Sigil of Satan Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠾⠟⠛⠛⠉⠉⠉⠉⠉⠉⠛⠛⠛⠷⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣡⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣤⣬⠛⢷⣄
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠉⣏⠀⠀⣀⣤⣶⠾⠟⠛⠛⠉⠉⠉⠉⠛⠛⠻⠷⣶⣤⣀⠀⠀⠉⡏⠀⠀⠙⢷⣄
⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠐⠛⣻⣴⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⣦⣄⠛⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⣠⡾⠛⠙⣿⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⣻⠏⠙⢷⣄⠀⠀⠀⠀⠈⢿⣆
⠀⠀⣰⡟⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⢹⡎⠻⣟⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠗⢁⡟⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⢻⣆
⠀⢰⡿⠀⠀⠀⠀⢠⡿⠃⠀⠀⠀⠀⠀⢿⣰⡫⡿⢿⣷⢦⣀⠀⠀⣀⣴⣾⡿⢛⢕⡀⣼⠁⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠀⢿⡆
⢀⣿⠃⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠘⣧⠱⣌⢳⣝⢷⣯⣷⣾⣩⠟⣩⠞⣡⠏⢰⠇⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠀⠀⠈⣿
⢸⡏⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠹⣆⣹⣷⡛⠃⠘⠛⣿⣅⣸⠏⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⢹
⣾⠇⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡴⣿⠹⣎⠃⣀⠀⠘⠁⡌⢻⣦⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠸
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⢹⣷⠏⠀⠙⣦⢸⡏⣠⠞⠁⠈⢷⣏⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣤⣾⡿⠶⣛⣫⣿⡆⢈⣷⣌⠃⠘⢁⣴⣏⢠⣿⣉⣛⠲⣭⣳⣤⡀⠀⠀⠀⠀⢸⡇
⢿⡇⠀⠀⠀⢸⣷⠀⢀⣠⠾⣫⠞⢻⣟⣋⣭⡿⠿⣿⡄⠙⡿⠆⠰⠿⠟⣡⣿⠧⢭⣉⢉⢛⡉⠳⣝⡷⣄⡀⠀⣼⡇⠀⠀⠀⢰
⢸⣇⣠⣤⣄⡀⣿⡶⠿⠿⠿⠭⠿⠿⠯⠯⣭⣤⢤⣿⣿⡦⣾⠧⠴⣷⢾⣿⡿⠤⠤⠭⠧⠬⠯⠼⠯⠿⠮⠿⢶⣿⠘⠿⢷⡆⣸
⠀⣿⡌⠉⠙⡇⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠘⠄⠐⠃⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⣠⣤⣴⣦⣿
⠀⠸⣷⠀⠀⠁⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⠀⠠⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠉⠉⠁⣾⠇
⠀⠀⠹⣧⠀⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠶⠶⢳⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣾⣇⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⢀⣾⠏
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠙⠿⣦⣄⡀⠀⠀⠀⠀⠀⠘⣧⣹⠃⠀⠀⠀⠀⠀⠀⣀⣴⡿⠋⠀⠀⠀⠀⠀⣠⡿⠃
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠈⠙⠻⢷⣦⣤⣤⣀⣀⣻⣏⣀⣀⣤⣤⣴⡶⠟⠋⠁⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⣿⢻⣯⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠜⠁⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⢶⣤⣤⣤⣀⣀⣙⣀⣀⣀⣤⣤⣤⡶⠾⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠶⠒⠒⡒⠒⠒⠲⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠉⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠉⠙⠢⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣃⣐⣀⣀⡀⠀⠀⠀⠀⠀⢠⡝⠦⡀
⠀⠀⠀⠀⠀⠀⣠⠎⠤⢤⣤⡀⠀⣠⠴⠚⠉⠉⠀⢠⣾⣦⠀⠈⠉⠑⠲⢄⡀⡴⠫⠄⠀⠙⢦⡀
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠉⠲⡔⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⡞⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⡀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠙⡄
⠀⠀⠀⡼⠁⠀⠀⠀⣰⠃⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢹⡀
⠀⠀⢠⡇⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⢀⣠⠄⣧
⠀⠀⢸⠀⣤⠴⠖⣸⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⢸⠈⠋⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⢹⠀⠀⠀⢀⣤⣶⣿⣿⣿⣶⣄⣸⣿⣇⣠⣶⣶⣶⣶⣶⣤⡀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠘⡇⠀⠀⠀⠘⡆⠀⢠⣿⣿⠟⠉⠉⠉⠻⢿⣿⣿⣿⣿⠟⠉⠉⠉⠛⣿⣿⡄⠀⢀⡟⠀⠀⠀⠀⡏
⠀⠀⠀⢳⡀⠀⠀⠀⠹⡄⢸⣿⣯⠀⠀⠀⠀⠀⢀⣿⣿⣿⣅⠀⠀⠀⠀⠀⣸⣿⡇⠀⡼⣄⡀⠀⠀⣸⠁
⠀⠀⠀⠀⢧⠀⢀⣤⠖⠙⢆⢻⣿⣶⣤⣤⣤⣶⣿⡿⠋⠻⣿⣷⣤⣀⣀⣴⣿⣿⢣⠞⠀⠀⠙⠳⣰⠃
⠀⠀⠀⠀⠀⢳⡉⢀⡴⠚⠈⠳⢍⡻⠿⠿⠿⠟⠋⠀⠀⠀⠈⠛⠿⠿⠿⠿⣻⠕⠁⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠙⢎⠀⠀⠀⠀⠀⠉⠲⠤⣀⣀⠀⠀⠀⠀⠀⢀⣀⡤⠴⠊⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠀⡯⠉⠉⠉⠉⠉⠉⠀⠈⠳⣄⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣀⡀⠠⢰⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠶⠤⠤⠤⠤⠤⠴⠖⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣟⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠹⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠈⢿⣦⡀⠀⠀⠀⠀⢀⣴⠟⠁⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠙⢷⣄⠀⠀⣰⡿⠃⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠙⣷⣾⠋⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⢀⣾⠟⠻⣦⡀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⠟⠁⠀⠀⠈⢻⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢿⡆⠀⠀⠀⠀⣰⡟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠈⢿⡄⠀⠀⣰⡟⠀⠀⠙⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠈⢿⡄⣰⡟⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠈⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⣰⡟⢿⡄⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣳⡟⠀⠈⢿⣟⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⣻⣿⠀⠀⠀⠈⣿⡏⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣏⠀⠀⢀⣴⠟⠹⣧⠀⢀⣼⡏⢻⣦⡀⠀⠈⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠾⠟⠁⠀⠀⠹⣧⣾⠏⠀⠀⠙⠻⠶⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⣧⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢹⡆⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⢡⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⢰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⢿⣆⠀⠀⠀⠀⠀⣰⠟⠁⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠙⢧⠀⠀⢀⡾⠁⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⢧⣴⠋⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⢠⣾⠿⡄⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡟⠁⠀⠘⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⣇⠀⠀⣴⠏⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠹⣷⣾⠏⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠀⢀⡀⣰⣿⣿⡀⢀⠀⠀⠁⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⣤⡀⢀⢼⣷⡟⠁⠈⣧⣾⣧⣐⡷⠀⠴⠛⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢿⠀⠀⠉⠉⠁⢸⣿⡀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠁⠀⠀⠀⢠⡏⠸⣇⠀⠀⣼⠻⣆⠀⠀⠀⠀⠀⢡⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⢀⣴⡟⠀⠀⢻⡄⢰⡏⠀⢻⣆⠀⠀⢠⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣤⡾⠋⠀⠀⠀⠈⣷⣾⠀⠀⠀⠙⢶⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠻⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣀⣀⣀⣈⣿⣇⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠈⠉⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠿⠛⠛⠻⣿⡟⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣦⣀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣨⣿⣇⣀⣀⣀⣀⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⡈⠛⠿⠋⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠛⠛⠛⠛⢻⣿⡟⠛⠛⠛⠛⠻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⢿⣷⠀⣾⠿⠛⠛⠻⠿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⢘⣿⡇⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⣶⣦⠈⠻⠄⡀⢀⣿⡟⠀⢣⡀⠀⢀⣸⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠿⠿⠿⠿⣶⣄⠨⣿⣧⣶⣿⣿⠿⠿⠿⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠸⢿⣀⣠⣤⣾⡿⠋⠀⠀⠀⠛⠿⠿⠟⢀⣴⣾⠇⠀⠀⠀⠀⠀⠀⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠙⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠀⢿⣆⡀⠀⠀⠀⢀⣠⣾⣿⠟⢿⣤⡀⠀⠀⠀⠀⢀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣾⣿⣿⠿⠟⠁⠀⠀⠉⠻⢿⣶⣶⣾⡿⠟⠁
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠹⡝⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡽⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡄⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠈⠳⣄⠀⠀⠀⠀⢀⡴⠋⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠶⠛⠉⠉⠁⠀⠈⠉⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠈⠢⡀⠀⣠⠊⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⠀⣹⢾⡁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠘⣧⠈⠻⢦⣀⠀⣀⡴⠟⠁⣼⠃⠀⠀⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠔⠁⠀⠙⢄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢹⡆⠀⢀⣽⠞⣯⡀⠀⢰⡏⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⡄⠀⠀⠀⡜⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢀⣿⡶⠋⠁⠀⠀⠙⠷⣝⡀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠹⡄⠀⡼⠀⠀⠈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣠⡴⠛⠙⣧⠀⠀⠀⠀⠀⣸⠏⠛⢶⣄⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠘⡾⠁⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠻⣂⣴⣟⣉⣀⣀⣀⣹⣂⣀⣀⣀⢠⣟⣀⣀⣀⣈⣻⣦⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡼⠹⡄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡉⠉⠉⠉⠉⠉⠉⢿⡉⠉⠉⣿⠉⠉⠉⠉⠉⠉⢉⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡜⠀⠀⠙⣼⠏⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄⠀⠀⠀⠀⠀⠘⣇⠀⣸⠇⠀⠀⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣎⠁⠀⢀⡼⢳⡀⠀⣰⠟⣄⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⢻⣤⡟⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠋⠀⠀⠳⣰⠏⠀⠈⠛⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢦⣤⣄⣿⣡⣤⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣃⠤⠾⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣤⣤⣼⣿⣤⣤⣤⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠘⠉⠀⠁⢹⣿⠀⠀⠈⠙⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠻⢿⡶⠊⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣾⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠉⠙⣶⡄⢰⣾⠿⠛⠿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⢰⣦⠘⠻⢄⠀⣠⣿⠃⠸⡄⠀⠀⣠⠀⠙⡆⠀⠀⠀⠀⠀⠀⣠⣶⡿⠿⠿⠷⣦⣄⢸⣿⣠⣶⣿⡿⠿⠿⣶⣄
⠀⠀⠀⠀⠀⠘⠿⣄⣠⣤⣾⠟⠃⠀⠀⠙⠿⠿⠟⢠⣴⡷⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠙⣿⣿⡿⠋⠁⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⢿⣄⣀⠀⢀⣀⣴⣾⡿⠛⢷⣤⣀⠀⠀⢀⣠⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠛⠋⠀⠀⠀⠉⠛⠿⠿⠿⠟⠋
⠀⠀⠀⠀⠠⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⡦
⠀⠀⠀⠀⠀⠹⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡟⠁
⠀⠀⠀⠀⠀⠀⠱⡘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠱⡀⠙⢄⠀⠀⠀⠀⠀⢠⠞⠁⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠞⠛⠛⠛⠛⠛⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠱⣄⠀⢀⠔⠁⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⢻⣦⣄⠀⠀⠀⠀⠀⠀⢀⣤⣾⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢈⡷⡋⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⢷⠈⠳⢦⡀⢀⣠⠾⠋⣸⠇⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡤⠋⠀⠈⢢⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠘⣇⠀⣠⡽⠻⣥⡀⢀⡟⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠱⡄⠀⠀⡼⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⣹⡞⠁⠀⠀⠀⠙⣾⣅⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠱⡀⡼⠁⠀⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣤⠞⠋⠀⢷⠀⠀⠀⠀⢸⠇⠈⠛⢦⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡶⠿⠶⠶⠶⠶⠾⡶⠶⠶⢦⡿⠶⠶⠶⠶⠾⠷⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣆⡜⠀⠱⣔⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⢻⡄⠀⣼⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⡰⠂⠀⠀⡼⡄⠀⢀⡿⡅⠀⠀⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠈⣷⣰⡇⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠊⠀⠘⣄⡾⠁⠈⠲⠤⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣤⣀⣸⣿⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠈⢠⣤⡶⠖⠛⠛⠛⠓⠲⢶⣤⠀⠀⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⠻⣝⠦⣄⠀⠀⣠⠖⣻⠋⠀⠀⠀⠀⠻⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢋⠀⠀⠀⠀⠀⠀⠘⢧⣈⡷⣞⢁⡾⠁⠀⠀⠀⠀⠀⠀⡌⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣧⠀⠀⠀⠀⠀⠀⢀⡼⠻⣄⣸⠿⣄⠀⠀⠀⠀⠀⠀⠀⢽⡿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣼⣿⡛⣏⣧⠀⠀⠀⠀⠴⠞⡀⢀⡽⢧⡀⢈⠳⠄⠀⠀⠀⠀⠀⠘⣷⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣿⡇⣿⠈⠀⠀⠀⢠⠤⠉⣷⣏⠀⠈⣳⣞⠡⢤⡀⠀⠀⠀⠀⠀⡟⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇⣿⣿⡆⢀⠠⡘⠒⠛⠁⠘⢦⠞⠁⠈⠓⠚⠁⠀⢀⠀⠖⣡⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠿⠛⡶⣏⣘⣷⠤⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠻⣌⠈⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⢁⣠⣼⣿⣿⣿⣿⣶⣿⣾⣦⣒⣶⣶⣺⠿⣳⣦⣾⣷⣷⣶⣽⡞⣿⣿⣿⡽
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢛⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⢽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⠃⠈⣿⣡⣽⣿⣿⣿⣿⣿⣿⣿⣿⠋⣸⣿
⠀⠀⠀⠀⠀⠀⠀⣠⡄⣾⡈⢿⣿⣿⣿⣿⣿⣿⡿⢟⣾⣿⣿⣿⣼⣏⣿⣿⣿⣿⣿⣿⣿⣿⡿⢠⡿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⢳⡁⠂⢀⣙⡛⠛⠛⠛⠋⠀⠀⣽⣿⣿⣿⣿⡳⠈⠉⠛⠿⢿⣿⡿⢋⣤⡼⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣏⣯⣛⣏⣭⣧⣼⡷⠻⣆⠀⢠⣿⣿⣿⣿⣿⡇⠀⢢⢬⣶⣿⡿⠉⠉⠉⣰⣻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⠿⡓⣂⣾⣏⠁⡘⢿⡯⠛⠻⣿⠏⠀⠀⢀⣾⣿⣷⡿⢶⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣗⣩⣿⠃⢹⣦⠰⣦⠖⠀⣼⣠⣼⢸⣿⡏⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣤⣾⣽⣇⣿⣸⣧⣼⣿⣵⠚⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠤⣿⠙⠉⣿⢉⡿⠇⠛⢻⢀⡏⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡜⠲⠒⠀⠉⠁⠐⠂⠘⠈⢢⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰

⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⡿⠿⠛⠛⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠛⠛⠿⢿⣿⣷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢿⣿⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠆⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⢿⣿⣦⡀
⠀⠀⠀⠀⠀⢀⣴⣿⡿⠋⠀⠀⠘⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠃⠀⠀⠙⢿⣿⣦⡀
⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠀⢹⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⡏⠀⠀⠀⠀⠀⠹⣿⣿⣄
⠀⠀⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣿⡈⠛⢿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⢁⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⠀⠉⠻⣿⣷⣤⡀⠀⠀⢀⣤⣾⣿⠟⠉⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇
⠀⣰⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⠀⠀⠀⠙⠻⣿⣶⣄⠻⠿⠋⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣆
⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⠀⠀⠀⣠⣴⣮⠙⢿⣿⣦⣄⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⢀⣴⣾⡿⠛⠁⠀⠀⠈⠛⢿⣷⣦⡀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠟⢋⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠠⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠋⠁⠀⠸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠈⠙⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⢿⣿⡇⠀⠀⠀⠀⠀⢀⣴⣾⡿⠛⠁⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠈⠛⢿⣷⣦⣀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣧⠀⠀⢀⣤⣾⣿⣟⣉⣀⣀⣀⣀⣀⣀⣀⣀⡈⣿⣷⠀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣉⣻⣿⣷⣤⡀⠀⠀⣼⣿
⠈⣿⠛⠀⠶⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗⠸⣿⣇⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠶⠀⠛⢿
⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠏
⠀⠀⠀⠹⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣇⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠘⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡄⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠃
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣾⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣶⣦⣤⣄⣀⣀⣀⠘⠃⣀⣀⣀⣠⣤⣴⣶⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠿⠿⣿⣿⣿⣧⣼⣿⣿⣿⠿⠿⠛⠛⠉⠁
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠻⣿⣿⣿⡿⣿⣿⣿⣿⡿⢛⣽⣿⣿⣿⡿⣿⣿⣿⡿⢃⣴
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣴⣼⣿⣿⠿⣿⣿⣿⡿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣯⣦⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡹⣧⡙⢷⡎⠻⣿⣿⡄⠻⣿⣿⢟⣧⡿⢫⣿⢏⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⡹⣿⣮⡻⣆⠘⣟⠃⠒⡿⢃⣾⢋⣴⣿⢣⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⣠⣿⣧⢹⣿⣷⣜⡆⢹⣧⣿⠁⣘⣵⣿⣿⢡⣿⣧⡀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠛⣻⣿⣧⢻⣷⣝⠿⠏⠻⠓⣾⣟⣵⡿⢣⣿⣿⠛⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⣻⣿⣧⠿⣤⣖⠀⣇⠀⢐⣌⣻⣱⣿⣿⡛⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⠏⣿⡏⣽⣧⡹⣛⡃⠀⠀⣩⡻⢣⣿⡏⣿⡟⠟⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⢠⣼⡟⣷⢟⣴⡹⣧⢄⡠⣿⢣⣮⡻⡿⢻⣥⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⢖⣫⣟⢛⣤⣿⣿⣷⡜⣾⢾⢇⣾⣿⣿⣤⡻⣫⣵⡲⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣿⣇⣠⣿⣿⢹⣿⢿⣿⡜⢏⣾⣿⢿⣏⢻⣿⣄⡈⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣿⣿⣿⣿⣿⣿⣼⣽⢏⡜⣿⣿⣼⣿⣾⣿⣿⣿⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣯⣍⢩⠏⣾⣿⡎⢯⠉⣭⣿⣿⣿⣿⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⢻⣿⣿⣿⠆⢾⣿⢿⣿⠆⢴⣿⣿⣿⡛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡜⢿⣿⠿⣫⣾⣎⢿⣾⠋⣼⣎⠻⣿⣿⢟⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣶⣴⣾⣿⣿⣿⣎⢃⣾⣿⣿⣿⣦⣶⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣶⣶⣶⣶⣶⣿⣷⣶⣶⣶⣶⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⢀⠀⠀⠀⠀⠀⠀⢻⣦⣀⡴⠚⠋⠉⠉⠉⠉⠓⠦⣀⣸⡄⣠⠔⠚⠉⠉⠉⠉⠙⠲⢤⣠⣾⠇
⢾⠀⠀⢾⠀⠀⢸⢸⡿⣟⢦⡀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⡿⣿⠀⠀⠀⢰⡆
⢸⠀⠀⣼⠀⠀⢸⡏⢳⠘⣧⠙⢦⡀⠀⠀⠀⠀⠀⣼⣿⣿⡄⠀⠀⠀⠀⠀⣠⠞⢁⡾⢡⡏⢧⠀⠀⢸⡇
⢸⠀⠀⢾⠀⠀⢸⠀⠘⣇⠘⢧⡀⠙⢦⡀⠀⠀⠀⡋⠘⠢⡕⠀⠀⠀⣠⠞⠁⢠⡟⠀⡾⠀⢸⡆⠀⢸⡇
⠘⢦⣀⣼⣀⣠⢿⡀⠀⢻⡀⠈⢷⡀⠀⠙⢦⡀⠀⢣⣀⣠⠇⠀⣠⠞⠁⠀⣠⠟⠀⢸⠇⠀⢸⠇⠀⢸⡇
⠀⠀⠀⢹⠀⠀⠈⢧⠀⠸⣇⠀⠈⢳⡄⠀⠀⠙⢶⡀⢻⡓⣠⠞⠁⠀⠀⣰⠏⠀⠀⣿⠀⢀⡞⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠈⠳⡀⢻⡄⠀⢦⢻⡄⠀⠀⠀⢙⣿⣟⠁⠀⠀⠀⣴⢋⡖⠀⣸⠇⣠⠞⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠻⣧⣄⣈⣷⣽⣄⠤⣴⠟⢹⠙⢷⣦⢤⣼⣷⣟⣀⣤⡿⠋⠁⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢿⣶⡾⠟⠃⣹⣿⣥⣀⢸⢀⣠⣝⣿⣇⠛⠿⣷⣼⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣠⠞⠁⠙⣮⠉⢸⠈⢉⡾⠁⠙⢷⣄⠀⠉⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠘⣧⢸⣀⡾⠁⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢹⠀⠀⠀⠀⡄⢀⣤⠞⠁⠀⣀⣀⣀⣀⠀⠘⣿⡟⠀⠀⣀⣀⣀⠀⠀⠙⢷⣄⠀⣄⠀⠀⠀⢸
⠀⠀⠀⣸⠀⠀⠀⣼⣷⣞⣁⠀⠀⠀⠀⠹⣿⡄⠀⣰⠟⢷⡀⠀⢠⡟⠁⠀⠀⠀⢀⣙⣷⣿⣆⠀⠀⢸
⠀⠀⠀⢸⠀⢀⣾⡿⠟⠋⠁⠀⠀⠀⠀⠀⠸⣿⣴⠏⠀⠈⢷⣄⡞⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣆⠀⢸
⠀⠀⠀⢸⠀⠚⠉⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⣹⣿⡀⠀⠀⢀⣿⡄⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠙⣂⣸⣀⣀
⠀⠀⠀⢺⠀⠀⠀⠀⠀⢠⡾⠋⠀⠀⠀⠀⣼⠃⠹⣷⡀⢀⡞⠀⢻⡄⠀⠀⠀⠈⠻⣧⠀⠀⢀⡞⠉⢹⠉⠈⢣
⠀⠀⠀⣿⠀⠀⠀⠀⠀⢼⠁⠀⠀⠀⢀⡾⠃⠀⠀⢹⣧⡜⠀⠀⠀⠹⣆⠀⠀⠀⠀⢸⡆⠀⢸⠀⠀⢸⠀⠀⢸
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⢧⣤⣤⣴⠟⠁⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠙⢷⣤⣤⣤⠞⠀⠀⢸⠀⠀⢸⠀⠀⢸
⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢸⠆⠀⢸⠆⠀⢸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⢛⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣧⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢉⣛⠻⠿⢰⡖⡠⠹⠆⢀⣈⠛⠛⣻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⢻⣼⠇⠶⠂⠉⠑⠒⠀⢿⣽⠇⠀⠰⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡉⡴⢦⡀⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⠙⣛⡴⠇⠉⠀⠀⠀⠀⠀⠀⠀⠄⠙⢷⣼⠞⢃⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢟⣥⣄⢻⣿⠇⡀⡀⢠⡀⠀⣠⠀⠐⢄⡈⢿⣿⢃⣼⡛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣜⠷⠞⣸⣿⣎⣤⠁⠀⠀⠀⠀⠀⣰⣷⣽⣿⣷⡾⠷⠟⣴⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⣈⡛⢿⣿⣿⡿⠷⡄⠀⠠⠾⠻⣾⣿⣿⣿⣀⠰⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠐⢯⡽⢠⣿⣿⣥⠀⠀⠀⠀⠠⣤⣻⣿⡿⢿⣽⠇⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣶⡄⠁⢸⣿⣷⡀⠀⣠⣾⣿⣿⠏⢀⣴⣶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⢸⣿⡏⢇⣰⣛⣿⣿⡇⠀⠀⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⠀⣿⣿⣧⣼⣽⣿⣿⣿⡇⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡟⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⠿⡆⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠾⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢂⡠⠄⠁⠉⠀⢀⣠⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡔⠢⠁⡂⠀⠀⢈⡄⢴⢦⣂⡀⠀⠈⠈⠁⢑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠄⢉⠐⣤⣤⣀⡀⣺⢧⠴⠤⠃⠀⠀⣀⣀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠡⡈⢃⣐⠜⠉⠀⠀⠈⠁⠁⡤⠀⠤⠄⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠇⡈⠒⡘⡶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢈⠠⠬⠁⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠗⠀⠐⠒⢘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠄⠈⠁⢨⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠒⠈⢁⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⠈⠠⠖⢈⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠠⠒⠁⠠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠠⠈⣀⠐⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠂⠠⠂⠉⡅
⠀⠀⠀⠀⠀⢀⡠⡴⠶⢏⠙⣍⢷⡶⡶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠩⠐⠂⠠⢴
⠀⠀⠀⠀⣐⡛⠣⠖⣉⠔⠊⠉⠉⠉⠉⠘⠾⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠐⠊⣉⡀⢼
⠀⠀⠀⡴⠁⠈⠌⡁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⡑⠤⠐⠺
⠀⠀⢰⡗⠂⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡍⢵⠀⠀⠀⠀⠀⠀⠀⠀⡿⠠⠔⠲⠈⢻
⠀⠀⢸⡋⠀⠈⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⡍⠀⠀⠀⠀⠀⠀⠀⢠⣁⠑⣈⠓⣈⠇
⠀⠀⠘⣏⠠⠤⢈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠄⠢⠀⢤⡜
⠀⠀⠀⢩⡠⠒⠠⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡞⡃⠥⡉⠂⢑⠞
⠀⠀⠀⠀⢢⡀⢁⡈⠤⢳⡤⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠚⡁⠄⢀⠈⠂⢠⣨⠋
⠀⠀⠀⠀⠀⠐⢆⠀⠂⡃⠘⢡⠉⠛⡂⡘⠒⠓⡛⢉⠩⡁⠢⢘⠐⢄⠐⠐⠁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠤⡄⠘⢀⠆⢠⠁⡆⢤⠠⠁⢢⠐⠀⠃⢀⠃⡄⢈⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⠶⣠⣆⣰⠀⣆⢀⡆⣼⣤⡡⠼⠒⠋⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⠭⣟⣛⣛⣛⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠲⠚⢀⠘⠀⠀⠠⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢰⣶⣤⠀⠀⠀⠀⠀⠉⠢⣤⢦⢄⣠⣤⠈⣄⣤⣤⣬⣀⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣸⠏⢿⠀⠀⢀⠇⠴⠆⠀⠠⢀⡴⠒⠳⡎⢁⠈⠀⠀⠀⠈⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠄⣘⣁⡴⠆⣀⣈⣆⢠⠀⠿⣟⠷⢾⡇⠸⡆⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣖⣸⣏⣉⠳⢾⣩⣾⣤⠋⠵⠈⢷⣤⣾⡇⠀⠻⣼⣦⡐⣦⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡍⡧⠤⡑⢦⣿⠟⠁⠀⠀⠀⠘⠿⡿⠁⠀⠀⠙⣿⡃⠛⡷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠉⡇⢏⠠⢭⣿⣿⡀⠀⣠⡄⠀⠀⢠⠁⠀⠀⠀⣸⠿⣠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣷⢰⣅⠈⠐⣿⢿⣿⣧⠀⣩⣥⣤⣤⣼⣠⣀⠀⣠⣯⠺⡇
⠀⠀⠀⠀⠀⠀⠀⢩⢁⣿⡿⠉⠓⠂⠁⠀⣡⣏⠛⠉⠉⠨⠍⠭⣿⣿⣿⣽⠿⣀⠁
⠀⠀⠀⠀⠀⠀⠐⣯⠇⠘⠎⠀⠀⠀⢰⣎⡅⠉⡙⠀⠀⠀⠀⡛⠛⡟⢿⣆⠑⢦⡀⣀
⠀⠀⠀⠀⠀⠀⢀⣯⠜⠀⠈⠀⠀⠀⢸⠀⠉⠹⠀⠄⠀⠀⠀⣇⠀⡏⠙⣿⠑⠀⠈⣳⣾⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⡀⠀⠀⠀⣻⠀⠀⠀⣂⣠⣄⠀⢀⣿⡧⠀⠀⢻⣄⠀⢠⢷⡟⣡
⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡇⢀⣴⣶⣾⠀⠀⠀⣸⣿⠛⠠⣿⣿⡆⠀⠀⠘⣿⣶⣿⡏⠗⡏
⠀⠀⠀⠀⠀⠀⢸⠀⣰⢏⣴⣿⣿⡿⡇⠀⠁⠀⣸⡏⠀⠀⠘⢿⣧⡐⠀⠀⠘⣿⠿⢤⠸⠃
⠀⠀⠀⠀⠀⠀⠈⠺⣿⡿⠛⣿⠋⠀⡇⠀⠀⠀⣿⡁⠀⠀⠀⣨⣝⣧⠡⠀⢀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢀⡇⠀⠀⢿⠓⢶⣶⢿⣿⠀⠀⡰⣿⢟⠻⡟⠛⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠋⠀⠀⢸⠏⣺⣿⢼⡏⠠⣴⣷⣁⡞⢡⡽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣀⣠⣼⡎⠛⢹⣾⣿⣶⣯⡭⣽⣀⣾⣶⠒⠒⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣴⣾⣿⣿⣿⣿⣠⣯⠽⣿⣿⣿⡀⠀⠉⠈⢿⣿⣿⣿⣦⣤⣄
⠀⠀⠀⠀⠀⣤⣖⣿⣿⣿⣿⣿⣿⣿⠿⠈⠉⠀⠀⢹⢟⣿⣇⡀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⡶
⠀⠀⠀⠀⠀⣠⣼⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣷⣤⣀⣸⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣀⣴⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣠⣴⣾⣿⣿⣿⡿⠇⠀⠀⠀⠀⠛⠿⠿⠿⠿⠟⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⢿⣿⣿⣿⡿⠿⠟⠛⠋⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⣦⡶⢶⣓⣶⣴⠒⠂⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣯⣞⡫⢩⡷⢿⣿⣿⣷⣖⣄⢂⣳⣾⣮⣢⡢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⡽⣿⣿⣿⣿⣿⣶⣮⣿⣿⣿⣿⣿⣦⣶⣿⣿⣿⣇⢿⠻⣻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣟⢎⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣧⣎⣥⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣭⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠉⠉⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣻⢿⣿⣷⡀
⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⢻⣯⣿⡿⠿⡿⣿⣿⣿⣽⠿⠻⠿⠟⠻⠟⢿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣄⣀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠈⠑⠚⠛⠒⠒⠊⠋⠈⠁⠈⡿⣟⡿⣿⣷⣦⣭⡉⣽⣿⣿⣿⣿⣿⣽⣟⣟⣿⣿⣿⣿⣷⣦⣄⠸⣿⣿⣻⣿⡟⡿⡹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣏⣿⣿⣿⣿⣷⡌⠿⣿⣿⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⡿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣷⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠻⠿⡿⣿⣿⣿⡼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠉⠉⠩⣯⣿⣇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⣻⣿⣿⣟⣽⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⣟⣿⢿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⡄⠋⡻
⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣧⣇⣹⣿⣿⣟⣜⣺⣻⣿⣿⣿⣿⠇⣲⡇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠫⡱⢰⣼⡟
⠀⠀⠀⢀⣿⣿⣿⣿⣿⢯⠻⡶⢿⠛⠙⣙⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠻⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣅⣶⣿
⠀⠀⠀⢸⣻⣿⡿⣿⣿⣗⠀⠀⢀⢄⣠⣟⢿⣅⢿⡻⣿⣿⣿⣿⣿⣿⣿⣿⣶⣌⠻⣿⣿⡯⠀⢿⣿⣿⣿⡯⠇⢛⢿⣫⣽⣿⣹⣿
⠀⠀⠀⣼⡣⠟⠋⡭⠩⠥⠀⠀⡶⠋⠉⠙⠻⣿⣶⣦⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣯⣿⣿⣭⣼⣷⣿⣿⣽⣿⣿⡏
⠀⠀⠀⢯⣙⡆⣤⠆⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠙⣿⣿⣇⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡾⡎⡍⠙⡍⠁
⠀⠀⠀⠸⣏⢃⡠⠀⠀⠀⠀⠀⠽⣄⠀⠀⠀⠀⠀⠈⠻⣿⣷⣾⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠇⠇⠀⠐⠁
⠀⠀⠀⠀⡽⡿⣴⠃⠀⠀⠀⠀⢊⠹⢿⣶⣤⣤⣀⣀⣀⣈⣹⣿⣿⣿⣬⣿⣽⢿⣿⣿⣿⣿⣿⣟⠵⣻⡿⡿⠛⢋⣀⣤⢉⣶⣄
⠀⠀⠀⠀⠘⡿⢛⠾⠀⣀⣤⣤⣤⣴⣧⡾⢶⣾⣿⣯⣭⣭⣹⣏⣟⠻⢿⣿⣿⣷⣿⣿⣪⣷⣏⣁⢀⣁⣠⣴⣮⠟⣩⢿⡾⢥⡿⢦
⠀⠀⠀⠀⠀⠀⠺⠀⣏⣧⣴⣿⣿⣿⣿⣿⣿⣿⣾⡇⣾⢦⣬⠉⠛⠶⣤⣌⣹⣿⣿⣿⣿⡿⠿⠿⠛⠛⠻⠧⣼⣴⣿⣾⡵⢀⡴⢿⡀
⠀⠀⠀⠀⠀⠀⣢⡶⠻⣙⣮⡶⠛⣛⠋⠉⢉⠛⡛⠻⢿⣿⣿⣴⡆⣤⠀⠈⠙⠳⣭⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡿⡏⠀⠉⠀⢊⡀
⠀⠀⠀⠀⣀⡜⢍⠰⠋⠉⠀⠈⠉⠑⠻⠴⠎⠽⠷⠾⠛⠛⠉⠙⠻⢿⣼⣇⣤⠀⠀⠉⠙⠲⢤⣀⣀⣀⣀⣀⣤⠞⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠈⠸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢴⣅⢀⠀⠀⠀⠈⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣾⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠶⠶⢤⣤⡄⠠⠄⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡴⢂
⠀⢰⣇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⣠⣶⠁
⠀⠘⣿⡇⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠧
⠀⢠⣀⣆⣻⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⣦⣤⡶⠁
⠀⠀⠙⣿⣿⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⣴⣿⣿⡟
⠀⠀⠀⠹⣿⣦⡠⡇⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⢰⣦⡂⢧⣼⣷⣿⠟
⠀⠀⠀⠀⠈⢿⣷⣷⠀⠀⢸⠀⠀⠀⢠⠀⠀⢀⡀⣐⢀⣄⣘⢿⣟⣾⠟⠛⣁
⠀⠀⠀⠀⠀⠀⠙⢿⣶⣁⢸⡄⠀⢠⣼⠀⠀⢴⢄⠸⣿⠁⢻⠖⠋⠁⣀⡼⠥
⠀⠀⠀⠀⠀⠀⡀⠈⢿⣿⣿⣇⣴⡹⣿⠀⠀⣤⢠⢀⠛⢀⡎⠀⢀⡶⠟⠀⠀⠀⢲⡶⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠉⠹⡿⣿⣾⣿⠄⣥⡇⠀⢆⠄⠸⠁⠀⣸⡁⠀⣦⣀⠀⢀⠀⠀⠀⣀⣴⠟
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠐⠤⢈⠻⣿⣷⣼⠇⣴⢸⠀⠀⠀⢰⠋⠀⠀⠘⠻⠿⠿⠷⠚⠛⠉⠁
⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠊⠙⢿⣷⣥⣼⠀⠀⠀⡇
⠀⢱⡄⠒⠻⠿⠷⢶⣶⣶⠶⠤⢠⣀⡀⠀⠀⠨⡻⣿⡆⠀⢸
⠀⠀⡡⠀⠀⣀⣠⣄⣈⡀⠀⠀⠐⣦⣌⣤⣀⠀⠈⠛⢧⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⢀⣀⣀⣀
⠀⢰⠃⠀⠀⠨⣽⣿⣿⣿⣶⣶⣲⣾⠟⢀⣽⣷⣆⣠⠾⠲⠃⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⠿⠿⠟⠉⠉⠉⠉⠁
⠀⠈⠙⢣⣴⡟⠀⠤⠞⠛⠉⠉⠁⠀⠀⡀⠴⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⠿⢿⣉⠉⠉⠁
⠀⠀⠀⠘⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠽⠒⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡤⣤⡶⠀⢀⣼⣿⣿⡿⠂⡀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠄⡌⠅⢨⡁⠀⣈⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⢀⣤⣬⡤⠄⠉⢣⣄⠀⠀⠀⠀⢀⠀⠀⠀⡠⠖⠸⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢠⠞⠛⢣⠀⠀⠀⠉⠀⠠⣄⣀⡘⠇⠀⣀⣀⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠂⠀⠀⠀⣣⠀⠀⢉⠀⢀⣾⠟⠛⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⢀⡇⠀⠀⠀⢸⡀⠀⠘⠀⢸⣿⣦⣴⣿⣿⣿⣷⡤⠤⡴⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⢀⠀⠀⠀⣼⠃⠀⠀⣷⠀⠻⣿⣿⣿⣿⡟⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⢀⡿⠀⠀⢄⠉⠀⠀⠀⠛⠀⠀⠀⠉⡉⠁⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣷⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⢮⣻⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣏⣄⡀⠘⣟⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣿⡟⣯⡀⠘⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⡿⠀⠘⢿⣷⡄⠀⠀⠀⠀⠀⢤⡀⠀⢀⠣⠀⠀⢠⢀⣀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢮⣤⣦⣽⣿⣧⡀⠀⠀⠀⠙⠀⠀⠀⣠⣤⠒⢤⠎⠃⠰⠍⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠐⠤⢀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠈⠁⠏⠠⠄⣆⠀⠐⠆⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠐⠖⠛⡶⢪⠔⠀⡀⡼⢣⠄⠀⢀⡈⠻⣿⣿⣿⣷⡄⠐⢒⠶⠐⠶⠓⡀⠘⡝⢀⡰⠊⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠐⠙⣀⣂⡀⡁⠀⠀⠀⠋⢠⠀⠙⣿⣟⣿⣿⣆⠀⣀⠄⢙⣿⣖⠺⣇⣀⣷⣠⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⡤⠀⢠⠘⠉⠈⠃⠀⠈⠀⠚⠙⠂⠀⠘⣷⡆⢻⣿⡆⢸⣷⡧⠔⠙⠲⣥⣘⡿⢋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⠶⠿⢿⢛⣛⠛⡓⠲⣶⣤⡘⣿⡐⢻⣿⡈⠀⡸⣁⢰⠦⠘⣿⣵⡶⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⣾⣿⣿⣷⢻⣧⠌⢡⠋⠛⠛⠛⢿⣿⣿⣿⣿⣿⢻⣿⣧⣬⣀⣀⠹⠇⠡⠋⠿⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡃⢿⣿⡿⠋⠀⠀⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⠀⣻⣷⢠⣬⠝⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣷⡾⣻⣿⣟⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⡿⠁⣿⡟⠛⠛⢿⣷⣮⣿⣷⣄⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⡿⣿⣶⣿⣿⣿⣿⣿⡛⠛⡛⠻⢶⣾⡃⢸⣿⣧⣤⣤⣤⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⣿⣿⣿⣿⣿⡿⠛⠛⠉⠉⣿⣿⣿⣿⠿⠿⣿⣿⣿⣴⡿⠇⠀⣽⣿⡀⠈⠁⢀⣿⣿⣿⣿⣯⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡅⠹⠿⣽⣿⣯⣀⡉⠒⠒⠂⠘⠿⣿⣿⣷⣤⣤⣾⣿⠏⠁⣤⣿⣿⣿⣿⣿⣿⣿⡾⠿⠛⠛⠉⠉⠁⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠈⢧⡤⠀⠁⠙⠻⠻⠟⣿⠿⣿⣿⡿⠿⠿⠿⠙⠉⢁⣠⣴⣾⣿⣿⣿⠿⠟⠋⠁⠀⠀⣀⠀⠐⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣦⣲⢄⣤⣭⣤⣤⣠⣄⣔⣤⣤⣴⣾⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠉⠛⠛⠛⠚⠛⠚⠚⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁

⣶⣶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠐⠾⠿⢶⢤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡝⣷⠦⠀⠀⠀⠀⠀⠀⣤⠤⠻⣶⢱⣶⣄⡤⢀⣀⣠⣴⣾⣿⢶⡾⠅
⠀⠀⢠⣄⠀⠀⠀⠀⢧⣿⠁⠀⠀⠀⠀⠀⠀⣤⣿⣿⡇⠀⢿⣿⣿⣏⣴⣿⣿⣿⡟⢿⠿⠁
⠀⠀⠀⠀⠘⢦⠀⠀⣼⣯⠀⠀⠀⢀⣤⣴⣾⣿⣿⣿⣷⣄⣼⣿⡏⠙⠛⠉⠙⣻⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣬⣿⣦⠲⠀⠀⠀⠉⢋⣉⡔⠀⠀⠀⠀⠀⠀⣴⣶⣦⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡙⡛⠿⠟⠟⠠⠄⠀⠀⠀⠀⠀⠉⠉⠒⠒⠶⠠⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐




⠀⠀⣀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠐⠂⠀⠤⣀⠐⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠈⡁⠀⠀⠀⠀⡤⠠⠴⣦⠀⠀⢀⣠⠈⠉⢀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠂⠁⠀⠀⠀⢀⣀⣀⣠⣤⣤⣤⣤⣤⣄⣀⠀⠐⠆⠽⠼⠿⠁⠀⠓⠶⡄⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⣠⡤⠖⠿⠟⠛⢻⠿⠟⡻⠿⢷⣭⡙⣷⠶⢀⠠⣤⣼⣷⣄⡀⢀⠤⡶⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠶⠶⠲⠶⠤⣍⡉⠙⠓⣄⡙⢄⢸⣿⡏⣿⠉⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢰⡯⠀⠀⠀⠀⠀⠶⠶⠤⣤⣀⣀⣀⠀⣬⠟⠋⠀⠙⣆⠹⣿⠇⠙⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣇⠀⠀⠀⠠⣄⣀⣀⣀⣀⣀⣠⣴⠞⠁⠀⠀⠀⠀⡼⢧⣹⡆⠃⠰
⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠈⡇⠀⠀⠀⠀⠈⠁⠀⠉⠉⠁⣰⠃⠀⠀⠀⠀⢀⠞⠀⣨⠿⡐⠇⠶⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠙⡄⠀⠀⠘⠷⠶⠶⠶⠒⢫⡟⠁⠀⠀⠀⣠⠊⣠⠞⠁⠀⠈⠉⠉⠀⠹⡦⣀
⠀⠀⣀⡀⠠⠤⠤⢀⣙⣆⠀⠀⠹⣄⠀⠀⢀⣀⣀⣠⠴⠋⢃⡠⠀⠀⠀⢁⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣀⠀⠘⣦⡀⠀⠉⠀⠀⠀⣠⢻⡅⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⣄⠁⠙⢦⠀⠺⠛⠉⠀⠀⠳⡇⡠⣿
⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⡄⠀⢀⡈⡳⣤⡈⠱⢦⡐⠶⠁⠀⠀⢸⢇⣿⠁
⠀⠀⠀⠀⠀⠀⠇⠀⢀⠀⠈⠙⠛⠛⠛⠛⠙⢦⣀⠙⢆⠀⣴⠖⠋⠀⢻⢦⡄
⠀⠀⠀⠀⠀⠀⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄⠁⠀⠠⠤⠖⠃⠈⢳⡀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠘⢦⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⢀⣀⠴⠙⣄⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⣄⡀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢀⡼⡆⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠉⠙⠲⣤⡀⠀⠀⠀⠈⣷⠀⠀⠒⠚⠉⠀⢹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠈⠙⠒⠦⣤⣤⣸⠀⠀⢀⣀⢀⣀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠙⠢⢄⡀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠈⠉⠁⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠒⠛⠁⠀⠀⠀⠀⠲⠤⠴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠤⣤⣠⣄⣀⣠⡤⠶⠖⠚⠉

