Deadline Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡩⠩⢑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢡⡌⠀⠡⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢖⢴⣏⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢘⠃⠸⠀⠀⢂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠊⢈⠎⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠷⠆⡸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢋⠃⢰⠁⠀⠀⠀⠀⠀⠀⡔⠁⠐⠆⠀⠀⣀⠤⠀⠀⡄⠀⠀⠠⡀⢄⡀⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⢀⠈⢤⠀⠉⢢⠖⠈⣧⡀⢀⠴⠒⠉⠀⠀⠀⠀⡜⠀⠀⠀⠨⠮⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⣇⢺⠠⢵⡄⠂⠀⠚⠁⠀⢀⠀⠀⠀⠀⢀⡠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⢠⠩⠍⡀⠀⠀⠀⠀⠘⡄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠂⠂⠹⡀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣀⡀⣤⢤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡀⠛⢀⡱⠊⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⡉⠓⠤⣀⠀⠀⠸⠴⠄⠀⣄⠀⠀⠀⠀⢀⠄⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠂⠈⠢⡀⡠⠻⣯⡇⠠⣀⡀⢸⠐⠂⠀⢁⣚⣐⠒⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠭⣆⢠⡊⠀⠈⠁⠀⣌⡰⡄⠑⠤⡉⠀⠀⠀⠨⠤⠘⠊⠢⠂
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⡊⠀⠀⢉⠊⠀⠀⠀⠀⡠⠁⣀⠴⡄⠐⡇
⠀⠀⠀⠀⢀⠔⣞⠄⡀⡐⠉⣽⡆⠀⠀⠋⠉⠉⠁⠈⠑⡾⣁⡔⠁⡘
⠀⠀⠀⡠⠋⠀⠫⠜⠒⠁⣰⡽⠁⠀⠈⠲⡀⠀⠀⠀⠴⡋⡷⠀⢠⠁
⠀⠀⠸⠀⠐⠲⠶⠤⢄⢴⡿⠁⠀⠀⢀⡴⢀⡀⠀⠀⠀⣍⡠⠴⢾⣦⠄⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠄⠤⠙⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠤⢄⠀⠀⠰⠴⣤⡤⠴⠀⠀⡠⠔⠒⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢀⡀⡆⠈⣿⣄⣀⢀⣸⣇⣀⣀⠾⠁⠀⠀⠀⢀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠘⠧⢘⣤⡶⣞⠉⢀⠺⠛⠉⠙⢄⡒⠒⠀⠀⢀⢼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢡⡶⠞⢋⣠⠠⠤⠤⠀⠴⠦⠌⢵⣦⡤⣕⠦⣤⣉⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⢀⠞⠉⠀⠀⠲⡀⠀⠀⠀⠀⠀⠈⠙⠺⣷⣄⠉⠋⠺⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠠⠂⠀⠀⠀⠀⣠⠱⣀⠷⠀⠀⠀⣀⠀⠀⠘⢿⣧⠀⠠⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⡆⠀⠀⠂⠁⠀⠸⡆⢻⣸⡇⠀⠀⠀⠉⠂⠀⠈⢿⣧⠀⠂⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠥⠀⠀⠁⠒⠀⠀⢹⣄⣿⡇⠀⠉⠉⠉⠁⠀⡀⠘⡿⡆⢘⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⣿⡏⠇⠀⠀⠀⡠⠢⠊⠀⠉⡿⣇⢰⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠳⡀⠀⠀⠀⠊⠀⠀⠈⠹⣾⠀⠀⠈⠀⠀⠀⠀⡸⢿⠇⡐⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⠡⡱⢄⠀⠀⠀⠀⠠⠄⠒⠒⠒⠂⠤⠀⠀⠀⣰⠟⡼⠲⣜⠆⠱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠊⠳⣌⠪⡳⣆⣀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡤⠞⣩⢆⣑⣎⡁⠀⠀⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⠀⠐⡎⣰⣮⡳⠿⠿⣶⣶⣶⣶⣶⣶⣿⣿⡔⢚⠁⣶⣯⣬⠁⢂⠐⡡⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠛⠁⡁⠐⡩⠓⠰⡻⢓⡗⠛⠗⢣⠟⡡⠊⠓⡬⣹⡏⠀⠀⠊⠔⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠠⠊⠀⢊⠔⠈⢀⠔⢁⠔⡡⠈⡠⠈⢀⠔⠊⠔⡡⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠃⠀⢊⠔⠁⠀⠀⣠⠚⡡⢊⡀⠈⢔⠁⡠⠂⠠⠊⢐⠧⠀⠀⠀⠄⠁⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣶⠀⠀⢀⠀⠐⢊⠀⡠⢊⠔⡀⢊⠄⡡⢊⠤⠀⠀⠔⠁⢰⠀⠀⠄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⣿⠀⠀⠠⠀⠀⡀⠈⠀⠠⢊⢔⡅⢊⠔⢁⢀⠀⠀⠀⠈⢸⠀⠀⢂⠄⡈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠔⡠⢊⠔⡡⢊⠔⡠⢊⠔⠁⢀⠄⡡⢂⠀⡠⠊⠈⠄⠀⠡⢊⠐⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⣾⠀⠀⠎⠐⡡⢊⠔⡡⠊⠐⢀⠔⠄⡡⢊⠔⡡⠊⡠⢊⠀⡆⠀⠐⠅⠂⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡡⢀⣿⠀⠀⠈⠔⡰⢃⠊⡠⢊⠔⡡⣢⢆⠔⣡⠄⡠⢊⢔⡡⡀⣧⠀⠀⣬⢅⡠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⠀⠀⠀⠀⠀⢊⠔⡡⠊⠐⣡⡾⠃⠀⠈⠀⠀⠈⠀⠁⠀⠀⠁⢻⠀⠀⠖⡡⢂⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠕⠀⢀⡁⠀⠠⠂⠈⡠⢀⠔⣰⠀⠀⢀⠀⠀⠐⡀⠊⠐⡁⢊⣴⣻⡇⠀⡫⢞⡅⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢀⠀⠀⣤⠀⡈⢡⠀⠀⢁⠈⡐⠁⠈⡘⠀⠀⢸⠀⠀⢊⠔⡡⠈⡀⠔⢹⢠⡀⠀⠈⠑⠈⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠘⠀⠀⠘⠀⢱⠀⠀⠀⠃⠌⠐⠡⠊⡁⠀⠀⢘⠀⠀⡁⠈⡠⡀⢀⠄⡁⠐⢀⡄⠀⠀⠀⠀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⠀⠀⠰⠀⠇⠀⠀⢊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⡡⢂⠐⠃⡇⢸⠃⢰⡇⣠⠀⢷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠳⠀⠀⠀⠀⢠⠀⠀⠂⠀⠐⡠⢠⠀⠀⠀⠀⠀⠐⡠⢊⠄⡡⠚⠀⠈⠀⡀⢸⠇⢻⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠁⡠⠀⠀⢸⠀⠀⠀⠀⠀⣉⠔⡀⠊⢀⡇⠀⠀⠀⠙⠀⠚⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⢁⠀⠀⢸⠀⠀⢠⠀⢀⠅⡀⠀⠀⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠀⠁⠀⢂⠔⠸⣄⡤⢴⠀⡠⢂⠔⠡⠊⠀⠳⢄⠄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠰⣅⣠⣀⣀⠀⢀⡰⣋⡱⠶⢉⠄⠈⠶⣤⣤⣀⡀⠀⠀⠀⠈⢻⠔⣁⢀⠀⠀⢈⠀⠀⠀⠀⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠩⠛⠩⠛⡩⠙⢀⠄⢊⠀⠈⠔⠠⢊⠐⡉⠛⡳⠀⡤⠥⠖⠈⠀⠁⠈⠊⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⣀⣠⣤⠤⠶⠶⠤⠴⠦⠀⠠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢴⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⣿⣥⠌⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣾⣧⣄⡘⣀⣬⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣴⣿⣿⣍⠉⢉⣽⣿⣷⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⣿⣿⣿⣿⣯⢙⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⣶⣆⣼⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⡮⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣉⣋⣻⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣯⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⡿⣻⡙⣯⣿⣿⡟⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⣿⣥⣄⠰⣶⠘⣿⡏⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢚⣋⢿⡿⠟⢿⣿⡟⢻⣿⡉⠛⣷⠀⢺⣿⣿⡆⠸⣿⣯⡿⣷⡝⠀⣿⣿⠛⢀
⠀⠀⠀⠀⢠⡶⠀⢶⣦⡘⣿⣏⠉⣾⣿⢢⠀⢻⣿⠠⢿⣧⠀⢻⠄⢸⣿⠟⡟⠘⢿⣹⣧⠙⠽⠆⠻⠟⠐⠛⠁
⠀⠀⠀⠀⠀⢻⡁⠈⢿⡇⣿⣿⠋⢸⡟⢨⡤⠀⠛⠀⣘⣋⣴⣿⣭⣤⣷⣶⣷⣶⢾⣠⣤⣤⢤⠤⠶⠶⠒⠚⠁
⠀⠀⠀⠀⠀⣿⡷⠴⠏⠀⠛⠛⠘⠋⢲⣾⣿⣿⡿⠿⠿⣛⣛⣛⣩⣥⣭⣴⣶⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠒⠤⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⢿⣟⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠘⢿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠈⢿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣟⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⠶⠶⠶⠶⠶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⣀⣀⣀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠈⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠛⠉⠉⢉⣿⠄⠀⠀⠘⣿⡏⠀⠀⠀⢰⡶⠶⠶⢶⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⣠⣾⣟⠁⢀⣀⣤⣴⣿⣷⣤⣄⡀⠈⠻⣦⣄⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⠁⢘⣿⡿⠟⣉⣥⣤⣤⣤⣌⣉⠛⠷⣶⣿⠛⢷⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠⡿⠋⣠⠞⠋⠁⠀⠀⠀⠀⠈⠙⠶⣌⠙⣷⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢁⣾⠋⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠙⣧⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⣼⠇⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⢸⡇⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⣿⠀⠀⠀⠀⠀⠈⠙⠿⣦⣀⠀⠀⠀⢠⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠘⢧⡀⠀⠀⠀⠀⠀⠀⠈⠛⢷⡄⠀⣸⠃⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠛⢦⣄⡀⠀⠀⠀⠀⠀⠀⣀⡴⠃⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣶⣄⡈⠉⠛⠒⠶⠶⠶⠚⠉⣀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣨⠟⠛⠛⠲⠶⠶⠶⠖⠚⠻⣯⡀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⣴⠾⠛⠛⠛⢻⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠻⢶⡄
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠃⠀⠛⠛⠛⠋⠀⠘⠛⠛⠛⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣿⣀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠛⠛⠛⠀⠈⠛⠛⠛⠃⠀⠛⠛⠛⠋⠀⢀⣴⠾⠛⠉⢩⡍⠉⠙⠻⣦⡀
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⢸⡇⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠛⠛⠛⠀⠈⠛⠛⠛⠃⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠻⠿⠛⠛⠛⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠷⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣦⣤⣤⣤⡶⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⠤⠤⠶⠒⠒⢋⣉⣉⡡⠥⠤⠙⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠴⠖⠒⠋⢉⣉⣩⡤⠤⠤⢒⠒⠚⠉⠉⠉⠀⠂⠀⠀⠇⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠉⠍⠛⣶⠊⠉⠩⠁⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⡠⠤⠤⠒⠒⡊⠉⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡄⠐⡿⠀⠀⠀⣀⣀⡠⠤⠤⠒⠒⠊⠉⠉⠁⠠⡀⠀⣁⡀⣤⠀⠄⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣿⢹⣿⣿⠛⠉⢉⣄⡀⠀⠀⠀⣠⣀⣤⠀⡶⡶⡆⣏⡟⠯⠄⠃⠀⠀⢼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣿⣿⣿⣿⠀⠀⠈⠇⢸⠰⣟⡀⠧⠈⠓⠂⠁⠁⠀⠀⣤⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⢿⠀⠀⠀⠉⠁⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⣰⢻⡇⠀⠀⠀⠘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⡇⢀⠀⠀⠀⠀⠀⢼⡇⠈⢻⠀⠀⠀⠐⠁⠀⣿⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⡇⣸⡇⠠⠀⠀⠀⡀⠀⠀⠀⠠⣯⣄⡀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠠⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢻⡇⣧⡇⣷⠀⠀⠀⠀⠁⠀⠀⣀⠀⠀⠈⣷⡀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠸⡇⣿⣧⣿⡄⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⣾⠁⠀⠀⠀⠀⠻⠀⠀⠐⠁⠀⣦⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⢹⢹⣸⣧⠀⠀⠀⠀⠀⠀⠑⠢⠔⠚⠁⠀⠀⠀⠀⠀⢀⠀⢀⢰⠋⠀⠘⣇⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢸⣿⢺⠸⡌⣿⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⣄⠀⢤⠀⠸⡈⣷⡈⣎⢷⠂⠀⠘⠄⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⢹⣾⠸⡇⡇⢃⢇⠀⢀⣀⠀⢲⠉⠁⢸⣧⡘⡄⠑⡌⡆⠀⢱⡘⡝⢾⡌⢇⠀⣀⠐⠂⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢸⢺⡄⡇⢩⢸⠸⡄⠘⡄⠙⣌⡶⠂⠀⡷⢳⡹⡀⢸⠸⡄⠀⢧⠹⠀⠛⠈⠋⠁⢀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢾⣼⡇⣷⢸⡄⣇⢧⠀⠸⡀⢸⡸⡄⠀⠸⠀⠓⠑⠊⠀⠉⠉⠀⠀⣀⣀⡤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⢸⢸⢸⠃⡟⢸⠆⢹⠈⡆⠄⠳⠼⠁⠉⠉⠀⣀⣀⡤⠤⠴⠒⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠮⠼⣼⣠⣇⣸⠀⣸⡆⠸⡄⠂⢀⣠⠔⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⢇⡠⠷⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⣠⣴⡾⠿⠛⠛⠛⠛⠿⢷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢀⣾⡟⠋⣀⠀⠀⠀⠀⠀⢀⣀⠙⢿⣧⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠹⣿⠆⠀⠀⠰⣿⠇⢀⣤⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠿⠷⣾⣿⠀⢤⣦⣤⣤⠀⣀⣀⠴⠺⠖⠉⠈⠈⣿⡆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠀⠃⠐⢸⣿⡀⠀⠀⠀⠈⠉⠻⠟⠀⠀⠀⠀⠀⢀⣿⠇⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⣤⣶⣾⠿⠛⠁⠀⠀⢾⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠺⠋⠁⠚⠒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⡀⠀⢀⡈⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⢸⠀⠀⠀⠸⠇⠀⠘⠃⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠸⡀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠀⠀⠀⠇⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠉⠐⠁⠀⠑⠘⠀⠀⢀⠜⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⣀⣀⣀⠠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⡄⠀⡄⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠙⣄⠛⡄⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡙⠛⢻⡌⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠈⡟⢇⢱⠀⠀⢀⠁⢀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⡇⠈⢧⢣⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢣⣇⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠃⠀⠀⠀⠻⡆⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣐⣲⣣⣼⠀⠀⠀⠀⠀⢹⡎
⠛⣿⡟⠛⠛⢷⣦⠀⠈⠻⣿⡟⠛⠛⢻⡇⠀⠀⠀⢸⣇⠀⠀⠀⠛⣿⡟⠛⠛⢷⣄⠀⠈⠻⣿⡟⠁⠀⠀⢸⢻⣿⠟⠛⢿⣿⡄⠀⠀⠙⢿⡟⠛⢻⣿⠛⠛⠛⣿
⠀⣿⡇⠀⠀⠀⢻⣷⠀⠀⣿⡇⠀⠀⠀⠇⠀⠀⠀⣿⣿⡀⠀⠀⠀⣿⡇⠀⠀⠈⢿⣧⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣷⡀⠀⠀⢸⡇⠀⢸⣿⠀⠀⠀⠸
⠀⢹⡇⠀⠀⠀⠈⣿⡇⠀⣿⡇⠀⢰⡆⠀⠀⠀⢸⠃⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⠸⣿⡆⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠘⡇⢿⣧⠀⠀⢸⡇⠀⢸⣿⠀⠀⢰
⠀⣼⡇⠀⠀⠀⠀⣿⡇⠀⣿⣧⣤⣾⡇⠀⠀⠀⡾⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢰⡇⠈⢿⣧⠀⢸⡇⠀⢸⣿⣤⣤⣿
⠀⣾⡇⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⢹⡇⠀⠀⢰⣧⣤⣬⣿⡆⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⢸⡇⠀⠘⣿⣆⢸⡇⠀⢸⣿⠀⠈⢿
⠀⣿⡇⠀⠀⠀⢀⣿⠇⠀⣿⡇⠀⠈⠁⢀⡄⣼⠉⠉⠙⢹⣷⠀⠀⣿⡇⠀⠀⠀⢠⣿⠃⠀⣿⡇⠀⠀⠀⣠⢸⣿⠀⠀⢸⡇⠀⠀⠸⣿⣼⡇⠀⢸⣿⠀⠀⠈
⠀⣼⡇⠀⠀⠀⣼⠟⠀⠀⣿⡇⠀⠀⠀⣼⢣⡇⠀⠀⠀⠘⣿⡄⠀⣿⡇⠀⠀⢀⣾⠏⠀⠀⣿⡇⠀⠀⢠⣿⢸⣿⠀⠀⢸⡇⠀⠀⠀⠹⣿⡇⠀⢸⣿⠀⠀⠀⢀
⣤⣿⣿⣦⣴⠾⠋⠀⠠⣴⣿⣷⣤⣤⣶⣿⣿⣷⡤⠀⠀⣴⣿⣷⣴⣿⣿⣦⡴⢾⠋⠀⠠⣼⣿⣷⣤⣴⣾⣿⣾⣿⣶⣴⣾⣿⣦⠀⠀⠀⢻⡇⢠⣾⣿⣶⣤⣴⣾
⠁⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠘⠀⠀⠀⠋⠀⠀⡃⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠃⠈⠀⠀⠀⠀⠀⠋⠘⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠸
⣿⣿⡟⠛⠻⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⡇⠀⡞⠃⣾⢹⠀⢺⠀⣖⡆⢰⢳⠀⡞⡆⣾⠢⢰⠂⢰⢻⠠⠻⠀⡏⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠒⠒⣇⣷⠒⣇⡗⢺⣼⠘⣿⢷⣧⣷⢺⣼⡚⣇⣷⡻⡜⢻⣸⣻⣼⣛⣼⣏⢧⣿⠉⠉⢹⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠀⠢⢀⠖⠐⡆⢀⠖⠒⠰⡀⠀⠏⠉⠑⠢⡀⠀⢀⠔⠢⠀⢰⠉⠃⢸⠉⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠈⠀⢠⡡⡄⠀⠃⠨⠀⣄⠀⢡⢰⠀⠘⠀⣠⠂⢀⠇⢰⣊⣀⣸⠀⢐⣚⠀⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⣀⡼⠀⢇⣀⡇⠃⣠⢄⠀⠈⣲⡆⠀⣤⣀⣑⡜⢦⣀⣠⠞⠡⠤⠼⠸⠤⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⡀⢀⠀⡀⠀⠤⢄⠠⠠⠀⠤⢠⠀⠀⠂⢠⠀⢲⠠⠄⢠⢀⠠⠄⠠⠤⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠠⠀⠅⠠⠤⠄⠠⠠⠜⠒⠐⠀⠄⠔⠘⠒⠃⠘⠄⠐⠸⠤⠜⠠⠤⠄⠠⡀⠄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡆⢰⠀⠂⠂⢠⠒⠄⠔⠰⡄⠖⢤⠀⠄⢠⠀⡔⢢⢠⠀⢄⠠⠄⢤⠠⠒⠆⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠑⠚⠈⠂⠂⠘⠂⠂⠈⠚⠁⠃⠊⠀⠂⠘⠀⠣⠞⠘⠂⠊⠈⠂⠂⠘⠒⠃⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠐⠒⠀⠒⠒⠄⡊⠱⠀⡎⠂⢘⠉⡆⠐⠉⡆⡒⢒⠀⠒⢲⠀⡒⢢⠀⠒⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠈⠚⠁⠉⠉⠁⠉⠈⠁⠁⠁⠈⠉⠁⠈⠉⠀⠉⠁⠀⠉⠁⠀⠉⠁⠀⠁⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢧⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣄⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠬⠭⠍⢁⣚⣛⣓⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⠒⠒⠒⠒⠀⠀⢉⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣂⣐⣂⣀⣀⡀⠀⠄⠒⠒⣒⣛⣛⣛⣛⣛⣒⣚⣓⣀⡀⠀⠀⡠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠒⠒⠒⠒⠒⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠓⠲⠞⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠿⣿⢿⠿⠿⠷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢟⠋⠉⠉⠁⠀⠸⠉⠉⠓⠢⢍⡛⠿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠲⣌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡈⢿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠳⡈⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠻⣑⢤⡀⠀⠀⠀⠀⠀⢀⣠⡶⠟⠁⠀⠀⠀⢣⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢬⡳⣄⣀⣠⠶⠛⠁⠀⠀⠀⠀⠀⠀⠨⡄⢿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⢀⠝⢠⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⡠⠋⣴⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣷⣤⡀⠠⠃⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡵⢊⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠿⣷⣖⣠⠤⠤⠤⠼⠤⠄⣒⣩⣥⣶⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠃⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⢿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⡄⣠⡀⢠⢠⣤⣤⣠⢠⡄⠀⢀⡀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢻⣿⣿⠿⡿⡿⢿⣿⣿⠿⢿⢿⣿
⠀⠀⠀⠀⠀⠀⠓⠛⠓⠚⠊⠛⠛⠛⠚⠃⠀⠀⠛⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣴⣿⣿⣾⣷⣶⣾⣿⣿⣿⣶⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⡀⠀⠀⠀⣀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣏⣈⣷⣀⣼⣉⣙⣆⣀⣾⣁⣹⣆⣠⣏⣉⣳⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢠⠟⠀⠀⠀⣤⠇⠀⠀⢀⡼⠃⠀⠀⢠⡞⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⠁⠀⠈⠁⠀⠈⠁⠀⠉⠀⠀⠉⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠆⠀⠠⠄⠀⠠⠄⠀⠴⠀⠀⠤⠀⢀⣀⣤⣄⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⢀⡀⠀⢀⡀⠀⢀⠀⠀⣠⠞⠉⠀⠰⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠁⠀⠈⠁⠀⠈⠀⠀⠈⠀⢰⠃⠀⠀⠀⢰⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠂⠀⠐⠂⠀⠐⠂⠀⠒⠀⣿⠐⠂⠀⢀⡼⠀⠀⠀⠒⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣆⠀⠀⠈⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣀⡘⣀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣆⠙⣧⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡈⢷⡀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢀⣾⠏⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠄⠱⠄⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⢀⡾⠁⠀⠀⣠⡶⠟⠉
⠀⠀⠀⠀⠀⠠⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⡀⠀⠘⠀⠞⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠓⢀⠎⠀⠀⣀⠶⠋⠁⠀⡈⠂⠀⠀⠀⠨⡀⠀⠉⠳⢎⡀⠀⠠⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡞⠀⠀⠀⠁⠀⣀⢄⣈⠤⠐⠒⠒⠒⠠⠌⣐⢄⡀⠀⠈⢂⠀⢳⠀⡇⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⢧⠀⠇⠀⡀⠂⠂⣡⠞⠁⠀⠀⠀⢀⢀⠀⠀⠠⠀⠑⠺⡀⠀⠈⢀⡞⢰⠃⠀⡰⠛⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⣄⠟⠁⠀⠀⠠⠶⠀⠘⠚⠀⠀⢰⠀⠀⠀⠘⢦⣀⡈⠀⠀⠀⠈⠒⠋⠉
⠀⠀⠀⠀⠀⢀⡀⢀⡀⠀⠀⠀⠈⠀⢀⣠⡏⠀⠄⢰⢴⠀⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⠈⡆⠉⠉⠉⠉⠉⠉⠀⠋
⠀⠀⠀⠀⠀⡸⠀⠀⠠⡀⠙⣄⠀⠉⠉⢠⠁⠘⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⢀⠔⠁⠀⠐⠈⢀
