Slurp Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣤⡴⣦⣄⢀⡀⠀⠀⠤⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀
⠀⠔⢀⡀⡔⠄⠑⠀⣸⠀⣿⡿⢧⣜⣿⠀⠀⠀⡔⢰⡀⣀⠀⠀⠀⠀⠐⠀⠐⠀⡀⠀⡈⠀⠀⠂⠐⠀⢨
⠘⠀⠀⠀⠀⠁⠄⠐⠀⠀⠻⣤⣾⡛⠁⠀⠀⠁⠁⠀⠈⠀⠀⠀⣂⣀⣁⣀⣁⣁⠀⠀⠉⠥⠷⠣⠧⠐
⠀⠀⠀⡀⠀⠐⢀⡁⢀⠁⢀⠀⠉⠉⡁⠀⠀⠀⠨⢐⠠⠀⣰⣿⡿⣿⣿⣿⣿⢿⣿⡆⠉⠀⣐
⠀⢰⠁⠄⠃⠑⠀⢀⠀⠀⠁⠐⢁⠀⡀⣀⣈⠀⠀⠰⠁⠀⣿⣿⣇⣻⣿⣿⣟⣼⣿⠇⠀⠀⠀⠄⠀⠄⡀
⠀⠈⡁⠀⠀⠀⠁⠐⠂⠀⠐⢰⡏⣷⣷⢾⢹⠀⠢⠀⠐⠚⠛⠛⠛⠛⠛⠛⠛⠋⠁⠂⡀⣤⠲⠂⢀⠄
⠀⠀⡇⠀⠤⠎⠀⠈⠒⠒⠀⢤⠉⠁⠈⠈⠀⠐⠀⠀⠀⠀⠀⠀⠂⠀⠀⢀⠀⠀⠀⠰⠤⠄⠃⠋⠉⢁⠈
⠀⣀⣇⡐⢆⠘⢰⡂⠐⢀⣅⠀⣀⣈⣈⠀⠥⠀⠀⠀⡀⡀⠀⠀⠈⠀⠠⢂⣀⣤⣤⡀⠀⠀⠒
⠀⢈⣫⣄⡉⠠⠔⠢⠀⠭⢠⡾⢿⡿⢿⣿⣦⠀⠀⠀⠘⠐⠊⠀⠀⠀⣴⣿⣿⣏⡹⣿⡆
⠀⣿⣿⣽⡗⠀⠀⣠⠀⠀⠸⣷⣦⣵⣾⣤⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⣹⣿⠟
⠀⠄⡉⢁⠀⣀⢂⠀⠐⠀⠀⠈⠛⣿⡿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⡰⢄⠈⠛⢿⠋⠄⠀⠀⠀⢀⡀⣤
⠀⠀⠀⠂⠈⠀⠃⠒⠀⠀⠀⠀⠐⠋⠀⠀⠌⣀⢀⡀⠀⣠⠄⡄⣌⣡⣤⣤⡄⢤⡄⠀⠈⠁⢀⢈⠀⠁⠠
⠀⠀⠀⠠⠃⢀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⢠⡀⠀⠳⠆⠋⠛⠛⠑⠋⠀⠋⠀⡐⠒⢅⠐⠂⡀⠠
⠀⠀⠀⠃⢡⣁⠀⣿⠀⠀⢐⣠⠄⠄⠀⠀⠀⣀⣤⠄⠀⠒⡆⢀⡀⠀⠂⠁⠐⠘⠀⢈⠐⠀⠀⡄⠀⠃
⠀⠁⣀⠀⣰⠀⢀⣛⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⢀⠠⠂⠀⠈⡠⠀⢋⠐⠐⠀⢀⠀⠀⡀⠀⠀⠐⠠⢁
⠀⠀⢘⠠⢤⠀⠠⢍⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⠀⠆⠰⢤⣰⠰⠀⢀⠀⠀⠀⠄⠉⠂⠁
⠀⡔⠁⣆⡷⠁⡄⠀⡃⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠬⢀⠀⢀⠌⠠⠁⠀⠀⠀⠀⠀⠀⡀
⠀⡄⢠⡫⠸⠆⢁⠔⣱⡟⠛⠋⢹⣦⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣀⣤⣤⣤⣤⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀
⠀⣬⣉⡉⢢⡰⠁⠀⠹⠿⣦⣴⠿⠟⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣽⣿⡿
⠀⠘⠋⠀⠀⠇⠀⠀⠀⠀⠈⠻⠀⠀⠘⠛⠀⠀⠄⠀⠀⠀⠀⠠⠞⠁

⣟⠀⠀⠀⠀⣴⢣⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⡸⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣟⠀⠀⣴⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠁⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⣠⣤⣶⣶⣶⣤⣤⣈⡉⠻⣿⣿⣿⣿⣿
⡿⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠸⣿⣤⣤⣤⣤⣤⣌⣉⣿⣄⣉⣉⡉⢻⣿
⣿⠀⢠⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡦⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⠛⣋⡁⢰⣿
⣿⠀⢸⡟⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢀⣾⡟⣟⣻⡿⠿⢷⣦⣴⠿⠛⠛⠁⣼⣿
⣿⠀⣿⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢈⣛⡛⣛⣛⣛⠛⠛⠛⠛⠛⠿⠷⠀⣿⣿
⣿⠀⡏⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠘⣛⣛⣛⡛⠛⠛⠿⠷⠶⠶⠶⣦⡀⢻⣿
⣿⠀⠃⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠙⠛⠛⠛⠛⠿⠶⠶⢶⣶⣶⠟⢁⣾⣿
⣿⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠘⠛⠿⠛⠛⠻⠶⠶⣶⣶⣤⣤⣄⠘⣿⣿
⣿⠀⢸⣿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠀⠛⠛⠿⠿⠶⢶⣦⣤⣤⣤⣴⡿⢁⣿⣿
⣿⠀⢸⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠶⠶⠶⠶⠶⠶⣦⣤⣤⣤⣉⣁⡀⠸⣿⣿
⣿⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⠶⠶⠶⣶⣤⣤⣤⣈⣉⣉⣹⡿⠀⣿⣿
⣿⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣴⣤⣤⣤⣤⣤⣄⣉⣉⣉⣉⠉⠀⢾⣿⣿
⣿⠀⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⢠⣤⣤⣤⣤⣄⣉⣉⡉⠛⠛⣻⡷⠈⣿⣿
⣿⡄⢸⡇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⣤⣄⣠⣤⣬⣉⣉⠙⠛⠛⠛⠛⠁⣼⣿⣿
⣿⡇⢸⠇⣿⣿⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢻⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⠀⣩⣭⣭⣉⣉⣉⠛⠛⠛⠛⠻⢿⡆⢸⣿⣿
⣿⣼⠀⠀⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣾⢲⢹⣆⣿⢿⣷⣿⣿⣿⣿⣿⣿⣿⡿⠀⣈⣁⣉⣉⣉⠛⠛⠛⠿⠶⠾⠟⠁⣼⣿⣿
⣯⠸⡄⠀⢸⣿⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣷⣿⣿⣿⣿⣷⣼⣷⣿⣾⣿⣿⣿⣿⣿⣷⠈⢛⣛⣛⡛⠛⠛⠿⠿⠶⠶⠶⣶⡄⢹⣿⣿
⣿⠀⣷⠀⠘⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⡙⠿⢁⣷⠀⠈⢿⣿⣿⣿⣄⣉⣉⠛⠛⠛⠻⠶⠶⢶⣦⣶⠿⠃⣼⣿⣿
⣿⠀⣿⡄⠀⢿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⠁⠀⠀⠀⠈⠛⠒⠋⠁⠀⠀⠈⢿⣿⡿⠛⠛⠛⠛⠻⠿⠷⠶⠶⠦⡤⠀⢲⡆⢹⣿⣿
⣿⡄⡇⢻⡀⠘⢿⡿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⠾⣿⡇⠘⠛⠻⠿⠷⠶⠶⠶⠶⠶⢶⣶⣾⡷⠈⣿⣿
⣿⢹⣇⠈⢧⠀⠸⣷⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⢳⡶⣿⠛⣿⠛⣍⠀⠀⢰⡟⠃⣴⡶⠶⠦⠀⣠⣴⣶⡶⠶⠶⢶⣆⠀⣾⣿⣿
⣿⠈⣿⡀⠈⣧⠀⢿⣿⣿⣿⣿⡇⢸⣿⠃⠙⠛⠻⣿⣿⣿⣿⣿⣟⣿⣿⣷⣿⣦⣿⣶⠟⢷⠀⣿⣷⡀⠹⠷⠶⠶⠿⠋⢴⣶⣶⣶⠶⠾⠟⢀⣿⣿⣿
⣿⠀⠘⠃⣸⠋⢧⠘⣏⣿⣿⣿⡇⠀⠿⠀⠀⠀⠀⠈⠹⣾⡿⣿⢻⣏⢰⣌⣿⣙⣯⣽⣷⣿⣶⣿⠿⣷⣲⣶⣶⣦⡄⠀⠀⢤⣤⡄⢴⣶⣶⣿⣿⣿⣿
⣿⠀⠀⣰⠃⠀⠈⠣⣼⣹⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢿⣿⠛⣿⣿⡟⢿⣟⣻⣤⣿⣾⣿⣾⠟⠛⠛⢿⣿⡇⠀⠀⢸⣿⡇⠸⣿⣿⣿⣿⣿⣿
⣿⠀⣰⣇⣴⠃⠀⠀⠙⢷⡽⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣿⠉⣧⣾⣿⡿⠛⠛⠛⠉⠁⠀⠀⠀⠸⣿⠀⠀⠀⢸⣿⠀⠀⣿⣿⡟⣿⣿⠋
⣿⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠈⠋⠀⠀⠸⠇⠀⣼⠏
⣿⠸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣀⣠
⣿⠀⠻⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⢹⣿⣿
⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿
⣿⢹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿
⣿⠀⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡖⠀⠀⠀⠀⠀⠾⢿⣿⡿
⣿⠀⠀⠙⣷⡚⠻⣶⣶⣦⣤⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⣿⠏
⣿⠀⠀⠀⠈⢿⣶⣿⡇⣿⠃⣿⣿⣿⢿⡿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⣿⠀⠀⠀⠀⠀⢹⣿⡇⣿⠀⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿
⣿⠀⠀⠀⠀⠀⢸⡿⢸⣿⠀⣿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿
⣿⠀⠀⠀⠀⠀⡼⠃⢸⣿⠀⠿⣿⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⣿⣿⣿⠿⠊⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⢸⣷⡆⢸⣿⠀⠀⣸⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⡿⠋⠈⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⢹⡇⢸⣿⠀⠀⣿⣿⢻⣇⣀⣀⣠⣤⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠫⠂⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⢀⣤⡄⢸⠇⣸⣿⠀⠀⣿⣿⣽⡏⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⢿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠙⠀⢸⠀⣿⡿⠀⠀⣿⢿⣿⡇⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡉⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⢠⡄⢸⠀⣿⡇⠀⠀⣿⠼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⣸⠀⣿⡇⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⢣⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢏⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⣠⡞⣡⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢋⣁⡸⣿⡿⢋⡼⠋⠀⠀⠀⠀⣴⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣸⣿⢰⣿⣿⣷⣝⠷⠿⣷⡄⠀⠀⠀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⠉⠛⠛⠙⠻⢿⣿⣿⣿⡉⣱⣿⣧⡀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠛⠉⠁⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣵⠟⠛⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣦⠀⠀⠀⣀⣿⣷⡶⠾⠿⢶⣦⣄⡀⠀⠉⠙⠻⠿⢷⣦⣄⠈⠙⣻⣧⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⣀⣿⡟⠛⠳⠶⠾⠿⠟⠿⢿⣷⣄⠀⠀⠐⢦⣄⣈⠀⠀⠈⣻⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠀⣿⡟⠛⠛⠛⠲⠶⠶⠤⠤⠀⠈⠙⠛⠛⠟⠛⣿⠛⠛⠛⠛⠛⣻⡟⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⢸⣧⢀⣤⣤⣀⣀⣀⣀⣀⣀⣀⣀⣠⣤⣶⣞⣋⣀⣀⣠⣤⢀⣿⠃⠀⣴⡿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣶⠿⣿⣦⡉⢩⣿⡿⠿⠻⠿⣿⣿⡉⢡⣾⡿⢷⣦⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠀⢻⡇⢸⡟⠁⠀⠀⠀⠈⣿⡇⣾⡏⠀⢸⣿⠉⠁⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣶⣦⡀⠀⠀⠈⣿⣄⠀⠸⣷⣿⠇⠀⠀⠀⠀⠀⠻⣷⡿⠃⠀⢸⡿⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠀⠀⢿⣯⣉⡀⠀⠀⠀⠀⣠⣤⣤⣤⣤⡀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⣀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢿⣿⣷⡄⢀⣾⣿⠿⠿⠿⠿⠇⠀⠀⢀⣿⠁⠀⠀⠀⠀⠴⠾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠹⣿⡇⢸⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢿⡇⠀⣿⣇⣸⡇⠀⠀⠀⠀⠀⢀⣠⠖⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⢿⠀⢸⣷⠀⢻⠙⠋⠀⠀⠀⠀⢀⣴⠟⠋⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠘⣿⠀⠸⠀⠀⠀⠀⠀⣠⠟⠁⢀⣤⣴⣿⠀⢰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡏⠀⠁⢀⡴⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣽⣇⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⡿⣿⣇⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠁⠀⠀⢸⣿⣀⠀⠠⠴⠶⣞⣿⣿⣿⣿⣿⣿⠇⠀⠀⠈⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣄⡀⠘⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠀⢀⣤⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡈⠙⠛⠷⠶⣤⣤⣀⣈⣉⣉⣉⣁⣠⣤⣴⠶⠟⠛⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠒⠂⠤⠤⠤⠤⠭⠭⠭⠤⠤⠤⠤⠐⠒⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣞⣋⢡⡆⣔⢢⢆⡖⢲⣰⢨⣝⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⢯⣁⡯⢤⠉⣠⣦⢹⣧⡞⠇⣸⢹⢀⣥⠎⠘⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣏⡐⢛⡃⣘⡇⢘⡓⢂⡓⣿⡇⡄⣿⢸⠊⣾⢱⡟⠒⡝⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢣⣎⡹⢿⣧⣿⢹⡄⢷⡯⠅⣿⡇⣿⣄⠻⠐⣿⣄⢻⠛⢇⡤⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⡂⠚⢳⠒⢋⣛⣼⣟⢎⡐⠳⣿⡇⢻⣿⣀⢊⣻⣿⠘⠷⡞⡔⠋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠩⠭⠭⠍⠙⣿⣧⠿⡿⠂⣡⢷⡿⣦⣿⢙⣯⢿⡏⢿⣾⠄⠯⢷⣿⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣿⠗⣟⠿⣣⣿⡟⣀⣿⡟⠯⣿⣲⣿⢋⡀⡙⣿⣇⠸⣿⠀⡷⠸⣿⠼⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢧⣷⡔⠫⢲⣽⣿⣁⡼⢹⡇⣦⣾⠃⠈⢸⣿⣿⣟⠻⣷⢿⡆⣳⢃⣿⡆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣦⡓⣿⣿⣿⡟⠁⠈⢿⡿⣿⠀⠀⠀⠸⠶⠣⠄⠈⢸⡇⠣⣿⣹⡇⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣁⣿⣧⢸⡟⠴⠶⠃⠀⠀⢸⡗⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣇⣿⡿⢧⠉⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢉⣿⣿⣺⡆⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠟⢹⣿⠿⣿⣸⠓⣹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⣷⣿⣿⣿⠦⠄⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⠟⣼⣿⣿⠰⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠌⢻⣿⢹⣦⡆⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡾⣿⡿⣧⣡⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⢶⡰⠻⣶⢹⣿⣆⠀⠀⠀⠀⠀⣿⡏⠉⠀⠀⠀⠀⠀⢀⣼⣿⡷⠿⢠⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠈⠳⣧⣿⣆⡻⣿⣷⣄⠀⠀⠀⣿⡇⠀⠀⠀⠀⢀⡴⠋⣼⡿⢻⣿⠾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠹⣏⠻⣿⣿⣿⣿⠻⣶⣄⣿⡇⠀⣀⣴⣾⠟⠀⢠⣿⣿⢿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠞⠉⡋⡞⠓⠌⣻⣿⣿⣿⣿⠟⠁⠀⠀⣸⣿⣿⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢀⣀⣀⣀⠀⠀⠀⣀⠀⡾⠀⠀⠀⠀⠈⣿⣿⣿⣁⡀⠀⣠⠴⣿⠻⠇
⠀⠀⠀⠀⠀⠀⡠⠒⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⢸⠀⣠⠴⢿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣦⣿⠛⡀
⠀⠀⠀⡀⢨⣥⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢮⣡⠀⠸⡿⢿⣿⣿⠿⡿⠟⣿⠛⠋⣿⣇⢡
⠀⠀⠰⠁⠀⠀⠈⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⡿⢀⣠⣀⠸⣿⢀⠘⢱⠐⠉⢈⡀⠉⣿⡈⠇⢀⡀
⠀⠀⢀⠀⠐⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣠⣠⢨⣿⣿⡠⣿⡆⠀⣘⣠⣶⣮⣿⠀⣿⡇⠈⠌⠫
⠀⠀⠘⡄⠀⠀⠈⢢⠀⠀⠀⠑⢦⡀⠀⠀⠀⠀⠀⠆⠀⢸⢿⣿⣾⣿⣿⣇⣿⣋⣴⣿⣿⣟⡩⣱⣦⣿⣇⠀⠈⢆⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⢐⡀⠸⢄⠀⠂⠀⠀⠀⠀⠳⠀⠀⠀⠀⠰⠄⢀⣸⣾⣿⢿⣿⣿⣿⣿⣿⣿⣿⡿⣋⣼⣿⡃⣿⣿⡀⠀⠀⠣⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢈⡅⠀⠀⠑⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⢀⠶⢾⣿⣿⣧⠟⣹⣿⣿⣿⡿⢟⣫⣾⣿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠡
⠀⠀⠀⡜⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠘⣆⢸⡿⠹⢋⣴⣿⡿⢛⣯⣶⣿⣿⣿⣿⣿⣿⢰⣿⠇⢸⠀⠀⢠⣤⠗
⠀⠀⠀⠻⣄⠀⠓⠄⠀⠀⠀⠀⠀⢆⠀⡇⠀⠀⠀⢿⣜⠃⠀⢸⣿⣿⠀⣻⣿⡿⣫⣿⣿⣿⣿⣿⣾⣿⠀⢸⡆⠀⠀⠀⠉⣶
⠀⠀⠀⠀⢈⠂⠈⠐⠢⣄⠀⠀⠀⠀⠐⢧⡀⠀⠀⠈⢿⣼⣶⣿⣿⣷⡾⢿⣿⣿⣿⣿⣿⡿⣟⢹⣿⡏⢀⣿⣧⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠹⣆⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢻⠀⠀⠀⢸⣹⣿⣿⣿⣿⣵⣿⣿⣿⣿⣿⣿⣷⣮⣾⣿⢃⣼⣿⣿⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠟⢻
⠀⠀⠀⠀⠀⠘⡆⠀⢀⠀⠀⠀⠀⠠⣈⠳⣼⡄⠀⠀⠈⢳⣿⣿⣿⣿⣿⣿⣿⢳⣿⣿⣿⣿⣿⣿⢏⣼⣷⡏⣿⡆⠀⠀⠀⡜⠀⠀⠀⠀⢸⠀⠃
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠑⢄⡀⠀⠀⠀⠉⠠⣿⠀⠀⡰⢻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣾⣿⡿⠀⣿⡇⠀⠀⠀⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠈⠀⠀⠄⠀⠀⠈⣣⡄⢷⠘⣎⠻⠿⣿⣿⣿⠏⠀⣿⣿⣿⣿⣿⣿⣿⣿⠃⢀⣿⣧⠀⠀⢰⡇⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠔⠀⠇⠀⠀⠈⠓⠢⠀⣠⣧⠀⠀⢸⣿⣿⣿⣿⡿⣻⠃⢀⣼⣿⣿⠀⠀⠀⠁⠀⠀⠀⠀⢸⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠴⠃⠀⠹⠀⠀⠂⠀⠃⠀⠀⠀⠀⠙⠷⠇⠀⠘⠿⠿⠿⠿⠿⠇⠀⠾⠿⠿⠿⠂⠀⠸⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⣀⣤⣤⣤⣤⣤⣤⣤⣄⣤⣤⣤⣤⠄⠤⣤⣤⣤⣠⣤⡤⢤⣤⣤⣤⡠⣤⣀⣤⣤⣤⣤⣤⠤⣤⣤⣀⣀⢀⡀
⠀⠀⠀⠀⠀⢸⣿⣿⣷⣼⣟⣙⣿⣿⣺⣿⠻⣻⣿⣻⢟⠻⡅⠈⣼⣸⡝⣿⣿⣯⣹⢻⣽⢿⢟⡽⠿⢼⣿⣧⣶⣿⣿⣿⣔⣈⡤⢚⣿⡿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠨⡄⠈⢿⣷⢿⠻⣫⣰⢻⡛⡉⡅⣤⡀⢠⡀⡆⣠⠀⣑⢘⣓⣾⠖⣶⠰⡶⠖⠒⠞⠦⠴⣆⣀⣿⣿⣿⣿⣟⢿⣟
⠀⠀⠀⠀⠀⢸⡧⣶⣏⠀⠈⢐⣆⣶⣷⣟⣯⣟⡿⠑⣟⢃⠋⠸⠐⣾⣻⠉⠓⢈⡟⠻⠧⢚⣿⣿⠆⣹⠛⢛⠤⠆⠒⣩⣽⣽⣿⣿⣻⡝⣵
⠀⠀⠀⠀⠀⠀⣳⣷⣦⣄⣠⡁⢿⣾⡿⣛⢡⣾⡟⡻⠿⣿⢼⠾⢌⢷⠋⠡⣾⠠⡶⣄⠀⠛⠛⠮⢑⣃⠀⠘⣟⣀⠰⠿⡧⣈⢾⣿⣿⣿⣾
⠀⠀⠀⠀⠀⣼⣿⣏⡙⢿⡿⠀⠀⢼⡸⠏⢃⢻⣯⠰⠤⠄⠩⠑⠋⢫⣆⠀⠭⠀⢀⣫⡒⠐⡔⠊⡟⢐⡞⣓⠂⢙⡀⠀⡡⠿⠚⣿⣿⣻⣿
⠀⠀⠀⠀⠀⢸⡿⣟⡅⠘⠃⡴⢶⣜⢳⠆⣽⣿⣅⣈⡍⢁⣁⡀⢼⣞⣽⠀⢀⣷⣼⣿⣧⠾⠓⠠⣺⣯⣁⠉⢘⢾⡁⠚⣳⣼⣱⡧⣾⣿⣿
⠀⠀⠀⠀⠀⢸⣷⣾⡇⠀⠀⠀⠈⠛⣘⣿⣿⡉⠡⡁⢰⣿⠉⠵⡈⠷⣛⢠⣿⠻⣿⣿⡧⠤⡤⡆⣿⡿⠬⡀⢠⣼⡏⢿⣷⣉⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⡖⠂⡏⠘⡿⡀⠒⣇⣸⣿⠄⣱⣿⣄⠭⠛⣄⠀⢟⣻⣷⣾⣶⣦⣽⡇⠔⣦⢾⢿⣄⠸⠿⠩⢽⣿⣿⣿⣹
⠀⠀⠀⠀⠀⢰⣿⣿⣇⣴⠖⠢⠹⠀⢻⣼⣿⣿⡆⠈⠿⠿⡇⠊⡉⠸⣾⠿⡉⠳⡿⢻⢿⣿⣿⣿⡿⣧⢿⡙⣌⢻⡏⢲⣶⣦⣀⠀⢿⣭⡿
⠀⠀⠀⠀⠀⢀⣿⡿⣿⠋⠚⠀⠀⠊⠛⣿⢿⣿⣿⠟⠈⡃⠀⢠⠁⠾⠿⣀⠁⡀⢠⠊⠈⠽⣿⣿⡗⢈⡚⢃⢛⡙⠿⡶⠿⣿⡏⠀⣾⣺⣷
⠀⠀⠀⠀⠀⢸⡧⢛⢱⣶⣶⡶⣾⡄⠀⢻⣿⡉⠁⠉⠁⠪⠀⠀⠀⢀⣀⠀⠈⣁⢀⡀⠀⠲⠆⢆⣀⣀⡁⢨⡀⣁⡈⣀⣀⣈⡯⢳⣺⣽⣿⠄
⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡾⣧⠀⢘⣿⠀⡀⠀⠻⠐⣿⡇⠸⢿⣿⠃⠀⣿⠀⡃⠖⣰⣀⣾⠐⢛⣖⣸⡇⣿⡆⢸⡇⡀⢘⣼⣭⢶⣾⠆
⠀⠀⠀⠀⠀⢰⣿⣟⢿⣿⣿⣿⣿⡿⣿⣟⣿⢳⣷⣖⣶⣾⣟⣷⣶⣮⣽⡿⣿⣽⣿⢹⣿⣿⣿⣻⣟⡈⡍⢹⣿⣯⣿⣟⣹⣿⣍⣾⣇⣏⣿
⠀⠀⠀⠀⠀⠀⣻⣿⣾⠿⣿⣿⣼⣿⣿⡿⢿⣶⠿⠿⣿⡿⣿⣿⣾⣻⢞⣿⣫⣿⠗⢸⣿⣿⣧⣼⢿⣗⣚⣟⣿⣿⡟⣉⡽⣥⡾⢓⣧⡷⣯⡀
⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡋⢙⣿⣒⣿⡃⢸⣿⡟⣣⣻⣿⣧⠙⣼⣓⣵⢻⠝⠹⣹⣿⣿⣯⣟⠿⠨⠿⠿⠃⠿⠿⠋⠛⠛⠟⠾⠿⠻⣾⣼⠇
⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉⠁⠉⠀⠉⠉⠉⠉⠉⠈⠉⠉⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠻⣿⣿⣿⣿⣿⠿⣋⣵⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⠿⠛⠋⠉⠁⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡛⠙⠂⠀⠀⠀⠀⠀⢈⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⢁⡤⣤⢤⡀⠀⠀⢀⣀⠐⣻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⠀⠳⠏⡼⠗⡀⢀⡿⠈⠟⣽⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⣠⣠⣄⢿⣿⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠙⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠼⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠶⠚⠛⠛⠛⠛⠛⠛⠳⠶⢦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠶⠒⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠏⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⣒⣶⠶⠄
⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠚⢋⡉
⠀⠀⠀⢀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣞⣋⠁⠀⠀⣉⠭⠭⣄⡀
⠀⠀⢀⣿⡄⡼⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠀⠀⠀⠉⠙⣲⠋⠀⠀⠀⠀⠙⣆
⠀⠀⢸⡏⣷⠃⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⢀⣠⡄⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠸⡇⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠉⣠⠖⠚⠒⠒⠦⣼⡠⢶⣦⠀⠀⠀⠀⣸⠓⠦⣄⡀
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠞⢋⡿⠁⠀⠀⢰⡇⠀⠀⠀⠀⠀⣈⣻⡿⠟⠀⠀⠀⡰⠃⠀⢀⣀⡙⠲⣄
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠰⠯⢤⣀⠀⠀⠈⣧⠀⠀⠀⠀⠀⢿⡿⣇⣀⡤⠶⠋⠠⠖⠋⠉⠀⠈⡧⠈⠻⣄
⠀⠀⢀⡴⠧⠀⠀⠀⠀⠀⠀⠀⠀⣬⣑⡄⠀⠈⢧⡀⠀⠀⠀⠀⢀⡏⠁⠀⠀⠀⠀⢀⣀⣀⡤⠞⠁⠀⠀⠘⣧
⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠁⠀⠀⠀⠭⠓⠚⠛⠉⠛⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⣸⠃⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠷⠊⣽⠀⣠⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟
⠀⠀⠸⣤⠉⢦⣀⢀⡤⠒⠋⣉⣉⣉⣉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠐⣞⠁
⠀⠀⠀⠀⠀⠀⠉⢻⡀⠐⠋⠑⠒⠋⠀⣀⡤⠤⣤⣄⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⣀⣀⣀⣄⠘⣧⠹⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠲⠖⠒⠚⠉⡽⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⡿⣍⠉⠉⠉⠉⠁⠈⠉⠉⢳⡈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠈⠑⠦⣄⡀⠀⡇⠈⠳⡄⣠⢤⠤⠤⣄⣀⡼⢁⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⢺⣦⠀⠀⠀⠀⢀⡭⡶⣷⣠⢶⡟⢹⠦⣕⡢⢄⣻⣤⠿⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⢳⡀⠀⣴⡉⠈⠑⠛⠶⢤⠗⠧⣀⡀⠉⠓⠺⠽⣶⣆⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣁⣀⡀⠀⠀⠀⠀⠙⠾⠊⠉⠓⠲⣤⣄⣀⠀⠀⠠⢬⡗⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠉⠉⠉⠛⠒⠒⢶⣶⠒⠒⣖⠋⠀⠘⣄⡀⢀⡤⠇⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⠀⠤⢤⣼⣤⡤⠄⢸⠀⠀⠀⠳⣌⡉⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠁⠀⣧⣀⠀⣿⣀⡀⠀⠀⠀⠉⠉⠛⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣟⡳⣖⡶⠚⡛⣯⠟⡧⢈⣿⢙⣆⣸⣦⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢾⣿⣿⣿⣷⣮⣭⣵⣲⣶⣶⣶⣿⣭⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⡿⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢻⡿⢷⣰⣶⣶⡄
⠀⠀⠀⣤⡐⣷⠀⠀⣷⠀⢿⡈⣧⢰⣏⢻⡍⢹⡄
⠀⠀⢠⣿⢷⠸⡇⠀⢸⡆⠘⣇⠸⣾⡏⠈⣧⢸⡇⠀⠀⣀⡀⠀⢠⣶⣄
⠀⠀⠈⣿⢈⠃⢿⠀⠘⣷⠀⢿⡄⣿⣷⡀⢸⣿⠇⢀⠀⣿⠛⠓⣼⠁⠉⠆⠀⠀⠠⡀
⠀⠀⠀⢻⣄⡀⠘⣇⠀⠹⡆⠸⣷⡸⣏⢻⣆⢻⡀⠋⣱⡞⠀⠀⠀⠀⠀⠈⠀⣿⡦
⠀⠀⠀⠀⠛⠿⣷⣿⡄⠀⢻⡔⣿⣧⠹⠆⢙⠊⢷⣰⠍⠀⠀⠀⠀⠀⠀⠀⠈⢁⣠⢷
⠀⠀⠀⠀⠀⣀⣠⣿⣿⣿⣆⢻⣿⠹⠆⠀⠀⠀⠨⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣨
⠀⠀⠀⠀⠀⠙⠿⠟⠉⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⡀⠓⠠⠀⠀⠀⠀⠀⣄⣈⣿⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠋⠁⠁⠸⠋⣸⠶⡀⠀⢀⣤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⡠⠢⠎⠀⡀⢾⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠴⢷⡴⠇⠀⠀⠘⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⡀⠀⠚⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⡀⠀⠀⠄⠀⠀⠀⠀⣧⣀⣰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⢲⣶⣖⠁⠈⠄⠀⠀⠀⠀⠀⠙⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠒⠒⠀⠤⠄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡀⠀⢠⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠳⠤⣄⡀⠌⠠⠬⠩⠩⠭⠉⠉
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠤⠤⠄⠀⠠⠀⠀⠀⣤⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂
⠗⠒⠒⠐⠂⠀⠲⡖⠒⠒⠈⠑⠒⠂⠖⠒⠚⠁⠒⠒⠢⠒⠂⠉⠁⠙⠣⠴⠤⠄⠂⠤⢤⡠⠀⣘⠂⠤⠄⣀
⠀⢠⡀⠈⣁⠀⠁⠀⢀⠀⠀⢀⠀⠁⠈⠀⣀⠉⠉⠈⠁⠑⢀⠀⠀⠐⠀⠃⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⢰⡟⠋⢉⠝⠋⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠁⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢋⡴⠒⠶⠬⡝⠿⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⣉⣉⣉⡉⢉⣩⠁⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠃⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢈⣛⣛⣓⣛⣛⠒⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⢀⣒⣒⡒⠢⠭⠽⢢⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂⣿⣿⣿⣿⣿⣿⠀⠶⠶⠬⠭⣭⡽⢪⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣱⡄⠛⣿⣿⣿⣿⣿⠈⠭⢭⣍⣓⣒⣲⢘⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣾⡿⣀⢻⣿⣿⣿⣿⠐⣖⣂⣒⡒⠊⣤⢸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⠎⣿⣿⣿⣯⢐⣒⡒⠢⠭⠭⣥⢺⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⣿⢸⣿⣿⣿⣿⣿⣟⡭⣙⣀⣠⣚⣅⣾⣿⣿⡏⠠⠤⠭⠭⠭⣙⣁⢾⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡰⡆⢸⡎⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠫⠉⠀⠈⢿⣿⠬⢭⣭⡀⠒⠒⢋⢾⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣇⠁⠀⢃⣿⣿⡟⣿⣿⣷⣦⣤⣀⡀⢀⣀⡀⠀⠚⠇⣈⡒⠒⣒⣒⣒⡚⣸⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠆⠰⠀⢿⣿⡇⡟⠉⠛⠿⣾⡋⠄⠖⠀⠀⠀⡐⠀⣓⡒⠋⠓⠒⠒⣃⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢈⣿⠃⠀⠀⠀⠀⠁⠺⠀⠅⣘⠃⣤⡷⠚⠙⣿⠀⠈⡟⢸⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⢸⢠⡾⠃⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠁⠈⠃⠘
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢲⣾
⠀⠀⠀⠀⠀⠀⠀⢸⠷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠹⣄⢤⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠇⠀⠀⠀⠹⠋
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠈⣿⠈⠸⣿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠁⠀⠀⠀⠀⣠⣶
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡛⠰⠀⢿⢰⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⠟⠀⠀⠀⠀⠀⣠⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠐⠇⢸⠀⣾⢻⠀⣀⣀⣤⣴⣾⣿⣿⡿⠛⠁⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠆⠁⣶⠀⣿⢸⠈⠉⠀⠀⠀⠙⣿⠋⠀⠀⠀⣀⠤⣺⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠂⠀⣿⠀⣿⢸⠀⠀⠀⠀⠀⠀⠈⢳⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠧⠤⠤⠤⠧⠤⠧⠼⠤⠤⠤⠤⠤⠤⠴⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⠔⣁⣐⣂⣀⡀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠡⠊⢱⢫⠛⠉⠗⠀⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣤⡄⠀⢀⠆⢀⡏⠘⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⡀⢄⣀⡤⠖⠊⠁⠈⢁⢶⠄⠃⡌⣧⢀⡄⠈⠂
⠀⠀⠀⠀⠀⠀⠀⢉⡾⠉⠀⠠⠀⠀⠀⠀⠀⡠⠃⠀⢇⠊⣧⣘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡤⠖⡛⠋⣉⣿⢩⢎⡁⠀⣢⠎⡏⣟⣩⣾⠀⢠⡐⢤⣀⡀
⠀⠀⠀⠀⠀⠀⢠⡾⢁⡐⠈⢋⣑⣰⡾⠍⠉⠉⠛⠤⣪⣾⣋⢀⡉⠓⡜⡴⣗⠂⠉⠑⠲⢄⡀
⠀⠀⠀⠀⠀⠀⠞⢇⡀⢁⢜⢉⠀⠈⠙⢔⣰⠆⠀⠀⠹⣫⣖⡶⠿⠶⠞⠛⢥⠂⢈⣁⣀⠆⡉⠢
⠀⠀⠀⠀⠀⠀⠀⡦⡁⠒⢤⣃⡠⠀⠀⠈⠦⠄⣐⡤⠵⡹⡔⣺⣥⡴⣖⠤⠚⠚⣥⠏⠁⣿⠔⡰
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠂⠅⠉⡓⠃⠧⠾⠥⠥⠤⣈⣄⣔⠃⠼⠼⠭⠵⠷⠖⠒⢉⣤⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠐⠂⠒⠐⠀⠀⠀⠀⠐⠒⠒⠒⠋⠉⢀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣀⣀⣀⣀⢀⠠⠤⠄⠒⠉






⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⡀⡀⠀⠀⠀⠀⡀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣶⣴⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢀⡁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠤⢤⣤⣤⣶⣤⣤⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⡈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣛⣛⣿⣛⣛⣿
⠀⠀⠀⠀⠀⠈⠀⢁⠀⠀⢀⠀⠃⡇⢹⣿⣿⣿⡿⠃⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣭⣭⣭⣭⠛⣭⣽⣭⣭
⠀⠀⠀⠀⠀⠀⣼⠀⠀⣤⠚⠢⢤⣶⣿⣿⣿⣿⣵⣶⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣾⣹⣟⣿⢰⣏⣿⣿⡗
⠀⠀⠀⠀⠀⢸⠉⠘⠀⣼⠖⠶⣄⣿⣿⣿⣿⡿⣣⣾⡿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠚⠻⡟⠛⠐⠛⢻⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡧⢀⠈⠋⠀⢻⡿⣿⣽⣷⣤⣄⣀⣠⣼⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⠄⠀⠀⠁⠛⢯⡏⠀⣱⣧⣾⣿⣿⣿⡿⢻⣿⣿⡇⠀⠀⣰⣂⣀⣠⣆⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠉⠁⠀⠀⠀⠙⠇⠈⢿⡿⠃⠀⠞⣿⠤⠤⢼⡥⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡆⠀⠀⣿⠒⠒⢺⠒⠒⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣿⠇⠀⠀⡉⠉⡉⠉⣉⠉⣉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠛⢼⠁⠀⠀⠀⠀⠀⠀⣀⣤⣶⠛⠁⠀⠀⣫⡆⠀⠾⠁⠀⠿⠀⠹⠀⠈⠷
⠀⠀⠀⠀⠀⢦⡀⢰⣦⡀⡀⠀⠀⠀⠆⣸⡀⢀⣀⣤⣴⣶⣿⣿⠟⠁⢀⣠⣴⣾⣿⡇
⠀⠀⠀⠀⠀⠈⢻⣧⠛⠛⠁⠀⠀⡇⠀⠛⢄⡀⠀⠈⢿⡿⠟⢁⣤⣶⣿⣿⣿⣿⣿⡇⠀⣤⠀⣤⡤⣤⣷⣤⣤⣤
⠀⠀⠀⠀⡀⠀⠀⠈⢷⢶⣶⣘⠀⠀⠀⡠⠀⠱⡀⠀⢨⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⠃⢰⡖⠒⣿⠒⠲⡆
⠀⠀⠀⠀⠈⡓⠒⠦⢄⡀⠻⣿⣴⠀⠀⠠⠖⠁⠙⡄⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡀⢸⣋⣉⣿⣉⣹⡇
⠀⠀⠀⠀⠀⠑⠤⠤⠤⠤⠤⢽⡻⣷⣶⠳⠒⠛⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣰⠏⠀⣠⠟⣿⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠠⢤⡤⠤⠾⠗⠿⡏⠀⢀⡄⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠋⠀⠋⠁⠀⠿⠀⠈⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠃⠐⠀⠃⠀⠚⠀⠐⠛⠛⠻⠿⠿⠿⠿⠟⠛⠛⠛⠛⠃

