣶⡦⠞⠛⠁⠀⠀⠛⠓⢤⣴⣞⣛⣁⣀⣠⠤⠤⠤⠤⢤⣿⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣷⣶⣶⣶⣦⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣿⣶⣶⣶⣶⣾⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠿⢿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡭⠃⠀⠀⠀⣿⣘⡇⠀⢀⠀⠀⠀⠀⡀⢻⠗
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⢈⠀⠀⠀⠐⠢⠀⢀⣴⣿⣟⠁⠸⠀⠀⠀⡀⠀⠀⠰⠶⠶⠶⠦⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⣀⠀⠀⠀⣀⣿⣿⣿⠟⢈⡅⠀⠀⠀⠀⣠⣍⠁⠀⠀⠀⠀⢠⣶⣶⣤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠘⠂⠀⠠⣀⠄⠛⠛⠻⠉⠀⠨⠤⠐⠀⠀⠤⡈⠙⠀⠀⠀⠀⠀⠘⢿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣂⣀⡀⠀⠀⠀⠀⠀⢀⠠⡐⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠠⠀⠀⢠⣿⣿⡿⣿⣿⣿⠁⠀⠀⠀⠀⢨⠀⡀⡁⠀⢠⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣾⣿⣿⣶⣿⣿⠃⠀⠀⠀⠀⠀⠀⠁⡐⡀⠴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣭⣤⣤⣴⣿⣿⣿⣿⣭⣭⣤⣤⣤⣤⣤⣤⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠛⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⢸⠀⠆⠀⠀⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣤⣿⣦⣤⠀⠀⠘⠟⠛⠿⣻⠛⠛⠇⡀⠀⠰⠀⠀⠐⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢠⣤⣤⣦⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢋⣴⣌⠻⣿⣿⣿⠟⣥⣌⠻⣷⡄⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣌⠻⣿⣷⣌⠛⣡⣿⡿⢋⣴⣿⣿⡄⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⣌⢻⣿⣿⣿⠋⣴⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠟⣡⣾⡿⠿⣿⣷⣌⠻⣿⣿⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⢡⣿⡿⢋⣴⣷⣌⠻⣿⣷⠈⣿⡿⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣍⣴⣿⣿⣿⣿⣷⣌⣡⣾⠟⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠀⠀⢀⣤⣬⣭⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⢡⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣿⣿⣿⣿⣿⣿⠏⣠⣭⣭⣭⣭⣭⣭⣭⣭⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣿⣿⡿⢛⣋⣥⡌⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⡏⢴⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⣌⢉⣉⣉⣉⣉⣉⣉⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠧⠸⠿⠿⠿⠿⠿⠿⠇




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

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣿⠉⢉⡉⢉⣉⣉⣉⣉⠉⣉⠉⣉⡉⣉⡉⢉⡉⢉⡉⢉⣉⠉⣉⠉⣉⣉⣉⣉⡉⢉⡉⠉⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⢸⣏⣉⣹⣿⠀⣿⠀⣿⡇⣸⡇⢸⡇⢸⡇⢸⣿⠀⣿⠀⣿⣏⣉⣹⡇⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣀⣈⣁⣈⣉⣉⣉⣉⣀⣉⣀⣉⣁⣈⣁⣈⣁⣈⣁⣈⣉⣀⣉⣀⣉⣉⣉⣉⣁⣈⣁⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠈⢿⡟⠛⣿⠛⠛⠛⠛⠛⢻⡟⠛⢻⣿⠛⠛⠛⠛⠛⠛⣿⡟⠛⢻⡟⠛⠛⠛⠛⠛⣿⠛⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠉⠀⠀⠀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⣿⠀⠀⠀⠀⠀⠀⣿⣇⣀⣸⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢀⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢶⡄⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⣇⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠇⠀⢀⡿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠛⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡟⠛⠛⠛⣧⢰⡟⠛⠛⢻⡆⣼⠛⠛⠛⢻⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇⣿⠀⠀⠀⣸⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢙⣛⣛⣛⠁⠈⢙⣛⣛⣋⠁⠈⣛⣛⣛⡋⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡟⠉⠉⠉⣷⢸⡏⠉⠉⢹⡇⣾⠉⠉⠉⢻⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣇⣀⣀⣀⡿⢸⣇⣀⣀⣸⡇⢿⣀⣀⣀⣸⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣈⣉⣉⣉⡁⠀⣉⣉⣉⣉⠀⢈⣉⣉⣉⣁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡏⠀⠀⠈⣷⢸⡇⠀⠀⢸⡇⣾⠁⠀⠀⢹⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣧⣀⣀⣠⡟⠸⣇⣀⣀⣸⠇⢿⣄⣀⣀⣼⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣠⣤⣤⣤⡄⢀⣤⣤⣤⣤⡀⢠⣤⣤⣤⣄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⡇⠀⠀⠀⣿⢸⡇⠀⠀⢸⡇⣿⠀⠀⠀⢹⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢷⣤⣤⣤⠟⠸⣧⣤⣤⣼⠇⠻⣤⣤⣤⡾⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁



⡿⠿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿

⠀⠀⠀⠀⠰⠂⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⢀⡄⡠⠀⠠⠠⠲⠀⠀⠀⠀⠀⢀⣐⣀⢴
⠀⠀⠀⠀⠀⠈⡢⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠆⠀⠐⣲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⡀
⠀⠀⠀⠀⠀⠀⣶⠇⠀⠀⠠⠀⠀⠀⢀⠄⠀⠀⢸⠀⠀⣿⠀⠀⠀⢀⠤⠤⣄⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠒⠂⠃⠀⠀⠀⢘⠀⠀⡙⠀⠀⠀⡄⠀⠶⠈⠃⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢺⠀⠀⠀⠀⠀⠁⡩⠇⠀⠀⠀⠈⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡠⠒⠂⠀⠀⠀⠀⠀⢨⢼⠀⠀⠀⠀⠀⣃⠔⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⢀⠄⣀⡘⢄⠀⠀⠀⠀⠀⢘⡡⠀⠀⠀⠀⠀⡥⠚⡁⠀⠀⠀⠲⢤⣤⣀⡀⠀⠀⠈⠉
⠀⠀⠈⠁⠀⠈⠂⠁⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠔⣪⠄⠀⠀⠀⠀⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣋⠴⠂⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠻⣶⣄⡀
⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠰⣪⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⢁⠂⠙⠂
⠀⠀⠀⠀⠀⠙⠛⠛⠀⠀⠀⠀⠀⠈⠢⡄⠀⠀⠀⠀⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠴⠶⠶⠶⠶⠶⠤⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣟⣫⡭⠷⢖⣛⣛⣉⣉⣉⣉⣉⣙⣛⡒⠾⠽⣿⣷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠾⣛⣭⠷⣾⣻⣭⠭⠶⠖⠒⠒⠒⠒⠲⠯⢭⣙⠓⠦⣌⠙⠻⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⢿⣫⡶⣛⣭⠖⣛⣭⡴⠖⠚⣋⣉⣉⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣼⣿⡷⢛⡽⢿⡿⢟⣩⠶⣋⣭⠶⠒⣋⣉⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⢀⣴⣿⣿⣿⣶⣿⣿⣋⣴⣋⣵⠞⣉⣤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠙⣦
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⡟⣿⠉⠉⠉⠉⠉⠉⢹⠋⠉⢉⠉⢙⡉⢻⠋⡟⢻⢻⣿⣿⣿⣿⠛⠛⠛⠛⠛⠚⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⠐⠀⠀⠀⠀⠀⢰⠄⠀⢠⠀⢈⡁⢸⠀⡇⢸⢸⣏⣧⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⢈⡀⠀⠀⠀⠀⢠⠄⠀⢀⠀⢈⡁⢸⠀⡇⢸⢸⠿⠇⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡏⢈⡁⠀⠀⠀⠀⢠⡀⠀⠈⠀⠸⠇⢸⠀⡇⢘⣴⣾⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢠⡄⠀⠀⠀⠀⢈⠁⠀⠘⠀⠐⠂⢸⠀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢠⡄⠀⠀⠀⠀⢸⠁⠀⢸⠀⠰⠆⠈⣸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢸⡇⠀⠀⠀⠀⠘⠀⠀⢰⠀⢰⡆⢠⢻⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⢀⡀⢸⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⡇⢨⠀⠀⠀⠀⠀⠀⠀⠀⣈⠀⢰⡇⢸⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⡇⠰⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⡇⢸⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣧⣼⣶⣤⣤⡀⠀⠀⠀⠀⠀⠀⢰⡄⣸⢸⡇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠀⠀⠉⠙⢳⣦⣀⡀⠀⠀⢠⡀⢼⢸⡇⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⣀⣤⣤⣴⠞⠋⠀⠉⠙⠶⢦⣬⣄⣿⢸⣇⣇⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⢀⣴⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣦⡀
⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠙⢿⣿⣿⣿⡿⠋⣠⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠙⢿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠙⠿⠋⣠⣾⣿⠟⢁⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡈⠻⣿⣷⣄⠙⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠰⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⠆⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠰⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠆⢀⣀⣀
⠀⠀⠀⠀⣠⣶⣄⠙⢿⣿⣦⡈⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⢁⣴⣿⡿⠋⣠⣶⣄
⠀⠀⣠⣾⣿⣿⣿⣷⣄⠙⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠋⣠⣾⣿⣿⣿⣷⣄
⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠈⠻⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠈⠻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠉⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡿⠟⡛⠀⠀⠀⠀⠀⠀⠀⠀⢈⢈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠐⣁⠀⠀⠀⠀⠀⠀⠀⠀⢨⢪⡅⠀⠰⢁⣤⡐⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡁⠀⢠⢣⡄⠈⠀⢻⣿⡿⠈⠆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢀⣀⣀⠀⡄⠁⢨⠃⠀⠰⠣⠄⠀⠀⠀⠄⠂⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡤⠀⠠⠴⣀⡉⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠘⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠐⠒⠒⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⣀⣀⣀⣀⡔⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠰⠆⠀⠀⠀⠀⡀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠂⠀⠀⠘⠂⠀⠀⠁⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠐⠀⠠⠅⠄⠀⠀⠀⢀⣈⣁⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠸⠀⠀⠀⣿⣿⣿⡿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢰⣾⣾⣷⣶⠀⢰⣿⣿⣿⠇⢿⣿⣿⡇⠀⠀⠀⢠⡄⣤⣤⢀⣤⡄⣠⣤⢠⣤⠄
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠇⠀⢸⣿⣿⡿⠀⠸⣿⣿⣧⠀⠀⠀⠈⠸⣿⠋⠺⠟⠰⠿⠃⠾⡟⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠀⠀⢻⣿⡿⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⠈⠤⠀⠀⠠⠄⠀⠠⠄⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠠⠇

⠀⠀⠀⠴⠀⢠⡀⢆⠀⢠⡁⢄⠂⠰⡀⠃⢼
⠑⠘⠆⠀⠂⠀⠀⠀⠀⠀⠙⢄⡂⢤⡙⡘⠻⢾⣶⣴⣿⣿⣶⣶⣂⣀⠶⠿⣿⣿⣿⡄
⡱⣀⣀⠀⠀⠈⠂⠀⢙⡤⠂⠉⠀⠀⢱⢯⣠⣾⣿⣿⣿⣿⣟⣿⣿⣿⡇⠀⠀⠉⠁⠀⠉⢣⠀⠠⠀⠂⠐⠒⠀⠀⢀⡀
⠉⠀⠈⠁⠈⠁⠀⠐⠈⠀⠀⠀⠀⠀⡞⢨⠉⠛⠿⢿⣛⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠐⠀⠁⠀⠀⠉⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠁⠈⢆⠀⠀⠀⠀⠉⠛⠻⠛⠉⠨⠤⢄⡀⠠⠔⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠘⣲⠊⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⣄⣀⣀⣀⣤⣴⣆⣈⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠉⠒⠠⢄⡀⠀⣀⣀⢀⣀⣠⣃⣀⠤⠊⠀⠀⣀⣠⣤⣶⣾⣷⣶⣦⣤⣈⣿⣿⣿⣿⣿⡟⠩⣸⣿⣆⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠄⠠⢀⡀⠈⠚⣅⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠂⢸⠀⡃⠖⠒⠈⠒⠹⠻⢿⣿⣿
