Bodega Emojis & Symbols

⠀⠀⠀⠀⠀⠀⢠⣤⣤⡀⢠⡀⠀⠀⠀⢀⣤⣤⣤⣤⠀⣤⣤⣤⣤⣤⡄⠀⣤⣤⣤⣤⣤⡄⠀⣤⣤
⠀⠀⠀⠀⠀⠀⢸⣿⡟⢻⡾⠻⣦⠀⣴⣿⡟⠛⣿⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⠀⣿⠀⣿⣿⡇⠀⢸⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⣤⣤⣤⣤⡄⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⢸⣿⠀⣿⣿⡇⠀⢸⣿⠀⠀⢸⣿⣿⠀⠀⠀⣿⠛⠛⠛⠛⠃⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⡇⢸⣿⠀⣿⣿⣧⣤⣾⠏⠀⠀⢸⣿⣿⠀⠀⠀⣿⣤⣤⣤⣤⡄⠀⢿⣿⣦⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠘⠛⠃⠘⠋⠘⠋⠀⠛⠛⠛⠋⠁⠀⠀⠀⠘⠛⠛⠀⠀⠀⠛⠛⠛⠛⠛⠃⠀⠀⠈⠛⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠐⠐⠶⠲⠶⡶⠂⠐⠂⠂⠐⠀⠘⠿⢿⣿⣿⣿⡿⣿⣇⣤⣤⠤⣴⢰⠆⢲⠛⠻⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠚⠹⠊⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠈⠁⠁⠁⠀⠄⠀⠀⠀⠀⠀⠀⠐⣠⡆⢠⢠⡄⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣃⣘⣚⠃⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢫⠠⢤⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣛⣶⣻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡅⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠠⠀⡇⠀⠀⠇
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⡂⠀⠀⢀⠀⠀⠂⠀⢀⠉⠉⠙⠛⠻⠿⢿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣾⡦⠿⠷⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠏⠻⠶⠀⠀⠝⠲⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠘⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⣒⣀⡀⠀⠀⣀⡀⠀⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠀⠀⠀⠀⠨⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡄⠠⠀⠀⠀⠀⠂⠀⠀⠀⠠⣔⠀⠂⢒⡀⠙⠻⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠉⠛⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⣀⡀⠀⣀⣀⣀⡀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⣷⡄⠀⣴⣿⡿⣿⡇⠈⣿⣿⡿⢿⣆⠀⢸⡏⠉⠉⠁⠀⣠⣾⡿⠿⠿⠿⠇⢀⣴⣿⠿⢿⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣼⡇⠀⣿⣿⠀⣿⡇⠀⣿⣿⠀⠀⣿⠀⢸⣇⣀⣀⡀⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣿⡿⠿⠿⣿⡄⠀⣿⣿⠀⣿⡇⠀⣿⣿⠀⠀⣿⠀⢸⡿⠿⠿⠇⠀⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣶⣾⡇
⠀⠀⠀⠀⠀⠀⢸⣿⣇⣀⣀⣼⡇⠀⣿⣿⣄⣿⠇⠀⣿⣿⣀⣠⡿⠀⢸⣇⣀⣀⡀⠀⣿⣿⣄⣾⣿⣿⠆⢸⣿⣿⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠋⠀⠀⠿⠿⠿⠃⠀⠀⠿⠿⠿⠟⠁⠀⠸⠿⠿⠿⠇⠀⠿⠿⠿⠿⠿⠁⠀⠸⠿⠿⠀⠸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣿⣿⣻⣿⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⠋⠉⠉⠉⠉⠉⠉⠉⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣯⣤⣤⣤⡠⠤⠤⠤⠴⠶⠶⠶⠶⠶⠒⠒⠒⠂⠚⢛⣿⣿⣿⣿⠛⠓⠛⠒⠚⠓⠒⠒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢸⠀⠀⢰⠐⠀⠰⠉⠆⠐⠉⡆⠰⠉⠂⠀⡏⡇⠀⡉⣿⣿⣿⣿⣀⣏⣽⡠⠬⠵⠤⠴⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⣶⣶⣶⣶⣶⡶⠶⠶⠶⠶⠶⠖⠒⠒⠉⠉⠉⠉⠉⠉⢙⣿⣿⣏⣀⣀⣀⣀⣀⣀⣀⣀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡏⠸⡁⠀⠀⠀⣀⣀⣀⣠⣤⠤⠤⠤⠶⠶⠒⠒⠛⠛⠛⢋⣹⣿⣿⡏⠉⢉⣍⠁⢠⢤⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣤⣭⣭⣭⣉⣀⣀⣐⣒⣀⣰⣐⣀⣰⣉⣂⣀⣏⣇⣀⣸⣿⣿⣿⣁⣀⣁⣈⡀⣸⢨⡀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠈⡏⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠁⠀⠁⠉⠉⠉⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⢹⠛⠛⢛⢚⠛⢋⣋⠉⠉⣉⡉⠉⣉⡉⠉⣉⡉⠉⢉⡉⠉⢉⡉⠉⠉⡉⠛⠚⣛⠛⠛⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣾⣀⣀⣘⠀⠀⢨⠀⠂⠈⠀⡇⢸⠀⠀⠀⡇⠁⠀⠁⢸⠀⡂⢸⠀⠸⠀⠀⢸⢈⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⠀⡄⢸⠀⠁⠀⠀⡇⠸⠀⠀⠀⡇⠀⠀⠀⢸⠀⡇⢸⠀⠀⢀⠀⠸⢸⣀⣠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡏⠀⠀⡀⡇⢸⠀⠀⠀⡆⡇⠀⠀⢠⠀⡇⠀⠀⠀⢸⠀⡇⠀⠀⠁⢸⠀⡆⠸⠇⠀⠉⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⡇⢁⠀⡀⢰⠀⡇⠁⠀⠀⢨⠀⡇⡀⠀⠀⠸⠀⠇⡆⢠⠀⠘⠀⣧⠴⠚⠧⣀⣴⡔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⢠⣸⠀⢇⣘⠀⠇⠰⠀⣀⡸⠀⣇⡇⠰⣀⡀⢠⡀⡇⢘⢀⠆⢸⢁⡶⡀⠘⢏⡀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡀⠁⠀⠀⠁⠀⠈⠁⠀⠈⠁⠀⠈⠁⠀⠉⠀⠀⠉⠀⠀⠉⠀⢢⠀⠉⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⢦⡀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠚⠛⠉


⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣄⠀⠀⣠⢤⣄⠀⣠⣤⣄⠀⢠⡤⣤⠀⢀⣤⣤⡀⠀⢠⣤⠀⠀⠀⠀⢀⣤⣤⡀⠀⢠⣤⠀⢠⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇⢸⠇⠀⠸⡆⣿⠀⠙⡇⢸⠁⠀⠀⣿⠀⠀⠁⠀⡿⠹⡇⠀⠀⠀⣾⠁⠀⠁⠀⡿⢸⡇⠀⠀⡏
⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠾⣇⢸⡄⠀⢰⡇⣿⠀⢀⡇⢸⣤⣤⠀⣿⢠⡶⣄⢸⣧⣤⣷⠀⠀⠀⣿⠀⠀⣀⢸⣧⣤⣷⠀⠀⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠧⠴⠋⠈⠳⠤⠞⠁⠻⠶⠞⠁⠸⠦⠦⠄⠙⠶⠶⠏⠸⠃⠀⠸⠂⠀⠀⠘⠷⠶⠃⠸⠁⠀⠸⠀⠐⠗


⠀⠈
⠀⠐
⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇⠀⡦⡖⡖⣞⡂⣶⣶⣓⢖⡲⣞⢐⣳⣴⡀⠈⠛⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⡷⠂⡢⣄⠐⠂⠀⢐⡂⡆⣆⠀⢀⣆⠠⣲⢰⡦⣒⢀⠂⣴⢰⡄⡄⠀⠶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⣇⡀⠂⢴⡦⣤⣤⣤⣦⣤⡤⢤⠤⠤⢤⢤⢤⣤⣤⣤⡥⣬⡤⠤⣤⡤⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷⣾⠀⡿⠛⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⡧⠀⠠⠤⠤⣤⡀⢡⣾⠀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⠏⠉⠙⣾⣿⡿⠛⠛⠛⠛⢹⡇⠀⠈⢀⠀⣌⡇⢸⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⣿⠀⢰⣞⣿⠍⠄⢄⣠⣀⣠⣸⡇⠀⠀⠐⣇⠏⣿⡎⢻⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢨⣿⡶⠶⢾⣿⣤⡤⢤⣦⣴⠘⢼⡇⠀⠀⠈⣿⠀⣛⡳⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⢸⣿⠀⠀⢀⣿⠉⠃⠀⢁⠛⣄⢸⡇⡀⠀⠀⣿⣷⡜⠀⢼⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⢸⣿⣯⣭⣽⣿⠀⠀⠀⠁⠀⠀⢸⡇⠀⢀⡤⠀⠉⣐⢖⣸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣼⡋⠉⠉⣉⣸⣀⢀⣀⣀⣀⣀⣸⣇⣀⣈⣉⣀⡀⠙⠋⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢄⣠⡄⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣋⡄⣠⣾⣡⣾⣏⣀⡀⣤⣄⣀⣤⡀⣠⣄⣠⡿⣤⢦⡄⢴⠆⣴⢾⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠸⣿⣟⣵⣿⣟⣽⣷⣫⣟⣝⣼⣡⣿⣠⣾⣟⣵⣿⣾⣷⢿⠶⠠⣿⡿⢿⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣿⣴⣟⡼⢋⣾⠃⠙⢉⣩⠴⢿⡷⣟⠳⠎⠁⠀⠁⢀⡀⠀⠀⠀⠀⠀⢸⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠛⠁⠈⢂⠀⠀⠀⢀⣾⣡⣶⠏⣰⠟⣠⣥⠀⣠⢤⡾⢡⡴⡄⢀⡤⣤⣾⣠⡶⣦⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⡦⠤⢦⠤⢤⢤⡽⣽⠿⢿⡧⣾⣷⣿⣾⣥⣾⣵⣿⣻⣶⣟⣶⣿⣾⣿⣾⣿⡖⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣄⢀⡀⢡⣿⣾⣿⣯⣶⣾⡇⠈⠥⠤⠬⢧⡭⠤⠭⠥⣤⣿⠯⠥⠼⠤⠤⠥⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡀⣀⣀⣙⣉⢁⡉⣉⣀⣀⣀⡀⡀⣀⣰⣓⣤⣤⡄⠈⣡⣤⣤⣶⣀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣼⣿⣿⡟⠉⢀⣴⣿⣿⡟⠉⠀⢠⣾⣿⣿⠟⠁⠀⣰⣿⣿⡟⢹⣿⣷⣄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⢠⣾⣿⣿⠋⠀⢠⣾⣿⣿⠏⠀⠀⣰⣿⣿⣿⠋⠀⠀⣴⣿⣿⡟⡀⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣤⣤⡿⠿⢿⣧⣤⣤⡿⠿⠿⣧⣤⣤⣴⠿⠿⢿⣧⣤⣤⣴⠿⠿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣯⠉⣿⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣯⡄⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠛⠁⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⠠⠥⠧⣿⣿⣷⣿⣿⣿⠀⣿⡟⠻⠿⠿⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⠦⣴⣻⣄⣿⠿⣿⠿⢿⣿⣿⣿⡿⣿⡿⣼⡿⢿⠿⢿⢿⣿⡟⢿⣿⢿⠀⣿⢉⣭⣉⡉⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢠⣿⡷⡞⣿⣿⢏⠉⢋⣿⣿⡟⣻⣿⣬⠋⠙⠋⠙⠛⠛⣿⣟⣛⣛⣻⡇⠯⠦⣤⢀⣎⡟⣸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠟⠀⣾⠿⠿⠴⠶⠤⠦⠤⣶⡦⣾⣤⣴⣄⣤⣬⣤⣌⣤⣿⣿⣿⣿⣿⣦⣰⣍⠻⣶⢲⠏⠉⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠉⠙⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣾⣿⣿⣷⠶⣶⣶⣶⣶⣶⣶⣶⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢀⡞⣝⠛⠛⠋⣻⢺⣻⣿⣿⣿⣻⣶⣾⢛⣿⣯⣯⣭⣽⣿⣭⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠉⠉⠛⠗
⠀⠀⠀⠀⠀⠀⢸⡏⠉⠀⠀⠀⣽⢹⣻⣻⣿⣟⣟⣿⡿⡻⣿⠗⡻⣿⡟⠿⢾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⡿⡆⠠⠀⠀⠀⠿⠸⣿⣿⠅⣿⡬⣿⣿⢙⣷⣟⣡⣿⣏⣻⣟⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⢰⢹⣯⣽⣀⣀⣀⣛⣺⣿⣿⣾⣿⣿⣿⣿⣿⣯⣿⣽⣿⣫⣝⣯⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢀⣀⢸⣿⣶⣶⣶⣶⣶⣿
⠀⠀⠀⠀⢠⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣭⣭⣽⣿⣛⣛⣛⣛⣛⣛⣛⣛⡛⠛⠻⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠤⣀⢀⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⣮⣼⣤⣼⣦⣬⣤⣤⣬⣼⣾⣿⣦⣤⣦⣤⣾⣇⣍⣀⣁⣼⣏⣉⣍⣹⣏⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠉⠟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠟⠟⠿⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠶⢾⡿⢿⢾⠷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣭⣭⣭⣭⣭⣽⣛⢻⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⠙⠛⠛⠛⠛⠛⠛⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⣿⣤⣠⡄⣼⡧⠀⠀⠀⣽⡅⠤⠄⣬⠀⠄⢸⡇⠀⢸⣿⣿⣿⣿⢸⢸⣿⢰⣶⣶⣶⣶⣤⣤⣤⣤⣼⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠁⠙⡗⢰⣶⣤⣿⡆⢀⣀⣤⣤⣄⣸⣷⣶⣾⣿⣷⡿⣿⢸⢸⣿⢸⣿⣿⢠⣦⢉⡉⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣴⣶⣶⣶⣷⣾⣶⣷⣾⣶⢸⣸⡇⣿⣿⣿⣿⣿⣿⡟⠋⠀⠛⣸⢸⣿⢸⣿⡀⢨⡟⢸⠇⣿⡟⠍⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣶⢨⣍⠛⣻⣏⠉⢹⣏⡉⣸⣹⡏⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⢸⣿⢼⣀⣁⣾⣧⣀⡂⣸⡇⠁⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⣼⣻⣀⣸⣿⢉⣉⡿⣿⢻⡇⢸⣿⣿⣿⢿⣿⣷⣦⣴⣤⣿⢸⣿⢸⣿⣶⡉⠴⠦⠀⣿⣏⣉⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣏⣿⣿⣿⠉⢸⣿⠉⠙⣿⡧⠘⠿⠮⠿⠶⠾⡏⠉⠛⠛⣿⢸⣿⢸⠉⠉⣗⠆⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣟⠛⠛⠛⠋⠉⠉⠛⣻⡟⠛⠛⠛⠛⠋⠙⠛⢻⠛⠛⠛⠛⠛⠛⣻⣼⣿⢸⠀⠀⣿⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣧⡄⠀⠀⠀⠀⠀⢺⣭⣷⣀⠀⡀⠀⠀⠀⠰⢾⠀⠀⠀⠀⠘⠃⣏⣸⣿⢸⣤⣤⣤⣬⣤⣤⣿⣿⣧⣼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠚⠉⢸⠂⠀⠒⣶⣚⣂⣉⡉⣉⣉⣭⣭⠙⢳⣾⣟⣲⣶⢒⣲⡒⣲⣄⢀⣤⡤⣤⠤⠤⠤⡀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⡆⠀⠸⠿⠏⠿⠻⢿⣷⣏⠘⠿⠃⢸⣿⡷⣿⣾⢹⣏⡷⣿⣈⢸⡿⡁⣿⣿⢹⣾⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⣼⡅⠀⠘⠛⠛⠻⠿⠿⠹⠾⠶⠶⠶⠶⠰⣶⣶⣾⣶⣭⢥⣬⣭⣬⣭⣥⣭⣭⣤⣯⣄⠀⣿
⠀⠀⠀⠀⣀⣀⣀⣀⣀⣙⣛⣉⣉⣡⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠀⢀⣀⣀⣉⣉⣉⣻⣿⣷⣏⣀
⠀⠀⣴⣿⣿⣿⣿⣿⣿⣻⣏⣻⣻⣛⣿⡏⠽⣿⣻⡿⢿⠷⡖⠲⠒⢶⠲⢶⣶⡶⣶⣶⣧⣶⣶⡤⠴⣼⣬⣿⣿⣿⣿⣿⡿⠿⣿⣟⠓
⠀⠀⠀⠛⠉⠀⠈⠃⠋⠀⠁⠀⢸⠉⠀⠀⣿⣉⠋⣹⡟⠛⠛⠁⢹⠉⡿⠉⠋⢳⠉⠹⣿⣿⢻⣿⡟⣿⣿⣿⣿⣿⣿⣿⡏⠙⠯⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⠀⠀⣀⡀⠀⠀⠀⢃⣤⣄⢠⡸⠀⣶⡆⠀⠀⢀⣀⡈⠀⠈⢉⡀⣀⣿⠯⠥⢭⣭⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⢸⡀⠀⠿⠁⠀⠀⣤⢸⣿⣿⣿⣇⠀⠋⡁⢤⡄⠈⠉⡇⠛⠀⠀⡀⠀⡛⠒⠀⠀⠈⠛⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠙⠀⠉⠙⡀⠈⢿⡾⣜⣤⠠⠅⠺⠸⠃⠐⡷⠀⠀⠀⢘⡉⡇⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⣶⣾⢿⣹⠀⠖⠂⠀⠀⠀⠀⣿⠀⠲⠀⢈⡗⣇⣀⣀⣀⠀⢨⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⢸⠀⠀⣶⣶⣶⣶⣾⡟⣿⣿⠾⢸⣿⣿⡇⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⢸⡀⠀⣿⣿⣿⣿⣿⣇⠈⠉⠀⢸⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠛⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⣿⣿⣆⣀⣀⡹⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠠⠤⠇⠀⢰⣿⣿⡯⠉⠈⠹⠇⣀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠀⣀⣴⣷⡀⠀⠀⡀⢀⡀⠀⠀⠠⣿⣿⣿⣿⡃⠀⣅⣰⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢿⣽⡏⣠⠀⣁⠀⢁⣄⢠⣌⣩⣿⣿⣷⣆⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠉⠉⠈⠉⠐⠘⠛⠊⠉⠈⠉⠉⠀⠀⠀⡚⠙⢿⣶⠣⢆⡆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠴⣠⠄⠀⠀⠀⠠⠂⠠⠀⡴⣿⡃⠘⠋⠛⠛⠛⠃⠂⠉⠀⠁⠁⢡⠁⠀⠁⠀⠈⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠷⠺⢡⣢⣶⢀⡔⣴⢟⠁⣠⠏⠀⣴⢀⠀⠀⡆⠀⠂⣀⣤⠔⠚⠁⠀⣀⣡⣤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⢠⣞⣀⡄⡀⠀⠘⠀⢀⣤⡼⠟⠋⠀⠀⠀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⠗⠶⠶⠾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠰⠗
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⢄⣼⠏⣥⣾⣷⠶⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠂⠄⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠆
⠀⠀⠀⠀⠀⢠⣾⠋⣼⠃⠀⣼⠇⣤⠀⡼⠃⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠂⠰⠃⠠⢀⣠⠔⠂⠀⠉
⠀⠀⠀⠀⠀⣾⠇⢸⡟⠀⢰⡏⢰⣿⣿⠁⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⢥⠒⢲⠄⡉⠀⠀⠀⠘
⠀⠀⠀⠀⠸⣿⠄⠈⡀⠀⠘⠃⠸⠀⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢄⠘⠄⢈⠃⠠⠀⠰⠀⢀
⠀⠀⠀⠀⢀⠿⠂⠠⡄⠀⠀⠀⠀⠀⢿⠃⠀⠀⣤⣴⠿⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠐⠄⠂⢨⡀⣠⠀⠠⠗⢄
⠀⠀⠀⠀⠀⡀⠀⠀⢣⠀⠀⣨⡀⣄⣸⣅⠀⠀⣠⣜⢂⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠴⢃⠀⠉⠀⠀⠀⡆⠈
⠀⠀⠀⠀⠀⠐⡀⠘⢾⣷⣄⢸⠀⠘⢿⡻⢧⡀⠙⢻⢦⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡑⠙⠂⠈⠁⠁⠂
⠀⠀⠀⠀⠀⠀⢀⣰⠾⠛⠋⠉⠀⠀⠀⠉⠒⠛⠛⠁⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡜⠂⠐⠀⠁
⠀⠀⠀⠀⠐⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠈⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⢾⣿⣿⡿⡿⢿⣿⣿⡿⣿⡿⣿⣿⣿⣿⠓⠒⠒⠒⠂⠈⠙⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣥⢣⣣⣇⣶⣾⣿⣷⣿⣿⣿⣿⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠰⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠷⠶
⠀⠀⠀⠀⠀⠀⠀⢠⣤⡤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⡤⣤⣤⣄⣄⣀⣀⢠⡄⣄⢤⣤⣤⣤⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⢦⣤⢠⣤⣤⡄⣠⣤⣤⣄⣤⡄⢠⣤⣤⣤⣤⢤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣴⣿⣿⣣⣿⣿⣿⣰⣿⣣⡇⢘⣿⡿⣻⠃⣾⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣿⣯⣭⡽⡅⢨⣭⣭⢱⡿⡇⢰⣿⣳⠃⣼⣿⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⢾⣿⢹⣿⢡⡇⢸⡿⢹⢿⣡⡅⣀⡙⢃⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣀⣦⡄⠡⠄⢠⣶⣄⣄⣸⣸⣿⢸⠇⣾⡇⣸⢃⣿⣨⣽⡏⠙⣿⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⠒⠒⢻⣿⣾⡟⠋⣿⠸⠾⠿⠇⠽⠾⠿⢹⣡⡦⢬⡅⠁
⠀⠀⠀⠀⠀⠰⠿⠿⡆⣀⣀⣀⣀⡾⠟⠛⠓⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⢉⡁⠸⡆⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢤⣀⣄⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⡀⠀⡀⠀⠀⠀⠈⠈⠉⠉⠉⠉⠉⠛⠛⠓⠒⠒⠒⠲⠶⠦⠤⠤⠤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠈⠂⠀⠀⠐⠀⠀⠀⠀⠤⠄⠀⠠⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⡖⡴⡅⠀⣀⢀⡄⢠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠬⠇⠼⠁⠃⢀⠞⣰⣃⠜⣱⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⢠⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢸⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢸⣿⣿⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠶⠶⠶⢶⠀⡴⠒⠐⠒⢲⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠈⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⢖⠀⠀⠀⠀⠀⢸⠀⡁⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠇⣿⠀⠀⠀⠸⠂⡇⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⠀⡷⠀⠀⠐⢺⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⡀⣀⣠⠀⡇⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣅⣸⣿⣿⠀⣇⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⢀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⢀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢸⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠌⠀⠀⠀⠀⠀⠁⠀⠀⠀⠙⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⣲⣿⣇⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋⠁⠉⠉⠉⠉⠉⠡⠀⠀⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⢀⡀⣀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⢺⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣿⢿⣾⣷⣿⣿⣿⣿⢿⢿⡿⠿⠿⠿⠿⢿⠿⠿⠿⠯⢉⠹⠭⠻⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣿⠛⢻⣿⠛⡿⠾⠶⠷⠶⠂⠒⠒⢺⠆⠀⠀⠀⠐⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢘⡇⠀⠈⠙⠒⣷⠚⠒⠒⡖⡶⣶⠶⠞⠀⠀⠀⠀⠀⠣⠶⣆⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢨⣇⠀⠀⠀⠀⣿⢶⠖⠖⢲⡞⠈⠒⠁⠤⠴⠶⠄⢠⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢰⡟⠀⠀⠀⠀⡷⠈⠀⠀⠐⢣⡤⣴⡦⠐⠀⠀⠀⡼⠴⠣⠠⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⣿⠚⠛⠛⠚⠚⢻⠻⣷⡖⠓⠓⠒⠛⠓⠚⠓⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠸⡇⠀⠀⠀⠀⣿⠀⣰⣶⣶⣿⣧⠀⠹⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠹⠿⠅⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⣿⣿⣿⣇⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠈⢹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿

⠀⠀⠀⣀⣠⣤⡶⣾⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣶⣷⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀
⠀⢀⣾⢋⣉⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣶⣀⠀⠀⠀⠀⠀⠀⠀⠼⡿⢿⣿⠛⢻⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⣤⣴⣿⣟⣿⢻⣿⣦⣄
⠀⣸⣿⣭⣥⣿⣿⣿⣿⣿⡇⠀⠀⠀⢰⣶⣿⣿⣿⣿⣟⣻⣿⣭⣽⣿⣷⣦⣄⠀⠀⠀⠀⢀⣧⣠⣤⣤⣼⣿⣿⣿⠍⠀⠀⠀⠀⠀⠀⠀⢸⣯⠀⣹⣿⣾⣷⣿⣏
⠈⠉⣻⢻⡋⣤⣄⣸⣏⣿⡇⠀⠀⠀⠀⠀⢻⡟⣿⣿⣿⠀⠒⠒⢻⠚⢹⣿⣿⡗⠀⠀⠀⠚⡟⠛⠛⠛⠛⣿⡿⣿⠂⠀⠀⠀⠀⠀⠀⠰⢾⠛⠛⠛⠛⢹⡟⣿⡏
⠀⠀⠸⠟⠿⠿⠿⠿⠻⠿⠇⠀⠀⠀⠀⠀⠼⠧⠿⠿⠿⠤⠤⠤⠜⠤⠼⠟⠛⠇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠿⠿⠿⠂⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠄⠀⠘⠿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙




⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⠞⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣦⣀
⠀⠀⢐⡾⣿⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣿⣿⣿⣛⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣡⢽⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣷⣄
⠀⣲⡟⠛⠛⣻⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣨⣿⣶⣿⠿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣼⣤⣿⣤⡿⢿⣿⣧⡀
⠀⣛⣷⣶⣶⣯⣼⣿⣭⣿⡃⠀⠀⠀⠀⠀⠀⠘⣫⣭⡵⠿⠒⠛⠛⣿⠇⠀⠀⠀⠀⠀⠀⠀⢈⣁⣠⣸⠀⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⣿⣿⡗
⠀⣿⡿⠿⡿⣿⣿⡿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠿⠛⡿⠛⠛⠉⠟⡿⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⣿⣿⣿⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢿⠚⠿⠿⣿⢿⣿⣿⡇
⠀⠐⠀⠀⠀⠐⠒⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂




⣶⣶⣤⣤⣤⣶⠾⡷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠛⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣤⣀
⣿⣿⢭⢿⠶⣿⢚⣟⠮⣽⣶⠀⠀⠀⠀⠀⠀⠀⣴⡋⠉⢉⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢴⣶⣿⣿⣿⣿⣿⣿⣀⣀⡀⠀⠀⠀⠀⠀⠐⣶⢿⣿⣿⣿⠛⠋⠙⠻⣷
⣿⣿⣚⣻⣛⣯⣩⣯⣛⣲⣿⠄⠀⠀⠀⠀⠀⠀⡟⠓⠛⠛⠋⣿⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⠟⣿⠿⣿⣿⣿⠿⢿⣿⡗⠀⠀⠀⠀⠀⠀⢸⡟⠻⣷⠀⠀⠀⠀⣿
⠿⠿⠉⠽⠵⠷⠴⠷⠿⠿⠿⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠿⠾⠿⠗⠀⠀⠀⠀⠀⠐⠶⠶⠿⠿⠿⠿⠿⠰⠾⠿⠇⠀⠀⠀⠀⠀⠀⠸⠃⠀⠿⠀⠀⠀⠠⠿
⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢼⠀⠀⢤⣤⣤⣤⣤⣴⣶⣶⣶⣶⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠆
⠀⠀⠀⡨⠀⠀⠠⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣼⣿⣿⠃⠀⣼⣥⣤⣤⣤⣤⣤⣤⣤⡄
⠀⠀⠀⣴⣶⣄⡀⠂⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⡟⠿⣛⠛⠛⠛⣛⣛⣛⣋⣭⣭⣭⣭⣭⣭⣽⣽⣍⢹⠇⠀⡼⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⣽⣷⣿⣿⣦⢤⠈⠉⠉⠉⠉⠉⠉⣉⣤⣅⣤⣤⣤⣦⣤⣴⣾⣿⣿⣿⡿⢿⣿⣛⣻⣿⣿⣿⣿⠋⠨⠤⠤⣷⡦⠤⠤⠤⠤⠤⢤⣤⣤⡄
⠀⠀⠀⢪⡙⠛⠛⢻⠀⡇⢀⡤⠶⠶⢤⡀⣿⣶⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠉⢹⣿⢰⡇⣴⡷⠜⢻⢠⠅⠼⠃⠀⡘⢷
⠀⠀⠀⠀⠄⠀⠀⠸⢀⣷⡏⠀⢀⠀⠈⣷⠈⠉⠉⠀⢸⣿⣿⣿⡏⣠⣧⠀⢠⠔⢟⠛⠻⣿⣿⡗⠿⠻⡆⢜⣂⠐⠇⠙
⠀⠀⠀⡀⠬⡀⠀⠀⣸⣏⢷⣄⣀⣀⣤⠇⠀⠀⡠⠁⠈⢿⣻⣇⣇⣿⣿⣦⣯⠀⠇⠀⣦⣏⣈⣦⡶⠀⣹⡶⠥⣤⢀⢤⣤⣴⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⣙⠟⣖⠀⠀⢰⡞⠀⠈⠉⠉⠀⠀⠀⠀⠁⠀⠀⠈⣉⣭⠉⠉⠉⠉⢠⠼⢀⣀⠻⣾⣷⣴⣦⣾⠛⠉⠉⠙⠋⠉⢛⣶⣿⡞⢼⣏⣙⣛⡃
⠀⠀⠀⣉⠀⠈⠀⠀⢨⣿⠇⢤⣤⣤⣤⣤⣤⣧⡀⠀⠰⠿⠏⣒⢧⣤⡄⣀⡀⠂⠀⠀⢰⡿⢿⢿⠙⢿⡄⠀⠀⠀⠀⠀⠘⠉⠚⠉⠑⠚⠓⠛⠁
⠀⠀⠀⠏⢉⢆⣒⠀⡈⠇⠀⢸⣿⣿⣿⣿⣿⣿⣷⠀⣟⠂⠐⠠⢿⣿⣷⣿⡇⠅⠀⠀⠘⣧⢼⣴⣀⣉⡅⠀⠀⠀⠀⢀⠘⠒⠒⠒⠒⡒⠒⠒⠂
⠀⠀⠀⣤⣾⣷⢿⠀⠇⠛⠋⢸⣿⣿⣿⣿⣿⠿⢓⡂⠀⣤⣿⣤⣦⣴⣦⣹⣼⡆⠘⠀⢀⣿⣯⣟⣿⡿⢧⠀⠀⠀⠒⠒⠰⡄⠀⠀⠀⠀⣖
⠀⠀⠀⠛⠛⠋⠉⠀⠀⠀⠀⠸⠳⠯⢻⡀⠀⠀⠀⠃⠀⠨⣅⣩⢚⣭⣸⣥⣯⡥⠴⠄⢸⣿⠏⠙⣿⢀⡀⣀⣀⡀⠀⢰⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⢃⡀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣦⣦⣤⣤⣤⣄⣈⡉⠉⣹⣿⠉⠉⠉⠁⠀⢸⣿⣽⣷⣆⣿⣀⣐⡀⠈⠉⣅⠘⣤⡇⠀⣿
⠀⠀⠀⠀⣙⠎⠰⢶⠶⢶⠶⠶⠶⠶⠶⠚⠛⠛⠛⠛⠛⠓⠒⠒⠚⠓⠒⢒⣛⣻⣿⣿⠛⡟⠉⢻⣍⠻⢋⣋⣃⡤⠿⢶⠿⠷⡶⠿⠆⢠
⠀⠀⠀⢲⢸⣀⣀⣈⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡏⢹⠉⠉⠁⠀⠀⠘⣿⠀⠀⠀⠀⢸⠀⠈⣇⣀⣼⡒⢚⣆⠀⡀⠨⣤⣀⡀
⠀⠀⠀⣸⠸⠿⠿⠿⠿⢿⠿⢿⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠟⠟⠷⠾⠒⠛⠿⠿⠶⠶⠶⠶⠶⠶⠶⠿⠤⠠⠿⠿⠿⠿⠿⢿⠿⣇⣤⣭⣭⡅
⠀⠀⠀⢿⠿⡿⢿⠿⡿⣯⣿⣭⡭⣅⡅⢀⡠⣀⡀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⡤⠤⣤⠿⠿⣿⠿⣿⠿⠿⠿⠿⠿⠿⠿⠿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⣤⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⣿⣿⣿⣿⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠋⢛⢷⡍⠙⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠛⢻⠛⣿⢻⡟⣻⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⣿⢸⣇⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣿⣿⠂⠉⢀⠈⠸⠇⢾⣿⣿⣷
⠀⠀⠀⠀⣀⣀⣀⣿⣀⣀⣠⣇⣸⣿⣀⣀⣸⣇⣀⡀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣛⣛⣛⣛⣛⣿⣿⣿⣿⣿⣿⣿⡷⠀⣀⣀
⠀⠀⠀⢠⡏⠉⠉⢿⣭⣍⠉⠉⢹⣿⣍⠉⠉⠉⣿⣿⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⣩⣍⠉⠉⠉⣩⣭⠉⠉⠉⢹⣿⡇⣿⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠛⠋⠀⠀⠀⠛⠛⠀⠀⠀⣿⣿⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣴⡟⠛⠀⠀⠀⣿⠛⠀⠀⠀⢸⣿⡇⣿⣿
⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⣿⡀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⣿⡇⣿⣿⡀
⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⣇⠀⠀⠀⢀⣿⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠻⣦⣀⣠⣼⠇⠀⠀⠀⠀⢸⣿⡇⣿⣿⡇
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⢸⣿⡧⣿⣿⡇
⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⣿⣿⣷
⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⣿⣿⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠇⢿⣿⣷⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⡻⣿⣿
⠀⠀⠸⠷⠶⠤⣤⣤⣤⣤⣄⣀⣀⣀⣀⡀⠀⠀⣿⣯⣿⣿⣿⠿⠀⠀⠀⠀⠀⠐⠷⠶⠤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⠀⠀⢸⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠙⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⢤⣤⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣶⣖⣒⣦⣄⣠⢯⠟⠋⠉⠉⠙⠺⣟⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠞⠉⠀⠀⠉⢹⣿⣯⠀⠀⠀⠀⠀⠀⠘⣮⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣧⡏⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⠀⠀⠀⢸⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣻⠀⠀⣀⣤⠤⣀⣸⣿⢸⣼⡄⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡴⠚⠁⢸⠀⠀⠉⠉⠺⣿⣧⣀⡀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⢠⠶⣒⡋⠉⢹⣿⠀⠀⠀⢸⠀⠀⠘⢿⣶⣿⣇⠀⠉⠙⠲⠤⣤⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠉⠓⢾⣿⡀⠀⠀⢸⠀⠀⠀⠀⠈⣿⡿⠿⣷⣦⣤⣀⣀⠈⠙⠒⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⢸⣿⠉⠑⠦⣼⡀⠀⠀⠀⠀⣿⡇⠀⠀⠈⠉⠉⠛⠃⠀⣀⣠⠤⠞⢻
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠀⠀⠀⠀⠈⠙⠲⢤⣀⣿⠇⠀⠀⠀⠀⠀⢀⡴⢿⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⠳⢤⣀⠀⢀⡴⠋⠀⣿⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠟⠀⠀⠈⢙⡋⠀⠀⠀⢻⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢾⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⢸⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡸⠠⠤⠤⠤⣤⡇
⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡼⠁⠀⠀⢀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡼⠁⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢤⡀⠀⠀⠀⠀⢸⠀⡜⠁⡠⠚⠁
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠦⣄⡀⢸⣞⠴⠋
⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠁
⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣞⣀⣀⣀⣈⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠍⠉⠉⠉⠉⠍⠉⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⣀⣠⣋⣇⣈⣆⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⠇⠘⠁⠈⠉⠋⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠛⠒⠓⠒⠒⠒⠒⠒⠚⠁



⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣴⣶⣾⡄⣶⣖⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡠
⠀⠀⠀⠀⠀⢀⣶⣶⠘⠿⠋⣀⠉⢀⠈⠛⠀⠀⠠⠀⢀⠀⠀⠀⠀⠀⠀⢐⡒⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⠥⠤⠖⠀⠠
⠀⠀⠈⠀⠐⢌⠻⢿⡿⠂⢸⣿⣿⠿⠃⠀⠀⠀⠀⠁⠀⠀⢁⠠⢀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠂⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠁⠀⠀⠀⠀⠀⡀⠀⢀
⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⢈⠐⡤⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⡁⡂⠐⡀⢐⠀⣂⢡⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡼⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠤⠀⠅⠤⠀⠒⠐⠀⠀⠀⠀⠀⠀⠀⢌⢂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⢰⡶⠄
⠀⠀⠀⠀⠀⢂⢐⠐⠂⠐⠀⡃⣈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⡀⠀⠀⠀⠙⠋⠈⣡⣴
⠀⠀⠀⠀⠀⠀⠈⠀⠁⠁⠀⡀⠁⠠⠤⠄⠀⠀⠀⠈⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⡿⠀⠀⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⣀⠠⠆⠀⠤⡀⠣⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠀⠀⠀⢨⣤⠀⠀⠀⠀⠀⠀⠰⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡅⢠⣳⡶⠀⠁⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⠀⠀⠀⠀⢀⣴⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠁⠤⡅⠛⣀⠈⢀⠁⠀⡕⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠢⠀⠀⠀⠀⠆⠀⠀⠲⠒⠀⠲⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣴⣶⠦⠀⠀⠠⠀⠀⠀⠀⠃⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⢀⣄⣸⣶⡟⣹⡏⢿⡇⠀⠠⠃⠀⠀⠀⢀⡀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⠂⠀⢸⡶⢋⣿⠛⣿⡍⢛⠂⠛⠃⠊⠁⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠸⠿⠀⠿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⠿⠛⠛⠛⠛⠿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⠆⢘⣛⠁⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠈⢛⡋⠐⣷⣶⣶⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⣴⣿⣿⣷⡀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣾⣿⣿⣷⠈⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇⠹⢿⣿⠿⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠻⣿⣿⠟⢠⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣷⣦⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣴⣶⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠄⠒⠒⠒⠒⠒⠂⠒⠆⠀⠀⣈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠤⠤⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡠⠄⠀⢀⠠⠄⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⡔⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⡚⠀⣀⣀⢘⢲⡀⠐⠀⠀⠀⢰⠒⢢⠤⠄⠀⠀⠠⠄⠀⠀⠀⠀⢃⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢸⡀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡔⠃⠊⠙⠂⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠘⢀⠄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠂⠘⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⡆⠀⠀⠀⠀⢀
⠀⠀⠀⠖⠲⠖⠒⠒⠒⠂⠒⠂⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⢄⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡆⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⡇⠀⡠⠒⠂⠐⠒⢤⠀⠀⡴⠂⠑⢢⠀⠖⠉⠙⢆⢸⠀⠀⢀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡐⠇
⠀⠀⠀⡇⡘⠀⠀⠀⠀⠀⠀⢱⡀⢣⡀⢀⡼⠀⢦⣀⣀⠎⢸⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⡇⠀⠀⠀⢀⠀⠀⠀⡃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⣠⠤⢤⠄
⠀⠀⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⡇⣠⠆⠠⢄⠀⡴⠒⠢⡄⢸⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠧⠀⠀⠀⠘⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁
⠀⠀⠀⡇⠱⡀⠀⠀⠀⠀⠀⣰⠁⣇⠀⠀⣸⠀⣄⠀⢀⡄⢸⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠰⠄⠀⠀⠠⠄⠄⠤⠀⠀⠐⠀⠀⠰⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⡇⠀⠈⠢⠤⠀⠤⠞⠁⠀⠈⠉⠈⠁⠀⠈⠉⠉⠀⢸⠀⠀⠀⠀⠀⠀⠀⠅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⢀⠀⠀⡀⠀⢀⣄
⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠳⡀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠠⠂⠁⠀⠀⠀⠀⠉⠒⢄⠀⠀⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⡼⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠢⢀⣀⣀⣀⣀⡀⠠⠊⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠸⣅⠼⠁⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠴⠊⢠⠘⠅⠉⠳⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠤⠀⡀⠤⢀
⠀⠀⠀⢠⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠄⡀⠀⠀⠀⠈⠀⠄⠀⠀⠤⠂⠀⠘⡆⠀⢀⡔⠚⠒⠒⠢⡀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠈⠀⠀⠀⢨⣀⠠⠀⠠⠐⠨⠀⠐⡇⠀⡎⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣀⠀⢀⡠⠀⠀⠀⠀⠑⢤⡐⣀⢶⠡⡈⠎⠀⠀⠡⡀⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⡆
⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠔⠉⠑⠀⠈⠁⠀⠉⠀⠀⠀⠀⠀⠈⠛⠂⠑⠉⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⡄⠀⠀⠀⡇⠀⢐⠀⠀⠁⢰⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡦⠀⠀⠠⠄⠀⢠⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠜⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⡇
⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⠀⠀⡇
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠢⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⠊⠀⠁⠀⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠿⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⢀⠀⠀⠀⠀⠙⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⢠⡏⠀⠀⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⠃⠀⠀⠀⠀⠀⠈⢿⣦⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣄⣀⣸⣿⠀⢹⠀⢀⡄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠛⠉⠉⠉⠉⠉⠉⠻⠀⠈⠀⡾⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠀⠀⠈⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠠⠴⠛⠉⠀⣠⡤⠶⠚⠛⣻⡆⠀⠈⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠁⣀⠤⠴⠚⠉⣀⣴⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣿⡀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⣀⣀⣀⣤⣤⡶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠘⠁⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠐⣷⡀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⣀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣆⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢷⡿
⠀⠀⠀⢀⣠⣴⡶⠿⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⢀⣴⠟⠋⠀⢀⣤⠴⠖⠒⠂⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠷⢶⣤⣤⣤⣤⣴⠟
⢠⡟⠁⠀⠀⠜⠉⠀⠀⠀⠀⠶⠋⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣨⣽⠟⠁
⠻⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠰⣶⣤⣤⣤⣤⣴⡶⠾⠛⠉
⠀⠈⠻⢷⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣸⠁⢀⡆⠀⠀⢿⡇
⠀⠀⠀⠀⠉⠉⠙⠛⠛⠿⣦⡀⠀⠀⠋⠀⣼⠁⢀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⢀⡏⠀⣼⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣄⠀⠀⠘⣇⣰⠏⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢷⣄⠀⠈⠁⠀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⣾⠏
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣬⣿⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣥⣤⣤⣈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