⠀⠀⠀⡀⠀⠀⠀⠀⣤⡀⢀⣠⣄⠀⠀⣀⠀⢀⠀⠀⠀⣠⠀⠀⡄⠀⢀⠠⠀⡐⠂⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⡀
⢀⠰⡁⢀⠆⠀⠀⢸⣿⢣⣿⠟⠋⠀⠀⣄⠀⠐⠠⠀⠀⣀⠀⠈⠄⠀⢸⠀⠀⠦⠀⠀⢀⠆⠀⠀⠘⠉⠀⠀⠀⠀⠉⠇⠀⠀⠁⡀⠀⢸⠁⠀⠈⠙⡇
⠐⠀⠘⠁⠀⠀⠀⠀⠛⠸⠿⠿⠀⠀⠀⠀⠀⠘⠈⠀⠀⠄⠀⠈⠀⠀⠸⠀⠀⠒⠀⠀⠀⡠⠚⠀⠀⡇⠀⠀⠀⠀⠐⡃⠀⢀⠀⠀⠀⠋⠀⠠⠀⠀⠑
⠀⢠⠤⠤⠤⠤⠴⠀⠀⠀⢀⡀⢧⠀⠁⠀⠀⠁⠈⠀⠀⠅⠀⠈⠀⠀⠁⠈⠀⠸⣁⠀⠀⠀⠈⠀⢠⣤⣴⣶⢶⣶⣤⣁⠈⠄⠂⠀⢀
⠀⠀⠂⠀⡂⠀⠈⢄⠀⠀⢅⢈⠁⡱⠀⠠⠀⠂⣀⠀⠀⡀⠀⠀⠀⠠⠀⠀⢠⢀⠀⢄⠀⠀⠈⠀⠰⣿⣿⣿⣿⣿⣿⣷⠀⠈⠃⠀⢸
⠀⠀⠀⠐⠀⠀⠀⠘⠅⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠂⠒⠊⠉⠐⠌⠂⠘⠈⢂⠀⠀⡇⠈⠂⠍⠉⠉⠉⠅⠃⠀⠀⣤⡆⢈⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠡⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⡀⡂⠀⠀⠀⠂⠀⠀⣀
⠀⠀⠀⠈⠀⠀⠀⠀⠏⠀⢰⠦⢤⣤⣴⣶⣠⣄⢠⠀⠐⠀⠂⠀⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠁⠁⠐⠂⠀⠱⠄⢠⡐⠂⢉⡉⠴⠀⠰
⠀⠈⠀⠀⠀⠀⡀⢠⡄⠀⠈⢦⣤⣄⣀⣀⣀⣤⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⡄⠀⠀⠀⢀⠈
⠀⡰⣻⢆⣤⢀⡛⠩⠤⠀⠀⠀⣿⣿⣻⣿⣿⣿⠀⠀⠀⠀⠀⠀⣀⣼⣤⣾⣦⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠠⠁⠀⠀⠀⠀⠡
⠀⠹⠟⢾⡁⣿⣹⣶⣷⣤⣤⠠⣿⣭⣿⣿⣿⣿⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⢀⠃⠀⠀⠔⢢⠀⠀⠀⢀⠀⠈⠀⠂⠀⢀⠃
⠀⠀⠀⠈⠀⠉⠁⠟⡃⡿⠛⠀⢽⣿⣿⣿⣾⡇⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⢘⠠⣠⠋⠀⠠⠀⠀⠀⠀⠂⠀⠀⡀⠀⠈
⠀⠐⠈⠀⠑⢀⠀⠀⠀⠀⠀⠀⠊⠻⠾⠷⢟⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⡿⣿⡿⡿⣿⣿⣿⣿⣿⡇⠀⠑⡧⢀⠀⢀⡆⠀⠀⠀⠀⢀⠠⠧⠀⡄
⢀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⡀⠀⠀⢀⠀⠘⣿⣿⣿⣿⣿⣧⣄⣿⣇⣄⣿⣿⣿⣿⡟⠀⠠⠀⠈⠈⠈⠁⠁⠀⠀⠀⠀⠈⠀⠀⠈
⠀⠀⠀⠀⠀⠀⡄⠀⠀⠓⠀⠀⠀⠀⠀⠀⠃⠀⠸⠼⠀⠀⠈⠙⠿⠿⣿⣿⣿⣿⣿⣿⡿⠿⠟⠋⠀⣀⠀⠀⠀⠀⢠⠀⠀⠀⠾⠋⠥⠀⠉⠡⢄
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠰⠀⠀⠠⠀⠀⠀⡅⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠍⠁⠈⠈⠍⠒⠀⠈⣀⡀⣀⣀⣀⠘
⠀⠂⢠⢤⠄⠘⠀⠠⠤⠀⠂⠀⠀⠀⠀⠀⡇⠀⠀⢣⠄⠜⠁⠁⠐⠠⢀⡁⠀⠀⢀⠀⣀⣀⡔⠀⠀⢸⡀⠐⢀⠐⠀⠀⠐⡀⢨⣿⣷⣿⣿⣿
⠀⠀⢸⢠⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡅⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠄⠁⠀⠀⠸⠀⠸⠁⠄⠀⠠⠀⠁⠠⠆⠀⣿⣿⣿⣿⡿
⠀⠀⠈⠁⠀⠀⠀⠐⠀⠀⠀⣄⢀⣀⣀⠠⠆⠀⠀⠠⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⣠⣶⣖⣲⣸⠀⠀⠁⠄⠂⠀⠐⣀⠌⠀⠀⣿⣿⣿⣿⡇⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⢼⣿⣿⣿⣿⣿⡄⢀⡀⠄⠈⠉⠉⠀⢀⠀⠘⠀⡉⠉⡁⠃⠀⢘⠿
⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⢰⠃⠀⠄⠀⠈⠰⠀⠀⢠⠀⠀⠂⠀⠀⠂⡠⠀⠀⠀⠸⣿⣿⣿⣿⡿⠁⠀⠑⠀⠀⠀⠀⠀⢀⡀⠀⡀⠀⠀⠐⠆
⠀⠰⣀⠀⡀⠠⠄⢠⠀⠀⠈⣰⣖⣒⣲⣶⠀⠀⠀⠀⠐⠁⠀⠖⠀⠀⢀⣀⠀⢠⡀⠀⠩⡭⠉⠈⠀⠐⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠂⠀⠎⠀⠐⠠⡀
⠀⠈⠀⠀⠀⠀⠀⠠⠁⠀⠀⢹⣿⣿⣿⣿⠀⠀⠀⠁⠆⠀⠐⠀⠀⠄⠈⠻⢧⢹⠇⢤⣐⣓⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡟⠀⠠⠀⠉⡁⠉⠙⡖⠃⠀⠀⠛⠃⣤⡄⢀⡀⡄⢀⠀⣀⡀⠀⠀⠀⠈⠀⠀⠀⠀⣆⠔⠀⠆⠀⢁
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠂⠀⠘⠀⠀⢸⡇⠛⢸⢳⡇⣿⣸⠩⠇⢀⠀⠀⠄⠀⠀⠀⢀⠃⠔⠀⠈⠀⠃⠀⠈⢀
⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⣅⠿⠿⠿⣃⠀⠘⠄⠀⠀⠀⠀⠀⢀⠃⠀⠀⠸⣇⣿⣻⠸⣧⡿⣿⢰⡆⠀⠀⣀⣀⣀⣀⣠⠂⡎⠀⠀⠀⡁⠠⣀⢀⡀
⠀⠀⠈⠀⠈⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⠋⠈⠉⠀⠀⠙⠟⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠐⠀⠂⠀⠀⠀⠁

⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⢸⢿⣿⣉⣽⣿⣿⣍⣿⣿⣻⣿⣿⣻⣿⣿⣿⢻⣿⣉⣿⣻⣿⣿⡀⠀⢀⣿⢿⣿⢛⣿⣿⠛⣿⣿⡛⣿⣿⣛⣿⢿⣿⣿⣿⡇
⠀⠀⠀⢸⣿⣉⢉⠉⣭⠉⡏⠉⣟⠛⡏⠻⠉⠉⡙⠉⡙⢉⠛⢻⠉⠹⡍⢹⣿⡛⠙⠛⠛⡏⣝⢿⠛⠛⣿⠛⠟⠋⠛⠛⠙⣟⡟⣻⣷
⠀⠀⠀⠘⢻⠏⢻⠋⠙⣯⡿⠟⠛⠉⠛⢿⣿⡿⠟⠿⣿⣟⣿⣯⣿⡟⢻⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣟⣿⣿⣉⣿⠛⢺⡟⠙⣿⡛⣿
⠀⠀⠀⢰⡾⠦⢼⣳⠾⠉⠀⠀⠀⠀⠀⠀⣿⠃⠀⠠⠈⢿⠏⠁⠈⡿⠛⠉⠀⠀⠀⠈⢻⠋⠉⠉⠉⠉⠻⠏⠉⠛⠛⢻⡇⠀⣟⣿⡏
⠀⠀⠀⢸⣿⣀⣾⠋⠀⠀⠀⠀⢀⠀⢀⡴⠁⢠⡆⠀⠀⣿⠀⠀⠀⡇⠀⢀⡄⠀⠀⠀⡘⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⣾⡏⠹⣏⣿⡇
⠀⠀⠀⢸⣹⠓⢿⣆⠀⠀⠀⠀⣿⣷⠋⠁⠀⣿⠀⠀⢰⡟⠀⠀⠀⠇⠀⠈⠁⠀⣠⣞⠀⠀⠀⠉⠀⠀⢀⠀⠀⢀⣾⣻⡇⢰⣟⣿⡇
⠀⠀⠀⢸⣿⠄⠘⠙⢷⣄⠀⠀⠹⡇⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣤⣤⣴⠁⠀⠀⠼⠿⣻⡇⠰⣿⣿⡇
⠀⠀⠀⢸⣿⠏⣱⣾⡿⠿⣧⠀⠀⢡⠀⠀⠘⠋⠈⡆⠀⠀⠀⠀⠌⠀⢠⣦⠀⠀⣠⠊⠀⠀⢀⣿⠾⠏⠤⢤⡌⠀⠀⣹⡗⢺⣭⣿⡇
⠀⠀⠀⢰⣿⣧⣸⠋⠀⠀⠉⠀⠀⠀⡆⠀⠀⠀⠀⡇⢀⣀⠴⠀⠀⠀⠘⠛⠛⠟⣻⡶⢶⢶⡿⠁⠀⠀⠀⠀⠘⡆⠀⢿⡟⠋⠁⠀⠁
⠀⠀⠀⢸⣿⢧⣼⣆⠀⠀⠀⠀⢀⠀⠁⠀⠀⣀⣴⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⢉⠀⠀⢿⡇⠀⠀⠀⠀⠀⠀⣷⡀⢈⡇⢸⣶⣴⡆
⠀⠀⠀⢚⢻⡷⣹⠿⠿⠦⠤⠶⠿⠤⠤⠶⠿⠿⠼⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠟⠷⠾⠿⠦⠀⠉⠀⢤⡾⣿⠿⣿⡇⢸⢿⣿⡇
⠀⠀⠀⢽⣿⠅⢻⣄⣄⣤⣀⣀⣂⣀⣃⣀⣀⣂⣀⣀⣀⣀⣄⣀⣀⣀⣀⣀⣀⣀⡀⢈⣀⣀⣘⣀⣓⣀⣀⣀⣒⣒⣀⣸⣟⣸⣶⢿⡇
⠀⠀⠀⣶⣿⡋⣹⡟⢁⣤⣤⠀⠴⢦⣤⣤⣤⣤⣤⣤⣤⡤⠄⠀⠀⠀⠀⢠⣤⣤⣤⣤⠤⠀⠀⠀⠀⠀⠀⠀⠉⠁⢸⣿⡇⢸⡿⢿⡇
⠀⠀⠀⢿⣿⣍⣿⠀⡟⠉⣻⡗⢶⣿⡟⠁⠀⢠⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⢻⡇⢸⣿⣾⡇
⠀⠀⠀⣠⣿⣀⣽⠀⣿⣾⡏⠀⠀⢹⣿⣶⣶⣾⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣄⣀⠀⡀⢀⢠⣀⣄⣀⣀⠀⠀⢀⡧⢼⣧⣿⡇
⠀⠀⠀⣿⣿⠀⣿⠀⢻⡿⣷⣤⡴⠟⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢺⡇⢻⣿⣹⡇
⠀⠀⠀⣿⣿⡆⢻⠀⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⣤⣠⣤⣤⣄⠺⣿⣶⠙⠟⢶⣤⠤⠀⠀⠀⠤⣤⣤⣼⡇⢸⡇⢸⣿⣿⡇
⠀⠀⠀⣟⣿⣷⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⢯⣾⢳⣿⣟⣿⣿⣷⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡇⢸⡏⢹⣻⣿⡇
⠀⠀⠀⣉⣿⣤⣿⠁⠀⠀⠀⠀⠀⠀⠀⣀⣀⣾⡿⣱⣿⢋⣿⣿⢻⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡇⢸⠟⠋⠃
⠀⠀⠀⣿⣿⠅⣿⠀⣷⣤⣤⣤⣤⣶⣿⣟⣿⠏⡼⠟⠋⠘⣿⢿⣿⡏⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣶⠟⠁⠀⠀⠀⢸⣇⢲⡀
⠀⠀⠀⢿⣿⡦⣿⠀⣿⠟⠻⢿⣿⣏⣹⠟⠁⠀⠀⠀⠀⠀⠀⠀⣾⣷⣶⠀⢠⣶⡶⣿⣿⣿⣿⣿⣿⣏⠀⠀⠀⣀⡄⢸⡀⢹⣿⣿⡇
⠀⠀⠀⣠⣿⠁⣿⠀⣿⣤⣼⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠘⠿⠿⠿⠿⣿⣿⣿⣿⣿⠶⠶⣾⠁⠀⢸⡇⢸⣿⣼⡇
⠀⠀⠀⣿⣿⣩⣿⠀⣿⣻⡿⠟⠃⠀⠀⠀⠠⣀⣀⣤⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠛⣿⣏⣁⠀⢰⣿⣶⡄⢸⠁⢹⡿⣿⡇
⠀⠀⠀⣟⣿⡭⣿⠀⠟⠉⣠⠴⠊⡁⠀⣴⣾⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣞⣛⣛⣛⣋⣟⣛⢛⡉⣿⡼⠿⠃⠸⠃⢸⣿⣿⡇
⠀⠀⠀⣿⣿⠷⠛⠀⠶⢋⣥⣶⣿⠿⠋⠉⠀⠀⠀⢀⡀⠀⣀⣠⣤⣶⢶⣾⣽⣹⣹⣹⣍⣩⣭⣹⣿⣯⣩⠋⠀⠀⠀⠀⠀⢸⣿⣾⡇
⠀⠀⠀⣿⢏⠀⣛⠀⣾⡿⠛⠉⠀⠀⠀⠀⠀⠰⠎⢿⠙⠀⠀⠉⠀⠘⣿⠋⠀⠀⠀⠀⠘⠋⠀⠀⠀⠛⣡⡄⠀⠀⠀⠀⠀⢸⡿⢿⡇
⠀⠀⠀⣵⡇⣶⠿⠀⠁⣀⣠⣄⠀⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣤⣀⠀⢠⡤⢼⣿⣿⡇
⠀⠀⠀⡿⠃⠀⣠⢤⡈⠛⠛⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠾⠿⠟⠋⣠⣾⣧⢼⡏⢹⡇
⠀⠀⠀⣤⡾⠛⢻⠛⠛⠞⠛⢛⠻⠛⠛⠛⢻⠻⠛⠛⢻⠛⠟⠛⣿⠛⢿⠛⠿⠟⠷⢖⢳⠒⠶⠖⠒⢶⠒⠒⠿⢻⠳⠛⠛⢻⣟⣿⡇
⠀⠀⠀⡿⢿⣿⣛⣿⣿⣿⡿⠋⠉⠉⠛⣿⣿⣿⣿⡛⣿⣳⣿⣿⣿⠿⣿⣿⣿⣶⣷⣿⣿⣾⣿⣷⣿⠿⣷⣴⣾⣾⣶⣗⣶⣾⣉⣽⡇
⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠒⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠿⠿⠟⠿⠿⠿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⡿⠋⣉⣉⡙⠻⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⡋⢠⣾⣿⣿⣿⣆⠘⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⢿⣿⣿⡆⠀⣀⣼⣿⣿⣿⣿⣿⣿⣷⣶⣶⣾⣿⣿⣿⢟⠵⠛⠉⠀⠀⣽⣿⣸⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣏⠄⠀⢸⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢠⣶⣿⡴⠞⣫⣽⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⢠⡦⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⣴⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡾⣷⣿⣿⡟⣼⣿⣿⣿⣿⣿⣿⠿⢛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⡿⢰⣿⣿⣿⣿⣿⡿⠃⢀⡈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡟⡿⡿⠁⠘⣿⡏⣿⣿⣿⢁⣀⣸⣇⣼⠟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠼⠷⠞⠁⠀⠀⠈⠳⣌⣛⡁⠘⠛⠛⠛⠛⠳⣝⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠿⠒⠒⠈⠉⠉⠙⠒⠢⠄⠀⠀⠈⠉⠉⠉⢦⠀⠀⠀⠈⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡁⠀⠀⢀⣀⣤⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⠉⠙⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡻⣿⣿⣿⡿⢋⣿⣃⠄⠀⠀⠀⠀⠀⢀⣠⣾⣋⣁⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢹⣿⣿⡇⠘⠉⠀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠙⠛⢷⡄⠀⠀⠀⠀⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡟⡛⠛⠓⠶⢤⣤⣤⣤⣶⡏⠀⠀⡀⠀⢀⣀⣤⣿⡄⠀⠀⠀⠃⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣇⠀⠀⠀⠀⠀⠘⡀⢸⣧⠀⢀⣵⠞⢋⣭⢾⡿⠁⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡻⡆⠙⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⣿⡶⠛⢁⡴⠋⢀⡞⠀⠀⠀⠀⠀⢀⣤⣾⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⢁⡴⠋⠀⢀⡞⠀⠀⠀⠀⢀⣴⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢠⠊⠀⠀⢀⡞⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⡠⠃⠀⠀⢠⡞⠀⠀⠀⢠⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⣰⣿⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⣰⢿⣿⣿⣷⣿⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⢀⡼⠃⠀⠉⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣰⠟
⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠁⠀⠀⠀⠀⣠⠞⠁
⠀⠈⠻⣶⣶⢤⣤⣤⣤⡤⠶⠶⠚⠋⠁⠀⠀⠀⠀⣀⡴⠛⠁
⠀⠀⠀⠀⠉⠓⠶⢤⣤⣄⣀⣀⣀⣀⣀⣤⡤⠖⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⢀⡴⢋⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣯⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠴⠖⠒⢛⣛⣉⣉⣉⣉⣉⣉⣉⣉⣙⡛⠒⠲⠦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠖⢛⣉⣥⠴⠖⠒⠛⠉⢉⣉⣉⣀⣀⣀⣀⣤⣤⣬⣭⣭⣽⣷⣦⣄⠈⠛⢦⡀
⠀⠀⠀⠀⢀⣠⠖⢋⣡⠴⠚⠉⢁⣠⠤⠶⠚⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠀⠀⢷
⠀⠀⠀⣰⠏⠁⡴⠋⠁⣠⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⢀⡟
⠀⠀⢰⠇⠀⢸⡁⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠁⢀⡴⠋
⠀⠀⠘⣧⠀⠈⠻⢧⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⠤⠶⠚⠉⠁⣀⡤⠞⠋
⠀⠀⠀⠈⠓⠦⣄⡀⠈⠉⠙⠛⠒⠒⠲⠶⠶⠖⠒⠒⠚⠛⠋⠉⠉⠀⣀⣀⡤⠴⠚⣫⠟
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⢶⡶⠶⠤⠤⠤⠤⠤⠤⠶⠶⠖⠒⠚⠋⠉⠉⢀⣠⡴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣤⣀⣀⣀⣀⣀⣀⣀⣤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠛⠛⠛⠉⠉⠉⠉⠛⠛⠛⠛⠋⠉⠉⠉⠙⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⡿⠟⠋⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠛⠻⢿⣿⣿⣿
⡿⠋⠁⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠉⠻⣿
⡇⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⣿
⣷⣄⠀⠀⠀⠀⠈⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠊⠀⠀⠀⠀⠠⣊⣴⣿
⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⣾⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣥⣤⣤⣄⣂⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣂⣤⣤⣤⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣴⣶⣾⣿⣿⣿⣿⣷⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⡿⠟⠋⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠙⠻⢿⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠁⠀⣀⣤⡴⠶⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣆
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠃⠀⠀⠾⠋⠀⠀⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠘⢿⣷⡀
⠀⠀⠀⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⣴⣶⣦⠀⠀⠀⠀⠀⠀⠀⠉⢀⣀⠀⠀⠀⠀⠉⠀⠀⠀⠈⢿⣷⡀
⠀⠀⠀⠀⠀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷
⠀⠀⠀⠀⢰⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡿⠏⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇
⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢸⣿⣿⣧⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⣼⣿⣿⡇⠀⠀⠈⠉⠉⠉⠉⠉⠛⠛⠛⠶⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⢿⣷⡀⠀⠀⣿⣿⣿⣿⣀⠀⢀⣀⣀⣠⣤⣤⣤⣴⣶⣶⣶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠈⢿⣷⡀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠈⢿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⢀⣀⣠⣴⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠛⠻⠿⢿⣿⣿⣿⣿⡿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠾⣆⠐⠙⢷⠀⠀⠴⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠠⠄⢠⡎⢁⡀⠉⡆⠀⠀⢸⡯⠑⢂⠈⠀⠀⡀⠀⠀⠀⠀⠀⢱⣄⠀⠀⠀⠀⡀⢀⣀⡀
⠀⠀⠀⠀⠀⢨⣁⣀⢀⠀⠀⠀⢘⢷⣺⣁⡰⠃⠀⠀⢸⣿⠴⢤⠠⠘⠇⠃⠀⠀⠀⠀⠀⠈⠻⣿⠆⢀⣤⠈⢈⠟
⠀⠀⠀⠀⠀⢸⣥⣌⠘⢃⠀⠀⠈⢰⡟⠁⠀⠀⠀⠀⠐⣿⠙⣄⠲⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⠌⡐⡲⠃
⠀⠀⠀⠀⠀⠈⣿⣻⡓⠓⠛⠳⠊⡈⡇⠀⠀⠀⠀⠀⢸⣷⣘⣿⢄⣶⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠘
⠀⠀⠀⠀⠀⠀⢻⣯⠀⠀⢸⡆⠘⠘⠀⠀⠀⠀⠀⠀⢈⣿⣷⣄⢀⠩⠄⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠆
⠀⠀⠀⠀⠀⠀⢸⢻⠇⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢅⡈⠱⣂⡄⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠤⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⡟⠰⠊⠀⡇⠀⠀⠀⠀⠀⠀⠀⠢⢤⣀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡛⠛⠒⠲⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡄⠀⠀⠀⠰⢲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣤⣠⠐⠘⠀⠀⠂⠘
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡀⠂⠂⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣾⡁⢀⠀⢠⡈⢆⣂
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣻⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⣤⣼⣇⠹⠾⠿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠿⠖⠚⠸⠇⠀⠀⠀⠀⢀⢰⡀⢀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠈⠻⣿⣶⠖⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠖⣠⡀⠄⠀⠀⠀⠀⠀⠀⢸⣅⡰⠇⠁⠘⠅⡅⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⢸⣏⣇⣀⠀⠂⡀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠁⠈⠈⠛⠁⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣀⠀⢰⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⣀⣈⣀⡀⠰⠀⠀⠀⠀⠀⠸⠏⡍⠉⠁⠀⠀⠃

