Goodluck Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠁⠀⠀⣤⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⢀⠀⢀⣠⠀⠀⠀⢰⡆⠀⠀⢀⠀⠀⠈⠃⠀⠀⢰⡄⠀⠀⠸⡵
⠀⠀⠈⠀⠀⠘⡾⠛⠃⠠⠀⠀⠀⠠⠀⠀⢀⠀⠀⠀⠒⠀⠀⠸⠀⠀⠀⠠
⠀⠀⠠⣢⠀⠀⢣⡤⠀⠀⠃⡀⠀⠀⠀⠀⡘⢗⢄⠀⠀⠀⣰⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠣⡣⡀⠀⠀⠀⠀⡘⠌⠂⠄⣀⠚⠁⠈⠓⠩⠐⠛⠁⠐⠚⠛⠱⡶⠒⠋
⠀⠀⠀⠀⠑⠪⢀⣐⡂⠞⠀⠄⠀⠀⡈⠀⠀⠀⠀⠀⠁⠀⡀⠀⢀⠄⠀⠁⠀⠀⠀⠤⠂⠀⢀
⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠜⠃⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⢇⢀⠀⠀⠀⠀⣠⠄⠀⠀⡘⠁⠀⠀⠀⢀⡠⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣑⠂⢐⡲⠃⠀⠑⠬⠤⠶⠊⠩⣰⣤⠤⠇⢀⠀⠀⠀⠠⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣐⡲⠦⠙⠚⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠍⠚⠰⡐⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠒⠉⠉⠲⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⢰⡇⢀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⣀⠤⢄⠀⠀⠀⠀⠘⢄⡀⠀⠀⢠⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⡀⠀⠀⢀⡾⠃⢀⡴⠀⠀⠀⠀⠀⠀⠀⢀⡠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⢰⠃⠁⠀⠀⢠⣿⠁⠀⠀⡀⠀⣠⡀⢀⡴⣷⢠⡟⢰⡿⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠤⠐⠀⣼⠇⠀⠀⣾⡇⡾⠑⠇⣾⠁⠈⠈⠃⢸⡇⠀⠀⣰⠊⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡠⢪⡿⠀⠓⠂⣀⡈⢁⠔⠈⠀⠈⠣⠜⣰⠇⠈⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡠⣾⠏⡩⢀⡾⢁⠁⣌⢀⡀⠀⠀⠀⢀⣿⢢⡄⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢰⡟⠀⠀⣼⠃⠂⣀⠈⣉⠀⣰⠋⠂⣼⣏⠜⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠋⠀⠀⢰⡟⠁⣰⡏⣼⡇⣰⡏⢀⢘⡿⢸⡆⠀⠉⡀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣿⠁⠐⠙⠃⠻⠈⠘⠃⠁⠈⢀⠀⠻⢦⣀⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡀⠔⠁⠀⠿⠈⠀⠀⠀⠀⠀⠀⡀⠄⢈⡠⠤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠈⠉⠁⢠⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢟⠛⠋⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⠋⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⣠⣄⡀⠀⠀⠀⣀⣠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣆⠀⢀⣤⣤⡤⠄⣼⣿⠋⣇⠀⣰⡟⢹⡷⠀⢠⣾⠟⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⢀⡿⠃⢸⣿⡏⢀⡟⠀⡟⢀⣾⡃⢠⡟⢁⣴⣿⣿⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣦⣠⡿⠁⠀⠘⣿⣥⠞⠁⠀⠳⣾⠟⠁⠸⠿⠟⢰⣿⣗⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠇⠀⠀⢀⡀⠀⠀⣀⠀⠀⢀⣠⣄⠀⢀⣶⡗⣠⠾⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⣿⣿⣤⣾⣿⢇⣶⡟⠛⠛⠁⣼⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⢿⣿⡏⣿⠃⢸⣿⣠⣴⠖⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠶⠖⠛⠃⠀⠉⠀⠛⠀⠀⠉⠁⠁⢀⣿⡟⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⢰⣢⣾⣷⣒⠀⠀⠀⣾⣿⠀⠀⠻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⣿⡳⠀⠘⠀⠰⣿⠇⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⣷⣦⡀⠀⠀⢀⣴⣿⠿⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣿⠋⠀⠀⠀⠹⣿⡄⢠⣿⠏⠀⠀⠀⠙⣿⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠛⠋⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠙⠛⢿⣷⡄
⠀⠀⠀⠀⠀⠀⢠⣴⣤⡀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠰⣶⣄⠀⠀⣿⣿⠀⠀⣠⣶⠆⠀⠀⠀⠀⣿⡷⠀⠀⠀⠀⠀⠀⣠⣶⣄
⠀⠀⠀⠀⠀⢀⣾⣿⠻⣿⣶⣿⣷⠀⠀⠘⠋⠀⠀⠀⠀⠀⠈⠻⣷⣄⠈⠁⣠⣾⠟⠁⠀⠀⢀⣠⣴⡿⠃⠀⠀⣾⣿⣿⣿⠟⢻⣿⡀
⠀⠀⠀⢠⣾⣿⠟⠃⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠘⢿⣷⣶⣦⣄⠈⠁⠀⠀⠈⠁⣠⣴⣶⣾⡿⠟⠋⠀⠀⠀⠀⢘⣿⡆⠀⠀⠈⠻⣿⣶⡀
⠀⠀⠀⠀⠙⠻⣿⡆⢀⣤⣌⣿⣧⠀⠀⠀⢀⣴⣾⡿⠟⠛⠋⠁⣠⣄⠀⠀⣠⣄⠈⠙⠛⠻⠿⣷⣦⣀⠀⠀⠀⣾⣿⣥⣤⡀⣴⣿⠟⠋
⠀⠀⠀⠀⠀⠀⢿⣿⡿⠛⠛⠛⠋⠀⠀⢰⣿⠋⠁⠀⠀⠀⣠⣾⠟⠁⣠⣄⠈⠻⣷⣆⡀⠀⠀⠈⠙⣿⣆⠀⠀⠙⠛⠋⠻⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢀⣀⣿⣿⡀⠀⠀⠀⠈⠛⠁⠀⢀⣿⣿⡀⠀⠈⠛⠃⠀⠀⠀⢀⣿⣿⣀⡀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⠋⠻⣿⣶⣦⠀⠀⠀⠀⠀⣼⡿⢿⣧⠀⠀⠀⠀⠀⣴⣶⣿⠟⠙⠻⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠘⣿⣦⡀⠀⣀⣼⡿⠁⠈⢿⣧⣀⠀⣀⣴⣿⠃⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⡀
⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠋⠀⠀⠀⠀⠙⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⡀
⠀⠀⠀⠀⣴⣿⠟⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⣰⣾⠿⠛⠻⠿⠆⠀⠀⣠⣾⡿⠛⠛⢿⣷⣄⠀⠀⣰⣿⠿⠛⠻⢿⣷⡀⠀⢸⣿⠛⠻⢿⣦⠀⠀⠀⠀⠈⢿⣧⡀
⠀⠀⣼⡿⠁⠀⠀⠀⠀⢰⣿⠃⠀⠀⣤⣤⣤⡄⢀⣿⡏⠀⠀⠀⠀⢹⣿⡄⢸⣿⠃⠀⠀⠀⠀⢻⣷⠀⢸⣿⠀⠀⠘⣿⡇⠀⠀⠀⠀⠈⢿⣧
⠀⢰⣿⠃⠀⠀⠀⠀⠀⠘⣿⣆⠀⠀⠉⣹⣿⠇⠀⢿⣧⡀⠀⠀⢀⣼⡿⠁⠘⣿⣆⠀⠀⠀⢀⣾⡟⠀⢸⣿⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠘⣿⡆
⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣶⣾⠿⠋⠀⠀⠈⠻⠿⣷⣾⡿⠟⠁⠀⠀⠈⠻⢿⣷⣾⠿⠋⠀⠀⠸⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⠀⠀⠀⠀⣾⡆⠀⠀⢰⣷⡀⠀⠀⣠⣶⣿⣿⣶⣄⠀⢰⣿⠄⣀⣴⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠇⠀⣾⡿⠁⠀⠀⠈⠁⠀⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠆⠀⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⡀⠀⢻⣷⣤⣤⣾⡿⠀⠀⠙⢿⣶⣤⣤⣶⡷⠀⢸⣿⡍⠻⣿⣦⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠈⠻⣿⣄⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠀⠀⠈⠉⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠀⠀⠺⠗⠀⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣿⣶⣶⣶⣶⣶⣶⣿⡿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀⡀⠀⡀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢳⠀⠀⠸⠀⠀⠈⠁⠀⡇⠀⢠⠃⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⡄⠀⡄⢃⠀⡄⠃⢸⠀⢸⠀⢀⠇⡈⢠⠀⡜⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⡌⠂⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⡇⠈⢠⠆⡰⠃⠀⠀⣠⠄
⠀⠀⠀⠀⠀⠒⢄⡀⠰⣌⠂⠈⠂⠀⠀⠀⠀⠀⠀⠣⡀⠀⢀⣿⡴⠃⠀⠁⠈⠠⠊⡠⡊⠁
⠀⠀⠀⢀⡀⠀⠀⢌⡂⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣿⠀⠀⠀⠀⣰⠃⢀⢸⣿⠠⠔⠊
⠀⠀⠀⠀⠈⠑⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡄⠀⠀⣰⠏⠀⠁⢘⣿⠠⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠶⣏⢹⣿⡿⠑⠤⠞⠁⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⣤⢶⡂⠀⣰⡿⢿⠱⣿⡀⡿⠈⠙⠁⣿⠀⠀⠀⠀⠀⠀⢸⣿⠉⠤⠤⠴⠒
⠀⠀⠀⠀⠀⠀⡤⠄⢀⣾⠃⣼⣧⢠⣿⡇⢸⠀⠉⠉⠁⠀⠀⠀⣿⣀⣶⡄⠀⠀⢀⠈⡋⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣠⠏⣿⡟⠈⣓⠋⠀⠀⠀⠀⣴⡿⡆⣿⣿⡟⠀⠀⠀⣸⠀⠧⢀⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠉⢀⣿⣿⠀⣿⡆⢠⣄⢠⡀⣿⡇⢸⢿⣯⢳⡄⠀⢠⡏⢀⡀⠀⠂⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠄⠒⠁⠀⠀⣼⠙⣿⡆⢼⣇⢸⣿⢸⡇⡼⠓⠋⠈⠛⠀⠻⣄⡾⢀⠀⠨⣁
⠀⠀⠀⠀⠀⣀⠔⢊⠴⠊⢰⠃⠀⣿⠃⢸⣿⣰⠿⠿⠙⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⡑⠂⠀⠉⠒⠄
⠀⠀⠀⠀⠈⠀⠀⢀⡠⠂⣿⠀⢠⡿⠀⠈⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⡘⠂⠈⠓⢄⡀
⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⢿⣠⠞⢁⠀⡌⠀⢀⠀⠀⡀⠀⢠⠀⠀⢆⠀⡌⠂⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⢀⠎⠘⠀⡄⠘⠀⡆⠇⠀⡈⠀⠰⡈⠀⠘⣆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⢰⠁⠀⢀⡇⠀⠀⡇⠀⠀⢣⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠀⠀⠀⠃⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡿⠿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡟⠀⠀⠀⠀⣀⣄⡀⠀⠀⠀⢀⣤⣶⡿⠟⣾⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠁⢀⣠⣴⣶⣾⣿⣿⢷⣄⢀⣾⣿⡿⠋⢸⣿⠀⢠⣾⣿⡿⣃⡀⠀⢠⣿⣿⡟⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢠⣾⣿⣿⡿⣿⣿⡿⠀⠀⠀⣼⣿⡟⠀⠀⣼⡟⢠⣿⣿⠟⠀⣿⡇⢠⣿⣿⠏⠀⢠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠃⠘⠻⠿⠋⣰⣿⣿⠃⠀⠀⢰⣿⣿⠁⠀⢰⣿⢃⣿⣿⡏⠀⢰⣿⠇⣸⣿⡿⠀⣠⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣼⣿⡿⠁⠀⠀⠀⢸⣿⡏⠀⣰⡿⠃⢸⣿⣿⠀⢀⣿⡟⠀⢻⣿⣇⣴⡿⠁⢿⣿⡇⠀⢀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡀⢀⣠⣾⡿⠟⠁⠀⠀⠀⠀⠀⠻⠷⠶⠋⠀⠀⢸⣿⣿⢀⣾⠟⠀⠀⠀⠉⠉⢉⣴⣦⡀⠛⠿⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠋⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠏⠀⣴⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣧⡀⠀⠀⣼⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃⠀⣰⣿⣿⣿⠃⢀⣾⣿⡇⠀⣠⣾⣿⡿⢿⣿⣿⠇⠀⣼⣿⣋⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⢠⣿⣿⣿⠃⢠⣾⣿⡿⠀⢰⣿⣿⠏⠀⢿⣿⠏⠠⣾⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠁⠀⠀⣾⣿⣿⠇⣴⣿⣿⣿⠃⢀⣿⣿⠏⠀⠀⠀⠁⠀⢠⣿⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠀⠀⢻⣿⣧⡾⠟⣿⣿⡏⠀⢸⣿⣿⠀⠀⢀⡄⠀⠀⣾⡿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⢀⣶⠉⠉⠀⠀⢿⣿⡇⠀⢀⠻⢿⣤⣶⠟⠃⠀⢠⣿⠃⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠙⠛⠞⠋⠀⠀⠀⠀⠀⠀⠀⠘⠿⠀⠀⠈⢿⣿⣷⡄⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣷⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡴⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣧⣾⠋⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠋⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⢀⡿⠁⠀⢀⣾⣿⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⣿⡇⠀⠀⠀⠀⣼⠇⣿⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢳⣹⣾⠁⠀⣠⣾⣿⣿⣿⣟⣛⡻⠷⢦⣤⣀⣀⣀⣠⡾⠟⣿⠀⣿⣇⠀⠀⠀⠀⣿⢿⡿⠀⠀⠀⢀⣼⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⡔⢯⣿⠃⠀⠀⠻⢿⣿⠏⠉⠉⠉⠙⠻⢶⣾⣍⣉⣛⣋⣮⣼⠏⠀⠘⣿⣦⣀⠀⢠⣿⣼⡇⢀⣀⣴⠟⢁⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠉⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠉⠉⣉⣭⣍⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣯⣶⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠰⠍⣿⠁⢀⣀⡀⠀⣠⣾⣿⠀⢸⣿⣿⠿⣷⡀⠀⠸⣿⣿⠿⣷⡄⠀⢠⣾⣟⣿⣦⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠐⣰⡿⠀⠈⢿⣿⣿⢿⣿⣿⢠⡿⢿⡿⢦⣼⣧⣤⣾⣿⡿⢦⣼⣧⣤⣿⣽⠟⠙⣿⢣⣿⠃⢰⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠹⣹⡇⠀⠀⠀⠈⠀⣼⣿⠇⣿⢁⡿⠁⠀⢸⡇⣾⣃⣾⠃⠀⢸⡟⣼⣿⡟⠀⠀⣿⢈⣿⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⣼⣇⠀⠀⠀⠀⣴⣿⠏⠐⣿⣾⠃⠀⠀⣾⠃⣿⣾⠇⠀⠀⣼⠇⣿⢽⡇⠀⣸⣟⢹⡇⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣦⣤⣴⣾⡿⠋⠀⠀⢿⣿⡀⢀⣼⠏⠀⢻⣿⡀⢀⣼⠏⠀⣿⣿⣷⣴⡟⢻⣿⣇⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⠙⠛⠿⠿⠛⠋⣠⣶⣶⣦⠈⠛⠿⠛⠁⠀⠀⠈⠛⠿⠛⠁⠀⠀⠘⠻⢛⡋⠀⠀⠛⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡀⠀⠀⢀⣼⣏⣽⢻⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣌⡹⢷⣶⣼⣿⣿⣷⠾⣋⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⣿⢰⡿⠀⠀⢀⣴⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⣿⣿⣿⠏⠀⢸⡿⣷⠀⠀⢀⣿⢻⡷⠀⢀⣴⣿⠟⢹⣷⠀⢰⣿⣼⡇⠀⣴⡟⠁⣴⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢼⡟⠀⢀⣿⢱⡿⠀⢀⣾⠿⣾⠃⢠⡟⣿⠏⢠⣿⡿⠀⣾⡇⣾⢣⡾⢋⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⢣⣿⠁⠀⣼⢯⣾⠃⢀⣾⣿⣾⠏⢀⣿⢡⡿⠀⠈⠻⠃⢠⣿⣈⢻⣿⣴⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣿⣿⣿⠃⠀⢰⣿⣿⣇⣴⠟⣿⣯⡟⠀⢸⣯⣼⡇⠀⠀⠀⢀⣾⡋⣥⣾⡛⠋⠁⢀⣠⠀⠀⣴⣷
⠀⠀⠀⠀⠀⠀⣰⣿⡟⠋⠉⢩⣿⣿⣿⣿⣿⣶⣤⣙⠿⠛⠁⢸⣿⣿⣁⣴⠾⣿⣿⣇⣠⣤⣶⣿⠉⢱⡿⠘⢿⣿⣿⣿⠟⣠⡾⢻⡿
⠀⠀⠀⠀⠀⠀⢻⣿⣄⣀⣴⠟⣩⡿⠋⠀⠈⠙⠻⢿⣿⣷⣶⣤⣙⡛⠋⠁⠀⠈⠛⠛⠛⠉⠀⢿⣧⣾⠃⠀⠀⠈⣉⣥⡾⠟⢁⣾⠃
⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠷⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣿⣛⠻⠶⠶⣶⣦⣤⣤⣤⣤⣬⣥⣤⣶⠶⠟⣫⠑⢃⣾⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⢶⣦⣤⣔⣎⣀⣠⣼⣸⣡⣼⣵⣾⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⣷⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣵⠟⠁⢰⡤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢯⡗⠀⠙⠉⠀⣆⣉⣴⠟
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣄⠀⠀⠀⠀⠀⠀⠙⠛⠁⢀⡀⠀⠀⠘⠉⠁
⠀⠀⠀⠀⠀⠀⣰⡿⠋⠉⠀⠀⣀⠀⠀⡀⠀⠀⠀⠀⣾⠇⠀⠀⠀⣄⠀⣀⠀⠀⠀⢀⣤⣤⠀⣾⡇
⠀⠀⠀⠀⠀⢰⢋⣠⣤⣤⣠⣾⢟⣄⣾⢟⡀⣠⠞⣻⡟⠀⠀⠀⣼⣿⢸⡿⢠⣿⢠⡿⢻⡟⣰⣟⣀⣄
⠀⠀⠀⠀⢠⣿⠈⠛⣿⠇⢸⡏⡼⢹⡏⣸⢻⣇⣼⣿⠇⢠⡆⢰⣿⠃⢿⡷⣿⡇⢿⡇⣀⢈⣿⡟⠋⠁
⠀⠀⠀⠀⠈⠻⠷⠚⠁⠀⠈⠉⣡⡌⠓⠃⠈⠛⠃⠈⠓⠉⠀⢸⡏⣠⠆⠀⠹⠇⠈⠉⠁⢸⠘⠿⣄⡀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣼⡿⠧⢀⣴⡆⠀⠀⣴⡆⢠⣄⢈⣩⣥⠀⣰⡄⢀⣤⠀⠀⠀⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢸⡿⠀⢠⡿⢁⡞⠀⢸⡟⣵⣿⢇⣾⠟⢱⢠⡿⢁⣾⠃⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠘⠧⠚⠁⠀⣙⠋⣸⡟⠸⣏⡰⠃⠺⠷⠛⣿⠀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠉⠀⣠⡿⠁⣱⡄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠚⠁⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠂⠀⠀⠀⠰⠷⠁⠀⠀⠀⡀⠀⠀⠀⢰⠲⡀
⠰⠀⠀⠀⣴⢠⡤⣲⡂⡀⠀⠀⠀⠘⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠃⠁
⠀⠀⠀⠀⠀⠟⠉⠉⠀⠐⡀⠀⠀⠀⠠⠀⠀⠀⢀⡂⠀⠀⠀⠉⠀⠀⠀⡘⠁⠀⠀⠀⠀⠂
⢠⠰⠀⠀⠈⢣⡤⠄⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠘⢧⠣⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⢣⠱⡀⠀⠀⠉⠁⠀⠀⠘⢎⠀⠄⢀⣀⣠⡴⠃⠈⠳⢌⣒⣐⠰⠛⠁⠰⠬⠝⣒⣒⣒⣲⠶⠊
⠀⠀⠳⣈⠢⢄⠀⢀⣀⢠⠃⠀⠉⠐⠒⠒⡏⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠈⠑⠂⡠⠄⠒⠁⠀⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠁⠀⢀⡤⡲⢄⣠⠎⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⠀⠀⡠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠆⠴⠀⠀⢠⡷⠃⠀⠀⣠⠖⠀⡖⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠊⠀⠀⠸⢡⠃⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠑⠜⠐⠀⠀⢀⠞⠂⠀⠛⠀⠀⢀⠃⠋⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⠀⠀⠀⢀⣠⠊⢆⡑⠄⣀⡀⣀⣀⠔⡏⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠒⢪⠥⢶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠧⢀⣉⣁⡉⠴⠃⠀⠀⠈⠒⠀⠂⠚⠁⠰⠦⠏⡂⠒⠃⠀⣀⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⣀⣒⣂⠭⠕⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠢⠄⠉⠁⢧⠑⢄⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⣠⠴⠦⡄⣠⠴⠶⢤⡜⠁⠀⣰⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠚⠁⢀⡙⠃⢀⣤⡙⠁⢠⣄⠈⢀⣤⣤⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢦⠙⠂⠀⣴⡟⢻⡄⢺⣧⢳⢀⣿⣸⢀⡿⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⠀⢰⡿⠀⢸⣇⢼⣏⡟⢹⣨⠟⠹⣇⠂⢿⡇⠀⣾⠀⢀⣴⠃⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⡆⠈⠛⠊⣸⡏⠈⠋⠀⠈⠁⠀⠀⠀⠀⠘⠿⡾⠃⠀⣋⠴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣸⠃⠀⠀⡸⢹⠇⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣾⡷⠃⠀⠀⠀⠙⠋⢀⠀⠀⢀⣴⣆⠀⢀⣿⣠⠞⣻⠀⠀⠀⠀⠀⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣶⣿⠁⠀⠀⠀⣴⠀⠀⣾⠁⢀⣿⠁⠉⢀⣼⣿⡥⠞⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠐⣆⠀⠀⠀⠀⣿⠀⠀⠀⢠⡿⠀⡼⣿⠀⣸⣿⠀⢠⠞⢸⡟⢷⣄⡀⠀⠀⠀⣀⣼⠟⢠⡆
⠀⠀⠀⠀⠀⠀⠈⠉⠉⣇⠀⢻⣆⢀⡴⠋⢧⠞⠀⠻⠾⠁⠙⠒⠁⠀⠘⠁⣀⠙⠛⠿⠶⠿⠛⠁⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠉⠉⢀⣤⡀⠀⠀⣠⢦⣀⣀⡴⢧⣀⣀⡴⠛⢦⣀⡀⠀⠀⣀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠉⠀⠙⠛⠛⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣤⡀⠀⠀⠀⢸⡧⢰⠟⣤⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠁⠀⢨⡗⠀⠙⠀⠈⠇⠈⠀⠀⠀⠀⣀⣸⠃⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⡤⠈⠀⠀⠀⣠⢶⡄⣠⡖⡆⣠⠟⣹⡏⠀⠀⠁⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⢀⡾⣯⣸⣶⠿⣼⣴⡏⣴⡿⠀⠀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣤⣤⣶⡄⣾⢇⡼⢱⣏⡜⠁⠸⠞⢹⡇⠀⣰⠃⠺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢀⡴⠁⠘⣋⡴⠞⢛⠓⢲⠀⠀⠸⣇⣴⠋⠀⠀⠀⢀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣦⡸⣿⡀⠀⢀⣤⠟⠀⢠⡞⠁⠀⢠⡟⣠⢞⡄⢀⡄⢈⡴⣦⠀⠀⠴⣋⣡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠙⠳⠶⠋⠁⠀⠀⡟⠀⠀⠀⣾⢿⡁⣾⡱⣻⢣⣿⠔⢁⡄⢀⠀⢽⡦
⠀⠀⠀⠀⠀⠀⠐⠻⢛⠃⠀⠀⣠⣴⠿⠛⠻⢷⣄⠀⢸⡏⢸⠇⠉⣠⡿⢻⡇⢀⡾⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠴⠟⠀⢀⣾⠟⠁⠀⠀⠀⢠⠉⠀⢿⣥⠏⢠⠞⣹⠃⢈⡛⠋⠀⠀⣾⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠠⠀⠀⠀⠀⡀⠁⢰⠃⢠⠏⢀⣷⡃⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⡀⠈⠒⢋⡀⠈⠉⢁⣀⣼⠏⠀⠂⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⣤⣶⡄⣠⡞⢹⠀⣰⡏⡟⢀⣴⠋⣹⡟
⠀⠀⠀⠀⠀⠀⠈⠀⠀⢹⣇⠀⠀⠀⠀⠀⢀⡾⣰⠟⢧⠾⢺⠏⡿⠗⣾⢃⡜⢹⠃⠀⡴⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⡀⣀⣠⣿⠃⣿⣠⠎⠀⢿⠞⠀⠀⠙⠋⠀⣼⠦⠞⠁⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⢡⡿⠀⠈⠁⠀⠀⠀⢀⠀⢀⡶⡶⣸⢇⣠⠆⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠸⠆⠀⠀⣿⠃⠀⠀⠀⢀⡾⢠⡟⢠⡟⠀⣴⣿⠟⠋⠀⠀⣰⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣸⡟⠀⠀⠀⠀⣸⡷⢻⣃⣾⣥⠊⣿⠻⣆⡀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⣿⠷⠶⠶⠿⠛⠈⠀⠘⣋⠀⠀⠀⠁⠀⠈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠂⠀⢰⠏⣷⠘⠀⠀⠰⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠒⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠑⠢⣄⡀⢠⡿⠀⠀⠀⠀⠀⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⢠⠎⠁⠈⠹⣆⠀⠀⠀⠉⢹⡗⠒⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢀⠀⠀⠀⠀⠀⠎⠀⡠⠶⢦⡀⠀⢸⠀⠀⠀⠀⢿⠀⠀⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠉⠒⠂⠈⠀⠀⢸⣠⠀⠀⣿⢀⠜⣷⡃⠀⠀⣾⠀⢠⡏⠀⣼⣇⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣀⣤⣆⠀⠀⠀⠀⢠⣿⡖⢰⠟⠁⢸⣿⠙⠲⢶⠟⠒⢹⣧⣰⠋⢿⣆⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢸⣿⠀⠀⢀⡴⠋⠙⢛⠁⠀⠀⠀⠻⠶⠴⠋⠀⠀⠀⠉⠀⡀⠀⠙⠷⢤⣤⠤⠞⠁
⠀⠀⠀⢀⡠⠴⠒⠒⠒⠒⠒⠒⠒⢺⣿⠒⠋⠁⢀⡴⠋⠉⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣀⡀
⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⣾⠁⠀⠀⠀⡸⠀⠀⠀⠀⢀⡴⠋⠙⠀⢸⣇⠈⢸⡇
⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠐⢼⣯⣄⠠⠄⢊⠀⠀⣴⡆⠀⣾⡇⠀⠀⠀⣼⣿⣴⡋⠀⠀⠀⣠⠖⠋⠉⠹⣦
⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⣿⡀⠀⣿⣧⣀⣠⠞⠹⡿⠀⠉⠛⣶⣮⣄⡀⠀⢀⣠⠏
⠀⠀⠈⠻⢶⣤⣄⣀⣤⡴⠞⠁⠀⠀⠀⠀⠀⠘⣿⡀⠀⢠⣿⣄⡰⠻⠷⠊⠀⠉⠉⠀⠀⠀⠀⠀⢀⠞⠀⠀⠉⠙⠋⠉
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠁⠈⠉⡔⠒⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠤⠤⠐⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠉⠛⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠉⠉⠉⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⡠⠀⠀⠀⠀⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠈⠓⠦⣀⡀⢰⡟⠀⠀⠀⠀⢀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡴⠉⠀⠈⠻⣦⠀⠀⠀⠀⠉⣿⡗⠒⠒⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⡠⠤⣤⡀⠀⢰⡇⠀⠀⠀⠀⢻⡀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠙⠲⠤⠤⠒⠊⠀⠀⡾⢀⠀⠈⣷⠀⢨⢷⡴⠀⠀⠀⢸⠃⠀⣴⠛⠉⣿⡇⠀⠀⠀⠀⠀⠀⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⣀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢹⣿⣀⣀⣿⠔⠁⣾⡿⢦⣄⣀⣿⣠⠴⣿⡀⣰⠿⣿⡀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⢻⣷⠀⠀⠀⠀⢠⠞⢿⣦⠜⠁⠀⠀⢿⣧⡀⢈⡽⠃⠀⠀⠻⠟⠋⠀⠹⣷⣄⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⢸⣿⣀⣀⡤⠖⠁⢀⣠⠤⣤⡀⠀⠀⠀⠙⠛⠉⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠉⠛⠓⠒⠋
⠀⢀⡤⠚⠋⠉⠉⠉⠉⠉⠉⠉⠉⢹⡿⠉⠀⠀⠀⣴⠏⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣸⡇⢀⢤⣄
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⣸⡏⠀⠀⠀⠀⡸⠁⠀⠀⠀⠀⣠⡞⠁⠈⡇⠀⣿⡷⠁⣸⠇⠀⠀⠀⠀⠀⢀⣀⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠑⢲⣿⣅⣠⠤⠔⢊⠀⠀⢠⣶⠀⠀⣿⠁⠀⠀⠁⢰⣿⣧⡾⠋⠀⠀⠀⢀⠴⠚⠉⠉⠹
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠃⠀⢸⡇⠀⢀⣿⣇⠀⢀⡰⠋⢿⡏⠉⠙⠻⢶⣴⣁⡀⠀⠀⠀⣠
⠈⠻⣷⣄⣀⠀⠀⢀⣀⣤⠞⠁⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⢀⣿⡀⢀⡾⣿⡤⠞⠈⠙⠛⠉⠀⠀⠀⠀⠀⢀⡴⠃⠈⠙⠛⠛⠛⠋⠁
⠀⠀⠀⠉⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢦⠤⠊⠙⠛⢋⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⠀⣀⠤⠊⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣽⣶⣶⣶⣾⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣬⣤⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⣀⡀⠀⢀⣀⠀⠀⠀⠀⣾⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⢰⠋⠀⢠⡟⠉⢳⢤⣶⠆⢿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⡀⣀⠘⠄⠤⠊⠣⣠⠜⠹⢇⡴⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣀⠴⠋⠀⣾⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⣰⣿⣀⣀⣀⣠⣤⣤⡤⠴⠂⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⢫⣿⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⢀⣠⡿⠃⣸⡇⠀⡀⠀⠀⠀⢀⣠⠄⠀⢸⡇⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⡇⠘⡇⠀⠀⡻⣾⠀⠀⠀⢸⣇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠱⠄⡠⠆⠈⠳⠴⠂⠀⣿⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣼⣃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠒⠋⠀⠀⠉⠉⠉⠓⠂⠛⠛⠛⠋⠙⠛⠉⠁






⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣶⣤⡀⠀⠀⠀⠀⢸⣿⠀⣾⢋⣤⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠉⠀⠀⢹⡇⠀⠰⠂⠀⠀⡿⠀⠃⠈⠉⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⢀⠀⠟⠀⠀⠀⠀⠀⣀⠀⠀⢀⣤⡀⠀⣠⡶⢶⣿⠁⠀⠠⠄⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠛⠁⠀⠀⠀⠀⣠⡾⢉⠇⣰⡿⢠⠇⣴⡟⢠⣿⠇⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⣧⡼⣾⡏⢻⡾⢾⡟⣠⢿⡿⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⣠⣤⣴⣶⡄⣼⡏⢀⡞⢠⣿⣠⠏⠀⠸⡷⠋⣾⡇⠀⢀⡾⠁⢾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠉⠉⢁⡼⠃⠹⠷⢋⣠⣬⡿⠷⠦⣄⠀⠀⠀⣿⡇⢀⡾⠁⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⣠⠞⠁⠀⣠⡶⠛⠉⠀⣴⠂⠀⡼⠀⠀⠀⠘⠳⠛⡁⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠸⣿⣄⠀⢀⣠⡾⠋⡀⠀⣼⠏⠀⠀⠀⢰⡏⣠⠞⣱⠆⢀⣾⠀⣰⠟⣹⠇⠀⠀⠛⣥⣴⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠈⠛⠛⠛⠁⠀⠀⠁⢸⡏⠀⠀⠀⢀⣿⢿⡅⢠⣿⡴⢻⡏⣸⣿⠞⠁⣠⠄⣀⠀⠰⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠟⠀⠀⠀⠀⣠⣴⡾⠿⠿⠿⣷⣄⠀⠀⣼⡇⠀⡟⠀⠉⢀⣾⠷⣿⠁⠀⣰⠏⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⠗⠀⠀⣠⣾⠟⠉⠀⠀⠀⠀⠈⠛⠀⠀⣿⠀⡼⠁⢀⡴⢻⡟⠀⠻⣶⠾⠋⠀⠀⣽⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠻⠟⠁⣠⠏⠀⣾⠁⢠⣶⡆⠀⠀⠀⢰⡿⠀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⠀⢠⡏⢀⣼⠃⢠⡞⣯⡀⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠛⠛⢁⡀⠀⠉⠀⢁⣀⣠⣿⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀⠀⣠⡞⢹⠄⠀⣴⠋⡷⠀⢀⣾⠟⢩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⣿⢃⣾⣿⠀⣜⣠⣾⢿⣴⣃⣠⣿⠁⣠⢿⡟⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⢀⣼⠇⣼⡇⠘⡿⠉⢸⡇⣴⠋⠉⢸⣇⡴⠃⢸⡇⠀⣠⡟⠀⠀⠛⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣦⣀⣀⣀⣤⣿⣿⠀⣿⣠⡞⠁⠀⠸⠟⠁⠀⠀⠀⠉⠀⠀⣼⠛⠞⠋⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠋⢁⣾⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡆⣸⠇⢀⣤⠄⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⢠⣦⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⣼⠀⣰⡇⢀⣾⠃⠀⣴⣿⣾⠿⠋⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⣼⢇⣼⣿⠀⣼⠇⣠⢞⣿⣿⡁⠀⠀⠀⢀⣾⠃⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⣾⡿⠀⠀⠀⢀⣀⡀⣿⠟⢸⣧⡾⠻⠶⠃⣸⠇⠘⢷⣤⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠠⣿⠷⠶⠿⠿⠟⠛⠁⠀⠀⠀⢉⡀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠠⠀⠀⢀⡾⢹⡇⠙⠀⠀⠀⠰⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠋


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣏⠆⢾⣋⣉⣛⣋⣩⣉⣉⣋⣉⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣦⣄⠀⠀⠀⠀⠀⠀⣠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣭⣭⣭⣿⣿⣿⣿⣿⣿⣭⣭⣭⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠹⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢾⣽⣽⣿⣿⣿⣿⣿⣿⣿⣷⣾⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡆⠀⠀⢰⣶⡶⠿⢿⢿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣰⣆⠘⠁⠀⠀⠀⠈⠳⣽⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣼⣽⣏⣿⣷⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣤⠀⡄⢀⡤⣄⠀⣄⣤⠠⣄⣠⠄⠀⣠⢤⡄⢀⣤⡀⠠⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⡀⢿⣤⠇⠸⣧⡤⠀⡟⢧⠀⢹⡏⠀⠀⢿⣤⡄⣼⠿⣧⠀⣿


