Skeletons Emojis & Symbols

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

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

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

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

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

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

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣦⣴⣿⣷⣍⣉⡙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡦⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢠⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣗⡀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⢱
⠀⠀⠀⠀⠀⢸⡏⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡛⢻⣿⣿⣽⣿⣦⣤⣤⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣇⣿⣿⢿⣿⣿⣿⣿⣿⣛⣯⣵⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠇
⠀⠀⠀⠀⠀⠀⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⡟⠻⣿⣿⡿⠞⠋
⠀⠀⠀⠀⠀⠀⠹⣨⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⠁⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⠤
⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡟⠟⠛⠋⣿⣿⣿⣿⣿⠇⣠⣶⣼⠇
⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣟⠿⣿⣿⣷⣶⣶⣿⣿⣿⣿⠅⠀⢰⣿⣿⣩⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀
⠀⠀⢴⣿⣿⣿⣿⣿⣿⣿⣯⣤⣤⣤⣌⡛⠿⣿⠿⠟⠋⠉⠀⠀⠸⠟⠃⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠉⠉⠁⠙⣿⣿⣿⣿⣿⣿⣿⣿⡏⢳⣶⡾⠓⣲⣦⣀⣀⠀⠀⢾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⣿⣇⠈⠡⡴⢠⣿⠻⣿⠻⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⣿⣿⣿⡆⠀⠘⠘⠛⠀⠉⠛⠗⠓⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣶⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠉⠉⠞⠂⠀⠀⠀⠀⠀⡀⢷⡀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⡀⠋⣻⢿⡟⠑⣤⣤⣀⠀⠀⠀⠈⠓⠺⡇⠀⠀⠀⠀⠀⢛⣨⣇⣈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣾⠀⠀⠀⠀⠀⠉⠲⠦⣤⣀⣠⠜⠀⠀⠀⠈⠁⠘⢯⠉⡇
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣷⡤⠈⠀⢲⣦⠌⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡵⣧⡖
⠀⠀⠀⠀⠀⣠⣿⠋⣴⣿⣷⡌⠀⠀⠁⠀⠀⠀⠛⣢⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠀⠈⡇⢸⡇⠀⣼⠂⡒
⠀⠀⠀⠀⡼⠟⢀⣤⣿⣿⣿⣿⡄⠀⠰⣿⣿⣶⣼⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠠⣧⣾⡷⢸⡟⣸⠇
⠀⠀⠀⠀⠀⠀⣿⢿⣿⣿⣿⣿⣿⣆⠀⣿⣿⠟⠉⢙⣟⢡⡄⠀⠀⠀⠀⠀⠀⠀⠀⢘⣳⠀⣸⣿⣿⣿⣟⡗⣿⠇
⠀⠀⠀⠀⠀⠀⣽⣷⣿⣿⣿⣿⣿⣿⣷⡈⠁⠀⠀⢸⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⣩⡿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠩⣷⣦⣍⣉⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣸⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢷⣈⣻⡟⠛⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⢾⡟⠉⣷⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⣶⣄⠀⠁⠘⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⡟⠇⠀⣼⣿⣿⣿⣶⣤⣄⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠈⠋⠙⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠟⠋⠛⠉⠛⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⡿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣥⣤⣥⣴⣤⣼⣧⣴⣿⣥⣥⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⢦⡩⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠿⡀⠈⠛⠀⠀⠖⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⣟⢲⣮⣇⠀⠀⠙⣫⣿⣾⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡓⢽⡫⣧⠀⠀⢿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢿⣿⡧⣼⣿⣬⡺⡿⡟⠘⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣩⣯⢹⣿⣿⣿⠿⣴⣧⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣷⣾⢿⣿⣿⣿⣽⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣬⡿⣟⣷⣿⣿⣛⣹⡟⣻⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣿⣿⡟⠻⠟⢿⣿⠿⣣⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠸⣻⣿⣟⡟⠙⢻⡁⠠⠀⠀⢹⢆⣽⣽⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⣿⣻⣤⣀⣸⠲⠿⠄⢳⣦⠘⣿⣭⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣽⣿⣿⣿⣾⣤⣶⣩⣄⠼⠃⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⡿⢿⣿⣿⠿⣿⡙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠃⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡰⠾⠛⠛⢛⣿⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⢰⣿⣯⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⢾⣶⣷⡾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⣻⣿⠉⠉⢁⣼⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣏⢉⣠⣿⣿⣿⡟⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠉⣛⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠠⣶⣿⣿⠄⠀⠀⢀⣀⣀⣀⡀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠙⣿⣤⣯⣀⣀⠒⣶⡶⠞⠉⠀⠁⣤⠸⠷⣿⣅⣀⣁⣤⠙⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠻⣿⢿⣿⣙⢿⠆⢀⣤⠆⠀⣧⠀⣀⠀⢹⣿⣿⣿⠿⠷⢤⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⣍⣻⠿⣦⣤⠀⠂⠀⢩⠀⠩⣶⠂⣉⡿⢿⠇⠀⠀⢨⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠾⠿⢿⣦⣤⠤⠀⠆⠀⢶⣟⠁⠶⠿⣛⣻⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠿⣦⣬⣤⠀⠀⢠⡀⢿⣯⣀⣤⣾⣿⣿⣾⡇⠀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠶⠮⢏⡁⠤⣀⢿⣿⣧⠉⡛⠠⣼⡏⠙⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⢳⣤⣴⠋⠁⠐⠁⠀⢛⣿⣏⣅⠀⣨⡿⠛⣿⡌⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⣤⠜⠉⣀⠴⠂⠠⣟⣿⠄⢈⠉⠀⠰⠛⣿⡄⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠚⢁⣴⠟⠁⠐⠂⠀⣿⣿⡗⠈⠙⠃⠀⠛⠛⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣭⣽⣍⠀⠀⠀⠀⠀⠀⢰⡄⠀⠈⠻⢿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⠾⣿⠿⢷⣄⣸⣏⣙⣿⡶⠛⠀⠀⠀⢠⡞⠁⠀⠀⠀⠈⢿⣿⡆⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠺⢿⣷⣄⠀⠀⣉⣽⣿⣿⣷⣶⣤⠀⠀⣄⠀⠀⠀⠀⠀⠀⠈⠋⣵⡆⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠛⣿⠶⠛⠛⢻⡟⠛⠛⠁⠀⠀⡀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠁⢼⠀⠀⠀⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠸⠿⢀⠀⠀⠀⠀⠉⠁⠀⠀⣄⡑⣶⣿⠷⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠀⢠⣤⣾⣶⡄⠠⡄⠀⣴⡄⠀⠙⠿⠀⠈⠀⣾⡟⠀⠀⠀⠀⠀⢡⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⠀⠀⠀⠀⠀⣿⠁⠀⠀⠁⠀⠀⠀⠈⠻⢷⣴⠆⠀⠰⣶⣿⠀⠀⠀⠀⠀⠀⠀⢳⣄⣿⡆⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢿⡀⠀⠀⠀⠈⠟⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣷⣄⠀⠀⠀⠰⠋⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠛⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⣙⡿⡉⠻⣦⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠃⠀⢸⡇⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣉⠀⣽⣄⠈⢷⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⠟⠃⠸⡇⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⢹⡟⣿⠹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⢈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⣿⣄⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣤⣼⣿⣿⠆⠀⠀⠀⠀⠀⠀⠟⠛⠛⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠛⢿⡧⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡶⣶⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⢀⣌⠿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢿⡆⠀⠀⠀⠀⠀⠸⣿⣦⣿⡇⠀⠀⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣷⣾⣿⠀⠀⠀⠀⠀⢠⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⣿⣧⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⠀⠠⣿⣿⠻⣿⠿⠁⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣿⣿⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠿⡿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠒⠒⠂⠀⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢠⢀⣤⣤⣤⣄⠀⠀⢀⣤⣶⣤⣀⡄⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠉⢸⣿⣿⣿⡏⠀⠀⢹⣿⣿⣿⡿⠠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠇⠘⠿⠿⠿⢋⣿⣷⡈⠿⠿⠛⠃⠸⣄⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⣀⣀⣀⠀⢸⣿⢿⡇⠀⠀⣀⣀⣀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⢿⣿⣀⠀⣀⣀⢀⠀⣿⡟⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣯⢭⡭⡧⣭⣽⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⣀⣀⠀⠀⠀⢰⡀⠀⠀⠉⠉⠉⠉⠉⠀⠀⡄
⠀⠀⠀⠀⠀⠀⢀⠤⠒⠉⠉⠁⠀⠀⠀⠀⠈⠉⠓⠤⡀⠙⠤⣀⠀⠀⠀⠀⣀⠤⠊⠀⢀⠤⠖⠊⠉⠁⠀⠀⠀⠀⠁⠒⠠
⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠈⠉⠉⠉⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⣀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠸⢳⣾⣿⣿⣿⡆⠀⣿⣿⡿⠀⠀⠀⠀⠀⠀⡰⣿⡄⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠘⣿⣿⣿⣿⠃⣴⣿⣿⣧⠀⠀⠀⠀⠀⢴⢰⣿⡿⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⢆⠀⠀⠀⢀⠀⠀⠘⠃⠀⠀⠈⠉⠀⣸⣿⣿⠀⠀⠆⠀⠀⠀⠀⢸⠈⠉⠀⠸⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠈⠣⡀⠀⠈⢹⣿⠟⠶⢶⡶⠂⠀⠀⠉⠋⠉⠖⠋⠀⠀⠀⠀⠀⠋⠀⠀⠢⣶⠖⣶⡷⣶⠖⠶⠆⠀⠀⠀⠀⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠀⠀⠈⠐⠒⠛⠻⠀⠀⠸⣇⣀⡀⠖⠖⠒⠒⢲⠀⠀⠀⠀⠀⠰⢤⠤⡤⢴⢻⠀⠀⠀⡁⠀⠈⠠⣄⣀⣀⠠⠤⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠈⠒⠂⠓⠻⠿⠿⢿⠀⠀⠀⠀⠀⢘⡯⠽⠼⠋⠁⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⡀⠠⠔⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠒⠒⠊⠀⠀⠀⠀⠀⠀⠲⠒⠊⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠃⠀⢠
⠀⠀⠀⠀⠀⠀⠀⡠⢤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣧⢀⣒⠢⡄⠐⠂⠀⠤⠄⢀⠰⢦⢶⣤⣩⠖
⠀⠀⠀⠀⢠⣀⣠⣿⠋⣃⡠⢄⣠⠀⡙⠃⠂⠀⠀⠀⢙⣛⠃⠚⠱⠦⣤⣀⠀⣠⠐⣺⡿⠋⠉
⠀⠀⠀⠀⠀⠙⠻⠷⠿⠋⠀⠚⠀⠀⢀⣠⡀⠒⠂⠀⠈⢉⣟⠁⠻⠆⡀⣭⠈⠁⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠆⣠⡞⢃⠍⠀⠈⠇⠀⠀⠈⠉⠛⠻⠖⣠⡄⢈⠳⠀⠀⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⢀⡖⢀⣢⢷⠀⡆⠂⠀⠀⠀⣒⠚⠶⢶⣤⠀⡀⠀⠀⠀⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠖⡆⡺⣜⢟⠛⠋⠐⣠⠆⠀⠀⢶⣤⣉⣓⣀⣠⣘⠻⠐⠧⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠈⣚⠳⠶⠘⣁⠀⠀⠀⣤⣈⠙⠛⠷⠦⠉⡁⠶⠆⣤
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠻⠄⠐⠂⠘⠄⠀⠀⢀⣝⠛⠿⢮⣄⡀⠉⠻⠖⣀⡁
⠀⠀⠀⠀⠀⠀⠈⣀⡀⠀⠀⠀⠀⠀⠀⢀⠀⣌⠀⠐⠋⠁⠀⠀⠈⠙⠁⣤⡀⠈⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢂⠀⠀⠐⠀⠙⠃⢀⡃⠀⠀⠐⢦⡤⠀⠈⢡⠄
⠀⠀⠀⠀⠀⠀⢀⡶⠒⠲⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠙⣀⠀⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣎⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢤⡀⠀⢀⠈⣿⠁⣇⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠁⠀⠀⠘⣆⠀⠸⠉⣄⣼⠄⠒⠨⢿⠿⠶⠉⠙⠷⣿⣿⡿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠠⡀⠀⠀⠀⢀⣀⢘⠀⣶⡟⢯⡐⢿⣿⡶⠊⣽⣿⡷⠒⠛⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡁⠀⣀⡰⠋⠉⢛⡀⢽⠀⡀⠙⠳⣝⢿⣟⡛⠋⠉⠈⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠄⠈⠃⠀⠀⠉⠶⢸⣃⣗⠀⠀⠈⠳⢬⣙⠶⣥⣞⣒⡚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣷⡀⠑⠄⠀⠀⠀⡸⠶⢋⠀⠀⠀⠀⠀⠉⣳⣶⣄⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠗⠀⠂⡦⢽⡷⠈⢶⢶⠀⠀⠀⠀⠈⠿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠈⢀⣰⠇⠈⠛⠐⡆⠀⠀⣄⣀⠀⢲⣴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠘⠁⠀⠀⠀⢠⠃⠀⠀⠈⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⡤⠴⢲⣄⣤⣤⠀⠀⡀⠀⢠⡶⢒⡒⢶⡄⠀⠀⡤⣴⣹⡇⠀⣺⣉⣽⡠⠜
⠀⠀⠀⠀⠀⠰⣏⣶⣶⠫⢄⡴⠃⠀⡼⢹⠇⠀⠳⠬⠷⠋⠀⠀⢸⢧⣯⡼⠁⠸⣥⠽⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠀⠀⢸⣉⡿⠀⠀⠀⠀⠀⠀⠀⠀⢈⠋
⠀⠀⠀⢰⣋⣿⣏⣭⠽⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢞⢋⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣄⡀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣧⣶⣼⡧⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣎⣁⣸⡤⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢶⠲⣜⢆⠚⣳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀⡀⠀⠀⠀⠀⣀⣤⡄
⠀⠀⢀⣀⠀⠀⠀⠘⢿⣋⡏⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢞⢉⡿⡍⡽⠀⠀⢰⠏⢓⡴⠁
⢰⡶⣿⣌⠛⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⣨⠽⢧⠞⠁⠀⠀⡤⣛⠯⡷
⠀⠙⠧⢬⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠗⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣄⣀⣠⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡤⣄⣷⣴⣣⡼⠓⠦⠭⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠶⢛⢶⡬⠵⠇⠀⠀⢾⠒⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠬⢷⢶⠾⠚⠁⠀⠀⠀⠀⠈⠳⣍⡾⠀⠀⠀⠀⠀⢀⡔⣋⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⢧⣞⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣤⠝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢼⣭⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠙⢛⡷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⠶⠟⠀⠀⢴⠒⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠲⢶⢬⡳⢄⡸⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣄⣘⡆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉
⣿⣿⠟⠟⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡟⠛⠛⠛⠛⠛⠛⢿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣿⣿⣀⣠⣴⡾⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠂⣁⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠶⠶⠶⠶⠛⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡄⠀⠀⠀⠀⣠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⢀⡴⠋⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠐⠈⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠠⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⢀⣾⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⣡⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⡵⠊⠁⠀⠀⠀⡠⠚⣱⡿⠁
⠀⠀⠀⠀⠀⠀⢠⣶⠶⠶⠶⠶⣶⣶⣤⡶⠟⠋⠀⠀⠀⠀⠀⠀⢠⠞⣡⣾⠋
⠀⠀⠀⠀⠀⠀⣿⠃⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣼⠟⠁
⠀⠀⠀⠀⠀⠀⣿⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⣿⣄⠁⠀⠀⠀⠀⢤⡀⠀⠀⠀⠀⠀⠀⠀⣼⡿
⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠲⠶⠤⣬⣿⡄⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⣸⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⣀⠀⢀⡿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⣠⠞⠁⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣤⣈⣉⣀⣠⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⠀⠀⠀⠰⣿⣿⡟⠋
⠀⠀⠀⠀⠀⢀⠨⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠘⠚⠒⠀⠀⠀⠸⠛⠁
⠀⠀⠀⠀⠀⡨⠀⠐⠀⠀⠀⢀⣀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⣬⣄⡀⠀⠀⠀⠈⠀⠀⠂⠀⠀⠀⠘⠄⠂⠀⠀⠀⠀⠀⠄⠒⠀⠠
⠀⠀⠀⣸⣿⣿⣿⣧⠀⠰⠽⠋⠀⠀⠈⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣃⠶
⠀⠀⣰⣿⣿⣿⣿⣿⣷⣄⠀⢰⠇⠀⠈⠉⠀⠀⢀⣴⡆⠀⢀⠪⠂⠀⠈⠓⡀⠀⠀⠀⢀⠔⠒⠒⢄⠀⠀⢀
⠀⠀⣿⡿⢿⣿⢿⣿⣿⣿⣧⣤⣄⣀⢠⠤⠠⠤⠦⣿⣇⡀⠀⠀⢀⡬⡶⠀⠀⡐⠀⠀⠂⠀⠀⠤⠄
⠀⠸⠿⠷⠖⠂⠊⠛⠻⠿⠟⠿⠋⠁⠀⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠀⢀⣴⡟⢠⠀⠤⡈⠁⠐⠀⠀⡄⠀⠃
⠀⠀⠀⠀⢀⡀⠤⠀⠀⠀⠀⠀⠀⠀⠀⠾⠛⠃⠀⠀⠀⠈⣿⣷⣾⣶⣾⣿⣿⠉⠈⠀⠀⠒⠀⠐⠂⠨⠀⠀⢠
⠀⠀⠀⢞⣽⠋⠀⠰⣴⠄⠀⠀⠀⠀⠀⠀⠀⠠⠚⠀⠀⠀⠈⣿⡟⠟⠿⢿⣿⣦⠀⠀⠁⠰⢗⣻⣷⠀⠀⠀⠈
⠀⠀⠀⠈⠁⠀⠀⠀⠩⠄⠀⠀⠀⠀⣷⢲⡀⠀⠿⠂⠀⠀⡸⠋⠀⠀⠀⠀⠈⠻⣷⡄⠀⠀⠀⠈
⠀⠀⠀⢭⡄⠀⢠⣴⡄⠀⠀⠀⠀⠀⠈⠉⠋⠀⠀⠀⢀⡄⠃⠀⠐⡀⣲⡀⠀⠀⠘⢿⣤⡀⠀⠀⠁
⠀⠀⠀⠘⠃⠈⠀⠈⠀⠀⠀⣰⣷⣄⡀⠀⠀⠀⣀⣴⠇⠀⠀⠁⠐⠤⠏⠀⠀⠐⠄⠀⠉⠉⠓
⠀⠀⠀⠀⠀⠺⠧⡀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠰⠒⣠⠀⠀⠀⠐⠿⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠟⠋⠀⠀⠀⠀⠀⠀⠈⠻⢧⡄⠀⠀⠀⠀⠈⣲⢄⡀
⠀⠀⠀⠀⠐⠶⣴⢆⠛⠀⠀⠀⠀⢀⠄⠐⠐⠀⠀⡀⠀⠙⣷⣄⠀⠘⠺⠋⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠉⠻⣦⣀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡀⠈⠀⠀⠘⠿⢿⣷⣶⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠣⠟⠀⠀⠀⠀⠀⠀⡘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠂
⣿⣿⠛⠛⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄⠀⠀⠀⠀⠀⠀⢀⠴⠋⠉⠉⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⡀⠀⠀⠀⠀⣿⣄⠀⡀⠀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄
⠀⠀⠀⠀⠀⠀⢠⣸⣛⣋⣹⡧⠀⠀⠀⠀⣿⠛⠋⠙⢻⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤
⠀⠀⠀⠀⠀⠀⠊⠄⠛⠿⠋⠡⠀⠀⠀⣸⡇⠀⠐⠀⢘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⡀⠀⠀⢻⡀
⠰⢤⡀⠀⠀⠀⠘⠋⠃⠈⡍⠀⣀⠴⠛⠉⠛⠛⠟⠛⠛⠻⡄⠀⠀⠀⢰⡊⡪⣐⡀⠀⠀⠈⠁⠀⠀⢸⠂
⠀⠀⠉⠳⢤⣀⣀⣀⣀⣤⡴⠊⠉⠀⠀⠀⠀⠀⡉⠀⠀⠀⠘⢆⠀⠀⠀⠚⠾⠋⠀⠀⠀⠀⠀⠀⢀⣸⣇
⠀⠀⠀⠀⠀⣿⠁⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠒⠉⠉⠈⠒⠲
⠀⠀⠀⠀⠀⢸⡀⢀⣀⣨⣅⠀⠀⠀⡀⠀⠀⠘⠀⡄⠀⠀⠀⠀⢀⡟⠁⠲⡶⠒⠉⠉⢀⡀
⠀⠀⠀⠀⠀⡸⠋⠉⠉⠀⠈⠣⣄⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣤⠼⣄⣀⣸⠃⠀⢰⣮⣿⣿⠀⠈⠞
⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⣀⣀⣠⠤⠒⠊⠀⠀⠀⠀⠈⠉⠣⣀⠈⠛⠛⠁⠀⠄
⠀⠀⡴⠊⡃⠀⠀⠀⣈⣋⠍⡄⠀⠀⠀⠀⠈⡌⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⠃⠀⠀⠀⠡⢯⠾⠁⠀⠀⠀⠂⠀⡇⠀⠀⠀⠀⠨⠀⠀⠠⠀⠐⠀⠀⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⠙⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠉⠓⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢡⠁⠈⠉⠀⠀⠘⣔⣤⠤⢶⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣵⠀⠀⠀⠀⠀⠀⠀⠂⠀⢠⡟⠁⠀⠀⠀⠀⠀⠉⠢⡆⠀⠀⡀⠀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⡘⠀⣄⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⡀⠀⠀⠀⠀⠓⠠⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⡀⢀⡟⠀⠀⠀⣌⠈⠃⠀⠀⠀⠀⢀⠖⠁⠀⠀⠀⠀⠀⠀⡇⠀⣿⠃⠀⠀⠀⢉⠀⠆⡄⠀⠀⢾⠀⠁⠀⠀⠀⠀⠀⢔⡤⠷⠀⠒
⠀⠀⠀⠀⠀⠀⠁⡞⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⢸⠀⠀⡅⣀⣿⠀⠀⠀⠀⠈⣁⠃⠀⠀⣠⠃⠀⢰⠀⠀⡂⢀⠔⠉
⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⣀⣴⠟⠀⠀⠠⠀⠀⠀⢀⠂⣀⢠⡏⢫⢿⠀⠀⠀⠀⠈⠁⠀⠀⣰⠋⠀⠀⠀⠌⣼⢀⠋⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠚⠲⠷⡶⠶⠿⠟⠛⡉⠀⠀⢀⠂⠁⠀⠀⢠⡿⢣⣿⠈⠈⣾⡀⠀⠀⠀⠀⠀⢀⡼⠣⠀⠀⡀⠀⠠⠃⠆⠀⠀⠀⠒⠻⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⡾⡉⠀⡸⠻⢀⠀⢩⠳⣄⡀⠀⢀⠴⠛⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⡁⢃⡔⠀⠀⠀⡇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⡤⡺⢁⠔⡅⢀⠏⣀⡄⡠⢡⡞⠉⠉⠪⣀⠀⠉⠉⡁⠀⠂⠐⠁⡡⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣱⠇
⠀⠀⠀⠀⠀⠠⢖⡤⣭⠥⠶⡾⣾⠛⠋⠁⠀⠼⡜⡏⠄⢊⠊⡵⢋⡴⢃⠔⣀⠀⡀⠠⢀⣲⠌⠁⠈⠒⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣀⣸⢻⡟
⠀⠀⠀⠀⠀⣴⠗⣫⡄⣾⡗⠊⢁⣄⣀⣀⠀⠺⡇⣆⢩⣤⡴⠗⢫⠔⠡⠈⠀⠑⠌⠂⢠⠃⠉⠒⠀⠈⢂⠀⣱⠀⡀⠐⢧⠀⠄⢀⣄⡂⣀⠬⢛⡥⢋
⠀⠀⠀⠀⠀⠇⣰⡥⠚⢁⠼⠛⠊⠉⠁⠀⠀⠠⢧⡞⢋⠡⠒⠂⡶⠀⠀⠀⠰⠀⠀⢡⠆⡴⠊⠑⢦⡀⠆⠀⡧⢰⠀⣼⠀⠈⠁⢌⠉⠉⠁⠂⠁⠐⠉⠁
⠀⠀⠀⠀⢈⡔⣠⣿⠜⠁⠀⢀⠤⠠⡄⠀⠀⠀⢼⠜⠃⠀⡠⠃⡿⠀⠀⠸⠀⠀⠀⡎⡼⠀⠀⠀⠀⠁⣌⠰⠇⠸⠀⠇⠂⠀⠀⠀⠑⠢⢤⡄⢀⠀⡤
⠀⠀⠀⠀⡘⠀⡿⠉⠀⠀⢰⠣⡦⡈⠁⠀⠀⠀⣸⠀⡀⠀⡅⠀⠀⠆⢀⠀⠀⠀⠀⢰⡇⠀⠀⢄⠀⠀⠈⢣⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠘⠁⠀⣷⡇⠀⠀⠘⢦⠴⠁⠀⠀⠀⢠⣯⠜⠀⣼⠂⠀⠀⠈⠏⡇⠀⠀⠀⡼⠀⠀⢀⡀⠑⠄⠀⠀⢧⢂
⠀⠀⠀⠀⠱⡀⣿⡇⠀⠀⠀⠀⠀⢀⣤⡤⢚⡹⠷⠢⢎⡤⠂⠀⢀⠀⠄⢡⠀⠀⣰⡇⠀⠀⢸⠘⠃⡼⠀⠀⢸⡌⡄
⠀⠀⠀⠀⠀⠁⠼⢣⡀⠀⠀⢀⣴⠯⡿⠏⠁⠀⠀⠀⡎⠀⠀⠀⠈⠀⠀⢸⠀⠀⠐⡄⠀⠈⠈⠓⠚⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⢠⡇⢺⣻⣿⡿⠡⠞⠋⠀⠀⠀⠄⠠⠞⠀⠀⠀⠀⠈⠀⠸⠀⠐⢀⠄⠙⢄⡀⠀⠀⠀⠀⠀⣠⡟⣠⡧