⠀⠀⠀⠀⠀⢠⡄⡄⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡄⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢸⡇⡇⡇⡇⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⣰⢛⣿⡇⠀⢀⡾⢋⣭⣿⣦
⠀⠀⠀⠀⠀⢸⡇⡇⡇⡇⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⠿⠟⠛⠿⠿⣿⣿⣿⣿⣿⣷⡄⠀⠀⠀⠀⣿⢸⣿⡇⠀⢸⣇⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢸⡷⠷⢷⡇⠀⠀⢀⣼⣿⣿⣿⡿⢋⣥⠶⢛⣛⣿⣿⣷⣦⣍⠻⣿⣿⣿⣿⣆⠀⠀⠀⣿⣿⣿⡇⠀⠘⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠈⠙⣿⠛⠁⠀⠀⣼⣿⣿⣿⠏⣰⠟⣵⣾⣿⣿⣿⣿⣿⣿⣿⣷⡌⢿⣿⣿⣿⡆⠀⠀⢿⣿⣿⡇⠀⠀⠈⠛⣿⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢰⣿⣿⣿⡟⢰⡏⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡈⣿⣿⣿⣿⠀⠀⠈⠻⢿⡇⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⣿⣿⣿⡇⢿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣿⣿⣿⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠸⣿⣿⣿⣧⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢁⣿⣿⣿⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢻⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢃⣾⣿⣿⣿⠇⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠈⢻⣿⣿⣿⣷⣌⡛⠿⣿⣿⣿⣿⡿⠟⣋⣴⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣷⣶⣦⣤⣴⣶⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣶⣶⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠾⠛⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠶⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⢰⣦⣤⣀⣀⠀⠀⣀⣴⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠈⠻⣏⠉⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⣿⡷⠶⣶⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠘⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⢠⡄
⠀⠀⠀⠀⢀⣼⠟⠳⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⡄
⠀⠀⠀⢀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠸⣧
⠀⠀⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⣠⣴⡿⠀⣠⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⣾⢃⣴⡟⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣠⡄⠀⠀⠀⣠⣦⠀⢀⣼⡟⠀⢠⣾⠇⠀⣠⡾⠋⣽⠃⣴⠿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣄⣤
⠀⣼⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣆⠀⣿⣇⠀⠀⢰⡿⣧⠀⣾⢻⡇⢠⡿⢻⣀⣾⠋⠀⠀⢻⣿⠃⠀⠈⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⣿
⠀⠛⢡⡟⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏⠙⠻⣿⡟⠻⢶⣾⡇⢻⣤⣿⠈⢷⣾⠃⢸⡿⠁⠀⠀⠀⠀⠁⠀⠀⠀⠈⢿⡄⡀⡀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⢀
⠀⠀⣾⠃⣀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣻⡏⠀⠀⠀⠀⠀⠀⣼⠇
⠀⢀⣿⣿⣿⠀⠀⠀⠀⠠⣴⡾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⢠⣤⣤⠶⠞⠛⠛⠻⠆⠀⠀⠀⠀⢹⡿⠀⢀⣴⠶⣤⣰⡟
⠀⠈⠋⢸⡇⠀⠀⠀⠀⠀⣼⡷⠀⠀⠀⠀⠰⣶⣤⣤⣤⣤⡶⠾⠛⠀⠀⠀⠘⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⠀⣿⣅⡀⠈⢻⡇
⠀⠀⠀⠘⣿⠀⣀⣤⣄⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠋⠉⢻⡆⠘⣿
⠀⠀⠀⠀⢻⣶⠋⠁⢹⣧⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣴⠇⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣼⠏⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢸⡇⢰⡿
⠀⠀⠀⠀⢸⡏⠀⣴⠋⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⣠⡿⢁⣾⠁
⠀⠀⠀⠀⠘⣿⡀⢿⡀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⡀⠀⠀⠀⠀⠹⣿⠋⣠⡞⠁
⠀⠀⠀⠀⠀⠙⣷⡈⢻⣄⡀⣿⠀⠀⠀⠀⠀⢀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟⠋
⠀⠀⠀⠀⠀⠀⠈⠻⣦⣈⢻⣿⠀⠀⠀⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠛⠉⠉⠉⠉⠉⠳⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠾⠛⣷⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⣤⣤⣤⣀⣀⣀⣀⣀⣀⣠⣤⣤⡶⢶⣿⣿⡟⠛⠛⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠹⣯⡉⠉⠉⠉⠉⠉⠀⠀⢀⣾⠋⠈⢿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠛⣿⡀⠀⠀⠈⠛⠶⣦⣤⣤⣤⡴⠾⠛⠁⠀⠀⢀⡿⠛⠷⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁⠀⠀⠀⠸⣧⠀⠀⠀⠀⢀⣴⣿⣤⣿⣦⠀⠀⠀⠀⠀⣾⠃⠀⠀⠀⠀⠉⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⣠⣾⠁⠀⠀⠀⠙⣷⣄⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⣙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠻⢶⣄⠀⠀⠀⢠⡄⠀⠀⢿⣀⣼⠋⢹⣇⠀⠀⠀⣸⠏⠹⣦⣀⡿⠀⠀⠀⠀⣠⡄⠀⣠⡾⠋⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠙⢷⣄⠀⠈⣷⡀⠀⠘⠟⠁⠀⠀⢿⣤⣤⣴⡟⠀⠀⠈⠻⠃⠀⠀⠀⣰⡟⢀⣼⠏⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠹⣧⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠁⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⣾⠃⠀⠀⠀⠀⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠹⣧⠀⣿⠀⠀⠀⠀⠀⢸⡟⠀⠀⠀⢿⡆⠀⠀⠀⠀⠀⠀⣸⡇⣸⡏⠀⠀⠀⠀⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢹⡆⣿⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠀⣿⠁⣿⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣄⣀⠀⠀⠀⣿⠁⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⣠⡞⢋⣩⣽⠟⠛⠛⠿⢿⣄⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠙⠟⣿⠏⣠⡾⢃⣤⠀⠀⠙⣷⣄⣀⣤⣽⣧⣴⣾⠛⠷⣦⡜⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠻⢿⣏⣴⣟⣡⡾⠿⠟⠋⠉⠻⢷⣄⠀⠀⠉⢻⣆⠈⢻⣿⢁⣠⣤⣤⣤⣤⣤⣤⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⣠⡶⠟⣹⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠙⣧⡀⠀⠀⢹⡆⣼⡟⣋⣉⣀⣀⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⣠⡾⠋⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⢸⣿⠋⠛⠉⠉⠉⠉⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡄⠀⠋⠀⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢷⣤⣀⠀⠀⠀⠀⢀⣀⣤⡶⠟⠋⠛⠻⠶⢶⣤⣤⣤⣤⣤⣤⣤⣴⠾⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠟⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠈⠉⠁

⠀⠀⠀⢀⢀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠠⠤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⢸⢸⢸⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠁⠀⠀⠑⠄
⠀⠀⠀⢸⢸⢸⢸⡇⠀⠀⠀⠀⠀⠀⡠⠂⠁⠀⠀⠀⠀⢀⡀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠑⠠⡀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⢸⢸⢸⡇⠀⠀⠀⠀⡠⠋⠀⠀⠀⢀⡤⠖⠈⠀⠀⠀⠀⠀⠀⠉⠀⠂⠄⠀⠀⠀⠀⠈⢄⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⣼⠾⠼⡇⠀⠀⠀⡼⠃⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠠⠀⢣⠀⠀⠈⢿⡀⠀⠀⠀⠀⡜
⠀⠀⠀⠘⣷⣀⣠⠇⠀⠀⣸⠁⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢃⠀⠀⠘⢷⣤⣆⣴⠞
⠀⠀⠀⠀⢹⣿⠏⠀⠀⢠⡏⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠠⠐⡀⠀⠀⠀⢻⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠇⠀⠀⠀⠨⡿
⠀⠀⠀⠀⢠⠀⠀⠀⠀⢸⡄⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠁⡇⠀⠀⠀⢀⡠
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠈⣇⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡿⣿⠇⠃⠀⠀⠀⢰
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢻⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠐⠐⠈⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠙⢆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⢀⠃⠀⠀⠀⠀⢸
⠀⠀⠀⠀⢸⠀⠃⠀⠀⠀⠀⠀⢲⡀⠀⠀⠈⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠁⠀⠀⠀⡠⠂⠀⠀⠀⠀⠀⣼⠀⡄
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠈⠉⠒⠂⠤⠤⠤⠀⠒⠂⠉⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⣯⠀⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠹⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⣀⠀⠀⠄⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠤⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⢰⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⢀⠀⠄⣀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢅⠆⠆⠺⠏⠳
⠀⠀⠀⠀⠀⠀⣦⣴⢰⣧⠸⠀⠀⠚⠊⠀⠀⠀⠀⠀⠔⠁⠀⠀⢀⣠⣤⢴⢶⢲
⠀⠀⠀⠀⠀⠀⠋⠈⠈⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⡈⢀⣀⣰⠺⡏⠻⠸⣸⣼⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠈⠈⠀⠀⠀⠀⠀⠁⢏⠂⠹⡄⣙⣀⣤⠿⠛⠁
⠀⠀⠀⠀⠀⠈⠀⠀⠁⠓⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠈⠓⠒⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡀⠀⡀⠠⡄⡄⡀⢀
⠀⠀⠀⠀⠀⠂⠤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢠⠰⠀⠁⠀⠀⠁⣤⠀⡄⠁
⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠰⠀⠈⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⢀⡀⠄⢀⢀⠀⠀⠠⢄⠰⡀⠤⢸⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢳⡏⣷⠊⠶⠳⠎⠘⠣⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠂⠀⠀⠀⠘⠀⠢
⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠒⠀⡴⠀⠰⢄⠘⠃⠐⠀⡀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠋⠀⠀⠁⠀⠀⠀⠀⡀⠀⢀⠀⡄⢰⢀⡓⠓⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⡿⠾⠘⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠈⠳⠄
⠀⠀⠀⠀⢀⡀⣀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⠄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉
⠀⠀⠀⠀⠳⠤⠿⠃⠾⠀⡯⡴⢾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⣉⡉⢦⠊⣍⢁⡔⢡⡍⢶⠁⠻⡤⠃⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡁
⠀⠀⠀⠀⠀⠀⠀⢰⣀⣄⣠⡄⠀⠀⠀⢸⠀⡋⢥⡇⠘⠘⢸⠃⠆⡀⣾⢠⠸⢃⡆⡎⠀⠀⠀⠀⠀⠀⠀⠀⡀⠢⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠰⣜⣋⡜⢣⡰⢀⠜⣄⠴⢠⢇⡘⢠⡾⡰⠇⠀⠀⠀⠀⡆⠙⠐⠂⠁⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠁⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠈⠀⠀⠁⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠉⢀⢋⠀⠰⠄
⠀⠀⠀⠀⠀⠀⢀⡀⢤⠐⢒⠊⠁⠀⢡⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠴⠒⠴⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠀⠀⢁⡈⡒⠀⠠
⠀⠀⠀⢠⠊⢹⠀⠀⣈⠠⠼⠐⠲⠄⠤⠆⠀⠀⠀⠀⠀⠑⠀⢀⠀⠂⠀⠀⡀⠀⠀⠀⠙⠲⠀⠀⠀⠀⠀⠀⠀⠀⠐⠖⠒⠛⠉⡟⠀⢀⠈
⠀⠀⠀⠈⠒⠊⠉⠁⠀⠀⠀⠀⠀⠧⠄⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠈⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣖⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⢤⠅⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣩⠀⠙⡤⠤⡤⠤⠤⡤⠤⠤⠄⢀⡀⣀⠀⠀⠠⠠⠤⠄⠐⠒⢌⡀⠀⠀⡀⠄⠀⠀⠠⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⢒⠨⠏⠌⠙⠒⠏⠉⢳⠉⠁⠈⠁⠉⠁⠐⠒⠒⠘⠋⠥⠹⠗⡠⢄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗
⠀⠀⠀⠀⠀⠀⠀⠀⣧⢝⡐⠒⠒⠂⢀⠼⠦⠴⠦⡤⠤⠤⡤⠤⠤⠤⠶⡂⠐⠒⠒⣂⠯⡼⡀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠈⠐⠢⣔⠁⠀⠀⠀⠀⣈⠗⢏⡀⠀⠀⠀⠀⠈⣢⠒⠉⠀⠀⠘⡇⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢸⠄⡀⠀⠠⠊⠀⠉⠢⢄⡠⠊⠁⠀⠀⠉⠢⡀⡠⠔⠉⠀⠑⢄⠀⠀⣀⡆⠀⠀⠀⠀⢄⠀⠀⠀⡄⠀⢈⠁
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⡑⠣⣀⠀⠀⢀⠊⠑⠦⡀⠀⠀⢀⠴⠋⠢⡀⠀⠀⢀⡨⢶⠉⠀⡇⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠏
⠀⠀⠀⠀⠀⠀⠀⢸⠀⡜⠀⠀⠀⠑⣔⠁⠀⠀⠀⠈⢢⣔⠁⠀⠀⠀⢙⣤⡔⠁⠀⠀⢣⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠈⡶⢄⠀⠀⢀⠊⠀⠑⢄⡀⠀⡔⠁⠀⠱⡀⢀⠔⠁⠀⠑⡄⠀⠀⡠⢶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠉⢢⡃⠀⠀⠀⠀⡝⠪⡀⠀⠀⢀⠜⢣⠀⠀⠀⠀⢘⡔⠈⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠃⠈⠒⢄⢀⠌⠀⠀⠈⢢⡔⠁⠀⠀⠱⡀⣠⠞⠁⢡⢀⠜⢃⠀⡀⠀⠀⠀⠀⠀⠀⡀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣀⠀⠀⡳⡀⠀⠀⠀⠜⠢⠀⠀⠀⢀⣽⠀⠀⣀⠔⠁⠀⠈⠅⠂⠅⠀⠀⠀⠨⠅⠂⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠚⠀⠬⠦⢀⣎⣀⣀⣱⡀⠠⠎⠄⠓⠉⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⠀⠐⠁


⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣶⡀⠀⠀⠀⠀⠀⠀⢠⠔⠈⠹⡄
⠀⠀⠀⠀⢸⣏⠢⣄⠀⠀⠀⢀⢿⣆⡀⠀⢣
⠀⠀⠀⠀⢸⣿⠂⠘⠲⠖⠒⠺⠈⢿⣾⡅⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡀⠀⠤⠤⠀⠀⠀⠀⠂⠀⠀⠠⠤⠀⣀⡀
⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⠘⣿⠇⠀⠟⠉⠘⠋⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠤⡀
⠀⠀⠀⠀⡻⠁⠀⠀⢲⣾⣷⡀⠀⠐⠈⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠦
⣀⣄⣤⡊⠀⠀⠀⠀⠀⠐⠉⠀⠀⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆
⣿⠗⣿⠏⢀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠸⠿⢶⣟⣛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⠀⠈⠙⠓⠒⢶⣀⠀⠀⢀⠄⠀⢀⢠⠴⡄⢀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⢤⡀⠁⠀⠌⣠⣁⣌⡀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠉⠀⠀⠀⠀⠀⠋⣡⡀⠁⠀⠀⠀⠀⢀⡄⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣤⣾⣧⣽⣿⣿⡘⠧⠀⠀⠀⠀⢰⡟⠈⣖⣡⣴⣾⣥⣤⡄⠀⠀⣠⣤⣿⣿⡄⠀⠀⠀⢀⠊⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⣿⣿⣿⠻⠂⠀⠀⠀⡰⠛⠛⠿⠿⠿⠟⠿⢿⣿⣴⡷⠟⠛⠉⢻⣿⡇⠀⠀⠀⢨
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣷⠿⢀⠀⠀⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠈⡇⢇⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢸⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠈⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠰⣨⠀⢸⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⢠⠀⠀⡰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠆⠀⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⡼⡟⢀⠃⠀⣆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⢠⡾⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠿⠟⠁⢠⣾⣦⡾⠉⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⣿⣧⣠⠾⠇
⣶⣶⡶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⣊⡥⠐⢲⠄⠠⡐⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⠎⠀⢊⢂⡈⠀⡠⠊⠢⡑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠠⠄⠀⠀⠒⠒⠒⠒⠒⠒⠒⠂⠀⡇⣸⠀⢀⡀⠂⠁⠡⠎⠄⠀⣁⢰
⠀⠀⠀⠀⠀⠀⠀⡟⠢⢄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣁⣀⣠⣼⠵⢳⡜⢇⠀⠀⡀⠸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠋⠁⠀⠀⢸⠈⠢⣢⠔⡡⠁
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⠂⣀⠥⠊
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢸⢰⠋⠒⠒⠒⠈⢁⠩⠳⡦⢄⡀⠀⣀⡠⠔⠂⠉⠉⠉⠲⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠈⡘⡀⠀⠀⠠⠂⠁⣠⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⡄⠀⠁⠀⠡⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢺
⠀⠀⠀⠀⠀⠀⠀⠀⢁⢁⠀⠀⠠⠀⠀⠀⠀⢀⠸⠁⢀⠀⠀⠀⠀⠀⠀⠀⢘⡈
⠀⠀⠀⠀⠀⠀⠀⠀⢸⢘⠀⠀⠀⠁⠁⠄⠂⠁⢠⠀⢀⠁⠀⠁⠀⠀⠀⠀⡉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠠⢈⣀⠀⠀⠀⠀⢠⠀⠀⠀⢷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⡰⠀⢀⠆⠀⠈⠓⠀⠀⠀⠀⠂⠀⢰⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢁⠀⠀⠀⡤⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⠀⠀⠀⠰⡀⠀⠀⢀⢀⠂⠀⠀⠀⠀⠀⢀⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠳⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⢬⡑⠒⠶⠤⠤⠄⠴⠂⢒⣩⠔⡺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⡈⠉⠁⠒⠒⠒⠊⠉⠀⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠒⠒⠒⠂⠉⠁