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

⠀⠀⠀⠀⠀⠀⢀⣶⡶⢶⣶⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⠉⠉⠉⠹⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⣽⣿⣿⣭⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⣿⣿⣻⣿⣉⣙⣀⣀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣿⡿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣽⣿⣿⣿⣿⣿⣷⣿⣿⣷⣷⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⣿⣿⡿⣏⡁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠋⠛⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠷⠾⠿⠏⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠉⠉⠉⠉⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣶⣾⣷⣄⠀⣴⣾⣶⡄
⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⠿⣿⣿⣿⢿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠀⠚⠻⣿⣿⣿⣿⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⢸⣯⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⡟⠻⣿⣿⣿⣿⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠿⠟⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠟⠁⢀⡾⠋⠈⠻⠋⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⣂⡄⡀⡔⠒⡀⠀⢀⠀⡤⠔⠒⠠⡀⠀⠀⢀⠀⡤⠔⠒⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡈⠀⡨⣜⢰⢅⠃⡰⢁⠊⠅⡄⠂⠄⠑⠀⡰⢡⠊⠁⡄⠢⡀⠱⠀⠀⠀⠀⠀⠀⠀⢸⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠡⡀⡁⡘⢹⠁⢀⠃⡌⠀⠃⡈⣒⠽⠀⢠⠁⡀⠀⢆⡈⣒⢽⠀⠇⠀⠀⢀⠠⠤⢤⣃⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠚⠊⠀⠀⠆⢘⠀⡇⠀⠀⠀⠀⠀⠀⣸⠀⠇⠀⠀⠀⠀⢸⠀⡀⢀⢊⠔⠁⡑⣦⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠈⠇⠀⠀⠀⠸⠀⠠⠀⠀⠀⠀⡘⠀⠿⠀⢠⠀⠀⠀⠀⡌⠀⠃⠁⢌⠀⠀⠇⠁⡬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠄⢀⠄⠄⠀⢣⠀⠑⠤⠤⠊⠀⡔⠀⢣⠀⠑⠤⠄⠊⠀⡌⠀⡀⠐⡀⠀⠈⢻⠧⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠤⠊⠀⠀⠀⠑⠠⠀⠠⠄⠊⠀⠀⠀⠑⠠⠀⠠⠔⠊⠀⠀⠑⠢⠄⠁⠩⠊⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢀⠔⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⡄⠀⠀⠀⠀⠀⢈⠀⢡⠀⠠⠪⢪⢢⠀⠀⠀⡠⡊⠥⢈⠑⢄⠀⠈⡀⡅⢀⠆⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢤⠀⠀⠈⡄⢸⠀⢀⢠⠌⡘⠀⠀⠌⠔⠀⠄⡂⠂⢘⠀⠀⢠⢠⡌⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠈⠀⠀⠀⠀⡈⠀⠈⠎⢀⠁⠀⡘⠘⠀⠀⠈⠢⠤⠊⠀⠀⢠⠌⡞⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠆⠁⠀⠌⠀⡌⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⡆⠃⠈⢢⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⠀⠀⠧⠊⢀⠌⠈⠀⠆⠁⠰⠀⠀⠀⠀⠀⠠⠀⢰⠀⡇⠀⠀⠀⢃⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢒⡇⣗⠂⠠⠤⠤⢲⡆⠀⠀⢀⠌⠀⠀⠀⢰⠐⠄⠑⠄⣀⠀⢠⠖⠀⠠⣀⡣⠁⠀⠈⢌⠒⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠜⠀⠈⠁⠒⠂⠁⠑⠤⠐⠁⠀⠀⠀⠐⠊⠀⠀⠑⠒⠒⠂⠁⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠥⠤⠤⠤⠶⢠⣏⣀⣀⣀⣤⡄⠀⠀⠀⡄⢀⣇⠴⠖⢠⠀⣿⡠⠖⠓
⠀⠀⠀⠀⠀⠀⠀⣉⣼⣉⣿⣉⣿⣀⡋⠣⡀⣼⠃⠀⠀⠀⠀⠾⠃⣸⠣⣤⡐⠛⢠⠟⢤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡿⠤⠯⢤⡯⠤⠄⣠⠝⠧⣄⣀⠀⠀⠀⡶⠤⠥⠤⠬⠧⠤⠥⠤⠤⠽⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⡚⡣⠒⠊⣡⠦⠀⠈⠙⠁⠀⠺⠃⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⣥⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣽⣧⣤⡮⠥⠤⠕⠶⣤⠀⠀⠀⠉⠉⠉⢉⡽⢻⡟⢭⡉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣠⠶⠀⢸⡇⠀⠒⠦⣤⡀⠀⠀⠀⠀⣠⠶⠋⠀⢸⡇⠀⠙⠳⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠀⠘⠃⠀⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣦⣴⣤⠴⣆⢴⡦⣴⣦⣴⢦⣴⣤⡤⡶⢤⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⠛⠘⠒⠋⠓⠛⠋⠘⠛⠛⠛⠚⠘⠁⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠚⢛⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣖⡒⠶⠶⣤⣀⣾⢁⠀⡀⠀⠀⢉⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⡒⠂⠨⣀⡢⢍⡻⣿⣆⠞⡴⢀⠀⠀⠔⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠠⠀⣀⡒⠭⣓⠯⣺⢎⡞⠐⢁⠄⡀⠐⡿⠀⣀⣤⣤⢤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣍⠀⠀⠬⣕⡪⢝⠊⢳⠀⠀⠀⠈⣠⡾⣡⠞⡙⠀⠁⠐⠀⠋⡻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣤⣤⣙⢦⣤⣙⡒⠍⠀⠀⢠⡆⡔⠀⣠⠏⡴⠏⠰⢁⠄⡠⢀⠀⠀⠀⢺⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡏⠁⠀⠂⠈⢢⡙⢷⣮⡙⠺⣄⡀⢀⢸⡏⡴⡴⢃⠞⠁⠀⠀⢈⠜⡡⢃⢎⡄⡀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⠢⣈⠢⠁⠀⠀⠉⡳⢤⡙⠦⣙⣏⣞⡵⠳⢂⣞⡰⠤⠤⣞⢼⣱⢿⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡁⠀⠐⠬⣑⠮⣑⠀⠀⠠⣑⣊⣑⣉⣓⣻⣿⣿⣖⠻⡛⠉⠀⠀⢤⡲⢝⠮⡓⢍⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣦⣽⣶⣹⢦⢟⢀⠀⠁⠉⠱⢫⠞⢩⠿⡽⢍⡓⢦⣀⢀⠢⡑⢌⠢⡑⢌⠢⠁⢈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠟⡵⡱⢣⢣⢎⠄⡄⡀⣴⠋⣰⢏⠜⣷⠒⠝⢦⣌⠳⣵⡘⠂⠁⠉⠀⠀⡐⣌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠘⠰⢡⢣⠏⡎⡜⣰⡼⢡⢞⠁⠀⠀⠺⣏⠣⠀⠈⠳⣮⡙⠦⣷⣼⣤⣦⠽⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣰⢀⠁⠀⠈⠀⠀⢠⣟⣵⠟⣩⢖⡄⡠⠀⣐⢀⠀⣀⠒⠤⢉⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣮⣴⣠⣥⡷⠞⠉⣾⠅⢈⠔⣫⢞⠔⡱⣯⡢⣕⠢⣉⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠁⠀⠀⠰⡷⠂⠀⢊⠵⣫⢎⢜⣵⢝⠮⡑⠀⠀⠀⠠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⣿⡥⢀⠁⢊⠔⠁⣾⠿⣧⠁⠀⠀⠀⣈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⡟⠀⠀⠀⠀⠀⠘⢶⣥⣔⣡⡴⠞⠁⠀⠈⠛⠶⠾⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠒⠓⠶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠈⠑⠦⣀⡀⣸⡇⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⣰⠋⠀⠈⢻⡆⠀⠀⠀⠉⣿⠓⠒⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⡠⠃⢀⠔⠒⢦⡀⠀⣿⠀⠀⠀⠀⣷⠀⠀⣀⣤⣿⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠉⠀⠈⠁⠀⠀⢺⣴⠀⠘⡇⡠⢻⣿⡀⠀⠀⡟⠀⣼⠃⢀⣿⡆⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣦⣤⣴⡆⠀⠀⠀⠀⣸⣿⣒⡺⠋⠀⢸⣏⠙⠓⣾⠓⠋⢿⣧⠞⠙⣿⣄⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⣠⠔⠁⢉⣉⡀⠀⠀⠈⠻⠶⠞⠁⠀⠀⠀⠀⢀⠀⠈⠛⠷⠦⠤⠔⠁
⠀⠀⠀⢀⠴⠒⠛⠛⠛⠛⠛⠛⠓⢻⡟⠋⠉⠀⢠⡞⠉⠀⠙⣧⠀⠀⠀⠀⠀⠀⣀⠀⠀⢰⡏⢀⣤⡀
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⢠⡿⠀⠀⠀⢀⠏⠀⠀⠀⠀⣰⠏⠉⢹⠀⣾⡗⠁⣾⠁⠀⠀⠀⠀⣀⣀⡀
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠑⣾⡧⠤⠄⠒⣡⠀⠀⣾⠀⠀⣿⠀⠀⠀⢠⣿⣷⣞⣁⡀⠀⢀⠔⠋⠁⠈⢻⡆
⠀⠀⢿⣄⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣿⡇⠀⠀⢰⣿⠀⢰⣿⠀⢰⢿⣦⣤⠴⠃⠸⠇⠀⠈⢙⠿⣧⣤⣀⣀⣠⠞⠁
⠀⠀⠀⠙⠷⣶⣤⣤⡴⠖⠋⠀⠀⠀⠀⠀⠀⠸⣿⣄⠀⡸⣿⣤⠞⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⢠⠊⠉⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠔⠒⠈⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⣠⣤⣤⣈⠒⢄⠀⠀⠀⠀⠀⠀⠀⢀⠄⢊⣡⣤⣤⡀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⡄⠑⢄⠀⠀⠀⢀⠔⢁⣴⣿⣿⣿⣿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⢿⣿⣿⣿⠏⠐⠈⠉⠁⠀⠀⠉⠉⠐⠈⢿⣿⣿⣿⠇⡘⠀⠀⠀⠠⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠘⡫⠋⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣠⠌⠫⡛⢀⠃⠀⠀⡆⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠑⡋⣿⢿⡅⠄⠀⠀⠀⠀⡊⡿⣿⡍⡗⠁⠐⡆⠀⠀⡠⠃⢄⠆⠄⠄⡀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠈⠉⠈⢀⣤⣤⣤⣄⠈⠈⠉⠈⠀⠀⠀⢠⠀⡐⠀⠀⠀⠉⠘⠁⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⢁⣀⠈⠥⢰⠁⠙⢿⠟⠁⢳⠨⠍⢁⡈⠉⠀⠸⠠⠁⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⠈⢁⠠⠒⠉⠹⠀⢀⠜⢄⠀⡸⠉⠐⠢⡀⠉⢀⠃⠁⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡁⠀⠀⠀⠀⠙⠢⠤⠤⠚⠀⠀⠀⠀⠀⣰⡇⠘⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⡟⡷⣦⣄⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣿⣿⠃⠃⠀⠀⠀⠀⠀⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠉⠑⠻⠾⠭⠿⢿⣿⣿⠿⠿⠿⠟⠛⠉⠢⡀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠈⠑⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠃⠀⠀⠀⠀⠀⠈⠲⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⢄⠀⠀⠀⠀⠀⠈⢦⣀⣠⣮⣐⢄⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠔⠉⠉⠂⠀⠑⠄⣠⡀⣀⣠⣿⣿⣿⣿⣿⣷⡑⡀⠀⠀⠠⠂⠉⠑⢄
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠈⢂⠠⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢀⠊⠀⠀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠇
⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠰⠀⡠⠄⠂⠀⠐⠠⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠣⠀⠀⠀⠀⠠⡀⠠
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⣨⠻⣿⣿⣿⣿⣿⣿⣿⣿⢫⡁⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠑⢄⡂⣂⣂⡣⠗⠓⠱⢍⠻⠿⣿⡿⠿⢛⠵⠁⠒⠦⣆⡆⡄⣄⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⢿⠿⠿⢿⠿⣿⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⢄⡽⠧⡤⣤⣭⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⢉⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣼⣙⣛⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⣠⠼⢧⣄⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠸⣇⠀⠀⠘⠂⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠈⠙⠓⢦⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠹⣄⡀⣀⣸⠇⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠹⠏⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⡿⣿⣿⣿⡿⡿⢿
⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠶⠤⣄⡉⠛⠲⠶⠶⠒⠋⢁⣠⠴⠶⠒⠒⠒⠒⡶⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠶⣄⡀⠀⢳⠀⠀⡏⠀⢀⡤⠖⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠹⣦⡟⠀⠀⢷⣴⠋⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠒⠖⠚⠉⢧⠀⢀⡞⠉⠓⠲⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠔⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠐⠀⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠸⠂⠀⠀⠛⢀⡄⠖⠀⢀⣀⢀⣠⡀⠀⠀⠀⠀⠀⣨⠀⠗⣈⠀⠀⠠⡦⠀⠀⠀⠀⠘⠋⠀⠃
⠀⠀⢀⣄⠄⠖⠀⠀⠛⠀⠀⠀⠖⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⢈⠀⢀⠀⣀⠀⠀⠀⠀⠀⣴⠄⢈
⠁⠀⢀⣠⠀⠀⠀⠘⠓⢀⠰⠆⠀⢠⡄⠀⠀⠀⠛⠀⠀⠀⠀⢉⡀⠀⠀⠀⢠⠀⠹⠀⠀⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⢰⠄⠀⠸⠂⠀⠀⣀⠀⠀⣀⠀⠀⠀⠀⠀⠴⠗⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⡄⢀⡈⠉
⡄⠀⠀⠀⠀⠀⢀⡀⠀⠀⠁⣙⠀⠀⡛⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠃⠰
⠀⠀⣄⠄⠀⠀⠈⠁⠀⠀⠀⣛⢰⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠛⠀⠀⠀⠋
⠀⠀⠉⢀⠀⠠⡤⠀⠐⠇⠀⣛⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠿⠀⠀⡀⢀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠶
⠘⠃⣤⠘⠃⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⣀⠀⠈⠃⠀⠈⠁⠀⠤⠾⠄⠀⢴
⠘⠃⠀⠘⠃⠀⠀⠀⠀⠀⠀⠒⠶⠀⠀⡀⠀⠀⠀⠐⢀⣌⣿⣤⣤⠀⠀⠀⠀⡤⠀⠀⠖⠀⠀⣀⠘⠗⢀
⠐⠇⣠⠘⡁⠨⠃⠀⠀⠀⢠⡄⠰⠆⠘⠃⠀⠀⠀⠀⠸⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠀⠀⠛
⠀⠗⠈⠀⠛⠀⠀⠀⠀⢀⠈⣤⠀⠶⠀⠀⠶⠀⠀⠀⠀⠙⢿⣿⠁⢀⠀⠀⠀⠚⠂⠀⠙⠈⢀⡶⠀⠀⣦
⠀⠀⠀⠀⠁⠐⠗⠀⠀⠛⠀⢉⣄⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠁⡀⣀⡀⠐⠀⠈⠀⠀⠀⠀⠀⢀⣠⡀
⠀⠀⠀⣰⠆⠀⠖⠀⠶⠀⢀⠈⠁⢠⡄⠀⠰⠆⠀⠀⠀⠀⠀⠐⠁⠉⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠰⠆⠀⠀⠀⠶⠀⠀⠀⠘⡃⠀⠀⣄⠀⠀⠀⠀⠀⠀⢠⠄⠠⠰⠆⠀⠰⠆⠀⠀⠰⠂⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠠⠀⠀⠴⠀⠀⢛⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠠⠄⠶⠈⠀⠀⠐⠀⠘⡁⠀⠈⠀⠀⠀⠀⠴⠠⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⢢⡤⠀⠄
⠲⠀⠀⠀⠗⠀⠐⠀⠀⠛⠂⠈⠃⠀⠂⠛⠀⠀⠂⠀⠶⠀⠀⠀⠰⠆⠀⠀⠀⣤⡄⠀⠀⢀⣄⠀⠀⠸⠂
⠈⢃⠘⠃⠀⠀⣠⡀⠀⠀⠀⠂⠻⠀⢠⡄⠀⢀⣄⠀⠀⠀⣤⠀⠀⠀⠀⣤⠀⠀⠀⣠⡄⠈⠁⠀⠀⣀
⠀⠉⠀⠰⠆⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣤⠀⠉⠀⠀⠀⠀⠀⠁
⠐⠁⠀⡀⠀⣛⠀⠀⢀⠀⠐⠇⢀⠀⠘⠃⠀⠰⢒⡂⠀⠀⠀⠀⠛⠀⢠⡉⢀⠁⣁⡘⠃⡀⠀⠀⢰⡆
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⡿⠿⠿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠿⠋⣫⣏⠁⡠⢿⣽⡿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣾⣿⡿⣹⠁⣰⠩⣾⠥⠁⠓⠼⣿⣜⢎⢻⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⣿⣿⠝⣽⡿⠁⠉⠀⠀⠀⠀⠀⠈⢻⣾⣿⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⢳⣿⢍⠠⣖⠀⠀⠀⠀⢐⡆⢋⠙⣟⣤⠌⠻⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣵⣿⡷⡏⡇⠈⠀⠈⠀⠀⠀⠀⠉⠀⠈⢀⠘⣷⡆⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡀⠀⣾⣿⣿⣿⡇⢳⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡖⢨⣷⡶⠂⠀⢸⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢰⣿⣿⣿⣿⠁⠊⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠎⠘⠱⡼⡛⠁⠀⠀⠈⠀⠉⠉⠓
⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠙⠃⠰⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣷⣾⠁
⠀⠴⠁⠈⠄⠀⠀⠀⠀⠀⠀⢠⣴⣾⣿⣷⡆⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣨⣷⣶⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀⠈⠛⢛⢿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠏⠀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠈⠂⣠⣾⣿⣿⣿⣿⣶⣿⣿⡿⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣻⣢⣾⣿⣿⣿⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⡇⢏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀
⣷⡀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶
⣿⣿⣶⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠀⢀⠄⠀⠀⠀⡠⠚⣝⠒⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣜⡾⢀⠀⠈⠙⠒⠦⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢚⣸⣗⠁⠀⠀⠄⠀⠀⠀⠀⠈⠙⠢⡀
⠀⠀⠀⠀⠀⠀⡠⢒⠉⠀⡌⢱⣤⡀⢀⣀⣀⣠⣤⣤⣀⣀⣀⡀⠀⠀⢞⢀⣧⢥⣾⢐⣠⣖⣓⠓⠓⠒⠂⢳⡀⠁⠘⣆
⠀⠀⠀⠀⡠⠊⠄⠀⠀⠀⢄⠀⡳⣞⠙⠉⠀⠀⠀⠀⠀⠈⠉⠙⠛⠲⠷⣬⣕⣺⣇⢔⣸⣶⡴⡇⠀⠀⠀⢸⡂⠄⠠⠈⣦
⠀⠀⠀⡜⠀⠁⠀⠀⠀⢔⠔⡀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠁⠁⡧⠀⠀⠀⠘⡕⠀⠴⠀⠠⠑⢤
⠀⠀⡘⠰⠀⠈⢀⠀⢃⠕⣪⣞⠄⠈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⣿⠀⠀⠀⠀⠘⣔⡡⡊⢀⠀⢀⠙⢦⡀
⠀⢀⠁⠀⠀⠄⠈⡢⠕⡨⢐⢕⠀⠂⢨⣇⣖⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢹⢛⢛⣝⣟⣙⠁⠮⠴⠥⠥⣅⠈⣸⡗⠢
⠀⢠⡈⠀⠀⢄⠩⢒⠡⠢⡁⢕⢄⡰⣌⢽⡈⠿⣏⣉⣏⠒⠤⢤⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⡾⢡⢤⣐⣰⡰⡴⠜⡲⠖⠒⠶⠏⠁⠀⠒⠚⡀⠀⠠⠤⠄
⠀⡆⠀⠠⢀⠅⢊⡵⠥⡒⣁⡉⠍⠍⢍⢀⢧⠀⣿⡒⢒⣶⠒⡒⠶⠭⠷⠒⢒⣶⠶⣶⠶⠍⡭⠉⠉⣉⣉⣁⣀
⠘⠀⠀⢀⠔⣊⣤⠶⣛⠓⠉⠭⠙⠋⠉⠙⠉⢈⠁⠀⠘⠢⠖⠚⠂⠘⠛⠀⠀⠈⠒⠛⠀⠀⢴⣒⠫⣉⠈⠁
⣀⠀⠐⢡⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠰⣁⣉⣆⠀⠀⠀⠒⠂⠀⠀⠀⠀⠈⠙⠁⣄⢤⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⣿⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠟⠛⠉⠉⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⠈⠉⠛⠻⢿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠿⠋⠁⢀⣠⣤⣶⣾⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣷⣶⣤⣄⡀⠈⠙⠿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⠁⢀⣤⣾⣿⣿⠿⠛⠋⠉⢹⣿⣿⠀⠀⣿⣿⡏⠉⠙⠛⠿⣿⣿⣷⣤⡀⠈⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡏⠀⣠⣴⣿⣿⠟⠉⠀⣀⣤⡄⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢰⣤⣀⠀⠉⠻⣿⣿⣦⣀⠀⢙⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⣿⣷⣿⡿⠋⠀⢀⣴⣾⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⣷⣦⡀⠀⠙⢿⣿⣶⣿⡿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⣠⣿⣿⠏⠀⢀⣾⣿⡿⠋⣻⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠙⢿⣿⣷⡀⠀⠻⣿⣿⣄⠀⠹⣿⣿⡄
⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏⠀⣰⣿⣿⠃⠀⠀⠉⠉⠉⠀⠀⢿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⡗⠀⠀⠉⠉⠉⠀⠀⠘⣿⣿⣆⠀⠹⣿⣷⡀
⠀⠀⠀⠀⠀⠀⣼⣿⡟⠀⢠⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⡄⠀⢻⣿⣧
⠀⠀⠀⠀⠀⢠⣿⣿⠃⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠁⠀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⠀⠀⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⢸⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⠀⠀⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣿⡀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠘⣿⣿⣿⢿⣿⣿⠿⠿⣿⣿⡿⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠿⣿⣿⠿⣿⣿⠿⡿⣿⣿⣿⠃⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣆⠀⠹⣿⣷⡄⠀⠀⣀⣀⣀⠀⠀⣾⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⠀⠀⣀⣀⣀⠀⠀⢠⣾⣿⠏⠀⣰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠙⣿⣿⣆⠀⠙⢿⣿⣷⣄⣻⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣟⣠⣾⣿⡿⠋⠀⣰⣿⣿⠋⠀⣰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣶⣾⣿⣿⣿⣷⣄⠀⠙⠻⣿⣿⣿⡇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⢸⣿⣿⣿⠟⠋⠀⣠⣾⣿⣿⣿⣷⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣏⠀⠙⢿⣿⣷⣦⣀⠀⠉⠛⠇⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠸⠛⠉⠀⣀⣴⣾⣿⡿⠋⠀⣹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠈⠻⢿⣿⣿⣶⣤⣀⡀⢸⣿⣿⠀⠀⣿⣿⡇⢀⣀⣤⣶⣾⣿⡿⠟⠁⢀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣦⣄⡀⠈⠙⠻⠿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⠿⠟⠋⠁⢀⣠⣴⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣶⣦⣄⣀⡀⠀⠉⠉⠉⠀⠀⠉⠉⠉⠀⢀⣀⣠⣴⣶⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⠿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠛⠛⠉⠉⠉⠁


