Decay Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⡻⠛⠛⠒⠦⣤⣀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠶⠶⠶⠶⠦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣞⣓⣚⣋⠉⢉⡥⠞⠁⠀⠀⠀⠀⠀⠉⠙⢓⣶⡶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⢠⡶⢉⢷⣠⠞⠁⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⠀⠀⢤⡤⠤⠶⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆
⠀⠀⠀⠀⠀⢀⢇⢃⡏⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⠀⠀⠀⡜⠘⡸⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠟⠃⠀⠀⠀⠀⠾⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠰⠁⠑⠁⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⡇⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠒⠲⣄⢀⡤⠖⠛⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣇⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⣠⣶⠦⢸⡞⠀⣠⣤⣤⡀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠸⡄⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠿⣿⠾⣸⡇⠀⣿⣿⣤⡗⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⢀⡽⠀⢀⣠⡼⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⠤⠴⠞⠁⢷⡀⠈⠙⠋⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏
⠀⠀⢀⡴⠋⣠⠞⠉⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⣠⠎⣠⠞⠁⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢹
⣰⠃⢰⣏⣀⣀⣤⠤⠴⠒⠋⣁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⣀⣀⣠⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇⣸
⠳⢤⣤⣤⡤⠤⠴⠖⠒⠛⠉⠁⣿⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠉⠉⠀⠀⠀⠈⢷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠓⠛⠛⠛⠀⠀⠀⠀⠉⠙⠛⢦⡀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⣿⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⢻⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠰⠃⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠛⠁⠀⠀⠀⠀⠀⠀⣿⠀⠀⣼⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⣿⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢿⠀⠀⠈⠙⢦⡀⡤⢦⡤⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⢀⣴⠞⠋⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠶⠶⣶⣄⠀⠀⠀⠀⠀⢠⡟⠀⠀⢸⠀⠀⠀⠀⠈⢻⡁⡞⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡿⠋⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠙⢷⣄⠀⠀⢀⡾⠁⠀⠀⠀⡇⠀⠀⠀⠀⠀⣷⠃⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⣀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡶⠟⠁⠀⠀⠀⠀⢧⣀⡀⠀⠀⣀⣀⣀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⣤⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠚⠁⠈⠉
⣶⣶⡶⠶⣶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⠀⣭⡍⠉⠁⠀⢒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢠⠀⠤⠀⠄⠄⢠⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⣤⣶⣶⡾⠺⣯⣅⢉⡛⡇⡏
⠀⠀⣀⠀⣀⡤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⣒⣺⣀⣐⢸⡃⡒⢚⠒⠓⣿⣛⡓⣘⡒⠚⣿⣿⣿⣿⣿⣷⣶⣿⣿⠿⠿⢶⣧
⠀⠘⣿⣤⣿⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣶⣷⣼⣿⣿⣿⣿⣾⣿⣿⣿⣼⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⡏⢛⠻⠟
⠀⠀⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⢠⢰⣦⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⢰⣿⣿⠇⢀⡈⠀⠀⠀⠀⢀⣴⣶⡷⠠⠨⢸⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⠽⣧⠀⠀⢀⣀⣈⡉⠉⠛⠛⠻⣿⣿⣿⣿⣿⡷
⢀⣿⣿⣟⣠⠏⠉⣩⠁⢨⣬⣿⠟⠩⡤⢨⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣶⣄⣈⣹⣯⣭⣄⣤⣤⣷⣦⣾⣿⣿⣿⣿⣯⣽⠁⠀⠂
⢸⣿⣿⣿⠇⣆⢠⣅⢠⣼⣿⢻⠒⢺⣿⢻⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣇⠇⠆
⣾⣿⣿⠏⠀⢺⣻⢩⣽⠽⢿⢾⣋⢹⣿⣾⣿⣿⣿⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢹⡇
⣿⣿⣿⠀⠰⠾⣾⡾⢿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⡏⢹⢿⡀
⣿⣿⣿⡆⠀⠨⣿⡯⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣏⣡⡼⠋⠋⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣦⣼⠘⣧
⣿⣿⣿⣇⠀⣸⣿⣽⣿⣿⣿⣿⣿⣿⣿⡿⠟⢿⣿⣿⠩⠛⠉⠁⠀⠠⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⢫⣿⡟⢸⢹⠘⡇⠀⠀⠀⣠
⢿⣿⢸⡟⠃⠛⣿⣿⣿⣿⣿⣿⣭⣵⣤⣴⠴⣿⠓⠔⢐⣱⠺⣿⣥⣤⣶⣿⣿⣿⣿⡿⣻⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣧⣿⣿⠃⣼⠀⠀⣿⣂⣠⢤⣾
⠘⣿⣿⡇⠀⣸⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣇⣿⣿⣯⡟⣹⣀⣿⣠⡶⢿⠁⠠⣿⣿
⠀⢹⣿⡧⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⡟⢻⣿⣿⣿⣿⡇⣲⣿⣿⡟⠁⠀⢸⠛⠋⣹⣿
⠀⢸⣿⠇⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⣟⣿⢻⣿⡛⢿⣇⠻⢿⡟⠿⡟⠋⠛⢻⣿⣷⡦⢸⣿⣿⣿⣿⣿⣿⣿⠚⠁⠀⠴⢻⠀⢠⣿⣿
⠀⢸⣇⡀⣸⣯⡹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡇⠉⠉⢹⢩⠛⠉⣯⠈⢸⣿⣿⣿⢟⣿⣿⣿⣿⡷⣶⣾⣿⣿⣿⣿⡿⣿⣼⠈⠁⠀⢸⣤⣾⣿⣿
⣤⣿⣿⠀⢸⣿⡆⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⢸⣇⠀⠀⢸⢐⠀⠀⣬⠐⣿⣿⣟⣿⠶⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡧⢸⣀⣴⣿⣿⣿⣿⣿⣿
⣿⡛⢿⣇⣾⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⡝⢸⡏⠀⠀⢸⢸⠀⠰⢾⣚⣿⣿⣿⢩⡕⠁⣄⣼⣿⣿⣯⡞⣻⣿⣿⣿⣿⣶⣿⠟⠋⣹⣿⣿⣿⣿⣿
⣷⣿⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⠀⠛⢸⡇⠀⠀⢸⡎⠰⠃⠈⣿⢫⣿⣷⣼⣦⣾⢻⡏⢹⣿⡃⢸⣿⣿⣿⡿⢿⠏⠀⠀⠀⣿⣿⡟⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⡆⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠆
⠀⠀⠀⠀⠀⠀⢀⠐⢿⣿⣿⣿⣿⣿⣿⡄⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⢀⣴⣷⣄
⠀⠀⠀⠀⠀⣤⣿⣷⣤⣝⡻⣿⣿⣿⣿⣷⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣳⣿⠟⣠⣴⣿⣿⣿⣿⣧⡀
⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠈⣿⣿⣿⣿⣿⣿⣿⡿⠋⣰⡿⣣⣾⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠰⢾⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢹⣿⣿⣿⣿⣿⣿⡇⣴⣿⣿⣿⣿⣿⣿⣿⣿⠟⢿⣿⣷⡀
⠀⢀⣶⣤⣄⣉⠙⠛⠿⢷⣶⣬⣭⣉⡉⢉⣻⣿⡎⣿⣿⣿⣿⣿⣯⢤⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣿⣿⣧
⠀⣼⣿⣿⣿⣿⣿⣷⣶⣤⣌⣉⠛⠻⠿⢹⣿⣿⣿⣿⠛⠟⠁⣾⣁⡸⠿⢿⣿⣿⣿⣿⣿⣟⣛⣛⣉⣉⣉⣁
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣴⣦⣬⣥⣬⠋⠀⣿⣝⣻⣿⣿⣶⣾⡝⣲⣾⡿⠛⠿⠿⠿⠿⢿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⠿⡻⢯⣓⡿⣴⣎⢦⣭⣛⠯⣙⡛⠟⠘⠘⢿⣿⣿⣶⣶⣶⣶⣶⣶⣦
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢟⣋⣭⣴⣶⢟⠟⣹⣾⣿⣿⣧⡈⢿⣿⣾⣿⣿⣷⣶⣤⣙⠻⢿⣿⣿⣿⣿⣶⡄
⠀⢻⣿⣿⣿⣿⣿⣿⣯⣷⣾⣿⣿⣿⣋⢁⣵⣾⣿⣿⣿⣿⣿⣿⣄⠻⣿⣿⢟⣵⣿⣿⣿⣿⣶⣮⣽⣟⠛⠻⠗
⠀⠀⢘⣿⣿⣿⣿⠿⠟⠛⠋⣩⣼⢋⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣬⡁⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢿⣿⡿⠋⠀⣰⣾⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣜⢿⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠈⠟⠁⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠙⢿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠙⢿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡿⣳⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⡿⣫⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠉⠉⠉⠁
⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣿⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣋⣹⣿⣷⣼⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣼⣿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣟⣉⡉⢿⣿⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠋⠉⠀⠈⡿⠁⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠈⠀⡀⠀⠀⠀⢢⡀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠿⠀⣷⠀⠀⠶⠈⡇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⠀⠇⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⠀⣀⣀⡀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⡟⠹⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⢠⣿⡇⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⣿⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠐⡗⠀⡇⠀⣿⣧⣼⣇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡆⠀⠀⠀⢨⠀⠀⡗⠀⡇⠀⣿⣿⣷⣾⣿⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠋⠁⠀⠀⠀⠶⠤⠤⠿⠷⠧⠠⠿⠿⠛⠋⠁⠀⣹⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢄⠄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡍⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠈⢿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣉⢳⠶⣿⣿⡏⢀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⣾⣶⣿⡿⣡⣾⣿⣤⣠⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣟⣋⡉⢹⣿⣳⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⢻⣿⣿⠋⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢭⣿⣿⡿⠇⠈⠙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠀⠀⢠⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠋⠀⠀⠀⠀⠉⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⣤⣤⣤⣤⠤⣤⣤⠄⠀⠀⠠⠄⠀⢻⣀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠒⠊⠉⠉⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠉⠀⠘⠀⠙⠛⠛⠛⠛⠙⠛⠢⣤
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣧
⠀⠀⠀⠀⠀⢹⣷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⣶⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣏⠁⠀⠀⠀⠀⠀⠀⠰⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠹⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠦⣤⣤⣀⣀⣀⣀⣀⣐⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉