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⣀⣀⡀⢸⣿⣿⡧
⠀⠀⠀⣤⣄⠀⠀⠀⠀⣶⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⣿⣿⡆⠀⠀⣿⣿⣷⢸⣿⣿⣷⣤
⠀⠀⢸⣿⣿⡇⠀⠀⠀⣿⣇⠀⠀⠀⢸⣿⠀⠀⠀⣰⣿⣿⣷⠀⠀⣿⡿⣿⣾⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⢸⣿⣿⡇⠀⠀⠀⢹⣿⠀⠀⠀⣾⣿⣇⠀⠀⢿⣿⣿⣿⡄⢸⣿⡇⢿⣿⣿⣿⠻⣿⣿⠇⠀⢠⠀⠀⣾⠀⢸⣿
⠀⠀⠸⣿⣿⣿⠀⢀⣀⣼⣿⡇⠀⢠⣿⠁⣿⡆⢠⣿⣿⢻⣿⣧⢸⣿⠇⠘⣿⣿⣯⠀⠙⠿⠀⠀⣺⡆⢠⣿⡇⠘⠛
⣠⣤⣠⣿⣿⣿⡿⠛⠛⠉⣿⣿⠀⢸⣿⠀⠸⣿⣾⣿⡏⠘⣿⣿⣼⣿⡇⠀⠈⠛⠃⠀⠀⠀⠀⠀⠉⠃⠈⠉⠁
⠉⠉⠉⢻⣿⣿⡇⠀⠀⠀⢹⣿⣇⣿⣿⣄⠀⠙⠛⠟⠀⠀⢽⣿⣿⢿⠃
⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⠸⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠈⠋⠋
⠀⠀⠀⠀⣿⡿⠿⠇⠀⠀⠀⠋⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠳⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠀⠀⣀⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠁⠀⠀⠀⢼⣿⣤⣿⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⣾⣧⣼⣷⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠟⠃⠀⠀⠀⠀⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⣀⣀⠀⠐⠛⠛⠛⠛⠿⠷⣦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⡟⠉⠉⢳⡄⠀⠀⠀⠀⠀⠀⠀⠉⠻⠇⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⢀⡇⠀⠀⠀⣿⠀⠀⠀⣀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠃⠀⢀⣴⣯⡴⠞⠛⠉⠁⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⠀⠀⠛⠉⠀⠀⠀⠀⢀⣀⣤⡴⠾⠋⠀⠀⠀⠀⠀⠀⠀⡾⠃
⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠞⠛⠛⣧⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣀⣀⣀⣠⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⢿⡉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣤⣀⣀⠀⠀⠀⠀⠀⢀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⣀⣀⠀⠀⠀⣀⡠⠤⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠠⠤⠤⣤⣄⣀⣀⡀
⠀⠀⠀⢰⡸⠭⠇⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠤⣀
⠀⠀⠀⠀⠉⡶⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢤⡀
⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⣦⣀⠠⠤⢺
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠄⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠋⠇⠠⡤⠔⠃
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⢀⡄⠠⠦⠠⢦⠁⣤⢟⢧⣷⢠⠴⠀⣤⢤⠤⠤⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠈⢦⠀⠀⠀⡀⠁⠞⠁⠐⡒⠄⠙⠸⠜⡧⠓⠯⠥⡈⠠⠭⠄⠀⠀⠠⠬⠐⠴⠤⠉⠂⠀⠾⠂⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠁⡸⣆⠀⠠⢸⠋⠒⠢⠤⠄⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠢⢠⣀⡀⠀⠊⠀⠀⠃⡀⠀⠳⣄
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠂⡴⠁⠸⡄⠀⢇⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⢸⠙⠀⠀⠀⢃⠀⠀⠉⠐⠒⠂⠶⠬⣀⢒⡞
⠀⠀⠀⠀⠀⠀⠀⡆⠈⣼⠀⠀⠀⠱⡄⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⡎⠳⡄⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⢧⠀⢧⠀⠀⠀⠀⠱⡈⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠐⣆⠀⠘⣄⠚⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠈⠓⠒⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⠉⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡖⠋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠶⠶⠞⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣄⣀⣀⠀⠀⠀⢀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠐⠿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠈⠑⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠉⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠲⠦⠤⠄⠀⠀⣀⣀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠙⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣄⠀⠀⠀⠀⠀⠀⠉⠓⠶⢤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠓⠒⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⡀⣀⡤⠶⡎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠉⠉⠀⣀⣿⣤⣤⣄⠀⠀⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠄⠀⠈⠷⣤⣄⠀⠈⠉⠁⠀⣠⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢓⠐⠒⠘⢖⠀⠀⢙⡟⣠⠶⢲⡶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠧⠀⠀⠤⠳⣰⠟⣰⢋⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠰⡀⠀⠀⠑⣄⣠⡞⠃⣠⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠀⠐⡄⠀⠀⠀⠙⠶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⢠⡈⢃⠀⠙⠂⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⢣⡈⠳⠐⠒⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢠⡀⠙⠁⢉⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠰⡄⠈⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠑⢦⠈⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣧⡴⠋

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣽⣏⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣀⡴⠟⠁⠀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣿⣿⣿⡯⢚⣭⠶⠞⣛⣉⣉⣩⣭⣄⢠⣿⣿⣿⣿⡿⠇⠀⠸⠀⠀⢠⠃⠀⠀⢀⣼⡿
⠀⠀⢀⣤⠞⠋⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣿⣿⢟⣩⢞⣩⣴⣾⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⠟⠉⠀⠀⠀⠃⠀⠀⠁⠀⠀⣠⡾⠟⠁
⣠⡶⠋⠁⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⡿⢟⡵⣋⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉
⠁⠀⠀⠀⠀⣠⣶⣶⣿⣿⣿⣿⣿⣿⣿⡿⣫⡶⣫⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉
⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡿⣫⠞⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤
⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⣫⠞⣡⣾⣿⡿⠟⠉⢁⣾⡟⣹⣿⣿⣿⠟⠁⠀⣠⠤⠶⠚⠛⠛⠛⠛⠻⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣥⠞⣡⣾⣿⡿⠋⠀⠀⠀⣼⣿⣿⣿⡿⠋⠀⣀⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠿⣫⡶⢟⣵⣾⣿⠟⠉⠀⠀⠀⢀⣼⣿⣿⡿⠋⡠⠒⠋⣁⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⣿⣿⣿⣿⣿
⣿⣿⣿⡿⣛⣵⠿⣫⣶⣿⡿⠛⠁⠀⠀⠀⠀⣠⣾⣿⣿⢋⠔⠈⠐⠉⠙⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣿⣶⣶⣶⣤⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿
⣿⢟⣭⡾⢟⣵⣾⣿⠿⠋⠀⠀⠀⠀⠀⣠⣾⣿⡿⠛⣑⣥⣴⣤⡻⠄⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣷⠿⣫⣶⣿⠿⠛⠁⠀⠀⠀⠀⢀⣴⣿⣿⢿⣯⡾⠛⠛⠛⠛⠛⠛⠳⠷⢦⣤⣈⠛⠶⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣵⣿⠿⠋⠁⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⡷⣋⠥⠒⠉⠉⠉⠉⠁⠀⠀⠀⠀⠈⠉⠛⠳⠦⣬⡓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣿
⠟⡃⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⠗⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠶⣭⣓⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠻⠿
⠀⠀⠀⢀⣴⣾⣿⣿⣿⣻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢭⣓⠦⢄⣀
⠀⢁⣴⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠶⠭⣕⣒⡠⠤⢀⣀⣀⣀
⣴⣿⣿⣿⡿⠋⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠲⠶
⣿⣿⠿⣋⠤⠂⠁⠀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣤⣤⣤⣤⣄⡀
⢟⡷⠊⣁⡤⠴⠚⠋⠉⠁⠉⠙⣿⣿⣿⡿⠿⠟⠛⠛⠛⠛⠉⠁⠀⠠⠃⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠆⠀⠀⠰⠀⠀⡌⠉⠉⠉⠉⠉⠉⠉⠉⣱⠀⠁⠀⠀⠀⠀⢗⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⠀⠀⠀⠀⠚⠴⠿
⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢨⠁⠂⠀⢠⡇⠀⠀⠀⠀⠡⠀⠀⠲⢦⠄⠀⠀⠀⠀⠶⣾
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠎⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢰⠀⠀⠘⣀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠦⡀⠀⠀⠂⠀⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠌⠀⠀⠀⠑⠀⢀⡀⠀⠁⠀⠀⠀⠀⠀⠀⢘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁
⠀⠀⠄⠄⠐⠄⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠈⠁⠀⡀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡄
⠀⠀⠀⠀⠰⠀⣀⣄⠀⢀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⡝⠃⠀⠀⡀⠰⠂⢀⠀⠀⠀⢀
⠀⠀⠀⠀⢀⠃⠄⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠸⠁⠆⠀⠀⠠⠄⠟⠀⠀⠀⠀⠀⠀⠀⢩⠄⠀⠠⡝⠓⡐⠀⠀⠀⠀⠀⠀⠖⠄⠀⠀⠀⠐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠂⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⣤⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠐⡦⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠔⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠐⠀⠄⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉
⣿⣿⣿⠿⣿⠿⢿⣿⣿⡿⢿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣧⣷⣮⣶⣷⣶⣿⣶⣿⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⡤⢤⡤⣤⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⡿⠿⡓⣛⣽⢥⣍⡩⣽⣭⢅⡉⣝⣚⡻⢿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠛⢉⠀⡰⢾⣷⢷⢿⣄⠸⠈⠃⠼⠎⠸⢀⣦⣼⡒⠦⠭⣟⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠿⠋⡴⠒⠆⣻⣮⠂⠀⠓⠊⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠋⠳⠀⠀⣻⣦⠉⢻⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⡹⢶⠞⣥⣠⠀⠘⠀⠀⠀⠀⠀⠀⠀⠠⠄⠠⠄⠀⠄⠀⠀⠀⠀⠀⠈⠓⠻⠉⢛⣶⢝⢻⣧⡀
⠀⠀⠀⠀⠀⠀⢀⣼⡿⢃⠖⠑⢾⣝⠃⠀⠀⠀⡀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠣⠟⢿⡶⢽⣿⣄
⠀⠀⠀⠀⠀⢀⣾⡟⣶⡋⣤⠛⠀⠀⠀⠀⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠤⡻⡌⣿⣦
⠀⠀⠀⠀⢀⣾⡟⣸⠻⠂⠉⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠘⡾⠞⢿⣇
⠀⠀⠀⠀⣾⡟⣭⣉⡹⠆⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠛⣼⣿⡄
⠀⠀⠀⢸⣿⠫⣽⣿⣝⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡯⠆⠳⣽⣷
⠀⠀⠀⣾⣟⣼⡗⠃⠚⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠚⣿⢿⣰⣿⡄
⠀⠀⠀⣿⡟⣄⣗⣆⡀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠘⢧⣿⡇
⠀⠀⠀⣿⡇⣼⡟⠩⠄⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⣶⠃⣿⡇
⠀⠀⠀⣿⡟⡇⠀⢀⡄⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠻⡏⢀⣿⡃
⠀⠀⠀⢿⣟⢹⣤⣿⡤⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠀⢀⡖⣤⣿⢨⣿
⠀⠀⠀⢸⣿⣧⡨⢶⣾⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⣽⣿⡇⣿⡟
⠀⠀⠀⠀⢿⣷⣧⣄⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘⠀⠀⠀⢾⣉⡙⢡⣿⠃
⠀⠀⠀⠀⠈⢿⣿⣻⣧⡓⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠊⠀⠀⢀⢿⣾⡿⢠⣿⠏
⠀⠀⠀⠀⠀⠈⢿⣿⡛⣽⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⣀⣤⡞⢿⡿⢱⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠻⣿⣤⣼⠿⣶⣛⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣒⣙⠃⡱⢋⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣟⠿⣯⣀⡲⡴⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠈⠉⣩⠿⢚⣱⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣷⣥⡀⠚⣞⠄⠀⠸⣆⡄⣦⠀⠀⠀⠀⡀⠀⠀⣀⢠⡒⢀⠀⡺⢷⠛⠒⣃⢴⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣶⢮⣆⡐⠀⠿⠇⠀⠀⣛⠩⢄⠤⣚⡁⢰⡞⠻⠶⠾⢤⣡⣴⣾⠿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠾⣯⣈⣦⣤⣬⣉⣁⣓⣉⣁⣡⣾⣭⣶⣶⡿⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠋⠉⠉

⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⣧⢀⣀⣠⣤⣤⣤⣤⣄⣀⣀⠀⠀⠀⣠⣤⣄
⠀⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⢹⠋⠉⠁⠀⠀⠀⠀⠈⠉⠙⠛⢶⣾⠁⠀⠙⠷⣤⡀
⠀⠀⠀⠀⢰⣿⡿⠛⡛⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠻⣄⠈⠛⠷⣦
⠀⠀⠀⠀⢸⣿⣧⡀⠈⣨⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡳⣴⠿⣦⠀⠀⣿
⠀⠀⠀⠀⢸⣟⢿⣟⣛⢉⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠳⢤⣄⣀⡀⠀⠀⢸⢇⡴⠆⠀⠀⠀⠀⢀⣠⡤⢤⣄⡀⠉⠉⡹⣦⠘⠷⣴⠏
⠀⠀⠀⠀⠀⠹⣦⡌⠙⠋⠉⠙⠛⠛⠛⠛⠛⠛⠟⢛⡟⠀⠀⠀⠀⠈⠉⠉⠉⠉⠠⢦⡄⠀⠀⣠⠞⠉⠀⠀⠀⠀⠉⠳⣦⡿⠘⢷⣄
⠀⠀⠀⠀⢀⣾⡋⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣾⡟⢠⠞⠁⠀⡴⠋⢷⠀⣾⠳⢄⠈⠳⡄⠀⠙⢷⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⡀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⣼⠁⠀⣿⠀⠹⡄⠈⡇⠀⠹⡄⠀⠈⣿
⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠢⠈⠀⠒⠈⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠹⣤⡴⠃⠀⠀⠉⠛⠁⠀⢠⠇⠀⠀⣾⠁
⠀⠀⠀⣿⠂⠁⠀⠀⠠⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣰⠏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠿⣄⠀⠀⠀⠈⠉⠓⠲⠶⠶⠶⠖⠊⠁⠀⠀⢀⡾⠋
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⢀⣠⡴⠟⠉
⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣶⣦⣤⣤⣤⣤⣤⠶⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠈⠻⣦⡀⠀⠀⠐⣤⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⡤⠀⣀⣤⡞⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⡏⠉⠓⣦⣤⣤⣀⣀⣠⡇⠀⠀⠀⣼⡗⠋⠁⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⣿⣦⣉⣽⡄⠀⠉⠉⠉⣧⠀⠀⢰⡿⣷⠄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⢙⣧⠉⠉⠀⠀⠀⠀⠀⢻⣅⠀⢸⡇⠿⠶⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢰⠟⠉⠉⠙⠻⠉⠀⠀⠈⠻⣆⣀⡤⢤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠋⠉⠉⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠁⠀⠀⠀⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠲⢦⣄
⠀⠀⠀⠀⠀⠀⣴⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷
⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⢈⣷⠄⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠺⣧⡀
⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⢸⣧⢰⣦⡀⢂⣄⣲⣦⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠛⠦⣤⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠈⠈⠈⢁⣠⣾⠟
⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡍
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠦⠰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⡀⠀⠀⠀⠀⠀⠀⣠⠞⠢⣄⣀⠀⠀⢀⣀⣠⡾⠲⢤⣤⣶⠿⢋⡴⠒⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⠶⠶⠶⠖⠛⠁⠀⠀⠀⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠈⠧⣀⣀⣼⢂⡤⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⡼

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠐⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠉⠢⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠆
⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢆⡀
⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠘⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡦⠄⢀⣀⠀⠄⠀⠢⠤⠄⠤⠄⠐⠈⠉⠁
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠐⠈⠉⠈⠉⠲⢄⠀⠐⠈
⠀⠀⠀⠀⠀⠀⡀⠀⡁⠂⠀⠑⡄
⢀⡂⠀⠀⠀⠘⠟⠀⠏⠀⠀⠀⢱
⢠⠑⡄⠢⠄⣀⠀⠀⠀⡀⠀⠀⠘
⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⡌⠑⠢⢄⣀⢀⠀⡄⠊
⠀⠀⡇⠀⠀⠀⠀⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠢⡀
⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠃⠀⠈⠒⢄
⠀⠀⣀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠗⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠂⠁
⣶⣶⡶⡶⠶⢶⣶⣶⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠈⠒⠒⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠁⡀⡀⠐⠈⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⠈⠉⠀⠀⠀⠀⡠⠈⠈⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠐⠀⡠⢅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠁⠤⠤⠠⠄⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠁⠀⠀⠀⠀⠀⠈⢀⠀⠀⠆⠀⡰⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣈⡂⠤⣀⣈⠀⠀⠀⡀⠈⠀⠀⠀⡊⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡈⠓⠀⠀⠄⠄⢀⢀⣱⠀⢠⠴⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠈⠤⠀⠰⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠈⠂⠀⠀⠀⠀⠄⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⡀⠀⠈⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠐⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⢂⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠐⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⡀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⢠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠠⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠐⢸⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣣⣊⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣛⣻⣟⣿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠤⠤⠤⠤⣤⣤⣄⣀⣀⣀⣀⣀
⠀⠀⠀⠀⡔⡻⢶⢀⣀⡤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠙⠓⠒⠦⢄⡀
⠀⠀⠀⠀⠳⣌⣊⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⣄
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠑⢄⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⣵⢤⠤⠐⠒⣻
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠫⠼⠀⢤⡤⠞⠁
⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠆⠀⢀⠸⠸⢀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠀⠈⠀⢀⠐⠂⠀⠀⠀⠈⠀⠀⠎⠃⠂⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⠀⠀⠐⠀⡀⠀⠀⠀⣰⡆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⡊⠀⣴⠀⠀⢐⢑⡶⠤⣄⣈⠀⠀⠀⠁⠀⡀⠀⢀⠀⢠⠀⠀⠀⢀⠀⡀⠅⠀⠀⠄⢀⠀⢠⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠀⡰⠉⣧⠀⠀⣼⠀⠀⠀⠀⠉⠉⠐⠒⠒⠒⠒⠒⠒⣶⢆⠀⠀⠈⢰⠋⠙⠒⠤⠀⣀⡀⢀⡑⠄⢀⡌⣻
⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢀⡜⠁⠀⠸⣄⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢠⣄⠀⠈⣆⠀⠀⠀⠀⠀⠀⠉⠉⠓⠢⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢹⠁⠀⠀⠀⠘⡆⢀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠁⢀⠇⠈⢣⡠⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢺⡀⠀⠀⠀⠀⠙⣌⣚⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠘⣦⠀⠀⠳⡌⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠓⠚⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠰⡀⢂⠐⠀⠀⢦⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠂⠐
⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠒⠀⠀⠐⠀⠈⡇⠀⠀⠀⠸⠀⠁⠀⠈⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠠⠀⠀⠈⠰⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⢢
⠀⣬⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⢀⡠⠿⡟⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠆
⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡘⢱⡐⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠨⠀⠀⠀⢈⠀⠀⠄⠀⠀⠀⠄⠀⠙⠀⢡⡀
⠀⠀⠀⢠⡈⠰⠄⠀⠀⠀⠀⠀⠀⠀⣀⣈⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀
⠀⠀⡠⠛⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢈⡀⠀⠈⠃⠀⠀⠀⠀⠃⡀⠓⠄⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⠀⠀⠹⢿
⠀⠀⢰⠇⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡔⢉⡈⠉⠀⠐⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠐⢷⣧⡀
⠀⡘⠈⠀⠀⠀⠈⡄⠁⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⡀⠈⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢳⠀⡀
⠀⣕⠢⠀⠀⠀⠘⠁⠀⣀⣀⠀⢀⠀⠄⠀⠀⠀⠀⡀⠀⢀⣀⢑⡂⡙⠃⡄⠀⠀⠀⠀⠘⠀⠀⠂⠀⠀⠀⡀⠀⠀⠃⠀⠈⣦⠁
⠀⠐⠦⠀⠀⠀⡠⠀⠀⠀⠁⠀⠀⡀⡆⠀⠀⢀⠲⠤⡄⠀⠀⢀⡄⣀⠀⢡⠀⠀⢀⢀⣀⠈⠃⠀⠀⠀⠀⠀⠀⠟⣶⡀⠀⠂
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠑⠀⠀⠀⣀⡈⢴⡄⠀⠈⠛⠁⣴⠏⠀⣀⡭⠃⣈⣿⠆⠀⠀⠀⠀⠀⠀⠠⠻⡧
⠀⠀⠀⠀⠂⣤⠀⠀⠀⠀⢰⣷⠰⡂⢠⠀⠀⠀⠿⣿⣆⣀⠀⠀⠉⠉⠁⡠⠀⡈⣑⢙⠝⡡⡃⠀⡀⠀⠄
⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣔⣤⣉⠄⠀⠀⠀⠈⢎⠻⣶⠎⢐⡠⣺⠁⡔⠃⠚⢹⠀⡇⠁⣴⡁
⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠆⠹⠁⠁⠀⠀⢀⠠⡀⢚⠓⡄⠑⠦⢀⡡⣾⣖⢀⡄⠈⠓⠾⣷⣄⣤⠄⠀⠀⠀⠀⠈
⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡗⢤⣀⣀⠀⠀⠈⠂⢐⠀⠀⠘⢦⡀⠀⠀⠀⠬⠕⢦⣀⡀⠀⣀⡙⠣
⠀⠀⠀⠀⠀⠀⠀⠐⠂⣶⠀⠀⢴⠈⠿⢴⣦⠃⠀⠀⠐⢄⣀⣀⡙⠢⣀⠀⠀⠀⠀⠈⠣⠀⠈⠙⠳⡀⠘⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠠⠆⠀⠀⠀⠀⠀⠘⠃⠀⠑⠤⠀⠀⠉⠀⢠⠀⠄⠋⡀⢨⡷⢀⡗⠰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠐⠲⠶⠷⠐⠁⠘⣧⣤⡥⠜⢁⠂⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁⠐⠈⠁⠀⠀⠆⠙⠛⠉⠁⠀⠁⠀⠰⠾⢷⡆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠶⠛⠛⡟⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠶⠚⠉⠀⠀⠀⣠⠞⠁⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠉⠀⠀⠀⠀⢀⣤⠞⢉⣠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⠀⢀⣠⠴⠞⣋⣤⠶⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⣠⡴⢞⣫⡴⠶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠋⠀⢀⣤⣶⣿⡷⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⢀⣤⣾⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢀⣠⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⢀⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⣠⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⣞⣫⡥⠶⠶⣦⡀⠀⠀⢀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⣚⣭⠶⠛⠉⠀⠀⠀⠀⢸⡇⠀⢠⡾⠁
⠀⠀⠀⠀⠀⣀⡴⣛⡵⠞⠉⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁⣰⡟⠁
⠀⠀⠀⣠⢞⣭⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⣠⢾⡟⠁
⠀⣠⢞⣵⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⣤⠾⢋⣴⠞⣡⠟
⣼⢫⡟⠁⠀⠀⠀⠀⠀⠀⣀⣤⠾⠋⣁⡴⠋⣡⡾⠃
⣿⠈⠷⣤⣤⣤⣤⠶⠾⠛⠉⣀⡴⠞⠉⣠⡾⠋
⠘⢷⣤⣄⣀⣠⣤⣤⠶⠞⠛⢉⣀⣴⠞⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠛⠓⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠏⢯⣭⢾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⣾⠿⠛⢉⣇⣴⣼⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⢀⣀⣾⡿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⣉⣠⣤⣤⠶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⣋⣥⣶⠾⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⣡⡴⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠒⠚⠉⠉⠻⣿⠉⢠⡟⠁
⠀⠀⠀⢀⣠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠀⣿⠃
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⣿⡿⠋
⠹⢴⣤⣤⣤⣤⣶⣶⣞⣻⠭⠭⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠡⠂⠘⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠂⠀⠀⠑⠀⠑⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠆⠀⢂⡕⠉⠉⠑⡤⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠀⢀⠤⠂⠁⠀⠀⠀⡠⠁⠸⠂⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⡠⠐⠈⠀⠀⢀⡀⠄⠐⠈⠠⠂⢀⠥⡔⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡓⢁⠀⠠⠶⠠⠬⠄⠒⠀⠈⢉⠰⠂⠁⢀⢆⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠔⠉⢀⠤⠂⠁⠀⠀⢀⠤⠐⠈⠀⢀⠠⠐⠁⡈⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⡠⠊⢁⠀⠠⠐⠈⢁⠀⠠⠐⠂⠁⠀⠀⢀⠴⠄⠒⠛⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⢀⠔⠁⡀⠄⠂⠁⠀⠀⠀⣀⡀⠤⠀⠀⠁⠀⠀⡠⠊⠀⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠔⠉⠀⢠⣊⠀⠀⢀⠀⣀⢀⡄⠀⠁⠀⠀⠀⠄⠀⣀⡀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⡄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠠⠄⠤⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⠠⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠄⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⠀⠠⡤⠀⢄⣄⠀⣀⡀⠀⠄⣤⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠢⣀⠈⠙⠢⢀⡀⠀⠠⠒⠂⢤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⠓⠲⣀⠉⠉⠑⡠⠤⠊⢁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠢⢤⠄⠑⠒⠄⢀⣀⠄⢚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⣀⠀⣈⠰⠤⢄⠈⠈⡡⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡁⠈⠑⠠⣀⡀⠠⠈⠉⢐⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡋⠑⠌⡉⠀⠉⡀⠖⠮⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠚⠛⠃⠀⠒⠂⠉
⣿⣿⣻⣻⣛⣟⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣛⣛⣿⣿⣛⣻⣛⣻