⣶⣶⡶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣟⡆⠀⠀⠀⠀⠀⠀⢰⢹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡄⠀⠀⠀⠀⠀⣴⡟⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⡀⠀⠀⢠⣿⠃⢀⡞⠁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠋⣤⣷⣄⠀⢸⣿⡄⢸⡄⠾⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣗⣤⣀⠀⠀⠀⠀⠀⠈⠻⢿⣧⠈⢻⣧⣼⣥⣼⣷⣦⡀⣸⡆⠀⣀⡄⠀⡠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣶⠒⢦⣀⣀⠀⠀⠙⢷⣼⣿⣿⣭⣿⣿⣯⡱⣬⡥⠔⠉⣠⡾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠿⢷⣶⣤⢿⣿⣿⣿⣿⣿⣿⣿⣴⣶⣄⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣶⣶⣶⣶⠿⠒⢛⣛⠻⠿⣿⣿⣿⣿⣿⡉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠄⠘⠛⠋⠉⠀⣴⢿⣭⣄⣄⣾⣿⣿⣿⣿⣿⣿⣏⠙⠛⠿⢿⣷⣆⠢⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠤⣶⡿⠛⠁⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣟⠉⠉⠉⠙⢿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠉⠀⠛⠛⠛⠛⠚⠛⠒⠤
⠀⠀⠀⠀⠀⢠⣷⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢟⣁⠀⠀⠀⠀⡨⣿⣿⣿⣿⣹⣿⡄
⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⣷⡈⠿⠓⣤⣤⠠⣧⣿⣿⣿⣿⣷⢻⣿⡄
⠀⠀⠀⣠⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡋⠺⢃⣉⣙⣠⣿⣿⣿⠟⠀⣿⣯⢿⣿⡀
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣶⣼⣿⣿⣿⣿⡿⠋⠀⠀⢸⡿⣾⠳⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⢠⣷⢄⠀⠘⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⡿⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠉⠛⠶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠈⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞

⠀⡀⣀⢀⣀⣠⣀⠠⣤⡀⣠⣀⢀⣠⣀⣀⣀
⠀⠀⠀⠈⠉⠀⠋⠀⠈⠈⠁⠈⠈⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⣀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡀⠀⠀⠀⢀⢻⣿⡿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣮⡑⡄⡈⠀⣴⠁⠀⠀⠀⢀⣀⣀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣛⣿⣧⠏⠀⣰⡆⠀⣠⡞⠿⡹⢡⣯⠶⢿⠋⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢟⣳⣿⣿⣴⣿⣇⣸⢻⣡⣧⣿⣛⣃⡤⠊⠀⠀⠀⠀⣾⢉⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠙⠺⢿⡟⢻⡿⢿⣿⣿⣾⡟⣧⣌⡵⠊⠀⠀⠀⠀⠀⢀⠙⠀⠰⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣦⣬⡉⣍⢦⠈⠱⣾⣿⣾⣧⠘⢿⡿⠿⢛⣉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡸⣿⣿⣾⣇⣇⢻⣿⠋⠉⣦⣶⣞⣩⡻⣿⠁⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣶⢷⣾⣿⣿⣾⣿⡂⠀⣿⢙⢿⣿⣭⠿⣷⠿⠀⠀⠀⠀⢀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠳⣾⡶⢷⣿⣷⣧⣠⠋⠘⠙⢿⣿⣜⠠⠖⢸⠋⡟⢰⢣⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣄⣨⣶⣿⣿⣿⣏⠀⠀⠀⠀⢉⡟⢻⢀⡟⣸⣥⣿⣟⡿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠋⠁⢠⣿⣿⠃⡿⢯⢰⡧⢄⣠⣿⣇⣿⣾⣿⣯⣿⣻⣟⠞⠁⠀⣴⣿⣴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡋⠄⠀⢸⡀⣷⠿⣿⡿⠿⠿⣿⣧⣽⣯⡕⠁⠀⠀⠀⢿⠷⠶⢿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣴⠲⡖⢶⡶⢤⣄⡀⠈⠀⠀⣤⡾⣇⣯⣞⣥⠀⢠⡶⠂⢠⡄⠀⠀⠀⠀⠀⠀⠈⢄
⠀⠀⠀⠀⠀⠤⠖⠾⠷⢿⣺⠷⣾⣦⣹⣄⢳⡝⣷⣀⣨⣿⣳⣿⣿⣭⡿⣿⠚⠛⠛⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣍⠴⠟⢛⣿⣷⣼⡏⢹⡯⠙⣿⣏⢻⣏⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠷⠿⠶⢛⣿⣿⣤⣁⣀⠇⠙⠾⠿⣿⣁⣤⢶⣶⠤⢀⣀⠀⠀⠀⢠⣿⣿⣿⡿
⠀⠀⠀⢤⣀⣀⣀⠀⠀⠀⠀⠀⢐⣟⣿⣶⣾⣿⣿⡿⡿⢯⠙⣷⣦⠶⢶⣿⡟⣠⡾⣡⣴⢋⣉⣱⣤⠀⠀⠙⠓⠛⠁
⠀⠀⠀⠈⣛⣢⣤⠘⡔⠂⢤⡀⠈⠙⠋⣿⣿⣿⡁⠃⡭⠖⡟⢑⣫⣶⢾⣿⣿⡿⢿⡿⢿⡿⢿⣷⣾⣧⡀
⠀⠀⠀⠀⢩⣭⣽⣷⣷⢸⠠⢱⣄⠀⠀⡟⠁⠀⠀⠀⠀⣼⣦⡽⠿⢿⡛⣧⠀⢳⡀⠑⣄⠙⢆⠑⣈⡿⠿⠆
⠀⠀⠀⠀⠀⣷⣶⠶⠿⢿⣤⡄⡏⡷⣤⣴⣤⣀⠀⣠⠞⠻⣅⣠⣤⣀⠈⠑⠻⠶⣿⣶⡬⠷⠟⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣗⣛⣛⣛⣿⣿⣥⠿⣿⣿⣷⡄⠉⠀⡼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠉⠻⡟⠉⢰⣮⣽⣶⡎⣧⠹⠁⠀⢸⣾⣷⣿⣿⣿⢷⡇⠀⠀⠀⠀⠀⠀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠥⣴⡏⡇⢸⣸⣾⠃⠀⠀⠀⣼⣿⣿⣿⣿⣿⣻⠃⠀⠀⣠⡼⡇⠀⠛⢸⠀⠁⠈⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣰⣿⡷⠿⠋⢸⡇⠀⠀⠀⠀⠹⣿⣻⢿⣿⡿⠃⠀⠀⠘⠓⣄⣀⣀⡤⢨⠄⣠⡀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠈⠉⠻⡟⠀⠀⠀⠀⠀⠀⢻⣮⡤⡇⡀⢀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠁⠀⣠⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣷⣀⣀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠃⠀⠀⢀⢈⡇⢐⢁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣫⡿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠈⠉⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣶⣤⣄⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠀⣼⣿⣿⣷⣦⣄⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⣠⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠠⠶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣄⣀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢩⣍⣉⣻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⠈⣿⣿⣿⠛⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⡿⢿⣿⠁⠀⠀⠸⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣄⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡾⠃⠛⢿⣿⡿⣿⣷⣸⠏
⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠉⠢⡀⣼⣾⣿⣹⣿⣿⠇⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⢀⢀⡀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢀⣸⣾⣷⡆⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⢀⣀⣾⣿⣿⣿⠉⢠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⣠⣾⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣻⣿⣿⣿⣿⣻⣿⣿⣿⠏
⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡟⢷⣮⣸⠇
⣿⣿⣿⣿⣿⣿⣿⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣨⡿⣶⣶⣄
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠻⠿⢿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⣀⢀
⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⣄⠤⢴⠶⢞⣁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣦⠴⣀⠸⡁⢃⣴⣿⠍⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢴⣀⠷⢴⣛⣈⠗⠋⣲⣿⡿⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣋⡽⠶⠥⣋⣉⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧⡌⠉⢳⡿⢁⣈⣙⠛⡿⠦⣥⣦⣤⣦⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣤⠎⡷⢻⠧⣈⣉⠋⠀⠉⠉⠒⠃⠈⠉⠉⠉⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡈⡘⠛⡟⡿⠢⠤⠬⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠿⠛⡞⢻⠗⠒⢒⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢏⠂⠤
⠀⠀⠀⠀⠀⠀⠀⣶⣏⣑⣳⣗⣋⣩⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒
⠀⠀⠀⠀⠀⠀⠀⣍⣍⡻⣟⣯⠤⠶⢾⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⡎⠉⠙⡇⠉⢲⠂⢄⡀
⠀⠀⠀⠀⠀⠀⠀⢿⢸⡛⡝⣿⡆⠚⣉⠷⣄⠀⠀⠀⠀⠀⠀⡠⢴⠉⠉⢃⠀⢹⣀⣀⣇⢀⣿⠀⡜⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠘⡿⡉⣏⣫⣹⣏⢀⡜⠁⡟⢰⡖⢲⡎⢹⡄⠈⡱⡀⣨⡷⢻⣇⣈⡿⢤⣽⣟⢳⣴⢁⡴⠧
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⢯⠙⢍⣿⡙⣦⡜⠁⣸⠀⢸⠃⠨⣇⣠⢐⣟⣋⣽⣟⡢⠼⣷⡶⠻⣼⡿⡏⢾⣀⠴
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣱⣾⣗⠻⣤⣹⣟⠛⢷⢣⣄⠘⢤⡀⢉⣽⢉⡭⠷⣤⣎⣷⠖⢎⢹⡦⡏⢷⠿⠿⢣⠔
⠀⠀⠀⢀⢴⣖⢋⠶⠞⣭⣿⣭⣶⣿⡶⣬⠟⠯⣄⡭⠶⢎⣉⣏⣠⡿⠛⠉⠀⠀⠈⣧⣼⢿⠁⣸⢻⣀⣧⣾⡦
⠀⢀⣎⢿⠞⢉⡿⠒⠒⣻⠤⢼⡛⠶⣽⣯⣙⣲⡊⠑⣲⣮⠴⢎⣁⡀⠀⠀⠀⣠⡴⣯⢶⣼⠟⣇⣠⠋⣸⢷⠓
⢰⣁⣎⣼⡟⠁⣹⠛⠛⣧⠤⣬⠽⢖⠂⠛⢾⠿⢟⡙⠳⣞⠳⢯⡉⠛⣍⡘⢿⣥⠴⣏⡤⢧⡞⢁⡿⢾⣥⡟⠻
⡦⢽⠛⣇⡿⡎⢻⠖⡻⠧⣴⡷⣾⣜⠛⢶⠛⠧⣄⠷⣶⠛⠳⢼⠛⠶⣋⠛⠛⢦⡴⠇⢴⠊⢉⡿⣶⣡⣿⠝⠁
⢇⣨⣶⢏⠀⡶⣿⠺⣇⣾⣇⠀⠀⠉⠓⠦⡷⣤⣸⣦⣄⣳⠦⢬⣷⠦⢬⣝⠛⠙⢶⡖⠋⠙⠧⠘⣋⡿⠃
⠸⡤⠞⢣⣘⣷⠼⡦⠳⡾⢮⣷⣖⢶⣶⠶⣿⢶⡿⢦⣄⣳⣦⣐⣚⡶⠤⠼⣟⣉⢉⣺⣏⣠⣼⠾⠋
⠀⠱⣴⠋⣤⠛⢠⠟⠾⡿⢻⠯⣽⠛⢏⠙⣷⡀⢛⡄⠉⣏⢛⣶⣮⣻⣶⠶⢿⣭⠿⠲⣿⠉
⠀⠀⠈⠛⠄⣴⠛⠀⣰⠀⡨⠀⡷⠀⢨⠀⢸⣇⠠⠖⠚⠉⠀⠀⠀⠈⠙⢠⢚⣁⡤⠖⢻⡆
⠀⠀⠀⠀⠀⠀⠁⠚⠃⠴⠇⠤⠗⠐⠚⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣋⣡⣀⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⢠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠦⣀⡤⣮⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⣿⡵⠾⠚⠚⠓⠷⠿⣿⣵⣖⣖⣖⡞⣏⣳⣘⣦⣤⠿⠋
⠀⠀⠀⠀⠀⠀⡤⢖⠁⣤⠵⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠛⠛⠉⠁
⣀⣀⣐⣒⣉⣁⣔⣊⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠟⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠦⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠘⠓⢄⡀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣿⣿⣧⡀⠀⠀⠀⠀⠈⠁⠀⠀⢤⡄⠈⠀⢀⣉⡀⠀⠁⠈⠛⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡖⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠲⠿⠇⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣿⣿⣿⡕⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣿⣷⣤⣀⡀⣤⣀⢤⣤⡄⢴⡖⣶⣤⣤⣈⠂⠀⢸⠀⠂⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡯⢩⣕⠰⣶⡉⠛⠢⣹⡁⣼⣄⣾⠵⠿⢿⣧⡀⠠⠆⠈⠀⡄⠀⠀⠈⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡙⡗⢄⣩⠳⣤⣈⠿⠆⣟⡁⠈⠃⠀⠀⠀⠀⠙⡇⠀⠀⠸⠀⡇⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿⠊⣭⣭⣤⣤⣴⣾⡯⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣷⠿⡛⠿⠿⢿⣿⣿⣷⣿⢆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⣀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⠞⣫⣾⣷⣦⡭⠳⣦⣀⡲⠾⠿⢿⣿⣿⣶⣦⣤⣤⢤⣀⣴⣾⣿⣿⣿⣷⣶⣠⢄⡀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢊⡴⣫⡵⠟⠉⠁⠀⠀⠀⠈⠉⠛⠛⠛⠓⠲⣆⡩⣭⣥⣤⣶⣿⣿⣿⣿⣿⠟⠟⠛⠿⢝⣿⣺⢶⣄⡀⡀⠂
⠀⠀⠀⠀⠀⡠⣢⣶⡿⠟⠋⢀⣀⡀⠠⠄⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠪⢍⠻⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠨⠉⠋⠒⠯⣄⡀
⠀⠀⠀⢀⣴⡼⠟⠉⠀⠀⠁⠀⠀⠀⠀⠀⢄⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡌⢻⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠢⢤⡄
⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄⠀⠀⠂⠄⡀⠀⠀⠀⡀⠀⠠⢿⠸⣿⣿⣿⣧⡄⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⠀⠀⠈⠓⠀⠀⠀⣦⠔⠘⣰⣿⣿⢹⡿⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⠈⠀⢀⣴⢿⣿⡇⠞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢴⠀⢀⣀⣠⠼⠛⠛⠏⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⢨⣿⣟⠋⠁⠀⠀⠀⠨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣶⣶⣦⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠻⣿⣿⣿⣿⣿⣿⡶⠄⠀⠀⠀⣤⣤⣶⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢠⡞⠂⠀⠀⠀⢀⠀⠀⠀⠀⠉⠁⠉⠛⠉⠀⠀⢀⡤⣦⡀⠿⢿⡿⠀⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠀⠀⠀⢠⣟⠁⠀⠀⠀⣼⣿⡿⠋⠀⠀⠀⠀⠀⠀⢀⣠⣶⢟⣽⣧⢰⣶⣿⣇⣠⡄⢀
⠀⠀⠀⠀⠀⠀⠀⣰⣿⡏⠀⠀⠀⢀⡼⠇⠀⠀⠀⠀⠹⣿⣷⡀⠀⠀⠀⠀⢀⣴⣿⠟⣡⢟⣾⣿⣿⡟⣯⢹⡟⡄
⠀⠀⠀⠀⡀⠀⠂⠉⢛⡓⠀⠂⠀⠘⠁⠀⠀⠀⠀⠀⠀⠹⣿⣿⣄⠀⠀⠠⣾⡿⢁⣼⢫⣾⡿⢿⣿⣧⢹⣼⢑⡶
⠀⠀⡀⠀⣀⠒⠉⠘⠥⣿⠷⢄⣞⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡄⠀⠀⠉⢠⡿⢡⣿⣿⡅⣿⠵⡹⠂⠟⢾⡅
⠀⠀⠛⡀⢭⣤⡔⠐⢠⠄⢷⣼⡁⠁⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⡿⣿⠀⠀⠀⠀⣰⣿⣿⢟⡁⣿⣿⣿⠀⠀⢸⣿⡄
⠀⡠⠋⢔⢘⡏⠁⠀⢀⣭⠉⠐⣇⠂⠀⣀⢀⢀⣤⣰⣼⣿⣿⣿⣿⣿⡀⠀⠀⠀⣹⡿⢿⣿⣿⣿⣿⡝⢦⡀⢸⣿⣇
⠀⡀⠐⠛⢋⣄⡞⡔⢻⡿⣤⠰⡿⣙⡟⢿⣻⣬⡻⣟⣷⣷⣿⣿⣿⡏⢁⣀⠀⠀⣿⣥⠿⠿⣿⣿⣿⣿⠘⡇⠈⠻⠛
⠀⠀⡀⡔⢡⣿⣇⠘⠈⠁⠉⠉⢵⣿⣮⣾⣬⣷⣷⣼⣿⡿⠿⠟⣩⡶⠋⠻⠛⠂⠀⠀⠐⠀⠘⢿⣿⣿⢸⡇⠀⠀⠀⠀⢀
⠀⣠⣿⠀⢛⢋⣟⡀⢀⢰⣠⠀⠀⢹⣿⣿⣿⡿⢿⣿⣿⣷⣖⡟⠁⠂⠀⠀⠀⠀⠀⢀⡘⠀⢀⣸⣿⣿⣸⠃⠀⠀⠀⣠⠎
⠀⠉⢻⡶⡌⠛⠙⣧⡷⢻⡿⠦⠀⢸⣿⣿⠋⠀⠀⠙⠛⠟⠋⠀⠀⡁⠀⠁⠴⠀⠂⠈⠀⠀⠀⢹⣿⣿⡿⠀⠀⠀⠀⡔
⠀⠐⢸⠛⣦⠆⠀⣿⡇⠱⠞⡄⢠⠀⢿⡏⠀⠀⢀⣤⣄⣤⣴⣷⣦⣦⣴⣤⠀⠀⠀⠀⠀⠀⠀⠻⠟⠛⠃⠀⠀⠀⠐⠁
⠀⠠⠈⢻⡅⠀⣧⡷⠟⣶⠀⡟⠁⠀⠈⢧⠀⠈⠉⠉⠉⠉⠉⠉⠃⠛⠛⠛⠀⠀⢀⣠⣷⣦⣀⠀⠄
⠀⠀⠠⠈⣿⢼⣘⠘⠚⠁⠳⣆⠀⠀⠀⠈⣧⣀⣀⠀⠀⠀⠀⣀⣄⢠⡄⠀⣠⣤⣘⢿⢿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⣶⠀⠳⠴⣌⠳⣦⠀⠀⠈⠀⠀⠀⠘⣿⠉⠀⠀⠀⠀⠀⡾⠯⣬⣄⣀⣀⣉⣉⣼⣼⣿⣿⣗⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠘⠢⢄⠀⠑⠀⠹⣏⣰⠆⠸⣦⣀⠀⠀⠀⠀⢀⣷⡆⠀⢀⣀⣿⠉⠁⢠⡈⠒⣤⠾⡁⠉⠉⠀⠀⠀⢀⠀⠴⠋
⠀⠀⠀⠀⠀⠙⢷⣄⠀⢢⠁⠁⠀⢸⠟⠻⡿⠶⠿⠿⠿⠛⠉⣿⢿⡿⢶⣶⣿⣄⣠⣞⣀⠤⠤⠴⠂⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠙⢻⡾⢦⡄⢀⡀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠉⠻⡁⠈⢻⣿⣿⣿⣻⣱⡆⠀⠀⣄⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠊⠀⠀⣌⣒⢤⣤⣤⡄⣸⢤⡤⡀⠀⢠⡉⠉⠉⠉⠙⠉⠉⠁⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠚⠳⣟⢑⠆⠙⠦⡛⠡⠼⠆⠀⠈⠁⠀⢤⣠⡶⣄⠂⡸⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠃⠀⠀⢨⠿⠛⠃⠀⠀⠆⠠⠀⠋⠁⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⣿⣿⣿⣿⣹⣿⣻⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⡟⢻⠉⡏⡽⢻⢻⣿⣟⠟⣿⠿⣊⣷⡄
⠀⠀⠀⠀⠀⠀⣰⣿⣽⣿⠻⣀⠣⠈⢦⣡⣷⣸⣸⢰⠉⣷⡿⣿⣿⢿⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⡑⠢⠄⢉⡵⠟⠋⠉⠉⠛⣧⢿⣿⣯⣟⢶⣽⡇
⠀⠀⠀⠀⠀⢸⣿⣿⢿⡦⣉⠘⢒⡻⠁⠀⠀⠀⠀⠀⠘⢿⣷⣿⣟⣧⢻⣿⣦⣤⣄⣠⡄⣤⡤⠄
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣗⠂⠬⠩⠇⠀⠀⠀⠀⠀⠀⠀⢺⣿⠿⣿⢼⡐⣿⣿⣿⢿⣿⣯⣦⣀⣀⣀⠀⢀⡀⠢
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣍⣀⣀⣱⡀⠀⠀⠀⠀⠀⠀⠰⣿⢷⣿⣯⣿⣾⣿⣟⣷⡿⠛⠞⣶⣯⣉⢿⡧⣿⡂⢀⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⡄⠤⠐⢓⡄⠀⠀⠀⠀⠀⠀⠈⠛⣦⣨⣬⣌⡇⢹⢿⣆⡇⣒⣦⣏⣋⣛⢤⣬⣥⠬⠖
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡍⠩⠥⢚⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠼⣄⡞⢿⣻⢸⡿⣿⠏⠁⠀⠨⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣆⣂⣉⡤⠋⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢬⡳⢬⣷⣼⣷⣤⠤⠠⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣯⣿⣿⢳⡴⠖⠊⣠⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠤⣽⣷⣬⣧⠀⠀⠀⠈⠹⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣎⠿⣿⣉⢁⡤⠋⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡢⡙⢧⡄⠔⠋⢀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣟⢶⣳⡖⠒⣁⡴⠧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣗⢿⣏⢉⣀⡼⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⢺⣿⣉⡁⣠⠾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣼⣿⣍⣉⣀⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣿⡿⣯⣿⠤⠤⠖⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⠶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⢾⣿⡟⠳⢿⡾⣯⡿⣾⡯⣴⢏⡿⠒⠒⠘⣯⣿⣧⣶⣄⣄⠀⠀⠀⠀⣼⢿⡇
⠀⠀⠀⠀⣠⡶⣿⢿⡿⣿⠟⣛⣡⣿⣯⢼⢮⣯⣿⠿⣣⣟⡷⠂⠒⢺⢷⣿⡿⠼⠾⣿⡿⣆⠀⠀⠘⢷⣿⣶⣦⣄⣀
⠀⠀⢀⣾⣿⣳⣿⣿⡾⣡⣾⢻⣿⠋⣵⣿⣵⣾⡶⣿⢋⣿⡋⠀⢙⣿⣿⣉⣿⣠⣶⡿⠳⢿⣇⠀⠀⠀⠈⠉⠛⠿⣿⣿⣦⡄
⠀⢀⠞⣼⡇⣿⣼⣏⣿⢯⣜⣧⣽⠿⠾⢻⣿⡓⣿⣳⣿⣯⠁⢀⡽⠻⣿⡿⠂⠃⣛⣱⢣⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠹⣯⣹⡆
⠀⣸⠓⣿⡇⠸⣧⣽⣯⢸⡟⠁⠀⣀⡴⢋⣿⢱⡿⣋⡾⠦⢠⣔⣠⣴⡿⣿⣿⠛⣭⢇⢺⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⢿⡇
⠀⣿⣷⡙⠿⠶⣠⠈⠁⠸⡿⣶⡟⣛⣴⠟⢡⣾⠟⣻⣉⡶⠟⠋⣱⠿⠿⠟⠿⠛⢉⣤⡿⢃⣤⡞⢛⣋⣹⣿⣿⣿⡼⣿⣿⣾⠃
⠀⢻⣌⣹⡖⢀⣼⡇⣰⢦⣀⣙⣛⣍⠑⠾⣻⣴⣾⡻⠉⣩⣟⣴⣾⣿⣟⢛⣿⡶⠞⠉⡰⠟⣿⣿⡿⢿⣷⠾⠿⠿⠶⠿⠛⠁
⠀⠘⡿⣿⣧⣾⣁⣰⠟⣲⢤⡞⠣⣭⢻⡿⡽⣰⣿⢿⠟⣹⣿⣾⣷⠿⠓⠋⠁⠀⠀⠀⢿⣼⣿⣯⠛⢶⡼⣷⣄
⠀⠀⠘⢮⣙⢿⣦⡸⣷⣉⣳⠶⣟⠳⣟⠄⣿⠛⢟⡲⣮⣿⡷⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣟⣷⣾⠛⣛⣽⣦
⠀⠀⠀⠀⠈⠑⠚⠷⠮⠯⠿⠶⠾⢷⣿⠈⠓⠀⣹⣴⡟⠽⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠼⠿⣬⣁⠁⣢⡌⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⣇⠺⣷⠶⠖⣶⣷⡞⢻⣟⠻⠏⡻⣿⣛⣿⡿⢿⣻⣟⡛⢿⣦⣿⣣⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⡟⣤⢻⣶⢶⣬⣭⣷⠤⠾⣿⣼⡇⢻⠿⣥⣼⣋⣉⣿⠿⢶⡿⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡘⠃⠆⣤⣶⡶⠾⣷⣶⣦⣤⣀⣀⠘⣉⠋⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠺⠿⠷⠶⢿⣯⣭⡿⠿⠽⠿⠟⠛⠉⠁
⠀⠀⢀⡀
⣿⡿⣛⣛⡿⠿⠿⠿⠿⠿⠿⢿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣷⣧⣼⣷⣶⣶⣾⣶⣶⣶⣷⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⡟⢆⠀⠀⠀⠀⢀⣤
⣠⢀⣀⠀⠀⡇⠀⠳⡀⡠⠖⢉⠂
⠈⠲⣌⠉⠓⠓⠀⠀⠉⠀⠀⣯⣤⣄⣀⣀
⠀⠀⠈⠢⡀⠀⠀⠀⣀⠀⠀⠀⠀⣠⠔⠉⠀⠀⠀⠀⠀⢀⡀
⠀⢀⡠⠖⠉⠀⠀⠀⠸⡓⢄⠀⠺⢤⣀⠀⠀⠀⠀⠀⢀⣸⠹⣤⢤⡄
⠀⠉⠉⠛⠛⠛⣻⠀⠀⣱⡀⠙⢶⡉⠉⠀⠀⠀⠀⠻⠭⣅⠀⠀⢯⡀
⠀⠀⠀⠀⠀⠀⣏⡴⠊⠁⢳⠀⠀⠉⠲⣄⠀⠀⠀⠀⠀⣼⠖⠉⠉⠛
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣⡀⠀⠀⡈⠳⣄⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠢⡈⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠹⣄⠀⠀⠈⠳⢄⠙⢦⡀
⠀⠀⠀⠀⢀⣀⢀⡞⠸⡄⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠑⠀⠉⠳⢄
⠀⠀⠀⠀⠈⠻⣍⠀⠀⠉⣳⡦⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠈⠢⣄⠙⢦⡀
⠀⠀⠀⠀⢀⣴⡧⠶⢤⣸⡁⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠈⠑⢄⡙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠳⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⡰⠀⢀⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠈⢀⡔⠁⠀⠉⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⡴⠛⢦⡀⠀⠀⠀⠈⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠚⠁⠀⠀⠙⢦⡀⠀⠀⠀⠀⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣓⣶⣦⣶
⣿⣿⣉⣉⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣧⣶⠿⠛⠳⢦⡀⠀⠀⠀⠀⠢⡀⡠⠊⠠⠀⠀⠀⠀⠀⠐⠄⠀⢢⠁⡀⠌⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣰⠾⠛⠛⠶⣄
⢿⣁⠀⠀⠀⣸⡇⠀⠀⠀⠀⡀⠘⠀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠑⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣀⣼
⠀⠉⠛⠛⠛⠋⠀⠀⠀⢀⠔⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⠀⠀⠀⠀⠔⢣⠗⢩⣍⡀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⢀⡀⠀⠀⠀⠀⠀⠁⠀⠀⢰⠀⠀⠀⠁⠀⠀⠀⢠⣤⣤⣼⣥⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⡢⢔⡙⠈⠂⠀⠀⠀⠀⡀⠀⢢⠀⢰⡏⠀⡀⠀⠀⢠⣈⣁⣀⣹⣅⣀⣈⣠⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠎⠈
⡀⢀⠁⢢⡄⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠂⠀⠀⠀⠘⠉⠉⠉⢹⡏⠉⠉⠙⠀⠀⠀⠀⠉⠂⡀⠀⠀⠀⢀⠔⡀⠀⠀⠀⠘⣠
⢠⡄⣀⠔⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠳⢦⣸⣥⡾⠿⠛⠳⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡙⡀⠀⠀⠀⠀⠀⢹⣄⠀⠀⢀⣴⡿⢯⣀⠀⠀⢀⣸⠇⠀⠀⠀⠂⡠⠛⢄
⡖⠀⠀⠀⠀⠀⠀⠤⢃⠜⠨⡍⡀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠙⠛⠛⠛⠉⠀⠀⠀⠀⠊⠁⠀⡀⠑⠄⠀⠀⠀⠀⠀⠀⠐
⣧⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⣇⣀⣈⣁⡄⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠀⢀⡼⠀⠀⠀⠜⠈⢅⠢⢂⠁⠐⡀⠀⠀⠀⠀⠀⠈⡆⠀⢰⠃⠀⡀⠀⠀⣄⣈⣀⣀
⡏⠉⠉⠉⠃⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⣠⢔⣩⣀⡸⡑⢤⠇⠀⠀⠀⠀⠒⠊⠀⢤⠃⠐⠒⠀⠀⠀⠋⠉⠉⠉
⣧⣾⠿⠛⠓⢦⡀⠀⠀⠀⠀⠠⡀⡠⠂⡠⠀⠀⠀⠀⠀⠐⠄⡀⢢⠀⡠⠘⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣰⠾⠛⠛⠳⣤
⢿⣅⠀⠀⠀⣸⡇⠀⠀⠀⠀⡀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⢑⡀⠀⠀⠀⠀⠀⢿⡀⠀⠀⣀⣼
⠀⠈⠛⠛⠛⠋⠀⠀⠀⠀⠀⠁⠀⠀⠁⢄⠀⠀⠀⠀⠀⠀⠀⢶⡆⠀⠀⠀⠀⠀⠀⠜⣡⠇⠩⢍⡀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠁
⠒⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠁⠀⠀⠀⢰⣤⣤⣼⣤⣤⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⡢⢄⡉⠈⠂⠀⠀⠀⠀⠀⠈⢱⠀⢰⡏⠀⠄⠀⠀⢠⣈⣀⣀⣸⣇⣀⣈⣠⠀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠎⠈
⡄⢀⠁⠢⡄⠀⠀⠀⠀⠐⠂⠀⠠⠎⠀⠒⠁⠀⠀⠘⠉⠉⠉⢹⡋⠉⠉⠙⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠘⡤⠂
⢠⡆⣀⠔⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⠳⢦⣼⣶⡾⠟⠛⠳⣦⠀⠀⠀⠁⠈⠢⠀⠔⠁⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢙⡀⠀⠀⠀⠀⠀⢹⣄⠀⠀⣀⣴⡿⢯⣀⠀⠀⢀⣼⠇⠀⠀⠀⢠⡠⠉⢄⡀
⡖⠀⠀⠀⠀⠀⠀⠤⢃⠞⠨⠌⡁⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠉⠛⠛⠛⠁⠀⠀⠀⡀⠊⠡⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠐
⣧⣤⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠷⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤
⣇⣀⣀⣀⡄⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠌⠈⢍⠢⢂⠁⠘⡄⠀⠀⠀⠀⠀⠈⡇⠀⢱⠁⠀⠀⠀⠀⣄⣈⣀⣀
⡏⠉⠉⠉⠃⠀⠀⠀⠀⡐⣀⠀⠀⠀⢀⡐⡐⠀⠀⠀⠀⣠⣄⣩⡀⣰⣁⢤⠃⠀⠀⠀⠀⠒⠊⠀⢠⠁⠁⠒⠀⠀⠀⠋⠉⠉⠉
⣟⣿⣿⣻⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣿⣻⣿⣿⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠶⠂⠀⢀⡄⠀⠀⠀⠘⢏⡴⢛⣯⠉⣯⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣠⣵⡌⢡⠀⢀⣤⢎⠿⣧⡈⠀⢂⡀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠯⣤⣿⣮⡀⠀⢻⡈⠔⠛⡽⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣼⠁⣦⠟⢸⡾⠃⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⣾⡀⠀⣠⣴⣾⡟⠛⠛⠛⡻⠖⠒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣖⠲⠞⠛⠻⠒⢶⣄⠀⠀⠀⠀⢈⡟⠀⡄⣈⣿⢁⣘⡛⠊⣡⣉⣨⣤⠤⠤⣶⣤⡄⢈⣳⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⣉⠻⠤⢦⡀⠙⢧⣀⣄⣀⢀⡀⠀⢻⣹⣿⣜⣽⢿⡟⠛⢏⠀⠈⠰⠄⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠷⣮⠵⣦⣀⣿⠟⠛⢻⡅⠀⠀⠙⢿⢿⣶⠐⠚⠳⠤⣤⣄⠠⠖⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠚⣡⠀⠀⢨⠛⢦⠀⣿⠀⠀⠀⣰⠟⠹⠛⠂⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣻⠀⡟⣈⠁⠁⠀⠈⣆⡿⠀⠀⣼⠋⠀⠀⠠⣘⣶⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠄⡛⣀⠙⠏⠀⠀⠀⠀⢻⠇⢀⡞⠁⣷⡀⢀⣤⠈⠙⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⢘⣁⠀⡀⢨⠀⠀⠀⠀⠀⠀⣾⣤⠂⠀⢀⡮⠁⢀⡁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣦⣄⠀⠀⠘⠟⠀⠻⡾⠀⠀⠀⠀⠀⢠⡿⠈⢤⣈⣿⢼⡀⣴⣤⣀⠜⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣿⣇⣈⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⣀⠉⠛⣻⡿⡟⠋⠉⠄⠀⠀⣴⢓⡟⢦⡀⠀⢀⡀⠀⢰⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⢿⣽⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠘⣿⠋⠰⡉⠛⠐⠄⣀⠀⠀⣼⠁⠸⠀⣸⠇⠀⡛⠛⠑⣇⠈⡙⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⢽⡆⡄⠀⢠⡙⢀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢣⡞⠀⠀⠈⠀⠀⠛⠋⠀⠜⠃⠰⢋⣿⣦⣿⣿⣿⣗⡲⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣀⠹⣧⠀⢰⠁⢸⣆⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⢀⣤⠞⠁⠀⢀⣼⠀⣬⠀⣸⢧⡄⠈⠻⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣿⣴⡏⠀⠀⡟⠀⠀⠀⢠⠤⡦⠲⢲⡶⠀⠀⣾⠀⣠⡴⠛⠓⠽⠛⢳⠀⠰⠟⠀⠉⠋⠉⣨⡿⢷⣴⣯⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⠀⠀⡅⠰⠀⣼⠀⡸⠂⡠⡞⠀⢘⣤⣿⡼⠷⢤⣀⠀⠘⣐⣧⣶⣶⠓⠄⣀⡀⠀⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠦⠆⠀⠙⣷⣦⠀⠀⡶⠛⣿⠀⠘⠁⠆⠀⣸⡿⠋⠀⠀⠰⣟⠁⢀⡝⣻⡁⠙⣦⣬⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⡀⢢⣀⣐⣿⣷⠀⠐⠂⢨⡿⠀⠻⠄⢀⣴⠋⠀⠀⠀⠀⠀⠀⢰⣧⣀⠰⠑⢠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠫⢍⡙⣿⡿⠀⡧⠀⡇⠀⠀⣠⠞⠛⢷⠀⠀⠀⢀⣴⡶⢾⣧⡿⢀⣤⠀⠉⠃
⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣇⡼⠁⢠⠁⢠⡜⠁⠀⠀⢨⣇⠀⠀⢘⠻⠇⢰⠀⠈⠋⢸
⣴⠋⠹⠿⣿⡷⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠁⠀⠘⠠⡍⠀⠀⠀⢰⡟⢹⡄⠀⠈⠠⣜⣿⣤⣟⠤⠊
⠿⢹⣧⡤⠟⠀⠀⡀⠿⣿⡟⠛⢳⣤⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢠⣾⢆⡑⠀⠀⠀⣰⠏⢀⡼⣇⠀⠀⠀⠀⣿⠁⡏
⠀⠀⠀⠀⢀⣠⡶⣧⣠⣼⡸⣦⡞⠙⣷⠀⠀⠀⠀⠀⠀⠀⢹⡆⢸⣧⡏⠁⠀⢠⠺⠯⠀⡬⠁⡉⠀⠀⠀⠀⢻⣨⡃⠂⠒⢶⣶⡶⢶⣦⣤⣤⣤⣤⣤⡄
⠀⠀⠀⢀⡾⢩⣄⠹⠏⡈⠁⠀⡟⠀⣹⠀⠀⠀⠀⠀⠀⠀⢈⡇⣿⣿⠀⠀⢰⡇⠀⡀⠀⣧⣤⡟⠀⢀⠴⠀⠈⣟⠀⠀⠀⠀⣹⣟⣹⣷⠳⠖⠊⢉⠇
⠀⠀⠀⠸⣧⣸⡟⠂⡘⢿⣤⡾⠁⠘⠛⠀⢀⣤⢤⣄⡀⠀⠀⢹⣙⡃⠀⠀⠸⣷⠞⠁⠀⠉⠁⠀⡐⠀⠀⠇⠀⢩⣄⣤⠶⠚⠛⠁⠀⢠⣦⣼⠿⠋
⠀⠀⠀⠀⠉⢹⣇⣰⡇⠈⠉⠀⠀⠀⠈⣠⠞⣧⢀⡍⠙⠆⠀⢘⡏⠙⠀⠀⠀⠀⡤⢀⡀⠀⣦⠒⠀⠀⣡⣤⡖⠛⠀⢀⣀⣀⡀⠠⣴⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠙⠋⠀⣀⣀⣀⡀⠀⠘⣇⠀⠀⢹⣄⠀⠀⠀⢸⠳⠄⠀⠀⢠⠎⠀⢰⠆⠰⣬⡁⢀⡴⠆⠀⠀⠀⠀⠀⠘⠋⠁⡼⠃
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣸⡋⠙⢦⠀⡇⠀⠳⣦⠘⠳⠃⠀⣿⠀⠀⠀⠀⣾⠐⠀⣯⠀⡠⢢⡶⠛⠀⠀⠉⠁⠂⣢⡀⣽⣤⡾⠁
⠀⠀⠀⠀⠀⠀⣾⠃⠺⠆⡮⠛⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢻⠀⢀⣿⠈⡴⠟⣿⣤⣀⣀⣠⣤⢊⣱⡾⠾⠋
⠀⠀⠀⠀⠀⠀⣿⣄⣴⠀⢷⣀⡀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢐⣷⣞⣴⣏⣀⠀⠀⢀⡀⠑⠆⢤⣻⡟
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠈⠉⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⢻⠈⠀⠀⠀⡘⢣⣞⣋⣈⣉⡛⠶⣾⣿⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡜⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⣰⣰⠟⠁⠀⠀⠈⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⢃⠀⠀⠀⠀⠀⠸⣿⡇⢀⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢦⠀⠀⠀⠀⠀⣿⣿⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⢃⠀⠀⠀⠀⢹⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣎⡀⠀⠀⠀⠘⣷⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆

⠀⠀⠀⠀⠀⢤⣤⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠀⠀⣤⡀⠀⠀⠀⠰⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣠⡄
⠀⢀⠀⠀⠀⢰⠾⠛⠻⠰⠄⠀⠀⠀⠀⠘⢂⠉⠁⠀⠀⠀⠀⠀⠠⠦⣄⠀⠘⣿⠀⠀⠀⠴⣿⡽⣿⣤⠀⠀⠈⣾⣧⡀⠀⠀⣰⣿⣿⣿⠗⠀⠀⠀⢈⣿⣶⡄
⠀⠚⢃⠀⠀⠀⠀⠀⢀⣤⠄⡁⠦⠀⠀⠀⠈⠘⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣷⣷⡀⠀⠀⠈⣿⠃⠀⠀⠿⣿⡟⠉⠀⠀⠀⠀⠩⠿⠞⠁
⠀⠀⠀⠀⠀⠀⡀⠀⠋⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣃⡄⠀⠀⠀⠀⠈⠑⠆⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠠⠀⠘⢶⢶⣤
⠀⠀⠀⠀⠀⡇⠠⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⡽⠄⠀⠀⢀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠸⡦⠀⠀⢈⡄⠬⠒⠒⠀⠀⠀⠘⠛⣯⣥⡂
⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠠⣤⣀⠀⠀⠀⠐⣂⠐⢾⣾⠆⠀⠀⠀⠀⠀⠀⢦⣤⠀⠀⣠⠆⠒⠻⠷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠜⣨⠀⠀⠀⡄
⠀⠀⠠⣶⠃⠀⠀⠀⠀⢠⠂⠆⠀⠀⣀⠀⠀⢐⣾⡽⣷⡄⠀⠐⢀⢑⣠⣤⡤⠀⢤⣄⡀⠀⠈⠿⢟⣶⢺⣿⡷⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢠⠀⢠⠄⣀⠜
⠀⠀⠀⣿⠀⠀⠀⠀⡰⢋⠃⠀⠀⠀⠔⡞⠂⠴⠽⢿⡟⡿⢀⡀⠀⠀⠘⢿⠗⠰⢺⣯⢿⣦⠀⣰⣮⣤⣬⡋⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣟⡿⡈⢺⠀⣬
⠀⠀⠀⠃⠀⠀⣠⠞⢡⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣶⣽⡁⠶⠄⡖⠠⣀⠰⣀⠀⠈⠛⠿⢿⡟⠋⠿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢁⣬⣿⡎
⢀⣀⣀⣀⣴⣋⡠⠄⠉⠀⠀⠀⠄⠀⠀⠀⠀⠐⠙⠛⠁⢡⣶⡾⣾⡷⢤⡀⠙⢄⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⡿⣽⡇
⠀⠉⢣⣠⠌⠙⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⡷⡿⠟⠉⠀⠙⢆⠘⣆⣰⠏⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⣤
⠀⠀⠀⠙⢤⡀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠤⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠱⡄⠂⠀⠀⠈⠓⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠖⠉⠀⠈
⠀⠀⠀⠀⠀⠀⠈⣗⠂⡀⠀⠀⠈⠙⣆⠀⠀⠀⠀⠀⠈⠀⠤⠠⡄⠀⠀⣀⠀⠀⠟⠀⠀⠀⠂⠀⠀⠀⠀⢰⡞⠉⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠈⢡⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠢⢴⠂⢄⠀⠀⠐⠀⠀⠀⠀⡼⠋⠀⠀⠀⡀⠂⢀⡤⠂⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠄⡂⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠁⠣⡀⠀⢀⡞⠁⠀⠀⢀⣆⡱⠂⠁⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠲⡀⢣⣠⠞⠀⠀⢈⣠⠚⠁⠀⠀⠀⣨⠄⠀⠀⠀⠀⠀⢀⢀⠆⠒⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢐⠀⠀⠀⡀⠀⠀⠀⠀⠙⢾⠋⣆⣀⣜⠁⠎⠀⡄⠀⣄⢀⣸⣤⡀⠠⣤⡴⠂⠉⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⠁⠸⡉⠘⠇⣄⠀⠀⢸⡦⠞⢉⣤⠼⠐⢲⠆⠙⢿⣤⢿⡄⢀⡴⠁⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣦⡀⢀⣴⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡙⢿⣿⣿⣿⣿⡿⢋⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣾⣿⣿⣿⣿⣷⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⡿⠋⠙⢿⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⢿⣿⣷⣄⣠⣾⣿⡿⢿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⠿⠋⠀⠀⠹⣿⣿⣿⣿⠏⠀⠀⠙⢿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⠁⠀⠀⠀⠀⠀⢘⣿⣿⡃⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋⠀⣀⣀⣀⣀⣀⠀⠀⣰⣿⡿⠃⠘⢿⣿⣆⠀⠀⣀⣀⣀⣀⣀⠀⠙⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⡄⠀⣴⣿⡟⠁⠀⠀⠈⢻⣿⣦⠀⢠⣿⣿⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣾⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣷⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣯⡀⠀⠀⠀⠀⠀⠀⢀⣽⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠻⣿⣷⡄⠀⠀⠀⠀⢠⣾⣿⠟⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣴⣿⡿⠃⠀⠙⣿⣿⣆⠀⠀⣰⣿⣿⠋⠀⠘⢿⣿⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⣿⣿⡿⠁⠀⠀⠀⠈⢿⣿⣧⣼⣿⡿⠁⠀⠀⠀⠈⢿⣿⣷⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⢀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠉⠤⢈⡟⠁⠀⡀⠀⠀⠀⠈⠁⡀⢠⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠄⡀⠀⠀⢀⠀⣄⠀⡀⠀⠂⠄⠀⠀⠀⠦⠔⠂
⠀⠀⠀⠀⠀⠀⠠⠖⡐⠀⠈⠀⡈⠀⠀⠀⠀⣀⡴⠚⠀⠀⠀⠉⠃⠀⠰⢣⣴⡀⠀⡁
⠀⠀⠀⠀⠀⠀⢀⡐⠀⠀⢁⡈⠀⠀⠀⢀⣀⡸⠐⣤⠔⠁⠀⠀⢂⠔⠁⠁⠀⠉⠈
⠀⠀⠀⠀⠐⠂⠐⠀⠀⠉⠀⠅⠁⠀⠀⠀⠈⠀⠀⠁⠸⠀⢀⠘
⠀⠀⠀⠒⠁⠐⠐⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⢤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⢀⡀⢀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠁⠠⠀⠁⠙

