Cheer up Emojis & Symbols

⠀⠀⠀⠀⠘⠐⣂⠒⢀⡶⢦⠀⠀⣀⣀⠀⢀⡴⢦⠀⢠⣤⡀⠂⢠⠾⣶⡖⣶⠆⢠⡖⢺⡆⢀⡶⣄⠠⠁⢰⡟⣷⠀⣠⠶⠶⣆⠠
⠀⠀⠀⠀⠀⡞⠉⠳⡌⣧⠘⡇⠐⣇⣸⠃⠘⠦⠞⠀⠸⠼⠃⠂⠌⢣⡙⣰⠃⠉⠀⢧⢸⣶⠏⣠⡼⢋⡐⢸⣧⠟⣼⠃⣀⣴⠃⢄
⠀⠀⠀⠀⠀⠻⣄⡀⠹⡜⣧⡇⠀⠄⠀⣀⣤⡶⠾⢿⢧⡶⠶⢶⡆⠀⠙⢋⣤⠶⡇⠘⣎⡥⠞⠁⠸⣯⠿⠀⠋⠸⠟⣻⡭⠥⢤⡄
⠀⠀⠀⠀⠀⠀⠌⠉⣛⣛⠈⢁⣤⡶⠟⠋⠁⠀⠀⣾⢸⡇⠀⢸⣧⡴⠞⠋⠀⢀⡇⠂⠉⣡⣾⡇⠁⣶⠛⢛⠛⠳⣦⡙⠛⠶⠶⠏
⠀⠀⠀⠀⠈⢠⠞⢋⡽⢋⣴⠟⠁⠀⠀⣀⣀⣀⣀⣿⠸⡇⠀⣿⣿⠀⢰⠶⣤⣬⣇⣠⠞⠁⢸⡇⠀⣿⠀⣿⠙⢧⠈⢷⠰⣟⡇
⠀⠀⠀⠀⠀⠻⠖⢋⢠⡟⠁⠀⣠⠾⠛⣫⣭⣭⣽⠉⠀⡇⠀⡿⢸⠀⣾⠀⠀⠀⢸⢣⡤⠴⠾⠓⠀⣿⠀⣧⣤⠾⠀⣾⠀⠀⠸⣿⠂
⠀⠀⠀⠀⢸⣙⡇⡄⣼⠁⠀⣸⠏⠀⠄⢿⠀⠀⢸⣇⣠⠇⢀⡇⢸⡇⠻⢺⡇⠀⢸⣿⣧⣤⠀⠀⠀⣿⠀⠀⠀⣠⡼⢋⣤⣼⢳⣦
⠀⠀⠀⠀⠀⢀⣤⡀⢿⡀⠀⢿⡀⠠⠠⢸⡆⠀⠈⣉⣠⣤⠀⣷⠘⡇⣴⠾⠇⠀⢸⣧⡶⠿⠀⠀⠀⣿⠀⣾⣎⢿⡄⠘⣏⣿⠏⠁
⠀⠀⠀⠀⠀⠘⠶⠃⢸⣇⠀⠸⣧⡀⠐⠀⣷⠀⢸⣏⣀⣸⡄⢿⠀⡇⢻⣀⣤⣤⣸⡿⣷⠴⠶⢦⡄⣿⠀⣿⢿⡈⢿⡀⢘⣃⣀⣶⡄
⠀⠀⠀⠀⢠⣦⣠⣤⡀⢻⡆⠀⠈⠛⠷⠶⠿⠶⠿⠛⠋⢻⣇⣸⡇⠳⣦⣄⡀⢸⠉⠷⠶⠶⢦⣼⣧⣯⣀⣿⠘⣧⣈⣷⠈⢯⢛⡼⠁
⠀⠀⠀⠀⢻⡘⣿⣸⠃⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⣤⡶⣬⣿⢛⣀⣀⣀⣀⠀⡴⠶⣾⠉⡉⠀⠈⠉⢁⣀⡈⢿⠁
⠀⠀⠀⠀⠀⠙⢦⡏⢸⣻⠇⠈⠉⠙⠛⠛⠛⢿⠛⢻⢟⡿⢶⡄⠀⢸⡀⠀⣿⡟⢩⡭⣭⡙⣧⣿⠀⡿⠐⢀⣠⠶⠛⠉⠉⠻⣦⠐
⠀⠀⠀⠀⢸⣛⡆⠁⠀⣀⡴⠶⠶⠶⣤⣄⡀⠸⣆⣸⡿⡇⢸⡇⠀⢸⡇⢸⣿⡇⠸⠧⠾⢷⡟⣿⢀⡇⠰⠿⠷⠶⢶⣦⣤⣴⡿⠈
⠀⠀⠀⠀⢀⣭⣠⣤⡤⣏⠀⠀⠀⠀⣀⣈⣙⣷⣦⣄⠀⣿⠈⣧⠀⣸⠇⣾⢹⠁⣤⡤⠶⠟⠁⠿⢸⡇⢰⣶⡶⠶⠶⢶⣤⣄⣾⣿⡆
⠀⠀⠀⠀⠙⣎⠟⡼⠁⠈⠛⠛⠛⢛⣉⣉⣉⣡⣭⣥⠀⢻⡄⠈⠛⠋⢠⡏⣾⠀⣿⡄⡴⠶⠀⢲⠾⠃⠶⠙⠻⢶⠀⠀⠈⠻⠆⠉
⠀⠀⠀⠀⠀⠘⢾⡃⠀⣠⠶⠛⠛⠉⠉⠉⣡⣴⠟⣩⡶⢈⠳⣄⣀⣠⠞⠀⢿⣤⣼⠏⠛⠫⠄⢯⡿⢂⡟⠻⣆⠈⠛⠛⠷⠟⣡⢦⡀
⠀⠀⠀⠀⠈⠐⣀⠀⣾⠁⠀⠀⠀⢀⣤⠞⣉⡴⠞⣹⢃⣈⠐⠀⠀⠀⣀⠀⣀⣀⣤⡶⢀⡃⢡⡤⣤⢸⡇⠀⠘⡇⢁⠀⠁⠀⠙⠛⠁
⠀⠀⠀⠀⠀⢾⣿⠇⠙⠳⠶⠶⠛⠋⢡⡞⠉⢀⡼⠃⠿⡽⠃⠸⠶⠟⠋⠀⠘⠉⠉⢀⡾⠇⠘⢦⣼⠋⢷⣤⡼⠃⣠⢶⣤⡟⢛⡿⠂
⠀⠀⠀⠀⢠⠀⠐⠆⢰⣷⣼⢛⡷⠀⠸⠦⠶⠋⢰⣿⣆⣀⠀⢀⡀⠀⠀⠙⢋⣠⡴⠟⠀⣠⠾⣦⡟⢻⡇⢀⣀⠀⠳⣄⢻⢇⡾⣁
⠀⠀⠀⠀⠀⣴⢻⡆⠈⢳⣄⠟⠀⢰⡞⣶⠀⠂⣼⣹⣯⡿⠄⠘⠛⢛⡿⢻⡟⠉⠀⠆⠠⠈⠳⣌⢁⡿⠀⣾⠉⢻⠄⠙⣦⣼⢃
⠀⠀⠀⠀⠀⠈⠋⠁⠁⠀⠉⠀⠀⠈⠉⠁⠀⠀⠉⠁⠀⠐⠀⠒⠀⠻⠞⠋⠀⠈⠁⠀⠀⠀⠀⠈⠻⠃⠀⠘⠛⠋⠀⠀⠘⠃