⠀⠀⠀⠀⠀⠀⢀⣔⠒⢢⠀⠀⠀⠀⠀⠀⠀⢀⣰⣦⠀⠀⠀⠀⠀⠀⣀⠤⣠⣀⢀⠀⠀⠀⠀⣠⣴⠀⢤⣄⣀
⠀⠀⣘⣷⡄⢰⠉⣠⠒⠚⠢⠀⠔⠒⠢⡀⣴⣿⣯⣼⡁⠀⠀⠀⠀⢰⣿⣾⡛⠟⢷⡀⠀⠀⢰⡇⠸⣆⠀⡽⠿⢷
⠀⣾⣛⢧⣿⠜⣲⠧⠀⢀⣐⠆⠆⠀⠤⡇⢿⣯⣿⣿⢁⣤⠀⠀⢰⡸⣿⣽⣇⣉⣬⣷⠀⠀⢺⣿⣦⡈⠉⣳⣾⣿
⠀⢸⣿⡿⡟⠀⠙⣷⣾⣴⠋⠀⢴⣶⡶⠁⠘⣿⣳⡟⢛⣉⣀⣀⣸⠀⠻⣿⣯⣿⣼⠏⠀⠀⢸⣿⣿⣻⣷⣿⣿⡟
⠀⠘⣿⣿⠃⠀⠀⢿⣻⡍⠀⠀⠘⡝⠁⠀⠀⢿⣿⠇⠀⢿⣭⣼⠃⠀⠀⣠⣿⢹⣀⠀⠀⠀⠀⣇⠀⠈⡉⠈⣿
⠀⠀⢹⡇⠀⠀⠀⠈⣭⠃⢀⠀⠀⠑⠀⠀⠀⢸⣻⠀⠀⢸⣗⣿⠀⠀⠀⠉⠁⠈⢉⣴⠖⠂⠀⠻⣄⡀⣀⣠⡿
⠀⠀⢈⠁⠀⠀⣀⣀⡙⠀⣿⢯⣿⠀⡞⢻⡆⢀⣭⣤⣀⠀⠃⠃⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⣨⣄⠀⠈⠉
⠀⠀⠈⠀⡄⢠⣿⣿⣽⡄⣿⣾⡟⠀⡃⢸⡇⣾⡏⢹⣿⣇⠀⠀⠀⠀⠀⠀⢀⡀⠸⣷⢦⣤⠿⠿⠿⠿⠿⠿⣶⡄
⠀⠀⠀⠀⢡⢸⣿⣿⡇⡁⣿⣽⣿⠀⡆⢸⡇⣿⡇⢈⣿⣿⠀⠀⠀⠀⠀⠀⢙⣇⣤⣶⣭⣁⡄
⠰⡟⠀⢬⣿⣾⣿⣿⡇⡅⣿⣿⣿⠀⡇⢸⡇⠻⣧⣠⣿⠏⢀⣀⣠⢤⣄⠀⢈⢻⣿⣿⣯⣿⠧⠄⣶⠀⠀⡀⡴⡄
⠀⠉⠉⠋⠉⠘⠻⠿⠓⠀⠙⠛⡏⠀⠳⠾⠃⠀⠀⠈⠀⣼⣿⣥⣿⣿⣿⠢⣀⣼⢀⣀⣘⣿⠀⣦⢸⣦⣶⣶⣶⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢿⣿⣿⣿⠇⠀⠈⣻⠿⠿⣿⠁⠀⢸⣯⣻⣸⣟⢻⡬⡀
⠀⠊⠁⠉⢳⠀⢠⠖⠒⢤⡀⡠⠒⠒⢦⢀⡴⠒⠒⢆⢹⣶⣾⣿⣬⣿⡆⠀⠀⠉⡀⣠⠃⠀⠀⠘⣿⣷⠿⣿⢿⣿⠇
⠀⣷⠀⣀⣼⠆⣗⠀⠀⢰⠇⣷⠀⢀⣸⠘⣆⠀⣠⣾⠰⣟⠀⠀⢹⣯⠁⠀⠀⢞⡓⣻⡄⠀⠀⠀⠈⣿⢿⢻⣿⡇
⠀⠈⠉⠉⠉⠀⠈⠁⠉⠉⠀⠈⠉⠉⠀⠀⠈⠈⠉⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠉⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠄⠒⠒⠂⠀⠀⢀⡉⠀⠇⠀⠘⠀⠀⠀⠈⠉⠙⠁⠒⠒⠲⠤⠤⣤⣀⡀
⠀⠀⣀⡤⠖⠛⠋⠁⠀⢀⡤⠖⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠉⠙⠒⠤⣄⠐⠒⠀⠀⠉⠻⡆
⠀⢸⡏⠀⠀⠀⠀⠀⠀⠙⠳⢤⣤⣀⣀⣀⣀⣀⣙⣉⣀⣉⣁⣀⣀⣀⣀⣤⣤⡤⠞⠁⠀⠀⢀⣀⣴⠇
⠀⣘⠳⣦⣤⣄⣀⣀⣀⣀⡀⠀⠀⢀⡉⠉⠉⠉⠉⠉⠉⠉⠉⣀⣀⣀⣀⣠⣤⡤⠴⠶⢶⠿⠛⠉⠀⠈
⠀⠈⢩⠘⠓⠶⠬⠭⠍⣉⣭⣉⣉⣉⠉⠀⣈⣍⣛⠉⠛⠋⠉⣩⡍⠉⠀⠀⠔⠀⠀⠀⠀⢀⣀⣠⡄⠂
⠀⠀⠸⠂⠠⢤⣄⣀⣀⣀⡀⠀⠉⠀⠀⠐⠁⠀⠀⡀⠀⣀⣀⣀⣀⣠⣤⣤⡴⠶⠖⠛⠋⠉⠁⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉⠉⠉⠉⠉⠉
⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⢸⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣾
⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⠠⠀⠀⠒⠒⠂⠀⠃⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⣀⠀⣹
⠀⠀⠀⠀⣷⠀⠈⠀⠀⠀⡠⠤⠤⠄⠀⠀⠀⢀⣠⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡿
⠀⠀⠀⠀⢿⡑⠢⠄⢀⣐⠀⠀⠀⠀⠀⠠⠊⠁⠀⠀⠀⠀⠈⠁⠂⠀⠆⠀⠀⠀⢀⡀⠀⠀⢈⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢀⣨⠓⠚⠛⠶⢤⠠⠤⢤⣤⡤⠶⠦⢶⣶⣖⣉⠉⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡄⠀⠀⣠⠊⠀⠀⠀⠐⠶⠀⠀⠉⠙⠢⠀⠀⠀⢸⠃
⠀⠀⠀⠀⠀⣿⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⢰⠀⢠⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⢻⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⢸⡇⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⣧⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⣯⠀⡀⠀⢀⡀⠹⣆⠀⠀⠀⠀⠀⠀⠘⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡀⠁⠀⠀⠇⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠄⠲⠀⢈⡓⣤⡤⣠⡤⠶⠀⠀⠀⠈⠀⠈⠁⠂⠀⠀⢠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠰⣯⡉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡱⠀⢠⡾⠃
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⠙⠻⠶⣤⣀⣉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠾⠋
⠀⠀⠀⠸⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠳⠶⠶⠶⠦⠤⠴⠖⠒⠋⠉⠁
⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠒⠤⠤⠄

⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠄⡀
⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⣦⣄⣀⡀⠠
⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠐⠦⠄⠀⠀⠀⠀⠀⠈⠢⣄⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⡠⠔⠒⠂
⠀⠀⠀⠀⠀⠀⠂⠀⠘⠢⡀⠀⠀⠀⠀⠉⠒⢄⠀⠀⠀⠙⠦⣤⠶⠋⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⣀⠀⡠
⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠀⠀⠀⠀⠀⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢀⡔
⠀⠀⡰⠲⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠸⠿⠁⠀⠀⠀⠀⠀⠀⡞
⢀⡴⠀⠀⠀⠈⠉⠛⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄
⣿⠇⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⣴⣿⡟
⠀⠀⠀⠀⠀⠀⠙⣷⣶⣦⡀⣦⣀⢀⡀⠀⠀⠀⠀⠀⢼⣿⢘⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣧⣿⣿⠏⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣴⣦⡀⠀⠀⠀⠀⣠⠸⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⣿⣿⣿⡗⠀⠀⠀⠀⠉⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⢟⣵⣢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡌⣿⢾⣿⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡟⢠⢎⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣯⣭⣶⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⣰⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡇⠻⣿⣿⣿⣿⣿⣿⡟⠀⠀⢠⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⠀⢸⣿⣿⡟⣿⣿⣷⠀⠀⢸⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡄⣼⣿⣿⢠⣿⣿⣿⠀⠀⢸⣿⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⠶⠿⠟⠿⠛⠛⠛⠛⠿⠅⠛⠿⠀⠒⠈⠛⠈⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠤⠴⠶⠒⠒⠒⠚⠛⠛⠛⠉⠉⠉⠉⠉⠉⠙⠛⠛⠛⠛⠒⠒⠶⠶⠦⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⣀⣠⠤⠖⠚⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠻⠶⣦⣤⣀⡀
⠀⠀⢀⣠⠶⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠶⠖⠒⠛⠋⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠓⠒⠲⠦⢤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣦⣄
⢀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⣄
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿
⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠤⢤⣄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠐⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠉⠛⠶⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣷⣶⣤⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣶⣾⣿⡿⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⠮⢭⣭⣛⣛⣛⡛⠲⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⢶⣿⣛⣛⣿⣿⣿⣿⠿⠿⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀
⠀⠀⠀⠰⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠀⠠⠆⠀⠠⠀⠐⠂⠀⢄⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠉⠁⣤⠀⠀⡀⠔⠀⠒⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠀⣀⠀⡄⠄⠐⠑⠀⡃⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠄⠀⠀⠀⠀⠀⠀⡘⠀⢈⡄⠀⠀⠀⠀⠀⠸⣁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠄⢀⡀⠄⠊⠀⠀⠠⠈⡀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠂⠐⠒⠠⠂⠀⠈
⠀⠀⠀⠳⠀⠀⠀⠀⢀⠎⠈⠀⠀⠉⠘⡄⠀⠠⣲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠡⠀⠎⠑⠂⠀⠀⡀⠀⠀⠀⠀⠀⣀⠄⠄⡀
⠀⠀⠀⠀⡀⠀⠀⠀⢪⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⢀⡀⠤⣠⠊⠁⠀⠉⠦⠠⠤⡀⠀⠀⠀⠀⠀⠀⠈⠀⡀⠀⢠⠊⠁⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⢄⠀⡀⠀⠰⠉⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠒⠢⡀⠀⠀⠀⠂⠀⠀⢸⠀⠀⠀⠀⠀⠀⢰
⠀⠀⡠⠤⠤⠤⠤⠤⠤⠤⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⢠⢲⢀⡔⢦⢠⡖⡄⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠈⢄⣀⡀⡀⡀⢀⠝
⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⠀⠀⠀⡇⠀⠀⠀⠀⠘⠁⠈⠣⠞⠘⠋⠐⠆⠀⠀⠀⠘⠀⠀⢀⠀⠄⠀⠀⠠⠀⠀⠉
⠀⠀⠀⠀⠀⣀⣀⢀⡀⡸⠀⠀⠀⠀⠀⠀⢠⠛⠀⠠⡆⠰⠗⠰⢒⠀⣂⠀⣖⠀⠠⠀⠆⠀⡜⠀⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⠀⠤⠀⢀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠈⠠⡀⠀⠀⠀⠀⠀⠔⠀⠀⠘⠀⠁
⠀⠀⠀⠀⢠⣀⡀⠀⠈⠀⠀⠀⠀⠀⠈⣄⠀⠑⠂⠰⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠊⠀⠀⠀⠀⠀⠉⠋
⠀⠀⠒⠘⠁⠀⠀⠀⠈⢰⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠈⠒⠀⠒⢄⣀⣀⠔⠓⠂⠂⠀⠀⠀⠀⡰⠤⠤⡀⠀⠀⠈⠂⠀⠉⠀⠉⠀⠩
⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠂⠀⠀⢀⡀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⡤⡠⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠐⡁
⠀⠀⢀⠂⠈⠀⠈⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠄⡀⠀⠀⢠⠋⠀⠀⠀⠀⠉⠠⠀⠀⠀⠐⠂⠁⠀⠠⠄⠄⡀⠀⠀⠀⣰⠁
⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠞⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠡⠀⠱⡀⠀⠀⠀⠀⢀⡪⠀⢀⠄⠐⠐⠀⠂⠠⡀⠀⠀⠀⠉
⠀⠀⠂⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠉⠓⠒⠢⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠁⠑⠂⠅⠉⠀⠀⠀⣴⠀⠀⠀⠀⠈⠢⠀⠀⣀⠄⠀⠀⠀⠀⣀⠀⠀⠀⣀⠀⠀⠀⠁⢀⠀⠀⠀⠀⡀⠈⠀⢀⠀⠀⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠀⠀⠀⠀⠀⠀⠀⠀⣒⠄⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢻⣛⣟⣙⣛⣟⣿⣿⣟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⢀⣀⡀⠤⠤⠤⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠘⣃⡄⣤⡄⠀⠀⠀⠀⠀⠀⠈⠉⠉⡲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⢀⡔⠋⠁⠒⠁⠀⠀⠀⣀⣀⣀⣀⡤⠤⠤⡤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠈⠀⣀⠲⢤⣄⣠⠶⠋⠉⠁⠈⠁⠀⠀⠀⠀⠀⠀⠀⢠⠋⠑⠦⣀
⠀⠀⠀⠀⠀⠀⠀⡠⠢⠀⠀⠑⠒⠋⠁⠀⠀⠈⠁⢀⣀⠤⠄⠠⠴⠦⢄⣈⡁⣸⠇⢠⠎⠀⠀⠀⠀⠑⢄
⠀⠀⠀⠀⠀⠀⢠⡇⠀⠑⢦⡀⠀⠀⢀⠤⠠⠤⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⡮⡀⠀⠀⠀⠀⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠰⣇⣀⣤⠤⠝⠀⡔⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⡸⠀⠈⠢⡀⠀⠀⠀⠀⠀⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢀⡀⠙⠓⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⢠⠇⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡤⠖⠉⠀⠘⠦⢄⠀⠙⢦⡀⠀⠐⠀⠀⠀⠀⠀⡰⠁⣠⠃⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠳⣄⠀⠙⢦⠀⠀⠀⠀⠀⡼⠁⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⡈⢦⡀⠀⢀⡞⢁⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⢏⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠘⣄⠙⢦⠞⠀⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⣄⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠠⠀⢀⠀⠀⠀⠈⠑⢄⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠁⠢⢀⠀⠀⠀⠀⠀⠀⠉⠑⠒⣆⠉⠑⡄⠀⠀⡠⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢮⡀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠒⣒⠀⠒⠚⠑⠉⠹⣄⡀⠳⣄⡀⠀⠀⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡙⠶⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⣠⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠓⠒⠒⠒⠒⠚⠋⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠒⠒⢢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣦⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⠁⠀⠀⠀⠀⣸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢴⣴⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⡿⢻⠻⣌⠀⠀⢀⠰⣾⠃⠀⠀⢀⣠⠤⢤⣀⠀⠀⠀⠀⣛⣛⣯⣟⣋⣬⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠩⣤⣁⣤⣄⢠⣦⣐⡎⠀⠀⣴⣯⣬⠁⠀⠈⣧⠀⠀⣿⣿⣟⣭⣭⣿⡿⣻⡦
⠀⠀⢀⡴⠖⠛⠲⣤⡀⢛⣿⡏⠙⢉⣹⡿⠁⡴⠏⠁⠀⣀⡵⠶⠴⢿⣀⠀⢾⣿⣶⣤⣿⣶⣾⡉⠁⠀⢀⡿⣦⣀
⠀⢠⡏⠀⠀⠁⠀⢈⣷⠈⡿⣿⡿⣿⣿⠇⢰⡝⠂⠀⣼⠇⠀⠀⠀⠀⢈⡄⠸⣯⣏⣻⡟⢋⣿⡏⠀⠰⣏⣆⢻⣿⢳⡀
⠀⠈⢏⠀⠀⠀⣠⣬⣟⠀⢹⣿⡷⣾⢿⠀⠈⢹⢲⡴⣯⡀⢀⠀⠀⠀⠠⣾⠀⢹⣿⣟⣛⣿⡿⠀⠴⣴⠿⠼⣿⢘⣿⣧⡄
⠀⠀⢋⡀⢀⢠⣀⢢⡼⠀⠸⣿⡷⢮⣿⠀⠀⠺⣼⣗⠀⠣⣠⡀⠤⠤⢶⡋⠀⠀⣿⢹⣿⣿⡇⠀⠼⣯⣾⣱⠚⣜⣿⣹⠃
⠀⠀⢠⡾⣷⡾⡿⡾⠀⠀⠀⠛⠓⠚⠃⠀⠀⠀⠈⢿⣿⣶⣶⣦⣴⡻⠉⠀⠀⠀⢽⢾⣤⣿⠃⠀⠀⢻⣿⣿⣾⣛⣿⡿
⠀⠀⠀⠜⣧⣻⠓⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⣧⠃⠀⠀⠀⠀⢸⣿⣩⣿⠀⠀⠀⠘⣟⣿⣹⣯⣽⠇
⠀⠀⠀⢸⣶⢋⡣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⣖⡏⠀⠀⠀⠀⠀⠀⣿⣻⡿⠀⠀⠀⠀⢻⣿⣙⣯⣿
⠀⠀⠀⠈⡟⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢫⣻⣽⣩⠁⠀⠀⠀⠀⠀⠀⠹⢿⠇⠀⠀⠀⠀⠈⣿⡟⣿⠇
⠀⠀⠀⠀⠙⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠒⠚⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⠛⠓⠒⠒⠶⠤⠤⠀⠀⠀⠠⠤⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢡⣾⡿⡫⣄
⠀⠀⠰⣌⡀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣈⡉⠀⠀⠘⠿⠞⠁
⠀⠀⠀⠈⠉⠉⣯⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠟⢩⡿⠁⠒⠲⠀⠈
⠀⠀⠀⠀⠀⠀⢻⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⣦
⠀⠀⠀⠀⠀⠀⠸⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣤⡤⠀⠀⠀⠀⠀⡐⠺⠦⣥⠃
⠀⠀⠀⠀⠀⠀⠀⠹⣿⠉⠀⠀⠀⠀⠀⢠⣴⡾⢏⣀⣀⡀⠠⠤⠀⠐⠠⢀⠀⠀⢀⢁⣨⡋⠀⠀⠀⠀⠀⠀⢺⣿⣿⡿⠿⣷⣶⣾⠿⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿⣷⣻⣿⣿⣿⣩⣿⣛⠀⠂⠀⠀⠄⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⢀⣾⣿⡿⠉⠉⠙⠛⠛⠛⠿⠿⠿⠛⠛⠛⠛⠿⠿⠿⠿⢇⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⢀⣾⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡂⠈⠻⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣟⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⠁⠀⠀⢾⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣸⣁⠀⠀⠸⢿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⣠⡀
⠀⠀⠀⠰⠒⠒⠲⠆⠐⠒⠎⠛⠉⠛⠚⠃⠈⠉⠉⠉⠙⠉⠙⠓⠂⠀⠀⠀⠉⠛⠉⠁⠉⠉⠉⠉⠉⠀⠈⠛⠒⠒⠒⠢⠤⠤


