Literature Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢉⣉⣉⡉⠉⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠁⠀⠀⠉⠉⠉⠁⠀⠂⠒⠒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡤⠤⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠾⠿⠿⠟⠉⠩⠏⠁⠀⠀⠐⡟⠓⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⡶⢶⣶⣶⠶⢲⣶⢢⡖⠒⠒⠒⠒⠒⢶⠶⡶⠒⠒⠒⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣥⣿⣤⣴⣦⣼⣄⢠⡷⢼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠓⠈⠛⠛⠿⠿⢿⣿⣿⣿⢿⣿⣾⡄⠀⠀⠀⠀⠀⠀⠘⠀⡀⠀⠀⢀⣠⣀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠷⢾⠟⠛⠉⠉⠉⠉⡥⠖⠒⠁⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠤⠄⠠⣀⣀⣀⣀⡀⠀⢀⡤⠖⢒⠉⠀⠀⠀⠀⠀⣼⣷⡤
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠋⠀⡏⠀⠀⠀⠀⢀⣠⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠑⠒⠒⠢⠤⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⣠⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠐⠒⠢⠤⠄⠖⠛⠉

⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠄⠀⠐⠒⠀⠀⠀⠀⠀⠈⠉⠀⠒⠂⠤⠤⢄⣀⣀
⠀⠀⠀⠀⠀⠈⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠐⠒⠒⠠⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠠⢄⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠶⠶⠶⠞⠻⠛⠛⠛⡏
⠀⠀⠀⠀⠀⠈⠓⠲⣤⣀⠀⠀⠀⠀⠀⠀⠘⠲⣶⣿⠿⠟⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣇
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠈⠙⠲⠦⣄⡀⠀⠀⠘⣿⠄⠀⠀⠀⠀⠀⢀⣀⣀⠤⠤⣶⣶⣒⣺⣿⣯⣽⠇
⠀⠀⠀⠀⠀⠈⠁⠢⠑⠢⢄⡀⠀⠀⠉⠙⠲⠤⣿⣿⣶⣶⣖⣺⣭⡭⠷⠶⠚⠛⠋⠉⠉⠉⠀⠀⠠⡀
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⣑⠢⢄⡀⠀⠀⠀⠈⣿⡯⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⡴⣖⣿⠂
⠀⠀⠀⠀⠀⠂⢄⠀⠀⠀⠀⠀⠈⠑⠢⣌⠑⠢⢀⠀⣿⣄⠀⢀⣀⣀⣤⣴⣶⣺⣿⣽⠷⠾⠛⠛⠉⡍
⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠀⠀⠀⠉⠒⢤⠈⣙⣻⣭⡽⠿⠟⠛⠋⠉⠁⠀⠀⠀⠀⠐⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀⠀⠀⠀⠀⠀⢠⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣌⣥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⣀⠀⠀⢸⣯⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢸⣷⣆⣀⣠⣤⡤⠴⠒⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡤⠴⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠶⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠚⠛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠢⢭⣙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣈⠒⠦⣌⡉⠒⠦⣍⡛⠿⢶⣤⣄⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡐⠤⢍⣒⠦⢌⣙⠲⠤⣀⡀⠀⠈⠙⠻⣦⣄⠠⠐⠚⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡵⠒⢬⣝⡲⠬⣉⡒⠤⢌⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠈⣙⣳⣦⣍⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣛⢻⣏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠈⠉⠉⠉⠉⠛⠷⢶⣤⣄⣀⠀⠀⢀⣀⣼⠇⢀⣀⣠⣤⣶⠶⠞⠛⠉⠉⠉⠉⠉⠉⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⡶⠶⠶⠶⠶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⢶⡶⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⡟⢧⡛⠷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⡶⠶⠶⠾⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠁⠸⠀⠈⢷⡀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⠇⠀⠀⢀⣀⡈⣻⣿⣤⣴⣶⣶⣶⠶⠶⠶⠾⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡄
⠀⠀⠀⠀⣀⣠⣴⠶⠞⠛⠋⢹⣿⣴⠾⠟⠛⠛⠛⢛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⠶⠶⠶⠶⣶⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣈⣿⣆
⠀⢀⣴⠿⠛⠉⠀⠀⠀⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⣛⣒⣒⣒⣒⣒⡒⠲⠶⠶⠤⠤⠤⠭⠭⠭⠭⠉⠉⠉⠉⢉⣿⡉⠉
⢠⡿⠁⠀⠀⠤⣄⡀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢒⣂⣀⣀⡀⠀⠀⠀⠀⠈⠭⠭⠭⣭⣭⣭⣭⣅⣒⠒⠀⠀⠀⠀⣾⠀⠉⠓⠦⣄⡀
⣿⠁⠀⠀⠀⠀⠀⠈⠉⠂⠀⠘⣷⡀⠀⠀⠀⠐⠒⠒⠂⠤⠤⠤⠤⠤⠤⠤⢭⣍⣉⣉⣉⣉⣒⡒⠒⠒⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠈⠙⠳⣦⣄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣦⣄⣐⡒⠲⠶⠶⠶⠶⣤⣤⣤⣤⣄⣀⣀⣀⣒⡒⠂⠉⠉⠉⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⢷⣀⣀⣤⣤⡴⠶⣶⠟⠛
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠻⠷⠶⠶⠶⠾⠿⠿⠿⠿⣿⣿⣽⣿⣭⣭⣤⣀⣀⣀⣀⣀⣀⣐⣒⣀⣀⣀⣙⣏⠁⠀⠀⢰⠇
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠀⠀⠀⠀⠀⠀⢀⣠⣤⡤⠶⠶⠚⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⣼
⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠠⠤⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿
⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⢀⣀⠤⠄⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠙⠳⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠄⠐⠂⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⡀⠤⠄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⣀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⡴⠶⠶⠶⠚⣛⣋⣉⣥⣤⠶⠶⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣌⣛⣬⣑⠒⠀⠀⠀⠀⠒⠚⢉⣉⣉⣩⣭⣤⡴⠶⠶⠚⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠉⠲⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢺⡿⠋⢄⡈⠑⠤⣀⠀⠉⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⡴⡫⢈⠔⠁⠀⠀⠀⠀⠉⠒⠄⡀⠉⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡫⠊⠋⠀⠤⡁⠀⢰⠀⠀⠀⠀⡀⠀⠀⢈⠑⠢⣄⠈⠑⠠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠀⠀⣠⡶⠶⣬⡑⠢⢄⠀⠠⠀⠀⠀⠄⠀⠈⡀⠀⠁⠀⢄⡀⠑⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠋⠀⡒⣾⠛⠀⠀⠀⠍⠓⢦⣉⠢⢄⠐⠂⠀⠀⠊⠀⢀⠔⠠⠀⠈⠒⠤⡀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⣠⡾⠋⠀⠀⠈⠑⢢⠀⠐⠤⠤⠌⠛⠦⣍⠀⢀⡐⠠⡴⡡⠊⠤⠀⠀⠐⠀⠈⣁⠀⡨⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡾⠋⠉⢀⣾⠏⠀⠀⠀⠰⣶⣤⠀⠀⠀⠠⠈⠁⠀⠀⠈⠑⠦⣉⠒⢄⡀⠀⠀⠀⠁⠠⢀⠞⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣪⠆⢀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣠⠀⠀⠀⠈⠓⠦⡈⠑⠤⡀⢂⠔⢁⣼⡿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣤⠞⠁⣰⠟⡅⡆⠀⠠⠂⢀⡆⠐⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢈⣴⡿⣫⢞⣽⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁⣠⠾⣱⠐⠿⢀⠸⣄⠒⠺⣵⣮⠀⠾⠂⡨⠀⢄⡠⠄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣴⡿⢋⢜⣵⣿⠿⣷⡿⠂
⠀⠀⠀⠀⠀⢀⠞⣹⠞⠁⣠⠞⠁⠀⠛⠃⢠⣦⠠⡌⠁⡠⢄⢀⠅⣠⠘⡀⡠⠶⢆⣀⡘⠀⢠⠀⣀⢀⣤⠞⢁⣴⡯⢋⢔⣵⣿⣯⣜⣷⣾⠁
⠀⠀⠀⠀⠀⢸⣿⣦⣀⠀⠈⠻⢦⣀⠠⢶⡂⠷⠖⢘⡏⢁⡤⠠⢤⢠⡶⡎⠀⠀⠘⠁⠈⠁⢻⢔⣵⠟⢁⡴⡯⢋⢔⣵⣿⡻⢝⢷⣽⡿⣻⡀
⠀⠀⠀⠀⠀⠈⣿⢨⡙⠳⣦⣄⠢⢝⠳⣤⡀⠘⠉⢻⡁⢞⣻⠁⠄⢿⡉⣳⠈⠑⠠⡄⢌⡴⣵⠟⢁⣴⠏⢀⢔⣵⣿⠻⢷⣌⣳⣿⠏⢈⣼⣿⠆
⠀⠀⠀⠀⠀⠀⣿⡀⢌⡑⢤⡙⠻⣦⣍⠪⢕⠒⠘⠟⠙⣼⡧⢂⠄⡼⢄⠢⠁⠐⠈⠀⣡⠞⢁⣴⢟⡵⠀⣰⣿⠿⣗⢍⣢⣽⣿⢇⢔⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠙⢿⣶⣌⡓⢌⡒⢄⡙⠳⣦⣉⠲⢄⡐⠋⡣⠉⠋⠃⢌⡗⠀⠀⢠⡞⠁⣰⣿⡿⠋⣠⣾⣿⣭⣿⢾⣿⣿⠟⠕⣱⣿⠏
⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣶⣌⡓⢬⡂⢌⠉⠙⠲⣍⡢⠀⠺⢦⣀⠀⠀⠀⡴⠋⣠⡾⠿⠉⣰⡾⠟⡧⣊⢝⣷⣿⡟⠁⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⣥⣬⣙⠻⣿⣿⣿⣿⣦⠈⠃⠉⣀⠈⠒⠝⣢⣌⠓⢍⡳⠆⠈⣠⡾⣫⠄⣠⣾⣟⡒⢕⡪⣠⣾⡻⢋⢤⣾⡿⠃
⠀⠀⠀⠀⠀⠀⢸⣏⣉⠻⢶⣽⣢⣭⣻⢿⣦⣄⡈⡑⠢⣐⠬⡙⠻⣦⣌⣠⡾⣫⠞⣡⣾⠟⢓⠬⡻⣵⣾⢿⢟⢔⣵⡿⠋
⠀⠀⠀⠀⠀⠀⠸⣏⠠⣉⠢⢌⡙⠿⣯⣻⢮⣝⠻⣿⣗⠤⡉⠢⢕⠀⡽⢋⠜⣡⣾⡟⢁⠪⣑⢉⣾⠟⠕⡱⣫⣿⠏
⠀⠀⠀⠀⠀⠀⠈⠻⢷⣦⣍⠢⢍⡢⢌⠙⠳⣮⣟⡒⠿⢿⣮⣕⠢⠀⡷⣡⣾⠟⡁⢌⡒⠙⣢⣿⠋⠀⣪⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⣍⠒⢭⡒⢄⡁⠉⠻⢷⣮⢝⠿⣶⣾⠟⢛⣶⣮⣕⣭⣾⡿⠋⢜⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣦⡈⠑⠉⢀⠀⠑⠪⢿⣿⣦⢤⡲⣿⣿⣿⣿⣿⠏⡰⣷⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢶⣄⡀⢉⠒⢤⡒⢬⡙⠿⣯⣄⣠⣿⠟⡡⣪⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⡢⣌⡑⠪⣕⠄⣽⠿⠃⢔⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣭⡓⠤⡁⡇⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣦⣴⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠴⠦⣤⣀⣀⣤⠤⠦⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣼⡏⠉⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠈⠉⢹⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⣀⣀⣤⣤⣀⢸⣇⣀⣤⣤⣀⣀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠛⠛⠉⠁⠀⠀⠉⠛⠛⠉⠀⠀⠈⠉⠛⠛⠃⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠒⠀⠠⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶⠦⢤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠐⠒⠦⠤⢍⣁⣀⣀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠬⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠒⠋⠉⠁⠀⢰⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠓⠒⠲⠤⢤⣤⣠⣤⠴⠖⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠘⣇⣀
⠀⠀⠀⠀⢀⡀⠨⠶⠒⠶⠶⠤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠀⠀⠠⠤⢀
⠀⠀⠀⠀⠀⢻⠓⠒⠤⢤⣀⡀⠀⠀⠈⠉⠙⠓⠲⠶⢤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⠀⠈⠉⠻⣿⣆⣠⣤⣶⡾⠿⠿⠓⠒⠊⠉⠁⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠙⠻⠶⢤⣄⡀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠲⠶⣤⣤⣀⣀⡀⠹⡆⣀⣤⣴⢾⣿⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⢦⣤⣤⣤⠴⠶⠒⠚⠋⠉⠉⠀⠀⠉⠉⠛⠛⠛⠉⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⢶⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⣮⣄⣀
⠀⢠⣶⣖⣋⣭⣉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠐⠒⠊⢉⣁⣠⣬⡵⣿⡞⠉
⢀⣿⠏⠀⠉⠉⠛⠻⠶⣤⣍⣛⡳⢦⣤⣀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⠤⠤⠖⠒⠚⣉⣉⣡⣤⠤⠴⠖⠒⠋⠉⠉⠉⠁⠀⠀⣾⠁
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣿⣶⣤⣀⡜⢁⣆⣠⣤⣤⣶⠶⠾⠛⢛⣋⣉⣭⣥⣶⡶⠶⠚⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⢻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣿⣯⣥⣶⣶⠾⠿⠟⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣐⣈⣧⣤
⠀⠀⠙⠛⠿⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡴⠶⠾⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠾⢷⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⡾⠶⠚⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⢶⣤⣀⣀⠀⢀⣄⠀⠀⠀⠀⠀⢀⣀⣠⣴⡶⠶⠾⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠷⣮⣤⡶⠶⠟⠛⠋⠉⠁






