Michael faraday Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⢠⣾⣷⣦⣶⠂⠀⠀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣆⣠⣾⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣃⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣷⣶⣶⣦⣀⠀⠁⡀⣦⣴⡀
⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢹⣿⣿⣿⣿⣿⣿⣿⣦⠀⢻⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠉⠻⢿⣿⣟⠈⠉⠁⠀⣸⣿⣿⣿⣿⡟⠇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⠀⠉⠋⠀⢀⣤⣼⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠌⠱⣦⣬⠶⠈⢹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢧⣀⣀⣈⣿⣦⡀⢾⣿⣛⣿⣷⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣹⣷⣿⡿⣟⣿⣿⣿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣈⣿⣿⣿⠟⣡⣿⣿⣿⣿⣶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣤⣀⠉⣽⣾⡿⠟⢁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠋⠻⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⢁⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣶⣶⣿⣿⣿⣿⣿⣏⣁⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⠹⡿⠋
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⣩⣴⡿⠋⠁⠀⠐
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢇⣨⣿⠃⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢨⡁⣀
⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠿⣿⣿⣿⡟⣼⡟⡇⠀⠘⢃
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠁⠀⠀⠀⠀⠉⠋⢰⡟⠁
⠀⠀⠀⠀⢹⡿⠿⣿⣿⡿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢾⠃⠀⠟⠋⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡶⠞⠛⠙⠛⠛⠷⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡴⠶⣶⣿⣁⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⠁⠀⠠⠚⠿⣿⣄⢀⠀⠀⠤⣤⣄⡀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠛⠁⠀⠀⠀⠀⠀⢀⠀⣿⠛⠛⠛⠲⠶⢾⣿⣿⣦⡀⠀⠀⠀⠀⠈⠳⣦⣀
⠀⠀⠀⠀⠀⠀⣠⡶⠛⠁⠀⠀⠀⢀⠄⠀⣠⣶⣾⡾⠋⠀⠀⠀⠀⠀⠀⠈⠳⣌⢷⡀⠀⠀⠀⠀⠈⠹⣯⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⣠⣿⡿⠛⠫⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣧⠀⠀⠀⢀⠀⠀⠹⢿⣦⡀
⠀⠀⠀⣼⡿⠁⠀⠀⠀⠀⠀⢰⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠈⣆⢠⡄⠀⢹⣿⡆
⠀⣠⡾⠛⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢸⡇⠀⠀⢰⣿⡿⠃⠀⢼⡏⠁
⢼⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⠀⠀⠀⠀⢀⣀⣠⣆⣠⢀⡄⠀⢀⡇⣰⣇⣀⣈⠛⣼⡄⡘⣷⠀⠀⠐⣾⡃
⠘⢯⡀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⣰⠿⠛⢛⣿⣿⣿⢟⢻⣿⠉⠓⣿⡟⣛⣭⣙⡛⠛⠿⣿⣿⣿⣦⣤⡄⠈⣷
⠀⠈⢧⣴⠀⠀⠀⣀⠀⠀⠀⣿⣿⠀⠁⠰⠾⠏⠙⠿⢿⣶⠿⠃⠀⠀⢻⣿⣍⣻⣿⠟⠳⣦⡄⢹⣿⠋⠀⢀⣿⠿⠁
⠀⠀⠀⠉⢷⣾⡶⠿⢿⣀⣄⣿⣿⠀⠀⠀⠐⠲⠶⠚⠋⠀⠀⠀⠀⠀⠐⣇⠈⠛⠷⠦⠄⠀⠀⢸⣿⡃⢸⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣏⠻⣿⡆⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠀⠀⣏⢳⣄⠀⠀⠀⠀⠀⣸⣿⣦⡴⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⣴⣿⡀⠀⠀⠀⠀⣠⠞⠉⠛⠉⠳⠶⣾⠟⠋⠘⢷⣄⠀⠀⢠⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣇⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢹⠆⢠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣆⠀⠘⠁⡴⠶⠖⠛⠛⠓⠶⠛⠶⢦⢶⠄⠘⢀⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣽⣿⣿⣆⠀⠐⠁⠀⠀⠲⠶⠶⠶⠶⠶⣏⠀⠀⢠⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⡻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣴⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣷⣌⠛⠶⣤⣀⣀⡀⠀⠀⢀⣠⣿⣿⠃⠀⠈⠻⣦⡀
⠀⠀⠀⠀⢀⣀⣤⣤⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠉⠉⠉⢻⣿⣿⣿⡿⠁⠀⣀⣤⣾⣿⣿⣦⣄⣀⡀
⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣯⠙⢿⣿⣿⣶⣦⣼⣿⣭⣶⣶⣶⣾⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿⣶⣤
⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⣿⣿⣿⣯⢉⣽⣿⡿⢻⣿⣿⣿⡿⣷⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠻⠿⢻⣿⣿⣾⣿⢿⣶⣾⡿⡿⢡⣾⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠙⢿⣿⣿⡏⠘⢻⡍⣀⣴⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣿⣿⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠃⠀⣠⣶⣿⣿⣿⣿⣿⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡏⠀⠀⢰⡿⢛⣿⣿⣿⡟⠋⠁⠉⠁⠘⢶
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠏⠀⠀⠀⠉⠀⠸⢿⣿⡿⠉⠁⠀⠀⠀⠀⡛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣤⣤⣄⠛
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⡿⢀⠀⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣟⣀⣤⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⡀⠀⠰⣿⣿⣿⣿⣛⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⣈⢽⠇⠀⠀⠈⠿⠟⠁⠉⠛⠁⠀⠀⠀⢸⣿⣿⣿⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡏⠁⠀⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⢰⣧⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠐⠃⣼⣿⡁⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢴⣿⣿⡷⠎⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣻⡯⢻⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠠⣼⣧⣤⣠⣤⣀⣀⢀⠀⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣿⣧⣬⣄⠀⠀⠀⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⠉⠉⠁⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣠⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣌⢿⣦⣠⣄⣀⣤⣶⣾⠏⠀⠀⣠⣾⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣟⣉⣩⣤⣴⣿⣿⢿⡞⢸⡆
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠉⠀⠈⠀⠘⡇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⢿⣿⡟⢹⣿⣿⠛⢛⣿⠿⠿⠛⠋⡴⢂⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠘⠛⠉⠀⣼⣿⣿⣿⣿⣿⣷⣿⣷⠲⣄⠀⠀⠘⡥⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣃⣙⢧⣼⡿⠉⠉⠛⠃⠙⣆⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣏⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣶⣷⣦⣶⣿⣿⠿⣷⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣶⣿⣿⣿⣿⢿⣿⣿⣿⣿⣯⣶⣿⣿⣷⣿⣿⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⢿⣿⣿⣿⣯⣿⣿⣿⣯⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣻⣿⣿⣿⡿⠟⠃⠀⠀⠙⠛⠉⠉⠀⠀⠈⠿⣿⣿⣿⣶⣾⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣋⣿⣶⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣭⣿⣷⣿⠿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣤⣴⠶⣶⣄⡀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠸⢿⣿⣾⣿⣭⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣽⣿⣤⣼⣿⡗⠀⠸⣿⣿⣿⣋⡉⢉⠀⠀⠀⢠⣿⣿⣿⣼⣿⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡈⠛⠿⠃⢩⠁⠀⠀⠉⠻⠏⠻⠛⠀⠀⠀⠀⣸⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡅⠀⣾⣇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢃⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠛⡛⣃⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠙⢷⣄⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣶⡶⢛⠗⠿⠶⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⠀⠀⠀⢀⢀⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡄⠀⢀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣧⠀⠀⠠⣾⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡟⣿⡿⢆⠀⠀⠠⡀⠀⠀⡄⢰⡟⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⢙⣿⣿⣿⣓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣤⠏⠀⠀⠀⠀⠀⠀⠀⠈⣠⡿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠘⠛⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⣄⣰⡟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣍⡻⣿⡿⠿⠿⠓
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣷⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠂⠀⠀⠀⠀⢀⣰⣼⣯⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣜⠧⠒⡄
⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⣿⣿⣿⣏⠀⠀⢠⡀⣀⣸⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡄
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⢼⣿⣿⡿⠀⣰⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⢿⡆
⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢾⣿⣿⣿⣠⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣡⣎⠁
⠀⠀⠀⠀⠀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠙⢯⠁
⠀⠀⠀⠀⢸⡿⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠑
⠀⠀⠀⠀⠀⠀⠀⣿⡏⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠩⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠀⠀⠀⠀⢀⣀⠿⣽⢻⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠋⠁⠀⠀⠀⠀⠀⠴⠦⠧⠸⡶⣶⣷⠿⠃⠛⠿⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠂⠈⠉⠉⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⠀⡀⠀⠀⠀⢦⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣡⣤⣴⣿⣄⣠⡤⠞⠩⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⠈⢿⣶⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡧⠐⠂⠀⠀⠀⠀⠀⠈⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣧⣤⣀⣀⣠⣀⣀⡀⠀⢸⣿⣁⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣷⡄⢹⣿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡿⢻⡇⠀⠉⠛⣯⡁⠉⠁⣘⣿⣿⣿⠟⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣆⣴⣿⣧⣠⣄⠲⣀⢀⣤⣾⣿⣿⠛⠃⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠠⠜⠓⠁⢂⣻⣿⣿⣿⣿⣿⡟⠉⠀⠈⠻⣿⣼⣿⣿⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠆⢘⡿⣿⣿⣿⣿⣿⣟⣓⠒⠻⣆⢙⣿⣿⣿⣧⡀⠚⠁⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣼⣿⣿⣿⣟⡛⠛⢷⣶⣽⠟⣡⣿⣿⣿⣿⣮⣁⠛⠒⢰⠉
⠀⠀⠀⠀⠀⠀⠀⣀⣶⠀⠚⢸⠁⠹⣿⣿⣿⣿⠿⠟⢋⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣔⡂
⠀⠀⠀⠀⠀⠀⣀⣥⣅⣀⣥⣶⣿⣷⣶⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⢄⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣆
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠉⠁
⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⠿⠟⠉⠁⠀⠀⠀⠀⠀⠋⠀⠈⠀⠉⠉⠉⠁⠉⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠿⠿⠋⣉⣁⣠⣿⣿⣷⣶⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣫⡽⠚⠗⠋⠀⢉⣥⣴⣿⣿⣿⣿⡍⠉⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣏⠁⠀⣀⠴⢛⣒⣿⡿⠋⠙⠻⢷⣶⣤⣀⠀⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⠋⣀⣤⣶⡿⠟⠛⠁⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⣾⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣷⣿⣿⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⠿⠿⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣷⠆⠀⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣏⠀⣀⣀⣀⠀⠀⡀⠀⢀⣀⣀⣀⣀⣀⢻⣿⣿⣿⣿⣦⡀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣩⣿⣿⣿⠃⢙⣷⣿⣿⣿⣿⣥⣙⢛⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢉⣩⣼⣏⠀⢸⠇⠙⠷⣄⠁⠀⠀⠀⣿⣿⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⢡⣿⠋⠀⢘⢲⣄⠀⠀⠀⠀⢀⣴⣿⢟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡼⠛⢿⠾⡟⠋⠀⠀⠀⠀⢀⣮⣿⣿⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡏⢀⣤⡴⠶⠶⣤⣤⠶⡀⠀⠘⢹⡟⢩⣵⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⢧⣄⣹⣿⠶⢶⣿⣛⡁⠀⠀⢀⡼⡾⣿⡿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣏⠀⠀⠀⠈⠉⣀⣴⣿⣿⠞⣱⡏⣸⣾⣿⢿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⠿⠋⠀⣰⣿⢱⣿⣿⢿⠏⢿⣿⣿⣿⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣷⣶⣾⣿⣿⣿⣿⣿⠀⣰⣿⡏⠀⠁⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣰⣿⡟⠀⠀⠀⠀⠀⠀⠀⠈⢿⡙⠻⠿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠟⠋⠉⠿⠛⠙⠛⠻⣿⣿⡿⣱⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠟⣱⣿⡿⠁⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⣿⠃⠀⠀⠙⢶⣤⣤⣀⣀⣀⣀⣤⣾⠁
⠀⠀⠀⢸⡧⢸⣶⢀⢀⣧⢀⣠⢀⡀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⡼⠃⠀⠀⠀⠀⠀⠉⠙⠛⠛⠿⣿⣭⠉
⠀⠀⠀⠘⠃⠈⠉⠉⠈⢁⣈⠙⠉⠁⠉⠀⠀⡀⠀⠀⠀⠀⠀⠘⠱⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣤⠠⣄⣠⡄⣤⣇⣤⡄⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠃⠀⠁⠀⠀⠈⠁⠈⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣶⣦⠝⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣇⣤⠁⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⢟⣽⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢇⣤⣀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⢟⡴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣶⣄⠀⠀⠄⢾⣦⣴⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣏⠀⠀⠠⡝⠈⠻⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣟⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⢲⣤⡄⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇⠐⢻⣿⣿⣿⣿⣿⡶⠆⡀⠀⠀⣼⣿⣿⣿⣿⣾⣿⣄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣟⣟⣙⣿⠇⠀⠀⢿⠿⣿⣿⣿⡿⠦⣄⠀⠀⠹⣿⣿⣿⣿⣿⣯⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡏⠀⠀⠈⠉⢹⣿⠀⠀⠀⠀⠀⠉⠛⠉⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⡇⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡀⠀⢠⣾⣿⣿⡀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣤⣿⣿⣿⣿⣿⣿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⡟⢸⣿⣿⣿⣿⡿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣬⣿⣿⣿⡟⠛⠛⠛⠒⠒⠚⠂⠀⠀⠀⠀⣹⣿⢛⣿⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⣿⣿⣿⣿⡿⣿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠛⢫⣾⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠜⠀⠈⣻⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣻⣿⣿⣿⣷⣤⣤⣄⣀⣤⣶⠞⠋⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠛⠛⠉⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣧⣀⣠⣴⣶⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣉⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⢠⡾⠒⠂⣠⣁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠿⠿⠏⠀⣻⢹⣿⣿⢿⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠄⠳⡗⣆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣟⠶⠄⠰⠂⠀⠀⠿⠀⠀⠀⠀⠋⡀⠀⠈⠁