⣶⣶⣶⣶⣶⡷⢾⣾⣿⡿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢾⡷⣶⠶⣶⣶⣾⡶⠶⠶⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⢟⣻⣙⣯⣍⠛⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣟⠋⠃⠀⠘⠛⣛⠛⠋⠀⠀⠀⣠⣄⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⣿⣷⠄⣀⣴⣾⠿⠿⠿⠿⠿⣷⣦⣈⠻⣿⣆⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣯⣿⣿⢁⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣊⠁⠀⠈⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠖⠚⣿⣿⣿⣻⣿⢯⡍⢉⣉⡉⠛⠒⠶⢤⣤⣀⠀⠀⠀⠀⠀⠀⢻⣿⡦⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡄⠀⠀⣿⣿⣷⣺⣇⣻⡆⣏⠀⣹⢰⠟⠳⡄⣠⣌⡉⠓⠶⢤⣀⡀⠀⣿⣯⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣸⡿⠛⣿⣿⣷⣤⡤⣌⣉⣁⡘⠦⠾⢡⡏⣀⡗⠀⢠⠆⠈⠙⠳⢿⣟⣺⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣤⣾⣋⡀⠙⣿⠁⢠⣿⣟⡺⠿⢠⣿⡏⠉⠉⠛⠓⠦⣌⣉⡀⠠⢯⡀⢠⡟⢀⡆⢈⡙⠧⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣿⡴⠋⠀⢸⣿⡒⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⢤⣘⠲⠟⣴⠋⠙⠆⣈⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠋⠀⠀⠀⣾⣿⢭⣀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣶⣌⡓⠃⡾⣷⠒⠂⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣹⣿⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠧⣿⡟⢳⣤⣙⠀⠀⣰⠏⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣷⢾⣿⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠋⠁⠀⢠⣿⠃⠙⢳⡞⠁⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠁⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣟⢣⣅⠀⣠⣿⠃⠀⢀⣾⣤⣤⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣤⣤⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢟⣲⣿⠏⣴⣿⠁⠀⠀⢸⡟⣫⡃⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠁⠀⠉⠀⠀⠀⠀⠀⢀⣴⣿⡿⠍⠈⠻⢃⣾⡿⠁⠀⠀⠀⠀⣿⠋⠉⠛⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣯⡀⠀⠰⣿⣿⠄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠒⠒⣄⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⢰⡏⠀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⣿⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⢠⡾⠁⠀⡕⠀⠀⠀⠀⠀⠈⠉⠉⡀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⠘⠀⠀⣰⡟⠀⠀⠉⠀⠀⠀⠀⢀⡤⡄⢀⡾⢡⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀⣀⠄⢪⣿⠁⠀⢀⣾⢢⡾⢹⣠⣾⠁⠗⠸⠇⢸⡇⠀⠀⢀⡴⡄⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡇⠀⠠⢺⡟⢸⣇⠀⠀⠓⠊⡠⠄⠒⠸⣇⠢⢠⡟⡀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠖⣁⣿⠓⡄⢀⣴⢫⠂⡞⠀⠀⠀⠀⠈⠁⣾⡇⠁⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⣹⠏⠈⠁⣼⢃⠋⠀⠳⠞⠁⢠⠤⠄⢸⡿⣹⠇⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣰⠏⠀⠀⢸⡿⠀⢠⡆⣰⡆⢠⡟⠀⢠⣿⢻⡁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣠⣿⠁⠠⣿⠑⣿⡀⢻⡃⠔⠘⠃⠈⣧⡀⠐⠀⠁⠉⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⡠⠊⢸⣏⠔⠀⠀⠀⠉⠀⠀⠀⢀⠠⠐⠀⠀⠙⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠐⠊⠀⠀⢈⠅⠀⠀⠀⠀⠀⠠⠤⠆⢁⡴⠎⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⡀⠀⠀⠐⠀⠁⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠶⢤⣀⠀⠀⠀⠀⠀⢀⣠⠴⠒⣶⠀⠀⠀⠀⠀⠤⠤⠄⢀⡀⠀⠀⠀⣠⢤⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠈⠙⠓⠒⠒⠒⠋⠀⠀⣠⠏⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠉⠒⠤⣰⠀⠈⢢
⠀⠀⠀⣀⣤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠈⠰⡌⠲⣄⠈⢆⠀⠀⠀⠚⠿⠢⡀
⠀⠖⣹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠊⡉⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⠈⢆⠀⠀⠀⣢⡀⠀⢱
⢰⣄⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠀⣀⣀⣠⣤⢤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠳⠤⠔⣁⣤⣀⡎
⠘⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠇⠀⠀⠀⠄⠀⠀⠀⡦⣸⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠙⠟⡁⠈⠙⡆
⢈⠀⢸⣿⣆⠀⠀⠀⠀⠀⠀⠰⠀⠀⠸⡀⡀⠀⠀⠀⠀⠀⠀⢀⡈⠀⠀⠀⠀⠀⠠⢖⡀⠀⠀⠀⠀⠀⠀⠀⠐⢡⠀⠠⠀⠀⠀⠀⣠⠔⠁
⢸⠀⢸⣿⣿⣷⣤⣤⡇⡀⠀⠀⠄⠀⡀⠁⠁⠀⠀⠀⠀⠀⠀⠀⢣⡔⣈⠔⠛⢀⡘⢀⣧⣄⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠘⢁⠒⠲⡶⣦⣤⣤⡀⡀
⠀⠓⢾⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠸⣿⣆⠰⠀⠀⠀⠀⠀⠀⠀⢸⡌⢹⠛⠉⠉⣖⠀⢁⠩⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠰⠷⡄⠘⠀⠀⠀⢰⣿⣿⡁⠻
⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⠀⠈⠁⠀⢡⠐⠀⠀⠤⠄⢀⣈⡓⢏⣠⣀⣀⣀⠠⢬⣀⣀⡀⢀⣀⠤⠂⠹⣄⠀⣀⠈⢀⠈⠀⠀⠀⠀⠀⠠⣿⣿⡇⠐
⠀⠄⠀⣿⣿⣿⣿⡿⣷⠀⠀⣴⡄⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠉⠉⠑⢀⠖⠒⠀⠀⠐⡰⠖⠉⠁⠀⠠⠂⠀⢹⠇⢠⠃⠀⠀⠀⠀⠀⠜⣸⣿⣿⠓⢹
⠀⠗⢠⢻⣿⣿⣿⣼⡏⣿⠀⠘⠿⠀⠀⠀⠀⠀⠀⢢⡀⠀⠠⠄⢀⡀⠔⠊⠀⠀⠀⡠⠔⠈⠀⠀⠀⢀⠔⠁⠀⠀⠸⣀⡯⡀⠀⠀⡀⢔⣮⣾⣿⣿⠇⠍⢘
⠀⠀⠀⢸⣿⣿⣯⣿⠃⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⢀⡒⠛⠁⠀⠀⠤⠴⢖⡉⣀⡤⢀⢀⣠⠶⠉⠀⠀⠀⠀⠀⠈⣿⣾⣿⣶⣿⣿⣿⣿⣿⡟⢸⢀⠛
⠀⠀⠀⠸⣿⣿⡿⢻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣟⣻⡿⠿⠿⠿⠿⣟⣛⡏⣽⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⡆⠠⠇
⠀⠘⢀⣸⣿⣿⡇⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⣿⣽⣿⣶⣮⢿⣿⣥⣿⣿⣿⣿⣾⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⣿⣿⣿⣿⣿⣿⡟⠨⢤⢸
⠀⠀⠀⣻⣿⣿⠀⢘⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾⣿⡻⠿⠛⣥⣿⣿⢷⡉⠛⢻⣿⠇⠀⠀⠀⠀⠀⠀⢰⢂⢻⣿⣿⣿⣿⣿⣿⡇⠗⠀⠂
⠀⠀⡀⢸⣿⣇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣀⡹⣿⣶⣎⡴⠿⢸⡧⣭⣽⢿⣳⡆⠀⠀⠀⠀⠀⠀⠈⠚⡸⣿⣿⣿⣿⣿⣿⢏⠠⠀⠇
⠀⠀⡁⠂⠙⠋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠤⠤⢭⠿⠿⠶⠾⠾⠧⠿⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⡄⠳⣹⣿⣿⣿⣿⢋⠂⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢻⣿⣿⣿⡿⢥⠈⠀⡈
⠀⠀⠀⠣⡈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡌⠋⠻⣿⡿⠁⠠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡐⠀⠀⠀⠀⢦⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠉⠲⠀⠈⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⡿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠋⣥⡄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠉⠁⠀⠀⠶⠶⠀⠀⠉⠙⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁⠀⠉⢁⣈⠙⠻⢿⣿⣿⣿
⣿⣿⣿⣿⠏⢠⡄⠀⠀⠀⠀⠀⠠⠶⠌⠻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣀⠀⠀⡀⠀⠀⣴⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢴⡆⠀⠀⠀⠉⣤⡄⠀⠉⣿⣿
⣿⣿⣿⠋⠀⠈⠁⠀⣤⡀⠀⠀⢀⣀⡄⣼⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⠿⠀⠘⠏⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠴⡦⠀⠀⠀⣤⡄⠀⡀⠰⣾⣿⣿
⣿⣿⣿⣌⠿⠀⠀⠀⠙⠁⠠⠖⢉⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⣿⣿⣿⣿⣿⣿⣿⣿⠁⠸⠷⠀⠀⠀⣶⡄⠀⠨⣇⣼⣿⣿⣿
⠿⠿⠿⡿⣷⡀⠸⡆⠀⣠⠤⢖⡩⡾⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢌⣋⠀⠀⠀⠀⠰⠗⠀⠀⣁⣀⠿⠿⠿⠿⠿⠿⠿⠿⠿⢶⣄⡀⠀⠾⠆⠀⠴⢀⣽⣿⠿⠿⠿
⠀⠀⠀⠀⠙⠿⣶⣄⣠⣴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⡶⢦⣄⣤⣴⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠺⠷⣤⣀⣤⡾⠷⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣰⣿⡄⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣯⡟⣉⣉⢰⠄⠀⠸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢹⠋⣠⡾⢿⠛⢻⣤⠀⡀⠉⠉⢻⣿⣿⡟⣽⣿⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣤⣿⠗⣿⣻⣇⣹⠫⠀⢺⣶⣾⣿⣿⣼⣿⡇⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⣷⣄⣾⠟⣣⡘⠷⣾⣥⡼⢃⣼⣴⣦⣅⡤⡝⢿⣥⣶⣶⣿⡿⣳⣿⣿⡄
⠀⠀⠀⠀⠀⢀⣄⠀⣿⣿⣿⣿⡴⢂⣽⣿⡯⣥⠏⡝⠏⣿⣿⣿⣿⣷⢳⣾⣿⣛⣫⣧⣿⣿⡿⣽⣷
⠀⠀⠀⠀⠀⢘⣿⣿⣦⣿⣿⣸⣾⡿⠛⠉⢻⣿⡿⣿⣷⣛⣿⣿⣿⣥⢾⣿⢟⣿⣿⣿⢿⣿⣾⣿⣿
⠀⠀⠀⢀⣾⣋⣭⣯⣼⢛⣿⣿⣿⣶⣶⣾⣿⣾⣿⣿⢋⣿⡟⢱⣿⣇⢹⣿⣿⣿⡿⠿⣿⡞⣿⡿⣿
⠀⠀⠀⠘⠉⠉⣿⢏⣽⣿⠟⠛⢉⣽⣿⣿⣿⣿⣿⣿⡘⢿⣧⠸⣿⣿⠀⣿⡿⣿⣾⣿⢸⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠈⣿⡿⠁⠀⢀⣾⣟⡿⠛⢻⡿⢮⣻⣇⢺⣿⣇⢸⣿⣄⢻⣿⣮⣭⣵⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⣸⣿⠁⣵⣿⣾⡌⢻⣷⢻⣦⣽⣿⣦⣽⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡷⣶⣿⣭⡟⠿⢿⠿⠓⣟⣿⣜⣻⣿⣿⣿⣿⣿⡛⠉⠁⢹⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⢿⣿⣶⣫⣶⣿⣿⣿⣿⣿⣿⣿⣟⠛⠛⠛⠀⠀⢸⣿⣿⣻⣟⣿⣯⣷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠉⠉⠁⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⢛⣽⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⣀⣀⣀⡠⠄⠀⠀⠀⠀⠀⠀⠤⠶⠄⠀⠀⡠⢊⣵⣿⣿⠷⣷⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣼⣱⣶⣯⡉⠳⢦⣀⣠⣤⠴⠶⠖⠚⠛⠓⠻⠷⠿⣿⣿⣋⢸⢸⡎⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣿⢙⠹⣿⣿⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⡇⠀⣠⠄⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣦⡰⠂⠘⠛⠓⠶⣍⡻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠦⢤⡀⠀⠀⠀⠈⢷⡶⡟⠛⠲⣤⡈⠻⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⢀⡴⠞⠛⠓⢦⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠉⠃⠀⠀⠀⠀⣷⢰⠀⠀⢀⢹⡆⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣸⠇⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⣠⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡷⠷⢸⣶⡇⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⢸⡿⡷⠿⠆⢺⠇⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠐⠈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⣾⠀⠀⠀⠀⢸⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠈⠓⣶⠆⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣴⠟⣹⣿⡄⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢮⣛⢶⣬⣑⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣋⣥⣪⢕⣿⠇⠀⠀⠀⠀⢰⡇⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⣼⢫⡑⠮⣷⣾⣿⣿⣿⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⢀⣾⠁⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠓⠦⣭⣻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⢿⡿⢴⣶⣿⡟⠛⢫⡍⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⣿⡷⣻⣿⢧⡀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⠖⠀⠀⠀⠀⠀⠀⠀⢸⡀⠰⢛⠿⣿⣻⣿⠿⠁⣩⠀⠀⣠⠊⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠁⠀⠿⠃⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⡇⠀⠀⠀⣄⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⡀⠀⢠⡏⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⣾⣷⣀⠀⠀⠀⠀⠀⠐⠏⢀⡾⠁⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢋⢿⡆⠀⠀⠀⠀⠀⠈⢿⡄⣼⠟⠉⠉⠉⠉⠀⡀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠈⢿⣏⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠹⣦⣰⠀⠀⡀⠐⣀⣼⠻⣆⡀⠀⠀⣒⣢⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠉⠀⠀⠈⠉⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣶⣶⣶⣦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⡿⠟⢯⣽⣭⣷⡌⠉⠙⠛⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣯⠁⠉⠀⢈⣉⣭⣍⣁⣀⠀⠀⣶⣦⡙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣻⣿⡿⢃⣤⣾⠿⠛⠋⠙⠛⠛⠿⢶⣤⡙⣿⠆⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣾⠿⠿⣾⣿⣥⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⠀⠀⠸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠋⠉⢹⣿⣶⣻⡿⢝⡗⣠⠶⢦⡀⣉⣙⠛⠲⠦⢄⣀⠀⠀⠀⠀⢻⣿⣆⣠⡀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣤⡀⣸⣿⣿⣿⣿⠞⠃⠻⣤⠼⢹⡃⢨⡇⡾⠲⡄⠈⠙⠓⠦⣤⣸⣿⣖⣿⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠻⣷⡿⠁⢹⣿⣛⣿⡟⣳⡶⠶⢦⣭⣉⡘⠧⠴⠃⢀⡾⠀⢀⡄⠉⠛⠶⣿⠇⢸⣿
⠀⠀⠀⠀⠀⠀⠐⠛⠳⢦⡄⣨⠏⠀⣿⣟⠊⠙⠁⣿⡇⠀⠀⠀⠈⠉⠓⠶⠦⣬⣓⠆⢾⣁⡾⣡⠶⣌⠙⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠁⠀⢠⣿⡷⠄⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣍⡀⢧⣤⢈⣴⣂⡌⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠸⣿⠯⣷⡆⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣻⣷⣤⣘⠹⡇⠀⠀⣸⠷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣙⣿⡇⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡳⠉⠛⢠⣿⠛⢦⣀⡞⠁⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡗⠉⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⣓⠀⠀⣠⣿⠃⠀⢠⡏⢀⣀⣠⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⣀⣀⣀⣸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣠⣿⠿⣵⣿⠇⣴⡿⠃⠀⠀⣿⢿⠻⢹⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⣠⣼⣿⡛⠹⣿⢋⣼⡿⠃⠀⠀⠀⢻⡾⠛⠲⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣓⠀⠀⢴⣿⣏⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣆⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠟

⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣷⡄
⠀⠀⠀⠀⠀⠀⠀⢀⣂⠀⠀⣤⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⢬⣀⣿⣿⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠈⠤⣄⣠⣤⣾⣿⡿⠃⠀⠀⢷⣌⡏⢶⠆⡄⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠘⠀⠀⠀⡄⡀⠆⣜⠐⢰⢠⠆⢀⢂
⠀⠀⠀⠀⠀⠀⠂⡖⠂⠀⠀⠀⢀⠄⠀⠀⠀⡀⢀⠂⠀⢰⠀⠇⠀⢁⠤⣄⡀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⢸⠁⠰⠲⠀⡄⡜⢠⡠⠠⠇⠈⠙⠀⠀⠀⠀⢀⠔⢡⡀⠀⠈⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⣠⡿⠀⠀⠀⠀⠀⠀⠈⠑⠤⡀
⠀⠀⠀⠀⣀⡀⠠⠄⠐⠂⠀⠀⠀⣧⡀⠀⠀⠀⠠⠤⣀⣀⣀⠾⢻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠘⢿⣶⣄⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⣠⣠⣤⡀⣠⣠⣶⣷⣿⠖
⠀⠀⠀⠀⠀⠈⠻⢿⣷⡴⣶⡄⢠⣇⣠⣤⣶⠲⠤⠤⢤⠤⠤⠒⠿⠷⢿⠿⠿⠟⠛⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣾⠿⢋⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠢⠄⡀⣄⠍⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠒⠉⣂⡬⣁⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⢀⡴⢊⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⠀⠀⡴⡫⠊⢁⠄⠠⢄⠀⠀⢀⠔⠉⠲⠀⠀⠀⠀⠀⠀⣎⡉⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢁⡜⡁⢠⡞⡔⠀⢀⡤⠀⠀⠀⢣⠀⡎⠀⠀⠐⠥⢀⠀⠀⠀⠀⠰⠀⢭⠤⠳
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢀⡾⡸⢠⡓⡜⠀⠐⠀⠀⠈⠢⣀⠀⣣⠅⢠⠎⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠈
⠀⠀⠀⠀⠀⠀⠀⡐⠀⡼⠳⠁⡞⠜⠀⠀⠐⢄⡀⠀⡀⠘⢣⣽⣷⢃⣀⣢⠔⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢠⠁⢰⡷⠃⢰⢍⠂⠀⠀⠀⠀⠀⡉⢭⠓⢿⢿⡿⢟⡉⡉⠉⠑⢆⠀⠀⠀⣘⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠄⠀⣼⣠⠀⢸⠪⠀⠀⠀⠀⢠⠊⠀⠈⠶⠉⢸⡗⠆⠙⠑⠀⢀⡼⠀⠀⣰⠃⢠⠆⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠆⠀⢯⣼⠀⢰⢽⠀⠀⠀⠀⠈⠢⠤⠄⠀⠀⡘⢃⠀⠀⠀⠀⡅⠀⠀⣰⠓⢀⠿⡔⠀⡜⠁
⠀⠀⠀⠀⠀⠰⡆⠀⢸⠙⡆⠈⣟⡆⠀⠀⠀⠀⠀⠀⠘⠂⠒⠁⠈⢳⣤⠔⠊⠀⢀⡜⠃⣠⠟⡩⠀⣼⡙
⠀⠀⠀⠀⠀⠀⣧⠀⠘⣷⠚⡄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡑⢄⡠⠋⢀⣴⢋⠟⢀⡜⡬⠁
⠀⠀⠀⠀⠀⠀⢈⡀⠀⣈⠀⠈⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠊⠀⣴⣿⡷⠃⢠⣾⡼⠁
⠀⠀⠀⠀⠀⠀⠈⠗⠾⡀⠐⢤⡀⠀⠀⠀⠈⠲⠄⡀⠀⣀⣀⡠⠔⠀⢁⣠⣶⠚⡲⠋⢀⡴⣿⣾⣷⡶⠶⣲
⠀⠀⠀⠀⠀⠀⠀⠈⢆⠈⠢⡀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⢀⣠⢤⠾⠛⣉⠕⠋⠀⡠⡛⡥⠋⠉⠉⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠈⠒⢄⡈⢙⠲⠦⣀⡀⠀⠀⠀⠀⠀⠓⠉⠀⣀⠰⢫⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣠⠈⠑⠢⠤⢀⣨⣇⡄⠀⢀⣀⡤⠔⣋⣥⣖⠉⠀⡀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠈⠓⠮⢦⣠⣠⡻⠲⠗⣚⣙⡡⠜⠚⠁⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠐⠈⠀⠂⠡⠀⠨⠐⡠⢀⠄⡉⢍⠡⡀⠄⡀⢔⠠⣐⠄⡂⠄⡂⠅⡂⢅⡪⠑⠌⠂⠅⠪⠂⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠉⠉⠙⠛⠏⠋⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⢠⡆⠀⣦⠰⣦⠀⢧⠈⢠⡀⠀⠀⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢈⢠⣾⣇⣸⣿⣆⣿⣷⣾⣦⣼⠉⢦⠀⣀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢚⠀⢸⡾⢋⣽⡟⠛⠋⠛⣩⣮⡝⡟⠀⠘⣾⡟⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⣇⠘⢿⡗⡠⠀⠀⠹⠿⠃⡇⠀⢰⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⡀⡀⠄⠀⠀⠀⠀⠀⣰⣥⠀⢸⣿⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣿⣿⣿⣇⠀⣻⡶⠛⠀⠈⣁⣼⡀⢺⡟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⣿⣿⢻⣦⣴⣾⣿⣿⡛⢧⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣾⣟⠲⣾⣿⣿⣿⠿⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⣼⣻⢰⡿⣿⠋⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣯⣿⣿⢿⣼⣿⣧⣽⣦⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⢸⣿⣿⣿⣤⡯⢿⣿⣿⣷⢃
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣀⣿⣿⡿⠋⣿⠏⠂⠻⣿⣿⣮⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⢀⣿⣿⠀⠀⣨⣄⠀⠀⣿⣧⠀⠀⠀⠀⠀⡆
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣱⣶⣿⣿⠀⠀⠀⠀⠠⠇⠀⢀⣤⣤⣤⡀
⠀⢰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⣿⣿⣿⣿⣿⣿⣿⡟⣻⣿⣄⡤⢤⠴⠁⠀⣤⣿⣿⣏⣭⣵⠄
⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡃⢻⣿⣿⣿⣯⠀⠀⠀⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⡏⣿⣿⣿⣿⣿⣿⠧⢞⣿⣿⣿⣿⣆⠀⠀⠈⣿⣿⣿⣿⡿
⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⡇⣼⣿⣿⣿⣿⠁⠀⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢀⡧⠀⠀⢀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠻⠛⠁⠀⠀⡠⠐⠊⠉⠙⠳⢦⡀⠀⠀⠀⠀⣸⣏⣚⢹⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢐⡄⠀⢹⡄⠀⠀⠛⠛⣮⢩⣬⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⡋⠸⠇⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢂⠴⠀⠄⠁⠀⠀⡷⠀⠀⠀⠀⢹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠘⣃⠘⠆⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⡃⢄⡀⢠⠃⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦⢛⣐⠓⠉
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠁⠀⠔⠁⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒
⠀⣠⠔⣊⡍⠋⠢⣄⣀⠤⠤⠴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣴⠄⠀⠀⠀⠀⠀⠀⠀⠂
⠀⡇⣉⠤⢒⣧⢄⠛⡉⠀⠠⠆⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡄⠀⠈⠛⠛
⠀⢡⡄⣀⠱⢆⡘⣄⡅⣠⠤⠀⣖⠋⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⢼⠧⠒⠱⡔⠱⠄⠑⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠶⢶⣶⣆⣻⠏⣀⣤⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠿⣟⠛⠋
⠀⠀⢀⠠⠤⠄⠀⠀⠀⣠⠖⠲⠄⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣿⠋⠀⠿⣷⣦⡀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠄
⠀⠀⡇⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠀⠀⠀⠀⠉⠙⠂⠀⠀⡖⠉⠰⢿⣿⡀
⡐⠞⠁⠁⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡐⢄⠂⡔⡆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠴⠋⠻⣦⡀
⠘⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⢌⣠⣱⠲⢮⠲⠵⠊⠁⠁⠀⠀⠀⠀⠀⠀⠀⢙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢈⠻⣄
⠀⣤⠲⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠲⠁⠀⠑⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠸⣷⠀⠀⠀⣴⣦⣿⡶⠀⠀⠀⠀⢀⢹
⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⢻⡆⠀⠀⢸⣿⡿⣿⠃⠀⠀⠀⢈⢘
⠀⠀⢰⠋⠀⠀⢀⡄⢰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠤⠻⣄⠀⠀⠈⠁⠃⠀⠀⠀⠀⠈⢸
⠀⠀⠈⠓⠀⢐⡫⠀⠌⠓⠄⠀⠂⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⠰⠙⠳⣤⣄⣀⡀⠀⠀⠀⠛⡠⠃
⠀⠀⠀⠀⠰⣿⡁⠀⠀⠀⠀⠀⠀⠀⠄⠆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⠯⠀⠀⠀⠀⠘⠀⠠⠄⠀⠈⠃⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠒⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠋⠉⠉⠉⠙⠻⣷⣄⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢠⡾⠛⣷⠀⠁⠀⠀⢠⣿⠇⠀⠀⣀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⡀⠀⠀⠀⣾⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⣿⠃⠀⡟⠀⣀⠀⢀⣾⣿⣴⣾⠿⠟⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⢠⣿⠁⢀⣠⡀⠀⠀⣿⣇⣼⠁⠀⠀⢀⣼⣿⠟⠉⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠤⠀⠀⠀⠀⢀⣿⡏⠀⣠⡿⠃⢠⡟⢁⡟⠀⠀⢻⣿⡁⠀⢀⡴⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⣼⣿⣥⡾⠛⠁⢀⣿⣡⠞⠁⠀⢠⡾⣿⣧⣴⠏⠀⣿⣿⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢠⣿⡟⠙⢳⣄⣀⣼⡿⠁⠀⢀⣴⠋⢀⣿⣿⠁⠀⢸⣿⡟⠀⠰⠂⠀⠀⢰⡆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠙⠿⣷⣦⣤⠆⣸⣿⠇⠀⠀⣿⠁⢸⣧⣀⣴⠟⠁⣠⡿⢻⣿⠀⠀⣼⣿⡇⠀⠀⠀⠀⢠⡿⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢠⣿⠀⠀⠙⢋⣁⠀⣴⠏⠀⣼⣿⠀⠀⢿⣿⡇⠀⠀⠀⣴⡿⠁⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⣿⣿⠀⠀⣼⠇⠀⠀⢀⣾⠏⣼⠃⠀⢀⣿⡏⠀⠀⠘⣿⣧⣤⣴⣾⠟⠀⠀⠉⠴⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣦⣄⠀⢻⣿⣀⣼⠟⠀⡤⣄⡼⠁⢰⡏⠀⢀⣾⠟⠀⠀⡀⠀⠈⠙⢛⣿⠃⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠃⠀⠛⠛⠉⠀⣼⣳⢯⣧⠀⠈⠷⠶⠟⠉⠀⠀⠀⠁⠀⠀⠀⣼⡏⠀⢀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠗⠀⠀⠀⢰⡏⠀⠛⠁⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⡀⢰⣿⠁⣠⠏⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⡄⠀⠀⢀⣿⠇⠀⠀⠀⢿⠃⠀⠀⠀⠉⠀⠀⠀⢠⣾⠟⠙⢃⣿⡏⣰⢃⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠟⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⣿⠀⢠⣿⠋⠀⠀⣹⡛⢳⡻⠟⠁⠀⢰⡆⠀⠀⢠⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⠀⠾⠇⠀⢀⡾⠀⢠⣿⡇⢀⣾⡏⠀⣠⣾⣻⣻⡛⠁⠀⠀⠀⠀⠀⠀⣰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⣾⠇⢠⢿⣿⢀⣾⣿⠁⣴⠏⢸⣿⡟⢿⣦⣀⠀⠀⢀⣠⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⣰⣿⣠⡟⢸⣿⡾⠁⠻⠟⠁⠀⢸⣿⠁⠀⠙⠻⠿⣿⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⢀⣤⣶⠿⠛⣿⠟⠀⠈⠛⠁⠀⠐⠟⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣠⣾⠿⠋⠁⠀⠀⠀⢀⠀⠀⠀⠀⠸⠂⠀⠈⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠋⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠄⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⠠⠤⠄⠤⠤⣀⡁⠂⠀⠀⠀⠀⠀⢰⡈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣶⡿⠛⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⡇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠹⠿⣾⣿⡻⠿⠁⠈⣻⠇⠀⠀⠀⠀⢠⡇⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠶⠖⠋⠁⠀⠀⠀⠀⠀⣼⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⡴⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⡿⠟⣡⠌⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠈⠙⠛⠿⢶⣶⣶⣶⣿⡿⠿⢛⣩⠴⣛⡅⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡝⡓⣒⣖⢖⡶⣠⢆⡴⡲⣖⢚⢛⣻⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣾⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣈⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣁⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠧⠿⠿⠿⠷⠿⠛⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣋⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⠶⠲⠀⠀⠀⠀⠀⠩⠃⠀⠀⠐⠐⠀⠀⠓⠀⠀⠟
⠀⠀⠀⠀⠀⢀⡀⠀⠀⣨⢿⣦⢀⣤⠀⡄⠀⠀⢄⠠⠀⠐⣶⣦⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⣤⣀⡀
⠀⠀⠀⠀⠀⠘⠿⣡⢨⣿⡄⢹⡄⢿⢇⣲⣧⡀⠀⡆⡀⠀⢠⡤⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣲⠆⠀⢐⡏
⠀⠀⠀⠀⠀⡀⢀⣍⢰⣿⣷⠈⡇⣴⣿⠗⢿⣵⠀⡷⠀⡄⣼⣦⣿⣄⠀⢠⣭⡀⢠⡀⢀⢀⠑⠀⣦⣤⠀⡀
⠀⠀⠒⢺⡎⠕⠻⣿⢪⣿⢿⡄⣡⣿⣿⣀⣀⠙⢰⢠⠀⠉⣿⣼⣿⡇⠀⠀⢟⠗⠈⠃⣿⡿⠻⠄⠻⢣⣤⠂
⠀⠀⠃⠀⠀⢀⣀⢻⣎⡿⠏⣠⣿⣿⣿⣿⣿⡇⠀⠘⢠⡀⡟⢺⡟⣧⢼⠃⠘⣷⡗⠁⣤⣆⣠⡄⣶⠈⣿⣆
⠀⠀⢠⢀⣬⣿⡏⣾⣿⣷⢀⣿⣿⣿⢻⣿⣿⠂⢠⠀⠸⢟⣾⣿⣷⣿⡐⡂⠰⠡⠃⢸⣿⣿⣌⣷⣿⡟⠀⣿⣤
⠀⠀⠀⢰⣋⣿⢰⣿⣿⡏⣼⣿⢿⣿⣶⣹⣇⣴⣸⡁⠠⢠⣿⣿⡛⣿⣏⡿⡀⠺⠢⣿⡿⢹⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⣿⣿⡎⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣦⡼⠃⠀⢸⣿⣿⡿⣿⡇⠀⢠⠀⣶⣿⣧⡿⢉⡉⣻⠋⣧⠐⠛⢀
⠀⠀⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⢟⣿⣷⣾⣿⣿⠛⢻⡿⠏⠐⠒⠒⠂⢹⣿⡉⣷⣸⣿⣿⡄⣠⣾⣶⣙⠓⣀
⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣸⣠⠀⠀⠀⠀⠀⠀⣿⣁⡁⣩⣽⣟⢻⣿⠻⣿⣿⣾⣇
⠀⠶⠿⢿⣿⣿⣿⣿⣿⣿⡷⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠛⠉⠀⢀⣀⣠⣤⣤⣿⣿⡇⢹⣿⣧⣾⢀⣤⣿⣿⣿⠟
⠀⠀⣠⣾⣿⣿⣦⣿⣿⣿⣿⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠈⠉⠉⢉⡉⠙⠿⠟⠋⠍⠛⠛⠛⠉⠁
⠀⠀⠉⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠆⠀⠀⣠⣴⣶⣶⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⢧⣶⠟⠛⠛⠟⠛⠛⠛⠍⠀⠀⣴⣶⣿⠁⠹⢿⠻⠆⠓⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⠛⠛⠧⠤⠴⠦⠐⠀⠀⡀⠦

