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⠀⠀⠀⠡⠀⠀⢃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⢨⣿⣿⣷⣶⣤⣶⣿⣿⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣘⣣⣱⣄⣻⠀⠀⠀⠀⠀⠀⠀⠀⠉⠸⣿⠿⣿⠻⢿
⠀⠀⠀⠀⠀⢠⣰⣿⡷⠞⡿⣷⣶⡄⠀⠀⠀⠘⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⢀⣷⣿⠁⠃⠀⡆⠸⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠖⢄
⣷⠀⠀⠀⣸⣽⢅⠠⠀⠀⡇⠀⠚⢿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢠⣾⠦
⣾⡆⠀⠀⢰⣾⣺⠀⠘⠿⠇⠀⢘⢻⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠐⣀⡄⠀⠀⠀⠀⠀⣼⣴⣦⡀⢠
⣿⡇⠀⠀⠈⢫⣿⣆⣀⣠⠄⡄⣮⣿⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⢿⣿⡿⠙⠘
⠈⠀⠀⠀⠀⠈⠈⠛⠻⠿⠾⠛⠋⠉⠀⠀⠀⡦⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣀⢀⣤⣄⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡟⠈⢿⣿⠃⠀⠀⠀⠀⠀⠀⡀⠀⢀⣀⡀⠀⡀
⠀⠀⠴⠂⠀⠀⠀⠒⢚⠆⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⣀⣄⡘⠃⠀⠀⠀⠀⠀⠀⠀⠸⣦⠼⠛⠵⣗⣀⡀
⠀⠀⠀⠀⡀⣀⡀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠠⣿⣿⣿
⠀⠀⠀⠀⠐⠙⢱⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡷⠀⠀⢐⣿⣻⣿
⠀⠀⠀⠀⠀⠀⠊⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢴⡶
⠀⠀⠀⠀⢰⠈⠠⠀⠄⠘⠀⠀⠀⠀⢠⣦⡄⠉⠈⠥⠁⠑⢶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⠿⠋⠈⠁
⠀⠀⠀⠈⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠈⣦⠀⢀⣤⡀⠀⠰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠸⠿⠿⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢀⣀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣾⣿⣷⣿⡇
⠀⠀⠀⠶⠢⣤⡤⠚⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⣿⣿⣿⡏⠛⠙⢹⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠿⣿⠂⢴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⣲⠀⠀⠀⠀⠀⠀⠀⣠⣬⣿⣿⠁⠘⠀⠸⣿⣿⣶⠄
⠀⠀⠀⡖⠢⣤⣴⠖⢻⠀⠀⠀⠀⠀⠀⠀⠘⠓⠂⠁⠉⠖⠚⠀⠀⠀⠀⠀⠀⣽⣿⣿⠀⠰⡶⠀⣿⣿⣷⡄

⠓⠚⠛⣿⣬⡟⠛⠀⠘⠋⣹⣿⣿⣛⣇⣠⡖⣇⠠⢉⣠⣀⠀⠀⣀⣠⡀⠐⣼⣿⡇⣀⡀⢰⣿⡀⠀⠀⣶⣷⠿⢳⣜⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⢻⣿⣿⡌
⠀⣠⣴⣿⣉⠁⢤⠀⠶⠀⠀⠼⣿⡛⢺⡇⠑⠿⠃⠘⠻⠟⣠⣄⠛⠻⠁⣠⠋⠌⠉⠉⠁⠘⠻⣷⢐⣴⢿⣶⣶⡛⠋⣠⢠⠔⠳⠿⠿⠛⠻⣿⣿⣼⠟⠋⠁
⠀⢿⣿⣿⣍⡀⠀⠀⠀⠀⠀⢶⡿⠛⢿⣄⡐⠊⣧⠄⠀⠀⠁⠁⠀⠀⢨⠧⠀⠀⠀⠀⠀⠀⠀⢻⣟⣵⠋⠈⠀⠀⠐⡧⠀⠀⠀⠀⠀⠀⠀⣿⣿⢿⡀
⣤⣼⣿⣿⣿⠷⣶⠀⣤⣀⣿⣿⣧⠀⠀⠛⠀⠀⡉⠐⣷⣶⡶⠶⠆⠈⡻⠀⣀⣀⢀⠀⠀⠀⠀⢠⡿⣁⠆⠴⠀⣀⣬⠁⠀⠀⠀⠀⠀⠀⠀⠈⣿⣷⣽⣷⡦
⠁⣀⠀⠉⢻⣡⣾⣿⣿⣛⠛⠉⠀⠀⠀⠀⠀⠈⠳⣦⡄⢹⠃⠂⠀⣾⠁⠀⠀⠉⠀⠀⢀⠀⠀⢸⠁⡉⠤⢐⠀⢸⣿⣄⠀⠀⠀⠀⠈⠀⢚⣦⠞⢫⣿⣏⠔
⠈⢛⠃⢠⡄⢹⣥⡈⣿⠇⠀⣤⣂⡄⠀⠀⢀⡀⠀⠈⢧⣼⠀⢁⠀⠹⠀⠀⠀⠀⠤⢤⠎⠀⠏⡸⠷⣄⣳⣶⣶⣿⡿⡟⢛⣶⣄⣀⣀⣠⣾⠀⠀⢿⡏⠴
⠀⠙⢣⣤⠀⢨⣿⣿⣿⠀⠀⠻⠿⠃⠀⠀⢿⣿⡿⠀⢸⣿⣧⣈⣠⠀⣣⡤⠂⠀⠀⠀⠀⠀⠀⠁⣴⣟⣛⣧⡜⣻⣤⡟⠋⢫⡼⠿⢏⣿⣿⣀⡶⠋⢀⡴⠦
⠀⠀⠈⢉⡴⣿⡟⣀⠻⣦⣄⣀⠀⠀⠀⠁⠀⠀⢀⣠⣞⢁⠀⠀⠄⢙⡟⠀⠀⡀⠀⠀⠀⣀⣀⠀⠈⣿⣅⣙⣡⣿⡿⢦⡶⢻⡷⠻⡟⢹⡿⠁⢀⡾⠋
⣠⠴⢻⣍⡄⠛⠋⢁⣃⢂⠈⣽⣿⣙⣀⣿⡟⣻⡏⠠⢮⠀⠀⠄⠀⠸⡇⠀⣾⣿⡷⠀⠘⠿⠿⠃⠀⢸⣿⡯⡿⢢⣤⠈⠗⠚⠣⠾⠻⠉⢳⣄⠀⠙⠀⢀⣴
⢤⡀⠠⡷⢤⠀⣸⠁⠬⣺⣤⣿⡱⣾⣣⣤⢮⣿⡇⣇⢀⡀⠀⠀⠀⢸⡇⠀⠀⠉⠀⠀⠀⠀⠀⠀⠈⠀⠯⠤⠾⠯⣭⡀⠀⢱⠀⠀⠀⠀⠀⠈⠁⢰⠁⠘⠧
⠀⠈⠛⢄⣠⣿⢿⣷⡴⠿⠿⠻⠏⠉⠉⠱⣿⡟⣧⠦⢠⡤⠴⠟⣷⡔⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⠄⠄⠀⠙⠋⠀⠀⠀⠀⠠⣀⠀⠀⠛⣄⠀⣰
⠂⢠⣦⡄⠉⢿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⠈⠙⡿⢋⡴⠒⣷⡆⢠⠾⠆⢂⠀⠀⢀⠀⠀⣠⡦⡤⠶⣾⢃⣀⠀⣤⣶⣶⣷⠲⣷⡄⠀⣀⠀⠀⠖⠶⣁⣤⠿
⣄⠀⠉⠀⠀⣸⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⣠⡐⠃⢈⣡⡁⢀⣥⠈⣶⡶⠋⠛⢈⠌⣉⣉⡀⠂⠀⣀⠁⢬⡋⠘⣿⡿⢿⣿⣧⣿⣿⣿⣯⣤⣄⣻⢦⠙⢦⣡
⣿⣵⣦⣄⣴⣿⣿⠋⠙⢤⠀⠀⠄⠀⡀⠺⣩⡿⠃⠈⠉⣀⡉⠛⡄⢸⣇⡄⠀⣿⠸⣏⢉⡿⠲⡟⠉⢳⡀⠁⠀⢈⠧⢠⠀⢀⠀⠀⢀⣀⠀⠉⠙⢳⣣⡈⠻
⠙⢿⡏⠉⠙⣿⣇⠀⢀⣴⣷⣤⣤⣤⣬⣔⢡⠸⣶⠂⠀⠙⢁⣰⣍⣀⣹⡯⢀⠙⠳⣭⣭⣤⣤⣽⣶⣿⠞⠁⠀⢱⠄⠀⠀⣀⠀⠀⠀⢀⡀⠀⠉⢙⡿⢷⣀
⣷⣾⣇⡀⠞⠀⠙⣿⣿⠃⣠⠤⠿⢏⣌⠁⣠⣦⢠⡴⠛⠓⠾⠿⠋⠁⠙⠙⣿⣦⣀⣠⣤⣴⣿⠛⠁⠱⠄⡴⢷⣾⠀⠀⠀⠉⠀⠀⠀⠈⠁⠀⠀⣽⣟⠻⠠
⣿⡽⣿⣧⡀⠀⠀⠈⢿⠏⠀⠀⠀⠀⢨⡗⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⠛⠋⢀⢹⣏⣤⣤⣀⢻⣶⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣴⡇
⣧⠞⠋⠀⠙⣤⠀⠀⢸⡀⠀⢠⣄⠀⢸⡀⠀⠀⠀⢠⣞⣳⡄⠀⠀⠀⠀⠀⠸⢇⠀⠀⡜⡦⢿⣍⢻⣿⣿⡟⠁⠘⠣⣀⣄⣀⣀⣀⣀⣀⣀⣀⡰⢋⣿⣿⠃
⠃⠀⣶⠀⠀⣸⣷⣭⣪⣇⠀⣰⠈⠙⠉⠳⣄⡀⠀⠈⠛⠋⠀⠀⠀⠀⣀⡀⠀⠀⢧⠀⢁⡷⠾⡿⠛⠛⠛⠁⣠⣤⠀⠀⠈⠛⢁⣼⠋⡛⠉⠉⣠⣿⡟⠁
⠀⠀⠛⠀⠂⢻⡥⠛⠁⠀⠀⠀⠙⢦⠀⠀⠀⣸⡶⢦⣀⠀⠜⠷⠀⢼⣿⣿⠀⠀⢮⣀⢞⣵⡟⠀⠀⠀⠀⠀⢹⠟⠁⠀⠠⠀⠀⠀⠀⠀⡴⢞⡼⠋⠀⠻⠃
⢀⣀⠀⢀⡴⠃⠀⠀⠀⢰⣼⣷⠀⠘⣄⠀⣾⠃⡀⠀⠈⠻⠢⢄⠀⠀⠉⠁⠀⠀⠀⠙⣮⠉⠹⡄⠀⠤⠀⢀⠞⠉⠁⠀⠐⡰⠀⠉⠳⢚⣛⠋⠀⠀⠀⠀⠘
⠀⠀⠈⠙⠀⢠⣤⡀⠀⠌⠙⠉⠀⠀⣹⠋⠉⠀⢀⡀⠀⠘⠃⠀⣼⠐⢤⣄⣀⣀⣀⢠⣿⠣⡉⢹⡴⠞⠛⢃⡈⠉⠁⠒⢦⣡⡀⠀⠀⠈⠧⣀⡀⠀⠀⠀⣠
⠀⠀⣶⣧⠀⢿⣿⠇⠀⠀⠀⠀⠀⠀⣯⢀⡀⠀⢈⡁⠀⢠⡆⠀⣀⡀⠀⢸⣿⣏⢹⣾⣿⣇⠀⠀⣺⣿⣿⡿⠿⠶⢶⣤⡴⣍⢈⡁⠀⠀⠂⠼⠃⢀⡀⠰⠏
⣀⣀⣿⣿⣧⣀⠀⠀⠂⠀⠀⠀⠀⠀⠸⣟⣣⣤⣭⢤⣤⣀⣀⠀⠛⠋⢤⣿⣿⡿⠻⣿⡞⣿⣿⢂⢩⡿⠛⠿⠿⠁⣤⣬⣕⠁⠈⢁⡠⠄⠀⠠⠤⠬⡧⣌
⣿⣿⠟⠻⣍⣿⡷⠈⠁⠀⠒⠢⣠⠔⠋⠉⠀⠀⠀⠀⠀⠈⠻⣤⣆⣠⣾⡽⠃⣠⠐⢬⠻⣿⣼⡟⢀⣀⠀⠀⠀⢀⠈⠉⢩⡀⢀⡎⠀⠀⠀⠀⠀⠀⠀⠉⠛
⡿⠁⣴⠁⢀⣠⣄⠰⣦⠛⠲⠞⠀⠀⠀⣠⣶⣄⠀⠀⢀⣄⡀⠈⣿⣿⠻⠇⠾⠁⣀⡤⠀⢨⡟⠀⢿⡿⠀⠀⠸⣿⡇⠀⣸⠁⠈⣀⠘⠀⠀⠀⠀⢀
⠉⠀⢿⣄⡈⣉⣙⡃⣾⣿⡁⠀⠀⠀⠀⠈⠙⠋⠀⠀⠻⣿⠟⢰⣿⡿⠓⢤⣀⠸⠋⢠⣄⠉⡿⣄⠀⠀⠀⠀⠀⠀⠀⣠⣶⣄⠴⠻⢦⠀⠀⢀⣤⠀⠙⠳⢦
⠀⢀⣼⠿⠟⠉⢁⡀⠉⠉⣁⡶⠒⠀⠤⢀⣀⠀⠀⠀⠀⣀⣴⠟⠛⢀⠀⠀⡹⢿⠯⡥⠭⠾⠃⠈⠙⣇⣠⡄⢀⡀⢸⢿⣿⡇⠀⠀⢸⣷⣤⣀⠁⠀⠀⢀⣤
⣴⣿⣿⣴⠀⠀⢼⣿⠂⢸⡿⠃⣾⠛⡄⠀⣿⢤⠋⢻⣤⣿⡇⠀⠀⠀⠀⢸⠀⣿⠀⡏⠀⠠⠀⠀⢈⡟⣹⡿⠛⣻⣟⣾⣿⡇⢄⠀⢀⠴⠛⠛⠛⠶⢦⣤⣀
⣿⠿⠛⣿⠀⠀⣀⡀⠖⢸⡄⠠⣿⠀⣇⠀⣿⠀⡆⠀⣿⢿⣇⠀⠈⢀⡤⠜⠀⠛⠀⠳⠤⣦⡀⢰⡿⡉⠂⠄⣱⡿⠛⠉⠉⠙⢮⣽⠉⣿⣷⡆⠀⠓⣀⠈⢿
⠿⢷⣤⣿⣿⠀⢿⡿⠀⣿⠇⠀⣿⣤⠟⠋⢿⣤⠇⠀⢸⣾⣻⡀⡼⢁⣤⣴⡄⠀⢀⣶⣵⡆⠙⣿⣷⡟⠀⢰⣿⠁⠀⢴⠆⠀⠀⢡⠤⠈⠉⣤⠀⢸⣿⡶⢸
⣀⠀⠈⢻⣿⣶⣤⣀⣀⡇⠀⠀⠘⠋⠀⠀⠈⠉⠀⠀⣼⠿⣿⠧⣶⣄⠛⠛⠹⠟⠇⠙⠋⢁⣴⣶⣿⣷⠾⡿⢿⣆⠀⠀⠀⠀⠀⢸⣄⠀⠀⠁⠀⠀⠈⢠⣿
⣷⠘⠀⠀⠀⣺⠿⠟⠉⢻⣦⡤⢆⣀⡀⢤⣤⡦⢤⣴⠱⠛⢻⣿⣿⣿⣿⣶⣄⣤⣠⣴⠞⠉⠀⠨⡷⠀⠀⠀⠀⠳⣠⣶⣶⣿⣿⡞⣥⢋⣲⣤⣤⡤⠛⣿⣿
⠛⢷⡤⠚⠉⠁⠀⠀⠀⠀⢻⣆⢻⣿⡅⢸⡟⢧⡈⠅⢴⡷⢈⠁⠠⠉⠉⡙⠛⠧⠿⣿⣷⣤⡰⠄⢣⢌⠿⠄⠃⣸⡏⠉⠀⣹⣿⣷⡶⠟⠉⠀⣠⣤⠀⠈⠻
⣀⡘⣧⠀⠐⠄⠀⠀⢀⣰⣾⠏⢼⣿⣿⠀⣷⢨⡇⡐⣆⣔⢠⠾⠀⡄⠈⠁⠀⠾⠀⢠⣩⡿⢛⣦⣈⠀⠋⠀⠛⠉⢻⡀⠀⣿⡿⠋⣀⡀⠀⠈⠿⠟
⠀⣀⠘⣧⣀⣀⣤⣶⣿⡿⣿⣄⣾⣿⢏⣠⡟⠦⣧⠁⡙⣋⡀⠀⢂⡰⠆⠈⠐⠀⠀⣾⡿⠃⠈⠀⡝⠓⠺⠖⠁⠂⠈⠛⢾⠋⠀⠸⣿⡿⠀⠁⠀⠀⢀⠤⠚
⢸⣶⣷⣄⣿⣿⣿⣿⡾⠛⠛⢿⣟⣿⠀⣉⠑⠃⢤⣾⡅⠙⢡⡆⢻⣧⣭⣀⡤⣶⣾⠟⠀⠀⠀⠀⢾⠀⠀⠀⠂⠐⢀⣀⣠⣷⡀⠀⠀⠀⠀⣀⠤⠊⠁⢠⡀
⠀⠉⠉⠀⡏⠙⠹⠿⠀⠀⢀⣾⣿⣿⣷⣄⠺⡷⠈⣥⡀⠛⢣⣴⡏⠰⣾⣿⠁⣿⠃⠠⠀⠀⢀⢠⡀⠀⣠⣤⡀⢀⣶⣿⣿⠿⣿⣶⣦⠶⠊⠀⠰⠒⣰⣿⣄
⠀⠀⠀⠸⠀⡄⠀⠀⠀⠰⢟⣿⢿⣿⣿⢟⣿⣶⣦⡉⢁⣈⠈⢋⣤⣴⣿⡿⢸⣿⣤⣤⣴⣾⣿⣄⡙⠻⢷⣄⠹⠏⣽⣿⠈⠀⠑⣯⣿⣧⠠⠀⢀⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⠛⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠁⠀⠀⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠀⡠⠐⢄⠀⠀⠸⣧
⢀⣶⠟⠛⠛⠷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁⢀⠊⠀⠀⠀⠆⠀⠀⣿
⣾⠃⠀⠀⡀⢀⠈⠻⢷⡄⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⡶⠶⠶⢶⠶⠶⠿⢤⣤⣁⠀⠀⠀⠀⢠⠀⠀⣿
⣿⠀⠀⢰⠀⠀⠑⢄⠈⠻⣦⡀⣀⣤⡶⠟⠋⠉⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠈⠙⠲⣄⠀⠘⠀⠀⣿
⣿⡀⠀⢸⠀⠀⠀⠀⠑⢀⣸⠟⠋⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣆⠀⢰⡟
⢸⡇⠀⠈⡀⠀⠀⢀⡴⠛⠁⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠘⠻⠿⠦⠀⠀⠘⣦⣾⠃
⠈⣿⡀⠀⢡⠀⣰⠋⠀⠀⠀⠀⠀⠀⠀⠈⠂⡀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⢠⠀⡀⠀⡀⠀⠀⠈⢿⡀
⠀⠸⣧⠀⢀⡾⠁⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠀⠀⠔⠁⠀⠀⠀⣠⣖⡊⠙⠓⢴⠁⠀⠀⠀⠘⣷⡀
⠀⠀⢻⡆⢸⠁⠀⢠⡿⠋⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⣠⡆⠀⢇⠀⠀⠀⠀⠸⣧
⠀⠀⠀⢿⣾⠀⠀⠈⠀⠀⢀⣱⢄⠒⠢⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡹⢿⣿⠿⠃⢀⡎⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠈⣿⠀⠀⠀⠐⠢⡞⣿⣿⣄⣠⡄⠱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⠤⠠⠴⠊⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⡇⢿⣿⣿⡿⠃⢀⠇⠀⠀⠀⣀⡠⣤⣄⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠘⢦⣈⣁⣀⡠⠊⠀⠀⣀⠾⣷⣶⣿⠏⠀⠀⢠⠀⠀⠉⠑⠢⡀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⠈⣧⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠘⢤⡀⢀⡴⠃⠈⠉⠉⠀⠀⠀⠀⠀⢀⣠⡾⠋⠁
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⠈⠁⠀⠄⠀⠠⠂⠀⠀⣀⡤⠞⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣦⣤⣄⣀⣁⡀⠀⠀⠀⠀⠀⣀⣀⣀⣠⠤⠶⠚⠋⠙⡄⠀⠀⠈⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡶⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⢻⡏⡙⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠘⢿⣄⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠹⣦⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠸⣧⣀⡀⢀⡠⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⠀⠹⣇⠀⠀⠀⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠡⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠈⠐⠄⡀⠀⠀⠀⠀⠀⢀⠤⠊⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡆⠀⠀⠈⠁⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣸⡏⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⣿⡇⠀⢀⣴⠟⠀⠀⠀⢀⣾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠸⡄⠀⠀⠀⠀⢸⡆⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣿⣇⣴⠟⠁⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣆⡀⠀⠀⠀⢷⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⡟⠋⠁⠀⠀⢀⣤⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡾⠿⠛⠒⠒⣾⣤⡀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⢰⣃⣀⣤⣴⠾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡅⡀⠀⠀⣰⢋⠁⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⣠⡿⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠿⠶⣿⣦⣤⣅⣀⣠⣤⣇⠀⠀⡄⠀⠠⠀⢀⣇⣀⣀⣤⣴⠾⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣉⣉⣉⣉⣛⣿⣶⣷⣶⣿⣾⣟⣉⣉⣉⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡿⠛⠛⠛⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣷⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠤⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⢁⣄⣠⠀⠈⣀⡀⠐⠀⠀⠀⠀⠁⡀⠡⠀⠀⠐⠀⠀⠒⠃⠀⣤⠀⠓⠒⠲⢄
⠀⠀⠀⣀⠀⠠⠈⠁⠀⣈⣁⣾⡀⠀⠿⠹⡿⢈⣀⢠⣿⡧⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠐⠀⠀⣀⠀⠀⠄
⠀⠀⠀⠁⠀⠀⠀⠀⠠⠿⠿⠿⠿⠄⠺⠆⠠⠈⠋⠈⠁⡀⠀⠀⠀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠚⠃⠤⣀⡀⠐⠉⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⣦⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠄⢀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣴⡄⠀⠀⠠⠀⠀⠐⠀⠀⠸⠂⠀⠀⠀⢀⠀⠀⢠⣶⠀⠀⢠⣄
⠀⠀⠀⠀⠈⠃⠀⠀⠐⠀⢠⣤⠀⠐⠀⠀⠸⠿⠃⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠀⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠈⠉⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠲⠶⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡿⠂⠀⠀⠀⠻⠟⠀⠀⠀⢀⣀⡀⠀⠃
⠀⠀⠀⠈⡁⠀⠀⠀⠈⢿⣿⠟⠀⠀⠀⠀⠀⡄⠀⠀⠀⠘⠿⠁⠀⠀⠀⠀⢀⣤⣀⣀⠀⠀⠀⠀⢰⣶⡆⠀⢴⣦⠐⠊
⠀⠀⠀⠁⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⢀⡀⢀⠄⡈⠫⠙⠏⠀⠀⠄⠀⠀⠉⠁⠀⠀⠉
⠀⠀⡀⠀⠀⠚⠂⠀⠀⣤⢠⡄⠀⢠⠀⢀⣤⣄⡀⠀⠀⠀⠀⠀⡇⠈⡆⠀⠀⡆⠀⠀⢲⠀⠀⠁
⠀⠀⠁⠠⠔⠒⠓⠀⠘⠃⠀⠉⠀⠈⠐⠻⢿⠏⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⢸
⠀⠀⢀⡀⠀⢤⡀⠀⠀⠀⠀⠀⣶⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠀⠀⠀⣴⣦⠀⠀⠀⣾⣷⠈⡆⠀⠀⡀⠀⠀⠚
⠀⠀⠀⠁⠀⠉⠃⠀⣠⣴⡄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠁⠂⠈⠛⠻⠀⠀⠀⠛⠋⠀⠀⡀⠙⠉⢠⡿⠆⠸⠧⢀⡄
⠀⠀⠢⠀⢤⣴⡀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⢀⣄⠀⠀⠀⠀⠐⠶⠾⠂⠀⢠⠀⠐⠀⠀⠀⠀⠰
⠀⠀⢀⡀⣠⠐⣀⢀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡷⢦⣤⡈⠀⠀⠛⠉⠢⣀⣀⡀⣀⡀⠀⠀⠀⠂⠀⠀⢀⡠⠀⠀⠀⢡
⠀⠀⠠⠀⠀⢶⠄⠈⠀⠀⠀⢠⠀⠠⠀⠳⣶⡤⠁⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⢀⡏⢠⠀⠀⢠⣠⠄⠀⢀⡀⣀⠀⠀⢸
⠀⠀⠄⡀⠀⠀⣠⡄⠀⠀⠀⠉⠀⠛⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠐⢿⣤⠄⠛⠀⠓⠀⠐⠃⠀⣼⡿⠛⡻⠀⢀⠃
⠀⠀⠀⠉⠀⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠐⠀⠐⠒⢶⣌⠀⠀⠀⠀⠀⠀⢀⣦⣤⣄⠀⠀⠈⠁⠉⢀⠐⠣
⠀⠀⠤⠶⠦⠴⠂⠀⠀⠀⠀⠀⠀⠀⠀⢑⣶⠎⠀⣤⠆⠀⠀⠀⠹⡟⠁⣠⣄⡀⠀⠀⢿⣿⣿⡇⠀⠀⣤⠀⠀⠀⠐
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠏⠀⠀⣀⠀⠀⢀⡄⠀⡁⠀⣿⣿⣷⠀⠀⠀⠌⠁⠀⠀⢰⠇
⠀⠀⠀⠀⢁⡀⠀⢸⣿⡦⠀⠀⣿⣿⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠼⠧⠀⠀⠉⣁⣀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⡁⠠⣴⠀⠀⠉⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠴⠁⠂⠀⠀⣤⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠂⡀
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣀⠀⡈⠉⠀⠀⠀⠀⠀⠘⠋⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢰⠀⠀⠠⠆⠀⠀⠀⠀⠀⠐⠆⠀⠀⢤⣤⡄⠀⢄⠀⠀⢀⡀⢤⠀⢀⣴⠒⢦⡀⠀⠀⠦⣀⣀⡄
⠀⢰⠰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠏⠐⠟⠘⠋⠃⠘⠛⠀⠀⠀⠀⠀⠀⠤⠁⠀⠀⣤
⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⣤⠀⠀⠀⢠⣦⠀⠀⢠⣀⡄⠀⠀⠀⠀⠀⢀⣀
⠀⢀⠔⠐⠷⠀⠀⠀⠰⠖⠒⠒⢾⡷⠀⠀⠀⣤⠀⠘⢿⠟⠛⠛⠿⠟⠀⢀⡀⠛⢋⠀⠀⠈⠉⠁⠀⠀⣀⠀⠠⠎⠁⠀⠆
⠀⠠⡀⠀⠀⠸⠓⠀⠀⠀⠀⣀⠀⠠⠀⠴⡀⠀⠀⡀⠀⠀⠀⠀⢀⡀⠘⠛⠁⠀⠈⣀⣀⣀⣠⠀⠀⠰⣧⣀⣀⣤⠋⠉⠂
⠀⠀⠀⠉⠩⡄⠀⠀⠀⠄⠀⠀⡀⢀⠀⢠⣿⠶⠂⠁⠠⣤⡤⠤⠤⠶⣾⠟⠂⠀⢶⠊⠁⠀⠉⠲⠀⠀⡍⠉⠉⢉⠐⢄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⠀⠴⠂⠀⠀⠈⠐⠀⠤⠀⠘⠂⠀⢀⣀⡀⢀⠀⠘