⠀⠀⠀⠈⠑⡄⠈⢳⡴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠉⠢⠀⠑⠲⠄⢀⠀⢀⣨⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⢈⠠⢴⣎⣩⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⡿⣷⣦⣀⣈⣠⣴⣾⣿⣿⣿⣿⣿⠁⠀⠀⠀⠈⠀⠀⠀⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠤⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠁⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣉⣩⣿⣿⡿⠟⠋⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡿⠻⢿⣿⣿⡿⠟⠛⠁⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠋⠉⠰⢺⠀⠀⠀⠀⢠⣶⡆⠀⠀⢀⣠⣵⣄⠉⠙⠚⠁⠈⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⠀⠀⠀⠳⣿⣿⣿⣆⠀⠀⠀⠈⠙⣡⣴⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⣤⣤⡀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣀⢀⣀⡀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣀⣀⢀⣀⡀⣀⣀⡀⠀⠀⠀⢸⣿⣿⠀⢀⣀⣀⢀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠹⣿⣿⣿⣿⠿⠿⠟⠛⢿⣿⡇⠀⠀⠀⠀⠀⣾⠟⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⠀⠀⢀⣀⡀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢻⣿⣿⣿⣦⣄⣠⣤⣿⣿⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⣀⡀⢀⠀⠀⠀⣿⣿⡇⠀⠀⠀⢀⣠⣤⣤⣶⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⣿⡇
⠀⠈⠉⠁⠈⠀⠀⠀⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠉⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⣿⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡟
⠀⠀⠀⠉⠉⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠁⠈
⠀⠆⠀⠀⠀⠀⠀⠀⣿⣿⡏⠙⠻⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠙⠛⠛⠛⠻⠿⠿⠿⢿⣿⣿⡿⢿⣿⠟⠋
⠉⡍⠉⠉⠈⠉⠁⠀⣿⣿⡇⠀⠀⠀⠀⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠉⠁⠈⠉⠁⠉⠉⠀⢸⣿⣿⠀⠈⠁⠀⠀⠉⠁⠉⠉
⠛⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⡀
⠀⠀⠀⠀⣠⠤⢖⣒⣒⣚⣛⣉⠉⠩⠭⠭⠭⠤⠤⠄⠀⠀⠐⠀⠀⠐⠒⠒⠒⢦⠀⢳
⠀⠀⠀⢰⢃⢸⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⡆⠸⡆
⠀⠀⠀⢸⢸⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠀⢀⡽⠂⠀⠀⠀⢰⠀⡇
⠀⠀⠀⠘⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠐⠀⢺⡀⢀⣤⡄⠀⠀⡀⢹
⠀⠀⠀⠀⡇⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣁⠀⠀⠄⠈⠉⠀⡗⠀⠀⠇⠘⡆
⠀⠀⠀⠀⢹⢸⠸⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⡴⢋⠀⠚⠃⣠⠔⠦⠤⠔⠋⠀⠀⠀⠘⡀⣇
⠀⠀⠀⠀⠸⡦⠄⡆⠀⠀⠀⢠⡞⠁⠀⠈⠉⠰⠕⢋⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢹
⠀⠀⠀⠀⠀⡇⠄⠇⠀⠀⠀⢸⡴⠒⠒⢦⠀⠀⠰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⡆
⠀⠀⠀⠀⠀⢹⢃⢰⠀⠀⠀⠀⠀⢠⡤⠋⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⢸⣸⠀⡄⠀⠀⠀⠀⠀⠙⠲⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠜⢀⡇
⠀⠀⠀⠀⠀⠘⡇⠀⢃⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠤⠴⣖⣒⣒⡫⠭⠥⠤⠔⠒⠻⡄
⠀⠀⠀⠀⠀⠀⠻⣦⣈⣓⣒⣒⣊⡩⠭⠥⠐⠒⠒⠂⠈⠉⠉⠀⠀⣀⣀⣠⠤⠴⡖⠒⢀⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡞⢇⠀⠀⣀⣀⣠⡤⠤⠤⡤⠖⠒⢖⠋⠙⣍⣉⠹⡄⠘⢶⠐⠈⣣⡀⠽⢦⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⢣⠈⢆⠀⠣⣀⣀⢵⡄⠀⠘⠆⠈⠉⣃⣀⠨⣤⠄⠘⢖⠈⢙⣀⠠⠱⠄⠒⠛⣦⡀⠱⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢦⠀⠙⣄⠠⢵⠔⠒⠘⣇⣈⠨⢦⠄⠘⢖⣀⣉⡧⠤⠼⢲⣒⣉⣉⣡⡤⠼⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⢣⠀⠘⢖⠚⢳⣉⡉⢨⡦⠤⠖⣓⣋⣉⡥⠤⠒⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢣⠀⠈⠻⠅⠒⠒⠛⠉⠉⠉⠁⠀⣀⣠⠤⠤⠒⠒⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