⠀⠀⠀⠀⠀⠉⠑⡴⠁⠀⠀⣽⠀⠀⠀⢸⠀⠀⠀⠛⠀⠀⠀⠀⢠⠂⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠉⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠚⠁⠀⠀⠀⠀⡆⠈⠀⠀⣄⠀⠀⢠⠏⠀⠀⠀⠀⠀⢤⠀⡀⠀⡌⠀⠀⠀⠉⠒⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⠀⠀⠘⡄⠀⠀⠀⠀⠄⠀⠀⡀⠀⠀⢤⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⣠⠖⢊⣠⣤⡴⠶
⠀⠀⠀⠀⠀⠀⢀⣠⣤⠴⢒⠀⢀⡏⠀⠀⠀⠈⢢⡀⠀⠀⠁⠀⠀⠓⠀⢀⠈⠀⠀⣠⠊⠀⠀⠀⠀⠀⢸⡅⢀⠀⠦⢤⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⣠⡴⠋⣠⠊⠀⢀⣀⠀⠀⢂⠜⠓⠤⠄⣀⣀⣁⣀⣀⠤⠔⠛⢄⢂⠀⠀⠀⢀⡀⠀⠑⢄⠙⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠈⣧⠀⠁⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠁⠀⢠⠏⠀⠀⠀⠀⠀⠁⠀⠈⠻⡷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡄⠀⠄⢰⡎⠈⢀⠀⢀⠈⠑⣄⠀⣠⠐⢚⣁⣠⠀⠀⠈⠀⠀⢘⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠰⣮⠀⠐⢸⡇⠀⠀⢸⡇⢰⡏⣄⠀⣤⠀⠜⣴⠛⠄⠈⠉⡁⠀⢠⡆⠀⢀⡄⠁⢀⠀⢀⠀⠀⡿⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠀⠸⡇⢠⡀⠈⠇⣼⢁⣛⡀⠘⠀⠾⠁⣾⠀⢘⠛⠁⠀⢸⡇⠀⢸⡇⢠⡟⠀⣶⡄⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠁⠘⣷⡀⢰⣇⡜⠸⣧⣀⡄⣲⡀⠹⠶⠶⠶⠞⠀⢸⡇⠀⢸⡇⢸⠇⢀⣍⣠⣼⡁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢀⣠⠈⠷⠴⠞⠀⠀⠀⠿⠤⠞⠷⠤⠖⠀⠀⠀⠀⠈⠻⠶⠤⠖⣿⣄⡀⠀⢠⣏⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲⠀⢸⢳⡆⠀⢰⠛⡆⠀⢰⠒⡆⢰⠓⡆⢸⠓⡆⢰⠛⡆⢠⠖⡆⢰⠟⡄⣾⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠤⢼⣈⠤⢼⠰⠧⠄⠸⡁⡷⠐⠾⡀⡳⠰⣂⡗⠺⣀⡗⠺⣀⡷⠲⡀⡗⢺⣀⡓⢻⢘⡛⠋⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⠙⠁⠘⠋⠀⠀⠀⠉⠁⠀⠈⠃⠁⠀⠉⠁⠀⠀⠁⠀⠁⠁⠀⠃⠃⠀⠁⠁⠈⠈⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠂⡀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⠋⠙⢐⠁⢐⠀⠁⠈⠀⡁⠈⠀⡏⠀⡅⠊⠀⠀⠈⠀⠈⠰⠁⠈⡀⠈⠀⡆⠉⠀⠁⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠈⠀⠈⠀⠁⠀⠀⠉⠉⠀⠀⠀⠁⠈⠉⠁⠈⠁⠀⠈⠁⠀⠀⠈⠈⠀⠙⠒⠋⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠁⡂⠀⠂⠈⠉⠂⡋⠉⠁⠈⠁⠆⢸⠁⠱⠐⠂⠐⠀⠂⠒⠀⢘⠐⠂⠈⠉⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠉⠉⠀⠑⠒⠀⠈⠀⠁⠈⠀⠀⠈⠉⠁⠈⠀⠀⠈⠀⠚⠀⠓⠒⠀⠈⠀⠁⠈⠀⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡖⠀⠆⠘⠀⠐⠰⠉⠉⠀⠏⠙⠀⠉⢹⠀⠋⠉⠃⡊⠉⠀⠘⠉⡀⠐⠂⣂⢰⠀⠒⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠁⠀⠃⠈⠀⠈⠁⠉⠀⠁⠈⠉⠀⠁⠚⠀⠉⠉⠁⠈⠉⠀⠈⠉⠁⠈⠁⠀⠈⠁⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢤⠄⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠀⣀⠀⠐⠒⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣠⣤⣤⣤⡤⠤⠤⠤⠤⠤⠄⠀⠀⠀⠀⠤⠤⠴⠴⠦⠤⠭⣭⡭⠭⠥⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡾⠿⠟⠛⠛⠛⠻⠿⠿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⣸⡽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠀⢃⡤⣄⢀⣣⣃⠀⠠⠟⠁⠀⠀⠀⠀⠀⠀⠻⡆⠀⣿⡇⠘⠃⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡀⠀⣸⠶⠛⠉⠉⠭⣭⡳⣄⠀⢠⡖⠲⡦⣄⣠⠴⠒⠒⠛⢿⣴⣛⣷⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⢀⣠⠤⢶⡾⠃⠀⠉⣻⡷⣾⡁⠀⠀⠀⠀⠀⠈⠛⠧⡀⢠⠶⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠀⣾⠀⢀⣠⠖⢻⣽⡶⠞⠓⠒⠛⠛⠻⠿⢯⣉⠉⠳⣄⠀⠀⠀⠀⠀⢢⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠯⣶⠞⠋⢀⣀⣤⡤⠤⠤⢤⣤⣀⠀⠈⠙⠦⣿⣿⣦⣄⡀⠀⢸⠂⡇⠐⠶⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠂⣠⠞⠁⣠⡾⠟⠉⠀⠀⠀⠀⠚⠀⠈⠙⠲⣄⠀⠈⠳⡄⠀⠙⠳⣦⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡴⠃⣠⢾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠙⣆⠀⢆⠘⣆⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣰⢣⠏⠀⠀⠿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠘⣆⠘⡆⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡇⡾⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢹⡀⢻⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⡇⠶⠀⠀⠀⠀⠀⠈⡶⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠘⡇⢸⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢸⡀⣇⠀⠀⠀⠀⠀⠀⠀⣱⠶⠃⠀⠀⠀⠀⠀⠀⠀⠒⣿⠀⠀⡇⢸⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠈⣇⢸⡀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢸⠃⡟⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠘⣆⢳⡀⠀⠀⠀⠿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⡞⡼⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢦⡙⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⣼⡽⣡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠉⠢⣍⡒⠤⣀⡀⠶⠀⠀⠀⢀⣠⠖⠋⠀⢀⣾⡿⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⣤⣀⠉⠛⠒⠲⠶⠒⠒⠛⠉⠀⣀⣤⡖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠁⠈⠙⠓⠶⠶⠶⠶⠶⠖⠚⠋⠙⣦⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀⣠⠶⠞⠛⣛⣛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢉⣤⡴⠶⠶⠶⢮⣍⠛⠂⢀⣤⠶⠛⢛⣉⣉⣙⠛⠶⣄⠠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠛⠛⠳⢶⣤⡀⠀⢈⣁⡴⠚⠋⠉⠉⠙⠛⠷⣦⡀⠈⢷⡀
⠀⠀⠀⠀⢠⡾⣰⠏⠀⠊⠀⠀⠀⠀⠀⠀⠀⠈⢻⣖⠋⠙⠻⣆⣀⣀⣀⠀⠀⠀⠈⢻⡄⠈⣷
⠀⠀⠀⠀⣾⢡⡟⢠⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⢿⣷⣶⠞⣩⣭⣭⣉⡛⢶⣄⠀⠀⢻⠀⣿
⠀⠀⠀⠀⣿⢸⡃⣿⠀⣤⡤⠴⠶⢚⣩⣤⡀⠀⠀⠴⠾⠿⠶⠿⠿⠿⢿⣿⣦⡙⣧⠀⠘⠀⣿
⠀⠀⠀⠀⠻⠸⣇⠸⣦⠀⠀⠀⠘⣆⢻⣿⠷⡠⢂⣀⣤⣤⣶⣶⣶⣶⣶⣶⣤⣭⣙⡻⢶⣤⡉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠙⢃⡸⠿⣦⣄⣀⣈⡤⢖⣫⠞⢋⡭⠶⠛⠉⢉⡉⣉⣉⠉⠛⠻⢿⣿⣷⣮⣙⢷⣄⠙⣦⡀
⠀⠀⠀⣀⠀⠀⠀⠉⠀⠀⠀⠈⠉⢁⠴⠯⢍⠝⠉⠠⡆⢠⡆⢸⡇⣿⣽⠀⠀⢠⡄⠈⠻⣿⣿⣷⡝⢷⡈⢷⡀
⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡇⠀⡇⠸⢿⠸⣷⠖⠀⢸⡇⠀⢀⡈⠻⣮⢻⣆⢻⡄
⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⣰⠶⡆⠃⠀⠁⠀⣾⠀⣷⠀⠀⠘⠁⠀⠏⣿⠀⠙⣧⢻⣆⢻⡄
⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡐⠀⠀⡇⢿⣀⡟⠀⠀⠀⠀⣿⢻⣿⠀⠀⠀⠀⢀⣴⣧⡆⠀⠸⣇⢻⡄⣧
⣿⠀⡄⠀⠀⠀⠀⠀⠀⠀⡎⠀⢠⡇⠀⠈⠁⠈⠛⠁⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⢶⡄⢿⠘⣧⢸
⠛⢠⠁⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⠀⢰⠛⣆⠀⠀⠀⠀⠀⠀⠀⣿⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⢸⡆⣿⠘
⠀⢸⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠸⠶⡟⠀⠀⠀⠀⠀⠀⠀⢻⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢸⠇⣿
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠈⣧⠀⢸⡆⢀⡼⠁⠀⡀⠀⠀⠀⠀⠀⠘⣧⠅⠀⠀⠀⠀⠀⠀⣀⡌⠁⠀⢸⠀⡿
⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⢻⡄⠈⣷⠀⠀⠀⡟⣻⠂⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢰⠏⡇⠀⠀⠆⢀⠇⠘
⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠈⢿⡄⠸⣧⠀⢠⡟⢳⠀⠀⣤⣤⣤⠀⠀⣠⠀⠀⢀⠶⠆⠘⠛⡟⠀⠀⠀⠘
⠀⠀⢀⠘⣯⣉⣉⣉⣉⣉⣉⡈⢿⣆⠹⣧⡈⠳⠞⠀⠀⠀⢰⠇⠀⣰⠃⠀⠀⠸⣆⠀⠀⠀⠃⡀
⠀⠀⠘⣧⠈⢻⣭⣭⣭⣭⣭⣥⣤⠻⣷⣌⢿⣦⡀⠀⠀⢀⡏⠀⠀⡏⢻⠀⢠⣤⠟⠀⢀⡠⠮⡭⠋
⠀⠀⠀⠈⠳⣄⠙⢷⣶⠶⠶⠚⠛⠶⣾⠿⢷⣯⣻⣶⣤⣈⠀⠀⠀⠻⠞⠀⠀⣀⣠⠖⣋⡴⢊⡕
⠀⠀⠀⠀⠀⠉⠳⠦⠉⠻⠃⠀⠀⠀⠘⣧⣀⣉⠛⠿⢿⣿⣿⣿⣷⣶⣾⣿⣿⠿⠞⣋⣥⠚⠉⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣀⠤⠾⣷⣶⣒⣒⡂⠀⠀⠀⠉⣉⣉⣀⣤⠴⠖⠋⠉⠙⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠖⠒⠒⠂⠉⠉⠉⠉⠉⠑⠒⠢⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⣀⣠⣤⡶⣲⠶⠻⢟⠟⠛⠒⢲⠶⠦⣤⣀⡀⠀⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠉⣀⡤⡾⢛⢹⣴⣿⠶⠿⠻⠿⠛⠿⠻⠳⠻⡿⢾⣵⣬⣝⡛⠲⢤⣀⡀⠀⠑⠢⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⢀⣴⡞⡙⣤⣷⠿⠋⠁⡀⣤⢰⡆⢰⣶⡄⢠⣦⠀⣠⡄⢀⡀⠊⠙⠫⢖⣬⣋⡳⢦⣤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠠⠊⢀⡴⡏⣨⣶⡟⢉⣥⡄⢶⡌⠟⠉⠀⠀⢸⣿⡇⠀⠁⠀⠉⠁⠘⠏⢴⡗⢀⣤⡈⠙⠿⣯⣭⢛⣶⣦⣄⣑⣤⡀
⠀⠀⠀⠀⠀⢀⠔⠁⣴⠻⣠⣾⠏⠁⣴⡹⣧⢻⣆⠀⠀⠀⠀⠀⠸⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⣤⡄⠀⠽⢗⣴⣚⡻⣮⡝⡻⣦⡠
⠀⠀⠀⠀⡠⠃⢀⡞⢀⣾⠟⢁⣀⠻⠎⠀⠸⣧⠹⡧⠀⠀⠀⠀⠀⡤⡄⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠉⠰⠿⢀⡀⠉⠿⢶⣛⡻⣦⡟⢷⣆⠒
⠀⠀⠀⡐⠁⢠⠏⢤⡿⠃⠀⣮⡙⠀⠀⠀⠀⠀⠡⠘⣆⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠙⢡⡶⠀⠙⢿⣯⣃⠙⣎⣿⣆⣐⡀
⠀⠀⡐⠀⣰⠏⣄⡞⠁⣰⣶⣄⡀⠀⠀⠀⠀⠀⠀⠁⡘⣆⠀⠀⠀⠇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣆⠀⠘⢿⠁⠈⢏⠹⣷⡤
⠀⠠⠀⢰⠿⠐⣾⠅⣀⠉⠻⠿⣿⠆⠀⠀⠀⠀⠀⠀⠀⡘⣧⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠟⠋⢡⣤⠀⠀⢂⠀⠘⣧⢸⣧⠄
⠀⡄⠀⡞⢂⣻⠇⢀⣙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠜⣆⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠦⠀⠈⠄⠀⠸⣧⢻⡇⠀⠅
⢰⠀⢸⣏⣾⡛⢀⣌⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⠐⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠀⠸⠀⠀⢳⢿⡿⠂
⡈⠀⣬⢋⣿⠃⣬⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⡆⠀⢸⣱⣿⣭⡄
⡇⠀⣯⢪⣿⠂⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣶⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣿⣿⣿⣷⠀⠁⠀⢸⢕⣿⡂
⡇⢰⡇⠐⣿⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⡛⠋⠀⠀⠀⢸⢺⣿⣒⡀
⠀⠐⢧⠪⣷⠀⢛⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠛⠁⠀⠆⠀⣞⢽⣿⠒
⢰⠀⠷⢌⣻⣆⠈⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡛⠃⠀⡸⠀⢀⠗⣹⡿
⠈⡆⠘⣗⠨⢧⡀⠀⡶⠆⢀⣠⣴⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣈⠋⠀⢠⠁⠀⣘⡩⣾⣯⡭⠩
⠀⢡⠀⢹⡬⢹⣧⡀⠰⣾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣶⣄⡈⠛⠀⠀⡌⠀⢠⠇⣺⡟⠉⠁⠄
⠀⠘⡄⠀⢻⣆⠻⣧⡀⠈⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⠏⠀⠀⡜⠀⢠⢛⣭⣟⣃⠈
⠀⠀⠈⢆⠸⢳⡌⠹⣯⡄⠀⠺⠓⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡦⠀⠀⢀⠈⠀⡰⢋⣿⡟⠓⠒
⠀⠀⠀⠈⠣⣞⡻⣶⡞⢷⡢⡀⠈⠋⣴⡆⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⣀⠘⠷⠀⠀⠀⡠⠋⢀⡜⢭⣵⡟⠒⠒⠂
⠀⠀⠀⠀⠀⠙⣧⡟⢷⣔⣻⣷⡀⠀⠀⠰⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⡀⢴⣆⠛⠃⠀⠀⣠⠞⠀⣰⢓⣰⡾⠯⠄⠄
⠀⠀⠀⠀⠀⠀⠀⠻⣎⣹⢶⡀⠛⠦⣀⠀⠈⠁⠾⢂⣶⠀⣀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢀⠀⣤⡙⠿⠟⠈⠁⠀⣀⠔⠊⠀⣠⠾⢉⣰⣟⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣅⠋⢦⣄⠈⠑⠢⢤⣀⠀⠁⠘⠛⠸⡷⠘⣿⣷⢰⣆⢸⡆⠙⠃⠉⠁⠀⣀⡤⠒⠊⠁⢀⣠⠾⢍⣽⠟⠓⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣉⢻⣧⡨⢱⠢⣄⡀⠀⠉⠐⠢⠤⠄⢀⣀⣈⣁⣀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠀⣀⣤⡶⣯⣥⡶⠿⠡⠬⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠤⠽⠷⢧⣡⢉⢉⠱⡰⡤⣄⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⢤⡔⠶⢙⣩⣤⡜⠛⠋⠑⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠆⢀⣙⣛⡳⣶⡤⣬⣌⣏⣫⡹⡙⢝⠏⡫⡙⠬⢑⣬⣦⠶⠽⠛⠛⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⣠⠖⠚⠛⠛⠛⣛⡛⠛⢛⣛⣛⣛⣛⡡⠬⠌⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢸⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣾⠀⠀⣷⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⠀⢹⡆
⠀⠀⠀⠀⠀⢰⠶⠶⠶⠶⢾⡇⠀⢸⡷⠶⠶⠶⣿⠀⢸⡷⠶⠶⠶⣿⠀⠀⣿⠶⠶⠶⢾⡅⠀⣿⠶⠶⠶⢾⡇⠀⢸⡷⠶⠶⠶⠶⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣸⡇⠀⢸⣇⠀⠀⢀⣿⠀⠸⣧⠀⠀⢀⣿⠀⠀⣿⡀⠀⠀⣸⡃⠀⣿⡀⠀⠀⣸⡇⠀⢸⣇⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡿⣇⠀⣸⢿⡆⠀⣿⣿⡀⣸⣿⡇⠀⣾⢿⡀⢀⡿⣷⠀⢸⣿⣇⢀⣿⣿⠀⢰⡿⣇⢀⣸⢿⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠻⢮⣭⣥⠟⠀⠀⠙⢮⣭⣭⠞⠁⠀⠘⠷⣭⣭⠾⠃⠀⠈⠳⣭⣭⡵⠋⠀⠀⠻⣬⣭⡵⠟⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠉⠉⠀⢹⠀⠀⣿⠉⠉⠉⠁⣿⠀⠀⣿⠉⠉⠉⠉⣿⠀⠀⣿⠀⠁⠀⠉⣿⠀⠀⡏⠀⠉⠉⠈⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠓⠒⠒⠒⠚⠀⠀⠙⠒⠒⠒⠒⠛⠀⠀⠛⠒⠒⠒⠒⠛⠀⠀⠛⠒⠒⠒⠒⠋⠀⠀⠛⠒⠒⠒⠚⠃⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢰⠛⠛⠛⠛⢻⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⣾⠛⠛⠛⠛⣷⠀⠀⡟⠛⠛⠛⠛⡆⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⣤⣤⣤⣤⣼⠀⠀⢿⣤⣤⣤⣤⣿⠀⠀⣿⣤⣤⣤⣤⣿⠀⠀⣿⣤⣤⣤⣤⡿⠀⠀⣧⣤⣤⣤⣤⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢠⣤⣤⣤⣤⣤⠀⠀⣠⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣤⠀⠀⣤⣤⣤⣤⣤⣄⠀⠀⣤⣤⣤⣤⣤⡄⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠛⠋⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢁⡴⠃⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠉⣴⣋⡥⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⡇⠀⢀⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠻⠾⠏⠀⠀⠀⠀⠈⠙⠛⠲⠤⣤⡴⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⢉⣠⠾⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡞⠁⠀⠈⠓⢦⣜⣧⣀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣸⠇⡄⠀⠀⠀⠆⠀⣷⠉⠛⢻⡆⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⣐⣀⣠⣤⡄⠉⣰⢦⡼⠷⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣬⡟⠉⠙⢦⣴⣾⣿⣯⣭⣷⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣶⢶⡄⠀⠀⠀⠀⠀⠀⢀⣤⡎⠀⠀⠀⣠⣿⢿⣻⡇⠀⣀⣀⣀⣀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⢈⡿⠻⣷⠶⣄⡀⠀⢀⡶⠋⠘⣿⡶⢤⣴⡿⠋⠘⣧⣿⠉⠉⠉⣯⢹⡆⠈⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣃⠀⠀⠉⠓⠉⠀⢀⡴⢯⣿⠙⠋⠀⠀⠀⠸⣜⣧⠀⠀⣿⢸⡧⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣤⠶⠋⠀⠘⢿⡀⠀⠀⠀⠀⠀⠈⢻⣀⡴⠿⣯⡀⠀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⢻⡀⠀⠈⠙⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⣀⣤⠞⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠿⠓⠶⠖⠛⠋⠉⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠋⠀⢀⣠⡴⠶⠒⠚⠛⠛⣷⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠘⣇⢀⣠⠴⢲⣶⡟⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⣠⡏⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠙⠋⠁⠀⣀⡿⢷⡀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣭⣀⠀⠈⠉⠙⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡤⠶⠛⠉⠀⠈⢷⣀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠦⣤⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡿⠟⠛⠉⠉⠀⣀⢀⡀⣀⠀⡉⠉⠛⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣤⣾⡿⠛⠁⢠⡀⠆⠘⠀⠁⠈⡘⣃⠁⠈⠀⠃⠰⢀⡄⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⢄⠙⠈⠓⡄⡄⠀⠀⠀⣇⣜⠁⠀⠀⠀⢠⠚⠀⠋⡠⠀⠙⢿⣷⣄
⠀⠀⠀⠀⣴⣿⠟⠀⣄⠑⠀⠀⠀⠀⠇⠇⠀⠀⣤⡀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠊⣠⠀⠻⣿⣦
⠀⠀⠀⣸⣿⠏⠀⠲⢦⢀⢀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⡤⠖⠀⠹⣿⣇
⠀⠀⢠⣿⡟⠀⣈⠁⠀⢸⠻⡸⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠁⠀⠈⣁⠀⢻⣿⡄
⠀⠀⣾⣿⠃⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠘⣿⣧
⠀⠀⣿⣿⠀⢘⣂⢠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣐⡃⠀⣿⣿
⠀⠀⣿⣿⠀⢨⠍⠀⠛⢀⣀⣠⣤⣤⡶⠶⠟⠛⠛⠻⠟⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠃⠩⡅⠀⣿⣿
⠀⠀⢿⣿⡄⠀⠂⠚⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⢠⣿⡿
⠀⠀⠸⣿⣧⠀⢉⡀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⣰⡆⠀⢀⡉⠀⣼⣿⠇
⠀⠀⠀⢹⣿⣆⠀⠴⠚⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠃⠓⠦⠀⣰⣿⡏
⠀⠀⠀⠀⠻⣿⣦⠀⠋⡠⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡀⠀⠀⠀⢄⠙⠀⣴⣿⠟
⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠊⣠⠀⡤⠁⠀⠀⠀⠀⣼⣅⠀⠀⠀⠀⠐⢥⠀⣄⠑⠀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠘⠁⠆⢠⠀⡀⢀⢨⡅⡀⢀⠀⡄⠰⠈⠃⢀⣤⣾⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⣤⣀⣀⠀⠉⠈⠁⠉⠀⣀⣀⣤⣴⣾⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⢱⣀⣤⠮⠀⠀⠤⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠤⠀⠀⠴⣤⣀⡎⠑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡾⠟⠉⠀⠀⠀⠀⠀⠈⠳⡀⠀⠨⢥⡬⠇⠀⢠⠞⠁⠀⠀⠀⠀⠀⠉⠛⢷⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⣸⡇⠀⠀⠈⠑⠢⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⢀⠠⣔⠩⠔⠐⠂⠉⠉⠁⠀⠀⠈⠉⠉⠐⠒⠤⠍⣲⠄⡀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⠄⢊⠗⠊⠁⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠈⠑⠺⣑⠠⣀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠔⠊⡠⠞⠁⠀⠀⢀⡠⠔⠂⠉⠀⠀⢀⠈⢁⡀⠀⠀⠉⠐⠢⢄⡀⠀⠀⠈⠳⢄⠁⠢⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⣠⠖⠁⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠈⠢⣄⠀⠀⠈⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠳⡀⠀⠀⠘⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠠⣧⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣼⠄⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢠⠃⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠀⡀⠁⠀⠘⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⡆⠀⠀⠘⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⠶⣤⣀⠀⠀⢀⣠⠶⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠒⠘⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠒⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡘⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠈⠄⣀⠐⢮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡀⠰⠁⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⢏⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡹⠃⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠲⠄⠀⠀⣠⠞⠁⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠈⠓⢄⡠⠀⠀⠀⠀⠀⠀⠏⠰⠀⠀⠀⠀⠀⠀⢄⡠⠚⠁⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠈⠑⠢⠄⣀⡀⠀⠠⠄⠀⣀⣀⠠⠄⠊⠁⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠗⠢⣀⡀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⣠⠔⠺⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡔⠁⠈⠁⠒⠠⠤⠄⣀⣀⣀⡀⠠⠤⠄⠒⠈⠁⠈⢢⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉


⣶⣶⣶⣶⣶⡾⣶⣶⣶⡶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣾⢷⠾⠷⠾⠷⣶⡶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣶⣾⣿⣷⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⣵⣿⣿⣿⠿⠿⠿⠿⢿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⣻⣿⣿⣿⠀⠀⠀⠀⠈⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠂⠀⠀⠀⠀⠀⢋⣿⣿⣧⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠠⠀⠈⣿⡏⠀⠀⠠⠅⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⠑⠈⠀⠀⠀⠄⠀⢠⢤⡀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠇⠀⠀⠀⠀⠉⠀⠀⠈⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢋⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠘⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠁⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⢀⠆⠀⠈⠒⠂⠀⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⡐⠁⢀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⠎⠑⠀⣀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⢀⠃⠀⠀⠘⠀⢀⡠⠔⢺⠛⠇⠆⣠⠄⠀⠀⠿⣀⢈⡦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠠⠀⢀⡞⠀⠀⠀⠀⠃⠀⢃⠀⣸⡏⡈⣰⠁⠀⠀⠀⠀⢈⣯⡇⠀⢋⠉⠙⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢀⡠⠁⠀⠀⠁⠀⠀⠀⠀⠀⢇⡈⢴⠃⡿⡨⢨⠁⠀⢀⡄⡄⠀⠀⠿⢒⡺⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣌⡑⠢⠄⢤⡀⠀⠀⠀⠀⠀⠀⠑⢌⠞⠐⠁⠀⠘⠦⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⣉⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠁⠐⠧⡀⠀⠀⠀⠀⠀⠀⠈⡛⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠒⠒⠴⠃⢀⡀⠀⠀⢠⠤⠉⠀⠂⠄⠀⠀⢀⡴⡇⠀⠀⠀⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠈⠛⠲⠭⣑⠲⠠⢾⣠⣇⠀⠀⠀⡼⠉⠁⠀⠉⠒⢶⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠉⠀⠂⠠⢈⡁⠂⠌⣱⢤⠀⠐⠈⠀⢹⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠉⠁⡼⢳⡆⡰⠚⠉⠀⠀⠀⠀⠀⢇⠀⠀⢀⠄⠀⢐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⡄⣰⠇⡞⠈⠲⡤⠄⠤⠀⠀⠀⠀⠈⠆⣴⠃⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠰⠐⡟⠸⠀⠀⠀⠰⠀⠀⠀⠀⠀⡀⢂⠊⠼⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠄⢀⠀⠀⠀⠀⠈⠀⠀⠀⡆⠘⠀⠀⠀⠀⠀⠀⠀⢐⣨⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠆⠀⠀⠀⠁⠀⠀⠀⠀⠘⠒⠂⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢒⣊⣉⣍⣍⣭⣭⣙⣒⡰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡛⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⢨⣀⣸⣇⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠽⠋⠋⠀⠀⠀⠀⠀⠀⠀⢀⡐⠈⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⡈⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⠒⠐⠀⡀⢀⠀⠀⠂⢀⠔⠂⠀⠀⠀⠀⣀⠔⠉⠀⠀⠀⠀⠈⠀⠀⠠⠁⠀⢹⣿⣟⣽⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⠰⠁⠀⠀⠔⠁⠄⠠⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⢼⢃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠐⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠀⠀⠂⠀⠘⡀⠀⢰⣽⠘⡷⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⢐⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠄⠀⠊⠀⠀⠀⠀⠀⠀⠠⠀⠸⠀⠀⢸⣿⢈⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡷⠊
⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⡇⢪⡀
⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠋⠉⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⣟⠃⡏⢩
⠀⠀⠀⠀⠀⠀⠀⠀⢒⠀⠀⡶⠶⠲⠲⠶⠖⠖⠖⢿⠄⠀⢿⡶⡲⠆⠆⠠⠄⠦⠄⠄⠦⠦⠤⠤⢤⡴⢿⠭⠀⡇⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠨⢸⠈⠀⠀⠀⠀⠀⠀⠀⡀⠘⠀⡀⣸⠁⠀⠀⠀⠁⠀⠀⠀⠀⠂⠀⠀⠀⠀⢁⢸⡋⠀⣗⠗⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠑⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⣆⠀⡇⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⣀⠄⠀⠠⢸⡵⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⢸⠖⠀⡇⠊⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠀⠨⠈⡧⠀⡇⡔⠋⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠟⠙⠛⠛⠃⠀⠀⠸⡧⠀⣿⢸⣿⡄⣿⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢍⡀⠀⠀⡁⠠⣿⢸⣿⠅⣿⣿⣿⠆⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡯⠀⣿⣃⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠈⠀⠈⠉⠁⠀⠒⠂⠠⠤⠤⠄⣀⡀⣀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⣻⡦⢯⠿⠙⣀
⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡁⠐⠒⠠⠆⠀⠤⠀⡤⡤⠄⠀⠄⠈⠅⠈⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⣀⣤⣤⣄⣀⣤⣤⣤⣀⣠⣖⣦⣄⣠⣤⣤⣀⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠚⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⣯⢽⣦
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇⢱⢻⡇
⠀⠀⠀⠀⠀⠀⠙⢦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣿⣫⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣀⠀⣉⡀⢀⣀⡀⣀⡀⢀⠀⡀⢀⣠⡀⢀⣀⢀⣉⡉⠉⠉⠈⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⢿⡼⠀⣷⡀⠸⣤⠀⢿⡄⢸⠟⢿⢸⣗⡇⢸⣂⢸⠷⡏⠀⠀⠀⡏⣿
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⢰⢒⣶⢲⠤⣤⠤⡤⠤⠤⣤⠤⠤⠤⣤⣄⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⣷⡿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠁⠒⢺⠒⠒⢲⠗⠛⠏⡦⠬⠭⠤⡇⢐⣒⣸⣀⣒⣈⡋⠯⢹⠇⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣸⢀⣀⣜⣀⡀⢸⠀⠀⢀⠇⠀⠀⢸⠀⠀⠀⡆⠀⠀⠸⠀⠈⣹⠀⢸⢿⠇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⡇⠀⠀⡇⠀⠀⡏⠉⠉⢸⠓⠒⠂⡞⠒⠒⢲⠇⠤⠤⡧⠤⢤⡇⠀⢸⣾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡧⢄⣰⣁⣀⣀⠃⠀⠀⡇⠀⠀⢠⠃⠀⠀⢸⠀⠀⢀⠃⠀⠀⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⢰⠃⠀⣸⠀⠀⡎⠀⠀⡜⠉⠉⢉⠋⠉⠉⡻⠀⠒⠒⠗⠤⠤⢼⠤⠤⢤⠇⢰⣿⠃
⠀⠀⠀⠀⠀⠀⡞⠀⠀⡗⠂⢰⠧⠤⠤⡧⠤⠤⣼⣀⣀⣀⡇⠀⠀⢸⠀⠀⠀⡇⠀⠀⣸⠀⢸⣿
⠀⠀⠀⠀⠀⣸⠁⠀⢸⠀⠀⠸⠀⠀⢰⠀⠀⠀⡇⠀⠀⢸⠉⠉⠉⡝⠉⠁⢸⠓⠒⠲⡇⠀⣿⡏
⠀⠀⠀⠀⢀⡇⠀⢀⡏⠉⠉⡏⠀⠀⡞⠒⠒⢲⠧⠤⠤⡦⠄⣀⣠⣇⣀⠀⡘⠀⠀⢸⠁⢠⣿⡇
⠀⠀⠀⠀⢸⠀⠀⣸⠀⠀⢸⠀⠀⢰⠁⠀⠀⡜⠀⠀⢠⠃⠀⠀⢸⠀⠀⠈⡏⠉⠉⡟⠀⣸⣿
⠀⠀⠀⠀⢿⣄⣀⡈⠉⠉⠛⠒⠒⠚⠒⠒⠒⠧⠤⠤⠼⠤⣄⣀⣌⣀⣀⣸⣀⣀⡰⢃⡤⣿⡏
⠀⠀⠀⠀⠀⠳⢶⣿⣭⣭⣽⣶⣒⣒⣒⣒⣒⣒⣲⣦⣤⣤⣄⣀⣀⣀⣀⣠⣤⡤⠔⠋⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠭⠭⠿⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣀⠤⠄⠒⣂⣀⣀⣐⡒⠠⠤⣀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⢀⠔⡫⠔⠂⠉⠀⠀⠀⠀⠀⠀⠉⠐⠬⣑⢦⡀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⢀⠔⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⢦⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⢠⢋⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢆⢳⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢀⠇⡎⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠈⡆⢇⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠙⠲⣤⡀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⢹⢸⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠂⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⢸⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⢸⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⢧⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢃⡇⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⢦⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢃⠞⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠳⣜⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠜⡡⠋⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠈⠓⢬⣓⠦⢤⣀⣀⣀⣀⣀⣀⡤⠔⣊⡵⠊⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⡀⠀⠈⠉⠒⠲⠮⠭⠭⠴⠖⠚⠉⠁⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⠦⣤⣀⣀⠀⠀⠀⠀⣀⣀⣠⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⡻⢿⢛⠿⢿⣿⣻⡿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠋⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⢣⣰⠾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠦⠞⣱⠃⠀⠀⣠⢤
⠀⠀⠀⠀⠀⠈⢳⠦⡀⠀⠀⠻⠀⣀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡖⡆⣀⠀⠠⡧⠴⢤⠇⣠⣒⠛⢁⠎
⠀⠀⠀⠀⠀⠈⠫⣀⣨⠆⠀⠀⠀⠙⠃⠹⣸⠀⠀⠀⣠⣠⢄⠀⠀⠳⠛⠛⠀⠀⠀⠀⠀⠚⠉⠁⠑⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⢠⠔⠊⠙⠲⢍⠀⣴⣒⠃⠁⢈⣍⡷⠀⡠⠚⠉⠉⠉⢢⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⡖⣄⠀⠀⡇⠀⠀⠀⠀⣨⠇⠈⠈⡇⠀⢸⠀⠀⠀⠧⣀⠀⠀⠀⠀⡇⠀⠘⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠧⠤⠴⡂⢫⡤⠴⠒⢲⡃⣠⣼⠋⣱⠖⣲⠏⢨⡓⠒⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠳⢀⠠⢀⣀⢀⠀⠀⠀⢀⠀⠀⣘⡆⠈⠉⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⢀⠄⠊⠁⠀⠀⠀⠈⡍⠁⠀⠀⠀⠀⠁⠀⢀⠀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⡔⠁⠀⠀⠓⠀⠀⠀⠀⣄⠀⠀⠀⠀⠒⠀⠀⠀⢢⠀⠀⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠈⠀⢠⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⡀⠀⢡⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⠓⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡇⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠏⡢⡀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⢣⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡐⠄⠀⢠⠀⢀⠂⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠑⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠁⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠉⠂⢄⣀⠀⠀⠀⠆⠀⠀⠀⢘⠁⠐⠀⢀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⡴⠤⠶⠤⣄⣉⣉⣉⣛⡤⠬⠥⠦⣄⡄⢖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠬⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⠀⢀⣀⡀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⣒⣦⣰⣋⣳⣤⡮⠭⡦⠾⠭⡷⠿⢿⡷⠿⢿⡿⠚⠓⣺⠗⠓⢲⠞⠛⢒⡾⡆
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣏⠩⠞⠀⠀⠚⠀⠀⠚⠁⠀⠉⣀⡀⢉⠀⡀⣨⣄⠀⣤⡀⡤⣄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⣸⠀⣿⣿⠀⠀⠀⠀⢰⠋⢠⡿⡄⣇⡀⢿⡄⢹⠓⠇⠳⠜⠐⠛⠃⠛⠙⠀⠀⠀⣀⣀⠈⡇
⠀⠀⠀⠀⠀⠀⡏⠀⢻⣿⡇⠀⢀⣀⣀⣩⢤⡤⠤⠤⠤⢤⠒⠒⠒⢲⡒⢒⣚⣻⣉⠉⢽⡭⠭⢽⠭⠭⡇⢹
⠀⠀⠀⠀⠀⣸⠁⠀⠸⣿⣿⠀⠸⡖⠒⠒⠒⠗⠒⠈⠉⢹⣉⣉⣉⣉⣇⣀⡠⢸⡤⠤⠤⡦⠤⠤⡦⠤⢿⠈⡇
⠀⠀⠀⠀⠀⡏⠀⠀⠀⣿⣿⡇⠀⢷⠒⠊⠉⢹⠉⠉⠉⢀⣇⣀⣀⡀⢥⠤⠤⠤⡧⠤⠐⢷⠒⠒⢻⠊⠙⡆⢹
⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⣿⣿⠀⠸⡖⠒⠈⠉⡏⠉⢀⣀⣹⣀⠠⠤⠼⡤⠤⠄⢲⠒⠒⢺⡂⣈⣙⣏⣉⣱⠈⡇
⠀⠀⠀⠀⡞⠀⠀⠀⠀⠘⣿⣿⡆⠀⢷⠒⠒⠈⢹⠉⣁⣀⡬⡦⠤⠔⠒⢗⠒⠒⠚⡏⠉⠉⣏⢉⣀⣣⣀⠬⡇⢸⡀
⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⢻⣿⣷⠀⢸⡊⠉⠉⢉⣅⣀⣀⣀⢸⠤⠤⠤⢺⠒⠒⠒⢻⠉⣉⣹⣀⡠⠼⠤⠤⠞⠀⣇
⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠏⠉⠉⠁⣣⣀⣀⣠⠤⠦⠴⠖⠚⠓⠋⠉⠉⠉⢀⣀⣀⣠⡤⠤⠤⠴⢶⠛
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢻⣷⣇⠀⠘⠚⠉⠉⠁⠀⢀⣀⣀⣀⡤⠤⠴⠖⠒⠚⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⢘⡇
⠀⠀⢰⠃⣀⣀⠤⠤⠖⠒⠒⠙⣿⡼⠤⠶⠖⠒⠒⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠖⠒⠒⠋⠉⠉
⠀⠀⠈⠙⠓⠶⠤⣄⣀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⠒⠒⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠾⠦⠖⠒⠚⠉⠉⠁


⣶⣶⠶⣶⠶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠞⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣟⣯⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣟⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⣀⣀⣀⣀⣀⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠉⠉⣿⡏⠉⠁⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠁⣀⣠⣤⣤⣶⣿⣷⣤⣤⣤⣀⡀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⣿⣧⣴⣾⣿⡿⣿⡛⠿⠿⠿⠿⠿⠟⣻⠿⣿⣿⣶⣤⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠿⠋⢁⣴⣿⡿⡿⠟⠂⠉⠀⠐⠀⠁⠸⠀⠁⠐⠀⠈⠁⠋⠟⡿⣿⣷⣄⠉⠻⢿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⢀⣴⣿⣟⠟⡈⠀⠘⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠈⠙⢟⣿⣷⣄⠀⠉⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡫⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠫⡻⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡝⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⡽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⡾⠷⢟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠟⠁⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠆⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⢆⠤⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠐⠦⢄⢼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣯⣢⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⡀⠠⣪⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣗⢄⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⢀⢔⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣮⣦⢜⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⠀⠀⠀⢘⢤⣪⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣶⣦⣤⣂⣠⣡⣬⢤⣥⣀⣲⣬⣶⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⠂

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡟⠀⠀⠀⠀⡿⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠘⢀⢀⣤⣤⣤⣤⣤⣄⣀⡁⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⢻⡄⠀⠀⣪⠒⠉⠉⠀⣄⠀⠉⠉⠙⠺⢕⢦⠀⠋⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⣁⠀⣸⠇⠸⠀⠀⠀⠏⠀⠀⢰⠀⠀⠀⣧⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣯⣯⣅⠢⡀⠀⡶⢦⢄⣈⡤⣖⣒⣿⣦⣦⣶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣀⠀⠀⢠⢫⣾⣿⣿⣿⣿⣯⡵⠃⣀⣈⣷⣾⣇⠾⣿⣿⣿⣿⣿⣷⡀⠀⠀⣠⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⢆⠻⣛⣽⣿⣽⣿⠿⠿⣿⣿⣿⣯⣭⣙⣳⣾⢿⣿⣿⣿⣿⡽⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⣀⠀⠀⠀⠈⢛⣽⠟⣫⣵⠞⠛⠋⠉⢹⠛⠛⢿⣟⠿⣷⣽⣿⣯⣟⡻⢃⠃⠀⠀⣀⣠⣤⠶⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠂⢠⡞⣡⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⢻⣿⡝⣿⣿⡅⠀⠛⠉⠉
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢠⡟⣰⢳⠃⠀⠺⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠹⣷⡈⣿⣿⡀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⣼⢡⠇⡇⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⢻⡇⢸⡇⡇⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⣿⢸⠀⡇⠁⠀⠀⠀⠀⠸⡋⡀⠀⠀⠀⠀⠀⠀⡀⣿⢸⣿⢸⡇⡄⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠖⠛⠉⠉⠀⠀⢻⠸⡆⣇⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⡿⢸⡏⢸⡇⡇⠀⠀⠉⠙⠛⠲⠦⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠘⣇⢳⡸⡄⠀⠀⠀⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⣾⢃⣾⡟⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠘⣦⠳⣜⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣼⢃⣾⠟⠀⠀⠀⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⠀⠀⠀⠀⣠⠄⠈⠳⣿⣷⣭⣒⠤⠀⠰⠀⠀⢀⣠⡴⢟⣥⣾⣷⠟⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⣈⣿⢿⣯⣭⣽⣿⣿⣿⣿⣿⣿⢿⣟⠉⠀⡀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠃⠀⠀⡉⠉⠉⠉⠉⠉⠀⠀⠹⣻⡄⠀⠙⣆⠀⠀⠀⠀⠙⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⣰⠁⠀⠀⠀⣧⠀⠀⠀⠀⠈⠋⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠀⠀⢀⡟⠀⠀⠀⠀⠟⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠠⠤⠤⠒⠶⠀⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⣠⠤⠤⠤⠤⠤⠤⣄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢸⠁⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸
⠀⠀⠀⠀⠀⡠⠤⠤⠤⠤⠀⠀⠀⠀⠒⠒⠒⠛⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠓⠒⠚⠒⠒⠒⠀⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⡁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⡇
⠀⠀⠀⠀⠀⡇⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⡔⠐⠒⠀⠒⠒⠒⠒⡄⠀⠀⣀⡤⠚⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠲⡇⠀⠀⠀⠀⠀⠀⠀⡱⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠰⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⣛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠀⠉⠛⢦⡀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⣀⡀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣲⣶⠶⠂⠀⠀⠀⠀⠀⢹⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡃⠀⠀⣰⠋⠠⠙⠃⠀⠀⠀⠀⠀⠛⣅⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠛⠛⠛⢹⠐⠀⠀⠀⠀⠀⢠⡙⠲⢀⡼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢻⡁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠁⡴⠉
⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠒⠚⢻⡄⠀⠀⢳⡀⠀⠀⢠⠀⠀⠀⠀⢀⠀⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠘⠁⢠⣀⠀⠀⠀⣷⠀⠀⠀⢳⡀⠀⡈⢦⡀⠀⠀⠼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⢸⡀⠀⠀⠈⣧⠞⢃⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢠⡟⠙⢣⡈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣀⣾⣀⣠⠚⠇⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠉⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⣿⠂⠀⢤⡄⠀⢸⡆⠀⠀⠀⠀⢠⠀⠀⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⣿⠀⠀⡠⣣⠊⠁⠀⠀⢀⣀⣸⣇⠀⠀⣀⠤⠬⠷⢔⣄⡀⠀⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠐⠚⠒⠛⠛⠛⠛⠛⠚⠋⠉⠉⢉⠀⢻⠁⠀⠀⠀⠈⠉⢛⠛⢦⡀⠀⠈⠑⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠈⠁⢸⡇⠀⠀⠀⠀⠀⠤⣍⣲⢽⣆⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣯⣤⣤⠞⠓⠲⠶⠶⢤⡭⠶⠋⠁⠀⠨⠇⢁⣠⠔⠂
⠀⠀⠀⠀⠀⠀⠘⠶⠶⠶⠶⠶⠶⠶⠶⠒⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠐⠂⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠲⠄⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⡠⠋⣤⠒⠐⢏⢆⠀⢉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⠈⠉⠲⣴⢁⢀⣇⠀⠀⢸⠘⠤⠋⢢⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠘⢿⣫⣿⠟⢀⡜⠀⠀⠀⠳⡄⠀⢸⠁
⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢘⣄⣏⡀
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡖⠀⢀⡀⠤⠤⠤⠤⣀⡀⠀⠀⠀⠀⠀⠀⢳⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠶⢺⠁⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠀⠀⠉⠉⠒⠢⢄⡀⠀⠀⠀⠀⢈⡆⠀⢸⣠⠤⢤⣀⣀⢀⢀
⠀⠀⠀⠀⠀⢠⠦⢤⡠⢄⣱⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⠤⠜⠁⠀⣸⡥⠦⣄⡀⡘⢋⡋⡗
⠀⠀⠀⠀⠀⣸⠂⠀⠀⣼⣉⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠙⡇⠀⠀⢉⠻⣤⠮⠇
⠀⠀⠀⠀⠀⢇⣀⣀⣤⡄⠁⠙⠢⣄⣀⣀⣠⡤⠤⠴⠒⠋⠉⠀⠀⠀⠀⢳⠀⠀⡟⠳⣄
⣠⣄⣀⠀⠀⠀⠀⢀⣤⣙⠢⣄⠀⠈⠑⢦⡄⣀⡠⠔⢛⡩⠔⠂⠀⠠⣦⣨⣧⠀⣳⡀⠈⠷⣄
⠳⡀⠈⠉⠲⢄⡀⣼⣿⣿⢿⣿⡛⠲⢄⡀⠈⣁⡤⠚⢻⣆⣀⣤⣶⣾⣿⣿⣿⢾⣿⣿⣦⡀⠈⠛⠶⢤⣀
⠀⠙⠢⣀⠈⠀⠙⢿⣻⣿⠀⢸⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⠿⢿⣿⣿⣧⣼⣿⣿⡗⠙⠢⣄⣀⣀⣀
