Bicycle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠘⠛⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⣩⡖⣦⣄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣉⣾⣿⡇⠂⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⣾⣿⣿⣽⠁⢰⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠃⢀⡿⣄⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠤⠤⠤⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢭⣭⣽⣗⡀⡼⠀⠈⢳⡞⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡁⠒⢉⣠⠤⠖⣒⣒⠒⠦⠤⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢠⠀⠀⠀⣸⢳⣽⣦⠀⠹⣌⢳⡀⠀⠀⠀⠀⠀⠀⡾⠀⣙⢦⡉⠔⠊⠁⠀⠀⠀⠁⡰⢅⢲⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠸⠀⠀⣰⠃⠀⠙⢿⣷⠀⠘⣆⢳⡄⠀⠀⠀⠀⣸⠃⣰⢃⠄⠙⢦⡀⠀⠀⠀⢀⠜⠀⠀⠱⡹⣄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⠀⢀⣘⡿⣇⠀⠘⣏⠈⠓⢒⣶⣖⡀⢰⠇⠊⠀⠀⠀⠙⢦⣀⡀⠁⠀⠀⠀⠀⠃⠙⠄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⡿⠂⠛⠉⠉⠁⠀⡀⠻⠀⠀⠈⠳⠔⠉⠠⠄⠈⠈⣤⠦⠤⠴⠶⠒⣀⣯⠿⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠀⠀⠀⠿⠟⠠⣀⠁⢀⡠⣇⠤⠶⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠳⣕⠄⡀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢥⣖⣤⣤⣀⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢦⣤⣤⣴⡤⠾⠛⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠔⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣼⠙⢦⡀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⣰⠋⢦⠀⢀⣠⣤⣴⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠉⠉⢹⠿⣿⣦⡙⢦⡀⠈⠓⢦⣀⠀⠀⠀⢠⠏⠀⣠⣿⣿⠛⠋⠉⠀⠉⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⢻⣷⡄⠙⢦⡀⠀⠈⠳⢄⣀⡞⠀⣴⣿⠋⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢻⣷⠀⠀⠙⢦⡀⠀⠀⡽⠁⢰⣿⠃⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢻⡗
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠿⠏⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⣹⢮⣹⠛⣆⢙⣿⣀⣀⡡⠤⠴⠾⠿⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢷⡴⠻⣎⡟⠉⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠠⠴⠏⠉⠛⠉⢹⣤⡌⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⣀⣀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⣀⣀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣒⣒⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢺⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⠒⠒⠒⠒⠒⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠯⠤⠤⠤⠤⠤⠤⠤⠀⠠⠬⠍⠲⠆⢀⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠲⠤⣤⡏⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠛⢄⠙⢄⣀⡤⠴⠒⠒⠒⠒⠤⢄⡀
⠀⠀⠀⢀⠔⠋⣀⠤⠖⠒⠒⠒⠒⠤⡄⠀⠺⡕⢄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡎⠀⢈⠗⠈⢁⡤⠒⠒⠒⠒⠒⠦⢄⡈⠓⢄
⠀⠀⡰⠃⣠⠊⠁⠀⠀⠀⠀⠀⠀⢰⠃⡴⢄⠈⢮⠳⡄⠑⢄⠀⠀⠀⠀⠀⢰⠁⡸⠀⣠⠋⣠⠚⢆⠑⣄⠀⠀⠀⠀⠀⠀⠉⢢⡀⠳⡀
⠀⢰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⡴⢃⡼⠁⠈⢣⠀⢧⠈⢢⡈⠳⣄⠀⠀⢀⠇⢠⠃⢠⠃⢰⠁⠀⠀⠳⡌⢢⡀⠀⠀⠀⠀⠀⠀⠱⡀⢱
⠀⡇⢀⡇⠀⠀⠀⠀⠀⢀⠴⢋⡤⠊⠀⠀⠀⠈⡆⢸⠀⠀⠙⢦⠈⠻⡏⡟⠀⡞⠀⢸⠀⡎⠀⠀⠀⠀⣘⣦⠑⢤⠀⠀⠀⠀⠀⠀⣇⠀⡇
⠀⡇⠘⡇⠀⠀⠀⠀⠀⠘⠦⠎⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠑⣄⠀⠀⠸⠥⠤⠾⠀⠓⠚⢉⣉⣉⡤⠤⠤⠜⠀⠀⠀⠀⠀⠀⡟⠀⡇
⠀⢱⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡼⠀⠀⠀⠀⠀⠈⣇⠀⣠⠤⠖⠲⡀⠸⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠁
⠀⠀⢣⡈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡰⠁⠀⠀⠀⠀⠀⠀⠈⢧⢧⠀⠀⠀⠱⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣠⠃
⠀⠀⠀⠑⢄⠈⠲⠤⣀⣀⣀⣀⣀⡤⠔⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠼⠆⠀⠀⠀⠘⢦⡈⠑⠦⣄⣀⡀⢀⣀⡠⠔⠊⣀⠔⠁
⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣉⣁⣀⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⣉⣁⣀⡠⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠚⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⣈⣠⠡⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⣧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠘⠲⡛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡢⠑⡢⠕⠐⡀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⢒⣨⠤⠴⠶⠤⢬⣁⡂⢄⣀⢞⡘⠄⠀⠀⠀⠀⠀⠀⡀⢔⠅⡪⢐⢅⠲⢛⣧⢣⠤⣶⠤⠤⣟⢀
⠀⠀⠀⠀⠀⠀⡔⣡⠞⠋⠡⠀⠀⠀⠀⡌⠀⡹⢟⣕⢣⠈⠌⢄⠀⠀⢀⡠⣊⠔⢁⢌⠔⡕⣡⣾⠋⠈⡈⡄⠐⠀⠀⠀⠑⣵⢤
⠀⠀⠀⠀⠀⠀⡹⠋⠀⠀⠀⠣⠀⠀⡰⠠⢌⠔⠁⡹⣆⢡⠈⢎⢡⠭⠐⠈⠀⡠⡡⠃⡌⣴⠋⠀⠑⢄⠰⡐⡄⠀⠀⡠⠊⠀⠑⢅
⠀⠀⠀⠀⠀⢰⠙⠐⠢⠄⣀⠀⠱⡠⢟⡴⠅⠊⠁⠀⢸⡄⡆⠈⢶⠦⢶⣤⢊⠔⠀⢸⢠⠇⠀⠀⠀⠀⠑⢧⢻⡠⠊⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡈⠽⡥⡍⠥⢤⡀⠀⠀⢨⡥⡭⠸⡪⡀⣤⡻⠁⠀⠀⢠⣸⠃⠐⠀⠀⠂⠒⡸⣮⢇⠂⠀⠒⠀⠒⠘⡇
⠀⠀⠀⠀⠀⠈⢆⡀⠄⠒⠉⠀⠰⠀⠱⡀⠀⠀⠁⠒⡾⠉⠁⠀⠈⠐⠊⢃⢀⠄⡀⠀⠈⢆⠀⠀⠀⡠⠊⠀⣀⠀⠑⢄⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠡⡣⡀⠀⠀⢀⠃⠀⠀⠱⡀⠀⢀⢔⠎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠱⡣⣠⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄⢔⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠪⡂⢄⡌⠀⠀⠀⢀⡱⡔⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢢⢄⠀⡀⣀⠀⢀⡠⡔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⡈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢧
⠀⠀⠀⠀⣀⣠⠤⠤⠤⢤⣄⣰⠃⠀⠉⠳⣄⡈⠙⠲⢤⣀⡀⠀⠀⠀⣀⣴⡟⠀⠈⢧⡀⠀⣀⣀⣀⣀⣀
⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⣸⠋⠳⣄⠀⠀⠀⠙⠦⣄⠀⠈⠉⠙⠛⠋⠁⡼⠁⠀⠀⣠⠟⢯⡁⠀⠀⠀⠉⠙⠲⣄
⢠⡞⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⢳⡀⠀⠀⠀⠈⠓⢦⡐⢮⡀⠀⢰⠃⠀⢀⡞⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⢳
⡾⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠉⢲⠟⠋⠻⣤⣤⣾⣄⣀⠀⠀⠀⠘⣆
⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣠⠏⠀⡇⠀⠈⠉⠛⠲⠤⣼
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣳⠄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠈⠳⢤⣀⡀⠀⠀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣀⣀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠓⠀⠐⢈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⣉⣈⣁⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⢏⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡩⠐⠂⠈⠉⠉⠙⣌⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⡠⠐⢀⣬⣆⡆⠀⡀
⠀⠀⠀⠀⠀⣀⣤⡾⠗⠟⠛⠛⠻⠻⢶⣤⣄⡀⠀⢀⣦⣆⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⡠⠊⣠⠞⠉⠀⢹⣍⠢⡽⣾⠿⠻⠿⠿⠶⣶⡀
⠀⠀⢀⣢⣾⠏⠁⠀⠀⠀⠐⠊⠙⠐⡢⠌⡙⢿⣦⠀⡈⣿⣦⠖⠁⠀⠀⠀⠀⠀⠀⢀⠎⢠⠞⠁⠀⠀⢀⣴⠿⣆⠱⣿⡷⠊⠉⠠⠒⠠⠉⠳⢦
⠀⠐⢷⠏⠀⡉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐⢠⡢⣝⣥⠘⠛⡄⠀⠀⠀⠀⠀⠀⠀⡠⠁⡴⠋⠀⠀⠀⣰⠟⠠⢊⠘⣷⡈⠁⠀⠀⠀⠀⠀⠑⢀⡄⠙⣄
⠀⢀⠏⠀⡰⠀⠀⠈⠛⢶⣄⡀⢀⡄⢄⣢⠅⠺⠯⠬⠷⠀⣼⡿⠀⠀⠐⢒⡀⠰⣀⠞⠁⠀⠀⠀⢠⡟⠑⡶⠄⡀⠘⢿⣌⢂⠀⠀⠀⠀⠀⠀⠩⣆⠈⡆
⠀⢨⠀⠘⠃⠀⠀⠀⠌⢁⣤⢘⡋⢾⣯⣤⠶⠤⢤⣤⣤⡄⠿⠲⠖⡤⣜⢡⡍⠳⡌⡀⠀⠀⠀⢈⣸⠀⢰⠃⠀⠈⢕⣢⣿⣦⣷⣀⣠⠀⠀⠀⠀⢊⠀⢺
⠀⢸⠀⠀⡀⠀⠀⠀⢢⡀⠉⠛⠛⠀⠈⠂⢶⠒⠛⠛⠛⠛⠈⠋⠀⠸⡈⠙⠋⢰⠇⡆⠀⠀⠀⠈⢸⠀⠸⡀⠀⠀⠀⠈⠙⠻⠿⠡⡀⠀⠀⠀⠠⠿⠀⢸
⠀⠈⣇⠀⠡⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠺⠶⢶⡶⠲⠤⠤⠤⣤⣌⣉⣀⠥⠚⢧⣤⡶⠀⠀⠈⣆⠀⠣⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠁⠀⡟
⠀⠀⠘⢇⠀⠐⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⠀⡼⠁
⠀⠀⠀⠈⠓⢄⡀⠙⠲⡤⢄⣠⣤⢤⠔⠋⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠐⠂⠤⠀⡄⣠⣤⠶⠛⠀⡤⠎
⠀⠀⠀⠀⠀⠀⠉⠒⠰⠤⠄⣄⠤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⣭⣥⣠⡤⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡋⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠉⣉⣿⡉⠉⠀⢀⢽⠀⠀⠀⠀⠀⠀⠀⠢⠘⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠑⡀⠀⠀⢈⡿⢊⠁⠀⠂⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠢⣘⣀⠞⣁⠀⢈⡇⠀⢐⣀⢀⣀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣽⣖⠂⠀⠲⢦⠀⣶⡈⡁⠀⠀⠀⠀⠀⠀⠽⢯⣿⡧⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠅⠈⠀⢀⠀⠀⠑⠬⠖⣁⣀⠀⠀⠀⠀⠀⠀⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠠⠂⠑⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡺⠁⠀⢨⠆⠀⠀⠀⠈⠀⠦⠐⠚
⠀⠀⢀⠠⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠩⠀⠀⠄
⠀⠀⠀⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⠀⠰⠂⠀⢀⡀
⠀⠀⠀⠰⡇⠶⢛⠄⠀⠀⠀⠀⠀⠀⢀⣼⣩⣑⠂⡀⣀⠀⣰⢶⣏⡇
⠀⠀⠀⠀⠉⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⡨⠛⠋⢤⠤⣈⡉⡉⣻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡐⢶⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⢈⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡶⠀⠀⠀⠀⠀⠈⠷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠄⠂⢀⠀⠖⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠔⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⣏⣛⠂⠀⠀⠠⠦⡶⣀⠀⠀⢸⠀⠀⠀⢸⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣟⣉⣉⠉⣑⣶⣿⡿⠀⠀⠘⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠀⠈⠛⠁⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣼⡟⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⡟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠶⠶⠦⠤⣄⣀⡿⢷⡀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣡⠊⢉⣉⡭⠭⠽⠯⠭⢭⣉⣙⠆
⠀⠀⠀⠀⠀⠀⠀⢚⣭⣶⣾⣿⣿⣿⣿⣷⣶⣿⣓⢦⡻⣎⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢶⡥⣞⣽⣶⣿⣿⣿⣿⣿⣷⣶⣿⣓⢤⡀
⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠉⠁⠈⠀⠁⠈⣿⠿⣿⣷⣝⢿⣷⡀⠙⣦⡀⠀⠀⠀⠀⠀⠀⣸⠏⡰⣫⣿⣿⡟⠛⠉⠁⠈⠀⠁⢉⡟⠿⣿⣷⣝⢆
⠀⠀⠀⠀⣰⣿⡟⠃⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠈⠻⣿⣧⢻⠻⣦⠜⠻⣆⡀⠀⢀⣀⣴⠏⣼⣿⣿⡟⠉⠳⣄⠀⠀⠀⠀⢀⡞⠀⠀⠈⠿⣿⣮⢷⡄
⠀⠀⠀⢰⣿⡯⠀⠀⠀⠀⠀⠀⠀⢁⣼⠃⠀⠀⠀⠀⠹⣿⣧⠀⠙⢷⡀⠬⣭⣍⠉⣹⠏⠘⣿⣿⡯⠀⠀⠀⠈⢷⡄⠀⢀⡞⠀⠀⠀⠀⠀⢻⣿⡎
⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣰⣾⣿⣧⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠈⠻⣄⣸⣀⣴⡏⢀⢠⣿⣿⠦⠤⠤⠤⠤⢤⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠘⢿⡿⠏⠀⠀⠀⠀⠀⠠⣿⣿⠀⠀⠀⢠⠛⣿⣶⣿⣷⣠⣤⣿⣿⣶⠶⠿⠛⢛⣻⣿⠿⠿⠏⠀⠀⠀⠀⠀⠠⣿⡿
⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠸⡙⢽⣿⢿⡹⠁⣀⣹⣿⣧⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⠇
⠀⠀⠀⠀⠹⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠀⠉⠛⢺⠚⠉⠁⠀⠀⠻⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⠀⢀⢀⣀⣤⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⠂⠀⠀⠀⠀⠙⠿⣿⣶⣤⣄⣀⠀⡀⢀⣀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⡿⠿⠛⠋


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

⠀⠀⠀⠀⠀⣤⣤⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠻⠶⠮⣽⣿⣟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡆⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣏⣿⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢷⣤⣤⣤⣀⡀⠀⣰⠇⡾⠙⠾⠋⠛⠓⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣟⣀⠀⠈⣿⣛⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡟⠋⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⠦⣤⣄⣀⠀⠀⠀⠀⢿⣿⣦⠤⣄⣀⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⡓⣶⠶⢭⣽⣛⡲⠦⣤⣄⣉⣿⢳⡶⠦⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠓⣺⠃⢳⡌⢷⡀⠀⠉⠉⠛⠲⠦⣽⢃⣾⣅⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠛⢻⡇⠀⣿⣄⡀⠻⡄⢳⡄⠀⠀⠀⠀⢰⡏⣼⠳⢮⡳⢦⣠⡴⠛⣉⣤⡴⠶⠶⡦⣤⣈⠙⠷⣄
⠀⠀⠀⠀⣠⠞⠋⣀⣤⠴⡶⣾⠀⢠⣇⡈⠙⢶⣽⣆⠻⣄⠀⠀⢀⡟⣰⠇⠀⠀⢹⣦⡙⢶⡟⣍⠀⢇⠀⢸⠁⢠⠏⠛⢦⡈⠻⣄
⠀⠀⠀⡾⠁⢠⡞⠉⢣⠀⢳⡏⠀⣸⣫⠛⣦⡀⠹⣿⣆⠙⣦⠀⣾⢡⡏⠀⠀⢠⡟⢠⣿⣦⣝⠿⣦⢸⡀⢸⠀⡞⠀⢀⡴⠻⡄⠘⣧
⠀⠀⣼⠁⢰⡏⠙⠢⣄⠳⣼⠃⠀⣿⡡⢊⡬⢷⡄⠸⣿⣦⣘⡿⠧⣾⡦⠤⠤⠾⣤⣿⣤⣬⣿⣷⣬⣿⣧⣟⣼⠀⡰⠋⠀⣀⢿⡄⠘⡇
⠀⠀⡏⠀⣾⠉⠙⠒⠢⣽⣿⢠⢸⡿⠿⠗⠒⠊⢿⠀⣿⢸⢯⢖⣒⢦⢻⡦⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣀⣈⠉⠻⣦⠒⠉⠀⠘⡇⠀⡇
⠀⠸⡇⠀⢿⠊⠉⢉⡩⠖⡿⡶⡟⢿⠫⢍⡉⠉⢹⠆⣿⢸⡌⣾⠿⠼⣰⠷⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠽⠃⢀⣿⡓⠒⠲⢴⡇⢀⡇
⠀⠀⢻⡄⠘⣷⠚⠉⢠⠞⢀⡇⢧⠀⠱⣄⠈⢑⡾⢠⡟⠀⣹⠟⠲⠾⠿⠶⠶⠶⣶⠶⣶⠶⡶⠶⢶⠶⢶⠶⢶⠛⣅⠙⠢⣤⡞⠀⣼⠃
⠀⠀⠀⠹⣆⠘⠷⣴⠃⠀⢸⠀⢸⡀⠀⢈⣳⠟⢁⡾⠉⠛⠷⠄⠀⠀⠀⠀⠀⠀⠹⣆⠙⢿⣄⡠⠋⠀⡼⠀⠸⡄⠈⣦⡴⠋⢀⡼⠃
⠀⠀⠀⠀⠘⠷⣄⡈⠛⠶⠷⣤⣤⠷⠖⠋⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡈⠙⠲⠦⠧⠤⠴⠷⠛⠉⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⣤⣤⡤⠶⠞⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠄⠙⠠⢍⣁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠒⠂⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⣰⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⢍⠪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠀⠁⢀⡤⠶⠶⠶⠶⠖⢲⠾⡟⠛⠋
⠀⠀⠀⠀⠀⠀⢀⠠⠄⣘⢒⡒⣛⠲⠶⣤⢏⡋⠢⡑⢝⢌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⡥⠤⢢⣿⡴⠶⠛⣛⣉⣛⡛⢿⢷⣤⡀
⠀⠀⠀⠀⡠⢊⠕⠚⡁⠀⠸⠀⠀⠈⠉⡾⣜⠫⡢⡙⢆⠱⡕⢄⢄⠀⠀⠀⠀⠀⠀⢀⢂⠊⢹⣶⣟⡡⠒⠈⠅⠀⠀⠸⠀⡏⠒⢬⡻⢶⢶⡄
⠀⠀⢠⢊⠔⠣⡀⠀⠰⡀⠀⡀⠀⠃⠠⠹⠀⠱⡜⢞⢆⠑⢌⠢⡑⠕⠀⠀⠀⠀⠀⡌⡌⣠⡟⡡⠛⢷⣄⠀⠐⡀⠀⠀⡸⠁⠀⡀⠙⢎⢻⣇
⠀⢠⠇⡌⠀⠀⠈⠢⢀⠰⡀⡇⡀⣠⣣⠃⠀⠀⠘⡌⣎⠆⠀⠳⡈⠪⡢⢀⠀⢀⡔⡐⢰⡉⢐⡁⠀⠀⠙⢷⣄⢡⠀⣴⠇⡠⠊⠀⠀⠈⣧⢳⡄
⠀⣼⠸⠀⠀⠀⠐⠠⠄⡑⢤⣴⢆⡟⠁⠀⠀⠀⠀⢳⢸⠇⠀⠀⠈⠲⡙⣆⣈⣩⣵⠁⠨⠇⠧⠤⠬⠤⠴⠶⠽⢷⣶⢿⡊⠠⠐⠂⠉⠀⢸⠘⡆
⠀⣧⢠⠀⠀⠀⠒⠂⢀⠀⢯⣳⠃⠀⠠⠄⠀⠀⠀⣸⢸⠀⠀⠀⠀⠀⢘⡿⠃⠀⠲⡀⡠⡤⡤⠤⢴⣶⡦⠀⠀⠀⣛⣃⠇⠐⠂⠤⠀⣀⢸⢀⡇
⠀⠸⡈⡄⢀⠠⠐⠈⠀⠈⠀⠀⠠⠈⠢⡀⠀⠀⢀⡏⡼⡄⠀⠀⠀⠀⠸⠠⡨⣩⢀⠏⠓⠓⢋⣉⣩⡤⢤⠴⠒⠚⠉⠉⠆⠈⠢⡀⠀⠀⡞⡸
⠀⠀⠳⡘⡄⠀⠀⠐⠁⠀⠂⠀⠀⠀⠀⠀⠑⢄⠞⡼⠁⠀⠀⠀⠀⠀⠀⢐⡿⠟⠁⠒⠉⠉⢍⠣⡀⠔⠁⠀⠰⠁⢠⠀⠘⡀⠀⠈⢢⠜⡰⠃
⠀⠀⠀⠙⢌⠢⣌⠀⠀⠘⠀⠀⠀⠀⠀⣀⠴⣫⠞⠀⠀⠀⠀⠀⠀⠀⠠⠶⡄⠀⠀⠀⠀⠀⠈⠳⣝⠢⣀⡰⠁⠀⠸⠀⠀⢡⣠⠔⣩⠞⠁