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠤⠄⠀⠀⠠⠀⠀⠄⠄⠀⠄⠀⠀⠀⠠⠀⠄⠄⠀⠀⠀⠀⠀⠐⠠⠂⠀⠆⠄⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠄⠒⠒⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠂⠐⠒⠒⠂⠤⠤⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢔⣾⠯⠷⠦⢰⠋⣤⣔⣒⣀⠀⠀⡀⠄⢀⣀⣀⣀⣀⣠⣤⣤⣔⣲⣬⣷⡿⠟⠃
⠀⠀⠀⠀⠀⢸⣼⠂⠀⠀⠀⠀⠀⠀⠤⠄⣉⡉⠉⠙⠓⠛⢛⡞⠛⢙⠩⠩⡫⡀⠀⠐⡨⡇
⠀⠀⠀⠀⠀⠀⡾⠶⢦⢤⣤⣄⣀⣀⣀⠉⠉⠉⠉⠀⠀⠀⢸⡈⢈⠀⣨⣪⣠⡦⣿⣿⠿⠿⠃
⠀⠀⠀⠀⠀⠀⠈⣷⣄⠀⠀⠀⠁⠉⠉⠙⠙⠒⠒⠛⠲⠲⠶⣿⣿⣿⣛⣛⣛⣭⣉⣓⣦⡀
⠀⠀⠀⠀⠀⠀⠴⠁⢑⣦⣁⣀⣀⣤⣤⣤⣤⣶⡶⡶⠲⡻⡻⣻⠞⠝⠝⡝⠝⠝⢝⢍⠻⣧
⠀⠀⠀⠀⠀⠀⠈⢔⠄⢫⠙⡯⡉⠁⠑⠑⠁⢔⠕⢔⢌⢔⠩⠪⠊⠂⠪⠂⠢⠢⡢⡀⣩⣽⡇⢀⡀
⠀⠀⠀⠀⢀⠠⠔⠚⢮⡄⠀⡏⡨⣠⣢⣢⣢⣠⣠⣡⣵⣰⣵⣵⣵⣵⣵⣴⢷⣶⣾⣾⣯⣿⣴⡿⠃
⠀⠀⣠⣞⣑⣧⣤⣤⢀⡙⠛⡻⠿⠿⠟⠛⠛⢙⣛⠹⠭⣍⡉⠈⠉⠅⠀⠉⠠⠀⢂⣴⡾⠛⢻
⠀⢰⣿⠉⠁⢨⢉⡉⠉⠒⠒⠓⠾⠭⠶⣶⣒⣒⡠⠤⠤⢀⡀⠀⠀⠀⠀⠀⣠⣴⠻⠏⠂⡄⣬⣶⣄⡀
⠀⠘⢿⣄⣀⡳⣾⠀⠠⠤⠀⣈⠀⠰⠂⡀⠀⠀⠉⠉⠛⠒⠒⠿⠽⣦⣶⢿⡏⠀⡁⣀⣶⣾⡛⠁⠀⠀⢉⣴
⠀⠀⠀⠀⢉⣩⣿⣿⠷⠶⢶⣦⣤⣀⣉⢀⡒⠀⠉⠁⠀⠀⠀⠀⠀⡇⠁⠨⠁⣤⣷⠟⢉⡀⢉⣠⣴⠞⡿⠁
⠀⠀⠀⢠⣻⠛⠛⠛⠯⣔⡂⠠⢀⡈⠉⠛⠛⠛⠲⠲⠶⣤⣄⣀⣀⣷⣠⣶⠟⠛⢂⣠⣶⢾⡿⠫⠁⠁⣧
⠀⠀⠀⢸⡧⠀⠀⠀⠀⠀⠌⡛⠓⠦⢅⣒⠠⠀⡀⠀⠀⠀⠈⠉⠉⠛⢋⡹⣖⡿⠟⢙⠍⡠⠁⠐⢐⣠⣿⠦⠍
⠀⠀⠀⢀⠻⣦⡀⠀⠀⠄⢀⠀⠑⠢⢀⠈⠉⠓⠤⢭⣐⡀⢐⣤⣴⢾⠻⠫⡁⠎⠀⠀⣂⣱⣴⣾⠟⠉⠁⠀⠭⠁
⠀⠀⠀⠀⠀⠐⠙⠶⣦⣀⠈⠀⠒⢤⣀⠀⠁⠀⠀⠀⠀⡝⠉⠩⡢⡁⡁⠀⢌⣂⣶⣻⡽⠚⠓⠂⠠⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠒⠶⢢⣄⡉⠙⠀⠀⠀⠀⡟⡑⠐⠔⣔⣴⣾⠿⠟⠉⠀⠀⠠⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠪⢤⣒⠤⢹⣤⣾⣾⠿⠋⠉⠁⠂⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠛⠁⢀⣀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠤⠄⠐⠂⠈⠉⠉⠁⠐⠒⠒⠤⠤⢤⣀⣀⣀⡀
⠀⠀⠀⢀⡤⠴⣒⣀⡉⠁⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠓⠒⠲⠦⠤⣤⣀⡀
⠀⠀⠀⡘⠀⠀⠀⠀⠈⠉⠑⠒⠒⠦⠤⢤⣀⣐⡒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⢙⡏
⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠲⠦⠤⣤⣀⣀⣀⣀⠤⠴⠚⠋⠉⠀⠀⠀⠀⠸⡄
⠀⠀⠀⣄⣭⠷⠶⠶⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠀⡀⠀⠀⠀⠀⠀⣀⣤⡴⠾⠛⢉⣒⡢⠤⡄
⠀⠀⠀⢸⡏⠙⠲⠤⣄⣀⠀⠀⠉⠉⠓⠲⠶⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⣿⠀⠉⠙⣷⣦⣤⠶⠟⠛⠛⠋⠉⠉⠀⠀⠀⠀⢹
⠀⠀⠀⢠⡇⠀⠀⠀⠀⠉⠙⠒⠦⣤⣀⡀⠀⠀⣀⣀⣉⣭⠽⠷⠶⠶⢤⣼⣤⡶⠟⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣼⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣎
⢀⣴⣶⣏⡙⠻⢶⣤⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡤⠤⠴⢒⣒⣫⣭⣽⣿⠟
⣾⠇⠀⠉⠙⠛⠷⢮⣿⣿⣶⣤⣀⡀⠀⠀⡾⠐⠀⠀⠀⠀⠀⣀⣀⣀⣠⡤⠴⠶⠒⣚⣋⣩⡥⠤⠶⠖⠛⠛⠉⠉⠁⠀⠀⡏
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⢷⣶⣥⣧⡴⠶⢒⣛⣛⣭⣭⣴⠶⠖⠚⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠘⠿⢦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⠶⠶⠚⠛
⠀⠀⠀⠀⠉⠛⠻⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡶⠶⠞⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣤⣀⣀⣧⠀⠀⣀⣀⣠⣤⣴⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⠾⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⠈⠉⠉⠀⠀⠒⠠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠠⠲⠠⠒⡂⠀⠄⡀⠀⠀⠉⠢⣄⡄⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠠⠄⠐⠂⠠⠀⡀⠀⠀⠀⠑⠀⠀⡐⠋⠀⢀⣠⢤⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠀⠀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⢀⣀⠀⠀⠂⠈⠴⣂⠥⣂⠍⡢⠄⠀⠀⠜⠀⠀⠘⠁⠐⠒⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⡼⢩⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠠⠁⠀⠂⠤⢐⡂⠄⡀⠀⠁⠀⠀⢀⠎⠀⠀⠈⠀⠈⠉⠈⠁⠠⠄⠀⠈⠁⡐⠂⠀⠀⠀⠀⡜⣡⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⡀⢀⣀⠀⠤⢈⡑⠂⠅⠂⠤⡑⠪⡑⠀⠀⢀⠊⠀⣀⠨⠀⠀⠀⠒⠒⠒⠀⠀⠀⠉⠁⠀⠀⠀⠀⢀⡜⣰⢿⣿⠁
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢀⠄⢐⡂⠤⢁⠁⠂⠍⡒⠤⡀⠈⠑⠀⠀⠠⠁⠀⠠⠐⠒⠀⠀⠀⠀⠀⠀⠐⠒⠂⠤⠀⠀⠀⠀⢀⢞⣼⣯⣿⡏
⠀⠀⠀⠀⠠⢎⣀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠬⡂⠀⠈⠀⠈⠂⠀⠀⡐⠁⠀⠀⢁⡀⠠⠤⠄⠉⠉⠁⠰⠆⢀⡂⠀⠀⠀⠀⢠⢮⢮⢯⣾⣿⢫⡇
⠀⠀⠀⠀⠠⣮⣌⣉⣉⠒⠢⠤⣀⠀⠤⠀⠀⠀⠈⠑⠄⠀⠀⠀⢀⠜⠀⢀⡄⠰⠀⠤⠤⠤⠤⠀⠀⠀⠉⠁⠒⠀⠀⠀⠀⡰⢣⢣⣣⣿⡿⣣⡻⠁
⠀⠀⠀⠀⢴⢥⡤⠤⢭⢭⣍⣒⡢⠝⡒⣤⡀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠁⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⡱⢳⣿⡿⣱⡗⠁
⠀⠀⠀⠀⢹⠥⡆⢀⡀⠀⠉⠁⠛⠿⠷⢿⣾⣷⣤⡀⢀⠀⡠⠗⠀⠀⣀⣀⣤⣤⡤⡤⡤⠤⣤⣤⣀⣀⠀⠀⠀⠀⠀⡰⡱⡽⣱⣿⡟⣴⠟
⠀⠀⠀⠀⠀⢱⢘⣕⠯⠕⡖⠚⠓⡒⠷⣶⣟⢛⣻⣿⡷⣴⣶⢶⡿⢟⣳⣾⣭⣭⣥⣤⡴⠍⡠⡔⠂⠌⣉⠓⠶⣬⡴⣼⡝⣵⢿⠏⣾⠿
⠀⠀⠀⠀⠀⢯⣙⣨⣭⢤⡴⣶⣶⣮⣷⣾⡿⣿⣛⣻⣛⣛⡛⡻⣿⣿⣟⢛⣿⡿⡭⣭⡴⣒⣠⠀⡉⠛⠓⢦⠴⣠⢜⣟⣾⣿⢋⣾⡟⠄
⠀⠀⠀⠀⠀⡎⠟⢹⣧⣼⣏⠁⣑⡁⠀⢉⠛⠒⠒⠲⠲⠭⢭⠭⠿⠿⠻⠿⢿⠟⣫⣭⣍⣑⣨⣍⣛⣲⣤⣄⣈⣦⣭⣿⣼⣡⣿⣓⣀⣅⣂⣬⣐⡠⢀⡀
⠀⠀⠀⠀⠀⣤⠱⠂⢻⡗⡇⢒⣚⡿⠯⠭⠽⠿⠿⠟⠛⠋⠉⠉⠙⠓⠒⢲⣖⠿⣿⣿⢟⡵⢿⣃⣠⣌⣁⠨⣍⢻⣭⣿⣷⣿⣿⣿⡽⣛⣯⣽⣷⣾⣿⠇
⠀⠀⠀⠀⠀⢿⠀⡟⠸⣏⢏⣥⡤⠁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⣸⣾⡟⣡⣟⠀⢽⡤⣤⡥⣼⣵⠤⣅⣬⣤⣭⣭⣤⣼⠿⠿⣿⢿⠿⠏
⠀⠀⠀⠀⠀⠹⡄⠋⠀⢺⣿⠻⠛⠿⠿⠶⡶⠾⢶⢴⣤⣀⣀⡤⢤⣒⡃⣽⡿⣣⠾⠷⡾⢀⠡⠂⢁⠩⠁⡀⠌⠀⣊⠍⠑⡀⠅⡀⠠⢁⠐
⠀⠀⠀⠀⠀⠀⣣⡀⠀⢀⠽⡬⢴⣖⣂⡤⠾⠾⠛⠻⠿⠋⠉⠉⠛⠛⡱⠶⠚⠉⠉⣼⢁⠀⠀⢄⠂⢉⡤⠶⠖⢃⡠⢤⣂⣤⣂⠜⠊
⠀⠀⠀⠀⠀⠀⠘⣯⢲⢪⢿⠉⠁⠤⠔⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢀⢹⣧⠆⠁⠐⢊⠰⣒⠧⠖⠋⢩⢁⡠⠍
⠀⠀⠀⠀⠀⠀⠀⠙⣭⢿⣽⠀⢒⣒⡊⠉⠀⠀⠀⠀⢀⣀⠀⠀⢀⢀⣀⡀⢒⠐⠉⠡⣟⠷⠀⢊⡉⠠⡐⣪⡤⠆⡅⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣛⡼⠶⠶⣒⢒⢂⣛⣙⣉⠭⠬⠽⠫⠖⠔⠃⢘⠗⡫⢑⠩⠑⡂⠌⢀⠌⢁⠭⢀⠠⢁⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠉⠈⠉⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⡆⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡗⠀⠀⠀⠀⠀⢸⡆⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⢀⠠⠊⢡
⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⠘⠛⠯⣕⣦⡀⢀⡄⣊⣀⣀⡀⡠⠂⠁⠀⢀⡾⠃
⠀⠀⢸⡷⠦⠤⢤⣀⠀⠀⠃⢀⣠⣤⣴⣒⡈⣿⠀⠀⠀⠀⠀⡩⠛⠿⠿⣟⡒⠤⣁⠀⠀⢀⠔⠉
⠀⠀⠺⠿⠿⠷⠶⠶⠿⠷⡾⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠉⠓⠾⡦⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣄⠀⠀⠀⠀⠀⠀⠐⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣬⡓⢤⡀⠀⠀⠀⠀⠀⢑⠥⠆
⠀⠀⠀⣶⣄⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⢠⣞⡂⠀⠙⠲⣌⡒⣤⡦⠐⣈⠤⠒⡀⠒⠂⠀⠠⠤
⠀⠀⢀⠱⡈⠑⢄⠀⠀⠙⠢⣄⠀⠀⠀⣴⣿⠽⣮⡄⠀⠀⠈⠙⠷⠛⡍⠀⣠⠞⠁⠀⠀⠀⠀⢀⡰⡯
⠀⠀⠙⢇⠈⠢⡀⠀⠤⠄⠀⢈⣻⣆⣼⣿⠿⠟⠛⠉⠀⠀⠀⠀⠀⠀⣠⠚⠁⠀⠀⠀⠀⠀⡠⢊⠞⠁
⠀⠀⠀⠀⠱⣄⠈⠀⠰⣾⣻⠿⠟⠛⠫⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠦⢉⣁⠒⠂⠤⢎⠔⠁
⠀⠀⠀⠀⠀⠈⠳⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠉⠉⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⢀⠀⠀⡀⠀⣀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠦⠞⠁⠜⠁⠀⠌⠀⠰⠁⠀⠸⠀⠇⠈⠘⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠊⠁⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡽⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠠⠤⠤⠀⢀⣹⡴⠟⣫⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣴⡿⠋⣠⠤
⠀⠀⠀⣀⣀⣀⣀⣀⡀⢀⣆⢀⡴⠛⠁⠀⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢁⣴⠿⢛⣵⠂
⠀⠀⠈⠉⠉⠉⠉⠉⠛⢻⣿⣾⣶⡭⠄⡀⢀⠀⡀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠛⢋⡡⠖⣉⣤⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠘⠋⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡋⣠⡿⠛⠁⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠛⠉⠈⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣿⣿⢶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⡿⠿⠛⠋⣉⣤⣴⡾⠿⢻⣷⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⠛⠋⢉⣀⣤⣴⡾⠟⠛⢉⣠⣤⣶⣿⣿⠿⠿⠿⣿⣷⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠿⠟⠛⠋⠉⣀⣠⣤⣶⠿⠟⠋⢉⣠⣤⣶⣾⠿⠛⠋⠉⣀⣠⣴⣶⣿⣿⣿⡿⢿
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣾⠿⠟⠛⠉⠁⠀⣀⣤⣴⣶⠿⠟⠋⢉⣠⣤⣶⠿⠟⠛⠉⣀⣠⣤⣶⣾⣿⡿⠿⠟⠛⠉⠁⠀⠀⢸
⠀⠀⠀⣀⣤⣶⡾⠿⠛⠋⠁⠀⢀⣠⣤⣶⣾⠿⠟⠋⢉⣠⣤⡶⠿⠛⠋⢁⣠⣤⣴⣾⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣾⣿⡋⠀⠀⣀⣤⣴⣶⣿⠿⠟⢋⣉⣤⣴⡶⠿⠛⣉⣡⣤⣶⣾⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣿⣿⣿⣿⣿⣿⣟⣋⣭⣤⣶⣾⣿⣿⣭⣴⣶⣿⠿⠟⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣿⡇⠀⠈⠙⠛⠿⠿⠿⠿⠟⠛⠛⢻⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⣿⣤⣀⠀⠀⠀⠀⠀⢀⣀⣠⣼⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⠉⠻⠿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⣿⣿⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢠⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠇
⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⡿
⣿⣿⠇⠀⠈⠉⠛⠛⠛⠛⠛⠛⠛⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠟⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠛⠋⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⡿⠿⠛⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣿⡿⠿⠛⠉⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⢀⣀⣤⣴⣾⣿⡿⠿⠛⠉⠁
⠻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣾⣿⣿⠿⠛⠉⠁
⠀⠈⠿⣷⣦⣤⣤⣤⣤⣤⣶⣾⣿⠿⠟⠋⠁
⣀⣀⣀⣈⣛⣿⣿⣿⣿⣟⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣀⣀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠟⠋⢉⣉⣀⣀⣀⣉⡉⠙⠛⣿⠛⠋⠉⠉⣉⣉⣉⣉⣉⣉⡙⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡟⠡⠖⠊⠉⠁⠀⠀⠀⠀⠀⠈⠁⠀⡟⠀⠀⠊⠉⠁⠀⠀⠀⠀⠀⠈⠉⠀⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢀⣿⣿⠁⢀⠤⠄⠒⠒⠉⠉⠉⠉⠉⠒⠀⢰⡇⠀⠤⠔⠒⠊⠉⠉⠉⠉⠉⠉⠂⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⢀⣀⡠⠤⠤⠤⠤⠤⣀⠀⠀⢸⠁⠀⠀⠀⣀⣀⠠⠤⠤⠤⠤⣀⡀⠀⢹⣿⣇
⠀⠀⠀⠀⠀⣸⣿⣿⠀⠐⠊⠁⠀⠀⠀⣀⣀⡀⠀⠀⠁⠀⡟⠀⠐⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⢀⣿⢧⡇⢀⡤⠔⠒⠉⠉⠉⠀⠀⠈⠉⠓⠀⢠⡇⠀⢀⠤⠔⠒⠊⠉⠉⠉⠉⠒⠒⠦⠄⠘⣿⣿
⠀⠀⠀⠀⣸⡿⣾⠀⠀⢀⣀⡠⠤⠔⠒⠒⠒⠒⠤⠄⠀⢸⠃⠀⠀⠀⣀⣀⠤⠤⠤⠤⠤⠄⣀⡀⠀⠀⣿⣿⡇
⠀⠀⠀⢠⣿⢷⡇⠀⠚⠉⠀⠀⠀⣀⣀⣀⣀⡀⠀⠄⠀⣼⠀⠀⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢻⣿⣇
⠀⠀⠀⣸⡿⣼⠀⣀⠤⠒⠊⠉⠉⠀⠀⠀⠀⠈⠉⠁⠀⡏⠀⠤⠒⠒⠉⠉⠉⠉⠉⠉⠉⠉⠐⠒⠢⠀⢸⣿⣿
⠀⠀⠀⣿⣿⡇⠀⠀⢀⣀⣤⣤⣴⣶⣶⣶⡦⢤⣄⡀⢠⡇⠀⢀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣤⣤⣄⣐⠂⠈⣿⢹⡇
⠀⠀⢰⣿⣿⢁⣤⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣾⣧⣾⣿⣿⠿⠿⠿⠷⠶⠶⠾⠿⠿⠿⣿⣿⣿⣶⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣭⣥⡶⠶⠶⠶⠶⠶⠶⠿⠿⠿⠿⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠿⣿⠇
⠀⠀⠘⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠟⠋⠉⠀⠀⠀⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣦⣄⡀
⠀⠀⠀⠀⢀⣠⣤⡶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⡀
⠀⠀⠀⠀⣿⣿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⠋⠁
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠉⢸⣿⡀
⠀⠀⠀⠀⢹⣿⣯⣿⣿⣷⣽⡻⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⠋⠁⣠⣴⣾⣿⣿⠟
⠀⠀⠀⠀⣿⣿⣿⣿⣷⣿⡻⣿⣾⣝⡿⣦⣄⠀⢀⣤⣶⠿⠛⠁⢀⣤⣶⣿⣿⣿⣿⣯⡀
⠀⠀⠀⠀⣿⣿⣷⡝⣿⣿⣿⣿⣿⣿⣽⡾⣝⣿⠟⠉⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⢿⣿⣿⣿⣮⢿⣿⣿⣿⣿⣿⣟⣿⣿⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿
⠀⢀⣤⣶⠾⢿⣿⣯⣿⣷⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠛⠛⠋⠉⠉⠁⣸⣿⣄⡀
⠀⣿⠿⣦⡀⠀⠻⣿⣿⣿⣿⣿⡿⠟⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⣙⣿⡦
⠀⣿⡆⠙⢿⣦⡀⠙⣿⣿⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣿⢿⣯⡀
⠀⠈⣿⣶⡀⠙⢿⣦⡈⢻⣿⣿⡇⠀⠀⠀⢀⣀⣀⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⡿⠿⠿⢋⣿⣿⠆
⠀⠀⣿⣿⣿⣦⡀⠙⢿⣦⡹⣿⣿⣾⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⠟⠋⠁⠀⠀⣀⣴⣾⣿⡋⠁
⠀⠀⣿⣜⢿⣿⣿⣦⡀⠙⢿⣮⡛⠛⠉⠉⠁⣀⣤⣶⠿⠛⠉⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢻⣿⣎⢿⣿⣿⣿⣦⡀⠙⢷⣄⣠⣶⠿⠛⠉⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠈⢻⣟⢯⢻⣿⣿⣿⣿⣦⡀⣿⠋⠁⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⣤⣶⣿⣿⣮⢧⡻⣿⣿⣿⣿⣿⣿⣆⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⠉⢉⣿⣷⣦⡀
⣼⣿⠿⣿⣿⣿⣧⢳⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠛⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣦
⣿⣿⢷⡈⠻⣿⣿⣷⡹⣼⡟⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿
⣿⣿⢦⡙⢦⡘⢿⣿⣷⣽⡇⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⠇
⢻⣿⣦⡙⢦⡙⢦⡙⢿⣿⣧⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⢰⣿
⠘⣿⣯⡳⣌⠻⣄⠻⣆⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠸⣿
⠀⠹⣿⣷⣌⠳⣌⠳⣌⠳⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷
⠀⠀⠈⠻⣿⣷⣌⠳⣈⠳⡌⠳⣌⠻⣿⣿⣿⣿⣿⣿⣿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟
⠀⠀⠀⠀⠘⢿⣿⣧⡙⢧⡘⢧⡘⢧⡘⢿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣦⡙⢦⡙⢦⡙⢶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡙⢦⡙⢾⠁⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣌⠳⣼⡂⠀⠀⠀⢀⣠⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣼⡇⢀⣤⣶⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⠟⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⣀⠤⠄⠒⣈⡡⢤⡐⠧⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⣚⠔⠁⠈⠑⠢⢀
⡔⠻⡤⣐⠾⠟⠊⣉⡠⠴⡖⠀⠀⠀⠀⠀⡠⡔⠒⠊⠉⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠉⠒⢄⡀
⡇⠀⡟⠉⢱⠂⠉⠀⢀⠴⢂⣁⣠⠀⠀⠀⣄⠑⣄⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠈⠒⣤
⡇⢀⡇⠀⢸⠀⠀⢰⡵⢉⣙⣯⡓⠀⠀⠀⠌⢢⡈⢢⡀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⢀⡠⠖⡻⣅
⡧⣼⢱⠀⡏⠀⣎⡉⠋⠀⠀⠉⠉⠓⢤⡀⢧⢀⡱⣄⠑⣄⠀⠀⠀⠀⠀⠀⠀⣀⣙⡎⢁⢔⡮⠊⠉⠲⢄
⠀⠘⠻⢤⡷⡚⢿⡨⡑⠦⣀⠀⠀⠀⡠⠜⠾⠣⣀⠈⢦⡈⢣⠤⠴⠒⠒⢉⡩⢅⡿⢮⡒⠁⠀⢀⣠⠔⡏
⠀⠀⠀⠀⠻⡫⡑⠪⣮⣄⠀⠙⣲⢉⡠⣶⠟⠢⣈⠑⢤⡑⢿⣂⣀⠤⠝⠒⠉⢁⣠⢴⣋⠴⠚⢉⠤⣪⠝
⠀⠀⠀⠀⠀⠈⠑⢕⡄⠉⢻⡖⢚⡭⠚⢇⠙⠢⣀⠑⢤⡈⠲⢄⡀⠀⣀⡤⠒⢉⢔⡮⢻⡤⣒⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠐⠓⠁⠀⠀⠀⠑⢤⡈⠓⢤⡈⠲⣄⢉⡎⠁⠀⣪⣶⠵⢊⡿⡊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠈⠳⢄⡉⢓⡶⠟⠉⡠⢔⠵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⢸⠑⠀⢔⡭⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣿⣿⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⣿⣿⠿⠟⠛⣉⣡⣴⣾⠿⠿⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⡿⠿⠛⠋⠉⣁⣤⣴⣶⠿⠛⠋⣁⣤⣶⣿⣿⣿⣿⠿⣿⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⠿⠟⠛⠋⠉⢀⣀⣤⣶⡾⠿⠛⠉⣁⣠⣤⣶⡿⠿⠛⠋⠉⣁⣠⣤⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⠿⠟⠛⠉⠁⠀⣀⣠⣴⣶⡿⠟⠛⠉⣀⣤⣴⣶⠿⠛⠋⠉⢀⣀⣤⣶⣾⣿⣿⠿⠿⠛⠋⠉⠁⠀⣿
⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⠛⠉⠁⠀⢀⣀⣤⣴⣾⡿⠟⠛⠉⣀⣤⣴⡾⠿⠛⠉⢁⣀⣤⣴⣾⣿⡿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⣾⡟⠛⠉⠀⠀⢀⣠⣤⣶⣾⡿⠟⠛⢉⣀⣤⣶⠾⠟⠛⢉⣀⣤⣴⣶⣿⠿⠿⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣿⣿⣷⣤⣴⣶⣿⡿⠟⢋⣉⣠⣴⣶⡿⢛⣋⣡⣤⣶⣶⡿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠀⠀⣿⣿⠉⠛⠿⢿⣿⣶⣿⣿⣿⣿⣿⠿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢀⣿⡇⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣿⣷⣤⣀⣀⣀⣀⣀⣀⣠⣤⣶⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⠉⠙⠛⠿⢿⣿⣿⣿⣿⡿⠿⠿⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿
⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⢀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⣿⣿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⡇
⣿⣿⡏⠉⠛⠻⠿⠿⣿⣿⠿⠿⠿⠿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣿⣿⠿⠋
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⠿⠛⠋⠉
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠛⠋⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⠿⠛⠋⠁
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣀⣤⣴⣶⣿⡿⠿⠛⠋⠁
⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣠⣤⣶⣾⣿⣿⠿⠛⠉⠁
⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣿⣿⣿⠿⠟⠋⠁
⠀⠀⠙⢿⣿⣷⣶⣶⣶⣾⣿⣿⡿⠟⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢤⠒⠶⠶⠒⠒⠦⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠂⠁⠀⠀⠀⠀⠀⡀⠀⡄⠀⠀⠠⠈⠟⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠭⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢠⠶⠱⠀⠐⠈⠁⠀⠀⠀⠉⠁⠀⢀⠀⠀⠀⠀⡀⠀⠀⠹⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⠓⠂⠸⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⢄⡠⠐⡄⠠⠆⠀⠂⠐⠈⠀⠀⠀⠁⠈⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⠒⠆⠀⡇⠀⠀⠀⢀⠊⠃⠁⠁⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡇⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⣷⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠰⠀⢀⣀⡀⢀⣤⣄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⣒⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⢀⡀⢀⡄⢠⡶⠤⠰⢖⣶⠘⣟⢸⡇⣿⠉⣿⢸⡄⢻⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠍⠀⠸⠀⠀⠀⠀⢰⣛⣦⢾⠉⣻⠈⣇⢘⡧⠹⣆⡴⠸⠯⠿⠇⠛⠂⠃⠈⠉⠁⠈⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣤⡄⢠⡄⠀⠀⠀⠘⠷⠗⠈⠛⠛⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡀⠠⠄⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣂⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⢀⠀⠠⠆⠠⠤⠀⠔⠀⠂⠀⠀⠀⠂⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠅⠀⣧⠀⠀⠈⡄⠀⠀⠜⠈⠈⠁⠁⠈⠀⠀⠀⢀⢀⢀⠀⠀⢀⠀⡄⠀⠀⠀⠐⠀⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣍⡁⢿⠀⠀⠀⠈⢆⠀⢠⡀⠀⡄⠀⠐⠰⠘⠁⠀⠀⠀⠀⠀⠀⡈⠀⠀⢀⠀⢀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡒⠂⢸⠀⠀⠀⠀⠀⠱⢄⠀⣀⠀⣀⢀⢀⠀⠀⡄⠠⠐⠀⠀⠀⠃⠀⠀⠈⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠷⠆⢸⡇⠀⠀⠀⠀⠀⠀⠙⠢⣈⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣁⣄⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠢⠤⠤⣤⣤⣤⣤⣤⣴⣶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢒⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣧⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠭⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣸⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⣤⣤⣤⣤⣤⣤⣤⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣤⣤⣶⣦⣤⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣤⣤⣤⣤⣴⣴⣴⣶⣴⣦
⣿⣿⣇⣀⣀⣀⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠄⠀⠀⠀⠀⠀⠁⠢⢀⡀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠐⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠊⢀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢎⡎
⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⠤⠀⠀⠀⠐⠀⠀⠀⠀⢠⢢⣾⡗
⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⡀⠀⠐⠠⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣣⣿⣿⡁
⠀⠀⠀⠐⠠⠤⢀⡀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠈⠁⠀⠀⠀⠀⡐⣿⢯⣾⣿⡿
⠀⠀⠀⣈⣋⣒⣒⣂⠬⢔⡠⢁⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⣼⢿⣿⢏⡾⠁
⠀⠀⠀⠹⠠⠀⣈⠈⠉⠛⠺⠿⣿⣦⣄⠀⠀⠴⠀⢀⣀⣠⣤⡤⡤⠤⢤⡄⣀⡀⠀⠀⠀⢜⡾⣡⣿⢫⡞
⠀⠀⠀⠀⠥⡿⠞⠚⠈⠉⢙⣻⣯⣭⣿⣿⣾⣾⣿⣯⣶⣶⣶⣶⠶⠒⠒⠂⢀⡈⢙⠲⢼⡾⣱⡿⣣⡟⠁
⠀⠀⠀⠀⢷⢶⣏⣹⠟⠛⠋⠹⠯⠝⡶⣓⣒⣹⣽⣿⣷⣿⡭⣟⣻⠕⢒⠶⢄⣈⠉⢊⡹⣿⡟⣰⠟⠀⠀⠀⢀
⠀⠀⠀⠀⡄⠄⢿⢿⠀⢜⢾⠢⠵⠿⠷⠶⠆⠁⠑⠒⠦⠄⢾⣿⣿⡫⠗⠦⠄⡈⠍⠲⣕⣟⣼⠋⠀⠀⠀⠠
⠀⠀⠀⠀⡇⢸⠘⣿⣬⣖⣀⡀⠤⢀⠀⠀⠀⠀⠀⢀⡀⣧⡾⣫⣮⡁⠀⠀⡀⠬⠑⠢⢼⡾⠁⠀⠄⠐
⠀⠀⠀⠀⢸⠀⠀⠙⣇⢉⠉⣉⣉⣽⡽⣿⠶⠶⠭⡿⣐⣫⠾⠍⡝⠁⠀⠀⠀⢀⣀⢤⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢷⢤⣾⡟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⣧⡄⠀⠀⠀⡡⠤⠚⠊⠉⠉⠉
⠀⠀⠀⠀⠀⠘⢧⢿⡇⠐⠴⠈⠀⣀⢀⣀⢀⡀⠀⣀⡀⢀⣀⢉⣸⠛⠀⠈⠀⡀⠤⠐
⠀⠀⠀⠀⠀⠀⠈⠛⠭⠍⠩⠤⠀⠓⠒⠒⠒⠉⠉⠈⠀⠉

⠀⠀⠀⡆⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠲⠀⡖⠂⠰⠒⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢲
⠀⠀⠈⡏⠉⡍⢁⠀⠀⢤⠀⠀⠀⠀⠀⣶⠾⠰⠆⡇⠶⠰⠰⢲⡆⠀⠀⠀⠀⠀⠈⡏⢉⠍⢅⠀⠀⠀⠀⠀⠀⠀⠘⠙⠀⡠⠀⠸⣀
⠀⠀⠀⡇⠸⣶⡶⠂⠀⠀⡀⠀⠀⠀⠀⣿⠘⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠠⡇⠸⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣤⣀⣀⣀⡇
⠀⠀⢠⣧⣤⣤⣤⡄⢾⡿⡧⠀⠀⠀⠀⠿⠚⠛⠒⠛⠓⠚⠛⠺⠇⠀⠀⠀⠀⠀⢠⣧⣤⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠠⠙⠿⣿⡿⠟⠁


⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⢰⠏⠁⠉⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⡏⠈⡇⡉⠈⡇⠀⠀⠀⠀⠀⠀⢀⡔⠉⣠⡄⠉⢢⡀
⠀⠀⠀⠀⠀⣸⣦⣄⣤⠟⠀⠀⠀⠀⠀⣏⠀⠀⢠⠦⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣷⣶⡇⣆⣀⡇⠀⠀⠀⠀⠀⠀⢼⠐⠳⠌⠁⠻⠇⡇
⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠈⠀⠒⠁⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠘⣄⠺⠆⠸⠗⣠⠃
⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠋


⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⣠⣶⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⣦⣤⢛⠁
⠀⠀⠀⢠⣾⢿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣛⡋⢙⡇⠀⠀⠀⠀⠀⠀⠀⠐⠢⠬⢿⣿⠧⠄⠆
⠀⠀⠠⡜⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣭⣽⣿⣿⣥⣄⠀⠀⠀⠀⠀⠀⠀⠰⠀⢡⡉



