Boodle Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣈⢍⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠶⠒⠒⠒⠒⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⢺⠀⠻⠇⠀⠀⠀⠀⠀⠀⡠⠔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠘⠀⠐⠀⠀⠀⠃⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠘⡓⠀⣀⢚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠰⠀⠠⠁⡅⠀⠈⠈⠀⠀⠈⠀⠀⠀⠄⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⡆⠀⢀⠀⢰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣠⠐⡄⠀⠃⠀⠐⠀⠞⡏⠀⠣⡀⠠⠛⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⡌⠀⠀⠀⠀⡠⠀⢀⠘⠂⠂⣂⡀⡻⠠⡅⠐⠠⠌⠶⡦⣄⣀⠀⢁⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⢀⠀⠉⠉⠉⣔⠋⠊⠔⡂⢀⡸⠄⠠⠀⠀⠈⠉⠉⠠⠤⠄⠀⠀⠀⡇
⠀⠀⠀⠀⡇⠀⠀⠀⢆⠡⠂⣤⠀⠩⠜⠢⠠⣤⡒⠉⠀⠀⣺⠀⠀⠄⠄⠴⠚⠀⡄⠀⠀⢀⠇
⠀⠀⠀⠀⠘⡄⠀⠘⠈⠉⠈⠀⠔⠐⠀⠂⠁⢈⡍⠓⠒⢾⠳⠄⠀⠀⠀⠀⠀⠈⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠶⡁⠀⡀⠠⠐⠀⠀⠀⡀⠐⠊⠉⠁⢂⠟⢀⡦⠀⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠐⠄⠀⠠⠈⠡⠈⠷⠂⠀⠀⠀⠁⠀⠀⠀⣠⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠶⢄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠐⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠠⠤⡀
⠀⠀⠀⢠⢤⣠⡄⠀⠀⠀⠀⠀⣠⣀
⠀⠀⠀⠉⠞⢱⠻⡽⠷⠯⠷⠯⠧⠧⠿⠵⠽⠆
⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡤⠀⣀⣠⠖⠁⠀⣠⣤⣤⣄⣀⢀⣁⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣐⣊⡐⠫⠲⢄⠙⠇⣠⠞⠋⠀⠀⢰⣶⣤⠄⠉⠙⠳⢬⡑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣤⣴⣯⣝⡙⠻⢹⡃⡀⠸⠀⣼⣟⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢉⡤⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣏⠩⠄⠘⣿⠈⠸⠇⠃⣈⠁⣥⡀⣻⣿⡟⢳⠅⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠗⠄⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠙⡄⠀⡀⠀⣿⣧⡄⢠⣁⠻⠧⣻⢵⡈⣿⡿⣥⣄⢠⡁⣠⡶⣿⣶⣄⠀⠀⠀⣰⠆⠂⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⢿⣀⠀⠀⠙⠿⠙⢧⣙⣄⣩⣭⣥⣄⢼⣽⣿⣿⣽⢈⣿⢾⣿⣤⣻⣿⣟⣠⡧⠞⢸⡗⢠⣄⠰⣘⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠀⠻⡏⠁⣀⣽⡆⢠⣬⠟⢩⡿⣿⣿⣴⣿⡏⠈⢻⣿⣿⣿⣿⠟⠋⠉⠉⢉⣀⠔⣾⡅⠀⠛⠶⣡⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣤⣤⠞⠛⣡⣂⡼⢉⣼⣯⢟⠟⣹⡾⠏⢁⡴⣫⣿⣼⣿⢿⡀⡤⠚⠋⠹⡇⣾⢿⣧⣰⢎⠀⠤⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⠛⠛⢲⠉⣿⢁⡼⣵⢿⠏⣾⠟⠁⡰⢋⣽⢏⣼⣷⡟⣼⡟⠐⠀⠀⠀⠳⠽⢁⠋⢀⣨⣄⠓⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⣅⠖⠾⢷⣿⢸⣷⣿⣹⢰⣿⠀⢰⣧⣽⠋⣼⢻⡏⣰⣿⣧⡀⠀⠀⠀⢀⡴⠛⣢⡤⠥⠠⠅⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣧⡧⠀⠀⣙⣷⢸⣾⣿⢿⠘⣿⠀⣾⠻⣿⢠⣧⣼⢳⣿⢓⣽⢻⠞⠓⠶⠞⠅⡈⠁⣀⡤⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⣶⣶⣿⣿⣦⣟⣻⣾⡄⣿⡀⣷⣴⣇⣺⢷⡏⣸⣏⢸⣁⡴⢶⣭⡀⣤⣴⡷⠛⠛⠠⣄⡀⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢿⣿⣏⣿⡿⢿⠿⠿⠷⣽⣇⡜⣼⣿⣿⣻⣇⣿⣿⣴⣿⡇⠀⠙⡟⢯⡙⢍⠛⠲⢄⠀⠙⠃⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠈⡧⠈⡛⠉⠀⠀⠀⠀⠍⠉⠙⠻⢿⢼⡿⠛⠉⠍⠁⢀⣾⡀⠀⢹⣾⣹⡛⢳⡤⢠⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡘⢰⢴⣾⢰⣶⣿⡍⢸⠇⠀⢾⠏⠁⠀⠨⠁⠀⢹⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⠿⠿⣷⣦⢤⡀⣰⢿⠶⡈⠻⢄⢶⡎⢀⡿⣀⢂⡞⠀⠀⠀⠀⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠙⠻⠻⠋⠻⣷⣯⣦⠴⠃⡀⠀⣀⡉⣉⣭⡵⠞⠀⢀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠋⠒⠒⠀⠐⠚⠛⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣀⣀⠀⠀⠀⠀⢀⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡠⠤⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢓⡂⠉⢐⣒⣛⢛⣉⢉⢉⡉⣁⢀⢀⠀⠀⠀⠄⠄⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢦⢊⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⡖⢦⡈⣄⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠲⠉⢘⠇⠀⣖⠀⣈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⣌⠐⠻⠀⢤⡤⡀⣾⠉⡆⠀⠳⠖⢤⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠡⡄⢦⡉⢀⡈⠁⢡⠃⣌⡋⠆⢳⡄⣠⠴⠈⠵⣦⠙⢢⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣔⣴⠂⠠⡆⠐⠛⠙⡍⠩⠐⠂⠨⣤⣠⢖⡃⢰⠙⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢓⠾⣬⣖⡑⠀⠽⢐⠺⢃⢀⣀⠰⡦⠀⢁⣁⣘⣃⣤⠶⢚⡍
⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⢭⣉⣛⡛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⡉⠁⠶⠚⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠠⠨⠉⠉⠙⠛⠋⠙⠉⠉⠉⡉⠑⠅⠉⡀⡨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⢀⣟⣶⣷⣤⡿⡭⡷⡤⠀⠀⠐⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠈⡿⣹⣯⣯⣯⣾⣷⣮⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⢙⠉⠙⠉⠉⠉⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⢠⠁⠀⠀⠀⣀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠔⢫⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣒⠟⠋⠁⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⡪⠐⠉⡀⠄⠒⣉⠤⠐⡂⢽⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡒⠕⢊⡁⠄⣊⠩⢐⡊⠥⠒⢉⣡⣤⠨⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠮⢒⣁⡤⠦⣁⠢⠅⠒⢉⣀⠤⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⣨⣴⣞⠯⠓⢂⣉⠤⠐⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢵⢾⣿⣽⡻⠐⠂⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣿⣟⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣾⣯⣷⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⠤⠶⠖⠒⠒⠒⠒⢺⣽⣽⣻⢷⠛⠒⠒⠒⠒⠒⠲⠶⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠅⢀⣠⡤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣤⣶⣶⣶⣶⣶⣶⣦⣤⣤⣀⣀⡈⠻⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣛⡻⠿⠿⣯⣿⣺⣿⣿⣿⣿⣿⣿⣻⡿⣝⡻⣟⣿⣟⣿⣿⣿⣻⣿⣿⣿⡿⠿⠾⢛⣋⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⢐⠈⠉⠐⠒⠰⠄⠠⠤⠭⠭⠭⠉⠩⠉⠉⠩⠍⠉⠉⠭⠉⠉⠠⠀⠖⠐⠒⠈⠉⢐⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣒⣒⡠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡱⢊⠄⠒⣂⡀⢑⠕⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢨⠀⣎⠀⠈⠂⢺⣼⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⢎⢆⠺⢤⣀⠀⡠⠂⡔⡜⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡠⣀⠀⠀⠀⠀⠀⠀⠑⠵⣂⡄⣀⡐⠭⠊⠀⠀⠀⠀⠀⠀⠤⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢥⣀⣒⣒⠀⠠⠤⠤⠤⠤⠤⠤⠤⠄⢐⣒⣂⣀⠥⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢠⠠⠄⠀⠀⠀⠀⠀⠀⣀⡠⠤⠴⠒⠒⠒⠛⠋⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠈⡇⠈⠻⠀⠀⡀⠀⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⠂⠀⣀⠂⠀⢈⣀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⢀⢹⠀⠠⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠛⢀⠃⠱⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⢀⠄⠈⠀⠀⠀⡀⠀⢸⠠⠀⠆⢀⣧⣀⡀⠀⠄⣀⡀⠀⠀⠀⠈⠑⢤
⠀⠀⠀⠀⡰⠁⠀⠀⠀⠐⠃⠘⡄⢀⢐⡀⢅⡌⠡⠀⠈⠲⣙⠁⠐⠀⠀⡀⠀⠀⠀⠀⠄
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠊⢀⣤⣍⠈⢱⢤⠝⠃⠿⠀⡀⠁⠋⠛⢳⡤⠤⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠠⠀⠀⠀⠃⠂⢆⠀⠀⠀⠄⣅⠈⠈⢀⡠⠮⠁⢠⡉⠀⠀⠁⢀⣨⠘⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢇⠀⠀⠘⠣⠬⠴⠃⣀⡉⢈⠥⠙⠷⣤⣄⣀⣼⠅⠐⠁⠁⡏⠀⠠⠂⠀⠀⡰
⠀⠀⠀⠀⠈⢦⠀⠀⠀⢀⡠⠈⠀⠁⢀⠀⠐⠚⠀⢠⡈⠤⠅⡀⣤⠄⡀⠀⠀⠀⠀⡴⠁
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠑⠀⣄⠀⠄⠀⠄⠀⢤⠄⠀⠀⠀⠀⠁⠘⠁⠀⠀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⢀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠢⡀⡠⠖⠒⠠⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣽⠖⠋⠑⠫⣍⣆⣸⣤⡠⣌⣣⡀⠀⣀⣀⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠒⠉⠉⠀⣀⣀⣀⠀⠈⠀⠀⠀⠘⠉⠀⠀⠁⠀⠈⠐⠄⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠐⠁⡔⡞⠀⡄⠠⠀⠁⢀⣀⠀⠈⠢⠀⠀⠀⢀⠔⠉⠀⠀⠈⠐⢄⠀⠀⠆
⠀⠀⠀⠀⡠⢂⡁⠳⠀⠑⢆⣸⢤⠁⠀⡴⣿⡟⠛⣆⠀⢳⠀⢠⠃⠀⡴⣻⡿⠷⡀⠀⠆⠀⢈⣁⣈⠑⠄
⠀⠀⠀⢠⠀⡁⠇⢀⠗⠠⡴⢃⢸⠀⠀⣿⣿⣷⣶⣿⠀⢸⠀⢸⠀⢰⣿⣿⣷⣴⡇⠀⢸⠸⣮⣍⠙⢆⡘
⠀⠀⠀⠀⢄⠈⢆⠈⠫⣹⠀⡏⠸⣆⠀⠙⢿⣿⡿⠏⢀⠞⠀⠘⣆⠈⢿⣿⣿⠟⠀⢠⠇⠀⣉⠂⠡⢘
⠀⠀⠀⠀⡠⠛⣉⡡⠌⠙⣦⣡⡀⠫⢗⠤⠤⡤⠤⠖⢋⡀⠀⢀⠈⠓⠤⢤⡤⠤⠖⠁⠈⠻⢳⡖⢁⠮⠄⡈⠑⡀
⠀⠀⠀⡰⠀⠠⠁⡠⠄⢄⠈⡖⠻⢒⡀⣙⡀⠀⢶⣄⡈⠟⠛⡛⠀⠀⢀⢀⠤⡴⠀⣧⠀⢶⣄⠈⢿⢊⣩⣭⣉⠓⢤
⠀⠀⡠⠁⣠⣇⣰⠋⣅⠙⠋⡟⢉⣀⠈⢣⡈⡛⡟⠀⡏⠻⠬⠜⣩⣆⠘⠓⠚⢁⠔⢰⠀⢸⡇⣧⠈⠃⠀⠀⠈⢳⣀⡀
⠀⠙⠒⠂⢹⠀⠙⠲⢿⣇⣨⣱⠁⡴⡇⠀⠻⣿⣄⡼⢹⣖⡖⠈⠁⠨⣿⣷⡛⢉⣠⠟⠀⡼⠀⣘⡤⠖⢛⡇⡆⠀⣆⠡
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠉⠙⠚⠓⠾⠦⠤⠤⣥⣤⣞⣉⣉⣀⣩⡖⠤⡉⠉⠉⠉⠳⠚⠛⠉⠁⠀⠄⢸⠏⡁⢴⢳⠀⡆
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢨⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⡏⠀⠐⢤⠉⠰
⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠸⠠⠔⠁⠀⠀⠀⠀⠀⠀⣸⠁⠘⠂⠁⠐⠁
⠀⠀⠀⠀⠀⠀⠘⣆⠀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⠄⠊⠀⡠⠃⢀⠀⠀⠉⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠯⠽⣒⣒⣒⣒⣒⣒⣒⣒⣪⠭⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠒⠊⠉⠉⠉⠒⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣊⠭⠥⠤⣀⡠⠔⠒⠈⠉⠒⠲⠤⣀⠀⠈⠑⠒⠒⠒⠦⣀
⠀⠀⠀⠀⣠⠔⠒⠋⠀⡞⠀⠀⠀⠀⠈⠓⠤⣀⣀⠀⠀⠀⠀⠀⠉⠈⠉⠑⠲⢄⠀⠀⠉⠒⠢⢄
⠀⠀⢀⡜⠁⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⢱⡀
⠀⡴⢻⠀⠀⠀⠀⠀⠀⠀⢉⠗⠒⠒⠈⠉⠀⠀⠀⠀⢸⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣗⠒⠒⠦⡀⢱⢄
⡞⠀⢈⡇⠀⠀⢀⠀⠀⠀⡎⠀⠀⠀⢀⡠⢤⡀⠀⠀⢸⠀⠀⠘⡶⠴⠀⠀⠀⠀⠉⠀⠀⠙⠂⠀⠈⢏⠀⠙⡄
⣧⠚⠉⠘⠦⠤⠞⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⡇⠀⠀⡈⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⡇
⡇⠀⠀⡤⡔⠋⠉⠉⠉⠓⠚⠉⣠⠚⠉⢳⠀⠑⠤⠴⠃⠀⠀⠀⠀⠉⠲⠤⣀⠴⠊⠉⢱⣀⡤⠃⠀⠀⢣⡀⠈⠢⡀
⠘⠦⣀⠀⠸⢠⠖⠒⡶⠒⠒⠊⣡⠴⠒⠁⢀⣀⡀⠤⠤⠀⢀⣀⣀⠠⠤⠚⠁⠀⠀⢀⠞⠀⠀⠀⠀⠀⡄⠘⢆⠀⡇
⠀⠀⠈⢢⡀⠇⠀⠀⠁⠀⡰⠋⠀⢀⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠚⠁⠀⠀⢲⠀⠀⠀⠁⠀⠘⡄⡇
⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠓⠒⢴⠁⠀⢀⠤⠀⠠⠤⠤⠤⠤⠤⠒⢆⠀⠀⠀⡌⠀⠀⠀⠀⠀⣇⡀⠀⠀⠀⢠⠃⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⢠⠖⠀⠀⠀⠀⠀⠀⠁⠀⢠⠃⠀⠀⠀⠀⣰⠁⠈⢳⠔⠊⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠙⠢⠤⠼⠤⢄⣀⠀⠀⠀⠀⠀⠒⠁⠀⠀⠀⠀⠀⠁⠀⠀⠸⠁⠀⢠⠃⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⣀⣀⣀⣀⠀⠀⠀⡱⠀⠀⠀⣠⠔⠒⠉⠓⣖⠒⠒⠉⠀⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠉⠓⠯⣀⡀⢀⡜⠁⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠀⠀⠀⠈⠉⠉⠑⠒⠤⣀⠀⠀⠈⠢⢄⣀⡠⠄⢀⡼⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢏⠑⠒⠤⣀⠀⠀⠀⠀⠀⠉⠓⠒⠒⠒⠒⡖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢹⠦⣀⣀⣠⠤⠤⠤⠤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠛⠁⠀⠀⣀⣤⣴⠶⠾⠟⠛⠛⠛⠶⠶⢦⣤⣀⡀⠀⠈⠙⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠛⠷⣤⡀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⢀⡴⠛⠁⠀⢀⣠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠛⠒⠶⣄⡀⠀⠈⠙⢦⣀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣴⠋⠀⠀⣠⡶⠋⠁⠀⠀⣀⣀⣠⣤⢤⣤⣤⣀⠀⠀⠀⠙⠷⣄⠀⠀⠙⢦⡀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⣼⠃⠀⢀⡾⠁⣠⣴⣾⣯⣀⠀⣠⡶⣿⡏⣼⡟⠁⡠⠀⠀⠉⠙⠲⣤⠖⣤⢬⣷⡄⢀⣈⣳⡀⠀⠘⣷
⠀⠀⠀⠀⢰⡏⠀⠀⣾⠁⢸⣿⣿⣿⣿⣿⣿⡁⢡⡈⣷⠜⠓⠚⢳⣾⣿⣷⡀⠀⢸⠀⠃⣜⣹⣿⣿⣿⣿⣷⡀⠀⠘⣧
⠀⠀⠀⠀⣿⠀⠀⣸⠇⢀⣀⣿⡅⠀⢰⠏⠉⠃⣰⡏⠓⢄⠀⠀⣿⠿⠛⠉⠙⣦⣞⡲⣤⣟⢸⣿⢿⡟⠋⠈⣧⠀⠀⢻⡀
⠀⠀⠀⢰⡇⠀⢀⣿⠚⣩⣼⣷⣶⣶⡏⠀⠀⣼⣫⠀⠀⠈⢣⠀⠁⠀⣠⠖⠛⠷⠶⣮⣵⡼⠟⠁⣈⡧⣀⠀⢻⠀⠀⠸⡇
⠀⠀⠀⢸⡇⠀⠀⡏⣼⣿⣿⣿⠋⢸⣧⣴⠖⣾⠃⠀⣄⢢⣠⠇⠄⠴⣁⡀⢰⡆⠀⣠⠙⠇⡆⠘⣇⠀⣸⣄⣼⠁⠀⢐⡇
⠀⠀⠀⢸⡇⠀⠀⣷⠹⣿⣿⠃⠔⠁⡿⠃⢀⠇⠀⠀⣿⠏⠀⠀⠀⠀⠀⠹⡄⠁⠀⠙⠘⠛⠧⠤⠞⠀⣏⣤⢘⠇⣴⠿⠋
⠀⠀⠀⠈⣷⠀⠀⢻⣀⡤⠘⠧⢤⣀⠿⠒⢺⡀⠀⡰⣿⠀⠀⠀⠀⠀⠠⢂⡇⠀⢀⣾⡋⠂⠀⣀⣤⠜⠸⢃⡜⢠⠞⣉⠙⢇⢀
⠀⠀⠀⢀⣽⡊⣠⠜⢿⠀⠀⠀⣰⠃⠀⠀⣾⡷⠛⠛⠻⡶⠛⠓⢄⠀⠀⣸⣥⡀⣼⠀⠻⠉⢭⠀⣸⣄⣠⡍⠙⠃⠾⠋⣠⠇⢸
⠀⠀⠀⠊⠀⡟⠁⠀⣨⠟⠚⠉⠉⠉⢙⣿⡈⣇⠀⠀⠀⠘⣧⡄⡼⠛⢱⡋⠀⠉⠋⠑⠲⣶⡞⢁⡀⠀⠙⡄⠶⠀⡴⠊⠁⠆
⠀⠀⠀⠀⠀⠁⡀⣸⠁⠀⣴⣶⡿⠿⠟⣛⠛⠉⠑⠠⢀⣀⠈⢿⣧⣤⣤⣃⠀⠀⠀⢰⡆⣽⣧⡘⠓⠀⣰⡇⠀⡰⠃
⠀⠀⠀⠀⡠⢸⠀⡏⠀⠀⣿⠁⠘⠓⠀⠉⢸⡆⠀⠈⠀⠉⠑⢄⠐⠀⠀⠈⢳⠤⣄⡈⠀⠙⢿⣿⡦⠚⠛⠶⣞⡥⠖⠲⡶⣄
⠀⠀⠀⠀⠱⠈⠀⣧⡀⠀⢻⡄⠀⢠⣤⣶⣾⣇⠰⠦⠀⠟⢀⢸⡆⠀⢀⠀⡄⠀⠀⣧⣀⣶⣞⡙⢿⠀⣠⡀⠉⠀⣸⣇⣼⡞
⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠾⢁⣄⡈⠧⣤⣶⣿⣤⣐⣲⣿⣡⡼⢇⠀⣸⢀⡇⢰⢀⣿⠟⠉⣍⡙⢮⠁⢉⣵⣏⠹⠿⠛⡉
⠀⠀⠀⠀⠀⢀⠀⢀⣼⣇⣀⣀⣠⡿⣳⣿⣿⣿⣿⣿⠛⣍⠁⢠⠘⡇⣿⣼⢁⡇⣼⣷⠛⡆⠙⠛⢸⡤⠋⠀⠈⣧⣿⠀⠇
⠀⠀⠀⠀⢀⡏⣰⣿⣿⣿⡟⠉⠀⠙⣾⠟⣿⡏⣤⡜⣧⠗⢒⠢⡄⣿⣿⣿⡿⣿⣿⣇⣰⣷⣦⣤⣼⠦⠈⠀⣠⠟⠁
⠀⠀⠀⠀⠈⠇⢿⣟⣻⠏⣰⣾⠡⣯⣉⣇⠉⠐⣦⣤⠿⠀⠈⠑⣇⠛⢉⣩⡿⢯⠠⠀⣿⣿⣿⣿⣿⣤⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⢨⠇⣼⣿⣿⠀⠻⡀⠉⠉⢻⠁⠀⣠⡀⠀⢀⡈⣻⢷⣎⠳⡖⣶⠀⡟⠉⠉⢠⣿⣿⠃⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠻⡿⠏⠀⠀⠙⣄⠀⢸⡀⠀⣿⢇⠀⢠⠋⠁⢀⣿⢀⡇⣯⡼⠃⠀⡰⠋⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣛⠓⢦⣑⠶⠃⠘⠶⠋⠀⠀⡞⡞⢸⠀⢀⣠⡴⢾⣷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣏⠀⠐⣧⣀⠀⠈⠙⠓⠲⠤⠤⠤⠼⣇⠣⣌⣻⠿⠛⣉⣩⣍⢹⣜⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣭⣉⣒⣒⠲⠤⠤⠄⠀⠀⠈⠙⠚⢛⣋⣉⣡⣤⠽⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠤⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⡀⠤⠤⠀⠒⠒⣒⣂⣉⣉⣡⡤⠤⠤⠤⠔⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣲⠖⣶⣖⣋⣉⣉⡉⠁⠀⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢺⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠀⣄⡀⠁⣁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠈⠩⠠⠤⠀⠀⠠⠦⣠⡀⢄⠀⠀⠤⡠⡀⠉⠑⠒⠢⠤⡀
⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀⠀⣀⠀⠐⠠⡔⠛⠠⢜⢲⡨⣭⠅⡍⠦⠀⠀⠄⡀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠗⠤⢄⡀⠈⠐⠂⢠⡱⠂⠂⠓⢬⣩⠷⡒⠂⡓⠭⢲⠤⢉⠉⢁⣈⡡⠖⡏
⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⠁⠒⠚⠒⠐⠰⠮⠲⠽⠴⠠⠔⠲⠀⠂⠓⠊⠉⠁⠈⠁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⠉⠙⠐⠒⠒⠒⠂⠉⠉⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠉⠉⠉⠉⠉⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⠞⠋⡁⠄⠊⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠢⠑⡁⠄⠂⠁⠐⠠⠁⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⣀⠤⠂⠁⠄⠑⠀⠤⠖⠒⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢔⣫⣵⠶⠛⡁⠁⠄⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠤⠤⠶⠒⣾⣿⣿⡟⠒⠲⠶⠤⠤⠤⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣃⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣤⣤⣄⣸⣿⡦
⠀⠀⠀⠀⠀⠀⠀⢸⠄⠉⠝⠛⢛⡛⠻⠿⠿⠿⠿⠿⠿⠽⠿⠿⠿⠿⠿⢛⡛⠛⠩⠉⠑⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠈⠀⢠⠁⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠔⠂⠄⢐⠀⠐⠕⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠃⠀⠀⠐⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⡀⠀⠀⠀⠀⠑⢆⠈⡀⡀⣅⠕⠁⠀⠀⠀⢀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠪⠴⣂⣀⣀⡀⠤⠬⠤⠤⠄⢀⣀⣀⣂⠤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠐⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠀⣀⣤⣤⠤⠴⢤⣀⣀⣀⣸⠦⢤⠤⠤⠐⠒⠒⠛⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠠⣤⡤⠤⠤⠤⠒⠒⡞⢛⣻⣧⣈⡏⠛⠿⣿⡛⣯⡄⠤⠀⠀⠠⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠐⠀⠀⠀⠀⠒⠒⢻⠓⢃⡻⠿⢄⡀⠀⠈⠒⠾⣿⠏⠉⠉⠉⠫⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢦⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⠀⠀⡀⠀⠂⠀⣀⣀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⣶⡎⠀⣀⢐⣨⠀⡀⢧⡑⠰⠂⢜⡿⣣⠴⣦⡀⠀⠗⠒⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢰⢻⣶⠷⡸⠿⡏⣀⠀⠉⣃⢠⠛⠻⡙⠺⣿⣃⠛⢤⠞⠁⣎⣴⣴⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡾⣤⣤⣢⣥⣀⣦⠤⢋⣠⣀⣴⡈⣰⣾⣿⣿⣷⣤⣤⣬⣭⣀⣶⣭⣷⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠈⠉⠉⣉⠛⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠠⣤⣶⣶⢶⣶⣶⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⣤⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡄⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡶⠿⢿⣿⣿⣿⣯⣿⣿⣽⣿⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠤⠤⠽⠿⠿⠿⠿⠿⠛⠛⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠠⠤⠄⠐⠒⢒⣂⣈⣉⣁⣁⠤⠤⠤⠤⠐⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣓⣲⠖⣖⠚⠉⠉⠉⠉⠁⠀⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⡄⡀⠀⡅⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠂⠉⠁⠀⠤⠀⠀⠠⠴⢠⡀⣢⠀⠀⠤⡠⠈⠉⠉⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⡊⠀⠀⠀⠀⣀⡀⡀⠀⠀⠠⡎⠉⠠⢜⠢⠨⣜⠑⠭⠦⠀⠀⢄⣄⠀⠀⢣
⠀⠀⠀⠀⠀⠀⡗⠦⢄⡀⡀⠐⠀⢨⡵⢂⡂⠳⢤⣬⠶⡒⠂⡒⠍⣱⠤⢄⠀⢀⣀⡠⠖⡏
⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠉⠉⠐⠚⠊⠒⠒⠮⠲⠵⠴⠀⠐⠒⠀⠊⠓⠈⠉⠉⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⣀⠉⠉⠀⠒⠒⠂⠀⠉⠁⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠶⠶⠶⢶⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠋⠉⠁⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⠉⠠⠐⡏⣀⠀⠀⠈⠉⠉⠉⠙⢧⣀⡀⠀⠳⠄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣄⡀⠀⠐⠀⠀⠇⠏⠀⠀⢀⠀⣀⠴⠀⠒⠀⢹⡄⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠏⣾⣿⣄⣀⡄⠂⠰⠀⠀⢀⡬⠁⡑⠀⠀⠂⠀⠀⠅⠀⠀⢇
⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⣞⠀⠀⠘⢛⡔⡀⠐⡂⣰⠟⢥⡆⠘⠀⠀⠀⡄⠀⢠⠁⠀⠸
⠀⠀⠀⠀⠀⠀⠘⣿⠟⡜⡇⠀⠀⠀⢂⡈⠒⡼⢮⠁⠀⠀⢈⣦⣀⠀⠴⠀⠀⠘⠀⡐
⠀⠀⠀⠀⠀⠀⠀⢿⣏⢞⠝⢦⣦⢲⠿⠿⣩⢿⣟⠀⠀⠀⠈⠟⣝⣹⢔⢧⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⣦⢮⣇⣇⠀⠀⠈⠁⠻⡇⡀⠀⠀⠀⠈⢻⠖⡇⣡⣀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢹⡼⢸⡙⣤⣄⡀⠐⡈⣏⢆⢀⡁⡀⠠⢚⠜⢁⠏⣷⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢲⡣⣜⣛⣛⠵⣹⢯⣖⠀⣥⣀⡀⠖⠚⡉⡫⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣭⣤⡠⠬⢿⣚⣭⣖⣒⣬⠵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠶⢤⣀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢁⡴⠞⢛⣶⠶⠛⢋⣉⣡⣤⣉⣉⡉⠛⠲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠛⠛⠛⠛⠿⣥⣴⠚⢉⣁⣤⣤⣤⣌⣉⠛⢦⣄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣤⠶⠖⠛⠒⠶⣤⡉⠻⢯⣅⡀⠀⠀⠀⠉⣳⣄⣈⣷⣤⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢀⡾⠋⢀⣠⡤⠶⠶⢤⣄⠙⠳⢤⣈⠉⠙⠛⠛⠋⠉⣁⣀⣀⣀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⢀⡟⢀⡴⠋⠁⠀⠀⣀⣀⣈⣛⣦⣄⣉⣙⣛⣓⣛⣛⣉⣉⣉⣉⣉⣙⣦⡀⠈⢷
⠀⠀⠀⠀⠀⠀⢠⣤⠴⠾⠷⠚⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠓⢲⡆
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠸⡟⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠶⠶⠶⠚⠛⠉