⠀⠀⠀⠀⠀⠈⠁⠃⠻⠀⠀⠀⠀⢀⡀⠒⢋⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⢣⠘⢿⡪⣲⣜⣷⣦⣤⣤⡾⠟⠈⠁⠀⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠉⠳⠌⠙⠳⠦⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣿⣿⡟⢹⣿
⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠁⠀⠀⠀⣠⣖⣠⣿⣻⣿⣿⣿⣟⣖⠠⣄
⠀⠀⠀⠀⠀⢸⣿⣿⣟⣛⣓⡒⠒⢒⠒⣶⠶⣶⣤⣬⣤⣤⣾⠿⠿⠾⠻⡅⠀⢀⡀⢠⣤⣤⡤⠤⠤⣤⣤⠤⣼⣷⣶⣤⡄
⠀⠀⠀⠀⠀⠈⢩⣿⣿⣿⣿⣿⣤⣾⣿⣿⣟⣋⣉⣿⣿⣏⣅⠀⠀⠀⠀⣬⡒⠛⠻⣿⡿⣿⣧⣤⣿⣶⣦⣶⢿⡿⠟⠛⠁
⠀⠀⠀⠀⠀⠀⣾⣿⡿⠃⠁⢹⣿⣿⣿⣿⣿⣿⣍⣀⠶⠶⠛⠆⠀⠀⠘⡟⢻⢛⣫⣥⣾⣿⡿⣻⣿⣿⣿⣿⡀⢻
⠀⠀⠀⠀⠀⠀⠈⠻⣧⠀⠀⢼⣿⣿⣿⣿⣻⡋⠉⠉⠙⠋⢿⣷⠄⠀⠀⣞⠻⢿⣿⠿⠛⣉⣴⣿⣿⣿⣿⣿⡿⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣴⣿⣿⣝⣿⣿⣿⣷⣶⣌⠁⠀⣉⡁⠀⠀⠀⠈⠰⣈⣤⣶⣿⣿⣿⠟⣸⣿⣿⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣾⣿⣿⣯⣿⡋⠁⢀⣀⣀⣼⠇⠀⠀⠀⢺⢦⣬⡿⠿⠟⢉⣠⣾⣿⡏⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⢿⣾⡿⣿⣿⣿⣿⠷⠀⠀⠀⣤⡀⠀⠀⢠⡖⠀⠂⢴⣶⣿⣿⣿⡿⠋⣴⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣸⣿⣿⣿⣟⣋⠀⣀⠤⠂⠀⠛⠁⠀⠀⠀⢈⢑⡂⠀⠙⠛⢋⣁⣴⣿⣤⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⢿⣯⡉⠛⠿⣟⣛⠿⠀⠀⠀⠀⣶⠂⠀⠀⠀⠈⣀⠁⠀⠛⠿⢿⣿⣿⠀⢉⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣤⡿⠿⣯⣽⣉⠀⠀⣠⣴⡾⢀⡴⣠⣆⣀⠁⠐⣌⡻⠶⠤⠀⠀⢉⡠⢴⣧⣄⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⢿⣤⣀⣀⣉⢿⡿⠇⠀⠀⡤⠈⣾⣿⣏⣙⡿⣧⠈⠀⠀⠐⠲⣾⣅⡀⠀⠈⣩⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣼⡿⠿⣿⣿⡠⠀⠀⠂⠘⣡⣤⣿⣿⠿⢿⣇⣀⠀⠐⢌⠲⢤⠀⠙⢉⣠⣞⡁⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢿⣦⣤⡴⠾⠓⠒⢂⣴⠾⢋⣹⣿⣿⣿⣺⡟⠛⢷⣄⠀⠑⠐⠒⢾⣧⣄⠙⢛⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣶⣟⠛⠷⠶⡤⡔⠋⣁⣴⠚⠫⣿⣿⣅⣸⣧⣀⠀⠙⠷⣦⡀⢀⠠⣩⡤⢴⣿⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣙⠛⠿⢐⠞⢡⡾⠋⠁⣠⣀⣿⣿⡿⢿⡇⠈⠻⣆⠀⠈⠻⣦⠀⠀⠀⢀⣉⣴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠁⠀⠈⠀⠀⠀⠈⢹⣿⣿⣷⢶⡟⠙⠀⠈⠧⠀⠀⠉⠈⠑⠄⠩⠥⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣶⣿⣿⡟⠙⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⣿⡉⢈⡿⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣶⣶⣶⣶⣶⣶⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⡾⠿⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⢀⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⡀
⠀⠀⠀⠀⣠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄
⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀
⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧
⠀⢸⣿⣿⡇⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿
⠀⣾⣿⣿⡇⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠀⠹⣿⠀⠀⠀⣀⣤⡤⠖⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⣿⣿⣿⣿
⠀⣿⣿⣿⠀⠀⠀⠹⠃⠰⠞⠉⠀⠀⢀⣀⣀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣄⣀⡙⠓⢦⡀⠘⣿⣸⣿⣿⣿⣿
⠀⣿⣿⡿⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⢹⡆⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡈⠀⠹⣿⣿⣿⣿⣿
⠀⢿⣿⣧⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠚⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⢻⣿⣿⣿⣿
⠀⠈⢿⣿⣦⣤⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠄⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⡿⠃
⠀⠀⠀⠙⣿⡁⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢀⡀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣹⠟⠁
⠀⠀⠀⠀⣾⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⢠⣿⣷⡀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⡄⢀⣿⡆
⠀⠀⠀⣰⡟⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⣠⣿⣿⣿⣿⡄⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⡇⠘⣿⣷⡀
⠀⠀⣰⡟⠀⠀⠀⣦⡀⠻⢿⣿⣿⣿⣿⣿⠿⠋⠁⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠙⠿⢿⣿⣿⣿⣿⣿⠿⠋⢀⣼⠇⢠⣿⣿⣷⡀
⠀⠀⣿⠀⠀⠀⠀⠘⠻⠶⣤⣭⣍⠉⠁⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠈⢉⣁⣀⣤⣶⠿⠁⠀⣼⣿⣿⣿⡇
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠛⠛⠉⠉⠉⠉⠁⠀⠀⣴⣿⣿⣿⣿⠃
⠀⠀⠈⢷⡀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠙⠳⢦⡄⢤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠻⣿⣿⣿⡏⣿⣿⣿⣿⠏⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢿⣿⡀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠈⠙⠋⠀⠈⠛⠋⠁⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣸⡿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⢸⣿⡇⠀⠀⠀⠀⠀⡼⢡⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠘⡆⠀⣤⠀⠀⢰⣿⠀⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣧⠀⠀⠀⠀⢠⣴⡿⠇⢠⣤⣀⠀⠀⠀⠀⠀⢻⣿⡶⠀⠀⠘⣻⡄⢨⡄⢠⣿⡟⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢹⣿⣷⡀⠀⠀⠈⠉⠀⠀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⡙⢸⡇⣸⣿⡇⠀⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠁⠀⠀⠈⠀⣿⣿⠃⢀⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠸⣿⣿⡆⢤⢀⣤⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢠⡤⠴⠢⣿⣿⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠉⠢⡀⠀⢁⠀⠉⠀⠀⠁⠀⠀⠀⠀⠀⠊⠀⠀⠁⠚⠈⠀⠀⢠⠄⠛⠛⠀⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠉⠉⠈⠀⠀⠀⠐⠀⠐⠂⠀⠂⠀⠂⠀⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠴⠖⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠒⠂⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠉⠻⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠺⠋⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣆⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠀⣠⣾⣿⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⣿⣶⡀⠹⣿⣄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡇⢰⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡀⢹⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⢸⣿⣿⣿⣿⣿⣿⡿⠀⣠⣿⣦⠂⠈⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠏⠀⠈⠿⠿⠿⠿⠿⠟⠁⠀⣹⣿⣿⡞⠀⠙⠻⠿⠿⠿⠿⠟⠀⠀⢀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠀⠀⠀⡀⢠⣴⡶⠁⢸⣿⣿⣿⣷⡄⠘⣷⣤⣀⠀⠀⠀⠠⢴⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢠⣀⠀⠀⠀⠀⠀⢹⣿⣶⣌⠁⠾⠛⠀⡄⠘⠛⠁⠙⠋⠛⠀⠘⢿⠉⠑⢀⣈⣵⡟⠀⠀⠀⠀⠀⢀⡤⠠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⣸⣧⠀⠀⠀⠀⠈⣿⡿⣯⠰⣤⣄⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢈⡶⠛⣿⣿⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠤⣤⣿⡇⠀⠀⠀⠀⠀⢹⡇⢹⣤⡿⡿⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⣸⡇⣰⡿⣿⠃⠀⠀⠀⠀⠀⢘⣿⣦⠤⣼⠟
⠀⠀⠀⠀⠀⠀⠀⣠⢴⣿⡃⠺⣿⣷⡄⠀⠀⠀⠀⢸⡁⠈⣿⡿⢿⣶⣶⢾⡶⢾⡶⢶⠶⣿⣾⣿⣿⡟⠁⢹⠀⠀⠀⠀⡀⠠⣾⣿⠛⢐⣿⡦⣄
⠀⠀⠀⠀⠀⠀⠀⡁⠀⣀⣀⣀⣀⢀⣤⣼⡀⠀⠀⠸⣇⠀⢿⣷⡸⠀⢱⠀⠀⢠⠀⢰⠀⡇⢸⣽⣿⠀⠀⠾⠀⠀⠀⠀⣧⣤⡀⣈⣀⣀⣀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠰⣤⣈⡙⠛⠛⠉⠙⠻⣷⣄⠀⠀⠹⢦⡈⠿⡇⠈⠉⠐⠋⠙⠁⠈⠘⠁⠀⠘⢃⣤⠞⠀⠀⠀⣠⣾⠟⠋⠙⢛⡛⢋⣁⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠲⠶⢦⣤⡈⠹⡷⢤⡀⠀⠙⢦⠈⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠟⠁⠀⣀⡠⠾⠟⢁⣤⣴⠾⠶⠂⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣙⠁⠻⠦⣀⠑⠢⠬⠧⠀⠀⠐⠀⠈⠡⠖⠉⣀⠴⠞⢁⣉⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢶⣤⣀⠈⠐⢤⣀⠀⠀⠀⠀⣀⡤⠒⠉⣀⣤⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠳⢶⣴⣈⠉⠲⠶⣉⣁⣦⡶⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠛⣿⣷⣦⣄⣀⠉⠛⠢⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠊⠁⣠⣴⣿⡿⠛⠉⠉⠛⢿⣷⣦⣄⠈⠑⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⣀⠀⠀⠤⢤⣴⠾⠯⠅⠐⢢⣬⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⢿⣤⣐⠂⠨⠹⠷⣦⡤⠤⠄⢀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⠛⠉⠉⠉⣠⣤⡓⠶⠄⠀⠀⣠⡴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣀⠀⠀⠠⠶⢚⣢⣄⠙⠉⠉⠛⣦
⠀⠀⠀⠀⠀⠀⠀⢿⡆⠀⢠⣤⣌⡡⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⢀⣡⣤⡄⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⢿⣿⠃⠀⣀⣰⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠈⣻⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣀⣀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣀⣀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠠⠄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⣩⣷⡏⠆⠎⣡⡈⢀⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⡀⠀⠠⣶⣄⠀⣠⣤⠀⠀⠀⠀⠀⠘⠀⣉⣻⣯⡌⠏⢿⣿⢺⠀⠰⡠⠄⢠⠀⠀⠀⢠
⠀⠀⠀⠄⠀⠀⠀⠑⠀⠈⠁⠄⠛⠛⠁⠂⠀⠁⠀⠀⠀⠉⢛⠛⠑⠂⠐⡀⢃⠀⠤⠐⡫⣀⡤⠐⠄⠀⠀⠀⠀⣀
⠀⠐⠀⠀⠀⠀⠀⠆⠀⠀⠀⡈⠒⢀⠖⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠖⠂⠈⠻⣿⣿⣶⡃⠀⠀⠀⢀
⠀⠀⠀⠀⠘⠃⢴⠠⠐⠀⠠⠀⠀⠘⡥⣩⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢈⠦⣐⠲⡛⡉⠽⠁⠠⠄⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠐⠒⠀⠂⠀⠀⠐⠂⠈⡀⠀⢀⣴⣢⣉⠱⠀⠜⠓⠚⠀⣁⠀⠈⠈⠀⡀⠀⠀⠠⠀⢉
⠀⠀⠀⠀⠌⠀⠀⠂⣀⡠⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠈⡻⡟⠠⠁⠀⠀⠀⠒⠀⠸⠀⡀⢠⣹⡆⠁
⠀⠀⠀⠀⠀⠀⠃⡘⢿⡿⠠⠀⠀⠀⠓⠀⠀⠀⠀⠀⠀⠀⢀⡀⠠⣿⡷⢀⠀⠀⠰⠀⠈⢈⡋⠜⢠⡌⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⣐⠂⡡⠤⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠰⣃⣀⡨⠊⠓⠀⠀⠐⠀⠃
⠀⡀⠀⠀⠀⠈⠐⠀⠁⠀⠀⠘⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀⠀⢀⣝⢾⣷⡞⣦⢠⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣾⣿⡷⢌⠠⡑⠀⠀⠀⠀⣄
⠀⢀⠐⠀⠀⠀⠐⠀⠀⠀⣀⠀⠄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡹⠯⣟⡡⠊⠀⠀⠀⠀⠀⠀⢀
⠈⡂⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠰⠀⠀⠀⠀⢉⠠⠀⠈
⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡂⢀⠀⠀⠶⠐⣀⠂⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢾⡇⡻⠿⠗⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⣠⣂⠀⠀⠀⠀⠀⢰⡈⣄
⠀⡀⠀⠁⠀⠀⠀⠄⣰⠈⠂⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡦⠸⣿⠟⠀⡆⠀⠀⠀⠄⠊
⠀⠀⠀⠀⠀⠀⠳⠀⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠠⠁⠠⠄⠊⠁⠀⠀⠀⠲⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠠⠀⠀⠀⠁⠀⠀⠔⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡔⣄⠀⠀⠂⠀⢤⠄⠀⠄⠀⢀⣀⡀⢦⣀⠀⠂⡀⢘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⢙⡛⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⣡⠯⢁⠠⠤⠁⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠸⡿⠃⠐⡂⠙⢡⣥⠈⠀⠀⠁⠀⠐⡀⠙⠞⢈⣁⠉⠙⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣺⠗⠀⠀⠀⠀⠀⠀⠀⠸⠿⠂⠐⡇⠃⠀⠀⠀⡀⠈⠄⠀⣀⠸⣟⠁⠀⠀⢀⠀⠁
⠀⠀⠀⠀⠀⡀⠀⠈⠄⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⡤⠄⠀⡀⠂⠀⠓⠀⠁⠉⡀⢦⠀⠈
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠛⢈⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠼⣘⢇⠀⠀⠀⠀⠈⠀⠙⠆⠠⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠇
⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⣀⣀⡀⣠⠀⠀⠀⢤⣤⣤⡆⠀⠀⠀⠀⠁⢀⣤⢴⣦⡰⠀⢀⣠⣷⣾⣶⠀⠀⠀⠌⠀⢘⡀⣤⠄⠀⣰⡄⠂⣀⢬⢠⣔⣄⣼
⢀⣀⣈⣽⣿⡿⠛⠀⠐⣶⣾⣿⠋⠀⠀⠀⠀⠀⠘⢿⣿⣿⠛⠁⠀⠈⢹⣟⡏⠀⠀⠀⠀⠰⢿⣾⣵⣿⣿⠟⠛⠀⠈⠻⢿⣿⣻⡏⠉
⠀⢰⣿⠟⣿⡁⠀⠀⠀⢹⠏⣿⠀⠀⠀⠀⠀⠀⠀⠸⡏⡿⠀⠀⠀⠀⠸⢸⠇⠀⠀⠀⠀⠀⠀⠙⡯⣿⡿⠀⠀⠀⠀⠀⠈⠻⣿
⠀⠀⠙⢰⣿⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠃⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁⢩⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣴⡏
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢁⣤⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣤⣆
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⣼⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠋⡇
⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠇⣿⠀⠀⠀⠀⠀⠀⠀⠠⢸⡇⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣆⣷⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⣳⣇⣿⡀⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⢤⡼⣇⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⢀⢭⡿⣧⡀⠀⠀⠀⠀⠀⢦⣷
⠀⣌⢁⣽⢭⣿⠀⠀⠀⠰⢿⡾⣆⠀⠀⠀⠀⣀⣸⣿⣇⣀⣷⠀⠀⢰⣿⡏⠳⡄⠀⠀⠀⠀⠀⠀⢀⣿⣿⣦⠀⠀⠀⠀⠈⢹⣿⣇
⠀⠙⠛⠉⠛⠃⠀⠀⠀⠷⠞⠿⠏⠀⠀⠀⠀⠈⠉⠀⠙⠛⠉⠀⠘⠛⠉⠻⠟⠃⠀⠀⠀⠀⠘⠛⠋⠀⠛⠁⠀⠀⠀⠰⠶⠋⠙⠛⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣯⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⢂⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠖⠒⠶⣦⡀
⠀⠀⠀⠀⢀⣠⠶⠾⠧⣄⡀⠀⠀⣸⠠⡏⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠀⠀⠀⠀⠈⢻⡄
⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⡡⢡⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⢰⡏⠀⠀⢄⣀⡀⢀⣀⠀⠀⠈⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠟⠛⠳⣆
⠀⠀⠸⣷⡀⠲⠶⠮⠉⠉⠡⠔⠋⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠐⢤⡶⡤⠤⠄⠀⠀⢹
⠀⠀⠀⠈⠛⠿⠶⠶⠶⠾⠿⠿⠷⣶⣄⢀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠛⠁⠀⠀⢀⣀⠀⠀⢀⡀⣄⣠⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣕⠢⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠀⠀⠀⠀⢀⡴⠋⢁⡤⢶⣤⣼⣥⠴⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⡀⠀⠀⠀⠈⠳⣄⠀⠀⣠⠞⠉⠀⠀⠀⠀⣀⣴⠞⢋⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⡢⡀⠀⠀⠈⠻⣏⠁⠀⠀⣀⣠⠤⠚⢉⡥⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡕⢄⠀⠀⠈⠳⣄⠀⠿⠙⣠⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⣹⣿⣦⣁⡀⡀⠀⠈⠻⣶⣎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⢰⡞⢿⣿⣿⣿⣌⠢⡀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠉⠀⠀⠀⠀⢈⣿⣼⣿⠟⠋⠙⢷⣇⡀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⢀⣀⣀⣤⣀⡀⢀⡤⠚⠉⠀⠀⠀⠀⣀⣤⡾⣻⡿⠋⠁⠀⠀⠀⠀⠙⠻⣦⡰⣄⠀⠀⠙⢦⡀
⠀⠀⣰⠞⠋⠉⠁⠀⠀⠉⠃⠀⠀⠀⢀⣀⣴⣿⢖⣯⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣾⣦⡦⡀⠀⠙⢦⣀
⠀⠀⣯⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢴⣿⡿⣫⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡌⢳⣄⠀⠈⠙⠛⠒⠒⠒⠒⠒⠒⡒⢦⣀
⠀⠀⠹⣦⣀⠀⠀⠀⢀⠀⠀⢀⡴⢪⡵⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡄⢣⠀⠀⢦⣀⣀⠀⠀⢱⡐⠘⡀⠹⣦
⠀⠀⠀⠈⠛⠿⣶⣮⢁⠀⠀⠀⣤⣽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣾⠀⠀⢦⣭⣵⠖⠀⠀⠁⠀⣣⠀⣿
⠀⠀⠀⠀⠀⠀⢻⡆⡉⠃⠀⠀⣿⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠹⣷⣄⠀⠒⢒⣩⣃⣼⠟
⠀⠀⠀⠀⠀⠀⠈⢿⣜⡲⠶⠿⣫⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢤⣤⠈⣿⡿⣶⡾⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠠⠚⠼⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢶⣶⠾⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠴⠒⠒⠉⠉⠉⠉⠉⠉⠒⠒⠦⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆
⠀⡇⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢹
⠀⡇⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢸
⠀⡇⠀⠀⠀⠀⡇⠀⠀⣀⣤⣶⣾⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣶⣶⣤⣀⠀⠀⠸⡀⠀⠀⠀⢸
⢰⠁⠀⠀⠀⡼⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢳⠀⠀⠀⠘
⠸⠀⠀⠀⢠⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⡇
⡇⠀⠀⠀⠘⡄⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⢠⣶⣶⣦⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢀⠇
⡇⠀⠀⠀⢀⡇⠀⠈⠛⠿⢿⡿⠿⠟⠛⠉⠀⠀⠀⠀⣿⣿⣿⣿⡄⠀⠀⠈⠉⠛⠿⠿⠿⡿⠿⠛⠀⠀⠸⡀
⢡⠀⢀⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀
⠸⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠹⣗⠦⣤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣿⣿⡟⠹⣿⣿⣷⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣤⣤⠤⢺⠏
⠀⠀⠘⡆⠘⡽⣿⣿⢹⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠹⠟⠁⠀⠻⠟⠁⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣟⠇⢰⠃
⠀⠀⠀⡏⠀⢱⠘⠿⠈⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇⠻⠋⡜⠀⢣
⠀⠀⠀⡇⠀⠈⡆⠀⠀⢹⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠁⠀⢠⠃⠀⢸
⠀⠀⠀⠙⢄⡀⡇⠀⠀⠈⣿⣿⣿⣇⣀⣀⣤⡤⣤⠲⡔⠒⡖⠒⣤⠲⡤⣤⣄⣀⣠⣿⣿⣿⠇⠀⠀⢸⠀⣠⠇
⠀⠀⠀⠀⠀⠉⡇⠀⠀⠀⠸⣿⣿⣿⣿⣸⣸⣇⣽⣠⣧⣠⣧⣠⣿⣠⣇⣴⣸⣨⣿⣿⣿⡟⠀⠀⠀⣾⠋⠁
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠻⠿⠿⠿⠧⠧⠧⠼⠼⣤⢇⠧⠼⠼⠦⠧⠷⠿⢿⠿⠟⠁⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⣀⠀⠀⠀⣀⣀⡀⠀⠀⣀⡤⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠐⠂⢘⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠛⠉⠀⠉⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢿⣁⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢏⣾⣿⣿⠇⠀⣀⡀⠀⠀⠀⠀⠀⣈⠂
⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣸⣿⠟⠃⣴⣿⣿⣿⣷⣤⠀⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠏⠀⠀⣿⣿⣿⠿⠿⠃⠀⠈⣻⠹⠁
⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣄⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠍⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠟⡵⠋
⠀⠀⠀⠀⠀⠀⠀⣟⠻⡄⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⢸⣷⣾⢿⠿⠓⡄⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⣿⠗⠲⢤⣄⣄⡤⡔⢲⢢⠈⡈
⠀⠀⠀⠀⠀⠀⠀⢠⠒⠮⣓⣤⡀⠀⠀⠀⢀⣧⠀⠀⠀⠉⠀⠂⠀⢠⢦⢷⢷⡜⠛⠊⣄
⠀⠀⠀⠀⠀⠀⣠⠇⠀⠀⠀⠉⠛⠀⠀⢣⠈⡟⣆⠀⠀⠀⠀⠀⠁⠈⠈⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠙⠛⠒⠮⢆⡇⠹⡶⣤⣤⠤⢀⣀⠀⠰⠀⠀⠀⠀⠀⢁⡀
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⣷⠀⢻⡼⠿⣿⡃⣼⠅⡑⠒⠤⠤⠴⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⢻⣿⠶⢹⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢤⣀
⠀⠀⠀⣴⠋⠀⠀⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠈⢳⡀
⠀⠀⠀⣯⠀⠀⠀⠀⠀⠀⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢨⡇
⠀⠀⠀⠘⣦⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡋
⠀⠀⠀⣴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⣷⠀⠀⠀⠀⠀⢀⡗⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⣇⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠈⠳⢤⣀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⣤⠴⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠀⠀⠄⠀⠀⠀⠀⠀⢀⡠⡔⠀⠀⠀⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⡞⠐⠈⠁⠰⠀⠀⠁⠀⠀⠠⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠑⠂⡐⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠐⡄
⠀⠀⠀⠀⠀⠀⠐⡤⠤⡀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠔⡐⠀⠐⢠⠈⠄⠀⠀⢀
⠀⠀⠀⡀⢀⠠⠀⢸⠀⡐⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠂⠀⠀⠀⡁⠐⢀⠀⠈⠁
⠀⠀⠀⠀⠀⢉⡠⢆⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⠁⠰⠁⢀
⠀⠀⠀⠀⠀⠀⠀⡀⣀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠠⠀⢀⣠⠔⠀⠀⠐⠠⠈
