Mary Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠴⠄⠀⠀⠰⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⣀⣀⣙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⢀⠐⣡⣴⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⢡⣾⢿⣿⠟⠿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⢰⡟⠛⠋⠁⠀⠀⠈⠙⢋⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠀⢸⠀⢀⡉⠒⠀⠀⠀⢀⣼⢡⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢊⠔⠀⠀⠀⠀⠀⣰⠁⣿⠀⠀⠈⠀⠀⠐⢲⡞⢡⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣇⡾⠀⠀⠀⠀⠀⠀⡉⠀⣿⠀⠀⠀⠀⢀⣠⡞⣰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠃⠀⠀⠀⠀⠀⡰⠀⢸⠿⣦⣀⣠⣿⠿⢋⣰⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡰⢻⡿⣼⡏⠀⠀⠀⠀⠀⢠⠃⠀⣾⣿⠟⠛⠙⠁⣠⣾⠏⣼⡇⢻⡈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⣾⠃⣿⣧⣿⠁⠀⠀⠆⠀⠀⡼⠀⢰⣏⠁⠀⠀⣰⣾⡿⠃⣰⡿⡇⠈⢷⡈⢎⠻⡄
⠀⠀⠀⠀⠀⢠⡟⣵⣿⣿⣿⣿⠀⠀⠈⠀⠀⣜⠇⠀⢸⡇⠁⠀⢼⣿⠏⠀⡔⣹⠁⡇⠀⢸⣧⠨⡄⢹⡄
⠀⠀⠀⠀⣴⢻⣷⢻⣿⣿⣿⣧⠀⠐⡂⠀⣸⡎⠀⠠⣿⣄⣠⡴⠛⡇⣀⠞⣰⡏⢘⣇⠀⠀⣿⡄⢧⠀⢷
⠀⠀⠀⢀⠹⣾⢿⣾⣿⣿⣿⡿⣦⣀⢧⠀⣿⣿⣿⣤⣯⠿⣳⠛⠛⠉⠉⠉⣩⡿⠋⠉⠤⢀⣉⠛⢾⡄⢸⡄
⠀⠀⢀⣬⡶⠿⣮⣿⣿⣿⢻⣷⢼⠿⠦⠷⢼⠟⣽⠏⠀⢠⠈⠐⡋⠝⠁⠀⠂⠀⠠⣴⣿⣿⣿⣿⣦⠈⢶⡇
⠀⠀⡾⠁⠀⠀⠈⣻⢿⣿⠋⠒⠀⠀⠀⠀⡃⠸⠁⠀⠀⣨⠀⢠⡎⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣯⠁⠀⣼⣧
⠀⣸⠁⠀⠀⢠⠞⢁⢾⠁⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⠹⣀⠻⢒⠒⠲⠠⠤⠤⠿⠿⣿⣿⣿⠿⠀⢰⣷⣿⡀
⠀⡟⠀⠀⢀⣿⡄⠀⣼⡄⠀⠀⠀⠀⠀⠀⠄⢉⠀⠀⠀⠰⡘⢿⣶⣬⣀⠀⠒⠂⠠⠄⠀⠁⡠⠀⢐⣥⣿⡟⢳
⠀⣇⠀⢠⢸⣿⢷⠀⣸⡀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⢦⡘⢦⣈⠙⠻⢽⣷⣦⡤⢀⣀⣀⡤⠀⢀⣾⡟⠁⢸
⠀⢿⡀⢨⢸⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⢤⡀⠀⠙⢷⣄⡀⠀⠀⠀⠀⠛⠃⢀⣴⣿⡟⢹⣰⡜
⠀⠀⠙⢦⣘⣿⢏⣾⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⣀⣪⣤⣽⣿⣷⣶⣤⣭⣙⣷⣶⡦⢶⣶⠟⠉⣼⣿⣣⣾⣿⡇
⠀⠀⠀⠀⢹⡿⣨⣿⡟⣿⣿⣻⣷⣶⣶⣾⠿⠟⠛⠛⠻⢿⡿⠁⠀⠁⠈⠻⣏⠉⢦⢘⣻⠃⣴⣿⣿⣿⡿⠁
⠀⠀⠀⠀⢸⡇⣿⣿⣼⠏⡟⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠙⢟⢦⠀⠀⠀⠈⢷⣌⢿⣿⢨⢹⣿⣿⣟⠁
⠀⠀⠀⠀⢸⣿⣿⣿⡟⠸⢰⣿⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠓⠸⠏⢸⠈⣿⡏⢿⣇
⠀⠀⠀⠀⠈⣿⠉⣿⠁⠁⣸⣧⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠀⡇
⠀⠀⠀⠀⠀⠁⢰⡇⠠⡆⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠈⠃
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡇⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠁⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⠀⣀⡈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⣀⡉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣠⠟⡁⠉⢵⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣠⠗⡉⠉⣵⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⠀⠁⣾⢻⣷⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠁⠀⠁⣾⢻⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣷⡄⣸⣿⣿⠇⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⣷⡀⣸⣿⣿⠟⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠎⠀⣿⡇⢉⡘⠟⠂⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠏⠀⣿⡇⢉⡸⠟⠂⠐⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⢠⡟⠁⢠⣡⠈⢃⡿⠛⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⢠⡟⠀⢠⣥⠈⢇⣿⠟⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⣆⢀⠟⠀⠀⢸⢡⣤⣼⣁⡀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠏⠰⣆⢠⠏⠀⠀⢸⢰⣤⣼⣁⡀⠸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢠⠃⠀⠞⠛⠘⠀⠆⣠⠋⣯⠙⠛⠙⡁⠀⠛⡀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠀⠞⠛⠘⠀⠆⣰⠋⣟⠙⠛⠛⡁⠀⠻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡆⠀⠈⠀⣄⣀⣤⣾⣿⣿⣿⣿⣦⠀⢣⡎⠆⣶⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠈⠀⣀⣄⣤⣿⣿⣿⣿⣯⣦⡀⢻⡘⠄⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⣿⡿⠛⠛⠋⠉⣸⣿⡷⢸⣷⡆⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⡿⠛⠛⠋⠉⣼⣿⡷⢸⣷⡄⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⡿⠉⠀⢀⡋⠀⣻⣿⣿⠏⣿⠇⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡿⠉⠀⢀⡃⠀⣿⣿⣿⠏⣿⡇⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣿⠇⠀⠀⠀⢰⠀⡿⢹⣿⢸⣿⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⠇⠀⠀⡄⢰⠀⡷⢹⣿⢸⣿⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣿⠀⠀⠀⠀⢸⡇⠀⣸⣿⠸⠏⢠⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⠀⠀⠀⡇⢸⡇⠅⣸⣿⠸⠏⢰⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡏⠀⠀⠀⠃⢸⡇⠀⣿⣿⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇⠃⠀⢰⡏⠀⠀⠀⠇⢸⡇⠀⣿⣿⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠀⠀⣼⠃⠀⢀⢠⠀⢸⡇⢠⢹⢿⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⣾⠇⠀⢀⢠⠀⢸⡇⠠⢹⢿⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠘⠀⢸⣿⠀⠀⢸⢸⡇⢸⡇⡇⠀⢸⡇⢀⣿⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⠀⠀⢸⢸⠀⢸⡇⠇⠀⢸⡆⢀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⡆⡟⠃⠀⠀⠸⢸⠁⣼⠃⠀⠀⢸⡇⢸⣿⠃⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣀⡟⠃⠀⠀⠸⣸⠀⣼⠙⠀⠀⣸⡇⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠘⠁⡀⠀⠀⢀⣿⠀⢿⠀⠀⢀⣿⡇⣼⢫⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠁⡀⠀⠀⢀⣿⠀⢿⠀⠀⢠⣿⡇⡼⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡀⠈⢸⡆⣸⠀⠀⢸⣿⣿⡁⢸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣇⠀⡇⠀⠀⠈⢸⠀⣸⠀⠀⢸⣿⣿⡁⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠸⠀⡇⠀⡇⠀⠘⢤⡟⠀⠀⣼⣿⣿⣇⠸⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⡇⠀⡇⠀⠸⢤⡇⠀⠀⣾⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡇⠀⡇⠀⠀⠀⠁⠀⢸⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⡇⠀⡇⠀⠀⢀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⡇⠀⠀⠘⠀⠀⢸⣿⣏⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⠀⠀⢸⢸⠀⢸⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢠⠀⠀⢸⡄⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⢠⠀⠀⢸⠀⠈⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⢀⠀⢸⠃⠀⢹⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⡇⢸⢠⠀⢸⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢸⢸⠀⢸⠀⠀⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣷⢸⢸⠀⢸⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠁⣈⢸⡄⢸⠀⠀⠝⣸⠷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠁⡘⢸⡇⢸⠀⢈⠍⣼⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠠⠚⠁⢸⡇⠸⠀⠀⠀⠁⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠛⠁⠠⠚⠁⢸⡇⠸⠀⠀⠈⠁⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠒⠤⠴⠾⠵⠯⠴⣄⣄⠀⢀⠴⠶⠖⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⣾⣵⣯⣴⣤⣄⣀⣀⣬⣶⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠠⠀⠤⠤⠤⠀⠀⢀⡀⠀⠀⠀⠀⠤⠒⢀⣁⣠⣤⣤⣤⣤⣤⣤⣤⣀⣈⠀⠂⠀⠀⠀⠀⠀⣀⠀⠀⢀⢀⣀⣀⣀⣀
⠀⠆⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⣀⣤⣶⠿⠟⠛⠉⠉⠉⠉⠀⠉⠉⠉⠉⠛⠻⠿⣶⣤⣀⠐⠄⡀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠟⠉⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢷⣆⡀⠀⠀⠉⢢⡀⠀⠀⢸
⠀⠀⠀⠠⠊⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢉⠐⠒⠄⠀⠀⠀⠀⠀⠈⢷⣶⡀⠀⡀⠑⠀⠀⠸
⠀⠀⠀⠀⠀⠀⣠⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣆⠈⢄
⠀⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠶⠤⣶⣦⣤⣶⡢⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⢂⠀⢠
⠀⠀⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⠀⣤⣶⠴⠿⠿⢿⣷⣼⣟⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠⣿⣀⠞⠋⡠⠊⠀⠐⢶⣦⣽⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⡀
⠀⠀⠀⣾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⢸⣿⠋⣠⡞⠀⠀⠀⠀⢀⣉⣿⣿⣹⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣺⣿⡄⢀
⠀⠀⢸⣿⡏⠀⠀⠀⢀⠀⠀⠀⠀⡀⢠⡈⡿⢸⡟⢀⠶⢿⣶⠀⣿⣿⣹⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠠⠀⢬⣿⣇⠈⡀
⠀⠀⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⡘⣿⡷⣿⡗⠈⢲⣾⡏⠀⢸⣿⠟⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢺⣿⣿⠀⢁
⠀⠀⣿⡛⠏⠀⠀⠀⠀⠀⠀⠀⠀⢨⠇⠸⣿⣿⡿⠄⠀⠀⠀⣠⣀⣻⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠍⠟⢿⠃⠘
⠀⢰⣿⣷⣴⡆⠀⠀⠀⠀⠀⠀⠀⡌⢧⡀⠹⣿⣤⡌⠀⠀⠀⣩⡽⢿⣿⣿⣿⣿⣿⢿⡀⠀⠀⠀⠀⠀⠀⠀⣼⣶⣾⡆
⠀⢸⣿⣿⣿⣿⡀⠀⠀⠀⠀⡀⢠⠇⠸⣷⢧⡹⣿⣿⣦⣶⣦⣄⢘⣋⣽⣿⣿⣿⣿⢸⡅⠀⠀⠀⠀⠀⠀⡀⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⡀⠀⠀⠀⠀⢠⣶⡄⠹⣿⣷⣾⣿⡟⠋⠛⢿⣿⣿⣿⣿⣿⠉⣿⢸⣧⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⢸⣿⣿⣿⣿⣿⣿⣃⡴⠀⠉⠙⠻⠷⡶⠶⠬⣙⠻⣷⣷⡀⣾⣷⣿⣿⣿⠏⣼⠟⢸⣿⡄⠀⠀⢀⢰⣿⣿⣿⣿⣿⡇
⠀⠘⣿⣿⣿⣿⣿⡿⣯⣴⣶⣦⣀⠀⠀⠈⠐⠶⣌⠳⠸⣿⣟⠿⣿⣿⣿⣿⣿⡏⣤⢸⣿⣷⡀⡔⢸⣿⣿⣿⣿⣿⣿⠇⢸
⠀⠀⢿⣿⣿⣿⠏⠙⢻⣿⠟⠛⠻⠿⣦⣀⠀⠀⠈⠙⣦⣻⢧⡀⠈⠋⠈⢿⣿⣿⡟⣾⣿⢹⣿⣷⣾⣿⣿⣿⣿⣿⣿⠀⡈
⠀⡄⢸⣿⣿⣿⡆⢰⠟⠁⠀⠀⠀⠀⠀⠈⠻⣦⣄⠀⠘⣿⣆⠙⢢⡀⠀⠀⠜⣋⣧⣿⣏⠸⣿⣿⣿⣿⣿⣿⣿⣿⡏⢀⠃
⠀⠀⠀⣿⣿⣿⣧⠏⠘⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣦⣽⣿⣆⡛⣿⣎⢁⣾⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⠁⡘
⠀⠀⠂⠸⣿⣿⠋⣰⡀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠙⠀⠈⠛⢿⣿⣟⠢⠽⣿⣿⣿⣿⣿⣿⣿⣿⠛⣿⡋⢸⣿⣿⠏⢠⠇
⠀⠀⠀⠀⢻⣿⠀⠻⣷⡄⠀⠀⠀⠀⣴⣿⣿⣷⡀⠀⠀⡀⠀⠙⣿⣷⣄⠈⠙⠛⣿⣿⣿⣿⣿⣾⡿⣰⣿⣿⡟⠀⡌
⠀⡤⠀⠀⠀⠻⣷⡄⢠⡝⠂⠀⠀⣸⣿⣿⣿⣿⣷⡄⠀⠸⣷⣄⠈⠹⣿⣷⣄⣲⠭⣭⣴⠶⣿⣿⢠⣿⣿⡟⠀⡜⠀⣴
⠀⠀⠀⠀⠀⠀⠹⣿⡜⣿⡄⠀⠀⠙⠿⣿⣿⣿⡟⢿⣆⠀⢻⣿⠀⠀⢻⣿⣿⠿⣷⣦⣤⣶⣿⣿⣿⣿⠏⢀⠎⠀⡀⢠
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣆⠀⠀⠀⢀⣄⣉⠻⣿⣄⠹⠀⣼⣿⡆⠀⠸⣿⣿⣷⣾⣿⣿⣿⣿⣿⠟⠁⡠⠁⢀⠎⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⠻⢷⣦⣤⣄⣉⠛⠻⣿⣿⣦⡀⠸⣿⣿⣆⠀⢿⣿⣿⣿⣿⣿⡿⠟⠁⡠⠊⢀⡴⠁⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠢⡀⠈⠂⢀⠉⠛⠿⣿⣿⣿⣿⣿⣿⣷⡄⢹⣿⣿⣆⣼⣿⣿⠿⠛⠉⡠⠐⠀⢀⠔⠋⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⠐⠠⠀⠉⠙⠛⠛⠻⠿⠿⠿⠿⠛⠛⠋⠉⡀⠤⠐⠉⣀⠤⠒⠁⠀⠀⠀⠀⠀⠀⢸
⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠈⠀⠉⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠉⠉⡉⠉⣉⡉⡉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⢀⡀⠀⠀⣀⣉⡉⠉⠉⠉⠉⢉⣉⣉⣉⡉⠉⣉⣉⣉⠉⡉⠉
⣿⣿⡿⠻⠿⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⠴⠤⡤⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡾⠛⠛⠛⠲⣄⢹⡟⣂⣈⣙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡏⠀⢀⣤⡶⠟⠛⠛⠻⠷⣦⡀⠈⠻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⢸⡇⣠⡿⢫⣤⣖⠹⠟⠳⣆⣈⢻⣆⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢴⣤⣹⣿⠃⣿⠋⠀⠀⠀⠀⠀⠙⣆⢻⣄⡼⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠘⣴⣿⠇⢸⠃⠠⡶⠅⠀⣤⣀⡀⣎⣼⣿⣠⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣿⡟⠀⡾⠀⠀⠀⠀⢸⠃⠉⠁⣿⠃⣿⡏⡽⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⣽⢃⢡⡿⠀⠀⠀⣀⡀⠀⠀⣰⣿⠀⣿⣷⠁⠀⡀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⣱⠇⡀⣼⣿⡆⠀⠀⠈⠀⢀⣼⢱⡄⠀⢹⣿⡀⠀⣿⣧⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⡟⢰⢠⢸⡿⠁⠀⠀⠐⠊⠁⢿⡹⡇⠀⠘⣿⣗⣞⠉⠋⠉⡧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷⣿⢃⠏⡄⣼⣖⣤⣀⣀⣀⣀⣠⣤⠧⡇⡆⠀⢻⣿⣉⡀⠶⠷⠠⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡟⣿⠊⢠⠇⣿⡝⢿⢷⢶⡶⢶⢿⠟⣿⣷⡇⠀⠈⣿⠿⢧⣠⠚⠶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡷⠃⣠⡿⢸⣯⡙⢎⠂⢀⠓⡡⣄⠸⢋⠾⣇⠀⠀⢹⡇⣨⡹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢿⣿⡿⠁⣈⠽⡧⣿⡶⢝⠂⠀⠘⢺⡶⢋⣠⡺⢞⣻⠀⠀⣦⣭⣍⠹⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣛⣊⣿⡷⠊⠁⠈⠉⠉⠉⠴⢄⣰⣿⣿⣿⣿⣿⡃⣫⣭⠐⢀⠀⠀⠂⠀⢹⡟⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣛⢭⣭⣿⠋⠀⠀⠀⠀⠀⠀⠠⠤⢋⠑⢾⣿⣿⡷⠚⠀⡐⢿⠀⠐⢷⢦⠀⠀⠘⣟⢿⣭⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡋⡴⠋⡇⢇⠀⢀⣠⢤⡶⢶⢲⣴⢞⡥⠠⠎⠛⠛⠁⠀⡠⣬⣛⡇⠀⢘⡞⣦⡀⠀⡏⠇⣹⢻⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⢸⡇⠀⡇⢘⠒⡾⣿⠸⡃⢨⡈⡷⣫⡆⡀⣀⢀⢠⢤⠰⡙⢦⣝⡇⠀⢀⢀⣿⡑⢞⠇⡇⢿⠀⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⢠⣿⣏⣾⡇⠀⠱⣌⣰⡇⣿⡷⡄⠘⠇⢹⣟⣽⢣⡏⣾⢸⡼⣧⢻⣮⣻⡇⢀⠈⣼⣿⡰⣄⣨⠃⣷⢠⢻⢿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⢿⡇⣤⢠⢻⣿⢇⣿⣿⣄⠀⠀⠀⣿⣯⣿⣧⣿⣸⣧⣿⣷⡿⢿⡇⢃⣼⣿⣿⣷⡍⠩⣴⠣⢃⣾⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⢹⢿⣷⣜⣮⣷⣭⣾⣿⡘⣿⣦⣄⠀⢻⣍⣉⣉⡉⠛⢋⣉⡁⠀⡊⠐⣹⢻⡿⢹⣿⣿⣶⣶⣾⣿⣿⠘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣾⣜⡏⢿⣿⣿⠉⣿⣿⣇⠹⣿⣿⣷⣤⣭⣷⣖⣈⣉⣉⡑⠒⠋⣡⣾⢇⣿⠇⣿⣿⡏⣿⢿⣿⡿⢣⡆⣼
⠀⠀⠀⠀⠀⠀⠀⠸⣿⡟⣿⣷⢸⣿⢿⡀⣿⡏⣿⡀⠹⣞⢿⣏⠻⣿⣤⣀⣠⣤⣤⠶⣛⣿⡏⣾⡟⣸⣿⣿⠇⣿⠸⣿⢡⣿⣷⠏
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡜⣿⡘⣿⡜⣧⢹⣇⠸⣧⠀⠹⣆⠻⣧⡈⢙⡻⢭⣭⡴⢚⣵⡟⣼⣿⠱⢋⣾⣿⠀⢹⡆⣃⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⢻⡜⢷⣿⠧⠹⣿⣿⠀⢻⡆⢣⠘⣧⡘⢿⣄⠙⢶⣶⣾⣿⡟⣸⣿⢃⣴⣿⣿⡟⢸⢸⡏⣿⡏⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠁⠛⠈⠛⠃⠀⠙⠛⠃⠀⠛⠀⠃⠈⠓⠀⠙⠃⠀⠈⠛⠛⠀⠛⠋⠘⠛⠛⠛⠃⠘⠘⠃⠙⠁⠛⠃
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠖⠚⠛⠉⠉⠉⠉⠛⠛⠒⠶⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⢀⣠⡴⠚⠉⠁⠀⠀⠈⠉⠙⠲⣄⠀⠀⠀⠀⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⠀⢰⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣄⠀⠀⢷⡀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⢸⠀⣰⡥⠖⠊⢁⣤⠴⠚⠛⠒⠶⣄⠀⠑⡄⢘⡇⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡴⠋⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠸⡇⠰⣇⢸⠇⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⡿⠁⢀⣴⠋⠀⢄⠀⠀⠀⠀⢀⠤⠀⠹⡄⠹⣿⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⡔⠩⠃⣠⢞⣿⠀⠀⠀⠋⠁⠀⠀⠀⠠⡄⡀⢇⡀⠹⡄⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⠀⢸⠁⣠⠞⢡⠘⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢁⠆⠈⠂⢿⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠠⢇⡾⡃⠀⠸⡄⠸⡆⠀⠀⠀⡀⠀⠀⠀⢀⡿⠋⠀⠀⣠⠞⠀⠀⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠄⠧⣀⡀⠀⣙⠀⠘⣦⡀⠀⠀⠀⠀⣠⣏⡴⠀⡤⣾⠃⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠙⢦⡉⠳⣤⡇⠉⠲⠦⠴⢚⡵⢋⡴⢚⠔⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠉⠢⠀⠙⢂⣀⣠⣼⣍⣠⣚⠑⢡⠖⠉⠙⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣁⣀⣀⣀⣀⠀⠀⢀⡴⠚⠉⠁⠀⠀⠀⠈⠉⠙⢦⣄⠶⢤⣄⡀⠹⣄
⠀⠀⠀⠀⠀⠀⢀⡤⠞⣁⣠⣤⣀⡀⠉⠉⠉⣰⠋⠀⠀⠀⣀⠄⠤⠤⢄⡀⠀⠀⠘⢦⠀⣈⣩⣥⠬⠷⢦⡀
⠀⠀⠀⠀⠀⣰⠿⠒⢛⣭⠿⠏⠉⠉⠉⠀⢰⠃⠀⠀⡤⡊⠁⠀⠀⠀⠀⠈⢢⠀⠀⠈⢯⠀⠉⠓⠄⠀⠀⠙⣆
⠀⠀⠀⠀⣼⢁⣠⠶⠛⠳⢶⣤⣤⣀⣀⠀⣻⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⣀⠀⢱⠀⠀⢸⠒⠋⣁⣤⣤⠶⠶⠚⢳⡆
⠀⠀⠀⢠⠗⢋⠁⠀⠀⠀⣸⠀⠀⠀⠉⠀⢹⠀⠀⠀⡇⠀⠀⠐⠀⠀⠁⢰⠇⠎⠀⠀⢸⠐⠋⠉⠀⠀⠀⠀⠀⣸⢻
⠀⠀⠀⡏⠀⡼⠀⠀⢀⡴⠃⠀⠀⠀⢀⠀⠈⣧⠀⠀⠙⡆⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⢀⣴⠃⢸⡆
⠀⡄⢀⡇⣼⠁⠀⣰⠏⠀⠀⠀⠀⢸⠀⠀⠐⢍⠳⢄⠀⢇⠋⠀⠀⣀⡴⠶⠾⢦⣀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⡼⠇
⠀⠀⢸⢰⠇⠀⠘⠁⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣠⣀⠁⢀⡭⢉⠟⠁⠀⠀⠀⠀⠉⢷⡀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢀⠃
⠀⠀⠘⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠠⣼⠃⠀⠙⠊⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠄⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣤⣀⡀⠀⠀⠀⠀⢀⠄⢀⡎⠛⢧⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠋⠁⠀⠉⠁⠀⠈