⠀⠀⠀⠀⠀⠙⠲⢭⣒⡒⠤⠤⢔⣒⡪⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢭⣁⣒⣚⣒⣋⠭⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣤⣤⣤⣶⣿⠋⠀⠀⣀⣀⣠⣼⡟⠁⢾⣿⡦⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠛⠉⠁⠀⣀⣤⣶⠿⠛⢋⣉⣩⣤⣤⣤⣿⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⣡⡴⠖⠛⠉⠁⠁⠀⠀⢸⣿⠀⠀⠀⠉⠉⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠽⣇⠀⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢷⣄⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⢍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⣡⣾⣀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣿⠃⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡉⠉⢭⣤⡐⠦⠌⠉⠀⠀⠐⠀⠀⠠⠄⠀⠀⠀⡀⢀
⠀⠀⢀⡴⠚⠉⠉⠀⠀⠙⠙⣾⣏⡀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠡⠰⣤⡶⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠽⣄⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⣼⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠼⡄⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⡟⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⢳⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠁
⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⠴⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣦⢄⣀⣀⣀⣀⣀⣀⣠⣀⣀⣤⡶⠶⠚⣻⣿⡯⠥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣟⠶⠮⠶⠄⠉⠉⠉⡻⢏⠉⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⠀⠀⠀⠀⢀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠟⡁⠀⠀⢠⠖⠀⠐⡟⢁⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣢⠴⠚⣉⣉⣉⣉⡓⠠⡀⠀⠀⢸⣿⠀⢀⡠⠚⠀⠀⠀⠉⠀⠀⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆⣀⡜⢁⣴⠯⠟⠒⠲⢿⣝⡳⣦⡢⡀⠀⣿⡇⠉⠀⠀⠀⠀⢰⣇⣤⡤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢴⡟⠠⠋⠁⠀⠀⠀⠀⠀⠈⠻⣿⣷⡌⢠⣽⡿⢆⣠⡄⠀⠀⡞⢿⣧⣤⡿⠟⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣒⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣎⢷⣴⡿⣿⣿⠀⢠⠁⠰⣉⠉⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠉⣿⢇⣌⢹⣷⣿⢫⢴⡿⢿⣿⣿⡆⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⣿⠇⡟⠀⠀⠀⠀⢠⣴⡄⣠⣾⣏⠀⠀⢀⠏⣾⣿⣾⣿⣿⢁⣾⠆⠀⢸⣿⡇⠈⠠⡀⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢰⡇⠀⠀⠀⡰⣿⣿⣿⣯⣿⣿⣿⡀⣫⣼⣿⣿⣿⣿⣟⣿⡿⠀⠀⠸⣿⡿⠿⠛⢳⣼
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⡀⠀⠀⢸⣧⣧⣾⣿⣿⣿⣿⣯⣽⣿⣥⣿⣿⣭⣽⣿⣿⠇⠀⠀⠀⣿⣿⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⢸⡁⠀⠀⠸⣟⠻⣿⣿⣿⣿⣿⣿⣟⡛⢹⣿⡿⡟⠀⢘⠟⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⢸⡇⠀⠀⠀⠙⠶⠫⠿⣿⣿⢿⣿⣿⣷⡎⢻⢳⡇⠔⠁⠀⠀⠀⠀⠀⠈⣿⣿⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄⣧⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡟⣿⠇⣼⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠸⡄⠀⠀⠀⠀⠀⠀⠁⠈⢻⣟⢛⣏⡴⢠⣏⡠⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⠹⣄⠀⠀⠀⠀⠀⠚⠀⣼⣿⢊⡼⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧⡘⢷⣄⠀⠀⠀⠀⠀⣠⡵⠏⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣍⡓⠶⠤⠴⠖⢋⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠉⠉⠙⢷⣷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⡁⢀⣀⣀⣠⣤⣤⡤⠤⠤⠄⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⢀⣤⣴⣶⡶⠶⠶⠶⠦⠄⠀⢰⣿⣿⡟⠛⠉⠁⠀⠀⠀⠀⠀⣀⡴⠊⣀⣼⣷⠟⣛⣉⣉⣉⠓⠒⠤
⠀⠀⠀⣠⣾⣿⣿⠿⠛⠋⠉⠉⠉⠑⠂⣤⣿⠟⠙⢷⡀⠀⠀⠀⠀⣀⣴⠟⠁⣠⣾⢟⡵⢻⡍⠁⠀⠀⠀⠈⠁⠢
⠀⢀⣾⣿⣿⠟⣡⠖⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠳⣤⣤⣤⣾⠟⠁⠀⣼⡿⣵⠋⠀⠀⢳⡀
⠀⠈⠙⢻⢏⣼⠃⠀⠀⠀⠀⠀⣰⣿⣥⡤⠶⠚⠋⠀⣾⠏⣿⣿⠇⠀⠀⣸⣿⣿⢃⡆⠀⠀⠀⢑⣤⣴⡦
⠀⠀⠀⣿⣸⡏⠀⠀⠀⣠⠀⣼⣯⣾⣄⣀⠤⠄⠀⠀⠻⣆⣙⣋⡴⠀⠀⠛⠛⣿⣼⡇⠀⠀⠀⢼⣿⠟⠁
⠀⠀⠀⣿⣿⣷⠀⠀⠀⢻⣦⣽⣿⣿⣀⣀⣤⡤⣶⣶⠀⠈⠉⠁⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠈⠛⠟⠋⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠸⣿⣿⣧⣀⠀⠀⠉⠉⠉⠉⠉⠀⠀⣠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁
⠀⠀⠀⠀⠹⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣏⡀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠷⣦⣤⣤⣤⣤⣴⣾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣶⣶⣶⣶⡾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠋⣽⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣖⣒⣒⣒⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠍⣿⠍⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⣶⠒⠖⠒⠒⠲⢦⠀⣸⣧⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣒⣮⣿⡿⠿⣯⣽⣓⢾⣿⠃⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡎⣠⡖⢉⣿⡝⣶⣿⠿⠿⢥⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠍⠁⠈⡇⠀⠀⠈⣻⣿⣯⣢⡀⠹⡄⠀⠀⠀⠀⠀⠀⣠⡞⣹⠏⢀⣿⠿⢿⡍⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⠀⠼⣦⣾⣟⠁⠀⠀⠀⢿⠀⢠⣿⠋⠀⠊⢷⣳⡀⢳⡀⠀⣠⣄⠀⡰⠏⣴⠏⣀⣾⠃⠀⠈⢷⣠⠀⡄⠀⠀⠀⠀⠻⢳⡀
⠀⠀⠀⠀⠀⠸⢸⠿⠉⠙⠛⠶⢦⣼⣷⣾⣥⣀⠀⡀⣸⣧⣧⢈⣿⣷⡟⠛⠙⡥⣺⠋⢰⣿⣇⣀⣀⣀⣨⣿⣴⣷⡴⠃⠀⠀⠀⠉⣧
⠀⠀⠀⠀⠀⠀⢸⡶⠄⠀⠀⠀⡬⠻⡿⠿⠿⣿⠿⠽⢷⣿⠧⣾⣿⣻⢿⠶⠞⠋⠀⠀⢸⣹⣏⠉⠉⠀⠀⢀⠽⡿⠓⠒⠀⠀⠀⠐⡿
⠀⠀⠀⠀⠀⠀⠈⣧⢇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠙⣿⠃⠂⢘⡶⠿⠛⠀⠀⠀⠀⠀⠀⠋⣿⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠠⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢷⣖⠀⡀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠐⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⣠⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⢤⣤⣦⡤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠶⣤⣤⣶⣤⡤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠳⣄⡈⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⢳⡀
⠀⠀⠀⣀⡤⠖⠛⠋⠉⠉⠛⢲⣾⣄⠀⠀⠀⠙⠦⣄⠀⠉⠓⠶⢤⣤⣀⣤⡤⠞⡿⠁⠀⠀⠙⣦⣤⠴⠖⠒⠒⠶⠤⣄⡀
⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⢀⡞⠀⠈⠳⣄⠀⠀⠀⠈⠓⢦⡀⠀⠀⣀⠀⠀⠀⣸⠃⠀⠀⣠⠞⠉⠳⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠉⠳⣌⡹⣄⣀⣠⠏⠀⠀⣴⠃⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⢻
⡟⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⢹⠏⠉⠉⢳⡴⢶⡧⢤⣄⣀⠀⠀⠀⢻⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⡼⠁⢸⡄⠀⠀⠈⠙⠓⠲⠬⡇
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢙⣦⠌⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⢀⣀⡤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣊⠩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠐⣒⣒⣥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠈⠁⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡗⢌⣉⣀⣒⣢⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⢶⠒⠒⠊⠀⠀⠀⠀⢀⡠⣔⡲⠝⣊⢽⣼
⠀⠀⠀⠀⠀⢀⠤⢔⣒⣒⣒⡠⠦⡃⠀⠀⠀⣀⢤⣲⣯⠓⠉⠀⣠⣾⠟⣁⠯⠦⠤⠤⠄⣀
⠀⠀⠀⡠⢪⠕⠉⠁⠀⠀⢀⣾⠟⠪⡑⣮⠛⠋⠁⠀⣿⠀⡠⡾⢋⢕⡩⠒⢻⢯⠉⠉⢉⠒⢍⡢⡀
⠀⠀⡐⡕⠁⠐⢄⠀⠄⢀⣼⠏⠠⠂⠈⢎⢆⢠⠔⠚⠛⢮⠊⢠⠣⠊⠀⠀⠈⣇⡇⠀⠂⠀⠀⠙⢜⠄
⠀⢠⢱⠀⠀⠐⠀⠁⢀⣾⣟⣈⣀⣀⠀⠨⡜⣇⢀⣖⡄⠀⣧⢇⠋⠐⠀⠀⠀⢸⣧⡀⠀⡀⠀⠐⠘⡜⡀
⠀⢸⢸⠀⠀⠀⠀⠄⠻⢴⠟⠛⢉⠉⠉⠉⣟⠦⡉⢫⣀⡴⢹⢸⠂⠀⠀⠀⠀⢼⣭⡗⢀⠀⠀⠀⠐⡇⡇
⠀⠀⢇⢆⠈⠀⠀⠂⠄⠀⠀⠀⢀⠀⠐⡼⡸⠀⠈⠉⠀⠀⠀⡏⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢱
⠀⠀⠈⢪⡲⡀⠀⠀⠀⠀⠀⠀⠀⣁⢞⠕⠁⠀⠀⠀⠀⠀⠀⠘⢌⢦⠀⠀⠐⠀⠀⠀⠀⠀⠀⡴⡣⠃
⠀⠀⠀⠀⠉⠪⠝⣒⡠⠤⢄⣚⠪⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⣒⠤⠄⠤⠀⠴⣒⠭⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣶⣶⣶⣦⣤⣀⠀⣠⣤⡉⠙⣿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣶⣦⣤⣀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣎⠁⣰⣿⣴⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠶⠖⠋⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣶⣿⣿⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣐⠾⠷⠾⣿⣮⡉⠉⠛⢿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣉⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⣿⡿⣿⣿⣦⡀⣬⣿⣿⣆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢰⣿⣿⣋⠀⣠⣾⣄⠈⠻⣿⣷⣈⣻⣿⣿⣼⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣽⣿⡿⣿⣿⣯⣭⡛⣿⣿⡄
⠀⠀⠀⠀⠀⣾⣿⡇⠈⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⡟⠻⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⡟⣼⣿⣿⣿⣿⣿⠛⣿⣿
⠀⠀⠀⠀⠀⢹⣿⣧⣀⡼⠿⠿⢿⡙⢛⣿⣿⠧⠴⠋⠙⠻⢦⣤⣄⣬⣿⣿⣿⣿⣿⣿⣿⠿⠸⡿⣷⡹⣿⣿⣿⣿⡿⣳⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣤⣀⣀⣠⣷⣿⣿⠏⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠋⠑⠓⠀⠲⢾⠿⢯⡛⠛⢻⠷⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣁⣴⠆⢠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠉⠙⠂⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠿⣌⣿⣟⢾⣯⣽⣿⣿⣿⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠠⣄⡀⠀⣀⡴⢏⣇⣀⠲⣾⣿⣿⣿⡿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⢘⣿⣞⣩⣵⣿⡏⢽⣿⣿⣿⣿⣿⣷⣿⣿⣿⠟⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡠⢬⣽⣿⣿⣿⡏⠀⠀⢽⣿⣿⣿⣿⢿⣿⣿⠝⠃⠛⠉⠙⢾⣦⡀⣀
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⢈⡉⠁⣤⢷⡀⠉⠉⠙⠛⢷⣦⣤⣄⣀⣠⣇⣠⣼⣶⣞⣷⣶⣤⣷⡀⠀⠉⠻⣮⣧⣸⢶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠳⣤⡻⢾⣿⣷⣦⣤⠄⠰⢎⣿⣿⣿⣿⣯⣽⣏⣟⡛⢿⣿⣿⣯⣵⡶⠦⠀⠈⣿⣧⢀
⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠈⡟⠀⠀⠀⠈⠻⣶⣯⡟⢿⣿⣾⣾⣿⣿⣿⢍⡵⠾⢛⣹⠿⢷⣟⡛⠿⣿⣿⣶⣄⣀⡘⢾⣏⣀⡇
⠀⠀⠀⠀⠸⣿⣶⣕⢄⡀⠀⠀⠀⠉⣶⣤⣀⣹⠝⣬⣿⣿⣿⣿⣿⢋⡾⠿⠛⠻⢥⣄⣀⡀⠨⠿⣿⣿⣯⣝⡉⢉⣛⣻⣯⣿⡧
⠀⠀⠀⠀⢀⠉⠛⠻⢶⣽⣣⣴⣶⣶⣼⣦⡻⢧⣈⣿⣿⡿⣿⣿⣏⡁⠄⠀⢀⣴⠟⠛⠉⠀⢀⡀⠠⣈⣭⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠶⠾⣿⣿⣶⣾⣿⣿⣿⣿⣉⣽⡀⡤⢿⣿⣿⡗⣰⡿⢿⣷⢀⠀⡾⠁⠀⠠⠖⠋⣠⣴⣾⢿⣿⡿⠿⢿⣙⣿⣷⣤
⠀⠀⠀⠀⠀⠈⠻⣿⠛⡿⣿⣿⣟⣺⡍⠘⢿⡱⣄⣽⣿⣵⠋⠀⢘⣧⣌⣁⠰⢄⣤⣄⣴⣾⣿⣿⣿⡴⠇⠠⠀⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠴⠿⠦⠀⠀⠁⠾⣥⣽⣿⣶⣷⣄⢻⣿⣿⣿⣿⣷⣤⣾⡿⢠⣽⣕⣾⣿⣿⣿⣿⣿⣿⣿⣷⣀⠀⢠⣿⠙⢿⣿⣷⠀⠀⡆
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢉⣽⣿⣿⣿⣿⣿⣟⠻⣻⣿⠿⣿⣿⢷⣄⠹⣿⣿⣿⣿⣿⣿⣿⢧⢳⣝⠏⢶⣿⣯⣤⡈⣿⣿⠀⢰⣿
⠀⠀⠀⠀⠀⠠⢤⣤⡀⠀⠀⠀⠸⢛⣻⣿⣿⣿⣿⣷⣤⣉⣿⣿⣷⣾⣿⣷⣾⣿⣿⣿⣿⣿⣧⠦⢚⣛⣅⣾⣿⣿⣿⠀⣿⣿⣤⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⠟⠁⠉⠙⠛⠛⠿⠿⠿⠿⠻⠿⣿⣿⣿⣿⡇⣽⣿⠛⠻⣿⣿⣿⡿⠃⣰⣿⣿⣽⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡛⢿⣯⣤⣀⣀⠀⠈⠳⠶⣦⣤⣤⣀⣀⠀⠀⠉⠛⠿⢳⣝⢿⣷⣄⠈⠉⠉⢀⡜⢛⣱⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡠⣬⣽⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⡀⠈⢻⣷⣿⣯⣗⣀⣀⣀⣵⣾⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠢⠤⠤⠽⢎⣛⠻⠿⢷⣤⣄⣍⡉⠛⠛⠿⠿⢿⣿⣿⣿⣿⣶⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣤⣶⣦⠀⣠⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⢀⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣷⣬⣽⣿⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢀⡾⠏⠁⠈⠙⢿⣿⣟⢿⣶⣶⣶⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⠋⠀⠀⣀⣤⣄⡀⠛⢛⡛⢿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣭⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣯⡻⣶⣿⣒⣚⠙⣿⣟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣞⣯⣿⣿⢻⣿⣦
⠀⠀⠀⠀⠀⢠⣿⡟⠁⣨⢟⡿⣿⣿⣽⣟⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢻⣿⠟⣨⢿⣿⢤⡙⣿⣷
⠀⠀⠀⠀⠀⠸⣿⣇⠀⢷⣛⣻⠛⢙⣿⣿⣿⣿⢗⣏⣙⣿⣿⣿⣿⣿⣿⣿⡀⢺⣿⡀⢿⡈⠃⣸⠇⣿⣿
⠀⠀⠀⠀⠀⠀⠻⣿⣦⣄⣀⣀⣤⣾⡿⠃⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠃⠘⢿⣷⣌⣙⣋⣡⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⢿⣷⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⣒⠠⠖⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢷⡔⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⣷⣄⣀⣀⣤⣤⣶⣶⠿⠿⠛⠛⠛⢿⣿⡆
⠀⠀⣾⡟⠒⣀⣀⣐⣚⣿⠿⠟⠒⠒⢦⣴⣿⣿⠛⠋⠉⠉⠀⠀⠀⠀⠀⡠⣾⠞⢿⠿⣄⠀⠀⠀⠄
⠀⠀⡨⣿⣖⡩⠄⠐⠒⠀⠀⠩⣒⣴⣾⠟⠹⡿⡀⠀⠀⠀⠀⠀⠀⢠⢮⡾⢁⣴⠞⢣⠹⡄⠐⠂
⢠⢎⢞⠔⠙⠄⠀⠀⠀⠀⠀⣀⡾⠛⠉⢂⠀⢻⣇⠀⠀⠀⠀⠀⣔⡵⠋⣰⡿⠑⠁⠀⢠⢳⠀⠀⠀⠀⠀⠀⠊⡂
⣤⣮⣎⣀⡀⠀⠢⡰⠀⣠⢾⠟⠁⠀⠀⠀⠃⠘⣿⡄⠀⠀⡠⣢⠟⠁⢐⡿⣄⣀⠀⠀⠀⢣⣣⡀⠀⠀⠀⠀⠀⠈
⠅⣼⠁⠈⠉⠙⠾⣿⣫⡕⣯⣤⣤⣤⣤⠤⠭⣤⡽⢧⣤⢎⡞⠁⠀⠀⣿⠀⠀⠀⠉⠉⠒⢢⣿⣷⡂⠀⠀⠀⠀⠀⠐
⠀⢿⠀⠀⠀⠀⠰⠙⣿⠿⠛⠓⠛⠛⠩⠿⠿⣟⡀⣹⣏⠔⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠉⠈⠋⠢
⠀⠘⠇⠀⠀⠀⠀⠀⠈⢹⡀⠀⡀⠀⢀⣐⣀⠘⣍⣶⣅⡈⠀⠛⠛⠃⠀⠀⠱⡀
⠀⠀⠀⠰⣀⠀⠀⠀⠀⠈⠉⠋⠉⠭⠉⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠍⡂⠀⠄⠠⠀⠂⠠⠜⠋
⣶⣶⠶⠶⠶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣾⣿⣿⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣶⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣀⡀⠀⠀⠿⣿⣻⠿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣤⣾⣿⣶⣌⣿⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⣷⣶⣶⡶⠂⠉⢩⡟⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢁⣼⣿⣮⣻⠋⠀⠀⣠⣿⣷⠀⠳⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣏⠃⠀⠀⠀⠘⣷⡀⠀⠈⠙⠢⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠈⠻⣦⡀⠀⠀⠘⣿⣿⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣷⠄⠀⠀⠀⠀⠙⢶⣤⣾⡙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⠻⢿⣿⣷⡀⠀⠀⠀⠀⠀⢠⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣴⣽⡿⠃⠀⠀⠀⠀⠀⠀⠘⣷⢶
⠀⠀⠀⠀⢀⣤⣴⠶⠶⠶⠶⣦⣄⡀⠀⢨⣿⡿⠟⠁⠀⠀⠀⠀⢀⠌⠀⠀⢘⣼⣆⣶⠶⣦⣤⣀
⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⣽⣻⣶⡛⠹⡄⠀⣿⠀⠀⠀⡐⠁⠀⢀⣴⠟⢱⣿⡏⠀⠀⠀⠙⠳⣄
⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣷⡀⢷⠀⢿⠇⡠⠊⠀⠀⠀⣾⠃⠀⠀⢻⣻⡄⠀⠀⠀⠀⠹⣧
⠀⣼⡇⠀⠀⠀⠀⢠⣾⣲⡄⠀⠀⢻⣿⣿⡇⢰⣶⠆⠀⣀⡀⠀⠀⠠⣿⠀⠀⠀⠘⣷⣷⡄⠀⠀⠀⠀⢹⡇