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣦⡙⣾⡷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣇⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⡌⣷⣴⣷⣤⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠻⢻⣿⡿⣷⡆⠀⠙⢶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠈⠋⠀⠀⠀⠀⠀⠀⣿⣿⣿⣦⣄⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡄⡄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣶⣄⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣷⣧⣀⣀⡀⠀⠀⠸⠿⠯⡃⠈⠉⠛⠿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡿⣿⡿⠋⠀⢈⡁⠀⠀⠈⠛⠆⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡂⠀⠀⠀⢙⡋⠀⠀⠀⢀⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⠷⢶⣤⣾⣷⣤⡦⢄⣸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⡿⢿⡿⢿⣿⠿⢿⢿⠟⠋⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⡏⠀⠈⠀⣠⢱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣇⢀⠀⠀⠀⠀⠑⠺⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣴⣆⢀⠀⣤⠆⠀⠈⠀⠀⠀⠀⠀⠆⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⡿⠉⠀⠙⢿⣿⣿⣵⣴⡏⠀⠀⠀⠀⠀⢀⣴⣀⣀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣧⠀⠀⠀⠈⣿⣿⣿⣿⠇⠀⠀⠀⣦⣠⣼⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⠀⠀⠘⢿⣿⣿⣦⠀⠀⠀⠈⢹⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠿⠿⠿⠿⢿⣄⣀⡀⠠⠼⠟⠛⠛⠲⠔⠂⠐⠊

⢡⡾⠟⠛⢶⣤⣿⣢⣤⣿⣿⣏⡻⢻⠿⣷⡴⡀⣿⣧⣽⣿⠀⢠⣿⠟⠐⠉⠓⠈⠻⠟⠉⠁⢨⣿⣿⣷⡀⢀⣄⠸⣏⡑⠀⠌⠁⠀⢼⣿⣿⣰⠆⣤⠀⠰⡟
⣿⡁⠀⠀⠀⢀⣾⣿⣿⣿⠿⡿⣿⣾⢃⣼⠿⠃⠈⠉⠉⠿⠀⠸⣯⡀⠀⠀⠀⢠⡆⣀⣤⣾⣿⣿⣿⣿⠿⠮⠩⣾⠟⠛⢦⣄⡀⠀⢀⠀⠈⢻⡀⠋⠦⣄⣀
⠀⠀⠀⠀⠀⠘⣷⣬⣶⣵⣶⣶⣿⢿⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣄⢀⡴⠋⠁⠉⣽⠿⢛⠛⢻⣿⣅⠀⢠⡇⠀⠠⠁⢉⣿⢣⠀⠀⠀⢸⡇⢠⣶⣤⡾
⠀⠀⠁⠀⠀⠀⠀⠉⠛⠛⠛⢻⣧⢸⣷⡄⠀⠀⢀⣶⠿⠦⠀⠀⠀⢸⣯⠎⢀⠠⠔⠛⠒⠚⠟⡛⢿⣿⣏⠿⢿⣧⣴⡃⢠⣿⣣⠆⠀⠀⣠⣾⠁⢀⣿⡀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣿⠀⠇⢤⠀⠀⠀⠀⠄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⠷⠚⠿⢷⡶⠷⢦⡹⠖⢋⠁⠀⢠⠀⣿⠃⢠⡟⠀⠑⢄
⠉⠀⠀⠀⠂⠉⠀⠀⠀⢠⣤⣴⣾⣍⣀⡥⠤⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⡶⠎⠀⠿⠉⣠⢀⡼⢀⡿⠟⠟⠛⢶⣦
⣳⢤⣞⠀⠀⠤⠀⠀⣴⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⣀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠉⠞⢡⣾⣀⡀⠀⠀⠈⣱⡄
⢇⠀⠙⠻⠟⣷⡶⠊⢿⣿⣿⣿⣿⣿⡟⠋⠀⢶⣤⣴⣶⣶⣾⣿⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⠶⠶⢶⣄⠂⡍⠹⣄⠀⠀⢸⡿
⡄⠰⠖⢀⣼⠏⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠘⠛⠛⠛⠉⣉⣩⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠉⠻⣧⡀⠙⢦⣀⡿
⣇⠀⢀⣼⡇⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⢻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠿
⣿⣆⣸⡏⠻⠟⠀⠀⠀⠀⠀⠀⠀⠀⣤⣶⡄⢀⣤⣤⣤⣤⣤⡴⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡴⣡⡾⠁⠀⠀⠀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⢹⡿⢷⣄
⠉⠉⢹⣷⡄⠀⠂⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⡞⠛⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⢿⣿⣿⡏⣴⡷⠴⡒⠊⢹⡁⢈⠀⣄⠒⠦⠤⣀⣀⠀⡟⠦⣹
⠀⠰⣿⣿⠃⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣶⣶⣶⣶⡆⠀⠀⠀⠀⠀⠀⡀⠀⠀⢉⣀⣀⡉⢻⡇⠿⠓⠚⠋⠉⠉⠉⠉⠉⠉⠒⠒⠒⠡⢼⡄⣿⠀⠈
⢷⣤⡐⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⢽⣄⠉⠉⢄⠈⠀⠀⠀⠀⠀⢠⣼⡏⠻⡶⠛⢻⣿⣧⣿⢣⣤⣶⣿⣷⣦⡀⠀⠀⠀⠀⠀⣠⣴⣶⣶⡇⢻⡀⢑
⠀⠉⠁⠀⠀⠈⠃⢀⣀⢀⢀⠀⡀⢀⠀⢀⣸⣿⣶⠀⢈⣦⡀⠀⣠⣶⠶⣿⣠⠖⠚⠀⢴⣿⣿⣿⣿⢸⡿⠛⠛⢿⣿⠇⠀⠀⠀⠀⢰⣿⡿⠟⣿⣇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠹⡟⢛⡿⢿⣿⡟⠻⠟⣻⣿⣷⠛⠙⣿⣿⠋⠁⣼⠛⠉⣈⠂⠀⠀⠹⣿⣿⣟⢸⣿⣿⣷⣾⠟⠀⣶⣶⡦⠀⠸⣿⣶⣿⡿⢃⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡉⠀⠀⠠⠤⠄⡀⢀⣾⣿⣿⠐⣺⢿⣯⡟⣿⣿⣶⠟⠛⣷⠶⢶⡄⢻⣿⡬⢷⣯⡛⠿⡃⠀⠀⠀⣏⠀⠀⠀⢙⠿⣫⡞⠋
⠀⠀⠀⠀⠀⠀⢀⣠⣶⣤⣤⣤⢶⣤⣴⣶⣿⡿⣿⣿⠀⣿⢸⣿⡇⢘⣿⣿⠀⢂⡇⠀⠘⣿⡌⢿⣤⠞⠿⣿⠇⠈⠙⠻⡄⠉⣀⠶⠛⠑⠀⢻⣇
⣤⠀⠀⠀⠀⠀⣿⢻⡟⠉⠀⠀⠀⠠⠀⠈⠛⢳⣿⠈⣆⣹⣾⣿⣿⣄⣸⣿⢄⡿⠀⠀⣀⠯⣷⡘⣏⢀⣼⢁⡄⠀⠀⠀⠐⠋⡝⠀⠀⠀⠀⠀⢻⡄
⠉⠤⠀⠀⡀⠀⢈⣿⠀⠀⠀⠀⠀⢀⠀⠀⣀⢘⣿⣦⣼⡷⠋⠁⠀⠀⠉⢣⣾⠃⠀⢠⠀⠀⢨⣷⡸⢟⣹⣿⠁⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠸⡛⢷⡀
⠁⠀⠀⠀⠀⡠⢚⣻⢀⣶⡆⠀⠀⠀⠀⢺⣟⣻⣭⡙⠏⠠⢤⡀⠀⠀⡀⠀⠛⠦⠴⠏⠀⢀⣎⠸⣷⡾⠿⡟⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣿⣦⣻⣦⡀
⣠⡶⣄⣰⠛⣠⣯⡉⣦⣽⠦⣤⣤⣤⣤⡄⠙⢛⡾⠳⡀⠀⠀⠄⠀⠀⠀⠁⣀⠐⠂⠀⠀⣿⣠⣴⣏⠿⣶⣄⣀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣠⣿⡽⣿⠈⠳
⣿⣵⡿⡇⠀⠀⢈⡁⢻⣄⡀⣉⠛⠛⠉⠤⣤⣾⠃⠀⠀⠀⠀⠐⠂⠀⠀⢀⠀⠄⠀⠁⠀⠀⢉⣈⣙⠿⣿⠿⢿⣿⣿⣶⣶⣾⣿⣖⣿⣛⣿⠽⣿⠿⢃⡀
⠟⠙⣿⣿⣷⣾⢾⣷⣾⠏⠀⢹⡆⢀⠀⠀⢹⡆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⡀⣀⡏⠁⠉⠀⣿⡆⠔⠀⠈⠉⠙⠛⠉⠉⠉⠈⠁⢀⣿⣷⣿⠇
⠁⠄⠙⡿⢨⡿⠟⠋⠁⠉⠀⠀⠀⣇⠀⠀⠈⢧⠈⠀⠀⠀⠀⠀⠀⠀⡔⢠⣄⣵⢶⣾⣫⡟⠱⢶⡤⠊⠹⡇⠀⢀⣀⠀⠀⠀⠀⠀⢀⣀⠀⢠⡏⠉⢿⣶⣾
⡔⠛⠂⠀⠀⠀⠀⠐⠆⠀⠀⠀⢻⡏⢷⡄⠀⠈⠳⠀⠀⠀⠀⣤⡀⠀⠁⠀⠀⠨⠉⢿⣿⣬⣡⠾⣶⠄⢠⣿⠀⠀⢹⠿⠟⠛⠛⠿⣟⠁⠀⢸⡧⠀⠒
⢹⠋⢤⠀⢀⠀⠀⠀⠀⢠⠀⡉⢸⠇⢨⡇⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠀⠀⢀⠰⣶⣦⣭⣌⡴⢿⡙⠆⠈⣿⠀⠀⢸⣷⠀⣀⠀⣀⣽⠀⠀⢸⡇
⣣⡀⠀⠀⠀⠀⠀⠀⠀⠈⢃⠀⡿⠦⢾⠃⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⡛⠃⢩⡿⠟⣛⡉⣙⠃⣄⠀⣙⠀⠀⣾⠁⠀⠀⠀⠀⢸⠀⠀⢸⣇⣀⠤⠤⠤
⠃⠀⠀⠤⠀⠀⠀⠀⠀⠐⠋⠀⠀⠀⠀⢰⠿⠷⠀⠀⠀⠀⠀⠀⠀⠠⠾⡟⠿⠞⣋⣄⣋⣉⣛⠻⣿⣌⠻⢿⡆⠀⣿⠀⠀⠀⠀⠀⢸⠀⠀⣿⠏⠁
⢶⡶⠶⠐⠆⠀⠀⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⣛⠀⢻⠋⠈⠙⣿⠻⣦⣙⠛⠾⡇⠀⣿⢦⡤⠶⠖⠊⢻⡆⠀⣿
⢦⣀⡀⢠⣴⠉⠀⢀⠀⠀⠀⠀⠀⠀⠀⠿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⠔⢱⣆⠹⣼⡿⠻⣿⢻⣿⣤⣿⠀⠀⠀⠀⠀⢸⣧⣶⡏
⣀⣝⢙⡛⣿⣶⣤⡀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠂⠰⠀⠆⢉⠆⠘⣿⣧⡌⠉⣿⠋⣾⣯⣴⣿⠁⠀⠀⠀⠀⣸⣶⣶⡇
⠁⠘⣷⣿⣯⠿⣄⣰⣿⢾⠗⣄⣀⣠⣴⠂⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠁⠀⣤⠃⣼⡄⢂⠺⣧⠉⠓⠬⣲⣼⡏⠤⣿⣇⠀⠀⠀⠀⣿⣶⡞⠷⣄
⠀⢠⣿⣿⡇⢤⡘⣻⣿⣷⣿⢿⣇⣹⣽⣷⠤⠆⠀⠀⢀⠀⠀⠀⠀⠐⠂⠀⣿⢠⣿⢿⣄⠳⠈⢷⠐⣠⡼⠏⣤⣾⠟⣿⠀⠀⠀⠀⣿⠻⣷⣶⠙⢳⣦⣀⡀
⠁⣜⣿⣼⡷⢤⡕⠛⢿⠀⣷⣚⣿⡯⢌⣷⣆⠘⣷⡶⢢⣠⣤⣄⢀⠓⠀⠀⣁⣦⣼⡁⠈⠳⡆⣨⠗⠛⠂⣴⣿⣿⣼⠟⠀⠀⠀⠀⢿⣄⡺⣿⣷⠆⠀⠀⢹
⠾⣿⠁⢙⠓⣀⠨⠀⠀⠀⠹⠟⠙⣿⡟⣟⣻⣿⢧⡀⠀⠹⣟⠸⣙⠑⠛⢀⠿⣼⢷⣹⣄⠞⣰⣿⡀⠀⠀⢘⡿⠾⠗⠀⠀⠀⠀⠀⠀⠈⠛⠲⠤⢀⣠⣤⡴
⠀⡇⠀⠙⠘⠋⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠓⠀⠹⣧⣤⣴⣿⣶⢍⣴⡇⠸⡶⣇⢦⢣⡟⡛⠁⠀⠈⠉⠉⠁
⢀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣿⢠⡏⣎⢳⡜⢧⡹⣭⣎⣾⠛⡇
⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢸⡇⣟⠓⠰⠎⣷⡘⣿⡟⠛
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢟⠠⠁⣆⡏⢸⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⣱⡸⡄⠀⠈⢀⡾⣾⣿⡄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣤⢀⣠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⣽⣻⡟⢿⣿⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣷⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢿⡇⠼⠃⢀⡀⠀⠀⣾⣿⣷⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡦⠐⠂⠀⠀⠀⢀⡩⠤⠤⡜⢸⡤⠒⢉⣉⡉⠉⠻⣿⣿⣏⠀⠀⢀⠲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡆⠤⠴⠅⠀⠀⢈⠰⠋⠱⠀⠹⠿⣿⢲⣄⣸⣿⣿⣿⠀⠈⠐⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠞⣿⠆⠈⢑⠊⠁⠐⠂⢀⡀⠔⠀⠃⠐⡤⢌⠉⠁⠘⡄⠀⠀⣠⣶⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣁⡴⡯⠒⢉⣈⡑⡄⠈⠈⢹⠤⣀⡠⠼⣀⠇⣸⣄⠀⡘⠀⠀⠰⠏⣩⠿⣶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣈⣉⣰⢱⣶⠞⢿⣧⣸⡆⠀⢈⠖⠁⠀⠀⠀⠀⡀⠙⠑⡌⠀⠀⠚⠛⢿⣿⡎⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⢁⠀⠀⢸⠀⢉⢻⢻⡿⠟⡓⠒⠚⠤⣀⠒⡶⠀⠘⢑⠀⠀⠈⠢⠤⠀⡄⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠸⠦⠓⢾⡀⠀⢡⠊⠀⠘⢷⣇⣤⠖⠈⡣⠄⡲⣾⡟⣠⠊⠑⠆⣀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠐⢡⠳⢄⣠⡠⢠⠼⠋⠹⠀⠀⣀⠧⠤⢏⡁⣤⣨⠑⢢⠃⢗⢀⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠡⢂⠩⡔⠸⣄⡀⠈⠀⠀⢀⠀⠓⣄⠎⠁⡀⢀⣀⣈⠫⡥⠒⣧⢡⢨⠋⢔⠀⡂⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠤⢊⢓⣾⡅⠘⠃⠀⠘⠁⠀⢸⠀⠈⠃⠚⠛⠋⠀⣇⠜⠀⠀⡿⠐⠠⣄⠄⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠨⠴⡈⢹⢁⠀⢶⠾⡿⠀⠀⠀⡄⡂⢐⢀⣾⠄⠀⠈⠀⢀⠀⣯⠀⠅⠈⠥⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⣬⣼⡦⠤⠤⠤⠤⠤⠤⠤⠧⠬⠥⠤⣥⣤⣤⣧⡤⠤⠤⠿⡐⠀⢂⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠃⠀⠀⠀⠀⢀⣤⣤⣄⠀⠀⠀⠹⢿⡿⠟⠁⠀⠀⢀⠗⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⣤⣄⡀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣦⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⡀⠀⠀⠀⠀⠀⣠⣴⣦⣄⢀⡠⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠈⠈⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠦
⠀⠀⠰⠂⠀⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠈⠒⠖⠃⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂
⠀⠀⠀⠀⣠⠀⠐⢦⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠠
⠀⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠸⠁⠉⠀⠈⡂
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⣰⠂⠀⠀⠀⠔⢖⠛⠁⣴⣧⡀
⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠟⠀⠀⠀⠀⠄⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠐⠛⠀⠀⢠⣿⣿⡀⠀⠀⣿⠿⠿
⠀⠸⢫⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠈⠈⠉⠁
⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠁⠀⠀⠀⠀⡀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠐⡁⠈⢐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠃⠛⠀⠀⣀⡀
⠀⠀⠀⠀⠐⠀⠀⠂⡃⠀⠈⠀⠂⠀⠀⠀⠀⠂⠀⠀⠘⡀⠆⠀⠀⠀⠀⠀⠈⠀⠈⢂⠀⡍⠀⠘⠦
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠰⡔⡂⡠⠀⠀⠠⠀⠀⠀⠄
⠀⠀⣴⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠦⠾⠧⠈⠘⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣠⣤⣦
⠀⠀⠀⢀⡀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⢉⠀⠀⢈⡁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠂
⠀⠀⠀⠈⠛⠛⠋⠀⠀⠀⠀⠀⠁⠀⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢠⠐⠁
⠀⠀⠀⠀⠀⢀⡀⠀⠀⢶⣄⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠀⠀⠊⡄
⠀⠀⠀⠀⠀⠘⠇⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⡃⠂
⠀⠀⠀⡠⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⣴⠿⢿⠟⠻⢦⠀⠀⠁
⠀⠀⠐⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠸⠋⠀⠀⠀⢀⠀⡃
⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠀⠀⠀⠔⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡷⠀⠀⠀⠀⢀⡰⠶⡚⡛⠈⣠⣷⣄
⠀⠀⠀⠈⠀⠀⠀⠐⠀⠀⠀⠐⠀⠀⠀⠰⣿⣿⡗⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠃⠀⠠⠀⠀⠀⠀⠀⠀⠠⣻⠇⠀⠀⠀⣰⣦⣤⠈⠀⠁⠸⣿⣿⣿⣆
⠀⠀⣠⣤⣶⡄⠀⠇⠀⠀⠁⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠈⠡⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠈⠀⠀⠀⢰⠿⣿⠿⡆⠀⠀⠀⠛⠙⠋⠉
⠀⠀⠛⣳⡍⠁⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⣦⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠒⠄⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠃⡀⠒⠀⠆⡀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢶⣶⢰⡾⠀⠀⠀⠀⠀⠉⠀⠀⠁⠁⠛⠁⠀⠈
⠀⠀⠀⠁⠀⠀⠐⠀⠀⡀⢀⠀⠒⡀⠄⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⢀⠐⢀⠀⠀⣀⠂⠀⢰⢠⠂⠀⡀
⠀⠀⠀⠀⠈⠂⠃⠈⠀⠉⠈⡀⠄⠀⠀⠘⠀⠀⠀⠐⠀⠀⠈⠀⠀⠀⠀⢩⡄⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢁⡀⠐⠅⠀⠸⠛⡒⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠘⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⠀⠠⡔⡒⠒⠠⠐⠀⠀⢂⠀⠀⠂⠀⠄
⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⢠⣸⡄⠐⠠⡄⠀⠀⠀⠂⠀⠀⢀⠀⢀
⠀⠀⠈⠋⠁⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠐⠗⠀⠉⡶⠈⠘⢓⢄⠀⠀⠀⠀⠀⠀⢠⢞⠤⠀⢆⠀⠀⢀⠁⠈⠁⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢢⠀⡁⢀⣤⠮⠦⣧⠀⠀⠀⠀⢠⠿⠃⠀⠈⠂⣧⡀⣀⠎⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠒⠂⣈⠉⠐⠂⢄⡀
⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠱⣷⣋⡁⠀⠀⢹⠀⠀⠀⠀⢸⠘⠢⠤⠤⠤⣈⠍⢓⠤⠈⠓⠂⢤⠘⡆⠀⠀⠀⠀⠀⠀⠀⠘⢁⠄⠀⠈⠀⠀⠉⠁⢀⡽⣶⡦
⠞⠉⠁⠀⠉⠉⠓⢦⣀⠀⠀⠈⢌⠑⠉⢯⢱⠀⠀⠀⠀⢸⡐⠀⠀⠀⠀⠨⡧⠑⠢⡄⠀⠀⠀⢀⠇⠀⠀⠀⠀⢠⠴⠅⢀⠀⠀⠀⠀⠀⠀⢀⡤⢋⠌⠸⣷⣄
⠃⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⢣⡀⠪⠃⠀⠀⠀⠀⠀⠱⣕⢄⠀⢰⠁⠇⠀⠈⢪⡑⠤⢆⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⠒⠀⢰⠋⠀⠀⡜⠀⠀⠈⠒⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣝⡁⠰⣀⣀⡀⠴⣃⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⡰⠁⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠓⠒⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⢀⡴⠁⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⡀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠹
⣄⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠀⠀⠀⠀⠀⠀⢀⡠⠤⣖⠂⢠⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠀⠀⣀⢄
⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⢀⡔⡩⡌⠁⠀⡇⠉⠑⣪⠆⠀⠀⢀⣠⠴⠒⠒⠛⠋⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⣠⣤⡄⠂⠁⠈⢸
⠀⠀⠀⠙⠢⢄⣐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠁⠈⠫⣆⠀⡏⡀⠊⠀⠀⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⣤⠔⠈⠁⡰⠉⣧⠈⠙⠪⡞⡙
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠴⠶⠂⢚⡿⠋⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠘⢦⡢⢼⠁⠀⠱⣧⣀⢔⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⢀⠔⠉⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠙⠲⠶⠶⠮⠤⠚⠁
⠋⠭⢍⢉⣙⠲⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠁⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤
⠀⠀⠈⠀⠀⠉⠲⡝⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢋⣀
⠓⣤⣈⠀⣀⠀⠀⢋⡈⣧⠀⠀⠀⠀⠀⠀⠀⢖⠚⠉⠛⣖⠲⢤⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⢰⠃⠎⠠
⢄⢺⡉⡖⠢⠁⠀⠁⡇⡿⠀⠀⠀⠀⠀⠀⠀⠀⠑⠀⠂⠀⠀⠀⠙⣆⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠁⠀⠀⠀⠀⢀⠞⠉⠈⢀⠀⠀⠀⠀⠀⣿⢸⣀⣈
⠀⠂⡆⡈⠁⠀⢀⣠⢣⠇⠀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠈⠄⠀⠀⠘⠀⠘⢧⡀⠀⣀⠀⠀⢀⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⣎⠐⠦⣄⡈⠄⠀⠀⠀⠀⢹⡀
⠀⡀⡇⢀⠀⢀⡴⣠⠏⠀⠀⠀⠀⠀⠀⣠⠎⠉⠀⠄⠀⠒⠄⠒⠸⠄⠁⠀⠀⠙⠒⠥⠤⠤⠴⠒⠉⠀⠀⠀⠀⠀⠀⠀⠀⣇⣀⣀⣠⡼⢿⡄⠀⠀⠀⠀⠳⡌⠢
⣉⣀⣑⣂⣠⠴⠚⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠐⠢⢄⡀⠀⠀⠀⠀⣉⠡⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣱⠋⠀⢀⡟⠈⡀⠀⠀⠀⠀⠈⠳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢕⠤⠴⡃⠀⣱⡀
⠀⠀⠐⠒⠒⠒⠒⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠖⢒⡰⠒⠶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠋⠁⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢹⡁⠀⠸⠂⠀⡔⠁⠀⠀⠀⢀⡠⠴⠚⠛⡛⠒⠒⢦⣄⠀⠀⠀⠀⠀⠀⠀⢀⡤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⡆⠀⠙⠦⢀⠐⠒⠀⠀⠀⠀⠀⠀⠀⣸⡎⠉⠛⠶⣄⣆⠎⠀⠀⠀⠀⣠⠏⡰⡀⠀⠀⢁⠈⠈⠐⠬⠳⣄⠀⠀⠀⠀⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢀⠇⠁⠀⠀⠀⢄⠀⠀⠀⠀⢅⢦⣀⡠⠴⢾⠋⠀⠀⠀⠀⣸⢃⠆⠈⠙⠢⣀⢸⠀⠀⡀⠀⣃⠘⡆⠀⠀⡸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢰⠂⠀⠀⠃⠡⠀⠂⠀⢄⠀⠀⠀⠘⡴⡁⠀⡐⠁⠀⠀⠀⠀⠀⡇⢀⡀⠒⡀⠨⣝⢛⠖⠑⠈⠁⠈⡇⡇⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⠀⢸⠀⠀⠠⠀⠀⡀⠀⠀⠀⣢⣀⠀⠀⠈⠪⡶⠀⠀⠀⠀⠀⠀⠀⢧⠈⡉⠅⠀⠠⠸⢈⠁⠂⠀⠐⢈⣳⠇⠀⢸⠇
⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠈⣆⣠⠀⢄⠀⠀⠁⠁⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡑⢄⡀⠀⠆⡆⠀⠄⠀⡎⣡⠏⠀⠀⣏⠀⡀
⠀⠀⣀⠤⠒⠉⠀⠀⠀⠀⠀⠀⠀⠈⠻⣄⡀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣉⣊⣀⡃⣐⣂⡭⠞⠁⠀⠀⠀⠈⠉⠛⢦⣀⡀
⠉⠉⠀⠀⠀⠀⠀⠀⢀⣀⠠⠤⢀⡀⠀⠀⠉⠚⠀⠀⠀⠀⠀⠀⣠⠖⠋⠉⠉⠉⡙⠒⢦⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙

⠀⠀⠀⠀⠀⢀⡄⠀⠀⣀
⠀⠀⠀⠀⠀⠘⡃⢠⣊⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠈⠑⠄⠀⠀⡸⠀⠀⡆⠀⠀⠀⠀⠀⠀⠱⡀⢰⠀⠀⠀⣀⠤⠄
⠀⠀⠀⠀⠀⡰⡥⢤⡀⠉⢣⣄⣀⠀⠀⠀⠀⠐⠒⡲⠒⠈⠉⠉⠉⠲⢞⠀⠀⠀⠀⠀⢉⠱⠦⠷⠯⠤⠤⣼⡡⠤⠤
⠀⠀⠀⠀⢸⠁⠦⠶⢣⡏⣻⠀⠀⠀⠀⠀⠀⠀⢀⡇⢀⣀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⣟⡃⣰⢀⣀⡀⡊⢟⢿⠉⠂
⠀⠀⠀⠀⠈⢇⠀⠴⠌⡩⠃⠀⠀⠀⠀⠀⠀⠀⢸⠠⣇⢆⠇⠀⣫⡱⠀⡴⠀⠀⠀⠀⠨⡉⠒⡄⠀⢤⠈⢑⠉
⠀⠀⠀⠀⣠⢴⠃⠀⠀⠁⢻⡒⢤⠀⠀⠀⠀⠀⠀⠙⢲⡄⠠⠦⠀⠀⡎⠁⠀⠀⠀⠀⠀⠉⡾⠀⠀⠸⡞⠉
⠀⠀⠀⠐⠃⢸⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⢸⠠⡄⢀⡴⠀⡃⠀⠀⠀⠀⠀⠀⠀⡅⠀⠀⢀⠇
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠻⣀⣘⣁⣀⡰⠁⠀⠀⠀⠀⠀⠀⠀⠘⡤⠤⡞
⠀⠀⠀⠀⠀⠀⣹⠒⠐⠂⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢠⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤
⠀⠀⠀⠀⠀⠀⠐⢄⠀⡀⠷⠼⠶⡁⠀⠀⠀⠀⠀⣀⠀⢄⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⣀⣀⣤⣏⣀⣀⢀⡀
⠀⠀⠀⠀⠤⠤⣱⠞⣡⡠⠄⣞⡷⢘⠆⠀⠀⠀⠀⢀⣽⡞⠓⢓⣠⡀⠀⠀⠀⠀⢠⢎⢥⡨⣏⠁⠀⠀⠀⠀⠙⢦
⠀⠀⠀⠀⠀⠀⢸⡀⠃⠑⠣⠴⢠⡎⠀⠀⠀⠀⢸⠸⡧⠞⠰⣟⡼⢳⠀⠀⠀⠀⢸⡙⠓⣡⢢⢰⠶⣄⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⢸⠇⢀⠀⠀⠀⡶⠑⡄⠀⠀⠀⠈⠲⡄⠐⠂⢀⠀⠘⣆⠀⠀⠀⢰⠋⡷⢺⠛⠉⢀⡌⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⡜⠀⠈⠦⣄⠠⠃⠀⡟⠀⠀⠀⠀⠀⣇⠙⠂⠘⠀⠀⡸⠀⠀⠀⠀⡄⢆⠈⡀⠀⡞⠀⠀⠀⠀⣸⠁
⠀⠀⠀⠀⠀⠀⢣⠀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠘⢦⣀⣀⡠⡔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢇⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠙⢖⠒⠒⢯⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠉⠉⠉⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠚⡡⠔⣀⡀⠀⠀⠀⠀⠀⡤⠤⢄⣈⣵⡦⢤⡧⠴⣎⣈⣁⠀⠀⠀⠀⡤⢰⡡⢤⡀⢀⢈⢊⡃
⠀⠀⠀⠀⠀⠀⡴⠋⠉⠉⠙⠢⡤⠄⠀⠀⠀⠀⣀⢤⠞⠱⠼⠇⠫⡸⠂⡈⢳⠈⠁⠀⠀⠀⠜⢩⠳⣂⠇⢦⣂⢶⡍⢀
⠀⠀⠀⠀⠀⡜⠀⣀⣀⡀⠀⢀⢹⠀⠀⠀⠀⠀⠁⢸⠀⠙⠋⠉⠉⢉⣉⡥⠞⠀⠀⠀⠀⠀⠀⢸⠀⠀⣀⡀⠀⠘⡧⠞
⠀⠀⠀⠀⡴⡇⠘⢥⠣⠇⠀⢷⠟⡧⡀⠀⠀⠀⢀⡟⡆⠀⠀⠀⢰⠧⠀⠀⠀⠀⠀⠀⠀⠀⢰⠉⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠁⠑⢤⣀⣀⣒⣒⣢⠔⠁⠃⠀⠀⠀⠀⠀⠈⢦⢀⣀⣨⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⣀⠀⠀⠀⣠⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⣀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀⡀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠑⠀⠰⠇⢺⠂⠀⣾⡏⠏⠀⠀⠲⠰⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣘⡆⠀⠀⠀⠀⠄
⠀⠈⠁⠀⠀⡄⠀⣤⡄⠀⠀⠀⠀⠀⠀⠸⠿⠟⠀⠐⠂⠀⠛⠛⠃⠀⠀⠀⠀⡉⠉⠹⣶⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠠⠙⠁⢠⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠁⠄⠀⢻⣸⠀⠘⠛⠿⠀⠀⢸⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠁⠀⠀⠂⡀⢈⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠁⠀⠀⠘
⠀⠳⠄⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡶⢴⡆⠀⠀⠀⠀⠀⠀⠰⠷⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠷⠶⠶⠴⠶
⠀⠘⠿⠛⠀⠻⢿⠷⠀⠡⠄⠌⠀⡀⣄⡂⡄⠀⠾⡿⠀⠀⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠉⠓⡄⠀⠀⢠⠚⠉⠑⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⡀⠀⢀⠆⠀⠀⢣⡀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠤⠤⠤⠤⣀⠀⠀⠉⠉⠉⢆⠀⠀⡰⠉⠉⠉⠀⠀⣀⠤⠤⠤⠤⠤⠄⣀
⠀⠀⠀⠀⢀⠔⠉⠀⠀⠀⣀⣀⣀⡀⠀⠉⠢⡀⠀⠀⠘⡄⢠⠃⠀⠀⢀⠔⠉⠀⣀⣀⣀⡀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⡰⠁⠀⠀⡠⠒⠉⠀⠀⠀⠀⠉⠒⢄⠈⢢⠀⠀⢡⡂⠀⠀⡔⠁⡠⠒⠉⠀⠀⠀⠀⠉⠒⢄⠀⠀⠈⢆
⠀⠀⢰⠁⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⣀⠗⠊⠉⠉⠑⢾⣀⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠈⡆
⠀⠀⠄⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⢀⡀⢀⡀⢀⡀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡆
⠀⠀⠈⢆⠀⠀⢣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠤⠄⣀⣀⣀⣀⠠⠤⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠀⠀⡰⠁
⠀⠀⠀⠈⠢⡀⠀⠙⠢⢄⣀⡀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⣀⡠⠔⠋⠀⢀⠔⠁
⠀⠀⠀⠀⠀⠈⠑⡲⠂⢀⠜⠁⠀⠀⠀⠀⠀⠸⠉⠐⠒⠒⠒⠒⠂⢉⠃⠀⠀⠀⠀⠀⠈⠳⡀⠐⢖⠊⠁
⠀⠀⠀⠀⠀⠀⡔⠁⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⣀⣀⣀⣀⠠⣼⠀⠀⠀⠀⠀⠀⠀⠀⠹⡀⠈⢣
⠀⠀⠀⠀⠀⢰⠁⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⢶⠀⠀⠀⠀⠀⠀⡦⡀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⡆
⠀⠀⠀⠀⠀⠘⡄⠀⠈⢆⠀⠀⠀⠀⠀⣀⠔⠁⡰⢗⠒⠒⠒⠒⡺⢆⠈⠢⣀⠀⠀⠀⠀⠀⡰⠁⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠱⡄⠀⠀⠑⠒⠒⠒⠉⠀⡠⠎⠀⠘⣄⣀⣀⣠⠃⠀⠳⢄⠀⠉⠒⠒⠒⠊⠀⠀⢀⠎
⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣀⣀⣀⠤⠒⠉⠀⠀⠀⠀⠘⢄⡠⠃⠀⠀⠀⠀⠉⠒⠤⢀⣀⣀⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⠶⠶⠶⣤⣤⣀⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣥⣴⣶⣶⣶⣶⣤⣌⡛⢿⣷⡀
⠀⠀⠀⢰⣿⣿⡿⠛⠛⠻⢿⣿⣿⡿⢿⣿⣿⠉⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡿⠛⢿⣿⣿⣿⣿⣿⣶⣿⣧
⠀⠀⠀⣾⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠉⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⢀⠿⣿⣿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⢟⣣
⠀⠀⢘⠋⡎⠉⠀⠀⠀⣦⠀⠀⠀⠀⢰⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠇⠀⠀⠀⠀⠶⠀⠀⠀⠀⠚⡉⠀⠀⡀⡀
⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⠆⠀⢀⠀⠀⠀⠘⣲⠺⠂⠀⠀⠀⠀⠀⣸⡇⠘⠂⠻⣗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠀⠀⠀⠈⠡⠀⠋⢀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠟⢋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠀⠰⠀⠀⠀⠔⠁⠀⠀⠀⠀⠀⠈⠑⢄⠙⠻⢿⣷⠀⢰⣿⠿⠛⢉⣠⣾⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠑⠒⣠⡐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⠀⠀⠀⠀⣠⣴⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⢀⠒⢄⢀⠆⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠈⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠄⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠂⠀⠀⠀⠀⠀⠸⠿⠗⠙⠛⠂
⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠰⠮⣹⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠘⠀⠢⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠐⠀⠀⠁⠀⠤⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠁⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⣄⣀⠀⠀⠐⠲⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠐⠴⠈⠄
⠀⠀⠀⠀⣠⠀⡠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠘⢇⠀⠀⠀⠀⠀⠀⢁⣠⠼⠀⠃
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⣿⣁⣩⣍⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⣄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⡟⠉⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⣿⣿⣯⣿⠇⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣜⠿⠿⡿⠀⠀⠀⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠳⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠋⠁⠠⣤⠀⠸⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⢹⣦⣜⣛⣛⣿⣿⣿⣿⣿⣿⡿⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⣴⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⠤⢀⣀⣻⣿⣿⣿⣿⣿⣿⡅⠀⠀⠀⠙⢋⣿⣿⣿
⠀⠀⠀⠀⠀⠀⣴⡄⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢉⣿⣿⣿⣿⣿⣦⠀⠀⣠⣾⣿⠟⠁
⠀⠀⠀⠀⠀⠺⠿⠃⠀⠄⠀⠀⠀⢠⣿⣄⡀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣇⢾⣿⡟⠁⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣾⣿⣿⣿⡄⠀⠀⡀⠉⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⣀⡀⢀⠘⠛⠃
⠀⠀⠀⠀⠀⠀⠐⠾⣿⣿⣿⣿⣿⡏⠉⠉⢻⣿⠀⠀⠃⢀⣿⡿⠛⠛⠛⣿⣿⣿⣿⣿⣿⣿⣿⣿⡖
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠿⣿⣿⣶⣤⣦⣿⣆⠀⠀⢸⣿⣧⣄⣠⣴⣿⣿⣿⣿⣿⠿⠛⠉⠁⠀⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠿⠿⠿⠿⠿⠿⠿⠻⠟⠛⠛⠛⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡐⡀⠀⠀⠀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⢂⢀⢤⠊⠀⠩⠐⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢿⠀⢀⠀⠀⣠⠐⠱⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠹⡨⠄⠉⠁⠀⠀⠀⠄⢠⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠆⠀⠈⠁⠀⠀⠀⠄⠀⠀⠀⠨⡆⠀⠀⠀⠀⠀⠀⠀⠀⣠⠠⡀
⠀⠀⠀⠀⠀⢀⡠⡀⠀⠀⠀⠀⠀⠃⠌⠠⠀⠀⣀⣀⣀⢠⡀⠀⠤⠆⢒⠀⠀⠀⠀⠔⡀⠌⠀⠀⠁⠘
⠀⠀⠀⢀⡌⡄⠀⠀⠀⠀⠂⠀⠀⠁⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⢀
⠀⠀⡠⠁⠐⠤⠀⠀⠀⠂⠀⠀⠠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⢀⠄⠀⠊⠐⠠⡀
⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⠁⠘⠀⠀⢂⣠⡀⢠⣀⠂⠀⢸⠀⡐⠀⠀⠄⠀⠀⢀⠈⢂
⠀⢜⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠇⠐⢩⣆⠀⠀⣶⡙⠀⡈⠀⠡⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⢃⠀⠀⡀⠀⠀⠀⠰⡦⠀⠀⠀⠀⢀⠀⠈⠉⠀⠀⠉⠁⢀⡃⠐⢤⠄⣀⣀⣀⣀⣀⣀⡠
⠀⠀⠀⠆⠈⠛⠀⠀⠀⠀⠀⠀⠈⠀⠀⠸⠑⢤⣀⣀⣀⣀⠴⢋⠀⠀⢸⠀⠀⠀⠀⠀⠀⢠⠁
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠘⠀⠀⢸⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠰⡀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⡆⠀⠀⢸⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠑⠒⠒⠀⠀⠉⠀⠀⠀⠀⠀⠐⠤⠤⠤⠤⠤⠔⠁⠀⠀⠀⠉⠁⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⣴⡄⠀⠀⠀⠀⠴⢿⣆⣠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡿⠤⠠⠀⠄⠀⠀⠈⠉⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢈⢁⡤⢏⠀⠀⠀⠀⠄⠀⠄⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⢠⠃⠀⠀⠈⠳⢄⠠⠤⠄⠆⠀⢂⡈⠁⢢⡀⠈⢧⡀⠀⡀⠠⣀⠁⡀⠀⠡⠤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢀⠀⠤⠀⠀⡀⠁⢈⠀⠀⠀⣂⢀⠂⠀⠀⠀⠒⠦⡀⠀⠀⠀⠀⠑⡌⢀⠊⠀⠀⠈⠀⣀⣀⠀⠀⠁⡄
⠀⠀⠀⠀⠀⡐⠁⠀⠀⠠⢀⠂⠐⡰⠇⠄⠀⠀⠈⠱⡀⠀⠀⠀⡀⠀⠀⣀⡀⠀⠀⢰⠀⠀⠀⡠⠤⠀⠠⣄⠑⢆⠀⠰
⠰⣄⡆⠀⠀⡅⠀⠀⠀⢀⡤⡄⠀⢤⡄⠀⠀⠀⠀⠀⡆⠀⠀⠚⡅⠀⠀⡄⠀⠀⠀⡄⠀⠀⠰⠘⠲⠠⠄⠉⠀⢸⠀⡘
⠚⠻⣿⠋⠀⠁⠀⠀⠀⠀⣤⡄⠀⣄⡌⠀⠀⠀⠀⠀⠃⠀⠀⡈⠀⠀⠀⢊⡀⠀⢰⢤⡀⠈⢠⣀⠀⠉⡡⠀⠀⠃⡐
⠀⠀⠈⠳⠦⠴⡀⠀⠀⠈⠘⠀⠐⠒⠀⠄⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣤⠿⠀⢰⣦⣤⡼⠁⡀⢡⡇
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⠤⢒⣿⠀⠀⠀⠀⢠⢷⠀⠀⠡⢄⠀⡠⠄⢀⣼⣿⣾⢇⠀⠘⠒⠊⠀⠀⢀⡞
⠀⠀⠀⠀⠀⠀⣴⣵⣤⡄⠀⠈⠋⠑⢀⡤⢄⡠⢴⣋⣿⠀⠀⠀⠀⣀⠀⠀⠀⣠⠋⢀⣨⠦⢀⣀⣀⡠⢒⡏⣀
⠀⠀⠐⢦⣤⣀⣻⠀⠉⠉⠒⠦⢖⠒⠉⡇⠀⠀⠀⠞⠟⠓⠂⠰⡎⡼⠁⠒⠈⠀⠰⠯⠥⠤⣤⡄⠀⢳⠈⠳⠟⠃
⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⢠⡟⢧⣰⡇⠀⠀⠀⠀⠀⠀⠀⣠⠿⠴⠶⢲⡆⠀⠀⠀⣀⣴⠟⠁⠀⠈⠷⠤⠤⢤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠉⠁⠀⠀⠀⠀⠀⢠⣾⣥⣤⡀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢡⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣀⣸⣿⣢⠀⢀⣄⠀⠀⢀⣴⡜⡉⣀⣤⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡿⠿⠛⢉⣻⠘⣻⣇⣻⣾⣯⣼⣻⠷⣠⣼⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠻⡟⠃⠈⡅⡀⢀⣼⣮⣹⠗⢈⣼⠦⢉⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⢤⡤⣐⠠⠸⣿⡟⣾⢸⠠⠋⠚⠟⢛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⡟⠉⠵⣬⣷⣍⢿⠀⠀⢷⡏⣴⠶⣹⢻⠿⣦⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠤⠀⠀⠈⠽⣿⣾⡇⠀⢻⣿⣯⣿⣿⣿⣁⣾⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠤⠀⠀⠐⠋⠙⠣⣤⣾⢿⣿⣎⠛⠫⢶⣾⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢛⠄⠀⠀⠀⠀⣭⡀⠀⠈⠉⣹⣿⣿⣯⣾⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣄⡀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣼⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣙⣿⣿⣿⣿⣮⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⢸⢁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠊⠀⣿⣿⣿⣹⣿⣿⣿⣿⣿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢴⢿⣽⣿⡿⢻⣿⣷⣿⡟⠟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠉⠉⠛⠉⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⡀
⠀⠀⠀⠀⡀⠀⠐⠀⠀⠀⠀⠀⣠⠞⠩⢀⡤⠶⣾⣷⠶⣄⠀⠀⠀⠀⠀⢀⠀⠌⠀⠁⡀
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⡾⠁⠀⣰⠋⠀⠀⠈⠻⣷⡌⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠁⠉⣼⠃⠀⠀⠀⠀⠀⠘⣿⡌⢧⠀⠀⠀⠠⠀⢀⠀⠀⢀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠐⡇⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠸⣷⢸⡄⠀⠀⠀⠀⠀⠑⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⢸⡇⠀⠀⠀⢰⣦⡀⠀⠀⣿⢸⡇⢀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠈⠈⣧⠀⠀⠀⢸⣿⡝⢦⡀⠏⢸⠇⡁⠀⡁⠀⠀⠐⠀⢸⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⢹⣄⠀⠀⣸⠘⣷⡀⢹⡇⢹⠀⠈⡊⠀⠀⠈⠀⠀⠘
⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠹⣎⣡⠙⢦⡀⡟⣶⣿⣧⣼⠀⠸⡆⠀⠀⢆⡀⠀⠀⠀⠀⠈⠐⠢⢀
⠀⠀⠀⠐⡆⠀⡀⠀⠀⠀⠀⠀⠈⣻⣷⣤⣉⣓⣟⣉⣵⡇⠀⠀⢧⠀⡓⠒⠒⡄⠀⣀⡀⠀⠀⠀⠀⠙⢀
⠀⠀⣰⣇⠈⡏⠀⠀⠀⡐⠈⠀⣤⣇⠼⣟⣿⣯⣿⡋⡜⠀⠀⠀⠘⣆⠀⠀⠀⡇⠉⣄⠙⠋⠐⠒⠲⠄⠤
⠐⢠⠇⠸⡄⣇⠥⡀⠇⣘⠀⠐⠿⠿⣽⣧⣟⡗⣏⠟⠁⠀⠀⠀⠀⠘⣆⠀⠀⡇⠀⠀⠀⠀⠀⣌⡉
⣠⡏⠀⠀⢹⡇⠃⠐⠀⠄⠾⠂⢼⡆⣼⣙⡮⠟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⡁⠀⠀⠉⠈⠒⠂
⢸⠀⠀⠀⢀⠇⠀⠀⠀⠀⠀⠃⠈⠋⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠗⢂⡡
⠈⠓⠤⠔⠊⠀⠀⠀⠀⠀⠐⠆⠠⡶⠖⠀⠀⠀⠀⠀⠰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⢶⣤⡼⢃⠈
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠁⠉⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣀⣀⣀⣀⣡⣤⣔⣀⣤⣤⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢧⣼⣿⣿⣿⣿⣧⣼⡿⠿⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡔⠒⠛⡹⢉⠽⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠙⠋⣉⠤⠔⠊⠉⢣⠚⣩⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠀⢡⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢳⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⢦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢒⣫⣽⠿⡆⢹⡟⣻⣶⣭⣓⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣺⣵⠾⢻⣿⣿⣿⣿⣉⣻⣿⣿⣿⣯⡻⣮⡑⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢫⣾⣿⣵⣶⣾⣿⠿⢛⠂⢹⡙⠯⣙⠻⢿⣿⣾⣿⣌⢳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⣱⣿⣿⡿⠟⠋⠉⠀⠀⠁⠀⠀⠂⠀⠈⠣⡀⠉⠻⢿⣿⡎⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣿⢟⠁⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢢⡙⢿⣸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⢹⡄⠁⠋⠀⡰⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠈⣿⡟⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⣌⡙⠒⠤⠴⣁⣀⣀⡀⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡇⢀⣀⣸⠧⠟⠃⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠐⠬⢭⣒⠲⠦⠤⢤⣬⣍⣉⣉⣉⣉⣉⣉⣉⣉⣭⣭⣥⠤⠤⠶⠒⠋⡝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⢍⣀⠀⠡⠄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⢊⡩⢄⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠸⣿⣿⣷⣶⣦⣤⣤⣄⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣤⣶⡾⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡇⣿⣿⡿⠁⣶⢻⣿⣿⣿⣿⢛⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⢃⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣹⡇⠀⣿⣿⠟⣻⣿⠇⢾⣿⠃⣾⣿⢿⣿⣿⣿⣟⡟⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠸⣿⣧⠀⢿⡿⠀⣿⣟⠀⣿⡇⠰⡟⢀⣿⠋⣾⣿⣿⢡⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⣿⣿⣷⣾⣿⣶⣾⣿⠐⣿⡇⢿⣿⣶⣿⣶⣿⣿⡏⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⢹⣿⡏⢹⣻⡟⢿⣿⡷⢼⣿⡶⠿⡿⢿⢫⣿⣿⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⣿⣇⢸⣿⣧⠸⣷⡇⢘⣿⡇⢠⣿⡮⢸⣿⠇⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⣿⣿⣶⣾⣿⣦⣿⣧⣼⣿⣧⣷⣾⣶⣿⡿⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⢹⣿⣿⡄⢹⢿⣿⠍⢸⣇⠰⣿⢰⣿⣿⠇⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡘⣿⣿⡇⢸⡏⣿⠀⢼⣿⡀⠋⣾⣏⡿⢰⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⢻⣿⣷⠘⢧⣿⢰⣾⣿⣷⢤⣿⣹⠃⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠘⣯⣿⣴⣿⣧⣸⣇⣿⣬⣾⢧⡟⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠦⣍⣛⡛⠿⠿⣿⣿⡿⠿⠟⣡⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠒⠒⠒⠛⠉

