Eater Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⡞⢛⣳⣶⣶⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣽⣿⣿⣿⠿⣻⣵⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠃⢿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠋⣵⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢤⡀⠀⠀⠀⠀⡤⠶⠢⠀⠀⠀⠻⠛⠿⠿⠟⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣠⣄⠀⠀⠀⠀⠠⣴⣤⠀⠀⠀⠀⠀⠀⠤⣚⢹⠀⠀⠀⡠⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠛⠃⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠘⡷⠘⠀⡠⠖⠁⣸⠀⢀⣀⣀⣀⠠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⡀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⡴⢫⢀⠠⠚⠉⠉⡉⣩⠄⢀⡴⠁
⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠉⢀⣀⠀⠀⠀⣀⡀⠀⠀⠀⠀⡄⠹⡁⣷⣥⣵⡶⠶⣒⣀⡸⠥⠖⠋
⠀⢠⠂⠀⠓⣺⣧⣀⡀⠀⠀⠀⠀⠀⠀⠳⡄⠘⢯⣥⠬⠽⡿⠁⠀⠀⠀⠀⡰⠀⠀⣹⣥⡏⠉⠉⠉⠁
⠀⡜⠂⠀⠛⣀⠠⠙⣿⡓⠦⠀⢀⣤⣾⣿⣿⣦⠘⠃⠀⠘⠁⠀⠀⠀⢀⣴⠁⢴⣾⣿⡇⣿⣿⣶⣶⣤⣄⣀⡀
⠀⢣⠐⠈⠀⠜⠂⢠⡊⠀⠀⣹⣶⣿⣿⣿⡆⢹⣇⠙⠛⠉⠀⠀⣠⣶⣋⠥⢒⡩⢞⠟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠈⠢⡀⠀⠐⠃⣠⠃⢀⣾⣿⣿⣿⣿⣿⣷⣾⣿⣷⢤⡤⠶⠛⠉⡁⠄⠚⠍⠂⢁⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⡄
⠀⠀⢀⠋⠒⠀⠀⠁⡸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢗⠭⠵⣶⠒⣒⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⡄
⠀⢀⠂⠀⠀⠀⠐⠁⠀⠀⠈⠙⢿⣿⣿⡟⣿⣿⣿⣿⣷⣿⣌⣢⡙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡎⢿⣿⣿⣿⣿⣿⣿⣿⣦
⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣀⣴⡿⠟⠉⢸⡿⠿⠿⠿⠿⠛⠻⠿⠿⠿⠿⣦⣌⠛⢿⣿⣿⣿⣿⣿⡿⠿⣿⣮⢿⣿⣿⣿⡿⠟⠋⠉⢧⡀
⡇⠀⠀⠀⠀⠀⢨⠀⢀⡠⠞⠁⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠄⠐⠂⠀⠀⢀⣼⣿⣿⣦⣴⣦⡭⡉⠁⠠⠀⠀⠙⠓⠻⠿⠥⢀⡀⠀⠀⣀⣙⠢⣄
⢧⠀⠀⠀⠀⠀⣂⠴⠋⠀⠀⠀⢰⣉⣉⣒⡦⢤⢤⣲⣖⣦⣔⣛⣭⣭⣽⣿⣾⠿⣿⣿⣿⣿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠙⠢⡀
⠸⠦⠤⠖⠒⠊⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⢿⠿⠟⡿⠿⣿⣛⣛⣻⣽⣋⡄⠀⠀⠀⠘⠦⣅⡀⠀⠀⠤⠀⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣯⣭⣭⣟⣿⣿⣿⣭⠽⣿⣷⡿⣶⣿⠽⣧⡦⠀⠀⠀⠀⡆⠉⠲⠀⠠⠄⠒⠒⠊⠁⠀⠉⠲⢄⡀
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢂⢱⢼⣻⣿⣿⡿⣿⣶⣾⣿⢻⣿⣿⣿⣿⣿⣿⣿⢋⠹⠓⠂⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠤⠤⠤⠖⠚
⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⠘⠛⠠⠟⢿⣿⠿⡿⣽⣿⣭⡭⣿⡿⠟⢿⠛⠋⣋⣀⡤⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠶⠶⠦⠤⠬⠤⠤⠤⠼⠵⠒⢒⣚⡋⠍⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⢉⣴⣿⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⣠⣾⣿⣿⠟⣵⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠿⠿⢿⡿⠟⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⣀⣀⡀⠀⠈⠿⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠜⠂⠉⠁⠄⠂⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⣀⠀⠀⠀⠀⠀⠀⢠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣧⠳⣤⡴⠂⠀⠀⣠⢿⠞⠀⠘⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠉⠘⢿⣇⠀⠀⢀⣰⣞⡅⠀⠀⣰⠃⠀⠈⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠙⢻⠁⣙⣿⡁⠀⢀⡰⠃⠀⠀⠀⠀⠀⠈⢳⠤⣄
⠀⠀⠀⠀⠀⠀⠠⢤⣄⣠⡤⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢿⡟⢱⡴⠋⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠠⣈⣵⠖⢷⠢⠄⣀⠀⠀⠀⠀⠀⠀⠀⣸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠰⡄⢚⣧⡜⣆⠀⢨⠳⣄⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⠇⠈⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣴⣛⣁⣤⡽⡇⠈⠳⠄⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠩⠉⠁⠀⠀⢧⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣦⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠈⢑⠂⣸⡦⠀⠀⠀⢀⣀⢀⢀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡇
⣀⣀⣀⣀⣀⣀⡄⣸⣿⣿⣷⣦⣴⣿⣿⣷⣮⣷⡼⢤⠀⢿⣿⣿⣿⡇⢀⣀⣀⣀⣀⣀⣸⣄⣀⣀⣀⣀⣀⣠⣇⣀⣀
⠀⠀⠀⠀⠀⠀⣶⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⡒⠂⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠠⠄
⠀⠀⠀⠀⢀⠔⠀⠖⠒⠉⠉⠒⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⡠⠠⠈⢀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀
⠀⠠⡀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⢦⡀⠀⠀⠀⠠⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⠄⡀⡀⠠⡄⠘⠒⠀⢀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠀⠀⡀⠀⡀
⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡌⠠⠁
⠀⠀⠌⠏⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡸⢎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠤⠄⠀⠀⠀⠀⠀⠀⡘⠋
⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠙⡶⠃⠀⠀⠀⠀⠈⠘⢝⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⢤⠀⡠⠃
⠀⠂⠀⠀⠀⡢⢮⠁⠃⡀⢀⣶⠐⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⡴⠉⠱⠶⠠⠂⠐⠢⡀
⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠈⠀⠳⠢⠄⠀⠄⠀⠀⠀⠠⠖⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠑⠄⠀⠈⢀⠀⢀⠀⠀⢀⠀⡈⢀⡴⡢⢄⢀⣀⣀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡂⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠂⠀⡀⠈⠁⠀⠒⠦⢀⠀⠁⢰⠀⠀⠀⢀⢖⠅⠈⠠⠐⠒⠂⡛⠱⠌⠁⠼⠀⠀⠂⠀⠐⢀⠀⠂
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⠔⠂⠈⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠽⠆⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠁


⡂
⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣿⣿⣻⣋⣛⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣻⣿⣟⣻⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⣄⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⠀⠀⠀⢀⡴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣯⣀⣈⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣏⣀⣨⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡏⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠉⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠹⢷⠙⢿⣿⡝⠿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠛⠉⢁⡀⠀⠀⠀⠀⠀⣀⡉⠀⠀⣿⡿⠹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢁⠟⣧⠀⠘⠁⠈⠀⠀⠀⠀⠈⠀⠙⠀⢀⣟⢣⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠠⠾⠟⣿⣿⡿⠘⢤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡼⠀⢸⣿⣿⡛⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣷⠀⠀⣿⠳⣄⠀⠀⠸⣋⠙⣙⠟⠀⠀⡰⢻⡿⠀⠀⢸⣿⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⢿⣓⣦⠿⣄⡈⠓⠤⣀⣈⠉⣁⣀⠤⠊⠀⢸⠇⠀⠀⠸⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠒⠶⢶⣿⣦⣖⡛⠒⠟⠒⠛⠢⣀⣀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠨⠭⣽⡿⠉⠛⠻⢷⣦⡤⠖⡋⠭⠉⠉⠙⢿⠒⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠤⠤⣼⡋⢰⠃⠀⠀⠀⠀⢸⢠⠊⠀⠀⠀⠀⠀⠀⠻⢭⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⢸⠁⡞⠀⠀⠀⠀⠀⠈⠳⣤⣄⣀⣀⣀⣀⣤⡴⣋⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣀⠞⠊⠓⠒⠒⠒⠒⠒⠒⠒⠚⠓⠬⠭⣥⣴⣞⣚⡥⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⠐⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣁⣀⡀⢀⡤⠬⣁⣬⡆⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⠁⠐⠋⠉⠉⠉⠈⠀⠀⢈⣉⣀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⡼⠀⠀⠠⠒⠁⠀⠀⢄⡉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⠀⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⡙⠇⠀⠀⢀⡠⠔⠀⣀⣠⠄⠀⠀⠀⠀⣀⣀⢀⣀⣀⣀⣀⢐⣒⣒⣒⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠣⢢⠆⠀⠀⠐⠋⠰⣄⣭⣤⣤⢤⣶⠆⠀⠀⣬⢿⠞⣶⢼⣿⢛⣧⣻⣻⣿⢱⠏⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣈⣈⡤⠀⠀⠀⠀⠙⠻⢟⠋⠉⠈⠀⠀⠀⠁⠒⠒⠒⠒⠒⠒⢲⡖⠒⠐⠛⠂⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⢉⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⣗⡄⠀⠀⠀⢠⡟⠤⠧
⠀⠀⠀⠀⠀⠀⠀⠁⣀⠔⠀⠀⠀⠀⠀⠀⠀⣠⠘⢍⠙⢿⠁⠘⠀⠀⢀⣀⠀⡀⠈⠁⠀⢀⡰⠋⠙⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠧⠤⠄⠁⠊⠀⠀⠀⠀⠂⠁⢀⠈⠑⠒⡒⠋⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠃⢀⣀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠤⠔⠁⠉⠩⠥⢠⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⢉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠷⣶⠶⠶⠃⠈⠉⡇⠉⠉⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⠒⠂⠀⠤⠄⠤⡤⠄⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⢀⠔⠹⣦⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⡰⠃⠀⠀⠹⣧⠀⠀⠀⢸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣀⣀⡄⠶⠊⠀⠀⠀⠀⠀⠉⠛⠛⠉⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⡀⠠⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠇⠀⢀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⣴⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠠⠋⠀⠀⠤⠀⡢⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⣿⡿⠡⠀⠀⠀⡐⠀⣀⡀⠉⠀⠉⠙⠛⢳⡶⠶⠶⠀⠀⠕⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠁⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠐⠈⢄⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡁⠀⠀⠀⡁⠀⡁⠀⠀⠠⢂⡀⠀⠀⠀⠀⠠⢂⠂⡀⠀⠀⠁⠀⡀⠀⠡⡐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠆⡠⠃⠀⠀⠀⠙⣄⠄⠀⠀⠐⠌⠁⠁⠀⠀⠀⠐⠌⠅⠁⠀⠀⠰⠜⠈⡀⠀⠐⠞⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠈⡆⠀⠀⠂⠀⢀⠀⠀⠀⠀⠀⣠⡤⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠠⠊⠁⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⡲⠠⡀⠀⠢⡀⠀⠈⠈⠣⠀⠀⠀⠀⢃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠊⠈⠀⠀⡠⠈⠀⠠⠠⡘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠜⠀⠀⠀⢄⠀⠈⠂⢄⡀⡀⠑⠀⠀⠀⠀⠑⠁⠅⠘⠁⠀⠀⠀⡠⠀⠀⢀⠄⠈⠀⠀⠊⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⠊⡠⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠄⠊⡀⠢⡈⠁⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡡⠊⠀⠀⠀⠀⠑⠀⠠⡍⠀⠈⠁⡧⠄⠠⠊⠀⠀⠈⠢⡈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠀⠀⠀⢇⢀⢀⠀⡑⠀⣠⡀⠀⠀⠀⡠⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠀⠀⠀⢠⡀⣀⠀⡔⠩⠉⡘⢠⡓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⠛⠀⠀⠀⠀⣦⠀⠀⢷⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⡀⣤⡄⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⣀⠀⣤⠀⠀⢿⠀⠀⡈⢷⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⣿⠤⠙⢷⠀⠛⠧⡀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⠒⠿⠐⠀⠐⢄⠀⠀⠈⠢⡀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠄⡀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠛⡄⠀⠀⠈⠁⠂⠀⠄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠚⢦⡀⠀⠀⢀⠔⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⣠⠁⢠⣻⠂⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠄⠀⠀⠉⠛⠳⠤⢀⢀⣀⡀⠋⠀⣀⡀⠀⣀⣀⣔⣀⣀⡤⠶⠿⠓⠋⠁⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠤⠄⠀⠈⠉⠀⠉⠥⠤⠤⠀⠀⠒⠂⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣙⠢⡀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢠⠆⠀⠉⠁⢈⣄⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠏⠀⣄⠶⣮⠠⠇⠉⢶⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠛⠃⡀⠀⠀⠀⠀⠀⣄⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⢠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⡍⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠐⠡⠤⠀⢠⠇⠀⠀⣀
⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⣠⡄⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠓⢆⠁⢀⡎⠀⠀⠐⠉⠑⢄
⠀⠀⠀⠀⠀⡧⠌⠀⠀⢰⠁⠿⠆⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠃⠈⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠀⡖⠊⠀⠀⠈⠉⠁⠀⢙⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⡀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠹⡄
⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢢⠀⠁⡘⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠄⡄⠀⠀⠇⠀⠀⠀⠀⠀⠀⢣⠀⠘⣆
⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⠀⠈⢀⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠃⠀⡀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⢦⠀⠘⡆
⡔⠀⠀⠀⡴⠛⠢⢤⠤⠤⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠘⠀⠀⡇⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠸⡄
⠀⠀⠀⠘⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠥⠬⢩⣀⣈⣁⣍⡁⠐⣄⠀⠀⠀⠀⠀⣀⡀⢀⡷⠀⠘⡄
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠞⠓⠠⠄
⠈⠀⢸⠀⠀⠂⠄⠀⠀⠀⢀⠊⢈⡲⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⠤⠠⠤⠒⠁⠀⢀⠀⠀⠀⡠⠎⢀⡠⠚⠁
⠀⠀⠘⡄⠀⠘⡈⢂⡠⠔⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢀⡄⡀⠉⠐⢊⠠⠊⠁
⠀⠀⠀⢱⠀⠀⠈⠁⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠡
⣶⣶⠶⠶⠶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠶⠞⠙⠳⠞⠉⠛⠖⠛⠙⠳⣄⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣄⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠇⠀⠀⠀⠀⣰⠶⠞⠉⠉⠉⠁⠀⠀⠀⠀⠉⠛⠛⠋⠻⣄⡀⢀⡀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣀⣀⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠳⢤⣤⣴⠋⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠻⣦⡴⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠀⠀⠀⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⢹⡆
⠀⠀⠀⠀⠀⠀⢰⠏⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⠃
⠀⠀⠀⠀⠀⠀⢻⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣀⠀⠀⢀⣼⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣚⣭⣭⣿⣟⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠀⢸⣿⣳⠿⣿⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣤⣀⠀⠀⢸⠻⣿⣯⣭⣵⡞⡇⠀⠀⢀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣟⠓⠾⣇⠀⠀⠀⠀⣰⠷⠖⠛⠋⠁⣸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢹⠏⠙⠓⠾⠷⡄⢠⡾⣥⠤⠴⠶⠚⠋⠙⣏⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⡿⠀⠀⠀⠀⠀⣏⢸⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⢰⡇⠀⣀⣤⡤⠶⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠙⠛⢻⡟⠉⠀⠀⠀⠿⠾⢶⡄⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠴⠾⢷⡆⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣤⣶⣾⣁⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢶⣤⣤⣼⣧⣄⣤⢤⣤⣤⣾⠁⠀⢹⣧⣀⣀⣤⣄⣀⡀⠘⡇⢀⡴⢶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⣤⣾⣤⡤⣶⡶⣿⠀⠀⢸⡇⠙⠶⠶⠞⠁⢨⡿⠛⠻⣯⠙⢷⣿⣿⣁⣼⠋⣧⡀
⠀⠀⠀⠀⠀⣰⠏⠉⠉⣉⡽⠿⢷⡾⢯⣿⠞⠋⠀⠈⠓⠒⠛⠁⠀⠀⠀⠀⠀⠸⣧⣀⣀⡿⠀⠀⢹⡿⢻⣿⡛⠿⢯⣉⠉⠉⠉⢻⡄
⠀⠀⠀⠀⢰⠏⠀⣠⠞⠉⠀⢠⣶⣿⣞⢻⣦⣤⡤⠞⣷⠀⠀⠀⣠⣤⣤⡀⠀⠀⠈⣩⣥⣄⣠⠞⠋⣀⣾⣽⠷⢶⡄⠉⠳⡄⠀⠀⢿⡀
⠀⠀⠀⢠⡟⠀⠀⣿⠀⠀⠀⢸⣏⠻⢯⣛⡻⠶⠶⠾⣿⣦⣤⣶⡯⠬⠿⠿⠳⠶⠛⠉⠀⣈⣩⡴⠞⢉⡾⣧⡴⠟⠀⠀⠀⣿⠀⠀⠈⣷
⠀⠀⢀⡟⠀⠀⠀⠙⢧⣀⠀⠀⠙⠷⣤⣈⠙⠛⠲⠶⠦⠤⣤⣤⣤⣤⣤⡤⠴⠶⠶⠚⠛⠉⣁⣤⠶⠛⠳⣼⠇⠀⠀⣠⡼⠃⠀⠀⠀⠘⣧
⠀⠀⡾⠁⠀⠀⠀⠀⠀⠉⠛⠶⣤⣄⣀⠉⠙⠓⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠛⠋⠉⠀⠀⠀⣀⣠⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠸⣆
⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠓⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠚⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⣸⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⡞⠁⠀⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠐⠦⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢀⠀⣀⣀⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠄⠒⠒⠚⠦⠤⣄⡀⠀⠀⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠋⠁⠀⠀⠀⠠⠀⠀⠀⠀⣀⠄⠈⠑⢦⡘⠿⠿⢿⡿⠟⠛⠿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠈⠀⠀⠐⠀⠀⠈⠀⠀⠀⢀⡀⠐⠀⠙⡄⠀⠐⢀⡠⠔⢂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠐⠀⠀⠔⠀⠀⢠⠴⣄⠀⠀⠀⠀⠀⠄⠀⠀⠀⡖⢺⡁⠉⠈⠁⣀⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠊⠁
⠀⠀⠀⠀⠀⢀⠎⢀⠄⠀⠀⠀⠀⠐⠐⠒⠊⢀⡠⠔⠀⠀⠄⡀⠀⢀⡴⣤⢱⣼⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⢀⠎⢀⠀⠀⠀⠃⠀⠀⠀⠐⠒⠈⢁⠀⠀⡠⠀⠀⠀⠀⠸⣄⠉⢛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⡜⠀⠂⠀⡀⠀⠠⠀⠀⠀⠠⠂⠀⡀⠀⠀⠁⢠⠋⠉⠳⣀⣸⡴⢯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⢷⡀⠀⠀⠈⠀⢀⡀⠀⠀⣀⣀⠀⢀⡀⠀⣠⠇⠀⢀⣠⠜⠉⠀⠀⠈⠙⠓⠒⠒⠒⠒⠚⠋⠉⠁
⠀⠀⠀⠀⠈⠛⢶⣊⠉⠲⢄⣀⠤⠊⠁⠀⠉⠒⠒⢋⣁⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠙⠫⢄⣀⣀⣀⣀⣀⠤⠤⠒⠊⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢠⠴⣶⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠉⣾⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣠⣤⣀⣙⣿⣿⣿⣀⣀⣤⣤⣤⣶⣶⠶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣶⣶⣿⣿⣿⣿⡿⢿⡿⠿⠿⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠿⠿⠿⠛⠋⠉⠉⠉⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣶⣶⣦⣦⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⠿⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠙⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠠⠄⠀⠀⠀⠀⠀⠀⢀⣄⣰⡄⠀⠀⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁⠐⠿⣦⡀⡀⠀⠃⢀⠘⣻⣿⠃⠘⠛⠻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢳⣶⣦⣤⣠⣼⣿⣠⠆⡀⠛⠘⠛⠛⠚⠓⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣇⠈⠹⠻⣿⣿⣿⣿⣧⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠈⠀⡈⢻⣿⣿⣿⣿⣿⣿⣰⣾⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠈⠁⠏⠐⠀⠀⣹⠟⠛⠿⢿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣠⡄⠀⠀⢠⡾⠃⠀⠀⠀⠀⠈⠉⠛⠻⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣀⠀⠀⠋⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠘⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢿⠀⠀⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⣆⣀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⢠⡿⣏⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢧⣿⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣿⢯⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠿⠛⠙⠿⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⡟⠉⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⠠⠛⠀⠄⠀⠀⠀⠀⠀⠉⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣀⠄⠄⠒⠈⠁⠑⠂⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⢀⠠⠰⣎⡁⠉⠒⠤⣄⣀⣈⣉⣉⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠬⠱⠦⠄⣀⠉⠓⠢⣄⣀⣀⣈⣉⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⣁⠔⢺⠁⠀⠀⠀⣀⡀⠀⠀⠈⠉⠉⠀⠀⠤⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠁⡠⠊⠀⢀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠵⣈⢟⡤⣀⠀⠺⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠈⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠈⢎⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣠⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⡹⡶⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⠁⢀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡐⠀⠑⠦⡍⡴⡢⡤⣄⢠⡀⢀⣠⡀⠔⠊⠁⠀⠀⠀⠀⠀⠀⡠⠤⠤⠐⠒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠊⠀⠀⠀⡠⠐⠁⠀⠑⢌⣒⣀⣪⠞⠙⡗⢄⠀⠀⠀⠀⠀⠀⠘⠲⠤⠤⠔⠒⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠚⠓⠈⠀⠐⠒⠠⠤⣀⠀⠀⠀⠀⠀⢀⠂⠈⡀⠀⠰⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠈⡀⢘⠀⠀⠀⢀⠔⠃⠤⠐⠠
⠀⠀⠀⠂⡀⠂⡧⢀⡔⢠⣆⠀⠀⠀⠐⢤⠐⠒⠀⠋⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠇⠀⠀⠁⠀⠀⠀⠨⠀⠤⠤⠤⠀⠄
⠀⠀⡄⠠⠄⣰⡇⠈⡇⠀⡈⠀⠀⠀⠀⠘⡀⠀⠀⠚⢄⡀⠀⠀⠀⠀⢀⠁⡒⢀⠀⠀⠀⠀⠒⠒⠀⠀⠉⠅⠠⠆
⡀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠒⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⡀⠀⠀⠁⠒⠠⠤⢄⡀⡐⠀⠀⠀⠀⠀⠀⢀⡀⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⢄⠀⠀⠀⠸⡉⠀⠈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⡧⠀⠀⠈⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠓⢄⣄⣐⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⢀⡤⠂⣉⡓⠲⠄⠀⠀⠒⠂⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠁⠀⠀⠠⠄⠤⠄⠀⠀⠀⠀⠔⠒
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠁
⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣛⣛⣿⣿⣿⣿⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⡿⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠋⠉⠉⠉⠉⠀⡀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠋⠀⠀⠀⠉⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢀⡽⢿⣿⠁⠀⠐⠒⠀⢰⠑⠂⠀⢷⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠈⢧⣣⡙⠃⠀⣠⠀⠒⠼⠀⠀⠀⣼⠀⠰⣿⡇
⠀⠀⠀⠀⠀⢀⡠⠤⢄⣷⡀⠀⠉⠒⠤⠤⠔⠂⢀⠇⠀⠀⢹⣀
⠀⠀⠀⠀⠀⢈⡜⠓⣦⠞⡗⣤⡀⠀⠀⠤⠄⣠⡎⠀⠀⠀⣾⢷⡁⣀
⠀⠀⠀⠀⠀⠈⢠⡟⣁⡀⠃⠀⣹⢛⠛⢻⠙⠎⣟⢆⠀⠐⡙⠇⣸⠗
⠀⠀⠀⠀⠀⣰⠏⡏⠀⡟⣧⣼⠃⠈⡀⠈⣔⡔⡜⢮⣱⠀⢠⢸⠁
⠀⠀⠀⠀⢠⡟⡇⠇⠀⡇⠇⣇⠀⠈⢏⠖⠈⠡⢇⠌⢣⠀⡎⠸
⠀⠀⠀⠀⣾⡇⢻⡆⣀⡧⠞⡋⠀⣸⡸⢀⠔⢚⣅⣀⠞⣼⠀⢠
⠀⠀⠀⠀⣿⠃⠀⡟⠻⡄⠀⠹⣛⡿⠁⢟⢻⣿⠃⠓⢄⠠⠄⠚⠢⠄⡀
⠀⠀⠀⠀⣧⣠⡀⡇⠀⠛⣄⠀⠋⠀⡼⠸⢸⡟⠀⣀⣮⠌⠁⠀⠰⠂⢸⢿⣶⣶⣤⣤⣀⡀
⠀⠀⠀⠀⣯⠉⠙⠛⠛⠓⠻⢦⡀⣰⣷⣿⣿⣿⣿⣏⠈⠢⢀⣀⠠⠔⢃⣼⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⠿⣿⠁
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠐⠚⠓⠲⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⣿
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⠻⣝⠻⢿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣤⣤⣤⣤⣿⣀⡀⠀⠈⢦⢸⣏⣹⣿⡟⠁⢸⡇⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢦⡀⢈⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣷⣿⡿⠿⠋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠿⣯⣾⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠈⠉⠉⠉⠉⠉⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠶⠶⢤⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡍⠀⠀⠀⠀⠀⠀⠈⢳⡀⣀⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⣀⠾⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠈⠉⠈⠀⠙⠃⠒⠀⠘⠃⠀⠀⡀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⣀⠀⢀⠂⡐⢁⠀⠉⠉⠑⠎⠁⠀⠉⠢⠀⢇⠀⠀⢀⣇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⠳⠞⠀⠀⢠⠀⠴⣶⠀⠀⠀⣶⠄⢀⠂⠈⠶⠂⠀⠀⠛⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠀⡐⠉⡃⠀⢀⠴⡄⠀⠀⠀⢸⡄⠀⠀⠀⢦⠀⠀⠀⣀⠀⠂⠀⠀⠐⠋⠉⠉⠰⡅⠀⠀⠀⠀⠞⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠁⢀⠃⠀⠈⠀⠃⠀⠆⢸⠀⠻⠤⢀⣀⡀⠀⣴⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠰⣤⣄⠰⣄⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠀⠀⣠
⠀⠀⠀⠘⠀⠸⠀⢀⠁⠸⠀⢰⠀⡄⠀⠀⠀⠀⢸⠁⠀⢈⠀⠄⠀⠀⠀⠀⠈⠁⠀⠀⠋⠀⢀⠛⠛⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⢀
⠀⠀⠀⠇⠰⣇⢀⠇⠀⠇⢠⡄⢠⠁⠀⠀⠀⠀⢸⠀⠀⢻⣷⣤⣁⠒⠀⠤⠀⠀⠀⠄⠀⣒⣡⣴⠆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⡌
⠀⠀⠀⠈⠢⣀⠁⠀⠀⠃⠚⢀⠌⠀⠀⠀⠀⠀⠘⣇⠀⠘⣿⡿⠿⠿⣿⠶⠲⠶⣶⣾⣿⣿⣿⠏⠀⢀⡟⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠘⠀⠀⡘⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠉⢀⡀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠋⠀⢀⡞⠀⠠⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡈
⠀⠀⠀⠤⠠⡧⠀⠉⠀⠀⠈⠑⡄⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠈⢠⣀⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⡀⠁
⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠈⠙⠛⠛⠒⠒⠀⠐⠀⢺⠋⠀⠙⢦⡀⠀⠀⡸⠉⠀⠀⠀⢡⠀⠀⠀⠔⠁⠀⠀⠙⠢⢄⡁
⠠⠈⠑⠂⠄⣤⠻⠋⠉⠁⠀⠠⠃⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠹⡄⠉⠳⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡄⠀⠀⠀⣸⠇
⡄⠢⠄⢀⣀⠰⡏⠀⠐⠀⠰⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠙⢄⠀⠀⠀⠀⠀⢀⡀⠀⠸⠗⠒⠈⠁⢀⡇
⠁⠄⣀⣀⠍⢠⠁⠀⡀⠀⠀⠸⡀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⢹⡄⠀⠀⢙⡆⠀⠀⣀⠎⠀⠀⡆⠀⠁⠀⠘⢀⡀
⠀⠀⡈⠀⠀⡈⠀⠉⢻⡀⠀⠀⠈⠢⢀⡀⢠⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠈⣧⠀⠀⠊⠁⠀⠉⠀⠀⡄⢫⠀⠀⠉⠀⠈⡨⠃
⠀⠀⠁⠀⠀⠁⠀⠀⠀⠱⣄⠀⠀⠀⠈⠁⣿⠀⢱⠀⠀⠀⠀⠀⠠⠤⠤⠤⠔⠊⠁⡀⠀⢀⠔⠁⠀⠀⠀⢹⡆⠀⠀⠀⠀⢀⡠⠈⠀⢈⠀⠀⠘
⠀⠐⣄⢀⠇⠀⠀⠀⠀⠀⠈⠳⢤⡀⣀⡜⠁⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⢀⠔⠁⠀⠀⠀⠀⠀⠘⡷⠀⠒⠀⠈⠁⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⡀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠙⠓⠠⠤⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠤⠀⠀⠁⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢤⣀⡀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣀⣀⠀⠤⠞⠉⠘⠒⠦⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠤⡀⠀⠀⠀⠀⠀⠀⠀⢁⠄⠀⠀⠈⠀⠈⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁⠂⠁⠄⠈⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠈⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⣀⠀⠠⢄⡊⠤⠶⠐⠒⢀⣀⠀⠮⠠⠤⠄⠀⠀⠀⠆⠀⠰⠤⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡄⢤⠀⠤⢤⠀⠀⠒⠒⠀⠈⣩⠀⠒⠂⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠉⠀⠈⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⣀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢠⠀⠈⠓⠀⠉⠟⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀⠀⠈⡌⢿⣦⣐⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⡿⣦⣤⠀⠀⠀⠀⠑⢦⡈⠻⢿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⠀⠀⠈⣿⡷⣼⡟⠀⠄⠀⠀⠀⠱⣄⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⡻⣶⡤⡿⣿⡟⠎⠉⢾⣷⣠⡀⠀⠀⠘⡆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡄⠀⠀⠀⠀⠀⠀⢀⣿⡿⠷⠶⣴⡌⠠⣅⠀⠀⠀⢹⣿⣿⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⣿⡆⠀⠀⠀⠋⠹⠶⢋⣀⣤⣤⡈⢳⠀⢉⣷⢧⡶⣺⣹⡿⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⣀⣴⣿⣷⣿⣆⡿⣦⡹⠿⣿⣧⢹⡿⠾⠛⠁⢀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀⠀⠀⠘⠀⠀⠀⠾⠟⠉⣸⠛⠛⠋⢰⣽⣯⠈⠋⠙⠋⠀⠀⠀⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⢠⣴⢆⡄⠘⠁⠀⠀⠀⠈⣿⡿⠀⠀⠀⢀⣀⡦⣀⣴⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⢦⡀⠀⠀⠟⣀⡀⠀⠀⠀⢀⡼⠛⠠⣄⠢⢴⣿⣿⣧⡿⡟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣷⠀⠀⠐⣌⠻⢶⣤⣀⣘⠁⠀⠀⢀⣀⣨⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⠸⣷⡀⠀⠉⠙⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⢉⣩⠽⠤⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠈⠉⠐⠁⠀⠀⠀⠀⠀⠀⠒⠒⠮⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢆⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡳⠊⠀⠀⠀⠀⠀⠀⣀⡤⠀⣤⠀⡄⢸⠢⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠂⢠⠀⠀⠀⠌⠁⠈⠁⢉⡹⠒⠙⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠒⠉⡴⠀⠀⡰⠋⠉⢡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣒⢴⠇⠀⠀⢇⡀⢴⡁⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠄⠀⠀⠀⠀⠀⠀⠙⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠃⠀⠀⠀⠀⡠⠦⢄⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡄⠀⠀⡠⠊⠀⠀⠀⠈⠐⠂⠠⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⢽⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠳⣍⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠀⠒⠀⠠⣠⡥⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡧⠒⠉⠁⠁⠀⠀⠇⠀⠘⠧⡀⠀⢀⠤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⠌⠀⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠱⠙⠘⠦⣎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠴⠴⠁⠈⡑⠀⠀⠀⣸⠁⠀⠀⢣⡀⢀⠐⠀⠧⠌⠍⠠⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡸⠉⢒⠚⡀⢠⠉⠐⡤⢰⣀⣀⣀⡀⠀⢼⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠐⠥⠄⡈⠉⠃⠁⠀⢰⠃⣆⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⣤⣄⣿⣄⢢⡀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⡏⠀⠹⠿⠏⠉⣶⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠜⠀⠀⠀⢀⠠⢪⢋⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠥⠄⠀⠵