⠀⠀⠀⠀⠀⣰⡊⠉⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢢
⠀⠀⢀⡴⢁⣀⠀⠉⠢⡀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠳⡀
⢀⡰⢅⡀⠀⠀⠙⢦⠤⠓⠒⠓⢤⠀⠀⠀⠀⣀⡠⠤⠤⠤⣀⣰⣅⠀⠀⠀⠀⣈⠇⠀⠀⠀⠀⣀
⡎⠀⠀⠈⠑⢄⢰⠁⠀⢀⡄⠀⢸⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⢀⠜⠓⠒⠒⢤⡴⠋⠀⣱
⡇⠀⠐⢄⡀⢈⠆⡹⠋⠁⠀⡠⢃⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⠀⠀⠀⡠⠊⠀⣠⠚⠁
⢱⡀⠀⠀⠈⠀⠀⠁⢀⡠⠊⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠊⢸⣀⡠⠤⠒
⠀⠑⡄⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⡖⠢⠤⣀⣀⣀⣀⣀⣀⡠⠤⠒⡄⠀⠀⢔⠉⠀⣀⣀⡠
⠀⠀⠇⠀⠀⠀⠀⠀⢰⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢀⠜⠉⠁
⠀⠀⢸⠀⠀⠀⠀⠀⠈⡄⠀⢸⠀⡠⠂⠉⠉⠒⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣎⠉⠉⠐⠒⠤
⠀⠀⠸⡀⠀⠀⠀⠀⠀⢃⠀⠀⢿⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢸⠀⠀⠉⠒⠢⢄⣠
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⡄⠀⢰⠀⠀⠀⠀⡇⠀⡄⠀⠀⠀⠀⡔⠒⠒⠊⠉⢨⠀⢸⠀⠀⠀⣀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢣⣀⣀⡣⣀⠀⠘⣀⠜⠀⠀⠀⠀⠀⠙⢄⣀⣀⠴⠃⢠⠃⠀⠀⡰⠁⢣
⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⢇⠀⡿⡄⠈⢁⢬⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⢰⡊⠀⠀⠀⢙⡆
⠀⠀⠀⠀⡸⡄⠀⠀⠀⠀⠀⠈⡾⠁⠘⡖⠁⠀⠙⢮⡑⠢⠤⠤⠤⠤⠔⠒⠁⠈⡷⡀⠀⠈⢦⠀⡰⠋
⠀⠀⠀⠀⡇⠘⡄⠀⠀⣀⡤⠊⠀⠀⠀⠘⢄⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⣀⠞⠀⠘⡄⠀⠀⠓⠁
⠀⠀⠀⠀⠳⡒⠊⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠉⠐⠒⠒⠊⠁⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠈⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣄⠀⠀⠀⢀⣴⡶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⠉⠉⣿⣷⠀⢠⣿⡏⠀⣼⠃⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⣠⣿⡿⢀⣿⣿⣠⡾⠃⠀⢴⣶⣶⣄⠂⢻⠇⣢⣤⡄⢠⣿⡏⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠉⠉⠁⣼⣿⡿⠋⣀⡀⠀⠀⠀⠉⣉⡓⠈⠰⠋⢁⣀⠘⣿⣷⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⣁⠀⠀⠀⠀⣠⣿⣿⣶⠟⢻⣿⡇⢠⣾⡿⠋⢹⡆⣰⣾⠟⠉⣿⢈⡿⢻⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢰⡆⠀⢀⣴⢿⣿⣿⠃⢀⣿⡟⢠⣿⣿⣤⡶⠟⣸⣿⣿⣴⠾⣫⡾⢁⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣷⣤⣤⣶⡿⠋⢸⣿⡟⠀⣼⣿⣡⣾⣿⣧⣀⣤⡾⢻⣿⣧⣠⣶⠏⠀⣼⣿⠏⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⠛⠻⠛⠛⠉⠀⠀⠈⠛⠃⠀⠻⠿⠋⠀⠉⠛⠋⠁⠀⠀⠙⠛⠉⠀⠀⠀⢿⣿⣤⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⡶⠖⣠⠀⠀⣶⣶⣆⠀⠀⣴⣶⡆⠀⢀⣴⣶⣦⣴⣶⣦⡀⠀⠀⡠⣀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣠⣴⣾⠃⣿⠀⣿⣿⡇⠀⢸⣿⣿⠁⠀⢸⣿⡿⠉⠀⠀⣿⣿⠀⢱⡝⣿⣍⠙⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠃⠻⠇⣿⣿⡇⠀⢸⣿⣿⠀⠀⢸⣿⡇⠀⠀⢠⣿⡿⠀⣼⡷⡼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢿⣿⣧⣴⡾⣿⣿⣤⣶⣿⣿⢠⣄⢀⣾⣵⢀⡀⠈⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡛⢋⣥⠀⠈⠉⠉⢀⣿⡿⠈⠿⠿⠿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣴⡿⠛⠻⣾⣿⣧⣄⣀⡀⠀⣉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⢿⣷⣶⣾⠟⠁⠉⠛⠿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠴⢞⢰⣌⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⣀⠀⠀⠀⣠⣤⣠⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⢾⣿⣿⣶⣴⣿⣿⠻⣿⠙⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣄⣼⣿⣿⡇⠀⢀
⠀⣀⣬⣿⣿⣆⠙⠻⠟⠀⠀⠀⠿⠛⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠻⣿⡟⣸⣿⣧⣾⣿⣿
⠸⣿⣿⡛⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠛⠻⣷⣀⡀⠀⢀⣤⡤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠈⢠⣿⠿⠛⣡⣿⣿⣤⡄
⣠⣼⣿⡿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣟⣛⢿⣶⣿⠿⢿⣷⣮⡻⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠉⠛⠀⠀⠀⠀⠀⠀⠀⠟⢛⣿⣿⣧⣄
⠻⣿⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡻⢿⣿⣿⣿⣿⣶⣌⠻⣿⣿⣿⣿⣧⣄⣀⣀⠀⠀⠀⢰⣿⡟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣽⣿⡿
⠀⣠⣿⡿⢿⡟⠀⠀⠀⠀⠀⠀⠀⣀⣠⣿⣿⣶⣌⣻⣷⣍⠻⣿⣷⣜⣿⣿⣿⣿⣿⣿⣝⠳⣄⠀⣀⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣷
⠰⣿⣿⣿⣿⣧⣶⡆⠀⣀⣠⣶⣤⣿⡿⣿⣯⣙⠿⣿⣿⣿⣷⣬⣿⣿⣿⡿⠟⠙⢿⣿⣿⣷⡹⣿⠋⣠⡾⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⡿⠿
⠀⠀⠙⠛⠛⣿⣿⣷⣿⣿⠿⣿⡿⠿⠿⣶⣬⡻⣷⣌⣿⣿⠏⣉⡉⠉⠁⠀⣠⣤⡄⣿⣿⣿⣷⣿⡾⠋⢀⣿⣥⣤⡄⠀⠀⠀⠀⢀⣀⣀⠙⢿⣿⣆
⠀⠀⠀⠀⠀⠙⠋⠉⠛⠋⠀⠙⢷⣄⠀⠈⠛⢿⣿⣿⣿⣿⣋⣉⠉⠀⡀⠀⢡⣤⠀⢹⣿⣿⡿⠋⠀⣠⡿⠛⠛⣿⣧⣶⣄⠀⣶⣼⣟⠿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠙⠻⣿⣿⡇⠉⢠⣶⣶⣶⡆⠀⠀⣸⣿⠟⠁⢀⡴⠋⠀⠀⠀⠛⠛⠙⣿⣷⣿⣿⣿⣷⡄⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄⡀⠈⠙⢿⣄⡀⠻⢯⡽⢁⣠⣴⠟⠁⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠀⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣄⠀⢸⡟⣿⠶⠟⠛⣿⠁⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡇⣿⠀⢀⣠⣿⣦⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣛⣯⣿⣿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣉⣠⠞⠛⣿⣿⣟⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠀⠛⠁⢀⣴⠟⠉⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣿⡉⠉⠀⠀⣹⣦⡀⠀⢀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡴⠋⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡴⣤⣄⠀⢀⡴⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⢸⣿⢠⣿⢃⡼⠁⣠⣤⡀⠸⡇⢄⣀⢀⣶⢶
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠙⠁⣾⡿⢋⣀⡀⠈⢉⣙⡂⠑⢋⣁⡘⣿⣾⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡁⣂⠀⠀⣴⣿⡿⠋⣿⠇⣼⡟⣁⡿⣴⣟⣁⣿⡼⢹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⣉⣴⠾⢹⣿⠃⣸⣟⣼⣿⣏⣩⠼⣿⣏⣩⠞⢁⣿⠏⢰⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣄⡉⠛⡋⠁⠀⠈⠙⠀⢙⡋⠁⠉⢉⡁⠀⠈⠉⠁⠀⠘⠿⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢛⣫⡵⣤⢸⣿⡇⠀⣿⡟⠀⢸⣿⠟⠛⢿⡆⢠⡰⣶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠑⠛⢸⣿⡇⠀⣿⡇⠀⣼⡟⠀⠀⣾⡇⠸⡇⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠷⠟⠻⠿⠟⣿⡇⣦⣼⡟⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢈⣡⡴⠶⣶⣿⣀⣀⠁⢚⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⢛⠁⣉⡛⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⡀⠀⡤⠒⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡴⠑⠊⠅⡒⠀⠀⢀⠙⠐⠢⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⠀⠀⡩⠀⠀⢸⢡⣀⣀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⢀⠉⡁⡉⠐⠈⠀⠄⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠨⢤⣁⠜⠀⡀⢰⠀⠀⠐⢁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣷⣶⡮⣵⣷⣮⡲⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⡿⠄⠀⠀⠀⠢⠤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡘⣛⢡⣿⣿⣿⡿⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢠⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⣿⣿⣿⢣⠃⡌⠀⠀⠀⠀⠀⠀⠀⠆⢠⢻⠀⠀⠀⠀⣀⣉⣋⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢌⡻⠟⣋⠁⠁⢀⠆⠀⠀⠀⠀⠀⠀⡘⠀⣸⣼⠀⠀⢀⢜⡿⠿⢿⡟⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⡀⠀⠌⠀⠀⠀⠀⠀⠀⡰⠁⡐⡏⢉⠀⣠⣾⢿⣤⣤⣴⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢀⠇⠀⠀⠀⠀⢀⠔⠀⡐⠁⠘⣾⣟⠡⠋⠞⠟⠻⠻⠀⢁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠈⠂⠀⠂⠁⡠⠊⠀⠀⠀⠀⠀⠀⢰⣤⣄⣠⣰⡄⢸⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡄⠀⠀⠀⠀⢐⠐⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠟⠛⠿⡇⠰⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⠂⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣄⠀⠀⠀⡇⠈⣏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡌⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠓⠅⠈⠢⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⡘⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⡥⠀⠀⠰⡒⠀⠀⡇⣆⠀⠀⠀⠀⠀⡄⢁⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⢀⠄⣂⠁⠀⠰⡰⠀⠀⢸⠀⠀⠀⡧⠀⠀⣧⡃⠀⠀⠀⠀⠀⠇⡜⠀⢀⠞⢢⠀⢁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⠀⢣⢧⠀⠘⠁⠀⠀⡇⠀⠀⠄⠸⠀⠀⠀⠀⠠⠀⠇⠀⣜⡰⠃⠀⡞⠀⠀⠐
⠀⠀⠀⠀⠀⠀⡠⠒⠀⠘⢆⠀⠀⠘⠄⠃⠀⡉⠀⠀⠄⠀⠀⠀⠀⠀⠠⠄⠀⠈⡚⠀⠠⠁⠀⠀⠸⠁⠀⠀⠓⠄⡀
⠀⠀⠀⠀⣀⠊⠀⠀⠡⠀⠀⢉⠡⠀⠀⠈⠀⠀⠀⠀⡀⠄⠀⠀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠐⠠⠀⡀⠀⠀⡀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⢊⠀⠀⠀⠀⠀⠀⢀⠤⣾⣤⠄⠀⢀⠀⠀⠨⢀⢕⢀⠀⠀⠀⠀⠀⠀⠀⡘⠒⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠁⠀⠀⠀⠀⠀⢠⣕⢔⣽⣟⠄⠠⠤⠤⢥⣮⣦⡁⠱⡄⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣯⣢⣿⣷⣤⣀⣐⣀⣆⡸⣧⣆⣵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠈⠙⢾⣿⡟⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⢃⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⡌⡡⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠣⡹⣦⡀⠘⠄⣀⣀⡀⠠⠃⢀⣴⣇⢌⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⠪⣙⠷⢦⡤⠤⣤⡴⡶⣻⠟⣡⡚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢅⠀⠀⠀⠁⠀⠀⠀⣸⡞⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⣤⣀⣤⢠⠁⠀⠉⠓⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠉⠙⠉⠉⠄⠀⠀⠀⢸⡇⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⡄⠀⠀⣿⡡⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠒⢲⠒⠖⠐⠚⠀⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠃⢾⠄⠗⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣷⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠀⠀⠀⣽⢿⢿⠆⠀⠀⠀⠀⢀⣤⣄⠒⠀⠀⣾⠃⡤⢀⣀⠀⠀⢸⢠⣇⠀⠀⠀⠀⠘⠆⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠈⠠⠊⠀⠀⠀⠀⠀⠸⠿⠏⡀⠀⠀⠟⠀⠘⣥⣭⠀⢀⠸⡏⠁⠀⢀⢀⠀⢠⠀⡰⠀⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠦⠀⣤⡀⠀⠀⠀⠘⢟⠀⠀⠀⠀⠀⠋⠑⢶⡄⠀⠀⠀⠀⠀⢘⠁⠀⠁⠀⣉⠀⠀⠂⣦⣤⢤⢸⣦⡀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⡿⠀⠒⠉⣀⠀⠀⠀⠉⡅⡦⢄⢀⣤⣤⣿⣷⣄⣶⣶⣩⡄⠐⠦⠹⣿⣠⣿⣿⣿⣾⠃
⠀⠀⠀⢀⠀⠀⠈⢁⣀⢰⡄⢠⣤⣴⣿⡧⠀⠀⢠⣶⠂⣀⣐⡊⠔⠃⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠤⠐⢂⣿⣿⣿⠿⠿⠟⠂
⠀⠀⠀⣼⣿⣿⣾⡌⠉⢸⣿⡾⣿⣿⣿⣿⡄⠰⠾⣿⣾⣿⣿⡶⢸⣶⡾⠁⣿⣿⣿⡿⠏⠛⠛⣉⡅⠀⠛⠚⠓⠿⠄⠀⠁⠀⠁
⠀⠀⠀⢹⣿⣿⣿⣷⡀⣿⣿⣷⣿⣿⣿⣿⠛⣦⣀⣿⣷⠈⠙⠳⡚⠛⠡⠶⠶⢰⡆⢰⠐⠂⠀⠀⢀⡁⢐⣺⡿⢿⣿⣶⣶⡀
⠀⠀⠀⠘⣿⠽⣦⣿⣷⡘⠻⣿⠿⠿⠟⠛⣯⡭⣉⠀⠀⠀⠒⡀⠁⣾⣧⣅⣐⣺⣧⣾⡦⠀⠀⠀⠈⠉⠈⠁⢁⣀⣙⡛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠀⠀⣶⣆⡀⠀⠀⠐⠀⠀⢠⣿⡿⣬⣬⣟⣿⣾⠹⡇⢀⣀⠰⠄⠈⠀⠐⠈⠀⠙⢻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣤⣬⣅⠀⠀⠀⠰⡲⠤⠪⠛⣄⣍⣉⣹⡄⢶⡶⠷⠖⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠒⠒⠃⠀⠐⠶⠋⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⣀⣀⣀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣩⣯⠽⠘⣧⠼⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⣰⡶⠊⡁⢿⢦⣘⠥⠀⠈⠂⢀⡀⠀⠛⠐⠙⠀⠀⢀⠀⢠⠂⠀⡀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢶⠖⠶⢦⣤⡼⠠⣿⠇⡆⣷⢸⣇⠦⠛⣠⠄⢀⡈⠁⠲⠟⠊⠉⠀⠀⠐⠀⣈⠁⠀⠈⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠖⣲⡄⠠⣤⡄⢻⡆⢰⣿⡟⣿⣿⣦⣽⡄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣰⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣰⠇⣼⡦⠁⠞⢹⣇⠹⣿⣏⠈⡀⠰⠆⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⢍⠀⡸⠁⢸⡅⣶⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⠿⡠⠐⢋⣀⠳⣀⠸⠿⠆⢿⡎⣠⡤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠟⠁⣴⠀⠀⢰⣏⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢤⣴⣇⣤⡀⠁⠀⠈⠀⠀⠀⠈⠠⠤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠓⠐⠦⠶⠶⠀⡷⢄⣿⡇⠀⣻⣿⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⢰⠸⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠛⠀⠰⠿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣶⢦⠀⣀⠀⠻⢦⡤⠚⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠄⠀⠀⠀⠀⠀⠀⡀⢀⠈⣐⣁⡀⣀⠇⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠁⡇⣰⠀⠠⠃⠀⠀⠀⠀⠀⣘⠉⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠰⠷⠾⠡⠘⡏⠮⣵
⠀⠀⠀⠀⠀⣰⣟⡡⠋⢐⡖⢛⠂⢿⣟⠀⠀⠀⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣈⣷⣴⡄⠀⠀⠄⡤⠀⡤⢀⢠⡄⢤
⠀⠀⠀⠀⠀⡻⠿⠁⠘⣙⣣⣤⣴⣾⡟⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⠀⠀⠇⠁⠰⠂⣤⠘⠃⣀⢀
⠀⠀⠀⠀⠘⣧⡼⠟⠛⠉⠉⠀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠓⠆⠰⠀⠋⠃⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠀⢰⡟⡾⣿⣟⢻⢿⣿⢿⣿⣷⠶⠆
⠀⠀⠀⠀⠀⠉⠻⣷⠇⠐⢶⣁⢀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⢹⣿⠛⠛⣿⠛⠀⠀⠙⠳⠿⠿⠾⠿⠿⣾⠿⠗⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣷⠀⠈⠿⠾⠋⠀⠐⠆⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⠈⡇⠀⠀⡼⠁⠀⠓⠢⠑⠠⠀⠀⢀⠀⠀⠀⠠⠊⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣀⣀⣀⣴⣄⠀⠐⠲⠀⠀⠀⠂⠠⠗⠒⠐⠣⠀⠈⠙⠃⠀⠀⠀⠰⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⠬⣉⠙⠿⠿⠃⢠⠀⠀⢠⠀⢀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠈⡀⢠⣭⠀⢒⡀⠐⠀⢀⠀⠀⠁⠈⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠈⠑⣶⢠⡘⠃
⠀⠀⠀⠀⠀⠀⢰⣤⡀⠀⠀⠀⠀⣈⠀⠀⠀⢂⡀⠘⠉⣅⠀⠈⠀⣠⠃⠀⠀⠀⢀⣴⣿⣷⣦⠀⠖⠀⠒⠊⠀⠀⠀⠀⠤⢀⠀⠂⣼⣿⢯⣄
⠀⠀⠀⠀⠀⠀⠈⡿⢃⠀⠀⠀⠀⠈⡅⠀⠀⠀⠹⣀⠀⠙⣄⠀⠀⠈⠁⠀⠀⠀⢸⣏⠾⠏⣽⠇⠀⠐⠷⠄⠀⠀⣤⠂⠀⢀⠀⠘⠛⠁⠀⡉⠐
⠀⠀⠀⠀⠀⠀⢀⡤⠈⢢⡄⠀⠀⠃⠐⠋⠀⠂⠀⠀⠁⠃⠐⠚⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠀⠀⠀⠀⣀⡈⢓⠀⢩⠄⠘⠄
⠀⠀⠀⠀⠀⢠⠊⠀⢀⡞⠀⢠⠀⢣⡿⠉⠄⡶⢠⢠⠀⠄⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠁⠀⠀⠀⠀⢠⠶⡆⢠⡉⠓⢂⣀⡀⠀⠀⠀⢰⠋⠙⣄
⠀⠀⠀⠀⠀⠸⠀⠀⢸⡽⠀⢀⡆⠀⠀⠀⠄⠠⠄⠀⢰⠀⠆⠛⣀⡠⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠙⠦⣷⣏⣸⡗⠈⠉⠑⠆⠀⢀⣾⡦⣴⣯
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⠜⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠨⢁⠈⠘⠈⠐⢤⠈⠀⠀⠀⠀⠀⡠⡀⠙⠀⢈⡀⣴⠦⣍⠀⠀⠸⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⣡⡶⠿⠿⢷⣤⡀⠀⠀⠁⠀⠀⠀⠀⢦⣄⠈⠀⠀⠣⢐⠎⡇⠀⠊⠀⠀⠀⠈⠛⢰⡤⣾⡇⠀⠀⠁⠀⠀⣀⠙⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠠⠄⠸⠟⠁⣤⣤⠀⠹⢗⠠⠀⠄⠐⠒⣶⣾⠦⠆⠀⠈⠑⠶⠆⡄⠀⠀⠀⢶⡄⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⡀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣥⣬⠀⢹⣿⢏⣠⣧⡴⠀⠀⠀⠀⣍⠁⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠴⠋⠻⣦⠀⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢷⣷⠾⠟⠁⠀⠀⢠⠀⠵⡿⣮⡀⡀⠀⢠⢟⣴⡦⣀⠀⠀⠀⠄⠀⠀⢀⠀⠀⠃⠀⢠⣾⣿⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠄⡀⠘⠋⠜⠁⠀⠀⠀⠀⠀⣀⡐⢉⣽⣷⠿⣾⡿⠿⣿⣿⡿⡳⠀⠀⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡁⢀⠁⡆⠐⠃⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⣠⠦⡁⢈⠁⠀⠤⠦⠤⠀⠐⠛⠃⢉⠐⣁⣠⠔⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠈⠀⡇⠀⠀⡀⢁⡁⠀⠀⠀⠀⠀⠀⠀⢰⠋⠁⠙⠛⠿⠟⠋⠀⢀⣀⣀⣀⣀⡀⠠⠤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠁⠀⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⣷⣶⣶⣿⣿⣿⡿⢛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⠟⠋⠐⠋⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠁⣀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠁⡀⠉⣸⣿⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠳⣤⣾⠟⣷⣀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣆⠸⠀⠀⠀⠀⢹⡟⠁⢸⡀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⣙⣶⠟⠁⠘⠶⠀⠀⠀⣰⡞⠀⠀⢸⣧⡀⠀⠀⢀⡾⠣⠊⠀⠀⠠⡲⠁⠀⠀⠰⠁⡀⠀⢀⡤⠖⠁
⠀⠀⠀⠀⠀⢀⣞⡽⠁⠀⠀⠀⠀⡀⠀⠀⣹⣇⠀⠀⣸⣿⠵⡀⡺⠋⠀⠀⠀⠀⢀⠈⠁⠀⠀⠀⠀⣀⣠⠾⠋⠀⠀⣀
⠀⠀⠀⠀⣡⡿⠋⠀⠀⠀⠀⠀⠀⠐⠀⠀⠿⣿⠀⢸⣿⣷⡖⡳⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠞⠁⠀⣀⠴⠚⠁
⢀⠀⢀⡶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠀⢸⣿⣷⣿⡅⢀⣔⠀⠀⢀⠀⠀⠀⢀⡤⠾⠁⠀⠀⡶⠚⠁
⠈⠐⢼⠗⢀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟⠉⠀⠀⢀⣿⣿⠿⠋⠠⠜⠁⠀⠀⠔⠀⣠⠶⠃⠀⠀⠀⢀⣠⡤⢂
⠀⠀⠀⠀⢸⡄⡀⠀⠀⠀⠀⠀⢀⣁⣠⣴⠟⠀⡾⠟⠁⠀⠀⠀⠀⠀⠀⣠⠂⠈⠀⠀⠀⠀⡰⠖⠋⠁⠂⠁⠀⠀⣀⢀⠄⠀⠀⠀⠀⠀⣀⠔
⠀⠀⠀⠀⠘⡇⠀⠀⠁⠐⠀⣠⢾⣹⣿⣅⢲⣤⡞⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⣀⠀⠊⠀⡀⠀⠀⠀⠀⣀⠰⠋⠀⠀⠀⠠⠐⠂⠈
⠀⠀⠀⠀⠀⠣⣀⢀⠀⢀⠔⠁⣾⣿⡝⣿⣿⣷⡀⠈⠀⢀⡤⠞⠋⢀⡀⠀⠐⠀⠉⠁⠀⣀⠔⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢸⣟⠀⠁⠀⢰⣿⣿⣏⣹⣿⣿⢲⣦⣶⡉⠀⣀⠐⠋⡠⠀⠁⠀⠀⠀⠀⠉⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣸⣏⡛⠓⠲⢦⣿⣿⣿⣼⠋⣰⡿⢑⠙⣷⠀⠀⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢠⣞⣿⡌⠳⢦⡀⣿⣿⣿⣽⣿⣦⠟⣱⠋⠁⣿⣶⡻⠂
⠀⠀⠀⢠⣿⡟⢻⠋⠠⣾⠿⣿⣿⣷⣯⣬⣥⣶⣿⣄⢀⡿⠉
⠀⠀⢠⣿⡟⢀⣿⣦⣏⣡⣰⣅⣽⣿⣿⣿⡻⠉⠀⠋⣽⠃
⠤⢤⣿⣟⢀⣿⣿⣶⣤⡴⢏⢍⣥⣼⣿⡁⠀⢀⣠⡾⠁
⠀⣼⣿⢿⡟⣿⣭⣟⣃⣭⡿⠟⠛⠉⠙⠛⠛⠛⠁
⢠⣿⣧⠞⣰⣿⣿⠿⡛⠉
⠘⣿⣇⡴⠛⠉⠉
⣶⡾⠿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣽⣷⣶⣶⣶⣦⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢀⣄⠈⣻⣍⠑⣄⢠⡶⠒⢶⢦⣶⡖⠲⣷⠶⠂⠀⠀⢿⢿⠃⡶⢶⡊⠉⠹⡿⡇⢠⠶⡂⠀⢸⣿⠌⢢
⠀⠀⡀⠉⣁⣀⠻⠟⣨⠋⡧⣾⣶⡘⡌⢰⣶⣶⡏⠸⢿⠇⢠⣶⡾⢠⠀⢿⡿⣀⣴⣶⠇⡆⠸⣿⠇⠀⢁⣤⡼
⠀⣜⣁⡀⠻⠟⠉⠙⠁⢠⡅⢩⣥⠇⢇⣨⡉⡇⣿⡧⢠⣄⡤⠴⠄⣸⣿⠆⣼⣥⠤⡆⢠⣿⣷⢀⣠⡄⠈⡿⠁
⢰⡙⠿⢇⠀⠀⠀⠀⠀⠘⣃⣘⠛⢋⡘⠿⠛⠀⢠⣤⡈⠛⠁⣾⠀⢇⣴⣄⠙⢋⣸⡇⠸⢈⣤⡈⠛⡃⡄
⢸⡿⠀⢿⠀⠀⠠⠀⠀⠀⠻⠟⠀⢿⡿⠀⡆⢠⠈⠛⠱⠭⠠⣤⢀⡀⠛⠉⠍⠀⢤⠄⣤⠈⠛⡁⢻⡿⢻⡄
⠘⡠⣾⣷⠑⠊⣴⣧⠀⢴⣶⠄⠀⢠⣤⡄⠁⡞⠰⣿⡷⢠⣶⣦⡘⠁⢾⣷⠀⣴⣦⠀⠛⠁⣾⠐⣤⣶⣜⡿
⠀⠘⠬⣁⡶⠷⠊⠁⣸⡮⠭⠔⢸⣈⠿⠟⢰⡿⠷⠤⠒⠚⠛⠋⣿⠿⠦⠐⠒⠛⠋⢸⡶⠶⠞⠀⠹⡟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⣠⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⣴⠀⡰⢁⠞⠁⠀⠀⠐⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡟⠀⠀⠀⠀⣠⣿⣿⣿⠇⣜⠵⠋⠀⠀⣀⢀⡷⣫⡤
⠀⠀⠀⢠⣶⣷⠀⣀⠀⠀⠀⠀⠀⠀⢀⢆⡿⣱⢇⢠⣾⣴⣿⣵⣿⣃⠴⠀⠀⠀⢀⠔⣱⣯⡾⠋⣀⡀
⠀⠀⠀⠸⣿⣧⠌⢿⣿⡀⠀⠈⠐⠂⠞⠼⣰⢧⣿⢿⣾⣿⣿⣿⣿⡟⠀⠀⢀⠔⣹⣿⣿⣫⣠⣴⠏⠁
⠀⠀⠀⠀⠈⣿⣶⣿⣿⣿⠀⡄⠀⠁⡀⠀⠙⣿⣿⣿⣿⣿⣿⣿⢟⢁⣤⠖⣡⣿⣿⣋⣉⢼⣯⠖⠁
⠀⠀⠀⠀⠀⣿⡟⠉⠙⠻⠀⢸⣶⣾⡷⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⢃⣴⣾⠟⣻⣿⢿⡥⣚⡡⠄
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⣿⣿⣿⡗⠀⢸⣿⣿⣿⣿⣿⡝⣻⣴⣿⢟⣷⣿⣿⡷⡿⠚⠉⣀⣤⡦
⠀⠀⠀⠀⢏⠀⠀⣴⠳⠀⠀⣿⣿⣿⡇⠀⣾⣿⣿⣿⣿⣾⣿⣿⢿⣷⣿⣟⡩⠟⠁⢀⣴⣛⣫⣥⠒
⠀⠀⠀⠀⠘⣠⣼⡉⠀⠀⣠⣿⣿⣿⣷⡀⠙⣿⡿⣻⣿⣿⣿⣿⣿⠿⠎⠉⣠⣴⣾⣿⣿⣿⠿⣩⡄
⠀⠀⠀⠀⠙⠃⣿⠦⣄⣠⣹⣿⢿⠻⣿⣿⠀⢘⣴⣿⡿⢷⣾⠋⢛⣠⣶⣿⣿⣿⡿⢛⣉⡤⠖⠋
⠀⠀⠀⠀⠀⠀⣿⠋⣻⣏⣻⡿⠤⣬⣟⣾⣇⠀⢄⣍⡻⣿⣿⡾⢿⣿⣿⣿⣿⣯⣶⠉⠁⠀⢤⣆
⠀⢀⣠⡴⠶⢾⣿⢤⣁⠊⣿⣇⠀⣿⣿⣿⣿⡄⠹⣟⣳⠘⣿⣟⣯⣿⣿⣿⣿⣭⣼⡿⠛⠛⠋⣀
⢠⡿⠁⠀⠀⢸⣿⠀⠉⢻⡿⢋⣳⣯⣿⣟⡻⠇⣼⣿⢑⡆⣿⣿⣿⣿⣟⣿⣿⣿⣿⠴⠔⠞⠁⠁
⣿⠁⠀⠀⠀⣸⣿⠀⠀⠀⢻⣿⣿⣿⡿⣿⡵⢞⣩⣌⣿⠇⣿⣽⡿⠟⠃⠈
⢿⡀⠀⠀⠀⠛⠈⠀⠀⠀⢸⣿⣿⣿⡿⣾⣿⣷⣽⣿⣥⠾⠋⠈
⠘⢷⣄⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⠿⠋⠉
⠀⠀⠙⠷⣶⣤⣤⣴⠾⠋⠀⠈⠉
⣿⡯⣟⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⠲⢾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣧
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⢀⡞⠁⠉⠳⣄⡈⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠘⢦
⠀⠀⣀⡴⠛⠉⠀⠀⠀⠀⢉⡟⠳⣄⠀⠀⠀⠙⠦⣄⠀⠈⠙⠒⠲⠶⠒⠋⣹⠁⠀⠀⢈⡷⣖⠋⠉⠉⠉⠙⠓⠦⣄
⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠈⢳⡄⠀⠀⠀⠈⠓⢦⡀⢠⡖⠀⠀⢰⠇⠀⠀⡴⠋⠀⠈⠳⡄⠀⠀⠀⠀⠀⠈⠳⡀
⣼⠁⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠙⠲⣽⡦⠶⢯⡀⣀⣼⠁⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⢹
⡇⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⡏⠉⡏⠉⠓⠲⠦⣤⣀⣸⡆⠀⠀⠀⠀⠀⠈
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠺⣅⠀⣧⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢰
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣒⣒⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⠸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢠⠃⠀⠀⠀⠀⠀⠀⠀⠠⣖⡒⠒⠚⠋⠉⢉⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠸⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠽⠂⢰⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠰⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡖⢀⡈⠢⡀
⠀⠀⠀⠀⠀⣠⠴⠒⠉⠉⡉⠉⠓⠲⠞⢀⣄⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⡜⠙⢆⠙⢤⠴⠒⠋⠉⣉⠉⠑⠲⢤⡀
⠀⠀⠀⡴⠋⣠⠔⠊⠉⠉⠉⠉⠙⢲⠀⠈⠢⡳⣄⠑⢄⠀⠀⠀⠀⠀⠀⢀⠇⢰⠃⢀⡴⠃⡀⢰⡊⠉⠁⠈⠉⠉⠒⢤⡈⠳⡄
⠀⢀⠎⢀⠞⠁⠀⠀⠀⠀⠀⠀⢠⠇⡰⠙⣄⠘⡌⠢⡈⠳⡄⠀⠀⠀⠀⡜⠀⡏⢀⡎⢠⠎⠉⢦⠑⣄⠀⠀⠀⠀⠀⠀⠙⢆⠈⢆
⠀⡜⠀⡎⠀⠀⠀⠀⠀⠀⢀⡴⢁⠜⠁⠀⠘⡆⠸⡀⠘⢦⡈⠢⣠⡤⣴⠁⡸⠀⡸⠀⡎⠀⠀⠀⠱⡌⢢⡀⠀⠀⠀⠀⠀⠈⡆⠘⡄
⠀⡇⢸⠀⠀⠀⠀⠀⠀⣞⢁⡔⠁⠀⠀⠀⠀⠃⠀⡇⠀⠀⠑⢄⠙⠇⠃⢠⠃⠀⡇⠀⣇⣀⡠⠤⠤⠞⠂⠙⡆⠀⠀⠀⠀⠀⢸⠀⡇
⠀⡇⠈⡆⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡇⢀⠇⠀⠀⠀⠈⠳⡄⠀⠘⠋⣉⣁⠀⣤⠤⠴⠒⠒⠋⠉⠉⠀⠀⠀⠀⠀⠀⡌⠀⡇
⠀⠸⡄⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⡜⠀⠀⠀⠀⠀⠀⠳⡄⡞⠉⠁⠘⡄⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⡼
⠀⠀⠙⢄⠘⠢⣀⠀⠀⠀⠀⠀⠀⢀⡠⠊⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⣵⢣⡀⠀⠀⠘⢆⠈⠢⣄⠀⠀⠀⠀⠀⠀⢀⡠⠚⢀⠜⠁
⠀⠀⠀⠀⠑⠦⣈⠉⠒⠒⠒⠒⠊⢁⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠙⠦⣀⠉⠒⠒⠒⠒⠊⢁⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠊⠉⠁