⠀⠀⠀⠀⠀⢘⣳⣷⠀⠀⠀⠀⠀⠀⠀⢀⣶⠶⢤⣤⣤⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣠⣤⣶⣶⡆⢀⠀⠀⠖
⠀⠀⠀⠀⢰⣿⠎⢹⡇⠀⠀⠀⠀⠀⠠⢿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⢿⣯⡠⣿⡍⠉⢹⣿⣿⣷⣤⣾⣿⡀
⠀⠀⠀⠀⣾⣿⣷⣼⠀⠀⠀⠐⠽⠇⠀⠀⠉⠉⠹⣿⣿⣿⢿⣿⣿⣿⣿⣽⣿⣿⡟⠀⠀⠀⠘⣿⡇⠛⠿⠿⠾⢿⣾⠟⠉⠋⠉⠃
⠀⠀⠀⢠⣿⣛⣟⠵⡶⠃⢀⡀⠀⠈⠀⠀⠀⠀⠀⢹⣿⣿⣿⠉⠉⢹⣿⣿⣿⣿⠃⠀⠀⠀⣼⠟⠀⠀⠀⠀⠀⢸⠃
⠀⠐⠘⠋⠈⠉⠁⠀⠀⠀⠰⢇⠀⠀⠀⠀⠀⠀⢀⣸⡟⢙⡿⠀⠀⣸⡟⠀⢙⣿⠆⠀⠀⠀⠻⡇⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡳⠄⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠛⠄⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡤⠃⠀⠀⠀⠀⠀⠒⢤⠀⠀⠀⠀⠀⠀⠀⠀⠞⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣴⣯⢞⡜⠃⠀⠀⠀⠀⠀⣄⠀⠑⠄⠀⠀⠀⠀⣠⣾⠇⠀⠀⠀⠀⠀⠘⢄⠀⠀⠀⠀⢀⣀⣠⣤⣴⣾⣿⡩
⠀⠀⠀⠀⣀⣤⣴⡿⣿⣷⢉⡆⠀⠀⠀⠀⠀⠀⣿⠀⠤⠈⢶⡄⠀⢰⣿⣿⠶⠂⠀⠀⠀⠀⣤⠏⠐⢤⣾⣿⣿⣿⣿⣿⡟⠋
⠀⠀⠀⣾⣿⢿⢻⡞⢛⣷⡌⣥⠀⠀⠀⣴⣦⣀⡏⠀⠀⠦⠈⠀⠀⢸⣿⣩⡴⠖⠀⠀⠀⠀⠳⠶⠶⠟⣫⠷⠉⠉⠁⠀⠉⠚
⠀⠀⠀⢻⡇⠀⠸⡇⢀⢹⠃⠀⠀⠀⢠⣅⣿⣿⣇⠀⠀⠀⠀⠀⠠⠾⠏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⢸⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣏⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣦
⠀⠀⢀⣄⣽⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⢁⡝⡛⢫⡭⠿⠀⠀⠀⠀⣠⣶⣾⣶⣶⣶⣶⣦⣶⡗⠀⠀⠀⠀⠀⣤⡀⣠⣾⣿⣿⠏
⠀⠀⠸⠇⠿⠀⠀⠀⠀⠀⠆⠀⠀⠈⠀⢸⣿⡁⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⠿⠿⠿⣿⣿⠋⠀⠀⠀⠀⠁⠀⠈⣰⣿⣿⣿⣿⡟⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠘⠹⡋⠁⠀⠀⠀⠈⠹⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠱⠄⠀⠀⠀⠚⠀⠰⠄⠀⠀⠀⢀⣿⣿⣻⣿⣿⣿⣿⠇
⠀⠀⠐⠀⠀⠀⠐⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣆⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣟⣿⣿⣿⠏
⠀⠀⠀⣠⣤⣶⣶⣶⣶⣶⣶⣶⣶⣤⣄⣀⠀⠀⠀⠰⠟⠛⠉⢷⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠟⠁
⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣻⣿⣿⣶⣆⠀⠀⠀⠀⠀⠋⠈⢿⡿⠟⠒⢶⠛⡿⠶⣦⣀⡐⠿⠻⠿⠟⠟⠁
⠀⠀⢿⣿⣿⣿⡟⢿⣿⣿⣿⣸⣿⣿⣿⣿⠿⢿⠿⢷⣄⠀⠀⠀⢠⣴⣿⣦⣀⡀⣠⣧⣀⠀⣾⠁⣿⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⢸⣿⣿⡿⠇⠠⠿⠿⠿⣿⢿⡟⠉⠉⠑⠘⠦⠼⠟⠀⠀⠀⢠⡟⠁⠉⠉⠉⠉⠀⠁⠹⡏⠀⠿⠀⠀⢠⣤⣾⣟⣿⣿⣿⣷⡀
⠀⠀⣿⣿⡟⠀⠀⠀⠀⠀⠀⢃⢻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠀⠀⠀⢰⢷⠀⠀⠀⠀⠀⠁⣿⠿⠿⠿⠿⢿⡋
⠀⠀⠀⠙⠛⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡆⠀⠀⠀⠀⠀⠀⠀⡸⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠒⠊⠉⠉⠉⠑⠢⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢊⣥⠤⠞⠓⠒⠒⠚⠓⠢⢄⣙⢢⠀⠀⠀⠀⡠⠐⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⣫⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢃⢀⡴⠋⠀⢀⡠⠄⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣉⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡈⠋⢀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⠲⠤⡄⣤⠠⠤⠒⠊⠁⠀⡼⢁⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢡⠎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡁⠋⠀⠀⡀⠠⠄⠀⠀⠠⠤⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⣀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠲⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆
⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆
⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⢛⠻⢷⣄⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⡀⠐⠄⣴⠋⣰⠊⠉⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⢠⠃⠀⠀⢹⣇⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠈⠀⣸⠇⠀⢇⡀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠘⢦⡀⢀⠄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠤⠤⠬⠷⣖⠊⢸⠀⠀⠀⠀⠀⠀⠀⣸⠀⢀⠠⠄⠐⠂⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠘⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠤⠤⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⡸⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠑⠄
⢀⡤⠤⠀⠶⣀⠠⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⢄⣀⣀⣀⣀⠴⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠑⠠⡰⠆⠠⠤⣄
⣏⠀⠀⠀⠀⢸⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠀⠀⢸⠀⠀⠀⠀⢈
⠈⠁⠀⠀⠐⠊⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠀⠀⠈⠒⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⠢⢍⠒⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢓⡰⢲⠶⢿⢷⡠⢌⠁⠀⠀⠀⠀⠁⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠴⣒⡁⠒⠈⠙⣢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠋⢰⠃⠀⠇⡎⠎⠞⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⢞⣠⡿⢻⠛⢷⠚⣙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠈⠒⠚⠀⠗⠁⣀⡠⠤⠐⢒⡈⠀⠀⠀⠘⠢⢀⡀⠀⠀⠈⢸⣰⠸⡀⠀⡇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢄⣀⠤⠒⢊⠩⠐⠂⠉⠀⠀⠀⠀⠀⠀⠀⠁⠒⠀⠍⣐⠢⠄⣉⠀⠉⠊⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠨⠙⠒⠤⠴⠋
⣶⣶⣶⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣽⣯⣭⣬⣭⣬⣭⣭⣥⣬⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣶⣾⣿⣶⣾⣿⣶⣶⣶⣾⣶⣶⣶⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠠⠂⠀⠉⠉⠉⠉⠉⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠀⠀⡉⠁⠀⠀⠀⠓⠀⠀⢀⠙⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⢰⠀⠘⠛⡁⠀⠀⠈⠻⠀⠀⠀⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⡆⠀⠀⠸⡀⢀⡀⠀⠀⠀⠀⢀⠀⠀⡼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢷⡘⢄⠀⣈⢌⣀⣀⡤⠞⢀⣼⢦⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠑⣌⠾⠀⠀⠀⡥⢊⠔⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠀⠀⠈⡇⠀⠀⠀⠗⠁⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⢇⠀⠀⢇⠀⠀⡠⠁⠀⠀⠀⠀⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢹⠀⠈⢆⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⢁⡇⠀⠀⠈⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠎⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡌⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠈⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞⠀⠀⠀⢀⠀⠀⠀⠀⡄⠀⠀⠀⡾⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⠱⡀⠀⠀⢡⠀⠀⠀⣇⠀⠀⠀⠀⢀⠎⠱⡀⠀⠀⠸⡄
⠀⣠⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⠀⡜⠀⠁⠀⠀⠘⡀⠀⠀⢸⠀⠀⠀⣴⠁⠀⠀⠱⡀⠀⠀⢇
⡎⠀⠀⠀⠀⠀⠀⠀⡇⠀⠆⠀⠀⢰⡇⠀⠀⠀⠀⠀⠇⠀⠀⠘⡄⠀⡜⢹⠀⠀⠀⠀⠙⡄⠀⠸
⡇⢰⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⠘⡄⠀⠀⢧⠈⠀⡼⠀⠀⠀⠀⠀⠘⡄⠀⢇
⠳⡀⠳⢄⡀⠀⠀⣀⡇⠀⠁⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠘⣖⣋⣄⡀⠀⠀⠀⠀⠀⠱⡀⠸⡄
⠀⠙⠢⣄⣈⠉⠉⠀⣘⡦⠀⡀⠐⠺⣄⣀⣀⣀⣀⣀⣀⣀⣧⠀⠀⠹⣄⡀⢨⡣⠀⠀⠀⠀⠀⠘⡄⠈⠒⠢⣄
⠀⠀⠀⠀⠀⠉⠉⠉⠀⠑⠲⠷⠤⠦⠃⠀⠀⠀⠀⠀⠀⠀⠹⣀⣰⠀⢈⣹⠉⠀⠀⠀⠀⠀⠀⠀⠘⢄⣀⣇⢈
⣶⣶⠶⠶⠶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠛⠉⠓⢄⠀⠀⠀⠀⡰⠚⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⢸⠀⠀⠀⢸⡃⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⣄⣀⣤⡎⠀⠀⠀⠀⢳⣄⣀⣠⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠒⠒⠒⠢⠤⣄⡀⠀⠀⠀⠀⠀⠱⡄⠀⠀⢠⠏⠀⠀⠀⠀⠀⢀⣠⠤⠔⠒⠒⠒⠒⠒⠢⠤⣄⡀
⠀⠀⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⢳⠀⢀⡏⠀⠀⠀⠀⣠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⣄
⠀⠀⠀⠀⢀⡜⠁⠀⠀⠀⣀⠴⠒⠉⠉⠉⠉⠉⠑⠲⢄⡀⠈⠳⡀⠀⠀⠈⡆⢸⠀⠀⠀⢀⠜⠁⠀⣠⠖⠊⠉⠉⠉⠉⠙⠒⠦⣀⠀⠀⠀⠈⢢⡀
⠀⠀⠀⢀⡎⠀⠀⠀⢀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⠀⠹⡄⢀⣠⠧⠤⣄⡀⢠⠏⠀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀⠀⠀⠀⢱⡀
⠀⠀⠀⡼⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣠⠟⠁⠀⠀⠀⠀⠉⠻⣀⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢧
⠀⠀⠀⡇⠀⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⣀⠀⠀⠀⠀⣀⠀⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠀⠀⠀⢸
⠀⠀⠀⣇⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⢯⡽⠀⠉⠀⠀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⣸
⠀⠀⠀⢹⡀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⢀⡇
⠀⠀⠀⠀⢣⡀⠀⠀⠙⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠒⠲⠤⠤⠤⠤⠤⠤⠖⠒⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠋⠀⠀⢀⡜
⠀⠀⠀⠀⠀⠑⢄⠀⠀⠈⠑⠦⢄⣀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠴⠊⠁⠀⠀⡠⠎
⠀⠀⠀⠀⠀⠀⠀⠙⠒⠦⣤⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⣿⠒⠲⠦⠤⠤⠤⠤⠴⠖⠒⡛⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⣤⠴⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠈⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡗⠒⠤⠤⠤⠤⠤⠤⠒⢺⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⢳⡀
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⢷⠀⠀⠀⠀⠀⠀⠀⠀⡿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠸⡄⠀⠀⠀⠀⠀⠀⠀⠀⣠⠜⠁⢨⣷⠲⠤⠤⠤⠤⠖⣾⡅⠈⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢧⠀⠀⠀⠘⠦⣀⣀⣀⣀⡠⠴⠚⠁⢀⡴⠃⠸⡄⠀⠀⠀⠀⢀⠇⠈⢦⡀⠈⠓⠦⣄⣀⡀⣀⣀⠴⠋⠀⠀⠀⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⠱⡦⠤⠤⢴⠎⠀⠀⠀⠙⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠲⠤⣤⣀⣀⣤⠤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠙⣄⡤⠋⠀⠀⠀⠀⠀⠀⠈⠙⠲⠤⣤⣀⣀⣤⠤⠖⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢲⠖⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠈⠭⠑⡄
⠀⡘⠩⠛⡃⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⡀⠰⠌⠈⠀⠀⠀⠀⠀⠀⠀⡔⠁⠀⠀⠀⠀⠀⢀⠀⠈⠁⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠓⠀⠀⠀⠀⠈⠄⠘⠁
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠰⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠
⠀⠀⢀⣄⡀⣷⣄⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡄⣠⣤⠐⠂⠀⢀⣄⣀⣄
⠀⠀⠈⠉⠁⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣠⡀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉
⠀⠀⡤⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄
⠀⠐⠃⠉⠀⠀⠀⠀⠀⠀⠔⣒⠠⡄⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⡀⠤⣀⠢⠙⠆⠀⠀⠀⠀⣀⠀⡎⣿⠿⠀⢀⡀
⠀⠀⠘⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⠀⠀⠀⡄⠻⠿⠁⠀⡌⠀⠀⠀⠰⠁⠀⠀⠈⠀⣶⡋
⠀⠀⠀⢀⠀⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⠄⢰⠍⠂⠀⠀⠀⠙⠢⠤⠈⠀⠀⢂⠀⠀⠀⠀⢀⣾⢠⠀⡀⠉⠂
⠀⠀⠀⣦⣥⣧⡄⠀⠀⠀⠀⠰⠄⠂⠐⠄⠀⠀⠀⠀⠉⠠⠜⠅⠉⠀⠀⠀⠀⠶⠀⠅⠇⠈⠉⠀⠀⠀⠀⠹⠈⠑⠃
⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⢤⢄
⠀⠀⠐⡋⠟⠤⠀⠀⠀⠀⠐⢀⡉⠀⠀⠀⠀⠀⠀⠀⢀⣩⣍⢃⠀⠀⠀⠀⠀⠒⠂⢊⡆⠀⠀⠀⠀⢀⠄⠠⠂
⠀⠀⠻⠟⠃⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡈⢀⠀⠀⠀⠀⠀⠠⠠⡷⠀⢀⠀⠀⠉⠀⠀⠀⣀⠘⠂
⠀⠀⠀⠀⠛⠃⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⡐⡀⢂⢔⡄⠀⠀⠀⠀⠁⠐⡀⠐⠀⠀⠀⠀⣞⠉
⠀⠀⠀⠐⠁⠰⠁⠀⠀⠀⠈⠀⠀⠃⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠈⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⢤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢠⡦⡤⠀⠀⠀⠀⠀⠀⠐⣢⠢
⠀⠀⠀⠁⠿⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⢁⠀⠀⠀⡀⠀⢄
⠀⠀⠀⠀⠈⠀⠀⡀⠀⠀⠀⠀⠀⣀⠁⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⢀⠀⠀⠀⠄⠁⠀⠀⠀⠀⠀⠀⠀⠰

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠶⠶⠶⠶⣦⣀⣠⠴⠶⠶⠶⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠈⠛⠉⠀⠀⠀⠀⠀⠉⠛⠶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⣠⡀⢀⡤⠖⠻⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣬⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⣠⠟⠹⣶⠋⠀⠀⠀⠻⣄⠀⢀⠀⠀⢠⡴⢰⣾⣇⡼⢧
⠀⠀⠀⠀⠀⠀⠀⡰⠋⠀⠀⢰⡄⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣾⡧⢠⣌⠳⠖⠋⠁⠀⠀⣦
⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⢠⠟⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢀⣿⠏⢀⡀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢠⠟⠀⠀⠀⢹⣷⣦⡄⠀⠀⠀⠀⠀⠀⠀⠈⠁⢸⣿⣧⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠀⣇⠸⡀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⣤⣤⣀⡀⠀⠘⠛⡇⠀⠀⠀⠈⡇
⠀⠀⠀⠀⠀⠀⣿⠀⠀⠘⢷⣿⡄⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⠀⠈⠙⠿⠁⠀⠀⢀⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠘⢧⡀⠀⢸⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢶⡄⢻⣆⠀⠀⠀⠀⠰⣤⡀⢀⣠⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⠀⠀⠀⠀⢳⣄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠙⢷⡀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⣀⠀⢠⣀⠀⢉⡻⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣄⣙⣳⣤⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠴⣾⣇⠀⠀⢿⣷⣤⣽⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡟⠉⠉⠛⢳⡶⠤⠶⠚⠛⠉⠀⠀⠘⣏⠀⠴⠤⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣷⣤⣀⣀⣼⣧⣀⣀⣀⣀⣀⣀⠤⠚⠉⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠛⣿⠃⠀⠀⣿⡉⣹⠏⠉⠉⠉⠉⢁⣀⠀⣼⠿⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⣼⠏⠀⠀⠀⠈⠙⠉⠀⠀⠀⠀⢰⡏⢉⣿⣿⠀⠀⠙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⣼⢋⡴⠲⡄⠀⠀⣠⠖⠲⡄⠀⠀⠀⠉⠊⢹⡟⠀⠀⠀⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣼⠃⠈⠳⠞⠃⠀⠀⠛⣤⠾⠁⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⠲⣄⠀⠘⢷⡀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⠀⠀⣼⠋⢙⡆⠀⠀⠀⠀⠀⠀⠘⢧⡴⠇⠀⠀⠀⢳⡀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣀⡀⠀⠀⠙⠓⠋⠀⠀⠀⢠⠶⢶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿
⠀⠀⠀⠀⠀⠀⠀⢀⣿⡅⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠳⣆⡸⠃⠀⠀⠀⠀⢀⣀⠀⠀⠀⠹⡆
⠀⠀⠀⠀⠀⠀⠀⢸⡏⠙⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣟⢀⡷⠀⠀⢀⣼⡀
⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢴⣶⠀⠀⠀⠀⠀⠀⣤⠶⣄⠀⠉⠉⠀⠀⠀⠺⣅⣿
⠀⠀⠀⠀⠀⠀⢸⡿⢠⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣤⠿⠀⠀⠀⠀⠀⠀⢀⣀⢼⡇
⠀⠀⠀⠀⠀⢠⡟⠀⠙⣿⡽⣟⠓⣂⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣤⣤⣤⣤⣤⣤⣤⣶⠾⣿⠛⠊⢳
⠀⠀⠀⠀⢠⡎⠀⠀⢰⡟⠀⡏⠀⠀⠀⠀⠈⠘⡿⢻⡄⠈⠉⠉⠉⠛⠛⢻⣿⠹⠉⢻⠀⢹⠀⠀⠘⡆
⠀⠀⠀⠀⠘⢧⣄⡀⠸⡇⠀⠁⠀⠀⠀⠀⠀⠀⣿⠀⡇⠀⠀⠀⠀⠀⠀⢸⡏⠀⢃⢸⡇⠈⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠉⠛⠷⣦⣤⣀⣀⣀⠀⠀⠀⠀⠹⠀⠇⠀⠀⠀⠀⠀⠀⠘⠁⠆⠀⠀⠀⣀⣠⣤⡶⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠙⠿⠛⠿⠶⠶⠶⠶⠾⠿⣶⣶⣶⣶⣶⣶⣶⠶⠾⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡆⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⢀⡀⣠⡀⢻⣄⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠙⠁⠀⠈⣻⠃⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠀⠀⠀⢸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠀⠀⣺⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⣀⣀⡜⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠉⢀⣨⠟⠛⠷⣤⣀⣀⣀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣅⣀⣀⢀⣀⣈⣉⣉⣬⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠉⠉

⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⣿⠿⣿⠿⠿⡿⠿⠿⢏⠹⠿⢿⡿⠿⠿⠿⣿⣿⣿⣿⡟⣻⣛⠛⠛⠛⣛⡛⣛⢻⣿⣿⠛⠛⠛⠿⠿⠟⢻⠛⠛⠻⣿
⣿⣷⣿⣶⣶⣷⣾⣶⣾⣶⣾⣶⣷⣶⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⠀⣤⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠟⠛⠛⠛⠛⠛⠛⠿⣶⣄⡀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠃⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠁⣰⣿⠏⠀⠀⠀⣴⣾⣿⣦⠀⠀⢠⣾⣿⣧⠀⠀⠀⠀⠀⠀⠘⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⢰⣿⡏⠀⠀⠀⢸⣿⣿⣿⡿⠀⠀⢿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠸⠿⠿⠛⢀⠶⠄⠘⢿⣉⠀⠀⠀⠀⠀⢠⠃⠀⣿⡟⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣄⣸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠃⢀⣼⡿⠁⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠁⠈⣿⠿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠟⠀⠐⡡⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣀⣴⡟⠋⠀⢀⣤⠟⠀⠹⣿⣿⣿⣶⣶⣶⠆⠀⣰⠀⢀⡷⠀⢸⣶⣶⡾⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⢿⣶⠿⠋⠉⣿
⠀⠀⠀⠀⢀⣠⣿⡋⠉⠀⢠⣾⣿⣄⣀⣠⣀⡙⠻⠿⠿⣿⣿⣦⣤⣿⣤⣼⣿⣤⣾⣿⡋⠀⢀⣀⣠⡾⠿⣦⡀⠀⠀⠀⠀⠀⣴⡿⣿⠀⠀⠀⠀⠀⣰⡟
⠀⢀⣴⠾⠛⠉⠉⠉⠻⣆⣸⡟⠀⠉⠉⠙⠛⢿⣦⡀⢀⣼⣿⣿⣿⣿⣿⣧⣤⣿⣿⣿⣿⣿⡿⠿⠿⣧⠀⠘⢷⣦⣤⣤⣤⣰⣿⠀⠘⠷⠶⣶⡶⠟⠋
⠀⣾⡇⠀⢀⣤⣤⡀⣀⣿⠟⠁⠀⠀⠀⢠⡄⠀⠙⢷⣿⣟⠛⠛⠛⠻⣿⡟⠹⣿⣛⣉⣀⣀⣠⣤⣶⣿⣶⣦⡄⠉⠁⢨⡿⠟⠃⠀⣠⣶⡾⠟⠀⠴⠂⠄
⠀⠈⠛⠛⠋⠉⠈⠙⡛⠁⡀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠿⠿⠿⠷⠶⣾⣿⠃⠀⣿⡛⠉⠉⠉⢉⣉⣉⣿⠇⠹⣿⣀⣴⣿⣦⠀⢀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠀⠁⠀⠀⠀⢠⠀⠻⢷⣦⣤⣄⣠⣤⣴⣿⠀⢸⣿⣟⡛⠛⠻⠿⣿⣿⡁⢀⠆⠙⠛⠋⠀⠹⣷⡿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠟⠉⠉⠉⣉⣽⣿⡄⢸⣿⡛⠿⣿⣷⣶⣤⣽⣿⢠⡇⠀⣀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢻⣿⡿⠟⢋⣩⣿⣷⣿⣿⣿⣦⣄⡙⢿⣯⡉⠀⠈⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠸⣯⣴⡾⠿⣿⣿⣁⣠⣿⣿⡿⠿⢿⣷⣿⡇⠀⠀⠀⠀⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠛⠿⢿⣶⣤⣶⠟⠙⠷⣦⡤⡈⠙⣿⠟⢁⣤⣶⠟⠉⢙⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣸⡷⣦⡀⠀⠀⠈⢀⣀⡀⣠⡿⠀⠀⣠⣿⠀⠈⢻⣇⠀⠀⢼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠛⠁⢸⣿⣤⣤⣾⠿⠛⠛⢿⣤⣴⠾⠋⠙⠷⣦⣤⣿⣦⠀⠈⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣄⡀⠈⠻⣿⣭⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⡇⠀⠀⠈⢻⣧⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢻⣦⠀⠈⠙⢻⣦⣤⣄⡀⠀⠀⠀⠘⠟⠀⠀⠀⠀⣿⡷⠾⢿⣷⣾⡏⠀⣯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⡾⠛⣿⠛⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⡄⠀⢈⣩⣿⠆⡼⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠙⢿⣿⠃⢰⣿⣶⠾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣾⠟⠁⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⠁⠀⣼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠁⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⡙⢷⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣧⣤⣄⡀⣻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠉⢳⡄⠈⢹⣿⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠿⠿⢿⣧⣀⠀⠙⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠿⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡛⠘⢏⠀⠀⠀⠀⠄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⠀⠀⠀⠡⠀⠈⠙⠢⢤⣀⠰⠜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡄⠀⠀⠀⢱⣄⠀⠀⠀⠙⣷⣾⠇⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⣠⡆⢀⠤⠀⣐⣈⡀⠐⢿⡇⠀⠀⠈⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠙⠰⠖⠉⠒⠓⠳⢰⠋⠉⣩⡶⠛⠁⠙⢦⠀⠀⣹⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠀⢸⠃⠀⠀⠀⠀⢨⣷⣾⣿⣯⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠤⠤⠤⠞⠃⠀⣼⠀⠀⠀⠀⠀⡸⠃⢸⡥⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠤⠞⢃⣀⣀⣤⠶⠛⠁⠀⣠⣵⢀⣾⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⣴⠖⠲⠤⠔⠒⠒⠒⠐⠂⣀⡤⠖⠛⠉⠉⠁⠀⢀⣀⣠⣤⣿⣷⣟⡻⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⠀⠀⠉⠢⣄⡀⠀⠀⠀⣠⣴⠟⠁⠀⠀⢀⣀⣤⣴⣾⡿⢟⡭⠏⠹⡋⣋⢱⣻⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠀⠀⠀⠀⠈⠙⠛⠛⠛⠻⠻⠟⠗⠖⠻⠟⠻⠋⠉⠁⢀⠈⢰⣞⣴⣷⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠹⡦⣤⣄⣠⡄⠀⣀⠀⣀⢀⣀⣀⣁⣘⢠⣂⣠⣤⣼⣾⢿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⢣⢹⡄⢹⡟⠈⢿⠛⣿⠛⢻⡟⢻⠟⢻⡟⢻⡏⢩⣇⣸⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠘⡎⣧⠀⣿⠀⢸⡀⢿⠀⣸⡁⢸⠃⢼⠀⢺⠇⢺⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⢠⣸⡆⢿⠂⠸⡇⢸⢢⣼⡇⢸⠂⣿⡀⣿⢓⣿⣿⣿⢿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠸⣿⣇⢺⣧⠀⣷⢾⡧⣿⣧⣿⢲⣿⢿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣶⣷⣤⣤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣾⣿⡸⣿⣿⡧⣿⡎⣿⣾⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣧⣤⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢜⣿⣿⣿⣧⣿⣿⣷⣿⣇⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣽⣶⠧⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢿⣿⣾⣿⣿⣿⣯⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠾⢟⡛⠓⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠿⠿⠿⠿⠿⠿⠛⠿⠟⠻⠛⠛⠙⠉⠉⠉⠁



⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⢶⠶⠶⠶⢶⣶⡶⠶⠶⢶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⣄⠀⣾⠏⣀⣤⠞⠋⠉⠉⠉⠉⠉⠛⠛⢶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣻⣿⣿⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠶⠞⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣦⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼⢿⣄⣀⣀⠀⢻⡀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣿⣀⣿⣤⡟⠿⠁⢼⡟⠛⠛⣧⣀⣿⣆⠀⠀⠀⠀⢵
⠀⠀⠀⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⣃⣼⠻⠟⠛⠋⠀⠀⠀⠀⠀⠀⠀⠛⠛⠻⠟⡀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⣿⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⡂⢠⠟⢸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⣴⠄⠀⢸⠃
⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⠀⣿⣷⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠙⡄⢀⡎
⠀⠀⠀⠀⠀⢹⣸⣇⠀⠀⠀⠀⠀⠀⢠⡿⠁⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⢿⠇⠀⠀⠀⠀⠀⠀⠀⣼⠆⣸⠃
⠀⠀⠀⠀⠀⠀⣿⢸⣇⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡴⠋
⠀⠀⠀⠀⠀⠀⢹⣆⠹⣦⠀⢀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠹⣧⠈⠀⠿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⡀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣀⣀⣠⡾⠃⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⠼⣷⣄⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⢀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠿⠛⠳⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⣿⣦⣤⣄⣀⡀⠀⠀⢀⣀⣠⣤⠞⠛⣿⡀⠀⠀⠀⠈⠙⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠟⠙⢿⣄⠈⠉⠙⠛⠚⠉⠉⠁⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠁⠀⠀⠀⠀⠛⠿⠶⢶⣤⣀⣀⣀⣤⡶⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⣠⡛⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣶⣾⣿⡿⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠻⠶⣄⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡿⠋⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡹⠟⠶⣦⣬⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⠀⠀⠀⠙⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⢹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⣠⡿⠀⠀⠀⠀⢾⡷⠆⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣴⣾⣿⣿⣿⣧⣄⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢀⣀⡀⢠⣴⠀⢶⣾⣿⣶⣦⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣦⣴⣶⣶⣦⣾⡿⠿⠛⠉⠁⣿⣿⣟⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄⠈⢻⡃⢸⡇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠶⠶⠄⢦⣤⠤⠴⠾⠋⠁⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣤⠶⠶⠶⢶⣤⡀⠀⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⣠⣤⠚⠛⠛⠛⠛⢳⣄⣼⠋⠁⠀⠀⠀⠀⠈⠙⣶⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⢾⣧⣄⠀⠀⠀⠀⠀⠀⣨⣿⣤⣤⣀⣤⣤⣤⡤⠶⢛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⢿⣿⣶⣶⣾⡛⠾⠉⠈⠙⠛⠋⠶⠶⠶⠶⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣶⣿⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⣶⣶⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢿⣿⡿⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⢿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣏⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣾⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⣶⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⡟⢛⡻⢿⣿⠇⠀⠀⠀⠘⣿⣿⠿⠿⢿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⣿⣿⣿⡿⠀⠀⠀⠀⠀⢿⣷⣿⣿⣶⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⡿⠛⠀⠀⣄⣠⡄⠀⠈⠻⢿⣿⣿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣈⣉⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣶⣦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣾⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⠀⠀⠀⠀⡴⠊⠀⠀⠀⠀⠈⠛⠿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⡿⠋⠁⠀⠉⠉⠐⠢⡀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⢀⠀⠀⠀⠀⠀⠀⣀⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠠⣄⡠⠊⠁


⣶⣶⣶⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⡶⢶⠶⠶⠶⠶⢶⣶⠶⠶⣶⠶⠶⠶⠶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⡞⢶⣣⠀⠀⠀⠀⠀⠠⣖⠤⠒⠒⠒⠒⠒⠒⢖⣲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣿⢿⡀⠠⠌⠳⣄⠀⠀⠀⠀⠈⠃⠘⠀⣬⣥⠈⠃⠸⢤⡤⠤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⠈⢷⡀⠀⠀⢈⣷⡄⠀⠀⠀⢧⡀⠀⠜⠅⠀⠀⣀⡀⢠⡄⠉⠀⠰⠟⠙⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣰⣿⣿⡉⠳⣄⠙⣦⣤⣿⣷⡇⠀⠀⠀⠀⠉⢲⡖⠒⣈⠁⠙⠁⣈⠀⠘⠛⠀⣤⠘⠓⠀⣷⡀
⠀⠀⠀⠀⢀⣠⠖⠹⡏⠁⢿⡀⠹⣇⠈⠳⣄⠈⠳⣇⠈⠉⠉⠀⠀⠀⠀⠀⠀⢸⡁⢀⠉⢠⣤⠀⠛⡀⠸⠇⢀⡀⠀⣶⠄⢴⢌⣑⡒⢠⣶⣶⠄
⠀⠀⠀⣰⡟⠹⣆⠀⢻⡀⠘⣧⠀⠙⣦⠀⠙⢦⡀⣣⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⡏⢳⠀⠞⠉⠉⠉⠉⠑⠺⡏⢸⠉⡄⣇⠀⠈⠉⠉⠉
⠀⠀⢰⠅⣿⣄⡙⢦⡀⢳⡀⠈⣧⣀⠈⠷⣤⣤⠉⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣶⠀⢸⣴⠇⠀⠀⠀⠀⠀⠀⢹⣾⠀⣷⡇
⠀⠀⠀⠀⣇⢹⢹⡈⡟⠉⠉⠀⠀⠉⢻⡀⡏⢸⢦⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠉⣀⢀⣀⣀⣀⣀⣠⣌⣡⣀⠈⣡⠴⢤⡀
⠀⠀⠀⠀⡟⣿⢸⣛⡇⠀⠀⠀⠀⠀⢨⣛⡇⠸⣦⡇⠀⠀⠀⠀⠀⣠⢤⣄⣶⢲⣮⠶⠭⠿⠾⠗⠚⠒⠛⠚⠓⠚⠓⠉⠉⠁⠈⠀⠉⠙⠓⠒⠒⠲⠦
⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠀⠀⠈⠙⠁⠀⠉⠁⠀⣠⠔⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣞⣙⣎⣱⣉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠋⠉⠉⠉⠉⠓⢦⡉⠒⠒⠒⠒⠲⠦⠤⢤⣄⡀⠀⣀⠀⠀⣠⣤⠤⠤⠤⠤⠤⣤⡀⢠⣄⠀⣤⠴⠶⠶⠶⠶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⣀⠀⢀⣀⣀⣀⣀⠀⠀⠙⣆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣤⡏⢧⡼⠀⠀⠀⠀⠀⠀⠀⠀⢧⠏⠸⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠈⠳⠍⣤⠀⠄⡀⢠⠉⡿⠃⢸⠀⠀⠀⠀⣀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣞⠝⠶⠒⠒⠢⢤⣠⢤⡀
⠀⠀⠀⣠⠴⠒⠛⠋⠉⢹⡇⠀⠘⡄⠀⠘⢿⠃⠀⠀⡇⠀⢸⠀⠀⢀⡠⠟⠛⢧⠀⠀⠀⠀⠀⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⣣⣏⡀
⠀⣠⢾⠁⠀⠀⠀⠀⠀⠀⢳⡀⠀⠑⠤⣈⣁⣁⡤⠞⠀⢠⠟⠀⣴⣋⣀⠀⠁⠘⡿⡆⠀⢀⡖⣋⠉⠀⠀⠀⠀⠀⠃⠀⠀⠀⠉⠉⠉⠉⠓⠲⢄⠀⢀
⣰⡇⣪⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⣀⣀⣀⣀⡤⠖⠁⠀⠀⠀⠀⠙⣧⠶⠆⢻⣳⡄⢸⡄⠋⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡼
⠉⠁⣵⠀⡴⡄⢀⣤⣤⣤⣤⣤⣤⣄⠀⡬⡍⢹⡃⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣾⡿⣆⠀⠳⣄⣀⠈⠁⢀⣀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⢸⢰⠁⡇⣿⠀⠀⠀⠀⠀⠀⢹⠀⠇⢇⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠀⠈⢟⣻⡀⠀⠀⠈⠙⡏⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿
⠀⠀⠸⠞⠀⠣⠇⠀⠀⠀⠀⠀⠀⠘⠾⠀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⢸⣭⣇⠀⠀⠀⠀⡇⠀⢀⡀⠀⠀⡤⠦⠤⠤⣤⠀⠀⣀⠀⠀⡇
⠀⠀⠀⢀⣠⣤⣄⢀⣀⣀⣀⣀⠀⢀⡠⠤⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢠⣄⡷⢼⡆⠀⠀⠀⣇⠀⣸⢻⠀⢀⠇⠀⠀⠀⢸⡀⣸⢿⠀⢀⠇
⠀⠀⢰⠋⠀⢀⡼⠉⠀⠀⠀⠈⢹⣏⠀⠀⠀⠈⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠈⠁⢻⡒⣇⠀⠀⠀⠉⠉⠁⠈⠓⠋⠀⠀⠀⠀⠀⠉⠁⠈⠉⠉
⠀⠀⡿⠀⠀⣼⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⢸⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⡀⠀⢹⣿⡿⠋⣭⣭⡉⠉⠉⠉⠋⢿⣿⡗⠲⣦⡀
⠀⠀⠹⣄⠀⢿⠙⠂⠀⠀⠉⠛⠀⠼⢤⡀⠀⣀⡼⠛⠓⠶⢤⡀⠀⠀⠀⠀⠀⠸⣿⠷⠀⠀⣤⡀⠀⠻⠟⠃⠀⣶⡆⠀⠀⢀⣶⣦⠀⢻⡀
⠀⠀⠀⣈⠉⠙⣆⠀⠀⠀⠀⢀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⢿⠀⠀⠘⠿⠟⠀⠰⣾⡦⠀⠈⠀⢰⣶⡄⠉⠁⢀⠾⡇
⠀⠀⠸⣍⠳⠶⠋⠀⠀⠀⣠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⠀⢸⢀⡶⡄⣾⠿⠤⠤⠬⠤⠶⠶⠖⠒⠻⡆⣴⡆⡜⠀⣧
⠀⠀⠀⠈⠑⠲⠤⠴⡖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠧⠀⠀⠀⣼⢸⠇⣇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⡇⡇⡇⠀⠛⠂
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⡄⠀⠀⣀⣀⣀⣠⣀⣀⡀⠀⢀⡀⠀⢸⠀⠀⠀⠀⣿⢸⠀⢻⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⡇⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⢳⠀⠀⡏⠀⠀⠀⠀⠀⢳⠀⢸⡇⠀⢸⠀⠀⠀⠀⢹⢸⠀⠸⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⢸⢠⣷⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⢸⢸⠀⠀⡇⠀⠀⠀⠀⠀⢸⠀⢸⡇⠀⣸⠀⠀⠀⠀⢸⣾⠀⠀⣇⣧⠀⠀⠀⠀⠀⠀⠀⠀⢸⣸⠗⡇⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠋⠈⠓⠚⠁⠀⠀⠀⠀⠀⠘⠒⠋⠘⠒⠃⠀⠀⠀⠀⠀⠁⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠀⠈⠋⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠘⠀⠀⠘⠁⠀⠀⠀⠐⠀⠂⠒⠀⠃⠐⠀⠒⠐⠐⠂⠀⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⡀⠀⠀⠀⠀⡠⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠤⠤⣄⠀⡼⡇⠀⡔⠄⢸⠀⠀⣠⠾⡀⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠈⣿⠀⢣⠀⠈⠉⠉⢉⡟⠁⠀⢈⣇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣠⠀⠀⠀⢿⡆⠈⢧⡀⠀⣰⠋⠘⠦⣄⡼⠀⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠸⡜⢆⠀⠙⢦⡑⠦⣄⡀⣰⠃⠀⠀⢈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡏⠀⠀⠀⠀⠀⠄⠙⢎⡓⢤⣀⠉⠓⠲⠮⠥⠶⠒⢛⣩⠼⣇⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⢬⣙⣒⣒⣒⣒⣺⠯⠭⠤⠔⠋⠁⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⢰⠖⠚⠲⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⣠⠤⠖⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⢀⠤⢤⣀⠀⠀⠀⠀⠀⢸⠀⠀⢳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣳⡤⠄⣀⡀⠀⠀⠀⠀⠘⠆⠴⡳⠀⠀⠀⠀⠀⠀⠈⡷⠀⠀⠀⠀⣸⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠀⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⢀⣠⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⢠⠇
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⢀⠔⠉⠀⠀⠀⠀⠉⠳⡄⠀⠀⠀⠀⠀⣠⠏⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆⠀⣀⡤⠞⢻⠀⣠⠃
⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⣏⣿⠀⠀⢻⠀⡿⢒⣒⠲⢄
⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠘⢿⠀⠀⢸⠀⠙⠮⠼⠁⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠚⠓⠶⣤⣼⣀⠀⠈⢧⠀⠈⣇⠀⠀⠀⠀⢀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠀⠈⠧⣀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢸⠉⠉⠉⢩⢵⣒⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠉⠓⠒⠂⠀⠀⠀⠀⡾⠶⠦⠤⣬⣬⡼⠁⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⠀⠀⢀⣜⠁⠀⠀⠀⠀⠀⠀⣠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⣤⣀⣀⣠⠤⠖⡯⠤⠤⣤⠤⠖⠀⠀⠤⢴⡞⠻⣌⠙⠒⠒⠒⠒⠚⠉⣱⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠀⢀⡇⠀⠀⢰⠀⠀⠀⠀⠀⣸⠱⣄⠀⠙⠓⠲⠦⠴⠖⣺⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⢤⡤⠞⠀⠀⠀⠘⢦⡀⠀⢀⡴⠇⠀⠈⠓⠦⢤⣤⡤⠴⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣷⠲⠙⠓⠉⠓⢶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⢀⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠀⠀⠀⠀⠀⠉⡻⠛⠲⢤⡀⠀⠀⠀⠀⠀⠠⠤⠀⢀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠈⠀⠀⠑⠀⡎⠀⢀⣀⣀⣀⠤⣤⣤⠤⢂⣁⣀⡀⠀⠹⡄⢠⠦⠀⠀⠠⠀⠈⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡗⢉⣉⠀⣤⣤⡄⠀⠀⠀⣤⡀⠀⣉⡑⠂⡗⠀⠋⠀⡀⠀⠑⠨⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⢀⠀⡚⢡⣿⣿⣿⣷⣶⣤⡄⢠⣤⣶⣾⣿⣿⣿⡆⠙⡄⠀⠞⣡⠀⠀⠀⠀⠀⠀⠐⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠧⠀⠻⣿⣿⣽⣿⡿⠀⠀⢿⣿⣯⣿⣿⠟⠠⢚⣷⠀⠀⠁⠀⠀⠀⠔
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡐⣚⠁⠈⠛⠛⠛⠑⠛⠚⠊⠛⠛⠋⠁⠀⣐⠃⡼⢂⡠⠀⠀⠀⠀⠀⠮⠄
⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠉⢧⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⣠⠏⠈⠂⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠠⣀⠀⠀⠀⠀⠀⠀⣀⠞⢉⠉⠒⠤⡙⢦⣄⣀⣀⠀⠀⠀⣀⣀⣤⠞⠁⠀⠀⠀⠊⠀⠀⢀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⢀⡀⠀⠀⠀⡍⠀⡀⠀⠀⠀⠈⣦⠤⠽⢿⣉⣉⣹⠿⠷⠦⢤⠀⡀⠤⢄⠄⡀⠀⠀⠀⢀⠀⠤⠤⠒⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢀⢀⠤⠄⠀⠀⠀⠀⠐⠋⢹⠀⠀⠀⠉⠉⠁⠀⠀⠀⡞⠘⠀⠠⠀⢀⠈⠆⠀⠀⠀⠳⣤⡀⠀⠀⠐⠴⠇
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀⠠⠀⠀⠀⠐⠁⠠⠒⢄⠀⠀⣸⡀⠀⠀⠀⠀⡀⠀⠀⠀⡇⠀⠀⠀⠀⠈⠋⠀⠀⠠⢹⣦⣌⠛⠿⠶⢶⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢀⡀⠀⠀⠀⠰⠀⠀⠀⠁⠐⠙⡇⠀⠀⢀⠘⠄⠀⠀⢠⡇⢀⠀⠀⠀⠂⠀⠀⠀⠁⠌⢉⡙⠁⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⢀⣄⠤⢤⣿⣄⣀⡴⠛⣄⢠⡇⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣤⠀⠀⢀⠢⣀⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢠⠘⠁⠀⠀⠀⠀⠉⠑⢄⠀⠘⣾⠇⠀⠀⠀⠀⠀⠀⠀⢸⡇⡞⣿⠃⠉⠁⠀⢬⠀⠀⠀⠀⣀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⠊⣾⠁⠈⠉⠉⠉⠉⠀⠀⠘⣿⣿⣉⠀⠀⠀⢀⡴⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣀⠐⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡷⠀⠀⠀⢠⣴⠀⠀⠀⠀⢿⠈⠻⢃⠂⠤⠞⠁⠀⠀⠤⠀⠴
⠀⠀⠀⠀⠀⡴⠊⠉⠉⠙⠺⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠃⠀⠀⠀⢸⢹⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⡂⠀⠀⠄⡀⠀⠈
⠀⠀⠀⠀⢸⠀⢀⠀⠈⠐⠄⣸⡆⠀⠀⠀⠀⠀⢀⣠⠊⣿⠀⠀⠀⠀⣾⠘⡆⠀⠀⠀⠀⢷⠘⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⢄⣸⣀⣀⣨⠉⠑⠉⠉⠀⠀⠉⡇⠀⠀⠖⠋⠀⢧⣤⡀⠤⠄⡞⠅⠀⠀⠀⠁⠘⠁⠀⠘⠀⡄
⠀⠀⠀⠀⠀⠈⠻⠤⣄⣠⠶⣹⠋⠉⣹⠉⡁⠀⠰⠀⠀⣠⣖⣠⠀⡄⠀⠀⠀⠀⠀⠀⠄⣴⡒⠀⠀⠀⠀⠀⠀⢀⡠⠈⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⠃⠀⢀⠈⠠⠴⠃⠠⠁⠀⠀⢀⣴⢋⡽⠁⠀⠀⠀⠀⠀⠀⡲⠘⠻⠶⠻⢦⣄⠀⠀⠃⠉⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠐⠀⣀⣷⡏⠀⠀⠀⠀⡀⠀⡜⠉⠀⠀⠀⠀⠀⡅⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠙⡆⠀⠀⠀⠐⠂⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⠣⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠡⠀⠀⠂⡠⠇