⡄⠈⠻⠿⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⠀⠀⠁⠺⣿⢛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⣀⠀⡄
⠀⢀⠀⠀⠈⠙⠿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⣯⣍⠹⠛⠛⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⢀⡀
⠀⠀⠃⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢀⣤⣿⣷⡀⠈⠙⠀⠀⠀⠀⢲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⣠⣄⢰⠀⡀⠘⢻⣿⣿⣷⠀⠀⠀⠀⠀⠀⢾⣿⣧⣀⠀⠀⢀⡄⠰⠀⣤⣄⡀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣷⠀⠀⠀⠀⢴⣿⠛⠘⠁⣿⣾⣿⣿⣿⡏⠀⠀⠀⡀⠀⣦⡈⢿⣿⣿⡆⠀⢺⣧⠀⠀⠹⣿⣿⣷⡄⠀⠀⠀⢉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⣀⠘⣿⣇⠀⠀⠻⢿⠟⠛⠟⠀⣴⠀⠸⠃⣸⣿⣿⡎⢿⣿⠁⠆⢘⣿⡆⠀⠀⠹⢿⣿⣧⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠈⠻⡄⢠⠀⠀⢀⣀⢤⡿⠁⠀⢀⢰⣿⣿⣿⣿⡞⠹⡷⣰⣿⣿⣿⣦⣄⠀⠀⠈⠉⠀⠀⣀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⢿⣇⠰⣶⠻⣿⣧⡀⠀⣿⣧⠙⢿⣿⣿⠀⠀⣵⡆⣿⣿⡶⢨⣭⠛⠲⣶⣶⡶⠾⠛⢣
⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⢻⣶⣤⣄⡀⢀⣷⣄⠹⣧⣼⣤⣿⡈⢿⣦⢿⣯⡔⣧⣸⢃⡀⢾⣟⡁⣠⣴⣶⣯⡷⣤⣿⣿⣿⣷⣦⣿⣿
⠀⠀⠀⠀⠀⠢⢄⠀⠀⠀⠀⠀⠻⠶⠀⢸⠝⠻⠻⣷⣯⣸⣧⡹⣿⣥⣤⠀⢶⣷⠈⣿⣤⣦⣧⡮⠠⣿⡿⠟⠛⠛⠉⢛⡛⣿⣏⣻⣿⣿⠿⠛⣿
⠀⠀⣀⡀⢲⣁⣀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠙⢿⣿⣷⣽⣿⢣⡿⠊⢻⣇⢹⣿⣿⣟⣡⣜⣿⣿⣿⣖⣻⠁⣀⣹⣿⡛⣽⣿⣿⡛⠂⣾
⠀⠘⣿⣿⣾⣿⣿⣿⣄⣿⣦⡀⠀⠉⣀⠀⢶⣾⣿⣿⡿⢾⣿⣿⣿⣧⣈⡀⠀⠌⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣯⡙⢻⠿⢿⣿⣿⣿⣿⣯⢙⣡⣿
⣷⣄⡙⠋⠘⢿⣿⣿⣿⣿⣿⣧⠙⠂⠉⠰⠟⣻⣿⣿⡿⣿⣿⣿⣿⣿⣿⡿⠗⠁⢀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣯⣡⣨⣴⣿⡿⠛⠛⢶⣤⡞⠙⣿
⣿⣿⣿⣿⣶⡄⢩⣿⣿⣿⣿⠃⠀⠀⠀⢀⠞⣫⣿⢿⣷⣿⣿⣿⣿⣿⡇⠁⢠⣤⣬⡃⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣡⣇⣠⣼⣿⣿⣿⣤⣿
⣿⣿⣿⣿⣯⣿⣷⣽⣿⣿⣿⡀⠀⠀⣴⢻⣾⣿⣷⣾⣿⣿⣿⢿⡿⣿⠀⠐⣿⣿⣿⣿⡄⠀⢠⣘⣿⠋⠛⠃⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣯⣉⠛⢿⣿⣆⣸⡅⠸⣷⣿⣿⣿⣿⣿⣿⣿⣥⠈⣇⢀⡻⣿⣿⣿⣧⠀⢠⣿⣿⡿⢶⣶⣦⡄⠀⠈⡿⢿⣿⣿⡰⠻⣿⣿⣿
⣿⣿⣿⣯⣍⠙⠙⠛⠻⠿⣦⠈⠙⠻⣿⣷⣽⣿⣿⣿⣿⣿⣷⣿⣿⣿⣜⣾⣿⣿⣬⣿⣿⣧⡸⣿⣯⣤⠄⢿⣿⣧⠄⠀⠀⠈⠉⠙⢿⣿⡄⠹⣿
⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠙⣷⣄⠀⠀⢉⡛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠻⣿⣿⣿⣿⢦⣤⣼⣿⠛⠻⣿⣿⣿⣷⣾⣿⡷⣶⣾
⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⠶⠟⠿⢷⣦⣈⠻⣶⣮⣿⣿⣿⣿⣿⣿⣿⣿⡿⣤⢠⣤⣼⣧⣀⣿⣿⡹⣿⣦⣿⣿⡷⣤⣀⠈⢿⣿⣿⣿⣿⣀⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠛⠟⠉⠀⠀⠀⠀⠙⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⣉⣽⣿⣿⣿⣿⣧⠘⣿⣿⣿⣿⣿⡆⢀⣋⣿⣟⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣯⣤⣤⣄⣈⣿⣿⣿⣿⣶⣾⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠛⠿⣿⣿⣿⣿⣿⣿⣿⣷⡈⠻⣿⣟⣿⣿⣾⣷⣻⣿⣿⡟⢻⠟⣿
⣿⣿⣿⣿⣿⠛⠷⠈⠻⣿⣿⣿⣿⣿⡿⠋⣴⡾⠿⣿⣿⣏⣻⣿⣿⣿⣿⣿⣥⣶⣄⡉⠛⠿⣿⣿⣿⣍⡛⣦⡙⢿⡿⢻⣾⢿⣿⣏⡙⢿⡇⠸⣿
⣿⣿⣿⣿⡋⠁⠀⠙⠀⠘⢷⣿⣿⣿⠃⠀⠀⢀⣴⠚⠛⠿⢿⡿⠿⠿⠻⢿⣿⣿⣿⡿⠧⠀⠀⠙⠛⠿⣧⡀⠙⠳⡄⠘⠧⠂⠻⠈⢉⠘⢿⣤⣿
⣾⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠹⣿⣿⣾⠀⠀⢈⣿⣷⡄⡠⠀⠒⣤⣴⣤⣾⣿⣿⣿⡟⢃⣀⣹⣿⣶⣄⡀⠙⠀⠀⠀⠁⠀⠀⠀⡀⠈⢰⡇⢿⡏
⣿⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣏⠀⠀⣤⠄⣿⣿⣿⣶⣾⣿⣿⠙⣷⡉⠁⠀⣤⣿⣿⢹⣿⣿⣿⣿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠐⠀⠃⣸⠆
⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠴⠀⢸⣿⣿⣄⡀⠐⢀⣿⠋⠙⢿⣿⣿⣯⢶⢼⣿⣾⣿⣿⣿⣶⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣶⣄⣀⠀⠉⠶⣄⣸⣷
⣿⣿⡏⠀⠀⠀⢀⣴⠄⠿⠀⠀⢈⣿⣿⣿⣿⣿⣛⣻⣇⣠⣾⣾⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⠋⣶⡀⠁⠀⠀⠙⢿⣿⡿⣿⡿⣷⡦⠀⠀⠙⠿
⣿⣿⣿⣷⣶⣶⡏⠁⠀⠀⠀⠀⠘⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⡿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣵⣶⠿⡤⠀⠀⠗⠀⠹⣇⠀⠙⣦⠉⠀⢸⣶⣠
⠘⢿⣿⣿⣿⣿⣿⡄⠀⣀⣤⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣙⠻⢻⣧⡀⠀⠀⠀⢀⣦⠘⣷⣤⣿⡇⠀⢀⡀
⠀⠈⠛⢿⡿⣿⣿⣷⣾⣿⣿⣿⣄⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡹⣿⡄⠈⠻⣷⠀⠀⠀⠀⠀⢦⡘⣿⡟⢠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢹⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣷⣆⠀⠀⠀⠀⠠⣼⡇⠀⣿⣿⡄
⠘⠃⠀⠀⠀⠀⠀⠛⣠⢸⣿⣿⣿⣿⣏⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡌⢻⣷⡀⠀⠀⢀⠉⠡⣤⣤⠘⠇⠀⠠⡇
⠀⢰⣆⠀⢀⠘⠐⢶⡿⠟⠛⠛⠛⢿⣿⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢠⡹⣷⡄⠀⠈⠀⠀⠘⣿⡀
⠀⢘⡿⠷⠈⠀⠀⠘⠁⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣿⣿⣿⡘⣇⠙⠛⠀⠑⣶⣦⠸⠋⠀⣄⠻⡆
⠀⢨⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣷⡀⣾⡦⢰⣆⣝⡷⠀⢀⡖⠉⢠⡄
⠀⠈⠻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡁⢀⡈⠃⣤⣶⣤⣬⣿⡄⠀⠈⠀⠈⠠⠃
⠀⠀⠀⠙⠷⣶⣿⣷⣀⡀⢀⣀⠀⢣⣴⣿⣿⣿⣿⡿⢹⣿⣿⣿⣽⣷⣿⣿⣿⣿⣿⣿⣿⡶⠏⠙⠣⠈⠁⠀⣿⣿⠟⣿⣿⣧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠫⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣴⣿⣶⣶⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⣴⣶⣶⣦⣤⡀⠀⠀⠀⢀⣶⣻⣿⣿⢿⣟⣽⣿⣿⣟⣻⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⢀⣤⣶⡶⣾⣿⢟⡿⢻⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣯⣹⣅⣽⣿⡿⠶⣻⣿⣿⣿⣿⣿⣿⣾⣦
⠀⠀⠀⠀⠀⡰⣻⣿⣷⣿⣿⣱⣿⣿⣿⣷⠾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⣿⣷⣾⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⣘⣿⣻⣟⣯⣿⡟⠋⠉⠋⠹⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⡋⠀⠀⠀⠀⠀⠙⠻⣿⡿⣿⣇
⠀⠀⠀⠀⣿⣿⣿⣿⠟⠁⠀⢀⠀⠀⠓⢺⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⡀⢻⣿⡿⣿
⠀⠀⠀⠀⣿⣿⣿⡏⠀⠀⠀⢸⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⢰⠺⠂⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⠀⠁⠸⣿⣿⣿
⠀⠀⠀⢀⣹⣿⣿⠀⠀⠀⠀⡜⠀⠀⠀⠀⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⢸⣯⢹⣧⢠⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏
⠀⠀⠀⠈⣿⣾⣿⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⡱⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⡀
⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢿⣿⣻⣿⣿⣶⣶⣶⣦⣴⣾⣿⣿⣿⣿⠻⠇⠐⠀⠸⠀⠐⠀⠀⠐⣿⣿⡇⠀⠀⣼⠁
⠀⠀⡄⠀⠸⣿⣿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⣿⣟⢿⣿⣿⣽⣿⡿⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿
⠀⠘⠃⠀⠀⢸⣿⡇⠀⠀⠀⠂⠐⠀⠀⠒⠐⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆
⠀⠀⠀⠀⠀⠀⢻⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣹⣿⣿⣷⣿⣿⣧⣆⣀⠀⠀⠀⡀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠂⠀⠀⠀⠀⠆⢸⡟
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⠿⢿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⠿⢿⣿⡄⠀⠂⠀⡀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣧⠀⠀⠀⠁⠀⢠⠁⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡎⠰⠖⠠⠄⢸⠀⠀⠀⣰⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠘⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⠀⠀⠀⢸⠀⠀⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣰⡏⠀⡀⠂⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡀⠀⠘⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠄⠀⣿⠃⡴⠁⠀⠀⡠⠊⠁⠀⠀⠀⠈⠙⢶⣄⠀⠙⠿⣦⠀⣿⡇⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡃⠀⠀⠀⠀⣿⡾⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠘⢷⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠈⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⣾⡿⢀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠘⣿⡙⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢸⣿⠁⠘⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⢿⣷⠀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⣼⡏⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠚⣿⡀⢸⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣀⣿⠂⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣲⠀⢿⣇⠀⠆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠘⣿⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⡇⠄⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⡏⠀⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⢸⣿⡄⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡿⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⢹⣇⠿⢶
⠀⠀⠀⠀⠀⠀⠀⠇⠀⣼⠃⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡞⠈⣿⣆⠈⡄
⠀⠀⠀⠀⠀⠀⢰⠀⣸⡏⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠘⣿⡀⢣
⠀⠀⠀⠀⠀⠀⠛⢰⡟⠀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠑⠉
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠚⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠲⢤⡀⠀⢀⡞⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⠓⡾⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⡇⠀⠀⠀⢠⠇⠀⠀⠀⠀⣀⠀⡠⠒⠒⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⡇⠀⠀⢀⡞⠀⠀⠀⠀⠀⠙⢤⡇⠀⠀⠀⢸⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣱⣦⣽⣀⣀⠞⠀⠀⠀⠀⠀⠀⠀⣸⠁⠉⠈⠂⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⢀⡞⠉⠙⠲⢤⣀⣀⡴⠒⠚⠕⠒⠦⠴⢄⡴⠯⠷⣀
⠀⠀⠀⠀⣠⢦⣠⢄⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣀⠀⡄⠀⠀⠈⠳⡀⠀⠀⠀⠀⢀⠤⣠⢶
⠀⠀⠀⢴⠻⠀⠁⠸⢤⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣿⣶⣿⠇⠀⠀⠀⠹⡄⠀⠀⢠⠼⠀⠁⠸⢳⠄
⠀⠀⠀⠈⠳⣄⠀⢠⠎⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢽⠟⠛⠀⠀⠀⠀⠀⢷⠀⠀⠘⢦⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠈⣧⠀⠳⡀⢰⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⢸⡄⠀⣰⠃⢠⡏
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠙⢾⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣷⠖⢸⣇⠞⠁⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠰⠷⠀⠀⠀⠀⠀⠀⠐⠛⡾⠧⠄⢸⠃⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡸⡄⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢀⣠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣼⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠋⠀⠀⠹⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡞⠁⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⢱⡀⠀⠀⠀⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣄⣀⣀⣠⠤⠞⠁⠀⠀⠀⠀⠀⠀⠙⢤⣀⣀⡤⠞⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⠀⣤⡄⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⡷⣼⡿⠛⠀⠀⠀⠀⠐⢾⣿⣷⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⠆⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⣿⡇⣠⠆⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⣶⡇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢀⣰⣾⣿⣣⣭⣀⠀⠀⠀⠀⠀⠀⠀⢈⣈⣿⣿⣿⣿⣿⣄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢽⣿⢋⣿⣿⡏⢿⣿⣿⣦⣴⠶⣦⣴⣾⣿⠟⣿⣿⣿⣯⣿⠏⣀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠠⣄⣀⣧⣿⣋⣾⣿⣿⣷⣌⣁⡉⠉⢡⣶⡄⠉⣉⢉⣴⣿⣝⣿⣿⣿⡛⠉⢠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⡏⡀⠈⠁⠁⠠⣿⣿⣿⣿⣿⣿⣿⣿⣆⢀⣿⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⣦⣴⣿⣿⣿⣿⣿⡝⣿⣿⣦⣼⣿⣿⣿⣾⢮⢲⢠⣾⣿⣟⣛⣿⣿⣿⣿⣿⣽⣟⣻⣿⣾⣥
⠀⠀⠀⠀⠀⠀⠀⠸⢹⡏⢹⠋⣽⠻⢿⣾⣯⣿⣿⣽⣿⣿⣿⣿⣸⣾⣿⣿⣿⣿⣿⣛⣛⣻⣿⣿⣿⣻⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢨⠀⠉⠀⡼⢯⣿⣿⣯⣙⣿⣟⣿⣿⣿⣿⣿⡶⣾⣿⣿⣿⣿⣿⢿⡭⠃⠋⠉⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣁⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⡿⠩⠀⠠⡏⠈⠇⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠘⠉⠛⠛⠛⣿⣿⡿⠟⠃⠋⠉⠀⠀⠀⠉⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠟⠛⠛⠛⠛⠛⠷⢶⣤⣀⠀⠀⠀⣀⣠⣴⠶⠟⠛⠋⠉⠉⠉⠙⠛⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⠙⢻⣾⣟⣋⣉⣠⣤⣴⣶⣶⣦⡄⠀⠀⠀⠀⠀⠈⠻⣷⡀
⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⢀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⠆⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⢠⣿⣿⣿⠿⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⢸⣿⡿⠿⢿⣿⣿⣿⠛⢻⣿⣿⣿⡿⠿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆
⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠸⣿⣿⣿⠀⠀⠿⠿⠟⠁⠀⠈⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠈⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿
⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠛⠛⠻⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⠀⠀⠈⢻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣤⣀⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠶⠶⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠴⠒⠚⠉⠉⠉⠛⠓⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣂⣈⣐⣒⡲⣖⣤⣤⣤⣠⣤⣤⠴⠖⠋⠉⣠⣤⡀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢦⣀
⠀⠀⠀⠀⣠⠶⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠋⠉⠁⠀⠶⢶⡴⣞⣧⣍⢳⠶⠖⡶⠂⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷
⠀⠀⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⠾⠯⣾⣿⣿⡿⢿⡁⣾⡁⠣⠀⠀⠀⠀⠀⠀⠀⠠⠄⠈
⠀⡜⠀⠄⠀⠔⠀⠀⠀⡠⠤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠘⠛⠛⠛⠒⠀⠀⠀⠀⠀⠀⠀⠀⠘
⡸⠀⠐⠀⡚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡄⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠔
⡇⠀⠀⠈⠁⠰⡇⠀⠰⠂⠙⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⠲⠖⠛⠉
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣽⣷⣦⣀⠀⠀⠀⠀⠀⠀⢀⢞⣽⣿⣧⡉⠙⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠁⢰⣿⣿⣷⡍⠛⠂⠀⠀⠀⢠⡏⠈⠿⡿⠟⠀⠀⠀⠙⡇⠀⠀⠀⠀⠀⠀⠀⢰⡜
⠈⢧⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⠈⠛⠋⠀⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠸⠁
⠀⠘⣇⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⠀⠀⠀⢠⠂
⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠻⢦⣄⣠⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣤⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠰⠤⠤⠤⠤⠔⠊⠀⠀⠀⠀⠀⠀⠀⠰⢤⣀⣀⣀⣀⠠⠄⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⠙⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠙⢦⡀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢀⣍⣾⡿⠀⠀⠀⠀⠀⠀⠀⠈⣿⡼⡀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⠰⡋⡾⢱⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿⣷⡇⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡔⡽⠋⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⡅⠹⣷⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⣨⢴⠃⠀⢨⡇⠀⠀⠀⠀⠀⠀⠀⡾⠂⠀⢿⡆⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠐⠁⣿⡀⠀⢈⣇⠀⠀⠀⠀⠀⠀⢠⠇⠀⢠⢿⡇⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⣼⠙⣇⠀⠐⡏⠀⠀⠀⠀⠀⠀⡸⠀⠀⡞⠀⣧⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣰⠃⠀⢻⡈⢣⡇⠀⠀⠀⠀⠀⢀⡇⢀⡜⠀⠀⢿⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⢰⠇⠀⠀⠘⣇⢸⠃⠀⠀⠀⠀⠀⢸⡇⡟⠀⠀⠀⢸⡆⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡟⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠀⠀⢸⡇⣰⠃
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⠋⠙⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣶⣶⣾⣿⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠚⠑⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠀⠘⢹⣦⡀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠤⢠⣶⡄⢀⣴⡶⠄⠀⢻⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠙⠙⡇⠈⠈⠉⠁⠀⣾⣿⠀⣴⣶⠆⠲⠰⠷⠒⠀⠒⢿⠆⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⡀⠈⠻⠃⠀⠀⣀⡼⢉⡏⠀⣿⢡⣶⣷⡄⢤⣿⡍⠀⠈⣧⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡅⠐⢺⡗⠂⠀⠀⣠⠎⠀⠀⣟⠀⢀⣿⣃⣀⠉⠀⠀⣴⠟⠀⢀⣀⣀⣀⣀⣇⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣄⣈⣀⡀⢀⣼⣿⡄⠀⠀⠙⢷⣇⠉⣙⣉⠁⠀⢾⠿⡿⠃⣾⠁⠉⢨⣟⣀⡈⠉⠀⢀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣿⡛⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⢸⡏⠈⠽⠟⠀⠀⣠⣾⠀⠀⠘⣷⣦⠌⠈⣋⡈⠳⣠⣾⣿⠁
⠀⠀⠀⠀⣀⡀⠀⣀⣀⠀⢰⡿⢿⡧⡉⠒⢫⣉⣏⣿⣿⣿⣿⣄⡀⢠⣿⣦⣶⣷⣤⣶⣿⣿⣄⠀⠀⠙⢿⡆⠉⠿⠋⠁⢈⣼⠃
⠀⠀⠀⢸⢣⠄⠀⠢⢄⡀⠀⠙⠦⠉⠀⠀⠈⠁⠡⠐⡴⠿⠒⠉⠉⢙⠛⠻⠟⠛⢿⣿⣿⣿⢽⡆⠀⢀⣼⣿⣿⣶⣦⣶⣾⣿⡄
⠀⠀⠀⢣⢨⣧⡲⣤⡑⠄⠑⢦⡖⠀⠀⠀⢠⡆⠘⠋⠀⠀⢀⠀⢀⣀⣄⡴⠤⠀⡈⢻⣿⣾⣯⣹⢷⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠘⠀⢳⣟⠂⠉⠀⠂⠀⠘⢦⡀⠀⢷⡆⠀⠈⣀⣤⠔⠁⠄⠀⠂⣰⠄⣑⠃⠀⣿⠳⠟⢉⠬⠀⠈⠀⠀⡀⠙⢿⣾⣿⣿⣿⣿⣄⠀⠀⢀⡀
⠀⠀⠀⡇⠀⠈⠛⣷⣄⠀⠀⠀⠀⠀⠉⠲⣼⣷⣶⡤⠟⠁⠀⠀⠂⠀⠡⠶⣚⣽⠃⠀⣿⣿⡓⠓⢀⣤⡾⠔⢛⣛⡭⢶⡜⢿⣛⣛⢛⡉⣉⣅⣀⡀⢁⣤⣄
⠀⠀⢠⡇⠀⠀⣸⣿⡝⠷⣦⣄⡀⠀⠂⠀⠻⣿⡃⡠⠂⠀⠀⠀⠀⣠⣶⣿⣿⢠⠀⠀⣼⣏⣷⡶⠋⠀⠀⠘⡩⢄⣠⣿⣡⣿⡿⠿⣿⡿⠷⠋⠡⢞⣩⣿⠘
⠀⠀⢸⠰⠀⠀⣿⣮⠿⣄⠘⣿⠿⣇⠀⢀⠀⠈⢻⣷⢀⣶⣶⠶⠛⠣⢿⣿⠋⠂⠀⢰⣿⣿⠟⠀⠀⠀⠔⡀⠴⢛⣫⠟⢹⣿⣿⣿⠏⠘⡱⡂⠀⣩⡾⡟⢡
⠀⠀⣼⠀⠀⠀⣿⣿⣟⠶⠒⠲⣷⠟⠻⣮⠀⠀⠀⠙⢿⣯⡀⠐⠛⣂⣼⣿⠐⠀⠀⢸⣟⠑⠂⠀⠀⢀⣀⣴⣿⣿⡏⠀⠈⣿⡿⠋⠀⠀⠀⠄⣨⣥⡾⠃⢸
⠀⢠⣏⠀⠀⠀⣿⣤⢾⣿⠷⠊⠁⠀⠀⣸⣧⡀⠀⠀⠈⠻⣧⢴⣉⡾⠋⣽⠀⠀⠀⢸⣿⣦⣀⣼⠟⢛⡯⠿⢿⠉⠂⠀⠀⣿⣥⣤⣀⣤⣤⣼⣿⣿⠃⠀⢸
⣴⣿⡇⠀⠀⢣⣿⣿⡿⠁⠀⠀⠀⣠⣾⣿⣿⣷⡘⠀⠀⠀⠙⣿⣯⡝⣹⣿⡄⠀⠀⢸⣿⣿⣻⣥⣀⣠⠟⢻⣿⢆⠀⠀⠀⣿⣿⣾⣿⣻⣛⣿⣿⡿⠀⠀⣿
⣿⣿⠀⠀⠀⣾⣿⠏⠀⠀⠀⢀⣾⣿⣿⣏⣿⣿⣿⣄⠀⠀⠀⠈⠻⢾⣣⣾⡇⠀⠀⠈⣿⢿⣿⣍⣉⣝⣶⣿⣿⠀⠑⠀⠀⣿⣿⣿⣿⣿⡿⣵⣿⢰⠀⠀⢸
⣿⣿⠀⠀⢀⡟⠁⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠉⢻⣇⠀⠀⠀⢿⡄⠙⠻⣵⡾⣿⣿⣿⣆⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠛⠀⠀⢸
⣿⣿⡌⡂⠀⠨⠃⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⣹⡄⠀⠀⢸⣷⡀⠀⠙⢿⣿⣟⣿⣿⡄⠀⠀⢸⡿⣙⠻⣿⣿⣿⣿⠀⠀⠀⠸
⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⢀⠀⠀⠀⠀⠀⠐⣿⣅⠠⠄⠀⠈⠻⠿⣿⣷⠀⠀⢸⣿⢄⠀⠘⢿⣿⣿⡇⠂⠀⢰
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣬⡀⠂⢈⣷⣼⣿⣿⣷⣦⡄⠀⠀⠀⠙⡏⡀⠀⠘⣿⡻⠣⠀⠀⠹⣿⣿⠀⠀⢰
⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣦⡀⠀⠀⠁⠠⣤⣠⣿⣷⣦⡀⠐⠄⠈⠻⠀⠀⣸
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣼⣿⣿⣿⣿⣿⣶⣄⠠⣀⡀⢸⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⣀⣠⣾⣿⣷⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣤⣀⠀⠀⠀⠀⠀⠀⣀⣼⣷⣀
⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⡟⠋⠀⢰⠋⢡⣶⣶⣶⣾⣿⣿⣿⣿⣿⠙⡞
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣝⣄⡁⠁⠀⣡⣾⣿⣿⣿⣿⣿⣿⣿⡟⣿⠃⢸⡇
⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⢥⣀⢹⡄
⠀⠀⠀⠸⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣧⣼⣿⣿⣿⣷⡶⠤
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⠏⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣋⠁⠀⠀⠀⠁⢦⡀
⠀⠀⠀⠐⢲⣶⣿⣿⣿⣿⣿⣿⣿⠟⣱⡟⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡷⠄
⠀⠀⠀⠞⢋⣻⣿⣿⣿⣿⣿⣿⠏⠀⣽⣆⠀⢛⢻⣿⣿⣷⣿⣿⡿⠋
⠀⠀⠂⣠⣽⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣻⣿⣿⠁
⠀⠘⠂⠉⠁⠀⠀⠀⣤⣶⣿⣿⣿⣿⣿⠿⠟⠛⠛⠛⠛⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣄⡀
⠀⠀⠀⠀⣠⣶⣿⣿⡿⠿⠛⠛⠉⠉⠉⠉⠉⠙⠛⠛⠻⠿⠿⠿⠿⠛⠛⠛⠉⠉⠉⠉⠉⠙⠛⠻⠿⣿⣿⣷⣤⡀
⠀⠀⢠⣾⣿⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣄
⠀⢠⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠿⠷⢶⣶⣶⣶⣶⠶⠿⠟⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣦
⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇
⣼⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⣀⣠⡴
⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⠃⠀⠀⠰⢶⣿⣿⠋
⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠟⠛⠛⠁⠀⠀⠀⢀⣾⡿⠿⠗
⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁
⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟
⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣺⣦⣄⡀
⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣿⠻⠿⠶
⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⡄⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡗⠀⠀⠀⠀⢿⣷⣤⣷⡀
⠀⠀⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃⠀⠀⠀⠀⠈⢿⡿⢿⣷⡀
⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⠀⠀⠀⠀⠀⠀⠈⠀⠀⠙⠷
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡿⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠁⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠀⠀⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣇⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⢠⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⡀⠀⠀⢀⣼⣿⡿⠀⠀⠀⠀⠀⢹⣿⣿⣄⠀⠀⠀⣠⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⣿⣿⣷⣾⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣶⣿⣿⡿⠋
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣿⣿⣿⣥⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣽⣿⣿⣿⣧⣤⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣇⣀⣀⣀⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣶⣶⣶⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⣄⣴⣿⣿⣿⡯⡉⠒⠛⢯⡛⠋⠉⠁⠀⣴⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠻⡏⣽⣿⣅⠀⠀⠀⠂⠈⠄⠠⢦⣄⣸⣶⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠀⠀⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⢨⠛⡙⣿⣿⣇⠀⣠
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡼⢀⣄⣦⣴⠥⢿⣿⢷⣶⣷⡶⢲⣴⣶⡆⠀⢼⣹⣿⣿⡟⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⣇⡘⣿⣯⣀⣤⣬⣿⣤⣭⣵⢼⣿⠈⣍⠀⢳⣆⢿⣿⣿⣇⣾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⡙⣳⣿⣾⣏⣵⡭⣭⣿⣿⣿⣿⣿⣄⡀⣈⣧⣦⠶⢟⡋⣥⠀⣿⢧⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠰⢤⣾⣿⣿⣿⣻⢁⣿⣷⣿⣿⣿⣿⣿⠯⢹⣿⣿⣾⣿⣇⢻⣷⣿⠟⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣿⣿⣿⣌⣿⣿⣿⢿⣿⣿⣿⣶⣿⢟⣥⣿⣿⡏⣿⣿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠬⣿⣿⣿⣿⢹⣿⣯⣷⣯⣿⣿⣟⣵⣣⣆⡿⢫⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣾⣿⣿⣿⣿⣻⣿⣿⣽⣻⣿⣷⣼⣿⡿⣫⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡿⢿⣿⣿⣿⣿⣿⣿⣿⣭⣿⣷⠦⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈⠻⢿⣿⣿⡟⠉⠁⠀⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⠀⣀⣤⡴⠶⠶⠶⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⡴⠛⢛⠿⠶⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣙⣷⣄⣼⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⠀⠀⠀⠀⣀⣻⡿⣇⠀⠀⣀⠤⠤⠿⣄
⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⠉⠉⠁⠀⠉⠽⠋⠀⠀⠀⠀⢀⣤⣖⣀⡀⠀⡀⠀⢀⣸⢇⣠⠖⠋⠁⠀⠀⠈⠉⣿⠁⠀⠀⢀⡜⢧
⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠉⠉⠉⠉⠛⠻⢿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠴⡏⠀⢸⡇
⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠁⠀⠀⠀⠀⣰⣶⡟⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠶⠊⢻⣄⡀
⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⢀⣠⣾⠟⠉⠲⣄⠀⠀⠀⠀⠤⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠙⢦⡀
⠀⠛⠶⣤⣀⣀⣀⣀⣀⣠⡤⠶⠛⠉⠀⠀⠀⠀⢀⣠⡴⠛⠉⠀⠀⠀⠀⠈⠳⡀⣠⠔⠒⠚⠛⠓⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣇⢀⡽
⠀⠀⠀⠀⠀⠉⠉⠉⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠃⠀⠀⠀⠀⠀⣀⣀⣀⣹⣁⡀⠀⠀⠀⠀⠀⠀⠘⡆⠁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠤⠒⠈⠙⠛⠁⠀⠀⢸⠁⠉⠉⠉⠉⠐⠒⠤⣴⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⡀⠀⠀⠀⣀⣴⠇⠑⢤⣀⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠑⠃⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠒⠛⠓⢶⣶⣲⣖⡊⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢤⣤⡤⠟⠪⢿⣯⢷⡦⣄⡀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⣟⡯⣷⡦⣀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⣧⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠯⣲⡯⣳⢦⣄⡀⠀⢀⣀
⠀⣴⠞⠳⣦⣀⡴⠚⠋⠃⠈⣿⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠛⠾⣭⣾⢽⣛⣩⣭⡹
⠈⢿⡀⠀⠈⠛⠀⠀⠀⠀⠀⢻⡄⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⠹⣇⠸⣿⣿⠟
⠀⣀⣽⣷⣦⠄⠀⠀⠀⠀⠀⣸⠃⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠈⠓⠶⠖⠚
⣾⠉⠀⠀⠀⢀⣠⡄⠀⢀⡾⠁⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠙⠷⣤⠴⠖⣻⠟⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⢸⣇⣀⣀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⣠⠶⠉⠳⣄⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢦⣤⣤⡴⠞⠁⠀⠀⠀⠈⠙⠶⢤⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⡆⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠉⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠺⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⠛⠙⠻⠿⠏⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣄⢘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡴⠞⠶⡄⠀⠀⠀⠘⠛⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠟⠀⠀⠀⠀⢸⠁⠀⠀⢹⠀⠀⠀⠀⠀⠸⣿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠿⠀⠀⠀⠀⠀⣸⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⣷⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⣸⠇⠀⠀⠀⠀⢿⣀⣤⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠆⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⢀⡁⠀⠀⠀⠀⠀⢠⠃⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⡀⠀⠀⡤⠀⠠⠀⠀⠀⡀⠀⠀⠀⠀⠀⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⡀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠀⠀⠁⠀⡖⠂⠀⢀⠖⡄⠀⠀⠀⠀⠀⠀⡠⠁⠀⠀⠀⠀⠀⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣄⠉⠀⠀⠀⠀⠱⡄⠀⠀⠐⠓⣄⠳⠯⡄⡎⠖⢃⡜⠂⠀⠀⢀⡖⠁⠀⠀⠐⠈⢀⠄⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡀⠈⠀⠀⠈⠣⣆⠀⣈⣎⠓⠶⠆⠶⠞⢻⡎⠀⣄⡶⠋⢀⠀⠈⢀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⣄⡼⢦⡼⠋⠀⠠⣤⡤⠀⠈⢻⣴⠾⡀⣄⣁⠤⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢤⣴⠇⠀⠉⠀⠀⠀⠈⠁⠀⣧⣤⠜⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢠⣄⣊⢙⡆⠀⠤⠀⠙⠀⠠⠄⢀⣟⢉⣆⣤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠆⠀⠠⠀⠠⠠⠤⠄⠐⠈⠁⠀⡩⣏⣙⠦⣀⣀⣀⣀⣀⡠⢎⣙⠯⡀⠀⠉⠐⠀⠤⠤⠀⠀⠤⠌⠠⠆⢀
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠟⠉⢁⣾⠿⠋⢻⠛⠻⢷⡄⠉⠙⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠀⢀⠞⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠢⣄⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡱⣤
⠀⠀⠀⣠⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠁⠀⠀⠐⢦⠀⠀⠀⠀⠁⠀⠀⠀⠀⢤⡀⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣧⡀
⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡊⣢⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠢⡐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛
⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠋⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠑⠁⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⢁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡀
⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⡀⠀⠀⠀⠀⡀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢬⣀⠀⠀⠀⢀⣩⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣦⡶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣾⣿⣿⠿⠳⠶⣿⠁⠆⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⠯⠉⠀⠀⠀⠉⠀⠀⠀⠀⠛⠛⠃⠀⠈⠉⠉⠿⣫⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠓⠶⠶⠤⠴⠴⠶⠤⠄⠀⠀⠀⠀⠰⠚⠛⠛⠉⠉


⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣾⣿⣿⡛⠙⢉⣉⡉⢉⠉⠉⠉⠉⠛⠛⠛⠳⢶⠤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣆⠉⠛⠛⠛⠻⠷⠷⢮⣤⣧⣤⣤⣤⣴⣤⣴⣴⡶⠶⣶⠶⠶⠛⠉⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡍⠙⠛⠲⠦⢄⣀⣀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣤⣤⣭⣤⡷⣶⠞⣫⣀⣀
⠀⠀⠀⠀⠀⠀⢠⡔⣶⣿⢀⠀⠀⢀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⢉⠉⢨⣿⠀⣼⠃⣻⠀⣿⡿⣫⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣍⠉⢋⣩⣤⣴⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢸⠀⢸⡿⠀⣿⠄⡼⠀⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⡇⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⡇⠀⡇⢀⠃⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠁⢰⠃⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⠃⢠⠀⡟⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⣤⠀⠅⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢸⡆⢸⠀⠀⠀⠀⠀⡇⠀⡀⠀⠀⠀⠬⠀⠀⢠⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠘⠁⢸⡇⠀⠀⠀⠰⡇⠀⠀⠀⠀⠀⠀⠀⠇⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠃⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⢀⡇⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⡧⠈⡇⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠐⠃⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⢡⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣌⡀⠛⠀⠠⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠠⢖⣀⣹⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠓⠒⠒⠒⠛⠛⠉⠉


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣭⣭⣽⣽⣿⣯⣭⣯⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣿⣷⣶⣷⣾⣶⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀

⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠂⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣈⢷⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⢏⣰⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢧⡙⢦⡀⡶⢖⠶⡄⣠⠞⣡⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣿⣧⠙⢡⣾⡷⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣄⣎⡻⢈⠸⢻⣇⡄⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠞⠋⠉⠀⢄⡨⠋⡄⢲⠀⡌⢫⣁⠀⠈⠉⠛⠦⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠁⠀⠀⢀⣀⠄⠀⣅⠂⡡⢬⢄⠁⢈⡂⠀⢄⣀⠀⠀⠀⠉⠳⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠀⣠⠟⠁⠀⢸⠄⣸⡄⢛⢠⣧⠁⡆⠀⠀⠙⢧⡀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⣏⠺⢵⢿⣇⠕⢫⠇⠀⠀⠀⠈⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡵⠀⠀⠀⠀⠀⠘⢦⡀⠉⢄⡼⠋⠀⠀⠀⠀⠀⢰⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠀⠉⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣟⣿⣻⣻⣛⣟⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣿⣻⣿⣟⣻⣟⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣦⢀⣀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣆⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠉⠁⠀⣀⣸⡟⠛⢿⣿⣿⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣀⣠⣿⠟⠁⠀⠀⢀⣀⣭⣽⣿⣿⣿⣿⠿⠿⠟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⠁⠀⠀⠀⠀⠈⠋⠉⠈⠉⠉⠁⠀⢀⣠⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠖⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠿⠶⠶⠶⠾⠶⠶⠶⠷⠶⠶⠶⠿⣶⠀⠀⢸⣿⢶⡿⠿⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⢷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡶⠒⠖⣶⠖⠒⠒⠶⠲⡶⠒⠒⣶⠟⠀⠀⠀⠹⣾⣷⠶⠲⠒⢶⠖⠖⠒⠒⠲⡶⠒⠖⢶⡞⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⣿⠀⠀⠀⠀⠀⠈⢿⡇⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⡿⠀⠀⠀⠀⠀⠀⢸⣗⡀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢠⣶⢶⣦⡀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⢸⡧⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⣰⣮⠉⠀⠉⣥⣄⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⣗⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠹⣧⣤⢀⣤⣽⠏⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠉⠉⠀⠉⠁⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⡀⠀⠀⠀⠀⠀⠀⢸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢹⡆⠀⠀⠀⠀⢠⡞⣩⢭⡝⢦⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿⢷⠀⢶⠀⢦⠀⢦⠀⢶⣿⡇⢸⡇⠀⠀⠀⠀⠸⣇⢷⣤⠟⣸⡅⠰⡄⠀⣆⠀⢦⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠞⠉⠛⠓⠚⠓⠚⠓⠚⠓⠚⠉⠳⠞⠁⠀⠀⠀⠀⠀⠙⠳⠶⠚⠛⠛⠒⠛⠒⠚⠓⠚⠁






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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⡄⡄⡀⠀⢊⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣿⣿⣿⣿⣿⣿⣷⣿⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠲⢦⣴⠾⠿⠟⠋⠉⠉⠉⠙⠛⠿⠿⢿⡷⠞⠒⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣦⣠⣦⣶⠒⠀⠐⣦⣦⣦⣠⣾⠀⠀⠀⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢘⣿⡿⠏⠀⠀⠀⠂⠀⠙⣿⣿⣿⠁⠀⣠⡀⣰⣣⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣄⢠⣶⣾⣿⡿⠗⢧⠀⠀⠀⢠⠿⢿⣿⣿⣶⣿⣿⣿⣿⣿⡄⡊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡿⠉⢳⢿⣾⣫⣶⣦⣾⠀⠀⠀⣸⣷⣶⣿⣿⣿⣿⣿⣿⣿⣥⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢈⠋⠀⢿⣿⣿⣿⣿⠁⠀⠀⣿⣿⣿⣿⣿⡿⡏⣹⣿⣿⡿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⠆⠀⠀⠈⠉⠉⣻⢃⣬⣴⣿⡿⠙⡿⠏⠰⠄⢺⣿⣟⢂⣴⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠠⣭⡀⠀⣤⣾⣷⣀⣨⠋⢀⣹⣿⣏⣾⣿⣯⣷⠄⣼⠩⠉⠨⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠐⣿⣷⠀⢿⣿⡏⠸⣿⡖⠿⠟⣿⣿⠝⣿⣿⡟⢳⣿⠛⠒⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠈⢿⣧⣀⢿⣇⠈⢿⣷⣴⣼⣿⠏⣠⣿⢿⣰⣿⠯⢄⣶⣆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢞⠹⣿⣿⣶⣿⣷⣦⣿⣿⣿⣤⣶⣿⣿⣿⣿⡿⢷⣾⣿⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢞⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⡿⢿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠸⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡿⠀⠀⠈⠉⠛⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⣿⣿⣟⠿⡿⠿⠛⠿⠿⡿⢟⣿⣿⣏⣻⡿⠃⠀⠀⠀⠀⠀⠀⠀⠈⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣿⣿⡟⡼⠁⠀⠀⠀⠀⢹⢘⣿⣿⠏⡌⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢷⠀⠘⢻⣟⠁⠀⠀⠀⠀⠀⠀⢛⣿⠳⠀⠰⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⣀⣁⠛⢇⠀⠀⠀⠀⠀⢠⠛⠋⣁⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠺⣰⣯⣸⠷⠞⠀⠀⠀⠀⠀⠘⠶⢿⣸⣧⡹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠉⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠒⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠠⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠊
⠀⠀⠀⠀⠀⠀⠀⠐⠂⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠃⠀⠒⠆⠀⠀⠤⠀⠀⠠⠀⠀⠰⠒⠀⠘⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⢉⡀⠀⠰⠀⠀⢠⢀⡀⣀⠀⠀⣀⢀⡀⠄⠀⠀⠄⠐⢀⡉⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠁⠐⠀⠒⠠⠀⠀⠤⠀⠀⠤⠠⠀⠄⠒⠀⠂⠈⠀⠁⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡁⠒⠀⠄⣄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣠⠠⠄⠒⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠠⠀⠤⡀⠀⢉⠉⠉⡉⠈⢀⠤⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠠⠄⢀⡁⠀⢸⠀⠀⡀⠀⢸⣀⠠⠄⠀⠊⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⢾⣿⡶⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣻⣶⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⡉⣸⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⡰⢦⣥⣾⡇⠠⢀⠾⠟⠋⡤⢁⢴⠦⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⡠⢤⣴⡿⠀⠈⢁⡠⢄⠀⠀⡵⢦⢸⣾⢢⣀⣀⡀⠀⢀⡀⢀⠀⠔⢀⠁⡄⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡁⢹⣏⢀⣚⠿⠗⠂⢰⡍⠀⠺⡃⠀⡇⢈⣿⣻⣿⣿⣿⣶⠃⠈⢀⣀⡠⣠⣾⣿⣿⡀⠀⠀⢀⡔⡋⠙⣿⣿⣀⡿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠈⠉⡁⠀⠀⠀⠘⣿⠶⢊⡇⠀⣇⣾⣷⣿⣿⡟⣹⣿⣴⣷⣿⣾⣿⣿⣿⣿⣿⠃⣠⡖⣡⢊⣴⠖⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠚⠁⢙⡏⠀⠢⡀⠀⢹⣿⣼⣳⣆⣙⣋⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢛⡁⠈⢿⣿⠿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣷⡄⢀⠤⠫⢀⣀⠉⣈⣶⢺⣦⣟⡩⢷⣷⣶⣬⣙⣿⣿⣿⣿⣿⣿⣿⡿⣛⠁⢾⣭⣶⣶⠏
⠀⠀⠀⠀⠀⠀⣠⡄⡸⡟⠁⠀⠀⠀⠉⢻⣟⣻⣾⣩⣟⣙⣼⣿⣧⣄⣽⣿⣎⢻⣿⣿⣿⡛⢚⡵⣣⡼⠁
⠀⠀⠀⠀⠀⢐⡁⠀⠟⠀⠀⠀⠀⢀⣠⣴⣿⣿⣏⣿⣿⣿⠿⣷⣿⣿⣿⣿⣿⣏⡏⣸⣿⣿⣿⣿⣯⢶⣖⡀
⠀⠀⠀⠀⣠⠈⢾⠗⠀⠀⠀⢀⡠⠾⢛⣿⣧⡟⣛⠛⢿⣿⣷⣿⡏⣿⣜⣿⣯⡿⢋⣌⣽⣿⡿⣿⡃⢿⣍⠿⠓⠄
⠀⢀⣤⣄⣈⠘⠃⠀⠀⠀⣴⡉⠀⠔⣲⣿⣿⣿⣿⣿⣶⣤⣈⠻⣧⣼⣿⣿⠿⡓⣼⢿⣿⡯⣥⠿⠓⠈⠛⠆⣾⣿⢷⡄
⠐⠩⠷⠋⠀⠀⠀⠀⣰⠊⠐⢹⡆⠀⠛⠗⣿⣿⣿⣿⣟⣿⣿⣦⢾⡿⠛⠧⢸⣿⣯⠉⠙⠮⢴⣦⠀⠀⠀⠀⠙⣿⡟⠁
⠈⠁⠀⠀⠀⠀⠀⣮⡀⠀⣴⠟⡠⠾⢿⣷⠈⠭⢾⣳⣿⣤⣽⣿⣿⣻⣿⣷⡌⠙⠛⠦⣄⠀⣤⢾⢀⠀⠀⠀⠀⠈⢣⣾⣣
⠀⠀⠀⠀⠀⣰⢧⢠⣽⠆⠁⢾⣷⠄⠈⠀⠀⠀⠀⣁⡙⢛⣿⠏⠁⠉⢿⡿⠁⠀⠀⠀⠀⢈⣷⣼⣄⡀⠀⠀⠀⠀⢸⠿⠟⠆
⠀⠀⠀⠀⣴⡡⠀⣸⠋⢀⢦⡄⠁⠀⠀⠀⠀⠀⠰⠛⠗⢼⣿⢫⣿⣶⠃⠀⠀⠀⠀⠀⠀⠀⠛⢀⠉⠇⠀⠀⠀⠀⠈⢤⣶⣶⢄
⠀⠀⠀⢠⠃⠁⢸⠃⠀⢈⡛⠀⠀⠀⠀⠀⠀⠀⠱⢴⡆⢨⡟⠈⢹⡏⠀⠀⠀⠀⠀⠀⠀⠀⣻⡏⢿⡆⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⡐⢲⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⠁⢸⢁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢹⢸⠃
⠀⠀⢠⠛⠓⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠾⠀⡼⣻⢷⡆⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠺
⠀⠀⣠⡀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⠀⡿⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⢱⡄
⠀⢰⢿⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⣱⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠛⠁
⠀⠰⠄⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣈⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣵⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠋⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⡣
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣾⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⡟⣙⣻⣉⣝⣛⣟⢋⣿⢻⣙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠐⠒⠀⠀⠐⠒⠒⠒⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⣠⡶⠿⡿⢿⠷⢶⣅⡀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⣠⡞⠋⠀⠀⠀⠀⠀⠀⠙⠹⣦⡀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⢀⡼⡿⠀⣤⡄⠀⠠⡄⠀⠀⠄⢀⠸⢻⣆⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣠⣾⠟⡿⠀⠈⡆⠀⢀⣆⠀⠀⡄⠀⢸⡄⣹⣧⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣴⠛⣇⠀⣠⣤⣾⣿⠻⠿⢿⢛⣿⣷⣦⣤⡁⠈⠿⢧⡀⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣟⠸⣇⢀⢘⣶⣿⣿⠿⠷⠀⠀⠀⠀⠈⠃⠥⣻⣷⡜⠀⠈⢧⢿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡼⣯⡜⡰⠆⠤⣀⣿⡿⠣⠂⠁⠀⠀⠀⠀⠀⠀⠁⠀⠈⢙⣻⣇⠀⣀⠈⣻⡟⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⣰⢾⡜⠓⠒⢢⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠙⢿⣅⠀⣀⢹⣽⡞⢧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣫⡼⢫⡎⠙⠛⣲⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠙⢷⡄⠰⠛⢝⣿⠓⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠓⣰⢳⡟⣿⢿⡿⡏⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠰⠟⢻⡍⢳⠄⠀⠀⠀⠈⢳⣾⢶⣿⡜⣧⠜⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⢰⠃⡏⢀⠀⠀⠸⠀⠀⣠⣿⠅⠀⠀⠀⢀⣴⣦⣤⡎⠀⣼⣧⠃⠀⢰⡀⠀⠀⠐⣿⠀⠘⣿⣹⣆⢵⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠄⠹⠸⣷⡆⠀⠀⣶⠀⢠⣿⡏⡀⢀⠀⠤⠾⠏⢊⣽⣷⣾⠿⡿⠁⠀⠀⢻⣆⠀⠀⣿⡄⠀⢹⡇⡯⣬⡇⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡈⢧⢻⣿⡀⢰⣿⣿⣾⢿⠟⠁⠀⠀⠀⠀⠐⠈⠉⠀⠀⠀⠀⠀⠀⠠⣼⣿⣱⣀⣿⣇⢠⣾⣹⡗⣰⢃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡈⢧⢻⡷⢾⠿⣹⣷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠿⠿⣿⣣⠿⢀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣀⣧⠴⠀⠠⠻⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠃⠀⢠⣱⣟⣀⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡌⣧⡷⡀⠐⠻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⢤⠄⣤⡾⢃⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣿⣵⣆⠠⠛⣷⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣴⣿⠛⠀⢀⣿⣟⣳⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠷⣽⣮⢆⠈⠉⠙⢻⣧⣾⣴⣶⣤⣤⣶⣴⣦⣾⡟⠛⠙⠂⣶⣿⣧⣿⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣽⠙⢷⣼⡀⠀⠁⠀⠉⠉⡙⠋⠙⠋⢉⠉⠀⢸⣧⣴⠛⠲⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⣿⣍⡊⠳⠤⠳⠠⠷⠄⠶⠆⠸⠛⠂⠋⣴⡏⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠈⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⡀⠁⠻⢧⣆⡀⠀⠀⡀⠁⠀⠀⢀⣴⠋⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⠙⠳⠦⠾⢷⣶⠶⠞⠋⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠶⠷⠿⠦⠾⠿⠿⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠄⠶⠶⠶⠒⠒⠒⠶⠦⠤⠤⡼⠀⠀⡇⠈⢙⠒⠲⠤⣤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡣⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦⢝⣒⣒⣣⠉⠙⢙⠇⠀⢹⡾⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠴⠺⡗⠪⠔⡨⢉⠓⠲⠤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠈⠉⠓⠒⠢⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢑⡦⢁⡀⠬⣁⠊⠀⠀⠀⠈⠁⠐⠒⠢⠤⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⢸⠳⠤⠤⠤⠥⠤⢬⢝⣒⠢⠤⣀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠒⠒⠦⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢀⡴⠒⠶⢬⣍⠀⠒⠩⠍⡐⠂⠠⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠤⠴⠾⣯⣿⣏
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⢒⣏⣉⣉⣙⣳⣦⡈⡇⠀⠀⠀⠀⢀⣩⠉⠒⠂⠤⡤⠄⣀⣀⣒⣒⠒⢒⠒⢒⠒⢀⣉⠭⠞
⠀⠀⠀⠀⠀⠀⠀⣾⢰⢋⠎⠁⠛⢿⠿⠿⠿⢦⢹⡇⠀⠀⠀⡞⠁⢘⢀⡤⢄⠘⣇⣀⠀⠀⢸⡁⠅⠙⠤⡀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⣅⢠⠀⣄⡀⠀⢸⠀⠀⣠⠞⣸⡇⠀⠀⢸⠀⠀⠘⠁⠀⢨⣴⠀⢹⠀⠀⣸⠀⠀⠀⠀⣿⢀⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠈⠳⠦⣭⣍⣉⣋⣩⠵⠚⢩⡇⠀⠀⢨⠀⠀⠀⠀⡇⢸⡇⠀⠈⢻⡆⡯⠀⠀⠀⢠⠏⡜
⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠇⠀⠀⢸⠀⠀⠀⠀⠇⢸⠀⠀⠀⡼⠳⠃⠀⢀⢠⠏⢠⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢸⠀⠀⠀⣾⠀⠀⠀⠀⡆⡞⠀⠀⢠⡇⠀⠀⠀⣼⡏⠀⡾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⣼⠀⠀⠀⡇⠀⠀⠀⢸⢰⡇⠀⠀⡾⠀⠀⠀⢰⢿⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⢰⡇⠀⠀⠀⢸⢸⠀⠀⠀⡇⠀⠀⢀⠏⡟⠀⣸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⡀⠀⡿⠀⠀⢸⠃⠀⠀⠀⡏⢸⠀⠀⣼⠁⠀⠀⡜⢠⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⢀⡇⢰⡇⠀⠀⢸⠀⠀⠀⢠⠇⣾⢰⠁⡇⠀⠀⣰⠃⣸⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡿⠀⠀⢸⠁⣸⠀⠀⠀⡟⠀⠀⠀⣼⠀⡟⠀⢸⡇⠀⢠⡏⠀⡏⠀⡾
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢠⡇⠀⢀⡏⠀⡟⠀⠀⠀⡇⠀⠀⢠⠇⣸⠁⠀⣼⠀⣠⠟⠀⣸⠁⢠⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⣸⠀⢰⡇⠀⠀⢀⡇⠀⢠⠏⢀⡏⠀⠀⣿⠚⠁⠀⢠⡏⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡁⣰⠃⠀⣾⣀⡠⠊⠀⣧⠔⠋⠀⡼⠀⠀⠀⠸⣆⠀⣠⠞⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢰⣇⡀⠀⠐⠂⠻⢥⣤⡤⠏⠀⠀⠀⠀⠹⣤⣀⡴⠃⠀⠀⠀⠀⠈⠋⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢇⡈⠑⠲⠤⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡤⠤⠴⢒⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠤⣤⣈⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣩⡥⠤⠤⠴⠶⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⡶⢾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠐⢀⣠⡄⠤⠴⠤⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠔⠁⠀⣠⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠈⠀⠂⠂
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⣿⣂⢄⡀⡀⠀⢄⡀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⣰⣾⣍⣹⣿⣿⣆⣱⡈⣦⠀⢉⣳⣤⠀⠀⠄⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠈⠢⢤⠄⠂⢠⣾⣿⣵⠿⠻⣿⣿⣿⠛⣹⣽⣿⣿⣛⣿⣏⣽⣉⡀⢠⣶⢿⣿⠛⢦⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡴⠟⠉⠁⠀⠀⠀⠀⢰⢏⣽⣿⡿⠹⣿⣍⡀⣀⡀⣸⣟⠉⠁⣸⠂⠀⠀⠈⢣⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⣺⣿⠏⠀⠀⠈⠙⠙⠋⠙⢏⢻⣆⠀⠁⠀⠀⠀⠀⠀⠙⡙⡆
⠀⠀⠀⠀⠀⠀⠀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⠤⠭⣭⣆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⡌⠀⣿⠀⡶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠒⠤⢤⠃⠠⠻⡀⠀⠀⢀⣼⡏⠺⠁⢧⢠⠁⢸⡇⡜⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⡀⠀⠀⣇⠀⣰⢿⠀⠓⣄⡀⣸⡼⢠⣼⡟⡰⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡉⠙⢦⡀⢀⣧⡔⢒⣾⡏⠈⠊⠑⠀⢠⠟⢹⠀⠀⡿⢲⠗⠤⣘⠓⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠒⠁⠉⠀⠈⠙⠚⣟⠉⠁⠀⡇⠀⠀⠀⣴⡁⠀⠈⠧⠴⠢⠤⣤⠞⣹⠀⣝⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⠋⠙⠒⡞⠀⢠⠾⠒⠒⠐⣻⣲⡶⠤⠼⠤⠤⠤⠤⠤⠄⣒⣚⡭⠷⣄⠈⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡁⠀⠀⠐⡤⠚⠛⠛⠛⠓⠒⠒⠂⠈⠉⠉⠉⠉⠀⢀⣤⣄⢸⣇⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢷⠀⠀⠙⠒⠦⠼⠀⠀⢀⣤⣤⣤⢠⡦⠴⣒⢦⢀⢺⣋⡝⣞⢿⣦⢾⠈⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣀⡉⠒⢦⠶⣦⡤⡄⠀⠈⠉⣿⠀⢸⣧⠾⢭⣣⠸⠞⠙⠳⠼⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉⠑⠲⠴⠛⠉⠁⠀⠀⠀⠛⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠉⡇⠂⠀⠉⠉⢲⠀⠀⠀⠀⠘⠙⠛⠐⠀⠀⠀⠋⠉⠉⡇⠉⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡇⠉⢳⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡇⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠃⢀⠏⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠾⡉⢳⡸⠖⢄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠛⢹⠘⠧⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⠀⠀⠀⠀⢸⠀⢰
⠀⠀⠀⠀⠀⣀⣠⡔⠦⢼⣟⣁⣦⡀⠀⠀⡆⠀⠈⠓⠤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⢀⠈⠀⠀⣤⣤⣤⣤⣤⣶⣒⡒⢤⣀⣀
⠀⠀⠀⠀⡏⠀⢀⡨⢁⣀⠀⠀⢀⣁⠀⠀⣳⣤⣤⣤⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⢸⣆⠀⢀⡇⠀⠀⠀⣀⣠⠋⠙⢬⡉⠉
⠀⠀⠀⠀⠑⡶⢞⠛⠃⠈⠉⡉⠉⠀⠉⠲⣼⡿⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡟⢩⠋⠛⠛⠭⢭⣉⣉⠵⣤⣀⡈⢢
⠀⠀⣀⣀⡜⢀⣀⢠⡄⠀⠀⠻⣄⠀⢀⠔⢈⠟⣦⠀⣀⡀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⡼⠈⣧⣈⡓⠒⠦⠴⠋⠀⠽⠶⠯⣀⠈⠉⠒
⡖⠚⠁⡴⠶⠋⠈⢳⡸⣄⠀⠀⠈⣃⠀⠀⠀⢀⣼⢤⢬⣉⡁⠀⠀⠀⠐⠒⠒⠒⠉⠀⠀⢈⣉⣠⡤⠞⠃⠀⠈⠉⠒⠢⠤⠤⠖⠲⠦⠤⣽⠷⠶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢲⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠿⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢁⣀⣤⣶⣥⡤⣠⡄⣀⣀⣀⢠⣤⣄⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢜⣴⣿⣿⣿⣿⣿⢄⣶⣼⣯⣾⣿⠸⣿⣿⣷⣶⣌⠃
⠀⠀⠐⠀⠀⠀⠀⠀⠠⣴⣿⣿⡷⠿⠿⢱⡿⣾⣿⣿⣷⢿⣿⢧⣿⡻⣿⣟⠛⢀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⣢⣍⠳⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠙⠟⠛⣛⡀⢠⣤⠤⠀⣀⣀⣀⣈⣀⡀⠀⠀⠠⢴⣿⣿⣿⣦⡝⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠿⣇⣩⣶⣿⠿⠿⠭⠩⠽⠿⢟⡷⣦⣄⠀⠙⢿⣿⣿⣿⣦⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢯⣄⣿⢛⠍⠠⣄⡀⣠⡀⠀⠀⠀⠈⠻⣝⣷⡀⠈⣿⠫⡽⠟⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣝⢻⣀⠙⢁⡹⠿⠿⠿⠷⠿⣣⢀⠂⠘⣿⡇⢰⣿⣾⢳⡟⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣯⡿⣼⣿⣛⠀⠀⠀⠀⠀⠙⣷⠆⠀⣿⣷⣿⣿⡿⠈⡿⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣺⡃⠙⣻⣿⡋⠈⠁⠀⠀⠀⣾⡀⢠⡟⣻⣿⣿⠇⣧⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣏⠇⣜⣼⣟⣿⢦⠰⠶⣧⠬⠋⢡⣿⢃⣿⣿⣿⠀⠉⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠻⢿⣷⣿⣿⣿⣿⡛⠓⠀⠈⠁⣼⣟⣿⡇⠸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢰⠀⣠⢾⣿⠇⣿⡇⢒⡀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⣾⣿⣿⡟⠀⣿⡇⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢾⣟⣛⡭⣧⢠⢸⡆⠠⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⣿⣿⣵⣟⡀⢝⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠙⠉⠀⠁⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣶⡦⢤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⢲⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣞⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢏⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣣⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⢾⣭⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠛⠛⠍⠉⠉⠛⠛⠋⠛⣻⣍⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⡀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢷⡤⠔⢉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⢀⣿⣏⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠯⠤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢓⡶⠀⠀⠀⠹⣆⡴⠚⠛⠛⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠿⣄⣀⣀⠴⢲
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉⣠⡴⠾⡅⠀⢹⢁⣴⠂⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⡿⠶⠤⣤⣁⡤⠛
⠀⠀⠀⠀⠀⣀⡠⠖⠉⠀⠀⠀⢎⣐⡷⢃⠀⠀⢸⡷⢤⣤⡴⠃⠀⠀⠀⠀⠀⠀⢀⡿⠁
⡴⠚⠓⣦⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡖⠚⠉
⢷⣀⣠⣟⡀⣀⣠⣤⣤⣀⣀⣀⣀⡤⠒⠑⣂⡴⠳⣤⣄⣀⣀⣤⠗⠚⠋⠉⠁
⠀⠈⠀⠀⢉⡟⢠⠃⡼⠀⠀⠀⠉⣩⠋⡽⠉⠀⣠⠃⣼
⠀⠀⠀⠀⠈⠉⠛⠒⠁⠀⠀⢀⠜⢁⡜⠀⢀⡴⠃⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠊⠀⢠⣏⣀⡴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣶⡎⠉⠉⠀⠚⠢⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠀⢀⣀⣀⠀⠒⠀⠀⠀⢉⣙⣛⣋⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣶⣿⣿⣿⣯⣥⣤⣶⣶⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠛⠛⠋⣡⣽⡿⠋⠁⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⡯⠛⣿⡟⠛⠀⠀⠁⢸⣿⣽⣿⠿⢷⣤⣄⠀⣠⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣻⣿⡭⢤⢿⡄⠈⡁⠀⠀⠀⠠⡴⠛⠛⠋⠁⠀⠀⠀⠘⠾⢟⡛⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⣿⣿⣿⣷⣶⣷⣤⣀⠀⣰⣶⠾⢿⣆⠀⠂⠀⠀⠀⢀⣏⡨⢵⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠻⣿⣿⣿⣿⣿⣿⣯⣭⣭⣍⣡⣤⠤⣿⣧⡀⠀⢀⣶⣿⢇⡸⠉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡸⡿⢿⣿⣿⣿⣾⣷⣆⠀⠀⠀⠀⠀⠐⣿⣧⡀⣿⠋⠁⢀⡁⣘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣶⣾⣿⣿⣿⣅⢤⣤⠀⣶⣦⢠⡿⣿⣹⢿⣧⠜⣷⡎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣋⡻⣿⣿⣿⣿⣿⣼⣿⠀⢹⡇⠀⣿⠀⠀⢻⡅⢸⣿⠀⣿⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⣿⣿⣿⣿⣽⣏⣿⡀⢸⡇⠀⣿⠁⠀⢸⡇⢸⣏⢐⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⣿⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⢿⠀⠀⣿⠃⢸⡏⢸⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣯⣿⣇⢸⡇⠀⢸⡇⠀⣿⠀⢸⡗⢸⡧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣏⣽⣿⢸⣇⠀⢸⡇⠀⣿⠀⣿⣇⣼⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⢠⣿⠀⢸⡇⠀⣿⠀⣿⡧⣿⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⣛⣛⣷⣿⣿⣿⣿⣿⣿⣾⣿⡌⣿⠀⢸⡇⠀⣿⡀⣿⣷⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⣤⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⡇⣿⠀⢸⣷⠀⣿⡇⣿⣽⡗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠴⣫⡿⣫⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⣿⡆⠀⡏⠀⣿⡇⣿⠚⣻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢐⠎⣠⢏⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇⠀⠇⠀⠿⠀⣿⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢁⣿⣿⣿⣿⣿⣿⣿⣿⣻⠿⣿⢿⣿⢘⠇⠀⡄⠀⠒⠂⣩⡀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⢸⣟⣿⣿⣿⣿⢿⣿⣿⣿⣷⣶⠀⠀⠀⢩⣼⡉⠛⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⢈⣿⣅⢈⣯⣴⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢿⣿⣿⣿⣶⣿⣶⣶⣶⣶⣾⠿⠿⠋⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁

⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣤⣀⠀⠀⠀⠀⣀⣤⣶⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⣾⣷⣽⣿⣿⣯⣾⣷⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⣭⣟⣻⣭⣾⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⣤⣤⡄⣠⣤⣄⢠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⢻⣟⣻⡟⣿⣿⣿⣿⣿⠁⢠⣾⣿⣿⡇⣿⣿⡇⢿⣿⡿⢸⣿⣿⢸⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣼⣽⣯⣥⣿⣿⣿⣿⠇⠀⣼⣿⣿⣿⣷⣼⣶⣾⣷⣶⣶⣷⣶⣧⣾⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠁⠈⣿⣿⣿⣿⣿⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⢸⣿⣿⣿⣿⠀⠀⠸⣿⣿⣿⣿⣱⣶⣍⣿⣿⣿⣩⣶⣎⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⠁⠀⠀⠈⣿⣿⣿⡟⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⠏⠀⠀⠀⠀⠹⣿⠟⠁⠀⠀⠀⠘⣿⣿⣿⣿⣿⣙⢿⣿⡿⢋⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠟⠀⠀⠀⠀⠀⠻⠿⠃





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

⠀⠀⠀⠀⠀⠀⠐⢦⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣹⢷⡦⢤⣄⡀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠒⠉⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⢀⠃⢠⢹⠆⠀⢸⠏⠀⠀⠀⠀⠀⠐⠒⠂⠉⠉⠀⠀⣀⣠⢴⢾⠀⠀⠈⠳⠄⣀⣀⣀⣀⣀⡤⣤⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢰⣏⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠁⢠⠏⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢾⠈⠁⠀⠈⠀⠀⣀⡴⠖⠉⠉⠀⢀⣠⢴⠇⠀⠀⠀⠀⠢⢤⣀⣀⣀⡠⠊⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢸⡆⠀⠀⣻⡷⡚⠁⠀⠀⠀⠒⠉⡉⠠⠁⠀⠀⠀⠀⠀⠀⠀⣈⢝⡟⠀⠀⠀⠈
⠀⠀⠈⠛⡓⠦⢤⣤⣇⣀⣈⣷⣼⣿⣄⣴⠏⠀⠈⠾⡉⠉⠉⠉⢀⡄⠀⠀⠀⠐⠦⣄⡠⠔⠉⠀⠘⠁
⠀⠀⠀⠀⠘⣆⠀⣀⣙⣿⣿⣿⣿⣿⣿⣯⡤⢴⠶⠓⠛⢯⣙⣉⠉⣄⠀⢱⣄⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⣠⠾⢛⣿⣿⣿⣿⣿⣿⣿⠉⠁⠘⣗⠲⡶⢧⣤⣯⣭⣶⣪⣍⣀⣄⠀⠀⠰⡄⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⢦⡀
⠀⠀⠀⠋⠁⠀⠸⣿⣿⣿⣿⣿⡿⠙⣎⡆⠀⡸⠀⢈⠹⣽⠉⣤⢀⡁⢠⣜⠁⠈⠉⠉⠛⠛⠒⠒⠒⠒⠬⠷⠤⣄⣀⣀⠑⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠉⢸⠃⠀⣸⣷⡀⠀⠀⠈⣷⣁⠄⠈⠋⠀⠀⠀⠀⣠⣼⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠐⠃
⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⢀⣾⠐⠁⠀⠀⠉⠀⠀⠀⠏⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢀⡼⠁⣀⡤⠒⠉⣿⠉⠢⠀⠀⠀⠀⠘⣦⣠⠄⠀⠀⢈⡾⠁⠀⠀⠀⠀⠀⠀⠠⠤⣾⣇
⠀⠀⠀⠶⠿⠾⠯⠭⣄⠀⠁⡗⠀⠀⠀⠀⠀⠀⠀⠟⠁⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠒⢤⣀⠀⠰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠇⠀⠀⠀⠘⡄⠀⠀⢀⠠⠤⠤⠤⣄⡀⣀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣌⡓⢄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠋⣰⠆⠀⠀⠀⠀⠀⠈⠙⢿⠃⠀⠀⠀⣀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢀⣞⣡⠀⠀⠀⠀⠀⠀⠀⣀⠈⢧⠀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣡⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⠶⠶⠶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣴⣿⡟⢉⣤⣤⣤⣶⣶⠆⣿⣷⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠿⠛⠛⠉⠉⠉⢩⡟⢋⣀⠾⠀⠀⠀⠈⠻⠦⣿⢹⣶⡌⠉⠉⠉⠙⠛⠻⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠿⠷⠶⠶⠶⠶⠶⠶⠶⠞⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡆⠙⠻⢷⣦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣠⣤⣤⡶⠾⠛⠉⢀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠛⢦⣀⠀⠀⠉⠉⠛⠛⠛⠓⠲⠶⠶⠶⠶⠶⠶⠶⠶⠟⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⢀⡠⠔⢋⣽⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⢳⣦⣤⣉⡙⠲⠲⠤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⡤⠤⠒⢒⣂⣩⣤⣶⣾⠋⠙⢻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠈⠉⠛⠻⠛⠋⠛⢛⣶⣶⠊⠉⠉⠉⠩⠭⠭⠭⠭⠭⠭⠥⠄⠀⠐⠒⠓⠒⠀⠈⠉⠈⠙⢧⡀⠀⢠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡶⠶⠶⠶⠾⠿⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠈⢿⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⢰⡄⠀⠀⢿⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣻⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡇⠀⠀⢸⡀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠸⣧⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⣿⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣿⡄⠀⢸⡇⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢠⣿⠀⠀⢰⣿⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢿⡇⠀⢸⣇⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⢸⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡇⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡿⠀⠀⢸⡇⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢸⡇⠀⠘⣿⠀⠀⠀⠘⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⢸⣧⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢸⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇⠀⠀⢸⡇⠀⠀⣾⡇⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⣿⡇⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⢸⡇⠀⠀⣿⠃⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⣿⡄⠀⣿⡆⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣼⡇⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⡇⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⠀⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⢻⡇⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢰⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⢸⡇⠀⣿⡇⠀⠀⠀⣿⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⢸⣿⠀⢹⡇⠀⠀⠀⣿⠄⠀⠀⠀⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⢸⣿⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣾⣿⡀⢸⡇⠀⠀⠀⣿⡆⠀⠀⠀⣿⠀⠀⠀⣿⠇⠀⠀⣿⡃⢀⣸⣿⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣶⣤⣤⣿⣇⣀⣀⣀⣿⣅⣀⣠⣿⣧⣴⣶⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⣤⣶⣦⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⡤⠴⠶⠶⠞⠛⠛⠛⠋⠉⠉⠉⢻⡆
⠀⠀⣸⠋⠀⠀⠀⠈⠙⠛⠻⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⣼⠇
⠀⠀⠸⢦⡀⠀⠀⠀⠤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢠⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣄⠈⠉⠃⠀⠀⠀⢻⡆
⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⣉⣉⠉⠀⢀⣀⣀⣀⡀⠀⢀⣾⠃
⠀⠀⠀⠀⣾⠀⠀⢰⡖⠒⠒⠒⠲⣶⠶⠒⠒⠒⠶⣦⠤⠤⠤⠤⢶⣶⠶⠶⠒⠲⣞⠋⠉⠉⢹⡏⠉⠉⠀⠈⣷⡾⠟⠁
⠀⠀⠀⠀⠙⠶⠶⢾⡇⠀⠀⠀⠀⣿⠀⠀⡄⠀⠀⢿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢹⠀⠀⠀⠀⢻⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⣧⣀⡀⠀⢀⣽⡀⠀⠈⠀⠀⣸⡀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢸⣧⡀⠀⢀⣼⣷⣤⣤⠾⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠋⠉⠉⠙⢿⠟⠶⠶⠶⠚⢿⣷⡶⠶⠶⢾⡿⠶⠶⠶⠒⠛⣿⠛⠛⠛⠙⣿⢀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⣀⣠⡗⠀⠄⡀⠀⢸⠀⠀⠀⠀⠀⠈⡏⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠆⠀⠘⡿⠛⢷⡄
⠀⠀⠀⠀⠀⢸⠋⠉⢿⣀⣀⣀⣀⣼⡀⠀⠀⢀⣀⣠⣿⡀⠀⠀⠀⣇⡀⠈⠀⠀⢀⣸⣆⣀⠀⢀⣸⣀⣀⣀⡴⠃⠀⠈⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠙⠛⠛⠋⠉⠉⠓⠒⠛⠉⠀⠀⠉⠉⠉⠁⠉⠉⠁⣀⡄⠀⠀⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠛⠛⠛⠛⠒⠶⠦⢤⣄⣀⣀⣀⣠⠴⣄⠀⠀⠀⠀⠀⠀⠰⠛⠳⣄⢀⣠⠴⠒⠖⠛⠁⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠘⠻⠶⣶⣤⣤⣤⣄⣀⣀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠘⠲⠋⠀⠰⣤⠆⠀⠀⠉⠉⠁⠀⢀⣀⣤⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠻⠷⠶⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡶⠶⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣙⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠆⠀⢶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠙⢷⡀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠻⡦⠀⠀⠀⠀⠈⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⣰⣷⠀⠀⠀⠀⠀⠀⠀⠀⢱⢡⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠸⡷⡀⠀⠀⠀⢠⢣⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⡇⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢳⣱⠀⠀⠀⣾⢾⡃⠀⠀⠀⢀⣀⠀⠀⠀⢀⠏⡹⠀⠀⣼⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣤⣀⠘⢦⠳⡀⢠⡾⠋⠉⠙⢦⢀⣞⠜⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠢⣌⢷⣹⣿⣠⣤⠤⠤⠬⠿⣎⣴⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⣿⠟⠉⠀⠀⠀⠀⠀⠀⠈⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢞⠡⢸⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡈⠛⠢⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⡏⢀⡏⠎⢻⡄⠀⠀⠀⠀⠀⠀⠀⢀⡏⠱⡌⣄⠈⢿⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡼⠀⢸⢻⡀⠀⢻⣆⠀⠀⠀⠀⠀⢀⡞⠀⠀⢻⠹⠂⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠈⣇⣣⠀⠀⠙⢷⣄⠀⠀⣠⠞⠀⠀⠀⢸⣸⡄⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠈⢯⡣⡀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⡟⡜⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣾⣄⠀⠀⠀⠀⠀⠀⠀⢀⣼⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⢀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠀⠀⠚⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⣴⡄⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠈⠃
⠀⠀⠀⠈⡔⠉⠉⠑⠚⠉⠉⢳⠀⠀⠀⠀⠎⠉⠉⠒⠊⢩⡉⢆⠀⠀⠀⠀⠁⣾⣿⣿⣶⣾⣿⣿⡆⠀⠀⠀⠀⣾⣿⣿⣶⣾⡟⢿⡆
⠀⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠄⡎⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣻⠇
⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⡇⢀⠋⠙⡄⢸⠀⠀⠀⠀⠀⢸⠀⢰⠉⢱⠀⢸⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⢹⣿⡧⠀⠀⠀⠀⠀⢸⣿⡿⠁⢹⣿⣿
⠀⠀⠀⠀⠀⢣⡼⠀⠀⢃⡜⠀⠀⠀⠀⠀⠈⢇⡎⠀⠀⣇⠎⠀⠀⠀⠀⠀⠀⠈⢿⠇⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⡄⠀⠀⠀⠀⢀⣀⠴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠎⠀⠀⠀⣠⠀⠖⠉⢁⡀⠀⠀⠀⣰⡴⠄⠀⠀⠀⢀⡀
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠔⠃⠁⠀⠀⣀⠴⠾⢁⡀⡴⠶⠈⠁⣀⠰⠚⠉⠀⠀⣀⣠⠔⠊⣱⠾
⠀⠀⠀⠉⠉⠙⠒⠲⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠞⠁⡀⠔⢋⢁⡐⠃⠁⡀⠤⠊⠁⢀⢰⠄⠋⣀⠠⠀⢁⡠⠀⠈⣡⡔⠈⠁
⠀⠀⠀⠀⠶⠖⠒⠒⠒⠒⠀⠀⠀⣀⣀⡄⠀⠀⠀⠉⠁⠰⠖⢈⠁⢀⡉⢹⣇⠐⠋⠀⢠⠤⠖⡉⡀⠘⠂⢁⠀⠚⠉⠀⠀⠈⠀⠀⣀⣀⡀
⠀⠀⣴⡶⣶⣶⣶⣶⡶⠶⠶⠾⠿⠛⢹⣿⡦⠀⠀⢠⠄⢤⣤⠀⢈⢹⡆⣘⢙⠋⣷⡐⠈⠀⠀⠁⠀⠠⡖⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦
⠀⠀⠐⠂⠈⠤⢤⡤⢤⠤⠬⠥⣤⣤⣤⠤⣴⣶⠶⠞⢻⡙⠿⡄⢹⢸⣆⡀⠘⢠⡟⢀⡀⠒⠂⢤⣤⣀⢠⣾⣧⡀⠀⠀⢀⠀⠀⠀⠀⠀⢚⣿⣧
⠀⠀⠀⠀⢰⠀⢸⠁⢰⠂⢀⣶⠋⢐⣧⠖⠉⠀⠀⢰⣾⣿⣆⠀⠈⠚⣃⣁⣡⣾⣷⠋⠀⠀⠀⠀⠈⢻⣷⣬⠁⠙⠦⣜⠉⠁⠛⠓⠚⠃⠛⣿⣿
⠀⠀⠀⠀⢸⠀⢠⡀⠸⣴⠏⢠⣤⠞⠀⠀⠀⠀⠀⠘⣿⡿⠋⢀⠖⠉⢁⣠⣭⣿⣿⡄⠀⠀⠀⠀⠀⠀⢹⣇⠱⢦⠘⣼⣧⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⡏⠀⠈⡇⠐⠃⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠐⡟⢶⡄⠁⠀⣤⠿⣿⠁⣤⠀⠀⠀⠀⠀⠀⢹⣏⢣⠙⢨⣿⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⣇⠀⠀⣿⠀⠀⢙⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠈⣠⣤⣶⣦⣤⢹⡀⢾⡄⠐⣆⠀⠀⠀⠀⠀⠀⠿⡞⢢⢮⣹⠃⠀⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⢹⡀⠀⠹⠀⠐⣾⠁⠀⠘⠰⠆⠴⠀⠰⠀⠀⢹⡀⣿⠙⠿⢿⡿⢸⠿⣦⣷⡀⠇⠒⠰⠀⠐⠂⣦⠘⣷⠖⣎⡏⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠈⠀⣧⠐⡂⠀⢴⡏⠀⠀⠀⠀⠂⠀⠀⠀⠀⣀⣽⠃⣉⣄⠀⠸⠁⡼⢀⣨⣿⡷⠀⢀⠀⠀⠀⠁⠀⠐⣿⠤⠈⡇⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠈⣤⡃⢄⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣏⡉⠉⠙⠋⠈⢁⣬⣿⡟⠡⣤⡏⠀⠀⠀⠀⠀⠐⢿⢴⡇⡇⠀⠀⠀⠀⠀⢀⣨⣿
⠀⠀⠀⠀⠀⠀⠻⠀⡊⢹⡁⢦⠀⠀⠄⠀⠀⠀⠀⢀⣼⠿⠻⠿⠿⢶⣿⣿⣿⣿⣿⡅⠰⠛⡃⠀⠀⠀⠀⠀⢰⣿⡀⢸⠇⠀⠀⠀⠀⠀⠼⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡇⡇⢿⡁⠀⠱⠀⠀⠀⣀⡤⠞⠋⠅⠦⠠⢠⡄⠴⠄⠶⠦⢄⣶⠽⢦⡀⠀⠀⠀⠀⠀⠀⢸⣇⢑⡨⠀⠤⠤⠤⠤⠶⠛⠛⠃
⠀⠀⠀⠀⠀⠀⢰⠁⠃⣾⡅⠀⢀⣼⠟⠋⠁⠀⠀⡀⠤⠔⠒⠛⠛⠛⣛⠷⠒⠊⠄⡀⠀⠛⠳⣦⣀⠀⠀⢀⣿⢫⠔⡇
⠀⠀⠀⠀⠀⠀⢸⡆⢇⣼⡇⣼⡏⠀⢀⡠⠐⠈⠁⢀⠀⠀⠀⣀⡴⠊⠀⠀⠀⢠⠀⠀⠀⠒⢄⣀⠍⢳⣄⣿⡟⠓⣶⠁
⠀⠀⠀⠀⠀⠀⢸⡇⡆⢸⣿⡿⢡⠖⠁⠰⠠⠄⠚⢁⡤⠖⠋⠹⢷⠐⠐⠂⠠⠰⢲⡂⠓⢀⡰⢉⡳⢿⣿⣿⠗⠙⡿
⠀⠀⠀⠀⠀⠀⠀⢷⣄⠹⣿⢡⠃⠀⢀⣀⣀⠴⠚⠿⠷⢴⣦⣤⣌⣀⣀⣂⣦⣼⣾⣧⣶⡾⣿⣿⣷⣎⣿⡿⠆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢿⣶⡏⣿⣠⠞⠉⠁⠀⠀⢀⠀⠀⣀⣈⡉⢙⠋⠉⡉⢛⠉⣉⡉⢀⠀⡀⠙⣛⣿⡟⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡧⡿⣿⣦⠞⠃⠙⠓⠺⢿⣀⣴⣄⠁⣀⣈⠀⢉⡈⠈⠉⣠⣤⣤⠾⠧⠾⣿⡇⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⢻⢿⠃⢠⠙⠂⢄⡀⠀⣄⠀⠀⠀⢉⠟⠉⠛⣟⠉⠙⣇⠀⠀⢀⡤⠚⠀⡇⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡸⢸⠗⠉⢦⡀⠀⠈⠀⠀⠀⠀⠀⡘⠀⠀⣠⠇⠀⠀⣼⠉⠛⠉⢰⣎⢰⠇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣾⣄⣠⡆⢙⣶⣆⣀⣠⡀⠀⣀⣇⣀⣠⣇⢀⣠⡞⡋⠑⠶⣤⣀⠉⡟⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠘⡏⠀⠑⠤⠄⠩⣄⣀⠙⠋⠙⢶⡤⠋⠀⢰⠛⠳⠁⠀⣄⠀⢙⡟⠃⢀⡆⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢻⡶⠀⠀⠀⠀⠀⠁⠀⠀⢰⣄⠀⠀⢀⡞⠀⠀⠀⠀⣹⠃⢈⠇⠀⣾⡇⠀⠸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠻⡄⠀⠘⣦⠀⢀⡀⢀⣼⠍⠉⠃⢉⣄⠀⠑⢶⡞⠁⢠⠎⠀⢀⡿⠀⢠⣶⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠙⣧⠀⢸⡄⠈⠻⠋⠀⡸⠉⠙⡏⠀⢱⠀⠘⣧⡴⠋⠀⠀⣼⠃⠀⢸⣟⠙⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⡖⠀⠀⠀⡔⠀⠀⡎⠀⢠⠇⠀⠔⡋⠀⠀⠀⡰⠀⠀⠀⠀⠈⠀⠐⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠟⢀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⠀⠄⠚⠀⠠⠉⠀⠀⠒⠀⠀⢀⣴⠃⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⢀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⢀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⣀⣀⡀⣀⣀⣀⣀⠀⠀⣀⣀
⣿⣿⣿⠟⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣤⣶⣦⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⡀⣠⠖⠙⠒⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⡤⠒⠚⠁⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⠉⢣⠀⠀⠀⣿⣿⣿⣿⡿⠛⠋⠀⠉⠀⠀⠀⠀⠀⠀⠈⠁⠁⠉⢻⣿⣿
⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣿⣿⣿⡟⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿
⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿
⠀⠀⠀⠀⣇⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⣿⣿⡆⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿
⠀⠀⠀⠀⡋⠁⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⣿⣿⣿⣿⡏⠁⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠙⠢⠤⠤⠔⠚⠛⠋⣿⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠑⠢⠤⠤⠖⠚⠛⠉⣿⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡜⠙⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⡎⠙⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰⢧⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⢇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⢸⠘⡄⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⡼⢸⡀⠀⠀⠀⠀⢰⣿⣿⣿⣿
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⡎⢀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⡇⢀⡇⠀⠀⠀⠀⣸⣿⣿⣿⣿
⠀⠀⠀⠀⢣⠀⠀⠀⠀⢰⠁⡜⢹⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⣸⠁⣾⣇⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠸⡀⠀⠀⠀⡇⢰⠁⢸⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⢀⡇⣸⣿⣿⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢇⠀⠀⢰⠁⡇⠀⠀⣇⠀⢀⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⢸⢀⣿⣿⣿⡆⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠘⣆⠀⡸⠛⠀⠀⠀⠘⢄⠞⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄⠀⣼⣿⣿⣿⣿⣿⣄⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁
⠀⠀⠀⠀⠙⣿⣦⡀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠀⠀⣦⠀⠀⠀⠀⠀⢠⡆⠀⠀⣠⣿⠏⠀⠀⠀⠀⠀⠀⣠⣾⡟⠁
⠀⠀⠀⠀⠀⠈⠻⢿⣦⣄⡀⠀⠀⠀⠀⢻⣷⣄⠀⢿⣠⣤⠀⢠⣤⣸⠇⢀⣴⣿⠃⠀⠀⠀⠀⣀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⣄⡀⠀⠙⢿⣷⣼⣿⣿⣿⣿⣿⣿⣴⣿⠟⠁⠀⣀⣤⣴⡾⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣄⣀
⠀⠀⠀⠀⠀⢀⣤⣶⣾⠿⠛⠛⠉⠉⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠈⠉⠙⠛⠿⢿⣶⣦⣄
⠀⠀⠀⣠⣶⡿⠛⠉⠀⠀⠀⠀⠀⠀⣠⣾⠟⢋⣤⣿⣿⣿⣿⣿⣿⣿⣷⣮⣙⢿⣦⡀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣦⡀
⠀⢀⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠁⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡝⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣄
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠁⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠙⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣿⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠄⡹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀
⠀⠀⠐⠀⠀⠑⠀⠂⠀⠊⠁⠐⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⡌⠐⠄⠀⠀⠀⠀⠂⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠠⠀⠐
⠀⠀⠀⠀⠠⠀⠨⠀⠠⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⡠⠐⠻⠇
⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⢀⠀⠀⠀⠀⠠⠄⠠⠀⠀⠀⠀⠛⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠰⠁
⠀⠀⠑⠒⠉⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠐⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣅⣯⣭⣭⣭⣽⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄
⡎⠀⠀⠈⠲⡀⠀⠀⠀⠀⠀⡔⠁⠈⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠱⡄
⢧⠀⠀⠀⣄⡇⣀⠀⠀⠀⣘⠀⠀⠀⠀⠈⣣⡤⠒⡐⠒⠢⣄⠀⢀⠞⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠈⠳⠤⠴⢯⡀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠈⠵⢀⣀⠁⠀⠀⠈⠳⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⡙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡷⠦⣄⡀⠀⠀⠀⠀⠀⣀⡠⢴⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠉⠙⢳⠒⠛⠙⠿⡤⠞⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⡜⠀⠀⠀⠀⠒⠚⠁⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢦⡀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠾⢇⠀⠉⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⣡⠤⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠦⣄⡉⠑⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠤
⠀⠀⠀⠀⠀⠀⠀⢰⡁⢰⡋⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠈⠉⢒⠄⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣈⢳⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠒⢈⡠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⢀⣀⠀⡰⠁⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠊⣁⠴
⠀⠀⠀⠀⠀⠀⠸⣈⠒⠃⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢄⣀⠀⢁⣠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⡇
⠀⠀⠀⢰⠋⠉⠉⠒⢤⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣅⡴⠚⠋⠉⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠸⡀
⠀⠀⠀⠸⡄⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠱⡄
⠀⠀⠀⠀⠙⢆⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠹⡀
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡯⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠆⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣈⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠙⠓⠲⢬⣉⠒⠦⣄⣀⠀⠀⠀⢀⣠⠜⢠⠃
⡠⠒⠒⠒⠂⠉⢉⣀⣀⡤⠤⠖⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣠⣞⣁⡠⠤⠤⠤⠤⡀⠉⠒⠦⣄⣈⣁⣉⣁⣠⠴⠃
⠱⢦⣠⣀⡞⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠋⠉⠛⠧⣤⣄⣀⣤⣤⣤⠤⠤⠤⣄⠀⢀⢀⣸

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠒⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠓⠚⠛⠓⠲⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣭⠞⠛⠲⣄⠀⠀⠀⣀⣀⣠⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⣀⣤⠖⠋⢁⡾⠁⠀⠀⠀⠘⣷⡞⠋⠉⠀⠀⠀⠀⠉⡿⠙⠓⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⣿⣯⣀⠀⠀⣸⠁⠀⠘⣇⠀⠀⠘⣷⣤⣄⣀⡀⠀⠀⣸⠇⠀⠀⠀⠀⠙⠳⣄⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⣿⠀⠉⠻⣿⣿⠀⠀⠀⢹⡄⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣷⣤⣄⣀⡀⠀⢀⣿⣧⡀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⢸⡆⠀⠀⠈⠻⢦⣀⠀⠀⢻⡄⠀⠀⠀⠈⠙⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⢷⡀⠀⠀⠀⠀⠙⠳⢦⣄⡙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣧⡀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠿⢶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⣀⠀⠀⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠒⠶⠦⢤⣤⣤⣄⣀⣀⣀⣀⣀⣨⣿⡿⠿⠛⠛⠛⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠲⠶⠦⠤⠤⠤⠶⠶⠖⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⠋⠉⠒⠠⡀⠀⣀⠄⠒⠉⠉⠉⠉⠑⠢⡀
⠀⠀⠀⠀⠀⢀⣀⣀⣀⡔⠁⠀⠀⠀⠀⠀⠀⠘⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠤⠒⠒⠢⢄
⠀⠀⠀⠀⠔⠉⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⡀
⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⢀⠀⠤⠄⣀⠀⠀⠀⠀⠀⠠⠤⠤⡀⠀⠀⢀⣀⡀⢀⡀⠤⠤⢼⣠⢚⠉⠙⠒⠲
⠀⠀⠀⢱⡀⠀⠀⠀⠢⢄⠞⠁⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠈⠠⡀⠀⠀⠘⠁⠀⠀⠀⠀⠈⢂⠀⠀⠀⡸
⡠⠚⠣⠚⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⣀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⡆⠀⠀⠀⣠⡄⠀⠀⠀⠀⠃⠀⢠⠃
⢣⠀⠀⠀⠀⢰⣉⠀⠀⠀⠀⠀⠀⠈⠊⠀⠀⠀⠀⠀⠀⠀⠸⠂⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠈⢀⠃
⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⢠⠄⠀⠀⡜
⠀⠈⢧⠀⠀⠀⢠⣤⠴⠒⠳⡀⠀⠀⠀⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠘⠐⠒⠒⣟⣀⣠⠼
⠀⠀⠈⢇⠀⠀⠈⡀⠀⠀⠘⣌⠂⠄⠤⠀⠀⡰⠻⡑⠀⠀⠀⠠⠀⢨⡇⠀⠀⠀⠀⠀⠀⠠⠞⠉⢱
⠀⠀⠀⠈⢆⠀⠀⣱⠀⡄⠀⠈⠑⠒⠀⠒⠊⠀⠀⠑⠢⠤⠤⠤⠒⠱⣇⠀⠀⠀⠀⠀⠀⢰⠼⠚⠁
⠀⠀⠀⠀⠈⠶⣉⡤⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠤⠜⠁⢠⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⠄⠠⠔⢦⡀⠀⠀⠀⣀⠔⠳⣄⠀⠀⠀⣠⡀⠀⣀⠜
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣶⣶⣶⣶⣿⣿⣿⣶⣾⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢂⣨⠕
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠀⠐⠀⠉⠢⢄⡤⢒⣒⠒⠊⠉⡹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣿⢿⣷⢦⣄⣀⣀⣰⢋⠈⠁⠀⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⢛⠻⠷⣿⣽⣿⣿⡀⠀⣀⢀⣠⡼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣍⠁⢃⠀⠀⠤⠤⠤⠤⠄⠐⡢⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⢆⠀⠀⠀⢀⡠⠒⠉
⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟⢻⠉⠙⢻⠋⠉⠹⠀⠈⣆⣨⡯⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⢀⡴⠋
⠀⠀⠀⠀⣴⣟⣛⠁⣣⡀⢸⣀⣀⣸⣄⡤⢼⠖⢾⠙⣇⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⠶⡖⠲⡤⢶⡒⠒⠒⣀⣉⣉⣀⣈⢢
⠀⠀⠀⠀⠘⣿⣿⣏⣻⠉⣿⠉⢹⠀⢸⠀⠘⣇⣸⣶⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⠀⠟⠀⣏⠉⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣶⣾⣶⣾⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣤⣇⢀⣇⣴⢿⣒⡖⠒⠒⠲⠿⠁⢺
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣿⣿⡿⣿⠿⠛⢁⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠣⣉⠾⡹⢠⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⡅⣿⣾⡦⣏⠓⢚⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⡗⠛⣧⠶⠒⣒⣒⡀⠛⠩⢭⠽⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡘⣇⣿⣿⡇⡏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣱⣄⣷⣒⠟⡃⡟⢄⠀⠀⠀⠀⠀⣦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣮⠿⠛⢧⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⢛⡡⢻⡇⣾⡇⣿⣾⡒⠒⢂⣉⡩⡿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⣰⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠾⢏⡡⠒⠁⠀⠀⣏⣇⡁⡇⢩⠍⠻⣛⣓⡠⠃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢟⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠹⠜⣷⢚⡆⠀⠀⠀⠀⣿⠚⡇⡿⢳⣶⣿⡷⠾⠷⠴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⡼⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⣞⡳⢈⢉⣀⠀⠀⠀⠀⣿⠤⡇⢩⣁⠠⠤⢤⠶⡒⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⣧⡟⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣯⣽⣼⣷⠀⠀⠀⠫⡻⡇⢈⠀⠀⠀⢀⣑⣮⢟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟⠃⠀⠀⠀⠀⠀⠀⠀⠈⡄⠉⠉⠉⠉⢀⡼⠀⠀⠀⠀⠑⢤⡘⡆⠩⠍⠒⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⢀⠔⠆⠀⠀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⢣⠎⠀⢸⢠⠋⠀⠙⡄⡇⠀⢠⠁⢹⠉⠀⣷⠊⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠘⠀⠀⢸⢸⠀⠀⡰⠁⢇⠀⣸⡀⠸⡀⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠑⠈⠀⠀⠀⠉⠀⠁⠀⠉⠁⠃⠀⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⢠⣦⣄⣿⡟⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠈⠛⣿⣿⠇⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠖⠛⢉⣽⣿⣿⣷⠄⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡄⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⣰⣾⣿⡉⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⡎⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠙⠿⣿⣿⣷⣄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡼⠁⠀⠘⣿⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣾⣋⠀⠀⠀⠀⠀⠀⣀⣨⣍⣿⣧⣀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠻⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠉⠉⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠻⠿⠿⠿⠿⣷⣆⠀⠀⠀⢣
⠀⠀⠀⠀⠀⣀⡀⢀⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠻⣿⣿⡟⠀⠀⠈⢣⡀⠀⠀⠀⠀⠀⢸⣇⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡐⣬⣷⣵⣿⠏⠀⠀⡇⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⣙⣿⠛⠛⠛⠷⠷⠟⠛⠛⠂⠀⠀⢠⣶⠾⠶⠷⠿⠿⠿⣿⣷⣄⠀⠀⠁⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣦⣄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⣠⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⣄⡀⡀⡄⢀⢀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣦⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⠶⠛⠛⠛⠛⠻⠿⡾⠾⠷⠶⠷⠴⠶⠾⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠻⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣻⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣶⣧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡂⣬⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣤⣶⣶⣶⣴⣤⣀⡀⢀⠀⣀⣠⣠⣀⡀⢀⢀⢀⣀⣤⣤⣀⡀⣀⣠⣠⣤⣤⣶⣴⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠛⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⡞⠉⠛⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠋⠙⡆
⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⢳⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣰⠃⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠁⠙⢦⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢠⠞⠁⠁⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡎⠀⠀⠀⠘⠋⠉⠳⡄⠀⢀⣀⣀⠀⠀⣀⣀⡀⠀⡴⠋⠉⠙⠀⠀⠀⠀⣆⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⢀⡾⡇⠀⠀⣠⣦⠀⠀⠀⠹⡞⠉⠀⠈⢳⠏⠁⠀⠙⠾⠁⠀⠀⢀⣦⣀⠀⠀⣻⣆⠀⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⡼⠀⠳⠖⠋⠁⠸⡄⠀⠀⢀⣦⠀⠀⠀⣠⡀⠀⠀⢀⣆⠀⠀⠀⣸⠁⠈⠓⠶⠃⠸⡄⠀⢸
⠀⠀⠀⠀⠀⠸⡇⣸⠃⡀⠀⠀⠀⠀⠀⠙⠦⠖⠋⠈⠳⠴⠚⠁⠙⠲⠴⠋⠈⠓⠦⠞⠁⠀⠀⠀⠀⠀⡀⢳⡀⡿
⠀⠀⠀⠀⠀⠀⣧⡏⠀⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡏⠈⣧⡇
⠀⠀⠀⠀⠀⠀⠈⠀⠀⣷⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢁⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠳⡄⠀⢀⣠⡤⠤⠴⠶⠖⠒⠒⠶⠦⠤⠤⣄⣀⠀⠀⣰⠃⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠙⣞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡞⠁⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠈⣿⠉⠙⢲⡤⠴⠤⣄⡤⠴⠤⣤⠚⠉⢉⣟⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠋⠈⠳⠤⠔⠲⢤⡤⠞⠦⣤⠴⠒⠦⠤⠋⠈⠛⠚⠋⠁





