Awkward laugh Emojis & Symbols

⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
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⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠃⠘⠹⠻⠀⠀⢠⣦⣇⣾⢠⣶⢠⠆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⠛⠟⠿⡘
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠤⠤⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡇⠀⠀⠀⢸⣀⠀⡆⣆⠐⢷⣢⡖⠀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢀⣦⣄⡤⣲⡖⡆
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣶⣴⢤⡀⠀⠀⠀⠀⠚⠣⣶⣦⡄
⠠⣴⠳⠂⠀⠀⠀⠀⢄⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣄⣸⡅⢠⠀⠀⠀⠀⠀⠀⠀⣰⠟⡭⠘⠋⢳⠀⡠⠄⠈⠁⠘⠿⣿⡧⠀⠐
⠀⠲⠄⠐⢀⣠⣖⣖⡒⢬⠀⠀⠀⠀⠀⢄⠀⣠⣀⣤⣤⣄⠈⣩⠛⢤⠀⠈⡧⠀⠀⠀⠀⡋⠀⠉⠀⠚⡉⠀⠀⡀⠘⠀⠀⠀⣻⠁
⠀⣦⡀⠀⣿⡴⢁⡀⠀⢻⠇⠀⠀⢸⠄⠈⢣⡽⣿⡎⢉⠉⡆⠀⠙⢦⠠⢀⡇⠀⠀⠀⠀⣠⣀⠑⠒⡀⣡⣦⡤⠀⢀⣠⣤⣊⣼⣦⣄⡀⠀⠀⠸⣦⡄
⠀⣿⣿⣶⣿⡁⠀⠀⠐⠇⠀⢀⠀⠚⠒⠒⠉⡇⠹⠃⠀⢠⢤⠖⠈⠈⠙⣟⠻⢲⡄⣤⢔⡍⢟⢷⢺⠙⣟⠋⡻⣻⠿⠟⠴⢿⣿⡿⠿⣻⠉⡁⠀⠈⠃
⢀⣼⣿⡿⡹⠷⣄⣨⡠⢄⠒⢅⠠⠘⣧⠀⠀⠱⡀⢩⠀⢰⣿⡄⠐⡂⠀⠠⠟⠁⠰⠇⢸⡇⣸⡗⡏⠀⢻⠀⣻⣡⣾⣿⣷⣶⣦⠐⢊⣕⣨⠔⢀
⢸⡿⢍⠁⠀⠀⠀⠙⠀⢆⣿⠏⠀⠀⢹⡀⠀⠀⠙⣿⣷⣮⣿⣷⠀⠀⠀⠀⠀⠇⢰⠀⠀⡇⣿⡇⡁⠀⠘⠐⢹⣿⡫⠉⠩⠍⣿⣿⡟⣿⡟⠀⠸
⠀⠈⠳⠀⠀⠀⠀⠁⠆⠀⣱⠀⣀⣤⣤⣷⣄⡀⠀⠹⣿⣿⣿⣧⠦⡀⠀⢀⠠⡆⠀⠄⠀⢁⣿⡇⠀⢠⢃⡄⢸⠃⠀⠀⠈⠁⢿⣿⣷⠀⠁
⠀⢀⡰⢶⡀⠀⠀⢀⡄⢰⢇⣼⠟⠋⠙⠿⣿⣿⣷⡀⠈⠀⠙⠉⢳⡀⠢⠀⣐⠦⡰⣦⡀⠸⣿⣇⡴⠃⢀⠄⡙⠇⠻⠟⠀⣠⣼⡿⠋⢀⠄⠀⠀⠀⠁
⠀⠘⠁⠀⠙⢶⡄⠁⠀⢸⣾⠋⣠⡀⠀⠀⠀⠀⠹⣿⠀⠀⠃⠀⠀⠑⢄⣁⢿⢷⣟⣋⡉⢛⠿⡟⠀⠰⠀⢰⠁⡈⠂⠀⡠⣟⡋⣁⣔⣁⣀
⠀⠀⠀⠀⠄⠀⣿⣅⡄⢿⡄⠀⠘⠣⠀⢰⡦⠀⣸⠏⠐⠊⠀⣀⠀⢠⣞⣵⠖⢉⣅⠀⠉⠙⠳⣕⣄⡥⠒⡻⢪⠛⠛⠋⡰⠫⠿⣫⣿⡿⠿⢧⡀
⠀⡰⢀⡄⠄⠠⠉⠀⠀⢸⠃⡄⠐⠀⠀⠀⠀⠀⡿⢦⠀⢠⣿⣿⣶⢋⡾⠋⠰⠦⠈⠀⠀⠠⢤⠈⡏⠀⢘⢁⡇⠄⡠⠊⠀⣶⠟⠉⠁⠀⠠⠖⠣⡄
⠠⡁⠈⠴⠆⠀⡀⠀⠀⢸⠀⠘⠦⣤⠄⠀⠠⠀⠁⢸⠀⣸⣿⣿⡿⣹⠃⠀⠀⠀⠀⠀⠀⠲⠄⣼⡟⠀⠈⢸⣢⠊⠀⠀⠀⢿⡄⠀⣩⠄⠀⡉⠀⠃
⠀⢧⠀⠴⢲⡄⠀⠀⣠⡞⣦⡈⠁⠀⠔⡠⠀⠀⠀⠘⡄⢻⣷⣿⣾⣇⠀⠀⣴⣶⣬⡆⠀⠀⣰⣯⡀⠀⢠⡿⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣉⡄⠁
⠀⠀⢆⠀⣸⡗⡾⣿⢋⠀⢹⣟⣒⡁⠀⣽⣄⠀⠀⠀⠃⠀⠙⣿⣿⣿⠀⠀⠻⠽⠆⠀⠀⣴⣡⣿⡇⢰⠸⠁⠀⠀⠀⡤⣡⠎⠉⠘⢄⠀⠉⠛⠁⣰⣤
⠀⠀⣿⠄⠨⣏⠀⠇⠸⠤⠤⠛⣧⡀⣸⣿⣿⣿⣦⣀⡀⠀⢀⣠⠽⡟⠈⢢⣀⣀⣀⡴⠾⡿⠿⠋⢘⣶⣦⣤⣦⡇⢜⡼⠃⠀⠀⠀⡾⠦⠀⠀⠚⠋⠘

⠀⢀⣠⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⣀⣤⣴⣄⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⡟
⢰⠿⠿⠿⠿⡆⠀⠀⠀⠀⣼⠟⠻⣿⣷⠀⠀⠀⠀⢠⡿⠛⢿⣿⡇⠀⠀⠀⠄⢰⠿⠿⠿⠻⡆⡀⠀⠀⠀⣼⣿⠻⢿⣿⡀
⢿⠀⣂⣀⠂⡿⠀⠀⠀⠀⣷⡈⣤⡍⡸⠀⠀⠀⠠⢸⣇⠁⡀⢍⡏⠆⠀⠐⠀⢿⠨⢅⡈⢁⠿⠐⠀⠀⠀⢿⡈⢠⠌⣸⠁
⠀⠑⠮⠽⠚⠀⠀⠀⠀⠀⠘⠛⠶⠺⠇⠀⠀⠀⠀⠀⠛⠷⠶⠿⠀⠀⠀⠀⠀⠀⠑⠲⠖⠋⠀⠀⠀⠀⠀⠘⠛⠲⠺⠇

⠀⠀⣀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⢠⡀⠀⣀⣀⣄
⢠⡾⣿⣿⡿⡅⠀⠀⠀⠀⣰⠟⢿⣿⣷⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠀⢀⣾⠿⣿⣿⡆⠀⠀⠀⠁⣼⢿⣿⡿⣧
⢾⡡⢂⣀⠆⣷⠀⠀⠀⠀⣿⠜⠀⠘⢹⠁⠀⠀⠀⢠⡏⠲⠀⠩⢘⡆⠀⠀⠀⢸⡇⠃⠈⠋⡇⠀⠀⠀⢸⡇⠛⡀⠗⢸⠆
⠀⠳⢽⡿⠗⠁⠀⠀⠀⠀⠘⠷⠦⢾⡇⠀⠀⠀⠀⠀⠙⢤⣀⡠⠎⠀⠀⠀⠀⠈⠿⣦⣤⣾⠀⠀⠀⠀⠀⠙⠦⣭⠴⠃

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⢀⣴⣿⣿⣷⡀⡾⡆⠀⠀⢠⣾⣿⣿⣦⠰⠋⠀⠀⠀⣠⣾⣿⣿⣆⠀⠘⠟⠁⢀⣾⢿⣿⣷⡀⠀⠀⠀⠀⣠⣾⣿⣷⡆
⣼⠫⡍⠩⡍⣧⠀⠀⠀⠀⣿⠱⠈⠻⢻⠀⠀⠀⠀⢠⡏⠬⠉⠥⢹⡄⠀⠀⠀⢸⢡⡆⠙⡟⡇⠀⠀⡐⢠⡏⣉⠉⣩⢹⡐
⠙⠂⣒⣓⡠⠋⠀⠀⠀⠀⢻⣦⣛⣤⡏⠀⠀⠀⠀⠈⠳⣴⣿⣧⠜⠁⠀⠀⠀⠘⣷⣌⣡⡼⠁⠀⠀⠀⠈⠣⣿⠀⣿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠋⠀⠀⠀⠀⠀⠀⠀⢀⣠⣟⡉⠱
⠀⠀⠀⠀⠀⠀⢀⣤⣶⠏⣠⣴⡤⠡⠀⠀⠀⠀⠀⠀⣴⣉⡍⠉⠃⠀⠆

⠀⠀⠀⠀⢠⣴⣿⣿⣿⡅⠙⠛⢦⣀⠀⠀⠀⠀⠀⠀⠉⠁⡆⠀⠀⣠⣶⡀
⠀⠀⠀⣴⣿⣿⣿⣿⡏⠡⣀⣀⣼⡿⠿⣷⣦⡀⠀⠀⠀⠀⢰⣴⣾⣿⣿⠧
⠀⠀⡼⢿⣿⣿⣿⣿⠃⠀⣿⠃⣿⣷⠒⢿⡿⠇⠀⢸⠛⢻⣾⣿⣿⠋⣠⠖⠉⠁⢈⡇⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⡀
⠀⠀⠛⠟⣽⣿⣿⡏⠀⢠⠹⠀⣿⣿⣿⡏⠀⠀⠀⠠⡏⢩⣻⣿⡏⡸⠁⠀⠀⢀⠿⠷⣀⠀⡀⠀⠀⠰⠏⠐⠄⠀⠈⢆
⠀⠀⠀⢸⣿⣿⣿⣇⣀⣘⣃⣀⣿⣿⣿⠀⠀⠀⠀⠀⣨⣿⣽⣿⠇⡇⠀⠀⣠⠋⢀⣠⠀⠀⠁⠀⠀⣂⣔⡢⣴⠀⠀⢨
⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠃⠀⠀⠀⣰⣿⣯⢿⣿⢸⠀⠀⢰⠇⠈⠙⡄⠀⠀⠀⠀⢀⢏⡨⠁⢡⡀⢀⠏
⠀⠀⠀⢰⣿⣿⣿⣿⡿⢻⣿⣿⣿⡿⠀⠀⠀⠀⠐⠿⢿⣿⣿⡏⡘⠀⠀⢸⣤⣀⠞⠁⠀⠀⠀⠀⠈⠈⠣⡀⢹⡖⠁
⠀⠀⠀⢸⣿⣿⣿⣿⠇⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⡉⠁⣾⡿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣠⠑⢀⣷⢆
⠀⠀⠀⣾⣿⣿⣿⡏⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡇⢸⣿⡀⢸⣿⣷⣄⣤⣄⠀⠀⠀⡰⠛⠒⠉⡇⢸⡆
⠀⠀⢠⣿⣿⣿⣿⢡⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⢸⣿⣇⢾⣿⣿⣻⡤⣜⠀⢠⠶⠧⠤⠤⢀⡦⢠⠁
⠀⠀⢸⣿⣿⣿⡟⣼⣿⣿⣿⣿⠃⠀⣀⡀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠈⣿⢿⣿⡿⠿⡧⣴⠊⠀⠈⡦⠚⣶⣒⣩⠔⢻
⠀⠀⣼⣿⣿⣿⡇⢿⣿⣿⣿⣿⠀⡀⢟⠣⡀⠀⠀⠀⠀⣿⣿⣿⣿⠀⡏⠀⠀⠀⠀⢻⠈⠀⠀⠀⠸⢿⣷⣶⣾⠀⠈
⠀⠀⠸⠿⢿⣿⠃⠀⠀⠀⠙⠻⠀⠻⣪⣦⠙⢄⠀⠀⢀⣿⣿⣿⣿⠸⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢨⠁⠀⣀⢱⡔⠛⠻⠻⠿⠛⠰⠀⡀⠀⠀⠀⣶⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠣⠄⣿⠀⡿⢦⡀⠀⠀⠀⢀⡼⣿⠀⠀⢠⡅⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠐⠦⢷⠈⠳⡄⠀⣀⠎⣰⣣⡿⠀⣸⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣆⠐⡆⠀⠘⠦⣀⢁⠞⠙⠀⠙⡻⠀⠀⡟⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢸⠀⡇⠀⠀⠀⣠⡾⠀⣇⠀⡰⢧⡀⢀⡇⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⡇⣸⢀⡇⣀⢀⡞⢣⣇⣠⠉⠉⠀⠀⢳⢸⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠛⠊⠉⢉⡞⢀⣠⠇⠀⢠⠀⢠⠀⣸⢸⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠤⡀⠠⠀⣼⠁⠀⠀⠀⠀⢸⠀⡏⠀⡏⠸⠇⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⡄⣀⡀⠀⠀⠀⢸⠀⡇⢸⣠⠀⠀⠀⠀⠀⠈⠉⠰⠟⠋⠬⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠓⠚⠦⠿⠏⠙⠃⠀⠀⠀⠀⠀⠀⠀⡰⡄⠤⠦⠸⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠂⠀⠐⠒⠆⠃⠀⠒⠄⠀⠃⠒⠀⠀⠃⠂⠐⠂⠀⠁⠒⠒⠀⠐⠂⠐⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠤⠄⠠⠄⠰⠤⠄⠄⠆⠠⠆⠠⠄⠢⠀⠤⠀⠄⠠⠤⠀⠤⠤⠤⡄⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⠿⠿⣿⣛⣛⣛⣛⡳⠶⢦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣛⣭⣶⣾⣛⣯⣽⣿⣿⣿⣿⣿⣿⣷⣮⣝⡻⣦⣄
⠀⠀⠀⠀⠀⢀⣴⣿⡿⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣝⢿⣦⡀
⠀⠀⠀⠀⢀⣾⣿⢏⣾⣿⣿⣿⣿⡿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢿⣿⣷⡹⣷⣄
⠀⠀⠀⠀⣾⣿⢣⣿⣿⣿⡇⣿⣿⡇⣿⣿⣿⡏⣆⣿⡿⠇⠿⠿⠇⢿⣿⣿⠏⡎⣿⣿⣿⣜⣿⣆
⠀⠀⠀⢸⣿⡟⣾⣿⣿⣿⡅⣽⣶⡆⣶⣿⣿⢡⣷⢸⣿⡇⣿⣿⡇⣿⣿⡟⣼⡆⣾⣿⣿⣿⡼⣿⡀
⠀⠀⠀⢸⣿⣇⢿⣿⣿⣿⡇⣿⣿⣿⢻⣿⣿⣸⣿⡏⣿⡇⣿⣿⢹⣿⣿⣱⣿⣿⣿⣿⣿⣿⢧⣿⡇
⠀⠀⠀⠀⢻⣿⣜⢿⣿⣿⣇⣿⣿⣿⣼⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣼⡿
⠀⠀⠀⠀⠀⠻⣿⣦⣝⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣻⣿⣿⣿⣿⣿⣿⡿⣋⣼⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣷⣶⣯⣭⣭⣭⣽⣟⣛⣛⡻⣿⡷⣛⣛⣛⣛⣿⣭⣭⡷⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠛⠛⢿⠙⣼⠉⠙⠋⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⢠⣤⣤⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢘⣿⠟⠉⢳⠗⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠻⣿⣦⠀⠀⣀⡀⣤⠖⠳⡄⢰⣿⣿⣿⣿⠟⠛⠉⠛⠻⣿⣿⣿⣧⠀⢀⣤⣄
⠀⠈⠙⠲⡀⠙⢦⣾⣿⣿⣿⣿⣿⣿⠟⠉⠐⠂⠈⢿⡇⠀⢹⡝⢹⠞⠉⠳⢄⣻⣿⠟⠀⠉⠀⠀⠈⠉⠀⠙⣿⠟⢀⡼⠦⣌⡇
⠀⠀⠀⠀⠙⣆⠀⢙⣿⠛⢿⠿⠛⠁⠀⠀⠀⠀⠀⠈⠓⡖⠋⢀⡼⢧⡀⠀⠀⠹⣏⠀⣤⣀⣀⣄⣀⡤⡶⠀⠉⣷⠉⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠸⣷⡏⠈⠑⢦⡀⠹⡗⠲⠦⠶⣶⠂⢠⣴⢃⡴⠋⠀⠀⠙⣄⠀⠀⠈⠳⣜⢿⣛⣛⣛⡿⠃⣠⠞⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣦⠀⠀⠙⢦⡙⠲⠤⠞⠁⣠⣿⡽⠋⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠈⠛⠛⠛⠻⠤⣞⠁⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣄⠀⠀⠙⠶⣶⣶⢾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⠋
⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣿⠀⠀⠀⠀⠀⠁⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⠄⠰⣶⣶⣦⣀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠤⣼⡟⠁⠀⠀⠀⣀⡀⠙⠳⡀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠑⠖⠀⠀⠚⠉⠀⠀⠀⠀⠀⠀⢷⣀⡤⠄⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⢀⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠤⢨⡀⠀⠀⢀⡀⠀⠀⠀⠀⢰⡏⠀⠀⠈⢆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡗⠲⠶⠚⠀⠀⠀⠈⢢⡤⣤⠈⠀⠀⠀⠀⠈⣽⠃⠀⠀⠎⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⢀⣠⡴⠋⠀⠘⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢤⣼⠁⠒⠰⠖⠛⠉⠁⠀⠀⠀⠀⣰⠀⠀⠀⠀⣤⣀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠆⠀⠀⠀⢀⣀⣠⣴⣿⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⣿⣶⣶⣾⣿⣿⣿⣿⡿⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⡿⠟⠛⢛⡏⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠋⠀⠀⠁⠀⠀⣀⡞⢀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠈⠑⠒⠒⠒⠚⠋⠉⡴⠃⠀⠀⢀⡯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡶⠦⠤⠤⠴⠶⠛⠁⠀⠀⠀⣸⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠉⢇⠀⠀⠀⠀⠀⣠⠞⠁⣠⠖⠁⠀⠀⣷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣻⡟⠀⠀⠘⢏⠙⠛⠛⣋⡥⠖⠋⠀⠀⠀⠀⠀⣿⠀⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⡿⠀⠀⢀⣤⡆⣣⡴⣫⠅⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⢀⠀⢧⡀⢀⣾⣿⣿⣿⣦⣤⣶⣿⣦⡀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠸⣆
⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢠⡟⠀⠀⠙⠛⠀⢹⣿⣿⣿⣿⡿⠟⠋⠱⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⣸⠃⠀⠀⣠⠟⠀⠀⠀⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠈⠉⠒⠛⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠁⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣷⣶⣶⣤⣀⡀⠀⠀⠀⠀⢻⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⢧
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⢰⠋⢹⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠛⠉⣀⡤⠖⢿⣶⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⢸⠀⠈⣿⣿⣿⠀⠀⣠⠴⠦⡄⠀⢀⡤⠤⠴⠒⠚⠉⠁⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠘⡆⠀⢻⣿⣿⣀⣀⣠⣤⣤⣤⣴⣏⣀⡀⠀⠀⠀⠀⢀⣠⣴⣿⡿⣿⣿⣿⣿⣿⡌⠁
⠀⠀⠀⠀⠀⠀⠙⢦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⠿⠛⠉⢠⢷⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠟⠛⠉⠉⠉⠀⣀⣤⣶⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⡝⢶⠦⣤⣤⣤⠤⠶⠶⠿⠿⠿⢿⡿⠟⢸⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆⠻⢭⣤⣤⣤⠤⠤⠖⠒⠋⠉⠀⢀⣼⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⣿⣿⣿⣿⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⡶⠟⠁⠈⠄⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠏⠁⠁⠀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⡸⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⢀⣠⡤⠤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡄⠐⠛⡉⠋⠀⠀⠀⣠⠇⠻⠇⠀⠀⠀⠌⠃⢰⠀⠀⠀⠀⠀⠀⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣄⡈⠀⠀⠀⠀⣴⠁⢀⣁⣠⠆⠒⣦⡀⠀⠈⠰⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡏⠀⠀⠀⠀⠀⡆⢀⣩⣭⡀⠀⠀⢈⣙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⣼⣁⡎⢰⣿⢸⡀⠠⣿⡦⠈⢳⠀⠀⣠⠀⠀⠀⠀⠀⠐⣤⡙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠃⠀⠀⢠⡿⡟⠟⠲⣴⣾⣇⢠⣙⢃⣠⠇⠀⢡⢳⠀⠀⠀⠀⠀⠀⠈⢻⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢡⠀⠀⢸⠀⠘⠦⣸⡏⠉⠙⢦⡀⠀⠀⠀⡠⠃⢘⠀⠀⠀⠀⠀⠀⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⡆⠱⢦⣄⣁⣲⠓⠚⠉⠲⠔⠊⠀⠀⠞⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣝⣄⢹⡄⠈⢷⡀⠉⠉⢁⣲⠶⠊⠁⢀⡁⠀⠀⠀⠀⡠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣿⣄⠀⠉⠛⠛⠉⠀⠀⣻⣄⣉⣁⣠⡤⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠛⠻⣦⡜⠻⣷⣦⣤⣤⣴⣶⠟⠋⢹⣯⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠘⣿⣷⡙⣿⠚⠛⠛⠉⠀⠀⣸⠏⣠⣤⣶⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⡘⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣀⠴⠚⠛⠛⠙⠻⢿⠿⢿⠐⠂⠀⠐⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⣨⠟⠁⠀⠀⠀⠠⢄⠑⢦⣳⠟⠲⢶⣆⠀⣰⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⡿⠀⠤⢤⣤⣌⡱⣄⠑⡀⢿⡛⠀⠀⠉⣷⣿⣿⣿⠇⣿⡇⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⢰⡏⢹⣿⡾⢷⣾⠦⠀⠀⠀⠀⢘⣿⣿⡟⠀⣿⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠠⡞⠀⠀⠀⢸⢡⣿⡿⠃⠰⣿⣷⣄⠀⠀⠀⠸⣿⡟⠀⢠⣿⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⣾⣿⡇⠀⠀⠻⣿⡟⢷⡀⠀⠀⠹⣷⠀⢸⡿⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠇⣿⣷⠀⠀⠀⣿⠀⠈⣧⠀⠀⠀⠘⣿⣸⣿⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠈⠀⢸⣿⣤⣆⡀⠈⢆⣰⣿⡄⠀⠀⠀⠘⢿⡇⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⣿⢿⣿⣿⣿⣤⣿⣋⣌⣿⡀⠀⠀⠀⠈⢻⣆⠀⠀⢠⣿
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣸⣿⠏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠻⣆⠀⣼⡏
⠀⠀⠀⠀⠀⠈⢷⣤⣤⡾⠛⠃⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢻⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⠀⢀⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠀⠈⠙⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠖⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⡄⣠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⣁⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⢠⣤⣄⣼⢿⣿⣳⣯⣶⣦
⠀⠀⣤⣤⣰⣷⣄⠀⣀⣀⠀⠀⠀⢠⡟⠁⣠⠾⠋⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠈⠻⣎⠉⠀⠀⠙⢱⣾⣥
⠀⣤⣽⣮⡻⠟⠹⠾⣻⠟⠀⠀⢀⡿⢀⡾⠃⠀⡠⠤⡀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⣠⡿⠂⠀⢀⣠⡴⠟⠋
⠀⢛⣾⣍⠁⠀⠀⠀⣿⡀⠀⠀⢸⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠋⠀⠀⢠⡾⠁
⠀⠻⠿⠧⣤⣤⡀⠀⠈⠙⠷⣤⡾⠿⠁⠀⠓⠲⢄⠀⠀⠀⠀⠀⠀⢀⡹⠷⣤⣀⢠⡶⠛⢶⣄⣴⠋⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⣿⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠉⠛⠃⢀⣨⡿⢷⣄⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠘⣷⣤⡀⠀⠀⢿⡶⢮⣭⣤⡴⠶⠶⠛⣻⠏⠀⠀⣠⡴⠛⠁⠀⠀⢹⣦⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣼⣏⢷⡀⠀⠈⢳⣄⠀⠀⠀⠀⢀⡼⠋⠀⣠⠞⠉⠀⠀⠀⠀⢠⡼⠋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⠀⠀⠉⠛⠶⠶⠞⠋⠀⣠⠞⠁⠀⠀⠀⠀⢀⣴⠋⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣝⠷⣤⣀⠀⠀⢀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣠⠟⠁⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡿⢻⣟⣻⣿⠟⠋⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣀⣩⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⡴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠾⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣶⠛⠃⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠶⠶⢶⡾⠋⣍⢤⠀⠐⠛⠁⠉⠙⠻⣦⣀⠀⢸⡇⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣼⠇⠀⠈⠻⣧⠀⠀⠀⠀⠀⠀⢸⡏⠳⣾⡏⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠰⣦⣀⣤⡴⠾⣋⠀⠀⠀⠀⠈⠰⣶⣤⣤⡀⠀⣼⠃⠀⠘⣧⣼⡇⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡟⠉⠁⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠙⢦⠁⠀⠀⠀⠀⠀⠈⣿⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⣿⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⡏⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⣠⣿⣿⣿⣷⣦⣄⣀⣤⣴⠞⠋⣷⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢹⡀⠀⠠⣶⡶⢿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠛⠶⢭⣭⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣴⠟⠳⣄⡀⠀⠀⠀⠀⠀⠀⣤⡾⠋⠁⠀⠀⠙⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢤⣀⠀⠀⢠⡾⠁⠀⠀⠀⠉⠒⠶⢤⣤⣠⡾⠋⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡞⢻⠄⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⡿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠘⢷⣤⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠏⠀⠀⠀⣼⠃
