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⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⠶⠶⠚⠛⠛⠛⠛⠛⠛⠲⠶⢦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠞⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⢶⣄⡀
⠀⠀⠀⠀⠀⢀⣠⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀
⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣼⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⡄
⠀⢀⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄
⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⢻⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣄⣀⣀⣀⣀⣠⣤⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁
⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃
⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠁
⠀⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⠶⠛⠉
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣉⣛⣛⣶⣶⣶⣶⣶⣶⣶⣖⣛⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⢋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⠀⢰⣇⣠⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣥⣤⣤⣄⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠋⠁⠀⠀⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⠟⠙⠉⠙⠉⠉⢉⣉⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠉⠙⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⣤⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠁⢀⡀⠀⠀⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡏⠀⠀⣀⡸⣆⣿⡀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢀⣰⣿⡴⢂⣤⠀⠀⠀⠉⢿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⠁⠀⠀⠈⣿⠟⢻⣿⡟⠛⣶⠀⠀⠀⢸⡇⠀⠀⠀⠠⣾⡛⢻⣿⡟⣻⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⢹⣿⠿⠿⠷⠾⠋⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⠿⠇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣰⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⡀⠀⠀⣰⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠶⠶⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡶⢦⣤⣤⣤⣤⣤⣴⠶⠞⠛⠉⠁⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡸⠷⠰⠆⡄⠀⠀⠀⠀⠀⠀⢠⠰⠖⠺⢇⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠎⣁⣺⣦⣶⣠⢳⡩⠧⠼⢍⡎⢄⢶⣴⣟⣈⠱⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⢬⢂⡝⢟⢉⢿⡜⣫⠖⢋⡙⠳⣍⢣⡴⡉⡿⢯⡐⡥⢄⣀
⠀⠀⠀⠀⠀⠀⣀⢠⡄⡄⡐⡶⣪⠭⠬⠭⢭⣚⣍⡞⣁⣶⣼⣯⣶⣈⢳⣩⣓⡭⠭⠤⠭⣝⢶⢃⣠⢠⡄⣀
⠀⠀⠀⠀⠺⠗⣉⡠⣦⠪⡽⣼⠁⢴⣦⣶⣠⣮⢋⡑⠿⢿⡟⢻⠿⠿⢚⡹⣡⣤⣶⣴⡦⠈⣧⢫⠵⣴⣄⣉⠺⠗
⠀⠀⠀⠀⣛⠀⣭⣿⢋⡖⡇⡇⢰⣿⣿⠻⢿⢋⡼⠉⠛⠲⢌⣡⠖⠛⠉⢳⡙⡿⠛⣿⣿⡆⢸⢸⣲⡉⣾⣭⠀⣛
⠀⠀⠀⠀⠈⢤⠐⡯⠧⣚⣒⡿⠜⠿⢙⣃⣃⣞⠤⢎⣑⣊⣹⢏⣙⣊⡱⠤⣻⣘⣈⡋⠿⠣⢿⣒⣓⠼⢵⡂⡤⠁
⠀⠀⠀⠀⡼⡴⣫⠖⠋⣅⢴⣮⢸⡏⠉⣀⡊⣻⠳⢮⡖⠲⢡⣎⠖⢲⡵⠞⣟⢑⡀⡉⢹⡇⣴⣦⣨⠉⠳⣝⢦⠧
⠀⠀⠀⠠⣓⡇⡇⠰⢷⣿⡿⠿⠸⡇⡸⢦⢟⡽⡌⡟⢮⣃⡏⢹⣘⡵⢻⠣⢯⡻⡤⢇⢸⡇⠿⢿⣿⡶⠆⢸⢼⣚⠄
⠀⣀⠶⢉⢯⢧⢳⡀⢻⣿⣗⢖⡥⠿⢷⡚⣚⣥⣥⣹⡄⡙⡣⢼⢋⢠⣏⣼⣤⣓⢓⡾⠿⢬⡲⣾⣽⡻⢀⡞⡼⡽⡉⠶⣀
⣤⡏⣠⣝⡯⢺⣱⢝⣮⠝⣡⠞⢀⠗⢴⢹⠮⣓⢦⣄⢜⣷⢿⡿⣞⡧⣠⡴⢚⠵⣏⡦⠦⡀⠳⣌⠫⣵⡫⣎⡗⢽⣯⣄⢹
⠉⣧⠈⢵⢯⢺⢜⡵⢫⣵⣌⢳⡀⠗⢹⡾⣫⣖⣋⣁⡘⢞⡿⢿⠳⢮⣈⣙⣲⣝⢧⡏⠺⢀⡞⣡⣮⡝⢮⡳⡖⡽⡮⠁⣼
⠀⠀⠛⠤⣯⡏⡞⠀⢿⣿⡟⠓⠍⡿⡟⢲⢲⡍⠇⡼⢁⣴⡏⢹⣦⡈⢧⠸⢉⡖⡖⢻⢿⠩⠚⢺⣿⣿⠀⢳⢹⡽⠤⠛
⠀⠀⠀⠈⡖⡇⣇⠘⠛⣿⢿⣿⢸⡇⠸⣉⡷⣽⠡⢗⡋⠆⢧⣼⠰⢝⡺⠎⣯⢼⣉⠇⢸⡇⣿⣿⣿⠛⠃⣸⢸⠲⠁
⠀⠀⠀⠀⠸⢙⠮⣓⠦⣭⣘⡯⠸⠧⢤⣀⣣⣾⠞⢫⣕⣚⣌⣡⣗⣚⡝⠳⣷⣔⣀⡤⠴⠇⢻⣋⣬⠴⢞⡵⡏⠏
⠀⠀⠀⠀⢠⠁⣰⣯⠟⡲⡔⡿⠱⣶⣶⠖⢂⢳⡁⠣⠔⠢⣼⣯⠴⠢⠜⢉⡾⡐⠐⣶⣶⠎⢿⢣⢖⡹⣽⣂⠉⡄
⠀⠀⠀⠀⠶⠀⣭⡿⣦⠓⡇⣇⠘⢿⣿⣶⢷⣦⢹⡤⠶⢚⡡⢌⡓⠶⢤⡎⣴⣿⣶⣿⡿⠃⣸⢸⠚⣤⡻⣭⠀⠶
⠀⠀⠀⠀⠺⠗⣤⢈⡋⣨⡽⡹⣄⠙⠃⠛⣈⣩⣧⡖⣿⣿⣷⣾⣿⢿⢲⣴⣌⣁⠛⠉⠋⣠⢏⢎⣕⢙⡁⣤⠺⠗
⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠐⣋⡮⢭⢭⡭⢭⢲⢥⡳⣄⠛⢹⡏⠛⣠⢞⡬⡖⡭⢭⠭⡭⢵⣛⠎⠈⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠨⠀⣵⣷⣤⣹⣱⢮⣓⠦⠴⢚⡵⢎⢹⣤⣾⣮⠐⠍⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡄⠺⠃⠛⡈⠟⠤⠯⠽⠤⠻⢁⠛⠘⠗⢠⣜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠟⠘⠃⠁⠀⠀⠀⠀⠀⠀⠈⠘⠓⠺⠇

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡗⢈⣉⣉⣉⣛⣛⡿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣀⣀⣄⠠⠆⠀⠀⠀⣄⣄⡀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠠⠾⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⠐⠒⠶⠶⠶⠶⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠐⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢈⣉⣭⣭⣤⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢨⠽⠶⢞⣛⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠰⠶⢛⣋⣭⣶⣿⣿⣿⣿⣝⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠰⣾⣿⠿⣛⣭⣶⣿⡿⣟⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣡⠴⢛⡽⣾⠟⣫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠻⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣥⢞⡭⠞⣡⢞⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠇⠃⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣤⣤⣬⣀⠚⠡⣯⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⣴⣾⣿⣿⣿⣦⡀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡉⢻⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⠃⢸⣿⣿⣿⣿⣿⣿⣷⠀⠛⠿⠛⠟⠛⠛⠛⠛⠻⠇
⢠⣤⣤⣤⣤⣤⠀⣴⣿⣿⣿⠏⢡⣶⣦⠙⣿⣿⣿⣿⣿⣿⣦⠹⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⡟
⣾⣿⣿⣿⣿⡇⣸⣿⣿⣿⣿⡄⠘⣿⡿⢁⣿⣿⣿⣿⣿⣿⣿⣇⢹⣿⣿⣿⣿⣿⣿⣿⢿⠀⠀⠈⠛⠿⠿⠟⠋
⣿⢿⣿⣿⣿⠀⣿⣿⣿⣿⣿⡿⢀⠈⢲⣿⣷⣤⢀⣤⣾⣿⣿⣿⠈⣿⣿⣿⣿⣿⣿⣿⣆
⣿⣾⣿⣿⣿⠀⣿⣿⣿⣿⠋⢰⣿⣷⡀⠙⢿⠇⣾⣿⣿⣿⣿⣿⠀⣿⣿⣿⣿⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠠
⠿⠛⠼⠟⠋⠀⢹⣿⣿⣿⡀⠘⢿⣿⣿⡆⠀⠸⣿⣿⣿⣿⣿⡏⣰⠿⠹⣿⢻⡏⠙⠟⠿⡿⣗⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣦⣄⣀⣤⣄⠻⣿⣿⣷⣤⣄⣀⣤⣾⣷⣤⣤⣭⣽⣿⠟⠀⠃⠀⠀⠃⠁⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⣿⣿⣿⣿⣿⣿⣿⣷⣌⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣡⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⢿⡿⣿⣷⣦⣍⣙⡛⠛⠛⠛⡋⠉⠀⠛⠛⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠃
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⠛⠛⠁⠀⢀⡀⠀⠀⠀⠀⠉⠉⠉⠻⠿⠛⣽⠋
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣈⣛⡁⠀⠀⠀⠀⢸⣧⣤⣀⣤⡄⠀⡄⠀⠀⠀⠘⠃⠐⠀⢠⣤⣦⣤⡄
⠻⠻⠿⠋⠙⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⣴⡆⠘⠻⠛⣛⡫⢌⣀⣠⢸⠟⠛⣿⠂⢀⡠⢤⠒⢋⣭⡁⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⣠⣄⣀⠀⢫⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣷⣾⣴⣧⣾⡇⠀⠹⣧⣼⣧⣴⣶⣿⣿⡇⠀⠀⠀⠀⠀⢸⣦⡀⠀⠀⠀⣴⣾⣿⣿⣦
⢸⣷⣿⣿⣇⣸⣷⣶⣾⣿⣿⣿⣽⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠹⣯⡿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠐⣿⣿⣿⣿⣿⡇
⠈⢿⣿⠿⣽⣿⣿⣿⣿⣿⣯⡹⠿⠿⢋⣿⣿⣿⡇⢸⣿⣿⣿⣿⡾⢻⣿⣿⣇⠀⠀⡀⠙⢿⣷⣾⣿⣿⡇⠀⠀⣦⣀⣴⣿⣿⣿⠀⠀⠀⢿⣿⣿⣿⣿⡇
⣴⣶⣷⣾⣿⣿⣿⣿⣿⡿⠿⠃⠤⠀⠸⣿⣿⣿⡇⠸⠟⠛⠋⠁⠀⣿⣿⣿⣿⡿⢿⣿⡀⠀⠈⠙⠛⠛⠃⠀⠀⣿⡟⣿⣿⣿⣿⠀⠀⠀⢸⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠙⠛⠟⠁⠀⠀⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⣽⣧⣿⣿⣿⣿⠀⠀⠀⠀⡇⣿⣹⣿⣧⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⢰⣿⣿⣷⡄⠀⠸⠇⢠⣤⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⢠⡄⠀⡇⡇⣿⢿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡞⣿⣿⣿⢃⣠⣶⡆⢸⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⢿⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣾⣿⡇⠀⣿⠁⡟⢸⣿⣿
⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣇⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣛⡓⢒⣻⣿⣿⣦⠀⠀⢠⡀⠀⠀⠺⡿⣿⣿⣿⠿⠋⠁⠀⠀⣿⣠⡧⠾⠟⠋
⠒⠂⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣁⣀⣰⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⡂⣆⣐⣤⣾⣾⣧⣄⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⢿⣿⣹⣿⣿⣿⣿⣿⡿⣿⣷⢮⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠻⢿⣿⣟⣿⣿⣿⣿⣿⢿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠷⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡟⢏⠿⢿⣿⣿⣟⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⢀
⠀⠀⢀⣤⣾⠏⠀⠀⣀⡼⠋⠀⣠⣦⡀⠀⠀⠀⠀⠀⢀⠔⣊⣉⠀⠨⣵⣿⣶⡄⠄⠀⠀⠀⠀⠠⠀⠀⠈
⠀⢠⣿⡿⠃⣠⡴⠛⡉⠄⠊⠉⣹⡟⠋⣤⣖⢖⣣⡤⠟⢋⠡⢀⠀⣀⢬⣿⠭⡛⣣⣄⠦
⣰⣿⠋⢠⡾⠁⠠⠀⠀⠀⠀⠀⠀⣤⢾⡖⢓⣨⣭⠖⣠⢏⡼⣡⢞⣵⢋⡴⡫⢛⣥⣭⡇⢠
⡿⠁⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⣾⡵⡋⢰⣻⠝⣊⣴⠃⡜⢱⠏⡼⢫⠄⡸⢡⡿⠛⠁⢹⣮⡃
⠁⢀⣿⠁⢀⠀⠀⠀⠀⠀⠀⢸⠏⢀⢔⡵⣿⢲⢲⡆⣸⠉⢛⢸⢃⣏⠁⠁⢉⠠⠊⠀⣩⣿⣢⢀
⠀⣼⡿⠀⢸⣇⠀⠀⠀⠀⡰⠇⡀⣀⡾⡴⣡⢣⡟⣠⢁⠀⡾⠀⣼⣿⡗⡇⠃⡔⠀⠀⣅⣿⣞⣧
⢠⣿⠃⠀⣼⣿⠟⣦⡀⠀⠁⢦⡕⢙⠐⡀⠛⣋⡴⣳⠃⢊⡵⢀⢿⢿⣻⠃⢸⣷⡆⠸⠝⣋⣡⠠
⣼⠃⠀⣴⣿⠋⣼⡿⠛⢲⣆⠋⠀⡉⢀⣵⢰⢟⢣⢁⠀⠸⡣⠀⢯⠔⣣⠀⠘⡟⢡⠖⠈⠛
⡇⢀⣾⡿⠁⣼⠟⢀⣴⡿⠋⠤⠂⣿⠈⡏⣼⢠⡞⡍⡴⠴⢃⠞⠀⡄⣘⣷⡦⠟⠧
⠣⣾⣿⡇⠚⠏⢀⣾⠏⠀⠀⠀⠀⠛⠌⠐⢡⡞⡜⡉⠗⠱⠃⢀⡞⢛⠹⠁⠁
⠀⢿⣿⣿⠀⠀⣿⠇⡔⠀⠀⠀⠀⠀⠓⠀⠈⠡⢸⢡⡀⢸⠰⣾⠃⡝⢰⠁
⣄⠀⠙⢿⣷⣤⣹⠰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢧⢷⠁⠀⣸⣩⠆⠸⠀⠀⠠
⣿⣦⡀⠀⠙⠛⠿⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠘⠈⠀⠐⠟⠸⠀⠄
⠹⣿⣿⣦⡀⠀⠀⠒⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⣷⡙⠿⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠠
⡟⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠐

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠠⠀⠒⠀⠀⠠⠘⠀⢠⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡀⠀⠀⡀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠈⠃⠀⠀⠀⠀⠀⣴⡄⠀⠀⠀⠀⠀⠃⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠁⠈⠀⠀⠀⣤⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠓⠀⢀⣤⣤⣠⣀⠀⠈⠃⠀⠀⠀⢀⠀⠠⠆
⠀⠀⠀⠀⠀⡉⠀⠀⠀⠀⢿⠁⠀⠀⢠⣴⣿⣿⡿⠿⠷⣿⡟⠲⣄⠀⠀⠀⠹⠇⠀⠀⠀⠈⡁
⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⢴⢿⡿⠍⣿⠻⠓⠻⢻⡏⢱⣾⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀⠀⡾⠸⣯⡶⠁⠀⠀⠀⠀⠳⣦⡇⢸⡆⠠⣀⡀⠀⠀⢀⠄⠂
⠀⠀⠀⠀⢀⠀⠁⠀⠀⠈⠉⠑⠀⢿⢩⡍⠻⣄⠀⠀⠀⢀⡞⢋⣯⣸⠇⠀⠈⠉⠀⠀⠀⠁⢄
⠀⠀⠀⠀⢈⡀⠀⠀⠀⠀⠀⠀⠀⠙⣾⠳⢄⣿⡴⡖⠶⣾⣧⡼⣻⠟⠀⠀⡀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⣾⠄⠀⠀⠈⠓⠤⣿⣿⣷⣷⣿⡿⠟⠉⠁⠀⠠⢽⠆⠀⠀⠀⢀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⢀⡤⠀⠀⠉⠉⠙⠀⠀⢠⣆⠀⠀⠀⠀⠀⠈⠇
⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⢀⣄⡀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⡀⠀⠠⡄⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠀⠀⡄⠀⠀⡀⠃
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠴⠀⠀⠀⠀⠀⠘⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢰⡖⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⢸⡇⢀⡤⢄⠀⣠⠤⡀⢸⡇⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡏⠀⣠⣤⠶⣄⣾⠃⢸⢴⡟⠀⣧⣿⠁⠀⠀⣰⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡇⠀⠉⠀⣠⠃⢿⣀⡼⠸⣇⣠⠇⢿⣤⠤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠒⡒⠊⠁⣀⠀⠀⠀⠀⠀⢀⠀⠀⢠⡀⠀⡠⠤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠔⣺⡇⢠⠎⣴⢃⡤⢄⠀⣴⠟⠀⠄⣸⣇⠊⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⣿⢠⢃⠈⢡⡿⠂⡾⠸⣿⠑⢒⠋⣿⡏⠀⢀⣴⡗⢲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢠⣿⠟⠉⠁⠘⠧⠔⡁⠀⠙⠒⠊⠘⠿⠀⡔⢡⡟⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠆⠀⠀⣿⠀⠀⠀⢀⠔⠈⡆⣖⠂⠁⢢⠀⠀⠈⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⡤⠤⣤⡀⠑⠄⡃⠬⡣⠐⠁⢰⡖⠒⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⢸⡇⢀⡤⢈⠈⣠⠄⡀⢸⡇⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⣠⣴⠶⣄⣾⠃⢸⣶⡟⠀⣿⣾⠇⠀⠀⣠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠉⠁⣠⠇⢻⣄⡼⠘⣧⣠⠇⢿⡤⠤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣝⠒⠒⢊⠁⢀⣄⠀⠀⠀⠀⠀⠀⢠⡄⢀⡠⠐⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠃⣸⠇⡔⣿⡗⢡⡏⣠⠄⡄⣠⠖⢰⣸⣧⠋⠀⣀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢀⣿⡜⣸⡟⠀⣿⣿⠓⢠⢷⡟⢂⠎⣿⣇⠔⠋⣼⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃⢸⡿⠀⣿⠁⢸⡇⠻⠤⠎⠀⠙⠊⠀⣹⠁⣠⠞⠁⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠷⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⠰⠄⠛⠸⡈⢃⠘⢁⡃⠓⠴⠆⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠰⠖⡉⠀⢒⡼⠂⠀⡴⠋⠀⢠⠶⠟⠀⠐⣂⡉⠰⠆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠂⠕⢰⠇⠀⠘⠁⠀⠀⢀⠀⠀⢀⠀⠀⠀⠋⠉⠁⢀⡪⠐⠣⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡐⢌⢄⡇⠀⠈⠀⡀⠀⣰⡆⢰⢞⡁⠶⡟⢠⣤⠶⠀⠀⠀⠉⠉⠉⠠⡡⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠠⡜⢁⡀⠈⠃⠀⡀⠀⣿⡀⠋⡉⠤⠄⠢⠔⠠⢄⢁⠐⢻⠚⠃⠀⠀⠐⠲⠦⠈⢣⠄
⠀⠀⠀⠀⠀⠀⢀⡐⠰⠸⣇⠀⠀⠀⣽⡠⢁⡴⠈⢥⠀⢒⣐⣂⡒⠀⡌⠃⢦⡈⠟⠛⠁⠀⠀⣀⡀⡄⢂⡀
⠀⠀⠀⠀⠀⠀⣨⠃⣤⡀⠀⠀⠘⣧⡬⠠⠖⣀⢋⠴⡊⢁⡀⠀⡈⠑⢢⡙⣀⠰⠀⢹⠷⠆⠀⠈⠙⠖⠈⣤
⠀⠀⠀⠀⠀⠀⡒⠰⠘⠧⠀⠠⣤⢉⠀⠎⢠⢀⠎⠀⠊⢀⠄⠤⡀⠑⡁⠱⡀⠄⠸⠀⡲⣆⡀⠀⢤⡀⠔⢒
⠀⠀⠀⠀⠀⠀⣩⠠⣄⠀⠀⠀⡘⢃⢈⠆⠴⢸⠀⢈⠀⡃⣿⣿⢨⠀⡁⠀⡇⠖⠰⡀⢡⡌⠀⠀⠀⠙⠂⣉
⠀⠀⠀⠀⠀⠀⠥⠔⠈⠓⠀⠈⠹⠮⠀⡆⠘⠈⢆⢀⢄⠈⠒⠐⠁⡠⠁⡰⠁⠃⡰⠀⣁⠛⠂⠀⢲⡄⠆⠬
⠀⠀⠀⠀⠀⠀⠛⡄⠴⣄⡀⠀⠰⢶⣇⠀⠆⠋⣬⠳⣄⡀⠁⠈⢀⡨⠞⣡⠉⠴⠂⡘⢻⡄⠀⠀⠈⠛⢀⡋
⠀⠀⠀⠀⠀⠀⠈⠡⠘⠈⠉⠀⠀⢀⣤⣴⡈⠳⢀⡘⠀⠬⠩⡍⠡⠀⢓⡀⠞⢁⠒⣟⠀⠀⠀⢹⡄⠆⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢃⡀⠲⠦⠄⠀⠀⢀⡤⣷⠄⣁⠑⠂⠔⠂⠐⠂⣀⣠⠊⣿⠀⠈⠀⢤⡀⠈⢁⡜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠡⢊⠂⣀⣀⣠⠀⠀⠀⠾⠛⠃⣼⠶⢈⡽⠇⠸⠏⠀⠈⠀⡀⠀⢸⠁⡑⠌⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢢⡄⡨⠁⢀⣀⣠⠀⠀⠀⠁⠀⠀⠁⠀⠀⢀⡄⠀⢰⠇⢔⠠⡶⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠰⠆⣈⠩⠥⠀⣴⠶⠃⠀⣠⠞⠀⠠⡞⠥⠀⣀⠴⠆⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠰⠆⢦⢨⢁⡄⢡⡈⡆⣤⠰⠆⠁⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣀⠀⠀⠀⠀⣠⠞⢳⡄⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⢀⣞⡵⠯⠿⢷⣶⣵⣼⡐⠂⣷⣦⣶⡿⠯⠽⢮⣣⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡊⡦⠭⣢⢀⣴⣿⢋⢴⣷⣦⣴⣨⢻⡽⡆⢲⢣⣟⢥⣦⣤⣶⣦⡙⣿⣦⡠⣐⠬⢶⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂⢇⢑⠪⢫⢻⣇⢶⣿⡿⣛⠿⣿⣷⣭⣻⣷⣭⣾⣿⠿⣛⢿⣿⡖⣸⡟⡞⡅⡊⡸⡐
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣼⣾⡢⣤⣙⣾⢣⣿⡟⣼⢉⣙⠦⣝⢿⡛⢻⡿⣫⠴⣋⡙⣇⢻⣿⡜⣿⣋⣴⢝⣷⣶⣤⣀⡀
⠀⠀⠀⠀⠀⢰⢻⡭⠟⡋⣉⣭⣝⣒⣭⣽⣿⣿⣱⣯⣿⣿⠿⠿⠟⠛⠛⠻⠿⠿⣿⣿⣽⣎⣿⣷⡫⣍⣒⣫⣭⣉⢙⠻⢭⡿⡆
⠀⠀⠀⠀⠀⢸⣽⠃⣽⣿⡿⠿⠿⠿⠿⢿⣥⣿⠟⣋⡥⢰⡶⢉⠻⠇⡼⠿⠉⢲⡕⢬⣙⠺⣿⣬⡿⠿⠿⠿⢿⣿⣿⣯⠘⣿⡇
⠀⠀⠀⠀⠀⢸⣿⠀⣿⣿⡇⣟⢉⣭⣽⡷⢋⣴⣋⠟⢧⠶⣾⠏⢩⣏⣱⡍⠻⣷⠶⡼⠻⣑⣤⡙⢿⣯⣭⡙⣿⢸⣿⣿⢀⣿⡇
⠀⡶⣥⠶⢚⢿⢟⣳⡽⣿⣷⢻⣸⣯⠟⣴⡉⣩⠖⣯⣱⡦⣐⢋⣬⡿⢿⣥⡙⣂⢴⣎⣽⠾⣍⢈⣦⠻⣿⣷⡏⣾⣿⢣⢾⡿⡟⡒⠶⣬⢢
⠀⠁⢘⢮⣒⠌⣈⣹⡷⡽⡿⣮⣿⢇⢌⢈⣵⡿⣢⠹⣽⡿⠛⣿⢯⣤⣬⡽⡟⠛⢻⣯⠅⣄⢿⣮⡋⡣⡹⣿⣼⢿⣯⢲⣋⣁⢡⣂⡵⠃⠁
⠀⠀⠀⣹⣾⡿⡿⣒⣵⣾⣥⣾⡇⡜⢹⣘⢥⣤⢥⣴⣿⠵⡿⣫⣯⣶⣾⣥⣝⢿⡦⣿⣦⡬⣤⠬⣃⡏⢣⢹⣷⣬⣿⣮⡳⡿⢿⣶⣆
⠀⠀⣴⣿⠋⣴⣾⣿⠟⣩⢶⣿⢑⠿⢣⠝⢶⠿⣯⠉⣾⢿⣽⣿⣽⢷⡿⣯⣿⣮⡷⣷⠉⣽⢿⡶⢫⡜⠻⡌⣿⡷⣍⠻⣿⣿⣦⡙⣿⣦
⠀⣞⢵⠁⣾⣿⣿⢳⡏⢱⣿⣿⢸⣶⢸⣷⢰⠈⢽⣧⣙⢸⣿⡿⡋⣰⡖⢙⢿⣾⡷⡟⢢⡯⠁⡆⢾⡇⡶⠂⣿⣿⡎⢻⡎⣿⣿⡷⢘⡮⣳
⠀⠈⢷⣷⡘⢻⣿⣷⣝⡳⣝⣿⢘⣤⡸⣅⣴⣬⡿⠁⣿⡼⢿⣿⢿⣴⣢⡿⣿⡾⣣⡿⠈⢿⣵⣦⣸⢇⣤⠃⣿⣭⠞⣩⣾⣿⡿⢁⣾⡾⠁
⠀⠀⠀⢻⣿⣦⣬⣙⢿⣿⠶⣿⡇⢏⢱⢛⠤⠿⠽⢿⣾⢼⣯⡻⡿⣷⣿⢿⢟⣽⡭⢷⡿⠯⠿⠤⡛⡎⡹⢸⣿⠷⣿⡿⣫⣥⣴⣿⡞
⠀⠀⣂⡴⣚⠍⠓⢺⣇⢯⣧⡾⣷⡔⡝⠘⢷⣮⡳⢠⢿⣦⣉⣫⡽⠶⠾⢩⣟⣉⣸⡿⡄⠞⣴⡾⠁⠫⢣⣿⢷⣼⡽⣹⡖⠟⠩⡒⢧⡀
⠀⢏⠾⣬⣐⣸⣜⣩⢆⣿⣿⣸⢻⣷⣌⠾⠃⠻⡄⡛⢻⡯⢛⡙⠿⣶⣶⡿⢋⡛⣽⣟⢗⣠⠟⢘⡷⣡⣿⡿⡎⣿⣿⡸⣝⣣⣇⣀⣥⠷⡵
⠀⠀⠁⠀⠉⢹⣿⠁⢿⣿⠇⣟⠘⠿⢿⣦⡑⢿⡒⣍⡧⣤⣤⡍⠸⡏⡻⠃⢩⣥⣤⢮⡩⢚⡿⢋⣴⡿⠿⠃⣿⢸⣿⣿⠘⣿⡏⠈⠀⠈
⠀⠀⠀⠀⠀⢸⣿⡀⣿⣿⣷⣯⣭⣭⣭⣽⠿⣶⣍⡻⠧⣰⣮⠉⣽⡼⢇⣝⠉⢵⣆⡼⢟⣠⣶⠿⣯⣭⣩⣭⣭⣾⣽⣿⢀⡿⡇
⠀⠀⠀⠀⠀⢸⣺⠧⣍⡋⠛⠿⢟⣛⠿⢽⣷⣽⢻⡿⣷⣾⣤⣭⣍⣁⣉⣭⣭⣤⣷⣾⢿⡟⣿⣾⡯⠿⣛⡻⠿⠛⢙⣩⠼⣿⡇
⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⢿⡿⡫⠿⢛⣾⣹⣿⣆⢿⡙⢛⡭⢟⣿⣋⣙⣯⡻⢭⡛⢋⡟⣸⣿⢏⢾⠛⠾⢝⣿⡿⠿⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈⡞⢐⡠⡹⣸⡗⣿⣿⣮⣛⣭⣶⣿⠟⣻⣻⠿⣿⣶⣭⣛⣵⣿⣯⢺⣏⣏⠆⡊⢣⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣧⠵⡞⠙⢿⣷⡐⢹⣿⠿⢿⠫⣲⢯⣏⢹⡽⣎⠝⡿⢿⣿⠯⢊⣾⡿⠊⠳⠬⣼⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂⠀⠀⠀⠀⠙⣟⢦⡤⣤⣴⣾⢿⣾⢀⡀⡗⡿⣷⣶⡤⢤⡴⣻⠏⠀⠀⠀⠀⠂⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⠛⠉⠁⠀⠁⠙⣄⣰⠟⠀⠀⠁⠉⠛⠾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠪⠕

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣠⠠⡮⠵⢄⣄⣠⡴⠮⢵⠄⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡐⠴⣰⣆⡧⣯⣶⠭⠛⠛⣿⡯⢽⣿⠛⠛⠭⣶⣽⢬⣰⣂⠧⢂⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⡲⣫⣖⠍⠚⢟⢝⠭⣙⢧⣀⡴⣫⠊⠑⣝⢦⣀⡴⣋⠯⡩⣻⠛⠭⣒⣍⢖⢄
⠀⠀⠀⠀⠀⠀⠀⠀⣔⠆⣛⣾⣹⢿⣆⣄⣀⡿⡎⢰⣄⡑⢿⣱⠱⡜⢣⠎⣞⡻⢊⣡⡆⢱⣻⣄⣠⡸⠿⣏⣷⣛⠰⣢
⠀⠀⠀⠀⠀⠀⠰⡝⡵⡻⠓⢿⣱⠉⡑⢒⠭⣸⡇⣷⣈⣿⣨⠧⠽⠧⠼⠯⠼⢥⣿⣁⣾⢸⣟⠽⡓⢉⠉⡾⡿⠛⢝⢮⢣⢆
⠀⠀⠀⠀⠀⡠⣛⣼⣿⣄⣀⣸⢹⠠⣽⠚⣶⡼⢓⣩⠴⣖⢲⡋⠯⠹⠯⠽⢙⡖⣲⠮⣍⡚⢧⣶⠃⣭⠄⡏⣇⣀⣠⣿⣯⣛⢄
⠀⠀⠀⡀⡜⣪⣷⡞⡕⠓⠒⡒⠯⣧⡻⢛⣥⡞⡯⣘⠥⠖⢚⢋⠉⠉⠉⢉⡙⡒⠪⢭⣃⢽⣓⣬⡳⣟⣼⠽⢒⠚⠒⢯⣳⣾⣕⢣⢀
⠀⠀⢘⡌⡜⠌⠈⢡⢡⠈⣃⠛⣻⢊⣴⠛⢖⠵⠋⠁⣰⢔⢬⠙⠮⡮⣵⠗⠻⡥⣢⣄⡄⠑⠮⡲⠛⣮⡳⣟⠛⣘⡁⡌⠏⠉⠳⣣⢡⡂
⠀⠀⢠⢽⣾⣤⣤⠼⣧⣣⡘⡟⣡⢻⢜⡕⡡⣐⣀⣷⣿⡱⣼⠾⣣⣟⡻⣜⠿⣧⠎⣿⣾⣠⣄⢌⠪⠡⡟⣜⢿⣃⣜⣸⢧⣤⣤⣿⡧⡆
⠀⣠⡥⣯⣟⡕⢊⣉⢭⣭⡝⣰⣋⢢⠏⢠⢫⣟⢡⡭⣯⡿⣷⣶⢿⣧⣼⡽⣶⣾⢯⣽⢫⡍⣻⡝⡄⠐⠐⣙⣎⢯⣭⡭⣉⡑⢪⢻⣽⢬⡄
⠀⡚⣜⡟⠻⣜⢆⠹⣦⣼⢰⢧⢡⠏⠠⢽⡃⢼⢟⣩⣟⡷⠋⠙⣾⡠⣄⣿⠋⠙⢾⣿⣍⡻⠧⢘⡯⠀⠀⠘⡼⡸⣧⣴⠏⡰⣣⠟⢻⣣⢓
⢎⡅⣿⠀⢀⡼⢗⣱⠦⡏⡜⠲⡾⠀⢌⣾⣽⣻⡽⢳⠗⠳⡴⢋⣠⣦⣴⣄⡙⢦⠞⠺⡖⢫⣝⣯⣷⡡⠀⠀⠖⢧⢻⠴⣞⡾⢧⡀⠀⣿⢭
⠈⢇⡷⡻⡪⠚⣁⠦⠷⡇⣏⡃⡇⢬⢲⣁⣾⠹⢿⡯⣀⣸⠠⢾⠟⣡⣌⠻⣷⠄⡇⣀⢵⣻⠟⣷⣚⡗⡥⢀⢸⣹⢸⡾⠴⣈⠓⢝⢟⢏⠎
⠠⡲⣷⣵⢕⢤⡉⠖⡶⡇⣏⡅⡇⢚⠾⡍⠿⣰⣿⣿⠉⢰⠘⢿⣦⡙⢋⣴⡿⠂⡇⠉⣾⣿⣶⠿⢩⠧⡓⢸⢸⣹⢸⢶⠲⢉⡤⣪⣮⢮⢖
⢎⣉⣿⠀⠈⢳⡮⣻⠖⡇⢳⠴⢧⠀⢊⣿⣿⣵⣳⡼⠦⡴⠳⣌⠙⠟⠻⠋⣡⠞⢦⠴⢧⣺⣯⣟⣿⡑⠀⣼⠦⣞⠸⠲⢝⣷⡞⠁⠀⣿⢸
⠀⢥⢣⣧⣴⣫⠎⣰⠟⢻⠘⡖⡈⡆⠀⣸⡇⢲⣮⣙⣿⡷⣄⣠⣯⠑⠊⣽⣅⣠⢾⣿⣋⣵⡖⢨⣗⠂⢰⢃⢳⠇⡝⠻⣆⠱⡝⣆⣼⡜⡬
⠀⠐⡓⣟⣇⡣⢌⣉⣚⣛⣳⠹⣍⠜⣄⠚⣝⣯⣘⣻⣟⣳⡿⠿⣿⡞⢳⣻⠿⢯⣶⣻⣞⣃⣽⣣⠓⣠⠣⣩⠏⣜⣛⣓⣉⡡⢜⣹⣻⢚⠊
⠀⠀⠠⢸⢿⠛⠛⢲⠍⡝⢩⣳⡙⣴⢊⡣⡐⠉⠏⡿⣿⡱⢻⣶⡟⣾⣵⢻⡶⡗⢎⣿⢿⠩⠉⢊⢜⡑⣧⢋⣼⡍⢣⠹⡶⠛⠛⡿⡗⠇
⠀⠀⠨⢃⢣⣦⣀⣸⡼⢀⡍⣤⣿⢌⠻⣤⠎⡲⣄⠀⠹⠪⢚⣶⢶⢿⡿⡆⣲⡓⠝⠏⠂⣠⢖⠵⣤⠟⡡⣿⣤⢯⡁⢇⢢⣀⣴⡝⡘⠅
⠀⠀⠀⠁⢣⢝⡿⢥⡣⢤⡤⠥⣖⡟⣵⣌⡛⢭⣔⢩⡒⠦⢬⣌⣀⣀⣀⣀⣡⡥⠴⢒⡍⣢⡭⢛⣥⣮⢻⡲⡬⢤⠤⣜⡸⢿⡫⡞⠌
⠀⠀⠀⠀⠀⠑⣽⢻⣿⠋⠉⢹⣸⠐⣛⢠⠿⢲⡬⣙⠲⠯⠼⣀⣖⣲⣖⣲⣈⠧⠽⠒⣋⢥⡞⠿⡄⣛⠂⣇⡏⠉⠙⣿⡏⣭⠊
⠀⠀⠀⠀⠀⠀⠰⢣⡳⣗⣄⣾⡾⣀⡁⠬⣖⢹⡇⡿⢉⣿⢐⡖⣲⡖⢲⣖⢲⡒⣿⡉⢿⢸⣯⣒⡥⣈⣠⢷⣷⣤⣮⢟⣜⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠪⠶⣯⢿⣱⣶⠟⠋⠉⣷⢇⠸⠋⡡⣺⡹⡰⢫⡝⢆⢯⣷⢌⡙⠇⡸⣼⠉⠙⠻⣷⣎⡿⣽⡲⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠽⣑⠵⣢⣤⣭⣎⣲⣩⡞⠉⠻⣝⢄⡠⣫⠞⠙⢓⣭⣒⣑⣽⡤⣔⠭⣋⠮⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠡⠲⠉⠫⡓⣟⠿⣆⣤⣤⣿⣗⣺⣟⣤⣤⣖⠿⣻⢚⠹⠍⠖⠌⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠐⣆⡲⠊⠃⠙⠓⢆⣒⠂⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢨⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⣶⣿⣿⣿⣿⣿⣷⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠠⣠⣀⣰⣘⢻⠿⣛⢽⣗⠻⠳⠻⢓⣿⣽⡻⠟⣛⣀⠐⣄⡠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⡺⠊⣦⣰⣶⠇⡽⠟⢝⠮⣉⠈⢃⠋⢁⡩⢞⠝⠿⡅⢷⣴⣠⡎⠪⡁
⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣺⢥⣿⠿⠻⢋⡾⣷⣤⣁⣥⣿⢿⣯⣿⢽⣦⣅⣡⣴⡿⣎⠫⠿⢿⣧⢽⣶⣦⣄
⠀⠀⠀⠀⢀⣄⣾⣿⣿⣯⣿⠖⣶⣶⣲⡿⢶⣿⣯⣼⠟⠳⡾⠋⠹⡾⠛⢻⣤⣿⣷⠾⣗⣲⣶⠖⠾⣮⣻⣿⣷⣦⣄
⠀⠀⠂⣤⣿⡿⣯⠿⣯⢸⢰⠃⢸⠽⣿⠟⢾⣇⣠⡭⠒⠲⡏⠹⠉⡷⠒⠪⣥⣀⣽⠞⢿⡿⢿⠀⣳⢸⠨⡿⢯⣿⣿⣧⡄⠂
⠀⠀⢰⣗⣏⣎⠟⠠⡍⢨⢔⢾⡿⠒⠿⡠⣖⠁⢀⣉⡤⡴⠈⣠⡈⠣⠤⣄⣁⠀⢑⡦⡸⠗⠿⣿⣖⣾⠀⢅⠘⢏⣏⣗⣆
⠀⠀⢸⣽⣿⣏⡰⠚⣰⣳⣿⡴⠷⣠⣊⠀⣨⠶⠿⠟⢼⢁⣴⠁⠱⣄⢹⠘⠿⠷⢮⡀⢈⣢⡰⠷⣾⣿⣷⡘⠓⣬⣿⣿⣽
⠀⠀⣸⣿⣿⡭⢔⡪⣿⣏⣽⡄⡼⠡⠙⠚⠑⠙⣓⣀⣸⡞⣠⣖⣦⡙⣾⣀⡜⡛⠁⠙⠚⠡⠑⡄⣼⣏⣿⡯⣒⠬⣽⣿⣿⡂
⠀⠀⠿⣫⠼⡗⠙⣀⡿⢿⠁⡰⠓⢺⡈⢫⡉⠉⢉⣭⢛⢵⢨⣯⣯⢲⢝⢫⣍⡉⠋⣩⠋⣰⠒⠳⡀⢹⡿⣏⡘⠙⡿⢾⡿⠇
⠀⠔⡵⣙⣷⠹⡿⢿⣷⡞⢳⢧⡄⣼⢵⡀⢷⡌⢻⣿⣿⡲⡾⣜⡼⡖⣺⣿⣿⠇⣴⠇⣴⢹⡄⣤⣇⠛⣶⣿⠿⡿⢳⣟⡹⡕⡄
⢔⣰⠤⠽⣿⠅⢹⣺⣋⣷⠄⠀⠀⣿⡧⡋⠈⣷⡤⡪⢱⣆⣛⠈⢘⡃⣶⠩⣲⣶⡏⠈⡣⣿⡇⠀⠈⢴⣟⣻⣺⠁⡿⣿⠥⣴⠐
⠀⢆⢧⢚⣿⢡⣯⣼⣿⣇⡘⡲⠂⢿⢵⠃⣼⠟⣶⣽⣷⣪⣯⢞⢮⣧⣶⣿⣷⡆⢿⡌⢷⢹⠇⠲⡒⣀⣿⣿⣬⣧⣹⣟⢢⢟⠆
⠀⠀⣥⣻⣥⣏⢉⠛⣧⣼⠁⢣⣀⢸⠃⡰⠋⡈⠛⠿⣙⠴⠸⡾⣾⠐⢜⡹⠟⠃⣐⠳⡀⢻⣀⣠⠋⢹⣤⡟⢋⢉⣧⠽⣫⡅
⠀⠀⣻⣿⣾⠯⢝⡪⣟⡟⣷⠖⢇⠨⢈⣉⡉⢉⣛⠟⢻⣇⠸⡿⡿⢀⣿⠛⢏⣋⠉⢉⣉⠠⢀⠗⢶⡟⣿⡧⣚⠭⢿⣾⣿⡃
⠀⠀⢺⢿⣿⡗⢦⣈⢻⣻⣿⣦⣬⠳⡛⠈⠳⣭⣥⣄⢰⠘⢧⠀⢤⠟⢰⢂⣤⣭⡵⠋⠘⡳⢫⣤⣾⣿⣻⢋⣀⠚⣿⣿⢿
⠀⠀⢸⣟⡗⡗⣄⠈⠆⢹⣝⣴⣧⣀⣬⠢⣞⠀⠙⠛⠮⠥⠀⠣⠃⣠⠭⠞⠛⠁⢘⡦⢪⣄⣠⣷⣞⣽⠁⠎⢀⡔⡏⣛⡏
⠀⠀⠀⠷⣿⣿⡿⣥⡯⢸⢡⡊⢹⢥⣿⣄⣼⠗⠲⠯⣀⣠⡏⢢⠊⣧⣀⡨⠷⠖⢯⣄⣼⣧⣽⠉⣢⢸⠨⣏⡼⣿⣾⡷⠇⡀
⠀⠀⠀⠀⠙⠿⣿⣿⣿⡾⣦⣄⣼⣿⣻⣯⣼⣷⡶⢿⣄⣠⣯⡉⣩⣧⣀⣸⠷⣶⣿⣬⣟⣻⣿⣄⣤⠾⣾⣿⣿⡿⠟
⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⢻⠴⣶⣤⢠⡙⣦⣿⠷⠚⠿⣿⢼⡿⣿⢼⡿⠟⠺⢿⣧⡞⣡⣠⣴⡶⢼⣘⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣣⡉⡿⢻⢿⡜⢧⣄⠕⡥⠚⡉⡙⡉⠛⠪⣕⢅⣤⠏⣼⢿⠿⡏⣡⣏⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠪⠰⠂⠀⢗⣉⣭⣛⢼⣟⣈⣡⣭⣘⣿⢼⣫⣍⣕⢓⠐⠲⠊⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣷⣿⣷⣿⣷⣿⣯⠿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠹⠙⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠠⠄⠀⠃⠀⠙⠀⠘⠁⠀⠃⠀⠤⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠲⠀⠈⢀⣀⡠⣤⣶⣲⣲⣖⣶⣲⣲⣦⡤⣄⣀⠈⠀⠰⠂⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠉⢀⡠⣴⡾⠝⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠽⣲⡤⣀⠈⠁⢠
⠀⠀⠀⠀⠀⠀⠰⠂⡀⣠⣔⡿⠛⠁⠀⠀⡄⡀⠀⠀⠀⡄⠊⠂⡄⠀⠀⠀⡀⡄⠀⠀⠙⠻⣗⣤⡂⠀⠶
⠀⠀⠀⠀⠀⠒⠀⢨⣾⠟⠉⠀⠀⠀⠀⠈⠀⢀⣁⡀⣊⣤⣤⣤⣌⡀⣀⣁⠀⠠⠀⠀⠀⠀⠈⠙⢿⣮⠠⠐⠂
⠀⠀⠀⠐⠂⢀⣴⡽⠃⠀⠀⠄⠄⠀⠤⢃⡾⠛⠋⠙⠋⠀⠀⠀⠈⠛⠉⠙⠻⣆⠃⠄⠀⠀⠄⠀⠀⠻⣵⣄⠀⠒
⠀⠀⠐⠂⢀⣮⡟⠀⠀⠀⠀⠀⢠⡶⠛⠛⢡⠀⠀⠀⢩⠀⠠⠀⢨⠀⠀⠈⢀⠙⠛⠳⣦⠀⠀⠀⠀⠀⠘⣿⣄⠀⠒
⠀⠰⠂⢀⣞⠎⠀⠀⠀⡀⠬⢠⣟⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠆⠀⠀⠀⢸⣇⠨⠄⡀⠀⠀⠈⣞⣆⠀⠲
⠠⠆⠀⡼⡝⠀⠀⢢⠁⠀⣼⠏⡡⠒⢠⠀⠀⠀⠀⠀⡀⢀⠀⢀⠀⠀⠀⠀⠀⠀⢠⠐⠠⡉⢻⡄⠀⢡⠂⠀⠸⡽⡀⠀⠆
⢠⠀⢀⣿⠇⠀⠀⠀⢀⢸⡇⠀⠀⠠⠀⠀⠀⠀⢐⡴⠋⠉⠝⠉⠍⢳⣄⠀⠀⠀⠀⠀⠀⠘⠀⣿⢀⠁⠀⠀⠀⢿⣇⠀⢠
⣀⠀⢸⢾⠀⠀⠀⡠⠂⣼⠏⠠⠂⠆⠀⠀⠀⢠⠏⠀⠁⠄⡐⡀⠄⠑⠈⣧⠀⠀⠀⠀⠆⠢⠈⢿⡄⠢⡀⠀⠀⢸⢾⠀⢀
⠀⠀⢸⣿⠀⠀⢎⠀⠰⣿⠀⠀⠀⠀⠀⠀⠀⢸⠇⠂⠰⡁⠰⠀⠱⠀⠀⢿⠀⠀⠀⠄⠀⠀⠀⢸⡇⠀⢈⠆⠀⢸⣿
⠉⠀⢸⢾⠀⠀⠀⠑⢄⢻⣆⠀⠄⠆⠀⠀⠀⠨⣆⠀⠀⠂⠡⠁⠂⡀⢀⡟⠂⠀⠀⠀⠆⠔⢀⣾⠃⠄⠁⠀⠀⢸⢾⠀⠈
⠘⠀⠈⣿⡆⠀⠀⠀⠈⢸⡇⠀⠀⠀⠀⠀⠀⠀⠨⠳⣄⣀⣠⣂⣂⡼⠫⠀⠀⠀⠀⠀⠀⠠⠀⣿⠈⠀⠀⠀⠀⣾⡏⠀⠘
⠐⠆⠠⢳⣳⠀⠀⠜⡀⠀⢿⣄⡑⠠⠘⠀⠀⠀⠀⠀⠁⠈⠀⠈⠀⠀⠀⠀⠀⠀⠘⠠⠀⣁⣼⠃⠀⡐⠄⠀⢰⣳⠁⠀⠆
⠀⠰⠀⠈⢯⢇⠀⠀⠀⠁⢒⡘⣧⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠄⢸⡏⢐⠂⠁⠀⠀⢀⢯⠏⠀⠴
⠀⠀⠠⠄⠈⢿⣧⠀⠀⠀⠀⠀⠘⠷⣤⣤⡈⢀⠀⠀⢘⠀⠐⠀⢘⠀⠁⠀⠈⣠⣤⡴⠟⠀⠀⠀⠀⠀⢠⣿⠏⠀⠤
⠀⠀⠀⠠⠄⠨⠻⣳⡄⠀⠀⠂⠐⠀⠂⡈⢷⣤⣀⣠⣄⠀⠐⠀⢀⣤⣈⣠⣴⠏⡀⠂⠀⠀⠢⠀⢀⣴⡻⠋⠀⠤
⠀⠀⠀⠀⠀⠤⠐⢙⢿⡦⣀⠀⠀⠀⠀⠐⠀⠈⡉⠅⢉⠛⠛⠛⢋⠁⠍⡉⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠰⠀⠁⠙⠫⣷⣤⡀⠀⠀⠃⠁⠀⠀⠀⠑⢄⠄⠃⠀⠀⠀⠁⠃⠀⠀⣠⣴⡯⠋⠁⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⠈⠕⠻⢷⣲⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣴⣲⠽⠓⠉⢀⡀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠴⠀⢀⠈⠉⠑⠛⠿⠽⠽⠯⠿⠽⠽⠛⠓⠉⠉⢀⠀⠰⠄⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠂⠀⡄⠀⣠⠀⢠⡀⠀⡄⠀⠒⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⠑⠀⠔⠐⠀⢀⣼⠋⢻⣄⠀⠀⠐⡀⠀⠀⡂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠠⡐⢼⡀⡦⣀⣦⢛⡡⠞⢦⣑⢣⡆⡀⡇⣠⠄⡄⠀⡢⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡞⠴⣦⣠⣠⢌⣙⡊⠁⣱⠏⡴⠋⣤⣾⣦⡌⠳⡌⣷⠁⠉⣚⣉⢤⣠⢠⡶⠝⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⡶⠒⠒⠒⠦⢌⠻⣦⣿⡔⣇⣛⡾⠁⠹⣾⣃⡇⣼⣧⡿⣫⠤⠶⠒⠒⢲⡆⣿
⠀⠀⠀⠀⠀⡀⡀⠈⠀⠀⢿⢰⢇⣸⡿⡟⡾⣊⣷⣩⠖⢲⠏⠉⠙⣦⡎⠉⠉⢷⠒⢮⣱⣇⢾⣞⡷⣿⡀⢧⢸⡇⠂⠀⠀⡀⡀
⠀⠀⠀⠀⠘⠤⢈⣄⢳⠄⡇⢸⢠⣽⢅⡀⢱⠁⠀⡈⢧⠸⡄⠙⠚⢃⠓⠚⢀⡞⢠⠏⡀⠀⢱⠁⣠⢿⣥⢸⠀⡇⡴⣃⣀⠠⠄
⠀⠀⠀⠀⡒⢒⡲⠮⢁⣀⢻⢌⠳⡴⠋⠉⢻⠓⠚⠁⣸⣀⢸⡲⣄⠻⢃⡴⣪⢀⣸⡀⠙⠒⢻⠋⠉⢳⡴⢋⢼⢃⣀⡭⠶⡒⠐⡀
⠀⠀⠀⠀⣈⣤⠦⢚⣛⣓⠻⢡⠞⠳⣄⠚⠋⠘⢷⠄⡇⢛⣤⡁⠪⠳⠋⢀⣥⠸⠋⡇⢴⠏⠈⠛⢀⡼⠛⢦⢻⢓⣛⣓⠣⣦⣌⡀
⠰⢂⡾⠛⣋⡥⠞⣉⢰⡎⡿⠛⠓⠦⡌⠙⢲⡖⠒⠺⢇⠄⢫⡁⠁⣜⣈⠡⡹⠠⢀⠿⠲⢒⣶⠛⠉⡤⠖⠛⠺⡍⣶⣨⡙⠦⣍⡛⠷⣆⠒
⠀⠈⠻⣦⠹⡄⢿⣿⠛⠃⢧⣀⣸⠀⠹⡼⠦⣶⡔⠂⢈⡴⢶⣠⣴⢫⢳⣤⣰⢴⡌⠈⠐⣼⠢⠾⡾⠀⢸⣀⣠⠇⠛⢻⣿⠃⡼⢡⡿⠃
⠀⡀⢄⢙⣆⢳⡈⣽⢟⡖⠛⢻⠐⢶⡦⢻⡅⠬⠐⢈⣀⢷⢃⠨⠛⠄⠘⠫⠐⢸⢇⡐⠀⠡⢄⣼⠡⣶⠖⣹⠛⠓⣞⢿⡍⣰⢃⡟⢀
⠀⠀⠄⣠⡙⣦⡙⠲⠾⣆⡀⠛⢀⡠⣖⠋⠩⠙⠼⣏⠭⠉⠇⠀⢥⠰⢠⠁⠀⠋⠩⢝⡿⠘⠉⠉⢓⣦⣀⠘⠂⣀⡾⡖⠚⣡⡞⣡⡄⠄
⠆⠀⡐⠶⣅⢈⣩⣿⢸⡉⣉⣉⡉⢈⣉⡲⠏⣀⠄⣼⡷⠆⡀⢉⡃⣞⠆⣋⠀⣀⠶⣻⠁⢀⣈⢶⣊⣉⠈⣉⣉⡁⣳⢸⣯⣉⢀⡵⠖⡀⠉
⠀⠀⠐⠛⢩⠟⣩⠭⢶⠏⠁⣄⠉⠓⢯⣀⡐⣀⢲⣏⣒⣀⡀⡡⠚⠰⠘⠢⠁⣀⣐⣊⣷⢀⡐⣀⣬⠗⠋⢁⡄⠙⢷⠦⢬⡙⣯⠙⠒⠂
⠀⠐⠀⢠⡏⡼⢁⣻⣎⠧⣄⣼⠠⠶⠗⣸⠋⢀⡈⢀⠙⣿⡁⢀⡤⠄⢠⣔⠠⣹⡞⢁⠈⡀⠈⣻⡐⠷⠆⢹⣄⡤⢏⡾⣃⠹⡌⣧
⠀⠀⢴⠟⣰⠃⣼⣿⣥⡄⡞⠉⢱⠀⢰⢷⠖⠼⠇⢂⢀⠷⠖⠫⢻⡘⡼⠻⠓⠾⢋⢀⠲⠿⠖⢷⢧⠀⢸⠉⠙⡆⣤⣽⢿⡄⢳⡘⣷
⠠⠈⠧⡤⣉⡳⢤⣛⠹⠇⣳⣤⣤⠴⢋⣀⣼⣯⣴⢴⡎⠠⣐⠀⠀⢉⠋⠈⣰⡠⠌⣶⢴⣤⣿⣤⣈⠳⢤⣤⣴⡃⠿⢘⣣⠔⣋⡡⣬⠏⠠
⠀⠀⠀⠀⠉⠻⠶⣬⣉⣉⣴⡘⢆⡰⠋⣠⡀⢠⣶⠊⡇⣔⠘⡃⢀⡴⣄⣈⡛⢱⣄⡇⢶⣆⢀⣠⡈⢳⣠⠞⣰⣌⣉⣉⡴⠞⠃⠁
⠀⠀⠀⠀⢄⡀⠠⠶⡊⠁⣼⢊⡴⢷⣀⣀⣽⡬⢤⡀⢹⠉⢘⡵⠋⣴⡌⠳⣝⠈⢹⠁⣠⠬⣼⣄⣀⡸⠷⣌⢿⡄⢉⡐⠦⠄⣀⠄
⠀⠀⠀⠀⢀⠀⠀⠙⡵⠢⡇⢸⠐⣿⡞⠉⣸⡀⠈⢁⡞⢠⠏⣠⢤⡙⣠⢤⡈⢧⠘⣆⠉⠀⣸⡈⠹⣾⡗⢸⠀⡇⠷⡍⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣳⠸⡌⢻⣷⣾⢷⡍⡿⡛⠦⠼⣄⡀⣀⠿⣇⡀⣀⡾⠤⠞⡻⡏⣵⢿⡶⣿⠋⡾⢸⡂⠀⠁⠈⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢘⡇⣷⠤⠤⠥⠶⢚⣱⡿⢷⢣⡏⣭⣽⡀⣸⣭⡍⡇⢢⡿⣷⣙⠲⠤⠥⠤⢴⡇⣟⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⢧⠶⠶⠒⠛⠙⢫⣥⡀⡹⣆⠳⣄⠟⣷⡟⢆⡴⢃⡿⡁⡀⣭⠛⠛⠛⠲⠶⢦⠷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠁⠀⠁⠚⠁⠓⠉⢗⣬⡓⢤⠔⣫⡴⠏⠁⠇⢙⠎⠣⠀⠁⠀⠀⠀⠉⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⡠⠀⠃⠠⠀⠉⢵⣄⣼⠝⠀⠨⠠⠃⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⠃
⣿⣟⣛⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⢀⣀⡀⠀⢀⣀⡀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠛⠿⠁⢠⣿⣿⣿⢰⣿⣿⣷⠀⣿⡿⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣿⣀⣸⣿⣿⢸⣿⣴⣿⢸⣿⣤⣿⠀⣿⣇⣸⣿⡇⢀⠛⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠿⠟⠉⠋⠀⠛⠿⠋⠀⠻⠿⠋⠀⠛⠿⠛⠋⠀⢸⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠠⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠗⢢⡔⠑⡆⠀⢠⡯⠬⢷⠤⠾⠌⢦⡤⡜⠛⠓⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⠇⠨⠇⠀⠣⣀⠜⠣⠤⠊⠀⠣⠤⠊⠀⠣⢄⠤⠓⠤⠤⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⢿⡗⣰⣿⣿⡆⣰⣿⣿⡆⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣯⠀⣀⡀⣿⣇⣽⡗⣿⣏⣸⣷⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠿⡿⠿⠇⠹⠿⠿⠃⠹⢿⠿⠃⠸⢿⡿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠴⢼⠁⡴⢲⡆⣴⢀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠖⠋⠛⠓⠚⠖⠛⢺⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⠿⠿⠿⠿⠿⢿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⠟⢉⣀⠀⠀⠀⢠⣤⣄⠀⠙⠻⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡟⠁⠀⠉⣭⣄⠀⠀⢰⣶⠀⠀⠀⠀⠘⢿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⡆⠀⠀⣀⣀⠘⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⢠⣖⠀⠀⠻⠟⠀⠀⠘⠿⠁⠀⣠⣾⠿⠀⢹⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣧⠈⠉⢷⣦⣄⣀⣀⣀⣀⣤⣶⢿⣿⠃⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣆⠀⠈⠻⣿⣹⠛⡟⢻⣍⣹⡿⠃⠀⠀⣰⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣦⡀⠀⠈⠛⠛⠿⠟⠛⠿⠁⠀⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣶⣤⣀⣀⠀⢀⣀⣀⣤⣶⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⠙⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢟⠁⠀⠀⠀⠇⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⣻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⡀⣾⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⡒⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣜⠈⠀⣀⣻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢕⠂⠀⢒⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠢⠀⠀⠐⣒⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠕⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠄⠀⠠⢀⡀⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠂⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠠⠄⡀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢐⠂⠀⠀⠀⠀⠀⠈⠥⠄⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠂⠀⠀⠀⠀⠤⠄⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⡸⡖⠅⠀⠀⠀⠀⠀⠈⠉⠁⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⡘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀⣀⡐⣲⣖⡂
⠀⠀⠀⠀⠀⢇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠍⠝⡋⠉⠍⣋⠍⣩⣝⣿⣿⣿⣷⡶⠾⠗
⠀⠀⠀⠀⠀⠀⠑⠂⠄⠀⡀⠀⠀⠀⠀⠀⠀⠀⠈⠭⠥⣐⣦⣭⣾⡿⠾⠟⠛⠉⠉
⣶⣶⠶⠶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣶⣾⣵⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠿⠛⠛⠉⠉⠉⠉⠛⠛⠿⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⡄
⠀⠀⠀⠀⠀⠀⣰⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⢠⣿⡏⠀⠀⠀⠀⢠⠞⣭⣿⢷⡄⠀⠀⠀⠀⡴⢫⣽⠿⢦⡀⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⢿⣸⣿⣇⣀⡿⠀⠀⠀⠀⡇⣿⣿⣄⣼⡇⠀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠈⠻⠿⠿⠟⠁⠀⠀⠀⠀⠙⠻⠿⠿⠋⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢰⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆⠀⠀⠀⠀⣿⡿
⠀⠀⠀⠀⠀⠘⣿⣇⠀⠀⠀⠈⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡟⠀⠀⠀⠀⣸⣿⠃
⠀⠀⠀⠀⠀⠀⠹⣿⣆⠀⠀⠀⠈⠻⣷⣤⣀⠀⠀⠀⠀⠀⢀⣠⣴⡿⠋⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⠀⠀⠀⠈⠙⠛⠿⠿⠿⠿⠿⠟⠛⠉⠀⠀⠀⠀⣠⣾⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣷⣶⣤⣤⣀⣀⣀⣀⣤⣤⣶⣾⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠋⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⠛⠿⢿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠛⠛⠛⠿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⢠⢟⣋⢛⠛⠉⠀⠀⠀⠀⠔⠒⠒⠒⠂⠀⠀⠀⠀⠀⠐⠒⠒⠒⠂⠀⠀⠀⢠⡇
⠀⠀⠀⠀⠀⠀⣇⢿⠿⣺⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠘⢪⣭⣥⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣄⣀⡤⠏⠀⠀⠀⠀⠀⠀⠀⢰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⠃⠀⠀⠀⠀⠀⢰⣶⣦⠤⢤⣤⣀⣀⣠⡤⠤⠴⣲⣶⠆⠀⢀⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣷⣶⣶⣶⣶⣿⣿⣿⠟⠀⢀⠞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⡀⠀⠀⠀⠀⠀⠙⠿⣿⠟⠛⠻⠛⠻⣿⠿⠃⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠒⠋⠁⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢄⣀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠒⠒⠚⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣶⣿⣿⣿⣭⣙⣻⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠿⠻⠷⣝⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣷⣿⣿⣏⠁⠀⠀⠈⠙⠻⡿⢿⣿⣿⣆⠙⠻⠌⣷⣷⣌⣿⣿⡷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣽⣿⣿⣶⣶⣶⣶⣤⣄⣀⠑⢬⣿⣿⣧⢰⣄⣿⣿⣿⣿⣿⣿⡎⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠿⠛⠛⠉⣉⣛⣛⣻⢿⣿⣿⣿⣷⣌⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⡏⠀⠠⠖⣻⡟⠛⠛⠿⠛⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣷⡄⢠⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠛⠛⠻⠿⠛⠛⠛⠉⠉⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣟⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠿⠛⣻⣿⣿⣿⣿⡎⣿⡿⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣠⣾⡿⠿⠶⢾⣿⣛⣫⣿⣿⠟⢠⣿⠁⢀⡴⠋⠉⠉⠛⠛⠖⠀⠀⠀⠀⠀⠀⠀⠰⠛⠋⠉⠓⢸⣿⣿⣿⣿
⠀⠀⠀⠀⣠⣾⠏⠀⠀⣰⣶⣶⣶⣬⣍⡁⠈⢁⣼⣿⠀⠀⢀⣤⣶⡶⣶⣦⠀⠀⠑⠀⠀⠀⢠⠎⣠⣶⡶⣶⣦⣿⣿⣿⣿⡟
⠀⠀⠀⣸⡟⠁⠀⣴⣾⣿⣿⡟⢋⣿⣿⣿⣿⣿⡿⠃⠀⠀⠙⠛⢢⣴⡿⠋⠛⠂⠀⠀⠀⠀⣟⠛⠻⠿⠶⠊⢹⣿⣿⣿⡿
⠀⠀⠀⣿⣰⠀⠀⢿⣿⣿⣿⠟⣿⣿⣿⢿⣿⠉⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠘⣿⣿⣿⣷⣄
⠀⠀⠀⣿⣿⡄⢲⣤⣤⣤⣤⣾⣿⣿⡏⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣆
⠀⠀⠀⢸⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣷⣜⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣦⠀⣀⡀⠀⢀⡼⠇⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⠀⠀⢸⣿⣿⡿⢷⣶⣾⣿⣿⣿⣿⣿⣿⣷⣶⣆⠀⠀⠀⠀⠀⠀⠀⢀⡜⠁⠈⠁⢉⡁⢀⣉⠀⠀⠀⠀⣀⣿⣿⡿⣿⣿⣿⡿
⠀⠀⠀⢸⣿⣿⣿⡆⠘⣋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠋⠀⢀⠴⠶⣿⣿⡿⠿⣶⣄⣠⡾⣽⢯⣻⡆⠈⣿⡿⠃
⠀⠀⠀⠀⠻⣿⣿⣷⣤⣬⣭⣾⣿⣿⣿⣿⣿⣿⣿⡟⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣦⣶⣤⣴⣦⣿⣽⣿⡾⠿⣻⣇⣴⠟
⠀⠀⠀⠀⠀⠈⠙⠻⣟⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠈⠻⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⣴⠆⠀⠀⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⣷⡀⠀⠈⠻⢦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⢠⣿⠇⠈⢿⡄⠀⠀⠀⠉⠛⠿⣶⣦⣤⣤⣤⣤⣶⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠏⠀⠀⠀⠹⣆⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠉⡇⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡿⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠶⠋⠁⠀⠀⢻⠀⠀⠀⠀⠀⠀⠈⢳⡀⠀⠀⠀⠀⠀⠀⣰⡟⠀⠀⠈⢿⡗⠶⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣀⡤⠞⠋⠁⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⢀⡟⠀⠀⠀⢀⣤⢽⡀⠀⠈⠙⠛
⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠈

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⠶⠾⠛⠛⠛⠛⠛⠻⠶⢶⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⣶⣄⡀
⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⢦⣄⠀⠀⠀⠙⠻⣦⡀
⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⢀⣠⡤⠤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠇⠀⠀⠀⠀⠈⠻⣦⡀
⠀⠀⢀⣼⠟⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠶⠶⣦⣄⠀⠀⠀⣀⣤⣶⣶⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆
⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⢠⠞⠉⠀⠀⠀⠀⠈⠳⣀⣼⠟⠁⠀⠀⠀⠙⢿⣄⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⢰⣿⠀⠀⠀⠀⠀⠀⠀⢠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠹⠃⠀⠀⠀⠀⠀⠀⠀⢹⡆⠀⠀⠀⠀⠀⠀⠀⢹⣷
⣾⡇⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢀⣴⡦⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠈⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄⠀⠀⠀⠀⠀⠀⢀⡾⣆⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⣤⣀⣀⡤⣶⡟⠁⠈⠳⢤⣀⣀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡿
⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣴⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠉⠉⠉⠀⠀⠀⠀⢺⠳⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠃
⠀⠀⠀⠻⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠉⢧⠀⠀⠀⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⡿⠃
⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣄⣀⣠⣤⠞⠃⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣿⣶⣦⣤⣀⣀⡀⠀⠀⠀⠀⣀⣀⣠⣤⣶⣾⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠿⠿⢿⣿⣿⣿⣿⠿⠿⠟⠛⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠖⠒⠒⠚⠛⠒⠒⠒⠶⠤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠚⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠳⡄
⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠙⣆
⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⣸⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⠀⢀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣇
⠀⠀⢸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸
⠀⠀⠈⡇⠀⠀⠀⠀⠀⠀⠸⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠀⠀⠀⠀⠀⠀⠀⡟
⠀⠀⠀⢻⡀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⡿⠁⠀⠀⠀⠀⠀⠀⣸⠃
⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣤⣀⣀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠻⠿⠿⠿⠿⠿⠿⠟⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⠦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠒⠦⠤⣤⣄⣀⣀⣀⣀⣀⣠⡤⠤⠶⠚⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡄⠀⠀⠀⠀⢠⣾⣿⡆⠀⣠⣤⣄
⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣄⠀⠀⠀⠀⣾⣿⣿⣿⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⠿⣿⣿⣿⡿⠃⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⢿⣿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡟⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⣿⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⣤⣿⠀⣠⠤⡄⣼⠀⠠⠦⡄⠀⠀⣗⡦⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⠀⣇⡀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⣉⣁⡀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡏⣀⣀⡇⠀⠀⠀⢸⠃⢰⣆⠀⢠⡆⠀⣰⣦⠀⠠⣆⠀⠀⣤⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠀⠻⣥⠟⠀⠀⠀⠀⡞⠀⠀⢿⡄⣾⠁⠀⣿⢻⡆⠀⠹⣦⣰⡏⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣸⠁⠀⠀⠘⣿⠏⠀⢰⡟⠘⣷⠀⠀⢹⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⡴⠃⠀⠀⠀⠀⣿⠀⠀⢸⡟⠛⣿⠀⠀⢸⣧⠀⠀⠀⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢆⡀⠀⠀⢀⡠⠞⠁⠀⠀⠀⠀⠀⣿⠀⠀⢸⡇⠀⠹⠇⠀⠀⣿⠀⠀⠀⠶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉

⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠤⠤⠤⠄⣀⡀
⠀⠀⠀⢰⠋⠀⠙⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠊⠉⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠉⠉⢢
⠀⠀⠀⢰⠀⠀⠀⢃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⡴⠚⠉⠓⢤⠀⠀⠀⣠⠤⢤⡀⠀⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⢠
⠀⠀⠀⠀⠣⣄⠀⠈⠢⣀⠀⠀⠀⠀⠀⡰⠁⠀⠀⡜⠀⠀⣤⡀⠈⢳⠀⡞⠀⢠⣀⠈⢦⠀⠀⠱⡀⠀⠀⠀⠀⠀⡠⠊⠀⢀⣠⠎
⠀⠀⠀⠀⠀⠈⣆⠀⠀⠈⠙⠢⡀⠀⣰⠁⠀⠀⠀⡇⠀⣿⣿⡿⠀⢸⠀⠀⠸⢿⡿⠀⢘⠆⠀⠀⢱⡀⠀⡠⠖⠉⠀⠀⢀⡎⠁
⠀⠀⢀⡤⠒⠉⢁⣀⣀⠀⠀⠀⠘⡄⡇⠀⠀⠀⠀⠱⣄⠀⠁⠀⢀⡜⠀⠢⡀⠀⠀⢀⠜⠠⣄⡀⠀⡇⡸⠀⠀⠀⠀⣀⡀⠉⠓⠦⡀
⠀⠀⢈⠀⠀⠀⠁⠀⠈⡇⠀⠀⠀⣿⠁⠞⠧⣄⡀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠉⠉⣁⣤⡖⠋⠃⠀⣿⡇⠀⠀⠀⡎⠀⠈⠉⠀⠀⠘
⠀⠀⠈⢢⣄⣀⣀⣠⠔⠑⣆⠀⠈⠃⡇⠀⠀⠈⠻⣗⠒⠦⢤⣤⣤⣤⣤⡤⠶⠚⠋⣡⡞⠀⠀⠀⠀⡇⠗⠀⢀⡔⠳⢄⣀⠀⣀⣠⠞
⠀⠀⠀⠘⢄⡀⠀⠀⣀⡴⢉⣦⣠⠂⢳⠀⠀⠀⠀⠈⠻⣶⣦⣤⣤⡤⠤⢤⣤⣶⣿⡿⠀⠀⠀⠀⢰⠃⢂⣀⣎⠳⣀⠀⠈⠉⢁⡸
⠀⠀⠀⠀⠸⣉⠉⠉⣁⡠⠞⣰⠁⠀⠀⢣⡀⠀⠀⠀⠀⠙⢏⠀⠀⠀⠀⠀⠀⣉⡿⠁⠀⠀⠀⣠⠃⠀⠀⢹⠘⢦⡈⠉⠉⠉⡹
⠀⠀⠀⠀⠀⠀⠙⠫⠥⠐⠊⠁⠀⠀⠀⠀⠱⣄⠀⠀⠀⠀⠀⠙⢥⣀⣀⣈⡽⠋⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠑⠢⠬⠭⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣀⠀⠀⠀⠀⠀⠉⠀⠀⠀⢀⣠⠖⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠶⠤⠴⠖⠒⠋⠉

⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠉⠁⠐⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠂⠈⠉⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠉⠁
⡀⠀⣶⠄⠀⢶⠆⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠤⠀⠀⠀⠤⠀⠂⠀⠀⠀⠀⠠⠄⠀⠀⠠
⠃⢀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠄⠀⠀⠤⠀⠀⠀⠀⠀⠀⠀⣠⠤⢤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄

⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⡄⠀⠶⠂⠀⠶⠆⠀⠀⠀⠀⠀⠀⠰⠆⠀⠰⠷⠀⠁⠀⠀⠀⠀⠐⠶⠀⠀⠶⠂⠀⠀⠀⠀⠈⠀⠾⠆⠀⠰⠆⠀⠀⠀⠀⠀⠀⠰⠶⠀⠐⠶
⠁⠀⠀⠀⡀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠠⡀⠀⠀⢀
⠀⠂⠀⢀⣀⠀⠀⠁⠀⠀⠀⠀⠐⠀⠀⣀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⣉⡉⠀⠀⠀⠀⠀⠀⠀⠁⠄⢁⣀⠈⠀⠂⠀⠀⠀⠀⠈⠀⠈⣀⡀⠁⠐

⠀⠀⠀⠀⡀⠐⠛⠓⠀⡀⠀⠀⠀⠀⣤⡬⢻⣿⡄⠀⠀⠀⢀⡐⠀⣈⠐⢀⠀⠀⠀⠀⡆⢸⣿⡷⢀⡆⠀⠀⠀⠀⢠⣼⣤⠀⠀⠀⠀⠀⢀⣴⠂⣀⢢⣄
⠀⠀⠀⠀⠳⣤⡹⣠⡴⠃⠀⠀⠀⠀⠀⣠⣾⣿⠏⠀⠀⠀⠀⠳⣦⢥⣴⠋⠀⠀⠀⠀⢿⣧⠉⣠⣿⠃⠀⠀⠀⠀⠈⣷⣏⡀⠀⠀⠀⠀⠈⠛⠈⠉⠞⣋
⠀⠀⠀⠀⠀⠸⠿⠿⠀⠀⠀⠀⠀⠀⠀⠿⠿⠉⠀⠀⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠀⠿⠀⠻⠇⠀⠀⠀⠀⠀⠀⠏⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙

⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⡄⣸⠫⠤⣀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⠀⠀⡏⠈⣿⣿⣆⠀⠀⠀⠀⠀⠐⢚⠳⠶⠆⠀⠀⠀⠰⣟⡦⣹⣿⡆⠀⠀⠀⠀⢠⣭⣬⣥
⠀⠀⠀⠀⠱⢬⡄⢰⠌⠆⠀⠀⠀⣠⣀⣈⣛⠋⠀⠀⠀⠀⠀⠈⢿⣿⣿⡏⠀⠀⠀⢀⣀⣀⡚⠂⠀⠀⠀⠀⠀⠀⠉⣶⣿⣿⠃⠀⠀⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠐⠒⠋⠀⠀⠀⠀⠀⠉⠈⠙⠛⠋⠁⠀⠀⠀⠀⠀⠸⠿⠃⠀⠀⠀⠀⠀⠁⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠻⠟⠀⠀⠀⠀⠀⠀⠀⠛⠛⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣠⡄⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠄⠠⠀⠀⠀⠀⠀⢼⣋⠈⢹⡇⠀⠀⠀⠀⠀⠀⢟⡟⠷⠀⠀⠀⠀⠀⣤⣿⣿⣿⣦⠀⠀⠀⠀⠀⠀⠀⣿⢻⡆⠀⠀⠀⠀⢠⣴⡿⠋⠁
⠀⠀⠀⠀⠠⢴⣶⣷⡄⠀⠀⠀⠀⠈⠱⡾⠏⢠⠀⠀⠀⠀⠀⠀⣮⣺⣧⠀⠀⠀⠀⠀⢸⣿⡆⣿⣿⠀⠀⠀⠀⣤⠀⠀⠉⠈⠁⠀⠀⠀⠀⡈⠛⢀
⠀⠀⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠛⠛⠃⠀⠀⠀⠀⠀⠘⠛⠃⠛⠛⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⢀⡄⠀⢠⡄⠀⠀⠀⠀⣼⣍⢃⡆⣩⣧⠀⠀⠀⠀⢀⢶⢷⢦⠀⠀⠀⠀⠀⠀⠀⠔⠛⠁⠀⠀⠀⠀⠀⢀⡸⣿⡇
⠀⠀⠀⠀⠹⠟⠁⠻⠏⠁⠀⠀⠀⠘⢻⡏⢹⡿⠃⠀⠀⠀⠀⣈⣽⡿⣌⡀⠀⠀⠀⠀⣀⡴⠖⠁⠀⠀⠀⠀⠀⠀⠈⢷⣿⡇⠀⠀⠀⠀⢀⣰⣶⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠈⠉⠁⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠈⠉⠉⠉

⠀⠀⠀⢀⠔⠈⠀⠀⠀⠀⠀⠀⡏⠉⠀⠀⠀⠀⢰⠋⠉⠂⠄⠀⠀⠞⠉⠑⢄⠀⠀⡐⠉⠉⠑⢀
⠀⠀⠀⡇⠀⠀⠘⠆⠀⠳⠀⠀⠢⠀⠐⢄⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⡄⠀⠁
⠀⠀⠀⠑⠄⠀⠀⠀⠀⠀⠐⠀⠀⠜⠀⠀⠀⠈⠀⠀⠀⠱⠈⡆⠐⠀⢰⠀⡇⠀⠇⠀⠇⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⡁⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡀⠀⡀
⠀⠀⠀⡔⠁⠀⠀⠘⠃⠀⠈⠃⠀⠀⠌⠀⠀⠂⢠⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⢀⡠⠄⠄⠤⡀
⠀⠀⠀⢧⣀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠠⠀⠠⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⢀⡄⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠀⠀⠀⠐⠢⠆⠀⠀⠀⢀⠀⠀⠈⠀⠀⡀⣀⠀⡀⠀⠴⠀⠈
⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠒⡠⠀⠀⠀⠀⠀⠀⠀⠁⠄⠀⠀⠀⣀⠀⠀⣀
⠀⠀⠀⡇⠀⠀⠂⢀⡀⠀⠀⠀⠓⠀⠀⢀⠀⠀⠀⠀⠀⡄⠀⠁⠀⠀⠀⠀⠀⠈⠀⣀⠀⠀⠀⠉⡆
⠀⠀⠀⠉⠒⠀⠈⠁⠈⠀⠀⡄⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠠⠃⠀⠀⠛⠀⠀⠉⠀⠀⢀⠔⠁
⠀⠀⠀⠀⠀⣠⠀⡆⠀⠀⠀⠃⠁⠀⠀⠸⠀⠄⠀⠀⢈⠀⠀⠀⠀⠀⠀⠀⠀⠤⠤⠤
⠀⠀⠀⠀⠀⠉⠀⠀⠀⠆⠀⡀⠀⠀⠐⠀⠀⠀⠀⠀⡈⠃⠘⠆⠀⠰⠁⠀⢠⡄⠀⢠⡄⠀⠈⠲⡄
⠀⠀⠀⠀⠀⠢⠔⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠠⠀⠐⠄⠀⠁⠀⠀⠁⡀⠀⠀⠇
⠀⠀⠀⠈⠢⠄⣀⠔⠁⠀⠈⠢⢄⡤⠃⠀⠀⠐⠤⣀⡼⠀⠀⠀⠀⠀⣀⠇⠀⠀⠀⠀⠀⠀⠀⠊



⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣯⣿⣿⣿⣿⣿

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

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

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣟⡛⠛⠋⠭⠭⣉⡉⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠉⠀⠀⠀⠀⠀⠉⢻⠿⠋⠉⠀⠀⠀⠈⠉⠒⠢⢄⡀⠈⠑⠒⢤⡀⠀⠉⠻⢏⡁⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⣀⣠⣄⡠⠚⢹⣗⠢⣄⡀⠀⠀⠀⠀⠀⠀⠀⠉⠲⢤⡀⠀⠈⠑⠦⣀⠀⠈⠒⢤⣤⣤⡀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⢠⣾⣿⡿⠋⠀⠀⡘⠘⣦⠀⠈⠓⠢⢄⡀⠀⠀⠀⠀⠀⠀⠉⠒⢤⣀⠀⠀⠑⠢⢄⡀⠈⠛⢿⣧⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⡿⠁⠀⠀⢀⠇⠀⠈⢳⣤⡀⠀⠀⠈⠑⠲⠤⣀⠀⠀⠀⠀⠀⠈⠉⠒⠤⠄⣀⡈⠑⢒⣤⣽⡄⠀⠀⠸⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⡿⠁⠀⠀⢠⠎⠀⠀⢠⠎⠉⠪⣓⠤⢀⡀⠀⠀⠀⠉⠑⠒⠠⠤⠀⢀⠀⢀⣀⣤⠴⠚⠋⣀⠼⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢀⠟⠁⠀⢀⠔⠁⠀⢀⠔⠁⠀⠀⠀⠀⠙⠒⠤⣉⡐⠒⠒⠒⠠⠤⠐⠒⠒⠒⠈⠙⣒⣒⣶⣛⣥⣴⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠠⣔⣁⣀⠤⠚⠁⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⠰⠒⠀⠀⠙⠏⠐⠒⠂⠀⠒⠒⠒⠒⠺⠭⣁⣀⠀⣀⣀⣨⡝⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⢸⣿⡿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠺⠆⠀⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣾⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣇⠀⠀⠀⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⠇⠀⠀⢠⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠀⠀⠀⠀⠀⠀⢐⡤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡀⠀⠀⢹⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⠏⠀⠀⢀⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣧⠀⠀⠈⢿⣿⣿⣿⣿
⣿⡿⠿⠟⠁⠀⠀⢀⣾⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣧⠀⠀⠈⠻⢿⣿⣿
⣧⣀⠀⠀⢀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣿⣧⣄⠀⠀⠀⠀⢈
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣤⣤⣀⣀⣑⣒⣒⣁⣀⣤⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⡶⠶⠿⠟⠛⠛⠛⠛⠿⠷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠾⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢶⣤⡀
⠀⠀⠀⠀⠀⠀⢀⣴⠿⠋⠀⠀⠀⠀⣠⣤⣤⣀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣄⡀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⢀⣴⠟⠁⠀⠀⠀⠀⠀⠚⠉⠀⠀⠈⠀⠀⠀⠀⠀⠀⠁⠀⠀⠈⠛⠀⠀⠀⠀⠈⠹⣷⣄
⠀⠀⠀⣠⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠿⠿⣷⣦⡀⠀⣠⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀
⠀⣰⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⠀⠀⠙⢷⣾⠟⠁⠀⠀⠀⠉⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀
⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠋⠀⠀⠀⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇
⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣰⣄⠀⠀⠀⠀⠀⠀⠀⣠⣤⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠛⠃⠀⠀⠀⠀⠀⠀⠀⠙⠋⠀⠀⢰⡿⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⣰⣷⡀⠀⠀⠀⠀⠀⢀⣾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣤⣄⣠⣤⡾⠋⠹⣿⣦⣀⣀⣠⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⣹⡏⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠈⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡏
⠀⠸⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿
⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡿⠁
⠀⠀⠀⠘⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⡿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟
⠀⠀⠀⠀⠀⠻⣿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣶⣦⣤⣀⣀⣀⡀⠀⠀⣀⣀⣀⣤⣤⣶⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⣿⣿⣿⡿⠿⠿⠟⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡾⠟⠛⠉⠉⠙⠛⠻⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠛⠛⠉⠉⠙⠛⠿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣄⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣷⡄
⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⣴⣶⠀⠀⢠⣶⡦⠀⠀⠈⣿⡆⠀⠀⠀⠀⣾⡏⠀⠀⠀⣶⣶⠀⠀⢰⣶⡆⠀⠀⠘⣿⡄
⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⠈⠉⠀⠀⠀⠉⠁⠀⠀⠀⢸⣷⠀⠀⠀⢠⣿⠀⠀⠀⠀⠉⠁⠀⠀⠈⠉⠁⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⣿⡆⠀⠀⢠⣄⠀⠀⠀⠀⢀⣤⠀⠀⠀⣸⡏⠀⠀⠀⠈⣿⡀⠀⠀⠀⢀⣠⣤⣤⣤⣀⠀⠀⠀⠀⣾⡇
⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠙⠷⢦⣤⡴⠞⠁⠀⠀⣠⣿⠃⠀⠀⠀⠀⢻⣧⠀⠀⢰⠟⠁⠀⠀⠀⠹⡧⠀⠀⣰⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠹⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⣠⣼⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢷⣶⣤⣤⣴⣶⠾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⢶⣶⣤⣤⣴⣶⠿⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢀⢤⣤⠦⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠤⡄⠐⢆⣀⠯⠔⠓⠈⣁⣴⠦⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠠⠤⠤⣀⠀⠀⣀⣀⠠⠀⠀⠶⠀⠀⠙⠅⠀⠀⠐⣄⠤⠑⠊⠁⠀⢀⡠⢴⡮⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣨⠑⠒⠀⠈⠂⠀⠀⠀⠐⡄⠀⡘⠔⠒⠉⠀⠀⢀⡀⠄⣺⡱⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠤⠔⠁⠀⠐⠀⠂⣄⠀⠀⠀⣆⡀⠔⠋⡁⠀⠀⢀⡠⠐⠊⠁⢠⡰⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠆⠃⡘⠭⡠⠔⠊⠁⠀⠀⢀⡠⠔⠈⠁⠀⠀⢠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠔⠁⠀⢸⠁⠁⠆⠐⠒⢲⠠⠍⠀⠈⠄⡀⠀⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⡠⠐⡀⢀⣡⠀⠀⣠⠖⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠈⠀⠀⠘⣞⠟⢿⠇⠁⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠐⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡿⠿⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠻⡧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⠖⠛⠋⠉⠛⠓⠦⣄⠀⠀⠀⠀⣀⣠⡤⠤⢤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄⣠⠞⠉⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢠⠒⣦⡄⠀⠀⠀⣿⣠⢲⣦⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠹⣽⣿⠇⠀⠀⠀⡟⠻⠽⠟⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⣀⣀⠀⠀⠻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⢧⡀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠃⠀⠀⠀⠀⣸
⠀⢀⡴⢻⡏⠉⠀⠀⠀⠙⢦⣀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠙⠶⣤⣀⣀⣀⣤⠶⠋⠀⠀⠀⠀⠀⢠⠇
⠀⠉⠀⠀⢳⡀⠀⠀⠀⠀⠀⠈⠙⠒⠒⠖⠒⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟
⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠙⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠹⡍⠛⠶⢤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⣀⣀⠀⠉⠉⠙⠛⠒⠒⠒⠒⠒⠒⠒⠛⣻⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⠔⠒⠒⠒⠶⡖⠉⠉⠓⣦⡴⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠲⢴⣁⣀⠀⠀⠀⣀⣀⣤⠴⠞⠉⠀⠿⠦
⣀⣀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣉⣉⡉⣉⣉⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣽⣭⣿⣿⣯⣿⣿

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣐⣿⣷⣆
⠀⢀⣤⣤⣤⣀⣠⣄⡀⢼⣿⣿⣿⣿⣿⣦⡀⢠⣤⣶⣿⣧⠄
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣌⢿⣿⣿⣿⣿⣿⡷⢳⣿⠟⠹⠛
⠀⢘⣿⣿⣿⡟⠛⠻⣿⣿⣞⣿⡟⠀⣿⣿⢃⡾⠛⠆⢀⣀⣠⣤⣤⣤⣶⣶⣶⣶⣤⣤⣤⣤⣀⣀⡀
⠀⢾⣿⣿⣿⣿⣦⡀⠈⢿⣿⣿⠀⣰⣿⣧⣾⣥⣶⣾⣿⣿⣿⣿⡿⠿⠿⠿⠿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣶⣤⣀⡀
⠀⠈⠻⠿⠿⠿⠿⣿⣦⡘⠟⣃⣼⣿⣿⣿⡿⢿⣿⣿⣿⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠿⣿⣿⣿⣶⣤⡀
⢀⣤⣶⣾⣿⣿⠷⠿⠛⠃⠛⠟⣋⣭⣤⣤⣤⣀⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣷⣄
⣿⣿⣿⣿⣿⠃⠀⠀⣠⣾⣿⣆⠙⢿⣭⡛⢿⣿⣿⣿⣿⡛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣿⣷⣄
⢈⣿⣿⣿⣿⣦⣶⣿⣿⡟⣿⣿⡆⠀⢻⣿⣷⡙⠻⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣷⡀
⣾⣿⣿⣿⣿⣿⣿⣿⡿⢸⣿⣿⣿⣶⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠘⠻⠿⠛⣿⣿⣿⠟⠁⣿⣿⣿⣿⣿⣿⣿⣿⠏⠀⠀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧
⠀⠀⠀⣼⣿⣿⠏⠀⠀⣿⣿⣿⣿⣿⣿⡗⠀⠀⣴⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⢠⣶⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣧
⠀⠀⢰⣿⣿⡏⠀⠀⠀⠈⠛⠉⠙⠛⠉⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡄⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣆
⠀⢀⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿
⠀⢸⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⠿⠿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠀⢿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿
⠀⢸⣿⣿⡆⠀⠀⠀⠀⢠⣴⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⣿⣿⣶⡀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠈⣿⣿⣷⠀⠀⠀⠀⠀⠀⠉⠙⢿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⡿⠋⠉⠙⠁⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⠀⠹⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⡟
⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⡿
⠀⠀⠀⠀⢻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠛⠛⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠹⢿⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣶⣿⣿⣿⡿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⢿⣿⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⡿⠿⠛⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡶⠟⠿⣶⡄⠀⠀⠀⢀⣠⣤⣾⠃⢻⣦⣤⣀⠀⠀⠀⠀⣴⡾⠟⠷⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣠⣼⣏⠀⠀⠀⠀⠈⠻⣤⠀⠀⠀⢠⡼⠋⠀⠀⠀⠀⢈⣿⣤⣸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠉⠋⠀⠀⠀⠀⠀⠀⣿⣀⣤⣄⣸⡇⠀⠀⠀⠀⠀⠈⠋⠁⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣄⠀⠀⠀⠀⠀⠀⠀⠛⠋⣁⡉⠛⠃⠀⠀⠀⠀⠀⠀⢀⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣼⠏⢻⣦⣤⣀⠀⠀⠀⠀⠀⣼⠏⢿⡄⠀⠀⠀⠀⢀⣠⣤⣾⠋⢻⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⢀⣼⠟⢀⣀⣀⣠⣼⠏⠀⠈⢿⣤⣀⣀⣀⠘⢿⣄⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣁⣤⣄⣸⡇⠘⢿⣍⠉⠉⠀⠀⠀⠀⠀⠈⠉⢉⣽⠟⠀⣿⣀⣤⣀⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠁⠉⠛⠋⠀⠀⠙⢷⡄⠀⢀⣤⣤⠀⠀⣴⠟⠁⠀⠀⠛⠋⠁⠉⠛⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠃⣰⡟⣡⡿⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡟⢳⣄⣼⠋⣴⠿⣄⠠⣤⣹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⡿⢁⣾⢋⣼⠿⣆⠈⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠃⠀⠀⠟⢡⡾⢃⣾⠿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⠀⠀⠀⠀⠈⠀⠿⢁⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⡋⠻⣦⡀⠀⠀⠀⠀⣠⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠂⠈⠙⠃⠀⠀⠐⠋




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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣴⣶⣿⣿⠿⠿⠿⠿⣿⣿⣶⣦⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⡿⠿⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠿⢿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣄
⠀⠀⠀⠀⠀⠀⣤⣿⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄
⠀⠀⠀⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡄
⠀⠀⠀⣴⣿⠋⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⣼⡿⠁⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣦
⠀⣸⣿⠃⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣆
⢀⣿⠇⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⣴⣶⣶⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡀
⣼⣿⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⡀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣧
⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟
⠈⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⠃⠀⠀⠀⠀⠀⢿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁
⠀⢻⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠃⠀⠀⠀⠀⠀⠀⠈⣽⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟
⠀⠀⢿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡟
⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟
⠀⠀⠀⠀⠙⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⣶⣿⠋
⠀⠀⠀⠀⠀⠈⠻⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⢿⣶⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣶⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⠿⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⠉⠉⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣴⣿⣏⡀⠀⠈⣿⣶⣾⠿⢷⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡾⠛⠋⠉⠉⠉⠛⢻⣶⣀⠙⣧⠀⠀⠀⠀⠙⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⠇⠀⠀⠀⠀⠀⠈⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣅⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣦⣴⣿⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡿⠛⠿⣦⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣭⡀⠀⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠃⠀⣠⣿⠿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⣿⣇
⠀⠀⠀⠀⣠⡾⠿⣷⣄⠀⢀⣿⠇⠀⢀⣿⡟⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣧⡀⠈⣿⡇
⠀⠀⠀⠀⣿⣇⠀⠈⠻⢷⣾⣯⠀⠀⣸⣿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣧⠀⠙⣿⡄
⠀⠀⠀⢰⣿⠁⠰⠖⠀⠀⠉⢁⠀⢲⣿⠃⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⣿⡇⠀⢿⡇
⠀⠀⠀⠘⣿⣆⣀⡴⠇⢰⡄⢀⡰⢆⣽⡇⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠙⣿⡄⠈⣧
⠀⠀⠀⠀⠈⣿⡿⠁⢀⡾⠏⠈⠀⣸⠿⢿⣆⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⠀⢻⣇⠀⢿⡇
⠀⠀⠀⠀⠀⠘⠿⣶⣎⢀⠀⠶⠈⠉⢀⣼⡿⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠈⣿⡄⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣄⣀⣠⣶⠿⠋⠀⠀⢀⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⣿⣧⣴⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠉⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⣴⡟⠋⠀⠀⠙⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇⠀⠠⠤⠄⣼⡗
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡏⠀⠀⠀⠀⠀⢹⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣦⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⣀⣀⣠⣴⣾⣿⡇⠀⠀⠀⠀⠀⠈⢻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡟⠙⠛⠛⠛⠿⠿⠿⠟⠛⠻⠿⠛⠉⠈⣿⣿⠁⢠⡄⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡟⠀⢸⣷⡀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠈⣷⣦⡿⣿⡇⠀⠀⣠⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢰⣿⠏⠙⣿⣆⠀⠀⠀⠀⠀⠻⣿⡀⢻⡇⠀⣰⣿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣾⠋⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⢿⣧⡻⣷⣤⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢰⣿⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⠈⣿⡇⠈⠙⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⢹⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠈⣿⣇⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡆⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⡄⠀⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢹⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣸⣿⠁⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠏⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⢸⣿⠁⠀⠀⠀⠀⠀⠀⠀⢿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⢻⣷⠀⠀⠀⠀⠀⢼⣟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣄⡀⠀⠀⠀⠀⣠⣤⣿⠟⠀⠀⠀⠀⠀⠈⣿⡷⣤⣤⣀⠀⢀⣀⣴⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣶⡿⠿⠿⠿⣿⣿⡾⠋⠙⣿⠀⠀⠀⠀⠀⠀⢀⣿⠃⠈⠉⠙⠻⢿⡿⠛⠉⠉⠙⢿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠋⠀⠀⠀⠀⠀⠀⠀⡶⠀⢹⣇⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠰⠂⠀⠀⢀⡄⠀⠾⢿⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⠶⠀⠀⠲⠗⢀⡀⠀⣄⠀⢸⣿⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠸⢀⡀⢀⡀⠋⠀⠀⠀⠈⠙⢿⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⠀⠉⠀⠉⢠⣸⡿⠀⠀⠀⠀⠀⣿⡇⠀⢀⡀⠀⠈⠁⠈⠁⠀⠀⠀⠀⠀⠀⠤⠛⠻⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠃⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠹⣷⣄⣸⣿⣆⠀⠀⠀⠀⠀⠀⢀⣀⠈⠀⠀⠀⠀⠘⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠉⠑⢦⡀⠀⢀⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⠛⠛⣷⣄⠀⠀⠀⠀⡌⠉⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣆⣀⡀⠀⠀⠀⠀⠀⠀⠀⢀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⡀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣿⣶⣦⣄⣀⣀⣤⣶⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⣦⣄⣀⣀⣤⣴⣾⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠾⠿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠿⠷⢶⣦⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦
⠀⠀⠀⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠆⠀⠀⠀⠀⠀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀
⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⡄
⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⠀⢀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠋⠙⣿⣷⠀⠀⠀⠀⡞⠈⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀
⠀⠀⣸⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣾⣿⣿⠀⠀⠀⠘⣿⣾⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠶⢶⣦⣄⠀⠀⢻⣇
⠀⢀⣿⠃⠀⠀⡴⢿⡟⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⠟⠀⠀⠀⠀⠹⣿⡿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠙⠀⠀⠘⣿
⠀⢸⣿⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⣿⡇
⠀⢸⣿⠀⠀⠀⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠟⠀⠀⠀⠀⠀⠀⠀⢠⣿
⠀⠀⢹⣧⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠋⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏
⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠁
⠀⠀⠀⠘⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠘⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⠶⠞⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⠃
⠀⠀⠀⠀⠀⠈⢿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣍⡉⠉⠉⣉⡅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⡿⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⢶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⡶⠿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠴⠶⠒⠛⠛⠛⠉⠙⠛⠛⠛⠒⠶⠦⢤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠖⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠲⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠓⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣄
⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⢀⡴⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⢠⡞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⠀⢠⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡄
⠀⢀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀
⠀⣾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⣽⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣞⣿⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣷
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡆
⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿
⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉
⢻
⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴⠞⠛⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠳⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⢷⠀⠀⠀⠀⠀⠀⠀⠘⠁⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡶⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠈⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠘⣦⠀⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃
⠀⠀⠀⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢠⡞⠁
⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⣿⣿⣿⣷⣶⣶⣶⣶⣾⣿⣿⡿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⣀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠛⠛⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠦⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠴⠚⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠲⠶⠤⣤⣤⣤⣀⣀⣠⣤⣤⠤⠶⠖⠚⠋⠉

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠶⢶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡃⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠷⣤⡴⠟⠁⠀⠀⣤⣤⣄⠀⢀⣤⣤⡄⠀⠀⠀⠀⠀⠀⠈⠷⣦⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠛⣀⣀⠀⠘⢷⡄⠀⣹⣦⣽⣷⣿⣥⣾⣁⠀⠀⠀⠀⠀⣴⠟⠀⣀⣀⠐⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⣏⠙⢷⡄⠸⡇⠀⠿⠉⠉⢹⡏⠉⠉⣿⠀⠀⠀⠀⠀⣿⠀⣸⡏⢹⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠙⣧⡀⠻⠶⠶⠶⠶⣦⡀⢾⣧⣤⣤⣿⠀⠀⠀⠀⠀⡿⠀⣿⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⠻⢦⣤⣤⣤⣤⠾⠃⢸⡇⠀⠀⣿⠀⠀⠀⠀⢀⣠⡶⠛⢀⣿⠁⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢠⡆⠀⠶⠶⠶⠾⠷⠶⠶⠿⠀⠀⠀⢰⡟⢁⣤⠾⠋⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡶⠶⣶⠶⢶⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠋⣥⠴⢶⣶⠶⣶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣧⣀⣿⣀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣄⣠⣿⣀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢰⡶⠿⠿⠿⠿⠿⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠿⠿⠿⠿⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛




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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡄⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠶⢦⣄⣰⣿⣴⣯⣷⣶⣿⣋⣤⣤⣀⣀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⣚⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣭⣝⡻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⠉⠛⠿⣿⣿⣿⣿⣿⡿⠟⠋⠉⠈⣿⣿⣿
⠀⠀⠀⢀⣴⣿⣿⣿⢿⣿⣿⠟⠛⠉⠉⠉⠉⠉⠙⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡏
⠀⠀⠀⣾⣿⣿⡟⣰⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⢠⣶⣶⣿⣶⠆⠀⠀⢿⣧
⠀⠀⠀⣿⣿⣿⠀⠛⠋⣠⣿⣿⡁⠀⠀⠀⢠⣶⡿⠿⠛⠛⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⢻⣿⣿⣧⢸⣿⣿⣿⠟⠃⠀⠀⠀⠀⠀⠀⠀⣀⣤⣀⠀⠀⠀⠀⡴⡡⠖⠢⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠈⣿⣿⣿⡘⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠀⠉⡆⠀⠀⢸⠋⠀⠀⠀⡇⠀⠀⠀⣼⠃
⠀⠀⠀⠀⢘⣿⣿⣇⢿⡿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣧⠀⠀⢸⡀⣤⣤⠾⠗⠀⠀⠀⠉⠉⠙⠛⠳⢦⣄
⠀⠀⠀⣰⡿⢛⠻⢿⡈⠿⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠛⢛⡩⠁⠀⠀⠈⢷⡑⠒⠀⠀⢀⣀⣤⣤⡤⣶⣦⠀⠀⢹⣦
⠀⠀⢀⣿⠡⠟⢻⡄⠀⠀⣠⣶⠶⠶⠶⢤⣤⣀⡀⠀⠀⠤⢶⡎⣀⣄⢀⣾⡿⠉⠁⠀⠀⠀⣀⣤⠆⢈⣿⠀⠀⠀⣿⡆
⠀⠀⠸⣿⡀⠀⢸⡁⠀⠀⣿⡇⢠⣤⣀⡀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠠⡌⠉⠀⠀⣠⣤⡄⣾⣟⡁⠀⢸⠇⠀⠀⢰⣿⠃
⠀⠀⠀⢿⣷⡀⠀⣠⠀⠀⠸⣇⠸⣿⡟⠿⠿⣴⣦⢠⣤⢀⣤⣤⣤⢰⣶⣶⠿⢸⣿⣙⡁⣿⣯⠞⢀⡟⠀⠀⠀⣾⡿
⠀⠀⠀⠈⠻⣿⣿⣿⠀⠀⠀⠹⡄⠙⢿⣶⡄⣿⡿⢿⣿⠘⣿⣤⡄⢸⣿⠶⠂⣴⣬⣿⡇⠛⠋⠀⠸⠀⠀⠀⢰⣿⠇
⠀⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⠹⡄⠀⠙⠃⠛⠃⢼⣿⠀⣿⣶⣶⠸⡿⠿⠟⠈⠉⠉⠀⠀⠀⠀⠀⠀⠀⣰⣿⠏
⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⣶⠄⠀⠀⠀⢀⠀⠀⠀⣠⣾⣿⠋
⠀⠀⠀⠀⠀⠀⠀⢿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⣾⣤⣶⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⣶⡄⠀⠀⠀⠀⠀⣀⣀⡀⠀⢀⣼⡿⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣶⣤⣤⣤⣤⣤⣤⣼⣿⣷⣤⣤⣴⣾⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠉⠉⠉⠈⠛⠻⠿⠛⠉