⠀⠀⠀⠀⠀⠰⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⡟⣹⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⡿⣿⣿⣿⢿⣿⣿⣿⣿⡿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠈⠻⢿⠿⠛⢠⡿⣧⠽⠶⠓
⠀⠀⠀⠀⠀⠀⠉⠉⣹⣿⢞⡻⠁⢆⠍⠙⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⢜⠂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠈⠁⠀⠈⠀⠀⣹⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⢙⠁
⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠘
⠀⢀
⠀⢀
⠀⠈
⠀⠘⠂
⠀⠸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⢉⡀⠑⠀⡈⠉⠉⠓⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠉⣰⣿⢾⡍⠉⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⣠⢞⡿⠋⠈⠉⢳⣤⣀⠀⠀⠀⠘⠃⡀
⠀⠀⠀⠀⠀⣤⠞⠁⠀⠀⠀⠀⠊⠖⠋⠀⠀⠀⠀⠀⠹⣦⡱⡄⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⢸⡇⠀⡀⠀⢠⣶⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⡄⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠘⣇⣸⠁⠃⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡄⢠⣄⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠘⢿⡘⣇⠀⠀⢀⣀⣀⣀⠀⢠⣀⣠⣤⠤⡄⠀⠀⢿⡄⢸⣏⢀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠑⣄⠗⠸⣋⡴⠿⢛⡇⠘⠉⠛⠙⠓⠄⠀⢀⡴⢀⣼⣥⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡄⠁⠀⠈⣁⡇⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢹⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⢿⠀⠀⢰⠃⣁⡠⠤⠀⢄⠀⠀⠀⣰⣿⠳⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠸⡇⢀⡟⠀⠈⠁⡀⠀⣀⠀⠀⠀⠉⠈⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣌⠀⠾⡚⣋⣉⡉⠉⠀⠀⠀⠀⣠⣧⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠜⠛⠉⠀⠀⠀⢀⣠⠞⣽⠃⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠙⣧⣀⣤⣤⣤⣶⠟⢁⡾⠁⠀⠀⠀⠀⠹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢴⠇⠀⠈⠏⠉⡩⠊⠀⣠⠟⠁⠀⠀⠀⠀⠀⠀⣿⠈⠓⠢⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠀⠸⣀⠀⠀⠀⠀⠱⢀⠾⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠈⠳⣀⢀⣴⠟⠉⠒⠂⢤⣤⡄⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⢀⡟⠁⠀⢀⠀⠀⢀⡴⠋⠉⠉⠁⠀⠀⠀⣷⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣸⠃⠀⠀⢟⠀⠀⣄⠀⠀⠀⡀⠐⠒⠒⠪⢇⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠾⠃
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠀⠀⠘⠟⠉⣭⠁⠀⠀⠀⣴⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣉⣍⣯⣬⣉⣽⣉⣽⣫⣍⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠖⢀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣠⠀⠀⠀⠻⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⢁⡴⡶⢰⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣆⠀⠀⣠⠆⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠙⠆⠀⠶⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⠛⠉⠀⠀⠀⠀⠀⠀⢀⣼⣷⢿⣶⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠖⢻⡿⢹⡇⠀⠀⠀⠀⣠⣴⣾⣿⣿⡟⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣁⣾⣧⠀⠀⢀⣼⣿⣿⣿⣿⣿⡇⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠋⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠉⠛⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⢀⠀⣤⢀⠼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣸⣼⣿⣿⢠⣿⣿⣿⣿⣿⣿⣿⡏⠀⠈⣻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣾⡿⡏⢹⣿⣿⣿⡿⠀⢀⢙⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡿⣿⣿⣿⣽⣐⢞⣿⣿⣿⣷⣶⣾⣿⠟⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢉⣴⣿⣿⣿⡟⠈⢘⣳⣿⣭⠭⠿⣿⣽⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠘⠿⢻⢿⠟⠻⠿⠿⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈
⠀⠀⠀⠀⠀⠀⢀⣴⢂⡄⠘⠊⠁
⠀⠀⠀⠀⠀⠀⣼⣷⠋⢀⡀⢠⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⢠⣖⣿⡿⣸⣿⡶⢢⣦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠔⢷⡶⠈
⠀⠀⠀⠀⡄⢼⣿⣿⣿⣿⣶⣿⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡀
⠀⠀⠀⠐⠕⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⣇⣀⠀⠀⠤
⠀⠀⠀⠀⠺⢿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣤⣤⡀⠄⢀⠀⣄⣠⣤⣤⣤⣄⡀⠀⠀⠀⠀⢀⢿⣴⣿⣇⢸⡆
⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⡿⢟⣉⣩⣤⣼⣿⣿⣷⠀⠘⠿⢿⣿⣿⣿⣿⣭⣙⠀⠀⠀⠀⣹⣮⣿⣍⣁⣩⣅⡀
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡇⢫⠿⡿⠌⠉⠉⠹⠏⠀⠀⠀⠀⠉⠉⠉⠉⠈⠉⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿⣿⣇⠀⠁⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣦⠀⠀⠀⠀⢀⠞⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣄⠀⡤⠃⣾⣿⣶⣦⠶⠦⠀⠀⠀⠀⠀⠀⠀⠰⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠃⠀⠼⣿⠟⠇⠀⠀⠀⠀⠀⠀⢤⡄⠂⠀⠀⠀⠉⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣬⣥⣄⣤⣤⣀⣀⣀⣀⠀⠈⠀⠀⠀⠀⣄⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⠛⢻⣯⣬⣌⡉⠀⠈⠉⠁⠀⠀⠀⠀⠀⠐⢺⣮⣿⣽⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡀⣾⣿⠿⠛⠛⠃⠀⠀⠀⠀⠀⠀⢀⣠⠄⣠⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠚⢁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣷⣶⣶⣶⣾⠶⠒⠁⢀⣴⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠹⢿⣿⡟⠋⠁⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣠⣟⣁⣀⡈⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢀⣠⣤⣀⠔⣠⣴⣶⣿⣿⣿⣷⣶⣶⣿⣿⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⢴⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⠛
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢋⠛⠣⠁
⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣿⣿⣿⣿⣿⣿⠿⡜⡂⠂
⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⠏⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠣⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠙⠛⠿⠛⢡⣿⣿⠑⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣤⣤⣤⣆⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠆⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠐⠔⠂⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠋⠀⡀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⡠⠠⠏⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣹⠜⠁⠀⠀⢀⡠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⣶⣭⡆⢀⣠⣰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡜⢻⢇⣀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣳⣖⣻⣃⣴⢚⣯⡽⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠈⠙⠃
⠀⠀⠀⠀⠀⠈⢶⣾⣻⣿⣿⣼⣯⣿⡿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢠⠇⣰⡀⢂⠀⣤
⠀⠀⠀⠀⠀⠸⣾⣿⣿⣿⣿⣿⣿⣿⣁⣀⣀⣀⣀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢰⣶⠀⡄⢣
⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⣛⣻⣿⣿⣎⠁⢺⣶⣿⣿⣿⣿⣿⡿⢷⡦⠀⠀⠀⢠⣏⢿⣾⠯⠂⡞
⠀⠀⠀⠀⠀⠀⠀⠛⢿⣿⣿⣿⣿⠨⢶⣾⣟⠿⠿⠛⢻⡿⠀⠀⠙⠙⠿⠿⠻⠿⠿⢶⡄⠀⠀⠀⢰⣟⣷⣶⣦⣤⣤⡤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣾⣿⣿⣿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣄⠀⠀⠀⢀⢘⣩⣆⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣷⣤⣴⠀⠘⣿⣿⣿⡿⠟⠛⠀⠀⠀⠀⠀⠀⠀⠘⠁⠠⠻⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢿⣿⣿⣯⣿⣯⠀⣘⠹⡟⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣻⣿⣿⣿⣿⣿⣷⡶⠶⣶⡶⠶⢦⣤⣤⠄⠀⠀⠀⠀⠀⠀⢶⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣯⠈⣿⣶⣦⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠘⠁⣄⣬⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡻⣿⣿⠟⠋⠙⠃⠀⠀⠀⠀⠀⠀⢀⡠⠀⢀⣶⣽⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠹⣿⣿⣿⣷⣶⣶⣶⡶⠞⠋⠁⠀⣴⣿⣿⣿⣿⣿⣷⣿⣿⣿⡿⢻⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠈⠻⣿⣿⡉⠉⠁⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣾⣿⣽⡏⣀⢠⡰⠴⢎⠝
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠄⣄⡠⢒⣴⣶⣾⣿⣿⣿⣶⣶⣾⣿⣷⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⠋⣷⢸⠋⣸⠟⣙⠋⠃
⣤⣤⣤⣤⣤⣶⣶⣷⣾⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣾⣘⣫⣾⠿⠃⠈⠁
⣼⠇⠀⠀⠀⢨⠉⠹⡏⢛⠉⠍⠉⠙⣯⢹⣝⢻⡟⢻⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣏⣶⣿⡿⡟⠏⠁⠋
⢎⠓⠀⠀⠒⠐⠞⠿⡿⣿⣿⢿⢿⣿⡿⠿⢿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⡟⣿⡏⢣⠀⠈
⠯⠴⠶⠶⠮⠤⠦⠼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣻⣿⣿⣿⣿⣼⣿⣿⣗⡙⡻⠋⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠏⠈⠉⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠈⠻⠿⡿⡿⢱⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣷⣯⠄⠀⠀⢀⡀⣀⣠⣤⣤⣤⣴⢀⣤⣤⣀⣀⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠟⣠⠀⠀⠂⣴⠾⢿⢿⣿⡟⠁⠸⣿⣿⣿⣿⣿⣿⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⣾⣶⣿⣿⠄⠀⠀⠀⠐⠿⠟⠉⠀⠀⠀⢻⡿⡿⠟⣿⣯⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⢿⣿⣷⡆⠀⠀⠀⠀⠠⠈⣀⣀⢀⣸⣷⣄⣾⣿⣿⣂⡄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠡⣶⣿⣿⡁⠀⠠⠀⠁⠀⠀⠉⣻⣿⣿⣟⣿⣿⠁⢿⣿⣿⣷⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡌⣀⠹⣿⣿⣇⠀⠀⣠⠴⠆⠀⠶⢾⣿⣿⣿⣿⣿⣤⢯⡿⢧⣭⠉⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⣳⢠⣼⢿⣿⡀⣾⡅⠀⠀⠑⠶⠶⣿⣿⣿⣿⣯⠰⢆⡠⠾⠟⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢻⣿⣿⣎⢿⣿⣿⣦⣀⣀⣀⣤⣶⣾⣿⣿⣿⣿⣧⣀⡈⠑⠘⠉⠁⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣿⣿⣿⣿⣦⡉⠻⣿⣿⢿⣿⡿⠟⣻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣌⠀⠀⠋⣁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⠀⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣄⢀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⣿⣿⢿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣷⠿⠁⠀⠚⠿⣽⡛⣿⣿⣿⣿⣆⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⡇⠃⠂⠀⠛⢾⣝⣿⣿⣿⣿⣿⡄⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⡀⠀⠀⡀⣰⣿⣿⣿⣿⣿⣿⣷⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠀⠈⠅⢺⣿⣿⣿⣿⣿⣧⠀⣸⣿⣿⣿⣿⡷⠅⠈⠓⠛⠉⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣀⣀⠀⢀⣠⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣻⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⠀⢀⣤⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣴⣶⣼⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⠿⠉⠋⠛⣛⣛⣺⡆⠀⠀⣴⣿⣋⣩⣈⣉⣽⢿⣿⣿⣿
⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠁⠠⠺⠿⠹⠿⠋⠉⠁⠀⠀⢿⠀⠁⠹⠟⠙⠛⠀⢿⣿⡟
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⢸⣿⣧
⠀⠀⠰⣆⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⢀⣾⣿⣿⣷⡄
⠀⠀⠀⠙⢿⣾⣿⣿⣿⣿⣿⣷⣈⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⣾⣿⣿⡿⢻⠟
⠀⠀⠀⠀⠀⠀⠉⢉⣿⣭⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠙⠋⠙⢷⣤⠾⠋⠁⠀⠀⢰⣿⣿⡋⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠁⢀⡀⠀⠀⠀⠘⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠐⠒⠀⠶⠞⠣⠤⠼⠿⠟⠷⠚⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣷⠦⠀⠀⠀⠀⠀⠀⠀⣠⠦⣴⣶⡆⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣷⣄⠀⠑⣄⠀⠀⠀⠁⠀⠀⠁⠀⠀⠰⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣸⣷⡄⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣤⣠⣴⣤⣴⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡉⠉⠙⠛⠃⠀⢰⣿⣿⣿⣿⡄
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠉⠉⠉⠁⠀⠐⠀⢼⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠉⠳⢀⣤⣶⣿⣿⣿⣷⣶⣀⣀⠴⣻⣿⣿⣇⣭⣭⣆
⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⣿⣏⢸⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠱⡶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣬⣷⣶⣾⠻⠶⢦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢂⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣷⣄⠈⠙⠻⢿⡿⠿⠿⣿⣿⣿⣿⣿⢿⣿⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⣿⣿⣷⣶⣦⣤⣄⣀⡠⣥⣿⣿⡦⠀⠀⠉⠹⢦⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣾⣿⣿⣿⠋⠀⠀⠀⢰⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⢾⣿⣿⣿⡿⠃⠀⠀⠀⠀⢻⣿⠿⢿⣶⣶⣶⣴⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠿⠾⠿⠷⠟⠛⠛⠉⠙⠟⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⠛⠛⠛⠛⠛⠛⠓⠛⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⠛⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⢿⣇⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣟⣿⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣿⠛⢋⣿⡿⠟⢿⠉⢉⣻⣀⣈⣿⢸⣿⣟⢻⣿⣿⣿⣿⣿⡾⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣭⣉⣾⣮⡏⠀⠂⠴⠟⠻⠟⠛⠛⠋⠀⠙⠃⠈⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣘⣿⢿⣿⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⣽⣿⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⣨⣿⣿⣮⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣯⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⠁⠀⠠⠖⠐⠓⠶⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣴⡦⠀⠀⠁⠀⠀⠀⠰⠋⣀⡈⠉⠙⢨⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⠉⠂⠀⢸⣿⣻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡏⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠇⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀⣀⡀⠀⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣉⡉⠉⠁⠀⠀⢠⡿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢠⡿⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⢸⣆⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⣶⡿⠆⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⣻⣆⠀⠀⠀⠀⠙⠓⠶⠶⠶⠶⠛⡅⠁⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣹⣽⣄⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⣾⣿⣿⣿⣿⣧