⠀⠀⣤⠶⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠉⠙⠛⠻⠶⣦⣤⣤⣀⣀⣀⣀⣠⣤⣤⠶⠟⠋⠀⠀⠀⠀⢀⡟
⠀⠀⢹⡄⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣿⡁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠈⣧⠀⠀⠀⠀⢳⣄⣤⡴⠶⠾⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠘⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢛⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⢤⣄⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠈⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⢿⣿⣶⣦⣤⣀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠀⠈⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢿⣷⣦⣀⠀⣼⣿⠃⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⢀⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⢀⣠⣾⡿
⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣦⣤⣀⡀⠀⠀⠈⠛⠿⢿⣿⣿⠋
⠀⠀⣰⣿⠏⠀⠀⠀⠠⣶⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣶⠄⠀⠀⠀⠹⣿⣆
⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⠛⠛⠛⠿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏
⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣄⢀⣠⣶⣾⡿⠿⠿⢿⣷⣶⣄⡀⣠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣋⡀⠀⠀⠀⠀⢀⣉⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⡿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⣿⠿⠿⢿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠁⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⡀⠈⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⢀⣴⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠙⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣷⡈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡄⢻⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣼⣿⠏⠀⢠⣿⠏⠀⠀⠀⠀⣀⣤⣤⣴⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠾⠿⠷⠶⣤⡀⠀⠹⣿⡄⢻⣿⣦
⠀⠀⠀⠀⠀⢀⣾⣿⠋⠀⠀⢸⠃⠀⠀⢀⣴⡿⠟⠋⠉⠀⠀⠉⠙⠻⣶⡄⠀⠀⠀⠀⠀⠀⣰⡟⠉⠀⠀⠀⠀⠀⠈⠙⣦⡀⠙⣧⠀⠹⣿⣧
⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠁⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠈⠀⠀⢹⣿⣧
⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢿⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⢻⣿⣇
⠀⠀⢰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀
⠀⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⠀⠀⠀⠀⠀⠀⠻⣿⡟⠻⠶⠶⠶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣇
⢠⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣤⡤⠶⠋⠁⡾⠀⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡞⠁⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡶⢾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣹⣧⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣦⣄⡀⠀⢀⣠⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡌⠙⠛⠛⠛⠋⢩⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⢷⣀⣰⣏⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠐⠛⠛⠛⢛⣛⣛⠛⠋⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠈⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠙⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⣄⣀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢀⣀⡀⣸⣇⣀⣀⣤⣤⣶⣾⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⠤⠤⡈⠷⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢈⣀⣀⡀⠀⠀⠀⠈⠷⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠊⠁⠀⠁⠀⠀⡠⠤⣄⠀⠾⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⢀⣠⠀⠀⠀⢀⠎⠀⠀⠀⠑⣄⠒⠺⣄⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡴⢊⠏⠁⠀⠠⠊⠁⠀⠀⠀⠀⠑⠈⢢⠀⢈⣧⠀⠀⠀⢀⠜⢁⡠⡙⢆⡠⢔⠂⠉⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡠⠀⡇⠀⠀⣀⣀⡀⣠⠒⠐⠀⢣⢀⠊⠀⠀⡠⠃⠴⠁⠁⣸⡤⠒⠁⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢘⠁⠀⢃⢰⠋⠀⠀⠈⠉⠁⠀⠀⠃⢺⣆⠀⡔⢡⠊⠀⣢⠴⡇⠀⠀⠀⠀⠀⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠠⠤⢄⣈⠉⠑⢄⡄⠄⠀⠀⠠⡶⠄⣀⠠⠃⠀⠓⠊⡔⡁⠖⠊⠁⠀⠈⢳⣈⣁⠀⠀⢀⡄⠈⠄
⠀⠀⠀⠀⠀⣀⣀⡀⡇⠀⠠⠒⠲⠤⣭⣐⡚⠷⠀⠀⠉⡉⠀⠀⠀⠈⠐⢄⣠⠊⠁⠀⠀⠀⠀⠀⠀⢀⡴⠁⢠⠊⠁⢁⠀⠠
⠀⠀⠀⠀⠘⠁⠀⠐⠛⠒⠲⢤⣤⣀⣆⣀⣸⣑⡒⢤⣔⠀⠀⠀⠀⠀⠀⣸⣿⣷⡄⠀⠀⠀⠀⠀⠀⢸⢱⠔⠋⠀⠀⠈⡆⠀⠇
⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⣀⠕⠀⠀⠀⠀⠀⠀⣿⣿⣄⣀⣀⣀⣠⣴⡿⠿⢿⡃⠀⠀⠀⠀⠀⠀⡘⢈⡠⠒⠁⠈⢣⠜⠀⡰
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣹⠉⢆⠀⠀⠀⠀⠀⠀⠹⣿⡿⠟⠛⠛⠛⠁⠀⠀⠀⠈⡆⠀⠀⠀⠀⢠⠃⠉⠀⠊⠉⢁⠆⢀⠜⠁
⠀⠀⠀⠀⠀⠈⡆⠠⡤⠭⡀⠉⢳⡳⡀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⢁⣀⡀⠀⣀⡔⢁⣀⣀⡀⣤⠎⣠⠣⢄⡠⠔⢋⡔⠁
⠀⠀⠀⠀⠀⠀⢡⠀⢇⠀⠘⠤⠜⠣⡈⠢⡀⠀⠀⠀⠑⣄⣀⣤⠚⠁⠀⠉⠙⠛⠡⠭⢀⡠⠖⠁⡰⣁⠔⠉⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠈⠆⠈⡆⢰⠚⢉⢑⣛⠂⠈⠲⢆⣛⣉⡱⠚⢀⡔⠀⠀⠀⠀⣀⠝⠓⠲⡤⠴⠚⢏⢀⡠⠚⠁⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠙⢄⠘⠂⠀⠀⡝⠒⠤⠤⠥⠤⠒⠁⠈⠉⠉⢏⠁⠱⡤⠦⠴⠁⠀⠀⠀⢣⠀⠀⠀⠀⣈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠅⢂⡀⠚⠢⠤⠤⣀⣀⣠⠒⠒⠢⠤⠖⠋⣄⠠⠋⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⡐⠁⠈⢢⣀⣀⡤⠒⠀⡖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠠⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⡰⠁⠀⠀⢠⡇⣇⢘⠒⢤⠆⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠄⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢲⠃⠀⠀⠀⢸⡇⡏⠁⠀⡸⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⠀⠀⠀⢸⢽⠁⠀⢠⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠁⠀⠀⠀⢀⠤⡀⣸⣿⡿⠀⢠⠃⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⣀⡤⠤⠔⠒⠊⢫⠀⢆⠀⠀⢀⠔⠁⠀⢳⡜⢿⠃⡠⢃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⢣⣈⡦⠒⠁⠀⠀⠀⠈⠧⣏⠔⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠑⢄⠀⠀⠀⠀⠀⠐⠚⠉⠱⡀⠀⠀⠀⠀⠀⠀⠈⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣠⢔⣳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠻⢿⣟⣛⠿⢿⡿⣿⣭⣭⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⠀⢘⡏⠒⠀⠈⡉⣉⣉⠛⠺⠿⠿⢛⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠄⠀⠠⠤⠤⠄⠒⠒⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣆⣠⣤⠶⠞⠛⠛⠉⠉⠉⠛⠛⠲⢦⣄⡀
⠀⠀⠀⠀⢀⣴⡟⠋⠁⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⣤⡈⠙⢦⡀
⠀⠀⠀⠀⢩⣿⠀⠀⠀⠀⠀⠀⠺⠋⠀⠀⣀⠀⢀⠀⠈⢿⡄⠀⠹⣆
⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⢀⠎⣡⢾⠁⠀⢨⢷⡄⢄⠀⠀⠀⠹⣆
⠀⠀⠀⢀⣿⣿⡀⠀⠀⡔⠀⠀⢜⣴⣿⠀⡇⠀⡌⢰⣿⣼⡄⠀⠀⠀⢻⡀
⠀⠀⠀⢸⣿⡄⠁⠀⣸⡇⠀⣠⡞⠿⠋⢰⠇⠀⢇⠈⢿⠿⣧⡀⠀⠀⠸⣧⡴⢦
⠀⠀⢀⣤⢿⡇⠀⢠⣻⠀⠀⡀⢧⠀⢀⡼⠀⠰⡾⡄⠀⢀⠏⠁⠀⡄⠀⡟⣴⢺⠇
⠀⠀⢾⠁⢄⠙⠆⠈⠋⢀⠀⠹⢦⣉⠋⠁⠀⠀⢻⡜⠛⣫⠞⠀⠀⣧⠀⣧⠻⡼
⠀⠀⠸⡾⠌⣧⠀⠀⠀⣼⠀⠘⠒⠀⠀⠘⡉⠀⠀⠙⢦⡐⠋⠀⠘⣿⠀⢹⣄⠇⠀⠀⠀⠚⣉⣡⣤⣄⡀
⠀⠀⠀⠹⣄⠻⠇⠀⢠⢻⠀⠀⠀⣰⠆⠘⠛⠛⠶⢤⣀⠙⠆⠀⠀⠉⠀⢸⡏⠀⠀⠀⠀⢈⣥⠴⠶⠦⣄
⠀⠀⠀⠀⠈⠳⣤⠆⠈⣽⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠈⠛⢋⠀⠀⠀⠀⢸⡇⠀⠀⣠⠞⠉⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⢸⡆⠈⠁⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣾⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⣏⣠⡤⠴⠶⠶⠦⠤⣤⣀⠀⣿⠀⢀⣼⠏⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⣗⠘⢷⡄⠀⠰⡍⠁⠀⠀⠘⠛⠛⠀⠀⠉⠓⠼⣠⣾⠏⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⣼⡇⠀⠈⠻⢦⣄⡐⢦⡀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠁⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠻⠁⠀⠰⡟⢢⠈⢉⡹⠿⣶⣦⣤⣤⣴⡾⠟⠀⠀⠀⠀⠀⠀⠈⣳⢶⡀⠀⠀⣀⡴⠟⢁⡤
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⠞⠋⣉⣠⣴⣴⣀⣠⣝⢍⡷⣤⠀⠀⠀⢰⡞⠳⣴⡷⢻⠏⠛⠋⢁⣠⡴⠋
⠀⠀⠀⠀⣠⠴⠶⢾⡇⠀⠀⠀⢹⠴⡿⢺⠷⣌⢩⡞⠀⣞⣧⡀⢰⣏⠳⡶⠎⠳⣾⡀⠀⣈⣩⡥
⠀⠀⠀⢸⡷⠚⠛⠋⠀⠀⠀⠀⠀⣨⠇⣸⠀⠈⠀⠀⠀⡉⠋⣻⣼⡏⠹⠃⠀⠀⠙⣿⠀⠀⠁
⠀⠀⠀⠸⠁⠀⠀⢀⣠⠶⠶⠖⣿⠁⢠⣿⣇⣀⠀⠀⠘⠳⡾⠁⠠⢷⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠰⢦⣤⠶⠛⠀⠀⠀⠀⣹⡶⣿⣿⣿⠭⠭⣭⣽⣿⡇⢸⠃⠀⠛⣦⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⢰⡟⠒⠶⠶⣿⣄⣈⠛⠁⠀⠀⠀⠀⣀⣹⣼⣧⡤⠴⠿⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⣌⠉⠃⠀⠀⠀⠀⠀⠉⠁⢠⡆⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠙⣷⠚⠛⠛⠛⠛⠛⠓⢶⡿⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢶⣤⣤⣿⠙⣿⡟⠛⠛⢿⡷⠈⣇⣀⣀⣀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠖⠾⠤⠽⣯⠀⣿⠁⠀⠀⢸⡇⠀⣹⠋⣯⡉⠁

⠀⠀⠀⠀⠀⠀⠈⠻⢤⣭⣤⠴⠲⠶⠟⠀⠈⠀⠀⠀⠈⠁⢸⣌⣉⠉⠙⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠦⣤⣬⣽⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣴⡋⣶⣳⣦⣴⢤⣶⣤⣴⡒⣀⣤⣄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⣾⣉⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣤⡤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣽⣴⡀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⢄⣴⣶⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣿⣆⣀
⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠿⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⡷⠄
⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡟⣶⡀
⠀⠀⠀⢀⣷⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⣀⣤⣴⣶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠓⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣭
⠀⠀⢰⣿⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⢀⣴⣾⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⡿⣿⡷
⠀⢀⣾⣷⣿⣿⣿⣿⠟⠁⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠋⠁⠀⠀⠉⠙⢿⣆⣠⣄⡀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣇
⠀⣿⣾⣿⣿⣿⣿⠃⠀⠀⠀⣰⣿⡿⠋⢀⣠⣤⣶⣦⣤⣄⠀⠀⠀⣾⡏⠀⠀⢀⣄⡀⠀⠀⠈⣿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡀
⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⢠⡿⠋⢀⣴⡿⠋⠉⠀⠀⠉⠻⣷⡄⠀⢻⣧⠀⠀⠘⠛⠃⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡇
⢠⣿⣿⣿⣿⠇⠀⠀⠀⠀⠘⠁⠀⣾⡟⠀⠀⠀⣠⣀⠀⠀⠸⣿⠀⠀⠙⢄⡀⠀⠀⠀⢀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⢠⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠛⠛⠀⠀⢀⡏⠀⠀⠀⠀⠙⢛⠶⣾⣿⣿⣷⣶⣄⣀⣤⠶⠖⢦⠀⠀⠀⠀⠀⠀⠸⣿⣿
⢈⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢒⣽⣿⣦⡀⠀⠀⠀⠀⣠⡾⣡⠀⠀⠀⠀⠀⠘⢷⣿⡟⢧⡀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠈⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠙⠻⠷⠶⣶⣟⡋⢰⣿⠀⠀⠀⠀⠀⠀⠈⢿⣄⢸⡇⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⣿⣿
⠀⠰⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡾⠋⣽⠟⢻⣿⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⠇⠀⠀⠀⠀⠀⠚⢡⡌⠁⠀⠀⠀⠀⢰⣿⣿
⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠋⠉⠀⠀⠻⣦⡘⣿⣇⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⡤⠶⣤⠀⠀⠀⠀⠈⠙⠻⣿⣦⣄⣀⣀⣠⣴⣿⣿⢿⠀⠀⠀⢀⣴⣶⣾⡿⠁⠀⠀⠀⠀⢰⣿⣿⠇
⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⢿⡿⠿⠿⠿⠟⠋⠁⢸⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏
⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⡀⢀⣠⣤⣤⣄⠀⠀⠀⠳⡄⠀⣀⣠⣤⣤⣼⣄⣀⣸⡟⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠁⠈⠙⣷⣄⠀⣀⣽⡿⠟⠋⠉⠉⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⢀⣼⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⣠⠶⠶⢶⣤⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠁⠀⢀⠀⠙⣿⡄⠀⠀⣀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣾⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣿⣿⣿⣿⣿⣿⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣏⡀⠛⡟⢠⣿⡿⢹⣿⣿⠃
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠿⠛⠛⠛⠛⠁⠘⠻⢣⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠔⠒⠒⠄⠀⠀⠠⠖⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠿⡉⣧⠀⠠⠾⠯⠀⠀⠀⠀⠚⠓⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣬⣿⣿⣿⣿⣶⡷⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⣇
⠀⠁⠈⠀⠀⠀⢠⣤⣼⡁⣻⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⣿⣀
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⢛⣧⣀⡀⠀⠀⠀⢠⣿⣶⣶⣾⠁⢰⡋⠁⠀⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠛⠧⢤⡀⠀⠘⣏⠀⠈⡟⠠⠛⠀⠀⠀⢄⠀⡀
⠀⠀⠀⠀⠀⠀⢀⡤⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠁⠐⣄⡙⠦⠴⠧⠀⠂⠀⠀⠀⠀⢐⡡
⠀⠀⠀⢀⣤⠾⠁⠀⢠⡟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢥⣀⣀⣤⠃⠀⠀⠀⢀⣴⠋
⠀⠀⣴⠋⠀⠀⠀⠀⠈⢷⣿⣿⡿⣿⡿⢿⣦⣤⡀⠀⠀⡠⠀⠙⠃⠀⠀⢀⡄⠉⠙⢧⡀
⣠⠞⠉⠀⢀⣠⣤⣄⣀⠀⠀⠀⠀⠘⢧⡞⠀⠀⠀⠀⠜⠀⠀⠀⢠⠀⠀⠀⢡⠀⠀⠀⢳⡀
⠻⡄⠀⡾⠉⠀⠀⠀⠀⠉⢆⠀⠀⠀⡸⠁⠀⠀⠀⢠⡀⠀⠀⠀⠈⡇⠀⠀⠈⢇⠀⠀⢀⡗
⠀⠙⣆⢻⡀⠀⠀⠀⠀⠀⠘⡀⠀⠠⠁⠀⠀⠀⢠⠃⠱⡀⠀⠀⠀⣿⠀⠀⠀⠈⢆⠀⡼⣧
⠀⠀⠈⠻⡇⠀⠀⠀⠀⠀⠀⢁⡠⠁⠀⠀⠀⢀⡆⠀⠀⠘⣄⠀⢀⡟⡄⠀⠀⠀⠀⢳⡃⢹⡀
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⡜⠀⠀⠀⣀⠘⣆⡜⢀⡇⠀⠀⠀⠀⠀⠳⡄⠃
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠐⠒⠚⢻⠀⠸⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡟⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⣀⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤⣀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠻⠿⣾⣶⣾⣿⣿⣿⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣰⡶⢟⣉⠉⠉⢻⣿⣿⣿⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡫⠿⢞⣓⣾⣿⣿⡿⣿⣿⣿⣿⡗⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣖⣚⡙⠩⢍⣛⠻⠙⠊⠓⠁⠁⠑⣄⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠉⢩⣤⣄⡙⠂⠉⠁⢶⣦⣄⣀⠁⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⡟⠀⠠⠿⠿⠬⠟⠀⠀⠀⠀⠙⠟⠋⡛⢣⡀⠨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠛⠒⠒⠲⢶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⢰⢾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢾⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠤⢺⡟⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢀⣠⡶⠂⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣠⣾⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣇⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠋⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡟⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠛⠛⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡿⢿⣦⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠃
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⢿⡄⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⢀⣿⠀⠀⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⡟⠀⠀⠠⣤⡄⢿
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠛⠛⠿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⡀⠀⣸
⠀⠀⠀⠀⠀⣾⡟⠀⣴⠶⣦⡄⠈⠙⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿
⠀⠀⠀⠀⢸⣿⡃⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢿⣧⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠿⣶⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⠉⠉⢹⡟⠙⠻⣷⡀⠀⠀⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⢻⡆⠀⠘⣷⠀⠀⣿⣧⠀⠀⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠹⠇⠀⠈⠉⠀⢸⣿⣶⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣿⠿⢿⣶⣤⣤⣄⣀⣀⠀⠀⠀⢈⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⠋⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠈⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢿⣷⡀⠀⠀⠀⠀⢻⣿⠀⢀⣠⣴⡿⠟⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⢿⣧⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠀⠀⠀⠙⢿⣿⠋⠁⠀⠀⠀⠹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠈⣿⣦⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠹⢷⡄⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⠟⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠘⣿⡇
⢀⣴⣶⣶⣶⣶⣶⣶⣾⠿⠟⠃⠀⠀⠀⠀⠀⣠⣾⠟⠋⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣄⣀⡀⣀⣠⣾⠋
⠀⠙⣿⡿⠿⠿⠿⠿⣿⣿⣆⠀⠀⠀⠀⢻⣷⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⡟⠉
⠀⠀⠹⣿⣦⣠⣤⣶⠿⠛⢿⣧⠀⠀⠀⣸⣿⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠈⠻⢿⣿⠿⠟⠁⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠋⠉⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠉⠉⠉⠉⢙⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⣏⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⢸⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⠚⢿⡏⠙⣿⡾⣿⣷⣾⣶⣿⣿⠀⢸⣿⣷⣶⣶⣾⣷⣾⠟⠉⡿⠗⢲⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠛⠛⣽⡷⣠⣿⣦⡾⠁⠀⠀⠀⠀⢹⣿⠀⢸⣿⠀⠀⠀⠀⠀⠹⣦⣼⣧⣴⣾⡿⠛⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣁⣀⣀⣀⣨⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣾⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣽⣁⣀⣀⣀⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇⣿⣯⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣿⡇
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤
⣿⣿⣿⣿⣿⠿⠿⢿⣿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⡀⠀⠀⠀⠀⣠⠖⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡤⠶⠦⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⣠⣶⣤⡈⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⠀⠀⠀⠀⢀⠀⠀⢹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢡⠇⢀⡀⠀⠀⠀⢀⡀⠘⡇⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⣸⣯⣭⣭⣄⠀⠘⢮⡉⠀⣿⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡸⠋⠀⠀⠀⠙⠳⣦⣄⣀⣼⢿⡆⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠘⠉⠙⠲⣤⣀⠀⠀⠉⠉⢁⣼⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⠀⠀⠀⠀⠙⠻⠶⣶⣾⡿⠟⠁⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣆⠰⠄⠀⠀⢀⣀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠹⣿⣷⡄⣠⠾⣿⣿⡇⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣌⠻⢿⡇⢀⣿⡿⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣟⠷⣤⣙⠻⣟⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⢊⣽⢷⣌⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⣿⠁⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⣄⢀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣟⢯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠸⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣷⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠐⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⣆⠀⠀⠐⠄⠀⠀⠀⠀⣠⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠠⣄⣀⣀⣤⣿⠁⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⡀⠀⠀⠀⠈⠛⠛⠛⠁⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠒⠦⢄⡀⠀⠀⠀⣠⢊⡈⡑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠰⠦⣀⠈⠐⡄⠀⠘⠦⡅⠰⣔⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⠀⠀⠀⡈⠉⠀⠀⡀⠠⠐⠈⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣄⠀⠇⠀⠀⠄⢠⡄⠀⠘⠀⠀⠀⠙⢢⡀⠂⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢈⠔⢉⠀⠀⠀⠀⢀⠄⠊⣀⢀⡀⠄⠀⠀⠂⡀⡀⠄⠤⡑⢀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤⠴⠷⠤⠤⢄⡤⢠⠀⠊⠀⠀⠀⠤⠤⢒⡉⠀⠂⠀⠀⠀⠀⠐⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⠀⢐⣃⠀⠀⡥⡵⠀⣀⠀⠀⠀⠀⢀⣩⠀⠔⢊⠀⠀⠀⢠⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠂⠉⠉⡄⠸⠤⠤⡄⠈⠑⠒⠒⠋⠁⠣⣭⡤⠔⣀⠔⢊⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣄⠀⢠⡀⠀⢠⠀⠀⠂⠠⡭⠉⠈⠀⠞⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⣾⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⣶⣦⣬⣹⣯⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠈⣾⣽⠿⠿⠟⠋⠀⢈⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠐⣿⠅⠀⠀⢸⠀⠐⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⢸⡄⠀⠀⠈⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠗⠀⣾⡇⠀⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡈⢹⡇⠀⠀⣸⠀⣸⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣁⣼⣇⢀⡤⠿⣶⡇⠀⣤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠄⠀⠠⡆⣈⡹⠉⠉⣉⣹⣿⡟⠂⠀⣄⠀⠀⡀⠸⠟⣻⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⢠⡀⠀⠃⠀⠀⠀⢛⣛⣳⣀⣤⣿⡿⠟⠿⠷⠾⠿⠀⢿⣥⡤⠶⠖⠋⠈⠁⠀⠀⠀⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢀⣠⢀⡀⢀⣀⡀

⣿⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠠⠀⣀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠁⠀⠀⠠
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀
⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⠉⢠⠄⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢀⠄⠀⠀⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠂⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠐⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⠂
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⢛⣿⣯⣥⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡴⠊⠙⠋⠉⠉⠙⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⡷
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⢤⣄⣀⣀⣀⣀⣀⡤⣶⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠠⠐⠒⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡼
⠀⠀⠀⠀⠀⠀⠀⠀⢷⣀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠚⠁⠉⠂⢹⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠋⠉⢉⠳⡀⡇⠀⠀⠀⠀⠀⠤⠤⠭⣕⣆⠀⠀⠀⠀⢀⣴⣂⣀⠀⣿
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢣⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠘⠉⠀⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⡴⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⢤⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡤⠴⡇⠀⠀⠀⠀⠀⠀⢴⣖⠒⠒⠶⠤⢤⣀⣀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⣀⡀⢹⡄⠀⠀⠀⠀⠀⠸⡿⠿⣷⣶⠶⠤⠤⠭⡿⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣋⠀⢀⣱⣄⠀⠀⠀⠀⠀⠹⣄⠈⢻⡀⠀⢀⡼⠁⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠢⣄⠀⠀⠀⠀⠈⠙⠒⠓⠒⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⣄⣀⣀⣀⣠⡤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣥⣾⣿⣿⣥⣴⣶⡤⠄
⠀⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⠈⠐⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢸⡿⠻⢿⡿⢿⣿⡿⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠈⢠⣶⣤⣀⣠⠀⠀⡀⠀⠀⣀⠀⠹⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠫⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠳⠛⠋⠁⠀⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠠⠖⠀⣀⠀⠀⠀⢠⠄⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⠅⠀⠀⠀⠀⠘⢄⠀⠀⠀⡀⠀⠀⡖⠀⠀⠁⠀⠀⡀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠐⢄⠀⠀⠐⠒⠿⠛⠀⠀⡀⠀⠀⠀⡀⠀⠀⠂⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠐⠈⠑⠲⠀⠤⠤⠄⠐⠈⠀⠀⠀⢐⠀⠀⠀⠄⠀⠈⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠄⠀⠁⢀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠄⠁⠒⠂⠂⠗⠀⠰⠁⠀⠀⠈⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠠⠀⡰⠊⠀⠀⠀⠀⣰⠃⠀⠉⠀⢀⠤⠀⠀⡀⠁⠄⡀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠀⠀⠈⠀⢀⡒⠢⠁⠀⠀⠀⠐⠊⠀⠀⠀⠀⠀⠀⠠⠆⡴⠂⠒⠀⠈⠉⠉⠁⠀⠀⠀⠐⠀⠀⠀⠀⠂⠁
⠀⠀⠀⡈⠀⠀⠠⠀⠀⠀⠀⠄⠐⠀⣠⠦⠐⠂⠈⡠⡄⠀⠀⠀⠀⠀⠀⣀⠤⠄⣀⠠⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢀⠌⡄⠷⢤⠤⡤⠒⠂⠹⡁⠶⠷⡖⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠄⠀⠀⠀⠀⢸⠰⡁⠠⣀⠔⠀⣰⠁⠀⠀⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠⠀⠀⠀⠀⠀⠘⠤⠉⠒⠀⢤⡭⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣠⣶⣶⡦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⠛⠓⠦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣒⣦⣤⣀⣠⣤⡤⢤⡀⠀⠀⠀⢀⡤⠖⠚⠃⠀⠀⠀⠀⠀⠈⠉⠙⠓⢦⡀
⠀⠀⠀⠀⢀⣴⣪⢏⢔⡀⠲⣾⣿⣿⣟⣋⠉⢢⠉⢢⡀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇
⠀⠀⠀⣠⣿⠿⢗⠕⢡⠞⠋⠉⠉⠉⠛⠻⢷⡄⢣⡀⠙⣆⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⡴⠋⠀⠔⣡⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣷⡄⡜⣦⣈⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⢸⡇⣧⣤⣾⢃⣀⣀⡀⠀⠀⠀⠀⢀⣀⡠⠤⡈⣿⣿⣿⡿⡆⠀⠀⠉⠒⠒⠒⠁⠉⠛⠛⠋⠁
⠀⠈⢧⡹⣿⣏⠈⠀⣈⣉⠁⠀⠀⠀⠉⣁⣀⣀⣀⢸⣿⡟⠁⡇
⠀⠀⠈⣿⣿⣿⠘⠿⣩⣯⠀⠀⠀⠀⠀⢹⡯⠝⠁⣿⣟⣜⡰⠁
⠀⠀⣰⡿⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇
⠀⢸⣿⡵⣣⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⢸⣯⡘⣿⣿⣧⠄⠀⠀⠀⠀⣀⣐⠀⠀⠀⠀⢰⣿⡟⣿⡇
⠀⠀⠛⢿⣮⣿⡇⠀⡐⠀⠀⠈⠛⠋⠁⠀⠀⣠⣿⡿⠁⡹⠁
⠀⠀⠀⠀⠙⣿⡇⠀⠃⢠⡄⠀⠀⠀⠀⣀⢴⣿⣿⠇⣼⠁
⠀⠀⠀⠀⠀⣻⡇⠀⠀⠈⢠⠗⠒⠒⠊⠁⠀⣿⣿⣿⣿⡄
⠀⠀⠀⣴⡿⣫⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⡛⡀
⠀⠀⢠⣿⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣧⡇
⠤⠶⢌⢿⣦⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣛⢹⣿⡿⣉⠴⠦⠤⣄
⢢⠀⠀⠑⢍⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⠖⠁⠀⠀⡜⠁
⠸⡄⠀⠀⠰⠁⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠉⠉⠉⠀⠀⠀⠀⡸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣛⣭⣭⣭⣛⡛⠿⠿⠿⠿⠟⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣼⣿⣿⠿⢿⣿⣿⡿⠿⣿⣿⣷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠲⢄⣿⣿⡸⠆⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿⣿⡿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣿⣿⣆⢿⣶⣭⣭⣭⣭⣶⡿⣸⣿⣿⣿⣷⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣮⡻⢿⣿⣿⡿⢛⣵⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣸⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⢹⣿⡿⠿⣛⣂⢿⣿⣿⣿⣿⣿⣿⡿⣑⣛⠿⢿⣿⡏⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢡⣾⣿⣿⣿⣎⢿⣿⣿⣿⣿⣿⢣⣿⣿⣿⣷⡎⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡎⢿⣿⣿⣿⣿⡜⣿⣿⣿⣿⢣⣿⣿⣿⣿⡿⢱⣿⡿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⣴⡄⠀⠀⠀⠀⠠⠮⣷⣀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠥⡀⠀⠄⠀⠀⠀⠉⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢒⡒⡠⣏⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠃⠀⠀⠈⠢⡀⢀⠀⠀⠄⠀⠀⠐⠒⠢⡀⠀⠹⣄⠀⠀⠀⣀⠀⠁⢀⠀⠒⣠⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⠀⠘⠀⡓⠀⠀⠀⠄⠀⠌⠀⠀⠀⠠⢤⡀⠊⠀⠀⠈⠐⢀⡙⢀⠔⠁⠀⠙⠄⠀⠀⠂⠀⠐⢄
⠀⠀⠀⠀⠀⢀⠀⠁⠀⠀⡀⠠⢀⠀⢸⠀⠀⠀⠉⠘⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣀⠀⢈⡉⠒⢄⠀⠙⡄
⢀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣁⠒⠈⣀⠀⠁⠀⠀⠀⠀⡁⠀⠀⠠⠞⠀⠀⠘⠋⠀⠀⠀⡇⠀⠀⢀⠡⣥⢀⡀⠙⠀⠈⡀⠀⠆
⠤⢵⣷⡴⠂⠀⠀⠀⠀⠀⠋⠉⠁⠀⠉⠣⠀⠀⠀⠀⠀⡇⠀⠀⠐⠍⠀⠀⠩⠄⠀⠀⣰⠀⠀⠠⡀⠁⠀⠪⠓⠀⠀⡰⠀⠌
⠀⠈⠻⣄⣀⣈⡀⠀⠀⠀⠬⠟⠀⢌⡻⠀⠀⠀⠀⠀⣠⠀⠀⠀⠙⠀⠀⠀⠀⠒⠀⠀⢾⢹⣦⠀⢨⣐⠒⣒⠎⠀⠔⣤⠊
⠀⠀⠀⠀⠉⠉⠡⠀⠀⠀⠀⡀⠀⠀⣀⣴⠁⠀⠀⠀⠈⣧⠀⠀⢀⠀⠀⠀⠀⠀⠀⣴⢟⣻⡁⠀⡸⣛⡻⠋⠀⠊⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠈⢬⠭⢚⠏⠀⠀⠀⢀⡴⣼⠆⠀⠀⠁⠂⠒⠉⠀⠚⢻⠟⠛⠣⡀⠀⠀⠀⠀⣀⣰⠋
⠀⠀⠀⡀⠀⠀⢸⡛⠾⠿⣄⡀⠀⠀⣀⢴⠉⠓⠚⠙⢲⣿⣆⡀⠀⢀⠔⡤⣀⣀⠠⠊⣀⡴⠞⠋⠐⠀⠒⣞⠁⣇⣤⣄
⠀⠀⠀⠙⠓⠶⢼⡇⠀⠀⠀⠉⢉⣭⠀⢸⡇⠀⠀⠀⠉⠃⠀⠀⠉⢷⡼⠁⢀⣀⠀⠀⠉⠙⠛⠛⣿⠇⠀⠹⡄⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠳⣼⠇⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠛⠉⣿⠀⠀⠀⠀⠴⠟⠋⠀⠀⠀⠙⠛⠛⢛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠷⠶⠦⠀⠰⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠒⠛⠛⠛⠛⠛⠛⠓⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⢤⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⠳⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠈⢳⡄⠉⠙⠶⣄
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠈⢷⣄
⠀⠀⠀⢀⣼⠋⠀⢠⡴⢫⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⢻⠲⣄⠀⠀⠻⣆
⠀⠀⢀⣾⠃⠀⣴⠏⢀⡾⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠾⠛⠉⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠸⡇⠘⣧⠀⠀⢻⡄