⠀⢿⡇⠀⠀⠀⠀⠘⠋⢭⣥⠀⠀⠀⠀⢰⡞⢿⣿⣷⣾⣿⣿⣿⡶⠆⣿⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢸⡇
⠀⠘⣷⡀⠀⠀⠀⠀⠀⢀⣇⠀⠀⢀⣠⡿⠁⠈⠛⠛⠉⠀⠉⠁⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁
⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠉⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣀⠀⠀⠀⠀⢀⣤⡾⠁
⠀⠀⠀⠀⠈⠛⠷⠶⣶⡶⠶⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣶⣦⡀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠾⠿⠛⠉⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠋⠀⠀⣸⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣴⣶⣿⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⢀⣶⣶⣿⣅⣼⡿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠛⠿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⢸⣿⣿⣿⣿⡿⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣌⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣆⠻⣿⠿⠛⠁⠀⣿⣿⡏⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣀⠀⠀⠀⣰⣿⡿⠀⠀⠘⠋⠀⠉⠉⠉⠉⠉⠉⠒⢿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡟⠿⠋⠩⠁⢪⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡾⠿⠿⢿⡶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣤⣤⣴⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠛⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡏⠀⠀⠙⡈⠄⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣷⣎⠀⠀⠀⠁⣰⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⣿⣿⣿⣥⣶⣿⣿⠃⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠿⠿⣿⡿⠟⠀⣷⠀⣴⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠡⠶⣾⡟⢻⠃⣠⡾⠟⣰⡟⢂⢀
⠀⠀⢀⣀⣀⠀⠀⠀⢸⢃⣄⠀⣹⣉⣀⣶⠞⠉⠀⠀⠉⠐⠻⠛⢦⣤⣀⣀⡀
⠐⠛⠛⠛⣻⣿⣄⠀⠀⠈⣩⣤⡿⠏⠋⠀⠀⠀⠀⠀⠘⢡⡖⣀⣠⠠⠀⠠⣀⣲⣿⣶⡆
⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣠⡆⠀⠸⣢⣭⣭⡄⠀⠚⢛⢷⣿⣿⣿⣧⡄⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢉⡽⠃⠀⠀⠈⠉⠋⢀⣶⠀⣾⣿⣿⣿⣿⣷⠐⢁
⠀⠀⠀⢠⠄⠘⠖⠀⠤⠄⠀⠀⠀⢀⣠⣾⣤⣤⣿⡆⠀⠠⢀⡠⠸⢿⣿⣿⣿⣿⣿⣿⣿⣦⣬⠏
⠀⠀⠀⠀⠐⠃⣿⣤⣶⣶⠀⣠⣤⣌⣛⣋⣉⡁⠀⠀⠀⠁⠀⠀⠀⠀⠸⠻⠿⣿⣿⣿⣿⣿⡗⠶⠤⠆
⠀⠀⠀⠀⠀⠀⠙⢿⡋⢥⣶⣿⣿⣯⣽⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠙⠁⠀⢿⣿⣿⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⢿⠿⠿⢿⣿⣿⣿⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⣨⣟⢿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡧⠴⠒⠒⢶⣌⠙⠃⠀⢸⣿⣆⠀⠀⠀⠀⠀⠈⠉⢹⣾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣼⣿⠀⠀⠀⢸⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⣿⡿⠇⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⣼⣿⣽⣿⣿⠏⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⡇⣠⠀⠀⠘⣿⣿⣿⡟⠀⠀⠀⠀⠠⣿⣝⣿⣿⣿⠛⢿⡀⠀⠸⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠸⢹⡇⢻⣀⣠⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠈⠁⠨⣿⣿⡄⠘⢻⡷⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⢿⣧⠘⣯⣷⣖⣄⣿⡏⠀⠀⠀⠀⣠⠞⠀⠀⠀⠛⣛⡀⢠⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣀⣿⡟⣄⠘⢿⠏⢺⣿⠀⠀⠀⠀⠈⠁⠀⠀⠀⣾⣛⠿⣿⣏⠀⠎⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣝⢿⣿⣷⣿⣆⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⣰⣮⣻⠄⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣝⠿⣿⢻⣷⣾⢻⣏⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣴⣿⣿⣿⣷⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣮⣄⡙⠁⠈⠉⠙⠛⠛⠓⠒⠚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡶⠁⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣯⡛⠁⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣷⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣷⣆⣀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣎⠀⠀⢠⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⢤⡧
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠀⠠⢾⣿⣯⡀⠀⢸⣟⡁⠀⡀⠀⠀⠀⠀⠀⢀⣿⠙⠣⡀
⠀⠀⣄⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡧⠀⠐⠀⠀⠙⠙⠛⠀⢠⡌⠓⠀⠀⠀⡀⠀⠀⢠⣼⣿⠄⠐⡋
⠞⠛⠋⠙⢷⡄⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⢸⡀⠒⠂⠀⠀⠀⠀⢀⣴⣿⣿⣤⣾⠇⠀⠀⠀⠁⠀⠀⠿⠿⣷⣄⠀⢼⡧⠀⠀⣤⣿⣿⣦⡄⠡
⠀⠀⠀⠀⠀⠻⣾⡁⠀⠈⠀⣰⠧⠀⠐⠀⠀⠀⢀⡜⠀⠈⢦⠀⠀⠀⠀⠉⠿⣿⣿⣿⣿⣷⣦⠄⠀⠀⠐⠀⠀⠀⠈⠉⢀⣿⠇⠢⠐⣾⣿⣃⡘⠟⠟
⣅⣰⢄⡀⠀⠀⢻⣶⣤⣠⣚⣀⠿⢿⣦⠀⠄⠀⣾⣄⢀⣄⣈⡆⠀⠀⣀⠀⠀⠘⠻⠛⣿⣿⡤⠀⠀⠀⠀⠡⡀⠀⠀⠀⣬⡟⡓⢡⣴⣟⣿⢿⣧⢺⣦
⢿⢿⡿⠧⠘⠂⠀⠉⠙⠗⠉⠈⣢⠀⢰⡄⢀⠈⡃⠉⢻⠟⢻⠟⠒⠀⠉⠡⠀⠀⠀⢀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣾⡿⠤⠤⢹⡏⣴⠆⢁⣆⢡
⣼⣮⡀⠀⠀⠀⠘⠲⠂⡀⠀⢺⣿⠀⠈⠰⠀⠀⠰⡀⠀⢀⡞⠀⠁⠀⠀⠀⠀⠤⠄⠈⠽⠿⣶⣀⣀⠀⠀⠒⠀⣄⣤⣿⢟⡉⠘⠻⣀⣾⣷⣴⣮⣽⡷
⢿⣿⡇⠆⠆⢘⣁⠠⠀⢆⠀⠀⠀⢀⡞⠁⠀⢀⡤⠴⠒⠁⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⢤⣀⡀⠀⣀⣄⢀⠀⠀⠀⠴⣿⣷⣶⣄⠀⠘⠛⠙⠛⠛⠛⠉
⠀⠈⠉⠁⠀⠘⠁⠀⠀⣈⣀⣀⡠⠊⠀⠀⠀⠈⠐⠆⠀⠀⠀⠀⠀⣀⠀⠀⠀⠠⡀⠀⠀⠉⠀⠠⢿⣿⣿⡇⠀⠀⠀⠈⠉⢹⣤⠀⠀⠀⠀⣀⡴⠆⡀
⠸⠗⠛⠉⠀⠀⠙⠠⠀⠠⠜⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠈⢉⡀⠀⠀⠈⠉⠀⠐⣤⠤⠄⠀⠈⠉⠙⠀⠠⢤⣦⣄⠙⠧⠔⠀⠀⣠⣴⣷⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣓⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠷⠇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣶⣾⣷⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠸⣾⣤⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠛⠋⠉⡇⠈⢛⠒⢒⠢⠄⠤⢀⡄⠀⠀⠀⠀⠀⠀⠀⡠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡿⣶⡆⠁⠀⠀⠻⠲⠓⠒⠶⠒⠒⠛⠛⠛⠻⣷⣦⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣁⡿⠿⢦⣀⣰⡖⢾⠑⠓⠞⠦⠀⠀⠀⠀⠀⠙⠛⠋⠛⠻⠶⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡿⣷⣿⣋⡾⠀⢀⠂⠀⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⣏⣽⣿⣿⣯⣿⣿⣿⣿⠃⠀⣾⣀⣀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣿⣿⣿⣿⠯⣤⣶⠇⠀⢸⣿⣿⣿⣿⡏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠽⠿⣿⣿⣿⣿⣿⠏⠀⢠⣿⣿⣿⣿⡟⠛⠻⠿⢿⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⡀⢈⣿⣿⣿⠁⠀⢠⠏⠨⣽⡏⠙⣤⡰⢄⡔⠊⣸⡗⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠀⠄⢂⣴⣿⣿⡏⢿⠁⢀⣿⢁⣤⣼⣷⣴⣿⣣⣤⣙⣷⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⡀⠂⡼⣿⣿⣿⣠⠏⢀⣟⣷⠋⣿⣿⣻⣟⣴⢠⣤⠘⣭⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⡀⠐⣼⠁⣻⠿⣯⡿⠀⣜⣿⣿⣦⣿⣿⣿⡿⡱⠿⠏⠸⣟⣾
⠀⠀⠀⠀⠀⠀⠀⠐⠤⡼⠀⠀⠀⠠⢰⠿⠶⠏⢰⡟⠀⣸⣿⣿⣿⣿⣿⡟⠈⣿⠆⠦⣵⡿⠻⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠈⢄⣾⣄⠀⣴⣿⠇⢰⣿⣿⣿⡿⠿⣿⣟⣃⣻⣄⣰⣿⣇⣴⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⢰⣿⣿⣧⣿⣯⣄⣿⢁⣿⣿⣿⣿⣿⣿⢃⣿⣥⣍⣀⢤⡅
⠀⠀⠀⠀⠀⠀⠀⣀⡌⢀⠀⠄⠀⢼⣿⣿⢀⠙⠱⠿⢿⣾⣿⣧⣿⣿⠿⣿⣽⣼⣯⡽⠏⠘⠃
⠀⣠⣴⣶⣿⣿⣷⣿⡟⠀⠀⡀⢳⣾⣿⣿⣀⣠⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⠿⠿⢿⣿⣀
⠈⢿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⢻⢿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣄⣀⣀⠀⠙⢷⡄
⠀⠀⠙⢿⣿⣿⣿⣿⣿⣷⣤⣄⣻⡈⠿⠋⠚⠁⠀⠈⠉⠁⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠈⠛⠿⢿⣿⣿⣿⣿⣿⡿⠷⠞⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣴⣦⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣤⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠁⠀⠀⢠⠿⠟⠉⠉⠙⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢠⡀⠀⣻⣶⡆⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀
⠀⠀⠀⠀⣤⣾⣧⡄⠈⠉⠁⠀⠀⠀⠀⠚⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃
⠀⠀⠀⠀⠼⠁⠸⠧⠀⠀⠀⠀⠀⠐⠦⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⢀⣾⠿⠿⠿⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠀⣠⡀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⡄⠠⡀⣤⣦⡀⠀⢾⡶⠀⠀⠀⠀⠀⠀⢀⠀⣾⡄⠀⠛⠀⣀⡩⣷⠀⠀⡇
⠀⠀⠀⠀⠠⠤⠀⢠⠍⠻⡿⠟⠀⠀⠀⠇⠀⠀⠀⠀⠀⠈⣻⠋⢻⡃⠀⠛⠿⢷⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠊⠉⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠛⣡⡦⣖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣗⣯⣭⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣇⡴⠪⡀⣨⡶⢋⡵⠊
⠀⠀⠀⠀⠀⣿⠗⣠⢟⡡⢞⠍⢀⠂⠀⠀⢈⠤⠀⢀⣤⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢺⣷⢷⣷
⠀⠀⠀⠀⠀⣿⣾⡫⠋⠈⣀⣤⡶⠞⢁⠆⢀⣤⠔⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢾⣿
⠀⠀⠀⠀⠀⣿⣿⣴⠞⢛⣉⣵⣶⣤⠾⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢸⣿⣺⣿
⠀⠀⠀⠀⠀⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⣿
⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣄⠀⠀⠀⠀⠀⢻⣿⣼⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠃⠀⠀⠀⠀⢹⣿⢺⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣿⣿
⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡏⢣⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠠⣤⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⢀⡤⢠⣶⡐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⠦⢤⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠘⠛⠛⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⣸⣿⡇⠈⣷⠀⠀⠀⠀⠀⠀⢀⣘⣙⣀⠀⠀⠀⠀⠀⠀⡰⢫⡴⠛⠛⠛⠲⣌⢣⡀
⠀⠀⠀⠀⢀⣿⣷⠀⠀⠀⠀⠀⢲⣿⡇⠀⠀⠀⠸⣧⠀⢻⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⣉⣉⡁⠀⠀⠀⠀⠀⢰⢡⡏⠀⠸⣦⡀⠀⠸⡆⣷
⠀⠀⠀⠀⠈⠿⠛⠀⠀⠀⠀⠀⠚⠻⠇⠀⠀⠀⠀⠘⢦⣸⣿⣁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⢸⡘⣇⠀⠀⠙⣇⠀⣰⠇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠿⠷⠀⠀⠀⠀⠀⠀⠳⣜⠳⢤⣤⣤⠶⣫⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠲⠟⠿⠛⠻⠷⠂⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠛⠛⢿⣧⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⢠⣄⣠⣤⣀⣀⡀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⣸⣿⠀⠀⠀⠀⢰⡏⠃⠀⠀⠀⠀⠀⠈⢱⡀⠀⠀⠀⠀⢀⣸⣿⣦⣤⣴⡶⠂
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠏⠀⠀⠀⠀⣿⠶⠴⠤⢤⣤⡄⠴⠤⠾⡇⠀⠈⠛⠿⢿⣯⢯⡷⣿⡋
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠻⣿⣿⠏⠀⠀⠀⠀⠀⠘⢄⣀⣀⠜⠈⢤⣀⣀⠜⠀⠀⠀⠀⠀⢠⣿⡿⠙⠿⣷⡄
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣲⣿⣦⣴⣿⣷⣂⠀⠀⠀⠀⠀⠀⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⣀⠀⣀⣀⣀⠀⠀⠀⠰⠛⣩⡿⠻⡛⢿⣯⡙⠓⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⡀
⠀⠀⠀⠀⢸⣿⢻⡏⢱⣿⠿⣷⡌⢹⣿⠀⠀⠀⠀⠀⣿⠁⠀⢘⠈⢿⠇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡃
⠀⠀⠀⠀⢸⣿⣿⣧⣜⢿⣿⠿⣣⣼⣿⠀⠀⠀⠀⠀⠙⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠋⠀⠙⠉⠉⠙⠀⠀⠀⠀⠀⠀⣠⣤⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣛⣋⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢷⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣶⡄⢿⠦⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢳⡀⠀⠀⠀⠀⠀⠀⣾⣿⠟⠋⠀⠀⠀⢹⣷⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣤⣤⣄⣀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⡜⠀⢸⣧⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⣸⣿⣿⠆⠀⠀⠀⠀⣼⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⡏⢷⣄
⠀⠀⠀⠀⠀⠀⡐⠀⣴⡈⣿⣧⡀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣰⣿⣿⡟⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠘⢿⣷⣤
⠀⠀⠀⠰⠀⠴⠶⠶⠿⠷⠿⠿⠷⠀⠲⠀⠀⠀⠀⠀⠀⣤⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⠹⠿⠛⠻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡇⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⢀⠠⠀⠂⠀⠀⢀⠀⠀⠀⠒⠒⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠉⢁⡕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢠⠔⠀⠀⠀⠛⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⢀⠠⠂⢀⡀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠣⠈⠠⠠⠬⠠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠀⠉⡠⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⡀⢄⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠲⠠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠉⠀⠁⠂⠀⡠⠀⠂⢉⣉⠓⠒⠀⢀
⠀⡀⠀⠀⠀⠀⠚⠀⠀⠀⠰⡚⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⣀⠀⠀⠈⠀⠀⠀⠀⠠⠂⡡⠋⠀⠊⠀⠀⠀⠀⠀⠐⢄⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢰⠃⠀⢀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠉⢢⠀⠀⠀⠆⠀⠀⠜⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠘⡄
⠀⠀⠀⠘⠀⠀⠀⠀⠘⠴⠃⠂⠂⠈⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠂⣀⢀⠀⠀⠀⠢⣄⠀⠀⡆⠀⠀⠀⠀⢀⡘⠀⢂⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠁⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠔⠒⠉⠁⣀⡀⠃⠀⠀⠀⠀⠄⠁⠂⠁⠄⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⣎⡥⠃⠘⠄⠡⡀⠀⠈⠀⠀⠀⠀⠀⡀⠂⡘⠁
⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠁⠀⠔⠀⠀⠀⠁⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢌⠐⠀⠄⠀⠠⠀⢊⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠲⠶⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣰⠂⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠛⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠶⠊⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢦⣴⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠀⠀⠠⠤⠀⢀⣼⠃⠀⠐⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢷⣦⣄⣠⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣄⡀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠁⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⠀⠰⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⠒⠎⠀⠀⡀⠐⠀⣁⠠⠤⠤⢀⡉⠑⢄⡀⠀⢀⠌⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢄⡤⠄⢀⡠⠒⠁⠀⠀⠀⠀⡊⠀⠐⢄⠑⡄⡌⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠠⠆⠁⠀⠀⠀⠀⡄⠂⠁⠀⠀⠀⠀⠀⠀⠀⢰⠄⠀⠈⠆⠘⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⣠⣄⢤⢠⠠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠘⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠀⠀⠉⠀⠀⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠓⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣀⡛⠀⢀⡀⠀⢈⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢀⢀⠠⠀⡏⠀⠈⠁⢀⠒⠀⡏⠀⢀⠁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢻⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠅⠀⠀⠀⠠⠂⠀⣁⠀⠀⢢⡤⠖⠚⠒⠒⠒⠒⠒⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠄⡄⠀⠀⠄⢀⠀⠀⢠⢹⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⢠⠉⡇⠀⠀⠀⠀⠀⠀⡇⢠⡇⠰⠋⠀⠀⠀⢀⣮⠚⡀⠀⠀⣠⠾⡟⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢡⠀⠀⡀⠤⢄⠀⡇⠈⠁⠀⠐⡀⠀⠀⡜⠁⠀⡇⠀⠟⠁⢰⠇⠀⠀⠀⠈⢳⡄⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠘⠀⠰⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⡞⠀⠀⢀⣷⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠘⡀⠀⣧⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⡾⠀⠀⠀⡞⠙⡅⠰⢠⡆⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠙⢄⣸⠀⠀⠀⠀⠀⢀⣇⣀⡴⠀⢰⡇⠀⠀⢰⡁⠀⢻⠐⠃⢇⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡀⠀⢹⡀⠤⠔⠂⠈⠁⠀⠐⡄⠁⢸⠀⠀⠀⣸⠅⢀⠸⡆⠐⠀⡆⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣿⠀⠀⠀⣿⠀⠀⠀⠩⡄⢤⡄⡄⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⡀⠠⣤⠒⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⢏⣀⣀⠜⠀⠀⠀⢸⡀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢸⡇⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠈⠳⣄⡀⠀⠀⠀⣀⡴⠞⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠉⠙⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⢀⣀⣤⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠰⠶⣶⣶⣶⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠷⠶⠶⠶⠶⠶⠶⠶⠶⢶⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣰⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣾⠋⠻⣆⣠⣤⡶⠶⠶⣦⣤⡀
⠀⠀⠀⠀⢀⣴⠟⠋⠁⠀⠀⠀⢩⣿⢷⡌⠻⣦⡀⠀⠀⠀⠀⣸⡏⠀⣰⡿⢿⡄⠀⠀⠀⠀⠉⠻⣦⡀
⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⣠⡿⠃⠈⢿⡆⠘⢿⣄⠀⠀⢠⡿⠀⣸⡟⠀⠀⠹⣦⡀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⢸⡟⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⢸⣷⠀⠀⠙⢷⣽⣿⠃⠀⣿⣇⣀⣠⣤⣼⣿⠆⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⢿⣿⠟⠛⢻⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠹⣷⣀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⣻⡀⠀⠀⠻⣦⣀⠀⠀⠀⠀⠀⢀⣠⡿⠃
⠀⠀⠀⠀⠀⠈⠛⠷⣶⣤⣤⣴⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⣶⡶⠾⠟⠋

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠄⠀⠀⠀⠄
⠀⠀⠀⠀⠈⢣⡄⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⡰⠁⠠
⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⠄⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠈⠀⠀⡂⢀⠢⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠂⠀⠀⠁⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠑⠠⢐⠠⠂⢁⠠⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡤⠀⠠⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠔⡀⠀⠈⠀⠀⠀⡐⠀⠀⢀⠔⠁
⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠂⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠠⠀⠀⡀⠐⠀⠀⢀⠔⠁
⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠌⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠐⠠⠀⠑⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢎⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⢩⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠂⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠈⢒⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠐⠀⠈⠀⡠⠊⠀⠀⠀⠀⠀⠁⠐⠀⢀⠠⠐⠈
⠀⠀⠁⠂⢀⠀⠀⠀⠠⠀⠀⠁⢄⠀⠀⠀⠀⠡⡂⠀⠁⠀⠀⠁⠀⠘⠀⡄⠀⡈⠀⠀⢀⠇⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠘⢆⠠⠄⠀⠄⠠⠠⠄⠁⠠⠀⠀⣠⠃⠀⠀⠀⡠⠈⠀⠀⠀⠀⠂⢀⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⢠⠄⣑⢄⠠⠀⠈⠁⠀⢀⠀⠁⠀⠀⠉⠀⠒⠤⢄⠀⠜⡢⠈⠀⠀⠠⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠐⠀⠀⠤⢀⠀⠀⠂⠀⠀⠀⠀⠂⢄⠈⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡢⡀⠐⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⣀⠀⠤
⠀⠄⠀⠀⠀⠀⠈⠐⠠⠄⣈⠀⠠⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⢀⣀⠀⠀⠀⠐
⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⠂⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡄
⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠠⠠⠤⠤⠠⠠⠤⠄⢄⢤⢠⢀⢠⢠⠠⣠⠀⡄⣤⣦⠧⠤⠤⠤⠤⠤⠤⠸⠠⠠⠤⠤⠤⠦⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⠒⠒⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⣉⣉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠈⠁
⢀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀
⠀⠀⠀⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣀⣀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⢤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢭⣩⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢃⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠠⣤⡏⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢻⡀⠀⠀⣀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠄⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡈⣿⠈⠁⠀⠀⠀⠀⠀⠈⠁⠂⢦
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⡄⠀⢻⡄⠀⠀⠀⠀⠀⠀⢠⡴⠋⠀⠀⠂⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣦
⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢻⡀⠈⢷⡀⢣⠀⠀⣠⡶⠋⠀⠀⢀⠁⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡀⢡⣶⣿⣤⣤⣤⠤⠤⠀⠠⢬⡧⠀⠼⣧⣼⣠⡶⠋⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠘⢷⣄⡀⠁⠀⠀⠀⠀⠀⠀⠀⢩
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠭⡉⡙⠛⠛⠟⠻⠶⡿⠶⣾⣼⣿⠻⡇⠀⠀⠀⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠹⡷⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⣺⠅⠀⠻⣥⣏⡳⠃⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣶⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡶⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣦⣤⣤⣤⣤⣶⣿⣿⣾⣦⡶⠒⢂⠀⡀⠀⡂⠀⢠⡤⢦⣐⠮⠤⠦⣤⡀⢿⣳⣮⣍⣒⣒⣒⣲⣬⣴⣿⣏⡤
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠋⠉⠉⠉⠉⠉⠉⠉⠃⠈⠐⠈⠉⠀⠀⠀⠄⠂⠀⠛⠃⠜⠁⠀⣀⣈⠡⠀⠙⠋⠉⠉⠉⠙⠉⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⡶⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠟⠏⢾⣦⡀⠀⠀⠀⠀⠐⡩⠒⠺⣡⣶⠿⠛⠻⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠂⡀⠀⠀⠙⣿⡀⠀⠀⠀⠀⡀⠠⣾⡟⢁⢂⠀⡀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠷⠇⠉⠁⣿⡇⠀⠀⠀⠀⡿⠷⣿⠶⠶⠚⠿⠗⠒⠂⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⢀⣠⡿⠁⠀⠀⠀⠚⠉⠀⠹⣧⣀⠄⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠶⠶⠶⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢰⢧⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠞⠘⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠾⠛⡿⢶⣍⠒⠄⡑⠢⠀⠀⠀⠀⠀⡞⣹⣴⠾⠛⠷⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⢀⡁⠀⠙⢷⡀⠀⠑⠠⣈⠒⠀⡘⣷⠋⠈⢦⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠷⠀⠀⠀⢸⡇⠀⠀⠀⡈⢻⡿⣿⣟⠉⠉⠈⠿⠀⠀⠀⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠘⠁⠉⠁⢿⡀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣄⣤⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⣤⣄⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣍⠀⠐⠒⠂⠤⢄⣀⡀⠉⢙⠗⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠈⢀⡘⢇⣀⣠⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠛⠉⠉⠉⠛⢷⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠈⣠⡾⢻⠉⠉⠉⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⠀⣠⣤⡄⠀⠀⢻⡄⠀⠀⠀⠀⠣⣀⣀⣤⣰⣏⣀⠀⡀⢤⣄⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠙⠛⠁⠀⠀⣸⡇⠀⠀⠀⠀⠀⢸⣶⡞⢠⣿⠉⠉⠰⠞⠋⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠓⠀⠀⠻⣦⣀⠀⠀⠀⣀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠭⣯⣭⠭⠽⣛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠒⠒⠤⠤⣄⠀⠀⠀⠲⠭⠷⢒⣊⠭⢿⡙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡖⢒⠉⠉⣀⡠⠤⠖⣚⡩⠥⠒⣪⠟⣣⠷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢸⣊⡩⠤⠒⠊⠉⠀⠀⣠⠞⡡⠊⠘⡖⢇
⠀⠀⢀⡤⠒⠊⠉⠉⠑⠒⠤⣀⠔⣡⡇⢸⠀⠀⠀⠀⠀⠀⡠⠞⣡⠞⠀⠀⣠⠼⡘⡏⠉⠑⠒⠤⣀
⠀⡴⠁⡠⠖⡏⠉⡏⢉⢖⠔⣡⠾⡁⡇⢸⠀⠀⠀⠀⣠⠎⣡⠞⠁⠀⣠⠊⢀⢴⣣⢻⡏⠉⡶⢤⡈⠳⡄
⡼⠀⡼⢅⡀⠸⡄⣧⠟⣡⢞⣱⡀⣱⣇⣸⠤⡤⢒⠾⣡⠞⠁⠀⠀⢰⠃⣰⢅⡀⠹⡎⣇⡜⠀⣀⡹⡄⠸
⡇⠠⡧⠤⠬⢷⡟⢻⣞⠛⠋⠉⠉⠀⠀⠀⠀⢧⣻⢳⡇⠀⠀⠀⠀⡎⠀⣇⣠⠬⢽⣾⢙⣮⡭⠤⠤⡇
⢧⠀⢇⣠⠔⢉⠿⡻⡙⠫⢍⣹⠋⢹⠋⠉⠉⠉⠹⠿⠛⠀⠀⠀⠀⢸⠀⢣⡠⠒⢩⠎⡟⡄⠉⠒⢤⠇⢠
⠈⢦⠈⠣⣄⡎⠀⡇⠙⣄⠔⠁⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠳⢤⡃⢠⠇⠸⣀⡠⠊⢀⠎
⠀⠀⠑⠦⣀⡉⠉⠉⢉⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡉⠉⠉⠉⢁⡠⠔⠁

⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⠀⣤⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠐⠺⢿⣷⣿⣞⣋⡀⠀⠀⠀⢰⣿⣿⣿⡇⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣯⡀⠀⠀⢀⣙⣿⣷⣿⣃⣀⠀⠀⣠⣤⣾⣿⣷⣿⠗
⠀⠀⡴⢛⡿⣿⣿⣿⡍⠉⠀⠀⠀⣸⣿⣿⣿⣷⠀⠀⠀⠀⠐⠿⢿⣿⣿⡿⠿⠃⠀⠀⠿⢻⣿⣿⣿⠟⠃⠀⠀⠈⣻⣿⣿⣿⣷
⠀⠀⠀⠈⢀⡟⠯⠃⠛⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠈⠻⡟⠀⠀⠀⠀⠀⠀⠘⠿⠙⠛⠓⠀⠀⠀⠀⠘⠿⣿⠉⠉
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⢸⡏⣿⢹⡏⡇⠀⠀⠀⠀⣠⣤⣄⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣰⣿⡄
⠀⠀⢸⣟⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠀⢠⣿⣿⣿⡇⠀⠀⢀⣴⣶⣿⣿⣶⣤⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⡄⠀⢰⣿⣿⣿⡀
⠀⠀⢸⣯⣿⣽⣯⡇⠀⠀⠀⠸⣿⣿⣿⡄⢸⣿⣿⣿⠃⠀⠀⣾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠸⣿⣿⣿⣿⣿⣿⡿⠀⠀⣼⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣟⡇⠀⠀⠀⠀⢿⣿⣿⠃⢹⣿⣿⠏⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⠃⠀⠀⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⠆⣿⣿⡿⠀⠀⠀⠀⠀⠙⠿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠿⢿⣿⡿⠟⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢀⠀⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⡤⠀⠠⢄⣀⡤⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡇
⠀⠀⣴⣮⣽⣭⣭⣭⣭⣭⣭⣯⣦⡄⠀⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⢃⠀⠀⠘⠙⡀⠀⢀⠀⠀⠀⠀⠀⢯⡆⠀⠀⠘⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠁⢹⠁⠉⠀⠀⠀⠀⠀⣤⣴⣶⣶⣦⣤⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠈⠙⠘⠉⠁⠀⠂⠀⠀⠀⠀⠀⣼⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⢉⠛⣡⠇⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠠⠀⠃⠀⠀⠀⠀⠀⠙⠿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠁⠀⠀⠀⠀⠀⡴⠐⠠⢄⣀⡠⠐⠒⣶⠀⠀⠀⠀⠀⠐⢿⣿⣿⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⣠⣀⣄⣄⡀⠀⠀⠀⠀⠀⠀⠐⢄⡀⠘⠉⠂⢀⡠⠃
⠀⢀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠇
⠀⠈⣿⡟⡿⡿⡄⠀⠀⠀⠀⠀⠀⠘⠻⠷⠶⠶⠶⠟⠃⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⢹⣷⣳⣷⡟⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣶⣧⡀⠀⣴⣶⠀⠀⠀⣠⣾⣿⣦⡀
⠀⠀⣸⠣⣿⣿⣷⠾⣆⠂⠀⠀⠀⠀⠀⠶⠶⠶⠶⠀⠀⠀⠀⠀⢿⣷⣦⣀⣀⣴⣾⣿⠀⠀⠀⣿⣾⣧⣼⣿⣿⠀⠀⠰⣿⣿⣿⣿⡇
⠀⠀⠛⠋⠉⢻⡄⠀⠘⣆⠀⠀⠿⠿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢳⣿⣿⣷⣿⡽⠀⠀⠀⣿⠋⠉⢻⡇
⠀⠀⠀⠀⠀⠀⠹⠂⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠻⠻⠿⢿⠇⠀⠀⠀⢿⣶⣶⣿⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠤⠖⢚⣹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣒⣓⣖⠒⠒⣒⣲⡶⠶⣞⣉⣩⡤⣶⠒⣿⡹⣧⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⠭⢿⢛⣉⡥⠤⢲⡾⣿⣧⢻⣧⣻⣧⠽⠗⣋⣹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡀⠀⠉⠉⠙⢓⣲⡆⠀⠀⠀⠈⠓⠒⣋⣩⠤⠖⢚⣉⣷⠸⣟⣩⣭⡴⣶⡞⢿⡏⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⠉⡏⠉⠀⢀⣀⡤⠤⠖⢛⣉⡥⠴⠒⠋⣩⠞⢉⡷⠻⣿⡸⣿⡘⣿⣘⣿⠼⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡷⠒⠋⣉⣠⠤⠖⠚⠉⠀⠀⠀⣠⠞⢁⡴⠋⠹⡴⢿⠷⣛⣛⠭⠵⠒⠋⠉