⠀⠀⢀⠴⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⣿⣆⠀⠀⠀⠀⠀⠄⠉⢀⣠⣿⣿⣿⣿⣿⣿⣧
⠀⠈⠀⠠⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠹⠺⣧⡎⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⢫⠋⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⠋⣻⣇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⡦⠁⠙⢇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣷⠿⣆
⠀⠀⠀⠀⠀⡄⠃⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣨⡿⣫⢆⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⣴⢻⡆
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⡴⠙⣮⡄⠀⠀⢀⠀⣿⣽⣿⣿⣿⣿⡿⣿⣿⠑
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢖⡉⢧⠀⠀⠀⡽⣻⣿⣿⣿⡿⠟⣿⢸⠀⠀⠂
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠹⣯⠿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣅⡟⠁⣾⡄⢀⠞⠹⠟⣿⣿⣿⡷⣻⣿⣲⠘⠃⠸
⠀⠀⠀⠀⠀⠧⡆⠀⠀⠀⠀⠻⠜⡿⢿⣿⣿⣿⣿⣿⡿⠋⠍⢹⣿⣿⣟⣀⠤⣸⣿⠋⠀⠀⠀⣿⣿⣿⡟⢋⠟⠣⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠿⠛⠛⠉⠋⠀⠈⢠⠁⣿⢿⡟⠁⠀⢠⠃⠀⠀⠀⠘⣹⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠁⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠘⠛⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠁⣩⡖⣦⣄⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣉⣾⣿⡇⠂⣼⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣴⣾⣿⣿⣽⠁⢰⠿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⡛⠛⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠃⢀⡿⣄⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠤⠤⠤⠄⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢭⣭⣽⣗⡀⡼⠀⠈⢳⡞⢧⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⡁⠒⢉⣠⠤⠖⣒⣒⠒⠦⠤⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⢠⠀⠀⠀⣸⢳⣽⣦⠀⠹⣌⢳⡀⠀⠀⠀⠀⠀⠀⡾⠀⣙⢦⡉⠔⠊⠁⠀⠀⠀⠁⡰⢅⢲⣠⡄
⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠸⠀⠀⣰⠃⠀⠙⢿⣷⠀⠘⣆⢳⡄⠀⠀⠀⠀⣸⠃⣰⢃⠄⠙⢦⡀⠀⠀⠀⢀⠜⠀⠀⠱⡹⣄
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⢀⣀⡼⠁⠀⠀⢀⣘⡿⣇⠀⠘⣏⠈⠓⢒⣶⣖⡀⢰⠇⠊⠀⠀⠀⠙⢦⣀⡀⠁⠀⠀⠀⠀⠃⠙⠄
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⡿⠂⠛⠉⠉⠁⠀⡀⠻⠀⠀⠈⠳⠔⠉⠠⠄⠈⠈⣤⠦⠤⠴⠶⠒⣀⣯⠿⠀⠀⠀⠀⠀⣀⡇
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡿⠀⠀⠀⠿⠟⠠⣀⠁⢀⡠⣇⠤⠶⠒⠊⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠳⣕⠄⡀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠺⢥⣖⣤⣤⣀⣶⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⢦⣤⣤⣴⡤⠾⠛⠉




⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⣉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣦
⠀⠀⠀⠀⣾⣿⡇⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⣿⣿
⠀⠀⠀⠀⣿⣿⡇⠀⣿⣟⣿⠟⣻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢹⣿⡇⠀⣿⣏⣴⠟⣡⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣏⣴⠞⣁⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢸⣿⡇⠀⣿⣏⣤⠞⢋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⡏⣡⠶⠋⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⡟⢁⡴⠛⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣀⣸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠀⣿⡿⢋⣤⠞⣻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠿⠿⠃
⠀⠀⠀⠀⢸⣿⡇⠀⣿⡷⠋⣡⠞⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠀⢸⣷⣾⣵⣾⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠸⣿⣿⣀⣀⣉⣉⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣿⣿
⠀⠁⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠂⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⠔⠉⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⣿⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣀⣼⠙⢦⡀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⣰⠋⢦⠀⢀⣠⣤⣴⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠉⠉⢹⠿⣿⣦⡙⢦⡀⠈⠓⢦⣀⠀⠀⠀⢠⠏⠀⣠⣿⣿⠛⠋⠉⠀⠉⠉⠙⠻⣿⣦⡀
⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⢠⠇⠀⠈⢻⣷⡄⠙⢦⡀⠀⠈⠳⢄⣀⡞⠀⣴⣿⠋⠈⠳⡀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⢻⣷⠀⠀⠙⢦⡀⠀⠀⡽⠁⢰⣿⠃⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⢻⡗
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠐⠿⠏⠀⠀⠀⠀⠀⢸⣿⠂⠀⠀⠀⣹⢮⣹⠛⣆⢙⣿⣀⣀⡡⠤⠴⠾⠿⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⢷⡴⠻⣎⡟⠉⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠠⠴⠏⠉⠛⠉⢹⣤⡌⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⣀⣀⣀⣤⡶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣄⣀⣀⣀⣀⣤⡶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⡆
⠀⠀⠀⠀⣿⣿⠀⢰⣶⣶⣶⣶⣶⡄⢠⣶⣶⣶⣶⣶⡄⠀⣶⣶⣶⣶⣶⣦⠀⢴⣶⣶⣶⣶⣶⠀⢰⣶⣶⣶⣶⣶⡄⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⢸⣿⢞⣥⠞⣿⡇⠰⣿⠞⣩⠞⣻⣗⠀⣿⡷⣫⡴⢛⣿⠀⣿⡷⢋⡵⢛⣿⡇⢸⣿⢞⣡⠞⣿⡇⠀⢸⣿⡇
⠀⠀⠀⠀⣿⣿⠀⢸⣿⢋⡵⠞⣿⡇⠀⣿⠞⣡⠞⢻⣧⠀⣿⡟⣩⠶⢛⣿⠀⢸⡷⢋⡴⠛⣿⡇⢸⣿⢛⣵⠞⣿⡇⠀⢸⣿⣷⣶⣶⣄
⠀⠀⠀⠀⣿⣿⠀⢸⣿⢋⣤⠞⣻⡇⠀⣿⢋⣡⠞⢻⣯⠀⣿⡟⣡⡴⢛⣿⠀⢸⡟⢋⡴⠛⣿⡇⢸⣿⢋⣤⠞⣻⡇⠀⢸⣿⡏⠉⣿⣿
⠀⠀⠀⠀⣿⣿⠀⢸⣿⢋⣤⠞⣻⡇⠀⣿⠋⣡⠞⢻⣿⠀⣿⡟⣡⡴⢛⣿⠀⢸⡟⢉⡴⠟⣿⡇⢸⣿⢋⣠⠞⣻⡇⠀⢸⣿⡇⠀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⢸⣿⠋⣠⠞⣻⡇⠀⣿⠟⣡⡶⢻⣿⠀⣿⡟⢁⡴⢛⣿⠀⢸⡿⢋⣤⠞⣿⡇⢸⣿⠋⣡⠞⢻⡇⠀⢸⣿⣇⣀⣿⣿
⠀⠀⠀⠀⣿⣿⠀⢸⣿⠛⣡⠞⣻⡇⠀⣿⠞⣡⡴⢻⣿⠀⣿⡟⢋⡴⢛⣿⠀⢸⡿⢋⣤⠞⣿⡇⢸⣿⠛⣡⠞⣻⡇⠀⢸⣿⡿⠿⠿⠟
⠀⠀⠀⠀⢿⣿⡄⢸⣿⠞⣡⠞⣿⡇⠀⣿⡞⣡⡾⣻⣿⠀⣿⡷⢋⡴⢋⣿⠀⢸⣿⢋⣴⢞⣿⡇⢸⣿⠞⣡⠞⣻⡇⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⡇⠈⣿⣾⣷⣿⣿⡇⠀⢿⣿⣷⣾⣿⣿⠀⢸⣷⣿⣾⣿⣿⠄⠸⣿⣿⣶⣿⣿⡇⠈⣿⣾⣷⣾⣿⡧⠀⢸⣿⡇
⠀⠀⠀⠀⢸⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠈⠛⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠤⠤⠦⠤⠤⠈⢉⣉⣉⣉⣋⣙⣛⣛⣛⣋⣉⣉⣉⣉⠉⠠⠤⠶⠤⠤⣀⣀⣀
⠀⠀⠀⠀⠀⢀⣤⡤⠖⠛⠊⣩⣭⣤⠔⠶⠶⠒⠒⠂⠉⠉⣩⣭⣩⣍⣉⣉⣩⣭⣭⣍⠉⠉⠙⠒⠒⠶⠶⠦⣬⣭⣉⠑⠛⠒⢤⣤⡀
⠀⢀⣠⠶⠛⣋⣠⠶⠞⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠳⠶⣄⣉⠛⠶⣄⡀
⣴⠟⠁⣠⠞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠉⠙⠒⠒⠖⠒⠒⠒⠒⠒⠒⠉⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⣄⠈⠻
⣿⡀⠀⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣍⠱⢆⣀⡀⠀⠀⠀⠀⠀⠀⢀⣀⡴⠏⣩⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟
⣿⠙⢦⡀⠙⠳⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠮⣭⣥⡶⠶⠒⠒⠿⠶⣬⣭⡴⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠔⠋⢀⣠⠐
⣿⠀⠀⠙⠳⠶⣀⣈⠙⠓⠶⢦⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡠⠶⠶⠛⠉⣀⣠⠶⠋⠁
⣿⠀⠀⠀⠀⠀⠈⠉⠛⠓⢤⣤⣄⡉⠉⠙⠛⠛⠒⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠖⠚⠛⠛⠉⢉⣉⣤⣤⡴⠚⠛⠉⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠒⠶⠶⠶⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠶⠶⠶⠶⠶⠒⠋⠉⠋⠁
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⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠉⢹⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣸⡇⠀⠀⢸⣇⣀⣀⣀⣀⣀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⣀⣀⣬⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇
⣿⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠
⣿⠈⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠁
⣿⡀⠀⠀⠈⠛⠳⠶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠶⠞⠛⠁⠀⠀⢀
⣿⠙⠷⣤⡀⠀⠀⠀⠀⠉⠉⠒⠶⠶⣤⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⠶⠶⠚⠉⠉⠀⠀⠀⠀⢀⣤⠞⠉
⣿⠀⠀⠀⠙⠛⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡴⠶⠛⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠶⠖⢤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡤⠶⠞⠋⠉⠉
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠋⠁
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⣿
⣿
⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠤⠶⠴⠶⠴⠦⠴⠶⠦⠴⠶⠶⠶⠴⣦
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠦⠴⠶⠶⠶⠶⠶⠶⠾
⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠉⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁
⠀⠀⠀⠀⠀⠈⠙⠓⠶⠶⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⠶⠶⠛⠋
⢀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⡉⠉⠙⠛⠛⢓⣒⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀⣒⡚⠛⠛⠛⠉⠉⣉⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠅⠀⠨⠁⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣶⣶⣶⣶⣶⣾⣶⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⠉⠉⠉⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠙⠉⠉⠉⠉⠉⠋⠋⠙⠉⠉⠉⠉⠉⠋⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⢳⡄⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⣀⣀⣀⣀⣈⣓⣋⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡨⠭⢄⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢀⠊⠀⠀⠀⠱⡠⠊⠉⠐⡄⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⢸⡀⠀⠀⢤⠠⡇⢤⠀⠀⡠⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠣⣀⣀⣠⠞⠱⣀⣀⣠⠂⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠉⠀⡀⠀⡈⠉⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡠⠒⠒⠐⠀⠀⠀⠈⠐⠒⠂⢄⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣀⡇⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⣇⣀⡀
⠀⠀⠀⠀⡠⠒⢉⣁⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⢄⣉⠑⠢⡀
⠀⠀⠀⡞⠀⡰⠋⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡇⠀⠈⠳⡀⠘⡆
⠀⠀⣸⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⡇⠀⢸⡀
⢀⠞⠁⠀⠀⢸⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣸⠀⠀⠀⠙⢆
⡎⠀⡜⢟⠢⣀⡼⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⣄⡠⢚⠟⡄⠈
⠑⠒⠓⠃⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠈⠓⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⣩⣽⣭⡉⠉⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠹⢶⠾⠁⠀⠀⠀⠀⠀⠀⠀⣸
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣳⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣃⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣤⣶⣲⣆⣴⣲⣆⣴⣒⣦⣀⣶⣦⣠⡶⣦⢠⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠘⡇⠀⠸⡇⠀⠀⣧⠀⠀⣿⠀⠀⠘⣏⠉⠹⣏⠉⢻⡉⠉⠉⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⡤⡤⢀⣀⡀⢀⣀⠀⠈⠀⠈⡁⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⣰⠁⠀⠀⠒⠒⠒⠒⠒⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠇
⠀⠀⠀⢠⠏⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠈⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⢀⡎⠀⠀⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠒⠒⠒⠛⠒⠒⠒⠒⠲⠆⠀⠀⢰⣿⣿⠁
⠀⠀⠘⡏⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠛⠓⠒⠒⠒⠲⠶⠶⠶⠶⠤⠤⠤⣤⣤⣾⢷⡇
⠀⠀⠀⠹⠶⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⣤⣤⣤⣭⣭⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡞

⠀⠀⠀⠀⢀⣀⣀⡀⢀⣀⣀⣀⠀⣀⠀⢀⠀⢀⣀⣀⡀⢀⣀⣀⣀⠀⣀⠀⣀⠀⢀⣀⣀⡀⢀⣀⣀⡀
⠀⠀⢀⠀⣼⡏⢉⡁⢈⡉⠉⣿⠀⣿⠀⣿⡇⣿⡏⢉⡁⢈⡉⢉⣿⠀⣿⠀⣿⠁⣿⡏⢉⡁⢈⡉⢩⣿
⠀⠀⢀⢀⣈⣅⣸⡇⢸⣯⣀⣉⣀⣿⠀⣿⣇⣉⣁⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣀⣈⣁⣸⡇⢸⣇⣀⣉⣀
⠀⠀⢠⠄⣉⣏⢉⠁⠈⣉⣉⣉⢉⡉⠀⢉⣉⣉⢉⣉⠁⠈⣉⢉⣉⣉⡉⠀⢉⣉⣉⡉⣉⡁⠈⣉⣉⡉⡉
⠀⠀⢠⠤⢛⡟⢻⡇⢸⣿⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⣟⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⡟⠛⠛⠛
⠀⠀⢠⠀⣼⡇⠘⠃⠘⠃⠀⣿⠀⣿⠀⣿⡆⣿⡇⠘⠃⠘⠋⢐⣿⠀⣿⠀⣿⠀⣻⡇⠘⠃⠘⠃⢰⣿
⠀⠀⢠⡀⠚⡿⠛⠃⠘⠻⠛⠛⠀⠛⠀⠛⠃⠛⠛⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠀⠻⠛⠛⠃⠘⠛⠛⠟
⠀⠀⠠⡀⣼⡿⠿⠆⠐⠿⠿⣷⠀⣿⠀⣾⠆⣾⡿⠿⠆⠰⠿⠿⣷⠀⣿⠀⣾⠀⣾⡿⠿⠇⠸⠿⠿⣷
⠀⠀⢠⡄⢼⡇⢰⡆⢰⡦⠀⠿⠀⣿⠀⣿⡇⠻⠇⢰⡆⢰⣦⠐⠿⠀⣿⠀⣿⠀⠻⠇⢰⡆⢰⡆⠘⠿
⠀⠀⢀⡤⣶⡶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠶⠶⠶⠾⠇⠸⠷⠶⠶⠶
⠀⠀⢠⣤⣦⡤⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⣦⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶
⠀⠀⠠⡄⣤⡤⢸⡇⢸⡯⠀⣤⠀⣿⠀⣿⡇⣤⡄⢸⡇⢸⡿⢀⣤⠀⣿⠀⣿⠀⣠⡄⢸⡇⢸⡇⢠⣤
⠀⠀⠀⠄⣼⣧⣤⡄⢠⣤⣤⣿⠀⣿⠀⣿⡇⣿⣧⣤⡄⢠⣤⣤⣿⠀⣿⠀⣿⠀⣿⣧⣤⡄⢠⣤⣤⣿
⠀⠀⠠⠤⣤⣤⣠⡄⢀⣤⣤⣤⠀⣤⠀⣨⡀⣨⣤⣤⡄⢀⣤⣤⣄⠀⣤⠀⣠⡀⣠⣤⣤⡄⢀⣤⣤⣄
⠀⠀⠀⠀⣼⡯⢉⡁⢀⡉⠉⣿⠀⣿⠀⣿⠇⣿⡏⢉⡁⢀⡉⠉⣿⠀⣿⠀⣿⠀⣽⡏⢉⡁⢈⡉⢩⣿
⠀⠀⠀⢀⣈⣥⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣇⣈⣁⣸⡇⢸⣿⣀⣉⣀⣿⠀⣿⣄⣉⣁⣸⡇⢸⣇⣀⣉⣀
⠀⠀⠀⣈⣉⣍⢉⠁⠈⣉⣉⣉⣉⣉⠀⢉⣉⣉⣉⣉⠁⠈⣉⢉⡉⣉⡉⠀⢉⣉⣉⣉⢉⠁⠈⣉⣉⣉⡉
⠀⠀⠀⠈⢛⡿⢻⡇⢸⡟⠛⠛⠛⣿⠀⣿⡛⠛⠛⢻⡇⢸⣟⠛⠛⠛⣿⠀⣿⠛⠛⠛⢻⡇⢸⡟⠛⠛⠛
⠀⠀⠀⠄⣼⡇⠘⠃⠘⠋⠀⣿⠀⣿⠀⣿⡆⣿⡇⠘⠃⠘⠋⢈⣿⠀⣿⠀⣿⠀⣿⡇⠘⠃⠘⠃⢸⣿
⠀⠀⢀⡄⠸⡿⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠁⠛⠛⠛⠃⠘⠛⠛⠛⠀⠛⠀⠛⠀⠛⠛⠛⠃⠘⠛⠛⠛
⠀⠀⢀⡄⣼⡿⠿⠇⠐⠿⠿⣿⠀⣿⠀⣾⡆⣾⡿⠿⠇⠐⠿⠿⣷⠀⣿⠀⣾⠀⣾⡿⠿⠇⠸⠿⠿⣷
⠀⠀⢀⡄⠼⡇⢰⡆⢰⡖⠀⠿⠀⣿⠀⣿⠇⠻⠇⢰⡆⢰⣶⠀⠿⠀⣿⠀⣿⠀⠻⠇⢰⡆⢰⡆⠘⠿
⠀⠀⢠⡴⣶⡶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠿⠶⠶⠶⠿⠀⠿⠷⠶⠶⠾⠇⠸⠷⠶⠶⠶
⠀⠀⠠⣤⣶⣶⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢠⣶⣶⣶⣶⣶⠀⣴⣶⣶⣶⣶⡆⢰⣶⣶⣶⣶
⠀⠀⢠⡄⣠⡄⢸⡇⢸⡿⠀⣤⠀⣿⠀⣿⡇⣤⡄⢸⡇⢸⡿⢀⣤⠀⣿⠀⣿⠀⣤⡄⢸⡇⢸⡇⢀⣤⠀⡄
⠀⠀⠀⠀⠹⠧⠤⠄⠠⠤⠴⠿⠀⠿⠀⠿⠇⠿⠧⠤⠄⠠⠤⠴⠿⠀⠿⠀⠿⠄⠿⠧⠤⠄⠠⠤⠴⠿⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⣒⣒⣲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇⢺⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⣔⠒⠒⠒⠒⠒⠒⢒⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⠯⠤⠤⠤⠤⠤⠤⠤⠀⠠⠬⠍⠲⠆⢀⠖⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠀⠈⢣⡀
⠀⠀⠀⠀⠀⠀⣀⠤⠔⠒⠒⠒⠲⠤⣤⡏⢀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢰⠛⢄⠙⢄⣀⡤⠴⠒⠒⠒⠒⠤⢄⡀
