Ethics Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⡇⣾⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣉⠉⣉⣉⣉⣉⣉⣉⡉⠉⠀⠈⠉⣉⣉⣉⣉⣉⣉⠉⣉⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁⢻⡄⠉⠉⠉⠉⠉⢹⡇⠀⠀⠀⠀⣿⠉⠉⠉⠉⠉⢠⡟⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠈⣷⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣾⠃⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠸⣧⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡏⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢿⡄⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⢸⡇⠀⣤⡄⠀⣿⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠘⠃⠀⠀⢸⣇⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⢠⡿⠀⠀⠀⠛⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠘⠃⠀⠀⠀⢻⡆⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣾⠃⠀⠀⠀⠛⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠘⠃⠀⠀⠀⠘⠃⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠛⠀⠀⠀⠀⠛⠀⠀⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣟⠛⠛⠛⠛⠛⠛⠛⠛⣻⡇⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣿⡛⠛⠛⠛⠛⠛⠛⠛⢛⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣦⣄⣀⣀⣤⣴⠾⠋⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠈⠻⢶⣤⣄⣀⣠⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠾⠿⠶⠿⠷⠾⠿⠶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠰⠆⠰⠆⠰⠶⠾⠿⠶⠶⠶⠶⠶⠶⠶⠶⠿⠶⠶⠀⠶⠀⠶⠀⠶



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⣀⣀⣀⣀⣀⣀⣤
⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠀⠰⠀⠀⠀⠀⠀⠐⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⢸⠀⠀⠓⠉⠃⠃⠛⠈⠙⠀⠀⠀⠉⠉⠀⠀⠀⣠⣄⠀⢠⣿⡀
⠀⠀⠀⣼⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠠⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠈⠓⠛⠙⠛⡉⠋⠋⠉⡉⠉⠉⠀⠰⠆⠀⠀⠘⠿⠏⠀⠘⠻⠃
⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠊⠀⠀⣤⡀⠀⠁⠀⠀⠀⠀⠀⠀⠁⣴⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⡇⠀⠓⠃⠓⠛⠙⠓⠑⠈⠒⠀⠁⠃⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠇⠈⠀⠀⠄⠀⠀⠀⠀⠀⠙⡿⠃⠀⠀⠀⣀
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠩⣭⣩⡍⠉⠉⡍⠉⢀⡄⠀⠈⠉⠉⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⣤⡠⣠⣀⢀⣄⠠⣴⡾⠏
⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⢈⡉⠉⠀⠀⠀⠀⠀⢸⠀⢀⣄⣠⣄⣠⣄⣠⡄⣀⢀⣀⡀⢀⢀⡀⣀⣠⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠈⢦⡘⠃⢀⠀⠈⠁⢀⢠⠳⠛⠀⠀⢸⠀⠿⠼⢧⠼⠗⠫⢿⠄⠘⠾⠼⠳⠼⠬⢧⠟⣿⠈⡱⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠀⠐⠚⠛⠛⠛⠂
⠀⠀⠀⠀⡄⠀⠉⠉⠉⠀⠀⠀⠈⢆⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⡀⠀⠀⠀⢀⡄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠖⠈⠦⠴⠊⠀⠘⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠁⢰⣀⣀⣀⣀⣀⣀⣀⡀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠉⠀⠀⠀⠠⢴⡂⡦⣠⠔⡄⠒⡄⠀⠤⡄⠀⠆⣤⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠐⠓⠒⠒⠚⠂
⠀⠀⠀⠀⠀⠈⠈⠛⠂⠒⠄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡡⠬⠔⠲⠊⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⠀⠀⠀⢀⣀⣀⣁⣀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠦⠀⠀⠒⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⠄⠐⠂⠚⠉⠀⠀⠀⠀⠀⠀⢀⣿⡧⠉⢠⣿⡀
⠀⠀⠀⠘⠛⠛⠛⠛⠛⠿⠋⠁⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⢠⠤⢄⢀⡀⠀⠀⠀⠀⡀⢠⡶⠀⠀⠀⡄⢠⡀⣠⡀⠠⢄⡄⠀⠀⠀⠀⠙⣿⣧⣀⣴⣿⠛
⠀⠀⠀⠐⠲⠆⠒⠔⠀⠀⠀⠀⠀⠘⣿⠟⠀⠳⠤⠜⠀⠀⠸⠤⠜⠙⠻⠶⣶⡶⠾⠃⠸⣷⣤⣤⣤⣥⣭⣬⣤⣭⣤⣤⣤⣤⣤⡄⠀⠀⠛⠙⠿⠋⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⢿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠀⠀⠈⠻⢿⣦⡀⠀⠀⠀⢠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⡀⠀⡀⠀⡀⠀⢀⠀⠀⠀⠀⣀⠀⠀⠀⣀⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠙⠻⣷⣦⣤⣾⣿⣿⠇
⠀⠀⣇⠀⠀⠀⠀⡇⠀⠀⡇⠀⢸⠀⡇⠀⡎⠀⠈⠀⠸⠄⣈⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋
⠀⠀⠧⠤⠤⠀⠀⠇⠀⠀⠇⠀⠸⠀⠅⠀⠣⠤⠔⠀⠰⠤⠬⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡄⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠿⠿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⣤⣾⣿⣿⣿⣿⣿⣶⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⠀⠀⠀⠀⠀⠘⠛⠛⠛⢛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⡛⠛⠃⠻⣿⡿⠃⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⢠⣧⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧
⠀⠀⠀⠀⠀⠀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡆
⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡀
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣧
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⣿⣿⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠛⣋⣛⠛⠛⠛⠛⠛⢛⣋⣡⣴⣿⣦⣉⣉⡛⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢸⣿⣿⣿⣿⣿⢿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⡃
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠈⠛⠻⠿⠿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⠿⠿⠛⠋
⠀⠀⠀⠐⠲⢶⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢿⣿⣿⣿⡿⠇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⠶⠒
⠀⠀⠀⠀⠀⠀⠀⢡⣴⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣦⣌⠻⢋⣴⣾⣿⣷⠄⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡄⠀⠀⢀⣤⣶⣿⣷⡌⢿⣿⣿⢋⣴⡌⢿⣿⣿⢋⣴⣿⣷⣦⣄⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⣴⣿⣿⣿⣿⣿⣿⣦⠙⣡⠘⠿⠟⣠⠙⢡⣾⣿⣿⣿⣿⣿⣷⠀⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣿⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⡏⣰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣧⠘⣿⣿⢸⣿⣿⣿⣿⡟⢸⣿⣷⠸⣿⣿⣿⣿⣿⢸⣿⡟⣰⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣧⣹⣿⢸⣿⣿⣿⣿⡇⢾⣿⣿⠄⣿⣿⣿⣿⣿⢸⣿⣠⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣦⡙⣡⣾⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⢤⡤⢠⡄⢠⡄⡄⢀⡤⠄⣠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣛⡀⢸⡇⢸⡟⢻⡇⡇⢿⣀⡀⣙⣳⠄
⣿⣿⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣶⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣀
⠀⠀⠀⠐⠒⠒⠲⣉⠖⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⢸⣷⣿⣾⣇⡷⠀⠀⠀⠀⠀⠀⠲⢦⠀⠿⠟⠋⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠀⠀⠻⢦
⠀⠀⢠⣤⣤⡄⠀⣿⠀⠀⣤⣤⡄⠀⠀⠀⠀⠐⢦⣸⣿⠿⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠓⢰⣶⣤⠄⣸
⠀⠀⠀⠉⠉⠀⠀⣿⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠘⢿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⣿⣿⣿⣿⣶⣦⣤⠀⠀⠀⠀⠀⠺⣦⣤⣿⣿⣶⣷⣦⣤
⠀⠀⠀⠀⠀⠴⠿⠿⠿⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠻⢿⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠻⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣰⣶⣶⡖⠒⠲⣆
⠀⠀⠀⣴⣶⣤⣴⣶⣦⣤⣴⣶⡀⠀⠀⠀⠀⣰⣿⣽⣿⢻⣿⣝⣧⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠈⣿
⠀⠀⠰⠿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⣿⣿⣯⠁⠀⢩⣿⣿⡆⠀⠀⠀⠀⢸⠀⠀⠀⠛⠁⣠⡀⠛⠀⠀⠀⠀⠀⢹⡟⠛⠛⢣⣤⣤⣤⡏
⠀⠀⠀⠀⠈⠻⢿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⢻⣟⢿⣶⣶⣾⣿⡿⠁⠀⠀⠀⠀⢸⡄⠀⣤⣤⣬⣭⣥⣤⠀⠀⠀⠀⠀⠀⠳⣄⠀⢸⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠷⠾⠻⠉⠀⠀⠀⠀⠀⠀⠀⠓⠶⠿⠿⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⠟⠁


⠀⠀⠀⠠⠀⢔⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⣛⡻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⣴⣶⡀
⠀⠀⠀⠶⠆⠀⠀⠰⠶⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣽⣅⣭⢻⣿⡄⠀⠀⠀⠀⠀⠀⣦⠀⣸⣿⣿⠧⢠⡆⠀⠀⠀⠀⠀⣠⣼⠟⠉⡀⡈⠻⣿⠆
⠀⠀⠀⠀⠠⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣶⣾⣿⡷⠀⠀⠀⠀⠀⠀⣿⣦⠛⣿⡿⢀⣾⡇⠀⠀⠀⠀⠀⠛⣿⠰⠆⢀⠰⠆⣻⣦
⠀⠀⠘⠻⢷⣾⣿⣿⣿⣿⣿⣶⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠹⣿⡆⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠐⣿⢦⣘⣛⣃⣴⡏⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠿⠀⠀⠀⠀⠀⠀⠠⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠈⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠏⠉⠋⠁


⠀⠀⠀⠠⠤⠤⣴⣶⡦⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠳⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣄⡀⠀⣀⣴⣶⡄⠀⠀⠀⠀⢰⣶⣶⣶⢶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣦⣠⣶⣶⣶⡄⠀⠀⠀⠀⠀⠻⣿⣿⣯⣷⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠘⠿⠿⠀⢸⣿⡇⠀⠿⠿⠃⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠙⠃⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣾⣿⣯
⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠋⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠉⠉⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡙⠛⠛⠛⠛⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⢋⡁
⠀⠀⠀⠀⠀⠀⠀⢀⡾⠻⣄⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠟⢷⡀
⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠙⣦⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣴⠋⠀⠀⠻⣄
⠀⠀⠀⠀⠀⣰⣯⣤⣤⣤⣤⣬⣷⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣾⣥⣤⣤⣤⣤⣽⣦
⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⣠⡟⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⣄⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠈⠛⠶⠶⠚⠋⠀⠀⢸⡇⠀⠀⠀⢰⡆⠀⠀⠀⢸⡇⠀⠀⠙⠓⠶⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣀⣤⠞⠳⣤⣀⣤⠟
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠊⡡⠤⠁⠉⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⠀⠀⠀⠀⠀⠀⢠⡃⠀⠐⢂⠂⠉⡂⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠐⠂⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠈⡆⢁⠇⠈⡀⠀⠉⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠁⠉⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⢠⠠⢂⠔⠁⠀⠀⠀⠀⠀⠉⠉⠛⠚⠂⠙⠒⢤⣤⣌⡦⠤⠤⠤⠄⠀⠀⣀⠀⠀⠀⠀⠑⠢⠄⣀⠠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣴⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁


⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠉⠉⠓⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠀⠒⠄⠂⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠖⠲⠲⡖⢤⡀
⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠈⠌⠆⠀⠀⠀⠀⠀⠀⠀⡄⠀⡌⠀⠰⠀⣈⣶⠛⣆
⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⡂⢹⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠀⢸⠉⠉⢹⡉⠀⠸⡄⢸
⠀⠀⠀⠀⠀⠸⡧⡂⠀⠀⠀⠀⠀⠀⢀⡏⠠⠀⠀⠀⠀⠀⠀⠧⠈⠁⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⠂⠁⢸⠁⠉⠒⣧⡸
⠀⠀⠀⠀⠀⠀⠱⣌⠢⠤⢀⡀⢤⣶⠭⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠳⠦⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣆⠀⠈⡀⠀⣸⠗⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠒⠒⠒⠉⠀⠀⠘⠢⠵⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠓⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠐⠒⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⡠⠄⠀⠀⠘⠆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⠀⠀⠒⠀⠀⡀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠑⠀⢠⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈⠙⠟⠋⠁⠀⠀⠀⠀⠀⠈⠙⡛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣑⣤⣤⠔⢁⠀⠀⠀⠀⠀⠀⠀⠀⠸⠐⠀⡂⠘⠈⠀⠁⢛⠀⠀⠀⠀⠀⠀⣤⠤⠬⡆⠀⠀⠀⠀⡀⠀⠀⢠⠥⠤⢦
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠀⢈⣀⡀⠀⡰⠃⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⡀⣸⠀⣇⢀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠒⠒⠒⠒⣣⠜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡤⢀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⢛⣀⣀⣀⠛⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡠⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠿⠿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢺⣿⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣶⡿⠿⠿⠛⠛⠉⠁⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠙⢿⣷⣤⣶⣶⡿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣼⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⡟⠁⠀⠀⢀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣦⠀⠀⠐⣿⡟⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⠿⠿⠛⠁⠀⠀⣼⣿⣿⡄⠀⠀⢀⣀⣠⣤⣴⣶⣿⠿⠿⠛⢛⣿⡟⠀⠀⠐⣿⣇
⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣶⣿⠿⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠀⠀⢀⣿⡟⠛⠉⠉⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠘⣿⣇
⠀⣠⣶⣿⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣿⣿⣦⣄⠀⠀⠀⢀⣤⣾⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⣄⠀⠀⠘⣿⣆
⠀⢻⣿⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⡿⠿⠛⠛⠉⠉⠀⠀⠀⢹⣿⠁⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⣾⣿⣄⠀⠀⠘⣿⣆
⠀⠘⣿⣧⣴⣶⣾⡗⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⢀⣾⣿⣿⣿⣆⠀⠀⠘⣿⣆
⠀⠀⠈⠉⠉⣰⣿⠇⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⠀⠀⠀⢸⣿⣶⣶⣄⠀⠀⠀⢀⣾⡟⠀⠀⢀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠘⣿⣆
⠀⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠈⢻⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⣰⣿⠇⠀⠀⢠⣦⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⠀⠀⠀⢠⣤⣤⣼⣿⠀⠀⠀⣿⣧⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⢠⣿
⠀⠀⣰⣿⠇⠀⠀⢠⣿⣿⣧⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⠙⠛⢻⣿⠀⠀⠀⢸⣿⠛⠛⠁⠀⠀⠀⠸⣿⣆⠀⠀⠈⠻⣿⣿⣿⣿⡿⠋⠀⠀⢀⣾⡿
⠀⣠⣿⠇⠀⠀⢠⣿⣿⣿⣿⣧⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠉⠉⠁⠀⠀⠀⣠⣾⡟⠁
⣰⣿⠋⠀⠀⠠⠿⠿⠿⠿⠿⠿⠧⠀⠀⠈⢿⣷⡀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣤⣤⣀⣀⣠⣤⣴⣿⠟⠋
⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⢀⣤⣤⣼⣿⠀⠀⠀⣸⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠋⠉
⢻⣷⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⠆⠀⠀⢸⣿⠃⠀⠀⢸⣿⠉⠉⠉⠀⠀⠀⠈⠉⠉⢹⣿
⠈⢿⣧⡀⠀⠀⠙⠻⠿⠿⠿⠛⠁⠀⠀⣠⣿⠏⠀⠀⠀⢸⣿⠀⠀⠀⣀⣀⣀⡀⠀⠀⢸⣿
⠀⠈⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⣼⡿⠋⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠈⠛⠿⣿⣶⣶⣶⣶⣾⡿⠟⠋⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣥⣤⣤⣄⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀
⠀⠀⠀⢠⣽⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⠔⠒⠢⡀⠀⠀⠀⠾⠿⠿⠿⠿⠧
⠀⠀⠀⠘⢿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⡿⣿⠁⠀⠀⢄⢀⣠⠃⠀⣀⠀⠀⢠⣴⣐⣂⣠⠞
⠀⢠⣴⡂⠐⡞⠀⣶⣤⠀⠀⠀⠀⠀⠀⠀⠛⢷⣿⢯⡀⠀⠀⢠⠁⠀⠀⠀⣿⣿⠿⠿⠟⠛⠉⠁
⠀⠀⠹⣧⡀⣇⢀⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⣖⣗⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣷⣦⣦⣶⢶⣶⣵⡄
⠀⠀⠀⠘⠺⡿⠟⠀⠀⠀⣀⡀⢀⡀⡀⣀⡀⢠⡄⣤⠀⡄⣤⠤⢀⡤⠀⣤⠄⠀⠛⠻⡿⣻⣿⣿⣿⣙⠋
⠀⢀⣄⣀⣴⣶⣭⠔⠁⠀⡷⣫⠘⡇⠇⠉⡶⠸⠇⠇⠱⠇⠼⠴⠤⠼⠇⠤⠗⠀⠀⠀⠀⠙⠛⠉⠉⡀
⠀⠀⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣀⣀⢀⡄⣠⢠⡄⣠⢤⢠⡤
⠀⠀⠀⡀⠀⠀⡠⠄⠀⠀⠀⠀⢨⣇⡄⢠⡇⠀⠇⠻⠠⠇⠣⠴⠠⠼⠂⠀⠀⠀⠀⠀⠐⠇
⠀⢀⡀⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢸⣦
⠀⢸⣆⣦⡇⠀⣀⠀⢀⠄⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀⡀⡜⠩⠼⠟⡆⣀
⠀⡞⢻⢻⡇⠀⠀⠀⢀⡠⠄⠀⠀⡀⠀⢿⡿⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⣈⣁⠁⢠⣤⠄⣡⠉⣤
⠀⠁⣾⠙⣇⠀⠀⠀⢸⠀⠀⡠⠂⡂⠀⢸⠃⠀⠀⠀⠀⠀⠀⢾⠿⠀⠀⠀⢸⡏⢸⣧⠀⠁⢀⡟⣦⡟⠆
⠀⠀⣿⠀⢿⠀⠀⠀⠈⢀⠈⠀⡀⠂⢀⣸⣐⣀⠀⠀⠀⠀⢀⣠⣬⣤⠀⠀⡇⢐⣾⠉⡄⠀⠘⢰⣿⣷⠈
⠀⠀⠃⠀⠘⠂⠀⠀⡄⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠃⠀⠈⠉⠉⠀⠀⠀⠀⢹⢹⡄⠀⠀⠀⠀⡆⡦
⠀⠀⠀⠀⠀⠀⢀⣼⡧⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⡇⢸
⠀⠀⠀⢰⣿⠀⣿⣯⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠃⠈
⠀⠀⠀⠘⠛⠀⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⡔⠒⡄⠉⠢⡀
⠀⠀⠀⠀⠀⠠⣤⠤⠤⠤⢤⣷⠤⠤⠤⢤⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⢱⣸⠀⠀⠀⢡⠀⠀⠀⠀⢰⠚⠳⡄⠀⠀⠀⢰⠚⠳⡄
⠀⠀⠀⠀⠀⠠⠁⡀⠀⠀⠀⣿⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠔⠁⠀⠀⢀⣥⠀⠀⠀⢸⠀⠀⠀⠀⠈⠶⠞⢁⠀⠀⢀⡈⠳⠞⢁⡀
⠀⠀⠀⠀⠀⠂⠀⠰⠀⠀⠀⣿⠀⠀⠀⠀⠀⢂⠀⠀⠀⠈⢢⠀⠀⠀⠈⠋⠀⠀⡠⠃⠀⠀⠀⢀⠀⠀⠀⡀⢂⠀⡘⣠⠀⠀⢀⡡
⠀⠀⠀⠀⡴⠀⠀⠀⢦⠀⠀⣿⠀⢠⡖⠀⠀⠐⡆⠀⠀⠀⠰⣄⢀⡀⠄⠀⠀⡏⠀⠀⠀⠀⢘⠘⠀⠀⠀⠛⢌⢠⡵⡻⢀⠀⢈⢳⡆
⠀⠀⠀⠀⠈⠲⠶⠒⠁⢀⣠⣿⣀⣀⡙⠲⠖⠊⠀⠀⠀⠀⢀⠀⠸⠆⠀⠀⠤⢇⠀⠀⠀⠀⠈⠀⡀⠸⠀⠀⠀⠀⠀⡇⠸⡇⢠
⠀⠀⠀⠀⠀⠀⢄⣐⣃⣠⣤⣤⣤⣤⣬⣦⣠⠀⠀⠀⠀⠐⠁⠶⠖⠦⠶⠖⠐⠲⠃⠀⠀⠀⠀⠀⡇⡄⢀⠀⠀⠀⠀⠃⠇⠀⢸
⠀⠀⠀⠀⠀⢀⢀⠀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠙⠁⠈⠛⣀⡀⢀⣛⣀⠘⠛
⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠁⠁⠁⠀⠀⢤⣶⣶⡄⠀⠀⣀⣀⣠⣄⡀⢀⡀⣀⢀⣀⣄⣀⣠⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣟⣹⡆⣿⣤⠟⣙⡷⢸⡇⣾⣧⡏⣿⣓⠈⣛⡆⢙⣳⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⢰⣾⡛⢢⣭⡤⣬⠀⡅⢠⠀⣩⡭⣀⣭⠤⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣭⣥⠀⡞⠀⡿⢶⡇⢾⠸⣯⣄⢈⡽⠂
⠀⠀⠀⠀⠀⡀⡀⢀⡀⡀⡀⣀⣀⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠈⠀⢀⣀⠀⢀⡉⠑⠢
⠀⠀⠀⠀⠀⠉⠁⠈⠁⢉⣉⣈⡈⠁⠁⠀⠀⠀⠠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⡤⠀⠀⠀⠀⠄⠈⠈⠈⠉⠉⢀⡁⠠
⠀⠀⠀⠀⢀⡀⣀⣀⠔⣉⢁⡀⠹⠓⣄⣠⡄⠀⠀⠙⣆⠀⠀⠡⢀⠀⠀⢀⠼⠉⠀⢀⡜⠁⠀⠀⠀⣀⠤⠔⠒⠒⠤⣄⡀
⠀⠀⠀⠀⠈⢡⠁⠆⠀⠄⠀⠄⠀⠆⢸⡇⢠⠀⠀⠀⠘⠦⠦⣀⠀⠁⠆⠀⢀⡠⠢⠊⠀⠀⠀⡠⢋⠤⠂⠉⠉⠉⠐⠢⡙⣆⠉
⠀⠀⠀⠀⠀⡎⠶⢾⣄⢠⠀⠆⠠⣆⣼⠗⠎⠄⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠁⠀⠀⠀⠀⠀⢰⣁⠀⠰⠁⠐⠤⠊⠐⡄⣅⠘⡄
⠀⠀⠀⠀⠀⡀⠀⠈⣄⠉⠓⠒⠚⢁⡌⠀⠀⡰⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠒⣄⠈⢄⠀⠀⠀⡠⠁⠈⣒
⠀⠀⠀⠀⠀⠑⠀⠀⠈⡆⠀⠀⠀⡞⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⡀⡀⠀⠀⠀⠀⠀⠈⡆⢨⠀⠈⠲⢀⠜⠀⠀⡐⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠳⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠘⢦⡑⠢⠄⠀⠠⠐⣈⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠠⣤⣤⠤⢠⣤⠀⣤⡀⢤⣤⠄⣠⡴⢤⡄⣤⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠒⠀⠀⣿⠀⢸⣿⠒⣿⠇⢸⣗⠀⣿⡀⠀⠀⠛⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠀⠀⠛⠀⠈⠋⠀⠙⠀⠛⠛⠃⠈⠛⠛⠁⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠁⠀⠀⠀⠀⠀⠘⢭⠂⠀⠀⠀⠀⠀⠰⠁⠄⠠⠀⠠⠀⠀⠀⠀⠀⠀⣐⣛⢉⢉⣳⠀⠀⠀⠀⠀⠀⣈⣀⡴
⠀⠀⠀⠀⠀⠀⠀⠜⢯⠀⠀⠀⠀⠀⠀⠀⠀⠤⠈⡄⠁⢤⠂⠀⠀⠀⢀⠀⢆⠥⢧⠂⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠠⠀⡀⠉
⠀⠀⠀⠀⠀⠐⡇⠀⠀⢉⠂⠀⠀⠀⠀⠀⠀⣄⡘⠈⠲⣀⡄⠀⠀⠀⠈⠀⠀⡁⢀⣴⡟⠀⠀⠀⠀⢀⡠⠀⠁⣤⠊⠙⠀⠀⠠⡍⢵⠀⠑⠆
⠀⠀⠀⠀⠀⠀⠐⠢⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠚⠁⠀⠀⠀⠀⠀⠀⠀⠠⠝⠁⠀⠀⠀⠀⠀⠁⠀⠒⠀⠀⠀⠀⠀⠀⠀⠣⠵⠤⠒⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠘⠃⣤⠀⢂⠀⠀⠀⠀⠀⠀⠶⢩⠽⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⡤⢤⡀⠀⠀⠀⠀⢀⠀⢾⡚⠉⢒⡷⠠⡀⠀⠀⠀⠘⣉⠸⢀⣝⠀⠀⠀⠀⠀⠀⠀⠰⠈⣉⠁⠀⠀⠀⠀⠀⢀⢄⠀⠁
⠀⠀⠀⠀⠀⢠⠀⠠⣀⡈⢠⠀⠀⠀⠀⠀⠀⢾⠣⣄⡤⣷⠒⠁⠀⠀⠀⢀⠈⠉⢉⡠⠒⠂⠀⠀⠀⠠⠤⠄⠀⢨⠀⠀⠀⠀⠀⠀⣠⣀⡃⠐⠂⠀⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠤⠌⠉⠁⠐⠀⠀⠀⠀⠄⠉⠠⠌⠌⠁⠀⠀⠀⠀⠐⠀⠀⠂⠀⠴⡄⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡂⢀⢀⠀⠀⠂⠀⠀⠀⠀⢀⣀⠒⠂⣀⡀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡢
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⢸⠀⣥⡞⠢⣄⠀⠀⠀⠀⠀⠀⠀⣤⡆⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⢀⠀⠀⠀⠨⠅⠴⡂⠂⠰⠕⡀
⠀⠀⠀⠀⠀⠠⠈⠀⢈⡃⠀⠀⠀⠀⠀⠀⢸⡀⠁⠨⠄⠀⠀⠀⠀⠀⠠⠤⠄⠀⡅⠠⢠⠀⠀⠀⠀⢀⡠⢌⡀⠸⠁⠀⠀⠀⠀⡇⢰⡄⠀⠀⡆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠩⠍⠁⠀⠀⠀⠀⠀⠸⠄⠀⠈⠈⠀⡄⠀⠀⠀⠀⠀⢀⡶⢆⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣘⠀⠀⠀⢁⡈⠀⠀⠀⢀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠁⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⡀⡀⠀⠀⠀⠀⠀⠀⢀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠔⠋⡈⠛⢥⠙⠣⠀⠀⠀⠀⠘⠀⠒⠆⠀⠀⠀⠀⠀⠀⢴⡆⠘⠟⠀⣶⠄⠀⠀⠀⠉⢉⡉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡉⢋⡀⠂⠀⠀⠀⠀⡇⢸⣇⠀⣸⠇⢸⡇⠀⠀⠀⢑⠀⠀⢀⣐⠁⠀⠀⠀⢠⣤⠀⠈⠉⠀⢠⣄⠀⠀⠀⠀⣥⣄⠩⢀⡄⠁⠁
⠀⠀⠀⠀⠀⢸⠠⠤⠰⠾⠤⠆⠀⠀⠀⠀⢰⡸⠧⣸⡰⠤⠚⠀⠀⠀⢯⠐⠀⡀⠁⡌⠉⡆⠀⠀⠀⠁⠐⠘⠛⠀⠂⠁⠀⠀⠀⡆⠀⠀⣀⡸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶⣦⠀⠠⠂⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠻⢿⡿⠛⠀⠀⠂⠀⠀⢀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠡⠉⠀⠰⢦⡰⠃⡀⠀⠀⠀⠀⠀⠀⣤⠀⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⢀⡀⠀⢁⠀⠴⡂⢰⠃⡀⠀⠔⠀⠄⠀⠀⠰⠀⠀⠠⡄⡂⠀⠠⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢰⠀⠀⠈⠁⠀⠙⠢⡑⠊⣿⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠋⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠘⠀⠘⠀⠀⠈⢳⣿⠀⠀⠀⢠⠄⠀⣀⠄⡀⠀⠷⠟⠀⠀⢂⡀⠀⠘⠑⢀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠂⠀⣴⡆⠀⠀⠈⠀⠀⠀⠘⠈⢿⡄⠀⠄⢹⣤⢚⣁⡀⢁⠀⠢⢰⡀⠀⢢⣰⣀⠀⠈⠉⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠀⠋⠉⠰⢉⠳⠀⠱⠦⢄⣀⣈⣧⠠⢁⡾⠁⠀⠙⠁⠜⢁⠀⠀⢀⣤⡔⠊⠁⣀⡀⠈⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠠⣄⠀⠀⠀⠄⠀⠀⠸⣆⣾⠀⠀⠀⠰⠀⢠⠞⢻⣴⡏⠀⠀⠀⢸⠭⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠠⠀⢢⠰⣄⣀⠇⠀⠀⠀⠀⡀⠀⠹⣇⠀⡀⠀⠙⠁⢨⠨⠸⢾⡇⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠑⠷⠤⠴⠶⢶⡶⠶⠿⣶⣤⣻⣦⠁⠐⠀⠀⠩⢛⠓⢾⡇⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠂⠔⠀⢀⠞⠁⢄⠀⠀⠀⡬⠙⢿⣷⡀⠀⠀⣐⡙⠁⢈⡇⢠⠀⠵⢀⠀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⠀⠀⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⡀⠰⣠⠇⠈⢉⡠⠃⠀⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠐⠕⠀⠄⢀⠂⠀⢁⠀⠀⠀⠀⠀⡀⠀⠀⠈⢿⣧⢋⢁⠁⢠⠿⢶⠶⠛⠁⠑⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠄⠀⠐⠃⢀⡀⠒⡀⠤⢢⣀⣀⣠⣄⣀⡛⢸⣿⣆⣡⡾⠃⢀⠄⠠⣎⣱⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠤⢐⠚⠉⠉⠉⠀⠈⠙⠻⢿⣿⡿⠋⠀⠐⠊⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣠⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠀⠛⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣰⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠈⠏⠑⢀⠰⣄⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⢀⣄⡰⠾⠂⠈⢠⠰⡴⠓⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠃⠀⠀⠀⠀⠁⠀⣤⠀⠉⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⠶⠶⠶⠿⠿⠿⠿⠶⠶⠒⠒⠂⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢖⠀⣤⠀⣥⣴⠀⢷⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢀⠙⣀⠀⠪⠠⡁⠄⢽⠗⣶⠊⠑⣟⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠃⠀⠉⠄⢬⠀⠀⠀⢤⡋⢘⡤⣿⡀⣀⣿⡜⠃⠠⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠞⠀⠰⠂⢠⡄⠆⣈⠀⠝⣿⡤⠘⣌⠓⢼⡀⠀⣶⣾⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠂⠉⠀⢖⠀⠀⠀⢬⡀⢼⣷⣰⠧⢿⢸⡆⢻⣀⠉⣸⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠐⠜⡳⢀⡀⠁⣀⠉⠁⠉⠻⣾⣿⠫⣄⠛⠔⢣⠸⢾⢤⠀⢸⡇⣖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡀⢰⠻⠆⢋⡦⢡⠈⠉⠠⠜⠀⠀⢹⣿⡅⠘⣦⡄⣽⠀⠚⠀⠓⠈⠁⠋⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⣀⢴⢛⠆⠀⠈⠁⠼⠪⠄⠒⢠⠸⣶⣷⠎⠹⠇⣾⠀⢁⡼⠉⢧⠹⡄⡁⢹⣞
⠀⠀⠀⠀⠀⠀⠀⢀⢀⡐⡃⢠⣠⡈⠙⢰⡌⠘⠂⢃⡀⠙⢀⢸⣿⡿⠀⣠⢆⠀⣲⡞⣀⣀⠸⡆⡿⡗⠃⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠄⡈⠛⡁⠰⠈⣤⡁⢀⡒⠀⠀⣀⢸⣿⣷⣿⣷⢻⡀⣹⠀⢸⣇⣽⣓⠲⠧⡤⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠺⢦⠚⡰⠚⠁⠒⡈⡀⠩⠠⠀⠙⢿⣿⣿⣯⠀⢷⣛⣿⠠⣸⠇⠳⢤⣖⣲⡇⠸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠩⠀⠙⠃⠄⠗⠀⠀⠸⣿⣿⡄⠀⠐⠜⠻⠴⠛⣾⣉⡉⠓⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠖⠃⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢿⣉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢃⡇
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣠⡶⣿⣿⣿⣧⣿⣿⣿⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠛⠿⣿⡿⠛⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⢶⣿⡶⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣉⣉⠤⣉⣉⣛⠻⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣤⣤⣤⣤⠄⠀⠀⢰⣿⡇⠀⠀⠠⢠⣤⣤⣤⠄⠀⠀⠀⣿⣿⣿⣿⠛⠛⠛⠛⣿⣿⣿⡏⠀⢸⣿⣿⣿⡟⠛⠛⠛⣻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⣶⣿⣶⣶⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠏⠉⠀⠉⠉⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣠⣴⣶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠘⣿⣿⠃⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣦⣄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⠀⠉⠛⠿⢿⣿⣷⣶⣶⣶⣶⣿⣿⣶⣶⣶⣶⣾⣿⡿⠿⠛⠉⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⣿⣿⣿⣿⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⢀⡟⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠀⢻⡀
⠀⠀⠀⠀⠀⣼⠁⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠈⣧
⠀⠀⠀⠀⣸⠇⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠸⣇
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠈⠆
⠀⢀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⡀
⠀⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠰⠿⠿⠿⠿⠆⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠻⠿⣶⣶⣶⣶⣶⣶⣶⣶⠿⠟⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠀⠀⠀⠀⠈⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠀⠒⢐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠂
⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⡠⠗⠁⠀⠀⠀⠀⠀⠀⠄⠀⠸⠘⡜⡆
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡆⠦⢸⣇
⠀⠀⠀⠀⠀⢼⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⢀⠀⠂⢀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⢢⢈⣽⡿
⠀⠀⠀⠀⢠⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠤⠇⠀⡤⡆⠁⠄⠠⠀⠀⠠⠐⠀⠀⠀⠐⠀⠀⠀⠐⡂⢸⠈⡈⣯⡇
⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠂⠀⠀⣚⣉⣍⣭⡅⠀⠀⠀⠡⠀⠠⠀⠀⠠⠀⡀⠀⠀⠀⠀⠂⢸⡏⡿⣾⡂
⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠨⠀⠀⠀⠀⢥⡤⣤⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄⠀⠀⠀⠀⣆⢷⢘⣿⠁
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⡃⣨⢸⡏
⠀⠀⠀⠀⢸⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠺⡅⣧⡇⠁
⠀⠀⠀⠀⢸⠘⢣⣖⣒⣒⣢⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⢄⣄⣔⡈⡇⣷⠃
⠀⠀⠀⠀⠀⢇⣉⠉⠉⠉⠉⠙⠉⠛⠛⠻⠿⣶⣦⣤⣄⣀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⠿⠿⠿⠛⣛⡃⡛⠉⠘⠋⠘⢑⣱⣿
⠀⠀⠀⠀⠀⠀⢀⣈⠙⠩⠮⠁⠀⠀⠀⠀⠀⠀⠒⣊⡹⠿⡿⣶⣶⣾⢟⡿⢝⡱⠶⠀⠒⠪⠍⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡟⡀
⠀⠀⠀⠀⠀⠐⠛⠒⠉⠲⠴⠶⠦⠤⠤⠤⠶⠖⠒⠲⠾⠷⢴⣿⣹⣽⡾⠿⠛⠒⠛⠓⠒⠂⠂⠄⠰⠰⠦⠶⠾⠛⠛⠻⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢦⡄⠀⠀⠀⠀⠀⡞⠻⣽⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡁⣶⣗⠀⠀⠀⠀⠀⢲⣿⠆⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡰⠋⠁⠀⠀⠉⣛⡼⢢⣀⠀⠩⣿⡄⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠐⢷⠙⢿⣯⡷⠶⣦⡁⢹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢒⣶⠶⠖⠉⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⡾⠿⢻⣢⡇⢸⡟⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠸⠀⠀⠀⠀⠀⠀⡇⠀⢼⣴⣋⣽⣟⠮⠇⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠃⠀⢠⣾⢎⣱⠆⢿⣦⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠷⠦⠤⠖⠀⠀⠶⣤⣤⣴⠄⣿⠸⠞⡋⣽⣸⢷⡄⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣼⡶⠟⣩⡥⢨⠃⠀⠀⠈⣧⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨⡧⢹⡟⢿⣇⢈⠀⠀⠀⠀⢹⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡹⢷⡎⢳⣸⡏⣼⠁⠀⢤⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠻⡏⢺⣿⢸⠃⣶⢻⣤⡀⢀⣿⠟⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡟⣎⢻⣎⡷⠁⣯⢸⣹⡷⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⣷⣿⢀⡿⡁⡄⣿⣤⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣷⢻⡿⢷⣄⢷⠙⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⣸⣿⣼⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢈⣡⣿⠏⣿⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣻⠛⠋⣻⠏⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠶⣾⣿⣷⣾⣿⣏⢠⣿⣟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣾⢻⡿⣰⣿⣏⣸⣿⡟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠉⠉⠈⠁

⡿⢿⡿⠿⡿⠟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣛⣛⣛⣿⣟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⠿⠿⠟⠿⠿⢷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⣤⣄⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣤⡄
⠀⠀⠀⠀⠀⣿⠉⠉⠉⠉⠛⠛⠛⠻⠿⠶⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡶⠾⠿⠛⠛⠛⠋⠉⠉⠉⢹⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⣶⠀⢰⠏⠁⠉⣧⠀⢰⡖⠀⢠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⢹⡇⢸⡄⠀⠀⣼⠀⣾⠇⢀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠈⣿⠛⠙⠒⠚⠉⢻⡿⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢹⣇⠀⠀⠀⠀⣾⠃⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣀⣈⣿⣀⣀⣀⣰⣟⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢛⣛⣛⣛⣛⣛⣛⣛⣛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣭⣭⣭⣭⣭⣭⣭⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⢶⡶⢶⠶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⠛⣿⡟⠿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠛⠛⠻⠿⠿⠶⠶⢶⣦⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⠶⠶⠶⠿⠿⠛⠛⢻⡇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢶⣤⣀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢀⣠⣴⠾⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠿⠶⣶⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣄⠀⣿⡇⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣴⡶⠾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠿⠶⢶⣦⣤⣤⣄⣀⡈⠻⣷⣿⣷⡿⠋⣀⣀⣤⣤⣤⡶⠶⠾⠟⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠟⠛⠛⠉⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠶⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠄⠀⠘⢳⠊⠀⠀⠀⠀⣴⠛⡆⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠉⡧⠀⠀⠀⠀⠀⢫⠂⠀⢨⠀⠀⠀⢸⠀⠀⢰⡆⠀⠈⡟⠁⠀⢱⠁⠀⢀⡴⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⢳⠀⠀⢸⠀⠀⢐⠀⠀⠀⣇⠀⢰⠁⠀⠀⠀⠳⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⣾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡆⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⢤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣴⠟⠛⠛⢷⡄⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠉⠉⠉⣿⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⢸⡗⠒⠒⣁
⠀⠀⠀⠀⠀⠀⡴⠲⣄⣀⣀⣀⣀⣿⠀⠀⠀⠀⠀⣤⣼⣿⣤⣤⣤⣼⣷⣤⡄⠀⠀⠀⠀⢸⣏⣉⣉⣛⣀⡴⠲⡄
⠀⠀⠀⠀⠀⠀⠓⠚⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⣴⡄⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⡙⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⡤⠀⠀⣿⠀⠀⠀⠀⠀⣿⠀⠀⠀⢹⠁⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠴⠋⠉⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢿⣤⣤⣤⣤⣤⣤⣤⣼⠇⠀⠀⠀⠀⢸⡗⠒⠒⠪⠂
⠀⠀⠀⠀⠀⠀⠀⣤⣀⣠⠔⠉⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠻⠶⢶⡶⠶⣶⠶⠶⢶⡶⢦⠶⢶⡶⠶⣶⠶⢶⡶⠦⠼⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⠇⠀⢸⠀⠀⢸⠀⢸⠀⢸⡄⠀⣷⠀⠘⢄⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⣹⠦⠤⠤⠴⠃⠀⠀⠿⠂⢰⠁⠀⢸⠀⠈⠁⠀⣿⠀⠀⠀⡇⠀⠀⠀⠸⣅⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠇⢠⠞⢦⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠋


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

⠀⠀⠀⠀⠀⠀⠀⠰⠶⠀⠶⠆⠰⠶⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠰⠶⠀⠶⠆⠠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢖⡲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⠀⣠⢤⡀⠀⠀⢸⢸⠉⠉⡆⡇⠀⠀⢠⠖⣆⠀⠀⠀⠀⡴⢦⠀⠀⠀⣾⡇⠀⠈⣿⠀⠀⠀⢰⢲⡄
⠀⠀⠀⠀⠘⠚⠧⣄⢰⠛⠛⢒⡒⠒⠓⡆⠤⠞⠉⠁⠀⠀⠀⠀⠙⠛⠦⡄⢰⣿⣷⡶⣾⣿⣶⠠⠤⠋⠉
⠀⠀⠀⠰⣉⡶⠶⠦⢸⠀⠀⢯⡝⠀⠀⡇⠶⠦⢾⣉⠆⠀⠀⢾⣹⠶⠶⠆⢸⣿⣯⡚⣼⣿⣿⠀⠒⠲⣏⡹
⠀⠀⠀⠀⢀⣄⡔⠒⠸⢤⣤⣬⣤⣤⡴⠃⠒⢦⣀⡀⠀⠀⠀⠀⣠⣤⠖⠃⠸⠿⠿⠿⠿⠿⠟⠐⠒⢄⣀
⠀⠀⠀⠀⠙⠜⠁⠀⠀⢀⡆⢰⠀⢲⡀⠀⠀⠘⠧⠇⠀⠀⠀⠀⠳⠞⠀⠀⠀⣰⠂⢲⠀⣇⠀⠀⠀⠸⠴⠃
⠀⠀⠀⠀⠀⠀⠀⢰⣉⡏⠀⡼⢦⠀⢹⣙⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣹⠁⢠⠞⡄⠈⣏⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⡀⣠⡀⡄⣠⡀⢀⠄⡤⢠⠄⢀⣠⡄⡠⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠃⠚⠂⠃⠋⠂⠀⠃⠋⠐⠂⠚⠈⠃⠃⠃⠈

⡟⠉⣿⣻⣛⣛⣻⣟⣟⣛⣻⡛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣻

⠀⠀⠀⠀⠀⠀⢀⣴⡶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠘⡷⠶⠶⠟⠀⠀⠀⢴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠛⠛⠻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⠃⠀⠀⢸⡇
⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⣠⣶⡄⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⣦⣤⡀⢾⣧⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣴⠾⠷⠤⣄⣀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣨⣿⣁
⠀⠀⠀⢀⣿⣿⢻⣿⣿⣿⣿⠻⣿⣿⠇⠀⠀⠀⠈⠃⠀⠀⠀⢸⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⠉⠉⠓⠦
⠀⠀⠀⢸⣿⠃⢈⣟⣛⣛⣏⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⢻⣿⣿⣿⡇⠻⠿⠃
⠀⠀⠀⠘⠛⠀⣼⣿⣿⠛⠛⠂⠀⠀⠀⠀⠀⠀⣤⣤⡄⠀⠀⣿⡇⢀⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣴⠀⠀⣿⣿⣿⣷⣶⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⢻⡇⢸⣿⣦⣤⣴⣿⡄⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠈⠀⢠⣿⢿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠐⢿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⢛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢷⡈⠻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡄⠙⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠘⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣌⢌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣦
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠂⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠦⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⠒⠒⠚⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣮⣉⠉⠁⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠋⠁⠀⢣⠈⠙⠻⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⠀⠀⠈⢧⠀⠀⠘⢿⣿⡄⠀⠀⢀⣀⠤⠒⠒⠉⠉⢱⡄⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⠀⣠⡔⠒⠒⠒⠒⠒⠒⢺⣿⣷⢶⠊⠉⠀⠀⠀⠀⠀⠀⢸⡽⣤⡄⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⡟⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡯⣿⡇⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠖⠉⠙⢷⡄⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⡿⠁⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡠⠓⢄⠀⠀⠙⢶⣿⣇⣀⣀⣀⣤⣾⣿⠟⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢸
⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠑⢄⠀⠀⣹⡿⠿⠿⠿⠟⠋⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣆⣀⣀⣸
⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢀⠜⠓⠊⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣿⡷⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁
⠀⠀⡎⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⣸⡿⣿⡇
⠀⠀⠈⠢⡀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣀⣤⠶⠯⠓⠒⠒⠚⠒⣿⡇
⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠷⠶⠶⠶⠶⡖⠒⢲⠶⣶⣾⣿⣷⠶⠶⠶⠶⠶⠶⠶⠶⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠒⠾⠀⠉⠉⠉⠁