⠀⠀⠀⠈⠑⠦⣄⡀⠉⢦⠀⠀⠉⠉⠉⠛⠋⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠉⠛⠻⠿⠟⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠶⠶⠶⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢋⣠⠴⠶⠚⠛⠳⠶⠶⣦⣄⡉⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢁⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡿⠲⠶⣦⡄⠀⡞⠉⡇⠀⣠⡶⠞⠛⠛⠷⠾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠻⠴⠞⠁⠀⠀⠀⢹⡆⣇⠀⡇⣼⠧⣤⣤⣀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣠⡾⠓⠛⠒⠳⠿⣤⣀⠀⠉⠛⠶⣄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣀⣤⠾⣋⣤⠴⠶⠚⠛⠛⠛⠛⠻⢦⣄⠀⠈⢳⣄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⡗⠚⠛⠋⣩⡶⢛⣩⠴⠖⠛⠛⠛⠛⠛⠶⣦⣄⠙⢷⣦⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⠟⢁⡴⠋⠁⠀⠀⢰⣉⡷⠀⠀⠀⠀⠉⠻⣦⡙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣾⠋⣰⠏⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⠃⡼⠁⠀⢠⣞⠉⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⣸⠃⠀⠀⠀⠈⠓⠦⣄⣙⣄⣀⠀⠀⠀⠀⠀⣀⣀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿⢠⡴⡄⠀⠀⠀⠀⠀⠻⣄⡼⣄⠀⠀⠀⠀⠳⠞⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢿⠀⢿⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢳⡈⢷⡀⠀⠀⠀⠀⢰⡟⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠘⣇⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣹⡆⠀⠀⣠⠏⠀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠘⢆⠙⢦⣀⠀⠀⠀⠀⠀⠀⡤⣤⠀⠀⠁⠀⢀⡼⠋⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⣀⠀⠙⠳⠶⣤⣤⣀⣀⣉⣉⣠⣤⣴⠞⠋⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠉⢻⡄⠀⠀⠀⠀⠉⠉⠉⠁⣀⣠⣤⡶⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡴⠋⠉⠛⠛⠛⠛⠛⠛⠛⠉⠉⠉⠳⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣾⣆⡤⠖⠚⠛⢳⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠾⠛⠁⠀⠀⠀⠀⠀⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣴⠋⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠉⠀⢠⣤⣶⣶⣾⡋⠉⠉⢉⡽⣿⣶⣄⠀⠀⠙⢯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠟⠃⠀⢀⣼⡵⠋⠹⠿⢿⠀⢠⡏⠀⠛⢛⡟⣇⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⡾⢹⠁⠀⠀⣠⠞⠀⠸⠷⣤⣠⡞⠀⢹⡄⠀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣥⡶⠀⠀⢀⡞⠁⠘⠷⠶⠚⠁⠀⠀⠀⠀⠀⠈⢳⡄⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣤⣶⠟⠀⠀⣇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀⠀⣾⠀⠀⠀⣠⠄⠀⠀⠀⠈⠉⠉⠉⠁⣴⣤⡀⣿⠟⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⠛⢲⡄⠀⠈⢻⣄⣿⠀⠀⠚⠉⠳⢦⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⢡⠖⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⣭⣽⣇⠀⢀⡏⡙⠻⠀⠀⠀⠀⠀⠀⠉⠛⠒⢀⣀⠜⠁⣀⣤⣤⣩⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣠⣿⠀⠀⣧⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣋⣁⣀⡀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣭⡤⣶⣿⠀⠀⠘⢷⣄⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢩⠽⠛⠛⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⡇⣿⢸⡇⢀⣤⠤⠄⠀⠿⠿⠧⣄⡀⠀⠀⠀⠀⠀⠀⢀⣿⡶⠚⠓⠀⣿⠙⠛⠛⠛⠛⠒⠲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⢸⡇⣸⠀⠀⠀⠀⠀⠀⣀⠠⠿⣦⡄⠀⠀⢀⣶⡛⣯⣤⡶⠆⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡇⢻⢸⡇⣿⠀⠀⠀⠀⣠⠞⠁⠀⠸⣿⡁⠀⠀⢸⣿⠛⣾⠁⢀⡀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣸⡇⣿⣠⡴⠖⠒⡯⠀⠀⠀⠀⠈⠛⠒⠚⠛⠋⠀⠹⣶⠟⠁⣸⣧⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣇⢸⣿⡷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠶⠞⠋⠈⢷⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠙⢻⣿⢸⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠻⣾⡟⠈⣿⠀⢰⠃⡠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡟⢳⣄⠀⠀⣿⡇⢀⡿⠀⣿⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣴⡞⢹⡇⠀⠙⣶⣾⢿⣿⡏⠀⣸⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠀⠀⠀⠀⠀⢹⡆⠀⠀⢸⠁
⠐⠛⠛⠓⠲⠶⣾⡏⠸⡇⠀⢷⡀⠀⣸⣿⠷⣿⡿⠶⠿⢤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⣦⣤⣤⣶⡿⠃⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠙⠷⠞⠻⢦⣼⠿⣶⡿⠛⣿⣿⠛⠲⠶⠤⣤⣄⣀⣉⣉⣉⣉⡉⠉⠉⠛⠛⠒⠒⠶⠶⠶⠤⠴⠿⠿⠧⠶⠤⠤⠤⢤⣿⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣹⠏
⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠉⠛⠛⠛⠓⠲⠶⠦⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠖⠚⠉⠉⠉⠉⠀⠀⠉⠉⠉⠙⠒⠲⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⢈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣞⡷⠖⠚⠛⠂⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠘⠳⠶⢤⡞⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⢈⡦⠀⠀⡊⠉⡆⠀⠠⡊⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⣀⠔⠊⢁⣀⣠⠠⠬⠯⠀⣀⣀⡈⠑⠢⣀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠤⠒⢫⠷⠒⠉⠁⣀⠀⣤⣤⢄⣤⢀⣀⠈⠉⠒⠴⣹⠂⢄⡀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠁⡠⠚⠁⣀⣔⠶⠙⠋⠉⠉⠁⣄⠉⠉⠘⠻⢷⣠⣀⠈⠓⢄⠈⠓⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⡠⡪⠋⠁⢰⡆⠀⠀⠠⡆⠙⡦⠀⡄⠀⣤⠈⠙⢕⢄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⢮⠊⠀⠀⠀⢸⡇⢹⠀⠀⠃⠲⠆⡰⠀⢻⠀⠀⠀⠀⠑⣷⡀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢠⣯⠃⠰⢆⠀⡠⡈⠁⠈⠀⠀⠀⠀⢠⠁⠀⠉⠀⠀⠰⡄⣤⠘⣵⡄⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣾⠃⠀⠀⢸⠸⣇⠇⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠮⠄⠀⠀⠘⣷⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⡏⠀⡀⢀⢠⠀⠀⠀⠀⠀⠀⠀⣀⣜⣀⣀⣀⣀⣀⣀⡀⠠⡄⠀⠀⢿⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣇⠀⠋⠈⠞⠀⠀⠀⠀⣀⣤⣶⠿⠿⠛⠛⠛⠛⠛⠛⠋⠈⠇⠛⠀⣸⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⠀⠀⠀⢠⠠⠀⢰⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣿⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠱⣧⠀⣠⠸⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠿⢀⡀⠀⣼⠏⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠹⣧⡀⠀⠀⠀⠀⣩⠁⠀⠀⢀⠀⠀⠀⢰⡂⠀⠀⠀⠀⢁⣼⠏⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠘⢝⢄⠀⠀⣤⠁⠀⠀⠀⡟⡦⠀⠀⠀⢋⣄⠀⠀⡠⡮⠃⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⡀⠙⠯⣦⣄⡀⠀⠀⠀⠰⠆⠀⠀⠀⢀⣡⣴⠟⠊⢀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢢⢄⡀⠁⠚⠳⠶⠦⠶⠦⠴⠶⠚⠓⠉⠀⣠⡔⠫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢁⠔⠁⠀⠉⠑⠒⠤⠤⢀⣀⣀⡀⠤⠤⠒⠊⠉⠀⠑⢄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣠⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣽⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣟⣵⣿⡿⠯⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠒⣫⠟⣵⣌⣛⡟⠁⠀⠀⠉⠓⠲⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⢫⡀⠀⣠⠞⣡⣾⣿⣿⣿⢠⡞⣠⠆⠀⠀⠀⠀⠀⠉⠓⠦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣀⣧⡞⣡⣾⣿⣿⣿⡿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣀
⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢀⠞⢃⣴⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⡀
⠀⠀⠀⠀⠀⢠⠏⢠⣄⡴⢃⠴⠋⠀⠘⣜⣆⠀⠀⠀⠀⠀⢀⢄⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡹⠃⣾⣷⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⡞⠀⢀⡏⠐⠣⡀⠀⠀⠀⢈⣨⣤⣶⣶⣶⣿⣿⡼⢟⣴⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⡸⢁⡴⣯⣀⣠⣐⣁⣤⣶⣾⣿⣿⣿⡿⣿⡏⢹⣿⣿⣷⣄⡀⠙⠲⢦⣤⣀⠀⠀⠀⠀⣠⡆⠐⠁⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢱
⠀⠀⠀⠀⢣⣞⣵⠟⢹⣿⠛⠛⠻⣿⢿⣿⣫⣿⣾⣿⣷⢘⣿⣿⣿⣿⣿⣷⣶⣦⣤⣭⣶⣦⣴⡾⢻⣿⠇⠀⢸⣿⣿⣿⡁⠟⣿⢹⣿⣿⠸⡸
⠀⠀⠀⣰⣿⠟⠁⠀⠈⠻⢶⡀⠢⡘⣷⠾⢿⣿⡿⠣⠬⣉⠛⢦⣌⠁⡹⣿⣿⣿⡿⣿⡿⠿⣟⢹⣿⡟⠀⠀⣿⣿⣿⣿⣷⣤⡇⣾⣿⡏⠀⡃
⠀⠀⠠⠛⠁⠀⠀⠀⠀⠀⠀⠙⠦⠴⣯⠀⠀⠙⢯⣀⠀⢀⡵⠀⠉⠛⠶⠶⣭⣭⡭⠿⠟⠲⠶⠤⢽⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢄⡀⣸⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⣿⣿⡷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢡⣤⣴⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⠉⠀⠀⠈⠁⠀⠁⠀⠀⠐⡤⠀⢸⣿⣿⣫⡭⣿⣿⣷⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⣧⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⢸⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⠄⠹⣄⠀⢠⡄⠀⠀⠢⡖⠀⠀⢸⣿⣿⢸⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠒⠊⠉⠈⠁⠒⠒⠂⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡵⣀⡤⠴⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠱⣤⣀⣰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠀⠀⠀⠀⢈⠆⠀⢄⢀⠀⠠⡊⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⡄⢊⠡⠀⠀⠀⠃⠀⠀⠄⠑⠠⣀⠀⠀⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠒⡁⠔⢉⣠⡤⠖⠒⠒⡒⠒⠲⢤⣄⠈⠂⠈⠂⠄⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡴⠋⠁⡆⠀⠀⡄⠱⠄⠀⠀⠌⠙⢦⡀⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⣠⠏⡀⠀⠀⠇⠸⠀⠁⠈⠀⠀⠇⠀⢀⠀⠹⣄⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⠀⢹⢰⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠉⠁⠹⡄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⡀⣠⡀⠀⠀⠀⠀⣠⣄⣀⣀⣀⣀⡀⢠⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠈⠀⠀⣠⣴⠾⠛⠋⠉⠉⠉⠉⠁⠘⠈⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⡆⠀⡈⠢⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣶⢀⠀⢠⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠹⣆⠁⠀⠀⢨⠁⠀⢀⠀⠀⠰⡂⠀⠀⠈⣠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠈⠳⣄⡐⠂⠀⠀⢛⡃⠀⠀⠰⢂⣠⠞⠁⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡑⢠⡈⠙⠲⠦⠤⠨⠄⠤⠴⠖⠋⢁⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠤⠤⠀⠀⠂⠈⠀⠑⢈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⠒⠒⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠒⠒⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡒⠀⠀⠀⠀⠐⢤⠤⠖⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⣦⡀
⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠉⠉⠉⢱⡴⠚⠉⠉⠉⠙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⡀⠀⣠⠀⠀⡀⣀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢫⠀⣿⡉⢹⠁⡇⢰⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡌⠻⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⢸⢀⡿⡇⢸⢠⡋⠸⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠑⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠈⠉⠇⢇⢸⠈⠉⠷⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣄⣀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢫⠓⢄⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠓⠒⠶⠶⠖⢺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡴⠃⠀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈
⠀⠀⣀⣀⣤⠤⠤⠔⠒⠒⠈⠉⠉⠈⠀⠀⠀⠀⠈⠑⠒⠂⠀⠉⠉⠀⠈⠑⠲⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡈⠑⢦⣠
⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠄⠚⠁⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣖⡉⣉⣀⠙⠐⠲⠖⠀⣀⡤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢉⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⡀⠤⠤⠒⠒⠒⠒⠤⠄⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⣸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡀⠀⠀⠀⠈⠙⢷⡀
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠣⣉⡡⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣉⡡⠃⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿
⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢠⠖⠶⠲⡆⠀⢠⠖⠶⠖⢦⠀⠀⡄⠲⠖⢦⠀⠀⡴⠲⠒⠲⡄⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠃⠀⠘⠓⠒⠒⠛⠀⠀⠓⠒⠒⠚⠀⠀⠛⠒⠒⠚⠁⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣠⠤⠤⢤⡄⠀⢠⠤⠤⠤⡄⠀⢀⡤⠤⠤⣄⠀⠀⡤⠤⠤⢤⠀⠀⣠⠤⠤⢤⡀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣇⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⡇⠀⠀⠀⡇⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⠓⠖⠶⠒⠃⠀⠘⠶⠶⠶⠃⠀⠘⠒⠶⠖⠞⠀⠀⠳⠶⠶⠞⠀⠀⠓⠲⠖⠖⠃⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⡀⠀⢀⣀⣀⣀⡀⠀⢀⣀⣀⣀⣀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⡇⠀⢸⠀⠀⠀⡇⠀⢸⠀⠀⠀⢸⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⣧⠀⠷⠤⠤⠤⠇⠀⠸⠤⠤⠤⠇⠀⠸⠤⠤⠤⠼⠀⠀⠧⠤⠤⠼⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠙⢷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣦⠀⠀⠿⠀⠀⢸⠇⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠳⣄⠀⠀⠀⠀⠀⠹⡆⠀⠤⠀⠀⠞⠀⠀⠸⠋⠀⠀⢀⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠘⠂⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡄⠀⠀⠀⢠⡴⠒⢦⣾⣹⣆⣀⡀⢾⣱⣄⣀⠀⠀⣾⣿⠀⠀⡼⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠘⣷⣄⣀⣿⠞⢹⣿⠇⠀⡹⢹⣿⠇⠀⣹⣿⡀⡼⠁⢻⣿⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⣿⡄⠀⢿⣇⢀⡿⠀⣼⣿⠀⢰⠃⣼⡿⠀⢰⠃⠈⠉⠀⢀⢼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡁⠀⠀⠀⠘⣿⠀⠈⠉⠉⠀⡀⠹⣧⣠⠏⠀⠻⣧⣠⠏⠀⣰⢊⠞⢣⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⣀⣠⡿⢀⠞⠛⠒⠋⠙⠦⠤⠤⠞⠓⢦⣄⣀⠀⢰⡇⠘⠠⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⡀⠀⢀⡀⠀⠀⣀⡀⠘⣇⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⣧⢀⡀⡿⠀⣿⠉⠀⠀⠀⠀⣿⣹⠆⠀⡏⠁⠀⠈⠓⠞⠋⢠⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⡿⠶⢂⡀⠂⠀⢹⣼⣷⡇⠀⣿⠉⠀⠀⠀⠀⣿⢳⡄⠀⡏⠁⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠘⠃⠙⠀⠀⠙⠒⠀⠀⠀⠀⠉⠀⠃⠀⠓⠒⠀⠀⠀⠶⠉⠡⣶⣶⣶⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢠⣶⡿⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠹⣷⣄⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⢸⡇⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢀⣾⢃⣀⣀⣀⣀⠀⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⣸⠇⠀⠀⠀⠀⠙⠒⠤⠤⠴⠶⠒⣾⡟⠉⠉⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⢠⡟⠀⣠⣤⣶⡶⢦⣄⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⣀⣤⠴⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⠟⠀⣼⣿⡟⠁⠀⣰⣿⣷⠀⠀⠀⣿⡿⠀⠀⢀⣾⣿⠃⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡾⠃⠀⠀⣿⣿⠀⢀⡼⣿⣿⣿⠀⠀⠀⣿⡇⠀⠀⣸⣿⡇⣀⡼⠃⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣼⠿⠶⠋⠀⣿⣿⡇⠀⠀⣸⣿⠁⠀⠀⣿⣿⡏⠁⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⢠⡇⠀⠀⠀⠀⢻⣿⡇⠀⣰⡟⢿⣄⣠⠞⢹⣿⡇⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⣸⠃⠀⠀⠀⠀⠸⣿⣇⣴⠏⠀⠈⠉⠉⠀⠀⠙⠿⣦⡤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣠⡏⠀⠀⠐⢦⣀⠀⠈⠉⢁⣰⠓⠦⠤⣤⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣶⣶⣶⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⣶⡄⠀⠀⠀⢸⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠻⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⢸⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿
⠀⠀⠀⢰⣶⣦⣬⣍⣉⣛⠛⠛⠃⣿⣿⡟⠛⠛⣛⣉⣉⣥⣤⣶⡆⠀⠀⠀⢸⣿⣿⣿⣿⠉⠙⠛⠶⠶⠦⣤⣤⣼⠀⠀⢸⣤⣤⡤⠶⠶⠞⠛⠋⠉⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣇⣛⣛⣱⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⢸⣿⣿⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣮⣵⣾⣯⣵⣶⣦⣄⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠉⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⢻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢻⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠀⠤⣄⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠄⠀⠄⠀⠀⢀⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⠀⠁⠀⠀⠀⠀⢠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠛⠀⠀⠂⢐⡶⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⡀⢀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣧⡀⠀⢀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⢿⣿⣷⣦⡄⠀⠴⠄⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠀⠀⠀⠈⠣⡉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠉⠘⠂⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠈⡷⠶⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠈⠁⠀⠐⠧⠄⣀⡀⠀⠈⠛⠛⣛⠋⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠈⣀⠋⠈⣩⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠈⣇
⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣆
⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠘⢧⡀
⠀⠀⠀⠀⢸⣇⣀⣀⣀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡹⠆
⠀⠀⠀⠀⠀⣹⠟⠛⠛⠿⢷⣦⣀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⠶⡾⠉
⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⡹⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿
⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⢸⡀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠁⣄⠐⠚⠆⠀⠀⠀⠀⠀⠀⠀⢸⠃⠘⣇
⠀⠀⠐⣇⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣀⠀⠀⠀⠀⠀⢛⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙
⠀⠀⠀⠈⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⢤⠎⠀⠁⠀⠀⠀⢢⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠉⠐⠊⠉⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠉⢹⠀⠀⠀⠀⠀⠒⠒⠒⢶⠚⠃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠓⠤⢄⠀⠀⠐⠒⡶⢋⠻⠇⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⠛⠉⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣶⣶⣟⣢⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠈⣏⡛⠻⠿⣿⣷⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠛⠃⠀⠀⠀⠈⠉⠛⠿⢿⣿⣷⣶⣤⣄⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠘⡶⠦⠀⣀⠀⠀⠀⠉⠛⠻⣿⣿⣟⣷⣷⣤⣀
⠀⠀⠀⠀⠀⢀⠏⢀⣤⣀⠀⠀⠀⠐⠒⠃⢠⠾⣽⠀⣰⣆⠀⡀⠀⢹⡦⠈⠉⠛⠻⢿⣿⣆
⠀⠀⠀⠀⠀⡞⠀⡜⠙⢺⠯⣷⣲⠤⣄⡀⠀⠀⠈⠀⠃⠘⡾⠁⠀⠀⠀⠀⠀⠀⠀⣼⠏⢸
⠀⠀⠀⠀⢰⠃⢰⠉⢑⠗⠤⣠⡏⠙⠳⣼⣉⣖⢢⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠈⣇
⠀⠀⠀⢀⡏⢀⣏⡉⡾⠤⢄⣜⠈⠙⢲⠧⢄⣈⠟⠚⠦⣽⣓⡶⢤⣀⡀⠀⠀⢰⣿⠁⠀⠀⢹
⠀⠀⠀⡼⠀⡼⣄⣸⠓⠲⣼⣉⠉⢒⠧⢤⣀⡎⠉⠒⣴⣅⡈⠉⡞⠺⢭⡇⢀⣿⠏⠀⠀⠀⠈⡇
⠀⠀⢰⠃⢰⠧⢄⣏⠙⢲⠧⢄⣉⡟⠢⢤⣜⠉⠑⢲⢧⣀⠈⡝⠓⠤⡼⠀⣾⡟⠀⠀⠀⠀⠀⢹
