Orangish Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣤⣠⣄⢠⡄⣄⣤⣠⠤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠚⠛⠓⠚⠓⠋⠛⠛⠟⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⡀⠈
⠀⠀⠀⠀⠀⢠⡎⠀⠀⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣾⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⡋⠀⠰⠀⣼⡔⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠁⠀⠂⠺⠃⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⡀
⠀⠀⠄⡀⠉⠁⢀⡀⠀⢹⣴⡴⠯⠉⠠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⢠⡄
⠈⠀⠀⠀⠈⠠⢾⠕⠂⠀⢆⡀⣐⡫⠂⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠰⠂⠀⠤⠂⠀⠀⠀⠀⠀⠔⠉⠙⠀⠉⠪⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠙⣄
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⠖⠃⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢄⣧⠀⠤⠀⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢀⡀⠲⡲⣄⣌⡟⠀⠀⠊⠈⠛⠄⠀⢀
⠀⢆⣦⣶⠄⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠁⣀⠀⠀⠀⢂⡢⠄⠈⢉⠁⠀⢔⣲⡀⠠⠈⠉⠁⠠
⠀⡮⢹⣗⠒⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠽⠋⠰⠞⠛⣘⠃⠀⠀⠀⠋⠁⡀⠀⠀⢀
⡰⣅⠠⠑⣤⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠬⣠⠏⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠁
⠁⠋⠀⠀⠈⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡴⠈⠀⠀⠀⠁⠀⠲⠄⠀⢀⠀⠀⠀⠀⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⡇⠀⠀⠻⠀⠀⠀⠀⠂⠀⢀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠸⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠙⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠘⣿⣾⢻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠹⠿⢷⣿⣿⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣶⣶⣤⣤⣀⠀⠀⠀⠴⠂⠀⠀⠀⠀⠈⢻⠀⠀⠀⢿⢧⠀⠀⠀⠀⢀⣴⣾⠿⣿⠿⠟⠉⠉⠙⠉⠛⠛⠻⣷⣦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠙⢿⣿⣿⣿⣿⣦⡀⠀⠀⠤⣤⣄⡀⠀⠘⠀⠀⠀⢀⠘⠀⠀⢠⠞⠿⠛⠉⢀⣀⡠⢄⣀⣤⣤⣤⣶⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⠀⠀⠀⠉⠻⣿⣿⣶⣄⢀⣶⣿⣷⣦⣁⣠⣤⣴⣴⣿⣿⠿⠿⠟⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡙⢿⣿⣿⣿⣧⠀⠀⠀⠢⣿⠉⢹⢟⣿⣿⣿⣿⣿⣧⡾⠿⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠀⠀⠀⠀⠁⢀⡼⣾⣿⣿⣿⣿⣿⣿⡷⣤⡤⠖
⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠑⠝⣿⠀⠀⠀⢀⣶⣀⠄⣿⡿⠿⠛⠋⠉⠁⠀⠈
⠀⠀⣠⣴⣿⣿⣿⣿⣷⣦⣤⣈⢿⣿⣷⡀⠀⠀⠀⠀⠀⡀⣠⡶⠞⠓⠦⡎⠁⠀⠀⠤⣀⠀⠀⠀⠑⠢⡀
⢀⣾⣿⣿⣿⡿⠿⠛⠋⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⣤⠴⠻⠷⢄⡀⠀⠀⠈⢢⡀⠀⠀⠈⣈⣁⣀⣀⣀⣈⣀⡀⠀⠀⠀⣀⣀⣀
⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣠⢯⡶⡄⠀⠀⠹⣷⣤⣀⣠⣧⡽⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⢀⣀⣠⣤⣤⠶⠶⠒⠛⠛⠓⢶⡄⢀⡴⡋⠂⠀⠈⣦⣴⣾⣿⣿⣿⣿⢛⠁⠁⠀⠀⠀⠈⠙⣿⣿⣿⡿⠟⠛⣁
⠀⠀⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⢼⣠⠟⠀⠀⠀⣀⣾⣿⣿⣿⣿⣿⣿⠥⠂⠀⠀⠀⠀⠀⠀⢠⠜⣯⡄⠀⠀⠀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢁⡀⡤⠴⠛⣿⣿⣿⣿⣿⣿⣿⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡰⣄⠄⠀⠀⠀⡐⢨⣾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣄⣶⣷⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁



⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡀⡆⠀⠀⣴⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣠⣄⣀⠢⢑⣯⢽⣿⡶⡻⠏⢁⠐⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠿⣿⡿⠷⠦⠀⠊⠉⢐⡁⠔⣲⣞⢛⣻⢷⠴⡆
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⠶⠆⠀⠀⠀⠚⠋⠙⠁⠌⠤⠼⠀⠄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⣿⠿⠟⠛⣿⣽⣶⣾⢏⣾⡞⠎⠑⢒⢷⣾⣤⣿⢗⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢰⢂⣴⢻⣿⣿⢿⢿⣿⣾⣷⡄⣀⢀⠑⠻⠿⣫⠏⠀⠀⡒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⣬⡄⠀⢬⣴⣋⡩⣷⣷⣾⢻⣿⡟⠻⢏⣘⠆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⠁⠿⠟⠟⠛⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣶⣤⣤⣀⣄⣀⣀⣠⣤⣶⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢲⣤⣶⢎⣺⠯⣏⡼⣶⡀⡔⣄⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠐⠀⠄⣀⣤⣴⣦⣭⣭⣬⣶⠖
⠀⠀⠀⠀⠀⣀⣤⠤⣽⡽⢻⣾⢥⣥⢟⣏⢹⣾⣶⣥⡮⣄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠁⢿⣿⡛⢻⣏⠉
⠀⠀⠀⠀⠀⠶⣿⣤⣷⣲⠻⣯⣳⣼⣧⣭⣿⣿⣽⣻⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣂⠈⠢
⠀⠀⠀⠴⣦⢾⣿⡿⢻⣯⢗⣛⣻⡜⢹⣿⣶⡿⣿⡥⡷⢴⣴⡄⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠉⠓⢿⣷⣤⣁⣄
⠀⠀⠀⠀⢘⡾⢧⠞⣏⣡⣼⡋⢿⣾⡟⣋⣾⠽⢿⣽⣷⢾⡾⠇⠀⠀⠘⢶⡄⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠈⠉⠙⠛⠉
⠀⠀⠀⣰⣭⣴⢾⣏⣽⣏⣿⡿⣯⢼⢻⣿⡿⢷⣿⡞⢻⣶⡗⠂⠀⠀⠀⠀⠉⠻⠦⠤⠠⠄⠈
⠀⠀⠀⠉⠈⠈⠻⠛⠋⠻⠤⣣⠘⡌⢀⡨⠁⠙⠻⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⡜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡃⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠳⢦⣠⠀⣠⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠁⠀⢤⢌⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢿⣿⡆⠀⠀⠈
⠀⠀⠀⠀⢰⢀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣾⣿⣷⡆
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡇⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠿⡿⠋⢛⡷⠘⠀⠀⡀
⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⢠⢄⡖⣱⣤⣤⣀⣤⣀⣦⠀⠀⠀⠀⢀⡀⠀⠂⠀⢤⢼⣩⢃⣀⢀⡘⠁⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣾⣳⢀⣀⣺⡇
⠀⠀⠀⠀⠀⠘⣿⣦⣶⡌⣠⢴⣾⣾⣾⣿⣿⣿⣿⣿⣿⠇⠀⢀⠠⠀⠸⠀⠀⠀⠒⠂⠀⠁⠚⠛⠛⠃⠠⠄⢀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⠄⠐⠄⠀⠀⠀⢾⡷⢷⣶⡄⠀⠀⠀⢤⡆⢘⠀⠀⠀⠀⠀⠀⠈⠻⠿⠋⣸⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⣿⣿⣿⡿⠿⠿⠿⠗⠂⠀⠀⠀⠀⠑⢶⣦⠀⠀⡄⠁⠳⣿⠲⢖⡂⢰⡌⢁⠄⠀⠀⠀⠀⠀⠀⠈⠲⠤⢴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣤⣀⣉⣉⣉⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠔⣶⢊⡝⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠛⠄⠈⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣀⡠⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠚⢁⣀⠢⣄⡴⢀⣤⢄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⢉⡛⠈⠛⠚⠡⠞⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⡀⢪⣌⣻⣿⠐⠿⡣⠿⠃⢄⠀⠀⡀⠀⠀⠀⠀⠸⣿⣤⣀⡀⠀⠀⠀⠀⢤⣉⣭⡅⠀⠀⠀⠀⣤⣤⡭⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡌⠀⠈⠬⣎⡿⠇⠈⠀⠀⠈⠛⡛⢛⡀⠀⡇⠀⠀⠀⠄⠠⠹⠿⠿⠿⠿⠟⠛⠒⠂⠉⢀⠹⣷⣶⣤⣀⢡⣍⠇⠀⠄
⠀⠀⠀⠀⣄⡀⠀⠀⠀⠀⣬⣿⡆⠀⠀⣠⣴⣶⢇⠀⣄⠰⡎⡥⢿⠝⠀⠀⡙⠀⠀⠀⠀⡀⠀⠒⠂⠀⣀⡄⠀⠀⠀⠘⢿⠀⣏⢘⣙⡋⠁⠀⢀⠄
⠀⠀⠀⠀⠀⠻⣶⣤⣿⣷⣿⣿⣿⡄⠀⠀⢋⢁⠚⣠⣩⣴⡙⡮⠀⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠐⠿⡫⣛⣿⡤⢔⣂⠘⠆⣰⣿⣿⣿⣿⠷⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠿⠿⠋⠠⡀⠀⠀⠀⠁⠈⠀⠀⠀⢊⣡⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⡉⠉⠑⠚⣉⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⢤⣤⣤⣤⣶⣾⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⡿⠿⠿⠿⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢺⣿⠻⠿⠿⢟⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⣷⠀⠀⠀⠀⢰⣻⠃⠹⢃⡄⠀⠀⠀⠀⠀⠙⠳⢦⣀⠀⠈⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠁⣿⠀⠀⠀⠀⠘⣇⠀⣴⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠉⠛⠐⠚⠛⠧⡀
⠀⠀⣀⣤⣴⣶⣤⣤⣄⡀⠀⠀⠀⠀⣾⡿⡏⠀⢘⡟⠀⠀⠀⠀⠀⠘⣉⡱⠂⢀⠘⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⢄
⠀⠀⠉⢻⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⣿⡟⠁⢀⣸⡇⠀⠀⠀⠀⠀⠀⠛⠀⠆⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠁⠀⠀⠀⠈⢆
⠀⠀⠀⠑⢮⠙⢻⣿⣿⣿⣿⣇⠀⠀⢻⣿⡆⢠⠞⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⢾⣿⣿⠶⠄⠀⠀⠀⠈⡄
⠀⠀⠀⠀⣀⡴⣶⣶⠿⣿⣿⣿⠀⠀⠀⢻⡟⠉⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠙⡖⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⣀⣿⣿⣯⠓⡂⠀⠀⠈⠀⡇
⠀⠀⢀⣞⣋⣀⣀⣈⣩⠟⠋⠁⠷⠂⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠄⠊⠀⠀⠀⠀⠀⢀⣴⣿⣇⣤⣽⣿⣿⣿⢿⡷⠷⠈⠀⠀⠀⠃
⠀⠀⠈⠙⠛⠲⠟⠉⠀⠂⠁⠀⠀⠀⠀⠈⠳⢄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢒⠿⣿⣿⣿⢿⣿⢿⠋⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠠⠋⠸⠛⠊⠙⠎⠀⠀⢀⡠⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⡀⠲⠄⠂⠀⠀⠀⠀⣀⣠⠀⠀⠀⢀⡤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠪⣍⡀⠀⠀⠉⢉⠉⠁⢀⣠⡤⠖⠉
⠀⠀⢀⣀⠠⠴⠶⠶⠶⢶⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠒⠂⠉⠉
⠀⠀⠈⠙⢿⣧⣄⠐⠒⠒⠶⣽⣿⣦⣲⣤⣢⠻⠷⠖⠀⢄⡀
⠀⠀⠀⠀⠀⠉⠛⠷⢶⣽⣿⣶⣿⣷⣾⣿⣿⣾⣭⣄⡀⠀⠈⠂⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠚⠋⠉⠉⠉⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠁⠀⠈⠑⠿⣿⣿⠗⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⡀⠀⠀⠀⠀⢱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⣿⠀⠄⠀⠀⠀⣀⣸⣶⣶⣿⣿⣕⡀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢽⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠐⠿⣿⣿⠛⠉⠸⣿⣿⡇⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⡟⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢠⠿⣿⣶⣤⣴⣿⣿⠟⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢻⣿⣤⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⠁⠀⠀⠀⠀⠀⠀⢷⡀⠁⠀⠀⠈⠋⠿⠿⡉⠛⡉⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢼⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣾⣶⣶⣤⣠⣴⣷⣶⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣤⡀⠁⠀⠀⠀⠀⠀⣀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠓⠒⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠂⣀⠀⠀⠀⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⢠⣤⣤⣾⣿⣿⠏⠀⠀⠀⠀⠀⢀⣀⠀⣤⡀⠀⠀⢸⡅⠀⠘⠻⣧⡄⠀⠀⠰⠿⠿⠿⠛
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⠁⠀⠀⠈⠻⣿⣤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⣶⠀⢿⣷
⠀⠀⢀⣤⠀⠀⠀⠀⢰⣦⠀⠀⠀⣠⠂⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢠⡤⠒⠀⢀⠀⠀⠀⢾⠿⣴⣾⠇
⠀⠀⢾⣿⣤⣤⣀⣄⣸⣿⡇⠀⢸⣿⣄⡀⣠⣴⣾⣿⡇⠀⠀⠀⠀⠀⣀⡐⠂⣿⡇
⠀⠀⠈⠻⣿⣿⣿⣿⡿⠏⠀⠀⠈⠿⢿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠛⢿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠉





⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢠⣶⣶⣶⣶⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠀⠈⠛⠟⠉⠻⠇⠀⣾⣿⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠛⢿⣿⣿⠿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣤⠟⠛⠋⡤⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⢀⣾⣟⣡⣤⣼⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣧⣴⣿⠁⠀⢀⡀⠀⢠⣀⣤⣀⣿⣇⣠⣿⣾⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠿⣏⣀⣀⣶⣦⡀⣵⣦⡀⠘⢿⠟⠀⠘⠻⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⢴⣶⣾⡟⠛⠿⠿⠿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠿⣿⣿⠿⠿⣿⣿⣯⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⣸⣿⣿⣷⠿⠶⢶⣾⣿⣿⣿⣷⣶⡄⠀⠠⠄⠀⠀⢀⠿⣿⣶⣦⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠻⣿⣟⣷⣄⡀⠀⢨⣿⣿⣿⡿⢻⣷⡄⢀⡀⢠⣤⠉⣦⣄⠉⠙⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠋⠁⢀⣀⡹⠿⣿⣥⣾⡿⠈⣿⣿⠉⣿⣿⣶⣶⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠻⣿⡿⠟⢧⡼⠛⠿⣿⣍⠿⣷⡌⠙⠿⠁⠀⠉⣿⣷⡀⠻⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣎⢁⠄⠀⣿⣿⡄⠿⠇⠀⠀⠀⠀⠀⢻⣿⣿⣶⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡟⢠⣤⣿⣿⣷⣄⣼⡆⢀⡀⢀⡀⠀⣿⣿⣟⣻⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⢀⣸⣿⠁⣼⣿⣼⣇⠀⣿⣿⣿⣿⣿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠈⠉⠉⠙⠻⠿⠿⠿⢿⣿⡿⠿⠚⠻⠟⠛⠉⠀⠀⠐






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢀⣤⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣇
⠀⠀⠀⠀⢰⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷
⠀⠀⠀⢰⣿⣿⣿⣿⡿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣿⣿⣿⣿⣧⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣠⣄⣀⣠⣤⡼⠟⠛⣦⣤⣴⡤⢠⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⣤⣴⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠉⠉⠉⠉⠁




⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣠⡄⠀⠘⢿⣿⡟⠁⠙⢿⣷
⠀⠀⠀⠘⣿⡇⠀⠀⠈⠻⣿⣦⣄⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣤⣤⣶⣾⣿⣿⣿⣶⣾⡇
⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠈⢻⣿⣆⠀⢰⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⢿⣿⣿⣏⣀⣸⣿⡿
⠀⠀⠀⢸⣿⠀⠀⠀⢦⡀⠀⠈⢿⣿⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣠⣾⣿⣿⣿⣿⡟⠛⠁
⠀⠀⠀⠸⢿⣶⣦⡀⠀⢿⣤⡀⠈⠋⣠⣌⠙⣛⠁⠀⠀⠀⠀⠀⠀⠈⠁⠛⠛⠋⠉⠉⠉⠉
⠀⠀⠀⢠⣄⠀⠈⠻⣷⣄⠛⠃⡀⠀⠉⢛⣿⡿
⠀⠀⠀⠀⠈⠻⢤⣄⡈⢻⣷⣾⣿⣶⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡦⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⣻⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣠⡄⠀⠻⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡗⠛⢿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣤⣼⡿⣿⠖⢶⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡟⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣁⣴⠿⠀⠀⠉⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⡾⠛⠋⠀⠉⠁⠀⡀⠀⣤⡄⠀⣤⠀⠀⠀⠀⠀⠀⠘⢿⣷⣴⣶⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡸⢿⣆⠀⡀⢀⣿⡇
⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣈⡉⠻⢷⡝⠋⣠⣿⣿⠁
⠀⠀⠀⠀⠀⠘⢿⣷⣄⠸⠿⢇⡀⢀⣡⣶⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣷⣤⣾⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠐⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣧
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣿
⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⡀
⠀⠀⠀⢪⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⣠⠔⠀⢰⠃
⠀⠀⠀⠈⢯⣿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⢀⡟
⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣤⣶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣋⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⣿⣯⣆⣄⠀⢀⠀⢀⣀⣀⣄⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⣠⡃⠂⠘⣆⣀⣤⢤⡤⠤⠤⠤⠀⠤⠴⠀⠀⠀⢰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⢀⠾⠋⠀⢀⢠⡋⠀⡔⠀⠀⠀⠀⢠⠊⠀⣀⣀⠄⢂⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠠⡴⠋⣠⠄⠂⢀⠞⣀⡼⠴⠖⠒⠛⠛⠿⠛⠍⠁⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢠⠗⠉⠀⢀⣐⡧⠚⠛⠒⠒⠦⢤⡀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢠⠀⠀⠰⢫⢏⡀⠀⠀⠀⠀⠀⠀⠀⠉⠈⢀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣬⣇⡐⣷⣏⣤⣤⣬⠑⠒⠒⠒⠚⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢖⢣⠈⠙⡟⠉⠉⠀⠀⠉⠉⠓⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠈⠆⠰⠀⠚⡈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠐⠚⠲⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⠶⠶⠤⠶⠶⠶⠚⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢠⡾⢛⡍⡨⠴⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⡄⠓⠀⠀⠀⠘⡟⣇⠀⠀⠀⠀⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀⢀⣀⣀⡤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢠⣀⠠⣷⠹⣧⡀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣌⠄
⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⢠⡿⠃⢸⣿⡆⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠈⠁
⠀⠀⠀⠀⠀⠛⠉⠀⡀⢀⡀⢀⣸⠁⠐⣿⣿⣿⡏⠤⠀⠀⠀⣤⣤⠀⠀⠀⠀⠁⠐⠒⠶⠲⠂⠐⠂
⠀⠀⠀⠀⠠⠄⠀⠠⠂⠀⠀⣈⣸⠀⠀⣼⣿⣿⣷⠀⠒⠀⢰⣿⣟⣛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣇⢀⠀⠀⠘⠏⠻⣿⣷⣤⡀
⠀⠀⠀⠀⡠⡤⠈⡓⠀⠀⡀⣄⠀⣿⣿⣿⣿⣇⣈⠛⠀⠀⠀⠀⠠⣿⣿⠿⠻⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠈⣠⠁⠀⠀⠀⠀⠛⠛⠹⠏⠉⣽⣿⣶⠰⠿⠆⠀⠾⠀⠀⠀⠀⡀⠉⠀⠀⠃⢠⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⡠⠖⠩⡴⣀⣀⣾⣿⠶⠶⠶⠶⠒⠛⠋⠉⠁⠠⠀⡴⠂⠀⠀⠀⣠⠀⢦⠀⠁⢀⣤⡏⠀⠒⣀⡀⠐⠂
⠀⠀⠀⠉⠉⠒⠾⢿⠉⡡⣠⡤⣶⣄⣀⢀⣠⣤⣤⣤⠀⢀⡀⠀⣀⡀⠀⠈⠁⠀⠌⢣⡀⠉⢹⡀⢀⡼⠋⠀⠸⠄
⠀⠀⠀⠀⣀⢀⠀⠀⣠⡄⠋⢀⣼⠟⢠⣬⠙⠧⢬⣻⢿⣻⣦⣶⣿⣧⣄⠀⠀⠀⠀⠀⠳⡄⢸⣷⡞⣀⣴⠴⠆⠀⠀⠂⠠
⠀⠀⠀⠀⢘⣿⣧⣄⣡⡁⠀⠀⠀⢀⣾⣽⡀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⢳⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⠰⣶⠙⠀⠙⠉⠀⠀⣴⣶⣿⣯⣭⢻⠟⣻⣿⣿⣿⡆⠀⣠⡄⢼⣿⡇
⠀⠀⠀⠀⠀⠐⣲⠈⢿⣿⡀⡝⠂⠀⠀⠀⠀⠀⠛⠘⣿⣿⣷⣴⣶⣿⣿⣿⣿⣷⣇⠁⠌⠀⢸⢿⣷
⠀⠀⠀⠀⠀⠘⢛⠁⠀⢀⡄⠳⢤⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⣱⢤⠀⡀⢐⡛⣸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠃⢸⡏⠀⠀⠀⠀⢀⢀⠙⠻⠋⠙⠛⠛⠿⠟⠛⢠⣟⡧⢸⠃⣿⠿⣿⣿⡇⠀⣠⣤⣰⣶⣶⣶⣦⡄
⠀⠀⠀⠀⡶⠶⣶⣶⡂⠂⡐⠂⣧⡄⠀⠀⠲⢼⡗⣰⠃⠠⠠⠀⠄⠀⠀⠀⣨⠙⢋⠛⠸⢇⢤⣿⣿⣷⢈⣿⣿⡿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠑⠶⠶⠿⢳⠘⠗⢠⡿⠿⢧⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣿⣾⣷⣄⣺⣿⣿⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠠⣆⣠⡖⣤⣠⡆⠀⠃⠈⠿⠀⠀⠈⠁⠀⠉⠐⠠⠀⠽⠿⠟⠛⠛⠻⠿⠛⢻⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡿⠿⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠁⣿⣿⡿⠿⣿⣿⣿⠿⢿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣒⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣤⡖⢻⡟⠃⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠚⢻⣿⢻⠧⠀⠀⠀⣿⢼⡿⠄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⣠⣶⣆⢰⣶⣿⣾⡿⠆⠀⠀⠀⠀⠀⠾⣶⣶⣆⣶⣦⠀⠀⠸⢀⠀⠀⠀⠈⣦⡀⠘⣇⠀⠀⢠⠏⠁⠁⢀⡤⠀⠀⠀⠀⣸⡀
⢸⣿⣹⣯⡙⠻⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠛⠉⠰⠓⠛⠀⠀⠰⠈⢿⣤⠀⠾⠿⠁⠀⠈⠧⠖⠁⠀⠀⠀⠡⠀⠀⠀⠀⠆⠳⢃
⠈⠈⠉⠉⠁⠀⠀⢀⠀⢀⣔⣰⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠈⠁⢀⣠⠆⠀⠀⠀⠀⢀⡴⠀⡖⠒⠋⠀⠀⠀⠀⠰⠖⣿
⠀⠀⠀⠀⠔⠀⡠⠙⠒⠟⠂⠙⠒⠃⢀⠉⢢⡀⠀⡜⠁⠀⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢸⠉⠙⠁⠠⢤⡀⠀⠀⠀⠘⠀⠻
⠀⠀⠀⡜⠀⡼⢡⣄⠀⠀⠀⠀⠀⣠⡄⢠⠀⢱⠀⠛⢀⢠⡀⠀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠁⠀⠀⠀⠀⠀⠀⠀⠬⡎
⠀⠀⠀⡇⠰⠇⠀⢹⡷⢀⣄⠀⣾⡿⠀⢸⡆⢸⠀⠀⠈⠛⠑⠦⠟⠈⡆⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⢀⠀⠀⠀⠀⠀⢠⡜⢀⠱⣄
⠀⠀⠀⢷⣴⡄⠀⢦⣀⠈⡙⢀⣈⡄⠀⢠⢇⡼⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⠀⠀⠀⠀⠀⠀⠊⠉⠉⠁⠀⠐⢶⣤⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠉⠛⠦⣾⠉⠻⠛⠏⣙⣀⣠⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠓⠄⠖⠊⠀⠀⠀⠀⠀⢀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠀⠈⠉⠀⠀⠀⠀⢀⡶⠰⢀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⢠⡤⣠⣾⣿⣿⣿⡆
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⣠⠼⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣤⣶⠄⠘⠻⢿⣿⣿⣿⣿⣷⣾⠆
⠀⢀⠂⠀⠀⠀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠈⠀⢦⡀⠀⠀⠀⠀⠠⣦⣾⣿⢿⣿⣯⠀⠀⠀⣾⡿⠋⢿⣿⠁
⠀⣾⠀⠀⢠⢨⣿⠄⣿⡇⠠⡆⠀⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⡹⣿⡽⢻⣿⣆⣀⣹⣿⣄⣀⣼⣯⣦⡀
⠀⢧⠀⢠⠈⠘⠛⢰⠌⠋⢀⠠⠀⠀⢠⢜⠇⠀⠀⠀⠀⠀⠀⣸⢇⠀⠀⠀⠲⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⠀⣠⡀
⠀⠈⠓⠶⠾⢠⣖⣐⣴⠀⠁⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⣿⣿⡄⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣾⠃⡄
⠀⠀⠀⠀⠀⠐⠀⠐⠒⠀⠀⠀⠀⠀⢀⡯⠀⠀⠀⠀⠀⠀⠀⣤⡿⠀⠀⠘⠿⠿⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠙⣇⠀⢁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⢤⡀⠀⠀⠘⠣⠠⠖⢦⢀⡀⣀⣸⣾⡇⠀⠀⠀⠀⠀⠀⢻⣿⢹⣿⡟⡋⣛⣿⡷⣾⣿⣥⡰⣏⣀⡽⣠⠂
⠀⠀⠀⢠⠔⢤⣀⠜⠀⠀⠀⠀⠑⢀⠀⠀⠀⠂⠀⡀⠐⠈⠁⠐⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⡿⠃⠒⠒⠘⣿⣿⣿⠛⣷⣏⣁⡴⠧⠤
⠀⢀⠀⡧⢦⡀⣭⠶⣦⡀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⢸⣿⣇⢀⣀⣀⣀⣿⣿⣿⣶⡈⠉⠫⠄⠀⠁
⠰⠛⠛⠁⠈⠋⠁⣰⣿⠛⠛⠏⠐⡒⠆⣄⣀⠤⠄⠐⢒⡤⠤⠄⠀⠀⠀⠀⠀⠀⢩⣿⣿⣿⣿⣿⣿⣿⣶⡍⠉⣻⣿⣿
⠀⢤⣀⣴⡶⠖⠛⠛⠛⠒⠐⠿⠓⢉⠁⢀⠀⣉⡿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⡟⠋⠉⠀⠻⠿⠿⣿⣶⣦⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠈⠉⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⢄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⢀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠘⠁⠀⠀⡉⠀⠆⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠐⠉⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠐⠊⠀⢀⣧⠀⠀⢠⠂⠀⠀⠀⠀⠀⠀⢉⡀⠤⠋⠀⠀⠀⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠁⠀⠀⠣⠀⢐⠹⡆⠀⠀⠀⠈⣄⠀⠀⠀⠀⢀⠷⠠⠤⠤⠄⠲⠏⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠶⠖⠛⠒⠛⠂⠛⠀⣐⣯⢒⠙⠀⠀⠀⠀⡎⠀⢻⠲⢄⡀⠁⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠤⠈⠙⢉⣶⠄⢠⡄⢘⡛⣱⢃⠖⠀⠀⠀⠀⠇⠀⠈⢦⡀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⣀⠀⠀⠀⣠⡆⠉⠉⢉⡠⠊⠀⠀⠀⠀⠒⠂⠉⠰⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢷⠂⠀⠁⠀⠀⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣦⣤⣤⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢀⠉⢈⣁⠾⠀⠀⠀⠀⠀⠀⢀⡴⢸⡎⡅⠐⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢤⡤⣬⠅⠐⠃⡢⠀⣀⠤⠒⠉⠀⠈⢷⡐⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⣀⣀⣀⣤⡶⢂⡴⠾⠓⠋⠁⠀⠀⠀⠀⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣏⣤⣀⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠻⢦⣬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁
⠀⠀⠀⠀⢠⠂⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢾⠁
⠀⠀⠀⠀⠘⢦⡈⠢⠴⢤⡀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸
⠀⠀⠀⠀⢀⡀⠧⡀⠀⠀⠀⠱⠀⠀⢠⡟⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢸
⠀⠀⢀⠔⢩⣆⠀⠀⠀⣸⢇⢰⠋⠲⡾⠁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢸
⠀⣖⢉⣴⠋⢨⠷⡀⡴⠁⠸⠃⠉⢶⠃⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⡆
⠀⠈⢠⠁⣰⡏⢸⡇⠃⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠈⠉⠀⠶⠊⠱⠆⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⡀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⠀⢰⡄⠀⣧⠀⡜⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢹
⠀⣀⣀⣠⣀⣤⡞⠁⠀⠀⠀⠀⠀⠀⠘⢧⡀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠑⢶⡄
⡞⠉⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣤⠽⠷⠒⠂⠀⠐⠒⠈⠉⠉⠉⠉⠉⣉⣉⣉⣙⡆
⠙⠳⢤⣤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠒⠒⠛⠉⠉⠉⠉⠉⠛⣆
⠀⠀⠀⠉⠃⠀⠤⠤⠤⠤⠤⠀⠉⠉⠁⠀⠀⠀⠀⠠⠤⠤⠤⠤⠄⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡴⠞⠃
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣃⣀⣀⣀⣀⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⣸⣿⣿⣿⣿⣇⣀
⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣦⣶⣭⣁⣤⣬⣿⣽⣿⣿⣿⣿⣿⣿⣶⣯⣍⡉⢙⡶⠒⠒⠤⢄⡀
⠀⠀⠀⠀⢀⡴⠊⠀⠼⠛⠉⠁⠀⣴⡿⠟⠛⠛⠛⣻⣿⣿⣿⣿⣿⢿⣿⣯⡙⠛⠶⢄⠀⠀⠉⠳⣄
⠀⠀⠀⣴⠏⠀⢀⡔⠁⠀⠀⢀⡴⠁⠀⠀⠀⠀⣴⡿⠟⠉⠉⠻⣿⠀⠈⠙⢧⡀⠀⠀⠉⠲⣄⠀⠈⢳⡄
⠀⢀⣾⡟⠀⣠⡏⠀⠀⢠⣀⠎⠀⠀⠀⠀⠀⢰⠟⡰⠀⠀⠀⠀⠘⡇⠀⠀⠀⠙⢦⠀⠀⠀⠈⢷⡀⠀⢻⣆
⢀⣾⣿⠁⣰⣿⠃⠀⠀⢉⣿⣷⣦⡀⠀⠀⢀⡏⠰⠁⠀⠀⠀⠀⠀⢱⠀⠀⠀⢀⣠⣶⡉⢂⠀⠨⣿⡄⠘⣿⡆
⣼⣿⣿⢠⣿⣿⠀⢀⣠⣾⣿⣿⣿⣿⣷⣄⢸⠁⠀⠀⠀⠀⠀⡄⠀⠀⢇⣠⣶⣿⣿⣿⣷⠈⠀⠀⢻⣿⠀⣿⣿
⣿⣿⣿⢸⣿⣿⠰⠟⣩⣤⣀⠉⠉⢉⣙⠛⣓⣦⡀⠀⠀⠀⠀⢁⣠⡞⠛⠉⠉⠉⠉⠉⠛⢳⣶⠀⣼⣿⡇⣿⣿
⣿⣿⣿⣿⣿⣿⣠⣤⣿⣿⡇⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠉⠉⢈⢹⡏⠀⠀⠀⢸⣿⣧⣀⣀⣿⣿⣿⣿⣿
⢻⢿⣿⣿⣿⡛⠛⠻⣿⣿⣿⣀⣤⣄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⣾⣿⠋⠉⣉⣹⣿⣿⣿⣏
⠸⡌⣿⣿⣿⣿⣿⡆⠘⣿⣿⣿⣿⣿⣦⡀⣿⣷⣷⣄⠀⠀⢠⣤⣄⡸⣿⣷⣿⣷⣶⣿⠃⡀⣰⣿⣿⢻⣿⡟⡜
⠀⢱⡘⣿⣿⡹⣿⣿⣄⣸⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣿⣿⠟⠛⠻⣧⣰⣷⣿⣿⢃⣿⡟⡱⠁
⠀⠀⠡⡘⣿⣷⡙⣿⣿⣿⡧⣤⣆⠈⢻⣿⣿⠿⢿⣿⣿⣿⣿⠟⠻⠿⣿⡃⣠⣿⣿⡟⣿⣿⣿⢃⣾⢟⡜⠁
⠀⠀⠀⠐⢌⠻⣿⣌⠻⣿⣷⡙⢿⣆⣼⣏⣀⣠⠀⠈⢻⣿⠁⣠⣦⢤⣼⣿⣿⣿⠟⣼⣿⡿⢡⣿⠕⠋
⠀⠀⠀⠀⠀⠑⠮⣻⣷⡈⠻⣿⣌⠻⣿⣿⣿⣿⡎⢦⣼⣿⣴⣿⢏⣾⣿⣿⡿⢋⣼⡿⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢬⣙⣳⣮⡙⠿⢿⣿⣌⢿⣿⠟⣣⣾⣿⠿⢋⠴⠾⠫⠐⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣵⣮⣭⣬⣷⣶⣾⣭⣭⣤⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣤⠄
⠀⠀⠀⠀⣄⡐⠂⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠃⠀⠀⢠⣤⠤⠀⠐
⣀⡀⠈⠀⠉⠸⠓⠂⠀⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠓⠀⠀⠀⠁⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠃⠄⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠒⣠⣐⠒
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⡄
⠀⠘⠁⠀⠀⣘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⠀⠀⠀⠀⠀⣴⣿⡿⢷
⠀⠀⠚⠛⠃⠀⠀⠀⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⣼⣿⡿⠧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⢶⡟⢛⡛⠛⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠈⠁⠀⠀⠈⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⠈⠀⠀⠠⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢄⠠⠀⠀⠀⠀⠀⠀⢐⡀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⡀⠀⠀⠀⠀⠘⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣁⡀⡈⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠙⠋⠀⠀⠀⠀⠀⠂⠀⠀⠁⠀⠽⠈⠏⠀⠀⠀⡀⠐⠄
⠀⠀⠀⠀⣀⠚⠓⢀⠀⠀⠀⠀⠀⠀⠠⠟⠛⠛⢛⠋⠰⠀⠀⠀⠀⠀⡀⠀⠀⠆⡤⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢹
⠀⠠⠀⠀⣛⣛⣛⣻⠀⠁⠀⠀⠀⠀⠢⡈⠠⠉⠀⠤⠀⡀⠀⠀⠀⠀⠾⠀⡀⠀⠔⠀⠀⠘⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠃⠀⠀⢸⡀
⠀⠀⠀⠒⡿⣿⣿⡗⠀⠀⠀⠀⠀⠈⠄⠀⡄⠀⠄⠨⡌⠠⠀⠀⠀⠀⠀⠰⠿⢾⣶⣿⡂⠀⠈⠈
⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⣀⡀⠀⠀⠀⠀⡄⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡶⢶⣤⣴⠀⠤⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⣀⠀⠄⠀⠀⠀⠀⠀⠃⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⣿⣧⡑⠀⠀⠀⠠⠔⠀⠰⢠⡃⠀⠀⠀⠀⠀⣴⣿⠿⢿⣿⡄⠂⠀⠀⠀⠀⠀⠀⠀⣠⣷⣀⢐⣢⣄⠀⠀⠀⠀⠀⠀⡠⠊⠀⠁
⠀⠀⠀⠐⠀⡀⠀⠀⠸⠦⠃⣼⣿⠀⠀⠀⠲⠠⠀⠀⠈⠁⠀⠀⠀⠀⠀⠰⠈⠀⠈⠸⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠉⠈⠛⠛⠃
⠀⠀⠐⠀⠄⠄⠀⠀⠀⠀⢠⡀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣦⠀⣴⣠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠤
⠀⠀⠀⠀⠀⡆⡱⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⠀⣀⠈⣉⠻⢿⣾⣿⣿⣴⣿⣿⡆⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠐
⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣦⡘⠀⠀⠀⣾⣇⣸⣿⡀⣸⣿⣧⠸⣿⣿⡧⣿⣻⡇⢿⣾⡟⠀⠀⠀⠉⠉⠋⠙⠋⢉⣠⡀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠄⠀⠀⡇
⠀⠠⠾⢿⣿⣿⣿⣷⡄⠀⠉⠁⠀⠀⠀⢿⣿⠙⣿⡇⣿⠛⢿⠆⣿⠉⠀⠿⠉⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣿⠿⠿⠓⠀⠀⡀
⠀⠁⠀⠑⠠⠾⠛⣿⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⣀⡀⢀⣀⠀⣀⣀⢀⣀⠀⢠⡄⠀⢀⣤⡀⣤⣤⣤⣤⣤⣶⣶⣶⡆⣾⣦⣿⠀⠀⠈⠂
⠀⠀⠠⠀⣀⠀⠤⠊⠀⠀⠀⠀⠀⠢⠀⠄⠀⠀⣿⣧⣼⣿⢰⣿⣿⣎⣿⣀⣼⣧⡄⣿⣭⡿⢿⣿⣿⡇⣿⣿⣿⡿⠧⠿⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠈⠀⠀⠀⠆⠀⠀⠿⠇⠸⠿⠘⠋⠙⠋⠛⠉⠉⠉⠁⠈⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⣀⢀⡀⠀⠶⠶⠖⡀⣀
⠀⠀⠀⠀⠀⠦⠀⠀⠀⢠⠀⠀⢁⠑⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⠀⣤⠄⣤⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠷⠐⠿⠀⠀⠀⠙⠥⠛⠃⠀⠀⠂⡀⡀⠈⠀⠀⠀⢠⠀⠠
⠀⠠⡄⠀⠠⢀⠃⠀⣀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢾⣄⣀⣤⡤⠢⠀⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠂⡎⠀⠀⠈⡇⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⣶⡄⠀⠀⠐⠀⠁⠀⠀⣠⣶⣶⣤⣤⣀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⢉⣉⡉⠁⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠢⠁⠀⠐⠊⠁⠁⠀⠀⠀⠠⡿⠁⢀⣉⠀⢹⡇⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡛⠉⠀⠀⠀⠀⠤⠠⡇⠀⠀⠀⠘⠿⠁⠀⠀⢀⠰⣣⡄⠂
⠀⠀⠀⠀⠀⢠⣴⣶⣤⣄⡀⠀⠀⠀⠀⠀⢀⠁⠀⠂⠙⠂⠸⠁⠂⠀⠀⣤⣠⣀⠟⠛⠛⠛⠻⣇⣀⡀⠀⠀⠀⠀⠊⠀⢀⠀⢀⣠⠦⣀⡀⠀⡈⠀⠈⠁
⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠻⢿⣬⣀⢘⣤⡿⠟⠀⠀⠀⢉⣩⠏⠉⠥⠀⢠⠈⠙⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣅⣀⡂
⠀⠀⠀⠀⠼⠟⢛⣛⣋⣛⣀⣠⣤⡀⢰⠉⡡⢼⠁⠛⠀⠷⠠⠚⡆⠀⠀⠀⠀⠢⣤⣄⣀⣀⣤⡄⠁⠀⠀⠀⠀⠀⠀⠀⠸⠛⠋⠀⠀⠀⠈⠉⠁
⠀⠰⢾⡿⠛⠛⠿⠿⠛⠉⠀⠉⠋⠀⠀⠀⣷⠀⠀⠛⠈⠀⣾⠀⠀⠀⠀⠀⢦⠰⣾⣿⣫⣳⣿⣄⠀⠀⠀⣾⣄⣼⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⢠⣶⡄⢀⠀⢶⣷⢀⡀⠀⠀⢸⡧⠀⣷⡿⣾⡄⣿⡆⠀⠀⠀⠀⡖⠀⣿⣿⣿⣿⣿⣄⠂⠀⢸⣽⣿⣻⡇⠀⠀⠀⠀⠀⠀⣴⡆⠀⣠⣄⠀⢀
⠀⠀⠅⢲⣌⣉⣁⡛⢁⣤⣤⣾⠃⠀⠀⠀⠁⣴⣿⠄⢿⣧⡙⠀⠀⠀⠀⠀⡇⠸⠿⠿⠿⠿⠿⠇⠀⠀⠀⣹⣿⣽⡏⠀⠀⠀⠠⠀⣷⣬⣀⣃⣉⣩⡄
⠀⠀⠀⠀⠙⠋⠙⠿⠛⠋⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠘⠾⠖⠀⠀⠀⠀⠀⠿⠿⠟⠁⠀⠀⠀⠀⠀⡈⠛⠛⠛⠛⠋⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠊⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠻⣝⠛⠒⠤⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠖⠚⠉⠉⠀⣠⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⣄⣼⠠⡀⢠⠏⠀⠀⠀⠁⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⣀⡤⠶⣻⠫⡀⠀⠀⠈⠓⠢⠤⠤⠤⠤⠤⠤⢖⣶⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠃⠀⢀⡿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡈⠉⠀⠀⡰⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢫⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣼⡄⡏⠀⠀⠀⠀⠀⢀⣠⠴⠒⠉⠉⠀⠀⣀⡤⣾⠀⠀⠀⠀⠀⠂⢤⣀⡀⠀⠀⣠⠖⠁⠀⠸
⠀⣀⣀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⢸⡇⠀⠀⠘⢷⣦⠶⠋⠁⠀⠀⠠⠔⠒⠋⠉⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⡽⡿⠁
⠀⠈⠉⠛⠲⡦⣤⣀⠀⠀⣠⠀⠀⠹⣇⣸⣧⠀⠀⣰⡟⠉⠳⣤⠤⠦⠤⠶⠒⠂⠀⠈⠀⠀⠀⢀⡀⠀⠀⢀⡤⠒⠁⢸⠃
⠀⠀⠀⠀⠀⠈⢢⡈⠙⠛⠻⣶⢶⣶⣿⣿⣿⣿⣾⠟⠀⠀⣀⣌⣷⣤⠤⢤⠤⠴⠃⠀⢠⡀⠀⠀⣉⠽⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣴⣾⣿⣿⣿⣿⣿⠿⢷⣞⣛⡯⠋⠁⣠⠈⠶⠤⢤⠀⢳⣄⡤⠽⠖⠉⠀⢀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⡴⠛⠉⣾⣿⣿⣿⣿⣿⣿⣿⡁⡀⠀⠉⣍⠙⢻⢓⣶⣾⣾⠾⠯⠥⠦⠤⣼⣄⣀⣀⣘⣆⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⡠⠞⠁⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠉⢷⣸⠀⠀⡼⠀⢈⠈⢦⠏⠉⡄⢀⡁⠠⣴⡁⠀⠀⠀⠀⠉⠉⠉⠑⠒⠒⠒⠲⠿⠦⠤⣤⡀⣀⡙⠢⣄
⠀⢠⠎⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠉⢸⠇⠀⢀⣿⣆⠁⠀⠀⠀⣷⠌⡀⠀⠙⡉⠀⠀⠀⠀⠀⣀⣸⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠉
⠀⠁⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣼⡠⠒⠁⠀⠈⠃⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣀⠠⠚⣿⠶⢄⠀⠀⠀⠀⠀⠢⡄⠀⠀⠀⠀⠀⠰⣤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣾
⠀⠀⠀⠀⠀⣀⣾⣣⠤⠶⠏⠑⠠⠄⡏⠀⠀⠁⠀⠀⠀⠀⠀⢻⡾⠋⠀⠀⠀⠙⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣩⠟⠳⢤⣀⠀⠀⠀⠀⢠
⠀⠀⠐⠚⠛⠉⠉⠛⠛⠛⠓⠤⡀⠀⡇⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠅⠀⠀⠀⠀⠸⡄⠀⠀⠀⠤⠒⠒⠒⠲⢦⣀⣤⣤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢤⣙⢮⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠙⢾⡏⠀⠀⠀⢀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⣠⠏⣀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢳⡀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠷⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣦⡀⣴⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣻⣿⣿⣿⣿⣎⠻⢻⡿⢁⣀⣤⡀⠀⢀⣶⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣯⠀⠐⠚⠋⠉⠉⢻⣆⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⡿⠋⠀⣠⣴⣾⣿⣿⢎⣾⣿⣿⣿⠀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣁⣤⣤⣴⣦⣭⣍⠛⣡⣿⣿⣿⣿⣿⣇⠻⢟⣵⣶⣷⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠟⣛⣩⣭⣉⠚⠛⣉⣤⣴⣤⣬⣛⠂⢈⣉⣉⡙⠻⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⢉⣴⣿⣿⣿⣿⣿⠇⣴⣿⣿⣿⣿⣿⣿⣷⡜⢿⣿⣿⣷⣦⡙⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠋⣴⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿⣿⣿⣌⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⢃⣾⣿⣿⣿⣿⣿⣿⡏⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢻⣿⣿⣿⣿⡆⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣼⣿⣿⣿⣿⣿⣿⡿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣿⣿⣿⣿⣿⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⢻⣿⠇
⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⣿⣿⣿⣿⣿⣿⣿⣇⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣇⢹⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⣿⣿⡿⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢻⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢣⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠈⠻⠿⠿⠿⠿⠟⠛⠁⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⡀⡀⠠⠀⠀⠐⠂⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⡄⠀⠀⠀⡀⢀⠔
⠀⠀⡁⠁⢨⡀⠒⠒⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋⠉⠀⠀⠐⢒⢂⣬⣤⠀⠉⡀⠘⠄
⠀⠀⠠⠈⠢⡀⠑⠢⣄⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⠀⠀⠠⠎⢉⡀⠄⠀⠀⢀⠲⢄⡀
⠀⠀⠀⠠⠀⠈⠄⠀⠀⠐⠒⠀⠘⠛⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠄⠀⠀⠀⢇⠀⠀⡐⠁
⠀⠀⠀⠀⠂⠀⠀⠁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠆⠤⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠸⡀
⠀⠀⠰⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⢢⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠙⠀⠀⠤⠀⣴⣦⡐⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠔⠒⠀⢨⡼⢤⠈⠁⠀⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠐⣀⠀⠀⠀⡄⢀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠦
⠀⠀⠀⠀⠈⠆⠀⠀⠈⢂⡄⠀⠠⠀⠀⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠨⠷⡄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⡀⠀⠀⠂⢀⡀⠸⡆
⠀⠀⠀⠀⠀⠠⠀⠂⠀⢀⠀⠀⠁⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠂⠀⠜⠠⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠐⠉⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⠤⠤⠶⢺⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢻⡄⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠘⢆⢰⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢻⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⠤⠴⢒⡾⢧⠀⣀⣘⣷⣿⡿⠖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⢀⡾⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠈⢿⡥⠼⣿⣿⣿⡙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⣠⠞⠁⠀⣼⣷⠶⠦⠤⠤⠤⠤⠶⠶⣺⠿⣆⠀⠀⠈⠳⡄⠉⠉⡇⢹⢄
⠀⠀⠀⠓⠲⢤⣀⡤⠊⠁⠀⠀⣀⡴⠋⠁⠀⣠⠜⠁⢈⣷⡤⣤⣤⣄⣀⣤⣴⡋⠀⠘⢷⡀⠀⠀⠘⠧⣄⠁⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⡏⠉⠛⢶⡶⣏⣁⠀⣀⡴⠞⠁⠀⣠⠞⠙⣧⠀⠀⠀⢀⡾⠉⠑⢤⡀⠀⠙⠦⣄⡀⢀⣈⣳⣤⠤⠤⠶⢻⡖⠒⠚⠛⠛⠛⠛
⠀⠀⠀⠀⠀⢠⡇⠀⠀⢸⡇⠀⠉⢻⡟⠦⣤⣴⣞⠁⠀⠀⠘⣧⠀⢠⡟⠁⠀⠀⠀⣙⣦⡤⠶⣾⠏⠉⠁⣼⠃⠀⠀⠀⡞
⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⠇⠀⢸⡏⠉⠛⠶⢦⣄⣼⠷⠛⣦⡤⠖⠛⠉⣹⠋⠀⣰⠏⠀⠀⡼⠃⠀⠀⠀⡜
⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⠇⠀⠀⣼⠀⢀⣾⣁⣤⣤⠴⠶⢺⣧⣄⣴⠿⣄⠀⠀⢰⡏⠀⣰⠏⠀⠀⡼⠁⠀⠀⢀⡜
⠀⠀⠀⠀⣠⡏⣀⣀⣠⣿⣤⠴⠖⠻⣟⠉⠙⢷⡀⠀⢀⡴⠟⠁⢸⣿⠀⠈⠛⣦⡟⠀⣸⡏⠀⠀⣼⠃⠀⠀⢀⡾
⠀⠉⠉⠉⠉⠛⢍⡀⠀⠈⠳⣄⠀⠀⠈⢷⡀⠀⣷⡶⠯⢤⣤⣀⢸⣿⣠⠴⠚⠉⠙⢦⣿⠀⠀⢠⡇⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⢧⡀⠀⢈⣷⣾⣭⣄⣀⠀⠀⠈⠻⣿⠁⢀⣠⠶⠛⠉⠙⢷⣄⡸⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⢿⣠⣞⣁⣀⡀⠀⠉⠙⠳⠦⣄⣿⡶⠋⠀⠀⢀⣀⡠⠴⠛⢷⣄⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⣠⡾⠋⠀⠉⠉⠉⠙⠓⠲⢤⣀⠈⣿⠀⣠⡴⠚⠉⠀⠀⠀⠀⣀⣨⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠒⠦⠤⣤⣀⡀⠀⠀⠀⠀⠈⠙⢻⠟⠁⠀⠀⢀⣠⠤⠒⠉⠁⠀⠀⠈⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠈⠉⠓⠢⢤⡀⠀⢸⡄⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢼⡷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡔⢀⢸⡇⠀⠆⠃⡄⠀⢀⠀⠀⠀⣄⠀⢠⢁⣊⠴⣫⠊⠀⠀⠀⠀⠀⠔⢠⠔⢠⠎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠈⡁⠀⠰⢠⠃⠀⠈⠀⠈⢠⠏⢀⢀⡞⢁⡔⠁⠀⠀⠀⠀⠀⣀⠖⠁⠊⠀⢀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⢻⠌⣇⠀⠀⡄⠀⠀⠘⠌⠠⢋⡰⢁⠊⠰⠋⠀⠀⠀⠂⠀⣠⡴⠋⡈⠀⠠⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠇⠀⢀⡄⠐⠉⢰⣶⣤⡁⠘⠑⠉⠀⢀⡄⠀⠀⠀⢀⡴⠁⢠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢿⢚⡄⢠⣶⣿⣾⣿⣿⡇⠀⠀⢀⠄⠋⠀⠀⢀⠔⠁⢠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠸⡆⠀⠀⠚⣩⣥⣿⣿⣿⣿⣿⡟⠁⠀⢀⢀⡤⠀⢀⡴⢃⣠⠒⡁⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⢣⡀⠘⣀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠇⢰⣦⡦⠎⢀⠴⠛⠊⠁⠀⠀⠀⠈⠂⠀⠈⡍⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠹⡀⢼⣤⡻⣿⣿⣿⣿⣿⣷⢿⣶⡗⢋⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡵⡼⡀⠙⢣⣿⣿⣿⣿⣿⣿⣼⣿⡟⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠘⠛⣵⠄⣸⣿⣿⣿⣿⣿⣿⣿⡿⠁⠔⣤⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠆⡴⢡⣾⣿⣧⣿⣿⣿⣿⣿⣿⣧⠀⠡⢄⣝⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣿⣿⣿⣿⡿⣿⣿⣿⡟⢫⡶⠻⡶⠶⡿⣶⢇⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⢿⣿⣿⢿⣿⣿⣿⣧⢸⣷⡾⢿⡂⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠈⣿⣹⣯⡿⣹⢾⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⡀⢹⣿⢿⣿⣿⣿⣿⡆⢓⣮⣤⡲⢎⣹⠤⢀⠤⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⡀⢻⡞⣹⣿⣿⣿⣿⣿⣷⠈⠒⣶⢲⡞⠁⠈⠀⠀⢴⠫⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⣿⣿⣿⣿⠷⣀⠘⠈⠃⣠⣬⣵⡀⠀⠇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣯⣿⣿⣿⡵⠋⠉⠉⠂⠐⠏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡌⣿⣿⣷⡆⢹⢀⣀⣀⠀⡸⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠀⠀⠀⣿⣿⣇⢻⣿⣿⣷⣬⠠⡀⠈⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⡇⠀⠀⢀⣿⣿⣿⢸⣿⣿⣿⣿⡁⠀⠄⠀⠀⠆⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠀⣴⣶⠎⢉⣿⣿⠈⣿⣿⣽⣿⡇⠀⢀⠢⠀⠀⠀⠀⠹⣧⡰⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⢸⣿⣿⡇⠀⢸⣿⣿⢼⣿⣿⣿⣿⣿⢠⡆⠀⡘⠤⣀⠀⠀⢿⣧⠜⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢑⡿⠤⠊⢿⣿⣿⡀⢸⣿⡿⣼⣿⣿⣿⣿⣿⣾⡇⠀⢦⠴⣽⠀⠐⡆⣟⠄⢀⡿⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠧⣚⣷⣾⣿⣿⣧⣼⣿⣧⣿⣿⣿⣿⣿⣿⣿⡇⢀⡟⠀⡈⠰⣶⣴⣿⣿⡏⡴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠈⢸⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣷⡟⠸⢁⡇⠀⢸⣿⣿⡟⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣶⡿⣶⣿⡿⣻⣿⣿⣿⣧⢻⡟⣿⡟⣿⣿⡟⢸⡀⢺⠁⣤⣾⣿⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠈⠃⠀⢨⠁⠐⣏⠉⠛⠻⠙⢧⡻⠙⠓⠹⣟⠷⣦⠁⡟⠀⠹⡈⠉⠁⠙⠲⣴⣄⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡘⠇⠀⠀⠀⠀⠀⢧⠀⠳⠀⠀⠐⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠐⠈⡉⢣⡿⣽⢤⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⢀⠀⠍⢈⡑⣆⡀⠀⣠⣄⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣬⡙⠛⠛⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⣠⣾⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⣰⣿⣿⣿⣿⣯⢀⣺⣿⣿⣿⣿⣿⣿⣿⣿⡏⣻⣿⣿⣿⣿⣿⣿⡿⣿⣷⡄
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢻⣿⣿⣷⣾⣿⣿⡄
⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠭⠥⣋⣉⣑⠦⠤⠝⠛⢿⣿
⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⢀⢀⣦⡀⠀⠀⠀⠀⣴⣄⠡⡀⢳⡀
⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠿⢿⣿⣿⡏⠀⡆⠚⠛⠃⠀⣲⣄⠈⠉⠉⠀⢳⠈⡇⠀⣀⡀⠀⠀⠀⣤⣤⡀
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⡿⠶⣇⢘⣿⣿⡇⠘⡇⢀⡀⠀⠙⠛⠛⢃⡀⢠⡄⢰⢀⡇⢘⡿⣿⡖⣖⢐⣻⣿⠇
⠀⠀⢀⣴⣿⣿⣿⣻⣿⡏⠀⠀⠈⡟⢿⣿⣧⡀⢣⠘⣿⣶⠿⣷⣶⡿⢿⠿⠃⠀⡼⢠⠃⠠⠂⣵⣠⠞⠁
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⡆⠀⠑⢸⣿⣿⣷⣤⡀⣠⡈⢠⠈⠉⠀⠀⢀⣠⣾⠁⢞⠃⠀⣤⣿⣥
⠀⠀⣿⣿⣍⢈⣻⣿⣇⣀⠂⢀⣈⣸⣿⡿⢿⢿⡟⠛⢻⣿⣿⣄⠰⢳⣾⣿⣿⠋⠴⡆⠙⠐⠛⠛⣻⡄
⠀⠀⠘⢿⣿⡎⣿⠶⣿⢾⡿⢿⠶⡿⠿⡗⢻⣿⣿⢰⣿⣿⣿⣿⡺⣁⠽⠿⣿⣶⣰⢁⠘⣿⣿⡏⣿⠇
⠀⠀⠀⠀⢹⢁⡿⢳⣿⢶⡷⢾⡶⣿⢾⡟⢻⣿⡏⢸⣿⣿⣿⣿⣿⣿⣷⣶⣉⣹⠿⢏⣙⣯⢿⠝⠁
⠀⠀⠀⠀⣿⠈⠀⡜⢹⣶⡷⢾⠶⣿⢺⡿⢻⠙⠅⣸⣿⣿⣿⣿⣿⣿⣿⣧⣽⠟⠛⣿⡿⡷⠃
⠀⠀⠀⠀⠹⣄⡜⢰⠃⠘⣷⢾⡷⠿⣃⡄⢸⣄⢀⣿⣿⣿⣿⡋⢻⣿⣿⣿⠶⣛⠷⣭⣾⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠙⠮⡛⢹⠟⣡⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠉⠤⣿⣿⣿⣿⡿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⣴⢺⠉⠻⣦⣢⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⢴⠏⣠⠿⣿⣿⣿⣷⣾⣿⣿⣿⡇
⠀⠀⠀⠀⢀⣠⣴⠚⣷⢛⡿⢘⡇⠀⠘⢻⡛⣟⣙⣿⡿⣿⣿⣿⣿⣿⡟⡉⣾⠊⠑⣤⣿⣿⣿⣿⣿⣿⣿⣿⡗
⠀⠀⠀⢰⣻⡟⣻⡛⣿⣹⠁⢈⡇⣠⠶⢿⡛⣿⣉⣿⣴⣿⣿⣿⣿⣿⠟⢿⠀⡷⡊⢸⣿⣿⢻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠈⢿⣏⣽⣯⣿⣉⠳⣼⣯⡥⢶⣾⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⠴⢹⡧⣀⠼⣿⣿⣯⣀⢺⣿⣿⣿⡿⠁
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣇⠉⠰⠛⠋⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠀⡸⡇⠉⡢⣿⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⢸⠀⠀⣿⣿⣿⣿⣿⣿⠉⠉⠉⠉⣧⠛⢄⣿⠊⠀⠈⢯⠉⠉⠉⠁
⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⢸⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⢰⢿⡀⠈⡟⣦⠊⢢⡸⣇
⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⡿⠀⢀⢧⡀⢡⡿⠀⠸⣧⡊⢁⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡄⢸⠀⠀⡿⠿⠿⠟⠛⠁⠀⣰⢲⣅⣸⢿⠃⠀⠀⢣⡙⠢⣸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⡆⠀⠁⠀⠀⠀⠀⠰⣟⡑⠯⣯⣟⡟⠀⠀⠀⢻⣶⣶⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⣤⡁⠇⢰⣀⡄⠀⠀⠀⠀⠀⠰⠖⠉⠉⠁⠀⠀⠀⠘⠿⠾⠳⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢘⣿⣦⣯⣬⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣿⠷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢶⠆
⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⣠
⠀⠀⠀⠀⠀⠿⠿⠀⠀⠀⠀⠀⠀⠶⠆⡀⠰⠆⠀⠀⠀⠀⣤⣤⣤⡀⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠶⠴⠥⠦⠶⠁⠀⠀⠀⠹⠛⠛⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠠⣀⣀⠀⠀⠀⠀⠀⣤⣤⣤⡤⠀⠀⠀⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠷⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣷
⠀⠀⠀⠀⣆⣸⣵⡎⠁⡀⠀⠀⠀⠀⣰⣿⣿⣿⣶⠀⠆⢻⣿⣿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠸⣿⣿⣿⡿
⠀⠀⠀⢰⠋⣾⠎⣄⡕⠄⠀⠀⠀⢰⣿⣿⣿⣿⣿⣶⣴⡾⠉⠁⠀⠀⠀⠀⠀⣠⠄⢀⣷⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠻⠶⣿⣿⣿⡇
⠀⠀⣠⣟⣹⠟⢺⡟⠛⠀⠀⠀⠀⠘⠛⠛⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣟⣡⡈⡇⠀⠀⠀⠀⠞⠳⠀⡇
⠀⠀⣿⡟⠁⠀⣠⢿⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡇⠀⠀⢰⡄⣄⣾⡧⠆⢸⡟⠿⠀⠀⠀⠀⠀⢀⡀⣽⡄
⠀⠀⢹⣧⣠⡾⡟⠋⣸⠀⠀⠀⠀⢠⡆⢸⡆⠀⠀⢸⡇⣧⢠⡇⢸⡷⠙⠛⠿⠃⠈⠁⠀⠀⠀⠁⠈⠈⠉⠁⠙⠋⠉⠉⠁⠀⠄
⠀⠀⠀⣿⠁⢀⡷⠚⠉⠀⠀⠀⠀⢸⣧⢾⡇⣾⣿⣼⣇⡿⠚⠓⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠒⠚⠃⠀⠀⠀⣤⡄
⠀⠀⠀⣿⣧⡞⠀⠀⠀⢀⡀⠀⠀⢸⡇⢸⡷⠏⠛⠘⠋⠁⠀⠀⠀⡀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢹⡇
⠀⠀⢀⣿⡏⠀⠀⠀⠀⠛⠋⠀⠀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰⣦⠀⣴⡆⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡈⠀⠀⣸⣇
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⢀⢒⣀⣸⠀⠀⠀⠀⠃⠀⠀⠛⠀⠈⠁⠀⠀⣹⣿⡄
⠀⠐⠛⠛⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠘⠛⠛⠋⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⣿⣻
⠀⠀⠀⠀⣠⣤⣤⣤⣦⣤⣤⣀⠀⡀⠀⠀⠀⠰⣶⢆⡰⣿⠦⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠠⠀⠀⠄⠠⠀⠀⠀⠻⠃
⠀⠀⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⣧⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠀⠡⠤⠴⠤⣟⢲⣄
⠀⠀⠘⣿⣿⣿⣿⡿⢿⣿⢿⣿⣇⠀⣿⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠠⠀⠐⠈⠻⡏⠉⡇
⠀⠀⠀⠈⣿⣼⠏⠰⠞⠁⠀⠀⢹⡇⠻⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠓⠤⡄⠀⠈⢹⠛⠛⢦⡀
⠀⠀⠀⠀⢩⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣤⣼⣷⡀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⡿
⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣀⣠⣀⣠⣴⣦⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⡆⢀⣼⡟⠁
⠀⠀⠀⠀⠀⠻⠿⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⢽⣿⣿⣿⣿⣟⢿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠙⠋⢻⣷⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠏⠉⠉⠻⠛⠛⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣈⣍⠀⠀⠀⢘⣁⡀⠀⠀⠀⠀⠀⠀⣤⣿⣿⡄⠀⠀⣿⣦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⣥⠂⠀⠀⠀⠀⠀⠀⠀⠐⣄⠀⠀⣌
⠀⠀⠀⠰⠋⠈⢁⣀⣀⠈⠉⠳⠀⠀⠀⠀⠀⢀⠀⠉⠙⠃⣀⠀⠛⠉⣀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⠠⠾⠏⢀⠀⠙⠿⠆
⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⢸⣦⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⣀⠀⠀⠴⠄⠀⠉⡀⠀⠀⠀⠀⠀⠈⠳⣄⠀⣈⣁⠀⣠⠞⠁
⠀⠀⠀⢸⣿⠇⠙⠛⠛⠁⢹⣿⡄⠀⠀⠀⠀⠀⠻⣷⣾⣿⣿⣷⣾⠟⠁⠀⠀⠀⠀⠀⠘⢶⣤⣤⣤⣤⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠁
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠒⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣨⡅⢈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⡄⠀⠀⣈⣉⠂⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣦⡄⠀⠀⠀⠀⠀⠀⠀⢢⡀⣐⠀⠀⢀⢀⠖
⠀⠀⠀⢀⠀⠀⣿⠃⠘⠟⠀⠀⡀⠀⠀⠀⠀⠀⢿⣿⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⡀⠀⠼⠿⠃⠈⠻⠏⠀⠠⡀⠀⠀⠀⢠⡀⠙⠉⠀⠀⠈⢉⡄
⠀⠀⠀⠸⣧⡀⡀⢈⠀⠀⣰⣶⠇⠀⠀⠀⠀⠀⠈⠛⠛⠁⠤⠈⠛⠁⠀⠀⠀⠀⠀⠰⣀⠀⠀⠁⠈⠀⠀⣀⡴⠁⠀⠀⠀⠀⣿⣶⣄⣰⣶⣶⣿⠃
⠀⠀⠀⠀⠙⠿⠻⢿⡼⠾⠛⠀⠀⠀⠀⠀⠀⠀⠙⣦⣤⣀⣀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠙⠋⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠
⠀⠀⠀⠠⢂⣀⠀⠀⠀⣁⣌⠑⠀⠀⠀⠀⠀⠀⢀⠠⢀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⣴⣦⡀
⠀⠀⠀⠀⣿⡋⠁⠀⠀⣹⣿⡇⠀⠀⠀⠀⠀⠀⠈⠻⠶⠂⠀⠲⠾⠋⠀⠀⠀⠀⠀⠀⡓⠀⠀⢀⣤⠀⠀⠀⡒⠀⠀⠀⠀⠀⠉⠈⠃⠀⠘⠁
⠀⠀⠀⠀⠈⡉⣀⢀⣀⣉⠉⠀⠀⠀⠀⠀⠀⠈⢳⣤⣄⣀⣁⣠⣤⡶⠁⠀⠀⠀⠀⠀⠛⣷⣤⣤⣀⡤⣤⡿⠃⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣶⡄
⠀⠀⠀⠐⠶⠿⠛⠛⠛⠻⠻⠧⠄⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠁⠈⠀⠀⠀⠀⠀⠀⠀⠿⠷⠀⠛⠨⠿⠏