⠀⠀⠀⢀⣔⠉⠟⠟⣑⡄⠀⠀⠀⠀⠀⠀⡀⠀⡭⠀⣀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠈⠉⠀⠀⠀⡅⠁⠀⠀⠀⠀⠀⠀⠁⠉⣄⠀⠈⠦⣐⣂⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢀⣁⣀⣀⡀⢠⣤⣴⡀⠀⠀⠀⠀⠀⠀⠚⠀⠈⣄⢀⠨⠄⡀⠀⠀⠀⠀⠀⠀⢠⣷⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⠂
⠀⠀⠈⠻⠿⠛⠀⠾⠿⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠤⠾⠿⠷⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠠⡌⢠⣤⢠⣥⠐⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠃
⠀⢠⠀⠈⠐⠀⠀⠀⢐⠀⢀⢰⠘⠃⣠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠀⠀⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢀⡀⠐⠀⠀⠀⠀⠈⠈⠀⠁⠀⠉⠀⠀⠀⠀⠀⠐⠃⠸⠄⡀⠠⠀⠀⠀⠀⠀⠐⠃⠀⠇⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠘⠇⠀⠐⠀⠀⠀⠀⠰⠀⠀⢤⡤⠀⡄⢀⠀⠀⢀⡆⢰⡂⠀⠀⠀⠀⠀⠀⠀⠀⡂⢀⡄⠀⠀⠦
⠐⠄⡀⠀⢀⠀⠈⠀⠀⠢⠀⠁⠀⠀⠌⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂⠴⠀⠆⢴⢤⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠢
⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢠⠀⡄⠀⠀⠀⠀⠀⠀⡠⡄⠀⠰⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠶⠂⠀⠀⠐
⠀⠀⠀⣀⠀⠀⠤⠀⠷⠀⠀⢀⡀⠀⠀⡇⠀⠀⢀⡀⡀⠀⠀⠀⡇⠀⠀⠘⠀⠀⠀⠀⠀⢰⠀⠀⠀⠓⠤⠤⠀⠀⠀⢀
⠀⠀⠈⠁⠀⠀⢰⡆⠀⠀⢀⡌⠀⠀⢠⢣⠤⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⠄⡀⠉⢁⡀⠄⠀⠠⠀⡀⠀⠀⠀⠀⠀⠸⣷⠀⠀⠀⠈⠃⠀⠀⠀⠀⡰
⠠⠀⠀⠀⡀⠀⠀⠃⠀⠀⢀⠀⠀⠀⠸⣄⣤⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠋⠁⡀⠀⠙⠄⠀⠀⠀⠀⠈⠳⠀⠀⠀⠀⠐⠀⠀⣄⠀⠠⢠⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂
⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠀⠘⢿⠂⠀⣼⠀⠂⢠⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁⠀⠀⠀⠀⠀⠄⠋⡀⠆⠑⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠃
⠤⠤⠀⠀⠠⠆⠀⠀⣤⠐⢀⠀⠘⣿⣤⠀⠀⠀⠀⠀⠀⠶⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠄⠀⢠⠄⢄⣀⡀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⡎⠂⠀⠀⠀⠀⠀⠠⠠⠐⠒⠒⠂⠈⠁⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⣀⡼⠁⠀⠀⠀⠀⠨⠆⠸⠠⠅⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂⠀⠀⣾
⠠⠤⠖⠀⠀⠈⠀⠀⠰⢤⣠⣄⠤⠚⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⠀⠀⠀⠆⠂⠀⡄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣄⠀⠀⠀⠀⢰⡀⠀⡀⠀⠀⠀⠁⡀⠀⠀⡁⠃⢀⠠⣭⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢁⠐⠀⠂⠀⠀⠀⠄⠀⠰
⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠦⠸⠄⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⢠⠀⠀⠀⡆⢀⠀⠀⠔⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠸⡦⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠱⠠⠤⠶⠇⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠆⠀⠠⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠚⠀⠀⠘⠒⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠰⠶⠃⠀⠀⠀⠀⠀⠀⠢⡈⠀⠀⠀⠀⠁⠀⠀⠠⠀⠀⠀⠀⠀⠀⠂
⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠒⠓⠒⠦⠀⠀⠀⡀⠀⠀⠀⢠⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠐⠁⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠈⡷⢸⠀⠀⠀⠀⠀⢠⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠐⠄⠀⠀⠘⠀⠀⡄⢀⠀⢠⠀⠀⢠⡄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⢀
⠅⡁⠀⠂⠀⠀⡀⠀⠘⠄⠀⠀⠀⠀⠀⠁⠀⠐⠦⠀⠀⠀⠀⠂⠀⠀⠇⠈⠠⠆⠀⠉⠠⠅⠀⠀⠀⠀⠀⠁
⢐⡦⠀⠀⡀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢤⡄⠀⠀⠀⠀⠀⠀⠀⢠⠐⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠂
⠀⠛⠀⠁⠓⠀⢀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⠀⢀⡁⠒⠀⠀⠀⠀⠜⠈⠀⠀⠀⠀⠀⡀
⠀⡄⠀⠤⠀⠀⠈⠄⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠖⠀⠀⠀⠐⠦⠀⠀⠀⠠⠀⠀⠀⢸⠔⡇⠀⡇⠁⠀⠤⠤⠤⠤⠤⠤⠤⠤⠴⠴⠭⠤⠤
⠀⠁⠀⠀⠀⠀⠀⡀⠀⠠⠂⠂⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠛⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⣤⡀⡀⠀⢀⣾⠛⠛⢤⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡿⠙⠃⠀⠈⠁⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡿⠿⣇⠂⠀⠀⠀⢀⡀⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⠀⠈⡅⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⢱⡀⠀⠀⢀⡐⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⢀⠀⠀⢐⡏⠉⠲⠄⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⠀⠀⠙⢦⠸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⢺⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣾⠀⠀⠀⠘⣃⣀⡀⠀⠀⠀⠀⠀⢀⣤⠶⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠛⠛⠋⢹⣿⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣩⣶⡿⣆⣀⣀⣀⣀⣀⣠⣤⣤⣤⠶⠶⠶⠄⠀⠀⠠⡔⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠠⠂⢀⣭⣭⣭⣵⣾⣿⡿⣛⡛⢛⣉⡭⠿⣛⣫⡄⠀⠀⠀⠀⠀⠀⣗⡀
⠀⠀⠀⠀⠀⠀⡀⢀⣠⣀⠀⠀⠀⢱⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⢀⣴⠉⠁
⠀⠀⠀⠀⠔⠫⠒⠕⣫⡵⢛⣥⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠏⣠⣤⣶⡿⡿⠟⠒⠂⠤⢤⣤⡤⢤⣤⣤⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠉⠈⠉⠉⠙⠺⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠿⠿⣅⣀⠸⣤⡿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⠖⠒⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡇⢸⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⠀⡴⠋⠀⢸⢸⠀⢉⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡇⡇⢸⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⣸⠁⠀⠀⢸⠘⢚⡥⠖⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢹⢸⠀⡇⡇⢸⠀⠀⠀⢀⢇⡀⠀⠀⣿⡀⢠⡀⠀⢣⠀⠀⠀⢹⠀⠀⢹⢀⣀⠀⠸⠚⠉⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⢹⡇⡇⢸⢸⠀⡇⡇⢸⠀⠀⠀⢸⢸⠁⠀⢠⡧⡵⣌⡏⢶⢬⣇⠀⢸⠈⡇⠀⠀⡏⠉⡇⠀⠀⠀⢀⡰⠋
⠀⠀⠀⠀⠀⠀⣀⣧⣸⣇⣇⣸⣸⣀⣇⣇⣸⣀⠀⠀⢸⢸⠀⠀⡾⠰⠀⠈⠉⠀⠆⢹⠀⢸⠀⡇⠀⣠⠕⠸⢵⡒⠒⠚⠉⠀⠀⠀⢀⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⡆⢸⡇⠀⠀⠠⠄⠀⠀⢸⡆⢸⢀⡇⠀⠛⠚⠒⠚⠃⠀⢀⣀⠀⠀⠀⢸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢤⣄⠀⠀⠀⠀⠀⠳⡙⣾⠙⢤⡀⠻⠟⢀⡴⢋⡹⣼⡞⠀⠀⠀⠀⠀⠀⠀⢀⡷⠾⢤⡀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡄⠸⠬⢷⠀⠀⠀⠀⠀⠈⠳⠞⢦⡏⠑⠊⢹⡤⠚⠓⠙⠀⠀⠀⠀⠀⠀⢠⢴⡏⠀⠀⠀⠹⡽
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⡇⢐⣒⣒⡂⢸⠀⠀⠀⠀⠀⣀⣠⡴⢻⠁⠀⠀⣈⣟⠦⣀⣀⡀⠀⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⢀⡞⠉⠀⠈⠙⡇⠈⠭⠭⠅⢸⠀⠀⡰⠋⠉⠁⢼⠀⡏⠀⠀⠈⠀⢸⠀⣹⠈⠉⠓⣄⠀⠀⠀⣤⠤⠤⠤⠤⠤⠤⣤⡤⢤⡄
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠷⠶⢶⠶⠖⠚⠀⠀⡇⠀⠀⢀⡜⠁⡗⠒⠒⠒⠒⢻⠈⢣⡀⠀⠀⢸⠀⠀⠀⡷⠒⠒⠒⠒⠒⠒⠛⠓⠲⡇
⠀⠀⠀⠀⢠⠜⠃⠀⣄⢸⢀⡄⠀⠸⢤⠀⠀⠀⢸⠁⠀⠀⠀⡇⠀⡇⠀⠀⠀⠀⡏⠀⡘⠀⠀⠀⠘⡃⠀⠀⡇⠀⠀⠠⡏⠉⡆⠀⠀⠀⡇
⠀⠀⠀⠀⢏⠀⠀⠀⠈⠛⠋⠀⠀⠀⠀⡇⠀⠀⡼⠀⠀⠀⠀⢸⡀⢹⠀⠀⠀⢠⠃⣰⠃⠀⡆⠀⠀⡇⠀⠀⡇⠀⣀⣀⣉⣉⣓⣀⡀⠀⡇
⠀⠀⠀⠀⠈⠳⠤⠤⠤⠤⠤⠤⠤⠤⠚⠁⠀⠀⡇⠀⠀⠀⠀⠀⢧⠈⡇⠀⠀⡼⢠⠏⠀⠀⡗⠀⠀⢻⠀⠀⡇⠀⠭⠭⠭⠭⠭⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠘⡆⢻⠀⢰⠇⡞⠀⠀⢰⡇⠀⠀⢸⠀⠀⠳⠶⠶⠶⠶⠶⠶⠶⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡀⣸⡀⣀⣀⣸⣸⣆⣜⣼⢀⣀⣀⣸⡀⣀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⢸⡇⠉⠈⠁⠉⠉⠈⠁⠉⠁⠁⠉⠁⠉⠉⠉⠉⢹⡏⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣏⡀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢤⣘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣾⠯⠥⢜⣦⣠⠎⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠣⣿⡹⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠉⢹⣿⣶⠟⠁⠀⠀⢀⡧⣙⣯⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠙⠚⠻⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠸⠤⠤⠤⠴⠋⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⢀⣀⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⣀⣴⡟⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⡏⠁⠀⠀⠀⠀⢠⠀⠀⠀⠀⢀⣤⣿⠂
⠀⠀⢢⠀⠀⠾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣿⣅⡀⠀⠀⠀⠀⡌⠀⠀⠀⠀⣿⠟⠁⡆
⠀⠀⢀⠱⡤⠀⠰⢖⡥⢐⠀⠠⠀⠀⠀⠀⣉⣭⣉⣭⣉⠀⠀⠀⠀⠑⠀⠀⠀⣄⠀⠀⠐⠁⠀⠀⢀⣷⣶⣤⣄⣦⣠
⠀⠈⢂⡩⠝⠒⠪⠥⠊⠁⠀⠀⠀⠀⠀⣸⡿⠛⣿⠛⢿⣇⠀⠀⠀⠀⠀⠀⠴⠿⠦⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣟⠿⣡⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⣹⣭⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣿⣿⡟⢿⡿⣹⠖⣺⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢀⣁⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣿⣿⣿⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣼⠏⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢿⣤⣾⣿⣷⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣾⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣿⣿⣷⠶⠄
⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣻⣛⣿⣟⣛⣿⣟⣻⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠿⠀⢃
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣩⣀⣏⣉⣇⣸⣁⣹⠀⠀⠀⠀⠀⠀⠀⢸⣿⣻⣶⢶⣶⡶⠿⣻
⠀⠄⠄⢤⣿⡧⠤⠠⠄⠀⠀⠀⠀⠀⠀⠀⠣⡼⢦⢧⢤⢧⠾⡤⠃⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠛⠾⠛⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠙⠛⠋⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣦⡀⠀⢸⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣯⣹⣿⣿⡿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣽⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⢤⣤⣄⠀⠀⣀⣸⣿⣿⣿⣿⣤⠀⠀⠀⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣦⣤⣈⣴⣿⡿⢿⣿⠟⢻⣿⠿⣿⣿⣧⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣿⣿⣿⣿⡛⠛⣛⣫⣤⣶⠁⠀⢿⣿⣿⣿⣿⡿⠻⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⢸⣿⣿⣷⡿⠛⣛⠛⢻⠇⢁⣀⢀⣈⣿⣿⣿⣿⡇⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠸⣿⣿⣿⠗⠖⠉⠉⣸⠃⣿⡉⠈⣉⣉⣩⣿⣿⣇⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⣿⠛⣿⡦⠞⠛⠁⠈⠀⣿⣇⢀⡀⠈⠛⠛⣿⣿⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⣿⣿⣧⣤⡞⠛⠁⢾⣆⣸⣿⡦⠀⠀⠛⠢⣬⣻⠄⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⣾⣷⣟⣉⣸⠷⠀⢂⣼⣿⣹⣿⡆⠀⠀⠻⠦⣈⣿⡄⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢹⣿⣾⣽⡿⠀⠀⠀⣨⣿⣿⣯⣄⡀⠀⠲⢦⣹⣿⠃⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⡿⠯⠀⢀⡘⠛⣽⣿⣿⣿⣝⠛⡀⠰⠆⠀⠿⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠸⠷⠶⠴⠛⠛⢉⣉⣿⣿⣟⠉⠙⠛⠢⢤⡴⠆⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣿⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⠀⢀⣀⣄⣀⠀⠀⠈⣻⣿⣧⡟⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⢻⡏⠀⠀⠀⣴⣿⣿⠿⢿⣷⣄⠈⣿⣿⣿⣿⣠⣶⣿⣿⣷⡆⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠃⠀⠀⠀⠀⠰⣟⠻⠟⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠣⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠟⠁⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⡶⠤⠛⢿⣿⣿⣿⣿⠿⠻⠿⣿⡿⠆⠀⠀⠀⠀⠀⢸⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢀⣾⣷⣾⡿⢷⣴⣿⡎⢹⣿⣿⣿⠶⣤⣾⣿⣷⣾⢆⣀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⡄⠀⠀⠈⣿⣿⣿⡁⠀⢿⣿⣇⣰⣿⠿⠋⠀⢻⣿⡏⠈⢿⣾⣿⡆⠀⢰⣾⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡇⠀⠀⠀⠘⣿⣿⣿⠀⠈⠿⠿⠋⠀⠀⠀⠘⠿⠋⠀⠀⣾⣿⡿⠁⢀⣾⣿⠃
⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⠇⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣠⣿⣿⡿
⠀⠀⠀⠀⣤⠟⠁⣿⠈⠙⣿⡄⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠃⢠⣿⡇
⠀⠀⠀⠈⠁⠀⣾⠉⣿⡇⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⣿⡟
⠀⠀⠀⠀⠀⢠⠛⣤⠘⠃⢸⠃⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢰⣿⠃
⠀⠀⠀⠀⠀⠙⠀⢹⠀⠸⠇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠆⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣀⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡏⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⣿⡇⠀⠀⠀⠀⠀⣠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⣿⡇⠀⠀⠀⠀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⣰⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⢠⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆⠀⢀⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣤⣾⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⣿⠿⠿⠿⠿⠿⠿⠿⠿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠉⠉⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠑⠠⠑⢀⠀⡄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠠⠀⢐⠀⠂⠀⡠⠀⠁⠀⠁⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠠⠤⠄⠀⠰⢁⠠⠀⠀⠂⠀⢀⠀⠜⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠠⠠⠂⠈⠁⠀⡇⠨⠭⠍⠀⠀⠂⠀⠀⠀⠂⠈⠀⠁⢄⠀⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠊⢁⠁⡀⠀⢀⡈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠂⠄⠄⠂⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢈⠤⠤⠄⢘⠃⠀⠘⡃
⠀⠀⠀⠀⠀⠐⠁⠀⠉⠀⠈⠂⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⢀⠄⠂⠀⢸
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠤⠠⠤⠀⠠⠄⠤⠄⠀⠀⠀⠀⢀⠘⠀⠀⠀⠀⢠⠀⢠
⠀⠀⠀⠀⠀⠈⠀⠀⠉⠀⠀⠁⠀⠊⠀⠒⠀⠘⠀⠒⠂⠀⠀⠀⡆⠙⠑⢄⠀⠀⠀⠀⠅⠄
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠐⠁⠀⠀⠁⢀⡀⠀⢠⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠠⢀⠄⠀⡀⡠⠠⡀⠀⢈⡁⡀⣉⡄⡄⡄⠀⠀⠀⠄⠃⠣⠜⡠⠠⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠈⡀⠁⡀⠀⠀⠀⠁⠀⠁⠄⠀⢠⡀⠄⠀⠀⠑⠤⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠅⠁⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠣⠄⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠒⠒⠁⣇⠈⠀⠀⠀⠀⠀⡀⠀⠑⠂⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠉⠉⠁⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⢸⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠻⣾⡇⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⣼⣧⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠉⠀⢹⣆⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢿⡄⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⣷⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠻⣇⢀⣀⣿⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣟⢉⣩⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉


⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⡇⢀⣴⢀⣿⣿⣿⣿⣆⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣾⣿⣿⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⢹⣿⣿⣿⠄⠉⠛⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠈⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⢴⣶⣿⡇⠀⠠⠆⠄⠤⠀⡀⣾⣤⣤⡄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠋⢸⣏⠶⠒⠀⣦⡀⠠⢽⣿⠃⢉⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠈⠛⠁⠀⠀⠀⠀⠀⠙⠛⠀⠘⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢄⣀⣠⣄⣀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠊⢰⣌⠉⣉⡶⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⢠⡈⠉⢉⣠⠤⠖
⠀⠀⠀⠀⠀⢀⣠⣄⢀⡀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠛⠶⢂⡉⠀⠈⠀⡶⠞⠛⠦⣄⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣠⣀⡀⣀⡀
⠀⢠⡆⠀⠩⠓⠲⠶⠶⢖⣒⠂⠉⠛⢦⣄⣀⣀⡤⢊⣀⠐⠶⠦⠖⠿⠛⠋⠛⠴⠴⠶⠂⣀⠁⢤⣀⢀⣠⣴⡚⠉⠀⣒⡲⠶⠶⢶⠚⠯⠁⢢⡄
⠀⠀⢡⡀⣀⡀⠀⢀⣠⢤⡌⢉⡲⠞⢛⡦⣍⣙⠓⠻⠤⠴⠖⠒⣧⣼⠶⠶⢲⡶⠒⠶⠶⠴⠷⠚⢛⣉⢴⡛⠷⠖⡋⢉⣤⣄⡀⠀⢀⣀⢀⡌
⢀⡴⠆⠉⠉⠙⢶⠉⠀⠀⢈⠁⠀⣠⠋⠰⠉⡙⠳⣄⠀⠀⠶⠞⠉⠉⠛⠛⠉⠉⠳⠦⠀⠀⣠⠞⠟⠉⠦⠙⣆⠀⠈⡉⠀⠀⠉⡷⠋⠉⠉⠱⢤
⢸⡇⠀⠀⠀⠀⢸⠋⠀⠰⠁⣠⠴⣿⡀⣇⠤⠗⢒⠉⠙⠲⣆⡀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⢙⡂⠰⠤⣰⠂⣿⠣⣄⠈⠆⠀⠘⠏⠀⠀⠀⠀⢀
⠈⢧⠀⠀⠀⣸⢿⠀⠀⠀⢰⢃⡤⠜⣷⣈⠲⣶⠀⠲⠴⠀⠈⢱⡀⠀⠀⠀⠀⢀⡞⠁⠀⠶⠖⠒⣸⠶⢃⣼⠣⣤⡘⡆⠀⠀⠀⡿⣦⠀⠀⠀⡼
⠀⠀⡆⠀⠸⠃⠀⠳⣄⠀⡎⢾⡵⠟⣀⠭⡷⠢⣍⣉⠻⠉⠛⠀⠷⠤⠴⠦⠤⡾⠀⠙⠁⡎⢉⣉⠥⢾⡍⣀⠻⢮⡷⢺⠀⢠⠜⠁⠘⠇⠀⢠⠁
⠀⠀⠇⠀⠀⠀⠀⠀⠘⠀⣷⣌⠷⣎⡁⠔⠛⢂⡀⠀⠛⠃⠀⢈⣷⠀⠀⠀⠀⣾⡁⠀⠘⠛⠀⢀⡐⠚⠆⢈⣡⡾⢃⣾⠀⠇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢰⠀⠀⠀⠀⢰⣷⠸⡙⢷⣌⡓⠾⠉⠀⠀⠠⢤⣤⡌⠛⣁⠀⠀⠀⠀⢈⠛⢡⣄⡠⢄⠀⠀⠉⠶⠞⣡⡴⢋⠃⣾⡄⠀⠀⠀⠀⡆
⠀⠀⠀⠸⠀⡇⠀⠀⣼⢹⡆⢱⡊⣉⠿⢲⣄⠉⠙⠛⢂⣁⠰⣦⣥⠀⠀⠀⠀⣼⡴⠆⣈⡙⠛⠛⠉⣠⡴⠿⣍⠑⡮⢡⡏⢷⠀⠀⠠⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠸⣽⣦⠙⢶⡞⠁⠀⠠⠶⠞⢋⣉⣡⡶⢣⡄⠀⠀⢠⡜⢶⣍⣉⠙⠲⠶⠅⠀⠉⢓⡴⠋⣠⣿⠇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢛⡀⢓⣄⡉⠛⠁⠴⣀⣈⠠⠔⣉⣴⠇⣀⠀⠀⢀⠸⣦⣉⠂⠄⣁⣐⠤⠌⠛⠋⣠⡚⢁⡻⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡜⢻⣄⠙⣁⠀⠙⢡⢤⣠⣩⡅⠖⢫⡾⠳⣿⠇⠀⠀⠸⣿⢎⢷⡝⠂⢀⣉⣀⣤⡄⠛⠠⡈⠋⢀⡎⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠉⢶⣌⠒⢮⣀⡔⠋⠁⣀⠴⣋⡞⡲⠃⠀⢀⡀⠀⠘⢞⠳⣝⠦⣀⠈⠙⠢⣀⡱⠖⣡⡴⠋⢧⠄⠀⠀⢰⡄
⠀⠀⠀⠀⢀⡄⠀⠀⠀⢰⡙⠇⢈⠿⠦⣀⠐⠀⣀⠀⠘⣩⠟⡀⢟⣴⠉⠙⢦⡹⢀⠻⣍⠃⠄⣀⠐⢂⣈⠴⠾⡉⠸⢏⡜⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⢳⡤⠑⢤⡔⢉⣉⠛⢁⣠⠞⡵⢺⠏⣼⠧⠶⠶⠤⣦⠰⡗⢬⡳⢄⡈⠛⣉⠙⢢⡤⠚⣡⠞⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⠇⠀⠀⠀⢸⡌⢦⠙⢠⡈⡹⠉⠋⣉⠔⠛⣰⢃⡼⡇⠀⢀⡀⠀⠘⢣⡙⣆⠛⠦⢉⠑⠈⢋⢁⡔⠛⣤⢃⡇⠀⠀⠀⠘⠇
⠀⠀⡀⠀⢰⠀⠀⠀⠀⣾⠟⢄⠳⣅⡸⠛⠛⠋⠀⣠⠞⠁⣼⠃⣾⠋⠉⠉⠉⣱⠈⣷⠈⠳⣄⠀⠙⠛⠛⠇⣈⠞⣡⠞⣷⠀⠀⠀⠀⡆⠀⢀
⠀⠰⡇⠀⣼⠀⠀⠀⠀⡟⡆⢎⡱⢌⠳⠮⠤⠴⠚⠁⠀⣸⠇⠀⣏⢀⣤⣤⣀⣻⠀⠙⣇⠀⠈⠓⠤⠤⠴⠖⠁⢎⡱⢠⣿⠀⠀⠀⠀⣷⠀⠸⠆
⠀⡀⠀⠀⣿⠀⠀⠀⠀⢇⣿⡌⢣⡜⠓⠄⡴⠖⣲⠀⣰⠋⠀⠔⢟⠉⠀⠀⠈⢻⠀⠆⠘⣆⠀⣦⠐⠦⠠⠚⢃⡜⢡⣯⡸⠀⠀⠀⠀⣿⠀⠀⣠
⠀⠃⠀⠀⢻⠀⠀⠀⠀⠈⣼⢉⣤⠳⣤⣦⠴⠊⣡⣾⠃⠀⠀⠀⣤⠀⠀⣀⣀⣸⡀⢀⠀⠈⣷⣈⠑⠦⢴⣤⠞⢠⡉⣯⠁⠀⠀⠀⠀⡟⠀⠀⠉
⣸⠀⠀⠀⠀⠰⡀⠀⠀⠀⠛⢾⡱⣁⠠⡤⠶⠛⠁⠀⠀⠀⠁⠀⠏⠀⠉⠋⠉⣠⠏⠠⠀⠀⠀⠉⠛⡶⠤⡤⣘⢌⣧⠻⠀⠀⠀⢀⠄⠀⠀⠀⠐
⣇⠀⠀⠀⠀⢤⡸⠀⠀⠀⠘⡈⢻⣎⣉⣉⡁⢀⠂⠀⠀⠀⠀⠀⣄⣀⣠⣤⣀⣈⠀⠀⠀⠀⠀⠀⡀⠀⣈⣉⣡⡞⢁⠃⠀⠀⠀⢎⡤
⠸⣧⣤⣦⣄⣸⡇⠀⠀⠀⠀⠹⣦⡙⠦⢞⡡⠃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠒⠋⠉⠀⠀⠀⠀⠀⠀⠑⢌⡳⠤⢛⣴⠏⠀⠀⠀⠀⢸⣇⣠⣴⣄⣼
⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣷⣾⣷⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣼⣿⢸⣿⢀⣿⣿⣿⣿⣶⣿⠿⢿⣄⠀⢸⠈⣀⠜⢳⣆⠹⣧⢰⣿⣿⢸⣿⢿⣿⣿⣿⡇⠀⢀⣼⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⣿⡇⣿⣿⣼⣿⣿⣿⣿⣿⣟⠀⢀⣿⣿⠈⢸⣿⣶⢿⣿⠀⣿⣾⡇⠀⢴⣿⣿⣿⣿⣿⣿⠀⣾⣿⣿⣿⡿⣻⣿⣿⣋⠀⠚⠿⣿⣿⣿⣇
⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⣾⣿⣿⠀⣾⣿⣯⣾⡿⠀⣿⣿⡇⠀⣼⣿⣿⣿⠛⣫⣽⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣷⡀⢠⣿⣿⣿⣿⡇
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣽⣿⣿⡀⣿⣿⡟⠀⢠⠈⣿⣿⡇⠀⣿⣿⣿⣻⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣹⣿⣿⣿⠃
⠀⠀⠀⢰⣿⣿⣿⣿⣿⡟⠉⣿⣿⡏⠀⢸⣿⣿⣿⢰⣿⣿⣇⠀⣿⣠⣿⣿⡇⢸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣗⣿⣿⣿⣿⢿⣿⣿⣿⣿⡇⣸⣿⣿⣿
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣏⠀⣿⣿⡿⠀⣸⣿⣿⣿⣾⣿⣿⣧⠀⣿⣿⣿⣿⡇⠘⣿⣹⣿⣿⣿⡇⣿⣿⣿⠟⣀⣿⣿⡇⢺⣿⣿⣿⣿⡇⣭⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡗⢰⣿⣿⡇⠀⣿⣿⣿⣿⡿⣿⣿⠏⢠⣷⠌⣿⣿⡇⢰⣿⣿⣿⣿⣿⡇⣿⣿⣿⠀⠐⣿⣿⡇⢰⣿⣿⣿⣿⣆⣽⣿⣿⡏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡟⢸⣿⣿⡗⠀⣿⣿⣿⣿⠂⣿⣿⠀⢸⣯⣖⣿⣿⡇⢸⣿⣿⣿⣿⣿⡇⢻⣿⣿⠂⢨⣿⣿⡇⠀⣿⣿⣿⣿⠃⣿⣿⣿⣇
⠀⠀⠀⠀⣽⣿⣿⣿⣿⠇⢸⣿⣿⡇⢸⣿⣿⣿⣿⠀⢻⣿⠀⣸⣿⠃⠛⣿⡇⢸⣿⣿⣿⣿⣿⡇⣿⣿⣿⡀⢸⣿⣿⡇⠀⣿⣿⣿⣿⠘⢿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⢸⣿⣿⡇⢸⣿⣿⣿⣏⠀⢸⡟⠀⣿⣿⠀⠀⣿⡇⢸⣿⠼⢿⠿⠫⢟⢫⢝⣻⡇⢸⣿⣿⡇⠀⣽⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⣇⠀⢸⣿⣿⡇⢸⣿⣿⣿⡏⠀⣿⡇⠀⣿⣿⡗⠀⢿⡇⢸⣿⣴⢀⣠⠈⠀⠀⠐⣼⡄⢸⣿⣿⡇⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⣾⣿⣿⡇⢼⣿⣿⣿⡇⠀⣿⡇⠀⣿⣿⡇⠀⢺⢣⢁⣿⠉⠀⠀⠄⣰⡇⠰⣛⡇⢸⣿⣿⡇⠀⣿⣿⣿⣿⠀⢺⣿⣿⣿⡅
⠀⠀⠀⠀⢾⣿⣿⣿⣏⠀⣼⣿⣿⣧⢸⡿⣿⣿⡇⠀⣿⡇⢨⣿⣿⡇⠀⢾⡧⠄⠸⢆⠀⠀⠈⠁⠀⠀⣨⡇⣸⣿⣿⡇⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣾⣿⣿⡗⠸⠗⣛⣸⡇⠀⣿⠇⢸⣿⣿⡄⠀⢸⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠿⣇⢺⣿⣿⡇⢠⣿⣿⣿⣿⠀⣼⣿⣿⣿⡎
⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⣿⢻⣿⡇⠀⠀⠀⠀⡇⠀⠋⠀⠈⡇⣿⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⠿⣿⣿⡇⢸⣿⣿⣿⡿⠀⢿⣿⣿⣿⡏
⠀⠀⠀⠀⢸⣿⣿⣽⣇⠀⢿⢸⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⡌⣿⡇⢸⣿⣿⣿⣧⠀⢺⣿⣿⣿⡗
⠀⠀⠀⠀⢿⣿⣿⣿⡇⠀⣾⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣸⣿⡇⢸⣿⣿⣿⣿⠀⢺⣿⣿⣿⡇
⠀⠀⠀⠀⣸⣿⣿⣿⣏⠀⣾⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢸⣿⣿⣿⡿⠀⣻⣿⣿⣿⡧
⠀⠀⠀⠀⢸⣿⣿⣿⡷⠀⢿⣾⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡇⢸⣿⣿⣿⡇⠀⣿⣿⣿⣿⡃
⠀⠀⠀⠀⣾⡇⢿⣿⡟⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⢀⣄⡤⣤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⢸⣿⣿⣿⠁⠀⢹⣿⣿⣿⡇
⠀⠀⠀⢸⣿⡇⢸⣿⡧⠀⢸⣿⠀⠄⠀⠀⠀⠀⠀⢠⣦⠀⠈⢣⢀⡀⠀⢀⣶⠁⢃⠀⠈⡀⠀⠀⠀⠀⠀⠀⣿⡟⠟⠀⢨⣿⣿⡇⠀⠀⣼⣿⣿⣿⡇
⠀⠀⠀⣺⣿⡧⠀⣿⡇⠀⠘⣿⢀⡆⠀⢠⠀⠀⠀⠀⠻⠿⢿⡟⠁⢻⣿⣆⠼⠆⠙⠁⡤⠧⠀⠀⠀⢀⡀⠀⡿⠁⢠⠀⠈⢻⣿⠁⠀⠀⢸⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣯⡀⠹⡅⠀⢠⡏⢹⠀⠀⠈⣧⣀⡀⠀⠀⠀⢸⡇⠀⠄⢿⠟⣰⡄⠀⠀⡇⠀⣠⠆⠠⠘⠁⠀⠰⡀⠀⠱⣤⣾⠋⠀⠀⠀⣧⢸⣿⣿⣧
⠀⠀⢀⣿⣿⡏⣷⣾⣧⣴⠿⣃⡜⠀⠀⠀⢻⣿⣿⣶⣶⣶⣾⣿⣶⣴⣶⡾⢋⡛⢠⡄⠉⠀⠁⠒⠒⠃⠀⠀⠀⢱⡦⠐⠊⠁⡂⠶⠀⠀⢻⣿⣿⡟⢢
⠀⠀⢸⣿⣿⡇⢿⢿⡿⢡⣾⠟⠄⠀⠀⠘⢿⡛⠛⠋⠉⠉⠉⣿⣿⣿⣿⡇⠉⣿⠋⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠘⠓
⠀⠀⢐⣿⡿⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⣿⠈⢿⣧⣸⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠈⢉⡀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⠀⠈⢿⣿⣠⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠰⠖⠂⠰⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡜⢃⣀⡀⣀⣋⠐⣤⣤⡤⢤⡶⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠉⠀⠀⠐⠁
⠀⠀⠐⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠀⠈⠉⠀⠉⠉⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⣤⠔⠀⠻⢷⣀⠀⠀⠈⠉⠛⢻⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⠶⠚⣯⡈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠂⡠⣻⠁⠀⠀⠀⠀⠀⠠⠀⠠⠤⠤⠤⠤⠤⠤⠤⣬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣏⣀⣀⣀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠤⠤⠤⠤⠤⠨⠴⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠉⠉⠀⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢬⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠐⠒⢷⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠛⠓⠶⠬⣤⡼⠤⠤⠶⠖⠒⠒⠒⠒⠠⠀⣤⡶⠶⢶⢶⠖⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠐⡖⠦⣰⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠠⣀⣭⣙⡉⢰⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠳⠦⢤⣤⡤⠤⠀⠐⠒⠒⠒⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⣽⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠬⡥⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡆
⠀⠀⠀⠀⠀⠀⠀⢰⡵⠿⣤⣶⣀⣀⣛⣀⡀⠀⠀⠀⠠⣤⣤⣤⣤⣤⣄⠤⠀⣠⣤⡤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠉⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⢐⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠒⠒⠒⠒⠒⠘⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠙⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣏⣻⡏⠀⠀⠀⢹⣿⣷⠆⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⠿⣿⣷⡀⠀⠀⣸⣿⡇⠀⢶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣇⠀⢘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠿⠿⣿⠿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣀⣄⡀⢤⣄⠀⠀⠀⠀⠀⣾⣿⣿⡿⢿⣿⡿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⢻⣿⣶⣾⣷⡀⠘⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⡿⠆⢻⣿⣿⣿⣷⣀⣶⣶⣶⣶⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠿⢿⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⣥⣾⣇⠀⠀⠀⠀⠀⠀⢠⣀⠀⣀⣼⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⣉⣹⣿⣧⣀⠀⠀⠀⠲⠀⠈⢹⣿⡿⢡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡈⢿⣿⣿⣿⡁⠀⠀⠀⠀⢰⣿⣿⣧⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢣⣤⣾⣿⣿⣿⣇⣠⣴⣿⣶⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣻⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣿⠋⠉⣽⣿⣿⣿⣦⣀⢀⠀⣀⣼⣿⣿⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣷⣶⣛⣭⣿⡍⠛⠿⠿⠟⠛⠛⠙⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣴⣶⣴⠆⢻⣩⣽⠿⣿⣿⣷⣦⣤⣄⣀⣤⣤⣶⣿⣿⣿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠈⢿⣯⣤⣾⣿⣿⠟⠛⠛⠛⠉⠙⠀⠀⢈⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠹⠃⠀⠘⠻⣿⣿⣿⣿⣶⣶⣶⣶⣴⣶⣶⣾⠿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠛⣿⣿⣿⣿⠟⠿⠛⠛⠋⠁⠀⢀⣿⣿⣿⡃
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟⠁⠀⠀⠈⠷⢠⣤⣴⣁⣼⣿⣿⣿⣿⠄⡐⢀⢀⣠⣴⡿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⢀⡤⠈⢙⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣴⡏⣿⡏
⠀⠀⠀⠀⠀⢸⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⠺⣥⣤⣄⣀⣤⣿⣿⣿⣿⡏⠉⢉⣿⣿⢿⣿⠇
⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⢻⠿⣿⣿⣷⣿⣿⣿⣷⣾⣿⠿⠷⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣟⠀⡾⢩⣿⣿⠿⠿⣿⣿⣄⡀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠙⠳⢾⣷⣤⣴⣿⣿⣿⣿⣿⣿⢏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⡆⠀⠐⠠⠝⠛⢿⣿⣿⣿⢿⣧⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢣⣠⡆⠀⠀⠀⣀⣉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠉⠛⠷⢾⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣄⣀⡀⠀⠀⠈⣻⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠓⠒⠒⠒⠒⠶⠤⠤⠤⠤⢄⣀⣀⣀
⠀⠀⠀⠀⠀⣀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⡋
⠀⠀⢠⣴⣎⣁⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⠄⠀⢸⡄
⠀⢠⠇⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠓⠒⠒⠶⠶⠶⠤⠤⣤⣤⣄⣄⣀⣀⣀⡠⠞⡁⠀⠀⢀⠔⡩⠃
⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠀⠈⠀⢀⠔⢁⠎⢀⠜
⠀⠀⡜⠉⠉⠓⠒⠒⠲⠦⠤⠤⢤⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣠⠔⠁⡰⠁⡠⠏⣀
⠀⠀⠘⡆⠤⠤⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠒⠒⠒⠦⠤⠤⠵⣾⣅⣠⠎⠀⡔⠁⢀⠔⡇
⠀⠀⠀⠉⠒⢤⡀⠀⠉⠙⠻⢭⣁⠒⠒⠂⠤⠤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⢎⡠⠚⠁⠀⢣
⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⠀⠀⠉⠙⠲⣤⡀⠀⠀⠀⠀⠀⠉⠉⠁⠒⠒⠀⠤⠤⣷⠗⠉⠀⠀⡠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠤⣀⠀⠀⠈⠑⠦⢄⡀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⡀⠀⠀⠈⠙⠢⢤⣀⡠⠒⠁⠀⠀⣠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠤⣀⠀⠀⠀⠍⠀⠀⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⣠⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⢘
⠀⠀⠀⠀⠀⠀⠔⠁⠀⠤⠤⠤⠄⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⡠⠃⠀⠁⠐⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡋⠀⡔⠁⠀⠀⢀⢤⠢⠓⠃
⠀⠀⠀⠀⠀⠐⢶⣤⣤⣠⣄⡀⣤⣄⣀⠀⠠⠤⡤⠤⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠎⠀⠀⢄⠎⠊⠁⠀⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠁⠂⠒⠦⣠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⡡⡰⠕⠈⠀⠀⣀⡴⠚⠉
⠀⠀⠀⠀⠀⠀⠐⡷⢆⣀⡄⣀⠀⠀⠀⠀⠉⠐⠉⠲⢀⡠⢀⡀⠀⢀⢄⠢⠑⠁⠀⢀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⣠⢊⣀⠀⠀⠀⠉⠒⠚⠴⢆⡠⢀⡀⠀⠀⠀⠈⠐⠺⠃⠁⠀⢀⡤⢖⠟⠁
⠀⠀⠀⠀⠈⡏⠈⠉⠉⠉⠉⠐⠒⠂⠙⠒⠚⠛⠛⠾⠥⠶⢄⣤⣀⣠⠴⠚⠉⡔⠁
⠀⠀⠀⠀⢠⣇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⡠⠊
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠁⠒⠒⠒⠒⠂⠠⠤⠤⠤⠤⠤⠤⠊





⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠸⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠛⠛⠛⠛⢿⡆⠀⠀⠀⠀⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⠶⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣤⣀⣼⣿⣷⡄
⠀⠀⠀⠀⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣆⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣍⣹⣿⣿⣿⡆⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠋⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠤⠤⠄⠀⠿⠿⠿⠿⠿⠿⠿⠀⠀⠘⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣿⣿⣿⣿⠆⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣿⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉

⡆⠠⢄⠀⠀⠠⠀⠀⢀⡀⣄⣀⣠⣠⣤⢀⣀⣀⣀⣤⣤⣦⣴⣦⣦⣶⣴⣶⣦⣴⣶⣦⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣄⣀⣀⣀⣀⠀⠠⠄⠠⠤⠀⢀⡤
⣇⢄⡀⠀⠒⠁⢡⣈⣤⣥⣼⣶⣾⣿⣿⣿⣽⣏⣿⣟⣿⣿⣿⢸⣿⣿⣿⣿⣿⣻⣿⢼⣿⣿⣿⣿⣿⣿⡯⣿⣿⣿⣿⣶⣶⣧⣬⣄⡑⠀⠒⠤⠄⢀⡀
⡟⠴⣁⣤⣶⣾⣿⣿⣿⣿⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⢿⣿⣿⣿⠟⢻⠛⠻⣿⡿⢿⣿⠉⢹⣿⣿⣾⣿⣏⣵⣤⣆⠀⢃
⢡⣿⣿⣿⣟⠿⢻⣿⣿⣿⠀⠀⠻⡆⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣏⣿⣿⠀⠚⠒⢶⡿⡀⣀⣿⠀⢸⠁⠠⠀⢻⡟⣿⡗⡾⣿⣦
⠙⣿⡭⠿⣿⢁⢻⣿⣿⣿⠀⠈⢻⠶⠒⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⣿⠀⠊⣻⣿⣿⠀⡋⠀⢺⣟⡏⣟⣿⠀⠈⠂⠀⠀⣿⣻⣿⡇⠀⣿⣻
⢸⣿⣧⣦⣼⣒⢻⣿⣿⣿⠰⠶⣾⣄⣀⣿⡛⠛⠋⠛⢻⠛⠛⠛⡛⠛⢻⣏⣀⡤⢷⠉⠈⣻⣷⢀⡃⠀⢺⣇⣏⣰⣿⠐⠺⠀⠀⠀⣿⣿⣿⡇⠀⣿⣿
⠀⣿⣧⣶⣿⣂⣻⣿⣿⣯⢀⣤⡌⣧⡿⣿⣷⣤⣶⣤⣸⣦⣤⣴⣧⣤⣾⣿⡤⠀⢚⣇⢀⠨⢷⣸⠀⠈⣿⡇⢯⡟⣿⠀⢠⠀⠀⠀⣿⣿⣾⡇⠀⣿⣿
⠀⣿⡧⠤⢿⢌⣿⠿⠿⣧⣀⣠⣼⢂⣳⣿⣿⣅⣀⣉⠉⣉⣀⣛⣛⣛⣻⣿⣿⣂⣩⣽⣦⣤⣼⣇⣀⣸⣿⣇⣴⣥⣿⢀⣸⡶⠶⠶⡿⣷⣿⣷⣶⣿⣿
⠀⡟⠰⠊⡰⠒⢒⢛⡂⣲⠙⠾⠒⠒⡖⠒⠲⠶⢖⣰⢒⡐⠒⠒⠒⠒⠒⠂⡖⠒⠂⠒⠒⠒⡒⢒⣖⣒⡴⠴⠒⢶⠂⠵⢄⣐⢀⣀⠀⠀⠀⠉⠒⠝⠙
⠈⠓⠚⠁⠙⠉⠉⠉⠉⠀⠀⠀⠐⠂⠁⠒⠂⠐⠂⠀⠊⠘⠃⠛⠂⠙⠀⠓⠓⠄⠒⠐⠒⠒⠱⠀⠃⠀⠐⠐⠒⠚⠀⠀⠤⠦⠈⠉⠋⠙⠒⠓⠒⠲⠔

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣄⣀⣀⣀⣀⣀⣠⣀⣀⣀⣄⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⢰⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠀⠀⠀⣿⣿⡇⠀⠀⠀⠙⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⣄⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠘⢿⣿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⡄⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣤⣴⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠁


⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣛⣟⠛⣛⣛⣛⣙⣛⣻⣛⡛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣿⣿⣟⣿⣿⣿⣿⣟⣛⣿⣛⣿⣟⣛⣛⣛⣛⣻