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

⠀⠀⠀⠀⢀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠘⣿⡆
⠀⠀⢰⢻⠋⠀⠀⠀⠀⠀⠀⣔⠔⡀⡀⣐⡎⢠⠤⠀⠀⠀⣁⣈⠀⠀⢀⣀⣀⡉⠁⠁⢈⣀⣁⣀⠈⠀⠀⢿⠇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠙⠛⠛⠋⠀⠀⠀⠀⠀⠈⠙⠋⠉⢡⠖⠀⢀⠄
⠀⠀⠰⠘⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣠⠆⢰⠀⡠⠄⡰⢃⠀⣀⣀⢤⡴⠋⢀⡄⢘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⡎⠀⠀⠀⡰⠃⠐⠁⠘⠃⠘⠀⠛⠀⠘⠁⡜⠁⠰⠉⢠⠏⠐
⠀⠀⠀⠈⠀⠀⠂⠂⠀⠠⠓⠂⠁⠂⠚⠁⠀⢰⠇⠤⠀⠰⠁⢠⡾⠹⡼⠢⠄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⣯⢿⢜⡜⣾⢽⠀⠉⠉⠐⠂⠐⠐⠢⠤⠴⣶⡀
⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⠀⠀⠀⠘⠫⢷⡮⠵⠋⠀⠠⠠⠄⠴⠆⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠁⣰⠂⠀⠀⠀⠁
⠀⠀⠀⢠⠀⠀⠀⠀⠀⣠⠞⣠⠀⠀⡰⠂⢀⡠⣄⡀⠀⣰⢋⡆⠀⢀⡤⠄⠀⣤⢀⡀⢀⡴⢁⡆⠀⠀⠀⠀⡀
⠀⠀⠀⠈⠀⠀⠀⢀⠜⠁⡼⠁⣠⠞⠁⠐⠎⠠⠏⢀⡞⣹⠉⠀⠰⠉⠀⠀⠀⠃⣏⠔⡼⠱⠏⠀⠀⠀⠀⠀⡆
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠠⠊⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⢀⡄⠀⠀⡀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⢀⡀⠀⠀⠀⠀⠀⠀⣧⣀⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠁⠀⠂⠀⣀⡀⠀⠀⡠⡄⠀⠀⠀⢀⣃⡀⠀⣀⡀⢀⠄⠀⠀⢀⣀⠀⠘⠂⠛
⠀⠀⠀⠘⡥⡄⠀⠀⡁⠀⠀⠀⡰⠉⠀⠴⠯⠍⠙⡜⠀⠀⠀⠀⠀⠒⠒⠒⠂⠔⠠⠆⠆⠴⠶⠲⠰⠶⠴⠆
⠀⠀⠀⢠⠀⡂⠀⠀⡁⠀⠀⡰⠣⡇⠀⠀⠀⡼⢰⢇⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠂
⠀⠀⠀⢸⠀⡅⠀⠀⠀⠀⢸⠇⡰⠁⠟⠺⠂⡗⢛⡏⠀⠀⠀⠀⠀⠀⠈⠉⢉⣉⠁⠀⠀⠀⠀⠘⣷⡶⠶⠶⠆
⠀⠀⠀⠈⠂⡅⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠁⠈⠁⠀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠤⠤⠤⠤⠅
⠀⠀⠀⠀⠀⡇⠀⣘⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⡀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⣄⣀⠀⡀
⠀⠀⠀⠠⠀⠇⠀⠿⠦⠤⠶⠶⠶⠶⠶⠾⠛⠁⠋⠉⠀⠄⠀⠈⠀⠀⠀⠠⠠⠊⠇⠂⠀⠀⢰⡤⠀⠉⠉⠐⠀⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠒⠒⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣖⠉⠉⠉⠉⠁⠀⠀⠈⠉⢉⠽⠄⠠⡃⠠⠦⡌⠙⣧⠂
⠀⠀⠀⠀⠀⠀⡄⣾⢿⡩⡐⠂⠤⣀⡀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠓⠒⠻⢬⠁⠒⠤⠤⣀
⠀⠀⠀⠀⠀⠀⢉⠦⠅⡀⠬⢀⠀⠀⠀⠉⠒⠒⠠⠤⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀
⠀⠀⠀⠀⠀⠀⡏⠒⠒⠒⠒⠒⢊⠡⠒⠀⠠⠀⣀⠀⠀⠀⠈⠉⠉⠒⠒⠂⠤⠄⠀⠀⠀⠀⣠⣤⠇
⠀⠀⠀⠀⠀⠀⡇⠀⢀⢀⠤⠾⠭⣍⡓⣄⠈⠁⠀⠀⠡⠤⢐⣒⣀⠀⠀⠀⠀⣀⣀⡈⠉⠛⠛⣿⠆
⠀⠀⠀⠀⠀⠀⡇⡔⡕⠸⠿⡿⢷⣯⡑⣾⠀⠀⢠⠖⠉⢀⣀⠀⣏⡈⠉⠑⡖⠒⠶⡀⠀⢠⠏⠁
⠀⠀⠀⠀⠀⠀⡷⡄⢄⣀⢀⡁⢀⡠⢃⣿⠀⠀⠶⠀⠘⠁⠀⣷⠇⢹⠀⢰⠃⠀⠀⢸⡆⠂
⠀⠀⠀⠀⠀⢸⡇⠙⠒⠮⢭⠭⠵⠒⠁⡇⠀⠀⡄⠀⠀⠀⠀⡇⠀⠈⢹⣸⠀⠀⢀⡞⡸
⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡇⠀⠀⠀⢀⡇⠀⢀⠏⠁⠀⢠⡞⢀⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡞⠀⠀⠀⢠⡇⠀⢰⠇⠀⠀⢸⢸⠀⠀⡼⠀⠀⠀⣿⠀⣸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⡜⢸⠀⠀⡇⠀⠀⡜⡟⠀⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡀⢸⠀⠀⢸⠀⠀⠀⡇⡟⠀⣼⠁⠀⢰⢠⡇⢸⠃
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡇⠀⢠⠃⡞⠀⠀⡌⠀⠀⢰⠁⡇⠃⡇⠀⢠⠇⣸⠀⡞
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠁⠀⡞⢠⡇⠀⠀⡇⠀⠀⡞⢸⠁⢰⠇⢀⠞⢀⡇⢠⡇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸⠀⢰⠁⢸⠀⠀⢀⡇⠀⡼⢁⡇⠀⢸⠖⠋⠀⣼⠀⢸
⠀⠀⠀⠀⠀⢸⠀⠀⠀⢺⣠⠇⠀⡿⠠⠎⠀⡷⠊⠀⡸⠀⠀⠈⢧⣀⡴⠃⠀⢸
⠀⠀⠀⠀⠀⢸⠠⢄⡀⠀⠓⠒⠚⠁⠀⠀⠀⠙⠲⠖⠁⠀⠀⠀⠀⠀⠀⢀⡠⡟
⠀⠀⠀⠀⠀⠈⠒⠢⠤⢭⣑⡒⠒⠒⠂⠀⠤⠤⠤⠤⠐⠒⢒⣒⣂⣉⠭⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢠⢀⠔⠂⠀⠀⠀⠀⠺⣇⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⢸⠠⡀⠀⠀⠀⠠⣺⠆⠿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠸⠀⠉⠛⠷⠿⠛⠁⠀⠀⠀⠈⠛⠀⠀⠀⠀⠀⠀⠐⠻⣦⠀⠈⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⢶⠈⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠃⠀⠀⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣐⠲⠶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠠⣿⠀⠀⣰⠋⠃⠒⠂⠀⠀⠀⠀⠀⠀⢶⣤⣀⡀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⠘⠁⢰⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠈⠙⠛⠿⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣐⣤⢀⣽⠃⠀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠸⠁⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡾⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⢀⡤⣤⣄⡈⠙⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀⣈⣷⡀⡯⠀⠀⠀⠙⠳⢤⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢯⡀⠰⡒⠲⠤⣍⣉⠛⠶⣄⠀⠀⠀⠀⠙⢦⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢄⠀⠀⠙⣟⠦⣌⠛⢦⡀⠀⠀⠀⢳⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡀⠘⢦⠈⢷⣦⡙⢦⡀⠀⠀⢷⡈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢦⡈⠳⣌⢧⡙⢦⡙⢷⡄⠈⢷⡈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣬⣷⣄⣹⣄⣘⣷⡙⢦⠹⡄⢻⡄⠈⣧⠸⡇⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠓⠿⠶⣿⣤⡼⠶⠷⠶⠶⠶⠟⠋⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣰⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⡇⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⢀⡇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⡀⡿⣀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⢀⣤⠞⠁⠀⠈⠛⢦⣀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠋⠀⠀⠀⠀⠀⠀⠀⠙⣃⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠷⠶⠶⠶⢶⣶⣶⣶⣾⣿⣤⣤⣴⣶⣶⠶⠶⠿⠿⠛⠛⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡚⠁⠀⠀⠈⠉⢹⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠄⠋⣹⠆⣰⣿⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠾⠷⠎⠉⠛⢹⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣑⣒⣒⡒⣒⢒⢚⠜
⠀⠀⠀⠀⠀⠀⠰⠒⠒⠒⠒⠒⠒⠒⠒⢲⡶⠶⠶⠶⢶⠖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⡆⢸⠀⣿
⠀⠀⠀⠀⠀⠀⠸⠒⠒⠒⠒⠒⠒⠒⠒⢿⠡⠤⡀⣠⠄⠸⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠺⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣀⣀⣀⣀⣀⣸⠠⠾⠜⠛⠃⢠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⢸⠀⣿
⠀⠀⠀⠀⠀⠀⢸⣉⣉⣉⣉⣍⣩⣍⣩⡩⣥⣤⣤⣤⣬⢬⣍⣉⣭⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣭⣭⣭⣩⡇⢸⠀⣿
⠀⠀⠀⠀⠀⣀⣸⣹⣉⡉⠉⠉⠉⣟⡷⠶⠾⠿⠿⠿⠷⠶⠶⣿⠀⠀⠀⠀⠀⠀⠀⢀⡀⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⢸⠀⣿
⠀⠀⠀⠀⠨⡇⠀⡯⠀⢸⠀⠀⠀⣯⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡇⢸⠀⣿
⠀⠀⠀⠀⢘⡇⠀⡇⠀⢸⠀⠀⠀⡷⡇⠀⠀⠀⣀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢠⢤⣤⡤⠤⢷⣧⡤⡤⡤⠤⢄⠀⠀⠀⠀⡗⡇⢸⠀⣿
⠀⠀⠀⠀⠀⡇⠀⡇⠀⢸⠀⠀⠀⣟⡇⠀⠀⠀⠛⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠾⠿⠿⣶⠾⠿⠷⡷⠷⠿⣶⡞⠻⡆⠀⣏⡇⢸⠀⣿
⠀⠀⠀⣰⢚⣷⣺⣿⣶⣷⣶⣶⣶⣯⣇⣀⣀⣀⣤⣀⢀⣀⣠⣿⣶⣶⣶⣶⣾⡀⠀⠀⣤⠀⢀⣀⣇⣀⡀⣼⣇⣀⣷⣾⣧⣗⢺⠀⣿⠲⡀
⠀⠀⢰⠃⠛⠛⠛⠛⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠊⠈⠉⠁⠀⠱⡀
⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠒⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢠⡖⢲⣤⣼⣤⣶⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣽⣿⡟⠛⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⣶⣾⣷⣶⣿⢻⣿⣿⣿⠀⡀⣘⠁⡂⠀⠀⠀⠀⢸⣿⣿⣿⠋⠉⠉⠉⠁⡄⠀⠀⠀⣿⣿⢿⣿⢿⣿⣿⣿⣿
⠀⠀⠀⣿⣿⣿⣿⣿⡌⣿⣿⣿⡀⣿⣿⣿⣿⡀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⢀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣷⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⢻⠀⠀⠀⠏⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠸⠿⠿⠿⠿⠿⠧⠚⠛⠛⠻⠿⠿⠿⠿⠇⠀⠀⠀⢸⣿⣿⣇⣀⣀⣀⣀⣀⣸⣧⣤⣤⣄⣀⣀⣀⣀⣹⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣻⡇
⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡇
⢸⡇⢠⣾⠛⣿⡆⠀⣠⣿⡇⠀⢸⡇
⢸⡇⠀⣠⡾⠟⠁⣾⣯⣼⣧⡄⢸⡇
⢸⡇⠸⠿⠿⠿⠇⠉⠉⠹⠏⠀⢸⡇
⢸⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⢿⡇
⠸⠿⠿⠿⠿⣿⠿⠿⣿⠿⠿⠷⠿⠇
⠀⠀⠀⠀⠀⣿⠀⠀⣿
⣶⠶⠶⠶⠾⠿⠷⠾⠿⠶⠶⠶⠶⠶⠶⠶⢶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶
⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⣾⠿⢿⣶⡀⠀⢀⣴⣿⡇⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣋⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣹⣿⠀⠀⠈⢉⣠⣾⡿⠁⣴⡿⠋⣿⡇⠀⠀⢸⣿⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣻
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣰⣿⣟⣁⣀⡸⠿⠿⠿⣿⡿⠇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⠛⠃⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸
⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⢻⡏⠉
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⣿⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⢸⣿⠛⠛⠛⠛⠛⠛⠛⣿⡟⠛⠛⠛⠛⠛⠛⢻⣿⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⢸⡇
⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⢰⡇⣿⡇⣿⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠘⠇⣿⡇⠟⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⣿⡀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⢸⣿⣀⣀⣀⣀⣀⣀⣀⣿⣇⢀⣀⣀⣀⣀⣀⣸⣿⠀⣸⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⣸⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⠤⠄⠠⠄⠉⠍⠉⠃⠧⠠⠤⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠐⣀⠁⠀⠀⠐⠀⠠⠊⠀⠀⠀⠀⠀⠈⠐⠀⠁⠀⢁⡐⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢰⠐⠀⡀⠀⠀⠨⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠀⠀⠀⠀⠀⢰⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠠⠤⠤⠀⠀⠈⠉⠉⠉⠉⠉⠀⠀⠀⠠⠤⠤⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠄⣀⠀⠀⢘⠀⠀⠊⠁⡆⠀⠰⠀⠀⠀⠀⠂⠉⡆⠀⠈⠀⢀⢀⡀⠄⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠇⢃⠑⢨⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠐⢁⠃⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢀⠀⢸⠀⠀⢸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠘⠀⢸⠀⠀⡇⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡇⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢸⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⡇⢸⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠃⢸⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢏⡂⠄⢀⠀⠐⠠⠂⠀⠀⠀⠠⠀⠀⠀⠔⠀⣀⠠⠐⣈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⢁⠂⠀⠀⢀⣀⠒⠒⠀⠐⠒⢂⡀⠀⠡⠐⢂⠡⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠒⠂⠀⠤⠤⠠⠠⠄⠀⠐⠂⠀⠈

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⠤⠀⠀⡤⣀⢐⠉⠂⠲⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⣀⣀⣀⡀⠠⣀⡀
⠀⣅⠳⢤⠔⢄⠀⠠⠹⢀⣀⣣⠞⢀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢘⠢⠏⠡⠡⠈⡀⠸⢹⡞⠀⢆
⡠⠃⠀⢀⣠⠛⠃⠀⠀⠢⠜⢸⠧⡄⢠⠸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢉⡁⣠⣘⢄⢀⢇⢀⣇⣰⡼⢻⡖⠊
⠃⠤⠊⠣⣀⡱⢦⡡⠌⣐⡰⠃⠑⠒⠂⠀⠀⠤⢀⠐⢌⣭⣔⡒⡒⢄⠀⠉⡧⠃⠁⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠁⠒⠂⠥⠤⠤⠴⠄⢻
⠀⠀⠀⠀⠀⠀⣀⠤⠤⢄⡀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠠⠤⢀
⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠈⠀⠰⠀⠤⠤⠤⠤⠖⠊⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠤⠅⢁
⠀⠀⠀⢸⡄⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠐⠊⠀⢄⠱⠄
⠀⠀⠀⠈⢌⠒⠄⢉⠒⠀⠐⢊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣀⠄⠂⠠⢀⣀⠀⠈⢈⡀⠀⠀⢀⢠⢖⡀
⠀⠀⠀⠀⠈⠱⠤⣀⣀⡩⠝⠁⠀⠀⠀⠀⡠⣄⣀⠀⠀⠀⠢⠀⢈⣁⠀⠀⡀⠐⠈⡄⠀⠀⠈⠀⡑⠏⣠⠲⠊
⠀⠀⠀⠀⢀⠃⡐⠀⠀⠀⠀⠀⠀⣀⠠⠚⠒⠀⠘⣤⣄⠀⠀⠀⡀⠈⠐⡄⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢀⠎⡘⠀⠀⠀⠀⠀⢴⡋⠄⠀⠀⠀⠀⡊⠀⠈⠉⠁⠚⡍⢣⠀⠷⠊⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠈⢄
⠀⠀⠀⢎⣠⠀⠀⠀⠀⠀⠀⠠⢈⠑⠈⠀⠙⠉⠈⠖⠉⠙⠆⠀⡠⠸⠁⠀⠀⢀⠔⠊⠁⠀⠀⠉⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠀⠚⠀⠈⠓⠀⠉⠓⠈⠀⠀⠀⠀⢀⠏⠀⢀⠔⢋⠑⡦⠂⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠑⣄⠀⠀⠀⠀⠀⠸⡀⠉⡇⠀⣭⠤⠣⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠲⢚⢉⠉⢫⡤⢀⠃⠀⠀⠀⠀⠀⠙⢤⡛⢮⣀⣀⡧⢏
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠄⠂⠉⠀⠑⠄⠀⠈⣎⡤⠊⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠈⢂⠠⡀
⠀⠀⠀⠀⢲⠠⠠⠅⠀⢂⠑⠄⡄⣀⡠⠴⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡘⢆
⠀⠀⠀⠇⠀⡑⢒⠔⠣⠤⠄⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠃

⠀⠀⠀⣠⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣾⣿⣿⣷⣶⣦⣀
⠀⠀⠀⣇⢿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣦⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠘⣞⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠸⠉⠉⠉⠉⠉⢃⠀⠀⠀⢐⠉⢹⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠘⣾⣟⣹⣿⣿⣦⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⢨⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠙⢦⡙⢿⣿⣿⣷⡄⠀⠀⠀⣶⠶⠶⠶⣶⣶⣶⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠈⠳⣄⠻⣿⡿⣿⣦⠀⠀⣿⣇⠀⣾⣿⣿⣿⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡃⠀⢿⣿⠷⢀⣩⣭⣶⣶⣶⣶⣶⣶⣤⣄⡸⢽⣿⠀⠀⠀⢀⣾⠛⠻⣿⣿⣿⣿⣿⣿⠟⠁
⣀⡤⠴⣶⣿⣷⣄⠀⠀⠀⠹⣄⢈⣥⣾⠿⠛⠉⠁⠀⠀⠀⠀⠈⠙⠻⢿⣶⣝⠀⠀⢀⣾⠃⣰⠃⠀⣽⣿⣿⡟⠁⠀⢰⠚⠿⣶⣄⡀
⣻⣷⡞⢹⣿⣿⡽⡆⠀⠀⠀⣰⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣧⣀⣾⢃⣼⠏⢀⣾⣿⣿⣿⠃⠀⠀⡄⠀⠀⠈⠙⢷⡀
⠹⣿⣿⡰⣿⣿⣿⡹⡄⠀⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⢣⣾⡟⢀⣾⣿⣿⣿⠃⠀⠀⣼⣴⡷⢀⠀⢀⠜⠁
⠀⢻⣿⣷⡹⣿⣿⣧⢻⣰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡟⠉⣾⣿⣿⣿⠃⠀⢀⣼⣿⠟⣰⣯⡴⠁
⠀⠀⢿⣿⣧⠹⣿⣿⣧⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⣅⣸⣿⣿⣿⠃⢀⣴⢿⠟⢁⣼⣿⡟⠀⠀⠀⠀⢀
⠀⠀⠈⣿⣿⣦⡜⣿⣿⣾⣿⡄⠀⠀⠀⠀⣀⣤⣶⣶⣿⡿⠿⠿⠿⠿⢿⣿⣶⣶⣿⡇⠀⠉⣿⣿⢃⣾⣟⠁⠈⢀⡞⣽⡟⠀⠀⣀⣰⣾⡏
⠀⠀⠀⠘⣿⣿⣷⠸⣿⣿⣿⣷⡀⣠⣶⣿⢿⡻⠝⠒⠋⠉⠉⠉⠉⠉⠉⠛⠛⠽⣿⣿⣦⣴⡿⢁⣾⣿⢿⣷⣦⡈⢠⣿⠁⠀⣴⠅⣽⣿⠁
⠀⠀⠀⠀⠹⣿⣟⢣⡹⣿⣿⣿⣿⣿⡫⠊⠁⢀⣠⠤⠖⠒⠒⠒⠒⠒⠒⠲⠤⢄⡀⠙⣿⣿⣧⣿⣿⡃⢸⣿⣿⣿⣿⡿⢀⡼⢋⣾⣿⠋
⠀⠀⠀⠀⠀⢻⣿⣠⠿⣮⣿⣿⣿⠋⢀⠴⠊⠁⠀⠀⠀⢠⣆⠀⢀⡀⠀⠀⠀⠀⠉⠂⢸⣿⣿⣿⡉⢉⣾⣿⣿⣿⡿⢡⡞⢠⣿⡿⠃
⠀⠀⠀⠀⠀⠀⢻⣷⣀⠈⣿⣿⡏⠰⠁⠀⠀⠀⠀⢙⣳⠶⢿⣞⣩⠐⣒⠶⠀⠀⠀⠀⣼⣿⣿⣿⡇⣾⣿⣿⣿⡟⠁⠀⠈⠉⣿⡅
⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⣿⣿⡹⡄⠀⠀⠀⢴⣞⣛⢤⣠⢼⣟⡵⠞⣋⡤⠂⠀⣠⣾⣿⣿⣿⣿⢗⢿⣿⣿⡟⣅⠀⠀⢀⢞⠞
⠀⠀⠀⠀⠀⠀⠀⠘⠿⠾⢿⣿⣿⣌⠓⠤⣀⠀⠈⠉⠉⠛⠛⠛⠒⢋⣁⣤⣶⣿⣿⣿⣿⣿⣿⢏⣿⣿⡿⢟⡞⣿⠀⢠⢮⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⡻⣿⣿⣿⣾⣿⣿⣿⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠓⠋⠉⠉⠀⢬⡙⠿⣶⣻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣮⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠫⣉⣻⣿⠟⠀⠀⠀⠀⠀⠀⡄⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣷⣮⣭⣿⣿⣿⣿⣿⣧⣤⣤⣶⣶⣾⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢀⢣⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⠀⠀⠀⠀⠀⡠⣫⣿⣿⢃⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠘⢦⣄⣠⣦⣿⣿⣿⠟⣡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣷⠎⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⠴⠶⠖⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠶⠶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠟⠋⠉⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣤⣀⣀⡀⠀⠀⠀⠀⠉⠛⠿⣿⡷⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣏⠀⠀⠀⠀⠀⠀⠀⢰⣟⠋⠉⠉⠉⠉⠙⠛⠶⢤⡈⠉⠙⣳⡄⠀⠀⠀⠀⠀⠈⠙⢷⡄⠉⠙⠛⠶⠶⣦⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠙⠷⣦⣄⡀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠶⠶⠶⠶⠶⠾⠷⠛⠋⠀⠀⠀⠀⠀⠀⠀⣠⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠉⠛⠳⠶⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⡴⠶⠛⠉⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⢸⡿⢦
⠀⠀⠀⠀⢀⣀⣤⣤⠶⠶⠞⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⣄⡀⠀⠀⠀⠀⠀⣼⠅⣠
⠀⣠⡴⠞⠋⠉⠀⠀⠀⠀⠀⣀⣤⡶⠾⠷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠉⠛⣛⣷⣶⣶⠿⠞⠋
⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⠁⠒⠦⢬⡙⠳⢦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡤⠖⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉
⣿⠻⣦⣀⠀⠀⠀⠀⠀⠀⠙⠛⠶⠶⢦⣤⣤⣬⣶⣤⠴⠿⠛⠛⠛⠛⠳⠶⠶⠶⠶⢶⣶⣶⠞⠛⠛⠋⠉
⣿⠀⠀⠙⠛⠶⣦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡴⠶⠛⠁⢸
⣿⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠞⠛⠋⠉⠁⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠞⠋
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠛⠳⠶⠶⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠚⠋⠉
⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠘⠷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠈⠛⠶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⠶⠶⠞⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⣲⡊⠙⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢠⠁⠙⡀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢀⠃⠀⠀⢡⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈⠀⠀⠀⠈⡄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢢⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⡰⠁⢣⠸
⠀⠀⠀⠀⢰⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⠀⠀⠎⠠⠁⠀⠀⡆⡇
⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⡿⠷⠶⢶⣶⢶⣶⣶⣶⣶⣴⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣯⣤⢿⣿⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⣸⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⡿⣿⣿⣷⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣶⣴⣿⣿⣿⣿⣬⣭⣭⣭⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⢠⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿
⠀⢸⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⡆
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⣿⣇
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿
⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠉⠙⠛⠻⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣽⣿⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠈⠉⠙⠛⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⣙⣛⣻⣿⣟⣛⣉⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠉⠁⠀⠀⠀⠉⠙⠢⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣳⡀
⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁⠹⡀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⣀⣀⡤⠴⠒⠋⠁⠀⠀⣀⡤⢾⡇
⠀⠀⠀⠀⠀⢸⠒⠚⠉⠉⠀⠀⠀⢀⣠⡤⠖⠋⠁⠀⢸⠁
⠀⠀⠀⠀⠀⠈⢦⣠⣤⠤⠔⠒⠋⠉⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠑⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠤⠤⠴⢶⠛⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⢝⢢⣀⡠⠴⢲⠶⠦⣄⡀⢧⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠠⡅⠉⠀⣰⠋⠀⣰⠃⠙⣎⣇⠹⣄⣀⣠⣄⣀⡀⠀⠀⣠⠴⠶⡄
⠀⠀⠀⠀⠀⠀⠀⢀⠞⠁⣰⠒⠀⠀⢀⡏⠀⢰⠇⠀⠀⠘⢻⡶⢫⠇⠀⣾⠀⠉⢲⠞⠔⠃⢀⡇
⠀⠀⠀⠀⠀⠀⠀⢼⣘⣿⣠⠴⢦⠀⢸⠀⠀⣾⡋⠉⠭⣛⠿⠀⢸⡀⠀⢹⠀⠀⠀⠐⣃⠀⠸⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣼⡄⠀⡏⢳⡄⠀⠒⠀⠀⠀⣇⠀⠸⡆⠀⠀⢀⠀⠩⢤⡘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⡟⠀⡪⠀⣄⠀⠸⣄⠀⠹⡄⣠⠏⠙⠲⠯⠥⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠾⣢⠞⠉⠙⠲⠾⠶⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢋⣍⠙⣿⣿⣿⠋⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢛⣠⣿⡟⠡⠿⠸⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣯⣀⣉⣉⣿⣷⣶⣶⣰⣾⡿
⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣻⣿⣿⣟⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠉⠉⠉⠉⠉⠉⠉⠁⠀⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠀⠈⠁⠈⠉⠉⠉⠉⠉⠉⠉⠈⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠉⠉⢹⣿⠉⠉⠉⠉⠉⠉⠉⠙⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⠉⠉⠉⠉⠉⠉⠉⠙⣿⡏⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⠛⠋⠙⠛⠉⠉⣿⡟⠛⠛⠉⠛⠛⠋⠉⢻⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣤⣿⣧⣤⣤⣤⣤⣤⣤⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⢀⡴⣷⠒⠚⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⢦⣤⣤⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⢀⡾⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿
⠀⢀⣾⠃⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢀⡾⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠁⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⣿⡁⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣇⠀⣿⣅⣴⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠦⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼
⣿⠀⠙⠋⠁⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣄⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠸⣿⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢹⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⠀⠀⠀⢰⡿⠤⠤⠤⢤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢸
⢸⡆⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢙⡿
⢸⡇⢀⡞⠀⠀⣴⣿⡿⠿⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⡀⠀⠀⣠⡾⠁
⢸⣧⠞⠀⠀⠀⠉⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣿⠁⢀⣴⠋
⠀⠻⠿⠶⠶⠶⠶⠾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡷⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣴⣿⢛⣉⣉⣡⣤⡍⢻⣦⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⠶⠞⠛⠛⠋⠉⣽⠟⣁⡼⠉⠉⠉⠙⠧⣿⢹⣯⠉⠉⠙⠛⠛⠷⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠒⠒⠒⠒⠒⠒⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦
⠀⠀⠀⠀⠀⠀⠀⢸⡟⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠞⣿⡇
⠀⠀⠀⠀⠀⠀⢠⣿⢻⡀⠉⠛⠻⠶⢦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⠴⠶⠾⠛⠋⠁⢀⡠⣿⣇
⠀⠀⠀⠀⠀⠀⢸⣿⣦⣍⣓⠲⠤⣄⣀⣀⡀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⢀⣀⣀⣀⣀⡠⠀⢀⣀⣭⡴⠿⣿⣄
⠀⠀⠀⠀⠀⠀⢸⣏⠀⠛⠻⠿⠶⠒⠶⣯⣭⠙⠛⠛⠛⠛⠛⠛⠛⠛⠋⠩⠉⠉⠿⠆⠀⠐⠛⠙⢿⡉⠀⢀⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠙⣿⡶⠶⠤⠶⠶⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢻⣦⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⠀⢰⡀⠀⢸⡇⠀⠀⠀⣷⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⡀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⢸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠆⠀⠀⡏⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠸⡇⠀⠸⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⢸⠀⠀⠀⡇⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⣇⠀⠀⡇⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣼⠀⠀⢠⡇⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⣿⠀⠀⣷⠀⠀⠀⣿⠀⠀⠀⠀⡇⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⠀⣿⠀⠀⠀⠀⣧⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⠀⣿⠀⠀⠀⢸⡄⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⡇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢸⡄⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⢸⠇⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⣿⠀⠀⠀⢸⡇⠀⠀⢰⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⢿⡆⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⣿⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⠘⣷⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⢸⡇⠀⢰⣿⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⣼⡇⠀⢸⡏⠀⢀⣿⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⠀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⢸⡏⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⡀⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⢸⡇⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣿⡇⢸⡇⠀⠀⢸⡇⠀⠀⢸⣿⠀⠀⣿⠇⠀⢸⡇⠀⢸⣧⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⢿⣷⣤⣄⣸⣇⣀⣀⣸⣿⣀⣀⣿⣤⣤⣼⣷⡾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣾⣿⡿⣫⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣿⣿⣿⠕⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⡿⠿⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡴⢀⣴⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣶⡿⣋⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢠⣟⡿⣿⣿⣿⠷⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣬⣶⣶⣾⣿⣿⡾⠿⠯⠁
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⢟⣽⣾⡿
⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⢳
⠀⠀⠀⠀⢸⣿⡿⣻⣿⣿⣿⣿⣿⢟⡟⡎
⠀⠀⠀⠀⠏⢊⣾⣿⣿⣿⣿⣿⣿⠞⠘
⠀⠀⠀⠀⢠⡿⣻⣿⢟⣽⣿⠋
⠀⠀⠀⠀⣭⣞⠋⠐⠝⠁⠁
⠀⠀⠀⠐⠋
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⣉⣉⣉⣉⣉⡉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠤⠤⠤⠧⠼⠥⠤⠤⠤⠤⠿⠤⠧⠤⢤⣤⣤⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠴⠒⣉⠤⠒⠀⣀⡤⠂⠀⢀⡴⠂⠀⠀⠀⠀⠢⡀⠀⠐⠦⣀⠀⠐⠢⢉⡑⠂⠄⡀
⠀⠀⠀⠀⠀⣀⣤⠖⠋⠡⠔⠂⠉⠀⠀⠴⠊⠁⠀⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠙⠢⡄⠀⠀⠁⠒⠈⠉⠒⢦⣄⡀
⠀⠀⠀⠀⣶⣋⣉⡉⠉⠉⠙⠛⠛⠓⠒⠒⠒⠒⠒⠒⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠖⠒⠒⠒⠒⠒⠒⠛⠛⠛⠉⠉⢉⣉⣈⣳⠆
⠀⠀⠀⠀⠀⢀⣯⣈⣉⣉⣉⣉⣉⠉⠉⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠈⠉⣉⣉⣉⣉⣉⣁⣨⡇
⠀⠀⠀⠀⠀⢸⣀⣀⣀⣀⡀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⢀⣀⣀⣀⣀⡼
⠀⠀⠀⠀⠀⠀⣍⠀⣠⠭⠭⣉⠉⠉⠉⠉⠉⡉⠉⠉⠉⠙⠒⠒⠒⠒⠒⠒⠒⠛⠉⠉⠉⠉⡉⠉⠉⠉⠉⢉⡩⠭⢤⡀⢨⠁
⢀⡤⢤⣀⡀⠀⢸⠀⡇⠀⠀⠈⠆⠀⠀⢰⠉⠈⠉⢳⠀⠀⠀⢰⠊⠉⠙⢦⠀⠀⠀⢰⠉⠁⠈⡇⠀⠀⠀⠏⠀⠀⢨⡇⢸⠀⠀⣀⡠⠤⣄
⡇⢀⡤⢤⡈⠓⣾⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡇⢸⠖⠋⣠⠤⣄
⢧⠈⣇⠀⠈⠳⣿⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⠸⠇⢸⡶⠋⠀⢠⠏⢠
⠈⢦⠈⠣⡀⠀⠸⠀⣿⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠸⠀⠀⡠⠋⢠⠎
⠀⠈⠳⣄⠈⠢⣴⠀⣟⠀⠀⠀⡇⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⢰⡤⠊⢀⡴⠃
⠀⠀⠀⠈⠑⠦⣀⡳⢡⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⢰⣃⡠⠖⠁
⠀⠀⠀⠀⠀⠀⢸⠁⢸⠀⠀⠀⠇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠘⠀⠀⠀⠻⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⠇⠀⠀⣼⠀⢙
⠀⠀⠀⠀⠀⠀⠘⡀⢸⠀⠀⠠⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢰⠀⠀⠀⣰⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢹⠀⡎
⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⡘⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⢸⡄⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⣿⠀⠀⠀⢸⠀⠀⢰⠁⠀⠀⠀⡇⠀⠀⡇⠀⣇
⠀⠀⠀⠀⠀⠀⢠⡇⢸⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡿⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡇⠀⡏
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡆⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⠀⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⠇⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⢸⡄⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠇⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⠀⠀⠀⢡⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡆⠀⠀⡇⠀⡇
⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⠀⢠⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢽⠀⢇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⠇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⡇⠀⢸⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⡾⢤⣈⠓⠚⠁⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⢸⡇⠀⠀⡇⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠑⠒⢃⣠⠾⡆
⠀⠀⠀⠀⠀⠀⠀⢣⢄⡈⠙⠒⠤⣀⡀⠀⠈⠒⠒⠊⠀⠀⠀⠸⣇⠀⢀⡇⠀⠀⠀⠘⠒⠒⠋⠀⢀⣀⠤⠔⠊⠁⣀⢴⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣄⡈⠑⠲⢤⣄⣈⠉⠓⠒⠶⠦⠤⢤⣤⣤⣬⣭⣥⣤⣤⣤⠤⠤⠶⠒⠒⠉⢉⣀⡤⠶⠚⠉⣀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠉⠉⠓⠒⠢⠤⠤⠤⢄⣀⣀⣀⣀⣀⡤⠤⠤⠤⠔⠒⠚⠉⠉⣀⣠⠤⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠒⠤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⠤⠤⠐⠒⠊⠉⠁
⣿⣿⡿⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⢀⡤⣲⣺⡯⢯⣛⣑⡛⣲⢦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⢀⡤⠤⠤⢤⣉⠛⠷⣉⠀⠀⠀⠀⠈⠉⠉⠓⢮⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⠞⠁⠀⠀⠀⢰⠌⠽⢦⡈⠳⣄⠀⠀⠀⠀⠀⠀⠈⢳⡸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⡎⠀⠀⠀⠀⠀⣿⠰⡇⠀⠱⣄⠈⣢⠀⠀⠀⠀⠀⠀⠈⣅⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢈⡤⠄⠒⠺⣀⠀⠉⢻⡆⠀⠀⠘⡄⠐⡂⠀⠀⠀⠀⠀⠀⢹⡌⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠋⠉⡀⠀⡇⡀⣀⠀⢃⠀⠀⠀⣅⠀⠀⠀⡾⠀⢹⠀⠀⡀⢀⡀⠀⠀⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠁⠀⠀⡅⠀⡆⢹⣏⡴⠘⠄⣠⠔⠻⠷⣤⠆⠀⡇⠀⡆⠘⠉⠁⠈⠭⢒⡓⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠣⡀⢀⠀⠇⠀⡇⠘⣿⣿⠆⣸⠟⠳⣾⣐⢰⡠⠤⡅⠀⣿⡇⠀⠀⠀⠀⠀⣸⢹⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠉⠻⢤⣠⠠⡇⠰⠁⢿⣷⡯⠤⡶⣿⣦⣹⠀⢀⡇⠀⡧⣱⠀⠀⠀⠀⢰⢭⡾⠚⠊⠋⢕⡢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠹⠐⣷⡆⢠⡩⢸⡗⠚⠋⠁⠈⠉⠛⠲⠇⢸⡏⢹⣹⠉⠋⠍⣛⠛⠳⣤⡀⠀⣾⠻⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⠑⠀⠀⣟⠈⠚⠧⣼⡁⠶⠆⠀⠀⠀⠰⢶⠀⢘⠉⢿⠇⠈⠀⠀⠈⢑⡦⠤⢽⣶⠃⣰⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣒⣸⠉⠢⠤⣀⠉⠒⠰⢶⣎⣙⡀⡋⠀⢞⣃⢀⣳⣄⠄⣀⣤⣦⣸⠊⠥⠛⠋⠉⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⢉⡓⠓⠮⣒⣲⠑⡇⡈⣻⠛⠉⠉⠉⠀⠁⠀⢀⡀⠀⢠⣐⠶⠆⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢔⡠⢄⡀⠩⠱⣦⡼⠀⠒⠈⠉⠉⠉⠉⠀⠴⠊⠁⢀⠄⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢄⣀⡀⠀⠀⠀⢂⠄⢊⠔⢁⠄⡠⠀⡀⡠⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡡⠊⡡⠊⡠⠊⡠⠪⠊⡀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠄⢊⠔⠉⡠⢊⠔⡡⠊⢀⠄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠠⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⢁⠔⢉⠔⡡⠊⢀⠔⡡⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠅⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠔⢁⠔⡡⠊⡀⠔⢁⠔⢀⠆⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠁⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠅⢀⠀⠀⠀⡠⠖⡡⠊⡠⠚⠁⡤⢃⠔⡡⠂⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡨⠄⠂⣉⠤⠂⠀⠈⠀⢊⡠⠊⢀⡠⢊⠔⢁⠀⢀⠊⠀⠘⣖
⠀⠀⠀⠀⠀⠀⢀⡸⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠐⠀⣉⠤⠒⣁⠀⢀⠔⠂⡠⠒⣡⠖⢁⠔⢁⠔⢁⠈⠔⠀⣖⡈⠨⠅⢒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠓⠤⣀⠠⠄⠀⠀⠀⠀⠀⢀⠀⠤⠄⢒⣊⡡⠄⢒⡉⠀⠀⡠⠔⢉⠤⠊⡠⠀⡁⠄⣁⠖⡡⢊⠐⠅⢸⣒⠀⠉⠠⡐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠦⣄⡀⠀⠀⠠⠄⠒⠂⠉⣁⠠⠔⢊⡡⠔⠀⢈⠠⠂⡁⠔⢁⠤⠊⡠⠀⡡⠊⡴⠃⠀⢂⣈⣇⡂⠩⠄⠐⢂⡈⠉⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠶⢤⣀⡂⠈⠁⠤⠒⣉⠥⢒⠀⠂⠁⠔⠈⡠⠚⠁⠔⠁⣀⣉⡀⠤⠒⠚⣉⠉⠅⠀⢀⡀⠨⠀⢂⣀⠠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠦⠤⢬⡥⢬⣤⣠⡴⣀⣤⠵⢒⠪⠙⢉⠡⠄⠒⣈⠁⠄⢐⠀⠁⠀⠂⠈⠥⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⠶⠶⠿⠿⠿⠿⠿⠿⠿⠿⠷⠶⢶⣤⣤⣀⡀
⠀⠀⢀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠷⣦⣄
⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣶⣦⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠉⠁⠀⠀⠀⠀⠀⠉⠙⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣄⣀⣀⣀⣀⣀⣤⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇
⣿⡟⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⣿⡇
⣿⡇⠀⠉⠻⠷⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣶⣟⣋⣁⣀⣿⣇⣤⣤⣤⣤⣤⣀⣀⣀⣀
⣿⡇⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠻⠿⠷⠾⠿⠿⠿⠛⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠿⢶
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠈⠙⠷⣦⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠷⠶⣶⣶⣶⣶⣶⣶⣶⣶⡶⠶⠶⠾⠿⠛⠛⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠻⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⠶⣶⠶⠶⣶⣶⠶⣶⣶⡶⠶⣶⣶⠶⠶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡛⢿⣷⢸⡟⣴⣿⡎⣿⠸⠿⢸⡇⢛⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣯⣭⣾⣿⣼⣿⣮⣭⣵⣿⣼⣷⣼⣧⣉⣩⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢹⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⡏⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⡆⣴⣶⣾⢡⣶⣶⡄⣶⣶⣦⢰⣶⣶⡆⣴⣶⣾⢡⣶⣶⡄⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⢱⣿⣿⡟⣼⣿⣿⠇⣿⣿⣿⢸⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⢸⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⠸⣿⣿⣿⢸⣿⣿⡿⢸⣿⣿⡇⣿⣿⣿⡇⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣫⣴⣮⣛⣫⣶⣭⣛⣭⣶⣝⣛⣥⣦⣝⣋⣵⣮⣝⣫⣵⣮⣛⣫⣶⣭⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⣩⣭⣭⣭⣭⣭⣭⣭⣭⣭⢋⡝⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⡵⣫⣾⣿⢸⣿⣿⠿⠿⠿⠿⠿⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢟⡵⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⢸⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⢸⢿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣼⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⣿⣿⣿⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠺⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠗

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤⠦⠄⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠈⡟⠋⠀⠐⠀⠀⠀⠒⠀⠀⢀⠒⠀⠀⠚⠂⠀⠆⠀⠂⠘⠀⠈⠑⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⣀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠄⠀⡠⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠰⠁⣺⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢀⠀⠘⣤⠄
⠀⠠⠆⠀⠀⠀⠀⠀⠀⠀⢰⢾⠃⠀⠀⠀⠀⠀⠀⢇⠀⣀⣀⢀⠀⠀⠁⠲⠤⣄⣀⣀⣀⣀⣠⠤⠒⠛⠁⠀⠀⠉
⠀⠀⠀⠀⠀⠀⡄⠀⡀⡄⢸⠈⢠⠀⠀⢠⠀⠀⢀⢠⢤⡂⠀⠀⠀⠀⠀⠀⠀⡆⠀⠠⠀⠀⠠⠄⠀⢀⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠃⠲⠷⢷⠂⠀⢸⢀⠀⡀⠀⠠⠥⠜⡇⢸⠀⡃⠀⠀⠐⠂⠀⠀⣤⢶⣄⢀⡀⠀⠀⠀⠀⢸⠀⠀⢠
⠀⠀⠀⠀⠀⠀⢠⠁⠀⠈⣧⠀⢸⠀⠀⠃⠀⠀⠛⢟⠿⢍⠀⠐⠀⠀⢀⠀⠀⠀⠿⠤⡿⠈⡇⠀⠀⢀⡿⠜⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⣿⠀⠺⠄⠀⢠⡆⡔⠀⠸⠀⢰⠀⡇⠀⣸⠸⠀⠀⢰⠒⠀⠀⠈⠀⠀⠀⢸⡇⠀⠀⠀⢸
⠀⠀⠈⠀⠀⠀⠀⠣⠀⢰⠏⡸⠇⠀⠀⠀⠁⡇⠀⠀⠀⣰⠀⠁⠀⠧⠺⡄⢀⡀⠧⠄⠰⠶⠀⠀⠀⠀⡇⠀⡇⠀⠘
⠀⠀⠀⠀⣀⠀⠀⠀⢐⣋⣀⠇⠀⢀⠀⠀⠀⠇⠄⡇⠀⡇⠀⢀⡴⠀⠀⠀⠈⠃⢀⠀⠐⡌⠁⠀⠀⠀⡇⠀⠁⠀⢰
⠀⠈⠉⠋⠁⠀⠀⠀⠘⡎⡘⢰⠀⠈⠃⠀⣴⠀⠀⠀⠖⠂⠐⠂⡇⠈⡲⠀⠀⠀⠈⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⠁⠀⠲⠶⠶⠀⡇⠀⠀⠀⢰⠀⠀⠀⠁⠀⡄⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠤
⠈⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⢱⡶⢶⡶⠀⡄⠀⠀⠀⢸⡄⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⠀⢨⠀⡄⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⣰⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠹⢦⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠶⠶⠶⣿⠀⠃⣰⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡠⠖⠋⣁⡤⠴⠒⡦⠀⢙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠁⠀⠀⠰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠠⠒⢋⣡⡴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠻⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⢲⡶⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢰⡄⠀⠀⠀⠀⠙⠳⣄⡀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠈⠛⠶⣄⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠈⠙⠳⠦⣴⡇
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠚⠛⠛⠃⠀⠀⠀⠀⠀⠉⠳⢦⣄⡀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⣰⠟⠁⢀⠴⠚⠉⢉⡤⠆⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⣶⡶⠋
⠀⠀⠀⠀⠀⠸⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢠⣤⣀⣀⣠⣤⠴⠞⠉
⠀⠀⠀⠀⠀⠀⠈⠻⢦⣤⣀⣀⡀⠀⠀⢠⠇⢠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣦⠀⠀⠀⡏⢀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠐⣇⡼⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠈⢀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠞⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠠⠤⠤⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠀⠒⠈⠉⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠉⠁⠒⠀⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⣀⠤⠀⠐⠂⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠐⠀⠀⠄⣀⠀⠀⠡
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠪⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠕⠶⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⠀⠀⠈⠁⠐⠒⠀⠀⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠐⠒⠂⠈⠁⠀⠀⣀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠉⢓⠂⠀⠀⡤⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢤⠄⠀⠐⠚⠉⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠘⠀⠀⠀⢠⠀⠀⠀⠃⠀⠈⠈⡏⠉⠉⠉⠉⠁⠀⢸⠀⠀⠀⡀⠀⠀⠀⡃⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⡀⣀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⣈⠑⠂⠀⠠⠄⢀⣀⣨⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⣄⣀⡀⠠⠄⠀⠐⠈⣁⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢉⠁⠂⠀⠄⠠⠤⢀⣈⣉⣉⣉⣉⣉⣉⣉⣉⠁⣀⠠⠄⠠⠄⠒⠀⠉⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠸⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⠀⠀⠀⠃⠀⠀⢠⠀⠀⠀⡇⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢸⠀⠀⠀⠀⠀⠀⠘⠀⠀⢠⠀⣀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣉⠑⠂⠠⠸⠄⣀⣀⡀⠀⠀⠀⠀⠀⢸⠀⠀⢀⣀⣀⠠⠤⠀⠒⠊⣉⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠙⠀⠒⠠⠄⠤⠄⣀⢉⣉⣉⣉⣉⣉⣉⡉⢀⠠⠄⠠⠄⠐⠀⠏⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⠠⢄⡀⠀⠁⠀⠀⠇⠀⠀⠠⠀⠀⠀⠀⠀⢸⠀⠀⠸⠀⢀⡠⠄⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠁⠙⠀⠀⠂⠠⠤⠼⠤⠄⠤⠤⠄⠼⠀⠀⠛⠈⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠐⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠒⠒⠒⠒⠒⠀⠉


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢤⣄⡄⣠⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣮⠟⠉⠁⠈⠉⢪⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⡟⡤⢤⡀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⠤⠤⠤⠤⠤⢤⣤⢞⣽⣶⣃⣉⡑⢿⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⠀⠀⢰⣯⠏⠀⠀⠉⠉⠉⠉⢻⣿⠒⠒⠛⠲⠶⠤⠤⢤⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢿⣷⡷⡶⠾⠶⢶⣶⣶⣿⣯⣾⣿⣧⣼⣧⣤⣦⣴⣿⣄⣰⣤⡦⣤⠀⣄⡀⣴⠶⢟⢟⡻⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢻⣋⡿⠃⣈⠀⠈⠫⠟⣿⢁⠀⠀⠀⣠⡤⣄⠉⠉⢹⣏⣿⣟⠛⠛⠛⣿⠿⣿⣧⠀⡫⢊⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣏⣻⠀⠀⠸⡭⠀⠀⠈⠸⡐⢔⡹⠀⠀⣜⡿⠩⠃⠐⠀⠘⢿⣢⠞⣿⠀⢐⠵⣺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⡍⣷⠀⠂⢠⣴⠮⠏⢶⣄⠂⣠⡄⢄⠈⠁⠄⠁⠀⠈⠀⠀⢠⡀⢀⡀⠀⠀⢸⣿⡇⠀⡪⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣸⣌⣱⣟⠀⢀⣿⠁⣶⢥⣤⠙⣷⠝⠶⢠⠃⠀⠀⡤⠤⠄⠀⢰⢡⢂⢲⢰⠈⠀⣠⡿⣿⠀⢒⠕⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡍⠉⠉⠉⠉⢉⣽⡿⠿⢿⣿⡛⠛⠛⠛⠛⠛⠓⠒⠶⠾⠾⠾⣧⡄⠐⡈⠠⢵⠏⠀⠈⠹⡯⢼⡇⠀⡪⠚⣧
⠀⠀⠀⠀⠀⠀⢀⣤⣴⣚⣫⣀⣀⢀⠰⣠⣿⠃⠰⠆⡀⢸⣷⡀⠰⠆⠀⠀⠀⠀⢀⣠⡴⢚⠙⡇⠀⠈⠉⠁⡀⠀⣠⣠⡿⠘⣿⠀⠐⡣⢻⡆
⠀⠀⠀⠀⠀⠀⢸⠏⠉⠉⠙⠛⠛⠛⠛⢿⡷⠶⠶⠶⠷⢿⣼⣷⣤⣼⣴⣤⣤⣤⣼⣆⠀⠑⡀⣻⠀⡀⠀⠀⠀⠀⡻⣿⣿⠄⢹⡇⠨⢆⠜⣷
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠾⠃⠀⠀⠀⠀⠀⢼⠇⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⡈⢘⡄⣴⡮⠰⡄⠠⠈⠈⣯⢁⠀⣿⠀⠡⠊⢻⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠈⣔⠅⣿⡹⠳⢗⡟⠀⠀⠀⣿⢄⡀⢹⡆⠈⡀⠍⣿⡀
⠀⠀⠀⠀⠀⢀⣿⠖⠶⣤⡤⠤⣤⣀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠄⡀⢿⠈⠉⠉⡄⠀⠀⠀⣿⠔⠠⠈⣇⠀⢤⠴⠘⣇
⠀⠀⠀⠀⠀⢸⡇⡸⢎⠀⠙⣤⠊⠉⠢⣠⠋⠙⢍⡩⠟⠛⡍⢉⠝⠛⡍⢉⠝⢛⡛⣻⣧⠀⠊⢀⠹⣀⡄⡁⠀⢠⠤⢤⣹⡈⠀⠀⢻⠀⠠⠊⠄⢹⡆
⠀⠀⠀⠀⠀⢸⣏⠀⠀⡱⣞⠀⠑⣤⠊⠉⠢⣠⠞⠑⣄⢀⡜⠣⡀⢀⠜⠣⡀⣠⠟⢽⣿⡄⠐⡁⢨⡿⠻⠃⠀⢹⡶⣶⢿⡇⠀⠀⣼⠷⠶⢷⣿⣿⠟
⠀⠀⠀⠀⠀⣸⡏⢣⡜⠀⠈⣢⡞⠀⠳⣠⠖⠉⢦⢀⡼⠣⡀⢀⡜⠣⡀⡠⠚⢦⠀⣸⣿⣧⠈⡪⠀⡧⠀⠀⠀⠀⡈⠡⠘⡇⠀⣸⠟⣨⣶⠿⠋
⠀⠀⠀⠀⠀⣿⡇⠁⠈⣦⠚⠁⠙⣤⠞⠉⠢⣀⡴⠛⢄⢀⠜⢯⠀⡠⠛⢤⠀⡠⠛⢼⣿⣿⡀⠰⠁⢿⣞⣕⡄⠀⠀⣴⠲⣗⣸⣿⡿⠛⠁
⠀⠀⠀⠀⠀⣿⠈⣢⡞⠁⠑⣤⠞⠁⠱⣄⠔⠉⢦⣀⠔⠳⡄⣠⠛⢄⠀⡰⠛⢄⢀⢼⣿⢾⣇⣀⡦⢸⠿⠿⠤⠥⠼⠿⢷⣿⠟⠉
⠀⠀⠀⠀⠀⡿⢎⠀⠈⢢⡔⠉⠣⣀⠔⠉⢢⣀⠔⠓⢄⡠⠞⢣⡀⡠⠻⡀⠀⡴⠻⣼⣯⣿⡏⡛⢻⣿⠄
⠀⠀⠀⠀⠘⠷⠤⣵⣶⣥⣌⣤⣔⣉⣢⣀⡔⠙⢄⡠⠊⠳⡀⡠⠛⢄⠀⡨⠫⡀⢀⢼⣿⡿⣈⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠛⠛⠚⠷⠶⠤⣼⣧⣀⣀⣿⡀⠀⡹⢇⢸⣿⣹⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠓⠶⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠻⣿⡇⠀⠀⠀⠀⣾⣟⣼⠄⠀⠀⠀⠀⢠⣴⣦⡄⠀⠀⠀⠀⣠⡐⡦⣀⠀⠀⠀⠀⠀⠀⠂⠐⢦
⠀⠀⠀⠀⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⢀⣾⣿⣿⣿⡄⠀⠀⠘⣷⢣⢠⡍⠀⠀⠀⠀⡀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠁⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⠒⠛⠁

⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⡰⣟⡲⠀⠀⠀⠀⠀⢠⣶⣶⣄⠀⠀⠀⠀⠀⣀⢸⡶⠀⠀⠀⠀⠀⢠⣾⣷⣄
⠀⠀⠀⠀⠀⢠⣿⣿⡗⠀⠀⠀⠀⠀⠿⠮⠙⠁⠀⠀⠀⢀⣿⣿⣿⣿⡆⠀⠀⠀⠰⣬⡳⠁⠀⠀⠀⠀⠀⠘⢿⣿⡿⠃
⠀⠀⠀⠀⠀⠾⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⡿⠃⠀⠀⠠⣿⠊⠁

⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠈⠉⢙⡇⠀⠀⠀⠀⠀⣾⣿⣿⣷⠀⠀⠀⠀⠀⢠⣴⣿⠃⠀⠀⠀⠀⢠⣿⣷⡆⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠻⢿⡿⠏⠀⠀⠀⠀⣸⣷⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠾⣟⡀⠀⠀⠀⠀⣴⢿⣿⣷⡄⠀⠀⠀⠀⣀⣾⣿⣦⠀⠀⠀⠀⢾⣿⣿⠇⠀⠀⠀⠀⣶⣤⣴⣦⣶
⠀⠀⠀⠀⠀⠠⣤⣤⠀⠀⠀⠀⠘⠁⣾⣿⣿⡿⠀⠀⠀⠀⠉⠛⣛⣽⠃⠀⠀⠀⠀⢹⠃⠀⠀⠀⠀⠀⠿⠿⠿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⠀⢠⡴⠀⠀⠀⠀⠀⢀⠄⢼⠷⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠆
⠀⠀⠀⠀⠀⢠⣾⣿⡿⠀⠀⠀⠀⢀⠮⠁⠀⠀⠀⠀⠀⠐⡃⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⢀⣯⡀
⠀⠀⠀⠀⠠⣾⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠇⠂⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠃⠀⠀⠀⢰⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⣿⣿⢛⢿⠛⡟⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢋⡟⢻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠁⠈⠙⣿⠀⠀⠀⠀⣀⣀⣀⣀⣸⣼⣤⣤⣯⣧⠤⢤⣤⡀
⠀⠀⠀⣤⣤⠤⠤⢤⡿⢧⣤⣤⣤⣿⣤⣤⣤⣼⣿⣿⣛⣷⣤⣿⠤⠶⠶⠗⠚⢋⡉⢹⣀⣴⠚⠳⡄
⠀⠀⢠⡿⠿⠤⠤⢤⣧⣤⣤⣤⣄⣼⣀⣠⣿⣽⣿⠋⠁⠈⢻⡇⠀⠀⢀⣤⣤⣼⠉⡿⠦⢿⣇⠀⣿
⠀⠀⢸⡇⠀⠀⠀⠀⠁⠀⠀⠀⢠⣿⠒⠲⠶⠾⡿⠖⠒⠒⠺⡗⠒⠒⠚⣻⢷⠾⣄⡇⠀⠀⣿⠉⠛⠦⣄⡀
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣼⣻⣇⣀⣀⣀⣇⣀⣀⣀⣠⣇⣀⣀⣀⣿⣿⡄⠈⠛⠲⢤⡟⠀⠀⢠⡤⠽⢷⡆
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠉⠉⠋⠉⠉⠁⠘⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠋⠓⢦⡟⡇⠀⢸⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⡇
⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⢸⡇
⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠓⢦⣀⠀⠀⠀⠀⡇⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⠳⢤⡀⠀⣇⣧⣀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡇⠀⠀⠀⠀⠀⠉⠳⠟⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠒⠛⠁⠽⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠀⠀⠀⠀⠀⠀⠐⠂⠀⠉⢛⣒⣒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣾⣿⣿⣥⣤⣴⣶⣾⣿⣿⣿⣿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⣿⣿⣿⣿⡿⠛⠛⢋⣤⣿⡟⢉⠀⠀⠙⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⡿⣿⡿⠛⡻⠟⠉⠀⠀⠁⣹⣯⣿⡿⠿⢦⣤⠀⣠⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡶⠾⠷⠈⠀⠀⠀⠀⠠⢞⡋⠉⠁⠀⠀⠀⠀⠛⢛⣋⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⣴⡶⠚⢻⣆⠈⠀⠀⠀⢀⡼⢤⣒⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⠿⣿⡶⠖⠶⠖⠒⠒⢿⣦⠀⢀⣾⡿⠛⠬⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣽⣟⣻⢻⣿⣿⣿⡷⠀⠀⠀⢀⠀⠈⣿⣷⣼⣇⠀⣤⠀⣤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⣿⣿⣿⣿⢿⣷⠛⢷⡄⠻⣧⠘⠹⣷⠋⣿⡆⣽⡟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⣽⣿⣿⣿⣿⡟⣿⡀⢸⡇⠀⣿⠀⠀⣿⠀⣿⠃⣽⡇⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣹⣿⣿⣿⣿⣿⣿⡇⢸⡇⠀⣿⠀⠀⣿⠀⣿⠂⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⢸⡇⠀⣿⠀⢀⡏⠀⣿⢠⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢻⣿⢸⡇⠀⣿⠀⢸⡇⢰⣿⢸⣯⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⢈⣿⠀⣿⡀⢸⡇⢸⣿⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠞⢷⣼⣿⣿⣿⣿⣿⣯⣿⡍⣿⠀⢸⡇⢸⣧⢸⣿⣼⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⢋⣭⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⠀⢸⡇⢀⣿⢸⣯⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠔⢫⡿⣻⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⢿⡆⠈⡇⠈⣿⢸⡗⢛⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠊⣴⣣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⡇⠀⠁⠀⠟⢸⡧⢐⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣻⣿⣿⣿⣿⣿⣿⣛⣻⡿⠿⠸⠃⢠⡄⣀⣙⣨⡥⢴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⣽⣿⣿⣿⡟⣿⣿⣿⣿⠿⠀⠀⠀⢰⣧⠈⢹⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⢻⣟⣿⣿⣿⣿⣿⣆⡀⠀⠀⣸⣿⣤⣼⡷⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⠿⠿⠿⠿⠿⠟⠛⠋⠉

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⠟⠻⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣴⣤⣶⣶⣄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣤⣤⠤⠤⠶⢿⣛⣻⢟⢦
⠀⠀⠀⠀⠀⠀⣠⡤⠶⠤⣤⡀⠀⢀⣴⣻⣟⡉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡻⣿⣏⣧
⠀⠀⠀⠀⢠⠞⢁⣴⣶⣶⣦⡙⢧⣼⣿⣿⣿⡎⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣿⣿
⠀⠀⠀⠀⣿⠀⣿⡏⣠⠖⠛⢿⣾⣿⣿⣟⣝⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠃
⠀⠀⠀⠀⢻⣠⣿⣿⣿⣦⣤⡀⠉⠉⠉⠉⣿⡇⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇
⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⣿⣷⣤⣤⣤⣤⣿⣷⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⣿⣿⡞⣿⡟⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠃⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣤⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⣀⣀⣀⣠⣤⣤⠤⠤⠴⠶⠶⠒⠚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⣠⡴⠶⣦⡀
⢸⡏⠀⠀⠈⢳⡄
⠈⠳⣄⠀⢄⠀⠙⣦⡀
⠀⠀⠈⠳⣄⠑⠄⠈⢳⣄
⠀⠀⠀⠀⠙⢷⡄⠀⢦⡙⢷⡀
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠙⢄⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⡈⠳⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡈⠂⠙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⠢⡈⠻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⢦⡙⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⢦⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡙⢤⡙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠉⠈⣿⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢾⣁⡴⠛⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣴⠚⢉⣹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣯⢴⡿⣿⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⢹⡜⢆⠙⠦⣍⠉⠛⠛⠶⢦⣤⣤⣀⣀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⣇⠘⣦⣄⠈⠑⠦⣤⠤⣤⣤⣀⣈⠉⠉⠹⢦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠷⠘⠗⠙⠿⠟⠲⢤⣈⠳⣆⡉⠻⣏⠙⠳⣶⢶⣤⣭⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠰⡶⠄⠔⢲⡆⠳⠄⠉⠷⠈⠙⠛⠛⠶⣦⠙⠶⠿⠿⠿⠾⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⡟⡗⣿⢿⠛⣆⠘⢦⠀⠛⠶⣄⠸⢿⣻⣀⣠⣤⢦⡴⣾⡛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢸⠃⣿⢹⠈⢣⠈⠳⣄⠻⣦⡙⢦⡙⠦⣙⠣⣤⠤⢼⣆⣈⡛⠓⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣟⠁⡟⠀⡿⢸⢧⡘⣷⣄⠈⢣⡈⠻⣦⣀⡀⠈⢻⣟⢦⡀⠻⣏⠉⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⡇⠀⡇⢸⠀⢳⡹⣌⢦⡀⠙⢦⣽⠎⠙⣷⡄⠙⢦⡙⢦⣘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⣇⠀⣯⣼⡀⠀⢷⠘⣆⠑⣄⠀⠙⣦⣀⠘⢿⣦⡀⠻⣆⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠶⠟⣷⢸⡇⢻⡄⣼⣀⠘⣦⡈⠳⡀⠈⢿⡛⢮⣷⡙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠁⢸⡇⢹⡏⢷⡘⢷⡄⢻⣦⣀⣿⠈⠻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⡿⠀⠈⣧⣼⠁⠈⢷⡼⢿⡌⢷⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠀⠀⣿⢸⡄⠀⠀⠀⠀⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣼⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣶⠶⠶⠦⢤⣤⣤⠶⠛⠛⠙⠉⠛⠳⠶⣤⣤⣀⣀⣴⠶⠦⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠘⣧⠀⠀⠸⠋⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠰⠛⠉⠀⠀⠀⠀⢹⣷⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣾⣃⠀⠳⢦⣄⠈⢀⣀⠀⣠⣠⣠⣬⣧⠀⠀⠀⠀⠀⠀⢀⠀⢀⣼⠃⠀⠀⣼⡿⢻⡄⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠸⣦⣈⠙⠂⠀⠙⣶⠟⣩⡿⠋⠉⠉⣧⠉⠙⠻⣶⣤⣤⣤⣄⣰⣟⣁⡀⢀⠦⠟⠀⠈⢻⣦⣤⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⣠⡿⠋⠛⠳⢦⣿⠀⠻⣆⠈⠳⣄⠈⢳⠀⢰⡏⠈⠓⠦⣌⠻⢭⣍⠙⢿⡤⡄⠀⢀⡾⠁⠀⠀⠀⠙⣦
⠀⠀⠀⠀⠀⠀⢰⡟⠀⠲⡄⠀⡾⠋⠀⠀⠀⠀⠀⣘⣷⡞⠋⠛⠳⠦⣤⣤⣤⣾⣧⣤⣄⣤⣤⣤⣤⣿⣤⣬⣿⠶⠶⠾⡇⠀⠀⠃⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⢀⣴⠆⠀⠀⠀⢀⡾⠋⠀⠻⠆⣤⠀⠀⠀⠀⠀⠀⣼⠃⠉⠛⠛⠛⠲⡄⠀⠀⠀⠀⣀⡀⢻⡄⠀⠀⠀⠀⣀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⡶⠋⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣦⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠐⢁⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠙⢧⡀⠀⠠⠀⡄⠀⠀⠀⠀⢹⡆⠀⠐⠡⠂⣠⠀⠀⠀⠀⠀⠄⠀⠘⣧⠀⠀⢀⡀⠀⠀⠀⠰⡆⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣦⣄⢸⡇⠀⠀⣁⣀⣬⣷⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⣠⠾⣧⠀⠀⣀⣤⣤⣤⣀⣿⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠘⢿⡷⠞⠋⠉⠀⠀⠻⠶⠶⠶⠶⠶⠤⣤⣤⣤⣤⡴⠞⠁⠀⠘⠓⠛⠉⠀⠀⠀⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠢⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠀⡀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠓⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⢠⣟⠀⠀⠛⠛⠳⠶⣄⠀⠀⠀⠀⢻⣆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⡇
⠀⠀⠀⣿⠣⠁⠀⠀⠀⠃⠀⢹⡆⠀⠀⠀⠘⢿⡳⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⣿⠁
⠀⠀⠀⠸⢦⣄⡀⠲⠄⠀⠀⣾⠇⠀⠀⠀⠀⠈⠛⢮⣝⠶⢦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠶⢛⣩⠾⠋⠀⢀⣤⠶⢶⠖⠚⠛⣶
⠀⠀⠀⠀⠀⠈⠙⢦⣤⣤⠾⠋⠀⠀⣀⣤⡀⠀⠀⠀⠈⠛⠲⢦⣭⣍⣉⣙⣛⠛⠛⠛⣛⣛⣩⣥⠴⠞⠋⠁⠀⠀⢰⡏⢹⡀⠘⠀⠠⠄⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠛⠉⠀⢻⣤⣤⣤⡀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠈⢡⣆⣤⠶⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣄⡀⠐⣄⠻⡀⠀⠟⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠞⠻⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣌⡛⢦⣄⣀⣀⠀⣀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣬⣭⣭⠿⠋

⠀⠀⠀⠀⠀⠀⢠⡶⠶⠶⠶⢾⡟⠛⠛⠛⢻⡷⠶⠶⠶⢶⡄⠀⠀⠀⠀⢠⣤⣤⣤⣤⣴⣶⢶⣶⣶⡆⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢸⡷⠶⠶⠶⢾⡇⣽⠢⢿⢸⡷⠶⠶⠶⢾⡇⠀⠀⠀⠀⢨⣭⣭⣭⣭⣹⡿⠣⣏⣘⡇⣭⣭⣭⣭⣭
⠀⠀⠀⠀⠀⠀⠘⣷⠶⠶⠶⢾⣧⣤⣤⣤⣼⡷⠶⠶⠶⣾⠃⠀⠀⠀⠀⠈⣭⣭⣭⣭⡙⠿⠿⠿⠿⠇⣩⣭⣭⣭⡉
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⡟⠛⣿⠛⢳⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡏⣭⣭⢩⣭⡍⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇⠀⣿⠀⢸⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠉⠉⠈⠉⠁⠉⠉⠉⠉⠁

⠀⠀⢠⣤⣤⣤⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣤⣄⠀⢀⡀
⠀⠀⠀⣿⣉⣉⣉⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡆⠀⠀⢸⣿⣿⠀⢸⡇
⠀⠀⠀⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⣀⡄⠀⣠⣤⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⢨⠁⠀⠀⠀⢰⣶⡆⡆
⠀⠀⠀⢸⣿⣿⡟⡇⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⠇⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠀⢸⠀⠀⠀⠀⢸⣿⡇⡇
⠀⠀⠀⢸⣿⣿⣇⡇⠀⠀⠀⠀⣴⡆⢸⣿⠀⠀⠀⠠⣿⡀⠀⣿⣿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣾⢰⣾⣶⠀⠀⠀⣘⣿⣧⣃
⠀⠀⠀⢸⣿⣿⢸⡇⠀⠀⠀⠀⣿⡇⢸⣿⠀⠀⠀⢸⣿⢸⠀⣿⢿⠀⠀⠀⣿⡇⠀⠀⢸⣿⣿⠸⣿⣿⠀⠀⠸⣿⣿⣿⡿
⠀⠀⠀⠸⣿⣿⣸⠇⠀⠀⠀⠀⠿⠗⠸⠟⠀⠀⠀⠸⠿⠺⠀⠿⠸⠀⠀⠀⠿⠃⠀⠀⠘⠿⠿⠀⠿⠿⠀⠀⠀⠿⠿⠿⠧
⠀⠀⠀⠀⣿⣿⠁
⠀⠀⠀⠀⠛⠛⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⢿⣿⣿⠷⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣬⣭⣭⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠿⠿⠿⠿⡇⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⣀⣄⣀
⠀⠀⢠⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣤⣤⣤⣴⡗⠀⠀⠀⠀⠀⣠⣾⣿⠟⢹⣶⣿⣿⣿⣿⣿⣶⣆
⠀⠀⢸⣿⣿⣿⣿⠀⠀⢀⣤⠶⠶⣤⡀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⢠⣾⣿⠟⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⢸⣿⣿⣿⣿⠀⢀⡞⠁⠀⠀⠈⢳⡀⠀⠀⠀⠶⠶⠶⠶⠶⠶⠶⠂⠀⠀⠀⠙⠁⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠁
⠀⠀⢸⣿⣟⣛⣛⠀⢸⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠄⠀⠠⠆
⠀⠀⢸⣿⣿⣿⣿⠀⠘⣇⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡶⠶⠤⠦⠦⠶
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠙⢦⣤⣶⡿⠋⠀⢀⣤⣤⣤
⠀⠀⢸⣿⡿⣿⠿⠀⠀⠀⠀⣸⣇⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠚⢛⡿⠿⢿⡛⠓⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣄⣀⡀
⠀⠀⢸⣿⣯⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⡔⠁⠀⠀⠀⠈⢢⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⢰⡀⠀⠀⠀⢀⣤⣶⡇⠀⠀⠀⠀⣏⠀⠀⠛⠟⠃⠀⠀⡇
⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠣⡀⠀⣰⣿⣿⣿⣿⡆⠀⠀⠀⠻⣄⡀⠀⠀⠀⢀⣼⠃
⠀⠀⢸⡿⠿⠿⠿⠀⠀⠀⠀⢻⡟⠀⠀⠀⠸⣿⣿⣿⠀⠀⠀⠀⠀⠈⠉⠛⠻⣿⡿⠟⠁⠀⠀⠀⠀⠉⠛⠓⠓⠛⠋⠁

⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⠂⠀⠀⠀⠰⣏
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⡞⠛⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠸⠡⠦⠰⠶⡀⠀⠀⠀⠀⠀⢠⠋⡍⠐⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⣤⣤⣤⡤⡔⠇⠀⠀⠀⠀⠀⢿⠦⣱⣦⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠳
⠀⠀⠀⢀⠃⡘⠀⠀⠀⢠⠀⠀⠀⠐⠃⠀⠀⠀⠙⣀⠇⠀⠀⠀⠀⠀⠀⠀⢰⡩⣤⡤⢄⣤⣭⣥⢦
⠀⠀⠀⠀⣽⡶⢤⣌⣄⢸⠆⢲⠔⠀⠀⠀⠀⠀⠸⠻⠀⠀⠀⠀⠀⠀⠀⢀⣾⣟⡟⣿⣿⣿⣿⢻⡍⡆
⠀⠀⠀⠘⠛⠳⠟⠚⠛⠊⠉⠀⠉⠙⠲⠖⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠾⠿⠛⠛⠛⠛⠛
⠀⠀⠀⠀⠀⠀⣤

⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⡀
⠀⠀⠀⠀⠀⢤⡇⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡉⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠈⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏
⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠛⠙⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⢀⣾⠻⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣶⣴⣦⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡃
⠀⠀⠀⠀⠈⠁⢀⠐⠄⠀⠀⠀⠀⠀⠀⢰⢷⡞⣿⡞⢻⣇⠀⠀⠀⢀⣠⠄⠀⠀⠐⠈⢛⠁
⠀⠀⠀⡠⠁⡀⠈⡀⠀⠀⠀⠀⠀⠀⡠⠃⠘⠀⠁⠀⠀⠉⣆⠀⠀⢿⣇
⠀⠀⣰⠁⠀⠀⠀⠀⠀⣦⠀⡀⠀⣰⢁⡀⢠⠀⠂⣐⣄⣤⣼⡆⠀⠈⠻⢷⣤
⠀⠘⠡⠤⠳⠶⠶⠤⠴⠟⠋⠁⠘⠿⣿⠳⠿⠟⠛⠛⠋⠉⠉⠁⠀⠀⠀⠀⠉⠓⠶⠶⠾⣤⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠯⢭⣉⣳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⢰⡏⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠤⢄⣀⡼⠁⡭⢦⠀⠀⠀⠀⣘⡟⠢⠏⢱⠀⡘⠀⣼⠇⡞⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠇⠀⠀⠸⡀⠀⣠⣾⠉⠷⡄⠀⢸⣀⡃⣀⣿⡜⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠒⠚⣄⠀⠀⢸⣀⣠⣤⣧⠖⠳⣜⠦⠀⠉⣵⠾⡉⠁⠹⣿⣴⣄⡉⢹⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠟⠒⠄⠈⠷⢤⣾⠿⠗⠚⢹⠀⠀⠀⠀⢀⡜⠁⠀⢧⢀⣘⡑⠁⢀⡬⢷⠀⠻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣀⡤⠤⣄⣀⠄⠀⢘⣆⣀⡀⣀⣧⣀⣀⠀⠹⡃⠀⠀⠈⠉⠀⢉⣉⡿⠤⡏⠀⠓⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠿⠀⠠⠋⡁⠀⣠⣤⣾⣾⣋⣉⣉⣋⣉⣉⣩⠭⠭⠤⠒⠒⠋⠙⢳⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢠⣀⠀⠀⡸⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⣠⣄⢴⣲⠶⢸⢳⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠻⣄⡀⠀⠈⠉⠒⠁⠀⠀⢰⣶⣶⠶⢸⣷⢛⣫⢹⠀⡎⣦⣼⠚⣎⣻⣯⡻⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡦⣄⡈⠙⠟⣟⣷⣒⠀⠀⠀⠰⣿⠀⠸⡿⠛⠓⠮⠖⠛⠁⠀⠙⠋⠉⠁⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠈⠙⠒⠋⠉⠁⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠈⢻⠐⠀⠈⠉⠉⢲⠀⠀⠀⠀⠈⠉⠉⠑⡆⠀⠀⠈⠉⠉⢩⠀⡃⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠉⢱⠄⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⠁⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠃⠀⡞⠀⢸⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠀⠀⢸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠓⢦⢸⠤⢄⢸⢰⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢸⢀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠁⠀⠳⠲⡜⢾⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⢸⠈⢸
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⠤⠦⡀⠀⢀⠀⡇⠀⠉⠢⣄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⡇⠇⠀⠀⠀⠀⢸⠀⠘⠀⠀⢀⣀⣀⣀⣀⣀⣀⣤⣀⡀
⠀⠀⠀⠀⣠⠚⠉⠙⠢⠄⠉⠛⠛⠛⠆⠈⠀⢰⡀⠀⠀⢀⣙⡆⢸⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢸⢸⡀⠀⠀⡟⠉⠉⠉⠉⠉⢹⠳⢖⠛⠛⠒
⠀⠀⠀⠀⠺⣀⣀⣾⣭⡑⠲⠤⠤⠖⠒⠦⣀⢸⣹⢹⠋⠋⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢸⡷⢤⡴⢧⣤⠤⠰⠲⠒⠧⣀⠀⠑⠦⡀
⠀⠀⠀⠀⠀⡜⠙⠂⠀⠀⠀⠀⣄⠀⠀⠀⠈⠹⣿⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⡇⢿⡇⢬⣀⡀⠀⠉⣹⠶⠲⣞⢙⠗⠒⠤⢭⣀
⠀⢀⠤⠴⠚⢠⠦⢄⡚⠄⠀⠀⠉⠦⡀⠐⠋⠠⠃⢹⠀⠠⠤⠤⠞⠀⠀⠀⢀⣀⣀⣀⣸⠸⠀⠀⡠⠤⠖⠃⣸⡗⠒⠤⢌⣉⠙⠁⠀⠘⠉⠙⠓⠢⣄⡀
⣏⣁⣀⣜⠉⠁⠀⣈⠷⠚⢆⠀⠀⣀⣤⣀⡀⠀⠀⠾⢿⠟⠦⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⡴⠶⠛⠉⠀⠀⠀⠀⠈⠉⠙⠚⠋⠉⠉⠓⠒⠛⠋⠉

