Summer vibes Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⢷⠀⢸⠃⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣄⡀⢘⡦⠶⠒⠺⢄⠊⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⡄⠀⠀⠀⣀⣠⠄⠾⠀⠀⠀⠀⠈⠇⢡⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡾⠁⢀⣀⡿⠀⠀⠀⣀⠀⢀⢀⣶⠀⣀⠀⢀⢀⣤
⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠁⠀⠀⣷⢰⣿⠀⣾⡾⢸⠌⣿⢠⣿⡟⣾⢔⡿⢀⣶⠷⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢛⠛⠶⣌⠙⠃⠘⠛⠉⢁⠀⠀⠙⠋⠈⠀⠀⠘⠷⠻⣇⣠⡾⠋⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠁⠀⡿⠀⠀⣆⢰⡇⣴⠀⣿⣽⠀⣾⡒⢠⡞⠦⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⢻⣼⠀⣿⠀⣿⢸⢷⡇⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠙⠂⠛⠋⠈⠓⠂⠘⠾⠃


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⢸⠀⢰⠇⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠤⢀⡵⠒⠋⠓⠢⡉⡠⠤⠖
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⢶⡄⠀⠀⠠⠴⠖⠘⠁⠀⠀⠀⠀⠛⠐⠶⠆
⠀⠀⠀⠀⠀⠀⢰⠋⠀⠠⠼⠃⢀⡀⣠⡀⢠⣆⣾⣇⣤⠀⢠⣤⢾⢂⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⠷⣤⣀⣀⡀⠼⣧⠿⣇⡼⣿⠘⠋⢾⣰⠿⠏⠻⢫⡏⣰⠟⠋⣠⣦⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠬⠉⢻⡆⠀⠀⠈⣀⠀⠆⣠⢤⡀⣀⣀⠀⣈⡉⠙⠲⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣤⠴⠁⠀⢸⣤⡇⢸⡇⣿⠛⡄⡿⠀⠸⣇⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⢸⡇⢿⡴⠃⣷⣀⢠⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠠⡏⠐⢁⢀⣰⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠘⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⣠⠀⢀⣰⣇⣴⣽⣦⣌⡖
⠀⠀⠀⠀⠀⠀⠀⣀⠙⢠⠻⠿⠋⣿⠀⢰⡇⠀⠀⠀⠀⣀⣰⣇⠱⡀⢀⡖⣯⣸⣷⣄⡠⢻⡄⠀⠀⠀⠀⠹⢄⡹⣧⣾⠏⢿⢋⢹⡁⡙⠁
⠀⠀⠀⠀⠀⠀⠀⣏⢀⠀⢠⠤⣖⢤⣠⢾⠁⠀⠀⠀⠘⠉⠀⠙⢆⠓⡏⠃⣹⠁⢹⡉⠃⡸⢁⠒⠖⠁⠀⠀⠀⠓⠀⣥⠘⣏⢾⢀⡜⣧
⠀⠀⠀⠀⠀⠀⠀⣿⢯⢸⡌⢳⡘⡏⠈⢹⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠆⠈⣦⠙⠛⠾⠀⠨⠇
⠀⠀⠀⠀⠀⠀⠀⠘⠘⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⡂⢀⠀⢰⡄⡆⣆⠰⢠⡆⠀⠀⠀⠀⠀⠀⠀⢁⠶⠙⢒⠸⠚⠆⠁⡀⠀⠀⠀⠀⢠⠀⠀⢠⢠⠀⠀⠀⠀⠀⠀⢠⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⠀⢱⡇⡈⡆⠈⠁⠀⠀⠀⠀⠀⠀⠀⡌⡈⠀⠀⢀⠰⡄⡈⠀⠀⠀⠀⠀⠀⢣⢨⢸⢸⢰⠀⢡⠐⢰⠀⠈⣦⢸⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠄⣤⡀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠏⠒⠀⡀⠮⠀⢠⠀⠀⠀⠀⠀⠀⠀⠁⢀⠁⢀⡀⢀⡁⣀⡀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡿⡀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠠⠀⢀⠅⣠⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢰⠂⠘⡅⡷⠁⡗
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠚⠃⠁⠙⠂⠈⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠃⣘⠀⢰⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠋⠁⠓⠁⠃⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢶⣿⠄⢈⡀⠀⠀⠀⠀⠀⢀⡴⢢⡄⠀⠀⣈⣉⣉⣉⣉⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠧⣄⣦⣤⢦⠂⡀⢀⠀⠀⠀⠀⠸⣧⡜⠁⣀⢀⣤⣤⣠⣤⣤⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⠯⢅⣿⣴⣟⠛⣇
⠀⠀⠀⠀⠀⠀⠀⡠⡀⡀⠀⠁⣡⣌⠀⠁⣸⠀⠀⠀⠀⠀⠀⠙⣷⠯⠻⠿⠘⠙⠋⠘⠹⠛⠄⢿⡹⠃⠀⠀⠀⠀⠛⠋⠈⠃⠈⣀⡉
⠀⠀⠀⠀⠀⠀⠀⠃⣿⠁⡜⠧⠻⡌⠟⠿⢯⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢒⣒⣒⣒⠊⠀⠀⠀⠀⠀⠀⠀⠀⢸⠳⢻⠀⣯⠁⣯⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠓⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠓⠀⠛⠙⠂

⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀
⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠉⢷
⠀⠀⠀⣾⠋⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⣟⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣱⣄⣤⣤⣤
⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⢀⣤⠀⠀⣀⠀⢀⡄⠀⢀⢴⡄⠀⢀⣴⠶⣆⠀⠀⡇⠀⣰⡟
⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⣤⡄⠀⠀⣤⠀⠀⣼⡇⡴⢹⣇⠔⢡⡟⠀⢰⡏⡰⣹⢇⡜⢡⡿⠀⢰⡟⢁⠠⠃⠀⡴⠁⣼⠏
⠀⠀⠀⠀⠀⠙⣷⡄⢀⠔⢉⣿⠁⣠⢺⡏⠀⠀⣾⡟⠀⣿⠏⢠⡿⠀⠀⣼⡟⠀⠿⠋⢀⣾⠁⠀⣿⡇⠀⠀⣠⠎⠀⣸⡏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢸⣿⣴⢃⣿⠀⠌⠀⠛⠁⠀⠋⢀⣾⠁⡔⠀⠿⠁⠀⠀⠀⣸⠇⢀⠂⠹⣧⡴⠚⠁⠀⠀⠹⢧⠤⠚⠁
⠀⠀⠀⠀⠀⢀⠎⠘⣿⠀⠈⠛⠁⢸⡇⡸⠀⠀⠀⠀⠀⠀⣸⢇⡜⠀⠀⠀⠀⢀⣀⠀⣟⡠⠃
⠀⠀⠀⠀⢠⠃⠀⠀⣿⠇⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⢀⣼⠋⢹⠀⠉⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⡇⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠴⠀⢀⣾⠃⢠⠋⠀⢀⣠⡤⣄⢀⣾⠋⣀⠟
⠀⠀⠀⠀⢧⣀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢏⣸⠁⢰⡖⠀⣼⣇⠔⠁⠀⢠⡿⠁⢀⠜⢸⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⠇⢀⣿⠁⢰⡿⢁⠖⢄⢀⣿⠃⠉⠁⣠⠃⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⡞⠀⣸⡇⠀⣼⡇⠀⠓⣺⠁⢿⣄⣀⡴⢁⣀⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣠⡟⠀⠀⢹⣇⡴⢻⡇⠀⣠⠃⠀⠀⠉⠁⠀⡎⠀⢁⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠘⠷⠞⠁⠀⠀⠀⠀⠀⠀⠳⠤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣤⣄⠀⠀⢠⣶⠂⠀⢲⣦⠀⠀⢀⣤⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠰⣖⠀⠉⣧⠀⠈⡟⢧⣀⡿⢹⠀⠀⢸⠻⣇⣠⠟⡿⠀⠀⣿⠛⢻⡆⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣙⣃⡀⢻⡄⠀⢸⠆⢀⣿⡄⠻⠁⠼⠷⠠⠾⠄⠹⠋⣰⣇⠀⣰⠏⠻⠁⠀⢠⡟⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⠉⢈⡷⠀⠙⠛⠋⠀⠀⠀⠀⠀⠀⢀⡔⣹⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠃⣠⡿⢦⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠋⠁⠀⠀⠀⠀⠀⠀⣀⡤⣴⣯⡿⠱⢿⣷⠶⢢⡄⣂⣁⠀⠠⣤⣄⡀⠈⠁⠸⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠀⢶⠋⣽⢣⡇⣼⢡⡾⢰⠇⣸⠿⢤⠍⠱⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⢠⣾⣿⡇⠀⠀⢰⢀⡡⠻⢈⢡⣬⣴⣿⠦⠔⢚⣒⣮⣅⣤⣄⣀⣹⣿⣿⣆⠀⣀⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⢀⣾⣿⣿⣠⣤⣤⣤⣤⣤⡀⠀⠀⠈⠉⠀⠀⠘⠋⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣰⣿⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⠋⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠠⠃⡨⠋⠉⠱⠦⢀⠀⠀⠀⢀⣠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣷⠦⠄⠀⡀⠇⠈⠀⢀⠀⡆⠀⠀⠀⢈⣡⣿⣿⡿⠛⣿⡿⠙⢻⡿⣿⣿⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢊⣿⣿⠟⠹⠃⠘⢿⠋⠙⠛⠿⡷⣄⠀⠙⠊⠐⠊⠁⠀⠐⠐⠃⠀⠛⠉⠁⠀⠀⠀⠛⠀⠀⢡⠀⠈⠋⠻⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠁⠐⠂⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡟⠛⠻⠿⠿⢿⣿⣿⣿⣿⡿⠿⢶⣶⠶⠂⠀⠉⠁⠀⠀⠀⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣀⡀⠀⠀⠀⠀⢀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠐⠂⡆⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠠⠒⠒⠣⠤⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠧⠤⠚⠒⠒⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡞⢹⠧⣄⠀⠀⠀⢀⡤⠼⡎⢳⣤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠗⠀⠀⠀⠺⣍⠀⣠⣄⠈⡽⠗⠀⠀⠀⠲⢏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⢺⢃⡞⢳⡹⡟⣸⠁⠘⡞⣻⢋⡞⢳⡘⡗⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠁⠀⣇⠸⡇⢰⠀⣿⠃⡼⠀⠀⠿⠃
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⣤⡴⣄⣤⣄⣠⡶⢻⣄⣧⣸⣠⠟⢶⣧⣄⣠⡴⠦⣴⢖⢶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢸⡁⢺⣧⣾⠀⣿⣧⢸⠙⢠⠀⠋⠀⡟⠛⣄⠘⠃⢸⡇⢐⣓⣸⣮⢁⡍⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠈⣿⢲⡌⢻⡀⣿⡇⣼⠀⣾⡆⣸⠀⣷⠀⣿⢀⣿⢸⡇⠸⡿⢻⣿⣼⡇⢸⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠶⠛⢳⡬⢶⣻⡇⣿⣿⠟⠀⠿⣴⡿⠛⠿⣼⠟⣓⢶⡏⠀⠈⠳⠤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣿⢹⡇⣾⡂⣿⠇⣿⠀⣻⣥⣇⠘⢯⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢁⣿⡇⡿⠁⣤⡘⣿⡀⠿⢏⣿⡟⠶⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠉⠓⠚⠷⠶⠞⠋⠙⠛⠋⠀⠙⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢹⡇⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⡷⠀⠀⠀⢰⠇⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠸⣷⢀⣠⠤⠒⠂⠈⠉⠁⠒⠲⢤⡀⠰⠋⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⣰⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⢲⠄⠀⠀⠀⠘⠀⠀⠀⠀⢀⣤⡀⣠⣄⠀⢀⠀⢀⡀⢀⣤⡀⠀⠀⢀⡀⠻⢼⢤⣄
⠀⠀⠀⠀⠀⡀⠀⢀⠽⣿⣇⣠⠞⣿⠀⢠⣶⠀⠀⣷⠎⢸⣷⢣⡿⠀⣿⡠⠋⣷⠎⣸⡇⠀⣼⣯⠇⢠⠏⣼⡟
⠀⠀⠀⣰⠁⠀⠀⠀⠀⣸⣿⡇⠀⣿⣀⠎⣿⠀⡘⣿⠀⢸⡇⣸⡇⡼⣿⠃⠀⠟⠀⣿⠃⣰⣿⠁⣠⠋⢸⣿⠀⣰⠂
⠀⠀⠀⠑⠀⠀⠀⢀⡾⢁⣿⡇⠀⠈⠋⠀⣿⡸⠀⠀⠀⠀⠀⣿⡿⠁⠈⠀⠀⠀⠀⢿⡴⠋⠙⠋⠁⠀⠘⠿⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠸⢧⠾⠟⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⠃⠀⣾⠀⡟⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢰⠗⢺⡇⢀⣧⣾⢤⣄⣴⣾⠹⠆⣰⣿⠘⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡼⠀⢸⡇⣸⡿⢁⣴⡏⢺⡇⠀⣠⠇⡼⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠈⠛⠋⠀⣎⠼⠁⠈⠛⠚⠁⠈⠓⠋

⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⡠⠤⡀⠀⡤⠤⣀
⠀⠀⠀⠀⠀⠀⡼⠀⡈⢢⢀⠎⠀⠀⠀⠈⢢⠀⣀⡀⠀⠀⠀⡤⠒⠦⡀⢰⠁⠀⠹⡄⠀⠑⠉⠁⠀⢨⡀⢠⠤⠤⠄⢀⡀
⠀⠀⠀⠀⡜⠉⠁⠀⠇⠀⠿⡀⠀⠀⠀⢀⠍⡜⠁⠈⠉⠐⢺⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠶⢅
⠀⠀⠀⢸⠀⠀⠀⠀⠈⠀⠀⠈⠒⢤⣰⠋⢚⠀⠀⠀⣴⠤⠏⠉⠉⠓⡦⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣁⣀
⠀⠀⠀⠈⡙⠀⠀⠃⠀⠀⠀⠑⠀⠀⣹⣀⡂⠁⡐⠠⡇⠀⠀⠐⠀⠠⣇⢀⡴⠚⠙⠒⢅⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠁⠀⢀⠇
⠀⠀⠀⠀⢣⡀⠀⠀⢀⢄⡀⠀⠀⠀⢹⠁⠐⠀⠉⢰⠁⢀⠀⠀⠀⡤⢨⣟⠂⠀⠀⠀⠀⣱⠀⠀⠀⠀⠀⢻⣇⠀⠒⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡎⠁⠀⢈⠥⣤⡁⠀⠈⢻⠶⠗⠒⡲⠂⢸⢰⡯⢦⡀⠀⠩⢿⡃⠀⠀⠀⠀⢠⢷⣀⠆⠀⠀⠀⠢⠀⢣⠀⠀⠀⢰⠴⠃
⠀⠀⠀⠀⣇⠀⠀⡆⠀⠀⠉⠢⣄⡼⠀⣠⣞⠃⠀⢸⡾⠁⠀⠀⡰⠃⡿⡟⠒⠢⠤⣴⠣⣻⡎⠀⠀⠀⠀⢀⢀⡽⠀⠀⠀⠈⠑⡄
⠀⠀⣄⣀⡈⣒⠖⡹⢆⠀⠀⠀⠨⠷⢆⠁⠘⡧⠄⠘⠃⢄⢀⡜⠁⠀⢈⠞⣆⠀⠀⢀⣠⡟⠓⠜⠀⢀⠀⠘⠁⢀⠀⠀⠀⠀⠀⠽
⠀⠀⡼⠃⠀⠀⠉⠚⠈⠳⡄⠀⠀⠀⢀⠆⠀⢃⠀⠀⠀⡄⠊⠀⠀⠀⠊⡜⢨⡖⠈⣌⠀⣼⡀⠤⢱⠉⠐⠄⣀⢅⠀⠀⠀⠀⡤⠃
⠀⠀⠱⣄⠀⠀⠀⣐⠦⠀⠈⠢⣀⠤⡙⢄⠀⢸⠀⠀⢠⡁⠀⠙⠄⠰⢱⠀⡈⠁⡞⠔⢀⣽⠛⠢⣄⠓⠛⠒⢅⠀⠀⠀⠀⢴⠁
⠀⠀⡖⠒⠛⠀⠀⠈⠀⠀⠠⠀⠀⠀⠈⠊⠀⠘⠢⠤⠞⠀⠈⠀⢠⠘⣾⠮⠉⣴⡽⠀⠈⠀⠀⠀⠀⣣⢀⠈⠀⢳⠀⠀⠀⠀⢱
⠀⠀⠈⣢⡴⠆⠀⠀⠀⣀⠀⠀⠀⡰⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢿⠀⢠⣟⠀⠊⠀⡠⠀⠀⠴⠁⠀⠀⠈⡀⠇⠐⠠⣔⠁
⠀⠀⠀⡏⠈⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⡠⠒⠉⠁⠐⢦⣰⠉⠉⠁⠛⠀⢰⢄⠀⠀⣠⠊⠀⡀⠀⠀⠀⢀⡏⠀⠀⠀⠱
⠀⠀⠀⠳⡀⠈⡀⠀⠄⠀⠀⢬⡀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⣸⠔⡡⠊⠉⠉⠄⠑⠦⣀⣀⣀⡠⠎⢡⢀⠀⠀⠈⠆
⠀⠀⠀⠀⢨⡖⠛⠁⠀⠀⠀⡀⢇⣀⢀⠠⢤⠞⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡧⠔⠁⠀⠀⠀⠀⠀⠘⠠⠀⠀⠐⠄⢩⠀⡀⠀⡠⠃
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢸⡦⠀⠀⠀⠀⢘⡤⡄⠀⢀⣀⣀⢴⠋⠀⠀⣸⠀⠀⠀⠀⡀⠀⠀⠃⠀⠀⠀⠀⠀⡈⠀⠘⢣⡀
⠀⠀⠀⢼⡄⠀⠄⠀⠀⢀⡘⠙⠦⣀⡠⢶⣯⣞⠏⠂⠁⠀⠀⠸⢤⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀⠀⠑⠀⠈⠠⣀⠀⠑⡀
⠀⠀⠈⠓⣅⣐⡀⣀⣀⠜⠀⠀⠀⠀⢠⠟⢉⡆⠀⠀⠀⠀⠀⢀⡀⠀⠀⠠⠴⢄⡀⠀⠀⠐⣲⣄⡴⣤⠐⠺⡀⠁⠀⣀⣅⠢⠕
⠀⠀⠀⠀⠀⠋⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠚⠑⠒⠊⠀⠁⠀⠁⠀⠀⠉⠑⠀⠉⠘⠀⠙⠚⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠉⢉⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣄⣀⡈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣠⠀⠈⢳⣤⡀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣶⡟⠀⠀⢶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣄⠀⠀⣿⣿⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⣿⡆⠀⢸⣺⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠃⢿⡀⢠⠏⢀⣾⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣙⠻⢦⠈⠇⢠⠟⠉⠉⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣧⠄⡀⠙⠀⠀⠋⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠯⠁⠀⠀⠀⠁⠐⠂⠺⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠁⢀⣤⠄⠀⠀⠀⠀⢈⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⡋⠀⢀⡄⢀⡠⠀⡀⠀⠲⣦⢀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠸⠁⠀⠰⣦⠈⢷⣌⢳⣶⣶⣾⣿⣿⣿⠛⢿⣿⣿⣿⡏⣰⡟⠁⣼⣧⡄⠀⢉⢸⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏⣆⠀⢰⡇⣿⡄⠘⣿⣬⣿⣿⣿⣿⣿⣏⠀⢸⣿⣿⣿⣷⣿⠃⣰⣿⡟⣧⡀⢸⡸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣿⣴⣿⡇⣿⣿⡆⣿⣿⣿⣿⣿⣿⣿⠇⠀⢸⣿⣿⣿⣿⣿⣴⣿⣿⡇⣿⣿⣿⡇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡀⢨⡟⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⠏⢀⣾⣿⣿⣿⣿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⣇⠀⠀⣀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢷⠀⠀⢿⠀⠀⡿⠀⢀⡎⠀⢠⠆⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡀⠀⠀⠐⣄⠀⠈⢧⠀⠘⣇⠀⠸⠄⠠⠧⠀⣼⡀⣠⠏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣾⡟⣿⡆
⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠈⢣⡠⠜⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢎⠀⠀⠀⠀⠀⣀⣤⣤⡄⠙⣿⡿⢛⣿⠟
⠀⠀⠀⠀⣴⡟⠁⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⣤⣤⣀⣴⣶⣦⠀⣴⣟⣽⠟⢀⣼⠟⢀⣾⠏
⠀⠀⠀⢸⣿⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀⠀⢀⣤⣤⡀⣴⠾⣿⡆⢀⣿⣿⡟⠀⣿⣿⠃⣿⡇⢰⣿⡿⠋⢠⣾⠋⠀⣾⡏⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠘⢿⣆⡀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⢰⣶⠆⠀⣼⣿⣷⡟⠁⣿⣿⠏⢸⣿⠀⣼⣿⡟⠀⢰⣿⠇⢰⣿⣁⣾⣿⣤⡴⠟⠁⠀⠀⣿⣇⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠙⠻⣦⣄⠀⠀⢀⣴⣿⣿⠀⢠⣿⣿⠀⣼⣿⣿⡟⠀⢸⣿⡏⠀⣿⣏⣼⣿⣿⠀⠀⠺⠛⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣶⠛⠁⣿⡇⢀⣾⣿⣟⣰⡟⢹⣿⠀⠀⠸⠟⠀⠀⠈⠉⠁⣠⡤⣤⡄⠀⠀⠀⠀⠀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⣿⣇⠀⠻⠷⠾⠋⠈⠛⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢡⣿⠇⠀⠀⠀⠀⣾⡟⣽⡟
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣼⣟⣰⡿⠋⠀⠀⠀⠀⠀⢘⣿⣿⣅⡀
⠀⠀⠀⢰⣿⠁⠀⠀⠀⣰⣿⠁⠀⠀⠀⠀⠀⠀⠀⣠⡄⢀⣾⡟⣿⡆⠈⠛⠀⢠⣿⣿⣋⠀⠀⣴⡟⣿⡇⢠⣾⠏⠀⠙⣷⡄
⠀⠀⠀⠈⢿⣦⣀⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠻⣶⣿⠷⣿⡟⠀⣾⣿⢿⣽⣷⣼⣿⠟⠋⣰⣿⣷⡶⠷⢄⣿⠇
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢀⣿⡏⠀⣿⡇⣸⣿⡇⢀⣿⡏⠸⣿⣤⣾⠟⠘⠿⣤⠴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢀⣾⡟⠀⠀⣿⣷⡟⠙⠻⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⠾⠋