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠁⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⢀⡞
⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⢠⣯⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠀⡠⠒⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡤⠀⡠⢀⠤⠀⣀⠄⣢⠄⢀⡠⠂⡾⠁⠀⠀⣼⠁⣠⠄⠀⣠⠄⢀⡤⡰⠄⣰⣏⠤⣶⠄
⠀⠀⠀⠀⠀⠻⢷⣤⣀⠠⢴⡞⠁⢾⡤⠜⠓⠸⠧⠤⠛⠂⠻⠖⠈⠃⠀⠀⣰⠇⠐⠿⠖⠈⠻⠄⠺⠧⠄⠸⠋⠐⠮⠅
⠀⠀⠀⠀⠤⠤⠤⠤⠤⣤⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣴⠿⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠄
⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⢀⠤⠒⠚⠛⠓⠒⠶⢤⣄⡀⠀⢀⣾⠋
⠀⠀⠀⠐⠒⠚⠉⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⣩⣿⠿⢧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⢀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠈⠉⠛⠓⠲⠶⠒⠋

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣤⣌⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠠⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣶⣀⠀⠀⠐⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⢠⠔⢨⠖⠀⠉⠈⠿⣅⠢⠁⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠁⢠⡃⠀⢀⣿⠀⠂⢹⡆⠘⡀⠀⠀⠀⠢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠸⠀⣿⣇⣀⠸⠿⠃⣌⣿⡏⢀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⡓⠈⢻⢁⡀⠀⢀⢸⡏⢐⠀⠀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⣴⣀⣩⠉⠁⠀⠉⠙⣧⣾⣶⡀⢀⣀⢠⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠈⠁⢹⠀⢲⣾⣿⣿⣿⣦⡀⠀⢠⡿⣻⣿⣿⣷⣄⠀⠸⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠄⠀⠨⠀⠀⠀⠤⠏⣰⣿⣿⣿⣿⢧⡟⠀⠀⠀⣿⣾⣿⣿⣿⣿⣶⠐
⠀⠀⠀⠀⢀⠠⡒⠀⠀⠀⠀⠀⣀⣰⣿⣿⣿⠁⡃⠘⣧⣠⣤⣀⣿⠹⠀⢻⣿⣿⣿⠀⡆⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠀⢻⣿⣿⡀⢫⠀⠙⠏⠣⠏⠠⠃⠀⡌⢿⣿⡇⠠⣇⠀⠀⠀⠀⠀⠈⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣨⣧⡀⠀⠁⠈⢻⣶⡍⠘⠂⠄⠊⣥⣴⣶⠁⠈⠿⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⢠⣿⡿⠃⠀⣀⣴⣿⣿⠃⠀⣴⡦⠀⣿⣿⣿⣦⠀⠀⠀⠀⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⡀⠀⢘⣭⠉⠉⠿⣀⠀⠉⠁⠀⠻⣿⣿⡇⠀⠠⠶⠅⡀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠋⢁⣿⠁⠈⠁⠀⠀⠀⠙⡓⠠⠄⠀⢐⣉⡛⠃⠠⠒⠰⡐⠀⣀⡁⠀⡀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠂⡀⠀⠀⠀⣤⠀⢀⣿⡇⣀⠀⠀⣶⠆⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠁⠐⠒⠺⣿⣥⣴⣟⡆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⣰⣇⣀⣹⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠠⡾⠀⠀⠀⠀⠀⠀⠧⠛⣿⣧⠻⣄
⠀⠀⠀⠀⠂⢀⠈⣹⠏⣻⣯⣬⠛⣄⠀⠀⠐⢻⠅⠀⠀⠉⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣦⡹⣇⠀⠠⠂⠀⠁
⠀⠀⠀⠀⠀⠀⢟⢁⣼⠏⣶⡜⡽⠻⣦⠀⠀⠀⠀⠀⠀⠀⠩⠀⠀⠀⠀⠀⠀⣀⣤⡖⢿⣆⠙⠿⡄⠀⠀⠀⠀⡤⠂
⠀⠀⠀⠀⢐⠀⠀⠚⢁⣾⠋⣼⢛⣾⢋⡿⠆⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢫⣿⣜⢿⡌⠹⡇
⠀⠀⠀⠀⠶⠀⠀⠀⠉⠉⠘⠋⠸⠻⡁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢀⠀⢀⠀⠀⠀⡈⠓⠒⠄⠄⠀⠀⠠⠤⠔⠂⠉⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢠⠄⠰⠖⠀⠈⠀⠀⠀⡠⢤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢤⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠦⣀⠉⠀⠀⠀⠀⠈⠑⡫⠋⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⠜⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠀⠀⠒⠈⠀⠀⠀⠂⠤⠄⠠⠤⠦⠂⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠤⠤⠤⢤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡀⢿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⠇⣰⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⢳⡘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⠇⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⢧⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⢠⠏⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠈⣇⠹⡄⠀⠀⠀⠀⠀⠀⠀⡼⢁⡞⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢻⡀⠀⠀⠀⠀⠀⣰⠃⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⢯⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢳⡀⠀⠀⠀⢰⠏⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠽
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠦⣬⣉⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢻⡈⣧⠀⠀⢠⡏⣰⠃⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⢋⣩⡤⠖⢻⡄
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠈⠙⠓⠦⣌⣙⠓⠦⣤⣀⠀⠀⠀⢳⡘⣇⢀⡞⢰⠇⠀⠀⢀⣀⡤⠖⢚⣉⡤⠖⠛⠉⠀⠀⠀⠀⣧⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⠦⣌⡙⠲⠦⣌⣧⠘⡾⢠⣏⡤⠶⠚⣉⡥⠶⠚⠉⠁⠀⠀⠀⠀⠀⠀⣀⡴⣛⡴⠚⢹
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠲⣤⡍⠀⠀⠈⣥⡴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⢚⣵⠊⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠖⣋⣡⣴⠀⡀⢰⣤⣍⡓⠲⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠮⣗⢦⣄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⢀⣠⠤⠖⢋⣩⠴⠖⠋⠉⢠⠏⡼⠻⡄⢷⠀⠉⠓⠶⢤⣉⡓⠶⢤⣀⠀⠀⠀⠀⠀⠀⠈⣿⠈⠙⠾
⠀⠀⠀⠀⠀⠀⠀⢸⡄⢀⣀⡤⠖⢛⣉⡤⠖⠛⠉⠀⠀⠀⠀⢠⡏⣰⠃⠀⢻⡈⢧⠀⠀⠀⠀⠈⠉⠓⠦⣬⣉⠓⠦⣤⣀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠛⣉⡤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢰⠏⠀⠀⠀⢳⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠦⣌⣙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢡⡏⠀⠀⠀⠀⠈⢧⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢁⡞⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⣰⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⢳⡀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⢠⠏⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢷⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣠⡏⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢧⣀⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣛⣶⠤⣤⣀⣀⣀⣀⣀⣀⣀⣤⠤⢶⡃⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠉⠉⠛⠓⠲⠶⠾⠶⠒⠚⠛⠉⠉⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁

⠀⠀⠀⠀⠀⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣆
⠀⠀⠀⠀⣼⣿⣿⡿⢿⣿⣷⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢤⣶⣾⣿⠿⣿⣿⣿⡄
⠀⠀⠀⠰⣿⣿⡗⡀⢨⣽⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⢾⡁⠛⢙⠆⠀⠀⠀⠘⣿⣿⣯⠀⡀⣿⣿⡿
⠀⠀⠀⠀⠘⣿⣿⣿⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣁⣀⣷⡖⠋⣉⡆⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⢠⡋⡀⠘⣿⢤⡬⠂⢀⣠⡄⠈⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠃⠀⣠⣶⡶⢶⣼⡟⠀⣀⣀⡀⠀⠀⠀⠉⠑⠚⢹⡄⠀⠀⣼⣿⠀⠀⢸⣿⣿⣿⣿⣿⣧
⠀⠀⠀⣼⣿⣿⣿⣿⣿⡟⠀⣼⣿⠏⠀⢀⣿⢃⣾⣿⣏⣻⣄⣠⣾⡿⢷⡀⣸⣇⣤⣤⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢠⣿⣿⡏⠙⣿⣿⡇⠀⣿⣿⣤⣴⣿⠇⣾⣿⠃⠀⢹⣿⣿⡏⠙⢺⡟⣿⣿⠏⢹⣿⠀⠀⠀⢠⣿⣿⣿⠉⢿⣿⣷
⠀⠀⣸⣿⣿⣀⣸⣿⣿⠁⠀⢈⣉⣭⣴⡿⠚⣿⣿⠀⣠⡟⣿⣿⠀⠀⣾⢹⣿⡏⠀⣾⡏⠀⠀⣠⠟⢹⣿⣿⣄⣸⣿⣿⡆
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣠⣾⡿⠋⢁⡿⠁⣀⣙⡛⠛⠋⠀⠻⣿⣦⠞⠁⠘⢿⣷⡴⠛⢷⣤⠾⠋⠀⠸⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣠⡾⢁⣾⠛⣿⣿⢀⣤⡆⢀⣤⠀⠀⣠⣤⣄⠀⢠⣷⠀⠀⣰⡿⠃⠀⣿⣿⣿⣿⣿⣿⣷
⠀⠀⣿⣿⣿⠉⢻⣿⣿⠛⠿⠟⠋⠀⣾⠃⢠⣿⠏⢸⣿⠃⢸⣿⠀⣼⡟⢰⣿⠀⣼⡏⢀⡴⠏⠀⠀⢀⣿⣿⡏⠹⣿⣿⣿
⠀⠀⣿⣿⣿⣀⣼⣿⣿⠀⠀⠀⠀⢰⣏⣴⡿⠋⠀⣿⡿⠀⣸⡟⢸⣿⠁⠀⠁⣠⣿⠿⣯⠀⠀⠀⠀⢸⣿⣿⣇⣰⣿⣿⣿
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣤⣾⡿⠋⠀⠀⢠⣿⣧⣤⣿⣇⣼⣿⠀⢀⣴⣿⡏⠀⣿⡄⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣄⣠⣴⠟⠛⠛⠋⠈⠛⠻⣿⣿⠟⠁⠿⠃⠀⢹⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⣿⣿⣿⡏⠹⣿⣿⣷⡀⠀⠀⠙⠛⠉⠑⠶⢤⠀⠀⢀⣶⡄⠀⠠⠶⠖⠂⠀⠀⠀⠉⠁⣠⣿⣿⣿⠋⢹⣿⣿⡟
⠀⠀⠀⠸⣿⣿⣧⣀⣿⣿⣿⣿⣄⠀⠀⠠⠤⢤⠤⠄⠀⠻⣏⠁⢉⡿⠃⠠⠶⠶⠶⠆⠀⢀⣴⣿⣿⣿⣇⣠⣿⣿⡿⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣠⠴⠂⠀⡷⠖⠾⡇⠐⠲⠦⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⠛⢿⣿⣿⣿⣿⣶⣦⣤⣤⣀⣀⣀⣀⣠⣤⣤⣶⣾⣿⣿⣿⣿⠟⢻⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣦⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣃⣠⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⠿⠿⠿⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⣤⣤⣀⠀⠀⢀⣀⣀⠀⠀⢀⣠⣤⠴⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣤⣄⡈⠙⠛⠛⠉⠉⠉⠙⠛⠋⠁⣠⣤⣄⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡟⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⣀⣄⣀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠘⠉⠉⠙⠁⢀⣀⣀⡀⠘⠉⠉⠙⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠐⠒⣃⠀⠀⣀⠀⠘⣻⡟⠁⢀⡄⠀⢀⣘⠲⢰⡿⠞⠛⠶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡈⠁⠀⠀⠈⠛⠛⠋⠙⠛⠋⠀⠀⠀⠉⣡⠟⠀⢀⡀⢀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣶⣤⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠞⠁⠀⠲⣼⣧⣼⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⢸⠃⢀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡆⠀⠘⠷⣽⡵⠞⠀⠀⠀⢠⣀⣀⣀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠳⣤⡴⠶⠒⠲⠶⢦⡀⠀⠀⠉⠙⢿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⣙⣦⠀⠀⠀⠀⢿⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠙⢦⣄⡀⠀⠰⣬⠟⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠉⠛⡟⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣠⡤⠶⠶⣤⡇⠀⠀⠀⠀⠀⠀⢸⣤⠴⠶⠦⣄⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡄⠀⡄⠀⢹⡄⠀⠀⠀⠀⢠⡟⠁⢠⡀⢠⡌⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠾⠷⠶⠷⠶⠟⠛⠳⠶⠶⠚⠛⠻⠶⠾⠷⠾⠷⠟⠁