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠲
⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⢚⡓⠀⠀⠀⣠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⣠⡆⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣷⡄
⠀⠀⠀⠈⠛⠛⠒⢠⡄⠴⠛⠛⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⡀⢨⡿⠻⠄⠀⠀⠀⠀⠈⠲⢦⣤⠄⠐⢶⠶⠋⡀⠀⠀⠀⠀⣠⣄⡉⠉⡤⠀⠛⠁⠀⣠
⠀⠀⠀⠠⣶⣶⣶⣶⣦⣀⣴⡆⠀⠀⠀⠀⠀⠀⢀⣀⠀⢈⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⣦⡀⣀⢘⣃⢀⣄⢾⠇⠀⠀⠀⠀⠁⠘⣿⣤⣴⣦⣄⣠⡟
⠀⠀⠀⠀⠀⠀⠘⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣿⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⢸⣿⢸⡿⠂⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠐⣶⣆⠀⡄
⠀⠀⠀⠀⠀⠀⣶⣶⣶⣿⣏⢠⡄⢄⣸⣧⣶⣿⣿⣿⣿⣶⣄⠀⣀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡏⠉⠀⢈⣿⣿⣿⣿⣿⣿⣿⠿⣿⡆⡰⠂⠀⣀⡀
⠀⠀⢀⣤⣿⣿⣿⣿⣿⣿⣿⡄⢘⣿⣿⣿⣿⣿⣿⣿⡝⣿⠿⠃⠁⠀⠀⠈
⠀⢰⣿⣿⡏⣽⣿⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⡀⠒⠀⠀⠀⠀⠀⢀⣀⡀⠀⣸⣦⡀
⠀⣺⣿⠿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⢹⣿⣷⡄⠀⠀⢠⣿⣿⡇⠀⠛⠉
⠀⣿⣿⡗⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⡿⠟⠿⢓⠀⠀⣿⣿⣿⡁⠀⠀⠀⠀⠀⣑
⠀⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣮⡓⢴⣿⡌⠀⣶⣾⣿⣿⣿⠀⠀⠀⠀⠘⢻
⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⣸⣟⣁⡀⠿⣿⣿⣿⡏⠀⠀⠀⠀⣴⣾
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣟⣿⣷⣾⣴⣿⣿⣧⣤⣴⣶⣤⡄
⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣷⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⣿⣿⣿⣋⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠘⢮⡓⠀⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢻⠿⠿⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢆⢀
⠀⠀⠀⠀⠱⣄⡀⠆⡤⠄⠀⠀⠀⣀⡄⢉⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠫⠑⠌⢃⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⢠⡓⠈⠀⢛⣡⣼⠎⠃⠰⡆⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣶⣶⣿⡇⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣄⠄⢸⠿⠋⣤⠀⠱⠀⠀⠀⠀⠀⠀⣠⢶⣶⣿⣿⣿⣿⣿⣿⣿⣿⠉⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⢣⣤⠟⠀⠀⠀⢘⣿⢀⠐⠀⠀⠀⠀⠀⠀⠟⠛⠟⠻⠿⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣈⣡⡞⢀⠀⠀⣦⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠴⠁⠢⠸⠄⠁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⡇⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠸⡇⠀⣇⠀⡇⠰⡆⠀⣧⠀⠁⢰⠀⠂⠀⠀⠀⠀⢸⡇⠀⠈⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡀⠿⠀⠀⣀⡴⠒⠀⢼⡃⠀⣿⠀⣇⠀⠁⠀⣿⠀⠀⠈⠀⡟⢀⠄⣀⠀⢼⡇⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢀⡀⠀⠉⠀⠖⠀⠈⣧⡀⢸⡄⢹⡤⣄⠴⢻⠈⢧⡀⣼⠀⣤⣀⠉⠒⢸⡇⡄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠰⠊⣧⠐⠰⣦⡔⢷⡄⠘⡇⢸⠁⠀⠀⠃⠀⠘⠀⠈⣧⢿⣄⠀⢉⠀⠙⣿⠀⣿⡀⠀⢠
⠀⠀⠀⢤⡀⣠⡆⠀⠀⠀⠀⠀⢸⡀⢀⠀⠈⢆⠀⠈⠇⠀⠃⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⣷⠘⠳⣤⣇⠴⢻⡇
⠀⠀⢠⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⡇⠋⠀⠀⠸⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⠀⠈⢷⡀
⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⡀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢞⡿⢿⡿⣿⢶⣦⡀⠀⠀⠀⠀⠀⠀⠋⠀⠀⢸⡇
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⣅⡈⠀⢀⠀⡈⣀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡠⠖⣤⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢦⠹⠿⠃⠀⠘⠻⠿⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡉⠑⣿⠛⠓
⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣞⢻⢿⣿⢿⡿⣣⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⠛⠐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣾⣿⣿⣷⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠄⠀⠀⠈⠉⠁⠀⢀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠒⠿⢏⣵⣟⢉⠍⠿⡍⢯⠉⡷⠂⢦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⠀⠀⠀⠀⠋⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠡⡴⠀⠐⢰⡀⠀⠀⠀⠀⠉⠉⠉⠒⠚⠃⠚⠚⠛⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⣀⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣩⡿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠛⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⠶⠦⠤⠤⠤⠼⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣞⡍⠀⢳⣹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣴⣎⠙⢿⣄⠈⡇⠉⣷⣦⣤⡤⣤⣄⣀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣯⣍⣛⣻⣿⣿⣿⡄⠈⣿⣆⠀⠀⢸⣿⣿⣷⣿⣿⡿⢿⣷⢯⡄⠶⠷⢿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡮⢵⣉⡥⡴⢶⣋⣯⡿⣿⣽⣿⣿⣿⣇⡸⣿⣿⣆⠀⠾⣿⣿⣿⣿⣿⠿⠿⠿⣿⣯⡻⢿⢲⢤⣙⠿⢷⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⡿⠶⠟⢛⣡⠾⢛⣉⣼⠟⣿⣿⡿⢛⡿⠗⡢⢤⣯⣹⡤⣼⣿⣉⢭⡛⠿⣷⡠⠡⡀⠈⠙⠲⢤⣀⠈⠙⠦⣍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⣰⣿⡵⢋⠀⢀⡴⠋⠀⢸⡥⣾⠿⡾⠟⠃⢠⡿⠡⣺⠁⢿⠿⢿⣇⠈⢿⣟⢷⡄⠉⠠⣢⡑⠲⣄⠀⠁⠀⠉⢗⢀⠀⠈⠳⣄⠹⣆
⠀⠀⠀⠀⠀⣼⡿⠋⠀⢉⡔⠋⢀⠔⣷⢋⠜⢁⣾⢟⡔⠃⡾⠀⢷⡇⠐⢸⠠⢸⣇⢧⠈⣏⢧⠹⣆⠀⠀⠙⠆⠢⡙⢦⡀⠀⠀⠙⠳⣄⠐⢌⢻⣿⡄
⠀⠀⠀⠀⣼⡏⠀⠀⠰⢋⡴⢁⠋⡾⢡⡏⢠⡾⠋⠎⠀⣼⠁⠰⡿⠀⠀⣼⠠⡌⣿⠀⠀⣄⠀⠱⡜⣧⡀⠀⠈⠀⠡⡀⢹⣆⠀⠈⢀⠘⣧⠀⠛⠻⢿
⠀⠀⠀⣼⡿⠁⢀⡾⠀⢘⠁⡌⡾⢁⡏⠀⡜⠠⠀⣼⠰⠃⠀⠘⡇⠀⠀⣿⠀⠀⣸⡄⠀⠸⡄⡁⠙⠘⣇⠀⠀⠀⠀⠙⡀⠙⣆⠀⠀⠂⠠⢷⡘⠆⠹⡇
⠀⠀⣸⡯⠀⢀⡾⠁⡠⠏⢸⣿⠀⡞⠀⠘⠀⠁⡸⢸⡇⠀⠀⢀⠇⠀⠀⣿⠀⠀⣿⠇⠀⠀⡁⢡⠀⠀⢹⡄⠀⠀⠀⠀⠈⠀⡈⣦⠀⠐⣧⠈⢳⡄⡀⢹
⠀⢀⢏⠀⠀⣼⠃⢰⠃⠀⣾⠁⠠⠁⠀⠔⠐⡗⢀⣾⠁⠀⠀⣼⠁⠀⠀⣟⢠⡄⣿⡆⠀⠀⢳⠀⠀⣠⠀⢻⡀⠀⠠⠠⣷⠀⠀⠘⣆⠀⠙⡧⠠⣿⡼⣼⡆
⠀⣼⠁⠀⢸⡇⠀⠠⣸⢠⠇⡆⠀⢀⡞⠀⢰⠁⢾⡟⠀⠀⢀⡏⠀⠀⠀⠛⢸⡇⢻⡇⠀⠀⠀⣷⠀⠸⡄⠈⡇⠀⠀⠀⠈⠀⠀⠀⠸⡆⠀⢻⡄⠻⢻⣼⡇
⢠⡿⠀⠀⣿⠀⠀⡇⠟⣾⠀⡇⠀⠚⠀⠄⠸⢰⣿⡁⠀⠀⠀⡀⠀⠀⠀⢿⠌⡇⢸⡇⠀⠀⠀⣿⡀⠀⣇⠀⢹⠀⠀⢀⠀⠀⢠⠠⠀⢻⡄⠈⢿⣦⡬⣿⡇
⢨⣇⢰⢠⣇⢀⠀⢱⢲⣿⠈⠁⠀⡆⢀⠀⡆⢸⡿⠀⠀⠀⠀⠃⠀⡄⠐⢸⡆⡇⢸⡇⠀⠀⠀⠈⠃⠀⢻⠀⠘⡆⠀⠈⡆⣀⠈⡆⠀⠘⣧⠀⡘⣿⢸⣿⠇
⠀⣿⡈⣸⡇⡇⠀⣼⣾⠛⠰⠀⢰⠀⡆⠃⠀⣏⡇⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⣏⣹⡇⠀⠀⠀⢀⠀⠀⠸⠀⠀⣦⠀⠀⠀⠸⢀⠿⢀⠀⣻⠀⢁⣿⠷⣿
⠀⢹⣇⢿⣧⣇⢀⠏⡿⠀⡆⠀⡎⠀⠁⠀⠀⢈⠁⠀⠇⠀⠀⠀⠐⠇⠀⢀⠀⣿⠹⡇⠀⠀⠀⠈⠀⠀⢰⠀⠀⣿⢰⠀⠀⢀⢇⢀⠂⠀⢹⠀⠘⣿⣿⡿
⠀⠀⢿⣿⣏⣿⠈⢸⣧⠇⡇⠰⡇⠀⠀⠀⠀⣘⡀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠀⢸⠀⡇⠀⠀⠀⠀⡆⠀⢸⡇⠐⠻⡎⠀⠀⠘⣿⡸⡸⢼⢸⡆⢡⣾⣿⠃
⠀⠀⠀⠻⣿⣿⣠⢸⣿⠀⢁⢠⣿⡀⠀⠀⣼⣿⡇⠀⠀⢸⠀⠀⠁⠸⠀⢸⠀⠸⢀⡇⢀⠀⠀⢸⡇⠀⠀⢡⠀⠀⡇⠀⣆⠀⢸⣧⠃⣾⣼⡇⣾⣿⠇
⠀⠀⠀⠀⠘⢾⣷⡀⣿⡃⢸⣸⡿⠇⠀⢀⡀⣷⣷⠀⠄⢸⠀⠀⠰⠄⠀⢸⠀⢠⣾⡇⢸⠀⠀⢘⣧⠀⠀⣿⠀⢀⡿⢸⠏⢆⢨⣿⢷⢻⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠈⠻⢿⣿⣧⣃⠹⣿⣿⡆⡿⠀⢻⣧⡆⠀⢸⠀⠀⠀⠆⠀⢀⡀⢸⢻⡇⣺⠀⠀⢸⡇⣇⠀⣿⣶⠘⡇⢸⡇⠾⠀⣿⣿⣼⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣆⡟⣯⢳⣥⠀⢸⣿⡇⢀⣿⡆⠈⠄⠀⡆⣿⠇⢸⢸⣇⡟⡇⠀⣾⠇⣯⢠⣿⠂⡇⣿⢀⣷⠀⣷⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣦⡟⢎⢼⣟⣇⢸⡟⡿⠀⢸⡀⣿⣸⠀⢸⣼⣿⣷⡇⠀⣿⣤⢩⣰⣟⡀⢏⣿⣸⡯⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣾⣜⣿⡾⣧⠇⣷⠃⡎⣧⣟⣧⣧⢦⣻⣿⢻⡇⡟⣻⡟⣾⡟⣿⡧⣿⣿⣿⣷⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣿⣧⣹⡆⢱⢿⣾⣯⣿⣴⣿⣿⣿⢸⣿⡏⣏⣿⣷⣏⣳⡿⠿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠾⠾⠛⠛⠛⠛⠛⠿⣿⣿⣿⣿⡿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠒⡟⠄⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠳⡈⠀⠀⠐⠒⠈⠀⠀⠀⣸⠁⠀⠀⠀⢀⡿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠳⠲⠤⢤⣤⣤⠤⠒⠋⡏⠀⠀⠀⣠⠏⠀⠀⠀⠀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠐⠀⠀⠀⡠⠊⠀⠀⢁⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⡰⠋⠁⠀⠀⣀⡴⠋⠃
⠀⠀⠀⠀⠀⠀⠀⢸⡙⠢⣤⠀⠒⠀⠀⠀⠀⠀⡠⠓⠀⠀⠀⠀⠂⠗⠀⢀⡜⠁⠀⠀⢀⠔⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡿⠉⠢⢄⣀⡠⠀⠈⠀⠈⠀⠀⠀⠀⠠⠀⢀⠎⠁⠀⣠⡞⠁⠀⠀⠀⡀⠀⠈⠢⣀
⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⡇⠀⠀⠀⡏⠢⣀⠀⠀⠀⠘⠀⠀⠀⠃⢠⠊⠀⡠⠊⠀⠀⠀⠀⠀⢀⣈⡵⡶⠒⣺⠏⠁
⠀⠀⠀⠀⠀⠀⠀⡌⠀⡸⠁⠀⠀⢀⠁⠀⠀⠛⢀⠀⠀⠴⠀⠰⢠⢀⡠⠚⠀⠀⣀⡠⢶⠖⠋⠁⠀⡸⠁⢰⠏
⠀⠀⠀⠀⠀⢀⡜⠀⣰⠃⠀⠀⢀⠌⠀⠀⠀⠀⠀⠙⢀⡀⠀⠂⠄⠈⢠⠀⠘⠋⠀⠠⠁⠀⠀⠀⠀⠃⠀⡟
⠀⠀⠀⠀⢠⠞⠀⡴⠁⠀⠀⣠⠊⠀⢀⣠⡎⠀⣤⠀⠤⠀⠰⠃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠶⢿⡒⠛⠻⢯⡉⠉⠉⠛⢥⡀⠀⠁⠀⠀⠱⠀⠒⠤⠐⠉⠀⠦⠀⣄⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠙⠢⡀⠀⠈⠑⢄⠀⠀⠙⣆⠀⠀⠀⠀⠃⠀⠤⠀⠘⠀⠀⠀⠁⠁⠐⣺⡦⢄⣀⠀⠀⣇⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠈⢢⠀⠀⢸⡀⠀⡸⠂⠀⠀⣆⠄⠐⠐⢆⠀⠀⠀⠈⠀⠀⠀⠀⣉⡱⠾⠦⢤⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠈⠆⠀⠀⡇⡔⠀⠀⠀⠀⡇⠀⠀⠀⠈⢂⠐⠀⠀⠀⢀⠀⠊⠀⠀⢀⠤⠒⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⢸⠀⢀⠟⠀⠀⠀⠠⣰⣇⠄⠀⠀⠉⠘⢣⡀⠰⢡⠃⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠸⣶⣃⣀⡀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠱⡆⠇⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⡞⠁⠀⠀⠀⠁⠢⡀⡧⢀⡀⠄⠀⠀⠀⠒⠒⠿⡄⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠒⠒⠂⠒⠤⣀⠀⢹⡟⢁⣀⠤⠀⠀⠐⠒⠲⠦⢽⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠐⢦⣴⣞⣷⣾⠛⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⡤⠄
⠀⢀⣴⣦⣤⣤⣼⣿⣿⣿⡿⠀⠀⠀⠀⠠⣤⠀⠀⠀⠀⠀⠀⢸⡟⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠃
⠀⣾⣿⡿⠟⠻⠁⠀⠙⠉⠀⠀⠀⢀⡀⠀⠀⠛⠳⠶⠀⠀⠀⣸⣧⠤⣿⣧⡀⠴⢢⡀⢰⡆⢶⠄⣴⠰⣦⠀⢀⡀⢠⡤⠀⠀⢀⡄⢠⣿⣿⣿⣿⡀
⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⣿⡏⠃⣤⢹⡇⢸⡇⢺⡆⣿⠀⣿⠀⠈⣧⡼⢡⣤⣤⣤⢶⣿⣿⣿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⢸⡟⠀⠾⠆⠀⠙⠘⠁⢿⠁⠛⠁⣿⠐⠋⠀⠀⣸⠁⠈⢿⣿⣿⣧⣝⠻⠿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡖⠀⣹⡇⠀⠀⠀⠘⠀⢀⠠⠀⠀⠀⠀⠀⠝⠁⠀⠀⠛⠀⠒⠥⠴⠃⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣶⣿⣶⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⣀⣿⡇⠀⠀⠀⠀⠘⣿⢠⣷⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢉⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡏⠉⣹⡿⣦⠰⢒⡆⠀⣿⠈⡏⢠⣶⢿⣿⣶⣶⡀⢺⠃⣸⠇⢸⡇⣴⠛⣦⠠⡾⢻⡆⢸⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⢴⡀⣷⠘⣿⢸⣿⢸⣿⣶⣿⣾⣿⠇⣿⠂⣹⡀⢸⡇⣿⠀⠈⢰⡇⠀⠁⢸⡇⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠠⠟⠳⠄⠈⠉⠁⢀⡏⢸⣟⠀⠙⠛⠛⠛⠁⠀⠀⠈⠉⠉⠉⠁⠈⠓⠁⠀⠙⠊⠀⠚⠇⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠛⠉⢹⡇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠀⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⢸⣿⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣄⣀⣀⠀⢻⣷⡀⠀⠀⢸⣿⣀⣤⣾⣿⣿⣟⠛⠛⠧⣄⡀⠀⠈⠙⠳⣍⠙⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣷⣤⡀⣰⣿⣿⠟⠉⠀⠀⠙⢳⡄⠀⠀⠉⠲⣄⠀⠀⠈⢧⡀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠟⠉⠁⠀⢀⡤⠖⠛⠋⣉⡭⢭⣿⠿⠿⣿⣿⡿⠋⢷⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠈⠳⡄⠀⠀⢳⡄⠀⢻⣿⡆
⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠀⣠⠞⠉⠀⣠⠴⠋⢁⡴⠋⠀⠀⢠⠏⠉⠀⠀⠈⣧⠀⠀⠀⠀⣀⣀⣻⠀⠀⠀⠀⠀⠙⣆⠀⠀⢳⡀⠈⢿⣿
⠀⠀⢠⣾⣿⠋⠀⠀⠀⢠⡞⠁⠀⢠⠞⢁⣀⣤⣏⣀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠸⡆⠀⣴⡿⠛⠉⠛⢷⣄⠀⠀⠀⠀⠈⣧⠀⠀⢧⠀⢸⣿
⠀⢠⣿⡿⠁⠀⠀⠀⣰⠟⠀⠀⣰⠋⣼⡟⠋⠉⠉⠉⠻⣦⠀⡇⠀⠀⠀⠀⠀⠀⢻⣰⣿⡀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠘⣇⠀⠸⡆⠀⣿
⠀⣾⣿⠃⠀⠀⠀⢰⡏⠀⠀⣰⠇⠘⣿⣄⠀⠀⠀⢀⣠⣿⢰⡇⠀⣼⣿⣆⠀⠀⠘⣟⠿⠿⠿⠿⣿⣿⡿⠗⠀⠀⠀⠀⠀⢹⡄⠀⢷⠀⣿
⢸⣿⡏⠀⠀⠀⠀⡿⠀⠀⢀⡟⠀⠀⠙⠛⣿⠛⠛⠛⠉⠁⢸⠀⣸⡟⠀⠹⣆⠀⠀⢻⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⣧⠀⢸⡇⣿
⣿⣿⠁⠀⠀⠀⢸⡇⠀⠀⢸⠇⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⢸⢀⣿⣅⣠⣤⣿⠆⠀⢸⡆⠀⠀⠀⠀⠀⠈⡇⠀⠀⢀⣠⣤⡆⢸⠀⠈⣷⣿
⣿⣿⠀⠀⠀⠀⢸⠁⠀⠀⣾⠀⣤⣤⣀⣠⣇⣀⠀⠀⠀⠀⢸⠀⠉⠉⠉⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣿⣴⣾⡿⠛⢹⣷⢸⡇⠀⣿⣿
⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⣿⡟⠛⠛⠿⠻⣿⡇⠀⣶⣾⣦⣤⣤⣤⣤⣤⣤⣤⣼⣷⣶⠀⠀⠀⢸⣿⠿⠋⠁⠀⠀⣸⡏⢸⡇⠀⣿⣿
⢹⣿⡇⠀⠀⠀⢿⠀⠀⠀⣿⠀⢿⣇⠀⠀⠀⠀⣿⣧⣄⣿⡇⠈⠉⠉⠉⠉⠛⠉⠉⠉⠉⣿⣇⣀⣤⣼⣿⠀⠀⠀⠀⣰⣿⠃⢸⡇⠀⣿⣿
⠘⣿⣷⡀⠀⠀⢸⡀⠀⠀⣿⠀⠘⣿⡄⠀⠀⠀⠉⠉⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠉⠉⠁⠀⠀⢀⣴⡿⠃⠀⢸⠃⢰⣿⡏
⠀⠸⣿⣧⠀⠀⢸⡇⠀⠀⢹⡄⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⢋⠀⠀⠀⣾⢠⣿⡿⠁
⠀⠀⠙⣿⣷⡀⠘⣧⠀⠀⠘⣇⠀⠀⠹⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⢟⣿⠟⠁⠀⠀⢠⣿⣿⣿⠃
⠀⠀⠀⠈⢿⣿⣦⣻⡄⠀⠀⢻⡄⠀⠀⠈⢻⡿⢷⣦⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⡿⠟⠛⠉⠀⠀⣿⠀⠀⠀⠀⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠙⢿⣿⣷⣀⠀⠀⢿⡀⠀⠀⠘⣧⠀⠈⠩⢛⣿⣻⣿⠿⠿⠿⠟⠛⠛⠋⠉⡇⠀⠀⠀⠀⠀⢠⡏⠀⠀⢠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣦⣄⡻⣄⠀⠀⠸⣆⠀⠀⠀⠀⠈⢯⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⡟⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣦⣄⣀⠘⢦⡀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⢀⣠⣿⣴⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣶⣤⣤⣤⣼⣦⣀⣀⣀⣀⣠⣿⣤⣴⣶⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢦⣸⣤⡄⣀⣀⠀⠀⠀⠀⠀⠉⠐⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢚⣭⣾⣿⡿⣟⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠛⠋⢡⠇⠸⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⡄⠀⠀⠀⠇⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⣼⠅⢁⣰⣇⡃⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣠⣴⣦⣄⠀⢀⠈⢻⣷⣦⣤⣤⣤⣀⠀⠀⣯⣦⠍⢧⡘⣷⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⢸⣿⣿⣿⣷⠊⠁⠰⢿⣿⣿⣿⣿⣿⣧⡀⢿⣿⣿⣿⡿⣿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⣿⣿⣿⣥⡀⠀⠠⣮⣿⣿⣿⣿⣿⡿⠀⠀⢿⣿⣿⠇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡛⣹⣿⣿⡄⠀⠈⠛⠻⠿⠿⠛⠀⠚⠃⠈⠉⠁⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⣀⣠⣾⣽⣿⣿⡇⠀⠀⠀⠐⢦⣄⠁⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠃⠚⠀⠀⠌⠀⠀⠉⠁⠀⣴⣾⣷⣦⣽⣦⣄⣠⡴⢾⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠰⠄⢠⢠⠠⡙⠿⠛⢿⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠤⠠⠤⠄⠊