⣿⠛⠛⣿⢿⠿⢿⠿⠿⣿⡿⠿⠿⡿⢿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠿⠿⠿⣿⠿⠿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡿⣿⣇⣤⣄⠀⠀⢠⣦⣰⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⢠⣤⡀⠀⣿⣿⡇⣿⣿⣿⣿⣧⠀⣿⣿⡇⣾⣿⣷⣷⡄⠀⠀⣀⣤⣴⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢻⣿⣆⠸⣿⣷⡀⢻⣿⣧⢹⣿⡇⣿⣿⢸⣿⣿⢰⣿⡟⣼⣿⡇⢠⣾⣿⠟⢡⣿⡇
⠀⠀⠀⠀⠀⢠⣾⡟⠻⣷⣄⠈⢿⣿⣄⢿⣿⣿⡎⠿⠿⠈⠛⠃⠙⢋⠈⠛⠃⠘⠿⠁⣿⡿⢠⣿⣿⣏⣴⣿⠟⠀⢀⣤⣦
⠀⠀⠀⠀⠀⠈⢿⣷⣤⣈⣛⠀⠀⠙⢿⣿⠃⠉⠁⠀⣠⡀⠀⠠⣴⣿⣦⡄⠀⢀⣀⠀⠀⠀⠘⣿⣇⣉⠛⠉⣠⣾⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣷⡀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠾⠛⠇⠀⠀⠀⠉⠀⠀⠀⠀⠈⠉⠁⢠⣾⣿⠟⠉⠀⢾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠛⠀⠀⣠⢖⣿⣿⣿⣿⣿⣶⣦⣤⠀⣤⣴⣶⣿⣿⣿⣿⣿⡲⣄⠀⠀⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣼⣿⣿⣿⣿⣿⣿⣿⠏⣀⡹⣿⣿⣿⣿⣿⣿⣿⣧⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⠟⢡⡾⠉⡇⠈⠻⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠿⠿⠶⠆⠀⠀⠉⠛⠋⠉⠀⠀⣾⠃⢰⠁⠀⠀⠀⠉⠙⠛⠉⠁⠀⠰⠾⠿⠟⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⠀⠀⠀⠀⢀⡀⠀⠤⢸⣿⢠⢃⣀⡀⠀⠀⣀⠀⣠⣄⠀⠀⠀⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠋⠉⢀⠀⠀⣼⠀⢿⣆⣴⠆⢸⣿⣡⠋⢸⡇⣠⡾⠍⠀⣿⡘⢀⠀⠀⣀⠈⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠋⠀⠀⢻⡀⡜⠀⣿⠀⡜⣿⠷⢒⣿⠋⢸⡇⣠⠆⣨⢿⡁⠀⠀⠈⠛⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠻⡴⢡⡏⣀⣾⠏⠀⠀⠉⠁⠀⠣⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⢀⣤⠀⠀⠀⠀⠀⣀⣄⠀⢀⣀⠀⢀⡀⠀⣀
⠀⠀⠀⠠⣦⠀⠀⣀⣄⡀⠀⠀⠀⣿⠀⣃⢤⡟⢀⣴⠂⢀⣾⡣⣻⡟⢡⡟⢠⣿⡐⣿⠖⢹⡇⣰⡿⠃⠀⣸⡧⠊⠙⠀⠀⣤⢤⣄⠀⠀⡤⢄⡀
⠀⠀⠀⠀⢻⣧⣼⣯⣹⡏⠀⠀⠀⣿⠊⠁⠸⡷⠋⣿⡴⠁⠿⠁⠛⠀⣼⢣⢿⠟⠰⠏⠀⣿⣠⠿⡤⠴⠊⡿⠁⠀⠀⠀⠘⡿⠛⠛⢠⡾⢁⠀⠁
⠀⠀⠀⢰⠛⠛⢛⡻⠻⡁⠀⠀⣮⡾⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣄⠘⠁⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⣠⠛⠒⠘
⠀⠀⠀⠈⠃⠈⠹⣧⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⢠⡄⢸⣿⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⣠⣤⣄⡀⠀⠀⠀⠀⢀⠀⠐⠃⠀⠁
⠀⠀⠀⠀⠄⠀⠀⠹⣇⠀⢃⠀⠀⠀⠀⠀⣿⡆⠀⢠⣿⠃⣴⠄⢸⣿⣴⣶⣦⡄⢠⡿⠋⢹⣿⢸⣿⣉⡙⠃⠀⠀⠀⠀⠀⠢⢀⠀⠀⡠⠂
⠀⠀⠀⠀⠐⡀⠀⠀⠹⠆⠈⠀⠀⠀⠀⠀⢸⣧⠀⣾⠇⠀⣿⠀⢸⣿⠁⢀⣼⡏⢸⣟⠛⠛⠁⠀⠉⠛⠻⣷⡀⠀⠀⠀⠀⠀⠐⠈
⠀⠀⠀⠀⠀⠐⠄⠀⠠⠐⠊⠀⠀⠀⠀⠀⠀⠻⡿⠋⠀⠀⣿⠀⢸⣿⠟⠛⠉⠀⠈⠻⣶⣶⠗⠈⠻⣶⣶⠟⠀⠀⠀⠀⠀⠔⠀⢠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⠀⠀⠀⡄⠀⢀⣀⠀⠀⣀⡀⠀⢀⡀⢪⣃⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⢠⢶⣶⣇⢀⣿⣿⣤⢠⡏⠿⣶⡼⠗⠁⢸⡧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡖⠁⠘⠋⠈⠉⠛⠀⠙⠁⠃⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡿⡰⠃⣀⢀⡤⠀⠀⣀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⣀⠀⡀⡁⢠⡗⠁⣸⢋⢼⣤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⢸⠀⠀⠀⣿⡘⠹⡷⢺⡇⢻⠉⢁⠜⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠁⠀⠀⠀⠹⡇⠀⠀⢸⣇⠇⠀⠎⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠋⠀⢸⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠋

⠀⠀⠀⢀⣤⣤⣤⠀⠀⠀⠀⠀⢰⣦⠀⠀⢀⡀⠀⠀⢠⣤⡀⠀⠀⣀⠀⠀⣶⠶⠆⠀⢰⣶⣶⣦⡀
⠀⠀⠀⢾⣇⠀⠹⠃⠀⠀⠀⠀⣾⡿⣧⣰⣿⣿⠀⠀⢸⡿⣷⣀⣾⣿⡄⠀⣿⡦⠀⠀⢸⣿⣿⡿⠁
⠀⠀⠀⠈⣉⠻⣧⠀⢠⣶⣶⠀⣿⠃⠹⠏⠀⣿⡇⠀⢸⡇⠘⠟⠁⢹⣇⠀⣿⡀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠛⠛⠃⠀⠈⠻⠃⠀⠉⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠈⠛⠀⠉⠉⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⣄⠀⢀⣤⣀⠀⢀⣀⣠⡀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⢸⡏⠀⣿⠀⢸⣿⡟⠀⢸⣏⡉⠀⣿⣁⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣼⠃⠀⣿⠀⢸⣿⣿⡄⢸⡏⠁⠀⢈⡙⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠐⠿⠀⠸⠿⠟⠁⠸⠷⠶⠀⠈⠛⠟⠁

⠀⠀⠀⠀⠀⢀⣴⠞⠛⢳⠀⠀⠀⠀⠀⣀⠀⣤⡄⢀⣄⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⣿⠃⠀⡠⠋⣠⡄⢰⣿⠀⣿⡾⢸⣿⢫⣿⠀⣾⣇⠎⣿⣧⢶⣦⠀⣀⡀⠐⣧⡇
⠀⠀⠀⠀⠀⢿⣦⣉⣀⠀⢿⣇⣸⣿⢀⢿⡇⠘⠃⣾⡏⢠⠿⠏⠸⠿⢡⣿⠃⣼⡿⠝⠀⡸⣳⣶
⠀⠀⠀⠀⠀⠀⠉⠙⠛⢷⣆⠉⠀⠙⠋⠀⢀⣀⡀⠙⠛⣡⣴⠶⡄⠀⣿⣃⡸⣿⣄⡠⠈⣾⠟⢡⡀
⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⡿⠀⠀⠀⠀⣰⠃⠠⣸⠆⢰⡿⠁⡰⠃⠀⠈⠁⠀⠀⠉⠀⢸⡏⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠑⠶⠒⠋⣰⠋⠻⣷⡀⡇⠒⢚⡋⠀⣿⡗⣪⢤⣄⠀⠀⣤⣦⠀⣴⠟⢦⠙⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢹⣷⣧⠀⣿⡇⠀⣿⡷⠥⣼⣿⠤⣾⡟⠊⢀⢿⣆⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢿⣿⠀⢿⣧⠜⠘⢥⡴⠟⠁⠀⠻⠧⢖⡁⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⡴⠶⠶⠒⠒⠒⠒⠒⠒⠒⠚⠹⣆⣀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢶⣄⠀⡀⠀⣦⠀⣿⢹⡿⢹⡇⣼⠏⣽⠎⣿⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠻⠀⣿⡀⢿⡄⣿⢸⡇⠸⡧⠟⢰⠏⢸⣇⣾⢏⣠⡾⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢶⣦⣽⠧⠜⠛⠁⠀⠀⠀⠀⠀⠀⠀⣈⣁⠙⠦⢤⡶⠟⠉⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⡀⢀⣄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠡⣄⠄⠀⠀⠀⢀⠀⠀⣠⣴⣶⣶⣤⣘⣿⣷⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⡀⠀⢸⢀⡾⢿⠛⠋⢉⣩⣿⣿⣿⣥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢄⠀⢱⡀⢸⡋⢠⠇⢀⣶⣿⡟⠁⢸⠹⣧⡙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠈⠣⠀⢃⣈⣁⠚⠠⠞⣿⣯⠆⠀⣿⠀⣿⡇⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣈⠓⣠⠞⠋⠉⠉⠙⠲⡄⢋⣻⠤⣼⡟⢸⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⢰⠃⠀⠀⠀⠀⠀⠀⠈⠀⢤⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⠘⠀⠀⠀⢀⣀⣀⣀⣀⣠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠥⣤⡀⢀⢾⡟⠛⢛⡉⠉⠙⠛⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⢠⡿⠀⢀⡾⠁⠆⣼⠇⠀⠀⠀⣀⡀⠈⣿⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⣿⠁⢠⡾⢁⡿⢠⣿⠋⢹⣦⡿⢁⡿⢠⡝⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢇⣴⠏⠀⢸⣇⠼⣇⢀⡿⠙⢯⣉⡤⠛⠧⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⣀⣠⣄⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠛⣷⡀⠰⠟⠛⠉⢹⣿⡆⠀⣿⡇⢀⡀⢀⣀⠀⣠⣤⣄⠀⢀⣀⡀⢠⡶⠶⠶⣦
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠿⠁⣾⠀⣰⡆⢸⡿⣿⣸⣿⣇⢸⣧⡿⢻⣿⠋⢰⡟⣰⣟⣽⠗⢸⣇⣠⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⣿⣀⣿⠁⢸⡧⠸⡿⠸⣿⠸⣿⠇⣿⠏⢀⣿⠁⣿⣉⣡⡶⢸⣿⠙⣿⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠻⣦⠈⠛⢁⣀⣼⠇⠀⠀⠀⠿⠀⠀⣴⣦⡀⢸⡏⠀⠈⠛⠉⠀⠈⠉⠀⢿⣄⠙⣿
⠀⠀⠀⠀⠀⠀⢠⣿⠙⢻⡆⢹⣇⠀⣈⡉⠁⢠⣤⠀⠀⠰⠄⢸⡏⣼⠇⠘⠛⠀⢀⣶⠿⣶⠀⣿⣦⡀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣄⠀⠀⣸⡏⢀⣿⣧⠀⢸⡿⣧⣤⣶⠀⢸⣿⢏⣤⣦⣤⣤⣾⣧⡶⠟⢠⣿⠈⣻⣶⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⣠⡾⠃⢻⡆⣿⠇⠀⠀⣿⠀⣾⣿⡿⢡⡾⢻⡇⠹⣧⡀⢀⣾⠃⣼⡏⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠈⣿⣿⠀⠀⠀⠙⠿⠃⠙⠃⠸⣧⡿⠁⠀⠉⠛⠛⠁⠀⠘⠷⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢦⣌⣿⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣿⣿⢿⣯⡌⠈⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠋⠉⢠⣾⠟⢸⣿⠀⠈⠹⠄⠀⠀⠀⠀⠀⠀⠀⠀⠖⠋⠉⡉⠙⠢⡐⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⢘⣿⣀⣀⠀⢀⣤⣤⠀⣀⣤⣀⠀⢀⣤⣄⢰⣿⢀⡀⠘⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⢸⣿⠟⢻⡇⣾⣯⡿⠃⣋⣭⣿⣆⣿⠋⠿⢸⣿⠿⣿⡄
⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠘⣿⣷⣿⠃⠻⣷⣾⠟⣿⣷⣿⡏⢿⣤⡶⢸⡿⠀⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣠⣤⣶⠄
⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢸⣷⠀⣴⡷⣚⠀⣿⡆⣀⠀⠀⣠⣄⠀⢀⣤⡔⠫⠿⠂
⠀⠀⠀⠀⠀⠀⠀⣼⢻⡄⠀⠘⣿⣷⣿⠃⣿⡇⣿⡿⢿⣧⣾⣯⣿⠇⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠙⣦⡑⡀⠀⢻⣿⡏⠀⣿⠇⣿⣷⣾⡏⢻⣿⣵⣶⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣄⠀⠉⡀⠀⠉⠀⠉⠈⠉⠀⠀⠈⠁⠀⠉⠀⠀⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣧⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣤⣤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⣸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣷⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡀⣀⠀⢇⢸
⠀⠀⠀⠀⠀⠀⠀⢠⣁⠀⠀⠈⢿⣆⠀⣀⠴⣿⠀⠀⣴⡀⠀⣴⣤⠏⢸⣧⠔⢻⡄⢀⣷⣰⠃⣿⣠⠞⣿⡆⠀⣼⠃⡼⠀⠀⡽⠃⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⡠⠻⣿⡄⠀⣿⡆⣰⣿⡇⠀⣸⡏⠀⢸⡟⠀⣾⡇⢸⣿⠃⠀⣿⠏⠀⣿⡇⢰⣿⠁⠀⢀⡼⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⢹⣿⠀⠙⠛⠁⠹⣧⡴⠛⠃⠀⠘⠃⠀⠹⣧⠜⠋⠀⠀⠀⠀⠀⠻⣧⠞⠿⣦⡴⠞⠀⠀⠘⢿⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⢠⡏⠀⡼⠀⠀⠀⠀⠀⠀⠀⢀⡤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⣀⠀⠀⣼⡇⠜⠁⠀⠀⢠⡾⢹⠆⠀⢿⣧⡟⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⢹⣷⠀⣸⠋⢩⣿⠀⠀⣿⡧⣆⠐⣆⡀⣿⡟⠁⠀⠀⢈⠟⢿⣦⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡿⠀⢸⣿⠀⢀⣿⡇⠀⢸⡏⠀⢿⡇⠀⢀⡴⠋⣠⠊⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠃⠀⠀⠛⠒⠋⠘⠿⠶⠋⠀⠀⠈⠙⠚⠉⠀⠘⢧⣤⣾⠟





⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣷⣶⣤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣶⣿⣿⣿⣿⠿⠾⠿⡯⣿⣿⣿⠙⠋⣿⠿⠇⠀⠀⣀⢤⣤⣤⣴⣤⡶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡄⠀⠀⠀⠀⠀⠀⢀⠙⠉⠀⡀⠀⠀⠰⣿⣷⣿⡏⠀⠀⠈⠙⡭⠆⠀⢸⠛⠛⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣶⣶⣶⣾⣿⡉⠛⠁⠀⠀⠀⢀⠂⢀⣠⣼⢇⣴⢀⡰⠛⠉⠁⢰⡄⠀⠀⠀⠀⠀⠚⠓⠿⣂⡄⠈
⠀⠀⠀⠀⠀⠀⠀⣤⢾⠟⠒⠛⠛⠓⢪⣿⣿⣿⡿⣡⣿⢷⡶⠶⣶⡶⠥⠴⢞⠛⠛⠿⠉⠁⣠⠤⠤⠞⠛⢓⣒⣒⢲⣄
⠀⠀⠀⠀⠀⣴⢽⣧⡜⠀⠀⠀⠀⠀⣾⣿⣿⣟⣡⢛⣫⣤⣭⣥⣼⣥⣤⠴⠿⠒⠒⠺⠯⠛⠉⠀⠀⠈⠉⠉⠉⠀⢸⣿⡄
⠀⠀⠀⠀⡾⠁⢸⣿⠀⠀⠀⠀⠀⢀⡟⢉⢘⣿⣿⡟⠛⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡅
⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⢠⣿⣧⢸⣿⣿⣿⡗⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⠃⠸⣶⠁⠈⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣸⣷⡄⢻⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡿⡇⠙⠢⢉⣢⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢷⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠘⢧

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠿⠻⡿⠛⠿⢿⠿⢻⡿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣖⡍⣠⣀⡀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⢀⠄⠊⠀⠀⢀⣠⡤⠭⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣱⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⣜⡵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠉⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⢀⡌
⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠠⠀⡀⠖⠱⠄
⠀⠀⠀⠑⣀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⣠⣄⠀⠠⠀⠀⠀⠐⠛⠃⠀⠄⡀⠀⠀⠀⠀⠁⠀⣤⣤⠄
⠀⠀⠀⢀⠁⠷⠀⣀⡀⠐⠀⠀⠀⡠⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡔⠀⠀⠀⡠⠊
⠀⠀⠀⠀⣤⣤⡀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠂⠀⣀⠤⠒⢢
⠀⠀⠀⠀⠈⠂⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠊⠉⠠⢤⡞⠁
⠀⠀⠀⠀⢠⠄⠠⠤⠤⠤⢼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠗⠒⠒⢓⠳
⠀⠀⠀⠀⠀⠙⠶⠖⢂⣀⣀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢿⡐⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠊⠥⢤⣴⠤⢺⡷⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠣⠔⠛⠁⠀⠈⠐⠂⠀⠠⠄⠀⠐⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠉⠻⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠺⠦⠶⣴⢀⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡄⢘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⣄⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣆⣿⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⡼⠃⠀⠀⡸⠳⠛⢯⣀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡏⠀⠈⣿⡆⠀⠀⠀⡸⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⡇⠀⠀⠀⠙⠿⠒⢀⡀⢀⠀⡀⢀⠠⠠⢤⢀⣀⣀⣀⠀⠀⠀⠏⠀⠀⢀⣿⡇⠀⠀⢀⣡⡇
⠀⠀⠀⠀⠀⢷⠃⠀⠀⣷⠀⠀⢀⡼⠄⢃⣈⠀⠀⠤⠤⠤⠶⠀⠴⠶⣿⣧⣭⣤⡳⠶⣷⣶⣤⣾⡿⠁⠀⠀⠈⢹
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠸⣤⣴⠛⠀⣰⠎⠁⠀⠀⣰⠃⠀⠀⠀⠀⠀⠸⣿⡋⠻⣿⣷⣍⢻⣿⠟⠁⠀⠀⠀⢠⡞⠀⠀⠀⠀⡀
⠀⠸⡆⠀⠀⠈⣦⠶⠀⠀⠈⠙⠂⣴⠃⠀⠀⠀⠀⣿⠀⠀⠀⡆⠀⠀⢀⣿⡃⠀⠈⣿⣿⠛⡇⠀⠀⢀⣼⣷⡟⠀⠀⠀⢀⡾
⠀⠀⢳⡴⢄⠀⠈⢦⠈⢠⡤⠀⢠⡿⢆⣀⢀⡀⣀⣼⡷⣄⠀⡇⢀⣀⣼⣷⡁⠀⢀⣿⣿⡆⢿⣤⡎⣠⣿⠋⠀⠀⢀⣜⡽⠁
⠀⠀⠀⠑⢆⠐⠄⠀⠉⠃⠇⠀⠸⠁⠀⠈⠉⠀⡼⠉⠀⠀⠀⠇⠉⠉⠙⢿⡛⠛⠛⣿⣿⢇⣼⠿⠛⠋⠁⠀⢀⠔⣼⡏⠁
⠀⠀⠀⠀⠀⠳⢤⣹⡇⠐⢾⣐⡆⡆⠀⠀⠀⠀⠳⣄⠀⠀⠀⡆⠀⠀⠀⡸⠁⠀⢀⣿⣿⣾⣿⡄⣤⣴⠒⠒⣿⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠂⡙⣄⠀⠀⠀⠀⠀⠛⠒⡄⡅⠀⠀⢾⣅⡀⢀⣾⣿⣿⠿⠿⠛⠋⠉⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡔⠊⠀⠉⣻⣮⡛⢶⣖⠀⠀⠀⢈⡅⠁⠀⠀⠀⠈⣹⣿⣿⣿⣧⠤⠤⣀
⠀⠀⠀⠀⠀⠀⣠⣅⡘⠉⠉⠉⠉⣀⡀⡬⣦⡙⢦⣄⣀⣺⡄⠆⠀⠀⣀⣤⣿⣿⡉⠛⠛⠻⠶⠾⣿⠶⡄
⠀⠀⠀⠀⠀⠀⠁⠘⠀⠀⠀⢰⣁⠀⠀⠐⠋⠙⠿⠶⢒⣉⣁⣀⣰⣿⠿⠿⠟⠻⠿⣍⣐⣄⠀⠀⠘⣷⠛
⠀⠀⠀⠀⠀⠰⣞⡀⠀⠀⠀⣈⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠇⠀⠀⠘⣧⡇
⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⠀⢠⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢹⠟⠀⠀⣠⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠁⠀⠀⠘⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠠⠈⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠈⠁⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠶⠀⠀⢨⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⢡⠀⠈⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⠁⠀⡀⠀⠀⠀⠈⠉⠀⠀⠈⠑⠀⠀⠀⠀⠀⠙⠒⠀⢻⠂⠀⠸⢠⡀
⠀⠀⠀⠀⠀⡒⠠⠀⠈⡍⠹⣤⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⣹⡏⠀⠠⠖⢠⠇
⠀⠀⠀⠘⠀⠘⣆⡠⢀⠑⡄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⢀⡔⠀⢀
⠀⠀⠀⠐⠂⠀⠈⠢⢤⣀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠴⠋⠠⢀⠉
⠀⠀⠀⠀⠀⣀⠀⢀⠀⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣂⣂⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠈⠀⠤⠵⠆⠀⠴⠶⢶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣵⣖⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠠⠔⠚⠋⢱⣄⣀⠀⠀⠀⠀⣀⣠⣤⢾⡑⠀⠀⠉⠉⠛⠓⠦⡂
⠀⠀⠀⠀⠀⠀⠀⢀⠰⠊⠀⠀⠀⡶⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠙⠓⠞⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⠆⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣰⠆⠀⠀⠀⠸⠁⠉⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⡼⠛⠰⣊⠉⢹⣄⠀⠀⠀⠀⠀⠀⠀⠉⢄⠀⠀⠀⠀⠀⠰⡀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡃⠀⠀⢠⡇⡀⣿⣦⣤⣼⠿⠽⠤⠤⠴⠦⢤⣀⡀⠈⣆⠀⠀⠀⠀⠀⣇⠀⠀⠀⠩⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⢘⠉⠀⣿⣄⠀⣾⠀⠀⢀⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠐⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠆⠀⣷⡀⠘⣯⣤⣿⡖⠟⠤⠷⠀⠀⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣲⡤⣤⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡔⠈⠳⣤⣈⣿⡯⢉⡂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢰⠉⠉⠄⢾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⣤⡀⢌⠙⣿⣇⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⡄⠀⠀⠺⣇⣠⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠨⡙⠶⣥⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢾⣻⣿⣧⣤⣴⣾⡿⠋⠀⠀⣀⡿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡖⠁⠀⠈⠳⢿⣶⣤⣽⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⢈⡷⢐⣿⣟⣛⠛⠉⠁⠀⢮⠴⠛⠁⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣴⠟⠀⠀⠀⠀⠀⠀⡈⠙⠛⣿⡽⠀⠰⠀⠀⠰⠂⡁⣷⣾⣿⡟⠙⠛⠶⠶⠶⠟⠛⠉⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠜⣶⠲⠶⠚⠛⠳⣅⠊⠀⣀⣢⣾⣿⣿⡟⠉⠙⠳⠶⠦⠤⠴⠴⠷⣦⣤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⡏⠀⠉⠙⠳⠶⢤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⣄⠄⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠰⠋⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⢀⡰⠟⠃⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠖⠂
⠀⠀⠀⠀⠀⠌⠀⠀⡀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠴⠓⠁⠀⠀⠀⠀⣀⡤⢴⠒⡖⣲⢦⣀
⠀⠀⠀⠀⠀⠀⢠⣞⡍⠱⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢂⣀⡀⠤⠊⠀⠀⠀⠀⠀⢠⠺⡅⠃⢰⠀⠀⠁⠈⡘⢹⢦
⠀⠀⠀⠀⠐⠀⠈⠙⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⠂⠱⠀⠀⠀⠀⠀⠐⢀⠊⡼⡇
⠀⠀⠀⠀⠀⠘⠒⠐⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢸⡢⡠⠀⠀⠀⠀⠀⠀⠀⠀⠈⢘⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠈⠀⠀⠠⢤⠄⠁⠀⠀⠀⠀⠀⠙⠦⣬⠁⢀⡠⣤⢄⡀⣤⠔⠛
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢰⡍⠀⠀⠤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠿⣤⣶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠛⠻⠿⢿⣿⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢹⣿⠛⠛⠛⠛⠂⠀⠀⠀⠉⠿⣆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣀⣀⣀⡀⡀⠀⢀⣀⣀⣠⣴⡟⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠀⡀⠢⢿⡿⠟⠛⠁⠀⠀⠙⠛⠿⠭⠄⢀⣲⣤⣠⡀⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⣤⣾⡗⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⠿⣿⣷⡟⠛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠀⠀⠀⠀⠀⢂⢲⣭⡖⠂⠂⢮⣶⠊⣄⠀⠀⠈⠉⠁⠀⣿⣾⠀⢀
⠀⠀⠀⢰⡀⠀⢸⣠⣧⣼⡇⠀⠠⢀⠰⠀⠁⢨⣌⠙⠀⠀⠀⠈⣯⣤⠀⠈⠰⣤⢤⡤⠄⣿⡏⠀⢸
⠀⠀⠀⠀⢇⠀⠀⠻⢿⡟⣤⣤⠣⡈⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠏⢿⡿⠋⠀⣰⠟
⠀⠀⠀⢀⠈⠲⣴⡀⡀⠈⠙⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠏⢀⣤⠾⠁⠀⡔
⠀⠀⠀⠈⠳⣄⡈⠛⠷⠷⠴⢶⢦⡀⣄⡀⠀⢤⣠⡀⠀⠀⠀⠀⣠⣤⣴⣦⣄⣀⡼⠿⡛⢡⣠⠾⠋
⠀⠀⠀⠀⠀⠈⠉⠛⠶⠦⠶⠶⠬⣿⡿⠿⣿⣿⡿⠟⠀⠀⠀⠀⠉⢉⠛⠫⠟⠛⠟⠚⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⣤⡶⠶⠦⠾⠿⠷⡄⠀⠀⠀⠀⠀⠀⣰⠛⠛⠓⠀⠉⠲⠦⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⠢⠷⠟⠒⠤⠤⣀⣀⢀⡈⠋⠙⠛⠺⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸

⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⣦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠂⢡⠀⠀⠀⠀⠠⠂⠄⠈⠀⠀⠜⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⡀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣐⣰⢁⠀⡈⠡⠔⠩⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠘⠈⠁⠒⢠⠃⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠳⣄⠀⠀⡀⢐⠀⠀⠀⠀⠀⠀⠀⠀⣓⡔⠀⠀⠀⠀⠀⠀⠒⠤⠀⠒⠁
⠀⠀⠀⠀⠀⠒⠂⠀⠁⠀⠀⠀⡅⠀⠀⠈⠠⠆⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠋⠇⡀
⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⣠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠓⠒⠀⠑⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠈⠀⠀⢀⢷⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠪⠥⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠐⡀⡀⠀⠀⠂⠀⠀⠄⠀⠀⠰⡙
⠀⠀⠀⠀⠀⠀⠀⠘⣠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⡆⠀⠀⠀⡀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠂⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡀⠈⠡⠤⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⡀⡇⠰⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⠎⡄⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠙⠒⠂⠂⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⢀⢤⣤⣤⢤⣤⣀⢀⡀⠀⠀⠠⠌⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡀
⠀⠀⠀⠀⠀⠀⠸⠟⠁⠁⠀⠀⠈⠓⠇⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠀⠀⠀⠀⢀⠀⡰⠊⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠸
⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠂⠠⡆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⢁⠜⠁⠀⠀⠁
⠀⠀⠐⡵⢋⠞⠡⠄⣀⢀⠀⣀⠀⡄⢢⡘⢮⡳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠼⠠⠇⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⢡⠏⢀⠉⠐⠃⢈⠀⠸⠀⢠⠀⢱⣌⡇⠀⠀⠀⠀⠀⠀⠀⠈⠘⢸⠸⠀⠀⢀⠄⡠⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠳⢾⡀⢼⠀⢸⠀⠠⡄⠀⣧⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢻⡇⠀⣀⠔⠁⠀⠀⠀⠀⠈⠀⢔⠢
⠀⠀⠀⠀⠀⠀⠑⠚⠐⠚⠀⠐⠃⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠦⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⣐⠄⠀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠤⠀⠀⠀⠀⠀⠄⢈⠀⠀⣀
⠀⠀⠀⠀⠀⠐⠊⢠⡔⠉⠀⠀⠀⠀⠀⠀⠈⢻⣽⣯⡉⠀⠎⠀⢀⠀⠀⠀⠀⠀⠶⠤⣀⡋⢀⡄⠀⠀⠀⠀⠀⠀⠤⣄⠒⠀⣱⠂
⠀⠀⠀⢈⠥⠒⠒⠦⠿⣶⠶⠀⣀⣀⡠⢔⠴⠯⡀⠑⠂⠴⠤⠔⠁⠀⠀⠀⠀⠀⠀⡠⠤⠨⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣦⡀
⠀⠀⡴⠉⠁⢠⠈⡀⠀⠇⠀⠁⡉⠉⠁⠨⢠⠂⠈⠀⢤⣀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠔⠁⠑⠒⠀⣄⠀⠀⠀⠀⠀⠀⢀⠰⠦⠈⠛⢯⠂
⠀⠀⠂⠀⠀⠁⠀⡏⠃⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡀⠀⠀⠀⠀⠀⠁⡀⠐⠀⣤⡤⢈⡁⠀⠀⠓⠀⠐⠋⠁⣀⠀⠈⡻⠉
⠀⠀⠀⠠⠁⠀⠀⢣⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠀⠸⠋⠓⠒⠀⢐⡒⠴⠛⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⢔⠉⡐⠀⢈⡦⠒⡂⠀⠀⠀⢰⢆⡴⠂⢶⡍⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⠀⠘⠀⠈⠀⠠⠉⠠⠤⠐⠁⠀⠑⠂⠾⠤⢾⢈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢘⠄⣰
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠁
⠀⠀⠀⡀⠀⠀⣀⢠⠀⠱⣰⡦⠂⠀⠀⠄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠐⠠⠤⣀⣀
⠀⠀⠀⠄⠀⠈⢤⡈⢢⣐⠋⠀⠀⠀⠀⠀⠘⣆⠀⠀⠋⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠚⠉
⠀⠈⠀⠀⠀⠀⡀⢁⣄⣻⠀⠀⠀⠀⠀⠀⠀⡷⠐⠐⠃⡇⡀⠀⠀⠀⠀⠀⠠⠜⢋⡁⠒⣳⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣂⡀⠳⠢⡀
⠀⠀⠀⠀⡀⠊⠀⠀⠈⠙⠄⠀⠀⠀⠀⢀⠼⠋⠁⡘⠈⠃⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁⠈⠀⠀⠀⠡⣤⠶⠤⠶⠶⣚⡂⠀⡄⡁⠙
⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠠⠯⠔⠂⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠘⠢⠀⠀⠀⠀⠀⠀⠀⠈⠀⣀⡀⡀⠁⠀⣀⣼⡐
⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠵⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠦⠁⠀⣀⢎⣠⠟⠁⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⠿⡻⠗⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡇⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⣿⣿⣿⠉⣨⣴⣿⣿⣿⣿⣿⡟⢿⣿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⣷⣶⣼⣾⣷⡟⢩⣿⣿⣿⣛⢛⣭⣿⠎⣉⡠⢧⡃⠉⢻⡆
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡿⠿⣟⣟⣭⣭⣽⢻⣬⣷⣾⣿⣿⣿⣿⢿⣿⣿⣿⣭⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠞⠁⢁⣬⣶⣿⣿⣿⣿⣿⢿⣯⢿⣿⢿⣿⣿⣦⡥⢿⣿⢿⣿⣿⣿⡿⠒⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⡟⣼⣿⣿⢿⣹⣿⣟⣻⣿⢹⢿⣿⣿⣿⣿⢸⢻⣿⣿⣷
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠷⣼⣿⣿⣿⣆⣻⣿⣿⣖⢿⡜⠘⠏⢹⣿⣿⣌⡀⠀⠹⠛
⠀⠀⠀⠀⠀⣼⠿⠟⠏⠈⢠⣿⣿⡝⠯⡧⢹⣿⣯⡓⠸⣷⠀⠀⠏⢿⢿⡓⠁
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⡿⡝⠆⠀⢸⣿⣦⡉⠀⣿⢀⣤⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠑⠈⠀⠀⠀⠻⠇⣩⣶⣿⡝⢿⣿⡿⣶⣤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣶⡿⣫⠵⢿⣹⡇⠀⠀⠹⡄⢹⠷⠄⠀⣀⣙⠿⣔⣲⣤⣀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠛⠛⠛⠛⣿⣷⣾⣖⣲⣤⣄⣿⡗⠛⠛⠓⠂⢲⣶⡼⠶⢦⣸⣷⣶⡿⠿⠿⠿⠛⠛⠛⠻
⠀⠀⠀⠀⠛⠉⠛⠒⠒⠒⠚⣁⣤⣤⠤⠤⠌⠈⠉⠒⠒⣿⡇⠉⢁⣷⡆⢠⠈⣡⠈⣋⣉⣣⠤⠴⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⡾⢸⢻⢃⣮⢞⢥⠀⠠⠤⠭⣭⣿⣿⣯⣈⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣬⡏⢀⡟⣯⣈⣬⡃⢻⣾⣯⣀⠀⠐⣾⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠄⠒⠊⠉⢛⡽⠋⠋⠀⠸⠘⠹⠉⠳⠈⠙⠛⠿⠿⢽⣿⠿⠿⠓⠒⠓⠂

⠀⠀⠀⢀⢀
⠀⠀⠊⠉⠳⢄⠈⠙⠋⠉⠉⠀⠁⠀⣁⠨⠜⠋⠁⠀⠀⠀⠀⣠⠀⠤⠈⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡀
⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⡀⠴⠒⠋⠁⠀⢀⠐⠊⠈⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠈⢃⣀⣀⣀⡴⠏
⠀⠀⠀⠈⠐⠠⠖⠋⠀⠉⠀⢀⡀⣀⠀⠀⠀⠈⠁⠀⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠚⠁
⠀⠠⢀⣀⠠⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⡠⠂⠛⠂⠃
⠀⠀⠀⠀⡠⠴⡄⠤⠠⢨⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠀⠸⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠃⠀⢅⠀⠀⡄⠀⠀⠀⡤⢀⡄⡄⡄⡀⠀⠀⠀⠀⠀⠘⠦⠀⠠⣔⣰
⠀⠀⠰⡀⠠⠆⠠⠤⠴⠊⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠉⠉⠋⠁⠉⠀⠁⠉⠙⠁⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁
⠀⠀⠀⠀⡀⠀⠐⠒⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠁⢀⡞
⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⢆⠀⠀⢠⠅
⠀⠀⠀⠀⢀⠐⠒⠂⠄⠀⠀⠀⠈⡄⠀⠄⡀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠐⠀⠀⠀⠈⡀⠀⠀⠀⠐⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠐⠄⠀⠀⠀⠀⠔⠀⠀⠀⠀⡄⠀⠀⡀⠀⠀⠀⠀⢀⡴⠞⠁⢠⣀⣀⣠⠤⠖⠒⠲⠶⠶⠶⠄⠀⠉⡑
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠒⠀⠀⠈⠀⠀⠀⠀⠉⠐⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⠽⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⣀⣀⣀⣤⡙⢛⣷⣶⣶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠒⠂⠋⠉⠑⢒⡒⠒⠓⣶⠀⢂⡒⠘⢽⣿⣿⣿⣆⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠻⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠞⠛⢓⣛⣫⣽⣿⣿⣿⣿⣽⣭⡿⠷⣦⢤⡀
⠀⠀⠀⠀⠀⢠⠖⠀⠄⠠⢄⠽⠠⠦⠤⠼⠬⣥⠬⣿⣿⣽⣿⠭⠿⠿⢿⣿⣟⣽⣿⣿⣷⣮⣄⣙⠋⠙⢷⣄⡀
⠀⠀⠀⠀⣰⠋⠁⠠⢤⣤⣀⣆⣀⣐⣀⣄⢤⣤⠠⠤⠤⢤⣴⣶⣶⣶⣿⠿⣾⣿⡷⣿⣿⣿⣿⣿⣷⣦⣄⠙⠻⣤⡀
⠀⠀⠀⢀⠎⠀⠂⠉⢁⣈⣈⣁⣀⣉⣁⣈⣉⣙⣏⣉⣉⣉⣉⣭⡤⠤⢶⣶⣿⡿⠗⣿⣿⣿⣿⣿⣿⡧⠀⠀⠀⢸⣷⣄
⠀⠀⠀⢠⠀⠀⢌⠐⠒⠂⠐⠳⠒⠒⠒⠒⠒⠲⠗⠒⠒⠒⠖⠛⢛⣋⣉⣯⣿⢷⣫⣿⣿⠛⠛⢿⣿⣻⠓⠂⠀⠐⠉⢻⡆
⠀⠀⠀⠘⢀⡠⢼⣒⣒⣉⣭⣭⣭⣭⣭⣭⣭⣭⣭⣬⢭⣭⡭⢽⠿⠿⢿⣟⣯⡥⣶⣿⣷⣦⣛⣿⣿⣖⠲⠤⠀⠀⠀⠀⢻⡄
⠀⠀⠀⢀⠀⠂⠹⢯⣀⡠⠤⠌⣃⠀⠀⠀⠠⠠⠟⠂⠀⠀⠒⢒⣚⣻⣿⠿⢷⣾⣿⣿⡿⣿⣿⣿⣿⣉⠄⣒⠀⠄⣀⠀⠘⡙
⠀⠀⠀⢨⠀⡄⠈⠻⡄⠀⣀⡠⠄⠀⠀⠀⠀⠀⣚⣉⣀⣉⣭⣥⣼⣗⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⠈⣀⠀⠀⠀⢈⡓⢱
⠀⠀⠀⠀⣅⠋⠘⠑⢨⡥⠤⠐⠂⣐⣀⣀⠈⠭⠭⣥⣀⣖⣾⣿⣯⣾⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡛⠛⠷⢮⣍⡒⠀⡀⠉⠹
⠀⠀⠀⠀⢨⡀⠼⠂⣀⠴⠒⠈⣉⣠⡤⠤⠶⣒⣺⣻⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣈⡁⠂⠤⡙⠳⢬⣑⠀⠂
⠀⠀⠀⠀⠀⠉⡼⠋⣠⡴⠚⣁⣭⠴⢒⡻⠭⣵⣶⣿⣿⣿⢿⣿⣿⣿⣿⣽⣭⣿⣿⣟⣿⣿⣿⡍⠉⠉⠛⠛⠶⣤⡑⠢⣙⠻⠃
⠀⠀⠀⠀⠀⠀⠐⢾⠁⡠⠚⣁⠴⣚⡽⣶⣿⣿⠿⠛⠉⣱⢫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠂⠤⢈⡒⠤⣀⠉⠛⢷⣄⡑⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢅⡼⣡⠞⣡⣮⢞⠁⢀⠀⢀⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣁⠀⠀⠈⠹⢷⣮⡛⠦⣄⡉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⣼⢵⠁⠈⠔⡁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣋⠉⠋⠙⣶⣤⣀⠠⢉⠛⣷⣬⡙⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⡀⠀⢐⢿⣿⣿⢿⣿⣿⣿⣿⠿⢿⣿⣿⣟⠋⠉⢷⠦⣄⠀⠙⠛⢿⣦⣁⠀⠈⠻⢦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⣾⡄⠹⣿⢸⣿⣿⣿⣿⣻⣿⣶⠬⠉⠁⠀⠀⠀⠀⠙⢶⣄⠀⠉⠛⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠄⡀⢿⣿⡆⠟⣾⣿⣿⣿⣿⣿⡿⠗⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⠀⢘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⡅⢸⣿⣿⢠⣿⣿⣿⣿⣿⣿⡗⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣈⠊⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡃⢸⣿⣿⣼⣿⣿⣿⣿⣯⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠛⡷⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣞⠻⣿⣿⣿⣿⡿⣿⡧⡉⠢⡀⠀⢢⡀⠀⢠⡀⠀⠳⣄⠀⠸⢓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⣿⣿⣷⡙⣮⠳⡙⠄⠈⢆⠀⠹⣆⠀⢹⣄⠀⠘⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠏⠉⠛⢿⡜⣧⢹⡈⢆⠈⢧⠀⠸⣧⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠱⢸⣎⡇⠘⣇⢸⡇⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣉⡉⡛⢿⣙⣠⡆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⢀⣠⠶⠛⠉⠀⠙⢀⣁⣒⣤⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣿⣽⣿⡆
⠀⠀⠀⠀⠀⠀⣴⣿⠀⢠⠾⠋⠨⠛⣤⠀⠀⠀⠁⣸⠛⢻⢿⢿⣎⢥⢴⣶⣶⣦⣤⣀⠈⠛⢿⣿⢛⡟⠂⠔⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⢰⣿⣉⣴⣿⠀⠀⢀⣤⠛⣷⠀⢻⡀⠛⣶⣿⣫⣬⠘⢾⡌⠉⠀⠀⠈⠙⢻⠒⣤⣽⣿⣿⡆⠀⠀⠀⠀⢰⣧
⠀⠀⠀⠀⠀⣿⡇⢿⣿⣿⣤⠴⣫⣀⣶⣼⠻⢮⠇⢈⣽⣿⣿⣷⡀⣨⡷⡀⠀⠀⠀⠓⢀⢤⡟⠈⠻⣿⣿⡄⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⣿⡬⣿⣿⠇⣄⣽⣿⣿⣿⢆⣈⣰⣦⠙⣿⣿⣿⠗⠎⠀⢨⣿⣧⡤⠀⠐⠮⡿⣴⠃⠈⠿⡧⠀⢀⡾⢘⠇
⠀⡤⠀⠀⠀⠀⢹⣧⡘⠷⢶⣾⣿⣿⣿⣧⣶⣿⠛⢀⣴⠟⠛⢿⣿⣄⣴⣿⣿⣿⡿⠀⠦⠀⠈⢫⠙⡋⠀⡰⢂⣾⢣⠈
⠀⢳⣾⣦⣤⣄⠀⠈⠻⡖⢶⣿⣿⣿⣿⣟⠛⠧⠸⠟⠁⡄⢠⣿⣿⠛⠙⣿⢿⣟⠛⢷⡀⠀⠀⠀⠀⠀⠀⣠⠾⠡⠎⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠹⣿⣿⣿⡷⠤⢠⣦⣀⣿⣿⣿⣿⣿⣞⠀⠀⠀⠀⠀⢸⣿⠘⠀⣾⢻⣼⡿⠀⠈⠀⠀⠀⠀⠁⢠⣄⠘⢂⣠⡔⠒⢈⣩⠄⠀⠀⠀⠁
⠀⠀⠀⠈⠛⠾⠿⣶⣤⣼⣿⠻⣶⣿⣛⠙⠇⢗⡤⠀⠀⠀⠛⢠⡄⠀⢸⠂⠙⡿⢶⡀⠀⠀⠀⢈⣡⢶⢿⡟⡋⠉⠄⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣭⣿⡟⠻⠿⠿⣿⣿⣦⡋⠁⠁⠀⠀⡀⣸⡇⠀⠾⠀⠀⠀⠁⠈⣀⣴⡾⠯⠄⡍⣌
⠀⠀⠀⠀⢠⠴⢋⣩⣤⡞⢡⣴⣦⣤⣄⣹⣿⣿⣄⣀⣠⠤⠉⠀⠀⠀⠄⢀⣀⣀⣠⣞⣋⣀⣴⡖⠚⠓⠛⢶⣌⠒⣄⡀
⠀⠀⠀⣴⠟⠚⠋⠁⠀⠀⣠⠰⠙⠉⠉⠛⢛⣏⣯⣻⠽⣿⣿⢿⣭⣿⣽⢿⣿⣾⣿⡀⠉⣩⣉⣛⣷⢆⠀⠀⠈⠓⠪⣱⣄
⠀⠀⠀⡷⠀⠀⠀⢀⣤⠾⠋⠛⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠙⠓⠌⠉⠉⠛⢧⠱⠄⠀⠀⠀⠀⢬⣃
⠀⠀⠀⠀⠀⢀⢤⡿⠉⡤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⣠⠿⠟⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⢄⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⣦⠀⣰⣿⠋⠀⣚⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢟⣩⣼⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠄⠹⣧⣿⡃⢀⣾⢿⠇⠈⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣾⣿⡿⢿⣿⣷⣷⢠⣤⣶⢂⣶⣦⣴⠖⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⠔⠀⠄⠀⣘⣬⣡⣘⠁⢉⡀⠀⢉⠉⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠟⣁⣤⠬⢿⣿⣿⣾⣿⢻⣕⠋⠁⠀⠀⠀⠀⠀⢀⡴⠯⠿⠻⢟⡁⠈⠀⠲⠟⡿⠃⠘⣷⠘⠧⠈⠃⠈⠈⠛⠃⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⣺⠋⡤⣶⣶⣾⣿⣿⣿⣿⢿⣭⠑⠦⣄⠀⠀⠀⠈⠀⠀⠀⠀⢸⡄⠀⠀⠁⢠⠇⠀⢸⡙⢧⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣾⢃⣽⠟⣿⡿⢿⠿⢿⣧⡉⠁⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠘⡇⠀⠁⠀⠈⠀⠀⢸⠆⠘⣇⠀⠀⠀⠀⠀⠀⢠⡏⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣽⠟⠀⠀⣿⡃⡾⠁⠈⢿⡁⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⢼⡄⠀⠀⠃⠠⡇⠀⠀⢸⡄⠀⠰⠃⡄⠀⢸⡏⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⢸⣿⡿⠁⠀⠀⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⠇⠀⠀⠀⣶⠀⠀⡎⢠⠀⡾⠀⠘⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠇⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢹⡆⠀⡄⠌⢰⠇⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢷⠀⠃⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⢠⡇
⠀⠀⠀⠀⢀⡀⢸⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣄⡀⠀⠀⠀⠘⡇⠀⢰⡇⠀⠀⠄
⠀⠀⠀⠀⣼⣰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠒⡶⢻⡶⠴⣶⣺⣻⡀⢼⠅⠀⠀⠠
⠀⠐⢤⣶⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡐⡡⢚⣙⢊⠴⢓⢃⡴⢚⣉⣡⣿⡆⣸⣃
⠀⠈⢩⣯⣿⣿⣾⣐⣤⣠⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠖⠛⠱⢭⠇⢉⠠⠟⢍⡲⠂⣀⣾⣿⣷⣿⣿⣵⣀⣤
⠈⢉⣹⣿⣿⣿⣛⣿⠷⠬⢿⣶⣾⣿⣿⣶⣄⣀⠀⣼⢻⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣶⣭⣾⣖⣏⣩⣍⣀⣒⣒⣛⣿⣾⣿⣿⣿⣿⣿⣿⣷⣶⠶⠄
⠀⢸⣿⣿⣿⠏⠉⠉⠛⣫⣉⣿⣿⣿⡿⣿⡟⠁⠀⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠹⠿⢿⣏⠹⣦⡄
⠀⠈⣿⡿⢃⣀⣤⣤⣤⣼⣿⣟⠟⣛⣿⢏⠁⠀⢀⡇⢀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⢈⣉⣉⠛⢻⣿⣿⣿⣾⣿⣿⡿⠿⣿⡿⣿⣟⣿⣿⣷⡟⣿⣆⠀⠡⣄
⠀⢛⡽⠂⠈⠉⠘⢩⣽⠛⢁⣍⣀⠀⠈⣀⡀⠀⠀⡅⢸⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣥⣼⣿⡿⢯⣿⣭⣬⣉⠳⠒⠛⠛⠉⠉⠉⠉⠀⠀⠀⢿⣆⠀⠙
⠺⠏⠀⠀⠀⠀⠀⠀⣀⡀⢉⣃⣠⡄⣈⣁⣀⠀⢈⣇⣿⢸⠁⠠⠀⠤⠤⠾⠟⠹⣯⠯⣭⣥⣶⡶⣿⣿⢌⡷⣟⣛⣷⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣏⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠿⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠒

⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⣤⡤⣀⡀
⠀⠀⠀⠀⡈⠘⢄⠠⢔⡲⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡞⠓⢺⣅⠀⠏⣷⣰⠉⢲
⠘⠯⠭⣉⠁⡷⡆⣰⠋⠀⠀⠀⣰⢹⣭⡭⠷⠦⢙⣶⣶⣶⠆⠀⠀⠀⣀⣀⠠⢤⣀⠀⠘⢯⡿⣿⠇⢰⢻⡁⠠⣁⣿⣿⣆⠘⡇
⠀⠀⠀⡜⣠⠖⠢⡘⣆⠀⠀⠈⠻⣇⣛⣛⣋⠉⣈⠿⠋⠀⠀⢀⠴⠿⠤⣍⢢⡀⣿⠈⢢⡀⠉⢿⡇⠘⣗⠀⢘⢠⡅⠙⢿⣶⣿⣄
⠀⠀⠈⠚⠁⠀⠀⠙⠚⠃⠀⠀⠀⠘⠛⠒⠂⠉⠀⠀⠀⠀⠀⢮⣿⣿⣷⣮⣿⣧⣇⡠⠏⠁⠀⠈⠋⠙⣯⠀⣰⣰⣀⡴⠶⠻⠿⣿⡧
⠀⠀⠀⣀⢤⠴⠶⡶⡤⣤⡀⠀⠀⠀⠀⠀⠀⢀⠠⡄⠤⢤⠠⠀⠀⠉⠙⢫⠞⠻⣟⠒⠒⡏⠀⠀⠀⠀⠈⠋⠉⠉⠁⢠⡴⠶⢤⡀
⠀⣴⠻⠹⠀⠀⡄⠀⢀⠃⠞⣳⣄⠀⠀⡠⠊⠣⡀⢣⡀⣹⠀⠇⠀⠀⠀⠘⠣⣧⠈⠹⠞⠀⠀⠀⠀⢀⡄⠀⠀⠀⡠⣿⢏⠢⠡⢹⡄
⠈⡎⢂⠁⠃⠀⢀⠀⠈⠈⡰⠡⢋⠆⡐⠙⠢⣤⠟⠉⡏⢻⢆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⣸⣷⠀⠀⠀⡇⡈⠘⡄⡂⠆⣧
⠀⠘⢆⢢⢠⣠⢘⠀⠀⠐⠐⡡⠋⠀⡗⠲⠴⡟⢒⣶⡷⣯⠤⡇⠀⠀⣰⠇⡢⠀⠄⡁⠨⢳⡄⠀⠀⡯⠰⡄⠀⠀⢻⢁⠀⠀⠀⠀⢻
⠀⠀⠀⠳⣕⢷⣿⣾⠌⣠⠊⠀⠀⠀⠘⢦⠒⢛⠥⣮⢶⠛⣤⠇⠀⠀⣥⠊⣠⠎⡔⢠⠁⡅⡇⠀⣘⣋⠉⢵⠀⠀⠸⡼⡠⡀⠀⠀⡿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠈⠙⠻⠤⠧⠜⠛⠁⠀⠀⠀⠋⣴⠃⣼⠁⣾⠀⣇⡿⠀⠫⣤⣺⣿⡷⠀⠀⢿⡵⣣⢇⢢⡇
⠀⠀⠀⠀⣠⡤⣀⢤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠒⠛⠴⠟⠒⠉⠁⠀⠀⠸⠿⠋⠁⠀⠀⠈⠻⣵⣯⠞
⠀⠀⡄⠊⠀⢳⡀⠈⢁⡆⠙⢦⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠻⣟⠛⠒⠚⠳⠲⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⣼⣀⣀⣠⡜⢋⠓⠺⢀⠠⠈⢳⣄⠀⠀⡼⣛⣫⡈⠛⣷⣄⠀⠀⠉⢢⡀⢀⠀⠃⠈⠙⠦⣄⠀⠀⠀⠀⠀⢴⣾⣿⣿⢧⣄
⠀⢱⠀⠏⣇⡼⣟⡖⠈⢹⠶⠛⠉⣿⠀⠀⡏⣴⣾⣿⣮⡟⣿⣆⠀⠀⠀⠙⠒⠒⠓⠒⠒⠒⠚⠓⠀⠀⠀⣠⡾⣿⣛⣫⣤⠛⠙⡦⡀
⠀⠘⣦⠚⠙⠑⡎⠉⠺⡇⢠⠀⣆⢺⠀⠀⢣⡻⣿⣿⣿⣿⡿⠛⠀⣀⠤⢤⠞⠛⠙⣟⠓⠦⣄⣀⠀⢠⢞⡙⠀⠈⠀⠀⠀⠐⡜⡄⢽
⠀⠀⠈⠙⠲⠶⠧⠖⠲⠶⠚⠒⠛⠛⠁⠀⠀⠙⠚⠛⠋⠉⠀⠀⢰⡡⠊⠀⢂⠀⠀⠘⣄⠳⣈⡟⠀⠸⡕⢡⢀⠁⠃⡛⡈⡁⡐⡜⣼
⠀⠀⠀⠀⢀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⣔⣚⠉⠓⢔⠄⡄⠈⣳⠋⠀⠀⠀⠘⠧⣼⢐⡃⡃⡇⢡⣱⡴⠋
⠀⠀⠀⣴⣿⣿⣛⡷⡄⠀⠀⢀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⠈⠳⣝⣛⢛⣿⣚⣷⠤⠾⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠛⠉⠁
⠀⢠⢚⢝⡉⠀⢉⣶⡿⡆⠀⡟⡩⠋⡥⠼⣧⡈⠉⢀⣴⣦⡌⢷⡀⠈⠈⠉⠀⠉⠀⠀⠀⠰⡒⡖⡲⠉⡟⠉⢹⠖⠘⠳⢦⣄
⠀⣇⢏⢮⠸⣼⣿⡿⣿⠁⢠⡋⢰⠺⢷⡄⠈⠻⣦⠈⢿⣿⣿⡎⣧⠀⠀⠀⢀⡖⠒⠦⣄⠀⠘⢶⠃⢀⠀⠀⠀⠀⢐⠈⠀⢘⢆
⠀⠈⠲⢽⣽⣿⣫⡞⠁⠀⢈⣶⡛⠳⣆⠙⠷⣄⠙⣦⢸⣿⣿⣧⣽⠀⠀⣀⡼⠅⠀⠀⠠⠛⣄⠈⠛⣇⠀⢠⠀⠀⡄⠀⠀⠀⢸
⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠘⢏⠙⠶⢌⣙⡓⢮⣿⣌⢣⣻⣿⣯⠏⠀⢸⠙⠌⠀⠀⠀⠀⠄⢹⠀⠀⠈⠛⣇⠀⠐⠀⠀⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣭⣭⣴⣍⣰⣇⠉⣿⡟⠀⠀⠈⠣⣀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠘⢶⣄⡀⠀⠀⠠⠀⢛⣨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠚⠓⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠢⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⠀⢠⠠⠃
⠀⠀⠀⠀⠀⢠⠔⠉⠉⠑⠆⠀⠐⠀⠈⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠁⠀⢠⣤⠆⠁⠀⠀⠀⠤⢀
⠀⠀⠀⠀⢀⠸⣄⠀⢀⣀⠜⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⡀⠠⠊⠀⠀⠀⠀⠀⠀⠀⡀⠱
⠀⠀⠀⠈⠀⠀⠈⠉⡉⠁⠀⠈⠢⠀⠰⠄⠄⠦⠀⢀⠴⡄⣀⡀⠤⠀⢠⣳⡠⠂⡠⡀⠤⠐⠋⠈
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠡⠀⠀⠀⢠⣏⣹⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⢀⠗⠊⠀⠀⣹⠆⣐⡄⠀⠀⠠⠤⠄⢔⠅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠦⠊⠀⠀⠀⠀⠀⠈⢻⣀⠠⡄⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠒⢒⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣖⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⡼⢤⡘⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠖⠁⠀⠀⠈⠐⠴⡉⠀⠈⠰⡶⠈⠐⢺⡇⠀⢠⣴⠀⠀⠠⢀⡴⠀⠀⠢⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠀⠀⠸⣧⠒⠐⠛⠠⢐⠂⡀⠀⠀⠀⠀⠃⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⣠⠀⠀⡤⠐⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢀⡀⠠⠗⠤⠀⠸⠅⡈⠾⠁
⠀⠀⢀⠄⠀⠐⠋⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⡀⠌⠉⠀⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠑⠀⠂⢄
⠼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⢐⡿⠀⠄⣴⡄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⣄⡀⠀⠈⠻⣆⡈⠀⠃⠀⠀⠛⠀⠁⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⢠⣿⠁⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠘⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⣿⠇⢸⣏⠀⠀⠸⣿⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⢻⣆⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠆⠘⣿⠀⠀⠀⡟⠀⠈⠻⠤⠀⠀⢿⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⢀⠠⣦⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢿⡆⠐⠀⢿⠃⠁⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⢠⡶⠃⠀⠹⣧⡀⠂⠈⠓⠀⠀⠀⠀⣠⣤⣴⣤⣀⠀⠀⣀⢀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠⣾⠛⠁⠀⠀⡈⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠄⢀⠁⠀⢠⣿⠁⠀⠀⠠⠂⠀⠀⠀⢐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣿⠇⠀⢸⡇⠀⠀⠀⠀⣾⠋⠁⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⡇⣴⠆⠀⠀⢿⡄⠀⠀⠀⠘⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠁⠀⠈⣷⠈⠳⠄⠀⠀⠙⠒⠀⠀⠈⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⠄⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠀⠀⠰⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠀⠀⠐⣄
⠀⠀⢰⡄⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠀⠀⠀⠀⠀⠀⡄⠀⢀⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠢⡄⠀⠀⠀⠀⠘⢆
⢠⣤⣬⣿⣼⣷⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠺⠃⠀⠀⠀⠀⢲⡄⠀⠛⠀⠘⠃⠀⡾⠁⠀⢀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠂⠤⠐⠘
⠀⣠⣾⣿⣿⡿⠶⣄⠀⠀⣠⡤⣤⣤⡀⠀⠀⠀⠀⠐⢦⣄⠀⢁⣴⣾⣿⣿⣿⣶⣄⠀⠞⠁⠀⠀⠀⠀⠀⣰⠃
⠐⠁⠸⢻⣏⠻⠄⠀⠀⣾⠃⠀⢸⣿⣷⠀⠀⠀⠀⣀⣀⠈⢠⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠶⠖⠂⠀⠀⠀⠁⠀⠀⢀⡴⡆
⠀⠀⠀⠈⣿⣄⠀⠀⢠⣿⠀⠠⣾⡿⠏⠀⠀⠀⠀⠉⠉⠁⠈⠉⠉⠉⠁⠀⠀⠀⢀⡀⠀⣀⠀⣀⣤⡀⠀⣠⣤⣤⠈⢷⣇
⠀⠀⠀⠀⠙⠿⠇⠀⠸⣿⣇⠀⠀⠀⣤⡀⠀⣤⡀⠀⢀⣴⣄⣴⣶⣆⡴⣿⡆⠀⣼⣷⢊⣿⡷⢫⣿⠇⣼⣿⣁⡾⢀⡞⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡆⠀⢠⣿⠇⠀⣿⡇⢀⡞⣿⡿⢱⣿⡟⢱⣿⠃⢠⣿⠃⣼⡿⢁⣿⠏⢰⣿⡏⠉⣠⠞⠀⣼⣿
⠀⠀⠀⢀⣤⣶⣶⣄⠀⢸⣿⣷⠀⣼⡿⠀⢸⣿⢁⡾⢸⣿⠁⣾⡿⠀⣿⡟⢀⣾⡏⠀⣿⠇⠸⣿⡴⠛⠿⡷⠞⠋⠀⣰⣿⠃
⠀⠀⠀⣾⠃⣠⣿⣿⠆⠀⣿⣿⠀⣿⣧⣴⢿⣯⡾⠁⠻⠟⠀⠻⠃⠀⠙⠛⠁⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣇⣀
⠀⠀⠀⣿⡄⠈⠛⠁⠀⢰⣿⡟⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣦⣀⣀⣀⣤⣴⡶⠛⠉⢹⡟⠀⠉⣷
⠀⠀⠀⠈⠳⣄⣀⣀⣠⡿⠋⠀⠀⠀⣶⣶⠀⠀⠀⣶⣶⢀⣤⣶⣖⠒⢶⣄⠀⠀⠀⠸⣿⣿⣿⣿⠿⠛⠁⠀⣶⣄⠘⢷⣄⡴⠏
⠀⠀⠀⠀⠀⠀⠉⠉⠀⣀⠀⠀⠀⢸⣿⠇⠀⠀⢰⣿⣯⣬⣰⣿⡏⠀⣸⣿⠀⣀⡀⢀⣀⠉⠉⠀⠀⠀⠀⣴⡿⠋⠀⢀
⠀⠀⠀⠀⣴⣶⣶⣾⠟⠁⠀⠀⢀⣿⡏⠀⠀⢠⣿⢏⣬⡉⣿⡿⠀⣰⡿⢣⣾⢃⡿⠹⣿⠀⠀⢀⣠⠴⠛⠉⠀⠀⢀⡀⠙⢷⣶⣦⣤
⠀⠀⠀⠀⠙⠛⠉⠀⢀⣤⡆⠀⢸⡿⠀⠀⣰⡿⠁⣾⡿⢸⣿⠷⢾⣅⢠⣿⣧⠞⠁⡸⢻⣷⡚⠉⠀⠀⠀⣀⣴⠀⠀⢷⣄⡀⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠀⠀⣿⠃⣠⠞⠋⠀⢰⣿⠃⣿⡏⠀⣸⣿⢻⣿⠁⢀⠄⡔⢹⣿⠃⠀⠈⠙⠛⠉⢻⡄⠀⠈⢻⣿⡆
⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣀⣀⣙⣋⣁⣀⣀⣀⣘⣿⣂⣘⣓⣒⣋⣁⣈⣛⣋⣁⣈⣓⣋⣁⣀⣀⣀⣀⣀⣀⣀⣙⣂⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣙⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⡶⠶⢶⣶⣶⣾⠿⣻⣿⣿⡟⣿⣿⣿⣿⡿⡻⣿⣿⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢾⣿⣷⡎⣿⡇⢾⣿⢻⡟⢐⣻⡇⣧⢹⡇⡇⡿⢡⢆⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⢸⠿⣫⡴⠿⢷⡈⢿⢸⡏⣼⡟⡇⣿⡘⣷⢀⠃⣴⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢰⣿⠋⣶⡿⢪⡷⢸⢸⡇⢿⢱⡇⣿⣇⠋⣾⡀⡿⣸⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣼⣿⣷⣭⣶⣿⣡⣾⣾⣿⣶⣿⣷⣿⣿⣿⣿⣷⣶⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠉⡉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡘⣇⢼⡆⡇⡏⠣⣀⡇⣇⡸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⡀⠀⡟⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣽⡆⡼⠀⡄⢠⠀⡆⡦⢆⡮⠅⠘⠢⠤⠀⡧⠺⣿⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠚⢁⡞⠀⢱⠀⠈⠃⠀⠁⠉⠁⠉⠁⠀⠉⠁⠐⡆⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢸⠀⢠⠀⠀⢰⢠⠯⠁⢸⠭⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠘⠒⠂⠀⠀⠀⠀⠈⠉⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠐⠂⠊⠂⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠁⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⢠⣶⡟⠋⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡀⠀⢠⣿⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⡇⠀⣿⡟⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⢰⠀⠀⠀⠀⣠⣴⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠀⠀⡜⠀⠀⣿⡇⠀⠀⠀⢀⠎⠀⠀⣀⡀⠀⠀⢀⣿⣿⠀⠀⠀⡆⠀⠀⣠⣾⡿⠋⠀⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⡠⠊⠀⠀⠀⣿⣇⠀⠀⡠⠊⠀⠀⠸⣿⡿⠀⠀⢸⣿⡏⠀⠀⡜⠀⠀⢰⣿⡿⠁⠀⡔⠁
⠀⠀⠀⠀⠀⠃⠀⠀⠀⢰⣿⣿⡠⠞⠀⠀⠀⠀⠀⣿⣿⣀⠔⠁⢀⠀⠀⠀⢀⣀⠀⠀⢸⣿⡇⠀⡜⠀⠀⢠⣿⡿⠁⡠⠊
⠀⠀⠀⠀⠀⢇⠀⠀⠀⣸⣿⡏⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠠⠊⠀⣿⠀⠰⣿⣿⠀⠀⠀⣿⣇⠜⠀⠀⠀⣼⣿⡧⠊
⠀⠀⠀⠀⠀⠀⠉⠉⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢀⠜⣿⣿⡃⢀⣾⡿⠀⠀⢿⣿⠀⠀⠀⣿⡏⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢀⠎⠀⠙⠿⠃⣾⣿⠃⠀⠀⠀⠻⢧⠠⠚⠹⣿⠀⠀⡰⢺⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢀⠎⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠦⠊⠀⢸⣿⠀⠀⠀⠀⠐⠒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠤⠔⠁⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢀⣴⠾⠛⠳⠀⠀⠀⠀⠀⣠⣼⣿⣶⣶⣶⣤⣤⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⡀⠀⠀⠀⠀⢠⣤⠀⢻⣆⠀⢀⣿⡏⠀⠀⢀⠀⠀⠀⠠⠋⠁⠀⠀⢀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⢢⠀⠀⢡⠀⠘⡄⠀⠀⣠⣤⡅⠀⠀⠈⠉⣼⣿⠀⠀⠀⡌⠀⠀⣀⣤⣄⠀⠀⠀⣿⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠆⠀⠀⠀⠑⠲⡦⠒⢸⣿⣿⠃⠀⠀⠀⠀⣿⣿⠀⠀⡘⠀⠀⣴⣿⠋⢨⠀⠀⠀⢿⣷⡀⡌
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⡏⠀⠀⠀⠀⠀⣿⣿⢀⠜⠀⠀⠀⣿⣏⠤⠃⠀⠀⠀⠈⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⡔⠹⣿⣇⠀⠀⠀⠀⠀⢀⠃⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⣿⠊⠠⢤⡤⠔⠻⣿⡀⠀⠀⢀⡠⠂⠁⠙⢿⣷⡀
⠀⠀⠀⠀⠀⢸⠀⠀⢻⣿⡄⠀⠀⠀⠀⡌⠀⠀⠈⢿⡇⠀⠀⠀⠀⡠⢿⣿⡆⠀⠀⠃⠀⠀⠈⠉⠀⠉⠁⢀⠔⠂⠀⠈⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡄⠀⠀⠠⠁⠀⠀⠀⠈⠻⠤⠤⠔⠉⠀⠈⢿⣧⠀⢀⠇⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠦⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠊⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⣀⡠⠔⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⣀⡀
⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣰⣶⣦⡀⠠⠄
⠰⠶⠲⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⣿⣿⡖⠐⠒⠂⠈⠉⠉
⠀⠤⠀⠀⠀⠀⠀⠠⠤⠶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠢⠽⠿⠟⠁⠀⠀⠀⠀⠀⠠⠤⠮⠤⠄⠀⠀⠀⠀⠀⠀⠤⡤⢤
⣤⣤⣤⣤⣄⣀⣀⡀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⢀⣀⡀⢀⠀⠀⣤⣤⡄⠀⣤⣤⣤⣤
⣿⣿⣿⣿⣶⣿⣿⣷⣶⣿⣷⣿⣿⣿⣿⣾⣿⣿⣿⣿⠄⠀⢤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⣤⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠹⣷⣤⣾⢿⠯⠿⠛⣿⣯⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⡄⠀⠠⢽⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣂⠀⡒⡺⠟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠴⣚⠶⠿⢦⣶⣮⣽⣿⣿⣿⣿⣿⣛⢛⣿⣶⣦⣌⣛⢷⠶⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⢤⣯⣛⣻⠿⣿⣟⣛⡛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠐⠒⠶⠒⠶⢶⣦⣴⡶⢦⣍⠉⢿⣿⣿⣿⡿⣛⢛⠉⣙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣛⣫⣾⣻⣿⣿⣿⣷⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠤⠥⢤⣄⣿⣷⣮⡥⢶⣶⣤⣬⣹⣿⣿⣿⠷⠖⠂⠙⢿⣿⣷⣶⣮⣝⡛⠿⢿⣿⣧⣶⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⣀⣀⣀⣤⣤⣍⣙⣛⣛⡛⠛⠀⠀⠀⣊⣉⣙⡛⠛⢛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⣠⡀⠀⠐⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠠⣀⠀⠀⣀⠤⠀⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⡘⠀⢀⠞⠳
⠀⠀⠀⠘⠀⠙⢦⡀⡇⠀⠀⠱⡀⠀⠀⠀⠀⡰⠉⢀⣴⣦⣄⠑⡎⢀⣴⣶⣤⠀⠈⢆⠀⠀⠀⠀⢠⠎⠀⠀⠀⣩⠔⠁⠀⠀⡇
⠀⠀⠀⢡⠀⠀⠀⠙⠃⠀⠀⠀⠘⡀⠀⠀⠀⡁⠀⣿⣿⠙⣿⡆⡇⢸⣿⡋⣻⡇⠀⢸⠀⠀⠀⠀⡈⠀⠀⠀⠀⠁⠀⠀⠀⡜
⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠱⣀⠘⢿⣿⠟⣡⠗⢌⡻⣿⠿⢁⡴⠃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠙⠢⠄⣀⡀⠀⠀⢇⠀⠀⠀⠀⠀⠈⠁⣒⡲⠈⠧⠤⠤⠏⠉⠏⠠⠄⣀⠀⠀⠀⢀⠎⠀⢀⠤⠒⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠙⠢⣄⣀⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠋⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠑⢤⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡄⠀⠀⠀⠀⢎⠀⠀⠀⢀⡠⠤⡄
⠀⠀⠀⠀⠀⠀⢀⠞⠁⣈⠑⠢⠤⣀⠇⠀⠀⠀⠀⠀⣀⣠⡤⠤⢴⠒⡎⢋⣹⠼⡟⠀⠀⠀⠀⠀⠈⠦⠔⠊⣡⠤⡄⠈⠢⡀
⠀⠀⠀⠀⠀⡴⠁⢀⠜⠀⠉⣒⣦⣄⠀⠀⠀⠀⠀⠀⠙⢗⠷⠺⠔⡖⢻⠙⣠⠞⠀⠀⠀⠀⠀⠀⢠⡴⠖⢧⠀⠀⠈⢆⠀⠱⡀
⠀⠀⠀⠀⡜⠀⢠⠎⠀⢀⠔⠁⡤⢉⣃⠀⠀⠀⠀⠀⠀⠀⠙⠲⠧⠵⠜⠊⠁⠀⠀⠀⠀⠀⠀⣀⠤⠐⠲⡀⠱⡀⠀⠈⢆⣀⠅
⠀⠀⠀⠀⠈⠒⠁⠀⢀⠎⠀⡜⠀⠀⡰⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡉⠱⠀⠀⠀⢱⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⠜⠀⠀⢠⠁⢠⠉⠉⠉⠉⠁⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⠀⠀⠀⢡⠀⢃⠀⠀⠀⠓⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠔⠒⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠿⢋⢀⠀⠀⢀⣀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢠⣦⠀⣸⣶⢸⣀⣴⡆⣁⣴⠟⢩⣿⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⠉⣿⣿⢴⣿⠁⡘⢸⣿⣀⢸⣏⡀⠉⣿⡇⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⢠⢾⣿⡟⠈⠿⠞⠀⠈⠙⣥⠘⢋⡴⢲⠈⠛⠁⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⢡⢾⣿⠇⣰⣿⣶⢲⠀⣴⡆⢀⣾⠐⡁⠀⢀⣴⠒⠘⢶⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣾⡟⠀⣿⣿⠀⢹⢹⣿⠁⢸⣧⠊⢹⣷⣸⣧⠀⢀⢢⠚⣿⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⣰⠟⠀⠀⢻⣿⡠⠃⠘⠿⠤⢻⣿⣜⣸⠟⠈⠛⠒⠁⠘⠶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡔⠲⣄⠀⠀⡀⠀⣠⣤⡀⠉⠀⣰⢎⠕⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⡆⢸⣀⣾⣿⡜⠀⣿⡇⠀⣸⡟⠀⣠⣿⠂⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠉⢸⠁⢿⣿⡇⢸⣿⠁⣼⣿⠁⢠⢹⣿⠀⡜⣿⡷⠿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢠⠎⠀⠸⣿⠇⠘⠿⠞⠘⣿⣤⠏⠀⠛⠛⡴⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠘⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢚⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠟⠉⠉⠀⠀⠀⢁⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠁⠐⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⢀⢀⠠⠀⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢀⠀⠀⠀⠐⠀⠀⠑⠈⢀⠀⠠⠀⠀⡤⢘⠢⠀⠀⠰
⠀⡘⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢟⠃⠃⡀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠠
⠀⠙⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⢄⠂⠀⠀⠀⠀⠀⠀⠑⠀⠄⠀⠀⠄
⠀⠴⠦⣖⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠿⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠈⠣⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⡀⠀⠈⠀⠈⠀⠂⠁
⠀⡆⢐⠀⠈⠀⠤⠀⠐⠀⠋⠀⡄⠂⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠘⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠂⡀⠀⢠⡀⠀⡄⠐⢀⠀⠀⠀⡐⠌⢿⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠈⠏⢂⡈⠀⠀⡠⠀⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠈⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠀⠠⡀⠀⠀⡀
⠀⢀⠀⢀⠀⠀⠀⠈⠁⠀⠀⠀⠁⠀⢀⠄⠁⠐⠄⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⡄⠊⠁
⠀⠀⡈⠉⠀⠀⠀⠀⠑⣄⠀⠀⠰⠀⡋⠀⠀⠀⡀⠐⠐⠁⠀⣐⠾⠛⠁⠀⢀⡀⠄⠀⢦⠢⣄⠀⣠⢀⠈⠈⠀⡄⠈⠓
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠑⠚⠓⠒⠂⠛⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠦⡀⢀⠀⠀⠐⢺⠗⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣡⣿⣿⡿⠁⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢹⣿⣆⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠉⠀⢀⠤⠀⠀⠀⢀⠤⠄⠀⠀⠀⠀⠀⠀⣿⣿⡿⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃⠀⢀⡀⠀⠀⣴⣾⡿⢳⠀⣠⣾⡿⢳⠀⢀⣤⡶⢾⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⡟⠀⠀⣸⣿⡆⢸⣿⡿⢄⣸⣸⣿⡿⢤⣸⢠⣿⡏⢀⣾⣿⣿⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢹⣿⣧⣤⡶⢿⣿⡇⠘⢿⣅⠴⠋⠙⢿⣇⠠⢋⠙⢿⡶⠋⠘⢿⣧⠴⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠋⠉⠀⠙⠻⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⣷⠀⠀⠀⠀⢀⡄⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⢠⣿⣿⡰⠃⠀⠀⠀⠀⠉⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣄⠀⠤⡄⠀⣌⣛⠂⠀⢼⣿⡏⠀⠀⠀⠠⠂⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⢦⣇⣀⣿⣿⠀⠀⣼⣿⣷⠚⢷⡄⣰⣿⢁⡿⠀⣿⣿⡴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⢁⡜⠀⠸⣿⣿⠀⡠⣿⣿⠇⢀⣿⠃⢿⣿⠋⣠⠞⡽⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠢⡉⠉⠉⠀⠈⠉⠈⠉⠀⣠⣀⠉⠋⠁⠘⣇⣿⠿⠕
⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⢸⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣿⣿⡷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢎⣴⣿⠟⠙⡆⠀⣿⣿⣦⠞⣿⡆⠀⣿⣿⠁⠀⢸⣷⠀⢸⣷⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢸⣿⡟⠳⢴⡧⠴⣿⣿⡏⢀⣿⠇⠀⣿⣿⠀⠀⣼⣿⠀⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⡤⠊⠀⠀⡙⠟⠀⠘⣿⠀⡰⠛⠿⠔⠊⠙⠛⠋⣿⣿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⡔⢉⣿⣿⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢴⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⡆⢸⣿⡇⠇⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠁⠘⠾⠿⠐



⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⣄
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂⡌⣫⠴⠒⠒⠒⠒⠠⡀⠠⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄⠀⠉⠉⠒⢌⡆⠂⠘⠃⢉⠉⠉⠉⠀⠂⠠
⠀⣀⠀⠰⣆⢦⠄⠈⠁⠀⢉⣾⣤⡀⣀⣴⡄⢀⠀⠈⠁⠀⠵⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢰⣇⡀⣬⣿⡒⠚⠛⠛⠴⡏⠴⠣⠄⠀⠈⠂
⠀⠀⠀⡞⣼⠃⠀⢀⠀⠴⣿⠶⣏⠙⣧⢌⠆⠰⠂⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡸⣗⡿⠛⠳⢛⢿⡷⣴⠖⠁⠀⠀⠄⠐⠂⡁⠄⠀⡂
⠀⠀⢸⡳⠎⠀⠰⢎⠂⠀⠈⠀⣼⡶⠌⠋⠀⠀⠀⠉⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⣠⣤⣶⠁⠈⣯⠞⢲⡯⠄⢐⠠⠀⠀⠀⠀⠱⠁
⠀⠀⠈⠀⠀⠀⢀⡈⠀⠀⠀⢰⣻⣇⣠⣠⣤⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠁⠀⢠⠁⠀⠠⢟⣛⠈⡶
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⢈⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠉⠀⠀⢀⡒⠀⠀⢠⣟⠁⠊⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡦⣴⠅⠷⡞⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠉⠐⠠⠄⠀⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡄⠀⠐⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⣇⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠟⠃⠒⠃⠀⠉⢀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⢄⣠⣭⣀⠈⣉⣉⢉⣛⣛⣛⣛⠛⠛⢿⣛⣿⣋⣟⣀⣀
⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⢀⠀⠈⠐⠀⠊⠀⠿⢯⠤⠤⣽⣤⡄⠀⠠⠠⠀⠀⠀⠐⠽⢬⢽⠀⠜⠀⢀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠈⠀⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢰⠂⠻⡆⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢂⠸⠄⠀⠀⠀⠀⢀⣠⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣙⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡆⠀⣠⣶⣿⣿⣷⡄⠀⢠⣾⣿⣿⣷⡀⠀⠀⣾⣿⣿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠿⠃⣰⣿⣿⣿⠉⣿⡇⠀⣾⣿⣿⠀⣿⡇⠀⢀⣿⣿⡿⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠁⠀⠀⠀⢠⣿⣿⣿⠃⢸⣿⡇⢰⣿⣿⠃⠀⣿⡇⠀⢸⣿⣿⡇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⣸⣿⣿⡟⠀⢸⣿⠀⣿⣿⣿⠀⢰⣿⡇⠀⢸⣿⣿⠇⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⣀⣠⣤⠄⢾⣿⣿⠁⠀⣿⡿⠀⣿⣿⡇⠀⣼⣿⠁⠀⣿⣿⣿⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠈⣿⡇⠀⠸⣿⣿⣇⣼⣿⠇⠀⢿⣿⣇⢰⣿⠏⠀⠀⣿⣿⣿⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣦⣾⠟⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠈⠛⠿⠟⠁⠀⠀⠀⠻⠿⠿⠟⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⢀⡀⠀⢀⣀⡀⠀⠀⣀⣄⣀⡀⠀⠀⣀⣀⣀⣀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⣿⠇⠀⣿⣿⣿⠀⢠⣿⣿⠙⣿⠀⠀⣿⣿⠛⠋⠀⠀⣼⣿⣿⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣸⡟⠀⠀⣿⣿⡏⠀⢸⣿⣿⣰⡟⠀⢰⣿⣿⠀⠀⠀⢰⣿⣿⠇⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣿⠁⠀⢠⣿⣿⡇⠀⢸⣿⣿⣿⡁⠀⢸⣿⣿⡀⠀⠀⠈⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣿⠇⠀⣼⣿⣿⢸⣿⠀⢸⣿⣿⠛⠃⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⣟⣼⠏⠀⢸⣿⡏⠀⠀⠀⣤⣀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠈⠛⠋⠀⠀⠙⠿⠟⠋⠀⠀⠘⠿⣷⣶⠄⠀⠹⢿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⡿⢶⡄⠀⢸⣿⣷⡄⠀⢸⡆⠀⢸⣿⣷⠀⠀⢰⣿⡇⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⢸⣧⠀⢸⣿⣿⣿⠀⣿⡇⠀⣾⣿⣿⠀⠀⢸⣿⣿⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⢸⣿⠀⢸⣿⡟⢻⡇⣿⡇⠀⣿⣿⣿⠀⠀⠸⣿⣿⡀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⣸⡿⠀⣼⣿⡇⠸⣿⣿⡅⠀⣿⣿⡇⠀⠀⠀⠙⠻⢿⣶⣿⣇⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣀⣿⠃⠀⣿⣿⡇⠀⢻⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣠⣴⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠟⠁⠀⠀⠻⠟⠁⠀⠀⠁⠀⠀⠛⠛⠛⠓⠀⠀⠀⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈





⣠⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⡿⠿⣿⣿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣠⣤⣼⣤⣠⣦⣤⣴⣦⣤⣾⣿⣱⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡄⠀⠀⢀⣀⣀⠄⠀⢀⣀⣀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠛⣿⢀⣾⣿⡿⣿⡆⢠⣿⡿⢻⡇⠀⢸⣿⡿⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠁⠀⠀⣼⣿⡿⠁⣿⠁⣼⣿⠃⢸⣿⠀⢸⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⣿⣿⠃⢠⣿⠀⣿⣿⠀⣼⡇⠀⣿⣿⠇⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠐⢶⡞⠀⣿⣿⠀⣼⡏⠘⣿⣏⢠⣿⠃⠀⣿⣿⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣤⠾⠁⠀⠙⠿⡿⠛⠁⠀⠻⠿⠾⠋⠀⠀⠿⡿⠿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⢰⡆⢠⣶⣤⠀⢰⣶⢤⡄⠀⣴⣶⣦⠀⠀⣴⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⣾⠁⢸⣿⡟⠀⣿⣿⢸⠇⠀⣿⡇⠀⠀⢸⣿⡏⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⠇⠀⢸⣿⡇⠀⣿⣿⣇⡀⢠⣿⣇⡀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⣸⣿⠇⠀⣿⣿⣸⡇⢸⣿⡏⠁⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠹⠿⠀⠀⢿⡿⠟⠀⠸⣿⣅⡀⠀⢷⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⡀⠀⠀⡀⠀⣠⡀⠀⠀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠙⡆⠀⣿⣿⡆⢸⡇⠀⣿⣿⠀⠀⣿⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⣿⠀⣿⡿⣿⢸⡇⠀⣿⣿⠀⠀⣿⣿⠀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⢸⡏⢀⣿⡇⢸⣿⡇⢀⣿⡏⠀⠀⠈⠛⠷⢾⣧⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⡿⠁⢸⣿⠇⠀⠻⠀⠸⣿⣧⡀⠀⠀⢠⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠋⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⣶⡾⠿⠿⠷⠆⠀⠀⢀⡀⢀⣠⣤⣴⣶⣶⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⣿⣥⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⡶⣦⣷⢛⣡⣶⣶⣶⡯⣮⣅⡀⣤⣢⣝⠿⢿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⣀⣹⣿⣧⠀⠀⠀⣀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣿⣛⠻⣿⣿⣼⣿⣿⣶⣿⣿⠛⠛⣻⣿⣿⡿⣽⡿⣿⣿⣿⣿⣷⣤⠀⡠⠄⣾⣀
⠀⠀⠀⠀⢺⣿⡋⡹⣿⣄⢀⣼⣧⡜⣟⣿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣿⣿⣿⣿⣿⢃⣄⣼⣿⣿⣏⣴⣿⣿⣜⣿⣿⣿⣟⣫⣦⠉⣼⠿⢹⣷⡆
⠀⠀⠀⠀⠈⢿⣧⢳⡙⠿⣾⣿⣿⣿⣿⡿⣾⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣊⣿⣝⣿⣿⣿⣧⣿⠆⣼⡿⠁
⠀⠀⠀⠀⠀⣈⣻⣷⣝⣀⣿⣿⣿⡾⣿⣧⣿⣿⣾⡀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣾⣿⣿⣿⠐⣨⢿⣿⣿⣟⣠⣼⢟⡁
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣶⢿⣿⣿⣿⡰⣿⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣼⣿⣧⢹⣏⢿⣯⡷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠻⢿⣯⣯⣽⠋⣙⡻⢿⣿⣿⣿⣧⡹⣿⣿⣿⣿⣿⣿⣿⡿⣱⣿⣿⣿⣿⣿⡿⣟⣿⣌⣽⣿⣭⣿⣥⣿⣭⣥⣤⣄⣀
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣶⣿⣶⣿⡿⢶⣶⣝⠷⣍⠙⢿⣿⣷⡈⢋⣙⣿⣟⣩⣋⣰⣿⣿⣿⡿⢋⡀⣻⣧⣘⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢠⣾⣿⣿⣿⡽⢿⣿⡿⠟⠀⠀⠘⢿⣿⣷⣿⣴⣦⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣴⣯⣿⣿⡿⠃⠀⠀⠉⠁⠀⠀⠀⠀⠈⠙⠻⠿⡟⢿⠇
⠀⠀⠀⢀⣿⡿⣻⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠈⠛⠃⠈⠙⠛⠻⠿⠿⠟⠛⠋⠉⠁⠛⠛⠛⠋⠁
⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢠⣤⣶⡶⠶⣤⡀⠀⠀⢒⣛⣛⣛⡻⣿⣶⣄⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣻⣿⣷⣶⣶⠶⠀⠀⠀⠀⠈⠙⠻⣿⣿⣳⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⢿⢟⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⠀⠀⠀⣾⣿⣿⡏⣿⣿⣿⠿⠋⠁⠀⢰⣤⣀⣄⣤⣤⣀⣀⣀⡶⠀⠀⠀⠉⠛⠿⢷⢺⣿⣿⣿⡄⠀⠀⠀⠀⣠⡼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠰⣿⣿⣿⣻⣭⣤⣄⠀⣠⣴⣿⣿⣽⣿⣿⣧⣿⣿⣿⣿⣿⣿⣦⣄⢀⣠⣤⣶⣮⣿⣿⡿⠁⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⡈⢛⣿⣿⣿⣿⠯⣾⣻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣷⠈⢀⣤⣾⡿⠋
⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣀⡈⠻⣷⣬⣝⡛⠿⡟⣾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⠿⣛⣻⣿⡖⠿⠟⠉⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢠⡾⠛⠛⠛⠛⠋⠙⠻⠿⠿⠷⣶⣮⣹⣿⣿⡧⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣛⣫⣭⣶⣿⢿⡿⠿⠟⠛⠻⠿⠟⠛⠛⣴⡄
⠀⠀⠛⠀⠀⠀⠀⠀⣀⣠⡄⣤⣤⣤⣤⣦⣬⣍⣛⣛⡿⠿⢿⣿⣟⣻⡿⣿⣹⣿⣿⠸⣿⣿⣿⣽⡿⣛⣵⣾⣿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⣰⠿⠿⠛⠉⠈⠉⠉⠀⠈⣉⣉⡙⣛⣻⡿⠋⠰⠋⣙⢿⣿⣿⣿⣿⣻⣷⠾⣻⣹⣿⣻⣻⣟⡛⣻⣟⠉⢉⡀⠀⠀⠉⠙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⣠⣶⣶⡆⢿⣿⠛⠿⢿⣿⡧⠿⠆⠙⠃⠩⠿⠟⠹⠿⠇⠘⠉⠻⠿⣿⡿⠿⠏⠛⣯⣾⣿⣿⣿⢶⣄⠀⠀⠈⠉⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡻⢿⡿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢟⣯⣿⣿⡄⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁

⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⢀⢀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠸⡄⠀⠀⢠⡆⠀⠀⢷⣀⣀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⠳⢄⣀⣀⣀⠀⠀⠀⠀⢧⠀⠀⠀⠀⢧⠀⠀⠀⠓⠒⢠⡄⢸⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠀⠀⠀⠀⢠⣦⣤⣭⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠁⠒⢶⣾⡿⣼⣿⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⢀⣤⣽⣿⣿⠋⠀⠠⡇⠀⠘⠲⣄⡀⠉⠓⣆⠀⡀⣠⣿⡇⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠠⡌⣿⣿⣿⣿⠀⢤⠀⠑⣄⠀⠀⢸⡇⠀⢠⣿⣿⢡⣿⣿⡧⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢹⣮⣿⣿⣿⠀⢠⡆⠀⢸⡇⠀⢿⣷⡆⣿⣿⣿⣤⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣈⣻⣿⣿⠇⠀⣾⣇⠀⠈⢷⡄⠘⣿⣇⠻⣿⣿⣽⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⢼⣿⣿⡿⠋⢠⡀⣼⣿⠂⠀⢠⡇⠀⢸⡿⢠⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢠⣶⣼⡿⠿⠟⠋⡴⢀⡜⠠⣿⣏⠀⢀⣾⠁⢠⡿⠁⣠⣾⣿⣻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡟⠋⠀⢠⠄⡗⠰⡃⠀⢸⠀⣿⠹⠀⢸⣧⡀⢸⣇⢠⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⡰⠃⠋⠀⠀⣇⢸⣆⣻⣢⠟⡤⠿⠶⠿⠧⣤⣾⡿⢃⣤⣿⠸⣧⣤⣤
⠀⠀⠀⠀⠀⢀⠊⠀⠀⢠⢸⣧⠿⠊⠉⠉⠀⠀⠀⠀⠀⠒⠒⠒⠈⠽⢿⣿⣿⠶⠿⠛⠛⠁
⠀⠀⠀⠀⠀⣧⣤⣴⣶⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⢀⠀⢀⣱⣴⣴⢛⣋⠐⠒⠢
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⣿⣿⠿⣭⡉⠉
⠀⠀⠀⠀⠀⠀⠀⡟⠈⣾⡗⣌⣷⡈⠙⠈⠀⠀⢀⣰⣿⠋⣓⠀⣠⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠘⠶⢻⡇⠈⢻⠙⠠⠀⠀⠀⡾⠁⢻⣷⡌⠸⢳⠏⣦⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠑⢤⡀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠑⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠎⠉⠠⢄⠀⠀⠀⠀⠉⠒⠦⢄⡀⠀⠈⠑⠦⡀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠓⠺⡞⠛⠛⠒⠂⠀⠀⠀⠀⠀⠀⠈⠑⠠⣀⠀⠈⠓⢄⡀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠑⢦⣀⠀⠙⢶⠀⠾⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⢐⠤⡹⣧⣠⣄⡈⠶⣬⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⣀⡤⠖⠋⠉⣀⠀⠚⢑⠻⣮⡾⠛⠓⠾⣧⡘⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠟⣹⢀⣴⠡⠜⡉⣤⣦⣾⣷⣿⣿⣿⣶⣄⠺⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠖⠟⠳⠿⠿⠾⠷⠷⠛⠁⠁⠀⠀⠐⠀⠐⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠠⠀⠀⠀⣠⠄⣸⡇⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⣸⡇⣶⠇⠠⣶⡆⢀⢴⡏⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⣿⠇⣿⠀⠀⢿⠀⠀⠸⡇⠘⣿⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⢀⣤⠦⠀⠀⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢾⠀⠀⢀⣾⠁⠀⠀⠀⠈⠀⢰⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⢠⡖⠀⢡⠀⢸⡏⢀⡀⠀⢀⡀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⣧⢠⡿⠀⣿⡅⠁⣿⣰⡏⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠸⡇⠀⣿⠁⢰⡿⠸⡇⠰⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠑⠀⠙⠀⠚⢁⡤⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢲⡄⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠂⠠⠀⣼⠇⢸⡇⠀⢠⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢰⡇⠀⠎⢹⡇⠀⣿⠀⢸⡇⠀⣿⠀⣾⡀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠁⡠⠀⠸⠇⠀⣿⠀⢸⡇⡸⠻⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⠀⠈⠉⠀⠀⡀⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠁⠀⠀⢀⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠙⢿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⣠⠶⠒⠒⠲⢤⡀⠀⠀⣀⣤⣄⡀⢘⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⠒⠋⠁⠀⠀⠉⠛⠋⠙⠻⣿⣿⣿⣄⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠶⢤⣀⡞⠙⢦⡀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⡴⠋⠉⠙⠳⣄⡼⠁⠀⠀⠀⠙⠓⠲⡄⠉⠙⠲⣤⠀⠀⠰⠿⣿⣿⡿⠟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠐⢦⠴⠃⠀⠀⡴⠋⠁⣀⣠⣤⣤⣤⠄⠀⠀⣿⠀⠀⠀⠈⢳⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⡀⠀⠀⣠⠞⠁⣠⠞⠉⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⢧⣠⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿
⠿⣍⣛⣉⣁⡴⠊⠀⣀⣀⣀⣀⣀⣤⠖⠋⠁⠀⡴⠚⠲⣄⠀⢀⣽⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠿⣿⡿⠟
⠀⢀⣨⣭⣤⣤⠴⠚⠉⠀⠀⠀⠀⠈⠙⢦⡀⠈⢧⣀⡀⣈⣩⣿⣿⣿⣿⣿⣿⣦⣄⠀⠀⣀⡴⠒⠒⠦⣀⠀⠈⠳⢤⣀⣀
⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣀⠀⠀⠙⢄⣀⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⣀⠀⠀⠈⠳⠦⣤⣤⠤⣴
⠀⢀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠿⠟⠛⠛⠶⠶⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡉⣻⣿⣿⣿⣿⣷⣶⣤⣤⣶⣿⣿
⠀⢸⣷⣶⣶⣶⠾⠟⣋⣡⣴⣶⣶⣿⣷⣶⣶⣤⣄⡈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡋⣿⣿⠿⠛⢉⣡⣤⣤⣤⣤⣤⣌⣉⠙
⠀⠀⠉⣉⣥⣤⣶⣿⣿⣿⣿⣿⣿⣿⢯⣭⣭⣤⣬⣥⣤⣤⣄⡉⠛⠻⠿⠿⠿⠟⠛⠉⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠘⠟⠛⠉⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⡿⠿⠿⠿⠿⢿⣿⠷⠶⠛⣣⣴⣿⣿⣿⣿⣿⣿⣦⠤⠤⠶⠿⠿⠟
⠀⠀⠀⠀⣠⣤⣶⣶⣿⣿⣿⣿⣶⣶⣤⣄⣀⠀⠙⠷⣶⣶⣶⣶⣶⣶⠶⠾⠿⠿⠿⠿⣿⠿⠛⣉⣤⣤⣶⣶⣶⣶⣤
⠀⠀⠀⠀⠙⢿⣿⣿⡿⠟⠛⠛⠻⣿⣿⣿⣿⣷⣦⣄⠀⠰⣶⣶⣶⣶⣶⣶⣶⠶⠟⣁⣤⣶⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⢳⣴⠶⠶⠶⢶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣬⣭⣭⣭⣭⣤⣶⣿⣿⣿⣿⣿⣿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣦⣤⣤⣀⡉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢋⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠳⠦⠄⠉⠛⠻⠿⠿⢿⣿⣿⠿⠿⠟⢛⣉⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⣶⣾⣿⣷⣦⣀⠛⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠠
⠀⠀⠀⢀⠄⠂⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠁
⠀⠠⡀⠂⠀⠀⠀⠀⡀⠀⠀⠀⢀⡶
⠀⠀⠁⠀⠀⠀⠀⠲⡇⠐⠟⢀⣾⣷⠂⠀⢠⣶
⠀⠂⡀⠀⠀⠀⡀⢀⠀⠀⣠⢞⣉⣦⡔⠀⠿⠄⠀⣀⡀⠐⠃⠀⠀⣀⣶⣄⡀
⠀⠀⡘⠁⠀⠀⢉⠀⠀⠀⠈⠀⠘⠙⠛⣽⣶⣆⠀⢴⠟⡿⠀⠀⠀⠈⠉⠛⢠⣤⡀
⠀⠀⠇⠸⠀⠐⠛⠂⠂⠀⠀⠀⠀⠀⡀⠈⡅⠁⠀⠋⠀⠀⠀⠀⠀⠀⠋⠀⢈⠙⠁
⠀⠀⠈⠀⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⡇⠀⡃⢀⡂⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠀⢰⠀⠀⠀⠀⠀⠐⠒⠁⠉⠁⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠂
⠀⠀⠀⠀⠀⡄⠈⠀⠀⠀⠀⣀⣠⣤⣶⣶⣶⣿⣿⠟⠿⠿⠯⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠐⠠⠁⠀⠀⣤⣶⣿⣿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠐⠀⠀⠉⠛⠈⠭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠁⠀⠤⣄⡀
⠀⠀⠠⠤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠀⠠
⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠒⠂⠒⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⠀⠀⠤⡄⠀⠀⠀⠀⢀
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣦⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣷⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣧⡄⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠐⡅⠈⠀⠐⠀⣀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠐⠝
⣤⣤⣤⣴⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠈⠉⠐⠂⠀⠌⠳⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⣀⣄
⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠤⢤⣄⡀⠀⢠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠚⠀⠠⣌⠹⣷⣧⡞⠁⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣆⣄⡀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢸⡘⢋⣴⣬⣦⢹⡆⣹⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠈⢻⣿⣦⠄⠂⠀⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠛⣿⢏⡾⠃⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣄⢢⡀⣹⣦⣀⣠⣿⣿⣾⣛⣁⣠⡴⠏
⠠⡄⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⠋⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠛⠉⠛⠿⠛⠛⠉
⠀⢇⠑⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⡀⠄
⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⣾⡧⢀⡀
⠀⠙⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡀⠀⠈⠀⠀⠀⠀⠸⠏⠀⢸⡿⢦⣄
⠀⠘⣦⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣧⠀⠀⠀⢰⡀⠀⠈⡄⢈⣄⡽⠶
⠀⠀⠘⢟⣀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⢀⣀⣿⡗⠠⠖⠛⠋
⠀⠈⠉⠉⢿⣧⣤⣴⠞⠉⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⠿⠃
⠀⠀⠀⠀⠈⠙⠛⣿⣤⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⡴⠀⢊
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠼⠁⠀⠁⢀⡀
⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢰⣦⠤⡄⠀⡀⠀⠀⠀⠀⠀⠀⠑⠒⠶⠦⠴⠾⢿⣷⣦⡤⠀⠀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡎⠀⣠⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣍⡉⠝⠉⠁
⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⣿⠀⠀⠀⠀⠐⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡿⠌⠑⠄⠀⠀⠀⣹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠷⠾⣤⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⣤⠤⠤⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⣴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠈⢷⡀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⣦⣈⣷⣶⡾⣃⢶
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠙⠷⠿⣥⡖⠀⠀⣠⠴⠚⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢛⠩⠍⠩⠥⠀⠩⠞⡇
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠱⣶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⠐⠀⠀⠀⣉⢹
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⡀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡆⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠀⠀⠄⠀⠀⢰⢸
⠀⠀⠀⡀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢬⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣿⣿⣿⠴⠤⢾⡶⢦⣿⣷⡷⣶⡆⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⢀⡀⠀⠸⣻
⠀⠀⠀⠁⢠⡇⡘⠀⠀⢀⣨⣔⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⡠⠺⠶⠋⠁⠀⠀⠀⠈⠀⠀⠀⠈⠳⠾⠢⡀⠀⠀⠀⠀⠀⠀⢀⡎⠉⢀⢀⠉⠳⢾⣿⠀⠀⠀⠀⡀
⠀⠀⡶⠀⢸⣿⣿⠞⠉⣅⣈⠙⡁⠀⠀⠀⠀⢀⠤⢤⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⣼⠇⠀⠀⠈⠀⠤⢼⡇⠀⠀⠀⠀⡇
⠀⠀⣿⣤⠘⣿⡇⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⡄⠀⠀⢿⡀⠠⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⡁
⠀⠀⢣⠀⢰⣿⠁⠀⠀⠀⠤⢼⡇⠀⢀⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡄⢸⣇⢀⡀⠀⠐⢀⡟⠀⠀⠀⠀⢁⡇
⡀⠀⠈⢧⠀⣿⣅⠠⠄⠀⢀⢸⡆⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⣿⠀⠀⠀⣛⡿⠁⠀⠀⠀⠘⢻
⡇⠀⠀⠘⣧⣾⣿⣄⡂⠀⡀⢠⣿⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣯⣿⣷⣭⣶⣿⠁⠀⠀⠀⠀⢠⠏
⡇⠀⠀⠀⠘⣆⠹⣿⣿⣷⣦⣿⡿⣿⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣟⠛⠛⠛⣫⠃⠀⣠⠀⣰⠟
⠸⣤⠄⠀⠀⠈⢧⡙⢿⣭⣭⣽⣾⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣭⣿⡿⢁⡀⠀⣭⡟⠋
⠀⠹⡄⠀⠀⠀⠀⠳⣄⠹⢿⣟⠿⢿⣿⡦⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⡿⠟⢉⠔⠉⢠⡴⠏
⠀⠀⠹⣦⣈⣄⣀⠀⠈⠻⢽⣯⣅⣲⣤⣿⣦⢄⡦⣤⣵⢀⠀⠀⠀⠀⣄⠀⠆⣠⡀⠠⠠⠂⠀⠀⠀⠀⠀⠀⠀⠠⠄⣹⣯⡥⢤⣴⣁⣠⠶⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠈⠻⣅⠀⢀⡈⢀⣈⣉⣽⡛⠛⡛⢷⣼⣿⣷⣯⣶⠗⠢⡠⣼⡾⢁⢸⣿⣿⠉⠙⣆⢀⠐⡠⣴⣤⣶⡯⢛⣫⣿⣏⣠⣬⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⡶⣧⡾⠁⡐⠂⠠⢹⡋⠀⠀⠈⢢⣷⣻⣼⠟⣩⣾⡟⣛⠋⠉⠁⠀⢠⡄⠀⠀⠀⠀⠀⣀⣼⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣾⣁⢌⡁⠁⠀⢒⢢⣌⣳⡹⣿⣿⣅⠣⠅⠉⠄⠛⠅⠀⠀⢀⣴⣿⣿⢏⣾⣿⠟⠛⠛⠛⠋⠙⠻⡋⠛⠻⠶⠶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢀⡐⠀⣠⡼⠛⠛⠛⠛⠛⠛⠿⠿⠿⢿⡿⣟⣿⣿⣾⣿⣾⣿⣿⣾⣧⣶⣿⣾⡿⢣⣾⣿⣷⣿⡶⠦⢦⣤⣶⣷⣟⠐⠠
⠀⠀⠀⠀⠀⠀⢾⣷⣿⠟⠀⠀⣀⣼⣯⣄⣄⢀⡀⣴⣦⣧⣿⣷⣮⣟⣻⣛⣿⣻⣛⣛⣛⣛⣭⣶⣿⣻⣇⣀⣀⣀⣠⣦⣼⡏⠢⡙⠷⣄⠀⠂
⠀⠀⠀⠀⠠⣃⣠⣿⠃⠀⠀⣰⠋⢸⡟⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠉⠉⠉⠉⠙⢷⡆⠀⡀⠘⣷⠶⠆
⠀⠀⠀⠀⣵⣿⠟⠁⠀⠀⢠⣟⢠⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣤⣴⠀⠹⡆⠠
⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⠈⣿⡯⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⡇⠀⠀⠹⣦⣥
⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⢻⣄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⡇⠀⠀⠀⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⢻⣿⣅⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠋⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠛⠁⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⣄⠀⠀⠀⣠⡤⣄⠀⠀⠀⠀⠀⠀⣼⡏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢳⣶⡆⠀⢰⡏⠀⠘⡇⠀⢠⡏⠀⠈⡇⠀⠀⣴⡟⠑⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⢀⡞⢹⡇⢀⣾⣿⠀⠀⡇⢀⣾⣿⠀⠀⡏⢀⣼⡿⠀⠀⣿⠏⠀⣰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⢀⡞⠁⢸⣿⠾⣿⡏⠁⣾⠟⠋⣿⡏⠑⢶⠟⠋⢿⡇⢀⡴⣿⡄⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⣠⣾⣿⠀⠻⢧⡴⠋⠀⠀⠹⢧⡴⠏⠀⠀⠈⠛⠛⠀⠙⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣸⡇⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢠⣿⠃⠀⢠⡄⠀⠀⣼⡟⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠁⠀⠀⠾⠁⠀⢰⣿⢁⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⠀⠀⣰⡆⠀⠀⣼⡿⠀⠀⢸⣿⡞⢀⡴⣦⡀⠀⠀⢠⣾⢟⡦⠀⣾⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⡿⠳⠤⢴⣿⡇⠀⠀⣼⣿⢀⡏⠀⣸⡇⠀⢠⣿⠗⠚⠁⠀⣸⣿⡀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⠇⠀⠀⢸⣿⠁⠀⣰⢿⣿⡼⢀⣤⣿⡧⠖⠛⣿⠀⠀⠀⣼⡧⢽⣷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠴⠏⠀⠀⠀⠀⠻⣦⡴⠃⠈⠿⠇⢯⣼⠟⠀⠀⠀⠙⠦⡤⠞⠙⠧⠼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠈⢷⠀⠀⣶⣦⠀⢠⣶⣦⠀⠀⣿⣇⡾⠁⠀⣶⡄⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡄⠀⢸⡇⢀⣿⡯⢀⡏⣽⣿⠀⢀⣿⡟⠁⠀⢸⣿⡇⠀⡼⢻⣧⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠓⢤⣾⠶⠛⣿⡇⣼⠀⣿⡇⢀⡾⣿⡇⠀⢀⣾⣿⠁⢰⠃⣿⣿⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣴⠃⠀⠀⢿⣿⠇⠀⢿⣧⡾⠁⠙⢿⣴⠟⠘⣿⣤⡏⠀⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⡴⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⣼⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣯⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⣿⡿⠷⢦⣕⠀⣀⣿⣿⢿⣍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⠇⠈⡿⠀⠀⠘⠆⢰⠟⢡⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⠀⠀⠰⡀⠀⠀⠈⠀⢸⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢶⣤⣿⡾⠦⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⢸⠀⠀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠟⠉⠓⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⣼⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⢀⣀⣄⣀⣻⣤⣴⣶⣿⣿⡿⣿⣷⣯⣷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣲⣦⣶⣿⣷⣾⣿⣾⣿⣽⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠁⠉⠲⠿⠿⠦⠬⠭⡍⢉⣉⣉⠉⠙⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⠔⠒⠀⠀⢀⣠⣤⡴⠂⠄⠀⠀⢀⣀⡠⠭⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡀⠁⠠⠤⠀⣀⣐⣲⣦⣤⣤⣤⣤⣄⣀⣤⣤⣤⣤⣤⣶⠶⠿⠛⠁⣠⢄⠀⠀⠀⠪⠹⠭
⠀⠀⠀⠀⣀⣀⢀⣀⣀⡠⠭⠵⠶⠶⢚⢛⡛⠛⠽⠟⠛⠋⠉⠉⠉⠉⠈⠉⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⣳⣷⠶⢦⠀⠀⠐⠄
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⢤⣤⣖⣲⡶⡺⠿⠿⠟⠛⠋⠀⢀⡞⠀⠀⠀⠀⠄⠀⠀⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡶⠾⠛⢛⣉⣁⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⠞⢫⡀⠀⠀⠀⢀⡠⠔⠋⡀
⠀⠀⠐⠒⢀⣉⣉⣉⣙⣉⣉⣭⠤⠤⠶⠟⠛⠛⠋⠉⠉⠈⠃⠛⣉⣀⢀⣙⣩⠽⠛⠛⠛⠛⢛⣫⠭⠉⠓⠋⠉⠒⠻⠧⠒⣒⠉⠁⠀⢀⣀⣱⠄
⠉⠁⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠴⠖⠒⡋⠉⠅⠐⠊⠉⠀⠀⢉⣠⠴⠒⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⢸⣶⣶⢷⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠒⠀⠀⠀⢀⣠⡤⠤⠒⢈⠄⠙⠂⠀⠀⠀⠀⠀⠀⢀⡤⢮⣱⣎⡿⢟⣓⣿⠉⢀⣉⡍⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠒⠒⠒⠒⠛⠛⠉⠁⠀⣠⡤⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠀⠊⠀⠐⠀⠂⠉⠰⠏⣥⠼⠃⠀⠉⠉⠁⠘⢋⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⢠⡞⠑⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡤⢀⠀⠀⢰⣀⣤⣆⡘⡤
⠀⠀⠀⠀⠀⠀⡀⠃⡔⠷⠟⢻⠁⢸⠀⠀⠀⠀⢀⣰⡀⢆⠀⣤⣖⣰⣶⣀⠐⣄⠀⠀⠀⠈⢧⡈⣦⡾⠟⡿⠉⡏⠙⠇
⠀⠀⠀⠀⠀⠀⡇⣀⠀⡄⢴⢠⣠⣾⠀⠀⠀⠘⠋⠈⠳⡜⢺⠁⢘⠁⠹⠙⢠⢃⠒⠒⠀⠀⠀⠙⠀⢥⠘⡀⡇⣤⢳⡀
⠀⠀⠀⠀⠀⠀⢟⢇⠷⠘⡆⠷⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⡆⠉⠑⠃⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠄⠀⠀⡀⠀⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡔⠤⢠⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⢀⠀⠙⡄⠈⡀⠘⠀⠀⠀⠀⠀⠀⢀⠀⠁⠈⠀⠠⢀⠀⠀⠀⠀⠀⠈⡀⠀⠈⢣⠀⠄⠂⡀⠈⢢⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠈⠂⢀⠮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⣄⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢄⠀⡐⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⡴⠁⣄⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⡈⡀⡃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢡⣶⡄⠐⠀⠀⠀⠀⠀⢀⡠⣄⠀⠀⢐⣒⣐⣒⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢤⣠⣌⢤⠆⡀⠀⠀⠀⠀⠸⣧⡞⢀⣈⣡⣄⣠⣤⣄⣀⡤⣄⡀⠀⠀⠀⠀⠾⢅⣇⣴⡗⢻
⠀⠀⠀⠀⠀⠀⡤⣀⠀⡀⢠⢤⢀⢀⡟⠀⠀⠀⠀⠀⠹⡞⠛⠛⠉⠙⠁⠉⠛⠁⠸⠽⠁⠀⠀⠀⡉⢉⢈⣁⢀⣈
⠀⠀⠀⠀⠀⠀⠀⣯⣰⠙⠈⠇⠋⠃⠃⠃⠀⠀⠀⠀⠀⠀⠀⠈⠩⠭⠭⠍⠀⠀⠀⠀⠀⠀⠀⢸⡟⢹⢸⣇⢸⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠂⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⡉⠓⢄⠀⠑⠀⢠⠀⠆⠀⡆⠸⠀⡄⢀⠏⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠖⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⠀⠀⠱⡄⢡⠈⡇⠀⡠⠴⠦⣄⠀⠌⢠⠂⢀⠂⠀⠀⠀⠀⠀⡴⠃⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⢹⠁⠀⠀⢳⠀⠣⢁⣾⠁⠈⠁⢸⠖⠒⠲⡄⠁⡴⠃⢀⠔⢠⠞⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⢸⡇⠀⠀⢸⠀⢠⢞⠇⠀⠀⠀⡏⠀⠀⠀⡼⠒⡒⢤⠀⡰⠃⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠢⠄⢸⡇⠀⠀⠈⢷⠋⣨⡴⢒⠒⣒⠲⣤⣀⠸⠁⠀⠀⢨⠟⠀⠈⢂⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢸⠀⠀⠉⠀⢠⠞⡥⣤⡏⣿⡏⣷⡖⡌⣧⡀⠀⡰⠋⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⢸⡀⠀⠀⠀⡸⡀⣷⡾⣧⠟⢷⡋⠏⣥⠬⡁⠼⠁⠀⢲⡼⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠠⠄⣇⠀⠀⠀⡗⢨⣓⣒⣒⡬⡅⣞⠙⢯⠀⣿⠀⠀⢀⠞⠠⠤⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⠔⠒⡀⢸⡄⠀⠀⠹⡌⢿⢹⣿⡿⣧⡟⠸⠾⣡⠃⠀⢠⠏⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⢁⠄⡞⠈⠂⠀⠀⠙⠲⣜⡛⠷⠛⣛⡡⠟⠁⢀⡴⠃⠤⡈⠑⠲⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠄⢧⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢀⡴⠋⢄⠀⠢⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠂⠙⢦⡀⠀⠀⠀⠀⠀⠀⠒⣲⠞⠉⢄⠈⢀⠑⠄⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠠⠉⠲⠤⣄⣀⣀⣠⠔⠁⡀⠡⠀⠣⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⡄⠀⠀⢰⠀⠁⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⢀⡀⡠⠐⣲⣷⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣯⠭⠁
⠀⠀⠀⠀⡴⠛⣻⣟⣿⣡⢾⠏⠚⣎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣷⠏⣯⡉⠀⠀⣠⣶⡒⣂⣤⣤⡀
⠀⠀⠀⠀⠀⢸⠟⠹⠓⡹⠋⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠁⡙⢀⠀⠀⠋⠘⢷⠈⠋⠗
⠀⠀⠀⠀⢀⠎⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠄⡀⠀⠑⠠⡀
⠀⠀⠀⡠⠃⠀⡠⠋⠀⠤⠀⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⡈⠑⠀⡀
⠀⠰⠞⢁⣴⠟⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠈⠗
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⢿⠿⡿⡿⣿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣿⣷⣆⠀⠀⠀⢀⠄⠀⠀⣴⡟⣹⠃⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⢠⠇⣤⠀⣼⣿⡱⠃⢀⣠⣄⠀⢀⣾⠋
⠀⠀⠀⠀⠀⣾⣁⣾⣿⣼⢹⣼⣿⡎⣿⣧⠀⡞⣠⡉⣰⣿⠟⢄⢠⣿⣯⡿⠁⢾⡏⢸⣿⡾⣯⢹⡇⣷⠃
⠀⠀⠀⠀⠀⠛⠛⠙⠛⠘⠞⠙⠛⠃⠸⣿⣸⢡⡿⡰⢹⡿⠀⣼⢺⡟⢋⡰⠁⠈⢿⡘⠛⠑⠘⠸⠗⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢸⡷⠁⢸⣇⡴⠃⠈⠓⠋⠀⣀⣠⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⣭⣭⣬⣭⣯⣿⣽⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⢦⡴⠶⣤⡴⠲⠲⠶⠶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠋⠉⠹⣿⠇⠀⠀⠀⣿⠇⠀⠀⠀⠀⢹⠍⠁⠈⠁⠉⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⢲⣿⡏⠁⠀⠀⠀⣿⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠿⠟⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⢸⣿⠁⠀⠀⠀⠀⢸⡆⢀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠃⠀⠀⠀⠈⣧⣄⡀
⠀⠀⠀⠀⠀⣠⠴⣾⡇⠀⠀⠘⡌⠀⠀⠀⠀⠀⢸⠀⠈⠃⠀⠈⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠇⠂⠀⠀⠀⣸⠿⠿⠉⢷⡀
⠀⠀⠀⠀⢰⡇⠀⠹⣧⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡀⠀⠀⠀⠈⡧⠀⠀⠀⣸⡀⠈⠀⠀⡼⠃⠀⠀⠀⠀⡟
⠀⠀⠀⢀⣯⠀⠀⠀⢱⠀⠀⠀⡀⢱⠀⠀⠀⠀⠀⣧⠄⠀⠀⠘⡇⠀⠀⠀⢸⡇⠀⠀⠀⡿⠁⠀⠐⣾⠓⠀⠀⠀⠀⣼⡿⢦
⠀⠀⢠⣟⠉⢦⠀⠀⠀⢣⠀⠀⠀⠘⡂⠀⠀⠀⠀⢸⡄⠀⠀⠀⡅⠀⠀⠀⢸⠀⠀⠀⣰⠀⠀⠀⢰⠉⠀⠀⠀⠀⠼⠋⠀⢈⡇
⠀⠀⢸⡇⠀⠈⢲⣄⠀⠠⢦⠀⠀⠀⠱⡀⠀⠀⠀⠈⣡⠀⠀⠀⣧⠀⠀⠀⣾⠀⠀⢀⠏⠀⠀⢠⠇⠀⠀⠀⢀⡜⠀⠀⢀⣼⣷⡄
⠀⠀⢨⢧⡀⠄⠲⡿⣆⠀⠉⢧⠀⠀⠀⢷⣖⠀⠀⠀⡏⡆⠀⠀⡿⠀⠀⢀⢿⠀⠀⡜⠂⠀⢀⣞⠀⠀⠀⢀⡜⠀⠀⢠⠎⠀⢀⡇
⠀⠀⠸⡆⠘⢄⠄⠈⠻⡆⠀⠀⢳⡆⠀⠘⣿⠀⠀⠀⢇⠃⠀⠀⠁⠀⠀⢸⠂⠀⣠⡇⠀⠀⡟⠄⠀⠂⢀⡞⠀⠀⡴⠃⠀⡠⠎⡇
⠀⠀⠀⣿⠳⣾⢷⡀⠀⠙⢶⣄⠀⢿⠀⠀⢹⠀⠀⠀⠸⡀⠀⠀⡀⠀⠀⣿⠀⠀⡿⠁⠀⣼⠇⠀⠀⡠⡣⠄⢀⠞⠀⣰⣯⠕⣾⠁
⠀⠀⠀⠘⣷⣈⣷⡙⢦⠀⠀⢿⣆⠀⢣⡀⠀⢧⠀⠀⠀⡇⠀⠀⠁⠀⠀⡏⠀⣸⠁⠀⣰⠏⠀⢠⡞⠁⢀⡴⢅⡠⣪⣞⣵⢴⠋
⠀⠀⠀⠀⠙⢮⡻⣝⢮⡳⣤⠀⠹⣆⠀⠹⡄⠈⡆⠀⠀⢣⣠⢀⢠⡤⢴⣁⢀⡧⠀⢰⠹⠀⢀⡏⠀⣠⢞⡱⣫⣺⠖⣫⡷⠋
⠀⠀⠀⠀⠀⠀⢹⣚⢷⣝⣮⡣⣄⠘⢷⡄⢹⣤⢿⠁⠁⠸⡅⠸⣂⠀⡾⠀⡼⠛⢲⣿⣀⣠⠋⢠⢞⡕⣫⢞⣩⢶⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⣬⡿⠿⣮⣢⣌⠿⣦⠙⡄⢣⠀⠀⡇⠰⠁⢠⠇⢠⡇⢀⡾⠉⡵⢷⣶⣗⣡⣎⣪⠭⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡶⣭⣻⢷⣬⠳⡘⣌⢇⠀⢿⢸⡇⢸⢀⣾⢀⣾⢀⣼⣵⣿⣿⣿⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⠷⢽⠿⢿⣦⡹⣞⢞⣆⢹⠸⣇⡏⢈⡟⣾⣫⣾⣿⣿⣿⠇⣡⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢲⣮⡟⣁⡀⠀⡿⣿⣯⣿⡌⡄⣹⡟⣾⡼⣱⣿⡟⠉⢡⣴⡏⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠶⠤⠽⠋⠉⠙⠳⠦⢌⣿⣿⣇⣿⣴⣿⣾⡟⢡⣷⠖⢚⣿⣛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⠷⠖⠶⠾⠗⠒⠒⠋⠉⠻⠿⠿⠟⠓⠛⠛⠛⠒⠿⠿⠶⠚