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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣀⠠⠤⠄⠂⡀⠒⠃⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠆⠄⠀⠀⠀⠀⠀⢀⠀⣠
⠀⠀⠠⠄⡶⢀⡋⠀⠀⡈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⠀⠀⠀⠀⠀⠀⡸⢆⠀⠀⣀⠠⠊⠁⡜⠁
⠀⠀⢀⡔⠀⠠⢦⡀⠉⠉⠑⠒⠒⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉⠉⠉⠀⠀⠀⠀⠄⠈⣩⢁⣠⣴⣤⡎⢀⢂⡀⠀⠱⡀
⠀⠀⠀⣀⠈⠱⡄⠀⠑⠦⣄⡀⠀⡀⠀⠀⠀⣴⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠔⠃⠀⠉⠀⣠⠊⠙⠁⢢⡀⢀⠀⠐⠊⠲⣄
⠀⠀⠀⡀⢁⠀⠈⠢⠀⠀⠀⢉⠠⣤⡀⠀⢈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠄⠠⠁⠀⠈⢾⠀⠁⠀⠀⡴⠊⠁⠈⠉⠂
⠀⠀⠀⡇⠀⠅⠀⠀⠁⢄⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⡔⠀⠢⡱⡅
⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡒⠒⠒⠊⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠣
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠈⢄⠀⠀⡇⠀⠄⡀⢀⢀⡀
⠀⠀⠀⠀⡄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠶⢴⠒⠮⠀⠀⢊⣀⡈⢆⣠⠀⠠⡤⠒⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⡄⠀⢈⣅⠀⠿⡿⣗⠀⢠⣄⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⢀⣤⠥⠚⠉⠁⠀⢠⡶⠞⠲⡁⢀⠀⠀⠀⠲⠦⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠄⠀⠁⠀⠀⠀⠀⠀⠡⡌⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠅
⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠⡉⢀
⠀⠀⠀⠀⠀⠐⣦⡀⠀⠀⠈⢆⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢄⡀⠀⠀⠋⠈⠀⢠
⠀⠀⠀⠀⢀⠀⠘⡄⠀⣀⠀⠀⢡⡢⠀⢀⠼⠁⠀⠁⠤⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⢤⡀⢳⢠
⠀⠀⠀⠀⠄⠀⠀⢰⠀⠈⠈⠀⢂⠱⢦⠁⠔⠂⠊⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠤⠒⠒⠂⡈⢸⣗⠘
⠀⠀⠀⠀⠀⠤⠄⣀⠀⠤⠄⠀⠀⠈⠀⠙⠤⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⡈⠁⠀⠀⠀⠀⣀⠤⠈⠉⡴⢉⠈⡆
⠀⠀⠀⠀⠀⠀⡀⠀⠑⡄⠀⠀⠔⠁⠀⡀⠤⠀⠓⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠈⠀⠀⠀⡜⠁
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠃⡀⢀⠠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠂⠁⠀⠀⠀⠀⠀⡼⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠠⠠⠎⠈⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⡔
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠔⠂⠁⠀⠑

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠁⢈⣿⡄⠀⠀⠀⢠⣾⠋⠁⠀⠀⠈⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡀⢠⣸⣿⣇⣀⣤⣴⡿⣷⣇⡠⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡟⠋⠉⡡⢾⡇⣿⣯⠄⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠿⠛⠋⠀⠀⠀⠀⣻⠁⣿⣿⣏⠀⠀⠀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⠈⠀⠀⠂⠀⠀⠀⠑⣱⣿⡿⠋⠀⠀⠀⣀⣼⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠟⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣡⡿⢿⣿⠿⠛⠉⠀⠁⠛⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿⠀⠁⠀⠀⢠⡶⣶⡄⠀⠀⠀⠀⠀⠀⠀⠉⠶⣋⠀⠂⠀⠀⠈⠙⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠘⠀⠀⠀⠀⠘⠿⠿⠗⠀⠀⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⢉⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢈⡺⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡗⠠⡄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡞⠁⠀⠀⠀⠒⠒⣧⣈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⣤⣤⣤⣴⣶⣤⣩⣍⣤⣴⣶⡶⠟⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢆⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠉⠀⠊⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⢀⣱⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⣴⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⣦⡀⢀⠠⠀⠀⣠⠂⠀⡠⠀⡀⢠⢌⣿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⢀⠀⠛⢧⣇⢀⣠⣴⠃⠀⠀⠄⠄⢁⣼⠿⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡄⠄⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣶⢀⣠⣰⣿⡞⠋⠁⠀⢀⣠⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⢀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠛⠷⣿⣿⡃⡀⣠⡴⠋⠉⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⢿⣿⣿⣟⠛⠱⠀⠀⠀⠀⢃⠀⠀⢀⠀⠀⠀⠂⠀⠀⠀⠁⠀⢈⣵⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣼⣾⣿⣵⣷⣌⡆⠀⠀⠀⠄⠀⠀⠀⠀⠀⠠⠀⠂⠀⢀⣌⠄⢀⣠⣿⡿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣹⣟⠋⠙⢛⣛⣳⣶⣶⢟⡛⠛⠿⠾⢶⠆⢀⣀⣤⣼⣴⣾⡿⢛⣩⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠿⠷⠿⠶⠷⠶⠶⠾⣿⣛⣛⣛⣫⣭⣶⠾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠋⠉⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⠤⠤⠔⡫⠋⠀⠉⠙⠲⠤⣄⣠⡴⠞⠛⠉⠉⠉⠙⠛⠲⢦⣄
⠀⠀⠀⠀⠀⢀⡔⡲⣤⠞⠁⠉⠉⠉⠒⠊⠀⠀⠀⠠⠤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⡄⢰⠁⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⡰⠀⠂⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠁⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠒⠲⡄⡤⠒⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢇⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⡇⢰⣾⣣⣼⠀⣶⡖⣂⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⢈⠆⢀⣠⢿⡀⠀⠀⠀⠀⠀⠀⠳⢤⣩⠕⠻⡀⠙⠛⢃⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇
⠀⠀⢀⠔⣡⠖⠁⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⢠⢋⡜⠁⠀⣀⣀⠤⢺⡄⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢸
⠀⠧⢤⣭⠭⠭⠴⠒⠒⠙⡇⠀⠀⠀⠀⢀⠔⠋⠉⠉⠉⠉⠉⠉⢣⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⡇⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠉⠉⢳⠀⠀⠀⠀⠀⢀⡟⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠸⠃⠀⠀⠀⠀⢸⡇⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠀⠀⠀⠀⢸⡇⠀⡇⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡇⠘⠢⣀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡀⢀⡴⠛⠉⠙⣆⠀⠀⠀⠀⠀⠀⠀⣀⡤⢤⣄⠀⠀⠀⠀⣸⠃⠀⡇⠀⠀⠈⢶⢹⠁⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠏⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⣠⠞⠉⠀⠀⠈⠳⣄⠀⣠⠏⠀⠀⢀⠀⠀⠀⠈⠏⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⣀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠘⠒⠢⠤⠖⠲⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠛⠋⠉⠛⠳⢤⣀⡀⣀⣠⡴⠞⠛⠉⠛⠒⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡴⠚⠛⠙⠛⠶⢤⣤⣤⣤⡤⠶⠛⠉⠉⠉⠉⠳⣄⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⢸⡄⠀⢠⠏⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣠⣄⡀⠀⠀⠀⠀⠀⠈⣧⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⢷⠀⡾⠀⠀⠀⠀⠀⡞⠁⠀⠉⠉⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⡆⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢷⡇⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⣷⠃
⠀⠀⠀⠀⠀⣀⣤⣤⣬⣧⠀⠀⠀⠀⠀⢸⡇⢀⣀⠤⢤⣀⠀⡇⠀⠀⠀⠀⠀⠀⣰⣧⣤⣤⣀
⣀⣤⠶⠚⠋⠁⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⣷⠋⠀⠀⠀⠈⠳⡇⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠈⠙⠓⠶⣤
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣼⠁⠀⠀⢀⡀⠀⠀⠹⡄⠀⠀⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢰⡇⠀⢠⠞⠉⠉⠳⡄⠀⢻⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣸⠀⢀⡏⠀⠀⠀⠀⢻⠀⠘⡇⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⣧⠀⠀⢸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⡟⠀⢸⠇⠀⠀⠀⠀⢸⡇⠀⣿⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⣧⠀⣸⠀⠀⠀⠀⠀⠈⣇⠀⣿⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⣿⣰⠃⠀⠀⠀⠀⠀⠀⠹⣄⣿⣰⠏
⣿⡿⠛⠟⢿⣶⣿⣿⣿⣿⣿⣿⣶⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣷⣾⣷⣷⣿⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣶⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠙⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣼⣿⣤⣤⣤⣤⣤⣤⣤⣤⣿⣷⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣤⡄⠀⠀⣠⣄⠀⠀⣠⣄⠀⠀⢠⣤⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⣿⣿⠀⠀⣿⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⢿⡿⠀⠀⢿⡿⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣤⣤⣤⣀⣴⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣶⡟⢻⠛⣿⠹⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠟⠁⠄⢈⣧⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢱⣤⢠⣾⡿⢸⡿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣣⠁⡇⢿⡸⡿⠃⢾⣿⡌⠈⠁⠒⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣋⣈⣀⠀⠀⠋⣀⣤⡄⢤⣿⣆⠠⠄⠀⠀⠀⠙⠒⠒⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠹⠉⢠⣼⠃⢀⠈⠋⣾⣧⡀⠉⢿⣷⣴⣆⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠀⢁⣴⣿⣿⡴⠿⠃⢸⠛⣿⣿⠶⢫⠉⣛⣼⡇⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠇⣠⣾⣿⣿⣿⠁⠀⠀⠈⠀⢾⣿⣷⣦⡡⠼⠛⣿⡀⠀⠀⣀⠀⠀⠀⢘⣇
⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⣿⣏⡴⠿⣏⣿⣿⣿⠄⠉⠀⠀⠀⠈⣿⣿⣿⡷⢦⢀⣿⣿⠀⣸⣿⠆⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣷⠀⣿⣿⡟⠁⣀⣿⣿⣿⣿⠀⠀⣠⡴⠂⠀⠸⣙⣿⣷⢰⣏⣿⣿⠶⣿⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢻⡀⣿⣿⣠⣾⠟⢻⣿⣿⣿⡆⡈⠉⠀⠀⠀⠀⢯⣿⣿⠊⢙⣿⣿⢸⣿⣧⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠘⣷⢾⣿⠋⢁⣴⠿⠟⠛⠛⠋⠈⠁⠉⠀⠠⠶⣿⠿⣿⣆⣸⣿⣬⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⣿⣷⡟⠁⣀⣀⣤⣽⣿⣶⣶⣶⣶⣶⠀⠀⠒⢶⣿⣿⣿⣿⣿⡏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⣡⣾⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⣿⣿⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⢨⣿⣿⠃⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠿⠋⠁⠈⠉⠀⠀⠀⠀⠀⠀⠀⣽⣿⡇⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⢻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡿⣿⣿⣿⢧⡄⠀⠀⠀⠀⡄⠀⠀⠀⠿⣿⣿⣿⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⢈⣿⡄⢢⡌⠙⢶⠂⠂⣤⠀⣤⣴⢶⣿⣿⢿⠏⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⠀⠀⣨⡄⠻⠀⠲⣮⠉⣉⣁⡠⠌⣻⣾⣿⢟⣿⢰⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢿⡆⠀⠠⢷⣄⠈⢀⠨⡄⠠⠀⠲⣾⣿⣿⡇⢸⡏⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡖⠀⢘⣷⡆⠈⢣⣠⣤⣄⡰⢟⣿⡿⢡⣿⠰⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⣷⡟⠠⠃⠠⡤⠤⠤⣴⠿⠟⠁⣘⡤⠃⣠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⠛⣠⡶⠀⢐⣶⣶⣏⢀⣰⣾⠌⠀⠀⢩⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢿⣷⠖⠈⠻⠋⠛⣫⠝⠁⠀⠀⠀⠈⠛⢿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠛⠁

⠀⠀⠀⠐⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠈⠀⠈
⠀⠀⢈⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠏⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠉⠋⠉⠀⠈⠙⠛⠙⠋⠉⠈⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡿⠋⠉⠉⠉⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠀⠉⠉⠛⢿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢀⣄⠀⠀⢰⠀⢠⣶⡀⠀⠀⠀⢠⣶⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⣿⣿⡆⠀⢸⡇⢸⣿⣇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⣿⣧⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⣿⡸⣧⠀⢸⡅⠈⣿⣿⡀⠀⠀⣿⣿⡄⠀⢠⡄⠀⠈⣿⣿⡀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⡇⣿⡄⣾⡇⡀⣿⣿⡇⠀⠀⣿⣿⡇⠀⣸⣧⠀⢰⣿⣿⡇⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣷⣿⣷⣿⣸⣿⣿⠀⡄⢹⣿⡇⢠⣿⣿⣶⣾⣿⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣔⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢛⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢺⣿⣄⠀⣤⣆⠀⠀⠀⠀⣤⡄⠀⣠⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣾⣦⠈⠟⢃⠀⠀⠘⠻⠁⣼⣷⠌
⠀⠀⢀⢠⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢈⣁⡇⠸⣟⣀⣀⣻⠇⢨⣌⠁⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠈⠈⠛⠂⠉⠉⠘⠿⢯⣽⣶⣄⡀⢸⣿⣧⣶⣸⣿⣿⣇⣶⣾⣿⠇⢀⣤⣾⣯⠽⠿⠚⠉⠉⠘⠛⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣅⣱⡀⣿⣽⣿⣿⣿⣿⣿⣿⣽⣿⢠⣎⣨⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⢤⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣯⠤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣠⠰⣶⡍⠿⠛⠸⠿⢷⢿⡿⢟⣡⣘⠻⢿⣿⡽⠿⠇⠛⠿⢹⣶⠦⣄⡀
⠀⠀⠀⠀⠀⡠⢾⡿⠉⠁⠁⠀⠀⠀⠀⢀⣸⣧⣰⣿⣯⣽⣿⣂⣼⣅⠀⠀⠀⠀⠀⠉⠈⠙⢿⡷⢄
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠻⠿⡿⣿⣿⣿⢿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣋⢭⣾⡿⠛⠀⠘⢇⠻⠿⣿⣷⡭⣙⣥
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠃⣾⣿⣷⡇⠀⣾⡇⠀⠘⣿⣿⣷⠘⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡔⠰⣿⣿⠱⠠⠀⠙⣿⠀⠠⠈⣿⣿⠀⢂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠁⠀⢿⣯⡦⠀⡀⢀⣿⠀⠀⢰⣿⡿⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⠀⠈⢿⣿⣦⡀⠉⠛⢁⣴⣿⡿⠁⠀⠀⢢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠀⠀⠙⠻⢷⣾⣿⡿⠟⠋⠀⠀⠀⠀⠈⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣖⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠉⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠸⢿⣿⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣧⡀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠟⠋⠁⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⣾⡿⠟⠋⣹⣿⣿⣿⡿⣿⣷⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠉⢀⣠⣴⡖⠀⠀⠀⢀⡀⠈⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡿⠁⠈⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⣼⣿⣦⣤⣤⣤⣾⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡇⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠸⣿⣿⣿⣿⣏⠀⣹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣧⠀⠀⣴⣶⣶⣶⣶⣶⡆⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢀⠀⠀⣿⣿⡏⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⣿⣿⡀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣬⣭⠁⠀⣼⡇⠀⢿⣿⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠀⢹⣿⡇⠀⠸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⠀⠀⣾⣦⠀⠰⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠈⣿⡿⠀⠀⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠀⢠⣿⣿⡄⠀⣿⣿⡄⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠋⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⣤⣤⣄⡀⠀⠉⠑⠲⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣻⣿⣇⣨⣿⣶⠀⠀⠀⢹⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢰⡆⠈⣓⣪⡼⡛⣩⣄⠃⢄⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡼⣷⣤⣨⣀⠘⣟⢁⣼⣃⣼⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢤⣄⠀⠀⠀⠈⢻⣿⣿⣿⣿⣦⣿⣼⣿⣿⣷⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣶⣄⡈⠻⣿⣿⣿⣿⣿⠟⠋⢀⣠⣤⣤⡴⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⣿⣿⣿⣿⣿⣷⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠿⠿⣿⣾⢿⣿⣿⣿⢿⣿⣿⣶⣶⣾⠿⠿⠿⠿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠋⠀⠀⠀⠀⣴⣿⣿⣿⣴⣾⣏⣽⢿⣶⣶⣶⣶⣄⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⠀⠀⠀⢸⣿⠋⣿⡝⠛⢿⣏⣼⠂⠀⠀⠈⠉⠛⣷⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠟⠁⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢠⡴⠛⠛⠳⣦⠀⠀⠀⠀⠀⠀⢀⣠⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⢸⠇⣀⣀⡔⠀⡴⠋⠀⠀⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠸⡇⢀⠀⠀⠠⠘⢋⡉⠉⠉⠛⠧⠀⠀⠀⠀⠀⢹
⠀⠀⠀⣀⠀⠀⠀⠀⢳⣄⠑⠒⣡⢐⠋⠁⠀⠘⠷⡄⡘⠦⠤⠞⢀⡟
⠀⢀⣰⠀⣥⠂⡆⠀⠀⠉⠛⠋⢹⡅⠶⡀⠸⣘⣆⢼⠛⠶⠴⠖⠋
⠀⢣⠘⠆⠁⢸⣀⢀⡀⠀⠀⠀⢸⠈⠉⠀⠀⢈⡀⣸
⠀⠀⢁⠀⠈⠉⡡⠤⠼⠀⠀⠀⢸⡄⠀⠀⠀⠀⠰⡏
⠀⠀⠀⠙⡆⡞⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠃⡇⠀⠀⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⢠⣇
⠀⠀⠀⢸⠀⡇⠀⠀⠀⠀⠀⢀⡞⠈⣇⠀⠀⠀⢸⡟⣆⣀⡀
⠀⠀⠀⣈⠀⠓⠒⠒⢒⡒⢒⡞⠀⠀⠸⡄⠀⠀⢸⠁⠸⣦⣉⣉⡛⠓⠒⠒⢒⠒⣶
⠀⠀⠀⠈⠉⠉⠉⠁⠀⠀⡞⠀⠀⠀⠀⢷⠀⠀⢸⡇⡀⠘⣆⠀⠀⠉⠉⠉⡿⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠘⡆⠀⢸⠃⠂⠀⠘⣆⠀⠀⠀⢀⡇⡸
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣷⠀⢸⡇⠀⠀⠀⠈⢧⡀⢀⠼⠰⢇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢀⠀⣸⠶⢾⣇⠀⠀⠀⠀⠀⢳⣁⠀⠀⠀⠇⡈
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⡠⠖⠈⠁⠉⠈⣿⣶⣿⣿⡇⠀⠀⠀⠀⠀⢻⡝⠓⠒⠈
⠀⠀⠀⠀⠀⠀⣸⠁⢠⠊⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁⠈⠢⡀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⢀⡇⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⢸⡇⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠘⣇⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠛⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡍⢿⠛⠒⠒⠒⠚⢻⢻⠉⠀⠀⠱⡄⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⣄⡀
⡠⠒⡉⠉⠉⠉⠓⠒⠲⠤⠤⢤⡽⠜⣆⠀⠀⠀⢠⠏⢾⣤⣤⣤⣤⡽⠶⠾⠗⠒⠒⠒⠦⣀⠀⠀⣀⠴⢉⠴⠋⠉⠑⠢⢵
⠓⠶⣅⣠⡆⣰⠶⠤⢤⣀⣀⣀⣀⣀⡼⠀⠀⠀⠘⠲⠶⠤⠤⠤⡤⣤⣤⣤⢤⡀⠙⡆⢑⣏⣉⣉⣡⠜⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⡀⠀⠀⢹⣧⡀⠀⠀⠀⡀⡀⠀⠀⠀⣠⣿⠃⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠀⠀⠙⢿⣦⣀⣾⣷⣿⣆⣠⣾⠟⠁⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠷⣶⣶⣮⣿⣿⣿⣿⣿⣿⣿⣧⣶⣶⡶⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠉⣩⣽⣿⢳⣿⣿⣿⣿⣿⣿⣿⣿⣭⡉⠙⠛⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⢀⣴⡾⠟⠹⣿⢸⣿⣿⣿⣿⣿⣿⣿⡿⠙⠿⣶⣄⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡏⠀⠀⠀⢻⣆⠻⣿⣿⣿⣿⣿⣿⠃⠀⠀⠈⣿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢿⠀⠀⠀⠀⠀⠙⢷⣦⣭⣭⣿⠟⠁⠀⠀⠀⠀⢸⠆⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠛⠉⠉⠉⠛⠳⢦⡀⠀⢀⣀⣀⣀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⢀⠠⠀⠀⠀⠀⠀⠄⠀⠙⠿⣋⠉⠖⣀⠉⠙⠻⣷⣀⣀⣴⣿⣷⡀
⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡄⠀⠈⠀⠰⣿⣠⡀⠀⠀⠉⠁⠀⠉⢹⠃
⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠄⠀⠀⠀⠄⠐⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⢰⡆⣹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣀⣀⣤⠴⠋⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⠗⠲⠤⠤⠴⠒⠊⠁⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⡼⠃⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⢸⢠⡏
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠸⡏⠻⢦⣀
⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⡀⠀⠀⠀⠹⣄⠀⢹⡆
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡷⢦⣄⡀⠘⣿⠾⠃
⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⠈⣷⣀⡿
⠀⠀⠀⠀⠀⣼⠃⢀⣤⣤⣤⣤⣤⣤⣤⣶⣊⣉⣍⣿⡆⠀⠀⠈⠉
⠀⠀⠀⠀⠀⣏⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⣋⣉⣭⣿⣦
⠀⠀⠀⠀⠀⠹⣆⡈⠛⠦⣤⣀⣀⠀⠀⠀⣀⣀⣤⠴⠞⣋⣥⠶⠛⠉⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⣉⠉⠉⠉⢉⣉⣡⡤⠖⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⣀⠤⠤⠤⠶⠶⠶⠶⠶⠦⠤⠤⠤⢤⣤⣀⡀
⠀⠀⠀⢰⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣙⣛⠒⠶⠤⢤⣤⣀⣀⡀
⠀⠀⠀⠈⠛⡮⢭⡵⣖⣒⣒⣒⣶⣶⣶⣒⣒⡯⡽⢶⠒⠒⠫⠭⠝⣛⣒⠶⠦⠭⣭⣍⣛⣛⣒⠲⠶⠶⠦⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠶⠖⠚⢛⣧
⠀⠀⠀⣀⢀⠗⠘⠇⣰⣧⡆⠉⠉⠙⣦⠀⣗⠉⠙⢺⣤⠴⠖⠒⠶⠦⣄⡉⠉⠙⠒⠒⠢⠭⠭⢭⣉⣛⣛⡛⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠒⠛⠛⢉⡟
⠀⢠⠋⠀⠸⣄⠀⡼⠁⢀⠀⢀⠀⢰⣿⠀⠈⡀⠒⢬⡇⠀⠀⠀⠀⠀⠀⠙⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠒⠒⠶⠶⠦⠤⠤⢤⣤⣤⣤⣤⡤⠶⠋
⠀⢧⡀⠀⠀⠀⣿⡇⠀⠀⢸⡁⠀⠀⠉⠃⠈⢵⡄⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⣸⠿⣶⣤⣤⡼⠿⣆⣀⣀⠽⢶⠀⠀⠀⢀⡾⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⣿⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⣄⠀⠀⠀⠀⠀⠀⠀⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⡿
⠀⢹⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠘⠢⣄⡀⠀⠀⠀⢀⣠⠤⠊⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⣷⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⢀⡟
⠀⠀⠀⠘⠧⣤⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠴⠋

⠀⠀⠀⠀⠀⡠⠔⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⡈
⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠂⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠈
⠀⢀⠔⠒⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠒⠲⡀⠈⠄⣀⠠⠄
⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠂⠀⠉⠉⠀⠀⠒⠀⠀⠀⠀⠄
⠀⠘⢦⡤⡤⡤⣤⠤⢤⣤⣤⣤⣤⣤⣤⣤⡤⠤⢤⣤⣤⢤⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀
⠀⠀⠀⡇⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⠀⠀⠀⠀⢁⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⡇⠀⠀⡇⠀⢠⠦⢠⠀⢠⣆⢰⠆⣴⠄⠀⢸⠀⢸⠀⠀⠀⠀⢠⠄⠒⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⢰
⠀⠀⠀⢡⠀⠀⡇⠀⢸⣀⢸⠀⡼⢿⢈⡆⡘⡆⠀⢸⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠸⡀⠀⠙⠤⠤⠉⠌⠁⢁⢈⠈⠁⠉⠥⠤⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠶⢶⣄⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⢀⣠⣧⣄⡀⠀⠀⢸⣥⠇⠀⠀⠫⡜⠀⠀⠀⣠⣼⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⢸⠀⠀⢠⡇⠀⠀⢸⡅⠀⠀⠀⠀⣿⠇⠀⢸⡅⠀⠀⡇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⢸⠀⠀⠰⣇⠀⠀⠈⠛⠗⠮⡹⠟⠋⠀⠀⣸⠇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠘⠦⠤⠼⠃⠉⠁⠒⠒⠒⠒⠒⠒⠒⠈⠉⠘⠧⠤⠴⠃⠀⠀⠀⠐⡀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⠂⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠀⠀⠀⠈⠀⠈⠉⠉⠉⠀⠁⠐⠤⢄⡀⠀⠀⠀⠀⢀⡠⠄⠀⠀⠈⠉⠉⠁⠀⠁⠐⠒⠂⠤⢀⣀
⠀⠀⠀⠀⠀⢀⡠⠔⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠂⠤⢀⡀
⠀⠀⠀⣴⠊⢤⠄⠀⢤⠆⠀⢤⢤⣤⣤⣀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡠⣤⡀⡤⠠⠤⡤⠤⠤⡬⠒⣦
⠀⠀⠀⠹⡑⢆⡀⠀⡎⠀⠀⠰⡏⠀⠀⠈⠆⠠⠚⠛⠒⢶⠀⠤⢤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡤⠤⠄⡶⠒⠛⠓⢄⡠⠉⠀⠀⢸⠆⠀⠀⢱⠀⠀⣰⡾⠃
⠀⠀⠀⠀⠑⡀⠙⢢⡅⠀⠀⢠⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⢃⠔⠉⠀⠀⠈⠑⡄⢠⠊⠁⠀⠀⠉⠢⡘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⡄⠀⠀⢸⡤⠊⡰⠁
⠀⠀⠀⠀⠀⠐⡄⠀⠈⠢⣄⣼⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣧⣀⡠⠊⠀⡜
⠀⠀⠀⠀⠀⠀⠈⢄⠀⠀⠈⠙⢄⡀⠤⠚⡇⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢸⠒⠤⢀⡠⠟⠁⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠈⠲⢄⡣⠄⢀⣀⣀⣰⠁⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠈⣆⣀⣀⡀⠠⢜⡠⠔⠈⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠈⠒⠤⣀⠀⠀⢆⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⡰⠀⢀⡀⠄⠊⠁⠀⠀⠀⠀⣠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⠬⠉⣀⡐⠒⠀⠁⠈⠐⢒⠀⠀⠨⠍⠒⠈⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠦⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠄⠒⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣀⣀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠢⠄⣀⠀⠀⣀⠠⠔⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂⠤⢀⡀
⠀⠀⠀⠀⢀⠠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠠⠄⡀
⠀⠀⠀⣾⡉⣚⣀⡀⠻⠃⠐⠛⡰⠖⠶⢤⠠⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠠⠤⣤⠄⠲⢄⠒⠐⠒⡖⢒⣒⡒⢉⡿
⠀⠀⠀⠘⡌⠻⢕⠛⣧⠄⣀⡘⠇⠀⠀⠀⢣⠔⠉⠉⠉⠻⡀⠒⡲⠤⠤⢤⡤⠤⠤⢤⡤⠤⠴⢖⠒⢀⠟⠉⠉⠉⠢⡐⠀⠀⠀⠸⢃⣀⠤⣼⠛⢊⠝⡟⠁
⠀⠀⠀⠀⠈⠄⠈⠑⢏⠉⠚⢹⢖⠒⠰⣄⡸⠀⠀⠀⠀⠀⣡⠊⠀⠀⠀⠀⠈⢆⡰⠁⠀⠀⠀⠀⠑⣌⠀⠀⠀⠀⠀⢇⣠⠆⠖⡲⡏⠃⠉⢹⠖⢁⠌
⠀⠀⠀⠀⠀⠈⢆⠀⠀⠑⠦⣼⠈⠀⠙⢁⡟⡎⡑⠒⢲⣀⡏⢀⣤⠀⠀⣀⠀⠘⠇⠀⡀⠀⠀⣠⡀⢩⣀⡖⠒⢊⢱⢻⡈⠉⠀⠁⣷⡤⠒⠁⢀⠎
⠀⠀⠀⠀⠀⠀⠀⢢⡀⠀⠀⠈⠲⢤⠂⠉⡇⠀⠀⠙⠂⠀⡍⢙⢀⡉⢉⠁⠦⠰⠆⠴⠈⡉⢉⣀⡏⢩⠀⠑⠋⠀⠀⢠⠉⠐⣠⠔⠋⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠑⣄⠀⠀⠀⠀⠙⠢⢕⠂⠠⠤⠤⠼⠀⠀⠁⠀⠈⠉⠀⢠⡄⠀⠉⠁⠀⠈⠀⠀⠧⠤⠤⠄⢐⡪⠔⠊⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠈⠒⠠⢀⡀⠣⢀⣀⠀⠀⠀⠀⡘⢃⠀⠀⠀⠀⣀⡀⠜⢀⠠⠔⠂⠁⠀⠀⠀⠀⢀⡔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠒⠀⠠⠤⠀⠀⠠⠀⠀⠀⠤⠀⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⢀⠴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠠⠤⢀⣀⣀⣀⣀⣀⣀⣀⡀⠠⠤⠐⠂⠉⠁

⣶⣶⣶⣶⣶⡾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣷⣶⣾⣿⣷⣶⣶⣶⣿⣿⣿⡿⣶⠿⠶⠷⠾⣶⡾⠷⠷⠾⠷⠶⠶⡶⠷⠶⠶⠷⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿⣶⣄⡀
⠀⠀⠀⢀⣴⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣿⣿⣦⡀
⠀⠀⢠⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⣶⣤⣤⣤⣤⣄⣠⣤⣤⣤⣤⣶⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⡄
⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡄
⠀⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⢸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣀⣠⡤
⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣿⠁⠀⠀⠰⣶⣿⡿⠋
⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠛⠛⠋⠁⠀⠀⠀⣠⣾⡿⠿⠃
⠀⠀⠈⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠚⠉⠁
⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡏
⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣀⣀⣿⣶⣄⡀
⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡟⠛⠛⠶
⠀⠀⠀⠀⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⡆⠀⠀⠀⢲⣄⠀⢀⡀
⠀⠀⠀⠀⠀⠸⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣶⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⢻⣿⣾⣷⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠈⠿⠋⠻⣷⡀
⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠇⠸⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠀⠀⢹⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡆⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣧⡀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡄⠀⠀⠀⢀⣼⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣶⣶⣿⣿⠟⠀⠀⠀⠀⠀⠀⠻⣿⣿⣶⣦⣶⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁
⣿⣿⣿⣿⣿⣿⢻⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣷⣿⣾⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣷⡿⣷⠿⠿⠿⠿⢿⣿⠷⠿⡿⠿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀⣂⣐⠒⠒⣢⢱⡀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⡂⠁⠀⡠⠀⠀⡠⠂⠀⠀⠐⢄⠀⠠⢀⠀⠀⢂⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⢛⠁⠒⠂⠀⠀⠀⠤⠤⠬⠤⠤⠤⠤⠤⠤⠥⠤⠤⠀⠀⠐⠒⠒⠈⡓⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠠⠀⠈⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠤⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⠄⡀⠀⠀⣀⡀⠀⠀⠐⠀⠐⠂⠀⠀⣀⣀⠀⠀⢀⠤⢤⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠂⡀⠂⢤⠀⠀⠀⡇⠀⠈⠀⠀⠇⠀⢸⠀⠀⠆⠀⠀⠀⠀⠃⠀⠀⠀⢠⠀⡤⠐⢂⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢃⠈⠲⠀⡀⠀⡇⠀⠀⠀⠀⠄⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠖⠁⡘⠠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡑⢤⡀⡇⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢊⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠄⠁⠀⡇⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠆⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⡇⠀⠀⠀⠀⡆⠀⢸⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠀⡇⠀⠀⠀⠀⡇⠀⠘⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⡄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡆⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡇⠀⠀⡇⠀⡁⠀⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⡘⠤⠇⠀⠀⡇⠀⡄⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠠⠄⢁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠄⣁⠒⠤⠀⠁⠒⠁⠀⠀⠄⠸⠀⠀⠈⠒⠁⡀⠤⠒⣈⠤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⢈⡑⠂⠀⠠⠌⠉⠉⠁⠈⠉⠉⠥⠄⠀⠐⢈⠡⠄⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠐⠒⠒⠒⠒⠒⠀⠀⠀⠀⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠤⣄⣴⣶⣶⣄⢺⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⡄⢰⠃⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⣀⣸⣿⣿⣿⣿⣏⣾⢛⣷⠀⠀⠀⠀⠀⠀⠀⠀⣀⣹⣾⣜⡁⠀⣀⣠⠤⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⢀⡾⠉⠙⣿⣿⣿⣿⣿⡷⠟⠙⠳⡄⠀⠀⢀⣠⠴⣾⣿⣿⣿⣿⣷⣛⣁⠀⠀⠀⠀⠀⣀⣠⢴⣶⡴⡀⡼⠁⠀⠀⢀⣠⠤⠒⠒⠉⠉
⠀⠈⢀⡶⠟⣿⣿⣿⣿⣿⡷⣦⡀⠀⠙⠤⢀⡾⠀⠀⣿⣿⣿⠟⠹⣆⠀⠈⠓⠦⡀⠀⠸⠁⠀⢃⣴⣷⣿⣿⣦⣴⠾⠭⠤⠴⠂⠀⠠⠄
⠀⢀⡞⠀⠀⣽⠉⡟⠋⠛⣇⠈⢿⡄⠀⠀⠘⠁⠀⠀⠀⣠⠟⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠛⢳
⠀⡞⠀⠀⠀⣿⠀⡇⠀⠀⣿⠀⠀⢿⠀⠀⠀⠀⠐⠛⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠛⢹⡆⠀⠳⡀
⠀⠀⠀⠀⠀⣟⠀⠀⠀⠀⠛⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠠⡆
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢀⣀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⡄⠀⠀⢰⠓⡀
⠀⠀⡰⢤⡈⣆⢰⣿⣿⣿⣆⢸⠀⡰⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣶⣿⣿⣧⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠈⡇⠀⠀⡿⠀⢱⡀
⠀⠀⠁⠀⠹⣼⡸⣿⣿⣿⠏⣜⡼⠁⠀⠀⠀⠀⠀⠀⠐⠢⢤⣀⣈⢿⣿⣿⣿⠇⣀⡤⠤⢄⣀⡀⠄⠀⠀⠀⠀⢀⡿⣅⠠⣃⣄⡄⡇⢀⡽⣇
⠀⠀⠀⢀⡴⣾⣿⣿⣿⣿⣾⣿⡴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣉⣻⣿⣿⣷⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⢈⣻⣿⣿⣿⣿⡋⢀⣼⠂
⠀⢀⠔⠁⠀⡇⢉⠟⠋⠙⢏⠀⡇⠈⠳⡀⠀⠀⠀⠀⠀⣠⠞⢋⡽⢿⣿⣿⠿⡏⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠡⣿⣿⣿⡏⢹⠛⠁
⠀⡜⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠀⠀⠀⣠⠞⠁⠀⣸⠃⠈⠀⠈⠀⡇⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠻⣿⠟⠀⠘⡄
⠀⠁⠀⠀⠀⠹⡀⠀⠀⠀⠀⢀⠁⠀⠀⠸⠀⠀⠈⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠸⡀⠀⠀⠈⢄⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠙⣆⣠⣤⣤⣤⣤⣤⣤⣤⣟⡀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⢀⣠⡶⠟⠋⠁⢠⠖⠲⠀⠔⢢⡈⠉⠛⠷⣦⣄⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠟⠁⠀⠀⠀⠀⣁⣤⣤⣄⡀⠀⠧⠀⠀⠀⠈⠻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣠⣤⠤⠼⠿⠤⣄⣿⣇⣄⡀⠀⠀⠀⠀⠈⢿⡆⠀⢀⣀⠠
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⣾⡇⠀⠀⠀⢸⣏⠀⢀⠀⠀⠀⠈⠁⠀⠀⣹⠆⠀⠀⠀⠀⠈⣿⠀⠉
⠀⠀⠀⢀⣀⠀⠉⠉⠉⠉⠉⢿⡏⠀⠀⠀⠀⠙⠷⣾⣤⣄⣠⣤⡾⠖⠛⠉⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⢿⠙⠲⠤⣄⣀⡀⢘⣿⡄⠀⠀⠀⠀⠀⠀⠀⣈⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢛⡿⠓⠒⣂⣤
⠀⠀⠀⠀⠘⡇⠀⡦⠤⣌⣉⡙⠛⠛⠶⠤⣀⡀⠀⠀⠀⠠⣭⠉⠁⢀⣀⡤⠤⠤⠴⠶⢶⠚⣛⣯⠉⣱⠃
⠀⠀⠀⠀⠀⢿⠀⠁⠀⠀⠈⣽⠋⠙⠒⠲⠶⢬⣝⣓⠶⢤⡴⠖⣻⣭⣠⡄⢲⢀⠟⣦⡸⠀⢿⠿⢀⡏
⠀⠀⠀⠀⡠⠛⠁⢧⣤⣡⣴⣃⠀⠀⠀⠀⠀⠠⢞⣿⠀⢸⡇⠘⠿⣮⠟⠗⢛⡀⡉⠩⢴⠖⢀⣀⣼⡀
⠀⠀⠀⢸⠁⠀⠴⠿⣤⠤⠤⡤⠉⠉⠉⣓⣛⠓⠺⠟⠀⢸⡇⠀⠔⠒⠚⢉⡽⠀⡇⠀⠀⢈⣼⠉⠀⠙⡆⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⢹⠀⢀⡀⠀⣸⠃⠉⠁⠀⠀⠀⠂⠄⠉⠍⠀⠀⢸⡇⠀⠀⠀⠠⠟⢠⠈⡇⠀⠀⢰⠋⠉⠀⠀⢸⠀⠀⠀⢠⡞⠉⠉⠙⣦
⠀⠀⠀⠈⢧⡀⠉⠙⣯⣍⣀⣀⠀⠀⠀⡠⠌⠉⠀⠀⠀⢸⡇⡃⠀⠠⠀⢀⠁⣼⡇⢀⡅⠺⣦⠄⠀⠀⡼⠀⠀⠀⠘⣇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠙⢲⠚⠡⣶⣀⣒⠄⠀⠀⢉⡒⠒⡀⠀⠀⢸⡇⡇⠀⠀⠀⠈⠾⡉⡇⠀⠢⢽⣇⠀⢀⡴⠃⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⡇⠀⣦⡭⣩⠍⠀⠀⢘⣚⡿⠯⠄⠀⢸⡇⢠⡤⠖⠒⠉⠁⢀⡀⠀⠩⣁⡟⢻⠛⢓⣒⣶⣶⣶⡾⠣⣀⠀⣀⣼⠳⣆
⠀⠀⠀⠀⠀⠀⠀⣧⠀⡒⠒⠉⠄⠀⠀⢈⣐⣒⡛⠀⠀⠘⡇⢀⡠⢤⣶⣾⣿⠿⠇⠀⡈⢤⡓⢸⣻⢻⠀⠹⡀⠀⠑⢄⢈⣡⣿⡷⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⢹⠀⣽⠃⠩⠁⠀⠀⠐⡦⠤⠔⠀⠀⠀⡇⠀⠊⣉⡩⠤⠒⣚⡉⠀⠈⠉⠀⢸⣿⡀⠑⣴⠿⠖⠒⠛⢋⡵⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠀⣠⠼⡀⠻⠫⠭⠤⠀⠀⢐⣓⣒⡈⠤⠀⠀⡇⠀⠅⠒⠚⠉⠉⢀⣀⡤⠴⠶⣶⠛⢹⡇⠀⢻⠖⠋⣉⠿⠋⣠⡾⡿⠁
⠀⠀⠀⠀⣠⠞⢁⣴⠟⣲⣶⣤⡤⣤⣀⣀⣀⣀⣉⣁⣀⡀⢹⠀⣀⣤⠴⠒⠋⠉⠀⠀⠀⣸⡏⢀⡼⠁⠀⣼⣠⠞⠁⣠⠞⠋⣼⠃
⠀⢀⣴⣋⣁⣴⣋⣵⣋⣀⣈⣿⣧⣀⣀⣀⣀⣀⣀⣀⣀⣬⣹⣯⣥⣤⣤⣤⣤⣤⣤⣤⣴⣿⣿⣭⣤⣤⣴⡟⠁⣠⠞⠁⠀⣰⠇
⠀⣿⡉⢉⠉⢉⣉⣉⣉⣉⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣨⣷⠞⠁⠀⠀⢠⡏
⠀⠉⠹⣏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠁⠀⠀⠀⣠⡟
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⣠⠾⠋
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⣀⡴⠟⠁
⠀⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⠾⠋
⠀⠀⠀⠀⠀⠀⠹⢶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢃⠀⠀⠀⠀⡠⠚⡟⢢⡀⠀⠀⠀⢠⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⣸⠀⠀⠀⣐⠁⠸⡿⠀⢑⠀⠀⠀⢸⡸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣻⢦⡀⠀⠹⣧⢨⣯⢠⡟⠁⠀⣠⠾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡙⢦⠀⢿⡘⡇⣸⠃⢠⠞⣡⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣮⣷⣼⣷⢴⣿⣴⣧⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⣀⣀⡸⣿⠛⡉⠩⡙⢻⡿⣀⣀⣀⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣆⠾⠙⠛⠙⠉⠀⠙⠉⣹⣿⣟⡶⠰⣍⣿⣿⠋⠙⠁⠈⠉⠙⠛⠹⢆⣶⣄
⠀⠀⠀⠀⠀⠀⠀⡄⠁⠀⠀⠀⠀⠀⠀⠀⢀⡤⢾⡾⣿⢛⡄⠀⣜⣻⡿⣾⠦⣀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠾⠋⢀⡿⢃⢈⣿⣿⣸⣿⠏⢀⠻⣇⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡴⠂⢀⡾⢡⠃⣾⠉⠻⡿⠋⢺⠆⢣⠹⣆⠀⠢⣼⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡋⠀⠀⠀⢨⣿⠋⠀⠘⠀⠀⠀⠀⠘⠀⠈⢳⣯⠀⠀⠀⠈⣣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⠉⠀⠀⠀⢀⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⠀⠈⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⣘⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣛⡀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸




⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠛⠳⢶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠉⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⢠⣤⣤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠈⠉⠙⠛⠷⠶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠶⣶⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⣶⣤⣀⠀⠀⠀⣀⣠⣤⡦⠀⠀⠀⠀⠉⠉⢻⣿⠷⢶⣦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⢸⠀⠉⠙⠛⠻⠿⠷⠾⢷⡶⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠈⣻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⣀⣀⣀⣤⠟⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⢀⣼⣏⣀⣤⣴⠶⠞⠛⠛⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠻⣅⡀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⠀⠘⠛⠋⠉⠁⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠒⠶⠶⢤⣀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠠⡿⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣈
⠾⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉
⠀⠀⢻⣿⣿⠉⠉⠉⣿⣿⡏⠉⠉⠉⢹⣿⡏⠉⠉⠉⠉⣿⣿⠉⠉⢉⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠛⠛⠛⠛⠿⠿⠿⠿⠶⠶⠶⣶
⠀⠀⢸⣿⣿⣦⣤⣤⣿⣿⣧⣤⣤⣤⣼⣿⣧⣤⣤⣤⣴⣿⣿⣤⣤⣼⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢸⣿⣿⡟⠛⠛⢿⣿⡿⠛⠛⠛⢻⣿⡿⠛⠛⠛⢻⣿⡿⠛⠛⢻⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠘⣿⣿⡇⠀⠀⢸⣿⣷⠀⠀⠀⢸⣿⡇⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⣿⣿⣿⣶⣶⣾⣿⣿⣶⣶⣶⣾⣿⣷⣶⣶⣶⣾⣿⣷⣶⣶⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⣿⣿⣿⠉⠉⠉⣿⣿⠉⠉⠉⢹⣿⡏⠉⠉⠉⣿⣿⠋⠉⠉⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⢻⣿⣿⡀⠀⠀⣿⣿⡄⠀⠀⢸⣿⣇⠀⠀⢀⣿⣿⡀⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢠⣦⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣾⢿⡇⠀⠀⠀⠀⣿⠁
⠀⠀⠀⢸⣿⣿⡇⠀⠀⢸⣿⡇⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣼⠇⠸⣷⠀⠀⠀⠀⣿
⠀⠀⠀⠘⣿⣿⣧⣤⣤⣼⣿⣧⣤⣤⣼⣿⣧⣤⣤⣼⣿⣧⣤⣤⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⣰⡟⠀⠀⢿⡆⠀⠀⠀⣿
⠀⠀⠀⠀⣿⣿⣿⠛⠛⠻⣿⣿⠛⠛⢻⣿⡟⠛⠛⣿⣿⡟⠛⠛⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢠⡿⠀⠀⠀⠸⣷⠀⠀⢸⡟
⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⣿⣿⠀⠀⢸⣿⡇⠀⠀⣿⣿⡇⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣾⠃⠀⠀⠀⠀⣿⡄⠀⢸⡇
⠀⠀⠀⠀⢻⣿⣿⣶⣶⣶⣿⣿⣶⣶⣾⣿⣷⣶⣶⣿⣿⣶⣶⣶⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣼⠏⠀⠀⠀⠀⠀⢸⣇⠀⢸⡇
⠀⠀⠀⠀⢸⣿⣿⡏⠉⠉⣿⣿⡏⠉⢹⣿⡟⠉⢹⣿⣿⠉⠉⢙⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢰⡟⠀⠀⠀⠀⠀⠀⠈⣿⠀⣿⠇
⠀⠀⠀⠀⢸⣿⣿⣇⣀⣀⣼⣿⣇⣀⣸⣿⣇⣀⣸⣿⣿⣀⣀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡿⠁⠀⠀⠀⠀⠀⠀⠀⢻⡇⣿
⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿
⠀⠀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣀⡀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⡉⠉⠛⠛⠷⢶⣦⣴⣟⣉⠁⠈⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠷⢶⣦⣤⣀⣈⠉⠙⠛⠻⠶⣶⣽⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠶⢶⣤⣤⣀⡉⠉⠛⠛⠷⠶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠷⢶⣦⣤⣀⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⣿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⢀⡀⠀⠀⣀⠀⠀⣀⡀⠀⢀⡀⠀⠀⣀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠸⣿⠀⠀⣿⠀⠀⣿⡇⠀⢸⡇⠀⢸⡟⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⣿⠀⠀⣿⠀⠀⣿⡇⠀⣸⡇⠀⢸⡇⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⡀⠀⣿⡀⠀⣿⡇⠀⣿⡇⠀⣼⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢻⡇⠀⣿⡇⠀⣿⡇⠀⣿⠃⠀⣿⠁⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⠀⠀⢸⡇⠀⢸⡇⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢸⣇⠀⢸⡇⠀⣿⡇⠀⣿⠀⢸⣭⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⣿⠀⢸⡇⠀⣿⡇⠀⣿⠀⢸⡗⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠀⣿⠀⢸⡇⠀⣿⡇⢀⣿⠀⢸⡇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠿⠀⠸⠗⠀⠿⠇⠸⠿⠀⠾⠇⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⢰⠊⠁⢨⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠐⣿⡿⠀⠀⣰⣿⣫⡃⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠠⠐⠼⠇⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⣰⡟⢠⣿⣤⣴⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠉⢇⣀⡈⣻⣿⣽⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⢈⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠿⠄⢀⣀⡀⣀⣼⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢻⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠾⣷⣶⡔⠲⠶⣶⠂
⠀⠀⠀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⣿⣀⣠⠾⣿⣉⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⢼⣶⣿⣯⠀⠀⠀⢸
⠀⠘⠉⠁⠀⠀⠈⠉⠉⠉⠙⠛⠛⠓⠒⠒⠛⠉⠉⠀⠙⠳⣄⣀⡀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣄⠘⣿⣿⠀⠿⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⡴⠶⠖⠛⠛⠻⣿⡿⠿⣟⣿⣷⣴⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠒⠒⠒⠲⠶⠶⠶⠶⠶⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠘⠿⠂⠀⠀⠉⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠖⠛⠒⠦⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠶⢶⣄⠀⠀⠀⣼⣿⠁⠀⠀⠀⢠⡶⢶⣯⠙⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⣴⣿⠟⢀⠀⠀⠀⠀⠻⢿⠇⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣦⣼⣦⣐⣧⡀⣤⠀⣄⣓⣻⣧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣓⠠⢀⣠⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣏⣉⠙⠛⠛⠛⠿⠿⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡙⠀⠛⠀⣼⣟⡀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢶⣦⠄⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣴⠆⠀⣀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣨⡯⢴⣿⡇⠀⠀⣀⣸⣶⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠛⠀⠉⠀⠀⠙⠻⣷⣿⡿⣿⣷⣿⣿⣷⣄⣀⣀⣀⣀⣠⠴⠛⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠶⠛⠛⣿⡿⠛⠋⠉⠉⠉⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠄⠐⠈⠉⠃⠠⠄⠤⠀⠈⠁⠉⠀⠀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⡀⠐⠀⠐⡀⠰⠀⠂⠐⠀⠠⠠⠀⡀⠈⢀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠦⠀⠀⠠⠈⠁⠀⠀⠔⠀⠀⠀⠀⢄⠀⠈⠲⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡘⠀⠀⠐⠁⠀⢀⠀⠀⣀⠀⠀⠘⣀⠀⠈⡄⠀⠠
⠀⠀⠀⠀⠀⠀⢀⠀⠐⡆⠀⠀⡁⡀⠸⠂⠠⡊⠀⠀⠀⠀⠈⠂⡀⠘⣀⠄⠐⠀⠘⠃
⠀⠀⠀⢀⠔⠀⠀⠀⠀⢡⠀⠀⡁⠁⡀⢀⠁⠐⠀⠀⠀⠀⠀⢀⠴⠀⡎⠀⡎⡀⡈⠀⠀⠀⠀⠂
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠄⠠⠈⠄⠁⢀⠀⢢⠔⠈⠘⠰⡄⠀⡀⠄⠂⠰⠄⡀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠘⠐⠂⠐⠀⠤⢀⠃⠀⠀⠀⠀⠰⠊⠀⣇⢀⠁⢠⠁⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠉⡀⠈⡇⡆⡒⠀⠀⠀⠀⠀⠀⢃⠀⢺⢈⠁⠆
⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠁⠤⣽⡤⠁⠀⠀⠀⠀⠀⠀⠈⠤⢺⡦⠊⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⣠⠀⠀⠀⠀⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⣠⠄⠛⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠁⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⢸⡇⠀⠀⠀⠀⣠⡇⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⣹⠆⠀⠀⠰⣏⠀⠰⡏
⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸⡁⠀⠀⠀⠀⠙⣦⠀⠉⠢
⠀⠀⠀⠀⠀⠀⣸⠃⠀⢠⠋⠀⠀⠀⠀⠀⣠⢸⡆⠀⠀⠀⠀⠈⡇⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⣦⠄
⠀⠀⠀⠀⠀⠠⠁⠀⠀⢸⠀⠀⠀⢀⣠⠞⠁⠀⠳⢤⣄⡀⠀⠀⠁⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⣀⣹⠦⢤⣄⡀⠀⠀⠈⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠙⠲⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⡅⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠚⠻⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡇⠘⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⢀⣴⢟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣀⣠⢿⣀
⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠲⠶⠤⠤⠤⠶⠶⠚⠛⠉⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⣏⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠄⢸⡆
⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⠀⠀⠄⠘⢀⣾⢁
⠀⠀⠀⠀⠀⠠⡂⢻⣦⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠟⡱⢃⠞⣠⠂
⠀⠀⠀⠀⢀⡩⠒⣁⠜⠛⢳⠶⣦⣤⣤⣤⣄⣀⣀⣠⣤⣤⣤⣤⠴⢶⠞⣻⠟⡱⢣⠎⡔⠁⠴⠁⠐
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠐⠁⠘⠁⠈⠀⠋⠀⠉⠀⠋⠐⠁⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⡿⣿⠓⠆
⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⢻⠁⠀⡟
⠐⠉⡟⠛⠶⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠛⢩⡇⢀⡼⠀⠀⡇
⠀⠀⣧⠀⠀⣧⠀⠉⢹⠛⠒⢶⡤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⠤⠶⡾⠛⠉⠀⠀⣿⠀⢠⡷⠋⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠘⠳⢤⣼⠀⠀⢸⠀⠀⠀⠈⣿⠉⠉⠉⠛⠛⡟⠛⠛⠛⠛⢻⠛⠛⠉⠉⠉⣿⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠶⠋⠀⠀⠀⠀⢸⠃
⠀⠀⢹⠀⠀⠀⠀⠀⠈⠳⠦⣼⡄⠀⠀⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⣇⣀⣤⠶⠛⠁⠀⠀⠀⠀⠀⠀⣾
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣿⡀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣸⣤⣤⡤⠴⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠉⠛⠛⠒⠷⢤⡤⠤⠤⠾⠲⡞⠛⠋⠉⠁⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠶⠋⠀⠀⠀⠀⣀⣀⣤⡤⠿⠶⠶⠶⠶⠶⠶⠦⠤⣤⣼⣁
⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣄
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⢀⣠⠴⠒⠋⠉⠉⠉⠉⠉⠁⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⣰⠞⠁⢀⡠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢹⠟⠁⣀⣰⣋⣀⣤⠤⠴⠤⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⣠⠤⠤⠴⢶⡞⠛⠛⠛⠛⣿⠉⠉⠉⢹⠛⠋⠉⢹⡖⠛⠛⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣟⠀⠀⠀⠘⡇⠀⠀⠀⠀⢻⠀⠀⠀⢸⡆⠀⠀⢸⡇⠀⢀⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣤⣀⣷⠀⠀⠀⠀⢸⡄⠀⠀⢈⣇⣀⣤⡼⠷⠛⠉⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠶⢤⣤⣤⣤⣤⣤⣤⡤⠶⠖⠋⠉
⣿⣿⡟⠛⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣾⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠇⠀⠀⣀⣤⣤⣤⣀⣀⣀⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⡻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡌⠛⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠉⢻⣿⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠀⡿⠁⡤⠀⠀⡍⢻⣿⠀⠀⠀⠀⣿⣧⠀⠀⠀⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣤⡟⠠⢀⣇⠀⠀⠉⠀⠸⡆⠹⣷⠲⠶⠴⠿⣧⣤⣀⡀⠈⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⠁⠀⢘⣧⠀⡇⠂⠀⢠⣼⠀⢻⠀⠀⠀⠀⠀⠈⠉⠙⠓⠻⠷⣦⣄⡀⠀⠀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠁⣄⣿⡄⠹⠀⣀⠾⣾⠁⢸⠇⠀⠀⠀⠅⠀⠀⠄⠀⠀⠀⠀⠉⠛⠻⠶⣤⣄⡔⡡⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠀⠠⠀⠐⠾⠳⣦⡀⠀⠀⠇⠀⠸⠧⠀⠁⠂⠀⢡⣤⡄⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠐⢂⡈⠅⠀⢹⡷⠀⠈⠳⠄⠀⠀⠀⠀⠀⠸⢿⣿⠟⠀⠀⠀⠀⠀⡀⠀⢤⣤⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⣿⠈⠍⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⠗⡆⣼⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⣹⣿⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠀⠀⠈⠀⠀⠀⠠⠀⠀⠰⠂⠠⠀⠀⠀⠀⠀⠀⠐⠀⠀⠂⣀⣤⡶⢾⣿⠛⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠁⠀⠀⠐⠆⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠⠀⢀⣠⣦⡶⠛⠋⢠⠀⢸⡇⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⡀⠀⢐⠀⠄⠀⠁⢀⠀⠀⡀⣀⣴⡴⠶⢲⣿⣿⣿⣿⡇⠠⠀⠺⠀⢾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣣⠀⠡⠂⠀⠀⣰⣷⣾⣿⣏⠁⠀⠀⡀⢸⣿⣿⣿⣿⣇⠆⠀⠀⠎⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣲⠠⠀⠘⢿⣿⣿⠿⢿⣆⠀⢠⠁⠀⢻⡉⠙⠋⠈⠙⠲⢤⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⣿⣛⣹⠂⠀⠀⠀⢉⣛⠳⠶⣬⠷⣌⠁⠄⠀⠙⠦⠬⠤⢬⣁⣀⣬⣽⡷⠶⠾⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠰⠯⣍⡤⠚⠷⠀⡟⢦⣨⢗⣶⣦⣷⣬⣷⠀⣤⠀⣤⡓⠲⡞⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣠⣤⣤⣴⠶⠾⣾⠟⠛⠛⠛⠋⠉⠉⢹⢡⡼⢻⡇⣸⠟⠛⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣾⡿⠿⢿⣷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣾⡿⠟⠛⠉⠀⠀⠀⠈⠉⠙⠛⠻⠿⠿⣿⣿⣿⠿⠿⠛⠋⠁⠀⠀⠀⠀⠀⠉⠛⢿⣷⡀
⠀⠀⠀⠀⣀⣴⣿⠿⠛⠛⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦
⠀⠀⢀⣼⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⢠⣿⡟⠁⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣾⣿⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇
⢀⣿⡟⠀⠀⠀⠀⠀⢶⣿⠿⠛⠛⠛⠙⠛⠛⠻⢿⣷⣦⣀⠀⠀⠀⣠⣴⣾⠿⠛⠋⠉⠀⠀⠀⠀⠈⠉⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣸⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣴⣿⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠃⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⣸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⣿⡟⠸⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡏⠀⠀⠈⣿⣇⠀⠀⠀⠀⠀⠀⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⢻⣿⠀⠀⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⠀⠀⠀⠀⣾⣿⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣿⡏⠀⠀⠀⢿⣷⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⣼⣿⠁⠀⠀⠀⠸⣿⡆⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⣰⣿⠇⠀⠀⠀⠀⠀⢿⣷⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⣄⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣶⣿⠿⠃⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⢀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⢟⠀⠀⣘⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣶⣶⡭⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢄⡠⠊⢉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠤⠎⠉⠁⠀⠀⠀⠪⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⢉⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣁⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠄⠦⠀⠀⠀⠀⠀⠀⢠⠋⠉⢉⣢⣤⡤⠤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠔⠊⠉⠁⡟⠁⠀⡰⡧⠄⠒
⠀⠀⠀⠀⠀⠂⠠⠴⢆⠀⠠⠤⣀⠀⠀⠀⠀⠀⠀⣠⠐⠀⠀⠐⠀⠀⠀⠀⠓⠤⠤⡼⠃⠊⠉⢀⣀⡀⠔
⠀⠀⠀⠠⠊⠀⠀⣠⠌⠢⡀⠀⠀⠈⠐⠀⠀⠀⠀⠇⠀⠑⠊⠀⡀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⡀⠀⠀⠈⠀⣜⠁⠔⠒⠒⠂⠀⠀⠰⠒⠀⠀⡈⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⠀⠀⠉⠳⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠋⠀⠂⠁⠀⠀⢠⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢸⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⡏⠀⢀⠀⠀⢠⢀⡀⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡇⣇⣀⠆⠀⠀⢸⠀⠘⡐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠘⡇⠝⠉⠀⣴⡶⠞⠘⠽⣷⠿⠁⢀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⡀⢡⣆⢠⡄⡶⠼⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠠⣄⣀⠀⠀⠀⣰⠁⠘⠒⠉⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⣠⡶⢺⡇⠀⠀⠀⡜⡸⠉⢹⠃⠀⠀⠀⠀⠀⠉⠒⠒⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⢶⠏⠃⠀⠐⣶⠷⠥⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠊⢹⠓⠤⠖⠋⡇⠀⠀⣀⣀⡧⣤⠤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠊⠀⠀⠀⠀⠻⠷⠶⣤⣿⣿⣷⠷⠾
⠀⠀⠀⢠⠀⢰⢰⢠⢠⢀⣠⢰⠐⠂⡄⡄⡆⠀⢠⠀⢠⢠⢠⢠⢠⢀⠀⢲⠠⠆⡀⠀⡄⠲⠀⠆⡀⢲⢀⡆⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠐⠒⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠀⠀⣀⡀⠁⠒⠀⠐⣒⣒⣂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣥⣶⣾⣿⣿⣿⣶⡶⢾⣿⡿⠿⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠟⠉⠉⣴⣿⣯⣄⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⣿⣏⣹⣏⠐⠉⠀⠀⢀⡴⠟⠛⠉⠉⠉⢣⠾⠿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣶⣶⣆⣀⣰⣖⡺⣦⡀⠀⠀⠀⣠⠤⢕⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⣿⣿⣿⣿⣿⣷⠶⠒⠒⠒⠻⣷⡀⢠⡾⠯⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣭⣿⣿⣿⣿⢀⣀⢀⣤⠀⣿⣿⣾⡄⣴⣆⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣚⣙⣿⣿⣿⣯⣿⡇⢹⡆⢹⡇⠀⣿⠀⣿⢸⡏⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣿⣿⣿⣿⣿⣧⢸⡇⢸⡇⠀⣿⢀⣿⢸⣧⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣛⣿⢸⡇⢸⡇⠀⡟⢸⣯⢸⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣟⣿⣸⡇⠀⡇⠀⡇⢸⣯⡼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⣿⣿⣿⣿⣿⣿⡶⡇⠀⣇⠀⡇⢸⡧⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢞⣫⣽⣿⣿⣿⣿⣿⣿⣿⡇⣷⠀⣿⠀⣷⢸⣷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠔⣿⢿⢿⣿⣿⣿⣿⣿⣿⣽⣧⣿⠀⢛⠀⣿⢼⣿⢺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠘⢡⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠈⠀⠗⣸⠎⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣯⣿⣿⣿⠍⠀⠄⣀⢨⣼⠖⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣿⣿⣿⣿⣿⣏⠀⠀⠘⣿⠀⣻⣐⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⣿⣷⣶⣶⣶⡿⠾⠛⠉⠁