⠀⠀⣾⠃⠀⣼⠃⢠⢾⡇⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⣷⠀⠸⣇⠀⠘⣧
⠀⢸⡟⠀⢠⠇⢠⠏⢸⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠞⠻⠆⠀⠀⠀⠀⠀⠀⢠⣴⣦⠀⠀⠀⠀⠈⠙⢶⡀⠀⠀⠀⣿⠀⠀⢻⡄⠀⢿
⠀⣿⡇⠀⠘⠀⡞⠀⢸⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⢀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⣿⠀⠀⢸⡇⠀⢸⡆
⠀⣿⠀⠀⠀⢠⡇⠀⢸⡄⠀⠀⡟⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠈⠻⡄⠀⢀⡴⠋⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⢻⠀⠀⠀⡏⠀⠀⢸⡇⠀⢸⡇
⠀⣿⡆⠀⠀⠀⡇⠀⠈⣇⠀⢰⡇⠀⠀⠀⡰⠁⠀⠀⢀⡀⠀⠀⠀⢳⠀⡞⠀⢀⣀⣀⡀⠀⠀⠱⠀⠀⠀⠀⢸⡄⠀⢸⠇⠀⠀⢸⡇⠀⢸⡇
⠀⣿⣇⠀⠀⠀⠁⠀⠀⠹⡄⢸⠃⠀⠀⠀⠁⠀⢠⣾⣿⣟⢁⡄⠀⠈⠀⠀⢀⣿⡿⢿⣉⡆⠀⠀⠀⠀⠀⠀⠘⡇⠀⡞⠀⠀⠀⢸⠁⠀⢸⠇
⠀⠸⣿⡀⠀⠀⠀⠀⣠⠶⠻⣼⠀⠀⠀⠀⠀⠀⠸⣿⡁⢈⣿⠇⠀⠁⠈⡀⠈⢿⣄⣀⣿⠃⠀⠀⠀⠀⠀⠀⠀⡗⠚⠳⣄⠀⠀⠀⠀⠀⣿
⠀⠀⢻⣧⠀⠀⠀⢠⡏⠠⣄⣿⠀⠀⠀⠀⠡⡀⠀⠈⠉⠉⠁⣠⠖⠒⠤⣐⢄⠀⠈⠉⠀⠠⠀⠀⠀⠀⠀⠀⠀⣇⣠⣄⢹⠀⠀⠀⠀⢀⡟
⠀⠀⠈⢿⣧⠀⠀⢸⡇⠀⠈⡟⠀⠀⠀⠀⠀⠈⢐⡦⠴⠒⢻⡏⠀⠀⠀⠀⠀⠁⠙⡛⢁⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⡎⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠈⢿⣆⠀⠀⠙⢦⣤⣇⠀⠀⠀⠀⠀⠠⢃⠄⠂⠀⠈⢧⡀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⠀⠀⠀⠀⠀⠀⣿⠴⠋⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢸⣇⠀⠀⢸⡄⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⣿⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⡀⠀⢰⡄⠀⠀⣿⡄
⢠⡾⠃⠀⠀⣼⠃⠀⠀⢠⣿⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⢠⡇⠀⠘⣧⡀⠀⢹⣷⣀
⢸⣷⣤⣤⠾⠃⣠⠄⠀⣼⡏⠀⠀⣿⢈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠰⣿⠀⢸⣧⠀⠀⠙⢿⣦⣤⣙⣻⣿⠞
⠈⠿⠿⠶⣶⣿⠟⠀⣰⠟⠀⠀⣴⣧⠞⠀⠈⠳⣄⡀⠀⠀⠀⠀⢀⡙⠚⠁⠀⠀⠀⠀⠀⣀⡴⠛⠁⠀⠀⢻⣷⡀⠻⣷⣄⡀⠀⠉⠙⣋⡽⠁
⠠⣶⣶⡿⢿⣡⣤⠾⣋⣀⣤⠾⠛⠁⠀⠀⠀⠀⠈⠙⠶⣤⣀⠀⠀⠉⠁⠀⠀⢀⣠⣴⠞⠋⠀⠀⠀⠀⠀⠀⠉⠛⠷⠾⠟⠻⠿⠟⠛⠉
⣀⣤⣤⣠⣠⣤⣠⣄⣭⣉⣄⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣍⣻⣷⣶⣶⣶⣿⣋⣩⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣠⣀⣄⣤⣤⣀⣤⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⢀⡇⢸⠀⠀⠀⠀⢞⣉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢲⠆⠀⠀⢸⣷⣄⠀⠀⠀⢸⠁⡜
⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢⠆⠂⠀⠀⢈⣿⠻⣷⣄⡀⠘⠀⠁⢀⣆⠀⠀⠀⠀⣀⣤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠈⠛⠿⢿⣶⠿⠛⠿⣶⣶⡶⢿⣿⠃⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣤⣤⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⣇⠀⠸⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣠⠞⠉⢀⣀⠈⠉⡿⣧⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⢁⣠⡿⠧⠤⣄⣰⣞⢧⡀⠀⢠⠇⠀⢻⣦⡾⢫⡽⠿⠚⠛⣿⡄⠀⣇⡀
⠀⠀⣀⣠⠀⠀⠀⠈⠙⢿⡟⠁⠀⣀⡀⠀⠙⢿⡇⠙⢆⣸⠀⠀⠈⢿⣧⣘⣧⣤⠀⢀⡿⡷⠒⢳⣀
⠀⠀⠧⢀⠅⠀⠀⠀⠀⣿⣿⣴⠟⠉⠙⢷⡀⠈⣷⣅⠀⠋⠀⣴⡀⠘⣧⠉⠉⠀⣠⣾⣥⣧⣤⡾⠋⠉⠉
⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⡀⠀⠀⢨⠇⠀⣿⣿⣷⡄⠀⡏⢧⣠⣼⠿⠛⠛⠉⢾⠁⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠠⠖⠚⠛⠛⢛⠿⣿⣿⡟⠒⠒⠉⠀⣰⣿⣿⡇⠙⠛⠁⠈⠁⠀⠀⣤⢦⡤⠬⠷⠀⠀⠀⠂⡠⠇
⠀⠀⠀⠀⠀⠀⢀⡴⠋⡰⠋⠻⣷⣦⣤⣶⠾⠿⣯⠀⠀⠀⠀⣀⣤⣄⡀⠈⠈⢦⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⣧⠀⣠⠞⢁⣠⣄⡈⠳⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠔⢀⠀⠀⠀⠀⢹⣰⠃⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡧⣄⡆⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠟⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈⠙⠛⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡿⠛⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠲⠀⠀⠈⠀⠀⠀⠀⠀⠳⠀⢨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⠄⢀⣤⠀⠀⠀⠀⣴⣶⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠣⣈⡿⠏⠀⠀⢀⠔⢻⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠀⢀⡤⢶⠁⠀⠈⡍⠉⠙⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠊⠀⠔⠊⠀⠠⠀⠈⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⡀⠀⠀⠀⡉⠡⠀⠈⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣇⢀⣤⣤⣛⠁⠢⠡⠄⠠⣶⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣀⠅⠀⣽⣿⠋⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⢀⣀⣀⡀⠀⠤⠐⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠐⢐⠒⠒⠒⠒⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠸⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆⠀⠀⠀⠀⠀⡀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡴⡇⠀⠀⠀⠀⠀⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠈⢁⣘⣿⡷⠴⠇⠀⠀⠀⣠⠀⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣹⣛⣛⣹⣋⣎⣛⣟⣋⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣭⣿⣭⣭⣽⣿⣭⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣷⣤⣄⣉⣉⡒⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⡿⣿⠟⠛⠏⢹⣿⡿⣿⣷⣦⣅⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢫⠟⣹⡿⠟⠁⠐⠁⠚⠀⠀⠈⠁⠀⠀⠈⠉⣉⣻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠁⣰⣠⢀⣠⡄⡀⠀⢠⠀⠀⠀⢰⣦⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢃⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⣷⡆⣸⣾⠇⢀⣯⢿⣿⠛⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⣿⣆⠀⡼⢀⣿⢿⣿⣿⣿⣿⣿⣿⣿⢻⡟⣴⢸⡏⢸⣿⢀⠀⠹⣿⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟⡞⣸⠃⣸⠏⣿⣿⣿⣿⣿⣿⣿⣧⡾⢱⠇⡾⢁⡷⢿⡿⠶⠄⢿⠈⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⢱⢃⡟⢀⣿⢸⣿⣻⣿⠿⠏⠙⠁⠋⠀⠋⠘⠁⠈⠀⠈⠀⣀⠀⠀⠀⢼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠇⣿⣸⠃⣸⡇⣾⠑⠋⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠿⠁⠂⠀⢨⣿⡇⠈⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⢠⡏⠀⠐⠛⠱⠃⠀⠀⠀⠀⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⢀⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠈⠱⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⠀⠀⠀⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡸⡏⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⢠⢣⡇⡀⣼⡦⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣶⣶⡖⠀⠀⠀⠀⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⡎⣼⠀⠁⣿⢁⣏⣦⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠘⢿⣿⣿⣿⣿⡟⠇⠀⠀⠌⠀⠀⠀⠀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢀⣿⣼⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠻⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⣀⡟⢀⠀⣼⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⢀⣿⠁⡾⣼⣿⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠢⡘⠿
⠀⠀⠀⠀⠀⢸⡇⣼⣿⣿⣿⣿⣿⣿⠿⠛⡿⠁⠀⠀⠀⠀⠀⢠⠏⡀⠁⠒⠠⢄⣀⣀⠀⣀⣠⡔⠉⠇⠀⠀⠀⠀⠀⠀⠈⢶⡀
⠀⠀⠀⠀⠀⢸⢠⣿⣿⡿⠟⠋⠁⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠈⠀⠈⠑⠲⠤⢀⡀⠈⠉⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠳⡤⣀
⠀⠀⠀⠀⠀⡿⣼⠛⠁⢄⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠋⠳⠶⠌⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠱⡜⣱
⠀⠀⠀⠀⠀⣾⠃⠑⢄⠀⠐⢄⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⡛⣄⠀⠀⠀⠀⠁⠀⠀⠀⢀⡈⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠢⡀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠂⠀⠀⡀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡆⠰⡄⣶⣶⣦⣳⣾⣾⣇⣷⣤⣿⣾⣴⣷⣼⣰⣀⣷⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣤⢍⢿⣳⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣷⣾⣀⡆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣼⣿⣮⣿⣿⣿⠿⠛⠛⠉⠉⠀⠀⢀⣀⣠⣤⠤⠤⣄⡀⠉⠉⠛⠻⢿⣿⣿⣿⣣⣠
⠀⠀⠀⠀⠀⠀⠐⢦⡐⢦⣾⣽⣿⣿⠟⠋⣡⣶⣾⣿⣿⣶⣶⣶⣶⣿⣿⠋⠀⠀⠀⠈⠙⣷⣄⠀⠀⠀⠈⠙⠻⣿⣿⣧⡄⢠
⠀⠀⠀⠀⠀⠐⢆⠐⣿⣷⣿⠟⠋⠀⣠⣾⠟⠋⠉⠉⠉⠛⠛⠿⠟⠛⠁⢀⡤⠴⠶⠤⣄⡈⠛⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣀
⠀⠀⠀⣀⡀⣙⣿⣿⣿⠟⠁⠀⠀⣰⠟⠁⠀⣠⠤⠖⠒⠦⢤⡀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠙⢦⡀⠀⣀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣆⣀
⠀⠀⢄⣤⣹⣿⣿⠟⠁⠀⠀⠀⠐⠁⠀⣠⠋⠀⠀⠀⠀⠀⠀⠈⠳⣼⠁⠀⠀⢀⣀⡀⠀⠀⠀⢿⣜⣵⡆⠀⠀⠀⠀⠀⠀⠈⢿⣿⣣
⠀⢀⢠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⣠⣤⡀⠀⠀⠀⣹⠀⠀⠀⠸⢿⠟⠀⠀⠀⣼⡟⠉⠉⠀⠀⠀⠀⠀⠀⠐⠨⢿⣿
⠀⣀⣽⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣤⣿⠀⠀⠀⠀⠻⠿⠃⠀⠀⠀⡿⣇⠀⠀⠀⠀⠀⠀⢀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⣿
⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⣩⠽⢿⡄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠉⠻⣶⣶⣴⣶⡶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠱⠀⠆⢻
⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣄⣀⣀⣤⡴⠛⠁⠀⠀⠀⠈⠻⣦⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠘⠀⢀⠂⢸
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⡀⢀⠀⠀⠀⠀⠀⠹⣿⠀⢹⠀⠀⠀⠀⠀⢐⠀⠀⠄⠀⠰⠄⢈⢈⣸
⣾⣿⠠⠀⠠⠂⢀⠀⠀⠀⠀⡄⢠⡨⠃⣀⠀⠀⠀⠀⠀⢰⡏⠉⠺⡀⠀⠀⠀⠀⠀⢹⣇⣼⠀⠀⠀⢀⡄⠢⡨⠆⠒⠂⢒⠒⠀⠃⢸
⣿⡧⠀⠀⠀⠂⠀⠀⢁⠂⠀⠀⠐⣦⠂⠀⠀⠀⣀⠀⠀⠄⠹⣄⡀⢳⡀⠀⠀⠀⠀⢸⣿⠋⠀⠀⢀⡀⢐⠀⠃⢶⠀⢠⠨⠉⡀⠂⣼
⣿⡇⠐⠀⠀⠀⠀⠄⡀⠀⠀⡄⠠⠋⠐⠀⠁⢀⠈⠁⠀⠨⠀⡘⠉⢟⠻⣦⣀⣀⣀⣼⠟⠀⠆⠀⣀⢀⢀⡁⢂⠈⡆⠁⠌⠠⢀⣠⣿
⢻⣿⢂⠂⠀⠀⠀⠀⠄⠃⠁⡇⠂⠂⠀⠁⠠⠀⠄⠁⠐⢢⠀⠰⠀⢸⡐⢾⡏⢉⠛⢁⠂⠤⠀⠁⠉⡀⢈⢘⠐⣠⡃⠠⠈⠀⠀⣼⡿
⠸⣿⡄⡠⠀⠈⠀⡀⠀⡅⠀⠙⢶⣄⣀⣈⣀⠀⠁⣀⣐⣤⣤⠶⠶⠟⠛⠛⠛⠓⠶⢧⣤⣁⡝⠠⢐⣀⣤⠴⠚⠁⡙⠂⢀⠀⣰⣿⠁
⠀⠹⣷⡀⠂⡀⠀⠠⠈⠀⢀⠀⠀⢀⠉⠉⠉⠛⢛⡋⠉⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⢳⣊⠉⠛⠛⠋⠉⠀⠄⠀⡂⠈⡀⠈⣰⣿⠃
⠀⠀⠹⣿⡁⠀⠀⢌⠘⢂⢀⠀⠄⢀⠀⡐⢀⠐⠀⠀⠀⠁⠀⡈⠀⠀⠀⣶⢦⣤⣤⠾⠣⠀⡀⠔⠨⢡⠄⠂⠑⠀⠠⢈⣼⡿⠁
⠀⠀⠀⠘⢿⣄⠠⠂⠀⠄⠀⠀⠀⠀⢀⣂⠀⠄⡠⠁⠐⠀⠀⢂⠀⠕⠆⠙⢘⠉⠓⢷⣠⠂⠀⠀⢀⡐⠈⠀⢠⠁⣠⣿⠟
⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠀⠐⡈⠁⠀⠠⠀⠀⠁⠀⠀⠀⠤⠤⡀⢨⠤⠈⠀⣤⠄⠂⢧⠄⢈⢈⡀⠀⠁⣋⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⣀⣀⣀⡘⠀⠀⡰⢀⠀⠁⠠⠐⠠⠂⠀⠂⠌⢩⠀⡨⠠⢸⣈⠘⠀⣀⣬⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢷⣷⣶⣦⣤⣤⣬⣄⣠⣠⣀⣀⣤⣦⣥⣦⣾⣶⣿⠿⠟⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣻⣟⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠛⠉⠉⠉⠉⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⣴⠾⠋⠂⠀⠀⠒⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⡀
⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠓⣤⣤⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⢀⡞⠁⠀⠂⠁⠀⣱⠏⡴⠃⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠁⠢⡀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⢸⠁⠎⠀⠀⠀⠊⡏⣸⠃⠀⢀⠤⠂⠈⠁⠀⠀⣀⣈⣐⣠⡀⠀⠀⠀⠀⠀⠁⠂⠄⣀⠀⠀⠀⢻
⠀⠀⠀⢸⡌⠀⠀⠀⠀⠀⠻⡇⡠⠂⠁⠀⠀⢀⡠⠖⠋⠉⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄
⠀⠀⠀⠀⢻⡀⠀⠀⠀⡆⠠⡏⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⣀⣀⣠⣤⣤⣀⠀⢸⡇
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠂⠀⣇⠀⡠⠐⠂⣿⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠘⠒⠀⠀⢸⡇⢸⠁
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡆⠀⢹⡎⠀⠀⠀⣇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡾
⢀⣀⠀⠀⠀⢸⠀⠀⠀⠁⠀⢸⢷⡀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡢⠀⢸⡽⠁
⡞⠻⣄⣠⡴⠃⠀⢀⠈⠀⠀⡞⠀⡷⠋⠙⠛⠂⠀⠀⠀⠀⣀⣠⠆⠀⠀⠀⠠⣄⠀⠀⠈⠉⠀⠀⠘⡇
⢳⡀⠀⠀⠤⠀⠂⠀⠀⢀⡞⠁⠸⡇⠆⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⠀⠀⠀⠀⣿
⠀⠙⠦⢤⣤⣤⣤⠤⠖⠋⠀⠀⠀⠙⠲⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣠⣤⣴⣾⡆⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡿⢿⡿⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠇⠖⠒⠋⠀⢀⣤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⠴⠦⢄⣀⣀⣀⣀⣤⡤⠤⠖⠛⢻⠉⢹⡗⠒⠒⠒⠒⠊⢉⠉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡯⢀⠀⢠⠁⠀⠀⠠⠤⠤⠒⠀⠀⠈⣧⣼⠦⠶⠒⠒⠒⠢⢄⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠛⠒⢦⠏⢀⡴⣄⣀⣄⠀⠠⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⠈⠉⠋⠀⠉⠴⡇⠈⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣉⣙⣲⣌⣿⣁⣀⣤⠤⠤⠴⢶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⠉⠀⠀⠀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣈⣹⣏⣁⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢳⠚⠒⢺⡗⠓⠚⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⣀⣸⣇⣀⣰⣇⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣺⣿⣿⣿⣿⣿⣿⣿⣿⣗⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⠀⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠉⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠐⠁⠈⠐⢤⠀⠂⠐⠀⠠⢀⠀⠀⠀⠑⢄⠀⠊⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡁⠠⣘⢁⠀⠀⠱⢀⠀⠀⠀⠑⠀⠀⠀⠀⢱⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⠉⠒⠷⠃⠈⠉⠀⠆⠐⠛⠦⠡⠀⠀⢣⠀⠀⠀⢀⡞⠀⠀⠀⠁⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡰⠁⠠⡀⢀⡰⠃⠀⠀⠀⡀⢀⡀⠀⢀⣠⠴⠯⠔⠒⢤⠀⢑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠛⠭⣉⠉⠀⠈⠂⠀⠒⣤⠀⠀⠀⠁⠀⠀⠈⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢇⠀⠀⣼⣀⢀⠄⠀⠂⡉⠜⠀⠀⠀⠃⠈⡠⠔⠉⠀⣀⣀
⠀⠀⠀⠀⠀⣠⠤⣀⠀⠸⡀⠈⢇⠀⠀⢀⡈⠧⡀⠈⠂⠀⠂⠀⠀⠈⢠⠀⢀⡴⢏⠁⠀⠀⠀⡜⠀⢠⠁⠀⣠⠤⡀
⠀⠀⠀⠀⠀⢣⡀⠈⠑⢤⣣⠀⠸⡄⠀⡤⠢⣔⢻⣦⡀⠠⠈⠃⠀⠋⣡⣔⣅⠒⡨⠷⢄⠀⣰⠀⠀⣎⡴⠊⠀⢀⠟
⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠉⠀⠀⢘⡾⠀⢠⠏⠉⠁⢈⠖⠦⠴⠖⠉⠁⢳⠈⠉⢳⡀⠈⢶⡁⠀⠀⠁⠀⡠⠞⠁
⠀⠀⠀⠀⢸⡉⠉⠙⠋⠀⠀⠀⠀⠁⠀⠀⣞⡠⠴⠒⢻⠀⠀⠀⠀⠁⠀⠈⣷⠦⣄⣧⠀⠀⠁⠀⠀⠀⠈⠓⠋⠉⢉⠆
⠀⠀⠀⠀⠀⠉⠑⠒⠒⢦⡀⠀⢀⡤⠴⠊⠁⠀⠀⠀⢳⢄⠀⠀⢀⡀⠀⣠⡇⠑⢄⠈⠓⠤⢄⠀⠀⢀⠴⠒⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⡏⠀⠑⠴⠁⠑⠊⠀⡇⠀⠀⠑⢄⠀⠈⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢲⢲⠂⠀⠀⡇⠀⢂⠀⠀⠙⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⠁⠀⠀⣾⢸⠀⠀⠀⢷⣠⠤⠷⢄⣀⣀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠉⠀⠀⠀⠀⠀⢻⢸⠂⠂⠀⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠋⠘⠒⠒⠲⠶⠶⠶⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠒⠋⠉⠉⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣷
⠀⠀⠀⣠⠴⠋⠀⠀⠀⢀⣀⣤⠴⠖⠒⠛⠋⠉⠉⠉⢙⠓⠲⢤⣹⡀
⠀⢀⣠⠟⠀⠀⠀⣠⠖⠋⠁⣠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠹⡇
⣠⣟⠁⠀⠀⢠⡞⠁⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢉⡟⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢸⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡤⠜⠂⠀⡇
⠘⣧⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠹⣆⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠆⠀⠀⠀⠀⠀⠙⣦
⠀⠀⢹⡄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸
⠀⠀⠀⠻⣄⠀⠈⣻⠄⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⣠⣤⣴⣾⡄⠀⠀⡾
⠀⠀⠀⢰⠏⠉⠛⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⡇