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

⠀⠀⡄⠀⠀⠀⣁⣤⠀⠀⠀⠀⠁⢺⡏⠋⠀⠀⠄⠱⣾⡆⠀⣻⣤⣄⠠⠀⠀⢈⣡⣤⣴⣦⣤⣄⣠⣽⣿⣆⣠⡀⠀⠀⠀⠀⠀⣶⣹⡂⠀⠀⠀⢰
⠀⠀⠁⢀⡤⠚⠉⢀⡰⠖⠀⠀⠀⢸⡇⠀⠀⠀⠀⣴⠻⠆⢀⣻⣄⠟⠀⢀⣴⢿⣍⢀⣤⠺⠋⠋⠁⢤⢉⠀⠉⠉⠻⣦⡀⢧⣴⡏⠛⢷⡖⢺⣄⣜
⠀⠀⡄⠀⠀⠠⠁⠈⠀⠀⠀⢠⡶⢼⡇⣳⠄⣴⣿⡏⢠⡾⠋⠉⣽⣟⠿⠛⠿⠾⢿⡾⠋⡀⠒⢀⡀⠠⡠⣤⠆⠰⠆⠘⠻⣦⠀⣠⣉⢸⠇⣤⣿⢹
⠀⠀⠀⠀⠈⠀⠀⠀⡀⠄⢠⠄⠙⠃⠠⡈⠛⣿⠃⠀⠸⠟⢀⣀⣤⡿⠂⠀⠀⣰⡿⠀⠈⠀⢀⠈⠁⢰⣷⡏⠀⠶⠀⠸⡄⢻⡞⢁⣹⣡⡖⢳⣎⣉
⠀⠀⡆⠀⠀⠀⠀⠀⡀⢀⠃⣠⣼⠂⡄⠥⠀⢻⣇⣤⠀⠀⢿⡄⠀⠃⣠⡔⠀⠾⠇⣤⡀⠀⣠⠀⠀⢹⣿⠃⠀⠀⠠⠀⠶⢸⡟⠀⣿⣧⣤⣺⣿⡿
⠀⠀⡆⠀⠀⠀⠀⡀⠀⣿⠷⠀⢲⣀⣾⡓⠂⠛⡿⢀⣴⠿⠋⠀⢀⠟⠁⢀⣀⣤⡄⢈⣿⡄⣀⠀⠤⠂⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⣹⠀⠀⠋⣠⣵
⠀⠀⠁⢀⡀⠀⢀⣴⢿⡇⠖⠀⠈⢹⡟⠹⠂⠀⡟⠈⠛⣤⠀⠠⣿⣀⣈⣺⣭⣟⣳⣈⣹⣇⠀⠊⠀⢀⠀⠀⠀⠀⠀⠙⠁⣾⢹⣷⣿⣾⡃⢰⡏⢹
⠀⠀⡰⢈⣤⢾⡟⠑⢈⡅⠀⠀⣰⣿⠿⣇⠀⠀⡏⢀⡴⠿⣠⠶⠮⠉⢀⡙⠛⠋⢉⡈⠻⠉⢛⣆⠀⢀⡉⠀⠀⠙⢂⢰⡟⠳⣻⢿⣈⢸⡟⠹⠷⠾
⠀⠀⣿⣿⡏⢘⣥⠾⢛⡟⣋⣜⣡⣾⡶⠃⠀⢠⣷⠟⠀⣏⠈⠱⢀⠘⠄⠀⠀⠀⠂⢠⣷⡀⠿⢁⣯⢈⣁⣀⠴⠀⠋⠀⠀⠀⠀⢊⡽⣩⡆⠖⠀⣠
⠀⠀⠈⠛⠁⣾⠁⣐⠂⣶⠿⡏⠀⠀⠁⠀⠀⠀⣿⠘⠤⠋⡡⢀⡿⠃⠀⠀⣀⢀⡀⠛⠈⣿⡴⢿⠙⡇⠀⠀⣀⠄⠶⠒⠖⡢⣄⠢⠠⣽⣧⡾⢯⢶
⠀⠀⠂⠀⠀⠉⠉⠙⣿⢧⣾⠃⠀⠀⠀⠀⠀⠀⣿⣾⠀⠠⣡⡤⢿⢾⠃⠀⠉⠉⢡⡘⠛⠿⢰⡟⣈⢳⣄⠼⠿⣄⡐⠆⣼⠁⡸⣧⠐⠈⢻⠃⡄⢸
⠀⠀⡇⣠⣤⡤⠶⠾⢠⣿⠛⢻⠛⠛⠛⠛⠛⠛⡻⢏⠁⠶⣫⡴⢀⣬⠀⠀⠀⠀⠀⠸⡆⡰⣈⠰⣉⢼⣏⠀⢰⣬⠝⠟⢠⡞⠁⣿⠀⠀⡌⡛⠇⠚
⠀⠀⡅⢻⣧⠀⠀⠀⠀⠀⠀⣹⠆⠁⠀⠀⠀⠺⣥⠀⢠⣶⠉⢲⡿⣯⢤⡀⢠⣄⢀⡄⢘⣶⢾⡿⣿⣨⠏⠀⣦⠁⢀⠀⣿⣆⣸⡟⠀⠰⠀⢻⡎
⠀⠀⡶⡏⢀⡴⠖⠙⠲⢦⣄⣿⠀⢀⡴⠆⣉⣹⠿⠶⢧⢹⡇⢪⣌⠻⣦⣀⠘⢋⣀⣴⠟⠛⡦⠛⣽⣿⠶⠈⡇⢀⡌⠶⡛⢛⠟⠀⢀⡴⢦⣇⢠⣤
⠀⠀⡂⢧⢾⡴⠋⣉⠙⢶⠋⠉⠉⠹⠷⠋⠀⢱⡶⠀⠀⠛⡧⠀⠉⠀⠀⠉⠛⠉⠉⠀⣤⡔⠀⠀⡿⠋⢤⣈⣇⡼⠃⡖⠉⠉⢠⣤⠚⠄⠸⢯⡞⣻
⠀⠀⠁⠘⠺⣇⠰⣿⣀⣸⣦⡀⢀⣾⡆⠀⢀⡴⠃⠀⠀⡎⠱⢶⡶⠆⠄⣱⢞⣡⣴⡆⠣⠤⠄⠉⠀⠃⠀⢹⡾⢇⣴⡟⢢⠀⢨⡯⣸⠃⠖⢳⣏⣻
⠀⠀⠃⢠⡄⢀⡐⢤⡄⢹⣻⠃⠈⣻⠁⠀⠸⠇⢀⠀⢀⣿⡀⠀⠀⠀⠀⠙⣸⣿⠛⠁⠀⠀⠀⢸⣦⡀⠀⣾⡟⠻⣿⠀⣠⡘⠟⣡⣽⢶⣆⠜⣋⢻
⠀⠀⠀⢈⣽⡻⣇⠀⢁⡴⢻⡗⠀⠋⠀⠆⡼⢦⣄⣀⡞⢃⣹⣶⠺⠋⢷⣉⢹⡟⣧⢱⡔⢶⣶⡿⠛⣧⢰⡶⡟⠃⣹⢾⣯⣥⣾⣥⣍⠀⠀⣾⢿⣴
⠀⠀⣟⡅⠉⠁⣌⢨⣋⡔⢹⡟⢦⠀⣀⢰⠀⠀⠉⣻⠙⠛⢧⢻⡇⠀⠘⡇⡄⠛⢠⡜⠃⢾⡇⢣⣤⡈⢿⡿⠷⢃⡿⣀⣭⡄⠾⠘⡗⢶⣛⡙⢀⣹
⠀⠀⡯⢤⠑⠀⢉⡀⣿⠉⠀⠠⠀⠀⠘⢧⡀⠀⠀⢯⡄⢠⣴⢘⣷⠀⠀⣿⣅⠀⣿⠇⠀⣼⡇⠀⣉⠟⢀⣙⠻⡾⠃⣿⠁⡐⠐⢀⡇⠐⠩⠟⠘⣿
⠀⠀⠄⠀⠀⠃⠀⢀⠘⠃⠀⠠⠆⠀⠀⠀⢙⣦⠒⠀⣹⣦⡈⢩⡿⠀⠀⡞⣣⡿⠙⢻⣧⡾⣧⠀⠿⠿⠿⠋⢳⡆⠀⠁⠘⠃⢀⡈⡇⠀⢀⠀⠦⠉
⠀⠀⡙⠃⣠⠀⠀⠚⠀⢈⠠⠀⣾⣦⡀⢠⣦⠙⠃⠶⣧⣴⠙⢻⣿⣠⣴⣟⠈⠁⠂⢠⡄⣰⣿⠉⠙⠀⡄⠐⢿⣥⣀⣠⡀⠀⣎⠀⣧⡀⢨⢃⡀
⠀⠀⣹⠂⣮⠐⢀⡇⢰⡦⣶⡆⠈⢳⣿⣿⣿⠀⠀⠀⠀⠉⠻⢶⣼⣏⣁⣀⣒⣤⣽⠋⢉⣄⢀⡀⠀⠐⠀⠉⠲⡙⠛⣿⠀⠀⢿⠀⣿⠀⠐⠏⣛
⠀⠀⣿⠀⣽⢀⢨⡇⠀⠋⢸⡇⢐⣀⣈⣿⣧⡬⠔⠒⠒⠶⠁⠀⠀⣮⡍⠉⠉⣷⠀⠉⠉⠉⠀⠙⢿⠓⠒⠟⠳⡹⢦⡽⠀⠀⠾⠀⣿⠦⣤⡄⣠
⠀⠀⣿⢄⣴⠉⢚⣡⢰⣾⣿⡇⢸⠉⠼⣿⣿⣦⣤⠀⣮⠤⣤⠄⢾⡯⠠⣈⣸⡇⠀⢀⣐⡀⢀⣀⣀⣧⠘⠻⣟⢥⠀⣿⡄⣛⣿⠿⢻⡖⠉⠃⡤⢀
⠀⠀⠃⣀⣿⡄⣻⡿⣷⠟⣫⣤⣼⡜⠢⠉⠉⠙⠛⠀⣏⠀⠀⠀⡀⡐⣊⣁⡸⣟⠷⢌⣥⣤⣤⣤⣬⣩⣧⠀⣿⢀⡴⠏⠙⠋⢁⣀⢷⣤⣄⣤⡛⠺
⠀⠀⠛⠀⢸⡇⠙⠿⣶⡄⠀⠀⠀⣴⢦⣤⣀⠀⠀⠈⠛⠛⠛⠛⠟⠛⠛⠙⠛⠛⠛⠋⠉⠉⠉⠉⠉⠀⠁⢀⠀⠘⠷⠂⢰⣞⣋⡹⢷⠉⠉⠁⠀⢼
⠀⠀⡇⢀⣿⠁⠀⠀⠈⣿⠀⠀⣰⢻⡄⢠⡿⠁⠀⠀⠀⣶⠷⠾⣶⠀⠀⠀⠀⣶⢿⠟⣷⡆⠀⠀⠀⣠⢾⡿⢧⠀⠀⠀⠈⣿⠀⠒⠼⣧⠀⠂⠀⢠
⠀⠀⠃⢘⣁⣸⣦⣴⠾⠋⠀⢠⣗⣠⣸⠟⠀⠀⠀⠀⣸⡇⢀⢸⣿⠂⠀⠀⢰⡏⠘⠃⣿⡇⠀⠀⡰⢻⣇⠑⠈⢧⡀⠀⠐⢻⣷⡤⠶⠟⠀⠀⠀⠸
⠀⠀⡿⣿⡟⣻⠁⠀⠂⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⢰⣷⣤⣤⣼⠗⠒⠀⠀⢺⣇⣠⣈⣿⠃⠀⠀⠀⠀⠸⣷⡖⠻⠇⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡇⠘⠿⠋⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠜⠟⠁⠈⠀⠀⠀⠀⠈⠋⠙⠙⠗⠴⠆⠀⠀⠀⠀⢀⣠⡄⠀⠀⠀⠀⠀⣴⠶⠶⣶⠀⠀⢠
⠀⠀⡿⠷⠶⢶⣆⠀⣠⡟⢹⢷⣶⠀⠀⠄⠀⠀⡄⣴⠶⠶⣤⠁⠄⠀⠂⡀⢠⣤⣶⣶⡆⠀⠀⠃⠀⢠⣴⠟⠛⣿⠀⠀⠈⠀⢼⡿⢂⠀⣾⠀⠀⠸
⠀⠀⠃⠀⠀⠀⠉⣰⠟⠀⠖⣾⠏⠀⠀⠀⠀⠀⣹⡿⢂⢡⣿⡇⠈⠂⢀⣦⣼⡏⣥⠸⣯⠕⠂⣤⣤⠨⣿⡈⠉⠸⣧⠀⠀⠀⣸⡇⠛⢠⡟⠀⠄⠰
⠀⠀⠃⠀⠀⠀⠀⠻⣦⣤⣼⡿⠀⠀⠀⠀⠀⠂⣽⣁⡀⣸⠿⠀⠉⠀⠘⠅⢸⡇⠈⠀⣿⡄⠘⣤⠂⠀⠸⣧⣀⣠⣿⠄⠀⢠⠍⠛⠛⠛⠃⠀⠀⠈
⠀⠀⡄⠀⠀⠀⠀⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠘⠛⠟⣋⠛⠀⠀⡀⠀⡀⠂⠸⣷⣶⠿⢿⠀⣋⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣠⡶⣛⣦⢀
⠀⠀⡀⠀⠀⢀⡾⢻⡛⣶⡄⠀⠀⠀⠀⠀⠀⣴⠾⣟⢻⣄⡀⠈⠀⠀⠀⡄⣠⡴⠟⢷⣀⡀⠀⠈⠀⠀⠀⢀⡴⠤⠴⣶⡀⠀⠀⠀⠘⢿⡴⡤⠀⢻
⠀⠀⡁⠀⠀⣼⠁⢬⠁⣿⠀⠀⠀⠀⠀⠀⣀⣿⡶⠋⠌⣷⢳⠐⡀⢀⠀⠁⣯⡠⡶⠘⣿⡏⠓⠀⠀⠀⠀⢻⡿⠰⠆⣻⠁⠀⠀⠀⠀⠉⢷⡀⣠⣴
⠀⠀⡇⠀⢾⣧⣄⣀⣼⠏⠀⠀⠀⠀⠀⠈⠀⢹⣧⣘⣀⣽⠆⠀⠘⠯⣷⠒⢹⣧⠀⢀⣹⡧⠀⠀⠀⠉⠀⢸⣇⠀⢐⡏⠀⠀⠀⠀⠀⠀⠘⠛⠛⢘
⠀⠀⡃⠀⠈⠈⠉⢉⡉⢀⠀⠀⠂⠀⠀⣀⠀⢀⣟⡉⠉⠁⠀⠀⠀⣀⣀⠀⠀⠙⠿⠛⠉⠀⠀⠀⢀⣀⣀⣈⠉⠛⠛⠉⠀⠀⠀⠀⢀⠀⠀⠀⠀⢈

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣉⣟⣛⣛⣙⣛⣻⣎⣙⣛⣛⣋⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣬⣯⣭⣭⣭⣿⣿⣯⣭⣭⣥⣭⣭⣭⣽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠉⢻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⣾⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⠟⠻⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧⣀⣠⣿⣿⣿⣿⣿⣿⣿⣤⣤⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⢿⣿⣿⣿⣿⣿⠿⣿⣿⠊⣿⡁⠀⣹⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣀⣸⣿⣿⣿⣿⣇⣀⣸⣿⡆⠻⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠋⠙⣿⣿⣿⣿⣿⡇⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣴⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠀⣸⣿⣿⣿⣿⣇⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢰⣾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⢀⣠⣿⣼⣙⢨⣴⡶⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⢀⣠⣀⢀⣀⣤⣶⠀⠀⠀⣴⣿⢿⣿⣥⢷⡾⠹⢄⠗⣿⣿⣆⠀⣸⣿⣷⣠⣾⣿⡷
⠀⢀⣤⣤⣴⣿⣿⣿⣿⣿⣿⠻⣟⣻⣴⣦⡀⣤⣬⡡⡴⠓⠚⠑⣄⡀⠀⣄⣽⣶⣌⠿⣻⣿⣟⡿⣿⢟⠃
⠀⢾⣿⡿⠛⣉⣉⠈⠻⣷⠀⠈⠡⠋⡤⠀⠀⠀⣐⠷⠛⣉⢂⠠⣭⡇⠸⠻⢯⡏⡃⢶⠏⢇⠈⠁⠤⣀⡀⣠⣄⠀⠚⢿⠟⠂
⢠⣲⡁⠀⠐⠟⡾⡿⡗⣼⡐⠢⢼⠃⠀⡇⡎⠁⠶⡖⣂⢭⣤⣤⠭⡵⠛⠛⠈⠁⣤⣠⠖⠘⠳⡀⢀⣉⣯⢿⣿⢘⠛⠒⢀⣄
⠘⣿⣿⣿⡮⣿⣿⢿⡵⠇⠃⠒⠈⠛⠓⠓⠀⠀⠀⠈⠉⠁⢀⣀⣠⢦⠰⣲⣎⢕⣮⣇⣰⢶⡲⡬⢼⣿⣿⣿⣴⣤⣤⣿⣟⣿⡛
⠀⠙⡎⣿⡒⣱⣾⣬⣶⠆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠋⠉⡃⠀⣀⠣⢸⣉⠁⡼⠻⡿⡷⢾⡯⣿⣿⣻⣿⣿⠿⠛⠟⠛
⠀⠿⠓⠉⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠀⠀⠢⠄⠠⠀⠑⠌⢁⢀⢀⠈⣟⣥⣄⠀⠀⡈⠀⠁⠉⠃⠁⠈
⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠊⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠁⠀⠀⠈⠉⠙⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢄⠀⠀⠀⠂⠀⠀⠀⠰⠷⠁⠀⠀⠀⡀⠀⠀⠀⢰⠲⡀
⠰⠀⠀⠀⣴⢠⡤⣲⡂⡀⠀⠀⠀⠘⠂⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⠃⠁
⠀⠀⠀⠀⠀⠟⠉⠉⠀⠐⡀⠀⠀⠀⠠⠀⠀⠀⢀⡂⠀⠀⠀⠉⠀⠀⠀⡘⠁⠀⠀⠀⠀⠂
⢠⠰⠀⠀⠈⢣⡤⠄⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠘⢧⠣⡀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⢣⠱⡀⠀⠀⠉⠁⠀⠀⠘⢎⠀⠄⢀⣀⣠⡴⠃⠈⠳⢌⣒⣐⠰⠛⠁⠰⠬⠝⣒⣒⣒⣲⠶⠊
⠀⠀⠳⣈⠢⢄⠀⢀⣀⢠⠃⠀⠉⠐⠒⠒⡏⠀⠀⠀⠀⠀⢠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠠⠀⠀⢀
⠀⠀⠀⠈⠑⠂⡠⠄⠒⠁⠀⠊⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠁⠀⢀⡤⡲⢄⣠⠎⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠⠀⠀⡠
⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠋⠀⠆⠴⠀⠀⢠⡷⠃⠀⠀⣠⠖⠀⡖⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠊⠀⠀⠸⢡⠃⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⠀⠀⠀⠀⠀⠑⠜⠐⠀⠀⢀⠞⠂⠀⠛⠀⠀⢀⠃⠋⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⣀⠀⠀⠀⢀⣠⠊⢆⡑⠄⣀⡀⣀⣀⠔⡏⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠒⢪⠥⢶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠧⢀⣉⣁⡉⠴⠃⠀⠀⠈⠒⠀⠂⠚⠁⠰⠦⠏⡂⠒⠃⠀⣀⡀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⣀⣒⣂⠭⠕⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠢⠄⠉⠁⢧⠑⢄⢀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⣶⣾⣉⣿⣿⣿⣿⣿⣿⣹⣷⣶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢺⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣓⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣱⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠈⠉⠛⢿⣿⣿⣿⣿⣿⣿⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣭⣿⣿⣿⣿⣿⣋⠁⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣇
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⠘⢿⣿⣿⣿⠁⣰⣿⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢸⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣌⢻⣿⣟⣴⣫⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣬⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡇
⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠉⠉⠁⠀⠀⣸⣿⣿⣷⣤⡀⠈⠉⠉⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠸⡟⢿⣟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣫⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣻⣿⢻⠇
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣯⣽⣿⡄⢀⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠀⣸⣿⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⣿⣿⡗⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⠟⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣷⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠾⣿⣟⣻⣿⣿⠿⣿⣿⡿⣿⣿⣟⣻⡿⠿⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣾⣿⣿⣷⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠙⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠶⣄⡔⠋⠙⢦⡴⠊⠉⠙⠶⠁⠀⠀⣾⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆⠈⠀⢀⣤⣌⠀⢠⡶⣄⠀⢠⡶⡄⠀⣴⠿⣶⡿⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⠻⠆⠁⠀⢠⣿⠋⢹⡄⠹⣿⣸⣠⡾⣧⣇⣼⡏⠀⣿⡇⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⣾⡇⢀⢸⣧⢾⣏⡝⠸⣇⡜⠉⠹⠔⠁⢿⡇⠀⣼⠃⠀⣠⠟⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣷⠀⠘⠛⢁⣼⡇⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠾⠋⠀⢊⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢁⡟⠀⠀⠀⡞⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣼⣿⠞⠀⠀⠀⠀⠳⠋⠀⠀⠀⠀⣠⣤⡀⠀⢰⡿⣠⠖⣳⠄⠀⠀⠀⠀⢠⣦
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣀⣤⣿⡇⠀⠀⠀⢀⡄⠀⠀⣼⠀⠀⣼⡏⠘⠃⢀⣿⣿⣡⠾⠋⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⢤⠀⠈⠉⠉⠁⣿⡇⠀⠀⠀⢸⡇⠀⣼⣿⠀⢠⣿⠀⠀⢠⠎⣿⢿⣧⡀⠀⠀⠀⠀⠀⢀⣾⠏⢠⡄
⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⡆⠀⢻⡇⠀⠀⡰⢻⣇⡼⠸⣷⣠⠏⠻⣦⠴⠋⠀⠿⠀⠙⠿⣶⣦⣤⣤⣶⠿⠋⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⠈⠻⠶⠞⠁⠈⠉⠀⠀⢈⡁⠀⠀⠀⣤⠀⠀⠀⣠⣷⡀⠀⠈⠉⠉⠀⠀⣀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣄⣀⣤⠞⠷⣤⣠⣤⠞⠙⠶⠶⠚⠉⠓⠒⠚⠉⠈⠙⠶⢦⣤⣤⠶⠞⠋