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢠⠦⣑⡠⢄⠀⠀⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠡⠜⣀⣱⡮⠴⢆⠄⠀⡸⣿⠂⡇⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠺⢙⡟⠁⢄⡈⠀⠀⠀⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠘⠳⡄⠈⠳⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⢀⡇⠀⢰⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢦⠹⣤⠼⢷⡁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠹⡄⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠧⠀⡀⠀⠀⠀⢸⠃⠸
⠀⠀⢀⠀⢀⠀⣀⡀⠀⠀⢀⡔⠀⠀⠠⠀⠀⢀⡔⠄⠀⠠⠀⠼⠀⠀⠀⠀⢀⠀⠈⠁⡀⠀⠀⣼
⠀⣶⣤⢢⠘⢦⣟⢍⠉⠉⠉⢣⢀⡤⠀⠐⠴⠋⠙⠀⠀⠀⢠⠀⠨⡀⠠⠀⢹⠀⠀⠈⢄⠀⣠⣿⣠⡄
⠀⣿⣟⠀⠃⠇⠈⠳⣽⡄⠀⠘⣯⡂⠀⠀⠘⣸⣿⣿⣿⡄⢸⡄⠀⠰⣠⡄⠀⣿⠤⠀⢰⢸⣿⣇⠀⠑
⠀⣿⠿⡉⣻⠁⠀⠀⠁⠀⠀⠀⢯⣷⠀⡰⠀⠘⢿⣿⠟⢁⠸⡆⠀⠄⢑⡈⠀⣩⠢⠄⢀⠈⠻⠿⢧⣰
⠀⠸⢤⣭⠿⡂⠀⠀⠀⠀⠀⠀⣭⠛⢿⣽⣇⣞⠀⠀⠉⠙⡄⠉⠀⢸⡀⠀⠀⠛⠀⠀⠈⢰
⠀⠀⠀⠸⠄⡇⠀⠀⠀⠀⠀⢨⠿⠿⢿⡪⣿⡟⠀⠀⠈⠤⠇⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠻⠆⠄
⠀⣐⠀⠀⢀⠉⠳⠶⠶⠶⠊⠉⣾⡀⣿⣿⣿⢿⠤⠤⠄⡄⣄⠀⢀⡄⠁⠐⠐⠠⠶⠴⠚⠡⠆⠀⠒⠚
⠀⠈⢹⠉⡇⢀⣤⣷⢠⣸⠀⠀⠿⡿⠿⠿⠋⠀⠀⢤⠀⢍⠉⠉⣉⠇
⠀⠀⠻⠷⠻⠎⠰⢿⣾⡿⠷⠦⠀⣿⣾⣅⠀⠀⡅⢀⠀⢤⣨⠀⢥⠐⡻⣷⣼⣂⡬⠿⠓⠿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠟⠍⠛⠿⠀⠓⠚⠖⠋⠀⠀⠈
⠀⠐⠁
⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⡏⢉⣉⣉⣉⣉⣉⣉⡉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⢰⡶⠶⠶⠶⠶⠶⠶⢶⠘⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢻⠘⣧⠀⣉⣁⣠⣤⣤⣤⡄⢻⡄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡍⣧⠸⣿⠿⠿⠿⠿⠛⠛⢻⡄⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⡆⢻⡄⢉⣀⣀⣤⣤⣤⠤⠌⢷⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⢻⡄⢻⣿⠿⠿⠟⠛⢛⣛⣳⡈⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢷⡈⢿⡄⠉⣁⣠⣤⡤⠶⠶⠊⠻⣄⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡘⣿⡌⢿⣾⠿⠿⠛⣛⣋⣩⣷⣌⠻⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠈⢷⡀⠻⣆⠉⢀⣠⣤⠶⠖⢛⣃⣩⣷⠦⠀⠀⠀⢸⣿⣿⣿⣿⣷⡘⣿⣄⠻⣾⡿⠟⢛⣋⣩⣥⠿⠗⢈⣛⣿⣿⣿
⠀⠀⠀⠀⠀⠈⢳⣄⠙⢷⣤⣤⠶⠾⠛⣋⣩⣥⠤⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣥⡌⢹⣧⣈⠛⢛⣋⣭⣤⡶⠾⢛⣛⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⣠⣤⠶⠞⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣦⡙⠿⠟⢛⣋⣭⣴⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠚⠛⠛⠛⠛⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢔⠏⠉⢀⣀⡤⢤⠶⠤⢤⣄⡀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣠⠖⣋⠥⠚⠈⠁⠁⠈⠘⠛⠶⣄⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⢠⠞⡥⠋⡠⠐⠈⠁⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⢰⣇⠋⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣾⡌⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢹⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⣸⠁⠀⢲⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⡴⠃⠀⢄⠿⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠑⢦⣀⠀⠀⠀⠀⠀⠀⣂⣠⠔⠋⠀⡠⣪⠛⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠉⠉⠛⠛⠉⠉⠁⠀⠀⢄⡮⠞⠉⣤⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⣠⠞⣳⠲⠤⣤⣀⣀⣀⣀⣤⠤⠔⢚⠉⢁⡸⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡪⠂⠀⢀⡠⢺⢀⠖⠉⠑⠲⠤⢦⣼⣄⣥⣰⡤⠦⠖⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠊⠀⢀⡴⠋⣡⠞⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⡴⢋⣠⠞⠁
⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣀⡔⢃⡤⠊⠁
⠀⠀⠀⢀⡴⠋⠀⠀⣠⠞⢁⡴⠋
⠀⢀⡴⣉⠀⠀⣠⠞⢁⡴⠋
⠀⢸⠀⠀⠙⡎⣡⠞⠁
⠀⠀⠓⢧⣠⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣶⣤⣶⣿⣿⣿⣿⣿⣿⡛⠂⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⣠⣧⣴⣶⣶⣶⣦⣤⣀
⠀⠀⠀⠀⣰⡟⠉⠙⣷⣦⡀⠀⢠⣿⣿⣿⣿⣿⣿⣿⢟⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⣿⣈⠓⠐⣿⣿⣿⣷⣿⣽⣿⣿⣿⢿⣩⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠘⢿⣷⣶⣿⣿⣿⡿⠽⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⢿⣿⣿⠿⠿⢻⣿⡿⢁⣀⡁⠀⢿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣷⣶⣤⠚⠿⠿⣿⣿⣿⣷⡜⢃⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⡄⠀⠈⣹⣿⠟⠀⠸⣧⣀⠀⠀⠹⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣏⢹⣏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⠿⢿⠉⠈⠁⠀⠀⠲⢶⣤⣤⢾⣿⠟⣴⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⡂⠒⠒⠢⠜⠿⠇⠀⠀⡼⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣤⣤⢤⢄⣀⣀⠀⠀⠂⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠦⣤⣀⡀⣀⣀⣠⣶⠿⣾⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣿⣿⣿⣿⣿⠖⣾⠟⠋⣰⣾⣷⡗⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⣨⡽⣿⢻⣿⣿⣟⣵⣟⣴⣿⣿⣿⢡⡇⢄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠤⣀⣈⣿⣿⣹⡇⣿⣷⣿⣿⣿⣾⣿⣿⡜⣿⢣⣿⣿⣧⣬⡭⠁⣒⠂⢄
⠀⠀⠀⠀⢀⣴⣿⣾⣿⣷⣿⡏⠙⢿⠿⢿⣿⣸⣿⣿⣿⣿⣿⣿⣁⣷⡀⢸⣿⡿⠙⠁⢰⣿⣿⣷⣆⠑⠄
⠀⠀⠀⠀⠀⠉⢾⣿⣿⡏⣠⣀⠀⣠⣶⣦⣿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿⣾⢟⣡⣴⣶⣄⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⠀⢯⡷⡎⣿⡽⣿⡿⠛⡛⢿⡇⠹⠋⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠅
⠀⠀⠀⠀⠀⠀⠉⠀⣿⣿⣿⣿⡇⢸⣷⣶⣿⣿⣿⣆⢻⣿⠚⠃⣠⣤⣉⠙⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⣿⣦⣙⠻⠿⣿⠿⠻⣛⣡⣴⣿⣿⣿⣿⣠⣿⣿⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣯⠉⠲⢿⣟⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠮⠅⠁⠀⠀⠈⠀⠉⠙⠋⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢠⡀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⡠⠤⢄⡀⠀⠀⠀⠀⡀⠀⠀⡀⢀⠀⠀⠀⠀⢄⠀⡀⠀⢀⠀⠀⢀
⠀⠀⣷⡀⠀⠀⠀⠀⡨⠛⢓⠦⣶⣶⠀⡎⠀⠀⠀⢱⠀⠀⠀⡠⠀⠠⠀⠀⢀⠀⠀⠀⠀⠀⠢⠀⡀⠀⠑⠀⠀⠂
⠀⠀⣿⠹⣆⡀⠀⢸⠀⠄⠄⠀⢸⠀⠀⠱⢄⣀⡠⣮⡀⠀⠀⣾⣿⣿⣏⣀⣀⡀⡀⢀⠀⠀⠀⠐⡔⠄⢀⢀⠀⠀⢂
⠀⠀⢻⠀⠘⠃⠀⠸⠣⣄⣀⠤⠃⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠈⠛⠛⢾⣿⣿⣿⠁⠀⠀⠀⠀⠀⠈⢂⠁⠀⠀⠀⠄
⠀⠀⠘⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠐⢀⡿⠀⠀⠀⠈⢁⣀⣀⠀⠀⠀⠀⢸⡗⠀⠀⠀⠀⠀⠀⠀⠡⠒⠈
⠀⠀⠀⠀⠀⠀⠀⢀⡴⢦⡄⠀⠀⠀⢁⣶⣶⣿⣿⡆⠀⣴⣿⣟⣿⡿⠏⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠁
⠀⢲⣶⣦⣤⣄⡀⠸⣦⣤⣧⣤⣀⠀⠀⠻⣿⣿⡿⠃⠈⢉⣿⣿⣿⣿⡄⢰⠀⣤⡀⠈⢡⡄⠀⠀⢀⠀⠀⠀⠐
⠀⠀⠙⡿⠟⠓⠁⠀⠀⢰⡏⠀⢘⡧⠀⠀⠀⢠⣄⡀⠀⠈⠋⠉⠉⠁⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠁
⠀⢀⣴⡇⠀⢀⣤⣤⡀⠀⠛⠗⠛⠁⢠⣦⣅⡀⠀⠈⢢⠀⠀⢀⠀⡴⠠⠀⣀⡀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠠⠠⠀⡄
⠀⠈⠉⠉⠀⢸⣧⣤⠟⡀⠀⠀⠀⠀⢹⣿⣿⡿⢀⣰⠈⡇⠀⠢⣀⣷⣾⣅⠀⠁⠀⢠⠀⠀⠀⠀⠈⠒⠠⠄⠄⠊
⠀⢤⣄⠀⠀⠀⠀⠡⠛⠃⠀⠀⠀⠀⠡⡈⠿⣧⠀⣽⡳⠁⠀⠚⠀⠟⠻⢥⡁⡄⠀⠀⠀⠸⠀⠀⠀⠀⢠⣦⣀
⠀⠈⠻⠇⠀⠀⠐⣾⣿⡇⠀⠀⠀⠀⠀⠀⠱⠶⡞⠊⠁⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⡀⠀⠙⠻⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠀⠰⠛⠛⠀⠀⠛⠀⠀⠀⠀⠀⢐⠂⠀⠀⢀⠂⠈⠀⠘⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠶⠀⠀⣞⡿⠃⡧⣄⠀⠀⠀⠀⠀⠶⠀⣤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣂⣀⣀⣀⣀⣉⣀⣀⡀
⠀⠀⡖⠀⠀⠀⠀⡀⠀⠀⠀⠀⠓⠋⠈⠁⠀⠀⠀⠀⠈⠻⠇⠘⠂⠀⠀⠀⢸⣿⡷⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢹
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢸
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠐⡄⠀⠰⠆⠘⠃⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⢧⣤⣤⣤⣤⣤⣤⣤⡼
⠀⠛⠒⠶⠤⠤⣄⣀⣀⠀⢸⠃⠀⢀⠤⠀⢤⣀⠀⢀⠠⠆⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠲⠶⠿⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⣄⣴⣦⡄⠘⢿⢧⠉⠀⠀⣠⡾⠁⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⠿⠇⢀⡴⠒⣄
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠦⠀⠀⡅⠙⠛⠀⣀⣄⠀⠀⢠⡶⠉⠀⠀⠀⠈⣥⠀⠀⢰⣶⣦⠀⠀⠀⢰⡋⣠⢞⣡⣦⡀
⠀⠀⠀⠀⢀⡄⠀⠋⠀⠁⠐⠀⠘⢷⡄⠐⢿⡟⠋⠀⠀⠀⠀⢠⡄⣤⠺⠂⠀⠀⠈⠙⠉⠀⠀⠀⠀⠉⠱⣿⣿⡿⠋
⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⠤⠤⠶⠶⠖⠒⠛⠛⠛⠉⢉⣉⣉⣛⣲⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⡒⠛⠛⠉⠉⠉⠉⠀⠀⣀⣀⣀⣠⣤⣤⠤⠶⠖⠛⠋⠉⠁⠀⠀⠀⠉⠛⠻⢯⣝⠳⠶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠉⡛⠶⢤⣄⡀⠰⢾⣋⡉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠤⠶⠖⠚⠛⠉⠉⣀⣀⠀⠈⠉⠛⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⡟⠓⠶⣬⣉⠓⠶⣬⣉⠓⠶⢤⣤⡤⠴⠶⠚⠛⠉⠁⠀⢀⣀⣠⡤⠶⠖⠚⠋⠉⠁⣀⣠⣤⠄⠀⠀⠀⠉⠙⠳⢦⣄⡀
⠀⠀⠀⠀⠀⠀⣀⣿⣄⡀⠧⣄⣀⠀⠉⠙⠲⢦⣌⡙⠓⠶⢤⣄⡀⠀⠀⠀⠀⠀⠉⠉⠁⠀⣀⣠⡤⠶⠖⠛⠉⠉⠀⠀⠀⠀⠀⢀⣀⣤⠤⣶⡞⠋
⠀⠀⠀⠀⢰⡟⠷⠦⣬⣽⣳⣦⣍⡛⠲⢤⣄⡀⠀⠉⠛⠶⢤⣌⡙⠓⠶⣤⣀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣀⣠⡤⠶⢖⣚⣻⣭⣷⣶⣿⣿⣅⡀
⠀⠀⠀⠀⢸⠇⠀⢰⠶⢤⣬⣉⣙⠛⠷⠶⣬⣙⠓⠶⣤⣀⠀⠈⠉⢻⠆⠀⠉⠛⠶⣤⣄⣀⣠⣤⣴⣖⠛⢯⣭⡴⣾⣻⣵⣾⣿⣿⠿⠛⣋⣩⠴⠛⠁
⠀⠀⠀⠀⢸⣀⠀⢸⣄⣀⠀⠈⠉⠙⠓⠶⢤⣉⣙⠳⠦⣍⡛⠶⢤⣼⠀⠀⢰⠖⢦⡀⢸⣟⣊⠩⣤⣲⣾⣷⣾⣿⡿⠿⢛⣋⣥⣴⣾⡿⠟⢓⣶⠄
⠀⠀⠀⠀⠀⠉⢹⡶⢦⣍⣙⠓⠶⢤⣄⣀⠀⠀⠉⠉⠛⠚⠿⢷⣦⣄⡀⠀⠹⣤⡼⠃⠈⣿⣲⣿⣷⠿⠟⠛⣉⣠⣴⣾⣿⣿⠿⢛⣩⠴⠚⠉⣉⡶
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣬⣍⡓⠶⣤⣉⣙⠓⠦⣤⣄⡀⠀⠀⠀⢸⡋⠛⠶⣤⣀⡀⠠⡟⢉⣁⣠⣴⣶⢿⣿⣿⠿⢛⣩⠴⠚⢉⣀⣤⢶⡏⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠈⠙⠛⠶⢬⣍⡛⠷⢶⣬⣍⣛⠳⠶⢼⠇⠀⢰⡏⢹⡏⠛⣿⣛⡿⢟⣯⠿⢛⣩⡴⠞⢋⣠⣴⣞⡯⠗⣊⣽⣷
⠀⠀⠀⠀⠀⠀⢸⡗⠶⣤⣙⠳⠦⣄⣀⠀⠈⠙⠓⠶⣬⣍⡛⠿⢷⣶⣦⣤⣌⡓⠚⠁⠀⣿⠾⢛⣩⡴⠞⣋⣠⣴⡾⠟⣛⣥⣴⣾⣿⣿⠿⠟⣛⡶⠄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣉⡙⠶⢤⣉⡛⠲⢦⣄⡀⠀⠈⠉⠛⠶⢤⠉⠙⠻⠯⣽⣓⠲⠷⣚⣫⣥⣶⢿⣛⠭⣔⣾⣿⣿⣿⠿⠛⣉⡤⢶⡋⠁
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠉⠓⠶⣬⣉⠛⠶⣬⣙⡓⠦⣤⣀⠀⢸⠀⠀⣠⠶⣄⠈⢹⡿⣝⣫⢗⣊⣥⣶⣿⠿⠟⢋⣡⠴⣞⣫⣵⣾⣿⣇
⠀⠀⠀⠀⠀⠀⠘⠧⣄⡀⠙⠶⣤⣀⠀⠉⠙⠳⢦⣌⣙⠳⠦⣍⡙⠛⠀⠀⣿⡀⣿⠀⢸⡿⣽⣾⡿⠟⠛⣉⣤⣴⣟⣹⣶⣿⡿⣟⣿⣿⡿⠟⣙⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡶⢤⣉⠙⠲⢤⣀⡀⠀⠉⠛⠲⢤⣍⡛⠲⢤⣈⡙⠁⠀⣽⠛⢉⣡⡴⣖⣿⣯⣿⢿⣻⣯⣵⣿⡿⠛⢋⣡⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⢈⣿⣷⣦⣌⡙⠓⠦⣄⡀⠀⠀⢹⠀⠀⣈⣉⡓⠲⣾⣶⣿⣷⣿⢿⣻⣽⣲⣿⠿⠛⢉⣀⡴⠞⠋⠉⠛⠦⣄⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣚⠉⢹⣄⠀⠸⣇⠀⠉⠛⢿⣟⡶⣤⣉⠛⠶⢾⠀⢰⡇⠈⣷⠀⣿⣟⣫⣵⣾⣿⠟⠛⢉⣀⡤⠖⠋⠁⠀⠀⠀⠀⢀⣀⣠⡭⠿⠂
⠀⠀⠀⠀⠀⢸⡇⠈⠉⣙⣳⣾⣿⣶⣬⣛⠶⣤⣀⠈⠙⠻⠿⣿⣷⣶⣤⣄⡙⠛⠁⠀⣿⡷⠿⠋⢉⣠⡴⠞⠋⠁⠀⠀⣀⣠⣤⣴⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠈⡇⠀⠀⡏⠉⠉⠛⠳⠮⣭⣙⡲⣭⣓⢦⣄⡀⠀⠉⠛⢿⠟⠿⣷⡶⢤⣿⡤⠶⠛⠉⢁⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⡿⠟⣉⡿⠖
⠀⠀⠀⠀⠀⠀⣧⣀⠀⠷⣤⣀⡀⠀⠀⠀⠈⠉⠛⠛⠷⢮⣽⣳⣦⣄⣸⠀⢀⣤⣝⠳⢦⣤⣴⣶⣿⣿⣿⣿⣿⣿⠿⣟⣻⡭⠟⣛⣡⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣬⣙⠓⠦⣤⣀⡀⠀⠀⠀⠀⠀⠉⠛⡾⢭⣀⠸⣇⣿⠀⢸⣿⢿⡿⢻⣯⣿⠿⠷⣚⣋⡭⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣬⣉⡛⠶⢤⣄⡀⠀⠀⡇⠀⡽⠳⣿⣅⡀⢸⡿⠖⢛⣉⣥⠴⠞⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢮⣽⣛⡶⠃⠀⠻⣴⠇⠈⣟⠻⢟⣻⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠓⠲⠦⣤⣀⣿⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⡠⢼⠙⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⢹⡤⡀
⠀⠀⠀⠀⠀⠀⠀⡆⡇⣾⠀⠀⠀⠈⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉⠀⠀⠀⢸⡇⡇⡆
⠀⠀⠀⠀⢀⣼⠉⡇⡇⡟⠀⠀⠀⠀⠀⠀⠈⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠈⡇⡇⡏⢹⣄
⠀⠀⣠⣤⣿⣽⢸⢇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⡀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣇⢷⢸⣿⣧⣤⡀
⠀⠀⣿⣿⣿⡟⢸⢸⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⢀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⢸⠸⣿⣿⣿⡇
⠀⠀⣿⡿⡏⡇⢸⢸⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢸⢸⠀⡏⣿⣿⡇
⠀⢸⣿⠰⠁⡇⣸⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⢸⠀⡇⠡⢹⣿
⠀⢸⣿⠀⠀⡇⡏⡘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡈⡇⡇⠀⢸⣿
⠀⣸⡇⠀⠀⠀⡇⡇⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⡇⡇⠀⠀⠀⣿⡀
⠀⣿⡇⠀⢀⠀⡇⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⡇⢈⠀⠀⣿⡇
⠀⣿⠇⠀⠈⠀⠃⠃⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃⠇⠈⠀⠀⢿⡇
⠀⣿⠀⠀⠀⠀⠀⠄⣐⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢐⡀⠄⠀⠀⠀⠀⢸⡇
⢸⣿⠀⠀⠀⠀⠀⠀⣀⡉⠑⠺⢽⣢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠟⢋⣁⣠⡤⠼⠘⡅⠀⢸⣿
⢸⣿⠀⢀⠄⠀⠀⠀⠠⠬⠭⣓⣲⠬⣝⡿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡾⣟⡭⢤⣚⡯⠭⠥⠶⠒⠒⠒⢧⠀⠸⣿
⣼⡇⢀⠂⠀⠀⢀⣐⡒⠤⠬⣑⡒⠯⢝⣛⠿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⢟⣛⠭⠗⣒⡩⠥⠔⣂⣀⠀⠀⠈⢂⠀⣿
⣾⣇⣒⠢⠭⢍⣒⠲⠬⢍⣓⠢⠬⣙⠒⠬⣙⠲⢍⡛⢿⣆⠀⠀⠀⢠⣾⠟⣋⠵⢚⡩⠔⢚⡩⠤⢒⣋⠭⠴⢒⣊⡭⠥⢒⣂⣿
⠙⠛⠿⠿⣷⣶⣤⣭⣙⡒⠢⠭⢑⠂⠌⠑⠂⠉⠂⠈⠂⠙⠳⡀⡰⠛⠁⠊⠀⠊⠁⠒⠉⠅⢒⡩⠥⠒⣚⣩⣥⣴⣶⡿⠿⠟⠛
⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⢀⣤⣽⣥⣄⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⡾⠿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠷⣶⣤⣴⣿⡿⠿⣿⣿⣷⣤⣴⡶⠿⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣤⡤⠶⠶⠖⠒⠒⠀⠀⠀⠀⠀⠀⠒⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠑⠢⢍⣛⠲⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⣈⡐⠦⣌⣙⠒⠤⣉⠛⠳⢶⣤⡀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠶⠚⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢌⣑⠢⢭⣑⠢⠌⡙⠒⠤⣀⠀⠀⠈⠙⢷⣄⠐⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠓⠮⣍⡒⠦⢍⡒⠢⠄⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢀⣤⣭⣷⣦⣌⣁⠂⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣾⣿⣯⣭⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣦⣤⣤⣀⣤⣴⣟⣁⣤⣤⠶⠾⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⢻⣿⢿⣿⠻⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⢛⣛⣛⣻⣿⣿⡿⠿⠿⠛⠛⠛⠛⠛⠛⠛⡿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⡜⠙⣧⠉⣿⠶⠶⠶⠶⠶⠾⠿⠛⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠤⠴⠖⠻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠁⠀⠈⢧⣿⠁⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⠤⠶⠖⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡴⢶⡟⣀⣠⣤⣤⣤⣬⣿⣿⣿⣿⣿⣿⣯⣉⣉⣉⣁⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀
⠀⠀⢀⣤⡶⠟⠛⠉⠁⠀⠀⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⣭⣉⣉⣉⣉⣉⣉⣛⣛⡛⠛⠛⠛⠷⠶⠶⠶⠶⠶⠶⠶⢶⡾⠿
⢀⣴⠟⠁⢀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠒⠒⠒⠒⠠⠬⠭⠭⠭⠭⠭⠭⠉⠁⠀⠀⠀⢠⡟⠳⢤⣀
⣼⠃⠀⠀⠀⠉⠑⠢⠄⠀⠀⢿⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣐⣒⡒⠒⠒⠦⠤⠤⠤⠤⢬⣭⣉⣉⣉⣉⣉⡉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠛⠶⣤⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⡀⠄⢀⣀⣀⣀⣀⣈⣉⣉⣉⠉⠉⠉⠉⠉⠉⣛⠒⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣀⣀⣤⣽⡷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠶⠶⣶⣦⣭⣭⣭⣭⣽⣿⣿⣿⣿⣿⣶⣶⣶⣶⡦⠤⠤⠄⠀⠀⠈⠉⠉⢁⣀⠀⠀⠙⣟⠛⠋⠉⠉⡟
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⢤⣀⠀⠀⠀⠀⠉⠉⠉⠉⣉⣉⣩⣽⡿⠿⠟⠛⠛⠛⠛⠛⠒⠶⠶⠿⠿⠛⠀⠀⠀⢸⠇
⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⣴⠞⠛⠉⠁⢀⡀⠤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⣸
⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⣀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠉⠳⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠉⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠀⠀⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢀⣀⣀⠤⠠⠄⠀⠀⠀⠀⠀⠀⠐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣄⡀⠀⠀⠈⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡤⠴⠶⠶⠒⣛⣉⣁⣤⣤⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠶⣤⣈⡳⣭⣑⠒⠀⠀⠀⠀⠒⠚⢋⣉⣉⣩⣭⣤⠴⠶⠶⠛⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠓⠲⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣀⠀⠀⠀⠉⢲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⣄⡈⠙⠲⢤⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⢀⣀⣀⣀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣰⠋⡀⠀⠀⠉⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⢿⠀⠢⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣀⠀⠀⠀⠈⠓⠤⣀⠀⠀⠀⠀⠀⠀⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠠⠉⡒⠤⣄⣀⣰⠙⢟⠢⡀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⠈⢀⡼⠃⠀⠈⢧⠙⢆⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠤⢤⣀⡴⠋⠀⠀⠀⠀⠈⡆⠈⡆⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⡒⠒⠒⠒⠒⠒⡄⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣒⣲⣒⣶⣒⣖⣲⠇⠀⠀⠀⠀⡸⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⣀⣀⣀⣠⠎⠀⠀⠀⠀⣠⠃⢀⠇⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠶⠒⠒⢺⠁⠀⠀⠀⠀⡗⠒⢒⡲⠖⠁⡠⠋⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⣤⣀⣀⠀⠀⢸⠀⠀⠀⠀⠀⢻⢉⣁⣠⠴⠊⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⠭⠟⠁⠀⠈⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠷⠶⣶⣶⣶⡶⠶⠶⠶⠶⠿⠽⠛⠛⠛⠉