⣿⣿⠛⢛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⠀⠑⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠁⠀⠀⠠⢊⣠⠤⠔⠒⠉⠁⠀⠀⠀⠘⢔⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡔⠉⠀⢀⡤⠤⠶⠤⢤⣀⣤⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⢊⢀⠔⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠚⠋⠛⠓⠲⢄⠁⠀⠞⠀⠀⠀⠈⢦⡈⠐⠚⠉⠉⠉⠉⠓⠂⢀
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠉⠉⠁⠀⠉⠈⠀⠂⠀⡀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⢄⣀⡀⠂⠈⠉⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⢀⠔⠩⠀⠀⠀⠀⠀⠀⠀⣠⠔⠒⠉⠁⠀⠀⠀⠀⠀⠀⠐⠒⠠⠤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⡀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢍⠢⣄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠌⢣⡀⠀⠀⠀⠀⠀⠳⡄
⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠹⡄
⠀⡼⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢳
⢰⠇⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠘
⣼⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⡟⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⡇⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⡇⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢀
⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣸
⠈⡇⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⢠⠇
⠀⠘⡄⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⢀⡞
⠀⠀⠘⢄⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⢀⠎
⠀⠀⠀⠈⠣⡀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠈⠂⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⢀⡀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠖⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⢏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠒⠒⡀⠶⣦⣤⢀⡎⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⣀⠀⠀⠀⠤⠛⢿⠓⢆⡈⠑⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⢸⡁⠑⠄⠀⠀⠀⠀⢀⡀⠈⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⢸⠀⠀⠤⠀⠙⠒⢂⠤⡀⢶⡊⠉⣱⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠇⢸⢀⡠⠤⠼⡄⡄⣓⡘⡆⣀⡳⠤⣂⢀⣈⠉⡁⠀⠀⠀⠀⡸
⠀⠀⠀⢠⠖⠀⠈⡉⠛⠚⠁⣀⡀⠀⠙⣮⢸⡑⣣⠤⠐⠚⠤⣌⡠⠀⠁⠀⠀⠀⡰⠃⠂⠄⡀
⠀⠀⢠⠃⡀⠀⠯⠬⢂⡤⡘⠓⠚⡤⠒⣶⣿⡏⠀⠀⠀⠀⠀⠘⣄⣀⣄⡨⠔⠊⠀⢀⡄⠀⠀⠢⡀
⠀⠀⢸⠀⠁⢰⠖⣆⡤⠭⠤⣄⡜⠀⢠⠄⣬⠁⢠⠀⠄⠤⠀⠀⡏⠉⡀⢀⡀⠀⡄⠠⠃⠀⠀⠀⢣
⠀⠀⠈⢧⡂⢀⠧⢀⠀⠀⠀⣀⡇⠀⢈⠭⢤⠤⠭⠀⠀⠀⢄⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠙⠾⡆⢈⠩⠛⠉⠉⠉⠉⠙⠲⡊⠑⠐⠚⠑⠒⠈⣼⠀⠀⠈⠠⠈⠁⠀⡀⠀⣼⢀⡼⠓⠒⠲⢤⡀
⠀⣠⠔⠊⠉⠁⠉⠬⠆⠀⠀⡀⠀⠀⠀⠀⠘⣄⣀⣀⣀⠤⠞⠉⠉⠑⢦⡁⠐⠂⠁⣀⡼⠛⠋⠀⠀⠀⠀⠀⠙
⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢃⠀⠀⠀⠀⣀⠠⠒⠂⠠⢹⠒⠚⠉⠁⠀⠀⠔⠊⠉⡂
⡇⠀⠀⠀⠀⢀⢀⠀⢠⠀⠤⠂⠀⠀⠀⠀⠀⡮⢉⠩⠉⢁⡑⠒⠋⣤⠀⢠⠂⠄⠀⠀⠐⠀⠈⠉⠉
⢇⠀⠀⠀⠀⠀⠈⡉⠀⠀⠄⠀⠀⠀⠀⠀⡰⠁⢀⢀⠀⠃⢞⢀⣀⡏⠀⡾⠀⠀⠀⢀⠀⠈⠠⠀⠀⠀⠀⠀⢀
⠘⢄⠀⠀⠀⠀⠀⠀⢀⢀⡀⠀⠀⣀⠠⠊⠱⢄⣀⠀⠄⢀⠡⠴⠋⣠⠞⢧⡀⠀⠁⠀⠠⠂⢀⡀⠁⣀⣀⠴⠋
⣀⣀⣑⣢⣤⣄⣠⣤⣒⣉⣉⣉⣁⣀⣀⣀⣀⣀⣉⣑⣒⣒⣒⣒⣉⣀⣀⣀⣉⣓⣶⣦⣴⣒⣉⣉⣉⣉⣀⣀⣀
⣿⣿⣉⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡀⢈⠀⠀⠀⢠⡤⢄⠖⠒⠒⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠄⠤⠤⠦⠴⠐⠢⢤⡁⠀⠠⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⢀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⡀⢡
⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠠⡀⠀⠀⡄⠀⠉⢁⠠⡀
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠠⣀⠀⠱⡀⡸⠀⡠⠔⠊⠀⠈⢆
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢄⡀⠀⢀⣀⠀⠀⢑⡆⠃⢳⡮⠤⠤⠤⠀⠀⢸
⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⡯⡏⢍⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠈⢱⣦⣀⠀⢀⠴⠃⠆⢣⠉⠒⢤⡀⠀⠀⢈
⠀⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⢠⡒⠉⠀⠠⠀⠡⠀⠉⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⡍⢉⠙⠡⡀⢸⠀⠀⠠⠀⠀⣀⠄⠁⠌
⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠐⠀⠀⠆⠀⠀⠆⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠹⠀⠀⠄⠒⠀⠀⠀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠒⠓⢄⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⠈⠀⠴⠀⠀⠀⠠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉




⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠠⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⢀⠔⠊⠁⠀⠈⠄⠀⠀⠀⣂⠉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠐⢤⠐⠒⠀⠂⠐⠀⠂⠀⠈
⠀⠀⠀⡔⠡⠐⠦⣄⠀⠀⠸⡀⠀⣰⠃⠀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⡜⠀⠀⠀⠀⠈⠙⢦⡀⢧⠀⡇⢀⡠⠖⠉⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡄
⠀⠀⣷⠀⠐⠒⠒⠦⠤⠤⢽⡊⠀⠀⣋⠀⠀⠀⣀⣀⠀⠃⢀⡤⠞⠈⢂⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠠⡀⠀⠀⠄⠀⠈⢁⠢⡀
⠀⠀⡏⡆⠀⠀⠀⣀⡤⠚⠁⡼⠱⡈⠦⡀⠀⠀⠀⢀⡵⠚⢁⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠠⣀⠀⢱⠀⡜⢀⡠⠔⠉⠀⠈⢂
⠀⠀⢣⠘⢄⠁⠉⠁⠀⠀⠜⠀⠀⢧⠀⠈⢑⡤⠖⠧⡫⢍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢑⡄⠒⢱⡭⠤⠤⠄⠀⠀⢸
⠀⠀⠈⣆⠀⠑⠤⣀⠀⠈⠀⠀⠀⢸⡤⠖⠉⠀⠸⠀⢡⠀⠉⠢⠄⠀⡘⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⣀⠔⢉⠊⢦⠉⠒⢤⡀⠀⠀⢡
⠀⠀⠀⠈⢦⡀⠀⠀⠉⠒⠒⠤⠤⡝⠑⠄⠀⠀⡄⠀⠀⢆⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⡀⠰⣄⠐⠊⠀⠀⡜⠀⠀⢢⠀⠀⡀⠔⠁⠌
⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⠀⠀⠀⣢⡀⠐⢄⠀⠁⠀⠀⠘⠄⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠑⢌⠐⠀⠤⠀⠀⠀⠄⠂⠀⠀⣀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠈⠒⠠⠁⠢⣤⠤⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠄⠀⠀⠀⠠⠐⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⣀⠀⠀⠀⡚⠃⠀⠀⠀⠀⠀⠀⢀⡰⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⣀⠀⠀⢠⣶⣶⡆⠀⠀⠸⣷⠀⠰⣁⣺⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠰⠶⢿⠟⠋⠀⠀⣼⣿⣿⣿⡀⢠⠞⠋⢁⡀⠉⡛⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣆⠀⣶⣄⠀⠀⠰⣄⣴⣿⣿⣿⣿⣷⣄⡆⠀⠸⣄⣾⡿⠀⠀⠀⠀⠀⠰⡇⡠⠁
⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⠈⣹⣿⣿⣦⣀⡀⠀⣿⣿⣿⣿⣿⣿⠀⢠⣤⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⡀⠀⡄⠀⠀⠆⠠⠴⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠢⠸⡇
⠀⠀⠀⢠⠁⠘⣇⡴⠋⠀⠐⠞⢻⣿⣿⣿⣿⡆⠀⣿⣿⣿⣿⣿⣿⡄⢸⣿⣿⣿⣿⡟⠆⠀⠀⠀⠀⠐⢧⣷⠠
⠀⠀⠀⢸⡀⢸⡟⠀⢰⣶⣶⣶⣾⣿⣿⣿⣿⣷⣸⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣧⣤⣤⡄⠀⠀⠀⠈⣿⡄⡀
⠀⠀⠀⠀⠑⢼⣇⠌⣼⣿⣿⣿⣿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠓⣿⠏
⠀⠀⠀⠀⠀⢸⣯⠘⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⣀⢰⡷⠀⣿
⠀⠀⠢⠘⢌⠙⣿⠀⠁⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠁⢀⣿⠀⠋
⠀⠀⠀⠀⣾⠀⣿⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⠀⠀⢸⡟
⠀⠀⣧⠄⢹⠀⢿⣇⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⢺⠃⠀⣾⡇⠀⢰⠇
⠀⠀⢈⣧⣼⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣼⣀
⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣶⣶⣶⣶⣶⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⡶⢶⣶⣶⡶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠐⡟⠁
⠀⠀⠀⠀⣀⣤⣤⣴⡒⢒⡴⠇⠙⠧⣤⡄⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⠠⢄⠇⠻⠣⣄⣠⠤⣀
⠀⠀⣤⡞⣡⠞⠁⠀⢩⠟⠁⠒⢶⡞⠉⠉⠻⡍⠉⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⢉⠔⠉⠀⢈⠝⠋⠀⠲⣶⠌⠁⠺⣿⠉⠑⣄
⢀⣾⢻⣿⠃⠀⠀⢀⡏⠀⠀⠀⠀⢻⠀⠀⠀⠘⢆⠀⠀⢣⠀⠀⠀⠀⠀⠀⡰⠁⢀⠏⠀⠀⠀⡌⠀⠀⠀⠀⠈⡆⠀⠀⠈⠇⠀⠈⢄
⣸⠁⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠆⠀⠀⠀⠈⣆⠀⢸⡆⠀⠀⠀⠀⠀⠃⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠘
⣿⡀⣿⡇⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⡀⢸⡇⠀⠀⠀⠀⠸⠀⠀⡁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡀
⢸⣷⣿⣿⡀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⡇⣼⠃⠀⠀⠀⠀⠈⣧⠀⢇⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⢠⠀⢠
⠀⠻⣿⣿⣧⣢⡀⠀⢿⡀⢀⠀⠀⠀⣸⣿⣦⡀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⠘⣄⠘⡄⢠⠀⠀⢇⠀⠀⠀⠀⠀⢠⠇⠀⢀⢀⡞⡠⠃
⠀⠀⠙⠻⠿⣿⣿⣷⣾⣿⣿⣿⣶⣾⣿⣿⣿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣦⣐⣶⣼⣦⡀⠀⠀⣀⣾⣥⣤⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠋⠉⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣿⡿
⠀⠀⠀⣀⡀⢀⣒⣚⠛⠃⣚⡿⠷⠄⠲⠶⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡤⠖⠒⠛⣿⡿⠿⠷⣿⣾⠿⢶⣤⣤⣄
⠀⢠⣾⢏⣴⣿⣿⣿⢠⣿⣿⣿⣷⡘⣿⣿⣦⡹⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢠⣾⣽⠏⢠⠂⣤⢃⣿⠄⠀⠀⠈⢷⡗⢀⠈⢿⡟⢳⡄
⢰⣿⠏⣾⣿⣿⣿⡇⣿⣿⣿⣿⣿⣷⠸⣿⣿⣿⣎⢻⣿⣇⠀⠀⠀⠀⠀⢀⡟⣹⠏⠠⠝⣷⡏⢸⣿⡇⠀⠀⠀⠘⣿⡀⢀⠀⢷⣇⢳
⣾⣿⢰⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⡇⢿⣿⣿⣿⠈⣿⠿⠀⠀⠀⠀⠀⢸⣽⣿⢀⢠⣵⣿⡇⣿⣿⣷⠄⠀⠀⠀⣿⣒⣣⠀⠈⣿⣿
⣿⣯⢸⣿⣿⣿⣿⢸⣿⣿⣿⣏⡿⢿⡇⢸⣵⠁⢱⡆⠸⠟⡅⠀⠀⠀⠀⠸⣿⣿⢸⢼⣿⣿⣇⣿⣿⡿⢺⣶⡇⠀⣿⣿⢯⢢⢀⣿⢻
⢻⣿⡘⣿⣿⣿⣿⠐⣿⣿⣯⡟⠟⣾⡗⢸⠁⠐⢔⠀⠀⡙⠀⠀⠀⠀⠀⠀⢿⣿⣆⢿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣿⣿⣯⢾⣼⣿⣾
⠈⢿⣧⡹⣿⣿⣿⣆⢻⣿⣏⡛⠚⡛⡇⠀⠀⡁⠈⠠⠉⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣮⣽⣿⣿⣿⣿⣿⣿⣿⣿⣢⣿⣿⡿⢠⣿⣿⠃
⠀⠀⠙⢷⣌⡻⣿⣿⡌⠻⡿⡿⠌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠷⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡟⢉⣿⣿⠋⠻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣧⣾⣻⣿⡀⣰⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣈⣉⠉⣻⡟⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⠶⠒⠚⣼⣯⣭⣭⣤⡛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⢀⣼⣿⣿⣿⣿⣿⣿⠀⠀⢈⣵⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣯⣶⢀⣼⡟⠉⠙⢿⠟⢠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢾⣟⣠⣴⣶⣶⠾⡷⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⠿⠛⢶⣿⣷⣴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠁⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡓⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡇⠀⠀⢻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣤⡾⣾⡅⠀⣹⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠀⠈⠀⠀⠀⢻⣿⠖

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡿⣿⡟⠻⠟⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠀⠀⠀⠀⠀⢠⣶⠖⢲⡦⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠘⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠏⣿⡇⠀⠀⠀⠀⢳⠛⠀⢀⠀⠪⣤⡀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀⣀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠛⠠⣿⠃⠀⠀⠀⠀⠈⢣⣤⠞⠀⠀⠺⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⡀
⠀⠀⢠⣶⣿⣿⣿⣿⣶⣶⣤⠀⠀⠀⠀⢸⣿⣍⠁⠀⢈⡟⠀⠀⠀⠀⠀⠀⠀⡰⠆⠁⣠⠀⢫⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⠈⣆
⠀⠀⠀⠈⠿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⡏⠀⠐⢼⠇⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁⠂⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠀⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠈⠛⠆⠉⣿⠿⣿⣿⣿⣿⠀⠀⠈⢿⣿⣆⣰⠋⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠈⣿⣿⣿⡟⠀⠀⠒⠀⡀⢰
⠀⠀⠀⠀⢀⣤⠶⣿⡿⠿⣿⣿⣿⡇⠀⠀⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⣳⠂⠀⠀⠀⠀⠀⠀⠀⡀⠖⠀⠀⣘⣿⣿⣏⠁⡌⠀⠀⠀⠐⢸⠃
⠀⠀⠀⡴⠛⣁⣀⣄⣈⣉⠿⠋⠁⠰⠶⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠤⠚⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣠⣤⣿⣿⣿⣿⣿⡷⠿⠄⠀⠀⠀⢸
⠀⠀⠈⠙⠳⠯⠽⠟⠉⠁⠐⠒⠀⠀⠀⠀⠁⠚⢦⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣿⣻⣿⣿⡿⣿⣿⡯⠁⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢱⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠄⡽⢹⠟⠲⠙⢷⠑⠀⠀⠀⠀⡀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣄⠀⠠⣄⠀⣤⠀⠀⠀⠁⠀⠀⢐⠀⠀⠐⠁⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⡲⠤⠀⠀⠀⠠⠄⠀⠐⠒⠁⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠦⣤⣤⣤⣀⣉⡉⠤⠶⠚⠉⠁
⠀⠀⠠⣤⣤⣚⡛⠋⠉⠛⠛⠻⠿⣦⣀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠙⠻⣷⣤⡀⠈⢉⣙⠛⠿⣟⠪⣿⣦⣴⢘⣉⡋⠀⠐⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠈⠙⠿⠿⠿⠿⣿⣿⣿⡷⡻⢯⣿⣿⣦⣤⣄⡀⠀⠈⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠚⠋⠉⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠙⠻⢿⣿⠟⠂⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⣿⡇⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠀⠀⠀⠀⠀⢠⡏⠀⢀⡀⠀⠀⠀⠀⣶⣤⣼⣿⣿⡅⡂⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣿⣿⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢶⣿⣿⡿⠏⠋⢻⣿⣿⣧⠀⠀⠀⠀⠀⢲
⠀⠀⠀⠀⠀⠀⢿⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣾⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣹⣿⣧⣄⣀⣼⣿⣿⡗⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣿⠇⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠈⠁⠻⢛⣿⡿⢿⣿⠟⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⠂⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣤⡀⣀⠀⠀⠀⠀⢀⣀⣀⢰⣶⣩⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢦⣄⣀⠀⠀⠀⣀⣀⣤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠁

⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢲⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢖⠒⠲⠒⠒⠒⠲⠒⠒⠒⠒⠒⣒⣒⠒⠒⠒⠒⠒⠒⠒⠒⡖⠒⠒⠒⠲
⣹⠀⠀⠈⠃⡀⣀⠖⠀⠀⠀⠀⠀⠐⠉⡐⠉⠉⠋⠃⠁⡜⠉⠀⠀⠀⠀⣼⣿⣿⣷⠀⠀⠀⠀⠀⠀⠄⠙⡁⠀⠀⠄
⢹⠀⠂⠀⣰⣇⠉⠀⢋⠀⠀⣀⠀⠀⠀⠀⠠⠔⠒⠒⠏⠥⠐⠀⠀⠀⣰⣿⣿⣿⣿⠀⠀⠀⠤⠂⠀⠀⠄⠑⠀⠎⠐
⢸⠉⠀⢲⣷⠆⠀⠀⠀⡤⠂⠉⠂⠄⠀⢀⡁⠀⠀⠠⣀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⡄⠀⠀⠀⠘⠀⠰⠇⠠⠀⢘⣄
⢸⠀⠀⣸⣷⠙⠂⠀⢛⠉⠈⠲⠀⠀⠐⠘⡅⠀⠀⠀⠈⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠐⢢⠞⢠⡊⢰⡬⠉
⢸⣤⠤⣿⣧⡀⠀⢠⡾⠆⢀⣀⡄⠤⠀⠀⣉⠈⠀⠀⠆⡀⠀⣠⣿⣿⣿⣿⣿⣿⣿⡇⢀⠀⠀⠐⠀⡐⣯⣉⠉
⢸⠀⡄⢘⣧⡄⣤⠞⢀⡴⠉⢁⠈⠐⠳⡀⢩⡶⠉⠙⠓⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡅⠀⠐⠀⠉⠀⠀⢁⡩⣦⠀⢂
⢸⣀⣷⡼⠷⠷⠥⠶⠊⢠⣄⠤⠐⠴⠃⠈⠉⠀⠀⠀⢀⣰⣯⣤⣀⡁⠀⠠⠈⠛⢿⡀⠀⠀⠀⠀⠀⠀⠈⢀⠑⡶⣩
⢸⠤⠤⠤⠤⣠⠌⠃⢒⠌⠁⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠠⠉⢬⠇⣽
⢸⠐⠂⠘⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡇⠐⠋⠁⢻⣿⠻⢿⠟⠩⠿⣿⣿⣦⣄⠀⠀⠀⢰⠀⠀⠀⣟
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠀⢐⡀⠀⢸⡿⠀⠐⠀⠀⠀⣿⣿⢿⣿⡷⠆⠀⠸⣇⠀⢀⡙
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠞⣀⣑⠄⡾⡇⠀⠀⠀⠀⣀⠈⠁⠘⠃⠠⢀⡀⠀⢹⣄⠛⢃
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⣵⠀⠀⣠⠀⠀⠁⠀⢀⠄⠀⠢⡀⠑⣄⠀⠻⣯⠺
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⠔⠾⢃⣤⡟⠁⠄⠀⠀⢠⠊⠀⠀⠀⠀⡀⠈⠂⠀⠀⠛
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠀⠀⠀⢀⠏⠀⠀⠀⠀⠀⢰⡄⠠⣠
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠐⢁⠀⠀⣠⣿⠀⠀⠀⠀⠀⢠⡾⣿⣶⣬⡢
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡘⠀⠀⡸⠋⢀⡼⠷⣏⢧⠀⠀⠀⠀⠘⡿⣿⣿⣿⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢣⡴⣯⡝⠷⣍⠛⠀⠀⠀⠀⢠⡟⣿⣿⣿⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡓⠀⠀⣴⡻⣦⡙⢶⣝⢃⡀⠀⠀⠀⢸⣛⣿⣿⣿⡇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠄⠨⠉⢭⣿⠀⣿⡟⠤⣛⡾⣬⠛⣁⡀⠀⠀⢸⣽⣿⣿⣿⣇
⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣍⡑⠄⠩⠶⠀⠄⣦⣤⣤⣼⣿⣷⣬⣿⣷⢪⡽⡷⠛⢲⠭⠳⣦⠀⢹⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⡏⠈⠀⢤⣖⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠁⢀⠉⠉⠉⠀⠀⠀⠀⠀⠢⠀⠁⢸⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⠀⠀⠀⢱⣾⣷⣶⠉⠡⠄⠠⠀⢾⣿⣿⣿⠀⠀⠀⠀⠣⡄⠀⠀⠀⠀⠠⠀⠀⠀⢀⡜⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⢠⣤⣶⠯⣵⣿⢿⣷⣤⣀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⢀⣤⣾⢇⣿⣿⣿⣿⡆
⢸⠀⠀⠀⠀⣮⣥⣬⣥⣥⣼⣿⣼⣿⣿⣿⣿⡂⣸⣿⣿⣿⡇⠤⠀⠤⠀⠀⠀⠠⠀⠀⣠⣶⣿⠟⣿⢸⣿⣿⣿⣿⠇
⢸⠀⠀⠀⠀⠸⣿⣿⣿⣦⢿⣯⣿⣿⣿⣿⣿⡅⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⣽⣧⣛⡿⢸⣿⣿⣿⣿
⢸⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣻⣿⣯⣿⣿⠀⠸⣿⣿⣿⣇⠀⠀⠀⠀⢀⣠⣾⣿⣣⡟⣼⣿⢊⡶⢸⣿⣿⣿⣿⠆
⢸⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣺⣻⣿⠿⣿⡟⠀⠀⢹⣿⣿⣿⣀⣤⣤⣶⣿⢿⣽⣿⣷⣿⣾⣟⣏⡷⢸⣿⣿⣿⣿⡆
⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣭⣿⣿⣻⣿⠇⠀⠀⠘⣿⣿⣿⣟⠉⢰⢎⡷⣿⣿⣿⡉⢹⣿⣿⣯⡗⢸⣿⣿⣿⣿⡇
⢸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣿⣿⣿⡇⠀⠀⠀⠙⠋⡏⠁⠀⢸⣯⡷⣻⣿⣯⣷⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⡇
⢸⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠋⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢘⣯⣴⢻⣿⣿⣿⣿⣿⣿⡿⡇⢸⣿⣿⣿⣿⣷
⢸⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣴⡧⠤⡤⡿⢮⠽⢧⠾⠿⠭⠿⠽⠧⠷⠦⠼⠿⠿⢿⣿⡧

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠖⢛⡛⠓⠲⠶⢤⣀
⠀⠀⠀⠀⠈⢻⣿⣶⣦⣤⣀⣀⣀⣀⣀⣀⣀⣤⣤⠶⠟⠋⠁⠒⢤⣀⣭⣽⠿⠶⠶⠶⠾⣿⣦⣄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣬⣉⠉⠉⠉⠉⠉⢉⡁⠀⠀⢒⣲⣶⢶⠟⠉⣡⠏⠀⠀⢀⣠⡴⠶⠾⢿⣷⣦⡀⣴⠿⠈⠙⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣍⠓⠲⠶⠶⠶⠾⢒⠿⠋⠉⡰⠋⠀⠰⠃⢀⣤⡾⠋⠁⠀⠀⠀⠀⠈⠙⣿⠏⠀⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢶⣤⣄⣀⠀⠁⠀⠀⠈⠀⢀⣀⣤⠾⠛⢁⣀⣀⣀⣀⣀⣀⣀⡀⣰⡟⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿⠿⠟⣛⣻⣿⣶⣾⣿⣿⣿⠿⠿⠿⠿⢿⣿⣿⣿⠁⢀⣾⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣼⣇⣠⡿⠻⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠛⠛⠦⠀⠈⠙⠿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣶⣦⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⢿⣿⣿⣿⡿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠤⠤⢤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠉⠀⠠⡶⠦⠀⠀⠀⠈⠉⣳⡤⠶⠒⠛⠛⠛⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⣰⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⢀⠄⠀⣄⠉⠉⠁⡄⠠⠉⠲⣄
⠀⠀⠀⠀⠀⡼⠁⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡟⠀⢀⠁⠀⠀⠘⣆⠀⠀⡇⠀⠀⠑⡈⠳⡄
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢠⡇⢀⠂⠓⢤⡀⠀⠘⣄⠀⡧⠀⢀⡞⠁⠐⠙⣆
⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡄⠠⠀⠀⠀⠈⠓⢤⡘⢆⣷⢠⠎⠀⠀⣀⣤⠘⣦
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠈⡀⠤⠤⠤⠤⠤⠭⡎⠁⠙⣖⡋⠉⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠂⠀⠹⡄⠂⠀⠀⠀⠀⢀⠴⢓⠖⡞⢦⡉⠉⠒⠦⠤⠂⢻
⠀⠀⠀⠀⣠⢷⣤⣀⣀⡀⠀⠀⠀⠀⠀⡀⢷⠀⠀⠀⠱⡀⠂⠀⢀⡔⠁⠀⣸⠀⠹⡀⠙⢦⡀⠀⠀⠄⢸⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠈⢉⣉⡻⢖⠲⢦⣤⣼⣇⣀⡀⠀⠙⣄⠀⡀⠀⠀⠀⡏⠀⠀⢳⡀⠀⠈⠲⠄⠀⣸
⠀⠀⠀⠀⢻⠀⠢⠖⠋⠉⠀⣠⠋⠟⡇⠙⠦⣄⠈⠉⠉⠛⠟⠲⣦⠢⢀⡰⠁⠀⠀⠀⢣⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠸⡇⠀⠀⠀⢠⠞⠁⢸⠁⢹⠀⠀⠈⠙⢦⣀⠀⠀⢠⢻⠳⢄⡀⠑⠀⠀⠀⠘⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⢻⡄⠠⠔⠁⠀⠀⡸⠀⠈⣇⠀⠀⠀⠀⠁⠀⡰⠃⣼⠀⠀⠉⠑⠒⠦⠤⠴⣲⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⡀⢀⡇⠀⠀⢸⡀⠀⠠⠊⣠⠞⠁⣰⠋⠙⠓⠒⠒⠒⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⡀⠐⠀⠀⠈⠈⣀⡤⢚⣁⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⠟⠛⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠒⠒⠢⠤⣀⡤⠴⠒⠲⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⢀⡞⠉⠑⣄⠀⠀⠀⠀⠈⠳⣄⠤⠄⠒⠒⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠈⣆⠀⠀⠀⠀⣴⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠻⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣠⠏⠀⠀⠀⠀⠹⠤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⢈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⢻⣿⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⣄⡀⠀⠉⠉⠉⡇⠘⡿⣇⠀⠀⣿⠆⠀⢈⣽⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⢠⣿⣷⣤⡀⠀⠇⢠⣿⢻⣆⠀⢋⣠⡴⠿⠛⠛⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⣠⠤⠒⠛⠉⣩⡴⠖⠁⠀⠀⠀⠀⢼⡻⢸⣿⡿⠿⠋⢀⣀⠀⠒⠚⠉⠛⠓⢦⣄⠙⢷⣄
⠀⡰⠋⢀⡴⠋⠀⠀⠀⢠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠁⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠈⢷⡈⣿⣧
⣰⠁⢠⠏⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⡀⠀⠀⠀⠈⣿⣽⣿⡇
⡇⢠⡏⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡄⠀⠀⠀⣿⣿⣹⡇
⣧⣼⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⣿⣿⡏
⢹⣿⡇⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⣼⣿⣿⡇
⠈⢿⣿⡀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣿⣿⣿⠃
⠀⠀⢹⣿⣄⠀⣠⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠻⣿⣿⣿⣿⣿⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⠸⠛⠁
⠀⠀⠀⠀⠈⠙⠿⢿⣿⣧⠻⣦⣄⡀⠀⢀⣀⣴⣿⣿⣿⡿⠀⠀⠀⠀⣀⣠⣾⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠘⠿⣿⣿⣿⣿⣿⣿⣿⠿⠋