⠀⠀⠀⠘⠷⡤⣦⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⠋⠙⣋⡡⠟⠁⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠧⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠛⠲⣤⣤⡤⠴⠶⠶⠚⠛⠉⠉⠈⢹⣆⠀⠀⡏⠉⠉⠉⠉⠉⠐⠒⠉⠁⢀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡶⠞⠋⠉⠉⠉⠉⠉⠉⠁⠛⠶⣤⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠈⡷⠤⣤⡞⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠃⢀⣿⣁⣀⡀⡴⠃⠱⠖⢊⡦⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⡀⠀⠀⢀⣉⣙⠆⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠈⠒⠒⠤⣩⠇⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡴⣶⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠰⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⠛⠛⡟⠛⠛⡿⠛⢻⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⣿⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡇⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢼⡇⠀⠀⣿⠤⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡇⠀⠀⣿⣿⣿⣷⣤
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿⣿⣿⣧⣤⣤⣿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡏⠉⠋⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢟⣹⣿⣿⣿⢷⣿⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠉⠉⠉⠛⢷⣮⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⢠⣆⣠⠀⠹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠇⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⣿⠀⠀⠀⠀⠀⢠⣴⡄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢻⣇⠀⠀⠀⢠⡟⣿⠃⠀⢀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣤⣿⣷⣄⡀⠈⠁⠀⣀⣴⡿⣿⣿⣿⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣴⡿⠋⢻⡆⢿⣿⣿⠿⠛⠛⠉⣼⡿⢸⣿⠀⠀⠀⣋⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⠀⢸⣇⠀⠉⠙⣧⡀⠀⣰⡿⠁⢸⣿⠀⡴⠟⣁⡀⢉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⠻⣶⡿⠁⠀⢸⣿⠀⢈⣛⣉⠁⢠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⣀⣀⣤⣤⣤⣼⣿⠀⠈⠉⠉⢡⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⣾⣿⠟⠛⠋⢻⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁⠸⣿⠀⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣶⣶⣿⠇⣶⣶⡿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠖⠒⢲⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢀⠜⠉⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣠⠖⠋⠉⠙⠢⣼
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡠⠒⠈⠁⠀⠀⠈⠙⠲⠁⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡇⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢹⡤⠤⢷⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣄⠀⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⡄⠀⠀⠀⠀⠀⠀⠀⢰⣤⣤⣴⣾⣿⣿⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣋⠉⠈⠁⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢄⣀⣀⡀⠀⠀⣀⣀⣀⠤⠖⠚⡅⠠⡤⠤⠤⠄⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢈⠉⠉⠉⠀⠀⠀⠀⠀⢹⣤⡧⠤⠤⠤⠤⣈⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠲⣞⢀⡴⣄⣀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⡈⣉⠀⢀⢲⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠤⠵⠶⠗⠒⠒⠒⠛⢩⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠠⣤⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⠶⠶⣿⠶⠶⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣆⣀⣿⣀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢸⣿⣿⣷⠀⠀⠀⠀⠀⢄⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⣿⣿⡏⣠⣝⠿⠋⠀⠀⠀⠀⠀⠈⢷⡄⠉⠒⠶⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠘⣿⣷⣿⣾⡇⠀⠀⠀⢀⡀⠀⢀⣲⣆⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣠⡀⠀⠀⢿⣿⣿⠁⠀⠀⠀⠀⠀⠑⠂⣼⣿⡏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⣿⣇⡀⢠⣿⠟⢧⡑⣭⣛⣋⣀⣤⣤⣌⠙⢻⡁⠀⢀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠘⢿⣷⣼⡇⠠⣀⠀⠉⠛⠛⠛⢛⠿⠿⢷⣄⢃⢠⣾⣿⣿⡇⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠈⠹⣿⣡⡶⢖⠮⠀⠀⠀⠀⠴⢶⠶⣤⣹⣿⡿⠟⠉⠁⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⠀⠀⠀⣸⠋⠒⠈⠁⠀⠀⠀⠀⠀⠘⠒⠉⡟⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⢷⣶⣦⣷⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣶⣦⣤⣶⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠙⠻⠟⠙⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠈⠛⠛⠛⠉⢉⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠁⢠⠀⢰⣼⣧⠀⠀⣠⣄⣤⡀⠀⠀⠀⢸⣧⣤⣧⣶⣶⣰⣿⢋⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⡀⠀⠈⠛⢃⡀⠈⠻⣛⠋⠋⠁⠀⠀⣸⠏⠛⠉⠛⢻⡿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⡓⣀⣤⣿⣷⡀⠀⠠⠄⠀⠀⢀⣼⣿⠷⠦⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠀⠀⠤⠠⠤⠞⠋⠁⢹⡶⡀⡀⠲⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡰⠀⠀⠀⠀⠻⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠒⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⡹⢉⠝⠀⠉⠁⠀⠐⠒⢤⡀
⠀⠀⠀⢀⠚⠉⠀⠀⠉⠂⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢧
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠨⠑⠂⠠⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠀⠀⠀⠤⠤⠄⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⢰⠀⠀⠀⠀⠈⠳⣄⠀⡀⠀⠀⠈⠲⡄⠀⠀⠀⣀⠤⡁⠈⣾⠀⠈⣦
⢂⠀⠈⠘⡄⠀⠀⠀⠀⠈⡆⠈⠐⠀⢀⠄⠉⠀⠀⠀⢡⠀⠈⠠⡤⠄⠐⠛⢄
⠘⢄⡀⠀⠙⢢⠀⠀⠘⣅⠈⠂⠀⠀⠀⠀⠀⡠⢀⡀⠸⢄⡤⠂⠀⠰⡄⠀⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⢧⡀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠔⢋⠜⠁⢀⢣⠀⠀⠀⢹⡖⠋⠀⠄⢀⢠⡤⠀⢠⠀⠀⢀⡴⠖⠒⠚⠉⠉⠉⠓⠒⢻⠇
⠀⠀⠀⠀⠈⢧⠀⢀⠔⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁⠈⠀⠀⠀⠀⠁⡟⠀⡆⠒⠂⡴⢋⠀⢠⡎⠤⠄⠐⠃⣁⡴⠋
⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢀⡗⠀⣴⠤⣤⠶⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠔⠉⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠋⠉⠑⢠⡬⠴⠚⠉⢲⡀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⢠⠞⠁⢀⠉⠈⠀⡐⠁⠘⠁⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡴⠃⠀⢦⣀⣀⣴⠈⠇⡅⠀⠉⡆
⠈⠂⠐⠲⢀⠀⡟⠀⡀⠄⠠⠤⠚⠉⠀⡤⠾⠶⠴⠀⠂⠀⠀⠘⠲⢤⡀⢀⠀⠀⠀⠈⠛⠋⢱⠀⢀⣠⠏⠀⠀⠈⠓⠒⠶⠶⠗
⠀⠑⠢⠤⢈⡿⡗⠀⠄⠀⢀⠀⠀⠈⠣⢄⠀⠀⠀⠀⠀⣀⡀⠀⠀⡠⠚⠘⠀⢡⠀⠀⠀⠀⠀⠈⠉
⠤⠀⠀⠂⡘⢡⣿⠀⠒⠂⢤⣠⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⠉⠻⡀
⠀⢀⡰⠊⢠⡞⠈⢧⡀⡄⢤⡑⠄⠑⠢⡀⢳⡀⠀⠀⠀⠀⠀⠀⠙⢆⠈⠉⠉⠃⢢⡤⠀⠀⠀⠀⠀⢂⠄⠤⠤⠤⠔⠒⠢⠤⠠⠤⡀
⠀⠊⢀⡴⢻⠀⠀⠀⠉⠀⠀⠙⠦⣄⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡴⠋⠀⠃⠁⠀⠀⠀⠉⡏⠦⡀⠒⠶⢂⠀⠀⠀⠀⠘⢦
⠀⠒⠁⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⢘⠈⢢⡀⠀⠈⠳⣄⠀⠀⠘⣇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠉⠀⠀⠀⠀⢄⠀⠀⠀⢀⠇⠀⠸⡀⠀⢻⡳⣦⡀⠉⠙⠆⡀⠈
⢀⡄⡴⠲⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⠁⠀⠀⢰⠀⠢⠈⡟⣵⡙⠲⣄⠀⢹
⠚⢀⠀⢀⡾⠉⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡠⠊⠀⠀⠀⠀⠀⢠⠐⠠⢄⡀⠀⢀⡂⠀⠀⡌⠡⡀⢰⠁⠘⣷⠀⠀⠉⠚⠒
⠀⣀⠐⠇⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠈⠁⣽⠃⡄⢠⠣⠀⠃⡺⠀⠀⢸⡃⢀
⠚⡏⣠⢄⡠⢤⠜⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠘⡅⠀⣃⠀⠁⠀⡖⢻⡍⠈⠳⠠⡀
⡼⢍⠀⠀⡇⠀⠁⢀⡆⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠃⠀⠀⡎⠀⠀⠛⠀⣠⠔⢧
⠀⠀⠑⠀⣉⡠⠴⠊⢁⡆⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡆⠀⠀⠀⠋⠀⠀⠀⠀⡽⠀⠐⢀⠛⣄⠀⡀
⢄⣠⡖⠉⠈⠀⠀⠀⢫⠱⡦⠀⠀⠙⢆⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⣼⣀⢤⡀⢀⡨⣦⠀⠀⣸⠀⠀⢹
⠀⠘⡁⢀⡀⠀⢸⠚⡅⠀⡓⣄⠀⠀⠈⠤⠀⠀⠀⠀⠀⠉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⢣⡞⠀⠈⠉⣶⠉⠉⠉⠁
⠀⠀⠈⠉⢀⠉⠁⠀⠘⢄⡁⠈⣧⡈⠀⠀⢣⠀⠀⠀⠀⠀⣠⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠎⢱⡄⠀⠀⠸⡄⠀⣠⣴⠁
⠀⠀⠀⠀⣼⠑⠀⠀⠀⠀⢿⠄⠀⠑⢄⠀⢿⠀⠀⠀⠀⢀⡇⠀⠀⠀⢘⠀⢀⠔⠁⣀⡤⠚⠁⡾⠀⠈⠂⢀⠃⠀⠀⠁⠈⢣⡄
⠀⠀⠀⡐⠙⡂⠀⠀⠀⠀⢸⡄⠀⠑⠀⠡⠟⠀⠀⠀⠀⢸⠀⠀⠀⢀⠏⡴⠁⠀⡜⠉⠀⠄⢸⡇⠀⠀⠀⡼⠀⠈⢣⡀⠠⠚
⠁⡀⠊⠀⠀⣿⡄⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠔⣨⠞⠀⠀⡰⢧⠀⠀⠈⡄⠉⠐⣦⠰⢣⡀⠀⠀⣳
⠀⠸⡀⠀⢀⡝⡉⢄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠵⡎⠁⠀⠀⡴⠁⠘⢆⠀⠀⠙⠄⠀⢸⡄⠀⠉⠁⠀⠉
⠀⠀⠀⠱⠈⠀⠰⠀⠈⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⣠⠞⠁⠀⠀⣿⠀⠀⠠⡆⠀⢸⡆
⠀⠀⠀⠀⠀⢠⡀⠆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢹⡀⠀⠀⠘⠄⠀⡷
⠀⠀⠀⠀⠀⠈⢇⠘⡀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞

⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠀⠀⠠⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠙⡛⡄⠀⠀⠀⠀⠀⠀⠀⠜⠟⠁⠀⠀⠀⠀⠄⢂⠀⠀⠀⠀⠀⠀⠀⡐⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠄⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⡄⠂⢄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠤⠤⠤⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠴⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠋⢠⣄⠌⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⡀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠁⠤⡀⠄⠀⢀⡁⠄⠀⠀⠀⠀⠀⡀⠇⠆⠈⠀⠀⠀⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⢀⠊⠀⠡⡀⠀⠀⠀⠀⠀⠀⠈⠢⠐
⠀⠀⠀⠀⠀⠀⡸⠀⢠⠂⠀⠀⠀⠀⠀⠐⠂⠀⠘⠀⠂⠀⠚⠀⠀⠀⠑⠂⠄⠀⠀⠀⠤⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠁⠁⠈⠉⠈⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠐⠃⠒⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣐⣀⣁⣀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠴⠿⠂⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⣴⠶⠀⠀⠀⠘⠶⠀⠀⠀⠀⠀⠀⢀⣀⣴⣆⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡄⠀⠀⠀⠀⠈⠙⠉⠁⠀⠈⢢⠊⠈⠉⠛
⠀⠀⠀⠀⠀⠠⠆⢀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠘⠀⢜⠃⠀⢀⠍⠀⠀⠀⠀⠀⡀⠀⠀⠲⠃⢘⠘⠃
⠀⠘⡀⠀⠀⠀⠀⠈⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠑⠀⠀⠈
⠀⠀⠈⠢⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠁⠂⠀⢀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠐⣒⠒⠠⢀⢀
⠀⠀⠀⠈⢄⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠊⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠢⠀⡁⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠉⠀⠀⠀⠘⠻⠆
⠀⠀⠀⠈⠉⠓⡆⠀⠐⠲⡄⠀⠀⠀⠀⢀⣤⣤⣀⡀⠀⠀⠀⢀⣀⣤⣦⠄⠀⠀⠀⠀⠀⠀⣡⣤⣉⠢⢀⣴⣤⠤
⠀⠀⠄⠀⠀⠀⠐⠀⡀⠀⠐⡀⠀⠀⠀⠀⢠⠀⠙⠛⠛⡿⠛⠛⠁⠀⡰⠀⠀⠀⠀⠀⠠⢀⣹⣷⣯⢀⢈⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⡠⠁⠀⠀⠚⡅⠀⠁⠂⠀⠊⠀⠈⠐⠒⠈⠀⢀⠂⠀⠈⠋⠀⠀⠈⠉⠁⠈⠑⠀⠑⠈⢀
⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠈⠦⢀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⠈⢢⣀⠀⠀⠀⠀⠀⣀⠠⠐
⠀⠀⢋⠂⠀⠄⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⢸⠁⠀⠈⠁
⠀⠀⠀⠓⣆⠁⠂⠀⠂⠈⠆⠀⠀⠀⠀⠀⠀⠈⠢⢀⠀⠀⠂⠁⢶⡀⠀⠀⠀⠀⠀⠀⠀⠈⠺⡆⡠
⠀⠀⠀⠀⠀⠈⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡦⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤
⠠⠶⠶⣤⣄⣀⣀⣀⣠⡤⠶⠿⠂⠀⠀⠀⠀⠈⠁⠀⠠⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠛⠂
⠀⠰⡀⠈⠉⢉⠋⢉⠉⠀⢀⠔⠀⠀⠀⡀⠀⠀⠀⠀⣤⠈⣤⠄⠀⠀⠀⣀⠀⠀⠀⠃⠀⠈⠉⠀⠾⠀⠉⠁⠀⠐
⠀⠀⠀⠁⠈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠅⠀⠀⠀⠀⡀⠔⠢⠀⠠⠊
⠀⠀⠀⠀⣀⠠⠤⠤⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⠤⠄
⠀⠀⠐⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⢀⠀⠀⢀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠘⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠷⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠤⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⣠⡾⠛⠉⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⣠⡾⠋⠀⠀⢀⣀⣀⣀⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣴⡟⠁⠀⣤⣾⣿⣿⣿⣿⡷⠀⠀⠉⠛⠳⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣼⠏⠀⠀⠈⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⣿⡛⠿⢶⡿⠛⠛⠛⠛⠻⠿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣧⠘⣧⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⢷⣀⠀⢀⣀⣄⠀⠀⢀⣠⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠈⠙⠛⠛⠋⠁⢀⣴⠟⠁⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠈⠋⢠⡏⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠻⣦⣀⠀⢀⣀⡀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⢀⣠⣤⣤⣾⠁
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣤⣴⡶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⠞⢛⣉⠀⠈⠹⣷
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⣷⣄⠀⠀⠀⠀⠀⠀⣴⡟⠉⠛⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠘⣯⡉⠙⠛⠛⠷⠶⠶⠿⠿⠟⠛⠛⣉⣽⣿⠈⠁⠀⠀⢀⡀⠀⢸⣧⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢿⠙⠳⠶⣤⣤⣤⣤⣤⣴⣶⣶⠟⠋⠉⣿⠀⠀⠀⢀⣿⡇⠐⠛⠁⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⣰⠏⠀⢠⡟⠁⠀⠀⣸⡟⠀⠀⠀⣾⠿⣷⣄⣀⣠⣴⠟⠃
⠀⠁⠀⠀⠀⠀⠀⠘⣷⡄⠀⠀⠀⠀⠀⢻⡄⠀⠀⢰⡏⠀⠀⠈⠀⣀⣤⡾⠟⠀⠀⢀⣾⠟⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠙⠶⢦⣼⣷⡶⠶⠾⠛⠋⠉⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⣄⣀⡀⠀⠀⠀⣀⣀⣤⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠉⣉⡇
⠀⠀⠀⠀⡠⠒⠉⠀⠉⠉⠑⢄⠀⠎⠀⠀⠀⠀⢀⠀⠀⠀⠀⡖⢒⡉⢉⠉⠑⠚⢆⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠸⢥⢄⣀⣀⣀⣀⢀⣸⠦⠀⠀⠀⠀⠀⠫⠴⠀⠀⠰⣳⠓⠑⠎⡷⢄⠀⢰⠁⠲⠆⠀⠀⢃⡼⠇
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⣧⠟⠑⢶⡍⠁⡏⠉⣉⣲⠄⠀⠀⢹⠈⠐⠀⠁⣄⠳⡺⠀⣃⣀⣀⣀⠂
⠀⠀⠀⠀⠀⢑⡤⠤⣤⠮⢼⡀⠀⢸⠃⠀⢱⣸⢑⢵⠀⠀⠀⠈⠦⣀⡒⠀⢠⠊⡡⢊⣴⡄⣤⡀⠉⡷⡄
⠀⠀⠀⢠⠾⠭⠭⠭⣽⠷⢤⡇⠀⠸⡀⠀⢳⠉⢆⣸⠀⠀⠀⠀⡴⠻⠥⠒⠉⠓⡇⡎⠄⠛⢠⠙⣄⢀⢸
⠀⠀⠀⣼⡄⠀⢀⠔⠁⠀⠀⢹⠀⠀⣷⣤⣼⡠⠼⠚⠀⠀⠀⢰⡅⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡙⡸⢸
⠀⠀⢸⢸⠈⠉⠁⠀⠀⡄⣠⡞⢀⠜⠉⠉⠚⠳⡄⠀⠀⠀⠀⠘⡇⠀⠀⠀⠘⡄⠀⡏⡳⢶⣀⣴⡟⠧⠎
⠀⠀⢣⢸⠀⠀⠀⠀⠀⡜⢡⢇⡘⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⠉⡦⣼⠃⡠⠀⠀⢹
⠀⠀⠀⢻⠀⠀⠀⠀⠀⠧⢤⡞⠇⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⣸⡗⠤⠤⢀⡼⠻⣄⠀⠹⠀⠀⣠⢾
⠀⠀⠀⡾⢭⣹⠑⣒⣒⣒⢺⡵⡤⢤⠤⣀⣸⠀⢸⠀⠀⠀⠀⣟⡇⢠⣀⢰⡓⢲⠎⡗⠢⠴⠊⠀⢸
⠀⠀⠀⡇⢸⣿⠀⡇⠀⠈⡟⡜⠀⡜⠀⢸⡞⡄⢸⢹⠀⠀⠀⠀⢹⠀⡍⠀⠓⠊⣸⡤⢤⡞⣏⠉⠉⣳
⠀⠀⠀⢹⠉⠀⢣⠈⠉⢹⠁⠁⠀⠁⠀⠀⡇⡧⢚⣬⠂⠀⠀⠀⠈⣦⠃⠀⠀⢠⢻⠁⠀⢇⣈⣉⣉⣹