⠀⠀⠀⠀⠀⠠⢊⢤⠀⢶⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠀⠂⠀⠀⠀⠀⠀⠀⡀⠉⠀⠀⠀⣀⠄⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠼⢆⠀⡐⠀⠀⡀⠀⠀⠀⠁⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠡⠊⠀⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢣⠥⠀⠀⠀⠀⠀⠀⠀⣀⡠⠀⠀⠀⢰⠊⠀⠈⠀⠐⡄⠠⠤⢀⡰⠀⠁⠀⠀⠢⣁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠲⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⠸⡄⠈⠀⠀⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠠⠣⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠄⠀⠀⢼⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢠⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣦⡀⠀⠀⠀⣀⣀⡀⠀⢀⣀⠀⠀⠀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡧⡴⣷⣤⢛⣿⡟⢠⣾⣿⣿⡆⠀⠘⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠐⠾⠙⠻⠿⠿⠇⠈⠿⣿⠟⠀⠀⣠⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⡿⢿⣶⣶⡄⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⡄⠹⢿⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠙⠋⠉⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣷⣄⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠙⣏⡉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠹⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡄⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣴⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⡇⠀⠠⣤⠏⠀⣤⡔⠀⢤⠦⠀⠈⠉⠁⣄⣠⣴⣷⡆
⠀⠀⠀⠀⠀⠀⣴⣷⠆⠉⠁⠉⠙⣿⠏⠙⠲⣶⣶⣶⣶⣤⣴⡆⠀⠀⠀⠀⠀⠀⠀⠈⠁⢀⣀⡀⠘⣤⡤⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⢹⣧⠀⢀⣀⠀⠀⠉⠀⠀⠀⠘⠃⠀⠀⣠⣿⣆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢉⣡⣶⣿⣧⠀⢠⣀
⠀⠀⠀⠀⠀⠀⣾⣧⣀⠈⠋⠀⠀⠀⢰⣶⣶⣤⣤⣔⣛⠋⠉⢼⡿⠇⠀⠀⠀⠀⠤⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠙⠛⠛⠷⡦
⠀⠀⠀⠀⠀⠀⢹⡆⠙⠀⡄⠀⠀⠀⠀⠀⠚⠁⢀⡹⠟⠃⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⣀⣿⠟⠋⣡⡄
⠀⠀⠀⠀⠀⠀⣺⠁⠀⠀⠀⠀⠰⣶⡬⣿⡿⠟⢻⣤⣤⣴⠿⢿⣿⠃⠀⠀⠀⠀⠀⣀⣀⠉⠀⣴⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠀⠆⠀⠙⢷⡿⠁⠠⠄⠉⠁⣀⣀⣿⣿⣿⠄⠀⠀⠀⠀⠛⠛⠁⠠⣤⣿⡟⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠿⠿⣿⠿⢿⣶⣶⣾⣿⡉⣉⣭⣅⠀⠀⠀⠀⠀⠠⣤⡀⠀⠉⣁⠀⠈⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⣤⡾⠁⠀⠊⠉⠀⠀⣠⠴⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⣷⣦⣤⣴⠟⢁⣀⣸⣶⣷⠀⠀⠀⠀⠀⠈⠓⠶⢀⣄⣀⣤⠀⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠐⠉⠉⠉⠉⠲⠦⠉⠙⠿⠃⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠛⠻⠂⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠛⠻⣶⣶⣶⣾⡉⠀⢀⡀⣠⡄⣠⣤⣀⣀⠀⠀⠀⢈⠉⢶⣶⠋⠀⣠⣀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡐⠋⠉⢈⣩⡿⠿⠿⠛⣉⣼⣿⣿⡁⠉⠀⠀⣀⣨⡀⣄⠉⠀⠀⠈⠉⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣿⣶⣾⣋⠁⠀⠄⢀⣴⣿⣿⣟⣿⣧⠀⠀⠀⠈⠛⠻⠾⠛⢶⣄⠀⠤⣄⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠉⣿⠿⠗⠲⡶⢫⡿⠿⣿⡿⠛⠃⠀⠀⠙⠷⢦⣤⣀⡹⣮⡉⠷⠄⠈⢡⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣊⠀⠀⠠⠛⡰⠇⠀⣠⣿⣇⠀⠾⠦⠀⠀⠀⠀⠈⠙⠛⢿⣧⠈⠛⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡆⠈⠛⠷⠶⠖⢉⠀⢠⡞⠉⣿⠋⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⢠⠐⢁⡔⠁⠀⠾⣿⣇⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠁⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠃⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣆⠀⢠⣼⣿⠀⢀⣤⡄⠀⠀⠀⠀⢤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣧⠀⣹⣿⠀⢈⣿⠁⢀⣴⣄⠀⠀⠈⢛⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⣿⣿⣧⣀⣽⡄⠈⠻⣿⠀⠀⠀⣀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⠀⢙⣿⣿⣿⣉⣹⣿⠀⣠⠀⠉⠀⠀⠀⠻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣯⣽⣿⣿⣿⡄⠀⠀⠀⠀⠘⢿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⣼⠿⣿⣿⣿⣿⣿⣿⣿⡿⢀⠀⠀⠀⠀⠀⠛⠛⣧⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠘⠛⠙⠟⢻⣿⠛⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⣼⣋⡉⢹⣇⠀⣀⡀⠀⠀⠀⠀⢀⣠⠂⣀⣴⣿⣿⣤⠟⠉⡹⢶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣟⣿⠟⠛⢻⣿⣾⠟⠋⠉⠑⣶⣦⣼⣿⣿⡿⠀⠉⠉⠛⠟⠛⠷⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠺⢿⣿⣧⣄⢀⣴⣿⠟⠋⠹⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡆⠀⡀⠈⠛⠿⠿⠟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⡟⠛⠙⠛⠛⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣇⣀⣀⣀⣀⣠⣄⡀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠤⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠳⡄⠀⡀
⠀⠀⢠⡴⠤⣄⡾⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠻⠋⠉⠛⣆
⠀⠀⣾⠀⠀⠀⠑⠀⠀⠀⢾⠀⠀⠀⠀⢰⠋⠁⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⣠⠼⠃⠀⠀⠀⠀⠀⠀⢀⡼
⠀⠀⠘⠦⣄⣤⣄⠀⠀⠀⠈⠻⢦⡀⠀⠸⣆⠀⠀⠀⠺⢤⣀⠀⠀⠀⣀⡴⠋⠁⠀⠀⠀⠀⢀⣠⡴⠞⠋
⢀⡖⠋⠉⠉⠓⣎⠳⣄⠀⠀⠀⠀⠙⢦⣾⠁⠀⠀⠐⠀⠀⠈⠙⣲⠞⠁⠀⠀⠀⠀⠀⣠⠔⠋⠁
⠸⡅⠀⠀⠀⠀⠀⠉⠙⠒⠒⠒⠲⠤⠤⠽⢿⣶⣶⣲⣤⣄⡴⠊⠀⠀⠀⠀⠀⢀⣤⢾⣥⣀⡀⠀⠀⠀⢀
⢠⠟⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠒⠲⠤⣄⣀⡠⠊⠁⠀⠀⠈⠆⠉⠛⠶⠋⠉⠉⠳
⡏⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣠
⢷⡀⠀⠀⠀⣰⠋⠉⠉⠛⠛⠒⠲⣦⠤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠧⣄⡀⠀⠀⠻⡏⠁
⠀⠙⠲⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠈⠓⣤⠞⠉⠉⠉⢛⠶⢶⡦⠤⣤⣄⠀⠀⠀⠀⠀⠈⡆⠀⠳⣤⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠶⠶⠊⠀⠀⠀⢀⡴⠁⠀⠀⠙⠢⡀⠻⣀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⠀⡶
⠀⠀⣰⠲⡄⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠰⡟⠳⣄⠁⠀⠀⠀⠙⠛⠻⡍⠀⠉⠁⠀⣀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠈⠉⠉⠀⠀⠀⠀⠘⠦⠤⠴⡆⠀⠀⠀⢀⡗⠀⣾⠀⠀⠀⣤⣀⣀⣤⠇⠀⠀⠀⣾⣉⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠖⠋⠀⠀⠙⢦⣀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⣿⣿⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⣠⣶⣶⣦⡀⠀⠀⠀⠀⣤⣤⣤⣄⠀⠀⠀⠀⢀⠐⠒⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠂⠒⢄
⠀⣀⣀⣿⣿⣿⣿⣿⠀⠀⠀⠸⠿⠿⠿⠟⡃⠀⢀⡀⣸⣤⣤⣀⣰⡇⠀⠀⠀⠀⣦⣠⣤⣠⡆⠀⣀⡀⡁⠀⠀⠀⠀⡀
⠘⢿⣿⣿⣿⣿⠟⠋⠀⠺⠿⠛⠇⠀⢠⠶⠁⠀⠢⣤⣤⣾⣿⣿⠝⠁⠀⠀⠤⣤⣴⣿⣿⠋⠁⠀⠄⠀⠀⠀⣠⡤⠒⠀⠀⠀⠀⢀⣠⠔
⠀⠘⣿⣿⣿⣿⠀⠀⠀⠀⢾⡀⣠⠀⠎⠀⠀⠀⠀⠱⢿⠟⣿⣇⠀⠀⠀⠀⠀⢿⡟⢻⡇⠀⠀⠀⠀⡀⠀⠀⡀⡇⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠹⣿⣿⡏⠀⠀⠀⠀⠘⣧⣿⠀⠃⠀⠀⠀⠀⠘⡞⢠⣿⠇⠀⠀⠀⠀⠀⠐⠁⣿⡇⠀⠀⠀⠀⢱⡀⠀⢡⠁
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⡀⠀⠈
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⢰⠀⠀⠀⠀⠀⠀⢸⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠸⠀⠀⠀⠀⠀⠀⢸⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠀⠀⠀⠀⠀⠇⠀⢠
⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⢸⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⠀⢸
⠀⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⢸⢿⠀⡆⠀⠀⠀⠀⠀⠰⡄⣿⠀⠀⠀⠀⠀⠀⠀⢰⣼⡆⠀⠀⠀⠀⠀⠇⠀⢸
⠀⠀⢨⣿⣿⣿⡀⠀⠀⠀⠀⢸⠰⡆⠇⠀⠀⠀⠀⠀⣿⣿⣻⣆⠀⠀⠀⠀⠀⢈⣿⣸⣇⠀⠀⠀⠀⢈⠃⠀⠀⣄
⠀⢰⣿⣿⣿⣿⡗⠀⠀⠀⡰⣾⠆⢐⠂⠄⠀⠀⠀⣎⠀⣽⠧⣿⠀⠀⠀⠀⢠⡁⣸⡿⣼⡇⠀⠀⢀⠁⠀⠀⠀⢸⠀⠀⠀⠀⠀⡀⠀⡀
⠀⠀⠛⠋⠉⠛⠁⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠙⠛⠉⠛⠋⠀⠀⠀⠀⠈⠛⠋⠉⠛⠀⠀⠀⠀⠃⠐⠙⠂⠈⠀⠀⠀⠀⠈⠉⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡌⠛⠛⠛⠛⢡⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣀⣤⣦⠀⠀⣴⣦⣠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠉⠉⢀⡀⠉⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣶⡄⠀⠈⠁⠀⢠⣶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢤⣤⣤⡤⠸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣄⡈⢁⣠⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣥⣬⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡖⠚⣿⡄⢀⣿⠓⠲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠚⠛⠛⠛⠛⣿⠟⢿⡿⢿⣿⠿⠿⣿⠿⢻⡿⠻⣿⠛⠛⠛⠛⠓⣔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢻⣿⣿⡟⠛⠛⣿⣿⣟⠛⠉⠛⣤⣤⠛⠉⠛⣻⣿⣿⠟⠛⢻⣿⣿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣶⡟⠻⣇⣀⣿⣏⣙⣛⠛⠛⣙⠟⠛⣛⠙⠛⣿⣋⣹⣿⣀⣸⠟⢻⣷⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢸⣿⣆⠙⣿⠟⠛⢛⣿⠀⠀⣿⠛⠛⠻⣿⢋⣴⣿⡇⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠘⣧⣿⣿⠿⠦⠀⢙⣿⠀⠀⣿⡃⠠⠶⠿⣿⣿⣾⡇⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⣰⡏⠻⠇⢴⣶⠖⢉⣻⣶⣴⣿⣁⠐⠶⠦⠺⠟⢹⡇⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⠀⣉⡙⠶⠶⢦⣤⣴⠟⠉⣿⣿⢿⣋⠰⣶⣶⡶⠶⢏⣉⠆⠀⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠙⠳⠆⢀⣀⣿⠟⠂⠀⢻⡿⠈⠛⢾⣿⣤⣤⣤⡜⠛⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠿⢧⡠⣾⡟⠁⠀⠐⠀⣿⣿⠀⠶⠀⣻⣿⣿⣯⣽⠿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⢋⡀⠀⢰⡟⠁⡘⠃⣀⡀⠒⠀⢿⣿⠀⠰⠂⣈⠻⣿⡉⠹⡖⠀⠀⠛⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⢸⡇⠀⠀⠻⠛⠀⠀⠉⠀⠀⠀⣿⣿⠀⠀⠀⠈⠂⠈⠛⠃⠀⠀⢸⡇⠈⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣴⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣶⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢸⡟⠀⠀⠀⠀⠀⣠⣤⣤⡀⠀⠀⢤⣿⣿⡄⠀⠀⠀⢠⣤⣤⠀⠀⠀⠀⠀⢹⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⢀⡌⠁⣄⠁⠀⠀⠀⢠⡞⠁⠀⠀⠛⣦⣤⣤⣿⢿⣤⣤⣤⡾⠋⠀⠀⠙⠇⠀⠀⠀⠈⣠⠈⢡⡀
⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠻⣿⠛⢥⡾⠛⢀⢿⣇⠀⠀⠀⢠⡖⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⢰⣿⠀⣼⠁⠀⠀⠀⠀⠀⠈⠳⣄⠀⢿⡀⠀⢿⡀⠀⣴⠋⠀⣉⠀⢀⡴⠋⠀⠀⠀⠀⠀⠈⣳⡀⣽⡆
⠀⠀⠀⠀⠀⢀⣼⠃⢸⡏⠀⠀⠀⠀⠀⢀⡀⠀⣹⣷⣀⣿⠀⠀⢿⣦⠋⠀⢈⣿⣤⣾⡁⠀⣀⠀⠀⠀⠀⠀⢻⡇⠸⣧⡀
⠀⠀⠀⠀⠀⣸⣿⣴⣦⠀⠀⠀⠀⠀⠀⠙⠳⠞⢻⠿⣿⠋⣶⣄⡀⠉⣀⣤⣝⠛⠻⢿⣳⠞⢻⡆⠀⠀⠀⠀⠀⣿⣤⣿⣇
⠀⠀⠀⠀⣠⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢸⡄⢰⡆⠀⠘⢀⣉⡛⠿⠿⠛⣋⡙⠗⠀⠈⡏⠀⢸⠇⠀⠀⠀⠀⠀⠘⢿⣿⣿⣄
⠀⠀⣀⠰⢿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠘⠃⢀⠳⠀⠀⠀⠉⠁⠀⠀⠀⠈⠁⠀⠀⠸⡇⢠⡇⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⠦
⠀⣀⠿⣷⣦⡈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⣵⣾⠟⣀
⡀⠙⣦⠈⠻⡇⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡷⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⢸⠟⢁⡴⠋
⠀⣦⠈⠳⡀⠃⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢀⠟⠁⣴
⠀⠀⠃⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠂⠀⣿⠀⠀⠀⠀⠀⠀⣀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣷⣶⣼⣷⠀⠀⠀⠀⣤⡛⣶⣶⣾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣀⣹⡿⡿⠂⠀⠀⠀⠹⣿⣏⣉⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠿⠟⠻⣿⠀⠀⠀⠀⠰⣿⠟⠟⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⠀⣾⠀⠀⠀⠀⠀⠀⠙⡆⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣹⣿⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⢸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠻⣧⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⣿⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢿⠀⠀⠀⠀⠀⠀⡆⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣰⣿⠀⠀⠀⠀⠀⣿⣴⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⠋⠀⠀⠀⠀⠀⢿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⣿⣿⣷⠀⠀⠀⠀⠰⣿⣿⣿⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⠛⣷⡟⠀⠀⠀⠀⠀⠀⠈⢻⣼⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣭⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣾⢻⡦
⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⡉⢀⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⠈⣡⣄⢀⣀⣀⣀⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠟⠉⠈⠛⠉⠋⠉⠉⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣀⣀⣀⣀⣀⣄⣀⣀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣿⣿⣿⣿⣿⣛⣿⣿⣻⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠠⠌⠤⡤⢀⢀⠀⠂⠀⠀⠀⠀⠁⠀⠂⠀⠼⣿⡁⣶⢖⠐⣂⡀⢀⠀⡤⠁⠀⠀⠴⠀⠀⠻⠦⠀⠀⠀⠓⢀
⠠⠤⠀⠀⠁⡄⠀⠀⠀⡀⠀⠤⠴⢤⠀⠀⢣⡄⠀⢁⠀⠘⠈⡭⠙⢃⠘⠉⠁⠈⠁⠾⢀⠆⠀⡔⠋⡀⠀⠀⢈⡀⠀⢀⠀⠰
⠀⠀⠐⠀⡄⣅⠀⠂⠀⡇⠀⠀⠒⠊⠀⠀⠀⣇⣄⢥⠉⠀⠂⠀⠰⢆⠀⠀⠀⠤⠒⠀⡬⠀⠾⠃⠀⠤⠀⠀⠀⠀⢨
⠠⠀⠂⠃⢀⡈⠳⠒⢚⡷⠖⠚⠁⠀⢐⡄⠀⠀⡁⢈⣀⡦⠀⠀⠀⠈⠀⠀⠠⠤⠀⠀⣷⡄⠀⡈⠲⠀⣠⠜⠃⠄⠠⠄⣠⠄⠁
⠀⠀⢀⣍⡀⠃⠀⢀⢺⡀⠀⠀⠀⠀⠈⢀⠀⠠⠀⠋⠈⠂⠀⠀⠀⠀⠰⡀⠈⠉⠀⠠⢟⠂⠀⠧⠖⠀⠐⢄⠀⡗⠀⣤⠀⠀⠇
⠀⠂⠐⠈⠋⠁⢁⠚⠉⠷⢦⣠⡄⠄⠚⠥⠀⠀⠄⠰⡄⠂⠃⠁⠀⠀⠀⠈⣀⡀⢉⡑⣀⠀⢀⠀⠀⠉⠁⠀⣦⠃⠐⢀⠄⠘
⠀⠀⢀⠳⠒⠌⠰⠶⢴⠂⢴⠄⣃⣀⠀⠠⢀⠒⡆⠨⠈⠀⢈⡀⠀⠁⠀⠂⠋⠀⠀⠀⣭⡀⠁⠒⢀⡀⡀⠰⠁⠀⣴⡀
⠀⣀⠁⢥⡔⣠⣤⠴⠒⠘⠋⢁⡀⠀⠉⢦⠒⡀⠀⢐⠤⠈⠀⠙⡒⠀⠇⠠⡆⣀⣀⡠⠋⢀⠂⠠⠀⢹⢅⠠⣠⠀⠀⠀⠶⠁
⢀⡸⢀⠀⣀⣞⠁⠀⠀⢠⠀⠧⣭⠂⠀⠀⢿⠁⢄⣸⠤⡥⠤⠯⣿⡆⢤⡀⠀⠉⠉⡀⠀⠀⠀⠀⡠⢈⠀⠀⠁⣰⠉⠉⠃
⠀⠒⠄⡀⠍⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢸⠋⠀⢀⣤⡀⠹⠦⠁⣀⠙⠀⠈⠀⠐⠑⠀⣒⣉⠀⠀⠁⡀⠀⡟⠀⠒⠀⠈
⠐⡄⠉⠀⠈⠛⠀⠀⡀⣄⣀⡤⠀⢀⣤⣂⠀⢼⡀⠀⠘⠓⠀⠀⠉⠀⠻⠄⢀⠀⠘⠀⠁⠀⠈⠀⢀⠀⠒⠏⠙⠤⠀⠀⠀⢀⠆
⠀⠉⢦⠃⣁⢷⡜⠀⠀⡄⠁⠐⠒⣀⡤⠥⣴⡉⠻⢆⠀⠀⢤⣤⣖⠜⠀⠂⠀⠀⠢⠀⠀⠀⠲⠖⠮⣗⠀⠈⠰⢀⠀⠤⠀⠀⡆
⠀⢠⠠⢂⡌⠨⠓⠀⠀⢀⠐⡽⠂⠀⠀⠀⠀⠉⠦⡀⢁⡀⠼⠟⠛⠶⡀⠂⣃⠱⠀⢠⠀⠘⠀⠠⣄⠈⢷⠀⢥⠘⡄⠀⠀⣸⠃
⠀⢂⡀⡀⠟⢛⣀⣩⠀⠈⡰⠀⠀⠀⢸⡙⠆⠀⠀⢃⡄⠀⠀⣠⡄⠄⠘⡁⠈⣠⠀⠀⠀⢀⠀⠈⠀⠀⢸⠀⣄⢀⠘⠁⠹⠈⡄
⠀⠊⡉⡐⡶⠂⠨⠡⠄⡠⣃⠀⠀⠀⠀⠁⠀⠀⠀⠏⠄⡀⠀⠈⠄⠀⢸⢙⡌⢃⡖⠀⠀⠈⠀⠀⠀⠴⠛⠠⠀⠈⢠
⠠⠀⢄⠒⣺⡶⣦⠠⠤⠄⣁⠰⡠⠀⠀⠤⠀⠠⢎⢀⠓⣜⠒⠰⢶⣤⠁⠀⠠⠘⠁⠀⠀⣁⡀⠀⠀⠐⠀⠐⣰⠀⠀⠆⡠⠀⠀⢀⠄
⠀⢠⠬⠟⠁⢀⣄⠒⠠⠀⠈⠳⣦⠀⠠⡅⠀⠀⢀⡀⢁⡀⠹⠄⠄⠦⠀⢠⡴⠀⢆⠀⠠⠌⠁⠀⠀⠆⠀⠀⠀⠣
⠀⠀⠀⠀⠀⠰⠀⠁⠀⠀⠀⢀⠁⢠⠠⠇⠀⡀⠀⠀⡠⠩⠀⣀⡨⠗⡀⠐⣀⠂⠈⡳⢀⠀⠀⠄⢁⠀⠀⠀⠀⠀⠨⠐⠀⠀⠄
⠀⠀⠀⡓⡤⣀⣀⡀⣠⡤⠖⣣⢏⠀⠔⠀⠂⡁⣰⠎⠉⠑⢜⡋⡙⢼⠋⠉⠐⠀⠈⠑⣒⢢⡄⠸⠀⠄⡀⠀⠡⢤⡀⠂⠌⢠⠀⠞
⠀⠀⢰⡄⢙⠣⠀⣤⠀⠒⠴⣯⡁⢠⠀⠀⠑⢡⡇⠀⠀⡀⠀⠉⠁⢀⠀⠰⠟⠂⠀⠀⠈⣱⡌⠀⠃⠲⡔⣄⢀⠀⠙⢄⠠⠄⠂⠐
⠀⠷⠚⢧⡠⠀⠙⠹⣓⡀⠀⠈⢹⠀⠰⠀⠀⠈⢤⠀⠀⠀⡀⠀⠀⠋⠑⠀⠀⠀⠀⠀⣠⠉⠀⠀⢀⠐⠠⠈⠀⠀⠀⠀⠀⠀⠋⠀⠁
⠀⠀⠈⠀⠉⠳⡄⠤⡈⠠⠄⠀⠛⢳⣔⠀⠈⢀⠈⠓⡤⠤⣴⡒⠚⠉⡀⠂⠴⣂⣠⠾⠋⢀⠀⣀⠚⠀⢄⠀⠁⠀⢀⠁⠀⡄⠀⠐
⠀⠀⠀⠂⠀⢀⠇⢀⠁⠀⡀⠀⠀⠀⡟⢰⡀⡙⢱⠁⠀⢼⢉⣭⡅⠀⢀⠠⣝⣀⣈⣶⠚⠋⢅⠀⠣⣄⠀⠀⠀⠡⠄⠀⡀⢳⠁⠰⡔
⠀⢒⡲⢤⡴⠊⠛⠸⠆⡀⣈⣁⣠⣼⢡⣶⠠⠀⠸⢆⡀⡀⠈⠁⠀⢀⢿⡛⠉⠀⠈⠊⢙⣍⠢⠈⡈⠚⠁⠰⠀⠀⠀⠀⠈⠂⠑⠀⠉
⠀⠀⠈⠁⠐⠐⠠⠖⠩⠆⠁⠂⡤⣠⡿⣤⡘⢣⠆⠄⢓⣀⢿⢓⣶⣦⡜⠅⠀⠚⠀⠀⠈⢃⢀⢠⡄⠠⠀⡐⠀⠂⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠙⠙⠋⠙⠙⢆⠀⠀⠀⠀⢀⡼⡡⠠⠀⠀⠀⠀⠀⠈
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣮⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⠄⠒⠒⠒⠒⠒⠠⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠢⢄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠙⢆
⠀⠀⠀⠀⠀⢀⡜⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⢀⡞⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⢀⠳⠆⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠸
⠀⠀⠀⢸⠀⣿⢿⡆⠰⠖⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠒⠒⠒⠀⠠⣟⣼⠀⡇
⠀⠀⠀⢸⠞⣿⡞⠀⢀⣤⣶⣶⣶⣶⣶⣔⠒⠒⠀⠀⠀⠀⠘⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⡀⠑⠀⠻⡗⠀⡇
⠀⠀⠀⢸⢲⡿⡤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⡀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⣄⣽⡄⠇
⠀⠀⠀⠈⣿⢧⠑⣿⣿⣯⣯⠍⠩⠝⠛⣙⢻⡇⠀⠀⠀⠀⠀⠀⠶⢠⣿⢟⣋⡙⠋⠻⢝⣻⣿⣿⡏⠻⠈⣷
⠀⠀⠀⠀⢿⢸⢸⣿⡯⣽⣏⠀⠄⠠⢙⢿⣿⡇⠀⢀⣀⣀⣤⣀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠙⣾⣿⣧⠀⠀⣿
⠀⠀⠀⢀⡇⣸⡼⣿⣯⡿⠻⣦⣾⣂⣦⣼⠟⠀⢠⣿⣿⣿⣿⣿⣄⢄⠻⢿⣍⣠⣶⡴⣤⣴⣿⣼⠟⠀⠀⢹⡀
⠀⠀⠀⣼⣸⠃⠀⠨⠛⠛⠛⠋⠛⠉⠁⠠⠶⠀⣿⣿⣿⣿⣻⣿⣿⡇⡕⠠⠄⠉⠉⠻⢛⡛⠛⠉⣠⣆⠀⠀⡇
⠀⠀⠀⢻⢹⡄⠀⠀⠌⠀⠀⠀⠀⣤⠄⠀⠀⣰⣿⣿⣏⣿⣟⢻⣿⣧⠈⠀⠀⠘⣿⠙⠀⠉⠳⣄⠸⡇⠀⢠⡗
⠀⠀⠀⠀⠳⣙⢻⡦⠀⠀⠀⣀⡀⠀⠀⠀⠀⢿⣧⣽⣿⡿⣿⣿⣿⣿⠇⠀⠀⠀⠀⣀⣀⣀⣀⣈⣀⣹⣦⡾⠃
⠀⠀⠀⠀⠀⠈⠙⣷⣿⣶⣦⡈⠻⣷⣦⡴⡗⠀⠙⠛⠛⣧⠙⠛⠛⠉⠁⠆⢠⣶⠿⣿⣽⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⢹⡟⢿⣿⣿⣆⣜⢿⣠⡃⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠁⣰⣾⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢋⠈⢿⣿⣿⣿⣿⣽⣡⡁⢸⠀⣤⠀⢤⠀⢆⡀⣠⣀⠀⠁⢿⠿⢿⡿⡟⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠸⢿⣧⠙⡛⢛⠉⢹⡏⢹⡟⠙⡟⠙⡿⠉⣿⠉⡟⢿⡏⢀⣾⢡⢱⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠘⠃⠙⢷⣥⣌⣣⢸⠀⣰⢁⣸⣇⣀⣀⡀⢹⠀⣷⣿⡥⣾⠟⢸⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡇⠀⠀⠈⠈⠙⢻⡟⣿⢻⣟⢻⠿⠟⣿⠿⣿⢏⣿⠀⠀⢀⠒⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢗⡸⠇⠀⠀⠀⠈⠓⠞⣧⠼⠔⢯⡬⠤⠢⠿⢬⠿⠌⠀⠀⠀⠨⢀⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣄⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⠀⠈⡇⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣴⣶⣶⣶⣦⣀⣀⣀⣀⣠⠔⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣤⣤⣦⠃⠈⣧⣤⡘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⣾⣿⠅⠀⠘⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⢿⣧⣻⠅⠀⢀⠀⢺⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣧⣼⠟⢃⡀⠀⠀⢠⠀⣿⣿⡛⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⡨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣀⠀⠀⠈⢠⠌⣰⣿⡇⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡀⠻⢦⡤⠯⢶⣟⡛⢻⡇⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠭⠄⠨⠤⢙⣷⣇⠀⠀⢸⣧⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠈⣿⣷⣺⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⠀⠀⠀⢸⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠛⠇⠀⠀⠀⠀⠀⠀⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⢠⡀⣠⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠀⠀⠀⠾⢏⠘⠃⡛⠗⠀⠀⠈⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⠁⠀⠛⢳⠛⠀⣀⣠⣼⣶⣼⣧⣂⣀⠀⠛⣛⡁⠁⠈⠲⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⣴⠋⠀⣸⠈⠃⠀⢈⣻⣿⡟⢻⣿⣿⡁⠀⠘⠉⣧⠀⠘⣦⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣼⢁⡀⠙⢓⠞⠋⢠⣄⠀⠛⢿⣿⢷⣾⣿⡿⠛⠁⢀⣤⠙⠓⠚⠋⣀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⢺⢱⡁⠈⠁⣀⣀⣹⣏⠁⠚⠉⢿⣿⠋⠛⠀⢉⣿⣁⡀⠈⠋⢙⡎⣗⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠈⠂⠛⠚⢛⡉⣁⡠⣀⠀⢠⣀⢸⡏⣀⡄⠀⣁⢉⣍⢙⡓⠶⠚⠡⠃⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⣀⣶⣾⡿⠿⠿⣿⣶⡆⠙⠻⠾⠛⢱⣶⣿⠿⠿⢿⣷⣶⣀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣨⣿⡿⠁⠀⠀⠀⠀⠹⣿⡗⠀⠀⢻⣿⠏⠀⠀⠀⠀⠈⢿⣿⣅⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⣽⣿⡇⠀⠀⠀⠀⠀⣰⣿⡟⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⢸⣿⣯⡀⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠿⣿⣶⣤⣤⣤⣾⣿⣏⠉⣼⣧⠉⣻⣿⣷⣤⣤⣤⣶⣿⠿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢣⢄⡛⠙⠿⠻⠟⠟⠁⠀⣼⣿⣿⣧⡈⠈⠿⠻⠟⠿⠋⠋⡤⢲⣘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢩⠆⡥⠄⠀⠀⠈⣿⣦⢸⣿⡿⢿⣿⡇⣰⣾⠃⠀⣀⣀⠒⡖⠷⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⡶⠗⠚⠛⣦⠀⠀⠛⠀⠉⠁⠈⠉⠀⠙⠉⠀⠘⠋⠉⠙⢛⡛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡄⠀⢀⡏⣰⡐⡦⢰⢐⡇⢸⡀⠇⢰⣒⣦⠀⠀⠀⢀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠃⣿⣿⠿⣿⢿⣿⣿⡷⣿⢿⣟⣫⠐⠃⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⣷⣤⡆⠉⠛⠷⣿⢶⣷⣾⡷⣿⠿⠛⠋⢰⣤⣾⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣝⣻⣿⣃⣠⣤⣤⡈⢱⡎⢁⣤⣤⣤⣘⣿⣿⣫⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⡉⢥⡰⠰⠈⠘⢃⠪⠄⣆⣤⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠰⡞⡷⠀⣼⢶⠻⡎⠱⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠙⠓⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠛⢶⡀
⠀⠀⠀⠀⠀⢀⡤⢤⣀⣼⠁⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠷⠚⠙⠳⣄
⠀⠀⠀⠀⠀⡟⠀⠀⠈⠛⠀⠀⠀⢸⡃⠀⠀⠀⠀⣴⠛⠉⠉⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡷⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⠀⠀⠻⣄⡀⢀⣀⠀⠀⠀⠈⠳⣄⠀⠀⠀⢿⠀⠀⠀⠀⢼⣀⠀⠀⠀⠀⠀⣀⡴⠚⠁⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁
⠀⠀⠀⠀⡤⠶⠚⠻⢯⡙⢦⡀⠀⠀⠀⠈⠛⢦⡀⡾⠋⠀⠀⠀⠀⠈⠙⠳⢤⣤⠞⠁⠀⠀⠀⠀⠀⢀⡠⠞⠋⠁
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠙⠲⠿⠤⣤⣀⣀⣀⣀⡙⢿⣦⣀⣠⣄⡀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠀⣀⣴⠋
⠀⠀⠀⠘⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠙⠓⠚⠿⠿⣏⣀⡀⠀⠀⠀⢀⠔⠊⠉⠙⢻⡓⠶⣤⣠⡴⠶⠶⢤⡀
⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠁⠀⠀⠀⠀⠀⠇⠀⠀⠈⠀⠀⠀⠀⢹
⠀⠀⢸⡇⠀⠀⠀⠀⠀⡴⠦⢤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⢀⠄⠀⠐⠀⠀⣤⡤⠴⠋
⠀⠀⠀⠳⣄⣀⣀⣠⠞⠁⠀⠀⠀⠀⠈⠉⠙⢯⡛⠒⣶⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⢯⠛⠲⣄⠀⠀⢈⡇
⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠚⠁⠀⠀⢀⡴⠋⠛⢿⡛⠒⢶⠀⠀⠀⠀⠀⠀⢸⠇⠀⠙⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠛⠛⠋⠀⠀⠀⠀⣠⡊⠀⠀⠀⠀⠙⢦⣈⣣⣄⣀⣀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⣾⣹⡄
⠀⠀⠀⠀⠀⣾⣙⣆⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠸⡍⠙⢦⡀⠀⠀⠀⠀⠉⠉⢳⠉⠉⠁⠀⣠⣤⡀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠓⠶⠶⢻⡀⠀⠀⠀⣰⠇⠀⣿⠀⠀⠀⣴⢤⣤⡤⠞⠀⠀⠀⠰⣯⣀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠶⠖⠚⠁⠀⠀⠘⠦⣤⡴⠃