⠀⠀⠀⠀⠀⣀⣠⠤⠤⠤⠤⢤⣀⡀⠀⠀⣠⠞⢹⠀⡗⠚⠉⠁⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⠀⢳⠘⣯⠤⠤⠤⣄⣀
⠀⠀⢀⡴⠊⠁⢀⣀⣤⣤⣄⣀⠀⠉⣲⠞⢁⠔⢻⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⡴⠋⠀⠀⢀⡴⠚⠉⣇⢳⣠⣤⣀⣀⠈⠙⠲⣄
⠀⣰⠋⠀⡠⠚⠹⡄⠀⡇⠀⡸⣻⠞⢁⠔⠻⡀⢸⠀⡇⠀⠀⠀⢠⣀⣠⠞⢁⡴⠋⠀⠀⠀⣰⠋⠀⡠⠾⡹⡌⣧⡇⠀⡸⠛⢦⡀⠈⢣⡀
⣰⠃⢀⡾⣄⠀⠀⢳⡀⡇⣼⠟⢁⠔⢙⣆⠀⢹⣸⣀⣷⣀⣀⡤⢽⢲⣭⡷⠋⠀⠀⠀⠀⣰⠃⢀⣞⡁⠀⠙⢷⠸⡇⡼⠁⠀⢀⣹⡆⠀⢳
⡏⠀⢸⣀⣀⠙⡲⣤⡷⠿⣁⣴⠿⠾⠿⠛⠚⠋⠉⠉⠀⠀⣏⢠⣏⡞⡏⠀⠀⠀⠀⠀⠀⡏⠀⣸⠀⠉⠙⠲⣬⡷⠿⡥⠖⠚⠉⠀⢹⠀⠈
⡇⠀⢸⠉⠉⠉⣉⠽⣧⣴⠷⣦⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⣬⡷⠿⠚⠁⠀⠀⠀⠀⠀⠀⡇⠀⢹⠛⠉⢉⡭⢟⣷⣴⠛⠯⣍⡉⠉⢹
⢹⡀⠈⢧⡴⠚⠁⣰⠁⡏⢣⡈⠙⠢⣤⠎⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⣦⠚⠉⢀⠎⢸⠉⢧⠀⠀⠉⢲⠏⠀⣸
⠀⠳⡀⠈⠳⣄⣰⠃⠀⣇⠀⠳⣀⡴⠋⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠈⠣⣤⠏⠀⢸⠀⠘⣆⢀⡴⠋⠀⡰⠃
⠀⠀⠘⠢⣄⠀⠉⠓⠒⠛⠒⠋⠁⢀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠈⠙⠒⠚⠒⠒⠋⠉⢀⡠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢖⠀⠀⠈⠉⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡄⠒⠒⠂⠈⠉⠹⡖⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⣠⠇⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⠀⠀⠀⠀⠀⢠⠧⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡏⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⣼⠤⠤⠒⠒⠊⠁⠀⢷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠤⣀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢷⠀⠀⠀⠀⠀⠀⠀⢸⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠸⡀⠀⢀⣤⢠⡀⠀⣸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⢀⡀⠀⠀⢀⣀⡤⠴⠒⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠙⠒⠛⢻⡇⠘⡏⢁⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣻⢿⠛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⣾⢿⡧⣼⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣀⣀⠀⡔⣱⠃⠸⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠚⠁⡗⠀⠀⢸
⠀⠀⠀⠀⠀⠀⡠⠔⠈⠁⠀⠀⠀⠀⠀⠀⠀⡼⢱⢧⡀⠀⢇⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠋⢉⡠⠔⡾⠃⣠⡟⢿⡡⢽⠦⠤⠤⠤⠤⠄⢀⡀
⠀⠀⠀⢀⠔⠋⠀⢀⡤⠖⠊⠉⠉⠉⠉⠑⡾⢡⠃⠀⠙⢦⠈⠀⢃⠀⠀⠀⠠⠆⠐⠈⢉⣀⠤⠒⠋⠁⣠⠎⢀⡞⠉⣇⠸⡇⣀⣤⣤⣤⣀⣀⠀⠀⠈⠑⢄
⢀⠔⣤⠊⠀⢀⠖⠁⠀⠀⢀⣤⠤⠤⢤⡜⢱⠋⠳⡀⠀⠈⠃⠀⠙⠂⠁⠀⢠⠤⠐⠊⠁⠀⠀⠀⢀⡔⠁⣰⠋⢠⠂⠍⠀⣟⠉⠀⠀⠀⠀⠉⠙⠲⣄⠀⠀⠱⡄
⡞⣸⠇⠀⢠⠋⠀⠀⢀⠔⠁⠀⠀⠀⡼⢠⠛⡄⠀⠙⣄⠀⠀⢳⠀⢰⠂⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⠜⠁⣰⠃⣠⢗⠀⢻⢀⡤⠖⠒⠒⠤⡀⠀⠈⠳⣄⣀⡽
⠳⣿⠀⢠⠇⠀⠀⣰⠃⠀⠀⠀⠀⡸⢡⠃⠀⠘⣆⠀⠘⣆⠀⠀⣇⠸⡀⠀⠀⠀⠀⠀⠀⠀⡴⠁⣠⠋⠀⢠⠃⢰⠃⢸⠀⢸⠏⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠈⣇
⠀⣇⠀⡟⠀⠀⢰⠃⠀⠀⠀⠀⡰⢡⠏⠀⠀⠀⠘⡄⠀⠘⡄⠀⢹⡀⡇⠀⠀⠀⠀⠀⢀⠊⠀⡀⠁⠀⠀⡎⢠⡇⠀⠈⡇⠘⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢸
⠀⠸⢴⡇⠀⠀⡞⠀⠀⠀⠀⣰⢣⠏⠀⠀⢀⣀⡤⠷⠐⠒⠛⠲⢌⣧⣸⣄⣀⡀⠀⡰⠋⢀⠞⠁⠀⠀⢀⠇⣼⠀⠀⢠⠿⡀⢇⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⡇⠀⠀⡇⠀⠀⣠⠴⣧⣯⣤⣴⣾⡁⢨⠆⠀⠀⠀⠀⠀⣼⢻⠁⠀⠈⠻⣾⠁⣠⠏⠀⠀⠀⠀⢸⠀⡏⠀⠀⣾⠀⢳⠘⡄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⡇⠀⠀⡇⠀⢰⡷⣉⢌⡇⠀⡏⠧⣼⠞⠀⠀⠀⣀⣤⣾⣿⠈⢇⠀⠀⠀⢸⡶⠁⠀⠀⠀⠀⠀⠈⣤⠃⠀⠀⡿⠀⠈⢧⠹⣄⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠇⠀⠀⣧⠀⠈⠣⠥⢼⣇⡀⠓⠾⠿⠶⠒⡿⠋⠉⠀⣿⠉⡇⢸⡦⡀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⣿⠀⠀⠀⠻⣽⣸⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠸⡄⠀⠀⠀⠀⠈⠉⠛⠶⣤⣀⢀⡇⠀⠀⢠⢻⠀⠘⢾⣡⠇⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢿⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⢆⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡻⠳⢤⣄⡾⠀⣇⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠘⢆⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⢸⠋⠓⠿⢦⣀⣀⣀⣴⠏⢱⣀⡀⠠⠤⠤⢤⡀⠀⠘⡄⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠈⢣⡀⠀⠈⠲⣄⣀⣀⣀⡤⠞⠁⠀⠀⡰⠃⠀⠀⠀⠀⠀⠉⠉⡴⢆⠈⡇⢳⠀⣀⣀⡤⠇⠀⠀⠹⡄⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠈⠉⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠹⣤⣀⣀⣀⣠⠞⠁⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⠤⠤⠤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⡀⠀⠀⠉⠉⠉⠁⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢤⣀⣀⠀⣀⣠⠔⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⣀⡠⠔⠁⠀⠀⠀⡀⠀⠤⠤⠤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠀⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⠤⠈⠀⠀⠰⠀⢠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣀⠄⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⣊⠐⠂⠀⢀⠀⢀⡤⠗⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠂⠉⣀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠘⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠂⠉⠀⠀⠀⠀⠰⠄⠀⢄⠀⠀⠀⢀⡀⠠⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠉⠁⣢⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠢⠀⠀⠀⠀⠀⠀⡠⢁⡤⠂⡀⠄⠐⠒⠂⠠⢄⠁⠢⡀
⠀⠀⠀⠀⠀⠠⠊⠀⣘⠀⠀⢉⡊⠀⢈⠳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⢦⣄⡀⠀⠀⠀⠀⠀⠀⠊⡠⠊⡠⠈⠀⠀⠀⠀⠀⠀⠀⠈⠢⠘⣆
⠀⠀⠀⠀⡀⠀⠀⠀⠘⣇⢀⡜⠀⠀⢢⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠙⠂⠀⠀⠉⠀⡀⠀⠃⠈⠀⠀⠀⠄⢀⠀⡁⠀⠀⠀⠀⢡⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠪⠁⠁⠁⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠠⠌⠀⠀⢀⣲⠀⠀⢸⠀⡃⠀⠀⠐⠀⡄⠀⢈⠄⠐⠀⠀⠀⠀⠆
⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⢀⠀⠂⠉⠀⠁⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠊⠉⠀⢀⣀⠈⡄⢀⠀⠀⠀⠄⠀⠒⢀⠠⠀⠀⠀⡘⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡀⠀⠀⠀⠀⠀⠀⢀⣸⡀⠀⠀⠀⠀⠄⢼⣩⠚⠀⠐⡀⠣⡀⠀⠈⠁⠀⠀⠀⠀⢀⠔⢀⠎
⠀⠀⠀⠀⠀⠑⠀⠀⠁⠀⠐⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣌⠐⠠⠄⠀⠨⠄⠀⣁⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠲⠴⠶⠚⠉
⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⣶⢶⣶⣶⣶⣶⣶⣶⣶⢶⠶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⣴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠦⣄⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⢦⣄⡀⠀⠀⠀⠀⣻⠀⠀⠀⠀⣸⠇⠀⠀⠀⢠⡴⠚⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠠⠔⠃⠀⠀⢀⠴⠋⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⡆⠀⠀⠀⠉⠉⠛⠛⠛⠶⠀⠀⠈⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠶⠚⠛⠛⠛⠳⠀⢀⡤⠶⠛⠋⠉⠉⠉⠉⠙⠛⠷⣦⣀⠀⠘⠉⠀⢀⡴⠋⠀⠙⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠀⠀⢠⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠀⠰⠋⢀⣠⠤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⠀⠀⣠⠞⠁⠀⠰⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⡴⠋⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠀⠀⠁⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠛⠒⠂
⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠒⠒⠒⠒⠒⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠓⠒⠛⠒⠒⠒⠲⠶⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⠚⠛⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣀⣀⣤⣤⠤⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⢛⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠞⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠶⠦⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⠶⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢢


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⢿⣿⣦
⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣦⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣦⣤⣤⣄⣀⣠⣶⣾⣧⡀⢙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣅⣸⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣯⡟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⢀⣼⣿⡿⠻⡉⢹⢉⣽⡻⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⣼⣿⡿⠑⣴⣟⣿⣝⢿⣿⣿⣿⣿⣻⣿⣿⣿⣽⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢼⣿⡿⢿⡿⢿⣿⡿⣦⠽⣿⣷⡀
⠀⠀⠀⠀⣿⣿⣟⠁⣻⡼⣿⡿⣸⡏⠙⣿⣿⣯⣿⣽⣿⣾⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⡇⢿⡀⣿⣯⠇⢸⡇⣿⣿⡇
⠀⠀⠀⠀⠸⣿⣿⣏⢀⠟⢻⠛⢏⢈⣿⣿⡿⠐⠾⠿⠿⠿⠟⠿⠿⠿⠟⠛⠿⠿⠿⠿⠿⠸⣿⣷⡞⢷⣮⣯⣶⠟⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣾⣶⣾⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣦⣧⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠐⢲⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣤⣤⣤⣀⠀⢀⣉⠙⢻⡶⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⡿⣇⠁⣼⣷⣾⣶⡄
⠀⠀⠀⠀⠀⠛⠉⠀⠈⠙⠻⠿⡿⣿⣿⢿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠟⠛
⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣶⣭⡉⠙⢿⣦⡄⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣀
⠀⠀⠀⠀⢠⣿⣿⠋⢉⣏⡉⠻⣿⣎⠻⣿⣷⣸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⢿⣷⣄
⠀⠀⠀⠀⣾⣿⠉⢻⣿⣿⣷⣾⣿⣿⣿⣿⣿⡟⠿⠾⣿⣿⣿⣿⣿⣿⣿⡇⣿⡟⣾⣿⣿⣿⣷⢻⣿
⠀⠀⠀⠀⢹⣿⣦⡼⠛⠛⢯⣉⣿⡿⠒⠋⠉⠛⠶⣤⣴⣿⣿⣿⣿⣿⢻⠃⢿⣷⣻⣿⣿⣿⣟⣾⣿
⠀⠀⠀⠀⠀⠙⢿⣷⣶⣶⣾⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠻⣿⣶⣭⣼⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⣿⣶⣦⡀
⠀⠀⠀⠀⠀⣤⡤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠻⢿⣿⣿⣿⣿⣮⣽⣿⡂
⠀⠀⠀⢠⣤⣽⣿⣿⣿⣿⣽⣦⣀⠀⠀⣠⣚⣿⣿⣿⣾⣿⡿⢿⣶⢶⣿⣻⡿⢿⣿⣿⣷⣿⣦⡠⡄
⠀⠀⠀⠘⢿⡀⢻⣿⣻⣿⣿⣿⣿⣿⣶⣿⢿⣿⣿⣿⣿⣿⣿⣿⠋⣻⠿⢿⣇⠀⣈⣙⡻⢿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⠳⣦⣝⢿⣿⣿⣿⣿⣿⢟⣁⣿⣿⢿⡋⠴⣞⣽⣶⣿⠿⣇⣀⣀⡁⠻⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠈⣻⣷⣜⢿⣿⣍⢿⣿⡿⢿⣧⣆⣀⣞⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣾⣿⣻⣿⣯⣽⠇
⠀⠀⠀⠀⠀⣀⣭⣍⠙⢻⣦⠙⠫⣾⣿⣷⣿⣿⣿⣿⣿⣿⡿⢛⣛⣻⣿⣿⣿⢿⣿⣷⣿⣿⡿⠋
⠀⠀⠀⠀⢠⣿⣿⣿⢧⣿⣿⣷⣶⣿⢛⣉⣻⢻⣿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣯⣤⣄
⠀⠀⠀⠀⢸⣿⣟⣾⣾⠿⢿⣿⣿⣿⣿⣿⣿⡆⢻⣷⣦⣌⡙⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠗⠣⡀
⠀⠀⠀⠀⠘⣿⣧⣿⣿⡿⣿⣿⡻⣿⣿⣿⣿⣷⡈⠉⠀⠙⠻⢿⣶⡞⣿⣿⣿⣿⢿⣿⣿⣿⣿⣷⣤⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢻⣷⣽⣟⣾⣿⣷⣌⣿⣿⣻⣿⣿⣄⣠⣤⣶⣷⣿⠏⣿⣿⣿⣿⡌⢛⣿⡏⠛⢿⣟⣷⡄⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠨⣙⣿⣿⣿⠟⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣟⣧⣴⣿⣿⣿⢽⣿⠹⣿⣿⣷⡀⢻⠟⣿⡄⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠿⢿⣿⣿⣿⣽⣿⣿⣼⣿⣿⣿⣿⣿⡿⠈⣛⣿⡧⣄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⢿⣿⣦⠛⢻⡿⠿⠛⠰⢿⣹⢧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⢿⣶⣤⣤⣤⣤⣾⢏⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢭⣙⣛⣛⣩⣵⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠤⢒⣊⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠭⢿⣭⣭⠭⠟⣛⣯⡥⣴⠲⣏⢻⣎⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠒⠦⠤⠤⣀⣀⠀⠀⠀⢰⣊⡽⠤⢒⣊⡭⢿⠻⡷⠽⠷⣛⣫⣭⣶⣾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⢤⠴⡖⠒⠉⠀⢀⣀⠤⠔⢒⣋⡭⠴⢒⠟⣦⢷⣏⢿⡌⣷⡸⣧⣻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠖⢒⣉⡥⠴⠒⠋⠉⠀⣠⠖⢁⠔⠹⡤⣿⠬⣗⣚⠯⠝⠒⠉
⠀⠀⠀⠀⣀⡤⠤⠴⠤⠤⢄⡀⠀⢀⠔⢹⠀⡗⠋⠉⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⠀⣱⠸⡭⠤⠤⠤⣀⡀
⠀⢀⡴⠋⢀⣠⠤⢶⠦⠤⣄⢈⠗⢁⠔⢹⠀⡇⠀⠀⠀⠀⠀⣠⠞⢁⠔⠁⠀⢀⡴⠋⢈⣇⢳⢤⠤⢤⣀⠉⠲⣄
⢠⠎⢀⡔⠁⠘⡆⢸⠀⣼⡵⢁⢔⠉⢣⢸⠀⡇⠀⠀⢰⣶⢾⣁⠔⠁⠀⠀⢠⠋⢀⠔⠙⢾⡜⣿⢀⡜⠈⠳⣄⠈⢆
⡎⠀⡞⠙⠢⢄⣸⣼⡾⢃⣴⣥⣾⠧⠤⠿⠒⠓⢲⠏⡹⡹⡟⠁⠀⠀⠀⠀⡇⠀⡞⠒⠤⣌⣳⣹⣞⣠⠤⠖⠛⡄⠘
⡇⠀⡟⠒⠒⢒⡿⣄⣽⣭⢥⣀⣀⣀⣀⣀⣀⣀⣈⣳⡽⠵⠃⠀⠀⠀⠀⠀⡇⠐⡷⠒⠒⣲⢿⣤⡽⢶⡒⠒⠒⡇
⢳⠀⠱⡤⠖⠁⡼⢹⠱⡌⠑⠦⣴⠃⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⢣⠖⠉⢠⠋⡇⢳⠀⠈⠑⡶⠁⢰
⠀⠳⡀⠙⠦⣴⡁⢸⡄⢘⣤⠖⠁⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠑⢴⣃⠀⡇⠀⣧⡠⠞⠁⡠⠃
⠀⠀⠈⠲⢤⣀⡉⠉⠉⢁⣀⠤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⡈⠉⠉⠉⢁⣀⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣀⠀⠀⠀⠀⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣭⣍⣹⣷⡀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⢻⣿⡛⠷⣤⡀⢀⣤⣄⣘⣆⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⠀⠀⣠⣀⡀⠀⠀⠀⠀⣀⠄⠀⠐⠒⠠⠤⣠⣤⣾⣿⣿⣷⠀⠀⠈⠀⠙⢿⣿⣿⣧⣄⣤⡿⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⡀⣶⣿⣿⣿⣦⡀⠀⣸⡇⣀⠗⢴⣶⣶⣿⡿⣿⠿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠹⣿⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣛⠿⣿⣵⣤⠁⢨⣿⣄⠀⡉⢁⠀⠤⣴⣲⣊⣉⠀⠀⣠⣶⣤⡀⠀⠀⠸⣿⣿⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣷⣄⠀⢀⣾⣿⡿⡇⠈⠀⠀⠈⠳⢤⣤⡀⠹⣿⠻⣿⣯⣿⣷⣦⣄⠸⣿⡀⠀⡁
⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣏⡐⣫⣷⣶⡅⠂⠠⣤⣾⣿⡿⠶⣤⣍⡻⣿⣿⣿⣿⡄⢻⢠⣾⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⢟⣫⣵⣶⡾⠋⠀⠀⠀⠈⠿⠋⠀⠀⢀⣭⣿⣿⣿⣿⣷⣶⣿⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠤⠉⠀⠀⠀⠃⢼⣿⡞⢯⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿⠿⠿⠿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠃⠀⠀⢴⣶⣴⣤⡴⠛⣧⠀⠙⠣⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣟⡟⠋⠉⠁⣠⣶⠛⠛⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠟⣾⣶⡀⢛⣿⣿⡧⣿⣿⣶⣿⠄⡀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣾⣿⣿⢧⣬⡿⣿⣀⣿⣿⣿⣟⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⢻⣿⣿⠷⡄⠀⣾⣆⠙⢿⣿⣧⠀⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣷⣿⣿⣿⡶⠿⠿⡛⢿⣛⠉⠀⠀⠀⢤⡀⠀⠀⠀⢸⠋⠿⣿⠃⠀⢻⣿⣾⣧⣼⣿⣿⡆⠀⠘⣿⡆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣁⣀⠀⠍⠠⠈⠛⠷⣶⣦⣤⣿⣆⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⣹⣿⣿⣿⡇⠀⠀⠟⡇⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣟⡁⣰⣿⣧⣀⣤⠀⣰⣦⡴⠈⠙⠻⣿⡿⠂⠀⠀⠀⠀⠀⢸⡏⢹⢿⣿⣿⣿⣿⣿⠇⠀⣼⣾⡧⠐⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠭⣩⣽⣿⣿⣿⣿⠟⠁⠀⠈⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢸⡷⡄⠈⠛⣿⣿⡿⠟⠀⢈⣽⡿⢀⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿⣛⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣘⣦⢀⣀⠀⠠⠶⣴⡿⣿⣷⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣈⠻⢿⣿⣤⣤⣴⠖⣲⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠿⢿⣿⣶⣶⣶⣶⣤⣀⣀⣀⣀⣀⡙⠳⣶⣬⣭⣥⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡇⠀⠀⢠⡄⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴
⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠘⣆⠀⠀⠀⢸⡇⠀⠀⡿⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠀⠀⠀⠐⢆⠀⠈⠳⡀⠀⠀⠘⣆⠀⡄⢸⡇⡆⢰⠇⡤⠀⣠⠋⠀⠀⢀⡴⠆⠀⠀⢀⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⠀⠀⠀⠑⢤⡀⠙⢄⠘⣆⠘⡄⡇⢸⠀⠃⡏⡼⢁⡴⢁⡔⢀⡴⠋⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄⠀⢠⡙⢮⠢⠣⡈⠂⠁⠀⢸⠀⠀⠐⢡⠎⡠⢋⠔⠋⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⣀⠀⠀⢉⠲⢌⠢⠀⠀⢈⠀⠂⣀⣈⣈⠀⠐⢅⠈⠀⢁⡤⠞⢁⡠⠀⠀⣀⣠⠴⠖
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠈⠉⠙⠲⠤⣉⠂⠍⢢⠔⢁⠔⠉⠀⠀⠀⠈⠒⢄⠡⡒⠁⠐⢊⠡⠤⠒⠉⠁⠀⠀⠀⠀⠀⢀⣀⣄⣀⡀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠓⠒⠶⠦⠬⠥⣀⣀⡀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠈⡆⠠⠀⢈⣀⠀⠤⠴⠖⣒⠿⠥⠖⠚⠓⠋⠀⠀⠉⠉⠉⠁⠀⠈⠉⠁
⠀⣀⣀⣀⣀⣄⣤⠴⠒⠒⠒⠒⠒⠶⢤⡀⠀⠀⠒⠀⠀⠀⠉⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣀⣀⡒⢀⣠⠼⠃
⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⡀⠀⠀⠈⠀⢣⣀⣤⣤⣄⣀⡤⠴⠚⠉⠉⠉⠉⠉⢠⡞⠁⣀⠴⠀⠐⠂⠖⠂⠀⠀⠀⠀⠀⢠⡄⠠⠄
⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠙⠚⠫⣉⢀⣀⠀⠀⠀⠀⠀⠀⡀⢀⣠⠤⠴⠚⢉⠰⠲⠁⢀⣀⠀⢀⡤⠴⠚⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠃⠒⠂⠠⠤⠄⢈⡓⠁⠐⠀⠓⡻⣲⣭⣀⠉⠉⠉⠉⠉⢩⠟⢁⠀⠀⠀⠀⠉⠀⠀⠈⠓⡂⢮⠖⠲⣤⠤⢤
⠀⠀⠐⠒⠀⠛⠃⠈⠃⠀⠐⠂⠈⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⡟⠉⡴⡀⢰⣐⣋⡂⠆⠀⠀⠐⢉⠀⠜⠅⠀⠐⠳⠶⠎⠡⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠚⣁⣀⣜⣫⡶⠛⠉⠙⠋⠁⠉⠉⠳⠶⠶⠶⠦⠖⠛⠒⠲⠦⠶⠶⠶⠶⠧⠿⢦
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⢀⡠⠤⠴⠶⠶⠚⠭⠍⠭⠽⡟⠉⠉⢰⡄⠒⠐⠒⠒⠋⠉⠒⠒⠊⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠰⠾⠋⠉⠉⠁⠀⠀⠀⠀⠉⠹⠦⢤⠀⠀⠀⠀⠐⠶⠞⠻⠭⠥⠥⠄⠠⠀⠀⠀⠀⠀⠀⠘⠛⠓⠒⠒⠒⠛⠋⠀⠀⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⣷⣦⣄⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠒⠬⠿⣷⣄⣈⡂⢄⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⡶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣀⣠⣿⣿⣿⣷⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣿⡏⠉⢡⣾⣿⣿⣿⣟⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣶⣶⣒⢿⣿⣿⣿⣿⣿⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣫⣾⣿⣿⣿⣿⣿⡼⡏⠁
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠼⠶⠿⠶⠤⣤⣼⣿⣿⣿⠟⢿⣍⣿⣿⣿⣿⣿⣿⣿⡿⠷
⠀⣀⠤⠀⠘⠛⡻⢫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣾⣿⣿⣿⣿⣿⣿⣿⠿⣳⣾⣧⣬⣽⣦⣤⣶⣤⣄⡀
⣀⣿⣆⢆⢰⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠙⣿⣿⣿⣿⣛⣽⣾⣿⣿⡯⣿⡇⣿⣿⣿⣿⣿⣿⣿⡇
⢹⣿⣿⢰⣾⣿⣿⣿⣿⣿⣿⣷⣤⣤⣭⡙⣿⡟⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠟⠛⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⡟⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⡇⠉⠉⠛⢿⣿⣿⣿⣿⣷⡄
⠀⠘⢳⣝⢾⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣧⣿⣿⣇⣀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡄
⠀⠀⣼⣿⣷⣿⣿⣟⣛⣛⣒⣂⡤⠖⠉⣠⣾⣿⣿⣿⣿⣿⣿⣿⡿⠁⡿⠟⢻⡉⠙⣧⣠⣯⠀⠈⢳⠀⠀⠀⠀⢹⣿⣿⣿⣿⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣶⣶⡾⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠑⠀⠻⣿⡿⠀⣀⠀⠀⠀⡶⣶⣶⣿⣿⣿⣿⡇
⠀⠀⢹⣝⣭⣿⣿⣿⣿⡟⠀⠀⠀⠉⠙⠛⠛⠿⠿⡿⢿⣿⣿⣿⡇⠀⠀⣀⠀⠑⠶⠶⠀⢀⡴⠃⠀⢀⣼⣤⣄⣀⣿⣿⣿⣿⡇
⠀⠀⠀⠙⠻⠿⠿⢿⣭⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣤⣾⠟⢳⣤⣀⠀⠀⠀⣀⣠⣴⣿⠟⠛⠛⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣇⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣖⣠⣾⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣖⣒⣒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⡾⠶⠅

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣤⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⣲⣯⡿⢿⠁⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣷⣾⣧⡀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⢰⠷⠰⡦⠀⠀⠀⠠⠤⠤⠤⢄⣀
⠀⠀⠀⠀⠀⢀⡀⠠⠀⠀⠀⠀⢦⣤⣤⣤⣴⣶⣶⣶⠾⢶⣾⠟⠋⠉⣁⠞⡛⢧⠀⠈⠙⠻⣇⡟⠀⠀⠀⣀⡀⣀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⣠⠞⢁⡤⣖⣶⣶⣶⡶⠐⣂⣿⣿⣿⣿⣿⣿⣿⣿⣁⢶⣾⣿⣿⣌⣛⣰⣷⣬⣽⡖⢹⠇⡀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⣼⣻⣾⣵⣿⣿⠿⠿⡿⢑⠾⠿⠿⣿⣿⣿⣿⣿⣿⣿⣟⢀⠻⣿⣿⡿⢸⣿⣿⣿⠿⡀⠻⠀⠀⢀⣴⣿⣿⢿⣏⠉⠉⠉⢻⣿⣿⣿⣷⡀
⠀⠀⣼⣿⣿⣿⢫⠞⠉⠉⠙⢧⢲⣶⣶⣶⡤⣤⣬⣽⣿⣿⣿⣆⠢⣠⣴⠚⠿⣷⣄⣶⣶⡇⠀⠀⢀⣾⣿⣿⠃⠀⢹⣷⡶⢤⠞⠙⣿⣿⣿⣷
⠀⠀⣿⣿⣿⡇⡇⠀⠰⡶⠀⠀⣃⣛⣛⣛⣓⣛⣛⡋⢹⣿⣿⡿⠀⣽⣿⣄⣀⣾⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⠛⠛⢻⠙⢷⣴⠃⠀⣸⣿⣿⣿
⠀⠀⢻⣿⣿⣷⡝⢤⣀⣀⣠⠜⣹⣧⣤⠤⠤⠤⠤⠤⠤⠤⡤⠶⠮⠿⠿⠿⠿⠿⠿⠿⠟⢠⠀⠀⠈⣿⣿⣿⣆⠀⠀⠁⠘⣇⠀⢠⣿⣿⣿⡏
⠀⠀⠈⠻⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠈⠁⠀⠀⠀⠘⢿⣿⣿⣿⣶⣤⣤⣼⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠈⠻⠿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠦⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠒⠉⠉⢒⡄⢉⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠒⠒⠒⠒⠒⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⡴⠋⠑⢦⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢠⡇⡜⠀⠀⠀⠀⢧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢲⠦⡴⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⡀⠀⠀⠀⢇⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⡞⣠⡐⢖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢢⠘⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⢉⣉⣡⣤⣍⣉⡙⠚⠴⣃⠱⡌⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠛⣉⣉⣥⣬⣍⣉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⣡⠔⠋⠁⠀⠀⡇⠀⠀⡝⢱⠦⡈⠳⣜⢆⠱⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⡠⠖⠹⡈⢇⠀⡇⠀⠀⠉⠒⢤⡉⠢⡀
⠀⠀⠀⠀⠀⠀⡜⢁⠞⠱⣄⠀⠀⠀⠀⡇⠀⡼⢠⠇⡠⠊⢢⡈⢦⠳⡘⢦⠀⠀⠀⠀⢠⠎⣠⠊⠳⣄⠀⢳⠘⡄⡇⠀⠀⠀⢀⡴⠛⢆⠙⡄
⠀⠀⠀⠀⠀⡼⢀⠎⠀⠀⠈⠳⣄⠀⠀⡇⣰⢁⡧⠊⠀⠀⠀⢳⠈⢇⠙⣄⠳⡀⠀⢀⠏⢰⠁⠀⠀⠈⠳⣌⡆⢹⡇⠀⢀⡴⠋⠀⠀⠈⢆⠘⡄
⠀⠀⠀⠀⢀⠇⣸⠀⠀⠀⠀⠀⠈⠳⣄⣿⣣⣞⣀⣀⣀⣀⣀⣀⣇⣸⣀⣈⣦⢙⠴⠚⢢⡇⠀⠀⠀⠀⠀⠈⠻⣄⣇⡴⠋⠀⠀⠀⠀⠀⢸⡀⢇
⠀⠀⠀⠀⠸⡀⢿⠒⠒⠒⠒⠒⠒⠒⡺⡿⣷⠶⠶⠶⠶⠶⠶⠶⡧⢼⠦⠤⣴⠃⠀⣠⠞⡗⠒⠒⠒⠒⠒⠒⢒⡿⡿⣖⠒⠒⠒⠒⠒⠒⢺⠇⡸
⠀⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⡠⠊⠀⡇⠈⠳⣄⠀⠀⠀⠀⢠⠃⡸⠀⠀⠙⠖⠚⠹⡀⢣⠀⠀⠀⠀⢀⡴⠋⠀⡇⠈⠳⣄⠀⠀⠀⠀⡸⢀⡇
⠀⠀⠀⠀⠀⠘⡄⠱⡀⠀⡠⠊⠀⠀⠀⡇⠀⠀⠈⠳⣄⠀⢠⠏⣰⠃⠀⠀⠀⠀⠀⠀⢳⡈⢧⠀⢀⡴⠋⠀⠀⠀⡇⠀⠀⠈⠣⡀⠀⡰⠁⡜
⠀⠀⠀⠀⠀⠀⠘⢦⡙⢮⣀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣈⠗⢁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠑⢏⡀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣨⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣈⠑⠲⠤⠤⠯⠤⠤⠖⢋⣡⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣉⠒⠢⠤⠤⠧⠤⠴⠒⢉⡠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠲⠶⠖⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠶⠶⠒⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣰⢶⣶⣶⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⣀⣤⣶⣼⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣬⣿⣦⣀⣄⠀⠀⠀⠀⠀⢀⣦⣴⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣴⠏⠀⠙⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠰⣿⣿⡟⠋⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟⠛⠛⠉⠋⠀⠀⠀⠀⠀⠸⢿⣿⣿⡟⠛⡗⠀⠀⠀⠀⠀⠀⠳⠾⢻⣿⣧⠀⠀⠿⣿⣿⣿⣇
⠀⠀⣴⠛⠛⠻⡍⢻⡗⣰⠞⠛⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⡆⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠻⣶⠀⠈⢹⣿⣏
⠀⠀⣇⠀⠀⢀⡿⣾⠃⢿⡀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⢸⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠻⠃⠀⠀⠉⠋
⠀⠀⠈⠛⠚⠋⠀⠉⠀⠈⠙⠚⠛⠁⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠛⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛


⠀⠀⠀⠘⢷⣄⣻⣇⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣾⣿⣷⣼⣤⠀⠀⠀⠀⠀⠀⠀⢻⣧⡈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣭⣿⣯⣽⣽⣯⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⣿⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠠⣄⠿⣠⡀
⠀⠀⠀⠀⠀⣿⣀⣸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠉⢧⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠟⠿⣿⣆⠀⠀⠀⠀⠈⠳⣍⠻⣷⣤⡶⠷⣦⣄
⠀⠀⠀⣤⣴⣾⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠐⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⠚⠻⠿⠿⢿⣿⠇⠀⠀⠙⠛⠂⠀⠀⠀⠀⠀⠈⠳⠾⠷⠶⠶⠶⢶⠶


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢬⣿⣿⡥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⣠⡀⠀⠀⠩⠟⡛⠍⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢩⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣤⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⠀⡀⢸
⠀⠀⠳⣝⢳⣦⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⢂⣿⣿⡀⠀⠀⠀⠀⠀⢰⣿⣷⣠⣾⣿⠟⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⡇⢏⡸
⠀⠀⠀⠈⠳⣝⠻⣷⠾⠛⠻⠷⠆⠀⠀⠀⠀⢴⣬⣭⣭⣤⣭⣭⣥⣴⠀⠀⠀⠀⣻⣿⣎⣿⡿⠋⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢰⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠈⠛⠻⠛⠛⠛⠛⠟⠃⠀⠀⠀⠴⠿⠿⢿⣿⣿⣿⣿⠇⠀⠀⠀⠈⠉⠻⠷⠿⠶⠶⠶⠿⠿⠿⠿⠷⠄⠀⠀⠀⠈⠻⠋⠀⠈⠉⠙⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠛⠻⢿⡿⠿⠟⢿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶
⠀⠀⠀⠀⠀⠀⠠⠾⢛⣻⣦⣴⣿⠆⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣼⣧⣤⣤⣼⣿⣿⡎⠉⠉⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣇
⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⠷⠶⠤⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣄⣀⣀⣠⣿⣿⢉⣛⣛⢁⣀⡄
⠀⠀⠀⠲⣤⣶⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⢿⣟⣻⠿⣟⣛⡿⠿⠿⠿⠿⠿⢿⣟⣻⡿⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⣿⡿⠿⠿⠿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋

⠀⠀⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣶⣾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⡇
⠀⠀⣿⡇⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⣿⡇⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⣿
⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⢸⣿
⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠰⠿⠿⠇⠀⠿⠿⠇⠀⠿⠿⠿⠿⠿⠿⠗⠀⢸⣿
⠀⣾⡇⠀⢀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⣿⡇⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⠀⠀⠀⣿⡆
⠀⣿⡇⠀⠀⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠀⠀⣿⡇⠀⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⡀⠀⣿⡇
⠀⣿⡇⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⣿⡇⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⢿⡇
⢰⣿⠀⠀⣶⣶⣶⣶⣶⡆⠀⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⣿⡇⠀⢰⣶⣶⣶⣶⣶⣶⠆⢰⣶⣶⠆⢰⣶⣶⡆⠀⢸⣷
⢸⣿⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⣿⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣾⡇⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⣿
⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿
⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠛⠛⢻⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡟⠛⢻⣟⠛⠛⠛⠛
⠀⠀⢸⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⣿⡇⠀⣼⡏
⠀⠀⣼⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣧⣾⣿⣥⣤⣤⣤⠄
⠀⠀⢩⣿⠟⠉⣿⡏⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣽⡟
⠀⠀⣾⡏⠀⠀⣿⡇⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡇
⠀⠀⢻⣧⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠻⣷⣦⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣞⠛⠛⣿⣤⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣷⣦⣿⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠞⠛⠛⠋⠙⠿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣷⣌
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣝⣻⣿⣿⣟⣿⣾⣿⡿⢿⠿⣿⣿⣿⣿⣿⣿⣿⣎⠻⠛⠃