⠀⠀⣞⠀⠘⠒⠼⣁⣉⠟⠢⢄⣼⠉⠒⡾⢥⣉⢓⠾⢤⣀⣹⠓⠦⣤⠃⢸⣿⠃⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠻⢽⣓⠦⢄⣀⠉⠑⠲⠤⣇⠉⢲⠧⢤⣀⡝⠓⠢⢤⣏⠑⢲⡎⢠⣿⣏⣀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠉⠓⠺⢽⣒⠦⣤⣀⠉⠛⠢⢄⣸⠉⠒⢢⠯⣀⠉⡞⠀⣾⣿⡀⠈⠉⠒⠦⢄⣀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠯⢽⣳⣦⣤⣀⠉⠓⠺⠤⣀⣹⠁⣼⣿⡇⠈⠑⠒⢤⣀⣠⠤⠭⠷⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠫⢽⣓⠦⣤⣀⠀⢠⣿⡿⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠺⢽⣛⣻⣧⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣝⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⣻⣥⣝⢆⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⣹⣇⠀⡉⠛⠻⢷⣤⡄⠈⠣⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡤⠄⠐⠉⢴⠿⠛⠻⢿⠀⡀⠀⠀⠈⠉⠳⢤⡙
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⣀⣠⣾⣄⠀⠀⠀⠀⠛⢷⣆⡆⣀⠀⠀⠙⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠙⢿⡓⠦⣄⡀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⣾⣟⠀⠀⠀⠱⣦⣄⠀⠀⠀⠈⠓⠺⢿⣦⣀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢈⡙⡛⠿⠿⣆⣠⡁⠙⢻⠶⣷⣂⡀⠀⢨⠟⢻⣟⠁
⠀⠀⠀⢀⣠⣤⣴⣶⣶⣽⣄⡻⣿⣧⣀⠀⠘⠿⣿⣦⣦⣔⣾⣿⣿⣿⣿⡶⠶⠛⢷⣄
⠀⠀⣸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣷⣿⣿⣿⣴⣤⣌⣏⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠈⠑⠚⠓⠚⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⣀⣤⣤⣀⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣀⣠⣀
⣿⣯⣤⣥⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠉⠉⠉⠙⠻⢷⣦⣀⣀⣤⣤⣤⣤⣀⣀⣠⣾⠟⠋⠉⠉⠉⠛⠿⣷⣄
⠀⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢿⣶⣤⡀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣠⣾⠟⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢹⣷⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠆⠀⠈⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣼⡿⠁⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠘⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⣿⡆
⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣴⣶⣶⣦⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⣹⣷⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣶⣶⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣷⣦⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠉⠙⠻⢿⣶⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣶⡿⠟⠋⠉⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣋⣥⠴⠶⠶⠦⣬⡛⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣷⣞⣉⣀⣀⣀⣀⣀⣀⣙⣶⣾⣿⣇⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⡴⠋⠳⡄⠀⠀⠀⢸⠀⡇⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣤⣤⣧⣼⣤⣤⣤⣤⣇⣠⣀⣧⣤⣤⣤⣼⣤⣧⣤⣤⣴⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⠁⠀⢹⣻⣟⡏⠁⠈⠹⣏⣉⣉⡿⠁⠀⠙⣟⣟⣻⠉⠀⢹⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⢸⡟⢻⡇⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⣿⠛⣿⠀⠀⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠸⡇⢸⡇⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⢿⠀⣿⠀⠀⣿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⠘⢦⣤⣤⣽⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣤⣤⠞⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠚⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠦⠶⠦⠳⠶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡌⠛⠁⣰⢶⣾⠊⢈⢈⣤⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣧⡀⢛⣉⠉⢨⡁⠛⠚⢺⡏⠉⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⣿⠉⠉⠛⠻⢧⠆⣄⠀⡅⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠀⠀⠀⠀⢀⠷⢧⡻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⢈⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠁⠀⣠⠞⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠉⢫⠀⠀⠀⣠⠟⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⠀⠀⠀⠉⠛⠲⢤⣀⠀⠀⠀⠀⣰⠏⠀⠀⢂⠈⠓⢲⠎⠀⠀⠀⠈⠑⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉⢀⣀⣴⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢶⡾⢫⠀⠀⠄⠈⢀⠀⠠⠁⢀⠀⢈⡀⠉⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⣠⣜⣹⠟⡉⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠎⠀⠀⠀⠀⠀⠐⢀⠀⠀⡀⠈⠀⠀⠀⢀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠋⠁⠀⣀⠤⠒⠈⣩⠿⠃⠢⡀⣠⡏⠝⠉⠛⠛⠓⠒⠀⠀⣤⡟⢀⡇⠀⠀⢀⠁⠀⠀⠀⠀⢀⠀⠰⣄⠀⠀⠀⠀⠀⠘⢣⡀
⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⢞⠱⣅⠀⠀⠀⠀⠀⠀⠈⠙⠶⠦⣀⡀⠀⠁⠀⠀⠂⠀⠀⡶⠀⠀⡁⠀⠀⠀⢀⠀⠀⠀⠐⢀⠟⠳⣄⠀⠄⠀⠠⠀⢳
⣀⣤⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠹⣆⠉⠓⠶⣀⠄⠈⣀⣼⠃⠀⠀⠀⠄⠂⠀⠀⠄⠀⠈⠀⡜⠀⠀⠈⠳⣄⠀⠀⠄
⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠊⠉⠣⡈⠢⡀⠀⠀⠀⠀⡀⠰⡅⠀⠀⠀⢈⣱⡿⢛⡿⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐
⠀⢸⡟⠻⣦⣄⠀⠀⠀⠀⢀⠀⠀⠀⠳⡀⠀⠙⣆⢹⣦⣠⠖⣉⠤⠚⠁⢀⣤⡶⠛⠁⣠⡞⠁⠉⠀⠀⠈⠐⠀⠈⠀⠀⠤⣴⠀⠀⠀⠀⠀⢀⡟
⠀⢸⡇⠀⠈⠛⢷⣄⡀⠀⢸⠘⡄⠀⠀⠹⣄⠀⠀⣡⠤⠒⠋⠁⣀⣴⠾⠛⠛⠒⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⢠⠞⠀⠀⠀⠐⡼
⠀⢸⡇⠀⠀⠀⠀⠉⠻⣦⡀⠁⢱⡄⠀⠀⠈⠲⠋⠀⢀⣠⣴⠟⠋⠁⠀⠀⠀⠀⢀⣤⠖⠀⠀⠙⠲⢦⣄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⣴⠋⠀⠀⠁⣠⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠀⠀⠀⠀⣀⣤⡶⠛⠉⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠐⢰⡾⢧⡔⢃⣀⠈⠉⣧⠰⠞⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣠⣴⠾⢻⡇⠀⠀⠀⠀⣈⣢⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠀⠀⢻⠖⠁⠀⡼⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠒⠋⠁⠀⠀⢀⣀⣴⠛⠦⠤⠚⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⣀⠢⣤⡤⠖⢺⣿⠋⠉⠛⠛⠛⠛⣿⠋⠉⣿⡆
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⢻⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⡄⠀⣿⣇
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⡿⣆⠀⠀⠀⠀⠀⠈⢧⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⡏
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⠁⠹⣆⠀⠀⠀⠀⠀⠘⡇⢸⡿⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⣷
⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⠀⠹⣆⠀⠀⠀⠀⠀⢻⣼⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⣀⡀⣀⣀⣼⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⢿⡇⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⣿⣀⣤⡶⠟⠋⠉⠉⠉⣿⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⣿⠉⠀⠀⢀⣀⣀⣀⣠⡿⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⣿⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⣠⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⠋⠉⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⢀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠀⠀⢰⡟⠋⠁
⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣶⡾⠋⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⡀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣤⣤⣶⣦⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠁⠀⠀⠈⠉⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⡴⠒⠋⠙⠛⠒⠲⢤⣠⠇
⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣔⡴⠶⣶⠴⢦⣠⣴⣲⠶⠤⡤⣿⣦⡤⠤⠤⢤⣤⣤⣴⣧⣤⣤⡴⣆⣀⣀⢶⣶⡤⠤⢤⣤⣤⡤⠤⣄
⠀⠀⠀⠀⠀⢰⠋⠉⠀⠀⣠⠞⠛⠁⠠⠾⠋⠉⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠙⠉⠁⢠⡾⠋⠡⡄⠀⠉⠁⠐⠢⡀⡆⢸⡆
⠀⠀⠀⠀⠀⢸⠑⢊⠔⠋⠁⠀⣇⠀⣀⣧⠀⠀⠀⠠⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⣀⠀⢸⡇⠀⠀⠙⢦⡀⣈⡦⠀⣧⡇⢸⡇
⠀⠀⠀⠀⠀⢸⣀⠏⠀⠀⠀⠀⢧⠤⠤⣤⠀⠀⠁⠀⠓⢄⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢤⡇⠀⠀⠀⠀⠙⢦⡀⠀⡿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠁⠀⠄⠀⢀⠠⠀⠀⠀⣿⠀⠀⠀⠑⢀⠈⠐⠉⠀⠀⠂⠂⠀⠀⠀⠀⡃⠀⠀⡼⠇⠀⠀⠀⠀⢀⢠⣙⠐⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⢠⠄⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡃⠀⠀⠑⠔⠉⠀⠀⠀⠀⠀⠐⣠⢦⠀⠀⡇⡇⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠠⣀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠⠰⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠀⢀⢀⡀⡀⠀⠸⠀⠀⠀⠀⡀⠀⠀⠀⠄⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠘⠀⢫⡀⠀⢀⠀⠀⠅⠀⠀⡇
⠀⠀⠀⠀⠀⢸⡄⠀⢀⠀⡈⠃⠀⠀⠀⢰⠀⠈⠁⠂⠀⠤⠤⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠚⡳⠠⠀⠀⠤⣤⠆⠀⡆⢀⡇
⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⢡⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⡆⠀⠀⡆⡀⠈⠑⠪⡥⢤⡌⠀⠀⠃⢸⠃
⠀⠀⠀⠀⠀⢠⠋⢢⠀⠀⠀⣡⡠⠀⠀⢸⠀⠀⠐⠀⠀⠀⠉⡀⠀⠀⠀⠈⠉⠀⠀⠀⡁⠁⠀⠀⠀⠈⢆⠀⠀⠀⠉⠀⠀⠀⠀⠼⡄
⠀⠀⠀⠀⠀⢸⡀⢀⠑⣄⠉⠀⢸⠀⠀⢸⠀⠀⠀⢀⠀⠀⢀⠄⠀⠀⢀⠀⠂⡘⡄⡀⠑⡇⠀⠀⡆⠀⡘⠀⠀⠀⠀⠂⠈⢶⡄⢸⡇
⠀⠀⠀⠀⠀⠸⣅⣨⣤⣈⣷⠤⢬⣀⡀⣸⣦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠊⠃⠀⠁⠁⠀⠀⡇⠀⠀⡗⠈⠀⠀⢀⠀⠁⡀⠀⢟⣧⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠐⠒⠒⠒⠀⠀⠐⠒⠒⠒⠒⠧⠤⠀⠷⠦⠠⠴⣁⣀⣐⣁⣠⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⠶⠶⠶⠶⠶⠶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠋⠉⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠉⠙⠛⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠋⠁⣀⡤⠖⠛⠉⠉⠀⠀⣄⢀⡀⠀⠉⠉⠛⠲⢤⣀⠈⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⡴⠚⠁⢀⢀⠀⠀⠀⠀⠀⠟⠰⠧⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀⠉⢻⣦⡀
⠀⠀⠀⠀⠀⢠⡾⠋⢀⡴⠋⠀⠀⠀⠛⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠙⢦⡀⠙⢷⡄
⠀⠀⠀⠀⣰⡿⠁⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄⠈⢿⡄
⠀⠀⠀⢠⡿⠁⢠⠏⢠⢠⣶⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦⠀⢻⡄⠈⣿⡄
⠀⠀⠀⣾⠃⢀⡟⠀⠈⠀⠉⠁⠀⠀⠀⠀⠘⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠀⢷⠀⠸⣷
⠀⠀⢰⡿⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⣿⡄
⠀⠀⢸⡇⠀⣾⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⡇⠀⢹⡇
⠀⠀⢸⡇⠀⢻⠀⣻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠾⠃⡇⠀⢸⡇
⠀⠀⠸⣿⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⠀⠀⠀⠀⠀⢰⡇⠀⣿⠃
⠀⠀⠀⢿⡆⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⡾⠀⢰⡟
⠀⠀⠀⠘⣿⡀⠘⣧⠀⢘⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⣿⡇⠀⣼⠃⢀⣿⠃
⠀⠀⠀⠀⠘⣷⡀⠘⢧⡀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢀⣾⠃
⠀⠀⠀⠀⠀⠘⢿⣄⠈⠳⣄⠀⠀⠀⣲⡖⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢀⣰⠟⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⠈⠻⢦⣀⠙⠁⠀⠀⠀⠀⢼⣧⡀⠀⠀⠀⠀⠈⠟⠀⣀⡴⠛⠁⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠉⠛⠶⢤⣄⣀⡀⠈⠉⠀⠀⢀⣀⣠⡤⠶⠛⠉⣀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣤⣀⡀⠉⠉⠉⠉⠉⠉⠉⠉⢀⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠⢴⣶⣶⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢡⢊⣀⣀⣉⣿⣿⣿⣿⣿⣿⣿⠷⡀⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠛⠻⠿⠿⣿⣿⣿⣿⣧⣌⣀⡀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠰⢉⡙⠠⠀⠘⠃⠀⠀⠆⠈⠉⠉⠙⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠈⠠⡇⠀⣀⠀⠀⠀⠀⠀⠀⠃⠀⢠⣿⠿⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠐⠘⠂⠀⠀⠐⣂⠀⠀⠀⠀⠐⠛⠂⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⣀⠀⠀⡀⠀⠀⡸⠁⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣀⣀⣀⡠⠔⠊⣉⡉⠀⠒⠢⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠂⠁⠀⢤⡼⣤⢀⠀⠀⠈⢆⠉⠁⠐⠂⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠉⠈⢹⠉⣃⠊⠘⠳⡠⡀⠈⠆⠀⠀⠀⠄⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠨⠥⠰⢳⢇⡁⠘⠄⠀⡀⡄⠒⠓⡄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠐⠘⠚⠃⠁⠐⣈⣐⣣⠤⠴⠟⠓⠒⡦⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⢀⠜⠓⢄⠀⡊⠉⠀⠀⠀⠀⠀⠀⠀⣹⠦⡈⠳⣄
⠀⠀⠀⠀⢀⡠⠄⠂⢤⡀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠣⡠⠀⠄⠀⠀⠀⠀⠀⠸⡆⠈⠢⡈⠳⡀
⠀⠀⠠⠊⠣⡀⠀⠀⠀⠘⢄⢀⡤⠊⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⢈⠦⠊
⠀⠀⠀⠈⠢⡈⠀⢄⠀⠀⠈⠃⣀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⡀⠀⠄⠀⠂⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⠑⢄⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡴⠟⡿⠀⠈⠉⠉⠉⠛⠛⠓⠲⠶⠶⠶⠦⢤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⢠⡏⠀⢠⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⢸⣧⣾⣿⡷⠶⠤⠭⠭⠭⣍⣉⣙⣒⣒⣒⣒⣒⣒⣒⠲⣶⣤⣤⣿⠀⠀⠀⠀⣀⣤⡴⠶⠚⠛⠋⢷⡀
⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⡤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠶⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣼⠿⣷⡀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⠀⠀⢿⠀⠀⠀⠀⢀⣤⣶⠿⢻⡿⠀⠀⠀⠉⠀⠹⣷⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣇⠀⠀⠀⠀⠁⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠻⣄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⣼⣿⠾⠛⠛⢷⣄⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠙⠷⣄
⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢿⡆⠀⠀⠀⠈⠻⡧⠀⠀⠀⠀⠀⠈⠛⠶⠶⡶
⠀⠀⠀⠀⠀⣿⢹⡟⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⡀⠀⠀⠀⣿⣿⣷⢏⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⣤⣤⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⣿⡟⣿⣹⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⣿⣧⢻⡜⣿⣽⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠉
⠀⠀⠀⠀⠀⢹⣼⢹⢱⢇⢧⣷⣬⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣄⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠚⠋⠁
⠀⠀⠀⠀⠀⠘⣇⣎⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠙⢷⣄⣠⣤⠶⠞⠋⠉
⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⠤⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡽⠒⠚⠉⠉⠉⠉⠉⠒⢿⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣞⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡵⡄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⡝⠶⠛⠻⠿⣷⣶⣄⠀⣤⣤⣀⡠⣴⡬⠾⠟⠚⠳⠾⣽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣿⠀⠀⣀⣤⡶⠾⢿⣿⠊⠛⣿⠿⢿⠿⠷⣦⣤⣀⠀⠀⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⣾⣿⣿⣿⣏⢻⡀⣨⣟⣁⡀⣴⣿⠄⣈⣳⣄⣼⡿⢛⣿⣶⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⣿⠛⡿⣿⣿⣦⣿⠿⠖⢀⣀⣁⣀⣀⡀⠐⠿⣿⣥⣿⣿⠋⠛⢻⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠚⡹⣫⣿⣋⡴⠞⠋⠉⠉⠀⠄⠉⠉⠙⠲⢦⣝⢻⡙⢯⠒⠺⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⣩⡾⠋⠀⠀⠀⠀⠀⣰⠺⠀⠀⢠⡞⠂⠈⠳⣄⠀⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣴⠋⠀⠀⠀⠀⠆⡇⠀⠈⠉⠀⢠⠽⠇⠀⠀⠀⡈⢧⡀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣸⠃⠀⠀⣾⣷⣀⡀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢸⡋⠑⠈⣧⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢀⡟⠀⡀⣀⠉⠁⠙⠦⠦⣀⣀⢠⡏⠀⠀⠀⠀⠀⢀⡀⣆⢹⡄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⢰⠂⡿⠀⠀⠀⠀⠀⠈⣷⣾⠀⠀⠀⠀⠀⠀⢈⡇⠛⢸⣷⣬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡝⢿⣿⠞⠀⠀⣀⠀⠀⠀⠀⠠⠋⠈⠣⡀⠀⠀⠀⢀⡄⠀⠂⣼⠿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠙⢦⠠⠄⠿⠀⠀⠀⡰⠁⠀⠀⠀⠑⣄⡀⠠⠺⣧⣄⡖⠉⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠠⠃⠐⢆⠀⠈⡅⠀⠀⢠⡆⠀⠀⢨⣦⣀⣶⣿⠋⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣴⣦⡀⠼⣦⣤⣀⣀⣘⡋⢀⣠⣢⣾⡿⢟⠕⡊⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣮⣭⣓⣻⣿⣿⣛⣛⣩⣵⣾⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣹⠟⠛⠿⢿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣄
⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄
⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠠⣦⠀⢠⡄⠀⣴⠀⠀⣦⠀⢠⡆⠀⢰⡄⠀⣤⠀⠠⣦⠀⠀⠀⠘⢿⣿⣆
⠀⠀⢀⣾⣿⠏⠀⠀⠀⣾⣾⣿⣶⣾⣷⣶⣿⣶⣶⣿⣶⣾⣷⣶⣿⣷⣷⣿⣶⣾⣿⣾⡆⠀⠀⠈⢿⣿⣆
⠀⠀⣼⣿⡏⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⢻⡟⠛⠛⡟⠛⠛⡟⠛⠛⣿⠛⠛⢻⠛⠛⢻⡇⠀⠀⠀⠈⣿⣿⡄
⠀⢰⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⣀⣠⣇⣀⣠⣧⣀⣠⣧⣀⣀⣧⣀⣀⣼⣀⣀⣼⡇⠀⠀⠀⠀⢸⣿⣷
⠀⢸⣿⡏⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⡇⠀⠀⣿⠀⠀⢹⠀⠀⢸⡇⠀⠀⠀⠀⠈⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⣿⠖⠒⢺⠖⠒⢺⡗⠒⠲⡗⠒⠒⡷⠒⠒⣿⠒⠒⢾⠒⠒⢾⡇⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⣧⠀⠀⠀⠀⠀⣿⣀⣀⣸⣀⣀⣸⣇⣀⣀⣇⣀⣀⣇⣀⣀⣿⣀⣀⣸⣀⣀⣸⡇⠀⠀⠀⠀⢠⣿⣿
⠀⠘⣿⣿⡀⠀⠀⠀⠀⣿⠉⠉⢹⠉⠉⢉⡏⠉⠉⡏⠉⠉⡏⠉⠉⣏⠉⠉⢹⠉⠉⢹⡇⠀⠀⠀⠀⣸⣿⡟
⠀⠀⢻⣿⣧⠀⠀⠀⠀⣿⠤⠤⢼⡦⠤⢼⡧⠤⢤⡧⠤⠤⣧⣤⣤⣿⣤⣤⣾⣤⣤⣼⡇⠀⠀⠀⢠⣿⣿⠃
⠀⠀⠀⢿⣿⣧⠀⠀⠀⣿⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠈⢻⣿⣷⡀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⣰⣿⣿⠋
⠀⠀⠀⠀⠀⠙⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣷⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⢠⡴⠒⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⢀⡤⠶⠶⠶⢤⡞⠋⠉⣉⣙⣯⡤⠴⠒⢻⠏⢧
⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⣹⣿⣠⣤⣤⣶⠾⣿⠛⠋⠉⢁⡿⠀⠀⠰⡿⠀⠈⢧
⠀⠀⠀⢀⣠⠼⠷⢒⣚⣛⣭⠭⠞⣟⠉⠁⠀⣀⡟⠀⠀⠀⠘⠃⠀⠀⢀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⣸⢠⠖⢻⠟⢻⡅⠀⠀⣴⣏⠀⠀⠀⠛⠁⠀⢀⣤⡀⠀⣷⢦⣸⣆⠀⠀⠀⠈⢷⡀
⠀⠀⠀⡏⢸⠀⢸⠀⠀⡇⠀⠀⠉⠁⠀⠀⠴⠶⣿⠋⢸⣯⡿⣄⠹⡆⠉⠛⠀⠀⢀⣀⡈⢳⡀
⠀⠀⢰⡇⢸⡇⠸⡇⠀⢻⠀⠀⠀⠀⠀⠀⠀⢀⣹⠀⠸⠇⠀⠉⢀⣀⣤⡴⠒⣻⡩⠄⢹⡄⠻⡄
⠀⠀⢸⡇⠸⡇⠀⣧⠀⠘⡇⠀⠀⠀⠀⠀⠀⠉⠁⢀⣠⡤⢶⠚⣙⣅⠄⢳⣈⠥⠿⣎⠉⠙⣆⠙⣆
⠀⠀⢸⡇⠀⡇⠀⢸⡀⠀⢹⡀⠀⠀⣀⡤⠶⡖⢋⡉⢧⠒⣊⣧⠴⠚⣏⠁⠙⣆⣀⠼⢶⡊⠉⢧⡈⢧⡀
⠀⠀⢸⡇⠀⣇⠀⠀⣇⠀⠀⢷⠀⠈⢇⠰⠒⣱⣬⠴⠚⢏⠁⠈⢧⣀⠼⢶⡊⠉⠣⡀⢀⣱⣤⠒⠛⣄⠙⢦
⠀⠀⢸⡇⠀⢻⠀⠀⢹⡄⠀⠈⢧⠀⠘⣞⠉⠁⠹⡄⣀⡬⢶⠚⠉⠳⡀⠀⣱⣤⠖⠛⢏⠀⠈⢣⣀⡬⠷⡀⠳⣄
⠀⠀⠘⡇⠀⢸⡄⠀⠀⢷⠀⠀⠈⢧⠀⠘⢦⠴⠚⠹⡅⠀⠈⢣⣠⠤⠛⢏⠁⠈⢢⣀⡨⠷⣞⠉⠙⢦⡀⢙⣦⡈⠳⣄
⠀⠀⠀⡇⠀⠀⣇⠀⠀⠈⣧⠀⠀⠈⢧⡀⠈⢧⡀⣀⡼⢶⠚⠉⠙⣄⠀⣈⡷⡔⠋⠙⢦⡀⠈⣳⣤⠔⠛⢯⡀⠙⣦⠈⠑⢄
⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠘⣆⠀⠀⠈⢳⡀⠀⠻⡄⠀⠈⢣⣠⠴⠚⠻⡁⠀⠈⢲⣄⠤⠛⢯⡁⠈⢲⣤⠴⠛⠋⠁⠀⢀⣠⠗
⠀⠀⠀⢸⠀⠀⠘⡇⠀⠀⠀⠘⣆⠀⠀⠀⠹⣄⠀⠙⢶⠊⠉⠙⢦⡀⣀⡼⠶⡚⠉⠈⠳⣄⡠⠽⠚⠉⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠘⡇⠀⠀⢳⠀⠀⠀⠀⠘⣆⠀⠀⠀⠘⢦⡀⠀⠳⣄⡤⠖⠛⢧⡀⠀⣈⡷⠖⠋⠁⠀⠀⠀⢀⡤⠖⠉
⠀⠀⠀⠀⣷⠀⠀⠘⣇⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⣦⠀⠈⠑⢦⣠⠴⠚⠉⠁⠀⠀⠀⢀⣠⠴⠊⠁