⠀⠠⠄⠀⠀⠤⠀⠄⠠⠤⠀⠀⠤⠀⠄⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⠋⠑⢄⠀⠀⠠⠄⠀⠠⠤⠀⠀⠤⠠⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠈⣦⣄⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢹⣿⣧⠀⢳⡀
⠀⠀⠀⠀⠀⠀⢀⠔⠚⠉⠉⠉⠉⠒⢤⠀⣀⡀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠻⣿⠀⠈⡇
⠀⠀⠀⠀⠀⠀⡏⣠⠤⣄⠀⠀⠀⢀⡜⠋⠀⣠⢅⠢⡠⣴⡃⠀⡤⠂⣄⠀⠀⠀⠀⠀⠙⣄⣀⡇
⠀⠀⠀⠀⠀⠀⠛⠀⠀⠈⠑⠤⡤⢼⠀⣀⠜⠁⠀⢃⢹⠱⢱⢴⡄⠀⣸⠓⠚⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠐⠦⠄⠜⣸⠃⠀⢠⠟⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⡤⣞⡥⠦⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠁⠀⠐⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣇⣠⣤⣤⣶⣾⣷⣶⣶⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⠿⠿⣏⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⢛⡿⠏⠁⠐⠒⠂⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣦⣤⣴⣶⡏⠉⠙⠛⠢⣄⣀⣀⣀⣀⣠⠞⠉⠁⠀⠈⢙⣦⣾⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣽⣿⣿⣿⣿⣷⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣦⡀
⠀⠀⢀⠾⠋⠁⠀⠀⠀⠈⠉⠁⠀⠀⠈⠉⠉⣽⡿⠿⡿⠛⠛⠛⠛⠛⠛⠋⠉⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⠷⣤⡀
⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⡆⠀⠀⠀⠀⠀⠛⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣹
⠀⠈⣇⠀⠀⠀⠀⣴⣶⣤⣿⡇⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⠒⣶⢤⣼⠀⠀⠀⠐⠓⢲⣤⣤⠀⠀⠀⣿⣿⠿⠀⠀⠀⣠⠏
⠀⠀⠘⣆⠀⠀⠀⠙⠛⢻⣿⡇⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢤⡤⣾⠀⣿⠀⠀⠀⢠⣤⣼⢿⣿⠀⠀⠀⠀⠀⢀⣤⠴⠞⠁
⠀⠀⠀⠈⠑⠦⢤⣤⣤⠞⠈⡇⠀⣀⣀⣠⣤⣤⡏⠉⢹⡀⠀⠀⠀⢀⣸⡄⢸⡇⠀⠀⠀⠀⣸⡆⢻⠀⠀⠀⢰⣀⣀⣱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠋⠉⠉⠻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡀⠀⢠⡀⠀⠙⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⡿⠉⠈⢻⣧⠀⠘⣿⠀⠀⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⢿⣿⡇⠀⣤⠀⢻⡆⠀⠙⠀⠀⣾⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢿⣿⡏⠀⠀⠘⡇⠘⡿⠀⢨⣿⡀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⢻⡆⠈⣿⠀⢰⡗⠀⢱⠀⠀⢀⣿⣿⣧⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡆⠀⢷⠀⠘⡆⠈⠀⣰⣿⣆⠀⠺⠏⠙⣿⡄⠀⠘⣄⠀⠘⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠁⢀⣀⣼⡄⠀⢀⠀⢻⡄⠐⠿⠋⢻⣧⡀⠀⠀⣿⣷⣀⣀⣿⣷⣀⣠⣽⣿⣿⣿
⠀⠀⠀⢠⣾⣿⣿⣿⠁⢰⣿⣿⣿⣷⠀⢻⣧⠀⢿⣦⣀⣀⣼⣿⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟
⢸⣿⣧⠈⢿⣿⣿⣿⣇⡀⠙⠋⠀⣼⣧⣴⣿⣷⣿⣿⣿⣿⣿⣿⠿⠿⢻⣿⡋⠉⠉⠉
⠀⢻⣿⣧⠈⢿⣿⣿⣿⣿⣷⣾⡿⠿⠿⣿⡟⠛⠉⠉⠈⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⢻⣿⡷⠈⠛⠉⠉⠉⠀⠀⠀⠀⠀⢿⣷⣄⡀⣀⣀⣀⣠⣤⣤⣶⣿⠟
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣼⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⣠⣤⡤⠄⠀⠀⠻⢿⣿⣻⠟⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣧⣀⣴⣿⣶⣦⣄⡹⣿⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⡿⣿⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢿⡻⣿⣿⣿⣿⡿⠋⠁⠀⢀⣟⠻⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠉⣠⣿⣿⣦⡙⡿⠋⠀⠀⠀⣠⣾⣿⡄⠀⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣿⣿⠻⡄⠀⠀⠀⠉⠙⣿⡅⠀⠀⠈⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣦⡁⠀⠀⠀⠀⠀⠙⣧⣄⠀⠀⠀⠀⠈⠳⣦⣰⣤⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠈⠻⢷⣄⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠙⢷⣤⣄⣴⡏⠹⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠉⠿⠹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠉⠛⢿⣿⣷⡄⠀⠀⠀⣀⠠⠄⠐⠚⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣾⣥⣿⠛⠃⠀⠈⠁⠀⠀⠀⠀⡠⠂⢻⣿⢿
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⠶⠶⣶⣦⣤⣀⠀⠀⠀⠀⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⢳⣸⣆⢥⣤⣤⣤⣄⣀
⠀⠀⠀⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⢟⣷⢖⡠⣾⠿⣯⠁⠀⢰⡆⠀⠀⠀⠀⢀⠜⠁⠀⠀⢀⣴⡿⣻⣿⣾⠓⠂⠘⠉⠛⡷⣦⡀
⠀⠀⠀⣴⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢞⣴⣟⢿⣆⠀⢹⡆⠀⢸⣧⠀⢀⠀⡰⠃⠀⠀⠀⢠⣿⠋⠀⠘⣿⢿⡄⠀⠀⠀⠀⠈⠪⡻⣦
⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣯⢻⣆⠈⡷⠀⠸⠿⠀⣁⠎⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠹⣎⢿⡀⠀⠀⠀⠀⠀⠀⠽⣧
⠀⢠⣿⡁⠀⠀⠀⠀⠀⠀⣠⣶⢂⣥⠀⠀⠀⠙⢿⣿⡿⣿⠄⢕⣦⡤⠀⠀⠁⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢿⣼⣧⣀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⢸⣿⠄⠀⠀⠀⠀⠀⠀⢿⡿⠿⠿⠀⠀⠀⠀⠙⠛⣋⣛⣶⣿⣿⣷⣤⣴⣿⣷⣶⣤⣤⣄⢸⣧⠀⠀⠀⠀⠀⠈⠻⠿⠟⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠂⠀⠛⢻⡇⠀⠀⠀⠀⠀⢰⣿⠀⢻⣿⣿⣿⡿⠿⣿⡿⠿⠟⠉⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣼⡇
⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⣀⠀⣀⣤⣿⠇⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠙⢿⣄⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⢀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠁
⠀⠀⠀⠀⠀⠙⠿⣶⣤⣄⣀⣀⣀⣀⣠⣤⣶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣤⣤⣶⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀⠀⠀⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠆⠠⠀⠀⠃⠀⠑⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⢰⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⢀⠀⠐⠂⡰⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠐⠚⠁⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡐⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠳⠤⢄⠀⢀⠠⠊⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡋⠀⠀⠀⢐⠂⠀⠀⠀⠀⠸⠶⢖⠒⠲⠦⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠠⠀⠐⠒⠦⢤⠁⠀⠀⠀⢀⡀⠀⠀⠀⢀⡃⡀⠀⠐⠄⢀⡀⢈⡒⣄⠀⠀⠀⠀⠀⠀⠀⠈⠒⣕⢄
⠀⠠⢂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠙⠢⡀⠀⣠⡁⠀⠀⠀⠀⠐⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠔⠃⣄⡄⠀⠉⠒⠤⢁⣴⠋⠀⠀⠀⠀⠐⠄⢀⠀⠀⠀⠀⠀⡄
⢰⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠹⠂⠀⠀⠘⠭⢶⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠈⡈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠊⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠱⡐⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠈⠃⡀⠀⠀⠀⠀⠀⢀⠀⡢⠂
⠀⠀⠈⠪⠐⠄⢀⠀⠀⠀⡀⠡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠒⠉
⣀⣀⣀⣀⣀⣀⣀⣐⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣭⣿⣯⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠀⢀⣜⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⠔⣠⣤⠊⣠⠀⠀⣨⡕⢶⣤⠤⣤⠣⠁⢨⣄⡀⢀⣀⣀⣀⣀⡄⢀⣀
⠀⠀⠀⠀⠈⢢⡉⠠⠈⠻⣄⠀⠙⢆⠀⠐⡄⠈⠉⠓⠀⠀⠀⠈⢰⢧⠀⠈⠀⠐⠈⠒⠀⡀⠀⠈⠁⠉⢸
⠀⠀⠀⠀⠀⠂⢀⣀⠈⠉⠀⠀⠁⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀
⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢘⠀⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠸⢶⡦
⠀⠀⠀⠀⠀⠀⢸⠐⠀⢀⡄⢃⠀⢐⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡀⠁⠀⢀⡀⠀⠰⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤
⠀⠀⠀⠀⠀⠀⠸⠤⠤⠤⠬⠥⠴⠤⠤⠴⠧⠤⠴⠾⠄⠀⣶⣾⡄⠀⠀⠤⠔⠒⡲⢀⠾⣆⠀⠀⠙⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠅⠀⠀⠀⠔⠉⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⡀⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣀
⠀⠀⣰⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣆
⠀⠀⠙⢿⣿⣿⠋⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠈⠻⣿⡿⠋
⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⢀⣤⣶⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢻⣷⣦⡔⡀⠀⠠⠂⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢄⡀⠀⢩⣿⠃⠀⠀⣸⣶⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣾⣇⠀⠀⠸⣯⡁⠀⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠶⠶⠶⠶⢶⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⠶⠶⠶⠶⠶⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠞⢸⠟⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⢻⠇⠳⠀⠀⠀⠀⢉⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡄⠀⠀⠀⢠⣄⣤⣆⣀⠀⠀⢠⡅⠀⠀⠀⠀⠀⠀⠀⢠⡄⡀⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢈⠏⢿⣿⠀⠀⠀⢀⣿⣿⣿⣿⡏⠀⠀⠀⠈⠐⠀⠂⠀⠀⠀⣹⣿⣿⣿⣷⠄⠀⠀⠀⠀⣿⡟⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠸⡟⠀⠀⠀⠐⠻⠿⣿⢿⠛⠶⠄⡀⠀⠀⠀⠀⣀⡤⠀⠺⢿⣿⣿⡿⠀⠀⠀⠀⠀⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠘⠃⠀⠀⠀⠐⠀⢂⠘⠉⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡋⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠜⣿⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣶⣿⣿⣷⣶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⢂
⠀⠀⠀⢀⠔⠁⠀⠈⠀⠀⠈⠣⠂⠐⠀⠀⠀⠄⠛⠛⠋⠉⡋⠉⠙⠛⠉⠉⠉⠉⡏⠛⠋⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠁⠀⠀⠑⢄
⠀⢀⡴⠓⠂⠀⠀⠀⠀⢄⡰⠓⠒⠂⠒⢠⡼⠐⠒⠀⠐⠔⠂⠒⠒⠒⠒⠒⠂⠀⠶⠒⠒⠒⠒⠒⠒⠂⠂⠃⠀⠐⠒⠒⠂⠒⠂⠂⠦⡀
⠠⠊⠀⠀⠀⠐⠠⡀⠀⠀⡧⠀⠀⠀⢠⠊⠑⠀⠀⠀⠀⠚⠂⠀⠀⠀⠂⠴⠆⠀⠈⠃⠀⠀⠀⠐⡠⠀⠀⣀⠒⠀⠀⠀⠐⠂⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠈⠂⠒⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢰⠀⠀⠀⠁⠁⠀⠀⠡⠀⠀⠀⠀⠈⠂⠀⠀⠤⡊
⢃⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠒⣾⠀⠀⠀⠊⢦⡄⠀⠐⠓⠀⠀⠀⢰⠃⠀⠀⠀⠈⡆
⠈⠳⢄⠀⠀⠀⠀⢀⡠⠴⢇⠀⠀⠀⠀⡰⠀⠀⠀⠀⠈⡒⠁⠀⠀⠀⠀⠈⠁⠀⢀⠁⠀⠀⠀⠀⠉⠉⠀⢀⡀⠀⠀⠀⠈⡲⢦⡀⠀⠀⢉
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠀⠀⠈⠉⠉⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠂⠒⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢷⠀⠀⢠⣴⣾⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠏⠀⢰⠀⠀⢸⠉⢿⣿⣿⣿⣿⣿⣶⣶⣶⣾⠁⠀⣴⣶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⡎⠀⠀⠀⡇⣼⡿⠿⠿⠿⠿⣿⠿⠟⠁⠀⢸⡟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⣿⣿⢀⡞⠀⠀⠀⠀⢹⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢸⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⡠⠔⣶⣿⡟⠚⠁⣿⣿⡎⢀⡴⣶⣦⡀⠀⠀⢀⣴⣦⡀⠀⢠⣾⡿⠛⡇⠈⢆⣠⢃⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢸⠁⢸⣿⡿⠀⠀⠀⢻⣿⣇⠎⠀⢸⣿⡇⠀⢠⣿⡏⢠⠃⠀⣿⡟⠀⣠⠃⠀⣠⠋⠉⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠸⠆⢸⣿⡇⠀⠀⡠⠛⣿⡿⠀⠀⣼⣿⠃⠀⢸⣿⣷⠋⠀⡠⢿⣏⠉⠀⢀⠔⠁⠀⢸⣿⠏⠀⠀⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⣀⠔⠁⠀⠈⠀⠀⠀⣿⡟⠀⠀⡸⠙⠻⠶⠚⠁⠈⠙⠒⠒⠁⠀⠀⠀⢿⣿⠀⢀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡭⠤⠤⠤⠤⠤⠤⠄⠀⠻⣧⣀⡴⠃⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠈⠛⡒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⣶⠀⢰⣖⠀⠀⡶⣆⠀⣿⠑⡄⢾⡆⢸⣆⡆⢠⡟⠀⠀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊⠁⠀⠿⠄⠘⠯⠄⠘⠉⠻⠀⠿⠔⠁⠸⠇⠐⠘⠇⠘⢧⠇⠀⠈⠑⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣍⡉⠉⠁⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⣷⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⢿⣿⣿⣖⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⢀⣨⣿⣿⣿⣿⠀⠈⠉⢛⡿⠿⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⢏⣿⣿⠟⣸⣿⣧⣀⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢻⣿⣿⣿⣿⡟⠁⣰⡿⣻⡿⣯⠉⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠠⢻⣿⣿⣿⢻⣿⣴⡿⢱⡟⣧⠸⡆⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣠⣅⢨⣿⣿⣹⣾⣻⡿⠁⣿⠀⢹⡄⢻⡀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⠿⠭⣿⣿⣿⣿⡿⠁⠠⣿⠀⠈⢧⣤⠧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠃⢠⣿⣿⣿⡿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⠀⢈⣿⣿⣿⡅⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠉⠉⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡀⣧⢰⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⠃⣿⢸⡇⣿⠿⠐⣦⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠀⣿⣿⡇⣹⣶⠀⣿⣄⢸⣏⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣰⠀⣿⣹⠃⣿⠉⢀⣿⡉⢸⣟⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡟⢰⠇⠟⠰⠟⠋⠘⠋⠁⠈⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⡠⣀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣷⢹⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠶⠶⠶⠶⠶
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⡟⠛⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⢀⠘⡇⢸⠉⣿⠻⢿⣿⣷⣶⣄⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⢸⣤⣿⢸⡆⢿⠀⢀⢿⡉⠛⣿⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⡇⢸⣿⣿⠀⠀⢾⡅⠚⢻⡇⠐⢿⡆⠤⠹⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⣼⣿⣿⡇⣼⢿⡏⢠⠀⣿⠁⠤⣼⡇⠀⣼⠃⡀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠋⢸⡇⣸⢀⡿⠀⢉⣿⣀⣉⣿⣠⣧⣼⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣤⣴⣿⣤⣿⣾⣷⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⢇⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⢫⡾⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⡀⠈⡂⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠂⠀⠀⠂⠖⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢤⠄⠛⠈⠐⢂⣀⣼⠯⣼⡀⠀⠌⠁⠀⣸⠆⠐⠴⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⢧⠄⣤⠀⠐⠎⠀⠀⠳⠀⠈⡁⠀⢀⡴⠔⠀⠈⠐⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣍⠀⠀⠛⠀⠀⢰⠂⢂⠩⡀⠀⠠⠀⠀⠰⠉⠂⢀⢓⠀⠀⠀⠀⠀⡁⠀⠀⠐⠘⠳⠦⠄⢀⠔⡀⠀⣠⠀⠀⢀⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡄⠰⠄⢊⠀⠂⠀⡇⠂⠁⣶⣀⠀⠠⠀⠀⠀⢠⠀⡀⠀⠐⠒⢦⣀⠀⠀⢀⡀⠀⢀⣀⡀⠼⢇⣀⡀⣈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡏⣠⣄⠈⠀⠀⣰⣾⠉⠀⠐⠠⡄⠦⠀⠀⠀⡠⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢸⡆⠈⡁⠁⠈⡠⠊⠁⠈⠑⠀⠀⢀⢁⡀⠋⠈⠩⠁⠀⠀⠥⠀⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠈⠂⠀⠀⠀⠅⡆⠀⠀⠀⠥⠀⠀⠄⠄⠀⠀⢤⡀⠠⠀⠀⠀⠀⠁⠀⠆⠀⠁
⠀⠀⠀⠀⠀⠀⠐⠒⠶⠦⣠⢀⠀⣡⡈⠀⠀⠀⢀⠀⣀⠀⠃⢀⠀⠀⣰⣀⠀⢀⡀⠙⠆⠀⠀⠀⠀⠧⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠆⠆⠈⠋⣃⠉⠀⡀⢰⡟⠀⠁⠀⢀⠍⡀⠊⠁⠀⠀⠚⠄⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠠⣗⠀⠀⠀⠀⠀⢭⡀⠰⠛⠸⠇⠀⠀⠀⠉⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⢠⢠⡖⠡⣄⠠⠀⠙⡠⠀⠀⠈⠪⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⡀
⠀⠀⠀⠀⠀⡄⠐⢶⠀⠱⠆⢸⠂⠀⠀⠉⢈⡁⠂⣴⠆⠀⠀⠐⠲⡈⠀⠐⡊⠄⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠀⠀⠤⣁⣀⣀⣀⠀⠀⠀⠀⢧⡀⠈⠫⠁⠀⠀⠀⠀⠀⠀⠕⠀⠀⠀⠀⠀⠀⠀⢀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠂⠀⠉⠙⠛⡦⠴⠖⠒⠋⠀⠀⠀⢠⠀⠀⠠⠐⠀⠀⠂⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠠⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠂⡀⢉⠴⠡⠄⠀⡠⠀⠑⠄⠀⡀⠀⠈⠀⠀⠤⠂⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠻⠄⠀⠀⠀⡤⠀⢀⡀⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠀⠀⣤⠿⠀⠄⠀⡁⢀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠈⡄⠁⠀⠄⢦⡰⠀⠙⠂⠉⠀⠴⢠⡀⣀⠀⠚⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⣈⠀⢢⡀⠀⠀⡀⣘⡄⠀⠀⠀⠀⠀⢆⡇⠀⡀⠀⡀⠸⠋⠶⠆⠀⠈⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠒⡐⠋⡉⠉⠐⠉⠑⢳⣦⣴⣤⣚⡉⠀⠀⢈⡁⢆⠀⠀⠀⡅⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠈⠋⠔⡲⠦⠄⢸⣇⢧⠙⡇⠐⠀⠉⠀⡈⡗⢀⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠂⠀⠀⠀⠀⠇⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣼⣀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣄⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡀⢰⣦⠀⡟⢦⣠⢄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠑⣤⡭⠧⣧⡚⠁⠀⢹⠁⠹⢄
⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⡀⠀⠀⠀⠲⣦⣄⡀⢇⠀⢰⠍⠀⠀⠀⠙⢆⠀⠀⠀⠀⢨⡦⡴⢤⣀⣀⢀⣀⠤⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡍⠓⠦⣀⠀⠈⢇⠈⢻⣄⣸⣆⠀⠀⠀⠀⠈⠀⣄⣆⣠⠋⠀⠀⠀⠋⠘⢿⣇⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⡀⣾⣧⡴⠚⠿⣿⣿⡿⠿⣄⣖⠙⠢⣄⠀⠉⠛⢿⣦⡀⣠⠖⠛⠛⠻⣿⠓⠒⠲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⡿⠁⠀⣠⠿⣯⠄⠀⠘⠫⠿⢶⢤⣿⢿⣇⠀⣠⣄⣠⣾⣥⠴⠚⠋⠉⣁⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠉⠛⢷⡀⠀⠀⠇⠀⠞⠁⠀⡼⠷⢖⣰⣋⠙⢻⡀⠁⠀⠈⣾⠃⠀⠈⠛⠖⣤⣴⡊⠉⣻⠟⠉
⠀⠀⠀⠀⠀⢀⣠⣤⠒⣾⠀⠀⠀⠀⠀⠁⠐⢓⡀⠀⣀⡤⠾⢧⠀⣜⠉⠉⠉⢢⠱⣄⠀⢀⡎⠀⠠⠄⠀⠈⠉⠛⣇⠴⠁
⠀⠀⠀⠀⠀⠈⠉⢉⣲⣿⣄⣀⡠⠲⡤⣥⣚⣛⣉⠵⠿⠀⠀⣰⠋⣹⣤⠶⣦⣸⣶⠿⣶⣿⣦⠀⠀⣀⣠⣤⣤⣴⣧⠤⣀⡀
⠀⠀⠀⢀⣤⣴⣞⣁⠬⠿⠛⠛⢣⡄⡠⠋⠁⠀⢀⠀⠀⢀⡾⠁⠀⠙⢦⣴⠄⠌⠃⠲⣶⣾⠻⡏⠉⠁⠀⠀⠹⣿⣏⣀⣀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠦⠔⠋⠉⠓⠋⠀⠀⠀⠀⠀⠙⠶⢦⣖⡆⠄⣅⣸⡧⠀⢀⣀⠀⠀⠀⣽⣿⣿⣋⠉⠉⠉⠷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠾⠿⢤⡀⠀⢘⡇⠀⠀⣿⣿⣯⠛⠻⣕⠒⠊⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠧⠴⣾⣿⣿⣷⡀⠀⠈⢳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠌⢿⡌⠙⠒⠒⠾⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⢀⣤⣶⠶⠛⠛⠛⠲⣄⠀⠀⢀⣼⠃⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿
⠀⢠⣾⣿⠟⠁⠀⠀⠀⠀⢀⣾⠄⠀⣾⡏⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⢠⣿⣿⠇⠀⠀⠀⠀⠀⠀⠻⠟⠀⣸⣿⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣴⢢⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠵⣶⡆⢀⣤⡶⠲⣦⠀⣠⡶⠲⣦⠈⢷⠷⣶⣆⠈⢿⣾⣀⠀⠀⢀⣀⣀⣀⠀⡟
⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠃⢰⣿⠁⣾⣿⣁⠴⠋⣼⣿⢀⡾⠋⢠⠀⢠⣯⡇⠀⣸⠻⣿⣿⣏⠁⠀⠀⢠⣿⣆
⠀⠈⠿⣿⣶⣤⣀⣠⣤⠶⠋⢀⣿⡿⠀⢸⣿⠀⢿⣿⣄⣀⣀⣿⣿⡉⢀⡠⠋⠀⣾⡗⣀⠔⢠⠂⠀⢻⣿⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠉⠁⠀⠈⢿⡀⠀⠈⠉⢉⡄⠀⠉⠉⠉⠀⠀⠀⠈⠉⠁⠀⠃⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⢶⣼⡒⢄⢠⠎⣇⢤⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣦⠀⠀⠀⢀⣤⠬⣧⠁⠀⠃⠀⠁⢸⣀⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⡷⣀⣲⠿⠏⠀⠀⢀⣀⣤⠀⠀⠈⠁⠀⡯⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⢿⢦⣤⣴⡶⢿⣿⡜⠁⠀⠈⠉⠓⠀⢀⣾⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣻⠀⠀⠈⠉⠉⠀⣠⣯⠋⠀⠀⠀⠀⠀⠙⢦⣾⣽⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⢦⡠⣤⣒⡿⣿⣛⠉⠁⠀⠀⠀⠀⠀⠀⣼⣝⠱⣤⢤⣄⣀⡀⠀⣠⣿⠋⣿⠠⣖⠛⠂
⠀⠀⠀⠀⠀⠀⣄⠀⢻⡛⠄⠻⠟⠻⠿⣷⣶⠀⠀⠀⠀⠀⠈⢻⣦⡈⠻⣿⡙⠛⠻⠿⠁⠀⣿⣦⣼⣆
⠀⠀⠀⠀⠠⣼⣯⣯⡉⠁⠀⠀⠀⠀⢀⢻⣿⠀⢀⣼⡿⣷⣶⣤⣽⣵⡤⠾⢛⡶⢋⠥⠒⢭⡻⣍⠙⠛⢛⣛⠶
⠀⠀⣠⠴⠚⠉⣿⠘⢷⣤⣤⡴⣶⡿⠋⢸⣿⣠⣯⣿⠿⠟⠛⠉⣁⣠⣤⣶⡟⡱⠃⠀⠀⠀⠙⡌⢶⣾⠁
⠀⢚⣓⣶⣶⠾⠟⠀⠀⠉⠀⣴⣟⣠⡤⠞⠛⠋⣁⣠⣤⣶⣾⣿⣿⣿⣿⣿⢡⠃⠀⠀⠀⠀⠀⢸⡘⣏⣇
⠈⡹⠚⠿⢦⣤⣀⢀⡤⡤⠚⣋⣉⣠⣤⣶⣾⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⡏⡸⠀⠀⠀⠀⠀⠀⠀⡇⢿⠙
⠐⡟⠁⠀⠀⢸⣿⡜⡟⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⡀
⢸⡥⠚⡇⠀⢸⣿⣧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⢸⠁
⠈⠀⠀⡇⠀⣰⢾⠙⠶⠿⢷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢸⠶⠶⠟⣛⣛⣛⣛⡇⣼
⠀⠀⠀⢧⢠⠃⠸⡄⢠⠀⠀⢹⣟⣯⣙⠛⠶⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠸⡌⣶⣿⣿⣿⣿⣿⣿⢰⡏
⠀⠀⠀⠈⠻⠄⠀⠙⣼⢧⠀⢀⠙⠳⢿⣽⣗⣶⢤⣍⣙⠛⣶⣦⣤⣀⣀⠀⠳⡘⣿⣿⣿⣿⡿⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠳⣜⣷⣀⣿⠏⠿⠋⠛⡷⠿⣽⡽⠟⠉⠙⠛⠿⠷⣿⣮⣙⣛⣋⡴⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢠⡇⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⠄⡂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣄⡄⠀⠀⢀⠀⠀⠀⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀⠀⠠⠄⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣤⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⠤⠀⠀⠀⠀⠄⠂⠒⠂⠤⢄⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠔⠂⠉⠉⠡⠂⢀⠑⢄⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠐⠁⠈⠁⠙⡁⠀⠉⠀⠀⠀⠀⠀⠈⡄⠀⠑⠈⠎⠐
⠀⠀⠀⠀⠀⠀⠀⢨⢀⣀⢠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠒⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⠀⢠⠤⠠⠤⠌⡀⢰⡀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠀⠀⠀⠐⡔⠈⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⢁⣀⣤⣤⣠⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⢸⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠐⡇⠀⠞⠉⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⢀⠚⡄⠀⠀⠀⠀⠀⡆⢠⠀⠔⠁⠌⠀⢀⡇⠇⠀⠀⣠⠴⠖⠒⠢⣄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⢁⠀⢀⠠⠠⡀⡇⠈⠁⠀⢂⠀⢀⠞⠁⠸⠀⠚⠁⢸⠆⠀⠀⠈⠳⡄⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⠀⠈⠀⡊⠀⠀⠸⡇⠀⠀⠀⠀⢄⡎⠀⠀⣴⡆⢠⠀⠘⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠈⢆⠀⣷⠀⠀⠀⠀⠰⠀⠀⠀⠀⡟⠀⠀⢰⠁⢱⠈⡐⡇⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠑⠾⠀⠀⠀⠠⠘⠓⠒⠁⠀⠇⠀⠀⣿⠀⠀⡆⢁⠓⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⡜⠀⠀⠀⠀⠀⠀⠘⡆⢸⠀⠀⠀⡏⠀⠀⠛⡀⣀⣁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠀⢸⠀⠀⢰⡇⠀⠀⠀⠑⠈⠈⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠪⣁⣠⠇⠀⠀⠘⡄⠀⠘⡇⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢧⡀⠀⠀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠉⠓⠒⠒⠚⠉⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⢀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⢘⡔⢁⠀⠀⡠⠃⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢰⣶⣾⣛⠀⠈⠢
⠀⠀⠀⠀⢀⡈⠀⡜⢀⠴⠡⣚⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠒⠂⠄⠐⠖⠋⠀⠘⣿⠋
⠀⠀⠀⢠⣼⣧⡄⠀⠀⠀⠉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠰⠋⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠏⠉⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡏
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠿⡗
⣶⣤⣾⠂⠀⠒⡄
⠉⢻⡿⠃
⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⢀⡀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠑⠦⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠤⣉⠊⡆
⣀⣴⣄⣠⠔⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⣁
⠀⣽⣿⡏⠁⠐⠄⠀⠀⡠⠊⠠⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⢀⣷⣶⡾⠆
⠴⠋⠀⠀⠀⢀⡁⢀⣌⠀⢐⠀⠀⠠⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⣿⠟
⠀⠀⠀⠀⡠⠜⢻⣿⡇⠀⠀⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠰⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣷⡀⠀⠀⢀⣴⣾⡿⠛⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⠀⠀⣸⠀⠀⣀⡀⠀⠀⢻⣿⡆
⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⣹⣿⣿⠃⠀⣠⣿⣿⡟⠁⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡉⠓⢶⡿⢶⣿⣿⡿⠂⠀⢸⣿⡇
⠀⠀⠀⠀⢀⣾⣿⣿⠃⢰⣿⣿⠏⠀⣰⣿⣿⡟⢀⡴⠋⠀⠀⠀⢀⣤⣶⡶⣄⠀⢠⣾⣿⡟⢀⡇⠀⡼⢁⣿⣿⡿⠁⠀⠀⣿⡿
⠀⠀⠀⢀⣾⣿⣿⡏⠀⢻⣿⠏⠀⢰⣿⣿⣿⡵⠋⠀⠀⠀⠀⢠⣿⣿⠟⢠⡟⠀⣾⣿⡟⢀⡞⠁⣼⠃⢸⣿⣿⠁⠀⠀⣼⡟⠁
⠀⠀⠀⣼⣿⣿⡿⠀⠀⠀⠙⠀⠀⣿⣿⣿⠃⢀⣴⣿⣿⡆⠀⣿⣿⣟⣰⠟⠀⢀⣿⣿⡷⠋⢀⡼⠃⠀⠈⢿⣿⠀⣀⡾⠋
⠀⠀⢀⣿⣿⣿⡇⠀⠀⠀⠀⢀⣼⣿⣿⡟⣠⠏⣾⣿⣿⠁⢠⣿⣿⡏⠁⢀⡴⠋⠈⠻⠦⠤⠚⠁⠀⠀⠀⠀⠉⣩⣍⡀
⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⣴⠟⢸⣿⣿⣷⠋⢸⣿⣿⢃⡴⠋⠈⠛⠧⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃
⠀⠀⠀⠹⣿⣿⡀⣀⣴⠟⠁⠀⠈⢿⣿⡇⠀⠈⠻⠿⠛⠁⠀⢀⣀⠀⠀⠀⠀⣿⣷⣄⢀⣴⠛⢻⣷⡄⠀⠀⣿⣿⠇
⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⠀⣀⣤⣶⡶⢶⣄⠀⠀⠀⣼⣿⣧⠀⠀⠀⣿⣿⣿⡟⠁⠀⢸⣿⣿⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟⠁⣠⣿⣿⠀⠀⣰⣿⣿⡿⠀⠀⢸⣿⣿⠟⠀⢀⣴⣿⣿⣿⠁⢈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⣰⣿⣿⡟⠀⢀⣿⣿⣿⠃⠀⢠⣾⣿⡟⠀⣴⠋⢰⣿⣿⡟⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⠀⢰⣿⣿⣿⠁⢀⣼⣿⣿⡏⠀⣰⢏⣿⣿⠁⢸⣇⣠⣿⣿⠟⠀⠀⢸⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⠃⢠⡞⢹⣿⡟⢠⡾⠁⢸⣿⡇⠀⠀⠉⠛⠉⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⡴⠋⠀⠀⠙⠛⠋⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠈⠙⢏⠛⠃⢤⠀⠀⠉⠉⠳⣄⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⣶⡟
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⠀⢄⣠⣄⡈⢦⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⢺⠝⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⢦⣀⣃⠀⠉⠋⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⣿⠞⠋⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠓⢄⣀⠀⣨⠟⠒⢀⣀⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠘⠁⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠿⠗⠒⠛⠉⠀⠈⠲⢝⢢⡀⠀⡀⠀⠀⣀⠀⡼⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠳⡑⡾⠁⠀⠀⠈⢏⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠂⠛⠐⠒⠢⢄⣀⠀⠀⠙⣜⡄⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀⠘⣞⡄⠐⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⡏
⠀⠀⢀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⢀⠀⠀⠀⢳⡀⠘⢂⡔⠓⠂⠀⠀⠀⠀⠠⠶⠊⠁⠀⣷
⠀⠀⠈⠑⠤⣄⣤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⢤⡀⢳⠀⣄⢹⠀⠀⠀⠀⡠⠞⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠉⠓⠦⣌⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⡇⢰⠸⠀⠀⣠⠚⠁⠀⡀⡠⠊⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠏⢲⣄⣀⠠⣄⡰⠖⠛⠁⠈⠛⢦⡀⡆⢰⠃⠀⢀⢀⠟⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⡤⠧⢤⣿⠀⠀⠈⠿⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠈⠑⠚⠛⠒⠦⣄⠀⠀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠛⠓⢦⡦⣾⠤⠖⠛⠛⠛⠲⣄⠀⣀⣀⣀⡈⢳⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⣠⠔⠋⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠚⠋⠉⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⡠⠤⣤⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠃⠀⠀⢀⡀⠀⠀⠀⣼⢰⡖⣪⡅⠪⣽⢃⠀⢀⡤⠀⠀⠀⣤⣟
⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⣆⠀⣀⡇⠉⠓⣤⡾⢻⣌⠳⢭⣠⠞⠉⢧⠞⠁⠀⠀⠀⠀⠈⠉⠑⡄
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣨⠟⠋⠀⠀⣼⠋⠀⠀⠉⠳⣄⠟⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠤⠐⠒⠠⡏⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⡏⠀⠀⣠⠞⠒⣒⢦⡄⠀⣠⡶⠶⠋
⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠹⣦⡀⠀⠀⣀⣉⠤⠞⠁⣠⠎⠹⣄⠀⢠⠏⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣄⣀⡠⠞⢋⠟⠉⠀⠀⠀⣀⡾⠁⠀⠀⠀⠉⠉⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠙⠒⠒⠒⢤⣄⠀⠀⠀⠀⠰⢟⣁⢀⣠⢞⠟⠒⠒⠉⢩⠑⢀⠀⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠳⣤⣀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠉⣉⡩⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⠴⠶⠚⠉⠂⠈⠙⠓⠛⠛⠲⠶⠴⠖⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠛⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣴⣿⣿⡆⠀⠀⣠⣾⣿⡿⠋⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣄⡀⣼⢀⣠⣴⣦⣄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣴⣿⣿⡟⢡⣿⣿⣿⠁⠀⣴⣿⣿⡿⠁⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠻⡆⠈⢹⡏⣹⣿⣿⡿⠁⠀⠀⣿⣿⠁
⠀⠀⠀⠀⢀⣾⣿⣿⡟⠀⣾⣿⡿⠃⠀⣼⣿⣿⡿⢁⡴⠋⠀⠀⠀⠀⣠⣶⣿⡟⢳⡄⢀⣾⣿⡿⠁⣸⠇⢀⡟⢠⣿⣿⡿⠁⠀⠀⣸⣿⠃
⠀⠀⠀⢀⣾⣿⣿⡿⠀⠀⢿⣿⠃⠀⢰⣿⣿⣿⠷⠋⢀⣀⡀⠀⠀⣼⣿⣿⠏⢠⡟⠀⢸⣿⣿⢃⣴⠋⢀⡾⠁⠸⣿⣿⠇⠀⠀⣰⣿⠃
⠀⠀⠀⣸⣿⣿⣿⠇⠀⠀⠀⠉⠀⠀⣿⣿⣿⡏⢀⣴⣿⣿⣿⠆⢰⣿⣿⣟⣴⠟⠀⢀⣼⣿⣿⠋⠁⢠⡞⠁⠀⠀⠻⣿⡀⣀⣴⠟⠁
⠀⠀⠀⣿⣿⣿⡿⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⢠⡞⣱⣿⣿⡟⠀⣰⣿⣿⡏⠀⠀⣠⠟⠀⠙⠿⠤⠖⠋⠀⠀⠀⠀⠀⠈⣩⣭⣀
⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⢀⣴⠏⢸⣿⣿⣷⡟⠀⣿⣿⡿⢁⡾⠁⠈⠻⠷⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟
⠀⠀⠀⠸⣿⣿⡇⠀⣀⣴⠟⠁⠀⠈⢿⣿⡟⠀⠀⠻⣿⡷⠋⠀⠀⢀⡀⠀⠀⠀⠀⢻⣿⣦⠀⢠⡶⠛⢿⣶⡄⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⢈⣡⣤⣴⡦⣄⠀⠀⠀⠀⣼⣿⣦⠀⠀⠀⣾⣿⣿⣷⠟⠀⠀⣸⣿⣿⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⣴⣿⣇⠀⠀⣸⣿⣿⣿⠀⠀⠀⣿⣿⣿⠏⠀⢀⣠⣿⣿⣿⡇⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠁⢀⣾⣿⣿⡇⠀⢠⣿⣿⣿⡇⠀⠀⣰⣿⣿⠋⠀⡰⠛⢹⣿⣿⣿⠁⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠁⠀⣾⣿⣿⡿⠀⠀⣼⣿⣿⡟⠀⢀⡾⣿⣿⡿⠀⣼⠉⢠⣿⣿⡿⠁⠀⢠⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢸⣿⣿⣿⠁⢀⡼⣿⣿⣿⠁⣠⡟⠁⣿⣿⠁⠀⠹⠶⠿⠿⠋⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠇⣠⠞⠁⠸⣿⣧⡼⠋⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠧

⠀⠀⠀⠀⠄⠀⠄⠄⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠤⠀⠠⠄

⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⢀⣾⣿⣿⣧⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣿⣿⣿⣆
⠀⠀⠀⠙⢿⡿⡃⠀⢀⣴⣿⣷⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣤⠀⠀⡹⣿⢿⠁
⠀⠀⠀⠀⠈⠀⠹⣦⠋⠙⣻⡇⠀⢠⣦⣦⣀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⣶⠀⠀⣿⡛⠉⢣⡾⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠿⠶⠶⣾⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⡶⠶⠾⠷⣾⣿
⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⠀⠘⠁⠋⡛⠃⠀⠀⣤⣄⠀⠀⠀⠀⠛⠋⠃⠙⠀⠀⠀⣸⣿⣦
⠀⠀⠀⠀⠀⠀⣻⣿⣿⡟⠀⠀⢰⣿⣿⣿⡆⠀⢸⣿⣿⣧⠀⠀⢰⣾⣷⣦⡀⠀⠀⠘⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣏⢹⡇⠀⠀⠸⠿⣿⢿⠧⢤⣜⠏⠉⢛⣠⡤⢼⣿⣿⡿⠁⠀⠀⠀⣿⠁⣿
⠀⠀⠀⠀⠀⠀⢸⡏⠸⣷⠀⠀⠀⠀⠉⠈⠀⠀⠈⠻⣿⠏⠉⠀⠘⠈⠁⠀⠀⠀⠀⢀⡿⠀⣿
⠀⠀⠀⠀⠀⢀⢼⡗⠈⢿⠒⡴⠒⠀⠒⠖⠀⠀⣲⣸⣿⣦⣀⠐⡆⠀⠀⠀⠀⠐⡒⢸⠇⠀⣿⠢⡀
⠀⠀⠀⠀⡰⠁⠈⠃⠀⠘⠄⠒⠒⠒⠞⠛⠛⠛⡟⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠓⠐⡘⠀⠀⠛⠀⠈⢂
⠀⠀⡠⠚⠀⠀⠐⠀⠠⡚⠂⠀⠒⣴⠒⠀⠐⢒⠂⠐⠒⠒⠂⠀⢲⡂⠀⠒⠒⠂⠀⠑⡐⠒⠒⠀⠀⠐⠓
⠀⡌⠀⠀⠊⠀⠐⢄⡀⢻⠀⠀⠸⠀⠀⠀⠀⡏⠃⠀⠀⠁⠉⠃⠘⠀⠀⠀⠠⠈⠃⢐⠈⠀⠀⠨⠅⠀⢀⡘
⠀⠇⠀⠀⠐⠄⠀⠠⠀⣸⠀⠀⠐⠀⠄⠀⠀⠇⠃⠀⠀⠀⣀⠀⠠⡇⠀⠀⠐⢀⡄⠀⢦⠀⠀⠀⠀⠀⠈⠱
⠀⠈⠢⢄⣀⢀⣀⠠⠼⢅⣀⣀⣈⠎⣀⣀⣀⣉⣀⣀⣀⣀⣉⣀⣀⣁⣀⣀⣈⣈⣀⢀⢌⣀⣀⣈⠾⢄⣀⣀⡱