⠀⠀⠀⠀⠀⠘⠁⠀⠀⣀⣤⣤⣤⣤⣤⣀⠀⠈⠻⣯⣿⣿⣿⣟⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣶
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠛⠛⠛⠿⢿⣷⣤⣞⡥⢿⣽⣿⣿⣿⣿⣿⢟⣥⣾⣿⣿⠿⣼⣿⡿⠁⢀⣤⡽⣧⢿⣿⢻⣿⣦⣀
⠀⠀⠀⠀⠀⠀⣾⣿⠋⠀⠀⣀⣤⣀⠀⠀⠙⢿⣣⣤⣾⣿⣿⣮⣿⣿⣻⣾⣿⣿⣿⢫⣾⡟⡭⠀⢠⣿⣟⢿⡽⣞⣿⡇⠈⠙⢿⣷⡀
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⣞⣉⣠⣭⣧⣶⣿⣿⣿⣿⠿⠻⣿⣿⣿⡇⣿⣽⣿⣿⣿⣷⠟⣸⠃⠀⣻⣽⠁⠀⢿⣽⣜⠗⢷⡀⠀⢿⣷
⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢯⣛⣋⣩⡭⠉⠉⣰⣾⣿⣿⣿⣿⣿⡿⢧⣬⣽⣿⣛⠉⢉⡴⠃⠀⠀⢿⣿⠀⠀⣧⠁⠛⠀⣸⠇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣄⡀⠀⠉⠉⠉⠀⢀⣰⣿⡟⠈⠙⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠘⣿⣆⠀⠈⠻⠶⠞⠋⠀⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣴⣤⣤⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣀⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢰⠚⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⢲
⠀⠀⠀⠀⠀⠀⠀⠈⣟⠙⣆⠀⠀⠀⢠⠴⠲⢤⡀⠀⠀⠀⠀⣰⠋⢹⠏⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠘⣆⠀⠀⣿⠞⠉⠉⡇⠀⢀⣠⠞⠁⢠⠏⠀⠀⠀⣴⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠈⠙⠲⢼⣦⡤⣾⡥⠞⠉⠀⣀⡴⠋⠀⠀⠀⣠⣿⢾⡽⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⠀⠈⢻⡅⠀⢰⡖⠋⠁⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⢠⣖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠈⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⡄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⢻⠀⢀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠓⠒⠒⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⡗⠒⠛⠛⠉⢷⠀⠀⠀⠀⠀⠀⣉⠀⢠⠏⠉⠳⡄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠁⠀⠀⠀⣧⠀⠀⣠⠀⠈⣧⠀⠀⠀⠀⠀⠻⠀⠘⠦⣤⠼⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠏⠀⠀⠀⠀⡿⠀⠀⡿⡄⠀⠘⡆⠀⠀⣀⡀⣾⠀⣠⠖⠒⠶⣄⠀⡇⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠇⠹⡄⠀⢹⡄⠠⠤⠅⣿⢰⡇⢰⣲⡆⢸⡆⡇⠨⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣼⠀⠀⠹⡄⠀⢷⠀⠖⠃⣿⠈⠳⣄⣉⣠⠞⠀⡇⠐⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠶⠆⢀⡟⠀⣰⠃⠰⠶⠀⢳⠀⢸⡄⠀⠰⠿⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⣰⠃⠀⠀⠀⠀⠘⡇⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠠⣤⣼⣧⣴⣧⣤⣤⣤⠀⠀⠰⠿⠶⠿⠶⠦

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣻⣿⣷⣦⣄⣀⡀
⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⡿⣶⣶⣿⣿⣿⠿⠿⠿⠿⣯⣍⠫⠭⠍⠹⠯⠽⠿⠿⠿⠿⠛⠟⠶⠶⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢢⣿⣿⣿⣦⡠⠤⠒⠀⠀⢀⣀⣀
⠰⠶⠴⠤⠤⠤⠤⠤⠤⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠨⣿⢿⣿⣶⡖⠒⠒⠚⠂⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢐⣀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠺⢿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠴⠂⠀⠀⠉⠁⠀⠀⠨⠥⠀⠀⢀⢀⣈⣉
⠐⠛⠋⠉⠉⠉⠀⠀⠀⠀⠉⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠈⠉⠀⠀⠀⠀⠐⠃⠙⠛⠛⠛
⢶⣶⡶⠶⢶⠦⢤⣤⣤⡤⠤⠤⠤⠤⠤⢤⣤⣀⣀⠀⠀⠀⢀⣀⣠⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣄⢠⣤⣤⣄⡀⢄⣀⣐⣲⢶⣶⣖⡀⢰⣶⣶⣶⣶⡶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡀⢀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠧⢀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠿⣿⠛⠛⠻⠿⠿⠿⣿⣿⣿⣿⣿⠿⠛⣛⣛⠻⠿⢿⣿⣽⣿⣿⣭⣿⡂⠀⠈⠉⠻⢿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠘⠿⠿⠿⠿⠻⠟⠛⠋⠁⠙⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠂⠀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣟⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⣀⠀⠙⠋⡿⠟⠿⡿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢀⣤⠴⢶⣦⣶⣄⣠⣤⣝⣻⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣯⣥⣤⣀⠘⠾⣽⣬⣤⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⢿⣿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠁⠀⠈⠐⠉⠙⠛⣿⢿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣷⣾⣯⣭⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠶⣶⣭⣭⣟⣻⣯⣵⣶⣤⣻⣝⢿⣿⣿⣿⣿⡻⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣻⣿⣿⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿
⠐⠒⠒⠒⠚⠲⠚⣿⣶⣦⣽⠷⢶⣤⡈⠙⢽⣿⣿⣿⣿⣿⢟⣉⠉⠀⢈⡛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢷⣾⣋⣉⣡⣾⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣄⣚⣁⣀⡀⣾⣿⣦⣌⣉⣤⣤⣤⣀⣀⡉⢿⣿⣿⣿⣿⣷⣦⠤⠭⠹⢿⣿⣿⣯⣤⣤⣌⡛⠿⢿⣿⣾⣿⣿⣥⣤⣶⣶⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠛⠛⠋⠋⠀⠙⠛⠉⠻⡿⢿⣿⣿⣿⣿⣶⠤⠤⠀⠀⡤⠿⠿⠿⣿⣿⣿⣿⣷⣶⣶⣾⣭⣟⣛⣛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠶⢦⣶⣶⡿⣿⣿⣶⣤⣮⣥⣤⣤⣀⠀⠀⠀⢠⣤⣶⣶⣦⣬⣤⣤⣤⣭⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⢀⣴⠁⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣠⡄⠁⢀⡼⠁⣠⠟⠁⠀⠄⠀⢀⠔⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⣤⠀⠀⠀⠀⠀⠀⠁⢀⣴⣿⢄⣼⣿⠃⢠⡾⢀⡾⠁⠀⠀⠀⠀⠐⢡⢆⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣴⡏⢀⠀⠀⠀⠀⠀⢠⣾⣿⢿⣾⣿⠟⢡⣋⠶⠋⠁⠀⠀⠀⢀⡄⢀⣷⢟⣡⣾⠆
⠀⠀⠀⠀⠀⢠⡶⢶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠅⢹⡏⢠⡾⠀⣀⣴⣎⣤⣿⡿⣡⣿⣿⠃⢀⡌⠀⠀⠀⠀⠀⠀⡴⠋⣴⢞⣡⣿⠟⠁
⠀⠀⠀⠀⠀⣿⣿⠿⠟⢀⣷⣶⣴⡄⠀⢀⠀⢠⢄⡀⠀⠀⡟⢰⠟⢠⡿⣱⢆⣽⣛⣿⣿⣿⣿⣿⣿⣯⣶⠜⠃⠀⠀⠠⠀⣠⢞⣡⣞⣵⡿⠟⢁⡠⣲⣶⠇
⠀⠀⠀⠀⠀⢛⣿⣿⣷⠆⠈⣿⣿⣧⠄⠀⠀⠀⠀⠘⡉⠟⠁⠿⢰⡿⢱⣧⡿⣿⣿⣿⣿⣿⣿⣟⣿⣿⡅⠀⠀⠀⢀⣴⠞⣠⣿⣿⣿⣟⣱⣆⣯⣾⠷
⠀⠀⠀⠀⠀⠀⠈⢹⣷⣦⣴⣿⣿⣿⣿⠀⠀⠀⠀⠐⠁⠀⡀⠀⠙⣷⣧⣿⣾⣿⣿⣿⣿⣿⣿⣿⡿⠛⠀⢀⣄⡴⠋⣵⣾⣿⣿⠟⠹⠍⣹⣏⢁⣤⠋
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⠀⣧⣀⣀⣠⣯⡀⠀⠀⢸⣿⣿⣿⣿⣟⣾⣿⣯⣿⣿⡅⣣⣞⠛⣉⢀⣼⣿⣿⣿⡷⢿⣶⠞⣹⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠉⡟⠀⠀⣿⣿⣿⣾⡿⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⡿⣯⢈⣼⣿⣿⠿⢋⣭⣿⣵⢟⣱⠞⣋⣤⠶⠂
⠀⠀⠀⠀⠀⠀⠀⡿⢀⠀⠀⠀⠀⢰⠁⠀⢀⣿⣿⣷⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣩⡿⣿⣩⣥⣿⣿⣿⣡⣾⣿⡿⣫⡶⣛⣴⠟⠛⠁⠀⢀⣠⡀
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢀⡠⠁⠀⠀⢸⣿⣿⣿⡿⡿⠀⠀⢰⣿⣿⣿⣿⣿⣿⡿⣻⣏⣴⣾⣿⣿⡿⢡⣾⣿⣿⢽⣾⡯⠙⠋⠀⠀⣠⣴⣿⠿⠋⢁⣀⠄
⠀⠀⠀⠀⠀⠀⢸⠀⠄⠀⢰⣯⠘⠃⠀⠀⢈⣿⣿⣿⢾⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣿⣿⣿⡇⣠⠞⠋⠀⠀⣠⣔⣟⢉⣋⠴⢖⡚⠁
⠀⠀⠀⠀⠀⠀⢸⠇⠀⣠⡞⠙⠀⠀⠀⢀⣸⣿⣿⡿⣿⣿⡀⠀⠙⣿⣿⣿⣿⢿⣾⣿⣿⣿⣻⣿⣿⣿⠿⣭⠉⠛⠀⢀⣠⣦⣽⣿⣿⣿⣿⣿⣟⠋⣁⡀
⠀⠀⠀⠀⠀⠀⢘⣴⡞⢻⡿⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣷⣤⠀⠈⣿⡟⣱⣿⣿⣿⣻⢛⣿⡿⣟⡭⠈⠁⣀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠟⠉⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠀⢸⠑⠷⣤⣄⣴⢈⣿⣿⡿⡏⠻⡿⣿⣿⡀⠀⢹⣥⣿⣿⣿⠝⣿⣿⡿⠐⢋⣡⣴⣾⣿⣿⣷⣿⣿⣿⠿⠋⣁⣤⡴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠓⢾⣿⠻⢿⣿⣿⣄⣀⠘⣷⠏⣼⣧⠀⠈⢉⡉⠛⢷⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣽⣟⣡⡖⠛⠉⠁⠄⢀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⡼⠛⢷⣴⢿⡏⠈⢹⠿⣿⣾⣿⣹⡄⠀⢍⣶⣿⣦⠙⢿⣿⣿⣟⣩⣿⣷⣷⣿⣿⣿⣿⣿⠿⡿⢤⣤⣤⣤⣬⡿⠛
⠀⠀⢀⣠⣶⠿⠟⠛⢛⢻⡷⣶⣄⡚⠃⣹⣿⣇⠀⢸⣿⣿⣿⣿⣿⣷⡦⠀⢻⣏⣻⣧⠘⣿⠿⣏⣭⣾⣿⣿⣿⣿⣿⣧⣤⣼⣿⠿⠍⠙⠛⠈⢀⣀⡤⠄
⠀⣰⣿⠋⠁⠀⠀⠀⢸⣿⡇⠀⠙⠻⣶⣿⣿⠛⢷⣼⢉⣻⣿⡾⠿⣿⠃⢠⣿⡿⣇⢤⡄⢸⣿⢿⣿⣿⣿⣿⡿⢟⣿⡟⣿⢿⣿⣀⣠⢴⣖⣊⠉⠁
⢰⣿⠁⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⡙⣿⣥⣤⣤⣼⣿⣽⣿⣷⡾⠉⣠⡾⡟⢾⣧⣾⡇⢸⣿⣿⣿⣟⣯⡕⠚⠻⠟⠒⠛⠛⠋⠉⠉⠁
⣿⡇⠀⠀⠀⠀⠀⠀⣸⢿⡀⠀⠀⠀⠁⢸⣿⣿⣿⣿⣿⣿⠿⣿⣎⠛⣋⠄⣴⣪⣽⣿⠃⣾⣭⣴⣿⠿⠟⠋
⣿⡇⠀⠀⠀⠀⠀⠸⠟⠈⠁⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣥⣽⣿⣷⣥⣛⠿⠿⠟⣡⣾⠿⠯⠙⠃
⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣨⣿⣿⡟⠟⠛⠿⠿⠟⠋⠁
⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⢿⣿⣿⠟⠋
⠀⠀⠈⠻⣷⣧⣤⣀⡀⠀⢀⣀⣤⣾⠟⠁⠀⠐⠛⠉
⣤⣤⣤⣤⣤⣍⣽⣿⣿⣿⣿⣿⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣯⠤⣟⣋⣛⣛⣋⣛⣙⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣷⠀⠀⠀⠀⠀⠀⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡆
⣴⣶⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⣆⠀⠀⠀⠀⠀⢠⣶⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⡀
⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡄⠀⠀⠀⠀⣾⢻⣿⣿⢿⡀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣧
⠛⣿⣿⣿⠛⠀⣰⣷⡀⠀⠀⣰⣗⡀⢛⣿⣿⣿⣿⠃⠀⠀⠀⠘⠃⢺⣿⣿⠘⢧⣾⣦⡶⣴⣷⡼⣻⣿⣿⣿⡟
⠀⣿⣿⣿⠀⠰⢻⣿⠻⠀⢼⣿⣿⡷⠈⢹⣿⣿⠉⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⢨⣿⣿⠀⢹⣿⠁⠉⣿⣿⡏⠁
⠀⣿⣿⣿⠀⠀⢸⣿⠀⠀⠈⣽⣿⠁⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠈⣿⡏⠀⢸⣿⠀⠀⣿⣿⡇
⠀⠈⠉⠁⠀⠀⠈⠉⠀⠀⠀⠈⠉⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠉⠁⠀⠈⠉⠀⠀⠈⠈⠁
⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⠀⠀⠀⣠⣤⡀
⠀⣀⣻⣟⡀⠀⠀⠀⠀⢀⣘⣿⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⣟⣀⠀⢀⣙⣿⣁
⢠⣿⣿⣿⣿⡀⠀⠀⠀⣼⣿⣿⣿⣷⠀⠀⠀⠀⢠⣶⣤⣴⣦⠀⠀⠀⠀⢠⣿⣿⣿⣿⣆⣾⣿⣿⣿⡆
⡾⣿⣿⣿⣿⣧⢠⣄⠰⠏⣿⣿⣿⠹⡆⣤⡄⠀⠸⣿⣿⣿⡿⠀⠀⣤⡄⡾⠁⣿⣿⡇⢿⣿⣿⣿⣿⣿⣠⣤
⠐⢻⣿⣿⠛⠘⣿⣿⣆⠀⣿⣿⣿⠀⢹⣿⣷⡄⠀⠈⠛⠋⠀⠀⣰⣿⡿⠁⠀⣿⣿⡇⠀⠛⣿⣿⡟⠃⢻⣿⣷
⠀⢸⣿⣿⠀⠀⣿⣿⠈⠀⣿⣿⣿⠀⠘⣿⡟⠁⠀⠀⠀⠀⠀⠀⠁⣿⡟⠀⠀⣿⣿⡇⠀⠀⣿⣿⡇⠀⢻⣿⠟
⠀⠘⠛⠛⠀⠀⠛⠛⠀⠀⠛⠛⠛⠀⠀⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠛⠛⠃⠀⠀⠛⠛⠃⠀⠘⠛⠁
⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⢿⡿⠂⠀⠀⠀⠀⠀⠸⣿⠗⠀⠀⠀⠀⢿⣿⠇⠀⠀⠸⣿⠇⠀⠀⠀⠀⢿⡿⠂⠀⠀⠀⠀⠀⠸⣿⡿
⠀⣾⣿⣿⣿⡄⠀⠀⠀⢰⣿⣿⣿⡆⠀⠀⣼⣿⣿⣿⡆⢰⣿⣿⣿⡆⠀⠀⣾⣿⣿⣿⡄⠀⠀⠀⢠⣿⣿⣿⣆
⣸⠏⣿⣿⡟⣧⠀⣀⢀⣿⣿⣿⣿⣿⡀⢰⡏⣿⣿⡏⣷⣿⣿⣿⣿⣿⡀⣸⠏⣿⣿⡏⣷⠀⣀⠀⣾⣿⣿⣿⣿
⠉⠀⣿⣿⡇⠘⣮⣿⡞⢽⣿⣿⣿⡏⠁⠈⠀⣿⣿⡇⠈⢽⣿⣿⣿⡏⠁⠉⠀⣿⣿⡇⠈⢯⣿⡽⢹⣿⣿⣿⡯
⠀⠀⣿⣿⡇⠀⣾⣿⡇⠀⢸⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⠀⣾⣿⡇⠀⠀⠀⠀⣿⣿⡇⠀⢸⣿⡇⠀⢸⣿⡇
⠀⠀⣿⢿⠇⠀⢸⣿⠀⠀⢸⢿⡇⠀⠀⠀⠀⣿⢿⡇⠀⠀⢿⢿⡇⠀⠀⠀⠀⣿⢿⡇⠀⢸⣿⡇⠀⢸⡿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡫⠛⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⡆
⠀⠀⠀⠀⠀⠠⠤⢤⣤⣤⣀⣀⠀⠀⠀⠀⠉⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⣿⡿⣿⡰⠶⠖⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠷⣦⡴⠚⠉⢿⣆⠀⠀⠀⠀⠀⠀⣠⣴⠞⢋⣴⡿⣋⡤⢿⣿⡀
⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠿⠿⢿⣶⣤⣀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⠏⢁⣴⡿⣫⣾⠏⠀⣠⣿⣿⡿⠿⠿⢿⣷⣦⣄⡀
⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠀⠈⠙⢿⣷⡄⠀⢹⣧⣤⡾⠟⢻⡏⣠⡿⢋⡼⠋⠁⣰⣿⠟⠙⣿⡇⠀⠀⠀⠀⠉⠻⣿⣆
⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡞⠛⣿⡀⠀⢠⣿⣾⠟⠁⠀⠀⠀⣼⡿⠁⠀⢀⠸⣿⡀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⣼⣿⠀⠀⠀⠀⣠⣼⣶⣦⣤⠶⠿⠭⠥⢿⣿⢤⣿⣷⣶⣾⠟⠁⠀⠀⠀⠀⢰⣿⡃⠀⠀⠘⣶⣿⣷⣶⣦⡀⠀⠀⠀⢘⣿⡆
⠀⠀⣿⣯⠀⠀⠀⠀⢿⣿⣿⣿⠿⠷⠿⠿⠷⢿⣿⢾⣟⣿⣿⣿⡶⢶⣶⠄⠀⠀⢸⣿⡀⠀⠀⠀⠘⣿⣿⣿⣿⡇⠀⠀⠀⢠⣿⡇
⠀⠀⠸⣿⡄⠀⠀⠀⠀⠿⠛⣿⠀⠀⠀⠀⢀⣿⣏⣈⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠈⠙⠛⠋⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠙⣿⣆⡀⠀⠀⠀⠀⠟⠓⠒⠂⣹⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠃
⠀⠀⠀⠀⠈⠻⢿⣶⣤⣤⣤⣤⣤⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣦⣤⣤⣤⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠋⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣨⣥⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣒⣾⣤⣭⡲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣀⣀⣸⣿⣿⣋⣿⡿⢯⡰⡀
⠀⠀⠀⠀⠀⠀⣠⣺⠀⠈⠿⠿⠿⣷⣶⣦⣀⠀⠀⠀⠀⣀⠞⠛⠛⠛⠒⢂⠤⠀⠘⠋⣀⡀⢹⣃⣨⢡⣹⡀
⠀⠀⠀⠀⠀⠀⢙⣿⢿⣶⣦⣌⣒⢄⠈⠙⠛⣛⠛⠶⠾⠇⡀⠀⠀⢐⡠⠚⠁⢀⣩⣶⡦⠀⢀⣀⣀⣤⣾⡇
⠀⠀⠀⠀⠀⠀⣾⡿⠛⠿⢿⣿⣿⣷⣷⣤⣤⣬⣿⣶⡶⠔⢊⣄⣰⠊⠀⠀⠶⣿⡿⠋⠀⠀⣰⣿⢻⠿⠉⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠋⢻⣶⣤⣤⣈⠉⠙⠛⠿⣿⣿⣿⢿⡧⣠⡿⣏⠻⡄⠀⠀⣠⣤⣤⣠⣄⣼⡟⣧⠿⣷⣿⣿⣟⠷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⡟⣳⠟⡫⠽⠿⢶⣦⣞⠫⣟⣉⣷⣸⣄⣹⠷⠙⢦⣜⠋⠀⠄⠈⢻⣿⣿⡿⠒⣷⣿⣿⣮⡳⣹⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣏⢹⡀⠻⠶⣻⡿⢻⣿⣿⠲⢏⣩⣍⠉⠁⠀⠀⠁⠙⣷⣄⣀⠁⣸⠸⣿⣳⡾⡿⣕⣊⢿⢹⣇⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣯⡢⠙⠶⡶⢛⢴⣿⣿⠇⠀⠀⠈⠉⠳⠦⠀⠀⠤⠴⠶⠭⠭⠭⠼⠆⢻⣧⡻⢿⡾⣿⡷⢟⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣶⣶⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣶⣶⣶⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⣶⣼⡥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⡀
⠀⠀⠀⠀⠀⠠⣤⣦⣤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣤⣤⣀⡀⠀⠀⠀⣈⠻⢿⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡾⣿⣿⡤⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠿⠿⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢈⣿⠟⠉⠁⠀⠀⠀⠀⠀⠹⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠸⠋⠀⠀⣀⣠⣤⣽⣶⣿⣾⠋⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣴⣾⣿⡿⠟⠿⣿⣿⣿⣿⣶⣦⣄⡸⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣼⣿⠟⠑⣳⣶⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢫⣿⣿⠋⣣⣿⡿⣷⣄⠙⣿⣿⡄
⠀⠀⠀⠀⠀⣿⣿⡀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢰⣿⣿⡈⣿⣿⣿⣯⣼⠄⣸⣿⡇
⠀⠀⠀⠀⠀⢿⣿⣇⠀⠙⠿⠟⠉⢉⣿⣿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⠋⢟⣻⣖⠙⠶⣿⡶⠋⣰⣿⣿⠇