⠀⠀⠀⢀⠔⠋⣀⠤⠖⠒⠒⠒⠒⠤⡄⠀⠺⡕⢄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡎⠀⢈⠗⠈⢁⡤⠒⠒⠒⠒⠒⠦⢄⡈⠓⢄
⠀⠀⡰⠃⣠⠊⠁⠀⠀⠀⠀⠀⠀⢰⠃⡴⢄⠈⢮⠳⡄⠑⢄⠀⠀⠀⠀⠀⢰⠁⡸⠀⣠⠋⣠⠚⢆⠑⣄⠀⠀⠀⠀⠀⠀⠉⢢⡀⠳⡀
⠀⢰⠁⡰⠁⠀⠀⠀⠀⠀⠀⠀⡴⢃⡼⠁⠈⢣⠀⢧⠈⢢⡈⠳⣄⠀⠀⢀⠇⢠⠃⢠⠃⢰⠁⠀⠀⠳⡌⢢⡀⠀⠀⠀⠀⠀⠀⠱⡀⢱
⠀⡇⢀⡇⠀⠀⠀⠀⠀⢀⠴⢋⡤⠊⠀⠀⠀⠈⡆⢸⠀⠀⠙⢦⠈⠻⡏⡟⠀⡞⠀⢸⠀⡎⠀⠀⠀⠀⣘⣦⠑⢤⠀⠀⠀⠀⠀⠀⣇⠀⡇
⠀⡇⠘⡇⠀⠀⠀⠀⠀⠘⠦⠎⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠑⣄⠀⠀⠸⠥⠤⠾⠀⠓⠚⢉⣉⣉⡤⠤⠤⠜⠀⠀⠀⠀⠀⠀⡟⠀⡇
⠀⢱⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⡼⠀⠀⠀⠀⠀⠈⣇⠀⣠⠤⠖⠲⡀⠸⡉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⠁
⠀⠀⢣⡈⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⡰⠁⠀⠀⠀⠀⠀⠀⠈⢧⢧⠀⠀⠀⠱⡀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⣠⠃
⠀⠀⠀⠑⢄⠈⠲⠤⣀⣀⣀⣀⣀⡤⠔⠉⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠾⠼⠆⠀⠀⠀⠘⢦⡈⠑⠦⣄⣀⡀⢀⣀⡠⠔⠊⣀⠔⠁
⠀⠀⠀⠀⠀⠙⠲⠤⣄⣀⣉⣁⣀⠤⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⢄⣀⣉⣁⣀⡠⠴⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢗⣻⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⢵⢟⣩⢿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⠏⣿⣳⣿⣿⣯⣞⣯⣿⢿⡿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠜⠁⠙⠙⠉⣾⠋⡩⣻⣟⢕⠿⣻⢻⢷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠺⠿⣷⠿⣻⣷⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢊⣠⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⣫⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣻⣿⣿⣠⡔⠁⠀⠀⠀⠀⠀⠀⣠⣾⣋⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢿⣿⣻⣿⣾⣿⣟⠕⠀⠀⠀⠀⢀⠀⡺⣫⣽⢟⢽⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⢛⣿⠟⠁⠀⠀⠀⠀⠁⠀⠈⢫⣴⣷⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⢇⣬⢠⡔⠀⠀⠀⠀⠀⠀⢀⣤⠚⠁⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣧⣾⣿⣿⣻⣷⠛⠔⠀⠀⠀⠠⢛⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⠿⢿⣿⡿⠃⠀⠀⠀⠀⠺⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⡞⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⢀⣴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣡⣴⡷⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠉⠉⠉⢹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠚⠲⠶⠶⠶⠶⠾⠷⠶⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣠⣤⣤⣤⣤⣤⣤⣄⣤⣀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣿⢕⡩⠊⣡⠞⢁⡴⣋⢼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣷⣯⣤⣮⣤⣶⣥⣾⣥⣾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣶⡶⣶⠶⣶⠶⣶⠶⡶⢦⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡯⢊⡥⢊⡡⢚⡥⢊⡴⣻⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠈⠻⠷⠶⠿⠶⠿⠶⠷⠾⠟⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣟⠿⣻⠟⣛⠟⣻⠟⣛⢿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡗⣫⠔⣋⡵⢊⡡⠞⣡⢾⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣉⣉⣉⣉⣙⣋⣙⣋⣉⣉⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣟⡵⣫⠜⣩⠞⣩⢞⡩⢿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣷⣪⣴⣋⣴⣋⣴⣫⣴⣻⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⡼⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠐⠒⠚⣀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⢀⣈⣠⠡⠵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠙⣧⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠘⠲⡛⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡢⠑⡢⠕⠐⡀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠠⢒⣨⠤⠴⠶⠤⢬⣁⡂⢄⣀⢞⡘⠄⠀⠀⠀⠀⠀⠀⡀⢔⠅⡪⢐⢅⠲⢛⣧⢣⠤⣶⠤⠤⣟⢀
⠀⠀⠀⠀⠀⠀⡔⣡⠞⠋⠡⠀⠀⠀⠀⡌⠀⡹⢟⣕⢣⠈⠌⢄⠀⠀⢀⡠⣊⠔⢁⢌⠔⡕⣡⣾⠋⠈⡈⡄⠐⠀⠀⠀⠑⣵⢤
⠀⠀⠀⠀⠀⠀⡹⠋⠀⠀⠀⠣⠀⠀⡰⠠⢌⠔⠁⡹⣆⢡⠈⢎⢡⠭⠐⠈⠀⡠⡡⠃⡌⣴⠋⠀⠑⢄⠰⡐⡄⠀⠀⡠⠊⠀⠑⢅
⠀⠀⠀⠀⠀⢰⠙⠐⠢⠄⣀⠀⠱⡠⢟⡴⠅⠊⠁⠀⢸⡄⡆⠈⢶⠦⢶⣤⢊⠔⠀⢸⢠⠇⠀⠀⠀⠀⠑⢧⢻⡠⠊⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡈⠽⡥⡍⠥⢤⡀⠀⠀⢨⡥⡭⠸⡪⡀⣤⡻⠁⠀⠀⢠⣸⠃⠐⠀⠀⠂⠒⡸⣮⢇⠂⠀⠒⠀⠒⠘⡇
⠀⠀⠀⠀⠀⠈⢆⡀⠄⠒⠉⠀⠰⠀⠱⡀⠀⠀⠁⠒⡾⠉⠁⠀⠈⠐⠊⢃⢀⠄⡀⠀⠈⢆⠀⠀⠀⡠⠊⠀⣀⠀⠑⢄⠀⠀⠀⣰⠁
⠀⠀⠀⠀⠀⠀⠡⡣⡀⠀⠀⢀⠃⠀⠀⠱⡀⠀⢀⢔⠎⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠱⡣⣠⠊⠀⠀⠀⠀⠀⠀⠀⠑⢄⢔⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠪⡂⢄⡌⠀⠀⠀⢀⡱⡔⠕⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⢢⢄⠀⡀⣀⠀⢀⡠⡔⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⢸⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⣾⣿⣿⠃⠀⠀⠀⠀⠀⢀⣼⠇⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠋⠀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⢰⣿⣿⠁⠀⠀⠀⠀⠀⢀⣾⣿⠀⠀⠀⠀⠀⠀⠠⢿⣿⣿⣿⡟⠀⠀⠀⠀⢀⣾⣿⡶
⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠀⢠⣾⣿⣿⣿⣿⠃⠀⠀⠀⠀⣾⣿⣿⡟⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠈⠉⠉⣿⡿⠃⠀⠀⠀⠀⠘⠛⣿⣿⠟⠀⠀⠀⠀⠀⠸⢿⣿⠋
⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⢸⡿⠋⠀⠀⠀⠀⠀⠀⠀⡿⠃
⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠼⠁⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠸⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⣠⡇⠀⠀⠀⠀⠀⢀⣿⣿⠏⠀⠀⠀⠀⠀⠀⣠⠆
⠀⠀⠀⠀⠀⢀⣼⡿⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⣼⣿⠏⠀⠀⠀⠀⠀⠀⣴⡿
⠀⠀⠀⠀⢠⣾⣿⡇⠀⠀⠀⠀⠀⢀⣼⣿⣿⣀⡀⠀⠀⠀⢰⣿⣿⡗⠀⠀⠀⠀⢀⣾⣿⣇⣀⣀
⠀⠀⠀⠀⠛⢿⣿⣷⣦⠀⠀⠀⠀⠘⣻⣿⣿⣟⡀⠀⠀⠀⢀⣿⣿⡀⠀⠀⠀⣠⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠘⢻⣿⣿⠟⠁⠀⠀⠀⠈⢹⡟⠁⠀⠀⠀⠉⠁⣸⣿⡟⠁
⠀⠀⠀⠀⠀⣾⠟⠀⠀⠀⠀⠀⠀⠀⣸⡿⠋⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣿⠏
⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠃⠀⠻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢿⣿⣿⣆⠀⠀⠀⠀⠈⠛⠿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢋⣿⠀⠀⠀⠀⠹⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠋⠀⢿⡁⠀⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠈⣿⠀⠀⠀⠀⢀⣿⡇⠹⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⢸⡏⠹⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⡤⠞⠁⠀⠀⢀⣴⡿⠟⠀⠀⠘⣿⣿⣧⡀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣄⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⣿⠁⢀⠜⠀⢀⣤⣤⣾⡟⠉⠀⠀⠀⠀⠀⠈⣿⣿⣷⡀⠀⠀⢠⠞⠋⠀⠀⠀⠀⠀⠀⠘⣷⣄⡀
⠀⠀⠀⣠⣤⣤⣤⣴⣶⡾⠟⠋⠁⠀⢸⡟⠀⠁⠀⠚⠁⠀⠀⢸⣧⠀⠀⠀⠀⠀⢀⣾⣿⡿⣿⣧⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠈⣇⣀⣀⡀
⠀⠀⢰⠟⠀⠀⠀⣠⡟⠁⠀⠀⠀⠠⣿⠁⠀⠀⠀⠀⠀⣠⣴⡿⠋⠀⠀⠀⠀⢠⣾⣿⡟⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢂⠈⠁⠈⢳
⠀⠀⡻⠀⠀⠀⣾⠁⠀⠀⠀⠀⣠⡤⡼⣷⣴⣶⡶⠿⠛⠋⠁⠀⠀⠀⠀⠀⣠⣿⣿⠏⠀⠀⠀⢻⣷⣴⣶⣶⠶⠶⠶⣦⣀⠀⠀⠀⠀⠈⠀⠀⠀⢸⠃
⡠⠞⠁⠀⠀⠀⢻⠀⠀⠀⢀⠞⠁⢀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⣿⣿⠃⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠈⠛⣷⠀⠀⠀⠀⠀⠀⠀⠈⣦
⠀⠀⠀⠀⠀⢀⠋⠀⠀⠀⠘⠀⠀⠎⠀⠀⢀⣀⣠⣤⡶⠶⠟⣿⡟⠋⠉⠙⣿⣿⣆⠀⠀⠀⠀⠙⠷⠶⣤⡀⠀⠀⠀⠀⢈⡿
⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⣴⠏⠀⠀⢠⣿⠀⠀⠀⠀⠘⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠈⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠘⢿⣿⣆⡀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠈⠛⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠈⣿⣿⣿⣷⣦⣄⣀⣀⣀⣀⣤⣤⠤⠤⢄⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⢹⣿⣇⠈⠉⠛⠿⢿⣧⣄⠀⠀⠓⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⠀⠀⠀⠀⠀⠀⣰⠚⠁⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⢻⣧⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠛⠉⠀⠀⠀⠀⠀⢀⡸⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣄⠀⠀⠀⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⣇⠀⠀⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠈⠿⢶⣶⣤⣤⣤⣄⣀⣤⣤⠾
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠉⠉⢠⡞⠁⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣿⣿⠀⢻⣆⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠟⣫⣿⣿⠟⠀⠀⣿⡆⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠋⠁⣠⣾⣿⣿⠁⠀⠀⠀⢹⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⢀⣾⣿⠟⠁⢻⣇⠀⠀⠀⠈⠻⣆⣠⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡼⠁⠀⠀⣴⣿⠟⠁⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡼⠉⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠘⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣸⣿⠇⠀⠀⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠛⠻⣆⠀⠀⠀⠀⠀⢸⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠙⠦⣀⠀⠀⠀⠀⠻⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⠞⢻⣿⠀⠀⠀⠀⠀⢽⠀⠀⠀⠀⠀⠀⠡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢻⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠷⠴⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡋⠉⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠺⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠿⢿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢯⡈⠓⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢧
⠀⠀⠀⠀⣀⣠⠤⠤⠤⢤⣄⣰⠃⠀⠉⠳⣄⡈⠙⠲⢤⣀⡀⠀⠀⠀⣀⣴⡟⠀⠈⢧⡀⠀⣀⣀⣀⣀⣀
⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⣸⠋⠳⣄⠀⠀⠀⠙⠦⣄⠀⠈⠉⠙⠛⠋⠁⡼⠁⠀⠀⣠⠟⢯⡁⠀⠀⠀⠉⠙⠲⣄
⢠⡞⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠈⢳⡀⠀⠀⠀⠈⠓⢦⡐⢮⡀⠀⢰⠃⠀⢀⡞⠁⠀⠀⠹⣄⠀⠀⠀⠀⠀⠈⢳
⡾⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠉⢲⠟⠋⠻⣤⣤⣾⣄⣀⠀⠀⠀⠘⣆
⡇⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⣠⠏⠀⡇⠀⠈⠉⠛⠲⠤⣼
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⣳⠄⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠈⠳⢤⣀⡀⠀⠀⣀⣀⡤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠤⣄⣀⣀⣀⣠⠴⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠶⠶⠄⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⡶⠒⠛⠛⠙⠉⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢫⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢣⠏
⠀⠀⠀⢠⡇⡰⣊⠩⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⢀⡟
⠀⠀⠀⢸⡇⠁⠁⠀⠙⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⢸⡇⠀⢀⣀⣀⡀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠘⠒⠒⠲⢦⡄
⠀⠀⠀⢸⠁⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠘⠛⠛⠉⠉⣹⠃⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⠉⣹⠇⠀⠀⠀⡜⠀⣀⣀⡀⠀⠀⠀⣴⠋
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⠛⢉⡟⠀⠀⠀⠀⣼⠃⠀⠀⢀⡀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣯⣤⣄⠀⠀⠀⢲⡏⠀⠀⠀⠀⠉⠉⢉⡽⠃⢀⢀⡼⠁
⠀⠀⠀⣿⣀⣠⣤⠀⠀⠀⢀⡞⠀⠀⠀⠀⢠⣏⣴⡶⠟⢋⡏⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⣢⡟
⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⡾⠁⠀⠀⠀⠀⠘⠋⠁⠀⠀⢸⡇⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⣁⣀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠟⢛⡿
⠀⠀⠀⠀⠀⠰⡇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⢠⠏⠀⠀⠀⠀⠿⠛⠒⡖⠀⢠⡟
⠀⠀⠀⠀⠀⢸⡧⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠤⡤⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⣰⠟
⠀⠀⠀⠀⠀⢸⡇⢡⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⠋
⠀⠀⠀⠀⠀⢸⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡞⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠃
⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠘⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⢠⡶⢶⡶⠶⠶⣦⠀⠀⠀⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠠⡇⢷⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⡴⠶⠖⠒⡒⣲⠆
⠀⠀⠀⠀⠀⢰⣶⠛⠀⠀⠀⠀⢺⡇⠀⠀⠀⠀⠀⠀⢀⡇⠈⢣⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⡶⠛⠁⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⢣⠀⠀⠀⠀⠀⡾⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣠⡏⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⢠⡶⠞⠃⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⡞⠉⠀⠀⢀⢂⡴⠛⠀⠀⠀⠀⠀⡇⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⣰⢏⠀⠀⠀⠀⢠⣿⣃⣀⣀⡀
⠀⠀⠀⠀⣿⡄⠀⠀⠀⣠⣭⠇⠀⠀⠀⠘⣇⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠐⠉⠉⠉⣩⠞⠁
⠀⠀⠀⠀⠸⡇⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣧⣤⡄⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠠⣞⠃⠀⠀⣈⣷⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⠓⢲⠀⠀⠀⠀⠀⣸⠀⢀⡀⠀⢠⠞⠀⠀⠀⢰⡯⠶⠖⠚⣿⠀⠀⠁⣴⠋
⠀⠀⠀⠀⠹⣆⠀⠀⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⡜⡀⢸⠀⠀⠀⠀⠀⠙⠋⢹⡟⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠹⣆⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢸⡄⠀⠀⠀⠀⠀⢀⡟⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠹⣆⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⡇⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡇⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣿⣿⣿⣿⣿⣿⣿⣷⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠉⠉⠀⠀⠀⠈⠉⠉⢹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣀⣀⣀⣀⣀⣀⣀
⠀⠀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣤⣤⣀⣀
⣴⣿⣿⣿⣿⣛⡛⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⠿⢿⠿⠿⠿⠿⠿⠿⠿⠿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⢛⣿⣿⣿⣷
⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⡿⢿⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⣷⣶⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣾⣿
⣿⣿⠛⠛⠛⠛⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠛⠛⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿
⠻⣿⣷⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿
⠀⠈⠛⠻⠿⣿⣿⣿⣶⣶⣶⣦⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⣶⣶⣿⣿⣿⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠟⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⠀⠀⠀⠀⠐⠫⠊
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⡤⠀⠀⠀⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠀⠀⣰⡟⢀⣀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠆⠀⠀⠀⠀⣰⣿⣿⡶⠛⠀⠀⢀⡄⣀
⠀⠀⠀⠀⠀⠀⠼⢿⣿⡿⠋⠀⠀⠀⠀⢀⣴⠟⠋⠀⠀⠀⠀⠀⢈⡽⠟⠁⠀⠀⠀⠀⡀⠔⠁
⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠠⠎
⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⢀⠐⡁⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠐⠋
⠀⠀⠀⠀⠀⠀⠁⠌⠁⠀⠀⠀⠀⠀⠐⠁⢠⢌⠁⠀⠀⠀⠀⠀⠀⠐⠐⠒⠀⠀⠀⠀⠀⢃⠔⠃
⠀⠀⠀⠀⢀⢀⡘⠂⡰⠀⠀⠀⠀⠈⠀⡪⠐⠁⠀⠀⠀⠀⠈⣱⠃⠀⠀⠀⠀⠀⠀⣀⠔⠁
⠀⠀⠀⠀⢀⠔⠁⠈⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⢴⡩⠇⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⡠⠏⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⠀⠀⠀⠠⣸⣁⣀⡀⠀⠀⠀⡠⠋⢠⣎⠁⠀⠀⠀⣠⠊⢘⡂⠖⠀⠀⠀⠀⢠⠎⣠⠄
⠀⠀⠀⠀⠰⢓⢖⡴⠋⠀⠀⠀⠈⣩⡿⠒⠁⠀⠀⠀⠀⣩⠏⠋⠀⠀⠀⠀⠀⠀⡠⠂⠁
⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠰⠎
⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠾⡄⠾⡆⠰⢷⠰⠦⡄⠸⣆⠰⢶⠰⢰⣀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠐⠞⠁⠶⠃⠰⠞⠰⠖⠁⠶⠃⠰⠊⠰⠞⢸⡇⠀⣼⠃⡝
⠀⠀⠀⢀⣤⣤⡶⢒⢶⠖⠲⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣼⠁⣼⣤⣄⣀
⠀⠀⠀⢸⣿⣿⠉⠉⠉⠉⠉⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⡰⠓⠚⣧⢻⡄
⠀⠀⠀⢸⣿⣻⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡰⠁⠀⠀⢸⢸⠇
⠀⠀⠀⢸⡟⢸⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠰⠁⠀⠀⠀⣸⢾
⠀⠀⠀⠘⣿⣾⡄⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⣰⠁⠀⠀⠀⠀⣟⣿
⠀⠀⠀⠀⣿⣽⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡠⣿⠀⠀⠀⠀⢀⡏⡏
⠀⠀⠀⠀⣿⢳⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⣾⠀⠀⠀⠀⢸⢧⡇
⠀⠀⠀⠀⢻⣧⣇⠀⠀⠀⠀⠈⣿⣧⡀⣖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⢸⡷⣿⠀⠀⠀⠀⠀⣿⣦⣝⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⣿⣾
⠀⠀⠀⠀⢸⡿⢹⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⡿
⠀⠀⠀⠀⢸⣇⣼⣧⣄⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣟⡇
⡤⠤⠒⠒⠐⠛⠛⠚⠓⠒⠾⠾⠷⠿⠿⠯⠶⠬⠾⠥⠼⠥⠼⠿⠿⠿⠤⠴⣷⣽⣿⡵⠫⡴⠇⠤⠤⠤⢄
⠙⠒⠂⠴⠤⠤⠤⠤⠤⠤⠤⠦⠴⠴⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⣤⣤⣤⣄⣠⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠓⠀⠐⢈⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠉⣉⣈⣁⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠘⢏⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⠿⢿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡩⠐⠂⠈⠉⠉⠙⣌⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⣀⠄⠊⠁⠀⠀⠀⠀⡠⠐⢀⣬⣆⡆⠀⡀
⠀⠀⠀⠀⠀⣀⣤⡾⠗⠟⠛⠛⠻⠻⢶⣤⣄⡀⠀⢀⣦⣆⠀⠀⢀⡠⠊⠁⠀⠀⠀⠀⠀⡠⠊⣠⠞⠉⠀⢹⣍⠢⡽⣾⠿⠻⠿⠿⠶⣶⡀
⠀⠀⢀⣢⣾⠏⠁⠀⠀⠀⠐⠊⠙⠐⡢⠌⡙⢿⣦⠀⡈⣿⣦⠖⠁⠀⠀⠀⠀⠀⠀⢀⠎⢠⠞⠁⠀⠀⢀⣴⠿⣆⠱⣿⡷⠊⠉⠠⠒⠠⠉⠳⢦
⠀⠐⢷⠏⠀⡉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠐⢠⡢⣝⣥⠘⠛⡄⠀⠀⠀⠀⠀⠀⠀⡠⠁⡴⠋⠀⠀⠀⣰⠟⠠⢊⠘⣷⡈⠁⠀⠀⠀⠀⠀⠑⢀⡄⠙⣄
⠀⢀⠏⠀⡰⠀⠀⠈⠛⢶⣄⡀⢀⡄⢄⣢⠅⠺⠯⠬⠷⠀⣼⡿⠀⠀⠐⢒⡀⠰⣀⠞⠁⠀⠀⠀⢠⡟⠑⡶⠄⡀⠘⢿⣌⢂⠀⠀⠀⠀⠀⠀⠩⣆⠈⡆
⠀⢨⠀⠘⠃⠀⠀⠀⠌⢁⣤⢘⡋⢾⣯⣤⠶⠤⢤⣤⣤⡄⠿⠲⠖⡤⣜⢡⡍⠳⡌⡀⠀⠀⠀⢈⣸⠀⢰⠃⠀⠈⢕⣢⣿⣦⣷⣀⣠⠀⠀⠀⠀⢊⠀⢺
⠀⢸⠀⠀⡀⠀⠀⠀⢢⡀⠉⠛⠛⠀⠈⠂⢶⠒⠛⠛⠛⠛⠈⠋⠀⠸⡈⠙⠋⢰⠇⡆⠀⠀⠀⠈⢸⠀⠸⡀⠀⠀⠀⠈⠙⠻⠿⠡⡀⠀⠀⠀⠠⠿⠀⢸
⠀⠈⣇⠀⠡⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠺⠶⢶⡶⠲⠤⠤⠤⣤⣌⣉⣀⠥⠚⢧⣤⡶⠀⠀⠈⣆⠀⠣⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠠⠁⠀⡟
⠀⠀⠘⢇⠀⠐⢄⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠂⠀⡼⠁
⠀⠀⠀⠈⠓⢄⡀⠙⠲⡤⢄⣠⣤⢤⠔⠋⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠐⠂⠤⠀⡄⣠⣤⠶⠛⠀⡤⠎
⠀⠀⠀⠀⠀⠀⠉⠒⠰⠤⠄⣄⠤⠤⠔⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⠤⣭⣥⣠⡤⠤⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣎⠁
⠀⠀⠀⠀⠀⡇⠀⠀⣀⣀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀
⠀⠀⠀⠀⠐⠛⠛⠋⠉⠉⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣤⣶⡶⠾⢿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⣀⣀⣀⡄⠀⠀⠀⣀⣴⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠛⠋⠉⠉⢀⡇⠀⠀⠈⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣿
⠀⡀⠀⣠⡶⢻⡄⠀⠀⢀⣴⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠈⠙⣦⠄⠀⠀⠀⠀⠀⠘⣿⠙⠛⠳⠿
⠀⠱⠞⠁⠀⠈⢷⣀⣴⠟⠉⣷⠀⠀⣠⠔⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⠁⠀⠀⢠⣤⣄⣀⡀⣿
⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠘⣧⠞⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠉⠙⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⡇
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣦⣤⣄⣀⣠⣾⡿⣿⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⣿⣧⣌⡉⠙⠛⠿⣿⣿⣀⠀⠈⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠙⠻⢿⣷⣦⣄⡙⠻⠿⣿⣶⣴⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠈⠙⠻⠿⣷⣶⣤⣈⡙⠋⠉⠛⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣿⣶⣦⣤⣬⣻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠙⠿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣦⣤⣀⡀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⠿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏
⠀⠀⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠁
⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠀⠀⠀⣰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⣸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠇
⠻⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠈⠻⢿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠉⠛⠿⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⠿⣿⣶⣦⣤⣄⣀⣀⡀⢀⣸⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⣤⣄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣿⣿⣿⣿⣶⣦⣿⣷⣶⣶⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣟⠿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣄⣙⠿⣿⣄⡈⠛⠿⣿⣿⣿⣇⣹⣿⣿⣿⣿⣿⣷⣦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣶⣎⠛⠻⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣶⣴⣾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣿⣿⣿⣭⣤⣤⣤⣽⣛⡻⢿⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡞⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣶⣿⣷⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡟⢿⣟⣷⣟⣿⣿⣿⣿⢹⡟⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⡻⢤⣈⡙⠻⣯⡻⣿⣿⣘⣶⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡷⣮⡙⠲⢜⣷⡘⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣯⣿⣿⣷⣾⡅⠄⢿⣷⡘⣿⣿⣿⣽⣿⣿⣿⣧⣶⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣦⣤⠠⣽⣷⡈⣿⣿⣿⢹⣿⣿⣿⣿⢛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢯⣽⣿⣿⣿⣦⣬⣿⣷⡘⣿⣿⣸⣿⣿⡿⢁⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⣿⣿⣿⣿⣿⣿⡛⠻⣷⡹⣿⣿⡿⢉⠁⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢻⣛⣿⣿⣿⣿⣿⣦⣾⣷⠙⣿⣧⣾⡜⣿⣷⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⣿⣿⠁⢹⣿⣿⣿⣿⡟⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡛⠛⠛⠉⠄⠙⠿⣧⢹⣿⣄⡎⠹⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣻⣿⣿⣿⣿⣿⣷⣶⣿⣧⢻⣿⣇⣰⡟⠈⠹⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣯⣍⣉⡉⠋⢛⣿⣿⣇⢻⣿⣿⠀⠀⠠⣼⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣏⢻⣿⣿⣟⣻⣿⣟⣿⣿⣿⣿⣆⢻⣿⣾⠆⣰⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣶⣿⣭⣭⣭⣿⣶⠶⣌⣻⣿⡎⣿⣯⣴⣷⠋⢩⣿⣿⣟⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠁⠲⢆⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⡥⠤⠀⢿⣿⣿⡌⢿⣿⡟⠉⣈⣽⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⣀⠉⠙⠛⠋⠙⢶⣤⡀⠀⠀⠀⠀⠀⠀⠘⠻⠛⠿⠿⣿⢷⣶⣶⣤⣉⣻⣿⡜⣿⣷⡞⢿⣿⣿⣿⣟⠁
⠀⠀⠀⠀⠀⣠⡄⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢀⣼⠇⠈⠙⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣯⣥⣉⠻⣿⡜⣿⣿⠻⣿⣿⡛⠉
⠀⠀⠀⠀⣰⣿⣿⣿⣷⣁⣄⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⢠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣧⡉⢻⣿⣿⣴⠟⡟
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣾⣿⠧⣠⡄⣀⡀⠀⠀⠠⣾⡏⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠟⠛⠛⠶⣯⣿⠻⣿⡹⣿⡟
⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣤⠄⣾⡟⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠸⠛⠓⠿⣷⡿⣏
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠏⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⣿⣆
⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣏⡶⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⢠⣿⣿⣿⣿⣿⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣧⣿⢿⡇⡀⢸⡇⠀⠀⠠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠸⣿⣿⣿⣿⣟⠀⠨⠴⢨⣿⢛⡩⢉⢻⣿⣿⣿⡟⠻⣸⢃⣿⡇⠀⠀⠀⠈⠛⠷⣶⣄⠀⠀⠀⠀⣀⣛⡠⠤⢾⠋⠀⠀⠐⠲⣆⣀⠀⠀⠀⠀⠀⠀⢃
⠀⠈⠉⠙⠛⠻⣦⣴⣾⣾⣇⠀⠙⠊⢠⣿⣿⣏⣀⣠⡶⠛⠫⣥⣆⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⡙⠁⠀⠀⠈⠁⠀⠘⠳⣤⢈⡋⠀⠀⠀⠀⠀⠘⣦
⠀⠀⠀⠀⠉⠙⠲⠦⣤⣌⣛⠳⠾⠷⠿⢿⡿⢟⡿⠁⠀⠉⠀⠊⠉⠑⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠀⠂⢤⣤⠀⠀⠀⠈⠻⡿⣤⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⢶⣤⡞⣁⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠃⠤⠀⢀⠀⡀⢟⡀⠄⠀⠀⡄⠁⠀⠀⠀⠀⠀⠘⠈⠻⠦⣄⠐⢀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢄⡀⠀⠄⠀⠴⠀⠀⠀⢠⡦⠀⠀⢠⠟⠁⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠠⠀⠐⠚⠋⠉⠀⠐⠉⠀⠀⠀⢸⣄⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣉⣀⣉⣿⡀⣸⣉⣉⣧⣉⣸⣿⣋⣇⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠰⣄⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠙⠛⣻⠖⠲⠶⢶⠶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⡛⡛⠟⠛⠛⣿⣦⣄⢰⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢰⡋⣹⣧⣯⣿⡇⠀⠀⠁⠠⠂⠀⢀⠀⠀⠀⠈⠀⠁⠀⠈⠀⠀⢀⡗⠘⣿⣿⣿⢿⡷⢶⡄
⠀⠀⠀⠀⠀⠀⢠⠾⠻⡀⠀⠀⢉⣿⡟⣙⣋⡀⠂⠀⠀⠀⠀⠀⠡⢂⡔⡠⢀⢀⠄⠠⢀⠔⠀⢸⠃⢸⡿⣿⣧⣸⡇
⠀⠀⠀⠀⠀⠐⣧⣤⣤⣷⠀⢠⣯⣸⣗⣿⣿⡷⠀⡴⢠⢀⠀⠀⠁⠉⠘⠀⠉⠈⠀⠀⠈⣠⠊⢸⠆⢸⡇⣿⡏⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡇⣿⡏⠀⠀⠉⣹⣟⠛⣻⡁⠞⠃⠓⠃⠀⠀⠐⠀⠀⠠⠀⢀⠀⣀⢀⠉⡡⢹⠀⣸⡇⣿⡃⣿⡇
⠀⠀⠀⠀⠀⢸⣿⡃⣿⣇⠀⠀⣾⣹⢫⢼⣿⣻⠂⢰⡖⠚⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠙⢷⣄⠟⡀⢹⣷⣿⣇⣿⠃
⠀⠀⠀⠀⠀⢸⣿⣴⣿⣿⠀⠀⠻⢿⣿⢟⣋⣍⡀⣼⡃⠀⠀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠈⣧⡴⠀⣾⣿⣿⢸⣿
⠀⠀⠀⠀⠀⢸⣿⢋⣼⣿⠀⠀⣴⢻⡟⢹⣿⣯⡿⢸⡇⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⣿⣿⣾⣿
⠀⠀⠀⠀⠀⢸⣿⡿⢿⣿⠀⠀⠻⢾⣿⠿⢋⣟⡀⢺⡄⠀⠚⠲⠒⠒⠚⠛⠛⠛⠛⠋⠀⢰⣿⣀⣂⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣿⣥⣿⣿⠀⠀⣴⡟⣿⡣⣟⡟⡗⢸⡇⠀⢀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠀⠈⣿⡧⠭⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣿⡗⣿⣿⠀⠀⠈⠻⣿⡷⢋⡩⠆⠸⡇⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⣿⣟⠃⠀⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⣿⣞⣿⣿⠀⠀⣲⡿⣿⠄⣿⣻⣿⠆⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣼⠿⠉⡤⠀⢿⣿⢸⡟
⠀⠀⠀⠀⠀⠀⣿⣯⣻⢾⠀⠀⠻⠷⣷⣶⡿⢛⣋⠀⠑⠹⠿⠶⢶⠒⡒⠒⠛⠟⠛⠋⠉⠋⡡⠀⣮⠀⢹⡿⢸⣧
⠀⠀⠀⠀⠀⠀⣿⣿⢿⢿⠀⠀⠀⣴⣷⡇⣰⣿⣷⣆⠀⠀⠀⠀⠀⡀⢀⠀⠄⠀⠄⠀⠈⠀⠔⠀⢾⠀⢸⣷⢺⣿
⠀⠀⠀⠀⠀⠀⣿⣿⣻⣿⠀⠀⠸⣷⣿⣧⡿⢿⠿⠟⠀⠂⠀⠀⠈⠀⠁⠀⠀⠀⠈⠀⠀⢠⠄⠀⢺⠀⢸⣿⣸⣿⠀⠀⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⠀⠀⠀⢀⣿⣟⠀⣠⢶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣔⣋⣴⢀⢸⠀⢸⣿⣿⣿⠀⠀⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣾⣿⠀⠀⠀⣿⣿⣿⣴⡷⡿⠃⠀⡠⠀⣀⣀⢀⡔⢀⠴⠤⣴⠾⣫⡿⠽⠭⠼⠀⠸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠿⠿⢿⠿⠀⠀⠀⠀⢿⣿⣏⣬⣤⣶⣶⣯⣽⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⢛⡛⠛⠻⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠛⠉⠙⢧⠐⠒⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

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

⠀⢀⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠒⡆
⢀⠟⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠈⣞⡵⡊⠉⠑⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⢘⢾⡀⠻⡀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⣞⣀⣷⠀⢷⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠈⠉⠉⢳⠙⢿⠑⢕⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⢀⡕⠈⣧⠘⣇⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⢸⠄⠀⢸⡆⢹⣄⠤⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠑⠚⢁⣽⡟⢻⡍⠦⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⡞⠁⢳⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠻⣆⡈⣧⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠈⠉⠙⡧⢚⢽⡒⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⡐⠐⣇⠈⢷⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⡅⠀⠘⣆⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠙⡦⢚⣡⠀⠉⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠐⡇⠈⢧⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡀⢸⡄⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⡤⠤⠶⠒⠛⠉⠁⠉⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢻⡈⢎⡦⢤⣒⣒⠒⠈⠉⠉⠉⠋⠉⣉⣉⣀⣤⣤⠤⠶⠶⠚⠛⠛⠛⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡸⢤⣤⣤⣀⣒⡐⠤⠬⠭⠒⠛⠋⠉⠉
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣴⣶⠂⠀⠀⠀⠀⠀⠀⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣃⣤⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇⠀⠀⠀⠀⠀⢸⣿⣥⣤⠄⠀⠀⠀⠀⢸⣿⣿⣶⣾⠂
⠀⠀⠀⠀⠀⠼⢋⣿⣿⣿⢃⣠⡄⠀⠀⠀⠀⠀⢠⣿⠇⢀⣠⡔⠀⠀⠀⠀⣰⣿⣿⣶⡗⠀⠀⠀⠀⠛⢛⣿⡟⠀⠀⠀⠀⠀⠟⢻⣿⣿⠏
⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⣾⣿⣾⣿⡟⠀⠀⠀⠀⠰⠿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⣼⠟⣽⣿⣿⠏⠀⠀⠀⠀⠀⢀⣾⠿⢛⣿⡟⠀⠀⠀⠀⠀⠀⣸⣿⣿⣡⣤⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⢰⣿⡿
⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠀⠀⠀⢀⣾⣯⣤⠆⠀⠀⠀⠀⠀⣿⣿⣷⡾⠁