⣿⠛⠃⠀⠀⠀⠀⠀⠀⠀⣀⠘⠚⡋⠀⢀⡀⠄⠐⠀⠃⠀⠀⢲⣨⠄⡾⢒⣻⣿⡖⣴⢸⢛⣉⠛⡞⣿⠁⡺⠁
⣿⡖⢲⠀⠠⠆⠐⠚⢀⡀⠁⠔⠒⡃⠜⠻⢁⠚⠡⠄⠀⠁⠐⠈⣟⣧⣆⣐⡂⣿⡐⠿⣚⠐⣂⣤⢠⡹⠀⠁
⣾⣟⠋⠠⠀⠰⢉⠀⠀⠈⠁⠀⠂⠁⣄⣀⠀⠉⠀⢄⡀⠀⠀⠀⠉⢽⢬⡛⡵⠋⡛⠋⠏⠋⠉⠛⠛⠻⠾⠠⠁
⡿⡄⠀⠀⠀⠃⣄⣅⢠⡦⢀⣀⣤⣤⠦⠴⠻⠶⠶⠥⠶⣤⣾⣿⣿⣿⣿⣿⣶⠂⠁
⢷⣻⠭⢽⠛⢫⡟⣻⣗⣟⠝⠋⠁⠀⠀⠀⠀⠀⠀⢠⣤⣼⠏⠉⠙⠛⢿⣿⣿⡷⠀⠀⠠
⣍⣿⣸⣧⢯⡸⣧⠻⠥⣿⡩⠄⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⠀⠀⠀⠀⠀⠀⢸⣷⣆
⡟⣻⢐⡽⣭⠋⢼⡷⣴⣰⣖⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⢿⡿
⣡⢻⣼⣥⡍⠐⠚⠿⠻⠿⠥⠀⠀⠀⠀⠀⠀⠀⣹⣿⣽⠇⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿
⡡⢸⣾⣳⡟⡶⢲⢦⡆⡴⠟⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆⠀⠀⢀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿
⠁⢺⡼⠉⠓⠙⠻⡜⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠈⠉⣿⣿
⢦⠈⡈⣀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠀⢨⡄⠀⠀⠀⠀⡤⠀⠀⠉⠀⢤⠺⣇⡂⠀⠈⠹
⣛⢢⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠱⣄⡀⠀⠸⠃⠀⠀⠐⢢⣎⠀⠀⠉⠀⠀⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⣂⣴⣛⡿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠿
⣿⣿⣿⣿⣿⣿⡿⡿⡿⠏⠀⠀⢀⠌⠂⠐⠀⠀⡠⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⠀⡰
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣖⣒⡊⠁⠀⠘⢧⢤⠾⣶⣿⣧⡄⠀⠦⣀⠂⠀⠀⣠⣴⡇⠀⠀⠀⠐⢼⠀⢀⣠⣶⡿
⠋⠁⠀⢈⣿⣿⣿⣿⣿⣿⣻⣖⠂⠀⠀⠠⡺⠿⡿⢿⣿⡟⠀⠀⠈⠀⠀⠀⢻⣟⠇⠀⠀⢀⢰⡌⣷⢿⠃⠉
⠂⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣾⠌⢦⣠⠀⠀⣁⣤⣌⣹⡧⠀⠀⠀⠀⠀⠀⠀⠻⣂⣰⣰⣿⣿⣿⡿⠋
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⡷⣼⣿⣯⣛⢿⣇⢸⣷⣄⠀⡀⠀⠀⠀⠈⠘⢋⣉⡹⠋⠀⠀⠀⠀⣰
⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣻⣿⣾⣿⣿⡆⣹⣆⣠⣰⣾⡁⠘⠉⠀⠀⠀⠀⢀⣾⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣤⡴⠶⠶⠚⠛⢛⡛⠛⠓⠶⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠐⠂⠀⠀⠚⡉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠦⢬⣙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⢿⡃⠀⠈⠀⢀⣀⣀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢌⡛⢦⣀
⠀⠀⠀⠀⠀⠀⡔⠉⠀⠬⠍⠁⠈⠀⣠⠞⠁⠀⠨⡳⣆⠈⣶⣶⣶⣦⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠙⠢⣝⢷⣄
⠀⠀⠀⠀⠀⣼⠀⠀⠀⣠⠴⢦⣄⡀⠈⠆⠀⠀⠀⢠⢸⡀⢸⠿⠓⠋⠉⠉⠉⣻⣿⣶⣤⣀⠀⠀⠀⠀⠈⠳⡙⢦⡀
⠀⠀⠀⠀⢠⡏⠀⢀⡞⠁⠀⠀⠠⠙⡎⠈⡄⠑⠀⡘⢸⡇⢸⣠⢾⣿⠿⠿⠛⡛⠛⢿⣿⣿⣷⣄⠀⠀⠀⠀⠙⢎⢻⡄
⠀⠀⠀⢠⢿⣧⠀⢸⠀⠀⠀⠀⠀⠀⢘⠀⡇⠀⠈⢀⡼⢠⡟⢷⠋⣠⣶⣺⣿⣿⠭⠽⠽⣿⣿⣿⣧⡀⠀⠀⠀⠈⢧⢹⡄
⠀⠀⢠⡟⡸⣿⡀⠘⣆⠀⠀⠀⠀⠀⣼⢡⠟⠒⠒⢍⣰⣿⣧⣋⣸⣷⡞⠉⠀⠤⣄⣀⣐⣒⣊⣻⣿⣷⡄⠀⠀⠀⠈⢧⢻⡄
⠀⠀⣼⢱⠃⠘⢷⡀⠈⠂⠀⣈⡤⠞⢡⡾⣷⠶⡾⢿⣫⡭⢯⣽⣇⠎⣀⣠⣴⣛⣯⠭⠿⠟⠛⠻⠿⣿⣿⡀⠀⠀⠀⠘⡎⣧⠀⠀⢀⣠⣴⢄
⠀⢠⡏⡞⠀⠀⠀⠛⢢⣤⡩⣡⣤⣶⣿⣗⡦⡘⢞⣻⣯⣻⣟⣫⣝⣟⢻⡟⣿⠁⠠⠀⣶⣦⠭⣿⣿⣿⣿⣇⠀⠀⠀⠀⢳⢹⣤⣶⣿⣿⢿⣾⠆
⠀⢸⡇⡇⠀⠀⢀⡤⡷⣘⣧⣇⡎⣙⣿⣠⣤⠞⢋⣉⣼⢿⡟⣥⡄⢹⣿⡔⠁⣠⣴⣪⠵⠞⠋⠉⠀⠀⠙⣿⠀⠀⢀⣤⣾⣿⣿⣿⣿⠗⠋⣁⣤⢄
⠀⢸⡗⡇⠀⠀⠈⣧⢀⣉⣿⣿⣏⣩⣿⣻⠋⣼⣯⡬⣙⣦⣭⣉⣨⣿⣷⠿⡿⢿⠋⡄⠈⠀⠀⢀⣠⣤⣶⣿⣤⣾⣿⣿⣿⣿⡞⢉⣠⣶⣿⣿⡿⣊⡆
⠀⢸⡇⡇⠀⠀⠀⠿⢿⣉⡝⣏⣡⡿⣿⣿⣴⠃⢈⡀⢻⡢⠀⢨⢉⡁⢻⣿⡃⣾⣀⣤⣤⣴⠾⠿⣿⠟⠋⠀⢹⣿⠽⠛⣹⣾⣿⣿⡿⢟⣯⠷⠛⠁
⠀⠈⣧⢷⠀⠀⠀⠈⠎⣿⣿⣟⣥⠘⠢⣽⣿⣆⢈⡉⢉⡷⢰⣷⣪⣁⣾⠃⢋⡻⢤⡣⢳⡶⠞⠃⠀⢀⣤⡞⢋⢥⣶⣿⣿⢟⣻⠵⠚⠉
⠀⠀⢹⡜⡄⠀⠀⠐⠋⡧⣹⣻⣷⣿⠶⢿⠿⣯⢋⣟⣿⡝⠋⣈⣙⠙⠓⢚⡛⣡⡾⢓⡩⠔⣊⣤⣾⡿⠟⣋⠥⢊⣹⣿⣾⡏
⠀⠀⠈⢷⣹⡀⠀⠀⠀⠈⢻⣽⠋⡠⠄⡤⣶⠟⠋⠉⠉⣹⣮⣶⣾⣷⢕⡤⠛⠡⠒⣩⣴⡾⠟⠋⢉⠅⣋⣥⠶⠛⢉⣏⡾
⠀⠀⠀⠈⢷⡳⡀⠀⠀⠀⢿⢯⠐⡃⢭⣿⠃⠰⠉⠈⠡⠈⠉⣟⣷⠟⠉⢀⣠⣶⡿⠟⠉⠀⣀⣴⡶⠟⠋⠁⠀⢀⣞⡾⠁
⠀⠀⠀⠀⠈⢳⡽⣄⠀⠀⠺⣼⡄⢘⢺⣟⡇⠘⢌⡂⣂⡠⠞⠉⣀⣤⡶⠋⠉⠡⢀⣤⣶⡿⠋⠁⠀⠀⠀⠀⣠⣯⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢮⡣⡀⠀⠈⠉⠲⠞⠛⠷⣤⡴⠛⠁⣀⣴⠾⠿⠅⢂⣩⣴⡾⠿⠛⠁⠀⠀⠀⠀⠀⢀⢜⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠛⢮⡲⣄⠀⠀⡀⠀⠊⢁⣠⡴⠟⠋⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡿⠟⠈⢀⣤⠾⠟⠁⣀⣤⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⡴⠞⠩⢀⣤⣶⣿⣯⣄⣀⣀⣀⣀⣀⣀⣠⣤⣴⣶⣾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣉⣠⠔⠛⣁⡠⠴⠚⠁⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠤⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⢰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⢷⠀⠀⠀⢠⣇⣀⣀⠀⠀⢀⡀⠀⠀⣀⣼⠀⢀⡿⢀⡞⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠘⣇⠀⠀⣸⣏⡤⠙⣇⣴⣏⣹⢀⡾⢹⡏⠀⣼⠇⢸⡏⢁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢹⡆⠀⡟⢹⠛⣹⠇⣿⣸⠟⠹⣧⠇⣧⡼⡿⠀⠀⢉⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⢿⣸⡇⠈⠛⠋⠀⠈⠁⠀⠀⠀⠀⠀⠀⠛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠈⣛⣀⣤⠤⠤⠤⠶⠶⠶⠶⠤⠤⢤⡤⢤⢤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⠐⠂⠉⠁⣀⣀⠈⡠⡄⠢⢷⢄⠙⢄⠀⠀⠈⠣⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⢠⠋⢉⠙⠒⢢⣮⠭⣼⠋⢣⡳⡕⢌⠳⡕⡌⢧⠀⢀⠀⠈⣍⠳⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⣀⣠⢤⣧⡀⢀⣀⡠⠾⢶⢶⡟⡄⢧⡑⢌⠢⡱⣜⣜⢎⠒⠁⢑⣚⡠⠳⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡾⠀⠁⠀⢾⠁⢀⣠⠟⢙⡭⠒⣒⣚⡡⠤⠼⣜⢆⠳⣀⣱⡝⣄⠈⢆⠣⡀⠙⠦⡄⠀⠘⡄⠀⠙⢦
⠀⠀⠀⠀⠀⢸⡇⡠⠴⣒⣚⣚⣕⣋⡭⢗⣲⣊⡥⢖⣋⡉⠭⢹⡜⡄⠸⣉⠉⠪⡳⣜⢏⠺⡑⢄⠳⢄⠴⡻⢰⡀⠈⢧
⠀⠀⠀⠀⠀⠘⣿⢖⠭⠤⢜⣫⠥⢴⡊⠁⣠⠔⡪⢕⣲⠬⢖⡪⢳⡸⡀⠀⠧⠠⠼⣜⠮⣓⣊⣿⠴⠶⠮⠤⣈⠇⡀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⣖⢉⡽⠋⠀⠀⠀⠘⢦⡃⢮⡀⠻⠤⣄⣑⢄⣻⠱⣌⣓⢒⣒⡨⡽⢋⡟⣹⠁⠀⠀⠀⠀⠈⣇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⢸⡹⣾⠀⡰⠛⠳⡄⠀⠘⡗⠤⣌⣉⠓⢦⠀⠱⡹⡗⠚⢉⣍⡤⢬⣐⣪⠴⠓⠓⠒⢦⡀⠀⣠⡏⢀⡾
⠀⠀⠀⠀⠀⠀⠈⣇⠈⠻⣧⡀⠀⣿⠀⠀⣟⠀⣭⣌⡽⣦⣍⣉⣉⡻⢦⡪⢊⠜⣉⣀⡍⠀⠀⢠⡄⠈⢷⠞⠋⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠈⠙⠶⣯⣀⢰⠇⣰⣮⣭⣥⣲⣥⠾⡮⠽⠦⢴⡞⠉⠁⠀⢲⡄⠀⠀⠀⠀⣰⡴⠞⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠉⠛⠳⠶⣤⣤⣐⣊⡕⠊⠀⠀⠀⡼⠀⠀⠈⠀⣀⣧⣠⡤⠶⠛⠉⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠉⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠲⠶⠶⠶⠶⠶⠖⠚⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀⠙⢛⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠻⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢿⣍⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢰⡿⣹⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡾⣳⢏⡟⡷⣝⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣞⡽⣣⣿⣼⣁⢈⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⢫⢞⡽⣡⠞⢛⣿⣷⣮⡳⣞⡛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣄⠲⣌⡓⠎⢿⢸⡸⢥⣷⢳⡼⢪⣷⠿⢯⣚⢫⠷⢂⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣿⣯⣬⣷⣭⣻⣿⣟⣵⣏⣽⣿⣶⣼⣷⡾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣦⣤⣄⣉⣉⣉⠉⠉⠛⠛⠋⠉⣉⣉⣉⣁⣤⣤⣶⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡀⡄⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠈⢁⠈⠋⠁⠸⢄⣀⢤⢄⢠⣐⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣄⣀⠄⠊⠃⠈⠈⢨⠐⡰⠤⣔⠀⠀⠀⠀⠐⠁⢡⠐⠂⠳⠤⡤⠤⠄⣀⡀⣀
⠀⠀⠀⠀⠀⢀⣠⡤⠴⠒⠛⠉⠁⠀⡀⠐⡱⠋⠉⠓⠄⠀⡔⠁⡀⢀⠀⢈⡠⠒⠒⠠⡠⠂⠉⠁⠆⣀⠉⠉⠢⡄⢀⡀⠈⠉⠒⠲⠤⢀⡀
⠀⠀⠀⢀⠖⠋⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠧⢀⡀⡀⠀⠹⢄⣐⠀⠁⠀⡅⢄⡁⠀⡘⠀⡀⠀⢀⡀⡀⢓⠑⠄⢬⣄⠽⠆⣀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⢫⡀⠀⠀⠀⠀⠀⠀⠄⢐⠄⢀⣀⡅⡀⠐⢄⠀⡤⠊⠀⠈⢉⠎⠀⠀⠈⠲⢅⡤⠐⠉⠀⠀⠀⢠⠀⠀⠄⠀⠙⠦⣀⠈⠂⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠑⠦⣄⠀⠀⢀⠰⠊⠀⠨⠁⠀⠒⠀⠁⠢⢏⡀⠠⠄⠂⣜⣀⠀⠀⡜⠈⠣⢄⠀⠀⠀⢂⠌⠁⠈⠀⠈⢰⠀⠀⠀⠁⠢⢄⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠒⠾⢦⣄⣀⡐⣀⠀⠀⠀⠀⠀⠀⠈⠁⡲⠞⠀⠈⠑⠊⠠⠄⠈⠊⠑⠂⠠⡂⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣟⡛⠓⠶⠶⠴⠤⢦⣴⣤⣤⣤⣬⣤⣤⣤⣤⣤⣤⣤⡤⠤⠽⠶⠒⢚⣟⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠁⠀⠀⠀⠀⠀⣀⣰⠶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⠶⠶⠶⠶⠶⠾⠷⠶⠶⠒⠒⠚⠋⠉

⣿⠿⠿⠿⣿⢟⢿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣛⣟⣻⣻⣿⣿⣟⣟⣟⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣦⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣸⠃⠀⣤⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣧⠀⠀⠀⠀⣿⣤⠤⣄⠀⠀⢀⣤⡀⠀⢀⣤⣴⡇⠀⢠⣿⠀⣾⣡⡼⠁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢹⡆⠀⠀⢸⣿⣡⠆⠘⣧⣴⣿⡀⡿⢀⡾⢡⣿⠁⢀⣾⠇⠀⣿⠁⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢿⡀⠀⣾⠉⣟⠛⢻⡏⢾⣀⡿⠛⢹⣇⡞⢸⣀⡞⣿⠀⠀⠈⠛⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠘⣷⢀⡿⠀⠻⠶⠛⠀⠈⠋⠁⠀⠀⠉⠀⠈⠉⠘⠷⠶⠶⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠹⣿⠇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⣀⣤⡤⠶⠒⢛⣋⣩⡍⠉⠉⠉⠍⠉⠉⠉⢩⠗⠋⡭⠛⠛⠛⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⠉⠀⢀⣀⡀⠀⠀⠀⢀⣀⣤⣀⣠⠒⡏⢹⠛⣟⢦⡈⢢⡀⠀⠀⠀⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⣰⠋⢉⡉⠉⠓⢲⣯⠔⢶⡞⠉⣦⡑⢌⠲⣌⠳⡙⣄⢹⡀⠀⣀⠀⠀⣧⡙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣀⣀⣤⣅⠀⠀⠀⢀⣠⠾⠾⠤⣾⢣⡘⢇⠙⢌⠳⣌⢣⡹⣌⢦⠳⠞⠉⠓⡚⢁⠛⢄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠏⠐⠃⠀⣰⠋⠉⠀⢘⡾⠳⢖⡫⠭⠤⠴⠊⣉⣉⣧⢳⡈⢣⠀⠙⢎⢧⠉⠙⣆⠳⡀⠀⠯⣀⡁⠀⠈⢣⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣀⣀⣸⣠⠴⣻⠿⠶⣒⣋⣠⠴⠋⣉⡭⠤⠤⠜⣧⢹⠀⢙⡷⠛⠦⡓⢄⠸⣗⢭⡢⢄⠀⢹⠀⠀⢀⢧⢀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠐⣇⡔⣋⡥⠤⠤⣴⡾⠕⢒⣊⡵⠒⠒⢒⣫⠥⢖⡒⠂⢁⣹⡆⢧⠈⠓⢦⠀⠈⢢⡱⡌⢆⠙⣦⡱⡌⠑⢒⡡⠊⠘⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⣿⠊⠥⠒⠒⣫⡥⠖⠛⠦⣄⠀⣠⠞⣡⠖⣫⠥⠒⢊⡭⠂⣹⡌⢧⡀⠘⠢⠒⠚⣳⣝⡢⣥⣤⡾⠞⠛⠛⠒⠒⢤⡇⠠⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡩⢤⡟⠁⠀⠀⠀⠀⠈⢳⡏⠐⣅⡀⠓⠤⢤⣈⡳⣄⢣⠘⢦⣉⣓⣒⣈⡭⢶⠎⣩⠋⣼⠀⠀⠀⠄⠀⠀⠀⢳⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠸⡏⢷⣜⠀⣠⠞⠛⢦⡀⠀⠈⣿⠦⣄⣉⡉⠓⢦⡀⠈⢧⠹⡗⠤⠏⢉⢏⣠⠬⠥⢖⣡⠾⠖⠓⠒⢦⡀⠀⠀⢀⡾⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠙⢦⣇⠀⠀⠘⡇⠀⠀⣿⠈⣀⠒⠙⢳⣄⡑⠒⠚⠒⢿⣄⡰⣫⠏⡴⠚⠉⢹⠁⠀⠀⠀⠀⠀⠙⣆⡠⠚⠁⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠙⠷⣤⣀⡇⠀⠀⣿⠀⣒⠛⠛⠋⣊⠽⢛⢝⡛⢦⣉⠙⣁⣾⠴⠒⠒⠺⡄⠀⠀⠀⠁⠀⠀⢻⢀⣠⠞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠀⠈⠙⠳⢦⣴⣏⠒⠚⠿⠭⠭⢒⡪⢟⡭⠒⠉⠉⠉⡿⠁⠀⠠⠀⠀⢹⡀⠂⠀⠀⣀⣤⠾⠛⢁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠶⣶⣭⣥⣖⣉⣀⣀⠀⠀⢰⡃⢀⣀⣀⣀⣀⣨⣧⡴⠞⠛⠉⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠠⠄⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠛⠉⠁⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⣀⣴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⠶⠚⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣴⠶⠷⠶⠷⠶⠶⠶⠾⢻⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⡴⢟⡋⣠⣐⣒⣤⣄⠐⣃⡀⣈⡁⠰⢶⣄⣉⡁⠙⠛⠻⢷⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⠟⢀⡄⠈⣴⠏⢉⣉⠀⠴⠀⠉⠁⠋⠉⠰⢶⠀⠀⠁⠖⠸⠳⢦⣀⠬⡉⢿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡟⢋⡉⡡⢖⡚⠁⣥⡾⢳⡄⢀⠀⠀⣛⡛⠀⠙⢁⡀⢀⡤⠶⠖⢲⡀⠲⠆⣈⠐⠀⣤⠀⡍⢿⣤⡀
⠀⠀⠀⣴⡶⠶⠶⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠶⣶⡄
⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠠⢀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠲⣼⣆⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠨⠁⠰⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⠀⠀⠀⠠⢀⡠⢤⣀⠀⠀⠘⢆⢑⠍⠅⠑⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠁⠀⠀⡠⠐⠉⢀⣠⠤⠤⠶⡶⠲⢤⡤⢮⣝⡒⢮⡑⠄⠀⠈⠣⡡⡂⠚⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⢀⠀⠀⠂⢀⠔⠋⠀⠁⠚⠁⡀⠀⡠⢎⡀⠱⢢⢌⠑⠮⡙⢦⡙⣄⢦⡀⠈⠄⢔⠐⣌⠳⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⢀⠂⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡄⢌⠑⢤⠀⠰⡀⠄⠌⢧⡘⢦⡙⠦⢀⠂⠓⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠤⢀⠀⠀⠈⢆⠐⢄⠡⡀⠱⡀⠢⡠⠙⠲⢬⡙⢦⡀⢄⠠⢀⠀⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠁⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⠑⡀⠀⠈⡄⠀⢆⠱⣄⠙⠠⢄⡀⠁⢦⡉⢦⠹⡄⢀⠀⠑⡀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢀⠂⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⡀⠀⠀⡁⠀⠀⠀⠌⡑⠢⢄⡈⠓⣄⠹⡌⢧⡙⢦⡑⠢⣔⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠔⠀⠀⢀⠃⠀⠀⠀⠀⠀⠑⠄⠙⢄⠈⢢⠙⡂⣙⠲⣬⠟⠁⠘⡆⠈⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⠊⠑⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⢣⡀⢅⡈⠣⣌⡁⠀⠀⠀⠀⠂⠀⢂
⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠈⠃⠠⣀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠣⡆⠀⠃⢄⡭⠃⠀⠸⠀⠀⠠⠀⠡⡀
⠀⠀⠀⠀⠀⢀⠂⠀⠞⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⣀⣉⣉⡡⠄⠀⢤⠤⢠⠄⠠⠤⣀⡀⠀⠀⠀⠀⠀⢀⡖⠀⣁⡲⠊⠀⠀⠀⠆⠀⠀⠀⠑⡀⠐⠄
⠀⠀⠀⠀⢀⠂⠀⠌⠀⠀⠀⠀⠀⠀⠀⢀⣀⠛⠆⠁⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠐⠀⠉⠉⠐⡦⢠⡞⠀⠔⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠐⠄⠈⠂
⠀⠀⠀⢀⠂⠀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠆⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠔
⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠠⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠔⠁⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁




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

⠀⠀⢤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠁⣿⣎⠇⠀⠐⠫⠐⠛⠀⠀⢀⠺⠹⠀⠀⠀⠰⠥⠀⠄⠀⠀⣄⣔⠒⠀⠀⠠⠤⠭⠤⠀⠀⠠⠌⡫⠄⠀⠀⠈⠁
⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠠⠀⠀⠀⠀⠀⠀⡔⠒⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⢀⣀⡀
⠀⠒⠒⠂⠂⠀⠀⠸⠿⠃⠀⠀⠈⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⠈⠒⠁⠀⠀⠘⠯⠽⠃⠀⠀⠀⠾⠥

⠀⠠⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡒⢄⠀⠀⣀⡀⢄⡀⠀⠀⢀⣀⣀⠀⠀⠀⣀⡀⡀⠀⠀⠀⠄⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⢀⠘⠀⠀⠈⠉⠉⠁⠀⠀⠛⠛⠛⠁⠀⠀⠙⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠁

⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⣀⣀⠄⠀⠀⠀⠀⠄
⠀⠒⠒⠒⠂⠀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠐⠀⠈⠀⠀⠁⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠁⠀⠁⠀⠀⠁⠈⠁⠁

⠀⠠⠠⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⣀⣈⡁⠀⠀⢀⠰⠀⠀⠀⠀⠐⠂⠒⠀⠀⠀⠐⠐⠂⠀⠀⣀⣀⣢⡀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⡿⡿⠿⡟⢛⣋⣹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⣤⣤⣴⣶⣶⣤⣤⣴⣶⣾⠿⠿⠛⣛⣋⣩⣭⣤⣴⣶⣿⡿⠿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⠿⣿⢿⢛⡏⡟⡇⣿⣿⣿⣿⣶⣶⠿⠿⠿⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠿⣿⡏⢹⢸⠀⡇⢸⢰⡇⡏⡇⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠿⡿⠿⠿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡇⢾⢸⠀⡇⢸⠸⡆⡇⡇⣿⣿⣿⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⣾⢸⠀⡇⢸⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡏⢸⠀⡇⢸⠸⣤⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠀⡇⢸⢸⣿⡅⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⢸⠀⡇⣼⢸⢻⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⡇⣾⢸⡇⡟⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⡇⣿⢸⡇⡇⢸⠀⡇⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣨⣿⡇⣇⣿⣸⣇⣧⣸⣀⣇⡇⣿⣧⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⣋⠭⢭⣉⣿⣿⡿⠿⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣯⣤⠤⠬⣝⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣾⣷⣾⣿⣿⣿⣥⣴⠚⣉⣉⣭⣉⣍⣉⣙⠒⠢⢬⡙⠿⣷⣶⣬⣷⣼⣿⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⣋⡭⠥⠒⠒⠒⠒⠢⠤⢍⣉⠛⢿⣷⣤⠤⠤⠤⠤⢬⣉⡓⢦⣬⣷⡿⢛⣉⣉⣉⣛⠻⢿⣦⣄
⠀⠀⠀⠀⠀⢠⣿⠟⣵⠚⣡⠖⣊⣉⠭⠭⠭⣉⡓⠒⠤⣉⠲⣌⢻⣷⡖⠒⠒⠒⠦⣍⣾⡟⣡⠞⣩⠤⠴⠤⢬⡙⢦⡙⣿⣆
⠀⠀⠀⠀⣠⣼⣿⣴⣧⣾⣵⣾⣷⣶⣶⣶⣶⣶⣿⣷⣶⣿⣷⣾⣷⣿⣿⣿⣿⣿⣶⣾⣿⣶⣿⣼⣧⣶⣿⣿⣦⣽⣤⣷⣼⣿⣄⡀
⠀⠀⠀⠀⣿⣟⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢻⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠈⣿⡆⠛⠃⠀⠛⠛⠀⠘⠛⠀⠛⠛⠀⠘⠛⠀⠛⠃⠀⠘⠛⠃⠘⠛⠃⠀⠛⠛⠀⠘⠛⠀⠘⠛⠃⠘⠛⠃⠛⢃⣿⡇
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣤⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⡿⠋
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠉⠛⠛⠛⠿⠿⠿⠿⠿⢿⣶⣾⡿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⣿⡇
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠿⠿⠿⠿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠃
⠀⢰
⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡠⠀⠀⢀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠤⠤⠰⠐⠒⠒⠒⠚⠈⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣂⣛⣘⠠⡄⣅⣉⣉⣉⠉⠅⠠⠀⠠⠀⠤⠤⠤⠤⠤⠤⠤⠤⠠⠤⠦⠦⠤⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠁⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢚⡡⢭⣝⠀⠘⠒⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢴⡏⠛⠀⠠⠭⠀⢔⣍⡐⠠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⣌⠄⡘⡇⠐⢤⡤⡀⢴⡤⠈⡦⠄⡈⡴⡂⠤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⣝⣀⡘⡉⠀⠄⠓⠞⠁⠗⢄⠻⢋⠤⠫⡄⠀⣀⣱⠘⢓⣤⡉⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢠⠭⢄⠁⠂⡀⠉⣀⡆⣮⠤⠶⠟⢀⣀⡀⠇⠉⢉⣈⠄⢍⡺⢿⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡐⠏⡫⡁⠬⣧⡀⢀⣀⡈⠋⢀⠁⣀⡢⠈⠬⠒⠛⠴⡄⡈⢀⠂⣇⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢍⠛⠶⣾⣤⣃⣀⣚⠂⠊⠀⠀⡐⠴⡄⢾⢗⣀⣀⣄⣠⣬⣥⠶⠞⢛⣡⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣬⣉⣉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⢉⣉⣠⡤⠴⠚⠉⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢈⠉⢉⠙⠛⠛⠛⠟⠛⠛⠛⢉⠩⠉⠉⠈⠀⠀⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣔⠡⡀⠀⠈⠅⢒⣤⣀⢤⠀⢐⡖⠖⣦⢀⠀⠠⠀⠄⡄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣊⢐⠀⢀⠀⢸⡷⡿⡿⣻⢯⢫⢣⠍⠹⡅⠀⠀⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⢸⡇⣼⡳⣻⣾⣺⣿⠿⠧⠆⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣀⢀⠠⠩⠄⠍⠈⠡⠓⠁⠀⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⡀⠀⠀⢀⡃⠀⠀⠀⠀⠀⢀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠶⠶⠶⠶⠶⠶⠶⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⣠⡶⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⠋⠉⠉⣉⣉⣁⣀⣀⣀⣀⣀⣈⣉⡉⠉⠉⠛⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠛⠋⢉⣀⣤⣤⠶⠶⠖⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠓⠶⢦⣄⡀⠈⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣀⣤⡴⠶⠛⠋⠉⠀⣀⣀⣠⣤⣤⠴⠶⠶⠶⠶⠶⠒⠛⠛⠛⠛⠛⠛⠋⠉⠉⠛⠛⠛⠿⢿⣷⣄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⢀⣠⡶⠛⠉⠀⠀⣀⣤⠴⠞⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠛⠁⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⣼⠃
⠀⠀⠀⢀⡾⠁⠀⢠⡟⠁⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⣠⡾⠃
⠀⠀⠀⢸⡇⠀⠀⠘⣧⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁⠀⢀⣠⡾⠋
⠀⠀⠀⠈⢷⡀⠀⠀⠈⠻⢧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⠶⠛⠋⠁⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠈⠙⠛⠳⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠶⠖⠛⠛⠋⠉⠁⠀⠀⠀⣀⣠⡴⠶⣻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⠶⠶⠛⠋⠉⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⣛⠛⠛⠓⠒⠲⠶⠖⠒⠛⠛⠛⠛⠛⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⡴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣠⣤⣴⣶⣶⢄⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⣩⠟⠛⠉⠓⢦⠤⢄⣀⡰⠋⣀⣀⡀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠈⢧⠀⣰⢃⡞⡱⠚⠙⠲⣬⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣄⣀⠀⡼⠀⣠⠔⠒⠒⢦⡀⠈⡶⢃⡜⡼⢁⡤⠤⣄⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⠟⢁⣀⣤⣤⣈⣙⡇⢠⡇⠀⡖⠀⠀⢳⠀⣷⠞⡴⢃⡞⠀⠀⠈⡇⢸⢿⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⡤⠛⠛⠛⠛⠛⠋⢁⡴⢋⡠⠤⠤⢄⡙⡇⠈⢧⡀⠉⠀⣠⠏⠀⡷⠊⡴⠋⡤⢄⠀⣠⠇⡼⡼⢳
⠀⠀⠀⠀⠀⠀⠀⣇⠀⢠⠏⢀⠔⠊⠉⠉⠛⠋⠁⣠⠎⣠⠤⠤⠤⣈⠙⣄⠀⠉⠉⠉⠁⢀⠼⡃⢠⠛⢚⢥⣀⣉⡀⣿⡱⠁⢸⠄
⠀⠀⠀⠀⠀⠀⠀⡟⢦⡞⢠⠏⢀⠔⠋⠉⣉⠉⠉⣀⠼⣉⠑⠦⣄⣀⣱⣤⣉⣉⠉⢹⣞⡓⣄⢑⡴⡓⢯⡎⠞⢳⢵⢀⠽⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢻⠀⠑⢾⣀⠏⣠⠞⢉⣩⠭⣉⡉⠲⡌⠓⠤⢄⠤⠤⠤⢄⡈⠙⢿⡀⠙⣼⣾⣋⠉⣶⠯⣒⡯⣊⣘⣦⠎⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⠈⠓⠧⣴⣋⡤⠦⢤⡙⢦⡈⠒⠶⠖⠒⠲⢤⡀⠈⢢⡀⠱⡴⣿⠕⢮⡆⠛⠉⢢⣼⠗⠋⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡙⠦⣀⠀⠀⠀⠈⠙⠒⠦⠽⢤⣉⣙⣛⠉⠑⠲⣄⠉⢦⠀⢳⣀⣱⣹⣤⠤⠷⠒⠊⠁⠀⠀⢀⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⠙⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠤⠚⠉⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠉⠉⠓⠒⠦⠤⠤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠖⠒⠋⠉⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣍⡒⠒⠶⠦⠤⠤⠤⠤⠶⠶⠒⠒⣩⠵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠦⠤⠤⠤⠲⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠐⠲⣶⡶⣶⣶⣶⣶⣶⠖⠒⠀⠒⠲⣶⣶⣶⣶⣶⣶⣶⠒⠂⠀⠒⢲⣶⣶⣶⣶⣶⣶⡶⠂⠂
⠀⠀⠀⠀⠹⣶⣾⣿⣿⣿⠏⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⢀⣠⣤⣴⣷⣤⣄⡀
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡟⠀⠁⠀⠈⠉⢿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠈⠀⢺⣷⣿⣿⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠻⢿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⡿⠛⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⣀⠤⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⡀⣶⠒⢠⡄⠠⡿⠒⡟⠉⡽⠤⣼⠶⠚⠁
⠀⠀⠀⠀⠀⠘⠋⣹⡥⠚⠁⠀⣿⠄⣸⢹⡀⡇⠀⡇⠸⣇⣠⠋⠀⠀⠀⠀⠀⢀⡴⢀⡄
⠀⠀⠀⠀⠀⠴⠾⣿⠒⣒⡿⠃⣇⣀⡟⠀⢿⡇⠀⠃⠀⣀⣀⡼⣀⢀⡆⠀⢠⢞⠔⡩
⠀⠀⠀⠀⠀⠀⣀⣿⠚⠁⠀⠀⠉⠁⢠⡆⡀⢸⣤⣴⢧⡿⡵⠿⠙⠋⠀⠀⠋⠁⠋⠀⠀⣀⢀⣀⣀⣄
⠀⠀⠀⠀⠀⠊⠁⠋⠀⠀⠀⡀⣤⣱⣿⣷⣧⣼⣨⠌⠈⠃⠀⠀⢀⣀⣤⠤⠄⠒⠢⢄⡰⠛⠋⠋⠉
⠀⠀⠀⠀⠀⠀⢸⣤⣼⣖⠿⠻⡟⡇⠸⠇⠁⠈⢀⠤⠂⠑⢎⣹⡋⢱⡌⢳⣉⠙⢦⠀⠈⠲⢄⠀⠀⠀⠄
⠀⠀⠀⠀⠀⡴⢸⠋⢹⠁⠀⠀⠃⢀⣀⣤⠖⢊⠏⠀⠀⠀⠈⠻⡫⣳⣄⡼⡌⡇⢸⢗⡒⢄⠀⠙⠦⣀
⠀⠀⠀⠀⠘⠦⠎⠀⠀⢀⣠⠔⠊⢹⠟⢛⢦⡄⠀⡠⠀⠒⠐⢆⠝⠀⠀⠑⢶⣃⣾⡄⡷⢸⡠⢒⢄⢌⠓⢤⡀
⠀⠀⠀⠀⠀⠀⢀⡤⠚⠉⠀⠀⢀⠨⠅⠌⠰⠌⠒⢦⣶⣆⠀⠌⠀⠀⠀⡀⠀⢏⡲⠷⢧⣟⠄⡼⢁⢮⡀⠤⡉⠲⡀
⠀⠀⠀⠀⠀⢠⡏⠴⢦⠀⠢⣈⠪⠑⠈⠀⠄⠸⠠⠄⠈⠙⢾⡀⠀⠁⠀⠀⢡⠊⠀⠀⠀⠙⢾⡡⡢⢊⡤⡪⢈⡦⢷
⠀⠀⠀⠀⠀⢸⠣⣀⠈⡇⠒⢂⠀⡀⡤⠠⡈⢢⡁⠂⣴⡶⢷⡌⣑⣾⣶⢀⠃⢠⠒⠒⠀⢢⠈⡟⣢⡭⠖⠋⠁⠀⡼
⠀⠀⠀⠀⠀⢸⠀⠈⠳⣇⡈⠈⠱⡈⣆⣧⣌⣀⠤⠖⠳⢄⡔⠉⠀⠉⢎⠛⠤⣸⢰⣰⡦⠼⠒⠉⠡⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠙⠢⣔⡙⠊⠉⡹⠁⢀⣀⠀⡜⢠⠊⠉⣆⣸⡤⠤⠚⠋⠁⠀⠀⠀⠀⢀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠙⠲⢤⣇⣠⣃⣀⣦⡥⠷⠒⠊⢉⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⢤⣤⡀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡤⠖⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⠀⠀⠀⠀⠤⠄⠀⠀⠀⣀⡤⠴⠚⠉⠀⠀⠀⢀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠙⠲⠤⣤⡤⠤⠖⠚⠉⠁⠀⠀⠀⣰⣆⡜⣧⢻⠎⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠟⠛⠉⠉⠉⠉⠁⠉⠉⠉⠙⠛⠛⠷⠶⣤⣀
⠀⠀⠀⠀⢀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠨⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢀⠶⠚⡡⣚⠒⠤⣒⠲⠄⣀⠀⠀⠀⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⢃⠀⠈⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⠀⠀⣀⡴⢾⡄⠉⠚⢪⡡⠺⢔⠿⣋⠀⠈⠤⡄⠀⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⡀⠖⠐⡾⠠⢂⠅⠆⢤⡔⠚⢀⡐⠒⢬⡀⣍⡛⠓⡀⢰⠤⠄⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⡄⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⡴⠁⢴⣯⣙⠕⠻⠆⢠⠛⡔⠂⠀⢒⠆⡐⢹⠖⡙⡖⣡⡆⠀⠀⠄⠠⡀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠸⡄⠀⠀⠀⢸⡇⠀⠀⡀⠚⢃⢘⠎⠀⠈⣢⠐⠀⠀⠒⢌⣯⢇⠠⠊⠠⣃⠀⠀⠦⠔⠀⠀⠀⡋⢂⠉⠄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⡸⠁⠀⣠⣏⠀⠸⠀⠄⢀⠀⠴⠀⠀⣨⡀⣳⢀⠠⡴⠄⠀⢶⡌⠔⡦⠏⣐⠃⠩⣀⣀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠱⡀⠀⢸⣇⡬⢂⢀⡞⠁⠘⢦⣀⠐⠌⠑⠘⢒⡔⡨⠀⠀⣹⠈⠙⢷⣄⣀⠈⡒⠁⠀⠇⠨⡻⡟⠋⠪⠁⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⢣⠀⠀⢿⣇⢴⣾⡆⠀⣠⠏⠹⠛⠷⠖⠒⡎⠀⡰⠉⡀⢻⠀⠀⠨⣤⣍⠘⡀⣠⠴⢂⠂⢀⠯⠉⡓⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢿⡜⢉⠚⣢⡌⢠⣄⠑⠀⠐⠄⢈⣶⡑⠐⠄⠚⢄⠀⠂⠠⢈⣛⡒⢁⢊⠅⠀⠸⠸⡌⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⢆⠀⠀⢻⣦⡰⠇⢌⡌⠀⠌⣡⣄⣂⠍⠁⠙⢮⣐⣐⡢⠛⠫⠋⠉⠑⡃⠁⢁⠚⢄⡀⢁⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠈⢆⠀⠀⠹⡻⢶⣔⡀⠐⡶⢊⠀⣀⣥⠀⠀⢠⣠⣠⢠⡀⠭⡄⢢⡭⠔⠠⠃⠀⢩⣵⠟⣵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠙⠻⢾⣤⣔⠋⠙⠃⠒⠡⠪⣤⠀⠉⠭⡣⠈⠒⠒⣈⣤⣽⠾⡋⢕⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠐⡄⠀⠀⠀⠳⡄⠀⡈⠉⠛⠿⠶⢷⣾⣶⣶⣴⣶⣾⣶⡷⠿⠟⠛⠉⢐⠠⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠘⠀⠀⠀⠀⠈⠳⣀⠀⠁⠀⠀⠀⠀⢀⣂⣀⡀⠀⠄⠀⠀⠀⠀⠌⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⢛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠱⡀⠀⠀⠀⠀⠀⣿⣶⣭⣙⡛⠶⠶⠾⠿⠶⠞⣛⣫⣥⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢆⢳⢀⣆⠀⠀⠀⠈⠻⣟⡻⠿⠿⢿⣿⡿⠿⠿⠿⣛⡯⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡝⣆⢣⠸⣆⠀⠀⠀⠀⠀⠉⠛⠒⠒⠶⠶⠒⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠸⡆⢇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⠿⠿⡿⢿⣟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣻⣛⣻⣿⣻⣻⣟⣟⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀
⠀⠀⢀⡤⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢼⡁⢡⠂
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡤⢀⠀⠀⠀⠐⠉⠁⠁⠏
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠁⠀⠀⡈⡂⠢⢀⠀⠀⢆⠀⡀
⠀⠀⠀⢢⠑⢄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠊⠀⠀⠀⠈⠐⠠
⠀⠀⠀⠐⠀⠂⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢀
⠀⠀⠀⠀⠐⠀⠈⠢⡈⠢⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠁⠠⡀
⠀⠀⠀⠀⠀⠈⠂⠀⠀⠑⠄⡁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠅⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⡀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⠀⠀⠁⠂⠄⡁⠂⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠦⢀⠀⠀⠀⠀⠁⠂⠠⠈⠐⠋⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠈⠐⠠⠀⡀⠀⠀⠀⠀⠀⠐⠠⠤⢁⣀⣐⡀⠤⠤⠤⣄⣀⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠜⠀⠠⠤⣄⡀⠀⠀⠂⠄⠄⠀⠀⠄⠀⠀⠀⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠅⡀⠀⡠⡈⡦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⡈⠄⢲⠁⠀⠁⡘⢍⠐⠄
⠀⠀⠀⠄⠀⠄⠀⠤⣖⠠⠔⠖⠁⠀⢤⠂⠶⣤⠰⠑⢶⢪⠒⢬⠒⠄⠠⠀⠠
⠀⠀⠀⡁⣏⡭⠐⠉⣍⠀⠀⣉⠀⢠⣯⠋⠀⣩⠁⠀⣉⢫⣡⢩⠉⠃⣽⣃⢈
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⢀⡤⠤⢴⡶⠶⠒⠒⠛⠓⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⢤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠉⠓⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡌⠙⠲⠦⣤⣉⣉⠛⠒⠶⠦⠤⢤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣼⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⡆⠀⠀⠈⠉⠉⠛⠓⠒⠒⠒⠒⠒⠒⠛⠛⠛⣛⣻⣿⠿⠟⠛⢛⣿⠿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⣙⣲⣦⣄⣀⣀⣀⣠⣤⡤⠤⠶⠶⠒⠒⢛⣉⣉⣤⡤⠶⠖⠛⣹⣇⣤⣷⡄
⠀⠀⠀⠀⠀⠀⣀⣠⡄⠙⢯⡉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⣉⣉⣁⣠⣤⠴⣒⣋⣽⣟⣩⠿⡟⠁
⠀⠀⠀⠀⠀⣾⣋⠀⠀⠀⠀⠈⠳⠦⠤⠤⠤⠤⢴⣶⣒⣚⣿⣭⣭⣿⣶⣚⣻⠯⠿⠟⠛⠉⠁⠀⣼⠃
⠀⠀⠀⠀⠀⢸⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠛⠻⡶⢦⡀
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣠⢃⣤⠏
⠀⠀⠀⠀⠀⠀⠀⠹⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣴⡶⠶⠛⠋⡽⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢶⣬⣿⠿⠿⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⢀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣶⡶⠶⣚⣛⣻⡏⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⣰⠃⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⣸⠃⠀⠀⡼⠃⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣴⠃⠀⢀⡞⠁⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⠃⠀⢠⠟⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⡼⠃⠀⣰⠏⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⡼⠁⠀⣼⠃⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⢀⣤⣤⡾⠁⠀⡾⠁⢀⡾⠁⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⡤⠤⣄⠀⠀⡴⠋⠉⠉⠙⠶⠋⠀⣼⠁⠀⣼⡁⢠⡟⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠙⠿⠁⢤⡀⠘⣧⣼⣧⡀⠠⠤⣄⡠⠤⣼⠃⠀⡼⠁⣻⠏⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⣠⠤⠀⠙⣄⣀⠀⠈⢻⣄⣀⣀⡠⢴⠇⠀⣼⠗⠺⠧⣴⠟⠉⠉⠙⣦
⠀⠀⠀⠀⠀⢀⡴⠋⠉⠀⢠⠤⢼⠁⢀⣀⠀⠁⠈⢣⠀⢸⡇⠀⠉⣴⠟⠛⠛⠃⣀⠤⣀⠀⠀⠀⠀⡀⠙⠛⠛⠲⣄
⠀⠀⠀⠀⢀⣼⣇⡠⠒⠒⠃⠀⣀⣰⠃⠀⣰⠞⠋⠛⠷⠞⠋⢳⣾⠃⠀⢀⣀⣰⠁⠀⠈⠉⠙⡆⢠⡈⠑⢆⡀⠀⢹⡆
⠀⠀⠀⣠⠟⠁⠈⠛⠦⣴⡒⠞⠀⠈⠀⢰⡇⠀⢀⡤⣤⠤⠄⠐⠟⠛⠻⣏⠀⠈⠀⡴⠒⠦⠀⠀⠀⠹⢄⠈⠉⡇⣸⡇
⠀⢀⡾⠁⠀⠰⡄⠀⠀⠈⠙⠳⢦⣄⣠⠟⠁⠀⡎⠀⠀⣀⠄⠀⠀⠀⠀⢸⠋⠉⠉⣀⣀⣠⣤⡤⠴⠶⠾⠛⠛⠛⠉⠻⣆
⣴⠋⠀⠈⢆⠣⠸⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⣍⡁⡠⠔⠃⢀⣀⣀⣤⠴⠿⠚⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⠀⡠⢀⠀⠘⢧⡀
⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⣟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠔⢡⠃⠀⠀⠈⠳
⠀⠀⠀⠀⠙⡷⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⢹⠌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⠶⠒⠛
⠀⠀⠀⠀⠀⣇⠀⠀⠉⠛⠶⢤⣄⡀⠀⢀⡼⠃⠀⢸⡇⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠒⠛⠉⠉⠀⠀⣿
⠀⠀⠀⠀⠀⣿⣤⣀⠀⠀⠀⠀⠈⠉⠛⠛⠀⠀⠀⢸⡇⠀⠀⠀⠻⣦⡤⠴⠶⠒⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿
⠀⠀⠀⠀⠀⢹⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⠉⠉⠀⣿
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣤⣧⣤⠴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠘⡇⠀⢠⣣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣧⠈⠓⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣡⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣿⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠁⡰⠃⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢸⡀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠖⠛⣻
⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣿⣤⣤⠴⠶⠚⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠴⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⣿⠀⠀⣀⣠⡤⠶⠚⠋⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣿⣞⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⡤⣦⣂⠤⣀⡀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⢖⠐⠢⣀⠀⠀⠀⠀⠁⠀⠈⠙⠿⣽⡻⠶⣭⣑⡲⠤⣀
⠀⠀⠀⠀⠈⠳⣅⠈⠑⢤⡀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢮⣝⠻⣷⢦⣍⣒⠤⣄
⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠈⠓⢦⣀⣤⠤⣤⣠⣶⣶⡦⠽⡻⡶⣭⣚⠿⣷⡦⣭⣑⠲⢤⣤⣤⢄⣀
⠀⠀⠀⠀⠀⠀⢀⣠⣼⣦⠀⠀⡤⠸⠿⣯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣭⣒⠭⣛⡲⢤⣍⡛⠶⢬⡯⢲⣦⣤⣀
⠀⠀⠀⠀⢀⢔⡿⠋⠀⠀⡇⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⢀⣀⣀⠀⣀⣀⣀⣀⣉⠛⢶⣭⣑⠪⢝⡓⠶⣬⣁⠒⠭⢙⣿⣦⡀
⠀⠀⠀⠀⣾⠘⠀⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⠯⡿⣿⣿⣟⡩⢋⡹⣼⣽⣭⡭⣻⢻⣻⣯⣻⣻⢶⣭⣑⠢⢍⡛⠷⣦⠬⣉⠓⠠⢄⡀
⠀⠀⠀⠀⠟⢾⣦⣀⣠⣔⣣⣀⣻⣀⣀⣀⡀⣼⡳⡆⠎⣷⣶⠿⣿⡽⣟⣡⣷⣿⢿⣿⢯⣽⣿⣶⣾⣿⣿⣶⡬⡉⠒⢭⣶⣬⣕⡢⠬⣉⠒⠢⠤⣀
⠀⠀⠀⠀⢸⠀⠉⠛⠿⣽⣻⡽⠧⢭⣯⣇⡋⣾⢨⠣⣘⢮⡳⣎⢕⡭⣶⣇⢣⢣⢮⢟⢾⡿⣿⣷⢹⣻⣿⣿⠿⢲⣭⡲⠤⣉⠓⠬⣕⡒⠭⢒⡢⠠⢉⠰⡆
⠀⠀⠀⠀⠀⣧⣄⡀⠀⠀⠀⠉⠉⠐⠒⠚⠹⠭⠤⠿⢾⣚⣿⣿⣿⣷⣿⣾⣿⣾⣞⣻⣛⣒⣋⡩⠭⠔⠒⠊⠉⠁⠀⢈⣱⡦⢉⡲⠤⣈⡑⠢⣄⡉⠛⠈⠁
⠀⠀⠀⠀⠀⠸⡄⠉⠓⠢⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢴⣺⣥⡼⠀⠀⠈⠑⠢⣍⢳⡔⣻
⠀⠀⠀⠀⠀⠀⠹⣾⣛⡛⢷⢤⡈⠉⠙⢓⣲⣶⣦⠠⠤⠤⠤⠤⠤⠤⣤⡤⠤⠤⠤⠤⠤⠤⣴⠶⡿⣍⡉⢁⡴⡿⣫⣴⠃⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠙⣶⠬⡳⣕⢌⣑⣊⢕⡪⠍⢭⠳⡂⠤⠤⣤⢶⣺⣙⠿⡒⢤⣀⣀⢔⡯⢚⠭⠭⢝⣮⣉⠞⡴⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠮⣺⣗⠾⣲⠷⠊⢩⣐⢽⣯⣶⣦⢕⡩⢄⡠⢽⣯⣖⠀⣒⡩⢖⣥⣮⣽⣿⣿⣿⣟⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡦⠤⠴⠯⠭⣍⠢⢍⣛⣛⣋⣱⠾⣓⣒⣻⣿⡯⠍⡿⣿⠿⢓⡭⠒⠒⠛⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠶⣤⣄⣀⡀⠉⠒⠒⠀⠒⠂⠉⠀⠀⠈⠉⠙⠒⠒⢒⢉⣁⡀⣤⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣭⣉⣉⡙⠛⠛⠛⠛⠛⠛⠛⠛⠉⢉⣉⠩⠤⣒⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⠭⠭⠭⠭⠭⠭⠭⠭⠽⠶⠒⠂⠉⠁



⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣇⣶⣽⣤⣦⣭⣵⣥⣿⣿⣧⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠀⢠⠠⣠⣤⣄⣀⣀⣠⣤⡖⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢤⣾⡛⣷⡶⣾⡿⢿⣂⣏⣩⠿⠭⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠪⣐⡛⠾⠯⠽⠧⠻⠿⠥⠂⠢⠀⢠⡠⠀⠀⠀⠀⢀⡀⣀⣀
⠀⠀⠀⠀⠀⢄⣀⡐⢲⣮⠩⠏⣈⠀⠀⠈⠑⠷⡶⡒⠂⠀⢀⠀⣠⠐⣿⣄⠀⠈⠉⠉⠁⡄⠉⠀⠀⠀⢄⡀
⠀⠀⠀⠀⠀⠼⢿⣶⣎⡉⠲⠤⠈⠁⠐⢄⠀⠀⠐⠌⣪⣵⢢⠊⡝⠛⠛⣿⣿⡾⠒⠀⠈⠁⠀⠀⠁⠀⠀⠈⢢
⠀⠀⠀⢀⣤⣕⢌⣙⠿⣷⣷⡆⡌⢓⣠⣦⡀⣁⡐⢧⣿⣯⣠⡐⠀⠠⡀⢸⣿⠙⠄⠀⠀⠀⠀⠤⠄⠀⠀⠀⢠⠧
⠀⠀⠀⠈⠉⠛⢿⣾⣭⣝⣛⣛⠓⠛⠻⠭⢁⠙⠛⠀⠉⠻⠿⣍⣂⠤⢄⣤⣿⣶⣦⣄⠀⠀⢀⣠⡄⠈⢁⣺⣅⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠆⣀⡒⣀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠉⠉⠉⠉⠉⠋⠉⠉⠉⠉⠁⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣴⣤⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠉⠉⠉⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠖⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⠚⠓⠒⠒⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢠⠞⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣧⣿⣤⣤⠴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡀
⠀⠀⠀⠀⠀⢀⣠⠤⠤⣽⣿⡤⠤⢤⣀⠀⠀⠀⢰⡟⠛⠛⡇⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⢰⠏⠀⠀⠀⠈⠉⠀⠀⠀⠈⢿⠉⠉⢹⣇⣀⣀⡏⠉⠉⠉⠉⠉⠉⠉⢹⡇
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⢉⣡⣤⣤⣤⣤⣤⠀⢠⣼⣇⡀
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣀⣀⡤⠾⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠷⣤
⠀⠀⠀⠀⠀⠈⠧⣄⣀⣀⣀⣀⣀⡴⠋⠀⠀⠀⠳⢦⡶⢶⣋⣽⣙⣷⣲⣶⣶⣖⣛⣓⣶⢶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣉⣈⣁⣈⣉⣀⣉⣩⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⣠⠅⣀⣀⠈⠁⠄⠬⢊⡁⠒⠐⠢⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠇⠀⠀⢀⡌⠉⣩⡉⣀⡈⠛⠀⣈⣠⡀⠀⡐⠚⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢠⠞⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠖⠊⢀⠀⠦⣤⠴⡀⢀⡈⣠⣄⠀⠚⣀⠀⠆⠩⢁⡀⠘⠄⠀⠰⠀⡀⢣⢄⠀⠀⡠⠁⣴⠋⣠⠋
⠀⠀⠀⠀⠀⠀⢠⠂⠉⠀⢈⡡⢈⣤⠬⠿⠦⢐⣊⠀⣉⡀⣤⡴⠀⢭⡍⣀⡀⠲⠊⡉⠀⠐⠀⠄⡀⠀⠲⡘⠓⠊⢀⡞⢁⡼⠁
⠀⠀⢀⠀⠀⣠⠈⠡⠆⠀⢋⡴⣡⠄⠴⠫⣂⡄⢁⡄⠉⠣⠈⡐⡠⡆⡀⠄⠀⠤⠙⢾⡀⠄⠐⠚⠛⠐⠆⡛⠐⠀⢫⣠⡎⠀⠀⠀⠠
⡀⣰⠁⢀⠐⣥⠒⠋⠈⣟⠡⣖⡀⠒⣃⠶⠌⡴⢩⡆⠐⢶⠠⠟⣰⠀⠓⠀⣉⠲⠀⠒⠂⠙⠜⠁⠴⠂⢀⠑⡤⢀⠠⠄⣆⠢⢄⠀⠐⡌⠆
⣇⠻⣀⣿⠃⠃⣴⠛⡩⠋⠗⠂⠀⠲⣬⣤⣀⣦⡈⠣⠰⠆⠓⣉⣉⡀⠣⣭⣥⣄⣙⠉⠀⡈⢰⡑⠂⠼⢀⢀⣬⣀⠄⢐⠈⠧⢀⢰⡠⠂⣼
⠹⡳⢌⡳⠶⣈⡉⡙⠲⢥⣶⡁⢦⠀⠦⠤⡀⠄⠉⣤⡐⢋⠴⠶⣦⣍⠂⠐⠂⠢⡀⠁⡍⠹⣌⣉⢉⠉⠟⣠⣶⢀⠳⠘⣚⣶⠥⢊⡡⢺⠇
⠀⢳⠀⠉⠒⠢⢍⣙⡓⠶⠤⢶⣌⡳⣀⠂⠤⣈⡐⠼⠆⠾⢉⡀⢶⡄⠈⠓⠀⠲⢞⣠⡀⠚⠀⣅⢠⣸⣀⡿⠥⠶⠒⣋⡩⠤⠒⠉⠀⡞
⠀⠘⡄⠀⠐⠀⠀⠀⠈⡉⠑⠒⠒⠠⠬⠭⠭⣈⣙⣒⣒⣒⣛⣛⣛⣒⣛⣂⣚⣒⣛⣛⣋⣀⡉⠭⠭⠤⠐⠒⠒⠉⠉⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠒⠂⠢⠤⠤⠤⠤⠄⠀⢀⣀⣀⣀⣀⠀⠀⠠⠤⠤⠤⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠏
⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡶⢶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣉⠙⠛⠲⠶⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠴⠖⠛⠋⠉⣀⠤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⢄⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠤⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃
⠀⠀⠀⠀⠀⠀⠀⢰⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠨⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠈⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠲⠊⢄⠀⠀⠀⠀⠀⠸⣉⡀⠠⠔⠀⠷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⠶⢦⡀⢠⠀⠀⠀⢨⣇⠀⠀⠀⠀⠀⠑⠦⢄
⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⠱⡄⠀⠠⠙⠂⠀⠀⠀⠀⠀⠀⠌⢧
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠈⢆⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠓⢄⣀⣜⠁⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⢤⣤⣤⣤⣤⠀⡤⠄⠀⠀⠀⢠⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⣰⢻⠃
⠀⠀⠀⠀⣶⣿⣿⣀⣿⣏⠷⠀⠘⣛⣩⡅⠰⣖⣤⠆⠀⢀⡏⣷⣄⠀⠀⠐⠀⠀⠀⠀⣠⡾⢃⡏
⠀⠀⠀⠀⠠⡹⣿⣿⣿⣿⣅⠐⠂⣾⡇⠁⠀⠈⢳⣿⠃⢰⠃⠘⣿⣦⡀⠀⠀⣀⣴⣿⠏⠀⢸
⠀⠀⠀⠀⠀⠉⠺⠿⠛⠟⠃⠞⠀⠁⠀⠀⠀⠀⢸⣏⣼⣿⠀⠀⠈⠻⣿⣶⣿⡿⠋⠁⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠮⡻⣇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠶⠶⢚⣛⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⣤⡤⠶⠚⣛⣉⣭⡴⠶⠚⠛⠉⠉⣀⣀⣤⣴⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⢛⣋⣩⣤⠴⠖⠛⠉⣁⣠⣤⣤⣶⡾⠿⣟⣛⣭⡭⠷⠖⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡞⣱⢞⣫⣭⢹⡶⣦⣤⡤⠶⢚⣛⣩⣥⠶⠖⠛⣉⣉⣤⣴⣶⠾⢿⣛⣛⣭⡭⠷⠖⠛⠋⠉⠁⠀⠀⠀⢀⣀⣀⡀
⠀⠀⡀⠀⣀⣤⣾⠿⢿⣷⣿⣼⣿⣿⣾⠿⣯⣶⠞⠛⣉⣉⣤⣤⣶⠾⢟⣛⣭⡭⠶⠚⠛⠉⠉⠁⠀⠀⣀⣀⣠⣤⡤⠶⠖⠛⠛⠉⠉⠉
⠀⠈⣠⢞⣥⣾⣿⣿⣟⣿⣿⡿⣯⡻⣎⢳⡙⣏⢿⣿⣻⣿⠿⠿⠛⠛⠉⠉⠀⣀⣀⣤⣤⠴⠶⠒⠛⠋⠉⠉
⠀⣼⣿⣿⡿⠟⢛⣋⣩⣿⣝⣿⣎⣷⢹⡄⣿⣿⢈⣏⣁⣀⣠⣤⡤⠶⠞⠛⠛⢉⠁
⢀⣿⣿⣴⣶⣿⣿⣿⡭⠷⣿⣿⡟⣿⢸⣷⣇⡟⣼⣿⡉⠉
⣸⢿⣿⣿⣋⣉⣤⣴⣶⣾⣿⡿⣱⢇⣾⣽⣿⣷⡟⢿⣻⡄
⣿⣾⠈⠉⠉⠹⣟⣿⣿⣿⣿⣳⢫⣿⡿⣿⢿⣟⣧⠈⠳⠛
⠈⠁⠀⠀⠀⠀⣿⣿⢸⣹⡇⡏⣾⢸⣿⣼⣻⣿⣾⡷⢦⡀
⠀⠀⠀⠀⠀⢠⢟⡿⣯⣿⣧⢷⣿⡿⣿⡏⣿⠉⣿⣿⣷⢻
⠀⠀⠀⠀⠀⢻⣸⣷⣿⣿⢹⣧⡻⣷⣧⢻⡘⣿⠵⠋⡟⣿
⠀⠀⠀⠀⢠⣞⣽⣾⣿⡿⣿⣏⣷⣜⣿⣿⣳⣼⢷⡄⢿⣻⣄⡀
⠀⠀⠀⠀⢸⣽⡇⠈⢻⣿⣿⣿⣿⣽⣞⣿⣿⢿⣷⣻⠀⠙⢾⣝⢶
⠀⠀⠀⠀⣸⣿⠃⠀⠸⣷⣿⣿⡻⣯⣹⣿⡿⣿⣿⣿⠀⠀⠀⠿⠾
⠀⠀⢀⠀⣿⢿⡀⠀⠀⣿⣿⡏⣿⣶⢿⣿⡟⠋⢿⣿⡀
⠀⠀⠘⠆⠘⢯⣷⡀⠀⡟⣿⡇⣿⣷⠞⣼⠃⠀⠈⢻⣝⣦
⠀⠀⠀⠀⠀⢸⣿⡇⠀⢷⡻⣷⣿⢣⡟⠁⠀⠀⠀⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠉⠀⢠⣿⣿⢫⣷⣿⣶⣤
⠀⠀⠀⠀⠀⠀⢀⣠⣿⡿⢣⢿⣿⣿⣿⣯⣴⣞⢲⡄
⠀⠀⠀⠀⠀⣾⣫⣷⣶⠖⠋⠀⢻⣷⡌⠉⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠛⠁⣿⣿⠀⠀⠀⠀⢻⡿⣦⡀⠠
⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⠙⠾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠔⢒⠒⢢⣤⢲⠀⠒⠲⢀⠝⡱⠦⠔⡲⢋⠖⠦⠄⡤⠐⡖⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠽⢄⡿⠁⠎⢁⠘⠀⠀⡇⡌⠀⠆⠸⢀⠁⠘⢸⠰⠀⢁⠀⡇⣔⣀⣠⣤⣤⣤⠤⠀⠠⠤⠄⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣁⠼⠱⠀⡆⢸⠀⡃⡀⠃⢡⠀⡆⢠⠘⠀⠰⠈⠀⡇⠈⠀⠃⡸⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠘⡄⠀⡄⠃⢀⡌⣰⢃⠁⠘⠈⡄⢂⠘⠀⡄⠘⢠⠀⠀⢀⠀⡀⠖⠒⠒⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⠈⢢⡇⠄⣞⠁⢂⢢⡷⢀⠇⠃⠈⠀⡇⡇⠀⠈⠀⡁⢸⠀⢃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢃⠜⠁⠀⢏⢱⢙⠀⣱⣻⠀⠇⢠⠀⠁⠀⡆⠸⢰⠁⠀⡄⡈⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠒⢠⠁⠘⡀⢃⠘⣇⠊⡔⡇⣿⠘⢀⠎⡌⠀⠀⣁⠀⡆⠂⢠⢠⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⣢⠼⠗⣤⡑⡈⠀⠫⢸⠀⣿⣻⠀⠊⠘⠀⠠⠀⢙⢢⡇⡆⠘⡄⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢆⣲⠃⠇⡁⠘⢦⠑⢟⣿⣎⠀⠰⡀⠀⢠⠂⡾⠀⡄⠠⢩⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠝⢇⡅⣇⠀⣄⣓⠉⢠⠃⠸⡏⡇⢂⠀⡇⠰⢸⡠⠁⢀⠇⡇⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠪⡑⣼⢣⠈⡢⡈⡨⠦⠁⣧⠃⠈⠀⠃⣰⠈⢠⢣⠈⢀⠃⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢘⡏⡉⢜⡱⠹⠁⡆⢠⡏⡀⢀⢨⢰⡰⣠⠠⣎⠔⡂⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠜⡖⣌⠒⢂⣀⠀⢠⡊⠆⢡⠘⠱⠋⡜⠀⣂⣧⡄⡌⢘⡇⢠⣃⠔⢿⠙⡉⠰⠎
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠰⣱⡿⠂⠁⠚⡁⠈⠒⠀⠜⠚⠁⠐⢪⢘⠁⠈⡼⡗⠗⢣⢰⡇⠸⠁⠀⠳⠒⡡⡊⣴⢃⡔
⢀⠤⠐⠂⢉⠥⢠⠞⠃⡩⠐⠊⢉⠉⠉⠉⠐⢛⠟⡩⠍⠉⠍⢐⠣⣀⡠⢓⣚⠒⡡⠪⡤⣿⣴⣤⣍⠒⡜⠀⡁⣾⠀⣀⣈⠉⠐⠒⠠⢀
⡗⠤⡄⢔⢸⠀⡁⠀⢈⠀⠠⡊⠠⣂⠀⠐⣠⡉⡃⢴⡡⢐⢑⠳⠳⣽⡏⠠⢠⡙⢮⠸⢈⠔⠸⣟⢿⣮⠄⠉⠠⠄⠀⠠⠄⣼⣀⢳⡤⠜⡃
⡇⠀⠀⠀⠈⠉⠉⠀⠒⠓⠂⠈⠶⠦⠬⠹⠿⠷⠬⠠⣽⣴⡒⣥⣴⣼⣱⡘⣿⣧⡀⠱⠼⠷⠵⠬⠧⠧⠽⠒⠐⠒⠚⠋⠉⠁⠀⠀⠀⠀⡇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠑⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠔⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠣⡄⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠟⠀⡸⠁⣠⠆
⠀⠀⠀⠀⠀⠀⠀⣸⠇⢀⡐⠀⠀⠀⡀⠈⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠆
⠀⠀⠀⠀⠀⠀⢰⡏⠠⣐⡈⠀⠀⠀⢂⠠⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⣠⠤⣤⡴⠒⢦⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⡿⢀⠭⢒⡂⠀⠀⠀⠐⠌⠁⠀⠀⠀⠈⠙⢶⣄⠀⠀⢀⣠⡇⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣾⠁⠀⢊⣁⣤⡤⠶⠖⠒⠛⢛⣛⣉⣉⣍⣉⠉⢹⡇⠀⠸⣄⡄⠀⣀⠀⢶⠁⢀⣀⣀⡤⠖⠒⠲⢤⡀
⠀⠀⠀⠀⣼⣧⡴⠞⣋⣡⡤⠤⠶⠒⠺⠛⠛⠋⠁⠀⠀⢹⠞⠋⠀⠀⢠⠟⠀⣴⠇⠠⠊⡴⠋⠉⠙⠂⠀⠀⠀⠀⢹⠤⣄
⠀⠀⠀⢰⣟⣇⣰⣟⠩⠔⣂⠨⠀⠀⠀⠥⠐⠀⠀⠀⠀⣸⠀⠤⡀⢰⠋⢠⠎⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠼⢄⡀
⠀⠀⠀⠀⠉⠉⠉⢿⡄⠀⢂⠩⠄⠀⠀⠀⠁⠀⠀⠀⠀⣿⠀⠀⠐⠃⢠⠀⠀⠀⠀⣀⠷⠒⠁⣀⣠⠤⠖⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠂⠀⠀⢻⠀⠀⠒⣈⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⠁⠀⠄⣀⡤⠒⢋⡠⠴⠒⠉⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⢷⠀⠤⢐⡂⠀⠀⠀⣠⢞⣩⡤⣹⢀⣠⠴⠚⠉⠅⠒⠉⠀⠠⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠾
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⣧⠊⡡⠄⠀⢀⡞⣱⣿⡿⠗⠛⠉⠀⠖⠒⣚⣉⣀⣤⡤⠤⠶⠖⢲⠒⠀⠀⠀⠀⠀⢀⠀⣸⠏
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⢀⠨⠀⣼⢰⣩⣤⣤⠴⠖⠒⠚⠉⠉⠉⠀⢠⠂⠀⠀⡤⠒⠲⡟⠲⣧⣀⣀⣠⠟⠉⠁
⠀⠀⢀⠑⠀⠀⠀⠀⠀⠀⠀⢿⡀⡊⠁⣿⢸⣿⣿⣮⠟⣶⡄⠀⠀⠀⠀⣄⣀⣈⣉⡉⠉⠁⠀⠀⠀⣸⠉⠁⠀⠀⠀⠀⢀
⠀⠐⠁⠄⠀⠀⠀⠀⠀⠀⠀⠘⣧⠐⡀⠘⠓⠒⠒⠛⠋⢹⡇⠀⠀⠀⠀⠀⠐⢁⠉⣽⠋⠛⠃⠀⠈⣧⠀⠀⠀⠀⠀⠠⣤⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⠠⠃⠹⠦⠴⠛⠦⠖⠃⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⠈⠁
⠀⢀⡤⠴⠶⡒⠂⠀⠀⢀⠤⠤⡀⢹⣦⡤⢄⠀⡀⠤⠤⠼⡇⠀⢀⣀⣠⡄⠀⠀⣀⡠⠤⠤⡄⠀⢀⣀⣀⠠⡄⠹⡆⠀⣠⠔⠐
⠀⢫⡇⠀⢸⢠⠀⠤⠀⢹⠀⠀⡇⢸⣽⠀⢸⡏⡇⠀⠀⠀⠸⡰⣿⠀⠀⡟⢠⡞⠁⠀⣀⣀⡼⢰⡇⠀⢸⢸⡇⠀⡅⠀⣾
⢸⢾⠀⠀⢸⠀⠀⠀⣼⢠⡇⠀⡇⢨⣿⠀⢸⡇⡄⠀⢰⡀⠀⣷⣻⠀⠀⣴⣽⠀⠀⡾⠓⠊⠀⢸⡇⠀⢸⣿⡇⠀⠇⠀⡯
⢸⡞⠀⠀⡇⠀⠀⠀⢸⢸⠀⠀⡗⢻⣿⠀⢸⡇⡗⠀⢸⣇⠀⢸⡿⠀⠀⣿⢸⠀⠀⡇⠀⠀⠀⢘⡇⠀⠈⠉⠁⠀⢠⠀⣿⡇
⡆⡇⠀⢠⠇⠀⠀⠀⢸⢸⠀⠀⡇⣸⡏⠀⢸⠧⢺⠀⢸⣿⡄⠀⠁⠀⠀⣟⢸⡀⠀⠳⣀⣀⣄⡐⡇⠀⢠⡟⡇⠀⢸⠀⣿⢷⣲
⡧⡇⠀⠈⠉⠉⠉⠉⢹⠘⡆⠀⠉⠉⠀⣠⠇⢸⢸⠀⠸⣧⢧⠀⠀⠀⢠⠙⡈⢧⡀⠀⠀⠀⠈⡇⡇⠀⠀⣧⡇⠀⢸⠀⢀⡔
⢯⠓⡖⠒⠶⠖⠲⠶⠞⠑⢝⡦⣤⡴⠞⠁⠀⢸⡾⢴⠶⠉⠿⠛⠻⠟⠟⠀⠈⠒⠬⠽⠿⠿⠟⠻⠟⠶⠾⠻⠟⠒⠞⠀⠺⠷⠤
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⡞⠉⠒⠾⣄⠀⠉⠱⣄⣀⣀⡤⠤⠶⠒⠒⠚⢋⣭⣭⡉⠉⠉⠉⠛⠒⠒⡶⣦⠤⢤⣀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣹⣦⡀⠀⠈⠛⠙⣄⣇⠈⢧⣀⠀⠀⠀⠀⣠⡿⣽⢿⡟⢿⣿⠶⣤⠀⢰⢃⡷⣴⢳⢠⠏⣼⠳⢤⣀
⠀⠀⠀⠀⠀⠀⠸⣀⠀⠈⣇⠀⠀⢀⠘⢾⠂⠈⠈⢧⠀⠀⢰⡟⢸⣇⣾⣳⣤⣿⠶⣾⡇⡜⣸⢠⢃⣾⢋⡼⢉⠗⡹⠊⠑⠦⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣹⡆⠀⠉⠓⠪⣕⢦⡳⢄⣦⡘⢦⡀⢿⣁⡞⢹⣿⡋⠁⢻⣄⣼⣧⣇⢇⡎⣸⢣⢞⡴⡡⢊⣤⣖⣽⠧⢚⣹⡿⠂
⠀⠀⠀⠀⠀⠀⠀⣼⠁⠉⣒⠶⣭⡳⣌⠳⡵⡄⠉⢙⠲⣍⢿⣿⡀⡸⢘⡏⠲⣤⠏⢠⣿⡸⣸⢱⣣⢫⢎⣾⣿⣿⢟⡫⠗⠋⢁⡀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢡⣖⡺⠟⢮⣧⠹⡿⣄⠀⠙⢮⣇⣟⠻⣤⣾⣟⣦⣾⣖⣻⣿⣷⣷⡿⡱⣳⢟⢽⣫⡗⢉⡴⢤⢴⠋⠉⢳⡇
⠀⠀⠀⠀⠀⠀⠀⣷⣰⢫⢮⣻⡻⣦⣀⣳⡳⣝⠮⣗⣶⣪⠽⡷⢚⠽⠀⠀⠀⠀⠀⠉⠛⢿⠿⢵⣵⣟⠵⢫⡏⠈⡹⠧⡤⢦⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠻⣿⢨⣧⣗⣿⡛⠻⣍⠛⡻⠛⠉⠀⠸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠇⠀⠀⠉⠲⣍⢠⠚⢁⡴⢦⡞⠀⢸⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠈⠺⣿⡏⠳⡉⢳⠈⣿⡁⠀⠀⠀⠀⠈⠙⠒⠲⠶⠶⠖⠒⠒⠋⠁⠀⠀⠀⠀⠀⢈⡏⠐⡅⢀⡞⢑⡶⠋⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠈⠙⠢⣌⣹⢀⡟⢷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠗⢲⣋⠭⠛⠉⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠦⣀⠀⠀⠀⠈⠙⠓⠾⠦⣝⣛⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣒⣺⣽⡿⠭⠖⠚⠉⠀⠀⠀⢀⡠⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠙⠋⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠈⠙⠒⠶⠤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠶⠒⠉⠁⠀⠀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢮⠙⠲⠤⣄⡀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢀⣀⠤⠖⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠈⠉⠓⠒⠶⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠴⠖⠒⠋⠉⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⡤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠴⣒⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣉⠉⠙⠛⠒⠒⠚⠛⠋⠉⢉⣀⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠒⠒⠒⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⠤⠀⠀⠀⠀⠋⠀⠀⠤⠄⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⡔⠒⠀⠡⠏⠁⠀⠔⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠓⠆⠠⢁⢸⣾⢠⠠⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⡔⠋⠀⣹⠇⢩⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣦⡏⣷⠚⡅⡝⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣀⡝⠛⣋⢸⢰⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣀⠀⠉⠈⣿⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⣜⣰⣇⠙⢁⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢅⡀⠜⣡⡏⣷⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇⢰⢋⡇⠉⢾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡒⣸⣽⡏⢸⣧⢖⣼⣿⡦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣻⣛⣛⣧⡀⣿⣃⠻⡷⠿⡘⢦⣴⣁⣄⣤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢈⣀⠀⣾⢉⣗⣄⣽⠵⠼⠁⠈⣁⣢⣍⣩⣮⣙⣻⠉⠿⡤⡖⢊⢢⡀
⠀⠀⠀⠀⠀⠀⠀⢘⢶⣦⣝⡶⠻⣪⡝⣻⣏⡐⠎⣻⢩⠐⡤⠞⣀⠲⠆⣼⠈⣦⣮⣽⣵⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡏⣎⣽⣿⣿⣿⣷⣷⣶⣶⣿⣬⣿⣤⣽⣦⣿⣶⣾⣷⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣾⡇⢾⣿⣯⣾⣿⣿⣿⣿⣿⣻⣿⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣅⡉⣯⣿⣿⣿⣟⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠿⠛⠻⡁⢤⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠿⠛⠙⠉⠛⠙⠉⡉⠁⠀⠀⠀⢀⡼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⠁⠀⠀⠀⠀⠐⠀⠀⠡⡄⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢙⡒⠲⠶⠴⠤⠶⠦⠖⠒⠚⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉
⣶⣶⣶⢶⡶⢶⣶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⣶⣶⣶⠶⣶⠶⡶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⠟⠁⠀⠻⢿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠉⠛⠻⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠁⠀⠀⠀⠀⠈⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣀⣤⣤⣤⣄⣠⣤⣤⣤⣀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠉⠋⠉⠉⠉⠉⠋⠉⢁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠀⠀⢀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠖⠒⠀⣬⣍⡛⢛⣩⣥⠀⠚⠶⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠉⣀⠀⠀⠀⠀⣿⡿⠟⠻⢿⣿⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣤⣤⣭⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⠉⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⣛⣛⠛⠛⣛⣛⣛⣛⠛⠛⠛⠛⠛⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢯⣛⡛⢉⣉⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠛⠁⠀⣰⠋⠈⠙⢳⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠸⣄⠀⠀⠀⠀⠀⡴⠛⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠙⣇⣠⣤⠞⠛⠆⢶⡰⠛⣧⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣾⣿⣿⣇
⠀⠀⠀⠀⠀⢸⡧⠤⠤⠰⢤⡤⠟⣤⠼⣏⡉⠒⠋⠁⠀⠹⣇⠀⠀⠀⠀⠀⠀⠹⠿⠿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⢻⡄⠀⠀⠰⢟⣠⠤⢼⢀⡞⠉⠳⣄⡀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠈⠷⣤⣴⡋⠀⠀⠀⢰⡶⠀⠀⠀⠀⣶⢀⣠⡾⠀⠀⠀⠀⡀
⠀⠀⣶⣤⣄⣀⣀⣘⠉⠛⠛⠛⠾⠿⠿⠿⠿⣿⣻⡛⠋⠉⠀⠀⠤⠶⢚⣋⣀⣀⣤⣤⡶
⠀⠀⠈⠙⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠋⠁






⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢴⠆⠀⠀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠁⠀⢘⠀⠀⠈⠁⠀⠈⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠒⠀⠉⠈⠁⠠⡀⠀⠀⠀⠀⠠⠀⠈⠈⠁⠐⠒⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⢀⣀⣑⡀⠀⠀⢀⣂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣀⣠⡤⠶⣾⡿⠿⠛⠋⠀⠀⠀⠈⠙⠻⠿⣿⡶⠦⣤⣄⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⣶⡯⠿⠟⢓⣈⠥⠔⠒⠚⠛⠛⠛⠛⠛⠋⠒⠒⠤⢬⣀⠛⠻⠿⣿⣶⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡤⠚⠉⠀⢀⡀⠀⢀⠣⡀⠐⠀⠲⢀⠂⡕⢀⡀⠈⠙⠲⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⢀⠔⠁⣀⠂⠠⠂⠀⠀⠀⠈⠀⠀⢀⡀⠀⠀⠀⠁⠀⠀⠀⢠⠂⠠⠙⢦⡀⢀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⡴⠃⠀⠀⠀⠁⠀⠀⠀⠰⠆⠀⠀⠀⢈⡃⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢧⠄⠙⢆⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠈⠁⠀⠤⠁⠠⠀⠐⠿⠂⠀⠀⠀⠀⠀⠀⢧⡀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⡾⠁⠀⠀⣶⠋⣭⡔⢄⢠⠄⢀⡀⢀⢰⠆⠀⠐⠀⠀⣰⠖⢢⠀⠀⢰⠀⣄⡤⢄⠀⠀⠈⢧⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠠⠀⢰⠃⠀⢠⡄⢻⡀⠒⠃⡌⢸⡀⠥⠟⡠⢸⡆⠀⢠⠀⢠⡏⠼⠇⢃⠀⣾⠀⠋⡐⢁⡀⠀⠀⠘⡆⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡜⢀⠀⠀⠀⢸⡇⠚⣳⡌⠺⡇⠺⡌⠀⢸⡅⠈⠀⢀⡾⢀⠀⢻⡀⢀⡏⢰⢦⠑⡈⠁⠀⠀⠀⢃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇⢸⠀⠀⠀⠀⠷⠈⢩⠐⠁⠓⠠⣩⡌⠈⠩⢩⡄⠄⢠⡤⠄⢠⠄⠀⠉⡅⠈⠉⠀⠀⠀⠀⠁⢸⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⢃⠈⠀⠐⢅⡐⣀⡀⠘⠂⣦⠀⠈⠁⢹⡆⠀⢸⡇⣀⠛⣠⠀⣾⠀⠁⠀⠀⠀⠰⡦⠀⠀⠀⠀⡆⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⢀⠸⡀⠀⠀⠀⠔⡦⢙⠆⠀⢿⠰⠀⠀⣸⡇⠀⢸⡇⠻⠒⣿⠀⣿⠀⠂⢁⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢳⠀⠀⠀⠑⠒⠋⠀⠀⠸⠀⠀⠀⠃⠀⠒⠈⠓⠀⠀⠛⠂⠛⠒⠂⠀⠀⠄⠑⠀⠀⠀⢀⠷⠀⠒⠒⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡧⠄⠒⠀⢉⣁⡉⣉⢀⡀⠐⠀⠠⢀⡀⠀⠀⠸⠇⠀⠀⠀⠀⢘⠡⠇⠰⠀⠀⠀⢠⣾⣤⣤⡀⠐⠃⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠑⠀⡀⠀⠀⠘⠛⠒⠒⠀⠀⠖⠖⣢⠄⣤⠈⠀⠀⠂⠠⠄⢀⣀⠈⠑⠒⢁⣀⡠⠴⠿⠛⠉⠀⢳⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢯⣠⣤⣶⣶⣾⣉⠉⠉⠉⠉⠐⠂⠀⠀⠉⢉⠀⠊⠽⠂⠀⠢⣀⡄⠀⣀⠀⡀⢠⠀⠀⠁⠀⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⣀⠀⠈⠛⠿⢿⣿⣿⡇⠙⠢⢤⣀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠄⣀⣈⢓⠀⠒⠘⠁⢈⣀⠠⠔⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠉⢀⠀⠐⠀⠀⠀⠁⣠⡀⠀⣤⠈⠉⠓⠒⠢⠤⠤⠤⠤⠤⠤⠔⠒⠊⠉⠁⣬⠅⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠩⠤⠤⠤⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠄⠤⠤⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠉⠉⠉⢉⠉⣉⣉⡉⠉⠀⡉⢀⢁⠈⡀⢁⠈⣉⠈⡁⣈⢉⢈⡁⡁⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⣀⣀⡀⠀⠉⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠁⠀⠁⠁⠈⠀⠁⠁⠀⠀⠀⠀⠔⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⠄⠀⣐⣂⣀⣀⣀⣠⣤⠤⠤⠤⠤⠶⠶⠒⠒⠒⠚⠚⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣛⣚⠀⡄⣇⣩⣉⣩⠩⠅⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠦⠤⠶⠧⠦⠶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⣄⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡘⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⡡⢭⣍⠂⠘⠒⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢰⡇⠛⠀⠠⠥⠀⢄⣍⡐⡠⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⣄⠄⠘⡇⠀⢤⡤⡄⢴⡄⠈⡆⠄⡈⡴⡂⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⣍⢀⡘⡙⠀⠄⠓⠟⠁⠓⢄⠻⢋⠤⠣⡀⠀⣀⣱⠈⢑⣤⡉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠀⢀⠭⢄⠁⠂⡀⠉⣐⡂⣮⠤⠶⠞⣠⣀⡀⠃⠉⢉⣀⡄⢍⡸⢼⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⢀⡐⠏⡣⡁⠬⢧⡀⠀⣀⡀⠋⢐⠁⣈⡢⠊⠬⠂⠛⠶⡄⡘⢂⠂⣇⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡛⢷⣾⣤⣃⣀⣚⠂⠊⠀⠀⡀⠴⡄⢾⢗⣀⣀⣄⣠⣬⣥⡶⠾⢛⣭⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠍⠳⢦⣭⣉⣙⡛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢋⣉⣉⣥⡤⠶⠛⠉⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⢉⠉⠉⠛⣿⠟⣿⡿⠟⠿⣿⠿⠿⢿⣬⠻⣷⡀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣴⠿⣛⣛⣛⠿⣾⣯⣼⢯⣼⢟⡟⠖⣾⠿⣶⣽⣧⣽⡇⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⢛⣉⣙⠻⢾⡷⡿⡿⣻⢿⢻⢣⠇⠹⣟⣉⣉⡙⢷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⣉⣙⢿⡬⡇⣼⡳⣛⣾⣺⣿⡿⣧⡞⣉⣉⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⢸⣧⣹⣟⢿⣮⣻⡷⠿⢃⣿⢰⡟⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⣮⣝⣧⣿⢏⣴⠞⣛⣡⡾⡹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⠿⠿⠿⠾⠿⠾⠟⠛⠋

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠔⠂⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⢒⢵⠒⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠁⡀⠀⠠⠄⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⣰⡤⡬⡯⠿⠃⠀⠀⠀⠀⠠⠄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣥⣔⡱⣴⣶⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢃⣿⢟⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠉⡀⣿⣼⡯
⠀⠀⠀⠀⠀⠀⡀⠤⠔⠒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⠲⣡⡋⡸⢣⠁⠀⠀⠀⠂⠀⠠⢀
⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⣀⣀⢠⣤⣜⠛⣿⡟⣿⣏⣠⣫⢷⣵⣷⣷⣦⠄⠀⣀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠘⠦⣄⣀⡠⠂⠀⣔⡿⢯⡽⡒⠟⣶⡽⣶⣻⣛⡷⢟⣋⠙⠣⡴⡾⣏⠴⣄⡈⠐⢄⠀⣀⠔
⠀⠀⠀⠀⠀⠈⢿⡿⣿⣾⣷⣿⣯⣼⣼⣼⣽⣽⣿⣯⣟⣰⣯⣔⣨⣿⣿⣬⣮⣿⣿⣾⣶⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣟⣾⣿⣿⣿⣵⡏⣿⣿⣟⣽⣿⣿⣽⣿⣿⣿⣿⢷⣮⣿⣿⠟⠛⠋⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⣽⣿⣿⣶⣾⣿⣿⣿⣿⠾⣻⣿⣿⣽⣿⣿⡿⠿⠟⢻⣀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣽⢟⣯⣯⣾⣯⣯⢻⣿⣿⡿⠉⠁⢉⣤⣴⠂⠛⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡭⣽⣯⣿⡿⠛⠉⣠⡴⠞⠉⢱⡶⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⢛⡟⢀⡴⠘⠋⣀⡶⢇⣩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⢶⣿⣥⣴⣾⣭⡽⢷⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⠶⠶⠖⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣠⣤⢀⣤⣶⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢖⣴⣿⠟⣴⣿⣿⡇⣉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⣾⣿⡿⢸⣿⣿⣿⠁⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⣿⣿⡇⢸⣿⣿⣿⠐⢟⣛⣠⣤⣴⣶⠾⠿⠟⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⣿⣿⡇⢸⣿⣿⣿⠈⣛⣋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡿⢿⣿⣿⣿⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢘⣿⣿⠀⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢻⣿⠇⠸⠀⣿⣿⠐⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⠸⠿⣿⣿⣿⣿⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⡜⢃⣾⣷⠀⣿⣿⠐⣿⣿⡇⢸⣿⣿⣿⠀⣿⣿⢰⣶⣶⡙⣿⣿⠇⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣮⣅⣘⡛⠿⠀⠿⣿⠘⣿⣿⡇⢸⣿⣿⣿⠰⣿⣿⠸⠿⠿⠃⢛⣋⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣝⡛⠿⠿⠿⠿⠿⠿⠟⣛⣫⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠳⠶⣤⡀
⠀⠀⠀⠀⡤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠀⠀⠀⣀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀
⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⢀⡠⠬⢍⣔⠵⢀⠊⢴⡈⠅⠠⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠂⠀⠘⠄⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⣀⢠⣀⠜⠫⠛⠁⠀⣨⡗⠁⡙⡉⠒⠤⣶⡍⠓⣀⡀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⢀⡿⠁⠀⠀⢀⠒⢀⡤⠧⠦⣮⠁⢉⡏⢀⣀⣋⢄⢈⠢⣑⠶⡼⢀⠋⢩⠀⣀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠈⠆⠀⠰⡀⠀⠀⠀⢸⠃⠀⠀⡠⣮⠀⡞⢋⠓⣄⡌⠀⢸⣌⢲⡂⢒⡡⠂⡎⠀⠀⣅⡫⠂⠈⠠⠊⠑⣀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⣿⠀⠀⡜⠀⠀⣸⠁⢰⠂⡉⠀⠀⡁⠀⠉⡋⣱⠄⠀⣩⡀⠀⣀⣀⢠⣶⠊⠔⠃⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⢢⠀⠀⢹⣄⠨⡇⣰⠛⠉⢆⢂⠠⠐⠢⠰⢆⠖⡉⢹⢹⡜⡲⢧⣀⡀⢉⠔⠊⢐⠈⣿⡶⠿⠩⠢⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⢣⠀⠘⣿⣨⣵⣏⠀⢠⡟⢹⠛⠲⡤⠀⡇⢀⠞⢣⢸⢄⠀⢸⣭⣉⣊⣀⡀⠴⠀⠈⢸⠑⡂⢄⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡏⣙⣲⣬⢥⣄⠑⠀⠰⢄⣼⣾⢂⠨⠘⣃⣀⠠⠨⢈⣛⡞⣐⡨⠀⠀⠖⠄⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠆⠀⠈⢷⣆⡎⠈⣄⠁⠬⣡⣄⣠⠉⠈⠳⣄⣦⡦⠛⠉⠍⢉⢘⠀⠉⡔⠣⣀⢈⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠈⠆⠀⠈⢋⠻⣶⣄⢙⠖⠋⣀⣈⡄⠔⠀⣥⠦⢤⣄⡐⢆⠲⠆⢂⠜⡀⢀⣴⠟⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠱⡄⠙⠻⢶⣬⣁⣩⢀⠁⢊⠲⠇⠀⢒⡑⣠⣋⣩⣥⡶⠟⡋⢐⡵⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠸⡀⠀⠀⠈⢦⡀⠀⠀⠉⠙⠛⠻⠿⠿⠷⠿⠿⠿⠛⠛⠉⡑⠂⠅⣪⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠁⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠀⠀⠉⠀⠈⠂⠀⠀⠀⠀⣂⡤⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡀⠀⠀⠀⠀⠀⣟⡳⠦⣤⣄⣀⣀⣤⣀⣀⣤⡴⢞⣩⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⢀⢳⠀⡄⠀⠀⠀⠻⣿⣷⣶⣬⣭⣭⣭⣭⣴⣶⣾⢿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡜⡆⢻⠹⡄⠀⠀⠀⠈⠑⠪⠭⣭⣭⣭⣭⡭⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠘⡄⢧⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⣀⣼⡆⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠈⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠓⢄⡀⠀⢙⣣⠄⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣤⡄⠠⣚⡉⠀⢀⡴⠂⠀⠀⠀⠀⠀⢸
⠀⠀⢀⣀⣀⡀⢤⣤⢄⣥⣔⣻⣿⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣛⣧⣶⣠⣤⠄⣀⣀⠀⠀⢸
⡶⢿⣿⣿⣿⣿⣿⣶⣨⣭⣟⣻⣯⣥⣭⣛⣛⡿⢿⣿⣿⣿⣿⡿⢿⣛⣻⣥⡭⣿⣛⣿⣭⣆⣶⣿⣿⣿⣿⠷⣾
⠉⠛⠒⠿⠯⠭⣽⣯⣙⣛⣙⣻⣷⣶⣍⠛⠛⠫⠭⠤⠘⠓⠖⠛⠻⠿⣛⣋⣞⣒⣒⣒⣹⣥⣭⠭⠽⠷⠚⠋⢹

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⢣⢣⢏⣞⠉⠙⢲⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣧⢻⣸⡸⡦⠀⠀⣠⠖⠒⠒⠶⠶⠶⠶⠶⠶⠶⠤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣚⠭⣟⣇⡇⡷⣸⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣯⣟⣼⢸⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠒⠛⡞⡞⣦⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠊⠁⠀⠀⠀⣴⣳⢳⢫⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣉⣳⡀⠀⠀⠀⠀⢳⣹⡸⡘⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡇⠀⣀⣤⣾⣭⢇⣇⡟⣸⣤⣌⣉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣼⣷⠿⣓⡺⢽⣛⡺⢾⣚⡯⢴⡒⠦⢉⣦⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠩⣛⣓⡺⠿⠤⠭⠭⠤⠿⢟⣒⣻⠍⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢨⣉⣛⠒⡒⠒⠒⠒⠒⣛⢉⡅⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣘⡛⠣⠟⠽⢯⠵⠯⠼⠗⢚⣁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠉⠉⠛⠒⠒⠒⠒⠛⠉⠉⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⢤⣤⣀⣀⣤⡤⠤⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⢻⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣟⠻⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⠟⢻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣦⣈⣿⣿⣿⣿⣿⣿⣿⢘⣿⣿⣿⣿⣿⣿⣏⣴⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⠟⢉⣄⡙⢿⣿⣿⣿⠿⠟⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠏⣰⣿⣿⣷⡄⢻⣿⡇⢰⣷⠀⣿⡆⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣴⣿⡿⢀⣿⣿⣿⣿⣷⠈⣿⡇⢸⣿⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢁⣴⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⢹⡇⢸⣿⢀⣿⡇⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡟⢁⣴⣿⣿⣿⣿⣿⣧⡈⢿⣿⣿⣿⠟⢀⣿⣷⡈⢿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⡇⣴⣿⣿⣿⣿⡆⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⡇⣿⣿⣿⣿⠏⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⡇⢸⣿⡇⣿⣿⠟⠁⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣠⣿⣿⣿⡇⢸⣿⡇⠙⠁⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠿⠛⠛⠃⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⢉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⢚⣉⠤⠖⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣚⡭⠔⠚⠉⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣒⣭⣵⣚⣩⡥⠤⠤⠴⠖⠒⢒⣛⣉⣉⣉⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⡿⡿⡷⡗⠒⠒⠛⠋⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠭⣛⢟⠞⡑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣒⣢⣄⣀⡠⢼⢼⠻⣸⠤⣤⡤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢞⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢠⠀⠠⠀⢀⠀⢠⠈⠀⠈⢸⠀⢸⠘⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠚⠚⠚⠚⠚⠚⠚⠚⠛⠚⠚⠛⠚⠛⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣀⡀⠀⣀⣀⠀⠀⠀⠀⣀⠤⠊⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠀⠀⠀⣀⠀⣉⠉⣁⢉⠀⣤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠳⡗⢅⠿⣀⠇⠷⠙⠄⠘⢂⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡠⠔⠒⠒⠒⠠⠤⠤⠠⠤⠒⠁⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⢤⣀⣀⣀⣀⣀⣀⣀⡤⠞