⠀⠀⢠⣤⣶⣶⣶⣶⣶⡐⠶⢶⣶⡶⠖⠲⠶⣶⣶⣶⠶⠖⠒⠂⠀⠒⠒⠲⠶⢶⣶⠄⠀⠀⢤⣤⠀⢴⣶⡶⣶⣶⣶⣶⣶⣶⣦
⠀⠀⢸⣿⡿⠟⠋⠁⠠⠀⠀⡸⣯⣼⣿⡇⠀⠸⠃⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠈⠁⠀⠀⠀⣤⣍⠀⠀⣹⣧⠼⣿⣿⣿⣿⣿⣿
⠀⠀⠸⠉⠀⠀⠀⠀⢠⣠⡎⣽⠿⠿⣿⠁⠀⠄⠀⠀⠀⠀⠉⠀⠀⠐⠃⠀⢠⠀⠀⢀⡀⠀⠘⠛⠃⢠⡜⢷⣶⣼⣿⣿⣿⣿⣿
⠀⠀⠐⠀⢀⣠⣰⣶⣿⣧⣿⣿⣖⠀⠀⠀⠓⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠀⠈⠃⠀⠀⠀⢀⠀⢡⣤⣿⣏⠚⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⠿⢿⣟⡋⠉⠀⠒⠠⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠸⢆⠀⣿⣯⣤⣀⡨⣿⡟
⠀⠀⢸⣿⣿⣿⣿⠟⠀⠀⠈⠙⠂⠀⠠⢃⠀⠀⠀⠀⠀⠀⠀⠀⣨⣝⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣌⡼⠛⢿⣿⣿⣿⣿⡇
⠀⠀⠸⣿⣿⣿⡇⣀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢰⣄⠆⠉⠈⣿⣷⠀⠀⠀⠀⠐⠒⠖⠀⠀⠈⠀⠀⠀⠐⢿⣿⣿⣿⡇
⠀⠀⢠⣿⡟⠉⠀⠀⠀⠀⠀⠰⡟⠀⡀⠀⠠⠀⠀⠀⠀⠘⣿⡦⠀⠠⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⡀⠀⠀⠀⠀⠀⢠⣿⣿⣶⣧
⠀⠀⢸⣿⡦⠴⠂⠀⠀⣠⠊⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠂⠙⠀⠀⠀⠉⢏⢻⣦⡀⠀⠀⠀⠀⠀⠘⠂⢀⡀⠀⠀⠰⣿⣿⣿⣷
⠀⠀⢸⠋⠀⠀⠀⡠⠊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢸⣆⢿⣿⡀⠀⠀⠀⠀⠀⠀⠈⣿⣶⡄⠀⡙⢿⣿⣿
⠀⠀⢸⣀⣀⣴⠾⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣰⣿⠀⠀⠀⣀⣤⣆⣰⣿⡎⢿⣷⠀⠀⠀⠀⠀⠀⠀⣀⣈⡫⠀⣌⠳⣌⡟
⠀⠀⢸⡿⠿⠄⠀⠛⠄⠀⠀⠀⠀⠀⠀⠀⠰⢁⣾⣿⣿⠇⠀⠐⠛⠺⠽⣿⣿⡏⣎⢻⡇⠀⠀⠀⠀⠀⠀⣙⡛⠛⣷⢤⣈⠁⡁
⠀⠀⢸⣧⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠈⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⣹⣷⠄⠚⢿⡀⠀⠀⠀⠀⠀⠈⠀⠈⠉⢀⠀⢠⡇
⠀⠀⢈⣿⠂⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣠⣶⡿⠃⠀⠀⠀⠀⠀⠀⣰⠏⠉⣄⣀⣠⣷⡄⠀⠀⠀⠀⠀⠀⠀⣤⡏⠀⣿⡇
⠀⠀⢈⣏⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠⣶⡿⢺⣿⠛⠀⠀⠒⠘⠀⠀⠀⡾⠋⣴⣄⣿⣭⣦⣼⣷⡄⠀⠀⠀⠀⠀⠀⢸⠧⠀⢻⡇
⠀⠀⢸⡇⠀⢌⠁⠀⠀⠀⠀⠀⠀⠀⠀⠃⡀⠘⠀⠀⠀⠀⠀⠀⠀⠊⠀⠊⢡⣾⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠁⠚⡦⢰⡇
⠀⠀⢸⣿⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣇⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⡇
⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠘⣋⠁⣼⡇
⠀⠀⢰⣿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠟⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢠⣿⡇
⠀⠀⢸⣿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⣰⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠠⠒⠇⠀⢸⣿⡇
⠀⠀⢸⣿⣇⠀⠀⠀⠀⠀⢳⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⣼⣿⣿⣿⠇⠀⠀⠀⠤⠄⠀⣰⠋⢰⠀⢠⣿⡿
⠀⠀⢸⣿⣿⣿⣿⣧⡀⠀⠀⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⣀⠀⠼⠃⠀⠸⠳⣾⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠠⠀⠀⠀⡀⠀⢀⣔⣼⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢢⠀⠀⠀⠀⠆⠀⠀⠀⠀⣿⣿⣿⣿⡉⠀⠀⠀⠐⠀⠀⠀⢠⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣇⣠⡀⠀⠀⠒⣤⡆⠁⠸⠀⢸⡄⠀⢠⡞⠀⠀⠀⠀⠀⣿⣿⣿⡇⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⣽⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄⠀⣽⡷⠀⢀⡀⢸⡇⠀⠀⡅⠀⠀⠀⠀⠀⣿⣿⣿⢃⠟⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠇⠘⠿⠀⠟⠶⢋⣬⣇⠸⡇⠀⠐⠀⠀⣰⠀⠀⢀⣿⣿⣧⠀⠀⠀⠀⠀⣠⣾⣷⣴⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⢿⣿⣿⣿⣷⠴⣦⣀⣀⠀⠁⠈⠁⠾⠒⠀⠀⣰⣧⠀⠀⣻⣿⣿⣿⠇⠀⢀⣀⣾⣿⠿⢧⣾⠿⢿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣧⣙⣿⣿⣿⣷⣾⣿⣿⣶⣶⣶⣤⣀⣤⠀⠚⠁⣀⣀⡄⣠⣼⣯⣩⣤⡴⠿⠛⠉⠉⡒⠿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠿⢿⣿⣿⣿⢿⣿⣿⡿⠋⢉⣉⣉⡉⠉⠻⣾⣿⣷⣿⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣠⡄⠈⠛⠛⠛⠻⣿⠁⠀⠛⠛⠻⠿⠆⠀⠙⠋⠉⡉⠛⠟⠉⠉⠉⠑⠊⠉⢿⣿⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣛⣿⣿⣆⠀⠀⠀⠉⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣩⣿⣿⣿⡇
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣶⣶⣶⣾⣿⣿⣿⣿⣿⣟⣿⣇
⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣶⣶⣶⣦⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠐⠈⠀⠀⠀⠙⢿⣿⣿⠿⠿⠿⢿⣿⣟⡋⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⡤⠴⣦⣤⣤⣤⣞⠣⣄⣀⣠⡾⠿⠿⠽⢿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣽⢿⡗⠐⠘⢿⣟⣿⡽⣷⠾⠋⠀⠀⠀⠀⠀⠀⠈⢻⡟⢧⣀⠀⠀⠀⠀⢀⣴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣈⣷⠀⠀⣾⣯⣹⣿⣷⣶⣶⣴⣤⡀⠀⠀⠀⠀⠀⢁⠀⠹⣿⡇⠀⢠⣾⣿⣿⣦
⠀⠀⠀⠀⠀⢠⣤⣤⣤⣤⣤⣤⣼⣷⣿⣦⣼⣿⣿⣿⣿⠏⠀⠀⠉⢿⣧⠀⠀⠀⠀⠀⠈⣤⣤⣿⣧⣴⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⣼⣭⣉⣉⣍⣩⣏⣽⣿⣿⣿⣘⣿⣿⣿⣿⠄⡀⠠⠄⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⣿⡇⠀⠀⠀⠠⠀⣀⠝
⠀⠀⠀⠀⠀⠈⠪⠛⣿⡋⠉⠿⣸⣿⣿⣿⣿⣿⣿⣿⣿⡷⠖⠀⠓⠛⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣷⣶⣶⣶⣶⣶⣧⣤⡆
⠀⠀⠀⠀⠀⠀⠀⠘⢿⢻⣬⡃⣹⠟⠁⢸⣿⣿⣿⡿⡷⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣛⡋⣹⠋⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢟⣿⣿⠀⠀⢸⣿⣿⠛⠃⠀⠀⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⢹⡖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⠀⢸⣿⣿⡀⠀⠀⠓⠀⢀⡀⣰⡏⠀⠀⠀⡆⠀⠀⠀⠀⠀⠸⠋
⠀⠀⠀⠀⠀⡄⠀⠀⠀⣼⣛⣟⣻⣄⠀⢸⣿⣿⣧⠀⠀⠀⠀⠈⣹⣿⣧⠀⣀⣼⣿⣶⣶⡶⠀⠀⠀⢠⡀
⠀⠀⠀⠀⠀⢻⣏⢋⢸⡏⠀⠀⢸⡇⠙⢿⣿⠏⠀⠀⠡⠀⣤⣾⣿⣿⣿⣿⣿⣿⣯⣥⣤⣀⣀⣀⣀⣀⡙⣷⣦⣀
⠀⠀⠀⠀⠀⠀⣿⡆⢸⡇⠀⢲⣾⡇⠀⠀⠛⢀⠀⠀⠘⡄⠈⠉⣻⣿⣿⣿⣿⣿⣿⣿⠟⣿⣿⣿⡿⠟⠃⣿⣿⣿⡗⠠⡄
⠀⠀⠀⠀⠀⠀⢸⣷⠸⣧⠀⠈⢻⡅⠀⠀⣃⠘⡄⠀⠀⠐⡀⡠⣿⣿⡿⠻⠛⣹⡿⠃⠀⠸⠛⠉⠀⠀⠀⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢿⡆⣿⡄⣠⣼⡇⠀⠀⠀⠀⠘⠄⠀⠀⠉⠈⠙⠛⠄⠀⣸⠟⠁⠀⠀⠀⠀⠀⢸⣶⣶⠟⠿⠿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⣿⣅⢻⡇⢠⣆⠀⠀⠀⣠⠖⠤⠄⢤⣤⣴⣶⣾⠋⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⠀⠀⠀⢻⡿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⠃⠊⠛⣿⣴⣿⣿⣧⠄⢈⠠⠀⣀⠀⠀⠙⢿⡿⢻⢷⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⡀⣠⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡀⢀⠉⣿⣿⡿⢁⠐⠡⠀⠀⠀⠀⠀⠤⠀⠀⠀⣿⢷⡀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠛⠛⠛⠛⢿⣿⣿⠟⡿⢿⠿⠿⣟⠿⠛⠛⠛⢛⡿⣿⢿⠿⣿⣿⣿⣿⡿⠿⠻⠿⠿⣿⣿⣿⡿⠟⠛⠛⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⠀⡗⠚⠐⣸⣯⣀⣀⢴⣾⣿⣿⢻⡐⠞⣿⠞⣿⣻⣿⡟⠠⠀⣸⣿⣿⣿⡗⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣽⠀⣿⠰⣴⣿⣿⣿⣿⣿⣿⣿⣿⣮⡿⠋⠁⠀⢸⣯⡄⠑⠀⢠⣿⣿⣿⣿⣇⠆⠰⡀
⠀⠀⠀⠀⠀⣤⣤⣶⣶⣿⣿⢽⠀⢩⣴⣿⣿⣿⣿⣿⣿⣿⡿⠛⠉⢦⣫⠀⠰⣤⣷⠁⢀⣴⣿⣿⣿⣿⣿⡷⢆⣀⢡
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣼⣆⣸⣿⣿⠟⣿⣿⡿⠟⠻⣄⠀⠀⠈⢿⠀⠀⠹⣿⣴⣾⣿⣿⣿⣿⣿⣿⡟⠨⢛⣚⣣⡀
⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⣀⣹⠿⢿⣿⣿⣿⠁⠀⡀⠀⠈⠳⢄⠀⠈⢳⣄⢀⣹⣿⣿⣿⣿⣿⣿⣿⣿⡟⠋⠉⠉⠉⠃
⠀⠀⠀⠀⢘⣿⣿⣿⣯⠉⢹⣿⣿⠀⠈⣿⣿⣿⠀⠀⠀⢀⣀⣀⣬⣷⣤⣴⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠈⠛⠛⠛⠛⠃⠘⠛⠙⠁⠀⠈⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠉⠉⠋⠛⠙⠉⠛⠛⠙⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣦⠄⠀⠙⠃⠀⢠⣶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⢀⣤⣤⣦⣄⠀⠉⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⢠⡿⡿⠛⠋⢿⣧⠈⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡆⣼⡗⠝⠆⠀⠛⣿⡶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣀⠠⠀⣰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣻⡋⠉⠉⢹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢿⣿⠉⠁⠒⠒⠘⢿⣟⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⣼⢹⡄⠀⢳⠖⠀⣼⣿⠘⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇⣿⢸⡀⠀⠀⠀⠀⣻⢹⠆⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣧⣿⢸⣧⡀⠂⠐⢀⣿⢸⢰⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡟⢸⣷⣿⠶⠶⣿⣿⠈⢸⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣍⠻⣿⡇⢸⡏⡏⠀⠀⠀⣿⠀⣿⢟⣥⣬⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡛⠿⡜⣿⢸⣿⡇⠀⠀⡆⣿⣸⢣⡿⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠛⠻⣆⠉⢽⢸⣿⡅⠀⠀⢃⡿⢹⠋⢀⡿⠛⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠛⢦⡙⠆⠈⣸⣧⠃⠀⠀⢘⣧⠈⠚⢋⣴⠚⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡑⠀⠈⢿⣷⡶⡏⢸⠀⠀⠀⢸⠘⣷⣶⣿⠡⡀⠐⡤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡂⢤⠃⢸⣿⡇⡇⢸⠀⠀⠀⠘⡇⡇⣿⡿⢀⠱⣄⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣧⣸⠘⣿⡇⡇⠀⠀⠀⠀⠀⠁⡇⣿⡇⠘⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣇⡇⠀⠀⠀⠀⠀⠀⡇⣿⡇⣾⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡁⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⡇⣿⢀⠅⣹⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⢸⣿⠰⠀⠀⠀⠀⠀⠀⡇⣿⢰⣶⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣐⠸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⣿⠀⠀⠀⠀⠀⠀⠀⢰⢇⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠘⠀⠀⠀⠀⠀⠀⠀⠈⠈⠻⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠤⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠉⠀⠀⠀⠀⠀⠀⠀⠉⠢⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⣀⠤⠤⠐⠒⠒⠂⠤⢄⡀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⠀⣠⠞⠉⠀⠤⠤⠀⠠⠤⠤⡀⠀⠈⠑⠬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢀⡞⠁⠀⠀⠀⠀⣀⡠⠴⠶⠶⡶⠿⠶⣄⠀⠐⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⡞⠀⠀⢀⣤⣶⣿⡥⠖⠛⠿⠛⠀⢠⡀⣬⣦⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⡸⠁⣠⣾⣿⣿⠏⠁⠀⠀⠀⣠⣴⣾⣧⣨⣿⣟⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠠⢡⠞⣱⣿⣿⢏⣀⣀⣀⣀⠈⠻⢿⣿⣿⣿⣿⣿⡏⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢀⠇⠁⣰⣿⣿⠁⠈⠙⢿⣽⣿⡷⢀⣾⣟⣛⣿⣿⣿⡇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⡜⠀⢰⣿⣿⡇⠀⠀⠀⠈⠙⠋⠁⢸⣿⣿⣿⣿⣿⣿⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⡜⣼⠁⢠⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣼⢻⣿⣿⡿⠉⣿⡇⡇
⠀⠀⠀⠀⠀⠀⠀⡘⣰⠇⢠⣿⢿⣿⣿⡇⠀⠀⠀⣀⠀⠢⣤⣿⣿⣿⣏⠀⠀⣽⡇⡇
⠀⠀⠀⠀⠀⠀⡔⠰⠃⢠⣿⡿⢸⣿⣿⣿⣷⡄⠀⠀⢼⣷⣶⣿⣿⠋⣿⠀⢰⣿⡆
⠀⠀⠀⠀⢀⠞⠀⠀⣠⣿⡿⠁⣼⣿⣿⣿⣿⢿⣶⣤⣤⣿⣿⣿⣿⡄⢹⡄⢸⠉⢸
⠀⠀⠀⢠⠂⠀⢀⣴⣿⠟⢀⣼⣿⣿⣿⡿⠃⠈⢿⣿⣿⣿⣿⣿⣿⣧⢸⣧⣼⠀⢸
⠀⡠⠔⢁⣠⣾⡿⣋⣥⣾⣿⣿⣿⣿⡿⠁⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣌⢿⣿⣆⠸⡀
⡾⠒⠿⠛⠛⣿⠟⠛⠛⠛⠛⠛⠛⠍⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣷⣽⣿⣦⡘⢄
⡇⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣇
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠄⠀⠀⠀⠀⠀⠀⠐⠉⠉⠻⣿⡿⠛⡟⠛⢿⣷⢄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠠⠀⠀⠳⡀
⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡆⠀⠀⢀⠀⠀⠀⠀⠀⠀⠈⢳⡈⠀⡆⠀⠀⠐⡄
⡷⡄⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⢰⠀⡀⠀⡀⠀⠀⠀⠀⢀⠀⠀⣁⠀⣴⠀⠀⠀⠸
⡇⡇⠀⠀⠀⠀⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠸⠀⠁⠀⠀⠀⠀⠀⣧⠀⠄⠀⢳⠀⠸⡆
⡇⡇⠀⠀⠀⠀⢀⡆⠀⣿⠀⠀⠀⠀⠀⡇⢹⠀⢰⠀⠀⡇⢠⠀⠀⠀⠀⠀⣿⠀⡖⠁⠈⠃⠀⣷
⣧⣿⣤⣤⣠⣤⣼⣤⣄⣿⣠⣤⣤⣤⣠⣧⣸⣤⣼⣄⣤⣧⣼⣠⣤⣤⣤⣄⣿⣄⣧⣤⣤⣤⣄⣿⣤⣤⣄
⣿⣛⣻⣛⣟⣙⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣟⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣤⣤⣶⣶⣶⣿⣷⣿⣶⣶⣤⣤⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣿⠿⠟⠋⠉⠉⠀⠀⠀⠀⠈⠉⠉⠙⠛⠻⠿⢿⣷⣶⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣿⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣧⣄
⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣧
⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡤⠤⠶⠖⠒⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠤
⠀⠀⠀⠀⠀⣤⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⢉⠔⢡⠊⠀⠀⠀⠀⠀⠀⠈⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣷⢄
⠀⠀⠀⠀⣹⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⡴⠃⡀⠦⠀⠀⣀⣠⣤⣶⣾⣿⣿⡄⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣧
⠀⠀⠀⠀⣻⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢃⡞⠁⡜⠀⠀⣾⣿⠟⠉⠉⠙⠛⠋⢿⣟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇
⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠋⢀⠼⢁⣀⣤⡿⠃⠐⠶⠄⠀⠀⠀⣿⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟
⠀⠀⠀⢽⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⡎⢸⣹⣿⠋⠀⠀⠶⠤⠀⠀⠀⣸⣏⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⡌⢠⠏⢠⠎⣿⡿⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡌
⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢁⡞⠀⠊⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⣀⠈⣴⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃
⠀⠀⠀⠙⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⣠⠎⡀⢀⠎⡐⠀⠀⣿⣷⣄⠀⠀⠀⠁⢈⣿⣿⣼⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠤
⠀⠀⠀⠘⢿⣷⠀⠀⠀⠀⠀⠀⠀⣠⡾⠗⠚⢁⠄⡠⠃⡴⠁⠀⣸⣿⠏⠻⣿⣦⣤⣤⣾⣿⣿⣿⣿⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟
⠀⠀⠀⠀⠸⣿⣇⠀⠀⠀⠀⠀⡰⠋⠀⠀⣠⣤⣰⠁⣼⠁⠀⢰⣿⡋⠀⠀⣾⣿⣿⣿⣿⣏⣿⣿⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⠇
⠀⠀⠀⠀⠀⠸⣿⣆⠀⠀⠀⢰⡇⠀⢀⣼⠿⠛⠏⢀⠸⡄⠀⣿⣿⠇⠀⠰⠉⡿⢿⡿⠛⢿⣿⣷⣾⣿⡆⠀⠀⢀⣀⣤⠀⠀⠀⠀⠀⣰⣿⠃
⠀⠀⠀⠀⠀⠀⠙⣿⣧⠀⠀⣿⠀⢀⣨⣷⡿⣷⣦⣀⢂⠱⠀⢻⡿⠤⠤⠤⠒⣡⠋⣰⡄⠺⣽⣿⣿⣿⡗⠴⠀⢀⣤⠈⢳⡄⠀⠀⣴⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⣤⣯⣴⡟⠁⠈⠀⠀⠈⠙⢻⡆⠀⢈⡗⢶⡖⠒⢿⠀⢷⠘⡁⠀⣿⣿⣿⣿⡇⢀⡄⣤⣶⣦⣾⣇⣀⣾⣿⡷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⡟⠀⠀⠀⠀⠀⠀⡀⠀⢿⡀⠜⠀⠀⠃⠀⠀⠀⠈⢀⣧⣄⠀⠈⠉⠉⠙⠾⠿⢿⣿⠿⠟⣿⣿⡋⠁⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠃⠀⠀⠀⠀⠀⠀⠰⡀⠈⣟⣆⢀⡠⠤⠤⠤⠴⠶⠾⢿⡏⠙⣒⡒⠀⠀⠀⠠⢤⠤⢴⣾⣿⣦⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣼⣿⠀⠀⠘⠀⠀⠀⠀⢀⠘⢄⡀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠈⡧⡀⠀⠀⠉⠲⢤⡄⠀⣰⡟⠉⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⠀⢀⠇⠀⠀⠀⠀⠀⠙⢦⣷⣄⣴⠃⠀⠀⠀⠀⢄⠀⠀⠇⠈⠀⠀⠀⠀⣀⣤⡾⠋⠀⠀⠀⠀⡝⠻⢿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠀⢸⠀⣧⠀⠔⠒⠒⠒⠦⠬⣿⢿⡀⡠⠀⠀⠐⠠⡙⣿⣾⣿⣿⣿⡟⠛⠋⠑⠦⣤⣤⠀⠀⠀⠀⠀⣼⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⢸⡄⠀⠀⠀⣠⣤⣴⣶⣶⡾⠧⠄⠐⠒⠂⠒⠻⡝⠋⠀⠈⠻⣿⢲⣄⠀⠀⠈⣨⠀⣤⠀⢰⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣼⣴⣾⣧⠀⠀⠀⣠⡼⢟⣋⡡⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠁⠀⠈⢆⠹⣧⢸⣧⣽⢀⣿⡄⣾⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣆⣀⠀⣻⣿⣿⠟⠁⠀⠀⠀⠀⣠⣴⡾⢿⡿⠀⠐⠀⠈⢤⣘⣇⢻⣿⣿⣿⠸⢿⣧⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⡜⢿⢟⣋⠁⢀⣀⠀⣠⣴⣾⠟⡰⠁⠀⠁⠈⠁⠀⠀⠈⠉⠹⣎⣿⣿⣿⡄⣡⣻⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣇⠙⠗⠛⠉⠉⣨⠿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠈⢹⣿⠂⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠚⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠘⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⣠⣤⣤⣤⣄⡀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡻⣿⣿⣿⣿⣿⣧⡀⠀⣦⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠀⠉⠉⠉⠙⢿⣿⡄⢿⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣿⠀⡀⠀⠐⠊⣉⡄⠈⠛⣿⢠⡀⣿⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣦⣥⠄⠀⠀⠉⠀⠀⠀⢹⡏⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠉⠁⠀⠀⠀⠀⠀⠀⢀⣾⡇⡇⠈⢿⣀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⣀⣴⣿⢿⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⠳⣄⠸⠄⠁⠀⠆⢀⠄⢹⣟⠋⠁⡏⠀⠀⠀⠀⠀⠈⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠂⠈⠁⠀⠀⢀⠴⠚⠋⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⢠⣾⣿⠀⣸⠁⠀⡴⠁⣀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⠀⠀⢀⠄⠠⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣴⡿⣻⡅⢀⡏⠀⡘⠀⠀⠀⢀⡀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡧⠂⠉⠉⠀⠀⠀⠁⠀⠘⠀⠀⠀⢸⠀⠀⠘⢶⣦⣶⣿⣿⠟⣱⣿⠁⢸⠁⠀⡴⠂⢠⡶⠋⠀⠀⣼⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⡆⠀⠀⠀⠀⠀⢰⣄⣒⡂⠀⠀⠈⠀⠀⠠⠿⠛⢿⣿⣿⣼⣿⣿⡀⣾⠀⠀⠀⣰⠏⡠⠀⢀⣼⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⠀⠀⠀⠀⡀⠀⠻⢿⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡾⠀⢠⣼⡡⠞⡡⣠⣾⡟⢹⣿⡟⠑⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⢁⣿⠀⠀⠀⠀⡉⠀⢿⠶⠶⣾⣿⡐⠛⠚⠋⠓⡐⠚⠿⣿⣿⣭⣝⠿⡧⢰⡏⣠⣴⣫⣿⠿⠿⢛⣭⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄⢸⣿⣷⣦⡀⠀⠈⠀⠀⢀⡠⠤⣌⣑⣄⠀⠀⠀⠰⡀⠄⢹⣿⣿⣯⣛⠲⣦⡑⢾⣥⣤⣴⣶⣾⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠀⠙⢿⣿⣭⣁⡀⣀⢀⡡⠴⠶⠶⠮⠛⠷⢄⡀⠀⠀⠲⠾⢿⠉⣿⣿⣷⡿⣿⣾⣿⣿⣿⣿⣿⡭
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⢪⡉⢛⣋⠉⠁⠀⠀⠀⠀⠐⠋⠋⠂⠉⠛⢲⠄⠀⢸⡄⣿⣿⣿⣿⣸⣝⠙⠻⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⠻⢆⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢿⣔⠾⠿⡇⢿⣿⣿⣿⣿⣿⣿⡖⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⣙⣀⠢⠀⠀⠀⣠⠄⠰⣦⣄⣹⣿⣿⣿⡘⣿⣿⣿⣹⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠓⠶⠒⠻⠥⠄⠸⣿⣿⣿⠻⡿⢻⣷⠘⠻⠗⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠃⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠀⠀⠠⠤⠤⢤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⢠⣠⠤⠒⠃⣡⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠄⠀⠀⠀⠀⠀⠒⠂⠠⠄⢠⠊⠀⡀⢀⡴⠋⠁⠀⣀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⣼⠏⠀⢀⡤⠞⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⠠⠤⠴⠞⠢⠴⠯⠀⣀⣠⡤⠔⠊⠉⠉⠉⠉⠉⠐⠂⠤⠤⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠤⠟⠒⠚⠀⠀⠈⠛⠓⠒⠒⠂⠵⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢡⠀⠀⠀⠀⠀⠀⣀⣀⣠⣎⣁⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠈⢀⠀⠀⠄⠀⠀⠀⠠⣤⣤⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡄⠀⢀⡴⠚⠉⣥⣶⠐⠐⠂⣤⡭⢙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡐⡀⠀⠲⣶⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣶⢋⡖⠁⡀⠐⠀⠁⠚⠉⢮⠙⠂⠀⢱⣬⡳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠵⣹⠒⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠃⡎⡇⠈⠀⢤⣄⠠⠄⠀⢀⡂⢀⡈⠀⢀⡹⣎⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⢈⠀⢉⠰⠆⢀⡀⢁⡀⠐⠊⠁⣼⠨⠀⠉⣀⠈⠳⠘⡄⠀⠀⠀⠀⠀⠀⠀⢀⣿⣆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣆⢃⡈⣤⠈⠀⠀⠈⠙⠀⢀⠉⠁⠀⠀⠻⠁⢀⣄⠀⢹⡀⠀⠀⠀⠀⢀⣠⣮⡟⣼⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡻⣌⢷⡀⠀⢣⠀⠶⠀⠤⠠⠀⠺⢇⠀⠀⠐⠒⠤⡾⢸⡇⠀⠀⢀⣠⣾⣿⣿⣳⢧⣬⣀⠀⢢⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣹⢦⡙⠦⢠⡀⠀⢰⣇⣀⡈⠁⠈⠓⠀⢀⣀⡔⢁⣾⠀⣠⣠⣿⣿⣿⣻⠟⠃⠀⠈⠉⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣶⡙⠳⣤⣙⠲⠀⠤⢬⣤⣤⣀⠹⠥⠚⣉⣴⣿⣿⣷⣿⣿⠿⠟⠋⣠⣄⠀⠀⠀⠀⠀⣠⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣦⣾⠉⠛⠲⠶⠤⠤⠤⠴⠶⠒⣋⠽⢯⣉⣉⠁⠀⠀⠀⠀⠘⣧⡼⠃⢀⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠽⠿⢿⣶⣤⣮⣤⣤⣤⡴⠞⠋⠁⠀⠀⠈⠉⠛⠛⠃⠐⠚⠛⠉⠈⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣄⣀⣠⣄⠠⠤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⠿⣿⠿⠿⠿⠿⠟⢛⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣦⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡶⠿⠛⠛⠋⠙⠛⠛⠿⠶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⢀⣀⣠⣤⣤⣤⣤⣄⣀⠀⠀⠙⠿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⢀⣴⢿⣿⠟⠛⢻⣇⣹⠟⢻⢻⣷⡀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢸⣷⣴⣿⣌⣁⣼⡿⠻⣆⣀⣼⡿⣷⡀⠀⠀⠈⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⣾⡏⢠⠘⣿⡿⠿⢯⣥⣿⣿⠿⣦⣾⣿⡄⠀⠀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⠀⢠⣿⠓⣶⡾⢿⣔⠂⣸⣧⣼⡀⠃⢸⣿⠿⢿⡄⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⣿⢟⡉⢻⣧⣣⣿⣿⡏⣻⡿⢿⡟⢻⣿⣄⣼⣷⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡿⢮⣥⡼⠛⢿⠁⢠⢹⣿⣌⣠⣿⢋⡉⢹⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠈⣷⡾⠛⠳⣥⣾⣷⡶⢿⡟⠛⠻⣿⣧⣥⣾⣿⣿⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣧⠀⠀⠸⣧⣋⣠⡟⠉⢉⢳⣿⣄⡃⣀⣿⠿⠛⢿⣴⡿⠀⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠻⣟⣙⣧⣀⣙⣴⣿⣿⣿⠿⣿⡀⠀⣻⣿⠇⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠈⠻⣿⣿⣿⣏⢤⠀⣻⣶⣿⣿⠿⠛⠁⠀⢀⣾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦⡀⠀⠀⠀⠙⢿⣿⣶⣾⡿⠟⠉⠀⠀⠀⢀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⡀⠀⠀⠉⠀⠀⠀⠀⣀⣤⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠶⠶⠿⠟⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠴⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀
⠀⠀⢀⢀⠀⠀⠀⠀⢀⠀⠀⣸⠀⠀⠐⠇⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀⢸⣆⣠⣿⣤⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣏⠔⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠈⣷⡄
⠀⠀⣼⠋⠃⠀⢸⣤⣾⣦⣶⣿⣤⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣤⣿⣿⣿⣿⣿⣿⣿⣥⡠⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⢄⠂⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⡇
⠀⠀⠘⠆⠀⣼⣾⢟⡟⠈⣹⠞⡿⢿⣧⡤⠤⢤⣄⡀⠐⣄⠀⣠⣟⢧⣿⡙⣼⣻⣿⣿⡿⡃⠀⠀⠀⠀⠀⠀⠀⢈⣿⠶⣾⡄⠀⠀⠀⠀⠀⠸⣿⣤⣤⣾⣿⠃
⠀⠀⠀⠀⠘⢿⣿⣧⣷⠞⡿⢿⠟⠿⣿⣧⣀⡴⠟⠁⠀⠘⠻⢿⣛⢟⣧⣟⣀⣙⣽⣿⣧⡠⡀⠀⠀⠀⠀⠀⣠⢾⠁⠘⡀⢣⡀⠐⣦⠀⠀⠀⠙⠻⠿⠛⠁
⠀⠀⣶⣂⣹⣿⣿⣼⠅⠛⠖⡉⣨⣷⣶⠀⠀⠀⠀⠀⠀⡄⠐⢻⣿⣾⣯⣼⣿⣿⠿⠿⠀⠙⢮⡆⠀⠀⢀⡾⠁⡇⡆⠀⡇⠱⣿⣆⠈⠳⠦⣤
⠀⠀⠈⠀⠀⢀⣿⣿⣾⣷⣄⣿⡿⣿⣿⠀⠀⠀⠀⠀⠀⢠⢀⡟⢻⡿⣿⣿⠛⣿⠇⠀⠀⠀⠈⠁⠀⠐⠋⣀⣼⢀⡇⣸⠇⠀⠙⠻⢷⣦
⠀⠀⠀⠀⢠⣆⢀⠟⠁⠸⡿⠛⡇⠈⠉⠙⠛⠀⠀⠀⠀⠀⢀⣀⠞⠰⢋⡟⠀⣸⠀⠀⠀⠀⠀⠀⠀⢀⠀⠉⠀⡾⠁⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠑⣤⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⠀⠐⠃⠀⡀⢔⣠⣶⣲⣾⠆⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠉⠉⠉⠓⡄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢀⣄⠀⢠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠛⠟⠛⣋⣡⢾⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⢀⣤⣾⡿
⠀⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡣⡄⣦⠋⠀⣀⣰⠃⠀⠀⠀⠀⠀⣀⠀⠀⠙⠿⣿⣉⡁⠔⠋⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠈⠻⢿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠚⢒⡤⢙⣛⣵⢿⣦⣶⡽⠛⣠⡟⠀⠀⠀⠀⠉⠙⡄⠀⠀⠀⢀⡠⡄⠀⠀⠀⠀⠀⢀⣸⣿⣿⣶⣄⠀⠀⠀⣀⣠⣤⣄⣀
⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠛⠌⣻⣦⢿⡉⠛⠛⠋⠀⠀⠀⠀⠀⠀⢰⣷⠀⠀⢠⢫⠊⠃⠀⠀⠀⠠⠶⠻⠟⠟⠛⠿⠿⠷⠤⠾⢿⣿⣿⠛⠛⠉
⠀⠀⠀⠀⠀⠱⣠⢤⣄⡀⠀⡀⠀⠀⠀⠀⠙⢼⠃⠹⠀⠀⠀⢠⣶⣦⣄⡀⠀⢸⢸⡀⠀⣡⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡶⣿⠛⣾⣿⣷⡟⢧⡙⢻⡆
⠀⠀⠀⠀⠀⠀⣧⠀⢹⣿⠛⠁⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠈⠀⠈⠛⢷⣄⠘⢸⣷⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠀⠀⣼⢰⠿⢹⣹⡌⠙⠳⠿
⠀⠀⠀⠀⠀⠀⠙⡇⢴⣏⠁⠒⢤⣤⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠷⢾⣿⣿⣧⣄⣀⣀⣀⠀⠀⠀⠀⠀⠘⠊⠀⢰⢃⡟⠀⢸⡏⠇
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⣫⡴⠶⠋⠀⠀⠉⠀⠀⠀⠉⢻⣇⠀⠀⠀⠀⠀⠰⣿⡜⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠤⠶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠟

⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀⠀⡀⣀⠀⠀⠀⡠⠀⠠⢀⠤⠀⠤⡀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀
⠀⠀⠀⠀⡀⡐⣀⠢⠀⠀⠀⠈⠁⠀⠀⠀⠀⡀⠀⠀⠁⢠⠠⡴⠀⡢⠀⡂⢀⠀⠈⠀⠤⠁⠈⠀⠀⠈⠁⠀⡀⢠⢀⠤⠀⠀⠠⡀⠄
⠀⠀⠀⠄⡑⠡⡀⣀⠀⠒⡔⠀⠀⠀⠀⠐⠆⠁⢸⠀⡀⢠⠀⣇⢠⡀⠰⡀⢀⠀⢎⠀⠀⡄⠀⠀⠀⠀⠀⡲⠀⠀⡈⠈⠁⢀⡀
⠀⠀⠰⣑⢀⢊⠰⠄⢂⢁⠀⠁⠀⠀⠐⠐⠀⠀⠈⠀⠇⠀⠀⠀⠂⠁⠀⠂⠀⠑⠀⠁⠀⠃⠀⠀⠀⠀⠃⠀⢢⠉⡄⢑⠐⢑⣁
⠀⠀⠀⠂⡁⠐⠁⠊⠀⣄⢂⠁⠀⠂⠐⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⡀⣀⠀⠄⠀⠀⠀⠀⠀⡀⠀⠀⠡⢌⠁⠒⠄⠊⡈⠨⠄
⠀⠀⠀⠀⠁⠡⠒⠄⠐⠂⠀⠀⠀⠁⠀⠀⢀⠠⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢀⠀⠀⠀⠀⠀⠀⠈⠄⢠⠰⣐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠢⡀
⠀⠀⠀⡰⠀⢀⠀⠀⠀⠁⠀⠀⠀⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠠⠀⠀⠈⠀⠀⠀⠀⠒⠀⡀
⠀⠀⠙⡄⡀⣂⠀⠀⠀⠀⠐⡀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠘⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠁⠀⠉⠂⠈
⠀⠀⠀⠉⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⡉⠀⠀⢀⠠⠐⠄⠀⢿⠁⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢀⠈⡀⠌
⠀⠀⠀⣄⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠄⠀⠀⠀⠀⢀⡂⠓⠀⠀⠀⢺⡍⠀⠀⠰⠈⠰⠂
⠀⠀⠶⢮⣹⠴⠄⠀⠀⠐⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⡠⠆⠀⠀⠀⠀⠀⠀⠧⡀⠀⠀⠁⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⠀⠄
⠀⠀⠀⠃⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠉⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣀⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠿⠀⠀⠀⠀⢸⠀⠀⠠⡊⣠⠀⠀⢸⠀⠂⠀⠀⠀⠠⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁
⠀⠄⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡘⠀⠐⡀⠀⢸⠀⡀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠐⠒⠒⠎⠀⠀⠀⠉⠒⠒⢺⠀⠀⠀⠀⠀⠀⠀⢸⠠⠄⠒⠲⠄⠀⠀⠸⠒⡒⠀⠀⠀⠰⡀⠡
⠀⠀⠀⡀⠀⠀⠀⠉⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠐
⠀⠀⠠⣀⠁⠀⠀⢀⠀⠀⠀⠈⠀⠢⡀⠀⠀⠀⠀⠈⡃⢀⠀⠀⠀⠀⠐⣰⠈⢐⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠀⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠊⠀⠀⠀⠰⠀⠈⠀⢀
⠀⢀⠤⠀⠀⠈⠃⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠨⠐⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠃⠀⠀⡀
⠀⠈⠈⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠐⠀⠘
⠀⠀⢀⠀⢀⠀⠀⢀⠄⠀⠀⠆⠘⠀⠃⠘⠀⣀⠐⠏⢹⠀⠀⠠⡄⠀⠀⠀⡦⠠⠀⡀⠀⠄⠀⠂⠀⠀⡀⠀⠀⠀⡀
⠀⡐⢀⠐⠄⢑⠀⠂⠀⠂⠀⡆⢀⢀⡀⠨⠀⠒⠀⡄⢸⠀⠑⠚⠁⠁⠀⠉⠁⠀⠡⠀⡀⢰⠀⢐⠆⠐⠄⠐⠒⠎⠀⠀⣀⢸⣄⠖
⠀⠠⣀⠀⢁⠌⠀⠀⠀⠀⠀⠠⠼⠀⠄⢨⢀⠆⠈⢣⣸⠑⠚⢀⠄⠀⠀⠀⠀⢻⠀⠠⠇⠘⡀⠂⠠⠀⣊⠓⠒⠀⠀⠀⠀⡜⢺⠙⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣁⣀⣀⣀⣀⣀⣈⣀⣀⣉⣁⣉⣁⣈⣉⣀⣀⣀⣀⣀⣈⣉⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⢿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⣀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢀⣤⠾⠟⣡⡾⠷⠋⣠⠧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⢛⡴⠚⢁⣤⡴⢻⣵⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⣿⣿⣷⣾⢟⣏⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣯⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠿⢿⣿⣿⣿⠿⠛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡆⠀⠀⠀⢠⣿⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⣠⠞⠉⠨⣰⠀⠀⠀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠰⣧⠀⠀⠙⡂⠀⠛⢏⠑⠛⣠⠟⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⢠⣶⣿⣿⣿⣶⣴⣻⣜⢺⠏⠀⠠⢄⣀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⠈⠷⢤⣿⣯⣿⣿⣿⣿⣆⣻⣿⣿⣤⡟⠑⠛⠉⠀⡀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢦⣄⡀⠛⣷⢾⣿⣿⣿⣿⣿⣿⣿⣿⣧⣀⠀⣤⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣗⠀⠀⠑⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢈⢣⣶⣿⢿⣿⣿⣿⠟⢿⣧⣀⣀⣴⡿⠏⠉
⠀⠀⠀⠀⠠⠄⠒⠲⠖⠲⢦⠤⣤⣄⠀⠀⠀⠀⣀⡀⣾⢁⡀⢸⡿⠋⠻⣧⠀⠀⠉⢋
⠀⠀⠀⠀⠀⠀⠀⠐⠊⠀⠀⠳⣤⡽⠙⢣⡀⠀⢿⡄⠘⠆⠁⣿⡇⠁⢀⠿⢷⡁⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡏⣹⠟⢦⡈⢿⡄⠀⠙⢸⡇⢀⠈⠀⠀⠀⠁⠒⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣀⣤⣤⣠⣵⠀⠻⣆⠀⢸⡇⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠻⣿⣿⣿⣿⣿⣿⣿⣷⣦⣜⣷⣼⡄⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣤⣤⣴⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣉⡉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⣾⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡿⠿⠟⠛⠛⠋⠛⡻⠟⣿⣿⣿⣍⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠟⠉⠀⠀⠀⠀⣴⣶⣿⣿⣾⣿⣿⣿⣿⣼⣷⣦
⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠃⠀⠀⠀⠀⠀⣴⣷⣾⣾⣿⣿⡿⠈⢿⣿⣿⣻⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⠇⠀⠀⠙⣿⣗⠹⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⢻⡿⢿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⡿⠁⠀⠀⠀⠀⢸⣇⠈⠙⢟⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡿⠁⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡿⠀⠀⠀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡧

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠖⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⡇⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢜⠁⡀⠀⢀⣸⣆⣳
⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠀⠀⠀⠰⠀⠀⠀⠀⢏⡀⠀⠀⠀⠀⠠⡆⡂⠀⣿⡇⠀⢀⣼⣿⠏⣩⢷⡄
⠀⠀⠀⠀⠀⠀⢈⡇⡆⠀⠀⠀⠀⠀⢠⣭⡄⠀⠀⠀⠐⣆⡄⠀⠀⠼⠃⢈⣅⠀⠀⠀⢰⡏⣵⠀⠈⠠⠀⣁⣾⣿⠿⠁⠈⣉⠄
⠀⠀⠀⠀⠀⢀⣴⡏⠃⠀⠀⠀⠀⢠⠟⠛⠀⠀⣀⠀⢐⣍⣡⡄⢀⢸⠇⠈⠋⠠⡆⠀⣀⡗⢻⣀⠀⠁⢰⢹⣿⠁⣀⡤⠞⠻⣇⠀⣖
⠀⠀⠀⠔⢠⣿⣯⠦⠄⠀⠾⠂⠀⣤⣶⠀⠀⢠⣙⡂⢸⣇⣼⠃⢿⠀⠀⣰⣦⣀⡽⠚⠉⠉⠈⠉⢑⣦⡔⠈⣤⡾⡉⠀⠀⡀⠃⠀⠞⢆
⠀⠀⡌⢠⣿⠟⢏⡔⠒⢠⠋⠳⣀⠹⠍⣴⣤⠚⠿⠀⡾⠉⢹⡆⠀⠀⡿⢻⡇⣹⠛⠛⠛⠛⠛⠋⠉⣩⣤⡈⢇⢸⣥⠔⠊⠁⠀⡾⠃⠙⠧
⠀⠀⢳⡌⣷⢆⢹⡅⠢⠆⡌⠘⢘⠁⠀⠉⠉⠁⠀⠘⠁⠀⠀⢿⠄⠀⠁⠀⠀⢿⣿⢳⡖⢲⣶⣋⢾⣽⣿⠃⠀⠻⠃⡠⠎⠀⠀⡖⠀⠒⢳⡀
⠀⠀⠈⠣⣿⣿⣧⡜⠒⠢⢞⡳⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⢨⣧⠆⠀⠀⠀⠈⢿⣛⣛⣛⣛⣉⣈⣭⠇⠀⠀⠀⠙⠃⠀⠀⣴⠷⠒⠒⠚⢷⡄
⠀⠀⠀⠀⠈⠙⠋⠉⠁⠀⠈⠀⠀⠀⣤⣾⣿⣷⡶⠶⠀⠀⠀⢠⠇⠐⠤⠤⠔⠁⠀⠈⠛⠍⠭⠄⠚⠁⠀⠀⠀⠀⠀⠀⠐⠿⠿⠿⢿⣿⠿⠿⠿⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠻⣇⠀⢧⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⣧⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡆⡖⢀⠀⢀⡀⣀⠀⠀⠀⢀⠀⠀⣀⡀⢀⠀⠀⠄⡀⠀⡀⡀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐⠡⠁⠇⠸⠁⠟⠀⢄⡃⠀⢈⠀⠐⠁⠐⠁⠈⠀⠇⠀⠏⠃⠃⠄⡇

⣿⣿⣾⣿⣿⡶⠟⠳⠖⢿⣷⣽⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣻⣷⣿⣿⣿⣿⣿⣶⣿⣿⣿⡟
⣿⣿⡿⣿⣭⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣻⣇⣿⣿⣿⣿⣿⣯⣽⣿
⠝⠿⠀⣴⡌⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡈⠀⢂⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⡿⠆⠈⠉⠙⠘⠷⢎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿
⣷⣼⠦⣰⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠰⣾⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⣟⠳⢖⢎⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠁⠀⠀⠐⠈⠉⠺⢶⣄⠀⠀⠈⣟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿
⣿⣷⣄⡀⠁⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠐⠋⠠⢌⣟⣿⣞⢿⣄⠀⠀⠺⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿
⣿⡿⣾⣿⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂⣸⡇⠀⣀⣈⣿⣿⣿⣞⢿⣧⠀⣶⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿
⠿⢷⡌⢻⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣷⠂⠉⠉⢸⣿⣿⣿⣿⡽⣿⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⣶⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠿⣿⢿⣿⣦⡒⠀⣼⣿⣿⣿⣿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⡿
⡿⠿⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣏⡥⠤⠊⢾⣻⣿⠋⠘⠿⢟⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⢿
⣷⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⡿⠉⢀⠀⠀⣾⣿⡿⠀⠀⠀⠈⠁⠀⠀⠙⠻⣿⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣾
⡿⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣿⣿⠏⠀⠀⠀⣈⣱⣾⣿⠿⠇⢀⣀⠀⠀⠲⠖⠂⠀⠀⠰⣿⣿⣿⡮⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡟
⡴⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠙⠋⠞⠀⠀⠷⠿⠛⠛⣧⠀⢀⡈⠁⠀⠀⢀⣠⠄⠀⠀⠀⢹⣿⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧
⣧⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣾⠛⢦⣀⣀⠀⠀⠀⣠⣤⣶⣿⣶⣾⣇⣀⢀⣴⣿⡿⠀⠀⠀⢠⣿⣿⡞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣠⣤⣬⣿⣿⠛⢻⡿⠿⠛⠛⢉⡀⠈⠉⠛⠻⠿⢿⣇⡀⠀⠀⢸⣿⣿⣿
⡗⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⣄⠠⠾⣿⠖⠛⢛⣿⣶⣄⠀⠀⠀⠀⠈⠳⣀⣶⣾⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⣿⣿⣿⠀⢋⠆⠀⠀⠀⠀⠀⠘⢿⣿⣿⣦⣤⣠⣀⣄⣿⣿⣿⣿⣿⣿⣿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡸⣿⡇⢀⢸⡄⠀⠀⠘⠳⣤⡀⢸⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡿⣃⠈⠈⠀⢻⣶⠀⠠⢿⣿⣿⣿⡟⠛⠛⠻⣿⡿⠛⠻⢿⣿⣿⡟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠋⠀⡀⣤⣾⣿⡷⠖⠻⣿⠛⠛⠛⠛⠷⢤⣏⡀⠀⠠⣤⢝⣿⣿⡲⣄⡀
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡏⢀⡿⢂⣰⡶⠊⠀⠀⠀⠀⠀⠀⠈⠛⢶⡀⠀⠈⣉⡿⣿⣿⣻⣦
⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⠀⢉⣿⢃⣾⣳⣾⠿⢋⣠⣾⡗⢠⠆⠀⠀⠀⠀⢸⣿⠀⠀⠈⠻⢿⣿⣿⣿⣿⡏⠛⡆
⣷⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡿⢀⣾⠃⣾⢿⣿⣧⣶⣿⣿⣯⠞⢁⣴⡯⠀⡔⠀⣼⣿⡇⠀⠀⠀⠘⢿⣿⣿⣿⡇⠀⢬⡅
⣿⠀⠀⠀⠐⠾⠀⢀⣄⠀⠉⠉⠋⠀⢻⣿⣼⣿⣿⠻⣏⢿⣅⣠⣬⣶⣿⡿⠁⠀⠀⣰⣿⣿⣧⠀⠀⠀⠀⢺⣿⣿⠟⠀⠠⡀⠁
⣯⠀⠀⠀⠀⠀⠀⠁⠀⠉⠁⡀⠀⠀⠙⢿⣿⣿⣿⣷⡈⠀⠻⣿⣿⣿⣿⠁⠀⠀⣰⣿⣿⡿⣿⡄⠀⠀⠀⠘⢿⣧⠀⠠⡴⠄⢀⠖⢀
⡇⠀⠀⠀⠀⠀⠀⠴⠷⣤⣴⣦⣀⣤⣴⣾⣿⢻⣿⣿⣿⣦⣄⠹⡿⠿⠄⠀⣀⣼⣿⣿⣿⡇⣿⣿⡄⢀⣾⠃⠈⣿⣷⡀⠉⠒
⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⢹⣿⢿⣿⣶⣶⣾⣿⣿⣿⣿⣿⣿⡇⣿⡟⣵⣿⣅⠋⢘⣿⣿⣷⡀
⠀⠀⠀⠐⠂⠐⠒⠺⣷⣿⡿⣿⡏⠉⠻⢿⣿⣿⣿⠟⣾⣭⡆⢹⣿⣿⣿⣿⣿⣿⣿⣿⡿⢁⣡⣾⣿⢻⡟⠀⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣽⢿⡇⢠⠘⣯⣿⣿⣿⣿⣿⣿⣧⣾⣿⣿⣿⠴⠁⠀⢠⣿⣿⣿⡇⠀⡀⠒⣀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠙⣿⣿⣿⠘⣇⣾⣄⣿⣿⣿⣯⣿⣿⣿⣿⣿⣿⣿⣿⣡⣠⣴⣾⣿⣿⣿⠃⠄⠀⠀⠉
⠀⠉⠉⠻⠿⠛⠃⠀⢠⡀⠀⠀⠀⠠⠀⠀⠰⣿⣿⣿⣿⣿⢳⠟⣛⣿⣿⠋⠙⢻⣿⣿⣿⣿⣿⣿⣿⣿⣥⠀⣾⣿⣿⡿⢰⡀
⠀⠀⠈⠉⠀⠀⠀⠀⣿⡀⠀⠀⠀⢠⣷⡀⢀⣼⣿⣿⡿⢋⡴⠟⠿⠁⠀⠈⠲⡶⣩⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⡿⣡⢾⠁
⠛⠿⠷⠒⠞⣻⣷⣶⣿⣷⣤⣀⣴⣏⣽⣧⣾⣿⣿⣿⠿⠏⡀⠀⣄⡀⠀⠀⠀⢀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⡀
⠀⠀⠀⠀⠀⠉⠉⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢧⡀⠀⠀⠀⠉⠀⠀⠀⠀⣫⣤⣿⡿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣠⣬⣽⣿⣿⣿⡿⠂⠀⠉⠛⣦⣤⠷⠀⠀⠀⠀⠻⣿⣿⣿⡌⢻⣿⣿⣿⣿⣿⣆⠘⣿⣿⣿⣿⣿⣿⣷⡶⡶⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠠⠿⣿⣀⠀⠀⠂⠀⠀⠀⠀⠀⣴⣦⣤⣴⣿⠟⠉⠙⠤⠷⡮⣿⡿⣿⣿⠗⠈⠧⣹⣿⣿⣿⣿⣶⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢠⣤⣤⣿⣿⣷⠀⢠⣄⠠⣀⣶⣉⣽⡉⢿⣟⠀⠀⠀⠀⠀⠀⠰⠇⠘⢿⣀⠀⠀⠐⠿⠛⢿⣿⡷⠁
⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⡀⢀⡉⠈⠀⠉⠉⠉⠘⠋⠀⠀⠀⠀⠈⠁⠀⠀⠠⠀⠀⠙⠠⡄⠀⠀⠀⠀⠀⠀⠐⣿⡀⠀⠀⠀⠃⠈⢉⡀⣀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⣀⡀⡀⢀⣠⣶⣟⠉⠉⣿⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣻⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⠤⠤⢤⣾⣹⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢁⡠⠄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠋⠁⡰⠋⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⢀⡞⠀⠀⠀⢀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⢀⡞⠀⠀⠀⠀⠈⠄⠀⠐⠊⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠒⠶⠦⠤⣄⡀
⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣠⠴⠚⠉⣀⣠⠤⠴⠶⠦⠤⢤⣀⣀⠈⠑⢦⡀
⠀⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⡠⠚⢁⡤⠖⠋⢁⣀⡠⣤⣤⣤⣀⣀⠀⠈⠙⠦⣄⠙⢦⣄
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇⣠⠎⡠⠚⢁⣤⡒⡿⣫⠉⡷⢡⡄⣣⡞⢉⠙⡖⢄⡀⠈⠳⣄⠙⢦⡀
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡴⠃⡜⠁⣴⢋⡄⣹⠱⠏⠜⡇⠛⣡⡟⢰⡟⣰⠗⠋⡿⢦⡀⠘⢆⠀⢳⡀
⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠁⠼⠀⡸⡇⠺⡁⢻⣤⣎⢀⡷⣶⠁⣇⣀⠘⠏⠸⢃⡥⠖⢷⡄⠈⢧⠀⢻
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡃⠀⡄⢰⠇⡷⢶⢀⠎⣤⢸⡎⣴⠌⣧⢊⡉⡿⣇⣠⡞⣰⠆⡸⢹⡄⠘⡆
⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢹⠁⢠⠁⢸⡟⣰⡆⡿⠘⠋⣾⠀⠉⢸⡇⠛⣡⠟⠙⣶⡀⢀⣴⡷⢦⡇⠀⡇
⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⣸⠃⣿⠀⢸⡀⠸⡇⠈⣐⣷⣤⢋⡜⠷⣩⠼⡧⢺⠃⡿⢠⣫⠟⢳⣾⣴⠘⡇⠀⡇
⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠂⠀⡀⠀⠁⠀⠀⡴⠁⠀⢸⠀⠘⡇⠀⡟⢺⢋⡄⢸⡎⣴⢀⡇⢠⡘⣾⠀⢀⢆⠏⡶⢸⡇⠈⢸⡇⠀⡇
⠀⠘⢦⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠈⣇⠀⢻⡀⠸⣾⠘⠃⣾⣁⣠⡶⢾⡌⠃⣏⡷⠛⣾⡀⣠⣯⣿⣤⢾⡇⢰⠁⢰
⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠸⣆⠈⢧⡀⢻⡤⢾⠋⣝⢶⡀⣦⠻⠦⠿⢡⡆⣏⡝⣩⠄⡇⢀⡞⢀⠇⠀⣾
⠀⠀⠀⠀⠀⠙⠲⢤⣄⣀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡀⠳⣄⠙⢦⡱⣍⠈⢿⡌⠀⣦⡆⠀⣠⡟⠀⢀⣞⡴⠋⣠⠏⠀⣴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⡈⠳⣄⠉⠛⠓⠻⠼⢶⣿⣧⡴⠥⠷⠞⠋⢁⣠⠞⠁⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢬⣙⠒⠲⠤⣤⣀⣀⣠⡤⠤⠴⠒⠚⠉⣀⣠⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠦⠤⣤⣤⣤⡤⠴⠖⠚⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠠⣿⠤
⠀⠀⠀⠀⠀⠀⠀⢰⠚⢦⢰⡖⣲⠀⠰⡇⠐⣇⠀⡇⠸⣛⠆⠸⠻⠆⠀⠀⠐⡟⠀⣦⠖⢠⡦⣆⠀⡇⠀⢹
⠀⠀⠀⠀⠀⠀⠀⢸⠋⠉⠀⠉⠈⠀⠉⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣸⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠙⠋⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠠⠄⠀⠀⠀⠀⠤⠂⠁⠀⠀⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⠀⠀⠀⠄⠀⢀⠀⠀⠀⠀⠁⠀⠀⠁⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠜⠀⠸⠿⠃⠀⢁⠀⠀⡀⠀⠈⡁⠀⠀⠀⠀⠀⠀⠃⠀⠂⠀⠀⢀⠀⠄
⠀⠀⠀⠀⠀⡀⠐⠒⠀⡀⠀⠄⠸⣳⡆⠠⠥⠀⠈⠁⡤⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠧
⠀⠀⠀⠀⣀⠀⠉⠉⠁⠀⠆⠀⠂⠀⠀⠀⢀⠀⠠⠀⡀⠁⠈⠃⠀⠀⣼⠀⠀⠘⠀⠀⠀⠂⠈⠋
⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠈⠀⠠⠀⠀⠀⠈⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂
⠀⠀⠀⠀⠘⣄⡀⠀⠀⢀⡀⠀⠀⡀⠐⠀⠈⠁⠀⠀⠀⠀⠂⠀⢲⠀⠀⠆⠀⠀⡄⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠤⠘⠃⠀⠀⠈⠉⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢀⡁⠀⠀⠀⡄
⠀⠀⠀⣶⠀⠀⠀⠀⠀⠰⠀⠠⣁⠀⠀⠄⠁⠧⠀⠀⠀⡀⠀⠆⠀⠀⠀⠀⡀⠀⡈⠀⠀⡀⠁
⠀⠀⠈⠒⠁⠀⠀⠀⠀⠀⠈⠠⠔⠀⢠⠂⠀⡀⠀⠀⠀⠇⣄⣀⣀⠀⢠⠀⠀⠂⠈⠃⠈⠀⢂⡄⠀⠀⠄
⠀⠀⠀⠐⠀⠄⠈⡄⠠⠄⠀⠂⠄⠀⠜⠀⠀⠁⠀⠈⢀⠀⠉⠁⠀⠹⠀⠀⢀⠀⠂⠩⠄⠀⣠⠄⠀⠂
⠀⠀⠀⠀⢀⠀⠂⠐⡄⠄⡀⠀⡈⠈⠀⠀⠀⠀⡀⠀⠁⠀⠀⠀⢀⠀⠀⠀⠈⠓⠦⠄⠰⠚⠉
⠀⠀⠀⠀⠈⠃⠀⢁⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠄⣀⡄⠄⢤⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠛⠀⠁⠀⠀⠠⠄⠀⠀⠀⠀⠀⠁⠠⠀⠄⢉⠀⠐⠎⠉⠋⠀⡀⡌⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠈⡀⠠⠄⠠⠀⠀⢀⠀⠁⠀⠈⠀⠀⠀⡀⠀⠀⠀⢠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⣠⣀⠀⠠⠄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠠⠐⠋⠉⠉⠐⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠰⠀⠂⠀⠈
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣶⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⢤⣀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⣤⣴⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡾⣿⠗⣤⡀⠀⣿⣿⣿⣿⣿⡿⠀⢀⣴⠹⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⢸⡿⢛⡄⢾⣽⣿⣿⣿⡗⢠⡛⢿⡇⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢁⡞⠁⣭⠟⡘⢋⣿⣿⡸⢀⠿⣭⠈⢷⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠞⢁⣾⣿⣿⣿⣿⣿⣿⣷⠙⠷⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠛⣿⣿⣿⣿⠛⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠋⠀⠀⠀⣰⣿⠃⠈⢿⣇⠀⠀⠀⠙⢷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣷⣶⣄⠀⣴⡿⠁⢀⡀⠀⢻⣆⠀⣠⣶⡶⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⢸⣿⠟⢠⣿⠁⣾⡿⠛⢿⣦⢿⡀⠙⣿⠃⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠸⣿⠀⢸⣇⠀⣿⠃⠀⣾⡟⢹⡇⠀⢸⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡟⢰⡇⠀⢰⡿⠀⠈⣇⠀⢸⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⢻⠀⣸⠀⢸⠀⠀⠘⣇⠀⠠⢿⠀⣿⠀⣸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣐⡾⡰⠃⠀⡜⠀⡶⠅⠘⠆⠀⠈⢆⢹⠀⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⠞⠁⠁⡠⠚⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠳⢿⣨⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠰⡌⠄⠞⠻⠀⢠⣆⣀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠈⠉⠁⠀⣀⣤⣠⣄⠀⠉⠉⠁⠀⠀⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠓⠀⠀⠀⠀⠀⣰⡿⠛⠫⣦⠱⡄⠀⠀⠀⠀⠞⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⢃⠁⠀⠀⠀⠀⠀⠀⣿⡗⠀⠀⠸⡧⡇⠀⠀⠀⠀⠀⠈⡐⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢙⠂⠀⠀⠀⠀⠀⠀⠀⣷⣿⡄⠀⢸⠃⢳⠀⠀⠀⠀⠀⠀⠰⡋⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢠⠺⠏⣽⢅⠀⠜⡃⠈⠳⡀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠁⠀⠀⠀⠀⠀⢠⠏⢠⣾⠁⠈⠀⠘⠁⡇⠀⠸⡀⠀⠀⠀⠀⠀⢶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⣟⡰⠀⠀⠀⠀⠀⣰⠏⠄⠁⡷⠄⣇⠀⣄⠘⠀⠀⠀⢡⠀⠀⠀⠀⠀⢆⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣏⡁⠀⠀⠀⠀⢠⣫⠇⠀⠀⡇⢾⣭⣷⡈⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⢈⣱⠄
⠀⠀⠀⠀⠀⠀⠈⠀⠈⠃⠀⠀⠀⠀⡾⠋⠈⢷⠦⢿⠂⠀⠳⣇⣀⡀⣰⠀⠀⡅⠀⠀⠀⠀⠘⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠃⠀⠀⠀⠀⠁⠀⡆⣦⠀⠀⠀⠀⠀⠀⠙⣿⣿⢠⢸⠁⠀⠀⠀⠀⠘⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠇⢀⡇⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⠈⡘⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⡀⠀⡀⢡⣼⡧⣿⠀⠀⠀⠀⠀⠀⠠⣿⣿⢸⠃⠀⠰⠀⢀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠂⢀⠨⠉⡡⠀⠸⠛⠛⢾⣿⠀⠀⠀⡇⠀⠀⢰⣿⣡⢺⠀⢠⠇⡀⢄⠉⠅⡀⠐
⠀⠀⠀⠀⠀⠀⠹⣄⠀⠛⠋⠀⠆⢀⠀⠀⠀⢸⠀⠀⠀⢹⡀⠀⠀⣿⠈⢸⠀⠃⡀⠰⠀⠉⠛⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⢈⠕⠂⠀⢀⠀⣌⣘⡿⠀⣻⡄⠀⠀⡀⢧⠀⡚⣿⢤⢰⢮⣇⣡⠀⡠⠤⠐⠪⡅
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⢰⠞⠻⣗⠻⠐⣧⠐⢄⢧⡈⠳⣍⡻⢸⣶⣶⡛⠻⠆⠈⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣀⠎⠀⢂⣌⠋⣸⢿⠯⠓⢄⡑⣉⣆⡹⢘⢿⡶⠀⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡮⠏⣹⣿⣶⡾⢶⣄⢉⡙⢯⣅⢺⠂⢶⢶⣂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠁⠀⢸⡟⢛⠎⠛⠁⣿⠉⠛⠳⠈⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡏⢠⣿⠀⠀⠀⣠⠤⡀⠀⣤⠶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⣼⡇⠀⠀⢸⣿⠀⡇⠀⣿⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⣶⡀⣸⡟⡏⠀⣿⠁⣠⡤⠄⠙⢳⣥⡄⠈⡿⣤⡄⢠⡆⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡆⢤⣿⣡⡿⢸⠁⢰⣿⢰⣿⠐⠁⡀⣾⡟⠀⡔⣸⡟⢠⡿⢡⣿⢡⣾⠋⠈⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠚⣰⠟⡅⠣⢀⡴⠟⢧⡯⠀⠈⣿⡸⠧⠠⠒⣀⠻⠤⠊⠀⣻⡇⠘⠃⣾⡇⠈⠁⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠻⠤⣋⡤⡀⠀⢀⡴⡗⠀⠀⠈⠛⠂⠀⣨⡁⠀⠀⢠⠃⢀⣿⠀⢰⡿⠀⢠⣾⠋⠙⢦⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⢲⡀⠀⣼⠏⠀⢠⡖⠚⠷⠃⠀⠀⠀⠀⠀⢰⣿⠀⢀⣀⣸⡴⠛⢁⣠⠟⠀⠀⠈⠻⠄⠔⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠈⣼⡟⠀⠀⢸⣷⠀⡀⠀⣘⠃⣠⡄⠀⣾⡇⠀⣀⠀⢀⡈⣀⡉⠀⣀⡄⣠⠀⣴⡆⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢠⣿⠄⢸⡇⢀⢉⣿⠃⣰⡿⠀⣿⡇⢠⣿⠁⢸⡿⠁⣾⠏⣸⡏⣴⡟⢰⡿⠀⢿⣇⡜
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠀⠀⢸⡟⠀⣿⢁⠄⣾⡇⢠⣿⡇⠀⣻⣿⢸⣿⢀⣿⠃⢸⡟⢰⣿⠰⣿⡀⢿⣇⠀⢸⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠘⠁⠀⠉⣡⠀⠈⠁⠀⠈⠁⡜⠀⣿⡆⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠤⠀⠀⠴⠚⠁⠀⠀⠀⠀⠀⠀⠃⠴⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠴⠛








⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⢀⣴⠶⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠀⠀⠀⠉⠳⡼⣟⢿⠈⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⡚⠁⣠⣄⣀⡤⠬⡟⠳⣄⣹⡈⠩⣿⠗⠉⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡇⠠⠆⠀⠉⣤⡀⠘⠁⣹⠀⠱⣄⡀⠀⠈⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⢶⣶⡄⠈⠛⣇⠐⠖⠀⢀⣠⢄⠀⢰⠁⢀⡠⠚⠃⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠻⠏⡇⠀⠀⠘⣆⠀⠘⢿⣀⡼⢠⡾⠚⡏⠀⠀⠀⢀⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠶⠿⢧⠀⣀⡴⢾⢓⡶⣶⣲⢶⢋⠿⣄⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠸⡋⠁⣀⡤⣼⣿⠿⢳⡿⢯⡠⠜⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⢀⣀⣈⡧⣄⣀⣠⣴⠟⣠⠴⣇⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠛⠛⠻⡖⠚⠁⠀⢨⠿⠟⠋⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⣰⠋⠂⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣔⠦⢄⣀⠀⠀⠈⠳⢶⠇⠉⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⣀⣼⠟⠒⣶⠦⣄⠙⠲⠤⣄⣀⣀⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⣠⠞⠁⠀⠈⢢⡀⠀⠀⠀⠀⠀⠹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⠾⠋⠀⠀⠀⠀⡴⠟⢦⡀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡎⢳⠀⠀⠈⠁⠀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠘⣆⣀⣤⠤⠴⠒⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣤⡽

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡤⠤⠤⠤⠤⠴⣽⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀
⠀⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⡀⠄⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⣰⠃⠀⠀⠀⠀⠀⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣀⡤⠖⠚⠛⠉⠉⠉⠙⠛⠒⠤⣄⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣠⠖⠋⣁⡤⠴⠒⠚⠉⠉⠛⠓⠲⠦⣄⡀⠙⢦⡀
⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢸⡇⠀⡠⠞⣡⠔⠋⣁⣠⡤⢶⣲⠖⢶⠒⡶⢤⣀⠀⠉⠲⣄⠙⢧⡀
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢃⡞⢡⠎⢀⡴⠛⣴⢣⡖⣸⠇⡿⢠⡏⣠⡄⣟⣙⣦⡀⠈⠳⡀⠙⣆
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠏⢠⠃⢠⡏⣼⠢⣏⢈⠔⠹⣆⡠⢻⠀⠛⣴⠋⣤⠘⢙⣦⡀⠙⣄⠘⣦
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣸⡟⠀⡁⢠⡟⣇⡡⠂⠼⠛⣦⡜⣉⢧⢈⠷⢦⣿⠀⢡⠟⣉⢈⢷⠀⠹⡄⠘
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⢰⢿⠇⠘⠃⢸⣰⢉⠹⡜⢰⠇⡾⠸⠏⣼⢣⡞⣠⡽⣞⡏⠘⢋⣜⣀⡇⠀⣇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⢠⡏⣼⠀⢸⠀⢸⡇⠛⢸⣇⣀⠎⣧⣠⠎⣫⣀⡴⢣⡆⡸⢧⣴⣩⢋⡈⣿⠀⡟
⠸⡇⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠞⠀⢹⠀⢸⡆⠈⡧⢴⠟⡉⢧⡌⡄⢹⡏⠙⢧⡇⠈⢠⢳⢣⡀⣿⠘⠃⡿⠀⡇
⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠘⡆⠀⢧⠀⢧⡌⠸⠃⣿⠘⢁⣞⣇⠿⠸⢧⡴⢇⡇⠙⣐⢹⡀⣸⡇⢠⠇⢀
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⢹⡄⠘⣄⠘⣧⣀⣼⠛⢶⣹⢠⡽⣦⣸⡏⣠⢸⢳⠞⡋⢹⠉⣱⠃⡼⠀⣸
⠀⠀⠀⠀⠙⠲⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠀⠀⠀⠀⠀⠀⠀⠹⣄⠘⢦⡈⢧⡘⢌⠓⠹⣎⠃⢱⣸⠃⠉⣸⠏⠈⣡⢋⡴⢁⡼⠁⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠶⠶⠶⠖⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠦⣀⠙⢄⠉⠒⠷⠦⣏⣦⣤⣿⣀⣜⣹⡤⠾⠋⢁⡠⠋⢀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢄⣙⠲⠤⢤⣀⣀⣈⡉⣁⣀⡀⠤⠔⠚⠉⣀⡤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠶⢤⣬⣭⣍⣀⣀⣤⠤⠶⠚⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⠀⠀⠀⢠⡧⠄
⠀⠀⠀⠀⠀⠀⠀⡶⠒⡆⣴⠒⡆⠀⢾⠀⢾⡀⣾⠀⣏⡇⠰⠞⡆⠀⠀⠐⡗⠀⣦⠖⢰⣠⡆⢸⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣟⠉⠁⠈⠉⠁⠀⠁⠀⠉⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠁
⣀⣀⣀⣀⣀⣀⣀⣏⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣟⣻⣛⣛⣿⣫⣛⣻⣛⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣿⣭⣭⣭⣿⣿⣿⣭⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⠀⣠⣤⣤⣤⡸⣷⡀⣼⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⡟⠁⠀⢹⣷⢹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⣰⡏⢻⣆⠀⢀⣾⠇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠁⠀⠙⠛⠛⠁⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠉⢁⣴⠶⢶⣦⠀⠀⠀⠀⠀⠀⢹⠉⣭⠀⠀⣶⠄⣴⠶⠶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡿⠁⠀⠀⣿⠆⠀⠀⠀⠀⠀⢸⡄⣿⣤⣶⣿⡀⣿⣤⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⣷⣄⣀⣼⠟⠀⠀⠀⠀⠀⠀⢸⡇⣿⠀⠀⣿⠀⣿⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠈⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⣀⣤⣴⣦⡄⠀⠀⠀⠀⠀⣀⣤⣤⣤⡀
⠀⠀⠀⠀⠀⠘⣿⠀⢠⣄⠀⣼⡇⣠⡾⠛⠻⣦⠀⣟⢀⣀⣼⡟⢰⡇⠀⠀⠀⣿⠀⠀⠹⣿⡄⣾
⠀⠀⠀⠀⠀⠀⢻⣇⣿⣿⣷⣿⢡⡟⠀⠀⠀⣿⠃⣿⣿⣟⠉⠀⢸⣿⠀⠀⠀⣿⠀⠀⣠⣿⠇⣿
⠀⠀⠀⠀⠀⠀⠸⣿⡟⠘⣿⡿⠈⢿⣤⣤⣾⠏⠀⣿⠀⠻⣷⠄⢸⣿⣤⣤⡀⣿⣴⡾⠟⠃⠀⡛



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠤⠤⠶⢒⡒⣛⣛⢋⠉⣉⠙⣦⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣼⠉⣶⣶⣴⣦⢸⣿⠃⢹⡇⠘⢾⡟⢠⡟⠉⠉⠉⢛⣲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠴⠋⠁⢸⡄⣟⠁⠛⠹⠆⠉⠓⢀⣁⣑⣛⣁⣸⣇⣀⣀⣀⣙⣳⣄
⠀⠀⠀⠀⠀⠀⠀⠠⡄⢀⠀⠛⣻⠂⢀⣸⣧⣤⠤⠶⠶⠚⠛⠉⠉⠉⠉⠉⠉⠉⡁⠀⠀⠀⠀⠉⠉⢁⣀⣴⠊⠀⢀⠴
⠀⠀⠀⠀⠀⠀⢰⢸⡇⢴⠀⢰⣧⠖⠋⠁⠀⠀⠀⠀⠀⠀⢀⣀⣤⢺⡇⠀⠀⠀⣿⡷⢲⣒⡶⣶⣻⠛⢆⣿⠀⣀⢸⣯
⠀⠀⠀⠀⢀⣀⣸⣸⣇⣿⠀⠈⠁⠀⠀⡶⣶⠲⡖⢲⢺⣿⣿⡇⠸⣮⢿⠀⠀⠀⠛⢀⡘⠟⠀⠈⠛⠀⠀⠁⠀⣿⢸⣿⢀⣶⣆
⠀⢀⡞⠩⠁⣀⡈⣉⣁⠉⠍⠙⢦⡀⠀⠯⡏⠐⡷⢾⠀⠿⠛⠇⠀⠁⣀⠀⣀⠀⢀⣼⡅⠀⢾⡆⠀⠰⣿⡆⠀⣿⡜⠋⠀⠈⠀⣀⡀
⠀⠀⡇⣤⣤⣴⢶⢠⡶⣄⣶⣦⢈⡇⠀⠀⠀⢀⠀⣤⠀⠀⠀⠀⢷⢴⠻⠔⣇⠀⣼⠽⡇⠀⣟⢳⡀⢀⣯⡿⣄⠙⢁⢠⣿⣆⡆⠘⠁
⠀⠀⢻⣇⠻⠿⢾⣼⠧⠿⠃⣿⣼⠃⠀⠀⢀⣸⣿⣿⣽⣃⠀⠀⠘⠄⠑⣒⠋⠘⠳⠛⠃⢸⣯⢈⣧⠸⠷⠿⠛⠀⠈⢳⣻⠯⠂
⠀⢠⠼⢯⣔⣆⣈⣩⣄⣀⣺⠴⠛⠑⣆⢏⣉⣩⣿⣧⠤⢼⡇⠈⣏⣧⣼⣹⠀⠀⢠⢒⠤⢠⡬⠉⡁⠀⣀⣀⠀⢀⡤⠠⡁⠀⢴⣤⠄
⠀⢸⠈⣀⠀⣠⡀⣰⣦⠀⣠⡀⢠⣄⢹⠈⡇⠀⣿⡇⠀⣸⠀⢺⡿⣿⣿⠿⡿⠀⠜⠬⠗⠠⠽⠖⠁⢸⡁⠈⠳⠏⠀⢀⡇⣀⠈⠁
⠀⠘⡟⢹⣾⠇⣿⣿⢸⣸⡏⢧⣾⠘⡾⠃⣇⠀⣿⣷⣀⡏⠀⠘⣷⠀⢐⢿⡇⠰⠀⠀⣠⡴⠲⣄⠀⠘⠧⣄⠀⠀⢀⡼⠁⣿⣿⠟
⠀⠀⣧⣠⢉⠠⡙⠇⡀⣙⣂⢬⡗⢔⡿⠀⢈⡅⢠⡀⢀⣀⡀⠀⠻⠼⠿⠾⠃⠀⢀⣠⡟⠀⠀⢸⣇⡤⢄⠀⠉⢷⡟⠁⠀⠈⣁⡀
⠀⠀⠛⠬⣏⣙⣙⣋⣹⡥⠽⠖⠚⠉⠀⠀⠘⢦⣾⣷⣯⡴⠃⢀⣤⢀⣀⠀⢀⣰⣋⠸⡇⠀⠀⢸⡏⠀⢸⡇⠀⠸⣅⠀⢠⠞⠁⠀⠉
⣀⠞⢦⡄⣀⠀⣀⣤⡤⡄⠀⡀⠀⢿⡷⠀⡏⠉⣿⠀⣿⠁⢹⣾⣿⣿⠟⣰⠏⠀⠉⣷⢳⣀⣠⣾⡞⠋⠉⢳⡄⠀⢸⡆⡏
⢨⡶⠼⠆⢹⡽⠛⠛⠉⠃⣸⣽⢠⣏⣷⠀⢻⠀⢿⢀⡯⠀⣼⢸⣿⢹⠇⣿⠀⠀⠀⣿⡏⠉⣷⣼⠁⠀⠀⢸⠇⢀⡾⠁⢹⣄⣀⣠⠼
⢠⣄⡄⠀⠀⠀⠴⣿⡆⠀⠧⠿⢀⠌⡄⠀⠘⠲⠺⠞⠓⠒⠋⠘⠛⠚⠀⠘⠦⣤⠞⣿⠀⠀⣿⡿⢄⣀⣠⠟⠀⠀⠀⠀⡾⠉⢠⣤⣤
⠚⢶⠛⠠⠴⣱⠴⢋⡀⣰⣆⢻⡿⠀⡗⣠⡀⠀⠀⠀⣀⢤⡎⣳⣀⠀⠀⠀⠀⢩⠀⡻⣄⣠⠏⡀⠘⢋⡴⠊⠉⠑⢦⠀⢇⠀⠮⣿⠿
⠴⠆⢀⣀⡉⠋⠀⠈⠙⠾⠋⠁⠀⢀⣔⣛⣓⣄⠀⢴⡙⠦⠟⠗⣚⣳⠀⠤⠀⠀⠄⢃⠘⠉⠠⠁⠌⢸⠀⠀⠀⠀⠈⡇⠈⡆⢀⣁
⠀⠀⣿⠁⡭⠉⠉⣉⣷⡆⠀⠀⢠⡗⠢⠗⡒⢚⡇⢸⣸⣿⢻⢫⣿⠇⠀⡿⡄⠀⠀⠱⣄⢀⢀⠊⠀⠸⣄⠀⠀⣀⡾⠁⣠⠇⠺⠿⠂
⠀⠀⣿⣀⣠⣴⠞⠛⣩⡿⡵⠀⢪⣤⣑⣭⣤⡭⣼⠀⠛⠛⠛⠛⠋⡤⠀⠛⠳⠤⡀⠀⠸⡶⠁⠀⠀⣠⠟⠛⠉⠁⣤⡀⠁⣤⣤⡄
⠀⠀⠈⠙⣯⣄⡴⢞⡥⠚⣁⢀⠈⢿⣿⣿⣶⣧⠏⢀⢀⣴⡄⠀⢺⡁⠀⠀⠀⠀⣈⠇⢰⣧⠀⠀⠀⡇⠀⣴⣶⠀⠉⢁⡄⠘⠋
⠀⠀⠀⠸⣯⠵⠚⠉⠀⠠⣻⣟⠄⠀⠀⠀⠀⠀⠀⢹⠾⡿⣿⠃⠀⠹⡟⡟⣿⣿⣿⠀⠸⠁⠀⠀⠀⢸⠀⠈⠈⢀⣽⡁⡷⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣁⣀⣀⣀⣧⣿⣟⣛⣛⣀⣀⣀⣀⣀⣴⣃⣀⣀⣀⣈⣉⣉⣃⣀⣀⣀⣀
⣿⣿⠛⡿⢛⠻⢻⡿⣻⢿⡟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣤⣤⣤⣤⣄⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣤⠶⠚⠋⠉⠉⢀⢀⣀⣀⣀⣀⡉⠉⠉⠓⠲⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠁⠀⠓⣀⣴⣶⡿⠟⠛⠛⠛⠛⠛⠿⢿⣶⣤⣀⠁⠈⠙⠲⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁⠀⠀⢀⣰⡾⠟⠉⠀⠠⠀⢀⣀⣈⣀⣀⠁⠀⠈⠙⠻⣷⣆⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⣿⠟⡀⠀⢀⡤⣖⣋⡉⠉⠉⠉⠉⠙⠲⢤⡀⠀⠈⠳⣷⡄⠀⠀⠀⠝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⣰⣿⠁⠀⠀⣴⠯⠊⣡⣿⣬⡑⠤⡀⠀⠀⠀⠀⡙⢦⡀⠀⠹⣿⣄⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠁⠀⠀⢰⣿⠃⠀⢀⡾⢁⣴⠇⡨⠛⠯⣍⠳⢬⡑⠤⡀⠀⠘⡄⢷⠀⠀⠸⣿⡆⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⣸⡿⠀⠀⣼⡵⣫⠖⠉⠀⠀⠀⠀⠉⢦⡙⣶⣌⠳⡄⠁⠸⡇⠐⠀⣿⣇⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠂⠀⠀⠀⠀⣿⡇⠀⢠⣿⢡⣃⠤⠄⠀⠀⠠⠶⠦⠄⢳⡈⠻⣷⡘⡆⠀⡇⠀⠀⣿⣟⠀⠀⠀⠀⠀⠀⡹⡆
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠒⢾⣿⠀⢹⢻⠈⢠⢤⣤⠄⠀⠀⠠⣖⠆⠀⠙⣦⡈⢧⡇⢰⡇⠀⠀⣿⡟⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠀⠀⠒⠺⣿⡇⢸⠀⣷⣾⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⣼⠇⠸⡇⠀⣸⣿⠛⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⢉⣉⣙⣛⣻⣾⠀⢨⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠀⠄⣿⣰⣟⣛⣉⣉⠁⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠠⠂⠀⠀⠀⠀⠠⠤⠤⠴⢿⡆⠈⠈⣷⣄⠀⠀⠀⠀⠀⠀⢀⠴⣿⣿⡇⢀⠃⢻⠿⠤⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠠⠴⢒⣫⣿⣧⠀⠀⢹⣿⣷⣄⣀⣠⡤⠖⠁⠀⣿⣿⠀⡌⠀⢸⣯⣛⡒⠦⠄⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⢋⠭⠖⡿⡈⡆⠀⠀⣿⣿⠇⠉⠃⠀⠀⠀⠀⣿⡏⢰⠀⠀⠸⢻⣶⢍⡙⠂⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠗⠁⠘⡄⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⢸⠁⡿⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⣣⠀⢸⣏⡒⠦⠤⠤⠤⠖⢚⡟⠰⠃⠀⠀⢠⣠⠀⠀⠙⡄⠀⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⣠⠋⠀⠀⠮⡁⠀⢸⠀⢸⠇⠉⠉⢻⡟⠉⠉⠀⡇⡇⠀⠀⠀⠀⠁⠀⠀⠀⠸⡄⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠄⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⠈⢀⡞⢠⡆⠀⠈⣃⠀⣠⠦⡇⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠠⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠙⢆⠀⢰⠃⠀⠀⠀⠀⠀⠀⢀⢆⠞⠀⠀⢀⣠⢞⢽⠽⣦⡁⠱⡘⢦⡀⠀⠀⢠⠶⠀⠀⡇⢸⡆⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡟⠀⠀⠀⠐⠃⠀⣠⣿⢋⡤⣤⠞⠕⡡⢣⠚⡐⢌⠃⠰⡟⠦⣍⡒⠤⣀⡀⠀⠀⢻⠀⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⣾⡟⠉⠀⢠⠃⠀⠀⢠⢻⡐⡜⠀⠀⠀⢷⠀⠀⠙⢶⣤⡉⠓⠢⣸⡄⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠇⠀⡠⢊⡿⠻⠁⠀⠀⢸⠀⠀⢀⠏⣼⢳⣜⠢⣀⠀⢸⠀⠀⠀⠀⢧⠉⠑⢄⠈⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣠⠊⠀⡜⠁⠃⣠⣀⠀⢸⣤⣐⣡⣾⠇⠀⠨⡗⢦⣤⣿⠀⢃⠀⠀⠘⠂⠀⠀⠑⡌⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡔⠁⠀⡜⠀⠀⠀⠈⢠⠀⢸⣿⡿⢀⡏⠀⠀⠀⣇⠸⣿⣿⠀⠘⡞⠄⠀⠀⠀⠀⠀⠸⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠘⠀⠀⠀⠀⠀⠈⠀⠈⠉⠁⠚⠀⠀⠀⠀⠘⠀⠉⠛⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡄⢠⡬⠙⠫⣅⣆⡞⣷⣽⠟⣶⢶⣿⡀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢀⡴⣁⡘⠖⣣⣞⣽⠿⠛⢟⣧⣿⣿⣷⡵⠃⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠺⢏⣿⠿⢿⠟⣹⡿⠠⣖⣶⣾⡿⠛⠉⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠘⢿⣿⣿⣿⢿⣿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡀⣀⠀⠀⠀⠀⠀⣀⣀⢀⡀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣄⣄⣀⣀⣀⣠⣀⡀⠀⠀⠀⠀⢀⠄⣠⣤⣄

⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣶⣾⠁⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢤⣾⣼⣷⣿⣼⣧⣶⣷⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣽⣿⣿⣿⣿⣿⣿⣿⣿⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⢿⣿⣿⣿⣿⠿⠍
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⢸⠋⡇⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠄⢰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡅⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⣄⠐⠂⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠈⠤⡄⡂⠀⢀⢀⣀⢄⡀⡀
⣀⠀⠀⠘⠆⠀⠀⠀⠰⣀⠀⠀⢢⣤⣤⠀⠀⠀⠀⠸⡜⠀⢀⡔⠀⠀⠀⢸⣙⣯⣸⣶⣠⡼⡈⠆⣄
⠈⠁⠀⠀⣀⣢⣄⣠⠁⠉⠤⠄⠀⠉⠁⣀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠉⠘⢳⢸⢟⡹⠿⠂⢫⡷⣄
⠀⠀⠀⠰⣿⢿⡿⣿⡷⠀⠀⠀⠠⠀⠚⠁⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠰⠂⠘⠀⠀⠋⠣⠒⠂⠞⠣⠼⣉⣦
⠒⠒⠈⠙⢻⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣷⡀⠀⠀⢻⡖⣊⡥⣘⠃⠐⠿⠷⠘⢴⣿⠂
⠀⠀⠀⠀⠐⠛⠛⣧⠀⠒⠲⠆⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠊⠥⠶⡼⣅⠀⢀⠂⣰⣆⣼⠋
⠀⠀⠀⠀⠀⠐⠀⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣦⣄⠈⠁⣿⠧⠼⠛⢙⡹⠃
⠀⠀⠀⠀⠀⡀⠀⢠⣀⡘⠀⠀⠀⠀⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣿⣶⣤⣄⣀⣀⣁⣤⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠻⣧⠜⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠛⠛⠋⠁
⣿⣿⠛⢛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠉⠀⣠⡄⡸⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠁⠀⣀⡠⣴⡟⠀⠀⢰⠇⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⢡⡾⠋⢠⣖⠦⡿⢀⡟⠀⠀⣸⡇⠀⠀⢠⡖⣰⢇⡖⣰⠊⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠻⠤⠺⠧⠺⠗⠰⠴⠛⠤⠀⠀⠸⠖⣿⠘⠓⠛⠘⠗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⡄⠀⠀⠳⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡺⠣⠀⠀⠀⠀⠀⠀⣼⣽⢷⠛⠀⠀⠀⠀⡟⣰⢂⡀⠀⠀⠀⢠⠞⢦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⣀⡀⠘⠃⠉⠈⠃⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⣷⠀⢀⣀⠀⢀⠞⢡⡟⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⣸⡏⡰⢡⡿⣰⠃⢀⣿⠃⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⣰⡟⡼⢁⣾⣷⠃⢠⣾⠇⣀⣀⠀⠀⣀⣀⡟⣀⢀⡤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢠⣿⢱⠃⢸⣿⠃⢀⣾⠏⣾⢃⣴⠆⣾⢁⣾⢣⣿⠞⢰⡯⢛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⠏⠀⠘⠋⠀⣾⡏⠀⠉⠉⠛⠈⠙⠁⡿⠈⠋⠀⠈⠓⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣀⡼⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⡤⡀⡀⢀⢠⣇⠗⣲⠀⡠⣆⢀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⡟⠷⡜⠃⡶⡻⢉⠄⢁⡾⠓⠈⠀⠀⠁⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠟⣛⢛⡛⠛⠷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢟⣵⠾⠿⠛⠛⠿⣿⣷⣤⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢧⣿⣵⠳⣦⣄⡀⠀⡹⣿⣿⣷⡌⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⢺⣯⡝⠉⠉⠐⢿⣄⠀⠹⣿⣿⡿⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⢸⣿⠳⣄⠄⠂⠀⢹⣆⡄⢻⣿⣿⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣌⢿⣄⢹⣄⡔⠀⠘⠟⡇⡄⣿⣏⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣾⢻⡆⣷⡿⠤⢤⡆⠃⣷⠈⢿⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠘⠃⡏⢀⡀⠈⠹⠃⠹⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⠠⠰⣸⣟⠻⡆⡆⠀⠀⢰⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⣄⠳⢢⡾⠋⠂⠙⣇⢀⡀⠁⡀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⢠⠊⣹⢡⣇⣳⡀⠀⣿⠉⠳⡄⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡰⠁⢈⡨⣿⠚⣿⡀⠀⡄⠀⠰⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡁⢀⢨⡟⣹⠀⡿⢿⠀⠃⠀⡄⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢰⠁⢸⢸⡇⡏⠀⣇⢸⠀⠀⡆⢠⣰⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⢸⢸⣿⠇⠀⢿⡇⠀⠀⡁⠄⠃⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⢰⠏⠀⠀⣸⢁⡀⠀⠋⠈⢧⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⢠⢀⠀⠀⠀⠉⠲⢧⡤⠀⠀⠀⠁⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⢸⢨⠀⠀⠀⠀⠀⠀⢹⣦⡀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠘⠀⠆⠀⠀⠂⠰⠇⠐⠃⠁⠀⠀⠻
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣟⣛⣟⣙⣻⣛⣿⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡁⠀⢠⡾⠃⠀⣰⡇
⠀⠀⠀⠀⠀⠀⢀⠔⠋⠉⠉⠙⠳⢦⣤⣤⠖⠀⠙⠿⢶⣿⠷⠶⠟⠋
⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⣦⡀⢰⡇⠀⠀⢀⣤⡀⢀⣿⠃⢀⡃⠀⡰⠋⣷
⠀⠀⠀⠀⠀⠀⠘⣄⣀⣠⣿⠇⣸⣇⣠⢠⣿⠗⠁⢸⡿⠀⣸⡏⢠⠁⣠⠟⠀⠈⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⣵⠋⢻⣯⠁⠸⣿⣀⡠⠛⢧⡴⢋⣛⣁⣈⣹⡇⢀⣠⠾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢀⣾⠇⠀⣀⡤⠥⠖⠒⠀⠀⢺⣀⣀⣴⠟⠉⣉⡤⠤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠙⣛⣡⣤⠤⢶⠒⡛⢋⡉⢉⢩⠉⡙⠢⠀⢀⣼⠏⠀⣠⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠂⣠⣄⠀⠒⠘⠛⣃⡼⠥⠴⠾⠦⠤⠀⢀⣾⢿⡾⠟⠋
⠀⠀⠀⠀⠀⠀⣴⠖⢶⣄⢸⠀⢹⣧⢠⢺⣦⠀⢠⡾⢃⣄⠀⣤⡄⣼⡏⢺⣅⢀⣤⡄⢀⡀
⠀⠀⠀⠀⠀⠀⣿⡄⠀⢻⣿⡀⠘⣿⡇⢸⣿⠀⣿⣇⣼⡏⢸⣿⢠⣽⠁⢠⠟⣿⡿⣽⣟⡽⠃
⠀⠀⠀⠀⠀⠀⠙⠷⠀⠸⣿⡇⠀⠻⠃⢸⣿⠀⠇⠀⢸⣿⠈⠉⠀⠻⡤⠊⢰⣿⠛⣿⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⢀⠀⠀⠸⣿⠜⢀⣀⡼⢿⣇⠉⠉⠉⠙⢦⠀⠻⠄⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠘⢷⣶⣶⡶⠟⠋⠁⠀⠈⠛⠷⣤⣤⡤⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⣠⣶⣾⣿⣶⣴⣁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⣀⣤⣤⡬⣅⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣊⣤⣽⣷⣬⢳⡀
⠀⢀⡞⠋⠁⠀⠙⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⣼⣟⡿⡯⣧⡟⣧⠰⠀⠀⠀⠀⠀⠀⠀⠈⣼⣿⣯⣿⣏⣻⢀⣿
⠀⣾⡇⠀⠀⠀⠀⠀⠹⣿⣿⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⣦⣟⣷⣻⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢜⢿⣿⣿⣿⡶⢋⣾⣿⠁
⠀⣿⣯⡀⠀⠀⠀⣠⣾⣿⡟⠀⠀⣼⠷⣭⣓⠶⣤⣥⣀⡀⡀⠠⠘⢮⡷⣿⣽⣯⠏⡘⠀⠀⠀⠀⠀⠀⠀⢬⡳⢿⣯⡵⠞⢋⡽⠋
⠀⠘⢿⣿⣿⣾⣿⣿⣿⠟⠁⠀⠀⢻⠀⠈⠉⢻⡶⣶⣦⣴⡾⠀⠐⠢⠍⠉⠩⠅⠊⢀⠦⠀⠀⠀⣴⠟⢉⣾⠃⠀⠉⠉⠉⠁
⠀⠀⠀⠉⠛⠛⠛⠋⠀⠀⠀⠀⠀⠈⠳⣤⣀⣀⣵⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠐⢸⡀⠀⡇⡀⠋⢠⡾⣡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠋⠁⣀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠸⢷⣼⡅⠀⣰⣯⠈⠁
⠀⠀⠀⣤⡠⠄⠀⠀⠀⠀⢾⣿⠀⠀⠀⠀⠀⢀⣀⣀⣀⣸⣆⡀⠀⠀⠶⢋⣾⠃⠀⠀⠈⣿⣿⣿⣿⣁⢁⠀⠀⠀⠐⣲⡶
⠀⠀⠀⢳⠙⢷⡂⠀⠀⠀⣾⠃⠀⠀⢀⡴⠛⣉⠄⠘⠉⠛⢿⣷⣤⡄⣠⡿⠁⠀⠀⠀⠀⠛⠙⣿⣿⣭⣤⣤⠤⠴⠛⠋
⠀⠀⠀⠈⠃⠀⠻⣧⣰⣷⣿⠀⠀⢠⣿⣴⣿⠉⠀⠀⣀⣤⣾⣭⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠄⠈⠉⠁⠀⢀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⢄⣮⣿⣿⡏⠄⠀⠸⡿⠻⣯⣀⣰⣾⣿⠟⡛⡛⠛⠻⢷⣶⣶⢦⣿⡀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠿⠿⢿⣶⣄
⠀⢠⣤⣤⠤⠔⠒⠛⠋⠉⠂⠀⠀⠀⠹⣦⡈⠛⢿⡿⣇⢺⣽⣻⣿⡟⣦⣬⣫⡀⠀⠀⠀⢀⠀⢠⣿⣿⣿⠁⠀⠀⠀⠀⢿⣿⣧
⠀⠀⠀⠙⠂⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠛⠻⠿⢇⠘⠷⣍⠻⠷⣿⣿⣻⠇⠀⠀⠀⠀⠈⠁⠸⣿⣿⣿⡄⠀⠀⠀⣀⣼⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠈⢻⣶⣶⣶⣶⡞⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⣶⣶⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⢿⣿⠭⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⠿⠿⠟⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⠂⠀⠀⠀⠂⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⢀⣿⢀⣤⡶⠶⣶⡄⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⢠⣸⡟⠁⠴⠂⣀⣹⠇⠀⠠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⠼⠞⣿⡀⠀⣄⢀⣽⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣡⡾⡀⢿⡟⢠⣴⠀⣿⡀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠈⢻⠙⢨⠃⠋⠛⣨⣧⡙⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡇⠀⠀⠀⡀⢇⠀⢷⣴⡌⠉⣴⠇⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃⠀⢤⣤⣸⡿⢷⣸⡟⠀⣸⢃⣤⠇⢹⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⢣⡞⠁⢘⡿⢋⣴⠋⡍⣰⣿⣧⢼⢻⣦⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⣼⠃⢀⡟⠀⣾⡇⠀⣷⣿⢿⣇⢆⠸⣇⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⣿⠀⣸⡇⠸⠛⣿⠀⢰⣿⣿⡟⣼⠀⣿⢸⣷⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣿⠆⣿⠀⠀⠀⣿⠀⠀⣿⣿⠀⣿⡆⣿⠸⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⠀⣿⠀⠀⠀⣿⠀⠀⢹⣿⠀⢹⡇⣿⢸⡿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⡿⠀⣿⠀⠀⠀⣿⡄⠀⠸⡏⠀⠀⡇⣿⢸⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠀⠀⠀⡿⠀⠀⠀⣿⡇⠀⠀⢣⠀⠀⡇⣿⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢹⣿⠀⠀⠘⡀⠀⡇⢹⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡆⢀⣀⡃⠀⡇⢸⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⡄⠀⠀⢸⡟⣠⣿⣿⠃⠀⡇⢸⠈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⡇⠀⡀⢈⣷⢹⣿⡅⠀⠀⡇⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⢸⡇⠀⠀⠂⡿⠈⢀⠀⠀⢀⠃⢸⢧⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠈⡇⠀⣸⠀⢁⡴⣿⠀⡀⢸⠀⠘⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣇⢴⣿⡀⢻⠁⣿⠀⠇⠘⠀⣷⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⢿⢸⡿⡇⢸⠀⣿⠀⡀⠀⢀⡟⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⢈⣸⠀⠃⣼⠀⣿⠀⡀⠀⣸⡇⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠰⠀⠞⠀⠀⣿⠀⣿⡇⣷⢠⣿⡁⣿⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢀⡞⢾⠁⠀⠀⠀⢸⠀⣿⡇⠁⣸⣿⢡⣿⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢿⡇⠃⢸⠀⠀⠀⣼⡇⡿⣷⠀⣿⢿⢼⡟⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⣸⠃⡇⠈⠄⠀⠀⣿⣏⡁⣹⡄⣿⢀⡈⣵⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⣿⢸⡇⡄⣗⡼⢡⣿⠹⠇⣉⣴⡼⣏⣼⠟⣡⡄⠔⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⣻⣬⣱⠷⠾⠥⢾⡅⢧⡤⠯⠍⣹⣍⠋⡼⠁⠿⢋⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠉⠀⠁⠀⠈⠁⠉⠁⠀⠀⠀⠀⠀⠉⠁⠈⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡦⠀⠀⠀⢀⡴⠒⠲⡄
⠀⠀⠀⠀⠀⠀⣀⣠⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⢠⣤⡀⠀⠀⣿⠀⠀⠀⢻⠀⢠⣾⠂⠀⢠⣿⣧
⠀⠀⠀⣠⠴⠋⣉⣐⣦⣀⢦⡈⠳⣄⠀⠀⠀⠀⠀⣿⡇⠀⠀⣹⣦⣀⣀⣸⣀⣸⣿⠀⠀⢸⠘⣿⡄⠀⠀⠀⠀⢀⣤⣄⡀⠀⣰⣶⣄
⠀⠀⠀⠈⠓⠦⣄⣩⡟⠋⣽⣫⠽⠋⠉⠉⠉⠉⠉⢹⣿⡴⢺⣿⠀⠀⠀⡼⠀⠈⣿⠀⠀⡞⠀⣿⣧⠴⠋⠉⠉⠉⠛⠛⠉⠉⢹⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢀⡤⢺⣿⠀⠘⣿⣄⣀⡼⠁⠀⠀⠙⠓⠚⢁⠴⢻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⣿⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⢀⠞⠁⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⢠⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⣠⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠒⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬⠽⠶⠿⠿⢿⣷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⡰⣻⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⢿⣿⣷⣄⠀⠀⠀⣠⣮⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠳⣿⣿⣿⣦⣴⡞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⡱⡹⣾⣿⣿⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⣤⣹⣿⣿⣿⣿⣧⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡐⢘⡿⢿⣿⢣⠹⣿⣿⣾
⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⡿⢀⢰⣟⠀⠀⠀⠀⠀⠀⠀⠀⢆⠀⣹⡗⠈⠘⣄⣃⣀⣾⡿⠿⣿⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡇⢎⠀⠓⠀⠀⠀⢀⡀⠀⢀⠀⠈⢀⣀⡴⠞⠛⠉⠉⠁⠀⠀⠀⠀⠀⠈⠉⠙⠳⣤⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣄⢆⡑⡄⣠⢀⠀⠀⠀⠦⠀⢠⣰⠞⠀⠀⠀⢀⡤⠀⠠⣶⠀⠀⠀⡀⠀⣀⠀⠀⠈⠛⠢⡀
⠀⠀⠀⠀⠀⠀⠀⣿⣻⣿⣿⣿⣿⣾⣮⣵⣼⣄⣩⢄⣄⣴⡏⠀⠀⠠⠚⠉⢁⣹⠷⠿⠀⠾⡦⢳⢦⣙⣃⣠⠀⠀⠀⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠹⢯⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠀⠀⠀⠲⠤⠀⡀⠉⠁⢈⣤⡑⣄⠊⣴⠛⣥⡀⢀⠘⡆⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢦⡉⠻⣟⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢐⣆⠘⢋⣥⡖⠀⢨⣍⠁⢉⡆⣠⠀⠉⠓⠉⠀⠛⠄⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠷⢶⣾⣿⣭⣽⠯⡤⢾⣿⣦⠀⠈⡉⡄⣠⣅⣰⠤⠺⢶⡤⣬⠿⠷⠾⢿⡧⠔⠲⠞⡤⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠈⣿⣿⣷⠀⠙⠿⠿⣋⠀⣴⣻⠆⢸⡇⢠⡴⠄⢀⣤⣶⡧⢠⠇⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣾⣣⡀⠀⠈⠻⣷⡎⡋⢁⠘⠋⢠⣽⡂⡀⠁⠀⣠⠎⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣷⣢⣀⠀⠀⠁⠉⠛⠷⢦⠀⠈⡀⠀⠀⠀⠉⠀⠀⣴⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⢿⣿⣿⣿⣧⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢍⡻⣿⣿⣿⣿⣿⣶⣶⣦⡤⢤⠤⠤⣶⣶⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠨⢿⣿⣿⣿⣿⣅⣀⡄⠐⡼⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣲⠖⠒⢲⡖⠶⣄⢀⡶⠒⣦⡀⣶⡆⠀⣶⡆
⠀⠀⠀⠀⠀⠀⢀⡤⠐⠀⠀⠀⢰⡞⠁⠀⠁⠀⣿⣄⡀⢸⣇⣴⠟⢸⡃⠀⢸⡇⣼⢷⣴⢿⡇
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⢻⠀⠀⢸⡏⢻⡄⠘⢧⣤⡾⠁⣿⠈⠋⢸⡇
⠀⠀⠀⠀⠀⠀⠈⠁⢠⠤⠤⠤⣿⢀⠴⣀⠀⠀⣀⠀⢠⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠀⢸⣿⠀⣸⡏⠀⠰⢸⡿⠀⢠⠀⠀⠀⠀⠂⠄⢀⣀⠕⠂⣐⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆⣀⠼⠃⠀⠘⠿⠔⠙⠧⠔⠁⠈⠷⠤⠃⠀⠀⠀⠀⠙⠀⠫⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢶⠀⠀⠀⡴⣶⠀⠀⢰⣶⡄⠀⢠⡶⣆⠀⠀⢠⢶⡆⠀⢀⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣧⠀⣼⣷⣿⠀⢀⣿⣿⣧⠀⢸⣷⣿⡄⢀⣿⣾⡇⠀⣸⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣶⣿⢻⣿⠀⣸⣿⠸⣿⡄⢸⣿⠿⣷⣾⡟⣿⡇⢠⣿⡇⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⠛⠃⢸⣿⢠⣿⠟⠛⢿⣇⠘⢿⡇⠛⠚⠀⣿⡇⣼⡽⠛⠻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡛⠀⠀⠀⠘⠛⠘⠛⠀⠀⠘⠛⣐⠛⠃⠀⠀⠀⠛⠃⠛⠃⠀⠀⠛⠃
⠀⠀⠀⠀⠀⠀⢠⠊⠁⠈⢹⡏⠉⠉⠉⠀⠀⠀⢠⠎⣷⠀⠀⣻⠀⠀⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢸⡇⠀⡀⠀⠀⠀⠀⠈⠁⣿⠠⠴⣋⠀⠀⡀⢀⠀⢸⡇⢀⠀⢀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣸⠁⢾⠁⢸⠀⠀⠀⢀⡀⣿⠀⠀⢈⡇⣾⠀⣿⠀⣾⡆⢸⡇⢾⡃⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠁⠀⠀⠁⢁⡀⠀⢀⠀⠉⣙⠀⣈⠉⠀⣈⠀⢈⡁⠈⠀⠁⠀⠈⠁⣺⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣀⣿⠀⢸⢿⠀⣿⣇⣸⣿⠀⡿⣇⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠟⢹⢀⡿⠾⣇⣿⠙⠏⣿⣰⠷⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢄⢰⠀⢀⡀⠲⡈⠁⠀⠈⠀⠁⠀⠁⠈⠀⠀⠉⠉⠀⢈⠷⢀⡀⠀⡆⣠
⠀⠀⠀⠀⠀⠀⠓⠀⠀⠌⠳⠄⠠⣠⣬⡷⠤⣤⣀⡀⠀⠀⠀⢀⣀⣠⠤⢾⣥⣄⠠⠤⠞⠡⠄⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠤⠉⠁⠠⠄⠀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠲⠀⠠⠄⠈⠉⠤⠀⠐⠃

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

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

⠈⣤⡄⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⡀⢀⣴⡄⠀⠀⠀⠀⠀⣀
⠖⠠⠀⠀⠀⠀⠀⠀⣿⠁⠀⡀⠄⠀⣤⢠⡀⠀⠀⠀⠉⠁⠀⢀⣤⣤⣾⠟⠃⠄⡀⠀⠀⡀⢰⠏⠛⠉⠀⠀⠀⣼⣾⠶⠋
⠀⠀⠀⠀⠀⠀⠀⠟⠛⠇⠀⠉⠀⢠⠃⠟⠁⠀⠀⠆⠀⠀⢰⡇⠀⠀⠙⣧⠀⠀⠉⠓⠘⠏⠀⠀⠀⠀⠀⠀⢰⣏⣠⣶⣿
⢺⡏⠀⠀⠀⠀⠀⠀⣾⣷⣶⣶⣆⢸⣶⣦⡀⢀⠀⠀⠀⠀⠈⠁⠀⡀⢀⣍⠀⠀⠀⠀⡹⠀⣤⠶⠀⠀⠀⠀⣿⢻⡟
⠘⠁⠀⠀⠀⠀⣠⢾⡿⠛⠋⠁⢙⣿⣭⣥⣿⡿⠿⢷⡀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⢻⡿⠇⠀⠀⠀⠀⠀⠻⠘⠃
⠀⠀⠀⠀⢀⣀⡛⠚⠃⠀⠀⠀⠈⠉⢨⣿⠁⠀⠀⠀⢹⠀⠀⠀⠀⠸⠟⠃⠀⠀⠀⠀⠘⠟⣀⣤⣶⣶⣦⣴⣄⡀⣦
⢀⡄⠀⠀⢀⣇⣿⠛⠂⠀⠀⠀⠀⣿⣿⢻⠀⠀⠀⠀⢸⣠⣤⣼⣷⣷⣶⡄⠀⠀⠀⠀⠀⣇⠝⠛⠿⣿⣿⣿⣿⣿⣿⣦⡀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠐
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠈⠠⡀⠀⠀⠠⠂⠒⠀⡄⠀⠁⠀⠀⠀⠀⠁⠠⡀⠀⠀⠀⠀⢀⠀⠀⠀⠘⡆⠀⠀⠀⠀⠀⠁⢰⠒⠂⠒⠀⠀⢂
⠀⠀⠀⠀⠀⠀⢐⠉⠐⠀⠀⠀⠀⠠⠁⠀⡀⠀⠐⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⡠⠀⠀⠠⠄⣄⠀⠀⠀⠤⠐⠀⠠⠀⠠⠠⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠄⠀⠃⠀⡐⠁⡀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠰⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⢠⡀⠀⠐⠅⣠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠊⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠠⠀⠐⠒⠃
⠀⠀⠀⠀⡀⠀⣀⠁⠄⠀⠀⠉⠀⢀⠄⠀⠀⠰⠀⠀⠘⡀⠀⢀⡀⠤⠒⠇⠀⠀⡄⣀⠈⠀⠀⠀⢡⠀⠀⡰⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠂⠀⠀⠀⠀⠑⠀⠂⠁⠀⡄⠀⠀⠀⠑⠂⠈⠀⠀⠀⠀⡀⠀⠀⠁⠀⠃⠒⠀⠃⠀⠂⠀⠀⠀⠀⠀⠀⠂
⠀⠀⡀⠀⠠⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠁⠀⠀⠀⢀⠀⡄⠒⢤⢀⠀⠀⠀⠀⠀⢐⣰
⠀⠈⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠃⠁⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠄⠤⢈⢉⠣⡀⠇
⠀⠐⢄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠒⠐⠤⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠈⠀⠀⠀⠢⢀⠂⠀⠀⠀⠀⠐⡄⠂⡀⠀⡡⢄⠀⠀⠀⠀⠀⡠⠤⠄⡀
⠀⠀⠀⠈⠀⠐⠁⠀⠀⠰⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠑⠰⢉⡈⠄⠐⠀⠀⠀⠀⠂⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠄⠀⢀
⠀⠀⠀⠀⠒⣤⠒⠀⠀⠀⠀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⡎⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠈⠐⠀⠒⠁
⠀⠀⢄⠈⢢⠦⣔⠁⡰⠀⠀⠀⠀⠂⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠈⠠⠄
⠀⠠⠌⣠⠝⠰⢂⣈⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⡀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠠⠀⠀⠀⠀⠀⠀⠁⠐⠁⠀⠀⠀⢀⡠⠣⠀⡀
⠀⠀⠀⠠⠄⠓⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⢄⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⢀⠔⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡄⢀⢐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠁⡀⠀⢢⠀⠀⠀⠈⠀⠀⠐⠄⠀⠀⠀⠀⠀⠀⠀⠀⠄⠁⠀⠈⠀⠀⠀⠀⠀⢀⠤⠄⡀⠀⠀⠀⠀⠀⠉⠀⠈
⠀⠀⠀⢀⠔⠀⠀⠀⠀⠀⢀⠀⢁⠀⠁⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠄
⠀⠀⠀⠐⢄⠀⠁⠀⠀⠀⠀⠢⠀⠀⠠⠊⠀⠀⠀⠀⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠠⠀⠀⠀⠀⡀⠆⠀⠀⠀⠄⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠡⠀⠐⠈⠀⠀⠀⠀⠀⠃⠄⠁
⠀⠀⠀⠀⠐⢠⠉⢒⢀⠆⢠⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠁⠀⢀⠠⠔⠠⠰⣅⠀⠀⠀⠀⡐⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⢄⡀⠄
⠀⠀⣀⡀⠁⢠⠄⠐⠢⢀⡈⡀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⡡⠊⠀⠀⠀⠀⠀⠈⠁⢀⠀⠒⠀⠀⡐⠀⠀⠀⠀⠀⡠⡅⠙⢠⠁⠀⡃⠜⠠
⠀⠈⠄⠈⠖⠀⠀⠀⠀⠈⢃⡀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⡅⠀⢀⠂⠉⠁⠢⠀⠀⠈⠀⠀⠀⠜⠀⠀⠀⠀⠀⡀⠴⠐⠦⡀⠆⠬⠁⣀⠈⠰
⠀⢀⠦⠠⢆⠀⠤⠀⠴⠀⡈⠀⠢⠈⠀⠀⠔⠁⠀⠂⠁⠀⢀⠃⠀⠐⢀⠀⠀⠠⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢈⠆⠠⠤⢄⡀⡀⢅⡀⠈⠱⠄
⠀⠈⠰⠁⢀⠱⠤⢤⡄⠊⢋⠐⡆⠀⠀⠀⠈⠢⠀⣀⡀⠀⠊⠡⡀⠀⠀⠈⠁⠀⠀⢀⠴⣀⡀⢠⠀⠀⠀⠀⠀⠀⠈⠚⢀⡠⠎⠀⣧⠀⠑⡈⠄
⠀⠀⠀⠀⠁⠼⡀⣤⠈⠦⠜⠀⠀⠀⠀⠥⠂⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⠤⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠐⢂⡠⠂⡐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⡿⢿⠿⡿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣛⣛⣛⣻⣿⣿⣿⣻⣟⣿⣛⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⣠⠖⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡻⡄⢰⡇⠀⣠⡶⠀⠀⠀⢀⡤⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢋⡭⢌⢹⣟⠉⠻⡟⠉⡼⠁⠀⢰⠆⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⢀⡈⢻⡏⣠⠖⠉⣺⡗⠁⣰⡆⠀⢨⣇⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡅⠈⠛⣸⠃⣿⠀⠀⣿⠀⠀⣿⡤⠐⡿⠝⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠧⣀⡴⢛⡀⠉⠛⡀⠘⠧⠔⠁⡰⠋⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠄⠠⢂⡀⣀⢸⡇⠀⢀⢸⡦⣤⠉⣠⢧⣤⢒⡯⠀⠀⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠆⠀⡐⠁⡀⢸⣿⡇⠜⠁⢸⣷⡟⢸⡧⣾⢿⣿⠀⢀⠀⣈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠃⢠⣁⣸⠇⠋⣸⠻⣄⣴⡿⠉⠓⡀⢰⣿⠀⠉⣠⠎⠀⡈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠻⠄⣠⠴⣫⣀⠀⣀⠀⠧⠿⠇⠀⠙⢲⠚⠁⣴⠏⢡⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠀⢀⣿⠜⢠⣿⠊⣼⠀⡆⡔⣆⠀⣦⣏⢰⠘⣿⠀⢀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢠⡖⠀⡀⣸⡟⢠⣿⠃⣼⢇⡠⢁⠉⣡⣰⡇⠻⣮⡀⠈⣡⡥⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠈⠓⠋⠀⣿⡇⠈⠁⣼⡟⣾⣠⡿⢼⣇⣿⠁⡔⢀⣽⣾⠏⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠤⠴⠊⠀⠸⠇⠀⣸⣿⢀⡈⠸⣇⡨⠁⠛⠊⢠⣾⠃⠘⠲⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠟⠀⠀⣨⠖⢱⡆⢀⣿⠟⠉⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⢀⡜⠸⣏⠀⣈⠔⠀⠻⠦⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠉⠉

⠀⠀⢀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠈⡃⢄⠀⠸⣧⠯⠃⠀⠀⢀⣀⣐⠀⠉⠁⠀⠀⠐⠀⠠⢀⠀⠀⠐⠻⡟⠂⠀⠀⠀⠀⠐
⠀⠐⢤⠃⠀⢀⠔⡅⠀⠀⠐⣋⠤⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄⠀⢀⣀⣂⣀⠀⣴⠆⠾⠇
⠀⠀⠀⢀⠔⢁⠢⡇⠀⠀⠀⠀⢀⡀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠈⠽⠒⠅
⠀⢀⡔⠁⢌⢢⢳⡇⠀⢹⡃⠀⣸⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠁⠈⠀⠤⠑⡄⠀⠀⠀⠀⣤⡞⠀⠹⠆⠬
⠀⠫⣉⠉⠚⢦⣻⠁⠀⡈⠳⠎⢸⣀⠄⡟⢃⡠⢧⣀⠀⠀⠀⠀⠀⠀⠘⠀⢀⡠⠒⠀⠋⠀⠑⢲⠊
⠀⠀⠈⠉⠓⠒⠛⠀⠀⡷⠴⢶⣺⢅⡜⠀⠀⠹⢪⠋⣹⠒⠶⠤⢤⠴⠶⡊⠟⣄⠴⠄⠈⠁⠲⠉⠀⢠⠄
⠀⠀⢀⣀⣀⡀⢀⣀⠀⠫⠻⣷⣮⡟⢠⢒⡒⢄⠀⠉⠀⠃⠰⠢⠌⠦⠆⠈⠃⠈⠀⠀⠀⢀⡈⠀⠀⠈⠂
⠀⠀⠙⢄⠀⢠⡋⠶⠥⡀⠀⠀⣀⠀⢧⡇⠈⢪⡄⣰⠞⠉⢢⠀⠀⢤⠀⠀⢰⠠⠂⠀⡠⢊⢁⠝⠑⠒
⠀⠀⣀⠀⠑⢼⢢⣔⠀⠉⠳⣄⣈⣯⠚⠿⣄⠌⣷⡟⠀⣀⠎⠀⠒⠈⣁⣠⠤⠒⠂⠀⠀⠁⠃⠔
⠀⡰⢤⡂⠐⣢⠉⠈⢳⡉⣉⠽⡏⠉⡑⠲⢬⡟⢙⣶⣚⣉⠙⢧⠀⣸⠑⠀⡗⠀⢀⠀⠀⡠⠀⠀⠀⠠⡀
⠀⠰⠔⠄⢀⡽⡄⠀⠀⠻⠴⡞⠁⠀⢦⠀⢠⠀⢀⡀⢀⡰⠀⠀⠀⢇⢀⡔⠱⢄⡜⠀⢠⠁⠀⠀⠀⠀⠄
⠀⠀⢀⣀⣸⢀⠇⢀⠀⠀⠀⢹⠀⠀⡆⠀⢸⠀⠀⠀⢠⠀⠀⠀⠀⡨⠉⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢠
⠀⠀⢘⢒⠢⡿⠤⠒⡃⠀⠀⢸⠀⠀⡇⠀⢸⠀⠃⠀⡆⠀⠀⠀⠴⡇⠀⠀⠀⢠⡼⠤⢀⡆⠄⠤⠒⠁
⠀⢀⣸⠂⠈⠑⠎⢸⠀⠀⢿⡛⣏⠀⠉⠉⠉⠘⠒⠒⠃⠀⢠⡤⠐⠀⠤⠶⠂⠀⠀⠀⠙⠂
⠀⠘⠤⣑⣦⣤⣔⡃⠴⠀⡰⠉⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡇⠃⡰⡟⣭⣕⡄⢯⡟⠀⠀⠀⢀⣸⠙⣳⠆
⠀⠀⠀⠈⣒⣒⡉⠀⠀⠀⠀⠀⠴⡀⠀⠀⠢⡁⢀⡀⢀⣀⢧⣆⠘⠷⢝⠿⠃⢀⠀⠂⠀⡠⠢⡀⢙
⠀⠀⢠⠊⡉⢳⣹⣀⠠⠰⣷⠄⠣⡁⠠⠀⢠⣌⠀⠲⠟⠉⠀⠉⠑⠒⠀⠀⠀⠀⠀⠀⢀⡫⠒⣠⠏
⠀⠀⠈⠦⣑⣊⣁⣀⠤⠒⠉⠀⠀⠈⠢⠘⠋⠈⠀⠈⠀⠈⠀⢠⡀⢢⠘⢑⡀⠀⠀⠀⢏⣋⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠉⠁⠁⠊⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⠤⢄⡀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⣼⠋⡄⠀⠀⢀⣈⣢⡀⠀⠀⠀⣨⠅⠀⠀⠄⠀⠀⠀⢠⡧⠆⢀⠀⠀⠀⠀⠀⠀⣴⠇⠢⡄⠀⠀⠀⠀⡎⠒⠢⡈⡆
⠀⠀⠀⠀⠀⣼⠇⡼⠀⢀⠴⠉⠛⣿⡇⠀⠀⠘⠛⢠⠜⡉⠆⠀⠀⠀⠎⢁⠀⣸⠁⠀⠀⠀⠀⠀⡏⠀⢠⠇⠀⠀⠀⢠⡛⣳⢀⠘⠃
⠀⠀⠀⠀⣰⣿⠀⣡⡶⠋⠰⠞⣰⣿⠇⠀⠀⠀⢰⠴⡂⢳⠀⠀⠀⢀⡄⠘⠁⠹⠂⠀⠀⠀⠀⣔⠠⠀⢈⣤⠄⠀⠀⠀⢁⡸⠋⠄
⠀⠀⠀⢠⣿⡿⢠⢡⣇⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠓⠀⠀⠒⠂⠀⠀⠀⠁
⠀⠀⢀⡿⠷⠀⣰⣼⠉⣶⣴⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠐⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢤⡀⢄
⠀⠀⠈⠀⠀⠈⠩⡻⣤⣝⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⢨⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⣸⡾
⠀⠀⠀⠀⠀⠐⣷⣿⡌⡛⢿⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣹⣧⢾⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡄⠈⠇
⠀⠀⠀⠀⠀⠀⠈⠙⠳⠘⠎⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⠏⠹⠁⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠧
⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣴⣶⣶⣶⣶⣷⣾⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠉⣠⠖⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⢤⣽⣤⣇⡠⣼⠁⠀⢀⡴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠞⠑⣼⡇⣀⠳⣴⡇⠀⡘⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠈⠗⣹⢰⡇⠀⣿⠀⢸⣇⠰⡿⠶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠴⢋⠈⠓⢅⠙⢄⡂⠁⠈⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⢀⠄⢠⣸⠀⡀⢸⡆⡄⣠⢶⣶⡿⠂⠀⢊⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡅⢠⠀⣤⠼⢻⣆⠀⣼⠿⡇⠛⣿⠛⠇⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢎⢁⡠⢏⠈⠋⠁⣆⣴⠀⠻⠤⠔⢃⡴⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡖⠁⠀⠀⣼⠀⢹⡇⢸⠂⣄⣴⡀⣤⡎⠀⢺⡁⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⡖⡠⢠⡿⢠⡟⢠⣏⡤⣡⣩⢤⡏⠙⠦⡀⣨⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⢉⠀⠘⣇⠀⢠⡟⠹⢴⡏⠿⢾⣁⠀⣴⠻⣇⣀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⢧⠖⡌⢓⠥⣄⠀⣾⠇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⣠⠱⡏⠀⡋⠀⠻⠤⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠉⠁



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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡰⢄⠈⠀⠀⢀⠀⠀⠅⠑⢶⣦⣄⣀⠀⠀⠀⠀⠀⠈⠀⠀⠰⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠈⠁⠀⠀⠈⣁⣀⡴⠖⢾⡟⠀⢀⠀⠀⢀⢀⡄
⠀⠀⠀⠃⠦⠈⢱⠚⠀⠀⠀⠈⠀⠀⠁⠀⠀⠀⠶⠈⠡⠈⢲⣶⠄⣠⠠⣤⡀⢠⡤⣀⣀⣀⢀⠄⣀⣤⠀⠄⠰⡶⠒⠄⠉⠏⠁⢀⠀⠀⠀⠈⠀⠐⠲⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠈⠀⠛⠀⠀⠀⠈⠟⠁⠁⠀⠙⠁⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⣈⠻⣤⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⣀⡸⠟⠀⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠠⠉⠀⠀⠀⠀⠀⠁⠀⠀⠊⠀⠈⠓⠖⠒⡀⠤⢤⠀⠀⢀⣀⠀⠀⢌⡀⢀⠀⠀⣠⠶⠤⡤⢀⠀⠰⠛⠀⠸⠁⠉⠉⠀⠅⠀⠀⠀⠈⠁⠀⠀⡏⡆
⠀⠀⠀⠀⠀⠀⢠⠄⠀⠀⠤⠀⠠⡖⢆⠀⠰⣴⡄⠀⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠶⠈⠀⠀⠀⠃⠀⠀⠀⡀⠀⠈⠀⠀⠀⠀⣀⠈⠀⠀⠈⠀⠀⠀⢱⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠽⠁⠀⠀⠀⠁⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠀⠀⠘⢶⡄⠀⠀⢀⣀⠀⢀⡈⠰⠆⠀⠀⠀⠁
⠀⠀⢤⣴⠾⠶⠶⠂⠀⠁⠀⠀⠀⠀⠀⠀⠀⠆⠀⠐⠀⠀⠘⣛⢂⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢺⠂⠄⠈⠁⡀⠈⠈⠒⢀⣀⠀⠰⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⣤⠀⠀⠀⠀⢸⡅⠀⠀⠀⠀⠀⠁⠈⡀⠀⠀⠈⠁⠀⠀⠆⠁⠀⠐⠦⡈⠉⢠⠀⠀⠀⠉
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⡰⠒⠇⠀⠰⠂⠀⡀⠀⢀⠀⠈⡧⢀⠀⠀⣀⠈⠉⠁⠀⠀⠀⣃⡦⠀⢀⠇⠐⠀⠀⠀⠺⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠈⠋⠀⠀⠀⠀⣀⣷⣄⢈⡧⠀⣧⠈⡷⡀⢨⣦⣀⠀⠀⠀⣠⡝⠀⠀⠀⠀⢀⠤⠈⠀⢠⡦⠀⢀⠀⠘⠛⠅
⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠈⢀⠊⠀⠀⠀⠀⠀⠀⠒⠈⠀⠐⠣⡾⢀⡨⠉⡍⢛⡍⣯⠉⡍⡈⢺⠀⠀⠁⠛⠀⠀⠀⠘⡐⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⢤⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⣇⡘⡄⢀⢁⠉⡍⣀⢠⢀⡁⣸⠀⠀⠀⣀⡀⠀⢀⠟⠀⠀⠀⠰⠆⠘⠇⡀⢀⠀⡀⠌
⠀⠀⠀⠀⠀⠈⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣘⣋⡶⣀⠏⡇⠀⢿⡜⠙⠇⠘⢿⣟⠃⢙⠛⡆⠀⠉⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠆⣁⠘⠀⠀⡀
⠀⠀⢀⠀⠀⠀⠀⠀⠉⠀⠸⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⡟⠘⠃⢀⠻⠃⢀⠓⠀⠘⠀⠐⠎⠁⠆⠉⠰⡧⠀⠀⠠⠀⠀⠂⠀⠀⠀⠀⢀⠀⠀⠀⣀
⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⢆⠂⠀⠀⡀⠀⠘⠋⠀⢀⠘⠁⠀⠘⠁⠀⠀⠖⠀⠛⠀⠀⠀⠀⠀⠀⠀⢀⢀⠀⠀⢢⣶⠱⠀⢿⠎⠀⣀⢠
⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢘⢥⠀⡛⠩⣀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡠⠀⠀⠀⠀⠀⣣⣾⠃⠀⠀⠀⠸⢛⠄⠀⠀⠶⠃⠚⠛⢠⣤⣿⠂⠁
⠀⠀⠀⠀⡀⠀⠄⠀⢀⠀⠀⠀⡠⣤⠀⠀⠀⠀⠘⣞⠀⣟⠢⠤⡉⣀⠒⠀⠀⠀⣀⠤⠤⠤⠐⠒⠀⠀⣈⠔⠀⡀⠀⠀⠀⠀⠀⠀⠚⠠⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠁⠀⠀⠈⠇⠀⠘⠛⠁⠀⠀⠀⡀⠀⠉⠀⢨⠙⠲⢤⣄⣀⣀⡀⠀⠀⠀⢀⡀⠠⠄⠒⠋⣀⠄⠨⢁⣀⠀⠀⠀⠀⢀⣀⠂
⠀⠀⠀⢠⣊⣅⡰⢄⡀⠀⠂⠀⠀⠀⠀⠀⠀⠛⢋⠉⠉⠀⠑⠢⠤⠄⢈⣉⣉⡉⠉⠉⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⢀⠀⠀⠉⠀⡀⠀⠀⠀⠄⠀⡄⠠
⠀⠀⠀⠘⠉⠞⠀⠙⢍⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠠⠄⠀⣰⣆⠀⡀⠀⠀⠀⠀⠠⠊⠀⠰⣤⠶⢎⡛⣠⣀⢠⡄⠀⠀⢈⠀⠈⠱⠀⠀⠠⡄⢸⠀⡇⢀⠆
⠀⠀⢠⡄⠀⠀⠀⠀⠀⠘⠌⡆⠀⠀⠈⠀⠀⠀⠀⢹⠀⠀⠀⠘⠳⠦⠴⠁⣁⣀⣴⣄⠀⠐⠀⠀⢬⣻⢨⣻⣯⢹⣶⢃⡢⠀⠠⠀⠁⢨⠀⠀⠀⡇⢸⠀⣇⢸
⠀⠀⠀⠀⠀⠈⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⠾⠿⡧⣥⢿⣼⡇⠀⠐⠐⠒⠐⠀⠀⠀⠁⠈⠀⠁⠈
⠀⠀⠀⠀⠠⠤⠀⠛⠠⠤⠤⠀⠀⠈⠑⣄⡀⠐⠀⠉⠁⠀⠁⠘⠈⠁⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⠀⠀⠀⠀⠁⢻⠶⠜⠃⠀⠰⣤⠖⡩⡀⠀⢴⣤⠀⠀⠠⠈⠓
⠀⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣏⢑⠑⡇⠀⠀⠁⣀⠀⢀⡤
⠀⠀⠀⠀⠠⠀⠀⠠⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠉⠀⠀⠀⠀⠾⢗⠀⠀⢤⡀⠀⠀⠀⠀⠄⡀⢸⣦⠀⠀⠈⠀⠀⡅⠀⠀⢀⠀⠑⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠙⠛⠉⠀⠒⠀⠀⠀⠄⠀⠀⠀⠀⠈⠙⠀⠀⠀⡐⠿⠁⠈⠋⠀⠀⠠⠀⠰⠀⠀⠀⠈⠀⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⠄⠀⠀⠠⠠⠄⠤⠀⠄⠀⠠⠀⠴⠤⠀⠐⠀⠀⠙⠿⠂⠀⠀⠀⠀⠀⠀⠀⢂⡟⠁⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠀⠀⠀⠁⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠉⠓⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣆⢀⡴⠋⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠈⠑⢂⣀⡤⠤⠖⠒⠒⠳⢄⡀⠀⠸⠞⠀⢰⡇⣀⡤⣶⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠀⣠⠞⠉⠀⣀⠴⣒⡩⠥⠔⠒⠛⠒⢦⡀⠀⠘⠋⠁⢠⠇⣀⡤⢖⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡍⠙⠛⠀⣰⠁⠀⢠⠾⠓⠉⠁⢀⣤⡀⠀⢠⠤⢄⣧⡀⠀⠀⠀⠚⠉⠁⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⢰⠃⠀⡰⠃⠀⠀⢀⠞⢁⣠⠟⠛⠛⣲⠤⢼⣷⣀⠀⠀⠀⠀⠰⠷⠶⣶⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⡼⠁⠀⣠⠴⣏⡉⢡⡷⣴⡄⠙⠻⢯⣚⣛⣮⣧⣀⠀⠀⠀⠀⠜⠁
⠀⠀⠙⢯⡉⠉⠉⠉⠉⠉⠀⣠⢧⣴⣃⡴⠋⠀⢰⣋⠤⠵⠛⠓⠳⢤⣀⡴⠋⢀⡴⠶⢤⠑⡄⠀⠀⠀⣀⣤⠤⣶⠞⠁
⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠎⠀⠀⠀⢹⡖⠀⣰⡟⠁⡤⠒⠲⢄⠀⢀⡞⢦⠀⢾⠀⢦⡸⠃⣸⣦⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠉⠢⣀⠀⢧⡀⠀⢀⡼⠁⠀⠸⡇⠸⣇⣸⠀⢈⣶⣯⣴⣶⠛⠚⠶⠤⠤⠞⢁⢸⡆⠀⠘⠷⣤⣀
⠀⠀⠀⠀⠀⠰⢶⣛⡉⠀⠀⣹⠉⠉⠀⠀⠀⠀⠙⠦⢄⣀⠴⠋⠘⣍⣉⡿⠀⠀⠀⠀⡄⠀⠘⣾⠳⣄⠀⠀⠊⠉
⠀⠀⠀⠀⣠⣴⡖⠋⠉⠀⢀⣿⠀⠀⠀⢀⠰⡆⠀⠀⠀⠀⠀⠀⠀⢄⣀⡤⠎⠀⠀⢀⡅⠀⠀⢹⠀⠈⢦
⠀⠀⠀⠀⠀⠀⠁⠀⢀⡴⠋⠈⢧⠀⠀⡏⠀⠙⢄⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠋⢠⠀⠀⡿⠄⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⣄⢙⠦⣇⠀⠀⢀⣭⠽⡋⡡⢬⣦⣀⣀⣀⡠⠎⠀⠀⢸⣠⠞⠁⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠉⠙⢪⠏⠀⠙⣶⠉⢹⡾⢦⡹⣡⠾⣎⠉⢹⠦⠤⣤⣤⣤⠾⠷⠦⢤⡀⠀⠀⣸
⠀⠀⠀⠀⠀⠀⢸⠀⢀⡤⠶⠒⢚⡦⠖⠛⠿⣶⡷⣜⣢⢿⣅⣚⡾⠶⢯⣄⣠⢴⡟⠿⠤⠤⢄⡀⠙⢆⢀⡏
⠀⠀⠀⠀⠀⠀⠸⡴⠃⢀⡤⢶⡯⠤⢤⣴⣚⣁⢀⡠⠖⠛⠛⠲⠦⣤⠴⠋⣿⣠⣵⣦⠆⠀⠀⠁⠀⠘⡞
⠀⠀⠀⠀⠀⠀⢰⡇⠀⡏⢠⣾⠁⠀⢸⡄⠀⠉⡟⠓⠒⠶⢤⡴⠋⢹⡇⠀⢻⠁⠀⠀⠀⠀⠀⣰⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠈⢧⠀⣇⠈⠛⠦⣄⣈⡇⠀⠀⡇⠀⠀⠀⠈⡇⠀⠀⣇⡴⢻⣄⠀⠀⢀⣠⠖⠁⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠈⠳⢼⣄⠀⠀⣿⠈⠉⡟⠓⢳⡤⣤⣀⡀⣇⣠⠾⡏⠀⢸⠈⢙⡟⠛⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠲⣇⠀⠀⣇⠀⠈⡇⠀⠈⠹⡏⠀⠀⡇⢀⡸⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠻⠦⣤⣇⡀⠀⠀⡇⠀⣠⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠒⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠔⣱⡱⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢻⣀⣹⡣⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠬⠔⠂⠢⢴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⡀⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡼⠢⣿⠀⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠀⠘⠧⠔⠻⠀⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⢠⡆⢠⠲⡶⠀⣻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⢀⠀⣿⢠⣾⠀⢋⢴⡄⠐⢻⠇⠐⢃⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⢛⣀⣀⣀⡁⢿⠀⢀⠞⢀⡐⡏⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠂⠀⢀⡀⠉⣤⠤⢨⢙⠢⡀⠀⣆⠘⣷⣣⡤⠆⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡨⠛⠀⢠⢞⣇⣴⣜⡀⢸⠘⡧⡡⢀⡟⣈⡙⠶⣅⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡄⠀⢊⠴⠃⠘⠁⣟⡏⠁⣸⠀⡇⠀⣸⠀⣹⠏⠙⢓⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡖⠙⢷⠀⣸⠁⠀⠇⡦⠄⠟⠠⠃⣰⢃⣖⠈⠳⣄⠛⠁⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠘⢠⡏⢰⢷⡖⡍⢵⣤⣦⡍⣡⠊⠉⠳⣤⠈⠁⠐⠛⢿⠇⢀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠻⣄⣀⡘⡽⢑⡏⣸⡟⠁⠀⢸⡇⠀⠻⠖⢛⡆⠀⣰⢰⣦⢶⢠⣰⣠⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠽⠃⠰⠆⠀⠀⠸⠁⠘⣇⣣⠹⠤⠤⣼⣤⡄⠠⠴⠋⠀⠀⡟⠽⢿⠎⠈⡿⠿⠃⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠂⢒⠚⢀⡀⢤⠀⣀⢤⡀⠀⠠⣤⡀⠀⢀⣠⠤⠄⡀⠀⢀⠀⠀⡀⠀⢄⡀⡆⠒⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠘⠉⠀⠀⠉⠀⠀⠰⠈⠀⢨⠅⠀⠀⠈⠀⠀⠈⠑⠃⠀⠀⠉




⣿⠿⠿⢿⠿⡿⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⢰⠛⠱⡀⠀⠀⠀⠀⠀⠔⠀⢢
⠀⢀⣤⠤⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⡆⠀⢸⠀⠀⢱⠀⠀⠀⠀⡘⠀⠀⢸
⠀⢸⢁⠀⠠⢌⠳⡄⠀⠀⠀⠀⢀⠤⠒⠉⠉⠈⠉⠑⠠⡀⠸⡴⣚⠧⢀⠼⣴⢤⣸⡀⠀⠀⢀⠃⠀⠀⡌
⠀⠈⢮⡈⠁⠀⣹⣼⣆⠀⠀⠰⠃⠀⡀⠀⠀⠀⠈⠀⠀⠈⢆⠹⡀⢀⠀⣀⠈⠓⠺⡇⠀⠀⢸⠀⠀⢸⠁
⠀⠀⠀⠑⣤⠀⣩⡅⠈⢆⠀⡅⠀⡀⠀⣠⠔⠢⣄⠀⠈⠀⠘⡀⠈⢦⠀⠀⠀⠉⢀⡇⠀⠀⡧⠂⠀⡎
⠀⠀⠀⠀⠈⠻⡻⣠⣤⣼⡄⡇⠀⠁⣰⠃⠀⠀⠈⡆⠀⠄⠀⡇⠀⠀⢏⠀⠀⠀⠨⠇⠀⢸⣣⡠⣰⠁
⠀⠀⠀⠀⠀⠀⠙⡟⠿⠟⢃⢸⠀⠀⠜⢦⡀⠀⡰⠁⠂⠆⢠⠁⠀⠀⢸⣤⣤⣍⣸⠀⠀⣇⣈⣍⡏
⢀⡤⡀⠀⠀⠀⠀⢸⡄⢒⢻⠀⢃⠀⠠⠀⠈⠉⠀⠠⠂⢀⠎⠀⠀⠀⠤⠸⠿⠿⠇⠀⠘⠯⠿⡾⠁
⣏⠀⠈⢂⠀⠀⠀⡸⠁⠀⠉⠀⠀⠳⢄⠀⠀⠀⠀⣀⠔⠁⠀⣀⠴⠚⠫⢲⡵⠃⠀⢀⣤⢬⡉
⠱⡀⠀⠀⡱⣲⢾⡅⠀⢠⠁⠀⠀⠀⠀⠉⠉⠈⢉⠔⠒⠊⠉⠀⠀⠀⣠⠞⠀⠀⢰⠁⢿⡄⢸
⠀⠑⠤⢀⡇⢘⠂⢉⠷⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣀⠀⠤⠔⠊⠀⠀⠀⠀⠈⠢⠿⠧⠞

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢁⡑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⠀⠀⠀⠀⣠⣚⢢⠀⠀⠀⠀⠀⣷⣤⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⣿⢧⠀⠀⠀⠀⢀⠼⣁⣪⠉⠀⠀⠀⠀⠶⠿⣿⠿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠻⠀⠀⠀⠀⠁⠀⠀⠀⠀⣠⡛⢄⠈⠁⠀⠀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣵⠃⠀⠀⠀⠀⠀⠀⣠⣜⠣⠙⠋⠀⠀⠠⠒⠂⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⡙⡅⠀⠀⠀⠀⠀⠀⠀⠛⠁⠙⠁⠀⠀⠀⠐⠄⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠞⠀⠀⡠⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣱⠀⠀⠀⠑⠊⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢰⣾⡧⠄⠀⠀⠀⠀⡠⠢⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⠈⠈⠃⠀⠀⠀⠀⠀⠀⠈⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡂⣣⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠠⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠢⣀⡀⠀⠀⠠⢄⠚⠈⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣄⠀⠈⠒⡤⠂⢄⠀⠀⠀⠀⠀⠀⣶⢑⣈⡄⠛⠁⠀⠀⠀⠀⣥⠐⠺⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡿⠉⠢⡀⠀⢄⣀⣴⠀⠀⠀⢴⡙⣄⣨⠜⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡤⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⠁⠀⠀⢙⣄⠀⠀⠀⠑⢶⡑⣤⡥⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⢠⣾⠉⠨⠿⠁⠈⠋⠁
⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠁⠀⠀⣰⣿⣿⣷⣄⠀⢠⣶⡷⡉⠀⠀⠀⠀⡊⠀⢐⠀⠀⠀⠀⠤⣾⠇⠀⠙⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⣠⣾⣿⣿⣿⠿⠋⠉⠢⡀⠀⠈⢆⠀⠀⠀⠈⠀⠁
⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣴⣿⣿⡿⠟⠁⠀⠀⣀⣤⣾⣷⣦⣤⠇
⠀⠀⠀⠀⠀⣸⡟⠀⣴⣿⣿⠟⠋⠀⢀⣠⣴⣿⣿⠿⠟⠋⠉
⠀⠀⠀⠀⢰⠟⣠⣾⡿⠛⢁⣠⣴⣾⡿⠟⠛⠉
⠀⠀⠀⢠⢏⣼⠟⣋⣤⡶⠟⠛⠉
⠀⠀⢀⣿⡿⠷⠛⠉
⠀⠀⠈⠁

⡿⢿⠿⠿⡿⠻⢿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⢿⣿⣿⡿⣿⣿⢿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣯⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠒⠒⠒⠒⠒⠷⢧⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦
⠀⠀⠀⠀⠀⠀⡴⠋⢀⠔⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢠⠞⠀⠀⠃⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿
⠀⠀⢰⠃⠀⠀⠀⠐⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣀⣠⠤⠤⠶⠶⠦⠤⠤⣀⡀
⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⣠⠔⠋⣉⣠⠤⠔⠒⠲⠦⠤⢄⣀⠉⠳⣄
⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢸⠀⠀⡠⠊⡠⠔⢋⣁⣠⣤⢰⠶⣴⢤⣤⣀⠈⠑⠢⣌⠳⣆
⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⢠⠎⡴⠋⣠⠶⢭⢯⡆⣸⢣⡖⣸⢃⣤⢸⣛⣦⡀⠈⢢⡈⠳⡀
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⡜⠀⣼⢱⡆⣾⢈⠐⢹⣀⡴⡏⠘⣣⡟⣡⢨⢉⣦⡀⠳⡄⠹⡄
⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⢰⠀⡼⢣⣠⠊⡜⠛⢦⢞⡙⣆⡹⠺⣴⡀⢁⠞⣡⢈⢷⠀⢱⠀⢹
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⢿⠀⠸⠀⣧⢋⡌⣶⠰⢇⡟⠘⢃⡟⠸⢃⡯⢿⣝⠈⢁⣮⣌⡇⠀⡇
⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇⣾⠀⢂⠀⣿⠘⢁⣷⡤⢊⠷⣔⣡⣧⢴⠏⡶⢸⣹⠶⣥⢣⡌⡇⠀⡇
⠘⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⣠⠃⠀⢻⠀⢸⡀⢸⠓⢏⡤⢱⢧⡆⣸⠁⡌⣾⠀⢠⢻⢣⡆⣿⠈⢡⡇⢠⠃⢰
⠀⠘⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠈⣇⠀⢇⠈⣾⠈⢃⣾⣀⢴⠳⣧⠁⢸⡟⠛⣾⣀⡼⢼⢦⢾⠇⡸⠀⣸
⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠀⠀⠀⠹⣆⠘⣆⠘⣗⠻⡁⣽⢾⡸⡞⡗⢻⠰⠇⣧⠋⠖⡸⢀⠞⣰⠃⢠⡏
⠀⠀⠀⠀⠀⠉⠓⠲⠤⣤⣀⣀⣀⣤⣤⠴⠖⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡈⠳⣌⠳⢜⣦⣈⡷⡀⢸⡇⢀⡼⣇⣠⡾⠗⢁⡴⠃⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣌⠳⢤⣀⣉⠉⠙⠛⠛⠋⠉⢉⣁⡤⠖⠉⣀⠼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠦⢬⣉⣙⣒⣋⣉⣉⣁⣤⠴⠖⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⡔⠂⠀⠀⠀⠀⠀⠀⢸⡧
⠀⠀⠀⠀⠀⠀⢰⠒⣦⢰⣒⡆⠀⣮⠀⣮⢠⠆⢾⡷⠀⠟⡆⠀⠀⢺⡗⢰⡴⠂⡆⡆⠐⡆⠀⡇
⠀⠀⠀⠀⠀⠀⢸⠉⠁⠀⠁⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠀⠈⠉⠉⠉⠉⠛⣷⣦⡀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠁⠀⠀⠀⠀⠀⠀⠀⡤⢻⣿⣷⣾⠃
⠀⠀⠀⠀⠀⠀⠀⣰⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⢠⣿⠄⣠⠀⠀⠀⠀⠀⡀⠠⡌⠟⡻⠁⣽⣿⡇
⠀⠀⠀⠀⠀⠀⣾⣿⠄⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⣨⣿⣧⠇
⠀⠀⠀⠀⠀⢸⣿⣿⣶⣷⣖⣀⡀⠈⣀⣀⣄⣜⣿⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⡟⣻⣿⠿⢋⣠⠤⠒⠒⠀⠀⠙⠓⠲⠠⡀
⠀⠀⠀⠀⠀⠀⠈⠀⠻⢯⣽⣟⣿⣛⣥⠆⠙⠉⠉⠀⣴⠋⠀⠠⠖⣤⡗⢠⣠⡀⢤⠀⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠠⣀⡈⠃⢠⣄⡥⠽⢣⣍⡐⡄⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠷⠈⡚⣀⢰⣀⣲⡦⣤⡄⠀⣍⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆⠈⠛⣋⡠⣶⢸⡅⠔⠀⠤⠶⡸⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣦⣀⠈⠛⠰⠦⢄⠛⠀⠂⠘⠁⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣷⣦⣄⣀⢀⡀⠀⣀⣠⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠹⢿⣿⣿⣋⡉⠱⠟⠋