⠀⠀⠀⢀⠀⣀⣀⣀⣀⣀⡀⠤⠤⠤⠄⢢⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠈⠍⠉⠉⠀⠀⠀⠒⠂⠀⠀⠀⠉⠉⠁⠀⣶⠲⠒⠾⠶⠶⠶⠶⡶⢦⠤⣤⣤⠤⡴⠷⠶⠶⣦⠀⣧⣤⡤⠤⠀⠀⠤⠤⠤⣄⣀⡀⡠⠃
⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠰⠀⠀⠁⢠⠨⢠⠡⠀⠂⠐⠇⠀⢠⢀⢪⣴⡇⠀⢰⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠆⠀⢀⠃⢸⠐⠀⠀⠠⡄⠄⡇⠀⢠⠀⡇⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢹⣠⣀⣋⣔⣘⣙⣂⣘⣀⣏⣀⠄⢀⠉⠠⢺⣀⡇⠀⠐⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣈⡀
⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⣁⣀⣀⣀⠀⢀⡀⠀⠈⠉⠉⣉⠉⠉⠉⠉⣉⠉⠉⡉⠉⠉⢉⠉⠁⢀⠀⠁⣌⠁
⠀⠀⠀⠀⡋⠉⠉⠉⠹⠀⠉⠀⣍⣧⠁⠀⣧⠴⠛⠤⢤⣀⣲⣀⡀⢠⠤⣹⠁⠤⣄⠀⣀⡠⠠⡴⡀⠀⣷⣙⡏⠉⠉⠁⢨⠉⠙⢡⠅
⠀⠀⠀⠀⠠⡆⣄⠀⠀⠁⠀⠀⣿⣟⢸⠈⠠⠄⠴⢧⠛⠈⣤⠁⠑⠴⠀⠈⠁⠀⠀⡴⠋⢱⡜⠀⠈⠀⣿⣿⡧⠀⠀⠀⢸⡀⣄⣱⡷
⠀⠀⠀⠀⡀⣇⣿⣄⠀⠀⠀⠀⢡⡬⢸⠀⠠⠀⠠⠀⢰⠀⢸⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠈⠱⡄⠀⠀⣟⣛⠇⠀⠀⠀⠘⠀⠀⠀⠂
⠀⠀⠀⠀⠁⠀⣀⣀⠀⠀⠀⠀⠀⠃⢸⠀⠀⠀⠈⠀⢀⠠⠈⡄⠀⠀⠀⡄⠀⡘⢣⡘⠒⠒⠒⠚⠀⠀⡇⠛⠀⠀⠀⠀⠀⢤⢠⡤
⠀⠀⠀⠀⢠⣤⣶⢳⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠂⠉⠀⠤⠀⡇⠀⠀⡠⠃⢞⠒⠒⠛⠀⠀⠤⣖⠀⠀⡇⠀⠀⠀⠀⠀⢀⡬⣽⡽⣁
⠀⠀⠀⠀⠸⡯⠽⠻⢤⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠃⡒⠀⡇⠀⠀⠉⠙⣞⢹⣒⣅⡤⣤⠈⢫⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠓⠆⠊⠒⠉⡍⠁⡇⠀⠀⠀⣴⠛⡎⠘⢍⢣⣠⡇⢀⢾⠃⡇⠀⠀⠀⠀⡆⠀⢰⣦⡶⢸
⠀⠀⠀⠀⢶⣰⡖⣦⠈⠀⠀⠀⠀⠀⠀⠀⠠⠏⢈⠀⠀⢘⠀⡇⠀⠀⠀⣿⠷⣯⣤⣴⣬⣠⣼⣤⣼⠀⡇⠀⠀⠀⣀⡇⠀⣸⠿⠿⣸
⠀⠀⠀⠀⠸⠻⠧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠰⠦⠦⠴⠔⠚⢸⡇⠀⠀⠸⡷⢾⡇⠀⠰⢰⢢⠄⣤⢴⠀⡇⠐⠀⢤⠂⡇⢸⠁⠠⠤⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠉⠉⠉⢹⡇⠀⠀⢸⣿⣿⣇⡀⠀⢘⢘⡠⣼⣸⡀⡇⠀⠚⣂⠸⡇⠀⢀⣤⣄⣼
⠀⠀⠀⠀⣁⣄⣤⢠⡀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡦⢡⠀⠀⢾⡽⣿⣏⠉⠉⣉⣉⠉⡉⣹⠀⡇⠙⠀⠁⠈⡇⢰⣼⡟⣁⣼
⠀⠀⠀⠨⠘⠿⣣⡼⣧⠀⠒⠒⠒⠐⣘⠀⠀⠀⢀⣀⡀⠀⡇⢸⡀⠀⢸⣤⣿⣓⢀⠀⠰⠘⢲⠇⣿⠀⡇⠀⠀⠀⠀⡇⢰⠺⣿⣶⡇
⠀⠀⠀⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⢰⣿⠍⡀⢰⠁⠀⡇⠀⠀⠾⣿⠶⠾⡶⠶⠾⠿⠛⢻⢠⡭⠭⢍⣉⣉⣁⣈⣉⣉⣉⣉
⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡀⣰⡘⣐⡀⡼⠒⠒⠃⠀⠀⠀⣿⠅⣽⡇⣻⡅⢸⣷⣿⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠨⠀⣏⣥⣤⣀⣹⡇⠀⠀⠀⠀⠀⠀⢯⣦⣼⣧⣽⣟⠾⠽⣿⢸⠄⠀⢀⣀⣀⣀⣀⣀⣠⣤⣼⡅
⠀⠀⠶⠶⠤⠤⠤⠶⠶⠤⠶⠶⠤⠀⠒⠪⠏⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣝⣀⣀⡀⠀⠀⠀⢀⣀⠐⠂⠀⠀⠨⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣎⠀⠀⠀⠀⠀⠀⠀⠀⠰⠎⠎⣱⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠆⡀⡀⠂⣠⡾⠣⠄⣀⠀⠀⠀⠀⠀⠐⣆⠀⠃⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⡀⠈⢀⣤⣀⠀⠀⠀⣎⣀⣀⠀⠈⢡⢰⣶⡆⠀⠀⠀⠉⠁⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⢾⠁⠀⢹⠶⡏⠀⠈⡷⠀⣼⣿⣿⣿⣿⣶⠟⠈⠋⠀⠀⠉⠉⠀⠀⠀⠀⠙⠷⠘⠿⠀⠀⠀⠀⠙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠾⠓⠖⠋⠀⠙⠲⠚⠷⠀⠸⣿⣿⣿⣿⡏⠀⠀⡀⠀⢀⡙⠳⣤⡀⢰⠇⠴⠰⠆⠄⠨⠀⠀⡠⠾⠷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡰⠶⢤⡇⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣔⣅⡈⠛⢁⣸⠇⠈⠻⣆⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⢀⣀⣛⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⢾⡄⠀⠀⣠⡄⠀⠀⢶⣲⡀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠀⠹⣆⠀⠀⠀⠀⠀⢀⣤⡶⠛⡋⠉⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠰⠤⠄⠹⠀⣀⣈⣁⣀⣀⣸⣿⣧⠀⠀⠀⠀⠀⣸⣿⣿⡆⠀⠀⢻⡄⢺⡗⢀⣴⠟⠁⢰⣿⡇⠀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⣦⡆⠀⢘⡷⠶⣤⠀⠘⠿⣿⣿⣿⣿⣇⣿⣿⡆⠀⣀⡀⠀⠘⠿⠿⠃⠀⠀⢸⡇⠀⣰⡟⠁⠀⠀⠈⠉⠁⠛⣩⣀⣈⣀⣄
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠾⠡⠤⠌⠧⠀⠺⠿⠿⠿⠿⠿⠸⠿⠷⠀⠛⠃⠀⢀⠐⠃⠐⠃⡀⢸⣯⢠⣿⠃⢸⣾⣾⠀⠀⢀⠀⠉⠀⠀⠈⠉⢄
⠀⠀⣀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠲⠶⢦⣄⡀⠘⠁⠀⠀⡀⠁⣼⡇⣼⡟⠀⢸⣿⣿⠀⣀⠻⠀⠀⠀⠀⠀⠀⠸
⠀⣴⣧⣄⣀⠘⠛⠀⡄⠰⠀⡔⢀⠀⠀⠘⣧⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈⠙⣦⡈⠃⠈⠁⢠⣿⢡⣿⠃⠀⠀⣉⣣⣄⣿⡆⠀⣤⡀⠀⢀⡄
⠀⠀⠀⣼⠋⢀⣈⠃⠀⠀⠀⠀⢋⡀⠀⠀⣿⠀⣴⣤⣴⡀⠀⠀⠀⡂⠀⠀⠈⣿⣄⠀⢀⣾⡏⣼⡿⠀⠀⠀⢽⣿⣿⡾⡷⠀⠉⠁⠀⠈⠁
⠀⠀⠘⠁⠀⠀⡤⠀⠀⠀⠀⠀⠡⠤⠀⠀⣿⠸⠿⠿⠿⠗⠀⠀⠀⠁⠂⠀⠀⢸⣿⣤⣾⡿⣱⣿⠇⣼⣇⡤⡄⠀⠀⣠⣤⣶⣿⣿⣿⣿⠶⠒
⠀⠀⣠⣤⠀⠀⠀⠀⢀⣤⣄⠀⠰⢛⡆⢰⣿⠀⣀⣤⣴⣶⣶⣦⣄⡀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠏⠈⠋⠉⡱⢣⠀⠀⠡⠀⠉⠉⠉⠉⡀⠂⠁
⠀⠸⠟⠁⠀⣷⣾⡄⠀⠻⠋⠀⠀⠈⠀⢸⣿⠀⠉⢳⣿⣿⣿⣿⠙⠉⠀⠀⢠⣿⣿⣿⣿⠟⠃⠀⠘⢲⣞⠁⠿⢦⡄
⠀⠁⣤⣦⣀⠈⡿⠀⠀⠀⠀⢰⣟⣠⢄⠈⣿⣆⠀⠀⠉⠙⢉⣵⡂⠀⠀⢠⣿⣿⡿⠟⢁⣠⠀⠀⠀⠉⠉⠁⠀⠎⠀⢀⡀⠐⠀⠀⠀⢀⡞
⠀⠀⠈⠉⠉⠀⠀⠈⠀⠀⠀⠛⠉⢹⠎⠀⠘⢿⣷⣤⣀⡀⠻⡿⠃⠀⣠⣿⣿⠋⠀⠈⠉⠀⠀⠀⠀⣴⣾⠀⠀⠀⠀⠀⠄⠚⠀⡄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠲⣶⠋⠀⠀⠀⡀⠉⠻⣿⣿⣷⣦⣀⣴⣿⡿⠁⠐⠀⠠⠆⠀⡜⠀⡀⠙⠛⠀⣀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠘⠉⠋⣀⡀⠀⠀⣀⠀⠀⠙⢿⣿⣿⣿⣿⡇⠮⠉⢭⡩⠨⠏⠀⠀⣷⠀⠀⠀⣍⠀⠀⠀⣤⣦
⠀⠀⠀⠀⠀⠀⠀⠻⠿⠃⠀⠀⣰⣶⣶⡄⠀⠀⠈⡆⠀⢀⣼⣿⣿⣿⣿⣷⣄⠀⠼⠧⠀⠀⠀⣸⡟⢠⡄⠈⠉⠑⢦⠈⠻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⠃⠀⠀⠀⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣴⡾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⢀⣾⣿⣿⣿⣿⣿⣿⡟⠁⠀⣀⣠⠤⠤⠖⠒⠶⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣀⡶⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠀⠀⠀⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢩⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⣰⠏⣀⣤⡤⠶⠖⠒⠒⠒⠒⠒⠒⠒⠶⠶⢤⣤⣀⠀⠀⢸⠀⠀⠀⠀⠀⣀⣀⣤⣤⠤⠴⠶⠦⠤⠤⣤⣤⣀⣀⡀⠘⢧
⠀⠀⠀⠀⣼⣿⣋⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡉⠛⠶⣼⣀⣤⣖⣚⣋⣩⣤⣤⣤⣤⣤⡤⠤⠤⠤⢤⣤⣤⣭⣭⣿⣾⣧
⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠛⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆
⠀⢀⣼⣼⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣶⣿⣿⣿⣷⣦⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀
⠀⠘⠛⠟⠻⠛⠻⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠿⠿⠛⠻⠿⠿⠿⠟⠻⠿⠿⠿⠟⠛⠛⠛⠻⠛⠟⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣛⣛⣛⣛⣛⣋⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⠿⠿⠿⠿⠿⠿⠻⠿⣿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣛⣿⣟⡛⣿⣿⣿⠿⠻⢻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢨⠀⠀⠈⠉⠉⠉⠀⠀⠒⠂⠠⠤⠠⣄⢀⡎⠀⠉⠑⠂⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠠⠖⣒⣒⠂⠀⢠⣤⠀⢀⣀⢠⢻⠀⠀⠀⠀⠀⠀⠀⠈⠑⢒⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠧⠭⢴⡄⠀⠀⣈⠭⠭⠅⠀⢸⠀⠢⣮⣂⢄⠔⡀⡀⠀⣠⣁
⠀⢃⢀⢖⠂⠤⢀⣀⠀⠀⠀⠀⢀⠌⠀⠬⠗⣒⠆⠀⠀⢀⣀⢈⡉⠁⢀⠆⠀⠀⠀⠈⠙⠪⢮⣔⠀⡗⠾⠋⠲⢤⡀
⠀⠀⠀⠈⠳⢕⡔⡠⡈⢉⠐⠒⠣⠤⢤⠿⠕⠂⠀⢄⣤⣭⢐⡂⠀⢀⢎⠄⠔⠀⠀⠀⠀⠀⠀⡨⠟⡳⣬⣋⣛⡵⡬⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠉⠛⠳⠥⣮⣊⣀⢔⠔⠵⢄⡠⠬⠽⡄⠈⢀⡠⠔⠁⠀⠈⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⠀⠑⠡⡪⢂⢔⠥⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⣉⠘⠛⠚⠛⠓⠈⠉⠁⠀⠀⢀⠔⠠⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⡀⠁⠑⡪⠣⢂
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠋⢉⠠⠦⡼⠟⠛⠓⠒⢲⠤⠤⠤⣐⣊⣡⣔⡷⠒⠀⢈⡉⠒⠤⠤⣀⠀⠀⠀⠀⠀⠈⠓⠉⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡘⠂⠤⠄⣀⠀⠈⠀⠀⡀⠄⣤⠍⡛⠍⠉⣂⣀⣉⣉⠉⠀⣠⠤⣺⢟⠻⠒⠀⢀⠏⡐⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢀⠈⠁⠀⢀⡤⠊⠉⠀⠀⢀⣀⣀⠉⠉⠒⠒⠤⠤⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡄⣀⠀⠀⠈⠑⠢⠊⠀⠀⠀⢀⡠⠊⣠⡤⣍⠉⠐⠒⠤⠤⣎⡉⣩⡩⠁⢒⠀⠄⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠂⠱⣶⠩⠲⢤⢀⠀⠀⣀⡠⠔⣉⣐⠾⠖⠂⠀⣀⡛⠍⠒⠂⢀⣽⣀⡀⠀⠒⠊⠉⣠⡤⠖⠗⡦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢥⣡⢢⠈⠝⡲⣖⠒⣢⠄⢀⡰⠤⢍⠑⠂⠀⢀⡤⣛⠭⠁⡤⣀⠀⢀⣾⣥⣀⡑⡀⡐⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠾⣜⡜⡻⢤⣚⣁⠊⠉⣀⣠⣤⠶⢛⠣⠁⠠⢼⡛⠀⢀⠎⠀⠐⠋⠉⢙⡒⠟⠒⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠒⡒⠦⣀⠈⠉⠓⠚⠚⠛⠛⠛⠭⢍⣋⡀⠀⠊⠍⠒⠂⡠⠋⠀⢀⠃⣘⡚⠉⠀⢀⢫⡅⠂⢈⡷⣄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠊⡀⢤⡒⠬⡐⠬⠒⠀⣉⣲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠦⠤⢊⣀⠀⠀⢸⢀⠒⠒⡄⢀⣐⠊⠀⠀⡿⣴⣩⣧⢱⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠒⢅⣈⠀⢀⠠⢄⡙⠲⣤⡤⠔⣛⣲⣤⣀⡀⠀⠀⠀⢯⠙⠒⠂⠤⠥⣀⣠⡃⠢⠼⠴⠁⢔⣒⠫⠄⠀⠇⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄⠀⠀⠀⠂⠭⠒⠌⢁⠴⠚⠛⠷⣔⡈⠈⡝⢦⡀⠀⠈⢷⡡⢂⢄⠀⡀⡀⠈⠉⢳⠀⢈⣁⠂⠉⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⡔⠁⠀⠀⠀⠀⠈⠙⠻⢴⣄⡇⠀⠀⠀⠉⠓⠧⢎⣔⡱⢡⠊⡸⣦⠥⠂⠉⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⠂⠀⠀⠀⠀⠀⠀⠉⠙⠛⠚⠾⠷⠦⠔⡪⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⠀⠀⡀⢀⡀⡀⣀⠄⡠⠄⡀⠀⠀⠀⠀⡀⢀⠀⡀⠀⠀⢀⠀⡀⢀⠄⡀⢀⠀⠀⠀⠐⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠈⠒⠕⠪⠔⠝⠩⠐⠕⠨⠊⠔⠊⠔⡩⠪⠔⠀⠠⠪⠎⠥⠊⠔⠡⠎⠔⠡⠚⠵⠨⠒⠅⡪⢊⠔⡠⡢⢀⠔⠔⡠⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠄⠐⠂⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠤⠐⠀⠉⠀⡠⠒⠀⠁⠀⠂⠄⡀⠉⠐⠠⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⢆⡀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⣀⠀⣀⠀⠠⠘⠂⠀⠀⠀⣀⠁⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠈⠒⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⠐⠐⠀⠈⠁⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⠀⠈⠑⠠⡀⠀⠀⢀⡀⠤⠄⠀⠒⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡠⠂⠁⢀⣈⡢⢄⠠⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠎⠁⠒⢠⡀
⠀⠀⠀⠐⠁⠀⡔⠁⠀⠈⠢⠑⠢⣀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠠⠄⠒⠂⠈⠁⠁⠀⣀⠠⠐⠊⠁⠄
⠀⠀⠀⠐⡀⠀⢄⠀⠀⠀⢀⠇⠀⠀⠑⠠⠰⠤⠤⠴⢒⡀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⡀⠠⠐⠊⠉⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠐⢄⡈⠐⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠒⠠⢤⠀⠤⠄⠒⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠄
⠀⠀⠀⠀⠀⠀⣨⡕⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠖⠋
⠀⠀⠀⠀⠀⡘⠀⠀⠀⠉⠘⠙⠒⠦⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠒⢁⠔⠁⠀⡀
⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠂⠠⠄⣀⠀⠀⠠⠀⠀⠀⠀⣀⠤⠐⠊⠁⠀⡠⠚⠁⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢈⠒⠓⠐⠚⠌⠁⠈⠀⡐⠈⠁⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠤⠄⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⠀⠀⢈⠀⠀⢀⠃⠀⠀⠀⠀⣀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠐⠒⠠⠤⠈⠒⡠⠄⠊⠀⠀⠈⡀⠀⢀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠒⠠⠥⠔⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⢸⣿⡟⠉⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢷⢯⣽⣧⣤⣾⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣺⢦⣤⣤⡿⢉⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⡋⠉⢩⣿⣿⣿⣿⣶⡝⠛⠳⢦⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡷⡠⠛⢻⡿⠿⣿⣿⡇⢀⣠⠞⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣰⡿⠁⣀⣨⣛⣴⣻⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣽⣿⣹⣿⣟⣋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠁⠀⠘⠻⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡆⠀⠰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣯⡁⠀⣭⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠁⠈⠀⠀⠈⣿⣷⠂

⠀⡤⢤⣤⡤⢤⡤⠤⢤⠄⢤⠴⠆⢦⠤⠤⡤⢤⡤⣤⣤⣤⡤⢤⠤⣤⠲⠶⠦⡤⣤⡤⠤⡤⠤⢤⣤⠤⡄
⢠⣤⣤⣤⣶⣾⣧⣦⣿⣦⣾⣤⣦⣿⠐⠶⠷⣼⠧⢼⠤⠾⠦⠴⠶⣣⣤⣤⣴⣷⣤⣦⣤⣦⣤⣤⢤⣤⣄
⢀⣀⣤⣀⣀⣦⣤⣤⣤⣤⣤⣾⣤⡄⠶⠀⣨⠔⠒⠚⠓⠠⣅⠀⠐⢻⣤⣤⣠⣄⣤⣤⣾⣾⣿⣇⣀⣸⡀
⠀⠈⠠⠈⣩⣅⣀⣄⣿⣿⣇⣈⣁⡽⢢⠊⠀⠀⠀⠀⠀⠀⠈⠓⡜⢈⣏⣭⣉⣹⣯⣭⣍⣉⣉⣩⡏⠩⠉
⠈⠟⠋⠛⠛⠉⠛⠛⠋⠉⠙⡟⢻⢡⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠘⠙⠛⢻⣿⠙⠛⠛⢻⣟⡿⢻⠻⣧
⠀⣀⣀⣤⣆⣀⣀⣲⣦⣄⣠⣇⣼⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆⣴⣀⣼⣿⣤⣴⣆⣸⣿⣧⣘⣃⣀
⠀⠀⠁⣿⢿⠉⠉⠉⡇⢽⡋⢸⠉⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⠈⠉⠁⠈⢹⣿⡏⠋⠉⠄⠀⢹⡇
⠰⠶⢶⣿⡿⠷⠶⣶⣷⠿⠶⠾⠶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡽⣾⠷⠶⢾⠿⠿⠿⠶⠷⠷⠶⠾⠧
⠀⠀⠈⠉⠉⠀⠀⠉⠉⠉⠀⠈⠁⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠉⠉⠉⠉⠀⠉⠉⠀⠉⠀⠈
⠀⡀⠀⠀⡀⠀⠀⠀⣀⢀⢻⣛⣛⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣛⡃⣀⣀⢀⡀⢀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠖⠒⠛⣋⠉⠉⠉⠉⠙⠛⠒⠲⠦⠤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢶⡿⢧⣀⠀⠀⠀⢿⠹⡇⠀⠀⠀⠀⠀⠀⣀⠤⢄⣀⠈⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⣠⡖⠏⡑⢯⣀⡞⢉⣷⡄⢀⠞⢿⣾⡀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠑⡄⠀⠀⠀⠈⠙⠳⣤⣀
⠀⠀⠀⠀⢀⣼⠯⡂⠁⠀⠀⠙⢦⡞⠀⠿⠈⢢⢻⡇⢱⠀⠀⠠⡞⠒⢣⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠈⣛⢿⣦⣄
⠀⠀⠀⢠⠟⡂⠅⠀⠀⠀⠀⠀⠈⢛⡀⠁⣠⣚⣚⡏⠉⠀⠀⠀⢇⠀⣠⣇⣶⣴⣷⣿⠁⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⣦⡀
⠀⠀⢀⡿⢑⠂⠀⠀⠀⠀⠀⣄⢲⢹⠃⡔⠋⠀⠀⡇⠀⠀⠀⠀⣀⣧⠉⣻⢿⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⡿⣿⣻⡽⣿⣦
⠀⠀⢸⠯⠐⠀⠀⠀⠀⡞⡧⠛⠄⠁⠀⡇⠀⠀⠀⠃⠀⣠⣴⠟⣽⣼⢊⡴⣩⠛⠢⣀⠀⠀⠀⠀⠀⣠⡾⢿⣛⠽⠗⠋⠉⠁⠨⡲⢕
⠀⠀⢸⠭⠀⠀⠀⠀⣀⣧⠀⠀⠀⠀⠀⡇⠀⠀⠰⠖⣩⣿⣿⢠⡿⠀⣫⢖⡁⠜⢴⣚⣦⣠⢴⣲⠿⠗⠋⠉⠀⠀⠀⠀⠀⠀⠀⠨⡑
⠀⠀⢸⡅⠀⠀⠀⢸⠦⠽⢧⠀⠀⠀⠀⢣⢀⣠⣴⣿⣿⣿⡇⢸⡵⠊⣀⡼⠀⠀⠚⢿⣿⡏⠉⠀⠀⠀⠀⠀⢀⣴⣿⢿⣶⣄
⠀⠀⢈⡧⠖⣒⣶⣬⣯⣉⣚⣷⠀⠀⠀⠘⢿⣿⠿⠟⣿⣿⣧⢊⡤⠊⠁⠀⠀⠀⠀⠀⢻⣷⠒⠒⢶⠒⠒⡞⡿⡈⠁⡀⠙⢮⠇
⠀⠀⡞⢸⣿⣿⡿⣿⢿⣿⣄⣽⣇⣀⣀⣀⠀⠀⠀⠀⣸⣿⠱⠋⠀⠀⠀⠀⢠⣆⠀⠀⣸⣸⠀⠀⠀⠀⠀⣇⣱⠣⡀⢉⣰⠞⢦⡀
⠀⠀⢣⡼⠟⢁⡀⠉⣸⡇⢣⠀⠀⠠⠞⠛⠳⣤⠞⣶⣿⡇⠀⠀⠀⠀⠀⢰⠁⣸⠀⡠⣿⣿⠀⠀⠀⠀⣷⠀⠈⣳⠋⢋⣡⣰⣴⣽⣄
⠀⣴⣿⣟⢳⣌⢀⣠⣿⢃⢸⠀⠀⠀⠀⠀⡾⠓⠚⠉⠉⣇⣀⣀⣤⠔⠒⠉⠉⢀⣠⡿⣟⣽⢶⣒⠒⠒⢿⠀⡴⠃⣠⡎⠳⣄⣀⣀⣀
⣼⣿⣿⡟⠛⢻⣿⣿⣻⠏⠛⠇⠐⡄⠀⠀⢹⠀⠀⢦⣤⡟⠀⣢⢰⠀⣠⣾⢿⡻⠕⠊⣿⣟⣩⡖⠒⢲⣾⠟⢀⣼⣻⠁⠀⣾⣿⡛⠛
⣿⣿⠏⠉⣿⠾⣿⣿⣿⠀⠀⠀⠀⢳⡀⠀⠈⡆⠀⠀⠛⠛⣿⣿⣿⣦⣷⣾⡇⠀⠀⠀⢿⣿⡝⡇⠀⢸⠁⣠⣾⣏⡇⠀⣸⣿⡦⡝⢦
⠸⣿⠀⢸⣿⣷⡼⣿⣿⡆⠀⠀⢀⣿⣿⣋⣥⣷⠖⣶⣶⣿⣿⣿⡿⠿⠛⠓⢻⠀⠀⣠⣿⣿⣷⡁⠀⣸⠾⠛⠋⡽⠀⢠⣿⣗⠭⠂
⠀⢿⠀⠘⣿⣿⣧⣦⣿⣧⣀⡤⠾⠛⠋⠉⠀⡟⠀⣿⠟⠋⠉⠀⠀⠀⠀⠀⢸⣀⣰⡿⠿⠛⠋⣇⠀⠙⢦⣄⠰⣇⠀⣼⣿⡪⠕⠀⢸
⠀⠘⣷⡶⠿⠿⢷⡿⣿⣉⣹⣀⣠⣤⣴⣶⣿⢟⣶⣷⠤⣄⣀⣀⣀⣀⣀⣀⡸⠏⠀⠑⢄⡀⠀⠈⠙⠛⢿⡿⠀⢳⣝⣿⡳⢮⣀⠀⣿
⠀⢠⣿⣁⣀⣀⣸⣄⣀⣀⣈⣉⣿⣿⣯⣽⣾⣿⣸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⣀⣀⣀⣀⣙⣦⣀⣀⣀⣀⣀⣀⣘⣿⣿⣿⣷⣭⣻⣿
⣿⣿⠟⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⣀⣙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣄⡀⠀⠀⠈⠉⠉⠉⠉⢹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⣿⣦⡀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠻⣿⣦⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠘⠛⠛⠛⠛⢻⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣷⣄⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠘⠻⣷⣄⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⠀⠀⠀⠸⠷⠶⠶⠾⢿⣷⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⣀⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠙⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⡶⣶⠀⠀⢸⣿⠀⠀⢀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠺⠿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⡶⡶⠖⠒⠒⠂⠤⠠⠄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⢞⡟⣱⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⠻⢻⠶⡶⣤⣄⣄⠠⠠⢄⣠⣴⣿⢋⠪⣎⡾⣞⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣄⠁⠀⠀⠀⠀⠀⠀⠉⠉⡿⡻⣫⢯⣗⣵⡺⠕⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠛⠷⢶⣤⣥⣘⡂⠀⠀⢸⣂⣢⣵⡿⢿⠋⠟⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡎⠻⣦⡀⡀⠀⠈⠉⠙⠛⠓⠺⠿⠻⠍⠈⠁⠀⠀⢀⣀⣉⡢⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠊⢻⣆⠈⠳⢶⠶⡶⢶⣿⡿⠻⠻⡻⣻⡛⣫⢛⢟⢛⠟⡝⠽⡝⡆⡂
⠀⠀⠀⠀⣀⢴⠎⡀⠀⠀⠀⠀⠙⢷⣦⢸⣫⢖⠃⠌⢈⢆⢘⢁⢀⢀⣑⢵⣕⣠⣢⣪⣾⡶⣣⣿⡿⠂
⠀⠀⠀⠀⢳⠤⠭⢴⣀⣠⠄⣀⠀⠈⣛⠿⢿⣿⡭⠭⡭⠭⠽⠾⠾⠭⠵⠲⠛⠒⠒⠟⠛⢉⣿⣿⡇
⠀⠀⠀⢠⠾⢤⣀⣀⠀⠘⠓⠒⠚⠐⠎⣇⠰⣯⢶⣿⣿⠻⣯⣤⣦⣐⣒⣂⣠⠤⢄⣀⣴⣿⣼⣿⠇
⠀⠀⠀⠀⠉⠁⠐⢫⢿⣷⣖⣲⢶⠤⢂⣧⣇⣿⡀⠠⠂⠀⠀⠀⠉⠉⠁⠀⠀⠀⣼⣭⣽⣿⣿⣯
⠀⠀⠀⠀⠀⠀⡠⡁⠀⠀⠀⠈⠉⠉⠉⠉⠉⠛⠿⠵⢶⣠⣬⠩⢉⣉⣃⠀⠄⠀⣏⣩⣿⣟⣏⣉⣻⠄
⠀⠀⠀⠀⠀⣰⣵⣷⣶⣖⣒⣶⡖⠒⠶⠦⠶⠶⠶⠤⠬⠬⠿⠿⠿⠛⢓⠺⠯⠶⣽⢞⠉⠙⠛⢋⢛⠕
⠀⠀⠀⠐⢴⡇⡏⡤⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠄⠀⢀⠀⠀⠀⠀⠈⠐⣤⠋⠀⠀⠀⠘⣗⠅
⠀⠀⠀⠀⠀⠣⣻⣤⣀⣂⣐⣀⣀⣀⣀⣤⡄⠤⠄⡠⠤⣤⡤⠀⣒⣒⣐⣒⣂⣤⣀⣒⣒⣒⣶⣶⣿⡗⡁⢆
⠀⠀⠀⠀⠀⠀⠈⠋⠙⠫⠛⠜⠛⠫⠛⠛⠛⠛⠛⠛⠑⠉⠋⠘⠁⠋⠋⠑⠑⠁⠁⠊⠊⠉⠀⠀⠈