⠀⠀⠀⢠⡎⠉⠙⠲⣄⣠⠤⠶⠶⠶⠶⠤⢤⣀
⠀⠀⠀⠘⣇⣠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠈⡆⠀⠀⢤
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⡇⠀⠀⠘⡆⠀⠀⠀⠀⠀⢰
⠀⢸⠃⠀⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⡜⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⡀
⠀⠘⣆⠀⠀⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⢀⣠⣞⣠⣤⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠛⠻⠿⣖⡦⣄⣀⡤⠤⠔⠒⣊⣩⡤⠟
⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡠⠖⠋⢁⣀⣠⣤⢤⡤⣤⡤⣤⣤⣤⣤⡤⠤⠴⠒⢊⣉⣡⢤⣶⠒⠛⠋⠉⢁⣀⡀⠤⠄⢒⣛
⠀⠀⠀⠹⣄⠀⢀⣠⠖⣋⡤⡶⣻⣿⣿⣻⣟⠿⠷⠛⠛⢋⣉⣡⣤⢶⣶⢞⣿⣹⣧⡤⠤⠟⠓⠒⢈⣉⣠⡤⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠈⢹⡻⣀⣾⣿⠥⠷⠛⢋⣉⣡⣤⣶⣶⣿⣻⣿⠽⠿⠟⠛⢋⣉⣉⣤⣤⡴⣶⣶⠟⠉⠉⠁
⠀⢀⣀⡤⠴⣒⣋⣩⣥⢴⣶⣿⡯⠿⠾⠛⣛⣉⣩⣥⣴⣶⡖⣞⡿⢻⣿⣿⣻⣮⢿⣺⡕⠋
⠮⢭⣶⡾⠿⢿⣓⣚⣭⠭⠴⠖⠶⠾⠿⠿⠿⠿⠷⠿⠿⠿⠿⠿⠿⠿⠛⠛⠓⠋⠉⢸⠃
⠀⠘⠒⠚⠋⠉⠉⠀⢧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡈⠙⠒⠦⠤⢤⣄⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠖⠒⠋⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⡴⣒⣒⣦⡀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⢀⣔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢽⡮⣖⡒⣝⣷⣄⡀⢀⡠⣪⢔⡯⢍⣪⡳⣄⣀⡠⢺⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡪⠥⢜⢞⠮⣓⠻⠟⣪⢝⡥⣒⣒⢽⡪⢵⣶⣾⣵⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⠋⠙⠪⣷⢵⣦⡭⣭⣴⣿⡫⠴⠮⣗⣭⣭⣽⣿⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠤⣄⡀⠈⠙⠛⠛⠛⠛⠉⠀⠀⠀⠀⠉⠉⠉⣀⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠉⠉⠓⠒⠲⠶⠦⠤⠤⠤⠤⠴⠒⠚⠉⢁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠥⠤⠤⠤⠬⠭⢉⣀⣒⡒⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠋⠂⢿⣍⡓⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡀⠐⠀⠀⠀⠀⠀⡴⠀⠀⢸⠧⡀⠀⠀⠀⠀⡠⢆⠠⠈⠁⠀⡀⢸⠹⡉⠳⢦⡉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠔⡹⠆⠀⠀⠀⠐⠈⡆⢿⠋⠉⡹⠋⠔⠈⣀⠄⠀⠀⠀⢸⠀⡇⠀⠀⠙⠳⣄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⡴⠁⠀⠀⠐⠂⠀⠀⠀⠀⡇⢸⢀⠼⠁⠀⠀⠀⠀⠀⠤⠀⠀⠘⣀⠇⠀⠀⠀⠀⠘⢧⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠚⠁⢤⠀⠀⠀⠀⠈⠀⣀⠀⢃⢿⠣⣀⠀⠋⠀⠀⠀⠀⠀⠐⠂⠀⡏⡷⡀⠀⠀⠀⠀⠀⢳⡄⠘⣦
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡏⠑⢤⡀⠀⠀⠀⣀⠀⠀⠀⠀⢸⢸⣦⣨⠑⢤⡀⠀⠀⠆⠀⠀⠀⠀⣇⢸⠘⡄⠀⠀⠀⠀⠀⢷⠀⠸⡆
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢣⡈⠲⣗⠈⠢⣄⠀⠉⠀⠀⠀⠀⠘⡄⡧⣼⠢⣄⠉⡢⣄⠀⠀⢠⠀⠀⢹⠈⡆⠘⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⡙⢦⡑⢄⡘⠑⢤⡀⠐⠃⠀⠀⡇⢳⡈⠑⢌⡑⢄⡀⠑⢤⡀⠀⠀⠸⡀⡇⠀⡄⠀⠀⠀⠀⢈⡇⠀⢸
⠀⠀⠀⠀⠈⡀⠁⠀⠀⠀⠀⠈⠣⣄⠉⠢⣌⠦⣄⠈⠢⣄⠀⡄⢷⢸⠙⠦⡀⠝⠣⣹⡢⢄⠉⠢⣌⠀⣇⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠹⡗⢏⠈⠑⢷⣝⠢⣀⠑⢦⣸⡇⠀⠀⠈⠲⣄⠈⠙⢦⡉⠢⣌⠑⢾⣇⠀⠀⠀⠀⠀⢀⡾⠀⢠⡟
⠀⠀⠀⠀⠀⠘⣄⠠⡀⠀⠀⠀⠀⠀⠙⢦⡑⠦⡠⠌⠓⢌⠱⡦⣈⡿⡄⠀⠀⠀⠀⠑⢬⡁⠈⠲⢜⣍⠺⢛⡆⠀⠀⠀⢀⡼⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠘⢦⠀⢄⠀⠀⠀⠀⠀⠀⠙⠢⣌⡒⢄⡈⠑⠻⣆⢥⡽⠀⠀⠀⠀⠀⠀⠈⠓⣄⡒⠩⠦⣽⠁⠀⠀⣠⡞⠁⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠑⢄⠀⠀⠀⠀⠀⠀⠀⠉⠓⠛⠦⢄⡉⠙⣿⣀⣀⡀⠠⠄⠐⠀⠀⠀⠈⠒⠄⣻⠆⣠⠾⠋⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡙⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡼⠛⢁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣉⡓⠢⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠞⠋⣁⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⣬⣉⣉⣛⡒⠒⠒⠒⠚⢛⣋⣉⣉⣤⣤⠶⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁




⡿⠿⠿⠿⡿⠿⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠖⠚⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠚⣋⣉⣤⡤⠶⠶⠚⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⢛⣋⣛⡫⠿⣧⣀⣤⠶⣛⣥⠶⠚⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⢃⣎⣋⠒⠄⠴⣈⡼⠁⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣇⣂⣵⠤⠬⠒⢊⣡⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠖⠚⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⢶⠖⠚⢛⠛⠓⠒⢚⠙⠶⠶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢛⠉⣂⡖⠶⠚⣃⠐⠂⠴⠦⣩⡓⠒⢄⡀⣭⡉⢙⠶⣶⡀
⠀⠀⠀⠀⠀⢀⣀⡶⣛⠛⠵⡞⢁⡼⢦⡈⡁⠀⣖⠂⠐⢂⠀⡠⠤⠴⡄⠠⢍⡃⠍⣌⢹⢻⣄
⠀⠀⠰⡶⠶⠾⠿⠶⠶⠶⠶⠶⠶⠶⠒⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠓⠒⠒⠲⠲⠶⠶⠶⠾⠷⢶⡆
⠀⠀⠀⠹⣦⡒⠐⠒⠐⠒⠀⠂⠐⠂⠒⠂⠐⠂⠐⠂⠐⠒⠐⠒⠂⠒⠀⠐⠂⠐⠂⠐⠂⠒⠀⢲⡞
⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠳⠦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠿⣿⡏⡟⢻⢻⠛⣿⡷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡇⡇⢸⢸⡇⣿⡇⠈⠻⣷⣦⣤⣤⣶⣶⣶⣶⠶⠿⠿⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⣿⡇⣿⢸⢸⡇⣿⡇⠀⠀⣀⣤⣤⣤⣤⣴⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⢸⢸⠸⡇⣿⣧⣤⣾⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⡇⢸⢸⠀⡇⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⢸⠀⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⣸⣧⣸⣸⣀⡇⢸⣇⣠⣤⣶⡶⠿⠿⠿⢶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠟⢛⣻⣿⡿⠟⢛⣛⣛⣛⣛⣛⠻⠿⣿⣿⣡⠖⢒⣛⣉⣛⡒⠲⢬⣙⣿⣶⣶⡶⠿⠿⠿⠿⠿⢶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⣾⡟⣡⢖⣿⡿⢋⡥⠖⣋⣩⠭⠤⠤⠤⠬⢭⣓⠢⣝⠻⣾⣉⡤⠤⠤⣍⣓⣶⡿⢋⡡⠴⢒⣒⣋⣉⣙⣓⣒⠢⢭⡛⢷⣦⡀
⠀⠀⠀⠀⠀⣸⣟⣰⣃⣿⣏⣴⣋⣔⣋⣥⣖⣚⣛⣛⣛⣓⣶⣌⣹⣎⣷⣽⣿⣴⣶⣶⣤⣿⣏⣴⣋⣴⣋⣡⣴⣒⣒⣒⡲⠬⣙⠲⣍⢢⡙⣿⡄
⠀⠀⠀⣿⡿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⢶⣶⡄
⠀⠀⠀⠘⢿⣦⡘⠛⠀⠚⠛⠃⠀⠒⠒⠀⠐⠶⠖⠀⠰⠶⠖⠀⠛⠛⠃⠀⠛⠛⠃⠀⠛⠛⠃⠘⠛⠃⠀⠛⠛⠀⠲⠶⠂⠀⠶⠶⠂⠴⠶⠆⣠⣾⠟
⠀⠀⠀⠀⠀⠙⠿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠾⠿⢿⣿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠴⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⣿⢻⡟⣿⢻⡿⡗⢿⢺⡗⣿⠒⠒⠒⠒⠒⠒⠒⠚⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⢸⡇⣿⢸⡇⣿⢸⡆⣧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⠇⡿⢸⢇⡏⣼⢠⡇⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⡇⢾⢸⡇⣿⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⡆⣷⢸⡜⣇⢻⠘⡇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢸⡇⣿⢸⡇⣿⢸⠇⡷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢸⠃⡟⢸⢡⡇⣾⢰⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⠀⡇⢹⢸⡇⣿⢸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⣇⣿⣸⣏⣧⣹⣌⣇⣻⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁


⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⠉⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⢿⣦⣀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⢈⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⢙⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢛⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣵⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣿⣿⠿⠿⠿⠿⠿⠿⠷⠶⠶⠶⣶⣶⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡶⠟⠛⠉⠉⠀⢀⣾⣿⡿⠃⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⢀⣀⣤⣤⣶⢶⣿⣿⠟⠛⠛⠛⢛⣛⡛⠛⠛⠛⠛⣛⣻⣿⠿⠶⢶⣦⣤⣄⣀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⣭⣄⣀⣀⠉⠁⠀⠿⢶⠆⠈⢩⡄⠀⠿⠃⠀⠉⠉⠻⠇⢀⣀⣀⣬⣽⡿⠿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣀⠀⠈⠉⠙⠛⠛⢛⡻⠿⢶⡶⠶⣿⣷⠶⠶⠶⠶⠾⠿⠟⠛⠛⠛⠉⠉⠀⢀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠶⣦⣤⣤⣄⣘⣋⣀⡘⠛⠷⠈⠋⠀⠀⠸⠇⣀⣀⣀⣀⣤⣤⣤⡶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠉⠙⠛⢿⣷⠀⠀⠀⠀⢰⣿⠋⠙⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⢀⣿⡇⠀⠀⠀⢠⣿⠃⠀⠀⠀⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣸⡿⠀⠀⠀⢠⣿⠃⠀⠀⠀⢠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢠⣿⠃⠀⠀⢠⣿⠏⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣾⡏⠀⠀⢠⣿⠏⠀⠀⠀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡟⠀⠀⠀⣸⣿⣀⣀⣠⣿⠏⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣶⣶⠿⠿⠿⠛⢻⣿⠃⠀⠀⢠⣿⠟⠛⢻⣿⠏⠀⠀⠀⣴⣿⣥⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡾⠿⠛⠋⣉⣀⣤⣤⣤⣴⣶⣶⣾⡟⠀⠀⠀⣾⣿⡷⣶⣿⡏⠀⠀⠀⣾⣿⣁⣉⠉⠛⠿⢷⣶⣤⣀
⠀⠀⠀⠀⢀⣴⡾⠟⠋⣀⣤⣶⣾⠿⠛⣻⣿⡿⠟⠋⠉⣠⣿⠃⠀⠀⢸⣿⠻⢷⣾⡏⠀⠀⠀⣼⡟⣉⣛⣿⣿⣷⣶⣤⣈⠙⠻⢷⣦⡀
⠀⠀⠀⣰⡿⠋⢀⣴⡿⣿⣿⠟⠁⣠⣾⠟⠁⠀⢀⣴⡿⣿⡟⠀⠀⢀⣿⡇⢀⣾⡟⠀⠀⢀⣾⣿⠿⠟⠛⠉⠉⣉⣉⣉⣻⣿⣦⡀⠙⢿⣆
⠀⠀⢰⣿⠁⠀⣿⡏⣼⡟⠁⢠⣾⠟⠀⠀⢀⣴⡿⠿⢷⣿⣧⡀⠀⣼⡟⠻⣿⣟⠀⠀⢀⣾⣿⣇⢀⣤⣶⠿⠿⠛⠛⠛⠛⠛⢻⣿⠀⠈⣿⡄
⠀⠀⠸⣿⣆⠀⢻⣿⣿⠀⢰⣿⠃⠀⠀⣠⣿⣿⣤⣀⠀⠈⠙⠻⣷⣿⠁⠀⠘⢿⣧⣠⣾⠏⠈⢿⡿⠋⠀⢀⣠⣤⣤⣶⣶⣦⣾⡟⠀⣰⣿⠃
⠀⠀⠀⣿⣿⣦⣀⠉⠻⢷⣿⣇⣀⠀⢰⣿⠁⠀⠉⠻⢿⣦⠀⠀⠘⢿⣆⠀⠀⠀⢻⣿⠋⠀⠀⠸⣿⣤⣾⠟⠛⣉⣥⣴⣿⠟⠋⣠⣴⣿⣿
⠀⠀⠀⢻⣇⠈⠻⢷⣦⣄⣈⠙⠛⠿⠿⣷⣶⣤⣤⣤⣌⣿⣧⣀⣀⣈⣿⣆⣀⣀⣈⣿⣆⣠⣤⣤⣿⣿⣶⠿⠿⠛⠋⣉⣠⣴⡾⠟⠋⣸⡟
⠀⠀⠀⠸⣿⡀⠀⠀⠈⠙⠻⠿⣷⣶⣤⣤⣀⣈⡉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⢉⣁⣀⣤⣤⣶⣾⠿⠟⠋⠁⠀⠀⢀⣿⠃
⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠿⠿⠿⠿⠿⣷⣶⣶⣶⣶⣶⠾⠿⠿⠿⠿⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢷⣶⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣶⡾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣿⣋⣙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠶⠶⠶⣶⡶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⢶⣶⣶⡾⠿⠿⣦⣶⠿⠻⢿⣶⡿⠻⢿⣭⣭⣉⣉⣩⣿
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡅⠀⠀⣿⠃⠀⠀⢰⣿⠀⠀⣾⣏⣉⣉⣉⣉⠁
⠀⠀⠀⠀⠀⠠⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣻⣇⠀⠀⢿⡇⠀⠀⣸⡇⠀⢠⣿⣋⣉⣉⣉⣹⣷
⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⡄⠀⢸⣷⠀⠀⣿⡇⠀⢸⣿⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⢸⡿⠀⠀⣿⡇⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⣾⡇⠀⠀⢸⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣿⡇⠀⢰⣿⡀⠀⠀⣼⡇⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠉⠁⠀⠀⣿⡇⠀⢸⣿⠻⣷⣴⡿⠁⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⣿⡿⠁⠀⠀⣠⣴⡾⠟⠻⣷⡀⠀⢿⡆⠈⢻⣧⠀⠀⣰⡿⠁⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⢰⡿⠁⠀⢠⣾⠟⠁⠀⣠⣴⣿⡇⠀⠸⣷⠀⠀⢻⣇⣴⣿⡿⠿⠛⠛⠛⠛⠛⠿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⣠⣾⡇⠀⠀⣿⠃⠀⢀⣾⠏⠁⣼⡇⠀⢀⣿⠀⠀⢸⣿⠋⠁⢀⣀⣤⣤⣤⣀⡀⠀⠈⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢰⡿⢿⡇⠀⠀⣿⠀⠀⠸⣿⠀⣰⡿⠀⠀⣼⡟⠀⠀⣼⣧⣴⠿⠛⠋⠉⠉⠉⠙⠻⢷⣄⠀⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠸⣿⡀⠀⠻⣷⣄⣠⣿⡿⠋⠀⢀⣼⠟⠀⠀⣰⡿⠋⠁⣀⣤⣶⠾⠿⣷⡄⠀⢸⣿⠀⠀⠀⠉⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢶⣾⡿⠶⠶⠿⠷⠶⠿⠿⠶⠶⠾⠿⠿⠿⠶⠶⠶⠿⠿⠶⠶⠾⠿⠷⠶⠾⠿⠿⠶⠶⠶⠿⠷⠶⠿⠿⠶⠶⠶⠶⠾⠿⣶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠆⠀⢀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠟⠀⠀⣠⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠋⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⡿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣴⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣻⣿⣿⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣛⣿⣿⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣤⣴⣲⣾⠿⢿⣛⣿⣿⣯⣿⣯⡭⡭⣍⣽⡿⣿⣯⣽⢿⣿⡿⣟⠿⠿⣷⣶⣦⣤⣀
⠀⠀⢀⡤⢖⣯⣷⣛⣩⣥⣶⣿⢿⣿⣿⣿⣯⣿⣿⣹⣾⡿⣗⣶⢖⢾⣻⣷⣛⣿⣵⡵⣶⣶⣮⣭⡛⠾⣽⡲⢄⡀
⠀⡴⣫⣾⢛⣽⣿⣿⣵⣿⡯⣺⣿⣿⣿⣿⡿⣿⣷⢿⡟⡟⣿⢿⡞⣟⣽⣵⢞⣯⣽⣿⣺⣿⣿⣿⣽⢢⣌⠛⣷⣝⢦
⡼⣱⣿⡱⡿⢿⡶⣿⣿⣿⣟⣽⣿⣿⣿⣯⣿⣿⣿⡜⣺⣿⣿⡞⣿⣻⡿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣷⢽⣣⣈⢿⣎
⡇⢿⣇⠓⠻⣿⣿⣶⣽⣿⢿⣿⣿⣿⣮⣛⣿⣿⣆⣿⣿⣟⣿⣞⣯⣯⢻⡜⣿⡿⢿⣿⣟⣿⡿⣿⣿⣯⣾⣿⣿⣸⡿
⠹⡜⢿⣄⠀⠻⠿⣿⢿⠿⣯⣿⡿⣿⣿⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢗⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠿⣫⡿⢣
⠀⠈⠢⣝⠷⣤⣀⠻⣿⣭⣷⣾⣿⣿⣾⣿⣿⣿⣿⣟⢭⣝⣿⣿⣿⣿⣽⣿⣷⣿⣿⣿⣻⣯⣿⡿⠟⣋⣤⠾⣫⠔⠁
⠀⠀⠀⠈⠙⠲⢽⣛⠶⣬⣍⣉⡉⠛⢿⣷⣾⣽⣿⣿⣛⣥⣿⢿⣿⣿⣿⣿⣽⣿⠿⢛⣛⣡⣤⠶⣛⡯⠗⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠚⠭⠭⣛⣓⣺⡿⠿⠶⠦⠤⠤⠤⠭⠥⠶⠶⢶⣖⣛⣛⠭⠭⠓⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⡭⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣠⣴⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣠⢤⣴⣾⣿⣿⣿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣛⣿⡽⣷⣿⢿⣻⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣷⣿⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⣶⣾⣿⣧⣿⣿⣯⣿⠽⠿⠽⠯⠭⣭⣭⣽⣿⣿⣖⣲⣶⣶⠤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣠⡤⣖⣺⣭⡷⠶⢛⣛⣭⣿⣿⣿⡶⣿⣷⣞⠳⡳⣭⠽⣟⣛⡿⢿⣾⠿⢗⣭⣻⢭⡛⠛⠷⢾⣭⣗⣲⢤⣀⡀
⠀⠀⠀⢀⡤⢞⣭⣷⣟⣋⣭⣴⢶⣿⣿⠿⣿⣛⣛⣿⢮⣾⣿⣇⣷⣧⡿⣟⣾⡿⡚⢯⡻⣯⣝⣓⢿⡵⠼⣶⣿⣷⣶⣭⡙⠻⣾⣭⡳⢤⡀
⠀⢀⡖⣩⣾⠟⣫⣿⣿⣿⣩⣿⢟⡭⣲⣿⣿⣿⣿⣿⣿⡽⡏⡆⣿⢿⡹⣻⢿⣪⢺⡿⣻⢳⣮⢭⣝⡻⣽⢺⣯⣯⣿⣿⣜⡦⣄⠙⠻⣷⣝⢲⡀
⢠⠏⣼⡿⢃⣾⣿⣯⣭⢟⡿⣗⣽⣞⣿⣿⣿⣛⣿⠿⣏⣿⢽⣇⣏⣾⢷⣿⣇⣟⢬⣚⣾⠿⠵⠿⢾⣿⢿⣿⣿⣿⣿⣿⣿⣾⣿⣷⣄⠘⣿⣷⣻
⡿⢸⣿⠳⡋⣯⣽⣿⣕⡻⣿⢟⣏⣾⡿⣿⣿⡸⣿⣿⢿⣯⢺⢸⣴⣿⣯⣷⡏⣿⠵⢿⣯⠲⠷⠷⡿⠷⠳⡛⠿⠟⠟⠟⢿⣿⢿⣿⣿⣿⣼⣿⡇
⣷⢸⣿⡄⠉⠙⣷⣿⣿⣿⣮⢻⣟⣿⢿⣿⣿⣿⡶⣶⢯⣿⣻⣸⣿⣟⢵⣿⣻⣽⣿⣷⣹⢻⢻⠿⣛⣛⣛⣻⣽⣿⣛⣿⣿⣿⣽⣿⣿⢿⢿⣿⡇
⠘⢦⠻⣿⣄⠀⠙⠫⢿⡶⢿⡿⢿⣾⣸⢿⣿⡾⣿⢟⣿⣿⣿⣕⣝⣿⣿⣿⢳⣾⣿⣿⡇⣿⡧⣫⣿⡾⣹⣿⣿⣿⣿⣿⣛⣽⣸⣻⠕⣣⣿⠟⡴
⠀⠈⠳⢌⡛⢷⣤⡀⠐⣿⡿⣟⣛⣦⣿⣻⡿⣽⣿⣾⣿⣿⣿⣿⢛⣟⣛⣻⣿⣿⣿⣟⣽⣿⢷⣾⣯⣶⣿⡿⣿⣛⣻⣿⡿⠟⢁⣤⡾⢛⡡⠞⠁
⠀⠀⠀⠀⠙⠲⢭⣛⠷⣦⣍⣙⡒⠒⠓⠻⢿⣮⣾⣭⣟⣻⣿⣿⡷⡯⢋⣿⣛⣿⣿⡿⡿⣻⢟⣿⣿⢿⣿⣿⠟⢉⣉⣥⣴⠾⣛⡭⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠬⣝⣛⡿⠶⢶⣤⣬⣛⣻⣿⡿⠛⠋⠑⠚⠛⠚⠯⠽⠛⠓⢛⣛⣛⣛⣯⣥⡶⠶⠿⣛⣫⠥⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠮⠭⠭⣭⣭⣽⣛⣛⣛⣛⣛⣛⣛⣛⣿⣭⣭⠭⠭⠵⠒⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⠒⠂⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢸⣃⠠⠤⡀⣬⣠⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠀⠐⠼⠃⡀⠊⢰⣷⡌⡰⠀⢀⡀
⢀⣀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠟⠉⠁⠁⢰⢈⠆⡴⠓⡆⢜⠩⠀⠁⠀⣆⠠⡀
⠀⠀⠀⡈⢉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠐⠁⠄⠎⠀⠚⢣⣨⠀⠄⠀⢰⣀⡰⢄
⠶⠞⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠶⢤⣀⠀⣀⣦⠀⠃⠃⠞⡀⠄⠺⢃⠔⢤⡓⠄⠙⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⡮⢊⠀⠔⠂⠄⢀⡌⡀⠀⠦⠀⠀⠀⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⣀⢤⡞⠁⢀⠀⠁⠀⠁⠀⢀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢴⠇⠸⠀⡴⠁⠀⠀⠀⣴⣦⢚⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢃⠀⠀⠀⠀⢡⠀⠀⠀⠀⠘⠋⠎⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⢁⠃⠀⠀⠀⠂⠳⠀⠰⠃⢈⣆⠈⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡆⠘⢯⠀⡁⠀⠀⠀⡇⠠⢀⠀⢠⠳⡷⣄⠈⠂⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⠛⠉⠁⠠⡇⢀⠆⡼⠀⠀⠀⢈⠆⠔⠉⠂⠈⡟⣀⠀⠆⠰⠈⡛⠻⠷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⢀⠐⠚⢩⠂⠀⠐⢱⠛⠚⡁⠊⡐⠁⠀⡂⠰⡇⠂⢠⢄⠤⢯⡠⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⡸⠃⠀⢀⠼⡄⡄⠀⡃⠀⠁⠀⡖⠡⢦⠘⠀⠀⠏⠎⡇⠉⠐⠄⣘⡁⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⡀⠀⣁⠆⠐⠉⠀⠍⢀⡐⢧⠰⡀⠀⠀⠘⠺⠀⠀⡜⠠⡄⠈⢶⡄⠸⠑⠐⠡⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⢳⢰⣏⡼⣨⢂⠀⠀⠀⣱⡈⡇⠡⠀⠀⡠⠘⡐⠞⡁⢠⠹⡀⢰⢥⠘⢠⠐⠀⡋⡄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⣾⠤⠊⠐⡉⠄⠀⠀⠈⠽⡇⡃⠅⠀⠈⠀⣆⠉⡞⠀⣠⣜⣀⣎⡤⠗⡄⢚⢸⠄⠈⠂⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠐⡄⢠⣡⡤⣤⠂⠀⢢⠀⠀⠀⡴⠡⠁⠐⠠⠠⢀⡿⠃⣷⢀⣐⡀⠀⡀⡠⠊⠀⠲⣄⣯⡰⠜⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣆⠘⢄⢲⠰⡇⠀⢀⢰⠆⠐⠀⠤⠀⢀⡋⠉⢁⡈⠀⠠⠟⢧⡞⠁⠁⣡⣛⠃⠀⠄⠨⣟⠏⣆⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⢈⠓⡹⠀⠚⠦⣀⡴⡲⠶⡒⢇⠚⠛⠂⢶⣈⠣⠀⠀⣀⡴⠊⠁⠩⠀⠀⠀⠓⢤⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢭⡳⢬⣀⡠⠶⢦⡈⠀⢀⣘⠃⠄⣀⡤⢈⠀⠈⠁⠀⠀⠀⠇⠀⠀⢈⣃⣠⣴⠾⢛⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠉⠛⠿⠶⣶⣤⣌⣁⣀⡀⡀⣇⢀⢀⢁⣐⣦⣤⣌⣤⣴⡶⠿⠛⠋⢑⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡠⢐⠀⠀⠀⠉⠉⠉⠙⠛⠛⠛⠛⠛⠉⠉⠁⠐⢂⠨⠀⠈⣂⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⣈⡑⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠴⠊⠁
⣿⢿⢿⠿⣿⢟⠿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣛⣛⣛⣻⣿⣟⣟⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣾⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣶⣾⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠛⣿⣿⡆
⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⢀⣠⣦⠀⠀⠀⠀⢸⣿⣷
⢰⣿⣿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣿⣿⡿⠟⠃⠀⠀⠀⠸⣿⣿
⣿⣿⡇⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣿⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⣿⣿⡇⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⢸⣿⣿
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⠿⠀⠀⠀⣾⣿⡇
⠘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⠿⠟⠋⠉⠀⠀⠀⠀⣰⣿⣿⠁
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣿⣿⣿⠿⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠁
⠀⣿⣿⡇⠀⠀⢰⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⣾⣿⣿⣿⣯⣤⣤⣀⣀
⠀⣿⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣤⣀
⠀⢹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣷⣦⣄
⠀⠀⠹⣿⣿⣦⣤⣀⣀⣀⣀⣠⣴⣶⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣦⡀
⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⣿⣿⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣾⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⡿⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⢻⣿⣧⣀⣠⣴⣶⣿⣿⣿⡿⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⡇⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣤⣤⣾⣿⣿⣿⡿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣿⣿⡿⠿⠛⠋⠉⠁⠀⠀⣀⣀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⡿⠛⠋⠉⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⣿⡿⠿⠛⠛⠉⠁⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡿⠀⠀⠀⠙⢿⣿⣦⡀⠀⢀⣴⣿⣿
⠀⠀⠀⠀⠀⣿⣿⡇⠹⣿⣿⡄⠀⠀⢀⣠⣤⣶⣶⣿⣿⣿⠿⠿⠛⠋⠉⠁⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣿⣿⡿⢿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⡀⠙⣿⣿⣆⠀⠀⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⡿⠿⠟⠛⠉⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠈⢿⣿⣧⡀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⣿⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡀⠀⠻⣿⣿⣆⠀⠀⢸⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣷⡄⠀⠘⢿⣿⣷⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠙⢿⣿⣷⣤⣄⣀⣠⣤⣴⣶⣿⣿⣿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⡀⠀⠀⠉⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣧⡀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣀⣀⣠⣤⣶⣾⣿⣿⡿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⠿⠛⠋⠉
⣿⣿⣿⠿⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠞⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⣤⣴⡶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡖⠁⢀⣾⣿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡁⠰⢘⣿⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠌⣿⡁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⡶⠤⢄⣀⣀⣀⣇⠘⠘⣿⣂⣁⣀⣀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣾⢸⠀⣴⣦⣌⣻⣶⣿⠀⠀⣿⣿⢟⡿⣯⢤⣱⣾⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣽⡾⠛⣉⣃⡻⡿⢻⣅⠀⠙⣆⣠⡿⣿⣿⡷⣇⣫⠟⠚⠺⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⡷⠤⠿⠿⢿⡇⢸⡓⢤⣤⣿⡷⢳⢽⡯⢗⠚⣓⣓⠚⠽⣆⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⣷⣿⢮⣭⣿⣧⠸⣇⡹⠻⡋⠣⡞⠁⣠⣴⣊⣈⡵⠯⠷⢛⢫⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢻⣿⣆⠀⣀⣉⣁⠉⠉⠉⠉⠈⠉⠉⢉⣉⡀⠀⠀⠀⠀⣖⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⢿⣿⣿⢹⣿⣾⡇⠀⠀⠀⠀⠀⢰⣿⣾⡿⠂⠀⠀⡼⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠻⣿⣦⣭⣽⣃⣻⣤⣤⣤⡟⠈⠫⠍⠕⢀⣴⡶⣳⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢌⣛⣿⣿⣿⣿⣾⣯⣥⣤⣤⣴⣴⣿⣟⡭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠒⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏⠀⢠⡶⠛⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⣠⡴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠾⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠖⠛⠛⠛⠉⠉⢉⣉⣉⣀⣀⣀⣀⣀⣉⣉⠉⠉⠉⠛⠛⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠛⠋⢉⣀⣠⣤⠴⠶⠶⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠳⠶⢦⣄⡀⠈⠙⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⣀⣠⡴⠶⠛⠋⠉⠀⢀⣀⣠⣤⣤⠤⠶⠶⠶⠶⠶⠖⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠾⢿⣷⣄⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⢀⣠⡶⠛⠉⠀⠀⣀⣤⡴⠖⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡄⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠛⠁⠀⢀⣠⠶⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⣼⠃
⠀⠀⠀⢀⡾⠁⠀⢠⡟⠁⠀⣠⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⠋⠀⠀⢀⡼⠃
⠀⠀⠀⢸⡇⠀⠀⠘⣧⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠛⠁⠀⠀⣠⡶⠋
⠀⠀⠀⠈⢷⡀⠀⠀⠈⠻⣧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠛⠋⠉⠀⠀⢀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠛⢦⣄⠀⠀⠀⠉⠙⠛⠶⠶⠶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠴⠶⠶⠶⠚⠛⠛⠉⠁⠀⠀⠀⣀⣠⣤⠶⢻⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⢦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠴⠶⠚⠛⠉⠁⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⣿⡛⠛⠒⠲⠶⠶⠶⠶⠶⠒⠒⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⢀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⢀⠀⠀⠠⡦⠥⡠⢄⣠⢤⡄
⡠⠶⠿⠇⢼⠺⢬⠤⢶⠬⠂⠁⠱⠹⠕⠤⡤⢤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣠⣤⣤⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⢠⠀⢸⡏⡄⡄⢹⡇⡄⡆⣿⡇⠀⠀⣀⣤⠤⣤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠒⠒⠚⠓⠒⠒⠛⠓⠒⠒⠋⠁⠐⠓⠂⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⡴⠦⠴⠖⠒⠒⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠈⠉⠀⢀⣀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⢀⣀⡠⠠⠀⣚⣉⡥⠤⠤⠒⠒⠂⠉⠉⠁
⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣺⣯⣽⣯⣿⣽⣽⣭⠿⣷⣦⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣝⡸⢿⡟⣽⣷⡯⣧⣫⡻⢿⠯⣿⣾⢽⣟⣕⡆
⠀⠀⠀⢀⣴⣶⡶⠀⣠⣤⣤⣤⣤⡀⠰⣶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣷⣿⣾⣿⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣿⠁⢀⣿⣿⣿⣿⣿⣧⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠈⠀⠼⢿⣿⣄⢸⣿⣿⣿⣿⣿⣿⠠⠾⠿⠟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⢀⣀⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠋⣉⣠⠤⠐⠂⠉⠁⠀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢶⢒⣋⡩⠤⠤⠄⠒⠒⠉⠉⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣯⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠄⠒⢀⢀⠔⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣆⠀⠀⠀⠀⢀⠖⢀⣀⣀⠒⢰⣤
⠀⠀⡂⠁⠀⠄⣀⡤⡏⡷⣶⣤⣄⣠⣶⣿⠑⠀⠀⠀⢸⢇⠀⠀⠤⠴⡲⢍⣅⣀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⣿⠏
⠀⠄⠻⣿⠖⣛⠱⣀⣍⡀⠎⠸⠫⠽⠟⢉⠔⠀⠀⣴⠃⡼⠀⠀⠀⠸⠒⠛⢄⢂⠅⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠇⠆
⠀⠀⠀⠈⢉⣺⡉⠁⢐⠎⠀⢒⡁⠄⠒⠁⣀⢔⠟⠁⠊⠂⠀⠀⠀⠀⠢⠠⠊⡀⠀⠀⠂⠀⡆⠀⢀⡠⠀⠀⢘⢀⢠
⠀⠀⠀⠁⠀⠀⠀⠀⠒⠀⠉⠀⠀⢀⡔⠞⠊⣠⠖⠋⠀⠀⠀⠀⠀⠁⠈⠀⠀⠠⠀⠀⡄⠀⢇⢀⡀⠀⣀⣀⣈⡤⠚
⠀⣄⠅⠀⠂⠀⠨⡄⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⢀⣰⠣⠀⠄⠀⠀⢀⠌⠎⡩⢥⡭⠆⠀⠀⠀⢀⣀⣀⡀
⠀⡗⠒⠐⠀⠀⠀⢹⠀⢀⠀⠀⠠⠤⠀⢀⠀⠀⠀⢚⠂⠀⠀⠀⠀⠅⠀⢈⣀⣀⢀⡌⠤⠦⠀⠐⠳⣢⠛⢛⣿⣶⣾
⠀⡿⡦⢀⣀⣀⠠⠺⡄⠀⢱⠔⣒⡩⣿⡭⡥⠀⠀⠡⠀⠀⠀⢀⣀⠂⠀⠘⠀⠀⠀⠐⠃⠀⠀⠈⠀⠻⠶⠟⠙⠋⠁
⠀⡇⡇⠀⠀⠀⢄⠀⡇⠀⠈⠎⢟⡿⣿⢯⡇⢡⠀⠀⠑⠠⠄⠘⠃⠀⠀⠀⠀⣐⠂⠀⠀⠀⠠⢀⡀⠀⠀⠀⢀⣴⠎
⠀⠙⢀⠀⠀⠀⠈⢢⢰⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⢀⠄⠀⠀⠀⡔⡀⠀⢠⡄⣄⠁⠂⠀⡬⠄⡠⠬⠟⡀
⠀⠀⢨⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⢤⣐⣂⣬⠴⠒⡺⠉⠉⢂⠀⢀⡊⠐⠸⠁⠀⠐⠀⠘⠁⠀⠇⢩⠴⠛⠃⠁
⠈⠒⠦⡄⠀⠀⠀⢀⡀⠇⠀⠀⠀⠀⠀⢈⢡⡎⠁⠀⢁⠀⠀⢨⠈⡀⠪⠌⠁⢒⡂⠀⠄⠌⠀⠀⠀⠀⠠⢒⡨⠸⡀
⠀⢀⣀⣀⡀⠄⠀⢀⠀⢀⠔⠀⠀⠠⡀⠀⠈⢷⡀⡐⠁⠀⠀⠀⡄⠡⠀⠀⠀⠀⠀⠀⠈⠁⠀⠠⠀⠘⠒⠒⠀⠈⠁
⢈⠔⣉⣡⢞⡋⠥⠐⠀⢈⠀⠀⠀⠀⡀⠀⠀⢸⡇⠀⠀⠀⡠⠊⢺⠀⢀⠀⢠⡴⠢⠄⠤⡆⠀⣀⣠⣒⠒⢲⣄⣀⣀
⠈⠂⠉⠀⠀⢀⠄⢲⡄⠈⠂⠤⠤⠔⠃⠀⠈⠀⠷⠄⠄⠀⠒⠃⠄⠂⠈⠀⠈⠻⠄⡠⠤⠗⠛⠒⠚⠒⠲⠾⢶⢦⠢
⠀⠀⢀⣴⠆⡖⠀⠀⣰⢒⣶⣤⣴⣖⢄⠀⠀⠀⠀⠁⠀⡠⡄⠤⡤⡀⠀⠀⠀⠠⠀⢀⡓⠀⠠⠌⠀⠀⠠⢧⠔
⢀⠰⣿⢇⣴⣿⣫⣿⡗⠐⡟⠿⣽⢿⢆⠡⡀⠀⡀⠮⠉⢉⠍⠒⠆⠉⠒⣄⠀⢀⠸⣿⣂⠠⢀⢠⠄⠀⠀⠈⠀⠀⢄

⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⢀⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠠⡍⠀⠀⠀⠀⠈⠉⠑⢄⠀⠋⢐⠃⡠⠀⠈⠳⡀⠀⠀⠀⡎⢁⣀⠆⡀⢀⡻⠌⠒⠀⠀⠀⠀⠀⠃⢔⠀⠀⠀⠈⠁⠈⡆⠘⡃⡑⠀⠀⣀⠤⠤⡄⣄
⠀⠀⠀⠀⠀⠀⡀⣀⠠⢔⣉⠃⠀⠸⡄⠀⠂⡄⡀⠉⡄⠀⢘⠀⠈⡂⠀⡀⠀⠀⠀⠐⡀⠀⠀⠀⠄⠀⠀⡠⠁⡀⠀⠂⡰⠀⢀⡈⠚⠀⢾⠥⠤⣈⠓⡎
⠀⠀⠀⢈⠄⠄⠀⠂⠀⡠⠌⡃⠀⠀⠙⢄⡀⠡⡈⠢⣸⠀⠀⢱⣝⡥⣩⠀⠀⠀⠀⠸⠀⠀⠀⡇⠀⠀⡔⡀⠀⠀⢂⡔⠁⠀⠈⠉⠣⡀⢠⠀⠀⠀⠀⡇
⠀⢀⡀⠐⣉⣁⠀⠄⠂⠂⠈⣀⡀⣀⣀⡀⠈⠒⠨⠆⠀⣠⠶⠀⣏⢝⡁⠀⠀⡀⢄⡧⠋⠒⡀⢄⠆⡐⣀⠀⠐⡠⠊⠀⠱⡴⣄⣀⣀⠁⠘⠀⠤⠤
⠀⠘⠇⠀⠀⣤⠀⠀⡠⠔⠋⠃⢠⢃⠂⡀⠉⢢⡀⠀⠀⡿⠛⠀⠸⣎⡆⠀⠠⠬⡞⠀⠀⠀⣍⠀⡀⠣⣀⠥⠊⠠⢀⠀⢀⡀⠀⡠⠁⠀⠀⣀⠀⠀⠐⠢
⠀⠀⢀⣄⡀⠁⢀⡊⢐⣁⣊⠠⠦⠌⢰⡖⡫⠵⠂⢀⣀⣤⣀⠄⠀⢳⠀⠀⠀⠈⢑⠂⠲⠔⠫⠕⠂⠀⠀⠀⢀⢁⠊⣀⡀⠈⠉⠁⠀⠒⠈⠀⠰
⠀⠀⢏⠀⢹⡄⠐⢴⠒⠠⠒⠚⠒⠊⣉⣀⡤⠜⠀⢸⠀⠀⠀⢉⠆⠀⡠⠂⠄⠀⢐⠀⠍⠀⠀⠀⠀⢠⣤⣠⣤⠁⠀⠑⠁⠀⠉⠂⠀⠀⠀⠀⣁⢀⡐
⠀⠀⣸⢀⠤⠱⡀⠀⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⡎⠀⠀⠀⣸⠇⠀⢇⠴⠀⠠⢈⠈⢃⠀⠀⢴⣿⣯⣹⣿⣔⣿⣆⠀⠤⠀⠀⢠⡀⠀⢤⣋⠀⡬⠁
⠀⠀⢱⠈⠒⢀⡇⠀⠀⠀⢶⠙⠒⠀⠤⠄⠒⡶⠀⠙⠃⠒⠖⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⢀⠀⢀⠿⠀⠀⠀⠀⠘⡄⠰⣀⣀⠔⠀⠀⠈⠉⠉
⠀⠀⠀⠙⠘⠉⢠⢴⡀⠀⢄⠁⠀⠐⠒⠂⠈⠁⠀⡀⢄⡢⣠⡀⠀⠀⣎⠀⠀⠀⠒⢲⠶⡄⠀⡉⠁⠀⠁⡛⠫⠈⠉⠀⠀⡸⠄⠀⠀⢀⠄⠐⡄⠐⠐⡄
⠀⣰⠀⣐⠀⢠⠁⠀⠉⠀⠉⠀⠀⠀⠠⠠⠤⠄⡖⣆⡿⠟⢁⠇⣀⠌⣰⠈⠉⠈⡉⠉⠂⠁⠀⠐⠀⠀⠄⠀⠀⠒⠀⡤⠉⠀⠊⠀⠀⠋⠀⠀⡸⠒⢴⡡
⠀⠈⠉⠀⢀⠂⠀⠀⠀⢀⠀⢠⢤⠈⠲⠂⠀⠋⠁⣀⠤⠖⢁⡊⠙⠀⢉⡐⣽⠟⠁⠀⠀⠀⢢⠀⠠⠖⠀⠀⣄⣀⠀⣠⠀⢀⡀⠀⡎⠐⢀⡎
⠀⠀⠀⠀⠂⠀⠀⠐⠀⠂⠀⠈⠁⠀⠈⠁⠀⠉⢁⠀⢀⣔⣡⠥⠴⠖⠚⢉⡠⡄⠈⠀⠀⠀⢸⠀⠀⠀⣄⢒⠄⠥⠠⠄⠨⠄⠋⢄⠈⠉⡠⠗⠀⢀⡀
⠀⢀⢀⠰⢈⠉⠣⣀⠀⠀⠀⢀⠢⠀⠤⠄⠀⠀⠓⠁⠀⠁⢀⣄⠀⠀⠀⡫⠭⠃⣀⠀⡀⠔⢁⡀⠀⢀⡀⠀⠄⠉⠉⠉⠈⠉⡁⢈⠀⠀⠀⢀⠋⠀⠈⡀
⠀⠘⢤⡀⠀⠀⠁⣁⠀⠀⠀⠐⠺⡞⠠⠀⠁⢠⣶⠶⠒⠃⠼⢻⠤⣄⠀⠉⠀⡐⠈⠁⠀⠀⠑⠁⠀⢰⡈⠀⠄⠀⠈⠁⠈⠀⠈⡋⠀⠀⠀⠸⢀⠄⣀⠃
⠀⠀⡀⠀⠶⢈⠜⠁⢀⠀⠂⠀⠀⠁⠄⠁⡠⠋⠀⠁⠀⠈⠂⠉⠉⠁⠀⠐⠡⡀⡄⢀⠀⠠⠂⣑⡄⠈⠒⠀⠤⠤⠤⠤⠤⠤⠒⣛⠆⠀⠀⠆⠁⠈⢨
⠀⢨⠁⠘⠀⠁⠆⠀⡨⢀⡠⠀⠤⢄⠀⠀⠀⠂⠸⠀⠀⣀⠂⢀⠔⠈⠙⠀⠀⠈⠀⠘⢠⠐⠒⠁⠀⠀⣀⠠⠀⠀⠀⠀⠀⠀⠁⠀⢰⣡⠀⠂⠤⡄⠆
⠀⠀⢇⢐⡀⡠⠐⠀⢠⡃⡴⠉⠠⠀⢄⠀⠓⠀⠀⠐⠚⠋⢁⠂⠀⠀⠀⢰⠀⠰⠀⠠⠊⠠⠊⠑⠀⠸⠃⢀⠠⠀⠀⢀⡠⠂⠄⠀⠈⠺⠀⢸⣀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢱⠈⠈⢀⠀⡠⠁⣎⠑⠒⠒⠀⡀⠀⠂⠀⠠⠀⣾⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠈⠐⠒⢧⠀⡀⢀⡱⠄⠀⠀⠀⢀
⠀⠀⡁⢨⢹⢸⡀⢲⡀⠘⠢⠚⠁⠀⠀⣐⠁⣀⡄⠀⠀⠀⠀⠈⠃⠀⠀⢣⠀⠨⠀⢉⠀⠀⠀⡠⠔⠋⠀⠀⠀⡠⠒⠀⠈⢉⡗⠁⠀⠀⠐⣧⣃⡠⠐
⠀⠈⠁⠀⠠⠀⠀⢈⡀⠀⠘⢁⣈⡢⠀⠈⠁⠘⠘⡀⠀⠢⡐⠀⠀⢀⣠⠃⠀⠀⠀⠘⠁⡠⡐⡁⠀⠀⠠⠤⡀⠁⠀⠀⣠⠎⢴⠇⠀⠐⠆⠉⠁
⠀⠀⠀⠃⠀⠃⢉⠉⠀⠀⠀⠁⠀⢀⠀⠀⡄⠀⠤⠃⠀⠀⠈⠁⠉⠁⠀⠀⠀⠀⠭⠦⠄⡓⠬⠌⣁⣒⣒⠛⡄⢉⠚⠩⠄⠒⠉⢠⠢⠀⠢⠀⠀⠀⣀⢀
⠀⠀⠀⢀⠀⠀⢈⠀⠀⠀⠉⡄⠀⠀⠒⠌⠀⣀⡤⡀⢰⠋⠉⠙⣇⠀⠉⡰⠀⠠⠀⠀⠀⠈⠄⠀⠀⢀⠤⡄⢀⠤⢆⢀⡠⢄⠀⠀⠁⠀⠀⠀⠀⠀⠁
⠀⢠⠀⡄⠢⠄⠤⠀⠀⠀⠲⢱⠊⠀⠀⠀⠀⠣⡤⠜⠰⠁⠀⢸⠀⠀⠀⠁⠀⠘⠄⠀⣤⠎⠀⠀⠀⠘⠜⠁⠨⠰⠉⠈⠀⠌⠀⢠⣤⠄⢈⠛⠑⠔⣀⣀
⠀⠀⠉⣀⢤⠐⢠⡀⠀⠀⢀⠰⠀⠀⠄⠒⠲⡄⠀⠀⠈⠉⠉⠉⠉⠁⠉⠀⠶⠂⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠠⠀⠀⠀⠀⢀⡜
⠀⠀⣜⠳⢉⡰⠟⠈⠀⠜⠄⢠⠔⠁⠀⠀⠂⡇⠀⢠⠊⢁⡠⠤⠴⡂⠀⠄⠠⠐⠂⡏⠀⠀⠀⠠⠂⠁⠀⠑⠞⠿⠯⠅⠀⠀⠀⠀⠑⡆⠠⠤⠤⠤⠊
⠀⠀⢻⡵⠒⠁⠀⠀⡀⠀⠀⠠⡀⠀⣀⠤⢊⣤⠀⡇⡴⠁⠀⠠⠄⠈⠐⠒⠒⠒⠉⢀⠀⠂⢁⣀⣀⠤⠤⠚⠀⠀⠑⠄⡀⠀⠀⠀⠤⠈⠀⠀⢀⣀⡐⡿
⠀⠀⠀⠈⠐⠀⠀⠊⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⠀⠀⠀⠀⠀⠈⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠟⠛⠛⠉⠉⠉⠛⠛⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⠀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⠛⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠇⠀⠼⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣦⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠉⠁⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠂⠀⠈⠀⠈⠀⠀⠁⠀⠈⠀⠀⠃⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠸⠆⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣶⣿⣷⣦⣤⣤⣀⡀⠀⠀⠀⠴⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠁⠀⣠⣴⣾⣿⣿⠟⠛⠋⠉⢹⡏⠉⢁⡿⠋⠙⠛⠿⢿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠟⠁⠀⠄⠀⠀⢀⣤⣾⡿⣿⠏⠁⢹⠀⠀⠀⠀⠸⠇⠰⠿⠥⠀⠀⠀⠀⠀⠈⣿⠻⣿⣶⣄⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣆
⠀⠀⠀⣼⣿⣿⣿⣿⣿⠋⠀⠠⠀⠀⢀⣴⣿⡿⠋⠀⢸⡀⠀⣼⠀⠀⠀⠀⠀⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠛⢿⣷⣄⠀⠀⠠⠀⠀⠹⣿⣿⣿⣿⣧
⠀⠀⣸⣿⣿⣿⣿⣿⠁⠀⢤⡄⠀⣠⣾⣿⣧⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢨⡇⠀⠀⠀⠀⣀⣴⣶⠀⠀⠀⠀⠈⢹⣿⣧⡀⠀⢠⡄⠀⠹⣿⣿⣿⣿⣆
⠀⢰⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⣸⣿⡿⣿⠁⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⢀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠛⠉⢻⣿⡄⠀⠀⠄⠀⠹⣿⣿⣿⣿⡆
⠀⣿⣿⣿⣿⣿⠃⠀⢀⠀⠀⢰⣿⣿⡇⢿⣄⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⠀⣰⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣻⣿⡀⠀⢀⡀⠀⢻⣿⣿⣿⣿
⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣿⡿⠈⠁⠀⠉⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠟⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⣿⣧⠀⠀⠀⠀⠈⣿⣿⣿⣿
⣼⣿⣿⣿⣿⡇⠀⠀⠁⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣰⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠉⠀⠀⢻⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⣿⣿⠛⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⣿⡇⠀⠀⠀⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠲⠂⠀⣿⣿⣤⣴⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢸⣿⣿⠀⠰⠆⠀⢸⣿⣿⣿
⣿⣿⣿⣿⣿⡄⠀⠠⠀⠀⣿⣿⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢶⣿⡇⠀⠠⠄⠀⢸⣿⣿⣿
⢻⣿⣿⣿⣿⡇⠀⠀⡀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⡀⠀⠀⣾⣿⣿⣿
⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⣿⣷⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⡝⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣿⡟⠀⠀⠀⠀⢠⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⡆⠀⠈⠀⠀⠘⣿⣿⣧⣤⡟⠀⠀⠀⠀⠀⠀⢰⡿⢋⠜⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢸⣷⣾⡿⠁⠀⠈⠀⠀⣼⣿⣿⣿⡿
⠀⠘⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠙⣿⣧⣀⡿⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⢤⣼⣿⡿⠃⠀⠀⠀⠀⣰⣿⣿⣿⣿⠃
⠀⠀⠹⣿⣿⣿⣿⣿⡄⠀⠘⠁⠀⠘⢿⣿⣅⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠙⠃⠀⣰⣿⣿⣿⣿⠏
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣆⠀⠐⠁⠀⠀⠙⢿⣷⣆⡀⠉⢉⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣭⡀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣿⣃⠀⠀⠀⠀⠀⢠⡾⣛⡛⠀⠀⠀⠀⠀⣀⣀⣬⣿⣿⠟⠋⠀⠀⠀⠁⠀⣠⣾⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣦⡀⠀⠁⠀⠄⠀⠈⠙⠿⢿⣿⣶⣦⣤⣀⣿⣋⣉⣹⣦⣠⣤⣶⣾⣿⠿⠟⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣦⣄⠀⠀⠰⠆⠀⡀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠛⠛⠛⠋⠉⠀⠀⠀⠀⠲⠀⠀⢀⣠⣾⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⠄⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⡀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⣒⣉⠉⢩⣉⣉⣉⣛⡒⠒⠢⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⣉⠤⡖⡏⡼⡰⢠⢠⢀⠀⠀⠀⠈⠉⠒⠢⠤⢍⡉⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⢁⢴⠏⡏⡜⡜⡸⢱⢁⢃⢎⠎⡜⡔⣠⠀⠀⠀⠀⠀⠀⠈⠙⠢⢌⠑⠢⣀
⠀⠀⠀⠀⠀⠀⡠⠊⢀⠔⠇⡜⡸⡐⡐⢰⢡⢃⣇⣎⣼⡼⠴⠵⠧⣧⣆⣄⣀⠀⠀⠀⠀⠀⠀⠉⠲⣌⠑⢦⡀
⠀⠀⠀⠀⢠⠞⢀⡔⡙⡜⡸⢰⠱⣱⠵⠓⣋⣉⣒⣒⡒⠒⠒⠒⠖⠒⠂⠀⠈⠉⠛⠶⣤⣀⠀⠃⠆⠈⠑⢄⡉⠢⣤
⠀⠀⠀⡴⠁⣠⢻⢡⢣⢱⣡⠗⣋⣀⣒⣒⣒⣒⣒⠐⠂⠐⠒⢂⣀⣀⡀⢀⠎⢭⡍⠒⡄⠘⢓⢦⡘⠀⠀⠀⠙⢦⠈⠻⡥⠤
⠀⠀⡜⠀⡔⡇⡎⣎⡶⠋⣑⣒⣒⣒⣤⣔⣉⠀⠂⠀⠀⠀⠀⢮⡐⠓⣨⠀⠳⣤⣸⠦⠃⠀⠈⠂⠉⠢⣄⠀⠀⠀⠑⡄⠘⢯⠍⠥⠄
⠀⡜⠀⣼⢃⢳⡼⠋⠀⢀⠀⣀⣰⡲⢒⠄⢺⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⢰⠉⠏⠿⢱⠄⠶⠀⡔⣈⢑⢮⠳⢄⠀⠀⡌⢆⠈⢯⠭⠭⠭⠠
⢰⠁⡜⠸⣠⠞⠠⠭⠥⠤⠤⠄⠒⠊⠑⢒⠗⠒⢤⠀⠀⢠⢊⣁⢉⡒⡌⠛⠒⠀⢚⣛⠀⢘⣂⣈⠓⠓⠊⠀⣀⡑⢄⠳⠈⠆⠈⢯⡉⠉⣀⣐⡀
⣼⢠⡇⣧⠋⠬⠩⠁⠀⣭⠍⠉⠁⠁⠀⢏⠛⠀⣣⢇⠀⠈⢮⣭⡬⣔⣅⠄⣀⠲⠀⡬⠀⣏⠆⠬⣱⢠⡀⠀⠀⠬⠀⠳⡉⢸⣀⠸⡖⠒⢒⣒⣂
⡇⢸⣧⠏⣒⣒⣒⣒⣒⡒⠂⠀⠀⠀⠀⠀⠈⢛⡣⠋⠀⢀⠀⠋⠙⠋⠀⠀⣉⣀⣀⣐⠂⠀⠹⠏⢁⡔⢂⠐⡲⡄⠀⠁⠙⣜⠁⠈⡧⠤⠤⠤⠤
⡇⢸⣿⢐⡒⠒⠀⠤⠤⠤⠀⠀⡠⠒⠤⢄⠀⠀⠀⠠⠎⡫⣁⠒⠠⣆⢵⠼⣠⡐⠯⢊⡗⠛⠀⡴⡘⣾⣉⡱⢇⡼⠀⠀⠀⢸⡇⠀⣟⣉⣉⣉⣉
⣧⠸⡇⠤⠤⠬⢭⠁⠈⠁⠀⠀⢇⠀⡬⢑⡇⡄⠀⠀⠸⡫⣀⡹⠸⡼⠞⢃⣀⡭⠉⣁⠤⢄⡀⠈⠊⢉⢽⣏⠛⠢⡀⠈⠛⢰⠀⢸⡯⠤⠤⠤⠤
⢽⣆⠱⡠⠭⠍⣐⣒⣀⠀⠀⠀⠀⠉⠉⠙⠊⠀⠀⠀⠀⣬⠒⠒⡟⠉⠙⠢⡀⠀⡜⠑⠶⠰⣌⢣⢀⡸⣀⠀⠺⣣⠇⠀⠀⣜⢠⠇⣟⣀⣀⣈⣉
⢸⠘⢆⠑⣄⠒⡒⣂⣀⣀⣀⠀⣀⣀⣀⠀⠀⠀⠀⡠⢀⡤⠼⣇⣹⢟⣀⡴⢁⣀⠳⣔⡇⣸⡭⠊⠈⠪⠮⠉⢉⣁⡀⢀⠞⢀⠯⣂⡗⠒⠒⠀⠤
⢸⠀⡈⠣⡈⠣⡐⢒⣒⣀⠀⢸⡷⠀⠬⣳⠀⠀⢡⡅⡏⢶⠄⢠⡈⢢⠀⢀⣿⠎⣑⠄⠀⠀⠀⠀⣠⡄⠀⠒⠈⠀⣠⠋⣠⢏⣘⣼⣉⣉⣉⣉⣉
⠈⡆⢸⡐⣙⢦⡈⠢⣄⠤⠄⠀⠉⠀⠨⠕⠁⠀⠈⢗⠕⠦⢄⡚⠥⠃⢰⠉⠾⠙⡄⡤⢚⠀⣠⣄⢤⠀⠀⠀⣠⠞⢁⡼⣻⣿⣷⡗⡒⠲⠶⠶⠏
⠀⢳⡀⢣⡐⠒⢙⣯⣘⠙⠫⣍⡂⠀⠀⠀⠀⠀⠀⠀⠑⠊⠒⠲⠄⠀⠀⠉⠀⠉⡸⣀⠻⠂⣦⠔⣼⣦⠴⠂⣠⠴⡄⣶⣾⢿⡿⠭⠭⠭⣭⡍⠃
⠀⠀⠱⣄⠱⣅⠒⣀⠈⡓⠢⢄⣉⠓⠲⠤⣴⣫⢭⡑⠲⣀⠀⠀⠀⢤⠀⠀⠀⠀⠛⢦⣭⡡⠔⠚⢉⡠⠔⡋⡅⡜⡐⡹⢁⡞⠒⢒⣒⣒⠂
⠀⠀⠀⠈⠢⣈⠳⢄⡺⠯⠀⠀⠈⠙⠒⠦⢄⣈⠍⠉⣉⠉⠒⠒⠒⠒⢒⡶⠆⠉⢉⣉⣠⠤⠖⢪⢁⠆⡜⡜⡴⣵⠟⣱⠯⠤⠤⠤⠤
⠀⠀⠀⠀⠀⠈⠳⢄⡉⠲⢄⡉⠒⠊⠀⣀⠀⠈⠉⠉⠛⠒⠒⠒⠒⠒⠒⠛⠋⠉⣡⡦⡲⡲⡡⢣⢂⢎⢜⡾⢖⣡⣞⣛⣛⣀⣘⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡈⠓⠮⢍⡉⠩⠍⠁⡠⢴⡂⠔⠁⠤⠀⠀⠀⠀⢠⢞⢝⠕⡕⡱⣡⡎⢋⣩⡴⠛⠐⢒⣒⣒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠈⠉⠒⠢⠤⣄⣀⣒⣓⡒⢂⠠⠀⠀⣴⣡⣯⡾⡖⢛⣡⣴⣞⣛⣒⣒⣒⡒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠲⠤⢄⣀⣀⠐⠀⠩⠉⠝⢑⣲⣫⣥⣥⣖⢚⡃⠀⠠⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠓⣒⣒⠒⠒⠂⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠐⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣾⣿⣿⠿⠿⠿⠶⠄⠀⠀⠀⠀⠁⠚⠝⡷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠛⠉⠉⠀⠀⣀⣀⣀⣤⣤⣤⣀⣀⣀⡀⠀⠀⠀⠉⠒⠫⢝⡳⢦⣀
⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠉⠀⠀⣀⣤⡶⠟⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠶⣦⣕⣤⣀⠀⠈⠑⠫⢗⡆
⠀⠀⠀⠀⠀⣴⣿⣿⡿⠋⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣮⡙⠲⢤⣀⠀⠈⠑⠦⣀
⠀⠀⠀⢀⣾⣿⣿⡟⠁⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠈⠙⠲⠦⣤⣀⣙⣢
⠀⠀⠀⠀⠙⠁⣿⠁⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠉⠛⠋
⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢠⣷⡠⣕⢮⣷⣦⢄⡀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⢸⣿⡟⢾⡿⠟⠁⠀⠈⠂⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⣿⣷⠀⠉⠓⠖⠚⠛⠳⣤⡈⢳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠘⣿⣧⠀⠈⢿⣄⠀⠀⠈⠙⢦⣌⠻⢗⡢⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⡄
⠀⠀⠀⠘⢿⣧⡀⠀⠙⢷⣄⠀⠀⠀⠙⠳⣤⡉⠲⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⠀⢠⠎
⠀⠀⠀⠀⠈⠻⣿⣦⡀⠀⠙⠳⣦⣀⠀⠀⠈⠻⢦⣄⠙⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣄⡀⠀⠉⠛⠷⢦⣤⣀⡙⢷⣄⠀⠙⠳⢦⣄⡀⠤⠤⠶⠖⠚⠉⠁⠀⠀⠀⢀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄⣀⠀⠀⠈⠉⠀⠙⠷⣦⡀⠀⠈⢿⡄⠀⠀⠀⠀⢀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣶⣤⣤⣤⣬⣝⡳⠶⢛⣤⣴⣶⠶⠟⠛⠉⠁

