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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⢰⡀⠀⣶⣠⡀⣠⣒⣠⣆⣴⢀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⣹⣼⣴⣾⣿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣧⣾⣅⣀⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⣸⣾⣿⣿⣿⣏⠙⢛⡀⣴⣿⡟⣉⣔⠚⠋⠀⢸⣿⠿⣿⣿⣿⣿⣶⣤⣄⣠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⣦⣾⣿⣿⠟⠋⠈⢛⣠⣼⣿⣿⣿⣄⠉⠟⠚⠚⠳⢠⣭⠀⠈⠉⢉⣽⣿⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣀⡀⣠⣴⣿⣿⣿⣿⣿⠿⠿⠿⠟⠛⢩⣿⣿⠙⠻⢿⣶⣶⣦⣤⣤⣤⣶⣶⡾⠟⢋⣁⣙⡋⠿⣿⣿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣨⣾⣿⡟⠛⣿⣿⢿⣿⠀⡿⢗⠖⣡⣿⡟⢙⠓⠶⡄⣄⡍⠉⣩⣽⣿⣿⡍⡴⡷⡛⢙⢉⢻⢶⡀⠙⣿⣿⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⣸⣿⣿⣿⣤⣾⠟⠅⣸⡟⠀⡶⠀⢀⣿⣟⢂⣍⠉⠀⢢⢪⣵⣿⠿⡋⠈⣿⣯⡠⡧⡤⠉⠩⢬⠍⠽⢦⡄⠘⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⣰⣿⣿⡟⠛⠛⢁⣴⠜⢻⣿⡀⢩⣠⣾⣿⡁⠹⠟⢛⢺⣴⣿⠟⣁⣴⡿⠇⠉⢻⣷⣤⠈⠉⣀⡹⡕⠙⠈⣡⣤⣼⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⣼⣿⣿⣿⣧⠐⠙⠘⢂⣴⣿⣿⠟⠛⢛⣿⡟⢿⣷⣶⣾⡿⣻⣥⠔⢻⢨⠸⠚⢣⣌⠻⣿⣶⣦⣈⣤⣶⣶⡿⠟⠛⠻⣿⣿⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⣴⠿⣿⣶⣼⣿⣷⣶⡾⠟⠛⢩⣿⠀⣶⣸⣿⠀⢠⣿⣿⢿⣿⡇⠱⡤⠀⢀⠌⠐⠈⢡⣤⣼⣿⣿⣿⣯⣍⠀⠀⠋⠉⠀⢻⣿⣿⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⣼⣿⡆⠘⣿⣿⠘⣿⣆⠰⢶⡖⢿⣿⠀⢹⣿⡟⢠⣾⡿⢁⡈⠹⣿⣬⡖⠉⢸⣵⣶⣿⣿⣿⣭⣴⣶⣷⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀
⣾⠟⠛⠃⣰⣿⣿⠀⠻⣿⣧⡼⠁⢸⣿⡀⢺⣿⢣⣿⣟⢰⡯⣷⠆⠈⣻⣿⣿⣿⣿⣿⣿⡿⠿⢿⡋⠉⠉⣿⣿⡿⠋⠉⠁⣶⣤⣤⠘⢿⣿⣿⡂⠀⠀⠀⠀⠀⠀⠀
⣿⣶⣶⣾⣿⣿⣿⣈⣀⣬⣿⣧⣧⣠⣿⣷⣾⣿⣿⡏⢈⣉⣀⣴⣷⣿⠟⠋⣹⣿⣿⡏⠀⠀⢠⣿⣹⢀⣼⣿⣿⣷⣶⣶⣤⣄⣁⠉⠋⠀⠹⣿⣿⣆⠀⠀⠀⠀⠀⠀
⣿⣿⣉⣿⣿⠉⠉⠉⣉⣿⣿⢟⠋⢉⣹⣿⠟⣻⡟⠛⠛⢛⠿⣿⣿⠿⣷⣦⣾⣿⣿⣿⣄⣀⣀⣩⣿⣿⣿⣏⡀⡐⢊⡍⣉⡙⠻⠿⣷⣶⣄⣾⣿⣿⣷⣄⠀⠀⠀⠀
⠈⣿⣿⣿⣶⣶⣶⣶⣿⣯⣁⣀⠛⣴⣿⠇⢀⣿⣧⣀⣡⣀⣀⣸⣿⣆⠁⠉⠉⠛⢿⣿⣿⣿⣿⡿⠟⢋⣈⠙⣿⣷⣜⠧⠍⣋⣀⣴⣶⣾⠿⣿⣿⣧⡟⢿⣷⡀⠀⠀
⠀⠈⣿⣧⡀⢀⣿⡿⠉⣉⡉⠻⢿⣿⡿⠿⠟⢻⣿⡟⢋⠛⠉⠉⢻⣿⡿⠿⢿⢿⣾⣿⣧⠀⠹⢿⣦⡉⠿⣥⡁⠙⢿⣷⣾⠿⠋⣉⣂⣴⣴⣄⠙⢿⣿⣯⣿⣷⡦⠀
⠀⠀⠈⢿⣿⣾⣟⣄⣀⣉⠑⠂⢸⣿⠁⡾⠗⠘⢻⣿⡘⣭⠲⠶⠀⢻⣿⡠⣶⣤⡀⢿⣿⣇⠀⡎⠻⣿⣄⠋⣁⣴⣾⡿⣿⣷⡈⢣⣬⢨⡘⠚⣳⣆⠙⢿⢿⣿⣿⡆
⠀⠀⠀⠀⠹⣿⣿⣉⠛⠛⠿⣶⣾⣿⣤⣶⣶⣶⣦⣿⣧⣀⣤⣴⣶⣶⣿⣧⡘⣙⣈⣈⣻⣿⣆⣀⣤⣼⣿⣿⡿⢋⣥⣤⠸⠻⣿⣆⣘⠾⠐⠆⢎⣴⣶⣾⡿⣿⣿⡇
⠀⠀⠀⠀⠀⠈⠻⣿⣷⣤⣀⡈⣿⡏⡉⢉⣰⣌⡉⢿⣿⡟⢋⡀⡀⡈⢽⣿⡿⠿⠟⠛⢻⣿⣿⠟⣋⣉⢹⣿⣷⠸⠏⠋⠦⣄⠈⢻⣷⡄⠉⣴⣿⠋⣡⣦⣀⣿⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠟⠻⢿⣿⣿⣿⣿⣿⣤⣍⠁⠀⠹⣿⡄⠛⢦⡆⠐⣿⣧⠘⣻⣷⡀⢿⣿⡌⢿⡻⠀⢘⣿⣧⠘⣟⢙⣁⣠⣤⣽⣿⣶⣿⢣⡾⡋⣃⠄⣝⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠛⢻⣿⣿⣶⣴⣿⣿⣮⣤⣤⣀⡙⣿⡀⠿⠋⢁⠀⢻⣇⣠⣶⣾⠿⢿⣿⣧⣴⡿⠟⠛⢋⣉⠛⠛⣿⣦⢁⠅⡂⠀⡲⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠁⠈⠉⢯⡙⡟⢻⠿⣿⣿⣷⡾⣿⠿⠿⢿⣿⠟⢉⣤⣡⠄⠙⢿⣿⡠⡿⣻⢭⠘⢾⢠⠌⢿⣷⣇⠸⣆⡯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⠃⠘⡇⠈⢻⠿⢿⣷⣤⣄