⠀⠀⠀⠀⠀⢀⠄⠀⠄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣠⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⡟⠃⠘⠀⠀⠀⠀⢀⣾⠟⠠⢡⣴⢦⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⢡⠀⢄⠈⠁⢀⡀⠀⠟⠀⢁⠀⠉⠸⠀⠀⠀⣼
⠀⠀⣸⡿⠾⠴⠤⠤⠀⣽⣿⠀⣤⡧⡂⢰⡿⢁⠆⠀⠀⣴⠛⢳⠀⢀⠔⠀⠀⠀⠀⠀⠁⠘⠄⠀⠀⠀⠀⠀⠂⢄⠀⠛⠀⡶⠀⠾⢀⣼⠇
⠀⠀⠁⠀⠀⠀⢀⣀⡀⣤⡌⠁⢯⣱⡜⠀⠃⠰⡇⠀⠀⠈⠛⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⢀⡄⠀⠀⠀⠀⠑⠠⢄⣀⡴⡻⠋
⠀⠠⠄⠒⠐⠈⠀⢠⢾⣿⡇⠀⠀⠉⣈⣀⣠⠀⠙⣦⠀⠀⠁⠀⠁⠀⡀⠀⡁⠀⠄⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡡⠔⠒⠂
⠀⠀⠀⠀⠀⣀⠚⣱⢫⠇⡃⠐⢲⣟⣿⣿⡝⠓⠀⢸⡇⠀⠂⠀⠀⠡⠠⠀⠀⠀⠄⠂⠀⠐⠀⡂⠠⣆⠀⠉⠉⠑⠆⠀⠀⣠⣴⠎⠁⠀⢀⡄
⠀⠀⠀⠠⠎⠀⡼⠁⡮⠀⡂⠀⠉⠀⣉⠁⣀⣀⠀⠾⠃⠀⠄⠀⠀⠀⠐⠀⠀⠀⡀⠈⠀⠀⠀⠑⢈⣊⠦⠐⠀⠈⠀⣴⣭⠀⠀⠀⣀⠀⠘
⠀⡠⠏⠁⠀⡜⠀⠰⠁⠀⠂⠀⠀⠀⠴⢿⣿⣿⣷⠀⠀⠐⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢠⡳⠀⠶⢸⡏⢷⣤⣀
⠀⠀⠀⠀⠉⠀⠀⢀⡤⠖⠉⣠⡄⠀⡀⠀⢉⠉⠀⠁⠀⠐⠀⠂⠀⠐⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠂⠀⠑⠀⠁⠐⠀⠐⡀⠐⠀⢘⡀⠨⡹⢿
⠀⠀⠀⠀⠀⣠⠶⠋⠀⢀⠀⢉⣥⠚⠇⠀⠈⠈⠁⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠘⠀⣃⠐⠀⠀⠀⠀⠐⠀⠐⠀⠀⠀⠀⠀⠈⢅⠐⢛⠴
⠀⠀⠀⣠⠞⠁⠀⠀⠀⢠⠀⠈⢃⡖⠀⢰⠂⠀⠠⠾⠇⠀⠀⠀⢀⠖⠁⠀⠄⠀⠀⠀⢀⡹⠇⠀⠠⠄⠀⠀⠀⠀⢀⣀⣤⣄⣀⠀⠀⢠
⠀⢠⣾⡥⢀⠀⠀⠀⠀⠘⠂⠀⠈⠷⠀⠁⠀⠀⠀⠀⠀⠀⢀⣴⣁⡀⠀⠋⠀⢀⠎⠀⠀⠠⡦⠀⠀⠷⠀⠀⠀⡐⠙⠃⠀⠟⠀⠳⠄⢮⡀⡇
⠀⠁⢋⡴⣣⢂⡔⢀⠀⠀⠀⠀⠀⡠⠛⠀⣴⣀⣀⠀⠀⡠⠾⠻⠿⠿⠿⢇⠀⠸⡀⠀⠀⠀⣠⣠⣤⣴⠄⠀⠀⠁⠰⠀⠠⠄⠀⠀⠀⢘⡖⢻
⠀⠀⠈⠘⠁⠚⠀⠎⡰⠀⠀⠀⠈⠀⠆⠀⣯⠙⠁⠠⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠣⡀⠀⢀⣄⡉⠉⠐⠂⠠⠊⢛⣓⢶⡤⠀⡰⠦⠀⠈⣿⢷
⠀⠠⠀⠀⢄⡀⠑⠀⡁⠀⠀⡀⠀⢀⡴⠋⠀⠀⢀⣴⣲⣄⠀⢰⠸⠂⠀⡀⠀⠀⠀⠀⣩⣤⡤⡀⠀⠀⣀⠀⠈⠁⡏⠈⠀⡈⠣⠄⠀⠀⠘⡞⠃
⠀⣿⡆⠀⢀⣇⠤⠤⠴⣊⣡⡄⠀⠁⢀⡔⠀⠀⠁⢸⣿⣿⠀⠈⠲⣤⡾⠃⢀⣀⡤⠼⠒⡈⠞⠃⠀⠘⢧⣻⠂⠀⠂⢄⠐⠀⠀⠀⠀⠀⠀⢀
⠀⠈⠓⢒⣉⡁⠐⠀⠙⠉⠀⣤⠀⢀⠞⠶⠆⠀⠀⢸⣿⣿⠀⠀⠀⢀⡴⢚⡭⠅⣀⡴⠊⠁⡡⢴⡖⠤⡄⠀⠀⠀⠀⢈⠂⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠈⠙⠉⠁⠀⠀⠐⠇⠀⠀⠀⠒⠀⠄⠀⠐⠀⠈⠿⠇⠀⠀⠀⠘⠧⠤⠴⠚⠉⠀⠀⠀⠑⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⣷⣤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⠉⠍⢉⡙⠙⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣦⣄⣤⣴⣦⣤⣶⠦⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⢾⣿⣥⣌⣉⠛⢿⣷⡐⠾⣿⡁⠀⢀⣴⠿⣛⣿⠿⠟⠻⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣻⠅⢀⡴⠖⠂⠀⠉⠛⠳⣦⣙⢷⡐⠋⠀⡴⠋⣀⣬⠤⠶⠶⠶⠶⠶⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⣴⣟⡜⠁⠀⠉⣀⡤⠶⠶⠒⠒⠲⠶⣯⡟⠀⠀⠀⣰⣋⣁⡀⠠⠖⠒⠂⠀⠀⠀⠬⣙⢿⣆
⠀⠀⠀⠀⠀⠀⢠⡟⡜⠀⢀⡴⡿⠋⢀⡤⠒⣀⣴⠶⠛⠉⣴⠋⠉⢿⠙⠻⣭⡉⠙⠛⠲⢤⣄⡀⠀⠀⠈⢣⣻⣧
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡴⣫⠊⠀⠀⠀⣠⡾⠋⠀⠀⢀⡾⠁⠀⠀⠸⡇⠀⠀⠻⣆⠢⡄⠀⠨⡻⢦⡀⠀⠀⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠸⣧⡞⡼⠁⠀⠀⢀⣾⡿⠁⠀⠀⠀⣾⠁⠀⠀⠀⠀⢿⠀⠀⠀⠹⣆⠈⠂⠀⠙⢄⠻⣆⠀⠃⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⢱⠇⠀⠀⢠⣿⡟⢀⡞⠀⠀⣼⠃⠀⡾⠀⠀⠀⢸⡀⠀⠀⠀⢻⡆⠀⠀⠀⠈⢧⢹⣆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠘⠀⠀⠀⣾⡟⠀⠘⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢫⣿⡀⠀⠀⠀⠈⡇⣿⣾⠏
⠀⠀⠀⠀⠀⠀⠀⢿⡄⠃⠀⢸⣿⠁⠀⠀⠀⠀⣾⢡⠀⠀⠘⠀⠀⠀⠘⡇⠀⠀⠀⠈⣟⣧⠀⠀⠀⢀⠇⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀⠸⣿⠀⠀⠀⠀⠀⣿⠈⠀⠀⠀⠀⠀⠀⡀⣿⠀⠀⠀⠀⢻⢿⠀⠀⠀⡜⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⣿⣧⠀⠀⠀⠀⣇⢣⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⡼⣼⠀⠀⢈⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⢻⡸⡄⠀⠀⠀⠀⠀⡇⡿⠀⠀⢀⠔⢡⣿⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⢷⣽⣲⣤⣤⣤⣾⣿⣧⣤⣤⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣆⠀⠀⠀⠀⠀⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⣶⠲⠂⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣿⣿⠀⠀⠀⠀⠘⠀⠀⠈⠑⠲⢤⣀⠀⠀⠀⠀⠀⣸⡏⠀⠈⠻⣧⠄⠂⠀⠀⠀⠀⠰⠀⠺⡶⣶⣿⡿⣽⡟⠁
⠀⠀⠀⠀⠀⠀⠀⢰⣦⣠⣦⣼⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣉⡓⣦⠀⠀⠀⣿⡃⠀⢤⡀⠹⣦⡚⣾⡆⠀⠀⠀⠂⣁⣶⣿⡿⠷⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠿⠿⣣⠟⠀⠀⠀⢌⠛⢣⣜⡻⣄⠘⠶⣡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣬⣽⣿⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⣁⣨⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢷⣶⡎⠈⢹⣿⣿⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠈⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠄⠀⠀⠀⠀⠀⠀⠻⡶⢹⠷⠚⢿⣶⣿⠃
⠀⠀⠀⣰⣿⣧⠀⠀⠀⠀⠀⠀⢺⣿⣧⠀⠀⢀⣼⣏⠀⠀⠀⠀⠀⢀⣸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠞⠛⠘⠀⣤⡠⡅⣠⠀⠀⠀⠙⠾⠶⠶⠼⠟⠉
⠀⠀⠀⣿⣿⣿⣷⣦⣤⣤⣤⣤⣽⣿⣿⠀⠀⢸⣿⣿⣦⣤⣀⣤⣶⣾⣿⣿⣿⠁⠀⠀⠀⠀⠀⢀⠀⠀⢰⡒⢝⠶⣰⣿
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢙⣭⣉⣇⣤⣾⣿⠇
⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠉⠛⠿⠻⠿⠿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢛⠃⠂⢀⣔⠉⠉⠐⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠤⠊⠀⠀⠀⣀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⠀⠠⠤⠄⠀⠀⠀⠀⠉⠐⠀⠀⠐⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢤⡉⠐⠒⠬⢧⡠⠒⢈⠀⠀⠀⡉⠐⠠⡀
⠀⠀⠀⠀⣠⢴⣂⣀⡀⢠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣀⡩⠋⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠠⡀
⠀⠀⡴⠊⠐⡡⠤⠒⠊⡟⠀⠀⣀⣂⣀⡀⠀⠀⣀⠤⠒⠊⠀⠀⡘⠁⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⢂⠡⠁
⢠⠊⠀⠀⠈⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⢀⠜⠀⠀⢀⡥⠖⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣉⣁⠀⠀⠀⠀⣀⣤⣴⠭⢶⡟⢀⡤⠾⠿⠶⢶⣶⡄⠀⠀⠀⡄⡰⠊
⠘⠢⣀⠀⠀⠀⠀⠈⠂⡤⣀⡀⣉⣉⣉⣉⠉⢀⣀⣤⣴⣾⠗⠉⣠⡤⠠⠀⠀⠙⠇⣰⡠⠞⠉
⠀⠀⠈⠑⠲⢤⣀⣀⠐⠀⠡⠍⠁⠀⠰⢐⣚⣋⣭⠿⠛⢁⣴⣾⣿⣿⣦⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠚⢿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠛⠻⠿⠛⠋⠁⠀⣾⠇⣿⠀⠀⠙⠻⠿⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⢇⠀⠀⠚⣆⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢛⣷⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣾⠃⡀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠷⣤⣆⣠⡀⢰⡃⣠⣀⣤⣤⡶⠞⡏⢀⣴⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣄⢀⣀⠀⠈⠁⠙⠉⢹⠉⠙⡩⠃⢀⣼⣾⡶⠟⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣤⣉⣉⣻⣶⣶⣷⣶⣾⣦⣺⣷⣞⣉⣉⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⣀⠀⠀⢀⣤⣶⣾⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣿⣿⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⡿⠿⠷⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣻⣿⣿⣿⣿⣿⣿⡯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⣿⣿⣿⣿⣿⣿⣷⡀⠙⢿⣿⣿⣿⣷⣶⣤⣤⣄⡀
⠈⠒⠒⠒⠤⠤⠄⣀⣀⣀⣀⣀⣠⠶⣿⠟⣁⣀⣀⣤⣽⣿⣿⣿⣿⣿⣿⣦⣀⠻⢿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠻⠿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢯⣀⣀⣰⣿⣿⣶⣶⣶⣶⣴⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠈⠉⠉⠛⠻⠿⠿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⢻⡏⠉⠙⠛⠛⠛⢿⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠙⠀⠀⠀⠀⣽⣦⠀⠀⠀⠀⠀⠉⠛⠙⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢀⣼⠋

⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠐⠁
⠀⠀⠄⢰
⠀⠀⠀⠈⣄⡀
⠀⠀⠀⡞
⠀⠀⠀⢧⡤⡄
⠀⠀⠀⢾⠁⠁⠀⢀⡀
⠀⠀⠀⠘⣦⣀⣶⣮⠖⠒⢤
⠀⠀⢠⠞⠙⡁⠈⠁⣦⡀⠈⡇
⠀⠀⢸⠀⠀⣛⣀⣤⣶⡾⢀⠇
⠀⠀⣈⢧⣄⡈⢉⣁⣥⣴⣊⣤⣀
⠀⡜⠁⠀⢈⠙⠟⠉⠙⠋⠙⢷⡀
⢸⡇⠀⡀⠿⠇⠀⠀⣴⡆⠀⠀⢿⠠
⠈⢇⠀⠻⣷⣷⣶⣴⣶⠆⠀⢀⣞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⢀⠳⣤⣄⣉⡀⢉⡁⣀⣤⣞⠛⠉⠙⢦⡀⢠⡀⠀⠠⡴⢄⣤⣀⣶⣊⣀
⠀⢈⣶⠋⠉⠟⠉⢉⠉⢹⠀⣠⡀⠀⠀⠀⢳⡍⢁⡴⠒⠲⠾⣿⠟⠛⠻⠉⠛⢦⡀
⠀⡿⠁⠀⠀⢰⣦⡀⠀⠀⡀⣿⣿⢄⡤⠦⢿⣷⣾⣀⣀⢀⣄⠀⠀⠀⣴⡄⠀⠀⢹⡀
⠀⣗⠀⠀⠀⠘⢛⣁⠀⣠⣤⣤⣴⡏⠀⠀⣈⠙⠉⠉⠋⠙⢿⣀⣀⢀⣭⣥⠀⠀⢸⠃
⠀⠻⣆⠀⠀⠘⠻⣿⣷⡟⢿⡿⠿⡇⢠⣜⣛⣀⠀⠾⠇⠀⠘⡿⠟⠿⠟⠁⠀⣠⠏⠀⠀⡔⠐⢢
⠀⠀⠙⢦⣄⣀⠀⢠⡀⠀⣠⣀⣀⣷⣄⠙⠻⠟⠾⠟⠃⢀⣼⣥⠴⠤⠴⠶⠚⠃⠲⠌⠤⠂
⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠉⠉⠁⠁⠈⠙⠒⠒⠶⠶⠖⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠶⠶⠒⠲⠶⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠦⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠊⢁⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠋⠀⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⢛⣉⣩⡭⣭⣍⣉⡙⠲⢤⣀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣋⣥⣶⣿⡛⣤⣿⡿⣿⢾⣿⣿⡗⠦⣌⠳⣄
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡡⢾⠏⠙⠙⣿⡹⠂⡾⠉⢻⣿⡿⢛⣏⣵⣬⣧⡈⢷⣄
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⣡⠚⣟⠛⠀⠀⠀⢿⡗⠀⠀⣠⣾⢽⡵⠛⠏⠀⠀⣼⣷⠈⣟⣆
⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⣰⣩⠽⣮⣷⣄⠀⠀⠘⣿⠀⢠⣿⠇⠀⢀⡶⣦⣀⣴⣶⡿⡇⢹⠘⡆
⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣼⢉⡶⣤⣌⣙⠛⣷⣤⣀⡻⢦⣿⠛⢀⣟⣯⠶⠟⠛⠋⠀⣹⡇⢸⠀⢻
⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⠟⢯⡀⠀⠈⠉⠁⠈⠉⡏⠀⠀⢨⣿⡋⠉⠀⢀⣀⣄⣀⢬⢷⡇⣼⠀⢸
⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡴⠒⠛⢀⣀⣠⣤⡴⠞⠛⣦⢶⡟⠷⣷⡀⠀⠈⠉⠣⣄⠀⡼⢠⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣉⣹⣾⡿⠛⠋⠉⠀⢀⣾⣇⢸⣷⠀⠈⠛⢿⣦⣀⣀⣨⡷⢡⠏⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠘⣏⣿⡇⢀⣤⣀⣀⣠⣾⡿⣿⠈⢻⡈⠀⠀⠈⠙⣯⢻⡞⣡⠏⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠦⠤⠤⠶⠖⠚⣇⠹⣾⠟⣻⠗⠀⠛⣻⣻⣾⡿⠀⠸⣇⢠⡀⢀⣏⡵⢋⡴⠁⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡈⠻⣅⠀⠀⣰⢣⡿⢹⡇⡾⣆⢿⣤⣯⠟⣫⠔⠋⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣌⡙⠚⠧⠬⠥⠾⠿⠴⠟⣋⣩⠴⠋⠁⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣓⡒⠒⠒⠒⠛⠉⠁⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠶⠒⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡀⣀⣤⣾⣿⣾⣏⣿⣿⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⢠⣟⣿⡿⣾⣿⣿⣿⣿⣷⣿⣿⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠘⢆⠀⠀⠀⠀⠀⠀⣠⣼⡿⠁
⠀⠀⠀⠀⣰⣿⣷⣽⣿⣃⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠄⠈⡄⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠈⠀⢧⣿⣿⣿⣭⣿⣿⠉⠉⣿⣿⣿⣿⣿⣿⣶⠆⠀⠀⠀⠀⠀⣠⢯⣀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡄⠠⡀
⠀⠀⠀⠀⢿⢿⣿⡿⣛⡿⣻⣦⣶⣿⣷⣿⣯⣽⣟⣛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠀⢀⠃⠀⠆⠁⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡄⢀
⠀⠀⠀⠀⠘⢏⢻⣿⣾⣻⣿⡏⣿⣿⣯⣿⣿⣷⣽⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠣⡼⣯⣿⣿⣷⣿⣿⣿⣿⣷⠍⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠀⠁⠀⠀⠀⢀⣠⣤⣶⡶⣀⣀⣸⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣟⣤⣿⡿⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠀⠀⠀⣀⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡃⠀⠀⠙⠿⠟⠸⣿⣿⡿⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠁⠘⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠈⠀⠛⠶⠦⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠡⠉⠘⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠁⠈⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡠⠤⠒⣀⣠⣤⣤⣄⣤⣤⣈⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢵⠋⡝⣷⠊⠀⠀⠤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⢶⣿⣿⣟⣵⣾⣿⣿⣿⣿⣿⡄⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠖⠋⠁⢀⠀⠀⢒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⢛⣯⣿⣿⣿⣿⣼⣿⣿⣿⡿⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠐⢚⡖⣄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣼⠟⠛⡛⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠠⠔⢀⣿⣯⣿⡋⣲⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣾⣿⣥⣴⣾⣿⣿⣿⢿⣿⠟⢋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣟⣮⣼⡟⣻⣫⣛⠀⠀⠂⠀⠀⠀⠀⠀⢻⡍⠀⣀⣠⣾⣿⣿⣿⣿⠇⢹⣿⣿⠞⠁⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠾⢾⠓⣟⣿⠏⡵⠏⠀⡀⠀⠀⠀⠀⠀⠈⢄⠙⠛⠻⢯⣭⣤⣤⡶⠿⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠏⠘⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣤⣠⣄⣀⣀⣀⣠⠐⡂⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⢿⡿⡦⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠁⠮⠷⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠒⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⢄⠔⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⢆⠀⣀⠤⠐⠒⠀⠐⠒⠒⠢⣴⡃⠀⠤⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡛⠓⠒⠶⠶⠦⣄⠀⡀⠂⠀⠀⠉⡑⠛⠐⠢⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡉⠂⠒⠒⠺⢾⠓⠐⠋⠉⠛⠓⠚⠷⢄⡀⠀⠀⠠⡢⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠚⠉⠒⠆⠈⢦⡕⠢⡀⠀⠀⠀⢀⡀⡀⠬⣢⡀⢸⣿⠀⠀⠀⢹⡆⠀⢢⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠃⠁⠀⠀⢰⣁⠀⣀⣩⠗⠃⠀⢀⣀⠀⢀⠙⠻⣦⡹⣿⣯⠀⠀⠀⢸⣿⠀⠈⣷⠘⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠂⠀⠀⠀⠀⠉⠉⠀⠐⣄⠀⠀⠀⠉⣿⣦⡑⢄⠙⢿⡜⣇⠸⠀⠀⢸⣿⠁⡆⣿⡆⢡
⠀⠀⠀⠀⠀⠀⠀⢇⠂⠀⠀⠄⠀⠀⡀⠀⠀⠀⠀⠹⣷⡄⠄⠀⠈⣿⣿⣆⢢⠘⣿⣨⡆⠀⢀⣿⣿⠀⠁⣿⡇⢸
⠀⠀⠀⠀⠀⠀⢰⠇⡇⠀⠈⠀⠀⢀⡅⠀⠀⠀⠀⠐⢿⣿⡈⠄⠀⠈⣿⣿⡆⠄⢹⡏⡃⢐⣽⣿⡏⠐⢸⣿⠁⡌
⠀⠀⠀⠀⠀⠀⢸⡇⡅⠀⡇⠀⠀⢠⡇⢠⠀⠀⠀⠀⢙⣿⣧⠘⡄⠀⢻⣿⣷⠐⠸⡇⡇⢲⣿⣿⢃⠂⣾⡏⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠇⡆⠀⠂⠀⠀⠐⣛⠐⠀⠀⠀⠀⠐⢿⣿⡇⢀⠀⠀⢿⣿⡄⣀⣗⣇⣿⣿⠇⠄⣼⡟⢠⠃⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠘⣵⠀⠘⡀⠀⠈⣹⣇⢃⠀⠀⠀⠀⢑⣿⣿⠀⡀⠀⢸⡿⠋⠁⠈⠉⣉⡏⢂⣼⠋⣰⣅⡴⢃⡀
⠀⠀⠀⠀⠀⠀⣀⢀⢥⡣⡀⠠⠀⠀⠈⣯⡀⡄⠀⠀⠀⠐⣽⣿⡄⢀⣤⣼⣧⡀⠀⠀⠀⠉⠉⠻⣥⣾⣿⣿⢟⣁
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠑⢅⠄⠠⠀⠀⠠⢤⠀⠀⠀⠀⠈⢨⣿⡟⠁⠀⠈⠿⢏⠀⠀⠀⢀⣀⡤⠔
⠀⠀⠀⠀⠀⠀⣢⣐⣾⣦⡀⠑⠤⣀⠀⠀⠀⣂⠀⠀⠀⠀⠱⢻⡃⠀⠀⡐⠀⠀⡕⠄⠀⠈⠙⢢⠀⠀⡤⡄
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠁⠀⠀⠀⠈⠒⠒⠒⠒⠐⠂⠐⠒⠈⠓⢚⠀⠀⠀⣤⠀⠈⠀⠀⠀⢸⠓⠚⠛⠠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣄⣄⣀⣄⣦⣠⢀⠀⠀⠀⠀⠈⠳⣄⡴⠋⢄⡀⣀⣀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢶⢿⢛⢛⢛⢛⢛⢿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⡟⡩⡡⡡⡡⡡⡡⡡⡡⡡⡡⡡⡩⠻⣦⡀
⠀⠀⠀⠀⠀⣠⣾⢟⢔⠔⢜⣔⣤⣤⡴⡶⠶⠶⠶⠶⠶⠴⠵⠜⠙⣦⣄⠀⢀⡾⢿⣷⠄
⠀⠀⠀⠀⣴⠟⠢⣢⡶⠞⠛⠉⠁⣀⣀⣤⣤⡦⠶⠶⠶⠞⠚⠛⠛⠛⠺⣿⠿⠉⣼⣏
⠀⠀⢀⡾⠓⠁⠀⠀⢀⣠⣴⢾⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⡾⠁⠼⢿⣿⣝⣶⢀⡀
⠀⣠⠟⣀⣠⣴⠶⠛⠋⢉⣶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⠓⣠⡶⠟⠛⠉⠉⠉⠛⠶⣄
⣰⠿⠋⠉⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⢀⣴⠞⠛⠉⠉⠉⠀⢻⣿⣶⣿⣇⣀⡀⠀⠀⠀⠀⠀⠈⢷⣄
⠁⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡴⠋⠀⠙⣌⠀⠉⠉⠻⣦⡀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠹⣇⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⣀⠀⠀⠀⢀⠀⠄⠀⠁⠀⠠⣀⣠⣤⣶⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⠿⠟⠋⠁