⠀⠀⠀⠀⠀⠉⠀⠁⠀⠀⠈⠁⠀⠀⠀⠀⡀⣀⡀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠠⠤⠄⠠⠄⠤⠤⠤⠄⠀⠀⠀⢈⣫⠸⣧⠷⣼⢨⡇⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢶⣴⠃⢲⡎⢾⡂⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠂
⠀⠀⠀⠠⠠⠤⠴⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⡁⢈⠉⠉⠉⣈⠉⠁⢄⠀⠀⠀⠤⠄⠀⠤⠴⠐⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠒⠂⠒⠒⠂⠀⠀⠐⠒⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐
⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠈⠁⠌⡉⠀⠁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀
⠀⠀⠀⠀⠀⠉⡉⠀⠀⣀⠁⠖⠓⣢⡶⡉⡀⡆⣤⡤⣠⣴⠀⢄⣀⣤⠤⠴⠆⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡥⠤⢶⣾⡏⣿⣤⣶⣴⣼⣷⢾⢁⠀⠀⠚⢼⠉⢀⡷⠺⢧⣼
⠀⠀⠀⠐⠁⠀⠀⡴⢒⣒⣲⣾⠟⠀⣿⣿⣿⣷⣼⠗⠳⣹⣿⣿⣿⡟⠈⠀⣯⡽⠿⣿⠁⠒⢁
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠈⠉⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠁⠈⠀⠀⠀⠈⠉⠀⠒⠂⠀⠀⠀⠀⠀⠀⢀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠨⠕⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠉⠀⢀⠀⠐⠠⠁⠀⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠁⣂⠑⠒⠤⢄⡀⠀⠀⠀⠀⠀⡀⠴⢿⢯⡮⠯⠁⣀⣈⣉⣩
⠀⠀⠀⠀⠀⠀⠀⠰⢂⠂⠉⡀⠀⠀⠀⠀⠀⠀⠀⠌⡀⠀⠀⠀⢀⠀⠀⠀⠨⠁⡀⠀⠀⠀⠀⣀⡴⡚⢅⠢⠐⠁⠀⠀⠀⠀⣢⠄⠊⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⡢⠄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠛⢅⣌⡢⠐⠀⠀⠀⠀⣀⣴⣿⣁⢀⡀⢀
⠀⠀⠀⠀⠀⠀⠘⠠⣘⠀⠀⠀⠌⠸⠑⡒⠤⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⠲⡙⢕⠈⡢⠑⠀⠀⠀⠀⣀⠔⣪⠞⠃⠁⢀⡠⢖⡫
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⣢⠤⣀⠀⠀⠐⠁⠠⠁⡁⡉⠑⡐⡒⠒⢗⢖⠫⡣⢑⠄⡢⠑⠈⠀⠀⢀⠤⠒⢉⡴⠎⠀⣀⠤⣚⠥⢖⡩⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠢⢙⠀⠀⠘⠉⠃⠢⣄⣀⡀⠁⠈⠈⠈⠢⠪⠢⢑⠵⠪⠂⠁⠀⢀⣠⠔⠊⣁⡤⠚⢉⣠⠔⣫⢕⠪⠓⠝⣩⣴⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠵⡲⠠⠄⡄⣀⡀⠉⠓⠒⠤⢤⣀⣀⠀⠀⠀⢀⡤⠴⠊⠉⣀⠴⠊⣁⠤⢒⡩⢒⠩⡀⢁⡠⠖⠫⣂⠅⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢴⣈⣐⡀⠀⠀⠀⠀⠈⠈⠉⠃⠒⠶⢦⢤⣌⣉⡉⠋⠀⠀⣀⠴⠊⡡⢔⣫⠕⢊⢅⠈⣀⠴⢚⠍⡂⠕⠨⣐⣬⡇
⠀⠀⠀⠀⠀⠀⠀⢀⡵⠲⠨⠬⠭⢉⣛⣒⠒⠤⠄⣀⣀⡀⠀⠁⠁⠐⠉⠉⠙⢈⡠⠖⡩⠖⢋⡔⢈⣡⠔⠋⠕⠠⣐⠄⢲⡳⠫⢓⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀⡉⠁⣐⠂⠌⠉⠉⣒⠒⠦⠭⢭⢒⡲⠤⠤⠤⢖⡩⢖⣩⠴⣍⡭⡖⠋⢊⢤⡮⢖⣫⠵⣛⡭⡴⢗⣫⢽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠬⢁⠀⠀⠀⠈⠉⠀⠐⠀⠤⢉⣓⠒⣡⢔⣭⣤⠊⢀⣂⠔⠟⣊⡽⡚⠕⣃⠽⣊⡭⢞⢏⡥⣾
⠀⠀⠀⠀⠀⠀⠀⠀⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠐⠠⠄⠀⣀⠀⡴⣛⠛⠁⣢⣡⡴⠖⣫⠅⠜⠐⣡⠴⠛⢁⣬⠖⣫⠃⣿⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠐⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡾⣧⣶⡾⣫⡩⡶⢚⠡⠲⢛⠊⣀⡡⣮⡪⠖⣈⠠⡢⣿⣈⠉⠉⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢀⠉⠐⠒⠠⠄⣀⢀⣿⡿⣋⠬⡊⢄⠠⡀⠅⣠⣔⡼⠊⠉⠀⡢⣀⣠⠾⠛⠃⠀⠀⠀⠠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠂⠤⠐⠢⢄⣀⠉⠐⠠⠄⠀⣈⣀⠷⠋⡀⠔⡈⢑⣨⢴⠾⠊⠑⠈⠠⣀⡼⠚⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠷⠤⣄⡀⠀⠀⡀⠀⠀⠀⠀⠐⠀⠁⠒⠀⠀⠂⠠⠤⡓⠥⢂⠡⠒⡩⡐⢁⡌⢂⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠦⣀⡀⠉⠀⠀⠀⠁⠀⠤⡀⠄⠀⠀⠀⠀⠤⢀⡤⠤⠔⢜⠡⡀⢁⠠⣖⣽⣞⠛⠓⠉⠉⠀⠄⠀⠀⠄⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠢⢄⣀⡀⠀⠈⠉⠂⠤⠀⠀⠀⠀⠐⠀⠠⠐⢀⣀⣷⠚⠋⠀⠀⠀⢀⡀⣀⣀⣀⡂⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠈⠐⢀⡁⠂⠀⠀⢰⠘⣨⡷⠛⠉⠉⠉⠀⠀⠂⠀⠀⠀⠀⡀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠂⠸⠿⠅⠤⠌⠭⠭⠭⠭⠍⠩⠶⣂⣂

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣷⣶⣤⡀⠀⣀⣠⣤⣴⣶⣿⣿⣿⣶⣦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉⠁⠀⠀⠈⠙⠻⣿⣦⡘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣄⣀⡀
⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠂
⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⢛⣉⣅
⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢤⣍⣙⠛⠿⠛⣋⣩⣤⡶⠶⠿⣿⣿⡧
⠀⠀⠀⠀⠀⠈⢿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢠⣶⣦⣬⣉⢋⣩⣥⣴⣶⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⣀⣠⣴⣾⡿⠏⣰⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠟⢛⣉⣴⣾⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⡿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢰⣶⣾⣿⣿⣿⣿⣷⣶⣶⣦⢠⣴⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣦⣼⣿

⠀⠀⠀⢀⣠⣤⣤⣤
⠀⠀⠀⡏⡴⠋⠉
⠀⠀⠀⢻⡸⡄
⠀⠀⠀⠀⢣⣳⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣦⣄⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠹⡅⢸⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⢹⠹⡄
⠀⠀⠀⠀⠀⡟⢿⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⢀⡞⠀⠉⠓⠦⣄
⠀⠀⠀⠀⠀⢱⡘⣧⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢠⠏⠀⠀⠀⠀⠀⢸⣆⢰⣆
⠀⠀⠀⠀⠀⠀⢧⢉⣁⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠁⠀⠀⠉⠛⣿⣿⣿⣿⣧⠀⠀⢸⠀⠀⠀⠀⠀⣀⣾⣿⢸⣿⣤
⠀⠀⠀⢀⣤⣶⣾⣿⣟⡁⢷⠀⠀⠀⠀⠀⠀⠀⢸⠉⠙⠟⠛⣫⠛⠷⠶⠚⢻⣿⣿⡿⠀⠀⠈⢳⣦⠰⠖⠛⢡⣿⣟⣸⣿⣿⡇
⠀⠀⢀⣼⣿⣿⣿⣿⣿⣧⣬⣇⠀⠀⠀⠀⠀⠀⢸⣀⡘⠛⡸⠁⣇⠸⠗⠀⣼⡙⡿⠁⠀⠀⠀⠸⣿⣧⠀⢠⣾⣿⣿⣿⣿⣿⠇
⠀⠀⠈⠿⣿⣿⣿⣿⣿⠛⠡⠼⡀⠀⠀⠀⠀⠀⠀⠉⢩⠉⢥⡄⠉⠉⠉⠛⣯⠜⠁⠀⠀⠀⠀⠀⣿⣿⣶⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠈⠻⣿⣿⣿⣆⠃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠈⡆⠤⣶⠴⠂⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣦⣤⡼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⣉⣁⣤⠞⠁⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣹⠀⠀⠀⢹⡍⠂⢄⡀⠀⠀⠀⢰⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⢀⡤⠦⠤⡄⠀⠀⠀⠀⠀⠀⢀⠀⠁⡼⠀⠀⠁⠀⠙⢦⠀⠙⣄⠀⠀⢸⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠓⠒⠚⠀⠀⢳⣄⣀⡀⠶⣒⣨⡇⢀⣜⠀⡟⠺⢄⣀⣸
⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⠀⠀⠀⢠⠇⠀⠀⠘⠂⢰⡄⣿⣿⣿⣿⣿⡟⡔⠁⠀⢹⠃⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡄⠀⢀⠎⠀⢀⡀⠀⣠⠏⠀⣿⣿⣿⣿⡟⠀⢹⡀⠀⡞⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣇⠀⡞⠀⠀⢸⡇⡰⠁⠀⠀⣿⣿⣿⡟⠀⠀⣠⠃⣸⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⡼⠁⠀⠀⣸⠃⠸⡀⠀⠀⣿⣿⡿⠀⢀⡜⠁⢠⡏⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠸⡇⠀⠀⢠⡏⠀⠀⠱⡀⠀⣿⣿⠁⣰⠋⠀⠀⡾⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣷⠀⢀⡞⠀⠀⠀⠀⠱⣘⣿⢇⡼⠁⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠸⡄⠀⠀⠀⠀⠀⢹⣿⠞⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⡿⠀⠀⣧⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⠀⠀⢰⠇⠀⣠⣿⠀⠀⠀⠀⠀⢸⡃⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠈⢀⡼⠃⣿⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⢀⡴⠋⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⠴⠚⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢷⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠞⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣈⣉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⢠⣾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⣿⣦⣄
⠀⠀⠀⠀⣿⣿⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⠀⣿⡇⠙⢿⣷⣄
⠀⠀⠀⠀⣿⣿⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣿⡇⠀⠀⠉⠻⣿⣦⡀
⠀⠀⠀⠀⣿⣿⠀⢠⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡄⠀⠀⣿⣇⠀⠀⠀⠀⠈⠻⢿⣦⣄
⠀⠀⠀⠀⣿⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣾⣿⣿⣷
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣿⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆⠀⢸⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⠀⠀⢀⣴⣶⣄
⠀⠀⠀⠀⣿⣿⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇⠀⠀⠀⢀⡐⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣦⡙⢿⡿⠋
⠀⠀⠀⠀⣿⣿⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⣿⣿⠀⠘⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⣿⣿⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⣿⣿⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠻⣿⣿⣿⣿⠟⠀⠀⣠
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣌⠻⠟⠁⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠘⢿⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⣿⣿⣿⣿⣛⣻⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣻⣛⣿⣿⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣐⣒⠚⠂⠀⠂⠤⠀⠀⠉⠈⠁⠐⣢⡀⠀⢀⣀⢀⡔⢒⠞⠑⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣀⡼⠄⠀⠀⠀⠀⡜⠁⠠⠀⢀⡀⠀⠀⠸⠓⡨⠀⠭⠘⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠠⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⡠⢞⠓⠊⠀⠀⠀⠀⠀⣸⡵⠟⠁⢑⠢⠄⠑⠂⠜⣁⣀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠄⠀⠀⠀⠀⠇
⠀⠀⠀⠀⣠⡔⠴⠛⠁⠀⠀⠀⠀⢀⣀⢀⢨⠀⠠⠴⠀⠂⢡⡯⠖⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡄⠀⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠤⠥⠤⠚⠪⡢⣀⣄⡄⠀⠚⠔⠀⠂⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⣀⣴⡂⠖⠓⠒⠁⠀⠀⠀⠀⠈⠀⠀⢐⡂⠀⢀⠀⠀⠀⠄⠀⠀⠃⠀⣀⠒⠀⠀⠀⠀⠲⠦⠤⣤⣤⠤⢀⠸
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠒⠏⠀⠀⣆⣆⢀⠂⠁⠀⠉
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠜⠃⠀⠀⢀⠀⠀⠀⠸⠀⠡⠄⠀⡀⣧⠈⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡤⠀⠀⠀⢀⣀⣀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠈⠂⠛⠉⣣⢤⠨⢀⠀⠀⢠⡆⠂⠀⠀⠐⡄⠀⠀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠊⠉⠐⠀⠀⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⢹⡀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠈⠵⠦⡀⠀⠐⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣇⣀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⠈⠀⢀⠀⠀⠀⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠌⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠃⡅⠀⠀⠀⠀⠒⠁⠀⠀⠘⠄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⢠⠃⠈⠁⠙⡁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢨⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠂⠬⠀⠀⠀⠄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⣀⡀⠀⣀⠤⠅⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⢀⠀⡀⠸⠴⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠠⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠞⠁⢀⠀⣈⡨⢷⣤⠄⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠴⠢⠶⠀⢲⠟⠸⡄⠁⠐⠧⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⡠⠞⠁⠀⠀⠀⠀⠲⠀⠂⠘⠁⠀⡀⢁⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠒⢍⣁⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠂⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⠀⠀⠀⠒⢜⠂⣀⠀⢠⣿⣿⣷⣦⣿
⠀⠀⠀⠀⠀⠀⢀⡤⠊⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡶⠈⠀⣠⣴⣌⠳⠀⠀⠀⡉⢀⠸⣿⣿⣿⡟⠛
⠀⠀⠀⠀⠀⠀⠁⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⡄⡀⣀⠻⣿⡿⠀⠀⢀⡸⣰⣼⡆⠀⠉⠉⠁⢹
⠀⠀⠀⠀⣼⠀⠜⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠋⠓⠈⠓⠀⠀⡠⠀⠉⠀⠉⠈⠁⠀⠠⢀⠉
⠀⠀⠀⠀⠈⠢⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⠐⣠⠄⠄⠤⠓⠟⠈⠄⠀⠀⠀⠄⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠺⠉⠁⣀⡠⢼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⢠⠘⠀⠀⠰⢤⠀⢀⣤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣀⣀⣀⡀⢤⠀⢸⡄⠀⢀⡀⠃⠀⣆⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⢀⠈⠤⠀⢺⡇⠀⠈⠆⠀⢦⡄⠀⡧⠤⢶⣧⣀⣰⠿⠏⠀⠀⠀⠐⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣀⣤⢸⠿⡆⠀⠐⣦⠶⠇⠀⡗⡀⠾⠇⠈⢀⡄⠀⠀⠀⠀⠀⣘⠁⠀⠄⠘⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠉⠀⠛⢇⣄⠀⠀⣇⠀⡂⠀⢂⠀⢰⡃⠀⣀⠙⠤⡀⠁⡀⣛⠃⠐⢀⣴⠶⠒⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⡈⠏⠀⠀⡏⠁⣠⠀⢠⠀⢠⡇⠨⠿⣄⠐⠒⠀⡒⠀⣰⠂⠹⠁⠀⠀⣀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡆⠀⠀⣿⠉⢉⠀⠈⠁⢤⠄⠐⡇⠈⠏⣉⠅⠁⠀⠀⠐⠀⠀⠈⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⣤⠈⠃⠀⠀⠈⠀⢘⠀⠘⠀⣉⠤⢚⠈⠰⠒⠁⠀⠚⠁⠀⢠⡄⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡅⢻⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠚⣿⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡏⠀⠀⠀⠀⠀⠀⠀⠘⡎⠀⡠⠀⠀⢀⡠⠂⢘⠉⠹⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⡞⠀⠀⠀⠀⠀⠀⠀⠂⡤⠟⠋⠀⡀⠀⠁⠀⠀⠸⠅⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡄⠂⠀⠀⠀⠀⠀⠀⠀⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣁⣀⣀⣀⣠⡇⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⣶⣾⣿⢢⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣾⣿⣿⣿⡟⢼⠿⢟⡛⠃⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠸⣿⣿⣿⣿⣷⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⠛⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠶⠶⠶⠖⠒⠀⠀⠐⠖⠒⠒⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠖⠲⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠁⠉⠈⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠇





⣿⣿⡿⠿⣿⡿⢿⢿⣿⡿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣶⣾⣶⣶⣴⣿⣵⣿⣴⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⢀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⡹⠟⣧⣦⣤⡺⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣤⣦⣜⣛⣠⡘⢳⡄⠈⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⠾⢿⣿⣿⣿⣿⡟⠉⢠⡜⠃⠀⢿⠋⠁⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣦⣿⡿⢧⠀⠀⠀⠀⠀⠀⡄⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⣼⠟⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⡏⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⢄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠻⠿⢿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣦⡤⠀⠀⣀⠀⠠⠘⠃⡄⠐⠀⠄⢀⣀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⣀⡩⠥⠄⠒⠒⠀⠀⠀⠀⠀⠒⠂⠤⠬⢉⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡉⠥⠀⠀⠀⠀⠀⠀⠠⠍⣉⡐⠂⠤⠠⠁
⠀⠀⠀⠀⠀⠀⣴⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠭⠤⠔⠒⢀⠀⠀⠀⠀⠀⡐⠒⠠⠆⡍⢈⡁
⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⣈⠩⠥⠄⠀⠀⠀⠀⠀⠠⠬⣷⣷⣷⣶⣦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠠⠥⠄⠒⢂⣉⠀⠀⠀⠀⢈⣉⡐⢻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠒⢂⣉⠭⠠⠄⠀⠀⠀⠀⠀⠠⠠⢽⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣠⣄⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣠⣀⣀⣀⣀⣀⣿⣿⣿⣿⣀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⢾⣆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠀⠤⠤⣦
⠀⢻⣆⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⣀⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣤⡀
⠀⠀⠻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⣠⠊⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣿⣿
⠀⠀⠀⠹⣧⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠔⠂⠈⠀⠀⠀⠀⢀⣿⣁⣤⠀⢀⣴⣶⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⠛⠛⠋
⠀⠀⠀⠀⠙⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡷⠛⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠘⣷⡄⠀⠁⠀⢀⠠⠐⠉⠀⠀⠀⢀⣠⣤⣶⣾⡿⠟⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢿⡄⠀⠘⠀⠀⠀⠀⣀⣴⣾⣿⠿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⡇⠀⣠⣴⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡷⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠉⠁⠉⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⡰⡠⢂⠘⠀⠠⢰⡄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠁⠀⠀⠀⠀⠠⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⣀⡠⡄⢰⡼⠁
⠀⠀⠀⠀⠀⠀⠰⠉⠀⠠⠀⠀⠀⢪⣾⣦⠀⠀⠸⣿⣷⠂⠸⠀⠀⠈⢠⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠁⠲⠀⠈⠛⠛⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠡⢠⡀⠀⠀⠀⠀⠀⠀⣠⣤⣀⠀⠀⠀⠀⠀⠀⣿⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢼⠠⠏⠤⠀⠲⠀⠀⠀⠀⠰⠂⠩⠙⡄⠀⣀⠀⠀⠀⣧⣴⣾⡀⠀⠀⠀⠀⠀⣿⢸⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⢦⢸⡴⠂⠈⠉⠀⠀⠂⠀⠰⠁⠀⠀⣰⠛⡆⠀⠀⠃⢤⠔⠃⠀⠀⠀⠀⠀⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡇⢠⡨⢶⠷⠞⠢⠂⠄⠊⠀⠀⠀⠸⠉⠉⠹⡄⠀⠀⠀⠀⢀⡠⠖⠆⠀⠀⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⣶⠞⠛⢻⡇⠈⠁⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠀⠀⠀⠀⢀⡇⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠑⠦⠐⠀⠉⠀⡤⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠓⠛⠁⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠔⠋⠉⠀⠤⠀⠈⠁⠀⠤⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣥⣠⣤⣀⣉⣀⣤⣠⣬⠵⠞⢃⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⣰⠀⢀⠀⠀⠀⠀⠄⠀⠀⠄⢀⠁⠁⠉⠈⣀⢠⠀⠀⠠⠀⡄⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡀⢴
⠀⠀⠁⠀⠀⠀⠚⠀⢲⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄
⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠈⠀⠈⡴⠀⠀⠀⠛⠠⠀⠟⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠋⠉⠀⠘⡿⢻⡏⠀⠉⠹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⣀⠁⠀⢢⡀⠀⢠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣠⣄⣼⣿⣿⣶⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡛⠉⢹⣟⣻⣿⣿⣿⠛⠛⢻⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠏⢷⣇⠸⣿⣿⣿⣿⡏⢠⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣻⡿⠿⡿⡿⣟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣍⣟⣛⣽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣭⣿⣭⣼⣓⣀
⠀⢠⣶⣢⣚⣉⣛⡋⠛⠢⢄⡀⠀⢀⡴⢿⣭⣉⡋⣭⣯⣛⣋⣹⡿⢦⡀⠀⢀⡠⠔⠛⢛⣛⣛⣓⣔⣦⡄
⠀⠀⣛⣛⠿⣟⣭⣝⣿⣴⣾⣭⣛⠲⣶⣶⣶⣆⣙⡠⠜⣁⢰⣶⣴⣶⠶⣛⣭⣶⣴⣿⣫⣽⣻⠿⣛⣛
⠀⣾⣿⣿⣧⣹⣿⣻⡿⡋⢼⠿⣟⣀⣬⡿⠳⣿⣿⣿⣿⣿⣿⡾⢿⣥⣀⣻⡿⡧⢹⢿⣟⣿⣏⣼⣿⣿⣷
⠀⢻⣿⡿⠁⣿⣿⡿⡻⣳⢌⠣⣍⡤⠤⠤⠦⡜⣿⣿⣿⣿⢣⡤⠤⠤⣄⣉⡜⡡⣞⢯⣿⣿⣿⠈⢿⣿⡟
⠀⢸⣿⡇⠀⠈⢹⡧⣇⢵⣺⡷⢈⣹⣷⠒⢾⣿⢨⣥⣬⡅⣿⡷⠔⣾⣏⡁⢾⣓⡮⣸⢼⡟⠁⠀⢸⣿⡇
⠀⠈⣿⡇⠀⠀⠘⢳⢈⡳⣝⡛⠉⠁⠀⠀⢀⣩⣾⣿⣿⣷⣍⡀⠀⠀⠈⠉⢛⣫⢞⣁⡞⠃⠀⠀⢸⣿⠁
⠀⠀⣿⡇⠀⠀⢰⠘⣎⢿⣢⠝⠳⢶⣼⠿⠋⡙⣹⣿⣿⣇⢋⠙⠾⣦⣶⠞⠫⣔⡮⣳⠃⡆⠀⠀⢸⣿
⠀⢰⣿⡇⠀⠀⠘⡄⣼⢷⣄⣀⣤⠶⠒⠋⣉⠔⣹⣿⣿⣏⠢⣉⠙⠲⠶⣤⣀⣠⡾⣧⢠⠃⠀⠀⢸⣿⡆
⠀⢸⣿⡇⠀⠀⢠⠻⣥⠶⠋⠛⢿⠶⠶⠛⢛⠔⢉⣿⣿⡉⢢⡛⠛⠶⠴⡿⠛⠙⠶⣨⠟⡄⠀⠀⢸⣿⡇
⠀⢸⣿⡇⠀⠀⠘⡴⠙⡳⣔⣟⠁⢀⣀⡴⠣⣤⢾⣿⢿⡷⣤⠜⢦⣀⠀⠈⣻⣢⢞⠋⢮⠃⠀⠀⢸⣿⡇
⠀⠘⣿⡇⠀⠀⠀⡝⢾⡋⠈⣙⡻⠛⠉⣀⠞⢡⢊⡿⢷⡵⡌⠳⡀⠉⠛⢻⣋⠁⢙⣷⢫⠀⠀⠀⢸⣿⠃
⠀⠀⣿⡇⠀⠀⠀⢹⡀⣽⢾⣭⣤⡴⠞⠁⣹⠋⢾⣿⣷⣖⠹⣍⠈⠳⢤⣤⣭⡷⣯⢀⡏⠀⠀⠀⢸⣿
⠀⢰⣿⡇⠀⠀⠀⢠⡿⣍⢠⡴⠀⠀⣀⣼⠇⠤⢴⣶⣾⡶⠴⠸⣦⡀⠀⠀⢦⡄⣩⢿⡄⠀⠀⠀⢸⣿⡆
⠀⢸⣿⠀⠀⠀⠀⡌⢇⣼⠓⠶⠶⡾⣿⡟⠀⠀⢙⣭⣭⣇⢀⠀⢻⣿⢷⠶⠶⠞⣧⡰⢡⠁⠀⠀⠈⣿⡇
⠀⣿⣿⡄⠀⠀⠀⠁⡼⢧⡀⢀⣺⡵⠋⠀⠀⠀⠼⠿⠿⠿⠀⠀⠀⠙⢮⣗⠄⢀⡼⢧⠈⠀⠀⠀⢠⣿⣿
⢰⣿⣿⣿⣄⠀⠀⠀⡇⢄⡹⠉⣡⡾⠀⠀⠈⠁⢻⣿⣾⣟⠙⠋⠀⠈⢷⣌⠙⠏⡠⢺⠀⠀⠀⣠⣿⣿⣿
⠘⢿⠿⣿⠍⠀⠀⠀⠙⣆⡭⢉⡿⠁⠀⠀⠀⠀⠉⠉⠍⠉⠀⠀⠀⠀⠈⢿⡉⢭⣰⠋⠀⠀⠀⢩⣿⠿⡿
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣬⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠐⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠀⢻⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠠⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠀⢸⡇⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠈⣧⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠀⢻⡄⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⡇⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢰⡇⠙⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠁





⣿⣿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠋⠻⡄⠀⣠⠴⠂⢲⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⠀⠀⣿⠟⠁⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡁⠀⠀⠀⠀⠴⣾⠁⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⡴⡿⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⢀⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⣼⢃⡴⠚⠙⠲⣤⡀⣾⡟⠀⠀⠀⠀⠀⠹⣆⠀⠚⠉⡉⠈⠁⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠀⠀⠀⠀⠈⠙⢯⡿⢦⡀⠀⠀⠀⠀⠙⢧⡀⡿⠙⡟⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣏⣿⡄⠀⠀⠀⢀⣀⣭⡇⢀⡇⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠶⠦⢤⣽⣷⣿⠤⠖⠚⠉⠉⠀⠙⠛⠓⠞⠷⢤⣤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠴⠶⠶⠶⢦⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠛⠛⠲⠶⠦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠷⢦⣤⣤⣤⣤⣤⣤⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣄⣀⠀⠀⠄⢀⡀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠛⠻⠶⢾⣷⣤⣬⣵⣤⣤⣵⡶⠶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠃⣙⢦⡀⠀⣠⡟⠀⠀⠶⠖⣻⣿⠟⠓⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠘⠁⣿⠂⢙⡾⠋⠀⠀⣠⡴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⣿⡶⠟⠁⣠⡴⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣾⠀⠀⠸⣧⠶⠻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⣿⣄⣤⠶⡿⢲⣤⣿⡄⠀⠀⠀⠀⠀⠀⠀⠰⣎⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⣽⣷⣭⡙⢲⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠀⣈⡝⢿⡀⠙⢿⡈⣿⡄⠀⠀⠀⠀⠀⠀⠀⢰⡿⣇⠈⠻⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣄⣠⣶⣶⠾⠷⠿⠿⣷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⡯⠀⣿⠷⢼⡇⢰⠜⠃⡇⢻⡀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣷⣄⢹⣦⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠠⣾⣥⣤⣤⣤⡴⠛⣻⣶⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⢳⠀⠸⡇⠈⡇⢸⠀⢸⡿⣾⡁⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠙⢷⣜⣧⡀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠸⡇⠀⢷⠀⣿⢸⠇⢸⣇⣿⢷⡀⠀⠀⠀⠀⠀⠀⠀⣟⠳⠦⣄⡈⠻⣷⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠸⡇⢸⡿⠀⠈⣿⢹⣆⣿⣤⢴⣶⡤⠶⣶⣆⢹⣷⠤⢀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⠛⠋⠛⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⢠⣤⢿⡀⠀⠸⡇⠀⣿⣸⣧⣤⠶⠛⠉⠉⠀⠀⢰⣿⠀⠀⣿⡽⠦⠿⠄⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠛⠶⠖⠲⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠈⠃⠀⠀⢿⡤⠼⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⣀⢹⡗⣶⠶⢆⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣒⣚⠛⠛⠛⠛⠛⣦
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠈⣧⣴⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣉⡚⣣⡇⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠁⠀⠈⠙⠓⠶⠶⠞⠁
⠀⠀⠀⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠸⣿⡃⠀⣀⣀⣀⣀⣀⣀⣀⡀⢸⣿⡟⠉⠉⠁⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⡏⠙⣿⠉⢻⣏⣉⣠⡤⠶⠚⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⢸⡇⠀⠀⣀⠀⠀⠀⠀⢀⣀⣤⠾⠃⠀⢿⣆⣠⣿⠀⠘⣏⠉⠀⠀⠀⠀⣿⣤⠤⠤⠴⠶⠶⠶⠖⠚⠛⠛⠛
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⢸⡇⠁⣿⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⢸⡇
⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⣿
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⠓⠶⠶⠂⠀⠀⠀⢸⡇⠀⣿⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠸⣧⣤⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢿⣤⠤⣤⠼⠟
⠀⠀⠀⠀⠀⠀⢸⡇⠈⠉⠉⠉⠉⠉⠛⢛⣻⣿⣿⣿⠿⠧⠤⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⢸⡁⠀⠀⠀⠒⠒⠂⣉⡿⠋⠀⣮⠈⠛⠓⠶⠶⢠⡿
⠀⠀⠀⠀⠀⠀⣾⡀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⣿⠀⠀⠀⠀⢀⡾⠁
⢀⣀⣀⣤⣤⣤⣿⣦⣀⣀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣀⣤⠴⠶⢿⡷⠻⠛⠒⠒⢠⡄
⠻⣤⣀⣀⣈⠹⣦⣹⣯⣭⡿⠿⠛⠛⠋⠉⠁⠀⠀⠉⠉⠐⠶⠾⠶⠧⠶⠶⠾⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠤⠴⠶⠶⣒⣒⣒⣛⣛⣛⣉⡋⢙⣓⣒⣓⣚⣒⣒⣒⣒⠒⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢧⡿⠋⢉⣉⣉⣉⣉⣉⠛⢛⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣿⠏⣿⢹⣿⡏⣽⠃⣼⠁⣿⡟⣸⣿⢸⡇⠀⠀⠀⠀⠤⠤⠤⠀⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⣿⠀⣧⠘⢫⠇⠏⡇⠃⡆⠛⡑⣿⡏⣿⠀⠀⠁⠀⠀⠀⠒⠐⠂⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣿⣷⣿⣶⣾⣶⣾⣷⣾⣷⣾⣷⣼⣷⢻⡇⠀⢀⣀⣀⣀⣀⣀⣀⣘⣷⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⣤⣤⡤⠴⠶⣿⣿⣭⣽⣿⣿⣿⣛⣛⣛⣛⣿⠿⠿⣿⣾⣷⣀⣬⣭⣭⣥⣤⣶⣶⣶⡶⠿⢃⡀
⠀⠀⠀⠀⠀⠀⠀⡾⢋⣉⣉⣉⣀⡀⠀⠀⠠⠀⢠⣭⣥⡀⢙⣋⡉⠙⠛⠛⠛⠙⠿⢛⣛⣋⣉⣩⣭⠍⠩⠄⣐⣀⣀⣹⣥⣿⣦⡽⠟
⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⡉⠉⢉⣉⡙⠛⠛⠛⠒⠶⠶⠶⠦⢤⣤⣭⣉⣀⣀⣈⣉⣭⣭⡴⠶⠶⠒⠛⠛⠋⠉⠉⠁⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢸⡇⢿⣒⣀⡀⠀⠀⠠⠤⠌⠉⣀⠀⠀⠀⠐⠒⠒⠀⠀⣼⠉⣁⡀⠀⠀⠈⢉⡉⠀⠠⠤⠀⠀⠀⣀⣀⠠⠦⠼⠲⣤⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠻⠦⣤⣤⣭⣉⣉⣐⡒⠒⠲⢦⡴⠞⠛⠛⠛⠻⠷⣬⣏⠀⠀⢀⣀⣀⠠⠤⠄⠐⢒⣂⣈⣭⣥⣤⣶⠶⠾⢿⣛⣻⣿⣿⣿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃⠄⠨⣛⠿⣭⣑⠢⠍⣛⠦⣥⣤⣴⣶⣶⣿⣿⣟⣛⣯⣭⣷⠶⠾⠟⠛⠛⠉⠉⠅⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⣸⣿⣿⠋⠿⢹⣿⣿⠃⡌⠀⠀⠀⠈⠐⠨⢝⡳⠦⣍⡒⠭⣛⠿⣿⣿⣿⠋⠉⠉⠀⠀⠀⠀⠈⠅⢀⡀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣹⣿⣠⣿⢰⣞⣿⠇⡐⠀⣄⢰⣠⢀⢀⡀⠀⠀⠑⠂⠍⡓⢦⣝⡲⣭⡻⢦⣄⠀⣀⠤⠔⠀⠀⠀⢀⣀⣭⠴⢞⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣿⣿⣿⣿⣿⣿⡏⠰⠀⠹⠏⠋⠙⠘⠋⡷⣟⡀⠀⠀⠀⠀⡙⢨⠏⠈⠏⣸⠏⢠⣐⣢⠤⠶⣚⣋⣭⣴⠶⠟⢿⣉
⠀⠀⢀⣠⣤⣶⣾⣿⣿⣇⣀⠀⠉⠉⣹⡟⠀⠁⠀⠀⢠⣠⠄⡀⠀⠀⠉⣴⢤⡄⠀⡰⢀⡏⡐⡜⣰⣿⣛⣭⣷⡾⠿⠛⠋⣉⣠⣤⣶⣾⣟⣿⣿⣿⣶
⢠⣾⡛⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⡄⣀⢀⠄⡈⡉⠛⠷⢏⣴⣠⠀⠻⠂⠠⢀⡞⠀⡹⢰⡿⠛⠋⣉⣠⣤⣶⠾⣟⣿⣽⣾⠿⢟⣋⣭⢶⡿⠋
⣿⣿⣿⣿⣿⣶⣶⣤⣭⣍⣛⣛⢿⣿⢁⠬⣶⡀⠀⠀⠁⠈⠋⠢⢤⡈⠁⠀⠀⠀⠀⡼⠁⢰⢡⣿⣶⣾⣿⣿⣿⣿⣿⠿⢟⣫⣥⡶⠾⠛⠉⠐⣘⡧⢤
⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⡞⠀⠹⣿⣦⣼⣦⠀⠀⡀⠀⠠⡀⠀⠀⠀⣼⠁⢠⢇⣾⣿⣿⣿⡿⢛⣫⣥⡶⠟⠛⠉⠀⣀⣤⠴⣚⣫⣽⠖⠋
⠀⠈⠉⠛⠛⠻⠿⣿⣿⣿⣿⣿⠏⡼⠁⠀⢀⢀⠉⢛⣿⢻⣿⣿⣀⠀⠈⡄⠘⣰⠃⢀⠎⣾⣿⣛⣺⣿⣿⠋⠉⠀⣀⣤⠴⢚⣫⣥⡶⠛⠋⠉
⠀⠀⠀⣠⣤⣤⣴⡏⣿⠦⣼⡟⠐⠀⠀⢀⠀⠈⠻⢾⣿⣁⠀⠙⠛⠛⠛⠡⢰⠇⠀⡜⣸⣿⣿⣿⣿⣿⣯⠴⢚⣫⣥⣶⠿⠟⣻⣿⣶⣦⣤⣄⣀
⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⡿⢡⠃⠀⠀⠈⣑⠂⢤⡀⠈⠉⠙⠂⠀⠀⢀⢣⡟⠀⡸⢠⣿⠿⠿⠿⣿⣿⣷⡿⠿⠛⣉⠠⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣷⣄⠀⠀⠀⠀⢀⣙⠒⠌⢑⠢⢄⠀⠀⠀⡌⡾⠁⢠⢃⣿⠟⠓⠒⣶⠛⣉⠡⠀⢀⡠⠄⢂⣩⠤⢞⣻⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣶⣄⡀⠀⠈⠉⠀⠀⠀⠀⠀⠀⡘⣼⠃⠀⡎⣾⣯⣁⠂⠈⢫⠅⠀⢀⣠⠥⠒⣈⣥⣶⣾⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⣷⣤⣤⣤⠴⢦⣄⡀⠀⢰⠏⠀⡸⣸⣿⣤⣄⣉⠉⠙⠓⢋⣩⣴⣶⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⡤⠶⠖⠛⠋⠉⠉⠀⠀⠀⠀⠈⠙⠶⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⡾⠿⠛⠋⠁
⠀⠀⢀⣤⠶⠚⠛⠋⠉⠀⠀⠀⠀⢀⡀⢻⠤⢦⡴⢜⡶⠛⠚⠁⠀⠀⠉⠻⢿⣿⣿⣿⠿⠟⠋⠁
⠀⢠⡟⠡⡀⠢⡀⠀⠀⢤⡐⡦⠽⠒⠋⠁⠀⠀⠀⠀⣄⠠⣄⠲⣤⣷⠤⠀⠀⠈⠙⠷⣦⣄⡀
⠀⣾⠃⡄⠈⠳⢄⠀⠀⡈⠁⠀⠀⢀⡀⢠⣌⣳⣬⠷⠚⠓⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣤⡀
⠀⣿⡀⠿⣇⣠⢀⠉⠢⣀⠀⠀⠀⠈⠟⠒⠋⠁⠀⠀⠀⡀⣤⠤⣒⣦⡬⠳⠖⠚⠛⠁⠀⠀⠀⠀⢈⡛⢦⡀
⠀⠈⢳⣄⡘⠹⢾⣱⡆⠈⠑⢄⡑⠢⠀⠀⠀⠀⠀⠀⠄⠒⠒⠈⡭⠶⠒⠛⠀⣀⣀⡤⠴⠖⢚⣋⣩⣤⡾⠃
⠀⠀⠀⠈⠻⣶⣌⠙⠀⣖⣠⢀⠈⠢⢄⠐⢦⡀⠀⠀⠀⠀⢀⠀⠀⠴⠀⣛⣩⣭⡴⠶⠚⠛⠉⠍⣴⡟
⠀⠀⠀⠀⠀⠀⠙⢷⣤⡉⠙⢿⣶⣠⠀⠑⠦⡀⣠⠀⣀⣭⣤⠶⠶⠛⠛⠉⠁⠀⠀⠈⠁⠰⠆⠀⢻⡿⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠈⢻⠀⣀⠀⣰⠃⣾⣏⣀⠀⠤⠐⠂⠀⠄⠀⠀⠀⣀⡀⠤⠔⢒⣋⣡⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⡀⠛⠁⡟⢸⣿⠤⠐⠒⠀⢀⡐⠤⠐⢒⣈⣭⣤⡶⠾⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣼⡇⠘⠿⠐⢒⣈⣥⣤⡶⠾⠛⠛⠉⠉

⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⢸⡏⠀⠀⠀⠀⠀⠉⠉⠙⠛⠲⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠶⠚⠛⠉⠉⠁⠀⠀⠀⠀⠈⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄⡀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢶⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠈⠓⡶⠛⠛⠓⠒⠒⠶⠦⣤⣄⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠖⠒⠒⠛⠛⠓⡖⠋
⠀⠀⠀⠷⠤⡶⠶⠶⠶⠤⠤⢤⣤⣉⣛⠲⢤⣀⠀⠀⢸⠀⠀⢀⣠⠴⢚⣋⣡⣤⠤⠤⠴⠶⠶⠶⡶⠴⠇
⠀⠀⠀⠀⠀⠛⠚⠛⠛⠛⠛⠓⠒⠚⠛⠿⠷⢿⣷⣦⣼⣤⣶⣿⠶⠿⠿⠛⠒⠚⠛⠛⠛⠛⠛⠚⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⣘⣳⡄
⠀⠀⠀⡠⠀⠀⠀⣠⣶⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⣰⣿⣿⡟⡇
⠀⠀⠀⢳⡀⠀⠐⠿⠟⠁⠀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣦⣄⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠻⠟⠋⠀⣧
⠀⠀⠈⢄⠱⡤⠴⠶⠴⠖⣉⠤⠒⣚⡀⠆⠀⠀⠀⠀⠀⢉⣩⣉⣉⣍⣉⠁⠀⠀⠀⠀⠀⠑⠂⠠⠀⢀⣄⠀⠠⠀⠒⠋⠀⠀⠀⠀⣿⣷⣶⣦⣤⣠⡆⣀
⠀⠀⠱⡈⢖⣚⣩⣍⣉⣛⡡⠚⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠾⣟⡆⢀⡄
⠀⠀⠀⠙⠉⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠏⠁⠐⠲⠷⠒⠈⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢺⢤⡖⣿⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡛⣿⠾⠾⡗⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠀⠀⠀⠀⠀⠀⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣦⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⡙⢛⣿⠇⣷⠟⢚⡝⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⡆⠉⠉⠑⠊⠉⠉⢰⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⣿⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡇⢠⣴⣿⣿⣶⡄⢸⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣇⡄⢀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠷⢴⡇⣿⣿⢸⡧⠾⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠸⣋⣠⠤⠤⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠼⠷⠿⠿⠾⠧⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣾⣷⣶⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⢠⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠛⠛⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣤⣶⣶⣦⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣶⣤⣄⡀
⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⡍⢹
⢸⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣹⣭⣭⣶⣮⣋⣵⣾⣭⣽⣏⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⢀⡘⠿⠿⠿⠟⠁⣘⡀
⠘⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣘⣇⣀⣇⣀⣸⣀⣸⣁⣀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⢿⣶⣶⣦⣶⣾⠿⠿⡇
⠸⠀⠀⢀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⣈⣣⣈⣇⣀⣹⣀⣽⣁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⣶⣾⣿⡇
⠈⠉⠀⠀⢉⣿⣿⡏⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⡫⣹⣉⣏⣝⣩⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠿⢿⣿⡿⣿⣿⣿⠿⠟⠃