⠀⠀⠀⠀⠀⢀⣶⠞⠻⢶⣄⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⣿⠃⣠⣤⡀⠙⣷⠀⠀⠀⠀⠀⣴⠟⠉⠀⠈⠙⣦
⠀⠀⠀⠀⠀⣿⡀⣿⠀⠹⡄⢸⡇⠀⠀⠀⢰⡏⠀⢀⣄⠀⠀⠘⡇
⠀⠀⠀⠀⠀⢹⣇⠸⡆⠀⢻⠀⣧⠀⠀⠀⢸⡇⢰⠏⠉⣻⠶⠞⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⢻⡄⢻⡄⢸⡆⣿⠀⠀⠀⢸⡇⣸⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠙⢷⡀⠀⠀⠀⠀⠀⠀⣰⢟⣉⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠻⡈⣷⠀⡇⢿⠀⠀⠀⢸⡇⡟⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠞⢦⠻⣄⠀⠀⠀⠀⡼⢣⠞⠉⣧⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⠃⠀⠘⠒⠋⠙⠛⠁⠀⠸⢃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⠀⠐⠃⠉⣾⡋⠛⠐⣶⠈⠈⠓⠋⠈⠃⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⠀⠄⠀⠀⠀⠀⠀⠀⢀⢠⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠁⠀⠀⠃⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⢀⣠⣤⡀⠀⠀⠀⠀⣀⣤⣄⡀⠀⠀⠀⡸⣱⢛⡶⢠
⠀⢀⠰⣖⢲⡻⡄⠀⠀⠐⠋⠉⠓⠀⠀⠀⠀⠉⠉⠙⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⣁⠀⠀⠀⢳⣿⣼⡅⢰
⠀⢱⠀⣽⣬⣷⠇⠠⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠂⠀⠀⠀⠀⢢⡄⠀⠀⠀⠀⠐⠀⠀⠀⢸⠃⠀⠀⢸
⠀⢸⠀⠀⠀⢿⡀⠀⠒⠀⠀⠀⠀⠦⠚⠳⠴⠂⠀⠀⠀⠉⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠙⠉⠈⠋⠀⠀⠀⠀⠀⠀⢠⠏⡠⠖⠒⣼
⠀⢸⠀⠀⠀⠈⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⣻⠀⣧⣀⣠⡟
⠀⠘⣇⠀⠀⠀⠐⣏⠛⠳⢶⡶⢦⡤⢤⠤⣴⠶⠖⠚⢛⣡⠞⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⢦⣍⣉⠛⠛⣳⠖⣒⣾⡛⠛⡉⢉⣠⠶⠃⠀⠈⢀⡟
⠀⠀⠘⢦⡀⠀⠀⠈⠛⠲⠶⠤⡜⠐⠒⠒⣻⠧⠴⠖⠋⠁⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠟⠉⠙⠛⢻⣈⣥⡠⠿⠏⠉⠻⣷⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠒⠛⠛⠉⠀⠀⠀⣤⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⢸⡇⠐⠒⢄⠀⣧⠊⠁⠀⢀⡌⠉⠀⢈⣉⠀⣠⠇⠀⠠⠏
⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⣠⠔⠀⠀⠒⠒⣊⣷⡶⣮⡴⠂⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⣠⠘⣮⠱⠶⢤⣤⣮⣤⣤⡬⠷⠚⠁⠀⠀⠀⠑⣄
⠀⠀⠀⠀⠀⢰⠇⠀⠀⡰⠁⠀⠀⠀⢠⣾⠟⢃⡀⠛⣾⠩⡤⢶⡟⢣⠀⠀⠀⠀⠀⠀⠀⠻⣧⠚⢁⢠⡼⠂⢀⣤⡈⢷⣧⡀⠀⠀⠀⠀⠀⠀⠀⢄⠈⣆
⠀⠀⠀⠀⠀⡿⠀⠀⠀⡄⠀⠀⠀⢠⢿⡏⢡⡗⡿⢠⣶⣛⡆⠠⣷⠘⡄⠀⠀⠀⠀⠀⠀⡴⠈⠛⡞⠋⣤⠄⢹⡌⡃⠀⢻⣧⠀⠀⠀⠀⡆⢆⠀⠀⣷⢸⡄
⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⣎⣿⣀⡾⠀⣾⣾⠉⢹⡇⠰⡏⠀⡇⠀⠀⠀⠀⠀⢠⡇⠐⢢⣧⠈⣹⡞⠀⢯⡿⠂⠉⢿⡇⠀⠀⠀⡙⢦⣀⡤⠟⠀⡇
⠀⠀⠀⠀⠘⡇⠀⠀⠀⢹⡀⠀⢰⣿⠇⠤⠄⠀⠀⠁⠀⠟⣀⣹⠃⢠⡇⠀⠀⠀⠀⠀⢸⡇⠈⡆⣿⡔⠀⠉⠀⠀⢒⢦⠄⢘⣿⠀⠀⢰⡇⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠹⢤⣸⣿⣉⠀⢴⠄⠐⣽⡟⠆⢠⡟⠀⣼⠁⠀⠀⠀⠀⠀⠈⣿⣮⡴⠋⢷⡂⠀⢷⣆⢙⣿⣷⠈⣿⠀⠀⠀⣧⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠀⠈⣿⠀⠤⣼⡂⠼⣿⠋⣈⡿⠁⣰⠇⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠘⢷⣀⠛⠏⢀⡉⠅⣺⠏⠀⠀⠀⡿⠀⠀⠀⢸⠏
⠀⠀⠀⠀⠀⢠⡎⣠⢀⡄⠀⡆⠀⠛⢦⣄⠀⠍⢙⡋⣰⠞⢡⠰⡝⡆⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⡶⢠⠄⠛⣶⣌⣀⣤⣾⣥⣤⣤⣄⣀⡇⠀⡆⢰⣸
⠀⠀⠀⠀⠀⠀⠙⠛⠛⠓⠚⠁⠀⠀⠀⠈⠉⠛⠉⠉⠙⠓⠛⠚⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠈⠓⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢸⣿⣿⣿⣦⠀⠀⠀⠀⣶⣮⣅⢀⣤⣤⣴⡄⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⠿⢿⣿⠀⠀⠉⠉⠛⠛⢻⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠋⠀⣀⣀⠀⠀⠀⠀⠀⠀⠊⠁⢀⣠⣤⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡁⠀⢙⣿⣿⣧⣤⣶⣴⡖⣀⡤⠖⠛⠉⠉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣏⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⡏⠙⠍⠿⠿⠏⠀⠀⠀⠀⠀⢠⡟⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⡿⠛⠛⣽⣿⣿⡟⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠸⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⢹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠁⢀⣀⠾⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠟⠁⠀⠀⠀⢀⣾⣿⣿⣶⡶⠋⣠⡤⠀⠀⠀⠀⠉⠀⢀⣶⣾⣿⣧⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣶⡿⠋⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⡟⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠛⣿⣿⣧⠀⠠⣀⡀⠠⢿⣿⣿⡇⠀⠀⠀⠹⣿⣖⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⠟⠛⣿⣿⠀⣿⠛⢿⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⡄⠀⣿⣿⣟⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠁⠀⢀⣿⡟⠀⢸⣄⢸⡀⠀⢠⠀⠀⠠⣿⣿⣿⠿⠛⠀⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢿⣤⣅⠀⠈⠶⠆⢀⡙⢿⠁⠀⠀⠀⣨⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠸⣿⣦⡀⠀⠀⠀⠘⠹⡇⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⢇⠀⠀⠀⠹⣿⠁⠀⠀⠀⠀⠠⢡⣄⣀⣰⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠺⠀⠀⠀⠀⠙⠷⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠛⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⣛⣛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠺⣦⣄⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠾⣿⡛⠁⠀⠀⠀⢉⠻⢷⣤⡀⠀⠀⠀⡀⠀⠀⡀⠊⠀⠀⣠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠤⣀⠀⠻⣿⣦⠈⠻⣷⣦⣀⠃⠀⠞⠁⠀⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠈⠙⠦⡀⠀⠀⠀⠀⢉⣿⡇⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠾⠟⠁⠀⠀⠀⠀⠀⠀⠈⠙⠦⣀⣴⡾⢏⣴⠔⠫⣴⣦⡄⢀⣭⣙⠑⠒⠰⠤⠄⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⣾⣿⣷⠀⠀⠀⢹⣭⠴⠛⠁⠀⠀⠈⠈⠀⠈⠙⠚⠃⠺⠿⠶⠀⣀⡤⠔⠓
⠀⠄⠁⠀⣠⣤⣄⠀⠀⢰⣿⢿⣿⡇⠀⠀⠙⠛⠊⠀⠀⢀⡼⠁⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡶⠛⠉⢀⠀⢀⡈
⠀⠀⠀⠸⣛⣛⡿⠀⠀⠈⠓⠒⠋⠀⠀⠀⠀⠀⠀⠀⢀⠞⠃⠀⣿⣿⣿⠀⠀⣴⣿⣷⡆⠀⢠⣤⣤⡀⠀⡴⢎⣤⡄⢺⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠙⠛⠉⠀⠀⠻⠟⠻⠃⠀⢿⣿⡿⠃⢰⣧⢸⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⠀⠀⠀⢀⠎⠀⠀⠀⢀⣴⣶⣄⠀⠀⣠⣤⣤⡀⠀⠀⢀⠀⠀⠀⣸⠀⠀⠁⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠋⠀⢀⡴⠃⠀⠀⠀⠀⠘⠿⠿⠋⠀⠀⣿⡿⢿⠇⠀⣾⣿⣿⠆⠀⣇⠀⣀⠀⢨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⢀⣤⣤⡄⠀⠀⠈⠁⠀⠀⠀⠉⠉⠀⠀⢠⠋⢸⢿⡇⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠃⠀⢰⣿⣿⡆⠀⣴⣿⣷⡀⠀⣼⠀⠌⢉⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⠁⠀⠙⠿⠟⠀⢠⡿⠟⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⣿⠀⣾⢃⡀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠉⠁⢹⡇⠀⠲⠀⠀⠿⠘⠃⠉⠁⠀⠀⠀⣾⠁⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁⠀⠀⣀⠘⠁⠀⠀⢀⣠⡀⠀⣠⢤⠀⣰⠞⣿⠇⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠁⠀⠀⠀⣰⣾⢁⣧⣾⣇⣎⣼⠏⣼⡟⠀⠀⣀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⣀⣀⡀⣸⡏⢙⠏⣿⢁⡟⢹⣯⡞⣿⠇⠀⢠⠇⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠛⠋⣩⠏⠻⡷⢋⣀⣿⣯⣤⡀⠉⠀⣿⠀⣠⠏⠀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣿⡇⠀⠀⠀⢀⡴⠋⠀⣠⠞⠋⠁⣰⠂⢠⠇⠀⠀⠘⠛⣋⠀⠀⠀⣠⡾⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡘⣿⣄⣠⡴⠋⠠⠀⣼⠃⠀⠀⢰⣏⡴⢫⡞⣠⣾⢀⡾⣩⠇⠀⠀⠩⢶⡛⠃
⠀⠀⠀⠀⠀⠀⠀⣠⣤⣌⠉⠀⠉⠁⣀⣤⣤⣤⣇⠀⠀⠀⣿⠙⡆⠸⠟⢹⣇⣾⠋⠁⣰⠁⠂⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠈⣩⡄⠀⠀⣠⣾⠟⠉⠁⠉⠙⠷⠀⢸⡇⣰⠇⢀⣤⣿⠉⣿⣤⡾⠁⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⢀⣼⠟⠁⠀⠀⠀⠀⠐⠂⠀⠸⠷⠋⣠⠋⢰⡇⢠⣤⠄⠀⠀⢠⡟⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠉⠀⢀⡀⠀⢤⠀⠠⡇⣠⠟⠀⣞⣧⡀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠉⢡⣤⠀⠀⢀⣤⣤⡿⠁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡀⠀⢸⣿⠀⠀⠀⠀⠀⠰⠶⢿⡆⣴⡟⢠⢃⣴⡏⡸⠀⣴⠟⢡⣿⠃⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⢠⡟⣼⠏⢳⠞⢹⡏⣽⠓⢻⣏⡰⠋⣿⠀⢀⡞⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣀⣀⣴⣿⠃⣿⣠⠏⠀⠸⠟⠁⠀⠀⠉⠀⢠⡟⠶⠋⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣸⡟⠀⠈⠁⠀⠀⡀⢀⡀⠀⣴⠿⢆⡿⣠⣴⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠿⠂⠀⢠⣿⠃⠀⠀⠀⠀⣼⢁⣾⠁⣾⠃⣠⣾⡿⠛⠁⠀⠀⣼⠃⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣼⡏⠀⠀⠀⠀⢸⣧⢻⣏⣼⣧⡔⢱⡟⢷⣄⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⣿⣧⡶⠶⠾⠟⠈⠁⠈⢛⠀⠀⠀⠘⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠄⠀⢀⡾⢹⡈⠃⠀⠀⢴⡦
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⡂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠖⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠒⠒⠲⠤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⣤⣀
⠀⠀⠀⠀⠀⣠⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄
⡀⠀⠀⠀⠈⠛⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠁⠀⠀⠀⠀⠀⠀⣠⣤⣶⣿⣿⡄⠀⠀⠀⠀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠙⢿⣿⣿⣧⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠈⠻⣿⣿⡄⠀⠀⠀⣴⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠙⢿⡷⠀⢀⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠛⠻⠿⠿⠿⣿⡷⠀⣠⣤⡀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⠀⠙⠟⠃⢠⣶⣶⣶⣤⣤⣤⣄⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣶⣿⣿⣿⣿⣿⠟⠉⣰⣦⠀⣶⣦⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⣴⣿⡇⠀⢸⣿⣷⣄⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⢀⣾⣿⣿⡇⠀⠈⣿⣿⣿⣦⡀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢠⣿⣿⣿⣿⠃⠀⠀⢹⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠈⠛⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠁⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤

⣴⣦⣶⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣴⣴⣶
⣿⣿⣿⡟⠉⠛⠛⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⣤⠄⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠳⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠀⠀⠀⢠⣤⠀⠰⠇⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣀⠩⣄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡆⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⢦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠨⣄⠀⢿⣧⣶⣿⣿⣶⣂⡄⠀⠀⠀⠘⠿⠂⠀⠀⠠⣶⠄⢠⣿⣄⠀⠀⠀⠀⠀⠀⠉⣽⣟⣿⣔⡶⣤⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣷⣤⣾⣿⣿⣿⠛⠭⠋⠀⠀⠀⠀⠀⠙⢿⣿⣷⣶⣶⠖⣤⣿⣿⣿⣦⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁⠉⠀⠈⠀⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡤⠠⠂⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠈⠉⠀⣀⡈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢰⡷⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⢀⡀⠙⠃⠀⡀⠀⠀⠀⠀⠀⠰⣶⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠘⠛⠀⠀⠘⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⢠⣾⣿⡷⠶⢦⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠰⠿⠆⣠⣶⣿⣦⠀⠀⠀⠀⠀⠀⠂⣀⡾⢿⣶⣶⣦⣶⣦⣶⣖⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣉⣀⣠⣶⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⠐⠾⠛⠿⠋⠄⠉⠀⠀⠀⠀⠀⠀⠀⠉⠽⠛⠿⠿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠉⠭⠛⠽⠛⠟⠛⠿⠛⠿⠛⠃⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈

⣶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣧⣂⣿⣭⣭⣭⣭⣭⣥⣭⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⡴⠶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠙⠲⠋⠁⠀⠀⠀⠉⠙⠋⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡄⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⢠⠖⠋⠁⠀⠀⠈⠙⠲⡄⠀⠀⠀⠓⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⢰⠃⠀⠀⣰⠚⠓⠂⠀⠀⠘⡆⠀⠀⠀⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡄⠀⠀⣿⠀⠀⠀⠙⠦⠤⡄⠀⠀⠀⣿⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⢹⡄⠀⠀⢀⣀⣠⠟⠀⠀⢠⡏⠀⠀⠈⠓⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⢤⡀⠀⠈⠁⠀⢀⡤⠋⠀⠀⠀⠀⠀⣀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠚⠉⠀⠀⠀⠀⢀⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⠚⠀⠙⢲⡀⠀⢀⡦⣤⢤⣀⡴⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡖⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣤⠤⠤⠤⡤⠤⠤⢤⡾⠤⣤⠤⢤⠷⠤⠤⣤⠤⠤⠤⡤⠤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣇⣀⣿⣀⣀⣠⣇⣀⣀⣸⣄⣀⣧⣀⣸⣄⣀⣀⣿⣀⣀⣠⣇⣀⣸⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣸⡀⠀⠀⣿⠀⠀⢀⣇⠀⣸⠀⠀⠀⠀⠀⢸⡀⠀⣇⠀⠀⢸⡇⠀⠀⣸⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠉⢹⠉⠉⠉⡏⠉⠉⢹⡏⠉⣿⠀⠀⠀⠈⣿⠉⢹⡏⠉⠉⢹⠉⠉⠉⡟⠉⢹
⠀⠀⠀⠀⠀⠀⢰⠛⠛⡞⠛⠛⢳⡟⠛⠛⢻⠛⠛⡞⠛⠛⢳⠛⠛⠛⢳⠛⠻⡞⠛⠛⢳⠛⠛⡟⠛⠛⢻⡞⠛⠛⢳⠛⠛⡆
⠀⠀⠀⠀⠀⠀⠸⠦⠶⠷⠴⠶⠾⠷⠶⠦⠾⠶⠶⠷⠶⠶⠼⠶⠶⠦⠾⠶⠴⠇⠀⠀⠸⠶⠴⠷⠶⠶⠾⠧⠶⠦⠾⠶⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠒⠤⠀⠀⠀⠀⠀⠀⣠⢤⡄⠀⢠⣄⠀⠀⠀⠀⢀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠋⢀⣀⡙⢛⣗⡒⢒⣺⡏⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣰⣗⣿⣙⡿⣿⣓⡷⣏⣸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢹⣏⣀⣍⠀⠈⠁⠀⣼⢦⠁⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⣿⠁⢋⡙⣷⠶⣶⢾⡟⣁⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣟⠀⢸⣇⣏⣾⡉⢹⡷⣏⠀⢛
⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠻⢮⣍⣉⠈⠉⠈⠁⠹⣧⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⡤⠴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣛⣛⣻⣛⣿⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠖⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⣀⣀⣤⣤⠤⣤⣤⣀⣀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠉⠉⠁⠀⠀⢌⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠔⣡⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢤⡞⠁⠀⠀⠀⠀⠀⠀⣰⢲⡔⡄⠑⡄⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠞⠁⠀⠀⢀⡠⢂⣠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⠀⠀⠱⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⠃⣀⡀⠀⠀⠎⣰⡟⠁⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⢀⡄⢸⣇⡠⠤⢄⣀⣀⡀⠀⢐⢠⡀
⠀⠀⠀⠀⠀⠀⣼⠇⠀⢈⣉⠓⠀⢠⡿⠀⠀⠀⠿⠀⠀⣀⣀⢤⠀⢀⣐⡒⡄⠀⠀⠀⢀⣇⡾⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡑⢝⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⣉⡉⠁⣼⡇⠀⠀⠀⠀⠀⣸⠧⣌⣧⣀⣿⢧⣹⡇⣀⡴⠚⠉⣿⠃⢀⣀⠐⠚⠋⣁⡀⠀⠀⠀⠀⠤⠀⠀⠈⢻⡜⠂
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠀⡠⠀⣿⡀⠈⠉⠉⢳⡆⢿⠀⢠⡏⠈⣏⠀⣸⠏⢹⡁⠀⣰⢿⣠⣟⡽⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠚⠀⠀⠘⣧⡀⠀⠀⣸⡇⠈⠑⠩⠄⠀⢙⠚⡩⠀⠈⢑⣊⠅⠀⠉⠻⠤⠄⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠪⣙⠒⣚⠏⠀⣠⠖⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠘⢀⠄⠀⠀⣀⣠⣄⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⣾⢸⠇⠀⠀⠀⠀⠀⠀⠻⠟⠃⠀⠀⢀⡿
⠀⠀⠀⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠀⠲⣄⡀⠀⠀⣿⠀⣠⠟⠁⠀⠀⣸⠃⠀⣾⣷⢠⡟⠉⠓⠀⣿⡿⢖⡦⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⠀⢀⠄⠀⠉⠓⠒⢿⡛⠁⠀⠀⠀⣴⢿⠀⠀⢷⡿⢸⣇⠀⠀⢀⣿⣏⠁⠀⢀⣄⡀⠀⠀⠀⣤⠤⠖⠋
⠀⠀⠀⠀⠀⠘⣧⠈⢆⠀⠀⠀⠀⠀⠀⠃⠆⠀⠘⣎⠳⢤⣀⣠⠼⠋⠈⠓⠤⠚⠓⠋⢉⡓⠚⠍⠃⠈⠙⠢⢿⡞⠁⢠⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⢀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠉⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣀⣀⠤⠖⠋⠀⠈⠳⣄⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠤⣌⣉⠉⠛⠛⠋⢉⣩⠴⠂⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡢⢊⣴⠟⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⠤⣄⣀⣀⣀⣀⣀⣀⠤⠖⠋⡡⢞⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⡦⢤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⡭⢖⡽⣉⢦⣿⣧⣧⡆⠙⢿⣟⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢊⣴⠋⢠⣟⢐⣢⣿⢋⡝⢻⣟⡁⠀⢻⣆⠹⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢸⡇⠀⣿⡇⠀⠀⠉⢳⣾⠟⠁⠀⠀⢸⣿⡆⢹⡹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢸⡇⠀⣿⡃⠀⠀⣀⡄⢣⣀⡀⠀⠀⠀⣿⡯⠈⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⢿⡀⢘⢷⠾⠟⠀⠁⠈⠁⠹⣷⣦⣤⣿⡇⢸⣇⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⣿⡏⢏⣤⣄⡀⠀⠀⣠⣤⣤⡉⠙⣿⢧⣟⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⢉⡃⠠⠾⠟⠏⡀⠂⢑⠖⠀⠀⠀⣧⠄⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣖⠒⠦⢾⣸⡧⢿⣀⠀⠁⢴⠃⢀⠈⡁⠀⠀⢠⡟⢥⣧⡼⠖⠒⢒⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠚⠛⠛⠿⢷⣦⣤⡙⠘⢿⡏⢓⢋⣶⣤⣼⣾⠁⠲⠚⢸⡟⣊⣫⡴⠞⠋⠉⠁⢉⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⢰⣣⣶⣾⣷⣶⣶⣄⡈⠛⢿⣶⣿⣷⡌⠛⠫⢥⣭⠙⠓⠂⣠⡿⣹⣿⠏⣠⣤⣴⣾⡿⠿⠿⠟⠻⣆
⠀⠀⠀⠀⠀⠀⠀⢸⡿⣛⠉⠉⠉⠙⠛⠿⢷⣤⣉⠳⢜⣿⣷⣤⣤⣤⣤⣶⣾⠟⣁⣿⢣⣾⣿⡿⠋⠀⠀⠀⠀⠀⠤⠿⣦
⠀⠀⠀⠀⠀⠀⢀⡞⣾⠃⠀⠀⠀⠀⠀⠀⠀⢈⣽⡗⢲⣿⣟⣿⣿⣿⣛⣋⣁⢀⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⡼⣼⡟⠀⠀⠀⠀⠀⢀⡤⠖⠛⠛⠓⠛⠩⣄⠚⠻⠷⣶⣿⣭⠍⠙⠛⠛⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⣼⣻⣟⠀⠀⠀⠀⠀⣰⠏⠀⠀⠒⠒⠦⣄⠊⠁⠀⠀⠀⠀⠉⡉⢱⠿⠒⠂⠀⠀⠘⢦⢠⡀⠀⠀⠀⠀⠀⠤⣽⣷
⠀⠀⠀⠀⣰⢣⡿⠅⠀⠀⠀⢤⣴⠇⠀⠀⠀⠒⠲⢤⠐⠋⠀⢀⣀⣀⣀⠀⠈⠫⢠⡴⠤⠖⠒⠀⠸⡄⢧⠀⠀⠀⠀⠀⠀⠈⢻⣇
⠀⠀⠀⣼⠃⠛⠀⢀⣀⣈⣩⣽⠿⠀⠀⠉⠑⠲⡤⠒⢃⡴⠊⢹⡿⣿⠩⠓⢦⠚⠰⢄⣠⠤⠶⠆⠀⢇⠸⣧⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⡼⠁⠔⣿⢿⡿⡿⢭⠋⠀⣀⡀⠀⣹⣷⢂⡚⠂⠉⠉⠙⠛⠛⠛⠓⠂⠉⠀⠰⠬⣅⣀⣤⡄⠀⠘⠲⠿⣧⣄⣀⡀⠤⣀⣀⠈⢻⡀
⠀⡾⠁⣠⠞⢁⡞⠀⢀⣯⣴⢟⣽⣾⠛⠉⣿⡜⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡀⠀⠀⠀⠀⢻⡺⡿⢟⣿⠛⠿⠿⠛⠳⣄
⣸⠇⣰⠃⢀⡞⠀⠀⢸⣿⣾⣿⣿⣟⡀⠀⠘⣷⡘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠘⠻⣦⣿⣛⠶⣼⣇⠀⠀⠸⡆⠀⠀⠀⠀⠘
⣿⠀⠀⠀⢸⡇⠀⠀⠈⣿⢻⣿⣿⡿⢒⡁⠀⠈⠻⣮⣛⠷⣄⠀⠀⠀⠀⠀⢀⣠⣴⡟⠁⠀⢀⣿⣿⣿⣿⣿⣾⣿⠀⠀⠀⣷
⢻⣄⠀⠀⠸⡇⠀⠀⠀⣠⣿⣿⣿⣿⣯⣖⣥⣆⣴⣯⣄⠉⡒⠒⠒⠒⠒⠉⢉⣀⣼⣷⣄⡢⣵⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿⠀⠀⠀⠀⢀
⠀⠙⠶⢤⣤⣵⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢣⠀⠉⠛⠒⠚⠓⠚⠛⠉⢁⠀⠈⡍⢹⣿⣿⣿⣿⣿⣧⣉⣀⣀⣀⣠⣧⣀⡀⣀⣤⣾
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⢤⡡⠞⢄⣀⠈⢁⠖⠒⢲⠀⠉⢁⣠⠴⠋⠓⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣄⣀⡼⠓⠉⠉
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠘⠃⠈⠆⢠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⣼⠟⠒⠶⠶⠶⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⢤⣄⠀⠀⣄⢀⢐⡆⢀⡀⢰⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣰⣯⠘⣟⠲⠶⠶⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢉⠀⠈⠉⢀⠈⠉⠁⣈⡁⠈⣁
⠀⠀⠀⠀⠀⠀⢦⡄⠀⠀⣠⠀⠆⠈⠙⢡⡆⠘⢦⣄⠀⠀⠀⣠⠶⢞⣛⣛⡛⠳⢦⡀⠀⠀⠋⠘⠆⠰⡹⠗⠄⠀⠿⠓⠈⠁⠑
⠀⠀⠀⠀⠀⠀⠀⠓⣄⠈⠡⠀⣈⢠⢶⡔⡧⠀⠀⠈⢁⣴⣿⡷⠛⠋⠉⠉⠁⠁⠀⠱⣄⠀⠀⠀⣠⠶⢴⠀⢴⡆⠀⠨⠟
⠀⠀⠀⠀⠀⣀⣴⢋⠤⣤⠴⠳⡯⢻⣼⡗⠃⣠⠀⣰⣿⣿⣿⠞⠃⠀⠀⠀⠀⠀⠀⣤⣙⠀⢰⣄⠀⠁⢠⠞⢤⠇⣵⡀
⠀⠀⠀⢠⣾⣉⠀⠳⢷⣌⡢⣴⢺⣉⡾⣡⠿⠇⢰⣿⣿⣿⡵⠀⠀⠀⠀⠀⡀⠀⠓⢴⠉⠀⢸⣿⡆⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⢻⣏⠳⣄⠀⠉⢡⡀⢀⡁⠴⠉⢀⢲⠟⠛⠉⠙⠂⢠⣔⣨⠡⠄⡄⢠⡄⠀⠀⠀⠘⣿⣿⣶⣶⣿⣾⠿⣿⡇
⠀⠀⠀⠀⢸⠘⣧⠘⢆⠐⣮⠉⠻⠀⠀⠈⣰⠏⠀⠀⠀⠀⠀⠀⣈⢳⣏⣬⠄⠸⠃⠀⢀⣀⣀⣿⣿⣾⡋⢀⣹⣿⣿⣿⡭
⠀⠀⠀⠀⢸⠀⠸⡇⠀⠀⠁⠀⠀⢠⣟⢰⡏⠀⣠⡦⣤⠀⠀⢋⣴⡄⠿⠀⠦⠀⠀⠀⠈⠉⢿⣿⡞⠉⢻⣿⡟⠉⣻⣿⣧⣴⣶⡀
⠀⠀⠀⠀⠘⡇⠀⢫⠀⠀⠀⠀⣀⠬⠿⠾⡇⠠⠶⠲⠦⠀⠀⡀⠤⣤⠀⠚⠛⢳⣶⣾⣿⣯⡬⢿⣿⠶⣯⢀⡽⢾⡿⢏⢻⣿⣿⡇
⠀⠀⠀⠀⠀⢹⠀⠈⠀⠀⠀⡠⡾⣡⡤⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⣤⣾⠀⡀⣿⣦⣼⡏⠀⠀⠙⠛⣿⣿⠛⠋⠀⣸⣻⣿⡿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠰⠱⢻⡄⠹⢾⣤⠀⠀⠀⢠⣀⣀⣸⣧⢘⣁⣼⡇⠉⣿⣷⣿⠀⣆⠀⠀⠻⠇⠀⡇⢠⡏⢾⣾⠁
⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣠⡶⢎⣻⣶⡆⠀⡄⠀⠀⠀⠉⠉⠉⢹⡎⣿⡿⣄⡀⠸⣯⣿⢀⡇⠀⠀⠀⠀⢠⣀⣸⣇⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢋⡅⠈⠉⠻⢃⡆⢡⣿⡀⠀⠘⣛⠋⠀⣸⠃⠈⠑⢌⢳⡄⢻⣾⢙⡇⢀⣴⡂⠀⣼⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣼⠋⠐⠉⠀⠀⠀⠀⢸⡇⢠⠉⠙⠦⣀⠉⣀⡞⠁⠀⠀⠀⠀⠛⣷⠈⣿⣷⣷⠀⢋⠿⢀⣯⢸⣟⣡⡾⠇
⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⡀⠀⠀⠀⢀⣿⠃⢸⠀⠀⠀⠿⠿⠛⠁⠀⠀⠀⠀⢀⣴⣿⡕⣿⣿⡹⣦⣄⣴⡿⢻⣾⠋⢿⣳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣙⣷⣄⠀⠸⣏⠀⢺⡀⠀⠀⠀⣰⡇⠀⠀⢀⣤⣾⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⠈⣷⣄⠙⢧⣄⠙⠂⠀⠀⠟⠀⣠⣴⡿⣟⢽⡿⠛⣿⣿⡿⣿⠿⠿⠿⢿⡿⢿⣿⡿⠟⣻⡿⢻⣿
⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⣿⣿⢳⣄⠙⠻⠶⠾⠋⢀⣼⠿⡉⣦⢉⣿⠃⠀⠿⢿⡇⠹⣆⠀⢠⠟⢡⡾⠿⠃⢰⡿⣾⡿⠁⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⣿⠀⢹⣷⣤⣶⠶⠶⡿⢿⠂⢧⠘⣾⣿⣀⣀⣀⡀⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣻⣇⣿⣡⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢸⣿⣦⡾⢸⢣⠲⡞⢢⢹⡘⠶⢶⣾⣿⠋⠉⡏⠉⠉⠉⣿⠉⠉⠉⣏⠉⠉⠉⠉⠉⠙⢻⡟⡄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣯⣤⣤⠘⣆⡀⠝⢋⡾⣷⢞⣿⣿⡟⠀⠀⢡⠀⠀⢀⡟⠀⠀⠀⣿⠀⠀⠀⠀⠀⠠⢏⡼⢡⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠈⣿⠰⢻⠷⣬⣽⠙⣩⣼⢏⣿⣿⣿⡇⠀⠀⠈⣧⠀⢸⡇⠀⠀⠀⡟⠀⠀⠀⠀⠀⢀⡾⢡⢎⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⣿⠀⡇⢰⠃⣿⡆⣿⣷⢸⣹⣷⣿⡇⠀⠀⠀⠸⣇⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠰⠋⣰⢋⡞⠁
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⣷⢸⠀⣿⣷⣼⡏⢸⢡⡿⣻⡇⠀⠀⠀⠀⠙⣿⢅⡀⢠⣸⡇⠀⠀⠀⠀⠀⠘⠁⠈
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⣟⣸⠀⡿⣿⣿⡇⡇⢿⣿⢻⡇⠀⠀⠀⠀⠀⢨⣿⠃⡞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣿⡇⣿⣿⠀⡁⣿⣿⡇⡇⢸⣿⢾⣷⠀⠀⠀⠀⠀⢸⡇⢸⠁⣿⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢹⡇⣼⢿⢠⡆⣿⡟⢁⡗⢸⠸⡞⢿⡄⠀⠀⠀⠀⠈⠁⠐⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⢸⣇⠸⣼⣼⠁⣹⡇⢸⣿⠈⡄⣿⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⣿⡄⡇⠋⠀⣸⠄⢸⡏⡄⠀⢸⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⢹⡇⠀⡃⠀⠈⠀⣸⣧⠃⠀⣼⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠃⢸⠀⠀⡟⢸⠀⡆⢻⡀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢸⠀⠃⠸⠃⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⣠⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⢠⣤⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⣠⠴⠒⠉⢉⠀⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠈⠋⠴⠁⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⣸⡿⣻⠿⣏⡇⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣴⣿⢿⣦⣾⣫⣟⢻⣾⡏⢿⣿⡯⣤⣄⠀⠀⢠⣶⣶⡀⠀⣀⣴⣤⡀
⠀⠀⠀⠀⠀⠀⣠⣤⣤⣴⣼⣶⣄⣤⣤⣿⣿⡷⠀⠀⠀⠼⢿⣿⣿⣿⢿⣿⢿⡟⠲⠋⢽⠚⢿⣯⣿⠄⢠⣾⣿⣿⣿⣾⣿⣿⣿⣁
⠀⠀⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣯⣧⣾⣷⣄⣠⣦⣤⡤⣼⢿⡛⠛⢝⣶⣤⡀⢲⣦⣾⣷⣾⣭⠿⣻⣿⣿⣻⣿⣿⡟⡛⠃⠀⠀⠀⠀⢀⡀
⢀⣼⣿⣿⡿⢟⠛⡋⠉⠈⠻⣿⣷⠀⠙⠫⠟⢫⡄⢀⠈⠀⣈⣾⣶⠟⢛⠛⡂⢌⣯⣽⠁⣾⢻⣽⣻⠩⠇⣷⡛⢻⡟⠛⠉⢡⡈⠉⠀⣀⣄⣀⠴⠿⣿⣿⠶⠂
⠀⣱⡽⠋⠃⠪⣹⣟⣿⣶⣦⣘⡏⣤⣁⣲⠿⢨⠸⡇⡼⠲⢨⣯⣄⣠⢸⠏⣆⣀⡿⣿⣲⡶⠶⠼⠛⠈⢁⡈⣡⣤⠿⣦⡀⠀⠙⢻⡿⣿⣿⡋⢾⣦⣬⡌⢀
⢸⣷⣷⣿⣤⡤⣬⣼⣷⣯⠷⣼⢻⣒⣉⣹⢾⣤⣤⣇⡟⠄⠸⠈⠧⠶⣬⠛⠋⠛⠋⠙⢁⣄⣀⣀⢤⣜⣷⡞⠋⣁⣠⣹⢾⢱⣾⣿⣿⣿⠻⠯⠓⠀⣤⣬⣿⣯⣤
⠀⢼⣿⢿⣷⡗⣿⠿⢟⣳⠯⡛⠂⠀⠉⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣼⡿⢧⠚⠷⣵⢏⣽⢿⣧⢼⣯⣿⣿⡟⢺⣾⣿⣿⣿⣿⣿⣶⣿⣿⣿⣽⣿⢥
⠀⠘⣻⡽⣟⡟⢩⣿⣯⣭⢷⡿⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡦⣛⠃⢀⣹⢄⡰⣀⡱⠖⣭⣿⣁⡶⢋⢽⣽⡿⢿⣿⢾⣽⢿⣻⣿⣿⣿⠯⡟⠟⠏⠋
⠀⢾⡽⠾⠉⠔⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠐⠂⠠⢤⣍⢶⡈⠚⢮⠙⠋⠀⢂⣀⣙⣿⣧⣄⢠⢈⠐⠉⠓⠈⠁⠛⠟⠛⠉⠃
⠐⡱⠏⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⡝⢿⠻⠉⠀⠀⠀⠘⠉⠁⠀⠠⣄⣀⡊⠟⠌⠙⠈⠙⠻⠛⢷⠷⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠻⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢄⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⢁⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣡⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠘⠻⣶⡀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⢀⠠⠤⠶⢦⣤⡀⠀⠀⠀⠀⠠⠴⠖⢶⡆⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⢠⣿⠃⠀⠀⠊⠁⡉⠙⢿⣆⠀⠀⠀⠀⢀⣤⡄⠀⢻⣷⠀⠀⠀⠀⣠⣤⡴⠆⠀⢀⣾⡟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⡄⠀⣼⠟⠀⠀⠀⣴⡿⠃⠀⢈⣿⣠⡆⠀⣰⣿⠋⠀⠀⣸⣿⠀⠀⢠⣾⡟⢁⣤⡆⠀⢸⣿⠠⠀⣨⣿⠇
⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⢠⣿⠀⠀⣡⣤⡶⠀⢰⣿⠁⠀⢀⣾⡟⠀⠀⠀⠻⠷⢀⣠⣾⠟⠁⠀⠀⠙⣁⣴⡿⠋⠀⠠⣈⣤⣴⠿⠋⠁
⠀⠀⠀⠀⠀⠀⣾⡇⠀⣠⣿⠏⠀⢰⣿⠁⠀⠀⠌⣉⣥⡾⠟⠋⠀⠀⠀⡀⠐⠚⠛⠉⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠻⠃⣰⣿⢃⡄⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢾⠃
⠀⠀⠀⠀⠀⠈⠙⠛⠉⢁⣾⠇⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢰⣿⠃⠀⠀⠀⠀⠀⠀⢀⣴⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⣿⠏⠀⠀⠀⠀⠀⠠⠿⠋⣼⡏⠀⠀⠀⠀⠀⢀⣴⣦⠀⣠⣾⡧⢀⣀⣴⠟⠉⢉⣤⣾⣯⠾⣿⣛⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠐⢟⣿⢃⡴⠋⣿⣧⠞⠹⣧⣤⠞⠋⠀⠋⠁⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣤⣴⠶⣶⣶⣦⠘⢿⡟⠁⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡏⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠖⠛⢋⣉⣥⠴⠖⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⠾⠟⠛⢋⣁⣤⣶⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠛⠉⢀⣤⡶⠾⣟⣛⣭⠤⠶⠶⠶⠂⠀⠀⠐⠂⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠀⢠⣾⠿⠿⠛⠛⠉⠁