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⣶⢶⣶⢷⣯⣄⣐
⠀⠀⢀⡀⠀⢀⣠⠈⠉⠐⠛⠿⣦⠂⠉⠉⢠⠶⠾⠿⠛⠛⠋⠉⠉⠛⠿⣿⣿⣤⡄⢀⣂⠀⠀⠀⠀⠈⠁⠀⠀⠑⠀⠂⡒⠂
⠀⠘⠋⠀⢀⢤⡀⠀⠀⠀⠀⠀⠘⢂⣄⠘⣾⣄⠀⠖⢀⣀⡀⢤⡀⠄⠀⠀⠀⠀⠀⠈⠉⠛⢻⣿⣿⣗⣦⠀⠁⠂⠀⠀⠀⠈⠀⠂
⠀⠀⠀⠀⢀⣨⣽⢶⡶⠶⢾⣭⠤⠶⠶⠳⡿⠷⢿⣿⣯⣥⣀⣀⣡⣤⡎⢁⣠⠔⢨⠷⠞⣛⣨⣽⣭⡽⠿⢿⣷⣶⣤⣤⣤⣀⡀⠀⠀⠯⠶
⠀⠀⠀⠚⠉⠀⠰⠿⠶⣶⣶⣶⣦⣤⣴⡦⠶⠶⠶⠾⠿⠻⣿⣟⣻⢿⣟⠛⠀⠈⠀⢘⣿⣮⡄⠀⠐⠐⠒⠙⠉⠉⠉⠛⠀⠭⢍⣉⡀
⠀⠀⠀⠀⠉⠀⣦⣬⣽⣍⣈⣙⢻⣿⣿⡟⡟⠿⠉⠉⣈⣭⣽⡿⠿⠾⠿⢮⣶⡒⠛⠁⠀⣈⣙⠳⢤⣄⡒⠂⠀⠀⠀⠀⠠⠤
⠀⢒⣛⠛⠞⠛⠻⣿⣿⣿⣿⣛⢉⣥⡞⠀⠁⣀⣄⣀⡉⣉⣤⡤⠆⠐⠀⠰⠁⠀⢄⣀⢀⡿⣿⣿⣦⣍⡛⠷⠂⠠⠤⠀⠠⠄
⠀⠠⠒⠒⠚⠉⠉⠉⠉⠙⠛⠛⠻⠟⠾⡍⠉⠉⣠⣬⠿⢟⠫⠄⠀⠀⠒⠂⢀⡀⠀⠈⠀⠉⠑⠉⠓⠈⠉⠀⠀⠀⠀⠀⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠀⠀⣘⣫⣬⡤⠖⠚⠁⠀⣀⣰⣶⠾⠟⠛⠛⠉⠉⣭⣤⡳⣢⣤⡀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠛⠻⠿⠛⠋⣉⠀⠀⠀⠀⠀⠐⠣⡖⠿⡄⠔⠈⠘⠿⠦⠌⠂⠄
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣀⣠⣤⡀⡀⢀⢀⣤⣤⣀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢰⣿⣿⣿⣿⢀⡀⣿⣿⣿⣿⠇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠉⠉⡁⢰⣿⣿⡏⢈⠉⠉⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡶⣦⠀⠘⠋⠙⠃⠀⣴⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⣀⢀⠀⠀⡀⢀⣿⠰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠛⢢⡐⢀⡔⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡄⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠒⠒⠀⠠⣀⠀⠀⠀⢀⠀⢀⣤⣼⠆⢰⣧⣤⣀⠠⡀⠀⠀⠀⣀⠄⠀⠐⠒⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣉⠉⢙⣛⠛⣷⣠⣭⣀⠚⠷⠾⠍⡁⢿⣄⣠⡿⢈⠩⠵⠾⠒⣀⣥⣄⣾⠛⢛⠋⠉⣉⣤⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠈⠴⠢⣦⣯⢀⡀⠈⢡⣴⠱⣿⠼⣻⢶⣤⠜⠀⠀⠀⠀⠣⣤⡶⣟⢧⣿⠎⣤⡈⠉⢀⡀⣽⣼⠔⠦⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠈⣿⢸⣧⢠⣿⣽⡀⠹⠗⠀⠈⠙⣷⣄⠀⠀⡠⣾⠋⠁⠀⡺⠏⢀⣿⣽⡄⣼⡇⣿⠁⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠉⠀⠉⠸⣯⣾⡿⠿⣷⡇⠡⣤⣿⠃⠀⠀⠘⣿⣤⠎⢰⣶⠿⢿⣷⣽⠇⠉⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⢸⢲⢣⡙⢷⠀⠀⠠⠀⠀⢀⡤⠁⠀⠀⠈⢤⡀⠀⠀⠀⠀⠀⡾⢋⡜⡖⠇⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⢿⢿⡽⢳⠝⠛⠁⠀⠈⡩⢶⡤⠀⠀⢤⡶⢍⠁⠀⠈⠛⠫⡞⢫⡿⡿⡀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸⣆⡛⢤⣈⠍⠛⠊⠉⠁⠘⡁⠀⠀⢈⠃⠈⠉⠑⠛⠫⣐⡤⢛⣴⡇⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⡌⠻⡙⠲⠥⣢⡄⠀⠠⠴⢋⡑⠀⠀⢊⡙⠦⣐⠀⢀⣰⡬⠔⢏⠟⢡⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢸⣟⢤⠀⠄⠀⠀⠈⠀⢀⡶⠀⠀⠀⠀⠀⠀⢶⡀⠀⠉⠀⠀⠠⠀⡠⣻⡇⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠈⣿⠣⡁⠂⠠⢤⠤⠔⠋⠔⠀⠙⡀⢠⠛⠀⠢⡙⠢⠤⡤⠄⠐⢈⠜⣿⠁⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⣿⡇⢤⡀⠀⠀⣀⣊⠄⢀⠀⣤⠁⠈⣤⠀⠀⠠⣑⡀⠀⠀⢀⡥⢺⣿⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⣿⣆⠀⠈⠈⠉⠉⢀⠐⢠⡞⠳⠆⠰⠟⢳⡄⠂⠀⠉⠉⠁⠁⠀⣰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⡂⠀⠀⠈⠁⠀⠀⣠⣀⠂⠁⠀⠋⠀⣻⡄⢠⣟⠀⠙⠀⠈⢂⣀⣄⠀⠀⠈⠁⠀⠀⢠⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠉⣶⠀⠀⣶⠉⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⠀⣱⠀⠀⠀⠀⠀⠚⠛⠋⠁⠀⠀⠀⠂⣍⠁⠈⣩⠐⠀⠀⠀⠈⠙⠛⠓⠀⠀⠀⠀⠀⣎⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠶⠿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠂⠐⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠦⠃⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠀⠉⠀⠀⠀⠈⠁⠐⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢡⠇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⣿⡇⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡁⢸⣷⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⣿⠄⠉⠁⣀⣀⡈⠁⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢿⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢟⢲⣶⣿⣿⣿⣿⣷⣾⠓⢳⢶⣿⣿⣿⣿⣷⣶⠺⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢹⢸⣿⣿⣿⣿⣿⠿⢿⡅⡝⠸⣿⣿⣿⣿⣿⣯⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠈⠳⠽⠝⠋⠉⠑⣢⣼⣿⣿⡜⠦⠍⠙⠉⠋⠕⠁⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⡀⠀⠀⠀⠀⠐⠀⠛⣿⣿⣿⡇⠀⠀⠀⠀⠀⡀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⡟⢳⡀⠀⠀⠨⡋⠫⠁⠀⠀⣴⠛⣿⣿⡗⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢹⣜⠧⠤⣄⣄⣀⢠⣤⣤⡤⢟⣼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠻⠗⠲⠤⢤⣠⣀⠤⠔⢓⢥⠋⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⠀⠀⠀⠀⢁⣐⣒⣀⠂⠁⠀⠀⠀⢀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄⠀⠀⠀⠉⠁⠀⠀⢀⡰⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⠤⠤⠤⠤⠔⠋