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⣞⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠈⠁⠀⠛⣀⣠⡤
⠀⠀⠀⠀⠀⠉⠍⠀⠄⡀⠀⢀⡀⠀⠀⠀⠠⠌⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢊⣉⠆⠀⠀⢉⣤⠖⠉⠛⠛
⠀⠀⠀⠀⠀⠒⣂⠛⣦⣮⣀⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠀⢀⠀⠢⠐⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠘⠻⢛⣾⣿⣯⣠⣫⡕⠂⡄⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢤⠃⠀⢣⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠠⠞⢡⡄⠀⢄⣀⡀⣀⠑
⠀⠀⠀⠀⠀⠀⠀⢫⣿⣿⡾⠛⢻⣏⠠⡀⠀⠔⠀⠀⠀⠀⠀⠐⢀⢀⡀⠀⠀⠀⠀⠁⡴⣦⠀⠀⠀⠀⠀⠀⠀⣀⡶⠉⡟⠧⠎⠙⠉
⠀⠀⠀⠀⠀⠀⠈⠁⣿⣠⣘⣟⡳⠏⠰⠄⠀⠔⣣⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠰⠷⡄⣀⠀⢤
⠀⠀⠀⠀⠀⠠⠀⠀⢈⣹⣦⣤⣿⡤⠀⠀⠄⠸⠁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠘⠀⠀⠀⠀⠀⠀⢂⣠⣴⡆⠀⠴⠷⠖⠒⠁
⠀⠀⠀⠀⠀⠰⠀⠀⠀⠹⢟⣳⣘⠻⠎⠀⠲⣄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠐⠉⠁⠀⠉⢀
⠀⠀⠀⠀⠀⠂⠀⠀⠰⠀⠈⠛⣛⣙⣃⠀⠰⢿⡎⠦⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⠀⠀⠀⠐⣉⠛⡉⠹⠂⠀⣸⣈⣀⡄⢠⠀⠀⡀⠀⠤
⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠙⣯⣽⠀⠀⡿⣷⡀⢂⠀⠀⠀⡀⠀⠀⠀⠿⠐⠧⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⡼⠃⠐
⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠀⠀⠀⢹⡆⠀⠐⠋⠁⠈⠀⠀⠘⡅⠀⠀⠀⡀⠀⡀⠀⡄⠀⢀⡀⢀⣀⠠⠤⠄⠒⣀⠀⡌⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠙⢦⣤⣤⠤⠤⠤⠒⠀⠀⠀⠠⠀⠓⣈⡁⠴⡿⠀⠈⠉⠫⡉⠁⠂⠠⣤⡴⠶⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⡠⠀⠀⠀⠀⠀⠘⠲⠆⠀⠀⢀⣀⠀⠊⢀⡜⠋⢠⠙⠋⠀⠦⡄⢀⡀⠉⠀⢀⠀⡀⡀⠤⠀⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⣀⠀⠀⠀⠀⠀⣀⠤⠒⠉⠁⢈⠀⠢⢉⡙⠏⡯⢉⣐⢀⠘⠄⠙⠒⢤⡀⠈⠁⢀⡀⠄⠂⠀⠀⢀⠀⠀⠁⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠄⠤⠊⠀⠀⠀⠀⠀⢀⠀⠀⢰⡏⣳⣆⣸⣿⠄⠀⠐⠄⠀⠀⠉⠀⠛⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⣉⣩⣠⣿⢀⠀⢀⠄⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠈⠀⠈⢀⡤⠀⠀⠀⠀⠐⢻⡿⠥⢄⣈⠛⠭⠉⠥⠀⢰⠎⠀⠀⠠⣤⡀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢠⠾⣷⢶⠶⠤⠛⡦⠰⠖⠀⠘⠂⠀⠀⠀⠀⠈⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠘⠦⠀⢀⡀⠐⢁⣠⠶⡄⠀⠐⠒⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠠⠞⠉⠉⠉⠁⠀⡔⣴⡄⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⣿⠉⣷⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⢠⡄⠀⠀⠀⠀⠀⠛⢠⣼⠋⢲⣄⡰⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡼⠀⠀⠀⢀⡀⠀⠀⢘⡃⠀⠀⢿⣧⢈⡀⠀⠀⠐⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢈⠁⠀⠀⢸⣏⠀⠀⢀⠀⠈⢣⠀⠀⠀⠂⠀⠀⠀⠀⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⣾⠀⠀⠀⢨⡻⠄⠀⠐⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⠀⠈⠀⠀⡿⠀⠀⠀⠘⣵⠆⠀⠀⣿⠀⠘⠠⠀⠀⢀⠊
⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠤⠆⠀⠦⠆⠴⠒⠠⠿⠀⡜⠀⠀⠀⠀⣇⠀⠀⠀⠀⢹⠆⠀⠀⢹⠄⠀⠴⠆⠛⠷⠀⠤⠀⠐⠠⠂⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠇⠀⠂⠀⢀⣬⠀⠀⠀⡀⣿⠇⠀⠀⠘⠃⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠋⠀⠀⠉⢁⠙⠋⠹⠀⠀⠀⢁⠀⢠⡀⠉⠁⠀⠀⠀⠀⢠⠀⠂⠀⠀⢆
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠂⠀⠘⠀⠾⠀⡀⢀⡆⠘⠆⢨⠧⠀⢀⠀⠀⠰⢜⡧
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⡠⠀⠀⠠⠂⠁⠠⢀⠁⠜⠁⠀⢀⡁⠀⠈⠃
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣄⠀⠀⠁⠀⠀⠀⠀⠈⠁⠀⠀⠇⠀⠄⠀⠀⠆⠀⠋⠀⠈⠡⠀⠘⠉⠀⠀⠀⠀⠠⠀⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠠⠀⠈⠠⠊⠀⠀⠀⠀⠀⠀⠀⣁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⢈⡷⠀⠀⠰⣿⡆⠀⠀⠀⠀⠀⠒⠛⠊⢠⡀⠲⠶⢄⠀⠁⠲⠊⠑⢀⡀⠀⣀⠴⠾⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠆⠀⠀⠸⡅
⠀⠘⠂⠀⠀⠀⠀⠀⠀⠀⠀⡺⠃⠀⣼⡟⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠔⠂⠘⠈⡄⠉⠡⢅⠀⠄⠈⢁⣀⡀⠀⠀⢸⣧⡀⠀⠛⠄
⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣠⣤⠘⠛⢻⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠃⢤⣄⡀⠘
⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠀⢸⠁
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣗⣀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠘⣃⠘⠋⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣐⡈⢷⣤⣀⠀⠀⠀⠀⠀⠀⠹⣇⣀⣠⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⠟⠿⠉⠛⠷⣬⡿⣿⣷⣿⣦⡷⣶⣄⣈⣿⡁⢸⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠀⠀⠀⠀⠀⢤⠚⡟⠋⠙⠻⣷⡌⠉⠙⠋⠀⢫⣼⣿⡿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢠⣤⣽⣶⣤⡶⠊⠉⠁⠀⣀⡀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡷⣄⠀⣸⣿⣷⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⣻⣿⣿⣿⣿⡷⠈⠲⢿⡇⠙⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣦⡉⠁⠀⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠀⠀⢀⣿⣿⣼⣿⣿⡿⠯⢇⠀⠀⠀⠀⠹⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠆⠀⢀⠀⣸⠀⠀⠀⠀⠀⣸⣿⣿⣿⣟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠻⣧⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣀⣧⣀⣀⣸⣿⡿⢷⣿⣶⣶⠾⣷⠀⢻⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠻⣿⣿⠟⠉⣿⠃⠚⠻⠟⠁⠀⢿⡀⣼⡿⠙⣿⠹⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⢠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠈⠁⠀⠘⢿⣤⠀⠀⠀⠀⠀⠘⢷⡿⠁⠀⣿⠀⢻⣿⡆⠀⠀⠰⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⢀⣿⡇⠀⢻⣧⠀⠀⠀⠀⠀⠀⣦⣄⢀⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠉⠀⠲⣾⣿⣷⠀⠀⠻⣆⠀⡇⠀⠀⠀⠀⠙⢟⣉⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⡀⠘⣿⣿⣿⣧⠀⠀⠈⠲⣧⡀⠀⠀⠀⣠⣿⠉⠙⠳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣸⣿⣿⣿⣇⠀⠀⠀⠀⠈⠓⠄⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠹⣿⣿⣿⣿⣿⣿⢆⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢹⣿⣿⣿⣿⣿⡎⢷⡀⠀⠀⣤⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠸⢿⣿⣿⣿⣿⣷⠈⣿⡄⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⢹⣿⣿⣿⣿⡆⣿⣷⠟⣼⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢿⣿⣿⣿⣷⣽⡏⣼⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣻⣼⣿⣿⣿⣿⡟⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡿⢃⣼⢿⣿⣿⣿⣿⡟⠘⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠻⠟⠡⠾⠍⠿⣿⠟⣶⢹⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⠄⠤⠶⢒⣶⡟⣼⡿⣿⠏⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢃⢤⣴⢁⣿⢟⣼⣿⠃⠙⠲⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠌⠱⣾⣏⠚⠛⠋⠀⠀⠀⠀⠳⠾⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⠑⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⣶⣶⣶⣦⣴⣶⣶⣶⣤⣶⣶⣴⣦⣴⣶⣶⣶⣦⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣶⣤⣶⣶
⣿⣿⣿⡻⠛⠛⡿⠋⢛⠛⠛⡛⠟⣛⡿⢟⡟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣾⣿⣿⣾⣿⣶⣿⣿⣶⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢲⣄⡀⠀⢀⡀⠀⢻⠁⠀⠀⠀⣼⣉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢸⡇⠰⣏⡽⠀⢸⠀⠀⣠⠞⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇⠀⣸⣇⣀⣼⣀⡀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠆⠀⠀⠀⠀⣸⠗⠋⠁⣀⣀⣀⡀⠉⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣠⠞⠀⠀⢠⡾⠋⠉⠉⠻⣦⠀⠀⠈⢦⣀⣀⣠⡖⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠉⠉⢩⠇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢧⠀⠀⠙⠋⢀⣰⣋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⢯⣹⠶⠶⠶⠶⠶⣾⠀⠀⠀⢰⣿⣷⣶⣶⣶⣶⣿⣷⡄⠀⠀⢸⡟⠛⠛⠛⠉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣽⠀⠀⠀⢸⣿⣿⣿⡁⣹⣿⣿⣿⡇⠀⠀⢸⠧⠤⠤⠤⠤⠤⣞⣹⠄
⠀⠀⠀⠀⠀⠀⠀⣴⠓⡶⠋⠉⣀⣀⠀⠘⣆⠀⠀⢸⣿⣿⣿⣇⣿⣿⣿⣿⡆⠀⢀⣞⣀⣀⣀⣀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠧⠼⠛⠛⠛⢦⡀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠁⣠⠞⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢲⣤⣀⣀⠀⢀⣀⣠⠴⡞⠁⠀⠀⠀⠀⢰⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢹⠉⠉⣽⠀⠀⡗⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⢸⠀⢠⠞⣆⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢲⠋⠀⠀⢀⡾⢦⠈⠛⠁⠀⡇⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠋⠀⠀⠀⠀⠉⠹⠤⠏



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢀⣀⣀⠀⠀⣶⣿⣦⣀⣀⣀⣀⣀⣀⣶⣿⣦⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢿⣿⣿⠆⠀⠻⠿⠛⠀⠀⠀⠀⠀⠀⠻⠿⠏⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦⡞⠁⠀⣀⣀⡀⠀⠉⠉⠙⠦⣄⣀⣀⣀⣀⣀⣀⣀⣴⣶⣦⠀⠀⣼⣿⣿⡆⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⠀⠐⣿⣿⣿⠖⠲⠦⣄⠀⠀⠉⠉⠉⠉⠉⠉⠉⠻⠿⠟⠀⣠⠞⠛⠛⠁⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠈⢱⡆⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠞⠁⣰⣾⣷⡄⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⢀⣴⣶⡄⠀⠀⠀⠀⢰⣿⣿⡆⠀⠀⠀⢀⣴⣶⣇⠀⠀⠀⠀⠘⢿⠟⠁⣴⣶⣶⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠉⠀⠘⢿⡿⠿⣄⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠘⠿⡿⠏⠀⠀⠀⠀⢠⠞⠀⠀⠻⠿⠟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠓⠢⣀⣀⠀⠀⠀⠀⠀⠀⢰⠋⠀⠀⠀⠀⠀⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⠤⠤⠤⠤⠤⠤⠤⠤⢤⡀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⣼⣿⣷⡄⢀⣤⣤⡀⠈⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⠀⠉⠛⠉⠀⠀⠀⠀⠙⠛⠛⠀⢸⣿⣿⠇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢋⣤⣦⡄⠀⠀⢠⣶⣶⣄⢀⣀⣠⠞⠁⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠟⠀⠀⠘⠿⡿⠏⠉⠉⠁⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⢤⣄⣀⠀⠀⠀⢠⠞⠛⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⢀⣼⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢀⢤⣄⡧⣩⣴⣥⣍⣄⢁⠒⠀⢂⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⣤⢨⣐⡲⢦⢹⣿⡿⠧⠼⠶⠎⠾⠾⠿⠫⠶⠶⢂⣄⣤⣀⡀⠄
⠀⠀⠀⠀⠀⢀⣐⢶⡸⣜⠎⠓⣑⢴⠦⠀⠀⠀⢠⣤⣤⡄⠀⠠⢀⡈⠉⠉⠢⣦⣴⠆⣚⢁⣂⡀
⠀⠀⠀⠀⡈⢀⠡⠙⠃⠛⠛⠉⢀⠄⠀⠀⠀⠀⠚⠛⠛⠃⠀⠰⠊⡁⠾⠋⣗⠒⠒⣒⠮⠍⠁
⠀⠀⠀⢐⠨⠲⠥⠖⠴⠶⠶⠦⢁⣠⢖⢄⣤⣤⣤⣤⣤⣤⣤⣔⣦⣉⣀⡐⣙⣛⢉⣥⡓⢈⡃⢒⣒⡂
⠀⠀⠀⠀⢁⣴⠳⡎⣿⡡⡋⠛⠛⠪⠓⢉⠨⣻⣾⢛⠛⡹⡛⣟⣒⠒⣲⠭⢤⣔⢮⢲⡆⣷⡽⣎⡛⠟
⠀⠀⠀⠀⠈⠉⠑⠛⠣⢪⡪⣉⠋⠧⣒⡵⣪⠵⠟⣸⠻⡝⣦⣤⡦⣿⡦⢹⡏⣿⡜⣦⢹⣬⣁⠈⣛⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢇⡟⣾⡞⣵⠊⣵⡇⣿⡿⢃⣧⢙⡏⡗⢿⣁⣿⣇⣿⡇⣹⣧⣉⡭⢳⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⢻⡇⢻⣆⢹⣧⠘⠿⠸⠿⠿⣤⠃⣮⡙⠛⠉⠉⠁⠉⠉⠉⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠈⠺⢉⣘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠶⣶⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠈⠛⢿⣻⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣠⡴⣶⣦⠀⢸⣿⡏⢹⣿⣿⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⠆⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣶⡀⣾⡏⣿⣷⢸⣿⡇⣸⣿⡆⢸⣿⣷⢸⣿⣿⡄⠀⠀⠀⢀⣴⣿⣿⠟⣡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣭⡄⣿⣇⢿⣿⠘⣿⣷⢹⣿⡇⢸⣿⣿⣸⣿⣿⠇⠀⠀⣴⣿⣿⡿⣫⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⣿⡇⠹⣿⣾⠟⠀⠛⠿⠿⠛⠁⠘⠛⠛⠋⠉⠉⠀⣠⣾⣿⡿⢋⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠉⠀⠀⠀⠀⠤⠤⠤⠀⠶⢒⣒⣒⡒⠒⠒⠈⠩⠭⠭⠍⠉⠉⠒⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠉⠉⠉⠀⠐⠒⠒⠒⠒⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠀⠀⢸⣿⣿⣿⠿⣷⣶⡄⠀⢀⣶⣶⣶⣆⠀⢰⣶⡶⣤⣤⠀⡆
⠀⠀⠀⣠⣾⣿⣿⡿⠋⠀⠀⠀⠀⢸⣿⣿⣿⣤⣿⣿⠟⠀⢸⣿⣿⢿⣿⡄⢸⣿⣧⢸⣿⡆⠇
⠀⠱⢌⡻⢿⣿⣿⣷⣄⠀⠀⠀⠀⠘⣿⣿⣿⡍⣿⣿⣿⡄⣼⣿⣿⣼⣿⣷⠘⣿⣿⣸⣿⡇
⠀⠀⠀⠙⠢⣝⠿⣿⣿⣿⣦⡀⠀⠀⣿⣿⣿⣷⣿⠿⠿⠁⠿⠿⠟⠈⠛⠛⠂⠛⠛⠉⠉⠁⠸
⠀⠀⠀⠀⠀⠈⠓⢬⡛⢿⣿⣿⣶⣄⣀⣀⢀⠤⠤⠤⢤⣀⣐⣒⡒⠂⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠇⠀⢘⠀⡀⣀
⠀⠀⠀⠀⠀⢠⡴⣻⣷⡿⣷⣯⣳⣤⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠘⠀⢪⡮⠖⠦⣮⠂⠘
⠀⠀⠀⠀⠀⠘⣿⣿⡟⢛⠛⣿⣿⡿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠐⡁⣶⣿⣷⡌⡗
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣤⣤⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢇⢻⣿⡿⢡⠃
⠀⠀⠀⠀⠀⠀⠀⠻⣟⢿⢿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⢜⠷⢬⠴⢓⠄⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠉


⠀⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄⠀⠀⠀⢰⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠀⠀⠀⠀⣴⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲
⠀⠀⠀⠀⢸⠀⠀⠀⢐⣊⣓⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠠⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⢤⣤⣤⡤⣤⣤⣤⠄⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠿⠯⠿⠇⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠴⠶⠶⠶⠶⠄⠀⢸⠀⠀⠀⠀⣿⠀⠀⠈⠁⠀⠀⠉⠀⠀⢸
⠀⠀⠀⠀⠸⠦⠤⠤⠤⣤⡴⠦⠤⠤⠼⠃⠀⠀⠀⠸⠦⠤⠤⠤⣤⡤⠦⠤⠤⠾⠀⠀⠀⠀⠻⠦⠤⠤⠤⣤⡴⠤⠤⠤⠾
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣤⣤⣤⣤⢤⡀
⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⢨⣭⣭⣭⣭⢸⡇⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣶⣶⣶⣶⣦⠀⠀⠀⠀⠀⣿⣿⣾⣿⣿⣿⣿⣾⣷⡆⠀⠀⠀⠀⢠⣿⣿⣿⣿⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⡿⠷⠿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⡛⢻⠟⢛⠛⢿⠛⣻⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣧⣬⣤⣿⣿⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣧⣤⣤⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣷⣾⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠠⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣠⡴⠞⢣⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠷⢏⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡿⠗⠚⠀⢵⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡷⠶⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡿⠿⠛⠶⠲⣤⣤⣤⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣤⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣥⣀⠶⣀⣴⠿⠟⠉⠀⣴⠋⠀⢀⡴⣀⣤⣤⣤⣀⣀⠉⠈⠁⣼⣿⡿⣄⡉⢛⣛⣋⣁⣈⡀⣀⡀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠉⠀⠀⠀⠀⠘⣷⠀⠀⠘⢤⣯⡍⠉⠉⠉⠉⠀⠀⠀⣿⣿⠀⣽⠁⠈⠉⠉⠉⠉⠁⣬⣧⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡷⠀⠶⣶⡶⠞⠁⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⢸⠏⠁⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣋⣿⣋⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⢰⡇⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣿⣿⣿⣷⡄⠀⣼⣶⣾⡟⢻⡶⠶⣶⡉⢹⡄⠀⠀⠀⠀⣾⠃⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⣰⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡶⣿⣿⣯⠙⠿⣤⣀⣀⠀⢀⣙⣃⣀⣸⣿⡇⠸⠇⣸⠋⣿⠛⢿⣦⠄⠀⣴⣿⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⢰⠆⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠈⣿⡟⣷⠀⠈⠙⠻⢿⣿⠏⠉⢉⣴⡿⠁⢦⡸⠇⢀⣿⣤⠀⢿⡀⠀⠈⠙⠃⠀⢀⡄⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⡗⡇⠀⠉⠀⠀⠀⠀⠴⠶⢿⣿⠁⣰⣿⠃⠀⣼⣿⣶⣦⣴⣧⣀⣀⣀⣀⣀⣼⠃⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⢠⣿⡏⣀⣠⣏⣉⠉⢉⣉⣹⣏⡉⠉⠉⠉⠁⠀⠀⠀⠀⣠⣟⣀⣀⣀⣀⣀⣀⣀⣘⣧
⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⢰⣿⣇⠀⠀⢀⣶⠀⠀⢀⣼⠏⠀⢠⡿⠿⣿⣿⣿⡟⠛⠛⠛⠛⠋⠙⠃⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⡟⠛⠛⠛⠛⣋⡹
⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⡟⢀⣤⡿⠁⣠⣴⠟⠁⢀⣴⣿⠃⠀⠀⠿⠛⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠃⠀⠃
⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⣿⣿⣿⣿⣯⣴⡾⠟⠁⣠⡴⠋⣽⠁⠀⢀⠇⢰⡇⠙
⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢰⣿⡿⠟⠋⣉⣁⠴⠖⠛⠁⢀⡼⠃⠀⣠⡟⠀⡆⢹⠄
⠀⠀⠀⠀⠀⢠⣿⠛⠃⠀⣸⣿⣤⣴⣿⠋⠁⠀⣀⣠⣴⣟⣁⣤⣾⣏⣤⣿⡼⠃
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⢻⡟⠋⠉⠉⠉⢙⡟⠙⠋⠛
⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⣿⣿⣟⣿⣿⡟⣠⡟⠀⠀⠀⠀⢀⣿⠀⠀⠀⣠
⠀⠀⠀⠀⠀⣿⡇⠀⠀⣼⣿⣿⣿⣿⣯⣼⡿⠀⠀⠀⠀⢠⣾⠃⠀⠇⠀⣿⡆
⠀⠀⠀⠀⠀⣿⡇⠀⢠⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⢀⣴⢿⡏⢀⣶⠀⢠⣿⡇
⣴⢓⣀⡀⢀⣿⡇⠀⣼⣿⣿⣿⣿⣿⠋⠀⠀⣀⣴⣿⣯⡆⢀⣾⠃⠀⣾⣿⡇
⠙⠿⣿⣿⣾⣿⣧⠸⣿⡅⠙⠛⠉⠀⣀⣤⣾⣿⣿⣿⠟⢀⣾⡟⠀⢠⣿⣿⠁
⠀⠀⠘⢿⣿⣿⣿⠀⠙⢿⣶⠶⣾⣿⣿⣿⣿⣿⣿⠋⣠⣾⣿⠁⠀⣸⡟⠃⢰⠃
⠀⠀⠀⠈⣿⣿⡏⠀⣰⣾⠟⣀⣤⣿⣿⣿⣿⠟⢁⣴⣿⣿⠇⠀⢸⣿⡇⠀⡂⢀
⠀⠀⠀⠀⠙⠿⣷⡤⢟⠁⠘⢿⣿⣿⣿⠟⢁⣴⣿⣿⠿⡏⠀⠀⣿⣿⠇⠀⡇⢸
⠀⠀⠀⠀⠀⢀⠛⢡⣾⣷⣄⠀⠙⢿⣁⣴⣿⡿⢃⠘⣦⠀⠀⣾⣿⣿⠀⢸⣧⠀⣶
⠀⠀⠀⠀⠀⠈⢰⣾⣿⣿⣿⣷⣄⠀⠙⣿⡏⠡⡎⢱⠁⠀⣸⣿⣿⡇⠀⢸⣿⠀⢻
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⡀⠈⠛⣧⣠⠇⠀⣰⣿⣿⣿⡇⠀⠀⣿⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠟⠋⠀⠛⢦⡄⠈⠻⣧⣄⢻⣾⣿⠘⠇⠀⠀⢿⠀⠀⠇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⣶⠟⠀⣰⠃⢙⣦⡀⠈⠻⣿⣡⠙⣤⠀⠀⠀⢸⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣁⣀⡾⠁⢠⣿⠋⡹⣧⡀⠈⢿⣦⡟⠀⠀⠀⣸⠀⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠟⠀⣰⣿⠋⡀⠙⢠⡿⢦⡀⠙⢷⡀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃⣀⣾⣿⠁⣠⠉⡄⣿⠁⣾⠛⢦⡀⠙⣷⣀⣿⠀⠀⠀⠈⣅
⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⡿⠁⣰⣏⣿⢱⡏⢀⠋⠀⢼⠙⢦⡀⠹⢿⡀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⠟⢀⣾⣿⣿⠇⣸⠃⣿⠀⠀⡻⠀⠀⠙⢦⡈⠛⣦⡀⢸⣿⡄
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠟⢁⣴⣿⣿⣿⡿⢠⡟⢀⠀⠀⣼⡇⠀⠀⠀⢸⠻⢦⡈⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣿⠟⣁⣴⡟⣿⣿⣿⣿⡇⣸⠃⢾⠀⢀⡿⠀⠀⠀⠀⣿⠀⠀⠙⢦⡘⢻⣷
⠀⠀⠀⠀⠀⠀⢾⣯⣾⣿⠟⣸⣿⣿⢹⡟⠀⣿⠀⢸⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⢨⡿⢦⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣶⣿⣿⠃⣿⠃⣼⡏⢀⡏⠀⢸⠃⠀⠀⠀⢀⠃⠀⠀⠀⢸⡇⠸⣿⣦⡙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⡇⣼⡟⠀⣿⠀⢸⡇⠀⡎⠀⠀⠀⠀⣸⠀⠀⠀⠀⣿⢱⠀⣿⣏⠙⢠⡙⢧⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡏⢰⣿⠁⢸⡟⠀⡘⠁⢠⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⡏⢸⠇⣿⢸⡄⠀⠙⢦⡙⠂
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣠⣿⡏⠀⣿⡇⢠⡇⠀⣸⡏⠀⠀⠀⠀⢻⠀⠀⠀⠀⡇⢸⠀⡇⢸⡇⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⠁⢸⡿⠀⠈⠇⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⡇⢸⡇
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⠀⣾⡇⠀⣴⠀⠀⣿⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⢸⡇⠀⡇⠈⣷
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⠀⠀⢀⣿⠀⠀⠀⣰⠸⡇⠀⠁⠀⠹
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣦⡀⠻⣿⣆⢸⣧⠀⣿⠀⠀⠀⠀⠘⢣⣄⠀⠀⠻⣄⠰⣆⠀⢇⠀⢿⡀
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠃⠀⢼⣿⣿⣯⣀⣼⡆⠀⠀⠀⠀⢀⣼⣷⣀⣀⣙⣶⣿⣆⣸⣧⣄⣻⣄
⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣤⣶⣶⣶⣶⣶⣤⣭⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣬⣥⣬⣭⣭⣤⣬⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⠿⠟⠻⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣀⣀⣀⣀⣀⣤⡀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠛⠛⠛⠓⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⢀⠀⠀⢸⠀⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠷⠀⢀⡀⠰⠏⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢲⡶⠾⠋⠻⠶⣶⠂⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣷⠀⠀⠀⢰⡇⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⢸⡆⠀⠀⣾⠀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⣧⠀⢠⡇⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⠀⢸⢱⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣋⣛⣉⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡉⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣤⣀⡀⠈⠲⠊⠀⣀⣠⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠰⠶⠶⠶⠒⠚⠛⠋⢉⣉⣠⣤⠤⠤⠤⣤⣈⡙⢶⡀⣰⠞⣉⣤⡤⠤⠤⢤⣤⣌⣉⠉⠛⠛⠒⠲⠶⠶⠶
⠀⠀⢀⣀⣀⣤⣤⠴⠶⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⢶⣿⡷⠞⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠲⠶⢤⣤⣄⣀⣀
⠀⣠⠟⠁⠀⠀⠀⠀⠠⠶⠶⠖⠒⠛⠛⠛⠛⠛⠓⠲⠦⣤⡉⣡⡤⠶⠒⠛⠛⠛⠛⠛⠛⠒⠶⠶⠦⠀⠀⠀⠀⠀⠙⢧⡀
⢜⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣾⠛⢻⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠓⠛⢻
⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠹⣇⠻⠶⠾⢃⡿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠁



⠀⢸⣏⣹⠆⣰⠏⠉⢷⠀⡾⠋⠙⣆⢸⣇⡴⠃⠀⠘⣇⢰⢿⠀⡿⢸⡇⢰⣏⠉⠀⡟⠙⢷⠀⣾⠋⠙⣦⢰⣿⡄⢰⢿
⠀⢸⣏⣹⠇⠸⠆⣀⠾⠀⢷⣄⣠⠏⢸⡏⠻⡄⠀⠀⢻⡾⠈⣷⠇⢸⡇⢀⣈⡷⠀⣇⣠⡾⠀⢿⣄⡠⠟⠘⠇⢳⡏⢸
⣿⣿⠟⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⠄⠒⠒⠂⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠉⠉⠐⠒⠢⠤⢤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠋⠑⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠒⠒⠂⠤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠘⠒⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣤⣤⣤⣶⠶⠶⠿⠿⠿⠿⡟⠁
⠀⠀⠀⠀⠀⠀⠱⠦⣄⣀⠀⠀⠀⠀⠀⠀⠉⠉⠰⠦⣤⣴⣶⣶⡶⠿⠿⠛⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠉⠙⠲⢤⣀⡀⠀⠀⠀⠀⠀⣿⡟⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠤⠶⣶⣖⠛⠂
⠀⠀⠀⠀⠀⠀⡠⣟⡂⢤⣀⠀⠀⠈⠙⠓⠦⢤⣀⠀⢹⣯⣄⣀⣀⣀⣠⡤⠤⠶⣒⣚⣩⣭⣽⡿⠷⠶⠟⠛⠛⠛⠁
⠀⠀⠀⠀⠀⢐⠀⠀⠉⠂⠈⠑⠢⢄⡀⠀⠀⠀⠈⠙⠚⠿⢿⣿⣿⠷⠶⠞⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⣘⣇⡀
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠠⣌⠑⠢⢄⡀⠀⠀⠀⢀⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⢶⣒⣛⣯⣽⠅
⠀⠀⠀⠀⠀⠈⠐⢄⡀⠀⠀⠀⠀⠀⠀⠈⠒⢤⣈⠑⠢⢀⣸⣿⣖⣀⣀⣀⣤⣤⣶⣖⣻⣽⣿⡶⠾⠛⠛⠉⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⡀⠀⠀⠀⠀⠀⠀⠈⠛⠦⡄⢈⣙⣻⣭⡽⠶⠿⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠂⠀⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⠀⠀⠀⠀⠀⠀⣰⣿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡊⣟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⣿⣛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠒⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⠀⣿⣧⠄⠀⠀⠀⢀⣀⣀⡤⠴⠖⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⡷⠶⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿⠁⠀⠀⠀⠀⢠⠀⠀⠀⢀⣤⣾⡇
⠀⠀⠣⠀⠀⠻⠋⢀⠀⢀⠀⠀⠀⠀⠀⣤⣾⣷⣤⡀⠀⠀⠀⣌⠀⠀⠀⠸⠟⠃⢰⠀⠀⠀⡀
⠀⢀⠠⣑⠦⠥⠯⢔⡨⠀⠐⠀⠀⠀⢠⣴⣶⣴⣶⡄⠀⠀⠀⠀⠀⠀⢀⣆⠀⠀⠁⠀⠀⢠⣿⣿⣿⣶⣧⣆
⠀⠀⠑⠊⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠟⠍⠩⠯⠩⠻⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⢾⡭⢾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣗⣛⡯
⠀⠀⠀⠀⠀⠀⠀⡀⢠⠀⠀⠄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠂⠀⠀⠀⠀⠘⠋⠁⠈⠁⣿⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣾⣿⣦⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣵⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⠀⠀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣾⣿⣷⠦
⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⣿⣯⣟⣿⣿⣿⣳⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⠿⠿⠿⢀⣃
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠤⠧⢼⠤⢤⢤⠧⢼⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⣷⣿⣛⣿
⠀⠀⠀⢺⣿⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠳⣚⢞⡒⣖⢞⡲⠃⠀⠀⠀⠀⠀⠀⠀⣿⡿⣿⣿⣿⣿⡿⠿
⣤⣤⣤⣬⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣭⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣥⣤⣭⣥⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡿⣿⣛⣻⣿⣿⣻⣛⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⢿⣿⣷⣿⣿⣉⡿⣷⣟⢹⣛⠿⣾⣿⣻⢿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣬⢿⣯⣟⢻⡹⣿⢷⣻⡿⣽⣿⣻⣾⢛⣯⣯⣙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣹⡟⣻⠟⣭⢯⣷⡿⣿⣿⣩⣷⣿⣿⣿⣿⣿⣵⢿⣜⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣼⣿⣛⣿⣟⣟⣿⣶⣿⢽⣿⢿⣾⣏⣥⣽⣿⣿⣭⣯⣹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣥⣭⣿⠿⣿⣿⡞⢿⢿⣿⣻⡿⢿⣽⣿⣿⣟⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣷⣿⣯⣿⣾⣿⣶⣿⣟⣿⣿⣶⡿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⣰⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⠀⠀⠀⠀⠀⠀⢧⡤⠶⠶⠶⠶⢲⠖⠛⠛⠛⠛⠛⠛⠁⠀⣼⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⢦⣄⠀⠀⠀⣰⠛⣆⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠙⠦⡴⠁⠀⠈⢦⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠛⠁⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⢀⣠⠴⠛⠉⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⠈⠙⠛⠛⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣾⡿⠟⠋⠉⠀⠀⠀⠀⠀⠉⠙⠻⢿⣿⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⠟⠉⠀⢀⡤⠖⠚⣋⣉⣉⠛⠒⢤⣀⠀⠉⠻⣿⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⠋⠀⢀⠖⣡⠴⢚⣩⠤⠤⢤⣍⡙⠦⣌⠳⣄⠀⠙⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⠃⠀⢰⠋⡼⢁⠖⠉⠀⠀⠠⠤⣄⠉⠲⡈⢣⡈⢦⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⡏⣼⢀⡏⠀⠀⢀⣤⣦⣤⡈⠱⡄⠹⡄⢷⠘⠆⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡁⡇⢸⡁⢸⠀⢸⣿⣿⣿⣷⠀⡇⠀⡇⢸⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⡇⢻⠈⣇⠈⢧⡈⠛⠛⠛⢁⡴⠃⣰⠁⡼⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠸⣌⢣⡈⠣⠀⠙⠓⠒⠒⠋⢀⠔⢁⡼⢁⠞⠀⢠⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣄⠀⠈⠦⣙⠦⣄⠀⠀⠀⠀⢀⣠⠴⢋⡴⠋⠀⢠⣿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣦⣀⠀⠈⠓⠦⢤⣄⣀⣀⣥⠤⠒⠉⠀⣀⣴⣿⣿⣿⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣦⣄⣀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠋⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣄⠁⠤⠤⠤⣶⣦⣉⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠈⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⣰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠻⣿⠁⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠚⠉⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⡀⠀⠘⠛⠢⠤⠤⠤⠤⠀⠚⠛⠛⠀⠀⠀⠀⠀⠀⠀⢀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠷⢦⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠟⠛⠛⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⣀⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢾⣷⡀⠀⠀⠀⠹⢿⡁⠈⠐⠙⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠐⠂⠈⠀⠰⠁⠹⢆⡀⠀⠀⠈⠙⢦⡀⠀⠄⠉⠖⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠚⠀⠀⠀⠀⠀⢠⡆⠀⠀⠠⠀⠀⠀⠀⠀⠀⠙⢤⠘⢣⡐⢣⠜⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠤⠀⣠⡄⠀⠀⠀⠀⠀⠉⠀⠉⠠⢶⡉⠹⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠴⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢇⡀⠓⠀⠀⡟⢠⡶⢀⡀⠀⠀⠀⠀⠈⠀⢀⠙⢦⡀⢾⣱⣄⡀
⡄⠀⠀⠀⢀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢆⡀⠀⠀⣶⠞⣿⠀⠈⣻⡆⠀⠀⠀⠀⠀⠈⠐⠈⠑⠀⠙⠊⣷⣆⡀
⠇⠀⠐⠀⠘⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠉⣴⠛⢀⡼⠇⣰⠗⠂⣀⠀⠀⠀⠀⠀⠀⠀⣄⠁⠄⠙⢷⣦⡀
⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⠀⠀⠀⠀⠀⠀⠀⢰⣾⠃⠀⣿⠀⣠⠛⠣⡄⠀⠀⠀⠀⠀⠈⠳⢶⣀⠀⢈⣿⣷⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠶⠉⣴⠋⢀⣤⣁⡴⢦⠀⠀⠀⠀⠀⠀⠉⠿⡏⠉⠁⡀⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦⣄⠀⠀⠀⣀⡀⠀⠸⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⣤⠾⣿⠏⠀⣼⠁⣴⠀⠀⠀⠀⠀⠐⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠳⣆⡀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡾⠃⣾⡏⢀⣴⠀⢀⡘⠁⣄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠻⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢠⡈⣿⣦⡿⠁⠠⡞⠉⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⡿⠀⢀⠞⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢠⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠿⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠷⣶⠒⠒⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠶⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⢰⡄⠈⠳⣄⠀⠀⠀⠀⠀⣠⡟⠀⡌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠘⠁⠀⠀⠀⠛⢤⡀⠀⣰⣿⡇⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠁⠿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠈⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⡀⠤⠤⠤⣤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣼⣷⣤⣴⣶⣾⣏⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣳⣤⣶⣤⣈⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡯⠀⠀⠉⠁⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⣼⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢛⣿⡿⠁⠁⢸⣿⣿⡄
⠀⠀⢿⣿⣿⣿⣿⡿⠤⠾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢡⣾⡟⠀⠀⠀⠀⣿⣿
⠀⠀⠸⣿⣿⣿⣿⢹⣦⣤⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣵⣿⣟⡄⠀⠀⠀⢸⣿⣿⡀
⠀⠀⢀⣼⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣯⣾⣿⣿⣿⠀⣾⣿⣦⣟⣿⣿⡇
⠀⠐⣿⣿⣿⣿⣿⣿⣿⣦⣘⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⡆⢽⣿⣿⡿⢿⣿⣿⡆
⢀⣄⣽⣿⣿⣿⣿⡿⢳⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⣀⣀⣤⢾⣿⣿⣿⣿⣿⣿⡿⠋⠀⢹⡟⠻⠾⠛⠀⠙⣿⣿⠻⠃
⠀⢹⣿⣿⣿⣿⣿⡇⣾⣿⡟⣼⡟⠀⠀⠀⠀⢀⣀⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠘⣿⣦⣤⡄
⢠⣿⣿⣿⣿⣿⡿⠧⢸⣿⣷⣿⡇⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⠀⣼⠈⡆⠀⠀⠀⠀⠀⠈⠈⢿⣷⣶⡀
⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢿⠂⢹⡀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⡆⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣰⣦⡀
⣿⣿⣿⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⡇⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠏
⣿⣿⣿⡗⠲⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠁⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⢿⣹⣿⣿⣿⣿⣿⡿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⡇⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⣸⠈⡇
⢿⣿⣿⣿⣿⣿⣿⣿⣟⡼⢿⣿⣿⣷⣼⣧⣿⣿⣿⣿⣹⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⣤⠀⠀⠀⠈⣆⠀⠀⠀⠀⠀⠀⠀⡏⠀⢻⡀
⠀⢹⣿⣿⣿⣿⣿⣿⡿⢇⣼⣿⣿⣿⣿⣿⣿⣯⣿⠛⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⢹⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⢰⠇⠀⢸⣇
⢀⣼⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢐⠃⠀⠀⠀⢸⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠸⠀⠀⠀⣿⡄
⠈⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⡆⠀⠀⠀⢙⣧
⠀⠀⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⠂⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢘⡆⠀⠀⢰⠁⠀⠀⠀⢸⡼
⠀⠀⠙⠻⣿⣿⣿⣿⣿⣾⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⡼⠀⠀⠀⠀⠀⡇⢀
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠇⠀⠀⠀⠀⠀⢹⠀⡀
⣤⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⣀⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣤⣬⣰⡀⠀⠀⠀⠀⠀⢸⡆⢁
⠑⢿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣴⣿⣦⣤⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⡇⠘⡆
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢊⣽⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣱⠀⠧
⢶⣟⣷⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣿⣿⡿⠿⠟⠋⠁⠀⠸⣿⣿⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠖⢒⣶⢄
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⢸⠄⠘⠿⢻⣾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠀⢦
⠀⠀⠐⠀⠀⠀⠇⠀⢀⡀⠐⢂⠠⠀⢨⢼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⡀⠀⡒⣇
⠀⠈⠀⢀⠀⠈⠀⠀⠊⠀⠀⠃⠀⠐⠀⠀⢹⣶⢶⣖⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⢲⠊⡀
⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⢑⡂⠜⠘⢠⡿⡄⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡠⠤⠄⡁⠰⢠
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡀⢸⠀⠀⠀⠁⠐⢠⡟⣧⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠸⠇⠀⠠⢡⠁⠸
⠀⠀⠀⠁⠠⠁⠀⠀⠀⠀⠀⠀⠀⠈⠸⠀⠈⠀⠘⡄⢸⣧⡻⣄⠀⠀⠀⠚⠀⡐⡤⢀⡇⠈⠊⢢⡀⣠⠧⠓⠀⡄
⢀⠀⠀⠈⢧⡀⠀⢀⠀⠀⠀⠀⠀⠀⠄⣧⡀⠀⢀⣧⡘⠟⣙⢻⡆⠀⠀⠀⠁⡀⠅⠁⠈⠁⠐⠦⡀⢈⠤⣶⡖⡇
⠀⠁⠂⠀⠀⠉⠐⡰⠃⠀⠀⠀⠀⠀⢸⠎⠿⣿⠩⠔⠓⠚⠁⠀⢿⠆⡼⡁⠀⠀⠀⠀⠀⠀⠒⡊⠁⠁⠈⠉⡦⠇
⠀⠀⡀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠐⠦⡘⠀⠀⢘⠁⠀⠀⠀⠀⠀⣼⡄⠈⠒⠓⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢀⠿⡂
⠀⠀⠀⠉⠀⠥⠀⠀⠀⡀⠠⠀⠐⢠⣨⠀⠀⠜⠂⠠⣼⠀⠀⠀⣧⠓⠀⢡⠀⢀⠀⠀⠀⠀⢀⡠⠚⠁⠀⡜⠀⠁
⠀⠀⠀⠀⠀⠀⠒⠃⠀⠀⠈⠀⠀⠠⡈⠀⠀⠀⠈⣛⠣⢷⡀⢀⠇⣂⠙⠐⠇⠂⠀⠀⡄⠀⠏⠀⡀⠀⠀⠲⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⢄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⡀⣀⠀⠀⠀⠀⠀⢱⡆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠂⠁⠠⠁⢠⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠴⠀⠺⢠⡄⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⡡⠀⠄⠀⣔⣭⣄⡁⡀⠀⠀⠀⠘⠳⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣞⣳⡀⢀⠀⠠⠄⠀⠀⠀⠀⢀⠀⢠⠉⠛⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⠃⠈⠠⠼⠀⠀⠓⠲⡖⣦⠀⠊⣴⠆⢀⠀⡔⡸⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠠⠂⠀⠀⠀⠀⢀⠂⠀⠀⠀⠠⣀⢉⠗⢦⠀⠁⠀⡠⠋⠀⠆⠀⢀⡀⠀⡀⠀⠄⠀⠀⠚⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⠐⠁⠀⠀⠀⠀⠧⠆⠀⡀⠀⠀⢈⠙⠒⠊⣐⣢⡞⣒⢂⣀⢈⡤⢦⡍⠀⠈⠁⠀⢰⠖⢺⡈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⣀⡠⠀⠀⢀⠀⠀⣤⠀⢀⠰⣾⡷⠄⢐⠉⠙⣇⣭⣒⣩⡀⠉⠁⠁⠀⠀⡔⠂⠘⠢⠜⠳⣆⢱⡦⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⠟⠑⢄⠀⠀⠛⠀⠃⢀⡀⠀⠠⢻⡍⠀⠀⣿⠂⠀⠀⠀⠀⠐⡇⠀⠢⢰⣦⣄⡀⢠⠀⠈⠀⠀⡂⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠁⢈⡉⠐⠀⣠⢁⠄⠤⠀⠀⠙⢆⢀⣏⡴⠶⢗⠀⠄⠀⠀⠀⠆⠈⠛⠋⠁⢀⡖⡚⠂⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⣀⠈⢉⡀⣲⢤⣾⡀⢘⣿⡋⠀⢀⣨⠟⠂⢀⡀⠀⠈⠊⡰⠀⡄⠜⠀⠀⠠⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠰⠥⠤⣀⠴⠛⠃⠣⠀⠀⢹⡟⠁⠉⠛⣠⣥⣄⠀⢀⣉⣀⡀⠀⠀⠀⠏⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⢦⠀⢀⣠⠖⠚⣧⡀⠀⠀⠀⠖⢮⡇⢠⠤⠤⡻⠿⠟⢀⡀⣸⠋⠙⣀⣠⡞⠀⠀⠀⠀⠀⠸⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣀⣀⡀⠈⢰⠊⠉⠓⢤⣾⡇⠀⢀⡇⢈⠂⠀⡧⠀⠀⠣⢸⢹⡞⠉⠁⠀⠈⡆⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣭⠿⣿⡇⠀⠀⠀⠀⠘⠛⠛⠛⣶⣾⣇⠹⠶⣶⣷⠶⠶⣤⠾⠋⠙⠦⠆⠁⠐⠁⠈⠠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠛⠋⠀⠀⠀⠀⠀⣠⠆⠄⠠⠈⢿⣿⣦⣼⠇⠀⠀⠀⠘⠓⠀⠀⠀⠀⣒⣦⡀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⢿⣿⣿⡄⠀⢼⣷⣦⣌⡙⠂⠄⠀⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡇⠀⠀⠀⠉⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠌
⠀⠀⠀⠀⠀⠀⠀⣐⠀⠈⠭⠍⠁⠀⠀⠀⠐⠀⠀⠀⠠⢄⠀⣀⣀⢀⡠⠼⠛⠛⠿⣧⣤⠀⠄⠀⠀⠀⠀⠀⠤⠤⠤⠀⠀⠐⠂⠀⠈⢉⣀⡠⢔⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⠀⠉⢉⣉⣁⣀⣀⣀⣀⣐⣒⣀⣒⣀⣤⣤⣤⣿⣗⣋⣐⣿⡿⠦⠁⠀⠀⠀⠉⠩⠭⠭⠭⠭⠥⠄⠙⠛⠛⠓⣒⠨⠇
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡿⠿⠿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠉⠉⠁⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠀⢾⣹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⠁⠀⠀⠈⠻⣿⣿⣄⣀⣀⣀⣀⣀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣯⣀⣀⣀⣀⣠⠖⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣯⠤⠤⠤⠤⠤⠭⠥⠤⠤⠤⣖⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣇⣀⣀⣀⣀⣀⣀⣸⣿⣷⣶⡒⠒⠒⠒⠒⠒⠒⢦⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠒⢒⣒⠒⠢⡀⠀⣙⡒⠒⠫⠽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⠀⠀⢹⣿⣿⣿⣿⣿⣿⣉⣙⣔⣁⣀⣈⣙⣣⣃⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⠀⢠⣿⣿⣿⣿⣿⣿⣿⣉⣉⣉⣉⠉⠉⠉⠉⠉⠉⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⠤⢴⢒⡌⠓⠤⠤⠤⢤⣒⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿⣿⠤⠤⠭⠤⠤⠤⠤⢤⡀⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠫⡽




⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣒⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⠀⠀⠀⠀⠀⣀⣐⣶⣶⣶⣾⣷⡄
⠀⠀⠀⣴⠟⠻⣷⣼⣿⣿⣿⣿⠀⠀⠀⣴⡟⠻⣷⣤⣴⣟⣿⣿⠀⠀⠀⣰⡟⠻⣷⣤⣴⣿⣿⣿⠀⠀⠀⣴⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢠⣿⣶⣶⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣦⣤⣿⣿⣿⣿⣿⣿⠀⠀⢠⣿⣧⣤⣿⣿⣿⣿⣿⣿⠂⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⣼⣿⣿⣿⣿⣿⣿⠀⠀⢸⣿⣿⢺⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠉⠙⠛⠙⠛⠛⠛⠛⠛⠋⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠋⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠉⠛⠛⠛⠛⠛⠛⠛⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠐⠒⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⢸⡷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠄⠠⢄⠀⢀⠤⠀⠠⢄⢀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢠⠊⠉⢨⠀⠀⠀⠀⢱⠃⠀⠀⠀⢀⠍⠉⠲⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⢼⢆⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⡴⠋⠈⠢⢄⡀⠀⠀⠀⡀⢸⠀⡄⠀⠀⠀⢠⠠⠔⠈⠑⣄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⢰⠀⠀⡄⠀⠀⠁⠀⠀⠀⠃⢸⠀⠁⠀⠀⠀⠊⠀⠀⡀⠀⠘⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠸⡀⠀⣈⢀⡀⠀⠴⠂⠀⠀⢸⠀⠀⠀⠐⠤⠀⢰⢀⡁⠀⢰⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠑⡌⠁⠀⠁⠀⠀⠀⠎⠠⢼⡦⠄⠢⠀⠀⠀⠉⠀⠈⡴⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠘⣄⠠⡀⠀⢀⡀⠀⣠⠋⠳⡀⠀⢀⠀⠀⣠⠄⡰⠁⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠈⠑⢢⣤⣤⠕⠊⠁⠀⠀⠈⠒⠤⠤⠤⠔⠊⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠤⢸⠇⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣄⣀⣀⣀⣀⣀⣀⣼⡀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⣠⠤⣒⣉⠡⠤⠄⠒⠒⠢⢤⡇⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⡴⠖⠛⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣶⡖⠒⠒⠒⠒⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠤⠤⠊⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠋

⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⣀⡤⠀⠀⠀⠀⣀⠄⢠⡀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⢀⡀⢀⣀
⠀⠀⠔⠉⠀⠙⠄⠀⠀⢈⠁⠀⠄⠀⠀⠀⠐⠁⠀⠀⠈⠀⠀⠀⠀⠨⠀⠀⠄



⠀⠀⠀⠀⠀⠀⢀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠐⠂⠠⣄⠀⠀⠀⡰⠒⠀⠢⡀
⠀⠀⠀⠀⠀⠐⠃⠀⠂⠁⠀⠀⠘⠉⠙⠒⠒⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠁⠀⠐⠲⠒⠒⠖⠂


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⣠⠖⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⢀⣤⠂⠀⠐⠉⠀⠈⠡⠀⠀⠀⠊⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⠀⠐⠐⠀⠐⠂⠀⠀⠈⠉⠃⠈⠀⠀⠀⠀⠀⠀⠀⠒


⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⡄⢠⡀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠊⠉⠇⠋⠩⠀⠀⠐⠁⠀⠠⠀⠀⠀⠀⠈⠉⠀⠉⠂⠀⠀⠘⠁⠈⠓⠄⠀⠀⢀⠴⠀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀⠀⢀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡆⢸⡇⡲⠂⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣢⡎⢯⢿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣉⣷⠞⢿⡟⠓⣾⣉⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡷⣎⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣟⣉⣉⣻⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢼⣿⡖⠛⢿⡟⢿⡧⢴⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣤⣤⣤⣤⣤⣤⣤⣤⡄⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⢠⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⣶⠒⣤⣷⣤⠛⢳⣾⠓⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠏⠉⠉⠉⠉⠉⠉⠉⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠉⠉⠉⠉⠉⠉⠉⠹⣿⡄⠀⠚⢁⡟⢹⡟⣫⠈⠟
⠀⠀⠀⠀⡀⢄⢰⢂⠆⣀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠉⠈⠁
⠀⠀⠀⠦⣽⡎⣻⣏⣻⡥⠶⠀⠀⢸⣿⠀⠀⢠⡞⠛⠃⣴⠛⢳⡄⢷⠀⣼⠀⡗⢸⡏⠛⣦⠀⠀⠀⢺⡆⢠⡞⢳⡄⠀⠀⣿⡇
⠀⠀⢀⣃⡛⣏⢉⣿⣹⣟⡛⠀⠀⢸⣿⠀⠀⠘⢧⣀⡀⢿⣀⡼⠇⠘⣷⠇⠀⡟⢸⣇⣠⠟⠀⠓⠀⢸⡇⠀⣉⡽⠃⠀⠀⣿⡇
⠀⠀⠀⠁⠞⡻⢿⠴⡛⠦⠁⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠉⠈⠀⠁⠀⠀⠀⠀⢸⣿⠀⠀⢠⣶⡔⡗⢸⢲⡄⡇⡴⠂⣐⣦⢸⠀⢰⠠⣖⠠⡆⢰⢰⣄⣴⡄⢴⡂⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⠁⠀⠉⠈⠈⠁⠁⠈⠁⠉⠉⠈⠀⠈⠀⠉⠀⠉⠈⠉⠀⠉⠀⠈⠁⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⢀⣤⢸⡇⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠹⢶⠟⠁⢸⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠷⠶⠶⠶⠶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢠⡟⠛⠛⠛⠛⢻⡆⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⣀⡠⣿⢦⣀⠀⠀⢸⣿⠀⢸⡇⢤⣀⣴⠟⢸⡇⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⡚⣛⡹⣿⠗⠂⠀⢸⣿⠀⢸⣇⣀⣉⣁⣀⣸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠬⠁⠃⠁⠀⠀⢸⣿⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢠⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢀⡴⠶⠶⠶⠶⢦⡀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣿⡇⠀⠀⠐⢪⣞⢻⣭⣮⣖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠻⣦⣴⠟⢸⡇⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⡇⠀⠀⠀⠞⢽⣤⢳⢟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⣴⠟⠻⣦⢸⡇⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿⡇⠀⠀⠀⠀⠘⠐⠘⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠈⠳⠶⠶⠶⠶⠞⠁⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣶⣦⠀⠀⠀⠀⢠⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡿⠿⠿⣿⣿⡿⠃⠀⠀⢀⣿⣿⣿⣿⣷
⠀⠀⢀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⠳⣴⠃⠈⠁⠀⡀⠀⣿⣿⡿⡟⣿⠛⠃
⠀⣀⣿⣃⣀⣘⣟⣀⢀⣹⣇⣀⣀⣻⣃⡀⣘⣟⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣼⡇⠀⠀⢀⣘⣟⣀⠉⠉⢿⣴⡏
⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣘⣛⣛⣋⣀⣀⣸⣿⣿⣿⣀⣀⣈⣉⣁⣀⣀⣀
⠀⠉⣿⡏⠀⢹⣿⡇⠈⣿⣿⠀⠉⣿⡏⠀⢹⣿⡏⠈⠉⠛⠲⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇
⢀⣀⣛⣃⣀⣘⣛⣃⣀⣛⣛⣀⣀⣛⣃⣀⣘⣛⡃⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣘⣛⣃⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠀⠀⠀⠙⣷⣀⣰⡟⠁⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠈⢿⣏⠀⠀⠀⠀⠀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⢀⣾⠆⢿⣆⠀⠀⠀⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⢀⣾⡏⠀⠈⣿⣆⠀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⢸⣿⠀⠀⠀⢸⣿⣤⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡌⡇⠀⠀⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣧⣤⣤⣤⣤⣄⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠉⠁⠀⣀⣀⠀⠈⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⢸⡇⢸⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠲⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢠⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⣾⠀⢸⠀⠀⠀⠀⠘⣿⣇⠀⣿⣿⣿⣿⣿⣷

⠀⠀⣠⠴⠒⠛⠉⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠴⠖⠒⠲⢤⣀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠛⠉⠙⠛⠲⣤⡀
⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⢠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠉⢳⡀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⡞⠀⠀⣤⠀⠀⠀⠀⢰⡆⠀⠀⢹⠀⢠⠃⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠸⡄⠀⢠⠏⠀⣀⣀⠀⠀⠀⢠⣴⠄⠀⢹⡄⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⣇
⡇⠀⢀⠉⠀⠀⠀⠀⠀⠀⡄⠀⢸⠀⢸⠀⠀⠀⠀⠘⠃⠀⠋⠀⢀⣴⠀⠀⡇⠀⢸⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡀⠀⣧⠀⢸⠀⠐⣖⠀⣠⡀⠀⣶⠆⢹⡄⠀⢹
⢳⠀⠙⢄⡀⠀⠀⠀⢀⡴⠁⠀⡟⠀⠸⡄⠐⢏⢀⡄⠀⣀⠀⣤⣠⠜⠀⢰⠇⠀⠸⡄⠀⣖⡖⢺⠛⣟⣙⣏⣿⠇⠀⡟⠀⠘⣇⠀⠸⣄⠙⠁⠀⠀⣠⡞⠀⢀⡟
⠀⠳⣀⠀⠉⠛⠒⠛⠉⠀⣀⠞⠀⠀⠀⠑⣄⠀⠀⠈⠉⠈⠉⠀⠀⠀⢠⠎⠀⠀⠀⠱⣄⠈⠻⠽⣭⣽⠤⠟⠁⢀⡼⠁⠀⠀⠘⢦⡀⠈⠙⠒⠒⠛⠁⠀⣠⠞
⠀⠀⠈⠑⠲⠤⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠈⠑⠢⠤⣀⣀⣀⡠⠤⠚⠁⠀⠀⠀⠀⠀⠈⠓⠦⢤⣄⣀⣤⠤⠖⠋⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠶⠚⠋⠁

⠀⠀⢀⠴⠒⠛⠉⠉⠛⠲⢤⡀⠀⠀⠀⠀⠀⢀⣠⠴⠒⠒⠒⠒⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠒⠒⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠛⠉⠙⠓⢦⡀
⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠻⡄⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠙⣦
⡼⠀⠀⠀⠠⡄⠀⠰⠄⠀⠀⢀⠸⡄⠀⣸⠁⠀⢀⡀⠀⠀⠀⣴⣀⠀⠀⢹⡄⠀⢠⠃⠀⠀⠀⠆⠀⠐⠀⠀⠀⠀⠘⡆⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⡇⠀⠀⠋⠁⠀⠀⠀⠀⠀⣀⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⢀⣀⣀⣞⡶⠀⠀⣷⠀⢸⠀⠀⢰⣶⠀⠀⠀⠀⣴⡦⠀⠀⡷
⢧⠘⣄⠀⠀⠀⠀⠀⠀⠀⢀⡜⣸⠃⠀⢧⠀⢶⣖⣒⣒⣛⣋⡭⣭⠏⠀⣸⠁⠀⠸⡀⠀⡠⢶⣙⡅⠀⠀⠈⠙⠂⠀⡟⠀⠘⡆⠀⠀⠀⢠⠒⠲⡄⠀⠀⠀⢰⠇
⠈⠳⣈⠓⠤⣀⣀⣀⣠⠴⢊⡴⠃⠀⠀⠘⢦⡀⠙⠦⢤⣤⠤⠞⠁⢀⡴⠃⠀⠀⠀⠱⣄⠀⠀⠉⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠈⠓⠲⠤⠤⠤⠖⠚⠁⠀⠀⠀⠀⠀⠀⠙⠲⠤⣤⣤⠤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠈⠓⠢⠤⣤⣤⠤⠴⠚⠁⠀⠀⠀⠀⠀⠀⠉⠓⠶⠤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⣠⠖⠋⠉⠉⠉⠉⠓⠦⣄⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠉⠉⠛⠦⣄⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⠉⠳⢦⡀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠉⠉⠉⠙⠓⢦⣄
⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⡾⠀⠀⢐⣶⡀⠀⢸⣞⠀⠀⠀⠈⣧⠀⢀⡏⠀⠀⣀⡀⠰⠆⢀⣀⡀⠀⢸⡆⠀⢸⠀⠀⠐⠂⠀⠀⠀⠃⠀⠀⠀⢹⠀⠀⢠⡇⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠈
⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⣀⠀⠀⣿⠀⢸⡅⠀⠘⠀⠈⠀⠀⠉⠀⠁⠀⢸⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡀⠀⢀⣀⣀⣀⣀⣠⡤⠤⠶⠂
⢹⡀⠀⠈⠉⠉⠉⠉⠻⡽⠉⠀⣰⠃⠀⠈⣇⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⡼⠀⠀⠈⣇⠀⠀⠀⠀⣠⠤⣄⠀⠀⢀⡞⠀⠀⠈⣧⠀⠈⢹⡙⢹⠀⠀⠀⠀⠀⠀⢠
⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⣠⠴⠃⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠈⠣⣄⠀⠚⠀⠀⠈⢁⣠⠞⠀⠀⠀⠀⠈⠳⣄⡀⠙⠋⠀⠀⠀⢀⣠⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠶⣤⣤⡀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⢈⣙⢷⡄⠀⣴⢟⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠻⠄⠀⠙⡾⠁⠸⠧⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⡴⢡⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠛⠿⣶⠞⠛⠛⠛⠒⠒⠛⠉⠀⢸⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣸⠇⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⣰⣏⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠥⠴⠒⠒
⠀⠀⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⠔⠉⠀⠐⠤⣀⡀⠀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⣴⣡⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠈⠉⢹⡏⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣌⣦
⠀⠀⠀⠀⠀⠒⠒⠶⠶⠮⠭⣭⣽⣟⣓⠶⢦⣄⡀⢸⡇⢀⣠⡴⠶⣚⣻⣯⣭⠭⠵⠶⠶⠒⠒
⠀⠀⠀⠀⠀⠈⠉⠉⠙⠛⠛⠓⠶⠾⢭⣝⣓⠶⣍⠻⠟⣩⠴⣚⣫⡭⠷⠶⠛⠛⠛⠋⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣁⣈⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣠⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠃⠀⠀⣠⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣾⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠟⢉⣁⣠⣤⣀⣉⠙⠻⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠏⢀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡈⢻⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣈⠙⠻⣿⣿⡏⢠⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⣷⡀⢻⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣷⡄⢹⣿⠀⢼⣿⡟⠻⣿⣿⣿⡿⠃⣰⣿⣿⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⢀⣿⡆⠸⣿⣿⣄⠘⢿⡿⠁⣼⣿⣿⣿⠃⣸⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠋⢀⣾⣿⣿⡄⠙⣿⣿⣷⡀⢀⣾⣿⣿⡿⠋⣠⣿⣿⡟⢁⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣦⣄⠙⣿⣿⣿⣦⠈⢿⣿⣿⣿⣿⣿⡟⢀⣼⣿⣿⣿⠀⣾⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⡄⠹⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⡇⠈⠋⠉⠉⠉⠉⠀⣸⣿⠟⢉⣀⣀⣀⣀⣉⣉⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣤⣤⣤⣤⣤⣤⡄⠹⣿⣿⣿⠀⣿⣿⣿⣿⣿⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⠿⠿⠿⠿⠿⠿⠋⠀⠿⠿⠿⠀⠙⠿⠿⠟⠋⠀⠿⠟⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⣠⣴⣶⣶⣦⣄⠀⣠⣴⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡏⣸⣿⣿⣉⡍⣿⣿⣧⢹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣾⣷⣿⣾⡇⢿⣿⣿⣯⣴⣿⣿⡿⢸⣷⣿⣾⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣀⠀⣼⡿⣿⣿⣿⡿⠟⠈⠻⢿⣿⣿⡿⠟⠁⠻⢿⣿⣿⣿⢿⣧⠀⣠⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣻⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⢙⣻⣟⡋⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠛⣿⣿⣿⣿⣿⣿⠻⣿⣿⣦⡀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣄⠀⠀⢠⣶⣿⡿⠋⣿⣿⣿⣿⣿⣿⠙⢿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠻⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⠀⠈⠻⠿⠇⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⠿⠋⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠙⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢠⣾⣿⡿⠋⣿⣿⣿⣿⣿⣿⠙⢿⣿⣷⡄⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⣿⠀⠀⠀⠙⠟⠋⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠙⠻⠋⠀⠀⠀⣿⣿⡏⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⡏⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⣬⣿
⠀⠀⢰⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⡀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢰⣿⡿⠉⡀
⠀⠀⢀⠱⣀⣠⣄⣤⠔⣂⠥⠤⢀⠀⠀⠀⠀⠘⢛⡛⢛⡛⠓⠀⠀⠀⠀⠸⠄⠀⠀⢀⡈⠁⠀⠠⠃⠀⠀⠀⣶⣤⣤⣀⣠⡄⡀
⠀⠈⢄⠣⣒⣒⣐⣒⣫⠔⠊⠁⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢾⣓⢀⡀
⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠒⠓⠀⠘⠓⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡻⠴⢻⡵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⣟⣛⣯⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⡀⠀⠀⡄⠀⠀⣀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠆⠀⠀⠀⠀⠀⠸⠛⠓⠈⠛⠁⣿⡥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡇⣀⣤⣶⣄⡀⣿⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠧⢸⢽⣽⢽⠤⠿⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣋⣤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⠀⠞⠻⠻⠛⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢷⣾⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣿⡷⢶
⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⣿⣛⣿⣛⣛⣿⣛⣻⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠇⠘
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣉⣉⣏⣹⣉⣉⣏⣩⣉⣉⠇⠀⠀⠀⠀⠀⠀⠀⢸⣾⢷⣶⣶⣦⣶⡾⠿⡇
⢀⡀⠀⣀⣾⣿⣀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠱⣀⣡⣨⣅⣨⣅⣼⣀⠜⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣷⣾⣿⡇
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠵⣷⣼⠮⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠿⠟⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠠⠄⠀⠈⠀⠉⠀⢠⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⡀⢀⠀⠹⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣷⠀⠀⠀⠀⠀⠗⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠂⠀⠀⠀⠀⢰⡗⠀⠀⠀⣀⠴⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟⣉⠀⠀⠀⠀⠀⠀⠃⠀⠀⢀⡘⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠀⠀⠜⠇⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠘⠁⠘⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡟⠁⠀⠁⠀⠀⣀⠀⠀⠠⡀⠁⠢⠞⠓⠀⠀⡀⠀⠄⠀⠀⠈⠉⠁⠀⠦⠀⠀⠀⡈⡘⠀⠀⠀⠀⠘⠿⠀⠀⠀⠠⠄⠀⠀⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣿⣿⠀⠀⢀⡀⠀⠤⠀⠸⡀⠁⢀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠾⠀⠀⠲⠄⢠⣴⡄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠞⣟⠉
⠀⠀⠈⠐⢰⡄⠀⠀⠀⠀⠀⢲⠀⠀⠄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠈⠀⠀⠠⠷⠀⠀⠀⠀⠀⠀⢠⠎⣰⠂
⠀⠀⠀⠀⠠⣿⣆⡄⠀⠀⠀⠀⠀⠀⠐⠀⣠⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠐⠃
⠀⠀⠀⠀⢰⠿⣿⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⠤⡄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠐⠃⠈⠀⠄⠀⠆⠀⠘⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⣤⣄⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⠤⠶⠖⢻⡏⢙⡟⠋⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠉⠈⠉⠀⠉⠉⠀⠑⠋⠉⠉⠈⠉⠈⠉⠁⠈⠉⠀⠋⠉⠈⠉⠈⠊⠉⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠒⠂⠰⠒⠒⠀⠂⠒⠒⠂⠐⠂⠀⠚⠂⠀⠂⠂⠐⠂⠃⠘⠒⠂⠐⠂⠂⠓⠂⠐⠒⠒⠓⠂⠐⠒⠐⠒⠂⠐⠐⠒⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⠿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⣿⣮⡻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣧⢿⣿⣧⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡞⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⣛⠿⣿⣿⣿⣿⣿⡜⣿⢻⣿⣿⡿⣫⣭⣭⡛⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⢿⣿⣮⢻⣿⣿⣿⠿⣀⢿⣿⡟⣾⣿⣿⣿⣿⣎⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⣭⣝⣇⢩⣭⣶⣿⣿⣷⣝⠸⣿⣿⣿⣿⣿⢏⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⢿⣿⣿⣿⣿⡟⣾⣶⣭⣭⣭⣵⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡼⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⡿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⢿⣿⣿⣾⣿⣿⣿⣧⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠤⠤⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠀⠀⠀⠀⠒⠀⠀⠀⠐⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠤⠀⠀⠤⠄⠀⠀⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡅⠉⠁⠀⠉⠀⢀⢀⡩
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢀⠀⠐⡒⠖⠈