⠀⠀⢠⡤⢤⠤⠤⠤⠤⡄⠀⠀⢠⣶⣶⡞⣟⡛⣶⣶⡆⠀⠀⡞⣟⣛⣆⡤⢤⢤⣤⣤⣤⣤⠀⠀⢰⣶⣶⢀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠸⡧⠾⠭⠬⠭⢭⠇⠀⠀⠘⡿⠛⣓⣒⣚⡛⢻⠇⠀⠀⠙⣶⡞⠹⡯⢭⣭⠯⣥⡥⡽⠀⠀⠘⢻⠋⢸⣿⣿⣿⢿⣿⢿⠇
⠀⠀⠀⡇⠀⢸⢹⢹⢸⠀⠀⠀⠀⡇⢸⠁⢹⠀⡇⢸⠀⠀⠀⠀⣿⡇⠀⡇⡏⡏⡷⣏⡷⡇⠀⠀⠀⢸⠀⢸⣿⢹⢸⠸⠬⢿
⠀⠀⠀⠛⠒⠛⠛⠚⠚⠀⠀⠀⠀⠛⠚⠓⠚⠒⠛⠚⠀⠀⠀⠀⠙⠁⠀⠛⠛⠛⠓⠒⠒⠃⠀⠀⠒⠚⠒⠚⠛⠛⠛⠒⠒⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⣶⠶⠶⢶⢖⣶⡄⠀⠀⢷⢚⣀⡤⠤⠤⢤⣤⣀⠀⠀⢰⣶⣶⢀⣀⣀⣀⣀⣀⣀⡀⠀⠀⢸⣿⡽⣴⠶⠶⣶⠶⠶⢶⡆
⠀⠀⠀⣛⠛⢛⣛⠛⢛⠁⠀⠀⢸⢸⡨⣭⡭⠍⣼⣿⣿⠇⠀⠈⢹⠉⠸⣿⢿⡿⡿⡿⣷⠇⠀⠀⠀⣿⠀⢹⡿⢿⠿⡿⠿⣿⠃
⠀⠀⠀⣿⠀⡏⢹⠀⢸⠀⠀⠀⢸⠀⡇⡿⣿⠀⣿⣿⣿⠀⠀⠀⢸⠀⠸⡇⠘⣧⣤⣤⣾⠀⠀⠀⠀⢿⠀⢸⡇⢸⠀⣧⣤⣿
⠀⠀⠀⠈⠛⠛⠛⠛⠋⠀⠀⠀⠈⠀⠛⠛⠛⠛⠛⠛⠋⠀⠀⠀⠈⠀⠈⠃⠈⠛⠛⠛⠛⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠉⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠘⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⠀⠀⠈⠛⠳⠛⠛⠛⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠹⠶⣤⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣤⣤⣿⣄⡀⠀⠀⠀⢀⣤⡶⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠟⣿⣁⡽⢀⣀⠈⠛⣦⡴⠞⠋⠁⠀⢀⣠⣤⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⢀⣀⠀⠀⠠⡏⣹⠇⢀⡼⣧⣠⣤⠶⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⣧⠀⣏⣈⡷⠀⠀⠉⠁⠀⠈⠓⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠶⢋⠀⠀⠉⡴⢤⡀⠠⡞⢳⡄⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠰⣏⣹⠀⠀⠙⠦⠇⠀⠉⠉⣀⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⡀⠈⠁⠀⠀⠀⣀⡤⣄⠀⢰⣣⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⢿⡤⠇⠀⣞⣹⠆⠐⢧⡤⠏⠀⣸⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠟⠉⠀⣸⣷⡀⠀⠀⢁⣤⣄⠀⢀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠉⠀⠀⣠⡾⠋⠀⠙⠷⢦⣼⡧⣾⠟⠋⠁
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠾⠋⠀⠀⠀⠀⠀⢸⡇⠀⢹⡄
⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠙⠳⠶⠶⠶⠶⢦⣄⠀⠀⠀⠀⠀⠸⠃⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡆⠀⠀⢀⣀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⡴⠛⠉⠻⣆⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋
⣿⣿⣿⣿⣛⣿⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣿⣛⣟⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⡤⠒⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡴⡇⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠲⡄
⠀⠀⠀⠀⠀⠀⢸⣀⣀⣀⣈⡆⠀⠀⠀⠀⠀⠀⡸⢭⣙⣳⠀⠀⠀⠀⠀⠀⣠⠞⠉⠉⢢⡀⠀⠀⠀⠀⠀⢸⠤⠖⠤⢾
⠀⠀⠀⠀⠀⠀⠀⢻⢮⣑⡏⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠘⡇⠀⠀⠀⠀⢸⡀⣀⠀⢀⡀⢈⠆⠀⠀⠀⠀⣏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⣷⡼⠀⠀⠀⠀⠀⠀⠀⠀⢻⠉⠉⢹⠁⠀⠀⠀⠀⠀⠙⣄⣍⣡⡸⠁⠀⠀⠀⠀⠀⠈⠓⡖⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠳⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢂⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⢀⡠⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⢀⣴⡁⣹⡀⠀⠀⠀⠀⠀⢠⠞⠓⠛⢤⠗⠀⠀⠀⠀⠀⠀⣀⣟⣆⡀
⠀⠀⠀⠀⢀⠴⠋⠀⠀⠀⠀⠙⠦⡀⠀⠀⢠⠞⠓⠒⠛⠓⢛⡆⠀⠀⠀⠀⢮⣀⣀⣀⣼⠄⠀⠀⠀⠀⠀⢸⠶⠤⠤⢧
⠀⠀⠀⠀⠈⠻⣍⣑⣉⣉⣊⣩⠗⠁⠀⠀⢸⠟⠛⠛⠛⣛⣛⡇⠀⠀⠀⠀⠀⢣⣶⣶⠃⠀⠀⠀⠀⠀⠺⢍⣛⣛⣛⡩⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⢀⡇⣃⠀⠀⠀⠀⠀⠀⠀⠀⠴⠿⠧⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⠤⢄⡀⠀⠀⠀⠀⠀⢀⢔⡒⣒⠢⡀⠀⠀⠀⠀⠀⢀⡤⠒⠢⢤⠀⠀⠀⠀⠀⠀⢴⣶⣶⣶⣶⡄
⠀⠀⠀⠀⠀⢠⢊⡩⣍⠲⡄⠘⡄⠀⠀⠀⠀⡇⡏⣞⠶⡱⣱⠀⠀⠀⠀⠀⣏⢀⡀⢀⠀⣱⠀⠀⠀⠀⠀⣺⣿⣽⣽⣿⡇
⠀⠀⠀⠀⠀⢤⢸⠰⣀⡧⢸⣀⠇⠀⠀⠀⠀⠱⣕⣮⣭⠷⠋⠀⠀⠀⠀⠸⡍⣿⢹⡏⡿⢸⠇⠀⠀⠀⠀⣿⣿⣯⣭⣽⡇
⠀⠀⠀⠀⠀⠈⠣⣳⣬⠤⠞⠁⠀⠀⠀⠀⠀⠀⠀⢹⡏⠀⠀⠀⠀⠀⠀⠀⢳⣻⣸⣧⣇⡟⠀⠀⠀⠀⠀⣗⣀⣈⣉⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⠉⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢄⡀
⠀⠀⠀⠀⠀⠀⡔⠉⠀⠀⡉⢢⠀⠀⠀⠀⠀⠀⠀⣠⡼⣝⢢⡀⠀⠀⠀⠀⠀⢺⢀⣀⡅⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⣄⡀
⠀⠀⠀⠀⠀⢸⠁⠏⠀⢀⠀⠈⡆⠀⠀⠀⠀⢠⠞⡄⢣⢸⠋⠁⠀⠀⠀⠀⢎⡉⠀⠈⢋⡱⠀⠀⠀⢀⡾⡀⢣⠀⣰⢃⡼⡆
⠀⠀⠀⠀⠀⠘⢄⣀⠀⢀⣁⡼⠁⠀⠀⠀⠘⣍⠢⡵⠜⠁⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠻⡀⠀⠀⠀⠀⠳⢼⠍⠉⠉⣽⠴⠁
⠀⠀⠀⠀⠀⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠋⠙⠦⠞

⠀⠀⠀⠀⠀⠀⠀⣴⣯⣭⡭⣿⣭⣽⣯⣭⣽⣦⡄⠀⢾⣿⣿⣿⣿⣯⣭⣭⣭⣭⣭⣭⣽⣧
⠀⠀⠀⠀⠀⠀⠀⢳⡖⠒⣶⠒⠒⠒⢶⠖⠒⡾⠃⠀⠈⢳⣷⡶⠒⢲⡖⠒⠒⠲⡶⠒⢲⡞
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠿⠀⠀⠀⠸⠀⠀⡇⠀⠀⠀⠀⢹⡇⠀⠸⠇⠀⠀⠀⠇⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣠⣄⣀⠀⠀⢠⡇⠀⠀⠀⠀⠘⣗⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣤⡝⠁⢛⣥⠀⢸⡇⠀⠀⠀⠀⠀⣗⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠹⣶⠄⣶⠟⠀⢸⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⣀⢸⠀⠀⠀⠀⠀⠀⠀⣾⣀⠀⠀⠀⠀⠀⢸⣆⡀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⢸⡏⣿⡀⠀⠀⠀⠀⠀⠀⣿⡏⡇⠀⠀⠀⢠⣞⡽⢯⢳⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠸⣇⡿⣿⣄⣧⣸⣄⣧⣸⠿⣇⡇⠀⠀⠀⠈⢧⣓⣫⣾⣄⣱⣀⣧⣸⡟




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣙⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⡀⣤⢤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⡙⡵⠛⠻⣿⡙⢎⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⢀⢱⠀⠀⠀⢣⢷⠀⠏⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠸⢼⠀⠀⠀⠈⣸⠀⢰⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡗⠀⢰⢈⠀⠀⠀⢀⣿⡀⢈⢠⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠴⠒⠀⠉⠉⠉⠉⠉⠉⢹⡼⠀⠉⠀⠀⠈⠁⠈⣮⠃⠀⠀⠀⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⢸⡁⣸⠀⠀⡇⠀⠀⠀⠀⠀⠀⢾⣿⠄⠀⠀⠀⠀⠀⠺⢿⠗⠀⠀⠀⠀⠐⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢻⠋⢽⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠔⠠⡀⠀⠀⠀⠀⠀⠰⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣾⢊⢽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⢊⢈⠔⡠⡀⠀⠀⠀⢊⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠨⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠡⠊⢊⠄⠔⠀⠀⠀⡥⡀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣪⠂⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠕⢼⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⡐⡄
⠀⠀⠀⠀⠀⠀⠀⡯⠐⣽⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⡈⡐⠄⢔⠆
⠀⠀⠀⠀⠀⠀⠀⡧⠂⣽⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⢀⠎⢈⢀⠌⠔⠈⣰⠢⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⡡⠠⠀⡀⠠⠪⠢⢂⠌⡼⠡⡡⠆⢰
⠀⠀⠀⠀⠀⠀⢰⡅⡂⢺⠀⠀⢀⠄⠀⠀⠀⠀⠀⢊⠄⠔⠐⠁⡠⡔⠐⡑⡑⠡⠂⡊⠡⡐⠌⢄
⠀⠀⠀⠀⠀⠀⢸⢗⠉⣺⠀⢘⠑⡁⡁⡠⠢⢀⠄⡀⡠⢀⠂⠊⢊⢊⢊⠄⡰⠑⡁⠀⠐⡀⠄⢈⠐
⠀⠀⠀⠀⠀⠀⢸⠕⠨⢺⠀⢠⠨⡪⢊⠔⡰⠡⢊⠌⠔⡑⡑⡱⡡⠢⢊⠜⠜⠔⡐⠀⠐⠀⠌⠀⠊
⠀⠀⠀⠀⠀⠀⠸⢵⣾⠿⡆⢸⠐⡔⡡⢊⠔⡁⡡⢂⢊⢎⢌⡔⡕⡡⢪⢂⢈⠄⡄⡰⠄⢄⢂⠠⡥⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠛⠖⠒⠒⠒⠲⠆⠁⠃⠤⠬⠬⠬⠐⠓⠓⠲⠒⠚⠚⠒⠐⠒⠒⠒⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⢀⠀⡀⢀⠄⠀⢀⠀⡠⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠈⠀⡀⢀⠀⣀⠄⡀⢐⠁⠀⢀⠩⢀⠁⠠⡈⠀⡁⡊⠔⡡⢂⠄⡨⢊⠔⡨⢀⠕⡠⢊⠔⡩⢂⠅⡠⠨⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠈⠀⠁⠐⠁⠊⠀⠁⠀⠀⠁⠈⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠂⡀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⡿⠀⠀⢀⠀⠀⠋⠤⢤⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣀⠂⠀⠀⠀⡀⠧⠠⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠
⠀⠀⠀⠂⠁⠀⠀⠀⠀⠘⣿⡄⠀⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⣇⠀⠀⠀⠀⡇⠈⠀⠇⠀⠀⠀⠀⠀⠀⠈⡀⠀⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠊
⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⢐
⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠚⠅⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⢀⡞⠑⠊⠀⠀⠰⠉
⠀⢡⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠈⠦⠃
⠀⠰⣅⠁⠀⠀⠀⠀⠀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠘⠀⢀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⢰⡄⠀⢀⢀⠀⠀⠀⠀⠀⠁⡀⠀⠃⠀⠄
⠀⠀⠐⠀⠀⠀⠀⠡⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠁⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠈⠉⠁⠁⠀⡄⠀⠀⠀⠀⠠⡄⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠂⠀⠀⠂⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣠⡤⡄⠁⠀⠀⡄
⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠴⠓⠀⠀⠀⠑⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⡱⠔⠠⠀⠀⠀⠀⠠⠀⠀⠢⢀⠀⠀⠀⠀⠀⠀⠀⠈
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠓⠂⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠊⠀⠀⠀⠀⠐⢄⠀⠀⠀⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄
⠀⠀⠀⠀⠀⠀⠑⢠⡴⠤⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠄⠀⠀⠁
⠀⡀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠰⣀⡢⠀⠀⢈⠈⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⣤⡄⠀⠀⠀⠀⠀⠀⡂⢀⠀⠀⠀⠀⠁⠀⠂⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠻⠅⠀⠀⠀⠀⡀⠐⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢄⡀
⠀⣂⠠⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⢢⠼⠁⠀⠀⠂⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠐⠀⢀⠄⠒⠀⠀⠀⠠
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⠦⣀⡀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⠂⠀⠀⠈⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠠⣤⠧⠄⠀⠀⠀⠀⠀⠀⠀⢜⠁
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⠀⠀⠀⠀⠀⠀⠌⢀⠒⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠠⠄⣀⠀⠂⢀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⠀⢀⡀
⠀⠀⠁⠀⠀⠠⡶⢢⠀⡀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⡎⠀⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠇⠀⠀⠀⠈⠉