⠀⠀⠀⠀⠀⠀⣀⡤
⠀⠀⣠⣴⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢆
⣿⣿⣿⣿⠛⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⣠⣿⡿⣄
⣿⣿⣿⣿⡗⠀⠀⠀⠀⠀⣀⣰⣿⢴⣦⣶⡿⢿⣿⠃⣏⠁⠘⢢⣀
⣿⣿⡿⠋⠀⣠⢤⣠⠄⡚⢻⡷⠾⠛⣩⣿⠱⠟⣼⠰⠸⠆⡀⠀⠉⠻⡒⠀⢄⡀⠀⡀
⠟⠋⠀⠀⣰⣿⣧⠚⢁⠴⠀⡠⢴⣿⣣⢀⣠⣾⢿⡆⠀⡀⠸⢆⠀⠹⠀⣀⠀⠉⡓⣧⡆
⠀⠀⠀⢸⣿⣿⡿⣻⠶⣛⣡⣼⡿⢣⢇⡞⡋⢡⠎⠹⡆⠀⠲⠎⠀⡄⣸⠀⣵⣇⣤⣧⣷⡀
⠀⠀⢠⣿⠟⠁⠀⣀⠛⣋⡙⠋⠀⣠⠎⢠⠉⢠⣆⠀⢀⠠⢠⡄⣀⣡⣬⣶⣿⣿⣿⣿⣿⣧
⠀⠀⠙⠳⠚⠛⠛⠛⠛⠛⢿⡄⡄⠁⠁⣾⣇⣸⣻⣤⣸⣤⣾⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⡄⢀⠀⠀⡅⣃⣣⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣾⣿⣧
⠀⠀⠀⠀⢰⡄⠀⣧⣎⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠛⠻⢿⣿⣿⣿⣽⣿⣿⣽⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⢸⢡⣀⣿⣿⣿⣿⣿⣿⣿⠿⠋⣿⣿⣿⡟⠀⠀⠀⣿⣿⣿⣿⣿⠏⠀⠘⢿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⣸⣾⣿⡿⠟⠛⠉⠉⠁⠀⠀⠀⣿⣿⣿⠃⠀⠀⠀⣿⣿⣿⣿⠃⠀⠀⠀⠀⠙⣿⣿⣿⡷
⠀⠀⠀⠘⠿⠿⠏⠀⠀⠀⠀⠀⠀⠀⠠⣶⣿⣿⠟⠀⢀⣤⣴⣿⣿⡿⠋⠀⠀⠀⠀⠀⢠⣾⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢔⠒⠂⠠⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠘⠃⢀⡀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠁⠀⢀⠈⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠋⠀⠀⡀⠈⠁⠀⠡
⠀⠀⠀⠀⠀⠀⠀⢀⠊⠀⠘⠂⠀⣤⡀⠈⠁⠀⢀⣀⠀⡗⢄
⠀⠀⠀⠀⠀⢠⠎⠀⠑⠄⣀⣀⣀⣀⡀⠀⠄⣀⡀⣉⠰⣡⢀⠦⡀
⠀⠀⠀⠀⠰⠁⢄⠐⠃⠀⠀⠀⠒⠀⠀⠠⠤⠀⠀⠀⢀⠤⢊⣀⠈⢆
⠀⠀⠀⠀⡆⠀⢀⠉⠒⠀⠤⠀⠀⠀⠀⠤⠄⠀⠒⠈⠀⢀⠀⠀⠀⢸
⠀⠀⠀⠀⠘⣄⠈⠁⠀⠀⠋⠀⠀⠀⠐⠒⠀⠀⠘⠁⠀⠉⢁⡀⡴⠁
⠀⠀⠀⠀⠀⠸⠈⠉⠀⠀⠉⠉⠀⠉⠉⠀⠉⠁⠀⠉⠁⠉⠀⢀⠃
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⠩⠶⠦⠀⠀⠀⠀⠤⠶⠭⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠈⠢⠴⠃⠀⠀⣀⢄⢶⠋⢉⠔⠖⢄⣀⡀⠤⠐⠈⠉⢡
⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠀⡤⠠⠤⠄⠀⢀⠔⢠⢂⡆⢀⠊⠘⠀⠱⠱⡤⠄⠒⠁⠀⠉⠑⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⣀⣀⠧⠀⠀⠀⠤⡜⠀⠂⢻⠃⣬⡀⠿⠆⠀⠀⡇⠠⠤⠀⠀⠀⢀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⢀⡀⢄⢰⠀⠸⠈⣻⠏⢀⠀⡜⡰⠁⠒⠠⢄⣀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⢤⠀⠈⢀⡠⠘⢢⡀⠀⠄⠰⡀⠈⣎⠔⠁⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠒⠁⠀⠀⠀⠈⠑⠚⠒⠚⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠿⠿⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⣤⣶⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠈⢀⢀⣾⡿⠿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠫⠷⠊⣴⡾⢣⡮⠟⣡⣿⡦⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢣⣶⣷⠊⠀⠁⢰⣼⢃⣤⠂⢈⡼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠋⢀⣸⢋⣿⣴⣶⡀⢺⡆⣸⠋⣄⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠔⣱⡟⡁⠀⠈⣾⣿⣈⣡⣾⡿⡿⢸⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣮⣥⣶⣿⡾⣷⢀⣐⣁⠙⠿⣟⡙⠇⡀⣇⣿⣯⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⣠⣄⣀⣼⣣⡶⣛⣉⡙⢷⡜⣶⣝⠆⢹⣿⡆⣿⡳
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠛⣱⠋⣾⡿⠛⡿⣸⠇⣿⣿⡄⢚⣳⡄⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡿⠁⠀⠁⠀⣿⣷⡀⠠⠴⠾⣿⣿⠇⠞⠛⠢⡜⢿⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠇⠀⠀⠀⣀⣈⠻⢿⣶⣶⣾⣿⣿⣷⣦⡈⢻⡟⣸⠧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣷⣦⣾⣿⣿⠁⠀⢘⣿⠀⠃⢿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⠶⠾⠏⣸⣧⡸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡿⠟⠻⢿⣿⣿⣿⣿⣿⣿⣦⣄⠙⢿⡉⣷⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠀⠀⠀⠀⢻⣿⣿⣿⡿⠋⠉⠉⢿⣎⢷⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⢠⠀⠀⠀⣿⣿⣿⡇⠀⢲⡀⠈⢿⡜⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇⠀⢸⡄⠀⢀⣿⣿⣿⣿⣶⣾⣿⣆⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⡀⠘⣿⣷⣿⡿⢻⣿⡿⠛⠛⢿⣿⡈⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧⠀⠹⣷⡀⠀⢸⣿⡇⠀⠀⠈⣿⠇⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⠀⠙⣿⣦⡈⣿⣇⠀⠀⢀⡿⠀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⠀⠈⠻⣷⣾⣿⡄⠀⠘⠁⠀⠈⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠈⠻⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠀⠀⠀⢰⠆⠐⠒⠢⢶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣞⠈⠙⢃⣄⠀⠀⠀⠄⠀⠀⡐⠀⢹⡀⠀⠀⠀⠀⠀⠀⡀⠄⣈⣤⡊
⠀⠀⠀⠀⠀⠀⢀⢀⡄⠀⢤⡀⢰⠋⣓⠄⠀⢀⣾⡇⠀⠀⢂⡀⠈⠃⢀⡩⠀⠀⠀⠠⣦⠀⠀⢡⡏⠉⣟⢳⠲⡄⠂
⠀⠀⠀⠀⠀⠒⠁⠀⠤⣝⡲⢭⡨⣝⡊⠓⠀⠈⠉⠀⠀⠀⠀⠘⠚⠁⠈⠁⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⡄⠹⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⡒⠦⠝⠲⠍⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠄⢀⠀⡄⠀⠀⠀⠀⠀⠀⡧⠰⡄
⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢰⢸⡁⠀⠀⠆⠈⠁⠀⠀⠀⠀⠀⠀⠀⠑⠚
⠀⠀⠀⠀⠀⠀⠀⠘⠃⠘⠋⠀⠀⠁⠀⠁⠀⠀⠀⠀⠀⠀⠀⠘⠞⠳⠃⠀⠙⠁⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠠⠘⠀⠂⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠉⠉⢈⠀⠀⠀⢀⠰⠔⠳⠄⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠀⠐⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⡇⠘⠋⢸⠀⠀⠀⠀⢁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠠⢀⡀⣠⠂⠀⠀⠈⢈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠃⠒⣁⠆⢀⡃⠐⢠⠀⠀⠀⠀⢸⠇⠗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡙⠁⠤⢠⠄⣀⡰⢀⡠⠀⠀⠀⠀⢀⠸⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣄⡸⠁⠀⠀⡀⠆
⠀⠀⠀⠀⠀⠀⠘⠈⠓⣐⡃⠀⣃⡀⠀⢨⡆⠀⢤⠄⠀⠀⠀⢸⡆⠀⢶⣶⢆⡀⡀⠀⢀⠀⠀⠀⠀⠁⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⢀⠀⠁⠾⢃⡈⠀⠉⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠃⠾⠂⠈⠃⠀⠀⠀⠀⡀⠈⠁⣄
⠀⠀⠀⠀⠀⠀⣠⠠⢄⣀⣤⠖⠒⢄⡠⠞⠀⠀⠂⠀⠀⠀⠀⠀⣀⢣⡄⠀⠠⠄⠖⠀⠀⠀⠀⠄⠀⠀⠈⠈⣀⡀⠀⠀⢄
⠀⠀⠀⠀⠀⠜⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠈⠙⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠶⠲⠒⠶⠏⠋⠝⠉⠋⠙⠉⠀⠠⣭⣥⣭⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠀⠂⢂⡈⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠁⠀⠙⠉⠓

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣼⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⠿⠅⠀⠀⠀⠀⠠⣿⣾⠀⠀⠈⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⣀⣄⡀⠀⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⣴⡤⠀⠀⠀⡿⠋⠻⣷⣄⣠⠤⢼⣿⡦⠤⣄⣠⣾⠟⠋⢻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⢿⣌⣓⠂⢉⡻⠀⠘⣿⠀⠀⢟⡉⠐⣒⣡⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠉⠉⠞⠛⠁⠀⠀⠉⠀⠀⠉⠛⠳⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠛⠛⠀⠀⡞⠿⠄⠀⠀⠀⢠⠿⢱⠀⠀⠛⠛⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠷⣦⡤⠤⠄⠀⠑⢐⠀⠀⠀⠀⠀⠒⠂⠀⠠⠤⢤⣶⡾⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⡀⢀⡤⠂⠀⠁⠐⠢⣎⡀⠀⠀⠀⢀⣣⠔⠂⠈⠁⠒⢄⠀⢀⣃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠉⠈⠀⠀⠠⡄⠈⠀⠘⢿⣿⣿⣿⡿⠃⠀⠀⡠⠄⠀⠀⡇⠉⠻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢄⠘⡄⠀⠀⠘⡄⠀⠀⠀⠀⣿⡀⠀⠀⠀⡰⠁⠀⠀⣠⠃⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠨⢦⡀⠀⠈⢲⠀⠐⠊⠀⠉⠀⠀⡍⠀⠀⢀⡴⠕⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠒⢼⡀⠀⠀⠀⠀⠀⢠⠧⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣒⠒⠒⠒⣂⣥⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣶⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⠛⠁⠀⠉⠻⠿⠏

⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠙⠛⠛⠁⠀⢀⠀⠀⠀⠀⠘⠑⠛⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠘⠙⠛
⠀⡀⣀⢂⠀⠀⠀⠁⠀⠀⠀⠀⢠⣠⠄⢙⡇⠀⠀⠀⠐⣦⠀⠀⠀⠀⠀⠀⠀⠀⡀⣀⢂⠀⠀⠈⠁⠀⠀⠀⠀⢠⣀⠄⢙⡇⠀⠀⠀⠐⣶
⠀⠱⠂⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡄⠀⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢚⠈⢡⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⡀⠈⠓⠇⠀⠀⠀⠻⠲⡰⠦⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠐⠓⠇⠀⠀⠀⠻⠲⡰⠶⠀⠀⠀⠀⠀⡀
⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⡄⠈⠃⠢⠤⠴⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠙⠶⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠦⠤⠴
⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠀⠂⠈⠁⠀⢀⡀⠠⠀⠀⠀⠉⠀⠁⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠁⠀⠀⠐⠂⠈⠁⠀⠀⡀
⠀⠀⠀⠀⠀⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠂⠀⠀⠀⠀⠀⠈⠁⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⢀⢤⠤⠤⣄⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠈⠁⡁⠀⠀⠀⠀⠀⠀⢐⠀⠀⡀⠀⢀⢤⠤⠤⣄
⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠐⠂⠋⠁⠀⡋⢠⠂⠂⠀⠆⠀⠀⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠉⠁⠀⡋⢠⠒⠂⠀⠆
⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡰⠃⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢤⣀⠀⠀⠀⠀⢣⡀⠀⢁⡠⠃
⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢠⢚⠶⢀⡆⠀⠀⢰⠁⡞⠛⡆⢀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠉⠈⠀⢰⠾⠶
⠀⠀⠀⠸⣦⣀⢀⣡⠟⠀⠀⠀⠀⠀⠀⢀⣶⡄⠀⠀⠀⠠⠀⠐⠀⠈⠁⠀⢸⠁⠀⠀⠸⣦⣀⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠠⠀⠂⠀⠈⠁
⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠘⣿⣧⠀⠀⠀⠈⠀⠀⠀⠀⠀⠠⠘⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠰⣶⡆⠀⠀⠀⣿⣧⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠉⠁⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣀⣀⣠⡄⠀⠀⠙⠁⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠉⠁⠀⠐⠀⠀⠸⢿⣿⡿⠃⠀⠀⠀⠀⠀⠀⢠⠤⣼⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠐⠀⠀⠘⢿⣿⡿⠇⠀⠀⠀⠀⠀⠀⣤⢤⣼⠁
⠀⠀⠀⡀⠀⠈⠱⠀⠀⠀⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠡⠆⠀⠀⠀⠀⠀⠀⠀⣤⡄⠀⠀⠀⢀⣀
⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠈⣛⣡⠰⠇⠀⠀⠀⡀⠟⠀⠀⠀⠀⠀⠀⠀⠀⢂⡒⡐⠀⠀⠀⠀⠀⠀⠀⠀⠘⣛⣡⡰⠆⠀⠀⠀⠀⠟
⠀⠀⠀⠑⠀⠀⠀⠠⣄⡄⠀⠀⠀⡯⣑⠀⣘⠃⠀⠀⠈⠀⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠑⠀⠀⠀⢀⣄⡄⠀⠀⠀⣯⣁⠀⣘⠃⠀⠀⠈
⠀⠀⠀⠀⠀⠐⢦⣀⡀⠁⠀⠀⠀⠈⠈⠘⠁⠀⠀⠤⡄⠀⠉⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢦⣀⡀⠁⠀⠀⠀⠈⠈⠊⠉⠀⠀⠤⡄⠀⠃⢀
⠶⠀⠄⠀⠀⠀⠀⠉⠛⠃⠀⠀⠀⠀⢀⠄⠀⠀⠃⡀⠠⠍⠉⠁⠀⠀⠀⠀⠀⠾⠀⠀⠀⠀⠀⠀⠉⠓⠃⠀⠀⠀⠀⢀⠄⠀⠀⠀⡀⠠⠉⠉⠁
⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠆⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈
⡀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠠⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢀⣀⠤⠆⠀⡴⠉⣉⡉⠃⡀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣀⠤⠇⠀⡴⠉⢉⡉⠓⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠀⠀⡅⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠘⠀⠄⠀⡅
⡆⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⢶⡽⠒⠀⠀⠀⠀⠘⠀⠤⠰⠊⠀⣠⣀⠀⠂⠀⠀⢀⠄⣠⣤⡀⠀⠀⠀⠀⠠⠶⠽⠖⠀⠀⠀⠀⠈⠀⠤⠰⠊⠀⣠⣀
⠃⠀⠀⢸⡀⠃⠀⠃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⣀⠀⠸⡄⠉⢰⠇⠀⠀⢸⡀⠃⠀⢃⣴⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⠀⡀⠀⠸⡔⠉
⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⢰⣿⡇⠀⠀⠀⠨⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠻⠶⠴⠞⠁⠀⢀⣀⡀⠀⠀⠀⣿⡇⠀⠀⠀⠨⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⠿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠀⢿⠟
⠀⠀⠀⠤⠄⠀⢀⣀⠀⢸⣶⣶⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⡀⠀⢰⣶⣶⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇

⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⣦⣴⣶⣦⣤⣤⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠤⠄⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠠⣶⣦⣾⣷⣤⣤⣤
⣠⡄⠈⠃⠉⠘⠁⣦⡄⠀⠀⠀⠀⠀⠀⠀⡸⠏⠀⠻⠙⠉⢧⠀⠀⠀⠀⠀⠀⠀⠸⢉⣤⡀⢀⣤⡙⠃⠀⠀⠀⠀⠀⠀⢻⡿⠋⠁⠀⠀⠈⠻⢿⣤
⢸⣏⠁⢐⣀⡀⢈⣹⠁⠀⠀⠀⠀⠀⠀⠠⣯⣁⣤⣤⣤⣄⣽⡆⠀⠀⠀⠀⠀⠀⠇⣾⠟⠃⠈⠻⣿⠸⠀⠀⠀⠀⠀⠘⣿⡇⠀⠀⣀⣀⠀⠀⢸⡟
⡀⠲⠂⠀⠉⠁⢰⠆⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣿⣿⣿⣾⠟⠀⠀⠀⠀⠀⠀⠀⠐⢄⠀⠙⠛⠀⡠⠃⠀⠀⠀⠀⠀⠰⣿⣷⣀⠀⠈⠉⠀⣠⣾⠿
⢛⠀⣘⠀⠐⠀⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠿⣿⡿⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⣷⣶⣶⣿⠛⠃
⠘⣴⡇⢠⠄⡤⢘⡆⠀⠀⠀⠀⠀⠀⠀⠀⠰⠯⠭⢭⢭⠽⡄⠀⠀⠀⠀⠀⠀⠀⢀⣆⠀⣶⢰⡆⣈⡀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⠠⠃
⠈⠛⠆⣸⡀⣇⠰⠛⠃⠀⠀⠀⠀⠀⠀⠀⣾⢉⣻⣛⣼⡉⠷⠀⠀⠀⠀⠀⠀⠀⠙⠿⢿⣿⢿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠻⠿⠿⣿⢿⡿⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⢶⣆⢀⣶⠆⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠈⠁⠀⠉⠙⢿⠟⠀⠀⠀⠀⠀⢀⣠⣄⠀⣀⡀
⠀⢀⠀⠐⠒⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠄⠀⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣀⣠⣤⣤⣤⣀⣈⠀⠀⠀⠀⠀⢠⠋⣸⣾⠿⠿⠿⣷⣄
⠣⠁⠀⠀⠀⠀⠀⠡⠇⠀⠀⠀⠀⠀⠀⣤⠆⠀⠀⠀⠀⠀⠰⣤⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⢰⣿⢻⡟⠁⠠⠀⠀⠀⢻⡛⣷
⠠⠀⠀⢿⣿⠿⠀⢀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⢀⣿⣧⠀⠀⠀⠀⠀⠀⠀⠈⣿⢩⠉⠀⠀⠀⠀⠀⠀⠀⠹⣏⠳⢄⠀⠀⠀⡀⠈⢛⡯
⠀⠑⠤⠀⣀⠠⡔⠁⠀⠀⠀⠀⠀⠸⢿⣿⣷⣄⠀⠀⠀⣠⣾⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠐⢂⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⡏⠉⠹⣷⣄⢿⣞
⠀⠀⠁⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡇⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠾⠈⣬⢠⠆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡄⠀⢀⣿⣜⡞
⢠⠂⠀⠀⠀⠀⠀⡓⡄⠀⠀⠀⠀⠀⠀⠀⢸⡿⢃⣴⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⡶⢘⠀⡄⣤⢚⠎⠀⠀⠀⠀⠀⠀⠐⢿⣿⣷⣤⣾⣿⣿⣿
⠈⠉⠺⠭⠶⠭⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠖⠲⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠿⠓⠻⠆⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠴⠖⠒⠒⠒⠲⠦⡤⠤⠒⠒⠉⠉⠉⠉⠉⠓⠲⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⡀
⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡄
⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄
⠀⠀⠀⠀⡟⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⣷
⠀⠀⠀⢸⠃⠀⡇⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢸⡄
⠀⠀⠀⢸⠀⠀⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠈⡇
⠀⠀⠀⠸⠀⠀⠘⢣⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⡾⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠆⠀⠀⠀⠱⡀⢀⣠⡶⠿⠿⠶⠖⠒⠒⠒⠓⠒⠒⠲⠦⠤⠤⣉⣛⠂⠀⠆⠀⠀⢀⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢀⡴⠖⠛⠓⠲⣤⡀⠀⠀⠀⣠⠶⠛⠛⠓⠒⢮⣍⠁⠐⠀⠀⠜⡇⠀⠀⠀⠀⣿
⠀⠀⠀⠀⣠⠦⢬⠀⠀⠀⢰⠏⠀⠀⠀⢀⡀⠀⢳⡀⠀⡾⠁⠀⠀⢀⣤⣤⣀⠙⣆⠀⠀⠀⢠⡗⠘⡆⠀⠀⡿
⠀⠀⠀⠀⠹⣄⠀⢇⠀⠀⣿⠀⠀⢠⣾⣿⡟⠓⣼⡇⢸⡇⠀⠀⣼⣿⣿⡟⠉⢳⣿⠀⠀⠀⣼⣀⠴⠃⠀⠀⡇
⠀⠀⠀⠀⠀⠈⡗⠛⡄⠀⠘⣆⠀⠸⣿⣿⣿⣾⡿⠀⠈⣧⠀⠀⢿⣿⣿⣾⣶⣿⠇⠀⠀⣼⠃⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠀⡇⠀⠙⣆⠀⠈⠙⠶⠾⠿⠟⢋⡄⠀⠀⠈⠳⣤⣌⣛⣿⣿⠿⠃⠀⢀⡼⠁⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⡼⠁⠀⠀⠘⢧⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠈⡇
⢢⣄⣀⠤⠞⠁⠀⠀⠀⠀⠀⠉⠳⢤⣀⠀⠀⠀⢄⣀⡤⠀⠀⠀⠀⢀⣠⡴⠚⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣄
⠀⠙⠦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⡷⠶⠶⢤⠤⠶⠶⠒⠛⠉⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡽
⠀⠀⠀⠈⠙⠲⠦⣤⣄⣀⣀⣠⣤⠴⠖⠋⠀⠀⠀⡸⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⠶⠤⢤⡤⠤⠶⠚⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⠣⠀⠔⠁⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡎⠀⠀⠀⠀⠀⢫⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠙⠀⠀⠀⠀⠀⠀⡀⠀⠑⠢⠀⠀⠀⠀⣀⡄
⠀⠀⠀⠀⠀⠀⠄⠀⣀⡀⠀⠀⠄⠂⠀⠀⠆⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢃⠀⠀⠀⡠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠁⠀⠀⠀⠆⢀⣀⣀⣀⠀⠴⠄⠀⠀⠀⠀⠀⠁⠀⠊⠁⠀⠐
⠀⠀⠀⠀⠀⠀⠢⠄⠠⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠆⣀⡀⢠⠤⡤⠤⠤⣄⠀⠀⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⢁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠹⣁⠁⠀⣀⣸⣄⠀⠀⡠⠃⠈⠑⡄
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣐⣄⣀⣠⣄⣑⣈⣀⣈⣙⣣⣤⣤⣤⣞⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠟⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣦⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠄⠈⠀⠀⠀⠀⠈⠐⠄⠀⠀⠀⠀⠀⠠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠂⢀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠐⡀⢀⠀⠀⢠⠀⡈⠀⠀⠀⠀⠀⠀⢀⡀⠈⠀⠀⠀⢀⠀⡐⠀⠀⠀⠀⠀⠀⠠⠄
⠀⠀⠈⠀⢰⣧⣷⠀⠀⢰⣮⣷⠀⠘⠀⠀⠀⠀⢼⣥⡧⠀⠀⣠⢠⡐⠄⠀⠀⠀⠀⠈⠀⢰⣧⣶⠀⠀⢠⣴⣦⠁⠰
⠀⠀⠀⠠⠶⠉⠉⠘⠛⠈⠉⠱⠦⠀⠀⠀⠀⠶⠂⠉⠁⠶⠆⠻⠿⡃⠀⠀⠀⠀⠀⠀⠠⠦⠙⠉⠰⠖⠘⠛⢫⡄
⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠄⠤⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⡄⡆⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⡀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⢒⣫⡀⠀⣜⡒⠌⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⢁⡀
⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣶⣶⣶⣤⣀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⢀⣤⡀⠀⠀⢤⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⠀⠀⠀⢠⣴⣶⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠈⠛⣿⠿⡆⠀⢠⣶⠾⠓⠶⠄
⠀⠀⠀⠀⠀⠀⠉⣿⡿⠙⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠇⢰⣄⡀⠁⠀⠃⣠⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⣼⣿⡟⡛⠿⣿⣿⣿⣿⡷⠀⠀⠀⢿⡟⠉⠙⠻⢿⣿⡿⠟⢫⣴⣶⣶⣾⠏⠁⠀⠴⠏⢹⣿⣿⡝⢦
⠀⠀⠀⠀⠀⠀⠀⠘⠧⠀⠸⣿⣿⠇⡏⠀⣸⡏⠈⢻⡇⠂⣀⡀⠸⣇⠀⠀⢀⡄⢿⡀⠀⣿⣿⣿⠋⢹⠀⠀⠀⠀⢠⣼⣿⣿⠷⣄
⠀⠀⠀⠀⠀⠀⣴⣀⣭⣆⠀⢸⠛⠁⣧⣀⣛⢁⣄⠘⣯⠄⠻⠀⠀⠀⠀⠀⢈⡷⢀⣀⣀⡟⠙⢿⣆⣀⡀⠀⢀⡐⠋⠘⠟⠉⠀⠛
⠀⠀⠀⠀⠀⠀⠉⠻⠋⢉⣥⡴⠁⠀⡟⢿⠇⠘⢿⣿⡷⠀⠈⠰⣦⣤⣶⠔⠊⠁⢠⣴⣶⣤⣤⣤⣤⡀⠀⠰⣿⣿⣿⣶⣄⣠⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠰⠒⢺⡿⠋⠀⢀⣠⣴⡶⠀⢀⣨⣭⣁⣀⣀⣀⣼⣿⣧⠀⠀⠀⣿⡿⠿⠿⢿⡟⠛⠿⠀⣼⠟⠻⣿⣿⣿⣿⣿⣿⡟⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⣤⣴⣿⠿⠛⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠿⠀⠀⠀⠘⡇⠀⠀⠀⠀⠀⠀⢸⣿⠛⠉⠁⢹⢻
⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣿⡿⠀⠀⠀⠈⣿⣿⡿⢿⣿⣿⣿⣿⡟⠁⠀⠀⢶⣶⣦⠀⢀⣴⣆⣀⣶⣄⠀⠀⠀⢸⠙⠂⠀⠀⢀⢷
⠀⠀⠀⠀⠀⠀⢠⡾⠿⠋⠀⠀⠀⢀⡀⠀⢸⢿⠋⠀⠀⠀⠘⡇⢸⡇⠀⠀⠀⠀⢿⣿⣶⣿⣿⣿⣿⣿⣿⣦⡀⠀⠞⠠⠃⠀⠀⠈⠈
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⣶⠶⢟⠂⠀⠸⠸⠄⠀⠀⠀⠀⠅⠀⠇⠀⣀⢠⣤⠈⠛⠛⢿⣿⣿⣿⣿⢿⣿⣷⠀⢀⣤⣤⣄⠀⢀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠀⠀⠀⣠⣴⣶⣶⣦⣤⣶⣾⣷⣦⣀⣄⠀⠀⠀⠈⢹⣿⢋⣀⠀⠀⣿⠀⠀⠀⢘⡏⣿⠀⠋⠟⠉⠉⠃⢈⣿⠁
⠀⠀⠀⠀⠀⠀⢹⣿⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣴⡠⠀⢲⡚⠛⠿⠃⠀⠀⣷⠆⠀⢠⠜⠁⡏⠀⣶⣶⣶⣾⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠈⢿⣦⠀⢰⡿⠋⠉⢻⣯⡙⠛⣿⢟⣿⠁⠈⠛⠛⠀⠴⠂⠁⠀⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠀⢠⡿⢋⡛⠋⢻⠁
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⣘⠃⠀⠀⠀⠉⠁⠀⠛⠂⠉⢀⣴⣿⡆⠀⠀⢸⣀⡠⠚⠀⠀⠀⢿⣿⣿⣿⣷⣶⣄⢸⠀⠀⡟⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣦⣀⣀⠀⠀⠀⢠⣴⣾⣿⣿⠿⣧⡀⠀⠈⢿⣷⣤⣦⣴⣤⠀⢿⣿⣿⣿⣿⣿⠀⠀⢤⣇⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿⣿⣆⠀⣻⡿⣿⣽⣿⠀⠹⢿⡄⠀⠈⠙⣿⠛⢿⣿⠀⠈⣿⣿⠉⢿⡟⠀⠋⠀⠉⠈⠻⠯⠽⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠿⠿⣿⣿⠏⠰⠿⠁⠘⠛⠃⠀⠀⠀⠁⠀⠀⠀⠽⠀⠜⢈⠀⠀⠿⠇⠀⠾⢇⣀⣠⣤⡊⢀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠘⣿⣴⢀⣴⣶⣶⣶⣶⣾⣿⣶⣄⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣷⣶⣄⡈⠉⠉⠙⠿⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠸⡀⠀⠀⢿⠁⠘⠟⢻⣿⣿⣿⡿⢻⡿⣿⡦⠄⠀⢾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠘⠧⠉⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣇⠀⣸⢸⡄⠀⠀⣼⠋⢻⠀⠀⣸⠇⠀⠓⠀⠀⢸⡟⣿⠛⠻⠛⣻⣿⠋⠛⢿⣿⡄⠀⠀⠀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⡉⠀⠃⠚⠀⠀⠘⠁⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠙⠀⠀⠀⠉⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⢧⣾⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⡄⠀⠀⣠⣤⣤⣤⣤⣄⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠛⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠄⢿⣿⣿⣿⣿⣿⣷⣦⡀⠀⠀⣸⡿⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⠿⣿⡿⣿⠛⣷⠀⠀⣼⠃⠉⠛⡿⢿⠿⢿⡇⠀⢀⣈⢁⣠⣤⡀⠙⣿⠙⠛⠟⠛⠁⠹⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿⡇⠀⠀⠰⠃⠀⠸⠀⠀⠻⡆⠀⢠⠃⢸⠀⠈⣡⡾⠛⠿⠿⣃⣨⡁⠀⣿⠀⢶⣶⣶⡄⠀⣼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠈⠁⠀⠈⠁⠀⠀⠀⠈⠁⠀⠀⠋⠀⠈⠉⠉⠁⠀⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠴⠁⢀⡀⠠⠤⢀⡀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⠀⠀⢠⠔⠊⢱⠀⢀⠔⠁⠀⠀⠀⠀⠈⠒⠉⠀⠈⠁⠒⠑⢄⡀⠀⣀⡀
⠀⠀⠀⠀⠀⠈⠢⢀⣸⠀⠜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⢀⢊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠘
⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠈⠀⠇
⠀⠀⠀⠀⠀⠀⠘⡜⠄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⡘
⠀⠀⠀⠀⠀⠀⠀⠈⠑⠠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠪⠊⠀⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⠄⠤⢀⠀⠡⡀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⠢⣀⡀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⣠⠃⠘⡀⢀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠠⠤⠤⠄⠊⠉⠢⠄⣀⡀⠤⠒⠁⠀⠀⠑⠉




⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠤⢤⣀⠀⠀⡔⠋⠱⠤⢶⠒⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⢠⠒⣄⠀⠀⢀⠚⠠⠂⢀⠤⣄⠓⠛⢢⡑⢄⣆⠀⡀⡱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠑⣄⠀⡴⢶⠲⡒⠒⡖⡚⢲⠲⣆⣠⠙⠚⡗
⠀⠱⡤⡑⠼⠟⢍⡑⣄⡀⠘⠂⠊⠀⢤⡠⣵⠋⠁⢌⣀⠧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡄⢠⣍⣘⢃⠀⢣⢸⠀⢰⢀⠀⡇⣿⣁⠀⠘⢆
⢀⠞⠀⠀⠀⠀⣠⠞⠳⠥⠀⠀⠀⢇⣠⠇⢸⡧⣵⠀⢠⠒⡾⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠉⠀⢀⡀⢹⣤⠂⢸⢀⠀⢸⡄⢠⢇⡿⢹⣱⠒⠋
⢇⠀⢀⡴⢾⡉⠙⠢⣄⣔⡇⠈⡂⢀⠤⣤⠋⠧⣈⣁⡜⠀⠀⠀⠀⢀⡀⠤⡤⠤⠤⢤⣤⣤⣀⡀⠀⠈⠓⢻⠃⣧⠜⠋⠈⠒⠞⠚⠒⠛⠛⠉⠉⠀⠀⠉
⠀⠉⠉⠀⠀⠑⠤⠴⠛⠾⣌⣉⣠⡧⠞⠁⠀⠀⠀⠀⠀⠀⠀⢠⡎⠑⠠⠀⣌⡶⡷⠷⢶⡀⠤⣍⣣⠀⠀⠘⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⡒⠲⠤⠤⡶⠤⠤⠼⠖⠚⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢁⠀⠀⠀⡆⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠉⠉⠉⠙⠲⢄⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⢸⠀⠀⠀⡇⠀⠀⢀⡄⢀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠉⠉⠒⡩⠤⢕⢤
⠀⠀⠀⠀⠀⡰⠲⡋⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⠀⠀⠀⠀⠀⠈⠐⠚⠠⠤⠤⠧⠤⠤⠚⠓⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⢡⠀⡠⢀⠑⠤⠚⠀⣣
⠀⠀⠀⠀⠀⣧⠀⠑⣄⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠇⠒⠁⠀⠢⠴⠃⢀⣀⠸⣄⡆
⠀⠀⠀⠀⠀⢏⠳⣄⠈⠓⠤⣀⡀⠀⠀⢀⣰⢆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠐⠒⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢧⠠⡄⠀⠐⢄⠸⣄⡨⠃⡠⠁
⠀⠀⠀⠀⠀⠘⢦⠀⠑⠢⠤⣀⣉⡉⢉⢉⡠⠞⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠖⠁⠀⠀⠀⠀⣀⡴⠂⠈⠉⠑⠢⠤⣀⠀⠀⠈⠣⣆⠀⠁⡁⠄⣀⣰⠻⠥⣀
⠀⠀⠀⠀⠀⠀⠀⢹⠦⣀⡀⠀⠈⣉⡡⠟⠀⠀⠀⠀⠀⠀⠀⢀⣀⠠⠂⠁⠀⠒⠀⠤⣠⠀⠈⠁⠀⠀⠀⠀⣀⠄⠊⠉⡄⠀⠀⠀⠀⠈⠈⡑⢟⡉⢠⠻⡖⠊
⠀⠀⠀⠀⠀⠀⠀⡼⠀⡰⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣎⣁⡘⠊⢇⡀⠀⠀⠀⠂⠀⠀⠀⠉⠓⢂⠔⠊⠀⠀⠀⡠⠃⠀⠀⠀⠀⠀⠀⠉⠁⠈⠙⡀⠸⡄
⠀⠀⠀⠀⠀⠀⣰⠃⡰⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⠊⠁⠀⠀⠈⠉⢀⠿⣓⣦⣀⠀⠀⢀⣤⣀⠂⠀⢸⠀⢀⡠⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠱⡀
⠀⠀⠀⠀⠀⣰⠃⣰⠃⠀⠀⠀⠀⠀⠀⠀⣞⡋⠤⡀⠀⠀⠀⠀⠀⠔⠅⠈⠀⠀⠀⢉⡹⠓⡄⠉⡆⠀⣸⠗⠁⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠱⣄⢑
⠀⠀⠀⠀⢰⠃⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⡆⠙⠲⢬⠑⠒⠒⢮⠒⠉⠓⢦⠒⠒⠲⣤⠐⠀⢠⠀⡾⠊⠀⠀⠀⠀⠀⣠⠔⠛⠉⡡⢈⠩⠛⢦⡀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠘⠲⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠤⣀⣆⠠⣀⣔⠠⠤⢄⣍⣀⠀⣀⣸⠀⡠⠞⠉⠁⠀⠀⠀⠀⢠⠞⠁⠀⠀⢀⣀⣀⡀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠉⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⢠⢏⠠⡀⢀⡴⢋⢀⢔⢹⡏⠩⢑⢿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠖⠐⡄⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠑⠚⡟⠀⠰⡉⢁⣼⢅⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠶⠤⣼⠗⠀⠈⠲⡀⠀⠀⠀⠀⠀⠀⠈⢷⡤⢀⣷⣀⠴⡻⠉⠈⠎⣆⡜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠔⠚⠉⠙⠢⡈⢆⠀⠈⣷⡖⠀⢞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠮⣄⣙⣺⣤⡤⠴⡟⠣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⣤⠐⠚⠉⠁⢄⠙⢆⠀⠀⠀⢹⠈⣄⡴⠙⠛⠶⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠘⢆
⠀⠀⠀⠀⡀⠠⣀⡀⠀⠀⠔⠉⢍⡉⠲⣌⠢⠀⠀⠀⠑⠘⣀⣠⠤⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠳⡄
⠀⠀⠀⠐⡀⠀⠸⡈⠑⠚⠒⢀⡀⠱⠀⠈⢆⣷⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⡄⠘⢆
⠀⠀⠀⠀⡝⠉⠀⡨⠛⢳⠔⠣⠤⠤⠤⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠢⠘
⠀⠀⠀⠀⢇⠀⣸⠽⠛⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⢦
⠀⠀⠀⠀⠀⠠⠤⣄⣀⡀⠀⠀⠀⠀⠀⠀⢀⠜⠁⢀⠏⡀⠀⢀⡀⠀⠀⠀⠀⢀⣀⡠⠤
⠀⠀⠀⢀⠩⠶⡄⠀⠀⠉⠓⢦⡀⠀⢀⠴⠋⠀⠀⠉⠀⠤⠖⢉⠁⠀⣠⠴⠋⠉⠀⠀⣠⠮⢅⠀⠀⡄
⠀⢸⡀⠀⠀⢀⣘⣆⠀⠀⠀⠀⠹⠃⠁⠀⠀⠀⠀⠀⠲⡲⠦⢤⣙⡾⠁⠀⠀⠀⠀⣜⣁⡀⠀⠀⢰⠃
⠀⠀⢳⡀⠀⠁⠀⠈⢆⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠐⠢⡘⡄⠀⣀⣀⠀⠀⠀⠀⡜⠁⠀⠙⠀⢠⠏
⠀⠀⠀⠱⣄⠀⠀⠀⣞⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⢚⡕⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠈⠓⣤⠀⠀⠀⠀⣠⡞⠋⠙⢦⡀⠀⠀⠀⠀⠀⣠⠖⠉⠙⣦⡀⠀⠀⠀⠀⡴⠋
⠀⠀⠀⠀⠀⠘⠲⢂⢀⠀⢰⣿⣧⣀⣀⣠⣷⡀⠀⠀⠀⢰⣿⣄⣀⣠⣿⣿⡀⢠⢀⢲⠘
⠀⠀⠀⠀⠀⠀⠀⠁⠙⡆⢸⣿⣿⠿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⡿⠿⣿⣿⠇⣸⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠈⢯⡣⣀⣀⣿⡿⠀⠀⠀⠀⠘⣿⣏⣀⡠⣫⠞⠀⢻
⠀⠀⠀⠀⠀⢀⣀⣀⣼⡀⠀⠀⠉⠙⠋⠉⠀⠀⣠⠲⡄⠀⠀⠉⠙⠋⠁⠀⠀⢈⣄⣀⣀
⠀⠀⡠⠶⠯⠭⠤⣀⠀⠉⠁⠒⠂⠤⠤⢤⡤⠀⠀⠀⠀⠠⢤⠤⠤⠄⠒⠂⠈⠁⢀⡠⠤⠭⠷⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠈⠑⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠒⠈⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠒⠀⠀⢀⠤⠄⠀⠤⢄⡀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠒⠒⠁⠀⠀⠀⠀⠀⠉⠒⠚⠚⠁

⠀⠀⠀⠀⠀⠀⠀⣀⡄⠐⡖⡦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠻⡆⠀⢁⢀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠀⠀⠀⣨⠛⠿⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠹⣶⠀⠀⡈⠀⡀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠤⠤⠤⢴⣿⣿⣯⣾⠄⠘⠱⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁
⠀⠀⠀⠀⠀⢦⡀⠀⠐⠄⢱⡾⠁⣸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢻⡟⣿⠏⠀⠀⠀⠀⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠈⣿⠀⠀⢰⡾⠁⢰⣿⠀⠀⠀⢰⠀⠂⠀⠀⣀⣀⠀⠀⠀⠀⠈⠀⣿⣆⢀⠀⣀⡔⠃⠀⠀⠀⠀⠀⢀⠀⢠⡀⠀⣤⠂⠁
⠀⠀⠀⠀⠀⠀⠸⡇⠀⠈⡀⢰⡿⠇⠀⠀⠀⠀⠀⠀⠀⠈⠟⠺⡕⠤⠄⠀⢤⡆⢹⣿⣿⣷⠏⠀⠀⠀⠀⠠⠀⠀⠸⠀⡘⠃⠒⢿
⠀⠀⠀⠀⢀⡀⢀⠛⣦⣄⢹⡏⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠠⡖⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠈⢇⡾⠀⣿⢿⡀⢻⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠼⠆⣰⠀⣸⣧⡠⠳⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠐⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡄⠀⠀⠀⠀⠐⠀⠘⠀⠀⠀⠐⠀⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠁⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠄⠖⠀⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠋⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢷⡃⣸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⡠⣠⡤⠀⠀⠀⠀⢀⣤⢤⠶⡖⠔⢾⠟⠉⠋⡀⢀⡤⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄⠀⢸⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠊⠉⠈⠀⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠁⠀⢨⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⠀⠀⠀⣦⡀⠀⠀⠀⢰⣿⣶⡎⢻⠁⠀⠀⠀⠀⠀⠠⢀⡄⠀⡀⢦⠀⠀⠀⣀⠀⢀⡀⠀⠀⠀⠀⣼⠀⠠⣾⡄⠀⠀⢠⠆
⠀⠀⠀⢀⠄⠀⡞⠉⠇⠈⠉⠉⠁⠈⢇⠀⣇⡖⠀⠀⠀⣰⣣⡾⢹⠀⠀⠈⠀⠈⠉⠉⢀⠻⠁⠀⠀⠀⠀⠟⢀⡞⠁⠉⠀⠀⠀⠀⠠
⠀⠀⠀⡌⢀⠾⡇⠀⠀⠀⠀⠀⠀⠀⢹⡤⣿⠃⠀⠀⣴⡿⠋⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠀⣿⣿⠀⠷⠼⠶⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠚⠒⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⣄⣀⠀⡄⠀⠂⠠⠄⠀⡄⠀⠀⠀⠀⠤⠀⡀⠆⠤⠀⠀⠀⠠⡀⠀⡠⠀⠀⠀⠀⢀⠀⠀⠠⠄⠂
⠀⠀⠀⠀⠀⡀⠁⠀⠄⠀⠠⠀⡀⠐⠓⠀⠀⡀⠰⣤⠾⣃⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠼⠂
⠀⠀⠀⠰⠆⣀⠀⠀⠀⠀⠀⠀⠤⠀⠀⡀⠀⠇⠘⠃⠐⠛⠀⢀⠀⠀⠠⠌⠀⠀⠈⠀⠀⠀⠀⠠⠀⠠⡄⠀⠠
⠀⠀⠀⠄⠈⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠀⣴⠌⣿⠆⠀⠀⠀⠀⠀⠈⠂⠀⠚⠀⡀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠒⠀⠒⠃⠁⠠⡀⠀⠁⠀⠀⠀⠀⠄⠰⠈⠁⠀⠀⠀⠄⠠⠄⠉⠉
⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠆⠀⠀⠀⢿⣿⣚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠠⠀⠀⠘⠀⠀⠀⠀⠃⠀⠈⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠀⠀⠀⠀⢀⣤⣤⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠐⠂⡀
⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⡿⡿⠿⠿⣿⣿⣦⠀⠀⠀⣰⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠠⠅⠠⠤⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠉
⠀⠀⠛⠂⠀⠀⢀⠀⠀⠀⠐⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⢰⠀⡀
⠀⠀⠀⠄⠀⡀⠀⠀⠀⠀⠀⠆⢠⣴⠀⠀⠘⢿⣤⣼⣥⣧⣼⣧⣿⡿⠁⠀⠀⠀⠀⠀⠀⢀⣠⣁
⠀⠀⠀⢀⠀⠀⢳⣤⡤⠄⠀⢀⠀⠀⢀⢀⡄⠀⠙⠛⠿⠿⠿⠛⠉⠀⠐⠀⠈⠉⠀⠀⣀⠀⣤⠄⡀⠀⣤
⠀⠀⠀⠉⠀⠀⣰⡋⢉⣓⠀⠀⢀⠠⠄⠈⠀⢠⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢈⠠⠀⣀⠁⠀⠘⠀⠠⠄
⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠤⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠸⠈⡄⠒⠀⢀⠖⡄
⠀⠀⠀⠀⠀⣄⠀⠀⠀⡈⠀⣇⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⢀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⠀⠂⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⢀⠐⠂⠠⣠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣥⡅⠀⠀⠀⠈⠀⠘⠈⡀⠈⠃
⠀⠀⠀⠀⠠⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠂⠀⠀⠀⠀⠀⢠⠠⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠐⠂⠀⠀⠀⠀⠈⠑⠀⠦⠄⠂⣀⠀⡈⠀⠀⡀⠄⢀⡀⢠⡄⠀⠀⠄

⠀⠀⠀⠀⠀⠀⠀⢀⢀
⠀⠀⠀⠀⠀⠀⡡⠞⠁⠛
⠀⠀⠀⠀⠀⠀⠚⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⡀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠐⠀⠡⠀⠀⢸⣯⣿⡇
⠀⠀⠀⠿⠿⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣶⡏⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠿⠛⠷⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⠀⠀⠀⠐⠶⠷
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⣤⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠖⠲⠂⠀⠀⠀⡀⠀⠀⠈⠀⠀⠀⠘⣿⠛⠛⠛⣿⠀⠄⠁⠄⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠄⠀⠀⠀⣿⡀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠂⠀⠀⠀⠠⠀⠀⠀⠛⠋⠀⠻⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⣨⣠⣤⣍⢥⠀⠐⠐⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⣠⡄⠀⠀⡄⠀⠈
⠀⠀⠀⠁⣠⣤⣤⣀⡀⠀⠀⠓⠊⠀⠐⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠇
⠀⠀⠀⠀⠋⠉⢙⣿⡗⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠁⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣤⢀⣤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠶⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠤⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠰⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢛⡻⡟⡛⣛⡛⣻⠿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⠖⠒⠲⡄⠀⠀⠀⠀⠀⠀⠀⣰⠲⡄⠀⠀⠀⠀⠀⠀⠀⢀⣞⡅⠀⠀⠀⠀⠀⠀⠀⡴⠂⠒⢦
⠀⠀⠀⠀⠀⠀⢮⣀⣄⣄⡼⠀⠀⠀⠀⠀⠀⡰⠣⠭⢽⡀⠀⠀⠀⠀⢀⡴⠋⠉⠉⠦⡀⠀⠀⠀⠀⢀⠷⠔⠒⢺⡄
⠀⠀⠀⠀⠀⠀⠈⢟⠢⡱⠁⠀⠀⠀⠀⠀⠀⣗⢒⠒⢒⡇⠀⠀⠀⠀⠪⣀⢤⣀⡠⣀⡼⠀⠀⠀⠀⠸⡀⠀⠀⢈⠇
⠀⠀⠀⠀⠀⠀⠀⠈⣷⠃⠀⠀⠀⠀⠀⠀⠀⠸⡁⠀⢸⠁⠀⠀⠀⠀⠀⠸⣴⣤⣦⠇⠀⠀⠀⠀⠀⠀⠈⢹⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄⠀⠀⠀⠀⠀⠀⠀⢠⢾⣱⠦⡀⠀⠀⠀⠀⠀⠀⢀⡖
⠀⠀⠀⠀⠀⢀⡤⠒⠒⠒⠢⡀⠀⠀⠀⠀⢀⡠⠚⣧⣼⣦⠀⠀⠀⠀⠀⡰⠁⠉⠉⡖⠁⠀⠀⠀⠀⠀⡴⠷⠞⢄
⠀⠀⠀⠀⢰⣫⢤⣀⠤⣄⡠⢌⣲⠀⠀⠀⣏⣩⣭⣤⣤⣤⡇⠀⠀⠀⠀⠳⡒⠒⢲⠞⠀⠀⠀⠀⢀⣠⣏⣙⣉⣹⣄⡀
⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠒⠊⠀⠀⠀⠀⠯⠭⠭⠭⠤⠴⠃⠀⠀⠀⠀⠀⢹⢻⠏⠀⠀⠀⠀⠀⠈⠑⠒⣢⣖⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾⠽⠦⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⢀⠤⠶⠭⡉⠑⣄⠀⠀⠀⠀⢰⠣⢢⢔⢮⢣⠀⠀⠀⠀⠀⡔⠉⠀⠈⠱⡄⠀⠀⠀⠀⢸⣿⣿⡿⣿⡇
⠀⠀⠀⠀⠀⡏⡞⡉⠳⡈⡆⢸⠀⠀⠀⠀⠸⡰⣐⣾⣮⡾⠀⠀⠀⠀⢰⠳⡖⢶⢲⡖⣷⠀⠀⠀⠀⢸⣿⣿⡛⢻⡇
⠀⠀⠀⠀⠀⠱⣱⢌⣒⣁⠷⠊⠀⠀⠀⠀⠀⠉⢲⣿⠃⠀⠀⠀⠀⠀⠈⣇⣧⢸⢸⢧⡏⠀⠀⠀⠀⢸⠋⠛⠻⠿⣯
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠟⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠛⠀⠀⠀⠀⠀⠘⠒⠒⠒⠒⠋
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⡔⠋⠀⢙⠲⡀⠀⠀⠀⠀⠀⠀⢀⣠⡓⢤⠀⠀⠀⠀⠀⠀⡞⡁⡙⡆⠀⠀⠀⠀⠀⠀⣀⣄⣤⣀
⠀⠀⠀⠀⠀⡜⠡⠆⠀⠈⠀⢳⠀⠀⠀⠀⠀⡴⡏⢣⠈⡞⠁⠀⠀⠀⠀⡖⠚⠈⠘⠒⡆⠀⠀⠀⣰⡎⢱⡀⢀⠎⢑⡆
⠀⠀⠀⠀⠀⢸⣀⠀⠀⠙⢀⠎⠀⠀⠀⠐⣮⠣⣸⡤⠋⠀⠀⠀⠀⠀⠀⢈⠇⠀⠀⢏⠀⠀⠀⠀⠙⢌⡏⠙⠋⢹⣡⠃
⠀⠀⠀⠀⠀⠀⠈⠙⣾⠉⠁⠀⠀⠀⠀⠀⠈⠑⠁⠀⠀⠀⠀⠀⠀⠀⠀⠯⠤⠋⠳⠬⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣶⣶⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⢀⣠⣤⢀⣀⣠⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢚⣉⣁⣤⣤⣤⣀⣉⡙⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⠄⠀⠀⠀⠀⢀⣾⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣹⣿⣧
⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⣼⣿⡿⠿⠛⠉⠈⠛⠻⠿⠛⠛⣿⣿⣿⣿⣿
⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠻⢿⣿⢿⣿⣿⣿⣿⣿⣯⠉⠀⠀⠀⠹⣿⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠘⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡿⠋⠀⠀⠀⠀⠉⠀⠉⠃⠈⠃⢹⡇⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⢋⠅⡀
⠀⠀⠀⠘⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠻⠀⠀⠀⠀⠀⠺⠀⠀⠀⠀⠀⠘⠈⠁⠀⢀
⠀⠀⠀⠀⡕⢢⠙⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⢀⡀⠀⢱⠿⠠⠂⠀⢀⠀⠀⠀⠀⢒⡶⢶⠀⠀⠀⠀⠀⠀⠀⣸⣇⠀⡗⠚⢾⡔
⠀⠀⠀⠀⠠⢈⠀⠀⠀⠀⠀⠘⠁⠀⠀⠀⠀⠈⠃⠀⠀⠆⠀⠁⢄⢀⣆⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣼⣿⠟⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠙⠁⡙⠿⣷⣦⣤⣄⠀⢀⣴⣶⣾⡿⠟⢉⣠⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠀⠁⠂⣀⠉⠙⠋⠀⠜⠙⠋⠀⢠⣴⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢂⠀⠀⡐⢂⠀⠄⠀⠀⠐⠀⠀⠀⠀⠀⠀⢹⣶⡄⠀⠀⠀⠀⠰⣾⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠃⠀⠀⠘⠁⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠹⣿⣿⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠄⢡⡀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠃⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠠⠊⠢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⣀⠄⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢤⠀⠥
⠀⠀⠀⠄⡀⠑⠂⠀⠌⠀⠀⠠⠀⠀⠀⠀⠠⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡉⠂⠂
⠀⠀⠈⠁⠜⠃⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠁⠐
⠀⠀⠀⠀⠀⠀⠀⠐⠀⠐⠀⠁⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠀⠙⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⡿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠓⡀⠀⠀⠐⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⡆⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠔⠈⠑⠂⠁⠀⠈⠂⠀⢀
⠀⠀⠀⣄⠠⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢠⠐⠈⠂⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠰⠦⠀⠀⠈⣐⠁
⠀⠀⠀⣏⠀⠟⠀⢠⢨⡁⠀⠀⠀⠈⠒⠁⠀⠀⠈⡈⠀⠀⠘⡐⢂⠀⠀⠐⠀⠄⠀⢀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀⢎
⠀⠀⠀⠀⠓⠀⠀⠀⠓⠉⠢⡀⠀⠀⠈⠀⠀⠀⠠⠦⠄⢀⣨⠠⠴⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⣀⡀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠊⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠄⠀⠀⠀⠄⡆⠀⠀⠀⠀⠸⠆⣄⠀⠀⡀⠀⢀⠀⠀⠤⠀⠉⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡦⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠁⠀⠀⠀⠠⣶⡄⢀⣒⣼⡓
⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠁⠀⠀⠀⠈⠂⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢙⡁⠀⠀⠰⠿⠀⢈⠋⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⠀⠀⣠⡁⠄⠀⠀⠀⠀⢅⠁⠢⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠈⣄⠉⠡⠋⠘⠌
⠀⠀⠀⠀⠉⠂⠂⠚⠀⠀⠀⠀⠊⠀⠰⠈⡄⠀⠀⠀⠀⠉⠃⡡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠔⠀⠂⠈⠀⠀⠀⠀⠈⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠠⠔⠀⠀⠀⠀⠀⠠⢀⠁⠀⠀⠀⠀⠀⡅⠀⠠⣶⠆⠀⠢⠤⠀⠀⠀⠀⡐⠁⠀⠀⠐⠂⠠⠀⢀⠁⠀⢀⡀⠀⠀⠐⠄⠐⠀⠤⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠀⠂⢀⠂⠀⠀⠀⠀⠀⠀⡇⡀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠈⠂⠀⠀⠠⠚⡔⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠈⠀⠀⠊⠁⠀⠀⡀⠀⠀⠀⠀⠇⡇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠰⢆⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⡐⡶⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠡⠐⠢⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠈⠀⠁⠢⡈⠢⡡⠀⠀⠀⠠⣹⠓⠂⠀⠃⠆
⠀⠀⠀⠀⠀⠀⠀⡀⠁⠱⠄⠀⠀⠀⠀⡅⠁⠁⠐⠀⠀⠀⠀⠀⢀⠀⢠⠀⠠⠀⠀⠠⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠀⢖⣈⠶⢌⠀⠀⠀⠈⠂
⠀⠀⠀⣂⡠⠐⠀⠃⢀⠚⠀⠀⠀⠀⠀⢈⠂⠒⠁⠀⠀⠀⢈⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠉⣀
⠀⠀⠈⠚⠢⢀⣀⠠⠊⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁
⠀⠀⠀⠀⠀⠄⠤