⠀⠀⠀⠸⣀⣀⣸⣀⣀⣘⣦⣤⣀⣀⣀⣀⣈⡱⠚⠁⠀⠀⠀⠀⠀⢸⣀⣀⣀⣸⢸⣀⣀⣀⣀⣀⣀⣈⣆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠶⠒⠒⠒⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⣀⠴⠒⠛⠳⠆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠒⠲⢤⣀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⡠⠊⢁⣠⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣦⣄⡈⠓⢄⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠠⠊⣠⡶⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠻⠿⣶⣄⠑⠄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠠⠚⠉⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠛⠆⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣠⣷⠶⠛⠓⠾⢷⣄⡤⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⡾⠷⠛⠻⠶⣾⣤⡄⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⢈⡿⠋⠀⠀⠀⠀⠀⠀⠈⠻⡦⡄⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢰⣿⠀⠀⠀⢀⣴⡟⠁⣀⠀⠀⢱⠘⠀⠀⠀⠀⠋⡎⠀⠀⣠⣶⡖⠂⡀⠀⠀⠀⣿⡆⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢿⠀⠀⠀⠘⣿⣻⣿⡿⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⣛⣷⣾⠇⠀⠀⠀⡿⠁⠀⠀⡇
⠀⠀⠀⠀⠀⣠⠖⠒⡇⠀⠀⠀⠘⢆⠀⠀⠀⠈⠉⠉⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠉⠉⠁⠀⠀⠀⡰⠃⠀⠀⠀⡗⠛⢦
⠀⠀⠀⠀⢀⢃⢤⡀⡇⠀⠀⠀⠀⠈⠓⠤⣀⣀⣀⣀⡠⠖⠁⢀⡤⠂⠀⠀⠠⢄⡈⢮⡢⢄⣀⣀⣀⣀⠤⠊⠁⠀⠀⠀⠀⡇⣠⠄⡇
⠀⠀⠀⠀⠘⡇⢰⡛⢳⠀⠀⠀⠀⠀⠉⠒⠦⠤⠄⠀⠀⠀⠀⣏⠀⡠⠀⠀⢀⠀⣳⠀⠉⠓⠶⢤⣤⡤⠶⠊⠁⠀⠀⠀⢀⠟⣳⠀⡇
⠀⠀⠀⠀⠀⠹⣄⠁⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⠒⠚⠁⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠁⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⣿⣤⣈⣷⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣿⣿⣿⣿⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠁⠙⠁⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣤⣄⣠⠟⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡑⢄⡀⠀⠈⠉⠁⠀⠀⠀⠀⣀⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠻⠶⣦⣤⡤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣴⡦⠀⠀⠀⠀⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠟⠀⠀⠀⠀⢶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⡈⠁⠀⠀⢄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢉⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣰⠑⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣌⠉⠁⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢀⣆⠈⠁⠀⢠⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠛⠻⡀⠀⠀⠀⠐
⠀⠐⠺⡖⠀⠑⠀⠤⠐⠊⠀⠁⠐⠂⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠐⠀⠀⠁⠒⠊⠁⠀⠀⠀⠀⠀⠀⠙⡀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠈⠦⢀⡀⠀⠀⠀⠀⠀⠀⣀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠈⠁⠈⠈⠁⠒⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠀⠀⠀⠀⠀⠀⢀⢀⠤⠊⠁
⠀⠀⠀⠀⠠⡄⠁⠀⡃⠀⠀⡃⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠘⠪⡁⠀⠃⠀⠉⠁⠀⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠐⠀⠀⠂⠉⠐⢆
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠓⠒⠒⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⢀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠁⠈⠉⠈⠉⠉⠙


⠀⠀⠀⠀⠀⠀⣴⠶⠖⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠂⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⠄⠀⠀⠀⠀⢴⣦⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠐⠀⠠⡀⠀⣀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡀⠈⠁
⠀⠀⢀⡀⠀⠀⠉⠙⠀⠀⠀⠀⠐⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢋⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣤⣠⣤⠑⣊⣤⣤⣌
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡈⠉⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠻⣿⣿⡿⢃⠛⢿⡿⠟⠀⠀⠀⠤
⠀⠀⠀⠀⠑⠂⠄⠀⠀⠀⠀⠀⣀⠠⠐⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⠀⠀⠠⠊⠉⠒⠀⠀⠂⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠠⠈⠉⠄⠊⠁⠂⠀⠐⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠑⠀⡀⠂⠐⠀⡱⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡦⠤⣠⠖⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⣀⠄⠒⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⢀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⠄⠀⠀⠀⠀⠀⠀⢀⣤⠾⠛⢃⡠⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠭⠤⢀⠀⠀⡀⠉⡓
⠀⠀⠀⠀⣉⠉⠙⠿⡷⠶⣶⡶⢶⢿⠛⠉⠁⠰⠀⠀⠀⠀⠀⠐⣊⠁⠠⣤⠄⠀⠇⠤⡤⠀⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠱⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠘⠦⡀⠀⠀⡠⠂⠘⠄⡀⠀⠀⠤⠁⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⣠⠧⢀⡀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠀⢀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⣀⡀⠠⠊⠓⠀⠤⠐
⠀⠀⠀⠀⠀⠀⣠⠔⠈⠉⠁⠀⠀⡇⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢶⣦⣀
⠀⠀⠀⠀⢠⡞⠐⠒⢲⠐⠀⠐⡎⠁⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠈⣁⠤⠄⠚⠒⠒⠀⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠻⠧
⠀⠀⠀⠀⠑
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣯⣥⣬⣧⣴⣠⣽⣮⣼⣯⣾⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠙⠛⠛⠛⠛⠛⣿⠿⢶⣶⣾⣿⠾⣿⡟⣻⠿⣻⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢦⡼⠟⠋⣿⠈⣿⠻⣧⣤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣀⣀⡀⢀⣀⣿⣤⣿⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠛⠳⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⣧⣠⣄⣤⣀⣰⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⡴⠞⠋⠁⠀⢸⣿⣧⣽⣧⣿⣽⡇⠈⠛⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠀⢀⣴⠞⠋⠉⣀⣀⣀⡀⠉⠙⠷⣦⡀⠈⠙⢿⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣰⠟⠁⣠⡾⠛⠉⠉⠉⠙⠻⢦⡀⠈⢻⣆⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢰⡲⠀⠀⠀⠀⠀⠰⣲⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣰⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣖⠿⠛⠛⢻⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡟⠛⠛⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡦⢀⡾⢹⣿⣿⣿⣿⡻⠶⣦⣀⣠⡶⠾⣻⣿⣿⣿⢿⣏⢷⡀⢴⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡟⠹⠿⡿⠷⠛⣡⣿⠃⠈⠻⣿⣿⣿⣾⣿⣿⣾⣿⣿⡿⠟⠁⠘⣿⣌⠛⠿⢿⠿⠏⢻⣧
⠀⠀⠀⠀⠀⠀⠀⣸⡿⢁⣀⣤⣴⡶⠿⠛⠁⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠈⠛⠿⢶⣦⣤⣀⠀⢿⣆
⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠑⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠊
⡶⡶⢶⢶⡶⠶⢶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⢾⣶⣶⡶⠶⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠉⠻⠟⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⠊⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⡴⠊⠀⠉⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠓⢀⡀⠀⠀⠀⠀⢀⡜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡏⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⢠⠀⡀⠀⠀⠀⠀⠀⡀⠈⠂⠀⠀⠀⠄⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⡾⠁⠀⠀⠀⠐⠚⠹⠆⠀⠀⠀⣠⣄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠙⢃⠀⠀⠀⠀⣶⣄⠒⠂⢠⡀⠁⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠏⢀⠟⡀⠀⠀⠀⠹⡙⢿⡆⠒⠚⡇⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡉⠀⠘⢤⡀⠀⠀⠑⠰⠧⠄⠂⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠊⠀⠀⠠⡁⠀⠀⠀⠽⠶⠦⢤⣤⣤⣤⡐⠋
⠀⠀⠀⠀⠀⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠈⠢⢀⣀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⠦⠴⠤⠤⠤⠤⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⢸⠁
⠀⠀⠀⢀⠸⣄⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠒⠂⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⡠⠊⠀⢀⠗⠐⠈⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠰⠁⣀⡔⠁⠀⠀⠀⠃⠀⠀⠀⠀⢀⣤⠄⠀⣀⠀⠀⠀⢠
⡌⠀⠀⠈⠀⠀⠀⠙⠛⠋⠁⠀⠀⠜⠁⠉⠻⣿⣿⠆⠀⠀⠈
⠓⠀⠀⠒⠒⠂⠀⠀⡞⠉⠉⠁⠀⠈⠁⠀⠚⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢡⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⡆⢰⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠒⠒⠶⠒⠀⠘⠲⠒⠒⠒⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠂⠘⠀⠀⠀⠀⠀⡆⠘⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠁⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠒⠋⠀⠀⠀⠀⠀⠀⠀⠇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⣸⡆⠀⠀⠀⠀⠀⠀⠀⣶⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣏⠑⢄⡀⠀⠀⠀⠀⢠⣌⠙⢩⣶⣶⠆⠀⢴⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⠎⠙⢂⠀⠀⠀⠀⠉⠛⠛⠿⠻⢤⣤⣌⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⣀⣀⡚
⣿⣿⣿⣿⠿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣇⣀⣀⣀⣠⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⢶⠋⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣀⣀⠀⠀⠀⠀⢸⣧⡀⠀⠀⠀⠀⠀⢸⡇⢀⡇⠀⠀⠀⠀⠀⠈⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠁⠁⢼⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⢸⠀⢸⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡞⠈⠀⠀⠀⠀⢠⣿⠙⢿⣿⣦⣄⠀⢸⠀⡈⠀⣠⣾⠀⠀⠀⠀⠀⢀⣠⣴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠻⢿⣿⣿⣶⣶⡿⠟⢿⣷⣦⣤⣶⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⠀⠀⢀⣠⣿⡟⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠉⠉⠉⠉⢠⣿⡏⠀⠀⣶⠚⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢿⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠾⠿⠧⣤⣀⣸⡆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠚⢳⣠⡞⠁⠀⣀⣀⣀⠀⠀⣹⠟⣧⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠚⠉⠀⢀⣼⣿⣀⣀⡀⠀⣶⢞⣧⡀⠀⠀⠀⡟⠀⠀⢿⣆⢠⡾⠟⣛⣿⠷⠾⠟⢲⣿⡀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢶⣶⣶⡾⠋⠉⠀⠀⠀⠉⠓⢿⣯⡅⠹⣦⠀⢰⡇⠀⠀⠈⣿⣺⡁⠘⣇⠀⠀⠀⠀⢸⣿⣇⣀⣸⡏⠒⠂
⠀⠀⠰⡦⠞⠆⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⣠⣤⣦⣤⡀⠀⠀⢻⡟⠄⠈⠳⣼⠁⠀⠀⠀⠘⣿⠳⣤⣿⡾⠟⠀⠀⣼⠛⣏⠁⠀⢻⣤⣤⣀⣀
⠀⠀⠰⠧⢄⡖⠀⠀⠀⠀⠀⠀⣿⣿⣟⣼⠟⠁⠀⠉⢻⡆⠀⠀⣿⣰⡄⠀⠉⠀⠀⣿⡀⠀⠸⣇⠀⠀⠀⢀⣠⡾⢫⡴⣿⣶⡶⠿⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣯⣿⠀⠀⠀⠀⠀⡿⠀⠀⣸⣿⣿⣦⣀⢀⣸⡏⣷⣀⣀⣽⣿⡿⠿⠟⠋⢰⡟
⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣽⢷⣤⣀⣠⠾⠁⠀⢀⠟⠻⠿⠝⢿⣷⡿⠇⠘⠟⠛⠋⢥⡐⢦⣀⣀⣈⢷⣄⠀⠀⠀⠀⣼⠛⢻⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⡤⠊⢻⣿⣟⣃⡀⠈⠀⠀⠀⣠⣿⣼⡴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣟⠻⣯⡉⠉⠉⠓⠀⠀⠀⠀⠳⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⡴⠋⠈⠛⣿⣷⣶⣶⡶⠟⢛⢿⣲⣀⠀⠀⠀⠀⢀⣀⣀⡄⢀⡀⢈⠀⠈⢅⡈⢁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠟⠀⠀⠀⠐⠉⠀⠀⠀⠀⠀⠈⠓⠙⢿⡆⠀⢀⡼⠛⠙⠛⣚⡙⠙⠻⡆⠁⠈⠛⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠘⡇⢀⡞⠁⢠⡞⠋⠉⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡞⠋⠈⢲⠄⠀⠀⠀⠀⠀⣿⡞⠁⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡧⢤⣰⠃⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠛⠣⢿⣿⢲
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡎⠌⢡⠀⢒⠲⠟⣫⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋⠃⠑⠂⠸⠇⠅⣤⣹⣯⡿⢀⣀⡠⣄⣤⣀⡀⠀⠀⢠⡾⣿⡾⢻⣦⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠉⣯⣍⣒⣤⡤⠀⠛⣿⣽⢛⠛⠀⠀⠀⠙⠻⢿⣷⣠⠾⠟⠛⠙⠯⢄⠙⡿⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⠠⡇⠀⢧⣈⡩⠋⠀⢀⠞⠁⠻⢬⠔⠀⢀⣘⢳⡄⠀⡼⣵⡘⠀⡇⢀⡔⠄⠘⡿⣻⢛⣿⣇
⠀⠀⠀⠀⠀⠀⢧⣀⣱⣀⠀⠀⠀⠀⣠⠃⢸⣒⠢⡈⠀⠀⠻⡀⠀⠁⣿⠁⡈⠱⠈⠀⠘⠓⠁⠀⣿⣿⣿⣿⣯
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠑⣦⡀⢠⠃⢠⠟⢻⣷⣌⣢⣰⣄⠀⠀⣤⡿⠘⣷⣒⣒⣒⣢⠗⠒⠂⠈⢹⠋⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⢸⠀⠘⠤⠤⠔⠋⠉⠀⠀⠀⠀⡬⢷⠀⢸⠿⠯⡽⠃⠀⠀⠀⢀⣾⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠎⢆⠀⠀⠀⠀⠀⠀⠀⠀⣇⣈⡠⢾⣆⠀⠑⠋⠀⠀⠀⠀⣠⢻⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠈⠣⡀⠀⠀⠀⢀⣠⠞⠉⠁⠀⠰⠋⠑⢦⣀⠀⠀⠀⣰⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢶⣴⠊⢉⡴⠂⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⣀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠻⠿⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠯⠿⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⢠⣶⠆⠀⠀⠀⠠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠟⠃⠀⠀⠀⠠⢷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠈⠀⠐⠂⢄⡀⠄⠀⠅⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠃
⠀⠀⠀⠀⠂⠀⠀⣀⡀⠈⢀⣄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⣄⠈⠁⣠⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⣘⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠁⡰⢄⠁⠀⢀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡰⢄⠈⠀⢀⠌⠀⠀⠀⠀⠀⢀⡀⠐⡀⠀⠀⠙⠻⠀⠀⠀⠠
⠀⠈⠙⣍⠀⠈⠀⠀⠉⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠈⡗⢤⠀⢀⠀⡀⣀⠀⢠⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠀⠀⠀⣀⠤⠒⠁
⠀⠀⠀⠀⠘⣆⢀⠨⢀⡀⠀⡀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⠺⠄⢀⡀⠀⣀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⢀⠈⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣆⣀⣀⣀⡀⠆⠀⠰⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠐⠋⠉⠀⠀⠀⠀⠙⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢋⠁⠀⠀⠀⠀⠙⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠛⠁⠀⠀⠀⠀⠛⠻⠄
⠀⠀⠀⠀⠀⠠⠶⠄⠀⠀⠀⠀⣤⡤⠄⠀⠀⠀⢀⠀⠀⠀⠀⠔⠋⠀⠀⠉⠢⡤⠂⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⢄⠀⠀⠀⠠⡀
⠀⠀⠘⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠆⡷⠂⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁⢾⣿⣿⣿⠄⣷⣿⣿⡇⠀⠀⢀⡀
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⢀⠔⠣⢀⣀⡀⠔⠀⠀⠀⠀⠀⠚⡇⠀⠡⠈⠛⠛⢋⠔⠢⢉⡉⠠⠂⠀⠀⠁
⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠓⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣀⣀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⠀⠤⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀⠀⠀⠀⠀⠀⢀⣠⠶⠛⢀⡰⠶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠋⠀⡀⠀⠀⢈⣛⠃