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⣤⣴⠶⠞⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟⠀⠀⢀⣀⣈⣁⣸⣛⡃⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠉⠉⠁⠀⠀⣿⣉⣉⣉⣉⣉⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉⠉⢹⠇⠀⣾⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⢰⡞⠛⠛⠻⣆⠀⠀⣿⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⢀⣾⡃⠀⠀⢸⡷⠶⠶⠾⣿⠀⠀⣿⠀⠀⢀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠘⣧⠈⠉⠛⠻⣇⠘⠷⠶⠶⠾⠋⠘⠛⠋⢰⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⠙⠻⠶⢶⡀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠌⠻⠶⠴⠾⠃⢤⣤⡀⠀⠀⠀⠀⠈⠻⢦⣤⣤⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⣰⠆⠀⠀⠀⢰⣄⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⢀⡾⠋⠀⠀⠀⠀⠀⠹⣦⡀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢦⣄⠈⠛⠶⣤⡀⠀⣀⣴⠾⠋⢁⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣶⣄⡈⠙⠿⣏⣀⣤⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠈⠙⠷⠶⠟⠋⠁⠹⠆




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠚⠉⠛⠉⠙⠛⠉⠳⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠃⠀⠀⠀⣀⠀⢀⣤⣀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠘⠲⡀
⠀⠀⠀⠀⠀⠀⠀⡤⠃⠀⠀⣀⡴⠚⠉⠉⠉⠀⠀⠀⠈⠉⠋⠙⢦⣤⣄⡀⠀⢀⣧⡀
⠀⠀⠀⠀⠀⢀⡴⣇⣀⡶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠻⡀⢹⡀
⠀⠀⠀⠀⠀⢸⢀⡞⠁⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠐⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⡀
⠀⠀⠀⠀⢠⠏⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡞⠁
⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣆⠀⢀⡰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⢀⡤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⢸⣿⣿⣟⣻⡇⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣸⠛⠛⠓⠚⣇⣀⣠⠤⣶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡾⠶⠭⣦⡀⢀⣜⣉⣠⡤⠴⢿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⢠⡇⠀⠀⠀⣹⠋⠀⠀⠀⠀⠀⠘⡇⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢠⣼⣀⡤⠖⠚⣇⣸⡀⠀⠀⠀⠀⠀⢹⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⡟⠀⠀⠀⠉⣩⡿⡄⠀⠀⠀⠀⠘⡇⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⠀⠀⠀⣇⠀⠀⠀⠀⣤⣶⣅⠀⠀⠀⠀⠀⢹⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⢹⡖⠒⣛⣶⠒⠲⡖⣇⣀⣸⠷⢶⣶⣶⢤⣈⣇⡴⢻⢷
⠀⠀⠀⠀⡴⠒⠒⣲⡾⣭⣿⣿⠽⠋⠙⠤⠴⠃⠀⠉⠁⠀⢯⠀⣸⡇⠈⢿⡷⣿⠾⣷⡒⠒⠒⢦
⠀⠀⠀⣸⠃⣠⠞⠁⢠⣾⣷⣛⣦⡤⢖⣦⠀⠀⣠⣤⣄⠀⢈⣭⣭⣤⠖⢋⣴⡿⠓⡆⠙⢢⡀⠈⢧
⠀⠀⣰⠃⠀⢻⡀⠀⠘⢯⡙⠻⢭⣛⣛⣛⠓⠛⠓⠛⢋⣉⣉⣠⡤⠴⠚⣩⣾⣳⠋⠁⠀⣰⠃⠀⠘⣆
⠀⢠⠇⠀⠀⠀⠙⠲⢤⣀⡉⠓⠶⠤⣬⣉⣉⣉⣉⣉⣉⣉⣥⠤⠴⠒⠋⠁⠀⣉⣠⠴⠚⠁⠀⠀⠀⠸⡆
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠴⠶⠒⠒⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠤⠤⠤⠤⢤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⢋⣉⡤⠤⠤⠐⠒⠂⠠⢤⣄⣀⡄⠈⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢛⡡⠔⠊⢉⡤⠴⢒⣉⣉⠉⠀⢀⣉⡄⣿⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠔⠉⢀⣤⠾⠕⠒⠉⠁⠀⢀⣀⡤⢶⡏⣸⢿⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢼⣅⣀⣔⠉⠀⠤⢤⣤⣴⡖⠚⠋⠉⠀⣾⢠⢛⢨⠳⣄⠀⠀⢀⡾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠉⠙⠛⡛⠓⠒⠛⠛⠛⠛⠃⢠⣇⢨⠘⡈⢦⡀⠉⠛⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠐⠋⠛⠀⠐⠛⠳⠀⠀⠀⢸⡟⣎⠣⣜⠢⣍⣲⣶⡶⢟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢠⣦⢰⠀⣠⡄⠀⠀⠀⠈⢷⣌⠓⢦⣭⢷⣆⠐⠒⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⢘⡷⠃⠘⠟⠁⠀⠀⠀⠀⠀⠙⠻⢋⡴⠂⢸⣆⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⡈⠛⠂⠀⠀⢀⡴⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣎⢿⣉⣉⣉⣉⡓⠦⡄⠘⣧⠀⠀⠀⠹⢤⣄⣤⠶⢻⠃⠀⠀⠀⠀⠀⠐⣦⣀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣴⣿⣮⣿⣿⣿⣿⣿⣷⣿⠀⠈⠳⣄⠀⠀⠀⢸⠀⣠⠇⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠿⠿⣿⣿⣿⡿⠿⠋⡇⠀⠀⠀⠈⠳⢦⡀⠈⠉⠁⠀⠀⠀⢀⣀⣤⠄⠀⢹⡓⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠙⠳⣤⣤⡴⠶⢚⣻⣿⠀⠀⠀⣸⠇⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⢋⡁⢸⡆⣴⠚⠁⠀⢠⠏⠉⠛⠛⠛⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣇⣠⠤⣄⡴⢤⣀⣤⡀⣰⠏⠁⢹⡀⣹⣼⣧⡆⠱⡄⢀⡏⠀⠀⢀⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢼⣄⡀⠀⠀⠀⠀⠀⢸⡯⣭⠗⠚⣉⣭⡿⢊⣽⣿⠠⡀⠘⠿⠓⠺⠿⠒⠚⣹⡾⠀⠀⣄⢸⠇
⠀⠀⠀⠀⠀⠀⡾⡏⠀⠀⠀⠈⠙⢦⡀⠀⢀⣠⠼⠿⣥⣶⠏⠩⠿⣏⠷⢶⡶⠾⢿⣖⢦⡈⠉⠒⠊⠉⠀⠀⠀⠀⠹⣾⡀
⠀⠀⠀⠀⠀⠀⣧⢳⡀⠀⠀⠀⠀⠀⠙⢦⣯⣾⣿⣿⣿⣮⣿⣗⡶⠦⣍⣻⡖⢮⡝⣻⣿⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⡞⢿⢧⡙⣆⠀⠀⠀⠀⠀⠈⢻⡍⠉⠉⠉⣹⣋⣧⣴⣿⡭⠷⠛⠋⠉⠉⠉⢳⡀⠀⢳⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠠⡇⠈⢳⡳⣌⠳⣄⠀⠀⠀⠀⠀⡷⣤⠞⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢛⡆⢸⡇⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢳⡀⠀⠹⣌⠳⣌⠳⢤⣤⣤⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠔⢚⡩⢔⣿⠔⠛⢛⡉⠓⣄⡀⢀⣀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠈⠳⢬⣳⣄⡀⢆⡽⢩⠷⢤⣤⣤⣤⡤⠶⣚⣫⡭⠶⢒⣫⡥⠾⠧⢴⡾⠛⠉⠀⠀⠈⢿⠛⢤⠙⠛⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠈⠉⠉⠉⠀⠈⠲⣤⡤⠖⣒⣚⣩⡭⠴⠒⠛⠉⠁⢀⣠⠶⢋⣠⠀⠀⠀⠀⠀⢸⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣄⣀⡀⠀⠀⠈⠉⠉⠉⠉⠀⠀⣀⣀⣤⠴⠾⣟⣛⣒⡯⠯⣁⣠⣤⠴⣶⣟⣁⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠚⠋⠉⠉⠉⠁⠀⠀⠰⣏⣭⣭⣤⠞⠉⠀⠀⠀⠀⠻⢬⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠉⢆⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡠⠐⠀⠒⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠋⠀⠀⠀⠀⡾⢈⡄⠔⠊⠙⠒⠺⠒⠐⠲⢤⡊⠀⣿⠀⠀⠀⠀⠙⠒⠠⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠗⠁⠀⠀⡊⠀⠀⠀⠀⣈⡂⠀⠉⠲⠃⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠱⠀⢀⠊⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢀⠀⢄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡇⣸⡷⡄
⠀⠀⠐⣆⠈⠇⠀⢀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠴⠿⣾⡇⢻⠷⠛⠀⠄⠀⡀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆
⠀⠀⠀⠃⠀⡀⢠⠈⠆⡄⠀⠑⢢⣄⣀⣀⢀⡠⢊⣀⠠⠤⠤⣀⠀⠀⠀⠀⠀⠀⠄⣑⣄⡀⠀⢀⠔⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢠⠋⠀⠀⠀⠀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡍⠉⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠆⠀⠀⠀⠀⢠⠃⠀⠀⠀⠠⣿⣿⡆⠀⢀⣾⣿⣦⠀⠀⠀⢸⠀⢆⠀⠀⠀⢠⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠠⠀⠀⠀⠀⡐⠀⢦⠀⠀⠀⠀⠙⠋⠁⠀⠀⠛⠛⠃⠀⠀⢀⠜⠀⡨⠓⠊⠀⠀⠇⢀⠃⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⠀⠈⠒⠈⠳⣄⠀⠉⠒⠀⠀⠂⠀⠀⠐⠀⠀⠤⠤⠐⠒⠁⡠⠊⠀⠀⠀⠀⠀⣧⣾⣾⣏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢺⡀⠇⠩⣅⡠⠚⠉⢁⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠊⠀⠀⠀⠀⠀⠀⣐⡘⠛⠻⠛
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠙⢂⠀⠑⠄⡉⠁⠔⠊⣷⣤⡀⠀⠀⠀⣀⡀⢀⣴⠟⠀⠀⠀⠀⠀⠀⡖⠉⠉⠉⠀⠂
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠺⠿⠛⠀⢀⣦⠀⠀⠘⣷⣦⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡴⠒⢩⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠀⡜
⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠀⠑⡄
⠀⠀⠀⠀⠀⠀⡿⢍⡐⠢⡀⠀⠈⠂⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠉⠢⡍⠐⡒⠂⠀⠀⠀⠈⠢⠄⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠄⠀⠀⠤⠄⠀⠉⠉⠁⠀⠀⢀⣤⠀⠊
⠀⠀⠀⠀⠀⠀⠀⡆⠆⠀⠀⠀⠈⠢⣄⣲⡀⠐⢄⡀⠀⠀⢀⣀⡀⠈⠑⠈⠁⠀⠀⠈⠀⠀⠀⣀⠤⠤⢀⣀⡠⢒⡬⢽⠋⡖
⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣤⣀⠀⠀⠀⠈⠂⠈⠑⠊⠉⠿⠒⡶⠨⢇⣀⣒⠭⠉⠩⠅⣂⣒⠮⠀⣶⠒⠦⠶⠂⠁⠀⢘⣠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣶⣤⣀⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣂⣨⣠⣤⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⠷⠶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⡿⠿⠿⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠲⠀⠆⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢬⠁⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢀⡔⠂⠀⠀⠒⠠⢄⡀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⢀⠎⠔⠊⠁⠀⠀⠤⡀⠉⢢⡀⠀⢥⠂
⠀⠀⠀⡐⢢⠀⡀⢄⠀⢀⠀⠀⠀⠀⠀⠇⠀⡌⠀⠀⠀⢠⠀⠀⠀⠀⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉
⠀⠀⢀⠁⡌⢀⠀⡌⢀⠁⠃⠀⠀⠀⠀⡇⠀⠃⠀⠀⣤⡀⠀⣀⠀⠀⡇⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢠
⠀⠀⠀⢠⠁⠀⢠⠀⡈⠘⠀⠀⠀⠀⢀⠗⢀⡄⠀⠀⠀⠀⠀⠉⠀⠀⠘⠀⣀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠐⠄⡉⠁⠐⠐⢁⠀⠀⠀⠀⠀⡎⠀⠸⣇⠀⠀⠀⠀⠄⠀⠰⠦⠀⠈⠉⠀⢀⡅⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⢁⠠⠅⠠⢀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠀⠀⢀⠀⠀⣁⣠⡄⢠⣏⣀⠂⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠘⡁⠀⠀⠀⠈⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢼⣿⣿⣿⣿⡿⠀⡼⠀⠀⡀⠈⠁⠀⡀⠀⠀⠀⠐⠣⢀⡒
⠀⠑⠀⠤⣜⠆⠈⠀⠠⠀⠀⠀⠀⢠⢷⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠜⠀⠃⠘⢄⠀⠀⠀⠀⠀⠀⢠⡀⠀⢀⠧⡀
⠊⠁⠀⡄⠛⠀⠉⠀⢃⠀⠀⠀⢀⡁⠀⠳⡀⠀⠀⠀⠀⠀⠀⠠⠊⠀⢀⡕⡄⠈⠢⠀⠀⠀⢀⠄⠀⠋⠉⠀⠀⠆
⠀⠈⠉⢀⠃⠐⢇⠀⠈⠢⡀⠀⣸⠇⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⢹⠀⠀⠖⠀⠂⠁⡰⠞⠀⠂⠀⢐⠃
⠀⠀⠀⡈⠀⠀⠈⢦⡀⠀⠈⢙⠇⠳⡀⠀⠀⠀⠀⠤⠄⠊⡁⠀⠀⠊⠀⠀⠈⡇⠀⠀⠀⡠⠈⠀⢇⣀⠈⠈⠁
⠀⠐⠒⠁⠀⠀⠀⠀⠉⠒⡤⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⡠⠊⠀⠀⠀⠀⠀⢹⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⠑⠒⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠓⠦⠤⠄⠄⠀⠲⠆⠀⠄⠀⠂⠈⠁⠉⠉⠑⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⡀⠀⠀⠄⠂⠅⠚⠁⠀⠀⠲⠃⠄⠀⠄⠠⠀⠄⠠⠟
⣀⣀⣀⣀⣀⣈⣓⣈⣄⣉⣅⣈⣀⣃⣀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠻⢿⠛⠿⠿⡿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠀⠠⠖⡡⠐⠒⠒⠠⡙⢦⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄
⠀⠀⠀⠀⠀⠀⡠⢂⠥⢂⠈⠌⢁⠠⠤⢀⠀⠀⠀⡀⠀⠀⠈⠂⢃⠀⢀⡀⢞⠃⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⡠⡪⢚⠕⠈⠀⠐⢊⡔⠂⠀⢨⡇⠢⠀⠀⠐⠄⠀⡀⠈⠀⠀⠹⡄⢣⠁⠀⠳⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢀⠌⠔⠔⠑⠀⠀⠀⡀⠈⠂⢊⠔⠋⠈⠓⠀⠀⠈⢆⠀⠡⠀⠀⠀⠀⠰⠀⠃⠳⡘⣆⣄⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂
⠀⠀⡌⠀⠀⠀⠀⠀⠁⠀⠙⢄⡉⠀⠀⠀⠆⠀⠆⠀⠀⠀⠂⠀⢀⠀⠀⠀⠀⢣⠀⠀⠈⠀⠁⠀⠀⠀⠒
⠀⠀⠀⢁⠀⣤⠀⠛⣀⡀⠀⠀⠉⠀⠀⠀⠁⠀⠈⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⡀⠙⠀⠘⠉⢼⣻⡆⠀⠀⠈⠀⠀⠀⠀⠈⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢴⢼⠰⠔⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⡀⠂⠀⠀⠀⠀⠀⢠⣻⠛⠀⠀⠀⠀⠀⠀⠒⢄⠀⡄⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢠⠀⠀⠀⢠⠀⠈
⠀⠀⢀⡀⠀⠄⠀⠀⠀⠈⠠⡄⠀⠇⣞⠄⠀⠂⠁⠀⠁⠐⢀⠀⠂⠐⠀⠆⠱⠠⠈⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠘⠀⠀⠀⠀⢄
⠀⠀⡜⠱⠀⢄⠀⠀⠀⠀⠀⠠⠖⠚⣿⢲⡦⢀⡄⢀⠄⠠⢂⠀⠀⠀⠀⡄⠃⠀⡀⠀⠀⠂⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠠
⠀⠀⡇⠀⠈⠀⠁⠀⡀⠀⠀⠀⠀⠀⠉⢋⠉⠅⢀⠠⢊⠔⠁⠀⠀⠀⠀⢟⠀⠰⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡄⠀⠀⡈
⠀⠀⠐⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⠀⠀⢒⣀⠠⠐⠈⠀⠀⢀⡀⠀⡀⠀⢀⣀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠌
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⢿⠀⠇⠈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠂⢉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠇⠀⡆⠀⢀⠈⠀⠀⢈⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠐⠢⢤⠂⠐⢄⠀⠀⠀⠇⠄⠀⠀⠂⠃⠀⠘⠂⠃⠀⠛⠀⠀⠇⠀⠀⠀⠀⠀⢀⡤⣤⢤⡤⠄⣲⠶⡄⣀
⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⢀⠆⠀⠀⠀⠀⠀⠐⠐⠠⠒⠀⠔⠂⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⡀⠀⠀⠀⠈⠢⢬⣎⡢⠄
⠀⠀⢀⠄⠢⠎⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠈⠈⠄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⣠⠎⠀⢢
⠀⠀⠠⠀⠀⡄⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⢢⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⢣⠀⠀⠀⠀⠀⠙⠲⣄⢸⡇
⠀⠀⠀⠀⠀⠐⠀⠀⠀⣀⠤⠄⠀⠀⠀⠀⠀⢠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠑⠊⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⢂⡵⢶⠒⢦⠀⠀⠀⠈⢹
⠀⠀⠀⠀⠀⡂⢡⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠰⠦⢴⣒⣐⠀⡠⠶⠆⠀⠄⠀⠀⡐⡀⣄⣘⠶⢞⡴⠀⠑⡄⠀⣠⠇
⠀⠀⠀⠀⠀⠘⠉⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡜⣯⠍⠄⠀⠀⠀⠃⠁⠀⠀⠀⠀⠀⠺⠀⠀⠏⠀⠀⠀⠈⡁⠀⠀⠀⡼⠜⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢼⠀⠀⠀⠀⠀⠀⢰⠄⠉⡄⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⢄⠞⠁
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠛⠄⠀⢌⠀⠀⠀⠀⠀⠀⢸⠀⠀⡁⡼⠇⠀⠀⢀⡠⠒⠉⣀⠤⠀⠀⠀⠀⠀⠉⠀⠀⠚⠛⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠔⠒⠒⠶⠤⢀⠀⠄⠀⠀⠀⠀⠀⢸⠀⡏⠁⡇⠀⠠⠚⠁⠀⢀⠄⠃⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⠃⢀⠈⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠺⠄⡰⠃⡀⠰⠅⠒⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣻⣿⠿⠿⢿⣶⣶⣤⣀⠀⠀⠀⠀⠀⢠⢤⣄⡀
⠀⠀⠀⢰⣷⣄⠀⠀⠀⢠⣴⣶⣶⣿⣟⡿⢾⣝⠻⣷⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣷⣄⠀⠀⠀⡏⡞⡞⡟
⠀⠀⠀⠀⢿⠘⣧⠀⢠⣿⢫⣿⣿⠉⠙⠻⣶⣝⡻⣶⣭⣝⣛⣛⣟⣛⣛⣿⡿⣟⣽⣿⣿⡀⠀⢸⠃⠛⢳⠃
⠀⠀⠀⠀⠘⣧⠘⣇⢸⣿⣾⢹⣿⠀⠀⠀⠀⠙⠿⣶⣮⣭⣽⣛⣻⣯⣭⣵⣾⠿⠛⠉⢿⣿⡄⢸⠀⣠⠏
⠀⠀⠀⠀⢀⣿⡄⢸⣿⣧⣿⣿⠏⢠⣤⣤⣄⠀⠀⠀⠉⠉⠛⠛⠛⢉⣉⣉⡀⠀⠀⠀⠸⣿⡷⣾⣼⡁
⠀⠀⠀⠀⢸⠀⣷⠀⢿⡟⠛⠁⠀⢀⡞⣉⡉⠻⡟⠛⠀⠀⠈⠛⣿⣋⡉⠙⣏⠁⠀⠀⠀⢿⢠⣧⡏⡷
⠀⠀⠀⠀⠸⣇⢸⡇⢸⡇⠀⠀⠀⢸⡟⣻⣿⡆⣿⠀⠀⠀⠀⢰⡟⣻⣿⡆⢹⠀⠀⠀⠀⠸⣿⣼⣰⠇
⠀⠀⠀⠀⠀⠘⢷⣿⠀⣧⠀⠀⠀⠘⣧⣽⣿⣇⣿⠀⢀⣀⡀⢈⣧⣼⣿⣃⣾⠀⠀⠀⠀⢠⣇⡿⠋
⠀⠀⠀⠀⠀⠀⢨⡿⡆⢿⠀⣠⠶⣀⣭⣉⡁⠀⠀⠀⠋⠐⠙⣀⣭⣍⣉⣩⣭⣘⢷⡄⠀⣸⣹⡄
⠀⠀⠀⠀⠀⠀⢸⡄⣷⢸⠀⠃⠈⠁⠙⢧⣹⠽⢿⢷⣶⣲⠞⠁⡠⢈⡧⠋⠀⠀⠀⠃⢀⣯⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣷⣿⠿⠶⠶⢾⣹⡆⠀⠈⠛⠶⠶⠾⠥⠴⠾⠛⠉⣦⣀⠀⣾⣹⠶⠾⠿⢿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣯⢠⠤⠶⠖⢻⣯⣄⣀⣀⣀⡀⠉⠛⠛⠋⠀⠀⣀⡙⠟⠀⡬⣿⠓⠶⠦⢤⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠒⠚⢋⣉⡷⠀⠈⠉⠻⢥⣀⣀⣀⣀⣠⡼⠟⠁⠀⠀⢘⣏⣙⡓⠲⠆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠛⢋⣉⣩⠟⠀⠀⠀⠀⢀⡈⠉⠉⢉⣁⡤⠖⢁⠀⠀⠸⢯⣉⣉⠛⠃⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢧⣉⣽⢿⣄⠀⠀⠀⠀⠀⠀⠉⠉⢭⠭⠤⠖⠚⠁⠀⠀⠀⢀⣸⣿⣍⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⢤⣄⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⣴⠾⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠒⠢⢤⣈⠛⠭⢉⠉⠲⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫⠤⢤⣀⡀⠀⠀⠀⠙⠫⠙⠓⠆⠀⠙⠪⣫⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⠉⠉⠉⠉⠉⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢨⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠈⢉⠩⠍⠛⠛⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⡀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠥⠤⠐⢲⣶⠖⠀⠀⠰⡀⠀⠀⠀⠉⠉⠀⠈⡂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⠙⡀⠀⠂⠀⠐⠄⠔⠑⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠋⠁⢀⣲⣴⣒⣚⣓⢢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠢⡤⠀⠚⠣⡥⢤⠤⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡠⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⣀⠀⠇⠀⠀⠘⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠖⠢⠈⢰⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠤⠖⢻⢸⠀⠀⠀⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠘⡆⠀⠀⠀⠀⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡤⠈⠓⠤⢄⣀⠀⠠⠖⠁⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡇⠀⢡⠊⠉⢱⠀⡁⠀⠀⠀⢀⠤⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠿⠀⠀⠀⠀⡃⠀⠘⣇⠔⠁⠀⢣⠀⢠⡂⣽⣿⠛⣳⠾⣣⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡇⠀⢀⡤⢤⣶⠀⠀⠈⠑⠀⠀⢸⣶⢿⠷⠧⠒⠉⠀⠄⠿⣽⡺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠉⣞⠿⢲⣶⠒⢢⣶⡖⠺⣥⣎⠀⠀⠀⠀⠘⢤⡀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡎⠀⠀⠀⡠⠉⠳⣜⠛⣤⣬⠛⢣⣤⡟⠁⠉⠢⡀⠀⠀⠀⠙⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢼⣀⠗⠢⡀⢎⡀⢀⠀⠈⠲⡉⢹⣶⣎⠟⠀⠄⠀⠀⣨⠆⠀⡔⠘⢗⠋⢑⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣀⣼⠀⡄⢅⡠⢿⠏⠀⠀⠀⠙⢿⢠⠋⠀⠀⠈⣶⣯⣀⣀⣀⠀⠂⢨⠃⠉⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠘⢄⣀⠨⡏⢠⠀⠠⠞⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠸⠦⠤⠔⠒⢦⢈⡏⣖⢈⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠐⠦⣖⡗⢊⠔⠂⣐⡀⠀⠀⠀⠀⠀⠀⠤⠀⢀⣐⣒⠒⠒⠒⠒⠜⠚⠆⠃⠀⠈⠙⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⢀⠔⠁⢰⣁⠰⠖⠢⠥⠄⠀⠀⠀⠀⠉⠉⠑⠆⢈⣱⠀⠀⠈⢆⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠛⠛⠛⠁⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⢀⠌⠀⠀⠀⠀⠀⠀⠐⠃⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠂⠀⠤⠤⠤⠤⠤⠄⠇⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠉⠉⠉⠛⠓⢤⣀
⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠀⠀⠀⠀⠤⡀⠀⠀⠀⠈⠳⡀
⠀⠀⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⢠⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠿⠃⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠈⣦⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡖⠤⢄⣀⣀⣀⣀⣀⣀⠤⠆⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⢁⠁
⠀⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡛⡺
⠢⣀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢶⣶⣬⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠈⡏⠗⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠓⠂⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⢊⠏⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⡠⠃
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⢸⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⢲⠏
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⡠⠒⠁⠀⠀⠀⠀⡌⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠁⠀⢀⠸
⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠂⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠉⠒⠂⠉


⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠂⣀⣀⣀⣀⣀⣀⣀⣈⡓⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢁⡤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠒⠤⣍⠳⣄⠀⠀⠀⠀⢠⠀⠈⠀⠀⠙⢲
⠀⠀⠀⠁⠀⠀⠀⢀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠘⡆⠀⠀⠀⢸⠉⠁⠐⠒⢀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠈⠑⠒⠦⠤⠤⠀⢀⣀⣀⠀⠀⠤⠤⠄⠀⠒⠁⣀⠜⠀⠀⠀⠀⢸⢀⠀⠀⠀⢈⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠤⠤⠄⣀⣀⣀⣀⣀⡀⠠⠤⠄⠒⠒⠈⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠈⣦⣀⣀⣤⣄⣀⣀⣠⠞⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣤⠶⠛⠉⠉⠀⠀⠀⠈⠉⠙⠳⢦⣄⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⣸⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠢⠀⠀⠀⠀⠀⢺⡗⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢿⡒⠀⠀⠀⠀⠀⣀⣀⣐⡷⠤⠿⠤⣤⡀⠀⢀⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠀⠀⠸⣅⡀⠀⠁⠀⠀⠀⢀⡿⣠⡿⣅⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣽⠦⣄⠐⠒⠈⠉⠛⠒⠶⢖⣛⡿⠟⠉⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⠀⠉⠀⠀⠀⠒⠒⠛⠉⣁⠤⠤⢄⣀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⢠⠀⠀⡈⠳⡄⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢀⡠⠷⠖⠶⢤⡀⠀⠀⠀⠀⠈⠉⠣⣀⣠⣇⣀⡟⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠉⠀⠀⠀⣀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⢈⡇
⣴⠚⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣤⠼⠦⡤⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡁⠈⢢⠱⡀
⣧⠀⠀⠘⢇⠀⠀⣀⡤⠶⢄⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠇⢠⠞⡠⠃
⠸⣦⡀⠀⢀⠏⡋⠀⣀⣠⠀⢧⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢸⠜⡆
⠀⠈⠻⠶⢥⡤⠷⠞⠋⠹⣆⠈⠳⠼⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠰⠉⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⣤⠼⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⢤⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢁⠀⠀⠉⠻⢶⣤⣄⣀⣀⣀⣀⣀⣀⣤⣴⠞⠉⠀⠀⢀⢹⡀⠀⠊⢈⡀⢀⠀⠀⠀⢀⡱⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⠀⡀⠀⢠⡞⠁⠈⠉⠉⠉⠉⠉⠉⠀⠈⢷⡀⠀⢀⠈⡟⠀⢠⡀⠀⠉⠐⠒⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠶⠟⠚⠁⠀⠀⠙⠢⢤⣀⣀⣀⣀⠤⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⢄
⠀⠀⠀⠀⢀⡀⠄⠤⠐⠒⠒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⣀⡠⠦⢀⣀⠀⠀⠐⢶⣦⠉⠀⠀⠐⠄
⠀⠀⠄⢀⡁⠀⢀⡀⠀⠀⢠⡄⠀⣀⠀⣶⠄⠀⠀⠠⠚⠉⢀⠀⠈⣍⡀⠈⠻⠂⡹⣄⠀⢤⣀
⠀⠀⠀⠘⣷⢰⣸⡇⡾⢷⣼⣧⣆⡏⣸⠏⠀⠀⠀⢰⠒⣄⡉⢀⠀⠈⢿⣿⡞⠀⢁⠘⣷⣌⣿⢿⣧⠀⠑
⠀⠀⠀⠀⠘⣿⢿⡇⣷⣼⠇⡟⣿⠰⠋⠀⠀⠀⣠⠂⠸⠟⠉⠪⡛⠙⠈⢫⡄⢀⣅⢰⣾⣿⣍⠠⡁⢘⣶⣆
⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⠀⠀⠋⢀⠀⣤⣾⡟⠀⠀⢀⠀⠀⠙⠓⣒⣃⠚⢻⣿⡿⠷⠦⣬⣕⣊⢶⣾⢿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠉⢛⣆⡤⢄⠀⠀⠀⠴⠋⠉⠀⠈⠀⠱⣮⡶⠶⠲⠿⢛⡳⣝⢷⣟⣿⠆⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠆⢼⡿⣷⣀⡌⠁⠀⢺⡐⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣦⣀⣀⠀⠀⠀⠀⠠⠐⠒⠶⡄⠀⠀⠀⠈⠆⠎⣷⠘⣿⣟⠂⢀⡼⣣⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣇⢤⠅⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢄⢀⡿⠀⢹⣿⣍⢷⠞⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣀⣀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⡞⣼⣾⠟⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⠘⢹⣿⣧⡀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠀⠀⠀⢀⡿⣄⡼⡋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⢏⡴⠻⡏⠙⣿⣦⡐⢦⣤⣤⢶⠋⡄⠀⠀⠀⢀⡞⣡⣿⡃⠁⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢰⣷⣆⠀⣼⠗⢿⢾⣿⢿⣆⠻⡽⡠⠀⡁⠀⠀⣠⡟⠀⢸⣿⣇⢁⣱⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣸⣻⣶⠀⠀⢾⣭⣤⡾⣻⣷⠐⠒⠊⠀⠀⠌⠉⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⣡⠁⠀⠸⢛⡉⣀⣡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⣽⣶⡶⡟⡿⠔⠭⠟⣤⠀⠀⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢰⠀⠠⣾⡇⡿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠉⠁⠀⢈⠛⠀⣿⡇⣷⡴⠓⠒⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣭⢉⠠⠀⠚⠁⣂⠀⣿⣇⠿⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠱⠂⠀⠄⠁⢄⠀⠐⠁⡂⠿⠃⠀⣀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣠⡄⠀⠀⢀⠈⠀⡀⢀⣀⠤⠀⠂⠁⠰⡀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠰⢀⠀⠀⠀⠐⡀⠀⠀⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣛⣭⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠻⣿⣿⡿⠿⠿⢟⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⢀⣀⡀⡀⠈⠛⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠀⢻⠿⡤⠀⠀⠀⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢀⠀⠀⠀⠀⠀⠀⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣑⡒⠊⢀⡀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣫⡓⠀⠉⠀⠀⢀⣤⠂⠘⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠤⢶⠿⣿⣧⣀⣀⣠⣴⠟⠁⢀⠀⢸⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢿⡇⠙⣿⡍⠁⢀⡄⠀⠀⡎⠀⠈⣏⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠟⠀⠀⢀⠀⠀⠀⠀⡀⠸⠇⠀⠈⣧⣤⠈⠀⠀⢸⠃⠀⠀⣿⠀⠀⠉⠓⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⡁⠀⠀⠀⠀⠀⠉⠀⠺⠁⠀⣾⠠⠞⣿⣿⠢⡀⠀⡌⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠋⠑⣄⠀⠀⠰⠀⠀⠀⡄⠀⠀⠐⠀⠀⢿⡄⢀⣿⣷⡀⠀⢰⠃⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠼⠃⠀⢀⠀⢠⡇⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⡇⠀⡎⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⠂⠀⠈⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⢰⠃⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠈⠘⡇
⠀⠀⠀⠀⠀⠀⠀⢠⠇⣄⠉⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠀⠀⠀⡜⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣞⡼⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠃⠀⠀⡼⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⡰⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠰⠁⠀⠀⠀⠀⣧⠀⣀⣀⣤⣤⣄⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⣀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⡏⢠⣿⣿⣿⣿⡋⣁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣀⣠⡼⠟⣿⠀⠀⠀⢰⡿⠀⠀⠀⠀⠸⠟⢠⡴⢀⣠⡄⠀⠀⢸⡇⠈⣿⣿⣿⡟⣼⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣈⣛⣉⣀⣀⣀⣿⠀⢀⠀⠈⠁⠀⠀⠀⠀⠀⠰⣿⣅⡉⠁⠀⠀⠀⡼⠧⠤⠬⠷⢤⡾⠿⠦⠴⠷⠶⠦⠤
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢧⣀⡈⠓⠶⠶⠶⠶⠒⠀⠀⠀⠊⣉⣁⡤⣤⠤⠊⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠷⠿⠿⠿⠿⠛⠛⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠞⠩⠖⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠓⠒⠒⠒⠒⠧⢄⡀
⠀⠀⠔⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡶⠆⠀⠀⠀⢠⣦⡀⠈⠳⣄
⠀⢘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡞⢰
⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⡈⠀⠀⡀⠀⠰⠀⡀⠀⠀⠂⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡄⡄
⠀⠃⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢃⠀⠠⡷⠀⢀⠀⡿⠀⠀⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢰⢠⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⠀⠀⠢⠀⠀⠠⠈⠀⠀⠀⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠈
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠙⠤⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⢠
⡴⠒⠶⡏⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⠈⡿⠖⠲
⢹⠤⣠⡇⠀⢀⠀⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⡄⠀⠀⣧⡠⢼
⠈⢦⣀⣑⣶⠋⢀⡤⢈⣉⠛⠒⠢⠤⢄⣀⣀⡀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣀⣀⣀⠤⠤⠒⠚⢋⣉⠠⣄⠈⢲⣖⢁⣠⠊
⠀⠸⢄⣀⣨⡟⠉⠀⠀⠀⠉⠉⠒⠒⠒⠤⠤⠤⠭⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠥⠤⠤⠔⠒⠒⠊⠉⠁⠀⠀⠈⠙⣧⣀⣁⠼
⠀⠀⠀⡏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠈⡇
⠀⠀⠀⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡀⠀⠀⠀⣀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢹⢹⡏⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠤⣤⣀⣀⡇⢸⢸⡇⣇⣀⣠⡤⠤⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠉⠀⠀⠀⠀⠀⠀⠉⠁⢸⢸⡃⠋⠁⠀⠀⠀⠀⠀⠀⠉⠲⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣭⣯⣿⣵⣭⣽⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣶⣶⣶⣾⣿⣷⣶⣶⣷⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠏⠻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣻⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣏⣉⣉⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⣠⣺⣿⣏⢻⣿⣿⣿⣿⣫⡷⣶⣦⡀
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷⣍⣛⣋⣵⣿⣿⣿⣿⣿⡄
⠀⠀⠀⣼⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠻⢿⣿⣿⣿⣷⢿⣿⣿⣟⣾⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢤⡤⠤⠄⢀⣀⠀⡸⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠣⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠀⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠂⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠈⡉⠉⠉⠁⠀⠀⠒⠂⠐⠈⠉⠉⠁⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠢⠤⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠔⢺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠃⠘⠉⠭⠽⠟⠛⠛⠛⠋⠈⠉⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠛⠻⠛⠛⠓⠒⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣠⠤⣄
⠀⠀⠀⠀⠀⢀⡴⢚⠁⡀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠈⠙⢦⡀⠀⢀⣠⡤⠖⠛⣉⠥⠄⠠⠜⠻⣦⡀
⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣞⠉⠀⠀⠀⠁⠀⠔⠊⠉⠴⠊⠩⡝⢳⣄⡀
⠠⢶⠤⣰⣇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄
⢣⡺⡤⣿⣌⠙⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣿⣠⢼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡲⠤⣄⣀
⡠⢀⠙⠺⠻⣷⣶⡤⠤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣩⡾⡃⣀⣚⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣀⣠⣤⣽⣦⣀⠤⢶⢺
⠀⠐⠣⡅⣙⢄⣋⣭⣛⣓⣶⠶⢶⣤⡤⣤⡴⠶⠶⠶⠶⠚⡟⠛⠶⠿⢱⣚⡏⠩⠈⠓⢄⡀⠀⠀⠀⠀⣀⣤⣤⣶⡶⢿⢿⠿⠿⠟⠛⠉⠉⢀⣀⣼⠿
⠀⠀⣾⣷⡔⠀⢨⠃⠂⠘⠑⠷⢶⣧⣲⡖⠃⢂⡵⠷⠒⠛⣓⣲⡒⠄⡽⣿⡀⠀⠀⠀⠀⠙⠢⣀⣴⡟⠛⢿⡟⠀⣠⣲⣷⣲⠶⢶⣒⠚⠿⢍⡳⢤⣤
⠀⠀⡟⠬⡙⠗⠦⣤⣀⠀⠀⣀⣀⠀⠀⣙⡉⠀⢀⠭⠒⣈⠡⠖⠴⠼⢂⣿⡁⢠⢠⡔⠀⠀⢠⣾⡇⠘⡳⡖⠀⡴⣱⡁⢸⡌⠉⢳⢦⣍⡒⠤⣉⣲⠖
⠀⢀⠛⣦⡘⠀⢀⡈⠉⠛⠖⠀⣀⣀⠀⠀⡠⠒⠉⠀⠀⠀⠀⠀⢀⣠⠎⢱⠙⢦⡸⢃⠄⠀⣾⣿⡇⠀⣷⡉⢙⡴⣝⢧⡀⣷⠀⠈⣶⣄⣹⣲⣇⡰
⠀⠐⡀⠈⠀⠩⠴⠿⠧⠤⠤⢤⣁⣉⣥⣌⡱⣶⡴⠭⢄⣒⣂⠭⠔⠋⠀⡸⠀⠀⠙⢼⠃⢠⡟⣿⣿⠀⠸⣿⡋⠵⢾⣭⣏⣻⡖⠐⡧⣄⣈⣉⣏⣹
⠀⠀⠑⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⠀⠙⢮⡇⠈⢻⠇⠀⢳⡉⠲⣤⣄⠤⣞⠛⢟⠿⣓⠊
⠀⠀⠀⠀⠈⠉⠒⠛⠛⠉⠁⠲⠶⠶⠶⠦⠤⠤⠄⠀⠐⠒⠂⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠈⢿⠭⡛⡘⣤⡉⢉⠇⠈⠑⢤⣙⡚⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⣈⣆⠱⣸⣌⣄⠉⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠛⠉⠛⠒⠦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⣀⡤⠔⠒⠒⠚⠛⠒⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢴⠋⠀⠁⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠛⠻⢗⡲⠦⢤⣄⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⢱⠦⢄⡼⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢽⣦⡀⠀⠙⠢⡀⠀⠈⠉⠓⢦⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠒⠤⡉⢱⠄
⠀⠀⠀⠀⠀⠀⢸⡄⣾⣤⠖⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠀⠙⣆⠀⠀⠀⠀⣷⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠏
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⡄⡟⠀⠀⠀⠀⠀⠀⢀⣴⣚⠉⠓⠦⡀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠋⠌⠀⠀⠀⠀⠀⠀⢀⠠⠊⡰⠃
⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⠗⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣄⠀⢹⠀⠀⠀⠀⠀⡇⠀⠀⠀⡌⠀⠀⠀⢀⠠⠔⠈⢁⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣹⡤⠶⠖⠒⠒⠶⠦⣤⣀⠀⠀⠘⢿⣿⣿⣿⡆⣸⠀⠀⠀⠀⠀⡇⠀⠀⠠⠅⠀⠐⠈⣀⡤⠴⠒⠉
⠀⠀⠀⠀⠀⢠⣾⡟⡛⣛⣻⢶⠶⣤⣀⠀⠈⠛⢦⡀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⣸⠃⠀⠀⢀⡤⠒⠊⠉
⠀⠀⠀⠀⢠⡏⠈⢷⡟⠿⣿⣿⢟⡵⠋⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⢀⡞
⠀⠀⠀⠀⢸⡇⠀⠀⠙⢶⣿⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⢀⡴⠃⠀⠀⢠⠟
⠀⢀⣤⣤⣼⣧⣄⣀⣀⡀⠯⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⣠⡴⠋⠀⠀⠀⢀⡏
⣰⢏⠖⠁⠀⠙⢿⡉⠹⡤⢺⠁⢡⡖⠂⢲⡆⠒⡖⡇⠀⢀⡼⣿⠿⣟⠦⡄⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⢿⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⢓⠚⠶⠿⠷⠾⣤⣏⣀⠞⠀⠀⠀⠀⠁⢹⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠉⠳⢄
⠘⣷⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠇⣿⣿⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣼⠃⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠳⡄
⣼⠃⠀⠀⣠⠀⢠⡦⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣾⣯⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⡄
⢿⡀⠉⠉⢁⣠⡟⠀⢫⡉⠳⢤⣼⣉⣒⡧⠝⠋⠉⢹⠟⢷⡀⢠⣀⣀⡼⢸⡆⠀⠀⠀⠀⠀⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢳
⠈⠛⠿⠿⠛⠋⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠈⢷⣄⠈⢉⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣀⠀⠀⠀⠀⡞⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠒⠋⢹⡀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⡐⠓⡄⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠈⠒⡀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣧⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠒⠲⠶⢤⣄⡀⠀⢳⡀⠈⡁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⢧⡀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢠⡄⢠⡟⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⣧⡸⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢹⣆⠀⠀⡟⠀⠀⠀⠀⠀⣼⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⠞⠙⢺⠇⠀⠀⠀⠀⢰⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢚⡋⠀⠀⠀⣰⠛⠧⢤⡟⠀⠀⠀⠀⢀⡟⠀⠀⣀⣻⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⡴⠉⡴⠊⢀⡼⠃⠀⠀⢸⠃⠀⠀⠀⢀⣾⠖⠛⠉⠉⠉⠉⠳⠆⠀⠀⠀⠀⣠⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢉⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢸⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⣃⢸⡀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠓⡏⠀⠀⠀⠀⢀⡠⢄⢴⠊⠉⢱⡀⡏⠉⠀⣁⢀⠖⠤⣄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠰⡁⠀⠈⢎⡆⠀⠀⢣⠡⠀⠀⢠⡌⠀⠀⡰⡰⠁⠉⢢
⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣼⠀⣧⣤⣤⣤⣀⣱⡀⢺⠁⠘⡀⠀⠀⣾⠀⠀⢸⠁⠀⢠⠛⠳⡄⢀⠎
⠀⠀⠀⢰⡉⢖⠉⠉⠀⠀⠀⠘⠤⠃⠀⠀⠀⠉⢁⡣⢉⣇⠀⢇⠀⠀⢹⠀⠀⡍⠀⠀⠸⠀⡸⠀⡎
⠀⠀⠀⢸⠪⠭⢉⣛⣒⣛⠛⠚⠶⠚⠛⣛⣚⣛⡉⠭⠑⡟⡀⢸⠀⠀⢸⠀⠀⠇⠀⠀⡇⠀⠇⣼
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠓⠧⣀⡇⠀⠸⠀⢸⠀⠀⠀⣃⡴⠊⠸
⠀⠀⠀⠀⡇⠀⠀⠀⣼⢷⡀⠀⠀⠀⢠⡾⣦⠀⠀⠀⢸⡀⠀⠈⠉⠒⠓⠒⠚⠒⠒⠉⠁⠀⠀⡇
⠀⠀⠀⠀⢡⠀⠀⠈⠁⠀⢁⣴⣶⣦⡉⠀⠈⠁⠀⠀⡏⡇⠀⠞⠋⠳⠀⠀⠀⠀⠞⠉⠳⠀⢰
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠘⣿⡟⣻⠁⠀⠀⠀⠀⢠⠄⢻⠀⠀⠀⠀⠀⣾⣷⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠛⠁⢀⣀⠀⣀⠀⢼⢀⣸⡀⠀⠀⠀⠀⠻⡿⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⢀⡀
⠀⠀⠀⠀⠀⢶⠀⠀⠀⠀⠀⠀⠀⡴⠋⠒⡝⡤⠖⢀⣉⣛⠫⠽⠿⣓⠒⠶⣶⣶⣶⣶⣤⣬⡿⣍⡓⠛⣉⣐⡶⣚⠭⠭⠓⢄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡃⠀⢀⡴⠛⢒⡒⢛⡒⠢⠤⠤⠒⢬⠲⣦⣄⣢⣶⠖⠥⠦⠶⠶⠛⠛⠛⠚⠲⢄⠀⠀⢸
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠉⢲⠋⠀⠀⠰⠃⠈⠁⠺⡄⠘⣀⠘⢣⡄⠰⠂⣰⠂⡤⢸⡇⣨⡃⠘⠁⠀⠀⠀⠱⡖⠁
⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠆⠤⠤⠤⢀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⡀⠤⠤⠄⠒⠒⠊⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣧⡻⣿⣿⣿⣿⣿⣿⣿⣿⢟⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣠⣭⣛⣛⡛⢛⣯⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣭⡍⣽⣾⣿⣿⣿⣿⣿⣷⣭⢩⣭⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⡿⠃⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⠿⣿⠿⣿⣿⣿⠀⠀⣷⠀⣹⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣵⣶⣭⣿⡿⠃⠀⣴⠋⣾⠁⢰⠋⠀⣾⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣛⣛⣛⠋⠀⠀⠀⣘⣁⠘⠃⠘⢃⣴⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⢤⣾⣿⣿⣿⠷⣀⣀⣾⡿⣫⣶⣿⣟⣮⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣝⡻⡿⢟⣺⣿⣿⣟⣱⣿⣾⣿⣿⣟⣸⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡀⠄⠀⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢌⡔⠊⣠⣶⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⠏⠀⢘⡕⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⠘⠁⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠐⡄⠀⠀⠀⠰⠀⡠⠄⡠⠐⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠃⣠⠈⢢⠀⠀⠀⢲⠀⠠⣅⠀⠀⣠⢃⠠⠄⠒⠒⠈⠁⠀⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠐⠀⢈⢹⣧⠀⠀⠀⠂⡼⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠄⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⣀⠀⠀⠀⠀⠀⡠⠞⠀⠀⠔⡒⢆⡉⢰⡄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠠⠂⠈⡀⠀⢠⠒⠖⠠⠤⢤⠠⣄⢀⡈⣙⣋⡉⠉⠀⠀⠀⣀⠰⠇⡾⠟⠣⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠐⠓⠶⠤⠴⠄⠤⠄⠥⠠⠃⠀⠀⣀⡤⢎⣀⣀⣀⣀⡀⣀⣀⣀⡀⣀⣈⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠘⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡪⠥⠠⢍⡹⠯⠸⣁⡆⢺⡟⠳⠿⣿⡉⠶⢂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢻⣮⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⢤⡤⠶⠚⠛⠛⠉⠙⠚⠾⠭⠶⣶⣿⣽⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠘⣿⣇⣃⡀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠐⠀⣤⣀⣀⣀⣀⡀⠠⠄⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢩⠈⠀⠀⠋⣨⠁⢃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⡆⢰⠉⢤⠌⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠘⡀⠘⡁⢡⠌⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠣⢄⡉⢳⠀⠑⠊⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⡠⠊⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠈⠐⠣⢀⡀⠀⠀⡐⠤⠀⠠⢄⡀⠔⢊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⠄⡀⠈⢹⢂⣍⠈⠁⠀⠈⡖⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠒⠼⠀⠛⢃⠀⢠⡄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢐⡀⠀⠀⠀⠀⠀⠘⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠄⠠⠐⣄⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⢵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠰⠴⠂⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡰⢁⠀⠀⠀⠀⠀⠀⠒⢺⠞⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠉⠿⠀⠐⠒⠒⠒⠒⠂⠀⠘⠒⠒⠾⠿⠽⠧⠤⠤⠤⠂

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠅⠐⠀⠁⠀⠀⠀⠀⠂⠀⣠
⠀⠀⠀⠀⠀⠀⠀⢈⣄⡀⠨⡶⠀⠀⣀⣀⣠⣴⠟⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣤⣿⣿⣷⣶⡌⢀⡁⠀⠈⠉⠋⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠟⠉⠀⠀⠀⠠⢔⠂⢀⣴⣾⣿⠆
⠀⠀⠀⠀⠀⠀⣦⣿⣿⣿⣿⣛⠿⣿⣶⣄⡀⠀⠈⡹⣦⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣠⠞⡽⠁⠀⠀⠀⠀⠀⠀⣢⣶⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⢶⣿⣿⣿⣿⣿⣿⡓⢬⠿⣏⡻⣦⡀⠘⣌⢷⣄⠀⠀⠾⣿⣿⣿⣿⣿⣿⣿⡿⣷⠀⡰⢛⠂⠀⠀⠀⠀⠀⡀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡀⠀⣦
⠀⠀⠀⠀⣠⣻⣿⣿⣿⣿⣿⣿⡷⠂⠀⠁⠙⠏⠳⠄⠀⠀⣽⢶⣄⢋⣿⠿⢿⣿⣿⡿⢿⣿⢹⢆⡄⠀⠀⠀⠀⠀⠌⠛⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣤⣿
⠀⠀⠀⢸⣿⢫⡿⠛⢿⣿⣿⣿⣿⣷⣤⣀⠀⠀⠂⠀⠀⠀⠈⠙⣿⢈⣿⣤⣼⡿⣿⣧⣠⣟⣺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣤⡈⠹⢿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⡿⠀⢀⠉⠟⢻⣿⣿⣿⣿⣿⣶⣄⡉⠀⠀⠀⠀⢸⣷⡾⢿⣿⡿⣿⡿⠷⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡶⠛⠉⠁⢛⣼⣷⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣯⣈⣻⠒⠀⠀⠈⣿⡿⢿⣿⡏⠛⠻⠛⠛⠳⠄⠀⠀⠘⠟⢿⠾⣿⣿⣿⣿⣿⣿⣻⠀⠀⠀⠀⣠⠂⠘⣹⣿⣿⣷⣾⢾⣿⣟⢻⣿⣿⣯⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣶⡀⠀⡀⢻⣧⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⣿⣿⣿⡷⢾⠀⠀⠀⠀⡿⠶⠿⣿⣿⣟⠁⠀⠸⣿⣿⣿⣿⣿⣿⣧⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣷⣀⣤⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠈⠀⠀⠀⠀⣤⠀⣦⠈⣻⠛⠓⠀⠀⠛⣿⣿⣿⣿⣿⡿⢿⣏
⠀⠀⠀⠘⠋⠛⢿⣿⠛⠛⠀⢟⣿⡜⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⠟⢿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⡄⠀⠀⠀⠀⢻⣟⣻⣿⣟⣕⣿⣟
⠀⠀⠀⠀⠀⠤⠀⡉⢡⣴⣿⣿⣿⡁⢺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣥⣤⣤⣼⣿⡇⠀⠀⠀⠀⠀⠘⠃⢠⣦⣸⣧⣤⡄⠀⠀⠈⠻⣿⣿⣻⡟⠉⠉
⠀⠀⠀⣰⡀⠀⠀⠀⣮⣿⣾⣿⣿⢧⢸⣿⡆⠀⣠⢀⣶⠀⠀⠀⠀⠀⠹⣿⣿⡟⢿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢀⣈⡁⢸⣿⣿⣶⡆⠀⠀⣸⣿⣿⣿⣷⡀
⠀⠀⠀⠘⣧⣾⣿⡿⢿⠟⣿⠋⠀⠂⠀⢿⣇⡰⠛⠾⠃⠀⠀⠀⠀⠀⠀⢹⣿⡇⣾⣿⠃⠀⠀⠀⠀⠀⠀⣼⡻⣿⣿⣿⠺⠟⠛⠟⠂⣠⡿⠋⠉⣿⣿⣿⣷
⠀⠀⠀⣾⣿⣿⣿⡄⠀⠚⠛⠶⣦⡀⠀⠸⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⢩⠀⢀⣀⣦⣀⣀⣬⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⡷⠀⠀⢀⣶⣆⣄⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡀⣙⡇⠀⠀⠤⠴⠂⠀⠀⠀⠀⠃⢀⣾⠛⣟⣉⣿⣿⣉⣦⠀⠀⢻⣿⣿⣿
⠀⠀⠀⠈⠻⣟⠙⠛⢦⣴⣈⣻⣿⣿⣤⣿⣿⠇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠸⣿⣿⡷
⠀⠀⠀⠀⢸⣿⣷⣦⣼⡟⢿⣿⣿⣿⣿⣿⡿⣴⣶⡿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣟⣄⠀⢀⣠⣄⣠⠀⠁⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠃⠀⠀⣹⢿⣷
⠀⠀⠀⠀⣾⣿⣿⣿⡟⣷⣄⠉⢻⡟⣻⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⢉⣏⣀⣿⡅⢰⣷⣤⡤⣤⢴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⠀⠀⣿⣾⡟
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶⣹⡆⠐⠻⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣶⣿⣷⣯⣤⣾⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⡀⠀⠀⣾⡟
⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣷⣄⣠⡄⠠⠚⠓⡶⠀⣤⣤⣴⣤⡄⠀⢀⣼⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⣿⠄⢸⣿⠧
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢀⣩⡴⠲⢂⣿⣿⣿⣧⡟⢀⡠⠉⠙⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠋⠀⢸⡿⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⢈⡉⠁⡰⢰⣿⣿⣿⣿⣿⣏⠀⢸⣷⣶⡾⠟⠉⢠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠈⠀⠘⣇⠐⠆
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠚⠛⠛⠁⠤⢿⣿⣿⣿⠿⠟⠀⢘⣿⣿⣷⡤⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⠻⠀⠂
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⣴⣶⣶⣶⣶⣶⣶⣶⣤⣶⣶⣦⣤⣴⣦⣤⣤⣿⡯⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⡆
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣁⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢀⠁
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣯⣿⣯⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⢉⣿
⠀⠀⠀⠀⢸⣿⣼⣿⣿⠿⠓⠏⣿⣯⣿⣭⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣾⠃
⠀⠀⠀⠀⠘⠁⠙⠛⠛⠀⢀⣴⣿⢿⣿⠿⣿⣿⣿⣿⣿⣿⣿⡏⢠⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠉⠀⠀⠀⠈⠉⠉⠈⠛⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⠈⠉⠋⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠆⠀⠐⠆⠰⠀⠀⠀⠐⠒⠂⠂⠀⠰⠂⠒⠒⠒⠂⠀⠂⠲⠠⠆⠲⠐⠂⠀⠒⠐⠀⠐⠀⠂⠒⠐⠂⠀⠒⠐⠂


⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡑⢄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠐⠂⢠⠀⠀⡈⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠊⢭⠿⠟⡦⣄⡀⠁⠐⡠⠕⣲⡀⠀⠀⢀⣀⡀⡀
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠋⠐⠁⡴⠻⠀⢀⠤⡄⠉⣆⠀⠐⣉⡀⠁⡀⠐⠛⢨⠡⠗⢤⡀
⠀⢀⠔⠁⠐⣄⠉⠒⠤⣄⡀⠀⠀⠀⠀⡇⠀⢀⡠⠢⣄⠀⠀⠀⠀⠁⠀⡇⣀⢰⡴⢽⣠⠀⠀⠀⠀⠀⠀⠀⠙
⢀⠂⠎⠀⠀⠈⠀⠀⠀⠈⠙⡳⣦⣄⡀⠁⡄⠀⠑⢤⠀⠁⠈⠓⠀⠀⢸⣻⠏⠇⠟⢈⠜⠀⠀⠀⠀⠀⠀⠀⢠
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢿⢿⡀⢧⣀⣀⠀⠀⠀⠀⠀⠀⠀⡨⠿⣀⠀⠀⢈⢠⡄⠀⠰⡀⠀⠀⠸⠋
⢧⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⠀⠀⠀⠀⠂⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⡀⡣⠀⠀⠀⠙⢆⠀⠀⡠⠒⠁
⠀⠉⡛⠲⢤⣄⣀⡘⣆⠘⡄⠀⠀⣠⠦⢤⣀⣠⠃⠀⠀⠀⠀⠀⠠⣄⠀⡀⠐⠁⠀⠀⢀⣠⣤⠦⠤⠼⡀
⠀⠀⢇⠀⠀⠋⠀⠀⠁⣠⠓⠶⠚⠁⡀⢰⠐⠤⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⡀⡤⠀⣅⡤⠴⢦⠇
⠀⠀⠀⠉⠂⠀⠠⠒⠈⠀⠀⠀⠄⠈⠀⠘⡆⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠠⠁⠣⠔⢇⠀⠐⠾⡀
⠀⠀⠀⠀⠴⠀⠆⠀⠀⠀⠀⠈⠁⠀⠀⠀⠹⡀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠘⡦⠆⠐
⠀⠀⠀⠀⠆⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡇⣼⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠙⠁⠒⠄
⠀⠀⠀⡜⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠼⢈⠟⠉⠔⠃⣄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠌⠀⢦⠴⢎⡠⠂
⠀⢠⡞⠀⠀⢀⠍⠳⡐⣄⠀⣀⢤⠿⣦⢬⡈⢀⠀⠀⡠⢼⡆⢀⠔⣦⣀⡤⠁⢀⡌
⠀⠈⠀⠀⠀⢸⠂⢠⠃⡿⢉⠹⡲⠋⠁⠀⠀⠀⢀⣘⠀⠀⡱⠀⢀⡄⠉⠀⠀⡼⠁
⠀⢠⠀⠀⠀⠏⠀⠂⣰⠃⠫⠔⠀⠀⠀⠀⠀⠀⠀⢸⡷⣄⠁⠄⠶⠀⠀⠀⣴⠂⠄
⠀⠈⠓⠺⣥⡄⠂⡰⠥⠄⠊⠀⠀⠀⠀⠰⡠⠔⠀⢸⠁⢠⢰⠰⢆⡀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠘⠾⣯⠞⠐⠤⠄⣀⡀⠀⠀⠀⠀⠸⡆⠀⠉⠑⠂⠙⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⠂⠀⠀⠉⠒⠲⣤⣀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢎⠻⡄⡔⠓⠒⠒⣆⣀⣀⡤⠤⠘⢻⠖
⠀⠀⠀⠀⠀⠀⢠⡓⠀⡃⢻⡊⠀⡏⠀⠀⠀⣀⠤⠘⢢
⠀⠀⠀⠀⠀⠀⢀⣍⡉⠑⠒⠒⠖⠛⠛⠋⠉⠀⠀⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠈⠉⠀⠉⠀⠀⠀⣠⠤⠀⠀⠐⠠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠐⠒⠤⡄⠴⡴⠆⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠞⠣⠒⠗⠒⠀⠀⠀⢄⢠⠤⠄⠤⣤⠲⠈⠀⡠⠶⠈⠑⠀⠤⣥⡔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠄⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣁⣂⣀⠁⠀⠈⠀⠀⠀⠀⠀⢂⠤⣎⣁⠠⠄
⠀⠣⠀⠐⠠⠄⠀⠀⠀⠤⠀⠀⠀⢠⠞⠀⠀⠀⢎⡀⠀⠀⠀⠀⠀⠀⠁⢀⡠⠐⠁⠀⢠⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠤⠄⠊⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠓⠦⢀⡀⠀⡠⠖⡩⠄⠀⠀⠀⢸⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠐⠈⠀⠈⠀⠀⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⠭⠔⠉⠀⠀⠀⠀⠀⠘⡇⡞⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠄⢠⡄⠀⠒⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⢀⡀⠀⠀⠀⠄⠠⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⠀⠀⠈⠀⢠⠘⣠⠀⠅⠤⠀⠃
⠀⠀⠀⠒⠂⠀⠀⠀⠀⠠⢀⡌⠀⠈⠀⠸⠁⠐⠀⠀⢀⠁⠉⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠉⠠⠴⠄⠊⠀⢀⠘⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠀⠀⠀⠀⠠⠃⠀⡜⠚⠀⠁⠂⠀⠀⢄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠄⠀⣴⠄⢀⡀⠙⠂⡀⢀⡀⠀⠀⠐⠧⡄⠣⡀⠀⠀⠀⠀⠀⠛⡠⠄⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⡀⠀⠰⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠄⢠⣄⣀⠀⠀⠀⢀⠀⠐⠠⠤⠀⠉⠈⠒⠈⠁⠀⠀⠀⠀⠀⣀⡀⠀⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠈⠠⢌⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⠀⠀⡐⠁⠀⠀⠀⠁⠀⠀⠀⢰⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠠⠄⠈⠀⢀⠀⠒⠂⠉
⠀⠀⠀⠀⠀⢆⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡀⠨⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⢀⠀⡀
⠰⢦⡀⠴⠄⠦⠔⠃⠀⠀⣀⠤⢠⡄⡶⠠⢄⠀⡆⠀⠀⠀⡀⠀⡝⠤⢀⡀⢠⡄⢤⣤⠀⣀⡠⠄⠀⡢⠄⠠⠄⠀⠀⣀⠤⠀⢨⣅⡰
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠐⢠⠀⡥⠀⣶⠀⠀⠀⠀⠀⠀⠀⠁⢀⢈⣛⡀⠀⢠⠀⠀⠘⠡⡤⠀⠁⠀⠕⠀⢀⡀⣤⠀⠀⠂⠁⠁
⠀⠀⠀⠀⠀⠀⠠⠄⠆⠀⠀⠐⠈⠀⠀⢠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⢩⠁⠀⣠⡀⠑⠓⠐⠆⠀⠀⠀⠀⠈⠉⠀⠉⠀⠂
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠥⠀⣀⣀⣈⡀⠴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠤⠀⠀⠀⠠⠀⠀⠃
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢬⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠉⠁⠀⠈⠁⠉⠀⠦⠤⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠷⠀⠀⠐⡶⠀⡀⠀⠀⣀⡀⠀⠁⠀⠄⠀⠺⢷⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠔⠀⠀⠠⠤⠀⡀⠠
⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠌⠓⠀⠀⠁⠀⠒⠀⠀⠀⠑⠠⠌⠂⠉⠐⠂⠉⠀⠀⠀⠀⠔⠀⠑⠒⠂⢀⣁⡀⠀⠈⠁⡠⠀⠀⠀⠄
⠀⣤⣄⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠊⠆⠀⠰⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠄⡀⠀⠀⠀⢈⠄⠠⠐
⡇⠀⠀⠁⢄⠘⠦⣄⢠⡄⠔⠀⠘⣰⠀⠀⠀⣀⣦⠄⠀⠈⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠡⠀⠀⣤⠀⠀⠀⢠⠁⠀⢠⡏⠀⠀⠀⠀⠀⡘⠂⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠄⡀⠁⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣥⡀⠘⢇⣀⠀⣀⠀⠔⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⢋⡤⠀⠀⠀⠀⠐⢀⠁⠀⠀⠠⠒⠈⠉⠉⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠉⠀⠀⠀⠇⠀⠀⠀⠀⢀⠄⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠤⡀⠀⢀⠀⠀⠀⡀⠀⡇⠀⠀⠀⠀⣦⠀⠀⠀⢰⠀⠀⠃⢄⣀⡀⢀⣀⡰⠞⠀⡈⠀⠀⠀⠀⠀⢀⠀⠘⠀⠀⠀⠊⠔⣈⠉⠋⣀⠲⠀⠀⡄
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⣊⠤⠖⠶⠤⡑⠀⠀⠡⠀⠀⠀⠀⠄⠀⠀⡐⠀⠀⣀⡀⣀⣀⠈⠁⠈⠣⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠊⠀⠀⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠐⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⡧⠼⣀⠀⠀⠀⠀⠀⠆⠀⠴⠀⠀⢀⠤
⠀⠀⠀⠀⠓⠈⠜⢉⠲⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⣰⠁⠀⠀⠀⠀⢠⠈⠁⠜⠂⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⢶⠀⠃⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣀⣠⡴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⣈⠁