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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⢀⠀⠀⠀⠀⠀⡀⠠⠠⢀⠀⠠⠠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⡔⠄⠀⡀⠀⠀⠎⠑⠀⠀⠀⡆⠀⠀⠆⠈⠀⠀⠃⠀⠀⠉⣑⠀⠄⡀⠉⠀⠠⢀⠀⢋⠑⢀⠌⢀⠂⠀⠀⠀⣀⢀⠇⠀⠀⠀⠶⡌⢏⡠
⠀⠀⠠⡊⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⡄⠘⠃⢠⠀⠀⠈⠉⠘⠃⠀⠁⠀⣠⠀⠈⠀⢀⠶⡀⠉⢀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢣⠚⠦
⠀⠀⠀⠈⠤⠄⠀⠃⠀⠀⠀⠀⠀⠀⠀⡇⠀⠙⡀⠀⠎⠀⢸⠀⠂⠘⠣⣀⠇⡀⠙⣀⡸⠀⠈⢢⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠈⠈⠀⠁⠈⠀⠉⠋⠀⠀⠀⠐⠁⠀⠈⠐⠁⠀⠀⠁⠊⠐⠂⠂⠀⠀⠀⠀⠤⣀⢄⠀⢀⡀
⠀⠀⠄⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⢀⠀⢀⠀⠀⠀⠀⠀⣠⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠴⡀⠀⠀⠀⠀⠀⢠⠘⠀⢀⡔⠀⢂⠘⠈⣆⠀⠀⢀⡀
⠀⠀⠀⠀⢀⠊⢀⠀⡈⠠⡀⠀⠀⠐⠤⠔⠀⠀⠀⠀⠦⢼⢲⣉⡀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠢⠀⠀⠀⠐⠄⢀⠜⠀⢡⢌⠉⢠⠂
⠀⠀⠀⠀⢠⠀⠂⣀⠔⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠋⢹⠀⠁⠀⠀⠀⠁⠀⠀⠀⠃⠀⢄⡎⠀⠀⠀⢌⠁⠀⠃⠘⠃⡀⠀⠰⠂⠤⠐⢉⠂
⠀⠀⠀⠀⠀⠑⠄⠀⠤⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠄⠁⢃⠚⠃⢈⠁⠴⢄⠤⡄⢠⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠙⠂⠄⠀⠀⠀⠀⠀⢀⠄⠀⡀⠀⠀⢀⢀⢤⠀⢆⠠⠀⠀⠀⠰⡀⠊⢁⠤⠍⢤⠐⠎⠀⠐⠈
⠀⠀⠐⠀⠀⠀⠀⠀⢀⢠⠀⠀⠀⠀⠀⠈⠑⡀⠤⠄⠀⠀⠀⠀⠀⠈⠂⠐⠁⠀⠀⠤⠊⡹⠤⢄⠈⢉⡀⠀⠀⠢⠤⢄⡁⢠⠂⣀⠀⠀⠁
⠀⠀⠀⠀⢀⠤⠀⠠⠁⠀⠡⠀⠀⢄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣢⠀⢃⡘⣠⢆⢑⣢⠀⠀⠀⠀⠀⠀⠀⠁⠀⠁⠀⠀⠀⠀⠑
⠀⠀⠀⢈⠁⠐⠤⢒⡈⠈⡒⢀⠐⠈⢀⠂⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⠀⠀⠃⠧⠀⢊⢁⢄⠘⠀⠀⠀⠀⣀⠀⠀⠀⠀⡀⠄⠤⢀
⠀⠀⠀⠀⠂⡲⠀⡈⢁⠈⠊⣀⠰⠂⠌⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣉⠂⠀⠀⠀⠀⠁⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠄⠠⡀⢡
⠀⠀⠀⡒⠠⠕⠈⠁⡈⢂⡀⠉⠠⠠⠀⠄⠀⠀⢎⡠⠀⠀⠀⠀⡀⠀⠀⠔⠉⠄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠄⠀⠀⠀⠀⠀⠀⢃⠀⠂⠀⠁⡰
⠀⠀⠀⢅⡀⢰⠑⠌⡁⢈⠨⠔⠂⢀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢠⡄⠀⠀⠑⠀⠀⠊
⠀⠀⠀⠀⠀⠊⢠⠂⡁⢈⠂⡄⠁⠂⠀⠀⡀⠀⠀⢀⠄⠂⠁⠄⠐⠁⠒⠀⠀⠀⠐⢆⠀⠠⢀⠀⠀⠀⠀⠀⠀⠈⡍⠁⢐⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠈⠀⠀⠀⠀⠀⠀⠀⠀⢁⢀⠄⢲⠜⠁⠀⠀⠀⠀⠀⠀⠲⢆⢄⠀⠐⠂⠀⠀⠀⠀⠀⠃⠒⠌⠀⠀⠀⠀⣀⠠⠔⢂
⠀⠀⡋⠈⡂⠀⠀⠀⢀⠀⠀⠀⠀⠀⡠⡄⠀⠀⠀⠀⠀⡐⠁⠀⠀⠂⣀⣀⡀⠀⠀⠀⠈⢂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠘⠂
⠀⠀⠈⠈⠀⠀⠀⠀⠸⠈⠂⡤⠰⣊⣀⣀⠀⠤⢄⠀⠀⠀⠀⠀⠀⠀⠄⠀⡠⠀⠀⠀⠀⠈⠀⠀⠠⠄⠀⠀⠀⣄⠀⠀⠀⠀⣀⠀⠀⠘⠂⠂⠤
⠀⠀⠀⠀⠀⠀⡆⠒⠈⠀⠈⡉⠀⠁⠀⠀⠀⠀⢘⠀⡸⢀⡀⠀⠀⠀⠠⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⢀⡆⣲⣈⣀⣀⣀
⠀⠀⢠⡄⠀⠀⢧⠄⠀⠀⠀⠗⠀⢺⠀⠀⠉⠉⠃⠀⡇⠀⠀⠀⠉⠉⠀⠀⠀⠂⠀⠀⠐⠒⡗⠢⡀⠀⢀⣀⣀⣀⣀⡀⣀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠈⠀⠀⠀⠀⠰⠀⢓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⢈⣀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⢉⠀⠀⠀⠀⠛
⠀⠀⠀⡀⠀⠀⠀⡄⠀⠀⢀⣈⡀⢠⠅⡤⠤⣤⠀⢀⡈⠤⠬⣄⢀⡀⠀⠀⡀⠀⢀⡂⣀⠜⠄⠎⠀⠀⢀⡀⠀⠀⢐⠀⢸⠀⠀⠀⠸
⠀⢀⠀⠀⠐⢀⠆⢈⠋⠁⣀⠉⠣⠘⠄⠄⠀⠀⠀⠄⠀⢣⠀⠛⢉⡉⠘⣏⡉⠉⠉⣉⠇⠀⠆⢠⠂⠀⠂⠠⠁⠀⠩⠉⠝⠉⠉⠙⡧
⠀⠙⠤⠀⢀⢘⠀⠆⠀⡊⠀⢁⡂⢁⠄⡄⠀⡀⠀⠂⠀⡐⠀⣄⠈⠈⠛⠄⠀⠀⠀⢸⠀⠀⠀⠆⢀⠀⢰⠁⠠⠥⠀⠘⠀⠈⠈⠛⡀
⠀⠀⠀⠋⠀⠀⠀⠐⢀⠈⠀⠁⡨⠹⠀⠀⢀⡄⢐⢆⠀⢳⣀⡛⠉⠙⠀⠄⠀⠀⠀⢸⠀⠘⡀⠀⠸⠀⠈⠠⠀⠀⠄⠈⡞⠁⠐⠀⡐⠀⠀⠀⠠⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠊⠀⠉⠀⠂⠁⠀⠈⠂⠊⠘⠂⠃⠐⠐⠁⠃⠚⠁⠀⠀⠈⠈⠉⠂⠐⠈⠀⠀⠀⠀⠈⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠙⠛⠛⠛⠛⠛⠛⠃⣤⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⠞⠛⢳⣶⠚⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⢦⣄
⠀⠀⠀⠀⠀⠀⢀⡾⠛⣼⠋⠀⢀⣴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄
⠀⠀⠀⠀⠀⣰⠟⠀⣴⠁⠀⢠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀
⠀⠀⠀⠀⠀⠟⠀⠀⠋⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣
⠀⠀⠀⠀⣿⠀⠀⣾⠀⢀⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⡟⠀⠀⠀⡄⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⢀⠀⠀⠀⠀⢿⣦⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠇⠙⢇⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⢸⠀⠀⠀⠀⢀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⠞⠁⠀⠀⠸⣇⠀⠀⠀⠀⠀⣰⡆
⠀⠀⠀⠸⠀⠀⠀⠀⣟⠈⣷⡀⡀⠀⠀⢀⣀⣀⣀⡀⠴⠶⠊⠉⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⠉⠉⠉⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⠦⣤⡟⠁
⠀⠀⠀⠀⡇⠀⠀⠀⠈⠛⢶⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡌⠁
⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠦⣠⡀⠀⠀⠀⠀⠀⣀⣠⠶⠏⠁⠀⠀⠀⠀⠀⠀⠀⣠⠾⠃⠐⠷⣄
⠀⠀⠀⠀⠀⠀⣆⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠈⠀⠚⠓⠺⣿⣿⣿⣇⣀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡏⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⢠⣿⡄⠐⠚⢷⡾⠛⠁⠀⠀⠀⠀⣠⣴⣶⠛⠋⢉⡉⠩⠁⣴⡿⠃⠀⣠⡴⠋⠀⢹⡇⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⣤⠿⢿⡀⠀⠀⠀⠀⠘⠻⠂⢀⣤⣤⣼⣿⡿⠛⠛⠛⣿⡇⠀⢠⡝⠛⢀⣼⠏⠀⠀⠀⠀⠘⢿⡄⠀⠀⠀⠀⠀⠀⠻⠻⣿⠆
⠀⠀⣴⠉⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⢿⣅⠀⠀⢸⡏⠙⢧⡀⠱⣦⣾⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⣷⠀⠀⠀⠀⣠⡴⣿⣤
⠀⢀⠛⠀⠀⠀⠉⡀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠃⠀⠈⢻⣆⠈⢳⡄⠘⢧⡀⠙⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⢿⠁
⠀⣿⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣠⡼⠃⠀⠀⢠⡄⠛⢣⡄⠙⢧⡌⢣⡄⠈⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠉⣶⠀⠀⣾⠀⠀⢀⣤
⠀⠋⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠸⠏⠀⠀⠀⠙⢦⡀⠘⠻⣿⣦⡈⠛⣧⣄⠀⠀⠀⢠⣴⣤⣤⠋⣤⠟⠁⢀⣴⠛⠁
⣧⠀⠀⠀⠀⠀⠀⣟⠀⠀⢀⡾⠋⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⣸⠿⢦⣄⠀⠈⠛⠛⠈⠛⠛⠓⠤⢿⣦⡄⠙⠛⠁⢀⣴⠛
⣿⠀⠀⠀⠀⠀⠀⠏⢀⡴⠋⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⢀⣠⠞⠁⠀⠀⠘⠛⢤⣤⡄⠀⠀⠀⠀⠀⠀⡽⠁⠀⢀⣴⠋
⣿⠀⠀⠀⠀⠀⢠⣿⠋⠀⠀⠀⠀⠀⢠⡼⠋⠀⠀⢠⡞⠛⢳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠆⠀⣴⠟⠁
⣿⠀⠀⠀⠀⠴⠋⠀⠀⠀⠀⠀⠀⣠⣾⠧⣤⢤⣶⣷⣤⣤⠈⢱⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠇⢀⣴⠃
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⣄⢀⣀⠀⠀⠀⠀⣠⡟⠁⢠⡟⠁
⣿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠛⠋⠉⠁⣴⣶⠦⣀⡀⠀⠀⠀⠀⠀⠀⢸⡇⠙⠋⠁⠀⣴⡶⠋⢀⣴⠏
⡿⠀⠀⠀⠀⠀⠀⢀⣤⠟⠁⠀⠀⣠⡾⠋⠀⠀⠉⠳⣦⣀⠀⠀⠀⢀⡸⠃⠀⢀⣠⠶⠉⢀⣴⠟⠁
⠀⣤⠀⠀⠀⣀⡴⢺⣿⠶⢦⣠⡾⠋⠀⠀⣀⡠⠶⠆⠈⠙⠳⠶⢶⡞⠣⠤⠀⠉⠁⣀⣴⠟⠁
⠀⠙⠶⠶⠞⠋⠀⣜⠻⣦⡀⠉⠀⢀⣠⠾⠉⠀⠀⠀⠀⠀⠀⠀⠀⢹⣄⣀⡀⠤⠈⠉
⠀⠀⠀⠀⠀⠀⠀⡿⠀⠈⠙⠶⠶⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠉⠁
⠀⠀⠀⠀⠀⠀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡄
⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⣤⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀
⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇
⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄
⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠈
⠀⠀⠀⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆
⠀⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁
⠀⠀⠀⠀⠙⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡞⠁
⠀⠀⠀⠀⠀⠀⠘⠿⠃⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠀⠛⠃⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣇⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠀⠀⢠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⡈⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣤⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⣿⡃⠀⠀⠀⠀⠀⠠⡏⢡⡄⠀⠀⠀⢿⡆
⠀⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⢀⣆⠘⡇⠀⠀⠀⠀⠀⠀⢀⡼⢃⡀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⢀⡄⠙⣦⠀⠀⠀⠀⠀⠀⠀⠘⢇⡸⢧⡀⠀⠀⢸⡿⢆
⠀⠀⠀⠀⣴⠉⠀⠿⠆⠰⠟⠁⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡈⠳⣄⠰⠟⠀⠸⡇
⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠈⠓⣀⣀⣀⣄⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢃⣀⣀⣀⡀⠛⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⡟⠛⠉⡿⠛⠛⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠁⠀⠀⠁⠀⠀⠀⡀⠀⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠁⢀⣠⠤⠴⠠⠤⣀⠀⠀⠘⠀⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠠
⠀⠠⠀⠀⠀⠀⡴⠉⠀⠀⠑⠀⠀⡈⠙⣄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⠤
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠐⠀⠂⠀⠄⠂⠀⠀⡄
⠀⠀⠀⠀⠀⢲⠐⠂⠀⠀⠀⠀⠀⠀⠋⠁⡇⠀⠄⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⣐⠀⠀⠀⢀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⢡⠀⡶⢁⡆
⠀⠀⠀⠀⠀⠀⡈⠗⠦⢠⣈⣀⡡⠔⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠰⠀⠀⠀⡀⣀⠃⠘⠅⠊⠠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠀⠀⠀⠀⠀⠀⠤⠄⠀⣠⠀⠀⠁⠁
⠀⠀⠀⠀⢀⡴⣀⠀⠀⠈⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⢠⠀⢋⠈⢀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀
⠀⠀⠀⠀⠀⠒⠡⠒⠀⠀⠁⠂⠀⠀⢀⠀⠀⠀⠀⠀⠀⠠⠀⠘⠂⠈⠀⠘⠁⠀⠀⠂⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⢐⣠⣤⠤⠖⠠⡔⠂⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⢣⠈⠌⠁⠀⠈⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠂⠀⠀⠀⡀⠠⠠
⠀⠀⠠⠈⠓⢉⢡⣤⡔⣶⠖⠵⠎⠀⠀⠀⠀⠀⠀⠐⠀⠈⠀⠡
⠀⠀⠀⠀⠀⠀⠨⠉⠀⠉⡤⠚⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠤⢂⠐⠉⠂⠠⣠⡄⠈⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⢢⣚⣂⣙⣶⡄⠺⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣃⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣼⡈⠷⠨⠈⠁
⠀⠀⠀⠀⠀⠄⠀⡀⠒⠉⠀⠉⠀⠀⠂⠀⠀⠄⠘⠀⠀⠀⠂⠘⠙⠀⠀⠁⠀⠀⠀⠄⠠⠄⢀⠤
⠀⠀⠀⠀⠀⠀⢀⡁⠠⠅⠀⠂⠀⠐⠀⠀⠀⠀⠀⠒⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠄
⠀⠀⠀⢈⠐⠂⣠⠄⠀⠀⠀⠀⠀⠀⠰⠀⢀⡀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠰⡄⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠈⠀⠘⠀⠀⠀⠈⠉⠻⠳⡄⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢸⡄⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠅⡀
⠀⠀⠀⠀⠀⢠⣆⡖⠀⠀⠈⠀⠀⢹⡤⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠈⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠐⠀⠐⠀⠀⠙⠀⠀⠂⡀⠀⠈⠈⠃⠀⠀⢂⡀⠂⠀⠀⠀⠠
⠀⠀⠀⠠⠁⠀⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⡆⠈⠷⠀⠀⠐⠀⠀⠈⠁⠀⠈⠀⢀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⢿⡟⣶⣀⠔⢻⣦⠃⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢸⡇⢸⡏⠀⠈⣿⠀⢸⡇⠀⠀⣠⡀⢰⡋⢀⡤⢠⣏⣠⡖⠀⢠⡦⠀⢀
⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠉⠀⢸⣷⠀⠀⣿⠀⢸⡇⠀⢸⡧⠀⢨⠛⣾⠁⠀⢁⡿⢀⢰⡿⢁⢴⡟
⠀⠀⠀⠀⢀⣾⠋⠙⡄⢠⣿⢡⠀⢸⣿⠀⠀⡿⠀⠸⣇⠀⡸⠷⠖⠁⠀⠳⢊⠀⠘⠓⡁⠘⠓⢡⡟⠀⢀⣰⠇
⠀⠀⠀⠀⣸⡏⢤⣰⠃⠀⣿⣸⠀⠘⠋⠀⠀⠀⣠⡄⠈⣩⠀⢠⡮⠍⠙⢻⡿⠖⠚⢉⠄⢂⣰⠟⠉⠉⠉⠁
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⠙⣤⣿⠀⢠⣤⠀⢸⡇⠀⠀⣾⠁⢀⡀⣘⠚⠋⠁⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢿⣆⠈⢻⡜⠷⡇⣿⠀⢸⣿⠀⠌⢿⣄⢴⡇⠀⣸⠋⣸⠊⣿⠀⣴⠋⣼⠇⢸⡧⠀⢰⡄
⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠘⣿⡀⠈⣷⠀⡇⠹⠦⠘⣿⣎⠴⠚⣿⠸⣧⠔⠟⠀⡟⢰⡇⢰⡏⣠⡟⢀⠘⣿⡠⠞
⠀⠀⠀⠀⠀⢻⣿⡄⠀⠀⠀⠀⡿⣧⠀⠹⠶⠁⠀⠀⣠⠊⠁⠀⢠⣿⠀⠀⠀⠀⠀⠀⠈⠉⠀⠉⠀⡩⠔⠊⣿⡇
⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⣰⠁⠁⠀⠀⠀⠀⠀⠀⣇⠀⠀⣠⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠀⢠⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠳⠞⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠞⠋⠉⠁⠀⠀⠀⠐⠢⠤⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠚⣽⣟⠄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⡳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠿⠃⠀⣀⣀⠀⠀⠀⠀⠀⠀⣠⡄⠤⠠⠤⠙⠙⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⡤⠖⢺⣿⣿⣿⣿⣿⣛⢦⣸⠏⠀⠀⠀⠐⣄⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⣼⠀⠀⠀⢠⠜⠃⢰⣇⠉⠋⡹⣿⡿⣿⡇⠻⡀⠀⠀⠀⣠⢻⠀⣼⣿
⠀⠀⠀⠀⠀⠀⢠⢻⠀⠀⡀⠁⠀⢀⠀⠀⠀⠀⢰⡟⢙⣫⢏⠆⠈⠒⠒⠚⣡⠞⠀⣿⣿
⠀⠀⠀⠀⠀⠀⢸⠠⠄⠀⢳⣄⣀⡀⢀⣠⣾⠇⢸⣆⢀⣤⣾⣶⣴⣶⣶⠟⠃⠀⢸⣿⡿
⠀⠀⠀⠀⠀⠀⠸⣦⠄⠈⢷⠝⠿⠿⠋⠉⠀⠀⠈⠙⣿⣿⣿⠟⠛⠻⣿⣷⡄⢀⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠳⣄⣦⣀⣤⣶⣾⣿⡟⠰⠊⠲⢿⣿⣧⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠙⠋⠀⠟⠀⠀⠀⠀⠀⠈⢸⣿⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠀⠀⢠⣀⣾⠏⣴⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⡇⠀⠀⠐⠦⠶⠚⢻⠀⠀⠀⠀⢀⣠⣶⣿⣚⣻⣿⣯⡦⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣄⠀⠀⠀⠀⣠⠇⠀⠀⠴⣾⠟⠋⠉⠀⣨⡿⠿⠋⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣷⣶⡖⠀⠁⠀⠀⠀⠑⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠋⣻⡯⢿⠀⠀⠀⠔⠀⠀⠐⣿⡧
⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⢰⡟⠀⠸⡌⠁⠀⠀⠀⠀⠀⠻⠙
⠀⠀⠀⠀⠀⠀⠀⣿⢻⡇⣾⡇⠀⠀⠉⠢⠤⠠⠤⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⢻⡀⢧⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠑⢾⡋⠛⠦⣤⡤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⣿⣿⣿⡿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡿⠃⠒⠂⣿⡏⠒⠂⠻⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣷⠀⠀⠀⣿⡇⠀⠀⢰⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣆⠉⠀⠤⠄⠈⢁⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣷⠀⠠⣤⣴⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡿⠛⠛⠛⠃⢀⣀⣀⠀⠛⠛⠛⠿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⠏⠀⠀⠀⢀⣶⣿⣿⣿⣷⣆⠀⠀⠀⠘⢿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡏⠀⠀⡐⠀⢸⢃⣈⢻⢃⣀⢻⠀⠠⠀⠀⠘⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⠃⠀⢀⠀⢀⠸⡇⠀⣈⡀⠀⡿⢀⠀⠀⠀⠀⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⠀⠀⠀⢀⠘⣱⠇⠒⠀⠐⠂⢧⡙⠀⠀⠀⠀⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠻⡄⠀⠀⢸⠦⣤⠀⠀⠀⠀⠀⢠⡤⢶⠀⠀⠀⡿⠛⠛⠛⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⢄⠑⢆⠀⣀⠖⢁⠄⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⠀⠈⠑⢦⡀⢀⠀⠘⠀⠀⠱⠁⠀⠈⠀⣀⠀⡰⠖⠉⠀⠀⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠀⠀⠙⢦⣌⣐⠂⠠⠶⠆⠈⢒⣈⣤⠎⠀⠀⠀⠀⠀⠀⣠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄⠀⠀⠀⠀⠀⠈⣿⠧⠄⢀⡴⣄⠀⠤⢿⠏⠀⠀⠀⠀⠀⠠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⣴⣏⢐⢈⣷⠄⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠛⠉⠙⠋⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢠⢀⡿⣦⡀⣄⡤⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⢆⡀⡀⠠⣄⡛⣴⢏⠀⠙⠛⠛⠃⠤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠷⡶⣿⡟⠉⠀⠀⠀⠀⠀⠀⠀⠈⠐⢄
⠀⠀⠀⠀⠀⡀⠰⢄⠐⢶⣄⢶⣄⠠⣀⢛⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠠⠤⠥⠭⠀⣘⣇⠸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⣄⠀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢉⠻⠾⠿⠧⠴⢤⣰⣤⡀⠀⠀⠀⠀⠘⠷⠀⢰⠀⠹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠈⠹⢄⠀⠀⠀⠉⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⠼⠃⠀⡀⠖⠀⡀⠀⣄⠀⠈⣷⣄⠀⠐⢀⠀⠐⣻⣽⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠆⠸⠢⣧⣱⠿⣤⣏⣀⠟⠀⠀⣾⡿⠛⣉⠉⠑⣚⣿⡅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣦⠄⢐⣿⡿⠶⢤⡉⢠⡆⠀⡰⣽⡛⣹⣤⣴⣿⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣿⠄⢀⣬⠹⣦⠨⣿⠎⢀⠄⢀⡿⠃⢿⣩⠭⠭⡿⠷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⡼⢄⣲⠝⠁⣀⠄⠁⣴⡟⠃⡟⣤⠄⡀⠄⠂⠰⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⢀⣀⡤⠞⣁⣠⡴⠛⠙⠿⣦⢾⠆⢻⡀⠐⠖⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠓⠚⠋⣿⣇⢈⠄⠀⠀⠙⠿⣧⣀⠠⠄⠓⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠴⠚⠋⠉⠉⢩⠙⡆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢀⡀
⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⢀⣠⡤⣴⠿⢤⠀⣸⠃⠋⠀⢀⡀⠀⠀⠀⠀⠀⣼⠁⠋
⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠐⠉⣀⡾⠋⣴⠻⢠⡏⣸⠇⠈⠉⣽⠀⠀⢠⡞⣹⡇⡾⣰⠋⣥⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠳⠖⠸⠄⠛⠁⠈⠛⠓⠦⠀⠘⠛⢹⡠⠛⠙⠊⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⡄⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢰⠇⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⠀⠰⡁⢰⡏⢣⠀⠀⠀⢀⡴⢠⡏⣴⡆⡀⠀⠀⡾⢀⡔⢀⠀⠀⠀⢀⠎⢳⠊⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⠸⠀⠀⠀⠘⠗⢹⠀⠣⠊⠀⠀⢠⣧⠾⠖⣿⠔⠀⠀⠈⡀⢀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⣀⠀⠀⠘⠂⠀⠀⣠⣄⠀⠁⠀⠀⠀⠀⠀⠀⠀⠡⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⡇⠀⣠⠀⢀⡞⢁⡟⠀⠀⠀⠀⠀⠀⡴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⣸⢣⢊⣾⢀⠎⢀⡾⠁⠀⠀⠀⠀⠀⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢰⢧⠃⣼⢃⠎⢠⡾⢡⡴⠆⠀⣠⡤⣴⡏⣰⣤⠞⣩⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⣾⠏⠀⠿⠎⢀⡿⠁⢿⡴⣿⠰⣿⡔⡿⢠⡿⠁⢸⣛⡡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⣸⠃⠀⠀⠀⠀⠀⠈⢸⡇⢀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡀⡀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢘⠀⠒⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠤⠀⠀⡄⠀⠈⠃⠀⠀⠀⠙⠁⠀⠀⠀⠹⠁⠀⠀⠀⣀⡀
⠀⠐⠀⠀⠀⢰⡄⢦⠀⣰⠀⣰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠈
⠀⠀⠀⣦⡀⢸⠃⠈⢳⣿⡖⠃⠐⡆⢠⣆⢀⡆⠐⡆⢠⣆⠀⠀⢰⡀⢠⡆⢸⠃⠀⣦⡀⢀⡄⢀⡀⢀⣤⠀⡴
⠀⠀⡴⣿⡖⠃⠀⠀⢸⡿⣇⠀⠀⠈⢺⣷⠋⠀⠀⠈⣾⣷⠒⠢⠀⠙⢺⣷⠁⠀⡠⣿⣶⠋⠀⠀⠙⣲⣿⠞⠁
⠀⠀⠓⣿⡇⠀⠀⣰⠏⠀⠘⢦⠀⠀⢸⣿⠀⠀⠀⢀⣿⣿⡇⠀⠀⠀⢸⣿⡇⠀⠓⢸⡿⡄⠀⠀⢠⣿⣿⡇⠈
⠀⠀⠀⣿⡇⠀⠀⠁⠀⠀⠀⠈⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡇⠀⠀⣸⢃⡗⠀⠀⠀⣿⣿
⠀⠀⠀⠁⠁
⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⣘⣿⡀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⢀
⠀⠀⠀⠸⢿⡿⣿⠶⠄⠀⠀⠀⠀⢰⣾⣿⣦⠶⠍⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠄⠀⡄
⠀⠀⠀⠀⢸⡗⣧⠀⠀⠀⠀⠀⠀⠸⣿⣟⠿⣄⠀⠅⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⢀⣠⡀⠀⢀⡗
⠀⠀⠀⠀⣸⠃⣼⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡟⠀⠀⠀⢠⡀⠀⠀⢰⡄⠀⠙⣇⡘⠿⣿⣿⡾⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⠀⠀⠀⣿⢷⡀⠸⣿⡷⢀⢰⡌⢻⣷⣿⣿⣶⣦⠈⠁⠀⣠⠄
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⢿⣿⣶⣿⡟⣶⣾⣾⣿⣿⣧⣹⣯⣤⡶⠟⠁
⠀⠀⢠⣧⠀⠀⠀⠀⠀⠀⠀⢰⣧⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣶⣿⣿⡷⣶⣷⣾⡟⣿⢸⣿⣿⣷
⠀⠀⢠⣧⣀⣀⡀⠀⢀⣀⣀⣸⣧⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⡾⣷⣿⣿⣿⡇⣿⢸⣿⣿⣿⣇
⠀⠀⢸⣿⠈⠁⠀⠀⠀⠀⠉⣿⣿⡄⠀⠀⠀⠀⠀⠀⣻⡿⢿⣿⣿⣿⣷⣿⣿⣿⣿⠇⠿⠛⢿⡟⣿⠛
⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠘⢿⠟⠃⠀⠀⠀⠀⣠⡾⠋⠀⣾⠇⠁⢿⠿⣿⣿⣿⣿⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠿⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣿⠀⠀⠀⠀⠀⣿⡿⣿⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢰⣮⢠⡶⣭⠍⢋⠠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠛⣳⠾⣃⣼⠃⣴⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢹⣿⢢⣷⣷⣉⠲⠘⢿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣛⣿⡇⠂⠈⣿⣏⢧⡀⠙⠺⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⣙⣛⣿⡶⢳⡟⡞⣿⣬⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⣟⡀⠀⣌⣽⣿⣻⣮⣑⠻⣯⢶⠃⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢛⣭⣅⡿⣿⣷⠕⢮⡙⢋⡻⢷⣤⣤⣽⣤⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢋⡾⣽⠟⣁⡀⣠⣿⣿⣶⠿⠛⠈⠉⠀⣠⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢯⠴⣟⡙⠏⠃⢈⠉⠁⠀⠀⠀⣀⠐⠈⠻⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⡁⢀⠀⠀⢁⣀⣤⡄⢘⣿⡄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣤⡄⠈⣱⠁⣾⣀⠀⣼⡿⠋⠀⠀⣽⠣⠺⣶⣤⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠾⠂⠀⠈⠀⠉⠀⠀⢀⣴⣿⣧⠀⠛⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⡴⡿⠟⠋⢉⣡⣄⠀⠻⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⢾⠠⣤⣤⣤⡦⠈⠉⠛⢍⠁⠀⠙⣶⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣦⡀⠈⠛⠿⣤⠀⠉⠛⠋⠀⠀⢀⣀⣠⣵⣶⠶⢘⣋⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠙⠛⠶⣴⡾⣷⠶⠶⢤⣄⣈⣉⣉⣭⣥⠶⠞⠛⠉⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⡀⠠⠄⠑⠀⢀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠒⠀⠀⠘⡗⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣄⡀⢀⣄⣀⠀⠀⠀⠐⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠘⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⢰⡿⠫⡋⢿⡾⢿⣿⠾⠷⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠀⠀⠐⠁
⠀⠀⠀⡀⠀⠠⡄⠀⣠⣾⣥⣤⣸⣷⣤⣧⣿⡿⠋⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡠⠦⡀⠀⠀⠈
⠀⠀⠀⢠⣤⡶⣾⡋⢹⣛⣤⡏⢿⡎⢉⡽⠋⠀⠀⠀⠀⠀⣀⡴⠋⠉⣁⡘⠉⠛⠳⣶⠁⠂⠉
⠀⠀⠀⠀⠙⢸⣿⣿⠟⢷⣷⣼⣿⡴⠋⠀⠀⠀⠀⠀⢠⡾⠿⠷⠻⠿⢿⡶⠂
⠀⠀⠀⠀⠀⠀⠈⠓⠚⠛⠛⠉⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⢀⣠⣾⡓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠄⠐⣀⣀⣤⣶⣏⡹⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⣴⣿⣏⠀⣿⣡⡯⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⡴⣿⡟⣬⢛⣷⠻⣦⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣤⡙⢿⣾⣿⣧⣖⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⡾⠋⠁⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⣾⠥⠤⠤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⠛⣙⢻⠋⣉⠛⢛⣟⠋⣿⡿⣻⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠋⠉⠀⠀⠀⠀⠀⠈⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⢠⣄⠀⢀⣄⠀⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⢠⡤⠀⠀⠀⠀⢳⡿⠀⠸⠿⠀⠀⠀⠀⠐⠒⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡃⠀⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠈⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠃⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡀⠀⠀⠑⠦⣀⠀⠀⠀⠀⠀⢀⡠⠚⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠉⠁⠤⠄⠉⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠒⠶⠒⠒⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠴⠶⠶⢤⡀⠀⠀⠀⠀⣠⠴⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠉⠀⠀⠀⠀⠀⠙⣆⠀⠀⡾⠁⠀⠀⠀⠀⠹⣆⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⢸⡆⢸⡇⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢻⡀⠀⠀⠀⠀⠀⠀⠸⣆⢀⡞⠀⠀⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠻⣄⠀⠀⠀⠀⠀⠀⠹⡾⠀⠀⠀⠀⠀⠀⢀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠙⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⢀⣀⣄⡀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣰⠋⠀⠀⠹⡆⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢻⠀⠀⠀⠀⣿⣤⣼⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠸⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠙⠦⣄⣀⠀⠀⠀⠀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⣀⠀⠀⠀⣀⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠄⠒⠀⠁⠐⠒⠒⠈⡉⠁⠒⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⡁⠔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⡀⠑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⡡⠊⡠⠔⠂⠁⢀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠈⢢⠀⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠐⢀⠀⠀⠐⠈⡠⠐⢀⠐⠢⡀⠠⡀⠀⠈⠂⠀⠣⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡀⠠⠀⠀⢁⠠⠒⠁⠀⠀⠈⠢⣀⠀⢄⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢤⡠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡤⢀⡀⠀⠀⠀⠀⠀⠀⢡⠤⢤⠀⠒⠀⠉⢀⡀⠀⠀⠀⠀⠀⢀⡀⠁⠂⠌⠁⣀⣀⣄⠀⠀⠀⠐⠰⠌⠸⠡⢗⠝
⠀⠀⠀⠀⠀⠀⣘⡑⠆⠏⢀⡠⡄⠀⠀⠀⠀⡇⠊⣺⠀⠀⠀⠀⠈⠁⠀⠀⠒⠀⠀⠀⠁⠀⠀⠀⠀⣜⠗⠉⠀⠀⠀⡰⢄⠀⠀⠀⢈⠝
⠀⠀⠀⠀⠀⠀⠩⡁⠀⠀⠀⢰⠀⠀⠀⠀⠀⢱⠀⡌⢃⠀⠀⢆⡰⠀⠀⠐⠤⠔⠀⠀⠀⠅⠅⠀⣔⠀⠀⠠⢄⠔⠊⠀⠀⠑⡄⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢢⠤⠐⠊⠢⡀⠀⠀⠀⡮⠀⢨⠞⠓⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣮⣀⡙⢆⠀⠈⢆⠀⠀⢠⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⡀⠀⠀⠀⠑⠂⠴⠁⢠⠂⢤⠂⡎⠁⠂⢤⠄⣀⠀⠀⡠⠄⡶⠁⠀⢡⠈⠂⠀⠙⡀⠠⢆⡔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⢠⠦⣠⠎⠀⠈⡀⠈⢲⠄⣠⠣⣈⠒⠒⣁⠜⣄⡠⠐⢩⠀⢸⠀⠀⠈⣖⠡⠒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⢀⡊⠁⠚⠀⠀⠀⡇⠀⢸⠀⠘⢄⠙⡀⢠⠋⡠⠃⠀⠀⠘⠀⢨⡀⠔⠉⠀⠢⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠐⠓⡖⠀⢰⠀⠀⡄⠀⠀⠀⠑⡜⠣⠊⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⠀⠰⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⡇⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠇⢀⠃⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⡇⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠰⠀⠎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢣⠎⠤⠀⠀⣀⣀⣀⣀⣇⣁⡀⠀⠀⠤⠤⠴⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡼⢄⣀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⠀⢀⣀⡠⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠠⠁⡠⠉⠹⠁⠐⠒⡖⠐⡆⠘⠉⢹⠀⢡⠈⡌⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⡰⠁⡐⠁⢠⠃⢀⠇⠀⡇⠀⡇⠀⡆⠈⠀⠀⠆⠈⠄⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢁⠌⢀⡜⠀⢠⠃⠀⡘⠀⢠⠁⠀⠃⠀⡇⠀⡇⠀⠘⡀⠈⢆⠈⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢎⡠⠁⢠⠊⠀⡰⠁⠀⢠⠁⠀⢸⠀⢰⠀⠀⡇⠀⢠⠀⠀⢃⠀⠈⢣⠀⠑⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢒⠧⢀⡜⠀⠀⢠⠃⠀⠀⡆⠀⢸⠀⠀⡇⠀⠘⡄⠀⠈⡆⠀⣀⠱⠂⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⠐⠁⠀⠀⠈⠁⡲⠃⠤⠀⢠⣁⣀⣈⣀⣀⣇⡀⠀⠧⡄⠒⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⡏⠀⠀⠀⠀⠀⢀⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⣡⣄⡠⠤⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠉⠺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⣀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠁⠂⠰⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢂⠀⠀⠀⠙⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢⢀⡀⢀⡱