⠀⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⡿⠁⠀⠀⠀⠀⠘⠛⣹⠏⠀⠀⠀⠀⠀⣸⣿⣿⡟⠁
⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠉⣼⡟
⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⢰⠏
⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠀⢀⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⣾⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⣸⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢰⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣤⡤
⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠀⠀⠀⠀⢼⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⢀⣿⣿⣯⣤⣤⣶⡶⠂⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣤⡄⠀⠀⠀⠀⠀⢠⣿⣿⣿⣶⠆⠀⠀⠀⠀⣸⣿⣿⣿⣏⣀⡀
⠀⠀⠀⠀⠸⠿⠟⢿⣿⣿⠟⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠃⠀⠀⠀⠿⠿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣼⣿⣿⠏⠀⠀⠀⠀⠠⠿⢿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⠃⠀⠀⠀⠀⠀⢰⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⢰⣿⣿⡟
⠀⠀⠀⠀⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠏⠀⠀⠀⠀⠀⠀⣸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡟⠁⠀⠀⠀⠀⠀⠀⣼⣿⡿⠁
⠀⠀⠀⠀⢠⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁
⠀⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠁⠈⠀⠀⠀⠈⠁⠁⠀⠀⠀⠁⠁⠀⠀⠀⠁⠀⠀⠀⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⣄⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⣿⡀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⠛⢿⡇⠀⢦⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡸⡿⣷⠀⠘⡇⠀⢀⠂⠀⢀⣀⣿⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣭⡷⡏⠀⠀⠑⣠⣄⣴⣾⠟⠁⠀⢀⣼⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣼⣷⣿⣦⡀⠀⣿⢿⣿⣿⡷⠉⢄⣸⣟⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⣭⣿⣿⣿⠙⠐⠋⠉⠉⠉⠃⠀⢸⣿⡿⢯⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣉⣿⣹⣿⣿⣿⣿⣦⢠⣾⠟⣿⣿⣿⡿⢿⡇⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡟⣿⣉⡟⠉⠉⠙⢻⢸⠋⠙⠉⢻⣿⡷⢾⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣍⠛⣙⢿⡄⠀⣠⡛⠘⣣⡀⠀⣿⣯⡁⢱⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠘⢿⡏⣷⠀⠏⣭⣭⡌⠃⣸⣿⡷⢠⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣠⡌⣇⣿⣷⡀⠈⠛⠀⣰⢿⣻⠄⣘⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⠂⢨⣭⢩⡄⡁⡅⠰⠛⠓⠒⠻⡅⠀⢠⠀⠀⠀⠶⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠖⠀⠀⠀⠀⠀⠀⠀⣠⠊⠉⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡋⡀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠉⣉⣿⡉⠉⠀⢀⢽⠀⠀⠀⠀⠀⠀⠀⠢⠘⢺⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠑⡀⠀⠀⢈⡿⢊⠁⠀⠂⠀⠀⠀⠠⠂⠀⠀⠀⠀⠁⠀⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠢⣘⣀⠞⣁⠀⢈⡇⠀⢐⣀⢀⣀⠀⠀⠀⡄⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⣽⣖⠂⠀⠲⢦⠀⣶⡈⡁⠀⠀⠀⠀⠀⠀⠽⢯⣿⡧⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠅⠈⠀⢀⠀⠀⠑⠬⠖⣁⣀⠀⠀⠀⠀⠀⠀⠁⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠁⠠⠂⠑⠀⠀⠀⠀⠙⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠐⣠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡺⠁⠀⢨⠆⠀⠀⠀⠈⠀⠦⠐⠚
⠀⠀⢀⠠⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠩⠀⠀⠄
⠀⠀⠀⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠰⠀⠀⠰⠂⠀⢀⡀
⠀⠀⠀⠰⡇⠶⢛⠄⠀⠀⠀⠀⠀⠀⢀⣼⣩⣑⠂⡀⣀⠀⣰⢶⣏⡇
⠀⠀⠀⠀⠉⠚⠉⠀⠀⠀⠀⠀⠀⠀⢀⡨⠛⠋⢤⠤⣈⡉⡉⣻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠈⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⡐⢶⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠁⢈⡟⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⡶⠀⠀⠀⠀⠀⠈⠷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠀⠀⠄⠂⢀⠀⠖⠒⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠔⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢸⣏⣛⠂⠀⠀⠠⠦⡶⣀⠀⠀⢸⠀⠀⠀⢸⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢹⣟⣉⣉⠉⣑⣶⣿⡿⠀⠀⠘⠀⠀⠀⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠉⠀⠈⠛⠁⠀⠀⠀⢀⢀⠀⠀⠀⠀⣀⠜

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠟⠛⠛⠉⠉⠙⠛⠛⠿⢶⣦⣶⠾⠟⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⣸⣿⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇
⠀⠀⠀⠀⠀⢠⣿⣯⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⣼⡿⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧
⠀⠀⠀⠀⢰⣿⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⣾⡟⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆
⠀⠀⠀⢰⣿⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧
⠀⠀⠀⣸⣿⣾⠁⠀⠀⠀⠀⣀⣀⣤⣤⣤⢤⣤⣤⣀⠀⠀⢸⡇⠀⠀⢀⣀⣤⣤⣤⣴⣦⣤⣤⣤⣤⣀⣀⠀⠤⠀⢹⡏⣿
⠀⠀⠀⣿⣿⡏⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣶⣼⣇⣴⣾⣿⣿⣿⣿⣿⣭⣭⣭⣽⣿⣿⣿⣿⣿⣷⣦⣼⣿⢻⡇
⠀⠀⢠⣿⣿⣷⣿⠿⢟⣛⣉⣉⣭⣤⣤⣤⣤⣤⣭⣭⣽⣿⣿⣿⣿⣿⣯⣭⣤⣤⣦⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⡿⠿⠿⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⠤⢄⡠⠤⢀⠤⠤⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⡆⠀⠀⠀⠢⠀⠀⠀⠈⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠖⢇⠈⠙⢀⠒⠀⠂⠚⠛⠛⠓⠒⠂
⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠈⠉⠉⠉⠀⠀⢀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⢀⣈⣶⣶⣞⣁⣀⣀⣙⡶⣶⣎⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⣤⡴⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠁⠀⠀⣆⠀⠀⣣⠀⠀⠈⠉⢠⠀⠀⠀⠀⠀⠀⠈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⣴⠒⢺⣟⣿⠓⢒⡄⠀⠀⠀⠁⠂⠰⠄⠀⠀⠀⠀⠤⠆⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠉⠙⠋⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⣴⢲⣶⣖⢢⡄⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⡀⠁⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢠⠀⠄⠈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠁⡀

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

⠀⠀⠀⠀⡏⢸⢹⠋⠙⠀⡟⠉⠉⢛⣹⣿⣿⠙⡏⠀⠀⢻⢿⡇⡆⢠⠐⠀⢰⠒⠖⢐⣖⠂⢰⢶⠒⡓⡖⣾⣿⣿⡀
⠀⠀⠀⠀⡗⠴⣰⡆⠀⡆⣇⢀⣴⣿⣿⣿⣿⣾⣇⣤⣀⣠⣬⣧⣏⠸⠜⠀⠈⢁⠃⡶⡏⣾⠠⣾⣿⠉⣼⣿⣿⣿⠁
⠀⠀⠀⠀⣧⣤⣼⣧⡎⢠⣿⣿⣽⣯⣾⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣷⣇⠀⠀⠈⠠⢠⡇⠘⢰⣟⣯⣾⣿⣿⣿⣿
⠀⠀⠀⠀⡿⣿⣿⣿⡿⠿⣿⢿⣏⣵⢟⣵⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣶⣾⣠⣶⣾⣿⣿⣿⣿⣯⣿⣿
⠀⠀⠀⠀⣿⣿⡏⣻⢃⣴⣏⣾⣿⢟⣿⣿⣿⣿⣿⣿⠟⠰⣾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣿⢾⣿⡟⡽⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⣰⣏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣷⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡿⢡⣿⡟⣿⠱⣻⣿⣿⣽⣿⣿⡿⣻⣿⣀⣴⣿⣿⣿⣿⣿⣿⣿⠟⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡟⢠⣏⡾⣿⣿⣿⣿⠉⣿⣿⣿⣞⣽⣿⣿⣿⡿⣽⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣷⣿⣽⡱⣿⣿⣿⣿⣇⣟⣽⣟⡿⢻⠟⣹⢋⣽⢿⣿⣿⣿⡏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠿⠯⠿⠵⠿⠿⠿⢽⣿⡿⢛⣿⣴⡿⢸⡯⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣐⣶⣬⡉⠁⠉⠙⠻⠧⠿⠿⠽⠿⠟⠋⠛⠛⠁⠀⠀⠉⠉⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣠⣶⣄⠚⠛⠛⠛⠿⠶⠶⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣠⣤⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⣤⣶⣶⣶⣿⣿⣿⣿⡿⠿⠛⠛⠋⢉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣟⠉⠉⠛⠛⠿⢿⣿⣿⣿⣿⣷⣶⣶⣦⠀⣀
⠀⣠⡾⠿⣿⣿⠟⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣄⡀⠀⠀⠀⢀⣀⣠⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠻⣿⣿⠿⣷⣄
⣾⠋⢹⣶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⠋⠉⠙⠛⠛⠛⠛⠉⠉⠙⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣶⡎⠹
⣿⣧⡌⠛⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣦⣤⣤⣤⣤⣤⣤⣴⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡟⢁⣼
⣿⡋⠹⣿⠏⣉⣿⣶⠶⢀⣀⣀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⠿⠿⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⣀⣀⡀⢰⣾⣿⣍⣉⣷⡿⢻
⣿⡇⠀⠀⠘⠛⠛⢡⣶⣾⣿⠿⠿⣃⣀⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣘⠛⠿⣿⣿⣷⣤⡜⠛⠛⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠿⠿⠿⠿⠿⠛⠛⢿⣿⣿⡟⠋⢻⣿⣿⣿⠟⠛⠛⠿⠿⠿⠿⠿⠗⠀⠀⠈⠉⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⠙⠛⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣰⣿⠀⠀⠀⠀⣿⣅⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠿⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠛⠛⠛⠿⢿⣷⠀⠀⠀⠀⣾⡿⠻⠿⠿⠿⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⢀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠿⠿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⣿⡟⠻⣷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣾⠟⢻
⣿⡇⠀⠀⠙⠛⠛⢦⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣶⡶⠛⠛⠋⠀⠀⢸
⣿⣿⣄⣀⠀⠀⠀⠀⠉⠉⠉⠙⠿⠿⠿⡿⠷⠶⠤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣴⠶⠶⠾⢿⡿⠿⠗⠈⠉⠉⠁⠀⠀⠀⠀⣀⣴⣿
⣿⡇⠘⠛⢳⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⡾⠛⠃⢸
⣿⡇⠀⠀⠀⠉⠉⠙⠿⠿⠿⢦⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⡴⠿⠿⠿⠛⠉⠉⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠛⠛⠓⠶⠶⠶⠶⠶⠶⠛⠛⠛⠛⠛⠛⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠓⠀⠚⠟⠒⠲⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠲⠚⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼
⠀⠈⠛⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⠟⠁
⠀⠀⠀⠀⠈⠙⠿⠿⢷⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⠼⠿⠿⠋⠁
⣀⣀⣀⣀⣀⣀⣀⣀⡀⠈⠉⠉⣉⣉⣡⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⣈⣉⠉⠉⢁⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠻⣿⠿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣯⣀⣠⣄⣀⣀⣀⣤⣀⣠⣤⡀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⣿⣛⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠷⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠓⠄⢀⡸⠂⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠀⠀⢢⡘⠋⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠘⠣⠀⣴⠴⠋⠀⠀⠀⡀⠀⠀⠀⠀⠈⠓⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡃⠐⡀⣠⠟⠀⠀⠀⢀⠤⠊⠀⠑⠄⠀⠀⠀⠀⠀⠙⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠮⢄⠀⢘⡬⠋⠁⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠉⠉⢆⠠⡶⠋⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢄⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠰⢏⠁⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠑⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⣠⡶⠠⠀⢀⡶⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⢀⡔⠁⠀⢡⣠⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⣠⠞⣇
⠀⠀⡠⠐⠺⡁⢱⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠⠞⠁⣠⡿
⢀⡾⠂⠄⢀⡼⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⡀⡠⠂⠀⠀⠀⠀⠀⡠⠚⢁⣴⡾⠋
⠈⠳⣄⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢀⡴⠋⢀⣤⠿⠋
⢾⡢⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠟⠁
⠀⠙⠦⡀⢌⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢀⣴⠟⠁
⠀⠀⠀⠈⠳⣕⢌⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⠋⡠⣠⠟⠁
⠀⠀⠀⠀⠀⠈⠳⣕⢌⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⡠⣮⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢕⡄⠑⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⢠⡾⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⠢⣄⠀⠀⠀⠀⠀⠀⠀⣠⠞⢁⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡈⠳⢄⡀⠀⠀⡠⠞⢁⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠙⠶⠋⠀⢔⡵⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡤⣠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠆⠀⠀⠀⠀⠈⠈⠀⠀⠀⠐⠱⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣷⣶⣦⣤⡀⠀⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠄⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠚⠴⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⠛⠋⠀⠈⠉⠀⠀⠁⠞⢛⠿⢿⣿⣿⣧⡀⠀⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠕⠀⠀⣾⣿⡥⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠴⢟⣿⣿⣷⠀⠀⢀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢸⣿⣿⠇⠀⠀⠀⣐⠙⠇⠘⠃⠘⠋⣡⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣒⠲⠦⠄⠀⢸⣿⣿⡀⠀⠀⣀⠙⡀⠃⠀⠀⢸⢀⠃⣠⠀⠈⠩⣿⣿⡇⠀⢀⠤⠶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠭⠍⠀⠘⣿⣿⡥⠀⠀⠈⢆⠰⠘⠀⠀⡄⠌⡰⠁⠀⢀⣼⣿⣿⠃⠀⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠹⣿⣇⠀⠀⠀⠀⠢⢢⢀⠀⣷⠐⠀⠀⠀⣀⣾⣿⠏⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢒⣒⡀⠀⠹⣿⣦⡀⠀⠀⠀⢱⣿⢾⡏⠀⠀⢠⣾⣿⣿⠏⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠘⢿⣷⠀⠀⠀⢠⠿⠽⡇⠀⠀⢤⣾⣿⠃⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠉⠐⠂⠀⠘⣿⡆⠀⠀⢰⡤⢤⣌⠀⠐⣻⣿⠃⠀⠤⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣑⠦⡀⠀⠀⣿⡇⠀⠀⠛⠓⢲⠛⠀⠸⣿⣿⠀⠀⠠⣪⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢝⡂⠀⠀⢿⣧⡀⠀⠀⠀⢰⠀⠀⢘⣿⣿⠀⠀⣨⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⢿⣿⣛⣛⣛⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣩⣭⣵⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣯⣤⡴⠶⣾⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣶⣶⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠿⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣌⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣛⣷⡔⣌⢷⣄⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⣾⠉⠉⠙⢛⠿⣧⣽⣯⠉⢉⠉⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⣠⣴⣿⣷⣤⠤⠀⠀⠀⠉⠛⠿⣿⣇⡀⠀⠘⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠄⠒⠉⠀⣴⣿⠿⠛⠛⠿⢷⡇⠀⡀⠀⠀⠀⠀⠀⠈⠙⠳⣤⣄⠙⠠