⢀⡀⠀⠠⠅⠀⠀⡠⠂⠈⠢⡈⠁⠠⠄⠄⠀⠂⠀⢀⠴⠀⠐⠛⠻⠀⠸⠇⠄⠄⢄⠀⠀⠊⠅⠀⠀⡠⠊⠉⠠⢀⠇⠘⡄⢀⠀⠀⠀⢀⠴⠀⠐⠛⠳⠁⠸⠔⠠
⠀⠀⠈⠀⠐⠀⠂⠁⠀⠀⢀⠜⠀⠀⠖⠀⠨⠀⠠⡀⠀⠀⡠⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠂⠑⠄⠀⢀⠘⠀⠀⠖⠂⠀⠀⠀⠀⠀⠀⠠⠊⠀⡀
⠺⠂⠳⣀⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠠⠀⠠⠃⠀⠈⠠⠈⠀⡔⠓⡜⢂⠀⣾⠃⠺⢆⠽⣠⠀⠈⠈⠒⠚⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠂⠔⢂⠀⣺
⠀⠘⠂⠁⠀⠀⡠⠐⠀⠁⠐⠠⠀⠀⠀⠀⠈⠀⠀⠀⡀⢀⠈⠲⢪⡖⡛⠀⢙⠒⠂⠛⠛⠏⠀⠀⠠⠐⠀⠀⠐⠢⠀⠀⠀⠀⠌⠀⠀⠀⢀⠀⠈⠶⢢⠔⡛⠀⢛
⠀⠀⠀⠀⠀⠀⡶⠤⣂⢴⡤⢴⡆⠀⡏⠀⠄⢩⠂⠀⠀⠆⠐⠈⡧⢾⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⡶⠄⣂⢠⡆⢶⡆⠀⡍⠀⠈⢩⠂⢰⠀⠀⠀⠈⡳⢮⠃⠀⠈
⠀⠀⠉⠀⠀⠀⠉⠉⠷⠘⣣⠍⠀⠀⠀⠀⠀⠐⠀⠀⠐⠀⠆⠀⠐⠂⣀⢀⡀⠁⠀⠀⠉⠀⠀⠀⠉⠉⠳⡜⣫⠝⠁⠀⢠⠀⠀⡀⠀⠀⠑⠀⠆⠀⠐⠂⢀⢀⡀
⠀⢀⠊⠀⠈⢢⠀⠀⠀⠈⠁⠀⠀⠀⠸⠄⠀⠄⠀⠀⢀⣲⠦⠀⠀⡁⠀⠀⡀⣥⠀⠠⠈⠀⠈⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⣀⣶⠦⡀⠰⡉⠀⢀⡀
⠀⢨⠀⠀⡠⠂⢀⢀⣑⠄⢀⣴⠤⠂⠀⠀⠀⣦⣡⡾⠟⠁⠀⡀⠀⠀⠂⠀⠀⡀⠀⢀⡀⠀⡀⠂⠀⢂⡐⠄⢀⡕⠴⠠⠀⠀⠀⣤⣱⡾⠟⠋⠀⠀⠀⠁
⡄⢢⠘⠈⠀⠀⢈⠠⡔⠲⠵⠷⣄⠒⠀⠀⠈⢜⠁⠀⠀⠠⠊⠀⠀⠀⠒⠐⠀⠀⠄⠡⠚⠈⠀⠀⠄⠀⡴⠲⠾⠶⣄⠂⠀⠀⠀⢎⠀⠀⠀⠠⠊⠀⠀⠈⠂
⢀⡁⠀⣀⠀⠀⠈⢐⠂⠠⠄⢂⠄⢀⠀⢠⣂⠀⠀⠀⠀⠀⠘⠀⠇⡈⣄⠀⠀⠀⢀⡀⠀⣀⠀⠀⠐⢐⠂⠠⠄⢂⠄⠀⠀⣘⣀⢀⠀⠀⠀⠀⠸⠁⠇⡘⡀
⠀⠀⠒⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⡯⠇⠀⠀⢀⠀⡰⠂⠺⠀⠀⠠⠀⠌⠢⠰⠁⠄⠀⠀⠀⠈⠀⠀⠀⠠⠀⠀⠀⣠⡿⠇⠀⠀⢀⠀⡯⠒⠲⡀
⠀⠃⢀⠀⠀⠀⣠⠄⠒⠀⠲⠀⠐⠂⠁⠂⠈⠁⠐⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢀⠀⠀⢀⠀⠃⠀⢠⢄⡒⣀⢶⣄⠐⠂⠑⠐⠈⠁⠐
⠀⠂⠁⠐⠊⠀⠃⠀⠀⠀⠀⠀⠆⠀⠀⡀⠀⠀⡀⡀⡀⠀⠄⢀⡀⠀⠀⢀⠀⠀⠀⠒⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⠀⡀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⡆⣀⠠⠀⠒⣀⠁⠀⠦⢷⠀⠘⡉⠁⡙⠀⠀⠀⠉⠀⠀⠉⠀⣀⠀⠀⠀⠀⠀⠀⣂⣀⠠⠀⠒⡀⠁⠀⢨⠷⠄⠘⠃⠉⠛⠀⠀⠀⠉⠀⠀⠉
⠉⠀⠉⢉⡄⠀⠀⠀⠀⢀⠨⠀⠁⢀⣀⡀⠄⠀⠠⠀⠀⡀⠴⣀⢦⠀⢀⡁⠉⠀⠉⠀⠋⢉⡆⠀⠀⠀⠀⠀⡤⠀⡃⠀⢌⡄⣄⠀⠃⠀⠂⠠⠴⣐⢦⠀⢀⡀⠉
⢉⣑⠈⢛⠄⠀⠀⡠⠊⠈⠐⡈⠃⠀⠖⠀⢀⠂⠀⣀⠖⠐⠒⠋⠺⠀⠰⡃⣀⣄⢢⣄⠨⢋⠖⠀⠀⡀⠊⠈⠀⢀⠃⠐⡔⠀⠀⠀⠀⠀⠔⠐⠒⠉⡳⠁⠘⡏⣑
⠀⠀⠀⠀⠀⠀⠪⠀⠀⠀⢀⠘⠀⠀⠔⠀⠰⠀⠨⡀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⢋⠐⠀⠀⢀⠜⠀⠀⠆⠂⢀⠀⠈⡀⠀⠀⡠⠈
⠿⣆⡵⣠⠀⠀⠈⠃⠈⠃⠀⠀⠀⠀⠂⠀⠀⠃⠀⠈⠂⠊⠀⡐⡁⣜⢂⠀⣷⢃⡿⢂⠵⣠⠀⠀⠈⠁⠈⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠈⠂⠂⢀⡜⡁⣄⢂⠀⣾
⠒⠛⠈⠁⠀⠀⡀⠂⠁⠀⠁⠢⠀⠀⢀⣀⣃⣀⠀⠀⠤⠀⡀⠒⠢⢔⡚⠁⠙⠐⠓⠛⠋⠇⠀⠀⠀⠀⠉⠉⠁⠢⠀⠀⢀⣀⣂⣀⠀⠀⠤⠄⡈⠖⠢⢔⡺⠀⢋
⠀⠀⢀⠀⠀⠀⡷⢒⣒⢒⡒⢶⡇⠐⡅⠀⠀⢨⠂⢀⠀⠆⠀⠈⡗⠺⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⣶⢒⣂⢒⡒⢶⡆⠐⠍⠀⠀⢨⠂⢰⠀⠀⢐⠈⡷⢮⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠐⠌⠶⡜⣣⠍⠀⠀⢀⠀⠀⠈⠀⠀⠁⠒⠒⠀⠐⢂⠀⠀⢀⠀⠀⠀⠀⣀⡀⠀⠙⠌⠷⠜⣯⠈⠁⠀⠀⠀⠀⡀⠀⠀⠁⠒⠂⠀⠐⠂⠀⠀⢀
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠀⠰⠀⠀⠆⠀⠀⣐⣶⠶⡄⢠⣁⡀⣀⡀⡤⠀⠠⠀⠀⠀⠃⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣶⠶⡀⠰⣁⠀⣀⣀
⠀⢠⠀⠀⡀⠂⢀⢀⡠⠄⠠⡔⠴⠂⠀⠀⠀⣤⣦⡿⠟⠁⠀⠈⠀⠀⠂⠀⠀⡀⠀⢠⠀⠀⡠⠂⠀⢀⡈⠄⠨⡔⢤⠢⠀⠀⢠⣔⣤⡿⠟⠁⠀⠄⠀⠁
⡔⢢⠊⠀⠀⠀⠀⠀⠌⠒⣋⠶⡴⠒⠀⠀⠈⢊⠀⠀⢀⠄⠊⢀⡀⢈⠁⠂⠉⠀⣔⠠⠈⠁⠀⠀⠁⠀⡌⠒⣟⠧⢴⠒⠀⠀⠀⢎⠀⠀⠀⠠⠊⣀⡀⢈⠁⠂⠉
⢀⡁⠀⣀⠀⠀⠈⠜⠂⠰⠄⢀⠄⠀⠀⢠⡇⠀⠉⠁⠀⠀⠘⠒⠃⡌⣄⠀⠀⠀⢀⡁⠀⣀⠀⠀⠀⠈⠊⠡⠤⢈⠄⠀⠀⢘⡄⠀⠉⠉⠀⠀⠘⠒⠂⡬⣄
⠁⠈⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣨⠯⠇⠀⠀⢀⠀⡸⠂⠸⠀⠀⠠⠁⡀⠃⠰⠀⠀⠀⠀⠈⠀⠀⠀⠀⠨⡀⠀⠀⣨⡿⠅⠀⠀⢠⠀⡯⠂⠒
⠀⠁⠀⠀⠀⠄⣠⢀⡔⠭⣳⠄⠘⠄⣉⣀⢈⡁⠜⠀⠀⠀⠁⠀⠀⠀⠀⢘⠀⠀⠀⠀⢀⠀⠃⠄⣠⢆⡔⢭⣳⢄⠘⠆⣉⣀⣈⡁⠜⠀⠀⠀⠁
⠀⠂⠀⠐⠀⠀⠁⠂⠀⠀⠀⠐⠇⠀⠀⡀⠂⠀⣀⣀⡀⠀⠄⢠⡀⠀⠀⢰⠀⠀⠀⠂⠀⠑⠀⠀⠡⠀⠀⠀⠀⠈⡆⠀⠀⠠⠂⠀⣀⣀⣀⠀⢀⢠⡀
⠀⣀⠀⠀⠀⠀⣎⠀⠄⠀⠀⢈⠀⠀⠲⢿⠀⠘⠅⠀⠜⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⣀⠀⠀⠀⠀⣂⠀⠄⠐⠀⣀⠀⠀⢰⡷⠀⠸⠁⠀⠘⠀⠀⠀⠀⠀⠈⢀
⣁⣀⣉⣈⡀⠀⠀⠀⠀⠀⡈⣀⣁⢀⣀⣀⣄⠀⠠⠠⠀⣀⡒⣑⢢⡀⢀⣀⣁⣀⣉⣀⣃⣀⡂⠀⠀⠀⠀⢀⡤⣀⣃⢀⣌⣄⣀⠀⠃⠀⠂⣀⠐⣑⣦⡀⢀⣀⣁
⣿⣿⡿⠻⠟⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⣀⣠⡤⠤⠶⠒⠒⠒⠲⠤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⣠⠔⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢄⠀⠀⠀⡰⠁⠀⢀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⠏⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣾⢚⠇⠱⡄⠀⠈⢆⠱⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢰⠋⠀⠀⠀⠙⢦⠀⠀⠑⠆⠀⠀⠀⢌⠢⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠶⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⣸⣄⣤⡀⠀⠀⠀⣙⣦⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣹⠁⠂⠘⢀⠀⠈⠡⠈⠻⠓⠦⣄⡀⠀⠀⣠⠤⠲⣄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠚⢡⡤⠀⢸⠀⠀⠀⢸⠁
⠀⠀⢰⠒⠦⢄⡀⠀⢸⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣸⠀⠀⢀⡏
⠀⠀⠈⠑⠦⣄⡉⠓⠾⣄⡣⠊⡹⠒⢦⣀⠀⠀⠀⠀⠀⠀⠘⡇⠙⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⣠⢣⠇⢀⡞
⠀⠀⠀⠀⠀⠀⠈⠓⠶⢮⣤⡖⢁⡴⠋⣬⠃⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠞⠁⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠈⠙⠻⣤⡚⠥⠴⠒⠲⠤⣀⣀⡠⣼⣄⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠠⢍⣑⣾⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠚⢉⠽⠉⡨⠽⠇⠀⢀⣈⠛⠲⡿⠺⠆⣜⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⢾⡏⢉⡓⠺⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⣀⣀⠄⠁⠀⠈⠒⠤⠍⠻⣿⢃⠞⢂⠿⡄⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⢸⠁⠸⢀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢏⡡⠃⠀⠱⣄⠀⠀⠀⢀⡤⠚⠁⠀⠀⢸⠀⠀⠈⠆⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⢹⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢼⡇⠀⠀⠀⠀⠈⠓⠤⣄⣀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⣝⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡰⡩⠃⠲⢼⡃
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⠁⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠐

⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠋⠉⠉⠉⠀⠈⠉⠻⠿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⡿⠟⢿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⡇⠀⠸⣿⡇⠀⠀⠐⠋⠉⠉⠛⠶⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠠⡀⠁⠀⠀⠀⣠⣤⣶⢤⡀⠀⠀⠀⠀⠀⠀⠋⠉⠙⡿⣏⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣼⣿⣿⡀⢠⠀⠀⠀⠀⠻⣿⣯⠀⠙⠀⠀⠀⠀⣤⣞⠛⣿⣾⣟⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠈⠿⢿⠤⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⡇⠀⠉⢹⣿⡻⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠱⡄⠀⠀⠈⢦⠄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⡏⠉⠒⠺⠧⠤⠤⣤⠭⠭⠬⠭⠿⠟⣿⠃⠀⠀⠈⢦⡀⠀⠀⠀⠀⠁⠀⠀⡶⠋⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡞⠁⠀⠀⠀⠀⠀⠑⢄⠀⠀⠐⠠⠤⢐⡴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⠉
⠀⡇⠀⠀⢀⡠⠤⠤⡔⠉⠓⠶⣉⠀⢡⠀⠀⠀⣀⠀⠀⠀⠀⠙⠢⠄⠠⠴⢿⢴⡛⠉⠙⠋⠉⠁⠀⠉⠛⠁
⠀⣧⠔⠊⠁⠀⠀⠀⠈⣗⢤⣀⠀⠉⠒⠦⣦⣎⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠒⢄⡀
⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠨⠀⠉⠢⢄⡠⠊⠁⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠀⠀⡠⠶⠖⢻⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢀⠠⡧⣉⡒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠈⠑⠻⣦⣀⣠⠸⡀⣀⠀⠀⠀⢸⣀⠀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠴⠚
⠀⡇⠀⠀⠀⠀⠀⡀⠀⢸⢰⢸⠀⠀⠀⠀⠀⠀⢠⠆⡜⠁⠀⠀⠀⣀⣈⣯⡙⠀⠀⠈⢦⣄⣤⣧⣬⣍⣥⣉⣛⠋⠉⠉⠉⠁⣀⡀
⠀⡇⠀⠀⠀⠀⢀⠃⠀⡌⠀⢸⠀⠀⠀⠀⠀⠀⢰⣋⣤⣴⠾⠛⠛⠉⠉⠉⠑⢄⠀⠀⠀⠡⡀⠀⠀⠀⠉⠉⠉⠙⠛⠷⣶⣥⡀⠈⠉⢉⡆
⠀⡇⠀⠀⠀⠀⢸⠀⢀⠇⠁⢸⠀⠀⠀⠀⠀⢀⡼⠛⠁⠀⠀⠀⢠⠀⠀⣀⡀⡀⢙⠡⠤⠄⣦⠀⠀⡀⠀⠀⣀⠀⠀⠀⠀⠉⢫⢇⠀⣁⠜
⠀⡇⠀⠀⠀⠀⢸⢖⠡⠂⡴⠁⠀⠀⠀⠀⣠⠾⡣⡀⡀⠅⠂⢐⠊⠤⠰⠄⠄⠈⠀⠀⠸⠃⢀⠃⠀⠁⠁⠈⢉⠗⠀⢐⣤⣴⡻⠏⢁⡠⠂
⠀⡇⠀⠀⠀⠀⣬⠀⡠⠊⠀⠀⠀⠀⢀⡜⠁⠀⠹⡓⠷⢦⣤⣤⣨⣄⣄⣀⣀⣀⡀⢈⣥⣠⣈⡀⠁⣀⣀⣲⡶⡶⠿⠛⠋⡡⠃
⠀⡇⠀⠀⠀⠀⣦⠞⠁⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠉⠉⠉⠘⠛⠛⠛⠒⠒⠊⠉⠉⠉⠀⠀⠀⠀⠀⢀⠜⠁
⠀⡇⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋
⠀⡇⣠⠞⠁⠀⠀⠀⠀⠀⠀⢠⠏⣀⠤⠤⠔⠒⠒⠒⠒⠒⠛⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠋⠉⠙⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⠤⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⡁⠀⠀⠀⣷⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⣟⡛⠷⣤⣴⠏⠉⠛⠛⠁⣀⣀⡠⠤⢤⠤⠤⠤⡤⢄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣿⢏⡋⠀⢴⠆⠀⢰⣺⡆⠀⡠⢒⡩⠷⣒⠰⠟⣹⣿⣉⣝⣽⣶⣖⠮⢽⡓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡿⠋⠀⠀⠀⠀⢒⡎⢼⠃⠀⠹⢬⣓⡊⠭⠉⣿⣧⣍⣹⣿⣦⡤⠿⠟⣚⡧⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠓⠶⠦⠦⢸⡇⠀⠀⠀⠈⠉⠙⠛⣿⠒⠒⣒⣲⣶⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⡷⠶⣒⡀⠀⢒⢒⠿⢥⣀⣀⠀⠀⣀⣠⠖⠋⢉⡟⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡴⠛⠁⠀⠈⠓⢛⢄⡙⠊⣥⣄⠈⠻⣏⠉⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⢠⡟⠲⣄⣴⠋⠀⠀⣠⠆⣀⣀⣬⠭⠥⠴⠾⠷⠶⠦⢼⣷⣀⣠⠞⠁
⠀⠀⠀⠀⠀⢸⣧⣤⣼⡇⠀⠀⣸⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁
⠀⠀⠀⠀⢶⡛⠉⣠⣿⠀⠀⢠⡟⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠈⠳⠞⠁⢸⡄⠀⢸⡇⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⡇⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠹⣦⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⣀⣼⡏⠀⠀⠸⡷⠶⣤⣤⣤⣤⠶⣶⠶⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⣰⡏⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣰⣿⣇⣠⣴⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠓⠛⠛⠛⠛⠛⠛⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣋⣮⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠿⣿⣶⠀⠀⠛⣉⣀⣈⡀⠀⠀⠙⢻⣿⣿⢋⣢⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢰⡚⢻⠇⠀⠘⠡⠉⠉⠀⠀⠀⠀⠘⠛⢁⠣⢸⠀⠇
⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⢀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⡜⠀⣀⣀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠨⢀⡾⠀⣀⣀⣠⠖⠐
⠀⠀⠀⠀⠘⢆⠑⡄⠀⠘⢛⠁⢀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠉⡉⣁⣀⠀⠀⡉⠀⠀⠀⠀⠀⠀⡳⠻⡃⠉⠉⠉⢀⡴⠁
⠀⠀⠀⠀⠀⠈⢆⡨⠦⢤⣿⣇⠀⣆⠐⣦⣾⣛⣬⣥⠀⠀⠀⣿⠀⠐⠯⠭⠥⠤⠘⠋⠀⠀⠀⠀⠀⠀⢠⣿⢾⡟⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⣬⣀⠀⢺⣿⣿⣄⣸⣷⣿⣿⣿⠿⠋⠀⠀⠀⠘⣧⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠘⣿⣦⡀
⠀⠀⠀⠀⠀⢀⡇⡀⠀⢧⢉⣤⣾⡿⠋⢰⠇⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⢰⣿⣿⡟⣷⣤⡀
⠀⠀⠀⠀⠀⠈⢿⡇⠀⠘⠊⠉⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣦⣤⠶⠀⠀⠀⠀⣠⠞⢡⣿⠇⣸⣿⣿⣷⢻⣿⣿⣶⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢨⠁⠀⠀⠀⠀⣀⡴⠃⠀⠀⠀⠀⠀⣀⣤⣴⣾⢻⣿⢘⠻⣿⣻⠗⠂⣀⡤⠞⣡⢢⣟⡟⢀⣿⣿⣿⣿⣿⣿⣿⠿⠻⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⠇⢸⣿⡜⢧⡀⠉⠉⠉⣁⣴⡞⢡⠟⡼⠁⣸⣿⣿⣿⣿⣿⣿⣿⣶⣆⢀⣀⠉⢉⠙⠓
⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⡟⡄⠀⠀⠀⣴⣿⣿⣿⣿⣿⠀⣾⡇⡇⠀⠙⠦⠴⢾⡿⠋⢠⠋⡼⠁⢰⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦⣾
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⣼⣿⣿⣿⣿⣿⡇⠀⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠁⢀⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠈⠀⣀⠀⠂⢙⠀⠀⠀⠀⣠⣴⠁⣠⣾⣿⣿⣿⣿⣿⣿⡇⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⢹⣿⣿⣿⣿⣿⣿⣿⣻⠇
⠀⠀⠸⠿⣤⣤⣼⣆⠀⣠⣾⡟⠋⢸⣿⣿⣿⣿⣿⠇⠙⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⠏⠏
⠀⠀⠀⠀⠀⢺⣿⣿⣷⣿⣿⡿⠁⠈⠉⠛⠋⠋⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⣿⣿⢟⣾⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠘⢿⣛⡭⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠱⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣭⣬⣯⣦⣯⣽⣭⣼⣯⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠀⠘⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠀⠀⠈⠐⠒⠚⠃⠀⠈⠙⠀⠀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢁⣢⠀⢀⠀⠐⡆⠀⠀⡀⠤⠀⣀⡒⠂⠡⠔⠓⡌⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣈⢀⣈⣉⠙⠃⠀⠈⠉⠁⠀⠀⢀⡀⣀⠠⠴⢞⠋⢀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⠁⠀⠀⠈⠉⡄⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⢀⠈⠘⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡢⢬⠀⠀⠐⠒⠒⢶⠀⠀⠒⠒⠒⠉⠉⢉⣀⡀⠠⠇⠂⢁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠺⢀⣉⠁⠂⠐⢻⠀⠒⠒⠒⠂⠉⠉⢁⣀⣀⡠⠔⠊⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡙⠲⠤⢀⠈⠉⠉⠙⠀⠀⠀⠀⠐⠊⠉⠁⢀⡨⠷⠄⢀⠄
⠀⠀⠀⠀⠲⣄⠀⠀⠠⠴⠀⠢⣑⣢⠔⠈⠉⣉⢘⠀⠒⠒⠒⠛⢉⣉⣉⡟⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠻⢀⠀⠀⠀⠋⠈⠉⠉⠉⠉⠉⢀⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠁⠀⠈⠁⠉⠁

⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⡆⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠠⢄⠀⠀⠀⢀⠀⠈⠄⠀⠀⠀⠀⠀⢀⠁⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠆
⠀⠀⠀⠈⠙⠤⠄⢠⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⠄⠀⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠁⠡⠂⠀⠀⠀⠀⣔⠃⠋⠓⡄⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠀⠠⠀⠂⣀⣀⡀⠀⢀⠀⠀⠀⠀⠀⠀⡀⢀⢈⠉⠈⠀⠀⠀⠀⡌⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡄
⠀⠀⡀⠀⠀⠀⠀⠈⠂⡀⠀⠂⡀⠈⠁⠁⠄⠀⡁⠀⢀⠠⡄⢀⠈⠀⠀⠠⠄⡀⣀⢀⠀⣼⣇⢄⠀⠀⠀⠈⠀⠀⢀⡀⠠⠲⠀⢀⡀⢀⠀⠀⠐⠁⢁
⠀⢰⡀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡘⠀⠀⠁⠐⢁⡄⠀⡀⠈⠑⡽⠋⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⣤⡠⡤⣤⡴⡀⠈⢤⡀⠔⢀⠠
⠀⠀⣦⣄⡠⠆⠀⠒⠁⠉⠀⠀⠀⠀⠡⠄⠠⡁⠀⠀⠁⠀⠀⠀⢀⠌⠀⠀⢨⡕⠀⠀⡀⢠⠀⠀⠀⠠⠀⠀⠠⠀⠦⠄⠘⣄⢋⠀⠤⠂⠀⠀⠀⠀⠆
⠀⠀⠀⡀⠂⠀⠀⢀⡀⠀⠀⠤⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠐⠀⠄⠀⠀⢀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠠⠄⡀⠀⠀⢘⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⠰
⠀⠀⠀⠈⠀⠀⠀⠀⠀⡌⠡⠒⢈⠀⠂⠀⠀⠀⠀⠐⠀⢀⣀⠈⠀⠀⠣⠊⠁⠆⠀⠀⠀⡀⠀⠀⠀⠔⠁⠀⠀⠀⠰⠀⠅⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠂⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠁⣀⣀⣀⣀⣄⣄⣀⣀⣀⠀⠀⠀⠈⠁⠀⠀⠠⠄⠀⠊⢀⡐⠊⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠂⢀⠀⠀⠀⢠⣶⣿⣿⣿⠛⢛⡛⡻⡟⣿⣿⣿⣶⡄⠀⠁⠀⠉⠀⠠⠸⡈⣈⠀⠀⠀⠀⠀⠡⠀⠀⠀⠠⡐⠂⠠⠁⠀⠀⠀⠀⠀⠑⠄
⠀⠘⡒⠐⠈⠀⠀⢀⠀⢀⣿⢛⡟⣿⢻⡛⢛⣿⢛⡛⠻⠟⠟⢻⣿⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠃⢀⠀⣂⢀⣀⣀⣀⠀⠀⡀⡂
⠀⠀⢡⠀⡄⠀⠀⠀⠀⠘⣿⢨⠁⢹⡜⣿⢸⣿⢈⡁⡇⠀⠀⡆⣿⠃⠀⠀⠀⠀⠐⠁⠀⠀⠀⠄⡂⠀⠀⡁⠀⠁⠀⠀⠇⠀⠍⠛⠋⠀⠛⠀⠐⠈⠁
⠀⠀⠀⠆⠃⠀⠀⢀⠀⠀⠹⣼⣼⣮⣭⣿⣼⣿⣼⣧⣡⣬⣤⣴⠏⠀⠀⠀⠀⠰⠀⠀⠀⡐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠈⠠⠀⠐⠀⠀⠀⠉⠛⠛⠛⠻⠛⠛⠛⠛⠛⠉⠀⠀⠀⠂⠀⠀⡀⠀⠠⠄⠁⠀⠀⠀⠀⠈⠀⠀⠀⢐
⠀⠀⢅⠀⠀⠀⠂⠀⠒⠠⠀⠀⢠⠐⢀⠀⠀⠠⢀⠀⠠⡀⠀⠀⠀⠀⠄⠀⠀⠠⡈⠀⡀⠔⠀⠄⠀⠀⣀⣀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⢉⠀⠸⠀⣕⠐⠘⠂⠀⠁⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠐⢀⠗⠁⠀⠀⠉⠄⠀⠀⠂⠈⠂⠀⣈⠀⠀⠀⠀⢙
⠀⠀⠀⠀⢀⠀⠁⠈⠀⠀⠀⡀⠀⠐⢀⠀⠀⠘⠕⠀⡀⠀⠀⢘⡂⠀⠀⠆⠀⠀⣀⠀⠀⠀⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠑⠂⠀⠘⠀⠰⣀⣀⠬⠃⢈⠀⠀⠀⠘⠀⠀⠀⠀⠀⠆⠀⠁⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠀⠐⠬⠄⠀⠀⠠⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⡄⠀⠀⠀⠀⠀⠀⢀⠰⠸⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠁⠀⠠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠢⠀⠀⠀⠀⠒⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⣀
⠀⠖⠃⢀⠄⢀⠀⠒⠑⢀⠀⠀⠠⠀⠀⠀⠃⠁⠦⠀⠐⠀⠅⠀⢸⠀⠁⠈⠀⠂⠀⠐⠀⢴⣊⠖⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢀⠀⠀⠀⢈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠂⠠⠤⠆⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠀⠀⠀⠠
⠀⠀⡀⠀⠀⠀⢀⠀⠀⠨⠂⠀⠀⠠⠊⠀⠀⢄⠀⠀⢀⡈⠀⠂⢰⣀⡠⡰⠱⢡⠄⠀⠀⠀⠀⡆⠀⠠⠀⠠⠀⠁⠒⠈⠀⠤⠀⠀⠀⠀⣠⠀⢀⠠
⠀⠈⠀⠀⠀⠄⠠⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠁⠀⠀⠠⢇⡀⠀⢸⢆⡀⠄⢠⠀⠀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⠀⠀⠀⠐
⠀⠢⠀⠠⠄⠂⠁⠀⠀⡀⠀⠀⡠⠔⠊⠁⠁⠀⠀⠈⠍⠀⡄⠀⠀⠈⠚⠚⠁⠀⠄⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠀⠀⡄⠀⠀⡀⠀⠀⡄⠀⠄
⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⠑⠂⢀⠉⠀⠐⠀⠄⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠁⠀⠠⠤⠄⠀⠁⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠓⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠆⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⠈⠀⢀⠀⠀⠀⢀⡀⠀⡴⠋⢀⡠⠄⠀⠀⣰⡏⠑⢄⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠀⣀⡀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠀⡰⠋⠀⣠⣴⣞⣉⣀⣀⣤⣾⡋⠀⠀⡈⣆⠰⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠐⠂⠀⠙⢶⠧⠦⣄⠀⠢⠀⠀⠀⠁⠀⣏⢠⡾⠋⠉⠛⠛⠛⠉⠀⠉⢻⣦⡀⢸⢸⠀⠃⠀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⣀⣠⣤⣤⡶⠶⠦⣤⣦⡈⣷⠀⡂⠀⠀⠀⢸⣿⣿⠁⠴⠛⠃⠀⠀⠀⠸⠿⠦⢹⣿⣾⢸⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡀⠀⣴⣿⢿⣿⠛⠉⠀⠀⣤⣤⠙⣿⣿⡆⠀⠀⠠⢀⣾⣿⣿⢀⠔⢒⠄⠀⠀⠀⠀⠒⠢⡀⣿⣿⣿⠃⡇
⠀⠀⢀⡀⢀⠀⢀⡴⠋⠀⣷⣿⣿⠁⠊⡉⠁⠀⠀⠀⠠⠤⡄⢸⣿⠇⠀⠀⠀⣿⢃⡿⠟⠘⠐⢾⠇⠀⠀⠀⠐⢿⠆⠀⢸⠟⡏⠀⡇⠀⠀⡇
⠀⠀⢸⣳⣄⠩⠭⠖⠋⢠⣿⣿⣿⢠⠋⣨⡇⠀⠀⠀⢰⣶⠈⠘⡟⣰⠀⠀⠀⣿⣾⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡼⡀⡇
⠀⠀⠸⣿⣿⣷⣶⣶⣶⣿⣿⠛⣿⠀⠀⠙⠁⠀⠀⠀⠀⠁⠀⠀⡇⣿⡀⠀⠀⠹⣿⣷⣾⡄⠀⠀⠀⢠⣦⣶⠀⠀⠀⠀⣼⣶⣧⡷⠁⠀⠀⡄
⠀⠀⢀⣈⣻⣿⣿⣿⣿⣿⣿⡄⠙⠀⠀⠀⠀⢀⣀⣠⠀⠀⠀⢠⣷⣿⣷⣄⠀⠀⠈⠻⢿⣿⣦⡀⠀⢸⣿⣿⠇⠀⣠⣾⣿⠟⠋⠀⠀⣠⠊
⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣾⣧⡀⠀⠀⢸⣿⣿⠇⠀⢠⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠉⠛⠳⠦⣍⣡⢴⡟⠋⠁⢰⡀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣿⣿⣿⣿⣦⣀⡈⠻⠟⣠⣴⠿⠿⠻⢿⣿⣿⣿⣿⠷⢤⡎⠀⢠⠀⡌⠀⠀⠠⠈⢁⠀⡀⠀⣳⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠏⠀⠀⠀⠉⠓⠋⠀⣠⠀⢀⠀⠈⢻⡿⠋⠁⠀⠀⠙⢆⡈⢠⠥⡤⠤⠤⢄⣈⡄⢣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⢿⠀⣀⢀⠀⠀⠀⡀⠍⠐⠓⢌⡄⠀⠈⡁⠀⠀⠀⠠⠤⠾⠤⢃⡀⠀⠀⠀⠘⢀⣩⠾⠃⠂⠤⠤⢀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠒⣀⡞⠀⠀⠀⠀⠈⠙⡞⠙⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠛⠚⠀⠀⠤⠀⣀⡀⢹⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠐⠀⠀⠀⠠⠤⠀⠸⠄⠀⠀⠄⠀⠐⠐⠘⠒⠓⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠱⡀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⡇⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠒⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣖⠶⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣠⣼⡇⠀⠀⠀⠀⠀⡇⠉⠐⠂⠠⠄⠬⠍⠙⠛⠓⠒⠲⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠶⡦⣤⠤⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣿⡿⠁⢹⣿⣿⣷⣶⡟⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢸⠁⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⢸⣿⣷⣶⣾⠇⠀⠀⠀⠀⠀⠀⠹⡇⠀⠀⠀⢸⠀⠀⣷⠀⠀⠀⠀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢻⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣸⠁⠀⠙⢦⣀⣠⣤⠤⠤⠤⠤⠤⠤⣤⣠⣞⠁⠀⠈⢧⣳
⠀⠀⠀⠀⠀⠀⣀⡴⠖⢛⣿⣿⢃⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣈⡿⠿⡟⠲⢤⡀
⠀⠀⠀⠀⢠⣞⣁⡤⢴⣾⣄⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⠿⣗⡲⠽⣧⡀
⠀⠀⠀⠀⡟⣿⠶⠚⠉⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠉⢙⡟⣷
⠀⠀⠀⠐⡇⠈⢧⡀⠀⡼⠀⠀⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣰⠏⠀⣸
⠀⠀⠀⠀⣇⠀⠀⠉⣳⠃⠀⠀⠀⠀⠀⣸⠙⢦⣄⡀⠘⢳⣤⣀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠸⣇⣀⠀⣿
⠀⠀⠀⠀⢻⠀⣠⠴⠃⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⠉⣉⡉⠁⠈⣉⡛⠛⠛⠋⠈⠓⢦⣀⠀⠀⠀⠀⣈⡿⢠⡇
⠀⠀⠀⠀⠘⣇⠙⣶⣤⡤⠤⠶⠚⠋⠀⠀⠀⠀⠀⠈⢡⡄⠀⠀⣠⡉⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⠹⡄⣼
⠀⠀⠀⠀⠀⢹⣾⢹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⡤⣄⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣿⠇
⠀⠀⠀⠀⠀⢘⡏⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⠀⠀⣀⠀⢀⡼⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣄⣀⣀⣴⠗⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠘⣆⠀⠀⡀
⠀⠀⠙⢯⡉⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣉⣁⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠈⢉⡿⠃
⠀⠀⠀⠀⠙⢦⣀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⠀⢰⣯⢷⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠉⠳⢤⣀⠀⠀⠀⢸⣿⣭⣭⠽⣾⡇⠀⠀⠀⠀⠀⢀⣠⠖⠋⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠦⣼⡘⠛⠛⠛⠋⣇⣠⣤⠤⠶⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢺⡟⠶⢤⣷⣄⠀⣠⣞⣁⣀⣤⠤⠖⢻⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⣸⠃⠀⠀⠀⢨⡄⡏⠉⠁⠀⠀⠀⠀⠘⡇⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⡟⠀⣀⣠⡤⢼⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠈⠛⢳⡞⠉⠀⠀⠸⠷⠷⣆⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠐⠞⢛⡦⠀⠀⠀⠀⠀⠀⢿⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⡾⠿⢥⡀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢙⡟⠒⠒⢛⣶⠚⠛⢶⠺⣇⠀⣠⡷⠶⣶⣶⣶⢤⣄⣻⣠⠞⢹⣿⡀
⠀⠀⠀⠀⠀⣤⠤⠤⠤⢤⣿⣙⣾⢷⣟⡿⠏⠻⣄⣠⡾⠀⠈⠉⠉⠀⢸⡋⠀⢙⡇⠈⢻⣿⣶⣿⣤⣷⠤⠤⠤⠤⣤
⠀⠀⠀⠀⣸⠃⠀⣠⠞⠋⢀⣹⣯⠿⣇⠀⢀⣤⣄⠀⠀⠀⢀⣀⠀⠀⠈⠓⢒⡚⠁⣠⡴⠋⣸⣻⣦⣌⠛⢦⡀⠀⠈⣧
⠀⠀⠀⢰⠇⠀⣼⠁⠀⠀⢸⠻⢿⡶⠿⣭⣭⣴⣿⣄⣠⣼⣛⣛⣿⣦⣤⠶⠛⠉⠛⣃⡤⠞⣻⣏⣠⠟⠀⠀⢹⡄⠀⠘⣆
⠀⠀⢠⠏⠀⠀⠘⢦⡀⠀⠈⠳⣤⣉⠛⠲⠦⢤⣭⣭⣉⣀⣀⣀⣤⣤⡤⠴⠶⠚⠋⣉⡤⠞⠧⣽⠀⠀⠀⣠⠞⠀⠀⠀⠹⡆
⠀⢀⡟⠀⠀⠀⠀⠀⠙⠳⠦⣤⣀⡉⠙⠒⠶⠤⣤⣤⣤⣄⣀⣤⣤⣤⡤⠴⠶⠚⠋⠁⠀⠀⣀⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⢻⡄
⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠶⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠶⠖⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⣼⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠈⠙⢦⠀⠀⣴⠀⠐⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠐⠳⠄⠘⠀⠀⢉⡉⢳⡄⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⡄⠀⠀⠀⠘⠉⠀⣸⠀⠓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠓⠀⠓⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠷⠶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠀⠀⠀⠀⠀⠀⠑⡈⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⠋⠀⠀⠀⠉⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠐⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⡈⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢀⣀⡀⣀⠠⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠀⠀⢤⠄⠂⠉⠀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢀⠠⢄⢀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣠⡴⢻⠑⠀⠰⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣠⠁⡀⠀⠙⣉⡿⠐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠘⢄⡠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠉⠉⠉⠀⠀⠈⠒⠊⠧⠄⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⣀⠴⠒⠄⡀⠀⡃⠐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⢀⠀⠼⠛⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⣰⣷⡶⠶⠛⠛⠛⣿⠀⠀⠀⠀⠀⡀
⠀⠀⣀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠠⠀⠒⠀⠉⠀⠀⣀⠠⠇⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠉⠐⠢⠤⣀⠀⠀⠉⠐⠂⠄⡀⠀⢠⠀⠀⠀⠀⠀⢄
⠂⠁⠀⠑⠂⠒⠈⢀⠘⠂⠁⠀⠀⠀⠀⣀⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠨⠦⠄⠄⠀⠀⠀⠀⠀⠐⠄⠰⠌⠒⠀⠀⠀⠀⡰⠂⠀⠁⠒⠀⠂⠁⠀⠁⠒
⠀⠀⠀⠀⠀⠀⣌⠀⠀⠀⠀⠀⠀⠀⠮⠠⠴⠡⠆⠤⠄⠐⠒⠒⠂⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠈⠅⠀⢄⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠇⠈⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠀⠈⠀⠀⠀⠀⡀⠀⡀⠃⠀⢘
⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠊⠁⠀⠀⡠⠂⠁
⠀⠀⠈⠐⠀⠄⢀⠀⠀⠁⠐⠒⠒⠒⠒⠀⠀⠤⠄⠠⠤⠤⠤⠄⠀⠀⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠄⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡀⠀⠀⠀⠠⠶⠆⠒⣦⠼⢾⡆⠰⡄⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢋⡉⢃⡈⢉⡀⢸⠑⢄⠗⢸⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⡠⠂⢹⠟⠑⠁⠐⠿⠿⠀⠀⠸⠀⡈⣁⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡀⠈⠓⠀⠈⠴⡀⠀⠀⠀⠘⠉⠠⡌⢣⡽⢀⣀⠠⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠐⠀⠀⠤⠡⠐⠣⠀⠀⠀⠀⠠⢤⠤⢋⠔⠉⠀⠀⠀⢀⡤⠊⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠥⠒⠚⢳⡎⢳⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢴⣋⠀⠐⠃⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢁⠤⠒⢉⣹⣦⡸⠃⢀⡤⠤⠤⠤⠖⡶⣷⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⡹⠤⢧⣤⡤⠶⠞⠋⠉⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠊⡡⠔⠻⡆⠉⣿⢀⣀⡤⠔⠋⠀⠻⢆⣘⡀⠀⠀⠀⠀⠀⢀⠠⠐⠈⢀⡌⠀⠀⠀⠈⠙⠲⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢧⠀⠀⠈⠀⠀⣠⡷⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⣀⣀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠈⠳⠤⠤⠶⣯⠿⠤⠤⠤⠤⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠂
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢀⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠘⣆⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⠤⢤⣤⡿⠀⢢⣠⡀⠀⠀⢀⡤⠞⠁⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣝⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠻⢶⠋⠁⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢀⣞⣡⢤⠴⠴⠒⠒⠉⠉⠉⠉⠑⠒⢒⡢⠠⠤⠤⠤⠤⢀⣙⢦⡊⠑⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸
⠁⠀⠀⠀⣀⠭⢻⢄⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠙⠋⢗⠒⠲⢬⡂⢀⡞⢃⣀⣀⣀⡀⢀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⢀⠊⠀⠀⣼⠁⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠵⣖⠂⠽⢦⡑⢤⡴⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⢈⠀⠀⠀⠙⢶⠠⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠂⢀⡟⢳⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠾⠥⢀⣀⡀
⠀⠀⠈⢢⡀⠀⠀⠀⡵⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⢐⡋⠉⠀⢠⠀⢯⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠳⢦⣀⠳⣄⣀⣀⣀⣀⣀⣈⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⠔⠁⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠦⠤⢤⣀⣈⣹⡆⠀⠀⠀⠀⠄⠀⠀⠀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠀⠐⢾⡉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠐⠂⠉⠁
⡀

⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠁⠀⠁⠀⠀⢀⠠⠤⠀⠀⠀⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠤⡀⢤⠰⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠦⠤⠐⠀⠈⠀⠠⠀⡀⣤⠀⠤⢤⡌⠛⠂⠀⠤⠆⠉⠉⠁⠠⠤⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠂⠀⠀⠀⠀⠀⣀⣡⡒⠦⠤
⠀⠂⠀⠀⠢⠄⠀⠀⠀⠄⠀⠀⠀⡀⠞⠀⠀⠀⠆⠀⠀⠀⠈⠀⠀⠀⠀⢀⡠⠘⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁
⠀⠀⠀⠐⠀⠀⠀⢀⣀⠀⠠⠄⠊⠁⠀⠀⠀⠀⠀⠁⠒⠦⢀⡀⢀⡤⢊⠡⠀⠀⠀⠀⢠⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠔⠈⠀⠀⠀⠂⠄⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠥⠖⠁⠀⠀⠀⠀⠀⠸⡄⠃⠀⠄⠀⠐⠀⠀⠀⠀⠀⠁⠀⠀⠀⡤⠄⠤⠀⢤⠄⠀⢂⠀⡰
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠉⠀⠀⠀⡆⠀⠀⠄⠂⢀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠐⠄⠃⡤⢀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠌⠀⢀⠀⠘⠀⠐⠄⠀⠀⠁⠀⠁⠠⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⠀⠀⠀⡐⠁⢠⡕⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠇
⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣀⠂⢀⠀⠙⠂⠀⡀⠀⠀⠀⠑⢧⠘⠀⠀⠀⠀⠀⠀⠐⢁⠤⠀⠀⠘⠀⠠⢀
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠄⣀⠀⠈⠀⠀⠀⠀⠢⠤⠀⠈⠀⠂⠈⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠
⠀⠀⠀⠈⠠⢌⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⠁⠀⡄⠀⠊⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠦⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠊⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⢀⠀⣀⢀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠘⠲⡀⠶⠄⠦⠐⠃⠀⠀⡀⠀⢠⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢀⠁⠤⠄⠀⠀⠀⠤⠄⠀⢀⡀⠀⢀⠠⠐⠠⠀⠀⠀⠠⠄⠀⠀⢀⠂
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⣁⡀⠠⠄⠄⠀⠈⠠⠄⠈⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠⠀⠠⢤⠄⠙⠃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠤⠀⠀⠠⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠀⠀⣀⠀⠀⠀⢰⠀⠀⠨⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠈⠉⠉⠀⠦⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠐⠖⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠐⠈⠁⠀⠂⠀⠀⠀⠀⠐⠒⠈⣀⣁⠀⠀⠀⡁⠀⠀⠀⠀⠃
⢀⠤⢄⡀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠉⠂⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠒⠂⢀⠀⠀⢀⠀⠀⠐⠢⠀⠀⠀⠀⠀⡄
⡇⠀⠀⠈⢂⠘⠢⠄⠀⠠⠒⠀⠈⡠⠀⠀⢀⡄⠆⠆⠀⠙⣂⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⠰⠀⠂⠀⠀⠀⠀⠁⠂
⠀⢀⠀⠀⠀⠱⡀⠀⠄⠒⠀⠀⢰⠁⠀⢰⠑⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠐⠠⠀⠀⠀⠀⠀⢂⠈⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⠀⠊
⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⠀⢆⣀⣀⡀⠀⠂⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠥⠄⠀⠀⠀⠀⡀⠀⠀⠀⠄⠐⠈⠉⠉⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠁⢀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠓⢠⠄⠀⠀⣤⠚⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠐⠢⠄⡈⠣⡙⠃⠀⠘
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠔⠒⣉⡐⠒⠤⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢀⣠⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⣤
⠀⠂⠀⠀⠀⡀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⡀⠁⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠽⠶⠾⡄⠀⠁⠀⢂⠐⠃⠐⡄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠑⠀⠈⣉⠲⠀⠀⠀⠀⠀⠀⢀⡉⠀⠭⠭⣝⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠁⠤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⣑⣤⣤⡤⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣶⣤⡀⠀⠀⢀⢀⡄⠀⢠⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⢯⡉⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣾⠛⣉⡘⠛⠿⠿⠿⣦⣀⢊⡾⠁⡰⡀⠈⠙⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠈⢩⡍⢹⣿⣿⣿⠀⠀⠀⠀⠀⣼⣿⠃⠰⠿⣿⠁⠀⠀⠀⢸⣧⡞⢁⢼⣄⠀⠉⢉⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⠀⣼⣧⢸⣿⣿⣿⠀⠀⠀⠀⢸⣿⠇⠐⣲⣏⠤⣠⠄⠶⣦⡼⠁⡐⠁⠀⠈⠳⣄⢿⡁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⡇⣿⢸⣿⣿⠿⠀⠀⢀⣀⡈⣿⢠⣶⢤⣄⠀⠃⢶⡗⣸⡇⠈⢆⠻⣄⠀⠀⢈⣓⣌⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡋⠀⠗⢋⣨⠽⠗⣶⣿⣿⣿⡿⢻⠟⠀⢹⣤⣿⣏⢳⠀⡰⠋⠀⠀⠘⢆⠙⢦⠰⡋⠉⢹⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠜⠁⢀⠴⣏⢸⡆⢠⣿⣿⣿⡿⣁⣸⡀⣤⣘⠿⠋⠀⣨⣾⣷⣦⣄⠀⠀⢈⠶⢮⣥⡵⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠳⡄⢿⣸⠁⣼⣿⣿⣿⣿⣿⣿⠟⡎⠹⣿⡷⠋⢹⣿⣿⣿⠍⠻⠠⡜⠂⣀⣀⣀⣀⣸⢸⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠓⣓⡾⣿⣿⠀⣿⠿⠛⢉⣦⠞⣱⠖⣧⣼⠟⡐⢲⣼⣿⣿⣿⠀⢸⡇⢹⡄⠒⠒⠒⠒⢲⢰⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣫⣀⠁⠉⠛⣾⣿⡴⠋⠀⣈⠛⡡⢞⡿⠀⠸⠇⠋⠁⢀⣿⣿⣿⣿⡄⢘⠃⠀⢳⡀⠀⠀⠀⢸⠘
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠷⠟⣃⣭⣽⠚⢩⡏⢰⣿⣀⣋⣥⡆⠰⡟⢀⡀⠀⡇⠀⠀⣸⣿⣿⣿⣿⡄⢸⠴⠁⠀⢳⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠾⠛⠁⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⡇⠀⣇⢸⠁⠀⢠⣿⣿⣿⣿⣿⡇⣷⡀⠀⠀⠀⣧⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⣦⣤⣄⡴⠮⠬⢴⣟⣿⣿⣿⣿⣿⣿⣿⣿⠁⢀⠁⢠⠀⣀⣾⣿⣿⣿⣿⣿⣷⡌⣧⠀⠀⠀⣿⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⠚⠉⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠄⠸⡀⢪⣿⣿⣿⣿⣿⣿⡇⠹⣽⡆⠀⠀⠈⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣀⣇⣀⣳⢀⣿⣿⣿⣿⣿⣿⠇⠀⠹⣶⣾⣿⣿⣿⣿⣃⠔⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⣿⣟⣷⣿⣿⣯⣤⣭⣘⣛⣿⣿⣿⣿⠟⠀⣀⣀⣙⢧⣤⣤⣴⠛⣶⣞⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠙⠛⠛⠛⠒⠒⠛⠛⠛⠛⠛⠀⠉⠀⠀⠀⠙⠛⠛⠒⠀⠀⠈⠉⠉⠉⠁