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⣴⣿⣿⠳⡄⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠠⣶⣯⡤⠤⣰⣾⣿⣿⣿⡏⢘⡟⠂⠈⠦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⢿⣟⣭⡵⠞⢝⣽⡿⣡⡿⣻⠀⠌⢷⡀⠀⠀⠈⠻⣶⠤⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣉⢛⣯⠔⢂⠰⠛⠉⣋⣴⡞⠻⠁⠉⣡⣿⡇⢁⠈⠈⠢⣄⠀⠰⣬⠣⠀⠈⠳⢦⣄⣰⢀
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠌⡀⠴⢋⣠⠴⢚⣿⡿⣵⠀⣤⣶⠟⢻⣿⡀⠒⠆⠀⠻⡔⠀⠈⢀⠲⣄⡀⡖⡤⢹⢾⠃
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡿⢋⣵⠟⣋⣡⣴⣿⡟⣠⠇⣼⢫⡁⢠⠞⠈⡿⡀⠀⠚⢶⠀⢠⡀⣸⡇⢠⣿⣇⣵⣾⣾⣧
⠀⠀⠀⠀⠀⢀⣼⣿⠟⠓⠀⠠⠴⠟⢋⡩⠟⠋⠀⣡⡎⠁⡈⠇⢨⡆⡀⠧⠀⢄⠀⡄⠀⡸⢁⣘⣴⣶⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠺⢿⣥⣶⠶⠶⢿⣥⡶⢿⣷⡀⢀⠠⠋⠆⢠⣿⡀⣼⡧⡇⡀⣷⠀⣰⣧⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡧⢸⠀⡄⠀⢺⣿⣴⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢟⣽⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⣶⠀⢰⠀⣀⣿⢸⣤⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⢻⡦⣈⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠛⠛⠿⣿⣿⣿⣿⣽⣿⣿⣯⡶⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢻⢠⡀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢹⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠋⠀⠘⢿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣼⣿⣿⣿⡿⠿⠿⠟⠛⠉⠉⠀⠀⠘⣿⣿⣿⡿⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣿⣿⣿⠇⠀⠀⠀⠀⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠿⠿⠿⠉⠀⠀⣰⣶⣾⣿⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠾⠿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⢠⠀⠀⣴⠛⠛⠀⠀⢠⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⢀⠔⠛⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣸⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠘⠀⠀⠀⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⠀⠀⣿⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⡇⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡧⠤⢾⠆⠀⣿⠤⠄⠀⠀⢸⠄⠀⠀⠀⠀⡇⠀⠀⠀⠀⣿⠀⠀⠀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠇⠀⠸⠁⠀⢷⣤⣤⡄⠀⠸⣤⣤⡤⠀⠀⢧⣤⣤⠀⠀⠘⢦⣤⠜⠁



⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠄⠠⢄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏⢀⣤⢤⣄⡈⠑⠦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠐⠊⣉⢤⣄⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⣞⠀⠀⠀⠙⠳⣄⠈⠳⣄⠀⢀⣀⣀⣀⣀⡀⢀⡤⠚⢉⡤⠖⠋⠁⠀⠈⡣⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠧⠀⠀⠀⠀⠀⣠⠗⠀⠈⠉⠁⠀⠀⠀⠀⠉⠉⠀⠰⣉⠀⠀⠀⠀⠀⢀⡄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠰⡆⠀⠀⣠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⡼⠁⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠲⣀⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢆⠀⣸⠃⡸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠶⠁⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⣤⢤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⣄⠀⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠿⠿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠟⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣶⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠘⢦⡤⠝⠦⣤⠏⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⠴⠃⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢑⠦⢄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢤⡖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠈⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠑⠒⠒⠊⠉⠀⠒⠒⠂⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣻⣽⣯⣿⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⣿⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣰⡃⣼
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡿⠻⠏⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⢿⣿⣿⣶⣤⣴⣦⣠⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣏⣠⡟⠛⠉⠉⠀⠐⠻⠛⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⢶⣾⠛⠻⠿⠻⣿⣿⣿⠿⢿⡿⠟⠉⠀⠉⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠉⠛⠿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠙⠂⠀⠀⠀⠙⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠟⠉⠛⠛⢿⠿⠿⡿⣿⣿⣿⣿⣿⣿⡛⠃
⠀⠀⠀⠀⠀⠰⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢿⡿⠿⠿⠛⠉⠉

⠀⠀⠀⠀⠚⠒⠒⠶⠴⠄
⡤⠀⠀⠐⠲⠖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣀⣀⣠⣤⣀⣀⣠⣤⣀⣤⣤⣀⠀⠶⢶⣶⣶⣶⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣄⣠⣤⣾⣿⣿⡿⠟⣛⣯⣿⠟⣩⣵⣿⣿⣿⣿⣿⣿⣉⣛⠛⠿⣤⠉⠛⠋⠩⠿⠿⠒
⣶⣶⣶⣶⡶⢾⡿⠷⠾⢋⡶⠖⠖⠀⠀⠀⣤⣠⣦⣾⣿⣿⣿⣿⣿⣿⣿⠛⠉⣀⣴⣿⡿⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣷⣶⣾⣿⣿⣿⣿
⠁⠀⠀⠀⠈⠈⠃⠀⠀⣠⣤⣠⣦⣄⣀⣰⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛⠁⢠⣾⣿⣿⡿⢋⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣼⡇⠀⢀⣤⣶⣾⢟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇⠀⠀⣼⣿⣿⠏⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⡆⠦⢹⡇⢳⣞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⣸⣿⣿⠇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⠿⣿⣿⣿⣿⣿⣿⡟⠋⠉
⠃⠀⣸⣿⣸⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠀⢨⣿⣿⣿⡏⠀⠀⠀⠀⠀⢿⣿⡈⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⡾⣿⣿⣿⣿⣿⣷⣿⡿⠖
⣿⢿⣿⣿⣧⣼⣿⣾⣿⣿⣿⣿⣿⣿⡿⢉⡀⢸⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠉⠹⣿⣿⣿⣿⣿⣿⣿⣿⣤⣀⣉⣉⡉⠉⠉⠉⠁
⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡁⠀⠰⠀⠀⠀⠈⣿⣦⡄⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄⠀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠈⠈⢻⣿⣞⠀⠀⠀⠀⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⢀⣽⡜⠋⢻⣿⡟⣿⣿⣿⣿⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⣿⣷⡀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡿⠛⠉⠀⠀⠀⠀⠀⠈⠉⠀⠀⣿⣿⣵⣿⣿⣿⣿⣿⣿⣿⣀⠀⠀⠀⠀⠀⠀⠻⣿⣿⡆⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⡿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡄⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣠⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣵⠀⠀⠀⠀⠀⠀⠀⠈⢻⣿⣿⣦⡄⠀⠀⠀⠀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣀⣀⣤⣠⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣤⡀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠿⣿⣿⣿
⣴⡆⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠈⠉⢻⠿⢿⠿⢷⣄⠀⠀⠀⠀⠀⠀⢤⡄⣈⣹⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢤⡀⠀⡀⠀⡀⠀⠀⠈⢿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⡿⠂⠁⢀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣵⣦⡿⣿⣿⣿⣧⣄⣀⣐⡀⠀⣴⣶⣶⣶⣿⣿⣶⣿⣤⣤⣦

⠀⠀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢤
⠀⢨⣿⣿⢟⢿⣿⣧⠀⠀⠀⣰⣞⡛⣳⣦⠀⠀⠀⠀⠀⠀⣤⣤⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠶⢤⠁⠆⠰⠆
⠀⠸⣿⣯⠈⠠⣿⡗⠀⠀⢠⡟⠛⠁⠙⠛⡇⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠉
⠀⠀⠀⢿⣤⣬⣇⠁⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⢶⢆⣶⣶⣇⠀⠀⠀⠀⠀⠀⣤⣤⣤⣄
⠀⠀⠀⠤⠤⠤⠆⠀⠀⠂⠀⠉⠉⠈⠉⠈⠀⠀⠀⠠⠄⠀⠤⠀⠿⠿⠿⠿⠿⠿⠀⠀⠀⠀⠀⠀⠻⡿⠿⠃
⠀⠀⠀⠀⢀⢀⠀⠀⣈⠥⠀⠀⠰⣠⡆⣴⢰⣴⢰⠀⣤⣤⢀⡆⣦⢰⠀⣆⣤⡄⠀⠀⠀⠀⠀⠀⠀⠓⠂
⠀⠀⠀⠐⠻⠳⠀⠀⠉⠁⠀⠀⠈⠁⠀⠈⠈⠈⡈⡀⣁⡈⡀⡅⠀⠈⠀⠉⠁⠁⠀⠀⠀⠀⠀⢠⡀⠀⠀⣠⡄
⠀⠀⢠⡀⣀⡀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⣤⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⣿⡶⣾⣏
⠀⠀⠀⠁⠈⠁⠁⠀⠀⣶⡄⠀⠀⢠⣶⠀⠀⣀⣻⠟⢉⠙⢿⣃⡀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⣿⠙⢹⣿
⠀⠀⠀⠏⠀⢒⣛⠀⠀⠘⡇⠀⠀⢸⠇⠀⠀⣭⣿⠀⠀⠀⢈⣯⡅⠀⠀⠀⠈⢹⣿⠀⠀⠀⠀⠀⣿⠀⢸⡟
⠀⠀⠀⣿⣿⣿⡗⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠈⢹⡷⣶⣶⣿⡍⠁⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢸⣶⣾
⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣡⣍⠀⠀⠀⠀⠶⠖⠀⣬⣿⡄⠘⠿⠗⠀⢸⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠐⠂⠀⠀⠒⠂⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠠⠴⠄⢠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣉⠀⠀⣤⡴⠀⠀⠀⠐⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠎⠀⠀⡘⡹⠁⠀⠀⠶⠶⢀⠁⠀⠀⠀⠀⠀⠀⠀⢦⠔⠀⠀⠀⠀⠹⣦⡀⠄⠀⠀⠈⠳⡄
⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⢐⣿⠇⠀⠀⢠⣶⣷⡆⠀⡄⠀⠀⠀⠀⠠⠀⠀⣴⣖⢲⣤⠁⠀⣿⣷⣄⠀⠀⠀⠀⠱⡄
⠀⠀⠀⠀⠀⣾⠀⠀⠀⠀⢰⣾⠀⢠⠀⠙⠿⠟⠋⠐⠀⠀⠀⠀⠀⠀⢰⡏⠛⠿⠿⠋⠁⠠⠸⣿⣟⠀⢰⡇⠀⠀⠈⠂
⠀⠀⠀⠀⢀⠟⣀⡆⠀⠀⢸⣿⠄⠀⢠⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢠⠀⠃⠀⢻⣿⡀⠘⠇⠀⠀⠀⠀⠲⡀
⠀⠀⠀⠀⣼⡞⠋⠁⠀⠀⠘⣿⠀⠠⡍⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠃⢾⣿⣷⢀⠀⠀⠀⠀⠀⠲⠳⢄
⠀⠀⠀⠀⣿⣿⣶⣠⠀⠀⠀⢻⡆⡦⢀⡀⠀⠀⠈⢀⣦⣴⣦⣴⣾⣿⣦⡀⢠⠀⠀⠀⡀⠈⢱⣴⢻⣿⡤⣷⣦⡀⠠⣶⢆⣰⣮
⠀⠀⠀⠀⣯⣿⣿⣿⠾⡼⣶⢸⣇⡇⠈⣾⡁⠀⠀⢸⣷⣿⣿⣿⣿⣿⣿⠇⠀⠡⠀⣀⣿⡿⢸⣿⢸⣿⡇⣿⣻⣷⡾⠿⣠⡟⠃
⠀⠀⠀⠀⢹⣿⣻⣇⣾⣿⣿⢸⣶⠙⢄⠸⣏⠊⠀⠀⠉⠛⢿⣿⠿⠟⠉⠀⠀⠀⢰⠟⠁⠴⢿⡿⠻⣿⡇⢻⣿⣿⠆⣴⢻
⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⡄⣸⠧⠀⠀⠀⢰⡄⠀⠀⠀⠀⣤⣁⠆⠀⠀⠀⠀⢠⡞⠀⠀⣀⣌⠀⢤⣿⣷⠸⣿⣫⣤⢋⡎
⠀⠀⠀⠀⠀⠀⠈⠿⣿⣿⢰⢏⣴⡗⠀⠀⠘⢿⣶⣤⣶⡴⠿⢿⣿⣶⣶⣤⡴⢏⡽⠋⠀⠀⠀⠀⠸⡛⢿⡆⣿⣻⣷⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠵⢿⠉⠁⠀⠀⠘⠓⠀⠻⢿⣯⠀⠀⠈⠀⠀⣸⣿⡽⠁⠀⠀⠀⠀⠢⠀⠀⠈⣊⣿⠀⠘⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⠀⠀⠀⠀⠀⠀⠈⢿⣦⣄⣀⣀⣴⣿⠟⠀⠀⠀⠀⠀⠀⡄⠀⠀⢿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⢀⣼⣦⠀⢰⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⢰⡦⠀⠘⣂⣀⣼⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢸⠇⠹⡇⣴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠰⠿⠟⠟⠛⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠈⠀⠀⠈⠀⠸⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⢌⡂⠀⡄⠀⠀⠀⢀⡀⠀⢹⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠈⠻⣷⣦⠘⣖⢠⣀⠀⣤⣰⣤⠀⠘⠛⠀⠀⠀⠀⠀⠀⠉⠐⢈⠙⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⡀⠀⠀⠉⠑⠛⠊⠟⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠀⡗⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⣀⠀⠀⠀⠀⠀⠀⣤⡀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡀⠀⠀⠀⠀⠀⠀⢸⡇⠀⡏⠀⠀⠀⠀⠐⢎⣀⡝⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⢦⣤⡄⠀⠀⠀⠈⣿⣄⠀⠀⠀⠀⠀⢸⠇⢰⠇⠀⠀⠀⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢶⢏⠀⡐⡅⠀⠀⠀⠀⣿⣿⣷⣄⠀⠀⠀⢸⠀⢸⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠁⠀⠀⠀⠀⠀⣿⡇⠙⢿⣷⣦⣄⣘⠀⣂⣀⣴⣷⡀⠀⠀⠀⣀⣤⣶⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠇⠀⠀⠉⠛⠿⠿⠿⠿⠛⠉⠻⣿⣿⣿⠿⢿⣿⡟⠀⠀⣀⣀⣀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢦⣤⣀⣠⣴⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠀⠀⠸⡏⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡄⠙⢿⣏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⣠⠖⠛⠉⠉⠉⠓⠶⣤⣿⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠇⠀⢸⣿⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⣾⠁⠹⣾⡣⠀⣠⣴⣶⣤⣀⣰⣟⣹⡆⠀⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣈⣉⣀⣀⣤⡾⠟⠓⠒⠶⣄⣹⣯⠝⢦⡀⠀⢀⡇⠀⠀⢻⡗⣾⠋⢰⡖⠛⠉⠉⠉⣿⣷⠀⠀⣧⣀⡀
⠀⠀⢀⣀⣠⠀⠀⠀⠀⠀⠉⠻⣿⡟⠁⠀⠀⣀⠀⠀⠀⠙⢿⡯⡈⠻⣄⢸⠃⠀⠀⠀⢻⣻⣄⠸⣇⣤⡄⠀⠀⡿⣿⠖⠒⢻⡀
⠀⠀⢈⡏⠀⡣⠀⠀⠀⠀⠀⢠⣿⣷⢅⣴⠿⠛⠻⢷⡄⠀⠈⣿⣔⠀⠈⠿⠀⠀⣆⠀⠈⢷⡉⠛⠛⠉⠀⢀⣼⢃⣸⣄⣀⣼⡿⠛⠓⠒⠂
⠀⠀⠈⠁⠙⠃⠀⠀⠀⠀⢀⣼⣿⣿⣺⡇⠀⠀⠀⠈⢿⠀⠀⢸⣿⣷⡄⠀⠀⢰⣿⡆⠀⠘⣷⣤⣤⣴⡶⠟⣱⠟⠉⠛⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⣿⣿⣿⣧⡀⠀⠀⣠⡞⠀⠀⣼⣷⣿⠻⣦⣀⣸⠀⢻⡶⠾⠿⠉⢉⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⡆⠠⡄
⠀⠀⠀⠀⠐⠚⠛⠛⠛⠉⠉⣹⠿⣿⣿⣾⠉⠛⠛⠁⠀⠀⣰⣿⣿⣿⠄⠘⠛⠋⠀⠀⠀⠀⠀⠐⣤⠷⣖⠒⠒⠛⠦⠀⠀⠀⠈⢠⡤⠓
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⡰⠛⠻⣷⣧⣤⣤⣤⣴⡾⢿⣯⡁⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⡎⠈⢳⡈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⣠⠞⠁⠀⠀⠜⠋⠉⠉⠉⠀⠀⠀⠙⣷⡀⠀⠀⣠⠞⠋⠉⠉⠉⠲⣇⠀⠀⠙⠆⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠸⡇⠀⡼⠃⢀⡴⠞⠛⠒⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣖⠒⠈⠦⡄⠀⠀⠀⠀⠀⣿⡼⠁⠀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣜⣠⣀⡎⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠉⠃⠀⠀⠀⠀⠀⠀⠏

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⣦⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⣡⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⢠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⣰⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠁⠀⢲⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠁⠀⢠⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣟⠀⠀⢨⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⣾⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠛⠀⣀⣼⡟
⠀⠀⠀⠀⠀⠀⠀⣰⠏⢷⡀⠀⠀⠀⠀⠀⠀⡘⠋⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⢀⣠⣶⠟⠋
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠈⠳⣄⡀⢠⡶⠛⠛⠻⠶⣦⣴⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣏⡀⠀⣾⡟⠁
⠀⠀⠀⠀⠀⠀⠀⣿⡄⠀⠀⠈⠙⡿⠁⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣀⠀⠀⠀⠀⠀⢀⣿⡁⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠘⢷⣤⣀⣀⠀⠀⣠⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠷⢦⣤⣴⠟⠁⠀⠀⢸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢉⡿⠁⠀⣙⣉⣁⠀⠀⠀⠀⠀⠀⠰⣦⡀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⣴⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⢰⠞⠉⠈⠙⢷⡀⢀⣤⣤⣤⣄⠘⣷⠀⠀⠀⠀⠀⠀⠀⣿⣶⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠀⣟⠀⠀⠀⠀⠈⣿⠏⠀⠀⠀⠙⢧⠀⠀⠀⠀⠀⠀⠀⢀⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣸⠃⠀⠀⣧⠀⣰⣾⢲⣀⡿⠀⠀⣀⣀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠘⢦⣿⣿⣿⣿⡇⠀⣾⣿⣋⣥⢸⡇⠀⠀⠀⠀⠀⣦⠀⠀⠘⣧
⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⠀⠀⠀⠉⠉⠉⠀⠹⣆⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⣰⣿⣆⠀⠀⠘⠓⣲⡶
⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⠿⠋⠀⠀⠀⠀⠀⢰⡏⠈⠻⠿⣶⣶⠾⠋⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠚⠛⠉⠉⠁⠀⠀⠀⠀⠈⠉⠳⣄
⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⢿⣗⡀⠀⠀⠀⣰⣏⣀⣀⣀⣀⣀⣠⡤⠶⠚⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠣⣀
⠀⠀⠀⠀⠀⠸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢹⣦⠀⢰⡟⠁⠈⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⡀
⠀⠀⠀⠀⠀⠀⣿⡀⠀⢴⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⣿⡇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⢸⣇⣀⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣧⣴⡿⠁⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠘
⠀⠀⠀⠀⠀⠀⠀⣿⡋⠀⠀⠀⠀⠀⣰⠋⠛⠀⠀⢠⣏⠉⠉⢀⣴⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀
⠀⠀⠀⠀⠀⢀⣤⠾⢷⣄⠀⠀⠀⠀⢻⡄⠀⢀⣴⠟⠙⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣥⡾
⠀⠀⠀⢀⡴⠋⠁⠀⠈⠛⢷⣦⣤⣤⣤⣿⡶⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡆⠀⠀⠀⢀⣿⠁
⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⣹⠃⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⢇⠀⠀⠀⣾⠁
⢠⡾⢀⣤⣾⠇⠀⠀⠀⠀⠀⠀⢀⡿⠀⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⢻⡏⢸⡆⠀⠀⣿
⣮⣴⠟⢡⡟⠀⠀⠀⠀⠀⠀⠀⢸⠇⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠾⣿⠀⢸⡇⢸⡇⠀⠀⢹
⠛⠁⠀⢸⡇⢀⡄⠀⠀⣤⠀⠀⣿⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡾⠟⠉⠀⡿⢀⡿⠀⠀⣿⠀⠀⣿
⠀⠀⠀⢸⡇⣼⢷⡀⠀⣿⡄⢠⡟⠀⠀⠀⠀⠀⠀⠙⣿⣦⣄⣀⡈⣃⠀⢳⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⡾⠟⠋⠀⠀⠀⢠⠇⣼⠃⠀⠀⣿⠀⢀⡟
⠀⠀⠀⠘⣷⡿⠘⣧⠀⡏⢳⣀⡇⠀⠀⠀⠀⠀⠀⢀⡿⠀⣿⠉⠛⢻⡄⠀⣿⡿⠶⠶⠶⠾⠿⠿⠛⠛⠉⠁⠀⠀⠀⠀⣀⣠⡏⣼⠏⠀⠀⢰⡏⠀⣸⠇
⠀⠀⠀⠀⠙⠗⠀⠘⢷⣧⠀⠛⠓⠀⠀⠀⠀⠀⠀⣸⠇⢸⡏⠀⠀⠸⣇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠋⠉⠛⣧⠀⠀⠀⣸⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⣀⣀⣰⠏⢠⡿⠀⠀⠀⠀⣿⠀⠘⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠴⣶⣠⠏⢠⡶⠾⠷⠶⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡏⠉⠛⠶⣿⡁⠀⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠛⠀⢸⠇⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣀⠀⢀⣾⠃⠀⠀⠀⢸⡏⠀⢰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣴⣷⣀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠛⠻⣧⡞⠁⠀⢀⣀⣀⣸⡃⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠸⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠀⠀⠀⢸⡏⠉⠛⠻⠿⣧⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠃⢀⣄⠀⠀⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣴⠟⢻⣄⡾⠁
⣀⣀⣀⣄⣀⣀⣠⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣤⣤⣈⣡⣀⣀⣿⣅⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⡿⢿⣿⡿⠿⣿⣿⣿⣿⡿⢿⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣧⣤⣤⣼⣧⣠⣧⣠⣶⣦⣥⣼⣿⣱⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