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⢀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢄⡀⣀⣠⣐⣬⣒⢥⡢⢕⠪⣔⢤⡀⣐⢬⡂⢄⡀⢀
⠀⠀⠀⠀⠀⠀⠀⢤⣠⣕⣾⡿⠟⠛⠉⠉⠉⠛⠛⢻⣿⣿⡛⠛⠛⠋⠙⠙⠛⠛⠛⠛⠲⠶⢦⣤⣄⣀
⠀⠀⠀⠀⠀⡀⢭⡳⢬⣿⠏⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣀⠀⠀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠳⢦⣤⣀
⠀⠀⠀⣀⠢⢌⣓⣿⣿⣿⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣾⣇⣾⠉⣨⣫⣽⣉⣗⠒⠒⢢⠤⢤⠀⠈⠙⡳⣦⣀
⠀⠀⠀⠠⣙⡶⠋⠁⣿⡟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠘⢿⣿⣴⡖⠋⢸⢩⢆⡑⢦⣼⡴⣇⣤⡴⠢⢿⠁⠙⢷⣄
⠀⠀⢀⣽⠏⠀⠀⠀⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠃⠀⠀⡀⠀⠀⠛⢿⣿⣦⣇⣸⢰⢋⠙⠻⣧⣨⢸⣿⣧⡾⢤⡴⣀⠙⢷⡀
⠀⠀⣿⠃⠀⠀⠀⠀⣿⣿⣿⣿⣦⣤⠀⠀⠀⠀⠤⠀⠀⠀⠀⠀⠀⠙⢿⣷⣶⣦⣭⣙⠹⢿⣾⡛⢛⣉⢷⣴⣯⣛⠿⣞⡁⠐⡣⣣⠎⣷⡀
⠀⣸⡇⠠⡙⠄⠀⠀⣿⣯⡙⢿⣯⡛⠷⢦⣤⣤⣀⠀⠀⠀⠀⣠⣤⣴⣴⠟⠛⠻⠯⣿⣷⣦⣍⠻⣿⡁⡸⠁⢿⡻⢀⡟⣹⡎⢜⡵⣫⢞⣧
⠀⢺⠁⡐⢜⡢⠀⠀⣿⣿⣿⣷⣿⠗⠀⠀⠀⠉⠉⠙⠻⠛⡛⠻⠿⠿⣶⣤⣴⣦⣤⣤⣉⣙⣛⣷⡾⠷⢿⠉⡿⠟⣩⠞⣡⣿⣗⡫⢖⣫⣿
⠀⢸⡇⢈⣒⠤⠂⠀⡙⢿⣿⣿⡧⣀⠀⠀⠀⠀⠀⠀⠐⠢⠀⠀⠀⠀⠀⠠⠈⠉⢉⣛⠿⣿⣋⡷⠤⠴⢼⡟⣷⣯⣴⡿⣏⠀⠡⢒⣩⣾⡿
⠀⠀⢻⡄⠠⡭⠭⠥⠀⠂⠙⠿⣽⣛⠂⠤⠀⠀⠀⠠⠠⢀⠀⠀⠀⠀⠀⢀⣤⠒⣏⠀⠿⡿⠙⡌⠳⢴⣲⢣⣿⣿⡿⠭⢑⣒⡾⢟⣯⣿⠅
⠀⠀⠀⠻⣦⠘⠭⠭⠥⠄⠀⠀⠬⠉⠛⠲⠦⠤⣤⣤⣀⣀⣀⣀⡀⢠⡤⠄⣯⠳⣝⠢⢾⠁⠀⣧⣀⡺⢃⡾⠭⠭⠭⠭⣃⣠⠬⣳⣿⠃
⠀⠀⠀⠀⠈⠳⣤⡙⠫⣭⣭⡍⠀⣀⣉⠄⢀⡰⣻⣏⣁⠀⢈⣙⣽⠟⠁⣠⣿⣷⣮⣕⣦⣝⣛⣉⣭⠶⠿⠿⠯⠿⠿⣟⣒⣺⣿⠫⣕
⠀⠀⠀⠀⠀⠀⠈⠙⠶⣤⣘⠛⣛⣓⣒⣒⣛⣶⣾⣋⣛⡶⡿⢩⡧⠀⠀⠀⣁⣡⠞⣛⣽⡛⠻⠯⠭⠭⠁⠀⠈⢉⣷⣴⣟⠯⣗⠭⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⣤⣌⣛⠛⠻⠋⠠⠍⠈⡹⠻⣧⣴⣿⠾⠯⠑⠛⠁⠈⠉⠉⠉⢀⣀⣤⣶⢾⡛⠯⣗⠮⡝⠦⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⢶⠶⢦⣴⣧⣽⣯⣤⣤⣤⣤⣤⣤⣤⢶⣶⠶⣛⠿⣝⠳⣝⡳⢬⡓⠤⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢉⠢⢌⠒⢬⡒⢭⡒⠍⡲⢌⡓⢬⡓⠬⣓⠮⡑⠢⠉⠂⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠨⡐⠈⠒⠌⠐⠈⠑⠈⠑
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣦⣵⣦⣤⣯⣽⣯⣽⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠂⠉⠉⠩⠙⢋⡀⡈⠉⠉⠩⠰⡒⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⢄⣠⠀⣠⠖⠋⠉⠀⠀⠀⠀⠉⠙⠲⢌⠀⠣⡀⠉⢙⠒⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⡤⠊⢀⠞⠁⠁⠀⠀⠀⠀⠀⠀⠀⠑⠀⠀⠱⡀⠈⡗⠀⠀⠀⠈⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⣠⠃⢁⠎⠀⠀⠀⣠⣴⠒⡶⣋⠶⢆⠀⠀⠀⠀⢱⠀⢣⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠖⠂⢸⠀⠀⠀⠘⡄⠓⠀⠓⠉⢁⠎⠀⢀⠀⠂⢈⠀⢀⠕⠀⠀⠀⠀⠀⠀⢀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⣔⠁⡀⢸⠀⠀⠀⡀⢸⡀⠀⠀⠀⡎⠀⠀⠀⠀⠁⡸⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠑⡄⠈⢆⠀⠀⠠⢀⡇⠀⠀⢸⠀⠀⠀⢠⠐⢠⠃⠄⠀⢣⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠴⠎⣀⠀⠈⢦⡀⠀⠀⠛⠀⠀⡃⡀⡄⡆⠀⡰⠃⠀⠀⡔⠋⠁⠀⠀⠀⠀⠀⠀⡀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠶⠤⠤⢄⠀⠸⠀⠀⠀⠀⠀⠀⠀⢱⣠⡄⡉⠢⢄⣸⡀⢑⣇⣠⠤⠔⠋⠀⠂⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠑⠀⢇⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠰⣎⡀⢘⡽⢟⡆⠈⠑⢦⡜⠁⠈⠺⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠤⢀⠀⠠⠔⠁⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣉⣉⣤⡴⢞⡧⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⡘
⠀⠀⠀⠀⠀⠀⣠⠤⠀⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⣴⡉⢠⣀⣴⡿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⡌⠀⡜
⠀⠀⠀⠀⠀⠰⡁⠀⢀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠠⠀⠀⠀⠀⠀⠈⠻⣙⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢸⠘⠀⢠⠊
⠀⠀⠀⠀⠀⠀⠈⠛⠀⠊⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠬⣀⠀⠀⢀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⠰⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⡔⠉⠉⠲⡀⠀⠀⠀⢻⡤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⢥⣌⣊⠔⠰⠀⠆⡔⡠⢠⢀⠎⡞⠠⠃⢋⣀⠤⠚⠁
⠀⠀⠀⠀⠀⠀⠑⠂⠄⠊⠀⠀⠀⠀⢸⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠐⠒⠢⠤⠤⠤⠤⠤⠔⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠰⣀⣱⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠉⢣⡸⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⢀⢸⡇⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢊⡠⢈⢴⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⡴⠆⣨⢔⠵⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡈⣷⠆⢐⠵⣪⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡈⠕⡪⢐⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣩⣂⠅⣻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⣩⣿⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⢘⣻⢹⠀⣺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢕⡏⢸⠀⢾⡄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⡠⢀⠴⣂⢔⡢⢊⠤⣂⠔⡪⠔⡡⢐⡠⢂⠴⣂⠔⣢⢔⡠⠠⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠥⢻⡰⣛⢕⡾⢳⣄⣿⢔⡠⢂⠄⡢⡤⠔⠔⠠⠀⠄⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⢀⣠⣤⣶⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣄⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⣶⣤⡀⠀⠀⠀⠀⣠⣴⣶⣶⣶⣦⣄
⠀⢀⣿⡿⣿⠏⣽⢹⣹⢸⣿⡇⠀⠀⠀⢰⣿⣿⣻⠻⣻⣿⣧⠀⠀⠀⢰⣿⡟⣿⣿⡯⣿⣿⣷⠀⠀⠀⢸⣿⣿⡿⣿⡿⣿⣿⣷⠀⠀⠀⠺⡿⠇
⠀⠀⢻⣿⣿⣿⣿⣿⣟⣼⠟⠁⠀⠀⠀⠘⢿⣿⣿⣿⣿⡿⠇⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠸⣿⣿⣾⣷⣿⣿⣿⡟
⠀⠀⢴⣿⣟⠛⠛⠛⠉⠁⠀⢀⣀⣀⣀⠀⠀⠙⠳⠀⢀⣀⣤⣤⣤⣤⣀⡀⠈⠻⢟⠛⠉⠁⠀⠀⠀⠀⠀⠈⠛⠛⢻⡿⠟⠁⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⡼⠋⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣦⠀⠀⢰⣿⣟⣿⣿⣿⣻⣿⣿⣷⡄⠀⠀⠀⣠⣾⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⣾⡟⣾⣿⣿⡿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣏⣾⣽⣿⣹⣳⣿⠂⠀⠘⣿⣯⣿⣽⣯⣯⣿⣼⣿⠇⠀⠀⠀⣿⣿⣿⣷⣼⣿⣿⡆⠀⠀⠀⠀⠀⢿⣷⣿⣾⣿⣧⣿⠇
⠀⠀⠀⣀⣠⣤⣀⡀⠀⠈⠻⣿⣿⣿⣿⠿⠋⠀⠀⠴⣿⡿⠻⠿⠿⠿⠿⠋⠁⠀⠀⠀⠀⠘⠻⠿⠿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠈⠉⠿⠋⠉⠁
⠀⠀⣾⣿⣿⢿⣿⣿⣆⠀⠀⠈⠙⠂⠀⠀⠀⠀⠀⠀⡼⠋⠀⠀⠀⣠⣴⣶⣶⣶⣄⡀⠀⠀⠀⠀⠐⢉⡀
⠀⠈⣿⣿⣿⣵⣿⣿⡿⠀⠀⣠⣤⣄⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣻⢽⣿⣿⣧⠀⠀⣠⣾⣿⡿⣿⣿⣷⡄⠀⠀⠀⢀⣴⣶⣾⣷⣶⣦⣄
⠀⠀⠈⠉⠛⢛⡿⠋⠀⠀⣴⣿⣿⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣾⣿⡿⠇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢰⣿⣿⡾⣿⡿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣮⣽⣯⠈⠛⠂⠀⠀⠀⠀⠀⠀⠀⠘⠷⠄⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⣿⠟⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⡿⠃
⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠈⠉⠙⠿⠿⠋⠀⠀⠀⠀⠀⣦⣤⣶⣿⣿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣀⣤⣤⣤⣀⠀⠈⠉⠉⠉⠿⠋
⠀⢀⣾⣿⣿⣿⢿⣿⣷⡄⠀⠀⠀⠀⣤⣶⣶⣶⣶⣄⠀⠸⣿⣿⣿⣇⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣷⡀
⠀⠘⣿⣿⣹⣿⣿⣿⣿⠇⠀⠀⠀⢸⣿⣿⣻⣿⣿⣿⡇⠀⠙⠿⣿⣿⣿⡿⠿⠋⢀⣤⣶⣶⣶⣶⣤⡀⠘⢿⣿⣾⣿⣿⣿⡿⠃⠀⢀⡀
⠀⠀⠈⠙⠛⢿⠛⠉⠁⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣇⡿⠻⢻⣿⣷⣷⠀⠈⠉⠛⣻⠟⠉⢰⣿⣾⣿⣿⣷⣤⣀
⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⣿⣿⣶⠀⠉⠛⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⣷⣿⣦⣾⣷⣵⠟⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣇⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣀⣼⣿⣿⠃⠀⠀⠀⠀⡀⠀⠀⠀⠀⣠⣶⣶⣶⣶⣶⣄⠈⠻⢿⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣿⠿⠿⠛⠁
⠀⠀⠀⠀⣀⠀⠀⠉⠛⠛⣿⠟⠛⠉⠀⠀⠀⠀⣼⣇⡀⠀⠀⢸⣿⡟⢿⣯⣯⣿⣿⡇⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠋
⠀⠀⣀⣿⣿⡻⣿⣷⡄⠀⠀⠀⠀⠀⣀⣤⣤⣶⣾⣿⡃⠀⠀⠘⠻⣿⣿⣿⣿⣿⠿⠁⠀⠀⣠⣶⣶⣾⣿⣿⣶⣆⠀⠀⣀⣤⣤⣶⣶⣤⣤⣀
⠀⠠⣿⣿⣿⣿⣿⡟⢡⡄⠀⠀⣰⣿⣿⣿⣿⣿⡿⢿⢻⡆⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⢰⣿⣟⡏⣿⣿⢿⣽⣿⠀⢰⣿⣿⢋⡿⣿⢿⠿⣿⣷⡄
⠀⠀⠙⠿⢿⣿⣿⠟⠈⠁⠂⢸⣯⢿⣸⣧⣿⣷⣇⣿⡿⠃⠀⠀⠀⠀⢀⣀⣀⣀⣀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⠏⠀⠈⢿⣾⣿⣿⣿⣿⣶⣿⡿⠃
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠙⠻⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣆⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠰⣿⣟⠛⠛⠿⠟⠛⠉
⠀⠀⠀⢠⣶⣴⣶⣶⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⢿⣿⣷⣿⣾⣾⣿⡏⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠞⠉⠀⣠⣄⣀⣀⡀
⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣆⠀⠀⠉⠙⠻⠏⠉⠁⠀⠸⣦⣶⣿⣿⣿⣿⣿⣦⡀⠀⠀⢀⣼⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠿⣿⣿⣿⣿⣿⣿⠿⠃⠀⠀⠀⠀⠀⣿⣿⣯⣽⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣷⣝⣹⣗⣿⣿⡿⠀⠀⢸⣿⣾⣯⣯⣻⡏⢶⠆
⠀⠀⠀⠀⠀⠀⠰⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠿⣿⠿⠿⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⡿⠿⠛⠁⠀⠀⠀⠙⠛⠻⡿⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠘⠂⠀⠶⠀⠠⠄⠠⣤⠀⣠⡀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂⠀⠴⠀⠠⡄⠀⣤⠀⢠⡄⠀⣈⠀⠀⠁⠀⠘⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠶⠀⠰⠆⠀⣤⠀⣠⡄⢀⣀⠀⣈⠁⠀⠛⠀⠘⠃⠀⠖⠀⠰⠄⠀⡄⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠰⠀⠠⡄⠀⣤⠀⢠⣄⣀⣈⣁⣈⡋⠀⠛⠃⠘⠟⠀⠶⠆⢰⡦⠀⣤⠀⢀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠐⠀⠰⠆⠀⣤⠀⣠⡄⢀⣤⠖⠉⠀⠀⠀⠀⠀⠉⠛⢿⡀⢾⡦⢠⣦⠀⣤⡄⢀⣀⠀⢈⠁⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠀⠐⠀⠀⠄⠀⣤⠀⢀⡄⠀⣈⠀⣸⣇⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣶⠖⠁⠀⠀⠀⠀⠘⣏⠀⠛⠁⠘⠇⠀⠶⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠤⠀⢠⡀⢀⣀⠀⢈⡁⠀⣻⠟⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢸⡶⠿⠀⢶⠆⢠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⡩⠊⠁⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠗⠂⠀⠀⠄⣠⡀⠀⡀⠀⠀⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠉⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠉⠀⠘
⠀⠀⠀⠀⠀⠈⠁⠀⠃⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠇⠀⠖⠀⠠
⠀⠀⠀⠀⠀⠛⠀⠰⠆⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡷⠄⢠⡄⠀⣀
⠀⠀⠀⠀⠰⠦⠀⣤⠄⢠⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣤⠀⢀⡀⠀⡀
⠀⠀⠀⠀⢠⡄⠀⣀⠀⣈⡉⢻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡟⢀⡁⠀⠈⠀⠀⠁
⠀⠀⠀⠀⣀⠀⠀⠁⠀⠙⠁⠘⠟⠿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢉⣉⠀⢈⠁⠀⠛⠀⠘⠂
⠀⠀⠀⠀⠉⠀⠘⠃⣴⠟⠋⠙⠣⢰⣶⠹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡎⠛⠀⠛⠃⠐⠗⠀⠰
⠀⠀⠀⠀⠀⠀⠺⢸⣏⠀⠀⠀⠀⣤⣤⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠿⠇⠐⠷⠀⢰⠄⠀⡀
⠀⠀⠀⠀⠀⠀⣦⠀⢻⣦⣤⣤⠖⣉⡁⢈⡛⠻⣷⣄⡀⠀⠀⠀⠀⠀⢀⣠⣾⡿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⠆⢰⡆⠀⣤
⠀⠀⠀⠀⠀⣾⠁⢀⣄⠀⢀⡁⠀⠙⠁⠘⠛⠐⠿⠛⢿⡿⢿⣿⡿⣿⣿⢩⣭⡁⣙⡻⣿⣶⣤⣤⣤⣤⣴⣶⣿⡟⢠⣦⠀⣠⡀⢀⡀
⠀⠀⠀⠀⠀⠈⠛⠋⠁⠀⠘⠃⠘⠛⠀⠾⠇⢰⣶⠀⣴⡄⣠⣭⢀⣭⡁⢈⣛⠀⠛⠃⠸⠿⠛⢿⠟⢻⣿⠁⣤⠀⢠⡄⠀⣈⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠚⠀⠰⠆⠀⣶⠀⣠⡄⢀⣭⠀⣈⡉⠈⡛⠁⠻⠟⠘⠿⠃⢾⡷⢠⣶⠀⣤⡄⢀⣌⠀⢈⠁⠀⠉⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣄⠀⣈⠁⢈⡋⠀⠛⠃⠘⠿⠀⠿⠇⢰⡶⠀⣴⡄⢠⣄⠀⣉⠁⠈⠉⠀⠛⠁⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠘⠋⠐⠿⠂⠲⠖⠠⣶⠀⣤⡄⢀⣌⠀⢈⡁⠀⠉⠀⠘⠃⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠂⠰⡶⠀⣤⡄⢠⣄⠀⣈⠀⠀⠉⠀⠙⠁⠘⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠲⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠗⠀⡀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⠋⠑⣤⠂⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠈⠀⠀⠉⠖⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⣠⠊⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⠧⠤⠤⢤⣤⡀
⠀⠀⠀⠀⠀⢀⠜⠁⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠖⠚⠋⠉⠀⢰⢯⡼⠛⠀⢀⣿⣧
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⢰⠋⠉⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⢀⣠⣶⣟⢾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠔⠋⠀⠀⣀⠀⠤⣄⠀⠀⣀⠠⠄⣾⢇⠉⢿⡿⣾⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠁⠙⠂⠺⠁⠀⡏⠀⠀⢸⠒⡉⠜⢆⣄⠙⠞⡾⡌⣵⢸⣿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡷⠶⣶⣿⠛⣇⠀⢁⠀⠀⣭⢁⠁⡸⣤⠯⡙⢶⠘⠀⣹⣸⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⣟⢳⡐⡿⡷⣄⡀⢰⠂⠃⠀⠱⠘⢆⡙⢦⢡⠇⢸⣼⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡏⢽⡢⠙⢦⠈⠈⠅⢫⠀⠠⠀⡧⣲⡌⢃⠀⠈⡞⣧⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢔⠿⣿⠀⣧⠀⢲⡀⠣⠀⡀⠀⢌⢷⠈⢰⡇⣣⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠮⣷⣝⠃⢤⡀⠲⣄⣈⢦⠺⠦⠁⡆⣯⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⣷⣶⣭⣄⣴⣦⣤⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡋⡛⣙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⣠⠴⠖⠒⠲⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⣿⣾⣷⣶⣦⠀⣀⣀
⠀⠀⠀⠀⠀⠀⢀⣤⡶⠶⠶⠶⢶⣄⡙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡄
⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠀⠀⠈⢷⡄⣠⡴⠶⠛⠛⠶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⠋⠀⠀⠀⠀⠀⠈⠻⣆⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣋⣹⣿⣿⣻⠟⣿⣿⣿⣿⣿⣿⠓
⠀⠀⠀⠀⣴⠟⢹⣏⣾⢸⣿⣼⡇⣼⣧⣿⢇⣿⣷⣿⠀⠀⠀⠙⣧⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯
⠀⠀⠀⣼⠃⠀⣼⢫⣿⡿⣿⢻⣷⣿⡿⣿⣾⢻⣿⣿⣄⣀⣀⡀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠉⣉⡭⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⡟
⠀⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣟⠛⠟⠛⠁
⠀⠀⠀⠈⠛⠶⠶⠶⠶⣷⠀⠀⠀⠀⠀⠀⠀⠀⣰⢦⡀⠀⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣼⣿⣿⣿⣿⣿⣿⣷⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⢿⡀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠉⠓⠤⣄⣈⣉⣉⠠⠐⠂⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣙⠶⠦⠤⠶⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠒⠒⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣯⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠉

⠀⠀⠀⠀⠐⠺⢿⣦⣿⣿⣿⢿⣱⣿⡃⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣠⣾⣿⣿⡇⣄⣾
⠀⠀⠀⠀⢸⠇⢙⣯⣽⣿⣿⣻⢿⣿⣧⡈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢴⣿⣿⣿⣿⡿⣥⣼⡏⣿⣻
⠀⠀⠀⠈⢳⠶⣿⣾⣿⡟⠉⠉⢘⣵⠽⠀⠀⠙⠳⣦⣤⣤⣄⣠⣴⣿⠛⠋⠛⣿⣿⣿⠏⡴⠋⠙⢿⣿⣿
⠀⠀⠀⠀⢸⣞⣿⣿⣿⡿⠀⠀⠈⠁⠀⠀⠀⠀⠈⠉⠛⢛⣉⡁⢀⣿⣒⣂⢺⣿⠟⢿⣸⣇⠀⠀⣾⣿⣿
⠀⠀⠀⠀⠓⠹⣿⢏⡏⠁⠀⠀⠀⠀⠐⠀⠀⠀⢀⣠⡦⠾⣻⣯⣟⠛⠁⠈⠉⣡⣴⠻⣿⣿⣖⣰⡿⢿⣿
⠀⠀⠀⠀⠀⡠⠔⠛⠁⠀⣠⡺⢀⣴⣦⠤⠶⠛⠉⢩⣴⢞⣽⠋⠉⠑⠒⠃⠉⠀⢸⡷⢻⣿⣽⣭⠒
⠠⠀⠀⠂⣈⣤⣀⣤⠴⠾⢛⣡⣾⠟⣡⡶⠟⣃⣤⣿⡾⠋⠸⣀⠀⠀⠀⠀⠀⠔⠋⣷⢇⣴⠶⣾⣦⡄
⣦⡤⢠⣵⣶⣮⡻⣤⣴⡾⢟⣩⡴⠞⣫⣴⣫⣿⣿⡾⢯⠀⢠⠛⠛⠒⠂⠀⠀⠀⠀⣻⣾⣥⠀⠀⠙⣿⠘
⣙⣉⣉⣠⣤⣌⣹⣿⠷⠞⠋⠁⢴⢿⠿⠿⠯⠿⡯⠉⠀⠀⡀⣳⣤⣠⡀⠀⠀⠀⡀⣿⣿⡀⢳⠀⠀⢹⡇
⣛⠛⠛⠉⠀⠀⠀⠜⠲⠤⠶⡶⠲⠶⠻⠿⣿⢆⣃⠀⠀⠀⠽⡍⡛⠩⠟⠳⡖⢿⣿⡿⣅⡀⣾⣴⠿⠀⠃
⠙⣳⠆⠀⢠⢔⣠⠤⠤⠖⣩⡴⢞⡛⠛⠓⢦⣝⠻⣿⣉⣒⠃⠘⠏⠁⠀⠀⠈⠀⠉⠁⡿⢰⢟⣯⡀⠀⠒
⣧⣙⣤⣠⠴⣿⠛⢶⣶⢟⣥⣀⣤⡤⣶⡶⠶⣶⣜⡳⢟⡚⠀⠀⠒⡀⠀⠀⠀⠀⡀⠘⡛⣴⢾⣏⣀⡠⠄
⠉⠉⠉⠁⠀⠀⠀⠈⠛⠳⠤⠴⣬⣿⣽⣤⠀⢈⡎⣰⡟⠷⠀⠀⠐⠛⠒⠤⠖⠛⠻⢿⣺⠋⣾⣯⣹⣦⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠉⠉⠀⠁⠈⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠆⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠀⠐⠀⠀⠀⢀⠀⠒⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠀⠀⠀⢀⠌⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⡈⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢄⣀⠀⠀⠀⠀⠀⠀⠁⠘⠘⠒⠓⠒⠒⠈⠋⠀⠀⠀⠀⠀⠘⠀⠀⠀⠐⠾
⠀⡠⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⢄
⢨⠄⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠢⠤⡀⠀⠀⠀⠘⢢⣤⣠⢤⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠈⢄⡀⠀⠀⠠⠐⢦⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠁
⠀⠀⠹⠂⠓⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢀⠠⠔⠉⠰⠄⠀⠀⠀⠀⠈⠓⢶⠀⠁⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠘⠁⠀⠀⢀⣰⠂⠀⠀⠀⠀⠀⠐⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠐⠀⠁⠈⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢐⠂⣤⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⣿⣿⢟⠿⡟⠿⢿⠟⣿⢿⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⡁⡜⣻
⠀⠀⠀⠀⢀⠤⠤⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣤⣤⠀⠀⠀⠀⠀⠀⢠⣤⡤⠀⠀⠀⠀⢧⡄
⠀⠀⠀⠀⠸⠄⠀⣀⡀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⡄⠈⠀⣸⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⠋⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣂⣮⣿⣿⣏⣱⡇⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⣿⢷⡄⠤⠋⠀⠀⠛⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢨⣿⢗⠁⠻⡸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⣀⣾⣁⠀⠀⢠⣷⡇⠀⠀⠀⠀⠀⠀⠀⠲⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠸⣿⣷⣭⣤⣾⡿⢇⠀⠀⠀⠀⠀⠀⠀⠀⢉⣸⣧⠀⠀⠀⢀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⣀⣠⠏
⠀⠀⠀⠀⢸⡇⠀⠈⠉⠉⠁⠰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧⡤⣄⣀⣸⡶⡾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⡟
⠀⠀⠀⠀⠀⠻⠶⠶⠶⠶⠶⠾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠤⠀⠀⠀⣀⣀⡀⠽⠃⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢰
⠀⠀⠀⠀⠀⢰⠂⢲⣶⣶⣶⠒⠢⡄⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⡀⠀⠤⠤⠆⠀⠀⠀⠀⠀⠀⠀⢠⠉⠀⠀⠀⠒⠒⠀⠀⠉⠆
⠀⠀⠀⠀⠀⢸⠀⠀⠛⠿⠁⠄⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠘⠛⠂⠶⢀⡻⠀⠀⠀⠀⠀⠀⡄⠀⢀⣀⣀⣀⣀⣀⢀⣀⣘⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⣠⡃⠀⠀⠀⠀⠀⢸⣷⣀⣀⣀⣄⣀⣉⣀⣭⣭⣽⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠘⣻⣿⠔⠂⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠨⢹⡏⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠀⠀⡀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⢸⣇⢀⠀⡀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⢸⣨⡤⠈⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⣼⠷⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⣸⣿⣷⡄⢠⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⠦⣀
⠀⠀⠀⠀⠀⠀⣿⣻⣗⣋⡈⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠀⠀⠘⠒⠦⠭⠍⠛⠀⠠⠴⠖⠃
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉

⠀⠀⡀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀
⠀⠀⠈⢿⣿⢷⣍⠑⣄⡀⠄⠒⠓⠀⠂⠀⡠⠂⢁⣴⡿⠃⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⢀⠠
⠀⠀⠀⠀⠐⢿⣿⣧⠌⠀⠀⠀⠀⠀⠀⠀⠀⢤⣾⣿⠀⠀⠀⠀⠀⠀⠀⠠⠚⠉⠀⣹⡏
⠀⠀⠀⠀⡀⠀⠹⡟⢀⡀⠀⠀⠀⠀⠀⠀⣀⢸⡿⠏⠀⡀⠀⠀⠀⠀⢀⠁⢀⣀⠀⠻⠠⠊⠀⠀⠂⠀⠀⠐⠢⣄⢀
⠀⠀⠀⠀⠀⠀⠒⠃⠘⠛⠀⠀⠀⠀⠀⠘⢻⠘⠃⠐⠀⠀⠀⠀⠀⠀⢞⠄⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣠⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠺⢖⠀⠀⠀⠀⠀⠃⠀⠀⢀⡀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢄⣤⣄⡤⠂⠀⢠⠃⠀⠰⣶⣤⠶⣿⡇⠀⠀⠀⠉⢫⢁⠀⠀⡀⠀⢸⠇⠀⢀⡇⠀⣸
⠀⠀⠀⠀⠀⠀⠀⠈⡲⢤⣀⠀⠉⢽⣂⡤⠖⠁⢀⣀⠀⡹⠛⢁⡌⠉⢻⣿⣶⣤⠀⢠⠹⠋⠃⠀⡼⠓⠒⡟⡇⠀⡗
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠠⣉⠁⠀⠉⢉⠀⠀⠀⣸⡟⠀⠉⠣⢾⣧⣾⣀⣿⣿⡟⠀⠈⠀⡀⠁⢸⠀⠀⣔⣱⠇⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⣶⣤⣶⡏⠀⠀⡰⠋⠀⠀⠀⠀⠈⣿⡟⠉⣻⣻⠀⠀⠘⠚⠀⠷⠊⠀⠀⠉⠙⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠉⠁⠀⠐⠁⠀⠀⠀⠀⠀⠈⠙⠁⠈⠁⠛

⣿⣿⣿⠀⠀⢸⣆⣠⢊⣿⣫⣶⣯⣤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⣿⣿⣿⠀⠀⢸⣿⣿⣿⡹⠀⠈⠯⠁⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠈
⣿⣿⣿⠀⠀⢸⣿⣿⡟⠀⠀⡰⠀⡇⠀⠠⠀⠀⠀⠰⠀⢄⡀
⣿⣿⣿⠀⠀⢸⣿⣿⣇⠀⢠⠃⠀⡅⠂⠀⠀⢀⣀⣠⡄
⣿⣿⢿⡄⠀⠀⣿⣿⣿⡀⣼⡄⠀⣱⡤⠖⠊⠉⠠⠊⠈⡆
⣿⣿⡄⠀⠀⠀⢹⣿⣿⣿⣿⣧⣼⡏⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣠⡄
⣿⣿⣷⡀⠄⠀⠀⢿⣿⣿⣿⣿⣇⢻⡀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢹⡇
⣿⣿⣿⣷⡄⠈⠀⠈⣿⣿⣿⣿⣿⣸⡇⠀⣀⡀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇
⣿⣿⣿⣿⣿⣦⡀⠀⢸⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⣛⣃⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣾⡀⠀⣀⣤
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠹⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣤⣸⣄⣦⣀⣠⡀⠀⣠⣴⣶⣶⣶⣤⡄
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣻⣿⣿⣿⣿⣿⣿⣦
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⠉⠙⠻⠿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣭⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⡤⠶⠒⠦⢤⣶⣶⣶⣶⣶⣶⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⠟⠋⠁⠀⠀⠀⢀⣀⠀⢈⣭⣽⣿⣯⣉⣉⡁
⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣷⣶⡶⣶⣾⣿⡿⠋⠁⠀⠀⠈⠙⠿⠿⠿⠿⠛⠛⠿⠶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋⠉⠛⠋⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣀⣀⣉⣛⠿⣿⠷⠄
⠀⠀⠀⣰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠶⠆⠀⠀⠀⠀⢾⣿⡿⠟⠉⠉⠀⠀⠀⠉
⠀⠀⣼⣿⣿⣿⠏⠀⠀⠀⠀⠀⢀⣀⡀⣀⡀⠀⠤⣤⣤⣄⠀⠀⠀⠀⠀⢀⣤⡄
⠀⣼⣿⣿⣿⠏⠛⠻⠿⠉⠨⣤⣿⣿⠿⣿⣿⣿⣷⣌⠻⠿⠀⠀⠀⠀⠀⠈⣁⡴⢾⣿⣿⣿⣶⣶⡄⠀⠀⠀⣠⣤⣀⡀
⢰⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠈⠙⠛⠀⠙⠿⠗⠀⠉⠀⠂⠀⠀⠀⠀⠀⠀⠉⠁⠈⠣⠭⠞⠙⠋⠀⠀⠀⠀⠙⢛⡋⠁
⣾⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣿⡄
⣿⣿⣿⣿⠶⢶⣶⣶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⣿⣿⣿⣿⠀⠀⠈⠉⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡷
⢻⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⠇
⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⣶⣶⠿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇
⠀⠹⣿⣿⣷⣤⣤⣤⣶⣶⣶⣶⣤⣠⡀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡟
⠀⠀⠹⣿⣿⣧⠀⠉⠙⠛⠿⢿⣿⣿⣧⣄⠀⠀⣴⣶⣾⣿⣷⣤⣤⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣿⣿⡁
⠀⠀⠀⠈⢻⣿⣧⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠷⢿⣿⣿⣍⣙⠛⠉⠉⠛⠛⠛⠀⠀⠀⠀⠀⠠⠀⠦⠾⠿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠉⠻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠶⣶⣶⣶⣶⣿⣿⣿⣿⣶⣂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠘⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣮⣶⣤⣤⣠⣤⣤⣤⣼⣿⣿⣿⣿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠛⠛⠛⠛⠉⠁

⣞⣏⣉⣉⣉⣉⣉⣉⣉⠉⣷⣶⡄
⠀⠀⠀⠀⠀⠈⠉⠉⠉⠈⠙⡏⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢚⣋⣭⣭⣿⣽⣟⣻⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣅⠀⢿⠀⢀⡿⠙⣷⣽⣻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆⢸⣄⡾⣡⡾⠋⢈⣷⡹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢉⡿⠟⠋⢈⣷⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣛⠛⠛⠉⣿⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆⠀⠰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡦⠀⣠⠞⠻⣿⣍⠛⠶⣶⡟⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⡼⠁⠀⠀⠙⣏⢷⣤⡟⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⡴⠋⠀⠀⠀⠀⠀⠈⢿⣫⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣄⠈⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡈⠻⣿⣿⣿⣿⣿⣿⣿⡿⠃⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠙⢿⣿⣿⣿⣿⠏⢠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠉⠉⠋⢁⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡦⠂⠰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠒⢺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠤⠤⠴⠲⡇⠀⢸⡗⠲⠤⠤⠤⢄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⡉⠀⠀⠀⠀⠀⠀⠀⠷⠤⠼⠀⠀⠀⠀⠀⠀⠀⢈⣱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⣿⣷⣶⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣿⡿⠟⠁
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⢁⠈⢉⣉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠄⠠⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡎⢠⣾⠏⠀⠀⠀⠀⠀⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡁⠀⠀⠀⠀⠀⠀⣿⠆⢸⡆
⠀⠀⠀⠀⠀⠰⡇⢸⡞⠀⠀⠀⠀⠀⢠⠏⠋⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣧⠀⠀⠀⠀⠀⠀⢸⠀⡸
⠀⠀⠀⠀⠀⠀⣱⠎⠀⠀⠀⠀⠀⠀⡿⠚⣠⣞⣠⣤⠴⠶⠖⠛⠛⠋⠉⠉⠉⠉⠉⠉⠛⠓⠒⠒⠦⠤⣄⣀⠀⠀⠀⠀⠀⠳⣄⠀⡍⠀⠀⠀⠀⠀⠀⠘⡶⠁
⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⢠⡼⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠲⢤⣀⡀⢠⠋⢀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⣄⡴⠂⠰⡄⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢤⣘⠀⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⢠⠛⠓⠈⡁⠁⣀⠀⠠⠊⠁⠀⠁⡠⠠⠀⠀⡀⠀⠀⠠⣀⣀⠀⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⣼⠁⠀⠀⠀⣦⠀⠀⠀⡠⠋⠀⠀⢀⡴⠊⢁⠉⠀⠀⠔⠀⠀⠀⠀⠀⠐⠉⠁⠀⠁⠀⠂⢇⠀⠀⠀⡀⠀⠀⣱⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡄
⠀⣸⠃⠀⠀⠀⢠⠀⠀⠀⠀⠇⠀⠀⠠⠋⢀⠄⠀⠀⢀⠀⡠⢠⠀⠀⠀⠀⠀⠀⢀⠀⠂⠠⠀⠄⠀⡀⠀⠀⠠⠀⠀⠀⠈⠠⡀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠈⣇
⠀⢿⡀⠀⠀⢦⡆⠈⠠⠀⠀⠁⠀⢠⠔⠈⠁⠀⠀⠀⠘⠁⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⢂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⡏⡇⠀⠀⠀⠀⣿
⠀⠈⢧⠀⠀⢘⠁⢆⠀⠀⡰⠀⠐⠁⠀⠀⠀⠀⠀⠀⠈⡇⠈⠂⠀⠀⠀⠀⢀⠀⠀⠄⠀⠀⣀⠀⠀⠀⠀⠀⠠⠴⠂⠰⠆⠀⠆⠀⡎⠂⠀⠀⠃⢷⠁⠀⠀⣠⠋
⠀⠀⠀⢳⡀⢸⠀⠀⠁⠀⠀⣠⢇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠠⡈⢋⢢⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠁⠒⠒⠁⠀⡌⠀⢀⡼⠁
⠀⠀⠀⠀⠉⠻⡆⠀⠀⠀⠀⡇⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠁⠈⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⡧⠤⠊
⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⢿⠀⠈⠀⠀⠀⠀⠀⡠⠃⠀⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠈⠳⠄⡈⣀⠀⠐⠺⣀⠀⠀⠀⡾⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠕⢀⠐⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠒⠐⠂⠁⠀⠀⢀⡀⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠈⠁⠐⠂⠀⠀⠀⠀⠀⠀⠀⣠⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⠦⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠑⠒⠒⠒⠚⠋⠉⠉⠉⠉

⠀⠀⠀⠀⠀⣠⠶⠈⠛⠛⠛⠋⠱⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⣀⠀⠀⣤⡄⠀⣴⡆
⠀⠀⠀⣰⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⣿⣇⠀⣿⣷⠀⣿⣿⡀
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⢹⣿⠀⣻⣿⡄⡏⢹⡇
⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⢸⣿⡆⢿⢿⡇⠀⠸⣿
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⡆⢸⢸⡇⠀⠈⣧⠀⠀⣿⡆
⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢹⡇⣼⢸⣧⠀⡸⣿⢀⡄⢹⡇
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢸⣇⣿⢸⣿⠀⡇⣿⢸⡇⠘⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⣿⡏⠘⣿⠀⡇⣿⣾⡇⠀⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⡇⡀⣿⡖⡇⢹⣿⠃⠀⢿
⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⣿⡇⡇⢻⡇⡇⢸⣿⢰⠀⢸
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢸⡇⡇⢸⣿⡇⠘⣿⢸⠀⢸
⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠸⣷⡇⠸⣿⡆⠀⠿⢻⠀⢸
⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏
⠀⠀⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⢠⠾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⠀⣸⡏
⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⣰⣯⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⣼⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⠀⠀⣰⢻⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⢰⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣰⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⢀⡘⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⣷⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⢿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⢸⢧⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣿⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠻⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠆⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠀⠀⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⢀⣤⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠀⠀⠀⠀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⣀⣀⣠⠞⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⠀⠀⠉⠉⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣾⣿⣾⣿⣿⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