⡀⣿⠷⢾⢧⠴⢽⠄⠛⣿⣇⠾⢸⠉⢩⠍⠩⢅⡀⢻⣿⣮⠁⡷⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠁⠀⠸⠀⠀⠙⠻⣿⣾⣿⣷⣄⡀⠛⣘⡃⣄⠙⣿⡄⢻⣀⡒⡇⠒⣀⣀⣀⣼⣿⣧⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⣠⣤⣿⣿⣿⣦⣘⠜⠃⣠⣹⣿⣦⠚⢑⣴⣿⠿⢛⣛⣻⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⣁⠀⠹⢿⣿⣿⡿⠟⠻⠻⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠟⣉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⣭⡴⡟⠉⠟⢴⢈⠻⣿⣷⣾⣿⣿⣿⡿⠿⢛⣋⣉⡥⡇⠞⠟⠃⠄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⢃⠐⠁⠒⠓⠂⠖⠒⠚⢀⠀⠉⠉⠉⠉⡀⠀⠰⠚⠹⠡⠀⠐⠆⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣯⣥⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⡤⠤⠤⠤⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣤⠶⠦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⢋⡽⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠙⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠉⠓⠦⣤⣤⣤⣤⣤⣤⣄⣀⠀⠀⠀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣷⡄⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆⠀
⠀⠀⣠⠞⠁⠀⠀⣀⣠⣏⡀⠀⢠⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⡃⠀⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⢀⡞⠁⠀⣠⠶⠛⠉⠉⠉⠙⢦⡸⣿⡿⠀⠀⠀⡄⢀⣀⣀⡶⠀⠀⠀⢀⡄⣀⠀⣢⠟⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⡞⠀⠀⠸⠁⠀⠀⠀⠀⠀⠀⠀⢳⢀⣠⠀⠀⠀⠉⠉⠀⠀⣀⠀⠀⠀⢀⣠⡴⠞⠁⠀⠀⠈⠓⠦⣄⣀⠀⠀⠀⠀⣀⣤⠞⠁⠀
⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⠁⠀⢀⣀⣀⡴⠋⢻⡉⠙⠾⡟⢿⣅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠛⠉⠉⠀⠀⠀⠀
⠘⣦⡀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢉⣹⣯⣍⣿⠉⠟⠀⠀⣸⠳⣄⡀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠈⠙⠒⠒⠒⠒⠚⠋⠁⠀⡴⠋⢀⡀⢠⡇⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⢀⡾⠋⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠀⠀⠉⠋⠻⣄⠀⠀⠀⠀⠀⣀⣠⣴⠞⠋⠳⠶⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠦⢤⠤⠶⠋⠙⠳⣆⣀⣈⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
★☆
⠀⠀⠀⠀⢞⠉⠳⡎⠁⢹⠆⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢀⡁⠬⠯⢉⡁⠠⡀⠀⠀⠀⠀
⠀⠤⢾⠖⠁⠀⠀⠀⠀⠈⠂⢎⠀⠀⠀⠀
⠻⠯⠏⠀⠀⢀⠔⠀⠀⠐⠢⢄⢁⠀⠀⠀
⢠⠘⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⣎⡄⠀⠀
⠸⠠⠀⠀⠀⠡⠀⠀⠀⠀⠀⢀⢿⣧⡀⠀
⢰⢰⠀⠀⠀⠀⠁⠒⠒⠒⠂⠁⠸⢛⣃⡀
⠘⡈⠀⠀⠀⠀⠀⡠⠄⠠⡀⠀⠀⠐⠁⢸
⠀⠨⢏⡖⠀⠀⠶⠀⠀⠀⠘⣍⡁⡇⢀⠎
⠀⠀⠀⠘⢶⡿⣀⣀⣀⣀⠜⠀⠀⠉⠁⠀