⠀⠀⠀⠉⡏⠙⠻⢿⡶⢦⣶⣴⣶⡟⠋⠉⢩⠆⠀⠀⠀⠀⠀⡎⠉⣀⣠⠄⠸⠅⢤⡄⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⠈⠀⠀⠈⠂
⠀⠀⠀⠀⠑⢄⠀⠀⢀⠐⠀⠢⢀⡀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⡰⠆⢀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠟⠀⢀⠛⠃⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠠⠊⠂⠤⠠⠔⠁
⠀⠀⠀⠀⠀⢀⢤⠂⠀⢉⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⢶⣤⣀
⠀⠀⠀⠀⢠⠛⠚⠒⡆⠉⠀⢸⠉⠉⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⢀⠸⠵⠄⠂⠋⠈⠀⠉⠀⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠁⠒⠠⠀⠀⠀⠀⠀⣠⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⢀⡤⠒⠉⠛⠀⠈⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠛⠶⠛⠋
⠀⡔⠛⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠛⠀⠀⢀⡀
⠀⢇⣠⠀⢿⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠁
⠀⠈⠙⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⢸⠉⠒⡂⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⠶⠤⠴⠤⡯⠔⠒⠀⠐⠒⠄
⠀⠀⠀⢸⠀⠀⢃⡀⣀⣀⣰⠖⠛⠋⠉⠉⠁
⠀⠀⠀⢸⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠘⠧⣀⣀⣀⡀⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠔⠊⠁⠀⠀⠀⠀⠀⠈⠉⠒⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⠄⠠⠠⠄⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠘⠒⠒⠀⠐⠀⠚⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠊⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡇⠀⠀⠀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠤⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢄⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⣡⠈⠛⠐⠤⡀⠀⢀⡤⠒⠘⠇⠀⠀⡀⢤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠘⣇⠀⠀⠀⠈⠑⠋⠀⠀⠀⢣⡔⠉⠀⠀⠂⣀⠔⠁⠀⠀⠈⠚⢢⡀⠀⠀⠑⢄
⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠋⠀⠀⠀⠔⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠘⣆
⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢁
⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠘
⠀⠀⢠⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠤⠄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠄⠀⠃
⠀⠀⡜⠀⡴
⠆⠈⠀⠀⠐⠔⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘
⣿⣿⡿⠿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣟⣋⠁⠀⠀⠀⠀⢀⣀⣀⣠⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄
⠀⠀⠀⠀⠀⠀⣠⣀⡀⠠⠴⠋⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⣿⣿⣿⣿⡿⢋⣿⣿⡿⠿⠋⣽⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⠀⠼⠛⠉⠀⠀⣼⣿⡿⠟⠋⠉⠀⣼⣿⣿⡿⠟⠋⠹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⣠⣼⣍⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠐⡏⠀⣨⠀⠀⡀⠀⠀⠀⡾⠛⠻⠆⠀⠀⠀⠀⠀⠈⠹⣇
⠀⠀⠀⠀⠀⣠⡴⠶⣤⡀⠀⠀⠑⠲⡄⠀⠀⠀⠀⠀⠈⠛⠁⠀⡼⠃⠀⠀⠀⠱⢤⠾⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⠄⡀⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠶⠚
⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⣠⠇⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⣤⣠⠟⠉⠁⢰⡄
⠀⠀⠀⠀⠀⠘⣧⡀⢀⣀⣿⣆⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢹⠈⠏⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠸⡟⠋⠉⠀⢹⡆⠀⠀⠀⠀⠀⠙⠷⣤⣄⣈⠻⣿⣿⠟⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⢠⡇⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⢀⠀⢉⡟⠛⠛⠷⠾⣿⣿⣿⣉⡁⠀⠀⠀⠀⠀⣰⠏⠉⠙⠒⢴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠈⡇⠀⢀⣠⢴⡞⠁⠀⠀⠑⠲⠖⠒⠚⠋⠁⠀⠉⣿⢄⠀⠀⢀⡾⠃⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠳⣴⠟⠁⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣠⣤⡴⠶⠾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠻⢶⣴⠟

⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢸⡇⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢄⠀⠀⠀⢸⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⢰⡄⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢠⠗⠘⠓⠒⢦⡒⠒⠓⠀⠀⠀⠀⠀⣀⣤⢄⣀⠬⠷⠶⠶⠻⢄⣀⠞⠀⠀⠀⠀⠀⠒⠲⣄⣈⣇⣼⡀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⣠⠃⡋⣉⡧⢀⣤⡟⠛⠒⠂⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⢠⣮⠉⣄⠀⠀⠉⢉⡽⣷⣮⢭⡉⠁
⠀⠀⠀⠀⠀⢳⠀⠙⠋⠁⢿⡤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣠⠶⢤⡀⠀⡠⣄⠀⠸⡀⠀⠀⠀⠀⠘⣔⠤⠞⠰⠤⠆⠰⠬⢜⡷
⠀⠀⠀⠀⠀⠈⢣⡀⠚⠓⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠧⡘⠴⠃⠀⠵⠬⠃⢠⠇⠀⠀⠀⠀⠀⢦⡈⢩⠆⠀⠲⡂⢈⡡
⠀⠀⠀⠀⠀⣤⢾⠇⠀⠀⠀⠘⢯⡙⢢⠀⠀⠀⠀⠀⠀⠀⠈⠓⣶⠀⠰⠦⠀⠀⢻⠁⠀⠀⠀⠀⠀⠀⠀⢉⡏⠀⠀⠀⣟⠉
⠀⠀⠀⠀⠸⠁⢸⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⡠⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠳⠋⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠈⢢⡤⠀⠠⠤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⠁⠒⣲⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⢱⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠒⠂
⠀⠀⠀⠀⠀⡀⢀⡈⢱⣀⡠⠔⠛⠛⠛⢦⡀⠀⠀⠀⠀⠀⠠⣀⠐⣆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⢀⣠⠤⠴⠯⠤⣄⡤⠤⠄
⠀⠀⠀⠀⠠⠤⠤⣬⠞⣡⣖⠲⠐⣏⣿⠆⡇⠀⠀⠀⠀⠀⠀⣼⢷⡋⠈⣓⣲⣄⠀⠀⠀⠀⠀⢀⣞⡤⣄⢾⡅⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠳⠭⠞⣀⠴⢀⡞⠁⠀⠀⠀⠀⢰⠱⣟⡠⠇⠸⣛⣨⠻⡀⠀⠀⠀⠀⢸⡙⠓⢚⡴⣦⢰⠦⢄⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠓⣄⠀⠀⠀⠀⠘⠦⣄⠀⠠⠄⠀⠀⠀⣇⠀⠀⠀⠀⠀⡟⢲⡬⣿⠷⠋⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢦⠀⠀⢀⡎⠀⢸⠄⠀⠀⠀⠀⠀⢸⠀⢦⡀⠸⠀⠀⢸⡆⠀⠀⠀⠀⡄⢸⠀⠇⠀⠀⣰⠃⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠉⠃⠈⠀⢀⡜⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠃⠘⠂⠘⠀⢠⡇⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠹⢄⡀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡶⠖⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣤⣀⡠⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠉⠉⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢋⡤⠴⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡀⣉⣵⡤⠤⡧⠤⣖⠉⠉⠀⠀⠀⠀⠀⢀⡤⢴⢡⠤⡀⠀⣀⣙⢞⠓
⠀⠀⠀⠀⠀⠀⠀⢠⠞⠓⠛⠓⠲⣍⣁⡀⠀⠀⠀⠀⠀⠀⢀⣠⠏⣬⣸⡇⢞⢩⡆⠀⠙⣏⠉⠃⠀⠀⠀⠀⡔⣛⢷⠄⣻⢰⡁⠌⣮⡙
⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠞⢩⡇⠰⠤⠴⠶⠶⠷⠶⠖⢀⡼⠀⠀⠀⠀⠀⠀⠀⢻⠀⠙⠁⠀⠉⠋⢸⡇⣠⠇
⠀⠀⠀⠀⠀⢠⡇⢀⡔⣻⢲⡀⢠⣖⢮⣇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡴⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⢼⠀⠀⠒⠒⠀⠀⠀⡏⠁
⠀⠄⠀⠀⢠⡟⡄⠀⠓⠒⣚⠀⠈⠛⠋⡜⢧⠀⠀⠀⠀⠀⠇⠱⣄⠀⠀⠀⠸⡏⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⡄⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠈⢲⡤⠤⣬⡭⠭⢴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⡦⠤⠤⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⣀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠙⠲⢤⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠁⠀⠈⣡⠋⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠉⠀⠀⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢰⠇⠀⣠⡀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⢃⠀⠀⡑⠒⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢋⡔⣦⠠⡀
⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⣾⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠱⠀⡃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⡴⢡⣋⣸⢁⣤⡌⠑⠦
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠰⠞⠂⢀⡌⠳⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠐⠁⠀⢸⠆⠀⠀⠀⠀⠀⢀⡜⢣⡈⠉⠟⢻⡞⡁⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡦⢉⣉⢣⠀⠀⠀⠀⢤⣄⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠳⣄⣹⣾⣣⡄⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡀⠈⠛⠃⠀⠙⠀⠀⠀⠀⢀⢒⡼⠀⠀⠠⠴⠖⠋⠀⠀⠀⠠⡀⣠⠏⢳⡄⠀⠀⠀⢀⠏⢰⢟⡥⠲⠌⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡔⠀⠀⠢⡀⠀⠀⠀⠀⢨⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⣠⠋⠀⠀⠀⠀⢸⡀⠀⢸⣼⢧⢞⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⣀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⢁⡼⠁⠀⠀⠀⠀⠀⠀⠉⠙⢦⠐⣨⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⢉⣉⣁⡀⠈⠑⡄⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠉⠱⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⢤⣤⠌⠀⢀⡏⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⢋⡾⣡⢆⡀⢱⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠤⢀⡀⠀⣧⠀⠙⠒⠤⠖⠚⠉⠀⡇⢸⠀⠀⠀⠀⠀⠀⡠⢡⠞⣴⢣⡞⠡⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⡈⣏⠀⠀⣀⠀⠀⢀⡴⠀⢸⣀⠴⠚⠳⡄⠠⡁⠈⠘⢡⠏⡐⠉⡂⠆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⡘⣎⠓⠋⣙⠓⠊⠁⠀⢀⠜⠁⠀⠀⠀⠷⠀⠙⣦⣀⠀⡸⢧⡞⠉⡆⢄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢻⣦⠞⠉⠉⠙⢦⡔⡁⠀⠀⠀⠀⠀⡎⢈⢦⠷⣈⠉⠀⢰⢀⡼⢁⠎
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⢀⣠⠶⢿⡀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⠀⡇⠀⠒⢂⢮⡉⣇⠈⠉⠓⠋⠛⢳⡆
⠀⠀⠀⠀⠀⠀⠈⠳⣄⣤⡶⢫⠁⠀⠈⠓⠦⡴⠾⠋⠲⡄⠀⠀⠀⠀⠀⡇⠀⢀⠃⠀⡗⠹⡄⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡂⠀⠀⢀⠹⣆⠀⠀⠀⢸⡇⠀⠘⡆⠀⡇⠀⠙⢦⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠘⠀⡈⠳⢦⣴⠟⠀⠀⠀⢱⠀⢣⠀⠀⠀⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠁⠀⠀⠰⠠⡀⠀⠀⢸⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠰⡀⠀⡟⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⢧⢀⡀⠀⠀⢧⠑⢤⡴⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠑⠦⡇⠀⠀⠀⠀⢸⡇⠀⠃⠀⠀⠀⢸⠀⠀⠀⠀⠸⣦⣄⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣇⢀⣀⣀⣀⣀⢸⠀⠆⠀⠀⠀⢧⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠤⠤⠤⠤⠤⠴⠒⠃⠀⠀⠀⠀⠛⠛⠛⠛⢛⠓⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣄⡀⠀⠀⠀⠀⠀⣀⡤⣞⠁
⣠⠤⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⡾⠉⠛⠓⠒⠛⠋⠉⡇⠘⡆⠀⠀⠀⠀⠀⣠⠴⠒⠒⠲⢦⡀
⡇⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⣾⢰⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⣠⠎⠁⠀⠀⠀⠀⢸⡇
⢹⡀⠀⠀⠀⠹⣆⡤⠶⠂⠀⠀⣛⠈⢃⡀⠀⠀⠀⠀⡴⠓⢧⠤⠷⠀⠑⢶⠁⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠳⣄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠉⡉⠉⠀⢧⠀⠀⢸⠀⠀⢀⣀⣀⡀⠀⠈⠀⠀⠢⣀⣀⡴⠛
⠀⠀⠙⠳⢶⣤⠾⠋⠉⠛⠲⠶⠾⣇⣀⣀⣼⠀⠀⠘⠦⠤⠤⠃⠈⠉⠙⠛⠛⠛⠛⠋⠉
⣿⣿⣿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠘⢍⡉⣑⣶⢄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡏⠉⠁⢀⠀⠀⠀⠀⠄⡀⢨⠀⠉⣹⠟⠓⠈⠁⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠤⠤⠤⠤⠤⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠐⠁⠀⠀⠀⠀⠐⢹⡛⠟⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡠⠊⠢⡀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢸⠋⣷⣦⣀⣀⣀⣀⡇⡜⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡠⠊⠀⠀⠀⠙⣄⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠗⠊⣉⣸⠿⢧⠄⠀⠀⠀⠈⠉⠙⠒⠢⢬⣆⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢲⡀⢩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠁⠀⢸⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢸⠀⢀⡴⠊⠓⢤⡀⠀⣲⠶⢀⣴⠅⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢠⣄⢃⣀⣬⣄⡀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡠⠚⠁⣀⡤⠔⠒⠒⠉⡿⢴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣸⣿⣿⡤⠇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡋⣀⣴⠊⠁⠀⠀⣀⡀⢰⠗⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠬⠭⠭⣬⣭⠤⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣉⣎⣀⣼⠀⠉⠛⠯⢅⡁⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⣀⠤⠤⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⣦⠀⡠⠞⡉⠠⠀⠂⠀⠀⠀⠉⠓⣄⠀⠀⠀⠀⠀⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣧⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠁⠀⠀⠀⠀⢸⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢂⡈⢃⠀⠀⠀⣠⠔⠋⢳⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸
⠀⠀⠀⠀⠀⠀⠀⠐⢇⣛⡩⢖⣿⠆⠀⣰⠃⠀⡀⣾⠒⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠀⢰⠃⠀⠀⡼⠧⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⢿⠇⢠⠿⣀⡠⠎⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⢰⠋⢺⡉⠉⡼⠀⡎⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡐⠧⠀⠀⢸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⠆⣠⡴⣅⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠱⡈⠑⠚⠉⣉⣉⣒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡘⡖⠤⣈⡑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣤⣤⣤⣤⣴⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⢛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡿⠋⠉⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠋⠈⠙⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠈⢿⣿⣿⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠸⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⣿⣿⠇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⢻⣿⣄⢀⣀
⠀⠀⣴⣿⣶⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣏⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⣿⠀⠸⢿⣿⣿⣿⣷⡀
⢠⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠀⠀⠁⢹⣿⣿⣿⣿
⠘⢿⣿⣿⣿⡟⠁⠶⠦⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟
⣴⣾⣿⣿⣿⣷⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠋⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠙⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿
⠀⠈⠛⠋⢻⣿⣿⣿⠿⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⢻⡿⢶⡄⠀⠀⠀⢠⣿⣿⣿⠏⠉⠁
⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⡇⠈⣇⠀⣷⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣤⣀⡀⠀⠀⠀⠀⠸⣧⠀⠑⠀⠉⠀⢹⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠛⠛⠶⣶⣶⣶⣿⠀⠀⠀⠀⣠⠟⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠙⣷⠀⠀⠀⠀⠀⠀⠉⠉⠻⣦⡀⠀⠘⢯⣠⡴⠟⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⠀⠀⢹⣧⡴⠛⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠙⢦⠀⠀⠈⣷