⠀⠀⠀⠀⢸⡄⠀⠀⢹⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠈⠛⢦⡀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠉
⠀⠀⠀⠀⠀⣧⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⢀⣹⢦⣠⠴⠚⠋⠁
⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⣣⠀⠀⠀⣀⡤⠴⠚⠳⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠳⠒⠛⠉⠉⠳⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⠿⠿⠿⢿⣶⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣾⠿⠛⠉⠀⠀⠀⣿⢸⠀⠀⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢎⠞⠁⠀⠲⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠖⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢳⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢧⠃⠈⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠤⠤⠀⠀⠀⣠⣤⣤⣤⣴⡟⣦⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠉⠁⠀⠀⠈⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢀⣎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⢎⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣄⠀⠀⠎⠀⠀⠀⢠⡀⠀⠀⠀⠈⠁⢀⡔⣡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣦⣤⣀⡀⠀⠘⠃⠀⣀⣠⣴⣾⡵⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣿⣿⣿⣿⣷⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⣿⣿⡿⠿⠿⠛⠛⠛⠛⠻⠿⠿⣿⣿⢷⣶
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣰⣿⣷⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣾⣿⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⣶⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣾
⠀⠀⠀⠀⠀⣿⣿⣦⡈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣴⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣉⣉⣉⣉⡙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⣠⣾⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠸⣿⣿⣿⣿⡇⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣡⣾⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡀⠛⠛⠛⠛⠃⣰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⢀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠳⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠟⠛⠛⠛⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣀⣠⠤⣴⣲⣽⠇⠀⠀⢀⡀⠀⠀⠀⠠⣤⠖⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⣀⣠⠤⣒⣺⣭⣷⣚⣻⣭⣥⣤⣤⣤⣤⣟⣙⢻⡆⠀⠀⢠⣴⡶⠶⢦⣄⡉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠘⣯⠵⠞⢋⣩⡼⠟⠋⠉⠀⠀⣀⣀⣀⣀⠀⠈⠉⠙⠳⢦⣄⠈⢷⡱⡄⠀⠈⠛⣦⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⣠⠞⠋⠁⢀⡠⠴⣒⣯⣭⠭⠶⠦⣬⣭⣙⡒⠦⣀⠀⠈⠳⣦⡻⣜⣆⠀⠀⠈⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠟⠁⢀⡠⣚⡥⠖⠋⠉⠀⠀⢠⡆⠀⠀⡤⢨⣝⠳⣬⡳⣄⠀⠈⢳⣿⣮⢦⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⡴⣫⠞⠉⠀⠀⠻⠂⠀⠀⠀⠀⠀⢈⡁⠶⡟⠀⠈⠻⣎⢣⡀⠀⠹⣏⢷⡳⡀⠀⣸⠀⡖
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⢏⡼⠃⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣿⠀⠀⠀⠀⣀⡘⢧⠹⡄⠀⢹⡎⢳⣝⣤⠏⠀⠉
⠀⠀⠀⠀⠀⠀⢀⡾⠁⢠⢣⡞⠁⠲⠞⠋⠀⠀⠀⠀⠀⠀⠀⢰⣏⡾⠁⠀⠀⠀⠀⠉⠁⠈⣧⢹⡀⠀⣿⠀⠙⠁
⠀⠀⠀⠀⠀⠀⣸⠃⠀⡏⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠠⡇⡇⢀⣠⠀⠀⠀⠀⢠⡾⠋⠛⣲⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢈⡇⡹⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⣧⠀⠁⠀⠀⠀⠀⠀⠛⠶⠛⠁⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⣸⢇⡇⠀⣼⠁⡶
⠀⠀⠀⠀⠀⡀⠸⣇⠀⠸⣽⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⣀⠀⢠⡟⡼⠀⢠⡟⠀⠃
⠀⠀⠀⠀⠀⠹⠀⢻⡄⠀⢳⣻⡄⠀⡴⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢀⣰⡿⢣⡟⡼⠁⢠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠀⠳⣽⣦⡀⠀⠀⣤⣶⠦⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⢉⣵⢫⠞⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡻⣄⠀⠈⠻⣿⢦⣀⠻⠽⠂⠀⠰⠇⠀⠀⠘⠃⣀⣠⠶⣋⠕⠁⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠐⣯⣀⣽⠟⠻⣦⡀⠈⠙⠯⣟⣓⠶⠶⠶⠦⠤⠶⠖⣛⡭⠔⠋⠁⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠙⠷⣤⣀⡀⠈⠉⠉⠉⠛⠛⠋⠉⠀⣀⣠⡴⠞⠉⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠶⠶⠶⢺⣿⠭⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣶⣶⡄⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢠⣶⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿
⠀⡀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⣿⡿⠏⣿⣿⣿
⠀⠁⠀⠀⠙⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣀⣀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⢠⡀⠀⠀⠀⢠⠤⠶⠲⣄⣠⠒⠐⢢⠌⢠⠄⠃⠐⢚⠀⢸⠛⠛⠻⠿⠿⠿⠻⠿⠿⠿⠿⠿⡇
⠀⠀⠀⠀⠈⠀⠀⠀⠀⢈⠨⠖⠂⡎⠀⣉⠖⢁⣴⠥⡒⠔⡦⠃⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣶⣶⡆
⠀⠀⠀⡠⠂⠉⠈⠀⠀⠀⡠⢖⠀⠈⠁⠀⠀⠋⢋⡠⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠑⠢⠤⠤⠤⠒⠈⠀⡐⠦⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⢰⡀⠀⠀⠀⠀⠐⡔⣮⡇⠀⠀⡀⠟⠛⠛⠿⠿⢿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠣⡀⠀⠀⠀⠀⡹⢉⠃⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⠛⠛⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣞⠀⠀⡠⠞⢀⠋⢂⠀⠁⠀⠀⠀⠀⠀⢊⡴⠈⠁
⢀⣀⡀⢀⣀⠀⠀⠀⠀⢄⠄⠀⣴⠹⢿⣤⡾⠗⠤⠎⠀⠀⠰⡀⠀⠀⠀⠠⢃⠎
⠈⠀⠉⠉⢹⡎⠀⢀⣬⣋⢌⣾⣅⡈⡾⡏⣀⠄⠀⢀⡄⠀⠀⠙⠀⠀⠀⠀⠎
⠀⠂⠤⠔⢃⢠⢼⠃⠀⠸⣔⣊⣛⣱⡵⣿⡦⡀⣠⠊⠀⠀⠀⠀⠈⢀⠀⠘
⠐⠦⣈⠀⠀⠀⠀⠉⠊⠋⠁⢃⡐⣦⠁⠀⠀⠈⠁⣀⡀⢀⣀⠀⠚⠙⠂⠂
⠀⠒⠬⣉⠐⠢⠄⠤⠔⠂⢉⡈⠵⠂⠉⠂⡀⠀⠀⠏⠁⠀⠩⡾
⠀⠀⠀⠀⠉⠒⠠⠠⢔⠈⠀⢀⠠⠐⠈⠁⠀⠀⠀⠂⠤⠤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠛⠉⣀⡴⠐⡖⠮⣄⡀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⠶⡿⠛⢀⣀⣉⣀⡘⠫⡓⠦⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⣠⠊⡁⠀⠀⠀⣩⠶⡈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⠁⠀⠙⢦⣤⠞⠉⠁⠘⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠸⡀⠀⠀⠈⠉⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠑⢄⠀⠀⠀⠀⢂⠄⢁⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠓⣦⣌⡁⣀⡉⢄⣶⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠈⠉⠉⠁⠀⠀⠑⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠖⠚⠋⢹⡄⠀⣀⡤⠖⠋⠉⣉⠛⠷⠶⠶⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⢀⣀⣀⠀⢷⣉⠀⣀⠴⠉⠊⠍⠀⠁⠀⠀⢀⣠⠼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⣰⠞⠉⠀⢘⣧⠤⠾⠋⠀⠀⠀⠀⢀⣀⡤⠴⠚⠋⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠇⠀⢹⡠⠴⠚⠉⠀⠀⠀⠀⣀⣤⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⠋⠁⣸⠤⠴⠒⠋⠀⠀⠀⢀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠚⣉⡤⠒⠺⢉⡍⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣰⣶⣏⣁⣀⡀⠼⠂⠡⠐⠊⠁⣀⣤⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠉⣟⠙⠛⠓⠶⢤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⣇⠀⠀⠀⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢻⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠇
⠀⠀⠀⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠈⠙⠛⠲⢿⣄⣀⠀⠀⣼⠀⣀⣤⠴⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠤⠤⠤⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠠⠀⠀⣀⠔⢊⠩⠔⠒⠃⠈⠘⠒⢄⡈⣧⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠈⠀⠈⠀⠰⣡⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣼⡆⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⢀⣠⣡⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣷⡀⠀⠤⢠⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠂⠉⠁⠀⠀⠀⠈⠢⡀⠀⠀⢀⠤⢤⠞⠉⠁⠀⠀⠉⠲⢤⡌⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⣼⣿⡇⠀⣀⣠⣤⢶⣶⣾⣿⡷⠀⠀⠀⠑⢲⡗⢤⣀⠀⠀⠀⠀⠀⠈⢑⣄⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣷⣾⣿⣆⢹⡘⣿⡽⠟⢶⢲⠲⠶⢶⣿⣶⣤⣬⣱⣦⣄⡀⠀⠀⣔⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⡿⢿⣛⡭⠖⠛⠋⠉⢉⡍⠛⠒⠯⣽⢿⣷⣿⣿⠛⠻⢿⡒⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡴⠊⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣟⢻⣅⠀⠀⠀⠙⢦⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢋⡴⠣⡄⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠐⠃⠈⢣⡙⣆⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢡⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⡧⠄⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠠⠄⢑⢻⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠅⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⢸⢠⡄⠀⠀⠀⢀⣠⣴⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⡇⣸⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠸⡄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⣼⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠘⢠⡇⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢷⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡖⣠⡎⡀⠀⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣽⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠴⢿⣷⣋⡄⠁⠀⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢳⣌⣒⣤⠼⣀⣀⣀⠠⠤⢤⠐⣐⣩⣶⠞⣋⣉⠁⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣤⣮⠟⠉⠛⠶⢿⣶⣾⣾⣿⣿⡻⠟⠛⠒⣰⠁⠿⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠛⠛⠛⡄⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⡈⣻⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⣀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢊⠈⠀⢀⠆⠃⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠄⠀⠀⣼⠀⠀⠠⢼
⠀⠀⠀⠀⠀⣠⡴⡶⣲⠖⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠂⠀⠘⠀⢀⠀⢀⠎
⠀⠀⠀⠀⠜⠃⠙⢁⠁⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠓⠬⠤⠥⠤⠚⠁
⠀⠀⠀⠰⠀⠀⠀⠙⠃⠐⠃⢸⠀⠀⠀⠀⠀⠀⠀⣔⣹⠋
⠀⠀⠀⠀⠣⣀⠀⠁⠉⢒⡠⠃⠀⠀⠀⠀⢀⠔⠋
⠀⠀⠀⠀⠀⠀⢹⣏⠉⠁⠀⢀⠀⢀⡠⠊⠁
⠀⠀⠀⠀⣀⠔⠉⡇⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⡠⠊⠁⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠙⢄⠀⠀⠉⠑⠒⠤⠤⠤⠤⢤⡀⠠
⠀⠀⠱⡀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⡌⢈⡁⠀⠉⠒⠤⢄⣀⣀⣀⡀⠀⠬⠛⠀⠀⠀⢀⣀
⠀⠀⠀⠈⠢⣀⢼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠱⡒⠤⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠐⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⡷⡶⢀⠀⠀⠤⠤⠤⣤⢰⡏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⣄⡑⠭⠤⠘⠃⠀⠀⠀⠀⠙⡇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢥⡂⠂⠤⣑⣄⠀⠀⠀⠀⠀⠣⢄⣀⣀⡌
⠀⠀⠀⠀⠀⢰⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡴⠁⠑⡀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⡄⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠛⢠⠤⢤⡞⡀⠀⠀⠑⣄⠀⠀⠀⠀⠉⠒⠒⢀⠄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠇⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠀⠁⠀⠀⠀⠈⠑⠢⠤⠄⠀⠠⣔⢋⣀⠤⠒⠉⠣⣵
⠀⠀⠀⠀⠸⠀⠀⠀⠈⡄
⠀⠀⠀⠀⡂⠀⠀⠀⠀⠡⣀⣀
⠀⠀⠰⠭⠭⠄⠀⠀⠀⠀⠳⠞⠁⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤

⠀⠀⠀⠀⠀⢀⣤⣤⣠⡖⢲⣄⠀⠀⠀⢠⡴⣦
⠀⠀⠀⠀⢠⡿⠀⣀⡀⠀⠀⠉⢛⣶⡴⠋⠀⢸⠇
⠀⠀⠀⢠⡞⣡⠞⠹⡇⠀⢀⣾⡿⠋⣴⡗⠀⣾
⠀⠀⢠⡟⣰⠃⠀⠀⢀⣴⣿⠟⣡⢞⡼⠁⣸⠇⠀⠀⣀⣤⣴⣤⡤⠤⠤⢤⣤⣶⠿⡆
⠀⢠⡿⠁⠏⠀⠀⣠⣿⡟⢁⡴⢁⡾⠁⣰⠏⣠⢶⣻⡿⠛⠉⠀⠀⠀⠀⠀⣾⡏⠘⠷⣦⣀
⠀⢿⣄⠀⠀⠀⣰⣿⠋⣠⠊⢀⡞⠀⡴⠣⢞⣵⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢿⡁⠀⠀⠈⠻⣷⡀
⠀⠀⠙⠳⢦⣴⡿⢁⡞⠁⣠⠏⢀⡞⢁⣵⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⣿⠁⣾⣀⡴⢃⡴⠋⠀⢪⢾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠞⣧⣀⣀⣀
⠀⠀⠀⠀⠀⣿⠀⠈⣁⡴⠋⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠙⡇⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⡤⠞⠁⠀⠀⠀⠀⣇⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢠⠞⣭⣥⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⣿⠀⣼
⠀⠾⣿⡿⠋⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠘⣧⠀⠀⠀⢸⡄⠀⢤⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣯⣀⣭⣹⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢀⠛⢦⣄⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⣸⠁⠀⢀⡀⠙⢳⡼⣄
⠀⠀⠘⠛⠛⠋⠀⠉⠩⠤⠤⠤⠴⠖⠒⠒⠛⠈⠙⢯⡀⠀⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⢀⣯⠀⠀⢸⣟⡖⢋⠇⡼
⠀⠀⠀⣀⣀⣀⣀⣀⣠⣤⣤⡤⠤⠤⠀⠒⠀⠀⠀⠀⠉⠓⠦⣤⣤⣤⣤⣄⠀⠀⠀⣀⣠⣤⠞⠉⠀⠀⠀⣀⣠⣴⣿⠀⠀⠀⠞⠁⣹⠳⠞⠁
⠀⠀⠘⠳⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣝⣲⢯⡻⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣦⣀⣰⡖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡗⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠲⠶⠖⠒⠒⠒⠒⠒⠒⢒⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣿⠉⠉⠉⣉⣭⣿⣷⣶⣶⣄⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣷⠀⣀⡤⢤⣀
⠀⠀⠀⠀⣴⣚⡶⠒⠒⣶⠋⠀⢻⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⢁⢔⡯⠛⣻
⠀⠀⠀⣰⠏⠉⢧⡀⣴⠁⠀⠀⠀⠙⠿⠿⡟⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡧⡶⠋⠀⢀⡿
⠀⣀⡾⠁⠀⠀⠈⢳⡙⣾⠀⠀⢠⠒⢦⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡿⠋⠀⢱⠁⠀⢀⣼⠃
⢺⣹⠀⠀⠀⠀⠀⠀⢳⡝⠻⠦⣨⠃⡾⡅⠀⠀⠀⠀⠀⣀⢤⡄⣀⠀⢙⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⣿⣿⣿⡿⠁⠀⣶⠃⢀⡴⠞⠁
⠀⠙⣧⡀⠀⠀⠀⠀⠀⠙⣦⠀⠙⢺⡀⠙⣆⡀⠀⠀⣞⡱⣋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⢹⣿⣿⠁⢠⢾⡋⣀⣹⡄
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢸⡟⡦⢼⠗⢺⠷⠁⢀⡠⢫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠻⢿⣰⠇⡼⠉⢁⡾⠁
⠀⠀⠀⠘⢷⡀⠀⠀⠀⢠⡞⠉⣡⠋⣰⠋⠀⣰⡏⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣧⣤⠞⠁
⠀⠀⠀⠀⠈⢷⡀⢀⡴⠋⢀⡔⣡⢺⡇⠀⢠⣿⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠸⣯⡽⠷⠚⠋⠁⠈⠉⠀⠀⠸⡟⡀⠀⠘⢛⡟⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣦⣤⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣥⣄⣀⡐⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡴⠟⠋⠉⠉⠉⠉⠛⠿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣸⠃⠀⠀⡀⠀⠐⢶⣶⡆⠤⣄⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣰⢄⠀⢀⢸⣹⡇⣶⣬⣇⠀⢨⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⢿⣿⣷⣿⣿⣿⣿⣿⣿⡿⠀⢂⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⡀⠀⣿⠛⣿⠛⢻⣿⣿⡏⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⣻⠀⣇⠀⢸⣿⣿⣇⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⢀⡋⣀⣐⣛⣟⣋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠟⠛⠻⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠿⠤⠐⠶⠖⠾⠷⠷⠤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⠉⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣤⣄
⠀⠀⠀⠀⠀⢀⢤⡶⠂⠀⠀⠀⢀⠤⠒⡉⠍⠄⠀⠠⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠳⣧⣤⣀⡀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⣠⠒⠁⢀⠀⠀⠀⠀⠀⠸⠀⢿⠀⠀⠀⠀⠀⠀⡀⠀⠛⢿⣇⢄⡀⠀⠀⠙⢿⣄
⠀⠀⠀⢠⠟⠀⠀⠀⣠⠀⡠⢊⠐⠀⠈⢦⡳⡀⠀⠀⠀⠀⣆⣸⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠙⢷⣕⣦⠀⠀⠈⢻⣦
⠀⠀⢀⠋⠀⠀⠀⣴⢃⠞⠁⠀⠀⠀⠀⠀⠑⡷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠄⠙⣿⣿⣄⡀⠈⢻⣧
⠀⠰⠏⠀⠀⢀⡞⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣠⣤⠀⠘⣿⣿⣿⣷⡈⢿⣆
⠀⡀⠀⠀⠀⠌⠀⠀⠀⠀⠈⠙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠋⡴⠟⠋⠀⠀⠀⠘⣿⣿⣿⡷⠸⣿
⠈⠃⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣟⠀⣻
⠆⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⣿⠁⢸
⠀⠀⠀⠄⣿⠀⠄⢠⣒⡒⠀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆⣸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿⢷⠀⠀⣿⡇⢹⣇⢨
⠀⠀⠀⢻⣹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣿⣟⡘⠇⢸
⢠⠀⠀⠈⣮⣇⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⣸⣿⠷⡏⠀⢸
⠀⡆⠀⠀⢿⠟⡄⠀⠀⣤⡲⡫⢵⠆⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠒⠄⠀⠀⠀⠀⠀⠰⢿⣷⣦⢦⡄⠀⢠⣿⡿⠾⠋⢀⣾
⠀⠈⠀⠀⢹⣧⢹⣆⣀⠙⣫⠔⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠉⠒⠙⠁⣠⡟⡟⢡⠋⢀⣾⠃
⠀⠀⠀⢀⠀⣻⣦⡛⡗⢀⠉⠀⠀⠀⠀⢀⣾⡃⠀⠀⣀⠀⣷⣤⠀⠠⠄⢀⢠⡀⠀⠀⠀⠀⠀⠀⣴⣏⠟⡴⠃⠀⣼⠋
⠀⠀⠀⠀⣦⠘⢿⣷⣌⢻⣷⣄⠂⠀⠠⡾⢉⠁⠀⠀⠉⠀⠉⠁⠀⠀⠀⡀⠀⠙⢆⠀⠀⢚⣤⣿⡷⣣⡾⠀⢀⡜⠁
⠀⠀⠀⠀⠈⠃⢈⠻⣿⣷⣍⡻⣷⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⡀⠀⠀⠈⠀⠀⠀⠀⣠⣼⣿⠿⣫⡴⠋⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢦⣼⣿⣿⣿⣧⣝⡻⠳⠲⠤⠀⣀⡀⠀⠀⠅⣀⣀⣀⣤⣴⣖⣭⠮⣋⣡⣾⠛⠀⣠⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠚⢻⣿⣿⣿⣿⣾⣯⣤⣄⣀⡀⠀⠽⠛⠛⠛⣛⣛⣋⣩⣶⣾⠿⠋⣁⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣛⣭⡵⠒⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⢦⡀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⣠⡤⣄⠀⠀⠀⢀⣤⠤⣄⠀⠀⠀⢀⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣾⣷⣶⠶⠿⠶⠶⣿⠶⠶⠾⠷⠶⠶⢾⠿⠶⠾⠷⠶⠶⣿⠷⠶⢾⣷⣶⣶⣿⣥⣤⣽⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠘⢷⠄⠀⠀⠀⠀⠹⠦⠀⠀⠀⠀⠀⠘⢷⠄⠀⠀⠀⠀⠙⢶⠀⠀⠀⠀⠀⠳⣤⠀⠀⠉⢹⡿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣧⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣸⡇⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⢀⠤⠤⠤⣄⢠⠤⠤⠤⡄⡤⠤⠤⢤⢀⠤⠤⠤⢄⡠⠤⠤⠤⣀⠤⠤⠤⡄⠀⢸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⡿⢤⠀⠀⢀⡇⣇⠀⠀⢸⠠⡀⠀⠀⡸⢇⠀⠀⢀⢿⡀⠀⠀⡧⠀⢸⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡎⠉⠀⠐⣴⠉⠉⠉⡇⠘⠉⠉⠙⡆⡎⠉⠉⢹⢠⠋⠉⠉⢱⡖⠉⠉⠙⣶⠋⠉⠉⡇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠧⠤⠤⠤⠛⠤⠤⠤⠇⠰⠤⠤⠤⠃⠣⠤⠤⠼⠀⠦⠤⠤⠜⠣⠤⠤⠤⠛⠤⠤⠤⠇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⡏⠀⠀⠈⣾⠉⠉⠀⡇⠈⠉⠉⠈⠃⡁⠈⠉⢹⢰⠁⠉⠉⢱⡏⠉⠉⠈⣾⠉⠉⠉⡇⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⡵⠶⠶⣖⡹⠶⠶⢶⢃⠴⠶⠶⡔⢁⠧⠴⠶⡞⣨⠦⠶⢶⠊⡱⠶⠶⡔⡹⠦⠶⢴⠃⠀⣼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⢸⡀⠀⠀⣿⡇⠀⠀⢸⢸⠀⠀⠀⣸⢠⠀⠀⠀⡇⣇⠀⠀⢰⢸⠀⠀⠀⡯⡇⠀⠀⠸⠀⢀⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢠⠊⠙⠋⡋⡔⠛⠛⢻⢥⠒⠛⠛⢫⡔⠒⠛⠛⡯⠔⠒⠛⢳⠁⠀⠉⠉⠉⠀⠉⠉⠉⠁⠀⣼⢹⣿⣿
⠀⠀⠀⠀⢠⣼⠏⠀⠀⠳⠤⠤⠤⠃⠧⠤⠄⠼⠸⠤⠀⢀⠜⠅⠀⣀⡠⠇⠄⣀⣀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⢸⣿⣿