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡋⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣳⠀⠀⠀⠀⠈⣻⠁⠀⠀⠀⠀⠀⢀⣠⠶⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠙⠷⣄⠀⠀⠀⣿⠀⢠⡞⢻⡆⠀⣿⠙⠦⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢹⠀⠀⢹⡟⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡤⠶⠿⠒⠲⠾⠧⣄⡀⣿⠀⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⢀⣠⠴⠶⠶⢤⡀⠀⠉⠻⣤⡀⠀⠀⠀⠀⠸⣤⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠳⣤⣤⣠⣄⡼⠃⠀⠀⢠⢞⡵⠛⠉⠙⠳⣝⢦⠀⠀⠈⠳⣄⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⣷⠀⠀⠛⠚⠁⠀⠀⡾⠁⠀⠀⠀⣾⢸⠁⠀⠀⠀⠀⢹⢸⡆⠀⠀⠀⠹⡏⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠳⠶⠶⠶⠶⢾⠇⠀⠀⠀⣴⠿⠾⠶⠶⠶⠶⠶⠾⠾⢧⡄⠀⠀⠀⣿⣄⣤⣤⣤⣤⣤⡴⠛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⣠⡀⠀⠀⠀⢸⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠙⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡴⠟⠛⠛⠛⠛⠛⠛⠻⡇⠀⠀⠀⣿⠀⠀⠀⠀⣻⠃⠀⠀⠀⢸⡇⠀⠀⠀⡿⠶⠶⠶⠶⢤⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣿⡄⠀⠀⣿⠀⠀⠀⠀⠛⠃⠀⠀⠀⢸⡇⠀⠀⣼⠃⠀⢀⡤⣄⠀⠈⠙⣟⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠹⣄⠀⠻⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠃⢀⡾⠛⠛⠛⠻⢦⡼⠁⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠳⡆⠀⠀⠀⠀⠀⠈⢻⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠚⠁⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⣿⠓⠒⢲⡟⠋⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⣉⣷⠀⢸⡇⠀⠀⠙⢶⣀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣄⢸⡇⠀⠈⠉⠁⢰⠞⢳⡄⠀⠀⠀⢹⡋⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠟⠉⠁⠀⠀⠀⠀⠈⠓⠛⠀⠀⠀⠀⠀⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣷⣤⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠆⣠⣶⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⢠⣤⣤⣤⣤⣀⣉⠙⠻⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠃⣾⣿⣿⣿⣿⣿⣿⣿⣧⣠⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠿⠿⢿⣿⣶⣤⣈⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠛⣁⣤⡄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣷⣶⣶⣦⣤⣀⣉⠙⠻⢷⣦⡈⠳⡄⠀⠀⠀⠀⠀⢀⡞⢋⣤⡾⠟⠋⣁⣈⠻⢿⣿⣿⣿⣿⣿⣇⣠⣿⣿⣧⣠⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠈⠛⠛⠉⠉⠛⠛⠛⠛⠛⠿⠿⢶⣦⣌⡙⠦⠈⠀⠀⠀⠀⠀⠉⡴⢛⣡⣴⡶⠿⠿⠛⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣦⣤⣤⣀⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣉⣁⣠⣤⣤⣶⣶⣄⠻⣿⣿⣿⣿⣿⣇⣀⣿⣿⣇⣀⣿⣿⣿⣿⣿⣿⠟
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣦⣤⣴⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⢻⣿⣿⣿⣿⣿⣿⠟⠘⢿⣿⣿⣿⣿⣿⠟
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣍⡛⠻⠟⢛⣡⣾⡇⠀⠙⠛⠛⠋⠁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠉⠉⠙⠛⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⠛⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠃⠈⢿⣿⡿⠃⠀⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⢠⡴⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁
⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⣀⡀⠀⠀⢸⡀⠀⣧⠀⠀⠀⢀⡴⢦⡀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠈⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⡆⠀⠈⡇⠀⢸⠀⠀⠀⣾⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉
⠀⠀⠀⠀⠀⠈⠳⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢹⡤⠤⡇⠀⠘⡧⠤⢲⡇⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣄⢠⠖⠒⠒⢒⠚⠛⠉⠹⡄⠀⢣⠀⡇⠀⠀⡇⠀⡸⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⣀⠰⡏⠙⢦⣀⣀⡇⠀⠈⢇⢸⠀⠀⠃⢀⠃⠀⢠⡇⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⠉⠙⣄⠀⠱⡀⠿⡄⠀⠈⠈⠀⠀⠐⠀⠀⠀⣸⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠘⣆⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⢀⡴⠋⢉⡳⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⡈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⠋⢉⡞⠁⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠉⠙⠋⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⢀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⢀⣠⠴⢻⠁⡤⠊⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⠦⢤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⡤⠖⠋⠀⠀⣠⠏⠀⠀⣠⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⣠⠎⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠫⣄⡀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⣠⢾⡁⢀⡞⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⠀⠀⠀⠙⠲⠤⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⡴⠃⠀⢣⡞⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠸⣄⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠑⠤⠼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⢡⠀⡀⠀⢸⠉⢹⠭⠍⠉⠉⣍⢹⠀⠀⢸⠉⠏⠭⠉⠉⠉⠉⡇
⠀⠀⠀⠀⡤⠄⠀⠄⡠⠀⢢⠂⠀⠀⢹⠀⠁⠀⢸⠃⠀⠀⠀⠀⠀⠦⢸⠀⠀⢸⢀⢠⠤⠄⠀⠂⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⢸⡄⠐⠒⠀⠀⠀⠖⢺⠀⠀⢸⡜⠋⠀⠉⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠂⢲⣤⡄⠀⠀⠀⠀⠉⠉⠉⠁⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠘⠿⠃⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⠄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠄⠐⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣺⠀⡀⢺⠂⠀⡖⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠏⢸⣇⡿⢠⣾⣷⣧⣍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠙⠛⠉⠇
⠀⠀⠀⠀⠀⠀⢀⣨⣿⣿⣿⣳⠛⣿⡏⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠎⣿⣿⣏⡯⢙⡿⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⡆⢄⠀⡀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣮⣉⠡⠡⠆⠌⠛⠉⠉⠀⠀⠀⠀⠀⠀⡁⣄⠀⢩⠀⠌⡰⠅⠀⠈⠁⢀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠺⠀⡘⢃⠀⠲⢦⡀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠴⢛⡀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠰⠂⢁⡌⠀⠁⠀⠐⠀⠲⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢨⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠈⠀⠠⠄⠀⠐⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣾⢅⠀⠀⠀⠀⠈⠀⡀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢄⢀⢠⡀

⠀⠀⠀⠀⠀⠈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣦⠀⠀⡀⠂⠈⠁⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⢀⠄⠂
⠀⠀⢨⠩⠉⠉⠉⠈⠂⠀⠀⠀⠀⠤⠀⠀⣿⣿⡆⠀⠀⠁⠀⠂⠀⠘⠀⠀⠀⠀⠰⣤⣶⡶⢶⣶⣶⠒
⠀⠀⠀⠲⣦⣤⣤⣶⡾⠀⠀⠀⡀⠀⠀⠀⡈⠉⢡⠀⠀⠀⢰⣶⣶⣶⣶⡄⠀⠀⠠⠙⠛⠁⢈⠉⠁
⠀⠀⠀⠀⠈⠁⠀⠂⠄⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⠀⠀⠈⠄⠀⠠⠀⠈⠀⠀⠀⠑⠠⠉⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⢠⣤⠀⠀⣀⣠⣤⣤⣤⣄⡀
⠀⠀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠈⠱⠀⠀⠀⠂⠀⠀⠀⠀⠐⠅⠀⣼⣿⣿⣿⣿⣿⣿⣿⣀
⠀⠀⠰⠀⠌⢀⣄⠄⠆⡀⠀⠠⠐⠂⡄⠠⠄⡈⠀⠀⠀⡆⠀⠻⠃⠻⠀⠀⠀⠀⠿⠻⠟⣛⣛⠿⠛⠛⣿
⠀⠀⠀⠄⠀⠸⠻⠀⠀⡀⠀⠀⢀⠀⠀⠀⠂⠀⢀⠀⠀⠀⡙⠶⣖⡒⠈⠁⠈⠀⠠⠀⠀⠋⠉⠀⠀⡀⠉
⠀⠀⠀⠀⠁⠂⠀⠀⠈⠀⠠⠦⠀⠰⠀⠠⠀⠐⠁⠀⠀⠀⠈⠐⠀⠀⠀⠈⠀⠀⠀⠀⠂⠠⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠒⡁⠀⠐⠀⠠⠀⠀⠀⠀⠀⠄⠀⠀⠠⢀
⠀⠀⠀⠀⣤⡄⢠⡶⠀⠁⠀⠀⠂⢀⡀⠀⢀⠠⡄⠀⠃⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡀⢀⠀⠀⠁
⠀⠀⠘⠀⡉⠁⠀⠀⡀⠀⠀⠘⠂⠈⠅⠀⠌⠀⠒⠀⠀⠀⠎⣀⣤⣤⡡⠄⠀⠀⠀⠀⡠⠑⠒⠄
⠀⠀⠀⠠⠀⠀⠀⠊⡡⠈⠀⠀⠠⣀⠀⠀⢀⣠⠆⠀⠀⠁⠀⠛⠉⠙⢛⠔⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⣄⠐⡤⠀⠉⠉⠰⠆⡀⡄⠀⡠⠀⠀⠀⠀⠠⡀
⠀⠀⡆⠀⢀⣀⣀⣀⠀⢠⠀⠀⠁⣀⠀⠀⠀⣀⠀⠀⢀⡈⠀⠀⠀⠀⠄⠀⢡⠇⠐⢀⠀⣀⡧⠤⠤
⠀⠀⣿⣿⠉⠉⢉⣉⣻⣿⡄⠀⡀⠄⢤⠀⢤⡀⠀⠄⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⢴⣟⠁
⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠦⡀⠀⠀⠤⠂⠀⠀⠈⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠳⠦
⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢠⣶⡆⠉⢡⣀⣴⡄⠀⢀⠔⢲⣶⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⡟⣿⣿⡦⡤⠄⠠⣼⢀⡎⠀⣠⠟⠒⠲⢬⡙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢸⣿⣿⣿⣇⢿⣿⣿⣿⣿⣿⣿⣾⠁⠀⠁⡀⠀⠀⠀⢹⡿⣏⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁⠀⢀⣝⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⡏⠀⠀⠀⣸⠇⣹⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⢤⠈⣽⣿⣿⣿⡾⠚⢻⣿⣿⣟⣿⣿⡿⠀⠐⢾⡁⠀⠀⠀⠉⢳⡟⠀⣀⣼⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⢸⣿⣏⣿⣿⣿⣟⣠⣤⠬⣧⡀⠀⠀⠀⢈⣇⠀⠈⠙⣷⡿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⣦⠀⠀⢸⣿⣿⣿⣿⠀⠀⢸⡟⣾⣿⣿⢻⡏⠀⠀⠀⠀⢀⣤⡴⠶⠾⠿⢄⡀⠀⠸⣇⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠀⠈⠙⠲⠾⠿⣿⣿⣿⠀⠀⢸⣱⣿⣿⠁⢨⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠂⠀⣿⢰⡿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠖⣀⡚⠚⢷⡤⣼⡇⠀⠀⢰⣿⣿⣿⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠈⠁⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣝⡟⠁⠀⠀⠀⠙⠃⣤⡈⠳⡀⠀⢸⣿⣿⣿⠀⢸⡷⠶⠶⠒⠚⠾⠅⠀⠀⠀⠀⢀⣦⡤⠟⠀⢠⡾⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣏⡛⠀⡰⠀⠐⢤⠀⠀⠀⠁⠀⠙⣦⡼⠿⢿⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠻⠛⠓⠀⠀⠀⢿⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡇⠾⢄⠔⠁⠀⣡⡀⠀⠀⠀⠰⣿⡀⠀⠈⣿⠀⢸⡇⠀⠀⠀⠀⠀⡀⢀⣠⠇⠀⠀⠀⢀⠀⠀⠈⠀⠀⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠠⡖⠁⠀⠐⠀⠀⠀⠀⢹⣧⠀⠀⢻⠀⢸⡇⠀⠀⠀⠀⢀⡿⠚⠁⠀⠀⠉⠋⠉⠉⠓⣆⠀⠚⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠰⣀⠤⠐⡇⠀⠀⠀⠀⠿⠀⠀⠸⠀⢸⣿⡶⠶⠶⠶⣿⡀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⡇⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⣀⠀⠀⠇⠀⠀⠀⢸⡟⠀⠀⠀⠀⠈⠙⠶⠦⠤⠤⠤⠞⠀⢠⣤⠟⢰⡆⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠸⠟⣟⣹⡆⠸⣀⡅⠀⢰⠀⠀⠀⢸⡇⠀⠀⠀⢸⣦⠀⠀⠀⢀⣄⠀⠀⠀⢨⣿⡀⠘⣷⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⡿⣿⣿⡇⠀⠉⠀⠀⢸⠀⢀⠀⢸⡇⠀⠀⠀⣸⠏⠙⠛⠛⠛⠉⠛⠚⠚⠋⠙⢿⣞⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠐⠧⣄⣹⠃⠀⢠⠀⠀⣼⣴⣾⣧⣸⡇⠀⠀⣼⠏⠀⠀⣰⣄⡀⡴⠖⠛⠳⣄⠀⢈⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡻⣄⣀⡜⠳⢤⡝⠀⢀⡄⠀⢀⠟⢿⣿⣿⣿⣇⣠⣿⣏⢀⣠⠴⠋⠈⢻⡅⠀⠀⠀⢹⣶⡾⠁
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣏⠻⢦⣾⠁⣠⡿⢀⢀⣾⣆⣼⣿⣿⣧⡟⠁⠀⠉⠉⠁⠀⢀⣰⣾⠃⠀⠀⢐⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⣙⣣⣏⣴⣿⣿⣿⣿⣿⣮⣻⡿⡇⠀⢀⢀⣹⣷⡾⠿⠋⠁⠀⣰⣀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣾⣟⢿⡿⣿⣿⣿⣿⠃⣟⠒⠛⠋⣉⠀⠀⢀⣆⣠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠽⢿⣿⠟⠃⠀⠘⢦⣤⣴⣿⣤⠶⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠉⠉⠉⡉⠉⠉⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⡇⠉⡝⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢻⠀⠀⠀⠅⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠈⠳⡌⢳⣄⡴⠒⠒⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⢀⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠤⠤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠋⠉⢻⣉⡟⠉⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠉⡇⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡀⠀⢸⠀⡇⠀⢀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡇⠀⢸⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⡇⠀⠘⠒⠃⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⢀⣀⣀⣀⣀⣀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢸⠀⠀⣇⣀⣀⣀⣀⡀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣸⠃⠀⠀⠀⠀⠀⢸⣀⣰⠃⠀⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⡇⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠚⠉⠲⠒⠋
⡆⢰⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⡇⢈⠐⠀⠁⢠⠹⡀⠆⠇⠀⠀⠢⠀⠀⠃⢄⠈⠁⠨⢌⢰⠀⢱⢸⢎⠀⠄
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣷⣷⣾⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣷⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠛⠛⢿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡏⠀⠀⠀⠀⢹⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⣠⡶⠶⠿⠿⠿⠷⠶⠶⠶⠶⠾⠿⠿⠿⠶⢦⣄⠀⠀⠀⠀⠀⢀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠏⢠⣤⣤⣤⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⣠⣤⣴⡆⠸⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣈⣁⣀⣀⢸⡇⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⢸⡇⣀⣀⣉⣁⣀⣸⡇
⠀⠀⠀⠀⠀⠀⣠⣴⣄⢀⡈⠉⢉⡉⠉⣉⠉⢸⡇⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⢈⠉⣉⡉⠉⣉⠁⢀⣤⣶⣄
⠀⠀⠀⠀⠀⠀⠻⣿⠿⠛⠛⠛⠛⠛⠛⠛⠋⢸⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⢸⡇⠙⠛⠛⠛⠛⠛⠛⠛⠻⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠛⠓⢸⡇⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠛⠛⠛⠛⠛⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⡄⢸⡿⠶⠆⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣸⡇⠲⠶⢾⠇⢠⣾⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠃⠀⠀⠀⠀⠀⠻⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⠟⠀⠀⠀⠀⠀⠘⠿⠿⠃






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⣛⣛⠿⠿⠛⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠠⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⢀⣴⣿⡇⠀⠀⠀⠀⠀⠀⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠉⢿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣷⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⣠⣾⡿⠀⣴⣶⣶⣶⣶⣾⣿⣿⣶⡄
⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣾⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣠⣀⣀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⣀⡀⠀⠀⠀⠀⢀⣤⣤⠀⠀⠀⠀⠀⠀⣀⣀⣄⣤⣀⡀
⠀⠀⠀⠀⢔⡴⠚⠉⠀⠀⠀⠀⠉⠙⠦⣐⢄⠀⠘⠛⠁⠀⡠⢀⡵⠚⠉⠀⠀⠀⠀⠉⠓⢮⡀⢄⠘⠿⠿⠀⠀⡠⢂⡴⠋⠉⠀⠀⠀⠀⠉⠓⢬⡂
⠀⠀⡔⣵⠋⠀⢀⡤⠤⠀⠀⠠⢤⣄⠀⠈⢧⡑⠀⠀⠀⠜⡴⠋⠀⡴⠛⣦⠀⠀⣴⠛⢦⠀⠙⢦⠡⠀⠀⢀⢈⡼⠃⠀⣠⢤⡀⠀⠀⣠⢤⡀⠀⠙⣮⠢
⠀⠘⣼⠁⠀⠀⢿⣟⣷⠀⠀⢾⣻⣿⠀⠀⠀⢳⠀⠀⠈⡾⠁⠀⠘⠃⠀⠛⠀⠀⠛⠀⠘⠃⠀⠈⢷⠁⠀⣀⡞⠀⠀⠀⢯⣴⡗⠀⠀⢿⣦⡿⠀⠀⠈⣧⠃
⠀⢀⡇⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠁⠀⠀⠀⢸⣷⠀⢰⡇⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⣇⠀⠀⠀⣠⠶⣦⣴⢶⣠⠶⣤⠀⠀⠀⢸⠏⠀⠸⣇⠀⠀⠀⠈⢧⣀⠀⠀⣀⡼⠁⠀⠀⠀⣸⠃⠀⠻⡇⠀⠀⠀⠐⠛⡟⠛⠛⣿⠛⠂⠀⠀⠀⣸⠁
⠀⠈⠘⢦⡀⠀⠀⠀⠈⠁⠀⠉⠀⠀⠀⠀⣠⠏⠀⠀⠀⠙⣆⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⡰⠋⠂⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠣⠁
⠀⠀⠀⠀⢙⠦⣄⣀⣀⠀⠀⢀⣀⣠⡴⠚⠁⠀⠀⠀⠀⠀⠈⠓⢦⣄⣀⡀⠀⠀⢀⣀⣠⡴⠛⠁⠀⠀⠀⠀⠀⠈⡓⢦⣄⣀⡀⠀⠀⢀⣀⣠⠴⠋
⠀⠀⠀⢀⣤⡁⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⣰⣶⠀⠀⢀⣀⠀⠀⠀⠐⠋⠉⠉⠉⠉⠙⠂⠈⠀⠀⢀⠀⠀⠀⠀⣀⡀⠁⠀⠉⠉⠉⠉⠉⠙⠂⠈⣠⡀
⠀⠀⠀⠘⠛⠁⢀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⡗⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠰⣿⣿⠀⠀⠀⠻⠋⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠛⠁
⠀⠀⠀⢀⠄⣨⠖⠋⠁⠀⠀⠀⠉⠓⢤⡂⢄⠀⠀⠀⠈⠉⢀⢔⡤⠖⠉⠁⠀⠀⠈⠉⠲⢥⡢⡀⠀⠀⠀⠀⠀⡀⢐⡤⠚⠉⠁⠀⠀⠈⠙⠲⣅⡠⡀
⠀⠀⡐⣡⠏⠀⣀⠀⢀⠀⠀⡀⠀⢀⠀⠙⢦⠠⡀⠀⠀⠔⡵⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢮⠢⠀⠀⠀⠔⡴⠋⠀⣠⢦⡄⠀⠀⣠⠦⣄⠀⠙⣎⢂
⠀⠐⣼⠃⠀⠀⠹⠦⠞⠁⠀⠻⠦⠞⠀⠀⠈⢷⠁⠀⠈⡼⠁⠀⠀⠚⠒⠒⠀⠀⠒⠚⠓⠀⠀⠈⢧⠁⠀⠀⡾⠁⠀⢸⣇⣿⡷⠀⠀⣿⣧⣹⠄⠀⠘⣧⠂
⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⡇⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣿⡇⠀⠀⠀⠉⠉⠁⢀⡀⠈⠉⠉⠀⠀⠀⢸⡀
⠀⠀⣇⠀⠀⠀⠘⢦⣤⣀⣀⣠⡴⠟⠀⠀⠀⢸⡟⠀⠘⣇⠀⠀⠀⠀⠀⣰⠟⠻⣆⠀⠀⠀⠀⠀⢸⠃⠀⢻⡇⠀⠀⠀⠀⠀⣾⠉⢙⡧⠀⠀⠀⠀⠀⣸⠁
⠀⠈⠜⣆⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣠⠟⠀⠀⠐⡹⣄⠀⠀⠀⠈⠁⠀⠀⠈⠁⠀⠀⠀⣠⢏⠀⠀⠀⠹⣄⠀⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⣰⠃⠁
⠀⠀⠀⠈⡓⢦⣀⡀⠀⠀⠀⠀⢀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠈⡳⢤⣀⡀⠀⠀⠀⠀⢀⣀⡤⢞⠁⠀⠀⣤⣄⠀⠈⠳⢤⣀⡀⠀⠀⠀⠀⢀⣀⡴⠚⠁
⠀⠀⠀⣀⡀⠁⠊⠍⠙⠛⠛⠛⠩⠉⠂⠁⠀⠀⠀⠀⣠⣤⡀⠈⠑⠈⠍⠙⠛⠛⠋⠩⠑⠊⠁⠀⠀⠈⠛⠋⠀⠀⠈⠐⠈⠍⠙⠛⠙⠉⠩⠑⣨⣤
⠀⠀⠀⠛⠃⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠸⠿⠀⠀⠙⠛⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠛⠛
⠀⠀⠀⢀⢔⡠⠞⠉⠁⠀⠀⠀⠉⠓⢦⡀⢄⠀⠀⠀⠀⠀⡀⢀⡤⠚⠉⠁⠀⠀⠈⠉⠓⢬⡂⠀⠀⠻⠗⠀⠀⡠⢀⡴⠚⠉⠁⠀⠀⠈⠉⠳⢤⡢⡀
⠀⠀⡐⣱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡡⡀⠀⠀⠔⡵⠋⠀⣀⠀⠀⠀⠀⠀⠀⣀⠀⠙⢮⠢⠀⠀⠀⢌⡴⠋⠀⢠⣄⡀⠀⠀⢀⣠⠄⠀⠙⣎⢂
⠀⠐⣼⠁⠀⠀⢲⣶⡦⠀⠀⠰⣶⡶⠀⠀⠈⢳⠁⠀⠈⡼⠁⠀⠀⠙⣷⣄⠀⠀⣠⣾⠋⠀⠀⠈⢧⠁⠀⠀⡞⠀⠀⠀⢀⡼⠟⠀⠘⠿⣤⠀⠀⠀⠘⣧⠂
⠀⠀⡇⠀⠀⠀⠀⠉⠀⠀⠀⠀⠉⠀⠀⠀⠀⢸⣿⠀⢰⡇⠀⠀⠀⠀⠈⠁⠀⠀⢈⠁⠀⠀⠀⠀⢸⡆⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⣇⠀⠀⠀⠀⢶⣄⠀⠀⣀⣴⠂⠀⠀⠀⢸⠟⠀⠘⣇⠀⠀⠀⢠⡞⢷⡼⢧⡾⢳⡄⠀⠀⠀⣸⠃⠀⠻⡇⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠀⠀⠀⠀⣸
⠀⠈⠜⣦⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⣠⠏⠀⡀⠀⡙⣆⠀⠀⠀⠀⠈⠁⠈⠁⠀⠀⠀⠀⣰⠋⠂⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⠁
⠀⠀⠀⠈⠛⢦⣄⣀⠀⠀⠀⠀⣀⣀⡤⢞⠁⠀⠸⣿⠇⠀⠈⡳⢦⣀⣀⠀⠀⠀⠀⣀⣀⡴⠞⠁⠀⠸⠗⠀⠀⠈⡳⢤⣀⣀⠀⠀⠀⠀⣀⣠⡴⢛⠁
⠀⠀⠀⠀⠀⠁⠐⠉⠉⠛⠋⠉⠉⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠉⠉⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠉⠉⠛⠋⠉⠉⠀⠈

⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂
⠀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡠⠓⠄⠀⠀⠀⠀⠀⠀⣅⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠
⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⣀⠄⠊⠀⢀⣞⠀⠀⠀⠀⠀⢈⠀⠈⠐⠂⠄⡀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠂⠉
⠀⠀⠀⠈⠁⠒⠂⠤⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡀⣀⣤⠊⡙⠁⠑⠠⡀⢀⡠⠁⠈⠐⠂⠠⡤⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠄⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠠⠄⠀⡀⠀⠀⠈⠑⡋⠉⠀⠀⠀⠀⣘⠁⠀⠀⠀⠀⠀⠀⡁⡀⢀⣀⠄⠠⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠒⠀⠀⠀⠚⡀⠀⠀⡔⠞⠁⣏⡄⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠒⠀⠤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠱⣀⣀⡇⢨⡆⢻⢠⠆⠤⠊⠀⡌⠀⠀⠀⠀⠀⠀⢈⣄⠀⠀⠀⡀⢀⡀⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠄⢀⠐⠂⠠⢀⠀⠀⠀⡐⢉⠈⣼⠈⡇⢸⠀⣤⠰⠇⡐⠀⠢⡀⡀⠔⠂⠈⠀⠈⠢⠄⠐⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀⡀⠔⠊⠉⠂⠔⢀⣯⠀⢶⠀⠇⠀⠄⠻⠀⡎⠀⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠒⠈⠈⢄⠀⠡⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⡨⠒⠢⣀⣀
⠀⠀⠀⣀⠠⠄⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠂⠀⢡⣀⣀⣀⠁⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠰⢀⡀⠀⢀⠊⠀⢀⠔⠀⠀⠁⠢⢀
⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠈⠀⢀⠌⠀⠀⠀⠀⠀⣢⠀⠀⠀⠀⠀⡇⠀⠈⠑⠢⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠄⠒⠂⠀⡠⠐⢷⠠⠀⠀⡀⠔⠈⠀⣡⠠⠄⠀⠤⠇⡀⠀⠀⠀⠑⢜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠀⠁⠀⠀⠀⠀⢀⠎⠀⠀⠀⠑⣔⠆⠀⠀⠀⠸⠀⣀⠀⠤⠤⠀⣠⠀⠀⠀⠀⠈⠐⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢀
⠀⠀⡀⠠⠄⠂⠁⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀
⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⡤⠄⣲⣆⡚⢿⡾⠿⠶⠂
⠀⡴⢦⣄⠀⠀⣀⠠⣤⣰⣖⡪⢝⡚⣖⣾⠕⠒⠛⠀⠉
⣸⡇⠀⠈⠙⠲⠄⠉⠛⠳⠌⠙⠀⠉
⢻⡆⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀
⠘⠻⣶⣀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠶⠚⠛⠋
⠀⠀⠈⠙⠿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠉⠒⢄⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠶⠾⠛⠉
⠀⠀⠀⠀⠀⠈⠛⢶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣦⣤⣴⡶⠞⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⣸⡿⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠠⠔⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⣿⡇⠀⢀⣀⠤⠶⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠷⠏⠛
⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣶⣿⣿⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠻⠻⠿⠯⣿⣿⡿⠿⠙⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣀⣀⡀⠀⠀⠸⠀⠀⠀⠀⠹⣷⣀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠰⡆⠀⠀⠀⠀⠈⠉⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣤⢴⣶⣿⣶⣶⠆⠀⠀⠀⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢈⣻⣿⣿⣭⠀⠀⠀⠀⠀⠈⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠂⠉⠉⠉⠱⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠁⣀⣤⠾⣦⣀⣙⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣨⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⠯⠶⠀⠰⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢻⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⡴⠀⢀⠶⢴⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⣿⣁⠀⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡈⢧⡀⠺⣂⣼⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡥⣚⡉⠄⠂⣿⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣸⠛⠶⢿⣿⠺⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡜⣿⡏⠀⠀⠀⢧⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣯⣿⠋⠀⣀⣨⡿⠗⠚⣷⣶⠖⠋⠉⠉⠉⢉⣉⣤⣴⣿⡿⠀⠀⠀⠀⠘⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠛⣿⡛⠦⣭⣥⡴⠶⢟⣽⣏⣴⣿⣿⣿⠿⠿⠟⢻⢛⠁⠀⠂⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣠⡇⠹⣮⡳⣤⡉⠀⠀⠚⠁⣿⡿⠋⠁⠀⠀⠀⠀⠀⠘⠀⠀⠈⠴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⠁⣰⡿⠁⠀⣌⠙⠦⣀⣤⣶⡀⠀⡟⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣾⠉⠻⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⣿⢠⣿⣧⣿⣄⢨⡳⣦⣙⢿⣿⣧⢸⠁⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⡿⠏
⠀⠀⠀⠀⠀⠀⢰⡏⣸⡟⢹⣿⣿⣆⢳⣬⣛⠷⣽⣿⣿
⠀⠀⠀⠀⠀⠀⡾⢡⣿⠀⠘⡇⢻⣿⡜⢿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⣼⢱⣿⠇⠀⠀⣧⠈⢿⣿⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢰⡇⢸⡿⠀⠀⣸⡟⡄⠢⡉⠉⠉⣩⣾⣿⠟⣇
⠀⣴⣶⡄⣿⢀⣾⠃⠀⠀⣿⣷⡜⠀⠙⣶⣾⣿⠟⢡⢸⣿⡆
⠀⠙⣿⢠⠇⣼⠃⠀⠀⢸⡏⠘⢿⡄⠀⠈⠉⠀⣴⡏⣼⣿⡀
⠀⠀⢸⠇⣼⠃⠀⠀⠀⣾⣿⣦⡀⢙⡁⠀⢠⣾⡟⢰⣿⣿⡇
⠀⠠⣋⣼⣿⠀⠀⠀⠀⣿⠋⠙⠿⠋⢀⣴⣿⠟⣰⣿⡿⣿⡇
⠀⠀⢻⣿⣿⣀⣠⣶⡀⣿⠀⠀⠀⣠⣾⡿⠃⣰⣿⣿⢣⣿⣿
⢷⠾⠿⢿⣿⡏⠉⠈⠁⣟⠀⢀⣼⣿⠟⢠⣾⣿⣿⡏⣼⣿⣿⡇
⠀⠀⠀⠘⣿⣧⠀⠀⢸⣿⣴⣿⡿⢃⣴⣿⣿⣿⡿⢠⣿⣿⣿⣿
⠀⠀⠀⠀⢻⣿⡄⠀⢸⣿⣿⡟⣵⠟⣹⣿⣿⣿⢃⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣧⠀⢸⣿⡟⢸⡏⢠⣿⣿⣿⠇⣾⣿⣿⣿⢻⣿
⠀⠀⠀⠀⠀⢹⡜⡄⢸⣿⣇⣿⠀⣾⣿⣿⠏⣼⣿⣿⣿⣿⠀⣿⡄
⠀⠀⠀⠀⠀⠈⣷⢁⢸⣿⣿⡏⢰⣿⣿⡟⣰⣿⣿⠿⣿⡟⠀⢿⡇
⠀⠀⠀⠀⠀⠀⢹⡎⣸⣿⣿⢃⣿⣿⡿⣱⣿⣿⡿⠀⣿⡇⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠈⣷⢿⣿⣿⣼⣿⠏⣰⣿⣿⣿⠇⠀⢹⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢹⡞⣿⣿⣿⡟⣴⣿⣿⣿⡟⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠈⣷⢻⣿⣿⣾⢿⣿⣿⣿⣡⠀⣧⢠⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⣿⡿⢃⣿⣿⣿⣷⡏⢸⣿⣿⣿⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⢀⣾⣿⣿⣿⣿⡇⣾⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿
⠀⠀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⣀
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣠⠀⠠⡄⠀⠀⡀
⠀⠰⠀⢰⠀⠀⠀⠀⠀⡄⠀⠸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣏⠱⠒⣎⣈⡅⠀⠅⠀⠀⠁
⠀⠀⡀⢸⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠔⢃⣼⠝⠁⢠⠖⡖
⠀⠀⡇⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠾⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡰⣅⣠⢿⠞⠀⠰⠁⠛
⠀⠀⡇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢀⣤⠦⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠠⡄⠀⠉⠒⠦⡹⢤⠔⡰
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠇⠲⠾⢻⠟⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⢂⢀⡤⣀⠀⠀⠀⠀⠈⠙⠉⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠐⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢹⣉⠛⠉⢳⡠⠁⠀⠀⠀⠀⠀⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠴⠄⠓⣄⣠⣄⣿⣷⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢨⡭⠁⢠⠏
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣀⡈⠀⠀⠀⠀⠒⠛⢡⠛⠒⠷⣆⣴⢛⣉⡉⠧⠢⠀⠘⢡⡿⠠⠴⠋⠉⠂⠀⠀⢸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠊⣄⡹⣯⡇⠀⢀⣀⠀⠀⢀⣾⠀⠀⠰⠁⢼⡋⠉⠓⠋⠀⡷⣇⣀⡋⢠⠎
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠐⠓⠈⠉⠀⠄⠐⢄⠀⠀⠀⢸⠁⠈⣴⣶⠃⠐⣉⣙⠂⠀⠀⠀⡰⠃⠈⣷⠋⠀⢀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠘⠂⠶⠎⡰⠈⠻⡁⣂⠀⠉⠉⠀⠀⠀⣐⣡⣄⣴⠃⠉⠈⠙⠃⠀⠸
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⣤⣤⣄⠀⠀⠀⡁⠀⠀⣿⡏⠀⠠⠄⢀⡠⢺⡏⠀⠈⠙⠛⣧⠀⢀⡤⠖⠁
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⣾⣿⣿⡟⠀⠀⠀⠃⠀⢠⣿⡇⠀⣼⣏⠀⠀⣼⡇⠀⠀⠀⠀⠈⠀⢹⣿⡈⡇⠀⠀⠀⠀⠀⠀⠀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⣿⣿⣿⣿⠁⠀⠀⠘⠀⠀⣼⣿⣷⣿⣦⣿⡖⣰⣿⣧⠀⠀⠀⢡⠀⠀⢸⣿⣧⠃⠄⠀⠀⠀⣠⢆⡥⢤⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣼⣿⣿⣿⠃⠀⠀⠀⠀⠈⠀⣿⣿⣿⣿⣘⣉⣤⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⡄⢌⡀⠤⣊⠵⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣃⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⡍⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⣄⣀⣠⣤⣤⣤⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣴⣶⣶⣾⣿⣿⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠐⠒⠂⠉⠉⠉⠑⠒⢸⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣿⣿⣿⣿⡇⢠⠀⠀⠀⠀⠀⡇⠀⠀⠠
⠀⠀⠀⠀⠀⢀⡠⠤⠒⢻⣿⣿⣿⣿⣿⣿⣿⠋⡜⠁⠀⠉⠙⠃⠤⣤⡀⡆⠀⠀⠀⢰⣄⠀⢈⡁⠀⠡⠈⢻⣿⣿⣿⡏⠈⠁⠂⠤⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠺⠁⠄⠀⠄⠷⠈⠿⢿⣿⣿⣿⡇⢠⡁⠀⠀⠀⣀⣤⠖⠛⠁⠇⠀⠀⠀⢸⢖⣐⠀⠀⠀⣀⠆⢈⣿⣿⠟⠀⠆⠀⠀⠀⠘⢦⡄⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⠚⡙⠁⠀⠀⢍⣀⣀⣀⣀⣇⣀⣀⣀⣸⣀⡘⢻⡀⠀⠀⠉⠁⢌⠁⠀⠀⠀⠀⠀⠀⣀⣤⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⣠⠊⠀⠀⠀⠀⣠⠀⠀⠀⢰⡇⠀⠀⠀⢹⠀⠀⠀⢷⡀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⣀⣈⣄⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡜⠁⠀⠀⠀⠀⠀⠁⠛⠒⠒⢺⡗⠒⠒⠒⢺⠗⠂⠀⠈⠱⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⠍⠻⠀⠀⠀⠀⠀⠀⠀⡤⣖⠀⢀⠸⠗⠀⠒⠒⠾⠒⠒⠒⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠄⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠂⠂⠀⠠⠽⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣶⢸⡇⢰⠃⢀⣀⡀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⡀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⣀⣳⣤⠀⢧⠀⢸⣷⠋⢀⣿⡯⠽⠆⣿⠁⠀⢠⡇⢸⡇⣼⠿⣵⣿⠀⣼⢰⡷⠙⠚⠁⣤⣞⣀
⠀⠀⠀⠀⠀⠀⢠⣭⣭⠗⠀⠀⠉⠉⢻⡄⡸⢿⣀⡀⠀⢿⣤⠎⠈⢧⣸⡗⢿⡠⠃⢿⡠⠿⡼⡇⠀⠀⠀⠺⣭⣭⡄
⠀⠀⠀⠀⠀⠀⠀⢉⣀⣀⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⡔⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠻⣷⣄⣴⡶⠷⣶⣄⢠⣴⣶⣶⣦⣶⣶⣶⣤⣄⣴⣶⣶⣶⣤⣶⠶⣶⣄⣴⡿⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⣧⣤⣼⡿⣿⣯⣤⣤⡀⣿⣧⣄⣼⣿⣿⡏⠀⢸⣿⡇⠀⢹⣿⢿⣷⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⡿⣿⡁⠀⣿⡏⠉⠉⠁⣿⡏⠉⢹⣿⣿⡇⠀⢸⣿⡇⠀⢸⣿⠀⠈⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣿⣿⡇⢹⣷⡀⣿⣧⣀⣀⣀⣿⡇⠀⢸⣿⣿⡇⠀⢸⣿⡇⠀⣹⣿⣀⡀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠾⠟⠁⠿⠃⠀⠛⠓⠛⠛⠋⠉⠉⠉⠁⠀⠈⠉⠙⠃⠀⠈⠛⠃⠀⠹⠟⠙⠿⢷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⣾⡇⣤⣴⡳⠄⠀⢸⠀⡇⡾⢧⣾⠀⡇⣷⠛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠘⢭⣉⣉⣉⣉⡉⠉⠉⠉⠉⠀⠻⠚⠋⠻⠄⠀⠀⠘⠰⡏⠓⠂⠙⠈⠓⠋⠀⠈⠉⠉⠉⢉⡉⠉⣉⣉⣭⠇
⠀⠀⠀⠀⠀⠀⠀⠉⢭⣙⣻⣭⡭⠭⠥⡀⠀⠀⠀⠾⠿⠀⠀⠀⠈⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡨⠭⠭⣭⣽⣛⣭⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂⢆⠀⣾⡇⠀⢠⣦⢀⣤⣤⡀⠀⢀⣴⡶⢲⣄⠀⢠⡶⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⠀⢀⣄⠀⠘⠳⣾⣼⣿⠇⠀⣼⣿⡿⢁⣿⡇⢀⣾⡿⠁⢸⣿⠀⠻⣷⣤⡤⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣧⠀⢸⣿⠀⠀⢠⡇⣾⣿⠀⠀⣿⣿⠃⢸⣿⡇⣼⣿⠇⠀⢸⣿⡖⢻⠏⢻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡴⠻⣿⡇⢀⡞⠀⢻⣿⠀⡼⣿⣿⠀⠸⣿⣇⣿⣿⠀⣠⣿⣿⠀⢻⡀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠈⠛⠛⠁⠛⠋⠀⠀⠈⠉⠀⠙⢻⡿⣿⣿⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣸⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠁⠀⠀⢀⣀⣀⠀⠀⠀⠉⠙⠲⢤⣀⠀⠀⠀⠀⢾⠀⠀⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠀⠀⠀⠀⢀⡴⠋⠻⢤⠈⠉⠳⣄⠀⠀⠀⠀⠈⠑⢦⡀⠀⠈⠓⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⡞⠲⠤⠞⠋⢀⡴⠏⠮⣇⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⡴⠋⠙⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⣠⣞⣓⣒⣒⣒⣒⣚⣓⣒⣒⣛⣢⣀⠀⠀⠀⠀⠀⠀⠙⣌⢧⡀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠘⣆⠉⠉
⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⣺⠉⡇⠀⡴⠊⠒⣄⡤⠤⠤⣄⡔⠊⠒⡄⠀⡏⢙⡆⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠻⢤⡇⠀⢧⣀⣠⠟⡇⠀⠀⡏⢧⣀⣠⠇⠀⣧⠴⠃⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠲⠲⠐⠔⣴⡃⠀⠀⢳⣔⠠⠴⠖⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣬⣷⣤⣤⣥⡸
⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢰⠖⠒⢶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡖⠒⢲
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⡞⢠⠦⣾⠀⠀⠀⢀⣆⢦⣦⢦⠀⠀⠀⢸⠦⢶⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⢀⡸⢁⠇⠀⢸⡀⠀⠀⠸⠜⠜⠞⠏⠀⠀⠀⣸⠀⠀⢇⠹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⢮⠀⢸⠂⠀⠀⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⢸⡀⢨⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡇⠀⠀⠀⠀⠀⠀⡼⣉⢣⠀⠀⠀⢸⢹⠀⠀⠀⠀⡇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠓⠓⢤⠙⠲⠤⣄⣸⢡⠛⡎⣇⠀⠀⣸⢸⠀⠀⢀⠴⠓⠛⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠈⡇⠀⠀⢠⠇⠿⠖⠃⠸⡆⠀⣿⢸⠀⠀⡏⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⠀⢀⣏⡞⠉⠉⠙⢦⣹⡀⣿⣸⠀⠀⠀⠀⠀⢀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠉⠁⠈⠉⠀⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠶⠦⣄⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡤⢬⣅⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⣤⠴⢤⣄⠀⠀⠀⠀⠀⠈⢳⡀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⢁⡴⠚⠋⠙⠃
⠀⠀⠀⠀⢸⡇⣤⣄⠈⢷⡀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠈⠉⠓⠲⠶⠦⠤⠴⠖⠒⠊⠉⠁⠀⠀⢠⠶⠒⢦
⠀⠀⠀⠀⠈⠛⠒⠋⠀⠈⣇⢀⡴⠚⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠉⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢀⡤⣄⠀⠀⣸⠀⢀⣄⠀⠀⢀⡴⠖⠛⠛⠳⢶⣄⠀⠀⢀⡾⠁⠀⠀⣾⡃⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⢾⠀⠞⠀⣠⠏⠀⣾⠟⠀⢰⠏⠀⠀⠀⠀⠀⠀⠈⢧⢠⡞⠀⠠⡄⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠈⠓⠒⠊⠁⢀⣰⢿⠀⠀⢸⠀⠀⣠⠤⡄⠀⠀⠀⢸⡟⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⢤⣤⣤⡤⠖⣫⣥⠘⣧⠀⠈⢧⡀⠛⢀⡼⠀⠀⠀⣸⠃⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢠⡞⠁⠸⣄⡘⢧⡀⠀⠉⠛⠋⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⢀⡟⠀⠀⠀⣸⠇⠀⠙⠶⣤⣄⣀⣤⡴⠞⠃⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⣠⠀⠀⠀⣸⢁⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⣿⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢰⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠸⡇⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⢿⡄⠰⣄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠻⣆⠈⢻⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⠀⠈⠻⣾⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣉⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠋⢻⣀
⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁⢀⣤⠞⠁
⠀⠀⠀⢀⣤⠶⠋⠁⠀⠀⠀⠈⣳⣤⡀⣀⣀⠀⠀⡀⠀⢀⣀⡀⢀⣀⣀⠀⣀⡀⠀⣀⣠⣴⣞⣉⣀⣀⣀⣀⣀⣀⣀⣀⣤⠾⠋⠀⣠⡴⠛⠁
⠀⠀⠺⢿⡶⠶⠶⠶⠶⠶⠶⣿⠁⢀⡿⠉⠉⣿⠋⢹⣿⠋⠉⣽⠋⠁⣹⠟⠉⣽⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡝⠁⢀⣤⠞⠋
⠀⠀⣠⣼⣧⣤⣤⣤⣤⣤⣤⣿⣄⣸⣧⡀⠀⣧⡀⢸⣿⣄⢀⣿⡀⠀⢻⡄⠀⢿⣄⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠙⠛⠉⠙⠋⠁⠉⠉⠈⠛⠶⠋⠙⠓⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠠⢀⠀⠄⠀⠀⠄⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⡿⠀⠀⠀⠀⠀⠀⢨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⣼⣧
⠀⠀⠀⠀⠀⢀⣤⡤⠀⢕⣚⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠡⠁⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⠀⠐⣽⡛⡙⠀⠙⠻⣟⡓⠈⢹⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠲⠶⣶⣿⣿⣶⠶⠒⠒⠀⠀⡀
⠀⠀⠀⠀⣤⠀⠛⠀⠀⠀⠈⠙⣋⠴⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⣸⣇
⠀⠀⢀⣠⠟⣠⡀⠀⠀⠀⠀⠉⠀⢰⡏⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⣠⣾⣧⠀⠀⠀⢰⢹⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢛⠌⡀
⠀⠀⢾⢸⠀⡏⣇⠀⠀⠈⠁⠒⠒⣹⡆⠀⠀⠀⠀⢀⡴⠋⠀⢠⣾⣿⠟⠁⠀⠀⢀⠃⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⢈
⠀⠀⠈⢄⠁⢀⡸⠂⠀⠀⠀⠀⢀⠏⠳⠀⣀⡠⠖⠁⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢠⣾⠀⣸⡇⠀⠀⠠⣴⣶⢋⡴⢚⠟⠉⠀⠀⠀⢀⣾⠟⠁
⠀⠀⡏⢋⣧⡹⡷⡀⠀⠀⡼⢣⠏⢠⠏⠀⠀⠀⠀⠀⠾⡡⠂
⠀⠀⠑⠀⢊⢻⡆⡅⠀⡼⢡⡟⢀⣿⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠐⢶⣶⣶⣶⣶⣶⣶⠶⠀⠀⠀⠀⠀⠐⠲⣶⣶⣶⣶⣶⣶⠶
⠀⠀⢀⡇⢌⡆⢻⢠⠀⡿⢸⠁⢸⢻⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⣀⠞⢡⡏⡇⣀⣠⢷⣇⠸⡀⠀⠈⣆⠀⠀⠀⠀⠃
⡾⠁⢠⠟⢰⡠⠵⣚⡿⢿⠀⢧⠀⠀⠘⣄⠀⠀⢸
⠁⠀⡞⠀⠸⠗⠊⠁⠀⢸⡆⠘⡆⠀⠀⠈⢦⡀⢸⡇
⠀⠀⠁⠀⠀⠀⠀⢀⡴⠋⢹⡄⠘⡄⠀⠀⡀⠻⣼⠁
⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⣠⢿⡄⠰⠀⠀⢷⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⢳⠀⠄⠀⠸⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⡞⠃⠐⠀⠈⠁⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⢀⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢶⣶⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⣩⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⡤⠚⠉⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡧⠤⡤⠞⠁⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⢠⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣧⣴⣯⣤⣄⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠈⠉⠙⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠘⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⣤⠤⠞⠁
⠀⠀⠀⠀⠀⢠⠴⠲⠖⠶⠶⠶⠶⠶⠶⠖⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠖⠶⠶⠶⠶⠶⠖⠶⠒⠦⣀
⠀⠀⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⢨⢷⡒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡄⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⠒⠒⣲⢾⠁
⠀⠀⠀⠀⣰⠋⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⡈⢳⡀
⠀⠀⣰⠛⠉⠙⠖⠋⠉⠳⡄⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢠⠞⣴⢻⡟⢦⠹⣄
⠀⣠⢿⡀⠀⠀⠀⠀⠀⢠⠿⡄⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⡴⠋⠀⠘⠒⡧⣄⠀⠈⢧⡀
⡴⠷⠶⠿⠦⣄⢀⣠⠶⠿⠶⠿⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⡾⠷⠶⠶⠷⣼⣇⡼⠶⠶⠶⠷
⠻⣄⠀⠀⠀⠈⠉⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⢧⡀⠀⠀⠀⠈⠁⠀⠀⠀⢀⡼
⠀⠈⠓⠦⣤⣀⣀⣀⣤⠴⠚⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⣀⣠⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⣿⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠚⠉⠓⠒⠚⠁
⠀⠨⠠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⢠⠀⡀⠀⠀⠀⡄⠀⢀⠀⠀⡄⢀⠠⢀⠀⠄⠀⠀⢠
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣃⣁⣁⣀⣀⣇⣀⣀⣑⣀⣇⣸⣘⣐⣀⣜⣈⣣⣸⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⢛⡛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣀⡀⠀⣸⣇⠀⢀⣀⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄
⠀⠀⠀⠈⠻⢤⣄⣀⣀⣀⣤⣴⣾⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⣿⣷⣦⣤⣀⣀⣀⣀⡤⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠉⢛⣿⠛⠛⠉⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠉⠛⠛⣿⡛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠻⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠟
⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⢧
⠀⠀⠀⠀⠀⠀⠀⡼⠁⡇⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢸⠈⢧
⠀⠀⠀⠀⠀⠀⡰⠁⠀⡇⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢸⠀⠈⢆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⡇⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢸⠀⠀⠘⣆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⡇⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢸⠀⠀⠀⠘⡆
⠀⠀⠀⢠⣇⣀⣀⣀⣀⣇⣀⣀⣀⣀⣳⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣞⣀⣀⣀⣀⣸⣀⣀⣀⣀⣸⡄
⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠙⠿⢿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠿⣿⣿⣿⡿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⣹⣦⣴⣿⣶⣶⣶⣶⣦⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣶⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠛
⠀⠀⠀⠀⠀⢠⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡅
⠀⠀⠀⠀⠀⠈⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⠿⣿⡿⠛⣿⡿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠉⢋⠉⠉⠈⠉⠀⠘⢿⣿⠋⠁⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠂⢸⡇⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣤⣶⣾⣿⣶⣦⣴⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣰⣿⠿⠋⠉⠀⠀⠈⠙⠻⣿⣦⣠⡄⠀⠀⠀⠀⢀⣠⣠⣤⣤⣄⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⣽⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠁⠀⠀⢠⣴⢞⣭⣷⣶⣶⣮⣝⡳⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⣏⣉⡉⠻⣶⢠⣤⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡄⢠⡿⣵⣿⣿⡿⠋⠙⢿⣿⣿⡜⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣟⠉⢹⠶⢻⡟⠉⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠁⢼⣧⣿⣿⠋⠀⠀⠀⠀⠙⣿⣿⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣄⣸⠛⠋⣠⡾⠃⢀⣼⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡄⠀⢘⣏⢿⣿⣷⣄⠀⠀⣠⣾⣿⡟⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣛⡟⣩⡁⣀⡀⠁⠈⠻⣿⣦⣤⣄⣠⣤⣴⣾⡿⠋⠀⠁⠀⠀⢛⣮⡻⣿⣿⣷⣾⣿⡿⣟⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣶⣾⣿⣿⣿⣿⣿⣿⣶⣶⣘⠋⠉⠛⢻⡟⠛⠉⣙⠃⠀⢀⣰⣆⡀⠀⠈⣙⠳⢯⢭⣭⠷⡞⣋⣠⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⡿⠛⠉⠁⠀⠀⠉⠙⠻⣿⣿⣇⡀⠀⠀⠁⠀⢀⣽⣷⣿⠿⠿⠿⠿⣿⣾⣯⡀⠀⠀⠀⣴⡿⠟⠋⠙⠿⣷⣄
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⠀⣠⣤⣤⣤⣤⡀⠀⠈⢿⣿⣿⡀⠀⣶⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠙⠻⣿⣦⣶⣸⣿⠀⣰⣾⣶⡄⢸⣿⡆
⠀⠀⠀⠀⠀⠈⣿⣿⡏⠀⠀⣰⣿⣿⠿⢿⣿⣷⡀⠀⠈⣿⣿⡏⠀⣸⣿⠟⢷⣦⣤⣶⠿⠿⣶⣤⣴⡾⠻⣿⣇⢸⣿⠀⠙⠿⠿⠁⢸⣿⠅
⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠘⢿⣿⣧⣀⣰⣿⣿⠇⠀⠀⣿⣿⡿⣤⣿⣿⠀⠀⢸⣿⠁⠀⠀⠈⣿⡇⠀⠀⣿⣿⣤⠹⣿⣶⣤⣴⣾⡿⠋
⠀⠀⠀⠀⠀⠘⢿⣿⣷⡀⠀⠈⢿⣿⣿⣿⣿⠏⠀⠀⣰⣿⣿⠗⠀⢿⣿⡀⠀⠈⢿⣦⣀⣀⣴⡿⠃⠀⢀⣿⡿⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠘⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠀⠀⣸⣿⣷⡀⠀⠀⠙⢻⡟⠋⠀⠀⢀⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣶⣶⣤⣴⣶⣾⣿⣿⠿⠃⠀⠀⠀⠉⠈⠻⣿⣶⣤⣀⣸⣇⣀⣤⣶⣿⠟⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠿⠟⢿⠟⠿⠟⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠛⠿⢿⡿⠿⠛⠻⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢒⣖⡶⠂⠒⠀⠀⠀⠀⠀⠤⠄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⣡⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⢒⡀⠤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⢁⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣴⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⣠⢖⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⡺⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⣠⢎⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⢌⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⣠⠞⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡡⡡⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⠎⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢌⢜⠌⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡠⢞⠕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣣⡯⡁⠀⠀⠀⠀⠀⠈⡗⡲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢀⢞⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⡴⠏⠊⠀⠀⠀⡠⠂⠠⢊⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⡑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡱⢟⠞⠁⠀⠀⡠⢊⠔⠔⡵⠃
⠀⠀⠀⠀⠀⠀⠀⣠⢊⡴⡡⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢮⡾⡳⠫⠊⠀⢀⢔⠔⠡⣪⠎
⠀⠀⠀⠀⠀⠀⡴⢡⣈⣾⢆⣠⡤⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢔⣵⡵⢊⠈⢀⠀⠔⡡⠃⠔⡵⠁
⠀⠀⠀⠀⠀⢠⣵⠟⢟⣿⡿⢻⡿⣿⣵⣶⣮⣴⣶⣶⣒⣂⠤⠄⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⣡⣪⢞⠔⢁⠔⠁⡨⠊⡠⢢⠎
⠀⠀⠀⠀⠀⣿⡟⠐⠢⠀⠀⠀⠤⠅⡊⠠⢋⡉⠙⣻⢛⠿⡿⢿⣶⣦⣬⣭⣐⣒⠀⠬⠙⡫⣫⡞⣛⠁⠀⡀⠀⠀⠊⢀⢌⡜⠁
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⣀⠀⠀⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠍⢈⡙⠚⠽⠾⣽⣿⣽⡷⡾⡫⣺⠋⡀⣈⠉⡠⢃⠐⡹⠋
⠀⠀⠀⠀⠈⡘⣿⡄⠁⠐⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⢀⠁⠐⢀⠀⠀⢨⠋⢌⠔⢀⣌⠉⣁⠈⡠⣋⠉
⠀⠀⠀⠀⠀⠘⢎⡛⠋⡃⠠⣄⣀⣠⢄⣠⡭⢀⣀⡐⠠⠀⡈⠀⠀⠀⠀⠀⠀⠀⠁⣽⠐⠁⢔⠕⢁⡔⢁⢊⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠂⠠⠌⠉⢒⣀⠭⡁⣂⡉⢉⣐⡀⠂⠂⠄⣒⠒⠤⢄⠀⢺⠀⠀⠃⢀⢏⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠐⠋⠡⠭⢿⣲⡆⣀⢈⠥⠀⡐⠈⣇⠀⢔⡑⡣⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠘⠿⠶⣒⡈⢐⡩⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠠⠐⡀⠅⡀⠄⠀⠨⠀⡀⠁⡀⠄⢐⠀⠒⠠⠀⡀⠄⡀⠄⣀⠄⠀⠄⡀⢄⠠⢀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠁⠐⠉⠀⠂⠡⠂⠁⠐⠉⠀⠂⠡⠂⠠⠂⠠⠔⠉⠔⠈⠔⠡⠂⠀⠊⠄⠊⠁⠊⠕⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠰⣖⣲⣀
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡀⠀⠀⣀⣀⣀⡀⣀⠀⣀⡤⠴⣶⡸⢿⡟⢿⡿⢾⠷⣲⠿⣳⣿⡿
⠀⠀⠀⠀⡀⠁⡀⠀⠀⠀⠀⠀⣀⠚⠛⠻⠻⠟⠛⡛⠛⠋⠀⠀⠁⠀⠉⠀⠃⠀⠀⠁⢀⠉
⠀⠀⠠⠄⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠑⠢⢄⠀⠀⠰⡀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠘⡀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⡀⢀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⢀⠀⠄⠀⠄⠠⠀⠀⢀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠄⢀⠻⠤⣀
⠀⠀⠀⠀⠀⠐⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡄⡾⠓⠲⡷⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠐⠀⠀⠀⠀⣿⣇⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⣿⣷⣆⠀⠀⠀⠀⠀⢻⣿⣶⣶⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠃⠀⢸⣿⣿⠀⠀⠀⠀⢀⠀⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⢀⢼⡿⠃⠀⠀⠀⠀⢀⡁⣀⣽⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠒⠋⠀⠀⠀⠀⠀⠀⠸⠷⠶⠿⠶⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠛⠲⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡀⠀⢸⡇⠀⢀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠈⠳⠀⢀⡀⠀⠞⠁⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⢰⡶⠴⠋⠙⠦⢶⡆⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⢿⡀⠀⠀⢀⡿⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠘⡇⠀⠀⢸⠇⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⢿⠀⠀⡿⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢸⡄⢠⡇⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⣛⣛⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠙⠳⠾⠋⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠠⣤⠤⠤⠶⠶⠖⠛⠛⢉⣉⣉⣤⣤⣤⣌⣉⠛⢶⡄⢠⡴⠛⣉⣡⣤⣤⣤⣉⡉⠉⠛⠛⠲⠶⠶⠤⠤⢤⡤
⠀⠀⢀⣀⣀⣀⣀⣤⡤⠶⠞⠛⠉⠉⠀⠀⠀⠀⠀⠉⠛⢦⣿⣿⡴⠛⠉⠀⠀⠀⠀⠀⠈⠉⠛⠲⠶⢤⣤⣀⣀⣀⣀⡀
⠀⢀⡾⠉⠉⠉⠀⠀⢠⣤⠤⠶⠶⠶⠶⠶⠶⠶⠶⠶⢤⣄⡙⢋⣠⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⣤⡄⠀⠀⠉⠉⠉⢷⡀
⣴⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢾⠿⠿⡷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠿
⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⡇⠸⠤⠤⠇⢸⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣶⣶⣶⣶⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣩⣩⣍⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠋⡥⠶⠲⠶⣬⠙⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⣀⠀⠓⢀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⡚⣛⢓⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣤⣀⣴⣿⠀⠀⠀⣀⣀⡀⠐⠀⣀⠂⠂⢀⢀⡀
⠀⠀⠀⠀⠀⠛⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠘⠿⠿⠿⠿⠟⠀⠀⠐⠛⠟⠃⠀⠀⠛⠀⠀⠘⠻⠛