⠀⢀⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣦⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⣠⣀
⠀⠀⠐⣷⡖⠀⠀⠀⠀⠀⢠⣄⠀⠀⠸⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣿⠃⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⣾⣿⣿⣿⡆⠀⠀⣼⣿⣿⣧⠀⠀⢀⠻⠟⢗
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⠛⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⡇⠀⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠻⠿⢿⣿⡖⠀⠀⠸⣾⣷⠇⠀⠀⠌⢸⠶⢿
⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⣾⡟⠀⠀⢰⡿⡏⠂⠀⠀⠀⠀⠀⠀⠀⠟⣿⢿⡟⣿⠛⡄⠀⠀⠀⠠⣧⣤⣤⡤⠤⢤⡀⠈⠓⠂⠀⠀⠀⠓⠚⠀⠀⠀⠀⢸⢰⢸
⠀⠀⠀⣿⡇⠀⠀⠀⠀⣼⣿⠁⠀⠀⠘⡄⣇⠀⠀⠀⠀⠀⠀⠀⢨⠀⢩⢾⡷⡭⠂⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⢠⣤⠀⠀⠀⠀⢀⣿⣀⠀⠈⠘⠈
⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⣿⠻⡇⠀⠀⠀⠀⠀⠀⡟⠀⠉⢸⡇⠈⠀⠻⠀⠀⠀⠐⣶⠀⠀⠀⣶⠂⠀⠁⠀⠀⡏⠀⠀⢀⠂⠚⣿⠓
⠀⠀⠀⣿⡇⠀⠀⠀⣺⣾⡇⠀⠀⠀⢠⣿⡄⠈⢆⠀⠀⠀⠀⣰⠀⠀⣿⢾⡷⣿⠆⠀⢣⠀⠀⠀⢹⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⠀⠀⢀⣀⣀⣿⣇⣀⣠
⠀⠀⠀⣿⡇⠀⠀⠀⢹⣻⡇⠀⠀⠀⠈⢹⡧⠀⠈⣇⠀⠀⠀⡇⠀⠰⡞⠖⠳⠻⡆⠀⢸⠄⠀⠀⣸⠀⠀⠀⠈⡇⠀⠀⠀⢰⡇⠀⠀⠈⣸⠏⣿⠙⢷⡀
⠀⠀⢀⣿⣯⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠈⡇⠀⠀⠈⠀⠀⠀⠃⠀⠀⡇⠀⠀⢰⠁⠀⠈⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⠀⣼⡇⠀⠀⠀⠁⠀⣿⡀⠀⠁
⠀⠀⣐⣿⣗⡀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⠀⠀⣼⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢸⣿⠀⠀⠀⢹⡇⠀⠀⠀⠀⢰⣿⣆
⠀⠀⠰⣿⡷⠄⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠂⠀⣿⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠸⡏⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⠿⠃
⠀⢠⣽⣿⣿⣅⠀⠀⠀⣿⡿⠃⠀⠀⠀⠈⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇⠀⠀⠀⢸⡇⠀⠀⢀⣠⣤⣤⣴⡖⢲⣒⠒
⠀⠘⢿⣿⣿⠏⠀⢀⣾⠏⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡏⠀⠀⠀⠀⠀⠀⣖⠂⠀⠀⢀⡇⠀⠀⠀⢸⡇⠀⠀⠛⠛⠛⠻⠟⠛⠛⠃
⠀⠂⠈⠻⠋⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⢠⣇⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⣇⠀⠀⠀⠺⢷⡄⠀⠀⣤⣶⣶⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠑⠂⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠐⠀⠀⠀⠁⠀⠀⠉⠁⠀⠉⠉⠙⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⣸⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠇⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⣛⡯⠽⢿⣟⣻⡶⣽⣉⣿⣿⣟⡿⣿⠟⠻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⡟⢶⣿⣟⠒⣺⣿⣟⣿⣏⣡⠘⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡟⢍⣻⢿⣿⣯⢿⣿⣿⣿⣯⡷⢶⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢢⣽⡿⣿⣾⣿⣿⣿⣿⣿⡶⣏⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡧⣻⣯⣻⣿⣿⠏⣸⣿⢿⣾⣿⣯⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡻⣿⣿⣿⡇⣴⣼⣿⡿⢿⣮⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣧⣿⣿⣿⣿⡃⢻⣿⣿⣟⣷⣽⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡅⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⠆⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡏⠙⠁⣽⣷⡝⣿⠛⠿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⣿⣿⣿⠇⠀⠿⠛⢿⣿⣾⠂⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣶⣾⣶⣿⢿⣿⠋⣘⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡌⣻⡿⣿⣿⣷⣟⣧⣾⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣷⣯⣥⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⣶⣶⡶⢿⡿⣿⣶⣶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⠖⠋⠙⠁⠀⠀⠀⠀⡀⢁⣀⣴⢿⣟⣉⢫⣝⠻⠿⢿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣹⣷⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⡴⠫⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄
⠀⠀⠀⢀⠞⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⢀⣮⣜⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⡞⠰⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢸⢷⣷⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⣿⣾⣿⣴⣆⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⢸⣿⠾⣿⣿⣿⣽⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠿⣿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣼⣇⠀⢘⣿⣿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠞⠁⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡍⠀⠈⢹⣿⣿⠏⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣀⣱⣠⡈⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡇⠀⠀⢸⡿⠃⠀⣀⣤⣴⣶⣾⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⢀⠏⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣙⠾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢿⣷⡀⢀⣽⠁⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢷⣤⣴⢧⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⡏⠒⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⠀⢻⡋⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⡇⠀⢸⣿⣯⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⢨⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⢹⣿⣿⡧⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠘⣠⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣽⡟⣿⣿⣿⣿⣿⣿⡇
⠀⢨⠀⠉⠀⠀⠀⢿⣿⣿⣿⣿⣿⣯⣽⣿⠿⠋⠁⠀⠀⠀⢰⣿⣿⣿⣿⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡿⣼⣿⣿⣿⣿⣿⡿⠁
⠀⠀⢧⣀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣾⣿⣿⣿⣧⣽⠛⠿⣿⣿⣿⡿⣟⣵⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠈⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⣀⢠⡄⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡐⠈⠑⢶⡾⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⢹⣿⣿⣦⣤⣀⣤⣴⣬⡢⣄⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣀⠀⠰⡎⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠸⣿⣿⣿⣿⣿⢿⣿⣧⡉⠛⠑⠂⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣯⣤⣶⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠻⣿⣿⡄⣿⠀⣿⣿⠻⣄⠀⠀⠀⠀⠀⠀⠀⠹⠿⠟⢡⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢻⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠙⠛⢿⢻⡇⣿⣿⡆⢹⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⡟⠀⣼⣿⣿⣿⣿⣿⣿⡿⠇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠇⢸⣿⣷⠈⣧⡀⠀⠀⢀⡀⢠⠀⠀⠀⠻⠙⠂⣍⢻⣿⣿⣷⣿⣾⡇⣼⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⣟⣧⣿⠙⠂⠇⢸⠃⢿⡆⠳⠚⣾⢰⡖⣿⠙⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢿⣿⡿⢦⣄⡀⣸⠀⣘⡀⣀⠀⣉⢸⡇⣙⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠈⢻⡛⡄⠀⠉⠙⠋⠉⠏⠉⠙⡏⠉⠃⠙⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠁⠀⠀⢀⣰⠆⠺⠄⠟⣼⠷⣼⣤⣷⣰⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠁⠀⠀⠁⠀⠀⠈⠉⠹⢿⣿⡏⡿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡄⣬⠚⢸⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠾⣷⡀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣅⣿⡇⣾⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⢤⣄⣀⣀⣀⣀⣴⣀⣴⣄⣴⣾⡿⠟⠋