⣶⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣴⣧⣼⣼⣷⣾⣼⣼⣥⣤⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣶⣶⣿⣾⣿⣿⣶⣶⣶⣾

⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⣶⡈⠄⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢿⡇⡀⢠⡆⠀⠀⠀⠀⠀⠀⠀⠀⠈⣤⣼⣿⣷⡿⠿⣿⣿
⠀⠀⢠⡖⠛⠓⢦⣄⠀⠀⠈⠻⢀⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠺⠏⡐⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠛⠉⠁⠀⠈⠻
⢀⡶⠺⣷⣄⠀⠀⢹⡆⠀⠀⠁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣦⠀⢀⡀
⢸⣇⠀⠈⢿⣟⢦⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠻⢼⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰
⠀⠹⢦⣄⣀⣹⣦⣽⣷⡀⠀⠀⠀⠀⠀⠀⠀⡢⠀⡢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠊⣩⡍⢞⠽⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⣦⡀⠀⠀⠀⠀⠀⢠⠂⠐⡑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢺⠇⡆⠠⠾⣲⣄⠀⠀⡀⠀⠀⠀⠀⠀⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠐⢄⠉⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣴⡗⢻⡤⢁⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⠖⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣿⣞⢈⢡⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌⢃⠄⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠎⠀⠱⠏⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡍⣂⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠱⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠔⠓⣒⡽⣳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⢩⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠴⡟⡍⢀⣵
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡄⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠐⠿⣦⣱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠱⡈⠁⠐⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣾⣿⣿⣿⠿⠟⠛⠛⠛⢿⣿⣿⣿⣶⡁⠀⠂⠈⠃⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣿⣿⣿⡿⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣆⢀⣠⣴⣾⣿⣿⣿⣿⣿⠿⠟⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠻⠿⠿⢿⣿⣿⣿⡿⠛⢉⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣁⡀⠴⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢆⠠⠾⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣶⠑⣅
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠞⠱⢍⠀⣘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⡀⢀⣀⣶⣿⣦⣶⡄⠀⠀⡀⠀⠀⠀⠀⠀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣷⣶⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⣠⣭⣽⡉⠙⣿⡛⣿⠟⢿⣿⣿⣿⣿⡿⡤⠀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⢀⡏⠁⠀⣹⣿⣷⣜⡇⠛⣠⣈⣠⣽⣿⣿⣿⠦⣀⠈⠀⠀⠀⠀⠱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣿⣽⣾⡿⣿⣍⣻⣿⡿⢄⣛⣿⡟⣻⣿⣿⣿⣿⡌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢠⠀⢻⣿⡎⣻⣿⣿⣿⣿⣿⣿⢻⡿⠿⣿⣿⣿⣷⣿⣿⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢀⣀⢳⣄⣉⣨⣾⣿⣽⣋⣹⣾⣿⣾⣾⣷⣿⣿⣿⣷⢀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠅⠠⡄⠘⢡⡜⠻⣿⣿⣿⢻⣳⣿⠿⠛⠛⣿⢿⣻⡿⢾⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡾⣷⡿⣿⣯⣻⡒⣻⡏⣴⠞⢁⡉⢞⣍⠃⣼⣽⣿⢃⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠈⢁⢘⣟⢻⣿⡏⢸⢫⠀⢸⢭⠾⣫⣾⣟⣋⠝⠁⠠⠀⠀⢀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠙⠿⠁⡀⠽⡅⠈⡿⠛⢿⢿⣿⣫⣽⣟⣁⣀⠀⠀⢈⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣤⣀⣽⣦⣤⣭⣤⣴⣶⣿⣿⣿⣿⢿⣿⣿⣿⡂⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢩⣿⡿⠛⣻⣿⢿⣿⠿⢻⣿⣿⣿⣿⣿⣿⡿⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠈⡀⠀⠐⠀⠉⠻⠣⠟⡦⠻⠿⠋⢸⣿⣿⠇⡼⠁⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⣈⡋⢉⣔⠁⠄⣊⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⠄⠑⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠀⠀⢻⡇⢸⡿⢤⡄⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠈⠀⢀⣴⣢⡇⠀⠀⠀⠀⠀⠢⠀⠀⠀⠀⠂⠀⢰⠏⠀⡿⠁⠚⢃⠀⡎⠀⠀⠄⠂
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⡤⠾⣿⠏⠀⠀⢠⣿⡯⣛⣄⠀⢀⣴⠀⠈⠀⠀⠠⠐⠁
⠀⠀⠀⠀⠀⠀⠠⠶⠿⠿⠿⠿⠿⠶⠆⠐⡀⠀⠡⢿⣷⣻⠞⠀⠘⠻⣙⣣⠀⠀⠀⠐⢾⣧⣾⡇⠀⠀⠀⠀⢀⠀⠈⢀⡀⠒⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠙⠢⠛⠐⠂⠲⢁⡄⠀⠴⠀⠉⠀⠉⡁⠀⠠⠀⠃⠀⠀⠀⠐⠛⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢐⣂⣈⣁⣀⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⣤⣤⣾⣤⣾⣶⣶⣶⣶⣤⠄⠀⢿⣿⣿⣿⡷⠥⠀⣈⣿⣿⣿⣿⣯⣉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⣿⡿⢿⢿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣫⣿⣿⣶⣯⣭⣻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⣷⣿⣿⣿⣮⠛⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡏⣾⣿⣿⣽⣷⣯⡜⠻⣿⣷⣝⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡿⠏⣯⡀⠈⣿⣿⡆⠈⠻⣿⣿⣆⡀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⢇⣾⣿⡿⠸⠛⠛⣿⣿⡄⠘⣿⣿⣯⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢰⣾⣿⠿⠃⠀⣸⣷⣦⣸⣿⣿⣤⡈⠨⢿⣿⣧⡀⠀⠀⠀⣿⣿⣿⣿⣿⢫⣿⣿⡛⠁⠈⣧⣄⠹⣿⣧⡀⠐⢻⣿⣷⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⣿⠀⢠⣾⣿⣿⣹⠋⣹⣇⠙⠿⣦⠸⣿⣿⣯⠀⠀⠀⣿⣿⣿⣿⣟⣿⣿⡏⠀⣰⣾⣿⢛⠞⢻⡏⠻⣷⡄⢻⣿⣿⣺⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢸⣿⣿⡃⢀⣿⡿⢿⣿⠃⢠⣿⡿⣿⣿⣿⡇⠹⣿⣿⡇⠀⠀⣿⣿⣿⣷⢻⣿⡿⠁⢰⣿⣿⣿⠊⢀⣾⣷⣶⣾⣿⡀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⣿⡧⠀⣼⣿⣿⣁⣵⣾⣾⣇⣻⣷⡞⢿⣿⠀⢾⣿⣿⠀⠀⣿⣿⣿⣿⣿⣷⠆⠀⣿⣿⣆⣹⣴⣬⡇⣲⣯⡹⣿⡇⠨⣿⣿⣺⣿⣿⣿
⠀⠀⠀⠀⢀⣿⣿⡅⠀⣿⣿⣿⢿⣿⡿⣿⡟⠻⣿⣧⢸⣿⠀⣹⣿⣿⡇⠀⣿⣿⣿⢗⣿⡿⠂⠘⣿⣿⠸⣿⣿⣿⡿⠿⣿⣇⢻⡇⠀⣻⣿⣷⣻⣿⣿
⠀⠀⠀⠀⢸⣿⣿⠷⠀⢹⣿⣿⣻⣿⡇⠈⡿⢻⡿⣿⣿⡏⠀⢸⣿⣿⡇⠀⣿⣿⣿⢼⣿⣿⡄⠀⣿⣿⣾⣿⣧⠈⡿⠲⡿⣿⣿⡇⠀⣽⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢨⣿⣿⡏⠀⢸⣿⣿⣿⣿⣇⡄⠀⢸⣿⣿⣿⠇⠀⢈⣿⣿⡇⠀⣿⣿⣿⢾⣿⣿⠁⠀⣿⣻⣿⣿⣟⡄⠁⢸⣿⣿⣿⠀⠀⣻⣿⣯⢿⣿⣿
⠀⠀⠀⠀⠐⣿⣿⣯⠀⢸⣿⣿⢸⣿⡟⠀⠀⠈⣷⣿⣿⠀⠀⢾⣿⣿⠁⠀⣿⣿⣿⣽⣿⣿⣁⠀⣿⣿⣿⣿⡟⠀⢀⠈⣷⣿⡏⠀⠐⣷⣿⡗⣿⣿⣿
⠀⠀⠀⠀⠈⢻⣿⣿⡀⢘⣿⣿⣼⣿⠰⣦⠞⠀⣿⣿⣿⡄⠐⣿⣿⡏⠀⠀⣿⣿⣿⣾⢻⣿⡷⠀⢻⣿⣿⣿⡇⣦⠞⠀⣿⣿⣷⠀⢘⣿⣿⢻⣿⣿⣿
⠀⠀⠀⠀⠀⢈⣿⣿⣇⠘⣿⣿⣿⣿⡄⡟⠀⢀⣿⣿⣿⣿⡤⣿⣿⡧⠀⠀⣿⣿⣿⣿⢯⣿⣿⠆⢸⣿⣿⣿⣇⡿⠀⢀⣿⣿⣿⣇⣸⣿⣿⣽⣿⣿⣿
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣀⢿⣿⣿⣿⡷⣇⠀⣸⡟⣿⡟⢿⣷⣿⣿⠇⠀⠀⣿⣿⣿⣿⣏⢿⣿⣦⠘⣿⣷⢿⣟⢧⠀⣸⡍⣿⠋⢻⣽⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣯⣾⣟⣣⠽⠿⢿⣶⣿⣇⣿⠀⣴⣿⣿⠏⠀⠀⠀⣿⣿⣿⣿⣿⢏⣿⣿⣾⣿⣥⠾⠛⣻⣷⠿⠠⣏⠀⣾⣿⣟⣾⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠙⢋⠻⡿⢻⣿⣶⣶⣯⣉⣤⣄⡈⠂⡿⢟⠁⠀⠀⠀⠀⣿⣿⣿⣿⣿⠷⠙⣈⢻⣋⣿⣿⣿⣿⣶⣶⣤⣀⡘⠛⡁⠺⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣦⢘⣃⣿⣫⣿⣝⣿⣿⠿⣿⣿⣿⡿⣷⣟⢠⡆⠀⠀⠀⣿⣿⣿⣿⣛⢃⢆⣭⣼⣭⢾⠷⣿⡟⢛⡿⢽⡿⢙⣿⣅⢎⡛⣻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢈⣳⣿⡿⡏⢯⣷⣦⣻⣷⣾⣿⣿⣿⠷⡿⣿⣿⣋⠀⠀⠀⣿⣿⣿⣿⣿⣯⡼⣯⢻⣘⣿⣟⣶⣿⣿⣾⣯⣾⣹⢛⣿⣮⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠹⠟⣓⠿⢻⣿⣷⠟⠿⠛⢻⣿⣿⠻⠗⡋⠷⠉⠀⠀⠀⣿⣿⣿⣿⣿⣷⣾⠦⠉⡡⣿⣿⣿⣿⣿⣟⣿⡯⢉⠨⢴⣷⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣀⣀⡀⢁⣚⣉⡉⣁⣀⣀⣀⣀⡉⡉⠉⣀⣁⣀⣀⣀⣀⣀⣿⣿⣿⣿⣿⣿⣿⣾⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⢸⣿⣿⣿⡄⢲⠀⡀⠒⣿⣿⣿⣿⣿⠀⡇⢀⡤⠸⣿⣿⣿⣿⣿⢘⠸⣿⣿⣿⣿⣿⣿⢑⣽⢀⡇⠶⣿⣿⣿⣿⣿⣿⣗⣬⠰⢿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠉⠉⠉⠁⠈⠉⠉⠀⠉⠉⠉⠉⠉⠁⠈⠉⠁⠈⠉⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢀⡖⠉⠀⠀⠈⠙⣄⠀⠀⠀⠀⠀⠀⣠⠴⠒⠋⠉⠉⠉⠉⠉⠓⠲⢤⡀⠀⠀⠀⢀⡞⠉⠀⠈⠙⢦⠀⠀⠀⠀⣠⠎⠁⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⡠⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⢸⡀⠀⠀⠀⠀⠈⢧⠀⠀⣰⠁⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⡞⠁⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠈⢧⡀⠀⠀⠀⠀⠈⢇⣰⠃⠀⠀⠀⠀⠀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠀⡞⠀⠀⠀⠀⠀⠀⢠⣻⡆⠀⣜⣦⠀⠀⠀⠀⢀⣀⠈⣇⠈⠳⡀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⢀⡞⠁
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢈⡇⢸⠁⠘⡏⠀⠀⠀⠀⠈⠛⠃⠀⠙⠋⠀⠀⠀⠀⠀⠈⠀⢹⠀⠀⠙⣄⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⢀⣀⣀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⢸⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⢸⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⢰⠏
⠀⢰⠋⠀⠈⣧⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⠸⡆⠀⠀⠱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⣼⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⢸⠀⠀⠀⠸⠶⠟⠀⠀⠀⠀⠀⠀⢸⠇⠀⢳⡀⠀⠀⠈⠲⢄⣀⠀⠀⠀⠀⢀⣠⠔⠋⠀⠀⠀⣰⠃⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠉⠩⠭⠉⠁⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⢸⠃
⠀⠀⠀⠑⢤⣀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠈⠳⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠘⣦⡀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠲⠦⠤⠤⠤⠶⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠒⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⢿⣶⣦⣄⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣠⣿⣿⡿⠿⣿⣮⠹⣿⣷⡄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣧⠛⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣰⣿⡿⠁⢳⣀⢻⣿⣄⠈⢿⣿⣆⠀⠀⠀⣿⣿⣿⣿⣿⣿⡿⠋⣅⠘⣿⣇⠘⠿⣿⣟⣿⣿⣿⣿
⠀⠀⠀⠀⣰⣿⡟⠀⣴⣿⣟⠟⣹⡙⢷⣄⢿⣿⡇⠀⠀⣿⣿⣿⣷⣿⡏⠀⣴⣿⢳⠚⡟⢷⣄⢹⣿⣯⣿⣿⣿
⠀⠀⠀⠀⣿⡿⠃⣼⣿⠿⣏⡰⠿⣿⢿⣿⡌⢿⣿⡄⠀⣿⣿⣿⣿⡿⠁⣼⡿⢿⠋⢸⢿⡶⣿⡆⢻⣿⣿⣿⣿
⠀⠀⠀⢠⣿⡗⠀⣿⣿⣶⣿⣿⣷⣿⣧⢿⡇⢹⣿⣧⠀⣿⣿⣿⣿⡇⠠⣿⡿⣾⣿⣿⣼⣷⢹⡇⠸⣿⣿⣿⣿
⠀⠀⠀⢼⣿⣧⠀⣿⣿⣿⣿⠙⡷⢾⣿⣿⠇⢸⣿⣿⠀⣿⣿⣻⣿⡆⠀⣿⣧⣿⡏⢹⠶⣿⣿⡇⢈⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣯⠀⣿⣿⣿⣿⡔⠀⢿⣿⣿⠀⢨⣿⡷⠀⣿⣿⣿⣿⣇⠀⣿⣿⣿⣧⠂⠸⣿⣿⠃⢀⣿⣿⣻⣿
⠀⠀⠀⠸⣿⣷⠀⢿⣿⣿⡟⣀⡄⢸⣿⣧⠀⣼⣿⠇⠀⣿⣿⣿⣿⣷⠀⣿⣿⣿⢇⣠⠀⣿⣿⠀⢸⣿⣯⣿⣿
⠀⠀⠀⠀⢹⣿⡇⢸⣿⣿⣇⣿⠀⢸⣿⣿⣆⣽⣿⠅⠀⣿⣿⣿⣿⣿⠄⢿⣿⣿⣾⡇⠀⣿⣿⣧⣸⣿⣿⣿⣿
⠀⠀⠀⠀⠸⣿⣧⣜⣿⡿⣿⣧⢀⣿⣿⠟⢿⣿⡿⠀⠀⣿⣿⣿⣿⣿⣧⢸⣿⢿⣿⣆⣰⢹⡏⢹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢺⠿⣿⣿⣶⣋⣽⠟⠓⠻⢠⣿⠟⠁⠀⠀⣿⣿⣿⣿⡿⠻⣿⢷⣯⣭⣟⣉⡘⠀⡾⠟⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢠⠲⣸⣾⣿⣿⣿⢿⣿⣷⣾⣴⢢⠄⠀⠀⣿⣿⣿⡿⡢⣉⣾⣻⡯⣿⠟⣿⣿⡟⣶⣃⡐⣻⣿⣿
⠀⠀⠀⠀⠀⢵⣿⣿⣻⣷⣯⣷⣿⣿⣿⣻⣿⡿⠇⠀⠀⣿⣿⣿⣿⢷⠿⢣⢿⣯⣾⣿⣿⣶⣯⡯⢿⢾⣿⣿⣿
⠀⠀⠀⠀⠀⠈⠡⠫⠽⠿⠋⠉⠉⠻⠿⠭⠹⠉⠀⠀⠀⣿⣿⣿⣿⣿⣧⣤⣾⣿⣿⣿⣿⣿⣧⣤⣽⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⡄⠄⠀⢰⣿⣿⣿⡇⡂⢀⠁⣿⣿⣿⣿⣙⢿⣿⣿⣿⣿⢋⡏⣻⢿⣿⣿⣿⡿⡟⣉⢿⣿⣿⣿
⠀⠀⠀⠈⠉⠉⠀⠁⠀⠀⠉⠉⠉⠁⠀⠉⠀⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠷⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⣀⠀⠀⠀⠀⠀⠰⠄⠀⠀⢀⣀⡀⠀⠀⠀⠀⣄⣠⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣧⠀⠀⠀⠻⠟
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢀⣤⡴⢶⣶⣶⣤⡀⠀⠀⠀⠘⡇⠀⠀⠀⠀⣠⣴⠶⠛⠛⠛⠂⠀⠀⢀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⠀⠀⠀⠀⠈⠙⢿⣧⡀⠀⠀⠹⠀⠀⢀⣾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⠀⠀⢀⡾⠁⠀⠀⢀⣤⣶⣶⣶⣶⣦⣄⡀
⠀⠀⠉⠀⠀⠀⠀⠀⠉⠁⣀⣴⠾⠟⠛⠻⠷⣦⣄⠀⠀⠀⠘⢷⡀⠀⠀⣸⠁⠀⢀⣴⠟⠋⠁⠀⠀⠀⠈⠉⠛⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠋⠀⠀⠀⠀⠀⠀⠀⠉⠳⣄⠀⠀⠘⣇⠀⠁⠇⠀⢠⠞⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⡀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠳⡀⠀⠸⡀⠀⠀⠀⠊⠀⢀⣴⡾⠿⠿⠿⢷⣦⣀⠀⠀⠀⠀⠀⠀⣀
⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠿⠿⠷⠶⣤⡀⠀⠘⠄⠀⠃⠀⠀⠀⢀⡴⠋⠁⠀⠀⠀⠀⠀⠈⠙⠳⣄⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠐⠂⠀⠀⠉⠲⠄⠀⠀⠀⣼⣾⣀⠞⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⣰⣶⡀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⣀⣤⣤⣀⡀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠴⠚⠉⠙⠛⢿⣦⠂⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠈⠃⠀⠀⠀⡼⠁⠀⠀⠀⠀⣠⣿⠟⠉⠀⠀⠉⠀⠀⠀⠸⠧⠀⠀⣠⣤⣄⡀⠀⠀⠀⠀⠹⣧⠀⠀⠀⡀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠸⠁⢠⣤⠀⠀⢰⡿⠁⠀⠀⠀⢠⣶⠶⠆⠀⠀⠀⠀⠈⠁⠈⠙⣷⡀⠀⠀⠀⠀⢹⡆⠀⢠⣿⣿⡄⠀⠀⠆
⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⠁⠀⠀⠀⢠⡟⠁⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⢇⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⣼⠁⠀⠀⠀⣿⠀⠀⠀⢀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⡀⡟⠀⠀⠀⠀⡇⠀⠀⠀⠼⡿⠆⠀⠀⠸⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠘⠀⠀⢀⡀⠀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠃⠀⠁⠀⠀⠀⠀⠁⠀⠀⣠⡄⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠘⠟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠆⢀⡶⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢻⡀⠀⠳⢄⡀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⢄⢸⡇⡄⢹⠑⡂⠀⣷⠀⢨⡏⢠⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢀⠀⠀⠀⠸⠁⠃⡟⢸⡁⠀⡟⠀⢸⡇⡸⢸⡇⢄⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⣿⠀⠀⢠⡞⠀⠀⠇⠀⠀⡠⠀⠀⡀⠁⠀⣸⠁⡼⠀⣿
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠀⠀⡾⢀⡔⠀⠠⠔⢸⡇⠐⢰⡇⠀⠐⠁⠀⠀⢠⠟⢸
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢀⠆⠀⢠⡇⢸⡀⡀⠀⠀⠀⢷⡄⣾⠀⠀⠀⠀⠀⠊⠁⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⠘⠀⠀⢸⡇⡎⠙⣷⠀⡄⠀⠘⢷⡻⠄⣷⣀⢠⠀⢸⢱⠂⠀⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠈⠁⠇⠀⠻⠐⠇⠀⠀⠘⣇⠀⠟⠾⣼⠀⣈⡈⠀⠀⠈⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣿⠀⠀⠂⡏⠀⠀⢹⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⢰⠇⢠⠏⠀⡆⠀⠉⠀⢀⡟⠀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠐⠄⠀⠀⠈⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⢸⡇⠶⠀⠀⠘⠁⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠁⠀⣴⣿⣿⣿⣿⣿⣶⡐⠀⠀⣸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣾⡿⠉⠉⠉⠉⠉⢻⣿⣤⠀⢹⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣰⣿⣷⠀⠀⠀⠀⠀⢸⣿⣿⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⠈⣿⣿⣿⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡇⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣧⠀⠀⠶⠀⠀⢸⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣷⣄⣀⣠⣴⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠞⣁⣠⠐⠂⠲⠖⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡏⢀⠰⢦⠃⠸⢀⠉⢿⣇⡀⠀⠀⠀⣿⡿⠿⢷⣶⣤⣀
⠀⠀⠀⢀⠀⠀⠀⠀⠀⠉⡀⠀⡏⠅⠂⠘⠀⠂⢠⣹⠿⠷⢤⣤⣌⠠⢠⠄⢼⠁⠂⠄
⠀⠀⠀⠀⢣⠈⠂⠠⡀⢰⣧⠒⠁⡠⢷⠦⠘⠄⠂⠀⠀⠀⠀⠀⠁⠀⢿⣤⠀⢻
⠀⠀⠀⠀⠀⠑⢤⣆⢀⣿⣀⡀⠋⠠⠊⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⢻⣦⣠⣤⣀⠀⠂
⠀⠀⢀⠄⢈⣃⣤⡬⣿⣿⣿⣧⢿⣗⣨⢭⣗⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⡈⠹⢿⣷⣄
⠀⢀⠄⣉⠀⢤⡠⠒⢩⢻⣿⡗⠢⠄⠸⡉⠄⠀⠀⠀⣀⣀⡄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⡲⠩⠏⠁⠆
⠀⠈⠀⢀⣀⣾⠒⡺⢋⠀⢻⠂⠏⠻⠀⠀⠀⠀⠀⠀⠎⠊⠲⠶⠓⠖⢶⠄⠀⠀⢸⡟⡂⠄⠂⠘⠰⢅
⠀⠀⠄⠈⣑⣠⠰⠁⢠⢆⢸⠠⠠⠀⠈⠁⢰⠉⣺⣄⣄⣀⣴⢢⣄⣠⣩⠀⠀⠀⠈⠅⠠⠥⡁⣑⡀
⠀⠀⠀⠂⠈⠹⡇⢲⣾⡲⠿⠳⠆⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠁⠈⡀⢀⠄⠀⠒⠤⠄⠠⠈⣧
⠀⠀⠀⠀⠀⢵⢂⠊⣉⡙⢓⣘⣾⡧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢠⠈⢢⠨⣽⣗⠑⢄⠰⡜
⠀⠀⠀⠀⠀⠈⢿⡗⠇⢪⣭⣽⢌⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣅⠐⠪⢬⠋⣓⠃⣉⠀⠚
⠀⠀⠀⠀⠀⠲⢦⣤⣼⣷⣾⣛⣾⣿⣿⣿⣿⣦⣤⣀⣀⣠⣤⣶⡄⠀⢿⡿⠀⣒⠐⠐⠂⠂
⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠿⠟⠁⠈⢀⠀⣈⣿⡿⡿⠛⣿⣿⠟⢙⠧⠠⡗⣠⡆⠈⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣤⣤⣄⣤⣴⣾⣇⡈⠿⢀⣲⠀⣹⠃⠀⣸⡶⠀⢠⣿⠏⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⣀⣴⣤⡄⠀⠉⢀⡿⠿⠋⠀⠠⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠠⠄⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠄⠒⠢⠢⣐⡀⠜⣀
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠈⠀⠀⠀⠀⠀⡀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠈⠉⠓
⠀⠀⠀⠀⠀⠰⢀⠐⠀⠤⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⢰⠀⠈⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⡰⢀⠀⡍⡄⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠠⡀⠀⠀⠀⠀⠀⠀⡀⠀⡠⠊⢀⠈⠊⠡⠘⢄⠀⠠⠀⠀⠔⠃⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⣶⡡⠔⠊⡠⠖⠁⠀⠀⠀⠑⢄⠳⣔⠀⠀⡀⠀⣀⡠⠄⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⢠⣖⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⡑⠃⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠐⡇
⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠂⠄⠀⠀⠁⠁⠈⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠤⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠤⠇⠁⠀⠀⠀⠀⠐⠢⠠⢤⡤⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢠⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠸⠉⠀⠘⠒⠁⠘⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣫⠆

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠋⢉⣡⡤⠉⠉⠙⠶⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠉⠉⠛⠻⣿⣦⣤⠀⠀⠀⠀⠀⠀⠀⠀⡰⠋⢀⠞⠋⠀⠀⠀⠈⠛⠢⡀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⢻⣷⠀⠀⠀⠀⠀⠀⣴⠃⠀⠀⠀⡘⠀⠁⠐⠇⡀⠀⢹⡀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⠀⢀⡁⠈⠈⠀⠃⠀⠠⠇⡾⠃
⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠞⡀⠀⢃⠀⠀⠠⠊⣼⠁⡇
⠀⠀⠀⡸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢿⠃⠀⠀⠀⠀⠀⢸⠂⠀⡞⠂⠤⠈⠘⠀⠀⠀⠀⡠⢞⢿⢠⠇
⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⢧⡀⠀⠀⠀⠀⢈⣀⣠⠴⠏⠂⢠⡾⠋
⠀⠀⠀⠀⠸⣷⣆⣀⣀⣀⣀⣤⣴⡿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢟⣛⣿⣿⣿⣷⣤⡤⠒⠊⠁
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠖⠶⠤⣤⡀
⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢀⠈⠪⣒⣦⡀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠒⠫⠟⢻⣉⣁⡿⢶⡲⠦⢤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠻⢛⣿⢹⠀⠈⡄⠀⠈⠛⠓⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠈⠉⠀⠀⠀⠷⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠃
⠀⠀⠀⠀⠀⠠⠠⠀⠤⠶⠚⠉⠀⠀⠀⠀⠀⣀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