⣿⣿⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡏⢹⢀⡴⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢳⢾⡇⢸⣻⠇⠀⣇⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⢤⣀⣸⣵⠃⢸⡇⣜⣻⢀⡾⠁⣿⠈⠙⢢⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣴⠚⣉⣿⣍⣙⣶⣄⠀⠀⠀⣀⠤⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢻⣾⠁⣠⠿⡛⠲⣾⡙⣆
⠀⠀⠀⠀⠀⢴⣿⣒⣋⣉⣁⡀⠀⠉⢧⠀⠳⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣉⣿⡴⠁⢰⠛⢦⣬⠿⣾
⠀⠀⣀⣶⠋⠉⢉⣹⣤⣄⣈⣭⡽⠶⣌⠳⣤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠼⣷⠖⠓⠲⠶⣽⡄⠈
⢠⢞⡥⠞⠋⠋⠉⠀⠀⠀⠀⠀⠙⢦⡀⠙⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡇⠀⠀⠀⣰⠃⠙⣶
⠿⠋⠀⠀⠀⠀⣀⣠⢤⡴⠶⠶⠶⠦⠬⠷⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀⠑⣾⠷⠤⠤⡾⠁⠀⠀⣸
⠀⠀⠀⠀⣰⠊⢁⣠⠞⠓⢒⣒⣶⣶⣶⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢐⣽⣿⣿⣄⠤⣈⡿⠃⠀⠀⣰⠛⠒⠒⠚⡏
⠀⢀⡴⠚⣉⡿⠋⠀⣰⣞⣉⣁⣀⡤⢴⡃⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⢞⣽⣿⣷⣼⡻⠿⠟⢨⡧⠋⠀⠀⠀⣠⡇⠀⠀⠀⢰⠃
⠀⣏⣤⠞⠋⠀⠀⡜⠁⢸⠁⠀⠀⠀⡴⠙⢾⣁⣰⠃⢠⠄⢀⡄⠀⡄⢸⡙⠿⠛⢈⡗⢢⣶⡋⠀⣀⠴⢚⣉⡞⠙⠢⣤⣴⠏
⠀⡇⢸⠀⠀⠀⢠⣧⢴⡇⠀⠀⠀⢰⠇⠀⠀⠉⣿⠚⠿⠦⠾⠧⠴⠧⠬⠿⠶⠒⢻⣿⠉⠙⣣⣾⠵⠚⣉⡿⠁⠀⢀⠞
⠀⢧⢸⠀⠀⠀⠸⡀⢰⡇⠀⠀⠀⠘⣦⠤⠖⠚⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⣀⡴⠏⢀⣠⠴⠋
⠀⠈⠻⡄⠀⠀⠀⠱⡞⢧⡀⠀⠀⠀⢷⠀⠀⠀⢀⡿⠒⠒⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠚⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣙⣦⣀⠀⠈⠳⠦⡖⠋⠀⠀⠀⠀⠀⣤⣭⣭⣭⣽⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠘⠢⢤⣀⣀⣀⣀⣀⣀⡤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⠀⠀⠈⣿⠀⠀⠀⡿⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⢹⠄⠀⢸⠇⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣤⣄⣀⡀⠀⣉⡴⠶⠚⠛⠒⢦⡐⠋⠀⣀⣠⣤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣼⠏⠀⠀⠀⠀⠀⠀⠹⡆⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠋⠙⠻⣷⠀⠀⠀⠀⠐⠚⠛⠋⠁⠙⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⣤⣠⡿⠀⠀⠀⠀⢀⣄⠀⠀⣠⡀⣴⣿⠀⣠⡄⠀⠀⣀⢀⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠉⠁⠀⠀⠀⣿⠀⣼⣿⠀⠀⣿⣴⠇⣿⡴⢻⡇⠀⣼⣿⡞⣹⣇⡴⣿⠇⢀⣴⢞⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢶⣤⣤⣤⣀⠀⠼⢿⣤⠇⢻⣆⡼⠻⡿⠀⠙⠁⢸⣧⡾⠛⠿⠁⠙⠋⢸⡟⣀⣾⠋⠉⠀⣠⣿⡥⠶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⠉⠛⢿⡆⠀⠀⠀⠀⠉⠀⠀⠠⡆⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠹⢦⠴⢾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠀⠀⢸⡇⠀⠀⢠⡆⢀⣿⠀⢰⡆⢰⣿⣩⡿⠀⣼⡟⠓⠂⣴⠟⢷⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠤⠴⠋⠀⠀⠀⠈⣿⢸⣿⠀⣾⡇⢸⣿⠉⢹⡆⣿⠉⠁⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠀⣿⣇⠈⣿⣠⡾⠁⣿⠀⠀⠀⡈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠇⠀⠘⠛⠀⠻⠛⠁⠀⠛⠓⠒⠀⠛⠾⠃



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⢻⢿⠿⢿⣿⡿⠿⡿⠿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣛⣛⣿⣿⣿⣛⣻⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣺⣿⣿⣗⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠘⠇⠀⠓⠠⠟⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⠀⠀⠀⠂⠀⠀⠀⡄⠐⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠞⠟⠃⠀⠀⠀⠀⠈⡀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡲⣤⣠⣤⣄⣷⣦⣞⣿⣿⣿⣾⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⣛⢛⡿⠛⠛⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠄⠀⠀⣀⡠⠤⠄⠀⠀⣀⡠⠍
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣶⠶⠶⠶⠶⠶⠶⠲⠖⠒⠒⠋⠉⠀⠂⢤⣄⣀⠀⠂
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠁⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣤⣤⢀⣀⣠⠤⠶⠶⠖⠚⠁⡸⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⢀⣀⡀⠤⠴⠶⣚⣩⣭⣤⠤⡶⣶⠶⠶⣶⣤⣤⣀⣀⣀⣤⣤⠴⠖⣋⣇⠀⢀⠠⠀⠈
⠀⠀⠀⠐⠒⠒⠒⠒⠒⠈⠉⠉⢀⣀⣀⡤⠤⠐⠂⠴⠒⠉⠀⣀⠤⠐⠉⠀⠀⠀⠀⠉⠀⣦⣠⣤⡖⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠤⠒⠀⢀⣀⠤⠀⠂⠀⠘⠀⠀⠀⠀⠀⡠⢢⣴⡷⢖⣽⡋⢈⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠒⠁⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⣾⡇⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡈⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢧⢿⠟⢬⠋⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⣸⠹⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⢸⠆⡀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠛⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⢾⣥⣷⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⠘⠛⢻⡟⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⣨⣿⡿⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠳⣶⣎⠁⢹⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠐⢯⠁⣾⡷⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⢙⣢⣄⣽⢀⡀⣀⢴⠒⣎⠏⢙⣫⠛⠂⢏⢿⡖⡻⡗⡶⠢⣤⣤⠼⢦⣿⣴⠀⣏⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠸⣿⣄⠀⠻⣌⣽⣿⣿⣤⣮⡷⠛⠛⠙⠿⣿⣵⠠⣶⣿⠽⠧⣤⠀⠉⠟⣳⣦⠏⠁⣴⢿⠃⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠹⣧⡴⠀⢈⣯⣉⡟⣿⣿⡛⠁⠀⠀⠀⢈⡆⠀⡟⠃⠀⠀⣐⣿⣿⣆⠙⣣⣤⢶⡧⠈⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡷⠄⠀⠀⠈⣧⣿⣿⣛⣿⣾⠻⠻⢻⣦⣤⠀⠀⠀⢡⠠⠁⠀⢀⠛⢁⡿⠇⠀⢏⣘⡻⠋⠀⢀⣀⣴⡦⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⢧⡠⡄⡀⠈⣷⠶⠉⠀⠀⠀⠀⠐⠟⠿⣷⣢⣤⣤⣴⠞⠛⠛⠋⠀⠀⠀⣾⢧⢘⣿⣾⠏⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⠻⠿⠻⢇⡀⠀⠈⠁⠠⡄⢀⠀⠀⢀⣥⣾⡆⠀⠀⡀⠴⠐⣉⣁⣉⣤⢞⠉⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣔⣶⣦⣼⠷⠾⠿⠛⠛⠛⠓⢶⣴⣷⠄⢹⡇⢈⣤⣢⠌⣀⣴⠋⠉⢉⣙⠐⠾⠒⠊⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣾⠛⠁⠀⡀⡀⠀⢤⣴⠾⢦⡬⠛⠋⠉⠛⠊⠓⠒⠉⠑⠚⠻⠷⠆⠆⠘⠏⠐⠀⠀⠀⠀⠈⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢰⠟⠉⠀⠀⠀⢀⣹⠓⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢴⣆⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠉⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠋⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⢀⣀⣀⣀⣀⡄⠀⣠⣤⢴⣦⡀⢀⡖⡫⠻⣶⡄⠀⢀⡴⢿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠰⣿⡟⠁⢉⣿⣁⣴⠟⣏⠀⣿⡇⡞⠀⢧⣠⣿⣇⣴⡟⠀⠀⡿⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠈⠉⠉⣿⠏⠀⣯⠀⠈⣹⡿⠋⡇⠀⢀⣽⠟⠁⢾⣇⢀⣼⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠤⠤⠞⠁⠀⠀⠙⠤⠾⠋⠀⠀⠙⠶⠛⠉⠀⠀⠘⠿⠟⠁⠳⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣴⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢸⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⢸⣿⡆⠀⣴⡿⠀⠀⠀⣿⡡⠋⠀⠀⠀⣠⡤⡄⠀⠀⣿⣣⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣸⣿⠀⠀⠀⡿⠖⢪⣿⠁⠀⠀⣸⣿⠒⣷⣄⢀⣾⣟⡴⠃⠀⠀⡹⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠸⣿⣄⣀⡼⠁⠀⠸⣯⠀⢀⡴⢻⡇⠀⣼⡇⣸⣿⡏⠀⢀⠀⠊⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠟⠀⠀⠀⠀⠙⠓⠛⠁⠟⠻⠶⠟⠒⠁⠉⠛⠛⠁⠀⢰⣷⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣴⡖⠛⠛⠛⠛⢶⣶⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⣶⡞⠉⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⢻⣶⠶⢦⣄
⠀⠀⠀⠀⢀⣠⣴⠁⠀⠀⠈⢿⡄⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣸⡇⠀⠀⠉⠳⣦
⠀⠀⠀⣰⠋⠀⠹⣆⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢹⡆⠀⠀⠀⠈⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣼⡷⢦⣄
⠀⠀⣰⠛⣆⠀⠀⢹⣆⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠏⠀⠀⠈⠻⣄
⠀⠀⣿⠀⠸⣆⠀⠀⢿⡄⠀⠀⠀⠸⣷⠀⠀⠀⠀⣿⠀⠀⠀⠀⡿⠀⠀⠀⣿⠁⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⢿⠀⠀⢻⡄⠀⠘⣷⠀⠀⠀⠀⢻⡆⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⢰⡏⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⢀⣴⣿⢷⡀
⠀⠀⠸⡆⠀⠀⢿⡀⠀⢻⡆⠀⠀⠀⢸⣇⠀⠀⠀⣿⠀⠀⠀⣾⠀⠀⠀⣿⠁⠀⢀⣼⠃⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠙⣆
⠀⠀⠀⢳⡀⠀⠈⣧⠀⠸⣇⠀⠀⠀⠀⣿⠀⠀⢠⡇⠀⠀⣸⠃⠀⠀⣼⠃⠀⢀⣾⠃⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠈⢧⠀⠀⠸⡆⠀⣿⠀⠀⠀⢀⡇⠀⠀⣼⠁⠀⢰⠏⠀⠀⣼⠃⠀⢀⡾⠁⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⣠⣾⣇
⠀⠀⠀⠀⠘⣧⠀⠀⢻⡀⢿⠀⠀⠀⢸⠇⠀⢠⡇⠀⢠⠏⠀⠀⣼⠃⠀⢠⡿⠁⠀⠀⣠⠟⠁⠀⠀⠀⠀⢀⣠⡶⠛⠉⠀⠘⣦
⠀⠀⠀⠀⠀⠸⣆⠀⠈⡇⢸⠀⠀⠀⣾⠀⠀⡾⠀⢠⠏⠀⢀⡼⠃⠀⢠⡿⠁⠀⢠⡾⠁⠀⠀⠀⢀⣠⣶⠟⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠹⡆⠀⢿⢸⠀⠀⢀⡏⠀⢰⠇⢀⡞⠀⢀⡼⠁⠀⣰⡟⠁⠀⣰⠏⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⠀⠀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⣾⠀⠀⢸⠃⠀⡟⢀⡞⠀⢠⡞⠁⠀⣰⡟⠀⢀⡼⠁⠀⠀⢀⣴⠟⠁⠀⠀⠀⣀⣤⡴⠾⠛⠋⠉⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣷⠘⣿⠀⠀⡿⠀⣸⢁⡞⠀⣠⠏⠀⠀⣼⠏⠀⢠⠟⠀⠀⢀⣴⠟⠁⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⣀⣠⡏
⠀⠀⠀⠀⠀⣠⠞⠋⠈⣇⣿⠀⢰⡇⢠⠇⡼⠀⡴⠃⠀⢀⣼⠃⢀⣴⠋⠀⢀⣴⠟⠁⠀⣠⠶⠋⠁⠀⠀⣀⣤⡴⠶⠞⠛⠛⣿⠁
⠀⠀⠀⠀⠀⠻⡀⠀⠀⢸⣽⠀⢸⠇⡞⡼⢁⡾⠁⠀⣠⡾⠁⣠⡞⠁⠀⣠⠟⠁⢀⡴⠋⠁⢀⣠⡴⠞⠛⠉⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⣿⡇⢸⣼⣱⢣⡞⠀⢀⣴⠋⢀⣼⠋⠀⣠⠞⠁⣠⠖⠉⣀⡴⠞⠋⠁⠀⣀⣀⣤⣤⣶⡶⠶⢾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢻⣷⢸⣿⣧⡟⠀⣠⠞⢁⣴⠟⠁⣠⠞⢁⡤⠚⣡⠴⢛⣁⡤⠴⠖⠛⠋⠉⠉⠁⠀⠀⠀⢀⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣾⣿⡿⢀⡜⣁⡴⠛⢁⡤⠞⣡⣴⣫⢔⡫⠔⣚⣉⣀⣀⣤⣤⣤⣤⡤⠶⠶⠶⠖⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⠿⠚⣉⡤⠖⣫⣴⣞⣻⡽⠾⣿⠟⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⢿⣵⠶⠟⠛⠋⠉⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⠲⠤⣼⠟

⠀⠀⠀⠀⠀⣀⣤⠶⠶⢤⣄
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠘⡆
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢷⠒⠒⣲⠄
⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⣠⡆⢀⡤⣶⠀⢀⡦⢀⢴⠆⣠⢖⡿⠀⢠⡶⠛⢹⠀⠀⡎⢀⣼⠏
⠀⠀⠀⠀⠙⣷⡀⠀⠀⢀⢴⡿⠀⢀⣼⠃⠀⢸⣇⠞⣸⣧⠋⣰⠇⠀⣸⣧⠂⣿⡜⠁⣾⠃⢀⣿⠅⠒⠁⢀⠜⢀⣾⠃
⠀⠀⠀⠀⠀⠈⢿⣄⠔⠁⢼⡇⣠⢣⡟⠀⠀⢸⡏⠀⡿⠃⣰⠏⠀⠂⣿⠃⠀⠉⠀⣸⠇⢀⠸⣯⠀⣀⡴⠃⠀⢸⡏⠀⣠⠆
⠀⠀⠀⠀⠀⠀⡐⢿⡆⠀⢺⡷⠃⣾⠁⡌⠀⠀⠀⠀⠀⢠⡟⢠⠂⠀⠉⠀⠀⠀⢠⡟⢀⠂⠀⠙⠛⠁⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⢀⠎⠀⠘⣷⠀⠀⠀⠀⣿⡜⠀⠀⠀⠀⠀⠀⢸⡥⠃⠀⠀⠀⣠⠖⢦⠸⠗⠁
⠀⠀⠀⠀⡎⠀⠀⢀⡿⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣴⠏⢀⠆⠀⠀⢀⣀⠀⢀⡴⠚⢳
⠀⠀⠀⠐⡄⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢑⡀⢀⡬⠀⣸⡏⡠⠊⠀⢠⡾⠋⢉⠇⣺⡧⠒⠁
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠈⡾⠁⣼⠃⢠⡿⢊⣠⡀⢀⣿⠅⠐⠋⡰⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠁⢸⡟⠀⣸⡇⠘⠦⡷⠸⣿⠀⢀⡔⠁⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠃⠀⢸⣇⣠⢿⡇⠀⢠⠃⠀⠙⠛⠉⢠⠋⠙⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠉⠁⠘⢧⠴⠃⠀⠀⠀⠀⠀⠘⢤⡤⠟