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

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

⠀⠀⠀⠀⠀⢸⣷⣦⣤⣄⣀⣀⡀⠀⠀⠀⢀⣀⣀⣀⣤⣴⣶⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢀⣄⣀⣀⣀⣤⣤⣤⣴⣶⣿⣿⡄
⠀⠀⠀⠀⠀⢸⣿⣿⣿⡿⠛⣛⠻⣿⣿⣿⠟⢛⠛⠿⣿⣿⣿⡇⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠈⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣿⣿⣿⠃⣼⢟⣿⣿⡿⢿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣴⣾⣿⣿⣿⣦⣤⣤⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣌⠙⠛⠛⠛⠋⢡⣾⣿⣿⠏⣰⣿⣿⣿⠿⠿⢿⣿⣿⣿⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣶⣤⣀⣠⣴⣿⣿⣿⠋⣰⣿⣿⢏⣠⣤⣤⣤⣙⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠉⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣿⣿⡿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⢷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣧⡀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣦⡀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣄⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣦⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣄⠀⣰⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⡿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⣿⡿⠿⢿⠿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣋⡀⠀⠀⠀⠀⠀⠹⣿⣿⣿⡿⡋⠒⢤⣺⡗⡔⠋⢿⣿⣿⣿⣟⣁⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⢟⠑⢠⡔⣿⣢⠚⠹⣿⣿⣿
⠀⠀⠀⠀⠀⣿⠏⡏⢡⢼⢦⡤⣿⠄⠈⢹⣿⣿⠁⠈⠀⢸⣭⣛⠇⠀⠀⢿⣿⡟⢹⠉⡤⣷⣤⢼⡇⠀⠙⣿⣿⡏⠀⠀⠈⣯⣟⡻⠀⠁⢸⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠁⠈⢀⡈⣁⠈⠁⠀⢸⣿⣿⠀⠀⢠⣟⣿⣻⣇⠀⠀⢸⣿⡇⠀⠀⠁⡁⢈⡈⠁⠀⠀⣿⣿⡇⠀⠀⣾⣻⣟⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⣿⠀⣄⣀⠘⠃⠃⠀⠀⠀⢸⣿⣿⠀⠀⠸⣯⣭⣽⠇⠀⠀⢸⣿⡇⢀⣀⡀⠋⠙⠀⠀⠀⠀⣿⣿⡇⠀⠀⢿⣭⣭⡿⠀⠀⢠⣿⣿
⠀⠀⠀⠀⠀⣿⡇⢹⢢⡆⣴⠶⣦⣿⠃⢸⣿⣿⠀⢀⣾⢉⣿⣏⣷⡄⡀⣿⣿⣟⠈⡗⣶⢠⡦⢶⣼⡗⠀⣿⣿⡇⠠⣰⡋⣻⣿⣹⡦⠀⢸⣿⣿
⠀⠀⠀⠀⠀⣿⠀⠀⠀⠁⠀⠀⠁⠈⠀⢸⣿⣿⣾⡼⣯⣉⣩⣍⡼⠳⣴⣿⣿⡇⠀⠀⠈⠀⠀⠈⠀⠁⠀⣿⣿⣷⣧⢿⣍⣉⣍⣡⠟⢬⣾⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⣤⣤⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣦⢀⡆⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣯⣭⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣼⣧⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡛⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡟⠈⠛⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠈⠙⠻⢿⣿⣮⡙⠛⠿⣿⣿⣿⡛⠿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⠿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠀⠀⠈⠉⠛⠀⠀⠉⠛⠻⠎⠻⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⠇⠀⢀⣀⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣤⣄⡀⠀⠀⠀⢸⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⢠⠾⠛⠛⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠛⠂⠀⠀⠘⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠅⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠼⢿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢋⠀⠀⠀⠀⠀⣠⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣄⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠘⠀⠀⠀⠐⠞⠁⠀⠀⠀⠈⠃⠀⠀⠀⠀⠀⠀⠀⠊⠉⠀⠀⠈⠙⠶⠀⠀⠀⠀⠀⠀⢠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠂⠤⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⢰⣯⣀⣒⠒⣒⣀⣡⣿⠇⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠀⠀⠀⠀⠀⠀⠀⠈⠻⣋⠉⠛⢿⣿⡿⠋⠀⠀⠀⠀⠀⠀⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⡠⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠐⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠤⢀⣀⡀⠤⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠈⠉⠛⢶⣄⠀⠀⠀⠀⠀⣠⣤⠶⠶⠒⠒⠶⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣄⢀⣴⠟⠉⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⢀⣠⣤⣄⡀⠀⢀⣤⣤⣤⣄⠀⠀⠹⣿⠁⢀⣠⣤⣤⡀⢀⣤⣤⣤⣄⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⣰⡿⠋⢹⣿⣿⣷⣿⣿⣿⣿⣿⣷⠀⠀⢻⣄⣿⠋⣹⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⣿⣷⠖⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣿⣇⣽⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⢸⡇⠻⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⣿⠀⠀⠀⠀⣀⣤⣤⣤⣄⡀
⠀⠀⢠⣴⣶⡶⣶⣄⠀⠀⠀⠀⢷⡀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠁⠀⢀⣿⠀⠀⠈⠙⠿⣿⣿⡿⠟⠁⠀⠀⠀⠀⣼⠇⠀⠀⠀⠀⠉⠀⢀⡏⠙⠿⡄
⠀⠰⠛⠉⣿⡆⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠉⠛⠿⢿⡿⠟⠋⠀⠀⠀⣠⡾⠛⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⢸⠇
⠀⠀⠀⠀⢹⣇⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠟⠀⠀⠈⠙⠷⣦⣤⣀⣀⣀⣀⣤⣴⠾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠶⣦⣤⣤⣤⣤⣴⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇
⠀⠀⠀⠀⠀⠸⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟
⠀⠀⠀⠀⠀⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⢶⣤⣤⣀⣀⣀⡀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣴⠶⠛⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⠛⠛⠛⠛⠛⠛⡉⠉⠉
⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⣶⣶⣶⠶⠟⠁

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣞⣳⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣛⡆⠀⠀⠀⠀⠀⠀⢄⢘⡅⡠⠀⠀⠀⠀⠀⣀⣼⠜⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⡿⡒⢄⠀⠀⠀⠀⡌⢻⠏⣠⠀⠀⢀⡴⢋⠖⡏⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⠸⡘⢆⠑⢄⠀⠀⡈⢺⠋⡠⠀⢠⠎⡠⠋⡜⠀⡀⠀⠆⣠⠯⠝
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠒⠓⢼⡄⢧⠀⡱⡄⠳⡈⢢⠀⠘⢾⠊⠀⡠⢃⡔⠁⡜⠀⠀⣤⠔⠛⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⡨⠿⠦⣇⠘⢆⠘⢄⢣⠀⢸⠀⡰⢡⠎⣠⢊⣠⠜⠫⠤⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠉⠙⠦⣑⣌⠢⢣⠸⣰⡱⣁⡴⠗⠉⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⣀⣠⠤⠤⠔⠒⠒⠒⠀⠤⠽⠿⣷⣿⣿⠾⠛⠂⠉⣉⣀⣉⣉⣀⣈⣉⣑⢒⣤⣤⡴⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠺⡹⠓⠿⣑⣒⠒⠀⠐⠒⠒⣒⣒⣒⣢⣴⢿⣿⡿⡻⣍⡉⠭⠥⠤⢈⡁⠤⠤⠀⠒⠋⠁⠀⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⢉⡉⠉⠉⢉⡤⢖⠟⣡⡿⢸⠣⠙⣌⠻⡓⠤⣀⣀⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠤⣬⠵⡞⢁⡴⠁⡴⢡⠁⢸⠀⢣⡈⢣⡙⢆⠘⡏⠢⣴⣒⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⣉⢶⡎⠹⠀⢃⠞⢀⠞⣠⠃⠀⢸⡀⠀⠑⣄⠱⡌⢢⠀⠀⠇⢹⠓⢴⢒⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠊⠈⠁⠀⢀⡎⣠⢋⡴⠁⢀⡴⠛⡏⠳⠀⠈⠢⡈⢦⢣⠀⠀⠈⠃⠈⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣜⡵⠋⠀⠀⠀⡠⠚⡟⠢⠄⠀⠀⠈⠑⢿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡖⢻⠉⠀⠀⠀⠀⠀⢠⠞⡏⠢⠀⠀⠀⠀⠀⠀⢯⡽
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠸⣉⠆

⠖⠀⠀⠀⡟⠀⠀⠀⠀⢷⠀⠀⠀⠦⠴⠀⠀⠀⡇⠀⠀⠀⠀⣇⠀⠦⣄⣀⣀⡤⠆⢀⡇⠀⠀⠀⠀⡇⠀⠀⠀⠿⠂⠀⠀⢰⠇⠀⠀⠀⠸⡆⠀⠀⠰
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⢧⣀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⣠⠏⠀⠀⠀⠀⠀⠙⢦⡀
⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠓⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠓⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛

⢴⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣴⠧⢷⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡴
⠀⠀⠉⢢⡀⠀⠀⠀⠀⢠⠞⠁⠁⠀⠀⠁⠙⢦⠀⠀⠀⠀⠀⢠⠞⢁⡀⠀⠀⣀⠙⢦⠀⠀⠀⠀⠀⢠⠞⢹⣷⣇⣇⡏⠙⢆⠀⠀⠀⠀⠀⣠⠊⠁
⡘⠃⠀⠀⣧⠀⠀⠀⠀⡾⠀⠶⠀⠀⠀⠀⠶⠀⡇⠀⠀⠀⠀⡏⠀⡈⠁⠀⠀⠉⡀⠈⡇⠀⠀⠀⠀⡇⢀⡀⠀⠀⠀⢀⡀⠸⡆⠀⠀⠀⢰⡇⠀⠀⠛
⠀⠀⠀⠀⡏⠀⠀⠀⠀⢷⠀⠀⠀⠒⠂⠀⠀⢀⡇⠀⠀⠀⠀⣇⠀⠹⢄⣀⣀⠼⠁⢀⡇⠀⠀⠀⠀⣇⠀⠀⠀⣤⠀⠀⠀⢰⠇⠀⠀⠀⠘⡇
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠙⢦⡀
⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙
⢲⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠼⣅⡀⠀⠀⠀⠀⠀⢀⠀⠀⢀⡤⠤⠤⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⠤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠬
⠀⠀⠉⢣⡀⠀⠀⠰⣅⢣⠞⠁⠀⠀⠀⠀⠙⢦⣇⡹⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠙⢦⠀⠀⠀⠀⠀⣠⠞⠹⣜⣉⠆⠀⠙⢆⠀⠀⠀⠀⠀⣠⠋
⠀⠀⢠⠀⣧⠀⠀⠀⠀⡿⠀⠘⠃⢀⠀⠛⠀⠀⡏⠀⠀⠀⠀⡏⠠⠖⠀⠀⠀⠐⠦⠈⡇⠀⠀⠀⠀⡇⠐⠒⠀⠀⠀⠐⠂⠸⡆⠀⠀⠀⢰⠇⢀
⠖⠀⠀⠀⡟⠀⠀⠀⠀⢷⠀⠀⠀⠺⠇⠀⠀⢀⡇⠀⠀⠀⠀⣇⠀⠢⣄⣀⣀⡤⠆⢀⡇⠀⠀⠀⠀⣇⠀⠀⠐⠦⠆⠀⠀⢰⠇⠀⠀⠀⠘⡇⠀⠀⠐
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠣⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠈⢦⣀⠀⠀⠀⢀⣠⠎⠀⠀⠀⠀⠀⠘⢦⡀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠙⢦⡀
⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠴⢴⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡶⠷⠷⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡤⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠴
⠀⠀⠈⢳⡀⠀⠀⠀⠀⢠⠞⠉⡷⡇⣿⣿⠉⢦⡀⠀⠀⠀⠀⢠⠞⢁⡀⠀⠀⣀⠙⢦⠀⠀⠀⠀⠀⣠⠎⠁⠁⠀⠀⠁⠙⢦⠀⠀⠀⠀⠀⣰⠋
⡘⠃⠀⠀⣷⠀⠀⠀⠀⡟⠀⣄⠀⠀⠀⠀⣄⠀⡇⠀⠀⠀⠀⡏⠀⡈⠁⠀⠀⢉⡀⠈⡇⠀⠀⠀⠀⡇⠐⠆⠀⠀⠀⠰⠆⠸⡇⠀⠀⠀⢰⡇⠀⠀⠛
⠀⠀⠀⠀⡞⠀⠀⠀⠀⢷⠀⠀⠀⢠⡄⠀⠀⢀⡇⠀⠀⠀⠀⣇⠀⠙⢤⣀⣠⠜⠁⢀⠇⠀⠀⠀⠀⣇⠀⠀⠐⠒⠂⠀⠀⢰⠇⠀⠀⠀⠘⡇
⠀⠀⣠⠞⠁⠀⠀⠀⠀⠈⠳⣀⠀⠀⠀⢀⣠⠞⠀⠀⠀⠀⠀⠈⠦⣀⠀⠀⠀⢀⡠⠎⠀⠀⠀⠀⠀⠘⢦⣀⠀⠀⠀⢀⡠⠋⠀⠀⠀⠀⠀⠙⢦⡀
⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙
⢲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴
⠭⢤⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⡤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠤⣄⡀⠀⠀⠀⠀⠀⣀⠀⠀⣀⡤⠤⠤⣄⡀⠀⢀⠀⠀⠀⠀⠀⠀⣠⠽
⠀⠀⠈⢳⡀⠀⠀⠀⠀⢠⠞⠁⠀⢪⣙⡼⠉⢦⡀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠙⢦⠀⠀⠀⠸⣉⣧⠎⢁⠀⠀⠀⠀⠙⢶⣃⡕⠀⠀⠀⣰⠋
⠀⠀⢠⠀⣷⠀⠀⠀⠀⡟⠀⠚⠀⠀⠀⠐⠓⠀⣇⠀⠀⠀⠀⡏⠀⠖⠀⠀⠀⠐⠆⠈⡇⠀⠀⠀⢀⡇⠀⠘⠃⢀⠈⠛⠀⠘⡇⠀⠀⠀⢰⡇⢠
⣿⢿⠿⠿⡿⢟⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣿⣻⣟⣛⣛⣿⣟⣻⣿⣿⣛⣻⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢦⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡆⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠰⢡⡀⡀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢰⣾⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠸⢿⡟⣿⠇⠀⠀⠀⠤⢰⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡧⠀⠀⠀⠀⠀⠁⠙⢡⡤⠀⢀⡠⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⢱⢹⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠃⠀⠁⠀⠀⠂⠀⠺⠃⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠈⢸⠇⠀⡄⠀⠀⠀⠀⣶⣠⣺⡆⣀⠀⢈⣷⣶⡆⢳⡄⠠⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣟⠀⡀⠆⣿⣿⣿⣿⣌⠀⠀⢐⣿⣵⣀⠀⠃⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⣠⣼⢙⣀⣃⢠⣿⣿⣿⣿⣿⣰⠎⠙⢿⣿⣧⣠⣀⣀⣀⡀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠻⠇⠀⠀⠀⠻⣿⠏⠟⠀⠸⣿⣿⣿⣿⣿⠇⠐⠒⠚⠛⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢐⠠⠀⠀⠀⠀⠀⠀⠀⠘⢀⣿⣿⣿⣿⣇⡀⣰⣶⣄⣠⣀⣀⣀⣀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣼⡇⡄⢸⠀⠘⠛⠋⠋⠀⠙⣿⣿⣿⣿⣿⣇⠻⣿⣿⡋⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⢲⣥⡄⠘⠀⠀⠈⠀⠀⠀⠀⢿⣿⣿⣿⣿⣹⠀⣼⠋⠀⠀⢻⣶⣼⠇⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣏⠀⠘⠀⠀⠠⢎⣿⣿⡄⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⣏⣀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠀⠀⢚⡲⢤⣾⣿⣿⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠭⠾⠿⠟⠿⠉⠉⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠐⠂⠀⠀⠄⠀⠄⠀⠄⠠⠀⠄⠀⠀⠠⠀⠀⠀⠠⠀⠀⠀⡀⠀⡀⡀⠀⠀⣀⠀⠀⠀⠀⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠒⠐⠂⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠛⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣶⣦⣀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⢿⠋⠉⠀⠀⠀⠈⠉⠛⢿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⠏⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⡉⢿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⢸⣿⣿⠃⠀⠶⣶⣶⣤⡄⠀⠀⠀⣠⣶⠿⠛⠀⢻⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠰⠚⠛⠛⠁⠀⠀⠀⠉⠙⠋⠓⠀⠈⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⢀⣤⣶⣶⣿⣿⣿⣿⣿⣿⣿⡆⠀⢀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⡀⠀⠸⣿⣿⣿⣿⠿⠿⠿⣿⣿⣿⠃⠀⣸⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣷⣄⠀⠈⠻⣅⡀⠀⠀⠀⣠⡿⠋⢀⣴⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣶⣄⡀⠈⠛⠻⠾⠛⠋⣀⣴⣿⣿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠙⠿⣿⣿⣿⣷⣶⣾⣿⣿⣿⠿⣿⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⢿⢻⠉⠉⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢸

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

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

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⢀⡤⣠⣼⣧⣦⢴⡦⣤⣼⣧⣀⢤⡀⡀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢰⡦⠴⡞⢻⡫⡝⣆⡓⠐⣧⣖⣲⣼⠢⢜⣰⢫⠝⡟⢱⡦⣴⡆⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⡝⠛⡁⣍⢫⠼⠑⡛⠓⠈⠉⠁⠀⠀⠂⢀⠀⠉⠁⠈⠛⠊⠵⡜⣹⢸⠛⢻⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣠⡴⣓⠰⣹⡬⠚⠉⠐⠀⣀⠤⠔⠂⠒⣒⢈⣁⣉⡁⣐⡂⠒⠠⠤⣀⠀⠀⠉⠓⢥⣏⠆⣊⢧⣄⡤
⠀⠀⠀⠀⠀⠀⠀⢀⡀⠾⢅⠝⡶⠗⠁⠈⣀⠵⢒⣉⣰⡺⢫⡾⠟⠛⠻⣮⣵⠟⠛⠻⢷⣝⢗⣂⣉⡐⠪⣀⠀⠈⠚⢶⠫⡨⠷⢀⡀
⠀⠀⠀⠀⠀⢠⣠⣼⢹⡜⡶⠋⠄⢀⠴⢊⡤⣴⡿⢛⠉⠛⢿⢁⣤⣱⣦⡸⢃⣴⣌⢶⠈⡿⠛⠉⡛⢿⣆⣦⡑⠮⡀⠀⢙⢷⢣⠏⣧⣄⡄
⠀⠀⠀⠀⠀⢰⡯⡨⢳⠏⠀⢀⠔⣡⣴⡾⠧⣿⠁⡶⣽⣿⡎⣬⠻⣉⡟⡵⢎⠻⣉⠟⣭⢱⣿⣯⢶⠈⣷⡾⢷⣦⡌⠢⡀⠀⡙⡞⢅⢽⡆
⠀⠀⠀⢀⣹⢓⡒⡾⠁⠀⡰⢋⢠⣿⠃⠄⢦⣤⣡⣝⠓⠴⢰⢶⡝⢢⡜⢸⡏⢣⡴⢫⡴⡎⠦⠚⣫⣌⣤⡴⠄⠘⣿⡆⡘⢎⠈⠈⢷⢒⡚⣏⡀
⠀⠀⠘⣺⠛⡒⡞⠀⠀⡔⢀⣭⣥⣿⡄⢿⣟⣉⡇⡆⣭⡙⢿⠤⣛⣂⡡⣩⣇⢌⣁⣚⠦⡷⢘⣭⢰⢸⣉⣻⡿⢠⣿⣯⣭⡀⢢⠀⠀⢳⢐⠙⣷⠃
⠀⠀⡴⡋⣍⡶⠀⠀⡜⢠⣿⠋⡠⣤⣮⠰⡒⣒⠢⢧⡜⣩⣴⣙⠿⠛⢲⡟⠻⡔⠛⠿⣏⡦⣝⢓⡼⠔⣒⢒⢦⣵⣤⢄⠙⣿⡄⢣⠀⠈⢶⣩⢙⣦
⢀⣠⣾⣜⣶⢁⠀⡸⢠⡾⣿⡀⢶⣾⣉⡆⢳⡙⡿⣣⣮⡴⡟⢍⣳⡿⢮⡾⢷⡵⢿⣞⡩⢻⢫⣵⣜⢿⢋⡞⢼⣉⣷⡶⢠⣿⢳⡆⢆⠀⠈⣶⣣⣷⣄
⠀⢽⡦⢜⡴⠠⢀⠃⣸⡾⠛⢋⣎⢽⡒⢏⣙⡟⣴⢻⣋⣰⡽⣿⡋⢧⡀⣷⣾⠀⣼⣙⣿⢭⣆⣙⠟⢦⢋⣉⡳⢒⣏⣱⡛⠛⢷⣅⠘⡀⠀⢆⡧⢴⡯
⠐⢮⢎⣞⡇⠀⢸⠀⣿⡁⢚⣹⡟⢳⡝⢯⣙⢋⢯⣟⠩⣷⡳⣄⣘⣏⢙⢃⡘⡛⢹⣃⣤⢞⣾⠍⣻⡽⡑⣏⡽⢫⡞⢻⣏⡑⢈⣿⠀⡁⠀⢸⣛⡱⣵
⣀⠾⠂⠚⡇⠂⠘⢰⡻⣷⣄⡑⢛⡵⢚⣉⣍⢸⢶⠉⣳⠝⠒⠺⣭⠡⡞⢩⡍⢳⠌⣭⠗⠒⠯⣞⠉⣶⡇⢹⣉⣓⢬⡛⢊⡠⣾⢏⡆⠁⠀⠈⢓⠒⢷
⠀⣻⢂⣲⡄⡀⢰⠘⣼⡟⢡⣔⡶⢍⡲⢄⣯⢸⢛⣴⢞⣭⡒⣺⣭⡌⢳⢤⡤⡞⣡⣭⣗⠒⣭⡳⣤⡛⡇⣬⡤⢖⡭⢶⣲⡌⢻⣧⠃⡆⠀⢰⣖⡐⣟
⠐⢻⣳⢲⡇⠀⢨⠀⣿⡄⠹⢼⣷⢞⡵⣣⣴⡦⡯⢷⣬⣿⡞⣁⡴⣣⠶⣆⣰⠦⣜⣦⣈⢳⡿⣥⠾⣽⢶⣦⡽⢮⡳⣾⡧⠍⢠⡿⠀⡏⠀⢸⡖⢞⡜
⢠⣼⣋⡱⣈⠀⠀⡆⢨⡻⣷⡶⢣⣽⣒⡳⢦⡶⡙⣾⣧⡈⡿⢻⣦⡏⢀⡟⠻⡀⢹⣴⡿⢿⢁⣼⣷⢃⢶⡤⢞⣒⣯⡔⢶⣾⢛⡅⢰⠁⠀⣏⣏⣙⢧
⠀⠈⡻⡱⣛⡄⡀⠘⡈⢳⣿⠀⡟⣻⣤⠋⡞⣴⠻⢎⡻⣽⣳⣴⠞⠿⢻⡻⢗⡝⠿⠳⣤⣞⣯⢝⡱⠟⣦⢳⣙⣤⣟⢻⠀⣿⡟⢁⠃⠀⢠⣓⢎⣟⠁
⠀⠀⠉⣦⠡⡛⡄⠀⠱⡈⢿⣦⣈⣉⡫⣙⣒⢢⡍⡏⣰⡮⢙⡲⠿⣶⡊⢷⡾⢓⣶⠿⠖⡊⢵⣇⢹⢩⡖⣒⣃⢝⣉⣁⣴⡿⢁⡎⠀⢠⢛⠌⣴⠉
⠀⠀⠰⠺⣶⢂⡻⡀⠀⠑⡄⢫⢭⣿⠀⠧⢿⣤⣧⣋⠭⠔⢻⢈⡶⢢⡄⢬⡥⢠⡄⢲⡉⡛⠢⠭⣙⣸⣤⡾⠿⠀⣿⡯⡽⢠⠊⠀⢀⢞⡐⣶⠗⠆
⠀⠀⠀⠀⠼⢞⣚⡻⢄⢀⠈⢦⠉⢿⣆⡁⢋⣩⠎⡵⣆⣙⡜⣉⠕⣫⡹⡾⢣⢎⣝⠪⣙⢣⣋⣰⢾⠠⣉⡙⢉⣰⡿⢉⡴⠃⠀⡠⢟⣒⡳⠧
⠀⠀⠀⠀⠀⢘⣿⣨⢞⡧⡀⢁⠑⢄⡙⢛⣟⣿⡄⢛⠼⠿⢣⠝⡾⣤⣷⡜⢣⣾⣤⢷⡍⡜⠿⠧⠙⢠⣿⣻⡟⢋⡠⠊⠀⢀⢾⡓⣅⣿⡃
⠀⠀⠀⠀⠀⠈⠈⢹⡸⠱⣻⠦⡀⠀⠙⠢⣍⠘⠿⣶⣤⡾⢿⣄⠛⠌⠋⣸⣆⠙⠠⠛⢠⡿⢶⣤⣶⠿⠃⡩⠔⠋⠀⢈⠴⣟⠆⢇⡏⠁⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⡵⣛⢗⣄⡀⠀⠉⠒⠤⢀⡛⠞⠛⠷⠶⢾⣫⣝⡷⠶⠾⠛⠱⢓⣀⠤⠒⠉⠀⢄⣠⡺⣛⢮⣸⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠈⠹⡼⣘⡜⠩⡰⡤⢀⡀⠀⠈⠉⠒⠂⠒⡐⠐⢠⠂⠒⠂⠒⠉⠁⠀⢀⡀⠤⢞⠝⣣⣁⢳⠏⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠶⢎⡧⢊⡹⢤⠶⡶⣢⡤⠤⣀⡀⢀⣠⠠⢤⡔⢶⠶⡠⢏⠱⣼⡸⠶⢾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⣯⠾⢮⡷⣣⣎⣁⣏⢛⡛⣩⣀⣳⣘⣾⡵⠷⣽⠋⠉⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⡤⠴⠶⠶⢤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠴⠞⠋⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⠀⢀⡾⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⠀⣰⠟⠀⠀⠀⠀⢀⣀⡠⠤⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠶⢲⠤⢄⡀⠀⠀⠈⢳⡀
⠀⠀⠀⣴⠏⠀⠀⠀⢠⠞⠉⠀⡠⠊⠁⠈⠙⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡴⠋⠀⢀⠔⠁⠀⠀⠉⢦⠀⠀⠀⢳⡀
⠀⠀⣰⠏⠀⠀⠀⢰⠃⠀⠀⣰⠁⠀⠀⠀⠀⡠⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⡞⠀⠀⠀⠀⠀⡴⣧⠀⠀⠀⢷
⠀⢠⡟⠀⠀⠀⠀⡏⠀⠀⠀⡇⠀⠀⠀⠀⢸⡁⢸⡆⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⡇⠀⠀⠀⠀⢸⠀⢸⠀⠀⠀⠘⡇
⠀⣼⠃⠀⠀⠀⠀⢸⡀⠀⠀⢳⠀⠀⠀⠀⠀⠑⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄⠀⠀⢡⠀⠀⠀⠀⠀⢱⠆⠀⠀⠀⠀⣷
⠀⣿⠀⠀⠀⠀⠀⠀⠱⣄⠀⠀⠳⣄⠀⠀⢀⠔⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠦⣀⠀⠑⢄⡀⢀⡴⠋⠀⠀⠀⠀⠀⢸
⠀⣟⠀⠀⠀⠀⠀⠀⠀⠀⠉⠒⠲⠾⠓⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠈⠉⠓⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⣿⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⡀⠀⠀⠀⠀⣠⠞⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⣼
⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⡟⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠒⠊⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢻⠀⠀⠀⠀⠀⠀⢀⡏
⠀⠀⢿⡀⠀⠀⠀⠀⠀⠀⢣⠘⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⡆⠀⠀⠀⠀⠀⠀⣼⠁
⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠈⢆⠈⠳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⡜⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠈⢣⡀⠈⠓⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠚⠁⢀⠜⠀⠀⠀⠀⠀⠀⣰⠏
⠀⠀⠀⠀⠈⠻⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠈⠉⠒⠲⠦⠤⠤⠤⠤⠖⠒⠉⠁⠀⢀⠴⠃⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠈⠓⠤⣄⡀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠤⠚⠁⠀⠀⠀⠀⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣤⡀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠒⠒⠒⠚⠉⠁⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⣤⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⢦⣤⣄⣀⣀⣀⣀⣀⣀⣤⣤⠶⠖⠛⠉
⣶⣶⡶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣤⣤⣦⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡖⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⣤⣀⡀⠀⠀⠀⠀⢰⢹⠱⠡⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⠀⠀⢏⠃⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠀⠳⡀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠟⢻⣿⡿⢛⣽⣿⣿⣿⡿⠛⢹⣿⣿⡇⠀⠀⠘⡀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡿⠁⠀⠋⠁⠰⠟⠛⠛⠉⠁⠀⠀⠀⢻⣿⣿⠀⠀⠀⢣⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣇⠀⠀⠒⠋⠁⠀⠀⠀⠀⠉⠑⠂⠀⠀⢿⣿⠀⠀⠀⠘⡆⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢻⠀⢰⠉⢢⡑⠀⠀⠀⠀⡎⠀⠌⠀⠀⣼⣿⠀⠀⠀⠀⢣⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠈⠒⠚⠀⠀⣀⡀⠀⠑⠒⠁⠀⠀⣿⣿⠀⠀⠀⠀⠈⡆⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⠀⠀⠀⠀⠀⢸⣿⣿⠆⠀⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⠀⢡⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠘⡆⠀⠀⢰⣦⣤⣉⣭⣤⣴⡆⠀⠀⣰⢸⣿⠀⠀⠀⠀⠀⠈⡆⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢤⡈⠦⡀⠀⠙⢿⣛⣛⡿⠋⠀⣀⠔⣡⣼⣿⠀⠀⠀⠀⠀⡠⠇⠀⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⢸⣿⣶⣬⣍⣒⣒⣒⣒⣒⣒⣭⣴⣾⣿⢿⣿⠀⠀⣠⣔⠉⠀⠀⠀⡠⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠿⠸⠿⠿⠿⠿⠿⠀⠀⠀⠸⠿⠿⠿⠿⠿⠸⢟⣴⣿⣿⣿⣷⡀⠔⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣾⣿⣿⣶⣶⣶⣾⣿⣿⢸⣿⢸⣿⣿⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠙⢿⣿⣿⠿⢛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⡨⠛⠁⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿
⠀⠀⠀⠀⠀⢸⠀⠀⠀⡠⠊⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⢃⠀⠀⠸⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠘⡄⠀⠀⢇⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠰⡀⠀⢸⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠣⠤⠤⠇⠀⠀⠀⠘⠛⠛⠛⠛⠚⠛⠛⠛⠛⠛⠛⠛⠛⠚⠛⠚⠛⠛

⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠉⣭⢈⣶⢰⡆⣶⡟⣿⢹⣿⢸⣿⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⣿⠁⣿⢸⣿⢸⡇⣿⡇⣿⠸⣿⢸⣿⢸⣿⣿
⠀⠀⠀⠀⣄⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⣀⣦⠀⠀⢸⡇⣿⠀⣿⢸⣿⢸⡇⣿⡇⣿⠀⣿⠸⣿⢸⣿⣿
⠀⠀⠀⠀⣿⣿⣏⣿⣿⣿⣶⣶⣶⣶⣾⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇⣿⠀⣿⢸⣿⢸⡇⣿⡇⣿⠀⣿⠀⣿⠘⣿⣿
⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢸⡇⣿⠀⣿⢸⣿⢸⡇⣿⡇⣿⠀⣿⠀⣿⠀⣿⣿
⠀⠀⠀⠀⠛⠛⠛⠛⠛⢿⡿⣿⢿⣿⣿⣿⠟⠛⠛⠛⠛⠁⠀⣼⡇⣿⠀⣿⢸⣿⢸⡇⣿⡇⣿⡀⣿⡀⣿⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⣿⠀⣿⢸⡇⢸⡇⣿⡇⣿⡇⣿⡇⣿⡄⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠛⠐⠛⠘⠃⠘⠃⠛⠃⠛⠃⠻⠃⠻⠃⠻⠋

⠀⠀⠀⠀⢀⣠⡴⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⣴⠟⠁⠀⠀⠀⢀⣀⠀⠀⠀⣀⠀⠀⠀⢀⡀⠀⣀⠀⣀⣀⡀⠀⢀⠀⠀⠀⢀⣀⣠⡿⠀⢀⣀⠀⠀⠀⣀⣤⠀⣠⡄⠀⣠⣤⡄
⠀⠀⣼⠃⠀⠀⠀⣀⢰⣏⣽⣧⣴⡟⢿⣄⡀⠀⠸⣥⢾⡇⣾⣉⡿⣿⣠⣿⣄⡄⠀⢿⣭⣿⣧⣰⣏⣽⠇⠀⣸⡟⢻⡿⠛⣧⣰⣿⣋⣁⡀
⠀⠀⠙⠛⠚⠛⠋⠉⠀⠈⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢸⡇⠈⠉⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⣄⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠴⠛⠋⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⣦⡄⠀⠀⠀⠀⠀⣸⡃⠀⠋⣠⣤⡀⣤⡀⣠⣀⣤⣤⣀⣠⡶⠶⠀⠀⠀⠘⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⠶⠟⠷⠶⠂⠀⠀⢛⡟⠛⠉⠸⣧⡾⢧⠜⠿⠋⠸⠷⠟⠉⠟⠀⠀⠀⠀⠀⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣤⣤⣤⣤⣤⡤⠤⠤⠴⠶⠶⠶⠦⠤⢤⣤⣤⣤⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣤⣤⡤⠶⠶⠖⠛⠛⠋⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠰⠟⠛⠋⠉⠉

⣿⠉⢹⡿⢻⠿⠿⠟⢿⡟⡿⡿⡿⡿⠿⠟⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⢿⣿⢿⠿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⡿⠿⠿⠷⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⢿⣿⡄
⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⢾⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠀⢀⡗
⠀⠀⠀⠀⠉⠉⠉⠉⠃⠀⠀⠀⢀⣤⣴⣶⣶⣤⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣄⣀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉
⠀⢐⣉⣀⠤⠀⠀⠀⠀⠀⠀⠚⠋⠉⠛⠋⠉⠉⠙⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠚⠛⠛⠛⠻⠟⠛⠶⠀⠀⠀⠀⠀⢀⡇⠀⠤⢀⣉⡂
⠀⠀⠀⢀⣤⣶⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼
⠀⠀⠴⠋⠀⢻⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠇
⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠈⢻⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⠞⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡍⠛⠶⣤⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠈⠉⠙⠛⠒⠶⠶⠤⠤⠤⠤⠤⠤⠴⠶⠶⣶⠛⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⡉⠙⠒⠒⠲⠶⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⣼⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣿⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡀⠀⠀⣠⠖⠋⠉⠉⠙⠛⢏⠁⠀⠀⢙⣶⠟⣻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⢶⣇⣀⠀⠀⠀⠀⣀⣀⣠⡤⠶⠛⠁⠀⠿⠷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣶⣶⣿⠿⠿⠿⠿⠿⠿⣿⣶⣶⣶⣤⣤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⠿⠟⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠛⠻⠿⣷⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⡿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠈⠛⢿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⡟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣷⡄⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⠀⣠⣾⡿⠋⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠾⠷⠶⣦⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⣄
⠀⠀⠀⠀⠀⣴⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠋⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣦
⠀⠀⠀⢀⣾⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣷⡀
⠀⠀⢀⣾⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⠿⠿⣶⣄⡀⠀⠀⠀⠀⣠⣴⣾⣿⣿⣿⣿⣶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷⡀
⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟⠁⠀⠀⠀⠀⠀⠀⠙⠻⣆⠀⢠⣾⡿⠋⠉⠀⠀⠀⠀⠉⠻⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣷
⠀⣸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⣿⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣇
⢀⣿⡿⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⢸⣿⡇⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⠀⣰⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡄⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿
⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠸⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠐⢿⣿⠇⠀⠀⠀⠀⠀⠀⢻⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⢸⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿
⠈⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⣄⡀⠀⠀⠀⠀⢀⣤⣾⡿⠃⠀⠙⢷⣄⡀⠀⠀⠀⠀⢀⣤⠞⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿
⠀⢹⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠶⠾⠛⢋⣿⡟⠀⠀⠀⠀⠀⠉⠛⠳⠶⠶⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⡏
⠀⠀⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣄⣀⣀⣤⡾⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿
⠀⠀⠈⢿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⠛⠛⠋⠀⠀⠀⠀⠀⠰⢶⢶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡿⠁
⠀⠀⠀⠈⢿⣿⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢿⡶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡟⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⡿⠁
⠀⠀⠀⠀⠀⠻⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠟
⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣷⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠿⠷⠶⠿⠿⠟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢿⣿⣿⣷⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣶⣿⣿⣿⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣷⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣾⣿⣿⣿⣿⡿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠟⠛⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⢀⣀⣀⣀⣀⣀⣀⢀⣀⣀⢀⣀⣀⠀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠊⠀⠀⠀⠈⢱⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⢠⠞⡿⡅⠂⠀⠀⠀⣼⠇⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠰⠿⣷⡿⠟⣙⣭
⠀⠀⠀⠀⠀⠀⠀⠘⢊⣁⢨⣀⢀⣠⡀⡜⠀⠀⣰⣏⡉⠀⠈⠱⡄⠀⢰⣾⣿⣛⣛⣿⣿⢿⡄
⠀⠀⢀⡤⠔⠶⢤⡄⠘⢿⣿⠛⠚⢻⣿⠇⣠⠞⠉⠀⣉⣰⣤⢼⣇⠀⠨⣿⣯⣭⡭⣭⣶⡛⠃⠀⠀⣦⣄⡀
⠀⢠⠏⠀⠀⠀⠀⢹⡆⢹⢿⣷⣿⣿⡇⢠⡓⠂⠀⡼⠅⠀⠀⠀⠈⣣⠀⣿⡽⢻⡿⠛⣿⡟⠀⢠⣌⡞⣻⡟⣆
⠀⠈⣇⠀⠀⠀⡤⣽⡃⠘⣿⣳⢼⣿⠇⠈⢱⢢⡼⣧⡀⠀⠀⠀⠀⢼⡆⠈⣿⣟⣛⣿⡿⠁⢠⣴⠯⢼⡧⢰⣿⣤
⠀⠀⢁⠀⡀⢀⡀⣠⠇⠀⢿⡵⢾⣿⠀⠀⠸⣼⣗⢈⠣⣤⡠⠴⡶⡞⠁⠀⢹⡟⣿⣿⡇⠀⠰⣯⣾⣔⢻⣞⣯⡟
⠀⠀⢰⠟⣶⢻⢷⠇⠀⠀⠈⠓⠚⠋⠀⠀⠀⠈⢿⣿⣾⣶⡦⣾⠏⠁⠀⠀⢸⡟⣷⣿⠇⠀⠀⢻⣿⣿⣾⣹⣿⠇
⠀⠀⠀⣏⢤⣟⠩⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⠾⡞⠀⠀⠀⠀⠀⣿⣯⣿⠀⠀⠀⠈⣟⣯⣹⣦⣿
⠀⠀⠀⢹⣞⣈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣟⣻⣽⠃⠀⠀⠀⠀⠀⢹⣭⣿⠀⠀⠀⠀⢻⣿⣭⣷⡏
⠀⠀⠀⠈⡉⡔⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡙⢻⡎⠀⠀⠀⠀⠀⠀⠘⠛⠃⠀⠀⠀⠀⠈⣿⣛⡿
⠀⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⣽⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣟⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻

⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣶⣶⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣋⢉⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠻⢻⣿⠏⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣶⣾⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠛⠉⠁⢀⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣯⣤⣴⣶⣶⣿⣿⠿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡤⠴⠿⢿⣿⣿⣛⠛⠉⠁⠀⣀⣼⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣧⣤⣤⣀⣀⣀⡉⠉⠛⠛⠿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⢿⣿⠯⣿⣭⣝⣿⣿⣷⣦⣤⣀⠈⠙⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⣷⣦⣠⣖⡛⠫⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡾⣋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠏⠙⠻⢿⣿⣿⡿⠿⠓⠙⠛⠿⢟⠟⠋⣉⣤⣾⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣩⣿⣿⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣶⣄⠀⠀⠀⠀⢀⣤⣤⣤⣤⣀⠀⠀⠹⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣶⣿⡷⡄⠀⢀⡌⢛⣭⣭⣍⠉⠃⠀⠈⣿⣿⢋⠵⢶⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠬⠝⠁⡿⠀⠙⠀⠉⢙⣛⣿⠛⠀⠀⠀⠈⠁⡌⠲⡷⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⠁⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡄⠀⡇⠿⢂⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣇⠀⡀⣿⠀⠀⣀⢠⠀⣀⠀⠀⠀⠀⢸⠇⢠⣧⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠖⠒⣶⠶⢦⡄⠀⢹⡜⢅⡘⠶⠚⠉⠁⠀⠈⠛⣦⢀⡖⠈⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⢸⡱⣉⢃⣿⢲⡎⣉⣉⣙⡷⠶⠶⠶⠒⠚⠛⠁⣸⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠓⠚⠛⠛⠋⠉⠉⣧⠳⣤⣠⣤⠖⠀⠀⠀⠀⠃⠀⣠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⣠⣆⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢾⣤⣀⣀⣤⡶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

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

⠀⠀⠀⠀⠀⠀⠀⢠⣿⠿⠿⠶⣶⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀⠀⠀⠀⠀⠈⠉⠙⠛⠷⣶⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠈⢹⣷⠀⠀⠀⢀⣀⣠⣤⣤⣶⡶⠿⠟⠛⠛⠋⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠛⢻⣿⠿⠿⠛⠛⠁⢀⡾⠛⠁⠀⣶⣟⠛⠋⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠋⠀⠀⠀⠀⣰⡟⠁⠀⠀⠀⠈⠻⣦⡀⠀⠴⣤⣤⣤⣤⣀⣀⣀⣤⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠘⢿⣆⠀⠀⠉⠙⢿⡛⠛⠋⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣷⡀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣇⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣷⡀⠀⠀⠘⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣦⡀⠘⣿⣀⣠⡶⠞⠛⠛⠛⠻⠶⣦⣄⡀⠀⠀⢸⣧⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⣶⣿⠟⢉⣴⠶⣶⡄⠀⠀⣠⣤⣉⠙⠷⣤⣈⣿⠀⢀⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠟⠁⣴⠟⠁⠀⠸⣧⠀⠸⡏⠉⠙⠳⣦⡈⠙⠿⣾⡛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⡼⠃⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠈⠳⠀⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠁⠀⠀⠀⢀⡀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠙⢦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⠀⢀⣠⣶⠿⠛⠻⠇⠀⠀⠺⠋⠙⠛⠻⢶⣄⠀⠀⠀⠀⠀⠀⠙⢷⡀
⠀⠀⠀⠀⠀⠀⠀⣠⡿⠃⠀⠀⠀⠀⠀⠰⠛⠉⠀⣀⣀⣀⣀⣀⣠⣤⣤⣤⣄⣀⣀⣉⣁⣀⠀⠀⠀⠀⠀⠀⠙⣦⡀
⠀⠀⠀⠀⠀⠀⣴⡟⠁⠀⠀⣀⣤⣴⠶⠾⠟⠛⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠿⣶⡄⠀⠀⠀⠈⠻⣦
⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⢺⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⠀⠀⠙⢷⡄
⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠈⢻⣆
⠀⠀⠀⣼⣿⠁⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⣼⠏⠁⠀⠀⠀⢠⣾⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡆
⠀⠀⢰⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣦⣄⣀⡿⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⢀⣴⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡄
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠈⠙⠻⢿⣦⣤⣀⣀⠀⠀⠀⣿⣆⣠⣶⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣧
⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣦⣄⡀⠀⠀⠈⠉⠛⠛⠿⠿⠿⠿⠛⠋⠁⠀⣀⣴⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠘⣿⣇⠀⠀⠀⠀⠀⠀⠀⢀⣿⢻⣀⡿⠻⠷⣦⣤⣄⣀⣀⠀⠀⢀⣀⣀⣤⣴⣾⠟⠉⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡇
⠀⠀⠀⠹⣿⣆⠀⠀⠀⠀⠀⠀⢸⣿⠀⢻⡃⠀⠀⣿⠉⠉⢙⡟⠛⠛⠛⢻⡋⠉⠀⣿⠀⣰⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠟
⠀⠀⠀⠀⠙⢿⣷⣄⡀⠀⠀⠀⢸⣿⠀⣿⠳⣄⣰⠃⠀⠀⢸⡇⠀⠀⠀⣾⠀⠀⢰⣧⡶⢻⡇⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⠿⠋
⠀⠀⠀⠀⠀⠀⠉⠻⣿⣶⣦⡄⠘⣿⣀⡏⠀⠀⣿⢦⣤⣀⣼⡃⠀⠀⢀⣿⣤⡴⣿⠋⢠⡿⠀⠀⠀⠀⣀⣤⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⢀⣾⡿⠻⣿⡇⠀⠹⣿⣇⠀⢀⡟⠀⠀⠉⡿⠛⠛⠛⢹⡏⠀⢠⣏⣴⠟⠀⢀⣴⣶⠿⠟⠛⠉⠻⣿⣦⡀
⠀⠀⠀⠀⠀⣰⣿⡟⠁⠀⢻⣿⣄⠀⠈⠻⣷⣼⣇⠀⠀⢸⡇⠀⠀⢀⣾⣥⣴⠾⠋⠁⠀⣠⣾⡟⠁⠀⠀⠀⠀⠀⠈⠻⣿⣄
⠀⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠻⣿⣷⣄⠀⠀⠉⠛⠛⠛⠿⠿⠿⠛⠛⠋⠉⠀⢀⣠⣴⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣧⡀
⠀⠀⠀⣰⣿⠟⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⣷⣶⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣿⠿⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣷⡀
⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀
⠀⠀⣾⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣧
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣶⠶⠶⠾⠶⢶⣶⠟⠛⠛⠛⠻⠶⢶⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡄
⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡿⠉⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷
⢠⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⢀⣀⣠⣤⣤⣤⣿⣧⡀⠀⠀⠀⠀⠀⠀⠈⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿
⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣽⡇⠀⠀⠀⠉⠉⠉⠀⠀⠈⣿⡟⠛⠛⠁⠀⠀⠀⠀⢸⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⣼⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠘⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠀⠀⠐⠲⠶⠶⠶⣶⣾⠿⠷⠶⠾⠛⠁⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣦⡀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⣸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣶⣤⣀⠀⠀⣿⣧⣤⣤⣤⣤⣤⣤⣤⣶⠿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⠿⠏⠉⠉⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣼
⢿⣿⣶⣦⣤⣄⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣤⣤⣶⣶⣿⡿⠿⠟⠛
⠀⠀⠉⠉⠙⠛⠻⠿⠿⢿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⠿⠿⠿⠟⠛⠛⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⢡⡀⠘⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠳⡀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢃⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠱⠀⠙⣀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠁⠘⠀⠈⡄⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠉⡄⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠀⠀⠄⠀⡀⠂⠁⠀⠀⠀⠀⠀⢀⠎⠀⠀⠀⠀⢀⡃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠖⣠⠞⠉⠀⠀⠀⠀⠀⠀⣀⠔⠁⠀⠀⠀⠀⢀⠎⠁⠐⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠃⢀⠀⠀⠀⠀⠀⠒⠉⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠁⢀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢀⡤⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⢀⡠⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠒⠉⠀⠀⠀⠀⠀⣾
⠀⠀⠀⠀⠀⠀⠀⣠⠀⣠⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⢛⡰⠃⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠖⠹
⠀⠀⠀⠀⠀⠀⠀⢈⠃⠀⠀⠀⠀⠀⡠⠖⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⡔⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠤⠄⠒⠉⢠⠃⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠣⣠⠎⠀⠀⠀⠀⠀⠀⣀⠠⠄⠂⠀⠁⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⢀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⢠⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⣠⠶⠋⠀⠀⠀⢀⡴⢫⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⠀⢀⡔⠁⠀⠀⠀⠀⠀⠀⢀⠀⠒⠉⠀⠀⠀⣀⠤⠒⠉⢠⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢋⠙⢶⡶⠠⢤⡴⠶⠚⠉⠛⣇⠀⠈⡙⠁⠈⠳⢄⡠⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄⠈⠀⢠⡀⡴⠀⠐⠀⠀⠈⠳⡄⠀⠀⠀⢀⠌⠁⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠠⠀⠘⡀⠳⡀⠀⠀⣠⠂⠈⠂⠀⡀⠀⠀⢀⡌
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡷⠀⢨⡇⠀⠀⠘⢠⠜⠁⠀⠀⢀⡜⠁⢄⠀⣼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠃⠐⠄⠀⠀⠀⠀⠀⢀⡴⠋⠀⠀⢀⠞⡛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢀⠜⠂⠀⠀⠀⠀⠉⠰⣄⠀⠀⠀⢰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡆⠁⠀⠀⠀⠆⠀⠀⠀⠀⣀⠝⠂⢀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡻⢀⡰⠊⠀⠀⠀⠀⣀⠞⠁⠀⢠⢊⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡏⠑⠀⠀⠀⠀⠀⠈⠓⢤⡀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⠀⠀⠒⢆⠀⠀⢀⡀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣇⠀⠀⠀⠈⡲⠊⠀⠀⠀⣴⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⢀⠀⠈⠀⠀⠈⠐⠞⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡗⠀⠀⠠⢀⢀⠀⠀⢀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀⠀⢀⡠⠉⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢟⠶⠈⠀⠀⠀⠸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⢀⠖⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄⠀⠀⢠⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡇⠀⠤⣸⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠁⠀⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⢠⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣴⡾⠟⠛⠋⠉⠉⠛⠻⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣾⡿⢻⡿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⣿⠛⠲⢶⣤⣴⠶⠾⠶⠶⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠟⣩⡾⠋⢠⡿⠁⠀⠀⠀⣠⣤⡶⠶⠶⣶⣄⠀⠀⢸⣦⠶⠋⠁⠀⠀⠀⠀⠀⠈⠙⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋⠁⣰⡟⠀⢀⣿⠁⠀⠀⣠⡾⠋⠀⠀⠀⠀⠀⠹⡇⠀⠚⢁⣠⠶⠞⠓⠀⠀⠀⠀⠀⠀⠈⢷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡟⠁⠀⢠⡟⠀⠀⣾⠃⠀⢀⣼⠏⠀⠀⠛⠓⠶⣤⡀⠀⣿⢀⣴⠟⠁⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡟⠀⠀⠀⢸⠃⠀⢸⠏⠀⢀⣾⡿⠶⠶⠶⠶⢶⣤⣤⣽⣦⣿⣾⡵⠞⠛⠉⠹⣏⠉⠛⠆⠀⠀⠀⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⡟⠀⠀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠙⣦⡀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣼⠃⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⣄⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⢰⡏⠀⣼⠏⠀⢀⣤⣤⣄⠀⠀⠀⠀⠀⠀⠀⠴⠶⣤⠀⠀⠀⠀⠀⢸⡇⠙⢷⣄⠀⠀⠀⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⠀⠀⠀⢠⡟⠀⣰⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⡀⠀⠙⣧⡀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢠⠀⠠⠟⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠈⢳⣄⠀⠈⢷⡀⠸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠀⢀⡞⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⢀⣤⣤⡀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⣦⠀⠘⣷⠀⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⢠⡞⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⠀⢻⣿⣿⠇⠀⠀⠀⠀⠸⣿⣿⡿⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⣿⠀⠈⢿⡄
⠀⠀⠀⠀⠀⠀⠀⣰⡿⠁⣠⠟⠀⠀⢀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⣿⡇⢀⣴⠿⠶⠶⢿⣄⠀⠘⣷⡀
⠀⠀⠀⠀⠀⠀⣴⠟⠀⢰⡏⠀⣠⠞⠛⠛⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⠋⠁⠀⣀⣀⡀⢹⡆⠀⠸⣷
⠀⠀⠀⠀⠀⣼⠃⠀⢀⡿⠀⢸⡏⣤⠦⣤⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⣿⡉⠙⠃⢸⡇⠀⠀⢻⣇
⠀⠀⠀⠀⣼⠇⠀⠀⣸⡇⠀⢸⡇⠁⢀⡞⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⣀⣀⣠⠀⠀⠀⠀⠀⠀⢀⡤⠀⠀⠀⣿⣴⠏⢸⡇⠀⢀⣾⠇⠀⠀⠈⣿⡀
⠀⠀⠀⠀⣿⠀⠀⠀⣿⠀⠀⠸⣇⠀⠸⠃⢸⡇⠀⠀⠀⢤⣄⣀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⣠⡾⠋⠀⠀⠀⠀⣿⣿⠀⠘⠇⣠⣾⠋⠀⠀⠀⠀⢻⡇
⠀⠀⠀⢰⣿⠀⠀⠀⣿⠀⠀⠀⠹⣦⡀⠀⠸⣧⠀⠀⠀⠀⠀⠉⢻⣶⡶⠦⠤⣤⣤⣤⣤⣴⣶⡟⠉⠀⠀⠀⠀⠀⢸⣯⣁⣠⣤⣶⠟⠁⠀⠀⠀⠀⠀⢸⣇
⠀⠀⠀⠈⣿⠀⠀⠀⣿⡀⠀⠀⠀⠈⠻⠶⣤⣿⡄⠀⠀⠀⠀⠀⠈⢿⣿⣷⣶⣦⣤⣴⣾⣿⠟⠀⠀⠀⠀⠀⠀⢠⡿⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⢿⡀⠀⠀⠘⣷⡀⠀⠀⠀⠀⠀⠀⠘⢷⡄⠀⠀⠀⠀⠀⠈⢿⣍⠉⠋⠉⣩⡿⠃⠀⠀⠀⠀⠀⠀⣠⡿⠁⢀⣤⣤⣤⡀⠀⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠘⣧⡀⠀⠀⠈⠻⣄⡀⠀⠀⠀⠲⣄⠈⠻⣶⣄⠀⠀⠀⠀⠀⠙⠳⠶⠟⠋⠀⠀⠀⠀⠀⠀⢀⣴⠟⠀⠀⣿⠀⠀⢀⡿⠀⠀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⠀⠙⣷⣄⠀⠀⠀⠈⠛⠷⡤⠀⠀⣽⠀⠀⠀⠙⢷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⠛⠁⠀⠀⠀⠻⣧⡈⠉⠀⠀⠀⠀⠀⠀⠀⣠⡿⠁
⠀⠀⠀⠀⠀⠀⠈⠻⣷⣄⡀⠀⠀⠀⠀⠀⣰⣟⠀⠀⠀⠀⠀⠈⠛⣷⣤⣄⡀⠀⠀⠀⣀⣠⡴⠟⠋⢿⣄⠀⠀⠀⠀⠀⠈⣻⣶⣤⣤⣤⣤⣤⡶⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠶⠶⠶⠟⠉⠈⠙⠶⢤⣤⣤⣤⡾⠋⠀⠉⠙⠛⠿⠛⠋⠁⠀⠀⠀⠀⠙⠻⠶⠶⠶⠟⠛⠁⠀⠀⠈⠉⠉


⣾⣶⣶⣾⣶⡶⢿⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣾⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣷⠾⣶⣶⡶⠶⣾⣶⡶⠷⢾

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⡶⠖⠂⠀⠀⠀⠘⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⢷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠊⠁⠀⠀⠀⠉⠲⣄⠀⠀⠀⠀⢀⠠⠐⠂⠀⠐⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠃⠀⠀⠀⠀⠀⢠⣶⣶⣮⣆⠀⠀⠠⠃⠀⠀⠀⠀⠀⠀⠀⠱⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⢸⣿⣿⣤⢿⠀⠀⢁⣴⣶⣦⡀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⡸⠀⠀⠘⣿⣿⣼⠇⠀⠀⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⢀⡔⠁⠀⠀⠘⢄⠉⠀⠀⠀⠀⠀⢀⠔
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠢⠤⠤⠄⠒⠁⠀⠀⠀⠀⠀⠀⠉⠒⠀⠀⠐⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⡩⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⠈⢢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠎⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⠔⢡⠎⠓⠦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠴⠛⢢⠑⠤⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢄⡀⡠⠋⡙⠙⠛⠲⠒⠲⠚⠛⠙⡉⠣⡀⣀⠴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠳⠤⢤⡼⠀⠸⣤⠤⠴⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠦⠤⠤⠴⠂



⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣛⣛⣛⣛⣛⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⡀⠀⢠⡄⠀⠀⡄
⠀⠀⠖⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⣀⠀⠉⢇⠀⣰⠃⢀⣐⣛⡀
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠉⠲⠤⠾⠟⠁⠀⡶⠿⠭⠥
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⠆⠀⠁⠉⠀⠤⢄⡀⠀⠭⠭⣭⠁
⠆⠀⠀⠀⠀⠀⠀⠀⠀⣴⣏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠰⠀⠃⠀⠀⠘⠳⢦⡄⠀⠄⢄
⠆⠀⠀⠀⠀⠀⠀⢸⠀⢩⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠂⠀⠁
⠰⡾⠀⠀⠀⠈⠂⠈⠀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡄⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⣀⠀⣤⣾⣦⡀⢻⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠹⣿⡛⠁⠈
⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠐⢡⡌⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⢀⣠⣔⠀⠶⠟⠁
⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⠟⠉⠁⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⢠⡄⡤⠀⢠⡄
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠆
⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⣶⣶⡷⠀⡔⢯⠤⠇⠀⠀⠀⠀⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠀⠈
⢀⣄⠀⢠⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⣀
⠾⠛⠁⠀⠀⢐⡝⡎⢹⠀⡤⠀⠄⠀⠀⠀⠀⠀⡇⠀⠀⡷⠔⠈⠒⡀⠀⠀⠀⠐⠀⠠⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠸⠜⠐⠞⠀⠼⠂⠀⠀⠀⠀⠀⠀⢈⣒⣈⣀⠢⣀⠠⠃⠀⠀⠀⢈⠀⠊
⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠈⠛⠋⠙⠷⠆⠀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠘⢣⡄⢠⡀⠀⠀⠉⠁⠀⠄⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠘⠀⢿⡇⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⢰⣿⣿⣿⡿⠃⠀⠀⢀⣠⡄⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⠟⠁⠐⠖⠀⠘⠘⠀⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠈⠡
⠀⠀⠀⠀⠀⠀⠀⣾⠙⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⡀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢠
⠀⠀⠀⠀⠀⠀⠀⡀⡀⠙⠀⠀⠀⠀⣠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⠜⠀⠂
⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠈⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡚⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⡔⡰⠃⠀⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⢡
⠀⠀⠀⠀⠀⠀⠙⠀⠂⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⢀⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠄⠟⠀⠯⠀⠀⠀⠠⡀
⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠏⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠔⢃⠀⠀⠀⠀⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠄⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠐⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⢠⣦⣿⡄⠀⠀⠀⠀⠀⠈⠁⠱⡆⠀⠀⠀⠀⢀⠃⠚⠁⠠⢤⠀⠀⠁⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣜⣥⠞⡄⠀⠀⠀⠀⠀⣀⠐⡻⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⢧⠀⠀⢠⠅
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠂⠀⠀⠀⡀⠀⠀⠀⣀⠀⠀⠝⠋⠀⠀⢐⡒⠀⠀⢒⣆⠁⢰⠃⠀⠀⠀⠀⢀⡃⠀⠀⠀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠈⠐⠀⠀⠀⠀⢀⣤⣾⠁⠛⠿⠿⠁⠀⠘⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠁⠀⠀⠀⠀⣀⡤⢘⡩⠿⣧⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣄⣀⣩⡤⢒⠋⠛⠀⠚⢆⠀⠀⠀⠀⠐⠀⠀⠀⠀⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⢀⠀⠀⠀⠘⡀⠀⠀⠐⠆⠀⠀⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢿⡀⢀⠀⠀⡀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠠⡀⠐⠉⠉⠻⠟⠀⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠒⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡬⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡤⠂⢠⠔⠀⠀⠀⠀⠀⠈⠁⠀⢸⠆⣠⠀⢀⡔⣠⠀⡤⠀⡰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋⠉⠉⠀⠁⠀⠈⠀⠀⠀⠀⠈⠀⠀⠀⠀⠉⠀⠉⠀⠈⠀⠉⠈⠁⠀⠁

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⡟⠛⠛⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