⣶⣦⣶⣶⣦⣶⣶⣶⣴⣶⣶⣦⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣍⣮⣫⣤⣍⣽⣽⣼⣏⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⠀⠀⣠⣎⣉⣳⡀⠀⠀⠀⠀⡤⡤⠒⡶⣠⠒⡦⡀
⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠉⢆⠀⠀⠀⠀⢠⠮⠭⠭⠭⠽⠄⠀⠀⠀⣸⠥⠄⠬⠦⠨⣆⠀⠀⠀⣜⠁⣼⡙⣧⢎⢫⣾
⠀⠀⠀⢠⡗⡒⡒⢖⠠⣒⡲⣶⠾⡀⠀⠀⠀⠘⡎⠉⠉⠉⢹⠁⠀⠀⠀⠙⣟⢻⡭⡛⢹⠁⠀⠀⠀⢹⠙⠓⠓⠚⠚⢉⡇
⠀⠀⠀⠀⢹⠒⠾⡉⣘⢿⠓⠞⡟⠀⠀⠀⠀⠀⢣⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⡟⢤⣺⠃⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠉⠁⠘⠳⣰⠞⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡁⡝⠀⠀⠀⠀⠀⠀⠧⠤⠤⠤⠄⠼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢀⡤⢒⣂⡉⣐⠲⡄⣠⢤⠀⠀⠀⠀⠀⣀⠤⢤⠒⠢⡄⠤⡀⠀⠀⢀⡠⠔⠒⠒⠠⢄⠀⠀⠀⠀⠀⢀⠤⠒⠢⢄
⠀⠀⠀⢠⢋⠖⠧⠎⢀⡠⡫⠞⠹⡆⠇⠀⠀⢠⠞⠁⠀⠀⠣⡀⢣⣀⣈⠆⠀⣎⣀⣀⣀⣀⣀⣠⡕⠀⠀⠀⢠⠋⠀⠀⠀⠀⢣
⠀⠀⠀⢸⠈⠆⠏⠋⠻⠒⡿⢭⢅⠙⠂⠀⠀⡜⠣⡀⠀⠀⢀⡝⠁⠀⠀⠀⠀⠈⡄⢠⠈⡄⠰⢠⠁⠀⠀⠀⠸⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⢆⠳⢖⢐⢢⣀⡰⢋⡞⠀⠀⠀⠀⡗⢄⣈⡶⠒⠁⠀⠀⠀⠀⠀⠀⠀⢡⣘⠀⢃⣃⠎⠀⠀⠀⠀⡼⢑⣤⠴⠊
⠀⠀⠀⠀⠀⠑⠢⠤⠭⠤⠒⠋⠀⠀⠀⠀⠀⢇⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠸⢭⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠠⠄⠀⠀⠀⠀⠀⠀⠀⣔⣒⣢⢠⡆⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢀⣴⢖⣂⠴⡒⣒⣄⠀⠀⠀⠀⠀⡤⠤⠤⠧⠧⠤⣄⣀⡀⠀⠀⠀⢸⠀⣇⠾⠼⠄⠀⠀⠀⢀⡔⡢⣄⡠⢚⠦⡀
⠀⠀⠀⠀⡎⡡⠒⠉⠒⠒⠒⢝⢦⠀⠀⠀⠀⢧⠀⠀⠀⠀⢀⣯⣗⠅⠀⠀⠀⡸⠀⡧⠋⠈⠣⡄⠀⢠⠊⠀⢱⠈⢳⠸⡄⠈⠆
⠀⠀⠀⢸⡴⠁⠀⠀⠀⠀⠀⠈⢿⠂⠀⠠⡆⠀⠷⠤⠄⠖⠋⠃⡲⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠸⡀⢀⠇⠀⢸⠀⡇⠀⡄
⠀⠀⠀⠸⠤⠤⠤⠀⠒⠀⠀⠤⠊⠀⠀⠀⠈⠑⠒⠒⠒⠒⠂⠉⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢸⠀⠀⠑⢤⣃⠴⠛⢄⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠣⠤⠤⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣄⡀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢤⣀
⠀⠀⠀⠀⠀⠀⣀⠀⣞⣵⣵⠗⠺⡄⠀⠀⠀⢠⠃⠘⡀⠀⠀⠀⠀⠀⠀⣠⠊⠱⢼⢌⢷⠀⠀⠀⣠⣀⡠⠤⣄⡀⢀⣀⣀⡀
⠀⠀⠀⢰⣊⠭⣔⡺⣽⠝⠁⠀⠀⡇⠀⠀⠀⡞⣚⠓⠙⡆⠀⠀⠀⠀⣴⠥⠔⠚⠉⢉⡏⠀⠀⢸⠀⠘⠀⣀⠈⠀⠀⠋⠀⢹
⠀⠀⠀⢎⣪⠛⠊⠉⠀⠀⠀⣠⠎⠀⠀⠀⠀⠹⠀⡇⢰⠃⠀⠀⡠⠊⠀⠀⠀⠀⡠⠋⠀⠀⠀⣞⡋⢛⡓⠒⡞⠉⣖⡛⢳⣋⡢
⠀⠀⠀⠈⠡⡀⠀⢀⣀⠤⠊⠁⠀⠀⠀⠀⠀⠸⠋⠁⠸⡄⠀⠀⠣⢄⡀⠀⠀⡔⠁⠀⠀⠀⠀⠘⠬⠅⠈⠫⠬⠍⠤⠬⠥⠜
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⡄⠠⠃⠀⠀⠀⠀⠈⠢⠞
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⠀⠀⠀⢀⡀⠀⢀⠀⡠⠄⠄⡀⠀⠀⠀⡠⠀⠀⠀⡀⡤⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠠⠄
⠀⠈⠀⠀⠀⠀⢀⠰⠀⠐⡀⠂⠀⠑⠀⠀⠐⠀⠊⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠲⠄⢠⣂⠁⢶⠂
⠀⠀⢀⡀⠀⠀⠀⠘⠆⠀⠑⢀⠀⡑⠜⠀⠀⢖⠧⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠈⠑⠀⠀⠀⠈
⠀⣄⠈⠀⠀⠀⢀⡠⠤⠠⠠⠠⡉⠀⠀⣴⡃⢸⣫⠀⠀⢄⠔⠁⠡⠀⠀⠀⠄⡠⠀⠀⠑⠢⠤⠀⠀⠀⠐⢀
⠀⠈⣀⠘⠀⠐⠁⡐⢀⡁⢠⣤⣪⠆⢀⣉⢀⠀⠟⠀⠀⠘⠄⠠⠼⠢⠀⠀⠀⠀⡀⠀⠘⠒⠀⠠
⠀⢸⠀⢣⠀⠒⠃⠐⣁⣒⡉⠤⠊⠀⡇⠀⠈⡄⠀⠀⠀⠀⡀⡀⠀⠀⣀⣤⣤⢆⡀⠉⠀⠀⠀⠀⠀⡠⢤⡀
⠀⢸⠰⡀⠁⠀⠀⣀⠄⡀⠀⢀⣀⠘⠄⢀⡰⠃⠀⠈⠄⠀⠀⠀⠀⠘⢻⡖⠛⠚⠳⠀⡃⠀⡐⠀⠸⠂
⠀⠀⠓⠐⢁⢄⠀⢈⠀⠀⠤⠀⠂⠀⢀⡤⣀⠀⠀⠀⠀⠀⠤⠤⡀⠁⠀⠀⠒⠂⠒⠀⠀⠏⠀⠀⠀⢂⠀⢀
⠀⢦⠀⠀⠂⠀⠁⠈⠀⠀⠐⠐⣢⠿⠜⢋⠘⢠⡄⠆⣀⡀⠈⠈⠀⠀⢀⠀⠀⠀⠀⠆⠀⠈⠀⠈⠀⠀⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐⠀⡁⢒⣠⠀⠖⢊⠀⠀⠀⠀⡆⠈⢀⢠⠒⠀⠂⡂⠦⡈⠄⢘⡄
⠀⡀⠰⠐⠃⡀⠀⠀⠀⠀⠄⠀⠀⠋⠀⠉⢡⡀⠀⢐⡭⠀⣀⠀⠊⡀⠀⠀⠉⠀⠀⠀⠀⠀⡁⠀⠀⠐
⠀⠐⠂⡀⢀⠔⠀⠀⠈⠿⠀⠀⡰⠛⠉⠁⠈⠷⠆⠈⠀⠂⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⣀⣅⠀⠀⠃⠀⠤
⠀⡒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠈⠀⠀⢀⠀⠊⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠠⡀⠄
⠀⠀⠀⠀⠀⠀⢈⢐⠀⠀⠀⠈⡀⠀⠈⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠠⠀⠂⢀⠙
⠀⠀⢀⠤⡔⠄⠀⠀⠀⠀⠂⡘⠀⡀⠀⠀⠀⠄⠀⠸⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠈⠦⡤⠀⠁⢠⡤⠐⡀
⠀⠈⠀⢀⠁⢈⠀⠐⢀⠠⠀⠁⠘⠓⠀⠀⠀⠀⣀⡔⠀⠀⠐⠀⠀⢊⠀⠀⠀⡀⠀⢀⡴⢡⠀⠀⠄⠋⠁
⠀⠀⠁⠈⠠⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⡀⢠⡄⠀⢀⠀⠀⠂⠀⠀⠂⠉⠉⠉⠂⠉⠁⠀⠁⠠⠀⠀⠀⠀⠄
⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⡁⠈⠆⠀⠚⠀⠈⠀⢀⠀⣀⠀⠀⠀⠢⠃⢂⠕⠐⢈⠂⢀⡀⠠⠄⠠⢀
⠀⠀⡡⡄⢠⡄⠀⠀⠂⠀⠀⠐⠄⠀⠀⠉⠉⠁⠀⠁⠰⠀⢀⠀⠀⠐⠀⠀⡀⢀⣀⡀⠀⠀⢠⠀⠀⠀⠀⣨⠁
⠀⢸⣍⠠⠋⠀⠘⠁⠀⠁⠀⠀⠃⠀⠂⠔⠂⠈⠁⢀⣀⡀⠍⠀⠀⠀⠀⠀⡨⠉⠉⠀⠀⠀⠀⠉⠀⠀⠐⢀⡀
⠀⠀⠁⠀⠀⠘⠀⠀⠈⠀⠈⠀⠉⠐⠊⠀⠀⠊⠀⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠊⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠈⠉⠉⠁⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⣀⣤⣶⡶⣶⣶⣤⣔⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠁⠀⠀⠀⣀⣽⠻⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠰⢰⡿⠁⠀⠀⢀⣤⠾⠁⠀⠀⠀⢹⣧⠀⠐⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⠋⠀⢠⡿⠓⠘⠁⠑⠀⠀⠀⠀⠀⠂⠠⢀⢹⡆⠀⠑⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⢸⡇⡰⠁⢠⣤⡄⠀⠀⠀⣤⣤⠀⠀⠘⡇⠀⠀⠈⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢆⡁⠀⠀⠀⠁⡗⠀⠀⠘⠛⠀⠀⠀⠀⠛⠋⠀⠀⢸⡇⠁⠀⠀⠀⢻⠗⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢅⠀⠀⠀⡀⠀⠀⠠⠴⡀⠀⠀⠀⠰⠖⠒⠢⠀⠀⠀⠀⠸⠄⠂⠀⠀⠀⠂⠀⢀⣌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠑⢄⠈⠐⠄⠀⠀⠐⡀⠀⠀⠀⠳⠖⠃⠀⠀⠀⠔⠀⠀⠀⠠⠂⠀⠔⠁⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠑⠄⡠⠋⠀⠐⢌⠂⠄⣀⠀⠀⢀⠀⠐⠀⡠⠐⠁⠙⢀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠁⠀⠂⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⡀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⢀⣀⠠⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠀⠀⠀⠉⠠⡀⠀⠈⠆⠀⠄⠀⠈⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠂⠐⢤⡀⠀⠁⠢⠀⠠⠀⠐⠀⢀⠀⠀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢬⠀⠀⠈⣷⠀⠀⠀⠀⢀⠂⠀⢀⣀⡥⠖⣀⠄⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠀⡐⠚⠛⠿⠿⠒⠌⠀⣤⠂⠠⣄⣹⣇⠀⠀⠀⢀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠐⠀⠀⢀⠀⠀⠀⠈⠞⡁⢰⣤⣿⣿⣏⠀⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⡉⠆⠀⠀⠀⠀⠀⠈⡄⠔⠂⠀⠉⠀⠀⠀⠀⠀⠀⢑⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠌⠀⢠⡴⣶⡶⢆⠀⡀⠀⠀⠀⠃⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡄⢁⠀⠐⠂⠠⠾⠚⠉⠀⠀⠞⠛⠛⠁⠀⢀⠀⠀⡠⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⠿⠿⠿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⠀⠀⠀⢀⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠘⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣀⣷⠔⠂⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⣟⠁⢿⠀⠀⠀⠀⠈⠙⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⠁⠀⠀⠀⠀⠀⠀⢠⠞⠀⠙⠂⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠹⡆⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠐⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢻⡀⠀⠀⠀⡇⣀⣤⣀⠀⠀⠀⠀⠀⠠⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢦⡈⢳⣌⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣴⠛⢳⡀⠀⠀⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠲⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣺⠀⠀⠻⡄⠀⢻⡀⠸⡿⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠖⠋⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠹⣆⠀⠳⡄⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠙⢦⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠷⢤⡤⢾⢳
⠀⠀⠀⠀⠀⣠⠴⣺⠿⠯⠽⢯⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠉⠓⠦⣌⣉⡙⠓⠦⢀⣀⣠⡴⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⠛
⠀⠀⠀⠀⣰⣣⠞⠁⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣤⣄⣀⣀⣈⣉⣉⣉⣉⣉⣀⢠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠉⠉⠉⠉⠉⡟⠛⠷⠶⣦⣤⣤⣀⣀⣀⣾⠋
⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠈⠉⠉⠙⠁
⠀⠀⠀⣟⣷⠀⠀⡶⠀⠀⠀⠶⣤⢤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢹⣿⣆⠀⠀⠐⠓⢛⡟⣛⣛⣉⣉⡉⠈⠀⠒⢶⡞⠲⣄⠀⠀⣀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⢠⡏
⠀⣤⠆⠀⠳⣽⣦⡀⡘⢋⣩⣭⣭⣥⣤⣤⣤⣤⣶⣦⣼⠷⠚⠛⠻⠛⢹⣧⠴⠚⠉⣻⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⢀⣠⠜⠻⢿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠘⠗⠀⠀⣠⠟⠐⠒⠓⠶⣶⡯⠤⣀⣀
⠀⠀⡔⠻⡭⠇⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⠀⣠⣄⠀⠀⠀⠈⢢⡔⠛⠋⣉⡣⠐⠒⢢⡄
⠀⠀⢶⣄⣨⠆⠀⠀⠉⠳⠆⠤⢤⡀⠀⠲⣦⢤⣤⡤⠤⠶⢦⣀⣘⣻⡏⠙⠉⠁⢠⣯⡶⣗⡽⢃⣄⠀⢀⡇⠀⠀⣹⠡⠴⠂⢸⠃
⠀⠀⠈⠷⠤⣀⠀⠀⢀⢀⡀⠀⠀⠛⠒⠒⠒⠛⢉⣀⣀⣠⠤⢤⣥⣄⠀⣤⡄⠀⠀⠀⠈⠻⠽⠁⣀⣤⠊⠀⠀⢀⣏⡠⠴⠒⠋
⠀⠀⠀⠀⠀⠈⠙⠢⠬⢚⣉⠀⠀⠀⠀⠀⠀⠈⠉⠰⠆⠤⠽⠒⠒⠉⠉⠁⠀⠀⣀⣀⡤⠤⠒⠉⠉⡇⠠⠀⠀⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁⠒⢲⡖⠒⠒⠒⢢⠖⠒⡶⠒⠒⣂⡉⠉⠉⢠⠀⠀⢰⠃⠀⣸⣁⠠⠄⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠁⠀⠓⠊⠉⠏⠀⠀⠀⣀⡠⠤⠞⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⠐⠒⠒⠠⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠩⠉⠩⠉⡉⢓⡄⢀⡴⠒⠉⠑⠒⠬⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣓⠒⠒⠒⠒⢚⡽⠁⡞⠀⠀⠠⡀⠀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣤⠖⠋⠉⠙⠓⠒⠲⠏⣉⢛⢧⠀⠀⠀⠀⠀⠘⠀⣀⡀⠀⠑⢶⠲⠤⠤⠤⣀⣀⡀
⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠁⠀⠀⠀⡼⠁⠀⠀⠀⠆⠀⠄⠀⠀⠈⠙⣆⠳⣄⠀⠈⠀⡔⠀⠀⠶⠀⠀⠈⡇⠀⠀⠀⠀⠀⠈⠉⠓⠢⢄⡀
⠀⠀⠀⡰⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠐⠒⢀⠀⢀⠀⠒⠀⠀⢀⣿⢤⡈⠑⠒⠤⠤⣤⠤⢄⠠⡤⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠋⠀⠘⢀⣀⣠⢔⡿⠋⠀⠈⠑⠢⣤⣄⣀⣀⣀⣀⣤⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠫⠭⢉⣉⠩⠭⣵⠒⠉⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠙⣆⢀⠙⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠘⠮⣓⣃⠇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠒⠚⢣
⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠎⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⢀⡇
⠀⠀⠠⠋⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⣞⠀⠀⠀⣠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁⠀⠀⠀⢀⡤⠖⠉⠙⡄
⠀⠀⠀⣇⠀⠀⠀⠈⠙⠛⠶⢶⣒⡒⠓⠒⠒⠒⠒⠒⠶⠤⢧⣘⣒⡂⣸⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡤⠤⠖⠒⠉⠀⠀⠀⠀⢀⣠⠴⠚⠁⠀⠀⠀⢀⠇
⠀⠀⢀⡞⠳⢄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠒⠢⠤⠤⠤⠄⣠⣤⣈⣉⣉⣉⣉⡉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⣀⡤⠔⠊⠉⠀⠀⠀⠀⣀⡤⠔⡏
⠀⠀⢸⡀⠀⠀⠈⠉⠐⠒⠒⠤⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢠⠇
⠀⠀⢠⠗⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠈⠋⠉⠀⠀⠀⣠⢾
⠀⠀⠸⣄⠀⠙⠲⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⠤⠤⢄⣀⣀⣀⣀⣀⣀⡤⠤⠤⠶⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣡⡇
⠀⠀⠀⠈⡳⢄⡀⠀⠀⠉⠓⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠶⠖⠒⠒⠊⠉⠀⣠⠞⣡⠇
⠀⠀⢠⠏⢇⠀⠻⡒⢤⣀⠀⠀⠀⠀⠈⠉⠑⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠊⠉⠉⠹⡄
⠀⠀⠘⣄⠈⠦⣀⠈⠒⢬⣙⡢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠊⣡⠏⠀⠀⠀⠀⡗
⠀⠀⠀⡏⠓⠤⣈⡑⠢⢄⣀⠉⠙⠒⠫⠭⣖⣢⣤⠤⣀⣀⣀⣀⣀⠀⣀⡀⣀⣀⣀⣀⣀⣀⠠⠤⠤⠤⠔⠒⠒⠚⠉⣉⣁⣀⠤⠚⠁⠀⠀⠀⢀⡠⠳⡄
⠀⠀⠀⣇⠀⠀⠀⠉⠓⠲⠬⣉⡒⠢⠤⢀⣀⡀⠉⠉⠙⠒⠒⠢⠤⠤⠤⢁⣀⣀⣀⣀⠀⠤⠤⠴⠒⠒⠛⠋⠉⠉⣉⣉⠤⠋⠀⠀⠀⣀⠤⠒⠉⠀⢀⡇
⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠢⠤⢌⣉⠉⠁⠚⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠛⠓⠂⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⣀⡴⠊⡇
⠀⠀⠀⠀⠘⢷⡦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠦⠤⢤⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠞⢁⡠⣤⠖⠋⠁⠀⣰⠃
⠀⠀⠀⠀⠀⠀⢻⡒⠭⣕⣒⠤⠤⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣥⠤⠒⢊⡥⠚⠁⠈⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡈⠉⠐⠒⠤⠤⣭⣉⣉⣑⠒⠒⠂⠤⠤⠤⠤⠀⣀⣤⣤⣤⣄⣀⠠⠤⠤⠤⠒⠒⠉⠉⠀⣀⡤⠖⠉⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⢤⣀⣀⠀⠀⠀⠉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠀⠠⠤⠤⠤⠤⠤⠤⠄⠒⠒⠉⠁⠀⠀⠀⣀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠒⠒⠒⠒⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠐⠒⠒⠒⠒⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠤⢊⠀⠂⠒⠒⠬⡑⢄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠀⢀⠔⡨⠄⠒⠈⠉⠈⠉⠁⡀⠀⠀⠀⠀⠀⠀⠐⡀⠇⠀⠀⡀⠖⠃⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⢠⢊⠔⡩⠂⠡⠀⠀⣠⠖⠄⠀⢹⠀⡄⠀⠀⠠⡀⠀⠀⠈⠀⠀⠱⡀⢢⠁⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡐⠁⠠⠊⠀⠀⠀⠀⠀⠣⠀⠀⠴⠉⠐⠇⠀⠀⠐⡄⠀⢁⠂⠀⠀⠀⠀⠀⠁⠡⡀⢆⢀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⡈⠀⠁⠀⠀⠀⠐⠀⠘⢇⠐⠈⠀⠀⢀⠀⠐⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⢀⠀⠶⠀⠀⠀⠀⠙⠂⠀⠀⠐⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠇⠘⠀⠀⡛⢉⡖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⢰⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡹⡷⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠋⡁⠀⠰⢠⠆⠀⠀⠀⠀⠕⠀⠁⠀⠀⠄⠠⠀⡀⢢⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠘⡀
⠀⠀⠰⢣⠀⠀⠀⠀⠀⠀⠀⠉⣀⠠⠾⡦⣤⡀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠌⠀⠀⠁⠀⠄⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠞⠄⠉⠀⢀⠀⠀⠂⠀⠀⠀⠀⣷⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠐⡀⠀⠀⠀⠁⠂⠄⠀⠀⠀⠀⠀⡈⠠⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠑⠂⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⠂⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⠀⠀⠒⠂⠁⠀⠀⠀⠀⠂⠂⠀⠂⠀⠄⠐⠀⣤⠀⡆⠀⠀⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⠀⠸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡇⠀⢀⡀⡄⠀⢰⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠊⠀⠀⠀⠉⠣⡀⠀⠡⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠀⠀⠀⠀⠀⠀⠶⠛⠋⠛⠋⠀⢏⠉⢣⡢⡀
⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠘⡣⠈
⠀⠀⠠⠈⠀⠁⠀⠀⠀⠀⠀⠄⠀⠠⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⠀⠀⠀⠀⠀⠀⢜⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⡢⢀⠀⣀⡀⠀⠀⠉⠢⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⣴⠛⡏⠉⠷⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⢠⢰⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⣤⢴⠆⠈⠁⠒⡭⠄⠀⠀⠚⠉⠀⠀⡀⠀⢀⡡⠀⠦⠌⢛⠡⠂⠀⠘⣄⢀⠾⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡗⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠊
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⢄⠀⡀⠄⡧⠀⠀⠀⠠⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⢀⠔⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠈⠑⠀⠐⠁⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡜⠃⠀⠀⣀⠄⠈⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠉⠒⠛⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠇⠀⠂⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠛⠚⠀⡠⠁⠀⠰⠇⠐⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣤⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠐⠀⢠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠝⢱⠀⠼⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠍⠱⠀⠼⢉⡀
⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⡀⠀⠀⠀⠀⡴⠋⡏⡩⠦⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡴⠫⡎
⠙⢣⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠆⠅⠜⢃⡶⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠄
⠀⠀⠀⠀⠀⠐⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠰⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠰⠤⠒
⡄⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⣀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠐⢷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠖⡼⣿⡆⢊⠀⠀⠀⠌⠄⢉⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣬⣖⣾⣻⡆⠊⠀⠀⠀⠌⠆⢙⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣰⠀⠀⠀⠀⠶⢋⠘⠃⢦⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡤⠀⠀⠀⠀⠶⠛⠘⠇⢦⡴⠂
⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠨⣁⠀⡤⠈⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣀⠂⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠈⠀⠀⡠⠦⡴⠐⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⡁⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢀⡠⠢⡤⠀⠀⠀⠀⠆
⠀⠀⠈⠛⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠖⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃
⠀⠀⠀⠀⢀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂
⠀⠀⠀⢀⠉⡰⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡰⠉⠆
⠀⠀⠀⠀⡶⠃⠘⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠃⠠⠀⢀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠜⠹⠀⠼⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⠀⠍⠹⠀⠨⢡⡀
⠦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠐⠀⠀⠀⣴⠁⡏⠩⡦⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠐⠀⠀⠀⡰⠉⡏
⠙⡃⡶⠀⠀⠔⠒⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⠔⠁⠜⠃⡴⠀⠀⠔⠒⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠆
⠀⠀⠀⠀⠀⠐⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠐⡤⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠒
⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀⣀⠀⠀⠐⠷⠆
⣗⠺⣻⡆⢂⠀⠀⠀⠌⠂⢉⠿⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡷⣊⣖⢾⣹⡖⠊⠀⠀⠀⠈⠂⢉⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠁⠀⣤⠀⠀⠀⠀⠖⢋⠘⠃⢦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡴⠀⠀⠀⠀⠷⢛⠘⠃⢢⠴
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠃⠀⠀⠈⣁⠐⠤⠐⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀⣀⠀⠀⠈⠀⠀⠃⠀⠀⠈⣁⠂⠤⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡰⠦⠶⡀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⠀⠀⠀⠀⠀⠀⠚⠀⠉⠀⠀⡤⠲⡴⠐⠀⠀⠀⠂
⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠈⠀⠉⠛⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⠀⠉⠛⠃⠀⠀⢀
⠀⠀⠀⠀⢀⠀⡠⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣠⠅⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠
⠀⠀⠀⠈⠉⠰⠉⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠉⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠘⠀⠃⠀⠊⠀⠀⠈⠀⠐⠀⠓⠀⠀⠀⠂⠐⠀⠀⠀⠐⠈⠒⠒⠀⠁⠁⠘⠘

⠀⠀⠀⠀⠀⠀⣒⣢⣄⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⢀⡀
⠀⠀⠀⠀⠀⠐⠁⠸⠙⠘⠶⠸⣇⡸⠄⢠⡄⠀⠠⠋⡹⢸⣶⢀⡀
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠊⠣⠀⠀⠀⠁⠈⠉⠳⠫⢰⠇⣰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠤⢄
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠋⠣⠀⠀⠀⠀⡀⠐⠉⠁⡠⠀⠀⢠⠄⠉⢢⠄
⠀⠀⠈⠀⠂⠀⡠⠄⠒⠠⠤⢄⣀⠀⠠⠋⠱⢲⣾⡆⣤⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⡪⠀⠀⠀⠈⠀⠀⠀⡇⠈⢢⠀⠑⡄
⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠀⠀⢌⢱⣄⠀⠀⠀⠀⠀⠉⠺⠄⠞⡂⠀⠀⠀⠀⠀⠀⢰⠁⢀⣀⣠⡂⠀⣀⠏⡅⠀⠀⣧⠀⡸
⠀⠀⢠⠋⠀⠀⢠⠃⢤⣬⣴⣡⠟⠠⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⢇⡸⠉⠓⠚⠓⠈⠀⠀⢴⠟⠁
⠀⠀⡜⠀⠀⡰⠇⠀⠐⠂⣈⡁⠀⣁⢰⠀⠀⠀⠀⠄⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡌⠠⢄⠀⢰⡤⠀⢀⡅
⠀⠀⡇⠀⢨⡐⠆⠀⡰⠊⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠹⠂⠒⠊⠁⠀⠀⠢⠀⠀⢀
⠀⢸⠀⠀⠀⠈⣉⣍⠓⡦⠤⢴⠒⠋⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠃⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⢸⠀⠀⠀⠀⣻⣼⠋⠁⠀⠈⠑⢤⡀⢃⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠉⠙⠒⠢⠤⠤⠤⠤⠖⠋⠒⣄
⠀⢸⠀⠀⡠⢚⠁⠀⢱⡄⠀⠀⠀⠀⠙⢾⠇⠀⠱⠀⠀⠀⠀⠀⠀⠀⡜⠁⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⠰⠶⠶⠃⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠚⠓⠒⠒⠓⠒⠒⠒⠲⠦⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⡆⠀⠀⢀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠘⡄⠀⡜⡀⡊
⠀⠁⠚⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡾⣤⣤⣀⡀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡠⢣⣀⢷⡗⠟⢢⡐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠈⠑⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⢏⠀⠀⠀⠀⣷⠀⢰⠑⢆⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⢹⠒⠤⣠⣀⡀⠀⠙⠢⠠⠤⠤⠄⢀⡠⣀⠀⡇⠀⠑⠒⠒⠉⠉⢯⠁⠀⢀⢛
⠉⠉⠉⠉⠉⠉⠉⠑⠢⡀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠢⡀⢸⠀⠀⠉⠑⠲⠤⣤⣄⢀⡥⠋⠛⠁⢷⠀⠀⠀⠀⠈⠀⠀⠑⠶⠶⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣶⣶⢷⠶⡄⠀⠀⡀⠀⠤⠄
⡍⠉⠉⠩⠭⠭⠭⣼⠁⠀⠀⠀⠀⠀⠘⣭⠭⠭⠭⠭⠭⠉⣉⣉⣉⡞⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⠻⠟⢯⢿⣭⡖⢻⠸⡁
⠇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠰⢾⠀⠘⣄⠀⠤⠀⠒⠀⠀⠈⠀⠀⣼⣀⣀⠘⢺⣃⣈⣀⡼
⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠈
⢀⠶⠦⠦⣴⠀⠀⢸⢰⠀⠀⠀⠀⠀⠀⢁⡇
⢸⠘⠺⠋⠯⠂⠀⢸⠸⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⢀⠘⠓⠀⠀⠀⠀⡀⠀⠀⠀⠀⡏⢸⡄⢻⠄⠀⠠⣎⠀⠀⠀⠀⠀⢀⣼⡵⣿⡶⠀⠀⠀⠀⠀⢀⣀
⠀⠘⠛⠙⠈⠼⠓⠇⠀⠼⠧⠴⠖⠀⠀⠀⢿⠈⠁⢀⡇⠀⠠⠰⠆⠢⡀⠀⠰⢣⣟⣿⠞⠀⠀⠀⠀⠀⢸⣇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠈⠁⢀
⠀⠀⢠⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠀⠀⠀⠠⣤⣤⣦⠄⠀⠀⡀⠒⠀⠁⠀⠀⠀⢠⣤⣤⣴⡀
⠀⠀⠈⠂⡀⠀⠀⠀⠀⠔⢰⡮⠃⠀⠀⠀⠶⠉⢀⠜⠁⠀⠀⠀⠉⠁⢀⠀⠀⣤⠞⣀⡈⠳⠀⠀⠈⢉⡻⠏⠁
⠀⠀⠀⡀⠃⠀⠀⠀⠘⠀⠤⠤⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠈⠡⢤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠋⠉⠂⢀⠀⢠⡄⠘⠓⢠⠀⠀⠀⠉⠙⠀⠀⠀⠀⣀⡀⠀⠹⠀⠀⠀⠀⠀⠀⠀⣙⡧
⠀⠀⠀⠀⣾⢴⡄⠀⠛⠀⢀⡀⠀⠘⠀⠈⠀⠀⠈⠀⠀⠀⢦⣤⣤⣤⡆⠀⠀⠐⠁⡠⡄⠀⠀⠀⠀⠀⠀⠊⠁
⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠁
⠀⠀⣠⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠄⠀⠀⠀⢠⠐⢄⣡⠀⠀⠀⢀⠀⢀⠀⠀⢀⣤⣤⣤⣀⡀
⠀⢀⠀⠀⢠⡀⠀⠀⠀⠀⠀⢂⡀⠀⠀⠀⣀⠀⢀⡀⡃⠀⠤⡥⢬⠖⠋⠀⠀⠅⡠⠀⣀⠀⠀⠀⠨⢉⠉⠁⠶⠋
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⡤⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡇⣀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡙⠁⠀⠘⢀⡐⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠠⠀⠀⢼⣵⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠐⠁⠀⠀⠀⢀⠀⢱⠀⠀⠀⠀⠀⠀⠀⢶⣴⣄⡇
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠈⠀⠀⠀⢠⣴⢶⠶⣦⠀⠀⠀⢀⣀⠀⠀⠀⠠⠶⠀⠀⠴⠆⠀⠀⣠⠀⡛⡀⠀⠀⠀⢠⣤⣤⡄⠰
⠀⠳⢶⣶⣶⠶⠃⠀⠈⠛⠾⠿⠛⠀⠀⢄⡈⠀⣀⡄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢚⠟⢑⣇⣱⠀⠛⠣⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠉
⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠶
⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠐⠓⠲⠂⠀⠀⠀⠙⢿⡉⠋⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠠⠾⠥⠀⠀⠀⠀⠰⣦⣄⣤⡾⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡦⠄
⠀⠀⢁⡐⠀⣨⠀⠀⢀⡠⠀⣤⠐⠀⠀⠀⢶⡌⠑⠄⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠸⠶⠂⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠈⠐⢎⠹⠀⠀⠀⠁⠈⠀⠁⠀⢀⣹⢿⡟⣿⡄⠀⠀⠠⠸⠠⡔⠁⠀⠀⠶⠆⡀⢠⡀⠀⠀⡘⠁⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠀⠐⠂⠐⠁⠀⠀⠀⠀⠘⠃⠈⠁⠀⠀⠠⢀⣠⣀
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢴⡄⠀⠀⣠⡀
⠀⠀⣩⣤⣦⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⡶⠟⠋⠁⠀⠀⠀⠀⠀⣦⡉⠀⠀⢀⠀⠀⡅⢠⠡⠀⠀⠀⠐⠀⠸⠃⠿
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠀⠋⠀⠀⠠⡿⠟⠛⠛⠃⠀⠀⠀⠀⠎⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡿⢿⡿⠻⡿⢿⢿⠿⣿⣿⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣷⣾⣶⣾⣷⣾⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠿⠛⠛⠛⠛⠛⠛⠋⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⢠⡶⠀⠀⠀⠀⠶⡄⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⠿⣿⡟⠀⠀⠀⠐⠚⠀⠀⢸⠈⠉⠀⠀⢸⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢷⡙⠿⠀⠀⠀⢠⠀⠶⣄⣽⠀⠀⠀⠀⢸⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⠀⠀⠀⠴⢿⡀⠀⠀⠀⠀⣠⠀⠀⡼⠀⠀⠀⢻⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡓⠚⠲⠼⣧⡀⠀⠀⠀⠉⠓⠒⠚⠋⠁⠀⢠⠇⠀⠀⠀⢸⣇⢦
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠛⠛⣢⡾⢻⠛⣦⡀⠀⠀⠠⢠⡄⠄⢀⣴⠋⠀⠀⠀⢄⢺⡽⢼⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠋⠉⢑⡾⠋⠀⢸⠀⠀⠉⣳⠲⠶⠶⡖⠚⢏⢻⢳⡄⠀⠀⢠⠻⣔⡉⠵⣫⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠶⠷⢶⣤⠀⠁⣼⠏⢰⠃⠀⢱⢀⠈⠀⢧⡙⣦⡀⠀⢱⠈⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠙⢸⠀⡇⣿⢘⠷⣼⠋⠀⠀⣠⡀⠀⢆⡷⠹⡎⢧⣧⠟⠀⢸⠀⣷
⠀⠀⠀⠀⠀⠀⡜⡟⢇⢸⠀⡇⣿⢸⠀⠿⠀⠀⠀⢳⠱⠜⠈⠑⢱⠡⠄⠱⡄⠀⡜⠀⢸
⠀⠀⠀⠀⠀⢀⡇⣇⢸⣦⠀⡀⣿⢈⣤⡾⠃⠀⣴⠀⡇⠀⣀⡤⠖⠋⠀⢠⣇⢰⠁⠀⢸
⠀⠀⠀⠀⠀⢸⡇⣿⢨⢻⣄⣥⡟⠋⠀⢷⣀⣠⡿⣴⠁⣼⣁⣠⢾⠳⡖⠋⠈⠛⠀⠀⣜
⠀⠀⠀⠀⠀⢾⡇⢟⢸⢸⡏⡙⢷⡀⠀⠘⣧⣁⡟⠁⡸⢹⢹⢱⡞⠀⠈⠳⡤⠔⠊⠈⠉⠓⠒⠤⡄
⠀⠀⠀⠀⠀⢻⣇⢸⢸⢸⠇⡇⢘⢳⡀⠀⠘⠋⠀⢰⡇⢸⢸⣼⠇⠀⠀⣰⢥⣉⣉⡡⠤⠤⠒⠊⢸⣿⣶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⢸⡟⠛⠿⠾⠶⣦⣤⣬⣿⣄⠀⠀⢀⣮⣤⣼⣾⣿⣴⣶⣿⠋⢣⡀⠀⠀⠀⠀⠀⣠⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⣤⣀⠉⠑⠒⠒⠊⠉⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿
⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠒⠺⠷⠶⢤⣭⣿⣿⣿⣿⣿⣿⣿⣧⣤⣬⣥⣭⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡟⠛⠿⣿⡿⢿⣿⣯⣭⣽⣼⣿⣯⣭⣿⣯⣿⣭⡿⠏⠉⠁⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠈⠻⣦⠈⠙⣿⢿⣿⣿⣿⣿⣿⠿⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⢦⣤⣤⣤⣤⣤⣿⣧⣀⡀⠀⠀⠈⢳⡀⢹⣀⣨⣿⣿⠋⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠛⢦⡀⠀⠀⢻⣾⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⢻⣄⣴⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⣿⡿⢋⣛⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠘⢿⣿⣶⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢶⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠃⠈⠙⢦⡀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⡴⠋⠀⠀⢻⣸⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⢖⣾⣷⠃⣀⡤⠶⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢤⣀⣻⡷⣶⠶⢤⣀
⠀⠀⠀⠀⠀⣠⠞⢁⣀⣤⣿⠉⣹⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⣼⣷⠦⣬⣳⣄
⠀⠀⠀⠀⢰⣿⣯⣭⠵⠛⢉⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⠙⠲⠾⣿⣇
⠀⠀⠀⠀⣿⠈⢧⡀⠀⠀⡾⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⣠⡞⠁⣿
⠀⠀⠀⠀⣿⠀⠀⠙⢦⣼⠃⠀⠀⠀⠀⠀⢸⠷⣄⠀⠀⠀⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠉⠀⠀⣿
⠀⠀⠀⠀⢹⡄⠀⠀⣰⠋⠀⠀⠀⠀⠀⢀⡾⠀⠈⠛⠲⠦⠤⠿⠛⠶⠤⣤⣤⣤⡿⠷⢦⡀⠀⠀⠀⠀⠀⠈⠓⣶⢀⡏
⠀⠀⠀⠀⠈⣧⠰⣞⠁⠀⠀⠀⢀⣠⡴⠛⠁⠀⠀⠀⠀⠶⠒⠀⠀⠀⠒⢦⠀⠀⠀⠀⠀⠙⠳⠦⣤⣤⣤⡤⣾⠁⣸⠃
⠀⠀⠀⠀⠀⢹⡄⡼⣿⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⣦⡏
⠀⠀⠀⠀⠀⠀⣿⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠓⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣿⠁
⠀⠀⠀⠀⠀⠀⣿⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿
⠀⠀⢠⣀⣀⡴⠃⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠘⢦⣀⣀⣄
⠀⠀⠈⠻⣍⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣨⡿⠁
⠀⠀⠀⠀⠈⠳⣤⣀⡀⠀⠘⢧⣀⠀⠀⠀⠀⠀⠀⠀⣿⣵⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠀⠀⣀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠉⠉⠳⢤⣀⠀⠀⠀⠀⣿⣯⣭⠿⢷⣻⡇⠀⠀⠀⠀⠀⢀⣠⡴⠋⠉⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⠦⣬⡇⠛⠛⠛⠛⢫⣇⣀⣤⡤⠶⣾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣿⠶⣤⣽⣄⠀⠀⣠⡞⣉⣀⣀⣤⠴⢿⡶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⡇⠀⠀⠀⠈⣷⢸⠉⠉⠉⠉⠁⠀⠀⠘⣧⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⣾⠀⠀⠀⣀⣀⣿⢸⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠐⠶⠷⣴⠞⠋⠉⠀⣿⣸⣦⡀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣠⣼⣧⡄⠀⠀⠀⠀⠀⠀⢻⡄⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣠⣤⣾⡃⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢦⣤⣄⣀⣿⣀⣠⣤⣄⣠⣼⠋⠉⢹⣆⣀⣀⣀⣀⡀⠀⢻⡀⢀⡴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣰⣏⣤⣤⣤⢿⡁⠀⢘⡇⠙⠦⠤⠾⠉⢩⡿⠛⢿⡍⠳⣾⣷⣏⢠⡟⢻⡄
⠀⠀⠀⠀⠀⣼⠛⠛⠛⣻⡿⠿⣴⠿⣿⡷⠛⠁⠈⠳⠶⠚⠁⠀⠀⠀⠀⠀⠸⣆⣀⣠⡟⠀⠈⣿⠟⣿⣟⠻⢿⣟⠛⠛⠛⢻⡄
⠀⠀⠀⠀⣰⠃⠀⣴⠟⠁⠀⣴⣾⣷⠛⣷⣄⣤⠶⢳⡄⠀⠀⣀⣤⣤⡀⠀⠀⢈⣭⣥⣀⣴⠞⠋⣰⣯⡿⠶⣦⠈⠻⣆⠀⠀⢻⡄
⠀⠀⠀⢰⠏⠀⠸⡇⠀⠀⠀⣿⡙⠿⣽⣛⠶⠶⠾⢿⣷⣤⣶⠿⠽⠿⠿⠳⠖⠛⠉⢀⣈⣥⠶⠛⣡⢿⣤⠞⠁⠀⠀⢸⠆⠀⠀⢷⡀
⠀⠀⢠⡟⠀⠀⠀⠙⢦⣀⠀⠈⠛⠦⣄⣉⠙⠛⠲⠶⠶⠦⠤⠤⠤⠴⠶⠶⠶⠞⠛⠋⣉⣠⠴⠛⠙⠷⡿⠀⠀⣀⡴⠋⠀⠀⠀⠈⣷
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⣄⣀⡉⠙⠓⠲⠶⠦⠤⣤⣤⣤⡤⠤⠴⠶⠖⠛⠋⠉⠀⠀⢀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠘⣧
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆
⣼⣅⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⢀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠐⣀
⠀⠀⢀⣠⢶⣿⡉⠉⠳⡀⠀⠀⠀⠀⠀⢀⡴⠋⠛⠶⣤⡀
⠀⣰⠞⣹⡍⠙⠁⠀⣴⣿⣶⣤⣤⢤⣶⠋⠀⠀⠀⠈⢀⠉⢷⡀
⢰⣇⡀⠈⡀⠀⠀⠐⣿⣟⣺⡿⠀⡀⠛⠁⣠⠠⣀⡀⠀⠀⠀⢹⡄
⣾⠉⠁⠀⠛⢠⡄⠀⠘⠛⠛⠉⠀⠁⠀⡈⢁⠤⠄⢒⡄⠀⠀⣴⡇
⣿⠂⠀⢀⠀⠈⠁⡀⠰⠆⠀⠀⠃⠀⠘⠁⣒⣶⢎⡕⣨⠀⠀⠛⣇
⡿⠀⢘⣿⣞⠀⠘⠟⠀⠀⠀⠠⣶⡆⠈⢖⠁⠢⠓⡂⠋⠐⠆⠀⣿
⣇⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠚⠃⠐⠁⠀⢠⡄⠀⡏
⣿⠀⠀⢠⣴⣄⠀⢴⠄⠀⠀⡤⠀⠀⠆⠀⡄⠀⠀⠀⠀⠀⠀⢤⣿
⣇⠀⠀⠘⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠛⠀⠀⠒⣯
⣿⡴⠀⠀⠀⠀⠀⠈⡂⠀⠀⠐⠃⢰⣤⢿⣤⡦⠈⠁⠀⠀⠀⠀⣯
⣿⠉⢠⣦⠘⠀⡀⠀⠀⠀⢰⠆⠀⣠⣿⣿⣿⣄⠀⠀⡁⡄⠒⠀⣿
⡟⠃⠀⠁⠀⠀⠁⣠⣀⠀⠀⠀⠀⠈⠀⠿⠀⠁⠀⠀⠀⠀⠁⠀⡏
⡧⢴⠀⠆⠀⠀⣾⣿⢿⡷⠀⠀⠀⢀⢠⠀⠀⠀⠶⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠰⢂⡄
⣿⠆⠚⠐⠀⠈⢿⣮⣿⣟⠀⢀⡀⢨⣿⣯⠁⠀⠀⢠⠀⠀⢈⠶⣿⠀⠀⠀⣀⠤⡏⠾⡿⣿⡦⣄⡀
⡇⠘⠃⠙⠀⠀⠀⠉⠋⠀⠀⠈⠁⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⣿⠀⣰⠛⡉⠀⣿⡖⠻⠁⠀⠈⠙⣆
⡷⠀⢠⡄⢀⠀⡌⠠⠀⠸⡷⠀⠛⠀⠀⢀⣤⣄⠀⠀⠀⠂⠀⠀⢷⢠⡏⢀⠁⢀⠉⠀⠀⠠⡤⠀⠐⣿⡆
⡟⠀⣈⠁⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⠀⠀⠀⠀⠰⣰⣿⣇⠈⠁⢀⠀⡈⠁⠀⠀⠀⠄⣿⣿
⣇⠐⠋⠀⢀⠈⠁⠀⢀⣄⠀⠀⠀⠀⠀⢿⣷⣿⠯⠀⠀⠘⠀⠘⡙⡋⢀⠀⠃⠀⠀⠁⢀⣄⠀⠀⠀⢻⣿
⢹⡀⢀⣘⣿⣞⠀⠀⢀⠁⠀⠘⠊⠀⢀⠀⠀⠀⠀⠀⠀⠀⣂⣰⠇⠔⡄⠐⠀⢀⡀⠀⠘⠟⠀⠀⠀⢘⡞
⠀⠙⠲⠤⣼⣬⣤⣦⣀⣀⣠⣤⣤⣤⣸⣇⣤⣤⣼⣧⠤⠾⠛⠁⠀⠀⠸⠷⠀⠛⠓⠀⠐⠂⠘⠀⠾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣲⣦⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣟⣿⣿⣿⣻⢻⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣛⣿⣿⣷⣾⣎⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢿⣿⡿⠻⣿⡿⠋⠁⠀⠉⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⣿⠁⠀⠈⠁⠀⠀⠀⢀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠈⣿⣿⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢤⠀⠀⠰⢾⣉⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠀⠁⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⢐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⣄⠰⠂⠀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⡀⢤⠀⠀⠀⡀⠈⠢⣄⣠⣤⡀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠠⠔⠂⠀⠀⡐⠀⠀⠀⠈⣿⣿⣿⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⡄⠀⣀⣀⠔⠀⠀⠀⠀⠀⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⢻⣿⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠐⡛⠂⠠⣄⠠⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⡤⠔⠋⢸⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠜⠁⠂⠄⠷⣵⡄⠀⢀⣴⣿⣿⣿⣿⣯⣿⣿⣿⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⢸⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⠍⡛⢮⠁⢹⣶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⠄⣸⡀⢸⣿⣦⣀⢈⣿⠿⢻⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⠋⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⣀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡗⠀⢀⣾⣿⣿⣿⠟⠁⢀⣿⣿⣿⣿⡿⠀⠀⠀⢀⣿⡇⠀⠀⠀⢀⣾⣿⡏⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⣴⣾⡿⠿⠿⠁⠀⠀⠘⠻⡿⠿⠿⣇⠀⠀⠀⣸⣿⢁⡇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣟⠁⠀⠀⠈⠉⠂⠉⢳⡔⠉⠉⢧⠀⢈⡇⠀⢀⣿⣿⠈⣇⠀⠰⣄⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠙⠓⠶⠦⠤⠬⠤⠦⠴⠒⠂⠐⠉⡽⠁⠀⢸⣿⣿⡆⣿⣦⣀⣬⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠓⠒⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⢀⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠤⠤⠄⠄⠀⡐⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠰⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠠⡀⡀⠀⠀⠀⠀⠀⠀⠀⠤⠐⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⢄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠉⠉⠙⠓⠒⠒⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠉⠛⠉⠙⠒⠲⢤⡀⠀⠀⠀⢀⣸⠤⠔⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⠖⠋⠁⠀⠠⠖⠡⠟⡷⣄⡀
⠀⠀⠀⠀⠀⠐⡒⡾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⠰⡄⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄
⠀⠀⠀⠀⠀⡈⠹⠿⢦⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣢⢟⣹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢉⣉⣲⣄⣀⢤⠄
⠀⠀⠀⠀⠀⠀⢁⣮⠆⢽⡍⢿⣖⢶⠶⢶⠒⣒⣚⣹⠉⠞⣳⡽⠉⠈⠢⣀⠀⣀⣤⣴⡶⢛⣟⢛⡛⣉⣁⣤⠞⠃
⠀⠀⠀⠀⠀⠀⠸⢛⠦⢄⣀⠀⠈⠉⠙⡓⠀⣉⠀⠘⠋⡥⢎⡟⢠⢀⠀⠀⣽⠉⢤⡏⠀⡎⢻⠛⢯⣄⡈⢙⣭⠗
⠀⠀⠀⠀⠀⠀⠘⠦⢀⣠⡌⠛⠁⠤⢄⢠⣈⣁⡀⠀⡀⣠⠎⠁⠣⣌⠄⢸⣿⡄⢸⣬⣖⡽⠄⣇⢀⠦⠽⢾⡜
⠀⠀⠀⠀⠀⠀⠐⢄⢀⠈⠉⠁⠈⠀⠈⠁⠉⠉⠉⠁⠈⢀⠄⠁⠀⠈⠢⣥⠙⣧⠈⣟⠬⣉⣁⡟⢴⣟⠓⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠀⠐⠒⠒⠒⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠙⡓⡁⢶⠨⠇⠈⠂⠭⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠱⠘⠏⠇