⣐⣂⣀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠿⠃⡀⠙⠁⠀⠀⠠⠈⠀⠻⠟⢋⠈⢿⡷⠀⠈⠁⢀⣾⣿⠇⠀⡀
⣿⣿⣿⣷⠀⠀⣀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⢰⠀⠠⠀⠀⠀⡆⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⠛⠛⠁⠀⠘⠋⠀⠀⠀⢨⠾⠀⠀⠀⣾
⠿⠿⠟⠁⠀⠀⠹⠀⠈⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⣄⠄⠀⠀⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠀⠠⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠂⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢈⠀⠂⠀⠀⠀⠀⠀⣤⣤⠀⢀⣀⢸⠀⠀⠀⠀⠀⠐⠂⠐⠂
⠀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠃⠀⠀⠀⠀⣄⠀⠂⠀⠀⠀⠀⢐⠀⠀⠀⠀⠰⠆⠀⠙⣿⣷⣿⣿⣿⠀⠀⢤⣤⡁⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢰⡆⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠄⠀⢂⠤⠀⠀⠀⠀⠀⠀⣿⡏⠀⠈⣿⡀⠀⠈⣹⣿⣦⡀⠀⠀⠀⣄
⣿⠀⠀⠀⡀⠀⠀⠀⠛⠛⣿⠋⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠃⠀⠀⠀⠉
⣿⣷⡀⠘⠛⠀⣀⠀⢀⡀⠀⠀⠀⠀⠀⣴⣾⠟⠓⠀⠀⠀⠄⠀⠜⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⠀⠀⠆⠀⠀⠀⢀
⠄⠙⠛⠀⠀⣠⣿⠀⠈⠋⠀⠀⠀⠀⢸⣿⣯⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢤⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢀⣿⡄⠀⠀⠛
⠐⠀⠀⠰⣿⠟⠻⣧⠀⠀⠀⠀⠀⠀⠀⠻⢿⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣲⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢿⣷⣄
⢈⠀⠀⠀⠀⠐⡠⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠂⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠹⠟⠀⠀⠉⠋
⠒⠂⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠛⠋
⠀⠀⠀⠐⠸⠂⠀⢰⣶⠀⠀⠀⠀⢰⠗⠀⠀⠀⠀⢀⠀⠀⠄⠀⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠈⡐
⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣡⣶⣦⣀⣿⣿⠀⠀⠀⠄⠀⠀⠀⠀⢸⠀⠀⠀⠀⣦⠄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢀⣀
⣒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⢿⣿⡇⠀⠀⠀⠰⠀⢠⠀⠀⠘⠀⠀⠀⠀⠀⠐⠈⠀⠀⠘⠃⠀⠀⠀⠀⠀⠈⠋
⠒⠀⠅⠀⠀⠀⠀⢤⠦⠀⠀⠀⠀⠰⡿⠇⠀⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣠⣴⡶⠄⠀⠀⠀⠀⠀⠀⠈⠄⡀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣀⠀⠀⠰⠶⠀⠀⠁⠀⠁⠀⠀⠀⣼⣿⡇⠀⠶⣦⣤⡀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠠⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠈⠉⠙⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣄⠀⠘⠋⠛⠀⠀⠠⠀⠁⠈⠀⠀⠀⡄⠀⢠⡄
⠀⠉⠀⠁⠀⠀⠀⡙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢰⣶⡄⠀⠀⠉⠉⠀⠀⠀⡀⠀⠀⠀⠄⠀⠀⠀⠀⠈
⠀⡀⠀⠀⢀⣤⣤⣧⣤⣄⠀⠀⠠⣤⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠄⠀⠀⠘⠃⠀⠀⠀⠐⠀⠠⠄⠀⠸⠀⠀⢀⣀
⠀⡇⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣷⠀⠀⣀⣰⣀⡀⠀⠀⠀⢀⡄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠈⠉
⡄⠀⡶⠀⠀⠈⠉⠙⠋⠁⠀⠀⣀⣼⣿⡟⠀⣾⣿⣿⣿⣿⠀⠀⡄⠀⠀⣰⡾⢿⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⠀⠂
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠈⠛⠛⠛⠋⠀⣰⣿⡄⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⣾⣇⣠⣶
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⣶⠀⠀⠀⣾⠟⠙⢻⣧⠀⠀⠀⣴⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠠⡿⡧⠀⠀⢹⣿⠿⠿
⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⣿⣷⣤⡀⠀⠀⡈⢠⠀⢀⣤⣾⣿⣧⠀⠀⣀⡀⠉⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⡀⠈⠀⣀⣠⣶⡄⠀⠀⢀⡴⠊⠀⠀⠂⡀⢀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠛⣾⢿⣤⡄⠙⠐⠤⣿⣿⣿⣷⣄⡀⢠⠀⠀⠀⠀⠀⢹⡦⢑⢦⣶⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠢⠀⠀⠀⠈⣷⣾⣿⡇⠀⠀⠀⡄⠀⢹⠏⠈⠀⢸⢀⠀⠀⠀⠀⠀⠙⠚⠀⠙⠻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠈⡈⠹⡅⠀⣠⠤⡿⠀⠀⠀⣠⡟⣸⣿⠽⠟⠛⢷⣦⣄⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⢉⡷⢞⡻⠁⠀⠀⠀⢀⣼⡟⠁⠘⠋⠀⠀⠀⠀⠙⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⣀⣔⣁⠀⠈⢹⣷⣶⣤⣀⣾⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⡄
⠀⠀⠀⢀⡔⢎⠩⡯⠥⣴⣀⠀⠀⠀⠀⠒⠛⠛⠛⠉⠉⠁⡈⢻⡙⢾⣿⠑⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣯⠿⡇
⠀⠀⣄⡠⠶⠶⠛⠛⠛⠋⢁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⢁⣸⡿⢦⠈⢳⣿⣾⡿⠷⠄⠀⠀⠀⠀⢀⣿⣿⣶⠇
⠠⠞⠁⠀⠀⠀⠀⠀⠠⡿⠻⠃⠀⠀⠀⠀⠀⠀⠀⠁⢰⢏⣠⠏⢸⣾⣧⠈⠻⣿⠀⠀⠀⠀⠀⣀⢾⣾⣿⣽⠋
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣀⣰⡄⠀⠀⠀⠀⠀⠀⢠⢰⣫⡞⠁⠀⠀⢹⣼⠀⠀⠉⠠⢤⣈⣁⣉⣴⣾⠿⠛⠁
⠀⠀⠀⠀⠀⢀⡫⠿⣾⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣞⠨⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⣤⣶⣶⣿⣿⡿⠟⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣛⣛⣛⣛⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣩⣛⣭⣛⣿⣩⣟⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠐⠛⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⢀⣀⣤⣀⡀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣤⣾⠿⠛⠛⢿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠾⠟⠋⠀⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣀⣀⠲⢶⣶⣦⢸⡟⣿⡇⢀⣤⣶⡶⠖⢂⣀⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣋⣙⠛⠿⣿⣿⣷⣦⡙⠿⠀⡷⣿⣷⠸⡿⢋⣴⣾⣿⣿⣿⣿⡿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠟⢛⣉⣉⣉⣀⠈⠙⢿⣿⡷⠀⣸⣷⣿⣿⣆⠠⣾⣿⠟⠋⠉⠤⠴⢶⣶⣦⣤⣉⡛⠿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⣡⣴⣿⣿⣿⣿⣿⡿⠟⣃⣠⣭⣤⣬⣉⠻⠿⠟⢛⣀⣈⣁⡀⠲⢶⣶⣶⣶⣦⣌⡙⠿⣿⣶⣌⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⢃⣼⣿⣿⣿⣿⣿⣿⠋⣰⣾⣿⣿⣿⣿⣿⣿⣷⠀⣾⣿⣿⣿⣿⣿⣷⣦⡙⢿⣿⣿⣿⣿⣶⡌⠻⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢁⣾⣿⣿⣿⣿⣿⡿⢡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣷⡌⢻⣿⣿⣿⣿⣿⣆⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⣼⣿⣿⣿⣿⣿⡟⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢻⣿⣿⣿⣿⣿⡆⢻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⣿⣧⠸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣿⣿⣿⣿⡏⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠈⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⣿⣿⣿⣿⣿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢿⣿⣿⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⣿⣿⣿⣿⡿⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠘⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣿⣿⣿⣿⠃⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⡿⠟⠁⠻⠿⠿⠿⠿⠿⠛⠋⠁



⠀⠀⣀⠀⠀⠀⢀⠀⠀⡀⠀⠀⡀
⣿⣟⣉⣿⠿⠿⠿⠿⠻⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣫⠽⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⡆⠀⠸⡆⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠒⠒⠚⠯⣍⠉⢻⠀⠀⣷⠋⢉⡭⠟⠛⠒⠒⠲⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠛⣛⣉⣓⣲⣄⠀⠈⢳⡾⠀⠀⢹⡴⠋⠀⣀⣠⠤⠤⠤⢤⣈⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣰⠊⢁⠴⠋⠉⠀⠀⢈⣩⠷⠶⠾⠥⣄⣀⣤⠷⠴⢾⣛⠛⠙⠓⠲⢤⡈⠙⢦⣷⡀
⠀⠀⠀⠀⠀⠀⡼⠁⣰⠋⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠙⢦⠀⠙⣇
⠀⠀⠀⠀⠀⢰⠃⢰⠇⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠈⢧⠀⠸⡆
⠀⠀⠀⠀⠀⢸⠀⡾⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⡆⠀⡇
⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠘⣇⢿⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢠⡇⣸⠃
⠀⠀⠀⠀⠀⠀⠙⣼⣆⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡼⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡼⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢼⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⢀⣴⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣀⣀⣠⢯⣄⣀⣀⣠⡤⠖⠋