⠀⠀⢀⣤⣤⣤⣤⣤⠾⠋⠀⠀⢀⣴⠟⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠠⣿⠁⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢀⣾⠁
⠀⠀⠙⣷⠶⠶⢶⣶⡄⠀⠈⢿⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢲⡶⠛⠁
⠀⠀⠀⠘⠳⠶⠟⠉⠻⣦⣤⡾⠁⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⣶⣶⠂⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⣤⣤⣼⡇⣿⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠒⡞⢳⡶⠶⠶⣾⡇⢹⡶⠶⠶⢶⠛⡖⢲⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣀⣙⣑⣉⣉⣀⣀⣀⣸⡇⣿⣀⣀⣀⣈⣊⡑⣟⣁⣈⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⠇⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡟
⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⢠⠖⠲⡄⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠈⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡞⢠⠇⠀⠀⢹⠀⠀⠀⠀⠀⢀⠔⠚⠓⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⠀⠀⢸⣿⠀⠀⠀⠀⢠⣇⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠘⣆⠀⢀⠞⠀⠀⠀⠀⢸⠿⠃⠀⠀⡸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠁⠊⠉⠑⠀⠀⠘⢤⣀⣀⡴⠁⠀⠀⠀⠀⠀⠀⠛⢿⠛⠉⠉⠙⢿
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⡤⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠔⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢰⠉⠣⠤⢤⣀⣈⣙⣢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⢦⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠘⢄⡀⢀⡠⠤⠤⠤⣀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣮⡀⠈⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠉⠉⠓⠲⢤⣤⡤⠞⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢤⣀⣀⣀⣀⣀⣀⣠⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠋⠁⠀⢠⡉⠉⠃⠀⠀⠀⠀⠀⠉⠙⢻⠐⠟⠛⢶
⠀⠀⠀⠀⠁⡀⠀⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠒
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠄⠠⠎⠘⠢⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠑⠂
⠀⢠⠀⣀⠀⡀⢀⠀⢀⡀⢀⠀⠀⣄⠀⠀⠀⣀⣀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⡀⡤
⠀⠀⠀⠉⠀⠁⠈⠁⠈⠀⠀⠀⠀⠀⠘⠁⠂⠀⠁⠉⠁⠂⠀⠀⠀⠉⠁⠁⠈⠁⠂⠃⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉
⠀⠀⠀⠀⢄⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠄⠠⠁⠀⠋⠀⠀⠀⠀⠀⠀⠀⠓⠄⠠⠴⠚⠉⠁⠀⠀⠈⠉⠑⠀⠈⠉⠄⠀⠀⠀⠀⠀⠴⠛⠁⠀⠈⠋⠙⠆
⠀⠀⠀⢶⡦⠀⠀⠰⡶⠀⠀⠀⠀⠀⠒⠒⠠⠀⠐⠚⠃
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠘⠃⠄⠀⠀⠀⢠⠠⠄⢀⠀⠀⡀⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠉⢀⣀⣀⠈⡓⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠁⠀⡏⠀⠀⠈⠁
⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓
⠀⠀⠀⠀⠈⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠁
⠀⠀⠀⠄⠀⢀⡀⠀⢀⠀⠀⠀⠀⠦⠀⢠⢠⢀⣠⠄⠀⠀⠀⢀⠀⡀⢀⡀⠀⣀⣀⠀⠀⠀⠀⠂⠀⣀⠂⠀⣀⡀⠤
⠀⠀⠀⠀⠀⠈⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠈⠈⠈⠈⠉⠀⠉⠁⠀⠈⠉⠀⠀⠀⠉⠉⠈⠀⠁⠀⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⡟⠾⠳⠶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⢤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⡠⡠⢀⣀⣄⣄⢣⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢎⡟⠲⠀⠀⠀⠠⠠⡄⢠⣇⡀⠀⠀⠀⠀⠀⠠⣄⡀⠠⣼⠔⠚⠀⢈⠀⠁⢁⠠⣸⠀⢩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠈⠁⠄⠀⠀⠈⠀⠁⢠⠞⠁⠀⠀⠀⠀⠀⠀⢾⡥⣠⣘⠀⠀⠃⠊⠀⠀⡘⠒⢹⠗⠚
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠒⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡀⠀⠀⠀⠀⠀⠄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣒⡤⡄⢠⠀⣤⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⣀⡀⠀⣀⡤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡁⠀⠈⠈⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠉⠑⠔⠉⠑⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠃⠀⠀⠀⠀⠢⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⣚⠀⠀⠀⠀⢠⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠁⠀⠀⠀⠀⠀⠱⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠙⠛⠘⠃⠈⠉⠧⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⢀⠀⠀⠀⠀⠀⢀⠁⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠈⠉⠀⠉⠑⠒⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠀⠀⠂⠐⢂⡿⢠⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢀⠀⠠⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠉⠉⠉⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⣛⠢⠀⠀⡤⢀⣀⡀⠠⠐⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠢⢤⠤⢤⠆⢰⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⠁⡙⠉⠒⡖⠀⣶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡎⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢀⡇⠀⠀⢃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣠⠾⠰⡇⠀⢀⡆⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢎⡠⣼⠇⠀⡰⠛⢐⡷⣾⣧⠀⠀⠀⠀⠀⠀⠀⠘⠻⠅⠓⠒⠊⠀⠐⠾⠦⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠁⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⣀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠊⠁⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢀⡔⠉⠉⠙⢦⠀⠀⠀⠀⠀⢀⠖⠉⠉⠓⣄⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡞⠀⢀⣴⣦⡄⣧⠀⠀⠀⠀⡏⣠⣤⣄⠀⠘⡆⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⣿⣿⣿⣼⠀⠀⠀⢸⣷⣿⣿⣿⡇⠀⡧⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢧⠀⢿⣿⣿⣿⡟⠀⠀⠀⠈⣿⣿⣿⣿⡇⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠙⣋⡝⠁⠀⠀⠀⠀⠙⣝⠛⠋⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢻⡿⠛⠿⠿⣿⣿⡿⠿⠿⣿⡟⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣀⣀⣠⠴⠋⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠒⠒⠒⠒⠒⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡤⣴⠄⠠⣤⣴⣀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠄⠐⢀⠉⠀⡉⢻⣷⡼⣿⡿⢦⣽⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⡰⣁⠀⠠⠬⣷⣑⠀⠈⣪⣠⡤⠿⡆⣵⣟⠉⠉⠻⠻⠷⣶⣤⣀⠀⠀⠀⣀⣼⢛⠀⠠⢠
⠀⠀⠀⠀⠀⠀⡴⠀⣿⡇⣠⡬⡿⢿⣹⣞⠉⠀⠀⠂⠙⠶⣭⣿⢶⣄⣀⣀⠀⠉⠛⣿⣯⣍⡉⠙⢻⡄⢀⠀⠃
⠀⠀⠀⠀⠀⢂⠣⡜⡛⠃⢼⢰⠱⠂⠈⠙⠛⠶⣤⣀⠀⠀⠀⠙⠻⠿⡿⣿⣿⣦⣤⠛⠿⣿⣿⡄⣌⣳⡀⠁
⠀⠀⢀⢀⠤⢀⡬⣱⣔⣾⡝⡎⠀⠰⢤⣄⠈⠐⠩⣹⠋⠉⠁⣀⢠⠤⣤⡶⠟⠻⣿⣿⠖⣾⣇⡉⠉⠉⠉⣀⢀
⠀⠄⠈⣿⡋⠋⠻⠅⠻⠾⠞⠀⠀⠀⠈⠉⠁⠀⠀⠀⣼⣿⡆⠀⡘⢦⣌⣳⣥⣀⣈⣿⣟⣛⣭⠿⠖⠚⠉⢁⣼⣿⣿⠦⡄⠀⡀⠂
⢤⣀⡄⠈⠙⠻⠻⣌⢽⡗⣀⢂⣤⣀⠈⢿⢷⠀⠀⠀⡿⠷⡇⢀⠀⠀⠈⠙⠉⠉⠉⢍⡡⠐⣀⣭⠥⠶⣆⣹⡟⣛⠫⠦⢀⡄⢬⣍
⠀⠀⠀⠀⠀⠀⡈⠉⠉⠱⢧⣿⣿⣿⢁⢴⡉⣀⠀⢠⡙⠿⣁⡿⣦⣀⠀⠀⠠⣶⡾⠿⢃⣸⢁⡒⣉⠌⣸⡟⠃⡙⠛⠒⣏⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣽⣄⣸⣿⣧⡼⢷⡺⣿⣻⣿⠔⠁⠉⣙⡓⢶⣶⡶⢿⡍⣭⡥⠖⣉⠉⡁⠈⠀⠓⠃⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠈⠁⠉⠙⠯⡴⢲⠒⡞⠏⢙⡛⡉⠛⠉⠹⠧⠐⠀⠉⢉⣁⠰⠰⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠓⠛⢀⠀⢤⣴⠶⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢛⣄⣼⣿⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⣿⢿⣿⣿⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⣠⣶⣿⣇⣀⣿⣷⠀⠋
⠀⠀⠀⠀⠀⠀⢠⡐⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠞⣽⣿⣿⣿⣿⣿⡟⠠⢤⠄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠘⣇⠈⠙⠀⠀⠀⠀⠀⠀⠀⠞⢡⣼⣇⣄⣴⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢐⠁
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣯⠁⢩⣿⡆⢰⡖⠀⣀⣴⠟
⠀⠀⠀⠀⠈⢲⠃⠀⠐⡀⣴⣦⡿⠂⠀⠀⠀⠀⠀⠀⠉⠋⠸⢿⣿⣿⣿⣿⠷⠋⠠⠞⠉⣡⡶
⠀⠀⠀⠀⠰⣷⢀⢰⣴⣿⣹⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢩⣼⣷⣮⣤⣶⣾⠏
⠀⠀⠀⠀⠀⢻⣦⣷⣽⣿⣷⡍⠀⢠⡶⠞⠛⣳⠀⠀⠀⠾⠷⢶⣶⡌⢸⣿⣿⣿⠛⣋⣽⠃
⠀⠀⠀⠀⠀⠀⣹⣿⡹⣿⣿⣿⢀⣏⣤⠷⢾⠟⠀⠀⢀⢒⡷⠿⣄⣹⠀⢻⣿⣿⣿⣻⠇
⠀⠀⠀⠀⠀⠀⢻⣿⡟⠻⢿⡏⠈⠉⠓⣾⠾⠇⠂⠀⠀⡜⢺⡗⠊⠉⠂⠸⠿⢛⡛⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠙⠊⢟⣦⡷⠀⠀⠀⠈⡠⠄⠀⠀⠀⢠⡈⠀⠀⠀⢀⢇⣴⡋⠥⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠆⠀⡔⠊⠓⠿⠦⡴⠾⠶⠓⠦⠀⠰⠃⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⣿⠈⠀⢦⣤⡤⠤⠤⠤⠤⠤⣤⡤⠀⠀⠀⣼⠥⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠈⢿⣿⣷⣶⣶⣶⣿⡟⠁⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⠀⠻⣄⡉⢋⣩⠟⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠢⢭⡭⠞⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠛⠉


⠀⡀⣀⠀⢀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣯⢹⣟⡛⣛⣛⡛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉⠐⠂⠀⠀⠀⠀⢀⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢠⠔⠈⠈⠃⢀⣸⡄
⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠁⠀⠠⠐⠛⠉⠉
⠀⡎⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠠⢄
⠀⠳⡶⠀⠩⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠸
⠀⠀⠈⡗⠤⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡴⡁⠤⠤⢀⡀
⠀⠀⠀⡇⠀⠸⡀⠀⠀⢀⣤⠤⠶⠶⠚⠛⠛⠉⠉⠉⠉⠙⠉
⠀⠀⠀⢁⠀⠀⠓⠂⠈⢹
⠀⠀⠀⠸⣀⡀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠁⠹⡁⠀⠀⠉⠉⠉⠉⠉⠀⠈⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⢀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⢀⣴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣄⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠈⡋⣄⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠑⠢⢄⡀⠀⡠⠔⠒⡆⠀⢀⡠⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢄
⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠙⡆⠀⠀⠀⠈⠋⠀⠀⠀⠱⠞⠁⠀⠈⠠⠔⠀⠀⠀⠀⠙⠢⡀⠀⠀⠱⡄
⠀⠀⠀⠀⡰⠃⠀⠀⡀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠈⠢⡀
⠀⠀⠀⡘⠀⠀⠀⠈⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⢱
⠀⠀⠀⠇⠀⢀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠈⡄
⠀⠀⢸⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⡀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⢁
⠀⠀⡆⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⡄⠐⠁⠀⠥⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣶⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠶⢤⣄⣀⣠⠤⠶⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠔⠒⣁⡀⢒⠒⠒⠲⠦⣆⢍⣍⡃⣵⠀⣠⠂⠙⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣁⣴⣶⣿⠟⣻⣉⣉⣁⣄⡤⣀⠡⡙⢻⣮⣞⡵⡣⢀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣷⣿⣿⣽⣿⣿⣿⣿⣿⣝⢮⣦⣻⣿⡾⣿⢶⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢣⢫⠙⠙⠙⠟⠛⠿⠻⢃⣼⢃⢀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣸⠀⠀⠀⠀⠀⠀⠰⢾⣯⠰⡈⢦⡻⣤⠤
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠽⠂⣭⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⣠⡾⣿⠟⡿⢿⡄⠹⣿⣿⠙⣿⠆⠀⡤⠖⠛⠛⠻⠿⣶⡤⡜⠀⣰⣠⣴⣶⡏⢷
⠀⠀⠀⠀⢰⣿⣾⣿⣤⣿⡀⢻⡄⠹⣿⠀⢸⡀⠀⠀⢀⣀⠀⠀⠰⣆⡉⠀⠀⡞⠁⠀⣠⠇⠸⡆
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣧⠀⢳⡀⠹⡇⠈⡇⠀⠀⠀⠙⠛⠛⠛⠉⠁⠀⠀⡏⠳⠶⢾⠖⠀⢧
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢧⠀⢷⠀⠱⡀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠸⡆⠀⠸⡄
⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣷⠸⣆⠈⢣⠀⢱⡀⠘⢦⡀⠀⠀⠀⠀⠀⠰⠖⢀⣴⠇⠀⢀⡇⠀⠀⣧
⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⣆⠘⡆⠈⠃⠀⠀⠀⠀⠻⡄⢐⡀⠀⢀⣀⡀⣀⡀⠀⠀⣸⠇⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⡄⢾⡄⠇⠀⠀⠀⠀⠀⢻⠚⠓⠿⣿⣿⢿⣿⠟⠀⣰⠏⣤⠒⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣇⠈⢻⡄⠀⠀⠀⠀⠀⠸⡇⠀⠀⠈⠉⠙⠉⣞⣴⡋⠀⠸⠆⠂⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣦⣰⠁⢠⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⡾⠋⠳⣴⣦⠀⠀⠀⢀⣤⣬⣷⣄
⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⡂⠏⣇⠀⠀⠀⠀⣿⣷⣶⣤⣤⣤⠾⠁⠀⠀⠈⠙⢳⡀⣰⣧⡽⠏⣀⣬⣷⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡏⠀⠀⠀⠀⠀⣼⠛⠋⠀⢠⣿⣷⣄⠀⠀⠀⠀⠀⠀⠹⣿⢃⣴⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⠀⠀⠀⠀⠀⢰⠇⠀⠀⢀⣾⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣟⣀⣀⡀⠀⠀⢀⡟⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⡿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠉⠃⠈⠉⠉

⠀⠀⠀⠀⣀⣤⢴⣺⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢟⣲⢤⣀⡀
⠀⠀⠰⣯⣿⣋⣡⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣬⣉⣻⢿⡗
⠀⠀⠀⠈⠛⠿⡿⠛⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠻⡿⠛⠋
⠀⠀⠀⠀⠀⠀⢣⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⡰
⠀⠀⠀⠀⠀⠀⠈⢇⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣀⠷⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠖⠴⣁⡀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠛⠉⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠉⠚⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⡤⠖⠉⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠗⠊⠉⠁⣀⣀⣀⡀⠉⠑⠲⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣼⣧⣤⣤⣿⡄⠀⠀⠀⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⣄⠀⠀⡤⠀⠀⠀⠀⠀⢠⡀⠀⢠⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡤⢤⡀⠀⠈⠓⠋⠁⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⣤⠤⣿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢐⠧⣼⠃⠸⣄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⢳⡠⠼⡂
⠀⠀⠀⠀⠀⢀⡴⠋⠀⢸⡄⠀⠀⠈⠉⠉⠒⠒⠂⠉⠁⠉⠑⠒⠒⠒⠒⠊⠁⠀⠀⣼⠁⠀⠙⢦⡀
⠀⠀⠀⠀⣠⠎⠀⠀⣠⠞⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠱⣄⠀⠀⠱⣄
⠀⠀⠀⡾⠁⠀⢀⠜⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠈⠳⡀⠀⠈⢷
⠀⠀⢰⡇⠀⢰⠃⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠙⡆⠀⢸⡆
⠀⠀⣸⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⣇
⠀⠀⡏⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢹
⣀⡴⠃⢸⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⡇⠘⠦
⣴⣶⣶⣶⣶⣾⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣷⣶⣶⣶⣶
⣿⣿⣏⣀⣁⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