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

⠀⠀⠀⠀⠀⠛⠻⣤

⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣶⣿⣿⣿⣿⣿⣷⣾⣍⠻⠿⢿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠟⠀⢀⣀⠀⠹⣿⣿
⠿⠿⠿⠛⠋⠉⠁⠀⠀⠀⣿⣿⠁⠀⣿⣿⡄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠃⠀⣰⣿⣿⣧⠀⠀⠀⠀⠀⣀⣀⠀⢀⠀⠀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⡿⠀⠀⠀⠀⠉⠙⣛⠛⣿⣶⣾⣿⣿⣿⣿⣷⣤
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⡍⢻⠀⠈⡇⠀⢘⣁⣸⣿⣿⣿⣿⣿⣿⣷⣽⣿⣏⢿⣯⣿⣧⡤⠀⢀⣀⣀⣀
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠃⠘⡇⠀⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣾⣸⠏⠀⠈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣥⣶⣿⣿⣿⣿⣙⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢛⢛⣛⣿⣹⣶⣄⡀⠀⠀⠌
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣄⣌⣯⠉⠉⣭⣏⣉⡟⠛⠟⠉⠉⠀⠤⠭⠉⠉⠉⠀⠁
⣿⣶⠒⠀⢆⠀⢰⡄⠀⠤⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣴⠶⠶⢶⣶⣶⣶⣶⣖⣾⣿⠿⠇⠀⠀⣀
⣏⡉⠳⠄⠀⠀⠘⠇⠀⠀⠀⢀⣿⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⡤⣾⣯⣤⣿⣿⣉⣈⣛⡡⠴⠶⠿⠿⠛⢟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠢
⠙⢏⠂⠀⠀⢄⠀⢠⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⣭⠄⠁⠠⣀⣀⣀⣤⣤⣤⣴⠶⢤⣄⡀⣀⣄⡀
⡄⠀⣤⣄⣱⡆⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⡿⠿⡿⠿⠛⠛⢻⠉⠉⠉⠀⠀⣀⣀⡀⠉⠃⣿⣿⣷⡆
⠷⣀⣻⣿⣿⣷⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣬⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣶⣿⣿⣷⠾⠶⢿⣿⠿⠟⠛⠁⠀⠀⠀⠘⠛⠋
⠋⠉⢉⣿⣿⣿⣷⣄⠀⠘⢿⣿⠿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣯⣉⣹⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⢀⣤⣀⡤⢀⣼⣿⣶⡆
⠛⠛⢻⣿⣿⣿⣿⣿⣿⣦⣄⠙⢿⣿⣿⣾⣿⣿⣯⣙⣛⣛⣛⣛⠻⣿⠯⠿⠟⢛⠛⢛⣿⣻⣯⣥⣶⡶⡿⠿⠿⠟⠁⠰⣿⣿⠟⠁
⠋⠉⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣙⡛⠿⠿⠿⣿⣿⣿⣿⣿⣷⣶⣶⣶⢿⣿⠿⠿⠿⠛⠛⠅⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔
⣤⣤⡟⠿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣟⣻⡶⠶⣤⣤⣬⣉⣉⣛⣛⠛⠛⠓⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠒⠈⣉⠀⠶
⣿⣿⣿⣷⣦⣍⠻⣿⣿⡷⢋⣴⠞⠉⢀⣴⡿⠛⢓⣶⣶⠦⣭⣭⣍⣉⣉⣛⣛⣛⡛⢛⣛⣛⠀⠈⠉⣁⢀⣠⣤⣤⣴⣦⡘⠓⠈⠁
⣿⣿⣿⣿⣿⣿⣦⠙⡟⣡⡿⠃⢀⣴⠿⠋⢀⣴⡿⠋⠀⣠⣾⠟⠉⣉⣽⡿⠛⠉⢛⠻⠿⠿⠉⠉⡻⣛⡁⠛⠿⣶⣄⡀⠉⠲⢤⡀
⠿⣿⣿⠿⣿⣿⣿⠀⢻⠉⢀⣴⡟⠃⠀⠴⢿⡏⠀⠀⢚⡟⠁⠀⠾⠿⠋⠀⠀⠺⠃⠀⠀⠀⠠⢾⣄⠉⠛⢷⣄⡀⠉⠛⠶⣄⠀⠉⠳