⠀⠀⠀⠀⠀⣠⠴⠤⣄⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠤⣄⡀⠀⢀⣀⣀⡀⠀⢀⣠⠤⢤⡀
⠀⠀⠀⠀⢀⠇⠀⠘⠊⠉⠉⠉⠶⣅⠸⠄⠀⠀⠀⠀⠀⡤⠤⣄⣰⠇⠀⠈⢣⢤⠴⠚⠉⣇⠀⠀⠀⠀⠀⠸⡈⢢⡀⠉⠉⠉⠙⠉⠉⠉⠁⢠⠊⢰⠃
⠀⠀⠀⠀⢸⠃⠀⠀⡀⠀⠀⠀⠀⠀⢹⠤⢤⡀⠀⠀⢸⡇⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠀⠀⠀⠀⠀⠙⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡶⠁
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⣀⣱⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⠀⠀⢻⠖⠲⢦⠀⠀⣠⠖⠳⡄⠁⠀⠀⠀⠀⠀⠀⠀⡰⠓⠦⡀
⠀⠀⠀⠀⡼⠢⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠇⠀⠀⡏⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢻⠀⠐⢇⡠⠖⢧⡀⡀⠀⠀⠀⠀⠀⢠⠟⠤⣀⠟
⠀⠀⡠⠞⠄⠀⠀⠐⠀⢠⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢣⠀⠀⠀⠈⠐⠠⠠⠤⠤⠤⠔⠺⠃⠀⠀⡸⠀⠀⠀⢧⡀⠈⢣⠀⠀⠈⠉⠀⢀⠏⠀⢠⡟
⠀⢸⣀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡀⢀⡴⠃⠀⠀⠀⠈⢳⠄⠀⠉⠂⠈⠤⠚⠁⠀⢤⠋
⠀⢸⠁⠍⠳⢉⡀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠈⣆⠀⡹⣹⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠀⠀⠀⠀⠀⠘⡇⠀⣤⡀⠀⠀⠀⠀⣠⡄⠀⡼
⠀⠀⠸⡄⢹⠟⠛⢀⣀⣀⣀⣠⠂⠀⠐⢦⠃⠀⠀⠀⠀⠀⢳⡀⠀⢣⡀⠀⠀⢀⡞⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⢹⠀⠀⠑⣄⣀⣀⡴⠉⠀⢰⠁
⠀⠀⠀⠙⠒⠒⠒⠉⠀⠀⠀⠉⠒⠶⠔⠋⠀⠀⠀⠀⠀⠀⠀⠓⠶⠒⠛⠒⠒⠚⠦⠦⠚⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⠲⠞⠃⠀⠀⠙⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢰⢩⡓⣄⠀⣠⣞⡍⡆⠀⠀⠀⠀⠀⠀⠀⢀⠤⣄⡀⠀⠀⣀⠤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠍⠓⠤⢤⣤⡤⠤⠔⠉⡇
⠀⠀⢀⡠⢄⡴⠛⠉⠠⠠⠄⠉⠛⢇⠀⠀⠀⠀⠀⠀⢠⠡⠀⠀⠈⠉⠉⠀⠀⠌⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠉⠀⠀⠀⠐⢇
⠀⠀⠸⣤⣴⠁⠂⠐⠀⠀⠀⠀⠐⠈⡆⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠤⣀⠀⠀⠀⠀⢀⢀⡀⠀⡴⡏⠀⠐⠒⠄⠀⠀⠔⠒⢂⢨⡄
⠀⠀⢀⠇⠇⠀⠂⠐⠀⠀⠄⠂⠀⠀⠸⠀⠀⠀⠀⠀⡇⠀⠐⠆⠀⠀⠀⠰⠂⠀⢹⡤⡈⢣⠀⠀⠰⡃⠀⠈⠻⠀⢠⡀⠀⠀⠀⠸⠀⠀⢐⢋⢉⢱
⠀⠀⡜⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠈⠀⠀⠀⠀⡸⠀⠀⢸⠀⣀⣀⡏⠈⢠⣶⣾⣿⣼⣶⡄⠄⠀⠀⠄⢼⠉⠉⢹
⠀⠀⢣⡀⠀⠀⠈⠀⠠⣈⣣⠆⠥⠄⠈⢆⠀⠀⠀⣠⠋⠓⠤⠀⠀⡀⠀⠀⠠⠞⠁⠀⣠⠃⢸⡀⠈⣁⣴⠻⡯⡙⠛⢋⢭⠻⣦⣜⣀⠔⠃⠰⣿⡼
⠀⠀⠀⡜⠀⠀⠀⣠⠊⣩⣹⡦⡤⣀⣀⡼⠀⠀⢰⠁⠀⠀⠀⠀⠘⠄⠅⠀⠀⠀⠀⡔⠁⠀⠀⡹⠃⢸⣏⠘⠷⠇⠀⠸⠾⢂⢻⡦⠹⡄⡴⠋⡈⡇
⠀⠀⠀⡇⠀⠀⠘⡇⢸⠶⢯⠀⠇⠀⠀⢸⠀⠀⠘⡶⠶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠛⠄⣾⢧⡙⠒⠀⠀⠄⠒⣃⡼⣿⡄⡿⠀⠸⠟⡰
⠀⠀⠀⢣⠀⠀⠨⡷⢿⣗⡇⣸⠀⠀⠀⡜⠀⠀⠀⢃⠀⠀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡸⠀⠀⠀⠀⠀⠸⣮⣿⣻⡟⠛⣿⣿⣧⣷⣋⣼⡷⠴⢦⠔⠁
⠀⠀⠀⠀⠙⠳⠄⠞⠂⠖⠚⠣⠰⠛⠋⠀⠀⠀⠀⠀⠓⠾⠿⠒⠒⠒⠒⠒⠻⠿⠒⠁⠀⠀⠀⠀⠀⠀⠈⠛⠻⠷⠤⠿⠿⠛⠁⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠋⠁⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠉⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⢻⣿⣿⠃⠀⠀⣰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⣄⠀⠘⣿⣿⠀⠀⣴⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⡀⢻⡟⢀⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⡈⠙⠷⠜⠃⠾⢋⣀⣠⣤⣴⣶⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠛⣋⡄⡀⣬⡙⠛⠛⠛⠿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⢀⣤⣾⠟⢰⣷⠈⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠏⠀⢸⣿⡆⠀⠙⣿⣿⣷⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠃⠀⠀⣾⣿⣷⠀⠀⠈⠻⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠆



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⣿⣿⣿⣿⣿⣓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠛⠶⣤⡶⠛⠛⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠤⣄⣀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⡠⠤⠤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠙⠳⣄⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣠⡾⠋⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠈⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡶⠋⠀⠀⠀⠀⠀⠀⠀⢸⣧⣀⡀
⠀⠀⠀⠀⠀⣴⠟⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⡄
⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣀⡟⠀⠀⠀⢷⣀⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⡞⠛⠛⠲⢦⣾⠁⠀⠀⢹⠃⠀⠀⠀⢸⡏⠀⠀⠀⣻⡴⠖⠛⠛⢳⡄⠀⠀⠀⠀⠀⢀⣴⠏
⠀⢀⣤⠴⠶⠶⠶⠿⠦⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠈⠛⢦⡀⣿⠀⠀⠀⠀⠀⡇⢀⡴⠛⠁⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠴⠿⠷⠶⠶⠶⢤⣄
⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠋⠁⠀⠙⢦⣄⠀⠀⠀⣠⠶⠶⢦⡀⠀⠀⠀⣠⡴⠋⠀⠉⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⢠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⢀⣨⡿⠶⠀⣇⠀⠀⠀⡇⠠⢶⣟⡁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣤⠶⠋⠁⠀⠀⠀⠙⠶⠶⠞⠁⠀⠀⠈⠙⢶⣄⣠⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠙⠳⠦⣤⣤⣤⣤⣤⡤⠄⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⣀⡴⣿⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠤⣤⣤⣤⣤⣤⡤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠰⡏⠀⠀⠀⢀⣠⡴⠞⠉⠀⣿⠀⠀⠀⠀⠀⡏⠙⠷⣄⡀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠈⠳⣦⡀
⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠙⠓⠚⠛⠉⠁⢿⡀⠀⢀⣹⡀⠀⠀⠀⢰⣇⡀⠀⢈⡿⠲⢤⣄⣀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠋⠙⣇⠀⠀⠀⣼⠉⠙⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠈⠳⠦⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⢀⡴⠋⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⣄⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⣀⣀⣠⡴⠋⠁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⠳⣤⣀⣀⣀⡴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⣠⡟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣙⣷⣶⣶⣟⣁⣉⣷⣶⣶⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠋⠀⠀⠀⣠⣴⣶⣤⡄⠙⠻⢿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⠿⠛⠉⠀⠀⣀⣤⣤⣀⠙⠻⠿⠟⠋⠀⠀⠀⠉⠛⠿⣷⣦⣀
⠀⠀⠀⠀⠀⢀⣴⡿⠟⠋⠁⠀⠀⣀⣀⠀⠻⢿⣿⡿⠃⠀⠀⠀⠀⠠⣾⣿⣿⣷⡄⠈⠙⠻⢿⣦⡀
⠀⠀⠀⠀⠀⣿⣿⣷⣤⣀⠀⠰⣿⣿⣿⣿⠀⠀⠀⠀⠀⢠⣶⣶⣶⣦⠈⠉⠉⠉⠀⠀⣀⣤⣾⢿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣄⡉⠉⠀⠀⢀⣤⣤⣤⡈⠙⠛⠛⠋⠀⠀⢀⣠⣴⡾⠟⠋⠁⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠋⠙⠻⣿⣿⣿⣶⣤⣀⠙⠿⠿⠿⠋⠀⠀⠀⢀⣤⣶⠿⠛⠉⣠⣶⡆⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣆⠀⠀⢹⣿⣿⣿⣿⣿⣷⣦⣄⡀⢀⣠⣴⣾⠟⠋⠁⠀⠀⣾⣿⣿⠃⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⡿⢷⣦⣾⣿⣿⡋⠈⠙⢿⣿⣿⣿⣿⠟⠉⢠⣶⣿⡆⠀⠀⠙⠋⣁⡀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠙⢿⣿⣿⣧⡀⠀⠘⣿⣿⣿⡏⠀⢠⣿⣿⡟⠁⠀⠀⣠⣾⣿⡇⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣄⡀⣸⣿⣿⡿⠿⢷⣾⣿⣿⣿⡇⠀⠀⠉⣁⣤⡀⠀⠀⢿⡿⠟⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⠋⠙⢿⣿⣿⣿⡆⠀⠀⢹⣿⣿⣿⡇⠀⠀⣼⣿⣿⠇⠀⠀⠀⣠⣶⡆⠀⢸⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣄⠀⠀⢹⣿⣿⣿⣦⣄⣠⣿⣿⣿⡇⠀⠘⠿⠟⠁⠀⠀⠀⣾⣿⣿⠃⠀⣸⣿
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣤⣾⣿⣿⡋⠉⠙⢿⣿⣿⣿⡇⠀⠀⢠⣶⣿⡆⠀⠀⠙⠋⣁⣴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣧⡀⠀⠘⣿⣿⣿⡇⠀⢠⣿⣿⡟⠁⢀⣠⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣷⣾⣿⣿⣿⡇⠀⠀⠉⣁⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣇⣠⣴⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠉⠛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢀⣤⡄⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠘⠿⠋⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢰⣿⡆⠀⠀⠀⠀⠀⠀⠀⢿⣿⠆⠀⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠛⠋⠛⠛⠛⠛⠛⠛⠛⠛⠛⢳⡄⠀⣤⣤⡀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣶⣶⠀⠀⠀⠀⠀⣴⣶⡄⢸⣧⡀⠙⠛⠁⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⠉⠀⠀⣀⠀⠀⠈⠉⠀⢸⡇⠙⢦⣠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⣿⡷⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⣤⠀⠀⠀⠀⠀⣠⣤⡀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠻⠟⠁⠀⠀⠀⠀⠻⠟⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠼⠃



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