⢀⣤⣦⣄⠀⣠⣶⣦⠀⣠⣤⣄⠀⠀⣠⣤⠄⠠⠀⠢⡀⠀⡀⠒⡆⠀⠀⠀⢄⠀⠀⠀⠀⡀⠀⠄⠀⢄⠀⠀⠔⠢
⠸⣿⣿⣿⣿⣿⣿⡿⠂⠿⠛⠯⠗⠘⠛⢋⡄⢈⣤⣜⣮⣿⣶⣶⡿⠀⢁⣦⣲⣨⣯⣶⡾⠇⠘⡀⠀⠀⠈⠁⣠⣄⠆⠀⠀⠀⠀⠀⢠⠤
⠀⠘⠛⣿⣿⣿⠁⠀⠀⠀⠃⠰⠄⠓⡜⠀⠀⠀⠙⠻⣿⣿⡿⠀⠀⠀⠀⠙⠻⣿⣿⡇⠀⠀⠀⠈⠒⠆⠐⣴⡏⠀⠀⠀⠁⠐⠀⢦⡍
⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣷⡆⠃⠀⠀⠀⠀⠀⡏⣻⡇⠀⠀⠀⠀⠀⠀⠘⣻⠀⠀⠀⠀⠀⠀⢸⠀⢹⠀⠀⠀⠀⠀⠀⠀⠈⡇
⠀⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⣿⠁⡀⠀⠀⠀⠀⠀⡇⢹⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⡇⡇⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⠁⠃⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⢰⡄⠀⠀⠀⠀⠀⡇⢺⡇⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⣿⢸⠇⠀⠀⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⣿⢸⡆⠀⠀⠀⠀⠀⡇⢸⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⢸
⠀⠀⢀⣾⣿⣿⠀⠀⠀⠀⠀⢰⣹⠨⡇⠀⠀⠀⠀⢠⠷⣼⣖⠀⠀⠀⠀⠀⠀⢦⣷⠂⠀⠀⠀⠀⠀⡄⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⣸⣿⣿⣿⣿⡆⠀⠀⠀⣽⡶⠂⠸⣷⠀⠀⠀⠜⢉⣿⣿⡉⠀⠀⠀⠀⠈⢹⣿⣯⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⠀⠀⠀⠀⠀⡀
⠀⠀⠿⠿⠉⠻⠋⠀⠀⠀⠘⢋⠅⠠⠠⠂⠀⠀⠰⠶⠟⠹⠿⠋⠀⠀⠀⠰⠶⠋⠹⠟⠁⠀⠀⠀⠄⠴⠟⠧⠄⠀⠀⠀⠀⠀⠊⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢱⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⢻⡆⣇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢸⣷⠸⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠈⠿⣤⣶⣾⣿⣶⣤⡶⠢⣴⣶⣶⣶⣦⡤⠟⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣗⢻⢹⣿⣿⣿⣿⠿⣇⣜⠸⢿⣿⣿⣿⡇⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠈⠓⠋⠉⠁⠀⣴⣿⣿⡞⠲⠈⠉⠉⠈⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⣤⣤⣄⠀⠀⠘⡿⢿⠇⠀⠀⣠⣤⣦⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⢻⡌⣇⣀⠀⠁⠈⠀⣀⣸⢣⠟⠻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠤⡀⠀⠀⡇⠀⠻⠶⠤⢉⣉⣉⡩⠭⣆⡟⠀⠀⠀⠀⠀⡠⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠇⠀⠀⢱⠀⠀⠳⡄⠀⠀⠁⠄⣀⣀⠀⠈⠀⠀⢠⠞⠀⠀⠜⠀⣄⡘⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣧⣠⣄⡀⠀⠡⠀⠀⠈⠑⠦⡀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠔⠀⠀⠀⠀⠳⠀⢙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠓⠠⡀⠀⠀⠢⠀⠀⠈⠁⠉⠉⠊⠁⢀⠠⠊⠀⠀⠀⠀⠀⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⢀⠠⠂⠁⠀⢀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⣀⠀⠀⠁⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠁⠉⠰⠄⠀⠀⠈⠀⠂⠄⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠀⠀⠀⠁⠀⠀⡀⠄⠀⠁⠀⠀⠀⠀⠠⢀⠀⠀⠀⠈⠁⠒⠖⠒⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣦⡀⢀⡀⠀⣀⠄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⣀⠀⠀⢰⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣶⣤⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⣀⠈⢠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⡀⢀⣀⠀⠀⠀⠀⠀⠒⠀⣀
⠀⠀⠀⡤⠀⠀⠀⣀⡀⠉⠁⠀⠀⣀⣀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠒⠉⠉⠀⠁⠀⠒⠶⠦⠄⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠤⠄⠀⠀⠀⣀⠀⠀⠀⠁⠀⠀⠤⢴⠦⠄⠠⠀⢀⠈⠉⣁⣀⠀⠲⠤⠀⠀⠀⠀⡀⠠⡀⠉⠓⠀⠀⠈⠶⠤
⠀⠠⠄⠀⠀⠠⠀⢀⡀⠒⣀⣀⡀⠀⡤⠐⠂⢖⣒⢤⠀⠀⠐⠉⣫⠈⢉⣤⡈⠁⠨⠄⣔⣭⡿⠵⣶⠏⣄⠀⣜⠓⡀
⠀⠀⠀⠂⠀⢀⠄⡀⠀⠀⠀⠈⠉⠀⠀⠐⢇⠳⠔⣠⣆⠴⠗⠀⢨⡐⠸⠶⡷⠞⠿⢋⡼⠁⢁⣾⡛⠓⡿⠁⠈⠁
⠀⠀⠀⠘⠂⠀⠐⠐⠃⠀⠀⠀⣀⠀⠈⠢⣀⡙⠉⠀⠴⠀⠀⠀⠉⠉⠀⠾⢖⠂⠀⠀⠙⠀⠹⠯⢙⣻⠛⠈⡀
⠀⠀⠀⠀⢀⣀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠒⣁⡀⠀⠀⢀⠂⠠⠤⠄⣊⠄⢐⣶⣶⡦⢀⠀⠉⠁
⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⢠⠀⠈⣁⣐⠒⢭⠐⣄⢀⡖⢁⣬⠌⠵⢤⣈⠉⢀⠀⢠⡉⠈⢰⢾⣦⣙⣆⣱⣄⣀⡀⣀⣀⠀⠀⠀⠀⠁⠆⠠
⠀⠤⠀⠐⠘⢉⡀⢼⡀⠀⠿⠀⣸⢡⣉⣶⡀⠃⡀⠈⣇⣱⢿⡾⣿⢉⣽⠆⠸⢃⣴⢧⡈⢸⠿⠏⠀⢏⠞⡡⡴⡺⢟⡀
⠀⣰⠁⢰⡛⡉⡄⣦⢡⢦⠀⠀⠈⠉⠨⠍⠋⠜⠉⠈⠒⠅⡀⢰⠾⠓⢀⠀⠀⠸⣭⣿⣜⣩⣽⣟⣃⠥⠾⢰⡀⢻⣩⠆
⠀⠉⠀⢸⣒⡅⠸⠃⢦⣀⡉⢒⣂⡈⠉⠩⠤⠤⠤⠂⠀⠀⣈⠄⢀⣤⡴⣄⠺⣑⠄⢋⣈⠩⠿⢛⣛⣒⡒⢞⠛⠞⠛⠃
⢰⠀⠀⠤⠀⠉⠂⠀⠒⠯⠍⠀⠈⠉⠀⠀⠀⠲⠶⠤⠀⣀⢇⡀⠻⢤⣂⣤⡿⢏⡴⠔⣢⢰⡦⢙⡋⠯⣭⡗⠀⠂⠅⠐⠀⠀⠀⠀⡀
⠈⡀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠉⠋⠀⠀⠀⠀⠧⣐⠒⠄⠈⠂⠋⠥⣬⡹⠯⠒⠁⢰⠎⡽⠀⢈⠭⡉⠂⢨⡀⠀⡄⣦⠀⠀⠀⠀⠀⠁⡀
⠀⠀⠀⠀⣀⣖⢀⠢⡄⠀⡀⠀⠞⠑⠋⣉⣉⡉⠉⠉⠁⠒⠀⠰⠿⡚⢟⠲⠄⣢⣄⠠⠠⣀⡀⠸⠗⢂⠉⢉⡽⢩⠍⠉
⠀⠀⠁⠘⠃⠀⠩⢐⡌⠀⠀⠀⢼⠀⠊⣁⡐⠀⠀⢀⡄⠀⠣⠰⡖⣸⣾⣥⠠⠮⠭⢕⢲⣯⣡⡝⠿⠟⢟⡻⠖⡀⢒⡀
⠀⢤⣤⠂⢀⠤⣤⠀⠚⠀⣀⡀⣠⣑⠂⢀⣉⣀⣀⣤⢀⣀⠀⢉⣚⡃⠈⠀⢶⣤⣀⠰⣇⣆⠂⢸⡆⠀⢠⠤⠅⣢⠅⠀⠀⠀⠀⠀⡀⡀⢀
⠀⣇⠒⠤⠤⠀⠀⠀⠀⠀⠖⠒⠒⠂⠀⠈⠩⣀⡭⠅⠀⠀⣀⢀⢀⣀⠈⠁⠐⠂⢈⡃⠉⠛⠛⠁⠄⠞⠀⠀⠠⠁⠛⠁⠀⠀⠀⠀⠁
⠀⠈⠉⠀⠀⠠⠤⠄⠀⣉⣁⠀⢀⣀⡀⠀⠀⠤⠭⠉⠤⠤⠄⠀⠨⠭⠥⢭⣙⣓⡒⠶⠒⠒⠲⢖⣂⣘⠉⠁⣓⠀⠐
⠀⠀⠀⠂⠒⠒⠀⠁⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⡀
⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠤⠄⠐⠒⠂⠀⠀⠄⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢋⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢡⡇⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⣿⡇⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢹⣿⠘⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⢸⣿⡇⢹⡿⠄⠀⠀⠀⠤⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠸⣿⡇⠀⢀⣠⣤⣤⣤⣀⠀⣀⡀⡀⠀⣀⣠⣤⣤⣀⣀⠀⠈⣿⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⢹⡑⣾⣿⣿⣿⣿⣿⣿⣿⣿⠉⢩⠫⣿⣿⣿⣿⣿⣿⣿⡗⠏⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⢫⡇⣿⣿⣿⣿⣿⣿⣿⠟⢟⡄⣜⠀⢿⣿⣿⣿⣿⣿⣿⡅⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠳⣬⣛⠛⠛⠋⠉⠕⡀⣾⣷⣿⡔⢦⣌⡙⠛⠛⠛⠫⠔⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⠀⠀⠀⠀⠀⠀⠠⠄⠸⢷⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣬⣤⣤⣤⣄⠀⠀⠀⠈⢿⡿⣿⠇⠀⠀⠀⢀⣤⣤⣴⣤⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⠿⣇⢸⡇⠀⠀⠀⠐⠆⠐⠀⠀⠀⠀⣿⢀⡿⠿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢻⣬⡓⠲⠶⡤⠤⢤⠤⡤⠦⠶⠾⣋⣼⠡⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢻⠏⡛⠦⠄⢄⣠⣀⠤⠤⠚⡛⢤⢃⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⠀⠀⠀⠉⠑⡂⠠⠴⠬⠐⠐⠀⠁⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⣄⠀⠀⠀⠈⠉⠛⠋⠁⠀⠀⠀⢠⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⣠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠉⠛⠛⠋⠁