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣁⣀⣀⣀⣀⣀⣀⣀⣈⣉⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡿⢷⣄⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠛⠛⠻⠷⣾⣟⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢙⣷⡾⠟⠛⠋⠛⠛⠿⣶⣄
⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⢠⣿⠛⣷⡌⠻⣧⡀⠀⠀⠀⠀⣸⡏⠀⣰⡿⠋⠻⣆⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠘⣿⡀⠈⠻⣦⡀⣀⣠⣿⠁⢠⣿⠁⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠰⡿⠋⠀⠀⠀⠀⣿⡇⠀⠀⠈⢻⣾⣿⡇⢀⣸⣯⣤⣤⣴⠶⠾⠿⠇⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢿⣿⠟⠛⠙⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣤⣤⣤⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠷⣶⣶⣶⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⣶⡶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠉⠉⠉⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠈⠢⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢠⠊⢀⡠⠴⠒⠒⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠙⣄⢣⠀⢸⠀⡏⠀⠀⡰⣡⠔⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢌⢇⠘⠀⠀⢀⣾⠞⠁⣀⠀⠀⠀⠀⠀⠀⠀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⡈⠓⠄⢀⠼⠚⠓⠒⠲⡟⠁⠠⠊⢀⣀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡏⠀⢛⢀⠘⠃⢸⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠒⢒⣒⠭⠛⠧⡀⠘⠚⠀⢀⡼⢍⠑⠒⠤⣄⣀⣀⣀⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠉⠁⢀⠤⠀⠀⢨⣷⠒⠒⠁⠀⠀⠙⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠴⠁⡰⠁⢀⣿⣿⠀⢰⠀⠳⡀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⠻⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤⢄⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⡿⠟⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⡸⢰⡇⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⣀⠤⠒⠉⠉⠀⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠈⠳⢄⡀⢀⡰⠁⠸⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁⡠⠊⢁⡠⠖⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⡀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⡇⠀⠣⡀⠀⢀⡠⠴⠊⠁⠀⣠⠊⢀⠴⠋⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠑⢄⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠉⠁⠀⠀⠀⢀⠜⢁⡔⠁⢀⡤⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠈⠣⡀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⡠⠃⠴⠋⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠈⠢⣄⠀⠳⡄⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⢀⠜⠁⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡈⠁⠀⠈⠢⡀⠀⠀⠀⠘⡌⡄⠀⠀⠀⢠⠎⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠈⢦⡀⠀⠀⢃⢱⠀⠀⢠⠃⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⠈⠢⣀⠸⡈⡆⢠⠇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢌⡓⢧⠱⣎⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⣤⠏

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢮⣉⣹⣉⣉⣉⡭⠿⢓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⢠⠖⣚⣯⡥⠴⢒⣛⣿⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⣀⣀⣠⠤⠭⠓⠀⠀⠀⢀⣉⠥⠴⠒⣋⡩⢤⡾⡌⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣀⣠⠤⠖⠒⣉⣡⠤⠖⠊⠉⢀⡴⠋⡠⢯⠉⡆
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⢀⡴⡇⢸⣡⠤⠖⠚⠉⠁⠀⠀⠀⢀⡴⠋⡠⠊⠀⠈⢏⢱
⠀⠀⠀⣠⠔⠚⠉⠉⠉⠉⠉⠓⠢⣄⡴⢋⡰⡇⢸⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠊⠀⠀⢀⡤⠾⡌⣏⠉⠉⠉⠓⠲⢄⡀
⠀⢠⠞⠁⣠⠔⣞⠉⣿⠉⢹⠶⡴⠋⡰⣎⠀⡇⢸⠀⠀⠀⠀⠀⢀⡴⠋⣠⠞⠁⠀⢀⡴⠋⢀⣤⢳⠸⣽⡏⠙⡶⢤⡀⠙⢦
⢰⠃⢀⣞⠁⠀⠸⡄⢿⢠⡿⠊⡠⠺⡀⠘⣆⡇⢸⠀⠀⣀⣤⣴⠋⣠⠞⠁⠀⠀⢀⡞⠀⣰⠋⠀⠳⣇⢻⠀⡜⠁⠀⠙⣄⠀⢣
⡏⠀⣼⠀⠙⠲⢄⣳⠼⣏⣠⣾⡶⠯⠷⠒⠛⠋⠉⠉⢹⠁⡶⣌⡟⠁⠀⠀⠀⠀⢸⠀⢰⠋⠙⠒⠦⣽⡼⢿⡤⠔⠚⠉⢹⡄⠈
⡇⠀⣯⠉⠉⢉⡩⢷⣤⡿⣤⠤⠤⣤⣀⣀⣀⣀⣀⣀⣈⣷⢿⣜⢧⡄⠀⠀⠀⠀⢸⠀⢸⠒⠒⢋⡩⢿⢦⣼⠻⢍⡉⠉⢹⡇
⢹⡀⠘⣤⠔⠋⢠⠏⣧⠳⡈⠙⠢⣤⠇⠀⡞⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⠀⠀⠀⠘⡆⠈⢧⠚⠉⢠⠏⢸⠘⡄⠀⠈⠓⡞⠀⣸
⠀⠳⡀⠈⠢⣄⡞⠀⢸⠀⢹⣤⠔⠁⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⠳⢤⡃⠀⡾⠀⢹⣀⡠⠊⠀⡰⠃
⠀⠀⠈⠢⣄⡀⠉⠉⠛⠋⠉⢀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠉⠙⠛⠚⠋⠁⣀⡤⠚⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡄⡀⢀
⠀⠀⠠⣿⡦⡀⣾⠉⢹⣿⡟⠓⢀⣴⣤⡀
⠀⢰⡖⠯⠀⠸⢿⣦⣰⣿⣾⣷⣿⠿⣿⣷⡄⢀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⣿⣷⡶⣶⡶⢾⠿⣯⣾⡿⠋⢀⣤⣌⢿⣿⠃⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⢤⡆⠀⠀⠀⠀⠀⠀⠀⡐⠉⣀
⠀⠉⠹⡀⣈⠁⣠⣾⣿⡟⠀⣴⢋⣼⠇⣾⠇⣰⠏⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃⣜⣾⢁⠀⠀⠀⠀⠀⠀⠠⠁⣰⠁⠀⠀⢀
⠀⠀⠀⣙⣤⣾⣿⣿⡿⠀⣸⡇⠾⢃⣼⡿⠐⠋⢠⠀⣠⢠⠀⡠⢸⠁⠔⠀⠀⡜⠁⡄⡷⠃⠈⡆⠀⠀⠁⢠⠏⢀⠄⠂⢰⠇⡀
⠀⣄⢾⣿⣿⣿⣿⣿⡇⠀⡿⠿⠷⠿⠟⠁⠀⢀⠆⢀⣡⠆⢀⣵⣟⡄⡰⠀⡼⠁⣈⡴⠁⠀⠀⡜⠀⠀⠀⡞⠀⣨⡾⡡⢠⢟⡾
⠀⠹⡎⠻⠛⠛⠛⠉⡄⢀⠀⠀⠀⠀⡀⢀⠇⢀⣄⠘⣣⣦⠘⣫⡾⡀⣱⡆⣡⡀⠛⣡⠀⡼⠀⣠⠀⣼⣀⣴⡀⢛⣽⢇⢨⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⠓⢐⣢⡾⠃⠛⠀⠛⠁⠙⠋⠀⠙⠋⠀⠙⠋⠈⠋⠈⠛⠁⠙⠀⠘⠁⠙⠁⠙⠃⠉⠛⠁⠈⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉







⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⡶⣶⣤⣄
⠀⠀⠀⠀⠀⢶⡦⢀⣀⣀⡀⠀⠀⠀⠠⠴⠋⢈⡉⣻⣷⠀⠀⠀⠀⠈⠿⢇⣤⣤⣤⠀⠀⠀⠄⠐⠋⢀⡤⣈⣿⠇
⠀⠀⠀⠀⠀⠀⣰⠿⠛⠙⢻⣦⠀⣀⠀⠐⢒⡿⣾⡋⢀⣀⡀⣀⣀⣀⣠⣟⣋⡉⢉⡟⠀⡒⠒⢢⣉⣹⠻⣯⣄⠀⠠⡄
⠀⠀⠀⣠⠠⠾⠿⠞⢉⡽⠛⠀⢴⣶⡆⢀⣿⣿⡇⣷⣬⣿⣿⢹⣹⣿⣛⠛⠉⠴⠛⢁⣴⣄⢀⠺⠛⠛⠒⠿⡈⠁⠀⢸
⠀⠀⣠⠴⠶⢿⣃⣴⣶⠀⣶⡷⠶⢭⣿⡿⠿⠛⠛⢿⡻⢿⣧⣿⣿⣿⣿⣴⣿⡇⠐⢿⣿⠿⠏⢀⡤⠶⠶⢶⡌⣷⠶⡅
⠀⢰⠃⠀⠀⠀⠙⡆⠀⢹⢇⠁⣠⡾⠋⣇⢀⣀⠀⣡⣿⠀⡞⠀⠀⠀⠀⠘⡆⠀⢄⠀⣀⠶⠛⢀⣇⡤⢖⣀⣾⠇
⠀⠸⡀⠀⠀⠀⢀⠇⠀⠘⢧⠚⠡⠶⣶⡿⣧⣴⡿⠟⠃⠀⠹⣄⠀⠀⠀⠜⠁⠀⠈⣂⣁⡚⢹⡿⠛⣶⣿⡟⢉⣀⣀⣀⡀
⠀⠀⠉⠲⠶⠂⠀⠀⢠⡴⠚⠛⠓⢶⡏⠀⠸⣿⣓⠶⡚⠋⠛⠻⢯⡉⠀⠀⠀⠴⠋⠉⠈⠉⠻⡅⠀⠸⣿⡽⠻⣅⠀⠈⠙⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠹⡆⠀⠙⣏⠀⠈⠂⠀⠀⠀⣧⠀⠀⢸⠃⠀⠀⠀⠀⠀⢸⡄⠀⢹⡇⠀⠀⠁⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠇⠀⠀⠃⠀⠀⠀⠀⠀⢀⠏⠀⠀⠸⣆⠀⠀⠀⠀⠀⡸⠁⠀⠀⠣⡀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⠄⠊⠀⠀⠀⠀⠑⠤⣤⡤⠴⠋⠀⠀⠀⠀⠈⠳⠦⠤⠖⠈⠀⠀⠀⠀⠀⠈⠒⠖⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡶⣄
⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠁⠉⢳⡄⠀⢀⣤⠴⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢹
⠀⠀⠀⠀⢠⣾⡿⠁⣠⡤⠀⠀⢸⡇⣴⣿⠃⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⢸⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⣾⣿⠁⠸⣿⡁⣀⣠⠟⢸⣿⠇⠀⢀⡞⠁⠀⣀⣀⣀⠀⢀⣠⡤⢬⡻⢿⣯⣀⡀⠀⢰⣿⣿⠋
⠀⠀⠀⠸⣿⣿⠀⠀⠀⠉⠉⠀⠀⣿⣿⠖⢚⣁⡀⠀⣼⣿⠋⢈⣇⣾⡿⠁⣰⠇⡼⢩⣿⡟⠁⠸⣿⣿
⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢀⠞⣿⣿⣠⢿⣿⡇⠀⣿⣿⠤⠊⢸⣿⡟⠉⠁⡴⢁⣿⡟⠀⠀⢠⡿⣿⣧
⠀⠀⠀⠀⠈⠻⢧⣀⣀⡤⠖⠁⠀⠸⣿⡏⣾⣿⠁⠀⡟⠿⠦⠴⠋⠙⠻⠶⠚⠁⢸⣿⠃⠀⢀⡞⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⣠⠎⣼⠁⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡤⣄⢠⣶⠀⢠⣤⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠃⢀⣧⠸⡞⢯⡴⢺⣉⣤⡼⠉⢡⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠲⠞⠈⠀⢁⠀⠀⠀⠀⠈⠁⠰⣬⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠀⠸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⣁⠠⠤⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠤⠤⣀⣘⡶⠤⠤⡀⠀⠀⠀⠙⠂⠁⠀⠀⠀⣀⣀⣼⣆⠠⠤⠄⡀⣀⣀
⠲⠶⠤⠤⢄⣀⣀⣀⢀⡠⠂⠉⡀⠈⠁⠰⠓⡶⣼⠁⠀⠀⠈⠑⠲⡀⠀⠀⠀⡤⠐⠋⠀⠀⠸⡴⡶⠲⠂⠉⠀⡀⠉⢢⣄⠀⣀⣀⣀⣠⣤⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠃⠀⠀⢹⣿⠀⡘⠒⣧⣷⠂⣤⣄⢀⡀⠀⠙⡶⣀⢸⡄⠀⠀⠀⣀⣤⣧⡷⠚⠀⢨⣿⠁⠀⠀⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⡜⠉⠓⠃⢶⣿⡏⢰⣿⣿⣿⣿⣿⢶⢃⢸⢭⠀⣶⣷⣿⣿⣿⣿⣷⠔⠒⠛⠉⣆⠀⠀⢦⡀
⣀⣀⣀⣀⣀⣀⣀⡀⢎⠀⠀⠐⣿⠉⠁⢠⡾⣼⢡⣿⣿⣿⣿⣿⡿⡞⠈⠀⡞⡇⢹⣿⣿⣿⣿⣿⣿⣦⠀⠉⢈⡗⠀⠀⠈⠇⣀⣀⣀⣀⣀⣀⣀
⠉⠉⠉⠉⠉⠉⠉⠉⠛⠿⠦⡤⠼⣯⡐⡺⢹⢇⣾⣿⣿⣿⣿⣿⣿⠁⠀⠀⢸⢻⠈⣿⣿⣿⣿⣿⣿⣿⠒⣨⡿⢤⠤⠾⠟⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡃⠈⠚⠿⢿⣿⣿⣿⣿⡟⠀⠀⠀⠈⡾⣇⢿⣿⣿⣿⣿⠿⠃⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠶⣶⣤⠀⠀⢠⠇⠀⠀⠀⠀⢧⠉⠘⠋⣉⣩⣤⡤⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠀⠀⠀⠀⠀⠘⠂⠒⠒⠉⠉
⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡥⠥⠤⠤⢤⣤⣤⣤⣤⣤⣤⠤⢤⣵⡞⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢵⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠘⣧⠐⠀⠀⠀⠀⠈⠐⠂⢄⣄
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠁⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠨⢷⡀
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣇⠀⢹⡄⢣⠀⠀⣠⠞⠁⠀⠀⠂⠀⠀⠀⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣤⣤⣄⣀⣀⠐⢾⠀⠀⣿⡼⣤⠞⠁⠀⠀⠀⣨⠀⠀⠀⠀⠀⠙⣶⣄⠀⠀⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠉⢉⣙⡛⠛⣺⠛⢿⡾⣿⡛⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠈⠻⠁⠂⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠏⠀⠘⠳⡧⠝⠁⠀⠀⠀⠀⢹⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠙⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⢹⣦⣄⣀⡀⠀⣀⣀⣠⣶⣯⣀⣀⣀⠀⠀⠈⢁⠀⠀⣀⡀⢀⣀⣀⠀⠈⣿⣴⣢⠤⠀⠀⠤⠤⣀⣶⠿⡁
⠀⠀⠀⠀⠀⠀⠐⠛⠿⠿⠟⠛⠛⠛⠟⠛⠛⠏⠛⠁⠐⠒⠐⠀⡂⠀⠘⠷⢏⡞⠉⠉⡙⢃⠙⠺⠿⠟⠛⠿⠟⠿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡖⠁⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⢰⣿⡟⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⠁⡠⠞⠀⣠⠖⠒⢦⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⣀⣠⣿⡿⠋⠀⠀⠀⣇⠀⠀⠈⡇⢀⣴⣶⠀⠀⡼⢀⣴⣶⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠀⣿⣷⠀⠀⢀⣴⣿⠓⠒⣾⠗⠋⣿⡟⢀⡼⠁⣼⣿⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡤⠀⠀⠀⠀⠀⠘⠻⠷⠖⠋⢻⣿⣀⡴⠃⠀⠀⠹⡧⠟⠁⠉⢻⣿⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣤⣶⠶⢦⡄⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣼⡿⢱⠋⠉⢹⠓⢤⣀⠀⠀⠀⠀⠀⣠⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠉⡆⠀⣿⡇⠈⠢⠔⠃⠀⠀⠉⠻⢿⣶⣶⣿⠟⠁⢰⡿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠴⠃⢠⣿⡇⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⢹⣇⡾⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠘⣿⡆⡞⢹⣿⡇⠀⣠⡿⠋⡇⠀⣿⡇⣠⠇⠀⡽⢓⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⣀⢿⣾⠀⢸⣿⡇⠀⣿⡗⠋⢀⡼⣿⣏⢁⡠⠎⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⢠⠇⠈⠛⠁⢸⣿⠀⠀⢸⠋⠋⠉⠀⠈⠉⠉⠀⠀⠀⣿⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠤⠴⠋⠀⠀⠀⠀⠘⠿⣦⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠐⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⠀⠀⠀⠀⣀⣀⣀⣠⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠠⠈⢉⣭⣄⠉⣦⢠⡶⠊⠙⢻⢳⣶⡾⠉⢻⠀⠒⠀⠀⠀⠸⠿⢿⢀⠿⢟⣉⠀⠈⠻⠿⡇⢰⠾⣋⡁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⢀⡘⠿⠟⢀⠟⣹⣴⣿⡆⡞⡜⢠⣶⣦⡏⠀⠀⠀⠀⠀⣴⣶⡾⢸⠐⣿⣿⢇⣠⣶⣶⠇⡇⠸⣿⡿
⠀⠀⠀⠁⠀⠀⠄⠛⠉⠒⠁⠀⣇⠈⣉⣡⢃⠇⣈⡛⠻⠁⠐⠄⠀⡀⢀⡠⠀⠀⣾⣷⡄⣠⣏⣀⠤⡄⢰⣷⣶⡄⠀⠀⠀⠀⠐
⠀⢰⠢⡀⡀⠀⠀⠀⠀⠀⠀⢠⠟⠀⢿⡿⠞⢸⣿⠟⠀⠀⠀⠀⠀⠡⠟⢠⠀⠀⠋⣉⡀⠻⠿⠃⣾⠇⣸⠙⠋⠀⠠⠨
⠀⣸⣦⠀⡠⠀⠀⠀⠀⠀⠀⠈⢘⠿⠆⠀⠐⠀⠀⠀⠀⠀⠀⠀⢠⠔⠂⠀⠀⠀⠀⠉⠛⠀⠒⠒⠋⠀⠁⠀⢀⠃⠀⠀⠀⢀
⠀⢻⠋⣀⣙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢀⣤⡈⠉⢀⡐⠀⠐⠀⠀⠀⠀⠀⠀⠁⠀⢀⠀⢀⠀⠀⠀⠁⢼⣇
⠀⠘⣄⢿⣿⠃⢀⠀⣮⡄⠀⠀⠀⠅⠀⠀⠀⣀⡂⡀⣰⣀⠸⠿⠟⠰⣿⣿⢂⣄⡀⠀⠐⠀⠀⠀⡆⠀⠀⠀⢤⠀⢳⣿⣿⡦⠟
⠀⠀⠈⠲⠤⠼⠿⠛⠁⠀⢸⠷⠤⠒⠂⡦⠤⠝⠓⠃⡿⠿⠓⠒⠊⠉⠉⠉⠸⠿⠟⠒⠒⠉⠁⠀⠀⡿⠿⠛⠚⠀⠈⠯⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⡛⠉⠉⣿⣿⣷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⣿⣿⣿⣿⣿⡙⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⢿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠉⢿⡿⠾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣀⣀⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⠽⠛⠉⣿⠁⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣏⣹⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣾⣿⣿⣿⢿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠈⠳⣄⠀⠀⠀⠀⠀⠀⠻⣄⣀⠟⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠁⠀⣠⠴⠖⠚⠛⠋⠙⠛⠓⠲⠦⣄⠀⠈⠓⠒⠋
⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⠻⠭⣉⠙⠒⠒⠶⠤⣤⣄⡀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⢀⣠⣤⠤⠴⠒⠒⠋⣉⠭⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣄⡀⠀⠀⣹⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⣏⠀⠀⢀⣠⠴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠈⠓⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⡀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢀⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⡸⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢇⠀⠀⠈⠙⠲⠦⣄⡀
⠀⠀⠀⠀⢀⣠⣴⣚⣉⠤⠤⠴⠒⠚⠋⠉⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠉⠙⠒⠒⠢⠤⠤⣉⣓⣦⣄⡀
⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⡀⠀⠉⠓⠲⠤⢤⡤⢤⡤⠤⠖⠚⠉⠀⢀⡴⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢀⠇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡴⠃⠀⠀⠀⠀⠀⠀⣼⠉⠉⣧⠀⠀⠀⠀⠀⠀⠘⢦⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣿⣿⡿⠻⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⣀⠀⠤⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣩⠶⡀⠰⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠆⠀⠀⠀⠀⡄⠀⠂⠀⠀⠀⠀⠀⡴⠀⠈⠐⣀⣧⢞⠁⢠⠀⠀⢀⠠⠀⠀⠄⠚⢢⡄⠊⠉⠹⡄
⠀⡲⠐⠄⡀⠀⢀⣼⣗⠇⠈⠀⠀⠈⠀⠀⢧⠀⢀⢸⠏⠜⢈⣶⡋⠈⠓⠀⠀⣠⠞⠀⠀⠀⡁⠀⠀⢠⢃⠄
⠀⠸⡂⠬⣟⣴⡯⠦⠯⠋⠀⠀⠀⠀⠀⠂⠈⠿⣿⣴⣮⠜⠉⠜⢁⠠⠀⠀⠀⠘⢄⡢⣀⠢⠁⣀⣀⡜⠁
⠀⠀⠄⠀⡐⡿⠗⠀⠀⠤⠀⠀⠀⠀⠀⢀⡴⠚⢹⠋⠟⣛⣿⢣⡀⠀⠀⠀⠀⠀⠀⣉⡻⠛⢻⠛⠓⡀
⠀⠀⠀⠈⠁⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠜⠀⠀⠀⠈⡆⠀⠉
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⡄⠁⠀⠀⠀⠀⠀⠀⠐⠄⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢀⡀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠄⡷⠀⠀⠀⠀⡈⠄⠴⠦⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠀⠀⠀⣁⣤
⠀⠀⠀⠀⠀⠀⠀⢠⣇⢀⡐⠡⠄⠒⡸⢏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⡘⠓⢄⠀⠠⠀⣠⠞⠀⠀⠐⢄⠀⠀⠀⠈⣽⠃⠀⠀⠀⠀⢀⢿
⠀⠀⠀⠠⠂⠀⠀⡼⠁⠑⡇⣇⢀⠎⠀⠈⢃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠨⡇⡄⠜⠁⠀⠀⠀⠀⠀⠇⠀⢀⡼⠁⠙⡄⠰⣠⠔⠁⠀⠂⠀⠈⠂
⠀⠀⠐⠀⠀⠀⠼⠁⠀⠀⢴⡤⠃⠀⠀⠀⡄⢦⠀⠀⠀⠃⠀⠀⠀⠀⠀⠾⠀⠀⠀⢐⣧⠟⠀⠀⠀⠀⠀⠀⠀⠈⠀⠚⠁⠀⠀⡇⣴⠛⡄⠀⠀⡈
⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠎⠑⠒⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⢀⡠⠊⠉⠒⠁⠀⠀⠀⠀⠀⢀⠜⠀⡀⠀⢀⣰⠃⠁⠈⠁⠀⠀⠀⡀⠔⠁
⠀⠀⠀⢀⣂⣤⣠⣤⣖⡉⠠⠤⠤⠤⠤⠤⠠⠀⠀⠊⠀⠀⠀⠀⠲⠴⠶⢒⣚⡉⠀⠀⠤⠤⠤⠀⠀⠒⠐⠊⡀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠀⠈⠈⠁