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠤⠤⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣱⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠘⢂
⠀⠀⠀⠀⣀⣠⣴⣿⣽⣿⣿⡿⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⠀⠀⠀⢲⡄⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠙⠛⠛⠛⠛⠻⠋⠨⡳⣝⢦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⢠⠠⠌⠁⠀⠀⠀⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡝⢎⣡⠶⢂⣤⣤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⣴⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⣶⢸⣶⡶⢒⡶⢮⣝⣛⠶⠆⠤⠶⣖⣒⡰⠾⠿⠶⠏⣙⣛⠻⣿⣷⣌⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣰⣮⢰⢸⣿⣇⢛⣛⡿⣛⣛⣫⡭⢉⠶⣾⠿⡛⣴⣿⣿⡇⣶⣝⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣄⣿⢰⡌⣿⣿⡎⣭⣝⣫⣭⠶⢟⣫⣵⡶⠮⠁⠈⣭⣏⢶⣜⢿⡆⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣘⠧⢻⡜⣿⡇⣯⡭⢩⡶⠟⠛⠉⠀⠀⠀⠀⠀⣿⣿⢇⡻⣆⢷⢹⣜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⠱⢹⣧⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⣾⠇⠈⠘⣠⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇⠀⠀⠀⠛⢧⠻⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠿⠀⠀⠀⠀⠀⠀⠀⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⣻⣽⣻⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣟⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣵⣿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⣿⣿⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣫⢟⣿⣿⣿⣿⣿⣧⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⣼⣿⣿⣿⣿⣿⡷⣝⣿⣦
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣧⢜⢿⣿⣿⣿⣿⣿⣿⡿⡃⣼⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢎⣿⣿⣿⣿⣿⣿⣎⢯⢻⣿⣿⣿⣿⡟⡽⣱⣿⣿⣿⣿⣿⢻⣿⣿⣿⣾⣟⣿⣇
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣎⣷⡹⣿⣿⢏⣾⣱⣿⣿⣿⣿⣿⣿⣿⣻⣾⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⢿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⡜⢫⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣱⣎⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣴⡼⢧⣦⣻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⣶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠘⣿⣿⣟⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⢹⣿⣿⡿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⠿⣿⣏⣾⣷⣹⣿⠿⣿⣿⣿⣿⣿⣿⣿⡱⣿⣿⢿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢻⣷⣿⣽⣿⣿⣿⣿⣿⣿⣟⢿⣿⣷⡛⣼⣿⣿⣧⢛⣾⣿⡿⣻⣿⣿⣿⣿⣿⣿⣯⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣝⣾⣿⣿⣿⣿⣿⢱⣷⣿⢟⣴⡹⣿⣿⢏⣦⡻⣿⣾⡎⣿⣿⣿⣿⣿⡷⣫⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⡻⣿⣿⣮⣛⣯⣾⣿⣷⡹⢏⣾⣿⣷⣽⣛⣵⣿⣿⢿⡿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣽⣿⣽⣻⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣟⣯⣿⣯⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣷⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄

⠀⠀⠀⠀⠀⠄⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠐⢁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⣠⠞⠀⠀⢀⡀⢀⣴⣿⠁⠀⢘⣾⡾⣆⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⠁
⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠉⠁⣾⢻⣿⣶⣴⣾⣿⠏⣸⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠛⣿⣧⡄⠀⠀⠠⠆⢻⡈⠻⢿⣿⠿⠛⣸⣿⠀⠀⠤⠀⠀⠀⢠⣿⠏⠑⠀⡀
⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠘⠛⠻⣷⣄⡀⠀⠈⠻⠷⡞⠟⢻⠿⣿⠡⠂⠀⠀⠀⠀⣠⡿⠟⠀⠀⠀⠛⠀⠀⢤
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣀⡀⠸⣴⣇⣷⣸⣧⣼⣦⣤⠄⠀⠀⠈⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠹⣿⣙⠛⠋⠘⢿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⠀⠀⠀⢀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠀⠀⣀⣀⣀⣀⠀⣀⠀⠀⢼⣿⣷⣤⡀⠈⠀⣀⡀⣠⣤⣤⣤⣄⠀⠀⠁⠀⠀⠀⠈⠆⠀⠈⠁
⠀⠀⠀⢰⠀⣷⠀⠀⢀⠀⢀⣴⣿⣿⠟⣿⣿⣭⡉⠀⠙⢷⠟⠁⠉⠉⠻⣿⡄⢀⣤⣄⠀⠀⠙⠿⢛⣷⣄⣀⣠⡄⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢸⣆⠀⠸⢶⣿⣿⡏⠀⣰⣿⣿⣿⣿⣦⣦⡄⣴⣦⡀⠀⠀⢨⣤⣾⣯⣤⣴⣄⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⠋
⠀⠄⠀⠀⠀⣮⣿⣷⣦⣾⣿⣿⣷⣿⣿⡿⠟⠛⣿⣿⣿⣿⣿⡿⠃⠀⠀⠘⠻⠟⢻⣿⣿⣿⠋⠙⢿⣿⣿⣿⣿⣿⠿⠋⢈⡿⠋
⠀⠀⠄⠀⢀⡼⢿⣿⣿⣿⣿⣿⣿⠟⠁⢀⣀⠀⠘⢿⣿⣿⣸⣦⣀⠀⠀⠖⢠⣾⣿⣿⣿⠏⢠⣦⣀⠘⢿⣿⡇⢉⣼⣷⣿⣿
⠀⠀⠀⠀⣼⠁⣾⣿⣿⣿⣿⣿⠁⣠⣾⣿⢿⣷⣄⠀⠉⠻⠿⣿⣏⡀⠀⠈⠻⠿⠟⠋⢀⣴⣿⠿⢿⣷⡄⢹⣶⣾⣿⣿⡏⢻⢻⡆
⠀⠀⠀⠀⢹⡀⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⡈⠻⢿⣿⠆⠀⠀⠙⠉⠀⠀⠀⠀⠀⠐⠾⠿⠟⠋⣀⠀⠻⠟⣿⣿⣿⣿⣿⡇⢀⠘⠃
⠀⠀⠀⠀⠈⣷⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡿⢂⡴⠀⠉⣀⣀⣀⣀⣀⢀⣄⣰⣾⣿⠶⣶⣶⡿⢿⡷⢿⡆⣿⣿⣿⣿⣿⣇⡘
⠀⠀⠀⠀⠀⠘⣿⣿⣟⣿⣿⣿⣿⣿⠟⠉⣴⣿⡿⣟⣯⢍⣭⣿⣿⣿⣻⣿⣿⣽⣿⣷⣲⣍⣤⣄⠁⢈⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⣿⢟⣿⢿⣿⣿⣿⣷⡀⠠⠻⢫⣾⣿⣷⣦⣭⣛⠿⣝⣿⠋⠹⣫⢟⣵⣿⣿⣿⣿⠀⣠⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠿⠛⠉⠘⣿⣿⣿⡟⠀⢣⡇⣿⣿⣿⣿⣿⣼⣿⣿⡾⣟⠀⠀⢓⣫⣵⣿⣇⠛⠉⢠⣿⣿⣿⡿⠛⠆⢻⡠⡆⠀⠀⠂
⠀⠀⠀⠀⡀⠀⠀⢀⡇⠀⠉⣿⡿⠗⠀⠘⠋⠻⣿⡿⠉⠻⣿⣻⣟⣿⡇⠀⠀⠨⣏⣼⣿⡟⢠⠀⢸⣿⣿⣿⣷⡆⠁⠘⠁⠙⠊
⠀⠀⠀⠀⠸⠆⢀⣾⡄⠀⠸⢿⡇⠐⠀⠀⣄⣰⣜⠁⠐⡀⠈⢿⣿⣿⣧⡀⠀⣰⣿⣟⣿⣷⣌⡀⢈⠇⠙⣿⣿⣷⠀⢇
⠀⠀⠀⣀⡄⠀⣸⡿⠀⠀⢀⣼⠧⠀⠀⠀⡿⠈⠛⠠⠀⠰⠆⠘⣿⣿⣿⣿⣾⣿⠙⣏⡙⣿⠟⣣⣤⡀⠀⢸⡟⢿⡆⠸⣿⣇⠀⠀⡄
⠀⠀⢴⡨⠀⡾⠿⠻⠲⠽⠿⠁⠀⢀⣀⣠⣤⡄⠀⢡⣾⡿⢺⣶⣿⣿⣿⣿⣯⣾⠀⠀⠐⠉⣶⣯⣿⣷⠀⠘⠃⠈⠣⠄⠹⢷⠀⠀⢣
⠀⠀⢀⠓⠀⠀⠀⠀⠴⠀⠀⠀⠈⠁⢀⡀⠀⠀⡀⠈⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣼⣿⣿⠃⢀⣉⠁⠀⠀⠀⠠⠀⠀⣠⣤
⣀⣀⡀⢀⡀⢤⠔⣀⣤⡄⡶⠃⠀⢀⢰⢇⣸⣿⣿⣿⣿⣿⣿⣿⡇⠙⣿⣿⡟⢿⣿⡻⣿⠿⠿⠿⠿⠿⠿⠿⣟⠋⠀⠄⢀⡀
⠉⠀⠈⢩⡇⠀⠀⠈⠀⣼⠁⠀⣀⢚⣉⣿⣟⣻⣿⡟⣸⣿⣿⣿⡇⠀⠘⠋⠀⠸⣿⣿⣿⣿⣿⣶⣤⠴⠈⠉⠉⠛⠳⠦⠤⠔⠀⠉
⡿⠟⠋⣼⣇⠀⠀⠀⠀⠿⠂⢀⣽⣿⡿⠿⠟⠛⠛⢠⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⢻⣿⣯⣄⠀⡈⠀⠀⠀⠀⠀⠀⠐⠀⠀⣠⣤⣤⣤
⣿⣦⣀⣀⣉⣠⣾⣿⣶⣶⠾⠟⠛⠉⠀⠀⠀⠀⠀⠸⠟⠿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠛⠃⣀⡀⣉⣿⣿⣿⣿⣿⣶⣶⣤⣤⣤⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀⣠⠀⠀⣶⠦⢤⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⠀⠀⣠⣄⣀⣀⣤⣀⡐⢀⣀⣰⣆⣿⣿⣆⡀⢈⣀⣀⣘⣷⣒⣀⡀⢂⣀⣀⣀⣀⣠⣄⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠺⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣶⣿⡿⠓⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⣷⣄⠀⠀⠀⠈⠉⠀⠀⠀⠉⢉⡟⣿⡇⠀⠈⠁⡉⣯⣤⠀⠀⠉⠁⠀⠀⠀⣠⣾⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⠘⣷⣆⠀⠀⣠⢾⣿⣿⣿⡁⣀⠀⠀⠀⣠⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢿⣿⣿⣟⢿⣷⣀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⣶⣿⣿⣿⣿⣿⣿⡇⢠⣾⡿⣻⣿⣿⣿⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⣿⣿⣧⡙⣿⣷⣄⠘⠀⠀⠀⠲⠀⠀⣤⠎⡽⢿⣿⣿⣿⣿⣿⣿⢏⣼⣿⣿⢫⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠘⣿⣿⣷⣼⣿⣿⣦⣁⠀⡄⠀⡇⠀⠀⠀⠀⠶⣼⣿⣿⣿⣿⣡⣾⣿⣿⠃⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⣿⣿⣿⣿⠙⠛⢿⣷⣿⢒⣳⡄⠀⠀⠀⢀⣼⡿⣿⢧⣿⣿⣿⣿⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠈⠛⢿⣿⣷⣄⠀⠻⣿⣜⣿⡄⠀⢀⣠⣾⠟⠀⢀⣾⣿⣿⡏⠃⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⣿⡿⠃⠈⠿⣷⣄⣠⣾⠟⠁⠀⢀⣿⣿⡏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆⠀⠄⣸⣿⣿⣅⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣧⣾⡿⣯⡝⠻⣷⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣹⣿⣿⣿⣧⣈⠀⢀⣿⣿⣿⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⣿⣿⣿⣧⣤⣾⣿⣿⢿⣿⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠋⠀⠁⠻⣿⣿⣿⣿⠟⠉⠀⠙⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⣦⣄⠀⠀⠀⠀⢀⣼⣿⣿⣧⣀⠀⠀⣤⡄⣠⣴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠸⣿⣷⡀⡄⣠⣿⣿⡟⢻⣿⣿⣷⣶⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⣦⣙⣿⣿⣴⣿⣿⠋⠀⠀⢻⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠀⠀⠀⢀⡀⣹⣿⣿⣿⣿⣿⠃⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⠀⢠⣤⠀⠀⠀⠀⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⡇⠀⠀⠂⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠘⢿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⢀⣽⣿⣿⡿⠋⠈⢿⣿⣷⣄⣰⣾⣿⡿⠋⠙⣿⣿⣿⣦⡀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣄⣐⣾⣿⣿⡿⠉⠀⠀⠀⠈⠻⣿⣿⣿⣿⠿⠁⠀⠀⠀⠉⢿⣿⣿⣷⣎⣶⣾⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠈⠈⢻⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠠⡀⠀⠀⠀⠀⡀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠸⠈⡠⠀⠀⡀⠶⠁⠰⡘⢃⣂⠀⠃⠀⠀⠈⠀⣀⢄⠀⠤⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⣡⣤⣶⠿⠛⠛⡛⠛⠛⣛⣻⣿⣿⣿⢷⣤⣄⣀⠑⠐⠁⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢶⠿⡏⠧⠘⣀⡰⠆⠈⠉⠁⠙⠻⠿⠿⢷⣾⣯⣭⣙⡛⠛⢻⡟⠉
⠀⠀⠀⠀⠀⢀⡀⠀⢀⠄⠀⢀⣠⣾⣧⣶⣴⣿⣶⣾⣿⡇⠀⠀⠀⠀⠀⢠⣌⠛⣿⣿⣿⣷⣎⡉⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠉⠔⡠⠈⢿⡿⣾⣿⣤⣶⣴⢶⠾⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦
⠀⠀⠀⠀⠀⠀⠀⣸⣦⣾⣿⣿⢿⣦⡀⠚⠂⣘⡇⢀⣿⡀⠁⠀⡀⠀⠀⠹⣿⣯⣉⣉⠛⣷⡿⠋⠈⠻⣿⠖
⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⢻⣿⣿⣟⠷⡀⠂⣠⠙⢈⢻⡇⠚⠀⠉⠂⢀⣴⣿⣿⣿⣏⣁⣉⠀⠀⠀⢠⣴⣄
⠀⠀⠀⠀⠀⠀⢩⣤⡀⢠⠀⠙⢻⣿⣷⣿⣀⠁⠚⠀⢺⣽⣶⣤⣶⣾⣿⣿⣿⣿⣿⣫⣽⣰⡄⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠈⡌⠃⠀⣄⠀⢹⣿⣿⡏⣿⣿⣿⣷⠛⢋⣉⠉⡍⢩⠡⠡⠈⠀⣿⣿⣿⣿⣿⡄⠘⠁⠀⠐⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢹⣦⣷⢀⣤⠟⠁⣽⣷⣿⣿⡟⠛⢀⢤⣁⡐⡛⠛⠻⠿⠷⠾⠿⠿⣿⣿⠿⠃⢀⡿⠛⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⢸⠟⠈⠉⠀⢀⠼⣿⣿⣿⣿⡇⢰⡿⠆⢿⣿⣿⣿⣿⣿⣿⠘⣷⠀⢹⠁⠀⠀⠛⣇⠀⠀⠀⠀⢀⠄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⣾⣏⡀⠀⢀⢠⣼⡄⣝⣻⣿⣡⣾⢀⣶⡟⢿⠿⠿⢿⢿⣿⣿⣆⠛⠀⣾⠀⠀⢀⠀⠈⠻⢦⣤⣶⣯⣤⣄⡀⠠⣾⠣⢦
⠀⠀⠀⠀⠀⠈⣿⣿⡷⠀⡿⠸⢿⡵⠛⡻⢿⣿⣿⠘⣿⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⡆⢸⡀⠀⠀⣷⣦⣤⣀⡈⠙⠛⠛⠛⠁⠀⠀⠀⠚
⠀⠀⠀⢠⣤⠀⠛⠋⢡⣾⠏⡌⠰⢮⣆⠐⢼⣿⣿⠀⣿⠿⠯⠿⠰⣯⡵⢸⣿⣿⣿⣇⢸⣿⣶⣶⣿⣿⣿⡟⠿⠅
⠀⠀⠀⠀⢿⡀⠀⣀⣾⡿⠀⠀⠂⠀⠉⠁⢌⣿⣿⠐⣿⠄⠀⡞⠀⢆⠀⣸⣿⣿⣿⣷⢸⣿⣿⣿⠿⣿⣿⣷⡜⢇⢀⡐⠀⠀⢀⠀⠉⠒⣦
⠀⠐⠄⠀⠉⠷⡄⠹⣿⣇⠀⠀⡄⠀⠀⠀⣼⣿⣿⠀⣡⣴⡈⢤⣤⡀⢠⣿⣿⡁⠤⠉⢼⣿⣿⡆⠀⠻⣿⣿⣿⠄⠉⠈⠇⢠⣉⡀
⠀⠀⠀⠀⢧⠰⠎⠛⠻⣿⣿⣀⣃⣀⣀⣤⡿⣿⣿⡀⠿⠿⠷⠄⠀⠰⠿⣿⣿⣿⡄⠓⢸⣿⣿⠇⠀⠀⠸⣿⣿⣧⡀⠀⠀⡀⠀⠉⠉
⠀⠀⠀⠀⠀⢈⡉⢒⣐⣂⣉⣽⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣶⢗⣦⣴⢲⣶⣶⣶⣶⣶⣾⣿⣿⣿⡖⣤⣴⣿⣿⣿⢿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠈⠛⠋⣌⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⠿⢻⣭⣤⡸⡇
⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⣠⡶⢸⣿⣿⣿⣿⣿⠀⠠⣾⣷⢀⣶⣀⠀⣿⣿⣿⣿⣿⣿⠋⠀⠀⣈⡉⠙⢧⡁
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡿⠀⠃⠀⣾⣿⣿⣿⣿⠏⠀⢉⡿⠁⢸⠯⠀⠀⣾⣿⣿⣿⣦⣉⡀⠘⠻⠋⠀⠀⣈⠁⣀
⠀⠀⠀⡐⢁⣿⠏⣿⣿⣿⣿⣿⣷⠀⢡⣾⣿⣿⣿⣿⣿⠀⣠⠜⠐⠇⢸⠀⠀⢰⣿⣿⡿⠟⡿⠉⠁⢠⣄⠀⠀⠈⠁⠈⠉⠳
⠀⠀⠀⠁⢸⡟⣼⣿⣿⡯⢁⡜⠃⠀⠀⠛⠛⣻⢿⣿⣿⣷⣄⠘⠿⠀⠈⢁⣠⣼⡿⠋⣰⠏⢡⠀⠀⠀⣿⡟⠒⣶⣫⠂⠀⠀⠀⢀⡴
⠀⠀⡆⠀⢸⣿⣿⣿⣿⣿⠛⠿⡄⠀⠀⣠⣾⡇⠀⢿⣿⣿⣿⣦⣤⣴⣿⣿⣿⡿⠁⠐⠃⠐⣿⡄⠀⠀⢹⠀⠀⢸⣿⠀⣠⡄⠀⠊
⠀⠀⠇⠀⠘⣿⣿⣿⣿⣿⣤⣤⡘⠶⢾⣅⣀⣿⡆⠸⣿⣿⣿⣿⡿⠋⠀⠈⢿⢀⠀⣤⠀⠀⠹⡇⠀⠀⠘⠀⠀⢺⣿⣿⡿⠗
⠀⠀⡇⠀⠀⠘⢯⠻⣿⣿⣿⣿⡆⠀⠀⠉⣻⣿⣧⠀⣻⣿⣿⡏⠁⠀⠀⠀⡄⠋⣰⢟⠀⡈⠢⠘⡀⠀⠰⠀⠀⢹⣿⠋
⠀⠀⠑⠀⣀⠐⢈⣳⣾⣿⣿⢹⡿⠗⠒⠋⠉⣿⢃⣴⣿⡇⠀⠐⠒⢁⢠⡆⠠⠊⠁⠀⢀⠑⠴⠶⢥⡀⠀⠉⠒⠘⠿⠆⣀⠀⣾⡂⠄
⠀⠀⠀⢦⡈⢀⠀⢀⡙⢿⡏⣸⣿⠀⠀⠀⠘⠁⠾⣿⣿⣧⡀⢨⠂⢸⡀⡠⠀⠀⠀⠀⠀⠰⠄⠠⢶⠄⡈⠉⠲⠄⠀⠀⠄⠀⠈⢁
⠀⠀⠀⠀⠱⣄⠀⠁⠆⠀⣈⠛⠛⢦⡤⢤⣤⠤⠴⠟⠛⢩⡇⠈⡁⠃⡜⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⡄⣀⠆⠀⠀⠀⠹⣿⡟
⠀⠀⠀⠀⠀⠈⠓⠄⠀⠠⠤⠇⢀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⡀⠒⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠖⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣤⣤⣽⣤⣦⣤⣿⣬⣼⣯⣴⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⣭⣿⣿⣿⠟⠋⠙⠛⢿⣿⣿⣽⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣽⣿⣿⣿⠁⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣷⣤⣀⣀⣤⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣗⣯⣿⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣟⡿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢯⢞⡽⣯⢾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣶⣦⡀⠀⠀⠀⠀⠀⠛⣟⣼⣫⣿⣝⣽⣳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠘⢻⣧⣿⣞⣷⣱⣏⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠁⠀⠀⠙⢿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠈⠻⣾⣿⡽⣿⢞⡵⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠋⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣿⣿⣦
⠀⠀⠀⠀⠀⢀⣠⣤⣤⣾⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⢿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣽⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣴⣻⣿⣿⣿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣟⣿⣮⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣴⣴⣦⡄
⠀⠀⠀⠛⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⢞⣫⢟⣭⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣫⣻⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣯⡶⢛⡵⣫⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣲⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡶⢋⣴⢿⣿⣿⣦⣀⠀⠀⠀⣀⣴⣿⣿⣯⣿⣳⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⣟⣵⢫⣟⣻⢿⣿⣿⣿⣿⣿⣿⣯⡛⣯⣲⣽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡟⣴⢏⣿⢿⢿⢿⣿⣿⡍⣶⡿⣾⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠇⠿⢸⡿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠊⢁⣤⠴⠶⠶⠖⠲⠶⠶⢦⣄⡈⠁⠀⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠋⠀⠀⠀⠈⢿⡳⣤⡀⠀⠀⣀⡴⢾⠟⠁⠀⠀⠀⠹⠷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠻⣄⠙⣦⣞⠋⣴⠋⠀⠀⠀⠀⠀⠀⢀⢻⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠃⠀⠀⠀⠀⠀⠀⠀⣸⢿⡁⢙⣿⡁⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⡟⣿⢷⡀⠀⠀⠀⠀⣠⡞⠁⠈⣻⣏⠀⠙⣦⣀⠀⠀⠀⠀⠀⠀⣿⣼⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⣷⢸⡿⠃⠀⠀⠀⢀⡐⠲⣦⡾⠁⠙⢦⣴⠖⣀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⢸⣧⣴⠀⠀⠀⢿⣭⡴⠋⠳⣄⣠⠞⠻⢦⣭⠇⠀⠀⠀⢀⠄⣼⣼⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣼⣿⢿⣡⡖⢨⣄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢰⡀⠊⢴⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠛⠁⣸⣿⣷⣾⣿⣙⣤⣢⡀⣀⣀⣀⣠⣴⣆⡀⣠⣤⣿⣀⡺⣟⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣼⣿⣿⣾⣯⣶⣿⣿⣿⣿⣿⣿⣿⣼⢿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⢻⡇⠘⢿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢁⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣞⣄⣤⢻⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⡏⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⢰⣾⢸⢧⠘⢿⣿⣿⣿⣿⣿⡿⠟⠻⣿⣿⣿⣿⣿⣷⠿⣿⣿⣿⣿⣿⣿⣿⡿⠁⣼⠹⡄
⠀⠀⠀⠀⠀⠀⠀⢸⡌⣧⣌⣁⣤⡬⠭⣍⢉⣉⠂⠀⠀⣾⣿⣿⣿⣿⣟⠄⢀⠋⠛⠛⠛⢻⣯⣴⣶⢟⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣽⣬⣽⣼⣶⣶⠿⡏⣹⣆⠀⢸⣿⣿⣿⣿⣿⣏⠀⠩⣿⢯⣯⣿⣥⣀⣈⣥⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠛⠛⢻⣭⣴⣿⣇⡄⡌⠻⠗⠋⠛⠿⢏⡄⢀⢠⣾⣿⣿⠿⠿⢿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣍⣼⡿⠁⣹⣦⠸⣷⠋⡀⣾⣰⡾⢆⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣴⣯⣼⣧⣿⣸⣧⣼⣿⣷⠏⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⡆⡏⠹⠹⡿⢈⣿⠇⠙⢻⡄⡟⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠓⠞⠒⠀⠉⠁⠑⠋⠘⠀⣃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡈⢿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⡌⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⠘⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠹⣷⠀⢀⣾⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣤⣿⣦⣾⣿⠏⠉⣣⡄⠀⠀⠀⢠⣶⢴⣿⣿⣿⡁⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠻⣿⣿⣿⣿⠀⠀⣴⣿⠓⡄⠀⠀⣿⣿⡾⠛⠉⣻⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣌⠙⣿⣿⣀⢸⣿⢿⠛⢧⡀⠀⢸⣏⡰⢷⣾⠿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣯⠈⠃⠸⠀⠀⢹⡆⠋⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠻⣿⣿⣿⣿⣦⣤⠀⠀⠀⠈⠁⠀⠀⠀⠀⢰⡟⢡⣾⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠁⣄⣤⣶⣾⣿⣿⠉⠿⢛⠁⠀⠀⠀⠀⢰⡀⢀⣠⣼⣻⣾⣿⣅⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣍⢠⣼⣿⣧⣤⣶⡟⠁⠘⠀⠀⠀⣸⠟⠈⠉⠀⠁⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠻⣏⣜⠛⣡⣼⣿⣅⠀⠲⠦⡤⢆⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⣤⣤⣀⠙⠛⠿⢿⣿⣿⣦⣀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣶⣄⣀⣀⣠⣴⣶⣶⣿⣶⣉⡛⠛⠛⠂⠀⠀⠉⠉⠻⣿⠀⠸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣝⣿⡟⠛⠿⣟⣥⣽⣿⣿⣦⡔⠛⠛⠛⢶⣤⠹
⠀⠀⠀⠀⠀⠀⢀⢄⡀⢻⣿⣿⣿⣿⣿⡖⣧⠰⠿⣿⣯⣈⣤⠟⠁⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠙⠎⠹⣶⣿⣿⣿⣿⣿⣿⣯⣼⣿⠟⢻⡿⠁⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⣿⣻⣿⡿⠛⠁⠀⠈⠣⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠻⢿⣿⠿⠛⠉⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣥⣭⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⠤⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⢁⣠⣶⣿⠟⣋⣉⣽⡽⢿⣿⣿⣿⣿⣿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠱⠿⠛⠻⢿⡉⠙⠻⡿⣧⡘⣿⣿⣿⣿⣿⣿⣿⣧⡈⠁
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⢁⠀⢀⣾⡏⢠⡙⣿⣿⣿⣿⣿⣿⡟⠛⠆
⠀⠀⠀⠀⠀⠀⢻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⠀⠀⠀⠀⠀⠀⠀⠀⣼⡀⢸⣿⠀⡾⠃⣿⢿⣿⣿⣿⣿⣧⣴⡇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⡃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡧⣿⠃⠀⠀⣿⡄⢻⣿⣿⣿⣿⣏⠁
⠀⠀⠀⠀⠀⠀⠘⣿⣻⣷⠀⣠⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣆⣣⡤⠂⢠⠛⣧⡀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣆⣻⠀⠀⠀⠀⠀⢠⡿⠋⠀⠀⠀⠀⠀⠀⠀⡼⣏⠙⣿⣿⣿⣿⣿⠁⣠⣿⠀⠘⢷⣿⣿⠟⠉⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣧⣀⣾⣿⣿⡮⠓⡼⢻⡆⠀⣼⣿⡿⠀⢰⣿⣯
⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⡿⠃⠀⠀⣠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡟⠀⣴⠁⠀⠻⠀⢻⣿⠃⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠻⢁⠻⣿⣿⣤⡀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠉⢹⣿⣹⣿⣿⢡⡾⠿⠀⠀⠀⠀⠘⣿⡇⢠⠟⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡘⣿⣿⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠃⠀⢻⠉⠃⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⢸⠛⣿⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢉⣿⢀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠙⣾⡇⠸⢄⣠⣄⠙⠃⠀⠀⠀⠀⠀⠀⣰⣿⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠁⠀⢀⣀⣀⣀⣀⣠⣤⣤⣄⡈⠉⠁⠀⠀⠻⠏⠁⠀⠀⠀⠀⠀⠴⣾⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠛⣿⣿⣿⣶⣿⣿⣿⡿⢿⣋⣉⣉⣉⣉⣁⠉⠁⠒⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣏⣀⣠⣄⣀⣈⡉⠙⠓
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠹⠷⢦⣄
⠀⠀⠀⠀⠀⠀⠀⢻⣦⡟⠀⠙⣿⡿⣿⣿⣿⡿⢹⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣷⡛⠻⣿⣿⣿⣷⣟⣿⣿⣿⣿⣿⣿⣿⣿⠿⠻⠿⠷⣶⣄
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠙⠻⣷⣄⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠉⠻⢆
⠀⠀⠀⠀⠀⠀⠸⣿⡻⠟⢢⡀⠙⠛⠳⠌⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⣰⣿⡀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿⣌⠉⠛⠃
⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠿⢿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠦⠄
⠀⠀⠀⠀⠀⠘⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠆⠰⠄⠀⠀⠦⠠⠶⠤⠤⠴⠆⠶⠠⠖⠀⠀⠴⠆⠶⠠⠀⠶⠆⠰⠶⠶⠶⠀⠦⠠⠖⠠⡶