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢔⣾⡿⣿⡿⣻⠟⢿⣿⣽⡣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⣶⡿⠿⢟⠛⢉⢥⣴⣻⡿⠻⢽⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢡⣿⡟⢡⣲⣾⣿⣮⣿⠟⠉⠁⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⣠⣾⣟⡥⢙⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⣀⢻⣿⣿⣿⣿⣀⣤⡶⡾⠒⠄⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣧⡄
⠀⠀⠀⠀⠀⠀⡠⠂⣀⣤⣿⣿⠿⠀⢐⣾⣿⣿⡿⠟⠟⠉⢁⣀⣴⣀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠐⠉⠀⠀⠛⠉⢋⡀⣄⣶⣾⣿⣿⢯⠙⠐⠊⠉⠛⠿⢿⣿⣦⣀⣀⣠⣴⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢺⣿⣧⣧⣾⣿⣿⣿⣿⣿⡻⠆⠀⠀⣠⣴⣄⣻⣿⣿⣿⣿⠟⠋⠀⠋⠙⢻⠟⡁
⠀⠀⠀⠀⠀⣠⣶⣾⣿⣛⣛⣿⣯⣭⣿⠉⢂⠁⠀⣀⣸⣿⣿⣿⣿⣿⡿⠛⠂⠀⠀⢀⡤⣴⣾⠿
⠀⠀⠀⠀⠀⢻⡿⠿⢿⣿⣿⣿⡿⣯⡉⠉⠀⠀⢺⣿⣿⣿⣿⣿⣏⠉⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⢠⣶⣦⣄⡀⠀⠅⠑⠀⠀⢀⣠⡀⣀⣦⣾⣿⣿⡟⠎⠙⠻⠷⠀⠀⠀⠀⢀⣤⣶⡿⠃
⠀⠀⠀⣿⣿⣿⣿⣷⣢⣦⣿⣼⣷⣾⣿⣷⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⣀⣚⣫⠭⠁
⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠛⠛⠉⠀⠈⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣾⠁⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢰⡆⣴⠀⢸⡆⠀⣰⡏⢐⣿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⣿⠚⡇⠀⠸⠇⠀⠸⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⡁⠀⠀⢀⡴⠒⠀⠘⠇⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠣⠀⠀⣼⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢸⡇⢠⡇⢰⡏⠐⢦⢀⡴⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠃⢸⠃⢸⡇⢀⣾⢻⡇⢀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠘⠃⠘⠃⠰⢃⣀⠁⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠐⡆⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠄⢰⡇⢸⡇⠀⣤⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠃⣿⠀⢸⡇⢸⡇⢀⡏⠰⣧⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠉⠀⢸⡇⠸⠧⠈⠑⠠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠐⠚⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡤⠊⢀⣼⣿⣛⣻⣋⣭⣭⠍⠀⠀⠀⠀⠀⠿⢶⣭⣙⠻⣻⡋⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⢀⣞⣀⣴⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠻⣶⣌⣷⣤⡘⢆⠀⠀⠀⠀⠀⣜
⠀⠀⠀⠀⠀⢸⡆⠀⠀⣧⣄⠀⢠⡏⠉⣹⣿⠿⣋⣠⣴⡶⠖⠛⠟⠿⠟⠿⠿⠿⠶⠶⢦⣤⣀⠈⠻⣿⣿⡿⠚⢓⡄⠀⠀⢨⡜
⠀⠀⠀⠀⠀⠈⢧⢦⠀⣷⢸⡀⣿⣷⣿⡟⣁⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣶⣌⠛⣦⣠⣼⡿⢀⡄⣸⠃⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠸⣏⢤⡘⣧⡳⠈⣿⡿⣧⡻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⠗⣿⣿⡿⣱⢛⣶⠇⠀⢠⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣎⢳⡝⠿⣶⣠⠿⣿⣯⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣷⡞⣻⣿⣮⡷⠛⢉⡠⢎⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣽⣓⠦⣭⣿⣿⣿⣿⣿⣦⣄⡀⢀⢀⢀⣤⠀⣠⣄⣠⡀⣠⣴⣾⣿⣿⣿⣿⢋⣭⠥⣄⣐⣠⣽⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢙⡛⢿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠵⢿⣛⠛⣛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣟⣯⠼⠟⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠾⠭⣗⣦⣕⡦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠋⠀⠀⣰⠚⣷⣷⣿⣭⣿⣿⡿⠿⠿⠿⣿⡿⡿⠿⣿⣷⣶⣶⣶⣷⣯⣀⡀⢠⣄⠈⠙⠻⢿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠁⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠙⠿⣿⣿⣷⡄⠀⠀⠀⠻⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⢀⡏⣹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣯⢽⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⢿⠟⠿⠙⠯⠹⠯⠻⢟⡻⢶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠏⠣⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠋⡝⠷⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠟⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠃⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡽⢳⣄
⠀⠀⠀⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⡀
⠀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢀⣠⣴⣶⡶⡾⣶⢶⣶⣤⣀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⡀
⠀⠀⠀⠀⢀⣿⠉⠁⠀⠀⣠⣾⡟⠿⠙⠀⠀⠀⠈⠉⠘⠻⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⢸⡏⠁⠀⢀⣼⠿⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⣾⠒⠀⠀⣾⡯⠁⠀⢀⣀⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡀
⠀⠀⠀⠀⢿⡄⠀⢰⣿⠄⢠⣾⠟⠋⠉⠉⠘⠙⢻⣧⡄⠀⠀⠀⠀⠀⢈⢾⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⡇
⠀⠀⠀⠀⢸⣟⠀⢸⣿⢠⣿⣡⡶⣶⠶⣶⡄⠄⠀⠙⣿⡄⠀⠀⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠈⢿⡆⠀⣿⣼⣿⣿⣿⣿⣦⣽⣿⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁
⠀⠀⠀⠀⠀⠈⢷⣄⠘⣿⣿⣿⣿⣿⣟⣰⡿⠀⠀⢠⣿⠁⠀⠀⠀⠀⣘⣿⣿⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣌⡻⢿⣿⣿⡛⢉⣀⣤⣼⡿⠁⠀⠀⠀⠠⢠⣼⠟⠛⠻⠶⠶⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣍⠙⠛⠛⠛⠋⠁⠀⠀⢠⡀⣠⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠷⣷⣦⣤⣦⣶⣦⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡀
⠀⠀⠀⠀⠤⠜⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣴⠂⣶⠆⣤⠤⢀
⠀⠀⢀⠀⠀⢠⣶⡶⢐⣀⡀⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡞⣿⡜⣿⡄⣿⢰⣿⢠⡾⢉⡶⡄
⠀⠀⠀⡅⠀⣿⠋⡾⣽⣿⣿⡦⡇⠀⢸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⡽⣿⣽⣷⣿⡇⣿⣼⠇⣾⢡⡟⣱⠿⡆
⠀⠀⠉⠇⠈⣿⣾⣗⢿⣿⡿⢣⠃⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣷⣿⡿⢸⢣⢟⡾⣡⢞⣩
⠀⠀⠐⠶⠛⠛⠳⢉⠀⠀⠀⠁⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⢇⣏⢏⣮⣞⣵⣯⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠃⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣹⠏⣜⣻⠿⠛⠋⠁
⠀⠀⠠⣠⣤⣶⣦⣀⣀⣤⣤⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣠⣿⣯
⠀⠀⠀⠙⢿⣟⠛⠛⠛⠛⠙⡋⢩⣙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠶⠽⠿⠿⠿⠿⠿⠿⠿⠆
⠀⠀⠀⠀⠀⠈⠙⠿⣩⠩⠉⡭⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣢⣄
⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⣈⠁⣽⡄⠀⠀⠲⢦⣤⣤⣀⣀⣸⣿⣄⣀⣤⣤⣀⠔⠂⠀⢠⣿⣯⣿⣶⡄
⠀⠀⠀⠀⠀⣾⣏⣻⡀⠀⠀⠀⠀⠀⢠⡿⠘⠙⣷⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⠿⠟⠉⠀⠀⠀⠀⣸⣿⣿⢿⡻⣿⡆
⠀⠀⠀⠀⣸⣿⣿⣧⣧⡀⠀⠀⠀⠀⣠⠀⣼⢢⣿⡆⠀⠀⠀⠀⠀⢀⣾⡿⠻⣿⣧⡀⠀⠀⠀⠀⠀⣿⣿⣏⡅⢎⡿⣿
⠀⠀⠀⢀⣿⣿⣿⠛⠼⣳⠀⠀⠀⢠⡿⠑⠈⢂⣿⡇⠀⠀⠀⠀⢠⡿⠛⠀⠀⠈⠻⣷⡀⠀⠀⠀⠀⣿⣿⡷⠜⡡⢞⡿
⠀⠀⠀⣼⣿⣿⣷⠛⡲⣿⡆⠀⠀⠨⠀⡀⠆⡿⢁⣷⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⢹⣿⣿⣣⢑⢹⡇
⠀⠀⣰⣿⣿⣿⣁⣼⠳⣄⣿⠀⠀⢈⣀⠦⠻⣹⡯⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⣬⠹
⠀⢠⣿⣿⣿⣿⣧⡄⠻⣀⢽⠆⠀⠈⠄⣘⢴⣿⡗⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣟⡷
⠀⢸⣿⣿⣿⡗⠒⢉⠗⠜⠺⡄⠀⠀⠠⠟⣿⣿⡷⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛
⠀⠀⠙⠋⠿⣿⡷⠺⡷⠚⠓⠁⠀⠀⠀⢰⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣷⣄⠙⣿⡆⠟⢁⣴⣾⣿⠿⢿⣿⣶⣤⣀⠀⠀⠀⣤⣶⡿⠿⠿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠀⠀⠀⢸⣿⡇⠘⠃⠰⣿⠟⣉⣀⣀⠀⠈⠙⠛⢿⣿⣶⣴⣿⣯⣤⣴⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⢸⣿⠇⢀⣀⣀⣀⠀⣡⣶⣿⣷⣆⢠⣼⡿⠛⣿⣧⠀⠀⢀⣠⣈⣽⡿⠛⠛⠛⠋⠁⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⡀⠹⢻⣿⠟⠁⢿⣿⣀⣿⡿⠀⣿⡇⢠⣿⡟⢀⣾⡿⠛⣿⣿⠃⢠⣴⡆⢠⣿⡄⠈⢀⣴⠆
⠀⠀⠀⠀⠀⠀⣶⣱⡀⢻⡇⠀⠘⣿⣧⠀⣾⡿⠀⠀⠘⠻⠟⠛⠁⠐⠻⣷⣾⡿⠁⣾⣿⣠⣾⡿⣿⣦⣾⡟⠁⢸⡟⠀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠉⠀⠀⠘⣿⣷⣿⠃⠀⣠⣄⠀⢀⣠⣤⣀⡀⠀⠀⠀⠀⠘⠛⠛⠉⠀⠉⠋⠋⠀⠀⠚⠤⠞⠛⢡⣀
⠀⠀⠀⠀⠀⠀⠀⠈⣙⣛⠋⢀⣄⡀⠈⠙⠁⣰⣿⡿⠋⠀⢸⣿⠛⠿⣿⣆⠀⠀⠀⣀⣀⡀⠀⠀⣀⣶⡿⣿⣦⠀⠲⢦⣤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠘⢻⣷⡀⠀⢰⣿⠏⢴⡶⠀⣸⣿⠀⠀⢸⣿⡆⢰⣾⡟⢻⣿⡄⠀⣿⡟⠀⠘⣿⡇⠀⡀⢻⣷⣷⡀
⠀⠀⠀⠀⠀⠀⣠⣾⣤⣼⣦⡀⠈⢿⣷⢠⣿⡟⠀⣴⣦⠀⣹⣿⠀⣠⣾⡿⠁⣾⣿⣶⣾⣿⠃⠀⣿⣇⠀⠀⠛⠁⣾⣿⠈⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⢿⣷⡀⠘⣿⣿⣿⠁⠀⣿⣿⠀⢸⣿⣿⣿⣿⡄⠀⣿⣿⢉⣽⡁⠀⠀⢿⣿⡄⠀⠀⣼⡿⠃⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⠈⠁⠀⠹⣿⡇⠀⠀⣿⣿⠀⢸⣿⣤⣴⣿⡇⠀⠸⣿⣿⡿⠁⢠⣾⡞⣿⣧⠀⠠⣤⣦⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⣠⣦⠀⠀⠁⠀⠀⠀⠙⠁⠀⠀⠛⠋⠙⠁⠀⠀⢀⠈⠉⠀⠀⠸⢿⣧⣿⡟⠀⢰⣦⡙⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⣿⠀⢠⣾⡿⢿⣷⡄⠀⣴⣆⠀⣀⣀⠀⠀⢰⣿⡇⠀⢠⣴⡄⠉⠉⠉⠀⠀⠀⣿⡌⡄⠻⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⠃⠀⠘⣿⣦⣿⣟⣀⣼⣿⣿⠀⢹⣿⣴⣿⢿⣿⠀⢸⣿⠇⠀⣿⣿⠀⠀⠀⠀⣠⣤⡄⣿⣷⡇⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣞⡢⡀⠀⠈⠛⣿⡿⠿⠻⣽⣿⠀⠹⣿⡿⠁⣼⣿⠀⢸⣿⠀⠀⣿⡇⠀⣠⣴⣿⣿⡿⠀⣿⣿⠇⠘⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡄⠀⠐⢿⣷⡀⣰⣿⠇⠀⠀⠛⠁⢀⣿⡟⠀⢸⣿⠀⠀⠻⣿⣿⠟⢫⣼⣿⠃⠀⠻⠋⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣦⣄⠈⢿⠿⠿⠋⢰⣦⠀⠀⠀⠸⠿⠃⠀⢸⣿⡇⠀⠀⠀⠀⠀⣾⣿⠃⢀⣴⣶⣴⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠠⢀⣠⣴⢀⡄⠰⠂⣀⣤⣶⣶⣶⣤⣄⣀⠈⠉⠀⣤⡀⠀⣠⣿⡿⠁⠀⠉⠀⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡿⠃⣾⡇⠀⢲⣿⠋⠉⠉⠉⠛⠻⢿⣿⣶⣤⣉⣠⣾⣿⠟⠀⣰⣶⡆⢹⣦⠈⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠯⠔⠁⠀⣿⣧⠀⠸⣿⣦⣀⠀⠀⠀⠀⠀⠀⣹⣻⣿⣿⣿⣷⣶⣾⡿⠏⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⠀⠀⠈⠙⠿⣿⣷⣶⣶⣶⣿⠿⠛⠉⠀⠉⠉⠉⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⡏⠭⣟⣛⣛⣛⡛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⣋⡴⠞⣡⡴⠋⠁
⠀⠀⠀⠀⠀⠠⣶⣋⠁⠀⣿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣶⡿⠴⣾⡯⢴⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣬⡙⠶⣌⠓⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠈⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⡇⠀⠀⢀⣠⣾⣿⣧⠀⠀⠀⣠⣾⣿⣷⡀⢠⣾⣿⡿⠿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⢠⣾⠟⠉⠈⢿⡆⠀⣾⡟⠉⠀⢻⡇⠈⠁⢸⡇⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢏⣴⣶⣦⡀⠀⣼⡏⠀⠀⠀⣾⡇⢸⣿⠀⠀⠀⢸⡇⠀⠀⢸⣧⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠈⠛⢛⣿⡇⠀⣿⡇⠀⠀⠀⣿⠃⢸⡇⠀⠀⠀⣾⡇⠀⠀⠸⣿⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣼⣿⡇⠀⢻⣧⠀⠀⣸⡿⠀⢸⣿⠀⠀⢠⣿⠁⠀⠀⠀⣿⠀⢀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⡿⠋⢸⡏⠀⠈⢿⣶⣶⡿⠁⠀⠀⢻⣷⣶⣿⠃⠀⠀⠀⢠⣿⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠈⠻⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⣄⠀⠀⣀⣤⣤⣤⠀⠀⣀⣤⣤⣴⣤⠀⠀⠀⣠⣤⣶⣦⠀⠀⠀⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⣼⡟⠀⠸⠛⢿⣿⠋⠀⠘⢻⣿⠉⢈⣿⠀⠀⣿⠟⠋⠉⠀⠀⢀⣾⡿⠛⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⢠⣿⠃⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⣀⣾⡇⠀⠀⣿⡀⠀⠀⠀⠀⢸⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⣼⡟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡿⠿⢿⡆⠀⣿⣷⣶⣾⠆⠀⠈⠻⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣰⣿⠃⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⣿⡏⠉⠀⠀⠀⠀⣀⠉⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⣀⣸⣿⣦⠀⠀⠀⣿⣇⣠⣾⠃⠀⢿⣇⣀⣤⣶⠄⠸⣿⣀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠸⠿⠿⠛⠁⠀⠀⠀⠙⠛⠛⠁⠀⠀⠸⠿⠿⠛⠉⠀⠀⠻⠟⠛⠋


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⡀⠀⣿⣧⠀⠀⠀⠀⢰⡆⠀⣿⠀⠀⠀⠀⠀⠀⢰⣧⡀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⢻⡇⠰⣿⣿⣧⠀⠀⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢸⡇⢸⣿⠀⢿⣇⠀⠀⣿⡇⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⠇⢸⣿⠀⠘⣿⡆⠀⣿⠁⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏
⠀⠐⠀⠀⠀⠀⠀⢸⣧⠀⠀⣰⡿⠀⢸⣿⠆⠀⠹⣿⣀⣿⠀⠀⣿⡇⠀⢀⣀⣤⡀⠀⠀⠀⣸⡿
⠀⠐⠀⠀⠀⠀⠀⠀⢿⣷⣶⡿⠃⠀⠀⠿⠀⠀⠀⠻⣿⣿⠀⠀⢻⣷⣾⣿⠿⠛⠁⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠏
⠀⠀⠀⠀⠀⠀⣤⡀⠠⣄⡀⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣌⡙⢦⣍⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠶⣤⣀
⠀⠐⠀⠀⠀⠀⠀⠀⣠⡼⢛⣩⠟⢋⡽⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⣠⡴⠟⠁
⠀⠰⠀⠀⠀⠠⡴⠛⢡⠶⠋⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣴⣧⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⣴⣿⢿⡿⠷⢦⡄⠀⣨⣿⡿⢿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠇⠀⡿⠀⠀⠘⠀⠸⠋⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠐⡀⠀⠀⠀⠀⡀⠀⠠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⣶⣿⡟⠂⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠞⠀⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣧⠀⣆⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠑⢦⣄⢠⣤⣤⣄⣿⢶⣴⠟⣽⣿⣿⣿⣿⣶⣿⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⣶⣽⣦⠾⠿⠿⠿⡿⣿⠷⣿⣿⠿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⠂⠀⠤⠄⠀⢉⡉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠒⠒⠃⠁⠀⣀⣤⡤⠒⠚⠐⠂⠀⣀⣤⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⣀⣤⣤⣤⣭⣽⣶⣶⠶⠶⠶⠤⠶⠶⠶⠶⠾⠖⠛⠋⠁⠀⠰⢀⣀⡀⠀⠑⠘⠂
⠀⠀⠀⠀⠀⠀⠤⠤⠴⠖⠒⠚⠛⠉⠭⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣄⣤⡽⠟⠉⡧⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⠶⠶⠶⠒⠒⠚⠒⠒⠛⠛⠛⠉⠤⠰⠖⠉⠉⠉⠉⠁⠀⠀⣀⡤⠃⠀⠀⠀⠀⠀⢀⡠⠚
⠀⠀⠀⠀⠀⠀⠤⢤⣤⣀⣠⠤⠶⠶⠛⣛⣩⣤⣤⣶⠶⠶⡶⢶⠿⠶⠖⢾⡶⣦⣤⣤⣤⣠⣤⣠⡤⠴⢶⡾⣏⣁⣲⡄⠀⣀⠠⠄⠂⠁⠠
⠀⠠⠀⡤⠠⠤⠤⠤⠴⠖⠒⠒⠒⠂⠉⠉⠁⠁⠀⠀⠀⣀⣀⡬⠤⠄⣒⡤⠖⠒⠉⠀⠀⣀⠠⠒⠋⠁⠀⠀⠀⠈⠉⠉⠀⣶⡀⣀⡀⣤⡤⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠒⠒⠚⢋⡉⠁⠀⠀⠀⠀⠀⢀⡀⠀⠠⡖⠋⠁⠀⠀⠀⠀⠀⠀⡠⢀⢠⣠⣀⠸⣿⠏⠿⠷⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⡄⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⢀⣠⡤⠖⠐⠉⠁⠀⠀⠀⠠⠀⠀⠀⢀⠀⠀⠤⠊⡴⠾⢷⡛⠒⠾⡧⠀⣲⡾⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠁⠙⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣽⣽⣭⣯⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢠⡿⠁⠀⠀⠀⠂⠀⠀⠀⠀⠀⣰⣿⠃⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⡇⠀⣿⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⡀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⡰⠁⢠⣿⡁⠀⠀⢀⠂⠀⣀⣀⠀⠀⢸⣿⠁⠀⢀⠃⠀⢠⣾⡟⠁⢨
⠀⠀⠀⠀⡀⠀⠀⠀⣸⣿⠇⢀⠊⠀⠀⠸⣿⡇⠀⡠⠁⠀⠀⠻⠟⠀⠀⢸⣿⠀⢀⠂⠀⢠⣿⡟⠀⡠⠁
⠀⠀⠀⠀⡄⠀⠀⠀⣿⣿⠐⠁⠀⠀⠀⠀⣿⣧⠊⡠⠀⣄⠀⣤⣤⠀⠀⢸⣿⢀⠂⠀⠀⣾⡿⡠⠊
⠀⠀⠀⠀⠀⠂⠀⠒⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣰⡟⠀⢹⣿⠀⠀⢸⣿⠂⠀⠀⢰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠌⠈⢿⠷⣰⣿⠃⠀⠀⠻⣇⡀⠘⣿⡀⠀⡠⢺⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⢀⠌⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⠧⠊⠀⢸⡇⠀⠀⠀⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠐⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣴⠟⠉⢡⠀⠀⠀⢀⡴⠾⠿⠶⠶⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⡄⠐⡀⠀⠀⠀⠺⠆⠘⠧⠀⣼⡟⠀⠀⠈⠀⠀⠀⠁⠀⠀⢠⡔⠐⠄
⠀⠀⠀⠀⠀⠀⢰⣿⠡⠀⠀⠑⠄⣀⡀⢠⣾⡿⠀⠀⠀⢀⣿⡇⠀⠠⠁⠀⣴⡾⠓⠀⠀⢸⣧⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⢸⣿⡇⡠⠁⠀⢸⣿⣁⠌⠀⠀⠈⢻⣷⣄
⠀⠀⠀⠀⠀⡔⢻⣿⡀⠀⠀⠀⠀⡀⠀⢸⣿⠀⠀⠀⠀⠸⣿⡗⠡⢤⡠⠘⢿⡅⠀⠀⣀⠔⠈⠙⢿⣦
⠀⠀⠀⠀⠈⠀⠈⢿⣧⠀⠀⠀⢠⠁⠀⠈⢿⡀⠀⠀⢀⠔⣿⣧⠀⠀⡃⠀⠀⠉⠀⠉⢀⠔⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⢀⠂⠀⠀⠀⠀⠙⠒⠈⠀⠀⠘⢿⣆⡠⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠸⢀⣠⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢢⠀⠀⠀⠀⠀⠀⠈⠄⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⣠⣤⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣴⣾⡿⣿⣶⣄⠀⠀⠀⠀⠀⢠
⠀⢻⣿⠀⢀⣾⣿⣿⣿⣦⠀⠀⣴⣿⣿⣷⣦⠀⠀⣿⣦⡀⠀⢀⣿⡟⠻⣇⡿⠛⣿⡆⠀⢤⣀⣀⡜⣇⡀⣀⡀
⠀⢸⡇⠀⠘⣿⣿⣿⢿⣿⠂⠀⠛⠛⠉⠛⠛⠀⠀⠻⣿⣿⠀⠘⣿⣿⠟⣹⡙⢿⣿⡇⠀⠀⠈⢳⣺⣓⡋⠁
⠀⠀⠃⠀⠀⠈⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⠿⣿⣧⣿⡿⠟⠁⠀⠀⢠⠓⠁⠙⠣⡀

⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⢀
⠀⣾⣿⠂⠀⣠⣤⣤⣤⡀⠀⠀⢀⣤⣤⣄⡀⠀⠀⣷⡀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣸⡆
⠀⢿⣿⠀⢸⣿⣿⣿⣿⣿⡄⢰⣿⣿⣿⣿⣿⡀⢀⣿⣿⣦⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠓⠶⣶⣿⣷⣶⠶⠒
⠀⠸⡏⠀⠈⠿⣿⡿⠛⠿⠇⠀⠉⠻⣿⠟⠉⠀⠀⠙⠻⡿⠀⠈⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣾⠿⢿⣇
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠛⠉⠀⠀⠀⠚⠁⠀⠀⠙⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠶⣤⣄⣀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡤⠰⠖⠒⠂⠀⠀⠀⠀⢠⣔⠂⠀⠒⢯⣿⣿⠃⠈⠙⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣻⠯⣶⣿⠿⠟⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣀⡀⠹⡄⠀⣠⠖⠁⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠙⢦⡄⢼⡁⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠣⣄⠈⠓⠋⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣦⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⡀⢀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠀⠀⣀⣼⠋⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠚⠀⣻⠀⣼⠡⣴⣉⠠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠾⠶⠄⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠈⠉⠛⠳⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⣾⠁⠁⠀⠀⠀⠀⢻⣷⣿⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⢠⡄⠀⠀⡄⠀⠀⠀⠀⠀⢼⡷⣦⠀⠁⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠰⣿⣿⡤⢤⣀⡀⠀⠀⠀⣤⡴⡆⠀⠀⠀⠀⠀⠀⣿⠇⠀⡀
⠀⠀⠀⠀⠀⠈⣟⠆⠀⠹⡀⠀⠀⠀⠀⢸⣧⡿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⢄⠀⠈⢻⣆⠀⠀⣿⣼⠃⠀⠀⠀⠀⣠⣾⠏⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠙⠦⣀⠀⠀⠐⣿⣽⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠸⠏⢠⣿⣿⣿⣆⣀⣠⣴⡿⠟⠁⠀⠰⠏
⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠉⠚⢧⡉⠻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣟⣿⠿⠋⠉⠉⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠤⣀⡀⢀⠀⠄⠈⠒⠘⠛⢶⣄⠀⠀⠀⠀⢦⡀⠀⠀⢀⡀⠀⠀⣠⣴⣶⡿⠋⠉⠉⠀⠠⢄⡄⠀⢀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⢉⡙⠛⠓⠒⠂⠉⢉⡿⣿⣶⣶⣦⣾⡿⣿⣶⣿⣷⣾⣿⣿⣿⡿⠉⠑⠒⠚⠒⠒⠚⣭⡉⢀⡀
⠀⠀⠀⣤⣤⣤⣤⣤⡀⠀⣈⣥⠤⠤⠛⠶⠶⠶⠖⠒⠂⠐⢻⣿⣼⢿⣿⣿⣶⣿⣻⣶⣿⣿⡿⣿⣿⣿⣒⡴⠶⠦⠤⠤⢤⠆⠄⢀⡉⠁⠀⠀⣲⣶⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⡠⠀⠀⠈⣧⢀⣠⢴⠊⠀⠀⠀⠀⣸⠖⠉⠀⠁⠀⠉⠉⠁⠈⠠⣄⠀⠀⠀⠉⠑⢠⣤⣤⣼⠀⠀⠲⣤⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠙⠄⠀⠀⡔⠉⠀⡼⠀⠒⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠀⢧⡀⠀⢠⡈⠀⠰⠆⠁
⠀⠀⠀⠀⠀⠀⠀⣴⡇⠀⠀⠀⠀⢀⠜⠋⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠙⠃⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⣦⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