⣷⣶⡶⠶⢾⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⣀⣀⣀⣤⣤⣤⣀⠀⢀⣤⣤⣤⣤⣤⣄⠀⢀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡀⠀⢿⣿⣿⠇⠘⣿⣿⡿⠀⠹⣿⣿⠟⠛⠻⣿⡆⠘⣿⣿⠋⠉⠙⢿⡇⠈⣿⣿⡟⠛⢻⣿⣆
⠀⠀⠀⠀⣾⣿⡟⠀⠈⠟⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⣿⣿⣤⣴⠀⠈⠀⠀⣿⣿⠿⠿⠀⣀⠀⠀⣿⣿⣷⣶⣾⣿⠃
⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⣦⣴⣿⣿⡇⠀⠀⣿⣿⠟⠛⢠⣶⡄⢰⣿⣿⣦⣴⣾⣿⡇⢠⣿⣿⣇⠀⢿⣿⣆
⠀⠀⠀⠸⣿⣿⣧⠀⠀⢀⡀⠀⢸⣿⣿⠋⠙⣿⣿⡇⠀⠀⣿⣿⣦⣴⣿⣿⠃⣿⣿⣿⣿⣿⡿⠟⠁⠻⢿⣿⣿⠀⢸⣿⣿⡄
⠀⠀⠀⠀⣿⣿⣿⣷⣶⣿⣿⠀⣸⣿⣿⠀⠀⣿⣿⣷⠀⢸⣿⣿⣿⣿⡿⠋⠀⠉⠉⠁⠀⠀⢀⣤⣤⣤⣀⠀⠀⠀⠈⠻⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⢀⣿⣿⣿⡆⠀⣿⣿⣿⡆⢻⣿⡿⠟⠉⣠⣶⣾⣷⣆⠀⠀⢀⣾⣿⣿⣿⣿⠇⠀⣾⣿⣶⣄⠉
⠀⠀⠀⠀⣤⣈⠻⠿⠿⠿⠋⠘⣿⣿⣿⠇⠀⢿⡿⠟⠁⢀⣤⣤⠀⠘⣿⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣿⡟⠀⠀⢹⣿⣿⣿⡇
⠀⠀⠀⠀⢻⣿⣿⣦⡄⠀⠀⠀⠈⣉⣡⣤⣤⣄⠀⢠⣾⣿⣿⡏⠀⠀⢹⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠸⣿⣿⣿⣿⡆⠀⢀⣼⣿⣿⣿⣿⠏⠀⠀⢿⣿⣿⠀⠀⠀⢸⣿⣿⡟⢻⣿⣿⣿⠋⢻⣿⣿⡇⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⢿⣿⣶⣾⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⠀⠀⠀⣸⣿⣿⣧⠀⢻⣿⡏⠀⣼⣿⣿⣇⠀⠀⢰⣿⣿⣧
⠀⠀⠀⠀⣼⣿⣿⣿⡀⠹⣿⡿⠃⢨⣿⣿⣿⠀⠀⢠⣿⣿⣿⣇⠀⢠⣿⣿⣿⣿⡆⠈⠉⠀⣰⣿⢿⣿⣿⡀⢠⣿⣿⣿⣿⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠈⠀⣰⣿⣿⣿⣿⣧⠀⣿⣿⣿⣿⣿⠆⢾⣿⣿⣿⣿⡇⠀⢼⣿⣭⣾⣿⢸⣿⡇⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠉⠙⠛⠋⠉⠀⠈⠉⠛⠛⠋⠁⠀⠸⣿⣿⠟⠃⠛⠋⠀⠀⠉⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣀⣀⡀⢀⡀⠠⠤⡄⠀⢀⠤⠤⠄⠀⠲⡀⠀⠤⠤⠤⠤⠤⠄⠀⠠⠄⣀⡀
⠀⠀⠀⡠⠒⠉⠁⢸⡆⠀⠀⠀⠀⢻⢸⠀⠀⠀⢱⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢀⣀⣰⢀⡇⠀⠀⠀⠉⠑⢦⡀⠀⠀⢀⡀⠀⣀
⠀⢠⠎⠀⠀⣀⠤⠼⢳⢁⠀⠀⠀⢸⢸⠀⠀⠀⢸⠀⢸⠀⠀⢐⣏⣙⠀⠀⠀⠀⠀⣧⣭⠋⣼⡇⠀⠀⡴⡄⠀⠀⢱⢠⡞⠁⠀⠀⠀⠑⠢
⠀⡎⠀⠀⢸⠡⠂⠀⠈⠸⠀⠀⠀⠈⠉⠀⠀⠀⢸⠀⢸⠀⠀⠀⠁⠉⡄⠀⠀⠀⠀⠀⣸⠀⣿⠀⠀⠈⠛⠁⠀⠀⣸⢻⠀⠀⠀⡤⠤⣀⠜
⠀⡇⠀⠀⢸⡀⠀⠀⠀⠀⡇⠀⠀⠀⡖⢆⠀⠀⢸⡆⢸⠀⠀⠀⣖⣒⡇⢸⠀⠀⢸⣛⣁⢰⡏⠀⠀⠀⠀⢀⡤⠚⢱⠙⣄⠀⠀⠙⠶⡀
⠀⢳⠀⠀⠈⠻⣄⣀⣄⠀⢃⠀⠀⠀⣗⢻⠀⠀⠈⡧⣼⠀⠀⠀⠀⠈⠈⢾⠀⠀⠀⠀⢸⣿⠁⠀⢀⡀⠀⠘⡆⠀⠀⠳⠬⡙⢆⠀⠀⠘⠂
⠀⠈⢧⡀⠀⠀⠀⠀⢸⢢⢸⣀⣀⣀⡼⠞⠦⠤⠞⡇⠙⠒⠒⠒⠒⠺⠀⠘⠒⠒⠒⠒⢺⡛⠒⠤⠞⣇⠀⠀⡇⣴⠋⠓⠒⠚⠃⠀⠀⡸
⠀⠀⠀⠙⠦⣤⡤⠴⠚⣳⠄⠡⠥⣤⠜⠃⠘⠀⠐⠁⠈⠁⠀⠁⠉⠉⠀⠈⠁⠀⠁⠁⠈⠙⠒⠈⠷⣁⣉⠙⡛⠯⣀⠀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠀⠃⠄⠤⡽⠐⠒⠚⠛⠛⠋⠁
⠀⠀⠀⠀⠠⠄⠀⠀⠀⠂⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢒⡩⠕⠋⠀⠱⡀⠀⡰⠁⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⡂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣄⠀⠀⣀⢔⣊⠥⠖⠊⠁⠀⠀⠀⠀⠀⠹⡄⠁⠒⠲⡄⠀⠀⠀⢀⣠⣄
⠀⠐⠒⠀⠀⠀⠀⠆⡁⠁⠀⠀⠀⡀⠳⠮⣀⡒⠤⢄⣉⠃⠛⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⡴⠊
⠠⠤⢤⠤⠀⠀⠀⠀⠈⠢⠀⠀⡼⠀⠀⠀⠀⠈⠉⠒⠤⠭⢑⣒⣤⣧⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⠑⢢⡄⠀⠀⠀⠈⠉⠉⠁
⢀⣀⣀⡀⠀⠀⠀⡤⠊⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⢾⣶⡄⠀⠈⢇⢸⠩⠈⠑⠂⠤⢀⡀⢀⠜⢉⠆⡿⡠⠒⠁⠀⠀⠀⠀⠐⠒⠶⠂
⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⡏⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡷⢠⠁⠘⢦⠑⢄⠀⠀⠀⠀⠀⠀⡠⠊⣰⠇⠈⠢⡀⠀⠀⠀⠀⠠⠤⡄
⠀⠀⠚⠛⠁⠀⠀⠀⡀⠁⠀⡇⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⣿⠇⡎⠀⠀⠀⠓⢤⡈⠐⠀⠀⠒⠉⢠⠴⠃⢀⠤⠐⠉⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠚⠁⠀⠀⠀⠀⢀⢷⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠏⡼⠁⠀⠀⠀⠀⠀⠉⠛⠒⠒⠛⢦⢸⡀⠀⠀⠀⡀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⠈⢷⣌⠻⠿⣿⣿⡿⠿⠛⣡⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢳⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⢀⡤⠤⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣇⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢀⡏⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡇⠈⢆⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢰⠧⠀⠂⢱⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠠⢀⠀⠇⠀⠀⠉⠀⠀⠘⣎⢧⢀⣠⠤⠴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡏⠀⠀⠀⠀⡧⠂⠁⠀⢄⠠⠁⠀⢀⣠⠁⠀⠙⠀⠀⠀⠀⠀⠀⣀⢼⠌⠃⣀⣤⡶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢀⠔⢋⣡⣤⡶⠟⠛⠉
⠀⠀⠀⠀⠀⠀⡤⠖⠤⡤⣄⡸⢱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣁⠘⠉⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠶⠶⠾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠒⠒⠀⠲⣖⡶⠒⢲⠖⠒⠒⠒⠒⠒⠒⢒⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⢀⠤⡄⠀⠀⠎⠂⠉⢦⠈⠀⢠⠛⣄⠠⠒⠦⣀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠉⠀⠀⠀⠀⢸⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠒⠦⡀⠀⠀⠀⢸
⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢡⡀⠐⠀⠀⠘⡄⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⢰⠀⠀⣠⠃⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⠒⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠣⣀⠤⠆⠒⠊⠉⠀⠈⠉⠉⠉⠉⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠉⠢⣄⡀⠀⠀⢀⣀⡇⠀⠀⣇⠀⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⡀⢸
⠀⣂⣀⣀⠀⠌⠀⡄⢨⣶⣀⡉⠦⢀⠀⣀⣀⡀⠀⠀⠉⠒⠉⠀⠀⡇⠀⠀⡏⠉⠁⠀⠈⠉⠁⠐⠒⠁⠀⠈⢣⣀⣀⠔⠒⠢⠖⠊⠁⠀⠀⠈⢾
⠀⠀⠀⠀⢈⠀⡘⢝⢸⣇⡜⢧⡤⠀⢩⠀⠀⠈⠉⠉⠁⠀⠀⠀⠈⡇⠀⠀⡏⠉⠉⠁⠉⠉⠀⠀⠐⠒⠒⠒⠂⠒⠒⠒⠒⠒⠒⠒⠒⠀⠀⠒⢸
⠀⠀⠀⠀⠈⡶⢣⡀⡱⡊⠈⢉⠇⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠙⠠⣈⣚⣒⣚⠩⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⡀⢀⢆⠀⣾⠅⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣶⣾⣷⣶⠶⠶⠞⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣀⡐⣌⣾⣈⠀⡯⠀⣜⣰⠕⢲⠆⣀⣤⣴⣾⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⡋⢰⡥⣟⡹⢒⡷⢻⣹⣏⣿⣝⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⣄⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠘⣏⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⢦⡘⢶⡿⢈⣷⣫⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣿⣿⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡒⣾⣆⠃⠋
⠀⠀⠀⢠⣴⣶⣶⣶⣾⣶⣦⣸⣿⣿⣿⣀⣀⠀⠀⠀⠀⠀⠀⠙⠶⠚⠀⠀⠀⠀⠀⠀⠀⣀⠛⠶⠟⠀⠀⠀⠀⠀⠀⢲⣤⡀⠀⣥⣭⣿⡆⠺⠟⠁
⠀⠀⠀⢼⣿⡟⣻⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣄⠘⠻⢋⢻⣷⣶⠆
⠀⠀⠀⢾⣿⣸⣿⣿⣿⣿⣃⣼⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣇⢰⣿⣿⣿⠟
⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠙⢿⡿⡾⠿⠛⠁
⠀⠀⢰⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠓⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣿⣿⣆⠀⠘⢀⣐⣒⢢⣄⡀
⠀⠀⠀⠙⠺⣬⣿⣿⣿⠟⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⣾⡟⢿⣿⣠⣴⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣶⣦⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⣭⣿⡟⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⡿⣏⡿⣿⣿⣿⣒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⣿⠃⣴⠀⠀⠀⢀⣴⣤⣄⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⡜⢿⣿⠟
⠀⠀⠀⠀⠠⣤⡀⠀⢀⢀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠓⠀⢸⡿⠀⠀⠀⠹⣿⣿⡿⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⡀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣷⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣾⠁⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠻⠿⢧⠄⠀⠀⠀⣾⣤⠀⠁⠀⠀⠀⠀⢀⣿⣿⠿⠃⢸⣿⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡄
⠀⠀⠀⠀⠀⢤⣤⣶⡚⣠⣶⠀⣠⠀⠀⠀⢠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⣾⣿⣿⣿
⠀⠀⠀⢴⣦⡈⢻⣿⣿⡟⠁⠈⠁⢠⡜⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣼⣿⣿⣿⡿⠀⠀⠀⢠⣼
⠀⠀⠀⠸⣿⣿⡄⠈⠛⢿⠟⠃⢤⣜⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⡟⣿⣿⣿⣿⠇⢠⢀⣴⠿⡟
⠀⠀⠀⠈⢻⣿⣿⡏⢀⣰⣾⣆⡰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠈⣿⣿⣿⣿⠟⢀⣼⣿⡿⠞
⠀⠀⠀⣀⣼⣿⣿⣷⡀⠈⠉⠉⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⢄⠹⣌⠛⢃⣴⣤⣤⣤
⠀⠀⠀⠉⠻⢿⣿⣿⣇⠀⢀⣴⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠂⠉⠁⠈⢉⣥⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠠⠼⠿⠂⢸⣿⣿⣿⣿⡿⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠿⠂⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⠻⠛⠀⠠⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠛⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠿⠿⠟⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣄⠀⠀⣠⡶⠆⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡉⠉⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠈⠀⠘⢿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢋⡗
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⢤⣤⡶⠲⣄⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⡀
⠀⠀⠀⠀⠀⣴⠟⠁⠀⢀⣾⡏⠱⢀⡇⠀⣼⣿⠁⢀⣀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⢀⣀⠀⠀⢸⣿⣇⠀⠀⣀⣴⡶⢀⣴⡶⢦⠀⠀⠀⢹
⠀⠀⠀⠀⢸⡏⠀⠀⢀⣾⡟⠀⢀⡞⠀⢠⣿⣿⡴⠛⣿⣇⠀⣰⣿⠋⢀⡇⠀⣠⣾⠋⣩⠇⣰⠋⠙⠛⠛⣻⣿⠏⠀⣼⣿⠀⡼⠀⠀⠀⡼
⠀⠀⠀⠀⠘⢷⣄⣀⣸⣿⣇⠴⠋⠀⠀⢸⣿⡏⠀⢀⣿⡟⢰⣿⡧⠖⠋⠀⢰⣿⠷⠚⠁⣰⠃⠀⠀⠀⣼⣿⠃⠀⠀⠘⣿⣿⡀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⣿⠇⢸⣿⡁⠀⢀⡴⣿⣿⠀⠀⡴⠁⠀⠀⠀⢰⣿⡇⠀⢀⡴⠊⠈⢻⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⠻⣿⡇⠀⣿⡿⠀⠀⠻⠷⠶⠋⠀⠘⠻⠶⠊⠀⠀⠀⠀⠀⠈⠻⠷⠞⠉⢀⡤⠖⠋⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⣀⠔⠁⠀⠻⠷⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠔⠒⠒⠒⠒⢦⡀⠀⠀⣠⠞⠁⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠹⣷⣤⣤⠴⠖⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⢰⠃⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⢿⡀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠚⠋

⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠎⠀⠴⣾⣿⣿⡇⠈⠃⠉⠛⠂⠀⠀⠂⠀⠀⠀⠰⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠤⠒⠚⣸⣿⣿⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇⠀⠈⡇⠀⠀⢀⠀⠀⠀⠈⠀⢀
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠞⡿⠋⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼
⠀⣀⠔⠂⠀⠦⢄⠀⠀⠀⠀⠀⢀⣠⠤⠄⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⠠⡀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⡄⢀⣾⡟
⠴⠁⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠈⠙⢂⣀⣀⣀⡀⢀⡠⠐⠒⠦⢄⠀⠀⠀⠑⠇⡀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢠⣤⣾⡿⠋
⠆⠤⠄⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢡⠀⠀⡀⠀⠐⢄⣀⣀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⢤⣶⣶⣼⣷⣷⣤⣾⣿⣿⣿⠖⠈
⠀⠀⠀⠀⠠⠀⠀⠐⠠⠀⠄⢀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠉⢳⡎⠀⠁⠈⠀⠀⠀⠤⢀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠛⢛⣟⣿⣿⣿⣷⣶⣖
⣦⡀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠎⢳⠀⠀⠀⠀⣠⠤⠀⠠⠤⢺⣿⣿⠘⠻⠟⠿
⠀⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⠊⠁⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢘⣆⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠄⠀⠀⠀⠀⠀⠀⡘⠁
⠀⠀⠀⠀⣀⠐⠀⠀⠀⠀⠹⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣻⣷⡀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣤⡄
⠀⠀⠀⠈⠀⣦⢄⡀⣀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣤⡄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠐
⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠤⠱⠶⣦⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣿⡇⠀⣾⡇⠀⢀⠔⠀⡠⠊⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠢⣀⣀⣀⠀⠉⠙⢿⣿⣿⣇⢰⣿⣷⣾⣿⡵⠄⠀⠀⠀⠀⠀⠀⠀⡚⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠘⠛⠛⠛⢻⣿⣾⣿⡿⢿⡅⠀⠀⠀⠀⠀⠀⠀⣀⠔⠀⠀⠀⠀⠀⠂⠸⡄⠀⠀⡼⠀⠀⣠⡄
⠀⣠⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣯⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠁⠈⠁
⠀⠠⢀⣸⣆⠀⠀⠀⠀⠀⢉⡂⠀⡀⠀⠀⠀⠀⠀⠀⠠⠶⠒⣈⡽⠟⠉⠉⠻⣧⠀⠉⠀⠀⠀⠶⠤⢄⠀⢀⠌⠀⠀⠀⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣠⠂
⠀⠀⠀⠀⠹⠞⠀⢀⠂⠀⠀⠀⠀⠀⠀⠐⠢⠤⠤⠴⢒⢀⠝⠉⠡⠚⠐⠆⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠂⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢠⠀⠀⠀⠀⠀⠀⣠⣄⡀⢸⠀⠀⠀⠀⠀⢀
⠀⣀⠀⠀⠀⠀⠀⠀⠐⠆⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠋⠉⠉⠻⢟⢿⣇⣠⡀⡀⠀⠀⠀⠀⠐
⠊⠁⠀⠀⠀⢀⠀⠀⢀⠈⠈⠲⠤⢶⡀⠀⣀⣀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡄⠀⠀⢀⠀⢠⠀⠀⢀⣴
⠀⠀⠀⠀⡤⣈⣀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠈⢳⣇⠀⢸⣧⣤⣤⣶⡿⠛
⣀⠀⠀⠀⡎⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠀⠀⠀⠀⠀⠹⣆⣧⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⣿⣿⠿⠿⠛⠂
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡄⠀⡌⠀⠀⠀⢀⣴⣾⡥⠂⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢿⣍⠀⠀⠀⠐⠊
⠀⠀⠀⣀⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠⠀⠀⠀⠤⣄⡀⠀⠀⠀⣹⣿⣿⣄⣄⡆⠀⣴⣾⣿⠯⢀⡄⠀⠂⠐⠀⠀⠲⠀⠠⠁⠈⠀⠈⠳⣍⡀⠒⠒
⠒⠠⢀⠀⠀⠀⠁⠀⠀⠢⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⠀⠀⠀⠙⢿⣿⡏⠃⢠⣿⣿⣯⠖⠁⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⠉⠐
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣷⣶⡅⣤⠛⠲⣹⣿⣣⠄⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠆⠀⠀⠀⠐⠚⠋⠉⠁⠘⠞⠛⠙⠋⢻⡿⢿⠅⣀⣶⠋⣵⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⢷⣄
⠀⠀⠀⠀⠀⣠⣴⣿⡿⠛⠋⠉⠙⠻⣷⣄⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠈⣿
⠀⠀⠀⣠⣾⣿⡿⠉⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⣠⣶⣿⠟⠉⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⣿⠃
⠀⢀⣼⣿⣿⠏⠀⠀⣠⣤⡄⠀⠀⠀⢀⣿⡇⢠⣾⣿⡿⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⠀⣿
⢀⣾⣿⣿⠏⠀⢠⣾⣿⠏⠀⠀⠀⠀⣼⡟⢠⣿⣿⡿⠁⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣀⡟⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⠟
⣸⣿⣿⡿⠀⠀⠸⣿⣿⡀⠀⢀⣠⡾⠋⠀⣿⣿⣿⠃⠀⠀⠀⣰⠟⠁⠀⠀⠀⢀⣠⣤⣄⡀⠀⠀⢀⣤⣶⣶⠖⢶⣌⠻⢿⣿⣧⣀⡀⠀⠀⠀⢠⣿⣿⣿⣿⠋
⣿⣿⣿⡇⠀⠀⠀⠈⠉⠙⠛⠉⠁⠀⠀⠀⣿⣿⣿⣀⣠⡴⠞⠁⠀⠀⠀⢀⣾⣿⣿⠋⠉⢻⣧⢠⣾⣿⡿⠁⠀⢨⡿⠀⢰⡏⠉⣻⣿⣿⡿⠃⠈⣿⣿⣿⡏
⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠀⠀⣤⣶⣶⡀⠀⠀⣾⣿⣿⡇⠀⠀⣼⠇⣾⣿⣿⠇⢀⣠⠞⠁⢀⡿⢀⣾⣿⣿⠋⠀⠀⠀⠹⣿⣿⣿⡀
⠸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣿⣿⣿⢀⣾⢻⣿⣿⡇⠀⠀⣿⣿⣿⣇⡤⠞⠁⠀⣿⣿⣿⠛⠉⠁⠀⢀⡾⠁⣼⣿⣿⠇⠀⠀⠀⠀⢰⡿⣿⣿⣿⡄
⠀⠻⣿⣿⡀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⣿⣿⣿⣾⠃⣿⣿⣿⠁⠀⠀⢻⣿⣿⡇⠀⠀⢀⣴⠿⣿⣿⣄⠀⢀⣠⠟⠁⢠⣿⣿⡟⠀⠀⠀⠀⢠⡿⠁⠘⢿⣿⣿
⠀⠀⠘⠻⣷⣤⣀⣀⣀⣤⠾⠋⠁⠀⠀⠀⢻⣿⣿⠇⢸⣿⣿⣿⠀⠀⠀⡼⠉⠻⠿⣶⠶⠟⠁⠀⠈⠙⠛⠛⠛⠁⠀⠀⢸⣿⣿⡇⠀⠀⠀⢀⡾⠃⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⣼⣿⣿⡇⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⢠⡾⢡⣶⠆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠁⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣷⣤⠴⠋⠀⢸⣿⡀⠀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠋⠁
⣶⣶⡶⠶⢶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠓⡄⠀⠀⢀⠤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠈⠀⢁⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊⠙⢀⠔⠁⠀⠘⢢⠀⠈⣉⣒⡶⠴⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⣄⠀⠀⢀⡼⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠥⠴⠃⠀⠉⠉⡁⠀⠰⣌⠣⡀⠀⠀⠀⠀⠀⠀⢀⠤⠔⠦⣤⠴⠶⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢸⡆⠀⢧⡆⠀⠈⠉⠉⠓⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠉⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⣅⠀⠀⠀⠀⢀⡤⠀⠉⢷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⡰⠉⠀⠈⠉⠉⠙⠉⠉⠀⠀⣠⠞
⠀⠀⠀⢀⠔⠋⠙⠲⡴⠋⠉⠉⠙⢦⡤⠶⠶⠶⠤⣄⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⢦⠀⠀⠀⠀⠀⢰⡇⠀⠀⡀⠀⠀⠀⢀⡟⠛⠉
⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠉⠛⠋⠛⠦⢤⠴⠚
⠀⠙⠒⠒⠻⣄⡀⢀⣀⡼⠲⠶⠶⠚⠀⠀⠀⠀⠀⠀⢀⣀⣀⡼⠃
⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⢧⣀⡀⢀⣤⣀⠀⢀⣠⠏⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣦⠤⣄⣀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⢀⠀⠀⠀⣀⠊⠀⠠⠀⠙⡅⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄⢀⡀⠀⡿⠉⠑⣶⠟⠀⠀⠀⠀⠀⣅⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⠀⠀⠁⠈⣾⠇⠀⠀⠀⠀⠀⠢⠐⠀⠉⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⡇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⢀⡀⢀⠂⠀⠀⢠⡟⣇⡀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠐⠀⠀⠀⠠⣧⡄⠀⣺⣵⣷⠆⠀⠀⢀⢁⠎⠀⠀⢠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠑⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠙⢯⠁⠀⠀⠘⣿⡏⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠺⠟⠯⡀⠀⠀⠈⠓⣄⣀⠀⠹⠧⢂⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠉⠉⠉⠉⠓⡄⢸⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠜⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠏⠉⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠤⠄⠀⠤⠀⠀⢀⡀⠀⢀⡀⢠⡔⠋⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠠⡀⡘⠁⠀⠼⣇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠁⢀⠀⠠⠋⠉⠦⠄⠀⠀⠀⠀⠀⢇⠀⠀⣀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⢲⣾⣿⣿⡟⠀⠈⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡟⠁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⠉⠀⠀⢀⠀⠀⠀⠀⢀⣰⡿⠁⢻⡏⠈⠉⢄⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠰⠚⠁⢁⠀⠀⠀⠐⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠖⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⡠⠒⡩⠈⣸⢏⡷⣄⢠⠖⢒⡾⢋⡇⢰⣶⢲⠞⢹⢧⡲⠂⣨⠏⢀⡟⣽⢹⢰⣶⣞⣽⢫⠟⣽⠋⣼⠁⡶⣚⢟⡸⢡⢃⡜⠉⡿⢦
⠀⠀⠀⠠⠊⠀⠈⣠⠔⠁⣼⢁⡿⣺⠟⢉⠀⣻⠀⡟⠁⡿⢠⡿⡌⠀⠀⣀⠤⠊⢰⡇⣾⣸⣵⠟⢠⡟⣼⢁⠀⣿⢸⢟⣵⠋⣰⢃⠎⠀⢸⡇⡾
⠀⠀⡴⠛⢁⣴⠟⢁⠴⠦⡿⠋⠀⡿⠞⢡⣶⡏⢰⡇⢰⠇⣼⠇⠋⣠⡾⢁⠂⠉⠉⠉⠁⡿⢃⠀⣾⡟⠋⠉⠉⠁⣾⡞⠁⢠⠟⠈⡰⠀⠈⣰⠃
⠀⣸⠷⢃⣾⠏⠔⠁⠀⠀⠀⠀⢸⡗⣡⣿⢿⠁⣸⢀⠀⣰⡿⢰⣴⡟⢀⣼⠔⣶⠒⡇⢸⣃⡾⢰⣿⢶⠶⡖⣷⢀⡟⣠⠇⣼⡠⡞⢁⢀⠞⠁
⢀⣏⣴⠿⣁⠎⠀⠀⠀⠀⠀⠀⢸⣴⠟⣣⢞⡿⠉⣿⠀⢻⠆⡼⢋⠔⢩⠏⢠⣇⣀⡇⢸⠟⢡⡟⣱⣿⡘⣧⡟⢸⣷⠋⢠⠏⣰⡁⢾⡁
⢸⠏⣡⡞⢹⠀⠀⠀⠀⠀⠀⠀⡎⣠⠞⠁⣈⢄⣠⠏⠀⣼⠀⡷⠁⢀⣴⠋⠉⠁⢀⠀⣅⡴⠉⢼⠛⠉⠉⢁⠀⡜⢁⠔⢁⣾⠈⡇⣼⢳⡀
⢸⣾⠋⢠⣾⣄⠀⢀⣠⡆⠀⢠⡿⠁⢠⡾⠀⠸⠋⠀⠀⡄⢰⢀⣴⠏⣹⠟⠉⣁⠈⢀⣃⠔⠁⢸⢁⠏⡍⢸⡃⠗⢁⠄⠉⢹⣼⢱⠃⡎⢳⡀
⠀⢧⣴⠏⡠⠋⣽⠏⣿⣻⠄⣸⢀⡴⠋⡄⠀⣶⠀⣾⢀⡇⣶⡞⢁⠞⣡⡞⢸⡇⢸⣾⣠⡆⡠⢡⠎⡰⠁⣿⣰⠊⣡⢄⡟⠀⢷⡎⢸⣃⡴⠃
⠀⠈⠳⣼⣁⣾⣋⡤⠟⠁⠀⣷⣿⡥⠾⠃⢰⣇⣸⠿⠼⢃⣏⣴⡧⠼⠟⠒⠛⠛⠉⣿⣿⡿⠵⠷⠚⠛⠛⠉⣿⣾⡿⠾⠃⠀⠈⣧⠞⠁
⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁








⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠂⠐⠂⠀⠀⠐⠀⠰⠐⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⢸⢆⢠⢣⢀⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢹⠒⠦⢈⣃⣀⣁⣈⡋⠸⠔⡹⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣏⠉⡠⠊⠁⠀⠀⠀⠀⠈⠑⠢⡉⢡⣃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣦⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⢠⣋⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠉⠉⠉⠉⠛⢶⡄⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⣔⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠱⠓⠒⠉⠉⠉⠑⠒⠢⢄⡀⠀⠀⠀⢸⠠⣔⠚
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢀⡌⢠⡐⠓
⠀⠀⠀⠀⢠⠔⠒⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⠊⠰⡒⠳
⡠⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣖⣓
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⣬⣵⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣯
⣿⣯⣤⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠸⡄⢸⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⡀⠁⠀⢁⣀⠀⠁⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠴⠋⠁⠀⠉⠲⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⡄⠀⠀⠀⠀⠀⡇⢀⡉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠢⡀⢀⠡⣄⡈⠋⣀⡴⠁⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠁⠀⠀⠀⠀⠁⢀⡉⠉⢁⡠⠤⢄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣀⣀⠀⠀⠐⠋⠀⠀⠁⠐⠀⠀⠙⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⢠⠔⠀⠀⣀⣠⣤⣄⣀⠀⢀⡤⠶⠒⠒⠦⢤⡀⢰⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠋⢀⡴⠊⠁⠀⠀⠀⠀⡴⠋⢀⠀⠀⠀⠀⠀⠙⠀⠉⠁⠉⠙⠢⠤⠔⠀⠈⠉⠁
⠀⠀⠀⠀⠀⣀⣀⡀⠀⠚⠀⢀⣀⡤⠤⠤⠤⢤⣀⡀⠀⠀⠀⢀⡠⠴⠒⠒⠲⢦⣄
⠀⠀⠀⣰⠛⠁⠀⠀⠀⣠⠞⢩⠂⡀⢐⠆⠀⠀⠀⡭⠗⣦⠰⠋⠀⠀⠀⢀⣀⠀⠈⠳⣄
⠀⠀⢰⠃⠀⠈⠀⠀⡰⠁⠀⠀⠀⢀⢨⠆⠀⠀⠀⠦⣀⠉⢳⠀⠀⠠⣆⢠⠚⣢⠀⠀⠘⣇
⠀⠀⢸⠀⠀⠀⡀⢠⠃⠀⠀⠀⠉⠁⠀⢀⠠⠀⠀⠀⠀⠀⠸⡆⠀⠀⠈⠢⠜⠁⠀⠀⠀⣿
⠀⠀⠘⣧⠀⠀⠀⢸⡀⠀⠠⠄⢠⠀⠀⢙⠀⠆⢐⠄⠀⠀⢰⡇⠀⠄⠀⠀⡠⠀⠀⠀⣰⠃
⠀⠀⠀⠈⠓⠦⢤⠀⢷⡀⢸⢀⡠⠃⠀⠈⠑⠁⠀⠀⠀⢀⡞⠀⠰⠦⢤⣤⣥⡥⠾⠛⠁
⠀⠀⠀⠀⠀⠀⢲⡀⢠⠙⢦⣉⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠀⣄⠸⡄⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⣷⠈⢰⠂⠸⡍⠛⠓⡚⠛⢁⠄⠀⠀⠀⠀⠸⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⡟⠀⣾⠀⠀⠙⡄⢠⠇⣰⠋⠀⠀⠀⠀⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢠⠇⠀⡇⠀⠀⠀⢸⠀⣸⠁⠀⠀⠀⠀⠀⠀⢠⡇⢸⡇
⠀⠀⠀⠀⣠⠶⠒⠒⠲⠦⡀⠀⠀⡏⠀⣿⢠⣀⣀⠀⢀⡤⠤⣸⠁⢸⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢫⡇⠀⣿⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠉⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠘⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣶⣤⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠂⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠀⠹⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⣀⣴⠾⠟⠛⠻⠷⣦⣄⠀⠀⠀⠘⢷⡀⠀⠀⣸⠁⠀⢀⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣇⠀⠁⠇⠀⢠⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀⠀⠸⡀⠀⠀⠀⠊⠀⢀⣴⡾⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⠶⣤⡀⠀⠘⠄⠀⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠐⠂⠀⠀⠉⠲⠄⠀⠀⠀⣼⣾⣀⠞⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⣰⣶⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠴⠚⠉⠙⠛⢿⣦⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠉⠀⠀⠀⠸⠧⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠸⠁⢠⣤⠀⠀⢰⡿⠁⠀⠀⠀⢠⣶⠶⠆⠀⠀⠀⠀⠈⠁⠈⠙⣷⡀⠀⠀⠀⠀⢹⡆⠀⢠⣿⣿⡄⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠁⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠼⡿⠆⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠀⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣠⢤⡀⠀⠀⠀⠀⣀⣀⠀⠀⡀⠀⢀⣀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⡰⠁⣤⠀⠀⣸⠃⢸⢀⡀⠁⢀⡟⢠⠀⠐⢏⡀⠀⠀⢙⣦⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠉⠀⠀⢰⣟⡔⠁⠀⢀⣀⠀⣿⠀⠌⡘⢁⣤⢼⡧⠂⢀⣀⣀⢉⡍⢉⡅⠈⡇⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢺⡗⣼⠿⣌⣆⣿⠀⣸⡇⣾⡇⣼⡇⢰⡿⢰⡏⣸⠇⣼⠇⣠⠃⠀⣀
⠀⠀⠀⠀⠀⢤⠄⠀⡔⠹⠇⣿⢁⠻⠧⠞⠀⢿⡰⠙⠓⠙⠚⠹⣇⠞⠷⠙⠗⠻⢶⣿⠚⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣀⠉⠁⢀⡴⠲⢰⠀⡆⣲⠂⢰⡒⢰⢶⡄⠀⣀⡄⠊⣼⠃⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠇⠉⠀⣿⣇⡀⢸⡇⠀⢹⠉⡇⢻⠁⢸⠉⢸⣯⠀⢸⠘⢊⡼⠃⣠⣿⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠛⣯⠁⢀⣙⣋⣈⣀⣁⣙⣃⣘⣛⣈⣈⣁⢀⡙⣉⣤⣤⣽⣛⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⣲⡤⢤⣤⣤⣤⣤⣼⣟⠃⠀⠀⠐⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣵⡆⣠⣶⡿⠋⠀⠀⠀⢀⣠⣾⣿⣿⣿⣄⠑⢌⠰⣌⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣾⣿⣿⣧⢳⡙⢿⣷⡈⢃⠈⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⡜⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣉⠓⠀⢒⣩⡃
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⣿⣿⣮⡛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⣿⣿⡿⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣞⣿⣿⣿⡦⠉⠁⠻⣿⣿⣿⣿⣧⠀⠀⠈⣿⠿⠛⠓⠀⡰⠿⣿⣿⡄⠀⢂
⠀⠀⢀⣴⣶⠶⠶⣶⣦⣙⣿⣿⣿⣿⣧⠄⠀⠈⢻⣿⣿⣯⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⢻⣿⡆⠀⠂
⠀⣴⡿⠋⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣅⣀⣀⣀⣿⣿⣿⣁⣀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄
⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⡻⢿⡟⠻⡀⠀⠀⠹⣷⣆⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠹⣷⠈⠳⣄⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⣢⣄⣀
⣿⡇⠀⠀⠀⠀⣀⣴⣶⠋⠀⠀⠀⢻⣇⠀⢨⣗⠀⠐⠀⡜⢻⡀⠀⠈⠳⣤⣀⠀⠀⣀⣀⠀⠀⢀⣤⣶⠾⠋⠹⣿⣷
⢻⡇⠀⠀⠀⢼⣿⣿⣿⠧⡀⠀⠀⠘⣿⠀⣾⡟⠀⠀⡰⠁⠀⠙⠀⠀⠀⠈⠻⣿⣶⣟⡛⣿⣶⡿⠋⠀⠀⢀⣤⣿⠿
⠘⣿⡀⠀⠀⠈⣿⡁⠙⠳⣬⡓⢤⡀⣿⢀⣿⠃⠀⡜⠁⠀⠀⢀⣀⠀⠳⣤⡀⠀⢉⢻⣿⣿⣯⠀⠀⠀⠀⠈⠉⠈
⠀⠹⣷⡀⠀⠀⠈⠑⠢⡀⠈⠛⢶⣽⣿⣼⡿⠤⣼⣤⣴⣶⣿⣿⣿⣇⣀⣬⣿⣇⠚⠙⠻⣿⣿
⠀⠀⠙⣷⣄⠀⠀⠀⠀⠈⠲⢄⠀⣽⣿⣿⠏⢹⣿⣿⣿⣿⣿⠿⠿⠛⠉⠁⠀⠙⣆⠀⠀⣿⣿⠿⣶⣤⡀
⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⢀⣠⣿⠿⣿⣿⡄⠘⢿⣿⠋⠁⠀⠀⠀⠀⠰⣦⣀⣼⣿⣤⣴⡟⣿⠀⠀⠙⠿⣷⡄
⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠟⠋⠁⠀⠨⣿⣷⣆⡈⠃⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠿⠏⢸⣿⣿⡆⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠿⠤⠖⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠈⣿⠹⣧⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢹⣇⢻⣆⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠻⣿⣿⡇⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⢀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣤⣤⣤⣴⡿⠋
⣶⣶⡶⡶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠆⢀⠀⢠⠄⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⡄⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⡔⠁⠚⣡⢎⠀⠀⣀⠀⣀⠀⣀⢀⡄⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⢈⡡⣾⡧⢦⢠⡴⢗⢠⡴⠟⢀⡴⠒⠠⣞⢃⡆
⠀⠀⠀⠀⠛⠋⠁⠙⠁⠀⢉⣁⣈⣁⣀⠈⠸⠋⠈⠁⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣬⣭⣥⣤⣭⡤⠤⠤⠭⠤⠤⠭⠥⠤⠤⠤⠤⠭⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠤⠤⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣋⡥⠶⠒⠒⠒⠒⠲⠤⣍⡙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⢋⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣵⡿⠀⣠⠞⠁⠀⠀⠀⠀⣠⡄⢠⣦⠀⡠⣾⢃⡐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠻⠋⢀⣴⣿⠖⣠⡾⢟⣴⣟⡫⠖⠹⠗⠋⠛⣋⡈⠁⠠⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⢀⣀⠴⠢⠖⠟⠁⠛⠊⢉⣉⣁⠠⠤⠄⠒⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣉⣁⡠⠤⠄⠒⠂⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⣫⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣍⣛⡒⠲⠶⠖⢒⣛⣩⠴⠞⠉
⠀⠀⠀⠀⣠⣾⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⡼⠡⠏⠀⣰⢑⡆⡠⣶⠀⡠⡶⠀⣰⡀⠀⣤⡄⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢉⡽⢣⡶⠟⢁⣀⠀⣀⡄⢀⣀⠀⢀⡀⠀⡼⠃
⠀⠀⠀⢇⣀⠔⣲⠃⢸⠸⡋⠥⠾⣋⠥⠔⢀⡏⣾⣠⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠴⠟⠛⠒⠿⠅⠺⠧⠂⠱⠟⠠⠾⠃⠼⠁
⠀⠀⠲⠶⠶⠶⠒⠒⠚⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⢸⣇⠀⣿⠀⢸⡇⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣄⠀⢻⡦⠘⣋⣤⠥⠴⠶⠦⠬⣥⣘⠃⢰⡟⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠙⠷⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣄⠾⠋⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⡈⠻⢃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡰⠟⠃⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⢃⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡰⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣃⣸⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠲⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡴⠚⠉⠉⠉⠙⢦⡙⠦⠤⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠰⠶⠶⠶
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠈⠙⠁⠀⠀⠀⠀⠀⠀⠙⠋⠉⠉⠛⠮⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡏⣤⣤⡀
⠀⠀⠀⠀⢀⣤⠴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢳⠀⠀⠀⠀⠀⠀⠀⣼⢡⣄⡉
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡆⠀⠀⠀⠀⢀⡼⢡⣄⠉⠛⠷⠂
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⣼⠀⠀⠀⣀⡴⠋⣴⣄⠙⠃
⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡕⠧⠴⠖⣋⡁⢲⣆⠈⠻⣷
⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢻⡆⠸⣧⠀⠉
⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠏⠀⠈⠀⠀⠹⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢄⠀⠀⠀⠀⠀⠰⢄⡀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠲⠂⠀⠀⠀⠀⠀⣠⠏⡐⠶⠃⠀⠀⠀⢀⡐⠂⠀⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠀⠀⠀⠀⠰⠊⠁⠀⠀⠀⢀⣞⠉⠳⠞⠀⠀⠀⠀⠀⠀⠀⠡⡌⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢬⡂⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⢦⡵⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢖⠸⠃⠀⠀⠀⠀⠀⠀⠀⣾⠤⣌⠆⠀⠀⠀⢰⠁⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢥⠊⠀⠀⢔⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣤⠂⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⡔⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⢠⡏⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡏⢩⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢢⣀⡀⠀⠀⠀⢤⡐⠀⠁⡘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠐⡀⠀⠈⠢⡤⠀⢄⠀⠀⠀⠀⠀⠀⠠⣎⢂⢈⡰⠂⠁⠀⠀⠀⠀⢈⡤⠂⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡐⠉⢆⠀⠀⢄⡀⣠⠀⠀⠀⠀⡖⢢⠀⣩⠜⠋⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡐⠀⠀⠀⠑⡄⠀⠀⠀⠑⢤⡈⢆⢙⡸⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡠⠒⠣⣰⣁⠁⠑⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⡐⠁⠀⠀⡠⠊⠈⠢⡀⠀⠀⡠⣅⡜⠉⠀⠀⠀⢀⠊⠈⠲⠀⠀⠀⠀⢀⣠⢉⠀⠐⠗⠃
⠀⠀⠀⠀⠀⠀⠀⢠⠃⡜⠀⠀⢀⠔⠁⠀⠀⢀⠜⠓⢄⠑⠉⠈⢢⠀⠀⠀⠀⠢⠄⠘⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠃⡜⠀⠀⡰⠃⠀⠀⡠⠖⠁⠀⠀⠀⣁⠦⢀⡀⠡
⠀⠀⠀⠀⠀⠀⡌⠌⠀⡠⠊⠀⢀⠤⠊⠀⠀⢀⠠⠒⠉⠀⣀⠠⠘⠋
⠀⠀⠀⠀⠀⡐⠌⢀⠔⠁⡠⠐⠁⢀⡠⠐⠈⣀⠠⠐⠊⠉
⠀⠀⠀⠀⢰⠏⡠⢃⠴⠊⡀⠄⢂⠡⠔⠒⠉
⠀⠀⠀⢠⣏⣨⢔⣡⠔⠍⠒⠉
⠀⠀⢠⡿⠟⠛⠉


⣿⢿⠿⡿⡿⠛⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣟⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢦⡀⠀⠀⡴⣣⠴⠣⠀⢠⡏⠂⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡴⠉⠀⢸⠀⠀⠈⠒⢼⣿⣅⠀⡼⠀⢸⢀⢃⠔⠉⠀⠙⢦⡀
⠀⠀⠀⠀⢀⡞⠀⠀⣠⢼⣲⠂⠀⠀⢸⣍⠻⠛⣦⡀⡎⢸⣷⠖⠲⣠⠤⢄⠘⣄⠀⠀⠀⢰⢆
⠀⠀⠀⠀⠠⡇⠀⠀⠀⢸⣷⡡⠀⠀⡘⢇⠀⢀⠉⠙⡿⠋⠁⠀⢀⠏⠀⠀⠻⣄⠱⡀⠀⠈⠀⡗⢄
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠸⢿⣿⠖⢩⣷⡮⠭⠉⢷⣦⣇⠀⠀⠀⡏⠛⠁⠀⠀⠈⡶⠃⢐⠀⢸⣷⡬⠖⠋⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠃⠀⠊⠉⢻⣷⣶⠳⠤⢍⠳⡄⢾⠀⡄⢀⣠⣶⠟⠓⠤⢄⣆⢸⠋⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣅⠀⢀⠄⠚⡝⢻⡠⠀⠀⠓⠛⢻⣿⣿⡿⠋⢻⠀⠀⠀⠀⢿⣬⡝⠛⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢲⡀⠀⠉⢸⠇⠀⢀⡄⢠⠞⠍⡞⠀⠀⢸⣖⣄⣀⣴⠋⠉⠓⢦⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⠔⠉⠉⣽⠁⠀⠈⠛⠚⡇⠀⠀⢘⣟⠛⣿⢸⣤⠢⠈⣢⡇⠀⠀⠀⢤⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠬⠷⣄⡀⣀⣧⣸⣄⠀⣾⣷⢰⣿⢸⣤⣟⠻⡁⠀⣀⡄⠀⠀⣧⠴⠂⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠂⠲⢿⡟⠳⡌⢙⠟⠀⠘⣿⠿⣷⢦⣿⣦⡃⣸⣿⣄⣠⡼⠯⠥⠶⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠁⠱⣄⣼⠸⠀⠀⡼⡇⢠⠃⢀⡿⠀⢻⢷⡄⣀⣀⡔⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠷⣹⠙⠻⣿⡁⠀⠾⠞⢿⣟⣉⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⢧⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⡴⠃⠀⠀⠀⠀⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡠⠤⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣽⣶⢶⣶⣆⡆⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣋⣬⣴⠺⡿⢷⠛⡉⠉⠉⠉⠛⠳⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠇⠉⢾⣇⣸⣶⠀⠀⠀⢰⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣠⠿⠛⠟⠈⢧⠀⠀⣿⣷⠦⣤⣄⡀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣀⣶⣶⣾⡆⠀⢹⣿⣷⣶⣶⣶⣶⠦⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⣿⠁⠉⠉⣧⠀⢸⡿⣿⣿⡿⠋⠀⠀⠀⣶⡄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⠀⢀⡞⠀⣸⣇⣾⡿⠁⠀⠀⠀⣼⣿⣷⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠧⢠⣿⡀⢀⠞⢀⡴⢋⣾⣿⠃⠀⠀⣠⣾⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢷⣶⣶⣴⠗⢋⡴⠋⢠⣿⣿⠇⠀⠀⠞⣿⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⡓⠤⢒⢿⡿⢋⣿⠾⣅⣀⣼⣷⠦⠿⠿⣿⡀⠀⠀⣼⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢱⢛⣟⡛⠊⢠⠞⠁⢠⡇⠀⢿⡛⣿⣿⣿⣿⣧⡀⠀⠹⣷⡆⠸⡇⠀⢀⡠⢴⣖⣒⣶⡶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⢧⣩⡿⠿⣷⣶⡾⣧⢲⣦⡙⢮⣿⣟⣿⣿⣧⠀⠀⢻⣇⢠⣬⡲⢯⣾⣿⣿⣿⣿⣿⣷⣮⣝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⣽⣾⣿⣿⣿⣿⣿⢰⣿⣻⣦⡉⠻⣿⣿⣿⣧⠀⠈⣿⣸⡏⣿⣶⣝⠿⣿⠿⠛⠻⠿⣿⣿⣷⡝⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢣⣾⣿⢟⡵⠞⠛⣿⣿⢸⣿⣿⣿⡽⣷⣌⠛⢿⣿⣧⠀⢸⣿⣽⣿⣿⣿⣷⣮⡓⢄⡀⠀⠀⠙⢿⣿⣞⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡏⣾⣿⢣⠎⠀⠀⢠⣿⡏⣿⠙⢿⣿⣿⣞⣟⢷⣄⡙⠻⣷⠈⢿⣿⣿⣿⡏⠻⣿⣝⣶⡝⢢⡀⠀⠀⢻⣿⡼⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⡌⠀⠀⠀⢸⣿⣷⡇⠀⠈⢿⣿⣿⡞⡆⠙⢿⣦⠜⠁⢀⣿⣿⣿⠷⠖⣛⣭⠷⠚⠉⠀⠀⠀⠀⣿⡇⣷
⠀⠀⠀⠀⠀⠀⠀⠈⡇⣿⣿⠁⠀⠀⠀⢿⡿⠿⠃⠀⠀⠀⠸⣿⣿⢹⠀⢾⣥⣴⣾⣿⣿⣩⣴⡾⢿⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⠀⠀⢿⠛⠿⠿⢻⢯⡹⣿⣮⡳⠀⠀⠀⠀⠀⠀⠀⣰⣿⢧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣸⠀⠀⠈⠙⠒⠒⠋⠀⠱⣌⠻⣿⣦⣤⣀⣀⣀⣤⣾⣿⢋⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣜⢿⣷⣄⡀⠀⠀⠀⠀⠀⣠⣿⣿⢣⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣍⣛⠻⠿⠿⢛⣫⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣍⠻⢿⣷⣶⣶⣶⣿⣿⢟⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠭⣭⠭⠵⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠠⠀⠀⠠


⠀⠀⠀⠀⣤⣦⣄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣴⣤
⠀⠀⠀⢼⣿⣿⣿⡿⠂⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠐⢿⣿⣿⣿⡥
⠀⠀⠀⠀⠹⢛⠿⣅⠀⠀⣤⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣦⠀⠀⣨⠿⡛⠏
⠀⠀⠀⠀⠀⠈⠀⠘⢧⡘⠋⠛⣻⡇⠀⠀⢤⣤⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡤⠀⠀⢸⣟⠛⠙⢃⡼⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣶⣬⣭⣤⣤⣴⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣤⣤⣤⣭⣤⣶⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠜⢹⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⡏⠣⠀⠀⠀⠈⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡄⠀⠀⠀⢠⣠⣴⣀⡀⠀⠀⣼⣿⣶⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠹⣿⠇⠀⠀⠀⣿⣿⣿⣿⠁⠀⠀⣾⢿⡿⡗⠀⠀⠈⣿⣿⣿⣷⡄⠀⠀⠀⢸⣿⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠈⠛⠻⡿⣻⠓⠶⣤⣛⢀⣀⣋⣤⡴⠖⡿⢿⠿⠿⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⡆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠙⢿⣿⠉⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢸⡏⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠐⠚⡧⠠⣠⠄⠀⠀⢠⠄⠀⠀⠠⣄⣾⣿⣄⠀⠐⠤⠀⠀⠀⠀⠀⠀⡤⠀⢼⠇⠀⣿⠠⡀
⠀⠀⠀⠀⠀⢀⠔⠁⠇⠀⠀⠿⡰⢡⣤⣤⣤⣶⣶⡶⠶⢾⠿⢿⣿⠿⠿⠶⢾⠶⡶⠦⠤⠤⢤⠜⡄⠿⠀⠀⠺⠀⠈⢆
⠀⠀⠀⢀⣔⡁⠀⣀⡀⠀⠀⣔⣀⣀⣀⠀⣐⣀⣀⣀⣀⣌⣀⣀⣀⣀⣀⡀⢰⣀⣀⣀⣀⣀⣀⣀⢨⡄⣀⣀⣀⣀⣀⣀⣑⣄
⠀⠀⡔⠉⠀⢀⠀⡀⠀⠈⢻⡄⠀⠀⢀⠜⢅⠀⠀⠀⡠⣄⠀⠀⢀⠄⣤⡀⠀⡧⠀⠀⠀⠠⢠⣄⠀⠈⢄⠀⠀⠀⠤⠀⠀⠈⢢
⠀⠈⠀⠀⠀⡁⠀⠈⠢⠤⠈⠀⠀⠀⠈⠂⠒⠀⠀⠀⡇⢘⠀⠀⠈⠘⠀⠁⠈⢁⡄⠀⠀⠠⠂⠈⠀⢇⢨⠀⠀⠀⠐⠀⠀⣠⠈
⠀⢠⠀⠀⠀⠑⠄⡀⠠⠔⢀⡄⠀⠀⢰⠂⠰⠀⠀⠀⡇⠘⠀⠀⠠⠀⣀⠆⠄⢾⡃⠀⠀⠐⢆⡠⠀⠘⣦⠀⠀⠀⠂⡀⠀⠀⠱
⠀⠀⠑⢄⡀⠀⠀⠀⣀⠤⢞⡁⠀⠀⠈⡲⡃⠀⠀⠀⣑⠃⠀⠀⠀⠀⠉⠁⠀⡈⠀⠀⠀⠈⠈⠉⠁⢀⡁⠀⠀⠀⢱⢧⡀⠀⠀⢑
⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠀⠉⠁⠁⠀⠉⠉⠉⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀

⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠿⠿⠀⣤⡄⠀⠀⠀⢠⣤⡄⢸⣿⠋⠉⠉⠛⠀⣀⣤⣤⣤⣤⣤⠀⣾⡿⠋⠉⠉⠉⢻⣿⡇⠀⢀⣠⣤⣤⡀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⢠⣿⡇⠀⠀⠀⣿⣿⠀⢸⣿⣀⣠⣤⣤⠀⢿⣿⠛⠉⠉⠛⠀⣿⡇⣀⣠⣤⣶⣿⡿⠃⢠⣿⠋⠉⠉⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠸⣿⣇⣤⣤⣶⣿⣿⠀⢸⣿⣿⠿⠿⠿⠀⣸⣿⣤⣤⣶⣶⠀⣿⣿⣿⣿⣿⠟⠋⠀⠀⠘⢿⣦⣤⣀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢻⣿⣿⣄⣤⣦⠀⣿⡟⠛⠛⠛⣿⣿⠀⢸⣿⠇⠀⠀⠀⠀⣿⣿⠛⠋⠉⠉⠀⣿⣿⠛⢿⣿⣦⡀⠀⠀⣠⡄⠈⠉⠛⣿⡄⢸⣿
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠀⣿⣿⠀⠀⠀⣿⣿⠀⢸⣿⣄⣠⣤⣤⠀⣿⣿⡀⣀⣀⣀⠀⣿⣿⠀⠀⠻⣿⣷⣤⠀⣿⣧⣀⣀⣴⣿⠃⠘⠋
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠛⠋⠀⠀⠀⠉⠉⠀⠘⠛⠛⠛⠛⠛⠃⠙⠛⠛⠛⠛⠛⠂⠙⠛⠀⠀⠀⠙⠋⠀⠀⠈⠛⠛⠛⠋⠁⠀⠘⠃










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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⠤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠲⢤⣄⡀
⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⡏⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⡀⠀⢀⡴⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⢸⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠁⠀⣾⠁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⡿⠚⠀⠀⠀⢀⣠⠀⠀⣠⠤⡀⠹⣤⠤⣄⠀⠀⢰⠒⠆⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠘⣇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⣾⡧⠊⣷⠀⢠⣿⠁⠀⢰⣿⠠⠀⠠⠃⢰⣿⠀⠀⠘⢧⣄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠘⣆⠀⠀⠀⠘⣿⣆⠀⠀⡠⢺⣿⠃⠀⣿⢀⡼⣿⠀⢀⢾⣿⢀⡠⠁⠀⢸⣇⣠⠄⣀⣠⣿⡆⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠛⠓⠋⠁⠈⠁⠀⠀⠉⠉⠀⠈⠉⠀⠀⠉⠉⠀⠀⠀⠀⠈⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠒⠊⠁
⠀⠀⢀⡤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠑⠢⢄⠀⠀⢈⠀⠀⠀⠀⣄
⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