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠉⠀⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢴⣶⣶⣶⣶⣶⣶⣶⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠀⢠⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣻⣿⠀⠛⢾⣶⣶⣦⣤⣀⡀⠀⢀⡄⣼⠁⠀⠀⢻⡄⢄⠀⢀⣀⣤⣴⣶⣶⣿⡿⠋⢸⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢺⣿⠀⠀⠀⢹⣿⣿⣿⣿⣿⡀⣾⣠⢿⣿⣿⢿⣿⢧⣼⠀⣸⣿⣿⣿⣿⣿⡟⠀⠀⢘⣿⠅⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢿⣿⠀⠀⠀⣸⣿⣿⣿⣿⣿⣇⢾⣿⢿⡛⠇⠟⣽⠿⡟⢠⣿⣿⣿⣟⣿⡿⣧⠀⠀⢘⣿⡆⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢼⣿⠀⠀⠀⠀⣉⣿⣷⣿⣿⣿⣦⠡⣼⣯⣿⣯⣿⡖⣰⣿⣿⣟⣿⣟⣥⡀⠀⠀⠀⠘⣿⠇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⣿⡿⢿⣿⣾⣿⣿⣿⣿⣗⣀⠀⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣸⣿⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣉⣁⢡⢀⢈⣃⣦⣿⣿⣿⣿⣿⣿⣿⣿⡗⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣹⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⢏⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠃⠀⠀⣿⠀⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢹⣯⢠⠟⠋⠙⠿⠉⠉⠉⠋⣼⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣩⣴⣿⠟⢿⡏⠀⣶⠀⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣸⣿⠈⠀⣤⣄⠀⠀⠃⠀⣴⣿⣿⢻⣿⣿⣿⣿⣿⣿⠃⣹⣿⣿⠞⠁⠀⠀⣤⣤⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢸⣿⠀⠀⣿⠈⠀⠀⢀⣼⣿⠟⠉⠀⣿⢿⣿⣿⣿⠿⡄⠈⢻⢏⡄⠀⠀⠀⠈⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢹⡯⠀⠈⠟⢲⡤⣾⡿⠋⠁⠀⣠⣿⣿⡌⠁⠉⠁⣯⣿⡄⠀⣴⣟⠀⠀⠀⠀⠠⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⢿⣯⠀⠀⠀⠘⠛⠛⠀⠀⣠⣾⣿⣿⣿⣷⡼⠳⣾⣿⣿⠿⣦⡞⠛⠃⠀⠀⠰⠀⠀⠀⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣾⣷⠀⡀⠀⠀⡀⠀⠰⢿⣯⣼⣾⠟⠋⠁⠐⣟⠉⠻⣿⣟⠹⣿⠆⠀⠀⠀⡀⢒⡀⠀⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⡇⠀⠀⠀⠐⢗⣿⣿⣿⣿⣧⡀⠀⠀⣀⠀⠀⠈⠹⢾⣿⢷⡀⠀⠐⠁⣴⠇⠀⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠁⢠⡀⠀⠋⠈⠋⠉⠉⢿⣿⢿⣷⣷⡏⠀⠀⢀⣴⢞⣿⣿⣭⢤⢀⠀⠉⠀⡄⣿⡇⠀⢸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⡀⠘⠁⠀⢄⠀⡆⠀⠀⠸⣧⣶⡌⣿⠳⢀⣤⢿⣿⠟⠋⢉⠁⢠⠋⠠⡀⠘⠁⣿⡇⠀⠸⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⠀⠁⣄⣴⣗⠸⣄⢠⣀⠠⠀⣞⠉⠀⠛⠃⢸⣿⡾⠫⠀⠀⢜⣶⡞⢀⢤⣟⠀⠀⣿⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣏⠀⢀⣼⣿⣛⣓⣿⡧⠾⠀⡂⠁⠀⠀⢦⡴⣿⠿⠃⠀⢵⣜⣿⣿⣑⡛⢽⣿⡄⠀⣿⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⣷⣾⣟⣶⡿⣋⢿⣥⣠⡶⠇⠀⠀⠀⣼⣟⢷⣀⠀⠻⣾⣿⢏⣻⣙⣽⡶⣼⣶⣷⣿⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⣹⣿⣇⣿⣳⣿⣾⣿⣿⣿⢡⠠⠀⠀⠙⣿⣾⡟⠀⠦⣽⣏⣻⣿⣿⣧⣹⣿⣹⣿⣿⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣯⣸⣿⣿⣿⣿⠀⡆⠀⠀⠀⢻⣄⠀⠀⠀⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⠿⠿⡿⠿⠽⠷⠻⡁⠀⠁⠀⡀⢀⠀⠉⠀⠀⠀⠈⠟⠌⠻⡽⠿⠿⠿⢿⣿⣿⡇⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠀⠀⠿⠿⠿⠟⠘⠀⠀⠀⠐⠀⠀⠀⠘⠐⠀⠂⠈⠀⠃⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠃⠀⠀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠤⠤⢤⣤⣀
⠀⠀⠀⠀⠀⣴⡾⠛⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠉⠁⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⢷⣦
⠀⠀⠀⠀⠸⣯⢠⡄⠀⢼⣷⡄⠀⠀⠀⠀⠀⠀⣠⢞⡵⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢦⡀⠀⠀⠀⠀⠀⠀⢠⣾⡧⠀⢠⡄⣽⠆
⠀⠀⠀⠀⣠⡿⠛⠀⠀⢸⠋⠻⣦⣀⠀⠀⠀⣰⣳⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣱⡄⠀⠀⠀⣀⣴⠟⠙⡇⠀⠀⠛⢿⣄
⠀⠀⠀⢸⡇⠀⠀⢀⣀⣀⣀⡀⠀⠙⠳⢤⣀⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢳⣀⣤⠞⠋⠀⢀⣀⣀⣀⡀⠀⠀⢸⡇
⠀⠀⠀⠀⠙⠒⠞⠛⠉⠉⠙⠻⣷⣦⣄⡀⢹⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢀⣠⣴⣾⠟⠋⠉⠉⠛⠛⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⡇⠀⠔⠒⠶⢄⡀⠀⠀⠀⡀⢀⠀⠀⠀⢀⡤⠖⠒⠄⠀⢸⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷⠀⠀⣤⣤⣀⠙⠲⠄⠀⠈⠋⠀⠠⠚⠉⢀⣠⣄⡀⠀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⣠⣴⣾⣿⣿⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠈⢿⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠭⠛⠋⠁⠀⢠⣾⣿⣄⠈⠉⠛⠛⠻⠁⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣞⡿⣦⡒⠀⠀⢀⣀⠀⠀⣿⣿⢿⣿⠆⠠⢀⣀⠀⠀⣀⣴⣿⣷⣦⣄⡀
⠀⠀⠀⠀⢀⣠⠤⠤⣤⣤⣤⣴⣾⡿⠿⠛⠉⣀⣨⠿⡟⢿⣿⠄⡀⡀⢀⠀⡀⡀⠀⠀⠐⢻⡿⣿⠿⣥⣀⠉⠛⠿⢿⣷⣦⣤⣤⣤⠤⠤⣄⡀
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠀⠀⢻⠀⣿⣶⢤⢤⢪⠠⡅⡤⠤⠴⢶⣾⠀⡏⠀⠀⠉⠓⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⠀⠀⠀⠑⠦⣄⡀⠀⠀⢸⣠⡶⠛⠁⠀⠀⠀⠀⠀⢸⡆⠨⣟⡻⢾⠾⢤⢧⠷⠶⢾⣫⠙⢰⡇⠀⠀⠀⠀⠀⠈⠛⢶⡄⠇⠀⠀⢀⣠⠴⠊
⠀⠀⠀⠀⠀⠀⠸⡟⠀⠀⢣⡟⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠉⠙⡛⠒⠂⠒⠋⠁⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡜⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠳⣤⣤⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢳⡠⢄⡀⠀⠃⠀⠤⢂⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣤⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠛⠛⠛⠉