⠀⣁⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠇⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⣷⣦⣿⣶⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣷⡼⢿⣯⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣌⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⣿⡟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⡇⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣶⣶⣶⣦⣥⠙⠋⠻⢉⣶⢶⣴⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠿⣿⣿⣿⣽⢃⣭⠠⣋⠈⣥⡾⣿⣿⠿⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠉⠀⠀⣿⣿⣿⣏⡴⠖⠀⠰⠶⠴⠾⢛⡿⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣟⣻⠏⠀⢐⠒⠲⠞⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠷⣖⠉⠅⣶⡘⠯⠀⠚⠂⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢄⣶⡶⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢷⡄⠀⢀⣻⡇⣰⠦⡄⠓⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠉⠛⠛⠒⠶⠦⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣿⠇⠀⠰⠋⣹⣟⡉⠰⣼⡆⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠙⠂⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡀⠀⠀⠀⠀⠉⠀⠀⠀⢠⣿⣯⡄⠀⠀⠁⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠈⣡⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⣅⠤⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠰⠋⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⣴⣾⣿⣤⣶⣾⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⢀⣼⣿⣿⣿⣯⣥⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣇⣴⢨⡹⣿⠟⢋⣭⠻⠆⡀
⠀⠀⠀⠀⣤⣠⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⡋⣿⣁⣿⠿⠏⠰⣿⠳⡆⠁
⠀⠀⠀⢰⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡟⠃⠈⠛⠁⠀⠀⠀⠋⠀⠀⡀
⠀⠄⠁⠤⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠆⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⣼⠆⠀⠀⠀⠀⠀⢠⣯⣉⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡄⢸⠆⠀⠀⠀⠀⠀⠘⣿⡋⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧⠈⠀⠀⠀⠀⠀⠀⠀⠘⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡌⡆⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣤⣄⣠⢴⣥⣿⣤⡤⠤⠤⠤⣤⣤⣤⣤⣤⣿⣿⣿⣶⣄⣦⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠟⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣤⣤⣄⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣿⣾⣖⣒
⠀⠀⠀⡠⠐⠀⠀⠛⠭⠉⠛⠿⣯⣸⠆⠉⠉⣤⢴⣶⠶⠾⠟⠛⠛⠛⠻⢿⣯⣭⣭⡥⠤⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠒⠒
⠀⠐⠋⠁⠀⣀⠀⠀⠀⠀⠀⠠⡀⠓⢀⠀⢀⣯⡀⣠⠆⠊⠀⣀⣀⣤⡤⡄⢠⠉⠉⠉⠉⠙⠛⠿⢿⣿⣶⣦⣄⠀⠐⠀⠀⠀⠁⠀⠁⠒⠂
⠀⠀⠀⠀⠈⠀⣳⢂⠤⠁⣦⢒⣂⣬⣥⠄⠐⣛⢥⡶⣾⣛⣋⠁⠈⣉⣀⡴⠛⠎⠁⢴⣭⡿⠷⠖⣒⣩⣿⣿⣿⣷⣶⣤⣤⣄⣀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠤⠀⠉⠀⢔⣣⡤⢬⣤⣀⣐⣀⣠⣤⣤⣤⣥⣥⡿⠽⣿⡻⠿⣥⡴⠜⠃⠒⠈⡡⢶⣪⣍⡉⠥⠀⠀⠰⠃⠉⠝⠛⠛⠭⢉⣀
⠀⠀⠀⠀⠀⠀⢍⢐⣂⠀⠈⠛⠛⣼⣶⡶⠶⠲⡒⠒⠂⠀⣛⣛⣋⣀⣀⠈⡀⢄⠀⠀⠀⠉⠀⠙⠳⣦⣄⠀⠤⠃⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠤⠤⠠⣴⣶⠾⣿⣿⣡⡤⠤⠖⠊⢉⠝⠐⠀⠀⠀⠐⠟⠛⠋⣉⡀⠀⠂⢨⡵⠁⡀⠀⠀⢹⣿⣶⣦⣉⠛⠶⣦⣄⢀⣀⠠⠀⡀
⢀⣐⣀⣤⠤⠶⠶⠛⠿⠿⢷⣷⣶⣿⣅⣶⠶⠖⠛⢛⣓⣤⣿⣟⡁⣀⣀⣄⣤⠀⠀⠀⠐⠂⠔⡪⠙⠻⢍⠻⠆⠀⠀⠀⠀⠀⢤⠀⠄⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⣀⠤⠤⠒⣠⣤⠼⠛⠉⠉⣘⣭⣥⣦⣴⡤⠶⠖⠚⡀⠦⣰⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠤⠭⠥⠤⠶⠶⠾⠿⠿⣷⣶⣤⣴⠶⠒⠛⠉⠀⠀⠀⢀⡔⢲⠟⡋⢀⡴⠎⣿⣤⣄⠀⡀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣁⣀⣨⣀⣠⣤⣀⣶⣄⣀⣈⣙⣉⣉⣁⣀⣀⣈⣀⣉⣀⣈⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣏⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢦⣄⠀⠔⢳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦⣹⣿⣴⣦⣭⣿⣶⣶⣶⣶⣶⣶⣶⣶⣤⣀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠺⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣦⣤⣍⣉⠉⠛⠋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣟⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣿⣿⠋⢀⣄⣴⣿⠉⠉⠉⠁⠀⠉⠉⠉⠛⠛⠻⠿⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⠟⢻⣾⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠟⠁⠀⠸⣿⣿⣿⣿⣿⣧⣤⣤⣀⡀⠀⠀⠀⠤
⠀⠀⠀⠀⠀⠀⠄⢲⣿⣿⣿⣿⣿⠟⠁⠀⠀⠀⠀⣻⣿⠟⠋⠉⠉⠉⠛⢿⣿⣦⡀
⠀⠀⠀⠀⠀⡠⠶⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⢰⣿⠏⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡄⠀⡆⣤⣦
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⣿⡌⣵⣾⣿⣶⠀⣷⠀⠀⠀⠀⢦
⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣺⣷⣿⣿⣿⣿⣶⠋⣠⣶⣶⣴⣾⠇
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⡀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠉⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣝⣿⣿⣿⣷⡆⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⠃⠈⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣷⣤⣤⣴⣿⣿⣿⡿⠿⠿⠿⠿⠛⠀⠈⠻⠿⠆⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣧⡀⡄⠀⠀⠀⣉⡛⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⡀⠀⠀⠀⠀⠀⠶⡄⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣀⣰⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠻⣿⣿⣿⣿⢿⣷⣄⠀⣀⣴⣾⡿⠀⠻⠿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡼⠀⡉⠉⡁⢀⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⡿⠛⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠛⠐⠁⡆⠹⡿⢿⡿⣶⣦⣴⣤⣤⡦⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⢟⠻⣿⣿⣷⡿⠀⠀⠀⠀⣸⠀⠐⠄⢠⡀⣸⣃⡈⢸⡀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣾⡉⠰⣾⣿⣿⡇⠀⠀⠀⠀⣿⣶⠛⠲⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡖⠂⠺⢶⣧⣿⣆⠀⠀⠀⠀⢻⠉⠀⢰⣧⣤⣋⡡⠶⠴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣦⣀⠀⢠⣤⡼⠐⠷⣶⣤⡞⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠪⠍⣛⠛⡹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣷⣦⣿⡆⠛⠶⠦⣤⠁⠀⣀⣈⣳⣄⠀⠀⠀⠀⠀⠀⠀⠉⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣾⠿⠶⣤⣤⣾⣿⣿⣿⣿⣿⣿⣿⣶⣤⣄⠀⢀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣶⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⠿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠍⠛⠻⢿⣿⣿⣿⣿⠟⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠈⢻⣿⣿⣿⠀⠉⠉⣿⣿⣿⣿⣷⡤⡀
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠃⠀⠀⢹⣿⣿⡋⠀⠀⠀⣿⣿⣿⣿⡿⠋⠀⣦
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠈⣿⣿⡇⠀⠀⢠⣿⣿⣿⣏⠁⣠⣞⠿
⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡄⠀⠀⠀⢹⣿⡇⠀⢀⣾⣿⣿⣿⣿⠞⡫⠁
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠛⠋⠻⣿⣿⣿⣿⣿⣿⣯⣅⠀⠀⠀⠀⢸⣿⠀⠀⣼⣿⣿⣿⣿⠣⠋
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⢀⣼⡿⠿⠟⠛⠋⠛⠻⣥⠄⠀⠀⠀⠸⡇⠀⢰⣿⣿⣿⣿⠣⠁
⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠟⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⢸⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡇⠀⠘⠉⣿⣟
⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣁⣀⣀⣀⣀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⢠⣄⢀⡀
⠀⠀⠀⠀⠀⣠⣞⣿⣿⣿⣧⣿⣟⣽⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣽⣯⣿⣿⡦⣿⢢⣾⣷⣷⣦⣄
⠀⠀⠀⠀⣴⣿⣿⢿⣿⢿⠷⣿⣿⣿⣿⣿⣿⣿⢦⠀⠀⠀⠀⠀⢰⣿⣻⣿⣿⠻⡿⣿⣿⡿⣿⠿⣿⢟⣻⣧⡀
⠀⠀⠀⢰⣿⣿⣷⣿⣿⣿⣿⣿⣽⣻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⣿⣻⣿⣿⣯⣿⣽⢿⣿⣿⣻⡿⣿⣿⣿⣿⣧
⠀⠀⠀⢾⣿⣽⣿⣾⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⢠⣼⣿⣿⣯⣜⣽⣿⢿⣽⣿⣿⣳⢿⣿⣽⣿⣿
⠀⠀⠀⣿⣿⣾⣿⣯⣿⣿⣿⣻⣿⢿⣧⣿⣿⣿⣿⣿⠇⠀⠀⠸⣿⣹⣿⣳⣿⡎⣷⣿⣷⣿⢿⣟⣿⣿⣼⣿⡟
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⡏⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡛⣿⣽⠿⣻⣧⣿⠿⣿⡿⠁
⠀⠀⠀⠀⢻⣿⣿⢻⣟⣿⣿⣿⣿⢿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠹⢾⣚⣙⣿⣿⣿⡿⠿⣿⡟⠛⡿⠟
⠀⠀⠀⠀⠀⠉⠛⠿⣿⢟⣿⣿⠿⣿⣾⠯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣧⣾⣿⣿⠛⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⢀⠀⠊⠁⠀⠀⠀⠀⠀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠉⠀⠀⠀⠀⠀⠈⠀⠂⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠁⢀⣀⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠒⠈⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠶⠛⠋⠙⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⠀⠀⠀⢰⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⡿⣿⣿⡿⢿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣹⣿⡍⢉⣁⣼⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢯⣀⣤⣿⣿⣿⡏⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠨⣬⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⡴⢤⠶⠄⢀⣀⣀⢘⣿⠿⠿⠃⢀⣀⣠⠴⠶⠤⢠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⣸⣷⣾⣦⣴⣬⡙⠛⠋⠀⠀⠰⠿⡍⠙⣿⣶⣦⣴⣾⡀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠉⠙⠛⣿⣷⣝⡛⠀⠞⠃⠀⠷⠄⢶⣀⡸⢿⣿⣿⡛⠉⠼⠯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⣿⣦⣟⠙⠛⠃⡀⠀⣘⣤⠜⣛⣥⣾⣯⣽⠀⠀⠀⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢈⣩⣍⠻⠟⠋⠀⠁⠀⣹⡷⠀⠉⣉⣴⣿⣹⡄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣍⣙⡿⠟⠁⠀⢸⣷⣝⠻⡶⠛⣻⡿⠿⣛⡃⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⠀⠀⠀⢾⣉⢉⡰⠖⢈⡿⠋⣿⣿⡴⠆⠉⠻⣧⣴⣟⠃⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⣠⡛⢛⠤⠄⢀⡤⠀⣶⣿⡏⠙⠶⠛⢣⢤⣟⠃⠀⠀⠀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⠉⣴⠖⢉⡀⠀⡿⢿⡇⠐⠲⠄⠀⢶⡟⠀⠀⠀⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠶⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠀⢿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢳⡄⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⣻⡗⠀⠀⠀⠀⠀⠀⢀⠀⠀⠹⢿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⣠⣶⣶⣦⣀⠀⡟⠻⣿⡤⠦⠀⠀⠀⢀⡼⠀⠀⠀⠀⠶⣾⣦⠀⠀⠀⠀⠀⠀⠀⠖
⠀⠀⢠⡤⠀⠀⠀⠀⠀⢻⠁⠀⠀⠀⠀⠸⣷⣿⣄⠀⠉⠙⣻⣿⣿⣷⣤⣄⠀⠀⡈⠀⠀⠀⠀⠀⠀⠻⠋⣤⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠈⠻⣿⣧⡼⠿⢿⣿⠿⠿⠋⠈⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠘⠉⡄⠀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠿⠄⠀⠀⠀⠉⠐⠂⠀⡀⣀⡳⣼⣿⣶⣦⡀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⣤⣼⣶⡄⠀⣄⠀⣰⡎⠁⠻⣷⠀⠙⢀⣿⠇⠀⠀⠀⠀⠙⡄⠙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⡄⠀⠀⠀⠀⣸⡏⠀⠁⠁⠀⠀⠀⠈⠿⣷⣴⠇⠀⢰⣼⡏⠀⠀⠀⠀⠀⠀⠙⣄⣻⣦⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠸⣧⠀⠀⠀⠀⠿⠃⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣦⡀⠀⠀⠐⠏⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⠃⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⢋⣿⣎⠻⣶⡀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⡚⠁⠀⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡁⠀⠀⠀⠀⠀⠀⠀⠀⠰⣷⣮⡁⣈⣦⠈⠻⡆⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⡿⠋⠀⠲⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠹⣿⣿⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠁⣼⣧⡀⠀⠀⠀⠀⠀⠀⣴⣴⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣾⣿⣿⠃⠀⠀⠀⠀⠀⠀⠉⡀⢠⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣉⢉⣹⡿⠀⠀⠀⠀⠀⠀⠀⠀⢣⡜⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢨⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⡐⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⠀⠀⢸⣿⣌⣿⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣬⣿⡄⠀⠀⠀⠀⢀⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠈⠉⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣿⣿⣿⡀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣿⠻⣿⡿⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣿⣿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⠿⢿⠟⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠈⠠⣤⣤⣤⣤⣤⣴⣶⠾⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣷⣄⡀⠀⠀⠻⣿⣿⡿⠃⣰⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣦⠀⢈⣿⡿⠁⣸⡏⣼⣿⠟⣻⣿⣧⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣯⣽⣿⠀⠈⠛⠀⢰⣿⣦⣿⡃⠈⠻⠈⣿⣆⠈⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⡦⠀⣿⣿⡏⠈⠃⠀⠀⠀⣤⣿⡆⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⠃⣸⡇⠀⠀⠀⣿⣿⠃⠀⠀⠀⠀⠀⠉⠻⡗⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠈⣿⣿⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣏⣹⣿⣿⣿⣾⣿⣿⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⡄⠀⣴⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣏⣼⣿⣿⣿⣿⠁⠀⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣭⣿⣿⣿⡟⠀⠐⠛⣤⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠸⡿⠀⠀⠀⠀⠀⠀⠀⠀⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⡀⠀⢰⣆⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⣰⡶⠱⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⡃⠁⠀⠈⠛⢁⣄⣀⣤⠀⠀⠀⢀⡾⠁
⠀⠀⢿⣧⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠀⠀⠀⣿⣿⣿⣿⣤⣤⣾⡿⠁⠀⠀⠀⢀⠄
⠀⠀⣿⡿⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡿⠀⠀⠀⢿⣿⣿⣿⡿⠟⠁⠀⠀⠀⠀⢠⣿
⠀⠀⣿⡧⠀⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣷⠀⠀⣀⣼⣿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⢢⡀
⠀⢰⣿⡇⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⠄⢈⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣦⠀⠹⣦
⠀⣼⣿⠀⣼⡇⠀⠀⢀⣀⣠⣤⣤⡤⣤⣄⡀⠀⠀⠀⢸⣿⣿⠀⢸⣿⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡄⠘
⢠⣿⡇⢰⣿⣀⣴⣾⡿⠛⠛⠛⠉⠀⠀⠈⠙⢦⠀⠀⢸⣿⡏⠀⠘⣿⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣄
⣼⣿⠃⢸⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⢠⣄⡀⠀⠀⢀⣿⡇⠀⠠⡏⠀⠀⠀⢀⢀⣴⣤⠘⡆⢠⡞⠁⠀⠀⠀⠀⢸⣿⣧
⣿⣿⣶⡿⠟⠁⠀⠀⣠⣦⣀⣀⣰⣶⣤⣿⣿⣿⣷⣄⠘⣿⡇⠀⠀⠀⠀⢀⣶⣿⣾⣿⣿⣷⣷⣯⣵⠟⠀⠀⠀⠀⠀⠸⣿
⣿⣿⣄⣀⣀⣤⣶⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⠀⣿⣷⡄⠀⡄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡆⠘⡇⠀⠀⠀⠀⠃
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢸⣿⡄⠀⡇⢸⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠘⣤⣶
⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠘⣿⣇⣤⠀⣾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⣴⣿⣿⡖⢁⣼
⠀⠸⣿⣿⣿⣿⣿⣿⣩⡿⢋⡟⢀⢿⡏⢙⣉⣿⣿⣿⣿⣆⠹⢿⠟⠀⠙⠻⠿⣿⣿⠟⠑⠀⠀⠀⠀⠀⠀⠈⠁⡟⢀⣾⠇
⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣦⣰⣿⢇⣽⣿⣿⣿⣿⣿⣿⣿⣄⣀⣠⣀⠀⠀⢠⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⡾⣿⣿⣿⣿⣿⣿⣿⡟⠛⠻⣿⣿⣷⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢰⣿
⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⠁⠀⠘⠀⢸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⠏
⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠈⠇⠀⠻⣿⣿⣷⡀⠀⠀⠀⠀⣤⣼⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠛⠀⠀⠙⠛⠻⠿⣦⡀⣴⣷⣼⣿⠹⠿⠛⢻⡿
⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠿⡿⢿⣿⡿⢿⣿⡿⠟⠀⠀⠀⠀⣀⣀⣀⣀⠀⠀⠙⠋⠁⠈⠃⠀⣠⣶⣿⡇
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣽⣟⣻⣿⣿⣿⣿⣿⣿⣿⡿⢿⣴⣿⣿⣿⠿⣿⣿⣿⣿⣷⣶⣄⠀⠀⠀⠀⣺⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⣴⣦⡏⠉⠁⠀⠀⠀⠀⠀⠉⠉⠙⠛⠳⠶⣶⣿⣿⣿⢹⡟
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⡀
⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣄⣀⠛⠋⡀⠀⠀⠀⠀⠀⠀⠀⣀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⡏⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠉⠛⠛⠁⠀⠀⠀⠀⠀⡰⠛⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣾⣿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⣡⡴⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⢀⡀⢀⣠⣴⣦⣶⠞⠋⣡⡾⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡿⠛⢋⣉⣉⣀⣤⣶⡿⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠚⠻⣿⣿⣿⡟⠛⠋⠁⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⠀⠀⠻⠿⠋⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⡛⠻⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣶⣬⣍⣁⣀⠀⠀⠀⢀⣀⣠⣴⣶⡿⠿⠋⢩⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠙⠿⠟⠛⠉⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣶⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣛⠛⠁⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⢿⠟⢋⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣶⡄⠀⠐⠿⠿⠛⠉⠁⠀⠀⣀⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣏⣙⣁⠀⠀⠀⠀⠀⣀⣤⣴⡿⠟⠋⠉⠛⠋⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠉⠀⠀⠀⠀⠀⠀⢠⡼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠟⠉
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣭⣥⣾⣶⣤⣤⣤⣤⣤⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⠟⢻⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⣤⠶⠖⠛⠛⠲⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠛⠛⠛⠲⣤⡀
⣰⠟⠀⠀⠀⠀⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠹⣆
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠻⣦⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣴⡋
⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠈⠷⣄⡀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⣀⣤⠟

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⢀⣠⣤⣤⣤⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣗⣞⣦⡆⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠀⢀⠠⣅⢠⣔⣁⣠⣿⠀⠀⠀⠀⠈⠀⠐⠀⢀⡀⠀⠀⠛⣿⠋
⠀⠠⢤⣶⣿⣿⠋⠁⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⠈⠙⣿⡿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⣿⣲⣿⢿⡟⠋⠉⠀⠀⠐⠀⡀⠀⡌⢰⠋⠁⠀⠀⠀⣿
⠀⠀⠻⡟⢻⣧⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢸⢱⡇⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⢹⢻⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⡧⠀⠀⠀⠀⠀⣉
⠀⠀⠀⠁⣾⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈⢸⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⣿⠆
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣤⡄
⠀⠀⠀⠁⣽⠁⠀⠀⠀⠀⠀⠀⣸⠳⠀⠀⠀⠀⠀⠀⣴⣾⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡇⠀⠀⠀⠀⠘⣿⠁
⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠹⢿⠀⠀⠀⠀⠀⠀⠀⢳⡆⠀⠀⠀⠀⠀⠀⠀⠀⠉⡟⠀⠀⠀⠀⠀⠀⠀⠀⢻⠗⠃⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⡀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣄
⠀⠀⠀⠐⣾⡇⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⢐⡿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢴⡧⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⢻⡇⠀⠀⠀⠀⠀⠸⠿⠀⠀⠀⠀⠀⠀⠀⠘⢷⠀⠀⠀⠀⠀⠀⠀⠸⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿
⠀⠀⠀⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⠆
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⣿⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢸⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⢨⣄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠀⠀⠀⠿
⠀⠀⢀⣽⣏⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣰⢦⠿⣇⠀⠀⠀⠀⠀⠀⠘⠲⠀⠀⠀⠀⠀⠀⠀⠶⢾⣻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⣦
⠀⠀⣌⢠⣿⢴⡇⠀⠀⠀⠀⠀⢀⠀⢀⠀⠀⢠⣤⣿⢿⣄⣤⣷⠀⠀⠀⢀⡤⡄⠀⠀⠆⠀⠀⠀⠀⣀⣀⡼⣿⣿⡤⠀⠀⠀⠀⠀⢀⡴⢤⡀⠀⠀⠀⠀⣠⣿⡀
⠀⠀⠙⠋⠉⠛⠁⠀⠀⠀⠀⠀⠉⠁⠈⠀⠀⠀⠉⠁⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠈⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠀⠀⠀⠀⠈⠉⠁
⠀⡟⠉⠓
⠀⠙⠒⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⠿⠛⠻⢿⣿⣿⣿⣿⣿⡿⠛⠛⠻⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢏⡿⠋⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢃⣾⠁⠀⠀⠀⠀⠀⡄⣼⣿⡟⠉⠙⣄⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣿⠀⠀⠀⢀⣀⣨⣴⣿⡟⠀⠘⠀⠘⣶⣦⣄⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠀⢸⣿⣽⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠘⣿⣦⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣷⣄⣠⣧⣤⣾⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⡴⠁⠀⠘⣿⡇⢠⡏
⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣼⡟⢀⣿⣿⣿⣛⠉⠛⢿⣶⣿⣿⣿⣿⣟⢻⡿⢿⡿⡻⡏⣿⠉⠉⠉⣅⠀⠀⢸⣇⠀⠀⣠⣿⠇⢸⣧⡀
⠀⠀⠀⠀⠀⠀⢹⡆⠀⢸⣿⠀⣸⣿⡟⣿⣿⡥⢤⣼⣿⣿⢡⡿⡟⠿⠘⠿⠘⠿⠻⢿⡇⠀⠀⠀⣿⣦⡀⠘⣿⣿⣿⣿⡏⠀⠈⢿⣷⡄
⠀⠀⠀⠀⠀⠀⣼⡇⠀⠸⣿⣆⢹⡏⠀⡟⠙⠁⠀⢸⣿⣾⣼⣷⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⢻⣿⣷⠀⠈⢿⣿⣿⣧⣀⣀⣾⡿⠁
⠀⠀⠀⢀⣴⣾⡿⠁⠀⠀⠙⣿⡜⣿⡄⡇⠀⠀⠀⠈⣿⢹⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡇⠀⠀⠈⣿⣿⣿⣿⣿⡟⠁
⠀⠀⠀⣿⣿⣿⡅⠀⠀⠀⠀⣿⣧⠿⣿⣧⠀⠀⠀⠀⠹⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⡄⠀⠀⣠⣿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣷⣤⣤⣤⣾⣿⣿⠀⣿⢹⠀⠀⠀⠀⠸⣿⠀⠀⣴⣠⣤⣤⣀⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⡧⢹⡀⣿⣾⣤⣠⣤⣶⣶⣿⣾⣷⣿⣿⣿⣿⣿⣷⣇⣀⣠⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡟⣵⣿⡇⠹⣌⢿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡿⣿⢟⠝⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⣾⣿⣿⠿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣽⣿⣿⣿⣿⡿⢿⣛⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣯⣿⣛⣻⠿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⢚⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⠶⠦⠭⠝⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠀⠒⠂⠀⠈⠀⠐⠒⠒⠲⢦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢎⡤⠓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣷⡄⢸⡆⠀⠀⠀⠀⠀⠀⠀⢼⣰⠆⠀⠀⠀⠀⠀⠀⠰⣦⣀⠀⢙⡟
⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿⠁⢀⣴⣶⣶⣦⡄⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣄⠙⣿⣀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡇⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣇⠘⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⣿⣿⣿⣿⣿⣿⣿⠀⣤⣷⡼⡀⢿⣿⣿⣿⣿⣿⣿⠀⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢤⠏⠀⠈⠛⠻⠟⠛⠛⠁⠀⣽⣿⣿⠃⠈⡛⠛⠻⠟⠓⠁⠂⢀⣴⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣧⣀⠀⣀⠔⣶⡗⠁⢸⣿⠿⣿⡧⠀⢻⣶⣄⡀⢀⣀⢔⡝⠂
⠀⢠⡖⠂⠒⢦⠀⠀⠀⠀⠘⣿⣷⡦⡘⠉⠀⠂⠀⠁⠀⠈⠈⠀⠀⠹⢀⣤⣤⣶⡏⠀⠀⠀⠀⣠⠖⠒⠒⢄
⠀⢸⣄⠀⢀⣾⠇⠀⠀⠀⠀⢿⡛⡇⢸⣳⠂⠂⢀⠀⠀⠀⠖⠀⠀⢲⡏⢸⣿⣿⠁⠀⠀⠀⠀⢿⣆⡀⢀⣼
⠀⣀⣽⠍⢽⣿⣄⠀⠀⠀⠀⢸⠃⢹⣾⣼⣤⣄⣬⣄⣤⣤⣀⣴⣤⣾⣷⡿⠉⡇⠀⠀⠀⠀⢀⣼⣿⠍⢽⣅⡀
⠘⠀⠙⠋⠉⠙⢃⣉⡆⠀⠀⢸⡆⠸⣿⡄⡏⢹⠉⡇⢹⠀⡇⢸⢹⣽⡯⠁⢀⡇⠀⠀⠀⣎⣉⠛⠉⠉⠋⠉⢸
⠸⣤⡈⠛⢟⠛⠛⠛⢷⣄⠀⠈⠳⣄⠹⣇⠀⠘⠢⠁⠛⠀⠃⠛⠈⠹⠃⣤⠎⠀⠀⢀⣼⠟⠛⠛⡻⠛⠉⣠⡜
⠀⠈⠉⠙⠛⠳⠾⣤⣌⠙⢷⣤⣀⠈⠳⡌⠀⢀⡀⠀⠀⠀⠀⣀⡀⢀⠞⠁⢀⣠⣶⠟⢁⣤⡼⠶⠚⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣭⡈⠓⠢⢌⡒⠴⠵⠀⠐⠒⠀⠁⠐⣉⠤⠚⠉⣨⡵⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢦⣄⡉⠓⠤⣀⡀⢀⡤⠔⠋⣁⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⣦⣄⡉⠑⠶⢵⡛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⣉⣤⣿⣿⠿⠶⣶⣤⣌⡑⠲⠤⣀
⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣖⡪⠭⢒⣲⢽⠿⠛⠉⠀⠀⠀⠈⠙⠻⢿⢵⣒⠮⠭⣒⣦⣤⣀⣀
⢠⡶⠛⠛⠻⣁⣐⣒⠪⠈⠀⢀⡠⠕⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢥⣀⠀⠈⠩⢒⣓⣀⡩⠛⠛⠳⣦
⢸⡇⢀⢀⣤⢉⠁⠀⠀⢠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢆⠀⠀⠀⢉⢡⡄⢀⠀⡿
⠀⠈⠙⢯⡟⠐⢀⣠⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣦⣀⠀⢘⣿⠟⠉
⠀⠀⠀⠘⢿⣠⣼⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢿⣄⣸⠃