⣾⠻⣆
⢻⡇⢻⡄
⠘⣷⡀⠻⣦⡀
⠀⠘⣿⣦⡈⠻⣦⣄
⠀⠀⠈⠙⠿⣷⣦⣝⡻⣦⣄
⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣧⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⣖⠶⢶⣾⣿⣿⠿⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣄⣀⣀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣾⣷⣦⣄⡀⠈⢷⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡿⢿⣿⡶⠏⠀⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡶⠶⠶⡿⠿⣿⠟⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡄⠸⠁⠀⠀⠈⠎⢿⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠃⠀⠀⠀⠀⠀⠠⣄⠀⡙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⣀⠈⡄⠀⠀⠀⠀⠈⢳⡈⠄⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣏⢆⠑⠀⠀⠀⠀⠀⠀⠙⣦⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣯⡹⡇⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠈⠹⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣮⣦⠑⢄⡀⠀⠀⢀⠀⠘⠇⠀⠀⣄⠀⠛⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢧⣀⠑⣤⣀⢀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠙⢿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣌⢿⣦⡱⡅⠀⠢⡈⠲⡤⡙⠶⣴⢤⡙⢿⠻⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣝⢷⣩⣬⣓⠪⢦⣔⣦⣷⣦⣤⣔⣽⣷⣾⣿⣿⣿⣷⠶
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣙⣋⣉⣀⣈⣉⣛⣛⣟⣛⣉⣉⣙⣿⣿⣿⣟⣋⣁⣀⣀
⣿⣿⡟⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⠤⠤⠔⠒⠒⠈⠉⠉⠉⠑⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣒⡚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠋⠉⠁⠀⠀⠈⠉⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣
⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⣼⠋⠠⡌⢳⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇
⣿⠀⠈⠁⣸⡀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⣀⠤⠤⣀⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⡞⠉⠛⠵⣶⡼⣀⣀⠀⠀⠀⠀⡜⣁⣤⣶⠿⢧⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⣿⠀⠀⠳⣄⠀⠀⠀⠉⠑⠒⠭⢵⣒⠮⠝⠋⠉⠀⠀⠈⡆⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⢻⡀⢠⣤⢸⠉⠒⠦⢄⣀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠒⠊⡇⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠈⠳⠤⠴⠋⠀⠀⠀⠀⢫⠉⠙⠒⠒⠒⠛⢅⣀⣀⣠⠞⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢃⣨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⣀⣠⡴⠟⠊⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⡀⡠⢊⡤⠒⢛⠀⠀⠀⡀⠀⠈⢡⢈⠀⠀⠐⠐⠂⡀⠀⢀⠐⢂⡀⢀⠂⠀⠀⣤⠀⠀⢠⣄⠀⡄⢠⣀⠀⢠⠀⠀⠴⠒⠒
⠀⠀⠀⣠⡤⡈⠊⠀⠃⠐⠈⡀⠀⠠⠄⠈⠶⣚⠩⠉⠁⠀⢸⣀⢀⣁⣀⣚⢀⣘⣃⣀⡃⠀⠀⢿⠀⢀⣸⠿⠀⠅⠀⡀⠀⠀⠀⠀⢰⠈⣆⠀⢰
⠀⠀⠘⢁⠀⡤⠀⢂⠀⠐⢉⠠⢨⡀⠀⠀⠀⠀⠀⠀⡁⣀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠈⠁⣀⣤⠤⠀⠀⠀⠐⠀⠀⠀⡆⠀⠡⠰
⠀⠀⡀⠈⠁⠤⠀⠁⢁⠐⣍⡐⠌⠓⡤⠀⠀⠀⠀⠎⢭⠀⠀⠤⣤⢅⡥⡥⢤⠄⠀⠰⢱⠋⠄⠀⠘⣁⢠⠰⡇⠀⠀⢐⠚⡀⠃⡇⠀⢈⠀⠀⡀
⠀⠀⠘⠊⠀⡠⠀⠂⠁⢀⠞⠀⠀⠀⠀⢈⡀⠀⠐⠄⡀⠀⠈⡀⠁⡀⢀⠀⠠⠀⠀⠀⠩⠀⠀⠀⠀⢫⠰⠚⠢⠀⠀⡰⡀⠀⠀⠇⠰⢀⣠⡖⣊⠅
⠀⠀⢀⡈⡁⣠⠴⠂⠀⢀⠀⠀⠐⢀⣁⣠⠀⠀⠀⠂⠀⢂⠀⠇⠀⠀⠀⢈⠁⠀⠀⠐⢀⠀⠀⠀⠀⠈⢄⣤⢤⠀⠀⠀⣀⠀⠀⠁⠘⡉⠦⠉⢀⠂
⠀⠀⠀⠈⠒⣀⣤⠴⠶⠈⠄⠈⡀⣘⣥⠐⠀⢀⠀⠠⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠐⠀⠀⠈⠃⠉⡀⢀⡈⠻⡄⢤⡄⢀⠀⠁⢉⣀
⠀⠸⡇⢰⠚⠉⠀⠀⠀⢀⠐⠀⠀⠀⣫⠐⡕⠀⠄⠠⡀⠄⠀⠀⢀⠀⠀⢄⠂⠀⢀⠀⢀⠀⠄⠀⠇⣠⡄⢀⠾⣽⣿⠇⡄⠹⡄⠀⠁⠋⢀⡐⠛
⠀⠀⣇⣈⠆⠰⣶⡆⡱⡄⠂⠀⠀⢂⠙⠋⠁⠀⠉⠀⠡⠀⠀⠀⢤⠀⠠⠠⠀⠐⠩⠄⠈⠆⠀⠊⠘⠛⠁⠉⠘⠃⠸⡼⠁⠀⠙⠀⠰⠙⠋⠹⢳⡄
⠀⠀⢸⣿⠀⠀⠁⠑⠐⡛⠀⠈⠀⠀⠀⠀⠀⡀⠤⠁⠀⠀⠀⠀⠀⠴⠖⠒⠒⠂⠀⠀⠀⠐⠒⠈⠁⠀⡸⠖⢶⡇⠀⠀⠀⡂⢂⠉⣧⠆⠀⢁⢀⣿⠁
⠀⠀⠸⣿⡂⠈⠂⠊⡀⣤⠤⢁⠐⠀⠐⠈⢁⣠⠀⠀⠀⠀⠁⡆⠐⢄⡄⣴⢀⡄⠀⢀⡀⣤⠄⠀⠀⠠⠛⠛⠛⠃⠀⣠⠞⠉⠊⠀⠙⠯⣚⣊⠕⠁
⠀⠀⠀⠀⠀⠠⠀⢁⠂⠀⠀⣉⡆⠀⢰⠊⠀⢘⠇⠀⠀⠀⠀⠁⢀⠀⣁⣉⣉⠀⠀⠁⠁⠀⠀⠀⠀⠘⠒⠒⠒⠂⠀⠛⢀⡚⠱⡄⠀⢀⡸⢄
⠀⠀⡄⠁⢀⠌⠀⢀⡡⠖⠉⢉⠀⠀⠀⠓⠈⠁⠀⠶⢦⠀⠠⠴⠀⢀⣤⣈⢉⣀⣀⡄⣠⡤⠁⢀⡤⣄⠀⠀⣤⠩⡏⠀⠀⣄⡀⠆⠀⠀⠀⠀⠀⠒
⠀⠀⢉⠉⠀⠀⡐⢀⠆⠀⡐⠁⠠⠀⠀⠀⡌⠉⡄⠘⠈⢁⠀⠀⠘⠀⠈⠑⠈⠁⠀⠁⠇⠀⠀⢿⣲⡭⠀⡠⠿⠟⠇⠀⠀⠐⠓⠀⠀⠊⢀⣀⣘⠠
⠀⠀⠸⢀⡣⠀⡁⠀⠀⠁⠈⠐⠀⠀⠀⠀⡀⠀⠠⠀⠕⠈⠁⠀⢰⠞⠓⠀⠀⠠⠂⢐⡄⠀⣀⣀⡀⠀⣸⣀⠶⠶⢐⣇⠀⠘⠚⠂⠠⠀⠉⠉⠉⠠⠃
⠀⠀⡀⠶⠄⠀⠈⢀⡴⠀⠀⠀⣰⠀⠃⠀⢷⠇⠀⠀⠀⢀⣀⠆⠀⠄⠀⠀⠀⡠⠀⡠⠁⢸⡅⣈⡇⠀⠏⣅⡔⠲⠏⢸⠀⠀⠀⠀⠀⠈⠉⢉⠉⠁
⠀⠀⢏⡩⠁⠀⠀⢀⣀⠀⢀⣀⡀⠠⢂⡀⠈⠀⣀⠀⠀⠈⠀⠀⠀⠈⠢⠜⢀⣁⠁⡘⠀⠀⠈⠉⠀⠀⣦⣟⣳⣶⡦⠶⠀⢠⣉⡄⠀⠀⠀⡸⠭⣄
⠀⣀⣤⣦⠹⠆⠀⠁⠀⡀⠁⠀⠀⠘⠉⠀⠀⢈⣉⠀⠀⠐⠀⠒⠤⢄⣀⣀⠠⠈⠉⠁⠀⢴⠀⠀⡀⡀⢀⣠⡤⣤⢶⠲⣶⣾⠟⢉⣿⠀⠸⡷⠿⢿
⠀⠸⡟⠫⠀⠐⡆⠘⠁⠀⠚⠃⠀⠀⠤⢀⠀⣺⠈⡄⠀⠀⠀⠀⠀⢘⡇⠤⠤⠀⠀⠀⠀⠘⠒⠂⠀⣀⢀⡀⠀⠀⠂⠂⠒⠲⠶⠶⠆⠀⠀⡇⠀⠸
⠀⠀⠀⠠⠀⠀⠈⠄⠀⠀⢀⠁⠁⠀⡀⣀⣠⢾⡄⢳⠀⢀⠀⡀⠔⠁⠀⢀⣠⠠⠄⢲⣆⠀⠀⡐⠁⡀⠲⠴⡁⡄⠀⢀⡀⠀⠀⠤
⠀⠀⠀⠐⠀⠀⠀⠐⢅⠀⠸⠰⠚⠈⠁⠁⠠⠀⠳⣾⡀⠀⡡⠠⠀⠔⠊⠹⢋⣁⠀⠘⢹⠀⠀⠠⠅⠀⢀⠀⣄⠆⠀⠑⠄⠀⠀⠒⢂⣶⠖⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠄⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⡀⠱⣎⡁⠀⠀⠀⠈⠀⢀⠀⠿⠂⠀⠐⠁⢀⣠⠁⠈⠀⠀⠀⠀⢂⠀⠀⠀⡄⠀⠈⢩⠁
⠀⠀⠀⠠⠢⣄⠐⣯⡔⠄⠀⠀⢀⠀⡐⠀⠀⠁⡄⠀⡇⢻⠀⠧⠡⠘⠀⠂⠈⠁⠀⢀⣀⠀⠀⠀⠠⡀⠀⣄⢀⣉⢀⣄⠀⠸⠄⠀⠀⠦⠀⠀⢼
⠀⠀⠀⢠⠃⡜⡤⡧⡶⢸⠀⠀⠐⡀⠀⣀⢀⡀⠂⠀⠁⠈⠀⣠⠔⠒⠒⢦⢀⠀⠈⠀⠛⠀⠄⠈⢀⡰⠂⠀⠘⡇⣀⢸⠀⡀⠦⠀⠀⡴⠀⠀⠰⡇
⠀⠀⠘⠢⣁⠰⠦⠀⡡⢸⠀⠀⠀⠀⠉⢀⣠⣤⠐⠢⠀⠀⢞⠁⠀⠀⢀⡰⠃⠀⠀⠀⠀⠀⠀⠅⠀⠀⠠⠀⠀⢧⠈⠸⡇⠀⠀⠠⡤⠷⠐⠒⡂⠁
⠀⠀⠀⠂⠄⠈⠈⠁⠄⠊⠀⠐⠀⠀⠀⠈⠙⠛⠋⠜⠀⠀⠈⠙⠒⠒⠋⠀⠀⢰⡀⠤⠤⠤⠤⠶⠀⠀⠂⠀⠀⠸⠒⠊⠁⠘⠄⠐⠡⠥⠤⠀⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠛⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢄⡀⣀⣀⣠⣤⣤⣴⣶⣶⢿⣿⣾⣿⣶⣷⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢫⣩⣴⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡜⣿⠟⣆⣨⣿⣿⣟⣿⣿⣿⣿⣿⣿⣿⣶⣦⣤⡀
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣻⣿⡟⣿⣧⣻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣼⣯⡿⣿⣿⠟⣽⣿⣿⢟⣿⣿⣿⣟⣿⣿⣿⣷⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⢿⡿⣛⣻⣽⢿⡄
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣴⣿⣷⡆⣾⣯⣯⣍⣸⣿⣿⣻⣙⣿⣏⣫⣩⢻⣿⣭⣭⡬⣿⣶⣾⣿⡿⢿⡿⢁⣩⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣿⢿⣿⣿⣿⣼⣿⣿⣿⣷⠾⠿⠾⠿⠶⢛⣟⣻⣭⣥⡟⣾⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⣿⣯⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⡂⠐⠀⢀⣘⣿⣿⣿⣿⡿⢿⣷⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⢹⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣏⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⣽⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡁⢸⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⠹⣷⠀⢠⠐⢶⡄⠀⠀⣿⡇⠀⢀⣿⡌⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⠀⣿⠀⠘⠀⢸⠀⠀⠀⣿⡇⠀⢸⣿⠃⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⠀⣿⣿⢸⣿⣿⡇⣿⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠈⣿⢘⢿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⣿⣿⡄⣿⣿⡇⣿⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⢸⣿⢸⣾⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⣿⣿⡇⣿⣿⣇⢻⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⢀⣿⡇⠀⢸⣏⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⢻⣿⡇⣿⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢼⠀⠀⢸⣿⡇⠀⢸⡏⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⡇⣿⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⢸⣿⠀⠀⢸⡇⣶⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⢸⣿⣧⢻⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⢸⣿⠀⠀⣸⡇⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡇⢸⣿⣿⢸⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⠸⣿⠀⠀⣿⡇⣻⡿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣿⢸⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢸⠀⠀⢰⣿⠀⠀⣿⠃⣸⣇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⣿⣿⢸⣿⣿⢸⣿⣿⣿⠀⣿⠀⠀⠀⢿⠀⠀⢸⣿⠀⠀⣿⠠⢻⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⣿⣿⣼⣿⣿⠸⣿⣿⣿⠀⣿⠀⠀⠀⢻⠀⠀⢸⡿⠀⠀⣿⢠⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢿⣿⡏⣿⣿⡄⣿⣿⣿⠀⣿⠀⠀⠀⣾⠀⠀⣸⡇⠀⢰⣿⠸⣿⣻⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀⢸⣿⡇⣿⣿⡇⣿⣿⣿⡀⣿⠀⠀⠀⡇⠀⠀⢿⡇⠀⢸⡇⠀⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⢸⣿⡇⣿⣿⡇⣿⣿⣿⡇⣿⠀⠀⠀⡇⠀⠀⢻⡇⠀⢸⡇⠈⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⢸⣿⡇⣿⣿⡇⣿⣿⣿⡇⣿⠀⠀⠀⡇⠀⠀⢸⡇⠀⢸⡇⣠⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⣀⣿⣿⢿⣿⡇⣿⣿⣿⡇⣿⠀⠀⠀⡇⠀⠀⢹⡇⠀⣸⠇⢻⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣝⣻⣾⣿⣧⣻⣿⣿⡇⣿⠀⠀⠀⡇⠀⠀⣸⣃⣀⣿⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⣿⣿⣟⣃⡘⡛⠓⠀⡟⢚⣉⣻⣭⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣀⣀⣀⣀⣀⣀⣠⣶⠶⠶⠶⠶⠶⠶⣶⣀⣀⣀⣀⣀⣀⣀⠀⣿⠀⠴⣦⢀⣴⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⠀⠀⠀⠘⠛⠻⠿⣶⣦⣀
⣿⠉⠉⠉⠉⠉⠉⣿⠀⡤⡄⠀⣤⠀⣿⠉⠉⠉⠉⠉⠉⣿⠀⣿⠀⠾⡥⠚⠺⠂⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⢋⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⣿⠛⠛⠛⠛⠛⠛⣿⢀⣔⡃⠾⢿⠄⣿⠛⠛⠛⠛⠛⠛⣿⠀⣿⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠋⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⣿⣤⣤⣤⣤⣤⣴⣿⣀⣀⣀⣀⣀⣀⣿⣤⣤⣤⣤⣤⣴⡿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⠶⣦⠀⢀⣴⠄⠀⣦⠀⠀⠀⠘⣿
⢸⡇⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠁⣠⡿⢠⡾⢳⡄⠀⣷⠶⣦⠀⠀⣿
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⢠⣾⣋⣀⠛⠓⢿⡟⠀⡇⠀⣿⠀⢀⣿
⢸⡇⠀⠀⠀⠀⠀⣾⠛⠛⢻⡟⠛⠛⣷⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠈⠉⠉⠉⠀⠀⠈⠀⠀⠁⠀⠉⢀⣼⡟
⢸⡇⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣿⠋
⢸⣇⣀⣀⣀⣀⣀⣿⣀⣀⣸⣇⣀⣠⣿⣀⣀⣀⣀⣀⣿⡇⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣤⣤⣤⣤⣴⣾⠿⠋⠁
⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢋⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠊⢱⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⡉⣆⡀
⠀⠀⠀⠀⠀⠀⠀⡠⣤⠤⣴⣷⣏⣴⢗⡄⠀⠀⠀⣀⠤⠤⠤⠇⠀⠀⠐⢢⠀⠀⠀⢀⣠⠤⢤⢼⠑⠀⠨⢳⡄
⠀⠀⠀⠀⠀⣴⠋⡔⢁⠞⢀⠼⡛⠓⠉⠀⠀⢠⠊⠀⠀⠀⠀⠀⡗⠒⠂⠁⠀⢀⡔⠫⢐⠠⢀⠀⢸⠒⠛⠈
⠀⠀⠀⠀⣸⠃⡜⡰⠃⡠⢋⡼⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⢀⠇⠀⠀⠀⠀⣜⠀⠈⠐⠨⢐⠀⡾
⠀⠠⢔⣶⢿⣜⣰⠁⣔⣡⠟⠁⠀⢀⠄⠒⠚⠂⡀⠀⠀⣀⡴⠋⠀⠀⣠⠰⢖⠳⡨⢀⢐⢨⡰⠟
⠀⠙⢵⠁⡜⡿⠉⠉⠉⠀⠀⠀⠀⠈⠢⠀⠀⢀⠏⠉⠉⠀⠀⠀⠀⠀⠓⣌⠐⡐⡎⠉⠉⠁
⠀⠀⠈⠙⠼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠧⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠉⠉⠉⠉⠉⠉⢱⠀⠀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠘⠈⠀⠀⠠⠄⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⢀⣠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠁⠁⠀⠀⠀⠔⠈⣤⠆⠀⠁⠠⢄⠀⡤⠠⠤⠠⡄⢄⡀⡀⡀⠀⣀⠀⠀⠀⠀⠇⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠠⠊⠀⠐⠀⠉⠉⠉⠁⠀⠈⡁⠇⠀⠀⠀⠀⢸⡀⠀⠀⠀⣸⠀⠀⠠⠀⡼⠂⠚⢧⠀⢀⡀⣀⣀⣀⣀⣀⣀⡸
⠀⠀⠀⠀⠀⢀⠠⠂⠀⠀⠀⠄⠐⠀⠀⠒⠀⠆⠒⠒⠒⠒⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⢴⠀⠀⠀⠀⣀
⠀⠀⢰⠒⠒⠲⠶⠶⠶⠶⠖⠂⠀⠀⠀⠤⠤⠀⠤⠤⡄⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠇⠀⠀⢸⠀⠀⡠⠋⠉⢳⡄⠀⠀⠀⠀⠀⢠
⠀⠀⢸⠀⣠⠒⠉⠉⠑⢦⡀⢠⠊⠉⠑⡀⠎⠉⠙⡆⡇⠀⠀⠤⠀⠀⠀⠄⠀⢸⠀⠀⠀⠀⠀⠀⢸⠀⠀⢅⠀⢀⠞⠃⠀⢀⠀⠀⠀⠀⡀
⠀⠀⢸⢠⠁⠀⠀⠀⠀⠀⢣⠈⠢⡤⠔⠁⠳⢤⠴⠃⡇⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⣠⠖⠲⣆⠁
⠀⠀⢸⠸⡀⠀⠀⠀⠀⠀⢸⢠⠊⠉⠱⣀⠎⠉⠑⡄⡇⠀⠀⠀⠀⠀⠀⡏⠀⠙⡀⠀⠀⠈⠀⠀⠈⠀⠀⡀⠀⢄⠀⠀⠀⠀⠧⡀⠀⠀⠅
⠀⠀⢸⠀⠱⣄⠀⠀⠀⡠⠃⠘⠦⠤⠔⠀⠳⠤⠔⠁⡇⠀⠀⠀⠀⠀⠀⠏⠉⠉⠁⠀⠀⠊⠉⠉⠁⠀⢠⠁⠀⠘⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠸⠤⠤⠤⠭⠭⠭⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠃⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡆⠀⢰⠀⢰⣷
⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⡴⡄⠀⠀⠀⠀⡄⢠⠀⠀⠀⠀⢀⠄⠂⠒⠒⠒⠀⠤⡀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡇⠀⢸⠀⠈⠛
⠀⠀⠀⢠⠀⠀⠈⡀⠀⠀⠀⡇⢹⠀⠀⠀⠀⡇⠸⠀⠀⠀⠀⠐⠄⢀⣀⠀⠀⣀⡀⠜⠀⠀⠑⠐⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸
⠀⠀⠀⠘⠀⠀⠀⠁⠀⠀⠺⣠⠏⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣴⡪⠩⡋⢩⡗⣤⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡇⠤⢼⠀⡀⠤⢀
⠀⠀⠀⠤⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⡀⠇⠠⠄⡀⠀⠀⢸⢢⡆⠈⠑⡕⢠⢊⡇⠀⢠⠔⠛⠚⠒⢄⠀⠀⠀⠀⠀⡇⠀⠐⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⢁⠀⠀⠸⡮⢒⠈⢑⢁⢗⣴⡇⠀⡇⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⡇⠀⢠⠀⡇⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠑⠄⠀⠄⠂⠀⠀⡠⡈⠳⠭⠴⠯⠾⠊⠀⠀⠘⢦⣀⣀⣠⠜⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠥⠀⡀⠀⠐⠀⠀⠂⠀⠀⢠⠊⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⡇
⠀⠀⠀⠸⠀⠀⠀⠀⠀⢰⠀⠀⠀⠃⠀⢀⢀⡀⢐⠀⠀⡇⠀⠀⠀⠀⢸⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠈⠉⡢⠀⠀⡇⠀⠈⠀⡇
⠀⠀⠀⠀⠆⠀⠀⠆⠀⠀⠀⠀⠠⠀⠀⢨⠀⠀⢨⠀⠀⢃⠀⠀⠀⢀⠜⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠐⠂⠀⠈⠁⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠠⠀⠰⠀⠀⠀⢂⠀⡀⠀⠀⠠⠀⠀⠄⠀⠀⠈⠢⢀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠔⠀⠣⠀⠴

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠁⠉⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⢨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠿⠿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣼⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠿⠛⠛⠋⠉⠙⠛⠻⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣨⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠿⠋⠀⠀⠀⠀⠀⠀⠀⠰⠿⠛⠛⠋⠉⠀⠀⠙⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⢷⣄
⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⢿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⣴⣿⡿⠛⠛⠛⠿⠿⢿⣶⣦⣄⣀⠀⢀⣴⣿⠏⠀⠀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠟⠋⠁⢀⣾⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁
⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⢿⣿⣿⡏⢀⣠⡾⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⣹⣿⡿⠶⢶⣶⣶⣶⣶⣶⠾⠿⠛⠁
⠀⠀⠀⠘⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣴⡿⠋⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⣰⣿⠋
⠀⠀⠀⠀⢸⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣀⣠⣴⡶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠸⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣋⣁⣀⣀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛
⠀⠀⠀⠀⠀⠹⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣟⠛⠋⠉⠁⠀⠀⢀⣀⣠⣤⣶⡿⠟⠋
⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠿⠿⢿⣿⣿⠿⠿⠿⠟⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠘⢿⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⣀⣤⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣦⣤⣤⣶⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡤⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣧⣀⠀⠀⠀⣀⣷
⠀⠀⠀⠀⢀⣀⡤⠤⠶⢖⣚⣿⡿⠿⣿⠷⠶⠲⢺⠒⡳⢖⠿⣿⡿⠛⠓⠶⠶⠤⣄⣀
⠀⠀⢀⣼⠭⠒⠀⠐⠉⠐⠈⠀⠂⠀⠀⠀⠔⠀⠸⠀⠘⣆⠁⠢⡈⠓⠬⣐⠢⢌⣐⠂⠍⢙⣶⡀
⠀⣴⠘⡟⠛⠷⢦⣤⣄⣀⣀⡀⠈⠀⠠⠂⠀⠀⢰⠀⠀⠈⢆⠀⠈⠢⣄⠀⣑⣢⣠⣥⡶⠾⣿⠇⣧
⠀⢿⠙⠒⠦⢤⣄⣀⡈⠉⠉⠛⠛⠛⠻⠶⠶⠶⠾⠶⠶⠶⠾⠷⠖⠛⠛⠛⠉⠉⣁⣀⣤⡽⠶⠞⣛⣦
⠀⠈⠛⣶⣤⣤⣀⡈⠉⠉⠓⠒⠲⠶⠦⠤⠤⠤⠤⠤⠤⠤⠤⠤⠠⠶⠒⠚⠛⠋⠉⣁⣠⣤⡴⠟⠟⠃
⠀⠀⠀⢹⠀⠀⠁⠉⠙⠛⠒⣾⣭⣿⣦⣤⣤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠶⠒⠛⠋⠋⢿⢿⡇
⠀⠀⠀⢸⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣾
⠀⠀⠀⠀⡏⢸⠃⣠⠤⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢠⠤⡔⢸⡏⣿
⠀⠀⠀⠀⡇⢸⠀⣿⠀⢹⡇⠈⢰⡏⠁⣀⡞⠋⣄⡰⠋⢡⣴⠋⠹⣼⠃⢸⡏⢸⡇⢸⢰⡇
⠀⠀⠀⠀⢓⢸⡇⢸⠀⢸⡇⠀⢿⠇⠀⣿⠀⠀⣻⡇⠀⠈⣿⠀⠀⣿⠀⢸⡇⢠⡇⢸⢨⠇
⠀⠀⠀⠀⢸⠘⡇⢸⡇⢸⣇⠀⣾⡆⠀⡿⠀⠀⢹⡇⠀⢠⣿⠀⠀⣿⠀⢸⡇⢸⠀⣿⢸
⠀⠀⠀⠀⠸⠀⡇⢸⡇⢸⡏⠀⣸⠇⠀⣿⡄⠀⢸⡇⠀⢸⣿⠀⠀⡟⠀⣾⠃⢸⠀⡏⠈
⠀⠀⠀⠀⠀⡇⣧⢸⡇⠸⡇⠀⢹⡆⠀⣿⡇⠀⢺⣇⠀⢰⣿⠀⢰⠃⠀⣼⠀⣿⠀⡇⡴
⠀⠀⠀⠀⠀⡇⢻⠈⣧⢠⡇⠀⢸⡇⠀⢹⡇⠀⣸⡇⠀⠸⣿⠀⢸⡇⠀⣿⠀⣿⢀⢷⡇
⠀⠀⠀⠀⠀⠁⢸⠀⡇⠈⣿⠀⠸⡅⠀⣸⠁⠀⣸⡇⠀⢀⡏⠀⢸⡇⠀⡿⠀⡟⢸⢸⡇
⠀⠀⠀⠀⠀⢸⢸⡇⣿⠀⣿⠀⢸⡇⠀⣹⡄⠀⣸⡇⠀⣼⡇⠀⢸⡃⢰⡇⢠⡇⢸⢸⠁
⠀⠀⠀⠀⠀⢸⡘⡇⣿⠀⣿⠀⢠⡇⠀⢿⡇⠀⢨⡇⠀⢿⡇⠀⣾⡁⢸⡇⢸⡇⡿⣸
⠀⠀⠀⠀⠀⢰⡆⡇⢹⠀⣿⠄⢠⡇⠀⢸⡇⠀⢼⡇⠀⣾⡇⠀⣿⠁⢸⡇⢸⠁⡇⡟
⠀⠀⠀⠀⠀⠈⡇⡷⢸⡀⣿⡄⢸⣧⠀⢸⡇⠀⢰⣷⠀⣿⡇⠀⣿⠀⢸⠃⢾⢠⡇⡇
⠀⠀⠀⠀⠀⠀⡇⣿⢸⡅⢹⡇⠸⣧⠀⢸⡇⠀⢹⠇⠀⢹⡇⠀⣿⠀⣿⠀⣼⢸⢷⡇
⠀⠀⠀⠀⠀⠀⡇⢹⢸⡇⢸⡇⠠⣿⠀⢸⡇⠀⢻⡇⠀⣿⡇⢰⡟⠀⣿⠀⡿⢸⢸⡇
⠀⠀⠀⠀⠀⠀⣿⢸⡀⡇⢸⡇⠀⣿⠀⢸⡇⠀⢻⡇⠀⣿⠇⢸⣧⠀⣿⠀⡇⣼⣼⠁
⠀⠀⠀⠀⠀⠀⠘⣾⡇⡇⢸⡇⠀⣿⠀⢸⡇⠀⢸⡇⠀⣿⠀⢸⡇⠀⡟⠐⣇⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠈⠢⣉⠘⠧⠀⣿⠀⢸⡇⠀⢺⡇⠀⣿⡀⢸⡧⠸⠟⣈⡵⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠀⠀⢈⣁⠀⠉⠁⢀⡉⠀⠠⠄⠒⠉⠁

⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⢸⣛⠍⠀⠀⠀⠀⠀⠀⠠⠀⣴⣶⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⢀
⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⢠⣦⣤⠀⢻⣿⣇⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠼⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡄⣰⠄⣬
⠀⠀⣀⡠⠤⠤⠤⢄⡀⠀⠀⠀⣼⣿⡇⠀⠸⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⡀⠀⠀⠀⠀⠀⠀⠀⣮⣄⡀⢬⣽
⠀⠸⠧⢄⣛⣛⣋⣀⠼⠀⠀⠐⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⢭⠙⠋⠳⠄⠀⠀⠈⠢⡢⡀⠀⠀⠀⠀⢩⣿⣷⣿⡏
⠀⠐⡒⠢⠤⠤⠤⠀⡺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠶⠞⠛⠒⣄⠀⠀⠀⠀⠀⠈⠪⡢⠀⠀⠀⢹⣿⣾⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⠀⠀⠀⠀⢀⣄⣠⠀⠀⠀⠀⠀⣇⢰⠒⠒⢒⡆⡏⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠸⠜⠉⡹
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡸⠀⠀⠀⠀⠀⠘⡄⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠡
⠀⠀⠀⠀⠀⠫⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⣀⣀⠌⠀⠀⠀⠀⠠⣦⡼⢹⡀
⠀⣀⢤⠀⠀⠀⠀⠀⣠⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠺⠃
⠈⠀⠀⡀⠀⠀⠀⠸⣿⣿⣿⠀⢀⠚⠉⠉⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤
⠀⠀⠀⠐⡀⠀⠀⠀⠈⠉⠛⢤⣘⣆⡀⢀⣜⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠆⠈⠁⡁⠀⠀⠀⠀⠀⢀⡴⣿⣿⣿⣆
⢂⠡⢄⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠀⣁⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⠏
⠀⠳⠀⠄⢈⡆⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢀⠛⠉⠉
⠀⠀⠙⢄⡣⠆⠀⠀⠀⠀⠀⢀⣄⣀⣀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠀⢀⠂⠀⠀⠀⠀⠀⠀⢰⢦⣠⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠂⠀⠐⠊⠀⠀⢀⡀⠀⠀⠀⠀⠈⠳⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡕⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠑⠒⠀⠁
⠀⢈⠀⠀⠀⠀⠐⡠⠂⠀⠀⠀⠀⠀⠀⡠⠊⠄⢄⠀⠃⠠⡐⢀⠀⠀⠀⠀⠀⠠⠟⠒⠛⠒⡦⡄⠁
⠀⠀⠁⠀⠀⣠⡀⠀⠄⠀⠀⠀⠀⠀⡔⠁⠐⠀⠀⠁⠀⢀⠠⠚⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣭⣽⣷⣤⡀
⠀⠀⠀⠠⠊⠀⠙⠃⠀⠀⠀⠀⠀⠊⠀⠁⣀⠘⢀⠤⠋⠁⠀⠀⠀⢰⢄⠀⠀⠀⢸⣿⡿⠿⠿⠿⢿⣿⡿⠀⠀⠀⠀⢀⣀
⠀⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡠⠕⢠⠴⠊⠁⠀⠀⠀⠀⠀⠘⠪⠬⠀⠀⠀⠸⣿⣷⣶⣶⣶⣾⣿⠇⠀⠀⠀⠹⡉⡸
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠉⠁
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⢠⡄⠀⠀⠄
⠀⡘⠀⠀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⢀⠂⢐⠀⠘⠗⠒⠛⠓⠒⠓⠂
⠀⠀⣤⣤⠄⠀⠀⠀⠀⠀⠀⠀⢠⠒⠓⠈⠖⠁⠜⠃⠒⠀⠆⠉⣀⡈⠠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⣲⡖⣄⡀⢠⠁
⠀⡆⢾⢿⠆⠀⠀⣠⢮⢣⡀⠀⠀⣸⡇⠠⢰⢻⠀⠀⠐⠀⠀⣕⣩⣊⢀⡀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠠⠶⠀⠀⡞⠚
⠀⠀⢿⣿⣿⣇⠀⣿⣿⠗⠁⢐⣀⠈⠁⠠⠾⠂⠈⠁⠄⠀⠀⠦⠶⣴⠞⠀⠔⢆⡀⠀⠀⠀⠀⠀⠀⠀⢌⡰⠱⡀⣺⠓⠒
⠀⢀⣻⣛⣛⣗⢀⠈⠉⠛⠓⠀⠀⢈⢁⡛⠒⠲⣴⣆⠒⢲⠀⠀⣴⠁⠀⣶⠒⢎⡄⠁⠀⠀⠀⠀⠀⠀⠨⠧⠴⠃⠛⡖⠶
⠀⠬⠤⠄⠀⠀⠀⠂⠀⠀⠀⡚⠩⠭⠽⠃⡅⠀⠀⠀⠈⠉⠀⠀⠉⡚⣞⣿⣧⢞⣃⠈⠀⠀⠀⠒⠀⠀⠀⢠⠁⡲⢶⢃⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣆⠀⠛⠠⠌⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠀⠄⢉⣁⡀⠀⠀⠁⠀⠀⠀⠀⠀⠢⠤⠂⠾⠠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠢⠤⢤⣀⣀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠊⠉⠑⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣄⣀⣀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡄⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣸⡿⢧⡀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣷⣦⠀⠀⠀⠀⣿⡇⠀⣇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⡏⠀⠀⣀⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢿⣇⠀⠀⠀⢸⣷⠀⢸⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⡇⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠀⠀⠈⣿⡀⠀⠀⠸⣿⡀⠈⡇⠀⠀⠀⠀⣿⣿⣟⠛⠿⠿⣿⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⢻⣧⠀⠀⠀⣿⣇⣀⣷⣤⡀⠀⢀⣿⣿⣿⠀⠀⠀⡏⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⡠⢼⣿⡆⠀⣿⣿⣿⠀⢸⣿⣇⠀⢸⣿⣿⡿⠀⠀⢸⠃⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣼⠟⢳⠀⢻⣿⣿⡏⠊⣻⣿⠀⢸⣿⣿⠇⠀⠀⡟⠀⢠⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣷⡀⣇⠸⣿⣿⣯⠁⠘⣿⡇⣸⣿⣿⠑⠒⢺⠁⢠⠏⣠⣿⣿⠃⠈⠙⢿⣿⣿⣿⣿⣿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢹⣧⣿⣿⣏⠀⢹⡏⢠⠏⣴⣿⣿⠏⠀⠀⠀⠀⡽⠛⠉⠁
⠀⠀⠀⠀⠀⢀⣤⣾⡿⠿⠿⠟⠁⠀⠀⠈⠉⠛⢿⣿⣿⣧⠀⢸⣿⣿⣿⡷⠍⣽⣧⢏⣼⣿⣿⣯⣄⡀⠀⢀⡞
⠀⠀⠀⣠⣾⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣷⠀⣿⣿⣿⡝⠒⣿⢏⣾⣿⣿⣿⣋⢀⣼⣷⠎⠀⠀⢐⡀⠀⠀⡂
⠀⢀⣼⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⣿⣿⠀⣿⣿⠏⠀⣸⣧⣿⣿⣿⣿⢟⣵⣿⣿⣿⣶⣶⣦⠸⣿⣄⣼⡇
⢀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡇⠀⢿⣿⡀⣠⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣿⡇⣿⣿⠀⡿⠛⠻⡇
⣼⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣇⠀⢸⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⢠⣿⣧⣿⣿⣄⣇⣀⣀⣇⠀⢀
⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣟⣻⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⢸⣿⣿⣿⣿⡏⠈⣿⡇⠈⢳
⢿⣿⣿⣆⠀⠀⠀⠀⠈⠀⠀⠀⣀⣴⣾⣿⣿⣿⡿⠿⠿⢿⣿⣦⣸⣿⣿⣿⣿⣿⠟⠛⣻⡟⠓⢂⠀⢸⣿⣿⣿⣿⡧⢸⣿⣧⠀⣾
⠘⣿⣿⣿⣷⣶⣤⣤⣤⣶⣾⢟⣫⣵⣿⠿⢿⣁⠀⢀⣠⠶⠛⠻⣿⣿⣿⣿⣿⣿⡗⠦⢾⣶⣾⣿⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠠⣿
⠀⠈⠿⣿⣿⣿⣿⣿⣿⣿⠿⢿⣿⡉⠀⢀⣠⡽⢿⣯⡁⢀⣠⠴⠚⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⢺
⠀⠀⠀⠀⠉⣵⣾⣿⠋⠀⠀⠀⣈⡿⣾⣋⠁⢀⣠⡼⠟⠻⣇⡀⣀⣤⣴⣿⣿⣿⡏⢱⣾⣳⣾⣿⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⣹
⠀⠀⠀⠀⠀⣿⣿⣿⣷⣶⣾⠋⠁⠀⣀⣽⢿⣟⡁⠀⢀⣤⣼⣿⣿⣿⣿⣿⠿⠛⡇⠀⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⡇⢸⣿⣿⠀⣿
⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣷⣶⡛⠉⠀⠀⣉⣿⣿⣿⣿⣿⣿⠿⠛⠉⠀⠀⠀⣿⡀⣿⣿⣿⡿⠀⠈⠛⠛⠛⠛⠋⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣷⣶⣿⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⡿⠟⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠤⠤⢀⣀⠀⣴⠋⠁⠈⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢶⡿⡛⠛⢲⣴⢿⠿⠷⣶⡀⠤⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢾⡟⠻⣅⣲⡶⠿⢤⣀⢨⣿⣷⠂⠠⠄⢸⣷⠀⠂⠀⠉⠑⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⡖⠠⠤⢤⣤⣶⢾⣿⡤⠤⠄⣿⡃⠘⠀⠀⢻⡍⢩⡉⠒⠠⠄⣠⡟⠉⠁⠒⠠⡞⡍⠉⠉⡇
⠀⠀⠀⠀⢀⡤⠐⠋⠀⢠⡿⠁⠠⡌⠙⣷⠀⠤⠿⣇⠉⠁⠐⣺⠧⠄⢀⡀⣴⠒⠋⢹⠀⠀⠀⠀⣇⢱⠀⠀⢹
⠀⠀⠀⠀⡇⠀⠀⠈⠉⢹⡗⠠⠄⠀⣀⣿⠀⠘⠀⠀⠀⢉⣐⣿⡄⠀⠀⠀⣿⡀⠀⢸⡇⠀⠀⠀⢹⠀⣇⠀⠈⣇
⠀⠀⠀⢰⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⢠⡏⠉⠐⠒⠠⢿⡄⠀⠤⣧⠀⠀⠀⢿⣇⠀⠀⣇⠀⠀⠀⠈⡇⠘⡄⠀⢸⡀
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣇⠀⠀⢻⠀⠀⠀⢸⢸⠀⠀⢸⠀⠀⠀⠀⢹⠀⢡⠀⠀⢧
⠀⠀⠀⡌⠀⠀⠀⠀⣩⣿⢿⣿⣿⡦⠀⠀⠀⠀⠀⠀⢸⢹⠀⠀⢸⡀⠀⠀⢸⡅⡇⠀⠘⡇⠀⠀⠀⠈⡇⠀⢆⠀⠸⡆
⠀⠀⠀⡇⠀⠀⠀⠀⢱⣿⣇⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡀⠀⠘⡇⠀⠀⠘⡇⢱⠀⠀⢻⠀⠀⠀⠀⢹⠀⠘⡀⠀⢻⡀
⠀⠀⠀⡇⠀⠀⠀⠀⢸⣿⣿⣆⠀⠰⠀⠀⠀⠀⠀⠀⣟⠀⡇⠀⠀⡇⠀⠀⠀⡇⠘⡄⠀⠈⣇⠀⠀⠀⠈⡇⢠⠛⣻⣶⠗
⠀⠀⠀⡇⠀⠀⠀⠀⢸⡇⠈⢿⣷⣄⣀⠀⠀⠀⠀⠀⡇⠀⡇⠀⠀⢻⠀⠀⠀⣷⠀⣣⣀⣀⢺⡉⠉⠒⠒⠺⠗⠋⠁
⠀⠀⠀⣇⣀⠀⠀⠀⠈⠃⠀⠸⣿⣿⣿⡷⠀⠀⠀⠀⡇⠀⢱⠀⠀⠸⣆⣀⠀⢹⡰⢋⡭⠽⠛⠁
⠀⠀⠀⠀⠉⠉⠛⠒⠦⠤⣄⣀⠀⠉⠉⠁⠀⠀⠀⠀⡇⢀⡜⠦⠤⣀⣧⠈⠙⠚⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠤⠤⣀⡀⠀⣧⢊⣠⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⠒⠒⠒⠒⠒⠒⠒⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣿⢠⠴⢦⠀⠀⣴⠀⢸⣇⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⢀⡜⢠⣞⣸⣀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡧⠤⠤⠤⠤⠤⠤⠤⠤⣿⠰⠿⠶⠆⠀⠸⠀⢸⡧⠤⠤⠤⠤⠤⠤⠤⠤⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠧⣤⠤⠤⠤⠤⠤⠤⠤⠿⠤⠤⠤⠤⠤⠤⠤⠼⠧⠤⠤⠤⠤⠤⠤⢤⡤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⣀⣀⣀⣀⣀⣀⠀⢠⠤⠤⢤⠤⠤⢤⠤⠤⢤⡤⠤⠤⡄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡏⠀⠀⣿⠈⠀⢹⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⣿⠀⠀⢸⠀⢸⠀⠀⢸⠀⠀⢸⠀⠀⢸⡇⠀⠀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⣿⠀⠀⢸⠀⢸⣀⣀⣸⣀⣀⣸⣀⣀⣸⣇⣀⣀⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣀⣇⣀⣀⣿⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣸⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⣧⠀⠀⢀⣤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠆⠀⠉⠑⠲⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⣠⠆⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠒⠀⠐⢀⡋⠋⠀⠀⠀⠀⠀⡎⠀⠀⡶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⣀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⣗⠲⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⡴⠁⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⣿⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⣷⠁⠀⠀⠀⠀⡀⣀⣀⡤⢀⠀⠀⠿⡄⠀⢠⠀⠀⠀⠀⠀⠄⠀⢰⠘⠀⠿⠅⣤⠀⣀⣀
⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠈⡇⣿⠀⠀⠈⠀⠀⠀⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠛⠀⣧⡟
⢺⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡆⠀⠀⠀⠇⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠃
⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⠖⠁⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡠⠁⠀⠀⠀⠀⠀⠇⢨⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⠜⠀⠀⠀⠀⠀⠀⠀⠄⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡶⠒⠂⢰⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⡆⠀⣀
⠀⠀⠀⠀⢆⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠈⢆⠀⠀⠀⢀⣀⡀⠤⠤⠛⠁⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢀⣸⣿⠿⠋⠁
⠀⠀⠀⠀⢸⣀⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠉⠁
⠀⠀⠀⠀⠈⠋⠁⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡝⠛⠿⣿⠿⠿⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡄⠙⠒⡄⠀⠀⠛⠀⠠⠂⣀⠀⢀⡀
⣀⡀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢼⣅⠀⠀⠀⠀⠀⠀⣇⠀⠀⠱⠀⠀⠀⡄⠀⠀⠀⠀⠀⣉⣴⠚⠛⠛
⠀⡥⠂⠀⠀⠀⠀⠘⠀⠃⠀⠀⠀⠀⠃⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠘⢈⠀⠀⡏⠀⠀⠀⠐⣿⣿⡟
⠀⡄⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⣿⣿⣇
⠀⠇⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣣⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢤⡇⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣾⡄
⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣯⡀⠀⠀⢠⣄⡄
⠀⣺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠏⠀⠀⣄⡾⠋⠁
⠘⣓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠄⠤⠤⠤⠤⣤⣤⣀⣀⣀⣀⡤⠤⠶⠶⠶⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⣿⣶⣤⣄⣀⠀⠀⠀⠉⠩⠿⣤⣿⠟⠛⢿⣏⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠿⠍⡀⠈⠂⠐⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠈⠉⠉⠉⠐⠂⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⢤⡴⠒⠻⠷⠶⢮⣥⣴⣄⣠⣄⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤⢤⣠⣉⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣟⠻⠿⣿⡿⢷⣶⠿⢿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣤⣴⣦⣤⣤⣬⣤⣾⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠿⣷⣿⣷⣦⣤⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣈⣤⣼⣿⣿⣿⣷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠋⠻⠯⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠉⠽⠃⠚⢻⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣷⢰⠀⡄⢲⣦⡀⠀⢂⠄⠠⣄⠰⠀⠠⠀⠤⡄⢀⠆⠠⠐⢦⡑⠀⠀⠛⣿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡛⠓⠿⠿⣀⣀⣏⣨⠿⠓⣒⣏⠀⠀⢸⡆⢑⡇⠀⠀⢹⠂⢸⡶⠀⠈⡇⠀⠀⠀⣿⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢦⣤⣄⣀⣀⣐⣼⡖⠉⣾⠀⠀⢸⡇⠈⡇⠀⠀⢸⠀⠀⠒⠀⠀⡇⠀⠀⠀⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠸⡏⠀⣿⡇⠀⣿⠀⠀⢸⡇⠀⡅⠀⠀⢸⠀⢀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟⠀⡇⠀⠿⡇⠀⢸⠀⠀⢸⡇⠀⡂⠀⠀⢸⠀⢸⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠇⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⡇⠀⡇⠀⠀⣸⠀⠘⠀⠀⢀⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡏⡇⠀⡇⠀⠀⡇⠀⢸⠀⠀⢸⡇⠀⡇⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⣧⢠⡇⠀⠀⡇⠀⢸⠀⠀⢸⡇⠀⠁⠀⠀⢿⠀⢀⠀⠀⢸⠃⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠸⡇⠀⠀⢶⠀⢸⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⢸⠀⠀⢸⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢹⠀⠇⠀⠀⢸⠀⠸⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⢈⠀⠀⢸⠀⢰⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⠀⠮⠀⠀⢸⠀⠠⠀⠀⢸⡇⠀⠀⠀⠀⢸⠀⠘⠀⠀⢸⠀⢈⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡃⠀⣆⠀⢰⠿⠀⠀⠂⠀⢸⡇⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⢸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠬⠀⠈⢸⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⣿⠀⡀⠀⠀⢸⠀⠸⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡄⠀⠐⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣿⠀⣷⠀⠀⣼⠀⠀⠀⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡋⠀⠀⠆⢣⡟⠠⠀⡄⠀⢸⡏⠀⠀⠀⠀⡿⠀⠹⠀⡀⠇⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⣒⣢⠤⠤⣈⣀⡀⠈⠀⠈⠀⠀⠀⠀⠀⠁⣁⣀⣀⣀⡤⠤⣖⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⢹⣿⣿⡷⢶⠶⠿⠬⠭⠭⢭⣭⣽⣯⣭⣭⠭⠷⠶⡾⢿⣿⣿⡅⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠿⣿⣇⣻⡀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣃⣼⣿⠿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠒⠲⠶⠶⠶⠶⠖⠒⠚⠛⠋⠉⠁


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⠹⣿⣿⣷⣶⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠉⠉⠉⠻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡘⣿⣿⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⣿⠿⠿⠛⠛⠛⠛⠿⢿⣿⣿⣿⣧⣤⣤⣤⣤⣤⣤⣴⣾⣿⡿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⠟⠋⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⡿⠿⠿⠿⠿⠿⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠋⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀⠀⠀⠀⠀⠀⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢢⡀⠀⠀⠀⠀⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠀⠀⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⠋⠉⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠑⠦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣤⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⣾⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣆⣤⣶⣿⡿⠟⠁
⠀⣴⣾⣿⡿⠿⠛⠛⠛⠛⠛⠻⢿⣿⣿⣦⣤⣄⣀⣀⣀⣤⣤⣶⣿⣿⡿⠟⠉
⠀⠛⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠛⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁
⠀⠀⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠘⣿⣿⣄⣀⣀⠀⠀⠀⢀⣿⣿
⠀⠀⠀⠀⠘⠻⠿⣿⣿⣿⣷⡄⣼⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛

⠀⠀⡀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠀⡀⢠⠀⠀⢸⠀⠀⢀⣀⡔⠀⠀⠀⢀⠀⠄⠐⠀⠀⠀⠀⠀⠀⡔⢦⡄
⠀⠀⠀⠀⠀⠀⠐⠀⠐⣇⠀⠀⠀⠀⢤⠈⠀⠀⡜⠀⠀⠈⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠔⠉⠀⡀
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠈⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢆⠘⠀⠀⢠⠄⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠑⠀⠀⠀⠀⠀⠒⠀⠈⠁
⠀⠀⢀⣰⡆⠀⠀⡆⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠂
⠀⠀⠈⠢⡠⠀⠀⢃⠀⠀⠠⠞⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠐⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⣤⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢠
⢄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢃⣠⡀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠠⠀⠀⠰⠏
⠈⠢⠀⠀⢀⠀⠀⠀⢠⡾⠁⣡⡿⢔⡶⠀⠀⢀⡀⠀⠀⣼⣣⡤⠀⣠⣤⣠⡄⢠⠄⣼⢡⣤
⠀⠀⠀⠀⠈⠂⠀⠠⠟⠀⣴⠋⣠⠟⣀⠴⣾⣯⡖⡠⢾⣿⡋⠤⢾⠫⠺⡿⢾⡯⣸⠟⣵⠏⠀⠀⣴⣄
⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⠊⠁⠀⠀⠀⠉⠀⠘⠀⠑⠶⠀⠀⠀⠀⠀⠀⠏⠈⠁⠀⠀⠀⠙⠻⠃
⠀⠀⠀⠲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⢀⠀⣦⡰⡀
⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠖⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⢠⣁⣀⡤⠇
⠤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⠀⠀⣀⠀⠀⠀⡴⢳⠖⠀⠀⠀⠀⠈
⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⡀⠀⠀⠂⠀⠀⠀⠀⠀⠸⠀⠐⠃⠀⠀⠈⠀⠀⣠⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣀⣀⣐⠀⠀⠀⠀⠀⠷⣤⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⢠⠀⠀⡀⠀⣀⠀⠀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠀⠀⠁⠀⠁⠀⠀⠀⠁⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠷⣶⣦⣤⣄⣀⡀⠀⠀⠀⣀⣴⣶⠾⠷⣶⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣯⣀⣀⠀⠉⠉⠛⠻⠿⢶⣾⣿⣥⣀⡀⠀⠀⠈⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢶⣶⣤⣄⣀⡀⠈⠉⠛⠛⠿⠷⣶⣦⣿⣧⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠿⠷⣶⣦⣤⣄⣀⠀⠉⠙⠛⠻⠿⢶⣶⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠻⠿⢶⣶⣤⣄⣀⡀⠈⠉⠙⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭⣭⣿⣿⣿⣷⣶⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⣀⡀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⠇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢿⡇⠀⠀⣿⡇⠀⠀⣿⡇⠀⠀⣿⠇⠀⢠⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⢸⣷⠀⠀⣿⡇⠀⠀⣿⡇⠀⢠⣿⠀⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⣿⠀⠀⢸⡟⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠘⣿⠀⠀⢸⡇⠀⠀⣿⡇⠀⢸⣿⠀⠀⣾⡇⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⣿⡄⠀⢸⣿⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡿⠀⠀⣿⠃⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⢻⡇⠀⢸⣿⠀⠀⣿⡇⠀⢸⡇⠀⢰⣿⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⢸⣷⠀⢸⣿⠀⠀⣿⡇⠀⣿⡇⠀⢸⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠘⣿⠀⠀⣿⡇⠀⣿⡇⠀⢸⡇⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠈⡿⠀⠀⢿⠀⠀⢿⠇⠀⢿⠇⠀⢿⠇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⡏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠶⠶⠶⠶⠶⠶⠶⠶⠄⣿⣿⣿⣿⣿⣿⣿⡇⠴⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣤⣤⣤⣤⣤⣤⣤⣤⡄⣿⣿⣿⣿⣿⣿⣿⡇⣤⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣶⣶⣶⠶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⣻⣿⢛⣿⡟⣻⣿⢛⣿⡟⣿⣿⣿⣟⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⢸⣿⡇⣿⣿⢸⣿⡇⣿⣿⣿⣿⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⣻⣿⣘⣿⣇⣻⣿⣘⣿⡃⣿⣾⣷⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣿⣿⣿⣿⣿⣿⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