⠀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣟⡔⠋⠉⠙⠢⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣿⣿⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⡀⠀⠠⢎⠁⠂⠘⢂⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣆⠀⠀⠀⠀⠀⠘⣿⣿⣿⡝⣿⠀⠰⣧⠇⠀⠀⠰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣦⡀⠀⠀⠀⠈⠛⠿⠯⠟⠚⠢⠆⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣿⣿⣿⣿⡻⠟⠁
⠀⠀⠀⣠⣞⣿⡿⣿⠿⠿⠛⠛⠻⠿⠿⠿⣿⣿⣻⣿⣿⠿⠿⠟⠛⠉
⠀⠀⢰⣿⣿⡴⠋⠀⣠⣄
⠀⠀⢸⣿⣿⣇⠀⠀⢹⣿⡇
⠀⠀⠈⠻⣟⣷⣷⣶⣿⠿⠁
⠀⠀⠀⠀⠀⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠈⠀⠀⠀⠘⠈⠓⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⢀⠀⢀⠄⠀⣄⠀⠀⠀⠀⠀⠀⠄⢀⠄⣤⠀⡀⠠⡀⠀⠀⣀⡀⢀⣀⡀⠀⣀⡀⢀⣀⠀⠀⠀⢀⠀⣄⢀⢀⢀⡄⢀
⠀⢾⠉⠃⠉⠀⠉⠀⠁⠋⢱⠆⠘⠞⠄⠘⠓⠀⠒⠃⠀⠛⠃⠠⢽⠻⢿⣿⢁⡀⢿⣿⠿⠹⠅⠀⢠⡎⠐⡛⠐⡴⡴⡻⠎⢲
⠀⢻⡆⠀⠀⣤⣤⣸⢄⠀⣾⠀⠀⢀⠀⠀⠘⠋⡗⠀⠀⠀⠀⠠⠒⠒⠤⡀⠘⠛⠀⡤⠔⠐⠆⢀⣾⠁⢰⡎⢳⣎⠰⡬⠀⢈⣧
⠀⢈⠁⠀⠀⠀⢸⣧⢸⠀⢉⠀⢠⡄⠀⠛⠚⠀⣷⠀⠀⠰⡆⠀⣳⠃⡄⢀⡀⢀⣀⠠⠈⣷⡁⠀⠟⠀⢀⡀⠈⠋⠀⠀⠰⠞⠋
⠀⣽⠆⠀⢀⠀⠈⠠⠀⠀⢿⡄⠈⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⢰⣿⡲⠆⠈⠽⠎⠍⢀⠗⣷⡧⠀⡟⠃⠀⠀⣤⣢⠄⠀⠀⠘⣏
⠀⢸⡃⠀⠀⠠⡂⠀⠀⠀⣟⠀⢠⠾⠸⡗⠒⣤⠔⠚⡏⠷⠆⠘⢦⠀⠤⠛⢦⠢⠟⠤⠄⢰⠃⠀⣶⠆⠀⢀⠘⣛⣀⠀⠀⢰⡆
⠀⢸⡃⠀⠀⠀⠁⠀⠀⠀⣿⠀⢠⠀⠀⢱⡂⠈⠐⡺⠀⠀⡆⠀⣄⢈⠂⢠⠘⡇⢰⠐⢂⢀⡄⠀⢛⡀⠀⠘⢿⣿⡏⠀⠀⢘⠃
⠀⠸⢗⣤⣴⠾⠻⣶⣤⣴⠷⠀⠈⢹⡴⠄⢭⠀⡠⡠⢦⠏⠁⠐⢩⡟⢀⡒⠲⠲⠴⡀⠚⡅⠁⠀⠀⠡⡿⠃⢼⠿⠐⠺⡧⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⣤⣦⡴⣴⣤⢤⢤⣤⡴⣤⣦⣄⠀⠀⠀⠠⣠⢴⡖⠂⠀⠀⠒⣲⠦⣄⡀⠀⠀⢀⣠⡄⣀⣤⡔⠒⠒⢢⣤⣀⢠⣄⡀
⠀⠀⢀⡌⠙⠋⠈⠑⠂⠒⠉⠈⠛⠉⣅⠀⠀⢀⣰⠂⠛⠛⠒⠃⠚⠜⠛⠛⠺⣏⡀⠀⢉⣹⠈⢈⡉⣽⠟⣿⣯⡉⠁⠁⣟⣏
⠀⠀⢘⣟⠀⠒⠀⢠⠶⠰⠀⠠⠀⢈⣯⠁⠀⠩⠄⠀⠀⠀⠀⠂⠺⠶⣆⠀⠀⡧⠅⠀⢸⣤⠀⣤⢀⣤⠄⠉⠽⡇⠀⠀⣿⡏
⠀⠀⢹⡇⠀⠀⠀⠀⠀⠴⠠⢤⠀⠀⣿⡁⠀⠘⡟⠀⠀⠀⢠⠴⠃⠠⡥⢸⠀⣟⠁⠀⠈⠸⠀⢀⠈⠀⠀⠀⠀⠈⠐⡀⠇⠁
⠀⠀⠘⠀⠈⠂⠁⠨⠀⠠⠀⠀⢱⠀⠨⠁⠀⢸⣋⣤⣤⠤⡄⠀⠔⢠⡌⠈⠀⣹⠆⠀⢰⢋⡀⠀⠀⠓⠛⠂⠉⠁⠀⢀⡐⡦
⠀⠀⠹⠦⠀⠀⠀⠀⠀⠀⠀⠀⠘⠠⢾⡁⠀⢸⡟⠈⢀⡀⠀⢀⠠⠈⠀⠁⠀⣻⠀⠀⠐⣼⠁⠀⠀⠀⠀⠀⠀⠀⡀⠈⢥
⠀⠀⠹⡽⠧⠦⠟⢿⣟⣿⠟⠧⠯⠾⡽⠁⠀⡐⣿⠧⢀⡥⠶⠦⢶⠶⢤⡠⢾⣿⠂⠀⢀⢺⣷⣆⠒⢰⣖⣶⡆⠐⣠⣶⠗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠾⠛⠛⠛⠛⠛⠛⠳⠶⠶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⢡⣄⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⣶⡎⠛⢶⣄
⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⣀⣧⠀⣀⣤⠶⠛⠋⠉⠉⠀⠀⠀⠀⠉⠉⠙⠛⠶⣦⣄⠀⠀⡇⠀⠀⠙⢷⣄
⠀⠀⠀⢀⣴⠏⠀⠀⠀⠀⢉⣵⢾⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⡿⣦⡀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⢀⡾⠃⠀⠀⠀⢀⣴⠟⠁⠀⢧⢙⡢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠋⣸⠁⠈⠻⣦⡀⠀⠀⠀⠈⢷⡄
⠀⢀⡿⠁⠀⠀⠀⢠⡾⠁⠀⠀⠀⠘⡆⠘⠮⣕⣦⣀⠀⠀⠀⠀⣀⣴⣪⠗⠋⢠⠇⠀⠀⠀⠈⢻⣄⠀⠀⠀⠈⢿⡀
⠀⣾⠃⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⢹⡐⢌⢪⡙⢮⣓⢤⡤⢚⡝⢋⠔⡡⠂⡞⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠈⣷
⢸⡏⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⣧⠈⢧⢙⣦⠟⠋⠹⠻⣴⡁⡼⠁⣸⠁⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⢸
⣾⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⢾⡏⣎⠃⠀⠀⠘⢀⡙⡷⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈
⣿⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⣠⠴⠋⣻⡞⠀⠘⢦⢸⡆⡠⠊⠀⠹⣾⡙⠦⣀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⣿⡀⠀⠀⠀⢀⡴⣺⠵⣚⣋⡽⣷⠀⣵⢆⠁⠈⡠⢾⠁⣼⢅⣉⣓⠮⣗⢦⡀⠀⠀⠀⠀⣿
⢿⡀⠀⠀⠀⢸⡇⢀⡤⢞⡩⣊⣘⡭⣭⡴⠚⠻⣧⡀⢌⠃⠘⠃⢀⣼⠟⠐⢤⡄⠈⡀⢑⠌⡳⢤⡀⢸⡇⣀⡀⠀⢀
⢸⣇⠰⠶⣦⠈⣿⠛⠓⠓⠒⠚⠒⠒⠒⠒⠒⠒⢿⣗⢺⠓⠚⡗⣿⡿⠓⠒⠒⠒⠒⠒⠒⠒⠛⠒⠛⣿⠁⠉⠛⠆⣸
⠀⢿⡄⠀⠸⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡌⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠰⠿⠿⢣⡿
⠀⠈⣷⡀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡗⣀⣀⢚⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠃⠀⠀⠀⢀⣾⠁
⠀⠀⠈⢷⡀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢷⢨⡄⡼⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⢀⡾⠃
⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠘⡇⢸⠃⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠉⠛⠶⣦⣄⣀⣀⠀⢹⡏⢀⣀⣀⣠⣴⠶⠟⠋⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⣟⢛⣟⠋⠉⠁⠀⠀⠀⠀⠀⠀⢀⣴⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⣀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⣀⣠⡴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠳⠶⢦⣤⣤⣬⣤⣤⣤⡴⠶⠾⠛⠋⠁

⣿⣌⢿⣿⡌⠻⣯⢷⠀⠙⠌⠄⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠻⣄⠀⠀⠀⠙⣆⠻⣄⠀⡣⡀⠀⢄⢻⣧⠳⡈⢿⣧⡠⡹⡄⢌⢻⣷⡹⣷
⢿⣿⣎⠻⣧⠀⠙⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⢢⡀⠀⠀⠀⠙⣦⡀⠀⠀⠈⢷⣽⣧⡙⣷⡄⠈⢢⡹⣷⡔⡄⢻⣷⡕⡜⢎⢢⠹⣿⣜
⡄⠻⠻⣆⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⡱⡄⠀⠀⠀⠈⢷⣤⢀⠀⠈⢻⣾⣷⡜⢿⣆⠀⠱⡜⢿⣎⢆⠹⣿⣌⢮⢣⡱⡘⢿
⠙⣆⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠀⠐⡄⠀⠀⠀⠀⠀⠀⠀⠱⡵⡜⢆⠀⠀⡀⠀⢻⣧⣳⡄⠀⠻⣿⣿⣎⢿⣷⡀⠘⢎⠻⣧⡣⡘⢿⣦⠣⡱⡜⢌
⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠘⢆⠠⠀⠀⠀⢀⢢⡀⠙⣿⣎⢣⡀⠀⠄⢦⡹⣿⣿⣆⠀⠹⣿⣿⣧⡻⣿⣄⢈⢧⡙⣷⡕⠌⢻⣷⡑⡜⢎
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠈⢷⡑⡀⠀⠀⠠⡻⣄⠘⢿⣷⡱⡄⠀⢪⢳⡜⢿⣿⣧⡀⠘⢿⣿⣷⡝⢿⣆⠀⠳⡌⢿⣎⢂⠹⣿⣌⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⡀⠀⠀⠀⠱⡄⠀⠀⠀⠈⢧⡀⡻⣌⢆⢢⠀⠱⡝⣧⡘⣿⣿⣜⢆⠀⠡⡻⣮⢻⣿⣷⡄⠈⢻⣿⣿⣎⠻⣷⡀⠙⢆⠻⣧⠀⠈⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢢⠀⠹⣜⢆⠀⠀⠀⠙⣆⠀⠀⡀⠀⢳⡔⢽⣦⡣⡱⡀⠘⢾⣷⡌⢿⣿⣮⣣⡠⡘⢽⣷⡹⣿⣿⣆⠀⠹⣿⣿⣧⡙⠟⣄⠈⢣⡘⢷⡄⠀⠹
⠀⢦⠀⠀⠀⠈⢄⠀⠱⡀⠘⢯⣣⡀⠠⠀⠈⢷⡠⠘⢄⠀⢻⣎⢿⣷⡕⡜⢆⠈⢻⣿⣎⢻⣿⣷⡵⡄⠈⢫⢿⡜⢿⣿⢆⠀⠈⢿⣟⢷⡄⠈⢢⠀⠱⡄⠻⣄
⠀⠀⠳⣤⠀⠀⠀⠣⡀⠙⣄⠈⢿⣷⡄⠑⡄⠈⢻⣶⡈⢣⡀⠹⣧⡻⣿⣎⢎⢣⡀⠹⣽⣧⡹⣿⣿⣞⢆⠀⠱⡹⣎⠻⣷⡀⠀⠀⠙⢦⠻⣆⠀⠡⡀⠘⢆⠈⠢
⠀⠀⠀⠹⣷⡄⢀⠀⠹⣄⠈⢧⡀⠻⣿⣆⢪⢆⠀⠹⣿⣆⠱⡄⠘⣷⡝⣿⣧⡣⡳⡄⠘⢮⢷⡘⢽⣿⣯⡣⠀⠘⠌⠆⠙⣷⡀⠀⠀⠈⠣⠙⣦⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠘⢿⣦⠢⠀⠘⢦⠀⠳⡄⠙⣿⣧⡹⣦⡀⠘⢿⣧⡘⢆⠈⢿⣎⢿⣷⡔⠘⣆⠈⢳⡹⡌⢪⢻⣷⡄⠀⠈⢂⠀⠈⢷⡄⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠑
⡄⠀⠐⡀⠀⠈⢻⣷⡵⡈⢮⢳⡀⠙⣆⢈⢿⣷⡜⣷⡄⠈⢻⣷⡌⢣⡀⠹⣧⠻⣿⣎⠈⢧⡀⠹⣌⠀⠡⡹⣿⣄⠀⠀⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠹⡄
⢻⣆⠰⣄⢄⠀⠠⡻⣿⣞⣌⢳⡽⣄⠈⢧⡡⡻⣿⣎⢿⣆⠀⠹⣿⣆⠱⡄⠙⢷⡙⣿⣧⡀⠳⡄⠘⢦⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠈
⠀⠻⣷⡘⣮⡢⠀⠘⣝⣿⣿⣦⠻⣾⣦⠀⠳⡕⡝⣿⣧⡻⣧⡀⠘⢿⣧⠘⢆⠈⢂⠈⢿⣷⡄⠙⢆⠀⠳⡀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠃
⠀⠀⠹⣿⣌⢷⣵⡀⢈⢮⢿⣿⣷⡙⣿⣷⡁⠙⣌⢎⢿⣷⡝⣷⡄⠈⢻⣇⡀⠂⠀⠡⠀⠻⡿⣄⠈⢣⡀⠑⠄⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣄⠀⠐⣜⢿⣮⢻⣾⢄⠀⠳⡿⣿⣿⣌⢿⣿⣄⠈⢦⠳⡹⣿⢎⢷⡀⠀⠙⣦⠀⠀⠀⠐⠀⠙⠝⢦⠀⠑⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢌⢷
⢚⣦⠠⠈⢮⣻⣷⡹⣿⣢⠁⠙⣜⢿⣿⣦⠻⣾⢦⠀⠱⡐⡜⢷⡂⠳⡄⠀⠈⢧⡀⠀⠀⠀⠀⠈⠊⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⢦⠑⣽⣄⠈⢿
⣧⡹⣷⡐⠀⠳⣽⣿⣌⢷⡕⡀⠈⢯⢻⣿⣧⡙⢧⡣⠀⠀⠈⢊⢳⡄⠙⣆⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠈⢳⡀⠀⢳⡌⢿⣦⠈
⠘⢷⡜⢿⣄⠀⠙⣞⢿⣦⠻⡌⢄⠀⠳⡙⣿⣆⠈⢳⡐⠀⠀⠀⠡⠙⣆⠈⢧⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⢳⣄⢠⡀⠻⣶⡀⠹⣎⢻⢷

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⡉⢭⡐⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⣏⡙⠿⣿⣮⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⣈⠮⢁⣮⠗⠢⠴⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣶⣿⡏⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢿⢟⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⢠⣀⣿⣏⡙⠋⣽⢿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣾⣏⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⢟⡋⡙⠻⠿⣽⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡇⣼⠀⠀⠤⢠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣇⡀⠀⢀⣾⠏⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⠺⠶⠶⠶⣶⢤⣀⡀⠀⠀⠀⠀⠀⣠⣿⡯⠃⢀⣶⣿⡿⠋⠀⢀⣤⣶⠟⠙⣫⠛⠙⠛⠿⢿⣿⣿⣶⡄
⠀⠀⠀⣠⣾⣿⣿⡻⢈⣠⣤⣼⣦⣤⣤⣁⣌⡉⣛⣲⣶⣶⣿⡿⠋⢮⣲⣿⣿⠟⣁⣤⡾⢋⣡⣶⣶⣶⣶⣿⣶⣶⣤⠀⠙⣿⣿⣿⡄
⠀⠀⣼⣿⣿⣿⠀⢢⣿⣻⣻⣭⣬⣅⡈⠉⣉⠙⠛⣛⣩⣭⣯⡀⣄⣠⣿⡿⠿⢛⣫⣽⣾⣿⡿⠿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿
⠀⠀⣿⣿⣿⡗⠀⢺⡿⠟⠿⣿⣿⣿⣿⣟⠛⠛⠙⣁⢻⣿⣻⣿⡿⠋⢁⣀⣼⣿⣿⠛⠉⣩⣼⠿⠋⠉⠙⠛⢿⣿⣿⣿⡇⢴⣿⣿⡿
⠀⠀⣿⣿⣿⣿⡍⠛⣧⠀⠀⠙⠛⠛⠛⠛⠛⠒⠛⠉⠉⢉⣽⠋⢃⣤⣾⢿⠯⢑⣠⣶⠟⠋⠀⠀⠀⠀⣀⣠⣶⣿⠿⣋⣼⣿⣿⡿⠁
⠀⠀⣿⣿⣿⣿⣷⣌⠁⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠛⣑⣾⡿⠟⠁⣠⣴⠟⠋⠀⢀⣀⣤⣶⣾⡿⠿⢟⣿⣿⡾⢟⣭⠾⠋
⠀⠀⢿⣿⣿⣿⣿⣿⣷⣮⣤⣭⣛⠲⠶⠶⠶⠖⢛⢩⣥⣿⣿⠻⢁⣤⡾⠛⠁⠀⣠⣶⣿⠟⢛⣉⣬⡶⣾⣟⣯⡿⠞⠋⠁
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣾⣿⣾⠿⢟⣡⣴⠟⠉⠀⢀⣴⣿⣿⢋⣠⣾⢟⣩⡴⠖⠛⠉
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣛⠿⡿⢿⡿⣿⣩⣉⣠⡾⠟⠁⠀⠀⢀⣾⣿⣿⡇⢸⣿⣷⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⡿⣶⠶⠷⠿⠈⠛⠛⠉⠀⠀⠀⠀⣠⣿⣿⣿⣿⣧⡈⢿⣿⣤⣤⣤⣴⠶⠿⠿⠟⠛⠛⡿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠈⠻⠛⠓⠃⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⢠⣿⣿⣿⣿⣿⣿⣷⣤⣤⣍⣍⣨⣤⣶⣴⣶⣾⣿⣿⣿⣶⣦⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⣿⣿⡿⣿⢿⠛⠻⠿⠿⠛⠛⠛⠉⠉⠉⠉⠉⠉⠙⣿⡧⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀⠀⠀⠀⠀⢀⣿⢇⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠛⠿⣷⣶⣶⣶⣶⣿⠿⠿⠿⠷⣷⣭⣍⣛⢿⢶⠶⠶⠿⣫⡾⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣿⣿⣭⣤⣤
⣿⣿⣅⣥⣤⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡄⠀⠀⠀⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣅⣶⣶⣿⣶⣾⣿⣿⣿⣿⣶⣦⣤⡄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣤⣼⣿⣿⠟⠛⠋⠛⠂⡏⠘⠁⠀⠉⣋⠉⢻⣿⣾⣤⣖⣒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⣠⣄⣠⣿⡿⠛⠊⡷⢂⠀⠀⠀⣠⡇⠐⠰⢣⠄⠀⠀⠉⠈⠉⠛⢿⣯⡅⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣴⣿⣿⠟⠣⢄⠀⠀⠀⣀⠂⢀⣼⣿⣿⡅⣹⢠⠀⠀⠀⠀⠀⠀⠀⢚⢹⣧⣧⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠛⠉⠁⠁⠀⠀⠌⠀⠀⠸⣇⣸⡿⠛⣉⠙⠻⣷⣴⣀⡶⠂⠀⢀⠀⠀⢀⠀⠃⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⣠⣿⡏⣀⣾⣿⣷⠀⠙⣿⣿⡿⢇⣔⠁⠀⠀⠈⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠑⠒⢆⠀⠈⣻⣿⣿⣀⣻⣏⣩⣿⣧⣼⠿⡻⣦⣺⠁⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⣾⣿⣿⣿⣿⣿⣯⣿⣿⠟⢣⠄⠓⠈⢻⠷⠂⠀⠀⠈⢀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⡻⢿⣿⣿⣿⣿⢿⣿⣿⠃⠀⣴⣶⣄⠀⠈⠀⠀⠀⠀⠀⠀⠗⠀⣼⣶⣾⣶⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣹⣿⣿⣿⣿⣿⣿⠻⣷⠀⢸⣿⣿⣿⠄⠠⣾⣤⣤⣴⠂⠚⢀⠀⠀⠗⠉⠙⡿⢿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣺⣿⠉⣿⣿⣿⣿⣿⣇⣸⣿⣿⣾⣿⣅⡶⠀⠁⠀⣠⣾⡿⠛⠉⢡⠉⠉⠃⠤⠒
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠙⠨⠥⠿⠛⠛⠛⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⠉⠙⢿⣿⡿⠉⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠒⢽⣧⠀⠈⣿⣛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⡌⢻⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢰⣠⡄⢿⣟⣹⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠱⠆⢀⣾⠇⡰⠰⠀⣾⣷⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡾⣠⣴⣿⢿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠂⠀⠀⠀⣤⣤⣨⣤⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣀⣀⣶⣝⣿⣿⣿⣿⣿⣿⣧⣴⣄⣀⣀⣤⡀⣀
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠿⠿⣾⡿⠿⢷⣯⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣮⣿⣿⠶⠶⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠙⠛⠉⠁⠨⠇⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠐⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡄⣠⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⢤⠀⡀⠀⡀
⠀⠀⠀⣀⣰⡨⠄⠲⠌⢠⣠⡀⠀⠀⠀⠀⠈⡒⣼⣇⢀⡀⡀⣸⡧⢛⠁⠀⠀⠀⠀⠠⢴⣶⠲⡆⠒⢠⠖⣲⡶⠆⠀⠀⠀⠀⠐⠾⠠⠬⠣⠄⠔⠣⠤⠳⠂
⠀⠀⢄⢹⡅⠐⣀⣄⢌⡤⣯⢀⠄⠀⠀⠠⢖⠂⠀⠁⢀⠠⣟⡈⠀⠀⣲⠆⠀⠀⠐⠴⡆⢠⡀⠈⣀⡁⠀⠀⢀⡶⠀⠀⠀⠀⠀⢀⠀⠀⠐⠂⠶
⠀⠀⢀⣺⡝⡻⠁⠸⢇⠇⣙⡄⠀⠀⠀⠀⠶⠀⠀⠀⢀⡠⠀⣘⠠⠀⠶⠀⠀⠀⠀⢾⡇⠀⠀⠀⠈⡁⢀⠁⢸⡧⠀⠀⠀⠀⢠⡆⠄⠒⠾⠀⠰⠀⠀⠰⡆
⠀⠈⡀⢿⡧⢉⠀⠀⠊⢸⡿⠂⠁⠀⠀⠀⣞⣀⣠⣄⡀⢠⠀⣀⠀⠀⣺⠀⠀⠀⠀⠛⠁⠣⠐⠈⠀⠀⠀⠠⠀⠙⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠄⠁
⠀⠀⡀⢸⡗⠘⠀⠁⠃⠌⣷⠀⡀⠀⠀⠀⣿⠋⢀⡀⠀⠀⠀⠁⠀⢂⢻⠀⠀⠀⠀⣱⣒⢀⢀⡀⣀⣀⡀⢀⣐⣊⠁⠀⠀⠀⠀⠖⠓⡟⠒⢤⡔⠚⣿⠣⠄
⠀⠀⢀⠞⠿⠶⠾⠛⠶⢾⠟⢆⠀⠀⠀⠀⢼⡷⢠⢴⠶⣖⠶⠦⢤⠺⠯⠀⠀⠀⠀⠈⠋⠘⠈⢛⡿⠛⠁⢓⠙⠋⠀⠀⠀⠀⢰⣆⠀⢹⣂⠈⠀⣸⠁⡰⡆
⠀⠀⠀⠀⠀⠁⠀⠠⢈⠄⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠰⠟⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠣⠹⡁⠯⠝⠷⠇
⠀⠀⠀⠐⠀⠈⠉⠈⠉⠀⠐⠀⠀⠀⠀⠀⠄⠠⠔⠒⠐⠂⠠⠠⠠⠄⠀⠀⠀⠀⠀⠀⠠⠂⠄⠠⠄⠠⠄⠐⠖⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢽⠷⣾⣿⠁⡀⣿⣷⠾⢹⠄⠀⠀⠀⠀⡶⣧⠐⢽⠀⠀⢄⢿⠖⡜⠳⠄⠀⠀⠀⠀⢰⠃⢂⡉⠐⠴⠔⣳⠆⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠖⠒⠠⡙⠾⠛⠏⡠⠖⠒⡆⠀⠀⠀⠀⣶⠅⠐⠈⠀⠀⠀⠱⠐⢀⣴⠂⠀⠀⠀⢰⣏⠀⠴⠞⣹⣧⠒⠦⠀⢈⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡠⣀⡀⢀⣀⢠⣈⣶⠀⠀⠀⠀⠀⠃⡆⠛⠐⠂⠀⠀⠐⢤⣸⡇⠀⠀⠀⠀⠀⣁⡀⠲⠀⠀⠀⠀⠀⠘⢋⡉
⠀⠄⠀⠀⠀⠀⠀⠰⡿⡛⢂⣬⠙⠋⢩⠀⠟⢻⡇⠀⠀⠀⠀⡄⢃⠈⠀⠤⠤⠀⠀⠀⠁⢠⠀⠀⠀⠀⠐⠋⠀⠀⠐⢦⡴⠂⠀⠀⠈⡑
⠀⡀⠀⠀⠀⠀⠀⠀⠲⣀⡓⠈⢖⡶⠉⠒⣃⠴⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⢸⠌⠀⠀⠀⠀⠀⣟⠀⠀⢠⣤⣤⡌⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠤⡥⢃⣠⡸⢃⣔⡈⢥⡠⡄⠀⠀⠀⠀⠀⠿⣦⠀⢰⣤⡆⠀⣰⡟⠂⠀⠀⠀⠀⠀⠐⠀⢶⠌⢾⡟⢡⢰⡆⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠮⠩⠵⣼⡚⠮⠬⠩⢦⠁⠀⠀⠀⠀⠈⠇⠽⠀⡰⠒⠂⡁⠼⠐⠁⠀⠀⠀⠀⠀⠈⠙⠋⠀⠙⠋⠀⠈⠝⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⡘⡛⠛⠳⡁⣀⠈⠀⠀⠀⠀⠀⠀⡄⠀⠤⠀⠀⠀⠠⡄⢠⡄⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠦⠀⢭⣤⣄⠐⠄⠦⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣵⣢⣿⣿⣿⣿⣋⣵⣿⣷⣿⣳⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣫⣽⣮⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠈⠙⠻⣿⣿⣿⣿⣿⣨⣯⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⣀⡀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⢫⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠶⠈⠚⠋⠉⠁⠲⠷⠶⢶⣮⣝⢹⣿⣿⣿⣿⡜⢟⣹⣭⢿⡗⢦⣄⠹⡛⢛⠿⠛⠁⠀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⢾⣿⣿⣧⠉⣿⣿⣿⣿⣸⣿⣿⣿⡗⠐⣛⣷⢠⣾⣿⡟⠟⠷⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠹⡎⢹⣿⣿⣿⣿⣿⣿⠟⣲⣿⣿⣿⠈⠛⢿⠗⠾⠁⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣿⣿⣿⣶⣦⣤⣀⠀⠀⠈⠛⠸⢽⣿⣿⣿⣯⠱⢺⣿⣿⣿⡏⣾⢿⣿⠦⣄⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⢿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣸⣿⣿⣿⣿⣷⣔⠙⣿⠟⣐⣈⡿⠷⡤⠟⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣼⠷⡟⣛⣟⡺⣿⣿⣿⣿⣿⣿⣿⣿⣷⣀⠹⣷⢲⣿⣿⣿⣿⣶⣷⣾⣿⡿⢷⣄⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠷⣶⣿⣾⡕⠈⠙⠁⠀⢾⠏⠻⣿⣿⣿⣿⣿⣿⣦⣜⢷⣽⣿⠻⢹⣿⣿⣿⠙⢷⡾⢟⡤⢋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣺⢽⣟⠩⠡⠀⢀⠞⠛⠳⢷⣤⣐⡛⢿⣿⣿⣿⣿⣿⣷⣯⣿⣾⣿⣿⣿⣻⣤⣶⠞⣋⣵⣮⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣽⡸⠶⠀⢄⠀⢾⡀⠀⠀⠀⠙⢿⣏⣍⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣖⣞⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⣵⢂⠄⠀⠀⢐⠳⣤⣀⠀⠀⠀⠙⠿⣶⣮⡝⠿⠿⣿⣿⣿⣿⣿⣿⡯⡿⣿⠿⠁⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠡⡄⠐⣈⣀⣈⣤⣭⣽⣿⣿⣛⣛⠛⠻⢿⣮⣭⣍⣉⠉⠛⠙⠋⠉⠀⣁⠀⠈⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣬⣿⢿⣿⣷⣿⣿⣇⣼⢈⡍⠓⠢⢬⣉⣛⡻⠿⠾⠛⠒⠚⠛⠤⣄⣀⡂⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠤⠛⡩⠷⠟⡉⢉⠉⠉⠉⢙⢛⢿⣶⣇⡄⢀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠀⢸⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠔⠁⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠈⠙⠳⣶⣀⠀⠀⠈⠑⠒⠒⠒⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠐⠒

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣠⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠀⠺⢿⣭⣭⣭⣭⣭⣤⣾⣿⠿⢦⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⢰⡿⠻⣷⡄
⠀⠀⠀⠀⠀⣀⣠⣤⣬⣭⣭⣩⣤⣄⠐⣧⣭⣽⣿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣄⣹⢦⡴⠟⠁⣠⣿⠇
⠀⠀⠀⣴⣿⠿⠛⠛⠉⠉⠉⡉⠈⡩⠊⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠛⠛⠿⣿⣶⣶⡿⠟⠋
⠀⠀⠀⢿⣿⣤⣤⣤⣤⣤⢤⣄⣼⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⣄⣀⣤⣶⣾⠿⢿⣦
⠀⠀⠀⠀⠉⠉⠛⠛⠛⠉⠀⠈⠀⠀⠀⠀⠀⠙⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉⣀⣤⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣆⠀⠀⠀⠀⢀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⣾⡿⠟⠉⠀⠀⠀⠀⢀⣾⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣽⣿⣦⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣤⠀⠀⢠⣾⣿⣿⣿⣟⣛⣛⣋⣽⣿⣿⠉⠉⠈⠙⠛⠶⠶⠶⢦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡏⠁⠀⠀⠀⠙⠿⢿⣿⣿⡿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣽⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣡⣴⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⢻⣿⣿⣛⣉⣭⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡇
⠀⠀⠀⠀⠰⠤⣄⠻⠿⠿⢿⣿⣿⡿⠿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣗⣀⣀⣀⣀⢈⣿⠇
⠀⠀⠀⠀⣴⡿⢿⣿⣶⣤⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⠤⠤⠶⠖⠛⠉⢿⣿⣭⣿⣿⡿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠙⠿⣷⣶⣾⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⢠⣦⣴⣤⣤⣤⣦⣴⣦⣴⣶⣶⣤⣤⣤⣶⣦⣤⣦⣤⣴⣶⣶⣶⣦⣤⣴⣶⣴⣦⣴⣶⣦⣴⣶⣤⣤⣦
⠀⠀⠈⢿⣷⣼⠀⠀⠀⠀⠘⡇⠀⠃⠀⠉⠁⠀⠀⠀⢹⠃⢹⠁⠁⠀⠀⠉⠈⠁⠀⠀⢘⠀⠀⠀⢠⣼⣿⡇
⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⣿⡇
⠀⠀⠀⠀⠀⢿⣎⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢡⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣗⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢠⣿⣿⠉⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣆⠀⠙⢿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏⠀⢀⡾⠃⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠈⠃⢻⣆⠀⠀⢿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠁⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⢘⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⡿⠀⢀⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠈⢿⣤⡀⠀⠀⠀⠀⠀⣤⣿⠟⠁⠀⠹⣠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆⠀⠀⠀⠘⠙⢷⣄⠀⠀⣠⣾⣿⡟⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢿⡆⠀⠀⠀⠀⠀⠹⣿⣾⣿⣿⢿⡇⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢻⣄⠀⠀⠀⢀⣶⠿⣿⣿⡇⢸⠃⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠀⣴⡿⠋⠀⠁⣿⣿⣾⣆⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠋⠀⠀⠀⠀⠀⠀⠙⢿⣿⡟⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣇⠀⠀⠀⠀⠀⠀⣠⣿⠿⣷⣜⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⠁⠀⠘⢿⣆⠀⠀⠀⠀⢰⡿⠃⠀⠈⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠋⡿⠀⠀⠀⠘⢿⣆⡀⠀⣠⡿⠁⠀⠀⠀⠀⢹⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠈⢻⣧⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣤⣤⣤⣤⡄⢠⡿⠃⠏⢻⣆⢀⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢿⣧⣰⡿⠁⠀⠀⠀⠹⣆⢩⡟⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⠸⢻⣿⠁⠀⠀⠀⠀⠀⣻⣟⠀⠀⠀⠀⠀⠀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠁⠀⠀⠀⠀⣰⡿⠻⣧⠀⠀⠀⠀⣴⡟⢹⣆⠀⠀⠀⠀⠀⢩⣹⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⢀⣠⣠⣶⡏⠀⠀⢹⣧⠀⠀⣸⣿⠀⠀⠙⢷⣄⣀⣦⣤⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠟⠋⠀⠀⠀⠀⠀⠹⣧⣸⠏⠁⠀⠀⠀⠀⠙⠛⠿⠛⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⠃⡀⠀⢀⣜⣈⣾⣇⣀⣀⣤⣤⣤⣬⣶⣤⣤⣤⣠⣿⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⣦⣤⣠⡤⣷⣤⣤⣾⣟⣛⣿⣿⣿⣿⠿⠟⠛⠟⠛⠊⠙⠛⠛⠛⠟⠛⠛⢻⣿⣽⣧⡀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣶⣶⣾⣿⣿⠿⠛⣿⢿⠋⠉⠉⠉⠉⠙⠆⠀⠀⠀⠀⠀⠀⢄⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡿⠟⠗
⠀⠀⠀⠀⣴⣶⣄⠔⠀⣴⣾⣿⣿⠟⠛⠁⠫⠑⠀⠈⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢦⡀⢀⣠⣤⣾⠿⣿⣿⣿⡷⠔
⠀⠀⠀⠈⢻⣿⣿⣿⣾⢃⣾⠟⠳⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣾⣅⣄⣿⣶⣿⣽⠯⠄
⠀⠀⠀⠀⠈⡷⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠒⢒⣿⣛⣹⣿⠿⣿⣿⣿⡁
⠀⠀⠀⠠⠈⣿⣿⣿⣿⣾⣁⡤⢶⣏⡄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⠀⣷⠀⣀⣾⣿⣿⡿⢣⡞⠁⠀⠀⠁
⠤⢤⣀⣀⠾⢛⡿⣿⣿⣿⣿⣀⣼⣯⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣰⠇⠀⠀⠤⢤⠆⢈⢛⡿⣿⣿⣿⡿⠟⠋⠛⢦⡀
⠀⣀⣀⣨⣿⣿⣽⢿⣿⠿⢿⣿⣿⣈⣧⠀⠀⠀⠀⠀⡇⠀⠀⣰⠇⡄⠀⠀⣀⣶⣖⠛⣛⠟⠻⠛⠃
⠀⠀⠀⠀⠀⠈⠀⠘⠿⠟⠻⢿⣟⣿⣿⣿⣶⣤⣴⣾⣷⣄⣤⣿⣧⣿⣶⡿⠛⣹⣿⡿⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠙⠛⢛⡿⠿⠿⠟⠛⠛⢯⠀⠁⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢂⣅⣤⣦⡤⣤⣌⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣅⢿⣿⠿⠺⠿⣿⣿⣧⡄⠑⢄⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣌⡂⠲⠠⣀⠙⢾⣿⣧⠀⠁⣄⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣷⣦⠀⠳⡌⠻⣯⣞⡤⡀⣁⠀⣀⡤⡾⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡂⢀⡀⠌⠙⢻⡇⠀⣷⠀⠈⠻⢟⡿⣻⣼⣿⢿⠹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢻⣶⠘⣳⣿⣿⣤⣄⣉⣠⣼⠇⠀⠀⠀⠀⠉⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡴⠆⢠⣹⡛⡋⠙⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢏⡆⣷⠀⣀⣩⣶⣿⣧
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠶⣾⣧⣾⡇⠀⠀⠸⣿⣿⣿⣿
⠀⠀⠀⠀⡞⢀⣤⣈⢢⣤⣿⣻⣿⣿⣷⣶⣯⣿⣿⠿⠃
⠀⠀⠀⢸⣧⡀⠀⠸⣯⠉⠓⠙⣾⣿⠿⠋
⠀⠀⠀⠈⣿⣿⣄⠀⢻⡧⣤⣼⣾⠃
⠀⠀⢀⠔⠙⢿⣻⣷⣯⣀⣠⠟⠁
⠀⠐⠁⠀⢠⣼⣿⡿⠋⠉⠁
⠀⠀⠀⠀⠿⣻⠟
⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡉⣉⣈⣉⣉⣉⣸⣋⣁⣹⣿⣿⣦⣿⣿⣟⣁⣉⣩⣉⣉⢉⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⠹⣧⣀⣤⣼⣧⣤⣤⣴⣿⣿⣿⣿⠟⠀⠀⠀⢀⣼⠃⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣏⣽⣿⣿⣿⣿⣿⣿⣿⠿⠿⣛⣥⣴⣶⣶⣶⣯⡏⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⢀⣤⣶⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣟⠋⣩⠞⠁⢠⢋⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⢺⡿⣿⣿⣷⣶⣦⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡄⠀⢀⡽⣏⣉⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⣿⣿⣿⣿⣿⣭⣿⣿⣯⣍⡉⢡⠃⠳⣴⣿⣿⣿⣿⣿⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⡿⣿⣿⣶⣴⣾⣿⣿⢟⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣼⣿⣿⣿⣿⣷⣤⣶⣿⣯⠈⠉⣽⣿⠏⣡⣼⣯⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣧⣠⣿⡿⣁⢼⡿⢡⣾⣿⠿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣯⣉⡀⢸⣿⡇⢾⣿⣿⣶⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢙⣿⣿⣷⣘⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣛⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠀⠤⠀⠀⠀⠖⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠠⠠⠒⠛⠉⠉⠀⡀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠉⠅⠁⠀⠀⠀⠀⠀⠀⠀⠀⢂⠀⠄⡠⠀⠀⠀⠈⠀⢈⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⡠⠒⠀⠀⠀⢀⣠⣤⣤⣽⣦⠀⠘⡷⠆⠬⢀⢀⣙⡓⠂⣀⠀⠀⠀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⡴⠚⠉⠼⡉⠀⠀⠉⢣⠀⠀⢀⠰⡾⠂⠀⠈⠝⠉⠉⢻⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣧⡀⡈⣇⡀⠀⢀⠀⡀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢢⠀⠀⠃⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢚⠩⡀⠙⣄⠈⣀⢠⣠⡁⡘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠀⢠⠀⠀⠂⢀⣇⠀⠀⠀⠀⠀⠀⠀⠀⡸⠋⡈⠀⠀⠰⠋⠈⡗⠨⡙⠉⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢷⠀⢿⠀⠀⠀⠀⠄⢸⠁⠀⠀⠀⠀⠀⠀⠀⠘⠡⠐⠣⣀⡤⢲⡀⢰⠀⠀⠀⠉⠀⠀⠁⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⣉⡀⠀⠈⠀⠀⣿⣠⠎⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠸⡅⠀⠀⠀⠀⡟⢉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠂⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⠀⠀⢣⢀⠀⠀⢀⠃⠘⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠲⣀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠙⠂⢺⣧⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠁⠂⠀⣠⠾⠥⠤⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠆⠀⡀⠈⢧⣠⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠃⠀⠀⡀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡀⠀⠂⠀⢀⡀⠀⠣⡀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⡀
⠀⠀⠀⠀⠀⠀⠈⢀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡘⣦⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠂⠀⠐⠀⢀⡁
⠀⠀⠀⠀⠀⠈⠀⠣⠄⠀⠀⠀⠄⣀⣀⡠⠤⠤⠄⠀⠄⠤⣈⣳⣼⣦⡀⠀⠀⠀⠀⠀⠀⠀⠦⠀⣀⠀⠀⠀⢀⡀⠁⢠
⠀⠀⠀⠀⠀⠘⠆⠀⠀⠀⣀⠴⠚⠉⠀⠰⠈⠀⠀⠀⠀⠀⠀⠀⠈⠷⣍⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠈⠀⠀⠀⠀⠀⡀⠀⠢⠀⠀⠀⠀⠀⠈⢷⢤⡲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠐⠅⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠆⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢼⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠤⡀⡀⠀⠂⠤⠀⠀⢨⠀⠈⢀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠣⢄⡀⠀⠉⠐⠢⠀⠀⠴⠂⠀⠀⠈⠁⠀⠀⠀⠀⠀⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⠀⢀⠈⠀⠀⣀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⡀⠂⠁⢀⡀⢘⣉⠤⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠆⠀⠠⠀⠐⠒⠂⠂⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠄⠠⠄⠀⠀⠤⠀⠠⠠⠀⠀⠠⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⡀⠐

⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⡿⠛⠉⠉⠀⠉⠉⠛⠛⢿
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠸⠿⡿⠟⠀⢰⣿⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠸⣿⣿⠆⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠁⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠳⢠⢀⢀⠀⠀⠀⠀⢀⠠⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡀⠀⠀⠀⠀⢠⣾⣿⣷⡄⢠⠀⠀⠀⠈⠛⢼⡃⣠⠀⠢⢠⡶⢩⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠘⣧⠈⠻⢷⡄⠸⠿⠿⠿⠃⢸⣇⠀⠐⡄⠀⠀⠤⠿⣦⣰⣿⣧⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⢹⣧⠀⠀⢻⡄⠀⠀⠀⠀⣸⣿⣷⣄⠉⢰⣿⡆⣶⣿⣿⣿⣿⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠀⣿⣿⣷⣄⢈⣿⡀⠸⡆⢀⣿⣿⣿⣿⣷⣬⣉⣼⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣦⣻⣷⣶⣨⣸⣿⣿⣿⣿⣿⣿⣯⣌⣿⣿⣿⣿⣿⣿⡶⠂
⠀⠀⠀⣤⣀⡀⠀⠀⠀⠀⣤⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣀⣤⡀
⠀⠀⠀⠘⣿⣿⣿⣶⣤⣼⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠭⠉⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⡉
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢾⣗⠀⠶
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣉⠱⠆⢉⢠⡁
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⠉⢻⣿⣿⢡⣄⣬⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢺⡁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡿⠹⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⣿⠿⠋⠇⡠⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣹⣯⣿⣿⣷⣶⣾⣿⣿⠁⣹⣿⣿⣿⣿⣿⣿⣿⠠⡄⠀⠉⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠯⠉⢐⡋⣡⣿⣿⣿⣿⣿⣿⠋⠉⣻⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⢋⡍⠈⢠⣦⣶⣿⣿⣿⣿⣿⣿⣿⣿⡗⠋⢰⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡟⠋⠀⠈⢀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣏⠀⠀⠀⠀⣨⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛
⠀⢤⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠇⢀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⡀
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⠃
⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠃
⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠃
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠲

⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅
⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠠⡀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂
⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⠀⠀⠈⢰⣿⣿⣿⣷⣶⠦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣣⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠
⠀⠀⠀⠀⠀⠠⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⡂⢐⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⢀⣤⣤⣤⣤⣀⣀⣀⣀⡀⠀⢀⣼⡟⠙⠛⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣶⣶⣦⣤⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⢻⣿⣿⣦⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡙⢿⣦⡀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡌⠻⣿⣦⠀⢀⣤⣾⡿⠟⣵⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡄⠈⣻⣿⣿⠟⠋⢠⣾⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢻⣿⣿⣾⡿⠛⢿⣷⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣼⣿⣿⣿⡄⠀⢠⣿⣿⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠟⠉⢿⣿⣿⣴⣿⣿⡟⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠤
⠀⠀⠀⠀⠀⢠⣤⣶⣶⣍⠁⠃⠀⠈⣿⣿⣿⣿⠋⠀⠀⠘⢿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣆⠉⢀⣴⣿⣿⣿⣿⡄⠀⠀⢀⣴⣷⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⣠⡀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣿⡿⠋⢻⣿⣿⣄⣴⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⣰⣿⡿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠁⠀⠀⢻⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⢸⣿⡇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⡀⠀⢀⣼⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠸⣿⣷⣄⣤⣴⣾⣿⡿⠟⠁⠘⢿⣿⣷⣴⣿⣿⠟⠻⣿⣿⣧⡀⠀⠀⠀⠀⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠙⠿⣿⣿⠿⠛⠉⠀⠀⠀⠀⠈⢿⣿⣿⠟⠁⠀⠀⠈⠻⣿⣿⣦⣄⣀⣠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙⠂⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠲⠰⠀⠤⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠌⠄⠀⠀⠀⠀⠈⠙
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