⠀⠀⠀⠀⠈⠉⠛⠛⢳⡶⠲⠶⠶⠤⢤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣇⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⣉⣉⣉⣉⣉⣉⣉⣀⣠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣿⣿⣭⣭⣭⣭⣭⣭⣽⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣀⣀⣀⠀⢀⣤⣤⣄⠀⣠⠤⠤⣄⣠⠤⠤⣄⢠⠶⠶⢦⡀⢠⠖⠛⠲⣄⣰⠚⣛⣳⣄⣞⣋⣙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣉⣉⣙⣦⣾⡥⠤⢼⡷⠿⠶⠶⣾⠷⠛⢒⣚⡟⠛⠛⣛⡟⠋⠉⣩⡿⠋⠉⠉⢉⣽⠋⠉⠉⢩⡟⠉⠉⢉⣽⠋⣧
⠀⠀⠀⠀⠀⠀⣠⣴⡟⣿⡉⢡⡴⠃⠀⠀⠰⠟⠀⠀⠀⠛⠃⠀⠀⠘⠁⠀⠀⠀⠉⠀⠀⠀⣈⣀⡀⠀⣤⡀⠀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⢀⡏⠸⣧⣿⣇⠀⠀⠀⠀⠀⠀⣠⠴⠆⢀⣶⡄⢰⡆⠀⢸⣯⠅⠀⣿⢦⣹⠀⣯⢈⡷⢰⡧⢿⠀⣷⠾⣇⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠇⠀⣿⣾⣿⠀⠀⠀⠀⠀⠀⢿⣀⡄⢸⠛⠷⠈⠧⠤⠘⠓⠒⠀⠛⠀⠙⠀⠙⠋⠁⠈⠀⠈⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡀⢸⡆
⠀⠀⠀⠀⠀⡿⠀⠀⢹⣿⡟⡇⠀⠀⠀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠤⠴⠶⡶⠖⠒⠒⠛⠛⣿⠛⠛⢉⣉⣹⣉⣉⣉⣭⣯⠤⠤⠤⢽⠤⠤⠼⡇⠀⣷
⠀⠀⠀⠀⢰⠇⠀⠀⠘⣿⣷⢻⠀⠀⢸⣍⣡⣤⡤⠤⠼⡦⠤⠴⠶⠒⠒⢻⠒⠒⠒⠒⠛⢻⠉⠉⠉⠉⠉⡇⠀⠀⠀⣸⣀⡀⣀⣘⣆⣀⣀⣻⠀⢸⡄
⠀⠀⠀⠀⡿⠀⠀⠀⠀⢿⣿⡿⡇⠀⠀⣧⠀⣀⣀⣀⣀⣷⣀⣤⣤⠤⠤⠼⡧⠤⠶⠒⠒⠚⡗⠒⠛⠋⠋⢻⠉⠉⠉⠉⡏⠉⠉⠉⣿⣉⣁⣘⡆⠀⣧
⠀⠀⠀⢸⠇⠀⠀⠀⠀⢸⣿⣿⢿⠀⠀⢸⡍⠉⠁⠀⠀⠸⡆⠀⠀⠀⣀⣀⣷⣀⣀⣠⡤⠤⢿⠤⠴⠶⠖⠺⡖⠒⠒⠛⢻⠉⠉⠉⠹⡍⠁⠀⢻⠀⢸⡄
⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⣿⣿⡿⡇⠀⠀⣷⠴⠒⠒⠒⠚⢿⠉⠉⠉⠉⠉⠸⡀⠀⠀⠀⢀⣸⣄⣀⣠⣤⠤⣧⠤⠤⠴⠾⡖⠒⠒⠚⢿⠉⠉⠙⣇⠀⣷
⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⢹⣿⣧⣷⠀⠀⢹⣀⣀⣀⣀⣠⢼⡤⠴⠒⠒⠒⠛⣟⠉⠉⠉⠉⠀⣇⣀⣀⣀⣀⣸⣤⠤⠤⠤⢷⠒⠒⠒⢻⡖⠛⠋⢻⡀⠸⡆
⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠸⣿⡿⣾⡆⠀⠈⣏⠉⠀⠀⢀⣀⣧⣀⣠⠤⠤⠶⠺⡖⠚⠉⠉⠉⢹⡉⠀⠀⠀⠈⣇⠀⠀⣀⣸⣀⣠⣤⠤⣧⠴⠶⠞⣇⠀⢷
⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣇⣧⠀⠀⢹⡛⠉⠉⠉⠁⢹⠀⠀⠀⠀⢀⣀⣧⣀⣤⠤⠤⠼⠷⠲⠒⠒⠚⢻⡋⠉⠁⢸⣇⣀⣀⣠⣽⣤⣤⣤⠿⠄⢸
⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⢸⡄⠀⠈⡷⠶⠶⠖⠒⠚⡏⠉⠉⠉⠉⠀⢹⡀⠀⣀⣀⣀⠶⠤⠴⠶⠶⠾⠓⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⣀⣀⣀⣤
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡎⣇⠀⠀⢻⠀⢀⣀⣀⣤⡽⠤⠴⠶⠒⠒⠛⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠶⠶⠶⠚⠛⠛⠋⠉⠉⠉⣧
⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⢻⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡤⠤⠶⠖⠒⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⢸⠁⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠖⢿⣿⡼⣇⣀⣀⣤⣤⡤⠤⠶⠶⠒⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠤⠶⠖⠚⠛⠉
⣿⣤⣶⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠸⡟⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠶⠖⠒⠛⠋⠉⠉
⠀⠀⠉⠙⠓⠲⠦⣤⣀⣀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⠤⠶⠖⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠶⠤⣼⣧⡤⠤⠶⠖⠚⠛⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢡⣶⣪⠃⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⡉⢁⣀⣀⣼⠏⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⠭⣅⢀⣴⠾⣧⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⡜⢡⣴⡄⠛⢸⡟⢀⡾⠀⠀⢩⢿⣿⡀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡷⢾⡟⠀⠀⣠⠝⠏⠓⠲⣾⡗⠚⠀⢹⣦⡀⣠⠖⢉⣠⡉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢾⠛⣿⣟⣷⣌⠁⣠⡴⡆⠀⠀⠙⣦⣶⣻⠟⣷⡏⠀⠃⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠑⢾⣿⢱⣿⣏⣹⠋⠶⠽⠞⠁⠀⠀⠀⠉⢧⠀⣀⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣀⡈⣋⣛⣣⣼⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡿⠿⢿⡏⠃⡀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠻⣟⢋⡟⢠⡮⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣸⡿⡀⠈⠳⢦⣤⣀⣀⣤⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⠽⠃⢸⣸⠇⠀⠁⠀⠀⠀⠀⠀⠀⢠⡶⢀⣻⢯⣤⠿⢇⠁⠀⠀⢀⠀⡠⢻⣦⣴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠳⠦⡀⠀⠀⠀⠀⠀⠀⠈⢙⠟⣫⣻⣶⡖⣠⣶⡿⠿⠀⠐⠣⢾⡛⣋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣾⣷⣦⡈⠀⠀⠀⠀⠀⠀⠀⢸⡀⠛⢛⣁⡉⠟⣇⠤⠤⠶⠖⠚⠛⢉⣩⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠋⠑⠶⠉⠙⠒⠿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠙⠻⢯⣹⠿⠃⠛⠛⠀⠀⠀⠀⠀⠻⠿⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠙⢀⡄⠀⠇⠀⠀⠀⠀⠘⢧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠈⠀⠀⠀⠀⠀⢀⡴⠒⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠘⠿⠞⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠘⠞⠀⠀⣶⣆⡴⠾⠗⠉⠈⠀⡅⠀⠀⠀⠈⠣⡀⠀⠀⠀⣠⣤⠀⢱⣤⢯⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡀⠀⠀⢴⠟⢿⠀⠀⠀⢠⡄⢠⠃⢀⠃⠀⢴⠶⢷⠛⠀⠈⠉⠉⢿⡟⣃⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠀⠀⠀⠀⠀⠸⣷⠀⠀⠈⣡⠃⠀⠀⠀⠀⠀⠂⢈⢣⠀⠀⠸⢻⣧⣿⣙⣛⡤⠀⠀⠀⠀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⣧⡀⠻⠀⢠⠀⠀⡀⠈⠀⣣⠞⠁⢀⣼⡀⠠⡀⠀⡷⢸⠠⢀⣠⠞⠋⢀⡟⢉⣽⣋⠳⡆⠀⠘⠿⢇⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡀⠈⠳⣄⣀⠀⠀⢀⣣⣴⡾⡁⢀⣴⣿⣿⣿⣶⡄⣷⣾⡏⠸⣇⣶⠄⢤⣶⣿⣟⣻⣿⣿⣥⠤⡴⠚⢉⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢥⣀⠈⢩⣛⡋⠯⢿⣽⣡⣾⣿⣿⣿⣿⣿⣬⠧⣿⡿⡱⠁⠘⠿⣟⠉⡺⢿⣤⣀⣄⣄⡀⣾⢺⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⡿⠁⣰⠿⡷⣹⠿⣧⣼⠟⠋⠉⠉⠀⠉⣈⠁⣌⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⠓⠊⠁⠀⢸⣏⠀⠀⠀⠀⣴⢋⣀⡀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⣷⣌⣋⡽⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣈⠓⣠⠏⠁⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣷⣶⣴⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢯⣝⣧⡶⣠⣞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠈⠉⠉⠓⠲⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠈⣎⢧⠠⠤⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⣠⠤⠴⠖⠀⠀⠸⡘⡀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⣫⠽⢭⡉⠙⠲⢤⡀⠀⢃⢇⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⢀⡴⠊⠀⢠⠋⠀⠀⠀⢹⠀⠀⠀⠙⠢⡐⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⡝⠔⡴⢋⡤⢄⠀⡇⠀⠀⡀⠀⢸⠀⠀⢀⡆⠀⠙⣿⡀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⣤⡉⠁⡞⠀⢸⠀⠀⠇⠱⣄⠀⡵⣊⠉⠉⢱⠈⡇⠀⠀⠘⡏⠈⠉⠉⠒⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣏⡙⠂⢀⡀⠁⠂⡄⠀⠀⣹⡀⠉⠁⢀⠜⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⣀⣀⢸⠙⢻⠁⠀⠀⠀⠀⠀⠉⠉⠒⠒⠉⠀⠀⠀⠰⣆⠉⠺⡀⠀⠀⠀⣧
⠀⠀⠀⢀⡠⠔⠊⠀⢯⠁⠀⠱⡤⣊⡥⢤⡼⠋⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠥⠤⠤⣇⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄
⠀⢠⠒⠁⠀⠀⠀⠀⠀⠣⣀⢆⠸⠁⠀⠘⠦⢄⣀⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠙⠯⠭⢦⠀⣿⢉⡋⠁⠀⠀⠀⠀⢀⡀⡠⠋⣢⠃
⠀⠀⢣⠀⠀⠀⠀⡀⣀⣀⡴⠞⠙⠀⣀⣤⠖⠮⣄⣀⠀⠀⠀⠉⠉⠉⠛⠛⠛⠉⠉⠙⢭⡉⠓⣦⣤⣤⠴⢧⢀⠈⠉⠓⢦⠀⠀⢸⡶⣁⠜⠁
⠀⠀⢠⠗⠒⠶⣄⠜⠁⠀⠙⢦⡴⢮⣉⠙⠻⢶⣦⣭⣙⠒⠦⢤⣀⣀⣔⣱⣄⡤⠴⠎⣭⣲⣶⡟⠂⠚⠋⡡⠛⡄⠀⠀⠈⡆⠠⣎⡹⠋
⠀⠀⢻⡈⠉⠉⢹⡆⠀⠀⠀⠀⠙⢦⡈⠛⠲⢤⣄⣉⠙⠛⠷⢶⡶⣿⡟⢁⣼⠤⠀⠰⠏⠀⠈⠑⡄⣠⠎⠀⠀⠐⡀⠀⡰⠃
⠀⠀⠘⣆⠀⠀⢸⡗⢀⠎⠙⠢⣤⠋⡨⠶⠤⣀⠀⠉⠙⠛⠒⡾⠁⠙⠃⡾⠉⡞⡓⠃⠒⠦⠤⠴⠟⠁⠀⠀⠀⡠⣷⣞⣀⣀⠤⢤
⠀⠀⠀⠘⣆⣀⣸⣷⠋⠀⠀⠀⠀⠙⠢⡀⠀⠀⠈⠉⠉⠉⠻⡥⣤⣀⠀⠑⢾⢱⣀⡠⢴⣶⠉⡑⢢⡀⠀⠀⠪⢄⡀⢌⡉⠉⠈⠉
⠀⠀⠀⠀⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢱⠝⡾⠻⣆⠈⠲⠥⠶⢅⣈⠉⠉⠉⠁⣀⡤⠔⠋⠀⠐⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢤⣤⣤⡤⠔⠻⡎⠑⠒⢻⢈⠖⠆⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠴⠋⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠑⣾⣿⣾⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠚⢌⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠈⠀⠁⠀⠀⠄⠀⢀⠔⡠⠀⡴⢿⡟⠁
⠀⠀⠀⠀⠀⠀⡄⡒⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢁⠈⣰⠎⡠⠋
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⢀⡀⠄⠤⠠⠤⠀⠒⠊⠁⠀⠀⠀⠠⠈⠀⢁⡴⢃⠜⠁
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠈⠋⠈⠉⠁⢀⣀⣀⠠⠄⠀⠐⠂⠀⠀⠀⠔⠁⢊⡠⠃⣠⠋
⠀⠀⠀⠀⠀⠀⠙⠀⠐⠀⠀⠒⠂⠁⠁⠀⢀⣀⣀⡀⡀⠤⠒⡰⠀⠔⣠⢟⢀⠞⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠒⠂⠊⠁⠀⠀⠀⠀⠀⢀⠀⠠⢲⡴⣏⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠤⠤⠤⠀⠂⠉⠁⠀⢂⠀⢠⡎⣠⠯⠤⠄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠤⠤⠄⠐⠂⠸⠣⣀⡴⣧⠞⢁⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠠⠤⠀⠐⡆⠀⡠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢆⠀⠀⠤⠤⠴⠟⠡⠀⠀⠀⠀⢀⢀⣀⣀⠀⠀⢰⠖⠒⠚⠻⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣦⡀⠀⠀⠀⣀⣀⠤⠔⠒⢛⣋⣉⣩⣥⣥⣤⣶⠾⡿⠽⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡷⣝⣒⣪⡭⠤⠖⢒⠚⠋⠭⢍⣀⣀⣠⣤⣤⣔⣶⣶⡾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣸⠛⢉⣐⣉⠠⠎⠁⠚⠋⠏⠋⠀⠂⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠄⠄⠤⠄⠀⠀⠀⠀⠀⠄⠔⠔⠐⠒⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠁⠀⠀⠀⠀⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠉⠉⠛⠶⣄⠀⠀⠀⣴⡶⢦⡀⠀⣠⡴⠖⠒⠒⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢣⠀⠀⠀⠀⠀⢀⣤⠟⠀⠀⢠⣾⡟⠉⠀⠀⠹⣄⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠙⠂⢀⣠⡴⣞⣩⡤⠶⠒⠚⠛⠛⠓⠒⠶⢤⣄⡈⠳⣤⡀⠀⠀⠀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⠶⠚⣩⡵⠛⣩⡤⠶⠚⠛⠉⠉⠉⠉⠙⠓⠲⢤⣍⠓⢦⣝⠲⣄⡈⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣌⠳⣄⠉⠓⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠹⡄⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡇⠀⠀⢹⡄⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣼⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣄⠀⠀⠀⠀⣠⣾⡿⠟⣿⠟⠀⠀⠀⠀⣧⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⢿⣷⣤⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⡄⢻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢇⡟⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡼⢁⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢧⡀⠹⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣌⠳⣄⠘⢮⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⣡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠓⢦⣙⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⣡⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠓⢦⣌⡙⠲⠦⢤⣤⣤⣤⣤⠤⠴⠖⢋⣡⢴⣞⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡴⠛⠉⠉⠛⠒⠶⠶⠦⠴⠶⠶⠒⠛⠉⠀⠀⠈⠳⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣫⣴⠾⠛⠛⠻⣿⣿⣤⠤⠴⠶⠤⢃⠈⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⠟⠁⠀⠀⠀⠀⠈⣏⢹⠦⣀⢀⠀⠠⠀⢙⣢⠀⠰⠊⠉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠙⠋⠀⠀⠀⠀⠀⠀⠠⢈⣁⡠⢤⠳⠒⠃⠉⠻⣿⠠⠰⠰⠿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠤⠈⠁⠀⠀⠀⠀⣀⠁⠀⢐⣀⣠⣱⡶⢻⠛⢹⠸⠃⠀⠀⠀⠀⢀⠴⣿⠀⠀⢀⠴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⡀⠀⠀⠀⠀⠀⢀⠀⠄⢐⣢⣄⠀⢱⣶⢿⡛⠋⠉⣈⠡⠀⠀⠘⠀⠀⠀⣤⠀⠀⠈⢦⣻⢀⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡿⣸⠀⠀⠀⢀⠔⢈⣥⣶⣿⣿⡿⡿⠐⠸⣏⠐⠁⠄⢈⣴⡶⠈⠀⠀⠀⠀⠀⠂⠑⠄⠀⠀⢹⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢽⠀⠀⠀⠎⢠⣿⣿⡿⣯⣿⣧⣿⠠⠀⡏⢀⠀⣰⣾⡿⠁⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠀⠀⠸⢸⣿⣿⣟⡿⠋⢿⣷⠶⠚⡇⠀⠀⠟⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠑⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠘⠀⠀⠀⠸⠀⣿⡿⡿⠇⠀⠈⢿⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣴⠖⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣿⣇⣟⡠⠒⠦⢼⠆⠰⠃⠀⠀⣟⣃⡒⠤⢀⠀⠀⠀⠉⡅⠀⠀⠀⡀⠀⢘⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⡄⣿⣿⡜⢱⡁⠀⢸⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠑⠃⠀⠀⠓⠀⠢⢄⠘⠆⢸⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡄⣿⢻⡆⠮⡁⠤⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⢸⠀⠀⠀⣤⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡀⢻⣨⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⢈⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⡁⢸⡇⡀⡀⠀⠀⠠⠀⠀⢄⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣤⣴⣿⣿⣟⠿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠀⠀⠀⠀⢸⣿⢁⡇⠀⠀⣰⡆⠀⢺⠉⠢⡄⠠⠐⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠟⠏⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⡀⠀⠀⢸⠈⣿⣿⣿⡶⠀⡿⡇⠀⢸⡄⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⡿⠛⠿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣶⣤⣀⠸⠅⣿⣻⣿⣇⡆⡇⣿⠀⢸⣷⣀⣠⣴⣶⣿⣿⣯⡛⠲⢭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡉⠛⠿⢿⣦⣿⣿⣿⣩⣛⣧⣽⣤⣾⣿⣿⢿⣿⣿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠦⢶⠝⠻⠿⣿⣿⣿⣿⣿⣿⡿⢿⡻⣷⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⢁⠀⠀⠀⠉⠋⠛⠙⠢⢄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⣇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢺⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠸⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠞⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐⠒⠂⠈⠑⠀⠀⠀⠢⠀⠉⠙⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⢰⠰⡇
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠐⢸⡇
⠀⠀⠀⠀⠀⠀⢸⠂⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⠁⠀⠀⠸⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⡁⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⢄⠀⠀⠀⢀⠀⠀⠠⢀⠀⠀⠀⡀⠀⠀⠄⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠁⠀⠀⠈⠁⢸⡁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⢀⠀⠀⡂⣀⠀⠀⢂⠀⠀⠐⢠⠀⣀⡐⣀⠀⠀⢆⠀⠀⠀⢀⡀⢸⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⢀⠀⠀⠂⣀⢀⣐⡨⠤⡀⢠⠄⠀⠀⠤⡄⠀⠠⢌⠀⠀⠀⠤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠀⠁⠀⠀⠀⡄⢈⢁⠴⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⢰⡇
⠀⠀⠀⠀⠀⠀⢀⠄⠀⠐⢀⠀⠀⠒⡀⠀⠀⢊⡀⠀⣀⣚⠄⠀⠄⠀⠀⠀⠀⠀⠀⠀⠄⡀⢸⡇
⠀⠀⠀⠀⠀⠰⣀⣀⣀⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⢈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⢸⡇
⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢞⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣰⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢞⡵⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⣵⣟⡓⠒⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠐⠂⢁⠜⣡⡾⠍⠉⠉⠑⠀⠀⠉⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⠛⠛⣿⠟⢿⣿⠿⡿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⢉⡠⠠⠐⠂⢉⠴⣡⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣀⠀⠀⠀⠀⠀⣀⠀⠀⣿⣿⣄⢀⢠⠀⠀⢩⠀⢘⡟⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣠⠞⠁⠀⠀⢀⡴⣢⡾⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⣤⣼⣿⠉⠁⠀⠈⢹⡞⠀⠀⠠⠀⡆⣞⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⡼⠁⠀⠀⣤⡴⢋⣴⣯⣷⣇⠂⠐⠀⢀⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢧⣀⠀⢃⡄⣿⣼⣿⠆
⠀⠀⠀⠀⠀⠀⠀⢀⣞⡵⠋⠀⠀⢀⡼⢋⣴⣿⣿⣿⣿⣿⠿⡿⣶⣿⣿⣴⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⠇⠀⠎⡇⣏⢹⣿⡄
⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⢠⣤⠞⣵⣿⡿⠿⣏⣧⣾⣯⣴⣵⣩⠥⣿⣿⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⣤⡷⣿⢸⣿⢧
⠀⠀⠀⠀⠀⢠⡏⠀⢀⣠⣾⢞⣵⠞⣛⠛⠿⠛⡛⢿⡯⢍⣙⡛⠿⢾⣿⣿⣿⣇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⣧⠀⣯⢱⡟⣿⢿⢸
⠀⠀⠀⠀⢠⣏⠀⠀⢀⡿⢫⣿⠁⠀⠀⣹⠁⢿⠋⠁⠈⠂⠉⠉⠉⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⡠⢤⠀⠀⢸⣿⣦⣽⡈⣧⣿⣾⡇
⠀⠀⠀⣠⠟⡘⠀⢀⣾⣵⠏⠸⣦⣖⡒⡛⠀⢟⣆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠣⠜⠀⠀⠸⣸⡿⣿⣇⣿⣿⡿⠅
⠀⠀⠀⢧⣊⣤⣶⡿⣿⣿⠁⠀⠙⢿⡙⠛⣿⣿⣿⣾⡅⠀⠒⠓⠒⠲⠤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⡴⢤⣦⣼⠀⠀⠀⠀⢠⣟⣿⣿⣾⣿⣷⡿⣳
⠀⠀⠀⣠⠟⣩⣎⣴⡿⠃⠀⠀⠀⠀⠉⠳⣬⣉⣿⣿⣷⣶⣶⣶⣴⣶⣶⣦⣽⣿⣿⣿⣮⣶⣶⣶⣾⣿⣿⣦⠤⡄⣄⠉⣉⣙⠛⣛⠿⢾⣾⣿⣽⣿⣯⣿⠾⠋
⠀⢀⣴⣷⠞⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⠿⣿⢿⡿⠧⠬⠉⠭⠿⠽⠏⠛⠛⣿⠛⠛⠛⠿⠉⠉
⢀⣾⠟⠁

⠀⠀⠀⠀⣀⡴⠖⠛⠋⠉⠙⠛⠲⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀
⠀⠀⢀⡾⠋⠀⠀⠀⠀⢠⠀⠀⠀⠀⠙⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠛⠛⢻⠛⠛⠛⡟⠋⠉⠻⠛⠉⠙⣿⡇
⠀⢀⡟⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣾⣿⣿⣿⣿⣇⠀⠀⠀⠀⣿⡇⢹⠀⢸⠈⣹⠂⡧⡎⢹⠤⢰⠉⡧⣽⡇
⠀⣸⠃⠀⠀⠀⠀⠀⢀⣬⡀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⣿⡇⠈⠀⣸⠘⠓⢀⣇⠙⠊⢀⡈⠓⠁⣾⡇
⠀⢹⡄⠀⠀⠀⠀⠀⡸⠛⠁⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⣿⣿⣿⡿⢿⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠘⣧⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⢀⣼⠟⠡⠀⠐⠀⠠⠉⠻⣆⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⣾⠁⠂⠀⠀⠀⡔⠀⠀⠂⠘⣇
⠀⠀⠀⠀⠙⠳⢦⣤⣄⣀⣠⣤⡴⠞⠋⠀⠀⠀⠀⣿⠐⠀⠀⠀⠚⠁⠀⠀⠐⠂⣿⠇⠀⠀⢀⡴⠚⠋⠉⠉⠉⠉⠛⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠂⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢠⡀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⣀
⠀⠀⠀⠀⠀⠀⡴⠚⠛⠓⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣌⡀⠀⠀⣈⣠⣾⠋⠀⠀⣨⣿⣿⣿⣷⣦⠀⢤⡤⠀⣠⣶⣶⣿⣾⡃
⠀⠀⠀⠀⠀⠀⠳⢴⣿⣿⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⢟⣩⣤⣼⣧⣤⣝⡻⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⢀⣠⣤⣿⣧⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠀⠀⠀⠘⠟⠋⣡⣿⠟⡋⠉⠉⠉⠉⢙⠿⣿⣍⠻⢿⠇
⠀⠀⠀⢠⣾⠟⠋⠁⠀⠀⠉⠙⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠇⠀⠀⠀⠀⢀⣸⡛⣅⠀⠉⠀⠀⢀⡴⠋⠀⣘⣿⣧
⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⠀⠀⠀⠀⠀⠀⡾⠁⠀⠈⠁⠘⣿⡇
⠀⠀⣿⠇⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠉⠁⠐⠒⠒⠚⠁⠀⠀⠀⠈⠁⣿⡇
⠀⠀⢿⡇⠀⠀⠀⢠⠎⠁⠀⠀⠀⠀⣸⡏⠀⣴⣶⣶⢰⡶⣶⡄⣶⢶⣦⣴⡶⣶⠀⠀⠘⣿⡄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠢⣼⡿
⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⣿⣾⣿⢸⣿⣼⡇⡗⣿⣿⣟⣿⢺⠀⠀⠀⠘⢿⣦⣀⠂⠀⢠⠀⠀⢣⣠⣾⠟⠁
⠀⠀⠀⠈⠛⢷⣦⣄⣀⣀⣤⣴⡾⠋⠀⠀⠀⠿⠿⠿⠸⠶⠾⠃⠿⠾⠟⠻⠷⠿⠀⠀⠀⢀⡼⠋⠻⠿⢶⣶⣶⡾⠿⠛⠻⣆
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠈⣿⠲⡀⠀⠀⠀⢰⠋⠀⢹⢧⠀⠀⠀⠀⡜⠁⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⡆⠀⢸⣀⣳⣀⣀⣀⣸⠀⠀⣞⣈⣇⣀⣀⣰⠃⠀⣼⣁⣇⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠼⠥⠥⢿⡀⢸⡽⠥⠼⠬⠧⢿⠀⢠⡯⠧⠬⠥⢤⡏⠀⡼⠥⢼⡧⣤⢤⣤⠀⡏⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣇⠈⣧⠀⠀⠀⠀⣸⠀⣼⡀⠀⠀⣠⣼⠁⡼⠁⠀⠀⡇⢸⠘⡟⣷⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠸⣿⣶⣿⠇⠀⠀⠐⣿⣷⣿⠇⠀⠀⢿⣿⣿⡇⠀⠀⠀⡇⢸⡆⡇⡏⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣷⢸⡇⡇⡟⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⡇⡿⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⡇⣇⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⣇⡏⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⣸⢸⢻⢣⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⡏⣼⣼⣾⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢰⢃⣇⣿⠇⡇⠀⡇