⠀⠀⠀⠀⠀⠀⡄⠀⢦⠀⠀⠁
⠀⠀⣠⠀⠀⣿⣼⣦⡈⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢈⣶⢄⣿⣷⣿⣴⡄⢳⡀⠀⠀⠀⡀⠀⠄⠠⡄
⠀⠀⠀⢹⠿⡙⣿⣿⣿⣍⠀⠀⠈⡇⠈⠀⠀⠀⢀⣿⡀
⠀⠀⠀⢻⣦⣿⣿⣿⣿⣭⢃⠐⠄⢀⠀⠲⠀⠀⢨⣿⣿⢦⠀⡆
⠀⠀⠀⢼⡿⡙⢍⣻⣿⣿⣿⣀⠀⠈⠀⠀⠀⡄⢰⡇⣿⣈
⠀⠀⢠⡼⣿⡏⠸⠛⠉⣛⣛⣛⢧⡀⠀⠀⢰⠘⠀⠑⠚⣋⠄
⠀⠀⠀⢳⣜⢷⡀⢰⡞⣃⣡⣞⣧⠁⠁⠰⠀⠀⠀⠀⢶⣶⡎⡴⠂⠈
⠀⠀⠘⡦⣆⠀⢳⡜⡯⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢀⠮⣙⣒⣦⡙
⠀⠀⠐⠀⢉⠀⣶⢿⣜⡻⠾⢛⠅⠀⠀⠀⠀⠀⠀⢂⠰⠿⠇⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠂⠐⠃
⠀⠀⠀⠀⢈⡒⣦⢻⡁⣽⠶⣖⠚⠀⠀⠀⠀⠀⠀⠈⢙⣲⣌⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠒
⠀⠀⠀⠀⠈⡁⣹⡄⢹⠀⠀⢮⠀⠀⢀⠀⠀⠀⠠⠀⠡⠙⠉
⠀⠀⠀⠀⠀⠀⠠⣿⡿⣧⡰⢿⠳⡄⢸⠀⠀⠀⠀⢷⡀⡄⠀⠠⠀⠆⠄
⠀⠀⠀⠀⠀⢱⡄⢈⣍⢻⣭⣦⣰⣮⡎⢀⢸⠢⡄⢀⠱⣴⣦⢀⡤
⠀⠀⠀⠀⠀⣟⠣⠹⣥⣼⣿⡀⣬⣿⠓⣶⠘⠀⠱⠀⠰⠟⠛⣇⠁
⠀⠀⠀⠀⣀⣿⡆⣧⠀⢹⣟⡙⢦⡟⢘⣻⡄⠀⢠⠁⠀⠰⠄⢨⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠸⣿⣷⠉⣷⠈⠈⠻⣛⣋⡀⠙⢻⠹⡈⡄⠀⣀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⠀⠀⣁⢀⣀⢈⢀⣀⠈⠀⠈
⠀⠀⠀⠀⠐⢀⡇⠰⠈⠀⠔⠋⣉⡀⠈⣆⠀⠀⠏⠐⢂⠀⠻⡀⠀⠄⠄⡿
⠀⠀⠀⠀⠀⢿⢻⡇⣳⠀⠆⡙⡿⠟⠴⠀⠀⠤⠀⠆⢾⡀⠀⠻⣀⡀⢈
⠀⠀⠀⠀⠀⠸⣿⠈⣿⢲⡀⠀⢹⣿⠇⡸⠀⣴⠰⡆⠈⠉⠀⠀⠸⢠⣬⡄
⠀⠀⠀⠀⠀⠀⢏⠀⢻⣆⡁⠀⠀⣆⠢⠀⠀⡘⡆⠆⢸⡰⢇⡆⣤⠀⠸⡄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢲⠙⠇⢠⠁⠉⠀⠀⠙⠃⣀⡄⢰⡃⡴⢠⡷⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠘⠛⠄⠀⠀⠀⢠⡄⠄⠀⢻⣿⣾⠁⣸⣿⠃⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣴⡆⠂⠈⠿⠛⢰⠀⣿⢠⡄⠺⠟⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠈⠀⠋

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

⠀⠀⠀⠺⡀⢀⡰⠄⠀⠁⠘⢄⠈⢲⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠳⢴⡆⠀⠀⠈⢵⢄⠑⢤⡀⠁⠀⠀⠀⠀⠀⠀⠀⡀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⣹⣶⡷⡄⠸⠀⠿⡄⠀⠁⠂⡀⠃⠀⠀⠀⠀⠀⠈⠀⠸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⢿⣻⣶⣿⣆⠀⠀⠘⢆⠀⢠⡀⠂⢀⠀⢐⣄⡠⣀⢀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡟⣆⠀⠀⠈⠓⠀⠑⠀⠀⠁⠀⠙⢺⡽⣷⡄⠀⡆⠀⠩⡀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠨⣛⠿⡈⢧⠀⠐⠄⠀⠀⠀⠀⠀⠸⠀⢿⡏⣩⡙⢀⡸⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢧⠀⣷⢸⡇⠀⠀⠀⡀⠀⠀⠀⠀⢱⢸⣧⡘⠓⢸⡄⠶⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠘⣆⠹⢹⣟⡄⠀⠀⠁⡀⠀⠀⠁⠸⠀⠀⠱⢤⡼⡇⠀⠀⢷⡀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠛⠀⠡⣿⣽⡄⠀⣂⡷⠢⠀⠀⠀⠀⠀⠈⠲⡗⣶⢄⠀⠀⢳⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠱⡆⠠⡽⣽⠓⠈⣱⢗⡀⢀⠐⠂⠀⠢⡀⠌⠻⣷⠀⠀⠈⢧⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠐⠽⣅⢀⠸⣆⠂⠠⠠⠀⠀⠀⠉⣄⠰⠐⣄⠀⠀⠈⡆⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⠘⣇⠘⠂⠈⢑⢠⣥⠀⠀⠀⠀⠘⣧⢡⢩⠂⠀⣀⠙⢆⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⠙⡄⠀⠀⠀⢻⡅⢦⠀⠀⠀⠀⢸⡆⢢⢺⣧⣮⡻⠀⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢹⡄⠀⠜⢆⠀⠀⠀⠉⠊⢧⠀⠀⠀⠀⣷⠘⣎⢣⡛⣡⡆⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⣿⣦⠀⠀⠣⠀⠀⠀⠀⠈⠆⠀⠀⢄⠸⣇⠹⣎⠉⢱⡌⠠⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⢹⠁⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠘⣆⠸⡆⠉⢻⡀⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⠀⠀⢰⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠄⠠⡄⢠⠀⠈⣷⠀⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠨⣿⡄⠀⠀⠔⠄⠀⠠⠀⠀⢄⡡⣄⠀⣿⡌⠡⣀⢿⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠘⠱⠀⠙⢄⠻⠘⠄⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠄⢀⠈⠀⠀⠘⣆⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠰⡁⠀⠀⠀⠈⡄⢀⢰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠐⠆⢀⣀⡠⡡⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣋⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠴⠖⠒⠛⠉⠉⠉⠉⠉⠉⠛⠒⠶⠦⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄
⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦
⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⢸⠁⠀⠀⠀⢰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⠀⠀⠘⡇
⠀⣿⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⣿⠀⠀⠀⠀⠀⢀⠏⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⠘⡆⠀⠀⠀⠀⠀⣿
⢀⡇⠀⠀⠀⠀⠀⡼⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀⠀⢹⡄⠀⠀⠀⠀⢸
⢸⠇⠀⠀⠀⠀⢸⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⣷⠀⠀⠀⠀⠸
⡜⠀⠀⠀⠀⠀⢸⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⢀⣀⣀⣀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿
⡇⠀⠀⠀⠀⠀⠸⡆⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⢠⣿⣿⣿⣿⣧⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⢀⡟
⡇⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠛⠿⣿⣿⣿⣿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠛⠻⠿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠸⣇
⢹⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀
⢸⡆⢠⠶⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⢰
⠀⣧⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿
⠀⠈⢻⡳⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡿⠹⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⡤⢖⡿⠁
⠀⠀⠀⠹⡄⠈⢿⢻⣿⣿⣿⢻⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠘⢿⣿⡿⠁⠀⠹⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢠⡞
⠀⠀⠀⠀⢹⠀⠘⡆⢻⣿⣿⠈⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⢿⣿⡿⢱⡏⠀⡟
⠀⠀⠀⠀⣸⠀⠀⢹⠀⠙⠛⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠇⠘⠛⠁⡞⠀⠀⣧
⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠀⠘⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠀⠀⠀⢠⠇⠀⠀⣼
⠀⠀⠀⠀⢹⡄⠀⠀⢧⠀⠀⠀⠀⢻⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⢀⣀⢀⣠⣄⣀⣤⣄⠀⣀⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠇⠀⠀⠀⣸⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠙⠦⣄⣸⠀⠀⠀⠀⠈⣿⣿⣿⣿⣷⣤⡴⣶⢲⡞⠹⡟⠈⢻⠀⠈⣿⠀⠈⣿⠁⢻⠋⢷⠚⡶⢦⣴⣿⣿⣿⣿⡟⠀⠀⠀⠀⡏⣀⣤⠞
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣇⣇⣿⣸⣇⣠⣧⣠⣾⣤⣤⣿⣤⡤⣿⣀⣼⣄⣼⣀⣇⣸⣿⣿⣿⣿⣿⠁⠀⠀⠀⢠⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣏⣿⣹⣇⣿⣨⣇⣸⣀⣯⢈⣇⢸⣀⣼⣀⣯⣨⣇⣿⣨⣏⣿⣿⣿⡿⠃⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠁⠈⠈⠉⠉⠉⠉⠉⠋⠉⠋⠉⠉⠉⠈⠉⠈⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣄⣀⣀⣠⠤⠴⠦⠤⢄⣀⣀⡠⠶⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⠒⠒⠂⠉⠉⠉⠉⠑⠒⠒⠤⠤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄
⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⡀
⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄
⠀⠀⠀⣠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦
⠀⡼⠁⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠈⢧
⢰⠁⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈
⠨⠀⠀⢠⠀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡄
⡇⠀⠀⠈⡄⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠘⢰
⡇⠀⠀⠀⠀⡼⠀⠀⠀⠀⣀⣤⣤⣴⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣦⣤⣤⣀⠀⠀⠀⠈⢧⠂
⡇⠀⠀⠀⡸⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀⠈⢇
⡇⠀⠀⠀⠇⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⢸
⢡⠀⠀⢸⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣴⣾⣷⣆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⡇
⠈⡄⠀⢸⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⡙⠀⢰⣿⣿⣿⣿⡇⠀⠃⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⡇⠀⢰
⠀⡇⠀⢸⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠐⠀⢀⣾⣿⣿⣿⣿⣷⠀⠀⠂⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⡇⠀⢸
⠀⣱⣀⡜⠀⠀⠀⠈⠛⠛⠿⠿⠛⠋⠁⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠉⠙⠛⠿⠿⠟⠛⠁⠀⠀⠀⢇⣀⣎
⠐⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠙⠲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⡏⢻⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠃
⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠇⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠈⢟⠒⠒⠛⠻⣿⣿⣷⣆⠀⠀⠀⠀⠀⠀⠛⠟⠋⠀⠀⠙⠻⠛⠀⠀⠀⠀⠀⠀⣴⣾⣿⣿⠟⠛⠒⠒⡻
⠀⠀⠀⠀⠈⢦⣀⣄⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠀⣰⣀⡴⠁
⠀⠀⠀⠀⠀⠀⠉⢹⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢸⣿⣿⣿⡇⢀⠋⠉
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⢿⣿⣿⡇⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸⣿⣿⡿⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⢿⣿⣿⣤⣦⡤⣠⠀⣀⠀⣤⡀⠀⣄⠀⣤⢀⣀⢤⣴⣦⣿⣿⡿⠁⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢹⡏⢿⣔⠐⡇⢹⠁⢿⠀⢇⡼⠀⡿⠈⡏⢸⠊⣢⡿⢹⠋⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠃⠆⠀⠀⠀⠷⣤⢙⢹⡷⢴⣤⢿⣤⢼⡧⣤⡯⣴⡧⢾⠟⠉⣤⠟⠀⠀⠀⡐⠘⡄
⠀⠀⠀⠀⠀⠀⠀⢨⠀⢸⠀⠀⠀⠀⠀⠻⢤⣥⣀⣧⣸⡇⢸⡆⡼⣃⣴⢀⣼⢤⠟⠈⠀⠀⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠳⣾⠀⠀⠀⠀⠀⠘⠀⠀⠈⠀⠉⠀⠋⠛⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⣷⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠊

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡗⠐⠶⢶⣶⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⢢⣿⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⡆⠈⠀⢻⣿⡿⠿⠿⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⣻⡟⠋⠉⣀⣼⡗⠤⠐⣿⣿⣷⣤⡀⢈⣿⣿⣧⡀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦⣾⣿⣿⡛⢀⣀⣀⣬⣥⣭⣿⣿⣿⣿⣿⣯⣤⣄⢹⣿⣿⣍⣀⣀⣀⣉⣉⣽⣴⣷⠾⠿⠿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢻⣷⣦⠜⣿⣿⣿⣿⣏⣹⣿⣿⣿⣿⣿⡿⠄⢠⣿⣿⣿⣿⣿⣿⣫⣿⣿⣿⣿⣶⣶⢟⣉⣤⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠹⢿⡿⢹⣿⣿⡿⠿⢿⣿⠟⠿⠿⠿⢿⣿⡇⣀⠀⢙⡿⠟⠛⠛⢶⣦⣿⣿⣟⣻⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡰⢟⣧⠈⣿⣿⡿⢶⣶⡄⠉⢉⣶⣶⣾⣿⣿⣿⣿⣿⣷⣶⣶⣾⣷⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣝⠛⠻⠾⣿⣿⣿⢿⣿⣿⣿⣿⣿⠙⠛⢿⣿⣿⡛⣩⣽⢶⣿⣿⣿⠿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣾⡿⠷⣤⣄⠀⠀⢶⣿⠯⢸⣿⣿⣁⣀⢨⣿⣟⣛⡻⠿⠛⢿⣿⣟⣷⣾⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⠻⣿⣦⣤⣾⣿⣾⣷⣿⣿⣿⣿⣿⣿⠉⠉⢈⣿⣿⣿⠓⠂⠀⠻⢿⣏⣻⣿⠋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣶⣤⠀⡉⢙⠋⣯⣽⣿⠿⢷⣶⣿⣿⣷⡄⢸⣿⠿⠛⢷⣾⣿⣷⣶⣿⡋⠀⢠⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠻⢏⣡⣤⣀⣼⣥⣭⣤⣦⡴⠛⢿⣿⣿⠟⠀⠙⣿⣿⣧⣤⣬⣉⡉⢿⣠⣠⣾⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠋⣰⣄⡉⠉⠛⠛⣿⣋⣭⡀⠰⣶⣿⣿⣿⠗⠀⠀⣯⡵⠻⣿⢻⣉⣽⣿⣿⣿⣿⣇⡀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⠟⠉⠁⠀⣓⣟⣩⣤⣴⣾⡛⣿⣿⣯⠀⠀⢸⣿⣿⣶⣤⣙⣛⡻⠟⠛⠉⠙⢩⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣦⡀⠉⠲⠶⣿⣿⣿⣿⠟⠁⣴⣿⢿⣿⣿⣶⣦⣾⣿⣿⣯⡉⠻⠗⣬⠑⣤⣤⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣃⣸⣷⣦⢠⣾⣿⣟⣩⣤⣶⣟⣁⣾⣿⡏⠀⣿⣿⣿⣿⣿⣉⠁⠂⢠⣽⣿⣿⣿⠿⣦⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡟⠻⠏⠀⢀⣴⣿⣿⠿⣋⣭⣿⠟⢹⣿⣿⣤⣿⣿⣇⠹⣿⣿⣧⣄⣀⠉⠋⡀⢀⣼⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠦⠀⠀⢹⣿⣧⠜⠒⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡽⠿⣿⣿⣯⣾⢿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⢀⣾⠿⠊⠉⠀⢤⢠⣿⠿⠋⠁⢠⣿⠛⢁⣿⣿⣇⠀⠙⢿⣿⣿⣷⣦⡀⠙⠛⠛⣾⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣀⡀⢠⣤⣾⠛⠁⢀⣤⣴⣿⡏⠀⢈⣹⣿⣿⣷⣦⠀⠈⠻⣿⣏⡿⣶⣤⣴⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⣉⢹⣽⣿⡿⣡⣴⣶⠟⠋⠉⢈⡿⠃⠙⣿⣿⣏⡀⠀⠀⠀⠀⠘⢿⣿⣿⠿⡾⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠾⠻⣿⡿⠟⠉⠀⠀⠀⠸⢿⣇⣀⣀⡙⣿⡿⠋⠀⠀⠙⠀⢀⣆⡙⢿⣾⣟⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⡟⠉⠀⠀⠀⠀⠀⠀⣀⣼⠛⠋⠻⣿⣿⣧⣄⠀⠀⠀⠀⠀⠹⠇⠀⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣤⣀⡀⠉⢨⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣤⣼⠏⠉⠁⣶⣾⣿⣤⡀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⣾⣿⣿⠿⡠⢄⣀⠀⠉⣷⣀⠀⠀⡀⠈⣿⠏⠁⣀⣤⣶⡟⠋⠁⠀⠐⢺⣴⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⢿⠉⠈⠉⢰⣷⠈⢻⣷⣶⣿⠛⠛⢶⣷⣿⣿⣿⣾⣿⣿⠋⠀⠀⠀⠀⠀⠀⢸⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⢹⣥⣾⣿⣿⡏⠀⠀⠀⠙⢻⣿⣷⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⢽⡾⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣄⠀⠀⠀⠀⢈⣿⣿⡇⠉⠻⠶⠶⢶⣶⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⣿⠆⠀⢀⣰⣿⣿⣿⣿⣶⣤⡀⠀⠀⢀⣉⣤⣿⣏⣿⣿⣆⣀⠀⠀⠀⠀⠀⣴⡟⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⠟⠁⠀⠘⣿⣿⡏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡟⣻⣿⣿⣿⣿⣿⣿⠃⠀⢀⡱⣤⡉⣩⣿⣿⣿⣿⣿⡏⠀⠈⣿⢄⣄⣤⣦⠈⢻⣷⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⢿⡿⢿⣿⣿⣧⡀⠀⢤⠀⠀⠀⣹⡷⣍⣩⡿⠀⠀⣴⠟⠀⠙⠁⢻⣶⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢳⠀⠀⠀⠉⡟⢻⣷⣦⣀⠙⢦⣴⣿⣿⣿⠋⠀⣠⠾⠋⠀⠀⠀⢀⡿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢹⣾⡟⠙⠛⢷⣤⣿⣿⣿⣿⣤⠞⠉⢀⡀⠀⣠⡴⣀⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠛⠛⠀⠀⣠⣉⣙⣿⣿⣋⣁⣤⡴⠞⠿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠛⢿⣿⡄⠀⢀⡰⠋⠙⠛⣿⣿⣿⣿⣧⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠙⠿⣶⡞⠁⠀⠴⠟⠛⠛⢿⡉⠛⠳⡶⠛⠉⠁⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⢀⣤⣤⡤⠈⠀⠀⠀⠀⠀⠻⠶⠀⠀⠀⣠⣿⣁⠤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠉⠉⠁⠉⠉