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣆
⠀⠀⠀⣠⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⢀⣀⣀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠁⡀
⠀⠀⠀⠀⠀⣴⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢰
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⠀⠀⠘
⠀⠀⠀⠀⡐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠙⠉⠉⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡉⠀⠀⠀⠁⢀⣀⣤⡤⠰⢂⡄
⠀⠀⠀⠄⠀⠀⠀⡀⠀⠀⢀⠀⠁⠀⢀⣶⣿⡆⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⢠⣈⡁⠱⠒⠂⢰⣿⣿⣦
⠀⠀⣀⣀⣄⠀⢠⠀⢀⠐⠀⠀⠀⠀⠀⠛⠛⠁⠀⠀⠀⠀⠒⡀⠐⠀⠀⠀⠀⠑⠀⠈⠀⠐⠀⠄⠀⠀⠁⠃⠀⠀⠀⣾⣿⣿⡄⠀⠀⠸⡿⠿⠛
⠀⠀⠿⣽⠛⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢠⡀⡀⠀⠀⠀⠈⠁⠉⠁⠁
⠀⠀⠀⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠨⢍
⠀⠀⠀⠐⠁⠀⠀⠀⠀⠐⢀⠰⠀⠄⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⡆⣥⡄⠀⠀⠀⠀⠘⠂⠁⠀⠀⠚
⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⡈⠐⠀⠤⠒⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⢀⠒⣀⠀⢀⡄⠀⠠⢄⠆⠀⡀
⠀⠀⠀⠀⠀⠈⠃⠀⠈⠀⣁⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠉⠀⠀⠀⠉⣤⠛⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⡼⠀⠀⠛⠖⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠈⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡣⡝⡒⡂⠄⠀⠀⢰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠁
⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢁⣧⣷⡀⠑⠶⠀⠀⠀⢀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣄⣀⡀
⠀⠀⣀⡀⠄⠒⠻⠿⠿⠽⠿⠻⠿⠿⢿⣿⣷⣶⣄
⠀⠘⠛⠿⣿⣯⣽⠓⠐⠶⠦⠤⠤⣤⣤⣠⣉⣙⡻⢷⣶⢢⣀⡂⠀⢀⡲⣲⣶⣶⣦⡄⢀⡀
⠀⠀⠀⠀⠈⠛⢿⣿⣶⣷⡀⠀⢐⠀⡀⢍⡙⡻⠛⣛⠮⢷⣽⠻⣷⣄⡷⠀⢉⣉⣙⣁⠄⠈⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣷⣾⣛⣿⣵⣶⣾⣽⣿⣶⣤⢿⡞⣿⣿⣞⣋⣉⡁⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠉⠉⠉⠉⠉⠻⠻⢿⣿⠿⢫⠴⠒⠹⠛⢼⡛⢻⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⣿⣿⣿⡟⠋⠑⠀⠀⠀⠐⢜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⠄⠀⠀⠀⠀⠀⠀⠈⡯⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣞⣴⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⡄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⢠⡳⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣽⢶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠣⢰⡽⣻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣄⣻⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣾⢆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣩⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⣿⢷⣔⠘⠒⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣐⣵⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣶⣥⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣤⢸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣾⣇⠀⣠⡀⠀⡀⠀⠀⠀⠀⢀⣤⢄⡀⠀⡘⣹⣶⣥⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣶⣿⣷⣷⡆⣵⣼⣿⣾⣿⣿⣿⣿⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⠛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⢷⣄⣀⠀⠀⠀⣀⣠⡤⠖⠋⠀⣰⢿⡿⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣸⢀⠛⣧⣈⠉⠉⠁⠀⢀⣀⡴⠚⢡⢿⣧⠀⠙⠳⠶⠤⣤⠤⠴⠖⣺⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⢀⡴⠃⣸⠀⢼⠻⣯⡉⠙⠋⠉⠁⣀⡴⠋⣸⡏⠳⣤⣀⣀⣀⣀⣤⡶⠋⡿
⠀⠀⠀⠀⠀⠘⠛⠲⠶⣾⣛⣉⣡⠤⠖⠋⣀⡴⠁⢠⠇⠀⡏⠻⣏⠛⠛⠋⢁⣠⠞⢹⡿⣖⠒⠛⠛⣉⢽⡟⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠶⣿⣥⣠⠔⠁⠀⡴⠁⢀⠏⢻⣏⠋⠉⠀⣠⢾⡇⠈⠉⢉⣽⠃⢸⡇⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠙⠳⡾⣤⣤⣞⡀⡼⠈⡻⣶⡊⠁⣼⠓⢲⣶⠃⢿⠀⠈⢧⡀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⢀⠃⠀⢸⠏⠛⠷⠾⣥⣜⡻⣧⣹⣴⡏⢿⡀⠘⠦⡀⠈⠳⢤⡀⠀⠙⢶⣤⣀⣀⣀⣀⣀⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⠋⠀⢀⣠⣎⣀⣠⣎⣀⣴⣧⣤⣦⣼⣿⣿⣿⣶⠷⢶⠿⠓⣲⠿⠛⢛⡻⠟⠛⢉⡽⠟⠋⠉⠉⠁
⠀⠀⠀⠰⠿⠛⠛⠛⠛⠛⠛⠛⠻⠿⣿⠋⣉⠉⠙⠻⣍⠉⠉⠻⣇⢦⡈⣧⣴⠛⠳⣿⣛⢷⡇⠀⡼⠁⠀⡴⠋⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠳⣄⠀⠘⣧⠀⠀⢸⣇⡷⢛⣋⡉⢻⡷⢺⡞⠻⣦⡇⠀⣼⠁⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠘⢦⠀⣹⠀⣠⡾⠋⣉⡉⠉⠙⢾⡷⠚⣧⡴⠚⠻⣄⣿⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⣧⣷⠾⠛⠋⣉⠉⠉⠙⠳⣼⡷⠊⠙⣧⢀⠴⠞⠻⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣴⠿⠗⠚⣉⣉⡉⠉⠛⠦⣄⣽⣀⡤⠶⢿⡇⠀⢀⣠⡼⢷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⠿⠷⠚⠛⠛⠲⠶⢤⣍⡑⠢⣌⢿⠁⣀⣤⢤⣿⣤⠋⠀⢀⣤⠾⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⡌⣿⡎⠁⠀⠀⠈⢿⡀⣰⠋⠀⣠⠖⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢻⣿⠃⠀⠀⠀⠀⠈⢷⡃⢀⡞⠁⢀⡴⠊⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⣿⠀⠀⠀⠀⠀⠀⠈⢧⡼⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠀⠀⠀⠀⠀⠀⠀⠈⣇⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣣⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⣰⣂⣀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠺⠇⠀⠉⠁⠀⠀⠠⠀⠀⠻⠛⡁⠙⢿⠆⠈⠉⠀⣼⣿⠇⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣿⣿⣿⡇⠀⠀⡀⠀⡀⠂⠐⠀⠀⠀⠀⠀⠀⠸⡈⠀⡀⠀⠀⠅⠀⠀⠀⡐⠂⠀⠀⠀⠀⠀⠚⠛⠁⠀⠘⠋⠀⠀⠀⠰⠳⠀⠀⠠⣿
⠛⠛⠋⠀⠀⠀⠇⠀⠀⠀⠀⢠⠀⠀⠶⠀⠀⠀⠀⢆⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠄⠀⠆⠀⠀⠀⠐⠷⠀⠀⠀⠀⠀⢀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⡀
⠀⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠨⠀⠄⠀⠀⠀⠀⢰⣶⡆⣠⣤⣀⡇⠀⠀⠀⠀⠈⠂⠈
⠀⠂⠀⠀⠀⠀⠀⠂⠀⠈⠀⠈⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠅⠨⠀⠀⠀⠀⠛⠀⠀⢹⣿⠿⠿⣿⡁⠀⠲⣶⣄
⣀⠀⠀⠘⠃⠀⠀⢀⣤⣴⣦⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠒⠀⠀⠀⠀⠀⠀⠨⠇⠀⠀⠟⠇⠀⠀⠾⠿⣷⡀⠀⠀⠰⠆
⣿⡀⠀⢠⣄⠀⠀⠀⠁⠙⠇⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠛⢿⣆⠀⠁⢰⡆⠀⢴⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⠀⠂⠈⠁⠠⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⡅⠀⠀⢀⣾
⢀⠀⠁⢀⣤⣾⣧⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣰⣿⣦⡀⠀⠁
⠀⠀⠀⠈⠋⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣶⣶⡄⠀⠀⠀⠀⠀⣾⡿⠋⠛⠿⠇
⡔⠄⠀⠀⠈⠀⠆⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⠇⠀⠀⠀⠀⡀⠈⠀⠀⠐⠂
⠀⠀⠀⠐⡄⡀⠀⣀⡄⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⢀
⠀⡀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⡇⣠⣄⡀⣴⣶⡀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⢀⣀⠀⠀⠀⠓
⠭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⠀⠀⠀⢠⠀⢀⠀⠀⠇⠀⠀⠀⠈⠉⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠘⠇⠀⠀⠀⢐
⠬⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⣿⡏⠀⠀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐
⠈⠴⠂⠀⠀⠀⠀⠁⠀⠀⠀⢀⣀⣀⡀⡀⠀⠀⢠⡦⠀⠀⠂⠀⠀⠀⠠⠀⣾⣿⠁⢠⣤⣀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠋⣿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣄⠀⠈⠟⠛⠂⠀⠀⠀⠀⠐⠀⠀⢀⠄⠀⣠⡀
⠀⠀⠀⠂⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⣰⣆⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢈⠀⠀⠈
⠀⠀⠀⠀⢀⣤⣼⣇⣤⡀⠀⠀⣤⣤⡀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠋⠀⠀⠀⠐⠀⠠⠀⠀⠰⠀⠀⢀⡀
⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇⠀⠀⠀⣻⣿⡀⠀⣀⣰⢀⡀⠀⠀⠀⢠⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠆⠀⠀⠌⠁
⡄⠀⠦⠀⠀⠈⠉⠉⠉⠀⠀⢀⣴⣿⡿⠀⢾⣿⣿⣿⣿⠀⠀⡆⠀⠀⡴⠿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠
⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠈⠉⠛⠛⠁⢀⣼⣷⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢿⣿⣀⣴⣦
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⢺⡀⠀⠀⠾⠋⠉⠻⠇⠀⠀⣰⡆⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠰⠿⠗⠀⠀⣿⡟⠛⢻
⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⣿⣶⣄⣀⣀⣀⣀⣀⣠⣾⣿⣿⣄⣀⣠⣤⣀⣹⣿⣆⣀⣀⣀⣀⣀⣀⣀⣛⣁⣀⣘
⣿⣿⡋⡛⣛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣀⣀⣀⠀⣀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠸⠛⠛⠛⠉⠛⠛⠋⠉⠩⠤⢡⣭⣯⣿⠿⣿⣻⣯⣻⣿⣽⣻⣿⣿⣟⣿⣿⣯⣭⠭⠭⠭⠭⠭⠉⠉⠅
⠀⠀⠀⢚⡛⠉⠉⠉⠑⠚⠁⠍⣹⣿⣿⢟⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⡛⢿⣯⠭⠭⠥⠭⠤⠅
⠀⠀⠀⠐⠂⠀⠀⠀⠒⠀⢀⣾⣿⢟⣴⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⣿⣿⠓⠌⠻⣿⣿⣿⣮⢻⣷⡁⠈⠭⠤⠅
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⣾⣿⠏⣾⣿⣿⣿⠟⢰⣁⣈⢿⣿⣿⣿⣿⣇⣤⣴⣧⣾⣿⣿⣿⡎⣿⣇⠀⠀⢀⡁
⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣿⣿⢸⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⢃⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠐⢒⣒⡆
⠀⠀⠀⠒⠀⠀⠒⠀⠀⢀⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣼⣶⣾⣿⣿⢿⣿⡟⠍⣻⣿⣿⣿⣿⠒⠐⠒⠂
⠀⠀⠀⠠⠤⠤⠀⠀⠀⠀⠹⣿⣷⣿⣿⣿⣏⢍⠻⠯⠉⠻⢿⠋⠋⠛⠋⠀⠁⢸⣤⣵⣿⣿⣿⣿⠧⠤⠄
⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣦⣠⣶⣅⢀⣠⣶⣦⣠⣾⣷⣦⣿⣿⣿⣿⣿⡿⢁⢀⣀⠀⠀⡄
⠀⠀⠀⣾⣿⣇⠀⣰⣿⣆⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⢂⡀⢐⡒⠒⠀⡂
⠀⠀⠀⣿⣿⣧⣾⣿⣿⣿⣧⡀⠀⠐⠂⠂⠒⠉⣻⠟⠛⠛⠛⠛⠛⠛⠛⠛⠉⠹⣅⠀⠀⠀⠀⣀⣲⠴⢐⡒⡃
⠀⠀⠀⣿⣿⣿⣿⡯⣭⢽⣿⣿⡄⠀⠈⠉⠡⡶⠁⠀⡄⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⡍⠈⠡⢾⠋⠀⢎⡠⢳⡀
⠀⠀⠀⣿⣿⣿⣿⣧⣭⡿⠿⠿⣿⣿⣿⣿⣿⠁⠀⢰⠃⠀⠀⠀⢰⠀⠸⡇⠀⠀⠘⣿⠭⠬⢉⣷⣾⣿⣦⣼⡇
⠀⠀⠀⣿⣿⣿⣿⡟⣡⡆⢠⠖⠘⠋⣿⣿⡏⠀⠀⣾⠀⠀⠀⠀⢸⡆⠀⣿⠀⠀⠀⣿⣧⣴⠟⠿⣿⣿⣿⣿⡇
⠀⠀⠀⢹⣿⣿⣿⠁⠿⠆⠀⠀⠀⠀⠉⣽⠀⠀⢀⣿⠀⠀⠀⠀⢸⡇⠀⠸⡆⠀⠀⢸⣿⡏⣈⠀⠀⠀⠉⣻⡇
⠀⠀⠀⢸⣿⣿⣿⡀⠐⡀⠀⠀⠀⠆⠀⡏⠀⠀⢸⣿⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⣿⣻⡄⠠⠀⠂⢰⣿⡁
⠀⠀⠀⢸⣿⣿⣷⣴⠀⠁⠀⠐⠀⡀⢸⡇⠀⠀⢸⣿⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⢿⣇⢻⡄⠀⣠⣞⣹⡆
⠀⠀⠀⢸⣿⣿⣿⣿⣄⣀⠙⠒⠛⡁⣼⠁⠀⠀⠘⣿⡇⠀⠀⠀⠀⢿⣇⠀⠀⠀⠀⠀⠸⣿⡄⢿⣏⢿⣾⡿⠃
⠀⠀⠀⢸⣿⣿⣿⣿⠻⣍⠣⣠⣟⣠⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⢻⣧⠀⢻⣿⣿⣿⣿
⠀⠀⠀⣼⣿⣿⡿⣬⣓⣾⠭⠭⠯⠶⣿⠀⠀⠀⠀⢹⣷⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⢿⣇⠀⠹⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣗⢶⣾⢿⣯⠏⣯⡭⢿⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠈⢿⣇⠀⠀⠀⠀⠀⠈⢿⡆⠀⠈⠛⠿
⠀⠀⠀⣾⣿⣿⣿⣯⣵⣻⡇⠀⢐⣛⣻⡄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠈⢻⡀
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣶⣭⣽⣇⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠙⠛⠛⠛⠛⠛⠉⠀⠀⠉⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶
⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠊⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡎⠙⡍⠑⢲⠇⠀⠀⠀⠀⠠⠤⠴⠒⠋⠁⢀⣠⠆⠈⠳⣄⠀⠀⠀⠀⢀⣀⡠⠀⠀⠀⠀⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣶⠃⢀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠚⢩⠋⠁⠀⠀⠀⠈⠉⠉⠉⡡⢺⠃⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⢸⠙⠉⠛⠂⠀⢀⡤⠐⠂⠀⢀⣠⠆⠀⠀⠀⠀⣀⠀⠀⡠⠊⠀⡏⠀⠀⠀⠀⠀⣿⣿
⢀⣀⡀⠀⠀⠀⠀⠀⠸⡄⢸⡀⠀⢱⣦⠞⠉⠀⠀⠐⠚⢉⠀⠀⠀⠀⠀⠀⠈⡩⣿⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿
⠀⠈⠉⠳⠦⣄⣠⣆⠀⢻⣼⣦⣀⡞⠁⠱⡖⠒⠒⠉⣁⠀⠀⠀⠠⣄⣀⠔⠉⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠱⡀⢈⣹⣿⣿⣿⣿⣿⣥⠤⠖⠚⢳⣘⡋⢡⡀⢱⣄⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢀⡤⠟⣿⣿⣿⣿⣿⡏⠉⠘⢓⠲⡞⣤⣯⣵⣆⣍⣁⣄⡀⠀⡄⠀⠀⠀⠈⢆⠀⠀⠀⢢⡀⠀⠀⠀⣿⣿⡀
⠀⠀⡴⠋⠀⠀⣿⣿⣿⣿⡿⠹⣴⠀⠸⠀⡉⣿⠉⡄⣈⠠⢜⠀⠈⠉⠉⠉⠑⠒⠒⠺⠷⠤⠤⠀⠙⢄⡀⠀⣿⣿⠁
⠀⠈⠀⠀⠀⠀⠈⠉⠉⢰⡇⠀⠿⢆⠀⠀⢸⡤⠀⠈⠀⠀⠀⠀⠤⡇⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣠⠀⠀⢀⣼⣏⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢀⣴⣡⠤⠮⠅⢸⠀⠁⠀⠀⠀⠘⣦⠆⠀⠀⢾⠁⠀⠀⠀⠀⠀⠒⢺⡷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠚⠉⠉⠉⠛⠢⠀⠘⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢱⢀⡀⠐⠈⠉⠓⢦⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⣵⡇⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⢠⠞⠀⠀⠀⠀⠀⠀⠹⡄⠀⢀⠴⠂⠀⠀⠀⠀⠀⠀⠀⠈⠹⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣠⠿⠊⠀⠀⠀⠀⠀⠐⠠⢵⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⡛⠉⡳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠐⠈⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⢸⡆⡄⡄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣖⡛⠉⠉⠉⠉⢙⢺⡇⡇⠀⢸⡤⠖⠛⠋⠉⠉⠉⣙⣲⣄⣀⡀
⠀⠀⠀⢀⣠⠖⠛⠉⠉⠀⠉⠉⠙⣲⣤⡀⢀⡼⠃⠃⠀⠀⠳⣄⠀⣀⡴⠚⠉⠉⠁⠀⠉⠉⠙⠲⣄⡀
⠀⠀⡴⠋⠀⠀⠐⠀⣀⡤⠖⠛⠛⡛⠻⣿⣿⡄⢀⣀⣤⣤⣤⣽⠟⠛⠉⠉⠉⠓⠶⣄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⡼⠁⠀⠀⠀⢀⠜⠁⢀⠔⠀⠀⠀⠀⠀⠉⣻⠋⠀⠈⠁⠀⠘⢦⠀⠀⠀⠀⠀⠀⠈⠱⣄⠀⠀⠀⠀⠀⠳⡄
⢸⠃⠀⠀⠀⢠⠏⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⢳
⣾⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠸
⡏⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⢠
⢿⢰⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣸
⠸⡆⡇⠨⠈⣇⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠄⠀⢠⠏
⠀⠹⣴⡀⠆⠹⡄⡀⠐⡀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠄⠀⠀⠀⠀⠄⢸⠃⠀⠀⠀⡀⠀⡀⢠⡿⠀⢀⣠⠟
⠀⠀⠙⢷⡌⢄⠻⣖⡀⠐⡀⠄⠀⠠⠀⠄⠹⡄⠐⡀⡆⠀⠘⠀⠀⠀⣼⠀⠀⠀⢰⠁⡸⢠⣿⠃⣢⣾⠋
⠀⠀⠀⠀⠙⠧⣤⣙⢿⣄⠐⣄⠀⠀⠡⡀⠲⣿⡀⠄⡇⠀⢀⠀⠀⢰⠇⠀⠀⣠⠃⡰⣡⢟⣡⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠳⢮⣷⣄⣐⢬⣂⣬⡿⣾⣽⡄⠈⣀⣴⣟⡴⣋⣔⣡⡴⠿⠛⠋⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣭⣤⣤⣤⣽⣿⣿⣯⣥⣭⣭⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡦⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢱⡀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠈⣷⣄⠀⠀⠀⠀⠀⠀⣠⡞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠈⣆⠙⠒⠲⠴⠒⠊⡝⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣤⠖⠋⠀⠀⠀⠀⠀⠈⠀⠸⠦⡀⠀⢀⣀⡼⠀⠀⠀⠐⢄⠈⠂⢄⣀⡀⣀⣀⠤⠔⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡈⠳⡤⢄⡀⠀⠀⠀⠀⠀⣰⡄⠉⠀⣰⠁⠀⠀⠀⠀⠀⢉⣢⠤⠖⡻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱⠀⠀⠁⠂⠀⠤⠀⠁⠡⠀⢠⠃⠀⣀⣀⠤⠖⠋⠉⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠅⠐⠊⠉⠸⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢀⠀⠀⠀⠀⡀⠤⡀⠣⠉⠒⢎⠑⠒⠠⣤⡀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢀⡮⠤⠒⠓⠤⡀⠀⢰⠀⠖⠒⠸⡄⠀⠀⡀⠈⠛⠲⠤⣐⡀⠀⢳
⠀⠀⠀⠀⠀⢀⣤⠾⠷⠯⠥⣄⡀⠀⠀⠀⢱⡀⢀⣆⣀⢀⣀⠹⣀⠎⠀⢀⡤⠔⠒⠈⢉⣓⣦⣧⡀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⡇⠜⠁⠀⠀⠀⠙⢿⠀⡰⠃⠀⢀⠔⠋⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⡿⠔⠒⠒⠲⠤⢤⣀⣷⡇⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢰⠅⠒⠒⠀⠀⠀⠤⢄⡙⣇⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠷⠒⠒⠒⠒⠂⠤⠀⡀⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣

⠀⠀⠀⠀⠀⢀⡀⢀⣴⣦⣀⡠⢴⣶⣤⣤⣶⠆⠀⠀⡀⠀⠀⢠⣶⣶⣶⣶⣴⣆⣶⣶⣾⠀⠀⣾⣾⣿⣷⣖⣾⣶⣶⣦⣶⣶⣶⣶⣶⣶⡶⣂⣠⡄
⠀⠀⠀⣴⣿⣶⠄⣼⣿⢿⣫⣾⣎⠛⠛⠋⠁⠀⣠⣶⡆⠁⢂⣿⣿⢿⢿⣷⣿⣾⣯⣽⣯⡀⠀⠈⠛⣫⣿⣿⣿⣿⣹⣿⣽⣿⣝⣿⣷⠛⢁⣽⣿⣷
⠀⠀⠸⣿⡿⢃⣾⣿⣿⡿⣿⣿⣿⡆⢠⣀⠂⢀⣙⠿⠣⡆⠸⢟⣵⣿⣮⠻⣿⣿⣿⣿⣿⣧⠀⠀⡘⢿⣿⣿⣿⣿⠈⣿⡿⣿⣿⣿⣧⡀⠙⣿⣿⣏
⠀⠀⢲⡈⠀⢸⣿⡟⢹⣾⣷⣎⠉⠀⠈⠉⣸⣿⣿⣿⣇⡁⠀⣾⣿⣿⣿⣿⣿⣿⣿⢨⣭⣤⡤⢁⣀⠈⠻⢻⣿⣷⣾⣿⣿⣿⣿⣿⣿⡅⠀⠸⣿⣯
⠀⠀⢸⡇⠀⢸⣿⣾⣜⣿⣿⣿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣤⠀⢨⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⡄⣿⣧⠀⣄⠹⣿⣿⣿⣿⠿⢛⠿⢿⡏⠀⢀⡘⠏
⠀⠀⢰⣿⡄⠘⣿⣿⣿⣿⣿⣯⡄⠀⢸⣿⣼⣿⣿⣿⣿⠉⠀⣼⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣾⣽⡈⠀⠀⠻⣧⣽⣿⣿⣿⣾⣿⣿⠆⠀⢀⣼⣿⡄
⠀⠀⢰⣿⡇⠄⢠⣾⣿⣽⣟⠉⠀⡀⠸⣿⣿⣿⣿⣟⠋⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠈⣿⣿⣿⣿⣿⣿⣿⡃⠀⠠⣾⣿⢏⣴
⠀⠀⢡⣝⠀⠀⠸⣿⣿⣿⣿⠀⠀⠃⠀⣿⣿⣿⢿⣿⠇⢸⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⢩⣿⣿⣿⣿⣿⣿⡄⠀⠐⢿⣿⣾⡇
⠀⡀⢼⣿⣧⠀⠀⢽⣿⣿⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⡀⢸⠀⣿⣿⣿⡿⢛⣙⣛⣻⢻⠿⣿⣿⣿⡏⠀⠂⠀⠀⠘⢿⣿⣿⡿⡿⡧⠀⠀⠈⣿⣿⡇
⠀⠀⣿⣿⣿⠆⠀⠨⠿⠿⠁⠰⡆⠀⠘⣿⣿⣿⣿⡿⠀⡇⠠⣿⣿⣿⣷⣿⣿⣿⡄⡈⠉⠀⠋⢽⡇⢀⣄⢀⣰⢀⣄⠈⢻⣿⣿⣇⠀⡀⠠⠸⣿⣧
⠀⠀⢿⣿⢋⣶⠘⢣⣾⣽⠇⣰⣿⣄⠀⡹⣿⣽⣿⣃⠀⠇⠀⠹⣿⣿⣿⣿⠿⠋⠀⠁⠀⠃⢰⣶⣶⣶⡈⣼⣿⣼⣿⣽⣮⣿⣿⠋⡏⠠⢀⡲⣟⣿
⠀⠀⠈⠉⠛⠛⠲⠟⠻⠀⠛⠛⠛⠿⠄⠙⠒⡉⢘⣂⡀⠄⠊⠀⠙⢛⠿⡷⢡⠅⠾⠀⢤⢸⡦⠙⠻⠿⠗⠽⠛⠛⠛⠫⠿⠟⠥⠶⠧⠴⠿⠟⠻⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠒⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⡴⠋⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⡼⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡤⠴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠓⠒⠢⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠄⠒⣒⠠⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⢀⠴⣊⠥⠒⠉⢧⢸⠉⠉⠉⠒⢬⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⡴⢃⠺⡁⠀⠀⠀⢸⣼⠀⠀⠀⣠⢊⠔⠲⡙⢆
⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⡘⣰⠉⠑⢬⣲⣀⠀⠀⣿⠀⠀⣴⡷⠁⠀⠀⠘⡌⢦
⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢰⢁⠇⠀⠀⠀⠈⠛⢷⣄⣹⣀⡾⠋⠀⠀⠀⣀⣀⠼⠸⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠸⠤⠤⠤⣤⣤⣤⠤⢾⡁⢈⡗⠚⠛⠛⠛⠒⠒⠒⡄⡇
⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠐⡆⠉⠉⠀⠀⠀⣠⡞⠙⡏⠙⢶⣄⡀⠀⠀⠀⢸⢠⡇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⢧⢳⡀⠀⠀⣠⣾⠋⠀⠀⣿⠀⠀⠙⢯⡲⢄⣠⠇⡜
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠈⢣⡑⢤⠜⡵⠁⠀⠀⠀⠋⠀⠀⠀⠀⠈⡦⢁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠙⠦⣘⠦⣄⣀⠀⠀⡇⣇⣀⡠⠔⣋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠤⠤⠤⠤⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠬⠭⠭⠥⠬⠵⠒⠋⠁





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢖⡽⠶⢶⣾⡆⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠀⠀⢀⣿⠇⢀⡄⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⠄⠀⣼⠟⣰⡿⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣣⠞⠋⠀⠀⠀⠀⢠⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⣭⣂⣤⠔⠀⠀⠀⢀⣸⣏⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠿⣟⣻⡍⢀⣀⣀⣀⠀⠀⢌⡛⢾⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠘⠛⠛⠛⠛⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣛⣉⣥⣤⣤⣤⣭⣭⣭⣭⣍⣉⣉⣉⣉⣭⣭⣥⣤⣤⣤⣬⣉⣛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣻⣵⣾⣿⣿⣿⣿⣿⡿⣿⡻⣿⠿⠿⣟⠿⠟⣻⠟⢛⣿⢟⣿⢿⣿⣿⣿⣿⣿⣷⣮⣟⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⡿⣮⣑⠀⠀⠍⠡⠀⠀⣪⡵⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⡼⣿⣿⣧⣻⣄⠙⢋⣷⣤⠆⠰⣴⣾⡙⢃⣸⣟⣼⣿⣿⢧⠈⢿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡟⠁⢸⢻⣿⣿⣇⠉⠛⠛⠿⠟⠉⣴⡦⠈⠿⠟⠛⠛⠁⣸⣿⣿⡞⡇⠈⢻⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡶⢰⢸⡸⣿⣿⣿⣶⣤⣀⠀⠀⠀⠈⠁⠀⠀⢀⣤⣤⣾⣿⣿⣿⢃⡇⡄⣿⣿⡟⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⡈⡇⠳⡹⣿⣿⣿⣿⡇⠠⠀⠤⠤⠤⠤⠀⠄⣹⣿⣿⣿⣿⢋⡞⢰⢃⣿⣿⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣴⡀⡹⡜⣿⣿⣿⣷⣄⠀⠀⠒⠒⠀⢀⣰⣿⣿⣿⡿⢡⢏⢠⢇⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⢁⣿⣻⡿⣿⣿⣿⣿⣶⣤⣤⣶⣿⣿⣿⡿⣿⡇⣾⡇⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣎⣿⣿⣿⢹⣿⣿⣿⣿⢿⡽⣾⣿⣿⣿⡇⣿⣿⣿⣧⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⢿⡏⠈⢿⡼⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡏⠈⢻⣿⣿⣿⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣇⠸⠀⠀⠋⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠈⠓⠀⠸⠃⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡟⠀⠀⠀⠀⠀⠀⣰⡄⢀⠀⠀⣀⣀⠘⠛⠓⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣷⣤⠲⠆⠤⠼⣾⠃⣷⣮⣴⠸⠍⠃⠀⣠⣄⣤⠀⠉⠓⢴⣶⠀⠀⠀⠀⠀⢰⢸⠀⡆⢠⣀⡀⣶⠀⠀⠐⠀⠀⠰⠀⠀⠀⠀⣾
⣿⠁⠀⠀⠀⠀⠀⠉⠁⠈⠁⠈⠀⢐⣾⢭⠙⡿⠀⠀⠀⠿⣿⠃⣤⠒⠾⢤⢼⣮⡟⠏⠛⡽⢋⡻⣆⣸⣤⣷⣖⣶⠀⣇⢠⠴⢿
⣿⡠⡗⠐⠃⠛⠐⠀⠀⠀⢀⠀⠀⠸⠟⠆⠙⠃⠀⠀⡀⠀⠀⠠⣤⠐⠀⠀⠀⠙⠃⠈⠈⠁⢸⠃⠀⠉⠁⠀⠀⠀⠀⠁⠀⠀⣾
⣿⠇⢀⣰⠀⣠⢤⣤⣆⣀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⡟⠀⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣴⡾⠁⠀⠀⠀⠀⠀⢀⢤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡄⣤⣴⣸⠟⠓⠀⣴⣿⣿⣿⣷⣦⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁⠀⠻⠻⠟⠛⠀⠀⣰⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣸⣇⢺⡇⠀⠰⣦⣤⣤⣤⣼⣦⣈⣉⠉⠉⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⣴⣼⡿⣿⣧⣦⠀⠈⠉⢩⠛⢻⢿⣿⣿⣿⣿⣿⣿⡿⠶⡴
⠀⠀⠀⠀⠀⠀⠀⣸⣶⠆⣀⣶⣼⣿⣿⣿⣿⣿⣿⣶⣧⣦⡆⠀⠻⠦⢀⡁⠠⠾⠇⠀⡇⠠⠄⠀⡀
⠀⠀⠀⠀⠀⠀⢠⣴⣿⣶⣿⠹⢹⡏⠹⡏⡍⠛⢭⢻⠹⢫⣧⠈⢷⣤⣬⣭⣤⣠⠞⢁⠇⢖⡄⠿⠿
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣹⣸⣣⣅⣇⣶⣯⣰⣿⣇⣿⣿⣷⣤⣤⣬⣭⣤⣤⣴⣿⣿⣷⣶⡶
⠀⠀⠀⠀⠀⠀⠘⢾⡇⢛⠀⣿⢁⢹⡇⢿⣯⢸⣿⢡⡌⢿⠸⠉⠇⣿⠐⢶⡇⠰⣾⠀⠻⢸⣿⠟
⠀⠀⠀⠀⠀⠀⠀⢹⡇⣿⣈⡇⣸⡘⡇⠘⣿⠘⢿⡘⢇⣿⡄⣶⢠⣿⠘⢻⡇⠻⢿⡀⣇⢸⣿⠒
⠀⠀⠀⠈⠀⠠⢂⠛⣿⡟⠿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣼⣥⠿⠖⠒⠒⠚⣿⣿⣿⣿⣿⣿⣴⣦⣶⣶⣶⣶⣶⣶⣶⣶⣿⠉⠉⠁⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠄⠄⠀⠀⠀⠈⠉⢉⠈⠙⢯⡛⢦⡤⠤⠗⣉⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠀⠀⠀⠈⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