⠀⠀⠀⠰⠖⠚⠛⠛⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣴⣟⠀⠀⠀⠀⠀⠁⠨⣿⣷⣦⡄⣀⠀⠀⠀⠀⠀⠙⢦⣔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣲⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠉⢻⣿⣮⣱⠷⠡⠂⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⡛⠛⡛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⡙⠓⢶⣤⡀⠀⠀⠀⠈⣳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣾⣿⠃⠀⠐⠀⠀⢲⣤⣀⡀⠀⠀⠀⠀⠀⠙⠳⠶⣽⣿⣶⣄⠀⡎⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠙⠿⣿⣿⣦⣠⣤⡀⢈⠌⢹⠛⢿⣧⣤⡀⣄⡄⡀⠀⠀⠀⠉⢿⣿⣿⣦⣤⡤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠁⢀⡈⡉⠙⠿⠇⢿⣆⣄⣆⠀⡄⢨⡟⠻⣿⣮⣡⣀⠀⠀⠀⡼⠉⠙⢷⣄
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣶⣦⣜⣿⠀⢟⢻⣿⣯⠀⠀⠀⠀⠟⣿⣿⣾⣤⣲⣀⣀⢦⣿⣿⣿⣿⣿⣿⣷⣶⣤⡾⠿⣷⣄⠑
⠀⠀⠀⠠⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣬⣙⣿⠿⣿⣤⡄⣈⠁⠛⣿⣽⣿⣿⣷⣾⣿⣿⡿⠿⠛⠛⠋⠉⠀⠀⠀⠈⠻⢷⣄
⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⢭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣶⣾⣥⣧⣼⣵⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠈⠡⠶⠶⠶⠶⠲⠟⠳⠯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡋⠉⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠉⠉⠉⠙⠛⠛⠉⠉⠉

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

⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⡀
⠀⠀⠀⢠⡶⠶⠿⠿⠿⠷⠶⢶⡄⠀⠀⡴⠶⠾⠿⠿⠿⠶⠶⡄⠀⠀⡴⠶⠾⠿⠿⠿⠶⠶⡄⠀⠀⣴⠶⠶⠿⠿⠿⠷⠶⣦
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢸⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⢈⠛⢿⣿⣿⣿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⣷⣤⡀⠀⠀⠀⡇⠀⠀⣿⢸⣷⣦⡈⠻⢿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣿⣿⣿⣿⣷⣄⡀⡇⠀⠀⣿⢸⣿⣿⣿⣷⣤⡁⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠻⣿⣿⣿⣿⣿⡇⡇⠀⠀⣿⠸⣿⣿⣿⣿⣿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⣦⣀⠀⠀⠀⠀⠀⡇⠀⠀⡇⣤⣌⠙⠿⣿⣿⡇⡇⠀⠀⣿⢰⣄⡙⠻⣿⣿⡇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⣿⣿⣷⣦⣀⠀⠀⡇⠀⠀⡇⣿⣿⣿⣦⣌⠙⠇⡇⠀⠀⣿⢸⣿⣿⣶⣌⠙⠇⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡇⢿⣿⣿⣿⣿⣷⠄⡇⠀⠀⡇⢿⣿⣿⣿⣿⣷⠆⡇⠀⠀⣿⢸⣿⣿⣿⣿⣿⡆⣿
⠀⠀⠀⠈⠓⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠓⠒⠒⠒⠒⠒⠒⠒⠃⠀⠀⠙⠒⠒⠒⠒⠒⠒⠒⠛




⣿⣿⠟⠻⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢆⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⣫⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡞⠫⣾⣿⠟⠓⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⢉⡴⢋⣾⣷⣬⠏⣀⠄⠀⠀⠈⠉⠒⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⢧⢀⡴⢋⣴⣿⣿⣿⣿⣴⣧⠞⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⢠⣾⢋⣴⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀
⠀⠀⠀⠀⠀⢀⡜⠁⠀⡠⠊⣠⠞⠋⡡⡄⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢦⡄⣠⣄⣀⣀⡀
⠀⠀⠀⠀⠀⡎⠀⢻⠟⢥⠊⠁⠀⠀⠑⠛⣀⣀⣀⣤⣤⣾⣵⣷⣢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢀⣿⣿⣿⣿⣿⣿⣷⣷⡄
⠀⠀⠀⠀⢰⠁⣠⡎⠀⢀⡱⠀⣀⣠⣴⣾⣿⣿⣿⡿⢿⣿⣧⣌⠛⠛⣧⣤⡀⠀⠀⠀⠀⠀⠀⡀⠀⠎⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷⡠
⠀⠀⠀⠀⣅⡼⣡⡿⣶⡶⡾⢿⣿⣿⣿⣿⣿⣿⣿⣏⠸⣿⣿⣿⣷⣶⣤⣄⣙⣳⣂⣤⢀⣠⣾⣷⡶⠀⢰⣿⣿⣿⠛⡿⣿⢿⣿⡿⡏⡇
⠀⠀⠀⣠⣿⡾⠋⠀⠻⣧⡄⢤⠙⢯⣽⣿⣿⣿⣟⡛⠲⢿⣝⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⠯⣼⣿⠃⠀⢸⣿⣿⣿⣇⣁⡇⣿⣿⡇⠇⠃
⠀⠀⠐⠿⠋⠀⠀⠀⠀⠈⠙⢦⣀⣼⠁⠈⠹⣷⠀⠈⠉⣦⠈⠻⠦⣤⣽⣿⣟⣻⣷⢤⣬⣀⣻⡿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⢸⠟⠒⠚⠁⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠊⠉⠉⠉⠉⠉⠓⠒⠦⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⣫⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣋⢔⡫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡴⢫⢞⡵⣫⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢌⢳⡀
⠀⠀⠀⠀⠀⠀⢀⡞⡱⡡⢪⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢣⡹⣄
⠀⠀⠀⠀⠀⢠⡏⡜⡕⣱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠡⡘⣆
⠀⠀⠀⠀⠀⡾⡸⡰⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⢣⢹
⠀⠀⠀⠀⢸⢣⢇⠃⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⡄
⠀⠀⠀⠀⣼⢸⠘⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠁
⠀⠀⠀⠀⣿⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⢻⢸⢠⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠸⡎⡏⡄⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢇⠁
⠀⠀⠀⠀⠀⢳⠹⡰⡘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⡌⣸
⠀⠀⠀⠀⠀⠈⢧⠳⡱⡘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⡘⣰⠃
⠀⠀⠀⠀⠀⠀⠈⢳⡹⣜⢎⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠈⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⠢⡑⢌⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡝⢆⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡌⢇⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡜⡎⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢰⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⡆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⢇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠸⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣟⠛⢛⡛⠛⢛⠛⣛⣛⣉⡉⢁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣍⣙⣻⣿⣿⡿⠿⠿⠿⠿⢿⡟⢋⣉⣩⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡛⢻⣿⣿⣟⠛⠛⠛⠛⠋⠉⠉⣉⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⢛⢿⣿⣿⡛⠛⠛⠛⠛⠛⠉⠛⣹⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣛⣛⣿⣿⣿⡗⠒⠒⠒⢛⣛⣻⣯⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣽⣿⣭⣉⣉⣉⣉⣡⣤⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⣿⣿⣿⣿⠟⠁

⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⢠⠖⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠳⣦⣄
⠀⠀⠀⡟⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄
⠀⠨⣿⣿⣿⣿⣿⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⢀⣿⡇
⠀⠀⠀⡇⠀⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⢨⣿⣿⣿⣿⣿⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢸⣿⡇
⠀⠀⠀⡇⠀⢉⡁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇
⠀⢘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⡇⠀⠉⠁⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡇
⠀⢐⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⣿⠀⠉⠁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⢸⣿
⠀⢐⣿⣿⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⡏⠀⠉⠁⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⠀⠀⣿⣿
⠀⢐⣿⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⡇⠀⠉⠁⠀⠀⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⡏
⠀⢐⣾⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⡏⠀⠉⠉⠀⠀⠛⠚⠛⠛⠛⠛⠛⠓⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠒⠒⠒⠒⠂⠀⠀⣿⡇
⠀⢐⣶⣿⣶⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⡏⠀⠉⠉⠀⠠⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⡥⠀⢠⣿⡇
⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡼⣻⠇
⠀⠀⠀⠘⢷⣦⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣒⣲⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⡿⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⣼⡟⠉⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢠⡟⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠶⠶⠦⠤⣄⣀⡿⢷⡀⠹⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⣡⠊⢉⣉⡭⠭⠽⠯⠭⢭⣉⣙⠆
⠀⠀⠀⠀⠀⠀⠀⢚⣭⣶⣾⣿⣿⣿⣿⣷⣶⣿⣓⢦⡻⣎⠉⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢶⡥⣞⣽⣶⣿⣿⣿⣿⣿⣷⣶⣿⣓⢤⡀
⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⠉⠁⠈⠀⠁⠈⣿⠿⣿⣷⣝⢿⣷⡀⠙⣦⡀⠀⠀⠀⠀⠀⠀⣸⠏⡰⣫⣿⣿⡟⠛⠉⠁⠈⠀⠁⢉⡟⠿⣿⣷⣝⢆
⠀⠀⠀⠀⣰⣿⡟⠃⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠈⠻⣿⣧⢻⠻⣦⠜⠻⣆⡀⠀⢀⣀⣴⠏⣼⣿⣿⡟⠉⠳⣄⠀⠀⠀⠀⢀⡞⠀⠀⠈⠿⣿⣮⢷⡄
⠀⠀⠀⢰⣿⡯⠀⠀⠀⠀⠀⠀⠀⢁⣼⠃⠀⠀⠀⠀⠹⣿⣧⠀⠙⢷⡀⠬⣭⣍⠉⣹⠏⠘⣿⣿⡯⠀⠀⠀⠈⢷⡄⠀⢀⡞⠀⠀⠀⠀⠀⢻⣿⡎
⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⣰⣾⣿⣧⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠈⠻⣄⣸⣀⣴⡏⢀⢠⣿⣿⠦⠤⠤⠤⠤⢤⣿⣿⣿⣧⠀⠀⠀⠀⠀⠈⣿⣿
⠀⠀⠀⣿⣿⡀⠀⠀⠀⠀⠀⠘⢿⡿⠏⠀⠀⠀⠀⠀⠠⣿⣿⠀⠀⠀⢠⠛⣿⣶⣿⣷⣠⣤⣿⣿⣶⠶⠿⠛⢛⣻⣿⠿⠿⠏⠀⠀⠀⠀⠀⠠⣿⡿
⠀⠀⠀⠸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡏⠀⠀⠀⠸⡙⢽⣿⢿⡹⠁⣀⣹⣿⣧⠒⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣺⣿⠇
⠀⠀⠀⠀⠹⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠟⠀⠀⠀⠀⠀⠉⠛⢺⠚⠉⠁⠀⠀⠻⣿⣷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠏
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣄⣀⠀⢀⢀⣀⣤⣾⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠷⠂⠀⠀⠀⠀⠙⠿⣿⣶⣤⣄⣀⠀⡀⢀⣀⣤⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⠿⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⢿⣿⣿⡿⠿⠛⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⢰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠶⠶⠶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠂⠀⣠⡴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠷⣦⡀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⡠⢄⡠⢤⡴⠤⣄⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⢰⡟⠀⠀⠀⠀⠀⢰⠀⢨⡅⠣⠇⢰⣩⠉⡆⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⢀⡿⠀⠀⠀⠤⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠖⠋⠀⠈⠻⣆⠀⠀⠀⠀⠈⡇⠀⠀⡸⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⢰⠀⠀⡇⠀⠀⢠⡾⠋⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉⠀⠀⠀⠙⢷⡀⠀⢸⠀⢀⡁⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣟⣉⣉⣉⣉⣉⣉⣛⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣤⣤⣤⠤⠤⠤⠤⠤⠤⠤⢤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣶⣶⡒⠶⣟⡛⠿⣟⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠪⣝⡻⢷⣲⠭⣶⢤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣙⠲⢽⣶⡬⣝⡶⢽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣌⡙⠲⢬⣙⠲⢭⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣌⡉⠓⠦⢬⣙⡒⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⣉⣛⣒⣦⣬⡽⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢯⣭⣭⡥⠴⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⢰⡄⠶⠆⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⡀⡀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠟⠉⠁⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⡤⠤⠄⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⢀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠐⠃⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢠⣄⣀⡀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⠛⠉⠁⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣶⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣄⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠋⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠠⠿⠉⠀⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⣤⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⡠⠀⣠⢀
⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠛⠐⠂⠻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠤⠤⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⡀⢠⣄⣀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠢⢀⠤⠺⠃⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠰⢿⣿⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣀⣰⣿⣿⡆⠳⠂⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⣤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⢯⣽⡅⠢⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⣿⡽⢠⡾⢋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⣶⡧⣤⡀⡆
⠀⠀⠀⠀⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠎⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧⡾⠁⠸⠇⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣜⢿⡄⠐⣠⣌⡅
⠀⠀⠀⠀⠤⠠⠶⠿⠆⠇⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⡀⣴⣆⣠⡀⡀⡀⡙⠿⢿⠛⠋⠀⢠⡖⠀⠋⠀⠀⠀⠀⠀⠸⠀⣀⡀⠀⠠⠆⡐⡿⣶⣷⠿⠄⠈⠆
⠀⠀⠀⠀⢄⡀⠄⣤⡀⣠⡀⠀⠀⠀⠀⢠⣿⡹⠯⠘⠓⢽⠟⠁⠰⢄⠈⢉⣀⡈⠀⠀⢣⣼⡇⠀⠀⠀⠀⠀⠀⢀⣶⢀⡟⠁⠀⠀⡀⢈⣧⣄⣁⣠⡀⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠈⠛⡀⢧⣄⢀⡀⠸⠟⠃⠚⠀⠀⣬⣧⣟⠀⠺⠖⣤⣤⡇⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠐⢾⠋⡠⠁⠀⠀⠀⠙⣾⣏⢡⡄⠈⠢⣠⡆
⠀⠀⠀⠠⠭⠤⠀⠿⠃⢀⡶⠚⠻⣄⠁⠀⠐⠂⠀⡀⢰⡄⠹⠃⠒⠢⢀⣀⢘⡗⠀⢰⠇⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡀⣀⢀⠀⢈⠀⠂⡶⠉⣰⣿⡄
⠀⠀⠀⠀⣠⠀⠀⣤⡀⠘⠣⢤⡾⠛⠇⠀⣴⣖⣿⡷⠄⠀⠀⠀⠀⠀⠀⠅⠘⠃⠀⠀⠀⠀⠀⢀⠚⡅⢹⡄⢀⡄⠀⠀⠀⢀⣙⣏⠀⠈⢺⠞⠁⣰⠏⠉⡃
⠀⠀⠀⠀⠁⢀⠘⢭⡗⢀⢀⡞⡃⠀⠀⠀⠿⣿⣿⡷⡆⠀⠀⣀⠆⣄⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⢠⣴⡄⡾⣾⣷⣘⢿⡧⠙⢿⣡⢒⡁⠀⣸⠏⠠⣶⡄
⠀⠀⠀⠐⠛⢻⣿⠿⠃⢀⡴⠞⢘⣷⡧⢀⣺⣿⡿⠀⢖⠀⢠⡇⠀⠘⢷⡀⠀⠀⠘⠀⠀⠀⡶⠄⠀⣺⣿⣧⣿⣿⣿⣿⣿⣦⡀⠸⠇⠻⠇⡸⠁⢀⣤⣍⠁
⠀⠀⠀⠠⣤⢸⣷⣴⣄⡾⠋⣸⠃⠙⣳⣿⣿⡿⠧⠐⢠⣄⢸⡇⠀⠀⠀⣷⡀⠀⣆⠀⠀⠀⠂⠀⠈⠛⢹⣿⣿⣿⡿⢻⣿⣿⣷⣤⠀⠀⣠⢷⣦⡼⠻⠠⠆
⠀⠀⠀⢀⣀⢀⣩⣿⡟⠁⢸⠇⠀⠐⠋⢦⣿⠀⠀⡇⠀⣙⣸⡇⠀⠀⣀⣈⣁⢼⣿⢁⣶⠀⠀⠳⠀⣼⣿⣿⣿⣿⣟⠛⢻⣯⣿⣿⣁⣰⡟⣷⠟⠀⠐⡦⡄
⠀⠀⠀⠈⠛⠂⢾⠿⣧⣀⣅⡀⠀⠀⣠⣤⣶⣶⣤⣴⡟⢿⣿⡷⢀⠐⠿⢟⣧⠀⣃⢸⣿⠀⢠⣤⣾⣛⢿⡿⣿⡿⢿⣤⣾⠘⡻⠚⣿⣿⣿⣿⣄⣠⣾⣤⡀
⠀⠀⠀⠠⡶⣤⣤⣶⣿⠁⢠⣷⣶⠋⢩⣽⡏⠙⠻⢿⣿⣿⣿⠁⠀⠀⠀⢠⣿⠇⠃⣸⣿⣦⡀⠀⢨⠿⠧⠸⠛⠁⠘⣿⠏⠀⢀⣾⣻⣴⣿⣶⣾⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠫⣹⡿⠧⢾⣿⡆⢀⣾⣿⠀⠀⠀⠀⢧⠀⣿⠀⠀⠀⠀⠀⢰⡼⠚⣹⠉⣿⠇⠀⠈⠀⠀⠀⠀⠀⠀⠋⠀⠀⠞⣼⡟⠿⣾⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⣤⣤⣀⣀⣠⣀