⠀⠀⠀⠀⡀⠰⠶⠶⠆
⠀⠀⠀⢳⠖⢛⠛⢶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄
⠀⠀⠠⢭⡀⠐⠁⡸⠥⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⣶⡦⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠌⢴⠬⠄
⠀⠀⠈⠉⠉⠒⢺⡍⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣧⣤⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠺⡄
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉


⠀⠀⠀⣀⢀⣤⣤⡄⢀⡀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⣀⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠈⣉⣹⣴⣷⣾⣉⡁⠀⠀⠀⣴⣿⣿⣿⣿⣷⡄⠿⠿⢿⡿⠿⠿⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷
⠀⠀⠀⠉⠀⡙⠛⣁⠙⠁⠀⠀⠀⠀⠀⠀⣁⡀⠀⠀⠀⠀⠈⢁⠀⠀⠀⠀⠀⣰⡘⠻⠛⣀⡆⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⠂
⠀⠀⠀⡒⣓⡓⣓⡓⢒⠀⠀⠀⠀⠀⠀⠒⢚⡀⠀⠀⠀⠀⠂⣚⠀⠀⠀⠀⠀⣿⣿⣷⣿⡿⠃⠀⠀⠀⠀⠀⢀⣧⣀⣀⡈⣿⡀
⠀⠀⠀⠀⠓⠃⠙⠋⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡶⠲⠆⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⡀
⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢐⡂⡅⣴⣿⠋⠣⠀⠀⠀⠀⠀⠥⢍⡩⠁⣀⠈⠬⢍⠉⠀⠀⠀⠀⠀⡧⠘⠫⡅
⠀⠀⢀⣴⡤⣿⣿⣿⣧⣤⡄⠀⠀⠀⠀⠀⠠⢧⣃⠉⠉⣷⡾⠀⠀⠀⠀⠀⣐⡒⡴⠔⠖⠨⣀⢲⠤⠀⠀⠀⠀⢀⡷⡀⡴⠇
⠀⠀⠈⠛⠁⠉⠛⠋⠈⠛⠃⠀⠀⠀⠀⠀⠀⠈⠙⠚⠁⠋⠀⠀⠀⠀⠀⠀⠒⠚⠀⠀⠀⠐⠂⠊⠀⠀⠀⠀⠀⠀⠒⠒⠃

⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠠⠒⠐⠀⠀⠀⠀⠀⠀⠐⠄⠢⢀⠠⠀⠠⢄⠀⠀⠀⡠⠀⢀⡀⠀⠀⠀⠀⠀⠁⣀⠄⠀⠀⠀⠠⡀⠈
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣉⠀⣈⠠⠀⠀⠁⠀⠀⠀⠀⠁⠀⠁⠐⠀⠀⠀⠀⠀⠂⠐⠈⡰⠁⠀⠀⠀⠀⠀⠈⠂
⠀⠐⠈⠁⠠⠑⠠⠄⠈⠀⠀⠀⠤⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠀⠈⠀⠀⠀⡇
⠀⢄⠀⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠐⠄⠀⠀⡀⠈⡀⠄⠀⠤⡀⠀⠀⠢⠀⢀⠌⡔⠀⠐⠱⡀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠁⠉⠁⠡⡄⠀⠄⠀⠀⢀⠐⠀⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠠⡀⠂⠉⢀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁⢀⠆⠀⠂⠁⠀⠀⠀⠁⢀⠀⠀⢠⠀⠠⡀
⠀⠀⢄⠠⠤⢄⠁⠈⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠀⡀⠀⠀⠀⡀⠀⠀⡄⠡⠀⠲⠖⡠⠂
⠀⠀⠉⠀⠀⠀⠂⠘⠄⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠀⢄
⠀⠀⠐⠦⠤⠔⠀⠀⠠⠀⠀⠄⢀⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠀⠀⠀⠉⢁⠀⠀⠀⠀⠀⢰⡀⣰⡀
⠀⠀⠀⢀⠄⠀⠀⠀⠀⠑⠄⠀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣀⣀⠀⠀⠀⠤⠠⠄⡁⠒⡋⢁
⠀⠀⠀⠃⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⣀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠉⠉⠢⠀⠮⠏⠀⠂⠀⠀⠀⠀⠈⢂⠰⠾
⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠀⠂⠀⠈⠀⢀⡁⠂⠀⠀⠋⠀⠁⠐⠄⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠐⠈⠁⠀⢁⠌⠀⠀⠻⠀⠘⠃⠘⠀⡁⠀⠀⠐⠀⠀⠈⢄⠘⠊⠀⡀⡠⠀⠠⠀⡀⠀⠂⠀⠂⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠐⠀⠀⠀⠐⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠐⣀⡠⠀⠈⠐⠀⠀⠀⠁
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠘⢿⣿⣿⣿⡿⡽⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⣿⡀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⣘⣿⣿⡿⠖⠁⠀⠀⠀⠀⠀⣀⣴⣿⣿⡟⠀⣸⠟⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⣤⣶⣶⣿⣿⣿⣷⣄⣤⣴⣶⣶⣿⣿⣿⣿⡿⠋⢀⡜⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣿⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠁⠀⠀⠈⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣸⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⢻⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⣼⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿
⠀⣶⣾⣾⣿⣿⣿⠟⣱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⣼⣿⣿⣿⠟⠁⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⢈⣿⠛⠉⠁⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠾⠃⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣡

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢠⠶⢦⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⢾⣉⣿⠀⠀⠘⢷⠟⠀⠀⢸⣏⣹⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣩⡇⠀⠀⠉⠉⢳⡄⠀⢸⠀⠀⣴⠋⠉⠁⠀⠀⣾⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⠈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⣄⣀⣀⠀⠻⣦⡀⠀⠀⠀⢸⡇⠀⣸⡀⠀⣿⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⣠⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠈⠳⠶⠋⠉⠉⢳⣄⠈⠻⣦⣠⡴⠞⠛⠋⠉⠉⠛⠛⠶⣤⣀⣠⠞⠁⣠⡶⠋⠋⠻⣤⠿
⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠘⢷⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡾⠋⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠸⣅⣸⡇⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⣠⡤⠤⠤⣤⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⣿⣀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠶⠶⠶⠶⣾⠁⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢹⡶⠶⠶⠶⠶⠞⠁
⠀⠀⠀⠀⢠⡶⢶⣄⣀⣀⣀⣀⣀⣠⡏⠀⠀⠀⠀⣰⠞⠛⠛⠛⠛⠛⠛⢶⡀⠀⠀⠀⠈⣧⣀⣀⣀⣀⣀⣀⣴⠶⣦
⠀⠀⠀⠀⠘⠳⠞⠉⠉⠉⠉⠉⠁⠙⣇⠀⠀⠀⠀⣿⠀⠀⠀⢿⠇⠀⠀⢸⡇⠀⠀⠀⢀⡟⠉⠉⠉⠉⠉⠉⠙⠶⠛
⠀⠀⠀⠀⠀⠀⣀⣀⣴⠟⠛⠛⠛⠛⢻⡄⠀⠀⠀⢿⡀⠀⠀⠘⠀⠀⠀⣸⠇⠀⠀⠀⣼⠛⠛⠛⠛⠛⢷⣄⣀⡀
⠀⠀⠀⠀⠀⢸⣏⣹⠇⠀⠀⠀⠀⠀⠀⢳⣄⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠉⠀⠀⢀⣼⠃⠀⠀⠀⠀⠀⠀⣿⣉⣿
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢀⡴⠋⢙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠙⢦⡀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢰⡟⢻⣦⣤⡴⠋⢀⣴⠟⠁⠈⠛⢶⡦⣤⣤⣤⡤⣶⠞⠋⠉⠳⣄⠀⠙⢦⣀⣠⡶⠳⣆
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠀⠀⠀⢰⡟⠁⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⠀⠀⠀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⡼⠃⠀⢀⣤⣠⠞⠁⠀⢸⠀⠀⠙⢦⣠⣄⠀⠐⣯⣨⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡿⠀⠀⢰⠟⢷⠀⠀⠸⣅⣼⠃⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠛

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠿⠻⠿⠷⣶⣄⡀⠀⠀⠀⠀⠀⠀⢸⡿⠻⣿⠟⣿⡿⢿⣿⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠉⠰⠆⠀⠀⠀⠴⠆⠙⣿⣆⠀⠀⠀⠀⠀⠸⠷⠾⠿⠶⠿⠷⠾⠿⠶⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⢻⣶⣤⣤⣤⣤⣤⣤⣤⣾⠟⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣼⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣤⡀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⣿⠀⣾⡟⢿⡆⢠⣶⠀⣴⡟⢻⣦⠀⣿⡋⢹⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⣉⣉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣾⣿⠀⠙⠿⠟⠁⠀⠿⠀⠘⠻⠟⠃⠀⣿⣦⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣴⣶⠄⠠⡦⠀⢶⠀⢰⣶⣦⡀⣿⠁⠀⣠⣴⠿⠟⠿⢶⣄⣠⣴⠾⠟⠿⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠃⠀⢠⣤⣤⣤⣤⣤⡀⠀⠙⣿⠏⠀⢰⡿⠁⠀⠀⠀⢀⡙⠛⠁⢀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣄⣸⡏⠉⠉⠉⣿⣇⣠⡾⠋⠀⠀⢸⣧⠀⠀⠀⣼⡟⢻⣧⠀⣿⠂⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡿⠷⣶⣶⡶⠿⢿⡏⠀⠀⠀⠀⠀⠻⣷⣄⠀⣿⡿⢿⣿⠀⣿⠁⢀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠶⠶⠶⢿⣇⠀⠀⠀⠀⠀⣼⡷⠶⠶⠶⢶⣦⡀⠈⠻⣶⣌⠀⠀⠁⠀⢈⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⢋⣤⣤⣄⠀⠀⠈⢻⣆⠀⢈⣿⣶⡀⢀⣴⡿⠛⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣶⣶⣾⣧⣼⡿⠛⠛⠛⢿⣿⠛⠛⠁⠈⢻⣿⠋⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠷⠶⣶⣶⡀⢸⣿⣤⣤⣤⣤⣤⣤⣤⣿⠏⣠⣶⠶⠶⠾⠟⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⠉⢿⣦⣉⣉⣉⣉⣉⣉⣉⣉⣁⣴⠿⣧⣄⠀⠀⠀⠀⠀⠀⢀⣼⣿⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠉⣿⡏⠉⠙⣿⠉⠙⠻⣿⣍⡀⠈⠙⠻⠛⠛⠛⠛⠛⠿⣯⣼⡿⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⣶⣄⣠⣤⣴⡿⠇⢀⣴⢿⣦⠀⠀⠀⠙⠻⣶⣄⣀⣀⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣴⡿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⡾⠏⠉⠁⠀⠀⠀⠘⢿⣶⡿⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣶⡿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⠿⠂⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄⢠⣾⣴⣾⣿⠀⠀⢀⣰⣿⠟⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣽⣿⣿⣅⣤⣾⣿⠿⡏⠀⠉⠀⠨⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⢽⣿⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⢻⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⠀⠀⢿⣿⣽⡿⡇⠀⢀⣀⢀⠀⠠⣤⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⣸⣿⣿⣿⣧⠈⠛⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢹⡇⠀⣼⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⡷⠀⢿⣿⣿⣿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢧⣠⢾⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⠙⢿⣾⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣻⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⢿⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⠃⠈⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⣀⠘⠿⠏⣽⣿⡻⢷⣄⠀⠀⢈⣀⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣷⣦⣬⣙⣛⣧⣤⣶⣿⣿⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⢿⣿⠿⠟⠛⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