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠉⠁⠀⠈⣹⡟⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⡟⠀⠀⠀⠀⠀⠀⠏⠀⠀⠈⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⣠⠴⠶⠶⠦⠄⠀⠀⣀⣀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁⠈⢀⢀⣀⡠⠄⠀⠀⣴⠏⠉⠛⣦⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢻⡿⠀⠀⠀⠁⠀⠈⠃⠀⠰⡎⠑⡲⣽⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠁⠀⠀⠀⠀⢠⠆⡤⢠⡄⣧⠀⠀⠀⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠛⡇⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢀⣴⣾⣭⣨⣙⣲⣄⣿⢀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣾⡿⢋⢉⣙⠓⠾⣿⡷⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⣿⣰⡟⢿⠿⣿⣦⣌⠙⢧⡺⣕⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢹⡟⣆⠀⠘⠟⠁⢸⡀⢸⡿⠻⣷⡄⢻⣎⢦⠁⣄⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⢧⠻⣷⣆⠀⠀⠈⣇⠈⢧⠀⣿⣷⡀⣿⣿⡇⣹⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠘⡇⠀⠘⢆⡀⠙⣦⣀⢀⡈⢳⣈⢁⠉⣋⣣⣿⣭⣷⡿⣿
⠀⠀⢀⡠⠄⠠⠒⠊⠁⠀⠀⠀⠀⠳⣄⠀⠀⠙⢦⡹⣟⣳⣿⠆⠨⠤⠤⠤⠀⠀⠀⠀⢤⠛⠦⢤⣄
⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⢠⡟⠻⣟⢻⡿⠳⣮⢥⣀⡀⠀⢀⣠⣀⠀⢢⡀⠀⠈⢧
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡿⠁⠀⢹⠏⠀⠀⠙⢦⣠⡟⢻⡛⠋⢁⡤⠋⢻⡄⠀⠈⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠴⠋⠀⠀⠀⠹⡄⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠁⠀⠀⠀⠀⠀⠀⠘⠆⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠘⠋⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣿⣇⣤
⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠐⠂⠈⠁⠙⠟⠿⠿⠀⠀⠘
⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣤⣄⣤⣀⣀⡀⠀⠀⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣄⣻⣳⠀⠀⠀⠀⠀⠀⠀⠚⠉⡹⢻⣿⣧⣿⣿⣷⣦⠼⣛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣷⡦⠀⠀⠀⠀⠀⠀⢀⣨⣠⣾⣿⣿⣿⣫⠔⠛⠿⠆⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢀⣴⣶⣶⣷⣄⢾⣿⣿⣿⣿⣿⣿⣷⡀⠄⠀⠀⠀⠁⠀⠈⠛⠻⣿⡿⠋⠀⠀⠀⠀⣤⣄⣄⣀⠀⢁
⠀⠀⢸⡳⣿⣟⢻⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠙⠻⠾⣽⣛⣻⠿⣿⠷⢿⣿⣿⣿⣿⣿⣧⣶⡦⡀⠀⠀⠀⠈⠉⠁⠀⠀⣀⣤⣴⣾⣟⣻⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠻⠷⠿⠿⠥⠤⢼⣭⣯⣷⣭⣶⠤⠀⠴⠷⠾⠿⠟⠛⠛⠉⠉⠉
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣻⣿⣻⣟⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠲⠞⠉⠛⠉⠙⠚⠋⠙⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠇⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⢠⡤⠞⠓⠒⠋⠉⠙⠉⠓⠶⠶⠲⣄⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⣀⣰⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠦⣤⣴⠋⢳
⠀⠀⠀⠀⠀⠀⡼⠉⡷⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡈⢧
⠀⠀⠀⠀⠀⢠⡇⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⢉⠀⠀⢀⡉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣈⡇
⠀⠀⠀⠀⠀⣞⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢁⣀⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⣀⣀⣀⡼⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣤⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⣿⢿⣿⠿⣿⡇⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⠀⠀⡟⣿⡿⡭⡽⡇⠀⠀⣀⣤⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⣏⣛⢣⡀⠀⠀⣠⠛⠛⠉⣁⣸⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠁⣼⠀⠈⠉⠛⡆⡞⠓⠛⠛⠉⠁⠀⣷⠉⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⢠⡇⠀⠀⣀⣠⡇⡇⠀⠀⠀⠀⠀⠀⠸⡆⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠘⠺⢶⠞⠉⠁⠈⣧⠷⣄⠀⠀⠀⠀⠀⠀⢷⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠶⠛⡆⠀⠀⠀⠀⠀⢸⡆⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⡀⠀⠀⢸⠀⠀⠀⠀⠀⡾⠿⢥⡀⠀⠀⠀⠀⠀⣧⠀⠀⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢩⡟⢓⣚⣳⡞⠛⢳⡛⣧⣀⣰⠗⢒⣶⢶⡶⢦⣸⣤⠞⢹⢿⡀
⠀⠀⠀⠀⢠⠖⠒⠒⣲⡿⣬⣿⣻⡽⠟⠉⠳⠤⠞⠁⠀⠀⠀⠀⢺⡀⢀⣿⠀⠙⣿⠷⣿⠾⣷⡒⠒⠒⠲⣆
⠀⠀⠀⢠⡏⠀⡴⠋⠁⢠⣼⣷⣛⣧⣠⠴⢲⡄⠀⠀⣠⣤⣄⠀⠀⣉⣭⣅⣰⠖⢃⣼⡽⠷⣦⠉⠳⡄⠀⠹⡄
⠀⠀⢀⡞⠀⠸⣇⠀⠀⠸⣍⠛⢯⣟⣒⠺⠿⠷⠶⠾⠷⠚⠛⠓⠚⢉⣀⣠⡥⠞⣋⣼⢷⠞⠁⠀⢀⡿⠀⠀⢹⡀
⠀⠀⡾⠀⠀⠀⠘⠲⣄⣀⠈⠓⠦⢤⣉⣉⠛⠛⠒⠒⠒⠒⠛⠛⢋⣉⣉⡤⠴⠚⠉⠉⠛⠀⣀⡴⠚⠁⠀⠀⠀⢳⡀
⠀⡼⠁⠀⠀⠀⠀⠀⠀⠉⠙⠒⠶⠤⣤⣭⣉⣉⣛⣛⣛⣛⣋⣉⣉⣉⣀⣀⣤⡤⠴⠖⠚⠉⠁⠀⠀⠀⠀⠀⠀⠈⢧
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠿⠛⠛⠛⠻⠿⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣇⠀⠀⠀⣠⡄⠀⠀⠀⢠⣄⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠉⠀⠀⠀⠠⠀⠈⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⠃⠀⠀⠀⠤⠆⠀⠀⡏⠲⠂⠀⠀⣷⠀⠀⠀⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⢰⡙⣿⡆⠀⠀⠀⠀⢀⣀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠦⣷⡀⠀⠀⢀⣸⡇⠀⠈⠙⠁⠀⠀⠀⠀⡿⠀⠀⠸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⢤⣈⣧⠀⠀⠀⠀⠙⠢⠤⠤⠤⠖⠁⠀⢸⠇⠀⠀⠀⢸⣅⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢉⡱⠦⣄⣀⡿⣷⣄⠀⠀⠀⠀⡀⢀⢀⠀⠀⣠⠟⠀⠀⠀⡠⣼⣷⣹
⠀⠀⠀⠀⠀⠀⠀⠰⠟⠠⣤⣼⠛⠀⡇⠘⠳⢦⣄⣀⣁⣈⣬⣴⠾⣧⡄⠀⠀⠀⢁⣿⣋⠛⢆⡔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣏⣨⣘⠀⠇⠀⠀⣼⠃⢹⠈⠹⡄⠈⠃⢻⡝⢦⡀⠀⠘⣌⠳⢃⣩⠞
⠀⠀⠀⠀⠀⠀⠀⢀⡾⡝⡏⡉⢹⠛⣧⢀⣼⠋⠀⠃⠀⠀⢃⢠⡠⣄⢻⣈⢓⡄⠀⢸⠀⡎
⠀⠀⠀⠀⠀⠀⢀⣾⢹⠀⡇⡇⢸⠀⡏⢻⡃⠀⠀⢰⡱⡀⡜⠸⢣⣸⠀⠻⣟⠁⠀⣸⠀⣇
⠀⠀⠀⠀⠀⠀⣶⢻⢸⡄⡇⡇⢸⠀⠇⣘⣧⠀⠀⡀⢧⠉⠀⠀⠀⣛⡼⠀⢸⡆⢠⠇⠀⢸
⠀⠀⠀⠀⠀⢠⡇⢸⠈⣷⡇⠀⢸⣤⠿⢛⡁⠀⢰⡇⢸⠀⣠⠖⠋⣡⣀⣀⡼⢣⣼⠀⠀⢸
⠀⠀⠀⠀⠀⢸⣇⢸⠀⡏⣷⣾⣏⠁⠀⠈⣷⠶⢾⣹⠃⡸⡷⡴⢟⡿⠙⢯⡀⠀⢁⡀⢠⢧⣀⡀
⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⡇⢸⠹⣦⠀⠀⠹⣧⠜⠁⢠⠃⡇⡇⣿⠃⠀⠀⢱⠖⣉⣀⣀⠀⢈⠉⢐⢢⣀⣀
⠀⠀⠀⠀⠀⢸⣧⣬⣄⡇⡇⠘⠐⠙⣧⡀⠀⠉⠀⠀⡿⠀⡇⣳⡿⠀⢀⣠⣶⠲⠤⠤⠔⠒⠒⠉⠀⡸⢿⣿⣿⣿⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢸⡏⠉⠙⠛⠛⠿⠾⠷⠿⢷⣄⠀⢀⣼⢶⣶⣿⣿⣿⣿⣿⡟⠈⠳⣀⠀⠀⠀⢀⣠⠴⠃⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣉⣉⣉⣁⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⣿⠇
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠒⠛⠛⠛⠶⠶⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠛⢿⣿⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠈⠻⣆⠈⠹⣿⠿⣿⣿⣿⣿⡿⠟⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⠶⢦⣤⣤⣤⣤⣼⣿⣤⣀⠀⠀⠀⠘⢧⡀⣿⣀⣬⣿⣿⠉⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠉⠻⣦⠀⠀⠈⣷⣿⣿⣿⣿⣿⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⢷⣀⣾⣿⣿⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⣿⡿⢛⡛⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠻⣿⣯⣶⣿⣿⣷⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣼⣛⡲⢤⡀
⠀⠀⠀⠀⢠⠖⠋⡍⠉⡉⢉⠙⠳⣄⠀⠀⠀⠀⠀⠀⣴⠶⠾⠷⠶⣦⠀⠀⠀⠀⠀⠀⠀⣼⠀⣿⠀⠀⠀⠀⠀⠀⢀⡴⢚⣩⣭⡇⡿⢟⡙⢦⣹⡄
⠀⠀⠀⠠⣏⣀⣃⣀⣁⣀⣁⣘⣀⣸⡆⠀⠀⠀⠀⢾⣉⣉⣉⣉⣉⣉⡇⠀⠀⠀⠀⠀⢠⡟⠒⠻⡆⠀⠀⠀⠀⢠⡟⣴⣋⣐⢺⡇⣷⡯⢟⣫⣍
⠀⠀⠀⠀⣯⠤⠤⣤⡤⠤⣤⡤⠤⣽⠇⠀⠀⠀⠀⠈⣯⠈⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⣾⠛⢳⠀⣿⠀⠀⠀⠀⣾⢰⡇⢯⡽⢸⣧⠵⣚⠉⡇⢹
⠀⠀⠀⠀⢳⠶⣶⡒⠒⠒⠒⣶⠶⣾⠁⠀⠀⠀⠀⠀⢻⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢸⡶⠛⠀⡟⠀⠀⠀⠀⢹⡌⣇⢰⣷⡌⠄⠺⠿⣰⢇⡿
⠀⠀⠀⠀⠘⠳⠾⠷⣦⣠⠾⠷⠖⠋⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣇⠀⠀⠀⠀⠀⠻⣌⠳⢭⣴⣏⡷⠞⣣⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠛⠀⠀⠀⠀⠀⠀⠀⠻⠤⠤⠤⠟⠀⠀⠀⠀⠀⠀⠈⠙⠲⠶⠶⠖⠛⠁
⠀⠀⠀⠀⠀⢀⣀⣠⢴⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⢀⣴⠻⡌⢿⢸⡟⣸⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢘⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡅⠻⣜⣳
⠀⠀⠀⠈⡏⠛⠛⠚⠛⠒⠛⠋⣿⠀⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⠶⣦⣀⡀⠀⠀⠀⢠⡞⠉⠠⠈⠠⠉⠄⠈⢳⡄⠀⠀⠀⠀⢀⡞⠰⠀⠄⠠⠈⡽
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣿⢁⡿⠀⠀⠀⠀⢹⡏⠹⡟⠉⢹⡏⢩⡟⠁⠀⠀⠀⠀⣸⣇⠐⠀⢂⣠⠞⠁
⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⣠⣤⣼⣧⣤⣤⣤⣤⣿⣭⣤⠀⠀⠀⠀⠀⢷⠀⣿⠀⣸⠁⣼⠁⠀⠀⢀⡤⡼⣃⡼⠓⠛⠉
⠀⠀⠀⠀⠀⠧⠤⠤⠤⠤⠤⠾⠀⠀⠀⠀⠀⠹⢦⣤⣤⣤⣤⣤⣤⣤⣤⠞⠀⠀⠀⠀⠀⠘⠧⠼⠤⠿⠴⠃⠀⠀⠀⠈⠳⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣉⣳⣆
⠀⠀⠀⠀⠀⠀⢀⣀⡴⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⣄⠀⠀⠀⠀⣾⣁⣈⣹⣄⣙⣦
⠀⠀⠀⠀⢀⣾⣩⣥⣀⣠⣩⣧⠀⠀⠀⠀⢀⡞⠉⠉⢹⠏⠉⠉⡟⠉⠉⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠲⢾⡀⠀⠀⠈⢷⠖⠲⠖⣶⣶⡞
⠀⠀⠀⠀⢻⣥⣄⣠⣤⣀⣤⣬⡇⠀⠀⠀⣼⢶⠶⠶⡶⠶⢶⠶⠶⡶⠶⢶⢾⡆⠀⠀⠀⠀⠀⢀⡴⠋⠳⢤⡀⢨⡇⠀⠀⠀⠘⣷⠖⢋⣡⣼⠁
⠀⠀⠀⠀⢸⡄⠈⡇⠀⣿⠀⣸⠃⠀⠀⠀⣿⣯⣛⣛⣟⣛⣋⣛⣫⣿⣚⣉⣿⡇⠀⠀⠀⢀⡴⠛⠳⣄⠀⠀⠙⣿⠀⠀⠀⠀⠀⢹⡞⢋⣠⡇
⠀⠀⠀⠀⠀⢷⠀⣷⠀⡟⢠⡏⠀⠀⠀⠀⠘⠲⠾⠭⠭⣬⣭⣭⣬⠭⠽⠷⠚⠀⠀⠀⢿⣉⠹⡄⠀⠘⣦⣠⡾⠃⠀⠀⠀⠀⠀⠈⣿⠉⡿
⠀⠀⠀⠀⠀⠘⠓⠛⠒⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣻⠙⠖⡦⠀⠀⠀⠀⣠⠞⠳⣄⢀⡴⠛⣦⠀⠀⠀⠀⠀⣯⠭⣽⡇⣴⠟⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⡴⣣⣏⡽⠷⣤
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠈⣷⠛⣯⣿⣹⠆⠀⠀⣰⠋⠐⣦⠈⣯⠀⢶⠈⢳⡀⠀⠀⠀⢸⠀⢸⠀⣹⠏⢹⣇⠀⠀⠀⠀⠀⠀⣠⢞⡵⣫⡞⠁⠀⠈⡇
⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡟⠛⢻⡾⠃⠀⠀⠀⢿⠀⣰⠃⠀⣽⢠⡏⠀⢸⠇⠀⠀⢀⡾⠀⠸⣾⠁⠀⠀⢸⡇⠀⠀⢠⡖⣫⡵⣫⡴⠋⠀⠀⢀⡼⠁
⠀⠀⠀⢰⡋⠀⠀⢀⣴⠏⢀⡴⠋⠀⠀⠀⠀⠀⠈⢧⡈⢁⡴⠳⣌⠃⣰⠏⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠸⣏⡽⠛⠁⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠙⢦⣴⠟⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠘⠛⠁⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⠀⠉⢳⣄⣀⣀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒⠒⠛⠓⠒⠒⠚⠃⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠟⣯⣵⣾⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠉⠁⠀⣸⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⠀⢀⣀⣀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣾⠂⠹⠥⣄⠀⠀⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣰⢀⠀⠀⠀⠀⠀⠀⠛⠋⢉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣈⣉⣉⠀⠀⠀⠀⠀⢀⣂⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢾⣽⢙⣪⣤⣤⣼⣼⣶⡶⠃⠀⠀⠀⠠⠎⠉⢸⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢒⣿⠦⠈⣿⣶⣯⡿⠋⣀⣀⣠⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡜⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⣗⡀⠍⠉⠁⢠⠞⠁⠀⠈⣫⣟⠲⣶⢷⣄⡀⢀⡠⠚⠁⢀⡴⠁⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣜⣿⣧⡴⠀⠀⢠⠋⠀⠀⠀⠀⠀⠉⡄⢈⠳⡏⣿⠱⡀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠓⢾⡿⠁⡀⠐⠃⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡜⠀⠁⠡⣸⠁⠀⠀⠀⠀⢠⠖⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠀⣀⣀⡤⠖⠋⢩⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠑⠦⢤⣀⣤⠶⠋⠁⠀⠀⠀⠀⠸⠇⠀⠀⠀⢰⢒⣄⠸⠶⣞⣒⣆⠤⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠄⠀⣤⣤⣤⣤⣤⠀⣀⡀⠀⠐⠂⠉⠁⠈⣶⡷⣵⣛⡀⣈⣉⢱⣤⡝⢶⣤⠼⠤⠤⠤⠤⠤⠤⠤⠤⠤⠧⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠶⠄⠶⠶⠀⠐⣒⢦⣦⢄⣭⣉⠄⢸⣯⢀⡀⣀⣠⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠰⠟⠁⢟⣘⣛⣼⣛⣷⠂⣸⡧⢜⡋⣰⡬⠙⠃⢙⠋⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⠠⢛⡱⠦⠩⢔⣿⡛⠁⠉⠁⠐⠘⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⡩⢣⠀⣤⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠓⣀⢦⠠⠤⡏⠇⠰⠰⡅⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠐⢆⡰⢌⡄⠹⡤⠿⠲⠁⣎⡹⠆⠡⠀⢂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⣄⢀⡸⠒⠎⠘⡀⡠⠞⢢⠀⠨⠁⠈⡀⢂⠈⠎⠖⣰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢹⠇⣤⠋⢄⠀⠑⢻⠲⠉⢇⠀⠁⠡⠐⠀⣃⣸⠞⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠶⣶⠀⠀⠀⠀⠀⣀⡇⠀⠈⢆⠈⡆⠀⠈⠓⠋⢇⢃⠘⡆⠀⠀⣣⡶⠟⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⣰⠃⠀⠀⠀⠀⢸⡗⠶⠦⢤⣬⣤⣽⣄⣦⣤⣥⣸⣬⣤⡴⣶⠞⠉⠀⠀⠀⠀⠀⡇
⠀⠀⣠⡤⠔⠒⠂⠀⣈⡧⡤⠓⠒⠤⢤⡀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⣠⣤⣤⣤⣤⣧⣤⣀⣀
⠀⣼⣏⠀⠀⠀⣠⣾⡿⠀⠀⠀⠀⠀⣀⢟⡆⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠛⠉⠉⠁⢀⠀⠀⠉⠉⠉⠛⠻⢷⣦⣤⡀
⠀⡷⠬⣍⣛⣒⠚⠛⠒⠒⠒⣒⣚⡉⠥⠚⡇⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⢉⡁⡄⠀⠀⠠⠀⠂⠀⠉⠊⠁⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣄
⠀⡏⠀⠀⠈⠉⠉⠉⠉⠉⠁⠁⠀⠀⠈⠀⡇⠀⠸⡆⠀⠀⣀⣴⠾⠋⠁⡠⢰⠄⠉⠀⠂⢂⣠⠤⠖⠒⡈⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠈⢿⣷
⢠⡇⠀⠀⠀⠀⠀⠠⠀⠀⠄⢀⡀⠀⠀⢀⡇⠀⠀⣇⣠⡾⠋⠁⠀⠀⢀⣈⣀⣤⠤⣤⠴⠿⠤⠤⣤⣉⣀⠀⢴⡴⠀⠉⠉⠠⡠⠆⠀⠀⠀⠈⣿
⢸⡇⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇⢀⣴⠟⠁⠀⢀⠀⠐⠊⠉⠁⣠⠔⠋⠀⠀⡄⠀⣀⠀⠈⠉⠛⠲⣄⠀⣆⠜⠀⠀⠀⠀⠀⠀⣸⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⡀⢸⣷⠟⠁⠀⠀⠺⠈⢀⡠⠂⣤⠞⠁⠀⢀⡀⠉⠀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃
⢸⡇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠄⡆⠀⣠⠞⠁⠐⠁⠐⠒⠁⢠⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⠟⢻⡇
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣰⡟⠀⠀⠀⠀⠀⠀⠋⠀⡴⠃⠀⠀⠸⠥⠛⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⢀⣤⡞
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠛⠉⣀⣤⠞⠋⣠⡇
⢸⡇⠀⠀⠁⠀⠀⠀⠀⡜⠀⠀⢨⣿⣦⣄⡀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⡿⠿⠛⠉⣀⣤⠶⠛⠉⢀⣤⠞⠉⡷⢤⡀
⣿⡇⠀⠀⠀⠀⠀⢠⠎⢀⡀⠔⠉⠳⣍⡛⠿⢿⣶⣶⣴⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⡿⠿⠟⠛⢉⣁⣤⠴⠞⠋⠉⠀⣀⣤⠾⠋⠁⣀⡼⠃⠀⣿
⣿⠇⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠸⣇⠙⠓⠦⢤⣈⣉⡉⠉⠉⠉⠉⠉⢉⣉⣁⣠⣤⠴⠖⠚⠋⠁⠀⢀⣠⣤⠶⠛⠉⠀⣀⣴⠞⠋⠀⠀⢠⣿
⣿⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⢳⢦⣄⣀⠀⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠀⠀⢀⣀⣤⡴⠶⠛⠋⠉⢀⣀⣤⠶⠛⠉⠀⠀⠀⢀⣴⡿⠁
⣿⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢦⣌⠉⠙⠛⠓⠒⠲⠶⠶⠶⠖⠒⠛⠛⠋⠉⠁⢀⣀⣤⡴⠶⠛⠋⠁⠀⠀⠀⠀⠀⢀⣤⡿⠋
⣿⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠉⠓⠲⠶⠤⢤⣤⣤⣤⣤⠤⠴⠶⠶⠞⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋
⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠁
⠘⣿⣦⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⢛⣻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⠿⠛⠁
⠀⢿⣝⡛⠻⠷⠶⠿⠿⠟⢛⣋⣩⡴⠖⠛⠁⠀⠉⠛⠷⠶⣤⣤⣤⣄⣀⣀⣀⣠⣤⣤⣤⣴⠶⠶⠟⠛⠋⠁
⣤⣤⣬⣭⣿⣿⣷⣾⣿⣿⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣭⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣭⣏⣼⣉⣯⣩⣿⣹⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢠⡄⠀⠀⠀⡠⠄⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠙⠀⠀⠀⠁⠀⠀⠀⠀⢠⣿⠀⠀⣙⠒⠖⠀⠐⠤⢖⡌⠀⢠⣧
⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⠉⠈⠐⠀⠀⠀⠈⠁⠀⠘⠛⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠃⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠈⠀⢈⠀⠀⠀⠀⠀⣐⠦⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠸⠟⠀⠀⠀⠀⣀⠤⢒⡀⣴⠖⠷⠤⠖⠒⠛⠺⣾⠀⠘⢦⡀⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢸⣷⠘⠳⠖⠖⠲⠲⠆⠚⠁⠀⣾⡀⠙⣦⢴⢪⣡⣾⣷⣿⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⢸⡝⣆⠀⠀⠀⠀⠀⠀⠀⢀⡼⣹⠁⣴⣧⢾⡹⣟⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠰⣆⠀⢸⡇⢻⡷⠄⡀⠀⢀⣤⠐⡿⢁⡏⣸⣯⢧⢃⡵⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢹⠀⠀⡇⠀⣧⠞⠻⣤⣼⠉⣻⠅⢸⢃⣿⣧⣿⡞⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⢸⠀⠀⢻⡀⢹⡄⠀⣿⣟⣷⣃⢀⣾⣾⣿⣷⠿⢸⠇⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣌⡇⠀⡾⣿⣻⣼⢋⢀⣽⣯⢀⢠⣏⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⢸⡧⠖⠦⢤⡀⡄⢸⠀⠀⠀⢸⠏⣷⠀⡿⢸⣿⠻⡟⠚⠉⠀⣼⢹⣸⣒⠉⠁⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⣰⠋⢧⡞⣦⠀⠀⢸⡀⣿⠀⡇⢸⡇⢰⠇⠀⢀⡖⣏⣸⠈⢦⢱⡄⠀⢸⡇
⠀⠀⠀⢀⣀⣀⣸⣇⠀⠀⣿⠀⠒⢧⣼⠀⠀⣘⣧⣿⣀⣇⣈⣧⣿⠤⢤⣼⣼⠉⠀⠀⢸⣇⡇⠀⣼⣀⣀⣀⣀⣀
⠀⠀⠀⣶⡯⠩⠷⠭⣿⣖⣿⣦⡉⢀⣼⣛⡟⣡⠖⠂⠀⠉⠁⠀⠀⠸⡆⠙⣿⣄⣀⣀⣼⣯⣴⣾⢏⡁⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢤⡉⣠⠎⠀⠉⠛⠛⢩⡟⢠⡏⠀⠀⠀⠀⠀⠀⠀⢀⠿⠀⣹⠏⠉⠉⠉⠉⠉⠙⣦⠐⠀⠀⡀
⠀⠀⠀⠀⠐⠩⢭⡞⠑⠃⠀⠀⠀⠰⠾⠿⢿⡶⣤⠭⢍⣉⣀⠀⠠⠖⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠼⣧⡀⠐⠂
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠒⠒⠢
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠸⠁⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠔⠀⠀⠀⠤⠀⠀⠀⠠
⠀⠀⠀⢠⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀
⠀⠀⠀⠀⠈⠑⠃⠀⠀⠓⠂⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠂⠀⠀⠌⠀⠰⠐⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠈
⠀⠀⠀⠰⠁⠠⠦⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠐⠓⢄⠠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠠⣀⠀⠤⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰
⠀⠀⠀⠀⡀⠀⠁⠀⠀⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠐⠂⠀⠀⠒⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⠐⠒⠒⠆⠀⠀⠀⠐⠒⠐⠒⠒⠒⠒⠒⠒⠒⠂⠉⠀⠉⠀⠈⠁
⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⠀⠀⠀⠀⠠⠴⠂⠐⠦⠄⠑
⠀⠀⠀⢐
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠐⠳⠤⠤⠄⠀⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠤⠤⣀⣀⠀⠀⣀⣀⠀⠀⠤⠀⠐⠂⠠⠔⠆
⠀⠀⠀⢀⠆⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢠
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠐⠀⠀⠒⠒⠂⠒⠤⠤⠤⠤⠠⠤⠤⠀⠤⠤⠀⠠⠤⠤⠤⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠤⠤⠄⠀⠀⠰⠒⠀⠀⠀⠀⠐⠈
⡠⠀⠀⠈⠳⠄⠀⠀⠀⠀⠠⠤⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⡠⠄⠠⠄⠤⠀⠠⠔⠋⠀⠀⠠
⠁⠀⠀⠀⠀⠀⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⣽⣀⡶⠃⠾⠐⠻⠀⠄⠀⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣆⠈⠀⠂⠘⣾⡿⠇⢸⢦⣎⡑⠻⠛⠧⢤⠀⠀⠀⠀⠀⢠⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡻⠿⠂⠀⠀⠀⣂⣽⣿⠉⠀⠘⠛⠷⢃⣒⠤⠄⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠮⡿⡜⠁⠀⠀⠀⠄⠀⠀⠈⠓⠒⠚⠂⠀⠀⠩⠅⠀⢰⠀⠰⠀⡀
⡤⠀⠂⣡⣴⡅⡄⣀⡀⢄⡀⣀⠄⡀⠈⠀⠀⢼⣿⢤⡄⠉⠁⣠⡄⠈⠀⠀⠀⢀⠀⠀⢠⠄⠀⡀⢿⣦⡀⢀⣴⡀⢀⢀⣆⣀⢀⣿⠤⢴⣄
⠣⣦⣆⡼⣃⡏⠰⣿⣧⣹⡭⠃⢀⡖⢹⢰⠰⣮⡁⠈⣻⠶⡄⢻⡅⣰⠲⠊⠀⠐⢁⡘⡿⠶⠀⢤⠀⡯⢉⡀⢸⠏⠎⠻⡿⢩⠀⣷⡀⠶⠋
⣾⣿⣿⢿⠿⣷⠂⢸⣿⣷⢎⣿⣧⡉⠁⡄⣭⢿⢁⢴⡗⠉⢠⣖⠉⠉⠁⠀⡦⣑⣿⣗⣿⠐⣷⣿⣟⠷⠉⢥⠀⢲⣦⡀⠀⢫⣿⣿⠁
⣵⣟⣿⣥⣾⣿⣿⢯⣉⣿⣿⣿⡮⢉⠈⣿⣁⡁⣠⣾⠷⣤⠸⢻⡟⢶⡄⠠⢸⣿⠙⢻⣿⡀⠙⠛⢿⢷⣶⡄⢠⣼⢃⠃⠀⠘⣿⠋⣧⣤⢶
⣿⣟⠙⢻⡍⣿⢤⣤⣻⣿⡿⠁⠀⠉⠉⣉⣿⡇⣿⣿⣤⣿⡿⠀⠑⠛⢧⣆⢨⣿⠀⠀⠈⡿⢳⡄⠈⢳⣬⡍⠿⣼⡏⠀⠀⣄⢿⡿⠃⠀⡄
⢁⡠⣥⠉⡿⠿⢚⣧⡾⣭⠠⠡⣰⣮⣿⡿⢃⢺⣿⡿⢿⣽⠇⠀⢀⣾⣿⢿⡏⠀⠘⠛⢳⣌⠋⣱⠀⡊⣿⡇⠚⣿⣧⣄⣀⢈⣈⢃⣶⡦⢶
⠟⣿⣿⣶⣷⠠⣿⣿⣧⠏⣶⣴⡛⣿⣿⣧⡈⣙⣻⣷⠾⠉⠀⠘⣿⠏⣷⣿⡇⠀⡀⠀⠘⣿⣯⠿⠠⣼⣿⠟⣎⠩⣿⡟⠋⠻⣿⣿⣵⡄⣀
⠀⠸⠒⠽⢹⣾⡆⣠⠀⠀⡏⡉⠓⣿⠛⣯⠿⣿⡏⡎⠀⠀⠀⠀⣙⢿⢸⡟⠇⡆⣇⠰⠀⢿⣿⡆⠘⣿⠳⣤⣿⡿⣃⢵⢨⣷⣈⢿⣿⢛⣿
⠰⠤⡅⠀⡏⠏⠤⠶⠶⠆⣥⣤⠾⠿⠿⢫⣾⣿⣹⠃⢰⡆⠀⠀⣿⡶⠆⣈⢿⣿⣯⣥⣴⣼⢿⣿⣷⣿⣦⢘⢿⡁⡤⡄⠸⠳⠇⣸⣿⠣⡵
⣀⣀⣀⣀⣷⣧⣴⣶⣶⣶⣦⣜⣦⣣⣾⣿⣮⣼⣿⣦⣬⣄⣀⣿⣿⣅⣻⣿⣿⣿⣿⣯⣼⣭⣋⣯⣴⣿⣏⣡⣿⣷⣼⣾⣖⣧⣴⣧⣼⣃⣀
⣿⣿⣉⣉⣉⡙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⡠⠖⠉⣤⡺⠭⢝⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣁⣀⠤⠒⠉⠀⠀⠀⢸⣽⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣛⡟⠴⠞⠂⠀⠚⠒⢃⠞⣶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠇⠠⠆⠀⠀⠠⠀⠹⢋⠁
⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⢀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢢⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠈⢄⠀⠉⠀⡨⠁⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠑⠤⠀⡄⢄⢀⠀⠀⠐⠀⠀⠀⠀⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢱⡄⢀⠷⠀⠀⠀⠀⠀⢀⡜⡀⡸⢀⠀⠉⢀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢘⡾⠄⠀⡀⠀⠀⢸⠀⠠⠀⣠⣴⣶⣦⠀⠐⠀⢫⠉⠂⠂⠚⠓⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠂⡀⠀⢀⠬⡤⣞⣒⣒⡺⢮⣠⣿⣇⢈⠖⠁⠀⠀⡀⠀⠀⠀⠀⠀⠄⠄⠈⠁⠄
⠀⠀⠀⠴⢳⠀⠀⠀⠐⡁⢘⣶⡄⣶⣶⣶⡄⠉⠉⠉⠀⠀⠈⠀⠄⠀⠀⡀⠀⡀⣉⠀⠀⠄⠁⠀⠘⢦⡀
⠀⠀⠂⠒⠺⠷⠀⠤⠀⠰⠏⠻⣿⣽⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠧⠉⠉⠀⠀⠀⠰⠃⠞
⠀⠀⠀⠀⠀⠀⠰⢴⢒⣲⡂⠀⠉⠛⠛⢁⣀⣀⣀⡤⡤⣤⣀⣀⣀⠠⠤
⠀⠀⠀⠀⠀⢰⣶⣿⠫⠈⢱⡆⠐⢸⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡷⠁⠄⡀
⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠙⢦⡄⡘⠉⠉⠉⠋⠋⠋⠋⠉⠉⢉⣁⣠⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠃⠢⠬⠤⠥⠜⠉⠉⠛⠛⠛⠋⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣀⡄⠀⠭⢅⣉⠉⠀⠀⠀⠀⢀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣒⣿⡯⠽⠛⠛⠛⠛⠻⣗⣦⣄⣀⡀⠀⠀⠀⠀⢀⣺⠼⠿⢿⣯⡹⢛⡲⠶⠤⠤
⠀⠀⠀⠀⠀⠀⠀⡘⠭⠿⡟⢿⣛⠛⣛⣦⡌⣿⠛⠛⠀⠀⠐⠂⢘⣿⠛⠓⠲⠤⣿⡕⣄⠀⠐⠒
⠀⠀⠀⠀⠀⢤⣤⣁⣀⠀⠀⠀⠀⠀⠀⠉⣿⣽⡄⠁⠀⠀⠀⠀⠈⢹⡄⠀⠀⠀⢸⣿⢹⣀⣈⣀⣀⣤⡄
⠀⠀⠀⠀⠠⠀⣤⣉⣉⣉⣉⣉⡉⠉⠙⠛⠛⠛⠛⠓⠀⠀⠀⠀⠚⢻⠛⠛⠋⠉⢹⣿⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠈⢻⣿⠿⠿⢿⣿⣿⣶⣶⣶⣶⡦⢀⠀⠀⠀⣠⣤⠸⣦⣶⣶⣶⣾⣿⣿⣿⡿⠿⠿⠋⠀⠈
⠀⠀⠀⠐⠀⠀⠀⠀⠐⠀⠀⠠⠄⠂⠈⠉⠉⠉⠀⠀⠀⠀⢀⡭⠁⠀⠈⠉⠦⠀⢸⡇⠀⠙⠇⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡠⣆⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡀⠀⠀⠁⠀⠀⢈⣧
⠀⡎⠐⢿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠁⠁⠀⢠⣤⣾⣿⣿⣿⣽⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⠿⣧
⠀⢻⣤⡀⠙⠿⢿⣿⣿⣶⣦⣤⣤⣤⣄⠀⠀⢠⣿⢁⠀⠀⠀⠈⠙⣿⣂⣀⡀⠀⣶⣆⣠⣤⣤⣶⣾⣿⣿⠿⢏⠴⡿
⠀⠈⠳⠮⠑⠲⢤⣍⡉⠙⠛⠛⠛⠻⠿⠿⣶⣿⣼⣶⣶⣶⣾⣿⡿⢿⣻⣿⣿⢿⣿⣿⠿⠿⠛⠛⠛⠭⠓⣈⡴⠾⠃
⠀⠀⠀⠀⠀⠓⠶⣶⡄⣉⣙⠚⣓⠒⠶⠶⠶⠤⠬⠭⠍⠡⠤⠤⠤⠤⠌⠶⠖⠀⠒⣚⠉⣠⣤⣶⣶⡄⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠻⠃⠾⠿⠷⠰⣶⡶⠶⣶⠀⣶⡇⢰⣿⣿⠿⠸⠿⠿⠁⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⡙⢿⣿⣿⣿⣿⣿⣿⡿⣋⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣣⣶⣮⣭⣭⣍⠩⣵⣾⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣽⡍⣶⣾⣿⣿⣿⣿⣿⣿⣿⣶⢩⣯⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠸⡿⠟⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⠀⠀⢲⡄⢀⣿⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣛⣤⣤⣟⣿⣿⠏⠀⢠⡾⢡⡟⠁⣴⠏⠀⣴⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠛⠁⠀⠀⠘⠇⠀⠳⠀⠻⠆⣠⡿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣄⠀⠀⣰⣾⣿⢆⣤⣶⢶⣜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣿⣿⣦⣙⢿⣿⣿⡿⣵⣾⣷⣿⣟⣵⣯⣿⣿⣿⡿⣰⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣏⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠮⠙⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡀
⠀⠀⠀⠀⣠⣴⣟⣋⣉⣉⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠙⠳⢦⣄⡀
⠀⠀⣰⠟⠙⣿⣾⣿⣿⣋⣿⠙⢦⣀⡀⠀⣀⣠⡴⠋⠁⠀⠀⠀⢠⣍⠙⢦⡀
⠀⡼⠃⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠠⠖⠻⠟⠛⠃⠹⡆
⢸⠇⠀⠀⠀⠛⣻⣿⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣴⣦⠈⠀⠀⠀⠀⢻⡀
⣼⠀⠀⠀⠀⢺⣿⣿⣯⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⣿⣷⠛⠀⠀⠀⠘⡇
⣿⠀⠀⠀⠀⠈⠙⣿⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⡇⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⣰⣿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠃⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠉⠐⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠖⠠⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⣉⣈⣉⣻⣋⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡄⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⢿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⢿⡗⣦⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⣷⣾⣧⣾⣷⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠺⣿⣿⢶⢽⡆⠀⠀⠀⡇
⣿⠀⠀⠀⠀⣿⣿⣿⣿⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣿⣿⣾⣿⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⣧⣯⣽⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⡟⢛⡛⠋⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠟⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠿⢫⡄
⣿⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣀⣖⢶⣶⣿⣷⣤⣄⡀
⣧⠀⠀⠀⢠⣴⢶⣶⣶⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡇⠀⠀⣠⢾⣿⣷⡟⠤⠨⠽⠛⠁⣹⡓⣷⡀
⣿⠀⠀⠀⠘⣿⣿⣷⣿⣶⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣧⣀⡀⠀⠀⡇⠀⣸⠁⠈⣿⣿⠃⠀⠀⠀⠀⣤⣌⠁⠈⣷⡀
⣿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡋⢉⠉⠉⠁⠀⠀⢧⣰⣿⠀⠀⢹⡏⠀⠀⠀⠀⢀⣿⣿⠀⠀⣿⣧
⣿⠀⠀⠀⠀⣩⣽⣿⣿⣏⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢂⣾⣷⡀⣤⠀⠀⠀⢀⣾⣿⡿⠀⠀⠈⠁⠀⠀⠀⠀⠀⠉⠁⠀⠀⣿⣿⣇
⣿⠀⠀⠀⠀⠹⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡅⠀⠀⠀⠾⡿⡟⠀⠀⢀⣠⣿⡀⠀⠀⠀⠀⣰⡇⠀⠀⣿⡿⣿
⢻⡀⠀⠀⠀⠀⠈⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⡇⠀⠀⠀⢠⡇⡰⠀⠀⢠⣿⣿⠇⠀⠀⠀⢰⣿⣿⡆⠀⠸⢻⠏
⠈⢷⣄⠶⠀⠀⢾⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⡿⠟⠁⠀⢆⣠⠟⠀⠐⠞⠀⠀⠿⠗⠀⠀⠀⠀⠀⠿⠟⠁⣠⣶⠏
⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠴⠴⠶⠶⠶⠾⠖⠛⠋⠁⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠻⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⣾⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿⣀⡀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⢘⣷⠊⢿⠀⠀⠀⠀⠒⠲⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⠁⠀⠀⠀⠀⠀⠀⡰⠋⠙⠆⠘⠂⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠈⠳⡀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⣷⠀⠀⠀⢸⢀⣤⣄⡀⠀⠀⠀⠀⠠⢤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢦⡙⢦⣹⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠖⠋⢧⡀⠀⢸⠋⠀⢀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠙⠚⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡷⠀⠈⢳⡀⠈⣆⠀⠿⠃⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠉⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠳⡄⠘⢦⡀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠙⢦⣴⠦⢤⣤⣀⡀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠓⠖⠻⣽
⠀⠀⠀⠀⢠⢞⡭⠟⠋⠉⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⠛⠲⠤⢬⣭⣥⡤⠴⠞⢻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠈
⠀⠀⠀⢠⣧⠏⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠲⠶⠶⠶⠤⠤⠤⠤⠤⣾⣤⣄⣀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠉⠉⠛⠛⠳⠶⢶⡶⠋
⠀⠀⠀⣿⡇⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⣷⣧⠀⠜⠀⠤⠤⢤⠾⠭⠽⠖⠒⠀⠀⢄⣠⢤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣸⠃
⠀⣀⡄⠘⣎⣧⡀⠠⠴⠒⣓⣒⣒⣒⣀⣀⣀⣀⣀⣽⡤⠾⠷⡶⢒⣆⣠⡴⠛⣷⠀⠀⠀⠀⠀⡟
⠀⠉⠀⠀⣈⠿⣿⣟⡲⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠁⠀⠀⠀⠀⠈⠯⠀⠀⢠⠟⠠⠴⠤⣤⣼⣃⡀
⠀⠀⡔⣿⡲⠄⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡟⠀⣀⡀⠀⠀⠉⠢⡀⠽⠛⠒⣒⠠⠤⢄⡀
⠀⠈⣤⣀⣰⠀⠀⠉⠳⠤⢤⣤⡀⠀⢲⣤⣤⣤⣤⠤⢦⣀⣀⣻⣟⠛⠋⠉⢠⣮⡴⣇⡷⢀⡀⠀⣅⠀⠀⣹⠁⠤⠆⢸
⠀⠀⠈⡷⢄⡀⠀⠀⡀⣀⠀⠀⠙⠒⠒⠒⠋⣁⣀⣀⠠⠤⣭⣍⡀⢤⡄⠀⠀⠀⠘⠯⠟⢉⣠⠔⠁⠀⠀⣏⡠⠴⠒⠃
⠀⠀⠀⠀⠀⠈⠑⠪⠗⣉⡀⠀⠀⠀⠀⠀⠈⠁⠶⠠⠬⠝⠒⠊⠉⠁⠀⢀⣀⡀⠤⢔⠊⠁⢸⠀⠀⠂⠹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠐⣲⠒⠒⠒⠒⣒⠒⢲⠖⢒⣡⡍⠉⠁⠰⠀⠀⡜⢀⣀⣧⠠⠤⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠁⠙⠀⢀⣀⠤⠔⠒⠛⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣠⣤⣤⡤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⡏⠓⠒⠯⠝⢻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠆⠀⠀⠹⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡇⢉⡐⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣾⣿⣥⣤⡀⡥⠀⠀⠀⠀⠀⠀⣐⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢰⠞⡃⠠⡠⢹⣦⣾⣟⣽⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣻⣿⣿⣻⣿⣿⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣯⠙⢋⡿⠣⣮⣉⣙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣏⡉⠙⠁⠀⠀⠒⠦⠈⠉⠉⠉⠿⣿⣿⣿⣿⡟⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡿⠛⠻⠛⠲⠶⠶⣄⣲⣀⡀⡒⠒⠺⠽⠛⠛⢋⣼⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣛⣉⣤⣶⣶⣮⠙⠉⠉⠉⠉⠉⠉⢉⣭⣶⣶⣤⣀⣾⣿⣿⠏⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡿⣭⣙⡻⣿⣿⣿⣷⡄⠀⠀⢀⣴⣿⢿⡟⣛⣹⣿⣿⣿⣿⢾⣾⡏⠎⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢷⠆⣈⡴⣋⠿⡝⢁⣙⠳⢶⡟⡇⠈⠀⢽⠋⡛⠞⢻⠁⢀⣰⡗⣵⠐⣯⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣷⡔⢡⠗⡂⣨⣍⠳⢮⠀⣀⣚⠋⠘⠀⠀⠀⢡⣴⣿⣉⣸⡏⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣏⡟⠳⠓⠜⠉⠉⠐⠚⠛⠿⢆⠍⣀⣎⣹⠷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠤⠄⠒⠂⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠘⠛⢶⣶⣿⡶⣦⣄⣀⣠⣴⣶⣦⣄⠐⠲⣶⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠘⢸⣿⣿⣿⣷⣾⠉⠻⣿⣿⣿⣿⣷⡄⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣜⢿⠿⢿⡿⠗⡴⢦⠀⠈⣿⣿⣿⣿⣄⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣄⠀⢠⣄⡀⠻⠂⠁⣸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣽⣶⠇⠻⢾⣶⣤⣼⣛⣫⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⠋⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣿⣿⣹⠾⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⢿⡟⠛⠛⠛⠛⠛⠛⠋⠉⠛⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣬⠛⠛⠊⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢁⣻⠁⠀⢀⡁⠐⠒⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠋⠃⠄⠸⠃⠀⠀⠀⠀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠉⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠚⠉⢉⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⣄⣀⣀⣀⣀⣀⣀⡀⠤⠄⠒⠈⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠒⠤⢀⣀⣀⣀⡀⠀⢀⠀⡀⣀⣀⡤⠤⠔⠒⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠙⠲⠤⣀⣀⣀⡉⠉⠉⠉⠁⠁⠀⣀⢀⣀⣀⣠⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢀⣀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣈⡉⠛⠓⠲⠶⠶⠶⠶⠶⠶⠖⠒⠚⠉⠉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⠀⠈⠉⠙⠛⠓⠒⠶⠶⠦⠤⠤⠄⠔⠒⠊⠉⣀⡤⠄
⠀⠀⠀⠀⠀⠈⠀⠤⠀⣀⠀⠀⠙⠓⠲⠤⠤⣤⣤⣄⣄⣀⣀⣀⣠⡤⠤⠖⠒⠚⠉⠀⠀⢀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠶⣶⣶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠴⠲⢒⡲⠿⠛⠉⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠁⠂⢴⠂
⠀⠀⠀⠀⠀⠠⠟⠂⡀⢰⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣷⣾⣿⣿⣿⣾⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢼⣿⠟⠛⠛⠛⠛⠛⠛⠻⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠙⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠻⢈⣠⣴⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣴⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠉⠛⣆⠀⠀⠀⠴⠾⠛⠛⠓⠦⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠶⢎⠀⠀⠀⢀⡤⢤⣤⡀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⣾⣿⣿⣿⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢸⡂⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠋⢁⠴⠀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣤⠒⣂⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠹⠿⠿⠿⣿⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣤⠀⠀⠀⠀⠀⠀⠀⢀⡈⢿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣴⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⠿⢄⣀⠀⠀⠀⢀⣰⡟⠃⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡬⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠶⣤⠄⢄⣀⡀⠈⠐⢿⡟⠀⠀⠀⣸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⣧⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣤⣠⢀⣄⣀⡀⠈⠢⡀⠀⠁⠄⣀⡴⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⣴⡄⠈⣧⠈⣦⡀⠀⢀⠀⠀⠀⢀⠀⠀⠀⣠⣾⣸⡟⠻⢦⡀⠈⠒⠀⠀⠈⠄⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠘⣿⡀⢻⡇⠈⢳⠶⣤⣤⣀⣀⣠⣤⣶⡿⢛⡽⠋⡀⢀⡈⠉⠉⢉⡀⠀⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢶⣄⠘⣿⠀⠹⣿⠟⠀⢀⡄⠢⣌⡉⠛⠛⠛⠉⡁⠔⣟⠁⡔⡁⠉⢨⣥⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠑⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡼⠫⠞⠀⠀⠀⡼⠁⠀⠈⠙⠻⠷⠭⠔⠒⢻⣯⣼⣿⡇⡠⠈⣿⠘⠃⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠳⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⠟⠃⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠰⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠂
⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣄⡀⠀⠹⡆⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡤⠒⠋⠉⠉⠉⠉⠉⠉⠛⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⣧⠀⠀⠀⠀⠀⠀⠀⠸⡅⡏⢀⣀⠀⣤⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡄⣴⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠸⡆⠀⠀⠀⠀⠀⠀⠀⢳⡀⠻⠿⠃⠉⠁⠀⠀⢀⣀⣤⡄⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⢀⡇⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢻⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠘⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢯⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⢾⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠁⠈⠙⣦⢠⠖⡶⢶⢶⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⠤⠿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⣠⡟⢿⠀⠳⢶⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⢰⠃⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣍⡧⠈⣳⠀⣊⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠹⡏⠓⠦⣄⢻⣀⠀⠀⣀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠈⡴⠋⢁⡤⠋⠀⡞⢲⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣷⠀⠀⠀⠀⡟⠉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣈⣓⣇⢀⡏⡴⣆⡴⠛⠉⣼⣑⣄⠤⠖⢛⡽⠋⠀⠀⡠⠋⠀⠀⠀⠀⢹⠀⠀⠀⠘⢧⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠙⢛⡶⠊⠀⠀⠀⠀⠀⠀⢸
⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣤⣤⣤⣤⣤⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