⢀⣀⣀⠤⠟⠀⠀⠀⠀⢀⣀⣀⡠⠤⠔⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣉⡀⠤⠄⣀⣀⣒⣢⡭⠭⠴⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡳⠤⠤⠤⠄⠒⣒⣈⣉⡩⠭⠴⠒⠚⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⢋⣉⡉⢉⡛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⣴⣿⠀⠀⠀⠈⢢⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⢎⣉⣩⠭⡷⠦⠀⠀⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠙⠃⠀⠐⠂⢺⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠠⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠚⢆⡑⢦⣤⡼⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⠀⠀⢧⠀⣩⣾⢋⢀⡇⠈⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠈⠛⠁⠙⠀⠋⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣰⠇⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢶⡈⠀⠀⠀⠀⢰⣤⣀⡘⣦⡄⡀⡀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢠⡏⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢀⡠⠟⠓⠤⢟⢩⠟⠁⠀⠉⠻⠶⠦⠾⣁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⣸⠁⠀⢀⣠⣾⣇⣤⡤⠤⡴⠊⠁⠀⠀⣀⢀⡿⠋⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⣿⠀⠀⠩⠟⠉⠁⠀⠀⠀⣇⣠⣤⣴⣶⣦⠟⠀⠀⠀⠀⠀⡄⠦⠐⠋⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢿⠀⠀⠀⠀⠀⠀⠰⣄⣀⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣇⠀⠔⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡘⢧⣀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣤⣤⠞⠁⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡉⠉⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢝⡦⠀⠀⠀⠀⠀⣶⣖⡂⠾⠦⠤⣄⡀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠔⠊⠀⠀⠀⠀⠀⠀⠀⠉⢻⠋⠒⡒⠒⠻⣦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣏⡛⢢⡀⠀⠀⠀⠀⠀⠀⢠⢎⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠤⠀⠁⠀⠀⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⠀⣠⠖⢰⡥⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠒⠂⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡖⠭⣈⢍⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⠶⠖⠋⣯⣴⠾⡿⣟⡿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣤⠚⠱⠮⣢⣄⡀⠀⠀⠀⠀⢀⡤⢶⠞⡝⢧⣀⣠⣼⢟⡹⢛⡴⢋⣥⡿⠟⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡔⠉⢣⠀⢧⠀⠙⢏⣹⢦⠀⣠⠟⢠⠋⢰⠁⠀⣰⡿⣡⠊⡱⠛⣶⠟⠁⣠⣤⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⠎⠀⠀⢸⠀⠈⢧⠀⢀⣹⠞⠋⠁⢀⡏⠀⠸⠦⣴⠟⣠⣁⡝⢃⣿⠋⢠⠞⠉⢿⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⢣⡇⠀⠀⠀⠘⡆⠀⠈⢿⠉⠹⡆⠀⠀⢸⠃⠀⠀⢰⣿⡒⠂⠾⢥⣿⠃⢠⠏⠀⠀⠈⣶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⢻⡀⠀⠈⠀⠀⢹⠀⣀⣼⠀⠀⠀⣾⢷⣭⡭⠀⠀⠹⣦⣏⠀⠀⢀⣀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣀⣠⠴⠞⠋⠁⢻⠀⠀⢠⡿⢺⡒⣒⠀⠀⠀⠻⠽⣦⣾⡉⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡟⠚⢬⢭⣷⠶⢤⣄⠀⠀⠀⠈⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠸⣯⡓⢤⢠⣿⠀⠘⠟⠳⣦⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠙⢷⣬⣐⣿⠀⠀⣀⡼⠋⠙⠳⠶⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠉⠛⠿⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢸⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣶⣤



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