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⡞⣸⣹⡟⢰⡇⠀⡇
⠀⠀⠀⣠⣀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢰⢃⣇⣿⢁⡟⡇⠀⡇
⠀⠀⠀⢷⠈⠙⠳⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢀⡏⡼⣼⠃⡾⠁⡇⠀⡇
⠀⠀⠀⠈⢳⣄⡀⠀⠀⠉⠙⠒⠦⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⡼⣰⣳⡏⣸⠃⢠⡇⠀⡇
⠀⠀⠀⣠⠏⠀⠉⠙⠓⠒⠲⠶⠤⠤⢤⣬⣉⣛⣒⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⢠⠏⣸⢡⢧⣿⢡⡏⢠⢏⡇⠀⡇
⠀⠀⠐⠓⠒⠒⠒⠒⠒⠒⢲⡖⢲⡖⠒⠒⣶⠲⠚⢻⢛⠛⣿⠛⠛⠛⣻⠓⣶⢻⠒⠓⠛⢻⠇⡞⢠⢏⢀⡇⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠛⠛⢻⣀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣥⣤⣶⡞⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠰⢒⡀⠀⠁⢍⠁⠀⠈⠉⠀⠁⠀⠀⠘⡤⠀⢸⣿⣿⣫⡭⣿⣿⣿⣶⣿⣿⣷⣾⣿⣾⣿⣿⣍⢻⣿
⠀⠀⠀⠈⢷⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⣿⣿⣷⢸⣿⣿⣏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⠀⡤⢄⠹⣄⠀⢠⡄⠀⠀⠣⠖⠀⠀⢸⣿⣿⢹⣿⣾⣿⣛⣿⣜⢿⣿⣿⣿⣿⣯⡟⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠁⠈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⡆⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠠⠴⠶⣶⣶⣶⣶⣶⣶⣒⣖⣒⣒⣒⠒⠲⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡆⠀⢀⡶⡄⢻⠋⢸⣛⠀⣟⣳⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⣠⣬⣭⣭⡭⠭⠭⠽⠿⠿⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠳⠦⠘⠉⠓⠸⠀⠸⠶⠄⠃⠙⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠐⠚⠛⠛⣛⣛⣛⣛⣛⣻⣿⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠘⢛⣛⣋⣉⣩⣭⣭⡥⠤⠴⠶⣶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⡤⣤⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠖⠃
⠀⠀⠀⠀⠀⠀⠈⡥⠴⠶⠶⣒⣚⣛⣛⣉⣉⣭⣭⣭⣭⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣭⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣘⣛⣛⣩⣭⣭⡭⠥⠤⠴⠶⠶⠶⠶⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡛⠛⠛⠲⣄⢠⣦⠙⠋
⠀⠀⠀⠀⠀⢠⡤⠭⠷⠶⠶⠒⢒⣛⣛⣛⣛⣛⣛⣛⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢏⡼⠛⠶⠶⢶⡟
⠀⠀⠀⠀⠀⢀⣀⣈⣙⣿⣭⣭⡭⠭⠭⠭⠭⠭⠭⠭⠥⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠘⠛⢃⡟⢻⠁
⠀⠀⠀⣀⣤⣬⠭⠥⠶⠖⢒⣒⣛⣛⣛⣛⣛⣋⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⢤⡭⢁⡾
⠀⠀⠀⠈⣛⣛⣛⣛⣋⣉⣭⣭⣭⣭⣭⣭⣭⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢾⡏⠻⣾⢻⣥⣀
⠀⠀⠀⠠⣭⡭⠤⠤⠶⠶⠶⢶⣶⣶⣶⣶⣶⡶⠖⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠉⠀⠀⠹⣾⡽⡿⠃⠉⠛⠶⣄
⠀⠀⠀⠀⠒⢲⣾⣿⣿⣭⣭⣭⣭⣭⣭⣭⠭⢥⣤⡄⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⢸⠗⣿⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠴⠶⠲⠒⠒⠚⣛⣛⣛⣛⣛⣛⣓⣒⣂⡀⠀⠀⠀⠀⠀⠀⡿⠀⠠⣄⠀⠀⠀⣾⣫⣽⠀⠀⢀⣤⠀⢹⡄
⠀⠀⠀⠀⠀⣬⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⡄⠀⠀⠀⠀⢰⡇⠀⣸⢿⠀⠀⢠⣿⣿⣽⡇⠀⢸⣿⡆⠀⣇
⠀⠀⠀⠀⠀⢒⣒⣲⣒⣛⣛⣛⣿⣿⣿⣭⣭⣭⣭⣽⡅⠀⠀⠀⠀⢸⡇⢠⡏⢸⡀⠀⠸⣿⡵⣾⠃⠀⢸⡏⡇⠀⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣷⣶⣒⣒⣒⣒⣲⣶⣶⣶⣶⣦⡄⠀⠀⠀⠀⢸⡇⠸⡇⢸⡇⠀⠀⠈⠛⠁⠀⠀⠘⣷⡇⠀⡟
⠀⠀⠀⠀⠀⠰⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠭⠭⢭⣭⠁⠀⠀⠀⠀⢷⠀⢿⡸⡇⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⣸⠃
⠀⠀⠀⠀⠀⠤⠿⠭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣿⣿⣿⡇⠠⠶⠶⠶⠾⢧⡈⠳⣞⣛⡛⠛⣿⣿⣟⣷⠟⢁⡴⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⢲⣖⣒⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠀⠀⠀⠀⠀⠀⠀⠈⠳⣴⣻⡟⠛⣿⢿⣦⣈⣿⡶⠋
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⣬⣙⣿⣛⣥⣤⣭⣭⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣃⣀⣀⣀⣀⣀⣀⡀⠀⡀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠤⠤⠤⠤⠴⠒⠒⠒⠒⠒⠲⠤⠤⠤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢩⠤⠤⠴⠒⠒⠒⠒⠒⠚⠉⠉⠉⠉⠉⠛⠒⠒⢢⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠸⡆⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣸⣄⠧⠤⠤⠤⠤⠴⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠓⠚⠋⠉⠉⠉⢹⠉⡇
⠀⠀⠀⠀⡏⢹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢹
⠀⠀⠀⠀⡇⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⠀⣷⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡸⡇
⠀⠀⠀⢀⡿⡀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠀⢠⠇⣧
⠀⠀⠀⢸⡄⡇⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠿⢿⣿⣀⣀⣀⣤⠤⠴⠒⠚⠉⣁⡤⠤⠚⠁⠀⢸
⠀⠀⠀⢸⡇⠙⢤⣀⡀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢿⡇⠀⠀⣿⣇⣀⡤⠤⠤⠤⠔⠒⠋⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢀⡇⠀⠀⠀⠈⠉⠉⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⢻⣷⣤⣤⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿
⠀⠀⠀⠀⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣿⠏
⠀⠀⠀⠀⢻⣻⡟⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⠤⠖⠒⠒⠉⠉
⠀⠀⠀⠀⠀⠙⠳⠿⠷⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠰⠒⠒⠒⠊⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⢤⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠤⠶⠖⠒⠚⠛⠉⠉⢉⣨⠷⠖⠒⠒⠒⣦⡤⠶⢛⣹⡷⠞⠁
⠀⠰⡖⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⣀⣠⠴⢛⣩⠴⢾⡉⠁
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⣀⣤⠖⠛⣉⣤⣶⠛⠁⠀⠀⠹⣄
⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠿⣺⠵⣦⣶⣻⣻⡿⠃⠀⠀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⢠⣾⣭⠶⠻⣤⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀
⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⢶⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⠄
⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠒⢻⡙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⢀⣠⡤⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⣿⡆⢀⣀⣀⣀⣀⣀⣀⠀⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠁⢹⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣾⣿⡐⠿⢃⣿⣿⣿⣿⣿⣿⣿⡄⠿⢃⣿⣷⡀⠀⠀⠀⢸⣿⣿⣿⣿⠋⠀⢸⣀⣸⠀⠀⠀⠀⠀⠀⠀⢸⣄⣸⠇⠈⢻⣿⣿⣿
⠀⠀⠀⠀⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⣀⣀⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⡀⢸⣿⣿⣿
⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⣷⢸⣿⡇⢸⣿⡇⣾⣿⡇⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⣿⣿⣿⠀⠀⣯⠀⠈⡏⠀⢸⡇⠀⢸⡇⢸⣿⣿⣿
⠀⠀⠀⠀⣿⢠⣤⣤⢠⣭⣥⢨⣭⡅⢨⣭⡅⣨⣭⡅⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⠛⠛⣿⠛⠛⣿⠛⠛⡟⠛⢻⡟⠛⢻⡇⢸⣿⣿⣿
⠀⠀⠀⠀⣿⠘⠛⠛⠘⠛⠛⠘⠛⠃⠘⠛⠃⠛⠛⠃⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⠤⠤⣿⣤⣤⣷⣤⣤⣧⣤⣼⣧⣤⣼⡇⢸⣿⣿⣿
⠀⠀⠀⠀⣿⢸⣿⣿⢸⣿⣿⢸⣿⡇⢸⣿⡇⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⠀⣷⠀⠀⣾⠀⠀⠷⠀⢸⣿⠀⢸⣿⣿⣿⡇⢸⣿⣿⣿
⠀⠀⠀⠀⠻⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠿⠁⠀⠀⠀⢸⣿⣿⣿⣿⣆⡉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⣁⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣷⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣭⠛⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠍⠀⠀⠀⠀⠀⡀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠿⢿⡃⠀⠀⠀⠀⠀⠈⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⠀⠀⠙⢷⣀⠀⠀⠈⠟
⠀⠀⠀⠀⠀⢀⡴⢧⠘⢯⡑⠀⠀⠈⡟⠓⠒⠒
⠀⠀⠀⠀⢠⠏⠀⠈⢳⡀⠙⢄⡀⢰⣧
⠀⠀⠀⣰⣏⠄⠀⠀⠀⠁⠀⠤⠕⢤⢿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠲
⠀⠀⠀⢻⠅⠀⠀⠀⠀⠘⢆⡀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠞⠃
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠈⢻⣧⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⣄⣤⡾⠋
⠀⠀⠈⢿⡀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⡀
⠀⠀⢠⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⢇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⠟
⠀⠀⠈⣇⠉⢷⡀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠿⠛⠁
⠀⠀⠀⢹⡄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣻⡶⠒⠋⠀⠀⠀⠀⠈⠂⠀⠈
⠀⠀⠀⠀⣷⠀⠘⣧⡀⠀⠀⠀⠔⢻⡇⣧⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠸⡆⠀⢸⣷⡀⠀⠀⠀⡾⠀⣿⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⢹⡀⠀⠛⡅⠀⠀⡸⠁⠀⣿⣠⡶⠛⡉
⠀⠀⠀⠀⠀⢠⡧⠀⠀⠘⠲⢦⣄⣀⣴⠟⠋⠀⠀⡇
⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⢀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣾⠟⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠏⠀⠀⣰⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⢨⡅⠀⢠⡉⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠉⠉⠀⠀⠀⠈⠉⠻⣦⠀⠀⠀⠀⠀⠀⠈⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣷⣤⣀⣀⣤⣢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣤⣤⣾⣟⣛⡒⠶⣤⡀⣤⣀⣐⡄⣰⠞⠉⣁⣤⣶⣶⣿⣿⣿⣁⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣷⠞⠃⠉⢻⡏⠁⠈⠓⠾⣿⣿⣿⣿⣿⣿⣿⣟⢦⠈⢳⡄⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠃⣰⠃⣰⠃⣿⣿⣿⣿⣿⡿⢋⣠⡴⢶⣚⣻⣿⣿⣿⣿⣶⣦⣭⣿⡿⣿⣿⣿⣿⢸⡇⠘⡇⢸⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⣿⠀⢿⣿⡿⢛⣿⠟⣫⣴⣾⠿⠟⠛⣛⠛⠛⠛⠿⣿⣿⣿⣷⣤⡉⠛⠿⣼⠇⠀⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠹⣤⠞⠋⣠⠋⣡⣾⠿⢋⡀⠀⠀⠀⣿⠀⠀⠀⠠⠆⠉⠻⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣾⡿⠁⠀⠈⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢈⠻⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣾⡟⢠⡄⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⣿⣧⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣾⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⢸⣿⠀⣀⡀⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠿⠇⢸⢿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢹⣼⣿⡄⠉⠁⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⡿⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢸⡆⠈⣿⣿⣷⡀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠠⠄⣸⢃⣿⠇⢀⡟⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢻⡄⠘⣿⣿⣷⡜⠃⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣾⠏⢀⡞⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⠻⣄⠈⢿⣿⣿⣦⣀⠀⣤⠀⠀⠀⣠⡀⠀⠀⠐⢂⣤⠞⣱⡿⠋⠀⠋⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠙⠿⣿⣿⣷⣦⣤⣀⣀⣘⣁⣠⡤⠶⣋⣵⣾⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢫⣿⠿⢿⣿⣿⣿⣿⣿⣶⣾⠿⠟⠉⠹⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠀⠀⠀⠈⠉⠉⠉⠀⠠⠤⠖


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⣠⠴⠒⠋⠉⠉⠉⢉⣁⠦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡿⠋⠀⠀⠀⠀⠀⠀⠙⣛⣒⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣠⡔⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡫⠖⠀⠑⠢⢄⡠⠔⠒⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⠆⠀⠀⣠⠔⠋⠀⠰⡴⠒⢤⡜⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⣇⢠⡄⠀⠀⠀⢈⡶⠀⠀⢰⡇⠀⠀⠘⣆⡠⠴⡁⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⣇⣶⠀⠀⠀⠑⢆⠀⠈⣇⠀⠀⠈⡸⠀⠀⠸⠊⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠤⠀⠀⠀⠀⠈⠉⢦⣹⣦⡀⠈⢧⠀⠈⠲⠤⠔⠋⠀⠀⠀⠒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢑⡦⠼⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠱⠞⠉⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠐⠊⠀⠀⠑⢦⡀⠀⠀⠀⠠⠊⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢠⡾⣿⠓⠒⠂⠀⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣧⡄⠒⠒⠒⠒⠒⠤⠼⠂⠚⠴⠯⠕⣢⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣋⢀⠀⢀⠾⠧⠒⠒⠀⠉⠉⡵⠂⠀⠀⠀⢠⡿⠄⣀⣀⣀⣀⣀⣄⡜⡈⢛⣩⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⡎⠉⠑⠂⠤⠄⣀⣸⡭⡅⠙⡭⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠁⠐⠊⠑⠊⠁
⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠒⡆⢀⠇⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢠⣒⡒⠋⡜⠀⠀⢀⠀⠠⡪⠔⠊⡝⠐⠧⠚⠢⠤⣄⣀⣀⡀⢀⣣⠀⠠⠤⠤⣄
⠀⠀⠀⣍⡴⠦⠆⠀⠙⠊⠁⠀⠀⠀⠐⠥⠤⠀⠀⠀⠤⠤⠼⠼⠧⠀⠀⠀⠠⡄⠀⣸⠔⡫⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠽⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠿⠟⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠻⠿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋⠉⠀⢀⣀⣤⡴⠶⠶⠶⠶⠖⠶⠶⠶⢤⣤⣀⡀⠀⠉⠛⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠋⠁⢀⣠⡶⠟⠋⠉⠀⠀⠀⠀⢠⡄⢠⣤⠀⠀⠀⠀⠉⠙⠳⢶⣄⡀⠈⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⣠⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠗⠰⢿⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⢶⣄⠀⠙⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⣠⡾⠋⠀⠀⠀⢸⠇⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠙⢷⣄⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢀⣾⡟⠁⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠹⣦⡀⠈⢻⣧
⠀⠀⠀⠀⠀⣾⡟⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢻⣧
⠀⠀⠀⠀⣼⡟⠀⢀⡿⠁⣶⠀⣿⣷⠀⠀⠀⠀⠀⠀⣶⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣇⠀⠈⣷⡀⠀⢿⣇
⠀⠀⠀⢰⣿⠃⠀⣼⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠘⣧⠀⠘⣿⡄
⠀⠀⠀⣼⡟⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢿⡇
⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠉⠛⠿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⢸⡇⠀⢸⣿
⠀⠀⠀⣿⡇⠀⢸⡇⠐⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠈⠙⠻⢷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠉⠁⢸⡇⠀⢸⡿
⠀⠀⠀⢹⣷⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⡄⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⡇
⠀⠀⠀⠘⣿⡄⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢰⣿⠁
⠀⠀⠀⠀⠹⣷⡀⠀⢿⡄⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡀⠀⢠⡟⠀⢀⣿⠇
⠀⠀⠀⠀⠀⢻⣷⡀⠈⢻⡄⠀⠻⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⣠⡟⠁⢀⣾⡏
⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠻⣦⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣴⠏⠀⢠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⠀⠈⠳⣄⡀⠀⠀⣭⣿⠇⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢸⣯⡉⠀⠀⠀⢀⣴⠟⠁⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠈⠻⢦⣄⠀⠋⠀⠀⠀⠀⠀⠀⣿⣷⡄⠀⠀⠀⠀⠀⠀⠚⠃⠀⣠⡴⠟⠁⢀⣤⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣦⣀⠀⠉⠛⠶⣤⣄⣀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⣀⣠⣴⠶⠛⠉⠀⣀⣴⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣄⣀⠀⠉⠉⠛⠛⠳⠶⠶⠶⠶⠛⠛⠛⠉⠁⠀⣀⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⡾⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⡟⠛⠎⣻⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣯⡛⠍⠀⢠⣬⡷⣿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⡿⢿⣿⣶⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡏⠀⠀⠀⠉⠛⠛⠉⢸⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡟⣋⠓⢆⠀⠀⣤⡶⣤⠀⠙⠛⠁⣼⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡇⠉⠛⠋⠀⠐⢤⣤⠀⠀⠀⠀⢠⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠠⣄⢀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⢴⣶⡦⠄⠀⠀⠀⣠⠎⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢶⣿⣿⣆⠀⠀⢀⣀⡴⠞⠁⠻⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠄⠺⠭⠀⢾⣧⡻⢿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⡟⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠈⡆⠀⠀⢸⠉⠁⢀⡹⠀⠀⠀⢀⣀⠀⠀⠀⢠⠄⠀⠈⠙⠋⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⢀⡇⠀⠀⠈⠀⣴⠏⠀⠀⣀⠀⠈⠿⢆⠀⢀⠟⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠘⠋⠀⠀⠀⠁⠀⠀⠀⠃⣀⠊⢀⡠⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⢀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⣿⣇⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⡄⠀⠀⠀⠀⠀⠀⠘⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢛⡛⠞⢤⣤⣿⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡆⠣⠀⠂⢤⡶⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠛⠛⠶⣬⣿⡟⢩⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣇⡠⠒⠒⠂⠠⢍⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢢⣿⣿⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡟⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢀⠀⣀⣄⠀⠀⠁⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⡄⠂⠐⠂⢄⠀⠘⣦⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁⠀⠀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⣄⠀⠀⠀⢠⡎⠠⢀⠒⢣⠀⠇⠀⠚⠧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⢮⣇⡀⠰⣿⠀⠀⠀⠀⠰⡘⠀⠁⠶⡦⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣌⠀⠀⠀⠀⠀⠘⠛⠙⢄⠀⠀⡀⣴⣄⠀⠈⠈⣷⠤⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠘⡄⠀⠁⢨⣤⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠠
⠀⠈⠀⠀⠀⠀⠀⢸⡇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠀⢡⡀⠁⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⣠⠶⣋⡉⠁⢠⣤⣬⣉⣉⣹⣿⣿⣋⣿⣦
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣽⢿⡷⠶⠿⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⢸⡿⢢⠄⢠⣀⣀⣀⣤⣤⣤⣶⣶⠶⠶⠿⠿⠛⠛⠛⠛⠋
⠀⠘⠀⠀⠀⠀⠀⠘⠿⠟⠛⠛⠛⠉⠉⠉⠁
⠀⢠
⠀⠘