⠀⠀⠀⠀⠀⠻⠶⠮⣽⣿⣟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢻⡆⣿⠀⠀⠀⠀⠀⠀⠀⣠⠞⣻⣏⣿⠶⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⢷⣤⣤⣤⣀⡀⠀⣰⠇⡾⠙⠾⠋⠛⠓⠶⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣟⣀⠀⠈⣿⣛⣡⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣹⡟⠋⠉⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣾⠦⣤⣄⣀⠀⠀⠀⠀⢿⣿⣦⠤⣄⣀⣉⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⢠⣿⡓⣶⠶⢭⣽⣛⡲⠦⣤⣄⣉⣿⢳⡶⠦⣤⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠓⣺⠃⢳⡌⢷⡀⠀⠉⠉⠛⠲⠦⣽⢃⣾⣅⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⢀⣤⠴⠖⠛⢻⡇⠀⣿⣄⡀⠻⡄⢳⡄⠀⠀⠀⠀⢰⡏⣼⠳⢮⡳⢦⣠⡴⠛⣉⣤⡴⠶⠶⡦⣤⣈⠙⠷⣄
⠀⠀⠀⠀⣠⠞⠋⣀⣤⠴⡶⣾⠀⢠⣇⡈⠙⢶⣽⣆⠻⣄⠀⠀⢀⡟⣰⠇⠀⠀⢹⣦⡙⢶⡟⣍⠀⢇⠀⢸⠁⢠⠏⠛⢦⡈⠻⣄
⠀⠀⠀⡾⠁⢠⡞⠉⢣⠀⢳⡏⠀⣸⣫⠛⣦⡀⠹⣿⣆⠙⣦⠀⣾⢡⡏⠀⠀⢠⡟⢠⣿⣦⣝⠿⣦⢸⡀⢸⠀⡞⠀⢀⡴⠻⡄⠘⣧
⠀⠀⣼⠁⢰⡏⠙⠢⣄⠳⣼⠃⠀⣿⡡⢊⡬⢷⡄⠸⣿⣦⣘⡿⠧⣾⡦⠤⠤⠾⣤⣿⣤⣬⣿⣷⣬⣿⣧⣟⣼⠀⡰⠋⠀⣀⢿⡄⠘⡇
⠀⠀⡏⠀⣾⠉⠙⠒⠢⣽⣿⢠⢸⡿⠿⠗⠒⠊⢿⠀⣿⢸⢯⢖⣒⢦⢻⡦⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣀⣈⠉⠻⣦⠒⠉⠀⠘⡇⠀⡇
⠀⠸⡇⠀⢿⠊⠉⢉⡩⠖⡿⡶⡟⢿⠫⢍⡉⠉⢹⠆⣿⢸⡌⣾⠿⠼⣰⠷⠶⠶⠶⠶⠶⠶⠤⠤⠤⠤⠤⠽⠃⢀⣿⡓⠒⠲⢴⡇⢀⡇
⠀⠀⢻⡄⠘⣷⠚⠉⢠⠞⢀⡇⢧⠀⠱⣄⠈⢑⡾⢠⡟⠀⣹⠟⠲⠾⠿⠶⠶⠶⣶⠶⣶⠶⡶⠶⢶⠶⢶⠶⢶⠛⣅⠙⠢⣤⡞⠀⣼⠃
⠀⠀⠀⠹⣆⠘⠷⣴⠃⠀⢸⠀⢸⡀⠀⢈⣳⠟⢁⡾⠉⠛⠷⠄⠀⠀⠀⠀⠀⠀⠹⣆⠙⢿⣄⡠⠋⠀⡼⠀⠸⡄⠈⣦⡴⠋⢀⡼⠃
⠀⠀⠀⠀⠘⠷⣄⡈⠛⠶⠷⣤⣤⠷⠖⠋⣁⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡈⠙⠲⠦⠧⠤⠴⠷⠛⠉⣀⡴⠛
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠶⠤⠤⠤⠶⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠦⣤⣤⡤⠶⠞⠛⠉

⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣠⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠠⠤⣾⣦⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠈⠉⢻⠋⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢐⣒⣿⣗⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠉⢹⠁⢉⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠨⣯⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢀⣀⣸⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠸⠿⢿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢠⣤⣼⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠐⠛⢻⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢰⣶⣾⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠉⣿⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣀⣿⣀⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠨⠿⣿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⢀⣀⣿⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠐⠓⣿⠚⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠠⣤⣿⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠙⣿⠉⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⣯⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⡇
⠀⠀⠀⠀⠀⠹⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠶⠶⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⡐⢀⡴⠚⠛⠛⠲⣄⠐⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠟⣩⣵⣮⣭⡻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢐⠀⡾⠀⢀⠀⢀⠀⠸⡆⢐⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣿⢿⣿⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⡀⠈⠀⠈⠀⣸⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣧⢻⣿⣿⣿⣿⡿⣰⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⢳⡄⣀⡀⣴⢃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣧⢹⣿⢿⡟⣰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢧⣿⣧⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣯⡷⢷⣿⠆⠇⢸⣣⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠮⠭⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⠶⢾⣿⣀⣀⣼⠷⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡶⠶⠛⠋⠉⠀⠐⠂⠀⢀⡉⠕⠋⠀⠀⠈⠉⠛⠷⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠭⢿⣠⣤⡴⠶⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣀⣨⡽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⠶⣿⣀⣸⡍⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢄⡀⠀⠈⢲⣄⠀⣄⠠⣐⣠⠴⠖⠋⠉⠀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢧⣄⡀⠛⠒⠚⠁⠀⠀⠀⣀⡠⠄⠒⠁⠀⠀⠀⠀⢀⣠⡽⠆⢀⣀⡭⠷⠚⠋⠁⢀⣀⣤⡶⠞⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣄⠈⠙⠶⣄⡀⠀⠀⠶⣌⡀⠀⠀⠀⠀⣀⡤⠶⢚⣉⡥⠶⠚⠉⠁⢀⣀⣤⡶⠞⠛⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠻⢦⣄⠈⠙⠲⣤⡀⠀⠻⢷⡖⣖⣩⡥⠖⠛⠉⠁⣀⣠⣤⠶⠟⠋⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠉⠛⢶⣄⡀⠙⢷⣶⡶⠟⠋⠁⣀⣤⣴⠶⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠈⠛⠶⣼⣿⣤⣶⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣄⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⡀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣄⢸⡇⠀⠀⣀⣴⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠶⠛⠉

⣿⣿⣿⣿⡿⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠿⠿⠿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠻⠋⠛⢿⣿⣿⣿⣿⣿⣿
⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⠿⢿⣿⣿
⣿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻
⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡴⠁⠀⢀⣾⠋⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣆⣠⠄⢠⡞⢁⣿⣶⣾⡶⣿⣉⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣾⠁⣠⣿⣟⣿⣿⣿⣿⡿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⣿⣿⣿⣿⢈⣿⢸⣏⣿⣿⣿⠃⣾⡷⣣⣔⣠⣤⠶
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⣿⣿⢻⣷⠿⠛⢸⢿⣿⠛⣿⣸⣿⣼⣿⣿⣥⣤⡼⢀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣼⢿⣛⣿⡆⠀⢸⣿⠟⣴⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣝⣿⠁⣸⣷⣴⢿⣿⣻⣿⣿⠛⣻⡟⣫⣽⣿⣿⣿⣿⣿⣀⡄
⠀⠀⠀⠀⠀⠀⠀⢰⣿⠞⠛⣿⣶⡟⠿⠿⠋⠉⠉⣵⣿⡿⢿⣾⣻⣿⣿⣿⣿⢿⡟⢋⣸⡷⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣴⣿⢛⠁⠀⠀⠀⢤⣿⣿⣯⣴⣿⣿⣿⣿⠿⠟⣻⡿⠛⣻⢟⡴
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣽⣭⡅⠁⠰⠂⠀⠀⠀⣹⣿⣯⣽⡾⠛⠙⢷⣴⠟⢡⣧⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⣐⣿⠀⠀⠀⣠⠊⢁⣰⣶⣿⣄⣠⣶⠾⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣻⡯⠇⠀⠸⠇⢂⣿⡿⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣻⠀⢠⡄⠼⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣯⡻⠇⠹⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣧⠀⡵⣸⣿⣿⣷⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣀⠀⣀⣀⣤⣴⣶⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⢰⡿⣿⣿⣿⣟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣽⣿⣶⣶⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣾⣷⣿⣿⣿⣿⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣈⣅⣈⡉⠶⠖⡂
⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⣿⡿⣿⢹⡰⣾⣿⣿⣿⠿⣧⣈⡻⠿⣿⣷⠾⡥⠭⠅⠀⠀⠉
⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣧⣿⣿⢻⡜⣿⣧⠈⠁⢈⠛⠻⣿⣃⠐⠒⠐⠒
⣿⣤⣤⣤⣤⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⡼⢎⡓⠫⢴⣤⡈⠑⠦⣭⠉⠁
⡄⠈⣉⡹⠛⠛⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⠿⣿⣿⣿⣿⢫⣉⡳⣮⣶⡠⢌⠻⠦⠀⠀⠀⠰⠶
⠏⠀⠀⠀⠀⠀⠹⢿⢿⡿⢿⢿⣿⡿⢻⣿⣿⢻⣟⡃⢸⡇⢹⣿⣿⣧⠙⣿⣛⣿⣿⠎⠁
⣤⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠿⠏⠁⣿⡏⢸⣿⡟⡉⢸⣿⢸⣿⣿⣿⣧⠜⠛⠉
⣿⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢋⡉⠙⠻⢈⠙⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠄⠙⠠⢍⣁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣄⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢌⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠒⠂⠀⠀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⣰⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢳⢍⠪⡢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠛⠀⠁⢀⡤⠶⠶⠶⠶⠖⢲⠾⡟⠛⠋
⠀⠀⠀⠀⠀⠀⢀⠠⠄⣘⢒⡒⣛⠲⠶⣤⢏⡋⠢⡑⢝⢌⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⡥⠤⢢⣿⡴⠶⠛⣛⣉⣛⡛⢿⢷⣤⡀
⠀⠀⠀⠀⡠⢊⠕⠚⡁⠀⠸⠀⠀⠈⠉⡾⣜⠫⡢⡙⢆⠱⡕⢄⢄⠀⠀⠀⠀⠀⠀⢀⢂⠊⢹⣶⣟⡡⠒⠈⠅⠀⠀⠸⠀⡏⠒⢬⡻⢶⢶⡄
⠀⠀⢠⢊⠔⠣⡀⠀⠰⡀⠀⡀⠀⠃⠠⠹⠀⠱⡜⢞⢆⠑⢌⠢⡑⠕⠀⠀⠀⠀⠀⡌⡌⣠⡟⡡⠛⢷⣄⠀⠐⡀⠀⠀⡸⠁⠀⡀⠙⢎⢻⣇
⠀⢠⠇⡌⠀⠀⠈⠢⢀⠰⡀⡇⡀⣠⣣⠃⠀⠀⠘⡌⣎⠆⠀⠳⡈⠪⡢⢀⠀⢀⡔⡐⢰⡉⢐⡁⠀⠀⠙⢷⣄⢡⠀⣴⠇⡠⠊⠀⠀⠈⣧⢳⡄
⠀⣼⠸⠀⠀⠀⠐⠠⠄⡑⢤⣴⢆⡟⠁⠀⠀⠀⠀⢳⢸⠇⠀⠀⠈⠲⡙⣆⣈⣩⣵⠁⠨⠇⠧⠤⠬⠤⠴⠶⠽⢷⣶⢿⡊⠠⠐⠂⠉⠀⢸⠘⡆
⠀⣧⢠⠀⠀⠀⠒⠂⢀⠀⢯⣳⠃⠀⠠⠄⠀⠀⠀⣸⢸⠀⠀⠀⠀⠀⢘⡿⠃⠀⠲⡀⡠⡤⡤⠤⢴⣶⡦⠀⠀⠀⣛⣃⠇⠐⠂⠤⠀⣀⢸⢀⡇
⠀⠸⡈⡄⢀⠠⠐⠈⠀⠈⠀⠀⠠⠈⠢⡀⠀⠀⢀⡏⡼⡄⠀⠀⠀⠀⠸⠠⡨⣩⢀⠏⠓⠓⢋⣉⣩⡤⢤⠴⠒⠚⠉⠉⠆⠈⠢⡀⠀⠀⡞⡸
⠀⠀⠳⡘⡄⠀⠀⠐⠁⠀⠂⠀⠀⠀⠀⠀⠑⢄⠞⡼⠁⠀⠀⠀⠀⠀⠀⢐⡿⠟⠁⠒⠉⠉⢍⠣⡀⠔⠁⠀⠰⠁⢠⠀⠘⡀⠀⠈⢢⠜⡰⠃
⠀⠀⠀⠙⢌⠢⣌⠀⠀⠘⠀⠀⠀⠀⠀⣀⠴⣫⠞⠀⠀⠀⠀⠀⠀⠀⠠⠶⡄⠀⠀⠀⠀⠀⠈⠳⣝⠢⣀⡰⠁⠀⠸⠀⠀⢡⣠⠔⣩⠞⠁
⠀⠀⠀⠀⠀⠙⠲⢭⣒⡒⠤⠤⢔⣒⡪⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢭⣁⣒⣚⣒⣋⠭⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠴⠶⠚⠛⠛⠛⠛⠛⠛⠓⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⠶⠶⢤⣤⣀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣄⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠙⣦
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠈⢷⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠈⣧
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⢸
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛
⢸
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞
⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡽⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢯⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠞⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⠀⢀⣀⣀⣤⠶⠋
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⡷⠤⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠷⠀⢀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣤⣤⣤⣶⣿⠋⠀⠀⣀⣀⣠⣼⡟⠁⢾⣿⡦⠀⠙⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠿⠛⠉⠁⠀⣀⣤⣶⠿⠛⢋⣉⣩⣤⣤⣤⣿⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⠋⣡⡴⠖⠛⠉⠁⠁⠀⠀⢸⣿⠀⠀⠀⠉⠉⠙⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⠽⣇⠀⠀⢹⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⢷⣄⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡿⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⡿⢍⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⢿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣌⣡⣾⣀⢀⣀
⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⣿⠃⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡉⠉⢭⣤⡐⠦⠌⠉⠀⠀⠐⠀⠀⠠⠄⠀⠀⠀⡀⢀
⠀⠀⢀⡴⠚⠉⠉⠀⠀⠙⠙⣾⣏⡀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠡⠰⣤⡶⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⣴⠃⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠽⣄⠀⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⣼⠁⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠼⡄⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⡟⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋
⢳⡀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁
⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠴⠋⠁
⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⢀⣠⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠶⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣠⠴⠖⠋⠁

⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⢳⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣿⡿⣿⣷⣶⣶⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢹
⠀⣸⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣄⣨⣿⣿⠟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣤⣄⣤⡀
⣰⡿⣡⠀⠀⠀⠀⠀⠀⠀⢀⣠⣮⣿⣾⣿⣿⠿⢿⣿⣿⣦⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⡿⢀⣞⡄⠀⠀⠐⠻⠿⣿⣿⣿⡅⢙⣿⣿⣅⣀⣈⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠁⢸⣿⡇⠀⠀⠀⠀⠀⠀⠠⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⠛⠻⠟⠻⠁⠀⠁⣿⣷⣶⣴⡶⠤
⠀⣼⢿⠁⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⠋⠀⠀⠀⠀⠹⣿⣿⣿⣾⣿⣿⣾⣿⣿⣿⣿⡏
⣶⣿⡍⠀⠀⠀⠀⠀⠀⠀⣾⢻⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣄⠀⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⡀
⡿⠋⠻⠟⠀⠀⠀⠀⠀⠀⣿⣼⣿⢇⠀⠹⣧⠀⢙⣿⣿⣿⣷⡔⢛⣿⢳⣿⢿⣿⣿⣿⡟⠋⠑⠛⣯⡹⣿⡿⠀⠀⠀⠀⠀⣷⡃
⠃⠀⢀⠄⠀⠀⠀⠀⠀⠀⠏⠈⠛⢿⣿⣷⣾⢟⣿⣽⣟⣴⣟⠛⠻⣿⣿⡭⣾⣿⣿⡾⢿⣷⣼⣿⣷⣏⣻⡁⠀⠀⠀⠀⠀⢻⣷⡄
⡧⠀⣨⡆⠀⠀⠀⠀⠀⠀⠨⣾⣾⣿⡿⣿⣾⣙⠃⠉⠹⣿⣿⣷⣬⣷⣿⣿⠿⢋⣡⣼⡶⢿⠋⢰⣿⣛⣻⡇⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠇⠀⣤⡇⠀⠀⠀⠀⠀⠀⠀⢺⣻⠽⣙⢹⠆⣹⢾⣶⣶⣷⣿⣿⣿⠿⢯⣿⣷⣾⣟⣯⣩⣜⣿⡭⣿⣩⢬⣧⡀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⣼⣿⡇⠀⠀⠀⠀⠀⣀⣤⣨⣶⣠⣷⠾⣿⠿⠛⣋⣉⠁⣾⣿⣿⣿⣷⡆⠉⠙⠻⠿⢷⣶⣷⣶⣿⡶⣀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠹
⢀⣿⣿⠃⣴⣷⠀⠀⠀⠈⣽⣿⣭⣭⣤⣿⠯⠀⠈⡿⢁⣀⣠⣭⣿⣿⣿⡿⣿⣦⠠⣤⣬⣿⣟⢿⡟⣿⣿⣿⡟
⣼⣿⡿⣼⣿⡟⠀⠀⠀⠀⠀⠉⠛⢯⣛⣿⣿⢟⣿⣀⣻⣿⣿⣿⣿⡿⠿⠛⠛⠉⠀⠶⢁⣠⣤⣤⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠠⣤
⣿⣿⣷⣿⣿⠃⠀⠀⠀⠀⣠⣦⠀⠀⠈⢻⣿⣷⢿⣿⣿⣿⣿⣿⣛⣁⢀⣀⣶⣾⣿⣿⢟⣻⣿⡇⠁⣾⣷⠄⠀⠀⠀⠀⠀⠀⠀⢶⡀
⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠙⠋⢠⡄⠀⢸⡏⠻⣼⣿⡿⣿⡟⠹⣿⣿⠛⣿⢏⣭⡿⢗⣿⣿⡏⠀⠴⠌⡁⠀⠀⠀⠀⠀⠀⠀⢻⣿⣇
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢠⣦⠀⠀⡀⢸⡇⠁⣸⣿⣾⣿⣧⡇⢹⠙⠀⣿⣿⣿⣿⡶⠟⣿⡇⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠘⢇⣽⣿⣿⣿⣿⣿⣿⣾⠀⣼⣿⣿⣿⣿⠄⠀⣿⣷⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿
⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢰⣿⣧⠀⢸⣿⣿⣿⠻⣿⣿⠀⣿⣿⠟⠉⠉⠀⠀⣿⡇⠙⠋⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣿
⣿⣿⣿⣿⢀⣾⣷⡄⠀⠀⠀⠀⠀⠀⠈⠛⢋⠀⠈⣿⣿⣿⠀⢹⣿⣿⣿⣏⣀⣀⡠⠀⠀⣿⠃⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯
⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⣿⣿⣿⠀⢸⡟⠟⠀⣿⣿⣿⠀⠀⠀⠃⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢀⠀⠛⠛⠻⠀⠘⠇⠀⠀⠉⠈⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