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣶⣶⣶⣶⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣯⣷⣤⣥⣵⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⣤⣼⣯⣿⣴⣤⣀⠀⠀⠀⠰⣤⠤⡅⠀⢉⠈⠙⣿⣿⣿⣿⣿⣿⣷⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣼⠏⣄⠈⠁⠈⠙⣆⠀⠀⠀⠀⠘⣻⣷⣴⡷⢛⣿⣿⣝⠿⣿⣿⣿⣿⣎⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⡉⠉⠉⠉⠙⠀⠀⠀⠻⣟⣭⣟⡿⣿⣟⣿⣿⣿⣧⡈⠙⠻⠿⣿⣿⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⣦⣤⣴⣬⣽⣶⠿⢸⣿⣿⠷⣿⣿⠋⠙⠻⠿⠦⢤⣄⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⢶⣤⠀⣈⣿⣿⣿⣿⡿⠗⠛⠻⢤⣠⣶⣬⡛⠿⣶⣶⣄⠀⠋⠀⠩⠁⣀⡀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠿⢯⠙⢻⣶⣿⣿⣿⣿⣿⣿⡿⠁⣀⠼⠣⣶⣌⣙⠂⢄⠀⠀⠀⣈⣧⣤⣶⣾⣿⠷⠿⠫⣄⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢏⣴⣿⣿⡗⠀⢹⣿⣿⣿⣿⣿⡿⠀⠀⢻⠷⣄⠘⠛⡻⠦⢐⣤⣶⢿⣿⣿⣿⡿⢿⣿⣷⣦⣤⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡿⣿⢿⣿⢿⡀⠛⢛⣫⣍⠉⠣⡀⠀⠀⠀⠨⠛⠳⣾⣶⣾⣿⠁⣺⣿⣿⣿⣿⣟⢯⠻⣏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⣮⣴⠶⣿⠻⢟⠿⣦⣽⣇⢰⠊⠙⠶⢄⣀⠀⠀⠀⣿⣿⣿⣃⣸⡏⠘⣿⡏⢁⣿⣷⣵⡸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣿⣟⢠⣄⣿⠋⠛⠠⣌⣥⣨⣿⣼⣿⣷⣦⣌⣉⠂⠀⣿⣿⣿⠛⡿⢨⣧⣽⣧⣼⣦⢻⣿⡇⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣞⢿⣿⣷⣏⠃⡄⠀⢸⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⠀⣇⡻⢈⣝⣹⣿⣿⡀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣿⣷⠿⠟⣫⠞⠁⣠⣿⠋⠻⠿⠶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣧⢻⣟⣷⣿⣽⢃⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣄⣉⣉⡙⠛⠛⠛⢻⡟⢧⡉⠻⠛⣠⣾⣿⠁⣼⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣷⣌⠛⠿⣿⠿⠟⢁⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠿⠿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣯⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠻⠿⠿⠿⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢿⣿⡿⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⢿⣿⡿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠨⣿⠇⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢀⣀⡀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢀⣀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⣘⣿⣇⣀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⣸⣿⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠙⠛⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠆⠀⠀⠀⠀⠀⠀⠰⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠛⠉⠉⠻⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣿⣿⣿⡇⠀⣾⣷⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⣶⡇⠀⠻⠃⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⡶⠿⣿⣿⠿⣶⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀⠉⠉⢀⣼⣿⣿⠏
⠀⠀⠀⠀⠀⢰⣿⡇⠀⠿⠿⠀⣽⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⠟⠃
⠀⠀⠀⠀⠀⢸⣿⡇⠀⣠⣄⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠻⣇⣀⣿⣿⣀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠁⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⣀⣤⣶⣶⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠟⠛⠛⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⢀⣶⣶⠀⢸⣿⣿⣷
⠀⢀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡇⢰⣿⣿⠀⢸⣿⣿⣿
⣰⣿⡇⠈⣿⡏⠈⣿⣦⠀⢰⣶⡧⠀⢻⣿⣿⣇⠀⠛⠋⢀⣼⣿⣿⡏
⣿⣿⡇⠀⠉⠁⠀⣿⣿⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣶⣿⣿⡿⠋
⢻⣿⡇⠀⣿⡇⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠁
⠀⠙⠳⣾⣿⣷⠞⠋
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣀⣀⣀⣀⣀⣠⡿⠀⠀⣴⠶⣦⠀⠀⠀⠀⠀⠀⣾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⣠⣽⡇⠀⠀⠀⠻⠦⣿⠀⠀⠀⠀⠀⣠⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣀⣸⡇⠀⠀⠀⠀⠀⠈⣷⠤⠤⠤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣀⣹⡇⠀⠀⠀⢀⣤⢼⡇⠀⠀⠀⢘⣧⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⢹⡇⠀⠀⠀⠸⣧⣤⡟⠀⠀⠀⢸⣧⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠃⠀⠀⠀⠘⢧⣄⠀⠀⠀⠀⠈⣷⣤⣤⣤⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀⣠⠞⠁⠀⠀⠀⠘⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣯⣀⣀⣀⣀⣤⡴⠶⠚⠛⠛⠓⠿⣿⠁⠀⠀⠀⠀⠀⠀⣿⣀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠉⠉⠉⠁⠀⠀⢀⣀⡀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⢠⣤⡄⠀⠈⠉⠁⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡙⠶⠃⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⠶⠶⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠓⠴⡀⠖⠂⠂⠔⡀⣰⠈⠀⣒⠤⡀⠴⠂⢄⡇⠰⠒⠔⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡠⠤⢊⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⡠⠔⠒⠒⢻⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣕⠢⠤⠤⠤⢽⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⡱⠋⠉⠉⠀⠀⠀⠀⢀⣀⠤⣒⠭⠓⢋⡵⡻⣸
⠀⠀⠀⠀⠀⠀⣀⠤⢐⣒⣒⣒⣒⠰⢵⡁⠀⠀⠀⣀⡠⣔⣪⢽⠊⠉⠀⢀⢔⡯⠊⣀⠧⠧⠤⠤⠤⢀⡀
⠀⠀⠀⠀⡴⣪⠔⠉⠅⠀⢠⠀⢠⣿⠓⢬⡲⣖⠭⠒⠉⠁⢸⢸⠀⢀⢔⡵⣫⢔⡩⠒⢺⢹⠉⠉⠉⠒⠪⣑⢤⡀
⠀⠀⢀⢎⠜⠁⠠⡀⠈⠀⠠⢠⣿⠃⡠⠊⠙⢌⢆⠀⣀⠤⠼⠾⣔⠕⠋⡔⡱⠛⢄⠀⠘⡏⣆⠀⠠⠁⢀⠌⠳⡑⡄
⠀⠀⡎⡜⠐⠠⠀⠈⠢⠐⢰⣿⡧⠊⠀⠀⠄⠚⡎⡾⠁⢀⣀⠀⠈⡆⡜⡼⠠⢀⠀⠑⢄⢹⢹⠀⢀⠔⠁⠀⠀⠸⡘⡄
⠀⠀⡇⡇⠄⠀⠀⠀⠁⢰⡛⣻⣶⣿⡶⠭⠭⠭⡧⡟⠲⢞⠖⠁⠀⣿⡇⡇⠀⠀⠀⠀⠂⢱⠿⢧⠁⠐⠀⠀⠀⠀⡇⡇
⠀⠀⡇⡇⠀⠀⠄⠂⡨⢊⠙⢉⠀⡐⠀⠄⡀⠀⡇⡏⠓⢄⡱⠔⠊⠀⡇⡇⠀⠀⠀⠄⢀⠘⠓⠛⢄⠀⠀⡀⠀⠀⡇⡇
⠀⠀⠸⡘⡄⠀⡠⠊⠀⠂⢀⠀⠄⠀⠠⡀⠀⡜⡸⠀⠀⠀⠀⠀⠀⠀⢱⠱⡈⠀⢀⠔⠁⠌⠀⠈⠀⠑⢄⠀⠀⡰⢱⠁
⠀⠀⠀⠘⢌⠪⣀⠀⠄⠀⢈⠀⠘⡀⢀⡠⢊⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⡑⢄⠁⠀⡐⠀⠀⠀⠐⠀⠀⢁⠔⡱⠁
⠀⠀⠀⠀⠀⠙⠪⢝⣒⠢⠴⠤⢒⣚⠭⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢝⡒⠤⢄⣠⣀⠤⠗⣚⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠃⠢⣀
⠀⠀⠀⢀⢀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠒⠒⢶⣿⣟⣛⣿⣥⣾⣶⣌⢳⡀
⠀⠀⠀⣿⣿⣧⣄⡈⠉⠒⣤⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣀⣀⣀⠀⠀⠀⠐⠄⣽⣿⣿⣿⣿⣿⣿⣿⣆
⢀⣀⠀⢿⣿⣿⣿⣿⣆⠀⠈⠉⠉⠑⢄⠀⢀⣼⠛⠛⠿⠿⣿⣿⣿⡻⠶⣦⣤⣼⠻⣿⣿⡿⠛⠉⣻⢿
⠛⠻⠛⠛⢷⡝⢿⣿⣿⣷⣦⣄⡀⠀⢀⣝⣿⣿⣀⠀⠀⣀⣤⠽⠿⠟⠒⠋⠁⠀⠀⢿⣿⣿⠀⢰⡏
⠀⠀⠀⠀⠀⠙⢦⠻⢿⣿⣿⣿⣿⣾⣿⢸⣿⣿⣿⠶⠋⠁⠀⠀⠀⢀⣤⣴⣶⠂⢀⠸⣿⣿⠀⣼
⠀⠀⡀⠀⢀⣴⣿⣷⣄⣙⣿⣿⣿⣿⣿⣷⣽⣿⣅⣀⣠⣶⣶⣤⣄⣀⠉⠉⢁⣴⣶⡀⣿⡿⠛⠋
⠳⣄⠙⢶⣿⡟⣿⠃⢻⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⢏⣴⣿⣾⣿⡇⢸⠁
⠀⠙⢷⣄⠹⣷⠉⢠⣿⣿⣿⣿⣷⣽⣿⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⢲⣼⣆⣀⣀
⠀⠀⣰⡝⣷⡖⠚⠿⣿⣿⣿⣿⣿⣿⣿⡷⠾⣻⣿⣿⣿⣿⣿⣿⢃⡞⣿⣿⡿⢛⣿⡼⡇⢫⠈⠉⠀⠀⠉⣀⣂⢀⡀
⠀⠀⢿⣿⣿⡻⢀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣷⡈⢶⡟⣿⣿⣿⢇⣾⠃⠿⢿⣵⣛⣁⡀⢰⠸⡀⣀⣠⣾⠿⣬⣛⢯⡉
⠀⠀⢸⣿⣿⣿⣿⠆⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣾⡇⣿⣽⡟⣼⡏⣰⣶⣿⣿⣿⣿⢿⣿⡆⡿⠿⠛⠙⠛⢦⣝⢶⡙⢦
⠀⠀⠀⢻⣿⣿⣳⣄⢀⣴⣄⡙⠿⣿⣷⣿⣿⣿⣿⣿⣿⡿⢱⡿⢠⣿⣿⣿⣿⡝⠙⣨⡿⣧⢹⡂⠀⠀⠀⠀⠙⣮⢻⡌⣧
⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠟⠓⠲⠿⡓⠦⢍⣙⠛⢛⣡⣿⢃⣿⣿⣿⣿⣿⡿⠛⠁⠈⠹⡼⠇⠃⠀⠀⠀⠀⠘⣏⣿⣿
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠉⠒⠊⠭⢍⣉⡡⠾⠷⠾⣿⣿⣿⡇⠀⠀⠀⠀⠔⠷⢀⣙⠀⠀⢰⣾⡏⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣯⢇⠀⠀⢠⡾⠆⠒⠈⠋⠉⠀⠉⢹⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣯⣪⣦⡤⣄⠉⠀⠀⠀⠀⠀⠀⡰⣳⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣭⡢⠌⢤⣀⣀⣀⣤⢴⣫⣾⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣿⣾⣿⣿⣧⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣟⡋⢐⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡻⣮⣿⡆⠙⣇⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢉⣱⣿⣿⣿⣼⣷⡄⡀⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡔⢮⣿⣶⣿⣿⣿⣿⣿⣵⠶⠀⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄⣠⣄⣶⣾⣿⣿⣶⡿⠟⣁⣼⣿⣿⣿⣿⠽⢖⣹⠟⣡⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⣟⣛⡟⡫⡬⠛⠛⣟⡛⢻⣿⣿⣷⣤⣭⣤⠀⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⠙⢧⣴⡿⠟⢼⣿⣓⠐⠈⡀⠀⠈⢿⣿⣿⣯⠀⠀⣀⣀⣠⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣴⠟⣷⣶⣦⣄⡈⠉⠙⠂⢉⠓⢤⣘⣿⣿⠟⢲⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢶⠖⠿⢿⣿⣿⣿⣶⣤⣀⣀⣑⡤⠉⢛⣙⠍⢉⡁⠀⠠⢄⡈⠈⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⣀⣤⣶⣤⡈⠐⠣⡟⡒⡾⣿⣿⣿⡿⣿⣿⣿⠟⠉⠀⠠⠼⠛⣿⣆⣁⣨⠶⢒⣘⣶⣾⣿⣦⣤⣄⡀⠀⠀⠈⢑⣦⣀⡀
⠀⠀⣠⢞⣯⣽⠿⣯⣠⣥⣽⣾⠻⡦⣿⣾⣏⣴⠟⡋⣀⠐⠀⠀⣠⣤⣬⡝⠋⠉⢀⣾⣿⣿⣿⣿⣿⣿⣿⡿⣿⣷⣶⠾⠟⠻⣿⣿
⠀⢰⣿⣿⠋⠀⣴⣿⡿⢹⣿⣾⣶⢡⣸⣷⠏⠁⢤⣗⣾⣤⣶⣾⣿⣿⣿⣿⣶⣾⣿⣿⣿⣿⡿⣫⣽⣿⣿⠿⠿⣿⠋⠀⠀⠀⠘⠇
⠀⣿⣿⣿⠡⣰⣿⣿⣅⣶⣿⢿⣿⣿⣿⡿⠀⠻⠦⣞⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⡿⠛⠛⠛⠄⢀⣾⣷⡀
⠀⣿⣿⡇⣵⣿⣿⡟⣿⣿⣇⢸⣿⣿⣿⠃⠀⠀⠀⠸⣡⢟⢹⣟⡻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣥⣾⡄⠀⠀⠀⢻⣿⣯⣇⡀⠀⠑⠀⠘⠆
⠀⢸⡿⢷⠹⡿⡻⡗⠿⠧⡈⢻⡛⣿⠿⠟⠀⠀⢠⣿⣿⣿⣜⠃⢹⢿⣻⣿⣿⣿⣿⣿⣿⡟⣿⡿⣿⣄⣠⣤⢤⣿⣛⣾⣿⡀
⠀⠀⢻⣨⠣⡈⢩⡄⢹⡤⠴⣾⣿⡜⠀⠀⠀⠀⠀⢹⣿⣿⣿⡖⠲⢟⣛⣿⣿⣿⣿⣿⣿⣇⣿⣦⣿⣿⣧⣤⢸⣿⣿⣿⣿⣷⡀⠀⠀⢸⡄
⠀⠀⠀⠻⣶⡈⠺⣤⣤⢷⣾⣿⡿⠃⠀⠀⠀⠀⠁⠘⢿⣿⡿⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⢿⡟⢿⣿⣿⡿⢃⡾⣿⣿⣿⣿⣿⣿⡇⠀⠈
⠀⠀⠀⠀⠈⠙⠻⠿⠿⠿⠿⠋⣀⠳⢤⣄⣀⠛⠻⠒⠲⠿⠤⠀⠰⢿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡶⣭⣭⣴⣯⡀⢻⣌⣿⣿⣿⣿⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣤⣾⣿⣿⣶⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣻⡯⠽⢿⣿⣽⣿⣿⣿⣿⢃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⠿⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣀⣿⢀⣾⠛⠻⣦⠰⣿⣛⡷⠀⣿⣛⣿⠆⠻⣦⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⣿⠈⢿⣄⣰⡿⢀⣿⣉⣿⠆⣿⣉⣹⡇⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠉⠁⠀⠀⠉⠉⠁⠀⠈⠉⠁⠀⠀⠈⠁

⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⡇⠀⠀⡄⠀⠄⠀⠀⠀⠀⠀⡠⠀⡇⠀⠀⠠⠀⢄⠀⠀⢠⠀⠂⢀⣄⢄⣀⠂⢰⠀⠀⠀⠀⠀⢴⠪⢔⠉⠣⠀⠀⠀⠀⢀⣴⣀⢀
⡇⠀⠀⠁⠈⠀⠀⢀⠀⠀⠀⠀⣂⡈⠙⢀⣁⠈⢰⠀⠀⢸⡀⠸⠉⠓⠚⠿⠀⣺⠀⠀⠀⢀⢰⠌⢗⠀⠀⠀⠁⠀⠀⠀⣸⠛⠢⠐⠌⢂
⠃⠂⠐⡐⠄⠀⠂⠒⠀⠀⠀⠠⠤⠤⠀⠠⡤⠤⠀⠀⠀⠀⠁⠁⠈⠀⠀⠁⠀⠁⠀⠀⢐⠋⠁⠈⠚⠂⡀⠀⠀⠀⠀⠀⠑⠈⡄⠀⠀⠈
⠀⠀⠀⠁⠀⠆⠀⠀⠀⠀⠐⠦⠢⠲⠴⠦⠤⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠐⠄⠀⠀⠁

⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣠⡄⠠⠄⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⢖⠔⠢⠀⠀⠀⠀⠀⢀⠠⠤⠀⣀⢄⠀⠀⠀⠀⠀⠀⡀⡄⢀
⠀⡐⢠⢦⡄⠀⡠⠊⠀⠀⠀⠈⠉⠐⡾⠂⠰⢮⡀⠀⠀⠀⢀⣴⢦⠀⢀⢎⠄⠀⠀⠀⠀⡐⠁⠀⠀⠈⡵⠃⠀⠀⠀⠀⠀⡰⣷⡗⡽
⠀⠀⣣⠍⡄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⡷⠀⠀⠀⠀⠀⠨⠋⠀⠈⡂⠀⠀⠀⠀⠀⠀⠁⣀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⢺⡧⣿⢙⡍
⠀⢸⣻⠁⠤⢤⠢⠀⠀⠀⠀⠀⠀⠀⠀⠉⠖⠀⠀⠀⠀⠀⠀⢪⡀⠀⠰⠀⠀⠀⠀⠀⠀⠜⠵⠊⠀⠂⠀⠊⠀⠀⠀⠀⠀⢘⠀⠀⢘
⠀⠉⠉⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡄
⠀⢀⡀⣀⢀⡀⡀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⡘⠀⠘⡀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⠀⠐⠂⠐⢀⢂⡀⠀⠓
⠀⠉⠑⡀⢀⡴⠉⠀⠀⠀⠀⣤⢀⠊⠫⡀⢄⡀⠀⠀⠀⢰⠁⠀⠀⠍⠁⠀⠀⣰⠀⠀⠀⡀⠂⠁⡨⠈⠁⠀⠄⠀⠀⠀⠀⡀⠀⢂⠀⠀⡄
⠇⠀⠰⠂⠀⠃⠈⠈⠀⠀⠀⠙⠺⢦⠀⠔⠀⠀⠁⠀⠀⢸⢴⡆⠈⠁⠀⠐⣶⢾⠀⠀⠀⣠⣊⠡⡤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠀⠀⠘⠛⠧⠀⠀⠀⠀⠟⠋⠀⠀⠐⠋⠋⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⢀⡌⡅⠀⠀⢰⡱⠀⠀⠀⠀⠀⠀⠀⡞⢡⠄⠀⠀⠀⠀⠀⡠⠒⠉⠉⡂⠀⠀⠀⠀⠀⠀⠈⠋⠉⠁⠀⠀⠀⠀⠀⠀⡤⠒⢉⣑⢦⡀
⠀⠐⢀⢀⡀⡪⠊⠀⠀⠀⠀⠀⠀⢐⢺⣀⠹⠀⠀⠀⠀⠀⠀⠣⡀⠄⢀⠁⡰⠂⠀⠀⠀⢰⡆⠀⠀⠰⡀⠀⠀⠀⠀⠘⠔⠁⢀⠀⡑⢱
⡀⠂⠁⡰⠻⠀⠀⠀⠀⠀⠀⠀⠀⡃⣸⠁⡀⠀⠀⠀⠀⠀⠀⠢⡄⢠⠟⠀⢴⠀⠀⠀⠀⢸⡆⠀⠀⠲⠀⠀⠀⠀⠀⠘⣄⠀⠢⠠⡁⡎
⠃⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⢠⡭⡁⠀⡃⠀⠀⠀⠀⠀⢀⠊⠉⠃⠜⠁⠁⠀⠀⠀⠀⠈⠁⠲⡀⠃⠀⠄⠀⠀⠀⠀⠀⠉⠒⠉⠈⠢⡀
⣶⣾⡿⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣷⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⣶⣶⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡂
⠀⠀⠀⠐⠀⠂⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣶⣦⣤⣤⣶⡶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠠⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀
⠀⠀⠀⠄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⠶⠀⠸⣟⣿⠀⣴⣿⣷⠀⠀⠄⠀⠀⠀⠈⠁
⠀⠀⠈⠀⠀⠀⠀⠀⠈⠂⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠻⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀⠉⢰⡉⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠲⠆⠀⠀⠀⢀⡀⢀⡀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠰⠶⠟⠋⠀⠀⠺⣿⠇⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠱⠄⠀⠀⠛⠀⠀⣤⠀⠀⠀⢀⣀⡀⠿⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡟⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⢀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠰⡲⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠣⣭⣠
⠀⠀⠀⠀⠀⡀⠐⠒⠲⣄⠀⠀⠀⠐⠂⠀⠤⠁⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠰⠉⠀⠀⠀⠠⣶⡄⠀⠀⠁⠀⢀⣤
⠀⠀⠀⠀⡜⠀⠀⠀⠀⢱⠀⠀⠉⢀⠀⠀⡀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠅⠀⠘⠓⠂⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠉
⠀⠀⡀⠀⢃⠀⠀⠀⣠⠇⠀⠀⢀⠈⠀⠀⠁⢀⠀⢀⡈⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠈⠋
⠀⠘⠛⠃⢘⠷⠒⠚⠁⠀⠀⣠⣤⡄⠀⢠⡄⠈⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠠⠀⠁⠀⠀⠘⡄⠀⣾⠋⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠰⢾⡷⢶⠀⠀⠀⠀⡀⠀⠀⠀⠰⠿
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⣇⠀⠀⠀⢀⢸⡿⢻⡆⣴⣿⡷⢺⡞⣷⠟⣦⢸⡇⢺⣇⠈⣇⢸⡟⠃⣿⢸⡇
⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠙⠶⠶⠾⠋⠸⠇⠀⠗⠘⠿⠗⠘⠇⠻⠀⠿⠸⠇⠰⠾⠃⠿⠸⠇⠀⠘⢻⡇
⠀⠙⠁⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠾⠁⠀⠀⠀⣀⣀⣠⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣹⣶⠖⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠶⠋⠀⠀⠀⠀⠸⠿⠲
⠀⢠⡄⢀⠀⠘⣻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠈⢷⡏⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⣴⡌
⠀⠀⠀⠈⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣖⡼⠥⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢹⡄⠀⠀⠃⠀⠀⠀⠀⠀⠄⠂⠀⠀⠄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠀⠀⠀⡠⠒⣠⡄⠐⠧⢼⡦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀
⣯⣿⣿⣽⣯⣭⣿⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣯⣭⣿