⠀⠁⠈⠉⠁⠀⠀⠀⠀⠒⠂⠀⠀⠀⠠⠄⠀⣀⣀⠀⠀⣀⠐⠀⠀⠀⠀⠃⠀⠒⠀⠀⠠⠤⠤⠀⠀⠀⠀⢀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢈⡀⠀⠀⠀⠀⢀⠂⠀⠢⠤⠄⠀⠀⠠⠤⠄⠀⣨⠋⠉⠉
⣀⡀⠀⠀⠤⠤⠄⠤⠐⠀⠐⠒⠒⠒⠀⠀⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⡑⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠍⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠐⠒⡦⠈⡠⠎⢽⠀⠀⠀⠔⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠌⠐⠀⠁⠀⠀⠀⠀⠐⠁⠐⡀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠒⠀⠁⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠅⠀⠀⠀⠀⠀⠀⠀⠠⠤⠶⡄⢸⠀⠹⠃⠀⠀⢡⡤⠆⢤⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡂⢄
⠀⠀⠀⠀⠀⠀⠀⠔⡠⠀⠀⠀⠀⠀⠀⠀⡴⠀⠒⠐⢀⠒⠋⠄⠘⠈⣂⠀⠀⠀⠑⡀⠢⠙⣖⠑⣒⡠⠀⡀⠀⠀⠀⠈⠂⡑⢄
⠀⠀⠀⠀⠀⢠⠆⠈⠀⠀⠀⠔⡀⢛⢉⡨⠤⠥⠄⠠⠆⠐⢀⠂⠙⡀⠘⠂⠀⠀⠀⢁⡀⠀⠈⠓⠈⠀⠑⢙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠒⠡⢠⡎⠐⢀⠤⠀⠀⡈⠀⠉⢉⠒⢁⣄⠀⠀⠀⡊⠈⠉⠓⠀⠀⠨⠐⠄⢨⠀⠡⣬⡄⡀⠀⢠
⠀⠀⠀⠀⠀⣀⠀⠀⠂⡀⣁⠈⠀⢂⡅⣄⣅⠀⠀⠐⠠⣄⠀⠊⣱⡲⠂⠛⠀⢤⠁⡀⢀⣠⠐⢭⡷⠒⠂⠺⣊⠐⠂⠄⢠⠀⡟⢐
⠀⠀⠀⠀⠀⢻⠄⠠⠈⠀⠀⠈⠀⠐⠁⠀⠀⠉⠁⡉⠐⢢⠐⠈⠐⢃⡈⡄⣀⠂⢈⣉⠉⠄⠀⠤⡈⠣⡄⡀⠀⠈⡡⠈⣻⠎⠀⠟
⠀⠀⠀⠀⠀⠀⡄⢢⡙⢦⣀⠀⠀⡌⠴⠨⠄⣁⠐⠒⠉⠁⠁⠆⠜⠋⠈⠁⠁⠛⠢⠤⠔⠐⠁⠒⢄⡔⠉⠈⠓⢲⣤⠞⢁⠔⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠉⠢⡈⠓⠾⣤⣀⠀⠎⠉⠛⠁⠁⠀⠀⠞⢀⠠⠀⠤⠞⠀⢠⠁⠀⠀⠀⠀⠈⣀⣠⡴⠞⢋⡠⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠘⡓⠀⡀⠀⠙⠢⠤⢉⡙⠳⠦⠤⣤⣠⣅⣀⣘⣁⣀⣀⣀⣈⣀⣅⣀⣤⠴⠶⠚⠋⢉⡠⠄⠀⠁⠀⢀⠤⢊⠉
⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠁⠂⠤⢀⠀⠀⠉⠀⠐⠀⠠⠤⠉⠉⠉⣉⣉⣉⣉⠁⠀⠤⠀⠀⠒⠂⠁⠀⠀⢀⠠⠐⠊⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⡁⠂⠠⠤⠀⢀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⢀⣀⠠⠄⠒⠉⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠄⠀⠀⠀⠀⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠆⡒⠀⠀⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢀⣀⠀⠀⠀⠠⠀⠀⠀⠰⠀⠀⠀⣀⣠⠤⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠷⠶⣶⣶⣶⣶⡶⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⢸⣁⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢰⣤⠄⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⣀⡗⠀⠀⠘⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⣿⠏⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡆⠀⠀⠘⢧⡀⠀⠀⢀⡤⠟⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠃⠠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡉⠀⠀⠀⢀⣀⡗⠀⠀⠀⠓⣆⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡆⠀⠀⠀⠀⠀⠀⠀⠠⠴⠞⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⡞
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣈⣉⡤⠖⠒⠲⢤⣀⣀⣀⣀⣀⡴⠚⠛⠓⢦⣀⣀⣼⠀⠀⣼⣁⣀⣀⡀
⠀⠀⠀⠀⡿⠿⠯⠭⠭⠤⠼⢥⣤⠄⢤⣬⣧⣤⣤⣤⣼⣥⣤⣤⣤⣤⣷⣤⠧⠤⠼⠧⠤⠭⠭⠿⠟⢷
⠀⠀⠀⠀⡇⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣸
⠀⠀⠀⠀⣷⠀⡞⠉⠉⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠈⢹⠀⡟
⠀⠀⠀⠀⢹⡀⠹⠤⠤⠏⠀⠀⠀⠀⠀⠀⠀⠀⡴⠒⢦⡴⠚⢲⡀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠞⢰⠇
⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠘⣆⠀⢰⠋⠻⡄⠀⠀⠀⠀⠀⠀⠈⠳⣄⢀⡴⠁⠀⠀⠀⠀⠀⠀⢰⠋⠛⡆⢀⡞
⠀⠀⠀⠀⠀⠀⠈⢧⡈⠲⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠚⢡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣀⠀⠀⠀⢰⣋⣳⠀⠀⠀⠀⠀⠀⠀⠀⢾⣉⡇⠀⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⢤⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⠗

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣷⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠟⠛⠛⠋⠉⠙⠛⠛⠻⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⢀⣠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠛⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣥⣠⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠉⠁⠀⠀⠀⠐⠀⠀⠓⠀⠐⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⠿⠋⢡⡀⠀⠂⠀⠀⠀⠀⠀⠀⣠⡀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠐⠀⣠⡉⠛⢿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⡿⠛⠁⠐⠀⠀⠁⣀⠀⠀⣀⠀⠀⠀⠈⢹⡇⠛⠀⣸⠇⠀⠀⠀⠀⢀⡀⠀⠁⠀⠐⠀⠙⠻⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠋⠀⠀⠃⠀⠀⠀⠘⢻⡇⠘⣿⠀⠀⠀⠀⢸⡇⢠⣼⣋⠀⠀⠀⠀⠀⠛⣗⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠁⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⠀⠈⠁⠈⢉⠉⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠁⠀⠙⢿⣿⣿⣿⣿⣷⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⠏⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠛⠀⠀⠀⠀⢀⠀⡀⢰⠀⠀⠀⠀⠀⠀⢀⣋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡈⢻⣿⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣿⣿⣿⣿⠏⠀⠙⠁⠀⢶⡆⣴⠟⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡇⣸⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⡼⠛⢻⡆⠀⠘⠁⠀⢻⣿⣿⣿⣿⣷⡀
⠀⣼⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⡇⣯⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⠀⠀⢀⣴⣿⣿⠋⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣧
⢠⣿⣿⣿⣿⣿⡟⠀⠐⠀⠀⠀⠀⠸⠇⠘⠷⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠾⠿⠶⠆⠀⠀⠀⠐⠀⠘⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⠇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠠⠀⠀⠀⣠⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⠠⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⣠⡀⠀⠸⣏⠀⢈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⢠⡄⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣈⣛⣹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⣀⣼⠇⠀⢀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠀⠈⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠀⢸⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⡇⠀⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣿⣿⣿⣿⣿
⠘⣿⣿⣿⣿⣿⣿⡀⠐⠀⠀⠀⠀⢰⣏⣙⡧⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⡣⠃⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢻⡇⠀⠀⠀⠀⠐⠀⢰⣿⣿⣿⣿⣿
⠀⢻⣿⣿⣿⣿⣿⣧⠀⠠⠀⠀⠀⢸⡛⠛⣷⠀⠀⠀⠀⠀⠀⠀⠸⢯⠜⠁⠀⠘⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣁⣸⡇⠀⠀⠀⠀⠄⢠⣿⣿⣿⣿⣿⡏
⠀⠀⢿⣿⣿⣿⣿⣿⣧⠀⡴⠆⠀⠈⠳⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠋⠀⠀⠰⡄⢠⣾⣿⣿⣿⣿⡿
⠀⠀⠈⢿⣿⣿⣿⣿⣿⣧⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠒⠒⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠒⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⣠⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⣠⠴⠦⠀⠀⠀⠀⠀⠈⠉⠙⣧⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣷⣄⠀⠂⠀⠀⠀⠀⠠⡏⠀⠀⠀⠀⠀⠀⣼⡷⠖⢶⡄⠀⠀⠀⠀⠶⣤⡴⠏⠀⠀⠀⠀⠂⢀⣴⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣷⣄⡐⠁⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⣀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠘⣀⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣀⣀⡀⠀⠀⠀⠻⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠤⠤⣒⣒⣒⣒⣒⡒⠒⠤⠤⢤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠋⠉⠀⣠⠴⠚⠉⠁⠀⠀⠉⢉⡿⣷⣦⠀⠀⠀⠀⠉⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⢀⠀⢠⠊⠀⢀⡕⠀⠈⠐⢈⣵⡿⠁⢹⠟⠠⡐⠲⣦⣄⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡔⠋⠁⢉⢧⡀⠀⣋⡀⢤⡴⠚⢭⢋⣷⣠⠊⠀⠀⠀⢀⢼⠟⣿⡆⡀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⡴⠋⢣⡀⣀⡬⠞⠿⣿⣋⢐⡾⢻⣢⠶⠴⠋⠐⠓⢀⣤⣴⠕⢳⣏⣟⣼⢃⡆⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⣿⡀⠀⠈⡃⠀⠀⣀⠘⡜⠋⠁⠈⠙⢶⡶⠒⠲⠿⣿⣋⣻⣧⡖⣩⡍⠵⣹⠇⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠈⠻⣄⡀⠣⣄⣮⣹⢿⠀⠀⠀⢀⡀⣼⣽⠗⠶⠶⠶⠻⠷⠚⣧⣈⣵⠞⠋⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠙⠛⠶⣶⣤⣤⣷⣷⣾⣶⣾⣋⣛⣒⣢⣭⣿⡿⠷⠞⠛⠉⠀⠀⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠦⠤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠔⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠶⠚⣛⣻⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡤⠶⠚⠛⣉⣡⣤⠶⠶⠛⢋⣁⣴⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡶⠶⠟⠛⠛⠛⠛⠛⠛⠉⠉⠛⣛⣻⣿⡶⠶⠞⠛⠉⣉⣠⣤⡶⠾⢛⣉⣉⣤⣤⠶⠞⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠖⠛⠋⠉⢁⣀⣤⣴⠶⠞⢛⣉⣡⣤⠶⣾⣯⣉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋⠉⣀⣠⣤⠶⠾⠛⢛⣉⣡⣤⠶⠛⠋⠉⠀⠀⠀⠀⠀⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⣀⣠⣤⠶⠛⠛⠉⠉⣉⣀⣤⡴⠶⠛⠋⢉⣀⣤⡶⣾⣿⣿⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⡿⠷⠚⠛⠉⢁⣀⣤⡴⠶⠛⠛⠋⠉⠀⣀⣠⡴⠾⠛⢉⣥⠶⣾⣿⠿⠿⣿⣟⠛⠳⢶⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠙⣧
⠀⢀⣀⣤⠴⠞⠛⣿⣿⣆⣀⣠⣴⠶⠟⠛⠉⠀⠀⠀⣀⣠⣤⠶⠟⣫⣷⣴⡇⣴⠟⣱⠞⠁⠀⠀⠀⠀⠘⣧⠀⠀⠙⣿⠿⣶⣤⣀⠀⠀⠀⠀⠀⢸⡆
⣴⠿⣷⢀⣀⣤⡶⠿⢻⣿⢿⡍⠀⠀⠀⢀⣠⣤⣶⣿⣯⣥⣽⣿⣧⢿⣿⣿⣿⡏⠀⣿⡀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⣿⣦⣿⣿⢟⣿⣦⠀⠀⠀⢀⣿
⠻⢾⣿⣯⡉⠀⠀⠀⠀⣿⣿⣧⡴⠶⢻⡿⢹⣶⣾⣿⡿⠿⢶⡶⠂⠸⣿⣿⣍⠻⣆⡙⢷⣤⣤⣤⠶⠛⠉⣀⣠⣴⣾⣿⣿⣯⡿⠋⢉⣿⢷⡄⠀⢸⣿
⠀⢾⣇⣸⣧⡤⠶⠚⢻⣿⣅⠀⠀⣴⢏⣿⣿⡿⣼⡿⣱⣶⣼⠃⠀⣠⣙⢿⣮⣻⣮⣿⣳⣶⣶⣶⣶⣶⣿⣿⣴⣿⡿⠻⠿⠿⣟⣻⣿⠷⣾⡇⣠⡾⣿
⠀⠀⠈⠉⠀⠀⠀⠀⢸⡏⠙⠷⣼⣏⠾⣿⡿⣹⣿⢣⣿⣷⡟⢠⣤⡜⢛⡀⠹⣿⣻⣿⣿⣿⢿⡿⣟⣱⡿⠟⠋⣉⣤⣶⣿⣿⣿⡟⣠⡾⣣⣿⠋⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡄⠀⠈⠙⠿⣾⣵⣿⣷⣯⣿⡟⣴⣮⣩⣤⠹⢟⣴⣾⡽⣿⣾⣾⣁⣾⢋⣿⣤⣶⣿⡿⠟⠋⢈⣽⠿⣻⣿⠿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣄⠀⠀⠀⠀⠙⠻⢿⣿⣿⡀⠻⠋⢹⢿⣶⢟⣛⠛⣿⣛⣛⠿⣯⣿⡜⣯⢿⣟⢷⡀⣠⣴⣯⣶⠟⠋⠀⠀⠀⢀⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣄⡀⠀⠀⠀⠀⠈⠙⠛⠿⠿⢶⣾⣧⣬⣿⣶⣿⣛⣿⣀⣈⣿⣧⣭⣬⣽⡿⠿⠟⠋⠉⠀⠀⠀⠀⣀⡴⣋⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣏⣿⣬⠿⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣚⣽⠋⣹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⡿⢻⣧⣿⣛⢿⣶⠶⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣀⣠⣤⣴⠶⠿⣿⣉⡙⣿⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣌⠙⠻⠷⢿⣿⣇⣘⣧⢸⣧⣦⠉⣿⣩⣿⢛⢻⣄⢠⡞⢋⠛⢦⣾⢩⣌⠻⣶⣏⣿⣤⠿⠟⢋⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠉⠉⠉⠉⠛⠓⠶⣾⣇⣙⣟⣁⣹⣎⣷⣾⣥⣬⣿⡿⠿⠞⠛⠉⠁⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⢞⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣬⣭⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣒⣚⣛⣋⣉⣥⡴⠞⠉